Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [294,5,Mod(97,294)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(294, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("294.97");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 294 = 2 \cdot 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 294.c (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(30.3907691467\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.339738624.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 7^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{8} - 4x^{6} + 14x^{4} - 8x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{6} + 20 ) / 14 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{6} + 7\nu^{4} - 28\nu^{2} + 9 ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -8\nu^{7} + 35\nu^{5} - 126\nu^{3} + 134\nu ) / 14 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 10\nu^{7} - 35\nu^{5} + 126\nu^{3} - 10\nu ) / 14 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} + 4\nu^{4} - 12\nu^{2} + 4 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5\nu^{7} - 28\nu^{5} + 91\nu^{3} - 96\nu ) / 7 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 8\nu^{7} - 28\nu^{5} + 91\nu^{3} - 8\nu ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{7} + \beta_{6} + 3\beta_{4} + 3\beta_{3} ) / 14 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - 2\beta_{2} - \beta _1 + 2 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5\beta_{7} + 8\beta_{4} ) / 7 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} - 3\beta_{2} + 2\beta _1 - 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -9\beta_{7} - 9\beta_{6} + 13\beta_{4} - 13\beta_{3} ) / 7 \)
|
| \(\nu^{6}\) | \(=\) |
\( 14\beta _1 - 20 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 31\beta_{7} - 31\beta_{6} - 44\beta_{4} - 44\beta_{3} ) / 7 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(199\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 97.1 |
|
−2.82843 | − | 5.19615i | 8.00000 | − | 43.8369i | 14.6969i | 0 | −22.6274 | −27.0000 | 123.989i | ||||||||||||||||||||||||||||||||||||||||
| 97.2 | −2.82843 | − | 5.19615i | 8.00000 | − | 10.0519i | 14.6969i | 0 | −22.6274 | −27.0000 | 28.4311i | |||||||||||||||||||||||||||||||||||||||||
| 97.3 | −2.82843 | 5.19615i | 8.00000 | 10.0519i | − | 14.6969i | 0 | −22.6274 | −27.0000 | − | 28.4311i | |||||||||||||||||||||||||||||||||||||||||
| 97.4 | −2.82843 | 5.19615i | 8.00000 | 43.8369i | − | 14.6969i | 0 | −22.6274 | −27.0000 | − | 123.989i | |||||||||||||||||||||||||||||||||||||||||
| 97.5 | 2.82843 | − | 5.19615i | 8.00000 | 12.2936i | − | 14.6969i | 0 | 22.6274 | −27.0000 | 34.7716i | |||||||||||||||||||||||||||||||||||||||||
| 97.6 | 2.82843 | − | 5.19615i | 8.00000 | 41.5951i | − | 14.6969i | 0 | 22.6274 | −27.0000 | 117.649i | |||||||||||||||||||||||||||||||||||||||||
| 97.7 | 2.82843 | 5.19615i | 8.00000 | − | 41.5951i | 14.6969i | 0 | 22.6274 | −27.0000 | − | 117.649i | |||||||||||||||||||||||||||||||||||||||||
| 97.8 | 2.82843 | 5.19615i | 8.00000 | − | 12.2936i | 14.6969i | 0 | 22.6274 | −27.0000 | − | 34.7716i | |||||||||||||||||||||||||||||||||||||||||
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 | 294.5.c.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 882.5.c.f | 8 | ||
| 7.b | odd | 2 | 1 | inner | 294.5.c.b | ✓ | 8 |
| 7.c | even | 3 | 1 | 294.5.g.e | 8 | ||
| 7.c | even | 3 | 1 | 294.5.g.g | 8 | ||
| 7.d | odd | 6 | 1 | 294.5.g.e | 8 | ||
| 7.d | odd | 6 | 1 | 294.5.g.g | 8 | ||
| 21.c | even | 2 | 1 | 882.5.c.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 294.5.c.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 294.5.c.b | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 294.5.g.e | 8 | 7.c | even | 3 | 1 | ||
| 294.5.g.e | 8 | 7.d | odd | 6 | 1 | ||
| 294.5.g.g | 8 | 7.c | even | 3 | 1 | ||
| 294.5.g.g | 8 | 7.d | odd | 6 | 1 | ||
| 882.5.c.f | 8 | 3.b | odd | 2 | 1 | ||
| 882.5.c.f | 8 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 3904T_{5}^{6} + 4260956T_{5}^{4} + 894192704T_{5}^{2} + 50771806276 \)
acting on \(S_{5}^{\mathrm{new}}(294, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 8)^{4} \)
|
| $3$ |
\( (T^{2} + 27)^{4} \)
|
| $5$ |
\( T^{8} + \cdots + 50771806276 \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( (T^{4} - 6844 T^{2} + \cdots + 10903132)^{2} \)
|
| $13$ |
\( T^{8} + \cdots + 29\!\cdots\!36 \)
|
| $17$ |
\( T^{8} + \cdots + 172220425465284 \)
|
| $19$ |
\( T^{8} + \cdots + 79\!\cdots\!76 \)
|
| $23$ |
\( (T^{4} + 136 T^{3} + \cdots + 361343836)^{2} \)
|
| $29$ |
\( (T^{4} - 200 T^{3} + \cdots - 54206402084)^{2} \)
|
| $31$ |
\( T^{8} + \cdots + 84\!\cdots\!84 \)
|
| $37$ |
\( (T^{4} + 1664 T^{3} + \cdots + 105864334084)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 21\!\cdots\!24 \)
|
| $43$ |
\( (T^{4} - 328 T^{3} + \cdots + 3375543568)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 17\!\cdots\!24 \)
|
| $53$ |
\( (T^{4} + \cdots + 7818957021712)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 55\!\cdots\!56 \)
|
| $61$ |
\( T^{8} + \cdots + 17\!\cdots\!44 \)
|
| $67$ |
\( (T^{4} + \cdots - 590477790180224)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots - 509861826592676)^{2} \)
|
| $73$ |
\( T^{8} + \cdots + 40\!\cdots\!64 \)
|
| $79$ |
\( (T^{4} + \cdots - 118271445421808)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 17\!\cdots\!04 \)
|
| $89$ |
\( T^{8} + \cdots + 48\!\cdots\!04 \)
|
| $97$ |
\( T^{8} + \cdots + 82\!\cdots\!76 \)
|
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