Properties

Label 4021.2.a.c
Level $4021$
Weight $2$
Character orbit 4021.a
Self dual yes
Analytic conductor $32.108$
Analytic rank $0$
Dimension $182$
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] = [4021,2,Mod(1,4021)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4021, 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("4021.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4021 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4021.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: \(32.1078466528\)
Analytic rank: \(0\)
Dimension: \(182\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 182 q + 18 q^{2} + 28 q^{3} + 208 q^{4} + 22 q^{5} + 18 q^{6} + 14 q^{7} + 54 q^{8} + 238 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 182 q + 18 q^{2} + 28 q^{3} + 208 q^{4} + 22 q^{5} + 18 q^{6} + 14 q^{7} + 54 q^{8} + 238 q^{9} + 7 q^{10} + 138 q^{11} + 47 q^{12} + 5 q^{13} + 72 q^{14} + 41 q^{15} + 256 q^{16} + 29 q^{17} + 35 q^{18} + 48 q^{19} + 56 q^{20} + 22 q^{21} + 19 q^{22} + 91 q^{23} + 46 q^{24} + 230 q^{25} + 88 q^{26} + 103 q^{27} + 15 q^{28} + 75 q^{29} + 18 q^{30} + 43 q^{31} + 116 q^{32} + 15 q^{33} + 13 q^{34} + 185 q^{35} + 364 q^{36} + 15 q^{37} + 53 q^{38} + 80 q^{39} - 13 q^{40} + 68 q^{41} + 32 q^{42} + 82 q^{43} + 259 q^{44} + 37 q^{45} + 13 q^{46} + 121 q^{47} + 53 q^{48} + 244 q^{49} + 93 q^{50} + 144 q^{51} - 16 q^{52} + 101 q^{53} + 47 q^{54} + 49 q^{55} + 199 q^{56} + 3 q^{57} + 4 q^{58} + 254 q^{59} + 24 q^{60} + 8 q^{61} + 37 q^{62} + 19 q^{63} + 326 q^{64} + 65 q^{65} + 41 q^{66} + 91 q^{67} + 50 q^{68} + 50 q^{69} + 5 q^{70} + 212 q^{71} + 77 q^{72} + 5 q^{73} + 101 q^{74} + 127 q^{75} + 22 q^{76} + 87 q^{77} - 20 q^{78} + 86 q^{79} + 71 q^{80} + 358 q^{81} - 20 q^{82} + 139 q^{83} - 30 q^{84} + 25 q^{85} + 82 q^{86} + 36 q^{87} - 8 q^{88} + 100 q^{89} - 87 q^{90} + 74 q^{91} + 171 q^{92} + 50 q^{93} - 13 q^{94} + 217 q^{95} + 42 q^{96} + 20 q^{97} + 47 q^{98} + 389 q^{99}+O(q^{100}) \) 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 −2.81003 −0.712978 5.89629 1.47211 2.00349 1.95505 −10.9487 −2.49166 −4.13669
1.2 −2.80453 3.34034 5.86536 −1.76340 −9.36806 −4.23059 −10.8405 8.15785 4.94549
1.3 −2.71768 −3.37653 5.38576 2.58064 9.17630 −1.70858 −9.20140 8.40093 −7.01334
1.4 −2.69235 −2.05539 5.24877 −1.71000 5.53383 −4.72268 −8.74684 1.22462 4.60392
1.5 −2.67927 −1.86893 5.17851 2.00029 5.00738 −3.62338 −8.51610 0.492900 −5.35934
1.6 −2.67278 1.10671 5.14375 3.24617 −2.95798 −3.43903 −8.40257 −1.77520 −8.67630
1.7 −2.60649 −2.80454 4.79381 −0.526210 7.31002 5.03978 −7.28206 4.86546 1.37156
1.8 −2.58741 0.104608 4.69469 −1.37313 −0.270664 −0.569668 −6.97226 −2.98906 3.55285
1.9 −2.58653 3.00156 4.69012 3.01819 −7.76362 1.43921 −6.95807 6.00937 −7.80662
1.10 −2.57672 −1.91417 4.63947 −2.65404 4.93228 −2.26992 −6.80118 0.664061 6.83870
1.11 −2.56968 2.97740 4.60327 3.17406 −7.65098 1.41939 −6.68959 5.86491 −8.15632
1.12 −2.53727 1.21389 4.43775 −4.40547 −3.07998 −1.31615 −6.18523 −1.52646 11.1779
1.13 −2.51665 0.669973 4.33353 4.29318 −1.68609 3.31221 −5.87268 −2.55114 −10.8044
1.14 −2.51316 2.70275 4.31595 0.484778 −6.79244 2.96790 −5.82035 4.30486 −1.21832
1.15 −2.46406 1.32162 4.07159 −2.41266 −3.25655 −2.12421 −5.10452 −1.25332 5.94493
1.16 −2.43939 2.42778 3.95064 −0.884079 −5.92232 −2.32249 −4.75837 2.89413 2.15662
1.17 −2.38633 −0.134750 3.69456 −0.899112 0.321557 −0.745765 −4.04378 −2.98184 2.14558
1.18 −2.37596 −0.498603 3.64520 −0.0295149 1.18466 0.657526 −3.90895 −2.75139 0.0701264
1.19 −2.35347 0.704714 3.53883 3.49583 −1.65853 1.05101 −3.62159 −2.50338 −8.22733
1.20 −2.33739 −2.27957 3.46338 1.15085 5.32823 1.11748 −3.42047 2.19642 −2.68998
See next 80 embeddings (of 182 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.182
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(4021\) \(-1\)

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 4021.2.a.c 182
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4021.2.a.c 182 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{182} - 18 T_{2}^{181} - 124 T_{2}^{180} + 4134 T_{2}^{179} - 1044 T_{2}^{178} - 455572 T_{2}^{177} + 1440385 T_{2}^{176} + 31854125 T_{2}^{175} - 167292476 T_{2}^{174} - 1567946638 T_{2}^{173} + \cdots - 450537346108 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4021))\). Copy content Toggle raw display