Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [32,9,Mod(15,32)]
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([1, 1]))
N = Newforms(chi, 9, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("32.15");
S:= CuspForms(chi, 9);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 32 = 2^{5} \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 32.d (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: | \(13.0361155220\) |
| 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: |
\( x^{6} - x^{5} + 78x^{4} - 514x^{3} + 4237x^{2} - 18333x + 238980 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{33} \) |
| 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
\( x^{6} - x^{5} + 78x^{4} - 514x^{3} + 4237x^{2} - 18333x + 238980 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{5} - 34\nu^{4} + 176\nu^{3} - 1202\nu^{2} + 12159\nu - 72444 ) / 1024 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -7\nu^{5} + 46\nu^{4} + 368\nu^{3} + 4766\nu^{2} - 5305\nu - 46620 ) / 512 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 29\nu^{5} + 38\nu^{4} + 2032\nu^{3} - 15402\nu^{2} - 12957\nu - 349324 ) / 512 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 31\nu^{5} + 162\nu^{4} + 784\nu^{3} + 18610\nu^{2} - 35679\nu + 900732 ) / 512 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 19\nu^{5} - 6\nu^{4} + 1424\nu^{3} - 6582\nu^{2} + 137709\nu - 198324 ) / 32 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - 2\beta_{4} - 8\beta_{3} - 2\beta_{2} - 48\beta _1 + 680 ) / 4096 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{5} + 66\beta_{4} - 56\beta_{3} + 66\beta_{2} + 688\beta _1 - 105832 ) / 4096 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -11\beta_{5} + 22\beta_{4} + 472\beta_{3} + 2070\beta_{2} + 6928\beta _1 + 893768 ) / 4096 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 343\beta_{5} - 1902\beta_{4} + 1672\beta_{3} + 4242\beta_{2} - 128208\beta _1 - 2071336 ) / 4096 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 237\beta_{5} + 35110\beta_{4} + 3736\beta_{3} - 116442\beta_{2} + 26512\beta _1 - 66476152 ) / 4096 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 15.1 |
|
0 | −117.102 | 0 | − | 816.841i | 0 | − | 1333.52i | 0 | 7151.84 | 0 | ||||||||||||||||||||||||||||||||||
| 15.2 | 0 | −117.102 | 0 | 816.841i | 0 | 1333.52i | 0 | 7151.84 | 0 | |||||||||||||||||||||||||||||||||||||
| 15.3 | 0 | 17.4864 | 0 | − | 796.785i | 0 | 1779.55i | 0 | −6255.23 | 0 | ||||||||||||||||||||||||||||||||||||
| 15.4 | 0 | 17.4864 | 0 | 796.785i | 0 | − | 1779.55i | 0 | −6255.23 | 0 | ||||||||||||||||||||||||||||||||||||
| 15.5 | 0 | 117.615 | 0 | − | 306.178i | 0 | − | 4321.70i | 0 | 7272.39 | 0 | |||||||||||||||||||||||||||||||||||
| 15.6 | 0 | 117.615 | 0 | 306.178i | 0 | 4321.70i | 0 | 7272.39 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 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 | 32.9.d.b | 6 | |
| 3.b | odd | 2 | 1 | 288.9.b.b | 6 | ||
| 4.b | odd | 2 | 1 | 8.9.d.b | ✓ | 6 | |
| 8.b | even | 2 | 1 | 8.9.d.b | ✓ | 6 | |
| 8.d | odd | 2 | 1 | inner | 32.9.d.b | 6 | |
| 12.b | even | 2 | 1 | 72.9.b.b | 6 | ||
| 16.e | even | 4 | 2 | 256.9.c.n | 12 | ||
| 16.f | odd | 4 | 2 | 256.9.c.n | 12 | ||
| 24.f | even | 2 | 1 | 288.9.b.b | 6 | ||
| 24.h | odd | 2 | 1 | 72.9.b.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8.9.d.b | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 8.9.d.b | ✓ | 6 | 8.b | even | 2 | 1 | |
| 32.9.d.b | 6 | 1.a | even | 1 | 1 | trivial | |
| 32.9.d.b | 6 | 8.d | odd | 2 | 1 | inner | |
| 72.9.b.b | 6 | 12.b | even | 2 | 1 | ||
| 72.9.b.b | 6 | 24.h | odd | 2 | 1 | ||
| 256.9.c.n | 12 | 16.e | even | 4 | 2 | ||
| 256.9.c.n | 12 | 16.f | odd | 4 | 2 | ||
| 288.9.b.b | 6 | 3.b | odd | 2 | 1 | ||
| 288.9.b.b | 6 | 24.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 18T_{3}^{2} - 13764T_{3} + 240840 \)
acting on \(S_{9}^{\mathrm{new}}(32, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( (T^{3} - 18 T^{2} + \cdots + 240840)^{2} \)
|
| $5$ |
\( T^{6} + \cdots + 39\!\cdots\!00 \)
|
| $7$ |
\( T^{6} + \cdots + 10\!\cdots\!40 \)
|
| $11$ |
\( (T^{3} + 23470 T^{2} + \cdots + 155265768392)^{2} \)
|
| $13$ |
\( T^{6} + \cdots + 54\!\cdots\!40 \)
|
| $17$ |
\( (T^{3} + \cdots - 13136750437640)^{2} \)
|
| $19$ |
\( (T^{3} + \cdots + 34\!\cdots\!08)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots + 21\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots + 11\!\cdots\!40 \)
|
| $41$ |
\( (T^{3} + \cdots + 26\!\cdots\!28)^{2} \)
|
| $43$ |
\( (T^{3} + \cdots - 30\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 35\!\cdots\!40 \)
|
| $53$ |
\( T^{6} + \cdots + 66\!\cdots\!40 \)
|
| $59$ |
\( (T^{3} + \cdots - 70\!\cdots\!48)^{2} \)
|
| $61$ |
\( T^{6} + \cdots + 12\!\cdots\!00 \)
|
| $67$ |
\( (T^{3} + \cdots - 12\!\cdots\!60)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $73$ |
\( (T^{3} + \cdots - 13\!\cdots\!20)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 16\!\cdots\!00 \)
|
| $83$ |
\( (T^{3} + \cdots - 15\!\cdots\!40)^{2} \)
|
| $89$ |
\( (T^{3} + \cdots + 46\!\cdots\!72)^{2} \)
|
| $97$ |
\( (T^{3} + \cdots + 47\!\cdots\!60)^{2} \)
|
| show more | |
| show less |