Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [32,10,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, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("32.17");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 32 = 2^{5} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| 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: | \(16.4811467572\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} + 59x^{6} - 313x^{5} - 315x^{4} - 92091x^{3} + 1261649x^{2} - 16074123x + 251007534 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{56}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - x^{7} + 59x^{6} - 313x^{5} - 315x^{4} - 92091x^{3} + 1261649x^{2} - 16074123x + 251007534 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{7} + 318 \nu^{6} - 2947 \nu^{5} - 70006 \nu^{4} - 49989 \nu^{3} + 742730 \nu^{2} + \cdots + 529423278 ) / 8912896 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 109 \nu^{7} - 4198 \nu^{6} - 48697 \nu^{5} - 446658 \nu^{4} - 6780319 \nu^{3} + \cdots + 235329770 ) / 8912896 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 130\nu^{6} - 1603\nu^{5} - 374\nu^{4} + 179003\nu^{3} + 1404810\nu^{2} + 15090631\nu - 50099410 ) / 65536 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 33 \nu^{7} + 318 \nu^{6} - 1187 \nu^{5} + 20426 \nu^{4} - 43365 \nu^{3} - 9678134 \nu^{2} + \cdots - 110482194 ) / 65536 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 12165 \nu^{7} - 73526 \nu^{6} - 45041 \nu^{5} - 1786802 \nu^{4} + 113520089 \nu^{3} + \cdots + 25150692506 ) / 8912896 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2943 \nu^{7} - 28094 \nu^{6} + 485763 \nu^{5} - 6462346 \nu^{4} + 71774917 \nu^{3} + \cdots - 95510259630 ) / 4456448 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 257 \nu^{7} + 642 \nu^{6} - 14013 \nu^{5} + 96118 \nu^{4} - 194107 \nu^{3} + 22185590 \nu^{2} + \cdots + 3840558802 ) / 65536 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + 2\beta_{6} - 2\beta_{5} + \beta_{3} + 10\beta_{2} - 196\beta _1 + 2048 ) / 16384 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{7} - 10\beta_{6} - 22\beta_{5} - 64\beta_{4} + 63\beta_{3} - 306\beta_{2} + 1588\beta _1 - 239616 ) / 16384 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 13 \beta_{7} + 86 \beta_{6} + 234 \beta_{5} + 128 \beta_{4} + 1907 \beta_{3} - 9298 \beta_{2} + \cdots + 1562624 ) / 16384 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 587 \beta_{7} - 1778 \beta_{6} - 430 \beta_{5} + 5312 \beta_{4} + 5515 \beta_{3} - 22458 \beta_{2} + \cdots + 18921472 ) / 16384 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 7493 \beta_{7} + 18886 \beta_{6} - 134 \beta_{5} - 85376 \beta_{4} - 282565 \beta_{3} + \cdots + 795383808 ) / 16384 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 158475 \beta_{7} + 5166 \beta_{6} - 175822 \beta_{5} + 540352 \beta_{4} - 1221557 \beta_{3} + \cdots - 15221901312 ) / 16384 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 2548773 \beta_{7} - 1246554 \beta_{6} - 4032102 \beta_{5} + 2370176 \beta_{4} + \cdots + 150053394432 ) / 16384 \)
|
