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: | \(36\) |
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 | −5.61297 | 3.00000 | 23.5055 | 13.7147 | −16.8389 | 34.2924 | −87.0318 | 9.00000 | −76.9805 | ||||||||||||||||||
1.2 | −5.53692 | 3.00000 | 22.6575 | 4.34863 | −16.6108 | −33.2757 | −81.1573 | 9.00000 | −24.0780 | ||||||||||||||||||
1.3 | −5.51047 | 3.00000 | 22.3653 | −14.2389 | −16.5314 | 3.43977 | −79.1594 | 9.00000 | 78.4630 | ||||||||||||||||||
1.4 | −4.75663 | 3.00000 | 14.6255 | 7.60303 | −14.2699 | −25.1377 | −31.5151 | 9.00000 | −36.1648 | ||||||||||||||||||
1.5 | −4.51714 | 3.00000 | 12.4046 | 19.2722 | −13.5514 | −8.39412 | −19.8961 | 9.00000 | −87.0554 | ||||||||||||||||||
1.6 | −4.38409 | 3.00000 | 11.2202 | −10.2931 | −13.1523 | 22.1567 | −14.1179 | 9.00000 | 45.1257 | ||||||||||||||||||
1.7 | −4.37392 | 3.00000 | 11.1312 | 9.50672 | −13.1218 | −4.68752 | −13.6956 | 9.00000 | −41.5816 | ||||||||||||||||||
1.8 | −4.25496 | 3.00000 | 10.1047 | −22.0823 | −12.7649 | 8.60879 | −8.95532 | 9.00000 | 93.9591 | ||||||||||||||||||
1.9 | −3.72734 | 3.00000 | 5.89305 | 10.0247 | −11.1820 | 19.7213 | 7.85332 | 9.00000 | −37.3656 | ||||||||||||||||||
1.10 | −3.33494 | 3.00000 | 3.12184 | 0.253130 | −10.0048 | −0.691781 | 16.2684 | 9.00000 | −0.844174 | ||||||||||||||||||
1.11 | −3.06792 | 3.00000 | 1.41213 | −10.3390 | −9.20376 | −32.3876 | 20.2111 | 9.00000 | 31.7193 | ||||||||||||||||||
1.12 | −3.06030 | 3.00000 | 1.36541 | −15.4248 | −9.18089 | 12.8287 | 20.3038 | 9.00000 | 47.2044 | ||||||||||||||||||
1.13 | −3.05325 | 3.00000 | 1.32235 | −6.59737 | −9.15976 | −7.71882 | 20.3886 | 9.00000 | 20.1434 | ||||||||||||||||||
1.14 | −1.93393 | 3.00000 | −4.25991 | 7.94786 | −5.80179 | −25.7974 | 23.7098 | 9.00000 | −15.3706 | ||||||||||||||||||
1.15 | −1.70047 | 3.00000 | −5.10841 | −3.95974 | −5.10140 | 33.3215 | 22.2904 | 9.00000 | 6.73341 | ||||||||||||||||||
1.16 | −1.31102 | 3.00000 | −6.28122 | 5.66783 | −3.93306 | −0.126112 | 18.7230 | 9.00000 | −7.43064 | ||||||||||||||||||
1.17 | −1.11376 | 3.00000 | −6.75954 | −13.8886 | −3.34127 | −3.42861 | 16.4385 | 9.00000 | 15.4685 | ||||||||||||||||||
1.18 | −0.942036 | 3.00000 | −7.11257 | 1.29514 | −2.82611 | 21.4949 | 14.2366 | 9.00000 | −1.22007 | ||||||||||||||||||
1.19 | −0.121017 | 3.00000 | −7.98535 | 12.2818 | −0.363050 | 1.74252 | 1.93450 | 9.00000 | −1.48630 | ||||||||||||||||||
1.20 | 0.439845 | 3.00000 | −7.80654 | −18.8775 | 1.31954 | −35.5540 | −6.95243 | 9.00000 | −8.30317 | ||||||||||||||||||
See all 36 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.a | ✓ | 36 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2013.4.a.a | ✓ | 36 | 1.a | even | 1 | 1 | trivial |