Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [102,12,Mod(1,102)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(102, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("102.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 102 = 2 \cdot 3 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 102.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: | \(78.3710044171\) |
| 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: |
\( x^{4} - x^{3} - 1088583x^{2} + 474043012x - 51746504200 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{2}\cdot 5 \) |
| 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
\( x^{4} - x^{3} - 1088583x^{2} + 474043012x - 51746504200 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} - 246\nu^{2} + 1060546\nu - 221087525 ) / 4675 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -13\nu^{3} - 5068\nu^{2} + 11874088\nu - 1855815300 ) / 23375 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} + 322\nu^{2} - 2040002\nu + 534679700 ) / 2125 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + 7\beta _1 + 33 ) / 120 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -341\beta_{3} - 159\beta_{2} - 1087\beta _1 + 21771047 ) / 40 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 328051\beta_{3} - 235801\beta_{2} + 1916257\beta _1 - 10640634417 ) / 30 \)
|
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 |
|
−32.0000 | 243.000 | 1024.00 | −10558.4 | −7776.00 | 84983.4 | −32768.0 | 59049.0 | 337868. | ||||||||||||||||||||||||||||||
| 1.2 | −32.0000 | 243.000 | 1024.00 | −5388.92 | −7776.00 | −4246.77 | −32768.0 | 59049.0 | 172446. | |||||||||||||||||||||||||||||||
| 1.3 | −32.0000 | 243.000 | 1024.00 | 8920.72 | −7776.00 | 29548.2 | −32768.0 | 59049.0 | −285463. | |||||||||||||||||||||||||||||||
| 1.4 | −32.0000 | 243.000 | 1024.00 | 9888.56 | −7776.00 | 23979.2 | −32768.0 | 59049.0 | −316434. | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(17\) | \( -1 \) |
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 | 102.12.a.e | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 102.12.a.e | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 2862T_{5}^{3} - 154845699T_{5}^{2} + 336545425260T_{5} + 5019166992381300 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(102))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 32)^{4} \)
|
| $3$ |
\( (T - 243)^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 50\!\cdots\!00 \)
|
| $7$ |
\( T^{4} + \cdots - 25\!\cdots\!52 \)
|
| $11$ |
\( T^{4} + \cdots + 26\!\cdots\!72 \)
|
| $13$ |
\( T^{4} + \cdots + 30\!\cdots\!40 \)
|
| $17$ |
\( (T - 1419857)^{4} \)
|
| $19$ |
\( T^{4} + \cdots - 11\!\cdots\!20 \)
|
| $23$ |
\( T^{4} + \cdots + 46\!\cdots\!00 \)
|
| $29$ |
\( T^{4} + \cdots + 16\!\cdots\!00 \)
|
| $31$ |
\( T^{4} + \cdots - 41\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots - 31\!\cdots\!08 \)
|
| $41$ |
\( T^{4} + \cdots + 54\!\cdots\!16 \)
|
| $43$ |
\( T^{4} + \cdots + 45\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots + 94\!\cdots\!24 \)
|
| $53$ |
\( T^{4} + \cdots + 51\!\cdots\!08 \)
|
| $59$ |
\( T^{4} + \cdots + 38\!\cdots\!00 \)
|
| $61$ |
\( T^{4} + \cdots + 58\!\cdots\!16 \)
|
| $67$ |
\( T^{4} + \cdots + 70\!\cdots\!28 \)
|
| $71$ |
\( T^{4} + \cdots - 26\!\cdots\!60 \)
|
| $73$ |
\( T^{4} + \cdots - 78\!\cdots\!52 \)
|
| $79$ |
\( T^{4} + \cdots - 29\!\cdots\!20 \)
|
| $83$ |
\( T^{4} + \cdots - 10\!\cdots\!00 \)
|
| $89$ |
\( T^{4} + \cdots - 22\!\cdots\!28 \)
|
| $97$ |
\( T^{4} + \cdots - 23\!\cdots\!40 \)
|
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