Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8281,2,Mod(1,8281)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8281, 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("8281.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8281 = 7^{2} \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8281.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: | \(66.1241179138\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.746052.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 7x^{3} + 8x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 91) |
| 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
\( x^{5} - x^{4} - 7x^{3} + 8x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 2\nu^{2} - 4\nu + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 4\nu^{2} + 3\nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} - 5\nu^{2} + 4\nu + 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{4} + 2\beta_{3} + \beta_{2} + 6\beta _1 + 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( -8\beta_{4} + 9\beta_{3} + 2\beta_{2} + 13\beta _1 + 19 \)
|
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 |
|
−2.72525 | 1.34642 | 5.42699 | 2.18716 | −3.66932 | 0 | −9.33940 | −1.18716 | −5.96057 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.21332 | −2.47443 | 2.89879 | −2.12280 | 5.47671 | 0 | −1.98932 | 3.12280 | 4.69843 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.26561 | 2.62728 | −0.398235 | −2.90260 | −3.32511 | 0 | 3.03523 | 3.90260 | 3.67356 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0.195656 | 0.259788 | −1.96172 | 3.93251 | 0.0508292 | 0 | −0.775135 | −2.93251 | 0.769420 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.00852 | −1.75906 | 2.03417 | 0.905722 | −3.53311 | 0 | 0.0686323 | 0.0942784 | 1.81916 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
| \(13\) | \(1\) |
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 | 8281.2.a.bx | 5 | |
| 7.b | odd | 2 | 1 | 8281.2.a.bw | 5 | ||
| 7.d | odd | 6 | 2 | 1183.2.e.f | 10 | ||
| 13.b | even | 2 | 1 | 637.2.a.k | 5 | ||
| 39.d | odd | 2 | 1 | 5733.2.a.bm | 5 | ||
| 91.b | odd | 2 | 1 | 637.2.a.l | 5 | ||
| 91.r | even | 6 | 2 | 637.2.e.m | 10 | ||
| 91.s | odd | 6 | 2 | 91.2.e.c | ✓ | 10 | |
| 273.g | even | 2 | 1 | 5733.2.a.bl | 5 | ||
| 273.ba | even | 6 | 2 | 819.2.j.h | 10 | ||
| 364.x | even | 6 | 2 | 1456.2.r.p | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 91.2.e.c | ✓ | 10 | 91.s | odd | 6 | 2 | |
| 637.2.a.k | 5 | 13.b | even | 2 | 1 | ||
| 637.2.a.l | 5 | 91.b | odd | 2 | 1 | ||
| 637.2.e.m | 10 | 91.r | even | 6 | 2 | ||
| 819.2.j.h | 10 | 273.ba | even | 6 | 2 | ||
| 1183.2.e.f | 10 | 7.d | odd | 6 | 2 | ||
| 1456.2.r.p | 10 | 364.x | even | 6 | 2 | ||
| 5733.2.a.bl | 5 | 273.g | even | 2 | 1 | ||
| 5733.2.a.bm | 5 | 39.d | odd | 2 | 1 | ||
| 8281.2.a.bw | 5 | 7.b | odd | 2 | 1 | ||
| 8281.2.a.bx | 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):
|
\( T_{2}^{5} + 4T_{2}^{4} - T_{2}^{3} - 17T_{2}^{2} - 12T_{2} + 3 \)
|
|
\( T_{3}^{5} - 9T_{3}^{3} + 16T_{3} - 4 \)
|
|
\( T_{5}^{5} - 2T_{5}^{4} - 15T_{5}^{3} + 20T_{5}^{2} + 48T_{5} - 48 \)
|
|
\( T_{11}^{5} + 11T_{11}^{4} + 36T_{11}^{3} + 22T_{11}^{2} - 45T_{11} - 33 \)
|
|
\( T_{17}^{5} - 5T_{17}^{4} - 22T_{17}^{3} + 106T_{17}^{2} + 93T_{17} - 429 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 4 T^{4} + \cdots + 3 \)
|
| $3$ |
\( T^{5} - 9 T^{3} + \cdots - 4 \)
|
| $5$ |
\( T^{5} - 2 T^{4} + \cdots - 48 \)
|
| $7$ |
\( T^{5} \)
|
| $11$ |
\( T^{5} + 11 T^{4} + \cdots - 33 \)
|
| $13$ |
\( T^{5} \)
|
| $17$ |
\( T^{5} - 5 T^{4} + \cdots - 429 \)
|
| $19$ |
\( T^{5} - 9 T^{4} + \cdots - 223 \)
|
| $23$ |
\( T^{5} - 10 T^{4} + \cdots + 12 \)
|
| $29$ |
\( T^{5} + 3 T^{4} + \cdots - 108 \)
|
| $31$ |
\( T^{5} + 6 T^{4} + \cdots - 356 \)
|
| $37$ |
\( T^{5} + 4 T^{4} + \cdots + 7036 \)
|
| $41$ |
\( T^{5} - 14 T^{4} + \cdots - 1584 \)
|
| $43$ |
\( T^{5} - 2 T^{4} + \cdots + 64 \)
|
| $47$ |
\( T^{5} - T^{4} + \cdots - 5169 \)
|
| $53$ |
\( T^{5} - 17 T^{4} + \cdots + 19959 \)
|
| $59$ |
\( T^{5} - 11 T^{4} + \cdots + 33 \)
|
| $61$ |
\( T^{5} - 11 T^{4} + \cdots + 8461 \)
|
| $67$ |
\( T^{5} + 13 T^{4} + \cdots + 22699 \)
|
| $71$ |
\( T^{5} + 15 T^{4} + \cdots + 6336 \)
|
| $73$ |
\( T^{5} - 75 T^{3} + \cdots - 712 \)
|
| $79$ |
\( T^{5} - 2 T^{4} + \cdots - 1000 \)
|
| $83$ |
\( T^{5} - 6 T^{4} + \cdots - 7488 \)
|
| $89$ |
\( T^{5} + 4 T^{4} + \cdots + 7692 \)
|
| $97$ |
\( T^{5} + 12 T^{4} + \cdots - 2384 \)
|
| show more | |
| show less |