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Properties

Label 1210.2.b
Level $1210$
Weight $2$
Character orbit 1210.b
Rep. character $\chi_{1210}(969,\cdot)$
Character field $\Q$
Dimension $54$
Newform subspaces $13$
Sturm bound $396$
Trace bound $6$

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Defining parameters

Level: \( N \) \(=\) \( 1210 = 2 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1210.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(396\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(3\), \(13\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1210, [\chi])\).

Total New Old
Modular forms 222 54 168
Cusp forms 174 54 120
Eisenstein series 48 0 48

Trace form

\( 54 q - 54 q^{4} + 4 q^{6} - 54 q^{9} + O(q^{10}) \) \( 54 q - 54 q^{4} + 4 q^{6} - 54 q^{9} - 4 q^{10} + 4 q^{14} + 4 q^{15} + 54 q^{16} - 16 q^{19} - 4 q^{24} - 12 q^{25} - 4 q^{26} - 4 q^{29} + 4 q^{30} + 24 q^{31} + 8 q^{34} + 8 q^{35} + 54 q^{36} - 16 q^{39} + 4 q^{40} + 36 q^{41} + 32 q^{45} + 12 q^{46} - 82 q^{49} - 24 q^{50} - 48 q^{51} - 16 q^{54} - 4 q^{56} - 4 q^{59} - 4 q^{60} + 36 q^{61} - 54 q^{64} - 16 q^{65} + 48 q^{69} - 24 q^{70} - 24 q^{75} + 16 q^{76} + 16 q^{79} + 54 q^{81} - 8 q^{85} - 8 q^{86} + 12 q^{89} + 20 q^{90} - 8 q^{91} + 20 q^{94} + 4 q^{96} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(1210, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1210.2.b.a 1210.b 5.b $2$ $9.662$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{2}+iq^{3}-q^{4}+(-2+i)q^{5}+\cdots\)
1210.2.b.b 1210.b 5.b $2$ $9.662$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+3iq^{3}-q^{4}+(-1-2i)q^{5}+\cdots\)
1210.2.b.c 1210.b 5.b $2$ $9.662$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{2}+3iq^{3}-q^{4}+(-1-2i)q^{5}+\cdots\)
1210.2.b.d 1210.b 5.b $2$ $9.662$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+2iq^{3}-q^{4}+(1+2i)q^{5}+\cdots\)
1210.2.b.e 1210.b 5.b $2$ $9.662$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{2}+3iq^{3}-q^{4}+(2-i)q^{5}+3q^{6}+\cdots\)
1210.2.b.f 1210.b 5.b $4$ $9.662$ \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{2}+2\beta _{1}q^{3}-q^{4}+(-2-\beta _{3})q^{5}+\cdots\)
1210.2.b.g 1210.b 5.b $4$ $9.662$ \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{2}+2\beta _{1}q^{3}-q^{4}+(-2-\beta _{3})q^{5}+\cdots\)
1210.2.b.h 1210.b 5.b $4$ $9.662$ \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+\beta _{2}q^{3}-q^{4}-\beta _{2}q^{5}-\beta _{3}q^{6}+\cdots\)
1210.2.b.i 1210.b 5.b $4$ $9.662$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{12}^{3}q^{2}+(1-2\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+\cdots\)
1210.2.b.j 1210.b 5.b $4$ $9.662$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{12}^{3}q^{2}+(1-2\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+\cdots\)
1210.2.b.k 1210.b 5.b $8$ $9.662$ 8.0.\(\cdots\).1 None \(0\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{2}+(-\beta _{3}+\beta _{4})q^{3}-q^{4}+(1+\cdots)q^{5}+\cdots\)
1210.2.b.l 1210.b 5.b $8$ $9.662$ 8.0.\(\cdots\).1 None \(0\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{2}+(-\beta _{3}+\beta _{4})q^{3}-q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
1210.2.b.m 1210.b 5.b $8$ $9.662$ 8.0.303595776.1 None \(0\) \(0\) \(6\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{5}q^{2}+(\beta _{2}-\beta _{4}+\beta _{6})q^{3}-q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1210, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1210, [\chi]) \cong \)