Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,6,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, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| 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: | \(70.7292645375\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{193}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 48 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 147) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{193})\). 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 |
|
−5.44622 | 0 | −2.33867 | 36.0000 | 0 | 0 | 187.016 | 0 | −196.064 | ||||||||||||||||||||||||
| 1.2 | 8.44622 | 0 | 39.3387 | 36.0000 | 0 | 0 | 61.9840 | 0 | 304.064 | |||||||||||||||||||||||||
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 | 441.6.a.r | 2 | |
| 3.b | odd | 2 | 1 | 147.6.a.j | yes | 2 | |
| 7.b | odd | 2 | 1 | 441.6.a.q | 2 | ||
| 21.c | even | 2 | 1 | 147.6.a.h | ✓ | 2 | |
| 21.g | even | 6 | 2 | 147.6.e.n | 4 | ||
| 21.h | odd | 6 | 2 | 147.6.e.m | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 147.6.a.h | ✓ | 2 | 21.c | even | 2 | 1 | |
| 147.6.a.j | yes | 2 | 3.b | odd | 2 | 1 | |
| 147.6.e.m | 4 | 21.h | odd | 6 | 2 | ||
| 147.6.e.n | 4 | 21.g | even | 6 | 2 | ||
| 441.6.a.q | 2 | 7.b | odd | 2 | 1 | ||
| 441.6.a.r | 2 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):
|
\( T_{2}^{2} - 3T_{2} - 46 \)
|
|
\( T_{5} - 36 \)
|
|
\( T_{13}^{2} + 1296T_{13} + 169776 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 3T - 46 \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( (T - 36)^{2} \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} + 480T + 54512 \)
|
| $13$ |
\( T^{2} + 1296 T + 169776 \)
|
| $17$ |
\( T^{2} - 936 T - 2032128 \)
|
| $19$ |
\( T^{2} - 216 T - 3990384 \)
|
| $23$ |
\( T^{2} - 504T - 87808 \)
|
| $29$ |
\( T^{2} + 6372 T - 9612604 \)
|
| $31$ |
\( T^{2} + 9936 T + 8672832 \)
|
| $37$ |
\( T^{2} - 11124 T - 22869468 \)
|
| $41$ |
\( T^{2} - 20952 T + 107495424 \)
|
| $43$ |
\( T^{2} + 6264 T - 268110576 \)
|
| $47$ |
\( T^{2} - 7920 T - 560613312 \)
|
| $53$ |
\( T^{2} + 2220 T - 66407452 \)
|
| $59$ |
\( T^{2} - 29736 T + 185038992 \)
|
| $61$ |
\( T^{2} - 17280 T - 982141200 \)
|
| $67$ |
\( T^{2} + \cdots - 1919121200 \)
|
| $71$ |
\( T^{2} + \cdots + 1814661632 \)
|
| $73$ |
\( T^{2} + 56592 T + 631577088 \)
|
| $79$ |
\( T^{2} + \cdots - 1239346496 \)
|
| $83$ |
\( T^{2} + \cdots - 3085453296 \)
|
| $89$ |
\( T^{2} + \cdots + 3143484288 \)
|
| $97$ |
\( T^{2} + \cdots - 18184056768 \)
|
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