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Properties

Label 8847.2.a.b
Level $8847$
Weight $2$
Character orbit 8847.a
Self dual yes
Analytic conductor $70.644$
Analytic rank $1$
Dimension $28$
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] = [8847,2,Mod(1,8847)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8847, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8847.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8847 = 3^{2} \cdot 983 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8847.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: \(70.6436506682\)
Analytic rank: \(1\)
Dimension: \(28\)
Twist minimal: no (minimal twist has level 983)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 7 q^{2} + 17 q^{4} + 7 q^{5} - 25 q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 7 q^{2} + 17 q^{4} + 7 q^{5} - 25 q^{7} + 15 q^{8} - 17 q^{10} + 10 q^{11} - 28 q^{13} - 5 q^{14} + 3 q^{16} + 24 q^{17} - 13 q^{19} + 4 q^{20} - 21 q^{22} + 9 q^{23} - 33 q^{25} - 2 q^{26} - 58 q^{28} + 14 q^{29} - 16 q^{31} + 27 q^{32} - 6 q^{34} + 2 q^{35} - 58 q^{37} - 6 q^{38} - 24 q^{40} + 24 q^{41} - 43 q^{43} - 28 q^{46} - 2 q^{47} - 21 q^{49} - 17 q^{50} - 47 q^{52} + 16 q^{53} - 16 q^{55} - 30 q^{56} - 31 q^{58} - 12 q^{59} - 22 q^{61} - 3 q^{64} + 24 q^{65} - 38 q^{67} - 11 q^{68} + 19 q^{70} - 2 q^{71} - 124 q^{73} - 14 q^{74} + 3 q^{76} + 20 q^{77} - 16 q^{79} - 10 q^{80} + 2 q^{82} - 12 q^{83} - 73 q^{85} - 41 q^{86} - 61 q^{88} + 3 q^{89} + 5 q^{91} - 44 q^{92} + 9 q^{94} - 17 q^{95} - 105 q^{97} - 16 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −2.50886 0 4.29437 2.58171 0 −2.53517 −5.75625 0 −6.47714
1.2 −2.14734 0 2.61107 −1.19756 0 −3.58025 −1.31217 0 2.57157
1.3 −2.00103 0 2.00413 3.16164 0 −1.56426 −0.00827143 0 −6.32654
1.4 −1.97659 0 1.90691 1.42423 0 1.52123 0.184004 0 −2.81512
1.5 −1.61868 0 0.620135 0.0106016 0 −5.20788 2.23356 0 −0.0171606
1.6 −1.45003 0 0.102585 −0.663561 0 −2.02015 2.75131 0 0.962183
1.7 −1.30501 0 −0.296942 −2.67718 0 1.19123 2.99754 0 3.49375
1.8 −1.12563 0 −0.732961 3.34100 0 0.00528609 3.07630 0 −3.76073
1.9 −0.912923 0 −1.16657 1.17872 0 3.45815 2.89084 0 −1.07608
1.10 −0.768290 0 −1.40973 1.02856 0 −1.07434 2.61966 0 −0.790236
1.11 −0.413754 0 −1.82881 −1.10128 0 −2.64498 1.58419 0 0.455658
1.12 −0.0562061 0 −1.99684 −0.503583 0 3.25094 0.224647 0 0.0283044
1.13 0.120414 0 −1.98550 −0.840169 0 −0.0546560 −0.479911 0 −0.101168
1.14 0.400086 0 −1.83993 −0.291591 0 −3.58224 −1.53630 0 −0.116661
1.15 0.434502 0 −1.81121 1.89683 0 1.45633 −1.65598 0 0.824176
1.16 0.468024 0 −1.78095 2.20940 0 0.238349 −1.76958 0 1.03405
1.17 0.875382 0 −1.23371 −0.141089 0 −0.646767 −2.83073 0 −0.123507
1.18 0.930689 0 −1.13382 −3.65614 0 −1.39504 −2.91661 0 −3.40273
1.19 0.971766 0 −1.05567 2.58229 0 −3.41775 −2.96940 0 2.50939
1.20 1.64741 0 0.713964 −2.48769 0 2.09423 −2.11863 0 −4.09824
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.28
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(983\) \(-1\)

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 8847.2.a.b 28
3.b odd 2 1 983.2.a.a ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
983.2.a.a ✓ 28 3.b odd 2 1
8847.2.a.b 28 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} - 7 T_{2}^{27} - 12 T_{2}^{26} + 184 T_{2}^{25} - 110 T_{2}^{24} - 2026 T_{2}^{23} + 3083 T_{2}^{22} + \cdots + 7 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8847))\). Copy content Toggle raw display