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Properties

Label 8042.2.a.c
Level $8042$
Weight $2$
Character orbit 8042.a
Self dual yes
Analytic conductor $64.216$
Analytic rank $0$
Dimension $86$
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] = [8042,2,Mod(1,8042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8042, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8042.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8042 = 2 \cdot 4021 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8042.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: \(64.2156933055\)
Analytic rank: \(0\)
Dimension: \(86\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 86 q - 86 q^{2} + 12 q^{3} + 86 q^{4} - 4 q^{5} - 12 q^{6} + 35 q^{7} - 86 q^{8} + 72 q^{9} + 4 q^{10} + 13 q^{11} + 12 q^{12} + 45 q^{13} - 35 q^{14} + 17 q^{15} + 86 q^{16} + 5 q^{17} - 72 q^{18} + 47 q^{19} - 4 q^{20} + 15 q^{21} - 13 q^{22} + 6 q^{23} - 12 q^{24} + 112 q^{25} - 45 q^{26} + 51 q^{27} + 35 q^{28} - 14 q^{29} - 17 q^{30} + 24 q^{31} - 86 q^{32} + 43 q^{33} - 5 q^{34} + 42 q^{35} + 72 q^{36} + 61 q^{37} - 47 q^{38} + 20 q^{39} + 4 q^{40} - 16 q^{41} - 15 q^{42} + 72 q^{43} + 13 q^{44} + 6 q^{45} - 6 q^{46} + 11 q^{47} + 12 q^{48} + 89 q^{49} - 112 q^{50} + 56 q^{51} + 45 q^{52} - 7 q^{53} - 51 q^{54} + 48 q^{55} - 35 q^{56} + 65 q^{57} + 14 q^{58} + 24 q^{59} + 17 q^{60} + 31 q^{61} - 24 q^{62} + 98 q^{63} + 86 q^{64} - 9 q^{65} - 43 q^{66} + 157 q^{67} + 5 q^{68} + q^{69} - 42 q^{70} - 11 q^{71} - 72 q^{72} + 74 q^{73} - 61 q^{74} + 76 q^{75} + 47 q^{76} - 13 q^{77} - 20 q^{78} + 57 q^{79} - 4 q^{80} + 34 q^{81} + 16 q^{82} + 65 q^{83} + 15 q^{84} + 102 q^{85} - 72 q^{86} + 49 q^{87} - 13 q^{88} - 34 q^{89} - 6 q^{90} + 91 q^{91} + 6 q^{92} + 57 q^{93} - 11 q^{94} - 13 q^{95} - 12 q^{96} + 64 q^{97} - 89 q^{98} + 56 q^{99}+O(q^{100}) \) 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 −1.00000 −3.40447 1.00000 −1.12046 3.40447 3.83806 −1.00000 8.59041 1.12046
1.2 −1.00000 −3.12962 1.00000 1.55366 3.12962 0.219811 −1.00000 6.79451 −1.55366
1.3 −1.00000 −3.00414 1.00000 −0.143261 3.00414 −0.146700 −1.00000 6.02484 0.143261
1.4 −1.00000 −2.84562 1.00000 0.429228 2.84562 3.51713 −1.00000 5.09754 −0.429228
1.5 −1.00000 −2.78277 1.00000 −4.36753 2.78277 −3.26500 −1.00000 4.74379 4.36753
1.6 −1.00000 −2.75315 1.00000 3.02853 2.75315 4.50146 −1.00000 4.57981 −3.02853
1.7 −1.00000 −2.59394 1.00000 −0.859547 2.59394 −3.29568 −1.00000 3.72855 0.859547
1.8 −1.00000 −2.58298 1.00000 0.255702 2.58298 −2.29966 −1.00000 3.67181 −0.255702
1.9 −1.00000 −2.54862 1.00000 −1.05969 2.54862 −0.366146 −1.00000 3.49545 1.05969
1.10 −1.00000 −2.53308 1.00000 −2.12930 2.53308 −1.20392 −1.00000 3.41648 2.12930
1.11 −1.00000 −2.49912 1.00000 −3.77840 2.49912 −0.508744 −1.00000 3.24558 3.77840
1.12 −1.00000 −2.38541 1.00000 2.54225 2.38541 3.05592 −1.00000 2.69019 −2.54225
1.13 −1.00000 −2.19026 1.00000 −2.77068 2.19026 3.06919 −1.00000 1.79726 2.77068
1.14 −1.00000 −2.17249 1.00000 −0.488966 2.17249 −0.203617 −1.00000 1.71971 0.488966
1.15 −1.00000 −2.17138 1.00000 3.37288 2.17138 0.0526160 −1.00000 1.71488 −3.37288
1.16 −1.00000 −2.10336 1.00000 −2.86600 2.10336 3.42965 −1.00000 1.42413 2.86600
1.17 −1.00000 −2.02152 1.00000 3.49725 2.02152 −2.15386 −1.00000 1.08655 −3.49725
1.18 −1.00000 −2.00326 1.00000 3.10974 2.00326 0.986750 −1.00000 1.01305 −3.10974
1.19 −1.00000 −1.95612 1.00000 0.359780 1.95612 −1.34656 −1.00000 0.826410 −0.359780
1.20 −1.00000 −1.75423 1.00000 −3.88795 1.75423 3.77690 −1.00000 0.0773276 3.88795
See all 86 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.86
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(4021\) \(-1\)

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 8042.2.a.c ✓ 86
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8042.2.a.c ✓ 86 1.a even 1 1 trivial