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Properties

Label 350.4.c
Level $350$
Weight $4$
Character orbit 350.c
Rep. character $\chi_{350}(99,\cdot)$
Character field $\Q$
Dimension $28$
Newform subspaces $14$
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.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
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 192 28 164
Cusp forms 168 28 140
Eisenstein series 24 0 24

Trace form

\( 28 q - 112 q^{4} - 96 q^{9} + O(q^{10}) \) \( 28 q - 112 q^{4} - 96 q^{9} + 28 q^{11} - 56 q^{14} + 448 q^{16} + 96 q^{19} + 308 q^{21} + 72 q^{26} - 784 q^{29} - 288 q^{31} + 968 q^{34} + 384 q^{36} + 1280 q^{39} - 1372 q^{41} - 112 q^{44} + 224 q^{46} - 1372 q^{49} + 2996 q^{51} + 648 q^{54} + 224 q^{56} + 460 q^{59} - 900 q^{61} - 1792 q^{64} - 712 q^{66} + 56 q^{69} + 2448 q^{71} + 1472 q^{74} - 384 q^{76} - 248 q^{79} + 3444 q^{81} - 1232 q^{84} - 464 q^{86} + 492 q^{89} - 644 q^{91} - 1664 q^{94} - 4392 q^{99} + O(q^{100}) \)

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.c.a 350.c 5.b $2$ $20.651$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}+10iq^{3}-4q^{4}-20q^{6}+\cdots\)
350.4.c.b 350.c 5.b $2$ $20.651$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}+8iq^{3}-4q^{4}-2^{4}q^{6}+7iq^{7}+\cdots\)
350.4.c.c 350.c 5.b $2$ $20.651$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}+4iq^{3}-4q^{4}-8q^{6}-7iq^{7}+\cdots\)
350.4.c.d 350.c 5.b $2$ $20.651$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}+4iq^{3}-4q^{4}-8q^{6}-7iq^{7}+\cdots\)
350.4.c.e 350.c 5.b $2$ $20.651$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}+3iq^{3}-4q^{4}-6q^{6}+7iq^{7}+\cdots\)
350.4.c.f 350.c 5.b $2$ $20.651$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}+2iq^{3}-4q^{4}-4q^{6}+7iq^{7}+\cdots\)
350.4.c.g 350.c 5.b $2$ $20.651$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}+2iq^{3}-4q^{4}-4q^{6}+7iq^{7}+\cdots\)
350.4.c.h 350.c 5.b $2$ $20.651$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2iq^{2}+iq^{3}-4q^{4}+2q^{6}+7iq^{7}+\cdots\)
350.4.c.i 350.c 5.b $2$ $20.651$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2iq^{2}+iq^{3}-4q^{4}+2q^{6}+7iq^{7}+\cdots\)
350.4.c.j 350.c 5.b $2$ $20.651$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2iq^{2}+3iq^{3}-4q^{4}+6q^{6}-7iq^{7}+\cdots\)
350.4.c.k 350.c 5.b $2$ $20.651$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2iq^{2}+5iq^{3}-4q^{4}+10q^{6}+7iq^{7}+\cdots\)
350.4.c.l 350.c 5.b $2$ $20.651$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2iq^{2}+7iq^{3}-4q^{4}+14q^{6}-7iq^{7}+\cdots\)
350.4.c.m 350.c 5.b $2$ $20.651$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2iq^{2}+8iq^{3}-4q^{4}+2^{4}q^{6}-7iq^{7}+\cdots\)
350.4.c.n 350.c 5.b $2$ $20.651$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2iq^{2}+8iq^{3}-4q^{4}+2^{4}q^{6}-7iq^{7}+\cdots\)

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

\( S_{4}^{\mathrm{old}}(350, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)