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Properties

Label 1050.3.l
Level $1050$
Weight $3$
Character orbit 1050.l
Rep. character $\chi_{1050}(43,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $72$
Newform subspaces $8$
Sturm bound $720$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1050.l (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 8 \)
Sturm bound: \(720\)
Trace bound: \(13\)
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 1008 72 936
Cusp forms 912 72 840
Eisenstein series 96 0 96

Trace form

\( 72 q + 8 q^{2} - 16 q^{8} + O(q^{10}) \) \( 72 q + 8 q^{2} - 16 q^{8} + 24 q^{13} - 288 q^{16} - 24 q^{17} + 24 q^{18} + 16 q^{22} + 16 q^{23} + 80 q^{26} - 64 q^{31} - 32 q^{32} - 144 q^{33} + 432 q^{36} - 8 q^{37} - 320 q^{41} + 32 q^{43} + 256 q^{46} + 64 q^{47} + 48 q^{52} + 152 q^{53} - 32 q^{58} + 672 q^{61} - 256 q^{62} - 384 q^{66} + 48 q^{68} - 512 q^{71} + 48 q^{72} - 312 q^{73} + 224 q^{77} - 648 q^{81} + 320 q^{82} + 576 q^{83} + 128 q^{86} + 192 q^{87} - 32 q^{88} + 32 q^{92} - 240 q^{93} - 664 q^{97} - 56 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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.l.a 1050.l 5.c $8$ $28.610$ 8.0.\(\cdots\).1 None \(-8\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-\beta _{3})q^{2}+\beta _{6}q^{3}+2\beta _{3}q^{4}+\cdots\)
1050.3.l.b 1050.l 5.c $8$ $28.610$ 8.0.\(\cdots\).1 None \(-8\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-\beta _{3})q^{2}-\beta _{6}q^{3}+2\beta _{3}q^{4}+\cdots\)
1050.3.l.c 1050.l 5.c $8$ $28.610$ 8.0.\(\cdots\).1 None \(-8\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\beta _{3})q^{2}+\beta _{1}q^{3}-2\beta _{3}q^{4}+\cdots\)
1050.3.l.d 1050.l 5.c $8$ $28.610$ 8.0.\(\cdots\).1 None \(-8\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\beta _{3})q^{2}+\beta _{1}q^{3}-2\beta _{3}q^{4}+\cdots\)
1050.3.l.e 1050.l 5.c $8$ $28.610$ 8.0.\(\cdots\).1 None \(8\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+\beta _{3})q^{2}-\beta _{6}q^{3}+2\beta _{3}q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots\)
1050.3.l.f 1050.l 5.c $8$ $28.610$ 8.0.\(\cdots\).1 None \(8\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-\beta _{3})q^{2}-\beta _{1}q^{3}-2\beta _{3}q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots\)
1050.3.l.g 1050.l 5.c $8$ $28.610$ 8.0.\(\cdots\).1 None \(8\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-\beta _{3})q^{2}-\beta _{1}q^{3}-2\beta _{3}q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots\)
1050.3.l.h 1050.l 5.c $16$ $28.610$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(16\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+\beta _{2})q^{2}+\beta _{4}q^{3}+2\beta _{2}q^{4}+(\beta _{3}+\cdots)q^{6}+\cdots\)

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

\( S_{3}^{\mathrm{old}}(1050, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(350, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(525, [\chi])\)\(^{\oplus 2}\)