Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [712,6,Mod(1,712)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(712, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("712.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 712 = 2^{3} \cdot 89 \) |
Weight: | \( k \) | \(=\) | \( 6 \) |
Character orbit: | \([\chi]\) | \(=\) | 712.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: | \(114.193279707\) |
Analytic rank: | \(1\) |
Dimension: | \(25\) |
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 | 0 | −29.3361 | 0 | −1.97169 | 0 | 111.555 | 0 | 617.608 | 0 | ||||||||||||||||||
1.2 | 0 | −25.0259 | 0 | 59.7843 | 0 | 236.571 | 0 | 383.295 | 0 | ||||||||||||||||||
1.3 | 0 | −23.3777 | 0 | 39.6322 | 0 | −68.9215 | 0 | 303.518 | 0 | ||||||||||||||||||
1.4 | 0 | −21.5806 | 0 | −50.0637 | 0 | −182.115 | 0 | 222.722 | 0 | ||||||||||||||||||
1.5 | 0 | −20.6268 | 0 | 7.99819 | 0 | 69.7533 | 0 | 182.464 | 0 | ||||||||||||||||||
1.6 | 0 | −18.5026 | 0 | −84.6330 | 0 | 91.0861 | 0 | 99.3462 | 0 | ||||||||||||||||||
1.7 | 0 | −12.4495 | 0 | −58.4861 | 0 | −182.524 | 0 | −88.0090 | 0 | ||||||||||||||||||
1.8 | 0 | −10.2652 | 0 | −86.1286 | 0 | −165.772 | 0 | −137.626 | 0 | ||||||||||||||||||
1.9 | 0 | −9.94465 | 0 | 67.8837 | 0 | −41.5444 | 0 | −144.104 | 0 | ||||||||||||||||||
1.10 | 0 | −9.27112 | 0 | 91.8955 | 0 | −245.668 | 0 | −157.046 | 0 | ||||||||||||||||||
1.11 | 0 | −8.72289 | 0 | 53.2082 | 0 | 92.5016 | 0 | −166.911 | 0 | ||||||||||||||||||
1.12 | 0 | −2.36191 | 0 | 2.62985 | 0 | −69.2763 | 0 | −237.421 | 0 | ||||||||||||||||||
1.13 | 0 | −0.00905606 | 0 | −66.8813 | 0 | 217.342 | 0 | −243.000 | 0 | ||||||||||||||||||
1.14 | 0 | 0.0172035 | 0 | 21.7119 | 0 | 208.181 | 0 | −243.000 | 0 | ||||||||||||||||||
1.15 | 0 | 2.51249 | 0 | −107.082 | 0 | 97.4143 | 0 | −236.687 | 0 | ||||||||||||||||||
1.16 | 0 | 11.1109 | 0 | 6.45498 | 0 | −18.5914 | 0 | −119.549 | 0 | ||||||||||||||||||
1.17 | 0 | 11.7629 | 0 | 88.2078 | 0 | 8.13456 | 0 | −104.635 | 0 | ||||||||||||||||||
1.18 | 0 | 13.1808 | 0 | 1.54086 | 0 | −82.4863 | 0 | −69.2666 | 0 | ||||||||||||||||||
1.19 | 0 | 13.2111 | 0 | −88.4774 | 0 | −178.726 | 0 | −68.4660 | 0 | ||||||||||||||||||
1.20 | 0 | 14.3948 | 0 | 76.9626 | 0 | −39.5728 | 0 | −35.7893 | 0 | ||||||||||||||||||
See all 25 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(1\) |
\(89\) | \(1\) |
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 | 712.6.a.a | ✓ | 25 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
712.6.a.a | ✓ | 25 | 1.a | even | 1 | 1 | trivial |