Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [847,2,Mod(1,847)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(847, 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("847.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 847 = 7 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 847.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: | \(6.76332905120\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.6.7674048.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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
\( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - \nu^{2} - 3\nu + 1 \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - \nu^{3} - 4\nu^{2} + 2\nu + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} - 2\nu^{4} - 4\nu^{3} + 6\nu^{2} + 4\nu - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + \beta_{2} + 4\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 5\beta_{2} + 6\beta _1 + 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + 6\beta_{3} + 8\beta_{2} + 18\beta _1 + 8 \)
|
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 |
|
−1.38595 | −0.122479 | −0.0791355 | −0.133004 | 0.169750 | 1.00000 | 2.88158 | −2.98500 | 0.184338 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −0.823556 | −0.960649 | −1.32176 | 2.98565 | 0.791148 | 1.00000 | 2.73565 | −2.07715 | −2.45885 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0.120360 | 2.76784 | −1.98551 | −2.80853 | 0.333137 | 1.00000 | −0.479696 | 4.66094 | −0.338034 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 1.27656 | −2.57603 | −0.370384 | −4.09144 | −3.28847 | 1.00000 | −3.02595 | 3.63595 | −5.22298 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.10939 | 1.69851 | 2.44952 | 0.492391 | 3.58282 | 1.00000 | 0.948212 | −0.115054 | 1.03864 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.70320 | −2.80719 | 5.30727 | −0.445072 | −7.58839 | 1.00000 | 8.94020 | 4.88032 | −1.20312 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
| \(11\) | \(1\) |
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 | 847.2.a.n | yes | 6 |
| 3.b | odd | 2 | 1 | 7623.2.a.cp | 6 | ||
| 7.b | odd | 2 | 1 | 5929.2.a.bm | 6 | ||
| 11.b | odd | 2 | 1 | 847.2.a.m | ✓ | 6 | |
| 11.c | even | 5 | 4 | 847.2.f.y | 24 | ||
| 11.d | odd | 10 | 4 | 847.2.f.z | 24 | ||
| 33.d | even | 2 | 1 | 7623.2.a.cs | 6 | ||
| 77.b | even | 2 | 1 | 5929.2.a.bj | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 847.2.a.m | ✓ | 6 | 11.b | odd | 2 | 1 | |
| 847.2.a.n | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 847.2.f.y | 24 | 11.c | even | 5 | 4 | ||
| 847.2.f.z | 24 | 11.d | odd | 10 | 4 | ||
| 5929.2.a.bj | 6 | 77.b | even | 2 | 1 | ||
| 5929.2.a.bm | 6 | 7.b | odd | 2 | 1 | ||
| 7623.2.a.cp | 6 | 3.b | odd | 2 | 1 | ||
| 7623.2.a.cs | 6 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(847))\):
|
\( T_{2}^{6} - 4T_{2}^{5} + 12T_{2}^{3} - 4T_{2}^{2} - 8T_{2} + 1 \)
|
|
\( T_{3}^{6} + 2T_{3}^{5} - 11T_{3}^{4} - 20T_{3}^{3} + 25T_{3}^{2} + 36T_{3} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 4 T^{5} + \cdots + 1 \)
|
| $3$ |
\( T^{6} + 2 T^{5} + \cdots + 4 \)
|
| $5$ |
\( T^{6} + 4 T^{5} + \cdots + 1 \)
|
| $7$ |
\( (T - 1)^{6} \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( T^{6} - 4 T^{5} + \cdots + 16 \)
|
| $17$ |
\( T^{6} - 22 T^{5} + \cdots - 2687 \)
|
| $19$ |
\( T^{6} - 6 T^{5} + \cdots + 592 \)
|
| $23$ |
\( T^{6} - 2 T^{5} + \cdots + 8656 \)
|
| $29$ |
\( T^{6} - 12 T^{5} + \cdots + 481 \)
|
| $31$ |
\( T^{6} + 2 T^{5} + \cdots + 6736 \)
|
| $37$ |
\( T^{6} - 14 T^{5} + \cdots + 2896 \)
|
| $41$ |
\( T^{6} - 26 T^{5} + \cdots - 3803 \)
|
| $43$ |
\( T^{6} + 4 T^{5} + \cdots - 2288 \)
|
| $47$ |
\( T^{6} + 16 T^{5} + \cdots + 9088 \)
|
| $53$ |
\( T^{6} - 4 T^{5} + \cdots + 2257 \)
|
| $59$ |
\( T^{6} + 4 T^{5} + \cdots - 313856 \)
|
| $61$ |
\( T^{6} + 8 T^{5} + \cdots - 971552 \)
|
| $67$ |
\( T^{6} - 6 T^{5} + \cdots - 48896 \)
|
| $71$ |
\( T^{6} - 22 T^{5} + \cdots - 2048 \)
|
| $73$ |
\( T^{6} - 14 T^{5} + \cdots - 23024 \)
|
| $79$ |
\( T^{6} + 28 T^{5} + \cdots - 183488 \)
|
| $83$ |
\( T^{6} - 22 T^{5} + \cdots - 44012 \)
|
| $89$ |
\( T^{6} - 254 T^{4} + \cdots + 62137 \)
|
| $97$ |
\( T^{6} + 4 T^{5} + \cdots - 62759 \)
|
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