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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/294\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(199\) |
| \(\chi(n)\) | \(1\) | \(-\zeta_{6}\) |
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 | −145.000 | − | 251.147i | 216.000 | 0 | −512.000 | −364.500 | − | 631.333i | 1160.00 | − | 2009.18i | ||||||||||||
| 79.1 | 4.00000 | − | 6.92820i | 13.5000 | + | 23.3827i | −32.0000 | − | 55.4256i | −145.000 | + | 251.147i | 216.000 | 0 | −512.000 | −364.500 | + | 631.333i | 1160.00 | + | 2009.18i | |||||||||||||
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.l | 2 | |
| 7.b | odd | 2 | 1 | 42.8.e.b | ✓ | 2 | |
| 7.c | even | 3 | 1 | 294.8.a.e | 1 | ||
| 7.c | even | 3 | 1 | inner | 294.8.e.l | 2 | |
| 7.d | odd | 6 | 1 | 42.8.e.b | ✓ | 2 | |
| 7.d | odd | 6 | 1 | 294.8.a.f | 1 | ||
| 21.c | even | 2 | 1 | 126.8.g.a | 2 | ||
| 21.g | even | 6 | 1 | 126.8.g.a | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.8.e.b | ✓ | 2 | 7.b | odd | 2 | 1 | |
| 42.8.e.b | ✓ | 2 | 7.d | odd | 6 | 1 | |
| 126.8.g.a | 2 | 21.c | even | 2 | 1 | ||
| 126.8.g.a | 2 | 21.g | even | 6 | 1 | ||
| 294.8.a.e | 1 | 7.c | even | 3 | 1 | ||
| 294.8.a.f | 1 | 7.d | odd | 6 | 1 | ||
| 294.8.e.l | 2 | 1.a | even | 1 | 1 | trivial | |
| 294.8.e.l | 2 | 7.c | even | 3 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 290T_{5} + 84100 \)
acting on \(S_{8}^{\mathrm{new}}(294, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 8T + 64 \)
|
| $3$ |
\( T^{2} - 27T + 729 \)
|
| $5$ |
\( T^{2} + 290T + 84100 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} + 1130 T + 1276900 \)
|
| $13$ |
\( (T - 7563)^{2} \)
|
| $17$ |
\( T^{2} - 14504 T + 210366016 \)
|
| $19$ |
\( T^{2} - 25043 T + 627151849 \)
|
| $23$ |
\( T^{2} - 6664 T + 44408896 \)
|
| $29$ |
\( (T - 6820)^{2} \)
|
| $31$ |
\( T^{2} + \cdots + 31003814241 \)
|
| $37$ |
\( T^{2} + \cdots + 17685010225 \)
|
| $41$ |
\( (T + 661206)^{2} \)
|
| $43$ |
\( (T - 147095)^{2} \)
|
| $47$ |
\( T^{2} + \cdots + 809341333956 \)
|
| $53$ |
\( T^{2} + \cdots + 1703369537424 \)
|
| $59$ |
\( T^{2} + \cdots + 5671847086096 \)
|
| $61$ |
\( T^{2} + \cdots + 6891160510404 \)
|
| $67$ |
\( T^{2} + \cdots + 15787851932449 \)
|
| $71$ |
\( (T + 4291762)^{2} \)
|
| $73$ |
\( T^{2} + \cdots + 35168375065849 \)
|
| $79$ |
\( T^{2} + \cdots + 39484747448721 \)
|
| $83$ |
\( (T + 4252250)^{2} \)
|
| $89$ |
\( T^{2} + \cdots + 2192768640000 \)
|
| $97$ |
\( (T + 9407434)^{2} \)
|
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