Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6724,2,Mod(1,6724)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6724, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("6724.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6724 = 2^{2} \cdot 41^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6724.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: | \(53.6914103191\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.40716288.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 10x^{4} + 28x^{2} - 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 164) |
| 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 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
\( x^{6} - 10x^{4} + 28x^{2} - 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 4\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 7\nu^{3} + 10\nu ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{4} - 6\nu^{2} + 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + 6\beta_{2} + 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( 3\beta_{4} + 7\beta_{3} + 18\beta_1 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −2.34822 | 0 | 1.51414 | 0 | 1.20731 | 0 | 2.51414 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.88997 | 0 | −0.428007 | 0 | −2.69889 | 0 | 0.571993 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −0.955965 | 0 | −3.08613 | 0 | −3.90620 | 0 | −2.08613 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 0.955965 | 0 | −3.08613 | 0 | 3.90620 | 0 | −2.08613 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.88997 | 0 | −0.428007 | 0 | 2.69889 | 0 | 0.571993 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.34822 | 0 | 1.51414 | 0 | −1.20731 | 0 | 2.51414 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(41\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 41.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 6724.2.a.e | 6 | |
| 41.b | even | 2 | 1 | inner | 6724.2.a.e | 6 | |
| 41.e | odd | 8 | 2 | 164.2.f.a | ✓ | 6 | |
| 123.i | even | 8 | 2 | 1476.2.k.a | 6 | ||
| 164.i | even | 8 | 2 | 656.2.l.f | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 164.2.f.a | ✓ | 6 | 41.e | odd | 8 | 2 | |
| 656.2.l.f | 6 | 164.i | even | 8 | 2 | ||
| 1476.2.k.a | 6 | 123.i | even | 8 | 2 | ||
| 6724.2.a.e | 6 | 1.a | even | 1 | 1 | trivial | |
| 6724.2.a.e | 6 | 41.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 10T_{3}^{4} + 28T_{3}^{2} - 18 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6724))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 10 T^{4} + \cdots - 18 \)
|
| $5$ |
\( (T^{3} + 2 T^{2} - 4 T - 2)^{2} \)
|
| $7$ |
\( T^{6} - 24 T^{4} + \cdots - 162 \)
|
| $11$ |
\( T^{6} - 12 T^{4} + \cdots - 2 \)
|
| $13$ |
\( T^{6} - 22 T^{4} + \cdots - 72 \)
|
| $17$ |
\( T^{6} - 86 T^{4} + \cdots - 72 \)
|
| $19$ |
\( T^{6} - 38 T^{4} + \cdots - 1458 \)
|
| $23$ |
\( (T^{3} + 2 T^{2} - 20 T - 24)^{2} \)
|
| $29$ |
\( T^{6} - 54 T^{4} + \cdots - 2312 \)
|
| $31$ |
\( (T^{3} + 4 T^{2} - 40 T - 16)^{2} \)
|
| $37$ |
\( (T^{3} + 18 T^{2} + \cdots - 82)^{2} \)
|
| $41$ |
\( T^{6} \)
|
| $43$ |
\( (T^{3} + 10 T^{2} + \cdots - 508)^{2} \)
|
| $47$ |
\( T^{6} - 150 T^{4} + \cdots - 7442 \)
|
| $53$ |
\( T^{6} - 214 T^{4} + \cdots - 20808 \)
|
| $59$ |
\( (T^{3} + 2 T^{2} - 20 T - 24)^{2} \)
|
| $61$ |
\( (T^{3} + 4 T^{2} - 4 T - 4)^{2} \)
|
| $67$ |
\( T^{6} - 428 T^{4} + \cdots - 2068578 \)
|
| $71$ |
\( T^{6} - 194 T^{4} + \cdots - 13122 \)
|
| $73$ |
\( (T^{3} - 12 T^{2} + \cdots + 1706)^{2} \)
|
| $79$ |
\( T^{6} - 222 T^{4} + \cdots - 8978 \)
|
| $83$ |
\( (T^{3} - 18 T^{2} + \cdots + 1304)^{2} \)
|
| $89$ |
\( T^{6} - 430 T^{4} + \cdots - 267912 \)
|
| $97$ |
\( T^{6} - 102 T^{4} + \cdots - 1352 \)
|
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