Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [64,15,Mod(63,64)]
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([1, 0]))
N = Newforms(chi, 15, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("64.63");
S:= CuspForms(chi, 15);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 64 = 2^{6} \) |
| Weight: | \( k \) | \(=\) | \( 15 \) |
| Character orbit: | \([\chi]\) | \(=\) | 64.c (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: | \(79.5705396172\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 9547x^{2} + 9546x + 91126116 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{25}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 16) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} + 9547x^{2} + 9546x + 91126116 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -25760\nu^{3} + 2902288\nu^{2} - 488959152\nu + 13606715664 ) / 15189277 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 73184\nu^{3} + 30397648\nu^{2} + 1427772944\nu + 145786588368 ) / 45567831 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -384\nu^{3} - 5498688 ) / 9547 \)
|
| \(\nu\) | \(=\) |
\( ( 32\beta_{3} + 171\beta_{2} - 597\beta _1 + 6144 ) / 24576 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -32\beta_{3} + 28809\beta_{2} + 28041\beta _1 - 117307392 ) / 24576 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -9547\beta_{3} - 5498688 ) / 384 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 63.1 |
|
0 | − | 4273.45i | 0 | 59268.7 | 0 | 1.04417e6i | 0 | −1.34794e7 | 0 | |||||||||||||||||||||||||||||
| 63.2 | 0 | − | 1141.90i | 0 | −15768.7 | 0 | 288003.i | 0 | 3.47902e6 | 0 | ||||||||||||||||||||||||||||||
| 63.3 | 0 | 1141.90i | 0 | −15768.7 | 0 | − | 288003.i | 0 | 3.47902e6 | 0 | ||||||||||||||||||||||||||||||
| 63.4 | 0 | 4273.45i | 0 | 59268.7 | 0 | − | 1.04417e6i | 0 | −1.34794e7 | 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 | 64.15.c.c | 4 | |
| 4.b | odd | 2 | 1 | inner | 64.15.c.c | 4 | |
| 8.b | even | 2 | 1 | 16.15.c.c | ✓ | 4 | |
| 8.d | odd | 2 | 1 | 16.15.c.c | ✓ | 4 | |
| 24.f | even | 2 | 1 | 144.15.g.g | 4 | ||
| 24.h | odd | 2 | 1 | 144.15.g.g | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 16.15.c.c | ✓ | 4 | 8.b | even | 2 | 1 | |
| 16.15.c.c | ✓ | 4 | 8.d | odd | 2 | 1 | |
| 64.15.c.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 64.15.c.c | 4 | 4.b | odd | 2 | 1 | inner | |
| 144.15.g.g | 4 | 24.f | even | 2 | 1 | ||
| 144.15.g.g | 4 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 19566336T_{3}^{2} + 23813150736384 \)
acting on \(S_{15}^{\mathrm{new}}(64, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + \cdots + 23813150736384 \)
|
| $5$ |
\( (T^{2} - 43500 T - 934589340)^{2} \)
|
| $7$ |
\( T^{4} + \cdots + 90\!\cdots\!44 \)
|
| $11$ |
\( T^{4} + \cdots + 49\!\cdots\!04 \)
|
| $13$ |
\( (T^{2} + \cdots - 15\!\cdots\!24)^{2} \)
|
| $17$ |
\( (T^{2} + \cdots - 69\!\cdots\!56)^{2} \)
|
| $19$ |
\( T^{4} + \cdots + 30\!\cdots\!24 \)
|
| $23$ |
\( T^{4} + \cdots + 37\!\cdots\!04 \)
|
| $29$ |
\( (T^{2} + \cdots - 21\!\cdots\!16)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 12\!\cdots\!44 \)
|
| $37$ |
\( (T^{2} + \cdots - 40\!\cdots\!40)^{2} \)
|
| $41$ |
\( (T^{2} + \cdots + 16\!\cdots\!04)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 39\!\cdots\!84 \)
|
| $47$ |
\( T^{4} + \cdots + 97\!\cdots\!04 \)
|
| $53$ |
\( (T^{2} + \cdots - 99\!\cdots\!16)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 33\!\cdots\!44 \)
|
| $61$ |
\( (T^{2} + \cdots + 12\!\cdots\!16)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 54\!\cdots\!84 \)
|
| $71$ |
\( T^{4} + \cdots + 26\!\cdots\!44 \)
|
| $73$ |
\( (T^{2} + \cdots - 16\!\cdots\!56)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 51\!\cdots\!00 \)
|
| $83$ |
\( T^{4} + \cdots + 26\!\cdots\!24 \)
|
| $89$ |
\( (T^{2} + \cdots - 80\!\cdots\!64)^{2} \)
|
| $97$ |
\( (T^{2} + \cdots - 16\!\cdots\!44)^{2} \)
|
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