Properties

Label 75.2.e
Level $75$
Weight $2$
Character orbit 75.e
Rep. character $\chi_{75}(32,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $8$
Newform subspaces $2$
Sturm bound $20$
Trace bound $6$

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Defining parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 75.e (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 15 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 2 \)
Sturm bound: \(20\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(75, [\chi])\).

Total New Old
Modular forms 32 16 16
Cusp forms 8 8 0
Eisenstein series 24 8 16

Trace form

\( 8 q - 12 q^{6} + O(q^{10}) \) \( 8 q - 12 q^{6} + 4 q^{16} - 12 q^{21} - 4 q^{31} + 36 q^{36} - 24 q^{46} + 48 q^{51} - 44 q^{61} - 8 q^{76} - 72 q^{81} + 36 q^{91} - 36 q^{96} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(75, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
75.2.e.a 75.e 15.e $4$ $0.599$ \(\Q(i, \sqrt{6})\) \(\Q(\sqrt{-15}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+\beta _{1}q^{2}+\beta _{3}q^{3}+\beta _{2}q^{4}-3q^{6}-\beta _{3}q^{8}+\cdots\)
75.2.e.b 75.e 15.e $4$ $0.599$ \(\Q(i, \sqrt{6})\) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+\beta _{1}q^{3}-2\beta _{2}q^{4}+\beta _{3}q^{7}+3\beta _{2}q^{9}+\cdots\)