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Properties

Label 400.3.p
Level $400$
Weight $3$
Character orbit 400.p
Rep. character $\chi_{400}(193,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $34$
Newform subspaces $12$
Sturm bound $180$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 400.p (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 12 \)
Sturm bound: \(180\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(400, [\chi])\).

Total New Old
Modular forms 276 38 238
Cusp forms 204 34 170
Eisenstein series 72 4 68

Trace form

\( 34 q - 2 q^{3} - 2 q^{7} + O(q^{10}) \) \( 34 q - 2 q^{3} - 2 q^{7} + 4 q^{11} + 14 q^{13} - 2 q^{17} - 36 q^{21} + 46 q^{23} + 112 q^{27} + 132 q^{31} + 68 q^{33} + 22 q^{37} - 36 q^{41} - 66 q^{43} - 242 q^{47} - 572 q^{51} - 26 q^{53} + 16 q^{57} + 60 q^{61} + 222 q^{63} + 334 q^{67} + 836 q^{71} - 170 q^{73} + 100 q^{77} - 134 q^{81} - 274 q^{83} - 496 q^{87} - 188 q^{91} + 116 q^{93} + 22 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(400, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
400.3.p.a 400.p 5.c $2$ $10.899$ \(\Q(\sqrt{-1}) \) None \(0\) \(-6\) \(0\) \(6\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-3+3i)q^{3}+(3+3i)q^{7}-9iq^{9}+\cdots\)
400.3.p.b 400.p 5.c $2$ $10.899$ \(\Q(\sqrt{-1}) \) None \(0\) \(-4\) \(0\) \(4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-2+2i)q^{3}+(2+2i)q^{7}+iq^{9}+\cdots\)
400.3.p.c 400.p 5.c $2$ $10.899$ \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(-14\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+i)q^{3}+(-7-7i)q^{7}+7iq^{9}+\cdots\)
400.3.p.d 400.p 5.c $2$ $10.899$ \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(-14\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-i)q^{3}+(-7-7i)q^{7}+7iq^{9}+\cdots\)
400.3.p.e 400.p 5.c $2$ $10.899$ \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-i)q^{3}+(-3-3i)q^{7}+7iq^{9}+\cdots\)
400.3.p.f 400.p 5.c $2$ $10.899$ \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(14\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-i)q^{3}+(7+7i)q^{7}+7iq^{9}+4q^{11}+\cdots\)
400.3.p.g 400.p 5.c $2$ $10.899$ \(\Q(\sqrt{-1}) \) None \(0\) \(6\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{4}]$ \(q+(3-3i)q^{3}+(-3-3i)q^{7}-9iq^{9}+\cdots\)
400.3.p.h 400.p 5.c $4$ $10.899$ \(\Q(i, \sqrt{6})\) None \(0\) \(-8\) \(0\) \(16\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-2+2\beta _{2}+\beta _{3})q^{3}+(4+4\beta _{1}+4\beta _{2}+\cdots)q^{7}+\cdots\)
400.3.p.i 400.p 5.c $4$ $10.899$ \(\Q(i, \sqrt{41})\) None \(0\) \(-2\) \(0\) \(14\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\beta _{2})q^{3}+(3-3\beta _{1}-\beta _{3})q^{7}+\cdots\)
400.3.p.j 400.p 5.c $4$ $10.899$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{3}-4\beta _{3}q^{7}-6\beta _{2}q^{9}+3q^{11}+\cdots\)
400.3.p.k 400.p 5.c $4$ $10.899$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+3\beta _{1}q^{3}-4\beta _{3}q^{7}+18\beta _{2}q^{9}+15q^{11}+\cdots\)
400.3.p.l 400.p 5.c $4$ $10.899$ \(\Q(i, \sqrt{6})\) None \(0\) \(8\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{4}]$ \(q+(2-2\beta _{2}+\beta _{3})q^{3}+(-4+4\beta _{1}-4\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(400, [\chi])\) into lower level spaces

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