Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [32,11,Mod(31,32)]
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, 0]))
N = Newforms(chi, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("32.31");
S:= CuspForms(chi, 11);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 32 = 2^{5} \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 32.c (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: | \(20.3314320856\) |
| 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: |
\( x^{6} - 218x^{3} + 4356x^{2} - 14388x + 23762 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{47} \) |
| 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
\( x^{6} - 218x^{3} + 4356x^{2} - 14388x + 23762 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 4288944 \nu^{5} - 7083256 \nu^{4} + 46565236 \nu^{3} + 1428840016 \nu^{2} - 14065184680 \nu + 25779052412 ) / 90126105 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 13952\nu^{5} + 557568\nu^{4} + 920832\nu^{3} - 1520768\nu^{2} + 1351522304 ) / 275615 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 17670576 \nu^{5} - 29183224 \nu^{4} + 10066804 \nu^{3} + 2887437904 \nu^{2} + \cdots + 126024842108 ) / 90126105 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 722048\nu^{5} + 8141312\nu^{4} + 47655168\nu^{3} - 78703232\nu^{2} + 9790876831 ) / 826845 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 248375424 \nu^{5} + 410195776 \nu^{4} + 1143551264 \nu^{3} - 65526726016 \nu^{2} + \cdots - 1911459888032 ) / 90126105 \)
|
| \(\nu\) | \(=\) |
\( ( 16\beta_{5} - 12\beta_{4} + 39\beta_{3} + 52\beta_{2} + 601\beta _1 + 4 ) / 65536 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -8\beta_{5} - 169\beta_{3} + 233\beta_1 ) / 4096 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 528\beta_{5} + 396\beta_{4} + 4775\beta_{3} - 1716\beta_{2} + 16345\beta _1 + 3571580 ) / 32768 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -327\beta_{4} + 5641\beta_{2} - 23789459 ) / 8192 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -20912\beta_{5} + 13068\beta_{4} - 231259\beta_{3} - 70580\beta_{2} - 437797\beta _1 + 196439804 ) / 16384 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
0 | − | 444.947i | 0 | −68.1771 | 0 | 29341.8i | 0 | −138929. | 0 | |||||||||||||||||||||||||||||||||||
| 31.2 | 0 | − | 169.359i | 0 | 4478.89 | 0 | − | 6985.88i | 0 | 30366.5 | 0 | |||||||||||||||||||||||||||||||||||
| 31.3 | 0 | − | 63.5882i | 0 | −5268.72 | 0 | 25023.7i | 0 | 55005.5 | 0 | ||||||||||||||||||||||||||||||||||||
| 31.4 | 0 | 63.5882i | 0 | −5268.72 | 0 | − | 25023.7i | 0 | 55005.5 | 0 | ||||||||||||||||||||||||||||||||||||
| 31.5 | 0 | 169.359i | 0 | 4478.89 | 0 | 6985.88i | 0 | 30366.5 | 0 | |||||||||||||||||||||||||||||||||||||
| 31.6 | 0 | 444.947i | 0 | −68.1771 | 0 | − | 29341.8i | 0 | −138929. | 0 | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 32.11.c.b | ✓ | 6 |
| 3.b | odd | 2 | 1 | 288.11.g.b | 6 | ||
| 4.b | odd | 2 | 1 | inner | 32.11.c.b | ✓ | 6 |
| 8.b | even | 2 | 1 | 64.11.c.e | 6 | ||
| 8.d | odd | 2 | 1 | 64.11.c.e | 6 | ||
| 12.b | even | 2 | 1 | 288.11.g.b | 6 | ||
| 16.e | even | 4 | 1 | 256.11.d.d | 6 | ||
| 16.e | even | 4 | 1 | 256.11.d.e | 6 | ||
| 16.f | odd | 4 | 1 | 256.11.d.d | 6 | ||
| 16.f | odd | 4 | 1 | 256.11.d.e | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 32.11.c.b | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 32.11.c.b | ✓ | 6 | 4.b | odd | 2 | 1 | inner |
| 64.11.c.e | 6 | 8.b | even | 2 | 1 | ||
| 64.11.c.e | 6 | 8.d | odd | 2 | 1 | ||
| 256.11.d.d | 6 | 16.e | even | 4 | 1 | ||
| 256.11.d.d | 6 | 16.f | odd | 4 | 1 | ||
| 256.11.d.e | 6 | 16.e | even | 4 | 1 | ||
| 256.11.d.e | 6 | 16.f | odd | 4 | 1 | ||
| 288.11.g.b | 6 | 3.b | odd | 2 | 1 | ||
| 288.11.g.b | 6 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 230704T_{3}^{4} + 6594994944T_{3}^{2} + 22960810561536 \)
acting on \(S_{11}^{\mathrm{new}}(32, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + \cdots + 22960810561536 \)
|
| $5$ |
\( (T^{3} + 858 T^{2} + \cdots - 1608845960)^{2} \)
|
| $7$ |
\( T^{6} + \cdots + 26\!\cdots\!44 \)
|
| $11$ |
\( T^{6} + \cdots + 94\!\cdots\!00 \)
|
| $13$ |
\( (T^{3} + \cdots + 10\!\cdots\!64)^{2} \)
|
| $17$ |
\( (T^{3} + \cdots - 41846545623240)^{2} \)
|
| $19$ |
\( T^{6} + \cdots + 19\!\cdots\!24 \)
|
| $23$ |
\( T^{6} + \cdots + 24\!\cdots\!00 \)
|
| $29$ |
\( (T^{3} + \cdots + 25\!\cdots\!36)^{2} \)
|
| $31$ |
\( T^{6} + \cdots + 48\!\cdots\!04 \)
|
| $37$ |
\( (T^{3} + \cdots + 76\!\cdots\!00)^{2} \)
|
| $41$ |
\( (T^{3} + \cdots + 25\!\cdots\!20)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 37\!\cdots\!84 \)
|
| $47$ |
\( T^{6} + \cdots + 25\!\cdots\!96 \)
|
| $53$ |
\( (T^{3} + \cdots - 84\!\cdots\!64)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 16\!\cdots\!44 \)
|
| $61$ |
\( (T^{3} + \cdots - 33\!\cdots\!12)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 15\!\cdots\!00 \)
|
| $71$ |
\( T^{6} + \cdots + 89\!\cdots\!16 \)
|
| $73$ |
\( (T^{3} + \cdots - 16\!\cdots\!44)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 97\!\cdots\!76 \)
|
| $83$ |
\( T^{6} + \cdots + 17\!\cdots\!00 \)
|
| $89$ |
\( (T^{3} + \cdots + 53\!\cdots\!00)^{2} \)
|
| $97$ |
\( (T^{3} + \cdots - 44\!\cdots\!16)^{2} \)
|
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