Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [448,4,Mod(65,448)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(448, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("448.65");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 448 = 2^{6} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 448.i (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: | \(26.4328556826\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.11163123.4 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 14x^{4} + 49x^{2} + 27 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | no (minimal twist has level 56) |
| 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
\( x^{6} + 14x^{4} + 49x^{2} + 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 7\nu + 3 ) / 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{2} + 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{5} + 23\nu^{3} + 6\nu^{2} + 63\nu + 27 ) / 6 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{5} - 4\nu^{4} - 23\nu^{3} - 34\nu^{2} - 51\nu - 27 ) / 6 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -2\nu^{5} + 4\nu^{4} - 23\nu^{3} + 34\nu^{2} - 51\nu + 27 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + 2\beta_{3} - \beta_{2} ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} - 9 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -7\beta_{5} - 7\beta_{4} - 14\beta_{3} + 7\beta_{2} + 24\beta _1 - 12 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 3\beta_{5} - 3\beta_{4} - 17\beta_{2} + 126 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 49\beta_{5} + 49\beta_{4} + 110\beta_{3} - 55\beta_{2} - 276\beta _1 + 138 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(129\) | \(197\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{1}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 65.1 |
|
0 | −4.25602 | − | 7.37164i | 0 | 4.23844 | − | 7.34119i | 0 | 17.7274 | − | 5.36106i | 0 | −22.7274 | + | 39.3649i | 0 | ||||||||||||||||||||||||||||
| 65.2 | 0 | −0.0691041 | − | 0.119692i | 0 | −5.04289 | + | 8.73453i | 0 | −18.4904 | − | 1.05037i | 0 | 13.4904 | − | 23.3661i | 0 | |||||||||||||||||||||||||||||
| 65.3 | 0 | 3.82512 | + | 6.62530i | 0 | 7.30445 | − | 12.6517i | 0 | 10.7631 | + | 15.0717i | 0 | −15.7631 | + | 27.3025i | 0 | |||||||||||||||||||||||||||||
| 193.1 | 0 | −4.25602 | + | 7.37164i | 0 | 4.23844 | + | 7.34119i | 0 | 17.7274 | + | 5.36106i | 0 | −22.7274 | − | 39.3649i | 0 | |||||||||||||||||||||||||||||
| 193.2 | 0 | −0.0691041 | + | 0.119692i | 0 | −5.04289 | − | 8.73453i | 0 | −18.4904 | + | 1.05037i | 0 | 13.4904 | + | 23.3661i | 0 | |||||||||||||||||||||||||||||
| 193.3 | 0 | 3.82512 | − | 6.62530i | 0 | 7.30445 | + | 12.6517i | 0 | 10.7631 | − | 15.0717i | 0 | −15.7631 | − | 27.3025i | 0 | |||||||||||||||||||||||||||||
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 | 448.4.i.k | 6 | |
| 4.b | odd | 2 | 1 | 448.4.i.l | 6 | ||
| 7.c | even | 3 | 1 | inner | 448.4.i.k | 6 | |
