Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2768,2,Mod(1729,2768)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2768, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2768.1729");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2768 = 2^{4} \cdot 173 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2768.b (of order \(2\), degree \(1\), not minimal) |
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: | no |
| Analytic conductor: | \(22.1025912795\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} + 20x^{10} + 154x^{8} + 568x^{6} + 997x^{4} + 665x^{2} + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 346) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} + 20x^{10} + 154x^{8} + 568x^{6} + 997x^{4} + 665x^{2} + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{11} + 12\nu^{9} + 34\nu^{7} - 52\nu^{5} - 291\nu^{3} - 235\nu ) / 36 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{11} + 16\nu^{9} + 86\nu^{7} + 160\nu^{5} - 11\nu^{3} - 183\nu ) / 12 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{10} + 15\nu^{8} + 82\nu^{6} + 206\nu^{4} + 225\nu^{2} + 38 ) / 9 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -2\nu^{10} - 30\nu^{8} - 155\nu^{6} - 322\nu^{4} - 225\nu^{2} - 4 ) / 9 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{11} + 16\nu^{9} + 92\nu^{7} + 232\nu^{5} + 253\nu^{3} + 105\nu ) / 6 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -5\nu^{11} - 78\nu^{9} - 422\nu^{7} - 892\nu^{5} - 435\nu^{3} + 401\nu ) / 18 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -5\nu^{11} - 84\nu^{9} - 518\nu^{7} - 1444\nu^{5} - 1773\nu^{3} - 721\nu ) / 36 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 2\nu^{10} + 33\nu^{8} + 194\nu^{6} + 481\nu^{4} + 435\nu^{2} + 34 ) / 9 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( -4\nu^{10} - 63\nu^{8} - 349\nu^{6} - 794\nu^{4} - 606\nu^{2} + 7 ) / 9 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} + \beta_{7} - \beta_{3} - 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{11} + \beta_{10} - \beta_{6} - 6\beta_{2} + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( -9\beta_{9} + \beta_{8} - 8\beta_{7} + \beta_{4} + 10\beta_{3} + 18\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -10\beta_{11} - 10\beta_{10} + 11\beta_{6} + 2\beta_{5} + 35\beta_{2} - 63 \)
|
| \(\nu^{7}\) | \(=\) |
\( 64\beta_{9} - 12\beta_{8} + 53\beta_{7} - 14\beta_{4} - 76\beta_{3} - 88\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 77\beta_{11} + 80\beta_{10} - 87\beta_{6} - 26\beta_{5} - 207\beta_{2} + 330 \)
|
| \(\nu^{9}\) | \(=\) |
\( -425\beta_{9} + 103\beta_{8} - 335\beta_{7} + 132\beta_{4} + 519\beta_{3} + 457\beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( -541\beta_{11} - 586\beta_{10} + 609\beta_{6} + 235\beta_{5} + 1246\beta_{2} - 1825 \)
|
| \(\nu^{11}\) | \(=\) |
\( 2747\beta_{9} - 776\beta_{8} + 2093\beta_{7} - 1056\beta_{4} - 3379\beta_{3} - 2485\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2768\mathbb{Z}\right)^\times\).
