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Properties

Label 75.3.c
Level $75$
Weight $3$
Character orbit 75.c
Rep. character $\chi_{75}(26,\cdot)$
Character field $\Q$
Dimension $10$
Newform subspaces $6$
Sturm bound $30$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 75.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(30\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(7\), \(13\)

Dimensions

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

Total New Old
Modular forms 26 16 10
Cusp forms 14 10 4
Eisenstein series 12 6 6

Trace form

\( 10 q + 4 q^{3} - 16 q^{4} - 18 q^{6} + 12 q^{7} + 12 q^{9} + O(q^{10}) \) \( 10 q + 4 q^{3} - 16 q^{4} - 18 q^{6} + 12 q^{7} + 12 q^{9} - 4 q^{12} - 32 q^{13} + 24 q^{16} + 40 q^{18} - 6 q^{19} - 42 q^{21} - 20 q^{22} + 54 q^{24} - 44 q^{27} - 12 q^{28} + 110 q^{31} + 20 q^{33} + 4 q^{34} - 78 q^{36} + 32 q^{37} + 42 q^{39} + 60 q^{42} - 32 q^{43} - 136 q^{46} - 76 q^{48} - 176 q^{49} + 42 q^{51} + 32 q^{52} - 228 q^{54} - 8 q^{57} + 140 q^{58} - 10 q^{61} - 12 q^{63} + 408 q^{64} + 510 q^{66} - 48 q^{67} + 12 q^{69} + 120 q^{72} + 148 q^{73} - 356 q^{76} - 160 q^{78} - 156 q^{79} - 60 q^{81} - 280 q^{82} - 288 q^{84} - 140 q^{87} - 60 q^{88} - 214 q^{91} - 72 q^{93} + 424 q^{94} - 114 q^{96} + 332 q^{97} + 630 q^{99} + O(q^{100}) \)

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.c.a 75.c 3.b $1$ $2.044$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(-3\) \(0\) \(11\) $\mathrm{U}(1)[D_{2}]$ \(q-3q^{3}+4q^{4}+11q^{7}+9q^{9}-12q^{12}+\cdots\)
75.3.c.b 75.c 3.b $1$ $2.044$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(3\) \(0\) \(-11\) $\mathrm{U}(1)[D_{2}]$ \(q+3q^{3}+4q^{4}-11q^{7}+9q^{9}+12q^{12}+\cdots\)
75.3.c.c 75.c 3.b $2$ $2.044$ \(\Q(\sqrt{-11}) \) None \(0\) \(-5\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1-2\beta )q^{2}+(-2-\beta )q^{3}-7q^{4}+\cdots\)
75.3.c.d 75.c 3.b $2$ $2.044$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-15}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+iq^{2}+3iq^{3}+3q^{4}-3q^{6}+7iq^{8}+\cdots\)
75.3.c.e 75.c 3.b $2$ $2.044$ \(\Q(\sqrt{-5}) \) None \(0\) \(4\) \(0\) \(12\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{2}+(2-\beta )q^{3}-q^{4}+(5+2\beta )q^{6}+\cdots\)
75.3.c.f 75.c 3.b $2$ $2.044$ \(\Q(\sqrt{-11}) \) None \(0\) \(5\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1-2\beta )q^{2}+(3-\beta )q^{3}-7q^{4}+(-3+\cdots)q^{6}+\cdots\)

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

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