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Properties

Label 4027.2.a.b
Level $4027$
Weight $2$
Character orbit 4027.a
Self dual yes
Analytic conductor $32.156$
Analytic rank $1$
Dimension $159$
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] = [4027,2,Mod(1,4027)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4027, 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("4027.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4027 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4027.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: \(32.1557568940\)
Analytic rank: \(1\)
Dimension: \(159\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 159 q - 22 q^{2} - 19 q^{3} + 148 q^{4} - 70 q^{5} - 23 q^{6} - 19 q^{7} - 66 q^{8} + 126 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 159 q - 22 q^{2} - 19 q^{3} + 148 q^{4} - 70 q^{5} - 23 q^{6} - 19 q^{7} - 66 q^{8} + 126 q^{9} - 23 q^{10} - 33 q^{11} - 57 q^{12} - 90 q^{13} - 28 q^{14} - 22 q^{15} + 130 q^{16} - 145 q^{17} - 50 q^{18} - 28 q^{19} - 121 q^{20} - 69 q^{21} - 26 q^{22} - 79 q^{23} - 62 q^{24} + 123 q^{25} - 40 q^{26} - 70 q^{27} - 43 q^{28} - 109 q^{29} - 43 q^{30} - 21 q^{31} - 139 q^{32} - 83 q^{33} - 93 q^{35} + 75 q^{36} - 65 q^{37} - 122 q^{38} - 18 q^{39} - 43 q^{40} - 71 q^{41} - 88 q^{42} - 72 q^{43} - 79 q^{44} - 181 q^{45} - 11 q^{46} - 114 q^{47} - 118 q^{48} + 118 q^{49} - 77 q^{50} - 29 q^{51} - 169 q^{52} - 220 q^{53} - 80 q^{54} - 37 q^{55} - 72 q^{56} - 90 q^{57} - 8 q^{58} - 60 q^{59} - 42 q^{60} - 108 q^{61} - 152 q^{62} - 65 q^{63} + 114 q^{64} - 81 q^{65} - 40 q^{66} - 50 q^{67} - 319 q^{68} - 103 q^{69} + 4 q^{70} - 7 q^{71} - 129 q^{72} - 94 q^{73} - 79 q^{74} - 59 q^{75} - 46 q^{76} - 329 q^{77} + 8 q^{78} - 18 q^{79} - 190 q^{80} + 59 q^{81} - 56 q^{82} - 201 q^{83} - 71 q^{84} - 26 q^{85} - 52 q^{86} - 126 q^{87} - 66 q^{88} - 114 q^{89} - 33 q^{90} - 30 q^{91} - 204 q^{92} - 125 q^{93} + 9 q^{94} - 84 q^{95} - 88 q^{96} - 56 q^{97} - 110 q^{98} - 46 q^{99}+O(q^{100}) \) 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 −2.82139 1.96833 5.96026 2.22438 −5.55342 0.144940 −11.1735 0.874306 −6.27587
1.2 −2.78842 −1.28715 5.77530 −4.20295 3.58911 −2.09011 −10.5271 −1.34326 11.7196
1.3 −2.73932 −1.57093 5.50386 0.430202 4.30328 4.35305 −9.59817 −0.532177 −1.17846
1.4 −2.73389 −3.04946 5.47414 −1.23983 8.33688 −1.91443 −9.49790 6.29922 3.38954
1.5 −2.72666 0.318995 5.43469 −1.22389 −0.869792 −4.59347 −9.36524 −2.89824 3.33713
1.6 −2.70494 2.29108 5.31668 −3.64034 −6.19723 2.95034 −8.97141 2.24905 9.84688
1.7 −2.69495 1.90961 5.26276 −0.957907 −5.14631 0.102731 −8.79297 0.646618 2.58151
1.8 −2.64324 2.50355 4.98674 1.52675 −6.61749 −4.42325 −7.89468 3.26775 −4.03556
1.9 −2.63499 −1.76075 4.94319 1.57936 4.63957 3.18044 −7.75529 0.100248 −4.16161
1.10 −2.59596 −2.97309 4.73903 −2.99704 7.71803 3.05793 −7.11041 5.83925 7.78021
1.11 −2.59401 −2.55671 4.72888 1.35267 6.63213 −3.48116 −7.07874 3.53677 −3.50885
1.12 −2.58046 0.749522 4.65878 −1.56103 −1.93411 3.58823 −6.86087 −2.43822 4.02819
1.13 −2.57109 −0.605717 4.61051 3.41025 1.55735 −4.02437 −6.71185 −2.63311 −8.76807
1.14 −2.54982 3.26459 4.50158 −1.30834 −8.32411 3.58480 −6.37858 7.65753 3.33603
1.15 −2.50942 −0.724063 4.29717 −3.46864 1.81698 2.45320 −5.76455 −2.47573 8.70427
1.16 −2.48250 −1.21941 4.16279 0.264971 3.02717 −2.37151 −5.36913 −1.51305 −0.657790
1.17 −2.46215 −2.74752 4.06218 4.08202 6.76481 −0.414263 −5.07738 4.54887 −10.0505
1.18 −2.44784 0.695688 3.99191 2.35878 −1.70293 2.06757 −4.87586 −2.51602 −5.77391
1.19 −2.37585 0.673958 3.64466 −1.12762 −1.60122 1.18808 −3.90747 −2.54578 2.67904
1.20 −2.29049 2.02403 3.24633 −4.01248 −4.63602 1.58118 −2.85471 1.09671 9.19054
See next 80 embeddings (of 159 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.159
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(4027\) \(1\)

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 4027.2.a.b ✓ 159
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4027.2.a.b ✓ 159 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{159} + 22 T_{2}^{158} + 9 T_{2}^{157} - 3300 T_{2}^{156} - 18924 T_{2}^{155} + 214717 T_{2}^{154} + \cdots - 17579932200 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4027))\). Copy content Toggle raw display