Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [144,10,Mod(1,144)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(144, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("144.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 144 = 2^{4} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 144.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: | \(74.1651604076\) |
| Analytic rank: | \(1\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2) |
| 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 | −870.000 | 0 | 952.000 | 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 | 144.10.a.d | 1 | |
| 3.b | odd | 2 | 1 | 16.10.a.d | 1 | ||
| 4.b | odd | 2 | 1 | 18.10.a.a | 1 | ||
| 12.b | even | 2 | 1 | 2.10.a.a | ✓ | 1 | |
| 15.d | odd | 2 | 1 | 400.10.a.b | 1 | ||
| 15.e | even | 4 | 2 | 400.10.c.d | 2 | ||
| 24.f | even | 2 | 1 | 64.10.a.h | 1 | ||
| 24.h | odd | 2 | 1 | 64.10.a.b | 1 | ||
| 36.f | odd | 6 | 2 | 162.10.c.i | 2 | ||
| 36.h | even | 6 | 2 | 162.10.c.b | 2 | ||
| 48.i | odd | 4 | 2 | 256.10.b.e | 2 | ||
| 48.k | even | 4 | 2 | 256.10.b.g | 2 | ||
| 60.h | even | 2 | 1 | 50.10.a.c | 1 | ||
| 60.l | odd | 4 | 2 | 50.10.b.a | 2 | ||
| 84.h | odd | 2 | 1 | 98.10.a.c | 1 | ||
| 84.j | odd | 6 | 2 | 98.10.c.b | 2 | ||
| 84.n | even | 6 | 2 | 98.10.c.c | 2 | ||
| 132.d | odd | 2 | 1 | 242.10.a.a | 1 | ||
| 156.h | even | 2 | 1 | 338.10.a.a | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2.10.a.a | ✓ | 1 | 12.b | even | 2 | 1 | |
| 16.10.a.d | 1 | 3.b | odd | 2 | 1 | ||
| 18.10.a.a | 1 | 4.b | odd | 2 | 1 | ||
| 50.10.a.c | 1 | 60.h | even | 2 | 1 | ||
| 50.10.b.a | 2 | 60.l | odd | 4 | 2 | ||
| 64.10.a.b | 1 | 24.h | odd | 2 | 1 | ||
| 64.10.a.h | 1 | 24.f | even | 2 | 1 | ||
| 98.10.a.c | 1 | 84.h | odd | 2 | 1 | ||
| 98.10.c.b | 2 | 84.j | odd | 6 | 2 | ||
| 98.10.c.c | 2 | 84.n | even | 6 | 2 | ||
| 144.10.a.d | 1 | 1.a | even | 1 | 1 | trivial | |
| 162.10.c.b | 2 | 36.h | even | 6 | 2 | ||
| 162.10.c.i | 2 | 36.f | odd | 6 | 2 | ||
| 242.10.a.a | 1 | 132.d | odd | 2 | 1 | ||
| 256.10.b.e | 2 | 48.i | odd | 4 | 2 | ||
| 256.10.b.g | 2 | 48.k | even | 4 | 2 | ||
| 338.10.a.a | 1 | 156.h | even | 2 | 1 | ||
| 400.10.a.b | 1 | 15.d | odd | 2 | 1 | ||
| 400.10.c.d | 2 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} + 870 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(144))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T \)
|
| $3$ |
\( T \)
|
| $5$ |
\( T + 870 \)
|
| $7$ |
\( T - 952 \)
|
| $11$ |
\( T + 56148 \)
|
| $13$ |
\( T - 178094 \)
|
| $17$ |
\( T - 247662 \)
|
| $19$ |
\( T + 315380 \)
|
| $23$ |
\( T - 204504 \)
|
| $29$ |
\( T - 3840450 \)
|
| $31$ |
\( T - 1309408 \)
|
| $37$ |
\( T - 4307078 \)
|
| $41$ |
\( T + 1512042 \)
|
| $43$ |
\( T + 33670604 \)
|
| $47$ |
\( T + 10581072 \)
|
| $53$ |
\( T + 16616214 \)
|
| $59$ |
\( T - 112235100 \)
|
| $61$ |
\( T + 33197218 \)
|
| $67$ |
\( T - 121372252 \)
|
| $71$ |
\( T + 387172728 \)
|
| $73$ |
\( T - 255240074 \)
|
| $79$ |
\( T + 492101840 \)
|
| $83$ |
\( T + 457420236 \)
|
| $89$ |
\( T - 31809510 \)
|
| $97$ |
\( T + 673532062 \)
|
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