Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2873,2,Mod(1,2873)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2873, 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("2873.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2873 = 13^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2873.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: | \(22.9410205007\) |
| 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: |
\( x^{11} - 18x^{9} - x^{8} + 113x^{7} + 11x^{6} - 286x^{5} - 30x^{4} + 240x^{3} + 19x^{2} - 14x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 221) |
| 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
\( x^{11} - 18x^{9} - x^{8} + 113x^{7} + 11x^{6} - 286x^{5} - 30x^{4} + 240x^{3} + 19x^{2} - 14x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4 \nu^{10} + \nu^{9} - 71 \nu^{8} - 18 \nu^{7} + 440 \nu^{6} + 109 \nu^{5} - 1101 \nu^{4} - 240 \nu^{3} + \cdots - 37 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 11 \nu^{10} + 2 \nu^{9} - 193 \nu^{8} - 39 \nu^{7} + 1180 \nu^{6} + 254 \nu^{5} - 2904 \nu^{4} + \cdots - 80 ) / 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( 2\nu^{10} - 36\nu^{8} - 3\nu^{7} + 225\nu^{6} + 34\nu^{5} - 562\nu^{4} - 101\nu^{3} + 455\nu^{2} + 77\nu - 14 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 16 \nu^{10} + \nu^{9} - 290 \nu^{8} - 39 \nu^{7} + 1829 \nu^{6} + 346 \nu^{5} - 4626 \nu^{4} + \cdots - 139 ) / 6 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 17 \nu^{10} + 2 \nu^{9} - 307 \nu^{8} - 51 \nu^{7} + 1930 \nu^{6} + 386 \nu^{5} - 4866 \nu^{4} + \cdots - 140 ) / 6 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 6 \nu^{10} + \nu^{9} - 108 \nu^{8} - 23 \nu^{7} + 677 \nu^{6} + 166 \nu^{5} - 1704 \nu^{4} - 414 \nu^{3} + \cdots - 53 ) / 2 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( - 6 \nu^{10} - \nu^{9} + 108 \nu^{8} + 23 \nu^{7} - 677 \nu^{6} - 166 \nu^{5} + 1704 \nu^{4} + \cdots + 53 ) / 2 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 13 \nu^{10} + \nu^{9} - 233 \nu^{8} - 33 \nu^{7} + 1454 \nu^{6} + 283 \nu^{5} - 3645 \nu^{4} + \cdots - 109 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} + \beta_{8} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} - 2\beta_{3} + 6\beta_{2} + \beta _1 + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{10} + 9\beta_{9} + 9\beta_{8} + \beta_{7} - 2\beta_{6} - \beta_{4} + 30\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 11 \beta_{9} + 10 \beta_{8} + 9 \beta_{7} + 3 \beta_{6} - 12 \beta_{5} + 11 \beta_{4} - 20 \beta_{3} + \cdots + 96 \)
|
| \(\nu^{7}\) | \(=\) |
\( 12 \beta_{10} + 68 \beta_{9} + 67 \beta_{8} + 13 \beta_{7} - 25 \beta_{6} + \beta_{5} - 13 \beta_{4} + \cdots - 1 \)
|
| \(\nu^{8}\) | \(=\) |
\( - \beta_{10} + 89 \beta_{9} + 77 \beta_{8} + 64 \beta_{7} + 40 \beta_{6} - 104 \beta_{5} + 94 \beta_{4} + \cdots + 610 \)
|
| \(\nu^{9}\) | \(=\) |
\( 104 \beta_{10} + 483 \beta_{9} + 473 \beta_{8} + 118 \beta_{7} - 228 \beta_{6} + 17 \beta_{5} + \cdots - 18 \)
|
| \(\nu^{10}\) | \(=\) |
\( - 17 \beta_{10} + 645 \beta_{9} + 540 \beta_{8} + 423 \beta_{7} + 379 \beta_{6} - 801 \beta_{5} + \cdots + 3999 \)
|
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 |
|
