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Properties

Label 8041.2.a.e
Level $8041$
Weight $2$
Character orbit 8041.a
Self dual yes
Analytic conductor $64.208$
Analytic rank $0$
Dimension $66$
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: \(66\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 66 q + 7 q^{2} + 3 q^{3} + 61 q^{4} + 4 q^{5} + 10 q^{6} + 14 q^{7} + 21 q^{8} + 65 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 66 q + 7 q^{2} + 3 q^{3} + 61 q^{4} + 4 q^{5} + 10 q^{6} + 14 q^{7} + 21 q^{8} + 65 q^{9} + 13 q^{10} - 66 q^{11} + 9 q^{12} - 12 q^{13} + 25 q^{14} + 13 q^{15} + 47 q^{16} - 66 q^{17} + 37 q^{18} + 19 q^{20} + 26 q^{21} - 7 q^{22} + 47 q^{23} + 15 q^{24} + 52 q^{25} + 16 q^{26} + 9 q^{27} + 3 q^{28} + 57 q^{29} + 2 q^{30} + 31 q^{31} + 39 q^{32} - 3 q^{33} - 7 q^{34} + 36 q^{35} + 39 q^{36} - 14 q^{37} + 18 q^{38} + 71 q^{39} + 29 q^{40} + 62 q^{41} - 3 q^{42} + 66 q^{43} - 61 q^{44} - 2 q^{45} + 19 q^{46} + 32 q^{47} + 26 q^{48} + 42 q^{49} + 10 q^{50} - 3 q^{51} - 7 q^{52} + 33 q^{53} + 100 q^{54} - 4 q^{55} + 61 q^{56} + 35 q^{57} - 16 q^{58} + 59 q^{59} + 50 q^{60} + 26 q^{61} + 29 q^{62} + 62 q^{63} + 29 q^{64} + 55 q^{65} - 10 q^{66} + 5 q^{67} - 61 q^{68} - 36 q^{69} - 35 q^{70} + 128 q^{71} + 87 q^{72} + 23 q^{73} + 64 q^{74} - 11 q^{75} + 74 q^{76} - 14 q^{77} + 45 q^{78} + 39 q^{79} + 95 q^{80} + 54 q^{81} - 6 q^{82} + 48 q^{83} + 38 q^{84} - 4 q^{85} + 7 q^{86} + 14 q^{87} - 21 q^{88} + 28 q^{89} + 135 q^{90} - 18 q^{91} + 108 q^{92} - 9 q^{93} + 37 q^{94} + 149 q^{95} + 104 q^{96} + 19 q^{97} + 30 q^{98} - 65 q^{99}+O(q^{100}) \) 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 −2.67289 −2.02637 5.14435 2.69749 5.41626 1.58680 −8.40451 1.10617 −7.21011
1.2 −2.55220 1.68778 4.51375 −1.92240 −4.30756 −1.71256 −6.41560 −0.151402 4.90636
1.3 −2.49759 2.42195 4.23793 −1.17913 −6.04903 1.05331 −5.58943 2.86584 2.94498
1.4 −2.44538 −0.190207 3.97990 3.44061 0.465129 3.54065 −4.84161 −2.96382 −8.41362
1.5 −2.44026 −2.32024 3.95484 −3.72812 5.66197 −3.99727 −4.77032 2.38350 9.09757
1.6 −2.41404 0.194701 3.82757 −3.15933 −0.470017 −0.226574 −4.41183 −2.96209 7.62675
1.7 −2.39048 1.48105 3.71439 1.30814 −3.54043 −2.95859 −4.09821 −0.806479 −3.12708
1.8 −2.38493 −1.31374 3.68787 1.21176 3.13318 −4.42296 −4.02545 −1.27408 −2.88995
1.9 −1.96819 1.90311 1.87379 3.64418 −3.74569 3.36449 0.248405 0.621828 −7.17246
1.10 −1.96076 −1.83181 1.84459 −1.45283 3.59175 0.993575 0.304726 0.355533 2.84866
1.11 −1.92404 0.667839 1.70195 1.74197 −1.28495 −1.15955 0.573464 −2.55399 −3.35162
1.12 −1.88145 −3.19345 1.53984 2.05646 6.00831 −0.224588 0.865759 7.19812 −3.86912
1.13 −1.79269 1.29476 1.21374 −0.440748 −2.32111 4.25701 1.40952 −1.32359 0.790124
1.14 −1.70598 0.804154 0.910358 −1.55515 −1.37187 1.24699 1.85890 −2.35334 2.65305
1.15 −1.64902 −1.36167 0.719266 −2.44108 2.24541 −1.20008 2.11196 −1.14587 4.02539
1.16 −1.40918 0.980513 −0.0142062 1.84997 −1.38172 −3.81355 2.83838 −2.03859 −2.60695
1.17 −1.35222 2.90367 −0.171490 −2.71204 −3.92642 −3.82442 2.93634 5.43132 3.66729
1.18 −1.32441 −2.65196 −0.245926 −0.221912 3.51229 0.607593 2.97454 4.03289 0.293903
1.19 −1.31054 3.05736 −0.282496 3.37199 −4.00679 1.80971 2.99129 6.34748 −4.41911
1.20 −1.19284 −3.20142 −0.577127 −3.80611 3.81880 4.55130 3.07411 7.24912 4.54009
See all 66 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.66
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.e ✓ 66
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8041.2.a.e ✓ 66 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{66} - 7 T_{2}^{65} - 72 T_{2}^{64} + 602 T_{2}^{63} + 2271 T_{2}^{62} - 24463 T_{2}^{61} + \cdots + 10656 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8041))\). Copy content Toggle raw display