Properties

Label 847.2.a.b
Level $847$
Weight $2$
Character orbit 847.a
Self dual yes
Analytic conductor $6.763$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 847.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.76332905120\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3 q^{3} - 2 q^{4} - q^{5} + q^{7} + 6 q^{9} + O(q^{10}) \) \( q - 3 q^{3} - 2 q^{4} - q^{5} + q^{7} + 6 q^{9} + 6 q^{12} + 4 q^{13} + 3 q^{15} + 4 q^{16} - 2 q^{17} + 6 q^{19} + 2 q^{20} - 3 q^{21} - 5 q^{23} - 4 q^{25} - 9 q^{27} - 2 q^{28} - 10 q^{29} + q^{31} - q^{35} - 12 q^{36} - 5 q^{37} - 12 q^{39} + 2 q^{41} + 8 q^{43} - 6 q^{45} + 8 q^{47} - 12 q^{48} + q^{49} + 6 q^{51} - 8 q^{52} - 6 q^{53} - 18 q^{57} + 3 q^{59} - 6 q^{60} + 2 q^{61} + 6 q^{63} - 8 q^{64} - 4 q^{65} - 3 q^{67} + 4 q^{68} + 15 q^{69} + q^{71} - 10 q^{73} + 12 q^{75} - 12 q^{76} - 6 q^{79} - 4 q^{80} + 9 q^{81} - 12 q^{83} + 6 q^{84} + 2 q^{85} + 30 q^{87} - 15 q^{89} + 4 q^{91} + 10 q^{92} - 3 q^{93} - 6 q^{95} - 5 q^{97} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 −3.00000 −2.00000 −1.00000 0 1.00000 0 6.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(-1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 847.2.a.b 1
3.b odd 2 1 7623.2.a.j 1
7.b odd 2 1 5929.2.a.f 1
11.b odd 2 1 77.2.a.a 1
11.c even 5 4 847.2.f.h 4
11.d odd 10 4 847.2.f.i 4
33.d even 2 1 693.2.a.c 1
44.c even 2 1 1232.2.a.l 1
55.d odd 2 1 1925.2.a.h 1
55.e even 4 2 1925.2.b.e 2
77.b even 2 1 539.2.a.c 1
77.h odd 6 2 539.2.e.f 2
77.i even 6 2 539.2.e.c 2
88.b odd 2 1 4928.2.a.bj 1
88.g even 2 1 4928.2.a.a 1
231.h odd 2 1 4851.2.a.j 1
308.g odd 2 1 8624.2.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.a 1 11.b odd 2 1
539.2.a.c 1 77.b even 2 1
539.2.e.c 2 77.i even 6 2
539.2.e.f 2 77.h odd 6 2
693.2.a.c 1 33.d even 2 1
847.2.a.b 1 1.a even 1 1 trivial
847.2.f.h 4 11.c even 5 4
847.2.f.i 4 11.d odd 10 4
1232.2.a.l 1 44.c even 2 1
1925.2.a.h 1 55.d odd 2 1
1925.2.b.e 2 55.e even 4 2
4851.2.a.j 1 231.h odd 2 1
4928.2.a.a 1 88.g even 2 1
4928.2.a.bj 1 88.b odd 2 1
5929.2.a.f 1 7.b odd 2 1
7623.2.a.j 1 3.b odd 2 1
8624.2.a.a 1 308.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(847))\):

\( T_{2} \)
\( T_{3} + 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( 3 + T \)
$5$ \( 1 + T \)
$7$ \( -1 + T \)
$11$ \( T \)
$13$ \( -4 + T \)
$17$ \( 2 + T \)
$19$ \( -6 + T \)
$23$ \( 5 + T \)
$29$ \( 10 + T \)
$31$ \( -1 + T \)
$37$ \( 5 + T \)
$41$ \( -2 + T \)
$43$ \( -8 + T \)
$47$ \( -8 + T \)
$53$ \( 6 + T \)
$59$ \( -3 + T \)
$61$ \( -2 + T \)
$67$ \( 3 + T \)
$71$ \( -1 + T \)
$73$ \( 10 + T \)
$79$ \( 6 + T \)
$83$ \( 12 + T \)
$89$ \( 15 + T \)
$97$ \( 5 + T \)
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