Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [209,2,Mod(1,209)]
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([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("209.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 209 = 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 209.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: | \(1.66887340224\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - 7\nu^{2} - 2\nu + 4 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{4} + 9\nu^{2} + 2\nu - 12 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 9\nu^{3} + 14\nu - 6 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - 10\nu^{4} + 23\nu^{2} - 4\nu - 10 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{6} + 12\nu^{4} + 4\nu^{3} - 41\nu^{2} - 20\nu + 34 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 7\beta_{3} + 9\beta_{2} + 2\beta _1 + 24 \)
|
| \(\nu^{5}\) | \(=\) |
\( 9\beta_{6} + 9\beta_{5} + 2\beta_{4} + 9\beta_{3} + 31\beta _1 + 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( 4\beta_{5} + 47\beta_{3} + 67\beta_{2} + 24\beta _1 + 158 \)
|
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 |
|
−2.78328 | −2.10880 | 5.74667 | −2.97131 | 5.86939 | −1.34513 | −10.4280 | 1.44705 | 8.27000 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.61330 | 1.19599 | 4.82936 | 4.07680 | −3.12549 | 3.61829 | −7.39397 | −1.56960 | −10.6539 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.19313 | 3.16232 | −0.576442 | 1.08235 | −3.77306 | −1.30958 | 3.07403 | 7.00029 | −1.29138 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −0.456669 | −0.835165 | −1.79145 | 0.221953 | 0.381394 | 4.69915 | 1.73144 | −2.30250 | −0.101359 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.45416 | 1.71116 | 0.114592 | 2.59296 | 2.48830 | −2.00933 | −2.74169 | −0.0719365 | 3.77059 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | 2.03821 | 1.87275 | 2.15429 | −3.24760 | 3.81704 | 1.92338 | 0.314472 | 0.507178 | −6.61928 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.55401 | −2.99825 | 4.52299 | 0.244850 | −7.65758 | 4.42321 | 6.44376 | 5.98952 | 0.625350 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(1\) |
| \(19\) | \(-1\) |
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 | 209.2.a.d | ✓ | 7 |
| 3.b | odd | 2 | 1 | 1881.2.a.p | 7 | ||
| 4.b | odd | 2 | 1 | 3344.2.a.ba | 7 | ||
| 5.b | even | 2 | 1 | 5225.2.a.n | 7 | ||
| 11.b | odd | 2 | 1 | 2299.2.a.q | 7 | ||
| 19.b | odd | 2 | 1 | 3971.2.a.i | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 209.2.a.d | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 1881.2.a.p | 7 | 3.b | odd | 2 | 1 | ||
| 2299.2.a.q | 7 | 11.b | odd | 2 | 1 | ||
| 3344.2.a.ba | 7 | 4.b | odd | 2 | 1 | ||
| 3971.2.a.i | 7 | 19.b | odd | 2 | 1 | ||
| 5225.2.a.n | 7 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{7} + T_{2}^{6} - 14T_{2}^{5} - 10T_{2}^{4} + 59T_{2}^{3} + 27T_{2}^{2} - 66T_{2} - 30 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(209))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{7} + T^{6} + \cdots - 30 \)
|
| $3$ |
\( T^{7} - 2 T^{6} + \cdots + 64 \)
|
| $5$ |
\( T^{7} - 2 T^{6} + \cdots - 6 \)
|
| $7$ |
\( T^{7} - 10 T^{6} + \cdots + 512 \)
|
| $11$ |
\( (T + 1)^{7} \)
|
| $13$ |
\( T^{7} + 4 T^{6} + \cdots - 5716 \)
|
| $17$ |
\( T^{7} - 2 T^{6} + \cdots - 17088 \)
|
| $19$ |
\( (T - 1)^{7} \)
|
| $23$ |
\( T^{7} - 10 T^{6} + \cdots + 1920 \)
|
| $29$ |
\( T^{7} + 18 T^{6} + \cdots - 276 \)
|
| $31$ |
\( T^{7} - 24 T^{6} + \cdots + 4 \)
|
| $37$ |
\( T^{7} - 121 T^{5} + \cdots - 8992 \)
|
| $41$ |
\( T^{7} + 12 T^{6} + \cdots - 1824 \)
|
| $43$ |
\( T^{7} - 2 T^{6} + \cdots + 4976 \)
|
| $47$ |
\( T^{7} - 8 T^{6} + \cdots + 79872 \)
|
| $53$ |
\( T^{7} - 2 T^{6} + \cdots + 768 \)
|
| $59$ |
\( T^{7} + 10 T^{6} + \cdots - 6552192 \)
|
| $61$ |
\( T^{7} - 14 T^{6} + \cdots - 36544 \)
|
| $67$ |
\( T^{7} - 8 T^{6} + \cdots + 13544 \)
|
| $71$ |
\( T^{7} - 10 T^{6} + \cdots + 39756 \)
|
| $73$ |
\( T^{7} + 6 T^{6} + \cdots + 67328 \)
|
| $79$ |
\( T^{7} - 52 T^{6} + \cdots - 203264 \)
|
| $83$ |
\( T^{7} + 10 T^{6} + \cdots + 576936 \)
|
| $89$ |
\( T^{7} - 401 T^{5} + \cdots - 8199552 \)
|
| $97$ |
\( T^{7} + 24 T^{6} + \cdots - 17393056 \)
|
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