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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

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

Display 100 coefficients 10 coefficients

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

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
75.4.e.a 75.e 15.e $4$ $4.425$ \(\Q(i, \sqrt{6})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+\beta _{1}q^{3}-8\beta _{2}q^{4}-6\beta _{3}q^{7}+3^{3}\beta _{2}q^{9}+\cdots\)
75.4.e.b 75.e 15.e $4$ $4.425$ \(\Q(i, \sqrt{6})\) \(\Q(\sqrt{-15}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+\beta _{1}q^{2}-\beta _{3}q^{3}+19\beta _{2}q^{4}+3^{3}q^{6}+\cdots\)
75.4.e.c 75.e 15.e $8$ $4.425$ 8.0.\(\cdots\).8 None \(0\) \(6\) \(0\) \(16\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{3}q^{2}+(1+\beta _{2}-\beta _{5}+\beta _{6}-\beta _{7})q^{3}+\cdots\)
75.4.e.d 75.e 15.e $16$ $4.425$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{3}q^{2}-\beta _{8}q^{3}+(6\beta _{1}+\beta _{5})q^{4}+(-6+\cdots)q^{6}+\cdots\)

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

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