Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [806,2,Mod(373,806)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(806, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("806.373");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 806 = 2 \cdot 13 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 806.g (of order \(3\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(6.43594240292\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 8.0.4601315889.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} + 6x^{6} - 3x^{5} + 26x^{4} - 14x^{3} + 31x^{2} + 12x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{8} - x^{7} + 6x^{6} - 3x^{5} + 26x^{4} - 14x^{3} + 31x^{2} + 12x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -26\nu^{7} - 189\nu^{6} + 729\nu^{5} - 911\nu^{4} + 3051\nu^{3} - 3618\nu^{2} + 14317\nu - 1215 ) / 4243 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 115\nu^{7} + 20\nu^{6} + 529\nu^{5} + 276\nu^{4} + 3314\nu^{3} + 989\nu^{2} + 483\nu + 1947 ) / 4243 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -135\nu^{7} + 161\nu^{6} - 621\nu^{5} - 324\nu^{4} - 2599\nu^{3} - 1161\nu^{2} - 567\nu - 11694 ) / 4243 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -649\nu^{7} + 994\nu^{6} - 3834\nu^{5} + 3534\nu^{4} - 16046\nu^{3} + 19028\nu^{2} - 17152\nu + 6390 ) / 12729 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 434\nu^{7} - 109\nu^{6} + 2845\nu^{5} + 193\nu^{4} + 12064\nu^{3} + 338\nu^{2} + 16249\nu + 6573 ) / 4243 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 514\nu^{7} - 833\nu^{6} + 3213\nu^{5} - 3858\nu^{4} + 13447\nu^{3} - 15946\nu^{2} + 16585\nu - 5355 ) / 4243 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + 3\beta_{5} - \beta_{4} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} + 4\beta_{3} + \beta_{2} \)
|
| \(\nu^{4}\) | \(=\) |
\( -5\beta_{7} - 12\beta_{5} + \beta_{2} \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{7} + 6\beta_{6} + \beta_{4} - 17\beta_{3} - 17\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 7\beta_{6} + 23\beta_{4} - \beta_{3} - 7\beta_{2} + 51 \)
|
| \(\nu^{7}\) | \(=\) |
\( 8\beta_{7} + 3\beta_{5} - 30\beta_{2} + 74\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/806\mathbb{Z}\right)^\times\).
| \(n\) | \(249\) | \(313\) |
| \(\chi(n)\) | \(-\beta_{5}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 373.1 |
|
0.500000 | − | 0.866025i | −0.946534 | + | 1.63944i | −0.500000 | − | 0.866025i | 0.0974383 | 0.946534 | + | 1.63944i | −1.73839 | − | 3.01097i | −1.00000 | −0.291852 | − | 0.505503i | 0.0487191 | − | 0.0843840i | ||||||||||||||||||||||||||||
| 373.2 | 0.500000 | − | 0.866025i | 0.304588 | − | 0.527561i | −0.500000 | − | 0.866025i | −2.19091 | −0.304588 | − | 0.527561i | 1.11904 | + | 1.93823i | −1.00000 | 1.31445 | + | 2.27670i | −1.09545 | + | 1.89738i | |||||||||||||||||||||||||||||
| 373.3 | 0.500000 | − | 0.866025i | 1.21763 | − | 2.10899i | −0.500000 | − | 0.866025i | −2.66454 | −1.21763 | − | 2.10899i | −0.747605 | − | 1.29489i | −1.00000 | −1.46523 | − | 2.53786i | −1.33227 | + | 2.30756i | |||||||||||||||||||||||||||||
