Properties

Label 8043.2.a.u
Level $8043$
Weight $2$
Character orbit 8043.a
Self dual yes
Analytic conductor $64.224$
Analytic rank $0$
Dimension $53$
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: \(0\)
Dimension: \(53\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 53 q + 11 q^{2} + 53 q^{3} + 63 q^{4} + 24 q^{5} + 11 q^{6} + 53 q^{7} + 30 q^{8} + 53 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 53 q + 11 q^{2} + 53 q^{3} + 63 q^{4} + 24 q^{5} + 11 q^{6} + 53 q^{7} + 30 q^{8} + 53 q^{9} + 2 q^{10} + 46 q^{11} + 63 q^{12} + 32 q^{13} + 11 q^{14} + 24 q^{15} + 67 q^{16} + 46 q^{17} + 11 q^{18} + 14 q^{19} + 53 q^{20} + 53 q^{21} + 13 q^{22} + 68 q^{23} + 30 q^{24} + 71 q^{25} + 11 q^{26} + 53 q^{27} + 63 q^{28} + 55 q^{29} + 2 q^{30} - 2 q^{31} + 51 q^{32} + 46 q^{33} - 7 q^{34} + 24 q^{35} + 63 q^{36} + 53 q^{37} + 16 q^{38} + 32 q^{39} - 20 q^{40} + 38 q^{41} + 11 q^{42} + 36 q^{43} + 70 q^{44} + 24 q^{45} + 4 q^{46} + 51 q^{47} + 67 q^{48} + 53 q^{49} + 32 q^{50} + 46 q^{51} + 10 q^{52} + 104 q^{53} + 11 q^{54} + 11 q^{55} + 30 q^{56} + 14 q^{57} + 4 q^{58} + 36 q^{59} + 53 q^{60} + 3 q^{61} + 25 q^{62} + 53 q^{63} + 82 q^{64} + 46 q^{65} + 13 q^{66} + 54 q^{67} + 88 q^{68} + 68 q^{69} + 2 q^{70} + 101 q^{71} + 30 q^{72} + q^{73} + 32 q^{74} + 71 q^{75} - 35 q^{76} + 46 q^{77} + 11 q^{78} + 14 q^{79} + 39 q^{80} + 53 q^{81} - 29 q^{82} + 38 q^{83} + 63 q^{84} + 16 q^{85} + 23 q^{86} + 55 q^{87} - 8 q^{88} + 52 q^{89} + 2 q^{90} + 32 q^{91} + 76 q^{92} - 2 q^{93} - 53 q^{94} + 46 q^{95} + 51 q^{96} - 3 q^{97} + 11 q^{98} + 46 q^{99}+O(q^{100}) \) 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 −2.66147 1.00000 5.08344 2.81521 −2.66147 1.00000 −8.20648 1.00000 −7.49261
1.2 −2.65959 1.00000 5.07341 1.50911 −2.65959 1.00000 −8.17400 1.00000 −4.01361
1.3 −2.63741 1.00000 4.95592 0.460886 −2.63741 1.00000 −7.79596 1.00000 −1.21554
1.4 −2.39422 1.00000 3.73230 −2.33465 −2.39422 1.00000 −4.14751 1.00000 5.58967
1.5 −2.29113 1.00000 3.24926 −2.33697 −2.29113 1.00000 −2.86221 1.00000 5.35430
1.6 −2.26724 1.00000 3.14038 4.07533 −2.26724 1.00000 −2.58552 1.00000 −9.23976
1.7 −2.15559 1.00000 2.64657 3.65015 −2.15559 1.00000 −1.39373 1.00000 −7.86823
1.8 −2.10439 1.00000 2.42847 1.35273 −2.10439 1.00000 −0.901667 1.00000 −2.84668
1.9 −2.03619 1.00000 2.14608 −0.995010 −2.03619 1.00000 −0.297449 1.00000 2.02603
1.10 −1.92293 1.00000 1.69768 1.88319 −1.92293 1.00000 0.581347 1.00000 −3.62124
1.11 −1.78404 1.00000 1.18281 −2.51827 −1.78404 1.00000 1.45789 1.00000 4.49270
1.12 −1.46877 1.00000 0.157298 −1.01808 −1.46877 1.00000 2.70651 1.00000 1.49533
1.13 −1.45637 1.00000 0.121016 −1.02950 −1.45637 1.00000 2.73650 1.00000 1.49933
1.14 −1.32463 1.00000 −0.245346 4.05110 −1.32463 1.00000 2.97426 1.00000 −5.36623
1.15 −1.31380 1.00000 −0.273938 −2.47121 −1.31380 1.00000 2.98749 1.00000 3.24666
1.16 −1.27919 1.00000 −0.363673 3.49939 −1.27919 1.00000 3.02359 1.00000 −4.47639
1.17 −0.969084 1.00000 −1.06088 −0.461084 −0.969084 1.00000 2.96625 1.00000 0.446829
1.18 −0.817713 1.00000 −1.33134 0.352247 −0.817713 1.00000 2.72409 1.00000 −0.288037
1.19 −0.805347 1.00000 −1.35142 3.29338 −0.805347 1.00000 2.69905 1.00000 −2.65231
1.20 −0.701767 1.00000 −1.50752 −0.804883 −0.701767 1.00000 2.46146 1.00000 0.564840
See all 53 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.53
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.u 53
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8043.2.a.u 53 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8043))\):

\( T_{2}^{53} - 11 T_{2}^{52} - 24 T_{2}^{51} + 683 T_{2}^{50} - 878 T_{2}^{49} - 18989 T_{2}^{48} + 54676 T_{2}^{47} + 306335 T_{2}^{46} - 1332519 T_{2}^{45} - 3033531 T_{2}^{44} + 20098274 T_{2}^{43} + 16130675 T_{2}^{42} + \cdots - 172032 \) Copy content Toggle raw display
\( T_{5}^{53} - 24 T_{5}^{52} + 120 T_{5}^{51} + 1622 T_{5}^{50} - 18843 T_{5}^{49} - 9119 T_{5}^{48} + 941340 T_{5}^{47} - 2721436 T_{5}^{46} - 23788076 T_{5}^{45} + 137432557 T_{5}^{44} + \cdots + 28597059876864 \) Copy content Toggle raw display
\( T_{11}^{53} - 46 T_{11}^{52} + 725 T_{11}^{51} - 1278 T_{11}^{50} - 96835 T_{11}^{49} + 1028345 T_{11}^{48} + 1499498 T_{11}^{47} - 90444216 T_{11}^{46} + 404561498 T_{11}^{45} + 3125575358 T_{11}^{44} + \cdots - 11\!\cdots\!64 \) Copy content Toggle raw display