Properties

Label 75.4.e
Level $75$
Weight $4$
Character orbit 75.e
Rep. character $\chi_{75}(32,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $32$
Newform subspaces $4$
Sturm bound $40$
Trace bound $6$

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Defining parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 75.e (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 15 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 4 \)
Sturm bound: \(40\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(75, [\chi])\).

Total New Old
Modular forms 72 40 32
Cusp forms 48 32 16
Eisenstein series 24 8 16

Trace form

\( 32q + 6q^{3} + 12q^{6} + 16q^{7} + O(q^{10}) \) \( 32q + 6q^{3} + 12q^{6} + 16q^{7} - 132q^{12} - 68q^{13} - 784q^{16} + 240q^{18} + 972q^{21} + 500q^{22} - 702q^{27} - 508q^{28} - 896q^{31} + 240q^{33} + 2364q^{36} + 1156q^{37} - 540q^{42} - 548q^{43} - 1496q^{46} + 1116q^{48} - 1128q^{51} - 224q^{52} - 684q^{57} - 60q^{58} - 2216q^{61} - 1428q^{63} - 1380q^{66} - 404q^{67} + 1800q^{72} + 2512q^{73} + 10248q^{76} + 360q^{78} + 1332q^{81} - 2800q^{82} + 1680q^{87} - 2460q^{88} - 3536q^{91} - 3408q^{93} - 10164q^{96} - 1904q^{97} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(75, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
75.4.e.a \(4\) \(4.425\) \(\Q(i, \sqrt{6})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{3}-8\beta _{2}q^{4}-6\beta _{3}q^{7}+3^{3}\beta _{2}q^{9}+\cdots\)
75.4.e.b \(4\) \(4.425\) \(\Q(i, \sqrt{6})\) \(\Q(\sqrt{-15}) \) \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}-\beta _{3}q^{3}+19\beta _{2}q^{4}+3^{3}q^{6}+\cdots\)
75.4.e.c \(8\) \(4.425\) 8.0.\(\cdots\).8 None \(0\) \(6\) \(0\) \(16\) \(q-\beta _{3}q^{2}+(1+\beta _{2}-\beta _{5}+\beta _{6}-\beta _{7})q^{3}+\cdots\)
75.4.e.d \(16\) \(4.425\) \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{2}-\beta _{8}q^{3}+(6\beta _{1}+\beta _{5})q^{4}+(-6+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(75, [\chi]) \cong \) \(S_{4}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 2}\)