Properties

Label 2015.2.a
Level $2015$
Weight $2$
Character orbit 2015.a
Rep. character $\chi_{2015}(1,\cdot)$
Character field $\Q$
Dimension $119$
Newform subspaces $12$
Sturm bound $448$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 2015 = 5 \cdot 13 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2015.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 12 \)
Sturm bound: \(448\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2015))\).

Total New Old
Modular forms 228 119 109
Cusp forms 221 119 102
Eisenstein series 7 0 7

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

\(5\)\(13\)\(31\)FrickeDim
\(+\)\(+\)\(+\)$+$\(9\)
\(+\)\(+\)\(-\)$-$\(20\)
\(+\)\(-\)\(+\)$-$\(20\)
\(+\)\(-\)\(-\)$+$\(9\)
\(-\)\(+\)\(+\)$-$\(23\)
\(-\)\(+\)\(-\)$+$\(8\)
\(-\)\(-\)\(+\)$+$\(8\)
\(-\)\(-\)\(-\)$-$\(22\)
Plus space\(+\)\(34\)
Minus space\(-\)\(85\)

Trace form

\( 119 q + 9 q^{2} + 4 q^{3} + 121 q^{4} + 3 q^{5} + 4 q^{6} + 21 q^{8} + 139 q^{9} + O(q^{10}) \) \( 119 q + 9 q^{2} + 4 q^{3} + 121 q^{4} + 3 q^{5} + 4 q^{6} + 21 q^{8} + 139 q^{9} - 3 q^{10} + 12 q^{11} + 36 q^{12} - q^{13} + 32 q^{14} - 4 q^{15} + 145 q^{16} + 14 q^{17} + 61 q^{18} + 12 q^{19} + 5 q^{20} + 36 q^{22} + 28 q^{24} + 119 q^{25} - 3 q^{26} - 8 q^{27} + 34 q^{29} - 4 q^{30} - q^{31} + 61 q^{32} + 48 q^{33} + 2 q^{34} - 8 q^{35} + 189 q^{36} + 34 q^{37} - 12 q^{38} - 4 q^{39} - 15 q^{40} + 30 q^{41} + 16 q^{42} + 4 q^{43} - 12 q^{44} + 23 q^{45} + 24 q^{47} + 52 q^{48} + 143 q^{49} + 9 q^{50} + 32 q^{51} - 7 q^{52} + 2 q^{53} - 16 q^{54} - 4 q^{55} + 48 q^{56} + 40 q^{57} - 10 q^{58} + 12 q^{59} - 28 q^{60} - 22 q^{61} - 3 q^{62} + 56 q^{63} + 161 q^{64} - q^{65} - 40 q^{66} + 84 q^{67} + 2 q^{68} + 32 q^{69} + 56 q^{70} + 65 q^{72} + 22 q^{73} - 82 q^{74} + 4 q^{75} + 12 q^{76} + 16 q^{77} + 36 q^{78} + 80 q^{79} - 3 q^{80} + 207 q^{81} - 54 q^{82} + 68 q^{83} - 56 q^{84} - 18 q^{85} - 28 q^{86} - 48 q^{87} + 92 q^{88} + 38 q^{89} - 39 q^{90} + 8 q^{92} + 4 q^{93} - 40 q^{94} - 4 q^{95} - 60 q^{96} + 86 q^{97} - 39 q^{98} + 68 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2015))\) 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 13 31
2015.2.a.a 2015.a 1.a $1$ $16.090$ \(\Q\) None \(-2\) \(-3\) \(1\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+2q^{4}+q^{5}+6q^{6}+\cdots\)
2015.2.a.b 2015.a 1.a $1$ $16.090$ \(\Q\) None \(0\) \(-2\) \(-1\) \(-4\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{3}-2q^{4}-q^{5}-4q^{7}+q^{9}+3q^{11}+\cdots\)
2015.2.a.c 2015.a 1.a $1$ $16.090$ \(\Q\) None \(0\) \(1\) \(-1\) \(2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}-q^{5}+2q^{7}-2q^{9}-2q^{12}+\cdots\)
2015.2.a.d 2015.a 1.a $3$ $16.090$ 3.3.316.1 None \(0\) \(0\) \(-3\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}-2q^{4}-q^{5}+(2+\beta _{2})q^{9}+\cdots\)
2015.2.a.e 2015.a 1.a $6$ $16.090$ 6.6.2501557.1 None \(0\) \(-1\) \(-6\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}-\beta _{1}q^{3}+(2-\beta _{5})q^{4}-q^{5}+\cdots\)
2015.2.a.f 2015.a 1.a $7$ $16.090$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-2\) \(-2\) \(-7\) \(-5\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{3}q^{2}-\beta _{1}q^{3}+(1+\beta _{4})q^{4}-q^{5}+\cdots\)
2015.2.a.g 2015.a 1.a $8$ $16.090$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-2\) \(-1\) \(8\) \(-7\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+\beta _{7}q^{3}+(\beta _{1}+\beta _{2})q^{4}+q^{5}+\cdots\)
2015.2.a.h 2015.a 1.a $8$ $16.090$ 8.8.1299600812.1 None \(0\) \(-3\) \(8\) \(-1\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+\beta _{6}q^{3}+(1-\beta _{1}+\beta _{3}+\beta _{6}+\cdots)q^{4}+\cdots\)
2015.2.a.i 2015.a 1.a $20$ $16.090$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(3\) \(7\) \(-20\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-\beta _{6}q^{3}+(1+\beta _{2})q^{4}-q^{5}+\cdots\)
2015.2.a.j 2015.a 1.a $20$ $16.090$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(5\) \(1\) \(-20\) \(11\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-\beta _{10}q^{3}+(1+\beta _{2})q^{4}-q^{5}+\cdots\)
2015.2.a.k 2015.a 1.a $22$ $16.090$ None \(2\) \(3\) \(22\) \(3\) $-$ $-$ $-$ $\mathrm{SU}(2)$
2015.2.a.l 2015.a 1.a $22$ $16.090$ None \(5\) \(4\) \(22\) \(5\) $-$ $+$ $+$ $\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2015))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2015)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(155))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(403))\)\(^{\oplus 2}\)