| \(n\) | \(693\) | \(1039\) | \(1905\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1729.1 |
|
0 | − | 3.34507i | 0 | − | 3.84981i | 0 | 3.54858i | 0 | −8.18949 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1729.2 | 0 | − | 2.81696i | 0 | − | 1.60330i | 0 | − | 3.40548i | 0 | −4.93525 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1729.3 | 0 | − | 2.62422i | 0 | 3.67910i | 0 | − | 0.165323i | 0 | −3.88651 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1729.4 | 0 | − | 1.77738i | 0 | − | 0.354332i | 0 | − | 1.90225i | 0 | −0.159069 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1729.5 | 0 | − | 0.783416i | 0 | 0.456735i | 0 | 3.87057i | 0 | 2.38626 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1729.6 | 0 | − | 0.464689i | 0 | 2.17680i | 0 | 1.08770i | 0 | 2.78406 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1729.7 | 0 | 0.464689i | 0 | − | 2.17680i | 0 | − | 1.08770i | 0 | 2.78406 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1729.8 | 0 | 0.783416i | 0 | − | 0.456735i | 0 | − | 3.87057i | 0 | 2.38626 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1729.9 | 0 | 1.77738i | 0 | 0.354332i | 0 | 1.90225i | 0 | −0.159069 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1729.10 | 0 | 2.62422i | 0 | − | 3.67910i | 0 | 0.165323i | 0 | −3.88651 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1729.11 | 0 | 2.81696i | 0 | 1.60330i | 0 | 3.40548i | 0 | −4.93525 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 1729.12 | 0 | 3.34507i | 0 | 3.84981i | 0 | − | 3.54858i | 0 | −8.18949 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 173.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2768.2.b.b | 12 | |
| 4.b | odd | 2 | 1 | 346.2.b.b | ✓ | 12 | |
| 12.b | even | 2 | 1 | 3114.2.b.e | 12 | ||
| 173.b | even | 2 | 1 | inner | 2768.2.b.b | 12 | |
| 692.d | odd | 2 | 1 | 346.2.b.b | ✓ | 12 | |
| 2076.h | even | 2 | 1 | 3114.2.b.e | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 346.2.b.b | ✓ | 12 | 4.b | odd | 2 | 1 | |
| 346.2.b.b | ✓ | 12 | 692.d | odd | 2 | 1 | |
| 2768.2.b.b | 12 | 1.a | even | 1 | 1 | trivial | |
| 2768.2.b.b | 12 | 173.b | even | 2 | 1 | inner | |
| 3114.2.b.e | 12 | 12.b | even | 2 | 1 | ||
| 3114.2.b.e | 12 | 2076.h | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{12} + 30T_{3}^{10} + 327T_{3}^{8} + 1563T_{3}^{6} + 3057T_{3}^{4} + 1776T_{3}^{2} + 256 \)
acting on \(S_{2}^{\mathrm{new}}(2768, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( T^{12} + 30 T^{10} + \cdots + 256 \)
|
| $5$ |
\( T^{12} + 36 T^{10} + \cdots + 64 \)
|
| $7$ |
\( T^{12} + 44 T^{10} + \cdots + 256 \)
|
| $11$ |
\( T^{12} + 77 T^{10} + \cdots + 73984 \)
|
| $13$ |
\( (T^{6} + 3 T^{5} + \cdots - 776)^{2} \)
|
| $17$ |
\( T^{12} + 80 T^{10} + \cdots + 65536 \)
|
| $19$ |
\( T^{12} + 98 T^{10} + \cdots + 250000 \)
|
| $23$ |
\( (T^{6} + 13 T^{5} + \cdots - 352)^{2} \)
|
| $29$ |
\( (T^{6} + T^{5} - 64 T^{4} + \cdots + 2448)^{2} \)
|
| $31$ |
\( (T^{6} + 5 T^{5} + \cdots + 8800)^{2} \)
|
| $37$ |
\( (T^{6} + 9 T^{5} + \cdots + 360)^{2} \)
|
| $41$ |
\( (T^{6} + 11 T^{5} + \cdots - 8261)^{2} \)
|
| $43$ |
\( (T^{6} - T^{5} - 107 T^{4} + \cdots + 5648)^{2} \)
|
| $47$ |
\( (T^{6} + 7 T^{5} + \cdots - 256)^{2} \)
|
| $53$ |
\( T^{12} + \cdots + 2402568256 \)
|
| $59$ |
\( T^{12} + \cdots + 190329616 \)
|
| $61$ |
\( T^{12} + 460 T^{10} + \cdots + 15681600 \)
|
| $67$ |
\( (T^{6} + 5 T^{5} + \cdots - 126496)^{2} \)
|
| $71$ |
\( T^{12} + \cdots + 60576038884 \)
|
| $73$ |
\( (T^{6} - 5 T^{5} + \cdots - 11099)^{2} \)
|
| $79$ |
\( T^{12} + \cdots + 4123437796 \)
|
| $83$ |
\( (T^{6} + 35 T^{5} + \cdots - 62512)^{2} \)
|
| $89$ |
\( (T^{6} - 16 T^{5} + \cdots - 478)^{2} \)
|
| $97$ |
\( T^{12} + \cdots + 22065319936 \)
|
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