Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [805,2,Mod(1,805)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(805, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("805.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 805 = 5 \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 805.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: | \(6.42795736271\) |
| 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: |
\( x^{8} - x^{7} - 11x^{6} + 9x^{5} + 36x^{4} - 23x^{3} - 30x^{2} + 17x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{8} - x^{7} - 11x^{6} + 9x^{5} + 36x^{4} - 23x^{3} - 30x^{2} + 17x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 6\nu^{2} - \nu + 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 7\nu^{3} + 9\nu - 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( \nu^{7} - \nu^{6} - 11\nu^{5} + 8\nu^{4} + 36\nu^{3} - 16\nu^{2} - 30\nu + 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( 2\nu^{7} - \nu^{6} - 22\nu^{5} + 8\nu^{4} + 72\nu^{3} - 17\nu^{2} - 62\nu + 12 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 6\beta_{2} + \beta _1 + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 7\beta_{3} + 26\beta _1 + 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{7} - 2\beta_{6} + 8\beta_{4} + 33\beta_{2} + 10\beta _1 + 71 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{7} - \beta_{6} + 11\beta_{5} + 41\beta_{3} + \beta_{2} + 138\beta _1 + 21 \)
|
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 |
|
−2.28279 | 0.445964 | 3.21111 | 1.00000 | −1.01804 | 1.00000 | −2.76472 | −2.80112 | −2.28279 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.94419 | 3.14297 | 1.77986 | 1.00000 | −6.11052 | 1.00000 | 0.427999 | 6.87827 | −1.94419 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.17189 | −1.97939 | −0.626674 | 1.00000 | 2.31962 | 1.00000 | 3.07817 | 0.917966 | −1.17189 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 0.181467 | 3.25071 | −1.96707 | 1.00000 | 0.589897 | 1.00000 | −0.719892 | 7.56714 | 0.181467 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 0.322285 | −1.07802 | −1.89613 | 1.00000 | −0.347431 | 1.00000 | −1.25566 | −1.83786 | 0.322285 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.18794 | 1.57054 | −0.588791 | 1.00000 | 1.86571 | 1.00000 | −3.07534 | −0.533418 | 1.18794 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.28656 | 2.13745 | 3.22836 | 1.00000 | 4.88741 | 1.00000 | 2.80872 | 1.56870 | 2.28656 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.42061 | −0.490228 | 3.85934 | 1.00000 | −1.18665 | 1.00000 | 4.50072 | −2.75968 | 2.42061 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(7\) | \(-1\) |
| \(23\) | \(-1\) |
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 | 805.2.a.m | ✓ | 8 |
| 3.b | odd | 2 | 1 | 7245.2.a.bp | 8 | ||
| 5.b | even | 2 | 1 | 4025.2.a.t | 8 | ||
| 7.b | odd | 2 | 1 | 5635.2.a.bb | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 805.2.a.m | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 4025.2.a.t | 8 | 5.b | even | 2 | 1 | ||
| 5635.2.a.bb | 8 | 7.b | odd | 2 | 1 | ||
| 7245.2.a.bp | 8 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(805))\):
|
\( T_{2}^{8} - T_{2}^{7} - 11T_{2}^{6} + 9T_{2}^{5} + 36T_{2}^{4} - 23T_{2}^{3} - 30T_{2}^{2} + 17T_{2} - 2 \)
|
|
\( T_{3}^{8} - 7T_{3}^{7} + 8T_{3}^{6} + 35T_{3}^{5} - 68T_{3}^{4} - 32T_{3}^{3} + 87T_{3}^{2} + 8T_{3} - 16 \)
|
|
\( T_{11}^{8} - 6T_{11}^{7} - 35T_{11}^{6} + 189T_{11}^{5} + 317T_{11}^{4} - 1352T_{11}^{3} - 64T_{11}^{2} + 1904T_{11} - 976 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - T^{7} - 11 T^{6} + \cdots - 2 \)
|
| $3$ |
\( T^{8} - 7 T^{7} + \cdots - 16 \)
|
| $5$ |
\( (T - 1)^{8} \)
|
| $7$ |
\( (T - 1)^{8} \)
|
| $11$ |
\( T^{8} - 6 T^{7} + \cdots - 976 \)
|
| $13$ |
\( T^{8} - 8 T^{7} + \cdots - 8 \)
|
| $17$ |
\( T^{8} - 4 T^{7} + \cdots - 7024 \)
|
| $19$ |
\( T^{8} - 63 T^{6} + \cdots + 512 \)
|
| $23$ |
\( (T - 1)^{8} \)
|
| $29$ |
\( T^{8} + 3 T^{7} + \cdots + 19168 \)
|
| $31$ |
\( T^{8} + 16 T^{7} + \cdots - 126296 \)
|
| $37$ |
\( T^{8} + 7 T^{7} + \cdots + 240256 \)
|
| $41$ |
\( T^{8} - 7 T^{7} + \cdots - 2416 \)
|
| $43$ |
\( T^{8} + 10 T^{7} + \cdots + 30976 \)
|
| $47$ |
\( T^{8} - 19 T^{7} + \cdots + 1514432 \)
|
| $53$ |
\( T^{8} + T^{7} + \cdots - 371648 \)
|
| $59$ |
\( T^{8} - 21 T^{7} + \cdots + 612736 \)
|
| $61$ |
\( T^{8} + 7 T^{7} + \cdots - 62944 \)
|
| $67$ |
\( T^{8} - 11 T^{7} + \cdots + 2473984 \)
|
| $71$ |
\( T^{8} - 8 T^{7} + \cdots - 2504704 \)
|
| $73$ |
\( T^{8} + 3 T^{7} + \cdots - 103216 \)
|
| $79$ |
\( T^{8} + 15 T^{7} + \cdots - 47599856 \)
|
| $83$ |
\( T^{8} - 25 T^{7} + \cdots - 227072 \)
|
| $89$ |
\( T^{8} - T^{7} + \cdots - 1934752 \)
|
| $97$ |
\( T^{8} - 10 T^{7} + \cdots - 46256 \)
|
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