Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [735,6,Mod(1,735)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(735, 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("735.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 735 = 3 \cdot 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 735.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: | \(117.882107563\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 15) |
| 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)
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 |
|
−2.00000 | 9.00000 | −28.0000 | 25.0000 | −18.0000 | 0 | 120.000 | 81.0000 | −50.0000 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(7\) | \(-1\) |
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 | 735.6.a.a | 1 | |
| 7.b | odd | 2 | 1 | 15.6.a.a | ✓ | 1 | |
| 21.c | even | 2 | 1 | 45.6.a.c | 1 | ||
| 28.d | even | 2 | 1 | 240.6.a.k | 1 | ||
| 35.c | odd | 2 | 1 | 75.6.a.c | 1 | ||
| 35.f | even | 4 | 2 | 75.6.b.d | 2 | ||
| 56.e | even | 2 | 1 | 960.6.a.m | 1 | ||
| 56.h | odd | 2 | 1 | 960.6.a.v | 1 | ||
| 84.h | odd | 2 | 1 | 720.6.a.w | 1 | ||
| 105.g | even | 2 | 1 | 225.6.a.c | 1 | ||
| 105.k | odd | 4 | 2 | 225.6.b.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 15.6.a.a | ✓ | 1 | 7.b | odd | 2 | 1 | |
| 45.6.a.c | 1 | 21.c | even | 2 | 1 | ||
| 75.6.a.c | 1 | 35.c | odd | 2 | 1 | ||
| 75.6.b.d | 2 | 35.f | even | 4 | 2 | ||
| 225.6.a.c | 1 | 105.g | even | 2 | 1 | ||
| 225.6.b.d | 2 | 105.k | odd | 4 | 2 | ||
| 240.6.a.k | 1 | 28.d | even | 2 | 1 | ||
| 720.6.a.w | 1 | 84.h | odd | 2 | 1 | ||
| 735.6.a.a | 1 | 1.a | even | 1 | 1 | trivial | |
| 960.6.a.m | 1 | 56.e | even | 2 | 1 | ||
| 960.6.a.v | 1 | 56.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(735))\):
|
\( T_{2} + 2 \)
|
|
\( T_{13} - 686 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T + 2 \)
|
| $3$ |
\( T - 9 \)
|
| $5$ |
\( T - 25 \)
|
| $7$ |
\( T \)
|
| $11$ |
\( T - 472 \)
|
| $13$ |
\( T - 686 \)
|
| $17$ |
\( T - 1562 \)
|
| $19$ |
\( T - 2180 \)
|
| $23$ |
\( T - 264 \)
|
| $29$ |
\( T - 170 \)
|
| $31$ |
\( T + 7272 \)
|
| $37$ |
\( T + 142 \)
|
| $41$ |
\( T - 16198 \)
|
| $43$ |
\( T + 10316 \)
|
| $47$ |
\( T + 18568 \)
|
| $53$ |
\( T - 21514 \)
|
| $59$ |
\( T + 34600 \)
|
| $61$ |
\( T - 35738 \)
|
| $67$ |
\( T + 5772 \)
|
| $71$ |
\( T + 69088 \)
|
| $73$ |
\( T - 70526 \)
|
| $79$ |
\( T - 47640 \)
|
| $83$ |
\( T + 74004 \)
|
| $89$ |
\( T - 90030 \)
|
| $97$ |
\( T - 33502 \)
|
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