Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [882,5,Mod(197,882)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(882, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("882.197");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 882.b (of order \(2\), degree \(1\), 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: | \(91.1723074400\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.224054542336.12 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 199x^{4} + 14641 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{8}\cdot 3^{4}\cdot 7^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{8} + 199x^{4} + 14641 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{6} - 640\nu^{2} ) / 363 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -10\nu^{7} + 121\nu^{5} - 659\nu^{3} + 10769\nu ) / 27951 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{6} + 726\nu^{4} + 2240\nu^{2} + 72237 ) / 2541 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 10\nu^{7} + 121\nu^{5} + 659\nu^{3} + 10769\nu ) / 3993 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -32\nu^{7} + 121\nu^{5} - 7699\nu^{3} + 66671\nu ) / 9317 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -42\nu^{6} - 3276\nu^{2} ) / 121 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 32\nu^{7} + 121\nu^{5} + 7699\nu^{3} + 66671\nu ) / 1331 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + 7\beta_{5} - 3\beta_{4} - 21\beta_{2} ) / 84 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} - 63\beta_1 ) / 84 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{7} - 35\beta_{5} - 48\beta_{4} + 336\beta_{2} ) / 42 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 14\beta_{3} + 7\beta _1 - 398 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -89\beta_{7} - 623\beta_{5} + 1653\beta_{4} + 11571\beta_{2} ) / 84 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -160\beta_{6} + 2457\beta_1 ) / 42 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -659\beta_{7} + 4613\beta_{5} + 23097\beta_{4} - 161679\beta_{2} ) / 84 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/882\mathbb{Z}\right)^\times\).
| \(n\) | \(199\) | \(785\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 197.1 |
|
− | 2.82843i | 0 | −8.00000 | − | 26.6723i | 0 | 0 | 22.6274i | 0 | −75.4407 | ||||||||||||||||||||||||||||||||||||||||
| 197.2 | − | 2.82843i | 0 | −8.00000 | − | 12.6723i | 0 | 0 | 22.6274i | 0 | −35.8427 | |||||||||||||||||||||||||||||||||||||||||
| 197.3 | − | 2.82843i | 0 | −8.00000 | 12.6723i | 0 | 0 | 22.6274i | 0 | 35.8427 | ||||||||||||||||||||||||||||||||||||||||||
| 197.4 | − | 2.82843i | 0 | −8.00000 | 26.6723i | 0 | 0 | 22.6274i | 0 | 75.4407 | ||||||||||||||||||||||||||||||||||||||||||
| 197.5 | 2.82843i | 0 | −8.00000 | − | 26.6723i | 0 | 0 | − | 22.6274i | 0 | 75.4407 | |||||||||||||||||||||||||||||||||||||||||
| 197.6 | 2.82843i | 0 | −8.00000 | − | 12.6723i | 0 | 0 | − | 22.6274i | 0 | 35.8427 | |||||||||||||||||||||||||||||||||||||||||
| 197.7 | 2.82843i | 0 | −8.00000 | 12.6723i | 0 | 0 | − | 22.6274i | 0 | −35.8427 | ||||||||||||||||||||||||||||||||||||||||||
| 197.8 | 2.82843i | 0 | −8.00000 | 26.6723i | 0 | 0 | − | 22.6274i | 0 | −75.4407 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 21.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 882.5.b.j | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 882.5.b.j | ✓ | 8 |
| 7.b | odd | 2 | 1 | inner | 882.5.b.j | ✓ | 8 |
| 21.c | even | 2 | 1 | inner | 882.5.b.j | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 882.5.b.j | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 882.5.b.j | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 882.5.b.j | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 882.5.b.j | ✓ | 8 | 21.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(882, [\chi])\):
|
\( T_{5}^{4} + 872T_{5}^{2} + 114244 \)
|
|
\( T_{13}^{4} - 26532T_{13}^{2} + 132296004 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8)^{4} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{4} + 872 T^{2} + 114244)^{2} \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( (T^{4} + 77452 T^{2} + 1378339876)^{2} \)
|
| $13$ |
\( (T^{4} - 26532 T^{2} + 132296004)^{2} \)
|
| $17$ |
\( (T^{4} + 224984 T^{2} + 335402596)^{2} \)
|
| $19$ |
\( (T^{4} - 149104 T^{2} + 2584298896)^{2} \)
|
| $23$ |
\( (T^{4} + 135388 T^{2} + 66552964)^{2} \)
|
| $29$ |
\( (T^{4} + 1573804 T^{2} + 10864726756)^{2} \)
|
| $31$ |
\( (T^{4} - 400752 T^{2} + 170772624)^{2} \)
|
| $37$ |
\( (T^{2} - 32 T - 1896044)^{4} \)
|
| $41$ |
\( (T^{4} + \cdots + 27977350367044)^{2} \)
|
| $43$ |
\( (T^{2} + 164 T - 1889576)^{4} \)
|
| $47$ |
\( (T^{4} + 2398568 T^{2} + 791537619856)^{2} \)
|
| $53$ |
\( (T^{4} + \cdots + 1845636365764)^{2} \)
|
| $59$ |
\( (T^{4} + \cdots + 81078715114384)^{2} \)
|
| $61$ |
\( (T^{4} + \cdots + 9636417730564)^{2} \)
|
| $67$ |
\( (T^{2} + 5760 T + 4577652)^{4} \)
|
| $71$ |
\( (T^{4} + \cdots + 496433068285284)^{2} \)
|
| $73$ |
\( (T^{4} + \cdots + 269328338823076)^{2} \)
|
| $79$ |
\( (T^{2} + 14780 T + 54308692)^{4} \)
|
| $83$ |
\( (T^{4} + \cdots + 10\!\cdots\!76)^{2} \)
|
| $89$ |
\( (T^{4} + \cdots + 56\!\cdots\!76)^{2} \)
|
| $97$ |
\( (T^{4} + \cdots + 15\!\cdots\!64)^{2} \)
|
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