Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [294,8,Mod(67,294)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(294, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("294.67");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 294 = 2 \cdot 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 294.e (of order \(3\), degree \(2\), not 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: | \(91.8411974923\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 2119x^{4} - 65706x^{3} + 4519836x^{2} - 71825616x + 1150023744 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2}\cdot 7^{4} \) |
| Twist minimal: | no (minimal twist has level 42) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{6} - x^{5} + 2119x^{4} - 65706x^{3} + 4519836x^{2} - 71825616x + 1150023744 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 748007 \nu^{5} + 742355 \nu^{4} - 1573050245 \nu^{3} + 23770157970 \nu^{2} + \cdots - 406150052256 ) / 53726255217936 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 728225 \nu^{5} + 41175703 \nu^{4} - 87251314657 \nu^{3} + 65641237782 \nu^{2} + \cdots + 29\!\cdots\!48 ) / 26863127608968 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 748007 \nu^{5} + 742355 \nu^{4} - 1573050245 \nu^{3} + 23770157970 \nu^{2} + \cdots - 406150052256 ) / 26863127608968 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 2095 \nu^{5} + 4439305 \nu^{4} + 98819573 \nu^{3} + 9469056420 \nu^{2} + \cdots + 9996388681824 ) / 1357958124 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 132754993 \nu^{5} + 97335322 \nu^{4} - 206253547318 \nu^{3} + 10861539866835 \nu^{2} + \cdots + 69\!\cdots\!52 ) / 959397414606 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - 2\beta_1 ) / 7 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 23\beta_{2} - 9894\beta _1 - 9894 ) / 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{4} - 2095\beta_{3} - 2095\beta_{2} + 223254 ) / 7 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -2119\beta_{5} + 2119\beta_{4} + 80531\beta_{3} + 20664414\beta_1 ) / 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 29675\beta_{5} + 5136655\beta_{2} - 787712886\beta _1 - 787712886 ) / 7 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(199\) |
| \(\chi(n)\) | \(1\) | \(\beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 67.1 |
|
4.00000 | + | 6.92820i | −13.5000 | + | 23.3827i | −32.0000 | + | 55.4256i | −201.921 | − | 349.738i | −216.000 | 0 | −512.000 | −364.500 | − | 631.333i | 1615.37 | − | 2797.90i | ||||||||||||||||||||||||
| 67.2 | 4.00000 | + | 6.92820i | −13.5000 | + | 23.3827i | −32.0000 | + | 55.4256i | 47.5526 | + | 82.3635i | −216.000 | 0 | −512.000 | −364.500 | − | 631.333i | −380.420 | + | 658.908i | |||||||||||||||||||||||||
| 67.3 | 4.00000 | + | 6.92820i | −13.5000 | + | 23.3827i | −32.0000 | + | 55.4256i | 99.3686 | + | 172.111i | −216.000 | 0 | −512.000 | −364.500 | − | 631.333i | −794.949 | + | 1376.89i | |||||||||||||||||||||||||
| 79.1 | 4.00000 | − | 6.92820i | −13.5000 | − | 23.3827i | −32.0000 | − | 55.4256i | −201.921 | + | 349.738i | −216.000 | 0 | −512.000 | −364.500 | + | 631.333i | 1615.37 | + | 2797.90i | |||||||||||||||||||||||||
| 79.2 | 4.00000 | − | 6.92820i | −13.5000 | − | 23.3827i | −32.0000 | − | 55.4256i | 47.5526 | − | 82.3635i | −216.000 | 0 | −512.000 | −364.500 | + | 631.333i | −380.420 | − | 658.908i | |||||||||||||||||||||||||
| 79.3 | 4.00000 | − | 6.92820i | −13.5000 | − | 23.3827i | −32.0000 | − | 55.4256i | 99.3686 | − | 172.111i | −216.000 | 0 | −512.000 | −364.500 | + | 631.333i | −794.949 | − | 1376.89i | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 294.8.e.ba | 6 | |
| 7.b | odd | 2 | 1 | 42.8.e.e | ✓ | 6 | |
| 7.c | even | 3 | 1 | 294.8.a.w | 3 | ||
| 7.c | even | 3 | 1 | inner | 294.8.e.ba | 6 | |
| 7.d | odd | 6 | 1 | 42.8.e.e | ✓ | 6 | |
| 7.d | odd | 6 | 1 | 294.8.a.v | 3 | ||
| 21.c | even | 2 | 1 | 126.8.g.h | 6 | ||
| 21.g | even | 6 | 1 | 126.8.g.h | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.8.e.e | ✓ | 6 | 7.b | odd | 2 | 1 | |
| 42.8.e.e | ✓ | 6 | 7.d | odd | 6 | 1 | |
| 126.8.g.h | 6 | 21.c | even | 2 | 1 | ||
| 126.8.g.h | 6 | 21.g | even | 6 | 1 | ||
| 294.8.a.v | 3 | 7.d | odd | 6 | 1 | ||
| 294.8.a.w | 3 | 7.c | even | 3 | 1 | ||
| 294.8.e.ba | 6 | 1.a | even | 1 | 1 | trivial | |
| 294.8.e.ba | 6 | 7.c | even | 3 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 110T_{5}^{5} + 111865T_{5}^{4} - 26240130T_{5}^{3} + 9113426325T_{5}^{2} - 761505247350T_{5} + 58262536340100 \)
acting on \(S_{8}^{\mathrm{new}}(294, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 8 T + 64)^{3} \)
|
| $3$ |
\( (T^{2} + 27 T + 729)^{3} \)
|
| $5$ |
\( T^{6} + \cdots + 58262536340100 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} + \cdots + 10\!\cdots\!36 \)
|
| $13$ |
\( (T^{3} + \cdots - 1384679947008)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 40\!\cdots\!36 \)
|
| $19$ |
\( T^{6} + \cdots + 14\!\cdots\!16 \)
|
| $23$ |
\( T^{6} + \cdots + 31\!\cdots\!96 \)
|
| $29$ |
\( (T^{3} + \cdots - 82770327433116)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 91\!\cdots\!25 \)
|
| $37$ |
\( T^{6} + \cdots + 20\!\cdots\!16 \)
|
| $41$ |
\( (T^{3} + \cdots - 78\!\cdots\!72)^{2} \)
|
| $43$ |
\( (T^{3} + \cdots + 23\!\cdots\!76)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 20\!\cdots\!44 \)
|
| $53$ |
\( T^{6} + \cdots + 89\!\cdots\!16 \)
|
| $59$ |
\( T^{6} + \cdots + 22\!\cdots\!96 \)
|
| $61$ |
\( T^{6} + \cdots + 28\!\cdots\!84 \)
|
| $67$ |
\( T^{6} + \cdots + 10\!\cdots\!96 \)
|
| $71$ |
\( (T^{3} + \cdots + 81\!\cdots\!64)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 26\!\cdots\!84 \)
|
| $79$ |
\( T^{6} + \cdots + 19\!\cdots\!29 \)
|
| $83$ |
\( (T^{3} + \cdots + 11\!\cdots\!98)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 12\!\cdots\!24 \)
|
| $97$ |
\( (T^{3} + \cdots - 14\!\cdots\!26)^{2} \)
|
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