Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1334,2,Mod(1,1334)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1334, 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("1334.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1334 = 2 \cdot 23 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1334.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: | \(10.6520436296\) |
| 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: |
\( x^{9} - 3x^{8} - 16x^{7} + 44x^{6} + 87x^{5} - 209x^{4} - 160x^{3} + 348x^{2} + 12x - 100 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{8}\) 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
\( x^{9} - 3x^{8} - 16x^{7} + 44x^{6} + 87x^{5} - 209x^{4} - 160x^{3} + 348x^{2} + 12x - 100 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} + \nu^{5} + 13\nu^{4} - 7\nu^{3} - 44\nu^{2} + 8\nu + 22 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - \nu^{5} - 13\nu^{4} + 9\nu^{3} + 42\nu^{2} - 22\nu - 18 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{8} - 3\nu^{7} - 12\nu^{6} + 34\nu^{5} + 43\nu^{4} - 105\nu^{3} - 48\nu^{2} + 62\nu + 28 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{8} + 5\nu^{7} + 6\nu^{6} - 54\nu^{5} + 21\nu^{4} + 143\nu^{3} - 134\nu^{2} - 18\nu + 44 ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{7} + 4\nu^{6} + 9\nu^{5} - 43\nu^{4} - 14\nu^{3} + 119\nu^{2} - 26\nu - 44 ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -\nu^{8} + 5\nu^{7} + 6\nu^{6} - 56\nu^{5} + 23\nu^{4} + 165\nu^{3} - 144\nu^{2} - 70\nu + 52 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + \beta_{3} + \beta_{2} + 8\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} + 9\beta_{2} + 14\beta _1 + 27 \)
|
| \(\nu^{5}\) | \(=\) |
\( -2\beta_{8} + \beta_{7} + 3\beta_{6} + \beta_{5} + 12\beta_{4} + 13\beta_{3} + 15\beta_{2} + 71\beta _1 + 33 \)
|
| \(\nu^{6}\) | \(=\) |
\( -2\beta_{8} + 14\beta_{7} + 16\beta_{6} + 14\beta_{5} + 18\beta_{4} + 30\beta_{3} + 81\beta_{2} + 161\beta _1 + 216 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 26 \beta_{8} + 20 \beta_{7} + 48 \beta_{6} + 22 \beta_{5} + 123 \beta_{4} + 137 \beta_{3} + \cdots + 404 \)
|
| \(\nu^{8}\) | \(=\) |
\( - 34 \beta_{8} + 151 \beta_{7} + 191 \beta_{6} + 161 \beta_{5} + 239 \beta_{4} + 348 \beta_{3} + \cdots + 1895 \)
|
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 | −3.18584 | 1.00000 | −0.979916 | 3.18584 | 4.69416 | −1.00000 | 7.14956 | 0.979916 | |||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −3.11529 | 1.00000 | 2.08342 | 3.11529 | −4.29295 | −1.00000 | 6.70504 | −2.08342 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −2.04237 | 1.00000 | 4.28519 | 2.04237 | 0.891555 | −1.00000 | 1.17127 | −4.28519 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | −1.07506 | 1.00000 | −2.43466 | 1.07506 | 1.40780 | −1.00000 | −1.84425 | 2.43466 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | −0.783963 | 1.00000 | −1.63370 | 0.783963 | −4.03607 | −1.00000 | −2.38540 | 1.63370 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 0.540845 | 1.00000 | 1.92844 | −0.540845 | 2.85963 | −1.00000 | −2.70749 | −1.92844 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 2.05766 | 1.00000 | 3.66144 | −2.05766 | 4.21443 | −1.00000 | 1.23398 | −3.66144 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 2.10346 | 1.00000 | −3.58097 | −2.10346 | 2.82290 | −1.00000 | 1.42454 | 3.58097 | ||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | −1.00000 | 2.50055 | 1.00000 | 1.67075 | −2.50055 | −2.56147 | −1.00000 | 3.25275 | −1.67075 | ||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(23\) | \(1\) |
| \(29\) | \(-1\) |
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 | 1334.2.a.j | ✓ | 9 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1334.2.a.j | ✓ | 9 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1334))\):
|
\( T_{3}^{9} + 3T_{3}^{8} - 16T_{3}^{7} - 44T_{3}^{6} + 87T_{3}^{5} + 209T_{3}^{4} - 160T_{3}^{3} - 348T_{3}^{2} + 12T_{3} + 100 \)
|
|
\( T_{5}^{9} - 5T_{5}^{8} - 20T_{5}^{7} + 114T_{5}^{6} + 95T_{5}^{5} - 779T_{5}^{4} - 10T_{5}^{3} + 1912T_{5}^{2} - 322T_{5} - 1470 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{9} \)
|
| $3$ |
\( T^{9} + 3 T^{8} + \cdots + 100 \)
|
| $5$ |
\( T^{9} - 5 T^{8} + \cdots - 1470 \)
|
| $7$ |
\( T^{9} - 6 T^{8} + \cdots + 8896 \)
|
| $11$ |
\( T^{9} + 3 T^{8} + \cdots + 240 \)
|
| $13$ |
\( T^{9} - 13 T^{8} + \cdots + 243056 \)
|
| $17$ |
\( T^{9} - 2 T^{8} + \cdots + 468 \)
|
| $19$ |
\( T^{9} - 16 T^{8} + \cdots - 61376 \)
|
| $23$ |
\( (T + 1)^{9} \)
|
| $29$ |
\( (T - 1)^{9} \)
|
| $31$ |
\( T^{9} - 15 T^{8} + \cdots + 33088 \)
|
| $37$ |
\( T^{9} - 12 T^{8} + \cdots - 40768 \)
|
| $41$ |
\( T^{9} + 6 T^{8} + \cdots - 34848 \)
|
| $43$ |
\( T^{9} + 3 T^{8} + \cdots + 132080 \)
|
| $47$ |
\( T^{9} + 19 T^{8} + \cdots + 273312 \)
|
| $53$ |
\( T^{9} - 5 T^{8} + \cdots + 3058362 \)
|
| $59$ |
\( T^{9} - 12 T^{8} + \cdots + 422400 \)
|
| $61$ |
\( T^{9} - 12 T^{8} + \cdots - 201344 \)
|
| $67$ |
\( T^{9} + 6 T^{8} + \cdots - 675484 \)
|
| $71$ |
\( T^{9} - 12 T^{8} + \cdots + 26575104 \)
|
| $73$ |
\( T^{9} - 354 T^{7} + \cdots - 1965088 \)
|
| $79$ |
\( T^{9} - 29 T^{8} + \cdots + 16643750 \)
|
| $83$ |
\( T^{9} - 24 T^{8} + \cdots + 672218736 \)
|
| $89$ |
\( T^{9} + 2 T^{8} + \cdots - 111132 \)
|
| $97$ |
\( T^{9} - 12 T^{8} + \cdots - 790352 \)
|
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