Properties

Label 400.6.c
Level $400$
Weight $6$
Character orbit 400.c
Rep. character $\chi_{400}(49,\cdot)$
Character field $\Q$
Dimension $44$
Newform subspaces $16$
Sturm bound $360$
Trace bound $9$

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Defining parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(360\)
Trace bound: \(9\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(400, [\chi])\).

Total New Old
Modular forms 318 46 272
Cusp forms 282 44 238
Eisenstein series 36 2 34

Trace form

\( 44 q - 3400 q^{9} + O(q^{10}) \) \( 44 q - 3400 q^{9} - 724 q^{11} - 196 q^{19} - 2128 q^{21} - 12240 q^{29} + 12928 q^{31} - 10376 q^{39} - 8404 q^{41} - 85604 q^{49} - 115444 q^{51} + 77136 q^{59} + 2288 q^{61} + 43648 q^{69} + 25016 q^{71} + 58888 q^{79} + 156724 q^{81} - 132588 q^{89} + 160104 q^{91} + 244272 q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(400, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
400.6.c.a 400.c 5.b $2$ $64.154$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+13iq^{3}-11iq^{7}-433q^{9}+768q^{11}+\cdots\)
400.6.c.b 400.c 5.b $2$ $64.154$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+12iq^{3}+86iq^{7}-333q^{9}-132q^{11}+\cdots\)
400.6.c.c 400.c 5.b $2$ $64.154$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+11iq^{3}-109iq^{7}-241q^{9}+480q^{11}+\cdots\)
400.6.c.d 400.c 5.b $2$ $64.154$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+10iq^{3}+12iq^{7}-157q^{9}-124q^{11}+\cdots\)
400.6.c.e 400.c 5.b $2$ $64.154$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+9iq^{3}+11^{2}iq^{7}-3^{4}q^{9}-656q^{11}+\cdots\)
400.6.c.f 400.c 5.b $2$ $64.154$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+6iq^{3}-44iq^{7}+99q^{9}-540q^{11}+\cdots\)
400.6.c.g 400.c 5.b $2$ $64.154$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+11iq^{3}-142iq^{7}+122q^{9}-777q^{11}+\cdots\)
400.6.c.h 400.c 5.b $2$ $64.154$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4iq^{3}-54iq^{7}+179q^{9}+604q^{11}+\cdots\)
400.6.c.i 400.c 5.b $2$ $64.154$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+3iq^{3}+59iq^{7}+207q^{9}-192q^{11}+\cdots\)
400.6.c.j 400.c 5.b $2$ $64.154$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{3}+96iq^{7}+227q^{9}+148q^{11}+\cdots\)
400.6.c.k 400.c 5.b $2$ $64.154$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}-31iq^{7}+239q^{9}+12^{2}q^{11}+\cdots\)
400.6.c.l 400.c 5.b $4$ $64.154$ \(\Q(i, \sqrt{129})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-3\beta _{1}+\beta _{2})q^{3}+(-13\beta _{1}-3\beta _{2}+\cdots)q^{7}+\cdots\)
400.6.c.m 400.c 5.b $4$ $64.154$ \(\Q(i, \sqrt{409})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\beta _{1}+2\beta _{2})q^{3}+(-6\beta _{1}+4\beta _{2}+\cdots)q^{7}+\cdots\)
400.6.c.n 400.c 5.b $4$ $64.154$ \(\Q(i, \sqrt{241})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{1}-2\beta _{2})q^{3}+(-2\beta _{1}+20\beta _{2})q^{7}+\cdots\)
400.6.c.o 400.c 5.b $4$ $64.154$ \(\Q(i, \sqrt{241})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-4\beta _{1}-\beta _{2})q^{3}+(-4\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots\)
400.6.c.p 400.c 5.b $6$ $64.154$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{3}+(-24\beta _{1}-\beta _{3}-\beta _{4})q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(400, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(400, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 10}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(20, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(40, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 2}\)