↑

Properties

Label 700.2.g
Level $700$
Weight $2$
Character orbit 700.g
Rep. character $\chi_{700}(251,\cdot)$
Character field $\Q$
Dimension $70$
Newform subspaces $12$
Sturm bound $240$
Trace bound $6$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 700 = 2^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 700.g (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 28 \)
Character field: \(\Q\)
Newform subspaces: \( 12 \)
Sturm bound: \(240\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(3\), \(11\), \(19\), \(37\)

Dimensions

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

Total New Old
Modular forms 132 82 50
Cusp forms 108 70 38
Eisenstein series 24 12 12

Trace form

\( 70 q + 3 q^{2} + q^{4} - 3 q^{8} + 62 q^{9} + O(q^{10}) \) \( 70 q + 3 q^{2} + q^{4} - 3 q^{8} + 62 q^{9} + 11 q^{14} - 19 q^{16} + 27 q^{18} - 18 q^{22} + 9 q^{28} + 20 q^{29} + 13 q^{32} - 19 q^{36} + 20 q^{37} + 4 q^{42} + 12 q^{44} - 32 q^{46} - 10 q^{49} - 12 q^{53} - 21 q^{56} + 48 q^{57} - 42 q^{58} - 53 q^{64} - 47 q^{72} - 4 q^{74} - 20 q^{77} + 36 q^{78} - 26 q^{81} - 32 q^{84} + 44 q^{86} - 82 q^{88} + 10 q^{92} + 16 q^{93} - 37 q^{98} + O(q^{100}) \)

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.g.a 700.g 28.d $2$ $5.590$ \(\Q(\sqrt{-7}) \) \(\Q(\sqrt{-7}) \) \(1\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta q^{2}+(-2+\beta )q^{4}+(1-2\beta )q^{7}+\cdots\)
700.2.g.b 700.g 28.d $4$ $5.590$ \(\Q(i, \sqrt{6})\) None \(-4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1+\beta _{1})q^{2}-\beta _{3}q^{3}-2\beta _{1}q^{4}+\cdots\)
700.2.g.c 700.g 28.d $4$ $5.590$ \(\Q(\sqrt{-3}, \sqrt{-7})\) \(\Q(\sqrt{-7}) \) \(-1\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(-\beta _{1}-\beta _{3})q^{2}+(1+\beta _{2})q^{4}+(1-2\beta _{3})q^{7}+\cdots\)
700.2.g.d 700.g 28.d $4$ $5.590$ \(\Q(\sqrt{-2}, \sqrt{-5})\) \(\Q(\sqrt{-5}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{2}q^{2}+(2\beta _{1}-\beta _{2})q^{3}-2q^{4}+2\beta _{3}q^{6}+\cdots\)
700.2.g.e 700.g 28.d $4$ $5.590$ \(\Q(\sqrt{-3}, \sqrt{-7})\) \(\Q(\sqrt{-7}) \) \(1\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(\beta _{1}+\beta _{3})q^{2}+(1+\beta _{2})q^{4}+(-1+2\beta _{3})q^{7}+\cdots\)
700.2.g.f 700.g 28.d $4$ $5.590$ \(\Q(\zeta_{12})\) None \(2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\zeta_{12}-\zeta_{12}^{2})q^{2}+(-\zeta_{12}+\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
700.2.g.g 700.g 28.d $4$ $5.590$ \(\Q(\zeta_{12})\) None \(2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\zeta_{12}-\zeta_{12}^{2})q^{2}+(\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+\cdots\)
700.2.g.h 700.g 28.d $4$ $5.590$ \(\Q(i, \sqrt{6})\) None \(4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1-\beta _{1})q^{2}-\beta _{3}q^{3}-2\beta _{1}q^{4}+(\beta _{2}+\cdots)q^{6}+\cdots\)
700.2.g.i 700.g 28.d $8$ $5.590$ 8.0.5473632256.3 None \(-4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1+\beta _{4})q^{2}+(-1+\beta _{1}-\beta _{2}+\beta _{5}+\cdots)q^{3}+\cdots\)
700.2.g.j 700.g 28.d $8$ $5.590$ 8.0.342102016.5 None \(-2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(\beta _{5}-\beta _{7})q^{3}+(-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
700.2.g.k 700.g 28.d $8$ $5.590$ 8.0.5473632256.3 None \(4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1-\beta _{4})q^{2}+(1-\beta _{1}+\beta _{2}-\beta _{5})q^{3}+\cdots\)
700.2.g.l 700.g 28.d $16$ $5.590$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{2}+\beta _{10}q^{3}+(1-\beta _{4})q^{4}+\beta _{14}q^{6}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(700, [\chi]) \cong \)