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Properties

Label 304.3.z
Level $304$
Weight $3$
Character orbit 304.z
Rep. character $\chi_{304}(33,\cdot)$
Character field $\Q(\zeta_{18})$
Dimension $114$
Newform subspaces $4$
Sturm bound $120$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 304.z (of order \(18\) and degree \(6\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 19 \)
Character field: \(\Q(\zeta_{18})\)
Newform subspaces: \( 4 \)
Sturm bound: \(120\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(304, [\chi])\).

Total New Old
Modular forms 516 126 390
Cusp forms 444 114 330
Eisenstein series 72 12 60

Trace form

\( 114 q + 6 q^{3} - 6 q^{5} + 3 q^{7} + 6 q^{9} + O(q^{10}) \) \( 114 q + 6 q^{3} - 6 q^{5} + 3 q^{7} + 6 q^{9} + 3 q^{11} - 6 q^{13} + 6 q^{15} - 6 q^{17} - 42 q^{19} + 21 q^{21} + 6 q^{23} - 6 q^{25} + 9 q^{27} - 6 q^{29} + 9 q^{31} + 81 q^{33} - 69 q^{35} + 12 q^{39} - 66 q^{41} + 198 q^{43} - 3 q^{45} + 150 q^{47} - 276 q^{49} + 150 q^{51} - 30 q^{53} + 75 q^{55} - 6 q^{57} + 6 q^{59} + 66 q^{61} + 219 q^{63} - 369 q^{65} + 486 q^{67} + 423 q^{69} + 150 q^{71} - 198 q^{73} + 282 q^{77} - 330 q^{79} + 99 q^{81} - 429 q^{83} - 438 q^{85} - 429 q^{87} + 354 q^{89} - 909 q^{91} - 33 q^{93} - 402 q^{95} - 258 q^{97} - 237 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(304, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
304.3.z.a 304.z 19.f $12$ $8.283$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(-6\) \(-6\) $\mathrm{SU}(2)[C_{18}]$ \(q+(-\beta _{1}+\beta _{4}+\beta _{5}-\beta _{6}+\beta _{7}+\beta _{8}+\cdots)q^{3}+\cdots\)
304.3.z.b 304.z 19.f $18$ $8.283$ \(\mathbb{Q}[x]/(x^{18} + \cdots)\) None \(0\) \(6\) \(0\) \(-9\) $\mathrm{SU}(2)[C_{18}]$ \(q+(1-\beta _{4}-\beta _{5}+\beta _{7}+\beta _{10}-\beta _{12}+\cdots)q^{3}+\cdots\)
304.3.z.c 304.z 19.f $24$ $8.283$ None \(0\) \(6\) \(0\) \(18\) $\mathrm{SU}(2)[C_{18}]$
304.3.z.d 304.z 19.f $60$ $8.283$ None \(0\) \(-6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{18}]$

Decomposition of \(S_{3}^{\mathrm{old}}(304, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(304, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(19, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(38, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(76, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(152, [\chi])\)\(^{\oplus 2}\)