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Properties

Label 1950.2.bc
Level $1950$
Weight $2$
Character orbit 1950.bc
Rep. character $\chi_{1950}(751,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $88$
Newform subspaces $11$
Sturm bound $840$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.bc (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 11 \)
Sturm bound: \(840\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1950, [\chi])\).

Total New Old
Modular forms 888 88 800
Cusp forms 792 88 704
Eisenstein series 96 0 96

Trace form

\( 88 q + 44 q^{4} - 44 q^{9} + O(q^{10}) \) \( 88 q + 44 q^{4} - 44 q^{9} - 24 q^{11} + 12 q^{13} - 44 q^{16} - 8 q^{17} + 36 q^{19} + 8 q^{23} - 20 q^{26} - 20 q^{29} - 12 q^{33} + 44 q^{36} - 12 q^{37} + 16 q^{39} + 12 q^{41} - 4 q^{42} + 8 q^{43} + 24 q^{46} + 32 q^{49} + 16 q^{53} + 36 q^{58} + 24 q^{59} + 16 q^{61} + 8 q^{62} - 88 q^{64} - 24 q^{66} + 24 q^{67} + 8 q^{68} - 4 q^{69} + 72 q^{71} + 28 q^{74} + 36 q^{76} + 32 q^{77} + 40 q^{79} - 44 q^{81} + 12 q^{82} - 24 q^{84} + 12 q^{87} + 48 q^{89} - 36 q^{91} + 16 q^{92} - 24 q^{93} + 16 q^{94} - 72 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(1950, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1950.2.bc.a 1950.bc 13.e $4$ $15.571$ \(\Q(\zeta_{12})\) None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\zeta_{12}-\zeta_{12}^{3})q^{2}-\zeta_{12}^{2}q^{3}+(1-\zeta_{12}^{2}+\cdots)q^{4}+\cdots\)
1950.2.bc.b 1950.bc 13.e $4$ $15.571$ \(\Q(\zeta_{12})\) None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\zeta_{12}-\zeta_{12}^{3})q^{2}-\zeta_{12}^{2}q^{3}+(1-\zeta_{12}^{2}+\cdots)q^{4}+\cdots\)
1950.2.bc.c 1950.bc 13.e $4$ $15.571$ \(\Q(\zeta_{12})\) None \(0\) \(-2\) \(0\) \(6\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}-\zeta_{12}^{2}q^{3}+(1+\cdots)q^{4}+\cdots\)
1950.2.bc.d 1950.bc 13.e $4$ $15.571$ \(\Q(\zeta_{12})\) None \(0\) \(2\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}+\zeta_{12}^{2}q^{3}+(1+\cdots)q^{4}+\cdots\)
1950.2.bc.e 1950.bc 13.e $8$ $15.571$ \(\Q(\zeta_{24})\) None \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\zeta_{24}^{2}-\zeta_{24}^{6})q^{2}-\zeta_{24}^{4}q^{3}+(1+\cdots)q^{4}+\cdots\)
1950.2.bc.f 1950.bc 13.e $8$ $15.571$ \(\Q(\zeta_{24})\) None \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\zeta_{24}^{2}q^{2}+(1-\zeta_{24}^{4})q^{3}+\zeta_{24}^{4}q^{4}+\cdots\)
1950.2.bc.g 1950.bc 13.e $8$ $15.571$ 8.0.\(\cdots\).1 None \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{2}-\beta _{5})q^{2}+(1-\beta _{6})q^{3}+\beta _{6}q^{4}+\cdots\)
1950.2.bc.h 1950.bc 13.e $12$ $15.571$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(-6\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{8}q^{2}-\beta _{4}q^{3}+(1-\beta _{4})q^{4}+\beta _{9}q^{6}+\cdots\)
1950.2.bc.i 1950.bc 13.e $12$ $15.571$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(-6\) \(0\) \(12\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{4}+\beta _{7})q^{2}+\beta _{6}q^{3}+(1+\beta _{6})q^{4}+\cdots\)
1950.2.bc.j 1950.bc 13.e $12$ $15.571$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(6\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{4}-\beta _{7})q^{2}-\beta _{6}q^{3}+(1+\beta _{6}+\cdots)q^{4}+\cdots\)
1950.2.bc.k 1950.bc 13.e $12$ $15.571$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(6\) \(0\) \(6\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{8}q^{2}+\beta _{4}q^{3}+(1-\beta _{4})q^{4}+\beta _{9}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1950, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1950, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(78, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(195, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(325, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(390, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(650, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(975, [\chi])\)\(^{\oplus 2}\)