Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [64,24,Mod(33,64)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(64, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 24, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("64.33");
S:= CuspForms(chi, 24);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 64 = 2^{6} \) |
| Weight: | \( k \) | \(=\) | \( 24 \) |
| Character orbit: | \([\chi]\) | \(=\) | 64.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: | \(214.530583901\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} + 1488736907456 x^{10} + \cdots + 28\!\cdots\!65 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{100}\cdot 3^{20}\cdot 5^{10}\cdot 11^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} + 1488736907456 x^{10} + \cdots + 28\!\cdots\!65 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 55\!\cdots\!36 \nu^{11} + \cdots - 22\!\cdots\!20 ) / 14\!\cdots\!49 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 55\!\cdots\!72 \nu^{11} + \cdots + 80\!\cdots\!60 ) / 28\!\cdots\!65 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 11\!\cdots\!36 \nu^{11} + \cdots - 43\!\cdots\!00 ) / 55\!\cdots\!45 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 77\!\cdots\!28 \nu^{11} + \cdots - 29\!\cdots\!00 ) / 55\!\cdots\!45 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 35\!\cdots\!66 \nu^{11} + \cdots - 40\!\cdots\!15 ) / 14\!\cdots\!90 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( - 11\!\cdots\!32 \nu^{11} + \cdots - 49\!\cdots\!40 ) / 69\!\cdots\!57 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 92\!\cdots\!52 \nu^{11} + \cdots - 39\!\cdots\!40 ) / 69\!\cdots\!57 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 18\!\cdots\!26 \nu^{11} + \cdots + 27\!\cdots\!65 ) / 21\!\cdots\!70 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 13\!\cdots\!24 \nu^{11} + \cdots - 14\!\cdots\!00 ) / 51\!\cdots\!37 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 76\!\cdots\!16 \nu^{11} + \cdots - 84\!\cdots\!00 ) / 51\!\cdots\!37 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 68\!\cdots\!48 \nu^{11} + \cdots - 13\!\cdots\!00 ) / 51\!\cdots\!37 \)
|
| \(\nu\) | \(=\) |
\( ( 5\beta_{3} - 90\beta_{2} - 192\beta_1 ) / 46080 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( - 90 \beta_{10} + 90 \beta_{9} + 1280 \beta_{7} - 235 \beta_{6} + 153600 \beta_{5} + \cdots - 13\!\cdots\!00 ) / 5529600 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 2560 \beta_{11} + 226230 \beta_{10} - 630710 \beta_{9} - 1825305984 \beta_{8} + \cdots + 34\!\cdots\!00 ) / 7372800 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 31106382000 \beta_{11} + 195353860023585 \beta_{10} + \cdots + 15\!\cdots\!00 ) / 11059200 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 301770708793680 \beta_{11} + \cdots - 22\!\cdots\!00 ) / 4423680 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 20\!\cdots\!80 \beta_{11} + \cdots - 64\!\cdots\!00 ) / 7372800 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( - 44\!\cdots\!60 \beta_{11} + \cdots + 10\!\cdots\!00 ) / 22118400 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( - 25\!\cdots\!60 \beta_{11} + \cdots + 62\!\cdots\!00 ) / 11059200 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 11\!\cdots\!00 \beta_{11} + \cdots - 27\!\cdots\!00 ) / 7372800 \)
|
| \(\nu^{10}\) | \(=\) |
\( ( 41\!\cdots\!80 \beta_{11} + \cdots - 80\!\cdots\!00 ) / 22118400 \)
|
| \(\nu^{11}\) | \(=\) |
\( ( - 33\!\cdots\!40 \beta_{11} + \cdots + 64\!\cdots\!00 ) / 22118400 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/64\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) | \(63\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 33.1 |
