Properties

Label 1050.3.e
Level $1050$
Weight $3$
Character orbit 1050.e
Rep. character $\chi_{1050}(701,\cdot)$
Character field $\Q$
Dimension $76$
Newform subspaces $5$
Sturm bound $720$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1050.e (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(720\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(11\), \(13\)

Dimensions

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

Total New Old
Modular forms 504 76 428
Cusp forms 456 76 380
Eisenstein series 48 0 48

Trace form

\( 76 q + 8 q^{3} - 152 q^{4} - 8 q^{6} - 12 q^{9} + O(q^{10}) \) \( 76 q + 8 q^{3} - 152 q^{4} - 8 q^{6} - 12 q^{9} - 16 q^{12} - 40 q^{13} + 304 q^{16} + 32 q^{18} + 16 q^{19} + 28 q^{21} + 56 q^{22} + 16 q^{24} - 64 q^{27} + 144 q^{31} + 48 q^{33} - 208 q^{34} + 24 q^{36} - 160 q^{37} - 56 q^{39} + 256 q^{43} + 40 q^{46} + 32 q^{48} + 532 q^{49} + 104 q^{51} + 80 q^{52} + 40 q^{54} + 264 q^{57} + 64 q^{58} - 56 q^{61} - 112 q^{63} - 608 q^{64} + 144 q^{66} - 480 q^{67} - 192 q^{69} - 64 q^{72} - 168 q^{73} - 32 q^{76} - 128 q^{78} + 208 q^{79} - 300 q^{81} + 368 q^{82} - 56 q^{84} + 280 q^{87} - 112 q^{88} - 224 q^{91} + 176 q^{93} - 208 q^{94} - 32 q^{96} + 552 q^{97} + 1352 q^{99} + O(q^{100}) \)

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.e.a 1050.e 3.b $4$ $28.610$ \(\Q(\sqrt{-2}, \sqrt{7})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{3})q^{3}-2q^{4}+(-2+\cdots)q^{6}+\cdots\)
1050.3.e.b 1050.e 3.b $16$ $28.610$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{2}+\beta _{2}q^{3}-2q^{4}-\beta _{9}q^{6}+\beta _{6}q^{7}+\cdots\)
1050.3.e.c 1050.e 3.b $16$ $28.610$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{2}-\beta _{2}q^{3}-2q^{4}-\beta _{9}q^{6}-\beta _{6}q^{7}+\cdots\)
1050.3.e.d 1050.e 3.b $16$ $28.610$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(8\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{2}+\beta _{6}q^{3}-2q^{4}+(1+\beta _{7})q^{6}+\cdots\)
1050.3.e.e 1050.e 3.b $24$ $28.610$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$

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

\( S_{3}^{\mathrm{old}}(1050, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(42, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(525, [\chi])\)\(^{\oplus 2}\)