Properties

Label 324.3.f
Level $324$
Weight $3$
Character orbit 324.f
Rep. character $\chi_{324}(55,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $92$
Newform subspaces $19$
Sturm bound $162$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 324.f (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 36 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 19 \)
Sturm bound: \(162\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(324, [\chi])\).

Total New Old
Modular forms 240 100 140
Cusp forms 192 92 100
Eisenstein series 48 8 40

Trace form

\( 92 q + 2 q^{4} + O(q^{10}) \) \( 92 q + 2 q^{4} - 20 q^{10} + 4 q^{13} + 2 q^{16} - 6 q^{22} - 186 q^{25} + 60 q^{28} + 40 q^{34} - 8 q^{37} - 98 q^{40} + 432 q^{46} + 242 q^{49} + 52 q^{52} - 80 q^{58} + 4 q^{61} - 412 q^{64} - 54 q^{70} - 56 q^{73} - 420 q^{76} - 416 q^{82} + 104 q^{85} - 186 q^{88} + 60 q^{94} + 124 q^{97} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(324, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
324.3.f.a 324.f 36.f $2$ $8.828$ \(\Q(\sqrt{-3}) \) None \(-4\) \(0\) \(-2\) \(12\) $\mathrm{SU}(2)[C_{6}]$ \(q-2q^{2}+4q^{4}+(-2+2\zeta_{6})q^{5}+(8+\cdots)q^{7}+\cdots\)
324.3.f.b 324.f 36.f $2$ $8.828$ \(\Q(\sqrt{-3}) \) None \(-4\) \(0\) \(7\) \(-15\) $\mathrm{SU}(2)[C_{6}]$ \(q-2q^{2}+4q^{4}+(7-7\zeta_{6})q^{5}+(-10+\cdots)q^{7}+\cdots\)
324.3.f.c 324.f 36.f $2$ $8.828$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(-7\) \(15\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(-7+7\zeta_{6})q^{5}+\cdots\)
324.3.f.d 324.f 36.f $2$ $8.828$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(2\) \(-12\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(2-2\zeta_{6})q^{5}+\cdots\)
324.3.f.e 324.f 36.f $2$ $8.828$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-1}) \) \(-2\) \(0\) \(8\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q-2\zeta_{6}q^{2}+(-4+4\zeta_{6})q^{4}+(8-8\zeta_{6})q^{5}+\cdots\)
324.3.f.f 324.f 36.f $2$ $8.828$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-1}) \) \(2\) \(0\) \(-8\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q+2\zeta_{6}q^{2}+(-4+4\zeta_{6})q^{4}+(-8+8\zeta_{6})q^{5}+\cdots\)
324.3.f.g 324.f 36.f $2$ $8.828$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(-2\) \(-12\) $\mathrm{SU}(2)[C_{6}]$ \(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(-2+2\zeta_{6})q^{5}+\cdots\)
324.3.f.h 324.f 36.f $2$ $8.828$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(7\) \(15\) $\mathrm{SU}(2)[C_{6}]$ \(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(7-7\zeta_{6})q^{5}+\cdots\)
324.3.f.i 324.f 36.f $2$ $8.828$ \(\Q(\sqrt{-3}) \) None \(4\) \(0\) \(-7\) \(-15\) $\mathrm{SU}(2)[C_{6}]$ \(q+2q^{2}+4q^{4}+(-7+7\zeta_{6})q^{5}+(-10+\cdots)q^{7}+\cdots\)
324.3.f.j 324.f 36.f $2$ $8.828$ \(\Q(\sqrt{-3}) \) None \(4\) \(0\) \(2\) \(12\) $\mathrm{SU}(2)[C_{6}]$ \(q+2q^{2}+4q^{4}+(2-2\zeta_{6})q^{5}+(8-4\zeta_{6})q^{7}+\cdots\)
324.3.f.k 324.f 36.f $4$ $8.828$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-1}) \) \(-4\) \(0\) \(-8\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q+(-2+2\zeta_{12})q^{2}-4\zeta_{12}q^{4}+(-4\zeta_{12}+\cdots)q^{5}+\cdots\)
324.3.f.l 324.f 36.f $4$ $8.828$ \(\Q(\sqrt{-3}, \sqrt{13})\) None \(-3\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1+\beta _{1})q^{2}+(-\beta _{1}+3\beta _{2}+\beta _{3})q^{4}+\cdots\)
324.3.f.m 324.f 36.f $4$ $8.828$ \(\Q(\sqrt{-3}, \sqrt{13})\) None \(3\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1-\beta _{1})q^{2}+(-\beta _{1}+3\beta _{2}+\beta _{3})q^{4}+\cdots\)
324.3.f.n 324.f 36.f $4$ $8.828$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-1}) \) \(4\) \(0\) \(8\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q+(2-2\zeta_{12})q^{2}-4\zeta_{12}q^{4}+(4\zeta_{12}+\cdots)q^{5}+\cdots\)
324.3.f.o 324.f 36.f $8$ $8.828$ 8.0.207360000.1 None \(0\) \(0\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{2}+\beta _{3})q^{2}+(-1+\beta _{1}+\beta _{7})q^{4}+\cdots\)
324.3.f.p 324.f 36.f $8$ $8.828$ 8.0.207360000.1 None \(0\) \(0\) \(0\) \(12\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{2}q^{2}+(1-\beta _{5}-\beta _{7})q^{4}+(\beta _{2}+\beta _{3}+\cdots)q^{5}+\cdots\)
324.3.f.q 324.f 36.f $12$ $8.828$ 12.0.\(\cdots\).2 None \(-1\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{3}+\beta _{6})q^{2}+(1+\beta _{2}+\beta _{4})q^{4}+\cdots\)
324.3.f.r 324.f 36.f $12$ $8.828$ 12.0.\(\cdots\).2 None \(1\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{3}-\beta _{6})q^{2}+(1+\beta _{2}+\beta _{4})q^{4}+(1+\cdots)q^{5}+\cdots\)
324.3.f.s 324.f 36.f $16$ $8.828$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{5}q^{2}+(1-\beta _{2}-\beta _{3}-\beta _{7})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(324, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(324, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 2}\)