Properties

Label 324.4.e
Level $324$
Weight $4$
Character orbit 324.e
Rep. character $\chi_{324}(109,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $24$
Newform subspaces $9$
Sturm bound $216$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 324.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 9 \)
Sturm bound: \(216\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(324, [\chi])\).

Total New Old
Modular forms 360 24 336
Cusp forms 288 24 264
Eisenstein series 72 0 72

Trace form

\( 24 q + 30 q^{7} + O(q^{10}) \) \( 24 q + 30 q^{7} - 60 q^{13} + 120 q^{19} - 444 q^{25} - 186 q^{31} + 696 q^{37} - 132 q^{43} - 792 q^{49} + 2268 q^{55} - 96 q^{61} - 204 q^{67} - 780 q^{73} - 1356 q^{79} - 1134 q^{85} - 2352 q^{91} - 384 q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(324, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
324.4.e.a 324.e 9.c $2$ $19.117$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-18\) \(-8\) $\mathrm{SU}(2)[C_{3}]$ \(q-18\zeta_{6}q^{5}+(-8+8\zeta_{6})q^{7}+(6^{2}-6^{2}\zeta_{6})q^{11}+\cdots\)
324.4.e.b 324.e 9.c $2$ $19.117$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-9\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q-9\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+(-63+63\zeta_{6})q^{11}+\cdots\)
324.4.e.c 324.e 9.c $2$ $19.117$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(-3\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q-3\zeta_{6}q^{5}+(4-4\zeta_{6})q^{7}+(24-24\zeta_{6})q^{11}+\cdots\)
324.4.e.d 324.e 9.c $2$ $19.117$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-17\) $\mathrm{U}(1)[D_{3}]$ \(q+(-17+17\zeta_{6})q^{7}-89\zeta_{6}q^{13}+107q^{19}+\cdots\)
324.4.e.e 324.e 9.c $2$ $19.117$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(37\) $\mathrm{U}(1)[D_{3}]$ \(q+(37-37\zeta_{6})q^{7}+19\zeta_{6}q^{13}-163q^{19}+\cdots\)
324.4.e.f 324.e 9.c $2$ $19.117$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(3\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+3\zeta_{6}q^{5}+(4-4\zeta_{6})q^{7}+(-24+24\zeta_{6})q^{11}+\cdots\)
324.4.e.g 324.e 9.c $2$ $19.117$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(9\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+9\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+(63-63\zeta_{6})q^{11}+\cdots\)
324.4.e.h 324.e 9.c $2$ $19.117$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(18\) \(-8\) $\mathrm{SU}(2)[C_{3}]$ \(q+18\zeta_{6}q^{5}+(-8+8\zeta_{6})q^{7}+(-6^{2}+\cdots)q^{11}+\cdots\)
324.4.e.i 324.e 9.c $8$ $19.117$ 8.0.49787136.1 None \(0\) \(0\) \(0\) \(16\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{3}q^{5}+(-4\beta _{1}+\beta _{4}-\beta _{7})q^{7}+(-\beta _{2}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(324, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(324, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(9, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(27, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(81, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(162, [\chi])\)\(^{\oplus 2}\)