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: |
\( x^{8} - 3 x^{7} - 118665 x^{6} - 683929 x^{5} + 4157106948 x^{4} - 12549505824 x^{3} + \cdots - 11\!\cdots\!08 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{28}\cdot 3^{3}\cdot 5^{2} \) |
| 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
\( x^{8} - 3 x^{7} - 118665 x^{6} - 683929 x^{5} + 4157106948 x^{4} - 12549505824 x^{3} + \cdots - 11\!\cdots\!08 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 13\nu - 29663 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 79993628273 \nu^{7} - 294823600324362 \nu^{6} + \cdots - 61\!\cdots\!72 ) / 24\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 167747694763 \nu^{7} - 91903197998778 \nu^{6} + \cdots - 45\!\cdots\!68 ) / 10\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 384699278273 \nu^{7} + 15768621076512 \nu^{6} + \cdots + 53\!\cdots\!72 ) / 17\!\cdots\!00 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 40484599547351 \nu^{7} + \cdots + 94\!\cdots\!64 ) / 72\!\cdots\!00 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1470935697068 \nu^{7} + 29151750532683 \nu^{6} + \cdots - 52\!\cdots\!52 ) / 25\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 13\beta _1 + 29663 \)
|
| \(\nu^{3}\) | \(=\) |
\( -3\beta_{6} + 11\beta_{5} - 42\beta_{4} + 29\beta_{3} + 95\beta_{2} + 48684\beta _1 + 371752 \)
|
| \(\nu^{4}\) | \(=\) |
\( 1573 \beta_{7} + 289 \beta_{6} + 3186 \beta_{5} + 2043 \beta_{4} + 3294 \beta_{3} + 69094 \beta_{2} + \cdots + 1442410399 \)
|
| \(\nu^{5}\) | \(=\) |
\( 83484 \beta_{7} - 195219 \beta_{6} + 1022283 \beta_{5} - 3800664 \beta_{4} + 2970171 \beta_{3} + \cdots + 76142006450 \)
|
| \(\nu^{6}\) | \(=\) |
\( 151963515 \beta_{7} + 31020657 \beta_{6} + 310803284 \beta_{5} + 55788573 \beta_{4} + \cdots + 82316489929573 \)
|
| \(\nu^{7}\) | \(=\) |
\( 10088866762 \beta_{7} - 10478502683 \beta_{6} + 79846311741 \beta_{5} - 273812396382 \beta_{4} + \cdots + 82\!\cdots\!92 \)
|
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 | −253.133 | 0 | −625.000 | 0 | −2401.00 | 0 | 44393.3 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −210.521 | 0 | −625.000 | 0 | −2401.00 | 0 | 24636.1 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −188.152 | 0 | −625.000 | 0 | −2401.00 | 0 | 15718.3 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | −19.7050 | 0 | −625.000 | 0 | −2401.00 | 0 | −19294.7 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | −0.675343 | 0 | −625.000 | 0 | −2401.00 | 0 | −19682.5 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 109.834 | 0 | −625.000 | 0 | −2401.00 | 0 | −7619.55 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 0 | 146.011 | 0 | −625.000 | 0 | −2401.00 | 0 | 1636.31 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 0 | 251.342 | 0 | −625.000 | 0 | −2401.00 | 0 | 43489.7 | 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.g | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 280.10.a.g | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} + 165 T_{3}^{7} - 106758 T_{3}^{6} - 15144886 T_{3}^{5} + 3312966693 T_{3}^{4} + \cdots - 537826007347200 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(280))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + \cdots - 537826007347200 \)
|
| $5$ |
\( (T + 625)^{8} \)
|
| $7$ |
\( (T + 2401)^{8} \)
|
| $11$ |
\( T^{8} + \cdots + 18\!\cdots\!00 \)
|
| $13$ |
\( T^{8} + \cdots + 13\!\cdots\!00 \)
|
| $17$ |
\( T^{8} + \cdots - 30\!\cdots\!56 \)
|
| $19$ |
\( T^{8} + \cdots + 42\!\cdots\!28 \)
|
| $23$ |
\( T^{8} + \cdots + 53\!\cdots\!00 \)
|
| $29$ |
\( T^{8} + \cdots - 25\!\cdots\!68 \)
|
| $31$ |
\( T^{8} + \cdots + 86\!\cdots\!48 \)
|
| $37$ |
\( T^{8} + \cdots - 22\!\cdots\!32 \)
|
| $41$ |
\( T^{8} + \cdots - 68\!\cdots\!00 \)
|
| $43$ |
\( T^{8} + \cdots + 15\!\cdots\!00 \)
|
| $47$ |
\( T^{8} + \cdots + 40\!\cdots\!64 \)
|
| $53$ |
\( T^{8} + \cdots + 96\!\cdots\!00 \)
|
| $59$ |
\( T^{8} + \cdots - 17\!\cdots\!00 \)
|
| $61$ |
\( T^{8} + \cdots - 13\!\cdots\!64 \)
|
| $67$ |
\( T^{8} + \cdots - 86\!\cdots\!00 \)
|
| $71$ |
\( T^{8} + \cdots + 42\!\cdots\!00 \)
|
| $73$ |
\( T^{8} + \cdots - 18\!\cdots\!00 \)
|
| $79$ |
\( T^{8} + \cdots - 86\!\cdots\!96 \)
|
| $83$ |
\( T^{8} + \cdots - 33\!\cdots\!56 \)
|
| $89$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
|
| $97$ |
\( T^{8} + \cdots + 95\!\cdots\!68 \)
|
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