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Properties

Label 1050.2.s
Level $1050$
Weight $2$
Character orbit 1050.s
Rep. character $\chi_{1050}(101,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $100$
Newform subspaces $10$
Sturm bound $480$
Trace bound $6$

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Defining parameters

Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.s (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 21 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 10 \)
Sturm bound: \(480\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(11\), \(37\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1050, [\chi])\).

Total New Old
Modular forms 528 100 428
Cusp forms 432 100 332
Eisenstein series 96 0 96

Trace form

\( 100 q + 50 q^{4} - 6 q^{7} + 6 q^{9} + O(q^{10}) \) \( 100 q + 50 q^{4} - 6 q^{7} + 6 q^{9} - 50 q^{16} + 8 q^{18} + 24 q^{19} + 16 q^{21} - 12 q^{22} - 6 q^{24} - 6 q^{28} - 18 q^{31} + 42 q^{33} + 12 q^{36} + 12 q^{37} + 12 q^{39} + 2 q^{42} + 32 q^{43} + 20 q^{46} - 2 q^{49} - 16 q^{51} + 12 q^{52} + 18 q^{54} + 72 q^{57} - 22 q^{58} + 60 q^{61} - 40 q^{63} - 100 q^{64} + 48 q^{66} + 12 q^{67} - 8 q^{72} - 24 q^{73} - 40 q^{78} + 2 q^{79} + 10 q^{81} + 24 q^{82} - 4 q^{84} - 54 q^{87} - 6 q^{88} - 8 q^{91} + 28 q^{93} + 24 q^{94} - 6 q^{96} - 128 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(1050, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1050.2.s.a 1050.s 21.g $4$ $8.384$ \(\Q(\zeta_{12})\) None \(0\) \(-6\) \(0\) \(10\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}+(-1-\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
1050.2.s.b 1050.s 21.g $4$ $8.384$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(10\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}+(\zeta_{12}-2\zeta_{12}^{3})q^{3}+\cdots\)
1050.2.s.c 1050.s 21.g $4$ $8.384$ \(\Q(\zeta_{12})\) None \(0\) \(6\) \(0\) \(-10\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\zeta_{12}-\zeta_{12}^{3})q^{2}+(1+\zeta_{12}^{2})q^{3}+\cdots\)
1050.2.s.d 1050.s 21.g $8$ $8.384$ 8.0.303595776.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{5}q^{2}+(-\beta _{1}+\beta _{5})q^{3}+(1+\beta _{4}+\cdots)q^{4}+\cdots\)
1050.2.s.e 1050.s 21.g $8$ $8.384$ 8.0.303595776.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{5}q^{2}+(-\beta _{1}+\beta _{7})q^{3}+(1+\beta _{4}+\cdots)q^{4}+\cdots\)
1050.2.s.f 1050.s 21.g $12$ $8.384$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(-2\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{4}q^{2}+\beta _{7}q^{3}-\beta _{6}q^{4}+(\beta _{2}+\beta _{10}+\cdots)q^{6}+\cdots\)
1050.2.s.g 1050.s 21.g $12$ $8.384$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(2\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{4}q^{2}+(\beta _{3}-\beta _{9})q^{3}-\beta _{6}q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots\)
1050.2.s.h 1050.s 21.g $16$ $8.384$ 16.0.\(\cdots\).8 None \(0\) \(-6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{7}q^{2}+\beta _{1}q^{3}+(1+\beta _{12})q^{4}+(-\beta _{11}+\cdots)q^{6}+\cdots\)
1050.2.s.i 1050.s 21.g $16$ $8.384$ 16.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{1}q^{2}+(\beta _{8}+\beta _{12}+\beta _{15})q^{3}+(1+\cdots)q^{4}+\cdots\)
1050.2.s.j 1050.s 21.g $16$ $8.384$ 16.0.\(\cdots\).8 None \(0\) \(6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{7}q^{2}-\beta _{1}q^{3}+(1+\beta _{12})q^{4}+(-\beta _{11}+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1050, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1050, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(525, [\chi])\)\(^{\oplus 2}\)