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Properties

Label 369.3.l
Level $369$
Weight $3$
Character orbit 369.l
Rep. character $\chi_{369}(55,\cdot)$
Character field $\Q(\zeta_{8})$
Dimension $136$
Newform subspaces $4$
Sturm bound $126$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 369 = 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 369.l (of order \(8\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 41 \)
Character field: \(\Q(\zeta_{8})\)
Newform subspaces: \( 4 \)
Sturm bound: \(126\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(369, [\chi])\).

Total New Old
Modular forms 352 144 208
Cusp forms 320 136 184
Eisenstein series 32 8 24

Trace form

\( 136 q + 4 q^{5} - 4 q^{7} + 12 q^{8} + O(q^{10}) \) \( 136 q + 4 q^{5} - 4 q^{7} + 12 q^{8} - 8 q^{10} + 4 q^{11} - 32 q^{13} + 96 q^{14} - 504 q^{16} - 60 q^{17} + 44 q^{19} - 128 q^{20} + 4 q^{22} + 48 q^{26} + 60 q^{28} + 32 q^{29} - 152 q^{32} + 20 q^{34} + 84 q^{35} + 24 q^{37} - 132 q^{38} - 108 q^{41} + 32 q^{43} - 268 q^{44} + 4 q^{46} - 196 q^{47} + 164 q^{49} - 68 q^{50} - 272 q^{52} + 72 q^{53} - 108 q^{55} - 104 q^{56} + 52 q^{58} + 584 q^{59} - 36 q^{61} - 344 q^{62} + 432 q^{65} + 356 q^{67} + 180 q^{68} - 148 q^{70} + 396 q^{71} + 40 q^{73} + 148 q^{74} - 680 q^{76} + 120 q^{77} + 160 q^{79} + 452 q^{80} - 88 q^{82} + 624 q^{83} - 124 q^{85} - 44 q^{88} - 264 q^{89} + 568 q^{91} - 1600 q^{92} - 740 q^{94} - 240 q^{95} + 328 q^{97} + 184 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(369, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
369.3.l.a 369.l 41.e $4$ $10.055$ \(\Q(\zeta_{8})\) None 41.3.e.a \(0\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{8}]$ \(q+\zeta_{8}q^{2}-3\zeta_{8}^{2}q^{4}+(-2+2\zeta_{8}^{2}+\cdots)q^{5}+\cdots\)
369.3.l.b 369.l 41.e $20$ $10.055$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None 41.3.e.b \(8\) \(0\) \(12\) \(-4\) $\mathrm{SU}(2)[C_{8}]$ \(q+\beta _{11}q^{2}+(\beta _{1}-\beta _{6}-\beta _{7}+\beta _{8}-\beta _{9}+\cdots)q^{4}+\cdots\)
369.3.l.c 369.l 41.e $56$ $10.055$ None 123.3.h.a \(-8\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{8}]$
369.3.l.d 369.l 41.e $56$ $10.055$ None 369.3.l.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{8}]$

Decomposition of \(S_{3}^{\mathrm{old}}(369, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(369, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(41, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(123, [\chi])\)\(^{\oplus 2}\)