Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8,9,Mod(3,8)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8.3");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8 = 2^{3} \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8.d (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: | \(3.25902888049\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 78x^{4} - 514x^{3} + 4237x^{2} - 18333x + 238980 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{15} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} + 78x^{4} - 514x^{3} + 4237x^{2} - 18333x + 238980 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} + 80\nu^{3} - 594\nu^{2} + 4831\nu - 19068 ) / 2048 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 66\nu^{4} + 272\nu^{3} - 1810\nu^{2} + 19487\nu - 125820 ) / 2048 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{5} - 10\nu^{4} - 192\nu^{3} - 570\nu^{2} + 3507\nu - 27340 ) / 512 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5\nu^{5} + 54\nu^{4} + 208\nu^{3} + 6438\nu^{2} + 17691\nu + 226452 ) / 2048 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -37\nu^{5} + 138\nu^{4} + 944\nu^{3} + 15002\nu^{2} + 7301\nu - 350996 ) / 2048 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} + 7\beta _1 + 8 ) / 32 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 7\beta_{4} - \beta_{3} + 8\beta_{2} - 55\beta _1 - 848 ) / 32 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 16\beta_{5} - 35\beta_{4} - 51\beta_{3} + 32\beta_{2} + 123\beta _1 + 7000 ) / 32 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 48\beta_{5} - 9\beta_{4} + 95\beta_{3} - 1080\beta_{2} + 4041\beta _1 - 14432 ) / 32 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -1184\beta_{5} + 2109\beta_{4} - 1155\beta_{3} + 32\beta_{2} - 2709\beta _1 - 521048 ) / 32 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(7\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3.1 |
|
−13.6648 | − | 8.32309i | −17.4864 | 117.452 | + | 227.466i | 796.785i | 238.948 | + | 145.541i | 1779.55i | 288.260 | − | 4085.84i | −6255.23 | 6631.71 | − | 10887.9i | ||||||||||||||||||||||||||
| 3.2 | −13.6648 | + | 8.32309i | −17.4864 | 117.452 | − | 227.466i | − | 796.785i | 238.948 | − | 145.541i | − | 1779.55i | 288.260 | + | 4085.84i | −6255.23 | 6631.71 | + | 10887.9i | |||||||||||||||||||||||||
| 3.3 | −0.413549 | − | 15.9947i | 117.102 | −255.658 | + | 13.2291i | − | 816.841i | −48.4273 | − | 1873.00i | 1333.52i | 317.322 | + | 4083.69i | 7151.84 | −13065.1 | + | 337.803i | ||||||||||||||||||||||||||
| 3.4 | −0.413549 | + | 15.9947i | 117.102 | −255.658 | − | 13.2291i | 816.841i | −48.4273 | + | 1873.00i | − | 1333.52i | 317.322 | − | 4083.69i | 7151.84 | −13065.1 | − | 337.803i | ||||||||||||||||||||||||||
| 3.5 | 7.07833 | − | 14.3491i | −117.615 | −155.795 | − | 203.136i | 306.178i | −832.521 | + | 1687.68i | − | 4321.70i | −4017.58 | + | 797.653i | 7272.39 | 4393.38 | + | 2167.23i | ||||||||||||||||||||||||||
| 3.6 | 7.07833 | + | 14.3491i | −117.615 | −155.795 | + | 203.136i | − | 306.178i | −832.521 | − | 1687.68i | 4321.70i | −4017.58 | − | 797.653i | 7272.39 | 4393.38 | − | 2167.23i | ||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 8.9.d.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 72.9.b.b | 6 | ||
| 4.b | odd | 2 | 1 | 32.9.d.b | 6 | ||
| 8.b | even | 2 | 1 | 32.9.d.b | 6 | ||
| 8.d | odd | 2 | 1 | inner | 8.9.d.b | ✓ | 6 |
| 12.b | even | 2 | 1 | 288.9.b.b | 6 | ||
| 16.e | even | 4 | 2 | 256.9.c.n | 12 | ||
| 16.f | odd | 4 | 2 | 256.9.c.n | 12 | ||
| 24.f | even | 2 | 1 | 72.9.b.b | 6 | ||
| 24.h | odd | 2 | 1 | 288.9.b.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8.9.d.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 8.9.d.b | ✓ | 6 | 8.d | odd | 2 | 1 | inner |
| 32.9.d.b | 6 | 4.b | odd | 2 | 1 | ||
| 32.9.d.b | 6 | 8.b | even | 2 | 1 | ||
| 72.9.b.b | 6 | 3.b | odd | 2 | 1 | ||
| 72.9.b.b | 6 | 24.f | even | 2 | 1 | ||
| 256.9.c.n | 12 | 16.e | even | 4 | 2 | ||
| 256.9.c.n | 12 | 16.f | odd | 4 | 2 | ||
| 288.9.b.b | 6 | 12.b | even | 2 | 1 | ||
| 288.9.b.b | 6 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 18T_{3}^{2} - 13764T_{3} - 240840 \)
acting on \(S_{9}^{\mathrm{new}}(8, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 14 T^{5} + \cdots + 16777216 \)
|
| $3$ |
\( (T^{3} + 18 T^{2} + \cdots - 240840)^{2} \)
|
| $5$ |
\( T^{6} + \cdots + 39\!\cdots\!00 \)
|
| $7$ |
\( T^{6} + \cdots + 10\!\cdots\!40 \)
|
| $11$ |
\( (T^{3} - 23470 T^{2} + \cdots - 155265768392)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 54\!\cdots\!40 \)
|
| $17$ |
\( (T^{3} + \cdots - 13136750437640)^{2} \)
|
| $19$ |
\( (T^{3} + \cdots - 34\!\cdots\!08)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots + 21\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots + 11\!\cdots\!40 \)
|
| $41$ |
\( (T^{3} + \cdots + 26\!\cdots\!28)^{2} \)
|
| $43$ |
\( (T^{3} + \cdots + 30\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 35\!\cdots\!40 \)
|
| $53$ |
\( T^{6} + \cdots + 66\!\cdots\!40 \)
|
| $59$ |
\( (T^{3} + \cdots + 70\!\cdots\!48)^{2} \)
|
| $61$ |
\( T^{6} + \cdots + 12\!\cdots\!00 \)
|
| $67$ |
\( (T^{3} + \cdots + 12\!\cdots\!60)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $73$ |
\( (T^{3} + \cdots - 13\!\cdots\!20)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 16\!\cdots\!00 \)
|
| $83$ |
\( (T^{3} + \cdots + 15\!\cdots\!40)^{2} \)
|
| $89$ |
\( (T^{3} + \cdots + 46\!\cdots\!72)^{2} \)
|
| $97$ |
\( (T^{3} + \cdots + 47\!\cdots\!60)^{2} \)
|
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