Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [98,8,Mod(67,98)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(98, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("98.67");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 98.c (of order \(3\), degree \(2\), not minimal) |
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: | no |
| Analytic conductor: | \(30.6137324974\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{2389})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 598x^{2} + 597x + 356409 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 14) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} - x^{3} + 598x^{2} + 597x + 356409 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 598\nu^{2} - 598\nu + 356409 ) / 357006 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 598\nu^{2} + 714610\nu - 356409 ) / 357006 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 896 ) / 299 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 1195\beta _1 - 1195 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 299\beta_{3} - 896 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 67.1 |
|
−4.00000 | − | 6.92820i | −38.4387 | + | 66.5778i | −32.0000 | + | 55.4256i | −10.6226 | − | 18.3989i | 615.019 | 0 | 512.000 | −1861.57 | − | 3224.33i | −84.9808 | + | 147.191i | ||||||||||||||||||
| 67.2 | −4.00000 | − | 6.92820i | 10.4387 | − | 18.0804i | −32.0000 | + | 55.4256i | −108.377 | − | 187.715i | −167.019 | 0 | 512.000 | 875.567 | + | 1516.53i | −867.019 | + | 1501.72i | |||||||||||||||||||
| 79.1 | −4.00000 | + | 6.92820i | −38.4387 | − | 66.5778i | −32.0000 | − | 55.4256i | −10.6226 | + | 18.3989i | 615.019 | 0 | 512.000 | −1861.57 | + | 3224.33i | −84.9808 | − | 147.191i | |||||||||||||||||||
| 79.2 | −4.00000 | + | 6.92820i | 10.4387 | + | 18.0804i | −32.0000 | − | 55.4256i | −108.377 | + | 187.715i | −167.019 | 0 | 512.000 | 875.567 | − | 1516.53i | −867.019 | − | 1501.72i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 98.8.c.h | 4 | |
| 7.b | odd | 2 | 1 | 14.8.c.a | ✓ | 4 | |
| 7.c | even | 3 | 1 | 98.8.a.k | 2 | ||
| 7.c | even | 3 | 1 | inner | 98.8.c.h | 4 | |
| 7.d | odd | 6 | 1 | 14.8.c.a | ✓ | 4 | |
| 7.d | odd | 6 | 1 | 98.8.a.h | 2 | ||
| 21.c | even | 2 | 1 | 126.8.g.e | 4 | ||
| 21.g | even | 6 | 1 | 126.8.g.e | 4 | ||
| 28.d | even | 2 | 1 | 112.8.i.a | 4 | ||
| 28.f | even | 6 | 1 | 112.8.i.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.8.c.a | ✓ | 4 | 7.b | odd | 2 | 1 | |
| 14.8.c.a | ✓ | 4 | 7.d | odd | 6 | 1 | |
| 98.8.a.h | 2 | 7.d | odd | 6 | 1 | ||
| 98.8.a.k | 2 | 7.c | even | 3 | 1 | ||
| 98.8.c.h | 4 | 1.a | even | 1 | 1 | trivial | |
| 98.8.c.h | 4 | 7.c | even | 3 | 1 | inner | |
| 112.8.i.a | 4 | 28.d | even | 2 | 1 | ||
| 112.8.i.a | 4 | 28.f | even | 6 | 1 | ||
| 126.8.g.e | 4 | 21.c | even | 2 | 1 | ||
| 126.8.g.e | 4 | 21.g | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 56T_{3}^{3} + 4741T_{3}^{2} - 89880T_{3} + 2576025 \)
acting on \(S_{8}^{\mathrm{new}}(98, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8 T + 64)^{2} \)
|
| $3$ |
\( T^{4} + 56 T^{3} + \cdots + 2576025 \)
|
| $5$ |
\( T^{4} + 238 T^{3} + \cdots + 21206025 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 71110682271225 \)
|
| $13$ |
\( (T^{2} + 1316 T - 91342860)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 16\!\cdots\!81 \)
|
| $19$ |
\( T^{4} + \cdots + 78\!\cdots\!25 \)
|
| $23$ |
\( T^{4} + \cdots + 12\!\cdots\!69 \)
|
| $29$ |
\( (T^{2} - 172820 T + 5666758164)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 74\!\cdots\!25 \)
|
| $37$ |
\( T^{4} + \cdots + 81\!\cdots\!29 \)
|
| $41$ |
\( (T^{2} - 1156260 T + 332194626036)^{2} \)
|
| $43$ |
\( (T^{2} - 57544 T - 154524293360)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 62\!\cdots\!81 \)
|
| $53$ |
\( T^{4} + \cdots + 19\!\cdots\!61 \)
|
| $59$ |
\( T^{4} + \cdots + 73\!\cdots\!29 \)
|
| $61$ |
\( T^{4} + \cdots + 32\!\cdots\!81 \)
|
| $67$ |
\( T^{4} + \cdots + 16\!\cdots\!61 \)
|
| $71$ |
\( (T^{2} + \cdots + 2760738723264)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 28\!\cdots\!25 \)
|
| $79$ |
\( T^{4} + \cdots + 43\!\cdots\!25 \)
|
| $83$ |
\( (T^{2} + \cdots - 11352739197744)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 24\!\cdots\!21 \)
|
| $97$ |
\( (T^{2} + \cdots + 6136617007220)^{2} \)
|
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