Properties

Label 847.2.a.a
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

\(f(q)\) \(=\) \( q - q^{2} + 2 q^{3} - q^{4} - 2 q^{5} - 2 q^{6} + q^{7} + 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + 2 q^{3} - q^{4} - 2 q^{5} - 2 q^{6} + q^{7} + 3 q^{8} + q^{9} + 2 q^{10} - 2 q^{12} - 4 q^{13} - q^{14} - 4 q^{15} - q^{16} - 4 q^{17} - q^{18} + 2 q^{20} + 2 q^{21} - 4 q^{23} + 6 q^{24} - q^{25} + 4 q^{26} - 4 q^{27} - q^{28} + 6 q^{29} + 4 q^{30} + 10 q^{31} - 5 q^{32} + 4 q^{34} - 2 q^{35} - q^{36} - 6 q^{37} - 8 q^{39} - 6 q^{40} - 4 q^{41} - 2 q^{42} - 12 q^{43} - 2 q^{45} + 4 q^{46} - 10 q^{47} - 2 q^{48} + q^{49} + q^{50} - 8 q^{51} + 4 q^{52} - 6 q^{53} + 4 q^{54} + 3 q^{56} - 6 q^{58} + 2 q^{59} + 4 q^{60} - 10 q^{62} + q^{63} + 7 q^{64} + 8 q^{65} + 8 q^{67} + 4 q^{68} - 8 q^{69} + 2 q^{70} - 12 q^{71} + 3 q^{72} + 8 q^{73} + 6 q^{74} - 2 q^{75} + 8 q^{78} - 8 q^{79} + 2 q^{80} - 11 q^{81} + 4 q^{82} - 2 q^{84} + 8 q^{85} + 12 q^{86} + 12 q^{87} - 6 q^{89} + 2 q^{90} - 4 q^{91} + 4 q^{92} + 20 q^{93} + 10 q^{94} - 10 q^{96} - 10 q^{97} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
−1.00000 2.00000 −1.00000 −2.00000 −2.00000 1.00000 3.00000 1.00000 2.00000
\(n\): 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.a 1
3.b odd 2 1 7623.2.a.n 1
7.b odd 2 1 5929.2.a.b 1
11.b odd 2 1 77.2.a.c 1
11.c even 5 4 847.2.f.k 4
11.d odd 10 4 847.2.f.e 4
33.d even 2 1 693.2.a.a 1
44.c even 2 1 1232.2.a.a 1
55.d odd 2 1 1925.2.a.c 1
55.e even 4 2 1925.2.b.d 2
77.b even 2 1 539.2.a.d 1
77.h odd 6 2 539.2.e.a 2
77.i even 6 2 539.2.e.b 2
88.b odd 2 1 4928.2.a.g 1
88.g even 2 1 4928.2.a.bi 1
231.h odd 2 1 4851.2.a.a 1
308.g odd 2 1 8624.2.a.bc 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
77.2.a.c 1 11.b odd 2 1
539.2.a.d 1 77.b even 2 1
539.2.e.a 2 77.h odd 6 2
539.2.e.b 2 77.i even 6 2
693.2.a.a 1 33.d even 2 1
847.2.a.a 1 1.a even 1 1 trivial
847.2.f.e 4 11.d odd 10 4
847.2.f.k 4 11.c even 5 4
1232.2.a.a 1 44.c even 2 1
1925.2.a.c 1 55.d odd 2 1
1925.2.b.d 2 55.e even 4 2
4851.2.a.a 1 231.h odd 2 1
4928.2.a.g 1 88.b odd 2 1
4928.2.a.bi 1 88.g even 2 1
5929.2.a.b 1 7.b odd 2 1
7623.2.a.n 1 3.b odd 2 1
8624.2.a.bc 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(847))\):

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{3} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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