Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2001,2,Mod(1,2001)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2001, 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("2001.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2001 = 3 \cdot 23 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2001.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: | \(15.9780654445\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - x^{9} - 17x^{8} + 23x^{7} + 69x^{6} - 88x^{5} - 106x^{4} + 101x^{3} + 60x^{2} - 23x - 11 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - x^{9} - 17x^{8} + 23x^{7} + 69x^{6} - 88x^{5} - 106x^{4} + 101x^{3} + 60x^{2} - 23x - 11 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 31 \nu^{9} - 14 \nu^{8} - 481 \nu^{7} + 466 \nu^{6} + 1581 \nu^{5} - 1705 \nu^{4} - 1465 \nu^{3} + \cdots - 357 ) / 52 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 11 \nu^{9} - 23 \nu^{8} - 182 \nu^{7} + 448 \nu^{6} + 587 \nu^{5} - 1710 \nu^{4} - 493 \nu^{3} + \cdots - 450 ) / 26 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 55 \nu^{9} + 24 \nu^{8} + 871 \nu^{7} - 810 \nu^{6} - 3065 \nu^{5} + 3051 \nu^{4} + 3297 \nu^{3} + \cdots + 703 ) / 52 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 33 \nu^{9} - 30 \nu^{8} - 533 \nu^{7} + 720 \nu^{6} + 1865 \nu^{5} - 2647 \nu^{4} - 1999 \nu^{3} + \cdots - 505 ) / 26 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{9} + 3\nu^{8} + 48\nu^{7} - 70\nu^{6} - 161\nu^{5} + 256\nu^{4} + 155\nu^{3} - 279\nu^{2} - 3\nu + 54 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 51 \nu^{9} - 148 \nu^{8} - 871 \nu^{7} + 2694 \nu^{6} + 2809 \nu^{5} - 10007 \nu^{4} - 2645 \nu^{3} + \cdots - 2227 ) / 52 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 41 \nu^{9} + 42 \nu^{8} + 663 \nu^{7} - 982 \nu^{6} - 2299 \nu^{5} + 3737 \nu^{4} + 2367 \nu^{3} + \cdots + 785 ) / 26 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 93 \nu^{9} - 120 \nu^{8} - 1521 \nu^{7} + 2594 \nu^{6} + 5263 \nu^{5} - 9717 \nu^{4} - 5435 \nu^{3} + \cdots - 2033 ) / 52 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{9} + \beta_{8} - \beta_{5} - \beta_{4} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{9} - 2\beta_{8} - \beta_{6} + 2\beta_{5} + \beta_{4} + \beta_{3} - 2\beta_{2} + 7\beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 14\beta_{9} + 11\beta_{8} - 14\beta_{5} - 10\beta_{4} - 4\beta_{3} + 2\beta_{2} - 6\beta _1 + 23 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 50 \beta_{9} - 34 \beta_{8} + 2 \beta_{7} - 12 \beta_{6} + 35 \beta_{5} + 17 \beta_{4} + 13 \beta_{3} + \cdots - 29 \)
|
| \(\nu^{6}\) | \(=\) |
\( 186 \beta_{9} + 135 \beta_{8} - \beta_{7} + 8 \beta_{6} - 179 \beta_{5} - 114 \beta_{4} - 61 \beta_{3} + \cdots + 241 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 695 \beta_{9} - 480 \beta_{8} + 29 \beta_{7} - 136 \beta_{6} + 516 \beta_{5} + 268 \beta_{4} + \cdots - 510 \)
|
| \(\nu^{8}\) | \(=\) |
\( 2492 \beta_{9} + 1766 \beta_{8} - 31 \beta_{7} + 192 \beta_{6} - 2293 \beta_{5} - 1391 \beta_{4} + \cdots + 2851 \)
|
| \(\nu^{9}\) | \(=\) |
\( - 9338 \beta_{9} - 6497 \beta_{8} + 349 \beta_{7} - 1579 \beta_{6} + 7263 \beta_{5} + 3936 \beta_{4} + \cdots - 7726 \)
|
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 |
|
