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: | \(1\) |
Dimension: | \(59\) |
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 | −2.77566 | 2.36943 | 5.70431 | 1.60665 | −6.57673 | 0.747628 | −10.2819 | 2.61418 | −4.45952 | ||||||||||||||||||
1.2 | −2.69273 | −0.478176 | 5.25077 | 0.135724 | 1.28760 | −4.02730 | −8.75344 | −2.77135 | −0.365467 | ||||||||||||||||||
1.3 | −2.49699 | −1.64487 | 4.23496 | 3.35969 | 4.10722 | 2.65574 | −5.58068 | −0.294408 | −8.38911 | ||||||||||||||||||
1.4 | −2.47917 | −2.68137 | 4.14631 | 1.95848 | 6.64759 | 0.410621 | −5.32107 | 4.18976 | −4.85541 | ||||||||||||||||||
1.5 | −2.43722 | 1.24735 | 3.94004 | 0.277998 | −3.04005 | 4.01805 | −4.72830 | −1.44413 | −0.677543 | ||||||||||||||||||
1.6 | −2.32696 | −0.944934 | 3.41472 | −1.62617 | 2.19882 | 3.30319 | −3.29200 | −2.10710 | 3.78402 | ||||||||||||||||||
1.7 | −2.24849 | 2.23227 | 3.05570 | −0.764282 | −5.01923 | −0.688314 | −2.37374 | 1.98301 | 1.71848 | ||||||||||||||||||
1.8 | −2.19361 | −1.71235 | 2.81195 | −2.23633 | 3.75624 | −3.04720 | −1.78110 | −0.0678453 | 4.90564 | ||||||||||||||||||
1.9 | −2.18971 | −0.875363 | 2.79481 | −1.80760 | 1.91679 | −0.359878 | −1.74040 | −2.23374 | 3.95812 | ||||||||||||||||||
1.10 | −2.00564 | 2.28170 | 2.02257 | −4.04874 | −4.57625 | −0.876531 | −0.0452719 | 2.20615 | 8.12030 | ||||||||||||||||||
1.11 | −1.84055 | 2.78518 | 1.38763 | 3.40108 | −5.12627 | −2.70641 | 1.12710 | 4.75723 | −6.25987 | ||||||||||||||||||
1.12 | −1.78339 | 0.526387 | 1.18048 | 0.564222 | −0.938754 | 1.86157 | 1.46152 | −2.72292 | −1.00623 | ||||||||||||||||||
1.13 | −1.69299 | −2.95831 | 0.866208 | −2.49874 | 5.00838 | 1.49318 | 1.91950 | 5.75160 | 4.23033 | ||||||||||||||||||
1.14 | −1.67397 | −0.759884 | 0.802190 | 2.08858 | 1.27203 | −1.20948 | 2.00510 | −2.42258 | −3.49623 | ||||||||||||||||||
1.15 | −1.54716 | −2.20924 | 0.393702 | 3.10213 | 3.41805 | −0.403081 | 2.48520 | 1.88074 | −4.79950 | ||||||||||||||||||
1.16 | −1.45467 | 0.452328 | 0.116052 | −2.57197 | −0.657986 | 3.66955 | 2.74051 | −2.79540 | 3.74136 | ||||||||||||||||||
1.17 | −1.40700 | 1.38079 | −0.0203573 | 0.687727 | −1.94276 | −3.68772 | 2.84264 | −1.09343 | −0.967630 | ||||||||||||||||||
1.18 | −1.33108 | 2.24902 | −0.228227 | −0.768051 | −2.99362 | 2.32195 | 2.96595 | 2.05807 | 1.02234 | ||||||||||||||||||
1.19 | −1.25853 | 0.116839 | −0.416106 | −2.24208 | −0.147046 | −3.42737 | 3.04074 | −2.98635 | 2.82172 | ||||||||||||||||||
1.20 | −1.17292 | −0.112413 | −0.624270 | 4.03608 | 0.131851 | 1.76356 | 3.07805 | −2.98736 | −4.73397 | ||||||||||||||||||
See all 59 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.b | ✓ | 59 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4031.2.a.b | ✓ | 59 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{59} + 5 T_{2}^{58} - 67 T_{2}^{57} - 366 T_{2}^{56} + 2065 T_{2}^{55} + 12599 T_{2}^{54} + \cdots + 83 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4031))\).