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

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

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

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
1210.2.b.a \(2\) \(9.662\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-4\) \(0\) \(q-iq^{2}+iq^{3}-q^{4}+(-2+i)q^{5}+\cdots\)
1210.2.b.b \(2\) \(9.662\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(0\) \(q+iq^{2}+3iq^{3}-q^{4}+(-1-2i)q^{5}+\cdots\)
1210.2.b.c \(2\) \(9.662\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(0\) \(q-iq^{2}+3iq^{3}-q^{4}+(-1-2i)q^{5}+\cdots\)
1210.2.b.d \(2\) \(9.662\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(2\) \(0\) \(q+iq^{2}+2iq^{3}-q^{4}+(1+2i)q^{5}+\cdots\)
1210.2.b.e \(2\) \(9.662\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(4\) \(0\) \(q-iq^{2}+3iq^{3}-q^{4}+(2-i)q^{5}+3q^{6}+\cdots\)
1210.2.b.f \(4\) \(9.662\) \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(-8\) \(0\) \(q-\beta _{3}q^{2}+2\beta _{1}q^{3}-q^{4}+(-2-\beta _{3})q^{5}+\cdots\)
1210.2.b.g \(4\) \(9.662\) \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(-8\) \(0\) \(q+\beta _{3}q^{2}+2\beta _{1}q^{3}-q^{4}+(-2-\beta _{3})q^{5}+\cdots\)
1210.2.b.h \(4\) \(9.662\) \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+\beta _{2}q^{3}-q^{4}-\beta _{2}q^{5}-\beta _{3}q^{6}+\cdots\)
1210.2.b.i \(4\) \(9.662\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(4\) \(0\) \(q+\zeta_{12}^{3}q^{2}+(1-2\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+\cdots\)
1210.2.b.j \(4\) \(9.662\) \(\Q(\zeta_{12})\) None \(0\) \(0\) \(4\) \(0\) \(q-\zeta_{12}^{3}q^{2}+(1-2\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+\cdots\)
1210.2.b.k \(8\) \(9.662\) 8.0.\(\cdots\).1 None \(0\) \(0\) \(2\) \(0\) \(q-\beta _{3}q^{2}+(-\beta _{3}+\beta _{4})q^{3}-q^{4}+(1+\cdots)q^{5}+\cdots\)
1210.2.b.l \(8\) \(9.662\) 8.0.\(\cdots\).1 None \(0\) \(0\) \(2\) \(0\) \(q+\beta _{3}q^{2}+(-\beta _{3}+\beta _{4})q^{3}-q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
1210.2.b.m \(8\) \(9.662\) 8.0.303595776.1 None \(0\) \(0\) \(6\) \(0\) \(q-\beta _{5}q^{2}+(\beta _{2}-\beta _{4}+\beta _{6})q^{3}-q^{4}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1210, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(55, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(110, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(605, [\chi])\)\(^{\oplus 2}\)