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Properties

Label 8043.2.a.s
Level $8043$
Weight $2$
Character orbit 8043.a
Self dual yes
Analytic conductor $64.224$
Analytic rank $0$
Dimension $50$
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: \(50\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 50 q - q^{2} - 50 q^{3} + 53 q^{4} + 11 q^{5} + q^{6} - 50 q^{7} - 6 q^{8} + 50 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 50 q - q^{2} - 50 q^{3} + 53 q^{4} + 11 q^{5} + q^{6} - 50 q^{7} - 6 q^{8} + 50 q^{9} + 16 q^{10} - 31 q^{11} - 53 q^{12} + 42 q^{13} + q^{14} - 11 q^{15} + 59 q^{16} + 44 q^{17} - q^{18} + 11 q^{19} + 7 q^{20} + 50 q^{21} + 19 q^{22} - 16 q^{23} + 6 q^{24} + 71 q^{25} + q^{26} - 50 q^{27} - 53 q^{28} + 3 q^{29} - 16 q^{30} + 13 q^{31} - 23 q^{32} + 31 q^{33} + q^{34} - 11 q^{35} + 53 q^{36} + 53 q^{37} + 28 q^{38} - 42 q^{39} + 50 q^{40} + 23 q^{41} - q^{42} + 9 q^{43} - 78 q^{44} + 11 q^{45} - 8 q^{46} + 26 q^{47} - 59 q^{48} + 50 q^{49} - 38 q^{50} - 44 q^{51} + 86 q^{52} + 58 q^{53} + q^{54} + 28 q^{55} + 6 q^{56} - 11 q^{57} - 4 q^{58} + 7 q^{59} - 7 q^{60} + 51 q^{61} + 7 q^{62} - 50 q^{63} + 74 q^{64} - 14 q^{65} - 19 q^{66} + 23 q^{67} + 98 q^{68} + 16 q^{69} - 16 q^{70} - 75 q^{71} - 6 q^{72} + 34 q^{73} - 68 q^{74} - 71 q^{75} + 31 q^{76} + 31 q^{77} - q^{78} - 18 q^{79} - 21 q^{80} + 50 q^{81} + 31 q^{82} + 40 q^{83} + 53 q^{84} + 30 q^{85} - 15 q^{86} - 3 q^{87} + 70 q^{88} + 63 q^{89} + 16 q^{90} - 42 q^{91} - 38 q^{92} - 13 q^{93} + q^{94} - 77 q^{95} + 23 q^{96} + 77 q^{97} - q^{98} - 31 q^{99}+O(q^{100}) \) 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 −2.82109 −1.00000 5.95855 −4.23433 2.82109 −1.00000 −11.1674 1.00000 11.9454
1.2 −2.63224 −1.00000 4.92870 3.86633 2.63224 −1.00000 −7.70903 1.00000 −10.1771
1.3 −2.63191 −1.00000 4.92693 0.271396 2.63191 −1.00000 −7.70340 1.00000 −0.714288
1.4 −2.62429 −1.00000 4.88687 −1.77661 2.62429 −1.00000 −7.57598 1.00000 4.66233
1.5 −2.39697 −1.00000 3.74548 1.84555 2.39697 −1.00000 −4.18388 1.00000 −4.42374
1.6 −2.33599 −1.00000 3.45686 4.00725 2.33599 −1.00000 −3.40322 1.00000 −9.36091
1.7 −2.33165 −1.00000 3.43661 −1.59780 2.33165 −1.00000 −3.34969 1.00000 3.72551
1.8 −2.30033 −1.00000 3.29153 −3.56175 2.30033 −1.00000 −2.97095 1.00000 8.19322
1.9 −2.16078 −1.00000 2.66897 −0.263100 2.16078 −1.00000 −1.44551 1.00000 0.568501
1.10 −1.95593 −1.00000 1.82566 0.355087 1.95593 −1.00000 0.340990 1.00000 −0.694525
1.11 −1.84012 −1.00000 1.38605 −2.07033 1.84012 −1.00000 1.12975 1.00000 3.80966
1.12 −1.76145 −1.00000 1.10269 2.11729 1.76145 −1.00000 1.58056 1.00000 −3.72950
1.13 −1.46962 −1.00000 0.159775 3.40283 1.46962 −1.00000 2.70443 1.00000 −5.00085
1.14 −1.46567 −1.00000 0.148201 0.335135 1.46567 −1.00000 2.71413 1.00000 −0.491199
1.15 −1.19941 −1.00000 −0.561407 −0.150113 1.19941 −1.00000 3.07219 1.00000 0.180048
1.16 −1.18848 −1.00000 −0.587510 −4.24260 1.18848 −1.00000 3.07521 1.00000 5.04226
1.17 −1.18460 −1.00000 −0.596731 3.79711 1.18460 −1.00000 3.07608 1.00000 −4.49805
1.18 −1.15299 −1.00000 −0.670610 0.489993 1.15299 −1.00000 3.07919 1.00000 −0.564958
1.19 −1.12728 −1.00000 −0.729231 −0.145556 1.12728 −1.00000 3.07662 1.00000 0.164083
1.20 −0.719015 −1.00000 −1.48302 −1.74243 0.719015 −1.00000 2.50434 1.00000 1.25284
See all 50 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.50
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.s ✓ 50
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8043.2.a.s ✓ 50 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8043))\):

\( T_{2}^{50} + T_{2}^{49} - 76 T_{2}^{48} - 73 T_{2}^{47} + 2697 T_{2}^{46} + 2484 T_{2}^{45} + \cdots + 1024 \) Copy content Toggle raw display
\( T_{5}^{50} - 11 T_{5}^{49} - 100 T_{5}^{48} + 1484 T_{5}^{47} + 3497 T_{5}^{46} - 91590 T_{5}^{45} + \cdots - 1449124352 \) Copy content Toggle raw display
\( T_{11}^{50} + 31 T_{11}^{49} + 169 T_{11}^{48} - 4399 T_{11}^{47} - 57153 T_{11}^{46} + \cdots - 30\!\cdots\!68 \) Copy content Toggle raw display