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Properties

Label 975.2.a
Level $975$
Weight $2$
Character orbit 975.a
Rep. character $\chi_{975}(1,\cdot)$
Character field $\Q$
Dimension $38$
Newform subspaces $19$
Sturm bound $280$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 19 \)
Sturm bound: \(280\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(975))\).

Total New Old
Modular forms 152 38 114
Cusp forms 129 38 91
Eisenstein series 23 0 23

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

\(3\)\(5\)\(13\)FrickeDim
\(+\)\(+\)\(+\)$+$\(5\)
\(+\)\(+\)\(-\)$-$\(4\)
\(+\)\(-\)\(+\)$-$\(6\)
\(+\)\(-\)\(-\)$+$\(4\)
\(-\)\(+\)\(+\)$-$\(7\)
\(-\)\(+\)\(-\)$+$\(2\)
\(-\)\(-\)\(+\)$+$\(2\)
\(-\)\(-\)\(-\)$-$\(8\)
Plus space\(+\)\(13\)
Minus space\(-\)\(25\)

Trace form

\( 38 q - 4 q^{2} + 38 q^{4} + 6 q^{6} + 4 q^{7} + 38 q^{9} + O(q^{10}) \) \( 38 q - 4 q^{2} + 38 q^{4} + 6 q^{6} + 4 q^{7} + 38 q^{9} - 4 q^{11} + 4 q^{12} - 2 q^{13} + 4 q^{14} + 42 q^{16} - 4 q^{17} - 4 q^{18} + 12 q^{19} + 4 q^{21} + 4 q^{22} + 8 q^{23} + 18 q^{24} + 44 q^{28} - 12 q^{29} + 28 q^{31} - 4 q^{32} + 4 q^{33} + 24 q^{34} + 38 q^{36} - 28 q^{37} + 44 q^{38} + 4 q^{39} - 4 q^{41} - 4 q^{42} + 16 q^{43} - 28 q^{44} - 40 q^{46} + 28 q^{47} + 24 q^{48} + 46 q^{49} + 8 q^{51} - 2 q^{52} - 44 q^{53} + 6 q^{54} - 76 q^{56} + 12 q^{57} - 8 q^{58} + 4 q^{59} + 4 q^{61} - 44 q^{62} + 4 q^{63} + 22 q^{64} + 20 q^{66} - 4 q^{67} + 20 q^{68} + 8 q^{69} + 12 q^{71} - 12 q^{73} - 32 q^{74} + 36 q^{76} - 24 q^{77} - 2 q^{78} + 16 q^{79} + 38 q^{81} + 4 q^{83} + 28 q^{84} - 48 q^{86} - 8 q^{87} + 4 q^{88} - 12 q^{89} + 12 q^{91} + 16 q^{92} + 20 q^{93} - 28 q^{94} - 18 q^{96} + 4 q^{97} - 20 q^{98} - 4 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(975))\) 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 13
975.2.a.a 975.a 1.a $1$ $7.785$ \(\Q\) None \(-2\) \(-1\) \(0\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-q^{3}+2q^{4}+2q^{6}+q^{7}+q^{9}+\cdots\)
975.2.a.b 975.a 1.a $1$ $7.785$ \(\Q\) None \(-2\) \(-1\) \(0\) \(3\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-q^{3}+2q^{4}+2q^{6}+3q^{7}+\cdots\)
975.2.a.c 975.a 1.a $1$ $7.785$ \(\Q\) None \(-2\) \(1\) \(0\) \(-3\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+q^{3}+2q^{4}-2q^{6}-3q^{7}+\cdots\)
975.2.a.d 975.a 1.a $1$ $7.785$ \(\Q\) None \(-1\) \(-1\) \(0\) \(3\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}-q^{4}+q^{6}+3q^{7}+3q^{8}+\cdots\)
975.2.a.e 975.a 1.a $1$ $7.785$ \(\Q\) None \(-1\) \(1\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}-q^{4}-q^{6}-q^{7}+3q^{8}+\cdots\)
975.2.a.f 975.a 1.a $1$ $7.785$ \(\Q\) None \(-1\) \(1\) \(0\) \(4\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}-q^{4}-q^{6}+4q^{7}+3q^{8}+\cdots\)
975.2.a.g 975.a 1.a $1$ $7.785$ \(\Q\) None \(0\) \(-1\) \(0\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{4}+q^{7}+q^{9}-q^{11}+2q^{12}+\cdots\)
975.2.a.h 975.a 1.a $1$ $7.785$ \(\Q\) None \(0\) \(1\) \(0\) \(-1\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}-q^{7}+q^{9}-q^{11}-2q^{12}+\cdots\)
975.2.a.i 975.a 1.a $1$ $7.785$ \(\Q\) None \(1\) \(-1\) \(0\) \(0\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}-q^{4}-q^{6}-3q^{8}+q^{9}+\cdots\)
975.2.a.j 975.a 1.a $1$ $7.785$ \(\Q\) None \(1\) \(-1\) \(0\) \(1\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}-q^{4}-q^{6}+q^{7}-3q^{8}+\cdots\)
975.2.a.k 975.a 1.a $1$ $7.785$ \(\Q\) None \(1\) \(1\) \(0\) \(-3\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}-q^{4}+q^{6}-3q^{7}-3q^{8}+\cdots\)
975.2.a.l 975.a 1.a $2$ $7.785$ \(\Q(\sqrt{2}) \) None \(2\) \(-2\) \(0\) \(0\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}-q^{3}+(1+2\beta )q^{4}+(-1+\cdots)q^{6}+\cdots\)
975.2.a.m 975.a 1.a $3$ $7.785$ 3.3.148.1 None \(-3\) \(-3\) \(0\) \(1\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{2})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
975.2.a.n 975.a 1.a $3$ $7.785$ 3.3.564.1 None \(-1\) \(-3\) \(0\) \(-5\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
975.2.a.o 975.a 1.a $3$ $7.785$ 3.3.316.1 None \(0\) \(3\) \(0\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(3+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
975.2.a.p 975.a 1.a $3$ $7.785$ 3.3.564.1 None \(1\) \(3\) \(0\) \(5\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
975.2.a.q 975.a 1.a $3$ $7.785$ 3.3.148.1 None \(3\) \(3\) \(0\) \(-1\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
975.2.a.r 975.a 1.a $5$ $7.785$ 5.5.1821184.1 None \(0\) \(-5\) \(0\) \(-5\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(1+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
975.2.a.s 975.a 1.a $5$ $7.785$ 5.5.1821184.1 None \(0\) \(5\) \(0\) \(5\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(975))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(975)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(195))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(325))\)\(^{\oplus 2}\)