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Properties

Label 196.2.f
Level $196$
Weight $2$
Character orbit 196.f
Rep. character $\chi_{196}(19,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $32$
Newform subspaces $4$
Sturm bound $56$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 196 = 2^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 196.f (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 28 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 4 \)
Sturm bound: \(56\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(196, [\chi])\).

Total New Old
Modular forms 72 48 24
Cusp forms 40 32 8
Eisenstein series 32 16 16

Trace form

\( 32 q + 3 q^{2} - q^{4} + 6 q^{5} - 6 q^{8} - 6 q^{9} + O(q^{10}) \) \( 32 q + 3 q^{2} - q^{4} + 6 q^{5} - 6 q^{8} - 6 q^{9} - 6 q^{10} - 12 q^{12} - 13 q^{16} + 6 q^{17} + 17 q^{18} + 12 q^{24} - 2 q^{25} + 12 q^{26} - 56 q^{29} + 34 q^{30} - 17 q^{32} - 6 q^{33} - 22 q^{36} + 14 q^{37} - 18 q^{38} - 12 q^{40} - 34 q^{44} - 8 q^{46} - 26 q^{50} - 10 q^{53} + 18 q^{54} + 28 q^{57} + 14 q^{58} - 20 q^{60} + 18 q^{61} - 22 q^{64} - 20 q^{65} + 6 q^{66} + 19 q^{72} - 30 q^{73} + 24 q^{74} + 40 q^{78} + 24 q^{80} + 20 q^{81} - 12 q^{82} - 52 q^{85} - 54 q^{86} + 54 q^{88} - 54 q^{89} + 92 q^{92} + 38 q^{93} + 30 q^{94} + 24 q^{96} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(196, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
196.2.f.a 196.f 28.f $4$ $1.565$ \(\Q(\zeta_{12})\) None \(-2\) \(0\) \(6\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(-\zeta_{12}+\cdots)q^{3}+\cdots\)
196.2.f.b 196.f 28.f $4$ $1.565$ \(\Q(\sqrt{-3}, \sqrt{-7})\) \(\Q(\sqrt{-7}) \) \(1\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q+\beta _{3}q^{2}+(1+\beta _{1}-\beta _{2})q^{4}+(3-\beta _{1}+\cdots)q^{8}+\cdots\)
196.2.f.c 196.f 28.f $8$ $1.565$ 8.0.339738624.1 \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q+(-\beta _{2}-\beta _{5})q^{2}+2\beta _{4}q^{4}-\beta _{3}q^{5}+\cdots\)
196.2.f.d 196.f 28.f $16$ $1.565$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{6}-\beta _{11})q^{2}+(2\beta _{1}+\beta _{3})q^{3}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(196, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(196, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 2}\)