Properties

Label 400.2.n
Level $400$
Weight $2$
Character orbit 400.n
Rep. character $\chi_{400}(143,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $18$
Newform subspaces $4$
Sturm bound $120$
Trace bound $13$

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Defining parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.n (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 20 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 4 \)
Sturm bound: \(120\)
Trace bound: \(13\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(400, [\chi])\).

Total New Old
Modular forms 156 18 138
Cusp forms 84 18 66
Eisenstein series 72 0 72

Trace form

\( 18 q + O(q^{10}) \) \( 18 q + 6 q^{13} + 6 q^{17} + 48 q^{21} - 24 q^{33} - 30 q^{37} - 48 q^{41} + 18 q^{53} + 48 q^{57} + 18 q^{73} - 24 q^{77} - 66 q^{81} + 24 q^{93} + 18 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(400, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
400.2.n.a 400.n 20.e $2$ $3.194$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q-3iq^{9}+(5-5i)q^{13}+(5+5i)q^{17}+\cdots\)
400.2.n.b 400.n 20.e $4$ $3.194$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\zeta_{12}^{2}q^{3}+\zeta_{12}^{3}q^{7}+3\zeta_{12}q^{9}+\cdots\)
400.2.n.c 400.n 20.e $4$ $3.194$ \(\Q(i, \sqrt{5})\) \(\Q(\sqrt{-5}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q-\beta _{3}q^{3}+\beta _{2}q^{7}-7\beta _{1}q^{9}+10q^{21}+\cdots\)
400.2.n.d 400.n 20.e $8$ $3.194$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{24}q^{3}+4\zeta_{24}^{5}q^{7}-2\zeta_{24}^{3}q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(400, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(400, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 3}\)