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Properties

Label 8037.2.a.x
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.

\(\operatorname{Tr}(f)(q) = \) \( 34 q + 5 q^{2} + 31 q^{4} + 14 q^{5} + 15 q^{8}+O(q^{10}) \) 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 −2.52876 0 4.39463 0.530546 0 2.49663 −6.05546 0 −1.34162
1.2 −2.51321 0 4.31621 0.193757 0 −2.83563 −5.82111 0 −0.486951
1.3 −2.30786 0 3.32621 3.90957 0 4.59532 −3.06071 0 −9.02274
1.4 −2.03967 0 2.16026 −0.819346 0 −1.85619 −0.326868 0 1.67120
1.5 −2.03839 0 2.15502 −0.454609 0 −0.471792 −0.315984 0 0.926669
1.6 −2.01001 0 2.04014 −2.87380 0 −2.42125 −0.0806874 0 5.77637
1.7 −2.00731 0 2.02928 3.98339 0 1.86470 −0.0587640 0 −7.99588
1.8 −1.77713 0 1.15820 0.780777 0 1.17580 1.49598 0 −1.38754
1.9 −1.26443 0 −0.401218 −2.22199 0 −5.27042 3.03617 0 2.80954
1.10 −1.13866 0 −0.703460 3.85215 0 −1.19429 3.07831 0 −4.38628
1.11 −1.05033 0 −0.896801 1.73879 0 −3.42874 3.04261 0 −1.82631
1.12 −0.845487 0 −1.28515 0.187502 0 1.30459 2.77755 0 −0.158530
1.13 −0.594197 0 −1.64693 3.23926 0 4.13510 2.16699 0 −1.92476
1.14 −0.437177 0 −1.80888 −4.32371 0 2.11390 1.66515 0 1.89023
1.15 −0.292375 0 −1.91452 −2.57586 0 −0.135286 1.14451 0 0.753118
1.16 −0.0794800 0 −1.99368 2.46391 0 −4.68518 0.317418 0 −0.195831
1.17 0.146324 0 −1.97859 −1.83482 0 −1.25452 −0.582164 0 −0.268479
1.18 0.229930 0 −1.94713 1.59514 0 3.20891 −0.907563 0 0.366771
1.19 0.281819 0 −1.92058 −0.192097 0 −2.48203 −1.10490 0 −0.0541367
1.20 0.625588 0 −1.60864 3.29167 0 −1.24186 −2.25752 0 2.05923
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.x yes 34
3.b odd 2 1 8037.2.a.u ✓ 34
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8037.2.a.u ✓ 34 3.b odd 2 1
8037.2.a.x 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8037))\):

\( T_{2}^{34} - 5 T_{2}^{33} - 37 T_{2}^{32} + 215 T_{2}^{31} + 571 T_{2}^{30} - 4141 T_{2}^{29} + \cdots + 76 \) 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