Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1670,2,Mod(1,1670)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1670, 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("1670.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1670 = 2 \cdot 5 \cdot 167 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1670.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: | \(13.3350171376\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 17x^{6} + 14x^{5} + 80x^{4} - 56x^{3} - 83x^{2} + 38x + 23 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} - 17x^{6} + 14x^{5} + 80x^{4} - 56x^{3} - 83x^{2} + 38x + 23 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 19\nu^{4} + 8\nu^{3} + 87\nu^{2} - 63\nu - 34 ) / 13 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{7} + \nu^{6} + 19\nu^{5} - 14\nu^{4} - 105\nu^{3} + 46\nu^{2} + 140\nu - 8 ) / 13 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{6} + 38\nu^{4} - 3\nu^{3} - 187\nu^{2} + 35\nu + 120 ) / 13 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} + 3\nu^{6} - 19\nu^{5} - 49\nu^{4} + 98\nu^{3} + 198\nu^{2} - 119\nu - 115 ) / 13 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5\nu^{7} - 2\nu^{6} - 82\nu^{5} + 13\nu^{4} + 354\nu^{3} + 5\nu^{2} - 252\nu - 62 ) / 13 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 2\beta_{3} + \beta_{2} + 7\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + 3\beta_{5} + \beta_{4} + 2\beta_{3} + 11\beta_{2} + 31 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{7} + 15\beta_{5} + 5\beta_{4} + 27\beta_{3} + 17\beta_{2} + 56\beta _1 + 8 \)
|
| \(\nu^{6}\) | \(=\) |
\( 19\beta_{6} + 49\beta_{5} + 19\beta_{4} + 35\beta_{3} + 114\beta_{2} + 7\beta _1 + 275 \)
|
| \(\nu^{7}\) | \(=\) |
\( 19\beta_{7} + 5\beta_{6} + 187\beta_{5} + 87\beta_{4} + 310\beta_{3} + 224\beta_{2} + 476\beta _1 + 169 \)
|
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 |
|
−1.00000 | −3.28148 | 1.00000 | −1.00000 | 3.28148 | −1.64278 | −1.00000 | 7.76809 | 1.00000 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.45989 | 1.00000 | −1.00000 | 2.45989 | 3.58986 | −1.00000 | 3.05108 | 1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −1.11866 | 1.00000 | −1.00000 | 1.11866 | −2.68452 | −1.00000 | −1.74860 | 1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −0.932332 | 1.00000 | −1.00000 | 0.932332 | 1.71088 | −1.00000 | −2.13076 | 1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 0.399020 | 1.00000 | −1.00000 | −0.399020 | 3.32748 | −1.00000 | −2.84078 | 1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 0.917346 | 1.00000 | −1.00000 | −0.917346 | −4.92439 | −1.00000 | −2.15848 | 1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 2.55611 | 1.00000 | −1.00000 | −2.55611 | 3.73448 | −1.00000 | 3.53371 | 1.00000 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 2.91989 | 1.00000 | −1.00000 | −2.91989 | −0.111013 | −1.00000 | 5.52574 | 1.00000 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(167\) | \( -1 \) |
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 | 1670.2.a.g | ✓ | 8 |
| 5.b | even | 2 | 1 | 8350.2.a.v | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1670.2.a.g | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 8350.2.a.v | 8 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + T_{3}^{7} - 17T_{3}^{6} - 14T_{3}^{5} + 80T_{3}^{4} + 56T_{3}^{3} - 83T_{3}^{2} - 38T_{3} + 23 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1670))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{8} \)
|
| $3$ |
\( T^{8} + T^{7} + \cdots + 23 \)
|
| $5$ |
\( (T + 1)^{8} \)
|
| $7$ |
\( T^{8} - 3 T^{7} + \cdots + 184 \)
|
| $11$ |
\( T^{8} + T^{7} + \cdots + 72 \)
|
| $13$ |
\( T^{8} - 11 T^{7} + \cdots + 4168 \)
|
| $17$ |
\( T^{8} + 12 T^{7} + \cdots - 231 \)
|
| $19$ |
\( T^{8} - 17 T^{7} + \cdots + 344 \)
|
| $23$ |
\( T^{8} + 10 T^{7} + \cdots - 27 \)
|
| $29$ |
\( T^{8} - 5 T^{7} + \cdots - 4392 \)
|
| $31$ |
\( T^{8} - 15 T^{7} + \cdots + 71 \)
|
| $37$ |
\( T^{8} + T^{7} + \cdots - 9976 \)
|
| $41$ |
\( T^{8} - 97 T^{6} + \cdots - 72 \)
|
| $43$ |
\( T^{8} - 5 T^{7} + \cdots - 376 \)
|
| $47$ |
\( T^{8} + 10 T^{7} + \cdots - 188616 \)
|
| $53$ |
\( T^{8} + 3 T^{7} + \cdots + 2664 \)
|
| $59$ |
\( T^{8} - 11 T^{7} + \cdots - 77781 \)
|
| $61$ |
\( T^{8} - 35 T^{7} + \cdots - 303608 \)
|
| $67$ |
\( T^{8} - 20 T^{7} + \cdots + 27136 \)
|
| $71$ |
\( T^{8} - 11 T^{7} + \cdots + 1369608 \)
|
| $73$ |
\( T^{8} - 9 T^{7} + \cdots + 119299 \)
|
| $79$ |
\( T^{8} - 38 T^{7} + \cdots + 5848576 \)
|
| $83$ |
\( T^{8} + 11 T^{7} + \cdots + 3912 \)
|
| $89$ |
\( T^{8} - 2 T^{7} + \cdots + 1948389 \)
|
| $97$ |
\( T^{8} - 35 T^{7} + \cdots - 12991144 \)
|
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