Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1053,2,Mod(649,1053)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1053, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("1053.649");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1053 = 3^{4} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1053.b (of order \(2\), degree \(1\), 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: | \(8.40824733284\) |
| 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: |
\( x^{10} + 16x^{8} + 91x^{6} + 222x^{4} + 228x^{2} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3^{3} \) |
| Twist minimal: | no (minimal twist has level 117) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} + 16x^{8} + 91x^{6} + 222x^{4} + 228x^{2} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} + 7\nu^{2} + 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{9} - 16\nu^{7} - 91\nu^{5} - 204\nu^{3} - 120\nu ) / 9 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{9} + 16\nu^{7} + 91\nu^{5} + 213\nu^{3} + 165\nu ) / 9 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{6} + 10\nu^{4} + 27\nu^{2} + 17 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -2\nu^{9} - 32\nu^{7} - 173\nu^{5} - 354\nu^{3} - 213\nu ) / 9 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 7\nu^{9} + 103\nu^{7} + 511\nu^{5} + 960\nu^{3} + 534\nu ) / 9 \)
|
| \(\beta_{9}\) | \(=\) |
\( \nu^{8} + 15\nu^{6} + 76\nu^{4} + 146\nu^{2} + 82 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} - 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} - 7\beta_{2} + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{7} - 6\beta_{5} - 8\beta_{4} + 27\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{6} - 10\beta_{3} + 43\beta_{2} - 66 \)
|
| \(\nu^{7}\) | \(=\) |
\( -\beta_{8} - 14\beta_{7} + 32\beta_{5} + 53\beta_{4} - 152\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( \beta_{9} - 15\beta_{6} + 74\beta_{3} - 259\beta_{2} + 358 \)
|
| \(\nu^{9}\) | \(=\) |
\( 16\beta_{8} + 133\beta_{7} - 170\beta_{5} - 333\beta_{4} + 875\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1053\mathbb{Z}\right)^\times\).
| \(n\) | \(326\) | \(730\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 649.1 |
|
− | 2.47728i | 0 | −4.13693 | 0.890732i | 0 | − | 0.982330i | 5.29379i | 0 | 2.20660 | ||||||||||||||||||||||||||||||||||||||||||||||
| 649.2 | − | 2.28298i | 0 | −3.21200 | − | 3.21586i | 0 | − | 2.42273i | 2.76698i | 0 | −7.34174 | ||||||||||||||||||||||||||||||||||||||||||||||
| 649.3 | − | 1.63641i | 0 | −0.677835 | 1.09738i | 0 | − | 3.20939i | − | 2.16360i | 0 | 1.79576 | ||||||||||||||||||||||||||||||||||||||||||||||
| 649.4 | − | 1.07384i | 0 | 0.846877 | 1.27644i | 0 | 1.02875i | − | 3.05708i | 0 | 1.37069 | |||||||||||||||||||||||||||||||||||||||||||||||
| 649.5 | − | 0.905597i | 0 | 1.17989 | − | 2.24306i | 0 | 3.43611i | − | 2.87970i | 0 | −2.03130 | ||||||||||||||||||||||||||||||||||||||||||||||
