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Properties

Label 70.3.f
Level $70$
Weight $3$
Character orbit 70.f
Rep. character $\chi_{70}(43,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $12$
Newform subspaces $2$
Sturm bound $36$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 70 = 2 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 70.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 2 \)
Sturm bound: \(36\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(70, [\chi])\).

Total New Old
Modular forms 56 12 44
Cusp forms 40 12 28
Eisenstein series 16 0 16

Trace form

\( 12 q + 4 q^{2} + 8 q^{3} + 8 q^{5} - 8 q^{8} + O(q^{10}) \) \( 12 q + 4 q^{2} + 8 q^{3} + 8 q^{5} - 8 q^{8} - 12 q^{10} + 24 q^{11} + 16 q^{12} - 12 q^{13} + 32 q^{15} - 48 q^{16} - 12 q^{17} + 28 q^{18} - 40 q^{20} - 56 q^{21} + 16 q^{22} - 16 q^{23} - 4 q^{25} - 40 q^{26} - 136 q^{27} - 48 q^{30} - 32 q^{31} - 16 q^{32} - 152 q^{33} + 56 q^{36} + 148 q^{37} + 144 q^{38} + 40 q^{40} + 256 q^{41} + 104 q^{43} + 116 q^{45} + 128 q^{46} + 8 q^{47} - 32 q^{48} + 92 q^{50} - 152 q^{51} - 24 q^{52} + 124 q^{53} - 328 q^{55} + 296 q^{57} - 128 q^{58} + 48 q^{60} - 192 q^{61} - 128 q^{62} - 112 q^{63} + 292 q^{65} + 128 q^{66} - 264 q^{67} + 24 q^{68} + 112 q^{70} + 16 q^{71} + 56 q^{72} - 12 q^{73} - 280 q^{75} + 112 q^{77} - 368 q^{78} - 32 q^{80} - 60 q^{81} + 224 q^{82} + 528 q^{83} - 60 q^{85} - 544 q^{86} - 248 q^{87} - 32 q^{88} - 44 q^{90} - 168 q^{91} - 32 q^{92} - 88 q^{93} - 184 q^{95} - 388 q^{97} - 28 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(70, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
70.3.f.a 70.f 5.c $4$ $1.907$ \(\Q(i, \sqrt{14})\) None \(-4\) \(4\) \(12\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\beta _{2})q^{2}+(1+\beta _{1}+\beta _{2})q^{3}+\cdots\)
70.3.f.b 70.f 5.c $8$ $1.907$ 8.0.\(\cdots\).20 None \(8\) \(4\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+\beta _{3})q^{2}+(1+\beta _{1}-\beta _{3})q^{3}+2\beta _{3}q^{4}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(70, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(70, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 2}\)