| 373.4 | 0.500000 | − | 0.866025i | 1.42432 | − | 2.46699i | −0.500000 | − | 0.866025i | 1.75802 | −1.42432 | − | 2.46699i | −1.63305 | − | 2.82852i | −1.00000 | −2.55737 | − | 4.42949i | 0.879008 | − | 1.52249i | |||||||||||||||||||||||||||||
| 497.1 | 0.500000 | + | 0.866025i | −0.946534 | − | 1.63944i | −0.500000 | + | 0.866025i | 0.0974383 | 0.946534 | − | 1.63944i | −1.73839 | + | 3.01097i | −1.00000 | −0.291852 | + | 0.505503i | 0.0487191 | + | 0.0843840i | |||||||||||||||||||||||||||||
| 497.2 | 0.500000 | + | 0.866025i | 0.304588 | + | 0.527561i | −0.500000 | + | 0.866025i | −2.19091 | −0.304588 | + | 0.527561i | 1.11904 | − | 1.93823i | −1.00000 | 1.31445 | − | 2.27670i | −1.09545 | − | 1.89738i | |||||||||||||||||||||||||||||
| 497.3 | 0.500000 | + | 0.866025i | 1.21763 | + | 2.10899i | −0.500000 | + | 0.866025i | −2.66454 | −1.21763 | + | 2.10899i | −0.747605 | + | 1.29489i | −1.00000 | −1.46523 | + | 2.53786i | −1.33227 | − | 2.30756i | |||||||||||||||||||||||||||||
| 497.4 | 0.500000 | + | 0.866025i | 1.42432 | + | 2.46699i | −0.500000 | + | 0.866025i | 1.75802 | −1.42432 | + | 2.46699i | −1.63305 | + | 2.82852i | −1.00000 | −2.55737 | + | 4.42949i | 0.879008 | + | 1.52249i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 806.2.g.f | ✓ | 8 |
| 13.c | even | 3 | 1 | inner | 806.2.g.f | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 806.2.g.f | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 806.2.g.f | ✓ | 8 | 13.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(806, [\chi])\):
|
\( T_{3}^{8} - 4T_{3}^{7} + 17T_{3}^{6} - 26T_{3}^{5} + 69T_{3}^{4} - 79T_{3}^{3} + 217T_{3}^{2} - 120T_{3} + 64 \)
|
|
\( T_{5}^{4} + 3T_{5}^{3} - 3T_{5}^{2} - 10T_{5} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T + 1)^{4} \)
|
| $3$ |
\( T^{8} - 4 T^{7} + \cdots + 64 \)
|
| $5$ |
\( (T^{4} + 3 T^{3} - 3 T^{2} + \cdots + 1)^{2} \)
|
| $7$ |
\( T^{8} + 6 T^{7} + \cdots + 1444 \)
|
| $11$ |
\( T^{8} + 7 T^{7} + \cdots + 64 \)
|
| $13$ |
\( T^{8} + 5 T^{7} + \cdots + 28561 \)
|
| $17$ |
\( T^{8} - 7 T^{7} + \cdots + 130321 \)
|
| $19$ |
\( T^{8} - 5 T^{7} + \cdots + 71824 \)
|
| $23$ |
\( T^{8} - T^{7} + \cdots + 144 \)
|
| $29$ |
\( T^{8} + 7 T^{7} + \cdots + 729 \)
|
| $31$ |
\( (T + 1)^{8} \)
|
| $37$ |
\( T^{8} + 9 T^{7} + \cdots + 17161 \)
|
| $41$ |
\( T^{8} + 55 T^{6} + \cdots + 17161 \)
|
| $43$ |
\( T^{8} - 9 T^{7} + \cdots + 2116 \)
|
| $47$ |
\( (T^{4} - T^{3} - 172 T^{2} + \cdots + 3862)^{2} \)
|
| $53$ |
\( (T^{4} - T^{3} - 94 T^{2} + \cdots - 472)^{2} \)
|
| $59$ |
\( T^{8} - 12 T^{7} + \cdots + 1602756 \)
|
| $61$ |
\( T^{8} + T^{7} + \cdots + 506944 \)
|
| $67$ |
\( T^{8} - 12 T^{7} + \cdots + 850084 \)
|
| $71$ |
\( T^{8} + 23 T^{7} + \cdots + 99856 \)
|
| $73$ |
\( (T^{4} + 5 T^{3} - 90 T^{2} + \cdots - 8)^{2} \)
|
| $79$ |
\( (T^{4} - 17 T^{3} + \cdots + 226)^{2} \)
|
| $83$ |
\( (T^{4} + 20 T^{3} + \cdots - 18198)^{2} \)
|
| $89$ |
\( T^{8} + 35 T^{7} + \cdots + 12830724 \)
|
| $97$ |
\( T^{8} + 9 T^{7} + \cdots + 2149156 \)
|
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