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Properties

Label 361.3.d
Level $361$
Weight $3$
Character orbit 361.d
Rep. character $\chi_{361}(69,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $96$
Newform subspaces $7$
Sturm bound $95$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 361 = 19^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 361.d (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 19 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 7 \)
Sturm bound: \(95\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(361, [\chi])\).

Total New Old
Modular forms 148 128 20
Cusp forms 108 96 12
Eisenstein series 40 32 8

Trace form

\( 96 q + 3 q^{2} + 9 q^{3} + 79 q^{4} + 3 q^{5} - q^{6} + 6 q^{7} + 85 q^{9} + O(q^{10}) \) \( 96 q + 3 q^{2} + 9 q^{3} + 79 q^{4} + 3 q^{5} - q^{6} + 6 q^{7} + 85 q^{9} + 60 q^{10} - 36 q^{11} - 30 q^{13} - 54 q^{14} + 18 q^{15} - 97 q^{16} + 49 q^{17} - 128 q^{20} + 102 q^{21} + 39 q^{22} + 8 q^{23} + 91 q^{24} - 49 q^{25} + 196 q^{26} - 8 q^{28} + 12 q^{29} - 384 q^{30} - 51 q^{32} - 123 q^{33} + 6 q^{34} + 17 q^{35} + 74 q^{36} + 32 q^{39} + 96 q^{40} - 63 q^{41} + 44 q^{42} + 39 q^{43} + 101 q^{44} + 58 q^{45} - 35 q^{47} + 147 q^{48} - 110 q^{49} - 132 q^{51} - 162 q^{52} + 12 q^{53} + 11 q^{54} + 15 q^{55} - 192 q^{58} + 147 q^{59} + 222 q^{60} - 57 q^{61} + 122 q^{62} - 101 q^{63} + 62 q^{64} - 253 q^{66} - 201 q^{67} + 136 q^{68} + 198 q^{70} + 102 q^{71} - 210 q^{72} + 16 q^{73} - 204 q^{74} + 346 q^{77} - 450 q^{78} - 286 q^{80} - 60 q^{81} + 159 q^{82} - 250 q^{83} + 85 q^{85} + 270 q^{86} + 696 q^{87} + 72 q^{89} + 438 q^{90} + 216 q^{91} + 202 q^{92} + 140 q^{93} - 370 q^{96} - 21 q^{97} - 411 q^{98} + 45 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(361, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
361.3.d.a 361.d 19.d $2$ $9.837$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-19}) \) \(0\) \(0\) \(9\) \(-10\) $\mathrm{U}(1)[D_{6}]$ \(q-4\zeta_{6}q^{4}+(9-9\zeta_{6})q^{5}-5q^{7}-9\zeta_{6}q^{9}+\cdots\)
361.3.d.b 361.d 19.d $4$ $9.837$ \(\Q(\sqrt{-3}, \sqrt{-13})\) None \(0\) \(0\) \(-8\) \(-20\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{2}-\beta _{1}q^{3}+9\beta _{2}q^{4}+(-4+4\beta _{2}+\cdots)q^{5}+\cdots\)
361.3.d.c 361.d 19.d $6$ $9.837$ 6.0.6967728.1 None \(3\) \(9\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1+\beta _{5})q^{2}+(-\beta _{1}-2\beta _{2}-\beta _{3}-\beta _{5})q^{3}+\cdots\)
361.3.d.d 361.d 19.d $12$ $9.837$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(-9\) \(-3\) \(-12\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{1}+\beta _{4}+\beta _{5}+\beta _{7})q^{2}+(-1+\cdots)q^{3}+\cdots\)
361.3.d.e 361.d 19.d $12$ $9.837$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(14\) \(44\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{4}+\beta _{6})q^{2}+(\beta _{9}+\beta _{11})q^{3}+(2+\cdots)q^{4}+\cdots\)
361.3.d.f 361.d 19.d $12$ $9.837$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(9\) \(-3\) \(-12\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{1}-\beta _{4}+\beta _{6}+\beta _{8})q^{2}+(\beta _{1}-\beta _{3}+\cdots)q^{3}+\cdots\)
361.3.d.g 361.d 19.d $48$ $9.837$ None \(0\) \(0\) \(-4\) \(16\) $\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{3}^{\mathrm{old}}(361, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(361, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(19, [\chi])\)\(^{\oplus 2}\)