Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [448,4,Mod(225,448)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(448, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("448.225");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 448 = 2^{6} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 448.b (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: | \(26.4328556826\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.819791424.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 169x^{2} - 312x + 288 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 169x^{2} - 312x + 288 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 195\nu^{4} - 15\nu^{3} - \nu^{2} + 24\nu + 23173 ) / 1447 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -52\nu^{5} + 11\nu^{4} + 667\nu^{3} - 1499\nu^{2} - 199\nu + 1092 ) / 8682 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -30\nu^{5} + 62\nu^{4} - 450\nu^{3} - 30\nu^{2} + 720\nu + 4971 ) / 1447 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -97\nu^{5} + 104\nu^{4} - 8\nu^{3} - 1544\nu^{2} - 16483\nu + 15060 ) / 4341 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -291\nu^{5} + 312\nu^{4} - 24\nu^{3} + 1156\nu^{2} - 49449\nu + 45180 ) / 2894 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{4} + \beta_{3} + 4\beta_{2} + 3 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{5} - 9\beta_{4} ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{5} - 20\beta_{4} - 13\beta_{3} + 52\beta_{2} + 4\beta _1 - 19 ) / 8 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -\beta_{3} + 30\beta _1 - 477 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -64\beta_{5} + 270\beta_{4} - 171\beta_{3} - 684\beta_{2} + 64\beta _1 - 289 ) / 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/448\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(129\) | \(197\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 225.1 |
|
0 | − | 7.04302i | 0 | − | 10.4751i | 0 | −7.00000 | 0 | −22.6041 | 0 | ||||||||||||||||||||||||||||||||||
| 225.2 | 0 | − | 3.21943i | 0 | 2.02303i | 0 | −7.00000 | 0 | 16.6353 | 0 | ||||||||||||||||||||||||||||||||||||
| 225.3 | 0 | − | 0.176409i | 0 | 18.4981i | 0 | −7.00000 | 0 | 26.9689 | 0 | ||||||||||||||||||||||||||||||||||||
| 225.4 | 0 | 0.176409i | 0 | − | 18.4981i | 0 | −7.00000 | 0 | 26.9689 | 0 | ||||||||||||||||||||||||||||||||||||
| 225.5 | 0 | 3.21943i | 0 | − | 2.02303i | 0 | −7.00000 | 0 | 16.6353 | 0 | ||||||||||||||||||||||||||||||||||||
| 225.6 | 0 | 7.04302i | 0 | 10.4751i | 0 | −7.00000 | 0 | −22.6041 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 448.4.b.c | ✓ | 6 |
| 4.b | odd | 2 | 1 | 448.4.b.d | yes | 6 | |
| 8.b | even | 2 | 1 | inner | 448.4.b.c | ✓ | 6 |
| 8.d | odd | 2 | 1 | 448.4.b.d | yes | 6 | |
| 16.e | even | 4 | 1 | 1792.4.a.e | 3 | ||
| 16.e | even | 4 | 1 | 1792.4.a.h | 3 | ||
| 16.f | odd | 4 | 1 | 1792.4.a.f | 3 | ||
| 16.f | odd | 4 | 1 | 1792.4.a.g | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 448.4.b.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 448.4.b.c | ✓ | 6 | 8.b | even | 2 | 1 | inner |
| 448.4.b.d | yes | 6 | 4.b | odd | 2 | 1 | |
| 448.4.b.d | yes | 6 | 8.d | odd | 2 | 1 | |
| 1792.4.a.e | 3 | 16.e | even | 4 | 1 | ||
| 1792.4.a.f | 3 | 16.f | odd | 4 | 1 | ||
| 1792.4.a.g | 3 | 16.f | odd | 4 | 1 | ||
| 1792.4.a.h | 3 | 16.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(448, [\chi])\):
|
\( T_{3}^{6} + 60T_{3}^{4} + 516T_{3}^{2} + 16 \)
|
|
\( T_{23}^{3} + 224T_{23}^{2} - 9996T_{23} - 3161088 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + 60 T^{4} + \cdots + 16 \)
|
| $5$ |
\( T^{6} + 456 T^{4} + \cdots + 153664 \)
|
| $7$ |
\( (T + 7)^{6} \)
|
| $11$ |
\( T^{6} + 3204 T^{4} + \cdots + 434639104 \)
|
| $13$ |
\( T^{6} + \cdots + 8631153216 \)
|
| $17$ |
\( (T^{3} - 22 T^{2} + \cdots - 13128)^{2} \)
|
| $19$ |
\( T^{6} + 1708 T^{4} + \cdots + 3549456 \)
|
| $23$ |
\( (T^{3} + 224 T^{2} + \cdots - 3161088)^{2} \)
|
| $29$ |
\( T^{6} + \cdots + 6500296986624 \)
|
| $31$ |
\( (T^{3} + 304 T^{2} + \cdots - 1423744)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 7122151837696 \)
|
| $41$ |
\( (T^{3} + 154 T^{2} + \cdots - 23491368)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 3035790491904 \)
|
| $47$ |
\( (T^{3} + 688 T^{2} + \cdots - 51518208)^{2} \)
|
| $53$ |
\( T^{6} + \cdots + 14\!\cdots\!36 \)
|
| $59$ |
\( T^{6} + \cdots + 182960142428176 \)
|
| $61$ |
\( T^{6} + \cdots + 23\!\cdots\!24 \)
|
| $67$ |
\( T^{6} + \cdots + 93\!\cdots\!64 \)
|
| $71$ |
\( (T^{3} + 832 T^{2} + \cdots - 692014848)^{2} \)
|
| $73$ |
\( (T^{3} - 342 T^{2} + \cdots + 1439352)^{2} \)
|
| $79$ |
\( (T^{3} + 1216 T^{2} + \cdots - 482780928)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 46\!\cdots\!76 \)
|
| $89$ |
\( (T^{3} + 1354 T^{2} + \cdots - 152926056)^{2} \)
|
| $97$ |
\( (T^{3} - 150 T^{2} + \cdots - 27498344)^{2} \)
|
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