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Properties

Label 637.4.a
Level $637$
Weight $4$
Character orbit 637.a
Rep. character $\chi_{637}(1,\cdot)$
Character field $\Q$
Dimension $123$
Newform subspaces $14$
Sturm bound $261$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 637.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(261\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(637))\).

Total New Old
Modular forms 204 123 81
Cusp forms 188 123 65
Eisenstein series 16 0 16

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
\(+\)\(+\)$+$\(31\)
\(+\)\(-\)$-$\(29\)
\(-\)\(+\)$-$\(30\)
\(-\)\(-\)$+$\(33\)
Plus space\(+\)\(64\)
Minus space\(-\)\(59\)

Trace form

\( 123 q - 2 q^{3} + 470 q^{4} + 26 q^{5} + 52 q^{6} + 1081 q^{9} + O(q^{10}) \) \( 123 q - 2 q^{3} + 470 q^{4} + 26 q^{5} + 52 q^{6} + 1081 q^{9} + 6 q^{10} - 42 q^{11} + 58 q^{12} + 13 q^{13} + 16 q^{15} + 2110 q^{16} + 68 q^{17} - 180 q^{18} - 58 q^{19} + 48 q^{20} - 128 q^{22} + 140 q^{23} + 568 q^{24} + 3175 q^{25} + 78 q^{26} + 406 q^{27} + 206 q^{29} + 1142 q^{30} + 106 q^{31} - 240 q^{32} + 76 q^{33} + 520 q^{34} + 4012 q^{36} - 2 q^{37} + 4 q^{38} - 156 q^{39} - 542 q^{40} - 606 q^{41} + 78 q^{43} + 408 q^{44} - 378 q^{45} + 156 q^{46} + 742 q^{47} + 1358 q^{48} - 1508 q^{50} + 1050 q^{51} + 624 q^{52} - 58 q^{53} + 964 q^{54} - 396 q^{55} + 160 q^{57} - 1656 q^{58} + 682 q^{59} + 1200 q^{60} - 1446 q^{61} - 2632 q^{62} + 9898 q^{64} - 468 q^{65} - 228 q^{66} + 626 q^{67} + 974 q^{68} - 3972 q^{69} - 3322 q^{71} - 4636 q^{72} - 2774 q^{73} + 2438 q^{74} + 276 q^{75} + 3616 q^{76} - 1066 q^{78} + 832 q^{79} + 180 q^{80} + 12059 q^{81} + 3172 q^{82} - 2678 q^{83} - 400 q^{85} - 12472 q^{86} - 2968 q^{87} - 8484 q^{88} + 3434 q^{89} - 1424 q^{90} - 7048 q^{92} + 5088 q^{93} - 2790 q^{94} - 3160 q^{95} + 1584 q^{96} + 2774 q^{97} - 3798 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(637))\) 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
637.4.a.a 637.a 1.a $1$ $37.584$ \(\Q\) None \(-5\) \(7\) \(7\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-5q^{2}+7q^{3}+17q^{4}+7q^{5}-35q^{6}+\cdots\)
637.4.a.b 637.a 1.a $2$ $37.584$ \(\Q(\sqrt{17}) \) None \(1\) \(-5\) \(3\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-4+3\beta )q^{3}+(-4+\beta )q^{4}+\cdots\)
637.4.a.c 637.a 1.a $3$ $37.584$ 3.3.1384.1 None \(1\) \(-1\) \(22\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{2})q^{3}+(-1+\cdots)q^{4}+\cdots\)
637.4.a.d 637.a 1.a $4$ $37.584$ 4.4.5364412.1 None \(-4\) \(5\) \(36\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(1+\beta _{1}+\beta _{2})q^{3}+\cdots\)
637.4.a.e 637.a 1.a $5$ $37.584$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(7\) \(5\) \(-16\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{4})q^{2}+(1+\beta _{3})q^{3}+(10-\beta _{2}+\cdots)q^{4}+\cdots\)
637.4.a.f 637.a 1.a $6$ $37.584$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(2\) \(-13\) \(-26\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(-2+\beta _{4})q^{3}+(2+\beta _{2})q^{4}+\cdots\)
637.4.a.g 637.a 1.a $9$ $37.584$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(3\) \(-12\) \(-12\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(-1-\beta _{3})q^{3}+(2+\beta _{2})q^{4}+\cdots\)
637.4.a.h 637.a 1.a $9$ $37.584$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(3\) \(12\) \(12\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{3})q^{3}+(2+\beta _{2})q^{4}+\cdots\)
637.4.a.i 637.a 1.a $11$ $37.584$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(-7\) \(0\) \(-2\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+\beta _{3}q^{3}+(4-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
637.4.a.j 637.a 1.a $11$ $37.584$ \(\mathbb{Q}[x]/(x^{11} - \cdots)\) None \(-7\) \(0\) \(2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-\beta _{3}q^{3}+(4-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
637.4.a.k 637.a 1.a $13$ $37.584$ \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(5\) \(0\) \(-14\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-\beta _{6}q^{3}+(5+\beta _{1}+\beta _{2})q^{4}+\cdots\)
637.4.a.l 637.a 1.a $13$ $37.584$ \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(5\) \(0\) \(14\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+\beta _{6}q^{3}+(5+\beta _{1}+\beta _{2})q^{4}+\cdots\)
637.4.a.m 637.a 1.a $18$ $37.584$ \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None \(-2\) \(-24\) \(-56\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(-1-\beta _{4})q^{3}+(3+\beta _{2})q^{4}+\cdots\)
637.4.a.n 637.a 1.a $18$ $37.584$ \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None \(-2\) \(24\) \(56\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1+\beta _{4})q^{3}+(3+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(637))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(637)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(91))\)\(^{\oplus 2}\)