Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8001,2,Mod(1,8001)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8001, 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("8001.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8001 = 3^{2} \cdot 7 \cdot 127 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8001.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: | \(63.8883066572\) |
| 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: |
\( x^{11} - 2 x^{10} - 15 x^{9} + 25 x^{8} + 88 x^{7} - 112 x^{6} - 247 x^{5} + 215 x^{4} + 313 x^{3} + \cdots + 57 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2667) |
| 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
\( x^{11} - 2 x^{10} - 15 x^{9} + 25 x^{8} + 88 x^{7} - 112 x^{6} - 247 x^{5} + 215 x^{4} + 313 x^{3} + \cdots + 57 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 4\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - \nu^{10} + 6 \nu^{9} + 5 \nu^{8} - 66 \nu^{7} + 8 \nu^{6} + 255 \nu^{5} - 59 \nu^{4} - 399 \nu^{3} + \cdots - 51 ) / 7 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 2 \nu^{10} - 5 \nu^{9} - 24 \nu^{8} + 55 \nu^{7} + 110 \nu^{6} - 209 \nu^{5} - 246 \nu^{4} + 308 \nu^{3} + \cdots - 45 ) / 7 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 3 \nu^{10} - 11 \nu^{9} - 29 \nu^{8} + 128 \nu^{7} + 81 \nu^{6} - 520 \nu^{5} - 19 \nu^{4} + 840 \nu^{3} + \cdots + 146 ) / 7 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 2 \nu^{10} - 12 \nu^{9} - 10 \nu^{8} + 139 \nu^{7} - 37 \nu^{6} - 559 \nu^{5} + 272 \nu^{4} + 882 \nu^{3} + \cdots + 193 ) / 7 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 4 \nu^{10} + 17 \nu^{9} + 34 \nu^{8} - 194 \nu^{7} - 66 \nu^{6} + 761 \nu^{5} - 89 \nu^{4} + \cdots - 204 ) / 7 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 8 \nu^{10} - 27 \nu^{9} - 82 \nu^{8} + 304 \nu^{7} + 279 \nu^{6} - 1172 \nu^{5} - 333 \nu^{4} + \cdots + 191 ) / 7 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 10 \nu^{10} - 32 \nu^{9} - 99 \nu^{8} + 352 \nu^{7} + 298 \nu^{6} - 1325 \nu^{5} - 180 \nu^{4} + \cdots + 321 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} + 2\beta_{8} + \beta_{7} - \beta_{5} + \beta_{3} + 6\beta_{2} + 2\beta _1 + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{9} + 4\beta_{8} + 3\beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + 9\beta_{3} + 9\beta_{2} + 21\beta _1 + 10 \)
|
| \(\nu^{6}\) | \(=\) |
\( 11 \beta_{9} + 23 \beta_{8} + 13 \beta_{7} - \beta_{6} - 9 \beta_{5} + \beta_{4} + 14 \beta_{3} + \cdots + 65 \)
|
| \(\nu^{7}\) | \(=\) |
\( 25 \beta_{9} + 53 \beta_{8} + 39 \beta_{7} - 10 \beta_{6} - 12 \beta_{5} + 12 \beta_{4} + 71 \beta_{3} + \cdots + 77 \)
|
| \(\nu^{8}\) | \(=\) |
\( \beta_{10} + 94 \beta_{9} + 206 \beta_{8} + 127 \beta_{7} - 15 \beta_{6} - 65 \beta_{5} + 17 \beta_{4} + \cdots + 359 \)
|
| \(\nu^{9}\) | \(=\) |
\( 2 \beta_{10} + 231 \beta_{9} + 513 \beta_{8} + 372 \beta_{7} - 79 \beta_{6} - 108 \beta_{5} + 107 \beta_{4} + \cdots + 561 \)
|
| \(\nu^{10}\) | \(=\) |
\( 17 \beta_{10} + 745 \beta_{9} + 1696 \beta_{8} + 1103 \beta_{7} - 152 \beta_{6} - 449 \beta_{5} + \cdots + 2148 \)
|
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 |
|
