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Properties

Label 8047.2.a.e
Level $8047$
Weight $2$
Character orbit 8047.a
Self dual yes
Analytic conductor $64.256$
Analytic rank $0$
Dimension $168$
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] = [8047,2,Mod(1,8047)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8047, 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("8047.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8047 = 13 \cdot 619 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8047.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: \(64.2556185065\)
Analytic rank: \(0\)
Dimension: \(168\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 168 q + 11 q^{2} + 26 q^{3} + 181 q^{4} + 41 q^{5} + 11 q^{6} + 12 q^{7} + 27 q^{8} + 220 q^{9} + 11 q^{10} + 23 q^{11} + 78 q^{12} + 168 q^{13} + 47 q^{14} + 10 q^{15} + 203 q^{16} + 147 q^{17} + 13 q^{18} + 17 q^{19} + 81 q^{20} + 13 q^{21} + 20 q^{22} + 85 q^{23} + 14 q^{24} + 225 q^{25} + 11 q^{26} + 89 q^{27} + 12 q^{28} + 137 q^{29} + 26 q^{30} + 13 q^{31} + 60 q^{32} + 78 q^{33} - 2 q^{34} + 77 q^{35} + 278 q^{36} + 41 q^{37} + 68 q^{38} + 26 q^{39} + 11 q^{40} + 107 q^{41} + 43 q^{42} + 27 q^{43} + 39 q^{44} + 88 q^{45} - 23 q^{46} + 112 q^{47} + 127 q^{48} + 236 q^{49} + 14 q^{50} + 55 q^{51} + 181 q^{52} + 149 q^{53} + 3 q^{54} + 40 q^{55} + 134 q^{56} + 55 q^{57} - q^{58} + 44 q^{59} - 13 q^{60} + 81 q^{61} + 106 q^{62} + 34 q^{63} + 197 q^{64} + 41 q^{65} - 20 q^{66} - q^{67} + 278 q^{68} + 75 q^{69} - 42 q^{70} + 48 q^{71} - 34 q^{72} + 107 q^{73} + 74 q^{74} + 93 q^{75} + 20 q^{76} + 206 q^{77} + 11 q^{78} + 14 q^{79} + 115 q^{80} + 328 q^{81} + 48 q^{82} + 62 q^{83} - 11 q^{84} + 6 q^{85} + 27 q^{86} + 51 q^{87} + 31 q^{88} + 173 q^{89} - 21 q^{90} + 12 q^{91} + 179 q^{92} + 73 q^{93} + 17 q^{94} + 90 q^{95} - 33 q^{96} + 110 q^{97} - 13 q^{98} + 24 q^{99}+O(q^{100}) \) 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 −2.81579 1.75700 5.92868 −0.878024 −4.94733 −1.13202 −11.0623 0.0870368 2.47233
1.2 −2.76928 −1.56618 5.66891 −1.99910 4.33718 −3.75052 −10.1602 −0.547092 5.53606
1.3 −2.76470 3.06859 5.64359 2.61743 −8.48374 4.52897 −10.0734 6.41624 −7.23642
1.4 −2.69715 2.56324 5.27461 3.51333 −6.91344 −4.43417 −8.83210 3.57021 −9.47597
1.5 −2.59593 −2.11181 4.73887 −0.212258 5.48213 0.982044 −7.10991 1.45976 0.551008
1.6 −2.59413 −3.39586 4.72950 2.81515 8.80929 −0.0603743 −7.08068 8.53185 −7.30287
1.7 −2.58760 −0.932881 4.69570 1.35951 2.41393 0.312733 −6.97540 −2.12973 −3.51787
1.8 −2.56793 0.140851 4.59428 −0.0428559 −0.361697 −5.13541 −6.66194 −2.98016 0.110051
1.9 −2.56640 1.62937 4.58642 4.01357 −4.18162 3.25555 −6.63781 −0.345155 −10.3004
1.10 −2.55763 0.266085 4.54145 −4.02410 −0.680547 −2.52095 −6.50007 −2.92920 10.2921
1.11 −2.52416 3.25012 4.37136 −2.90640 −8.20382 1.38708 −5.98570 7.56330 7.33621
1.12 −2.51705 1.87666 4.33553 0.537447 −4.72364 0.939568 −5.87865 0.521853 −1.35278
1.13 −2.51491 3.41977 4.32478 −1.23152 −8.60042 −3.67991 −5.84661 8.69483 3.09716
1.14 −2.50460 −2.26782 4.27304 0.233077 5.68000 −0.679959 −5.69306 2.14302 −0.583765
1.15 −2.43925 −2.47662 3.94994 4.18680 6.04110 3.85857 −4.75640 3.13366 −10.2127
1.16 −2.43547 −1.20585 3.93153 2.83456 2.93681 −2.65946 −4.70420 −1.54594 −6.90350
1.17 −2.39466 −2.35552 3.73441 0.950970 5.64068 −0.373318 −4.15331 2.54848 −2.27725
1.18 −2.38728 1.74228 3.69910 −3.20056 −4.15931 −0.986874 −4.05622 0.0355448 7.64064
1.19 −2.35312 −0.357062 3.53719 1.67497 0.840212 4.28147 −3.61721 −2.87251 −3.94141
1.20 −2.33973 −0.0830321 3.47431 −2.47504 0.194272 1.67237 −3.44949 −2.99311 5.79091
See next 80 embeddings (of 168 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.168
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \(-1\)
\(619\) \(1\)

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 8047.2.a.e ✓ 168
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8047.2.a.e ✓ 168 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{168} - 11 T_{2}^{167} - 198 T_{2}^{166} + 2598 T_{2}^{165} + 17984 T_{2}^{164} + \cdots - 21530657600 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8047))\). Copy content Toggle raw display