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Properties

Label 37.4.a
Level $37$
Weight $4$
Character orbit 37.a
Rep. character $\chi_{37}(1,\cdot)$
Character field $\Q$
Dimension $9$
Newform subspaces $2$
Sturm bound $12$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 37.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(12\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(37))\).

Total New Old
Modular forms 11 9 2
Cusp forms 9 9 0
Eisenstein series 2 0 2

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

\(37\)Dim
\(+\)\(5\)
\(-\)\(4\)

Trace form

\( 9 q - 2 q^{2} + 2 q^{3} + 24 q^{4} - 18 q^{5} - 18 q^{6} - 8 q^{7} - 60 q^{8} + 47 q^{9} + O(q^{10}) \) \( 9 q - 2 q^{2} + 2 q^{3} + 24 q^{4} - 18 q^{5} - 18 q^{6} - 8 q^{7} - 60 q^{8} + 47 q^{9} + 6 q^{10} + 50 q^{11} + 78 q^{12} - 82 q^{13} - 46 q^{14} - 132 q^{15} + 196 q^{16} + 74 q^{17} + 96 q^{18} - 166 q^{19} - 224 q^{20} + 64 q^{21} - 22 q^{22} - 202 q^{23} - 96 q^{24} + 117 q^{25} + 294 q^{26} + 200 q^{27} - 204 q^{28} + 242 q^{29} - 100 q^{30} + 474 q^{31} - 872 q^{32} - 52 q^{33} + 484 q^{34} - 32 q^{35} - 974 q^{36} - 37 q^{37} + 500 q^{38} + 296 q^{39} + 1034 q^{40} + 372 q^{41} + 70 q^{42} - 598 q^{43} + 446 q^{44} + 314 q^{45} - 594 q^{46} - 196 q^{47} - 686 q^{48} + 637 q^{49} + 498 q^{50} - 132 q^{51} - 1568 q^{52} + 474 q^{53} - 762 q^{54} - 64 q^{55} + 1096 q^{56} - 288 q^{57} - 946 q^{58} - 1430 q^{59} - 1792 q^{60} - 1742 q^{61} + 558 q^{62} - 1096 q^{63} + 3860 q^{64} - 320 q^{65} + 2006 q^{66} + 782 q^{67} + 2060 q^{68} - 800 q^{69} - 1160 q^{70} + 1968 q^{71} + 1008 q^{72} - 1280 q^{73} - 370 q^{74} + 364 q^{75} + 1372 q^{76} - 1392 q^{77} + 5118 q^{78} + 586 q^{79} - 4464 q^{80} + 2273 q^{81} - 34 q^{82} - 716 q^{83} - 2176 q^{84} - 1908 q^{85} + 2684 q^{86} + 3396 q^{87} - 2600 q^{88} - 2242 q^{89} - 3076 q^{90} - 3148 q^{91} + 3252 q^{92} - 1600 q^{93} + 6 q^{94} + 2176 q^{95} - 6188 q^{96} - 4938 q^{97} - 288 q^{98} + 4912 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(37))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 37
37.4.a.a 37.a 1.a $4$ $2.183$ 4.4.21208.1 None \(-6\) \(-11\) \(-29\) \(-32\) $-$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{2}-\beta _{3})q^{2}+(-2-2\beta _{2}+\cdots)q^{3}+\cdots\)
37.4.a.b 37.a 1.a $5$ $2.183$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(4\) \(13\) \(11\) \(24\) $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(3-\beta _{2})q^{3}+(3-\beta _{1}+\cdots)q^{4}+\cdots\)