Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [72,14,Mod(1,72)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(72, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("72.1");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 72 = 2^{3} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 72.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: | \(77.2062688454\) |
| Analytic rank: | \(1\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 8) |
| 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 |
|
0 | 0 | 0 | 4330.00 | 0 | −139992. | 0 | 0 | 0 | |||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
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 | 72.14.a.a | 1 | |
| 3.b | odd | 2 | 1 | 8.14.a.a | ✓ | 1 | |
| 4.b | odd | 2 | 1 | 144.14.a.f | 1 | ||
| 12.b | even | 2 | 1 | 16.14.a.c | 1 | ||
| 15.d | odd | 2 | 1 | 200.14.a.a | 1 | ||
| 15.e | even | 4 | 2 | 200.14.c.a | 2 | ||
| 24.f | even | 2 | 1 | 64.14.a.d | 1 | ||
| 24.h | odd | 2 | 1 | 64.14.a.f | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8.14.a.a | ✓ | 1 | 3.b | odd | 2 | 1 | |
| 16.14.a.c | 1 | 12.b | even | 2 | 1 | ||
| 64.14.a.d | 1 | 24.f | even | 2 | 1 | ||
| 64.14.a.f | 1 | 24.h | odd | 2 | 1 | ||
| 72.14.a.a | 1 | 1.a | even | 1 | 1 | trivial | |
| 144.14.a.f | 1 | 4.b | odd | 2 | 1 | ||
| 200.14.a.a | 1 | 15.d | odd | 2 | 1 | ||
| 200.14.c.a | 2 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} - 4330 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(72))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T \)
|
| $3$ |
\( T \)
|
| $5$ |
\( T - 4330 \)
|
| $7$ |
\( T + 139992 \)
|
| $11$ |
\( T - 6484324 \)
|
| $13$ |
\( T + 22588034 \)
|
| $17$ |
\( T - 23732270 \)
|
| $19$ |
\( T - 325344836 \)
|
| $23$ |
\( T + 921600632 \)
|
| $29$ |
\( T - 3865879218 \)
|
| $31$ |
\( T + 2253401440 \)
|
| $37$ |
\( T - 18250384566 \)
|
| $41$ |
\( T + 34422845322 \)
|
| $43$ |
\( T + 17192501444 \)
|
| $47$ |
\( T - 67371749904 \)
|
| $53$ |
\( T - 87281218426 \)
|
| $59$ |
\( T + 540214518668 \)
|
| $61$ |
\( T + 51276568850 \)
|
| $67$ |
\( T - 25519930676 \)
|
| $71$ |
\( T - 1387500699032 \)
|
| $73$ |
\( T + 819049441238 \)
|
| $79$ |
\( T + 4030935615344 \)
|
| $83$ |
\( T + 4180823831428 \)
|
| $89$ |
\( T + 2677027798266 \)
|
| $97$ |
\( T + 14039464316446 \)
|
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