| 649.6 | 0.905597i | 0 | 1.17989 | 2.24306i | 0 | − | 3.43611i | 2.87970i | 0 | −2.03130 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 649.7 | 1.07384i | 0 | 0.846877 | − | 1.27644i | 0 | − | 1.02875i | 3.05708i | 0 | 1.37069 | |||||||||||||||||||||||||||||||||||||||||||||||
| 649.8 | 1.63641i | 0 | −0.677835 | − | 1.09738i | 0 | 3.20939i | 2.16360i | 0 | 1.79576 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 649.9 | 2.28298i | 0 | −3.21200 | 3.21586i | 0 | 2.42273i | − | 2.76698i | 0 | −7.34174 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 649.10 | 2.47728i | 0 | −4.13693 | − | 0.890732i | 0 | 0.982330i | − | 5.29379i | 0 | 2.20660 | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1053.2.b.i | 10 | |
| 3.b | odd | 2 | 1 | 1053.2.b.j | 10 | ||
| 9.c | even | 3 | 2 | 351.2.t.c | 20 | ||
| 9.d | odd | 6 | 2 | 117.2.t.c | ✓ | 20 | |
| 13.b | even | 2 | 1 | inner | 1053.2.b.i | 10 | |
| 39.d | odd | 2 | 1 | 1053.2.b.j | 10 | ||
| 117.n | odd | 6 | 2 | 117.2.t.c | ✓ | 20 | |
| 117.t | even | 6 | 2 | 351.2.t.c | 20 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 117.2.t.c | ✓ | 20 | 9.d | odd | 6 | 2 | |
| 117.2.t.c | ✓ | 20 | 117.n | odd | 6 | 2 | |
| 351.2.t.c | 20 | 9.c | even | 3 | 2 | ||
| 351.2.t.c | 20 | 117.t | even | 6 | 2 | ||
| 1053.2.b.i | 10 | 1.a | even | 1 | 1 | trivial | |
| 1053.2.b.i | 10 | 13.b | even | 2 | 1 | inner | |
| 1053.2.b.j | 10 | 3.b | odd | 2 | 1 | ||
| 1053.2.b.j | 10 | 39.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1053, [\chi])\):
|
\( T_{2}^{10} + 16T_{2}^{8} + 91T_{2}^{6} + 222T_{2}^{4} + 228T_{2}^{2} + 81 \)
|
|
\( T_{5}^{10} + 19T_{5}^{8} + 112T_{5}^{6} + 255T_{5}^{4} + 243T_{5}^{2} + 81 \)
|
|
\( T_{17}^{5} - 3T_{17}^{4} - 33T_{17}^{3} + 90T_{17}^{2} + 135T_{17} - 81 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 16 T^{8} + \cdots + 81 \)
|
| $3$ |
\( T^{10} \)
|
| $5$ |
\( T^{10} + 19 T^{8} + \cdots + 81 \)
|
| $7$ |
\( T^{10} + 30 T^{8} + \cdots + 729 \)
|
| $11$ |
\( T^{10} + 49 T^{8} + \cdots + 16641 \)
|
| $13$ |
\( T^{10} - 4 T^{9} + \cdots + 371293 \)
|
| $17$ |
\( (T^{5} - 3 T^{4} - 33 T^{3} + \cdots - 81)^{2} \)
|
| $19$ |
\( T^{10} + 129 T^{8} + \cdots + 700569 \)
|
| $23$ |
\( (T^{5} - 12 T^{4} + \cdots - 729)^{2} \)
|
| $29$ |
\( (T^{5} - 6 T^{4} - 33 T^{3} + \cdots + 81)^{2} \)
|
| $31$ |
\( T^{10} + 216 T^{8} + \cdots + 11758041 \)
|
| $37$ |
\( T^{10} + 231 T^{8} + \cdots + 41641209 \)
|
| $41$ |
\( T^{10} + 118 T^{8} + \cdots + 522729 \)
|
| $43$ |
\( (T^{5} + 2 T^{4} + \cdots - 179)^{2} \)
|
| $47$ |
\( T^{10} + 205 T^{8} + \cdots + 7017201 \)
|
| $53$ |
\( (T^{5} + 27 T^{4} + \cdots + 243)^{2} \)
|
| $59$ |
\( T^{10} + 145 T^{8} + \cdots + 8661249 \)
|
| $61$ |
\( (T^{5} - T^{4} + \cdots - 10867)^{2} \)
|
| $67$ |
\( T^{10} + 270 T^{8} + \cdots + 531441 \)
|
| $71$ |
\( T^{10} + 97 T^{8} + \cdots + 178929 \)
|
| $73$ |
\( T^{10} + 369 T^{8} + \cdots + 56746089 \)
|
| $79$ |
\( (T^{5} - 7 T^{4} + \cdots + 1217)^{2} \)
|
| $83$ |
\( T^{10} + 580 T^{8} + \cdots + 45279441 \)
|
| $89$ |
\( T^{10} + \cdots + 5851791009 \)
|
| $97$ |
\( T^{10} + 612 T^{8} + \cdots + 6859161 \)
|
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