Properties

Label 1650.2.f
Level $1650$
Weight $2$
Character orbit 1650.f
Rep. character $\chi_{1650}(1649,\cdot)$
Character field $\Q$
Dimension $72$
Newform subspaces $8$
Sturm bound $720$
Trace bound $6$

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Defining parameters

Level: \( N \) \(=\) \( 1650 = 2 \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1650.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 165 \)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(720\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(7\), \(23\), \(29\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1650, [\chi])\).

Total New Old
Modular forms 384 72 312
Cusp forms 336 72 264
Eisenstein series 48 0 48

Trace form

\( 72 q - 72 q^{4} - 20 q^{9} + O(q^{10}) \) \( 72 q - 72 q^{4} - 20 q^{9} + 72 q^{16} - 16 q^{31} + 16 q^{34} + 20 q^{36} + 96 q^{49} - 72 q^{64} - 26 q^{66} + 48 q^{69} - 28 q^{81} - 144 q^{91} + 90 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1650, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1650.2.f.a 1650.f 165.d $4$ $13.175$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{8}q^{2}+(\zeta_{8}-\zeta_{8}^{3})q^{3}-q^{4}+(-1+\cdots)q^{6}+\cdots\)
1650.2.f.b 1650.f 165.d $4$ $13.175$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{8}q^{2}+(\zeta_{8}+\zeta_{8}^{3})q^{3}-q^{4}+(1-\zeta_{8}^{2}+\cdots)q^{6}+\cdots\)
1650.2.f.c 1650.f 165.d $8$ $13.175$ 8.0.2051727616.3 None \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{4}q^{2}+(-1-\beta _{2})q^{3}-q^{4}+\beta _{1}q^{6}+\cdots\)
1650.2.f.d 1650.f 165.d $8$ $13.175$ 8.0.2051727616.3 None \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{4}q^{2}+(-1-\beta _{6})q^{3}-q^{4}+(\beta _{3}+\cdots)q^{6}+\cdots\)
1650.2.f.e 1650.f 165.d $8$ $13.175$ 8.0.2051727616.3 None \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{4}q^{2}+(1+\beta _{2})q^{3}-q^{4}+\beta _{1}q^{6}+\cdots\)
1650.2.f.f 1650.f 165.d $8$ $13.175$ 8.0.2051727616.3 None \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{4}q^{2}+(1+\beta _{6})q^{3}-q^{4}+(\beta _{3}+\beta _{4}+\cdots)q^{6}+\cdots\)
1650.2.f.g 1650.f 165.d $16$ $13.175$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{6}q^{2}+\beta _{1}q^{3}-q^{4}+\beta _{4}q^{6}+\beta _{15}q^{7}+\cdots\)
1650.2.f.h 1650.f 165.d $16$ $13.175$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{6}q^{2}+\beta _{1}q^{3}-q^{4}-\beta _{4}q^{6}-\beta _{15}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1650, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1650, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(165, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(330, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(825, [\chi])\)\(^{\oplus 2}\)