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Properties

Label 348.2.m
Level $348$
Weight $2$
Character orbit 348.m
Rep. character $\chi_{348}(25,\cdot)$
Character field $\Q(\zeta_{7})$
Dimension $24$
Newform subspaces $2$
Sturm bound $120$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 348 = 2^{2} \cdot 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 348.m (of order \(7\) and degree \(6\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 29 \)
Character field: \(\Q(\zeta_{7})\)
Newform subspaces: \( 2 \)
Sturm bound: \(120\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(348, [\chi])\).

Total New Old
Modular forms 396 24 372
Cusp forms 324 24 300
Eisenstein series 72 0 72

Trace form

\( 24 q + 4 q^{5} + 4 q^{7} - 4 q^{9} + O(q^{10}) \) \( 24 q + 4 q^{5} + 4 q^{7} - 4 q^{9} + 18 q^{13} - 14 q^{15} + 8 q^{17} - 8 q^{19} + 10 q^{23} + 10 q^{25} + 28 q^{29} + 36 q^{31} - 40 q^{35} + 48 q^{37} + 8 q^{39} - 8 q^{41} + 22 q^{43} - 10 q^{45} - 16 q^{47} - 20 q^{49} - 6 q^{51} - 58 q^{53} - 42 q^{55} - 32 q^{57} - 28 q^{59} + 16 q^{61} - 10 q^{63} - 66 q^{65} + 4 q^{67} - 28 q^{69} + 10 q^{71} - 16 q^{73} - 16 q^{75} - 2 q^{77} + 14 q^{79} - 4 q^{81} - 24 q^{83} + 54 q^{85} - 28 q^{87} + 6 q^{89} + 22 q^{91} + 4 q^{93} + 24 q^{95} + 8 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(348, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
348.2.m.a 348.m 29.d $12$ $2.779$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(-2\) \(-5\) \(2\) $\mathrm{SU}(2)[C_{7}]$ \(q+\beta _{11}q^{3}+(-1+\beta _{2}-\beta _{3}+\beta _{5}+\beta _{6}+\cdots)q^{5}+\cdots\)
348.2.m.b 348.m 29.d $12$ $2.779$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(2\) \(9\) \(2\) $\mathrm{SU}(2)[C_{7}]$ \(q+\beta _{8}q^{3}+(1+\beta _{3}-\beta _{5}+\beta _{7})q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(348, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(348, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(29, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(58, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(87, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(116, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(174, [\chi])\)\(^{\oplus 2}\)