Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [32,8,Mod(17,32)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(32, 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("32.17");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 32 = 2^{5} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 32.b (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: | \(9.99632081549\) |
| 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: | \( 2^{30} \) |
| Twist minimal: | no (minimal twist has level 8) |
| 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
\( x^{6} - 3x^{5} - 10x^{4} - 24x^{3} - 320x^{2} - 3072x + 32768 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{4} + 3\nu^{3} + 42\nu^{2} - 72\nu + 106 ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -3\nu^{5} - 7\nu^{4} - 50\nu^{3} + 104\nu^{2} + 1088\nu + 15872 ) / 256 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -11\nu^{5} - 47\nu^{4} + 222\nu^{3} - 1112\nu^{2} + 3136\nu + 60928 ) / 256 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} + 16\nu^{3} + 108\nu^{2} + 1200\nu + 2260 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 93\nu^{5} - 487\nu^{4} - 1458\nu^{3} - 3352\nu^{2} + 101952\nu - 205312 ) / 256 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} + 2\beta_{4} - \beta_{3} - 8\beta_{2} - 2\beta _1 + 512 ) / 1024 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{5} + 6\beta_{4} - 63\beta_{3} + 72\beta_{2} + 26\beta _1 + 4960 ) / 1024 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -9\beta_{5} + 38\beta_{4} + 329\beta_{3} - 2296\beta_{2} + 58\beta _1 + 32288 ) / 1024 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -141\beta_{5} + 222\beta_{4} - 1587\beta_{3} - 3288\beta_{2} - 638\beta _1 + 376864 ) / 1024 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 807\beta_{5} - 218\beta_{4} - 4327\beta_{3} - 41848\beta_{2} + 698\beta _1 + 4357792 ) / 1024 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(31\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 17.1 |
|
0 | − | 76.9497i | 0 | − | 338.443i | 0 | 438.996 | 0 | −3734.25 | 0 | ||||||||||||||||||||||||||||||||||
| 17.2 | 0 | − | 40.2163i | 0 | 324.492i | 0 | 956.960 | 0 | 569.651 | 0 | ||||||||||||||||||||||||||||||||||||
| 17.3 | 0 | − | 21.9408i | 0 | 184.916i | 0 | −1051.96 | 0 | 1705.60 | 0 | ||||||||||||||||||||||||||||||||||||
| 17.4 | 0 | 21.9408i | 0 | − | 184.916i | 0 | −1051.96 | 0 | 1705.60 | 0 | ||||||||||||||||||||||||||||||||||||
| 17.5 | 0 | 40.2163i | 0 | − | 324.492i | 0 | 956.960 | 0 | 569.651 | 0 | ||||||||||||||||||||||||||||||||||||
| 17.6 | 0 | 76.9497i | 0 | 338.443i | 0 | 438.996 | 0 | −3734.25 | 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 | 32.8.b.a | 6 | |
| 3.b | odd | 2 | 1 | 288.8.d.b | 6 | ||
| 4.b | odd | 2 | 1 | 8.8.b.a | ✓ | 6 | |
| 8.b | even | 2 | 1 | inner | 32.8.b.a | 6 | |
| 8.d | odd | 2 | 1 | 8.8.b.a | ✓ | 6 | |
| 12.b | even | 2 | 1 | 72.8.d.b | 6 | ||
| 16.e | even | 4 | 2 | 256.8.a.q | 6 | ||
| 16.f | odd | 4 | 2 | 256.8.a.r | 6 | ||
| 24.f | even | 2 | 1 | 72.8.d.b | 6 | ||
| 24.h | odd | 2 | 1 | 288.8.d.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8.8.b.a | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 8.8.b.a | ✓ | 6 | 8.d | odd | 2 | 1 | |
| 32.8.b.a | 6 | 1.a | even | 1 | 1 | trivial | |
| 32.8.b.a | 6 | 8.b | even | 2 | 1 | inner | |
| 72.8.d.b | 6 | 12.b | even | 2 | 1 | ||
| 72.8.d.b | 6 | 24.f | even | 2 | 1 | ||
| 256.8.a.q | 6 | 16.e | even | 4 | 2 | ||
| 256.8.a.r | 6 | 16.f | odd | 4 | 2 | ||
| 288.8.d.b | 6 | 3.b | odd | 2 | 1 | ||
| 288.8.d.b | 6 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(32, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + \cdots + 4610229696 \)
|
| $5$ |
\( T^{6} + \cdots + 412405245440000 \)
|
| $7$ |
\( (T^{3} - 344 T^{2} + \cdots + 441929216)^{2} \)
|
| $11$ |
\( T^{6} + \cdots + 21\!\cdots\!00 \)
|
| $13$ |
\( T^{6} + \cdots + 24\!\cdots\!00 \)
|
| $17$ |
\( (T^{3} + \cdots - 9112197964104)^{2} \)
|
| $19$ |
\( T^{6} + \cdots + 47\!\cdots\!04 \)
|
| $23$ |
\( (T^{3} + \cdots - 2134822184448)^{2} \)
|
| $29$ |
\( T^{6} + \cdots + 42\!\cdots\!00 \)
|
| $31$ |
\( (T^{3} + \cdots - 18\!\cdots\!28)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 61\!\cdots\!64 \)
|
| $41$ |
\( (T^{3} + \cdots + 17\!\cdots\!00)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 77\!\cdots\!56 \)
|
| $47$ |
\( (T^{3} + \cdots - 15\!\cdots\!96)^{2} \)
|
| $53$ |
\( T^{6} + \cdots + 12\!\cdots\!36 \)
|
| $59$ |
\( T^{6} + \cdots + 55\!\cdots\!04 \)
|
| $61$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots + 75\!\cdots\!24 \)
|
| $71$ |
\( (T^{3} + \cdots - 38\!\cdots\!92)^{2} \)
|
| $73$ |
\( (T^{3} + \cdots + 21\!\cdots\!72)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots - 49\!\cdots\!40)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 63\!\cdots\!16 \)
|
| $89$ |
\( (T^{3} + \cdots + 11\!\cdots\!20)^{2} \)
|
| $97$ |
\( (T^{3} + \cdots - 16\!\cdots\!24)^{2} \)
|
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