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Properties

Label 75.9.d
Level $75$
Weight $9$
Character orbit 75.d
Rep. character $\chi_{75}(74,\cdot)$
Character field $\Q$
Dimension $46$
Newform subspaces $4$
Sturm bound $90$
Trace bound $4$

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Defining parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 75.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 15 \)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(90\)
Trace bound: \(4\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(75, [\chi])\).

Total New Old
Modular forms 86 50 36
Cusp forms 74 46 28
Eisenstein series 12 4 8

Trace form

\( 46 q + 5160 q^{4} - 2 q^{6} + 11492 q^{9} + O(q^{10}) \) \( 46 q + 5160 q^{4} - 2 q^{6} + 11492 q^{9} + 511032 q^{16} + 359214 q^{19} + 918774 q^{21} - 287334 q^{24} - 3218858 q^{31} - 1006364 q^{34} + 13513874 q^{36} - 2402834 q^{39} - 36479184 q^{46} - 50793772 q^{49} - 22870682 q^{51} + 24424268 q^{54} + 50274102 q^{61} - 41461504 q^{64} - 62032210 q^{66} + 36826668 q^{69} + 400879388 q^{76} - 212023636 q^{79} - 177577364 q^{81} + 767549976 q^{84} + 11622338 q^{91} - 894613664 q^{94} - 66741466 q^{96} + 664033870 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{9}^{\mathrm{new}}(75, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
75.9.d.a 75.d 15.d $2$ $30.553$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+3^{4}iq^{3}-2^{8}q^{4}+4273iq^{7}-3^{8}q^{9}+\cdots\)
75.9.d.b 75.d 15.d $4$ $30.553$ \(\Q(i, \sqrt{14})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{2}+(-9\beta _{1}+3\beta _{2})q^{3}+248q^{4}+\cdots\)
75.9.d.c 75.d 15.d $20$ $30.553$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-2\beta _{1}-\beta _{4})q^{3}+(79-\beta _{2}+\cdots)q^{4}+\cdots\)
75.9.d.d 75.d 15.d $20$ $30.553$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+(-\beta _{1}-\beta _{3})q^{3}+(155-\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(75, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 2}\)