Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [336,3,Mod(79,336)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(336, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 0, 2]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("336.79");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 336 = 2^{4} \cdot 3 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 336.be (of order \(6\), degree \(2\), minimal) |
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: | no |
| Analytic conductor: | \(9.15533688251\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | 6.0.1364138928.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 35x^{4} + 364x^{2} + 972 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{6} + 35x^{4} + 364x^{2} + 972 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} + 17\nu^{3} + 22\nu + 36 ) / 72 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{5} - 121\nu^{3} - 36\nu^{2} - 542\nu - 396 ) / 72 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 12 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + 23\nu^{3} + 6\nu^{2} + 118\nu + 72 ) / 12 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} + 36\nu^{4} + 17\nu^{3} + 684\nu^{2} + 94\nu + 1980 ) / 72 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( -6\beta_{4} - \beta_{3} - 8\beta_{2} - 4\beta _1 + 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} - \beta_{4} - 19\beta_{3} - \beta_{2} - \beta _1 + 174 \)
|
| \(\nu^{5}\) | \(=\) |
\( 91\beta_{4} + 17\beta_{3} + 125\beta_{2} + 151\beta _1 - 138 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).
| \(n\) | \(85\) | \(113\) | \(127\) | \(241\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 79.1 |
|
0 | 1.50000 | − | 0.866025i | 0 | −2.29832 | + | 3.98081i | 0 | 4.95955 | − | 4.93992i | 0 | 1.50000 | − | 2.59808i | 0 | ||||||||||||||||||||||||||||
| 79.2 | 0 | 1.50000 | − | 0.866025i | 0 | −1.15548 | + | 2.00135i | 0 | −6.36328 | + | 2.91696i | 0 | 1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||||||
| 79.3 | 0 | 1.50000 | − | 0.866025i | 0 | 3.95380 | − | 6.84819i | 0 | 6.90373 | + | 1.15694i | 0 | 1.50000 | − | 2.59808i | 0 | |||||||||||||||||||||||||||||
| 319.1 | 0 | 1.50000 | + | 0.866025i | 0 | −2.29832 | − | 3.98081i | 0 | 4.95955 | + | 4.93992i | 0 | 1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||||||
| 319.2 | 0 | 1.50000 | + | 0.866025i | 0 | −1.15548 | − | 2.00135i | 0 | −6.36328 | − | 2.91696i | 0 | 1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||||||
| 319.3 | 0 | 1.50000 | + | 0.866025i | 0 | 3.95380 | + | 6.84819i | 0 | 6.90373 | − | 1.15694i | 0 | 1.50000 | + | 2.59808i | 0 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 28.g | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 336.3.be.f | yes | 6 |
| 3.b | odd | 2 | 1 | 1008.3.cd.i | 6 | ||
| 4.b | odd | 2 | 1 | 336.3.be.d | ✓ | 6 | |
| 7.c | even | 3 | 1 | 336.3.be.d | ✓ | 6 | |
| 7.c | even | 3 | 1 | 2352.3.m.n | 6 | ||
| 7.d | odd | 6 | 1 | 2352.3.m.o | 6 | ||
| 12.b | even | 2 | 1 | 1008.3.cd.h | 6 | ||
| 21.h | odd | 6 | 1 | 1008.3.cd.h | 6 | ||
| 28.f | even | 6 | 1 | 2352.3.m.o | 6 | ||
| 28.g | odd | 6 | 1 | inner | 336.3.be.f | yes | 6 |
| 28.g | odd | 6 | 1 | 2352.3.m.n | 6 | ||
| 84.n | even | 6 | 1 | 1008.3.cd.i | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 336.3.be.d | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 336.3.be.d | ✓ | 6 | 7.c | even | 3 | 1 | |
| 336.3.be.f | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 336.3.be.f | yes | 6 | 28.g | odd | 6 | 1 | inner |
| 1008.3.cd.h | 6 | 12.b | even | 2 | 1 | ||
| 1008.3.cd.h | 6 | 21.h | odd | 6 | 1 | ||
| 1008.3.cd.i | 6 | 3.b | odd | 2 | 1 | ||
| 1008.3.cd.i | 6 | 84.n | even | 6 | 1 | ||
| 2352.3.m.n | 6 | 7.c | even | 3 | 1 | ||
| 2352.3.m.n | 6 | 28.g | odd | 6 | 1 | ||
| 2352.3.m.o | 6 | 7.d | odd | 6 | 1 | ||
| 2352.3.m.o | 6 | 28.f | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(336, [\chi])\):
|
\( T_{5}^{6} - T_{5}^{5} + 45T_{5}^{4} + 212T_{5}^{3} + 1852T_{5}^{2} + 3696T_{5} + 7056 \)
|
|
\( T_{11}^{6} + 3T_{11}^{5} - 109T_{11}^{4} - 336T_{11}^{3} + 12580T_{11}^{2} - 4032T_{11} + 432 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{2} - 3 T + 3)^{3} \)
|
| $5$ |
\( T^{6} - T^{5} + \cdots + 7056 \)
|
| $7$ |
\( T^{6} - 11 T^{5} + \cdots + 117649 \)
|
| $11$ |
\( T^{6} + 3 T^{5} + \cdots + 432 \)
|
| $13$ |
\( (T^{3} + 22 T^{2} + \cdots - 4312)^{2} \)
|
| $17$ |
\( T^{6} - 8 T^{5} + \cdots + 20358144 \)
|
| $19$ |
\( T^{6} - 30 T^{5} + \cdots + 1138368 \)
|
| $23$ |
\( T^{6} + 24 T^{5} + \cdots + 66382848 \)
|
| $29$ |
\( (T^{3} - 17 T^{2} + \cdots + 35952)^{2} \)
|
| $31$ |
\( T^{6} - 39 T^{5} + \cdots + 3756483 \)
|
| $37$ |
\( T^{6} - 6 T^{5} + \cdots + 183115024 \)
|
| $41$ |
\( (T^{3} + 68 T^{2} + \cdots - 37152)^{2} \)
|
| $43$ |
\( T^{6} + 4010 T^{4} + \cdots + 7660812 \)
|
| $47$ |
\( T^{6} + \cdots + 7468832448 \)
|
| $53$ |
\( T^{6} + 89 T^{5} + \cdots + 609892416 \)
|
| $59$ |
\( T^{6} + 63 T^{5} + \cdots + 84672 \)
|
| $61$ |
\( T^{6} + \cdots + 73796982336 \)
|
| $67$ |
\( T^{6} + \cdots + 84665952108 \)
|
| $71$ |
\( T^{6} + 6164 T^{4} + \cdots + 615874752 \)
|
| $73$ |
\( T^{6} - 36 T^{5} + \cdots + 688642564 \)
|
| $79$ |
\( T^{6} + 3 T^{5} + \cdots + 140726403 \)
|
| $83$ |
\( T^{6} + \cdots + 22314082608 \)
|
| $89$ |
\( T^{6} + \cdots + 10788229287936 \)
|
| $97$ |
\( (T^{3} + 133 T^{2} + \cdots + 35252)^{2} \)
|
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