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

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

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(75))\) 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
75.6.a.a 75.a 1.a $1$ $12.029$ \(\Q\) None \(-7\) \(-9\) \(0\) \(-12\) $+$ $+$ $\mathrm{SU}(2)$ \(q-7q^{2}-9q^{3}+17q^{4}+63q^{6}-12q^{7}+\cdots\)
75.6.a.b 75.a 1.a $1$ $12.029$ \(\Q\) None \(-4\) \(-9\) \(0\) \(225\) $+$ $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-9q^{3}-2^{4}q^{4}+6^{2}q^{6}+15^{2}q^{7}+\cdots\)
75.6.a.c 75.a 1.a $1$ $12.029$ \(\Q\) None \(2\) \(9\) \(0\) \(132\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+9q^{3}-28q^{4}+18q^{6}+132q^{7}+\cdots\)
75.6.a.d 75.a 1.a $1$ $12.029$ \(\Q\) None \(4\) \(9\) \(0\) \(-225\) $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+9q^{3}-2^{4}q^{4}+6^{2}q^{6}-15^{2}q^{7}+\cdots\)
75.6.a.e 75.a 1.a $1$ $12.029$ \(\Q\) None \(6\) \(-9\) \(0\) \(40\) $+$ $+$ $\mathrm{SU}(2)$ \(q+6q^{2}-9q^{3}+4q^{4}-54q^{6}+40q^{7}+\cdots\)
75.6.a.f 75.a 1.a $2$ $12.029$ \(\Q(\sqrt{89}) \) None \(-9\) \(18\) \(0\) \(-108\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-4-\beta )q^{2}+9q^{3}+(6+9\beta )q^{4}+\cdots\)
75.6.a.g 75.a 1.a $2$ $12.029$ \(\Q(\sqrt{31}) \) None \(-6\) \(-18\) \(0\) \(-102\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-3+\beta )q^{2}-9q^{3}+(8-6\beta )q^{4}+\cdots\)
75.6.a.h 75.a 1.a $2$ $12.029$ \(\Q(\sqrt{409}) \) None \(1\) \(18\) \(0\) \(112\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+9q^{3}+(70+\beta )q^{4}+9\beta q^{6}+\cdots\)
75.6.a.i 75.a 1.a $2$ $12.029$ \(\Q(\sqrt{31}) \) None \(6\) \(18\) \(0\) \(102\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(3+\beta )q^{2}+9q^{3}+(8+6\beta )q^{4}+(3^{3}+\cdots)q^{6}+\cdots\)
75.6.a.j 75.a 1.a $2$ $12.029$ \(\Q(\sqrt{89}) \) None \(9\) \(-18\) \(0\) \(108\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(5-\beta )q^{2}-9q^{3}+(15-9\beta )q^{4}+(-45+\cdots)q^{6}+\cdots\)

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

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