Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [209,2,Mod(208,209)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(209, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("209.208");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 209 = 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 209.d (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: | \(1.66887340224\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.484000000.9 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + x^{6} + 16x^{4} + 66x^{2} + 121 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + x^{6} + 16x^{4} + 66x^{2} + 121 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -28\nu^{7} + 148\nu^{5} - 525\nu^{3} - 539\nu ) / 1991 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 40\nu^{7} + 73\nu^{5} + 750\nu^{3} + 4752\nu ) / 1991 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3\nu^{6} + 10\nu^{4} + 11\nu^{2} + 284 ) / 181 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 54\nu^{7} - \nu^{5} + 17\nu^{3} + 2035\nu ) / 1991 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -68\nu^{7} + 75\nu^{5} - 1275\nu^{3} - 1309\nu ) / 1991 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -47\nu^{6} - 36\nu^{4} - 1379\nu^{2} - 2398 ) / 1991 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 89\nu^{6} - 186\nu^{4} + 1171\nu^{2} + 2211 ) / 1991 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{7} - 4\beta_{6} - 3\beta_{3} + 1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{5} - 4\beta_{4} + \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -10\beta_{7} - 7\beta_{6} + 17\beta_{3} - 24 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( -7\beta_{5} + 17\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 37\beta_{7} + 38\beta_{6} + 75\beta_{3} - 113 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -37\beta_{5} + 75\beta_{4} - 38\beta_{2} + 38\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/209\mathbb{Z}\right)^\times\).
| \(n\) | \(78\) | \(134\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 208.1 |
|
−2.49721 | − | 2.93565i | 4.23607 | −2.85410 | 7.33094i | − | 4.25325i | −5.58394 | −5.61803 | 7.12730 | ||||||||||||||||||||||||||||||||||||||||
| 208.2 | −2.49721 | 2.93565i | 4.23607 | −2.85410 | − | 7.33094i | 4.25325i | −5.58394 | −5.61803 | 7.12730 | ||||||||||||||||||||||||||||||||||||||||||
| 208.3 | −1.32813 | − | 2.52626i | −0.236068 | 3.85410 | 3.35520i | − | 2.62866i | 2.96979 | −3.38197 | −5.11875 | |||||||||||||||||||||||||||||||||||||||||
| 208.4 | −1.32813 | 2.52626i | −0.236068 | 3.85410 | − | 3.35520i | 2.62866i | 2.96979 | −3.38197 | −5.11875 | ||||||||||||||||||||||||||||||||||||||||||
| 208.5 | 1.32813 | − | 2.52626i | −0.236068 | 3.85410 | − | 3.35520i | 2.62866i | −2.96979 | −3.38197 | 5.11875 | |||||||||||||||||||||||||||||||||||||||||
| 208.6 | 1.32813 | 2.52626i | −0.236068 | 3.85410 | 3.35520i | − | 2.62866i | −2.96979 | −3.38197 | 5.11875 | ||||||||||||||||||||||||||||||||||||||||||
| 208.7 | 2.49721 | − | 2.93565i | 4.23607 | −2.85410 | − | 7.33094i | 4.25325i | 5.58394 | −5.61803 | −7.12730 | |||||||||||||||||||||||||||||||||||||||||
| 208.8 | 2.49721 | 2.93565i | 4.23607 | −2.85410 | 7.33094i | − | 4.25325i | 5.58394 | −5.61803 | −7.12730 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
| 19.b | odd | 2 | 1 | inner |
| 209.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 209.2.d.c | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1881.2.h.e | 8 | ||
| 11.b | odd | 2 | 1 | inner | 209.2.d.c | ✓ | 8 |
| 19.b | odd | 2 | 1 | inner | 209.2.d.c | ✓ | 8 |
| 33.d | even | 2 | 1 | 1881.2.h.e | 8 | ||
| 57.d | even | 2 | 1 | 1881.2.h.e | 8 | ||
| 209.d | even | 2 | 1 | inner | 209.2.d.c | ✓ | 8 |
| 627.b | odd | 2 | 1 | 1881.2.h.e | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 209.2.d.c | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 209.2.d.c | ✓ | 8 | 11.b | odd | 2 | 1 | inner |
| 209.2.d.c | ✓ | 8 | 19.b | odd | 2 | 1 | inner |
| 209.2.d.c | ✓ | 8 | 209.d | even | 2 | 1 | inner |
| 1881.2.h.e | 8 | 3.b | odd | 2 | 1 | ||
| 1881.2.h.e | 8 | 33.d | even | 2 | 1 | ||
| 1881.2.h.e | 8 | 57.d | even | 2 | 1 | ||
| 1881.2.h.e | 8 | 627.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 8T_{2}^{2} + 11 \)
acting on \(S_{2}^{\mathrm{new}}(209, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 8 T^{2} + 11)^{2} \)
|
| $3$ |
\( (T^{4} + 15 T^{2} + 55)^{2} \)
|
| $5$ |
\( (T^{2} - T - 11)^{4} \)
|
| $7$ |
\( (T^{4} + 25 T^{2} + 125)^{2} \)
|
| $11$ |
\( (T^{4} - 4 T^{3} + \cdots + 121)^{2} \)
|
| $13$ |
\( (T^{4} - 7 T^{2} + 11)^{2} \)
|
| $17$ |
\( (T^{4} + 40 T^{2} + 80)^{2} \)
|
| $19$ |
\( T^{8} - 36 T^{6} + \cdots + 130321 \)
|
| $23$ |
\( (T + 4)^{8} \)
|
| $29$ |
\( (T^{4} - 17 T^{2} + 11)^{2} \)
|
| $31$ |
\( (T^{4} + 45 T^{2} + 55)^{2} \)
|
| $37$ |
\( (T^{4} + 60 T^{2} + 880)^{2} \)
|
| $41$ |
\( (T^{4} - 35 T^{2} + 275)^{2} \)
|
| $43$ |
\( (T^{4} + 85 T^{2} + 1805)^{2} \)
|
| $47$ |
\( (T - 2)^{8} \)
|
| $53$ |
\( (T^{4} + 100 T^{2} + 880)^{2} \)
|
| $59$ |
\( (T^{4} + 60 T^{2} + 880)^{2} \)
|
| $61$ |
\( (T^{4} + 80 T^{2} + 1280)^{2} \)
|
| $67$ |
\( (T^{4} + 185 T^{2} + 6655)^{2} \)
|
| $71$ |
\( (T^{4} + 155 T^{2} + 55)^{2} \)
|
| $73$ |
\( (T^{4} + 100 T^{2} + 80)^{2} \)
|
| $79$ |
\( (T^{4} - 260 T^{2} + 4400)^{2} \)
|
| $83$ |
\( (T^{4} + 5 T^{2} + 5)^{2} \)
|
| $89$ |
\( (T^{4} + 180 T^{2} + 880)^{2} \)
|
| $97$ |
\( (T^{4} + 400 T^{2} + 22000)^{2} \)
|
| show more | |
| show less |