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Properties

Label 8847.2.a.g
Level $8847$
Weight $2$
Character orbit 8847.a
Self dual yes
Analytic conductor $70.644$
Analytic rank $0$
Dimension $54$
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: \(0\)
Dimension: \(54\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 54 q - 8 q^{2} + 64 q^{4} - 7 q^{5} + 31 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 54 q - 8 q^{2} + 64 q^{4} - 7 q^{5} + 31 q^{7} - 24 q^{8} + 15 q^{10} - 8 q^{11} + 34 q^{13} - q^{14} + 80 q^{16} - 32 q^{17} + 15 q^{19} - 4 q^{20} + 25 q^{22} - 15 q^{23} + 125 q^{25} + 14 q^{26} + 78 q^{28} - 8 q^{29} + 16 q^{31} - 36 q^{32} - 8 q^{34} + 2 q^{35} + 80 q^{37} + 14 q^{38} + 36 q^{40} - 30 q^{41} + 53 q^{43} - 6 q^{44} + 24 q^{46} - 22 q^{47} + 111 q^{49} - 14 q^{50} + 45 q^{52} - 10 q^{53} + 12 q^{55} + 30 q^{56} + 61 q^{58} + 4 q^{59} + 24 q^{61} + 12 q^{62} + 86 q^{64} - 32 q^{65} + 54 q^{67} - 33 q^{68} - 21 q^{70} + 6 q^{71} + 172 q^{73} + 32 q^{74} + 7 q^{76} + 12 q^{77} + 28 q^{79} + 66 q^{80} + 4 q^{82} - 14 q^{83} + 99 q^{85} + 31 q^{86} + 5 q^{88} + 5 q^{89} - 11 q^{91} - 20 q^{92} - 39 q^{94} - 5 q^{95} + 127 q^{97} + 29 q^{98}+O(q^{100}) \) 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 −2.74377 0 5.52830 4.11958 0 4.42313 −9.68085 0 −11.3032
1.2 −2.74028 0 5.50913 0.684911 0 −1.23535 −9.61600 0 −1.87685
1.3 −2.73579 0 5.48453 −3.19952 0 1.97538 −9.53294 0 8.75320
1.4 −2.65099 0 5.02774 −1.59044 0 0.277687 −8.02650 0 4.21624
1.5 −2.63841 0 4.96120 −0.0609686 0 −4.46047 −7.81284 0 0.160860
1.6 −2.42765 0 3.89350 −4.39830 0 −3.14623 −4.59676 0 10.6776
1.7 −2.38010 0 3.66489 −0.874242 0 3.22267 −3.96262 0 2.08079
1.8 −2.35411 0 3.54185 3.80188 0 3.13231 −3.62968 0 −8.95004
1.9 −2.33214 0 3.43890 −1.77614 0 5.24768 −3.35572 0 4.14221
1.10 −2.28980 0 3.24320 2.18718 0 −0.652196 −2.84668 0 −5.00820
1.11 −2.08879 0 2.36303 −1.85457 0 1.45975 −0.758281 0 3.87380
1.12 −2.08691 0 2.35519 2.03710 0 1.75745 −0.741253 0 −4.25124
1.13 −2.05643 0 2.22892 −3.73063 0 −0.348785 −0.470757 0 7.67180
1.14 −1.98254 0 1.93048 0.425094 0 0.736757 0.137820 0 −0.842768
1.15 −1.83931 0 1.38307 −0.900035 0 −1.47356 1.13473 0 1.65544
1.16 −1.41786 0 0.0103341 3.78661 0 −3.64106 2.82107 0 −5.36890
1.17 −1.38686 0 −0.0766077 −4.36721 0 2.06000 2.87997 0 6.05673
1.18 −1.26622 0 −0.396677 −2.54580 0 4.97214 3.03473 0 3.22356
1.19 −1.24247 0 −0.456258 −1.68565 0 0.816106 3.05184 0 2.09438
1.20 −1.14642 0 −0.685725 −3.10824 0 −2.62406 3.07896 0 3.56334
See all 54 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.54
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.g 54
3.b odd 2 1 983.2.a.b ✓ 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
983.2.a.b ✓ 54 3.b odd 2 1
8847.2.a.g 54 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{54} + 8 T_{2}^{53} - 54 T_{2}^{52} - 584 T_{2}^{51} + 1042 T_{2}^{50} + 19796 T_{2}^{49} + \cdots - 983 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8847))\). Copy content Toggle raw display