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Properties

Label 8037.2.a.w
Level $8037$
Weight $2$
Character orbit 8037.a
Self dual yes
Analytic conductor $64.176$
Analytic rank $0$
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: \(0\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 34 q + 5 q^{2} + 31 q^{4} + 6 q^{5} + 15 q^{8} + 4 q^{10} + 6 q^{11} + 2 q^{13} + 12 q^{14} + 21 q^{16} + 4 q^{17} - 34 q^{19} + 20 q^{20} - 8 q^{22} + 26 q^{23} + 32 q^{25} + 29 q^{26} - 4 q^{28} + 14 q^{29} + 2 q^{31} + 35 q^{32} - 18 q^{34} + 50 q^{35} - 10 q^{37} - 5 q^{38} + 17 q^{40} + 18 q^{41} + 6 q^{43} + 6 q^{44} + 18 q^{46} + 34 q^{47} + 28 q^{49} + 41 q^{50} + 10 q^{52} + 40 q^{53} - 8 q^{55} + 76 q^{56} + 4 q^{58} + 62 q^{59} - 2 q^{61} + 50 q^{62} + 11 q^{64} + 32 q^{65} + 20 q^{67} + 28 q^{68} + 22 q^{70} + 52 q^{71} - 8 q^{73} + 10 q^{74} - 31 q^{76} + 36 q^{77} - 12 q^{79} + 92 q^{80} + 10 q^{82} + 82 q^{83} - 4 q^{85} + 40 q^{86} - 16 q^{88} + 58 q^{89} + 100 q^{92} + 5 q^{94} - 6 q^{95} - 6 q^{97} + 45 q^{98}+O(q^{100}) \) 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 −2.52382 0 4.36965 0.683629 0 −2.85264 −5.98057 0 −1.72536
1.2 −2.45233 0 4.01392 −0.994616 0 −2.94476 −4.93878 0 2.43912
1.3 −2.42394 0 3.87547 3.99749 0 −0.305320 −4.54602 0 −9.68967
1.4 −2.30668 0 3.32079 0.443697 0 2.32803 −3.04665 0 −1.02347
1.5 −2.07430 0 2.30273 −3.91523 0 −3.22805 −0.627946 0 8.12136
1.6 −1.77726 0 1.15866 0.968051 0 3.87038 1.49528 0 −1.72048
1.7 −1.71289 0 0.934006 1.20458 0 0.0246048 1.82594 0 −2.06332
1.8 −1.58115 0 0.500024 −1.67156 0 1.85517 2.37168 0 2.64299
1.9 −1.47414 0 0.173088 0.241796 0 −3.39435 2.69312 0 −0.356441
1.10 −1.33363 0 −0.221428 −2.42545 0 2.95075 2.96257 0 3.23466
1.11 −1.08584 0 −0.820961 3.80339 0 1.68411 3.06310 0 −4.12986
1.12 −0.711079 0 −1.49437 −2.24243 0 −1.96953 2.48477 0 1.59455
1.13 −0.531889 0 −1.71709 3.75672 0 −1.03505 1.97708 0 −1.99816
1.14 −0.487118 0 −1.76272 −1.61184 0 0.600609 1.83289 0 0.785156
1.15 −0.356944 0 −1.87259 2.18998 0 5.04345 1.38230 0 −0.781702
1.16 −0.247945 0 −1.93852 1.08814 0 −3.59478 0.976539 0 −0.269799
1.17 0.0583690 0 −1.99659 −3.92229 0 −2.69203 −0.233277 0 −0.228940
1.18 0.0899584 0 −1.99191 −1.20146 0 3.43586 −0.359105 0 −0.108081
1.19 0.598453 0 −1.64185 −0.731998 0 −0.147095 −2.17948 0 −0.438066
1.20 0.805016 0 −1.35195 1.59878 0 1.80711 −2.69837 0 1.28705
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.w yes 34
3.b odd 2 1 8037.2.a.v ✓ 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8037.2.a.v ✓ 34 3.b odd 2 1
8037.2.a.w yes 34 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8037))\):

\( T_{2}^{34} - 5 T_{2}^{33} - 37 T_{2}^{32} + 215 T_{2}^{31} + 571 T_{2}^{30} - 4141 T_{2}^{29} + \cdots + 108 \) Copy content Toggle raw display
\( T_{5}^{34} - 6 T_{5}^{33} - 83 T_{5}^{32} + 536 T_{5}^{31} + 2971 T_{5}^{30} - 21186 T_{5}^{29} + \cdots - 304931 \) Copy content Toggle raw display