Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [9016,2,Mod(1,9016)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(9016, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("9016.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 9016 = 2^{3} \cdot 7^{2} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 9016.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: | \(71.9931224624\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{11} - 2 x^{10} - 20 x^{9} + 37 x^{8} + 125 x^{7} - 215 x^{6} - 278 x^{5} + 443 x^{4} + 256 x^{3} + \cdots + 99 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 1288) |
| 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
\( x^{11} - 2 x^{10} - 20 x^{9} + 37 x^{8} + 125 x^{7} - 215 x^{6} - 278 x^{5} + 443 x^{4} + 256 x^{3} + \cdots + 99 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 63 \nu^{10} + 433 \nu^{9} + 784 \nu^{8} - 8424 \nu^{7} + 599 \nu^{6} + 50081 \nu^{5} + \cdots - 44740 ) / 2147 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 33 \nu^{10} - 851 \nu^{9} + 983 \nu^{8} + 15987 \nu^{7} - 24598 \nu^{6} - 88017 \nu^{5} + \cdots + 82658 ) / 2147 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 135 \nu^{10} - 831 \nu^{9} - 1341 \nu^{8} + 15404 \nu^{7} - 7547 \nu^{6} - 84329 \nu^{5} + \cdots + 70172 ) / 2147 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 168 \nu^{10} - 465 \nu^{9} + 4652 \nu^{8} + 9402 \nu^{7} - 43000 \nu^{6} - 55236 \nu^{5} + \cdots + 68311 ) / 2147 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 209 \nu^{10} - 606 \nu^{9} - 3869 \nu^{8} + 11529 \nu^{7} + 20568 \nu^{6} - 68603 \nu^{5} + \cdots + 12364 ) / 2147 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 244 \nu^{10} - 43 \nu^{9} - 5309 \nu^{8} + 50 \nu^{7} + 37135 \nu^{6} + 3077 \nu^{5} - 97536 \nu^{4} + \cdots - 32062 ) / 2147 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 303 \nu^{10} - 366 \nu^{9} - 5993 \nu^{8} + 6002 \nu^{7} + 35453 \nu^{6} - 29093 \nu^{5} + \cdots + 10449 ) / 2147 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 703 \nu^{10} + 1422 \nu^{9} + 13394 \nu^{8} - 25055 \nu^{7} - 75388 \nu^{6} + 130730 \nu^{5} + \cdots + 8358 ) / 2147 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( - 887 \nu^{10} + 1623 \nu^{9} + 17090 \nu^{8} - 28140 \nu^{7} - 97973 \nu^{6} + 143522 \nu^{5} + \cdots + 9219 ) / 2147 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{8} - \beta_{5} + \beta_{4} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{8} + 2\beta_{7} - \beta_{6} - 2\beta_{5} + \beta_{4} + \beta_{3} - 2\beta_{2} + 7\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{10} - \beta_{9} - 12\beta_{8} - \beta_{6} - 14\beta_{5} + 11\beta_{4} + 2\beta_{3} - 2\beta_{2} + 2\beta _1 + 31 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4 \beta_{10} - 5 \beta_{9} - 43 \beta_{8} + 26 \beta_{7} - 16 \beta_{6} - 32 \beta_{5} + 18 \beta_{4} + \cdots + 17 \)
|
| \(\nu^{6}\) | \(=\) |
\( 22 \beta_{10} - 23 \beta_{9} - 139 \beta_{8} + 5 \beta_{7} - 19 \beta_{6} - 172 \beta_{5} + 117 \beta_{4} + \cdots + 291 \)
|
| \(\nu^{7}\) | \(=\) |
\( 83 \beta_{10} - 100 \beta_{9} - 539 \beta_{8} + 305 \beta_{7} - 205 \beta_{6} - 433 \beta_{5} + \cdots + 249 \)
|
| \(\nu^{8}\) | \(=\) |
\( 341 \beta_{10} - 361 \beta_{9} - 1626 \beta_{8} + 134 \beta_{7} - 286 \beta_{6} - 2049 \beta_{5} + \cdots + 2926 \)
|
| \(\nu^{9}\) | \(=\) |
\( 1245 \beta_{10} - 1467 \beta_{9} - 6483 \beta_{8} + 3476 \beta_{7} - 2458 \beta_{6} - 5515 \beta_{5} + \cdots + 3399 \)
|
