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Properties

Label 8041.2.a.j
Level $8041$
Weight $2$
Character orbit 8041.a
Self dual yes
Analytic conductor $64.208$
Analytic rank $0$
Dimension $82$
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] = [8041,2,Mod(1,8041)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8041, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8041.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8041 = 11 \cdot 17 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8041.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: \(64.2077082653\)
Analytic rank: \(0\)
Dimension: \(82\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 82 q + 8 q^{2} + 6 q^{3} + 98 q^{4} + 11 q^{5} + 10 q^{6} + 8 q^{7} + 30 q^{8} + 108 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 82 q + 8 q^{2} + 6 q^{3} + 98 q^{4} + 11 q^{5} + 10 q^{6} + 8 q^{7} + 30 q^{8} + 108 q^{9} + q^{10} + 82 q^{11} + 3 q^{12} + 26 q^{13} + 17 q^{14} + 66 q^{15} + 122 q^{16} + 82 q^{17} + 18 q^{18} + 12 q^{19} + 9 q^{20} + 22 q^{21} + 8 q^{22} + 50 q^{23} + 15 q^{24} + 117 q^{25} + 36 q^{26} + 30 q^{27} + 11 q^{28} + 33 q^{29} - 26 q^{30} + 40 q^{31} + 58 q^{32} + 6 q^{33} + 8 q^{34} + 16 q^{35} + 160 q^{36} + 31 q^{37} + 18 q^{38} + 41 q^{39} - 29 q^{40} + 42 q^{41} - 51 q^{42} + 82 q^{43} + 98 q^{44} - 2 q^{45} - 19 q^{46} + 84 q^{47} - 46 q^{48} + 136 q^{49} + 59 q^{50} + 6 q^{51} + 45 q^{52} + 83 q^{53} + 24 q^{54} + 11 q^{55} + 21 q^{56} + 23 q^{57} + 14 q^{58} + 96 q^{59} + 184 q^{60} - 6 q^{61} - 23 q^{62} + 8 q^{63} + 148 q^{64} + 5 q^{65} + 10 q^{66} + 78 q^{67} + 98 q^{68} + 61 q^{69} - 3 q^{70} + 155 q^{71} + 50 q^{72} - 23 q^{73} + 10 q^{74} - 19 q^{75} + 44 q^{76} + 8 q^{77} - 27 q^{78} + 31 q^{79} + 19 q^{80} + 150 q^{81} - 12 q^{82} + 54 q^{83} + 8 q^{84} + 11 q^{85} + 8 q^{86} + 20 q^{87} + 30 q^{88} + 25 q^{89} - 81 q^{90} - 14 q^{91} + 60 q^{92} + 36 q^{93} + 19 q^{94} + 111 q^{95} - 6 q^{96} + 2 q^{97} - 5 q^{98} + 108 q^{99}+O(q^{100}) \) 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 −2.77655 −0.377305 5.70922 −1.07900 1.04760 −3.82470 −10.2988 −2.85764 2.99589
1.2 −2.70525 −2.86444 5.31840 1.46004 7.74904 −3.90191 −8.97712 5.20501 −3.94978
1.3 −2.65190 1.08303 5.03256 0.484312 −2.87208 1.56821 −8.04203 −1.82705 −1.28435
1.4 −2.62978 1.21378 4.91575 1.03630 −3.19199 4.97586 −7.66780 −1.52673 −2.72523
1.5 −2.57949 2.85867 4.65376 3.17304 −7.37390 −3.48304 −6.84534 5.17198 −8.18483
1.6 −2.55417 −3.23448 4.52378 −3.35627 8.26140 1.37818 −6.44616 7.46183 8.57249
1.7 −2.40536 −1.47389 3.78577 −0.987275 3.54525 2.06145 −4.29541 −0.827635 2.37475
1.8 −2.34655 3.28622 3.50630 3.73933 −7.71129 3.13477 −3.53461 7.79927 −8.77452
1.9 −2.30854 −0.184135 3.32934 −0.542742 0.425082 −0.304747 −3.06883 −2.96609 1.25294
1.10 −2.30649 −1.79768 3.31992 3.97246 4.14634 2.36407 −3.04439 0.231661 −9.16247
1.11 −2.26888 0.654424 3.14780 −0.968068 −1.48481 1.81018 −2.60421 −2.57173 2.19643
1.12 −2.25270 2.53862 3.07467 −2.69802 −5.71876 −2.07248 −2.42091 3.44461 6.07784
1.13 −2.22059 −3.23881 2.93104 −1.33112 7.19209 −0.932335 −2.06746 7.48990 2.95587
1.14 −2.00119 −0.510214 2.00475 2.33538 1.02103 −3.93653 −0.00949577 −2.73968 −4.67353
1.15 −1.99568 2.01372 1.98276 2.98897 −4.01875 −3.65858 0.0344145 1.05506 −5.96504
1.16 −1.88314 −1.39410 1.54622 −3.71860 2.62529 0.268017 0.854526 −1.05649 7.00266
1.17 −1.73828 2.71256 1.02162 −1.00305 −4.71518 1.91856 1.70071 4.35796 1.74357
1.18 −1.70166 2.90991 0.895633 −0.762355 −4.95167 4.85107 1.87925 5.46758 1.29727
1.19 −1.64446 −1.94286 0.704260 −2.95674 3.19496 4.77511 2.13080 0.774694 4.86225
1.20 −1.62861 −2.08850 0.652359 −3.32951 3.40134 −5.07571 2.19478 1.36181 5.42246
See all 82 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.82
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(11\) \(-1\)
\(17\) \(-1\)
\(43\) \(-1\)

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 8041.2.a.j ✓ 82
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8041.2.a.j ✓ 82 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{82} - 8 T_{2}^{81} - 99 T_{2}^{80} + 942 T_{2}^{79} + 4383 T_{2}^{78} - 53052 T_{2}^{77} + \cdots + 37225728 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8041))\). Copy content Toggle raw display