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Properties

Label 8043.2.a.q
Level $8043$
Weight $2$
Character orbit 8043.a
Self dual yes
Analytic conductor $64.224$
Analytic rank $1$
Dimension $44$
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: \(44\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 44 q - 4 q^{2} + 44 q^{3} + 44 q^{4} - 16 q^{5} - 4 q^{6} - 44 q^{7} - 15 q^{8} + 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 44 q - 4 q^{2} + 44 q^{3} + 44 q^{4} - 16 q^{5} - 4 q^{6} - 44 q^{7} - 15 q^{8} + 44 q^{9} - 16 q^{10} - 2 q^{11} + 44 q^{12} - 34 q^{13} + 4 q^{14} - 16 q^{15} + 24 q^{16} - 4 q^{17} - 4 q^{18} - 22 q^{19} - 39 q^{20} - 44 q^{21} - 23 q^{22} - 56 q^{23} - 15 q^{24} + 32 q^{25} - 17 q^{26} + 44 q^{27} - 44 q^{28} - 33 q^{29} - 16 q^{30} - 32 q^{31} - 34 q^{32} - 2 q^{33} - 25 q^{34} + 16 q^{35} + 44 q^{36} - 47 q^{37} - 40 q^{38} - 34 q^{39} - 50 q^{40} + 2 q^{41} + 4 q^{42} - 12 q^{43} - 22 q^{44} - 16 q^{45} + 8 q^{46} - 27 q^{47} + 24 q^{48} + 44 q^{49} - 21 q^{50} - 4 q^{51} - 82 q^{52} - 114 q^{53} - 4 q^{54} - 29 q^{55} + 15 q^{56} - 22 q^{57} - 26 q^{58} - 40 q^{59} - 39 q^{60} - 47 q^{61} - 37 q^{62} - 44 q^{63} - 5 q^{64} - 20 q^{65} - 23 q^{66} - 14 q^{67} - 72 q^{68} - 56 q^{69} + 16 q^{70} - 65 q^{71} - 15 q^{72} - 21 q^{73} - 26 q^{74} + 32 q^{75} - 15 q^{76} + 2 q^{77} - 17 q^{78} + 6 q^{79} - 77 q^{80} + 44 q^{81} - 51 q^{82} - 30 q^{83} - 44 q^{84} - 26 q^{85} - 65 q^{86} - 33 q^{87} - 84 q^{88} - 32 q^{89} - 16 q^{90} + 34 q^{91} - 140 q^{92} - 32 q^{93} - 35 q^{94} - 50 q^{95} - 34 q^{96} - 83 q^{97} - 4 q^{98} - 2 q^{99}+O(q^{100}) \) 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 −2.76678 1.00000 5.65509 −0.763136 −2.76678 −1.00000 −10.1129 1.00000 2.11143
1.2 −2.62597 1.00000 4.89573 −3.97585 −2.62597 −1.00000 −7.60410 1.00000 10.4405
1.3 −2.55412 1.00000 4.52354 −2.76084 −2.55412 −1.00000 −6.44543 1.00000 7.05153
1.4 −2.54621 1.00000 4.48317 3.86614 −2.54621 −1.00000 −6.32268 1.00000 −9.84400
1.5 −2.45063 1.00000 4.00561 2.55159 −2.45063 −1.00000 −4.91502 1.00000 −6.25302
1.6 −2.33621 1.00000 3.45788 0.0158996 −2.33621 −1.00000 −3.40590 1.00000 −0.0371448
1.7 −2.16996 1.00000 2.70871 −0.0283937 −2.16996 −1.00000 −1.53788 1.00000 0.0616131
1.8 −2.14351 1.00000 2.59465 0.772092 −2.14351 −1.00000 −1.27463 1.00000 −1.65499
1.9 −2.03170 1.00000 2.12780 −1.52866 −2.03170 −1.00000 −0.259645 1.00000 3.10578
1.10 −2.01632 1.00000 2.06555 2.51738 −2.01632 −1.00000 −0.132166 1.00000 −5.07585
1.11 −1.73189 1.00000 0.999432 2.64661 −1.73189 −1.00000 1.73287 1.00000 −4.58362
1.12 −1.53916 1.00000 0.369005 0.549666 −1.53916 −1.00000 2.51036 1.00000 −0.846023
1.13 −1.50421 1.00000 0.262658 −3.49239 −1.50421 −1.00000 2.61333 1.00000 5.25330
1.14 −1.28577 1.00000 −0.346791 −1.72364 −1.28577 −1.00000 3.01744 1.00000 2.21621
1.15 −1.26807 1.00000 −0.392010 −2.94433 −1.26807 −1.00000 3.03323 1.00000 3.73361
1.16 −1.20214 1.00000 −0.554856 −0.710682 −1.20214 −1.00000 3.07130 1.00000 0.854341
1.17 −1.07517 1.00000 −0.844015 −3.93529 −1.07517 −1.00000 3.05779 1.00000 4.23110
1.18 −0.810573 1.00000 −1.34297 0.143356 −0.810573 −1.00000 2.70972 1.00000 −0.116200
1.19 −0.630759 1.00000 −1.60214 2.80904 −0.630759 −1.00000 2.27208 1.00000 −1.77183
1.20 −0.518887 1.00000 −1.73076 −2.15979 −0.518887 −1.00000 1.93584 1.00000 1.12069
See all 44 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.44
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.q ✓ 44
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8043.2.a.q ✓ 44 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8043))\):

\( T_{2}^{44} + 4 T_{2}^{43} - 58 T_{2}^{42} - 243 T_{2}^{41} + 1542 T_{2}^{40} + 6824 T_{2}^{39} + \cdots - 59976 \) Copy content Toggle raw display
\( T_{5}^{44} + 16 T_{5}^{43} + 2 T_{5}^{42} - 1256 T_{5}^{41} - 4923 T_{5}^{40} + 39337 T_{5}^{39} + \cdots - 675216 \) Copy content Toggle raw display
\( T_{11}^{44} + 2 T_{11}^{43} - 247 T_{11}^{42} - 496 T_{11}^{41} + 27792 T_{11}^{40} + \cdots + 65778230676096 \) Copy content Toggle raw display