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Properties

Label 75.9.f
Level $75$
Weight $9$
Character orbit 75.f
Rep. character $\chi_{75}(7,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $48$
Newform subspaces $5$
Sturm bound $90$
Trace bound $11$

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Defining parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 75.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 5 \)
Sturm bound: \(90\)
Trace bound: \(11\)
Distinguishing \(T_p\): \(2\)

Dimensions

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

Total New Old
Modular forms 172 48 124
Cusp forms 148 48 100
Eisenstein series 24 0 24

Trace form

\( 48 q + 4536 q^{6} - 4540 q^{7} - 17460 q^{8} + O(q^{10}) \) \( 48 q + 4536 q^{6} - 4540 q^{7} - 17460 q^{8} + 47232 q^{11} - 22680 q^{12} - 133420 q^{13} - 367960 q^{16} - 573300 q^{17} + 81972 q^{21} + 234700 q^{22} - 651480 q^{23} - 4585824 q^{26} + 3567940 q^{28} + 566908 q^{31} - 641460 q^{32} + 3815100 q^{33} + 11354904 q^{36} + 3607340 q^{37} - 8139840 q^{38} + 3231888 q^{41} + 9643860 q^{42} + 4805480 q^{43} + 51342728 q^{46} - 26529600 q^{47} - 3661200 q^{48} + 12336624 q^{51} + 15861080 q^{52} - 16612140 q^{53} + 100370760 q^{56} - 4714200 q^{57} - 63562980 q^{58} - 21166260 q^{61} + 35190840 q^{62} - 9928980 q^{63} - 93917232 q^{66} - 46836760 q^{67} + 197811840 q^{68} + 171363936 q^{71} + 38185020 q^{72} + 50835800 q^{73} - 447468656 q^{76} - 97175880 q^{77} - 131709240 q^{78} - 229582512 q^{81} + 181542400 q^{82} + 208234800 q^{83} + 305840712 q^{86} + 74298060 q^{87} - 138207420 q^{88} - 462480412 q^{91} - 652331400 q^{92} - 159787080 q^{93} - 71070048 q^{96} + 138370520 q^{97} + 50186520 q^{98} + O(q^{100}) \)

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.f.a 75.f 5.c $4$ $30.553$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+6\beta _{1}q^{2}-3^{3}\beta _{3}q^{3}-148\beta _{2}q^{4}+\cdots\)
75.9.f.b 75.f 5.c $8$ $30.553$ 8.0.\(\cdots\).2 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-2\beta _{1}-\beta _{5})q^{2}+3^{3}\beta _{3}q^{3}+(-106\beta _{2}+\cdots)q^{4}+\cdots\)
75.9.f.c 75.f 5.c $8$ $30.553$ 8.0.\(\cdots\).2 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(2\beta _{3}-\beta _{5})q^{2}+3^{3}\beta _{2}q^{3}+(224\beta _{1}+\cdots)q^{4}+\cdots\)
75.9.f.d 75.f 5.c $12$ $30.553$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{2}q^{2}+\beta _{9}q^{3}+(193\beta _{1}-\beta _{5})q^{4}+\cdots\)
75.9.f.e 75.f 5.c $16$ $30.553$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(-4540\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{2}q^{2}+\beta _{6}q^{3}+(-158\beta _{1}+3\beta _{2}+\cdots)q^{4}+\cdots\)

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

\( S_{9}^{\mathrm{old}}(75, [\chi]) \cong \) \(S_{9}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)