|
0 | − | 393660.i | 0 | − | 5.20473e7i | 0 | 4.90178e9 | 0 | −6.08248e10 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.2 | 0 | − | 393660.i | 0 | 5.20473e7i | 0 | −4.90178e9 | 0 | −6.08248e10 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.3 | 0 | − | 247122.i | 0 | − | 5.67554e7i | 0 | −6.58005e9 | 0 | 3.30740e10 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.4 | 0 | − | 247122.i | 0 | 5.67554e7i | 0 | 6.58005e9 | 0 | 3.30740e10 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.5 | 0 | − | 6191.92i | 0 | − | 1.92218e8i | 0 | 4.10941e9 | 0 | 9.41048e10 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.6 | 0 | − | 6191.92i | 0 | 1.92218e8i | 0 | −4.10941e9 | 0 | 9.41048e10 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.7 | 0 | 6191.92i | 0 | − | 1.92218e8i | 0 | −4.10941e9 | 0 | 9.41048e10 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.8 | 0 | 6191.92i | 0 | 1.92218e8i | 0 | 4.10941e9 | 0 | 9.41048e10 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.9 | 0 | 247122.i | 0 | − | 5.67554e7i | 0 | 6.58005e9 | 0 | 3.30740e10 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.10 | 0 | 247122.i | 0 | 5.67554e7i | 0 | −6.58005e9 | 0 | 3.30740e10 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.11 | 0 | 393660.i | 0 | − | 5.20473e7i | 0 | −4.90178e9 | 0 | −6.08248e10 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 33.12 | 0 | 393660.i | 0 | 5.20473e7i | 0 | 4.90178e9 | 0 | −6.08248e10 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
| 8.b | even | 2 | 1 | inner |
| 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 | 64.24.b.b | ✓ | 12 |
| 4.b | odd | 2 | 1 | inner | 64.24.b.b | ✓ | 12 |
| 8.b | even | 2 | 1 | inner | 64.24.b.b | ✓ | 12 |
| 8.d | odd | 2 | 1 | inner | 64.24.b.b | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 64.24.b.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 64.24.b.b | ✓ | 12 | 4.b | odd | 2 | 1 | inner |
| 64.24.b.b | ✓ | 12 | 8.b | even | 2 | 1 | inner |
| 64.24.b.b | ✓ | 12 | 8.d | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} + 216075541356T_{3}^{4} + 9472054753236006673200T_{3}^{2} + 362839792013755384155206760000 \)
acting on \(S_{24}^{\mathrm{new}}(64, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{12} \)
|
| $3$ |
\( (T^{6} + \cdots + 36\!\cdots\!00)^{2} \)
|
| $5$ |
\( (T^{6} + \cdots + 32\!\cdots\!00)^{2} \)
|
| $7$ |
\( (T^{6} + \cdots - 17\!\cdots\!00)^{2} \)
|
| $11$ |
\( (T^{6} + \cdots + 87\!\cdots\!56)^{2} \)
|
| $13$ |
\( (T^{6} + \cdots + 13\!\cdots\!00)^{2} \)
|
| $17$ |
\( (T^{3} + \cdots + 10\!\cdots\!00)^{4} \)
|
| $19$ |
\( (T^{6} + \cdots + 99\!\cdots\!64)^{2} \)
|
| $23$ |
\( (T^{6} + \cdots - 38\!\cdots\!00)^{2} \)
|
| $29$ |
\( (T^{6} + \cdots + 34\!\cdots\!00)^{2} \)
|
| $31$ |
\( (T^{6} + \cdots - 69\!\cdots\!00)^{2} \)
|
| $37$ |
\( (T^{6} + \cdots + 15\!\cdots\!00)^{2} \)
|
| $41$ |
\( (T^{3} + \cdots + 33\!\cdots\!24)^{4} \)
|
| $43$ |
\( (T^{6} + \cdots + 82\!\cdots\!00)^{2} \)
|
| $47$ |
\( (T^{6} + \cdots - 73\!\cdots\!00)^{2} \)
|
| $53$ |
\( (T^{6} + \cdots + 35\!\cdots\!00)^{2} \)
|
| $59$ |
\( (T^{6} + \cdots + 35\!\cdots\!96)^{2} \)
|
| $61$ |
\( (T^{6} + \cdots + 24\!\cdots\!00)^{2} \)
|
| $67$ |
\( (T^{6} + \cdots + 35\!\cdots\!00)^{2} \)
|
| $71$ |
\( (T^{6} + \cdots - 16\!\cdots\!00)^{2} \)
|
| $73$ |
\( (T^{3} + \cdots - 99\!\cdots\!00)^{4} \)
|
| $79$ |
\( (T^{6} + \cdots - 78\!\cdots\!00)^{2} \)
|
| $83$ |
\( (T^{6} + \cdots + 11\!\cdots\!00)^{2} \)
|
| $89$ |
\( (T^{3} + \cdots - 40\!\cdots\!12)^{4} \)
|
| $97$ |
\( (T^{3} + \cdots + 23\!\cdots\!00)^{4} \)
|
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