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Properties

Label 975.2.i
Level $975$
Weight $2$
Character orbit 975.i
Rep. character $\chi_{975}(451,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $86$
Newform subspaces $17$
Sturm bound $280$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 17 \)
Sturm bound: \(280\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(975, [\chi])\).

Total New Old
Modular forms 304 86 218
Cusp forms 256 86 170
Eisenstein series 48 0 48

Trace form

\( 86 q + 2 q^{2} - q^{3} - 40 q^{4} + q^{7} - 12 q^{8} - 43 q^{9} + O(q^{10}) \) \( 86 q + 2 q^{2} - q^{3} - 40 q^{4} + q^{7} - 12 q^{8} - 43 q^{9} - 2 q^{11} - 4 q^{12} - 5 q^{13} + 32 q^{14} - 38 q^{16} - 10 q^{17} - 4 q^{18} + 2 q^{19} - 14 q^{21} - 8 q^{22} + 10 q^{23} - 12 q^{24} - 18 q^{26} + 2 q^{27} + 18 q^{28} + 12 q^{29} + 34 q^{31} - 6 q^{32} - 2 q^{33} - 108 q^{34} - 40 q^{36} + 4 q^{37} - 40 q^{38} + 22 q^{41} - 8 q^{42} + 9 q^{43} + 64 q^{44} - 8 q^{46} + 36 q^{47} + 12 q^{48} - 34 q^{49} - 32 q^{51} + 2 q^{52} + 60 q^{53} - 32 q^{56} + 24 q^{57} - 18 q^{58} + 2 q^{59} + 7 q^{61} - 16 q^{62} + q^{63} + 132 q^{64} - 32 q^{66} - 3 q^{67} - 54 q^{68} + 10 q^{69} + 54 q^{71} + 6 q^{72} - 50 q^{73} - 46 q^{74} + 56 q^{76} + 36 q^{77} + 12 q^{78} - 74 q^{79} - 43 q^{81} + 10 q^{82} + 56 q^{83} + 22 q^{84} + 48 q^{86} - 30 q^{87} - 28 q^{88} - 12 q^{89} - 7 q^{91} - 136 q^{92} + 3 q^{93} + 4 q^{94} + 48 q^{96} + 11 q^{97} + 94 q^{98} + 4 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(975, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
975.2.i.a 975.i 13.c $2$ $7.785$ \(\Q(\sqrt{-3}) \) None \(-2\) \(-1\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}-2\zeta_{6}q^{4}+\cdots\)
975.2.i.b 975.i 13.c $2$ $7.785$ \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
975.2.i.c 975.i 13.c $2$ $7.785$ \(\Q(\sqrt{-3}) \) None \(-1\) \(1\) \(0\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
975.2.i.d 975.i 13.c $2$ $7.785$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+2\zeta_{6}q^{4}-\zeta_{6}q^{7}-\zeta_{6}q^{9}+\cdots\)
975.2.i.e 975.i 13.c $2$ $7.785$ \(\Q(\sqrt{-3}) \) None \(1\) \(-1\) \(0\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
975.2.i.f 975.i 13.c $2$ $7.785$ \(\Q(\sqrt{-3}) \) None \(1\) \(-1\) \(0\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
975.2.i.g 975.i 13.c $2$ $7.785$ \(\Q(\sqrt{-3}) \) None \(1\) \(-1\) \(0\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
975.2.i.h 975.i 13.c $2$ $7.785$ \(\Q(\sqrt{-3}) \) None \(2\) \(1\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-2\zeta_{6}q^{4}+\cdots\)
975.2.i.i 975.i 13.c $2$ $7.785$ \(\Q(\sqrt{-3}) \) None \(2\) \(1\) \(0\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-2\zeta_{6}q^{4}+\cdots\)
975.2.i.j 975.i 13.c $4$ $7.785$ \(\Q(\zeta_{12})\) None \(-2\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\zeta_{12}+\zeta_{12}^{2})q^{2}+\zeta_{12}q^{3}+(-2+\cdots)q^{4}+\cdots\)
975.2.i.k 975.i 13.c $4$ $7.785$ \(\Q(\sqrt{-3}, \sqrt{17})\) None \(1\) \(2\) \(0\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+\beta _{2}q^{3}+(-2+\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots\)
975.2.i.l 975.i 13.c $6$ $7.785$ 6.0.1714608.1 None \(0\) \(-3\) \(0\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{3}-\beta _{5})q^{2}+(-1-\beta _{4})q^{3}+(-2\beta _{1}+\cdots)q^{4}+\cdots\)
975.2.i.m 975.i 13.c $6$ $7.785$ 6.0.591408.1 None \(0\) \(-3\) \(0\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{5}q^{2}+(-1+\beta _{4})q^{3}+(\beta _{1}-\beta _{3}+\cdots)q^{4}+\cdots\)
975.2.i.n 975.i 13.c $12$ $7.785$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(-6\) \(0\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{1}-\beta _{3})q^{2}+(-1-\beta _{6})q^{3}+(\beta _{6}+\cdots)q^{4}+\cdots\)
975.2.i.o 975.i 13.c $12$ $7.785$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(-6\) \(0\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{5}q^{2}-\beta _{3}q^{3}+(-1+\beta _{3}-\beta _{6}+\cdots)q^{4}+\cdots\)
975.2.i.p 975.i 13.c $12$ $7.785$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(6\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}+\beta _{3})q^{2}+(1+\beta _{6})q^{3}+(\beta _{6}-\beta _{9}+\cdots)q^{4}+\cdots\)
975.2.i.q 975.i 13.c $12$ $7.785$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(6\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{5}q^{2}+\beta _{3}q^{3}+(-1+\beta _{3}-\beta _{6}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(975, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(975, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 3}\)