Properties

Label 8037.2.a.p
Level $8037$
Weight $2$
Character orbit 8037.a
Self dual yes
Analytic conductor $64.176$
Analytic rank $1$
Dimension $23$
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] = [8037,2,Mod(1,8037)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8037, 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("8037.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8037.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: \(64.1757681045\)
Analytic rank: \(1\)
Dimension: \(23\)
Twist minimal: no (minimal twist has level 2679)
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.

\(\operatorname{Tr}(f)(q) = \) \( 23 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 2 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 23 q - 5 q^{2} + 25 q^{4} - 12 q^{5} - 2 q^{7} - 12 q^{8} + q^{10} - 13 q^{11} + 5 q^{13} - 11 q^{14} + 33 q^{16} - 22 q^{17} + 23 q^{19} - 23 q^{20} + q^{22} - 23 q^{23} + 31 q^{25} - 21 q^{26} - 22 q^{28} - 24 q^{29} + 20 q^{31} - 19 q^{32} - 14 q^{34} - 15 q^{35} + 8 q^{37} - 5 q^{38} - 15 q^{40} - 28 q^{41} - 6 q^{43} - q^{44} + 29 q^{46} - 23 q^{47} + 39 q^{49} - 27 q^{50} + 12 q^{52} - 22 q^{53} - 28 q^{55} - 30 q^{56} - 5 q^{58} - 53 q^{59} + 4 q^{61} - 38 q^{62} + 30 q^{64} - 27 q^{65} + 6 q^{68} + 30 q^{70} - 54 q^{71} + 15 q^{73} + 12 q^{74} + 25 q^{76} - 33 q^{77} - 8 q^{79} - 45 q^{80} - 69 q^{82} - 20 q^{83} - 25 q^{85} - 50 q^{86} + 48 q^{88} - 26 q^{89} + 11 q^{91} - 81 q^{92} + 5 q^{94} - 12 q^{95} - 4 q^{97} + 55 q^{98}+O(q^{100}) \) 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 −2.70026 0 5.29141 −2.20508 0 0.381908 −8.88768 0 5.95430
1.2 −2.65726 0 5.06101 3.68922 0 −1.66641 −8.13390 0 −9.80320
1.3 −2.46332 0 4.06796 −1.49096 0 0.898858 −5.09404 0 3.67270
1.4 −2.43689 0 3.93845 −3.39444 0 −2.76712 −4.72380 0 8.27188
1.5 −2.28716 0 3.23110 0.184427 0 3.46017 −2.81571 0 −0.421813
1.6 −1.73912 0 1.02455 −3.12749 0 2.42300 1.69643 0 5.43909
1.7 −1.60510 0 0.576359 2.10169 0 1.01773 2.28509 0 −3.37344
1.8 −1.45453 0 0.115648 −0.945028 0 −4.13147 2.74084 0 1.37457
1.9 −1.07420 0 −0.846102 3.82265 0 −2.96171 3.05727 0 −4.10628
1.10 −0.803522 0 −1.35435 −2.59980 0 4.06920 2.69530 0 2.08899
1.11 −0.641477 0 −1.58851 −2.95137 0 1.16825 2.30195 0 1.89324
1.12 −0.553572 0 −1.69356 0.688386 0 3.33451 2.04465 0 −0.381072
1.13 0.176164 0 −1.96897 −4.18334 0 −4.24699 −0.699188 0 −0.736953
1.14 0.395557 0 −1.84353 0.554972 0 −0.565101 −1.52034 0 0.219523
1.15 0.539644 0 −1.70878 1.68281 0 −4.41391 −2.00142 0 0.908118
1.16 0.844930 0 −1.28609 2.16056 0 4.75097 −2.77652 0 1.82552
1.17 1.11388 0 −0.759262 −2.97800 0 1.49100 −3.07350 0 −3.31715
1.18 1.29424 0 −0.324956 2.49851 0 −3.56737 −3.00904 0 3.23365
1.19 1.60314 0 0.570052 −3.51993 0 3.73543 −2.29240 0 −5.64293
1.20 2.13087 0 2.54061 2.40433 0 0.918790 1.15198 0 5.12332
See all 23 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.23
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(19\) \(-1\)
\(47\) \(1\)

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 8037.2.a.p 23
3.b odd 2 1 2679.2.a.n 23
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2679.2.a.n 23 3.b odd 2 1
8037.2.a.p 23 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8037))\):

\( T_{2}^{23} + 5 T_{2}^{22} - 23 T_{2}^{21} - 146 T_{2}^{20} + 174 T_{2}^{19} + 1776 T_{2}^{18} - 135 T_{2}^{17} - 11712 T_{2}^{16} - 5445 T_{2}^{15} + 45722 T_{2}^{14} + 35387 T_{2}^{13} - 108725 T_{2}^{12} - 104907 T_{2}^{11} + \cdots - 276 \) Copy content Toggle raw display
\( T_{5}^{23} + 12 T_{5}^{22} - T_{5}^{21} - 544 T_{5}^{20} - 1571 T_{5}^{19} + 8829 T_{5}^{18} + 44230 T_{5}^{17} - 50758 T_{5}^{16} - 548297 T_{5}^{15} - 187286 T_{5}^{14} + 3667575 T_{5}^{13} + 4208544 T_{5}^{12} + \cdots + 137661 \) Copy content Toggle raw display