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: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{949}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 237 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(\beta = \sqrt{949}\). We also show the integral \(q\)-expansion of the trace form.
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 | −58.8058 | 64.0000 | −424.282 | 470.447 | 0 | −512.000 | 1271.13 | 3394.25 | ||||||||||||||||||||||||
| 1.2 | −8.00000 | 2.80584 | 64.0000 | 438.282 | −22.4467 | 0 | −512.000 | −2179.13 | −3506.25 | |||||||||||||||||||||||||
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 | 98.8.a.d | 2 | |
| 7.b | odd | 2 | 1 | 98.8.a.f | 2 | ||
| 7.c | even | 3 | 2 | 98.8.c.m | 4 | ||
| 7.d | odd | 6 | 2 | 14.8.c.b | ✓ | 4 | |
| 21.g | even | 6 | 2 | 126.8.g.d | 4 | ||
| 28.f | even | 6 | 2 | 112.8.i.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.8.c.b | ✓ | 4 | 7.d | odd | 6 | 2 | |
| 98.8.a.d | 2 | 1.a | even | 1 | 1 | trivial | |
| 98.8.a.f | 2 | 7.b | odd | 2 | 1 | ||
| 98.8.c.m | 4 | 7.c | even | 3 | 2 | ||
| 112.8.i.b | 4 | 28.f | even | 6 | 2 | ||
| 126.8.g.d | 4 | 21.g | even | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 56T_{3} - 165 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(98))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 8)^{2} \)
|
| $3$ |
\( T^{2} + 56T - 165 \)
|
| $5$ |
\( T^{2} - 14T - 185955 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} - 2408 T - 43237845 \)
|
| $13$ |
\( T^{2} + 10724 T + 26913780 \)
|
| $17$ |
\( T^{2} - 35098 T + 198263001 \)
|
| $19$ |
\( T^{2} - 2408 T - 625809965 \)
|
| $23$ |
\( T^{2} + 61684 T - 890257137 \)
|
| $29$ |
\( T^{2} + 95660 T - 302210796 \)
|
| $31$ |
\( T^{2} + \cdots + 5628386895 \)
|
| $37$ |
\( T^{2} + \cdots + 46003879093 \)
|
| $41$ |
\( T^{2} + \cdots + 154982022516 \)
|
| $43$ |
\( T^{2} + \cdots - 9474621680 \)
|
| $47$ |
\( T^{2} + \cdots + 776500057791 \)
|
| $53$ |
\( T^{2} + \cdots + 880940835909 \)
|
| $59$ |
\( T^{2} + \cdots - 9792650253 \)
|
| $61$ |
\( T^{2} + \cdots - 1149292144659 \)
|
| $67$ |
\( T^{2} + \cdots - 3517340463061 \)
|
| $71$ |
\( T^{2} + \cdots - 20052968986176 \)
|
| $73$ |
\( T^{2} + \cdots + 4575230600825 \)
|
| $79$ |
\( T^{2} + \cdots + 12232151046375 \)
|
| $83$ |
\( T^{2} + \cdots + 14206197364176 \)
|
| $89$ |
\( T^{2} + \cdots - 61691712832359 \)
|
| $97$ |
\( T^{2} + \cdots - 89006519008780 \)
|
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