Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [98,12,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, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("98.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| 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: | \(75.2976316948\) |
| 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: |
\( x^{4} - x^{3} - 101802x^{2} - 3237299x + 1820791991 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3\cdot 7^{2} \) |
| Twist minimal: | no (minimal twist has level 14) |
| 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
\( x^{4} - x^{3} - 101802x^{2} - 3237299x + 1820791991 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -8\nu^{3} + 2260\nu^{2} + 419174\nu - 95108320 ) / 2619 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8\nu^{3} - 1096\nu^{2} - 475046\nu + 35871778 ) / 291 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 9\beta_{2} + 96\beta _1 + 203562 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 565\beta_{3} + 2466\beta_{2} + 263827\beta _1 + 19904210 ) / 8 \)
|
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 |
|
32.0000 | −530.349 | 1024.00 | −13244.8 | −16971.2 | 0 | 32768.0 | 104123. | −423833. | ||||||||||||||||||||||||||||||
| 1.2 | 32.0000 | −468.588 | 1024.00 | 6652.75 | −14994.8 | 0 | 32768.0 | 42428.1 | 212888. | |||||||||||||||||||||||||||||||
| 1.3 | 32.0000 | 189.461 | 1024.00 | 6422.27 | 6062.75 | 0 | 32768.0 | −141252. | 205512. | |||||||||||||||||||||||||||||||
| 1.4 | 32.0000 | 543.477 | 1024.00 | −7334.23 | 17391.3 | 0 | 32768.0 | 118220. | −234695. | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(1\) |
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 | 98.12.a.j | 4 | |
| 7.b | odd | 2 | 1 | 98.12.a.l | 4 | ||
| 7.c | even | 3 | 2 | 14.12.c.a | ✓ | 8 | |
| 7.d | odd | 6 | 2 | 98.12.c.l | 8 | ||
| 21.h | odd | 6 | 2 | 126.12.g.e | 8 | ||
| 28.g | odd | 6 | 2 | 112.12.i.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.12.c.a | ✓ | 8 | 7.c | even | 3 | 2 | |
| 98.12.a.j | 4 | 1.a | even | 1 | 1 | trivial | |
| 98.12.a.l | 4 | 7.b | odd | 2 | 1 | ||
| 98.12.c.l | 8 | 7.d | odd | 6 | 2 | ||
| 112.12.i.a | 8 | 28.g | odd | 6 | 2 | ||
| 126.12.g.e | 8 | 21.h | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 266T_{3}^{3} - 380676T_{3}^{2} - 79288146T_{3} + 25589072475 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(98))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 32)^{4} \)
|
| $3$ |
\( T^{4} + \cdots + 25589072475 \)
|
| $5$ |
\( T^{4} + \cdots + 41\!\cdots\!25 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots - 29\!\cdots\!25 \)
|
| $13$ |
\( T^{4} + \cdots + 45\!\cdots\!00 \)
|
| $17$ |
\( T^{4} + \cdots - 22\!\cdots\!71 \)
|
| $19$ |
\( T^{4} + \cdots + 12\!\cdots\!35 \)
|
| $23$ |
\( T^{4} + \cdots + 50\!\cdots\!03 \)
|
| $29$ |
\( T^{4} + \cdots - 37\!\cdots\!16 \)
|
| $31$ |
\( T^{4} + \cdots - 72\!\cdots\!45 \)
|
| $37$ |
\( T^{4} + \cdots - 26\!\cdots\!83 \)
|
| $41$ |
\( T^{4} + \cdots - 17\!\cdots\!96 \)
|
| $43$ |
\( T^{4} + \cdots + 31\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots + 15\!\cdots\!91 \)
|
| $53$ |
\( T^{4} + \cdots - 55\!\cdots\!51 \)
|
| $59$ |
\( T^{4} + \cdots + 12\!\cdots\!55 \)
|
| $61$ |
\( T^{4} + \cdots - 29\!\cdots\!59 \)
|
| $67$ |
\( T^{4} + \cdots - 18\!\cdots\!29 \)
|
| $71$ |
\( T^{4} + \cdots + 29\!\cdots\!96 \)
|
| $73$ |
\( T^{4} + \cdots - 63\!\cdots\!15 \)
|
| $79$ |
\( T^{4} + \cdots - 27\!\cdots\!25 \)
|
| $83$ |
\( T^{4} + \cdots - 47\!\cdots\!84 \)
|
| $89$ |
\( T^{4} + \cdots + 76\!\cdots\!61 \)
|
| $97$ |
\( T^{4} + \cdots - 18\!\cdots\!00 \)
|
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