Properties

Label 786.2.a
Level 786
Weight 2
Character orbit a
Rep. character \(\chi_{786}(1,\cdot)\)
Character field \(\Q\)
Dimension 23
Newform subspaces 16
Sturm bound 264
Trace bound 7

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Defining parameters

Level: \( N \) = \( 786 = 2 \cdot 3 \cdot 131 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 786.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(264\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(5\), \(7\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(786))\).

Total New Old
Modular forms 136 23 113
Cusp forms 129 23 106
Eisenstein series 7 0 7

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

\(2\)\(3\)\(131\)FrickeDim.
\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(2\)
\(+\)\(-\)\(+\)\(-\)\(4\)
\(+\)\(-\)\(-\)\(+\)\(2\)
\(-\)\(+\)\(+\)\(-\)\(4\)
\(-\)\(+\)\(-\)\(+\)\(2\)
\(-\)\(-\)\(+\)\(+\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(5\)
Plus space\(+\)\(8\)
Minus space\(-\)\(15\)

Trace form

\( 23q + q^{2} + q^{3} + 23q^{4} - 2q^{5} - q^{6} - 4q^{7} + q^{8} + 23q^{9} + O(q^{10}) \) \( 23q + q^{2} + q^{3} + 23q^{4} - 2q^{5} - q^{6} - 4q^{7} + q^{8} + 23q^{9} - 2q^{10} - 4q^{11} + q^{12} + 6q^{13} - 6q^{15} + 23q^{16} + 2q^{17} + q^{18} - 8q^{19} - 2q^{20} + 8q^{21} + 4q^{22} - q^{24} + 33q^{25} - 2q^{26} + q^{27} - 4q^{28} + 6q^{29} + 2q^{30} - 4q^{31} + q^{32} + 10q^{34} - 16q^{35} + 23q^{36} + 2q^{37} + 4q^{38} + 6q^{39} - 2q^{40} - 22q^{41} + 12q^{43} - 4q^{44} - 2q^{45} - 8q^{46} - 24q^{47} + q^{48} + 27q^{49} + 31q^{50} + 6q^{51} + 6q^{52} + 22q^{53} - q^{54} - 16q^{55} + 2q^{58} - 4q^{59} - 6q^{60} + 6q^{61} + 24q^{62} - 4q^{63} + 23q^{64} + 4q^{65} - 12q^{66} - 32q^{67} + 2q^{68} + 8q^{69} - 32q^{71} + q^{72} - 2q^{73} - 2q^{74} - 17q^{75} - 8q^{76} - 8q^{77} - 14q^{78} + 12q^{79} - 2q^{80} + 23q^{81} - 14q^{82} + 20q^{83} + 8q^{84} - 44q^{85} - 20q^{86} + 10q^{87} + 4q^{88} - 54q^{89} - 2q^{90} + 4q^{93} - 16q^{94} + 8q^{95} - q^{96} + 46q^{97} + 9q^{98} - 4q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(786))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 3 131
786.2.a.a \(1\) \(6.276\) \(\Q\) None \(-1\) \(-1\) \(-2\) \(2\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}-2q^{5}+q^{6}+2q^{7}+\cdots\)
786.2.a.b \(1\) \(6.276\) \(\Q\) None \(-1\) \(-1\) \(-1\) \(-3\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-3q^{7}+\cdots\)
786.2.a.c \(1\) \(6.276\) \(\Q\) None \(-1\) \(-1\) \(-1\) \(-1\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-q^{7}+\cdots\)
786.2.a.d \(1\) \(6.276\) \(\Q\) None \(-1\) \(-1\) \(2\) \(0\) \(+\) \(+\) \(-\) \(q-q^{2}-q^{3}+q^{4}+2q^{5}+q^{6}-q^{8}+\cdots\)
786.2.a.e \(1\) \(6.276\) \(\Q\) None \(-1\) \(-1\) \(4\) \(-1\) \(+\) \(+\) \(+\) \(q-q^{2}-q^{3}+q^{4}+4q^{5}+q^{6}-q^{7}+\cdots\)
786.2.a.f \(1\) \(6.276\) \(\Q\) None \(-1\) \(1\) \(-3\) \(5\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}-3q^{5}-q^{6}+5q^{7}+\cdots\)
786.2.a.g \(1\) \(6.276\) \(\Q\) None \(-1\) \(1\) \(-2\) \(-2\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}-2q^{5}-q^{6}-2q^{7}+\cdots\)
786.2.a.h \(1\) \(6.276\) \(\Q\) None \(-1\) \(1\) \(1\) \(-5\) \(+\) \(-\) \(-\) \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-5q^{7}+\cdots\)
786.2.a.i \(1\) \(6.276\) \(\Q\) None \(1\) \(-1\) \(-4\) \(3\) \(-\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-4q^{5}-q^{6}+3q^{7}+\cdots\)
786.2.a.j \(1\) \(6.276\) \(\Q\) None \(1\) \(-1\) \(-1\) \(-3\) \(-\) \(+\) \(-\) \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}-3q^{7}+\cdots\)
786.2.a.k \(1\) \(6.276\) \(\Q\) None \(1\) \(-1\) \(4\) \(-4\) \(-\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}+4q^{5}-q^{6}-4q^{7}+\cdots\)
786.2.a.l \(1\) \(6.276\) \(\Q\) None \(1\) \(1\) \(-3\) \(-3\) \(-\) \(-\) \(+\) \(q+q^{2}+q^{3}+q^{4}-3q^{5}+q^{6}-3q^{7}+\cdots\)
786.2.a.m \(1\) \(6.276\) \(\Q\) None \(1\) \(1\) \(1\) \(3\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+3q^{7}+\cdots\)
786.2.a.n \(3\) \(6.276\) 3.3.316.1 None \(-3\) \(3\) \(2\) \(3\) \(+\) \(-\) \(+\) \(q-q^{2}+q^{3}+q^{4}+(1-\beta _{1}+\beta _{2})q^{5}+\cdots\)
786.2.a.o \(3\) \(6.276\) 3.3.961.1 None \(3\) \(-3\) \(1\) \(1\) \(-\) \(+\) \(+\) \(q+q^{2}-q^{3}+q^{4}+\beta _{1}q^{5}-q^{6}+\beta _{1}q^{7}+\cdots\)
786.2.a.p \(4\) \(6.276\) 4.4.64268.1 None \(4\) \(4\) \(0\) \(1\) \(-\) \(-\) \(-\) \(q+q^{2}+q^{3}+q^{4}-\beta _{3}q^{5}+q^{6}+(\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(786))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(786)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(131))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(262))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(393))\)\(^{\oplus 2}\)