Properties

Label 8043.2.a.n
Level $8043$
Weight $2$
Character orbit 8043.a
Self dual yes
Analytic conductor $64.224$
Analytic rank $1$
Dimension $40$
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] = [8043,2,Mod(1,8043)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8043, 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("8043.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8043 = 3 \cdot 7 \cdot 383 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8043.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: \(64.2236783457\)
Analytic rank: \(1\)
Dimension: \(40\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 40 q - 9 q^{2} + 40 q^{3} + 33 q^{4} - 27 q^{5} - 9 q^{6} + 40 q^{7} - 18 q^{8} + 40 q^{9} - 14 q^{10} - 29 q^{11} + 33 q^{12} - 40 q^{13} - 9 q^{14} - 27 q^{15} + 23 q^{16} - 46 q^{17} - 9 q^{18} + 17 q^{19} - 53 q^{20} + 40 q^{21} - 15 q^{22} - 62 q^{23} - 18 q^{24} + 35 q^{25} - 29 q^{26} + 40 q^{27} + 33 q^{28} - 47 q^{29} - 14 q^{30} + q^{31} - 45 q^{32} - 29 q^{33} + 9 q^{34} - 27 q^{35} + 33 q^{36} - 49 q^{37} - 52 q^{38} - 40 q^{39} - 12 q^{40} - 49 q^{41} - 9 q^{42} - 45 q^{43} - 68 q^{44} - 27 q^{45} - 36 q^{46} - 88 q^{47} + 23 q^{48} + 40 q^{49} - 32 q^{50} - 46 q^{51} - 34 q^{52} - 96 q^{53} - 9 q^{54} + 26 q^{55} - 18 q^{56} + 17 q^{57} + 12 q^{58} - 45 q^{59} - 53 q^{60} - 23 q^{61} - 15 q^{62} + 40 q^{63} + 50 q^{64} - 32 q^{65} - 15 q^{66} - 31 q^{67} - 76 q^{68} - 62 q^{69} - 14 q^{70} - 89 q^{71} - 18 q^{72} + 4 q^{73} + 10 q^{74} + 35 q^{75} + 51 q^{76} - 29 q^{77} - 29 q^{78} - 44 q^{79} - 53 q^{80} + 40 q^{81} - 15 q^{82} - 60 q^{83} + 33 q^{84} - 24 q^{85} - 9 q^{86} - 47 q^{87} - 64 q^{88} - 39 q^{89} - 14 q^{90} - 40 q^{91} - 80 q^{92} + q^{93} + 51 q^{94} - 71 q^{95} - 45 q^{96} - 21 q^{97} - 9 q^{98} - 29 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −2.75032 1.00000 5.56428 0.854023 −2.75032 1.00000 −9.80291 1.00000 −2.34884
1.2 −2.74138 1.00000 5.51515 −1.10050 −2.74138 1.00000 −9.63635 1.00000 3.01688
1.3 −2.64192 1.00000 4.97976 −3.51496 −2.64192 1.00000 −7.87228 1.00000 9.28626
1.4 −2.58104 1.00000 4.66178 2.25631 −2.58104 1.00000 −6.87016 1.00000 −5.82363
1.5 −2.32186 1.00000 3.39104 0.609544 −2.32186 1.00000 −3.22980 1.00000 −1.41528
1.6 −2.16260 1.00000 2.67684 −4.31110 −2.16260 1.00000 −1.46374 1.00000 9.32319
1.7 −2.08373 1.00000 2.34192 −2.93682 −2.08373 1.00000 −0.712460 1.00000 6.11952
1.8 −1.97104 1.00000 1.88501 1.28397 −1.97104 1.00000 0.226654 1.00000 −2.53076
1.9 −1.85383 1.00000 1.43669 −2.52734 −1.85383 1.00000 1.04428 1.00000 4.68526
1.10 −1.83203 1.00000 1.35635 1.36968 −1.83203 1.00000 1.17919 1.00000 −2.50930
1.11 −1.65595 1.00000 0.742162 −4.41540 −1.65595 1.00000 2.08291 1.00000 7.31167
1.12 −1.59287 1.00000 0.537230 2.23699 −1.59287 1.00000 2.33000 1.00000 −3.56323
1.13 −1.43635 1.00000 0.0630943 3.31438 −1.43635 1.00000 2.78207 1.00000 −4.76060
1.14 −1.36925 1.00000 −0.125168 0.412992 −1.36925 1.00000 2.90988 1.00000 −0.565488
1.15 −1.12599 1.00000 −0.732150 2.85700 −1.12599 1.00000 3.07637 1.00000 −3.21695
1.16 −1.09091 1.00000 −0.809919 −1.12988 −1.09091 1.00000 3.06536 1.00000 1.23260
1.17 −0.765993 1.00000 −1.41325 −3.34403 −0.765993 1.00000 2.61453 1.00000 2.56151
1.18 −0.717010 1.00000 −1.48590 1.86418 −0.717010 1.00000 2.49942 1.00000 −1.33664
1.19 −0.619181 1.00000 −1.61661 −0.937229 −0.619181 1.00000 2.23934 1.00000 0.580315
1.20 −0.506041 1.00000 −1.74392 −0.0764488 −0.506041 1.00000 1.89458 1.00000 0.0386862
See all 40 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.40
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(-1\)
\(383\) \(1\)

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 8043.2.a.n 40
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8043.2.a.n 40 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8043))\):

\( T_{2}^{40} + 9 T_{2}^{39} - 16 T_{2}^{38} - 369 T_{2}^{37} - 392 T_{2}^{36} + 6561 T_{2}^{35} + 15388 T_{2}^{34} - 65043 T_{2}^{33} - 230012 T_{2}^{32} + 372678 T_{2}^{31} + 2050662 T_{2}^{30} - 967682 T_{2}^{29} + \cdots - 112 \) Copy content Toggle raw display
\( T_{5}^{40} + 27 T_{5}^{39} + 247 T_{5}^{38} + 245 T_{5}^{37} - 10748 T_{5}^{36} - 62279 T_{5}^{35} + 71897 T_{5}^{34} + 1754561 T_{5}^{33} + 3575503 T_{5}^{32} - 20685092 T_{5}^{31} - 95797377 T_{5}^{30} + \cdots + 507904 \) Copy content Toggle raw display
\( T_{11}^{40} + 29 T_{11}^{39} + 198 T_{11}^{38} - 2150 T_{11}^{37} - 34978 T_{11}^{36} - 36899 T_{11}^{35} + 1900314 T_{11}^{34} + 9394822 T_{11}^{33} - 42601700 T_{11}^{32} - 443917469 T_{11}^{31} + \cdots + 221970893321008 \) Copy content Toggle raw display