Properties

Label 1040.2.bg
Level $1040$
Weight $2$
Character orbit 1040.bg
Rep. character $\chi_{1040}(577,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $80$
Newform subspaces $16$
Sturm bound $336$
Trace bound $15$

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Defining parameters

Level: \( N \) \(=\) \( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1040.bg (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 65 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 16 \)
Sturm bound: \(336\)
Trace bound: \(15\)
Distinguishing \(T_p\): \(3\), \(7\), \(11\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1040, [\chi])\).

Total New Old
Modular forms 360 88 272
Cusp forms 312 80 232
Eisenstein series 48 8 40

Trace form

\( 80 q + 4 q^{3} - 4 q^{5} + O(q^{10}) \) \( 80 q + 4 q^{3} - 4 q^{5} + 4 q^{11} - 4 q^{15} - 4 q^{17} - 4 q^{21} - 8 q^{27} + 4 q^{31} - 16 q^{33} + 4 q^{35} + 4 q^{41} - 16 q^{43} + 4 q^{45} - 40 q^{49} - 8 q^{53} + 52 q^{55} + 40 q^{57} - 16 q^{59} - 8 q^{61} - 12 q^{63} - 44 q^{67} + 44 q^{71} - 24 q^{73} + 16 q^{75} + 28 q^{77} - 56 q^{81} - 24 q^{85} - 8 q^{87} + 8 q^{89} - 12 q^{91} - 8 q^{97} + 68 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1040, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1040.2.bg.a 1040.bg 65.k $2$ $8.304$ \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{3}+(-1-2i)q^{5}-2iq^{7}+\cdots\)
1040.2.bg.b 1040.bg 65.k $2$ $8.304$ \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{3}+(-1+2i)q^{5}-2iq^{7}+\cdots\)
1040.2.bg.c 1040.bg 65.k $2$ $8.304$ \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(2\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{3}+(1-2i)q^{5}+2iq^{7}+\cdots\)
1040.2.bg.d 1040.bg 65.k $2$ $8.304$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-2-i)q^{5}-4iq^{7}-3iq^{9}+(-4+\cdots)q^{11}+\cdots\)
1040.2.bg.e 1040.bg 65.k $2$ $8.304$ \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+i)q^{3}+(-2-i)q^{5}-iq^{9}+(1+\cdots)q^{11}+\cdots\)
1040.2.bg.f 1040.bg 65.k $2$ $8.304$ \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(2\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+i)q^{3}+(1-2i)q^{5}-2iq^{7}-iq^{9}+\cdots\)
1040.2.bg.g 1040.bg 65.k $2$ $8.304$ \(\Q(\sqrt{-1}) \) None \(0\) \(4\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(2+2i)q^{3}+(-2-i)q^{5}-4iq^{7}+\cdots\)
1040.2.bg.h 1040.bg 65.k $2$ $8.304$ \(\Q(\sqrt{-1}) \) None \(0\) \(4\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(2+2i)q^{3}+(-1+2i)q^{5}+4iq^{7}+\cdots\)
1040.2.bg.i 1040.bg 65.k $4$ $8.304$ \(\Q(\zeta_{12})\) None \(0\) \(-2\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(\zeta_{12}-\zeta_{12}^{2})q^{3}+(2\zeta_{12}-\zeta_{12}^{2}+\cdots)q^{5}+\cdots\)
1040.2.bg.j 1040.bg 65.k $4$ $8.304$ \(\Q(i, \sqrt{5})\) None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{2}-\beta _{3})q^{5}+(-\beta _{1}+\cdots)q^{9}+\cdots\)
1040.2.bg.k 1040.bg 65.k $4$ $8.304$ \(\Q(i, \sqrt{11})\) None \(0\) \(-2\) \(6\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\beta _{1}-\beta _{3})q^{3}+(1-\beta _{3})q^{5}+\cdots\)
1040.2.bg.l 1040.bg 65.k $4$ $8.304$ \(\Q(\zeta_{8})\) None \(0\) \(4\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+\zeta_{8}+\zeta_{8}^{2})q^{3}+(-1+2\zeta_{8}^{2}+\cdots)q^{5}+\cdots\)
1040.2.bg.m 1040.bg 65.k $8$ $8.304$ 8.0.\(\cdots\).2 None \(0\) \(-2\) \(2\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{3}q^{3}+(\beta _{3}-\beta _{6})q^{5}+(-\beta _{4}-\beta _{5}+\cdots)q^{7}+\cdots\)
1040.2.bg.n 1040.bg 65.k $8$ $8.304$ 8.0.619810816.2 None \(0\) \(6\) \(2\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-\beta _{2}-\beta _{4}+\beta _{5})q^{3}+(\beta _{6}+\beta _{7})q^{5}+\cdots\)
1040.2.bg.o 1040.bg 65.k $12$ $8.304$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(-4\) \(8\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{1}q^{3}+(1-\beta _{7})q^{5}+\beta _{10}q^{7}+(\beta _{1}+\cdots)q^{9}+\cdots\)
1040.2.bg.p 1040.bg 65.k $20$ $8.304$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{3}q^{3}+\beta _{7}q^{5}+\beta _{11}q^{7}+(-\beta _{1}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1040, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1040, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(260, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(520, [\chi])\)\(^{\oplus 2}\)