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: | \(98\) |
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 | −2.78351 | 2.95190 | 5.74795 | −3.19044 | −8.21665 | −4.40114 | −10.4325 | 5.71371 | 8.88064 | ||||||||||||||||||
1.2 | −2.76750 | 0.603367 | 5.65907 | 3.54828 | −1.66982 | 1.63115 | −10.1265 | −2.63595 | −9.81988 | ||||||||||||||||||
1.3 | −2.75511 | −2.40315 | 5.59062 | −0.846757 | 6.62092 | 3.63959 | −9.89254 | 2.77511 | 2.33291 | ||||||||||||||||||
1.4 | −2.74332 | −0.716173 | 5.52580 | −3.18087 | 1.96469 | 1.27957 | −9.67240 | −2.48710 | 8.72614 | ||||||||||||||||||
1.5 | −2.61634 | −3.34235 | 4.84525 | −3.10216 | 8.74473 | −4.97241 | −7.44414 | 8.17130 | 8.11632 | ||||||||||||||||||
1.6 | −2.59643 | 0.964956 | 4.74146 | 3.18498 | −2.50544 | −0.304804 | −7.11802 | −2.06886 | −8.26958 | ||||||||||||||||||
1.7 | −2.58392 | 3.19893 | 4.67665 | 1.67511 | −8.26577 | 1.22424 | −6.91625 | 7.23312 | −4.32836 | ||||||||||||||||||
1.8 | −2.57508 | 1.48970 | 4.63103 | −2.75202 | −3.83609 | 2.46323 | −6.77510 | −0.780799 | 7.08668 | ||||||||||||||||||
1.9 | −2.48847 | −3.00036 | 4.19251 | 0.297405 | 7.46633 | −2.60645 | −5.45599 | 6.00219 | −0.740084 | ||||||||||||||||||
1.10 | −2.44621 | 0.244458 | 3.98396 | −3.48270 | −0.597997 | 1.44143 | −4.85320 | −2.94024 | 8.51943 | ||||||||||||||||||
1.11 | −2.43943 | −2.23361 | 3.95082 | 1.05551 | 5.44873 | −1.84487 | −4.75890 | 1.98900 | −2.57484 | ||||||||||||||||||
1.12 | −2.22708 | 1.27662 | 2.95990 | 0.197788 | −2.84313 | −3.89763 | −2.13778 | −1.37025 | −0.440491 | ||||||||||||||||||
1.13 | −2.11438 | 2.42067 | 2.47061 | 1.02780 | −5.11822 | 0.853130 | −0.995051 | 2.85965 | −2.17316 | ||||||||||||||||||
1.14 | −2.08450 | 0.675318 | 2.34516 | −1.05546 | −1.40770 | −4.46025 | −0.719483 | −2.54395 | 2.20012 | ||||||||||||||||||
1.15 | −2.03151 | −1.34287 | 2.12703 | 3.37426 | 2.72805 | 0.525143 | −0.258068 | −1.19670 | −6.85484 | ||||||||||||||||||
1.16 | −1.99223 | −2.07417 | 1.96900 | 0.414014 | 4.13224 | 4.11338 | 0.0617619 | 1.30220 | −0.824814 | ||||||||||||||||||
1.17 | −1.97068 | 0.422540 | 1.88358 | 2.63171 | −0.832693 | −1.79791 | 0.229417 | −2.82146 | −5.18627 | ||||||||||||||||||
1.18 | −1.96419 | −0.913998 | 1.85806 | −4.29424 | 1.79527 | −2.01487 | 0.278799 | −2.16461 | 8.43472 | ||||||||||||||||||
1.19 | −1.93664 | 0.437969 | 1.75056 | 3.14696 | −0.848187 | 5.11562 | 0.483081 | −2.80818 | −6.09452 | ||||||||||||||||||
1.20 | −1.86328 | 0.307270 | 1.47182 | 0.176749 | −0.572530 | −0.402020 | 0.984149 | −2.90559 | −0.329334 | ||||||||||||||||||
See all 98 embeddings |
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.d | ✓ | 98 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4031.2.a.d | ✓ | 98 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{98} - 6 T_{2}^{97} - 138 T_{2}^{96} + 887 T_{2}^{95} + 9088 T_{2}^{94} - 63290 T_{2}^{93} + \cdots - 142987209 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4031))\).