Properties

Label 342.4.g
Level $342$
Weight $4$
Character orbit 342.g
Rep. character $\chi_{342}(163,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $50$
Newform subspaces $10$
Sturm bound $240$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 342.g (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 19 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 10 \)
Sturm bound: \(240\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(342, [\chi])\).

Total New Old
Modular forms 376 50 326
Cusp forms 344 50 294
Eisenstein series 32 0 32

Trace form

\( 50 q - 2 q^{2} - 100 q^{4} + 8 q^{5} + 60 q^{7} + 16 q^{8} + O(q^{10}) \) \( 50 q - 2 q^{2} - 100 q^{4} + 8 q^{5} + 60 q^{7} + 16 q^{8} + 20 q^{10} + 134 q^{11} - 72 q^{13} - 12 q^{14} - 400 q^{16} - 12 q^{17} + 243 q^{19} - 64 q^{20} + 126 q^{22} - 46 q^{23} - 861 q^{25} + 344 q^{26} - 120 q^{28} - 80 q^{29} + 368 q^{31} - 32 q^{32} + 76 q^{34} - 444 q^{35} - 248 q^{37} - 88 q^{38} + 80 q^{40} - 959 q^{41} - 874 q^{43} - 268 q^{44} - 1584 q^{46} - 1192 q^{47} + 3578 q^{49} + 2060 q^{50} - 288 q^{52} - 964 q^{53} + 1210 q^{55} + 96 q^{56} + 424 q^{58} + 375 q^{59} - 1304 q^{61} + 280 q^{62} + 3200 q^{64} + 1780 q^{65} - 905 q^{67} + 96 q^{68} + 600 q^{70} - 580 q^{71} - 2007 q^{73} - 56 q^{74} + 252 q^{76} - 2652 q^{77} + 1706 q^{79} + 128 q^{80} + 2018 q^{82} + 5598 q^{83} + 1566 q^{85} + 424 q^{86} - 1008 q^{88} + 300 q^{89} - 256 q^{91} - 184 q^{92} - 3192 q^{94} - 4874 q^{95} - 3381 q^{97} - 2434 q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(342, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
342.4.g.a $2$ $20.179$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(-6\) \(38\) \(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(-6+6\zeta_{6})q^{5}+\cdots\)
342.4.g.b $2$ $20.179$ \(\Q(\sqrt{-3}) \) None \(-2\) \(0\) \(2\) \(-42\) \(q+(-2+2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(2-2\zeta_{6})q^{5}+\cdots\)
342.4.g.c $2$ $20.179$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(-12\) \(16\) \(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(-12+12\zeta_{6})q^{5}+\cdots\)
342.4.g.d $2$ $20.179$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(3\) \(-64\) \(q+(2-2\zeta_{6})q^{2}-4\zeta_{6}q^{4}+(3-3\zeta_{6})q^{5}+\cdots\)
342.4.g.e $4$ $20.179$ \(\Q(\sqrt{-3}, \sqrt{-10})\) None \(4\) \(0\) \(8\) \(36\) \(q+2\beta _{1}q^{2}+(-4+4\beta _{1})q^{4}+(4\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
342.4.g.f $6$ $20.179$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-6\) \(0\) \(1\) \(52\) \(q-2\beta _{4}q^{2}+(-4+4\beta _{4})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
342.4.g.g $6$ $20.179$ 6.0.6967728.1 None \(-6\) \(0\) \(2\) \(-34\) \(q+2\beta _{3}q^{2}+(-4-4\beta _{3})q^{4}+(-\beta _{3}+\cdots)q^{5}+\cdots\)
342.4.g.h $6$ $20.179$ 6.0.627014547.1 None \(6\) \(0\) \(10\) \(14\) \(q+2\beta _{1}q^{2}+(-4+4\beta _{1})q^{4}+(3\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
342.4.g.i $10$ $20.179$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-10\) \(0\) \(10\) \(22\) \(q+(-2+2\beta _{1})q^{2}-4\beta _{1}q^{4}+(2-2\beta _{1}+\cdots)q^{5}+\cdots\)
342.4.g.j $10$ $20.179$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(10\) \(0\) \(-10\) \(22\) \(q+(2-2\beta _{1})q^{2}-4\beta _{1}q^{4}+(-2+2\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(342, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(342, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(19, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(38, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(57, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(114, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(171, [\chi])\)\(^{\oplus 2}\)