Properties

Label 400.4.c
Level $400$
Weight $4$
Character orbit 400.c
Rep. character $\chi_{400}(49,\cdot)$
Character field $\Q$
Dimension $26$
Newform subspaces $13$
Sturm bound $240$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 400.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(240\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(400, [\chi])\).

Total New Old
Modular forms 198 28 170
Cusp forms 162 26 136
Eisenstein series 36 2 34

Trace form

\( 26 q - 214 q^{9} + O(q^{10}) \) \( 26 q - 214 q^{9} + 68 q^{11} - 140 q^{19} + 80 q^{21} - 84 q^{29} - 448 q^{31} + 808 q^{39} + 176 q^{41} - 1490 q^{49} + 236 q^{51} - 1752 q^{59} - 524 q^{61} + 1600 q^{69} + 872 q^{71} + 1448 q^{79} + 1090 q^{81} + 1392 q^{89} + 1256 q^{91} - 9720 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(400, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
400.4.c.a 400.c 5.b $2$ $23.601$ \(\Q(\sqrt{-1}) \) None 40.4.a.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+5iq^{3}+9iq^{7}-73q^{9}+2^{4}q^{11}+\cdots\)
400.4.c.b 400.c 5.b $2$ $23.601$ \(\Q(\sqrt{-1}) \) None 200.4.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+9iq^{3}-26iq^{7}-54q^{9}+59q^{11}+\cdots\)
400.4.c.c 400.c 5.b $2$ $23.601$ \(\Q(\sqrt{-1}) \) None 10.4.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4iq^{3}-2iq^{7}-37q^{9}-12q^{11}+\cdots\)
400.4.c.d 400.c 5.b $2$ $23.601$ \(\Q(\sqrt{-1}) \) None 50.4.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+7iq^{3}+34iq^{7}-22q^{9}-3^{3}q^{11}+\cdots\)
400.4.c.e 400.c 5.b $2$ $23.601$ \(\Q(\sqrt{-1}) \) None 25.4.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+7iq^{3}-6iq^{7}-22q^{9}+43q^{11}+\cdots\)
400.4.c.f 400.c 5.b $2$ $23.601$ \(\Q(\sqrt{-1}) \) None 40.4.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+3iq^{3}-17iq^{7}-9q^{9}-2^{4}q^{11}+\cdots\)
400.4.c.g 400.c 5.b $2$ $23.601$ \(\Q(\sqrt{-1}) \) None 200.4.a.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+5iq^{3}-2iq^{7}+2q^{9}-39q^{11}+\cdots\)
400.4.c.h 400.c 5.b $2$ $23.601$ \(\Q(\sqrt{-1}) \) None 40.4.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{3}-8iq^{7}+11q^{9}-6^{2}q^{11}+\cdots\)
400.4.c.i 400.c 5.b $2$ $23.601$ \(\Q(\sqrt{-1}) \) None 8.4.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{3}+12iq^{7}+11q^{9}+44q^{11}+\cdots\)
400.4.c.j 400.c 5.b $2$ $23.601$ \(\Q(\sqrt{-1}) \) None 20.4.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{3}+8iq^{7}+11q^{9}+60q^{11}+\cdots\)
400.4.c.k 400.c 5.b $2$ $23.601$ \(\Q(\sqrt{-1}) \) None 5.4.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}-3iq^{7}+23q^{9}-2^{5}q^{11}+\cdots\)
400.4.c.l 400.c 5.b $2$ $23.601$ \(\Q(\sqrt{-1}) \) None 100.4.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}-26iq^{7}+26q^{9}-45q^{11}+\cdots\)
400.4.c.m 400.c 5.b $2$ $23.601$ \(\Q(\sqrt{-1}) \) None 200.4.a.e \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+6iq^{7}+26q^{9}+19q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(400, [\chi])\) into lower level spaces

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