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Properties

Label 637.2.e
Level $637$
Weight $2$
Character orbit 637.e
Rep. character $\chi_{637}(79,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $80$
Newform subspaces $15$
Sturm bound $130$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 15 \)
Sturm bound: \(130\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(637, [\chi])\).

Total New Old
Modular forms 148 80 68
Cusp forms 116 80 36
Eisenstein series 32 0 32

Trace form

\( 80 q + 4 q^{2} - 36 q^{4} + 2 q^{5} - 12 q^{8} - 44 q^{9} + O(q^{10}) \) \( 80 q + 4 q^{2} - 36 q^{4} + 2 q^{5} - 12 q^{8} - 44 q^{9} - 10 q^{10} + 6 q^{11} - 10 q^{12} - 4 q^{13} + 20 q^{15} - 12 q^{16} - 4 q^{17} + 10 q^{18} + 8 q^{19} - 32 q^{20} - 68 q^{22} - 2 q^{23} + 20 q^{24} - 34 q^{25} + 6 q^{26} - 4 q^{29} + 30 q^{30} + 4 q^{31} + 46 q^{32} + 8 q^{33} + 32 q^{34} + 4 q^{36} + 24 q^{37} - 26 q^{38} + 8 q^{40} - 28 q^{41} + 12 q^{43} + 18 q^{44} - 32 q^{45} + 52 q^{46} + 2 q^{47} + 76 q^{48} - 76 q^{50} - 20 q^{51} + 6 q^{52} + 26 q^{53} + 18 q^{54} - 30 q^{58} + 10 q^{59} - 24 q^{60} - 12 q^{61} + 76 q^{62} - 28 q^{64} + 4 q^{65} + 6 q^{66} + 24 q^{67} - 18 q^{68} - 56 q^{69} - 60 q^{71} + 28 q^{72} + 22 q^{73} - 10 q^{74} - 20 q^{75} - 48 q^{76} + 20 q^{78} + 34 q^{79} + 70 q^{80} - 72 q^{81} + 44 q^{82} - 12 q^{83} - 60 q^{85} + 14 q^{86} - 28 q^{87} - 4 q^{89} + 88 q^{90} + 140 q^{92} + 66 q^{93} + 46 q^{94} + 22 q^{95} - 22 q^{96} + 68 q^{97} + 72 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(637, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
637.2.e.a 637.e 7.c $2$ $5.086$ \(\Q(\sqrt{-3}) \) None \(-1\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{4}-3q^{8}+3\zeta_{6}q^{9}+\cdots\)
637.2.e.b 637.e 7.c $2$ $5.086$ \(\Q(\sqrt{-3}) \) None \(0\) \(-2\) \(-3\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}-3\zeta_{6}q^{5}+\cdots\)
637.2.e.c 637.e 7.c $2$ $5.086$ \(\Q(\sqrt{-3}) \) None \(0\) \(2\) \(3\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}+3\zeta_{6}q^{5}+\cdots\)
637.2.e.d 637.e 7.c $2$ $5.086$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(-3\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{4}-3\zeta_{6}q^{5}+\cdots\)
637.2.e.e 637.e 7.c $2$ $5.086$ \(\Q(\sqrt{-3}) \) None \(2\) \(0\) \(3\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{4}+3\zeta_{6}q^{5}+\cdots\)
637.2.e.f 637.e 7.c $4$ $5.086$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{3})q^{3}+(-3+\beta _{1}+\cdots)q^{5}+\cdots\)
637.2.e.g 637.e 7.c $4$ $5.086$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(6\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+(-\beta _{1}-\beta _{3})q^{3}+(3-\beta _{1}+\cdots)q^{5}+\cdots\)
637.2.e.h 637.e 7.c $4$ $5.086$ \(\Q(\sqrt{-3}, \sqrt{5})\) None \(3\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{1}+\beta _{3})q^{2}+(2\beta _{1}+2\beta _{2}-\beta _{3})q^{3}+\cdots\)
637.2.e.i 637.e 7.c $6$ $5.086$ 6.0.2696112.1 None \(-1\) \(-2\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(-\beta _{1}-\beta _{2}-\beta _{3}-\beta _{4}+\beta _{5})q^{3}+\cdots\)
637.2.e.j 637.e 7.c $6$ $5.086$ 6.0.2696112.1 None \(-1\) \(2\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2}+\beta _{3}+\beta _{4}-\beta _{5})q^{3}+\cdots\)
637.2.e.k 637.e 7.c $6$ $5.086$ 6.0.4406832.1 None \(2\) \(-4\) \(-5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}-\beta _{2}+\beta _{4})q^{2}+(-1+\beta _{3}+\beta _{4}+\cdots)q^{3}+\cdots\)
637.2.e.l 637.e 7.c $6$ $5.086$ 6.0.4406832.1 None \(2\) \(4\) \(5\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}-\beta _{2}+\beta _{4})q^{2}+(1-\beta _{3}-\beta _{4}+\cdots)q^{3}+\cdots\)
637.2.e.m 637.e 7.c $10$ $5.086$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-4\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}-\beta _{7})q^{2}+(\beta _{4}-\beta _{9})q^{3}+(-2+\cdots)q^{4}+\cdots\)
637.2.e.n 637.e 7.c $12$ $5.086$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(-8\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{3}+\beta _{11})q^{2}+(\beta _{1}+\beta _{2}-\beta _{8}+\cdots)q^{3}+\cdots\)
637.2.e.o 637.e 7.c $12$ $5.086$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(8\) \(6\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{3}+\beta _{11})q^{2}+(-\beta _{1}-\beta _{2}+\beta _{8}+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(637, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(637, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(91, [\chi])\)\(^{\oplus 2}\)