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: | \(61\) |
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 | −2.62831 | 0.935172 | 4.90799 | −2.36331 | −2.45792 | 0.343928 | −7.64309 | −2.12545 | 6.21151 | ||||||||||||||||||
1.2 | −2.62306 | 1.48834 | 4.88043 | 0.915377 | −3.90400 | −2.34390 | −7.55555 | −0.784847 | −2.40109 | ||||||||||||||||||
1.3 | −2.53428 | −2.04027 | 4.42257 | −2.47983 | 5.17061 | −0.629929 | −6.13948 | 1.16269 | 6.28458 | ||||||||||||||||||
1.4 | −2.49843 | −0.867725 | 4.24214 | 3.27534 | 2.16795 | 1.98132 | −5.60183 | −2.24705 | −8.18321 | ||||||||||||||||||
1.5 | −2.42000 | −2.29529 | 3.85639 | 1.32254 | 5.55461 | 0.993943 | −4.49245 | 2.26837 | −3.20054 | ||||||||||||||||||
1.6 | −2.31412 | 0.311751 | 3.35513 | 1.29585 | −0.721429 | −1.13716 | −3.13593 | −2.90281 | −2.99874 | ||||||||||||||||||
1.7 | −2.25610 | 3.35950 | 3.09001 | −1.55616 | −7.57937 | −0.413231 | −2.45917 | 8.28621 | 3.51085 | ||||||||||||||||||
1.8 | −2.23009 | 1.62667 | 2.97329 | 2.06861 | −3.62762 | 3.32810 | −2.17052 | −0.353941 | −4.61317 | ||||||||||||||||||
1.9 | −2.10281 | −1.04272 | 2.42182 | 3.50696 | 2.19264 | −3.91864 | −0.887016 | −1.91274 | −7.37448 | ||||||||||||||||||
1.10 | −1.99079 | −3.03517 | 1.96324 | −1.96489 | 6.04239 | 0.618316 | 0.0731745 | 6.21228 | 3.91168 | ||||||||||||||||||
1.11 | −1.87723 | 1.73503 | 1.52398 | −2.87174 | −3.25705 | −0.555991 | 0.893598 | 0.0103383 | 5.39090 | ||||||||||||||||||
1.12 | −1.83395 | 2.22148 | 1.36337 | 1.24306 | −4.07408 | 2.01007 | 1.16755 | 1.93497 | −2.27970 | ||||||||||||||||||
1.13 | −1.77273 | −1.39503 | 1.14258 | −2.16266 | 2.47302 | 3.73562 | 1.51997 | −1.05388 | 3.83382 | ||||||||||||||||||
1.14 | −1.73344 | −1.78807 | 1.00480 | −2.76512 | 3.09951 | −4.11682 | 1.72512 | 0.197207 | 4.79315 | ||||||||||||||||||
1.15 | −1.55145 | 0.350675 | 0.406987 | −2.80587 | −0.544053 | −3.32540 | 2.47147 | −2.87703 | 4.35316 | ||||||||||||||||||
1.16 | −1.50584 | −0.996538 | 0.267567 | 0.533194 | 1.50063 | 3.56628 | 2.60877 | −2.00691 | −0.802908 | ||||||||||||||||||
1.17 | −1.44140 | 2.16928 | 0.0776255 | 0.0237256 | −3.12680 | 1.29161 | 2.77090 | 1.70580 | −0.0341981 | ||||||||||||||||||
1.18 | −1.44025 | 1.49462 | 0.0743258 | 2.03878 | −2.15264 | −3.27699 | 2.77346 | −0.766098 | −2.93636 | ||||||||||||||||||
1.19 | −1.19064 | −0.642478 | −0.582384 | −0.355693 | 0.764958 | 0.205071 | 3.07468 | −2.58722 | 0.423501 | ||||||||||||||||||
1.20 | −1.13471 | 1.15618 | −0.712442 | 3.10962 | −1.31192 | −1.02298 | 3.07782 | −1.66325 | −3.52850 | ||||||||||||||||||
See all 61 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.c | ✓ | 61 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
4031.2.a.c | ✓ | 61 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{61} + T_{2}^{60} - 82 T_{2}^{59} - 79 T_{2}^{58} + 3176 T_{2}^{57} + 2941 T_{2}^{56} - 77297 T_{2}^{55} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4031))\).