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.
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) |
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 |