Properties

Label 8021.2.a.d
Level $8021$
Weight $2$
Character orbit 8021.a
Self dual yes
Analytic conductor $64.048$
Analytic rank $0$
Dimension $174$
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] = [8021,2,Mod(1,8021)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8021, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8021.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8021 = 13 \cdot 617 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8021.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: \(64.0480074613\)
Analytic rank: \(0\)
Dimension: \(174\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 174 q + 6 q^{2} + 37 q^{3} + 214 q^{4} + 10 q^{5} + 12 q^{6} + 28 q^{7} + 15 q^{8} + 211 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 174 q + 6 q^{2} + 37 q^{3} + 214 q^{4} + 10 q^{5} + 12 q^{6} + 28 q^{7} + 15 q^{8} + 211 q^{9} + 47 q^{10} + 47 q^{11} + 81 q^{12} + 174 q^{13} + 22 q^{14} + 26 q^{15} + 286 q^{16} + 27 q^{17} + 22 q^{18} + 91 q^{19} + 18 q^{20} + 8 q^{21} + 58 q^{22} + 62 q^{23} + 24 q^{24} + 244 q^{25} + 6 q^{26} + 139 q^{27} + 43 q^{28} + 42 q^{29} + 31 q^{30} + 82 q^{31} + 11 q^{32} + 12 q^{33} + 50 q^{34} + 74 q^{35} + 310 q^{36} + 47 q^{37} + 10 q^{38} + 37 q^{39} + 118 q^{40} + 16 q^{41} + 26 q^{42} + 136 q^{43} + 74 q^{44} + 18 q^{45} + 53 q^{46} + 15 q^{47} + 132 q^{48} + 254 q^{49} - 5 q^{50} + 121 q^{51} + 214 q^{52} + 39 q^{53} + 30 q^{54} + 188 q^{55} + 55 q^{56} + 11 q^{57} + 32 q^{58} + 58 q^{59} + 16 q^{60} + 128 q^{61} + 27 q^{62} + 42 q^{63} + 423 q^{64} + 10 q^{65} + 4 q^{66} + 132 q^{67} + 52 q^{68} + 63 q^{69} - 8 q^{70} + 78 q^{71} + 2 q^{72} + 21 q^{73} - 16 q^{74} + 188 q^{75} + 160 q^{76} + 20 q^{77} + 12 q^{78} + 232 q^{79} + 2 q^{80} + 302 q^{81} + 115 q^{82} + 18 q^{83} - 26 q^{84} + 47 q^{85} + 27 q^{86} + 127 q^{87} + 163 q^{88} + 14 q^{90} + 28 q^{91} + 68 q^{92} + 15 q^{93} + 91 q^{94} + 75 q^{95} - 26 q^{96} + 34 q^{97} - 60 q^{98} + 181 q^{99}+O(q^{100}) \) 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 −2.81586 1.36738 5.92908 −2.81839 −3.85036 −0.890414 −11.0638 −1.13027 7.93620
1.2 −2.77267 1.50862 5.68769 1.04517 −4.18289 −5.21770 −10.2247 −0.724079 −2.89792
1.3 −2.75277 3.20644 5.57777 3.17240 −8.82662 4.29647 −9.84879 7.28128 −8.73291
1.4 −2.74939 −3.36521 5.55917 −2.17409 9.25228 0.588042 −9.78555 8.32462 5.97744
1.5 −2.69931 −1.31016 5.28630 −0.0823096 3.53653 −3.58591 −8.87075 −1.28348 0.222180
1.6 −2.69663 −1.80678 5.27183 −0.883800 4.87223 4.78301 −8.82294 0.264464 2.38328
1.7 −2.66552 1.95620 5.10502 −0.546887 −5.21429 5.08856 −8.27650 0.826707 1.45774
1.8 −2.65450 3.02354 5.04638 −2.29392 −8.02600 −0.907848 −8.08663 6.14182 6.08922
1.9 −2.65348 3.32664 5.04093 −4.27882 −8.82716 −1.52713 −8.06904 8.06653 11.3537
1.10 −2.64790 −2.80956 5.01138 2.16330 7.43944 3.49626 −7.97383 4.89364 −5.72819
1.11 −2.64300 −0.434399 4.98547 −0.633166 1.14812 1.76480 −7.89059 −2.81130 1.67346
1.12 −2.64088 0.716800 4.97424 −3.63503 −1.89298 1.69747 −7.85462 −2.48620 9.59969
1.13 −2.63723 −2.22050 4.95500 3.72301 5.85596 −3.62454 −7.79302 1.93060 −9.81845
1.14 −2.63597 −1.22103 4.94831 −4.05948 3.21858 −2.69195 −7.77165 −1.50910 10.7007
1.15 −2.62193 2.84190 4.87451 3.72911 −7.45125 −2.25507 −7.53678 5.07638 −9.77745
1.16 −2.58537 −0.157570 4.68414 0.713446 0.407377 −2.31899 −6.93951 −2.97517 −1.84452
1.17 −2.55071 −1.16628 4.50614 1.55643 2.97485 2.04272 −6.39246 −1.63979 −3.97002
1.18 −2.41415 1.51661 3.82811 −1.37172 −3.66132 −4.25980 −4.41333 −0.699894 3.31154
1.19 −2.39849 2.44398 3.75277 −3.07733 −5.86187 3.93094 −4.20400 2.97305 7.38096
1.20 −2.31194 −2.08272 3.34506 −3.48244 4.81513 3.73510 −3.10970 1.33774 8.05119
See next 80 embeddings (of 174 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.174
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(13\) \(-1\)
\(617\) \(1\)

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 8021.2.a.d 174
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8021.2.a.d 174 1.a even 1 1 trivial