Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2013,4,Mod(1,2013)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2013, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2013.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 2013 = 3 \cdot 11 \cdot 61 \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 2013.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: | \(118.770844842\) |
Analytic rank: | \(1\) |
Dimension: | \(37\) |
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 | −5.57224 | −3.00000 | 23.0498 | −16.5315 | 16.7167 | −30.4189 | −83.8611 | 9.00000 | 92.1176 | ||||||||||||||||||
1.2 | −5.34535 | −3.00000 | 20.5728 | 5.60041 | 16.0360 | 11.9095 | −67.2058 | 9.00000 | −29.9362 | ||||||||||||||||||
1.3 | −5.20750 | −3.00000 | 19.1180 | 13.8219 | 15.6225 | −32.6173 | −57.8972 | 9.00000 | −71.9777 | ||||||||||||||||||
1.4 | −4.94279 | −3.00000 | 16.4312 | 0.994625 | 14.8284 | 23.9386 | −41.6735 | 9.00000 | −4.91622 | ||||||||||||||||||
1.5 | −4.85102 | −3.00000 | 15.5324 | −6.76910 | 14.5530 | −14.4227 | −36.5396 | 9.00000 | 32.8370 | ||||||||||||||||||
1.6 | −4.27346 | −3.00000 | 10.2624 | −10.6639 | 12.8204 | −5.91849 | −9.66837 | 9.00000 | 45.5715 | ||||||||||||||||||
1.7 | −4.00724 | −3.00000 | 8.05801 | 20.5241 | 12.0217 | −10.4521 | −0.232470 | 9.00000 | −82.2450 | ||||||||||||||||||
1.8 | −3.99170 | −3.00000 | 7.93363 | 2.24192 | 11.9751 | 2.67076 | 0.264912 | 9.00000 | −8.94905 | ||||||||||||||||||
1.9 | −3.73288 | −3.00000 | 5.93441 | −18.1575 | 11.1986 | 21.4866 | 7.71060 | 9.00000 | 67.7799 | ||||||||||||||||||
1.10 | −2.84685 | −3.00000 | 0.104547 | −9.28768 | 8.54055 | 6.39681 | 22.4772 | 9.00000 | 26.4406 | ||||||||||||||||||
1.11 | −2.75819 | −3.00000 | −0.392374 | 6.83862 | 8.27458 | −1.91645 | 23.1478 | 9.00000 | −18.8622 | ||||||||||||||||||
1.12 | −2.70850 | −3.00000 | −0.664006 | 8.28724 | 8.12551 | −34.8516 | 23.4665 | 9.00000 | −22.4460 | ||||||||||||||||||
1.13 | −2.29940 | −3.00000 | −2.71276 | −16.8558 | 6.89820 | −2.53212 | 24.6329 | 9.00000 | 38.7581 | ||||||||||||||||||
1.14 | −1.94841 | −3.00000 | −4.20370 | 15.7553 | 5.84523 | 3.79858 | 23.7778 | 9.00000 | −30.6978 | ||||||||||||||||||
1.15 | −1.80549 | −3.00000 | −4.74021 | −6.98611 | 5.41646 | 21.1508 | 23.0023 | 9.00000 | 12.6133 | ||||||||||||||||||
1.16 | −1.54125 | −3.00000 | −5.62456 | −4.20975 | 4.62374 | −32.1268 | 20.9988 | 9.00000 | 6.48826 | ||||||||||||||||||
1.17 | −1.30641 | −3.00000 | −6.29329 | −3.39016 | 3.91923 | 25.4059 | 18.6729 | 9.00000 | 4.42894 | ||||||||||||||||||
1.18 | −0.441122 | −3.00000 | −7.80541 | 14.6691 | 1.32336 | −18.4288 | 6.97211 | 9.00000 | −6.47085 | ||||||||||||||||||
1.19 | 0.197712 | −3.00000 | −7.96091 | 13.9720 | −0.593137 | 33.5020 | −3.15567 | 9.00000 | 2.76244 | ||||||||||||||||||
1.20 | 0.250375 | −3.00000 | −7.93731 | 9.04215 | −0.751125 | 10.1810 | −3.99031 | 9.00000 | 2.26393 | ||||||||||||||||||
See all 37 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(11\) | \(-1\) |
\(61\) | \(1\) |
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 | 2013.4.a.d | ✓ | 37 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2013.4.a.d | ✓ | 37 | 1.a | even | 1 | 1 | trivial |