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Properties

Label 75.2.g
Level $75$
Weight $2$
Character orbit 75.g
Rep. character $\chi_{75}(16,\cdot)$
Character field $\Q(\zeta_{5})$
Dimension $24$
Newform subspaces $3$
Sturm bound $20$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 75.g (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 25 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 3 \)
Sturm bound: \(20\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

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

Total New Old
Modular forms 48 24 24
Cusp forms 32 24 8
Eisenstein series 16 0 16

Trace form

\( 24 q - 2 q^{2} - 8 q^{4} - 6 q^{5} - 2 q^{6} - 8 q^{7} + 12 q^{8} - 6 q^{9} + O(q^{10}) \) \( 24 q - 2 q^{2} - 8 q^{4} - 6 q^{5} - 2 q^{6} - 8 q^{7} + 12 q^{8} - 6 q^{9} - 14 q^{10} + 6 q^{11} - 8 q^{12} - 12 q^{13} - 12 q^{14} - 4 q^{15} - 10 q^{17} + 8 q^{18} - 2 q^{19} + 16 q^{20} - 4 q^{21} + 30 q^{22} + 36 q^{23} + 24 q^{24} - 16 q^{25} - 32 q^{26} - 26 q^{28} - 4 q^{29} + 44 q^{30} - 6 q^{31} + 12 q^{32} - 6 q^{33} - 6 q^{34} - 30 q^{35} - 8 q^{36} - 8 q^{37} - 26 q^{38} - 8 q^{39} + 12 q^{40} - 6 q^{41} - 14 q^{42} + 32 q^{43} + 26 q^{44} - 6 q^{45} - 16 q^{46} - 16 q^{47} - 32 q^{48} + 40 q^{49} + 86 q^{50} + 32 q^{51} + 40 q^{52} + 36 q^{53} - 2 q^{54} + 14 q^{55} - 16 q^{57} + 38 q^{58} + 12 q^{59} - 6 q^{60} - 20 q^{61} + 14 q^{62} + 2 q^{63} - 2 q^{64} + 12 q^{65} - 16 q^{66} - 124 q^{68} - 12 q^{69} - 70 q^{70} + 8 q^{71} + 12 q^{72} - 24 q^{73} - 72 q^{74} - 24 q^{75} + 32 q^{76} - 56 q^{77} - 40 q^{78} - 20 q^{79} - 154 q^{80} - 6 q^{81} + 4 q^{82} - 26 q^{83} + 12 q^{84} - 2 q^{85} + 36 q^{86} + 8 q^{87} + 72 q^{88} + 48 q^{89} + 16 q^{90} - 26 q^{91} + 22 q^{92} + 88 q^{93} - 58 q^{94} + 96 q^{95} + 6 q^{96} + 42 q^{98} - 4 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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.g.a 75.g 25.d $4$ $0.599$ \(\Q(\zeta_{10})\) None \(-1\) \(-1\) \(5\) \(0\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
75.2.g.b 75.g 25.d $8$ $0.599$ 8.0.26265625.1 None \(-1\) \(-2\) \(-5\) \(4\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-\beta _{3}-\beta _{7})q^{2}+(-1+\beta _{1}+\beta _{3}+\cdots)q^{3}+\cdots\)
75.2.g.c 75.g 25.d $12$ $0.599$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(3\) \(-6\) \(-12\) $\mathrm{SU}(2)[C_{5}]$ \(q-\beta _{2}q^{2}+\beta _{8}q^{3}+(-1+\beta _{1}-\beta _{5}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

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