Properties

Label 75.6.a
Level $75$
Weight $6$
Character orbit 75.a
Rep. character $\chi_{75}(1,\cdot)$
Character field $\Q$
Dimension $15$
Newform subspaces $10$
Sturm bound $60$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 10 \)
Sturm bound: \(60\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(75))\).

Total New Old
Modular forms 56 15 41
Cusp forms 44 15 29
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.
\(+\)\(+\)\(+\)\(3\)
\(+\)\(-\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(5\)
\(-\)\(-\)\(+\)\(3\)
Plus space\(+\)\(6\)
Minus space\(-\)\(9\)

Trace form

\( 15q + 2q^{2} + 9q^{3} + 176q^{4} + 54q^{6} + 272q^{7} + 60q^{8} + 1215q^{9} + O(q^{10}) \) \( 15q + 2q^{2} + 9q^{3} + 176q^{4} + 54q^{6} + 272q^{7} + 60q^{8} + 1215q^{9} - 288q^{11} + 828q^{12} + 146q^{13} + 2268q^{14} + 3124q^{16} - 3538q^{17} + 162q^{18} + 4442q^{19} - 2214q^{21} + 3444q^{22} + 576q^{23} - 1404q^{24} + 20880q^{26} + 729q^{27} + 884q^{28} - 11526q^{29} - 15486q^{31} + 25612q^{32} + 10548q^{33} - 23908q^{34} + 14256q^{36} - 12598q^{37} - 59440q^{38} + 1944q^{39} - 23286q^{41} - 29484q^{42} + 38876q^{43} + 48720q^{44} + 3904q^{46} + 19832q^{47} + 59904q^{48} + 71693q^{49} - 33750q^{51} - 63688q^{52} - 53494q^{53} + 4374q^{54} + 16920q^{56} - 14220q^{57} + 71280q^{58} - 33432q^{59} - 87072q^{61} + 46104q^{62} + 22032q^{63} - 172340q^{64} - 124200q^{66} + 54932q^{67} - 244256q^{68} - 15012q^{69} + 210120q^{71} + 4860q^{72} - 20674q^{73} - 282372q^{74} + 23728q^{76} - 52416q^{77} + 133668q^{78} - 32320q^{79} + 98415q^{81} - 3996q^{82} - 103404q^{83} + 55512q^{84} - 203148q^{86} + 14670q^{87} + 335220q^{88} + 234762q^{89} + 29834q^{91} + 172992q^{92} + 101088q^{93} + 640304q^{94} + 107676q^{96} - 134218q^{97} - 327326q^{98} - 23328q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(75))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5
75.6.a.a \(1\) \(12.029\) \(\Q\) None \(-7\) \(-9\) \(0\) \(-12\) \(+\) \(+\) \(q-7q^{2}-9q^{3}+17q^{4}+63q^{6}-12q^{7}+\cdots\)
75.6.a.b \(1\) \(12.029\) \(\Q\) None \(-4\) \(-9\) \(0\) \(225\) \(+\) \(+\) \(q-4q^{2}-9q^{3}-2^{4}q^{4}+6^{2}q^{6}+15^{2}q^{7}+\cdots\)
75.6.a.c \(1\) \(12.029\) \(\Q\) None \(2\) \(9\) \(0\) \(132\) \(-\) \(+\) \(q+2q^{2}+9q^{3}-28q^{4}+18q^{6}+132q^{7}+\cdots\)
75.6.a.d \(1\) \(12.029\) \(\Q\) None \(4\) \(9\) \(0\) \(-225\) \(-\) \(-\) \(q+4q^{2}+9q^{3}-2^{4}q^{4}+6^{2}q^{6}-15^{2}q^{7}+\cdots\)
75.6.a.e \(1\) \(12.029\) \(\Q\) None \(6\) \(-9\) \(0\) \(40\) \(+\) \(+\) \(q+6q^{2}-9q^{3}+4q^{4}-54q^{6}+40q^{7}+\cdots\)
75.6.a.f \(2\) \(12.029\) \(\Q(\sqrt{89}) \) None \(-9\) \(18\) \(0\) \(-108\) \(-\) \(-\) \(q+(-4-\beta )q^{2}+9q^{3}+(6+9\beta )q^{4}+\cdots\)
75.6.a.g \(2\) \(12.029\) \(\Q(\sqrt{31}) \) None \(-6\) \(-18\) \(0\) \(-102\) \(+\) \(-\) \(q+(-3+\beta )q^{2}-9q^{3}+(8-6\beta )q^{4}+\cdots\)
75.6.a.h \(2\) \(12.029\) \(\Q(\sqrt{409}) \) None \(1\) \(18\) \(0\) \(112\) \(-\) \(+\) \(q+\beta q^{2}+9q^{3}+(70+\beta )q^{4}+9\beta q^{6}+\cdots\)
75.6.a.i \(2\) \(12.029\) \(\Q(\sqrt{31}) \) None \(6\) \(18\) \(0\) \(102\) \(-\) \(+\) \(q+(3+\beta )q^{2}+9q^{3}+(8+6\beta )q^{4}+(3^{3}+\cdots)q^{6}+\cdots\)
75.6.a.j \(2\) \(12.029\) \(\Q(\sqrt{89}) \) None \(9\) \(-18\) \(0\) \(108\) \(+\) \(-\) \(q+(5-\beta )q^{2}-9q^{3}+(15-9\beta )q^{4}+(-45+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(75))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(75)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)