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: | \(0\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - x^{8} - 16x^{7} + 16x^{6} + 78x^{5} - 75x^{4} - 110x^{3} + 83x^{2} + 18x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 663) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{9} - x^{8} - 16x^{7} + 16x^{6} + 78x^{5} - 75x^{4} - 110x^{3} + 83x^{2} + 18x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -25\nu^{8} - 39\nu^{7} + 354\nu^{6} + 587\nu^{5} - 1551\nu^{4} - 2284\nu^{3} + 2610\nu^{2} + 1107\nu - 1681 ) / 673 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 25\nu^{8} + 39\nu^{7} - 354\nu^{6} - 587\nu^{5} + 1551\nu^{4} + 2284\nu^{3} - 1937\nu^{2} - 1107\nu - 1011 ) / 673 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -28\nu^{8} + 64\nu^{7} + 558\nu^{6} - 877\nu^{5} - 3460\nu^{4} + 3472\nu^{3} + 6692\nu^{2} - 3525\nu - 1452 ) / 673 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 77 \nu^{8} + 176 \nu^{7} + 1198 \nu^{6} - 2580 \nu^{5} - 5477 \nu^{4} + 10894 \nu^{3} + 6289 \nu^{2} + \cdots - 628 ) / 673 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 127 \nu^{8} - 98 \nu^{7} - 1906 \nu^{6} + 1406 \nu^{5} + 8579 \nu^{4} - 5653 \nu^{3} - 10836 \nu^{2} + \cdots + 1298 ) / 673 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 141 \nu^{8} - 130 \nu^{7} - 2185 \nu^{6} + 2181 \nu^{5} + 10309 \nu^{4} - 10754 \nu^{3} - 14182 \nu^{2} + \cdots + 2024 ) / 673 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 258 \nu^{8} + 109 \nu^{7} + 4084 \nu^{6} - 2072 \nu^{5} - 19479 \nu^{4} + 11129 \nu^{3} + \cdots - 2515 ) / 673 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{8} + 2\beta_{7} + \beta_{6} + 2\beta_{5} - \beta_{4} + 7\beta_{3} + 8\beta_{2} + 26 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{7} + 8\beta_{6} + 10\beta_{5} + \beta_{4} - 10\beta_{3} + 10\beta_{2} + 41\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 12\beta_{8} + 23\beta_{7} + 12\beta_{6} + 21\beta_{5} - 7\beta_{4} + 49\beta_{3} + 59\beta_{2} + \beta _1 + 178 \)
|
| \(\nu^{7}\) | \(=\) |
\( -2\beta_{8} + 26\beta_{7} + 56\beta_{6} + 87\beta_{5} + 15\beta_{4} - 83\beta_{3} + 84\beta_{2} + 291\beta _1 + 5 \)
|
| \(\nu^{8}\) | \(=\) |
\( 111 \beta_{8} + 208 \beta_{7} + 117 \beta_{6} + 181 \beta_{5} - 37 \beta_{4} + 350 \beta_{3} + \cdots + 1250 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.74152 | −1.00000 | 5.51592 | −0.0695055 | 2.74152 | −0.0285206 | −9.63895 | 1.00000 | 0.190551 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.54162 | −1.00000 | 4.45982 | 2.88279 | 2.54162 | 4.59425 | −6.25191 | 1.00000 | −7.32695 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.34438 | −1.00000 | −0.192655 | −3.86620 | 1.34438 | −5.14964 | 2.94775 | 1.00000 | 5.19763 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.248892 | −1.00000 | −1.93805 | 4.26511 | 0.248892 | −0.620237 | 0.980151 | 