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Properties

Label 8037.2.a.u
Level $8037$
Weight $2$
Character orbit 8037.a
Self dual yes
Analytic conductor $64.176$
Analytic rank $1$
Dimension $34$
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: \(34\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 34 q - 5 q^{2} + 31 q^{4} - 14 q^{5} - 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 34 q - 5 q^{2} + 31 q^{4} - 14 q^{5} - 15 q^{8} - 18 q^{11} - 6 q^{13} - 12 q^{14} + 21 q^{16} - 36 q^{17} + 34 q^{19} - 40 q^{20} + 12 q^{22} - 38 q^{23} + 32 q^{25} - 15 q^{26} + 28 q^{28} - 14 q^{29} - 6 q^{31} - 35 q^{32} + 10 q^{34} - 46 q^{35} - 2 q^{37} - 5 q^{38} + 31 q^{40} - 18 q^{41} - 6 q^{43} - 42 q^{44} - 14 q^{46} + 34 q^{47} + 44 q^{49} - 9 q^{50} + 2 q^{52} - 32 q^{53} + 8 q^{55} + 4 q^{56} + 8 q^{58} - 62 q^{59} - 10 q^{61} - 30 q^{62} - 37 q^{64} - 8 q^{65} - 92 q^{68} - 62 q^{70} - 4 q^{71} - 8 q^{73} - 34 q^{74} + 31 q^{76} - 52 q^{77} + 40 q^{79} - 48 q^{80} - 2 q^{82} - 110 q^{83} - 12 q^{85} - 16 q^{86} - 44 q^{88} - 2 q^{89} - 28 q^{91} - 60 q^{92} - 5 q^{94} - 14 q^{95} + 2 q^{97} - 45 q^{98}+O(q^{100}) \) 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 −2.69099 0 5.24145 0.119657 0 4.57407 −8.72272 0 −0.321996
1.2 −2.63945 0 4.96672 −2.52997 0 1.04693 −7.83051 0 6.67775
1.3 −2.54574 0 4.48077 −2.04236 0 −5.22189 −6.31539 0 5.19930
1.4 −2.48310 0 4.16581 −4.02387 0 2.71666 −5.37793 0 9.99168
1.5 −2.29036 0 3.24574 1.89255 0 0.151615 −2.85320 0 −4.33463
1.6 −2.27733 0 3.18623 2.62055 0 −0.449363 −2.70143 0 −5.96786
1.7 −1.94888 0 1.79814 −3.28240 0 −0.265314 0.393400 0 6.39700
1.8 −1.90848 0 1.64229 1.10387 0 3.60723 0.682679 0 −2.10672
1.9 −1.80842 0 1.27038 2.80582 0 −4.37193 1.31945 0 −5.07411
1.10 −1.53643 0 0.360613 −3.08058 0 0.219575 2.51880 0 4.73310
1.11 −1.29963 0 −0.310950 −0.475094 0 −2.76700 3.00339 0 0.617448
1.12 −1.23022 0 −0.486568 −1.27126 0 5.00391 3.05902 0 1.56392
1.13 −1.01631 0 −0.967116 2.07833 0 2.14831 3.01551 0 −2.11223
1.14 −0.965460 0 −1.06789 2.55498 0 −0.0105762 2.96192 0 −2.46673
1.15 −0.625588 0 −1.60864 −3.29167 0 −1.24186 2.25752 0 2.05923
1.16 −0.281819 0 −1.92058 0.192097 0 −2.48203 1.10490 0 −0.0541367
1.17 −0.229930 0 −1.94713 −1.59514 0 3.20891 0.907563 0 0.366771
1.18 −0.146324 0 −1.97859 1.83482 0 −1.25452 0.582164 0 −0.268479
1.19 0.0794800 0 −1.99368 −2.46391 0 −4.68518 −0.317418 0 −0.195831
1.20 0.292375 0 −1.91452 2.57586 0 −0.135286 −1.14451 0 0.753118
See all 34 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.34
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.u ✓ 34
3.b odd 2 1 8037.2.a.x yes 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8037.2.a.u ✓ 34 1.a even 1 1 trivial
8037.2.a.x yes 34 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8037))\):

\( T_{2}^{34} + 5 T_{2}^{33} - 37 T_{2}^{32} - 215 T_{2}^{31} + 571 T_{2}^{30} + 4141 T_{2}^{29} + \cdots + 76 \) Copy content Toggle raw display
\( T_{5}^{34} + 14 T_{5}^{33} - 3 T_{5}^{32} - 884 T_{5}^{31} - 2729 T_{5}^{30} + 22410 T_{5}^{29} + \cdots - 761235 \) Copy content Toggle raw display