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 | −5.20141 | 3.00000 | 19.0547 | −1.81138 | −15.6042 | 3.78751 | −57.5001 | 9.00000 | 9.42176 | ||||||||||||||||||
1.2 | −5.11974 | 3.00000 | 18.2117 | −8.33272 | −15.3592 | −0.123891 | −52.2812 | 9.00000 | 42.6613 | ||||||||||||||||||
1.3 | −4.92345 | 3.00000 | 16.2404 | 19.7094 | −14.7704 | 4.38553 | −40.5713 | 9.00000 | −97.0385 | ||||||||||||||||||
1.4 | −4.79323 | 3.00000 | 14.9751 | 12.3054 | −14.3797 | 26.5090 | −33.4332 | 9.00000 | −58.9827 | ||||||||||||||||||
1.5 | −4.44696 | 3.00000 | 11.7755 | −6.52314 | −13.3409 | −19.5475 | −16.7894 | 9.00000 | 29.0081 | ||||||||||||||||||
1.6 | −4.08981 | 3.00000 | 8.72654 | −14.1681 | −12.2694 | −18.6667 | −2.97140 | 9.00000 | 57.9450 | ||||||||||||||||||
1.7 | −4.00574 | 3.00000 | 8.04594 | −13.8167 | −12.0172 | 32.5179 | −0.184008 | 9.00000 | 55.3460 | ||||||||||||||||||
1.8 | −3.50711 | 3.00000 | 4.29983 | 10.2444 | −10.5213 | 1.41773 | 12.9769 | 9.00000 | −35.9282 | ||||||||||||||||||
1.9 | −3.25945 | 3.00000 | 2.62403 | 10.3368 | −9.77836 | −26.6613 | 17.5227 | 9.00000 | −33.6924 | ||||||||||||||||||
1.10 | −2.86674 | 3.00000 | 0.218214 | 1.45116 | −8.60023 | 27.7506 | 22.3084 | 9.00000 | −4.16011 | ||||||||||||||||||
1.11 | −2.38315 | 3.00000 | −2.32059 | 14.5269 | −7.14945 | 22.4384 | 24.5955 | 9.00000 | −34.6198 | ||||||||||||||||||
1.12 | −2.31501 | 3.00000 | −2.64074 | −15.6387 | −6.94502 | −20.9070 | 24.6334 | 9.00000 | 36.2036 | ||||||||||||||||||
1.13 | −1.93536 | 3.00000 | −4.25439 | −18.7593 | −5.80607 | 6.86594 | 23.7166 | 9.00000 | 36.3060 | ||||||||||||||||||
1.14 | −1.15474 | 3.00000 | −6.66657 | −7.99519 | −3.46423 | 12.3786 | 16.9361 | 9.00000 | 9.23239 | ||||||||||||||||||
1.15 | −0.933922 | 3.00000 | −7.12779 | 16.8720 | −2.80177 | −20.1785 | 14.1282 | 9.00000 | −15.7571 | ||||||||||||||||||
1.16 | −0.740360 | 3.00000 | −7.45187 | 7.63307 | −2.22108 | 1.52940 | 11.4399 | 9.00000 | −5.65122 | ||||||||||||||||||
1.17 | −0.477206 | 3.00000 | −7.77227 | 5.50183 | −1.43162 | −27.2643 | 7.52663 | 9.00000 | −2.62551 | ||||||||||||||||||
1.18 | −0.0387605 | 3.00000 | −7.99850 | −3.27851 | −0.116282 | −8.05520 | 0.620110 | 9.00000 | 0.127077 | ||||||||||||||||||
1.19 | 0.801460 | 3.00000 | −7.35766 | 19.8509 | 2.40438 | 26.4285 | −12.3086 | 9.00000 | 15.9097 | ||||||||||||||||||
1.20 | 0.812552 | 3.00000 | −7.33976 | −6.24435 | 2.43766 | 23.9419 | −12.4644 | 9.00000 | −5.07386 | ||||||||||||||||||
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.f | ✓ | 38 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
2013.4.a.f | ✓ | 38 | 1.a | even | 1 | 1 | trivial |