Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [5408,2,Mod(1,5408)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5408, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("5408.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 5408 = 2^{5} \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 5408.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: | \(43.1830974131\) |
| Analytic rank: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.134509248.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 15x^{4} + 60x^{2} - 48 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 416) |
| 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
\( x^{6} - 15x^{4} + 60x^{2} - 48 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} - 11\nu^{3} + 24\nu ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 9\nu^{2} + 10 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 19\nu^{3} - 80\nu ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 5 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{3} + 7\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} + 9\beta_{2} + 35 \)
|
| \(\nu^{5}\) | \(=\) |
\( 11\beta_{5} + 19\beta_{3} + 53\beta_1 \)
|
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 | −2.96724 | 0 | 1.23519 | 0 | −4.80451 | 0 | 5.80451 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −2.26572 | 0 | 3.99777 | 0 | −1.13349 | 0 | 2.13349 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.03053 | 0 | −0.701519 | 0 | 2.93800 | 0 | −1.93800 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 1.03053 | 0 | 0.701519 | 0 | 2.93800 | 0 | −1.93800 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 2.26572 | 0 | −3.99777 | 0 | −1.13349 | 0 | 2.13349 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.96724 | 0 | −1.23519 | 0 | −4.80451 | 0 | 5.80451 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(13\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 52.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 5408.2.a.bm | 6 | |
| 4.b | odd | 2 | 1 | 5408.2.a.bn | 6 | ||
| 13.b | even | 2 | 1 | 5408.2.a.bn | 6 | ||
| 13.f | odd | 12 | 2 | 416.2.w.d | ✓ | 12 | |
| 52.b | odd | 2 | 1 | inner | 5408.2.a.bm | 6 | |
| 52.l | even | 12 | 2 | 416.2.w.d | ✓ | 12 | |
| 104.u | even | 12 | 2 | 832.2.w.j | 12 | ||
| 104.x | odd | 12 | 2 | 832.2.w.j | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 416.2.w.d | ✓ | 12 | 13.f | odd | 12 | 2 | |
| 416.2.w.d | ✓ | 12 | 52.l | even | 12 | 2 | |
| 832.2.w.j | 12 | 104.u | even | 12 | 2 | ||
| 832.2.w.j | 12 | 104.x | odd | 12 | 2 | ||
| 5408.2.a.bm | 6 | 1.a | even | 1 | 1 | trivial | |
| 5408.2.a.bm | 6 | 52.b | odd | 2 | 1 | inner | |
| 5408.2.a.bn | 6 | 4.b | odd | 2 | 1 | ||
| 5408.2.a.bn | 6 | 13.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5408))\):
|
\( T_{3}^{6} - 15T_{3}^{4} + 60T_{3}^{2} - 48 \)
|
|
\( T_{5}^{6} - 18T_{5}^{4} + 33T_{5}^{2} - 12 \)
|
|
\( T_{7}^{3} + 3T_{7}^{2} - 12T_{7} - 16 \)
|
|
\( T_{37}^{6} - 81T_{37}^{4} + 1275T_{37}^{2} - 5043 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - 15 T^{4} + \cdots - 48 \)
|
| $5$ |
\( T^{6} - 18 T^{4} + \cdots - 12 \)
|
| $7$ |
\( (T^{3} + 3 T^{2} - 12 T - 16)^{2} \)
|
| $11$ |
\( (T^{3} - 3 T^{2} - 12 T + 16)^{2} \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( (T + 1)^{6} \)
|
| $19$ |
\( (T^{3} + 3 T^{2} - 18 T - 36)^{2} \)
|
| $23$ |
\( T^{6} - 135 T^{4} + \cdots - 46128 \)
|
| $29$ |
\( (T^{3} - 3 T^{2} - 21 T + 31)^{2} \)
|
| $31$ |
\( (T^{3} + 12 T^{2} + \cdots - 24)^{2} \)
|
| $37$ |
\( T^{6} - 81 T^{4} + \cdots - 5043 \)
|
| $41$ |
\( T^{6} - 81 T^{4} + \cdots - 6627 \)
|
| $43$ |
\( T^{6} - 99 T^{4} + \cdots - 12288 \)
|
| $47$ |
\( (T^{3} - 6 T^{2} + \cdots + 208)^{2} \)
|
| $53$ |
\( (T^{3} + 12 T^{2} + 27 T - 4)^{2} \)
|
| $59$ |
\( (T^{3} + 3 T^{2} + \cdots - 320)^{2} \)
|
| $61$ |
\( (T^{3} + 9 T^{2} + \cdots - 237)^{2} \)
|
| $67$ |
\( (T^{3} + 15 T^{2} + \cdots - 788)^{2} \)
|
| $71$ |
\( (T^{3} + 9 T^{2} + 6 T - 20)^{2} \)
|
| $73$ |
\( T^{6} - 234 T^{4} + \cdots - 288300 \)
|
| $79$ |
\( T^{6} - 396 T^{4} + \cdots - 786432 \)
|
| $83$ |
\( (T^{3} + 18 T^{2} + \cdots - 128)^{2} \)
|
| $89$ |
\( T^{6} - 435 T^{4} + \cdots - 2157312 \)
|
| $97$ |
\( T^{6} - 315 T^{4} + \cdots - 62208 \)
|
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