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Properties

Label 304.3.r
Level $304$
Weight $3$
Character orbit 304.r
Rep. character $\chi_{304}(65,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $38$
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.r (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 19 \)
Character field: \(\Q(\zeta_{6})\)
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 172 42 130
Cusp forms 148 38 110
Eisenstein series 24 4 20

Trace form

\( 38 q + 3 q^{3} - q^{5} + 20 q^{7} + 46 q^{9} + O(q^{10}) \) \( 38 q + 3 q^{3} - q^{5} + 20 q^{7} + 46 q^{9} + 4 q^{11} - 3 q^{13} + 3 q^{15} + 23 q^{17} + 18 q^{19} - 30 q^{21} - 23 q^{23} - 76 q^{25} - 3 q^{29} - 90 q^{33} + 122 q^{35} - 26 q^{39} + 57 q^{41} - 63 q^{43} + 28 q^{45} - 47 q^{47} + 146 q^{49} - 141 q^{51} + 21 q^{53} + 50 q^{55} + 37 q^{57} + 3 q^{59} + 7 q^{61} + 236 q^{63} + 387 q^{67} + 75 q^{71} + 27 q^{73} - 120 q^{77} - 165 q^{79} - 187 q^{81} - 76 q^{83} - 47 q^{85} - 330 q^{87} + 141 q^{89} - 234 q^{91} + 104 q^{93} + 201 q^{95} + 249 q^{97} + 164 q^{99} + O(q^{100}) \)

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.r.a 304.r 19.d $4$ $8.283$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(-6\) \(2\) \(-8\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-2+\beta _{1}+\beta _{2}-\beta _{3})q^{3}+\beta _{2}q^{5}+\cdots\)
304.3.r.b 304.r 19.d $6$ $8.283$ 6.0.6967728.1 None \(0\) \(9\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(2+\beta _{1}-\beta _{2}+\beta _{3}+\beta _{4})q^{3}+(2-\beta _{1}+\cdots)q^{5}+\cdots\)
304.3.r.c 304.r 19.d $8$ $8.283$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(-6\) \(-1\) \(12\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{3}+\beta _{4})q^{3}+(\beta _{5}-\beta _{6})q^{5}+(2-\beta _{2}+\cdots)q^{7}+\cdots\)
304.3.r.d 304.r 19.d $20$ $8.283$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(0\) \(6\) \(0\) \(16\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{1}q^{3}+(\beta _{4}-\beta _{19})q^{5}+(1+\beta _{3})q^{7}+\cdots\)

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

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