Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [128,5,Mod(63,128)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(128, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 5, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("128.63");
S:= CuspForms(chi, 5);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 128 = 2^{7} \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 128.d (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: | \(13.2313552747\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.1871773696.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 31x^{4} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{38} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 31x^{4} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -4\nu^{7} - 18\nu^{5} - 286\nu^{3} + 414\nu ) / 189 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -44\nu^{6} - 752\nu^{2} ) / 63 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 80\nu^{7} + 72\nu^{5} + 2696\nu^{3} + 4392\nu ) / 189 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 12\nu^{6} + 368\nu^{2} ) / 7 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -272\nu^{7} + 504\nu^{5} - 7352\nu^{3} + 12600\nu ) / 189 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 64\nu^{4} + 992 ) / 7 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -1024\nu^{7} - 2304\nu^{5} - 24832\nu^{3} - 43776\nu ) / 189 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + 4\beta_{5} + 32\beta_{3} + 112\beta_1 ) / 1024 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 11\beta_{4} + 27\beta_{2} ) / 256 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 5\beta_{7} + 4\beta_{5} + 64\beta_{3} - 272\beta_1 ) / 512 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 7\beta_{6} - 992 ) / 64 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -61\beta_{7} + 92\beta_{5} - 608\beta_{3} - 2800\beta_1 ) / 1024 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -47\beta_{4} - 207\beta_{2} ) / 64 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -337\beta_{7} - 572\beta_{5} - 3104\beta_{3} + 14704\beta_1 ) / 1024 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(127\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 63.1 |
|
0 | −15.8549 | 0 | − | 40.8444i | 0 | − | 40.7922i | 0 | 170.378 | 0 | ||||||||||||||||||||||||||||||||||||||||
| 63.2 | 0 | −15.8549 | 0 | 40.8444i | 0 | 40.7922i | 0 | 170.378 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 63.3 | 0 | −4.54118 | 0 | − | 16.8444i | 0 | 40.7922i | 0 | −60.3776 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 63.4 | 0 | −4.54118 | 0 | 16.8444i | 0 | − | 40.7922i | 0 | −60.3776 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 63.5 | 0 | 4.54118 | 0 | − | 16.8444i | 0 | − | 40.7922i | 0 | −60.3776 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 63.6 | 0 | 4.54118 | 0 | 16.8444i | 0 | 40.7922i | 0 | −60.3776 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 63.7 | 0 | 15.8549 | 0 | − | 40.8444i | 0 | 40.7922i | 0 | 170.378 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 63.8 | 0 | 15.8549 | 0 | 40.8444i | 0 | − | 40.7922i | 0 | 170.378 | 0 | ||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 8.b | even | 2 | 1 | inner |
| 8.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 128.5.d.d | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1152.5.b.l | 8 | ||
| 4.b | odd | 2 | 1 | inner | 128.5.d.d | ✓ | 8 |
| 8.b | even | 2 | 1 | inner | 128.5.d.d | ✓ | 8 |
| 8.d | odd | 2 | 1 | inner | 128.5.d.d | ✓ | 8 |
| 12.b | even | 2 | 1 | 1152.5.b.l | 8 | ||
| 16.e | even | 4 | 1 | 256.5.c.g | 4 | ||
| 16.e | even | 4 | 1 | 256.5.c.k | 4 | ||
| 16.f | odd | 4 | 1 | 256.5.c.g | 4 | ||
| 16.f | odd | 4 | 1 | 256.5.c.k | 4 | ||
| 24.f | even | 2 | 1 | 1152.5.b.l | 8 | ||
| 24.h | odd | 2 | 1 | 1152.5.b.l | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 128.5.d.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 128.5.d.d | ✓ | 8 | 4.b | odd | 2 | 1 | inner |
| 128.5.d.d | ✓ | 8 | 8.b | even | 2 | 1 | inner |
| 128.5.d.d | ✓ | 8 | 8.d | odd | 2 | 1 | inner |
| 256.5.c.g | 4 | 16.e | even | 4 | 1 | ||
| 256.5.c.g | 4 | 16.f | odd | 4 | 1 | ||
| 256.5.c.k | 4 | 16.e | even | 4 | 1 | ||
| 256.5.c.k | 4 | 16.f | odd | 4 | 1 | ||
| 1152.5.b.l | 8 | 3.b | odd | 2 | 1 | ||
| 1152.5.b.l | 8 | 12.b | even | 2 | 1 | ||
| 1152.5.b.l | 8 | 24.f | even | 2 | 1 | ||
| 1152.5.b.l | 8 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 272T_{3}^{2} + 5184 \)
acting on \(S_{5}^{\mathrm{new}}(128, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( (T^{4} - 272 T^{2} + 5184)^{2} \)
|
| $5$ |
\( (T^{4} + 1952 T^{2} + 473344)^{2} \)
|
| $7$ |
\( (T^{2} + 1664)^{4} \)
|
| $11$ |
\( (T^{4} - 12688 T^{2} + 19998784)^{2} \)
|
| $13$ |
\( (T^{4} + 34976 T^{2} + 6310144)^{2} \)
|
| $17$ |
\( (T^{2} + 84 T - 11548)^{4} \)
|
| $19$ |
\( (T^{4} - 329872 T^{2} + 19535093824)^{2} \)
|
| $23$ |
\( (T^{4} + 1010944 T^{2} + 59208515584)^{2} \)
|
| $29$ |
\( (T^{4} + 1752224 T^{2} + 273855449344)^{2} \)
|
| $31$ |
\( (T^{2} + 1179648)^{4} \)
|
| $37$ |
\( (T^{4} + \cdots + 2103056836864)^{2} \)
|
| $41$ |
\( (T^{2} - 996 T - 231228)^{4} \)
|
| $43$ |
\( (T^{4} + \cdots + 16563532508224)^{2} \)
|
| $47$ |
\( (T^{4} + \cdots + 3082299899904)^{2} \)
|
| $53$ |
\( (T^{4} + \cdots + 136280781732096)^{2} \)
|
| $59$ |
\( (T^{4} + \cdots + 896505638423616)^{2} \)
|
| $61$ |
\( (T^{4} + \cdots + 68799991019776)^{2} \)
|
| $67$ |
\( (T^{4} + \cdots + 1240942528576)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots + 897452045910016)^{2} \)
|
| $73$ |
\( (T^{2} + 4916 T - 46793564)^{4} \)
|
| $79$ |
\( (T^{2} + 29878784)^{4} \)
|
| $83$ |
\( (T^{4} + \cdots + 17\!\cdots\!64)^{2} \)
|
| $89$ |
\( (T^{2} + 6708 T + 4207268)^{4} \)
|
| $97$ |
\( (T^{2} + 5460 T - 93305628)^{4} \)
|
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