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Properties

Label 350.4.j
Level $350$
Weight $4$
Character orbit 350.j
Rep. character $\chi_{350}(149,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $72$
Newform subspaces $10$
Sturm bound $240$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 350.j (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 35 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 10 \)
Sturm bound: \(240\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(350, [\chi])\).

Total New Old
Modular forms 384 72 312
Cusp forms 336 72 264
Eisenstein series 48 0 48

Trace form

\( 72 q + 144 q^{4} - 64 q^{6} + 416 q^{9} + O(q^{10}) \) \( 72 q + 144 q^{4} - 64 q^{6} + 416 q^{9} + 68 q^{11} + 168 q^{14} - 576 q^{16} + 64 q^{19} - 412 q^{21} - 128 q^{24} + 160 q^{26} + 1288 q^{29} + 308 q^{31} + 544 q^{34} + 3328 q^{36} + 128 q^{39} - 656 q^{41} - 272 q^{44} - 320 q^{46} - 1488 q^{49} + 2308 q^{51} - 2216 q^{54} + 288 q^{56} - 1636 q^{59} + 1400 q^{61} - 4608 q^{64} + 1600 q^{66} - 104 q^{69} - 3696 q^{71} + 1608 q^{74} + 512 q^{76} + 3972 q^{79} - 5372 q^{81} + 160 q^{84} - 960 q^{86} - 7332 q^{89} - 968 q^{91} - 5608 q^{94} + 512 q^{96} + 24408 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(350, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
350.4.j.a 350.j 35.j $4$ $20.651$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-2\zeta_{12}+2\zeta_{12}^{3})q^{2}+10\zeta_{12}q^{3}+\cdots\)
350.4.j.b 350.j 35.j $4$ $20.651$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-2\zeta_{12}+2\zeta_{12}^{3})q^{2}+5\zeta_{12}q^{3}+\cdots\)
350.4.j.c 350.j 35.j $4$ $20.651$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-2\zeta_{12}+2\zeta_{12}^{3})q^{2}+4\zeta_{12}q^{3}+\cdots\)
350.4.j.d 350.j 35.j $4$ $20.651$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(2\zeta_{12}-2\zeta_{12}^{3})q^{2}+\zeta_{12}q^{3}+(4+\cdots)q^{4}+\cdots\)
350.4.j.e 350.j 35.j $4$ $20.651$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(2\zeta_{12}-2\zeta_{12}^{3})q^{2}+\zeta_{12}q^{3}+(4+\cdots)q^{4}+\cdots\)
350.4.j.f 350.j 35.j $4$ $20.651$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(2\zeta_{12}-2\zeta_{12}^{3})q^{2}+\zeta_{12}q^{3}+(4+\cdots)q^{4}+\cdots\)
350.4.j.g 350.j 35.j $8$ $20.651$ 8.0.\(\cdots\).9 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{2}-\beta _{4}q^{3}+4\beta _{2}q^{4}+(2-\beta _{6}+\cdots)q^{6}+\cdots\)
350.4.j.h 350.j 35.j $12$ $20.651$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-2\beta _{5}q^{2}+(-\beta _{5}-\beta _{8}+\beta _{9})q^{3}+4\beta _{2}q^{4}+\cdots\)
350.4.j.i 350.j 35.j $12$ $20.651$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{3}q^{2}+(-\beta _{7}-\beta _{9})q^{3}-4\beta _{2}q^{4}+\cdots\)
350.4.j.j 350.j 35.j $16$ $20.651$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-2\beta _{8}+2\beta _{11})q^{2}+\beta _{10}q^{3}+4\beta _{1}q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(350, [\chi])\) into lower level spaces

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