Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [98,8,Mod(1,98)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(98, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("98.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 98.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: | \(30.6137324974\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{2}, \sqrt{1801})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 903x^{2} + 904x + 200702 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 7^{2} \) |
| 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
\( x^{4} - 2x^{3} - 903x^{2} + 904x + 200702 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -14\nu^{3} + 21\nu^{2} + 6377\nu - 3192 ) / 1793 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -56\nu^{3} + 84\nu^{2} + 75712\nu - 37870 ) / 1793 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 452 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 4\beta _1 + 14 ) / 28 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 28\beta_{3} + \beta_{2} - 4\beta _1 + 12670 ) / 28 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 42\beta_{3} + 457\beta_{2} - 5414\beta _1 + 18998 ) / 28 \)
|
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 |
|
−8.00000 | −79.8157 | 64.0000 | −98.9949 | 638.525 | 0 | −512.000 | 4183.54 | 791.960 | ||||||||||||||||||||||||||||||
| 1.2 | −8.00000 | −40.2177 | 64.0000 | 98.9949 | 321.741 | 0 | −512.000 | −569.539 | −791.960 | |||||||||||||||||||||||||||||||
| 1.3 | −8.00000 | 40.2177 | 64.0000 | −98.9949 | −321.741 | 0 | −512.000 | −569.539 | 791.960 | |||||||||||||||||||||||||||||||
| 1.4 | −8.00000 | 79.8157 | 64.0000 | 98.9949 | −638.525 | 0 | −512.000 | 4183.54 | −791.960 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(7\) | \( +1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 98.8.a.l | ✓ | 4 |
| 7.b | odd | 2 | 1 | inner | 98.8.a.l | ✓ | 4 |
| 7.c | even | 3 | 2 | 98.8.c.n | 8 | ||
| 7.d | odd | 6 | 2 | 98.8.c.n | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 98.8.a.l | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 98.8.a.l | ✓ | 4 | 7.b | odd | 2 | 1 | inner |
| 98.8.c.n | 8 | 7.c | even | 3 | 2 | ||
| 98.8.c.n | 8 | 7.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 7988T_{3}^{2} + 10304100 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(98))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 8)^{4} \)
|
| $3$ |
\( T^{4} - 7988 T^{2} + 10304100 \)
|
| $5$ |
\( (T^{2} - 9800)^{2} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} + 3136 T - 26134052)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 45\!\cdots\!00 \)
|
| $17$ |
\( T^{4} + \cdots + 31\!\cdots\!96 \)
|
| $19$ |
\( T^{4} + \cdots + 23\!\cdots\!00 \)
|
| $23$ |
\( (T^{2} + 6272 T - 1019501840)^{2} \)
|
| $29$ |
\( (T^{2} + 37868 T - 28920403868)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 87\!\cdots\!04 \)
|
| $37$ |
\( (T^{2} - 617348 T + 87959913220)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 77\!\cdots\!96 \)
|
| $43$ |
\( (T^{2} - 1410880 T + 470168031964)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 76\!\cdots\!00 \)
|
| $53$ |
\( (T^{2} + \cdots - 2138410258844)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 14\!\cdots\!76 \)
|
| $61$ |
\( T^{4} + \cdots + 65\!\cdots\!00 \)
|
| $67$ |
\( (T^{2} - 2671080 T + 815633452944)^{2} \)
|
| $71$ |
\( (T^{2} + \cdots - 2982923301456)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 15\!\cdots\!76 \)
|
| $79$ |
\( (T^{2} + \cdots - 19980575976080)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 13\!\cdots\!04 \)
|
| $89$ |
\( T^{4} + \cdots + 22\!\cdots\!00 \)
|
| $97$ |
\( T^{4} + \cdots + 33\!\cdots\!24 \)
|
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