Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [8,8,Mod(5,8)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(8, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("8.5");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 8 = 2^{3} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 8.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: | \(2.49908020387\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} - 10x^{4} - 24x^{3} - 320x^{2} - 3072x + 32768 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{15} \) |
| 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
\( x^{6} - 3x^{5} - 10x^{4} - 24x^{3} - 320x^{2} - 3072x + 32768 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 3\nu^{4} + 10\nu^{3} + 24\nu^{2} + 320\nu + 2560 ) / 512 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 3\nu^{4} - 10\nu^{3} - 24\nu^{2} + 7872\nu - 6656 ) / 256 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -5\nu^{5} - 49\nu^{4} + 242\nu^{3} + 760\nu^{2} + 3136\nu + 26624 ) / 512 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 3\nu^{5} + 7\nu^{4} + 50\nu^{3} - 104\nu^{2} - 1088\nu - 15872 ) / 256 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 15\nu^{5} + 51\nu^{4} - 182\nu^{3} + 3032\nu^{2} - 2496\nu - 90112 ) / 512 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta _1 + 16 ) / 32 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{5} - 4\beta_{4} + 4\beta_{3} - \beta_{2} + 14\beta _1 + 152 ) / 32 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -4\beta_{5} + 68\beta_{4} + 28\beta_{3} - \beta_{2} + 206\beta _1 + 1000 ) / 32 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 28\beta_{5} + 164\beta_{4} - 132\beta_{3} + 11\beta_{2} + 2086\beta _1 + 11816 ) / 32 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 140\beta_{5} + 1076\beta_{4} - 20\beta_{3} + 319\beta_{2} - 7090\beta _1 + 136136 ) / 32 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(7\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 5.1 |
|
−9.70536 | − | 5.81430i | 40.2163i | 60.3879 | + | 112.860i | 324.492i | 233.829 | − | 390.313i | −956.960 | 70.1132 | − | 1446.46i | 569.651 | 1886.69 | − | 3149.31i | ||||||||||||||||||||||||||
| 5.2 | −9.70536 | + | 5.81430i | − | 40.2163i | 60.3879 | − | 112.860i | − | 324.492i | 233.829 | + | 390.313i | −956.960 | 70.1132 | + | 1446.46i | 569.651 | 1886.69 | + | 3149.31i | |||||||||||||||||||||||||
| 5.3 | 1.55200 | − | 11.2068i | − | 21.9408i | −123.183 | − | 34.7858i | − | 184.916i | −245.885 | − | 34.0522i | 1051.96 | −581.015 | + | 1326.49i | 1705.60 | −2072.30 | − | 286.989i | |||||||||||||||||||||||||
| 5.4 | 1.55200 | + | 11.2068i | 21.9408i | −123.183 | + | 34.7858i | 184.916i | −245.885 | + | 34.0522i | 1051.96 | −581.015 | − | 1326.49i | 1705.60 | −2072.30 | + | 286.989i | |||||||||||||||||||||||||||
| 5.5 | 11.1534 | − | 1.89807i | 76.9497i | 120.795 | − | 42.3397i | − | 338.443i | 146.056 | + | 858.247i | −438.996 | 1266.90 | − | 701.506i | −3734.25 | −642.387 | − | 3774.77i | ||||||||||||||||||||||||||
| 5.6 | 11.1534 | + | 1.89807i | − | 76.9497i | 120.795 | + | 42.3397i | 338.443i | 146.056 | − | 858.247i | −438.996 | 1266.90 | + | 701.506i | −3734.25 | −642.387 | + | 3774.77i | ||||||||||||||||||||||||||
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 | 8.8.b.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 72.8.d.b | 6 | ||
| 4.b | odd | 2 | 1 | 32.8.b.a | 6 | ||
| 8.b | even | 2 | 1 | inner | 8.8.b.a | ✓ | 6 |
| 8.d | odd | 2 | 1 | 32.8.b.a | 6 | ||
| 12.b | even | 2 | 1 | 288.8.d.b | 6 | ||
| 16.e | even | 4 | 2 | 256.8.a.r | 6 | ||
| 16.f | odd | 4 | 2 | 256.8.a.q | 6 | ||
| 24.f | even | 2 | 1 | 288.8.d.b | 6 | ||
| 24.h | odd | 2 | 1 | 72.8.d.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8.8.b.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 8.8.b.a | ✓ | 6 | 8.b | even | 2 | 1 | inner |
| 32.8.b.a | 6 | 4.b | odd | 2 | 1 | ||
| 32.8.b.a | 6 | 8.d | odd | 2 | 1 | ||
| 72.8.d.b | 6 | 3.b | odd | 2 | 1 | ||
| 72.8.d.b | 6 | 24.h | odd | 2 | 1 | ||
| 256.8.a.q | 6 | 16.f | odd | 4 | 2 | ||
| 256.8.a.r | 6 | 16.e | even | 4 | 2 | ||
| 288.8.d.b | 6 | 12.b | even | 2 | 1 | ||
| 288.8.d.b | 6 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(8, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 6 T^{5} + \cdots + 2097152 \)
|
| $3$ |
\( T^{6} + \cdots + 4610229696 \)
|
| $5$ |
\( T^{6} + \cdots + 412405245440000 \)
|
| $7$ |
\( (T^{3} + 344 T^{2} + \cdots - 441929216)^{2} \)
|
| $11$ |
\( T^{6} + \cdots + 21\!\cdots\!00 \)
|
| $13$ |
\( T^{6} + \cdots + 24\!\cdots\!00 \)
|
| $17$ |
\( (T^{3} + \cdots - 9112197964104)^{2} \)
|
| $19$ |
\( T^{6} + \cdots + 47\!\cdots\!04 \)
|
| $23$ |
\( (T^{3} + \cdots + 2134822184448)^{2} \)
|
| $29$ |
\( T^{6} + \cdots + 42\!\cdots\!00 \)
|
| $31$ |
\( (T^{3} + \cdots + 18\!\cdots\!28)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 61\!\cdots\!64 \)
|
| $41$ |
\( (T^{3} + \cdots + 17\!\cdots\!00)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 77\!\cdots\!56 \)
|
| $47$ |
\( (T^{3} + \cdots + 15\!\cdots\!96)^{2} \)
|
| $53$ |
\( T^{6} + \cdots + 12\!\cdots\!36 \)
|
| $59$ |
\( T^{6} + \cdots + 55\!\cdots\!04 \)
|
| $61$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots + 75\!\cdots\!24 \)
|
| $71$ |
\( (T^{3} + \cdots + 38\!\cdots\!92)^{2} \)
|
| $73$ |
\( (T^{3} + \cdots + 21\!\cdots\!72)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots + 49\!\cdots\!40)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 63\!\cdots\!16 \)
|
| $89$ |
\( (T^{3} + \cdots + 11\!\cdots\!20)^{2} \)
|
| $97$ |
\( (T^{3} + \cdots - 16\!\cdots\!24)^{2} \)
|
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