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Properties

Label 1050.2.o
Level $1050$
Weight $2$
Character orbit 1050.o
Rep. character $\chi_{1050}(499,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $48$
Newform subspaces $12$
Sturm bound $480$
Trace bound $14$

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Defining parameters

Level: \( N \) \(=\) \( 1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1050.o (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 35 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 12 \)
Sturm bound: \(480\)
Trace bound: \(14\)
Distinguishing \(T_p\): \(11\), \(13\)

Dimensions

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

Total New Old
Modular forms 528 48 480
Cusp forms 432 48 384
Eisenstein series 96 0 96

Trace form

\( 48 q + 24 q^{4} - 8 q^{6} + 24 q^{9} + O(q^{10}) \) \( 48 q + 24 q^{4} - 8 q^{6} + 24 q^{9} + 8 q^{14} - 24 q^{16} - 12 q^{19} + 4 q^{21} - 4 q^{24} + 8 q^{26} + 64 q^{29} + 20 q^{31} + 32 q^{34} + 48 q^{36} + 4 q^{39} + 96 q^{41} + 16 q^{46} - 48 q^{49} + 16 q^{51} - 4 q^{54} + 16 q^{56} + 16 q^{59} - 20 q^{61} - 48 q^{64} + 16 q^{66} + 64 q^{69} - 64 q^{71} - 24 q^{74} - 24 q^{76} - 8 q^{79} - 24 q^{81} + 8 q^{84} - 40 q^{86} - 16 q^{89} - 28 q^{91} - 24 q^{94} + 4 q^{96} + O(q^{100}) \)

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.o.a 1050.o 35.j $4$ $8.384$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}+(-\zeta_{12}+\zeta_{12}^{3})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
1050.2.o.b 1050.o 35.j $4$ $8.384$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\zeta_{12}-\zeta_{12}^{3})q^{2}-\zeta_{12}q^{3}+(1-\zeta_{12}^{2}+\cdots)q^{4}+\cdots\)
1050.2.o.c 1050.o 35.j $4$ $8.384$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}+\zeta_{12}q^{3}+(1+\cdots)q^{4}+\cdots\)
1050.2.o.d 1050.o 35.j $4$ $8.384$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\zeta_{12}-\zeta_{12}^{3})q^{2}-\zeta_{12}q^{3}+(1-\zeta_{12}^{2}+\cdots)q^{4}+\cdots\)
1050.2.o.e 1050.o 35.j $4$ $8.384$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}+\zeta_{12}q^{3}+(1+\cdots)q^{4}+\cdots\)
1050.2.o.f 1050.o 35.j $4$ $8.384$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}+\zeta_{12}q^{3}+(1+\cdots)q^{4}+\cdots\)
1050.2.o.g 1050.o 35.j $4$ $8.384$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}+\zeta_{12}q^{3}+(1+\cdots)q^{4}+\cdots\)
1050.2.o.h 1050.o 35.j $4$ $8.384$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\zeta_{12}q^{2}+(-\zeta_{12}+\zeta_{12}^{3})q^{3}+\zeta_{12}^{2}q^{4}+\cdots\)
1050.2.o.i 1050.o 35.j $4$ $8.384$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\zeta_{12}-\zeta_{12}^{3})q^{2}+\zeta_{12}q^{3}+(1-\zeta_{12}^{2}+\cdots)q^{4}+\cdots\)
1050.2.o.j 1050.o 35.j $4$ $8.384$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}-\zeta_{12}q^{3}+(1+\cdots)q^{4}+\cdots\)
1050.2.o.k 1050.o 35.j $4$ $8.384$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\zeta_{12}+\zeta_{12}^{3})q^{2}-\zeta_{12}q^{3}+(1+\cdots)q^{4}+\cdots\)
1050.2.o.l 1050.o 35.j $4$ $8.384$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\zeta_{12}-\zeta_{12}^{3})q^{2}+\zeta_{12}q^{3}+(1-\zeta_{12}^{2}+\cdots)q^{4}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(1050, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(105, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(210, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(350, [\chi])\)\(^{\oplus 2}\)