Properties

Label 1296.3.e
Level $1296$
Weight $3$
Character orbit 1296.e
Rep. character $\chi_{1296}(161,\cdot)$
Character field $\Q$
Dimension $46$
Newform subspaces $10$
Sturm bound $648$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1296.e (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(648\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1296, [\chi])\).

Total New Old
Modular forms 468 50 418
Cusp forms 396 46 350
Eisenstein series 72 4 68

Trace form

\( 46 q - 2 q^{7} + O(q^{10}) \) \( 46 q - 2 q^{7} + 2 q^{13} + 4 q^{19} - 188 q^{25} - 50 q^{31} + 20 q^{37} + 94 q^{43} + 240 q^{49} + 54 q^{55} - 118 q^{61} - 98 q^{67} - 28 q^{73} + 190 q^{79} + 120 q^{85} - 286 q^{91} + 122 q^{97} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1296, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1296.3.e.a 1296.e 3.b $2$ $35.313$ \(\Q(\sqrt{-3}) \) None 9.3.d.a \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{2}]$ \(q-2\zeta_{6}q^{5}-2q^{7}+\zeta_{6}q^{11}-4q^{13}+\cdots\)
1296.3.e.b 1296.e 3.b $4$ $35.313$ \(\Q(\sqrt{-2}, \sqrt{3})\) None 648.3.e.a \(0\) \(0\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{5}+(-3+\beta _{2})q^{7}+(-5\beta _{1}-\beta _{3})q^{11}+\cdots\)
1296.3.e.c 1296.e 3.b $4$ $35.313$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None 72.3.m.a \(0\) \(0\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{1}-\beta _{2})q^{5}+(-3-\beta _{3})q^{7}+(2\beta _{1}+\cdots)q^{11}+\cdots\)
1296.3.e.d 1296.e 3.b $4$ $35.313$ \(\Q(\sqrt{-2}, \sqrt{3})\) None 162.3.b.b \(0\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{5}+(-2-2\beta _{3})q^{7}+(-2\beta _{1}+\cdots)q^{11}+\cdots\)
1296.3.e.e 1296.e 3.b $4$ $35.313$ \(\Q(\sqrt{-3}, \sqrt{-11})\) None 36.3.g.a \(0\) \(0\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{5}+(-1+\beta _{2})q^{7}+(\beta _{1}-\beta _{3})q^{11}+\cdots\)
1296.3.e.f 1296.e 3.b $4$ $35.313$ \(\Q(\sqrt{-2}, \sqrt{3})\) None 324.3.c.a \(0\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{5}+(1-\beta _{2})q^{7}+\beta _{3}q^{11}+(-1+\cdots)q^{13}+\cdots\)
1296.3.e.g 1296.e 3.b $4$ $35.313$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None 18.3.d.a \(0\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{5}+(1+\beta _{2})q^{7}+(\beta _{1}-\beta _{3})q^{11}+\cdots\)
1296.3.e.h 1296.e 3.b $4$ $35.313$ \(\Q(\sqrt{-2}, \sqrt{3})\) None 81.3.b.b \(0\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{5}+(1+\beta _{3})q^{7}+(-\beta _{1}-3\beta _{2}+\cdots)q^{11}+\cdots\)
1296.3.e.i 1296.e 3.b $8$ $35.313$ 8.0.\(\cdots\).9 None 72.3.m.b \(0\) \(0\) \(0\) \(12\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{1}+\beta _{3})q^{5}+(2-\beta _{5})q^{7}+(-2\beta _{1}+\cdots)q^{11}+\cdots\)
1296.3.e.j 1296.e 3.b $8$ $35.313$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 648.3.e.d \(0\) \(0\) \(0\) \(12\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\beta _{1}+\beta _{3})q^{5}+(2+\beta _{5})q^{7}+(\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(1296, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1296, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(48, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(216, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(324, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(432, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(648, [\chi])\)\(^{\oplus 2}\)