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Properties

Label 115.4.a
Level $115$
Weight $4$
Character orbit 115.a
Rep. character $\chi_{115}(1,\cdot)$
Character field $\Q$
Dimension $22$
Newform subspaces $6$
Sturm bound $48$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 115.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(48\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(115))\).

Total New Old
Modular forms 38 22 16
Cusp forms 34 22 12
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.

\(5\)\(23\)FrickeDim
\(+\)\(+\)$+$\(5\)
\(+\)\(-\)$-$\(6\)
\(-\)\(+\)$-$\(3\)
\(-\)\(-\)$+$\(8\)
Plus space\(+\)\(13\)
Minus space\(-\)\(9\)

Trace form

\( 22 q + 4 q^{2} - 4 q^{3} + 88 q^{4} - 10 q^{6} - 40 q^{7} + 42 q^{8} + 290 q^{9} + O(q^{10}) \) \( 22 q + 4 q^{2} - 4 q^{3} + 88 q^{4} - 10 q^{6} - 40 q^{7} + 42 q^{8} + 290 q^{9} - 92 q^{11} + 46 q^{12} + 60 q^{13} + 284 q^{14} - 40 q^{15} + 408 q^{16} - 220 q^{17} - 402 q^{18} - 196 q^{19} - 40 q^{20} - 376 q^{21} + 464 q^{22} + 138 q^{23} - 12 q^{24} + 550 q^{25} - 242 q^{26} - 268 q^{27} - 280 q^{28} - 432 q^{29} + 80 q^{30} + 460 q^{31} + 40 q^{32} + 200 q^{33} - 596 q^{34} + 300 q^{35} + 1146 q^{36} - 856 q^{37} - 660 q^{38} - 596 q^{39} + 284 q^{41} - 712 q^{42} - 1044 q^{43} - 840 q^{44} + 1396 q^{47} + 398 q^{48} + 1158 q^{49} + 100 q^{50} + 88 q^{51} - 1430 q^{52} + 1416 q^{53} - 1786 q^{54} + 440 q^{55} + 4544 q^{56} - 2704 q^{57} + 2198 q^{58} + 156 q^{59} + 340 q^{60} - 864 q^{61} - 1670 q^{62} + 192 q^{63} + 1118 q^{64} - 640 q^{65} + 1044 q^{66} - 2196 q^{67} - 4640 q^{68} + 120 q^{70} + 204 q^{71} - 5838 q^{72} + 1636 q^{73} - 432 q^{74} - 100 q^{75} + 1092 q^{76} + 784 q^{77} + 1406 q^{78} + 1696 q^{79} - 2640 q^{80} + 3574 q^{81} + 1938 q^{82} - 740 q^{83} - 10392 q^{84} + 780 q^{85} + 3488 q^{86} + 2068 q^{87} + 2372 q^{88} - 396 q^{89} + 2340 q^{90} + 1176 q^{91} + 1104 q^{92} + 3016 q^{93} - 6534 q^{94} + 1480 q^{95} - 5166 q^{96} - 2196 q^{97} - 6772 q^{98} + 2212 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(115))\) 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 23
115.4.a.a 115.a 1.a $1$ $6.785$ \(\Q\) None \(1\) \(4\) \(-5\) \(-32\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+4q^{3}-7q^{4}-5q^{5}+4q^{6}+\cdots\)
115.4.a.b 115.a 1.a $1$ $6.785$ \(\Q\) None \(2\) \(-3\) \(5\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}-4q^{4}+5q^{5}-6q^{6}+\cdots\)
115.4.a.c 115.a 1.a $2$ $6.785$ \(\Q(\sqrt{109}) \) None \(-6\) \(-3\) \(10\) \(1\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{2}+(-1-\beta )q^{3}+q^{4}+5q^{5}+\cdots\)
115.4.a.d 115.a 1.a $5$ $6.785$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(-5\) \(-6\) \(-25\) \(-15\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(-1-\beta _{2}-\beta _{3}-\beta _{4})q^{3}+\cdots\)
115.4.a.e 115.a 1.a $5$ $6.785$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(6\) \(4\) \(-25\) \(-3\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{2}+(1+\beta _{1}-\beta _{2}+\beta _{3})q^{3}+\cdots\)
115.4.a.f 115.a 1.a $8$ $6.785$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(6\) \(0\) \(40\) \(11\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+\beta _{6}q^{3}+(5+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(115))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(115)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 2}\)