Properties

Label 75.4.a
Level $75$
Weight $4$
Character orbit 75.a
Rep. character $\chi_{75}(1,\cdot)$
Character field $\Q$
Dimension $10$
Newform subspaces $6$
Sturm bound $40$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(40\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(75))\).

Total New Old
Modular forms 36 10 26
Cusp forms 24 10 14
Eisenstein series 12 0 12

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)\(5\)FrickeDim.
\(+\)\(+\)\(+\)\(3\)
\(+\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(1\)
\(-\)\(-\)\(+\)\(4\)
Plus space\(+\)\(7\)
Minus space\(-\)\(3\)

Trace form

\( 10 q - 4 q^{2} + 60 q^{4} + 4 q^{7} + 36 q^{8} + 90 q^{9} + O(q^{10}) \) \( 10 q - 4 q^{2} + 60 q^{4} + 4 q^{7} + 36 q^{8} + 90 q^{9} + 24 q^{12} - 96 q^{13} + 60 q^{14} + 260 q^{16} - 40 q^{17} - 36 q^{18} - 20 q^{19} + 60 q^{21} + 20 q^{22} + 288 q^{23} - 180 q^{24} - 1020 q^{26} - 188 q^{28} - 300 q^{29} + 260 q^{31} - 116 q^{32} - 228 q^{33} - 520 q^{34} + 540 q^{36} + 104 q^{37} + 392 q^{38} + 420 q^{39} + 420 q^{41} + 252 q^{42} + 96 q^{43} - 1440 q^{44} - 1520 q^{46} - 232 q^{47} - 336 q^{48} + 950 q^{49} + 80 q^{52} - 112 q^{53} - 360 q^{56} - 312 q^{57} + 4 q^{58} + 2600 q^{61} - 312 q^{62} + 36 q^{63} + 2060 q^{64} + 280 q^{67} - 152 q^{68} + 360 q^{69} + 960 q^{71} + 324 q^{72} + 468 q^{73} + 5280 q^{74} - 1680 q^{76} + 1728 q^{77} + 600 q^{78} - 1760 q^{79} + 810 q^{81} - 1112 q^{82} - 1368 q^{83} - 1560 q^{84} + 2700 q^{86} - 924 q^{87} + 276 q^{88} + 1860 q^{89} - 1500 q^{91} - 1056 q^{92} + 1464 q^{93} - 4240 q^{94} - 5220 q^{96} - 580 q^{97} - 404 q^{98} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(75))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5
75.4.a.a 75.a 1.a $1$ $4.425$ \(\Q\) None \(-3\) \(3\) \(0\) \(-20\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{2}+3q^{3}+q^{4}-9q^{6}-20q^{7}+\cdots\)
75.4.a.b 75.a 1.a $1$ $4.425$ \(\Q\) None \(-1\) \(-3\) \(0\) \(24\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-3q^{3}-7q^{4}+3q^{6}+24q^{7}+\cdots\)
75.4.a.c 75.a 1.a $2$ $4.425$ \(\Q(\sqrt{41}) \) None \(-3\) \(-6\) \(0\) \(-6\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}-3q^{3}+(3+3\beta )q^{4}+\cdots\)
75.4.a.d 75.a 1.a $2$ $4.425$ \(\Q(\sqrt{19}) \) None \(-2\) \(6\) \(0\) \(26\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+3q^{3}+(12-2\beta )q^{4}+\cdots\)
75.4.a.e 75.a 1.a $2$ $4.425$ \(\Q(\sqrt{19}) \) None \(2\) \(-6\) \(0\) \(-26\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}-3q^{3}+(12+2\beta )q^{4}+(-3+\cdots)q^{6}+\cdots\)
75.4.a.f 75.a 1.a $2$ $4.425$ \(\Q(\sqrt{41}) \) None \(3\) \(6\) \(0\) \(6\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+3q^{3}+(3+3\beta )q^{4}+(3+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(75))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(75)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)