Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [280,10,Mod(1,280)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(280, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("280.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 280 = 2^{3} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 280.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: | \(144.210034126\) |
| 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: |
\( x^{8} - 3 x^{7} - 99674 x^{6} + 1019290 x^{5} + 2669616237 x^{4} - 73226219607 x^{3} + \cdots + 32\!\cdots\!48 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{28}\cdot 3^{2}\cdot 5^{2} \) |
| 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
\( x^{8} - 3 x^{7} - 99674 x^{6} + 1019290 x^{5} + 2669616237 x^{4} - 73226219607 x^{3} + \cdots + 32\!\cdots\!48 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 11\nu - 24924 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 14352235 \nu^{7} - 52272051579 \nu^{6} - 5098442152894 \nu^{5} + \cdots + 13\!\cdots\!48 ) / 19\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1305406307 \nu^{7} - 108277089387 \nu^{6} + 117223695715282 \nu^{5} + \cdots + 32\!\cdots\!00 ) / 16\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 90412135 \nu^{7} + 5968943823 \nu^{6} - 8020077287402 \nu^{5} - 379563302223938 \nu^{4} + \cdots + 12\!\cdots\!04 ) / 97\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 24590923123 \nu^{7} + 4561098114813 \nu^{6} + \cdots - 33\!\cdots\!52 ) / 23\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 3672128497 \nu^{7} - 27169706823 \nu^{6} - 339622951070822 \nu^{5} + \cdots + 63\!\cdots\!00 ) / 16\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 11\beta _1 + 24924 \)
|
| \(\nu^{3}\) | \(=\) |
\( -5\beta_{7} - 4\beta_{6} - 2\beta_{5} - 11\beta_{4} - 4\beta_{3} + 33\beta_{2} + 46074\beta _1 - 287368 \)
|
| \(\nu^{4}\) | \(=\) |
\( - 488 \beta_{7} - 224 \beta_{6} + 1221 \beta_{5} + 361 \beta_{4} + 560 \beta_{3} + 61997 \beta_{2} + \cdots + 1147917010 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 343879 \beta_{7} - 324404 \beta_{6} - 219931 \beta_{5} - 782650 \beta_{4} - 757046 \beta_{3} + \cdots - 5039781560 \)
|
| \(\nu^{6}\) | \(=\) |
\( - 44049824 \beta_{7} - 14909760 \beta_{6} + 113240942 \beta_{5} + 28840486 \beta_{4} + \cdots + 61589163777480 \)
|
| \(\nu^{7}\) | \(=\) |
\( - 20906737905 \beta_{7} - 21742262372 \beta_{6} - 17281946192 \beta_{5} - 50590751853 \beta_{4} + \cdots + 884843303987344 \)
|
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 |
|
0 | −250.054 | 0 | 625.000 | 0 | 2401.00 | 0 | 42843.8 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −130.386 | 0 | 625.000 | 0 | 2401.00 | 0 | −2682.49 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −116.306 | 0 | 625.000 | 0 | 2401.00 | 0 | −6155.90 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −57.1567 | 0 | 625.000 | 0 | 2401.00 | 0 | −16416.1 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 46.9898 | 0 | 625.000 | 0 | 2401.00 | 0 | −17475.0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 66.0094 | 0 | 625.000 | 0 | 2401.00 | 0 | −15325.8 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 201.764 | 0 | 625.000 | 0 | 2401.00 | 0 | 21025.6 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 236.139 | 0 | 625.000 | 0 | 2401.00 | 0 | 36078.8 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(-1\) |
| \(7\) | \(-1\) |
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 | 280.10.a.h | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 280.10.a.h | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 3 T_{3}^{7} - 99674 T_{3}^{6} - 1019290 T_{3}^{5} + 2669616237 T_{3}^{4} + \cdots + 32\!\cdots\!48 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(280))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + \cdots + 32\!\cdots\!48 \)
|
| $5$ |
\( (T - 625)^{8} \)
|
| $7$ |
\( (T - 2401)^{8} \)
|
| $11$ |
\( T^{8} + \cdots - 26\!\cdots\!40 \)
|
| $13$ |
\( T^{8} + \cdots - 79\!\cdots\!12 \)
|
| $17$ |
\( T^{8} + \cdots - 55\!\cdots\!20 \)
|
| $19$ |
\( T^{8} + \cdots - 25\!\cdots\!48 \)
|
| $23$ |
\( T^{8} + \cdots - 34\!\cdots\!68 \)
|
| $29$ |
\( T^{8} + \cdots - 44\!\cdots\!36 \)
|
| $31$ |
\( T^{8} + \cdots - 14\!\cdots\!00 \)
|
| $37$ |
\( T^{8} + \cdots + 14\!\cdots\!68 \)
|
| $41$ |
\( T^{8} + \cdots + 92\!\cdots\!52 \)
|
| $43$ |
\( T^{8} + \cdots - 14\!\cdots\!12 \)
|
| $47$ |
\( T^{8} + \cdots - 47\!\cdots\!52 \)
|
| $53$ |
\( T^{8} + \cdots + 20\!\cdots\!28 \)
|
| $59$ |
\( T^{8} + \cdots - 15\!\cdots\!24 \)
|
| $61$ |
\( T^{8} + \cdots + 88\!\cdots\!20 \)
|
| $67$ |
\( T^{8} + \cdots - 10\!\cdots\!32 \)
|
| $71$ |
\( T^{8} + \cdots + 92\!\cdots\!00 \)
|
| $73$ |
\( T^{8} + \cdots - 61\!\cdots\!00 \)
|
| $79$ |
\( T^{8} + \cdots + 86\!\cdots\!56 \)
|
| $83$ |
\( T^{8} + \cdots + 51\!\cdots\!16 \)
|
| $89$ |
\( T^{8} + \cdots + 35\!\cdots\!16 \)
|
| $97$ |
\( T^{8} + \cdots - 91\!\cdots\!24 \)
|
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