Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8619,2,Mod(1,8619)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8619, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("8619.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 8619 = 3 \cdot 13^{2} \cdot 17 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 8619.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: | \(68.8230615021\) |
Analytic rank: | \(1\) |
Dimension: | \(24\) |
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 | −2.62635 | −1.00000 | 4.89769 | 0.512917 | 2.62635 | −2.19487 | −7.61034 | 1.00000 | −1.34710 | ||||||||||||||||||
1.2 | −2.16723 | −1.00000 | 2.69690 | 0.362944 | 2.16723 | 2.98172 | −1.51034 | 1.00000 | −0.786585 | ||||||||||||||||||
1.3 | −1.86500 | −1.00000 | 1.47821 | −4.02513 | 1.86500 | 2.21617 | 0.973140 | 1.00000 | 7.50685 | ||||||||||||||||||
1.4 | −1.69539 | −1.00000 | 0.874348 | 2.79728 | 1.69539 | 0.609432 | 1.90842 | 1.00000 | −4.74247 | ||||||||||||||||||
1.5 | −1.57761 | −1.00000 | 0.488848 | −2.21771 | 1.57761 | −4.61789 | 2.38401 | 1.00000 | 3.49868 | ||||||||||||||||||
1.6 | −1.17199 | −1.00000 | −0.626431 | −3.45670 | 1.17199 | −1.76134 | 3.07816 | 1.00000 | 4.05124 | ||||||||||||||||||
1.7 | −0.936172 | −1.00000 | −1.12358 | −1.31747 | 0.936172 | 0.0103583 | 2.92421 | 1.00000 | 1.23338 | ||||||||||||||||||
1.8 | −0.470840 | −1.00000 | −1.77831 | −2.64995 | 0.470840 | 3.75195 | 1.77898 | 1.00000 | 1.24770 | ||||||||||||||||||
1.9 | −0.433070 | −1.00000 | −1.81245 | 2.79890 | 0.433070 | −3.33739 | 1.65106 | 1.00000 | −1.21212 | ||||||||||||||||||
1.10 | −0.294893 | −1.00000 | −1.91304 | 2.91911 | 0.294893 | 1.40550 | 1.15393 | 1.00000 | −0.860824 | ||||||||||||||||||
1.11 | −0.271243 | −1.00000 | −1.92643 | 0.00688268 | 0.271243 | −0.318502 | 1.06502 | 1.00000 | −0.00186688 | ||||||||||||||||||
1.12 | 0.324966 | −1.00000 | −1.89440 | 0.225941 | −0.324966 | 3.77265 | −1.26555 | 1.00000 | 0.0734231 | ||||||||||||||||||
1.13 | 0.342241 | −1.00000 | −1.88287 | −4.18392 | −0.342241 | −3.67826 | −1.32888 | 1.00000 | −1.43191 | ||||||||||||||||||
1.14 | 0.671598 | −1.00000 | −1.54896 | −1.50727 | −0.671598 | −4.47027 | −2.38347 | 1.00000 | −1.01228 | ||||||||||||||||||
1.15 | 0.963570 | −1.00000 | −1.07153 | −1.50390 | −0.963570 | 2.76951 | −2.95964 | 1.00000 | −1.44911 | ||||||||||||||||||
1.16 | 1.10514 | −1.00000 | −0.778661 | −0.625764 | −1.10514 | −4.27315 | −3.07082 | 1.00000 | −0.691558 | ||||||||||||||||||
1.17 | 1.16683 | −1.00000 | −0.638517 | 0.385677 | −1.16683 | 3.99174 | −3.07869 | 1.00000 | 0.450018 | ||||||||||||||||||
1.18 | 1.46048 | −1.00000 | 0.133014 | 3.73069 | −1.46048 | −2.14359 | −2.72670 | 1.00000 | 5.44861 | ||||||||||||||||||
1.19 | 2.10278 | −1.00000 | 2.42170 | −3.44599 | −2.10278 | 1.48926 | 0.886753 | 1.00000 | −7.24618 | ||||||||||||||||||
1.20 | 2.12481 | −1.00000 | 2.51480 | 1.58682 | −2.12481 | −2.67484 | 1.09385 | 1.00000 | 3.37169 | ||||||||||||||||||
See all 24 embeddings |
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(13\) | \(1\) |
\(17\) | \(1\) |
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 | 8619.2.a.bw | yes | 24 |
13.b | even | 2 | 1 | 8619.2.a.bt | ✓ | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
8619.2.a.bt | ✓ | 24 | 13.b | even | 2 | 1 | |
8619.2.a.bw | yes | 24 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8619))\):
\( T_{2}^{24} - 7 T_{2}^{23} - 8 T_{2}^{22} + 154 T_{2}^{21} - 136 T_{2}^{20} - 1343 T_{2}^{19} + 2302 T_{2}^{18} + 5900 T_{2}^{17} - 14226 T_{2}^{16} - 13305 T_{2}^{15} + 46628 T_{2}^{14} + 11811 T_{2}^{13} - 87948 T_{2}^{12} + \cdots - 13 \) |
\( T_{5}^{24} + 13 T_{5}^{23} + 13 T_{5}^{22} - 498 T_{5}^{21} - 1853 T_{5}^{20} + 6344 T_{5}^{19} + 41405 T_{5}^{18} - 15884 T_{5}^{17} - 421795 T_{5}^{16} - 336736 T_{5}^{15} + 2227486 T_{5}^{14} + 3387472 T_{5}^{13} + \cdots + 343 \) |
\( T_{7}^{24} + 12 T_{7}^{23} - 27 T_{7}^{22} - 818 T_{7}^{21} - 1075 T_{7}^{20} + 22840 T_{7}^{19} + 62844 T_{7}^{18} - 333768 T_{7}^{17} - 1291597 T_{7}^{16} + 2667478 T_{7}^{15} + 14325310 T_{7}^{14} + \cdots - 810641 \) |