| \(\nu^{10}\) | \(=\) |
\( 4623 \beta_{10} - 4929 \beta_{9} - 19190 \beta_{8} + 2445 \beta_{7} - 3950 \beta_{6} - 24131 \beta_{5} + \cdots + 30624 \)
|
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 |
|
0 | −3.12265 | 0 | −3.87723 | 0 | 0 | 0 | 6.75093 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.54118 | 0 | 2.00363 | 0 | 0 | 0 | 3.45759 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.20281 | 0 | 1.85689 | 0 | 0 | 0 | −1.55326 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −0.968614 | 0 | −4.14269 | 0 | 0 | 0 | −2.06179 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.852282 | 0 | −1.25440 | 0 | 0 | 0 | −2.27361 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 0.630680 | 0 | 0.354326 | 0 | 0 | 0 | −2.60224 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 1.03171 | 0 | −3.07991 | 0 | 0 | 0 | −1.93557 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 1.27162 | 0 | 3.38170 | 0 | 0 | 0 | −1.38298 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 0 | 1.66463 | 0 | −0.919906 | 0 | 0 | 0 | −0.229023 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 0 | 2.66192 | 0 | 0.382846 | 0 | 0 | 0 | 4.08581 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 0 | 3.42697 | 0 | −3.70527 | 0 | 0 | 0 | 8.74415 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
| \(23\) | \(1\) |
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 | 9016.2.a.bo | 11 | |
| 7.b | odd | 2 | 1 | 9016.2.a.bn | 11 | ||
| 7.d | odd | 6 | 2 | 1288.2.q.b | ✓ | 22 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1288.2.q.b | ✓ | 22 | 7.d | odd | 6 | 2 | |
| 9016.2.a.bn | 11 | 7.b | odd | 2 | 1 | ||
| 9016.2.a.bo | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9016))\):
|
\( T_{3}^{11} - 2 T_{3}^{10} - 20 T_{3}^{9} + 37 T_{3}^{8} + 125 T_{3}^{7} - 215 T_{3}^{6} - 278 T_{3}^{5} + \cdots + 99 \)
|
|
\( T_{5}^{11} + 9 T_{5}^{10} + 2 T_{5}^{9} - 172 T_{5}^{8} - 308 T_{5}^{7} + 944 T_{5}^{6} + 2109 T_{5}^{5} + \cdots - 361 \)
|
|
\( T_{11}^{11} - 45 T_{11}^{9} - 23 T_{11}^{8} + 570 T_{11}^{7} + 269 T_{11}^{6} - 2626 T_{11}^{5} + \cdots + 1349 \)
|
|
\( T_{13}^{11} + 15 T_{13}^{10} + 30 T_{13}^{9} - 589 T_{13}^{8} - 3654 T_{13}^{7} - 2484 T_{13}^{6} + \cdots - 188779 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} \)
|
| $3$ |
\( T^{11} - 2 T^{10} + \cdots + 99 \)
|
| $5$ |
\( T^{11} + 9 T^{10} + \cdots - 361 \)
|
| $7$ |
\( T^{11} \)
|
| $11$ |
\( T^{11} - 45 T^{9} + \cdots + 1349 \)
|
| $13$ |
\( T^{11} + 15 T^{10} + \cdots - 188779 \)
|
| $17$ |
\( T^{11} + 5 T^{10} + \cdots + 295431 \)
|
| $19$ |
\( T^{11} - 109 T^{9} + \cdots - 3971 \)
|
| $23$ |
\( (T + 1)^{11} \)
|
| $29$ |
\( T^{11} - 5 T^{10} + \cdots + 3071547 \)
|
| $31$ |
\( T^{11} + 6 T^{10} + \cdots + 27799763 \)
|
| $37$ |
\( T^{11} + T^{10} + \cdots - 6314911 \)
|
| $41$ |
\( T^{11} + 20 T^{10} + \cdots - 4275 \)
|
| $43$ |
\( T^{11} + 7 T^{10} + \cdots + 1052839 \)
|
| $47$ |
\( T^{11} - 3 T^{10} + \cdots - 99779471 \)
|
| $53$ |
\( T^{11} - 19 T^{10} + \cdots + 2010171 \)
|
| $59$ |
\( T^{11} + \cdots + 698463501 \)
|
| $61$ |
\( T^{11} + \cdots + 427307967 \)
|
| $67$ |
\( T^{11} - 7 T^{10} + \cdots - 30479671 \)
|
| $71$ |
\( T^{11} + \cdots + 1104596825 \)
|
| $73$ |
\( T^{11} + \cdots + 858292547 \)
|
| $79$ |
\( T^{11} - 11 T^{10} + \cdots + 42508957 \)
|
| $83$ |
\( T^{11} + \cdots + 1185916559 \)
|
| $89$ |
\( T^{11} + \cdots - 2678187261 \)
|
| $97$ |
\( T^{11} + \cdots + 1520122425 \)
|
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