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Properties

Label 8037.2.a.t
Level $8037$
Weight $2$
Character orbit 8037.a
Self dual yes
Analytic conductor $64.176$
Analytic rank $0$
Dimension $24$
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: \(24\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} + 32 q^{4} + 2 q^{5} + 6 q^{7} - 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 2 q^{2} + 32 q^{4} + 2 q^{5} + 6 q^{7} - 9 q^{8} + 7 q^{10} + 3 q^{11} + 11 q^{13} - 9 q^{14} + 40 q^{16} + 6 q^{17} + 24 q^{19} + 17 q^{20} + 15 q^{22} + 19 q^{23} + 54 q^{25} + q^{26} + 26 q^{28} - 32 q^{29} + 12 q^{31} - 30 q^{32} + 38 q^{34} + 35 q^{35} + 14 q^{37} - 2 q^{38} + 47 q^{40} - 30 q^{41} + 10 q^{43} - q^{44} - 3 q^{46} + 24 q^{47} + 56 q^{49} - 44 q^{50} - 10 q^{53} + 24 q^{55} + 6 q^{56} + 41 q^{58} - 11 q^{59} + 10 q^{61} - 2 q^{62} + 61 q^{64} - 35 q^{65} + 12 q^{67} + 22 q^{68} - 20 q^{70} - 44 q^{71} + 33 q^{73} - 36 q^{74} + 32 q^{76} + 23 q^{77} + 50 q^{79} + 13 q^{80} + 5 q^{82} + 72 q^{83} + 33 q^{85} - 44 q^{86} + 38 q^{88} - 48 q^{89} + 47 q^{91} + 29 q^{92} - 2 q^{94} + 2 q^{95} + 50 q^{97} - 22 q^{98}+O(q^{100}) \) 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 −2.78191 0 5.73903 −3.90653 0 −2.42186 −10.4016 0 10.8676
1.2 −2.74422 0 5.53073 4.04567 0 4.57970 −9.68911 0 −11.1022
1.3 −2.51690 0 4.33477 0.426695 0 0.519010 −5.87637 0 −1.07395
1.4 −2.31568 0 3.36235 −3.30552 0 0.545822 −3.15477 0 7.65452
1.5 −2.14138 0 2.58551 −1.17754 0 4.42929 −1.25380 0 2.52155
1.6 −2.07004 0 2.28507 2.79648 0 −4.09198 −0.590104 0 −5.78884
1.7 −1.40720 0 −0.0197875 4.17114 0 −0.442129 2.84225 0 −5.86964
1.8 −1.32507 0 −0.244181 1.47425 0 5.14619 2.97370 0 −1.95349
1.9 −1.20851 0 −0.539511 −1.29566 0 −3.90722 3.06902 0 1.56582
1.10 −1.17517 0 −0.618977 0.0159136 0 −0.631968 3.07774 0 −0.0187011
1.11 −0.816976 0 −1.33255 −3.91014 0 0.225930 2.72261 0 3.19449
1.12 −0.00508776 0 −1.99997 2.83570 0 3.05778 0.0203509 0 −0.0144274
1.13 0.277424 0 −1.92304 −3.91933 0 1.78457 −1.08834 0 −1.08732
1.14 0.382471 0 −1.85372 −0.0305530 0 2.60660 −1.47393 0 −0.0116856
1.15 0.533191 0 −1.71571 2.88214 0 −1.29872 −1.98118 0 1.53673
1.16 0.773241 0 −1.40210 −0.230990 0 −4.37125 −2.63064 0 −0.178611
1.17 1.37653 0 −0.105163 −2.90547 0 −3.22368 −2.89782 0 −3.99947
1.18 1.63192 0 0.663147 2.08410 0 −1.57374 −2.18163 0 3.40108
1.19 1.66864 0 0.784364 −2.66351 0 3.88200 −2.02846 0 −4.44445
1.20 1.93681 0 1.75122 −1.58674 0 −4.84388 −0.481832 0 −3.07321
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.24
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.t 24
3.b odd 2 1 2679.2.a.o ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2679.2.a.o ✓ 24 3.b odd 2 1
8037.2.a.t 24 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8037))\):

\( T_{2}^{24} + 2 T_{2}^{23} - 38 T_{2}^{22} - 73 T_{2}^{21} + 626 T_{2}^{20} + 1148 T_{2}^{19} - 5867 T_{2}^{18} + \cdots + 26 \) Copy content Toggle raw display
\( T_{5}^{24} - 2 T_{5}^{23} - 85 T_{5}^{22} + 154 T_{5}^{21} + 3101 T_{5}^{20} - 5015 T_{5}^{19} + \cdots + 3178 \) Copy content Toggle raw display