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Properties

Label 185.4.a
Level $185$
Weight $4$
Character orbit 185.a
Rep. character $\chi_{185}(1,\cdot)$
Character field $\Q$
Dimension $36$
Newform subspaces $4$
Sturm bound $76$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 185 = 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 185.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(76\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(185))\).

Total New Old
Modular forms 60 36 24
Cusp forms 56 36 20
Eisenstein series 4 0 4

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

\(5\)\(37\)FrickeDim
\(+\)\(+\)$+$\(10\)
\(+\)\(-\)$-$\(7\)
\(-\)\(+\)$-$\(7\)
\(-\)\(-\)$+$\(12\)
Plus space\(+\)\(22\)
Minus space\(-\)\(14\)

Trace form

\( 36 q + 4 q^{2} - 4 q^{3} + 156 q^{4} + 10 q^{5} + 16 q^{6} + 36 q^{8} + 388 q^{9} + O(q^{10}) \) \( 36 q + 4 q^{2} - 4 q^{3} + 156 q^{4} + 10 q^{5} + 16 q^{6} + 36 q^{8} + 388 q^{9} - 60 q^{10} - 52 q^{11} - 204 q^{12} + 68 q^{13} + 96 q^{14} + 20 q^{15} + 596 q^{16} - 116 q^{17} - 224 q^{18} + 204 q^{19} + 80 q^{20} + 552 q^{22} + 324 q^{23} + 132 q^{24} + 900 q^{25} - 424 q^{26} - 136 q^{27} + 452 q^{28} - 572 q^{29} - 280 q^{30} - 504 q^{31} + 492 q^{32} - 488 q^{33} - 916 q^{34} + 280 q^{35} + 2844 q^{36} + 74 q^{37} - 52 q^{38} - 560 q^{39} - 240 q^{40} - 332 q^{41} + 956 q^{42} + 420 q^{43} + 492 q^{44} - 390 q^{45} + 1224 q^{46} + 528 q^{47} - 928 q^{48} + 1720 q^{49} + 100 q^{50} - 776 q^{51} + 1924 q^{52} + 156 q^{53} + 128 q^{54} + 440 q^{55} - 2468 q^{56} - 784 q^{57} + 1556 q^{58} - 28 q^{59} - 820 q^{60} + 2324 q^{61} - 4016 q^{62} + 984 q^{63} - 428 q^{64} - 480 q^{65} - 5032 q^{66} - 2292 q^{67} - 2932 q^{68} + 2256 q^{69} - 1600 q^{70} - 1488 q^{71} - 3680 q^{72} + 3496 q^{73} - 100 q^{75} - 1428 q^{76} + 176 q^{77} - 10756 q^{78} - 1288 q^{79} + 800 q^{80} - 692 q^{81} - 2096 q^{82} - 460 q^{83} + 1856 q^{84} + 480 q^{85} - 716 q^{86} - 880 q^{87} + 5040 q^{88} + 2240 q^{89} + 180 q^{90} + 3232 q^{91} - 2952 q^{92} + 4008 q^{93} - 7540 q^{94} + 560 q^{95} + 5864 q^{96} + 4644 q^{97} - 3160 q^{98} - 1612 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(185))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 5 37
185.4.a.a 185.a 1.a $7$ $10.915$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-5\) \(-6\) \(35\) \(-49\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(-1-\beta _{1}+\beta _{3})q^{3}+\cdots\)
185.4.a.b 185.a 1.a $7$ $10.915$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(1\) \(-8\) \(-35\) \(-77\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(-1-\beta _{5})q^{3}+(2+\beta _{5}-\beta _{6})q^{4}+\cdots\)
185.4.a.c 185.a 1.a $10$ $10.915$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(7\) \(4\) \(-50\) \(49\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-\beta _{5}q^{3}+(5-\beta _{1}+\beta _{4}+\cdots)q^{4}+\cdots\)
185.4.a.d 185.a 1.a $12$ $10.915$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(1\) \(6\) \(60\) \(77\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+\beta _{3}q^{3}+(7+\beta _{2}-\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(185))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(185)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 2}\)