−2.70414 | −1.00000 | 5.31240 | 0.910949 | 2.70414 | −2.94138 | −8.95720 | 1.00000 | −2.46334 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.24431 | −1.00000 | 3.03693 | −1.31370 | 2.24431 | −2.96709 | −2.32720 | 1.00000 | 2.94835 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −2.09108 | −1.00000 | 2.37262 | 4.27000 | 2.09108 | 3.25035 | −0.779182 | 1.00000 | −8.92891 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.161383 | −1.00000 | −1.97396 | 2.08087 | −0.161383 | 3.66994 | −0.641331 | 1.00000 | 0.335818 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.611391 | −1.00000 | −1.62620 | 0.834863 | −0.611391 | −0.685491 | −2.21703 | 1.00000 | 0.510427 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.676519 | −1.00000 | −1.54232 | −1.80585 | −0.676519 | −4.44627 | −2.39645 | 1.00000 | −1.22169 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.39479 | −1.00000 | −0.0545700 | −2.50233 | −1.39479 | 4.61826 | −2.86569 | 1.00000 | −3.49022 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.08881 | −1.00000 | 2.36311 | 0.182168 | −2.08881 | 1.65593 | 0.758471 | 1.00000 | 0.380514 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.36221 | −1.00000 | 3.58005 | 3.25906 | −2.36221 | 3.14721 | 3.73241 | 1.00000 | 7.69859 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.74444 | −1.00000 | 5.53194 | 0.0839771 | −2.74444 | −2.30146 | 9.69318 | 1.00000 | 0.230470 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(23\) | \(1\) |
| \(29\) | \(-1\) |
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 | 2001.2.a.k | ✓ | 10 |
| 3.b | odd | 2 | 1 | 6003.2.a.k | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2001.2.a.k | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 6003.2.a.k | 10 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2001))\):
|
\( T_{2}^{10} - 3 T_{2}^{9} - 14 T_{2}^{8} + 49 T_{2}^{7} + 45 T_{2}^{6} - 254 T_{2}^{5} + 89 T_{2}^{4} + \cdots - 16 \)
|
|
\( T_{5}^{10} - 6T_{5}^{9} - 5T_{5}^{8} + 69T_{5}^{7} - 27T_{5}^{6} - 223T_{5}^{5} + 158T_{5}^{4} + 180T_{5}^{3} - 185T_{5}^{2} + 38T_{5} - 2 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} - 3 T^{9} + \cdots - 16 \)
|
| $3$ |
\( (T + 1)^{10} \)
|
| $5$ |
\( T^{10} - 6 T^{9} + \cdots - 2 \)
|
| $7$ |
\( T^{10} - 3 T^{9} + \cdots - 17576 \)
|
| $11$ |
\( T^{10} - 9 T^{9} + \cdots + 10016 \)
|
| $13$ |
\( T^{10} + 16 T^{9} + \cdots + 43153 \)
|
| $17$ |
\( T^{10} - 80 T^{8} + \cdots - 256 \)
|
| $19$ |
\( T^{10} - T^{9} + \cdots - 704 \)
|
| $23$ |
\( (T + 1)^{10} \)
|
| $29$ |
\( (T - 1)^{10} \)
|
| $31$ |
\( T^{10} - 17 T^{9} + \cdots + 3195478 \)
|
| $37$ |
\( T^{10} + \cdots + 263470441 \)
|
| $41$ |
\( T^{10} - 163 T^{8} + \cdots - 3146 \)
|
| $43$ |
\( T^{10} + 5 T^{9} + \cdots - 68296 \)
|
| $47$ |
\( T^{10} - 15 T^{9} + \cdots - 4432 \)
|
| $53$ |
\( T^{10} - 35 T^{9} + \cdots - 22759984 \)
|
| $59$ |
\( T^{10} - 49 T^{9} + \cdots + 29485024 \)
|
| $61$ |
\( T^{10} + \cdots - 105224032 \)
|
| $67$ |
\( T^{10} - 35 T^{9} + \cdots - 787384 \)
|
| $71$ |
\( T^{10} + \cdots - 1158168607 \)
|
| $73$ |
\( T^{10} + 15 T^{9} + \cdots - 32 \)
|
| $79$ |
\( T^{10} - 24 T^{9} + \cdots + 10721896 \)
|
| $83$ |
\( T^{10} - T^{9} + \cdots + 21233456 \)
|
| $89$ |
\( T^{10} - 15 T^{9} + \cdots - 12055048 \)
|
| $97$ |
\( T^{10} + \cdots - 139418176 \)
|
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