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Properties

Label 4022.2.a.c
Level $4022$
Weight $2$
Character orbit 4022.a
Self dual yes
Analytic conductor $32.116$
Analytic rank $1$
Dimension $31$
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: \(1\)
Dimension: \(31\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 31 q + 31 q^{2} - 14 q^{3} + 31 q^{4} - 13 q^{5} - 14 q^{6} - 29 q^{7} + 31 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 31 q + 31 q^{2} - 14 q^{3} + 31 q^{4} - 13 q^{5} - 14 q^{6} - 29 q^{7} + 31 q^{8} + 27 q^{9} - 13 q^{10} - 29 q^{11} - 14 q^{12} - 23 q^{13} - 29 q^{14} - 14 q^{15} + 31 q^{16} - 19 q^{17} + 27 q^{18} - 36 q^{19} - 13 q^{20} - 29 q^{22} - 24 q^{23} - 14 q^{24} + 4 q^{25} - 23 q^{26} - 65 q^{27} - 29 q^{28} + 18 q^{29} - 14 q^{30} - 41 q^{31} + 31 q^{32} - 18 q^{33} - 19 q^{34} - 28 q^{35} + 27 q^{36} - 31 q^{37} - 36 q^{38} - 31 q^{39} - 13 q^{40} - 51 q^{41} - 14 q^{43} - 29 q^{44} - 34 q^{45} - 24 q^{46} - 64 q^{47} - 14 q^{48} + 8 q^{49} + 4 q^{50} - 17 q^{51} - 23 q^{52} + 3 q^{53} - 65 q^{54} - 50 q^{55} - 29 q^{56} - 19 q^{57} + 18 q^{58} - 58 q^{59} - 14 q^{60} - 14 q^{61} - 41 q^{62} - 66 q^{63} + 31 q^{64} + 6 q^{65} - 18 q^{66} - 55 q^{67} - 19 q^{68} + q^{69} - 28 q^{70} - 42 q^{71} + 27 q^{72} - 83 q^{73} - 31 q^{74} - 49 q^{75} - 36 q^{76} + 18 q^{77} - 31 q^{78} - 56 q^{79} - 13 q^{80} + 3 q^{81} - 51 q^{82} - 43 q^{83} + 4 q^{85} - 14 q^{86} - 76 q^{87} - 29 q^{88} + 17 q^{89} - 34 q^{90} - 49 q^{91} - 24 q^{92} - 24 q^{93} - 64 q^{94} - 43 q^{95} - 14 q^{96} - 98 q^{97} + 8 q^{98} - q^{99}+O(q^{100}) \) 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 1.00000 −3.32689 1.00000 −3.79290 −3.32689 −3.92698 1.00000 8.06821 −3.79290
1.2 1.00000 −3.17886 1.00000 2.94122 −3.17886 −2.37061 1.00000 7.10517 2.94122
1.3 1.00000 −3.04963 1.00000 −0.453463 −3.04963 2.87005 1.00000 6.30022 −0.453463
1.4 1.00000 −2.91271 1.00000 3.14140 −2.91271 −3.38046 1.00000 5.48388 3.14140
1.5 1.00000 −2.85499 1.00000 1.11825 −2.85499 −3.01039 1.00000 5.15094 1.11825
1.6 1.00000 −2.62663 1.00000 −2.01799 −2.62663 0.869262 1.00000 3.89919 −2.01799
1.7 1.00000 −2.46221 1.00000 −2.36990 −2.46221 2.92726 1.00000 3.06248 −2.36990
1.8 1.00000 −2.43557 1.00000 0.846052 −2.43557 3.32195 1.00000 2.93199 0.846052
1.9 1.00000 −2.37527 1.00000 −1.97700 −2.37527 −4.20609 1.00000 2.64192 −1.97700
1.10 1.00000 −1.80853 1.00000 −3.10189 −1.80853 −2.67852 1.00000 0.270780 −3.10189
1.11 1.00000 −1.75046 1.00000 1.28935 −1.75046 −0.876800 1.00000 0.0641056 1.28935
1.12 1.00000 −1.49586 1.00000 3.62778 −1.49586 −1.50310 1.00000 −0.762393 3.62778
1.13 1.00000 −1.32273 1.00000 −4.14420 −1.32273 −0.0466766 1.00000 −1.25038 −4.14420
1.14 1.00000 −0.917670 1.00000 −0.382051 −0.917670 1.33651 1.00000 −2.15788 −0.382051
1.15 1.00000 −0.691780 1.00000 −1.18527 −0.691780 3.22162 1.00000 −2.52144 −1.18527
1.16 1.00000 −0.653164 1.00000 1.42104 −0.653164 −3.96008 1.00000 −2.57338 1.42104
1.17 1.00000 −0.0253042 1.00000 0.720155 −0.0253042 −3.77805 1.00000 −2.99936 0.720155
1.18 1.00000 −0.0246012 1.00000 2.47730 −0.0246012 1.25791 1.00000 −2.99939 2.47730
1.19 1.00000 0.524728 1.00000 0.579848 0.524728 1.74824 1.00000 −2.72466 0.579848
1.20 1.00000 0.689428 1.00000 −0.213014 0.689428 −0.113005 1.00000 −2.52469 −0.213014
See all 31 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.31
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.c ✓ 31
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4022.2.a.c ✓ 31 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{31} + 14 T_{3}^{30} + 38 T_{3}^{29} - 333 T_{3}^{28} - 1985 T_{3}^{27} + 1593 T_{3}^{26} + \cdots - 1093 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4022))\). Copy content Toggle raw display