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 | − | 247.414i | 0 | 1417.55i | 0 | −5087.57 | 0 | −41530.8 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 17.2 | 0 | − | 150.106i | 0 | − | 292.339i | 0 | 9955.46 | 0 | −2848.66 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 17.3 | 0 | − | 100.481i | 0 | − | 2583.09i | 0 | −6967.65 | 0 | 9586.48 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 17.4 | 0 | − | 67.6316i | 0 | 506.862i | 0 | −300.249 | 0 | 15109.0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 17.5 | 0 | 67.6316i | 0 | − | 506.862i | 0 | −300.249 | 0 | 15109.0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 17.6 | 0 | 100.481i | 0 | 2583.09i | 0 | −6967.65 | 0 | 9586.48 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 17.7 | 0 | 150.106i | 0 | 292.339i | 0 | 9955.46 | 0 | −2848.66 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 17.8 | 0 | 247.414i | 0 | − | 1417.55i | 0 | −5087.57 | 0 | −41530.8 | 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 | 32.10.b.a | 8 | |
| 3.b | odd | 2 | 1 | 288.10.d.b | 8 | ||
| 4.b | odd | 2 | 1 | 8.10.b.a | ✓ | 8 | |
| 8.b | even | 2 | 1 | inner | 32.10.b.a | 8 | |
| 8.d | odd | 2 | 1 | 8.10.b.a | ✓ | 8 | |
| 12.b | even | 2 | 1 | 72.10.d.b | 8 | ||
| 16.e | even | 4 | 2 | 256.10.a.s | 8 | ||
| 16.f | odd | 4 | 2 | 256.10.a.p | 8 | ||
| 24.f | even | 2 | 1 | 72.10.d.b | 8 | ||
| 24.h | odd | 2 | 1 | 288.10.d.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8.10.b.a | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 8.10.b.a | ✓ | 8 | 8.d | odd | 2 | 1 | |
| 32.10.b.a | 8 | 1.a | even | 1 | 1 | trivial | |
| 32.10.b.a | 8 | 8.b | even | 2 | 1 | inner | |
| 72.10.d.b | 8 | 12.b | even | 2 | 1 | ||
| 72.10.d.b | 8 | 24.f | even | 2 | 1 | ||
| 256.10.a.p | 8 | 16.f | odd | 4 | 2 | ||
| 256.10.a.s | 8 | 16.e | even | 4 | 2 | ||
| 288.10.d.b | 8 | 3.b | odd | 2 | 1 | ||
| 288.10.d.b | 8 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(32, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + \cdots + 63\!\cdots\!88 \)
|
| $5$ |
\( T^{8} + \cdots + 29\!\cdots\!00 \)
|
| $7$ |
\( (T^{4} + \cdots - 105959154151424)^{2} \)
|
| $11$ |
\( T^{8} + \cdots + 18\!\cdots\!00 \)
|
| $13$ |
\( T^{8} + \cdots + 18\!\cdots\!00 \)
|
| $17$ |
\( (T^{4} + \cdots + 49\!\cdots\!48)^{2} \)
|
| $19$ |
\( T^{8} + \cdots + 48\!\cdots\!72 \)
|
| $23$ |
\( (T^{4} + \cdots - 42\!\cdots\!76)^{2} \)
|
| $29$ |
\( T^{8} + \cdots + 46\!\cdots\!00 \)
|
| $31$ |
\( (T^{4} + \cdots + 67\!\cdots\!56)^{2} \)
|
| $37$ |
\( T^{8} + \cdots + 17\!\cdots\!48 \)
|
| $41$ |
\( (T^{4} + \cdots + 74\!\cdots\!80)^{2} \)
|
| $43$ |
\( T^{8} + \cdots + 94\!\cdots\!08 \)
|
| $47$ |
\( (T^{4} + \cdots + 54\!\cdots\!12)^{2} \)
|
| $53$ |
\( T^{8} + \cdots + 11\!\cdots\!88 \)
|
| $59$ |
\( T^{8} + \cdots + 77\!\cdots\!08 \)
|
| $61$ |
\( T^{8} + \cdots + 10\!\cdots\!00 \)
|
| $67$ |
\( T^{8} + \cdots + 13\!\cdots\!68 \)
|
| $71$ |
\( (T^{4} + \cdots + 60\!\cdots\!52)^{2} \)
|
| $73$ |
\( (T^{4} + \cdots - 27\!\cdots\!12)^{2} \)
|
| $79$ |
\( (T^{4} + \cdots + 14\!\cdots\!40)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 37\!\cdots\!68 \)
|
| $89$ |
\( (T^{4} + \cdots + 17\!\cdots\!20)^{2} \)
|
| $97$ |
\( (T^{4} + \cdots + 41\!\cdots\!72)^{2} \)
|
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