Properties

Label 400.2.l
Level $400$
Weight $2$
Character orbit 400.l
Rep. character $\chi_{400}(101,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $70$
Newform subspaces $9$
Sturm bound $120$
Trace bound $6$

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Defining parameters

Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.l (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 16 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 9 \)
Sturm bound: \(120\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(3\), \(7\)

Dimensions

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

Total New Old
Modular forms 132 82 50
Cusp forms 108 70 38
Eisenstein series 24 12 12

Trace form

\( 70q + 2q^{2} + 2q^{3} + 4q^{4} - 4q^{8} + O(q^{10}) \) \( 70q + 2q^{2} + 2q^{3} + 4q^{4} - 4q^{8} - 2q^{11} + 16q^{12} + 2q^{13} + 24q^{14} - 20q^{16} + 4q^{17} - 2q^{18} - 6q^{19} - 12q^{21} + 20q^{22} - 20q^{24} - 12q^{26} - 16q^{27} - 4q^{28} + 10q^{29} - 8q^{32} + 4q^{33} - 40q^{34} + 16q^{36} + 10q^{37} - 8q^{38} - 60q^{42} - 18q^{43} - 44q^{44} - 32q^{46} + 24q^{47} + 32q^{48} - 30q^{49} + 28q^{51} - 52q^{52} - 6q^{53} - 44q^{54} - 52q^{56} + 24q^{58} + 30q^{59} + 10q^{61} - 8q^{62} - 44q^{63} + 76q^{64} + 28q^{66} - 30q^{67} + 48q^{68} + 36q^{69} + 36q^{72} + 36q^{74} + 12q^{76} - 20q^{77} + 20q^{78} - 34q^{81} + 76q^{82} - 38q^{83} + 156q^{84} - 28q^{86} - 8q^{88} - 60q^{91} + 76q^{92} + 32q^{93} - 56q^{94} - 20q^{96} + 4q^{97} - 54q^{98} + 18q^{99} + O(q^{100}) \)

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

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
400.2.l.a \(2\) \(3.194\) \(\Q(\sqrt{-1}) \) None \(-2\) \(-2\) \(0\) \(0\) \(q+(-1+i)q^{2}+(-1+i)q^{3}-2iq^{4}+\cdots\)
400.2.l.b \(2\) \(3.194\) \(\Q(\sqrt{-1}) \) None \(2\) \(2\) \(0\) \(0\) \(q+(1-i)q^{2}+(1-i)q^{3}-2iq^{4}-2iq^{6}+\cdots\)
400.2.l.c \(2\) \(3.194\) \(\Q(\sqrt{-1}) \) None \(2\) \(2\) \(0\) \(0\) \(q+(1+i)q^{2}+(1-i)q^{3}+2iq^{4}+2q^{6}+\cdots\)
400.2.l.d \(4\) \(3.194\) \(\Q(i, \sqrt{11})\) None \(-4\) \(-2\) \(0\) \(0\) \(q+(-1-\beta _{1})q^{2}+(-\beta _{1}+\beta _{2})q^{3}+2\beta _{1}q^{4}+\cdots\)
400.2.l.e \(4\) \(3.194\) \(\Q(i, \sqrt{11})\) None \(4\) \(2\) \(0\) \(0\) \(q+(1+\beta _{1})q^{2}+(\beta _{1}-\beta _{2})q^{3}+2\beta _{1}q^{4}+\cdots\)
400.2.l.f \(12\) \(3.194\) 12.0.\(\cdots\).1 None \(-4\) \(-2\) \(0\) \(0\) \(q-\beta _{1}q^{2}-\beta _{6}q^{3}+\beta _{2}q^{4}+(\beta _{1}-\beta _{2}+\cdots)q^{6}+\cdots\)
400.2.l.g \(12\) \(3.194\) 12.0.\(\cdots\).1 None \(4\) \(2\) \(0\) \(0\) \(q+\beta _{1}q^{2}+\beta _{6}q^{3}+\beta _{2}q^{4}+(\beta _{1}-\beta _{2}+\cdots)q^{6}+\cdots\)
400.2.l.h \(16\) \(3.194\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{8}q^{2}+(\beta _{4}+\beta _{11})q^{3}+(\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)
400.2.l.i \(16\) \(3.194\) 16.0.\(\cdots\).1 None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{10}q^{2}+(-\beta _{2}+\beta _{9}+\beta _{12}-\beta _{14}+\cdots)q^{3}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(400, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(80, [\chi])\)\(^{\oplus 2}\)