Properties

Label 747.2.a
Level $747$
Weight $2$
Character orbit 747.a
Rep. character $\chi_{747}(1,\cdot)$
Character field $\Q$
Dimension $34$
Newform subspaces $11$
Sturm bound $168$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 747 = 3^{2} \cdot 83 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 747.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 11 \)
Sturm bound: \(168\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(2\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(747))\).

Total New Old
Modular forms 88 34 54
Cusp forms 81 34 47
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.

\(3\)\(83\)FrickeDim
\(+\)\(+\)$+$\(7\)
\(+\)\(-\)$-$\(7\)
\(-\)\(+\)$-$\(13\)
\(-\)\(-\)$+$\(7\)
Plus space\(+\)\(14\)
Minus space\(-\)\(20\)

Trace form

\( 34 q + 3 q^{2} + 35 q^{4} + 2 q^{5} + 9 q^{8} + O(q^{10}) \) \( 34 q + 3 q^{2} + 35 q^{4} + 2 q^{5} + 9 q^{8} + 10 q^{10} + 4 q^{11} - 6 q^{13} + 12 q^{14} + 37 q^{16} - 2 q^{17} - 6 q^{19} + 8 q^{20} - 16 q^{22} - 7 q^{23} + 34 q^{25} - 26 q^{26} - 20 q^{28} + 10 q^{29} + 11 q^{32} - 28 q^{34} - 12 q^{35} - 10 q^{37} + 16 q^{38} + 22 q^{40} + 17 q^{41} - 4 q^{43} + 12 q^{44} - 8 q^{46} - 4 q^{47} + 16 q^{49} + 5 q^{50} - 18 q^{53} - 30 q^{55} + 32 q^{56} - 18 q^{58} - 24 q^{59} + 2 q^{61} + 16 q^{62} + 53 q^{64} + 48 q^{65} + 18 q^{67} - 18 q^{68} - 24 q^{70} - 8 q^{73} - 20 q^{74} - 34 q^{76} - 29 q^{77} - 30 q^{79} + 56 q^{80} + 30 q^{82} - 6 q^{83} - 28 q^{85} - 8 q^{86} - 76 q^{88} + 28 q^{89} - 48 q^{91} - 22 q^{92} - 32 q^{94} - 8 q^{95} - 12 q^{97} + 11 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(747))\) 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 83
747.2.a.a 747.a 1.a $1$ $5.965$ \(\Q\) None \(-1\) \(0\) \(-2\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}-2q^{5}+3q^{8}+2q^{10}+\cdots\)
747.2.a.b 747.a 1.a $1$ $5.965$ \(\Q\) None \(-1\) \(0\) \(1\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+q^{5}-4q^{7}+3q^{8}-q^{10}+\cdots\)
747.2.a.c 747.a 1.a $1$ $5.965$ \(\Q\) None \(1\) \(0\) \(-1\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-q^{5}-3q^{8}-q^{10}+3q^{11}+\cdots\)
747.2.a.d 747.a 1.a $1$ $5.965$ \(\Q\) None \(1\) \(0\) \(2\) \(-3\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+2q^{5}-3q^{7}-3q^{8}+2q^{10}+\cdots\)
747.2.a.e 747.a 1.a $1$ $5.965$ \(\Q\) None \(1\) \(0\) \(2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+2q^{5}-3q^{8}+2q^{10}+\cdots\)
747.2.a.f 747.a 1.a $2$ $5.965$ \(\Q(\sqrt{2}) \) None \(2\) \(0\) \(6\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+(3-\beta )q^{5}+\cdots\)
747.2.a.g 747.a 1.a $4$ $5.965$ 4.4.6224.1 None \(-2\) \(0\) \(-6\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{3})q^{2}+(1-\beta _{1}-\beta _{2})q^{4}+\cdots\)
747.2.a.h 747.a 1.a $5$ $5.965$ 5.5.368464.1 None \(3\) \(0\) \(2\) \(8\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2}-\beta _{3})q^{4}+\cdots\)
747.2.a.i 747.a 1.a $6$ $5.965$ 6.6.33681152.1 None \(-4\) \(0\) \(-8\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{3})q^{2}+(2-\beta _{1}-\beta _{3})q^{4}+\cdots\)
747.2.a.j 747.a 1.a $6$ $5.965$ 6.6.9059636.1 None \(-1\) \(0\) \(-2\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(\beta _{1}+\beta _{2})q^{2}+(1-\beta _{5})q^{4}+(-1-\beta _{2}+\cdots)q^{5}+\cdots\)
747.2.a.k 747.a 1.a $6$ $5.965$ 6.6.33681152.1 None \(4\) \(0\) \(8\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{3})q^{2}+(2-\beta _{1}-\beta _{3})q^{4}+(1+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(747))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(747)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(83))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(249))\)\(^{\oplus 2}\)