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Properties

Label 700.2.p
Level $700$
Weight $2$
Character orbit 700.p
Rep. character $\chi_{700}(451,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $140$
Newform subspaces $6$
Sturm bound $240$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.p (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 28 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 6 \)
Sturm bound: \(240\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(3\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(700, [\chi])\).

Total New Old
Modular forms 264 164 100
Cusp forms 216 140 76
Eisenstein series 48 24 24

Trace form

\( 140 q + 2 q^{4} + 12 q^{8} - 56 q^{9} + O(q^{10}) \) \( 140 q + 2 q^{4} + 12 q^{8} - 56 q^{9} + 18 q^{12} - 20 q^{14} - 2 q^{16} + 6 q^{17} - 6 q^{21} + 12 q^{22} + 12 q^{24} - 30 q^{26} + 18 q^{28} + 16 q^{29} - 10 q^{32} - 6 q^{33} - 44 q^{36} - 2 q^{37} + 42 q^{38} + 50 q^{42} + 36 q^{44} - 22 q^{46} + 28 q^{49} + 36 q^{52} + 6 q^{53} + 24 q^{54} - 42 q^{56} - 12 q^{57} + 6 q^{58} - 30 q^{61} - 16 q^{64} - 138 q^{66} - 60 q^{68} - 4 q^{72} + 42 q^{73} + 16 q^{74} + 50 q^{77} - 144 q^{78} - 34 q^{81} - 54 q^{82} + 98 q^{84} + 4 q^{86} - 8 q^{88} - 54 q^{89} - 4 q^{92} + 2 q^{93} + 48 q^{94} + 114 q^{96} - 56 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(700, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
700.2.p.a 700.p 28.f $4$ $5.590$ \(\Q(\zeta_{12})\) None \(2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(-\zeta_{12}+\cdots)q^{3}+\cdots\)
700.2.p.b 700.p 28.f $8$ $5.590$ 8.0.3317760000.8 \(\Q(\sqrt{-5}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q+\beta _{5}q^{2}+(\beta _{1}+\beta _{5})q^{3}+2\beta _{3}q^{4}+(-1+\cdots)q^{6}+\cdots\)
700.2.p.c 700.p 28.f $32$ $5.590$ None \(-2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
700.2.p.d 700.p 28.f $32$ $5.590$ None \(-1\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
700.2.p.e 700.p 28.f $32$ $5.590$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
700.2.p.f 700.p 28.f $32$ $5.590$ None \(1\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$

Decomposition of \(S_{2}^{\mathrm{old}}(700, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(700, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(140, [\chi])\)\(^{\oplus 2}\)