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Properties

Label 1242.2.e
Level $1242$
Weight $2$
Character orbit 1242.e
Rep. character $\chi_{1242}(415,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $44$
Newform subspaces $5$
Sturm bound $432$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 1242 = 2 \cdot 3^{3} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1242.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 5 \)
Sturm bound: \(432\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1242, [\chi])\).

Total New Old
Modular forms 456 44 412
Cusp forms 408 44 364
Eisenstein series 48 0 48

Trace form

\( 44 q - 2 q^{2} - 22 q^{4} + 4 q^{7} + 4 q^{8} + O(q^{10}) \) \( 44 q - 2 q^{2} - 22 q^{4} + 4 q^{7} + 4 q^{8} + 10 q^{11} + 4 q^{13} - 22 q^{16} - 28 q^{17} + 4 q^{19} - 6 q^{22} - 22 q^{25} + 24 q^{26} - 8 q^{28} - 28 q^{29} + 4 q^{31} - 2 q^{32} - 6 q^{34} - 8 q^{37} + 14 q^{38} - 14 q^{41} - 2 q^{43} - 20 q^{44} + 20 q^{47} - 18 q^{49} + 2 q^{50} + 4 q^{52} + 80 q^{53} - 24 q^{55} + 12 q^{58} + 2 q^{59} + 28 q^{61} + 8 q^{62} + 44 q^{64} - 20 q^{65} + 22 q^{67} + 14 q^{68} + 12 q^{70} - 72 q^{71} - 44 q^{73} - 2 q^{76} + 16 q^{77} - 8 q^{79} - 36 q^{82} + 12 q^{83} + 30 q^{86} - 6 q^{88} - 72 q^{89} - 16 q^{91} + 12 q^{95} + 22 q^{97} + 4 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(1242, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1242.2.e.a 1242.e 9.c $2$ $9.917$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}-\zeta_{6}q^{4}+4\zeta_{6}q^{5}+\cdots\)
1242.2.e.b 1242.e 9.c $10$ $9.917$ 10.0.\(\cdots\).1 None \(-5\) \(0\) \(1\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{5})q^{2}+\beta _{5}q^{4}+\beta _{2}q^{5}+(1+\cdots)q^{7}+\cdots\)
1242.2.e.c 1242.e 9.c $10$ $9.917$ 10.0.\(\cdots\).1 None \(5\) \(0\) \(-5\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{1})q^{2}+\beta _{1}q^{4}+(\beta _{1}+\beta _{6})q^{5}+\cdots\)
1242.2.e.d 1242.e 9.c $10$ $9.917$ 10.0.\(\cdots\).1 None \(5\) \(0\) \(5\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\beta _{2})q^{2}-\beta _{2}q^{4}+(\beta _{2}+\beta _{4}+\beta _{7}+\cdots)q^{5}+\cdots\)
1242.2.e.e 1242.e 9.c $12$ $9.917$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(-6\) \(0\) \(-5\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{2})q^{2}-\beta _{2}q^{4}+(-\beta _{2}-\beta _{8}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1242, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1242, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(207, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(414, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(621, [\chi])\)\(^{\oplus 2}\)