| 8.b | even | 2 | 1 | 112.4.i.f | 6 | ||
| 8.d | odd | 2 | 1 | 56.4.i.a | ✓ | 6 | |
| 24.f | even | 2 | 1 | 504.4.s.i | 6 | ||
| 28.g | odd | 6 | 1 | 448.4.i.l | 6 | ||
| 56.e | even | 2 | 1 | 392.4.i.n | 6 | ||
| 56.j | odd | 6 | 1 | 784.4.a.bd | 3 | ||
| 56.k | odd | 6 | 1 | 56.4.i.a | ✓ | 6 | |
| 56.k | odd | 6 | 1 | 392.4.a.k | 3 | ||
| 56.m | even | 6 | 1 | 392.4.a.j | 3 | ||
| 56.m | even | 6 | 1 | 392.4.i.n | 6 | ||
| 56.p | even | 6 | 1 | 112.4.i.f | 6 | ||
| 56.p | even | 6 | 1 | 784.4.a.bc | 3 | ||
| 168.v | even | 6 | 1 | 504.4.s.i | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 56.4.i.a | ✓ | 6 | 8.d | odd | 2 | 1 | |
| 56.4.i.a | ✓ | 6 | 56.k | odd | 6 | 1 | |
| 112.4.i.f | 6 | 8.b | even | 2 | 1 | ||
| 112.4.i.f | 6 | 56.p | even | 6 | 1 | ||
| 392.4.a.j | 3 | 56.m | even | 6 | 1 | ||
| 392.4.a.k | 3 | 56.k | odd | 6 | 1 | ||
| 392.4.i.n | 6 | 56.e | even | 2 | 1 | ||
| 392.4.i.n | 6 | 56.m | even | 6 | 1 | ||
| 448.4.i.k | 6 | 1.a | even | 1 | 1 | trivial | |
| 448.4.i.k | 6 | 7.c | even | 3 | 1 | inner | |
| 448.4.i.l | 6 | 4.b | odd | 2 | 1 | ||
| 448.4.i.l | 6 | 28.g | odd | 6 | 1 | ||
| 504.4.s.i | 6 | 24.f | even | 2 | 1 | ||
| 504.4.s.i | 6 | 168.v | even | 6 | 1 | ||
| 784.4.a.bc | 3 | 56.p | even | 6 | 1 | ||
| 784.4.a.bd | 3 | 56.j | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(448, [\chi])\):
|
\( T_{3}^{6} + T_{3}^{5} + 66T_{3}^{4} - 47T_{3}^{3} + 4234T_{3}^{2} + 585T_{3} + 81 \)
|
|
\( T_{11}^{6} - 11T_{11}^{5} + 698T_{11}^{4} - 9851T_{11}^{3} + 422018T_{11}^{2} - 4673123T_{11} + 65593801 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + T^{5} + \cdots + 81 \)
|
| $5$ |
\( T^{6} - 13 T^{5} + \cdots + 1560001 \)
|
| $7$ |
\( T^{6} - 20 T^{5} + \cdots + 40353607 \)
|
| $11$ |
\( T^{6} - 11 T^{5} + \cdots + 65593801 \)
|
| $13$ |
\( (T^{3} + 70 T^{2} + \cdots - 17752)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 344049114249 \)
|
| $19$ |
\( T^{6} + \cdots + 15221884129 \)
|
| $23$ |
\( T^{6} + \cdots + 12196012613841 \)
|
| $29$ |
\( (T^{3} - 162 T^{2} + \cdots + 1324296)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 21744865333321 \)
|
| $37$ |
\( T^{6} + \cdots + 228323372199129 \)
|
| $41$ |
\( (T^{3} - 698 T^{2} + \cdots + 6465192)^{2} \)
|
| $43$ |
\( (T^{3} + 308 T^{2} + \cdots + 38848)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 350054034092361 \)
|
| $53$ |
\( T^{6} + \cdots + 8496286395921 \)
|
| $59$ |
\( T^{6} + \cdots + 13\!\cdots\!69 \)
|
| $61$ |
\( T^{6} + \cdots + 98846286969609 \)
|
| $67$ |
\( T^{6} + \cdots + 454704744973249 \)
|
| $71$ |
\( (T^{3} + 224 T^{2} + \cdots - 7551488)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 52\!\cdots\!61 \)
|
| $79$ |
\( T^{6} + \cdots + 88\!\cdots\!69 \)
|
| $83$ |
\( (T^{3} + 1380 T^{2} + \cdots + 4797376)^{2} \)
|
| $89$ |
\( T^{6} + \cdots + 24\!\cdots\!41 \)
|
| $97$ |
\( (T^{3} - 2602 T^{2} + \cdots - 528741144)^{2} \)
|
| show more | |
| show less |