Properties

Label 637.2.a
Level $637$
Weight $2$
Character orbit 637.a
Rep. character $\chi_{637}(1,\cdot)$
Character field $\Q$
Dimension $41$
Newform subspaces $14$
Sturm bound $130$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(130\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(2\), \(3\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(637))\).

Total New Old
Modular forms 72 41 31
Cusp forms 57 41 16
Eisenstein series 15 0 15

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(7\)\(13\)FrickeDim.
\(+\)\(+\)\(+\)\(9\)
\(+\)\(-\)\(-\)\(11\)
\(-\)\(+\)\(-\)\(12\)
\(-\)\(-\)\(+\)\(9\)
Plus space\(+\)\(18\)
Minus space\(-\)\(23\)

Trace form

\( 41 q - q^{2} + 4 q^{3} + 43 q^{4} - 2 q^{5} - 9 q^{8} + 45 q^{9} + O(q^{10}) \) \( 41 q - q^{2} + 4 q^{3} + 43 q^{4} - 2 q^{5} - 9 q^{8} + 45 q^{9} - 2 q^{10} + 8 q^{12} - q^{13} + 4 q^{15} + 43 q^{16} - 2 q^{17} - q^{18} + 12 q^{19} + 2 q^{20} - 16 q^{22} - 16 q^{23} + 4 q^{24} + 23 q^{25} - 3 q^{26} + 4 q^{27} - 26 q^{29} - 36 q^{30} + 4 q^{31} - 49 q^{32} + 4 q^{33} - 2 q^{34} + 27 q^{36} - 10 q^{37} + 8 q^{38} + 4 q^{39} + 10 q^{40} - 2 q^{41} - 4 q^{43} - 24 q^{44} - 22 q^{45} - 16 q^{46} - 8 q^{47} - 12 q^{48} - 35 q^{50} - 16 q^{51} + q^{52} - 2 q^{53} - 48 q^{54} - 12 q^{55} - 16 q^{57} - 42 q^{58} - 16 q^{59} + 14 q^{61} + 8 q^{62} + 23 q^{64} + 2 q^{65} - 12 q^{66} - 4 q^{67} - 42 q^{68} + 20 q^{69} + 12 q^{71} - 49 q^{72} + 22 q^{73} - 50 q^{74} + 48 q^{75} - 16 q^{76} - 8 q^{78} - 32 q^{79} + 38 q^{80} + 49 q^{81} - 14 q^{82} - 24 q^{83} - 24 q^{85} + 64 q^{86} + 16 q^{87} - 12 q^{88} - 26 q^{89} + 50 q^{90} + 76 q^{92} - 28 q^{93} + 20 q^{94} - 40 q^{95} + 4 q^{96} + 6 q^{97} + 12 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(637))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 7 13
637.2.a.a 637.a 1.a $1$ $5.086$ \(\Q\) None \(-2\) \(0\) \(3\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}+3q^{5}-3q^{9}-6q^{10}+\cdots\)
637.2.a.b 637.a 1.a $1$ $5.086$ \(\Q\) None \(0\) \(2\) \(3\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-2q^{4}+3q^{5}+q^{9}-4q^{12}+\cdots\)
637.2.a.c 637.a 1.a $1$ $5.086$ \(\Q\) None \(1\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-3q^{8}-3q^{9}-3q^{11}+\cdots\)
637.2.a.d 637.a 1.a $1$ $5.086$ \(\Q\) None \(1\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-3q^{8}-3q^{9}-3q^{11}+\cdots\)
637.2.a.e 637.a 1.a $2$ $5.086$ \(\Q(\sqrt{5}) \) None \(-3\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+(-1+2\beta )q^{3}+3\beta q^{4}+\cdots\)
637.2.a.f 637.a 1.a $2$ $5.086$ \(\Q(\sqrt{5}) \) None \(-3\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+(1-2\beta )q^{3}+3\beta q^{4}+\cdots\)
637.2.a.g 637.a 1.a $2$ $5.086$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(-6\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+\beta q^{3}+(-3-\beta )q^{5}+2q^{6}+\cdots\)
637.2.a.h 637.a 1.a $3$ $5.086$ 3.3.404.1 None \(-2\) \(-4\) \(-5\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(-1-\beta _{2})q^{3}+(2+\cdots)q^{4}+\cdots\)
637.2.a.i 637.a 1.a $3$ $5.086$ 3.3.404.1 None \(-2\) \(4\) \(5\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(1+\beta _{2})q^{3}+(2-\beta _{1}+\cdots)q^{4}+\cdots\)
637.2.a.j 637.a 1.a $3$ $5.086$ 3.3.316.1 None \(1\) \(2\) \(-2\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1-\beta _{1}+\beta _{2})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
637.2.a.k 637.a 1.a $5$ $5.086$ 5.5.746052.1 None \(4\) \(0\) \(-2\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+\beta _{4}q^{3}+(2-\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)
637.2.a.l 637.a 1.a $5$ $5.086$ 5.5.746052.1 None \(4\) \(0\) \(2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-\beta _{4}q^{3}+(2-\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)
637.2.a.m 637.a 1.a $6$ $5.086$ 6.6.4507648.1 None \(0\) \(-8\) \(-6\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+(-1-\beta _{1})q^{3}+(1-\beta _{1}-\beta _{3}+\cdots)q^{4}+\cdots\)
637.2.a.n 637.a 1.a $6$ $5.086$ 6.6.4507648.1 None \(0\) \(8\) \(6\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+(1+\beta _{1})q^{3}+(1-\beta _{1}-\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(637))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(637)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(91))\)\(^{\oplus 2}\)