↑

Properties

Label 350.3.k
Level $350$
Weight $3$
Character orbit 350.k
Rep. character $\chi_{350}(101,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $52$
Newform subspaces $5$
Sturm bound $180$
Trace bound $3$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 350.k (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 5 \)
Sturm bound: \(180\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(350, [\chi])\).

Total New Old
Modular forms 264 52 212
Cusp forms 216 52 164
Eisenstein series 48 0 48

Trace form

\( 52 q - 6 q^{3} - 52 q^{4} + 4 q^{7} + 96 q^{9} + O(q^{10}) \) \( 52 q - 6 q^{3} - 52 q^{4} + 4 q^{7} + 96 q^{9} - 18 q^{11} + 12 q^{12} + 28 q^{14} - 104 q^{16} + 78 q^{17} + 8 q^{18} - 78 q^{19} - 50 q^{21} - 72 q^{22} + 6 q^{23} - 24 q^{24} + 120 q^{26} + 44 q^{28} + 16 q^{29} + 150 q^{31} - 102 q^{33} - 384 q^{36} - 34 q^{37} - 108 q^{38} - 84 q^{39} - 136 q^{42} - 96 q^{43} - 36 q^{44} - 68 q^{46} + 18 q^{47} - 68 q^{49} - 154 q^{51} - 72 q^{52} + 254 q^{53} - 252 q^{54} - 64 q^{56} + 228 q^{57} - 8 q^{58} - 102 q^{59} + 582 q^{61} + 368 q^{63} + 416 q^{64} + 336 q^{66} - 74 q^{67} - 156 q^{68} + 144 q^{71} + 16 q^{72} - 534 q^{73} + 32 q^{74} - 102 q^{77} + 32 q^{78} + 206 q^{79} - 782 q^{81} + 264 q^{82} - 76 q^{84} + 40 q^{86} + 468 q^{87} + 72 q^{88} + 498 q^{89} - 4 q^{91} - 24 q^{92} + 250 q^{93} + 228 q^{94} + 48 q^{96} + 360 q^{98} - 344 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(350, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
350.3.k.a 350.k 7.d $4$ $9.537$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(6\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{1}q^{2}+(2+\beta _{1}+\beta _{2}-\beta _{3})q^{3}+2\beta _{2}q^{4}+\cdots\)
350.3.k.b 350.k 7.d $8$ $9.537$ 8.0.3317760000.3 None \(0\) \(-12\) \(0\) \(12\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{3}+\beta _{5})q^{2}+(-1+\beta _{3}-\beta _{4}+\beta _{6}+\cdots)q^{3}+\cdots\)
350.3.k.c 350.k 7.d $12$ $9.537$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(-6\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{3}q^{2}+(-1-\beta _{1}+\beta _{4}-\beta _{6}+\beta _{7}+\cdots)q^{3}+\cdots\)
350.3.k.d 350.k 7.d $12$ $9.537$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(6\) \(0\) \(4\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{2}q^{2}-\beta _{4}q^{3}+(-2+2\beta _{7})q^{4}+\cdots\)
350.3.k.e 350.k 7.d $16$ $9.537$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{11}q^{2}+\beta _{1}q^{3}+2\beta _{4}q^{4}+(\beta _{3}-\beta _{4}+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(350, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(350, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 2}\)