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 | −5.41306 | 3.00000 | 21.3012 | −13.4657 | −16.2392 | 11.1590 | −72.0002 | 9.00000 | 72.8907 | ||||||||||||||||||
1.2 | −5.39202 | 3.00000 | 21.0739 | 2.00966 | −16.1761 | −20.9762 | −70.4947 | 9.00000 | −10.8361 | ||||||||||||||||||
1.3 | −5.14443 | 3.00000 | 18.4652 | 9.21599 | −15.4333 | 10.6441 | −53.8373 | 9.00000 | −47.4110 | ||||||||||||||||||
1.4 | −5.05961 | 3.00000 | 17.5996 | −15.9011 | −15.1788 | 34.5776 | −48.5703 | 9.00000 | 80.4535 | ||||||||||||||||||
1.5 | −4.48960 | 3.00000 | 12.1565 | 5.33009 | −13.4688 | 16.1084 | −18.6612 | 9.00000 | −23.9300 | ||||||||||||||||||
1.6 | −4.28216 | 3.00000 | 10.3369 | 12.4880 | −12.8465 | −32.6569 | −10.0068 | 9.00000 | −53.4757 | ||||||||||||||||||
1.7 | −4.20330 | 3.00000 | 9.66773 | −8.42122 | −12.6099 | −13.3084 | −7.00999 | 9.00000 | 35.3969 | ||||||||||||||||||
1.8 | −3.67739 | 3.00000 | 5.52321 | 10.7113 | −11.0322 | −9.56351 | 9.10812 | 9.00000 | −39.3898 | ||||||||||||||||||
1.9 | −3.61193 | 3.00000 | 5.04607 | −17.1653 | −10.8358 | 13.1780 | 10.6694 | 9.00000 | 61.9998 | ||||||||||||||||||
1.10 | −3.59523 | 3.00000 | 4.92567 | 14.8330 | −10.7857 | 14.0315 | 11.0529 | 9.00000 | −53.3279 | ||||||||||||||||||
1.11 | −3.15750 | 3.00000 | 1.96979 | −18.8998 | −9.47249 | −24.1141 | 19.0404 | 9.00000 | 59.6762 | ||||||||||||||||||
1.12 | −2.45234 | 3.00000 | −1.98602 | −1.35025 | −7.35703 | −17.8806 | 24.4891 | 9.00000 | 3.31128 | ||||||||||||||||||
1.13 | −2.43775 | 3.00000 | −2.05739 | −9.35530 | −7.31324 | 19.3013 | 24.5174 | 9.00000 | 22.8059 | ||||||||||||||||||
1.14 | −1.88493 | 3.00000 | −4.44705 | 7.51572 | −5.65478 | 1.38168 | 23.4618 | 9.00000 | −14.1666 | ||||||||||||||||||
1.15 | −1.41755 | 3.00000 | −5.99056 | 8.60687 | −4.25264 | 2.21724 | 19.8323 | 9.00000 | −12.2006 | ||||||||||||||||||
1.16 | −0.951961 | 3.00000 | −7.09377 | 20.9675 | −2.85588 | 2.74716 | 14.3687 | 9.00000 | −19.9602 | ||||||||||||||||||
1.17 | −0.624353 | 3.00000 | −7.61018 | −20.5187 | −1.87306 | −20.9935 | 9.74626 | 9.00000 | 12.8109 | ||||||||||||||||||
1.18 | −0.476963 | 3.00000 | −7.77251 | 17.0373 | −1.43089 | −27.8002 | 7.52291 | 9.00000 | −8.12615 | ||||||||||||||||||
1.19 | −0.169683 | 3.00000 | −7.97121 | 0.479910 | −0.509049 | 32.7035 | 2.71004 | 9.00000 | −0.0814325 | ||||||||||||||||||
1.20 | −0.108103 | 3.00000 | −7.98831 | −6.73266 | −0.324309 | −18.3209 | 1.72838 | 9.00000 | 0.727821 | ||||||||||||||||||
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.c | ✓ | 37 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2013.4.a.c | ✓ | 37 | 1.a | even | 1 | 1 | trivial |