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: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 22x^{6} + 151x^{4} - 332x^{2} + 6 \)
|
| 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
\( x^{8} - 22x^{6} + 151x^{4} - 332x^{2} + 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} - 15\nu^{4} + 46\nu^{2} + 6 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 15\nu^{5} + 38\nu^{3} + 70\nu ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5\nu^{6} - 83\nu^{4} + 310\nu^{2} - 34 ) / 16 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} - 5\nu^{6} + 23\nu^{5} + 83\nu^{4} - 174\nu^{3} - 294\nu^{2} + 442\nu - 62 ) / 32 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - 5\nu^{6} - 23\nu^{5} + 83\nu^{4} + 174\nu^{3} - 294\nu^{2} - 442\nu - 62 ) / 32 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -3\nu^{7} + 53\nu^{5} - 234\nu^{3} + 174\nu ) / 16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + \beta_{5} + \beta_{4} + 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + 8\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 10\beta_{6} + 10\beta_{5} + 8\beta_{4} + 5\beta_{2} + 52 \)
|
| \(\nu^{5}\) | \(=\) |
\( 17\beta_{7} + 15\beta_{6} - 15\beta_{5} + 18\beta_{3} + 72\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 104\beta_{6} + 104\beta_{5} + 74\beta_{4} + 83\beta_{2} + 498 \)
|
| \(\nu^{7}\) | \(=\) |
\( 217\beta_{7} + 187\beta_{6} - 187\beta_{5} + 240\beta_{3} + 706\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
1.00000 | −2.52651 | 1.00000 | −2.27484 | −2.52651 | 4.02057 | 1.00000 | 3.38328 | −2.27484 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.52651 | 1.00000 | 2.27484 | −2.52651 | 1.33106 | 1.00000 | 3.38328 | 2.27484 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −0.153561 | 1.00000 | −3.32381 | −0.153561 | −3.68051 | 1.00000 | −2.97642 | −3.32381 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | −0.153561 | 1.00000 | 3.32381 | −0.153561 | 0.786348 | 1.00000 | −2.97642 | 3.32381 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 1.77365 | 1.00000 | −0.134992 | 1.77365 | 4.95530 | 1.00000 | 0.145840 | −0.134992 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 1.77365 | 1.00000 | 0.134992 | 1.77365 | 1.25283 | 1.00000 | 0.145840 | 0.134992 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 2.90642 | 1.00000 | −2.39983 | 2.90642 | 0.548309 | 1.00000 | 5.44730 | −2.39983 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 2.90642 | 1.00000 | 2.39983 | 2.90642 | −1.21390 | 1.00000 | 5.44730 | 2.39983 | |||||||||||||||||||||||||||||||||||||||||||
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.i | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1334.2.a.i | ✓ | 8 | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1334))\):
|
\( T_{3}^{4} - 2T_{3}^{3} - 7T_{3}^{2} + 12T_{3} + 2 \)
|
|
\( T_{5}^{8} - 22T_{5}^{6} + 151T_{5}^{4} - 332T_{5}^{2} + 6 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{8} \)
|
| $3$ |
\( (T^{4} - 2 T^{3} - 7 T^{2} + \cdots + 2)^{2} \)
|
| $5$ |
\( T^{8} - 22 T^{6} + \cdots + 6 \)
|
| $7$ |
\( T^{8} - 8 T^{7} + \cdots + 64 \)
|
| $11$ |
\( T^{8} - 8 T^{7} + \cdots - 1392 \)
|
| $13$ |
\( T^{8} - 4 T^{7} + \cdots - 656 \)
|
| $17$ |
\( T^{8} - 8 T^{7} + \cdots - 93348 \)
|
| $19$ |
\( T^{8} - 16 T^{7} + \cdots - 448 \)
|
| $23$ |
\( (T + 1)^{8} \)
|
| $29$ |
\( (T + 1)^{8} \)
|
| $31$ |
\( T^{8} - 8 T^{7} + \cdots - 306112 \)
|
| $37$ |
\( T^{8} - 8 T^{7} + \cdots + 56896 \)
|
| $41$ |
\( T^{8} - 8 T^{7} + \cdots - 1493664 \)
|
| $43$ |
\( T^{8} - 32 T^{7} + \cdots + 2677264 \)
|
| $47$ |
\( T^{8} - 16 T^{7} + \cdots - 180384 \)
|
| $53$ |
\( T^{8} - 150 T^{6} + \cdots + 3174 \)
|
| $59$ |
\( T^{8} + 4 T^{7} + \cdots + 8418816 \)
|
| $61$ |
\( T^{8} - 8 T^{7} + \cdots + 636544 \)
|
| $67$ |
\( T^{8} - 20 T^{7} + \cdots - 10372 \)
|
| $71$ |
\( T^{8} + 24 T^{7} + \cdots - 10564032 \)
|
| $73$ |
\( T^{8} + 4 T^{7} + \cdots + 12154528 \)
|
| $79$ |
\( T^{8} - 4 T^{7} + \cdots + 654854 \)
|
| $83$ |
\( T^{8} + 4 T^{7} + \cdots + 3431664 \)
|
| $89$ |
\( T^{8} - 20 T^{7} + \cdots + 1308 \)
|
| $97$ |
\( T^{8} - 4 T^{7} + \cdots + 2118256 \)
|
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