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: | \(\Q(\sqrt{505}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 126 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| 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 \(\beta = 3\sqrt{505}\). 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 | −28.4166 | 0 | −49.0000 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | −9.00000 | 0 | 106.417 | 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.u | 2 | |
| 3.b | odd | 2 | 1 | 1008.6.a.bf | 2 | ||
| 4.b | odd | 2 | 1 | 84.6.a.d | ✓ | 2 | |
| 12.b | even | 2 | 1 | 252.6.a.e | 2 | ||
| 28.d | even | 2 | 1 | 588.6.a.g | 2 | ||
| 28.f | even | 6 | 2 | 588.6.i.n | 4 | ||
| 28.g | odd | 6 | 2 | 588.6.i.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 84.6.a.d | ✓ | 2 | 4.b | odd | 2 | 1 | |
| 252.6.a.e | 2 | 12.b | even | 2 | 1 | ||
| 336.6.a.u | 2 | 1.a | even | 1 | 1 | trivial | |
| 588.6.a.g | 2 | 28.d | even | 2 | 1 | ||
| 588.6.i.h | 4 | 28.g | odd | 6 | 2 | ||
| 588.6.i.n | 4 | 28.f | even | 6 | 2 | ||
| 1008.6.a.bf | 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(336))\):
|
\( T_{5}^{2} - 78T_{5} - 3024 \)
|
|
\( T_{11}^{2} + 174T_{11} - 106056 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( (T + 9)^{2} \)
|
| $5$ |
\( T^{2} - 78T - 3024 \)
|
| $7$ |
\( (T + 49)^{2} \)
|
| $11$ |
\( T^{2} + 174T - 106056 \)
|
| $13$ |
\( T^{2} - 208T - 152804 \)
|
| $17$ |
\( T^{2} - 1482 T - 1091664 \)
|
| $19$ |
\( T^{2} + 352 T - 5859344 \)
|
| $23$ |
\( T^{2} + 3354 T + 807984 \)
|
| $29$ |
\( T^{2} - 276 T - 45430956 \)
|
| $31$ |
\( T^{2} + 6520 T + 4737280 \)
|
| $37$ |
\( T^{2} - 13864 T + 43962124 \)
|
| $41$ |
\( T^{2} + 12930 T + 41246280 \)
|
| $43$ |
\( T^{2} + 12712 T - 25049264 \)
|
| $47$ |
\( T^{2} - 28116 T + 182337984 \)
|
| $53$ |
\( T^{2} + 46992 T + 492995196 \)
|
| $59$ |
\( T^{2} + \cdots + 1037582784 \)
|
| $61$ |
\( T^{2} + 13148 T - 365832524 \)
|
| $67$ |
\( T^{2} + \cdots + 1367827744 \)
|
| $71$ |
\( T^{2} - 66042 T + 931452336 \)
|
| $73$ |
\( T^{2} - 60496 T + 842785084 \)
|
| $79$ |
\( T^{2} + \cdots - 2170625216 \)
|
| $83$ |
\( T^{2} + \cdots + 1671979536 \)
|
| $89$ |
\( T^{2} + \cdots - 6775324920 \)
|
| $97$ |
\( T^{2} + \cdots + 11297373964 \)
|
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