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Properties

Label 396.2.r
Level $396$
Weight $2$
Character orbit 396.r
Rep. character $\chi_{396}(19,\cdot)$
Character field $\Q(\zeta_{10})$
Dimension $112$
Newform subspaces $3$
Sturm bound $144$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 396 = 2^{2} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 396.r (of order \(10\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 44 \)
Character field: \(\Q(\zeta_{10})\)
Newform subspaces: \( 3 \)
Sturm bound: \(144\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(396, [\chi])\).

Total New Old
Modular forms 320 128 192
Cusp forms 256 112 144
Eisenstein series 64 16 48

Trace form

\( 112 q + 5 q^{2} - q^{4} + 6 q^{5} + 5 q^{8} + O(q^{10}) \) \( 112 q + 5 q^{2} - q^{4} + 6 q^{5} + 5 q^{8} - 10 q^{13} + 6 q^{14} - 9 q^{16} + 10 q^{17} + 6 q^{20} - 19 q^{22} - 34 q^{25} + 42 q^{26} + 10 q^{29} - 30 q^{34} + 18 q^{37} - 20 q^{38} - 40 q^{40} + 30 q^{41} - 6 q^{44} - 30 q^{46} - 2 q^{49} - 55 q^{50} - 50 q^{52} + 26 q^{53} - 48 q^{56} - 34 q^{58} - 10 q^{61} - 70 q^{62} - q^{64} - 60 q^{68} - 84 q^{70} + 10 q^{73} - 90 q^{74} + 6 q^{77} - 30 q^{80} + 25 q^{82} + 30 q^{85} + 31 q^{86} - 45 q^{88} - 12 q^{89} - 78 q^{92} + 90 q^{94} + 12 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(396, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
396.2.r.a 396.r 44.g $16$ $3.162$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 44.2.g.a \(5\) \(0\) \(6\) \(0\) $\mathrm{SU}(2)[C_{10}]$ \(q-\beta _{2}q^{2}+(-\beta _{2}-\beta _{4}-\beta _{5}+\beta _{6}+\beta _{10}+\cdots)q^{4}+\cdots\)
396.2.r.b 396.r 44.g $48$ $3.162$ None 132.2.j.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{10}]$
396.2.r.c 396.r 44.g $48$ $3.162$ None 396.2.r.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{10}]$

Decomposition of \(S_{2}^{\mathrm{old}}(396, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(396, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(44, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(132, [\chi])\)\(^{\oplus 2}\)