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Properties

Label 2415.2.a
Level $2415$
Weight $2$
Character orbit 2415.a
Rep. character $\chi_{2415}(1,\cdot)$
Character field $\Q$
Dimension $89$
Newform subspaces $23$
Sturm bound $768$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2415.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 23 \)
Sturm bound: \(768\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2415))\).

Total New Old
Modular forms 392 89 303
Cusp forms 377 89 288
Eisenstein series 15 0 15

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

\(3\)\(5\)\(7\)\(23\)FrickeDim
\(+\)\(+\)\(+\)\(+\)$+$\(6\)
\(+\)\(+\)\(+\)\(-\)$-$\(6\)
\(+\)\(+\)\(-\)\(+\)$-$\(6\)
\(+\)\(+\)\(-\)\(-\)$+$\(4\)
\(+\)\(-\)\(+\)\(+\)$-$\(7\)
\(+\)\(-\)\(+\)\(-\)$+$\(3\)
\(+\)\(-\)\(-\)\(+\)$+$\(3\)
\(+\)\(-\)\(-\)\(-\)$-$\(9\)
\(-\)\(+\)\(+\)\(+\)$-$\(7\)
\(-\)\(+\)\(+\)\(-\)$+$\(3\)
\(-\)\(+\)\(-\)\(+\)$+$\(3\)
\(-\)\(+\)\(-\)\(-\)$-$\(9\)
\(-\)\(-\)\(+\)\(+\)$+$\(2\)
\(-\)\(-\)\(+\)\(-\)$-$\(10\)
\(-\)\(-\)\(-\)\(+\)$-$\(10\)
\(-\)\(-\)\(-\)\(-\)$+$\(1\)
Plus space\(+\)\(25\)
Minus space\(-\)\(64\)

