Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2704,2,Mod(337,2704)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2704, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2704.337");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2704 = 2^{4} \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2704.f (of order \(2\), degree \(1\), not 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: | \(21.5915487066\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.153664.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 5x^{4} + 6x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 169) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 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
\( x^{6} + 5x^{4} + 6x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} + 3\nu^{2} + 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} + 4\nu^{3} + 3\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} - 3\beta_{2} + 5 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} - 4\beta_{3} + 9\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2704\mathbb{Z}\right)^\times\).
| \(n\) | \(677\) | \(1185\) | \(2367\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 337.1 |
|
0 | −0.801938 | 0 | − | 2.80194i | 0 | − | 2.69202i | 0 | −2.35690 | 0 | ||||||||||||||||||||||||||||||||||
| 337.2 | 0 | −0.801938 | 0 | 2.80194i | 0 | 2.69202i | 0 | −2.35690 | 0 | |||||||||||||||||||||||||||||||||||||
| 337.3 | 0 | 0.554958 | 0 | − | 1.44504i | 0 | 2.04892i | 0 | −2.69202 | 0 | ||||||||||||||||||||||||||||||||||||
| 337.4 | 0 | 0.554958 | 0 | 1.44504i | 0 | − | 2.04892i | 0 | −2.69202 | 0 | ||||||||||||||||||||||||||||||||||||
| 337.5 | 0 | 2.24698 | 0 | − | 0.246980i | 0 | 2.35690i | 0 | 2.04892 | 0 | ||||||||||||||||||||||||||||||||||||
| 337.6 | 0 | 2.24698 | 0 | 0.246980i | 0 | − | 2.35690i | 0 | 2.04892 | 0 | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2704.2.f.o | 6 | |
| 4.b | odd | 2 | 1 | 169.2.b.b | 6 | ||
| 12.b | even | 2 | 1 | 1521.2.b.l | 6 | ||
| 13.b | even | 2 | 1 | inner | 2704.2.f.o | 6 | |
| 13.d | odd | 4 | 1 | 2704.2.a.z | 3 | ||
| 13.d | odd | 4 | 1 | 2704.2.a.ba | 3 | ||
| 52.b | odd | 2 | 1 | 169.2.b.b | 6 | ||
| 52.f | even | 4 | 1 | 169.2.a.b | ✓ | 3 | |
| 52.f | even | 4 | 1 | 169.2.a.c | yes | 3 | |
| 52.i | odd | 6 | 2 | 169.2.e.b | 12 | ||
| 52.j | odd | 6 | 2 | 169.2.e.b | 12 | ||
| 52.l | even | 12 | 2 | 169.2.c.b | 6 | ||
| 52.l | even | 12 | 2 | 169.2.c.c | 6 | ||
| 156.h | even | 2 | 1 | 1521.2.b.l | 6 | ||
| 156.l | odd | 4 | 1 | 1521.2.a.o | 3 | ||
| 156.l | odd | 4 | 1 | 1521.2.a.r | 3 | ||
| 260.u | even | 4 | 1 | 4225.2.a.bb | 3 | ||
| 260.u | even | 4 | 1 | 4225.2.a.bg | 3 | ||
| 364.p | odd | 4 | 1 | 8281.2.a.bf | 3 | ||
| 364.p | odd | 4 | 1 | 8281.2.a.bj | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 169.2.a.b | ✓ | 3 | 52.f | even | 4 | 1 | |
| 169.2.a.c | yes | 3 | 52.f | even | 4 | 1 | |
| 169.2.b.b | 6 | 4.b | odd | 2 | 1 | ||
| 169.2.b.b | 6 | 52.b | odd | 2 | 1 | ||
| 169.2.c.b | 6 | 52.l | even | 12 | 2 | ||
| 169.2.c.c | 6 | 52.l | even | 12 | 2 | ||
| 169.2.e.b | 12 | 52.i | odd | 6 | 2 | ||
| 169.2.e.b | 12 | 52.j | odd | 6 | 2 | ||
| 1521.2.a.o | 3 | 156.l | odd | 4 | 1 | ||
| 1521.2.a.r | 3 | 156.l | odd | 4 | 1 | ||
| 1521.2.b.l | 6 | 12.b | even | 2 | 1 | ||
| 1521.2.b.l | 6 | 156.h | even | 2 | 1 | ||
| 2704.2.a.z | 3 | 13.d | odd | 4 | 1 | ||
| 2704.2.a.ba | 3 | 13.d | odd | 4 | 1 | ||
| 2704.2.f.o | 6 | 1.a | even | 1 | 1 | trivial | |
| 2704.2.f.o | 6 | 13.b | even | 2 | 1 | inner | |
| 4225.2.a.bb | 3 | 260.u | even | 4 | 1 | ||
| 4225.2.a.bg | 3 | 260.u | even | 4 | 1 | ||
| 8281.2.a.bf | 3 | 364.p | odd | 4 | 1 | ||
| 8281.2.a.bj | 3 | 364.p | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2704, [\chi])\):
|
\( T_{3}^{3} - 2T_{3}^{2} - T_{3} + 1 \)
|
|
\( T_{5}^{6} + 10T_{5}^{4} + 17T_{5}^{2} + 1 \)
|
|
\( T_{11}^{6} + 26T_{11}^{4} + 153T_{11}^{2} + 169 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{3} - 2 T^{2} - T + 1)^{2} \)
|
| $5$ |
\( T^{6} + 10 T^{4} + \cdots + 1 \)
|
| $7$ |
\( T^{6} + 17 T^{4} + \cdots + 169 \)
|
| $11$ |
\( T^{6} + 26 T^{4} + \cdots + 169 \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( (T^{3} - 2 T^{2} - 15 T - 13)^{2} \)
|
| $19$ |
\( T^{6} + 38 T^{4} + \cdots + 1 \)
|
| $23$ |
\( (T^{3} + 5 T^{2} - T - 13)^{2} \)
|
| $29$ |
\( (T^{3} + T^{2} - 44 T + 83)^{2} \)
|
| $31$ |
\( T^{6} + 97 T^{4} + \cdots + 27889 \)
|
| $37$ |
\( T^{6} + 62 T^{4} + \cdots + 841 \)
|
| $41$ |
\( T^{6} + 147 T^{4} + \cdots + 2401 \)
|
| $43$ |
\( (T^{3} - 13 T^{2} + \cdots + 13)^{2} \)
|
| $47$ |
\( T^{6} + 122 T^{4} + \cdots + 27889 \)
|
| $53$ |
\( (T^{3} - T^{2} - 86 T + 337)^{2} \)
|
| $59$ |
\( T^{6} + 195 T^{4} + \cdots + 1 \)
|
| $61$ |
\( (T^{3} - 4 T^{2} + \cdots + 239)^{2} \)
|
| $67$ |
\( T^{6} + 145 T^{4} + \cdots + 1681 \)
|
| $71$ |
\( T^{6} + 285 T^{4} + \cdots + 299209 \)
|
| $73$ |
\( T^{6} + 321 T^{4} + \cdots + 829921 \)
|
| $79$ |
\( (T^{3} - 5 T^{2} + \cdots - 127)^{2} \)
|
| $83$ |
\( T^{6} + 329 T^{4} + \cdots + 41209 \)
|
| $89$ |
\( T^{6} + 269 T^{4} + \cdots + 78961 \)
|
| $97$ |
\( T^{6} + 217 T^{4} + \cdots + 90601 \)
|
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