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: | \(39\) |
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 | −5.58742 | 3.00000 | 23.2193 | 7.47722 | −16.7623 | 8.26209 | −85.0366 | 9.00000 | −41.7784 | ||||||||||||||||||
1.2 | −5.19520 | 3.00000 | 18.9901 | 20.2319 | −15.5856 | −14.1923 | −57.0959 | 9.00000 | −105.109 | ||||||||||||||||||
1.3 | −4.97376 | 3.00000 | 16.7383 | −4.79568 | −14.9213 | −12.7682 | −43.4620 | 9.00000 | 23.8525 | ||||||||||||||||||
1.4 | −4.73713 | 3.00000 | 14.4404 | 3.01069 | −14.2114 | 21.8287 | −30.5090 | 9.00000 | −14.2620 | ||||||||||||||||||
1.5 | −4.26095 | 3.00000 | 10.1557 | −7.59416 | −12.7828 | −6.39265 | −9.18529 | 9.00000 | 32.3583 | ||||||||||||||||||
1.6 | −3.99434 | 3.00000 | 7.95479 | 10.8725 | −11.9830 | 34.9110 | 0.180577 | 9.00000 | −43.4284 | ||||||||||||||||||
1.7 | −3.90025 | 3.00000 | 7.21198 | −4.45780 | −11.7008 | 14.5889 | 3.07349 | 9.00000 | 17.3866 | ||||||||||||||||||
1.8 | −3.67115 | 3.00000 | 5.47734 | 14.7390 | −11.0134 | −28.2167 | 9.26107 | 9.00000 | −54.1090 | ||||||||||||||||||
1.9 | −3.32411 | 3.00000 | 3.04969 | −5.05305 | −9.97232 | −19.7829 | 16.4554 | 9.00000 | 16.7969 | ||||||||||||||||||
1.10 | −2.98423 | 3.00000 | 0.905618 | 18.2801 | −8.95268 | −1.48333 | 21.1713 | 9.00000 | −54.5520 | ||||||||||||||||||
1.11 | −2.88889 | 3.00000 | 0.345688 | −8.48381 | −8.66667 | 20.9827 | 22.1125 | 9.00000 | 24.5088 | ||||||||||||||||||
1.12 | −2.16141 | 3.00000 | −3.32830 | 7.38660 | −6.48423 | −25.4500 | 24.4851 | 9.00000 | −15.9655 | ||||||||||||||||||
1.13 | −2.09003 | 3.00000 | −3.63175 | −21.1504 | −6.27010 | 13.8715 | 24.3108 | 9.00000 | 44.2051 | ||||||||||||||||||
1.14 | −1.70549 | 3.00000 | −5.09131 | −1.03329 | −5.11647 | 1.91271 | 22.3271 | 9.00000 | 1.76227 | ||||||||||||||||||
1.15 | −1.69454 | 3.00000 | −5.12852 | −5.42092 | −5.08363 | −29.9078 | 22.2469 | 9.00000 | 9.18599 | ||||||||||||||||||
1.16 | −1.01355 | 3.00000 | −6.97272 | 12.1717 | −3.04064 | 28.0678 | 15.1756 | 9.00000 | −12.3366 | ||||||||||||||||||
1.17 | −0.878549 | 3.00000 | −7.22815 | −10.2662 | −2.63565 | 2.69154 | 13.3787 | 9.00000 | 9.01940 | ||||||||||||||||||
1.18 | −0.382791 | 3.00000 | −7.85347 | 10.1790 | −1.14837 | 17.4416 | 6.06857 | 9.00000 | −3.89641 | ||||||||||||||||||
1.19 | 0.0220344 | 3.00000 | −7.99951 | −17.2328 | 0.0661031 | −5.99920 | −0.352539 | 9.00000 | −0.379714 | ||||||||||||||||||
1.20 | 0.170804 | 3.00000 | −7.97083 | −3.72511 | 0.512411 | −19.4372 | −2.72788 | 9.00000 | −0.636262 | ||||||||||||||||||
See all 39 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.h | ✓ | 39 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2013.4.a.h | ✓ | 39 | 1.a | even | 1 | 1 | trivial |