−2.59327 | 1.40923 | 4.72506 | −3.24762 | −3.65453 | 4.17425 | −7.06682 | −1.01406 | 8.42196 | |||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.57783 | 2.64375 | 4.64523 | 4.28065 | −6.81514 | −2.18159 | −6.81896 | 3.98940 | −11.0348 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.86003 | −2.74868 | 1.45970 | −3.18753 | 5.11261 | −2.42385 | 1.00497 | 4.55521 | 5.92889 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.20175 | 0.493822 | −0.555799 | −0.949196 | −0.593450 | −2.48241 | 3.07143 | −2.75614 | 1.14070 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.125230 | −2.53841 | −1.98432 | 1.39880 | 0.317884 | −1.06142 | 0.498954 | 3.44352 | −0.175171 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.110264 | 1.95059 | −1.98784 | 3.14982 | −0.215080 | 2.77179 | 0.439716 | 0.804793 | −0.347313 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0.324759 | 3.21365 | −1.89453 | −1.65060 | 1.04366 | −2.78474 | −1.26478 | 7.32752 | −0.536047 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.25668 | −0.899758 | −0.420763 | −1.90948 | −1.13071 | −1.95011 | −3.04212 | −2.19044 | −2.39960 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.05344 | 2.67077 | 2.21664 | 0.413404 | 5.48427 | 3.04046 | 0.444849 | 4.13299 | 0.848902 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.17665 | −1.30219 | 2.73781 | −2.68244 | −2.83441 | 2.83529 | 1.60596 | −1.30431 | −5.83873 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1.11 | 2.65684 | 0.107227 | 5.05882 | 3.38418 | 0.284887 | 1.06232 | 8.12680 | −2.98850 | 8.99124 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(13\) | \(1\) |
| \(17\) | \(-1\) |
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 | 2873.2.a.s | 11 | |
| 13.b | even | 2 | 1 | 2873.2.a.t | 11 | ||
| 13.c | even | 3 | 2 | 221.2.e.b | ✓ | 22 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 221.2.e.b | ✓ | 22 | 13.c | even | 3 | 2 | |
| 2873.2.a.s | 11 | 1.a | even | 1 | 1 | trivial | |
| 2873.2.a.t | 11 | 13.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(2873))\):
|
\( T_{2}^{11} - 18T_{2}^{9} + T_{2}^{8} + 113T_{2}^{7} - 11T_{2}^{6} - 286T_{2}^{5} + 30T_{2}^{4} + 240T_{2}^{3} - 19T_{2}^{2} - 14T_{2} - 1 \)
|
|
\( T_{3}^{11} - 5 T_{3}^{10} - 11 T_{3}^{9} + 83 T_{3}^{8} - 5 T_{3}^{7} - 433 T_{3}^{6} + 316 T_{3}^{5} + \cdots - 27 \)
|
|
\( T_{5}^{11} + T_{5}^{10} - 38 T_{5}^{9} - 56 T_{5}^{8} + 490 T_{5}^{7} + 926 T_{5}^{6} - 2317 T_{5}^{5} + \cdots - 2192 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{11} - 18 T^{9} + \cdots - 1 \)
|
| $3$ |
\( T^{11} - 5 T^{10} + \cdots - 27 \)
|
| $5$ |
\( T^{11} + T^{10} + \cdots - 2192 \)
|
| $7$ |
\( T^{11} - T^{10} + \cdots - 8017 \)
|
| $11$ |
\( T^{11} + 6 T^{10} + \cdots - 10476 \)
|
| $13$ |
\( T^{11} \)
|
| $17$ |
\( (T - 1)^{11} \)
|
| $19$ |
\( T^{11} + T^{10} + \cdots - 7664 \)
|
| $23$ |
\( T^{11} - 11 T^{10} + \cdots + 461763 \)
|
| $29$ |
\( T^{11} - T^{10} + \cdots + 69984 \)
|
| $31$ |
\( T^{11} + 6 T^{10} + \cdots - 7747129 \)
|
| $37$ |
\( T^{11} + \cdots + 220530448 \)
|
| $41$ |
\( T^{11} - 27 T^{10} + \cdots - 104976 \)
|
| $43$ |
\( T^{11} - 24 T^{10} + \cdots + 1674864 \)
|
| $47$ |
\( T^{11} + \cdots - 111406896 \)
|
| $53$ |
\( T^{11} + 10 T^{10} + \cdots + 82371 \)
|
| $59$ |
\( T^{11} + \cdots + 132591168 \)
|
| $61$ |
\( T^{11} - 17 T^{10} + \cdots - 541648 \)
|
| $67$ |
\( T^{11} - 3 T^{10} + \cdots - 6573888 \)
|
| $71$ |
\( T^{11} + \cdots + 2188047956 \)
|
| $73$ |
\( T^{11} + \cdots + 155967024 \)
|
| $79$ |
\( T^{11} + \cdots - 1005302847 \)
|
| $83$ |
\( T^{11} + \cdots - 6827945328 \)
|
| $89$ |
\( T^{11} + 15 T^{10} + \cdots + 955089 \)
|
| $97$ |
\( T^{11} + 14 T^{10} + \cdots - 72909952 \)
|
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