Properties

Label 396.2.h
Level $396$
Weight $2$
Character orbit 396.h
Rep. character $\chi_{396}(307,\cdot)$
Character field $\Q$
Dimension $28$
Newform subspaces $4$
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.h (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 44 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
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 80 32 48
Cusp forms 64 28 36
Eisenstein series 16 4 12

Trace form

\( 28 q - 4 q^{4} + 4 q^{5} + O(q^{10}) \) \( 28 q - 4 q^{4} + 4 q^{5} + 4 q^{14} - 16 q^{16} + 24 q^{20} - 16 q^{22} + 24 q^{25} - 12 q^{26} + 12 q^{37} - 24 q^{44} + 12 q^{49} + 24 q^{53} - 12 q^{56} - 16 q^{58} - 64 q^{64} - 16 q^{70} - 16 q^{77} - 60 q^{80} - 56 q^{86} + 12 q^{89} + 48 q^{92} - 52 q^{97} + O(q^{100}) \)

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.h.a 396.h 44.c $4$ $3.162$ \(\Q(\sqrt{-2}, \sqrt{-11})\) \(\Q(\sqrt{-33}) \) 396.2.h.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{2}q^{2}-2q^{4}-\beta _{1}q^{7}-2\beta _{2}q^{8}+\cdots\)
396.2.h.b 396.h 44.c $4$ $3.162$ \(\Q(\sqrt{2}, \sqrt{-3})\) None 44.2.c.a \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-1+\beta _{2})q^{4}+q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)
396.2.h.c 396.h 44.c $8$ $3.162$ 8.0.207360000.1 None 396.2.h.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{2}+(1+\beta _{5})q^{4}+(-\beta _{4}+\beta _{7})q^{5}+\cdots\)
396.2.h.d 396.h 44.c $12$ $3.162$ 12.0.\(\cdots\).2 None 132.2.h.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{1}-\beta _{4}-\beta _{5}+\beta _{7}+\cdots)q^{5}+\cdots\)

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

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