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Properties

Label 396.2.j
Level $396$
Weight $2$
Character orbit 396.j
Rep. character $\chi_{396}(37,\cdot)$
Character field $\Q(\zeta_{5})$
Dimension $20$
Newform subspaces $4$
Sturm bound $144$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 396 = 2^{2} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 396.j (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 11 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 4 \)
Sturm bound: \(144\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\)

Dimensions

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

Total New Old
Modular forms 336 20 316
Cusp forms 240 20 220
Eisenstein series 96 0 96

Trace form

\( 20 q + q^{5} + q^{7} + O(q^{10}) \) \( 20 q + q^{5} + q^{7} - 6 q^{11} - 9 q^{13} + 11 q^{17} - q^{19} + 4 q^{23} - 2 q^{25} + 17 q^{29} + 11 q^{31} + 29 q^{35} + 27 q^{37} + 33 q^{41} + 16 q^{43} - 13 q^{47} - 4 q^{49} - 29 q^{53} + 7 q^{55} - 47 q^{59} - 25 q^{61} - 50 q^{65} - 48 q^{67} - 37 q^{71} - 33 q^{73} - 51 q^{77} - 29 q^{79} + 33 q^{83} - 27 q^{85} - 12 q^{89} - 27 q^{91} + 15 q^{95} + 23 q^{97} + O(q^{100}) \)

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.j.a 396.j 11.c $4$ $3.162$ \(\Q(\zeta_{10})\) None 44.2.e.a \(0\) \(0\) \(-3\) \(-7\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1-\zeta_{10}^{2})q^{5}+(-3+3\zeta_{10}+2\zeta_{10}^{3})q^{7}+\cdots\)
396.2.j.b 396.j 11.c $4$ $3.162$ \(\Q(\zeta_{10})\) None 132.2.i.a \(0\) \(0\) \(-3\) \(7\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1-\zeta_{10}^{2})q^{5}+(2-2\zeta_{10}+\zeta_{10}^{3})q^{7}+\cdots\)
396.2.j.c 396.j 11.c $4$ $3.162$ \(\Q(\zeta_{10})\) None 132.2.i.b \(0\) \(0\) \(7\) \(3\) $\mathrm{SU}(2)[C_{5}]$ \(q+(3-2\zeta_{10}+3\zeta_{10}^{2})q^{5}+(2-2\zeta_{10}+\cdots)q^{7}+\cdots\)
396.2.j.d 396.j 11.c $8$ $3.162$ 8.0.484000000.6 None 396.2.j.d \(0\) \(0\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-2\beta _{1}+\beta _{6}-\beta _{7})q^{5}+(-1-\beta _{2}+\cdots)q^{7}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(396, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(22, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(33, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(44, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(66, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(99, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(132, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(198, [\chi])\)\(^{\oplus 2}\)