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Properties

Label 75.3.f
Level $75$
Weight $3$
Character orbit 75.f
Rep. character $\chi_{75}(7,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $12$
Newform subspaces $3$
Sturm bound $30$
Trace bound $6$

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Defining parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 75.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 3 \)
Sturm bound: \(30\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(75, [\chi])\).

Total New Old
Modular forms 52 12 40
Cusp forms 28 12 16
Eisenstein series 24 0 24

Trace form

\( 12 q + 4 q^{2} + 24 q^{6} - 4 q^{7} - 12 q^{8} + O(q^{10}) \) \( 12 q + 4 q^{2} + 24 q^{6} - 4 q^{7} - 12 q^{8} - 32 q^{11} - 24 q^{12} + 32 q^{13} - 40 q^{16} + 40 q^{17} + 12 q^{18} - 12 q^{21} - 20 q^{22} - 56 q^{23} - 56 q^{26} - 44 q^{28} + 172 q^{31} + 76 q^{32} + 36 q^{33} + 96 q^{36} - 64 q^{37} + 96 q^{38} - 368 q^{41} - 12 q^{42} + 8 q^{43} - 88 q^{46} - 128 q^{47} - 48 q^{48} - 144 q^{51} + 80 q^{52} - 56 q^{53} + 840 q^{56} + 72 q^{57} + 12 q^{58} + 60 q^{61} - 88 q^{62} - 12 q^{63} - 48 q^{66} + 200 q^{67} + 104 q^{68} + 544 q^{71} + 36 q^{72} - 76 q^{73} - 464 q^{76} - 88 q^{77} - 120 q^{78} - 108 q^{81} - 128 q^{82} + 16 q^{83} - 1112 q^{86} + 84 q^{87} - 12 q^{88} - 268 q^{91} - 104 q^{92} + 72 q^{93} - 192 q^{96} + 20 q^{97} + 188 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(75, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
75.3.f.a 75.f 5.c $4$ $2.044$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{2}-\beta _{3}q^{3}-\beta _{2}q^{4}+3q^{6}+6\beta _{1}q^{7}+\cdots\)
75.3.f.b 75.f 5.c $4$ $2.044$ \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+2\beta _{1}q^{2}-\beta _{3}q^{3}+8\beta _{2}q^{4}+6q^{6}+\cdots\)
75.3.f.c 75.f 5.c $4$ $2.044$ \(\Q(i, \sqrt{6})\) None \(4\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+\beta _{1}+\beta _{2})q^{2}+\beta _{3}q^{3}+(2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(75, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)