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: | \(0\) |
Dimension: | \(38\) |
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 | −5.60876 | −3.00000 | 23.4582 | −9.03443 | 16.8263 | 14.8484 | −86.7015 | 9.00000 | 50.6720 | ||||||||||||||||||
1.2 | −5.27624 | −3.00000 | 19.8387 | −1.30000 | 15.8287 | 18.3929 | −62.4637 | 9.00000 | 6.85913 | ||||||||||||||||||
1.3 | −5.14871 | −3.00000 | 18.5092 | 14.6569 | 15.4461 | −22.4177 | −54.1088 | 9.00000 | −75.4639 | ||||||||||||||||||
1.4 | −4.79323 | −3.00000 | 14.9751 | −18.8616 | 14.3797 | −18.5258 | −33.4332 | 9.00000 | 90.4081 | ||||||||||||||||||
1.5 | −4.76869 | −3.00000 | 14.7404 | 6.42496 | 14.3061 | −27.3976 | −32.1428 | 9.00000 | −30.6386 | ||||||||||||||||||
1.6 | −4.20688 | −3.00000 | 9.69783 | −5.61420 | 12.6206 | 32.2171 | −7.14255 | 9.00000 | 23.6183 | ||||||||||||||||||
1.7 | −3.95658 | −3.00000 | 7.65452 | 8.88627 | 11.8697 | −1.22023 | 1.36690 | 9.00000 | −35.1592 | ||||||||||||||||||
1.8 | −3.35127 | −3.00000 | 3.23102 | −13.0308 | 10.0538 | 23.8563 | 15.9822 | 9.00000 | 43.6697 | ||||||||||||||||||
1.9 | −3.18056 | −3.00000 | 2.11597 | 11.4382 | 9.54169 | 18.3064 | 18.7145 | 9.00000 | −36.3799 | ||||||||||||||||||
1.10 | −3.08546 | −3.00000 | 1.52008 | 1.21167 | 9.25639 | −4.33357 | 19.9936 | 9.00000 | −3.73856 | ||||||||||||||||||
1.11 | −3.03304 | −3.00000 | 1.19935 | −11.5262 | 9.09913 | −17.2788 | 20.6267 | 9.00000 | 34.9594 | ||||||||||||||||||
1.12 | −2.90911 | −3.00000 | 0.462945 | 17.7603 | 8.72734 | −6.04713 | 21.9262 | 9.00000 | −51.6667 | ||||||||||||||||||
1.13 | −2.30626 | −3.00000 | −2.68118 | −4.34363 | 6.91877 | −19.3387 | 24.6335 | 9.00000 | 10.0175 | ||||||||||||||||||
1.14 | −2.21634 | −3.00000 | −3.08782 | −20.1616 | 6.64903 | 13.2153 | 24.5744 | 9.00000 | 44.6850 | ||||||||||||||||||
1.15 | −1.82980 | −3.00000 | −4.65183 | 19.9990 | 5.48940 | 32.4689 | 23.1503 | 9.00000 | −36.5943 | ||||||||||||||||||
1.16 | −0.789172 | −3.00000 | −7.37721 | 13.9316 | 2.36752 | −21.1583 | 12.1353 | 9.00000 | −10.9944 | ||||||||||||||||||
1.17 | −0.747694 | −3.00000 | −7.44095 | −10.4401 | 2.24308 | −14.9879 | 11.5451 | 9.00000 | 7.80602 | ||||||||||||||||||
1.18 | −0.565755 | −3.00000 | −7.67992 | 11.9466 | 1.69726 | 27.9962 | 8.87099 | 9.00000 | −6.75887 | ||||||||||||||||||
1.19 | −0.495609 | −3.00000 | −7.75437 | −15.1035 | 1.48683 | 2.90433 | 7.80801 | 9.00000 | 7.48545 | ||||||||||||||||||
1.20 | −0.449187 | −3.00000 | −7.79823 | 5.67323 | 1.34756 | 0.979067 | 7.09636 | 9.00000 | −2.54834 | ||||||||||||||||||
See all 38 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.e | ✓ | 38 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2013.4.a.e | ✓ | 38 | 1.a | even | 1 | 1 | trivial |