Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [336,6,Mod(1,336)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(336, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("336.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 336 = 2^{4} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 336.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: | \(53.8889634572\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{5569}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 1392 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 84) |
| 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 \(\beta = \sqrt{5569}\). 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 |
|
0 | 9.00000 | 0 | −77.6257 | 0 | 49.0000 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 9.00000 | 0 | 71.6257 | 0 | 49.0000 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
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 | 336.6.a.x | 2 | |
| 3.b | odd | 2 | 1 | 1008.6.a.bo | 2 | ||
| 4.b | odd | 2 | 1 | 84.6.a.c | ✓ | 2 | |
| 12.b | even | 2 | 1 | 252.6.a.h | 2 | ||
| 28.d | even | 2 | 1 | 588.6.a.k | 2 | ||
| 28.f | even | 6 | 2 | 588.6.i.i | 4 | ||
| 28.g | odd | 6 | 2 | 588.6.i.l | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 84.6.a.c | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 252.6.a.h | 2 | 12.b | even | 2 | 1 | ||
| 336.6.a.x | 2 | 1.a | even | 1 | 1 | trivial | |
| 588.6.a.k | 2 | 28.d | even | 2 | 1 | ||
| 588.6.i.i | 4 | 28.f | even | 6 | 2 | ||
| 588.6.i.l | 4 | 28.g | odd | 6 | 2 | ||
| 1008.6.a.bo | 2 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(336))\):
|
\( T_{5}^{2} + 6T_{5} - 5560 \)
|
|
\( T_{11}^{2} - 90T_{11} - 270856 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( (T - 9)^{2} \)
|
| $5$ |
\( T^{2} + 6T - 5560 \)
|
| $7$ |
\( (T - 49)^{2} \)
|
| $11$ |
\( T^{2} - 90T - 270856 \)
|
| $13$ |
\( T^{2} - 768T - 53028 \)
|
| $17$ |
\( T^{2} - 1926 T + 921800 \)
|
| $19$ |
\( T^{2} + 2248 T - 1944368 \)
|
| $23$ |
\( T^{2} + 6354 T + 10087760 \)
|
| $29$ |
\( T^{2} - 10572 T + 19031396 \)
|
| $31$ |
\( T^{2} - 3312 T - 26127360 \)
|
| $37$ |
\( T^{2} - 2104 T - 124195796 \)
|
| $41$ |
\( T^{2} - 1266 T - 5663952 \)
|
| $43$ |
\( T^{2} - 5768 T - 312456944 \)
|
| $47$ |
\( T^{2} + 15612 T - 85219200 \)
|
| $53$ |
\( T^{2} - 16512 T - 236774628 \)
|
| $59$ |
\( T^{2} - 13140 T + 29242400 \)
|
| $61$ |
\( T^{2} + 5796 T - 379738620 \)
|
| $67$ |
\( T^{2} - 56116 T - 401418272 \)
|
| $71$ |
\( T^{2} + \cdots - 4430537104 \)
|
| $73$ |
\( T^{2} + \cdots + 1653999820 \)
|
| $79$ |
\( T^{2} - 19620 T + 62354304 \)
|
| $83$ |
\( T^{2} - 44424 T - 76892656 \)
|
| $89$ |
\( T^{2} + \cdots + 10266670512 \)
|
| $97$ |
\( T^{2} + \cdots - 14004028100 \)
|
| show more | |
| show less |