Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [448,7,Mod(321,448)]
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, 0, 1]))
N = Newforms(chi, 7, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("448.321");
S:= CuspForms(chi, 7);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 448 = 2^{6} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 7 \) |
| Character orbit: | \([\chi]\) | \(=\) | 448.c (of order \(2\), degree \(1\), 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: | \(103.064229462\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.211968.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 30x^{2} + 207 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 14) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 30x^{2} + 207 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{3} + 18\nu ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{3} + 34\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( 4\nu^{2} + 60 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 3\beta_1 ) / 16 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 60 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -9\beta_{2} + 51\beta_1 ) / 16 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 321.1 |
|
0 | − | 29.9539i | 0 | − | 98.3767i | 0 | 195.794 | − | 281.627i | 0 | −168.235 | 0 | ||||||||||||||||||||||||||
| 321.2 | 0 | − | 3.84257i | 0 | 204.749i | 0 | −41.7939 | − | 340.444i | 0 | 714.235 | 0 | ||||||||||||||||||||||||||||
| 321.3 | 0 | 3.84257i | 0 | − | 204.749i | 0 | −41.7939 | + | 340.444i | 0 | 714.235 | 0 | ||||||||||||||||||||||||||||
| 321.4 | 0 | 29.9539i | 0 | 98.3767i | 0 | 195.794 | + | 281.627i | 0 | −168.235 | 0 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 448.7.c.h | 4 | |
| 4.b | odd | 2 | 1 | 448.7.c.e | 4 | ||
| 7.b | odd | 2 | 1 | inner | 448.7.c.h | 4 | |
| 8.b | even | 2 | 1 | 14.7.b.a | ✓ | 4 | |
| 8.d | odd | 2 | 1 | 112.7.c.c | 4 | ||
| 24.h | odd | 2 | 1 | 126.7.c.a | 4 | ||
| 28.d | even | 2 | 1 | 448.7.c.e | 4 | ||
| 40.f | even | 2 | 1 | 350.7.b.a | 4 | ||
| 40.i | odd | 4 | 2 | 350.7.d.a | 8 | ||
| 56.e | even | 2 | 1 | 112.7.c.c | 4 | ||
| 56.h | odd | 2 | 1 | 14.7.b.a | ✓ | 4 | |
| 56.j | odd | 6 | 2 | 98.7.d.b | 8 | ||
| 56.p | even | 6 | 2 | 98.7.d.b | 8 | ||
| 168.i | even | 2 | 1 | 126.7.c.a | 4 | ||
| 280.c | odd | 2 | 1 | 350.7.b.a | 4 | ||
| 280.s | even | 4 | 2 | 350.7.d.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.7.b.a | ✓ | 4 | 8.b | even | 2 | 1 | |
| 14.7.b.a | ✓ | 4 | 56.h | odd | 2 | 1 | |
| 98.7.d.b | 8 | 56.j | odd | 6 | 2 | ||
| 98.7.d.b | 8 | 56.p | even | 6 | 2 | ||
| 112.7.c.c | 4 | 8.d | odd | 2 | 1 | ||
| 112.7.c.c | 4 | 56.e | even | 2 | 1 | ||
| 126.7.c.a | 4 | 24.h | odd | 2 | 1 | ||
| 126.7.c.a | 4 | 168.i | even | 2 | 1 | ||
| 350.7.b.a | 4 | 40.f | even | 2 | 1 | ||
| 350.7.b.a | 4 | 280.c | odd | 2 | 1 | ||
| 350.7.d.a | 8 | 40.i | odd | 4 | 2 | ||
| 350.7.d.a | 8 | 280.s | even | 4 | 2 | ||
| 448.7.c.e | 4 | 4.b | odd | 2 | 1 | ||
| 448.7.c.e | 4 | 28.d | even | 2 | 1 | ||
| 448.7.c.h | 4 | 1.a | even | 1 | 1 | trivial | |
| 448.7.c.h | 4 | 7.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(448, [\chi])\):
|
\( T_{3}^{4} + 912T_{3}^{2} + 13248 \)
|
|
\( T_{11}^{2} - 2220T_{11} + 1227492 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 912 T^{2} + 13248 \)
|
| $5$ |
\( T^{4} + 51600 T^{2} + 405720000 \)
|
| $7$ |
\( T^{4} + \cdots + 13841287201 \)
|
| $11$ |
\( (T^{2} - 2220 T + 1227492)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 26266196601792 \)
|
| $17$ |
\( T^{4} + \cdots + 126735731638272 \)
|
| $19$ |
\( T^{4} + \cdots + 551809934313408 \)
|
| $23$ |
\( (T^{2} - 20292 T + 100711044)^{2} \)
|
| $29$ |
\( (T^{2} - 9132 T - 1357906716)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 86\!\cdots\!12 \)
|
| $37$ |
\( (T^{2} - 11596 T - 1847926364)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 28\!\cdots\!32 \)
|
| $43$ |
\( (T^{2} - 22348 T - 10521464924)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 12\!\cdots\!88 \)
|
| $53$ |
\( (T^{2} + 124308 T + 13614948)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 78\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 82\!\cdots\!48 \)
|
| $67$ |
\( (T^{2} - 217388 T + 11809058788)^{2} \)
|
| $71$ |
\( (T^{2} + 225804 T - 95443772508)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 26\!\cdots\!72 \)
|
| $79$ |
\( (T^{2} - 1046452 T + 268612291876)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 21\!\cdots\!08 \)
|
| $89$ |
\( T^{4} + \cdots + 15\!\cdots\!52 \)
|
| $97$ |
\( T^{4} + \cdots + 39\!\cdots\!12 \)
|
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