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Properties

Label 1274.2.d
Level $1274$
Weight $2$
Character orbit 1274.d
Rep. character $\chi_{1274}(883,\cdot)$
Character field $\Q$
Dimension $46$
Newform subspaces $13$
Sturm bound $392$
Trace bound $10$

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Defining parameters

Level: \( N \) \(=\) \( 1274 = 2 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1274.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(392\)
Trace bound: \(10\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1274, [\chi])\).

Total New Old
Modular forms 212 46 166
Cusp forms 180 46 134
Eisenstein series 32 0 32

Trace form

\( 46 q + 2 q^{3} - 46 q^{4} + 48 q^{9} + O(q^{10}) \) \( 46 q + 2 q^{3} - 46 q^{4} + 48 q^{9} + 2 q^{10} - 2 q^{12} + 12 q^{13} + 46 q^{16} + 2 q^{17} - 4 q^{22} + 12 q^{23} - 28 q^{25} + 6 q^{26} + 14 q^{27} - 28 q^{29} - 10 q^{30} - 48 q^{36} + 4 q^{38} + 28 q^{39} - 2 q^{40} + 46 q^{43} + 2 q^{48} + 6 q^{51} - 12 q^{52} - 24 q^{53} - 40 q^{55} + 32 q^{61} - 24 q^{62} - 46 q^{64} - 26 q^{65} + 32 q^{66} - 2 q^{68} - 28 q^{69} + 34 q^{74} + 72 q^{75} + 38 q^{78} - 60 q^{79} + 14 q^{81} - 40 q^{82} - 88 q^{87} + 4 q^{88} - 20 q^{90} - 12 q^{92} - 2 q^{94} + 28 q^{95} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(1274, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1274.2.d.a 1274.d 13.b $2$ $10.173$ \(\Q(\sqrt{-1}) \) None 1274.2.d.a \(0\) \(-6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{2}-3q^{3}-q^{4}+4iq^{5}+3iq^{6}+\cdots\)
1274.2.d.b 1274.d 13.b $2$ $10.173$ \(\Q(\sqrt{-1}) \) None 182.2.n.a \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}-2q^{3}-q^{4}+4iq^{5}-2iq^{6}+\cdots\)
1274.2.d.c 1274.d 13.b $2$ $10.173$ \(\Q(\sqrt{-1}) \) None 26.2.b.a \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+q^{3}-q^{4}+3iq^{5}+iq^{6}+\cdots\)
1274.2.d.d 1274.d 13.b $2$ $10.173$ \(\Q(\sqrt{-1}) \) None 182.2.d.a \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{2}+q^{3}-q^{4}+2iq^{5}-iq^{6}+\cdots\)
1274.2.d.e 1274.d 13.b $2$ $10.173$ \(\Q(\sqrt{-1}) \) None 182.2.n.a \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{2}+2q^{3}-q^{4}+4iq^{5}-2iq^{6}+\cdots\)
1274.2.d.f 1274.d 13.b $2$ $10.173$ \(\Q(\sqrt{-1}) \) None 1274.2.d.a \(0\) \(6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+3q^{3}-q^{4}+4iq^{5}+3iq^{6}+\cdots\)
1274.2.d.g 1274.d 13.b $4$ $10.173$ \(\Q(\zeta_{8})\) None 1274.2.d.g \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{8}q^{2}+(-1+\zeta_{8}^{3})q^{3}-q^{4}+\zeta_{8}^{2}q^{5}+\cdots\)
1274.2.d.h 1274.d 13.b $4$ $10.173$ \(\Q(\zeta_{8})\) None 1274.2.d.h \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{8}^{2}q^{2}+(\zeta_{8}-\zeta_{8}^{3})q^{3}-q^{4}+(-\zeta_{8}+\cdots)q^{5}+\cdots\)
1274.2.d.i 1274.d 13.b $4$ $10.173$ \(\Q(\zeta_{8})\) None 1274.2.d.i \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{8}^{2}q^{2}+(2\zeta_{8}-2\zeta_{8}^{3})q^{3}-q^{4}+\cdots\)
1274.2.d.j 1274.d 13.b $4$ $10.173$ \(\Q(\zeta_{8})\) None 1274.2.d.g \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{8}q^{2}+(1+\zeta_{8}^{3})q^{3}-q^{4}-\zeta_{8}^{2}q^{5}+\cdots\)
1274.2.d.k 1274.d 13.b $6$ $10.173$ 6.0.153664.1 None 182.2.n.b \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{5}q^{2}+(-1+\beta _{2})q^{3}-q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
1274.2.d.l 1274.d 13.b $6$ $10.173$ 6.0.30647296.1 None 182.2.d.b \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{4}q^{2}-\beta _{3}q^{3}-q^{4}+(\beta _{1}+\beta _{2})q^{5}+\cdots\)
1274.2.d.m 1274.d 13.b $6$ $10.173$ 6.0.153664.1 None 182.2.n.b \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{5}q^{2}+(1-\beta _{2})q^{3}-q^{4}+(\beta _{1}-\beta _{5})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1274, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1274, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(91, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(182, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(637, [\chi])\)\(^{\oplus 2}\)