Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,8,Mod(1,441)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 441.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: | \(137.761796238\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{865}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 216 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 7) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{865})\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−13.2054 | 0 | 46.3837 | 17.9456 | 0 | 0 | 1077.78 | 0 | −236.979 | ||||||||||||||||||||||||
| 1.2 | 16.2054 | 0 | 134.616 | 312.054 | 0 | 0 | 107.220 | 0 | 5056.98 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 441.8.a.l | 2 | |
| 3.b | odd | 2 | 1 | 49.8.a.c | 2 | ||
| 7.b | odd | 2 | 1 | 63.8.a.e | 2 | ||
| 21.c | even | 2 | 1 | 7.8.a.b | ✓ | 2 | |
| 21.g | even | 6 | 2 | 49.8.c.e | 4 | ||
| 21.h | odd | 6 | 2 | 49.8.c.f | 4 | ||
| 84.h | odd | 2 | 1 | 112.8.a.f | 2 | ||
| 105.g | even | 2 | 1 | 175.8.a.c | 2 | ||
| 105.k | odd | 4 | 2 | 175.8.b.b | 4 | ||
| 168.e | odd | 2 | 1 | 448.8.a.t | 2 | ||
| 168.i | even | 2 | 1 | 448.8.a.k | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 7.8.a.b | ✓ | 2 | 21.c | even | 2 | 1 | |
| 49.8.a.c | 2 | 3.b | odd | 2 | 1 | ||
| 49.8.c.e | 4 | 21.g | even | 6 | 2 | ||
| 49.8.c.f | 4 | 21.h | odd | 6 | 2 | ||
| 63.8.a.e | 2 | 7.b | odd | 2 | 1 | ||
| 112.8.a.f | 2 | 84.h | odd | 2 | 1 | ||
| 175.8.a.c | 2 | 105.g | even | 2 | 1 | ||
| 175.8.b.b | 4 | 105.k | odd | 4 | 2 | ||
| 441.8.a.l | 2 | 1.a | even | 1 | 1 | trivial | |
| 448.8.a.k | 2 | 168.i | even | 2 | 1 | ||
| 448.8.a.t | 2 | 168.e | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(441))\):
|
\( T_{2}^{2} - 3T_{2} - 214 \)
|
|
\( T_{5}^{2} - 330T_{5} + 5600 \)
|
|
\( T_{13}^{2} + 2534T_{13} - 166620776 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 3T - 214 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} - 330T + 5600 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} + 2844 T - 887776 \)
|
| $13$ |
\( T^{2} + 2534 T - 166620776 \)
|
| $17$ |
\( T^{2} + 1488 T - 22147524 \)
|
| $19$ |
\( T^{2} + 32810 T + 109928560 \)
|
| $23$ |
\( T^{2} - 6576 T + 10312704 \)
|
| $29$ |
\( T^{2} + \cdots - 18920124100 \)
|
| $31$ |
\( T^{2} + \cdots + 37023636384 \)
|
| $37$ |
\( T^{2} + \cdots - 126010986084 \)
|
| $41$ |
\( T^{2} + \cdots + 13303276364 \)
|
| $43$ |
\( T^{2} + \cdots + 50864711104 \)
|
| $47$ |
\( T^{2} + \cdots + 2090896416 \)
|
| $53$ |
\( T^{2} + \cdots + 2037435782724 \)
|
| $59$ |
\( T^{2} + \cdots - 615374101440 \)
|
| $61$ |
\( T^{2} + \cdots - 529516501136 \)
|
| $67$ |
\( T^{2} + \cdots - 533876854064 \)
|
| $71$ |
\( T^{2} + \cdots + 10359492378624 \)
|
| $73$ |
\( T^{2} + \cdots - 3340687254156 \)
|
| $79$ |
\( T^{2} + \cdots - 6335206025600 \)
|
| $83$ |
\( T^{2} + \cdots + 30181573873584 \)
|
| $89$ |
\( T^{2} + \cdots - 4649674734460 \)
|
| $97$ |
\( T^{2} + \cdots + 27021168617436 \)
|
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