Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [104,8,Mod(1,104)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(104, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("104.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 104 = 2^{3} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 104.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: | \(32.4880426503\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{4} - x^{3} - 2027x^{2} + 26505x + 42050 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | yes |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 2027x^{2} + 26505x + 42050 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -7\nu^{3} - 48\nu^{2} + 12629\nu - 83350 ) / 1380 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -7\nu^{3} - 48\nu^{2} + 23669\nu - 86110 ) / 1380 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4\nu^{3} - 126\nu^{2} + 6428\nu + 52520 ) / 345 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta _1 + 2 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -7\beta_{3} - 2\beta_{2} + 18\beta _1 + 2028 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 192\beta_{3} + 1859\beta_{2} - 3875\beta _1 - 147274 ) / 8 \)
|
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 | −75.8673 | 0 | −316.618 | 0 | 373.907 | 0 | 3568.85 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −44.9077 | 0 | 206.816 | 0 | −717.689 | 0 | −170.294 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | −5.76922 | 0 | 111.215 | 0 | 617.212 | 0 | −2153.72 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 55.5443 | 0 | −264.414 | 0 | −42.4292 | 0 | 898.164 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(13\) | \( -1 \) |
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 | 104.8.a.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | 208.8.a.l | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 104.8.a.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 208.8.a.l | 4 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 71T_{3}^{3} - 2925T_{3}^{2} - 208287T_{3} - 1091772 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(104))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 71 T^{3} + \cdots - 1091772 \)
|
| $5$ |
\( T^{4} + \cdots + 1925605150 \)
|
| $7$ |
\( T^{4} + \cdots + 7027465320 \)
|
| $11$ |
\( T^{4} + \cdots - 125650436544864 \)
|
| $13$ |
\( (T - 2197)^{4} \)
|
| $17$ |
\( T^{4} + \cdots + 14\!\cdots\!70 \)
|
| $19$ |
\( T^{4} + \cdots + 13\!\cdots\!48 \)
|
| $23$ |
\( T^{4} + \cdots - 29\!\cdots\!24 \)
|
| $29$ |
\( T^{4} + \cdots - 16\!\cdots\!44 \)
|
| $31$ |
\( T^{4} + \cdots - 60\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots + 55\!\cdots\!94 \)
|
| $41$ |
\( T^{4} + \cdots - 14\!\cdots\!04 \)
|
| $43$ |
\( T^{4} + \cdots - 23\!\cdots\!52 \)
|
| $47$ |
\( T^{4} + \cdots + 50\!\cdots\!76 \)
|
| $53$ |
\( T^{4} + \cdots - 84\!\cdots\!68 \)
|
| $59$ |
\( T^{4} + \cdots - 16\!\cdots\!48 \)
|
| $61$ |
\( T^{4} + \cdots - 34\!\cdots\!00 \)
|
| $67$ |
\( T^{4} + \cdots - 26\!\cdots\!52 \)
|
| $71$ |
\( T^{4} + \cdots - 75\!\cdots\!76 \)
|
| $73$ |
\( T^{4} + \cdots - 50\!\cdots\!72 \)
|
| $79$ |
\( T^{4} + \cdots + 18\!\cdots\!44 \)
|
| $83$ |
\( T^{4} + \cdots + 27\!\cdots\!32 \)
|
| $89$ |
\( T^{4} + \cdots - 16\!\cdots\!60 \)
|
| $97$ |
\( T^{4} + \cdots + 44\!\cdots\!60 \)
|
| show more | |
| show less |