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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 2) |
| 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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(-\zeta_{6}\) |
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 | −6.00000 | + | 10.3923i | −32.0000 | + | 55.4256i | 105.000 | + | 181.865i | −96.0000 | 0 | −512.000 | 1021.50 | + | 1769.29i | −840.000 | + | 1454.92i | ||||||||||||
| 79.1 | 4.00000 | − | 6.92820i | −6.00000 | − | 10.3923i | −32.0000 | − | 55.4256i | 105.000 | − | 181.865i | −96.0000 | 0 | −512.000 | 1021.50 | − | 1769.29i | −840.000 | − | 1454.92i | |||||||||||||
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.d | 2 | |
| 7.b | odd | 2 | 1 | 98.8.c.e | 2 | ||
| 7.c | even | 3 | 1 | 2.8.a.a | ✓ | 1 | |
| 7.c | even | 3 | 1 | inner | 98.8.c.d | 2 | |
| 7.d | odd | 6 | 1 | 98.8.a.a | 1 | ||
| 7.d | odd | 6 | 1 | 98.8.c.e | 2 | ||
| 21.h | odd | 6 | 1 | 18.8.a.b | 1 | ||
| 28.g | odd | 6 | 1 | 16.8.a.b | 1 | ||
| 35.j | even | 6 | 1 | 50.8.a.g | 1 | ||
| 35.l | odd | 12 | 2 | 50.8.b.c | 2 | ||
| 56.k | odd | 6 | 1 | 64.8.a.e | 1 | ||
| 56.p | even | 6 | 1 | 64.8.a.c | 1 | ||
| 63.g | even | 3 | 1 | 162.8.c.l | 2 | ||
| 63.h | even | 3 | 1 | 162.8.c.l | 2 | ||
| 63.j | odd | 6 | 1 | 162.8.c.a | 2 | ||
| 63.n | odd | 6 | 1 | 162.8.c.a | 2 | ||
| 77.h | odd | 6 | 1 | 242.8.a.e | 1 | ||
| 84.n | even | 6 | 1 | 144.8.a.i | 1 | ||
| 91.r | even | 6 | 1 | 338.8.a.d | 1 | ||
| 91.z | odd | 12 | 2 | 338.8.b.d | 2 | ||
| 105.o | odd | 6 | 1 | 450.8.a.c | 1 | ||
| 105.x | even | 12 | 2 | 450.8.c.g | 2 | ||
| 112.u | odd | 12 | 2 | 256.8.b.f | 2 | ||
| 112.w | even | 12 | 2 | 256.8.b.b | 2 | ||
| 119.j | even | 6 | 1 | 578.8.a.b | 1 | ||
| 140.p | odd | 6 | 1 | 400.8.a.l | 1 | ||
| 140.w | even | 12 | 2 | 400.8.c.j | 2 | ||
| 168.s | odd | 6 | 1 | 576.8.a.g | 1 | ||
| 168.v | even | 6 | 1 | 576.8.a.f | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2.8.a.a | ✓ | 1 | 7.c | even | 3 | 1 | |
| 16.8.a.b | 1 | 28.g | odd | 6 | 1 | ||
| 18.8.a.b | 1 | 21.h | odd | 6 | 1 | ||
| 50.8.a.g | 1 | 35.j | even | 6 | 1 | ||
| 50.8.b.c | 2 | 35.l | odd | 12 | 2 | ||
| 64.8.a.c | 1 | 56.p | even | 6 | 1 | ||
| 64.8.a.e | 1 | 56.k | odd | 6 | 1 | ||
| 98.8.a.a | 1 | 7.d | odd | 6 | 1 | ||
| 98.8.c.d | 2 | 1.a | even | 1 | 1 | trivial | |
| 98.8.c.d | 2 | 7.c | even | 3 | 1 | inner | |
| 98.8.c.e | 2 | 7.b | odd | 2 | 1 | ||
| 98.8.c.e | 2 | 7.d | odd | 6 | 1 | ||
| 144.8.a.i | 1 | 84.n | even | 6 | 1 | ||
| 162.8.c.a | 2 | 63.j | odd | 6 | 1 | ||
| 162.8.c.a | 2 | 63.n | odd | 6 | 1 | ||
| 162.8.c.l | 2 | 63.g | even | 3 | 1 | ||
| 162.8.c.l | 2 | 63.h | even | 3 | 1 | ||
| 242.8.a.e | 1 | 77.h | odd | 6 | 1 | ||
| 256.8.b.b | 2 | 112.w | even | 12 | 2 | ||
| 256.8.b.f | 2 | 112.u | odd | 12 | 2 | ||
| 338.8.a.d | 1 | 91.r | even | 6 | 1 | ||
| 338.8.b.d | 2 | 91.z | odd | 12 | 2 | ||
| 400.8.a.l | 1 | 140.p | odd | 6 | 1 | ||
| 400.8.c.j | 2 | 140.w | even | 12 | 2 | ||
| 450.8.a.c | 1 | 105.o | odd | 6 | 1 | ||
| 450.8.c.g | 2 | 105.x | even | 12 | 2 | ||
| 576.8.a.f | 1 | 168.v | even | 6 | 1 | ||
| 576.8.a.g | 1 | 168.s | odd | 6 | 1 | ||
| 578.8.a.b | 1 | 119.j | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 12T_{3} + 144 \)
acting on \(S_{8}^{\mathrm{new}}(98, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 8T + 64 \)
|
| $3$ |
\( T^{2} + 12T + 144 \)
|
| $5$ |
\( T^{2} - 210T + 44100 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} + 1092 T + 1192464 \)
|
| $13$ |
\( (T - 1382)^{2} \)
|
| $17$ |
\( T^{2} + 14706 T + 216266436 \)
|
| $19$ |
\( T^{2} + \cdots + 1595203600 \)
|
| $23$ |
\( T^{2} + \cdots + 4721338944 \)
|
| $29$ |
\( (T + 102570)^{2} \)
|
| $31$ |
\( T^{2} + \cdots + 51779912704 \)
|
| $37$ |
\( T^{2} + \cdots + 25768596676 \)
|
| $41$ |
\( (T - 10842)^{2} \)
|
| $43$ |
\( (T + 630748)^{2} \)
|
| $47$ |
\( T^{2} + \cdots + 223403694336 \)
|
| $53$ |
\( T^{2} + \cdots + 2232089784324 \)
|
| $59$ |
\( T^{2} + \cdots + 6973085235600 \)
|
| $61$ |
\( T^{2} + \cdots + 685090600804 \)
|
| $67$ |
\( T^{2} + \cdots + 15877008016 \)
|
| $71$ |
\( (T + 1414728)^{2} \)
|
| $73$ |
\( T^{2} + \cdots + 960952799524 \)
|
| $79$ |
\( T^{2} + \cdots + 12722062240000 \)
|
| $83$ |
\( (T - 5672892)^{2} \)
|
| $89$ |
\( T^{2} + \cdots + 142830942416100 \)
|
| $97$ |
\( (T - 8682146)^{2} \)
|
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