Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8993,2,Mod(1,8993)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8993, 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("8993.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8993 = 17 \cdot 23^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8993.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: | \(71.8094665377\) |
| Analytic rank: | \(0\) |
| Dimension: | \(11\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{11} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{11} - x^{10} - 17x^{9} + 14x^{8} + 99x^{7} - 55x^{6} - 248x^{5} + 55x^{4} + 276x^{3} + 25x^{2} - 114x - 39 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{10}\) 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
\( x^{11} - x^{10} - 17x^{9} + 14x^{8} + 99x^{7} - 55x^{6} - 248x^{5} + 55x^{4} + 276x^{3} + 25x^{2} - 114x - 39 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{10} - \nu^{9} - 17 \nu^{8} + 15 \nu^{7} + 99 \nu^{6} - 70 \nu^{5} - 248 \nu^{4} + 124 \nu^{3} + \cdots - 108 ) / 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 2 \nu^{10} + 3 \nu^{9} + 33 \nu^{8} - 45 \nu^{7} - 183 \nu^{6} + 209 \nu^{5} + 426 \nu^{4} + \cdots + 152 ) / 5 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 3 \nu^{10} + 2 \nu^{9} + 50 \nu^{8} - 25 \nu^{7} - 277 \nu^{6} + 71 \nu^{5} + 606 \nu^{4} + 15 \nu^{3} + \cdots + 50 ) / 5 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 4 \nu^{10} + 6 \nu^{9} + 63 \nu^{8} - 85 \nu^{7} - 321 \nu^{6} + 348 \nu^{5} + 645 \nu^{4} + \cdots + 147 ) / 5 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5 \nu^{10} - 8 \nu^{9} - 78 \nu^{8} + 115 \nu^{7} + 390 \nu^{6} - 487 \nu^{5} - 754 \nu^{4} + 671 \nu^{3} + \cdots - 157 ) / 5 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 8 \nu^{10} + 8 \nu^{9} + 131 \nu^{8} - 110 \nu^{7} - 707 \nu^{6} + 420 \nu^{5} + 1504 \nu^{4} + \cdots + 219 ) / 5 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 9 \nu^{10} - 12 \nu^{9} - 143 \nu^{8} + 170 \nu^{7} + 736 \nu^{6} - 697 \nu^{5} - 1474 \nu^{4} + \cdots - 272 ) / 5 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 11 \nu^{10} - 22 \nu^{9} - 172 \nu^{8} + 330 \nu^{7} + 874 \nu^{6} - 1524 \nu^{5} - 1822 \nu^{4} + \cdots - 778 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{9} - \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + \beta_{3} + 5\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{9} + 2\beta_{7} + \beta_{6} - \beta_{5} + 9\beta_{2} - \beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{10} - 10\beta_{9} - 10\beta_{8} + 8\beta_{7} + 8\beta_{5} + 13\beta_{4} + 9\beta_{3} - \beta_{2} + 31\beta _1 + 17 \)
|
| \(\nu^{6}\) | \(=\) |
\( -12\beta_{9} + 2\beta_{8} + 25\beta_{7} + 12\beta_{6} - 15\beta_{5} + 2\beta_{3} + 73\beta_{2} - 12\beta _1 + 104 \)
|
| \(\nu^{7}\) | \(=\) |
\( 14 \beta_{10} - 84 \beta_{9} - 84 \beta_{8} + 59 \beta_{7} + 4 \beta_{6} + 56 \beta_{5} + 127 \beta_{4} + \cdots + 131 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 107 \beta_{9} + 34 \beta_{8} + 238 \beta_{7} + 116 \beta_{6} - 160 \beta_{5} + \beta_{4} + 31 \beta_{3} + \cdots + 732 \)
|
| \(\nu^{9}\) | \(=\) |
\( 141 \beta_{10} - 672 \beta_{9} - 671 \beta_{8} + 441 \beta_{7} + 66 \beta_{6} + 388 \beta_{5} + \cdots + 992 \)
|
| \(\nu^{10}\) | \(=\) |
\( \beta_{10} - 867 \beta_{9} + 393 \beta_{8} + 2059 \beta_{7} + 1038 \beta_{6} - 1499 \beta_{5} + \cdots + 5368 \)
|
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 |
|
