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: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.8.8446345216.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 8x^{6} + 19x^{4} - 14x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 637) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 8x^{6} + 19x^{4} - 14x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{5} - 6\nu^{3} + 7\nu \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} - 6\nu^{4} + 7\nu^{2} \)
|
| \(\beta_{6}\) | \(=\) |
\( -\nu^{6} + 7\nu^{4} - 12\nu^{2} + 3 \)
|
| \(\beta_{7}\) | \(=\) |
\( \nu^{7} - 7\nu^{5} + 13\nu^{3} - 6\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + \beta_{5} + 5\beta_{2} + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{4} + 6\beta_{3} + 11\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 6\beta_{6} + 7\beta_{5} + 23\beta_{2} + 28 \)
|
| \(\nu^{7}\) | \(=\) |
\( \beta_{7} + 7\beta_{4} + 29\beta_{3} + 44\beta_1 \)
|
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 |
|
−2.33152 | −2.30901 | 3.43596 | −3.37112 | 5.38349 | 0 | −3.34797 | 2.33152 | 7.85981 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.33152 | 2.30901 | 3.43596 | 3.37112 | −5.38349 | 0 | −3.34797 | 2.33152 | −7.85981 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.52077 | −2.12621 | 0.312752 | 0.589391 | 3.23349 | 0 | 2.56592 | 1.52077 | −0.896331 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.52077 | 2.12621 | 0.312752 | −0.589391 | −3.23349 | 0 | 2.56592 | 1.52077 | 0.896331 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.579810 | −1.89204 | −1.66382 | 1.47362 | 1.09702 | 0 | 2.12432 | 0.579810 | −0.854419 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.579810 | 1.89204 | −1.66382 | −1.47362 | −1.09702 | 0 | 2.12432 | 0.579810 | 0.854419 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.43210 | −0.753592 | 3.91511 | −0.341537 | −1.83281 | 0 | 4.65773 | −2.43210 | −0.830652 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.43210 | 0.753592 | 3.91511 | 0.341537 | 1.83281 | 0 | 4.65773 | −2.43210 | 0.830652 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \( +1 \) |
| \(13\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 8281.2.a.ci | 8 | |
| 7.b | odd | 2 | 1 | inner | 8281.2.a.ci | 8 | |
| 13.b | even | 2 | 1 | 8281.2.a.cl | 8 | ||
| 13.c | even | 3 | 2 | 637.2.f.l | ✓ | 16 | |
| 91.b | odd | 2 | 1 | 8281.2.a.cl | 8 | ||
| 91.g | even | 3 | 2 | 637.2.g.m | 16 | ||
| 91.h | even | 3 | 2 | 637.2.h.m | 16 | ||
| 91.m | odd | 6 | 2 | 637.2.g.m | 16 | ||
| 91.n | odd | 6 | 2 | 637.2.f.l | ✓ | 16 | |
| 91.v | odd | 6 | 2 | 637.2.h.m | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 637.2.f.l | ✓ | 16 | 13.c | even | 3 | 2 | |
| 637.2.f.l | ✓ | 16 | 91.n | odd | 6 | 2 | |
| 637.2.g.m | 16 | 91.g | even | 3 | 2 | ||
| 637.2.g.m | 16 | 91.m | odd | 6 | 2 | ||
| 637.2.h.m | 16 | 91.h | even | 3 | 2 | ||
| 637.2.h.m | 16 | 91.v | odd | 6 | 2 | ||
| 8281.2.a.ci | 8 | 1.a | even | 1 | 1 | trivial | |
| 8281.2.a.ci | 8 | 7.b | odd | 2 | 1 | inner | |
| 8281.2.a.cl | 8 | 13.b | even | 2 | 1 | ||
| 8281.2.a.cl | 8 | 91.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8281))\):
|
\( T_{2}^{4} + 2T_{2}^{3} - 5T_{2}^{2} - 12T_{2} - 5 \)
|
|
\( T_{3}^{8} - 14T_{3}^{6} + 67T_{3}^{4} - 120T_{3}^{2} + 49 \)
|
|
\( T_{5}^{8} - 14T_{5}^{6} + 31T_{5}^{4} - 12T_{5}^{2} + 1 \)
|
|
\( T_{11}^{4} - 2T_{11}^{3} - 5T_{11}^{2} + 12T_{11} - 5 \)
|
|
\( T_{17}^{8} - 58T_{17}^{6} + 966T_{17}^{4} - 3866T_{17}^{2} + 3721 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 2 T^{3} - 5 T^{2} + \cdots - 5)^{2} \)
|
| $3$ |
\( T^{8} - 14 T^{6} + \cdots + 49 \)
|
| $5$ |
\( T^{8} - 14 T^{6} + \cdots + 1 \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( (T^{4} - 2 T^{3} - 5 T^{2} + \cdots - 5)^{2} \)
|
| $13$ |
\( T^{8} \)
|
| $17$ |
\( T^{8} - 58 T^{6} + \cdots + 3721 \)
|
| $19$ |
\( T^{8} - 94 T^{6} + \cdots + 1225 \)
|
| $23$ |
\( (T^{4} + 6 T^{3} + \cdots + 100)^{2} \)
|
| $29$ |
\( (T^{4} + 4 T^{3} + \cdots + 1781)^{2} \)
|
| $31$ |
\( T^{8} - 70 T^{6} + \cdots + 26569 \)
|
| $37$ |
\( (T^{4} - 4 T^{3} + \cdots - 380)^{2} \)
|
| $41$ |
\( T^{8} - 176 T^{6} + \cdots + 4096 \)
|
| $43$ |
\( (T^{4} + 16 T^{3} + \cdots - 3205)^{2} \)
|
| $47$ |
\( T^{8} - 226 T^{6} + \cdots + 27889 \)
|
| $53$ |
\( (T^{4} + 2 T^{3} + \cdots + 271)^{2} \)
|
| $59$ |
\( T^{8} - 194 T^{6} + \cdots + 169 \)
|
| $61$ |
\( T^{8} - 108 T^{6} + \cdots + 19600 \)
|
| $67$ |
\( (T^{4} + 10 T^{3} + \cdots - 283)^{2} \)
|
| $71$ |
\( (T^{4} + 4 T^{3} + \cdots + 5956)^{2} \)
|
| $73$ |
\( T^{8} - 428 T^{6} + \cdots + 2226064 \)
|
| $79$ |
\( (T^{4} + 2 T^{3} + \cdots + 8164)^{2} \)
|
| $83$ |
\( T^{8} - 350 T^{6} + \cdots + 405769 \)
|
| $89$ |
\( T^{8} - 606 T^{6} + \cdots + 45225625 \)
|
| $97$ |
\( T^{8} - 364 T^{6} + \cdots + 310249 \)
|
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