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Properties

Label 304.5.r
Level $304$
Weight $5$
Character orbit 304.r
Rep. character $\chi_{304}(65,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $78$
Newform subspaces $4$
Sturm bound $200$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
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: \(200\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(304, [\chi])\).

Total New Old
Modular forms 332 82 250
Cusp forms 308 78 230
Eisenstein series 24 4 20

Trace form

\( 78 q + 3 q^{3} - q^{5} + 36 q^{7} + 1054 q^{9} + O(q^{10}) \) \( 78 q + 3 q^{3} - q^{5} + 36 q^{7} + 1054 q^{9} + 4 q^{11} - 3 q^{13} + 3 q^{15} + 47 q^{17} + 1186 q^{19} - 246 q^{21} + 721 q^{23} - 4376 q^{25} - 3 q^{29} - 2286 q^{33} - 2254 q^{35} - 3002 q^{39} - 3531 q^{41} + 1409 q^{43} + 1084 q^{45} + 1441 q^{47} + 24106 q^{49} - 6045 q^{51} - 2451 q^{53} + 7586 q^{55} - 1139 q^{57} + 3 q^{59} - 1713 q^{61} - 3628 q^{63} - 8445 q^{67} - 9069 q^{71} - 137 q^{73} - 13928 q^{77} + 23859 q^{79} - 16183 q^{81} + 48868 q^{83} + 9505 q^{85} - 7194 q^{87} + 7197 q^{89} - 21210 q^{91} - 14320 q^{93} - 15023 q^{95} - 28299 q^{97} - 4924 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{5}^{\mathrm{new}}(304, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
304.5.r.a 304.r 19.d $10$ $31.424$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(0\) \(-9\) \(8\) \(24\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{3}-\beta _{5})q^{3}+(2+\beta _{1}+\beta _{2}-3\beta _{3}+\cdots)q^{5}+\cdots\)
304.5.r.b 304.r 19.d $12$ $31.424$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(12\) \(9\) \(52\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{1}-\beta _{3})q^{3}+(1-\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)
304.5.r.c 304.r 19.d $16$ $31.424$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(12\) \(-18\) \(-72\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1-\beta _{3})q^{3}+(-2-2\beta _{5}+\beta _{11})q^{5}+\cdots\)
304.5.r.d 304.r 19.d $40$ $31.424$ None \(0\) \(-12\) \(0\) \(32\) $\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{5}^{\mathrm{old}}(304, [\chi])\) into lower level spaces

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