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Properties

Label 1240.2.a
Level $1240$
Weight $2$
Character orbit 1240.a
Rep. character $\chi_{1240}(1,\cdot)$
Character field $\Q$
Dimension $30$
Newform subspaces $13$
Sturm bound $384$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1240 = 2^{3} \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1240.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(384\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(3\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1240))\).

Total New Old
Modular forms 200 30 170
Cusp forms 185 30 155
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.

\(2\)\(5\)\(31\)FrickeDim
\(+\)\(+\)\(+\)$+$\(6\)
\(+\)\(+\)\(-\)$-$\(2\)
\(+\)\(-\)\(+\)$-$\(3\)
\(+\)\(-\)\(-\)$+$\(3\)
\(-\)\(+\)\(+\)$-$\(5\)
\(-\)\(+\)\(-\)$+$\(3\)
\(-\)\(-\)\(+\)$+$\(1\)
\(-\)\(-\)\(-\)$-$\(7\)
Plus space\(+\)\(13\)
Minus space\(-\)\(17\)

Trace form

\( 30 q - 4 q^{3} - 2 q^{5} + 38 q^{9} + O(q^{10}) \) \( 30 q - 4 q^{3} - 2 q^{5} + 38 q^{9} - 8 q^{11} + 12 q^{17} - 16 q^{19} + 8 q^{21} + 30 q^{25} - 16 q^{27} + 12 q^{29} + 16 q^{33} + 8 q^{35} - 16 q^{37} + 20 q^{41} - 12 q^{43} + 6 q^{45} - 24 q^{47} + 14 q^{49} + 16 q^{51} + 24 q^{57} - 8 q^{59} - 12 q^{61} - 40 q^{63} + 12 q^{65} - 8 q^{69} + 24 q^{71} + 36 q^{73} - 4 q^{75} - 24 q^{79} + 78 q^{81} + 4 q^{83} - 56 q^{87} + 36 q^{89} - 40 q^{91} - 4 q^{93} - 8 q^{95} + 4 q^{97} - 48 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1240))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 5 31
1240.2.a.a 1240.a 1.a $1$ $9.901$ \(\Q\) None \(0\) \(-1\) \(-1\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-q^{5}-2q^{9}-2q^{13}+q^{15}+\cdots\)
1240.2.a.b 1240.a 1.a $1$ $9.901$ \(\Q\) None \(0\) \(-1\) \(1\) \(2\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}+2q^{7}-2q^{9}-2q^{11}+\cdots\)
1240.2.a.c 1240.a 1.a $1$ $9.901$ \(\Q\) None \(0\) \(0\) \(-1\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}-3q^{9}+2q^{11}+2q^{13}-8q^{19}+\cdots\)
1240.2.a.d 1240.a 1.a $1$ $9.901$ \(\Q\) None \(0\) \(0\) \(1\) \(0\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}-3q^{9}-4q^{11}+2q^{13}+6q^{17}+\cdots\)
1240.2.a.e 1240.a 1.a $1$ $9.901$ \(\Q\) None \(0\) \(2\) \(-1\) \(-4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-q^{5}-4q^{7}+q^{9}+4q^{11}+\cdots\)
1240.2.a.f 1240.a 1.a $1$ $9.901$ \(\Q\) None \(0\) \(2\) \(-1\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{3}-q^{5}+q^{9}+4q^{13}-2q^{15}+\cdots\)
1240.2.a.g 1240.a 1.a $1$ $9.901$ \(\Q\) None \(0\) \(3\) \(1\) \(2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+3q^{3}+q^{5}+2q^{7}+6q^{9}-2q^{11}+\cdots\)
1240.2.a.h 1240.a 1.a $2$ $9.901$ \(\Q(\sqrt{17}) \) None \(0\) \(-1\) \(-2\) \(0\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{3}-q^{5}+(1+\beta )q^{9}+(-2+2\beta )q^{11}+\cdots\)
1240.2.a.i 1240.a 1.a $2$ $9.901$ \(\Q(\sqrt{2}) \) None \(0\) \(0\) \(2\) \(4\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+q^{5}+2q^{7}-q^{9}+(2+\beta )q^{11}+\cdots\)
1240.2.a.j 1240.a 1.a $3$ $9.901$ 3.3.148.1 None \(0\) \(-3\) \(3\) \(-2\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{3}+q^{5}+(-1+\beta _{1}+\beta _{2})q^{7}+\cdots\)
1240.2.a.k 1240.a 1.a $4$ $9.901$ 4.4.112820.1 None \(0\) \(-1\) \(-4\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}-q^{5}+(1-\beta _{1}+\beta _{3})q^{7}+(2+\cdots)q^{9}+\cdots\)
1240.2.a.l 1240.a 1.a $6$ $9.901$ 6.6.473125168.1 None \(0\) \(-3\) \(-6\) \(-4\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}-q^{5}+(\beta _{2}+\beta _{4})q^{7}+(\beta _{1}-\beta _{2}+\cdots)q^{9}+\cdots\)
1240.2.a.m 1240.a 1.a $6$ $9.901$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(-1\) \(6\) \(-2\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+q^{5}-\beta _{2}q^{7}+(4-\beta _{3}+\beta _{4}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1240))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1240)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(62))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(124))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(155))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(248))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(310))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(620))\)\(^{\oplus 2}\)