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

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

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

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
400.3.p.a \(2\) \(10.899\) \(\Q(\sqrt{-1}) \) None \(0\) \(-6\) \(0\) \(6\) \(q+(-3+3i)q^{3}+(3+3i)q^{7}-9iq^{9}+\cdots\)
400.3.p.b \(2\) \(10.899\) \(\Q(\sqrt{-1}) \) None \(0\) \(-4\) \(0\) \(4\) \(q+(-2+2i)q^{3}+(2+2i)q^{7}+iq^{9}+\cdots\)
400.3.p.c \(2\) \(10.899\) \(\Q(\sqrt{-1}) \) None \(0\) \(-2\) \(0\) \(-14\) \(q+(-1+i)q^{3}+(-7-7i)q^{7}+7iq^{9}+\cdots\)
400.3.p.d \(2\) \(10.899\) \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(-14\) \(q+(1-i)q^{3}+(-7-7i)q^{7}+7iq^{9}+\cdots\)
400.3.p.e \(2\) \(10.899\) \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(-6\) \(q+(1-i)q^{3}+(-3-3i)q^{7}+7iq^{9}+\cdots\)
400.3.p.f \(2\) \(10.899\) \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(0\) \(14\) \(q+(1-i)q^{3}+(7+7i)q^{7}+7iq^{9}+4q^{11}+\cdots\)
400.3.p.g \(2\) \(10.899\) \(\Q(\sqrt{-1}) \) None \(0\) \(6\) \(0\) \(-6\) \(q+(3-3i)q^{3}+(-3-3i)q^{7}-9iq^{9}+\cdots\)
400.3.p.h \(4\) \(10.899\) \(\Q(i, \sqrt{6})\) None \(0\) \(-8\) \(0\) \(16\) \(q+(-2+2\beta _{2}+\beta _{3})q^{3}+(4+4\beta _{1}+4\beta _{2}+\cdots)q^{7}+\cdots\)
400.3.p.i \(4\) \(10.899\) \(\Q(i, \sqrt{41})\) None \(0\) \(-2\) \(0\) \(14\) \(q+(-1+\beta _{2})q^{3}+(3-3\beta _{1}-\beta _{3})q^{7}+\cdots\)
400.3.p.j \(4\) \(10.899\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}-4\beta _{3}q^{7}-6\beta _{2}q^{9}+3q^{11}+\cdots\)
400.3.p.k \(4\) \(10.899\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) \(q+3\beta _{1}q^{3}-4\beta _{3}q^{7}+18\beta _{2}q^{9}+15q^{11}+\cdots\)
400.3.p.l \(4\) \(10.899\) \(\Q(i, \sqrt{6})\) None \(0\) \(8\) \(0\) \(-16\) \(q+(2-2\beta _{2}+\beta _{3})q^{3}+(-4+4\beta _{1}-4\beta _{2}+\cdots)q^{7}+\cdots\)

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

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