Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1911,4,Mod(1,1911)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1911, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1911.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1911 = 3 \cdot 7^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1911.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: | \(112.752650021\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 35x^{3} - 26x^{2} + 236x + 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 273) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 35x^{3} - 26x^{2} + 236x + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 20\nu - 2 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 14 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 29\nu^{2} + 30\nu + 136 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 + 14 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 2\beta_{2} + 21\beta _1 + 16 \)
|
| \(\nu^{4}\) | \(=\) |
\( 4\beta_{4} + 31\beta_{3} + 4\beta_{2} + 41\beta _1 + 302 \)
|
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 |
|
−4.51667 | 3.00000 | 12.4003 | −6.18018 | −13.5500 | 0 | −19.8746 | 9.00000 | 27.9138 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.06033 | 3.00000 | −3.75502 | −9.41094 | −6.18100 | 0 | 24.2193 | 9.00000 | 19.3897 | ||||||||||||||||||||||||||||||||||
| 1.3 | 2.15869 | 3.00000 | −3.34004 | 14.9633 | 6.47608 | 0 | −24.4797 | 9.00000 | 32.3012 | ||||||||||||||||||||||||||||||||||
| 1.4 | 3.88985 | 3.00000 | 7.13095 | −18.8278 | 11.6696 | 0 | −3.38047 | 9.00000 | −73.2374 | ||||||||||||||||||||||||||||||||||
| 1.5 | 5.52846 | 3.00000 | 22.5638 | 4.45563 | 16.5854 | 0 | 80.5155 | 9.00000 | 24.6328 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
| \(13\) | \(1\) |
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 | 1911.4.a.n | 5 | |
| 7.b | odd | 2 | 1 | 273.4.a.g | ✓ | 5 | |
| 21.c | even | 2 | 1 | 819.4.a.j | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.4.a.g | ✓ | 5 | 7.b | odd | 2 | 1 | |
| 819.4.a.j | 5 | 21.c | even | 2 | 1 | ||
| 1911.4.a.n | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1911))\):
|
\( T_{2}^{5} - 5T_{2}^{4} - 25T_{2}^{3} + 121T_{2}^{2} + 84T_{2} - 432 \)
|
|
\( T_{5}^{5} + 15T_{5}^{4} - 250T_{5}^{3} - 3440T_{5}^{2} + 2184T_{5} + 73008 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 5 T^{4} + \cdots - 432 \)
|
| $3$ |
\( (T - 3)^{5} \)
|
| $5$ |
\( T^{5} + 15 T^{4} + \cdots + 73008 \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( T^{5} - 83 T^{4} + \cdots + 36889344 \)
|
| $13$ |
\( (T + 13)^{5} \)
|
| $17$ |
\( T^{5} - 43 T^{4} + \cdots + 17505216 \)
|
| $19$ |
\( T^{5} - 9 T^{4} + \cdots - 2593024 \)
|
| $23$ |
\( T^{5} + \cdots + 7334736192 \)
|
| $29$ |
\( T^{5} + \cdots - 1837651536 \)
|
| $31$ |
\( T^{5} + \cdots - 72098840576 \)
|
| $37$ |
\( T^{5} + \cdots + 5547405808 \)
|
| $41$ |
\( T^{5} + \cdots + 2806088576832 \)
|
| $43$ |
\( T^{5} + \cdots + 379769238272 \)
|
| $47$ |
\( T^{5} + \cdots - 186559575552 \)
|
| $53$ |
\( T^{5} + \cdots + 15529993056 \)
|
| $59$ |
\( T^{5} + \cdots - 1337206205952 \)
|
| $61$ |
\( T^{5} + \cdots + 33033189554672 \)
|
| $67$ |
\( T^{5} + \cdots + 1388829363712 \)
|
| $71$ |
\( T^{5} + \cdots - 57670949500416 \)
|
| $73$ |
\( T^{5} + \cdots + 46790014365392 \)
|
| $79$ |
\( T^{5} + \cdots + 66701258016768 \)
|
| $83$ |
\( T^{5} + \cdots + 355610054016 \)
|
| $89$ |
\( T^{5} + \cdots - 440272394839008 \)
|
| $97$ |
\( T^{5} + \cdots - 325748074208 \)
|
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