Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(26.0198423125\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.57516.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 24x + 6 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 21) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 24x + 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 16 \)
|
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 |
|
−5.30829 | 0 | 20.1780 | −5.56140 | 0 | 0 | −64.6443 | 0 | 29.5215 | |||||||||||||||||||||||||||
| 1.2 | −0.248072 | 0 | −7.93846 | 12.4346 | 0 | 0 | 3.95388 | 0 | −3.08468 | ||||||||||||||||||||||||||||
| 1.3 | 4.55637 | 0 | 12.7605 | −17.8732 | 0 | 0 | 21.6905 | 0 | −81.4369 | ||||||||||||||||||||||||||||
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 | 441.4.a.s | 3 | |
| 3.b | odd | 2 | 1 | 147.4.a.l | 3 | ||
| 7.b | odd | 2 | 1 | 441.4.a.t | 3 | ||
| 7.c | even | 3 | 2 | 63.4.e.c | 6 | ||
| 7.d | odd | 6 | 2 | 441.4.e.w | 6 | ||
| 12.b | even | 2 | 1 | 2352.4.a.ci | 3 | ||
| 21.c | even | 2 | 1 | 147.4.a.m | 3 | ||
| 21.g | even | 6 | 2 | 147.4.e.n | 6 | ||
| 21.h | odd | 6 | 2 | 21.4.e.b | ✓ | 6 | |
| 84.h | odd | 2 | 1 | 2352.4.a.cg | 3 | ||
| 84.n | even | 6 | 2 | 336.4.q.k | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.4.e.b | ✓ | 6 | 21.h | odd | 6 | 2 | |
| 63.4.e.c | 6 | 7.c | even | 3 | 2 | ||
| 147.4.a.l | 3 | 3.b | odd | 2 | 1 | ||
| 147.4.a.m | 3 | 21.c | even | 2 | 1 | ||
| 147.4.e.n | 6 | 21.g | even | 6 | 2 | ||
| 336.4.q.k | 6 | 84.n | even | 6 | 2 | ||
| 441.4.a.s | 3 | 1.a | even | 1 | 1 | trivial | |
| 441.4.a.t | 3 | 7.b | odd | 2 | 1 | ||
| 441.4.e.w | 6 | 7.d | odd | 6 | 2 | ||
| 2352.4.a.cg | 3 | 84.h | odd | 2 | 1 | ||
| 2352.4.a.ci | 3 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(441))\):
|
\( T_{2}^{3} + T_{2}^{2} - 24T_{2} - 6 \)
|
|
\( T_{5}^{3} + 11T_{5}^{2} - 192T_{5} - 1236 \)
|
|
\( T_{13}^{3} - 62T_{13}^{2} + 425T_{13} + 18452 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + T^{2} - 24T - 6 \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + 11 T^{2} + \cdots - 1236 \)
|
| $7$ |
\( T^{3} \)
|
| $11$ |
\( T^{3} + 35 T^{2} + \cdots + 9564 \)
|
| $13$ |
\( T^{3} - 62 T^{2} + \cdots + 18452 \)
|
| $17$ |
\( T^{3} + 48 T^{2} + \cdots - 112896 \)
|
| $19$ |
\( T^{3} + 202 T^{2} + \cdots + 233804 \)
|
| $23$ |
\( T^{3} + 216 T^{2} + \cdots - 1580544 \)
|
| $29$ |
\( T^{3} + 53 T^{2} + \cdots + 824976 \)
|
| $31$ |
\( T^{3} + 95 T^{2} + \cdots - 11823 \)
|
| $37$ |
\( T^{3} - 262 T^{2} + \cdots - 49152 \)
|
| $41$ |
\( T^{3} + 244 T^{2} + \cdots + 300384 \)
|
| $43$ |
\( T^{3} - 360 T^{2} + \cdots + 18269746 \)
|
| $47$ |
\( T^{3} - 210 T^{2} + \cdots - 5119128 \)
|
| $53$ |
\( T^{3} + 393 T^{2} + \cdots - 33169392 \)
|
| $59$ |
\( T^{3} + 1143 T^{2} + \cdots - 100468944 \)
|
| $61$ |
\( T^{3} + 70 T^{2} + \cdots - 84631000 \)
|
| $67$ |
\( T^{3} + 628 T^{2} + \cdots + 27993002 \)
|
| $71$ |
\( T^{3} + 318 T^{2} + \cdots + 28535976 \)
|
| $73$ |
\( T^{3} - 988 T^{2} + \cdots + 143207118 \)
|
| $79$ |
\( T^{3} - 861 T^{2} + \cdots + 193956337 \)
|
| $83$ |
\( T^{3} + 519 T^{2} + \cdots - 47916036 \)
|
| $89$ |
\( T^{3} + 1766 T^{2} + \cdots - 13004544 \)
|
| $97$ |
\( T^{3} - 19 T^{2} + \cdots + 44776452 \)
|
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