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Properties

Label 115.6.a
Level $115$
Weight $6$
Character orbit 115.a
Rep. character $\chi_{115}(1,\cdot)$
Character field $\Q$
Dimension $38$
Newform subspaces $5$
Sturm bound $72$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 115.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(72\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(115))\).

Total New Old
Modular forms 62 38 24
Cusp forms 58 38 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.

\(5\)\(23\)FrickeDim
\(+\)\(+\)$+$\(10\)
\(+\)\(-\)$-$\(9\)
\(-\)\(+\)$-$\(12\)
\(-\)\(-\)$+$\(7\)
Plus space\(+\)\(17\)
Minus space\(-\)\(21\)

Trace form

\( 38 q - 8 q^{2} + 8 q^{3} + 672 q^{4} - 130 q^{6} - 188 q^{7} + 42 q^{8} + 3314 q^{9} + O(q^{10}) \) \( 38 q - 8 q^{2} + 8 q^{3} + 672 q^{4} - 130 q^{6} - 188 q^{7} + 42 q^{8} + 3314 q^{9} - 536 q^{11} - 1010 q^{12} - 976 q^{13} + 428 q^{14} + 1100 q^{15} + 13560 q^{16} + 5180 q^{17} + 1746 q^{18} - 6344 q^{19} + 3800 q^{20} + 5588 q^{21} - 1496 q^{22} - 3174 q^{23} - 1692 q^{24} + 23750 q^{25} - 434 q^{26} - 7756 q^{27} + 584 q^{28} + 2118 q^{29} + 11600 q^{30} - 5786 q^{31} - 15464 q^{32} - 36352 q^{33} + 47876 q^{34} - 8250 q^{35} + 74322 q^{36} - 22844 q^{37} - 7788 q^{38} - 16160 q^{39} + 38282 q^{41} - 50152 q^{42} + 70936 q^{43} + 16776 q^{44} - 57968 q^{47} - 39922 q^{48} + 6020 q^{49} - 5000 q^{50} - 14348 q^{51} + 17770 q^{52} - 37884 q^{53} - 44986 q^{54} - 24200 q^{55} - 149248 q^{56} + 74480 q^{57} - 59570 q^{58} + 13758 q^{59} + 78100 q^{60} - 60620 q^{61} + 97930 q^{62} + 6396 q^{63} + 248318 q^{64} - 43900 q^{65} - 78684 q^{66} - 40780 q^{67} + 112720 q^{68} - 92400 q^{70} + 188346 q^{71} + 344274 q^{72} + 32140 q^{73} + 182064 q^{74} + 5000 q^{75} - 252268 q^{76} - 70556 q^{77} - 452098 q^{78} - 278164 q^{79} + 73200 q^{80} + 132886 q^{81} - 97270 q^{82} + 109420 q^{83} + 269832 q^{84} - 47550 q^{85} - 71464 q^{86} - 144272 q^{87} + 221348 q^{88} - 276492 q^{89} - 71100 q^{90} - 89852 q^{91} - 101568 q^{92} - 475700 q^{93} + 46730 q^{94} - 68600 q^{95} + 273186 q^{96} + 419120 q^{97} + 1032704 q^{98} - 796232 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(115))\) 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
115.6.a.a 115.a 1.a $2$ $18.444$ \(\Q(\sqrt{1821}) \) None \(4\) \(3\) \(-50\) \(53\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+(2-\beta )q^{3}-28q^{4}-5^{2}q^{5}+\cdots\)
115.6.a.b 115.a 1.a $7$ $18.444$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-12\) \(4\) \(175\) \(-275\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{2}+(1-\beta _{1}-\beta _{6})q^{3}+\cdots\)
115.6.a.c 115.a 1.a $7$ $18.444$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(4\) \(-3\) \(-175\) \(33\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(-1+\beta _{1}-\beta _{2})q^{3}+\cdots\)
115.6.a.d 115.a 1.a $10$ $18.444$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-12\) \(-18\) \(-250\) \(-15\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{2}+(-2+\beta _{3})q^{3}+(14+\cdots)q^{4}+\cdots\)
115.6.a.e 115.a 1.a $12$ $18.444$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(8\) \(22\) \(300\) \(16\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(2-\beta _{1}-\beta _{4})q^{3}+(5^{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(115))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(115)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 2}\)