Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [294,6,Mod(1,294)]
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, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("294.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 294 = 2 \cdot 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 294.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: | \(47.1528430250\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{9601}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 2400 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 42) |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{9601})\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
4.00000 | 9.00000 | 16.0000 | −22.4923 | 36.0000 | 0 | 64.0000 | 81.0000 | −89.9694 | ||||||||||||||||||||||||
| 1.2 | 4.00000 | 9.00000 | 16.0000 | 75.4923 | 36.0000 | 0 | 64.0000 | 81.0000 | 301.969 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
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 | 294.6.a.w | 2 | |
| 3.b | odd | 2 | 1 | 882.6.a.bb | 2 | ||
| 7.b | odd | 2 | 1 | 294.6.a.r | 2 | ||
| 7.c | even | 3 | 2 | 294.6.e.s | 4 | ||
| 7.d | odd | 6 | 2 | 42.6.e.c | ✓ | 4 | |
| 21.c | even | 2 | 1 | 882.6.a.bh | 2 | ||
| 21.g | even | 6 | 2 | 126.6.g.h | 4 | ||
| 28.f | even | 6 | 2 | 336.6.q.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 42.6.e.c | ✓ | 4 | 7.d | odd | 6 | 2 | |
| 126.6.g.h | 4 | 21.g | even | 6 | 2 | ||
| 294.6.a.r | 2 | 7.b | odd | 2 | 1 | ||
| 294.6.a.w | 2 | 1.a | even | 1 | 1 | trivial | |
| 294.6.e.s | 4 | 7.c | even | 3 | 2 | ||
| 336.6.q.f | 4 | 28.f | even | 6 | 2 | ||
| 882.6.a.bb | 2 | 3.b | odd | 2 | 1 | ||
| 882.6.a.bh | 2 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(294))\):
|
\( T_{5}^{2} - 53T_{5} - 1698 \)
|
|
\( T_{11}^{2} - 191T_{11} - 50886 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 4)^{2} \)
|
| $3$ |
\( (T - 9)^{2} \)
|
| $5$ |
\( T^{2} - 53T - 1698 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} - 191T - 50886 \)
|
| $13$ |
\( T^{2} - 379T - 254520 \)
|
| $17$ |
\( T^{2} - 340T - 931200 \)
|
| $19$ |
\( T^{2} - 1769 T - 2868440 \)
|
| $23$ |
\( T^{2} - 3236 T + 2579520 \)
|
| $29$ |
\( T^{2} - 4459 T - 3960660 \)
|
| $31$ |
\( T^{2} + 1994 T - 63563115 \)
|
| $37$ |
\( T^{2} - 20587 T + 99713092 \)
|
| $41$ |
\( T^{2} + 8814 T + 8966160 \)
|
| $43$ |
\( T^{2} - 15853 T + 44661910 \)
|
| $47$ |
\( T^{2} + 33912 T + 229093452 \)
|
| $53$ |
\( T^{2} - 49239 T + 484607124 \)
|
| $59$ |
\( T^{2} - 56735 T + 800680236 \)
|
| $61$ |
\( T^{2} + \cdots + 1136874660 \)
|
| $67$ |
\( T^{2} + \cdots + 1160899190 \)
|
| $71$ |
\( T^{2} + \cdots - 4289674884 \)
|
| $73$ |
\( T^{2} + \cdots - 1835129806 \)
|
| $79$ |
\( T^{2} + \cdots - 2156463813 \)
|
| $83$ |
\( T^{2} + \cdots - 7511023590 \)
|
| $89$ |
\( T^{2} + \cdots + 7687683936 \)
|
| $97$ |
\( T^{2} + \cdots - 5231869258 \)
|
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