Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [784,3,Mod(97,784)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(784, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("784.97");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 784 = 2^{4} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 784.c (of order \(2\), degree \(1\), not 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: | \(21.3624527258\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.2048.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 4x^{2} + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 98) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 4x^{2} + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 3\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 3\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(687\) | \(689\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 97.1 |
|
0 | − | 5.22625i | 0 | − | 1.21371i | 0 | 0 | 0 | −18.3137 | 0 | ||||||||||||||||||||||||||||
| 97.2 | 0 | − | 2.16478i | 0 | − | 8.15640i | 0 | 0 | 0 | 4.31371 | 0 | |||||||||||||||||||||||||||||
| 97.3 | 0 | 2.16478i | 0 | 8.15640i | 0 | 0 | 0 | 4.31371 | 0 | |||||||||||||||||||||||||||||||
| 97.4 | 0 | 5.22625i | 0 | 1.21371i | 0 | 0 | 0 | −18.3137 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 784.3.c.b | 4 | |
| 4.b | odd | 2 | 1 | 98.3.b.a | ✓ | 4 | |
| 7.b | odd | 2 | 1 | inner | 784.3.c.b | 4 | |
| 7.c | even | 3 | 2 | 784.3.s.j | 8 | ||
| 7.d | odd | 6 | 2 | 784.3.s.j | 8 | ||
| 12.b | even | 2 | 1 | 882.3.c.a | 4 | ||
| 28.d | even | 2 | 1 | 98.3.b.a | ✓ | 4 | |
| 28.f | even | 6 | 2 | 98.3.d.b | 8 | ||
| 28.g | odd | 6 | 2 | 98.3.d.b | 8 | ||
| 84.h | odd | 2 | 1 | 882.3.c.a | 4 | ||
| 84.j | odd | 6 | 2 | 882.3.n.j | 8 | ||
| 84.n | even | 6 | 2 | 882.3.n.j | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 98.3.b.a | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 98.3.b.a | ✓ | 4 | 28.d | even | 2 | 1 | |
| 98.3.d.b | 8 | 28.f | even | 6 | 2 | ||
| 98.3.d.b | 8 | 28.g | odd | 6 | 2 | ||
| 784.3.c.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 784.3.c.b | 4 | 7.b | odd | 2 | 1 | inner | |
| 784.3.s.j | 8 | 7.c | even | 3 | 2 | ||
| 784.3.s.j | 8 | 7.d | odd | 6 | 2 | ||
| 882.3.c.a | 4 | 12.b | even | 2 | 1 | ||
| 882.3.c.a | 4 | 84.h | odd | 2 | 1 | ||
| 882.3.n.j | 8 | 84.j | odd | 6 | 2 | ||
| 882.3.n.j | 8 | 84.n | even | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 32T_{3}^{2} + 128 \)
acting on \(S_{3}^{\mathrm{new}}(784, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 32T^{2} + 128 \)
|
| $5$ |
\( T^{4} + 68T^{2} + 98 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} + 24 T + 112)^{2} \)
|
| $13$ |
\( T^{4} + 68T^{2} + 98 \)
|
| $17$ |
\( T^{4} + 324 T^{2} + 13122 \)
|
| $19$ |
\( T^{4} + 928T^{2} + 128 \)
|
| $23$ |
\( (T^{2} + 40 T + 112)^{2} \)
|
| $29$ |
\( (T^{2} + 32 T - 82)^{2} \)
|
| $31$ |
\( T^{4} + 320T^{2} + 512 \)
|
| $37$ |
\( (T^{2} + 64 T + 1006)^{2} \)
|
| $41$ |
\( T^{4} + 1028 T^{2} + 164738 \)
|
| $43$ |
\( (T^{2} - 88 T + 1808)^{2} \)
|
| $47$ |
\( T^{4} + 4736 T^{2} + 4524032 \)
|
| $53$ |
\( (T^{2} + 48 T + 64)^{2} \)
|
| $59$ |
\( T^{4} + 9760 T^{2} + 36992 \)
|
| $61$ |
\( T^{4} + 12932 T^{2} + 41714978 \)
|
| $67$ |
\( (T^{2} + 128 T + 2944)^{2} \)
|
| $71$ |
\( (T^{2} + 16 T - 3136)^{2} \)
|
| $73$ |
\( T^{4} + 13444 T^{2} + 42154562 \)
|
| $79$ |
\( (T^{2} - 96 T + 1504)^{2} \)
|
| $83$ |
\( T^{4} + 2848 T^{2} + 1812608 \)
|
| $89$ |
\( T^{4} + 4420 T^{2} + 269378 \)
|
| $97$ |
\( T^{4} + 26500 T^{2} + 54100802 \)
|
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