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Properties

Label 4022.2.a.e
Level $4022$
Weight $2$
Character orbit 4022.a
Self dual yes
Analytic conductor $32.116$
Analytic rank $0$
Dimension $46$
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] = [4022,2,Mod(1,4022)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4022, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4022.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4022 = 2 \cdot 2011 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4022.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: \(32.1158316930\)
Analytic rank: \(0\)
Dimension: \(46\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 46 q - 46 q^{2} + 8 q^{3} + 46 q^{4} + 14 q^{5} - 8 q^{6} + 28 q^{7} - 46 q^{8} + 58 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 46 q - 46 q^{2} + 8 q^{3} + 46 q^{4} + 14 q^{5} - 8 q^{6} + 28 q^{7} - 46 q^{8} + 58 q^{9} - 14 q^{10} - 6 q^{11} + 8 q^{12} + 37 q^{13} - 28 q^{14} + 9 q^{15} + 46 q^{16} + 6 q^{17} - 58 q^{18} + 18 q^{19} + 14 q^{20} + 19 q^{21} + 6 q^{22} - 4 q^{23} - 8 q^{24} + 86 q^{25} - 37 q^{26} + 32 q^{27} + 28 q^{28} + 15 q^{29} - 9 q^{30} + 18 q^{31} - 46 q^{32} + 37 q^{33} - 6 q^{34} - 2 q^{35} + 58 q^{36} + 74 q^{37} - 18 q^{38} - 3 q^{39} - 14 q^{40} - 18 q^{41} - 19 q^{42} + 25 q^{43} - 6 q^{44} + 94 q^{45} + 4 q^{46} + 18 q^{47} + 8 q^{48} + 92 q^{49} - 86 q^{50} - 10 q^{51} + 37 q^{52} + 17 q^{53} - 32 q^{54} + 37 q^{55} - 28 q^{56} + 43 q^{57} - 15 q^{58} - 24 q^{59} + 9 q^{60} + 46 q^{61} - 18 q^{62} + 80 q^{63} + 46 q^{64} + 24 q^{65} - 37 q^{66} + 61 q^{67} + 6 q^{68} + 59 q^{69} + 2 q^{70} - 8 q^{71} - 58 q^{72} + 101 q^{73} - 74 q^{74} + 34 q^{75} + 18 q^{76} + 40 q^{77} + 3 q^{78} + 9 q^{79} + 14 q^{80} + 58 q^{81} + 18 q^{82} + 18 q^{83} + 19 q^{84} + 60 q^{85} - 25 q^{86} + 20 q^{87} + 6 q^{88} - 25 q^{89} - 94 q^{90} + 51 q^{91} - 4 q^{92} + 63 q^{93} - 18 q^{94} - 31 q^{95} - 8 q^{96} + 76 q^{97} - 92 q^{98} + 21 q^{99}+O(q^{100}) \) 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 −1.00000 −3.17262 1.00000 4.25415 3.17262 4.60982 −1.00000 7.06553 −4.25415
1.2 −1.00000 −3.08058 1.00000 −0.00834233 3.08058 −0.451365 −1.00000 6.48998 0.00834233
1.3 −1.00000 −2.98506 1.00000 2.19729 2.98506 −1.49134 −1.00000 5.91056 −2.19729
1.4 −1.00000 −2.89591 1.00000 −0.0804536 2.89591 2.75347 −1.00000 5.38632 0.0804536
1.5 −1.00000 −2.71162 1.00000 −3.39235 2.71162 −1.76888 −1.00000 4.35289 3.39235
1.6 −1.00000 −2.56761 1.00000 4.10115 2.56761 −1.16645 −1.00000 3.59263 −4.10115
1.7 −1.00000 −2.56517 1.00000 1.93763 2.56517 −4.25391 −1.00000 3.58009 −1.93763
1.8 −1.00000 −2.21023 1.00000 −3.23048 2.21023 2.26977 −1.00000 1.88514 3.23048
1.9 −1.00000 −2.06796 1.00000 2.09562 2.06796 5.11749 −1.00000 1.27644 −2.09562
1.10 −1.00000 −2.03510 1.00000 −3.10204 2.03510 −0.000158603 0 −1.00000 1.14162 3.10204
1.11 −1.00000 −1.80060 1.00000 −1.50418 1.80060 4.15262 −1.00000 0.242152 1.50418
1.12 −1.00000 −1.79779 1.00000 3.03526 1.79779 −2.51987 −1.00000 0.232040 −3.03526
1.13 −1.00000 −1.60420 1.00000 −3.64067 1.60420 4.45692 −1.00000 −0.426530 3.64067
1.14 −1.00000 −1.26909 1.00000 1.05177 1.26909 0.0899900 −1.00000 −1.38940 −1.05177
1.15 −1.00000 −1.02851 1.00000 −1.32303 1.02851 −0.0172071 −1.00000 −1.94217 1.32303
1.16 −1.00000 −0.961528 1.00000 0.960632 0.961528 −1.91758 −1.00000 −2.07546 −0.960632
1.17 −1.00000 −0.923967 1.00000 −1.00517 0.923967 −1.65407 −1.00000 −2.14628 1.00517
1.18 −1.00000 −0.848137 1.00000 2.52415 0.848137 4.28843 −1.00000 −2.28066 −2.52415
1.19 −1.00000 −0.498264 1.00000 0.121384 0.498264 −4.42409 −1.00000 −2.75173 −0.121384
1.20 −1.00000 −0.469086 1.00000 0.405337 0.469086 2.00530 −1.00000 −2.77996 −0.405337
See all 46 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.46
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(2011\) \(-1\)

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 4022.2.a.e ✓ 46
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4022.2.a.e ✓ 46 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{46} - 8 T_{3}^{45} - 66 T_{3}^{44} + 672 T_{3}^{43} + 1698 T_{3}^{42} - 25918 T_{3}^{41} + \cdots + 1936765 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4022))\). Copy content Toggle raw display