Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8026,2,Mod(1,8026)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8026, 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("8026.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8026 = 2 \cdot 4013 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8026.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: | \(64.0879326623\) |
Analytic rank: | \(1\) |
Dimension: | \(81\) |
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 | −3.26929 | 1.00000 | −1.66866 | 3.26929 | −1.36572 | −1.00000 | 7.68829 | 1.66866 | ||||||||||||||||||
1.2 | −1.00000 | −3.25240 | 1.00000 | −3.99675 | 3.25240 | 3.80728 | −1.00000 | 7.57814 | 3.99675 | ||||||||||||||||||
1.3 | −1.00000 | −3.23458 | 1.00000 | −3.62121 | 3.23458 | −0.933365 | −1.00000 | 7.46248 | 3.62121 | ||||||||||||||||||
1.4 | −1.00000 | −3.21582 | 1.00000 | 3.05469 | 3.21582 | −0.0968570 | −1.00000 | 7.34152 | −3.05469 | ||||||||||||||||||
1.5 | −1.00000 | −3.01169 | 1.00000 | 2.74703 | 3.01169 | −2.66211 | −1.00000 | 6.07025 | −2.74703 | ||||||||||||||||||
1.6 | −1.00000 | −2.83489 | 1.00000 | −3.31707 | 2.83489 | 3.86348 | −1.00000 | 5.03658 | 3.31707 | ||||||||||||||||||
1.7 | −1.00000 | −2.80063 | 1.00000 | −1.23643 | 2.80063 | −3.84630 | −1.00000 | 4.84350 | 1.23643 | ||||||||||||||||||
1.8 | −1.00000 | −2.77693 | 1.00000 | 2.15661 | 2.77693 | 3.32117 | −1.00000 | 4.71136 | −2.15661 | ||||||||||||||||||
1.9 | −1.00000 | −2.70930 | 1.00000 | 1.99036 | 2.70930 | −3.73074 | −1.00000 | 4.34031 | −1.99036 | ||||||||||||||||||
1.10 | −1.00000 | −2.56900 | 1.00000 | −2.27157 | 2.56900 | 3.80635 | −1.00000 | 3.59977 | 2.27157 | ||||||||||||||||||
1.11 | −1.00000 | −2.53693 | 1.00000 | 0.430224 | 2.53693 | 2.70951 | −1.00000 | 3.43604 | −0.430224 | ||||||||||||||||||
1.12 | −1.00000 | −2.51028 | 1.00000 | 0.572444 | 2.51028 | −2.83976 | −1.00000 | 3.30151 | −0.572444 | ||||||||||||||||||
1.13 | −1.00000 | −2.45434 | 1.00000 | −0.137052 | 2.45434 | 1.42878 | −1.00000 | 3.02378 | 0.137052 | ||||||||||||||||||
1.14 | −1.00000 | −2.44759 | 1.00000 | 0.910070 | 2.44759 | −1.13462 | −1.00000 | 2.99071 | −0.910070 | ||||||||||||||||||
1.15 | −1.00000 | −2.39342 | 1.00000 | 1.94313 | 2.39342 | 4.16913 | −1.00000 | 2.72848 | −1.94313 | ||||||||||||||||||
1.16 | −1.00000 | −2.34885 | 1.00000 | 3.93410 | 2.34885 | −0.279699 | −1.00000 | 2.51708 | −3.93410 | ||||||||||||||||||
1.17 | −1.00000 | −2.22289 | 1.00000 | −1.72959 | 2.22289 | 1.27645 | −1.00000 | 1.94125 | 1.72959 | ||||||||||||||||||
1.18 | −1.00000 | −2.20617 | 1.00000 | −2.45356 | 2.20617 | 0.571999 | −1.00000 | 1.86717 | 2.45356 | ||||||||||||||||||
1.19 | −1.00000 | −1.96246 | 1.00000 | −3.42476 | 1.96246 | −2.06086 | −1.00000 | 0.851259 | 3.42476 | ||||||||||||||||||
1.20 | −1.00000 | −1.92493 | 1.00000 | −0.857375 | 1.92493 | 1.26508 | −1.00000 | 0.705338 | 0.857375 | ||||||||||||||||||
See all 81 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(4013\) | \(1\) |
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 | 8026.2.a.b | ✓ | 81 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8026.2.a.b | ✓ | 81 | 1.a | even | 1 | 1 | trivial |