Properties

Label 847.4.a.b
Level $847$
Weight $4$
Character orbit 847.a
Self dual yes
Analytic conductor $49.975$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 847.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(49.9746177749\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + q^{2} - 2q^{3} - 7q^{4} + 16q^{5} - 2q^{6} + 7q^{7} - 15q^{8} - 23q^{9} + O(q^{10}) \) \( q + q^{2} - 2q^{3} - 7q^{4} + 16q^{5} - 2q^{6} + 7q^{7} - 15q^{8} - 23q^{9} + 16q^{10} + 14q^{12} - 28q^{13} + 7q^{14} - 32q^{15} + 41q^{16} - 54q^{17} - 23q^{18} + 110q^{19} - 112q^{20} - 14q^{21} + 48q^{23} + 30q^{24} + 131q^{25} - 28q^{26} + 100q^{27} - 49q^{28} + 110q^{29} - 32q^{30} + 12q^{31} + 161q^{32} - 54q^{34} + 112q^{35} + 161q^{36} - 246q^{37} + 110q^{38} + 56q^{39} - 240q^{40} - 182q^{41} - 14q^{42} - 128q^{43} - 368q^{45} + 48q^{46} + 324q^{47} - 82q^{48} + 49q^{49} + 131q^{50} + 108q^{51} + 196q^{52} - 162q^{53} + 100q^{54} - 105q^{56} - 220q^{57} + 110q^{58} + 810q^{59} + 224q^{60} + 488q^{61} + 12q^{62} - 161q^{63} - 167q^{64} - 448q^{65} + 244q^{67} + 378q^{68} - 96q^{69} + 112q^{70} - 768q^{71} + 345q^{72} + 702q^{73} - 246q^{74} - 262q^{75} - 770q^{76} + 56q^{78} - 440q^{79} + 656q^{80} + 421q^{81} - 182q^{82} + 1302q^{83} + 98q^{84} - 864q^{85} - 128q^{86} - 220q^{87} + 730q^{89} - 368q^{90} - 196q^{91} - 336q^{92} - 24q^{93} + 324q^{94} + 1760q^{95} - 322q^{96} + 294q^{97} + 49q^{98} + O(q^{100}) \)

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
1.00000 −2.00000 −7.00000 16.0000 −2.00000 7.00000 −15.0000 −23.0000 16.0000
\(n\): 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 847.4.a.b 1
11.b odd 2 1 7.4.a.a 1
33.d even 2 1 63.4.a.b 1
44.c even 2 1 112.4.a.f 1
55.d odd 2 1 175.4.a.b 1
55.e even 4 2 175.4.b.b 2
77.b even 2 1 49.4.a.b 1
77.h odd 6 2 49.4.c.c 2
77.i even 6 2 49.4.c.b 2
88.b odd 2 1 448.4.a.i 1
88.g even 2 1 448.4.a.e 1
132.d odd 2 1 1008.4.a.c 1
143.d odd 2 1 1183.4.a.b 1
165.d even 2 1 1575.4.a.e 1
187.b odd 2 1 2023.4.a.a 1
231.h odd 2 1 441.4.a.i 1
231.k odd 6 2 441.4.e.e 2
231.l even 6 2 441.4.e.h 2
308.g odd 2 1 784.4.a.g 1
385.h even 2 1 1225.4.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.4.a.a 1 11.b odd 2 1
49.4.a.b 1 77.b even 2 1
49.4.c.b 2 77.i even 6 2
49.4.c.c 2 77.h odd 6 2
63.4.a.b 1 33.d even 2 1
112.4.a.f 1 44.c even 2 1
175.4.a.b 1 55.d odd 2 1
175.4.b.b 2 55.e even 4 2
441.4.a.i 1 231.h odd 2 1
441.4.e.e 2 231.k odd 6 2
441.4.e.h 2 231.l even 6 2
448.4.a.e 1 88.g even 2 1
448.4.a.i 1 88.b odd 2 1
784.4.a.g 1 308.g odd 2 1
847.4.a.b 1 1.a even 1 1 trivial
1008.4.a.c 1 132.d odd 2 1
1183.4.a.b 1 143.d odd 2 1
1225.4.a.j 1 385.h even 2 1
1575.4.a.e 1 165.d even 2 1
2023.4.a.a 1 187.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} - 1 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(847))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -1 + T \)
$3$ \( 2 + T \)
$5$ \( -16 + T \)
$7$ \( -7 + T \)
$11$ \( T \)
$13$ \( 28 + T \)
$17$ \( 54 + T \)
$19$ \( -110 + T \)
$23$ \( -48 + T \)
$29$ \( -110 + T \)
$31$ \( -12 + T \)
$37$ \( 246 + T \)
$41$ \( 182 + T \)
$43$ \( 128 + T \)
$47$ \( -324 + T \)
$53$ \( 162 + T \)
$59$ \( -810 + T \)
$61$ \( -488 + T \)
$67$ \( -244 + T \)
$71$ \( 768 + T \)
$73$ \( -702 + T \)
$79$ \( 440 + T \)
$83$ \( -1302 + T \)
$89$ \( -730 + T \)
$97$ \( -294 + T \)
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