Properties

Label 32.8.a
Level $32$
Weight $8$
Character orbit 32.a
Rep. character $\chi_{32}(1,\cdot)$
Character field $\Q$
Dimension $7$
Newform subspaces $4$
Sturm bound $32$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 32.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(32\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(32))\).

Total New Old
Modular forms 32 7 25
Cusp forms 24 7 17
Eisenstein series 8 0 8

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)Dim
\(+\)\(4\)
\(-\)\(3\)

Trace form

\( 7 q - 278 q^{5} + 2867 q^{9} + O(q^{10}) \) \( 7 q - 278 q^{5} + 2867 q^{9} + 13618 q^{13} + 28334 q^{17} - 25088 q^{21} + 154049 q^{25} - 188446 q^{29} - 275200 q^{33} - 427094 q^{37} + 494230 q^{41} - 1109774 q^{45} + 288063 q^{49} + 3543482 q^{53} + 1442560 q^{57} - 1688670 q^{61} - 6938196 q^{65} + 9017856 q^{69} + 11399142 q^{73} - 15644160 q^{77} - 26117809 q^{81} + 21265812 q^{85} + 18861046 q^{89} - 33433600 q^{93} - 36299714 q^{97} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(32))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2
32.8.a.a 32.a 1.a $1$ $9.996$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-58\) \(0\) $-$ $N(\mathrm{U}(1))$ \(q-58q^{5}-3^{7}q^{9}-8898q^{13}-40094q^{17}+\cdots\)
32.8.a.b 32.a 1.a $2$ $9.996$ \(\Q(\sqrt{10}) \) None \(0\) \(-16\) \(-180\) \(1248\) $+$ $\mathrm{SU}(2)$ \(q+(-8+\beta )q^{3}+(-90+8\beta )q^{5}+(624+\cdots)q^{7}+\cdots\)
32.8.a.c 32.a 1.a $2$ $9.996$ \(\Q(\sqrt{15}) \) None \(0\) \(0\) \(140\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+70q^{5}+18\beta q^{7}+1653q^{9}+\cdots\)
32.8.a.d 32.a 1.a $2$ $9.996$ \(\Q(\sqrt{10}) \) None \(0\) \(16\) \(-180\) \(-1248\) $-$ $\mathrm{SU}(2)$ \(q+(8+\beta )q^{3}+(-90-8\beta )q^{5}+(-624+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(32))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(32)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 5}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)