Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [16,48,Mod(1,16)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(16, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 48, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("16.1");
S:= CuspForms(chi, 48);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 16 = 2^{4} \) |
| Weight: | \( k \) | \(=\) | \( 48 \) |
| Character orbit: | \([\chi]\) | \(=\) | 16.a (trivial) |
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: | yes |
| Analytic conductor: | \(223.852260248\) |
| 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: |
\( x^{6} - 2 x^{5} + \cdots + 25\!\cdots\!91 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{109}\cdot 3^{10}\cdot 5^{5}\cdot 7^{4} \) |
| Twist minimal: | no (minimal twist has level 8) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{6} - 2 x^{5} + \cdots + 25\!\cdots\!91 \)
:
| \(\beta_{1}\) | \(=\) |
\( 128\nu - 43 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 22\!\cdots\!04 \nu^{5} + \cdots - 34\!\cdots\!39 ) / 64\!\cdots\!75 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 73\!\cdots\!44 \nu^{5} + \cdots + 61\!\cdots\!29 ) / 64\!\cdots\!75 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 26\!\cdots\!56 \nu^{5} + \cdots - 99\!\cdots\!71 ) / 21\!\cdots\!25 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 25\!\cdots\!84 \nu^{5} + \cdots + 36\!\cdots\!19 ) / 64\!\cdots\!75 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 43 ) / 128 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + 80\beta_{3} - 89512\beta_{2} - 31637927613\beta _1 + 34142212290169219605044 ) / 16384 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 112272183 \beta_{5} + 30866400748 \beta_{4} - 29938405916863 \beta_{3} + \cdots - 10\!\cdots\!58 ) / 2097152 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 17\!\cdots\!40 \beta_{5} + \cdots + 26\!\cdots\!02 ) / 33554432 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 22\!\cdots\!33 \beta_{5} + \cdots - 10\!\cdots\!64 ) / 4294967296 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −2.71088e11 | 0 | 3.86336e16 | 0 | −1.07727e20 | 0 | 4.69001e22 | 0 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.73567e11 | 0 | −1.60606e16 | 0 | 5.32568e19 | 0 | 3.53682e21 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 0 | 2.12477e10 | 0 | −2.20168e16 | 0 | −4.33893e19 | 0 | −2.61374e22 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0 | 6.47897e10 | 0 | 4.93920e16 | 0 | 1.17429e20 | 0 | −2.23911e22 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 0 | 1.38357e11 | 0 | 1.96700e16 | 0 | −8.56099e19 | 0 | −7.44611e21 | 0 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 0 | 2.79336e11 | 0 | −1.98305e16 | 0 | 4.53429e19 | 0 | 5.14396e22 | 0 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 16.48.a.e | 6 | |
| 4.b | odd | 2 | 1 | 8.48.a.b | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 8.48.a.b | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 16.48.a.e | 6 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{6} - 59074429624 T_{3}^{5} + \cdots + 25\!\cdots\!00 \)
acting on \(S_{48}^{\mathrm{new}}(\Gamma_0(16))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} + \cdots + 25\!\cdots\!00 \)
|
| $5$ |
\( T^{6} + \cdots - 26\!\cdots\!00 \)
|
| $7$ |
\( T^{6} + \cdots - 11\!\cdots\!36 \)
|
| $11$ |
\( T^{6} + \cdots + 42\!\cdots\!36 \)
|
| $13$ |
\( T^{6} + \cdots + 77\!\cdots\!56 \)
|
| $17$ |
\( T^{6} + \cdots - 26\!\cdots\!00 \)
|
| $19$ |
\( T^{6} + \cdots - 39\!\cdots\!24 \)
|
| $23$ |
\( T^{6} + \cdots - 11\!\cdots\!76 \)
|
| $29$ |
\( T^{6} + \cdots + 13\!\cdots\!04 \)
|
| $31$ |
\( T^{6} + \cdots + 50\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots + 50\!\cdots\!44 \)
|
| $41$ |
\( T^{6} + \cdots - 22\!\cdots\!24 \)
|
| $43$ |
\( T^{6} + \cdots - 11\!\cdots\!24 \)
|
| $47$ |
\( T^{6} + \cdots + 16\!\cdots\!64 \)
|
| $53$ |
\( T^{6} + \cdots + 10\!\cdots\!64 \)
|
| $59$ |
\( T^{6} + \cdots + 11\!\cdots\!36 \)
|
| $61$ |
\( T^{6} + \cdots + 72\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots - 56\!\cdots\!84 \)
|
| $71$ |
\( T^{6} + \cdots - 19\!\cdots\!00 \)
|
| $73$ |
\( T^{6} + \cdots + 76\!\cdots\!44 \)
|
| $79$ |
\( T^{6} + \cdots - 17\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 15\!\cdots\!24 \)
|
| $89$ |
\( T^{6} + \cdots - 10\!\cdots\!24 \)
|
| $97$ |
\( T^{6} + \cdots - 13\!\cdots\!96 \)
|
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