−2.79113 | 2.59566 | 5.79039 | 0.921592 | −7.24481 | −2.76316 | −10.5794 | 3.73744 | −2.57228 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.39541 | 0.653767 | 3.73800 | 3.89009 | −1.56604 | 4.21437 | −4.16322 | −2.57259 | −9.31837 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.86672 | −3.31348 | 1.48464 | −2.67010 | 6.18534 | 4.60839 | 0.962029 | 7.97917 | 4.98433 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.13994 | −1.86678 | −0.700529 | 1.21652 | 2.12802 | −1.79719 | 3.07845 | 0.484865 | −1.38676 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.10029 | −0.107771 | −0.789367 | 1.34435 | 0.118579 | 2.78376 | 3.06911 | −2.98839 | −1.47917 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.547710 | 2.40839 | −1.70001 | 3.61217 | 1.31910 | −0.381710 | −2.02653 | 2.80034 | 1.97842 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.730083 | 1.14432 | −1.46698 | −1.11734 | 0.835446 | 4.94188 | −2.53118 | −1.69054 | −0.815753 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0.824684 | −2.51089 | −1.31990 | −2.44156 | −2.07069 | 0.237400 | −2.73787 | 3.30456 | −2.01352 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 1.28610 | 0.484800 | −0.345943 | −2.10511 | 0.623502 | −3.29753 | −3.01712 | −2.76497 | −2.70738 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.07787 | −3.14926 | 2.31755 | 0.415601 | −6.54377 | 0.157634 | 0.659826 | 6.91787 | 0.863565 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.82704 | −1.33874 | 5.99215 | 1.93380 | −3.78468 | 3.29616 | 11.2860 | −1.20776 | 5.46692 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(17\) | \(1\) |
| \(23\) | \(-1\) |
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 | 8993.2.a.i | yes | 11 |
| 23.b | odd | 2 | 1 | 8993.2.a.h | ✓ | 11 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8993.2.a.h | ✓ | 11 | 23.b | odd | 2 | 1 | |
| 8993.2.a.i | yes | 11 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8993))\):
|
\( T_{2}^{11} + T_{2}^{10} - 17 T_{2}^{9} - 14 T_{2}^{8} + 99 T_{2}^{7} + 55 T_{2}^{6} - 248 T_{2}^{5} + \cdots + 39 \)
|
|
\( T_{5}^{11} - 5 T_{5}^{10} - 15 T_{5}^{9} + 92 T_{5}^{8} + 52 T_{5}^{7} - 570 T_{5}^{6} + 175 T_{5}^{5} + \cdots - 261 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} + T^{10} + \cdots + 39 \)
|
| $3$ |
\( T^{11} + 5 T^{10} + \cdots - 16 \)
|
| $5$ |
\( T^{11} - 5 T^{10} + \cdots - 261 \)
|
| $7$ |
\( T^{11} - 12 T^{10} + \cdots - 206 \)
|
| $11$ |
\( T^{11} + 9 T^{10} + \cdots - 882 \)
|
| $13$ |
\( T^{11} - 4 T^{10} + \cdots + 326897 \)
|
| $17$ |
\( (T + 1)^{11} \)
|
| $19$ |
\( T^{11} + 2 T^{10} + \cdots + 49784 \)
|
| $23$ |
\( T^{11} \)
|
| $29$ |
\( T^{11} - 2 T^{10} + \cdots + 20397 \)
|
| $31$ |
\( T^{11} + 11 T^{10} + \cdots - 28019498 \)
|
| $37$ |
\( T^{11} - 15 T^{10} + \cdots - 3567394 \)
|
| $41$ |
\( T^{11} - 6 T^{10} + \cdots - 2523 \)
|
| $43$ |
\( T^{11} - 30 T^{10} + \cdots + 25705562 \)
|
| $47$ |
\( T^{11} + 17 T^{10} + \cdots + 1374894 \)
|
| $53$ |
\( T^{11} - 29 T^{10} + \cdots - 9441213 \)
|
| $59$ |
\( T^{11} + \cdots - 1630750272 \)
|
| $61$ |
\( T^{11} + 20 T^{10} + \cdots - 15679159 \)
|
| $67$ |
\( T^{11} + \cdots - 266741632 \)
|
| $71$ |
\( T^{11} - 28 T^{10} + \cdots - 6566496 \)
|
| $73$ |
\( T^{11} + 33 T^{10} + \cdots + 6219 \)
|
| $79$ |
\( T^{11} - 31 T^{10} + \cdots + 76799304 \)
|
| $83$ |
\( T^{11} + \cdots - 7912142814 \)
|
| $89$ |
\( T^{11} - 18 T^{10} + \cdots - 6824607 \)
|
| $97$ |
\( T^{11} + 5 T^{10} + \cdots + 175561 \)
|
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