Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [784,6,Mod(1,784)]
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, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("784.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 784 = 2^{4} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 784.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: | \(125.740914733\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{86}, \sqrt{134})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 110x^{2} + 144 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{7} \) |
| Twist minimal: | no (minimal twist has level 392) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 110x^{2} + 144 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{3} - 214\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( 4\nu^{2} - 220 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 220 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{2} + 107\beta_1 ) / 2 \)
|
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 |
|
0 | −20.8495 | 0 | 11.6406 | 0 | 0 | 0 | 191.700 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.30222 | 0 | −81.0956 | 0 | 0 | 0 | −237.700 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 2.30222 | 0 | 81.0956 | 0 | 0 | 0 | −237.700 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 20.8495 | 0 | −11.6406 | 0 | 0 | 0 | 191.700 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(7\) | \(-1\) |
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 | 784.6.a.be | 4 | |
| 4.b | odd | 2 | 1 | 392.6.a.g | ✓ | 4 | |
| 7.b | odd | 2 | 1 | inner | 784.6.a.be | 4 | |
| 28.d | even | 2 | 1 | 392.6.a.g | ✓ | 4 | |
| 28.f | even | 6 | 2 | 392.6.i.n | 8 | ||
| 28.g | odd | 6 | 2 | 392.6.i.n | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 392.6.a.g | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 392.6.a.g | ✓ | 4 | 28.d | even | 2 | 1 | |
| 392.6.i.n | 8 | 28.f | even | 6 | 2 | ||
| 392.6.i.n | 8 | 28.g | odd | 6 | 2 | ||
| 784.6.a.be | 4 | 1.a | even | 1 | 1 | trivial | |
| 784.6.a.be | 4 | 7.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 440T_{3}^{2} + 2304 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(784))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 440T^{2} + 2304 \)
|
| $5$ |
\( T^{4} - 6712 T^{2} + 891136 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} - 176 T - 407120)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 12619376896 \)
|
| $17$ |
\( T^{4} + \cdots + 1710445465600 \)
|
| $19$ |
\( T^{4} + \cdots + 9146414684416 \)
|
| $23$ |
\( (T^{2} + 984 T - 3491712)^{2} \)
|
| $29$ |
\( (T - 1010)^{4} \)
|
| $31$ |
\( T^{4} + \cdots + 38\!\cdots\!00 \)
|
| $37$ |
\( (T^{2} - 5252 T - 8039228)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 37\!\cdots\!04 \)
|
| $43$ |
\( (T^{2} + 14368 T + 18005872)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 58\!\cdots\!84 \)
|
| $53$ |
\( (T^{2} + 14148 T - 549022140)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 55\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 56\!\cdots\!04 \)
|
| $67$ |
\( (T^{2} - 36288 T - 368181648)^{2} \)
|
| $71$ |
\( (T^{2} + 88368 T + 1220405760)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 60\!\cdots\!04 \)
|
| $79$ |
\( (T^{2} + 67984 T + 1128904768)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 81\!\cdots\!16 \)
|
| $89$ |
\( T^{4} + \cdots + 16\!\cdots\!36 \)
|
| $97$ |
\( T^{4} + \cdots + 44\!\cdots\!56 \)
|
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