−2.12999 | 0 | 2.53686 | 1.18187 | 0 | −1.00000 | −1.14352 | 0 | −2.51737 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.81808 | 0 | 1.30541 | 1.39765 | 0 | −1.00000 | 1.26282 | 0 | −2.54104 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.57007 | 0 | 0.465107 | 2.53976 | 0 | −1.00000 | 2.40988 | 0 | −3.98759 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.50746 | 0 | 0.272426 | −0.476857 | 0 | −1.00000 | 2.60424 | 0 | 0.718841 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.910036 | 0 | −1.17183 | −2.84869 | 0 | −1.00000 | 2.88648 | 0 | 2.59241 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.455441 | 0 | −1.79257 | 0.749481 | 0 | −1.00000 | −1.72729 | 0 | 0.341344 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.678644 | 0 | −1.53944 | −2.84434 | 0 | −1.00000 | −2.40202 | 0 | −1.93030 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.68813 | 0 | 0.849768 | 4.40419 | 0 | −1.00000 | −1.94174 | 0 | 7.43483 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.14674 | 0 | 2.60851 | −3.76087 | 0 | −1.00000 | 1.30632 | 0 | −8.07362 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.22659 | 0 | 2.95768 | 0.700608 | 0 | −1.00000 | 2.13237 | 0 | 1.55996 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.74009 | 0 | 5.50808 | −2.04281 | 0 | −1.00000 | 9.61245 | 0 | −5.59747 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(1\) |
| \(127\) | \(-1\) |
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 | 8001.2.a.m | 11 | |
| 3.b | odd | 2 | 1 | 2667.2.a.k | ✓ | 11 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2667.2.a.k | ✓ | 11 | 3.b | odd | 2 | 1 | |
| 8001.2.a.m | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):
|
\( T_{2}^{11} - 2 T_{2}^{10} - 15 T_{2}^{9} + 25 T_{2}^{8} + 88 T_{2}^{7} - 112 T_{2}^{6} - 247 T_{2}^{5} + \cdots + 57 \)
|
|
\( T_{5}^{11} + T_{5}^{10} - 32 T_{5}^{9} - 32 T_{5}^{8} + 319 T_{5}^{7} + 197 T_{5}^{6} - 1331 T_{5}^{5} + \cdots + 288 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} - 2 T^{10} + \cdots + 57 \)
|
| $3$ |
\( T^{11} \)
|
| $5$ |
\( T^{11} + T^{10} + \cdots + 288 \)
|
| $7$ |
\( (T + 1)^{11} \)
|
| $11$ |
\( T^{11} - 7 T^{10} + \cdots - 36 \)
|
| $13$ |
\( T^{11} + 24 T^{10} + \cdots - 4206 \)
|
| $17$ |
\( T^{11} - 15 T^{10} + \cdots - 1027734 \)
|
| $19$ |
\( T^{11} + 19 T^{10} + \cdots + 5452108 \)
|
| $23$ |
\( T^{11} - 11 T^{10} + \cdots + 255744 \)
|
| $29$ |
\( T^{11} - 10 T^{10} + \cdots - 9221472 \)
|
| $31$ |
\( T^{11} + \cdots + 260432712 \)
|
| $37$ |
\( T^{11} + 22 T^{10} + \cdots - 1099454 \)
|
| $41$ |
\( T^{11} + \cdots - 357543498 \)
|
| $43$ |
\( T^{11} + \cdots + 926056896 \)
|
| $47$ |
\( T^{11} - 7 T^{10} + \cdots - 2774616 \)
|
| $53$ |
\( T^{11} - 28 T^{10} + \cdots - 416 \)
|
| $59$ |
\( T^{11} + 35 T^{10} + \cdots - 1303104 \)
|
| $61$ |
\( T^{11} + 6 T^{10} + \cdots - 1347574 \)
|
| $67$ |
\( T^{11} + \cdots + 9891955904 \)
|
| $71$ |
\( T^{11} + \cdots + 125943744 \)
|
| $73$ |
\( T^{11} + \cdots + 13520209618 \)
|
| $79$ |
\( T^{11} + \cdots - 11637350848 \)
|
| $83$ |
\( T^{11} + \cdots + 205464384 \)
|
| $89$ |
\( T^{11} + \cdots - 518712672 \)
|
| $97$ |
\( T^{11} + 45 T^{10} + \cdots + 350624 \)
|
| show more | |
| show less |