Properties

Label 1050.3.f
Level $1050$
Weight $3$
Character orbit 1050.f
Rep. character $\chi_{1050}(601,\cdot)$
Character field $\Q$
Dimension $52$
Newform subspaces $5$
Sturm bound $720$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1050.f (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(720\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(11\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(1050, [\chi])\).

Total New Old
Modular forms 504 52 452
Cusp forms 456 52 404
Eisenstein series 48 0 48

Trace form

\( 52 q + 104 q^{4} - 8 q^{7} - 156 q^{9} + O(q^{10}) \) \( 52 q + 104 q^{4} - 8 q^{7} - 156 q^{9} + 24 q^{11} + 8 q^{14} + 208 q^{16} + 24 q^{21} - 72 q^{22} - 24 q^{23} - 16 q^{28} + 8 q^{29} - 312 q^{36} + 128 q^{37} + 168 q^{39} + 192 q^{43} + 48 q^{44} + 40 q^{46} - 32 q^{49} + 72 q^{51} + 88 q^{53} + 16 q^{56} - 120 q^{57} - 224 q^{58} + 24 q^{63} + 416 q^{64} + 16 q^{67} + 168 q^{71} - 224 q^{74} - 216 q^{77} + 144 q^{78} + 32 q^{79} + 468 q^{81} + 48 q^{84} + 368 q^{86} - 144 q^{88} - 28 q^{91} - 48 q^{92} - 192 q^{93} + 192 q^{98} - 72 q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(1050, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1050.3.f.a 1050.f 7.b $4$ $28.610$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+\beta _{2}q^{3}+2q^{4}+\beta _{3}q^{6}+(-2+\cdots)q^{7}+\cdots\)
1050.3.f.b 1050.f 7.b $8$ $28.610$ 8.0.3317760000.3 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+\beta _{5}q^{3}+2q^{4}-\beta _{3}q^{6}+(2\beta _{1}+\cdots)q^{7}+\cdots\)
1050.3.f.c 1050.f 7.b $12$ $28.610$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(-10\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{5}q^{2}-\beta _{2}q^{3}+2q^{4}-\beta _{4}q^{6}+(-1+\cdots)q^{7}+\cdots\)
1050.3.f.d 1050.f 7.b $12$ $28.610$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(10\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{5}q^{2}+\beta _{2}q^{3}+2q^{4}-\beta _{4}q^{6}+(1+\cdots)q^{7}+\cdots\)
1050.3.f.e 1050.f 7.b $16$ $28.610$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{5}q^{2}-\beta _{1}q^{3}+2q^{4}-\beta _{8}q^{6}+(\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(1050, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(1050, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 12}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(350, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(525, [\chi])\)\(^{\oplus 2}\)