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Properties

Label 1296.3.q
Level $1296$
Weight $3$
Character orbit 1296.q
Rep. character $\chi_{1296}(593,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $94$
Newform subspaces $17$
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.q (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 17 \)
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 936 98 838
Cusp forms 792 94 698
Eisenstein series 144 4 140

Trace form

\( 94 q - 2 q^{7} + 2 q^{13} + 4 q^{19} + 217 q^{25} + 118 q^{31} - 4 q^{37} - 242 q^{43} - 285 q^{49} - 96 q^{55} + 2 q^{61} - 2 q^{67} + 44 q^{73} - 2 q^{79} - 48 q^{85} - 760 q^{91} + 122 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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.q.a 1296.q 9.d $2$ $35.313$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) 27.3.b.a \(0\) \(0\) \(0\) \(-13\) $\mathrm{U}(1)[D_{6}]$ \(q+(-13+13\zeta_{6})q^{7}+\zeta_{6}q^{13}-11q^{19}+\cdots\)
1296.3.q.b 1296.q 9.d $2$ $35.313$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) 12.3.c.a \(0\) \(0\) \(0\) \(2\) $\mathrm{U}(1)[D_{6}]$ \(q+(2-2\zeta_{6})q^{7}+22\zeta_{6}q^{13}-26q^{19}+\cdots\)
1296.3.q.c 1296.q 9.d $2$ $35.313$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) 108.3.c.a \(0\) \(0\) \(0\) \(11\) $\mathrm{U}(1)[D_{6}]$ \(q+(11-11\zeta_{6})q^{7}-23\zeta_{6}q^{13}+37q^{19}+\cdots\)
1296.3.q.d 1296.q 9.d $4$ $35.313$ \(\Q(\zeta_{12})\) None 108.3.c.b \(0\) \(0\) \(0\) \(-14\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta_1 q^{5}-7\beta_{2} q^{7}+(-\beta_{3}+\beta_1)q^{11}+\cdots\)
1296.3.q.e 1296.q 9.d $4$ $35.313$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None 24.3.e.a \(0\) \(0\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{5}-6\beta _{2}q^{7}+(\beta _{1}-\beta _{3})q^{11}+\cdots\)
1296.3.q.f 1296.q 9.d $4$ $35.313$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None 18.3.b.a \(0\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{5}-4\beta _{2}q^{7}+(-4\beta _{1}+4\beta _{3})q^{11}+\cdots\)
1296.3.q.g 1296.q 9.d $4$ $35.313$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None 216.3.e.a \(0\) \(0\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{5}-3\beta _{2}q^{7}+(7\beta _{1}-7\beta _{3})q^{11}+\cdots\)
1296.3.q.h 1296.q 9.d $4$ $35.313$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None 216.3.e.b \(0\) \(0\) \(0\) \(6\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{5}+3\beta _{2}q^{7}+(\beta _{1}-\beta _{3})q^{11}+\cdots\)
1296.3.q.i 1296.q 9.d $4$ $35.313$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None 54.3.b.a \(0\) \(0\) \(0\) \(10\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{5}+5\beta _{2}q^{7}+(-\beta _{1}+\beta _{3})q^{11}+\cdots\)
1296.3.q.j 1296.q 9.d $4$ $35.313$ \(\Q(\zeta_{12})\) None 27.3.b.b \(0\) \(0\) \(0\) \(10\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta_1 q^{5}+5\beta_{2} q^{7}+(-5\beta_{3}+5\beta_1)q^{11}+\cdots\)
1296.3.q.k 1296.q 9.d $4$ $35.313$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None 72.3.e.a \(0\) \(0\) \(0\) \(24\) $\mathrm{SU}(2)[C_{6}]$ \(q+5\beta _{1}q^{5}+12\beta _{2}q^{7}+(-4\beta _{1}+4\beta _{3})q^{11}+\cdots\)
1296.3.q.l 1296.q 9.d $8$ $35.313$ \(\Q(\zeta_{24})\) None 216.3.e.c \(0\) \(0\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta_{4}-\beta_{3}+\beta_1)q^{5}+(\beta_{7}-\beta_{6}+3\beta_{2}-3)q^{7}+\cdots\)
1296.3.q.m 1296.q 9.d $8$ $35.313$ \(\Q(\zeta_{24})\) None 81.3.b.b \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta_{6} q^{5}+(-\beta_{4}+\beta_{2}+\beta_1-1)q^{7}+\cdots\)
1296.3.q.n 1296.q 9.d $8$ $35.313$ \(\Q(\zeta_{24})\) None 324.3.c.a \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta_{5} q^{5}+(\beta_{2}-\beta_1)q^{7}-\beta_{7} q^{11}+\cdots\)
1296.3.q.o 1296.q 9.d $8$ $35.313$ \(\Q(\zeta_{24})\) None 162.3.b.b \(0\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta_{6} q^{5}+(2\beta_{3}+2\beta_{2})q^{7}+(2\beta_{7}-2\beta_{6}+\cdots+2\beta_1)q^{11}+\cdots\)
1296.3.q.p 1296.q 9.d $8$ $35.313$ \(\Q(\zeta_{24})\) None 648.3.e.a \(0\) \(0\) \(0\) \(12\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta_{6} q^{5}+(-\beta_{3}+3\beta_{2})q^{7}+(\beta_{7}-\beta_{6}-6\beta_{5}+\beta_1)q^{11}+\cdots\)
1296.3.q.q 1296.q 9.d $16$ $35.313$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None 648.3.e.d \(0\) \(0\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{11}+\beta _{12})q^{5}+(-2-2\beta _{2}+\beta _{7}+\cdots)q^{7}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(1296, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 15}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(72, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(144, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(216, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(324, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(432, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(648, [\chi])\)\(^{\oplus 2}\)