Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [802,2,Mod(1,802)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(802, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("802.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 802 = 2 \cdot 401 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 802.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: | \(6.40400224211\) |
| Analytic rank: | \(0\) |
| Dimension: | \(7\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{7} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{7} - x^{6} - 10x^{5} + 6x^{4} + 15x^{3} - 2x^{2} - 6x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{6}\) 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
\( x^{7} - x^{6} - 10x^{5} + 6x^{4} + 15x^{3} - 2x^{2} - 6x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 3\nu^{6} - 5\nu^{5} - 26\nu^{4} + 35\nu^{3} + 15\nu^{2} - 15\nu - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( -3\nu^{6} + 6\nu^{5} + 25\nu^{4} - 44\nu^{3} - 10\nu^{2} + 22\nu + 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( 3\nu^{6} - 5\nu^{5} - 27\nu^{4} + 36\nu^{3} + 24\nu^{2} - 21\nu - 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( 6\nu^{6} - 10\nu^{5} - 53\nu^{4} + 71\nu^{3} + 40\nu^{2} - 37\nu - 10 \)
|
| \(\beta_{6}\) | \(=\) |
\( 8\nu^{6} - 14\nu^{5} - 70\nu^{4} + 101\nu^{3} + 49\nu^{2} - 55\nu - 12 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - \beta_{4} - \beta_{2} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{6} - \beta_{5} - 3\beta_{4} + 2\beta_{3} - \beta_{2} + 6\beta _1 + 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{6} + 8\beta_{5} - 13\beta_{4} + 2\beta_{3} - 9\beta_{2} + 9\beta _1 + 25 \)
|
| \(\nu^{5}\) | \(=\) |
\( 30\beta_{6} - 6\beta_{5} - 35\beta_{4} + 21\beta_{3} - 12\beta_{2} + 51\beta _1 + 36 \)
|
| \(\nu^{6}\) | \(=\) |
\( 41\beta_{6} + 66\beta_{5} - 131\beta_{4} + 29\beta_{3} - 81\beta_{2} + 93\beta _1 + 227 \)
|
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 | −2.26428 | 1.00000 | 0.509131 | 2.26428 | 1.11603 | −1.00000 | 2.12696 | −0.509131 | |||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −0.692007 | 1.00000 | −2.58041 | 0.692007 | −2.90466 | −1.00000 | −2.52113 | 2.58041 | ||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −0.265216 | 1.00000 | 1.78121 | 0.265216 | 1.93788 | −1.00000 | −2.92966 | −1.78121 | ||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0.969858 | 1.00000 | 3.63215 | −0.969858 | −1.87179 | −1.00000 | −2.05938 | −3.63215 | ||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 2.14284 | 1.00000 | −1.35870 | −2.14284 | 3.94497 | −1.00000 | 1.59175 | 1.35870 | ||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 2.79683 | 1.00000 | 3.42155 | −2.79683 | 1.08416 | −1.00000 | 4.82224 | −3.42155 | ||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 3.31198 | 1.00000 | −0.404924 | −3.31198 | −2.30658 | −1.00000 | 7.96921 | 0.404924 | ||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(401\) | \(-1\) |
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 | 802.2.a.d | ✓ | 7 |
| 3.b | odd | 2 | 1 | 7218.2.a.r | 7 | ||
| 4.b | odd | 2 | 1 | 6416.2.a.g | 7 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 802.2.a.d | ✓ | 7 | 1.a | even | 1 | 1 | trivial |
| 6416.2.a.g | 7 | 4.b | odd | 2 | 1 | ||
| 7218.2.a.r | 7 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{7} - 6T_{3}^{6} + 3T_{3}^{5} + 35T_{3}^{4} - 47T_{3}^{3} - 23T_{3}^{2} + 28T_{3} + 8 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(802))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{7} \)
|
| $3$ |
\( T^{7} - 6 T^{6} + \cdots + 8 \)
|
| $5$ |
\( T^{7} - 5 T^{6} + \cdots + 16 \)
|
| $7$ |
\( T^{7} - T^{6} + \cdots + 116 \)
|
| $11$ |
\( T^{7} - 8 T^{6} + \cdots + 139 \)
|
| $13$ |
\( T^{7} - T^{6} + \cdots + 67 \)
|
| $17$ |
\( T^{7} - 12 T^{6} + \cdots - 3176 \)
|
| $19$ |
\( T^{7} + 6 T^{6} + \cdots - 1168 \)
|
| $23$ |
\( T^{7} - 25 T^{6} + \cdots - 59788 \)
|
| $29$ |
\( T^{7} - 4 T^{6} + \cdots - 9236 \)
|
| $31$ |
\( T^{7} - 16 T^{6} + \cdots + 277523 \)
|
| $37$ |
\( T^{7} + 2 T^{6} + \cdots - 64 \)
|
| $41$ |
\( T^{7} - 7 T^{6} + \cdots + 34817 \)
|
| $43$ |
\( T^{7} + 9 T^{6} + \cdots + 1343 \)
|
| $47$ |
\( T^{7} - 25 T^{6} + \cdots + 57404 \)
|
| $53$ |
\( T^{7} - 15 T^{6} + \cdots - 143867 \)
|
| $59$ |
\( T^{7} - 8 T^{6} + \cdots + 32132 \)
|
| $61$ |
\( T^{7} + 25 T^{6} + \cdots + 4219 \)
|
| $67$ |
\( T^{7} - T^{6} + \cdots - 615652 \)
|
| $71$ |
\( T^{7} - 25 T^{6} + \cdots + 4249 \)
|
| $73$ |
\( T^{7} + 15 T^{6} + \cdots - 45769 \)
|
| $79$ |
\( T^{7} - 11 T^{6} + \cdots - 465433 \)
|
| $83$ |
\( T^{7} - 26 T^{6} + \cdots - 1176001 \)
|
| $89$ |
\( T^{7} - 16 T^{6} + \cdots - 104209 \)
|
| $97$ |
\( T^{7} - 7 T^{6} + \cdots - 168812 \)
|
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