Properties

Label 847.4.a
Level $847$
Weight $4$
Character orbit 847.a
Rep. character $\chi_{847}(1,\cdot)$
Character field $\Q$
Dimension $163$
Newform subspaces $18$
Sturm bound $352$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 847.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 18 \)
Sturm bound: \(352\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(847))\).

Total New Old
Modular forms 276 163 113
Cusp forms 252 163 89
Eisenstein series 24 0 24

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
\(+\)\(+\)\(+\)\(41\)
\(+\)\(-\)\(-\)\(40\)
\(-\)\(+\)\(-\)\(37\)
\(-\)\(-\)\(+\)\(45\)
Plus space\(+\)\(86\)
Minus space\(-\)\(77\)

Trace form

\( 163 q + 5 q^{2} + 2 q^{3} + 631 q^{4} - 38 q^{6} + 7 q^{7} + 45 q^{8} + 1463 q^{9} + 128 q^{10} + 142 q^{12} + 140 q^{13} - 63 q^{14} - 176 q^{15} + 2559 q^{16} - 206 q^{17} + 37 q^{18} + 22 q^{19} + 12 q^{20}+ \cdots + 245 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(847))\) 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
847.4.a.a 847.a 1.a $1$ $49.975$ \(\Q\) None 77.4.a.a \(-3\) \(4\) \(12\) \(-7\) $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{2}+4q^{3}+q^{4}+12q^{5}-12q^{6}+\cdots\)
847.4.a.b 847.a 1.a $1$ $49.975$ \(\Q\) None 7.4.a.a \(1\) \(-2\) \(16\) \(7\) $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}-7q^{4}+2^{4}q^{5}-2q^{6}+\cdots\)
847.4.a.c 847.a 1.a $2$ $49.975$ \(\Q(\sqrt{2}) \) None 77.4.a.b \(2\) \(-4\) \(-4\) \(-14\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}-2q^{3}+(1+2\beta )q^{4}+(-2+\cdots)q^{5}+\cdots\)
847.4.a.d 847.a 1.a $4$ $49.975$ 4.4.522072.1 None 77.4.a.d \(2\) \(14\) \(10\) \(28\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{2}+(4+\beta _{1})q^{3}+(6-\beta _{2}-2\beta _{3})q^{4}+\cdots\)
847.4.a.e 847.a 1.a $4$ $49.975$ 4.4.509800.1 None 77.4.a.c \(4\) \(-12\) \(-18\) \(28\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{2})q^{2}+(-3+\beta _{3})q^{3}+(6-\beta _{1}+\cdots)q^{4}+\cdots\)
847.4.a.f 847.a 1.a $5$ $49.975$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None 77.4.a.e \(-1\) \(2\) \(-24\) \(-35\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-\beta _{3}q^{3}+(9+\beta _{1}+\beta _{2})q^{4}+\cdots\)
847.4.a.g 847.a 1.a $7$ $49.975$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 847.4.a.g \(-4\) \(2\) \(-18\) \(49\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+\beta _{3}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
847.4.a.h 847.a 1.a $7$ $49.975$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None 847.4.a.g \(4\) \(2\) \(-18\) \(-49\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+\beta _{3}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
847.4.a.i 847.a 1.a $8$ $49.975$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 847.4.a.i \(-6\) \(2\) \(-2\) \(-56\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-\beta _{3}q^{3}+(4-\beta _{1}-\beta _{3}+\cdots)q^{4}+\cdots\)
847.4.a.j 847.a 1.a $8$ $49.975$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 847.4.a.j \(-2\) \(2\) \(2\) \(-56\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{3})q^{3}+(4+\beta _{2})q^{4}+\cdots\)
847.4.a.k 847.a 1.a $8$ $49.975$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 847.4.a.j \(2\) \(2\) \(2\) \(56\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{3})q^{3}+(4+\beta _{2})q^{4}+\cdots\)
847.4.a.l 847.a 1.a $8$ $49.975$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 847.4.a.i \(6\) \(2\) \(-2\) \(56\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-\beta _{3}q^{3}+(4-\beta _{1}-\beta _{3}+\cdots)q^{4}+\cdots\)
847.4.a.m 847.a 1.a $14$ $49.975$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None 847.4.a.m \(-8\) \(4\) \(6\) \(98\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+\beta _{5}q^{3}+(5-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
847.4.a.n 847.a 1.a $14$ $49.975$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None 847.4.a.m \(8\) \(4\) \(6\) \(-98\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+\beta _{5}q^{3}+(5-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
847.4.a.o 847.a 1.a $16$ $49.975$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 77.4.f.a \(-6\) \(-21\) \(-32\) \(112\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(-1+\beta _{4})q^{3}+(3+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
847.4.a.p 847.a 1.a $16$ $49.975$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None 77.4.f.a \(6\) \(-21\) \(-32\) \(-112\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(-1+\beta _{4})q^{3}+(3+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
847.4.a.q 847.a 1.a $20$ $49.975$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None 77.4.f.b \(-1\) \(11\) \(48\) \(-140\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1-\beta _{4})q^{3}+(5+\beta _{2})q^{4}+\cdots\)
847.4.a.r 847.a 1.a $20$ $49.975$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None 77.4.f.b \(1\) \(11\) \(48\) \(140\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1-\beta _{4})q^{3}+(5+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(847))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(847)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 2}\)