Properties

Label 37.10.a
Level $37$
Weight $10$
Character orbit 37.a
Rep. character $\chi_{37}(1,\cdot)$
Character field $\Q$
Dimension $27$
Newform subspaces $2$
Sturm bound $31$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 37.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(31\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(37))\).

Total New Old
Modular forms 29 27 2
Cusp forms 27 27 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
\(+\)\(13\)
\(-\)\(14\)

Trace form

\( 27 q + 16 q^{2} + 146 q^{3} + 6228 q^{4} + 682 q^{5} - 1026 q^{6} - 5944 q^{7} + 20652 q^{8} + 171029 q^{9} + O(q^{10}) \) \( 27 q + 16 q^{2} + 146 q^{3} + 6228 q^{4} + 682 q^{5} - 1026 q^{6} - 5944 q^{7} + 20652 q^{8} + 171029 q^{9} + 22398 q^{10} - 67502 q^{11} + 97302 q^{12} + 46042 q^{13} + 311258 q^{14} + 56172 q^{15} + 1005140 q^{16} + 2334 q^{17} - 1232886 q^{18} + 96250 q^{19} + 1781624 q^{20} - 2893904 q^{21} - 1155590 q^{22} + 3345974 q^{23} - 7400688 q^{24} + 8150487 q^{25} + 3091342 q^{26} + 2970632 q^{27} - 14874924 q^{28} + 5772054 q^{29} - 3799924 q^{30} + 4946682 q^{31} + 1662440 q^{32} - 8680780 q^{33} + 34960880 q^{34} + 2680352 q^{35} + 13889878 q^{36} + 1874161 q^{37} - 18832884 q^{38} - 38574904 q^{39} - 18143174 q^{40} - 28821324 q^{41} + 61850278 q^{42} - 16363046 q^{43} - 138939226 q^{44} + 91766798 q^{45} + 134103738 q^{46} + 25726476 q^{47} + 71725762 q^{48} + 42814247 q^{49} + 86236928 q^{50} + 23842956 q^{51} - 125315992 q^{52} - 27649250 q^{53} - 182517450 q^{54} - 88480496 q^{55} + 67483896 q^{56} + 106572144 q^{57} - 320653370 q^{58} - 195773062 q^{59} + 594543632 q^{60} + 24339926 q^{61} - 61772014 q^{62} - 519127144 q^{63} - 55098572 q^{64} - 515725280 q^{65} - 163989898 q^{66} + 274804990 q^{67} - 23669004 q^{68} - 848191856 q^{69} + 425554880 q^{70} - 86596416 q^{71} - 482632776 q^{72} + 21688712 q^{73} + 149932880 q^{74} + 383254108 q^{75} + 135538460 q^{76} - 472665808 q^{77} + 1351400514 q^{78} + 409557674 q^{79} + 2884890224 q^{80} + 2016034115 q^{81} - 641479790 q^{82} - 895705452 q^{83} - 1166545792 q^{84} - 1309336620 q^{85} - 853281220 q^{86} + 2391354228 q^{87} + 792033800 q^{88} + 229869178 q^{89} + 340369616 q^{90} - 28062620 q^{91} + 792007540 q^{92} + 1495344896 q^{93} + 1048041606 q^{94} - 1289752608 q^{95} - 2726873948 q^{96} + 1956160578 q^{97} - 4585688766 q^{98} + 763076464 q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(37))\) 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
37.10.a.a 37.a 1.a $13$ $19.056$ \(\mathbb{Q}[x]/(x^{13} - \cdots)\) None \(-32\) \(-251\) \(-2159\) \(-12576\) $+$ $\mathrm{SU}(2)$ \(q+(-2-\beta _{1})q^{2}+(-20+\beta _{1}-\beta _{5}+\cdots)q^{3}+\cdots\)
37.10.a.b 37.a 1.a $14$ $19.056$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(48\) \(397\) \(2841\) \(6632\) $-$ $\mathrm{SU}(2)$ \(q+(3+\beta _{1})q^{2}+(28+\beta _{3})q^{3}+(248+5\beta _{1}+\cdots)q^{4}+\cdots\)