Properties

Label 400.2.c
Level $400$
Weight $2$
Character orbit 400.c
Rep. character $\chi_{400}(49,\cdot)$
Character field $\Q$
Dimension $8$
Newform subspaces $4$
Sturm bound $120$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(120\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\)

Dimensions

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

Total New Old
Modular forms 78 10 68
Cusp forms 42 8 34
Eisenstein series 36 2 34

Trace form

\( 8 q - 4 q^{9} + O(q^{10}) \) \( 8 q - 4 q^{9} - 4 q^{11} + 12 q^{19} - 16 q^{21} + 8 q^{29} - 40 q^{39} - 12 q^{41} + 12 q^{51} + 32 q^{59} + 24 q^{61} + 24 q^{71} + 8 q^{79} + 16 q^{81} + 12 q^{89} - 24 q^{91} + O(q^{100}) \)

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

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
400.2.c.a 400.c 5.b $2$ $3.194$ \(\Q(\sqrt{-1}) \) None 200.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+3iq^{3}+2iq^{7}-6q^{9}-q^{11}+4iq^{13}+\cdots\)
400.2.c.b 400.c 5.b $2$ $3.194$ \(\Q(\sqrt{-1}) \) None 20.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+iq^{7}-q^{9}+iq^{13}+3iq^{17}+\cdots\)
400.2.c.c 400.c 5.b $2$ $3.194$ \(\Q(\sqrt{-1}) \) None 50.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}-2iq^{7}+2q^{9}+3q^{11}+4iq^{13}+\cdots\)
400.2.c.d 400.c 5.b $2$ $3.194$ \(\Q(\sqrt{-1}) \) None 40.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{7}+3q^{9}-4q^{11}+iq^{13}+iq^{17}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(400, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 2}\)