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Properties

Label 8043.2.a.o
Level $8043$
Weight $2$
Character orbit 8043.a
Self dual yes
Analytic conductor $64.224$
Analytic rank $1$
Dimension $41$
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: \(41\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 41 q - 4 q^{2} - 41 q^{3} + 34 q^{4} + 5 q^{5} + 4 q^{6} + 41 q^{7} - 15 q^{8} + 41 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 41 q - 4 q^{2} - 41 q^{3} + 34 q^{4} + 5 q^{5} + 4 q^{6} + 41 q^{7} - 15 q^{8} + 41 q^{9} - 18 q^{10} - 17 q^{11} - 34 q^{12} - 38 q^{13} - 4 q^{14} - 5 q^{15} + 16 q^{16} + 4 q^{17} - 4 q^{18} - 15 q^{19} + 23 q^{20} - 41 q^{21} - 31 q^{22} - 8 q^{23} + 15 q^{24} + 16 q^{25} + 15 q^{26} - 41 q^{27} + 34 q^{28} - 27 q^{29} + 18 q^{30} - 23 q^{31} - 24 q^{32} + 17 q^{33} - 9 q^{34} + 5 q^{35} + 34 q^{36} - 81 q^{37} + 38 q^{39} - 52 q^{40} + 29 q^{41} + 4 q^{42} - 47 q^{43} - 20 q^{44} + 5 q^{45} - 40 q^{46} + 26 q^{47} - 16 q^{48} + 41 q^{49} - 23 q^{50} - 4 q^{51} - 64 q^{52} - 66 q^{53} + 4 q^{54} - 14 q^{55} - 15 q^{56} + 15 q^{57} - 50 q^{58} + 41 q^{59} - 23 q^{60} - 47 q^{61} + 5 q^{62} + 41 q^{63} - 37 q^{64} - 24 q^{65} + 31 q^{66} - 73 q^{67} + 10 q^{68} + 8 q^{69} - 18 q^{70} + q^{71} - 15 q^{72} - 14 q^{73} + 18 q^{74} - 16 q^{75} - 37 q^{76} - 17 q^{77} - 15 q^{78} - 80 q^{79} + 63 q^{80} + 41 q^{81} - 31 q^{82} + 20 q^{83} - 34 q^{84} - 82 q^{85} - 39 q^{86} + 27 q^{87} - 132 q^{88} + 17 q^{89} - 18 q^{90} - 38 q^{91} - 26 q^{92} + 23 q^{93} - 9 q^{94} - q^{95} + 24 q^{96} - 73 q^{97} - 4 q^{98} - 17 q^{99}+O(q^{100}) \) 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 −2.77034 −1.00000 5.67480 3.96125 2.77034 1.00000 −10.1805 1.00000 −10.9740
1.2 −2.55146 −1.00000 4.50996 0.568399 2.55146 1.00000 −6.40408 1.00000 −1.45025
1.3 −2.48498 −1.00000 4.17514 −2.72732 2.48498 1.00000 −5.40520 1.00000 6.77735
1.4 −2.47042 −1.00000 4.10295 −0.537322 2.47042 1.00000 −5.19517 1.00000 1.32741
1.5 −2.38443 −1.00000 3.68550 2.70331 2.38443 1.00000 −4.01897 1.00000 −6.44585
1.6 −2.34884 −1.00000 3.51704 1.78966 2.34884 1.00000 −3.56329 1.00000 −4.20363
1.7 −2.07017 −1.00000 2.28561 2.60848 2.07017 1.00000 −0.591253 1.00000 −5.40001
1.8 −1.93313 −1.00000 1.73699 −1.84501 1.93313 1.00000 0.508439 1.00000 3.56663
1.9 −1.86200 −1.00000 1.46705 0.268615 1.86200 1.00000 0.992360 1.00000 −0.500162
1.10 −1.81957 −1.00000 1.31084 0.0545380 1.81957 1.00000 1.25397 1.00000 −0.0992358
1.11 −1.72607 −1.00000 0.979322 −3.72360 1.72607 1.00000 1.76176 1.00000 6.42720
1.12 −1.36181 −1.00000 −0.145472 2.48570 1.36181 1.00000 2.92173 1.00000 −3.38505
1.13 −1.34769 −1.00000 −0.183736 1.68166 1.34769 1.00000 2.94300 1.00000 −2.26635
1.14 −1.32430 −1.00000 −0.246218 −1.21552 1.32430 1.00000 2.97468 1.00000 1.60972
1.15 −0.980315 −1.00000 −1.03898 −1.72725 0.980315 1.00000 2.97916 1.00000 1.69325
1.16 −0.973977 −1.00000 −1.05137 4.11702 0.973977 1.00000 2.97196 1.00000 −4.00988
1.17 −0.605577 −1.00000 −1.63328 2.30178 0.605577 1.00000 2.20023 1.00000 −1.39391
1.18 −0.555636 −1.00000 −1.69127 −1.46250 0.555636 1.00000 2.05100 1.00000 0.812618
1.19 −0.426737 −1.00000 −1.81790 −3.28382 0.426737 1.00000 1.62924 1.00000 1.40133
1.20 −0.386440 −1.00000 −1.85066 −2.37424 0.386440 1.00000 1.48805 1.00000 0.917500
See all 41 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.41
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.o ✓ 41
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8043.2.a.o ✓ 41 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8043))\):

\( T_{2}^{41} + 4 T_{2}^{40} - 50 T_{2}^{39} - 211 T_{2}^{38} + 1133 T_{2}^{37} + 5098 T_{2}^{36} + \cdots - 464 \) Copy content Toggle raw display
\( T_{5}^{41} - 5 T_{5}^{40} - 98 T_{5}^{39} + 512 T_{5}^{38} + 4263 T_{5}^{37} - 23488 T_{5}^{36} + \cdots - 52960 \) Copy content Toggle raw display
\( T_{11}^{41} + 17 T_{11}^{40} - 69 T_{11}^{39} - 2675 T_{11}^{38} - 4572 T_{11}^{37} + \cdots + 6008010630760 \) Copy content Toggle raw display