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Properties

Label 1260.2.a
Level $1260$
Weight $2$
Character orbit 1260.a
Rep. character $\chi_{1260}(1,\cdot)$
Character field $\Q$
Dimension $10$
Newform subspaces $10$
Sturm bound $576$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1260 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1260.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(576\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(11\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1260))\).

Total New Old
Modular forms 312 10 302
Cusp forms 265 10 255
Eisenstein series 47 0 47

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

\(2\)\(3\)\(5\)\(7\)FrickeDim
\(-\)\(+\)\(+\)\(+\)$-$\(1\)
\(-\)\(+\)\(+\)\(-\)$+$\(1\)
\(-\)\(+\)\(-\)\(+\)$+$\(1\)
\(-\)\(+\)\(-\)\(-\)$-$\(1\)
\(-\)\(-\)\(+\)\(+\)$+$\(1\)
\(-\)\(-\)\(+\)\(-\)$-$\(2\)
\(-\)\(-\)\(-\)\(+\)$-$\(2\)
\(-\)\(-\)\(-\)\(-\)$+$\(1\)
Plus space\(+\)\(4\)
Minus space\(-\)\(6\)

Trace form

\( 10 q + O(q^{10}) \) \( 10 q - 6 q^{11} - 4 q^{13} - 12 q^{17} + 8 q^{19} + 10 q^{25} + 10 q^{29} + 4 q^{31} - 2 q^{35} + 8 q^{37} + 4 q^{41} + 12 q^{43} + 20 q^{47} + 10 q^{49} - 4 q^{53} - 8 q^{55} + 12 q^{59} - 16 q^{61} - 10 q^{65} + 20 q^{67} + 16 q^{71} + 16 q^{73} - 8 q^{77} - 14 q^{79} + 32 q^{83} - 2 q^{85} - 14 q^{91} + 12 q^{95} + 4 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1260))\) 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 3 5 7
1260.2.a.a 1260.a 1.a $1$ $10.061$ \(\Q\) None \(0\) \(0\) \(-1\) \(-1\) $-$ $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-q^{7}-2q^{11}+4q^{13}-2q^{17}+\cdots\)
1260.2.a.b 1260.a 1.a $1$ $10.061$ \(\Q\) None \(0\) \(0\) \(-1\) \(-1\) $-$ $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{5}-q^{7}+4q^{11}+2q^{17}-6q^{19}+\cdots\)
1260.2.a.c 1260.a 1.a $1$ $10.061$ \(\Q\) None \(0\) \(0\) \(-1\) \(1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+q^{7}-3q^{11}-q^{13}+3q^{17}+\cdots\)
1260.2.a.d 1260.a 1.a $1$ $10.061$ \(\Q\) None \(0\) \(0\) \(-1\) \(1\) $-$ $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+q^{7}-4q^{13}-6q^{17}+2q^{19}+\cdots\)
1260.2.a.e 1260.a 1.a $1$ $10.061$ \(\Q\) None \(0\) \(0\) \(-1\) \(1\) $-$ $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{5}+q^{7}+2q^{11}+4q^{13}-2q^{17}+\cdots\)
1260.2.a.f 1260.a 1.a $1$ $10.061$ \(\Q\) None \(0\) \(0\) \(1\) \(-1\) $-$ $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}-q^{7}-4q^{11}-2q^{17}-6q^{19}+\cdots\)
1260.2.a.g 1260.a 1.a $1$ $10.061$ \(\Q\) None \(0\) \(0\) \(1\) \(-1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}-q^{7}-2q^{11}+4q^{13}-6q^{17}+\cdots\)
1260.2.a.h 1260.a 1.a $1$ $10.061$ \(\Q\) None \(0\) \(0\) \(1\) \(-1\) $-$ $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{5}-q^{7}+5q^{11}-3q^{13}+q^{17}+\cdots\)
1260.2.a.i 1260.a 1.a $1$ $10.061$ \(\Q\) None \(0\) \(0\) \(1\) \(1\) $-$ $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+q^{7}-6q^{11}-4q^{13}-6q^{17}+\cdots\)
1260.2.a.j 1260.a 1.a $1$ $10.061$ \(\Q\) None \(0\) \(0\) \(1\) \(1\) $-$ $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{5}+q^{7}-4q^{13}+6q^{17}+2q^{19}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1260))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1260)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(42))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(84))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(126))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(140))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(180))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(210))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(252))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(315))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(420))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(630))\)\(^{\oplus 2}\)