1.00000 | −1.06155 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.0830122 | −1.00000 | −1.99311 | −1.15934 | −0.0830122 | 3.41331 | −0.331477 | 1.00000 | −0.0962390 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.882952 | −1.00000 | −1.22039 | −1.41987 | −0.882952 | −2.20717 | −2.84346 | 1.00000 | −1.25368 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.82898 | −1.00000 | 1.34517 | 2.41736 | −1.82898 | −5.18563 | −1.19768 | 1.00000 | 4.42131 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.30334 | −1.00000 | 3.30539 | −1.07964 | −2.30334 | −1.27371 | 3.00677 | 1.00000 | −2.48678 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.77811 | −1.00000 | 5.71792 | 3.02929 | −2.77811 | 3.45734 | 10.3288 | 1.00000 | 8.41571 | ||||||||||||||||||||||||||||||||||||||||||||||
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.bi | 9 | |
| 13.b | even | 2 | 1 | 8619.2.a.bh | 9 | ||
| 13.e | even | 6 | 2 | 663.2.i.g | ✓ | 18 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 663.2.i.g | ✓ | 18 | 13.e | even | 6 | 2 | |
| 8619.2.a.bh | 9 | 13.b | even | 2 | 1 | ||
| 8619.2.a.bi | 9 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8619))\):
|
\( T_{2}^{9} - T_{2}^{8} - 16T_{2}^{7} + 16T_{2}^{6} + 78T_{2}^{5} - 75T_{2}^{4} - 110T_{2}^{3} + 83T_{2}^{2} + 18T_{2} - 2 \)
|
|
\( T_{5}^{9} - 5T_{5}^{8} - 18T_{5}^{7} + 105T_{5}^{6} + 56T_{5}^{5} - 515T_{5}^{4} - 217T_{5}^{3} + 870T_{5}^{2} + 680T_{5} + 43 \)
|
|
\( T_{7}^{9} + 3T_{7}^{8} - 48T_{7}^{7} - 116T_{7}^{6} + 712T_{7}^{5} + 1376T_{7}^{4} - 2775T_{7}^{3} - 6234T_{7}^{2} - 2700T_{7} - 72 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} - T^{8} - 16 T^{7} + \cdots - 2 \)
|
| $3$ |
\( (T + 1)^{9} \)
|
| $5$ |
\( T^{9} - 5 T^{8} + \cdots + 43 \)
|
| $7$ |
\( T^{9} + 3 T^{8} + \cdots - 72 \)
|
| $11$ |
\( T^{9} - 8 T^{8} + \cdots + 3424 \)
|
| $13$ |
\( T^{9} \)
|
| $17$ |
\( (T - 1)^{9} \)
|
| $19$ |
\( T^{9} + 5 T^{8} + \cdots + 2268 \)
|
| $23$ |
\( T^{9} + 21 T^{8} + \cdots - 126091 \)
|
| $29$ |
\( T^{9} + 3 T^{8} + \cdots + 123239 \)
|
| $31$ |
\( T^{9} - T^{8} + \cdots + 11048 \)
|
| $37$ |
\( T^{9} + 12 T^{8} + \cdots - 752 \)
|
| $41$ |
\( T^{9} - 29 T^{8} + \cdots - 32720522 \)
|
| $43$ |
\( T^{9} + T^{8} + \cdots + 37836 \)
|
| $47$ |
\( T^{9} + 5 T^{8} + \cdots + 44259392 \)
|
| $53$ |
\( T^{9} - 5 T^{8} + \cdots - 15181336 \)
|
| $59$ |
\( T^{9} - 7 T^{8} + \cdots + 61963060 \)
|
| $61$ |
\( T^{9} - 2 T^{8} + \cdots + 4049024 \)
|
| $67$ |
\( T^{9} - 6 T^{8} + \cdots + 3197861 \)
|
| $71$ |
\( T^{9} + 7 T^{8} + \cdots - 8780944 \)
|
| $73$ |
\( T^{9} - 34 T^{8} + \cdots - 29636612 \)
|
| $79$ |
\( T^{9} - 28 T^{8} + \cdots - 10656 \)
|
| $83$ |
\( T^{9} + 9 T^{8} + \cdots + 182476652 \)
|
| $89$ |
\( T^{9} + 45 T^{8} + \cdots + 73488364 \)
|
| $97$ |
\( T^{9} + 17 T^{8} + \cdots - 144448 \)
|
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