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Properties

Label 1560.2.bg
Level $1560$
Weight $2$
Character orbit 1560.bg
Rep. character $\chi_{1560}(601,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $56$
Newform subspaces $12$
Sturm bound $672$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1560.bg (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 12 \)
Sturm bound: \(672\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1560, [\chi])\).

Total New Old
Modular forms 704 56 648
Cusp forms 640 56 584
Eisenstein series 64 0 64

Trace form

\( 56 q - 4 q^{3} + 4 q^{7} - 28 q^{9} + O(q^{10}) \) \( 56 q - 4 q^{3} + 4 q^{7} - 28 q^{9} - 4 q^{11} - 4 q^{13} + 8 q^{17} - 12 q^{19} - 8 q^{21} - 8 q^{23} + 56 q^{25} + 8 q^{27} + 8 q^{29} - 32 q^{31} - 4 q^{35} - 8 q^{37} + 12 q^{39} - 24 q^{41} - 28 q^{43} + 112 q^{47} - 4 q^{49} + 16 q^{51} + 48 q^{53} - 4 q^{55} + 16 q^{57} + 32 q^{59} - 20 q^{61} + 4 q^{63} - 12 q^{65} - 4 q^{67} - 8 q^{71} - 8 q^{73} - 4 q^{75} + 32 q^{77} - 64 q^{79} - 28 q^{81} + 32 q^{83} + 8 q^{87} + 36 q^{89} + 4 q^{93} + 28 q^{97} + 8 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(1560, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1560.2.bg.a 1560.bg 13.c $2$ $12.457$ \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(-2\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}-q^{5}-2\zeta_{6}q^{7}-\zeta_{6}q^{9}+\cdots\)
1560.2.bg.b 1560.bg 13.c $2$ $12.457$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(-2\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}-q^{5}+\zeta_{6}q^{7}-\zeta_{6}q^{9}+\cdots\)
1560.2.bg.c 1560.bg 13.c $2$ $12.457$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(2\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+q^{5}+\zeta_{6}q^{7}-\zeta_{6}q^{9}+\cdots\)
1560.2.bg.d 1560.bg 13.c $2$ $12.457$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(2\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+q^{5}+3\zeta_{6}q^{7}-\zeta_{6}q^{9}+\cdots\)
1560.2.bg.e 1560.bg 13.c $4$ $12.457$ \(\Q(\sqrt{-3}, \sqrt{10})\) None \(0\) \(2\) \(-4\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{2}q^{3}-q^{5}+(-1+\beta _{1}-\beta _{2})q^{7}+\cdots\)
1560.2.bg.f 1560.bg 13.c $4$ $12.457$ \(\Q(\sqrt{-3}, \sqrt{17})\) None \(0\) \(2\) \(4\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{2}q^{3}+q^{5}+(-2+\beta _{1}+2\beta _{2}+\beta _{3})q^{7}+\cdots\)
1560.2.bg.g 1560.bg 13.c $4$ $12.457$ \(\Q(\sqrt{-3}, \sqrt{-19})\) None \(0\) \(2\) \(4\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{1})q^{3}+q^{5}+\beta _{1}q^{7}+\beta _{1}q^{9}+\cdots\)
1560.2.bg.h 1560.bg 13.c $6$ $12.457$ 6.0.591408.1 None \(0\) \(-3\) \(-6\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{4})q^{3}-q^{5}+\beta _{1}q^{7}-\beta _{4}q^{9}+\cdots\)
1560.2.bg.i 1560.bg 13.c $6$ $12.457$ 6.0.954288.1 None \(0\) \(-3\) \(6\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{3})q^{3}+q^{5}+(-\beta _{4}+\beta _{5})q^{7}+\cdots\)
1560.2.bg.j 1560.bg 13.c $6$ $12.457$ 6.0.27870912.1 None \(0\) \(3\) \(-6\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{4}q^{3}-q^{5}+(1-\beta _{1}-\beta _{2}-\beta _{4}+\cdots)q^{7}+\cdots\)
1560.2.bg.k 1560.bg 13.c $8$ $12.457$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(-4\) \(-8\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{3}q^{3}-q^{5}+(1-\beta _{1}-\beta _{3})q^{7}+(-1+\cdots)q^{9}+\cdots\)
1560.2.bg.l 1560.bg 13.c $10$ $12.457$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(0\) \(-5\) \(10\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{6})q^{3}+q^{5}+\beta _{1}q^{7}-\beta _{6}q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1560, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1560, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(78, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(104, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(156, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(312, [\chi])\)\(^{\oplus 2}\)