Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [209,2,Mod(18,209)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(209, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([7, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("209.18");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 209 = 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 209.k (of order \(10\), degree \(4\), 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: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{10})\) |
| 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: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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
\( x^{8} + x^{6} + 16x^{4} + 66x^{2} + 121 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 7\nu^{6} - 37\nu^{4} + 629\nu^{2} - 363 ) / 1991 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -28\nu^{6} + 148\nu^{4} - 525\nu^{2} - 539 ) / 1991 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -28\nu^{7} + 148\nu^{5} - 525\nu^{3} - 539\nu ) / 1991 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 40\nu^{6} + 73\nu^{4} + 750\nu^{2} + 2761 ) / 1991 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -61\nu^{7} + 38\nu^{5} - 646\nu^{3} - 1672\nu ) / 1991 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 68\nu^{7} - 75\nu^{5} + 1275\nu^{3} + 3300\nu ) / 1991 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 4\beta_{2} + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{7} + 4\beta_{6} + \beta_{4} - 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{5} + 10\beta_{3} - 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( 7\beta_{7} + 17\beta_{4} - 7\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 37\beta_{5} - 37\beta_{3} - 75\beta_{2} - 75 \)
|
| \(\nu^{7}\) | \(=\) |
\( -38\beta_{7} - 75\beta_{6} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/209\mathbb{Z}\right)^\times\).
| \(n\) | \(78\) | \(134\) |
| \(\chi(n)\) | \(-1\) | \(1 + \beta_{2} + \beta_{3} - \beta_{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 18.1 |
|
−2.02029 | − | 1.46782i | 0 | 1.30902 | + | 4.02874i | 0.500000 | − | 0.363271i | 0 | −1.80902 | + | 0.587785i | 1.72553 | − | 5.31064i | −2.42705 | − | 1.76336i | −1.54336 | ||||||||||||||||||||||||||||||
| 18.2 | 2.02029 | + | 1.46782i | 0 | 1.30902 | + | 4.02874i | 0.500000 | − | 0.363271i | 0 | −1.80902 | + | 0.587785i | −1.72553 | + | 5.31064i | −2.42705 | − | 1.76336i | 1.54336 | |||||||||||||||||||||||||||||||
| 94.1 | −0.410415 | − | 1.26313i | 0 | 0.190983 | − | 0.138757i | 0.500000 | − | 1.53884i | 0 | −0.690983 | − | 0.951057i | −2.40261 | − | 1.74560i | 0.927051 | + | 2.85317i | −2.14896 | |||||||||||||||||||||||||||||||
| 94.2 | 0.410415 | + | 1.26313i | 0 | 0.190983 | − | 0.138757i | 0.500000 | − | 1.53884i | 0 | −0.690983 | − | 0.951057i | 2.40261 | + | 1.74560i | 0.927051 | + | 2.85317i | 2.14896 | |||||||||||||||||||||||||||||||
| 151.1 | −2.02029 | + | 1.46782i | 0 | 1.30902 | − | 4.02874i | 0.500000 | + | 0.363271i | 0 | −1.80902 | − | 0.587785i | 1.72553 | + | 5.31064i | −2.42705 | + | 1.76336i | −1.54336 | |||||||||||||||||||||||||||||||
| 151.2 | 2.02029 | − | 1.46782i | 0 | 1.30902 | − | 4.02874i | 0.500000 | + | 0.363271i | 0 | −1.80902 | − | 0.587785i | −1.72553 | − | 5.31064i | −2.42705 | + | 1.76336i | 1.54336 | |||||||||||||||||||||||||||||||
| 189.1 | −0.410415 | + | 1.26313i | 0 | 0.190983 | + | 0.138757i | 0.500000 | + | 1.53884i | 0 | −0.690983 | + | 0.951057i | −2.40261 | + | 1.74560i | 0.927051 | − | 2.85317i | −2.14896 | |||||||||||||||||||||||||||||||
| 189.2 | 0.410415 | − | 1.26313i | 0 | 0.190983 | + | 0.138757i | 0.500000 | + | 1.53884i | 0 | −0.690983 | + | 0.951057i | 2.40261 | − | 1.74560i | 0.927051 | − | 2.85317i | 2.14896 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.d | odd | 10 | 1 | inner |
| 19.b | odd | 2 | 1 | inner |
| 209.k | even | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 209.2.k.b | ✓ | 8 |
| 11.d | odd | 10 | 1 | inner | 209.2.k.b | ✓ | 8 |
| 19.b | odd | 2 | 1 | inner | 209.2.k.b | ✓ | 8 |
| 209.k | even | 10 | 1 | inner | 209.2.k.b | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 209.2.k.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 209.2.k.b | ✓ | 8 | 11.d | odd | 10 | 1 | inner |
| 209.2.k.b | ✓ | 8 | 19.b | odd | 2 | 1 | inner |
| 209.2.k.b | ✓ | 8 | 209.k | even | 10 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - T_{2}^{6} + 31T_{2}^{4} + 99T_{2}^{2} + 121 \)
acting on \(S_{2}^{\mathrm{new}}(209, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - T^{6} + \cdots + 121 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{4} - 2 T^{3} + 4 T^{2} + \cdots + 1)^{2} \)
|
| $7$ |
\( (T^{4} + 5 T^{3} + 10 T^{2} + \cdots + 5)^{2} \)
|
| $11$ |
\( (T^{4} - 9 T^{3} + \cdots + 121)^{2} \)
|
| $13$ |
\( T^{8} - 4 T^{6} + \cdots + 30976 \)
|
| $17$ |
\( (T^{4} + 15 T^{3} + \cdots + 405)^{2} \)
|
| $19$ |
\( T^{8} + 10 T^{7} + \cdots + 130321 \)
|
| $23$ |
\( (T^{2} + 3 T + 1)^{4} \)
|
| $29$ |
\( T^{8} - 24 T^{6} + \cdots + 30976 \)
|
| $31$ |
\( T^{8} - 20 T^{6} + \cdots + 774400 \)
|
| $37$ |
\( T^{8} + 20 T^{6} + \cdots + 774400 \)
|
| $41$ |
\( T^{8} + 20 T^{6} + \cdots + 19360000 \)
|
| $43$ |
\( (T^{4} + 145 T^{2} + 4805)^{2} \)
|
| $47$ |
\( (T^{4} - 8 T^{3} + \cdots + 121)^{2} \)
|
| $53$ |
\( T^{8} - 280 T^{6} + \cdots + 774400 \)
|
| $59$ |
\( T^{8} - 140 T^{6} + \cdots + 774400 \)
|
| $61$ |
\( (T^{4} - 15 T^{3} + \cdots + 125)^{2} \)
|
| $67$ |
\( (T^{4} + 180 T^{2} + 880)^{2} \)
|
| $71$ |
\( T^{8} - 100 T^{6} + \cdots + 484000000 \)
|
| $73$ |
\( (T^{4} + 320 T + 1280)^{2} \)
|
| $79$ |
\( T^{8} - 20 T^{6} + \cdots + 19360000 \)
|
| $83$ |
\( (T^{4} + 20 T^{3} + \cdots + 125)^{2} \)
|
| $89$ |
\( (T^{4} + 60 T^{2} + 880)^{2} \)
|
| $97$ |
\( T^{8} + 40 T^{6} + \cdots + 774400 \)
|
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