• LfunctionsAndModularFormsII
• CentralValues
• 32A

See the Products page for definitions of the A,C,D,F functions.

First Symmetric Power

When twisting by odds we get

• \eta(q^8)^{-1}\eta(q^{16})^6\eta(q^{32})^{-2} \>\text{and}\> \eta(q^8)\eta(q^{32})^2
  A_6(q^8)\cdot A_1(q^8)\cdot A_1(q^{16}) \>\text{and}\> A_1(q^8)\cdot A_1(q^{32})\cdot A_1(q^{32})

When twisting by evens we get

• \eta(q^8)^{-1}\eta(q^{16})^2\eta(q^{32})^4\eta(q^{64})^{-2} \>\text{and}\> \eta(q^8)^{-1}\eta(q^{16})^4\eta(q^{32})^{-2}\eta(q^{64})^2
  A_3(q^8)\cdot A_2(q^{32})\cdot A_2(q^{32}) \>\text{and}\> A_3(q^8)\cdot A_4(q^{16})\cdot A_4(q^{16})

As with 27A, there are other forms that differ only on squares.

Third Symmetric Power

Weight 5/2 lifts for real/imaginary twists by odds of Sym^3(\psi_{32A}):

• \eta(q^8)^3\eta(q^{16})^4\eta(q^{32})^{-2} \>\text{and}\> \eta(q^8)^5\eta(q^{16})^{-2}\eta(q^{32})^{2}
  C_1(q^8)\cdot \Bigl[A_2(q^{16})\cdot A_2(q^{16})\Bigr] \>\text{and}\> C_3(q^8)\cdot\Bigl[A_1(q^{32})\cdot A_1(q^{32})\Bigr]

Weight 5/2 lifts for 3,7 mod 8 twists of Sym^3(\psi_{64A}):

• \eta(q^8)^1\eta(q^{32})^{6}\eta(q^{64})^{-2} \>\text{and}\> \eta(q^8)^1\eta(q^{16})^{2}\eta(q^{64})^{2}
  C_3(q^{32})\cdot\Bigl[A_1(q^8)\cdot A_1(q^{32})\Bigr]\>\text{and}\> C_4(q^{16})\cdot\Bigl[A_1(q^8)\cdot A_1(q^{32})\Bigr]

Fifth Symmetric Power

Weight 7/2 lift for real/imaginary twists by odds of Sym^5(\psi_{32A}):

• \eta(q^8)^{-5}\eta(q^{16})^{14}\eta(q^{32})^{-2} \>\text{and}\> \eta(q^8)^{-3}\eta(q^{16})^{8}\eta(q^{32})^{2}
  C_6(q^8)\cdot\Bigl[A_1(q^{16})\cdot C_1(q^{32})\bigr]\>\text{and}\> C_5(q^8)\cdot\Bigl[A_1(q^{16})\cdot C_2(q^{16})\Bigr]

Weight 7/2 lift for 1,5 mod 8 twists of Sym^5(\psi_{64A}):

• \eta(q^8)^{-9}\eta(q^{16})^{22}\eta(q^{32})^{-4}\eta(q^{64})^{-2} \>\text{and}\> \eta(q^8)^{-9}\eta(q^{16})^{24}\eta(q^{32})^{-10}\eta(q^{64})^{2}
  C_6(q^8)\cdot D_1(q^8)\>\text{and}\> C_6(q^8)\cdot D_2(q^8)

Seventh Symmetric Power

Weight 9/2 lift for real/imaginary twists by odds of Sym^7(\psi_{32A}):

• \eta(q^8)^{-1}\eta(q^{16})^{12}\eta(q^{32})^{-2} \>\text{and}\> \eta(q^8)^{1}\eta(q^{16})^{6}\eta(q^{32})^{2}
  F_1(q^8)\cdot C_3(q^{16})\>\text{and}\> F_1(q^8)\cdot C_4(q^8)

Weight 9/2 lift for 3,7 mod 8 twists of Sym^7(\psi_{64A}):

• \eta(q^8)^{-7}\eta(q^{16})^{20}\eta(q^{32})^{-2}\eta(q^{64})^{-2} \>\text{and}\> \eta(q^8)^{-7}\eta(q^{16})^{22}\eta(q^{32})^{-8}\eta(q^{64})^{2}
  F_2(q^8)\cdot C_3(q^{32})\>\text{and}\> F_2(q^8)\cdot C_3(q^{16})

Higher Symmetric Powers

Calculations with Sym^9 for twists-by-odds showed that the lifts of weight 11/2 do not correspond to an \eta-quotient as with the above.