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

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

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

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5
75.4.a.a \(1\) \(4.425\) \(\Q\) None \(-3\) \(3\) \(0\) \(-20\) \(-\) \(+\) \(q-3q^{2}+3q^{3}+q^{4}-9q^{6}-20q^{7}+\cdots\)
75.4.a.b \(1\) \(4.425\) \(\Q\) None \(-1\) \(-3\) \(0\) \(24\) \(+\) \(+\) \(q-q^{2}-3q^{3}-7q^{4}+3q^{6}+24q^{7}+\cdots\)
75.4.a.c \(2\) \(4.425\) \(\Q(\sqrt{41}) \) None \(-3\) \(-6\) \(0\) \(-6\) \(+\) \(-\) \(q+(-1-\beta )q^{2}-3q^{3}+(3+3\beta )q^{4}+\cdots\)
75.4.a.d \(2\) \(4.425\) \(\Q(\sqrt{19}) \) None \(-2\) \(6\) \(0\) \(26\) \(-\) \(-\) \(q+(-1+\beta )q^{2}+3q^{3}+(12-2\beta )q^{4}+\cdots\)
75.4.a.e \(2\) \(4.425\) \(\Q(\sqrt{19}) \) None \(2\) \(-6\) \(0\) \(-26\) \(+\) \(+\) \(q+(1+\beta )q^{2}-3q^{3}+(12+2\beta )q^{4}+(-3+\cdots)q^{6}+\cdots\)
75.4.a.f \(2\) \(4.425\) \(\Q(\sqrt{41}) \) None \(3\) \(6\) \(0\) \(6\) \(-\) \(-\) \(q+(1+\beta )q^{2}+3q^{3}+(3+3\beta )q^{4}+(3+\cdots)q^{6}+\cdots\)

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

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