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Properties

Label 4024.2.a.g
Level $4024$
Weight $2$
Character orbit 4024.a
Self dual yes
Analytic conductor $32.132$
Analytic rank $0$
Dimension $33$
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: \(0\)
Dimension: \(33\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) \) 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 −3.28143 0 −3.08568 0 0.442605 0 7.76776 0
1.2 0 −2.94941 0 3.18881 0 1.60634 0 5.69904 0
1.3 0 −2.71136 0 −1.51865 0 −2.20669 0 4.35148 0
1.4 0 −2.52320 0 −3.79071 0 1.89288 0 3.36655 0
1.5 0 −2.39336 0 0.556589 0 4.72926 0 2.72817 0
1.6 0 −2.15460 0 −0.0166844 0 −4.51155 0 1.64231 0
1.7 0 −2.14143 0 2.09880 0 3.79183 0 1.58570 0
1.8 0 −2.13657 0 −1.27952 0 1.02417 0 1.56495 0
1.9 0 −1.65797 0 −0.641790 0 −1.04800 0 −0.251130 0
1.10 0 −1.02127 0 0.309055 0 −3.45255 0 −1.95700 0
1.11 0 −0.950380 0 3.01565 0 0.279316 0 −2.09678 0
1.12 0 −0.741660 0 −2.54437 0 −0.0722362 0 −2.44994 0
1.13 0 −0.648270 0 2.57860 0 −1.91468 0 −2.57975 0
1.14 0 −0.105343 0 −1.39521 0 −1.93153 0 −2.98890 0
1.15 0 0.221515 0 4.40155 0 5.16895 0 −2.95093 0
1.16 0 0.281703 0 −0.580455 0 2.82149 0 −2.92064 0
1.17 0 0.410265 0 −4.21207 0 3.36092 0 −2.83168 0
1.18 0 0.649284 0 −2.92377 0 −1.90965 0 −2.57843 0
1.19 0 0.885832 0 −0.282055 0 −2.95093 0 −2.21530 0
1.20 0 0.940347 0 2.53897 0 0.182218 0 −2.11575 0
See all 33 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.33
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.g ✓ 33
4.b odd 2 1 8048.2.a.x 33
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4024.2.a.g ✓ 33 1.a even 1 1 trivial
8048.2.a.x 33 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4024))\):

\( T_{3}^{33} - 10 T_{3}^{32} - 23 T_{3}^{31} + 532 T_{3}^{30} - 529 T_{3}^{29} - 12061 T_{3}^{28} + \cdots - 116224 \) Copy content Toggle raw display
\( T_{5}^{33} - 106 T_{5}^{31} - 5 T_{5}^{30} + 5010 T_{5}^{29} + 446 T_{5}^{28} - 139884 T_{5}^{27} + \cdots - 9535488 \) Copy content Toggle raw display
\( T_{7}^{33} - 12 T_{7}^{32} - 71 T_{7}^{31} + 1361 T_{7}^{30} + 486 T_{7}^{29} - 66228 T_{7}^{28} + \cdots - 1527027904 \) Copy content Toggle raw display