Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4031,2,Mod(1,4031)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4031, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4031.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 4031 = 29 \cdot 139 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 4031.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: | \(32.1876970548\) |
Analytic rank: | \(0\) |
Dimension: | \(103\) |
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 | −2.80660 | −2.96734 | 5.87699 | 4.35928 | 8.32812 | −3.03197 | −10.8811 | 5.80510 | −12.2348 | ||||||||||||||||||
1.2 | −2.80086 | −1.28750 | 5.84483 | −1.26767 | 3.60610 | −2.24450 | −10.7688 | −1.34235 | 3.55056 | ||||||||||||||||||
1.3 | −2.75205 | 2.78767 | 5.57380 | −3.50027 | −7.67183 | 2.04758 | −9.83529 | 4.77113 | 9.63294 | ||||||||||||||||||
1.4 | −2.71060 | −0.763863 | 5.34738 | 1.53299 | 2.07053 | 3.02090 | −9.07341 | −2.41651 | −4.15533 | ||||||||||||||||||
1.5 | −2.68510 | −0.634380 | 5.20976 | 1.13239 | 1.70337 | 4.26000 | −8.61852 | −2.59756 | −3.04058 | ||||||||||||||||||
1.6 | −2.63961 | 1.15663 | 4.96757 | −1.20001 | −3.05306 | −2.18899 | −7.83323 | −1.66221 | 3.16755 | ||||||||||||||||||
1.7 | −2.62778 | 2.50964 | 4.90520 | −0.626861 | −6.59477 | 4.88608 | −7.63422 | 3.29828 | 1.64725 | ||||||||||||||||||
1.8 | −2.59576 | −0.589470 | 4.73795 | 2.97822 | 1.53012 | −2.97611 | −7.10705 | −2.65253 | −7.73074 | ||||||||||||||||||
1.9 | −2.54769 | −2.43102 | 4.49072 | −4.16655 | 6.19349 | −1.47765 | −6.34558 | 2.90987 | 10.6151 | ||||||||||||||||||
1.10 | −2.50341 | −3.31200 | 4.26704 | −1.06782 | 8.29127 | 3.43750 | −5.67533 | 7.96931 | 2.67318 | ||||||||||||||||||
1.11 | −2.41155 | 2.97695 | 3.81557 | 2.55399 | −7.17907 | 0.984137 | −4.37835 | 5.86223 | −6.15908 | ||||||||||||||||||
1.12 | −2.38730 | −1.73906 | 3.69919 | 0.845395 | 4.15166 | −4.68935 | −4.05648 | 0.0243354 | −2.01821 | ||||||||||||||||||
1.13 | −2.35149 | 2.74768 | 3.52949 | 4.05795 | −6.46113 | 3.42169 | −3.59657 | 4.54974 | −9.54222 | ||||||||||||||||||
1.14 | −2.35028 | 1.87633 | 3.52380 | −3.27377 | −4.40989 | −2.87617 | −3.58134 | 0.520602 | 7.69425 | ||||||||||||||||||
1.15 | −2.34575 | −2.20403 | 3.50256 | −4.15589 | 5.17012 | 5.11824 | −3.52465 | 1.85776 | 9.74869 | ||||||||||||||||||
1.16 | −2.20982 | −3.43245 | 2.88331 | 3.18051 | 7.58510 | 4.63810 | −1.95196 | 8.78169 | −7.02836 | ||||||||||||||||||
1.17 | −2.11397 | 0.494442 | 2.46887 | 0.938762 | −1.04524 | 1.45468 | −0.991174 | −2.75553 | −1.98451 | ||||||||||||||||||
1.18 | −2.09508 | −0.524599 | 2.38936 | −2.74536 | 1.09908 | −3.42586 | −0.815732 | −2.72480 | 5.75174 | ||||||||||||||||||
1.19 | −2.06403 | 0.822287 | 2.26023 | −2.41369 | −1.69723 | 3.09601 | −0.537124 | −2.32384 | 4.98194 | ||||||||||||||||||
1.20 | −2.04048 | 0.724757 | 2.16355 | −3.93757 | −1.47885 | 3.69332 | −0.333725 | −2.47473 | 8.03452 | ||||||||||||||||||
See next 80 embeddings (of 103 total) |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(29\) | \(1\) |
\(139\) | \(-1\) |
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 | 4031.2.a.e | ✓ | 103 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4031.2.a.e | ✓ | 103 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{103} - T_{2}^{102} - 166 T_{2}^{101} + 165 T_{2}^{100} + 13338 T_{2}^{99} - 13177 T_{2}^{98} + \cdots - 16810773 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4031))\).