Properties

Label 400.2.y
Level $400$
Weight $2$
Character orbit 400.y
Rep. character $\chi_{400}(129,\cdot)$
Character field $\Q(\zeta_{10})$
Dimension $56$
Newform subspaces $4$
Sturm bound $120$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.y (of order \(10\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 25 \)
Character field: \(\Q(\zeta_{10})\)
Newform subspaces: \( 4 \)
Sturm bound: \(120\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\)

Dimensions

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

Total New Old
Modular forms 264 64 200
Cusp forms 216 56 160
Eisenstein series 48 8 40

Trace form

\( 56 q + 5 q^{3} - 3 q^{5} + 9 q^{9} + O(q^{10}) \) \( 56 q + 5 q^{3} - 3 q^{5} + 9 q^{9} - 3 q^{11} - 5 q^{13} + 13 q^{15} - 5 q^{17} + 9 q^{19} + 6 q^{21} + 35 q^{23} + q^{25} + 5 q^{27} - 5 q^{29} + 9 q^{31} - 5 q^{33} - 28 q^{35} - 30 q^{37} + q^{39} - 5 q^{41} - 7 q^{45} + 5 q^{47} - 48 q^{49} + 62 q^{51} - 10 q^{53} + 13 q^{55} + 21 q^{59} + 7 q^{61} - 30 q^{63} + 4 q^{65} - 25 q^{67} - 9 q^{69} + 17 q^{71} - 5 q^{73} - 11 q^{75} + 30 q^{77} - 7 q^{79} - 39 q^{81} - 25 q^{83} + 20 q^{85} - 55 q^{87} - 18 q^{89} - 30 q^{91} - 113 q^{95} - 25 q^{97} - 76 q^{99} + O(q^{100}) \)

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.y.a 400.y 25.e $8$ $3.194$ \(\Q(\zeta_{20})\) None \(0\) \(0\) \(-10\) \(0\) $\mathrm{SU}(2)[C_{10}]$ \(q+(\zeta_{20}-\zeta_{20}^{2}-\zeta_{20}^{3}+\zeta_{20}^{5}+\zeta_{20}^{6}+\cdots)q^{3}+\cdots\)
400.2.y.b 400.y 25.e $8$ $3.194$ 8.0.58140625.2 None \(0\) \(0\) \(5\) \(0\) $\mathrm{SU}(2)[C_{10}]$ \(q+(\beta _{1}+\beta _{5}+\beta _{7})q^{3}+(-\beta _{3}+\beta _{4}+\beta _{5}+\cdots)q^{5}+\cdots\)
400.2.y.c 400.y 25.e $8$ $3.194$ 8.0.58140625.2 None \(0\) \(5\) \(0\) \(0\) $\mathrm{SU}(2)[C_{10}]$ \(q+(1+\beta _{3}+\beta _{7})q^{3}+(\beta _{3}-\beta _{4}-\beta _{5}+\cdots)q^{5}+\cdots\)
400.2.y.d 400.y 25.e $32$ $3.194$ None \(0\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{10}]$

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

\( S_{2}^{\mathrm{old}}(400, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 2}\)