Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [98,12,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, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("98.67");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| 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: | \(75.2976316948\) |
| 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 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 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 |
|
−16.0000 | − | 27.7128i | 45.0000 | − | 77.9423i | −512.000 | + | 886.810i | 3740.00 | + | 6477.87i | −2880.00 | 0 | 32768.0 | 84523.5 | + | 146399.i | 119680. | − | 207292.i | ||||||||||||
| 79.1 | −16.0000 | + | 27.7128i | 45.0000 | + | 77.9423i | −512.000 | − | 886.810i | 3740.00 | − | 6477.87i | −2880.00 | 0 | 32768.0 | 84523.5 | − | 146399.i | 119680. | + | 207292.i | |||||||||||||
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 | 98.12.c.b | 2 | |
| 7.b | odd | 2 | 1 | 98.12.c.a | 2 | ||
| 7.c | even | 3 | 1 | 14.12.a.b | ✓ | 1 | |
| 7.c | even | 3 | 1 | inner | 98.12.c.b | 2 | |
| 7.d | odd | 6 | 1 | 98.12.a.b | 1 | ||
| 7.d | odd | 6 | 1 | 98.12.c.a | 2 | ||
| 21.h | odd | 6 | 1 | 126.12.a.b | 1 | ||
| 28.g | odd | 6 | 1 | 112.12.a.a | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.12.a.b | ✓ | 1 | 7.c | even | 3 | 1 | |
| 98.12.a.b | 1 | 7.d | odd | 6 | 1 | ||
| 98.12.c.a | 2 | 7.b | odd | 2 | 1 | ||
| 98.12.c.a | 2 | 7.d | odd | 6 | 1 | ||
| 98.12.c.b | 2 | 1.a | even | 1 | 1 | trivial | |
| 98.12.c.b | 2 | 7.c | even | 3 | 1 | inner | |
| 112.12.a.a | 1 | 28.g | odd | 6 | 1 | ||
| 126.12.a.b | 1 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 90T_{3} + 8100 \)
acting on \(S_{12}^{\mathrm{new}}(98, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 32T + 1024 \)
|
| $3$ |
\( T^{2} - 90T + 8100 \)
|
| $5$ |
\( T^{2} - 7480 T + 55950400 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} + \cdots + 86751455296 \)
|
| $13$ |
\( (T + 210588)^{2} \)
|
| $17$ |
\( T^{2} + \cdots + 48482059964836 \)
|
| $19$ |
\( T^{2} + \cdots + 87355006032100 \)
|
| $23$ |
\( T^{2} + \cdots + 26\!\cdots\!00 \)
|
| $29$ |
\( (T - 166196354)^{2} \)
|
| $31$ |
\( T^{2} + \cdots + 14\!\cdots\!44 \)
|
| $37$ |
\( T^{2} + \cdots + 75\!\cdots\!00 \)
|
| $41$ |
\( (T + 197988378)^{2} \)
|
| $43$ |
\( (T + 809489728)^{2} \)
|
| $47$ |
\( T^{2} + \cdots + 67\!\cdots\!16 \)
|
| $53$ |
\( T^{2} + \cdots + 53\!\cdots\!16 \)
|
| $59$ |
\( T^{2} + \cdots + 21\!\cdots\!64 \)
|
| $61$ |
\( T^{2} + \cdots + 26\!\cdots\!76 \)
|
| $67$ |
\( T^{2} + \cdots + 77\!\cdots\!96 \)
|
| $71$ |
\( (T + 3849006656)^{2} \)
|
| $73$ |
\( T^{2} + \cdots + 34\!\cdots\!16 \)
|
| $79$ |
\( T^{2} + \cdots + 89\!\cdots\!64 \)
|
| $83$ |
\( (T + 5875980446)^{2} \)
|
| $89$ |
\( T^{2} + \cdots + 68\!\cdots\!00 \)
|
| $97$ |
\( (T - 149400800374)^{2} \)
|
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