Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8038,2,Mod(1,8038)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8038, 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("8038.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8038 = 2 \cdot 4019 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8038.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: | \(64.1837531447\) |
Analytic rank: | \(1\) |
Dimension: | \(75\) |
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 | 1.00000 | −3.39091 | 1.00000 | 3.77047 | −3.39091 | −1.27110 | 1.00000 | 8.49828 | 3.77047 | ||||||||||||||||||
1.2 | 1.00000 | −3.38305 | 1.00000 | 1.91300 | −3.38305 | −4.34480 | 1.00000 | 8.44501 | 1.91300 | ||||||||||||||||||
1.3 | 1.00000 | −3.32095 | 1.00000 | −3.93433 | −3.32095 | −4.89708 | 1.00000 | 8.02870 | −3.93433 | ||||||||||||||||||
1.4 | 1.00000 | −3.27949 | 1.00000 | −1.91910 | −3.27949 | 3.63095 | 1.00000 | 7.75505 | −1.91910 | ||||||||||||||||||
1.5 | 1.00000 | −3.25539 | 1.00000 | −2.85398 | −3.25539 | 2.81527 | 1.00000 | 7.59759 | −2.85398 | ||||||||||||||||||
1.6 | 1.00000 | −3.20148 | 1.00000 | 0.754953 | −3.20148 | 1.04626 | 1.00000 | 7.24947 | 0.754953 | ||||||||||||||||||
1.7 | 1.00000 | −3.17284 | 1.00000 | −1.87632 | −3.17284 | −0.162262 | 1.00000 | 7.06690 | −1.87632 | ||||||||||||||||||
1.8 | 1.00000 | −2.93359 | 1.00000 | 2.13721 | −2.93359 | 0.512181 | 1.00000 | 5.60596 | 2.13721 | ||||||||||||||||||
1.9 | 1.00000 | −2.85626 | 1.00000 | 0.772797 | −2.85626 | −4.64843 | 1.00000 | 5.15824 | 0.772797 | ||||||||||||||||||
1.10 | 1.00000 | −2.79660 | 1.00000 | −1.17558 | −2.79660 | 1.04192 | 1.00000 | 4.82097 | −1.17558 | ||||||||||||||||||
1.11 | 1.00000 | −2.59529 | 1.00000 | 3.17530 | −2.59529 | −0.203775 | 1.00000 | 3.73551 | 3.17530 | ||||||||||||||||||
1.12 | 1.00000 | −2.54835 | 1.00000 | −2.91194 | −2.54835 | −2.80949 | 1.00000 | 3.49408 | −2.91194 | ||||||||||||||||||
1.13 | 1.00000 | −2.53842 | 1.00000 | 2.78579 | −2.53842 | −2.33236 | 1.00000 | 3.44358 | 2.78579 | ||||||||||||||||||
1.14 | 1.00000 | −2.46931 | 1.00000 | 0.696459 | −2.46931 | −2.94189 | 1.00000 | 3.09749 | 0.696459 | ||||||||||||||||||
1.15 | 1.00000 | −2.32361 | 1.00000 | 0.467112 | −2.32361 | 3.12912 | 1.00000 | 2.39917 | 0.467112 | ||||||||||||||||||
1.16 | 1.00000 | −2.28450 | 1.00000 | −2.56431 | −2.28450 | 0.255204 | 1.00000 | 2.21893 | −2.56431 | ||||||||||||||||||
1.17 | 1.00000 | −2.22920 | 1.00000 | −4.17184 | −2.22920 | −0.836281 | 1.00000 | 1.96935 | −4.17184 | ||||||||||||||||||
1.18 | 1.00000 | −2.12788 | 1.00000 | 3.19034 | −2.12788 | 2.73143 | 1.00000 | 1.52787 | 3.19034 | ||||||||||||||||||
1.19 | 1.00000 | −2.10614 | 1.00000 | −0.0733427 | −2.10614 | 4.17239 | 1.00000 | 1.43582 | −0.0733427 | ||||||||||||||||||
1.20 | 1.00000 | −1.99460 | 1.00000 | 1.10571 | −1.99460 | 2.73220 | 1.00000 | 0.978446 | 1.10571 | ||||||||||||||||||
See all 75 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(4019\) | \(-1\) |
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 | 8038.2.a.a | ✓ | 75 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8038.2.a.a | ✓ | 75 | 1.a | even | 1 | 1 | trivial |