Trace form

\( 89 q + 3 q^{2} + q^{3} + 95 q^{4} + q^{5} + 3 q^{6} + q^{7} + 15 q^{8} + 89 q^{9} + O(q^{10}) \) \( 89 q + 3 q^{2} + q^{3} + 95 q^{4} + q^{5} + 3 q^{6} + q^{7} + 15 q^{8} + 89 q^{9} + 3 q^{10} + 12 q^{11} + 7 q^{12} + 14 q^{13} + 3 q^{14} + q^{15} + 119 q^{16} + 18 q^{17} + 3 q^{18} + 20 q^{19} + 7 q^{20} + q^{21} + 36 q^{22} + q^{23} + 15 q^{24} + 89 q^{25} + 42 q^{26} + q^{27} + 7 q^{28} + 30 q^{29} + 3 q^{30} + 16 q^{31} - 17 q^{32} - 4 q^{33} - 26 q^{34} + q^{35} + 95 q^{36} + 22 q^{37} - 20 q^{38} + 14 q^{39} + 15 q^{40} + 26 q^{41} + 3 q^{42} + 12 q^{43} + 4 q^{44} + q^{45} + 3 q^{46} + 32 q^{47} + 31 q^{48} + 89 q^{49} + 3 q^{50} + 2 q^{51} + 18 q^{52} + 38 q^{53} + 3 q^{54} + 12 q^{55} + 15 q^{56} + 20 q^{57} + 74 q^{58} + 28 q^{59} + 7 q^{60} + 30 q^{61} - 16 q^{62} + q^{63} + 215 q^{64} + 14 q^{65} + 36 q^{66} + 36 q^{67} + 126 q^{68} + q^{69} - 5 q^{70} + 40 q^{71} + 15 q^{72} + 58 q^{73} - 14 q^{74} + q^{75} + 60 q^{76} + 12 q^{77} - 54 q^{78} + 32 q^{79} + 31 q^{80} + 89 q^{81} + 14 q^{82} + 36 q^{83} + 7 q^{84} + 2 q^{85} + 84 q^{86} - 18 q^{87} + 20 q^{88} + 58 q^{89} + 3 q^{90} + 14 q^{91} + 7 q^{92} + 16 q^{93} + 16 q^{94} + 4 q^{95} - 17 q^{96} + 50 q^{97} + 3 q^{98} + 12 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2415))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5 7 23
2415.2.a.a 2415.a 1.a $1$ $19.284$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(-1\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}-q^{4}-q^{5}+q^{6}-q^{7}+\cdots\)
2415.2.a.b 2415.a 1.a $1$ $19.284$ \(\Q\) None \(-1\) \(1\) \(-1\) \(1\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}-q^{4}-q^{5}-q^{6}+q^{7}+\cdots\)
2415.2.a.c 2415.a 1.a $1$ $19.284$ \(\Q\) None \(-1\) \(1\) \(1\) \(-1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}-q^{4}+q^{5}-q^{6}-q^{7}+\cdots\)
2415.2.a.d 2415.a 1.a $1$ $19.284$ \(\Q\) None \(-1\) \(1\) \(1\) \(1\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}-q^{4}+q^{5}-q^{6}+q^{7}+\cdots\)
2415.2.a.e 2415.a 1.a $1$ $19.284$ \(\Q\) None \(0\) \(-1\) \(-1\) \(1\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{4}-q^{5}+q^{7}+q^{9}-3q^{11}+\cdots\)
2415.2.a.f 2415.a 1.a $1$ $19.284$ \(\Q\) None \(0\) \(1\) \(-1\) \(-1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}-q^{5}-q^{7}+q^{9}+q^{11}+\cdots\)
2415.2.a.g 2415.a 1.a $1$ $19.284$ \(\Q\) None \(0\) \(1\) \(-1\) \(1\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}-q^{5}+q^{7}+q^{9}+3q^{11}+\cdots\)
2415.2.a.h 2415.a 1.a $1$ $19.284$ \(\Q\) None \(1\) \(1\) \(-1\) \(1\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}-q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
2415.2.a.i 2415.a 1.a $1$ $19.284$ \(\Q\) None \(1\) \(1\) \(1\) \(-1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}-q^{4}+q^{5}+q^{6}-q^{7}+\cdots\)
2415.2.a.j 2415.a 1.a $2$ $19.284$ \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(-2\) \(-2\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{3}+q^{4}-q^{5}+\beta q^{6}-q^{7}+\cdots\)
2415.2.a.k 2415.a 1.a $2$ $19.284$ \(\Q(\sqrt{3}) \) None \(0\) \(2\) \(-2\) \(2\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{3}+q^{4}-q^{5}+\beta q^{6}+q^{7}+\cdots\)
2415.2.a.l 2415.a 1.a $3$ $19.284$ 3.3.148.1 None \(-3\) \(-3\) \(-3\) \(3\) $+$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
2415.2.a.m 2415.a 1.a $3$ $19.284$ 3.3.148.1 None \(-1\) \(-3\) \(3\) \(3\) $+$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+q^{5}+\cdots\)
2415.2.a.n 2415.a 1.a $3$ $19.284$ 3.3.148.1 None \(1\) \(-3\) \(3\) \(-3\) $+$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+q^{5}+\cdots\)
2415.2.a.o 2415.a 1.a $5$ $19.284$ 5.5.2508628.1 None \(0\) \(-5\) \(-5\) \(-5\) $+$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)
2415.2.a.p 2415.a 1.a $6$ $19.284$ 6.6.42978136.1 None \(1\) \(-6\) \(-6\) \(-6\) $+$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
2415.2.a.q 2415.a 1.a $6$ $19.284$ 6.6.15751800.1 None \(3\) \(-6\) \(-6\) \(6\) $+$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}-q^{3}+(\beta _{2}+\beta _{5})q^{4}-q^{5}+\cdots\)
2415.2.a.r 2415.a 1.a $7$ $19.284$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-2\) \(7\) \(-7\) \(-7\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}-q^{5}-\beta _{1}q^{6}+\cdots\)
2415.2.a.s 2415.a 1.a $7$ $19.284$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-1\) \(-7\) \(7\) \(-7\) $+$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+q^{5}+\beta _{1}q^{6}+\cdots\)
2415.2.a.t 2415.a 1.a $7$ $19.284$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(2\) \(7\) \(-7\) \(7\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)
2415.2.a.u 2415.a 1.a $9$ $19.284$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(1\) \(-9\) \(9\) \(9\) $+$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}+q^{5}-\beta _{1}q^{6}+\cdots\)
2415.2.a.v 2415.a 1.a $10$ $19.284$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(2\) \(10\) \(10\) \(-10\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+q^{5}+\beta _{1}q^{6}+\cdots\)
2415.2.a.w 2415.a 1.a $10$ $19.284$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(2\) \(10\) \(10\) \(10\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+q^{5}+\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2415))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2415)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(161))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(345))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(483))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(805))\)\(^{\oplus 2}\)