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Properties

Label 37.6.a
Level $37$
Weight $6$
Character orbit 37.a
Rep. character $\chi_{37}(1,\cdot)$
Character field $\Q$
Dimension $15$
Newform subspaces $2$
Sturm bound $19$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 37.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(19\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(37))\).

Total New Old
Modular forms 17 15 2
Cusp forms 15 15 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
\(+\)\(7\)
\(-\)\(8\)

Trace form

\( 15 q + 4 q^{2} - 22 q^{3} + 260 q^{4} + 72 q^{5} - 66 q^{6} - 194 q^{7} + 12 q^{8} + 989 q^{9} + O(q^{10}) \) \( 15 q + 4 q^{2} - 22 q^{3} + 260 q^{4} + 72 q^{5} - 66 q^{6} - 194 q^{7} + 12 q^{8} + 989 q^{9} - 562 q^{10} + 98 q^{11} - 2634 q^{12} + 660 q^{13} + 746 q^{14} + 582 q^{15} + 2388 q^{16} - 2308 q^{17} + 2094 q^{18} - 1778 q^{19} + 2584 q^{20} - 2042 q^{21} + 58 q^{22} - 2632 q^{23} + 3216 q^{24} + 13217 q^{25} - 8898 q^{26} - 8854 q^{27} - 9548 q^{28} - 10834 q^{29} + 27116 q^{30} + 864 q^{31} - 14072 q^{32} - 730 q^{33} - 4760 q^{34} - 4238 q^{35} + 44710 q^{36} + 1369 q^{37} - 3364 q^{38} - 13006 q^{39} - 55174 q^{40} + 4440 q^{41} - 24122 q^{42} + 26654 q^{43} + 22022 q^{44} - 27862 q^{45} + 17314 q^{46} - 7534 q^{47} - 61310 q^{48} + 18109 q^{49} + 67308 q^{50} + 53790 q^{51} + 119272 q^{52} - 3984 q^{53} - 88650 q^{54} + 42154 q^{55} - 47240 q^{56} + 87678 q^{57} - 73450 q^{58} + 10408 q^{59} + 20432 q^{60} + 39086 q^{61} + 49866 q^{62} - 14956 q^{63} - 37708 q^{64} - 12200 q^{65} - 62698 q^{66} - 50656 q^{67} - 260428 q^{68} + 231328 q^{69} + 90800 q^{70} - 135990 q^{71} - 194952 q^{72} + 74068 q^{73} + 27380 q^{74} - 397142 q^{75} - 103428 q^{76} - 73242 q^{77} + 139194 q^{78} + 30986 q^{79} + 179184 q^{80} + 70511 q^{81} + 122234 q^{82} + 186478 q^{83} + 80000 q^{84} - 149460 q^{85} + 281468 q^{86} + 158880 q^{87} + 120520 q^{88} + 364376 q^{89} - 670984 q^{90} - 202970 q^{91} - 305964 q^{92} - 215080 q^{93} + 291750 q^{94} + 96712 q^{95} + 691012 q^{96} + 2494 q^{97} + 460470 q^{98} - 77864 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(37))\) 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
37.6.a.a 37.a 1.a $7$ $5.934$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-8\) \(-47\) \(-64\) \(-293\) $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{2}+(-7+\beta _{1}+\beta _{2})q^{3}+\cdots\)
37.6.a.b 37.a 1.a $8$ $5.934$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(12\) \(25\) \(136\) \(99\) $-$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1})q^{2}+(3-\beta _{5})q^{3}+(20-2\beta _{1}+\cdots)q^{4}+\cdots\)