Properties

Label 847.2.a
Level $847$
Weight $2$
Character orbit 847.a
Rep. character $\chi_{847}(1,\cdot)$
Character field $\Q$
Dimension $55$
Newform subspaces $16$
Sturm bound $176$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(176\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(847))\).

Total New Old
Modular forms 100 55 45
Cusp forms 77 55 22
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.

\(7\)\(11\)FrickeDim
\(+\)\(+\)\(+\)\(13\)
\(+\)\(-\)\(-\)\(15\)
\(-\)\(+\)\(-\)\(17\)
\(-\)\(-\)\(+\)\(10\)
Plus space\(+\)\(23\)
Minus space\(-\)\(32\)

Trace form

\( 55 q - q^{2} - 4 q^{3} + 59 q^{4} + 2 q^{5} + 8 q^{6} - q^{7} + 3 q^{8} + 47 q^{9} + O(q^{10}) \) \( 55 q - q^{2} - 4 q^{3} + 59 q^{4} + 2 q^{5} + 8 q^{6} - q^{7} + 3 q^{8} + 47 q^{9} + 2 q^{10} - 16 q^{12} + 2 q^{13} + q^{14} - 8 q^{15} + 55 q^{16} + 2 q^{17} + 19 q^{18} + 6 q^{20} - 4 q^{21} + 4 q^{23} + 16 q^{24} + 57 q^{25} - 18 q^{26} - 16 q^{27} - 7 q^{28} - 6 q^{29} - 16 q^{30} - 16 q^{31} - 5 q^{32} + 6 q^{34} - 2 q^{35} + 43 q^{36} + 2 q^{37} - 24 q^{38} - 8 q^{39} - 6 q^{40} + 10 q^{41} + 12 q^{42} - 28 q^{43} - 2 q^{45} - 16 q^{46} - 20 q^{47} - 8 q^{48} + 55 q^{49} + q^{50} - 4 q^{51} - 18 q^{52} - 10 q^{53} + 24 q^{54} - 3 q^{56} - 4 q^{57} - 30 q^{58} - 20 q^{60} + 22 q^{61} - 20 q^{62} + 3 q^{63} + 83 q^{64} + 20 q^{65} - 36 q^{67} + 2 q^{68} - 16 q^{69} - 14 q^{70} + 23 q^{72} + 2 q^{73} + 26 q^{74} - 32 q^{75} - 20 q^{76} - 28 q^{78} - 4 q^{79} - 54 q^{80} + 15 q^{81} - 14 q^{82} - 28 q^{83} + 24 q^{85} - 44 q^{86} + 60 q^{87} - 2 q^{89} - 38 q^{90} + 2 q^{91} - 36 q^{92} - 32 q^{93} + 20 q^{94} - 4 q^{95} - 40 q^{96} - 10 q^{97} - q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(847))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 7 11
847.2.a.a 847.a 1.a $1$ $6.763$ \(\Q\) None 77.2.a.c \(-1\) \(2\) \(-2\) \(1\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}-q^{4}-2q^{5}-2q^{6}+q^{7}+\cdots\)
847.2.a.b 847.a 1.a $1$ $6.763$ \(\Q\) None 77.2.a.a \(0\) \(-3\) \(-1\) \(1\) $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-2q^{4}-q^{5}+q^{7}+6q^{9}+6q^{12}+\cdots\)
847.2.a.c 847.a 1.a $1$ $6.763$ \(\Q\) None 77.2.a.b \(0\) \(1\) \(3\) \(-1\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}+3q^{5}-q^{7}-2q^{9}-2q^{12}+\cdots\)
847.2.a.d 847.a 1.a $2$ $6.763$ \(\Q(\sqrt{5}) \) None 847.2.a.d \(-3\) \(-1\) \(2\) \(2\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+(-1+\beta )q^{3}+3\beta q^{4}+\cdots\)
847.2.a.e 847.a 1.a $2$ $6.763$ \(\Q(\sqrt{13}) \) None 847.2.a.e \(-1\) \(-1\) \(0\) \(2\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-\beta q^{3}+(1+\beta )q^{4}+(-1+2\beta )q^{5}+\cdots\)
847.2.a.f 847.a 1.a $2$ $6.763$ \(\Q(\sqrt{5}) \) None 77.2.a.d \(0\) \(2\) \(-4\) \(-2\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(1-\beta )q^{3}+3q^{4}-2q^{5}+(5+\cdots)q^{6}+\cdots\)
847.2.a.g 847.a 1.a $2$ $6.763$ \(\Q(\sqrt{13}) \) None 847.2.a.e \(1\) \(-1\) \(0\) \(-2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-\beta q^{3}+(1+\beta )q^{4}+(-1+2\beta )q^{5}+\cdots\)
847.2.a.h 847.a 1.a $2$ $6.763$ \(\Q(\sqrt{5}) \) None 847.2.a.d \(3\) \(-1\) \(2\) \(-2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(-1+\beta )q^{3}+3\beta q^{4}+\cdots\)
847.2.a.i 847.a 1.a $3$ $6.763$ 3.3.568.1 None 847.2.a.i \(-2\) \(-1\) \(1\) \(-3\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-\beta _{1}q^{3}+(3+\beta _{2})q^{4}+\cdots\)
847.2.a.j 847.a 1.a $3$ $6.763$ 3.3.568.1 None 847.2.a.i \(2\) \(-1\) \(1\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-\beta _{1}q^{3}+(3+\beta _{2})q^{4}+\cdots\)
847.2.a.k 847.a 1.a $4$ $6.763$ 4.4.2525.1 None 77.2.f.a \(-2\) \(-2\) \(-6\) \(4\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(-1-\beta _{2})q^{3}+(1+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
847.2.a.l 847.a 1.a $4$ $6.763$ 4.4.2525.1 None 77.2.f.a \(2\) \(-2\) \(-6\) \(-4\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(-1-\beta _{2})q^{3}+(1+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
847.2.a.m 847.a 1.a $6$ $6.763$ 6.6.7674048.1 None 847.2.a.m \(-4\) \(-2\) \(-4\) \(-6\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+\beta _{5}q^{3}+(1-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
847.2.a.n 847.a 1.a $6$ $6.763$ 6.6.7674048.1 None 847.2.a.m \(4\) \(-2\) \(-4\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+\beta _{5}q^{3}+(1-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
847.2.a.o 847.a 1.a $8$ $6.763$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 77.2.f.b \(-1\) \(4\) \(10\) \(-8\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1-\beta _{4}-\beta _{7})q^{3}+(1+\beta _{5}+\cdots)q^{4}+\cdots\)
847.2.a.p 847.a 1.a $8$ $6.763$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 77.2.f.b \(1\) \(4\) \(10\) \(8\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1-\beta _{4}-\beta _{7})q^{3}+(1+\beta _{5}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(847))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(847)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 2}\)