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Properties

Label 4024.2.a.f
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.

\(\operatorname{Tr}(f)(q) = \) \( 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9} + 22 q^{11} + 25 q^{13} - 4 q^{15} + 17 q^{17} + 6 q^{19} + 18 q^{21} + 16 q^{23} + 47 q^{25} - 20 q^{27} + 47 q^{29} - 7 q^{31} - 6 q^{33} + 19 q^{35} + 75 q^{37} + 21 q^{39} + 22 q^{41} - 5 q^{43} + 33 q^{45} + 10 q^{47} + 31 q^{49} + 9 q^{51} + 64 q^{53} - 3 q^{55} + 5 q^{57} + 28 q^{59} + 49 q^{61} - 10 q^{63} + 46 q^{65} - 14 q^{67} + 30 q^{69} + 35 q^{71} + 19 q^{73} - 33 q^{75} + 32 q^{77} - 12 q^{79} + 57 q^{81} + 82 q^{85} - 5 q^{87} + 42 q^{89} - 15 q^{91} + 55 q^{93} + 33 q^{95} + 4 q^{97} + 22 q^{99}+O(q^{100}) \) 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 −3.25521 0 −2.40628 0 −4.10646 0 7.59642 0
1.2 0 −3.20822 0 2.09579 0 −1.61648 0 7.29270 0
1.3 0 −3.01200 0 3.98796 0 4.01800 0 6.07216 0
1.4 0 −2.91979 0 2.60605 0 −3.20806 0 5.52520 0
1.5 0 −2.88194 0 −1.00235 0 −1.34540 0 5.30561 0
1.6 0 −2.48200 0 −3.41370 0 3.75697 0 3.16033 0
1.7 0 −2.27275 0 2.37900 0 1.75584 0 2.16537 0
1.8 0 −2.23369 0 −1.90057 0 0.0272535 0 1.98935 0
1.9 0 −2.05859 0 3.84987 0 −4.00015 0 1.23779 0
1.10 0 −2.05780 0 −1.22779 0 1.00021 0 1.23452 0
1.11 0 −1.25385 0 2.99234 0 2.30060 0 −1.42787 0
1.12 0 −1.19201 0 −0.00829431 0 3.59968 0 −1.57911 0
1.13 0 −0.710131 0 −2.92884 0 −1.90024 0 −2.49571 0
1.14 0 −0.502871 0 −1.85170 0 1.22131 0 −2.74712 0
1.15 0 −0.254882 0 3.76559 0 −2.29648 0 −2.93504 0
1.16 0 −0.224625 0 −2.01216 0 1.98253 0 −2.94954 0
1.17 0 0.0147975 0 −2.88853 0 −1.61248 0 −2.99978 0
1.18 0 0.176667 0 2.93614 0 −0.0214699 0 −2.96879 0
1.19 0 0.220191 0 1.20642 0 −3.77303 0 −2.95152 0
1.20 0 0.368053 0 2.18272 0 4.72026 0 −2.86454 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.f ✓ 33
4.b odd 2 1 8048.2.a.y 33
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4024.2.a.f ✓ 33 1.a even 1 1 trivial
8048.2.a.y 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4024))\):

\( T_{3}^{33} + 2 T_{3}^{32} - 69 T_{3}^{31} - 130 T_{3}^{30} + 2141 T_{3}^{29} + 3761 T_{3}^{28} + \cdots - 704 \) Copy content Toggle raw display
\( T_{5}^{33} - 12 T_{5}^{32} - 34 T_{5}^{31} + 915 T_{5}^{30} - 878 T_{5}^{29} - 30526 T_{5}^{28} + \cdots - 11206656 \) Copy content Toggle raw display
\( T_{7}^{33} - 4 T_{7}^{32} - 123 T_{7}^{31} + 513 T_{7}^{30} + 6578 T_{7}^{29} - 28866 T_{7}^{28} + \cdots + 56757896 \) Copy content Toggle raw display