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Properties

Label 4024.2.a.d
Level $4024$
Weight $2$
Character orbit 4024.a
Self dual yes
Analytic conductor $32.132$
Analytic rank $1$
Dimension $28$
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] = [4024,2,Mod(1,4024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4024, 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("4024.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4024 = 2^{3} \cdot 503 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4024.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: \(32.1318017734\)
Analytic rank: \(1\)
Dimension: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 2 q^{3} - 12 q^{5} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q + 2 q^{3} - 12 q^{5} + 18 q^{9} - 14 q^{11} - 31 q^{13} - 2 q^{15} - 9 q^{17} - 8 q^{19} - 28 q^{21} - 4 q^{23} + 22 q^{25} - 4 q^{27} - 47 q^{29} - 5 q^{31} - 26 q^{33} - 13 q^{35} - 67 q^{37} - 9 q^{39} - 28 q^{41} + 15 q^{43} - 57 q^{45} - 10 q^{47} + 20 q^{49} - 11 q^{51} - 58 q^{53} + 15 q^{55} - 31 q^{57} - 32 q^{59} - 55 q^{61} - 16 q^{63} - 44 q^{65} + 22 q^{67} - 44 q^{69} - 47 q^{71} - 5 q^{73} - 25 q^{75} - 50 q^{77} - 14 q^{79} - 28 q^{81} - 16 q^{83} - 78 q^{85} - 11 q^{87} - 20 q^{89} - 15 q^{91} - 83 q^{93} - 27 q^{95} - 8 q^{97} - 70 q^{99}+O(q^{100}) \) 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 0 −3.02746 0 −1.92479 0 4.18734 0 6.16549 0
1.2 0 −2.92644 0 0.124991 0 2.19824 0 5.56407 0
1.3 0 −2.80686 0 −3.33080 0 −2.85725 0 4.87847 0
1.4 0 −2.25613 0 2.34171 0 2.23682 0 2.09012 0
1.5 0 −2.22675 0 −3.90847 0 −0.127234 0 1.95840 0
1.6 0 −1.98770 0 0.688613 0 −1.63548 0 0.950962 0
1.7 0 −1.73728 0 2.58604 0 2.10260 0 0.0181326 0
1.8 0 −1.72455 0 0.586565 0 −2.46074 0 −0.0259209 0
1.9 0 −1.32661 0 −1.90684 0 −4.79236 0 −1.24010 0
1.10 0 −0.893789 0 −3.85908 0 3.77625 0 −2.20114 0
1.11 0 −0.667261 0 0.307121 0 4.25072 0 −2.55476 0
1.12 0 −0.596863 0 3.74289 0 1.10831 0 −2.64375 0
1.13 0 −0.447624 0 0.901345 0 −1.57888 0 −2.79963 0
1.14 0 −0.0730681 0 −4.10981 0 −1.52145 0 −2.99466 0
1.15 0 0.659121 0 0.997881 0 0.366743 0 −2.56556 0
1.16 0 0.682343 0 3.69511 0 −3.46637 0 −2.53441 0
1.17 0 0.799349 0 0.626001 0 −0.555593 0 −2.36104 0
1.18 0 0.823496 0 −2.12271 0 4.15456 0 −2.32185 0
1.19 0 1.49100 0 −1.09730 0 −0.799854 0 −0.776913 0
1.20 0 1.59308 0 2.75433 0 1.46360 0 −0.462108 0
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.28
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(503\) \(-1\)

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 4024.2.a.d ✓ 28
4.b odd 2 1 8048.2.a.v 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4024.2.a.d ✓ 28 1.a even 1 1 trivial
8048.2.a.v 28 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4024))\):

\( T_{3}^{28} - 2 T_{3}^{27} - 49 T_{3}^{26} + 98 T_{3}^{25} + 1045 T_{3}^{24} - 2099 T_{3}^{23} + \cdots + 5657 \) Copy content Toggle raw display
\( T_{5}^{28} + 12 T_{5}^{27} - 9 T_{5}^{26} - 625 T_{5}^{25} - 1416 T_{5}^{24} + 12843 T_{5}^{23} + \cdots - 212992 \) Copy content Toggle raw display
\( T_{7}^{28} - 108 T_{7}^{26} - 21 T_{7}^{25} + 5077 T_{7}^{24} + 2013 T_{7}^{23} - 136595 T_{7}^{22} + \cdots - 58309973 \) Copy content Toggle raw display