Properties

Label 4024.2.a.e
Level $4024$
Weight $2$
Character orbit 4024.a
Self dual yes
Analytic conductor $32.132$
Analytic rank $1$
Dimension $29$
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: \(29\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 29 q - 7 q^{3} - 4 q^{5} - 13 q^{7} + 20 q^{9} - 27 q^{11} + 16 q^{13} - 14 q^{15} - 15 q^{17} - 14 q^{19} + q^{21} - 25 q^{23} + 21 q^{25} - 25 q^{27} - 13 q^{29} - 27 q^{31} - 9 q^{33} - 29 q^{35} + 35 q^{37} - 38 q^{39} - 30 q^{41} - 38 q^{43} + q^{45} - 35 q^{47} + 14 q^{49} - 21 q^{51} + 2 q^{53} - 25 q^{55} - 25 q^{57} - 40 q^{59} + 10 q^{61} - 56 q^{63} - 50 q^{65} - 31 q^{67} + 11 q^{69} - 65 q^{71} - 23 q^{73} - 32 q^{75} + 13 q^{77} - 44 q^{79} - 7 q^{81} - 41 q^{83} + 26 q^{85} - 25 q^{87} - 48 q^{89} - 44 q^{91} + 25 q^{93} - 75 q^{95} - 18 q^{97} - 80 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 −3.34526 0 −0.499712 0 1.77499 0 8.19075 0
1.2 0 −3.05929 0 0.773308 0 −3.93634 0 6.35923 0
1.3 0 −2.89251 0 3.16304 0 −0.0867661 0 5.36660 0
1.4 0 −2.70977 0 2.99505 0 −2.81033 0 4.34283 0
1.5 0 −2.46401 0 −4.08257 0 −4.62698 0 3.07134 0
1.6 0 −2.34298 0 −1.32410 0 0.439798 0 2.48955 0
1.7 0 −2.20984 0 −0.810761 0 4.21793 0 1.88340 0
1.8 0 −1.85667 0 2.88879 0 1.11301 0 0.447227 0
1.9 0 −1.79909 0 0.0213618 0 −3.47965 0 0.236734 0
1.10 0 −1.15550 0 −3.26045 0 −1.92982 0 −1.66482 0
1.11 0 −1.10456 0 −2.70098 0 3.17910 0 −1.77994 0
1.12 0 −0.795174 0 3.83818 0 0.297725 0 −2.36770 0
1.13 0 −0.782310 0 2.74195 0 −3.13445 0 −2.38799 0
1.14 0 −0.582422 0 −3.18976 0 −3.00819 0 −2.66078 0
1.15 0 −0.502218 0 −3.38571 0 3.39706 0 −2.74778 0
1.16 0 −0.278789 0 −0.533193 0 1.93823 0 −2.92228 0
1.17 0 −0.137144 0 1.57958 0 4.44075 0 −2.98119 0
1.18 0 0.454325 0 0.622756 0 −4.14589 0 −2.79359 0
1.19 0 0.936010 0 1.25557 0 0.290471 0 −2.12388 0
1.20 0 1.28849 0 −1.09309 0 0.799248 0 −1.33979 0
See all 29 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.29
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.e 29
4.b odd 2 1 8048.2.a.w 29
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4024.2.a.e 29 1.a even 1 1 trivial
8048.2.a.w 29 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4024))\):

\( T_{3}^{29} + 7 T_{3}^{28} - 29 T_{3}^{27} - 295 T_{3}^{26} + 203 T_{3}^{25} + 5370 T_{3}^{24} + 2583 T_{3}^{23} - 55568 T_{3}^{22} - 60915 T_{3}^{21} + 361198 T_{3}^{20} + 547243 T_{3}^{19} - 1532820 T_{3}^{18} - 2880795 T_{3}^{17} + \cdots + 6975 \) Copy content Toggle raw display
\( T_{5}^{29} + 4 T_{5}^{28} - 75 T_{5}^{27} - 297 T_{5}^{26} + 2432 T_{5}^{25} + 9505 T_{5}^{24} - 44755 T_{5}^{23} - 171998 T_{5}^{22} + 515671 T_{5}^{21} + 1941171 T_{5}^{20} - 3870424 T_{5}^{19} - 14229831 T_{5}^{18} + \cdots + 16384 \) Copy content Toggle raw display
\( T_{7}^{29} + 13 T_{7}^{28} - 24 T_{7}^{27} - 969 T_{7}^{26} - 1832 T_{7}^{25} + 29380 T_{7}^{24} + 104502 T_{7}^{23} - 465151 T_{7}^{22} - 2376115 T_{7}^{21} + 4047703 T_{7}^{20} + 30762437 T_{7}^{19} + \cdots - 64747 \) Copy content Toggle raw display