Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [211,2,Mod(1,211)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(211, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("211.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 211 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 211.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: | \(1.68484348265\) |
| Analytic rank: | \(0\) |
| Dimension: | \(9\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{9} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{9} - x^{8} - 14x^{7} + 11x^{6} + 66x^{5} - 36x^{4} - 123x^{3} + 38x^{2} + 72x - 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - x^{8} - 14x^{7} + 11x^{6} + 66x^{5} - 36x^{4} - 123x^{3} + 38x^{2} + 72x - 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{8} - 31\nu^{7} - 58\nu^{6} + 309\nu^{5} + 82\nu^{4} - 732\nu^{3} + 91\nu^{2} + 186\nu - 200 ) / 116 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5\nu^{8} + 11\nu^{7} - 58\nu^{6} - 177\nu^{5} + 158\nu^{4} + 836\nu^{3} + 65\nu^{2} - 1110\nu - 292 ) / 116 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -7\nu^{8} + 31\nu^{7} + 58\nu^{6} - 309\nu^{5} - 82\nu^{4} + 848\nu^{3} - 91\nu^{2} - 766\nu + 84 ) / 116 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5\nu^{8} - 47\nu^{7} + 519\nu^{5} - 364\nu^{4} - 1600\nu^{3} + 1109\nu^{2} + 1384\nu - 524 ) / 116 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 9\nu^{8} - 15\nu^{7} - 116\nu^{6} + 157\nu^{5} + 470\nu^{4} - 444\nu^{3} - 637\nu^{2} + 322\nu + 124 ) / 58 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -21\nu^{8} + 35\nu^{7} + 232\nu^{6} - 347\nu^{5} - 768\nu^{4} + 920\nu^{3} + 1003\nu^{2} - 616\nu - 560 ) / 116 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{3} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{8} - \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + 7\beta_{2} + \beta _1 + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{8} + \beta_{6} + 10\beta_{5} - \beta_{4} + 7\beta_{3} + 2\beta_{2} + 28\beta _1 + 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( -12\beta_{8} - 12\beta_{7} - 8\beta_{6} + 2\beta_{5} - 9\beta_{4} + 9\beta_{3} + 48\beta_{2} + 11\beta _1 + 67 \)
|
| \(\nu^{7}\) | \(=\) |
\( -15\beta_{8} - 3\beta_{7} + 11\beta_{6} + 80\beta_{5} - 10\beta_{4} + 42\beta_{3} + 27\beta_{2} + 171\beta _1 + 54 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 110 \beta_{8} - 101 \beta_{7} - 50 \beta_{6} + 34 \beta_{5} - 63 \beta_{4} + 61 \beta_{3} + \cdots + 383 \)
|
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 |
|
−2.67629 | −1.87149 | 5.16253 | 3.51131 | 5.00866 | −4.24372 | −8.46385 | 0.502488 | −9.39729 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.16500 | 3.06548 | 2.68723 | 3.67947 | −6.63678 | 0.600695 | −1.48786 | 6.39720 | −7.96606 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.76769 | 1.01743 | 1.12474 | −0.330474 | −1.79850 | 1.14456 | 1.54719 | −1.96484 | 0.584177 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.09167 | −2.69353 | −0.808256 | −1.33282 | 2.94045 | −2.43218 | 3.06569 | 4.25511 | 1.45500 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.107203 | 2.59855 | −1.98851 | 0.449128 | −0.278573 | 0.845532 | 0.427581 | 3.75245 | −0.0481480 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0.959048 | −3.12841 | −1.08023 | 3.31637 | −3.00030 | 2.28069 | −2.95409 | 6.78696 | 3.18056 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.54891 | 1.20044 | 0.399122 | 4.25492 | 1.85937 | −4.34069 | −2.47962 | −1.55895 | 6.59049 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.79023 | 0.155219 | 1.20493 | 0.0597430 | 0.277877 | 4.81622 | −1.42336 | −2.97591 | 0.106954 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.50967 | −1.34369 | 4.29843 | 1.39234 | −3.37220 | −0.671123 | 5.76831 | −1.19451 | 3.49432 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(211\) | \(-1\) |
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 | 211.2.a.d | ✓ | 9 |
| 3.b | odd | 2 | 1 | 1899.2.a.j | 9 | ||
| 4.b | odd | 2 | 1 | 3376.2.a.s | 9 | ||
| 5.b | even | 2 | 1 | 5275.2.a.o | 9 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 211.2.a.d | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
| 1899.2.a.j | 9 | 3.b | odd | 2 | 1 | ||
| 3376.2.a.s | 9 | 4.b | odd | 2 | 1 | ||
| 5275.2.a.o | 9 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{9} + T_{2}^{8} - 14T_{2}^{7} - 11T_{2}^{6} + 66T_{2}^{5} + 36T_{2}^{4} - 123T_{2}^{3} - 38T_{2}^{2} + 72T_{2} + 8 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(211))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{9} + T^{8} - 14 T^{7} + \cdots + 8 \)
|
| $3$ |
\( T^{9} + T^{8} + \cdots - 32 \)
|
| $5$ |
\( T^{9} - 15 T^{8} + \cdots - 3 \)
|
| $7$ |
\( T^{9} + 2 T^{8} + \cdots - 192 \)
|
| $11$ |
\( T^{9} - 13 T^{8} + \cdots - 333 \)
|
| $13$ |
\( T^{9} + 4 T^{8} + \cdots - 931 \)
|
| $17$ |
\( T^{9} - 4 T^{8} + \cdots - 768 \)
|
| $19$ |
\( T^{9} + 2 T^{8} + \cdots + 92579 \)
|
| $23$ |
\( T^{9} + 3 T^{8} + \cdots - 512 \)
|
| $29$ |
\( T^{9} - 26 T^{8} + \cdots + 102912 \)
|
| $31$ |
\( T^{9} - 5 T^{8} + \cdots + 4064 \)
|
| $37$ |
\( T^{9} - 5 T^{8} + \cdots - 70173 \)
|
| $41$ |
\( T^{9} - 20 T^{8} + \cdots + 34048 \)
|
| $43$ |
\( T^{9} + 37 T^{8} + \cdots + 385587 \)
|
| $47$ |
\( T^{9} - 4 T^{8} + \cdots + 4961361 \)
|
| $53$ |
\( T^{9} - 13 T^{8} + \cdots - 101352 \)
|
| $59$ |
\( T^{9} - 14 T^{8} + \cdots - 48901984 \)
|
| $61$ |
\( T^{9} - 23 T^{8} + \cdots - 1016544 \)
|
| $67$ |
\( T^{9} + 3 T^{8} + \cdots + 45391648 \)
|
| $71$ |
\( T^{9} - 19 T^{8} + \cdots + 10015233 \)
|
| $73$ |
\( T^{9} - 17 T^{8} + \cdots - 35767104 \)
|
| $79$ |
\( T^{9} - 7 T^{8} + \cdots + 6812632 \)
|
| $83$ |
\( T^{9} - 6 T^{8} + \cdots - 81789792 \)
|
| $89$ |
\( T^{9} - 33 T^{8} + \cdots + 45488928 \)
|
| $97$ |
\( T^{9} + 11 T^{8} + \cdots - 1454432 \)
|
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