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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [847,4,Mod(1,847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(847, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("847.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 847 = 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 847.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
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

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

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 \(S_{4}^{\mathrm{new}}(\Gamma_0(847))\). 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
$5$ \( T - 16 \) Copy content Toggle raw display
$7$ \( T - 7 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 28 \) Copy content Toggle raw display
$17$ \( T + 54 \) Copy content Toggle raw display
$19$ \( T - 110 \) Copy content Toggle raw display
$23$ \( T - 48 \) Copy content Toggle raw display
$29$ \( T - 110 \) Copy content Toggle raw display
$31$ \( T - 12 \) Copy content Toggle raw display
$37$ \( T + 246 \) Copy content Toggle raw display
$41$ \( T + 182 \) Copy content Toggle raw display
$43$ \( T + 128 \) Copy content Toggle raw display
$47$ \( T - 324 \) Copy content Toggle raw display
$53$ \( T + 162 \) Copy content Toggle raw display
$59$ \( T - 810 \) Copy content Toggle raw display
$61$ \( T - 488 \) Copy content Toggle raw display
$67$ \( T - 244 \) Copy content Toggle raw display
$71$ \( T + 768 \) Copy content Toggle raw display
$73$ \( T - 702 \) Copy content Toggle raw display
$79$ \( T + 440 \) Copy content Toggle raw display
$83$ \( T - 1302 \) Copy content Toggle raw display
$89$ \( T - 730 \) Copy content Toggle raw display
$97$ \( T - 294 \) Copy content Toggle raw display
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