Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [98,14,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, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("98.67");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| 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: | \(105.086310373\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 25033x^{2} + 25032x + 626601024 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| 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
\( x^{4} - x^{3} + 25033x^{2} + 25032x + 626601024 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 25033\nu^{2} - 25033\nu + 626601024 ) / 626626056 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 25033\nu^{2} + 1253277145\nu - 626601024 ) / 313313028 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 4\nu^{3} + 150194 ) / 25033 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 100130\beta _1 - 100130 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 25033\beta_{3} - 150194 ) / 4 \)
|
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 |
|
32.0000 | + | 55.4256i | −554.432 | + | 960.304i | −2048.00 | + | 3547.24i | −27936.2 | − | 48386.9i | −70967.3 | 0 | −262144. | 182373. | + | 315879.i | 1.78792e6 | − | 3.09676e6i | ||||||||||||||||||
| 67.2 | 32.0000 | + | 55.4256i | 78.4317 | − | 135.848i | −2048.00 | + | 3547.24i | 11934.2 | + | 20670.6i | 10039.3 | 0 | −262144. | 784858. | + | 1.35941e6i | −763788. | + | 1.32292e6i | |||||||||||||||||||
| 79.1 | 32.0000 | − | 55.4256i | −554.432 | − | 960.304i | −2048.00 | − | 3547.24i | −27936.2 | + | 48386.9i | −70967.3 | 0 | −262144. | 182373. | − | 315879.i | 1.78792e6 | + | 3.09676e6i | |||||||||||||||||||
| 79.2 | 32.0000 | − | 55.4256i | 78.4317 | + | 135.848i | −2048.00 | − | 3547.24i | 11934.2 | − | 20670.6i | 10039.3 | 0 | −262144. | 784858. | − | 1.35941e6i | −763788. | − | 1.32292e6i | |||||||||||||||||||
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 | 98.14.c.l | 4 | |
| 7.b | odd | 2 | 1 | 98.14.c.m | 4 | ||
| 7.c | even | 3 | 1 | 14.14.a.c | ✓ | 2 | |
| 7.c | even | 3 | 1 | inner | 98.14.c.l | 4 | |
| 7.d | odd | 6 | 1 | 98.14.a.e | 2 | ||
| 7.d | odd | 6 | 1 | 98.14.c.m | 4 | ||
| 21.h | odd | 6 | 1 | 126.14.a.l | 2 | ||
| 28.g | odd | 6 | 1 | 112.14.a.d | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.14.a.c | ✓ | 2 | 7.c | even | 3 | 1 | |
| 98.14.a.e | 2 | 7.d | odd | 6 | 1 | ||
| 98.14.c.l | 4 | 1.a | even | 1 | 1 | trivial | |
| 98.14.c.l | 4 | 7.c | even | 3 | 1 | inner | |
| 98.14.c.m | 4 | 7.b | odd | 2 | 1 | ||
| 98.14.c.m | 4 | 7.d | odd | 6 | 1 | ||
| 112.14.a.d | 2 | 28.g | odd | 6 | 1 | ||
| 126.14.a.l | 2 | 21.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 952T_{3}^{3} + 1080244T_{3}^{2} - 165590880T_{3} + 30255123600 \)
acting on \(S_{14}^{\mathrm{new}}(98, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 64 T + 4096)^{2} \)
|
| $3$ |
\( T^{4} + \cdots + 30255123600 \)
|
| $5$ |
\( T^{4} + \cdots + 17\!\cdots\!00 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 23\!\cdots\!00 \)
|
| $13$ |
\( (T^{2} + \cdots - 10\!\cdots\!64)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 27\!\cdots\!04 \)
|
| $19$ |
\( T^{4} + \cdots + 67\!\cdots\!00 \)
|
| $23$ |
\( T^{4} + \cdots + 94\!\cdots\!00 \)
|
| $29$ |
\( (T^{2} + \cdots - 22\!\cdots\!36)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 13\!\cdots\!24 \)
|
| $37$ |
\( T^{4} + \cdots + 15\!\cdots\!00 \)
|
| $41$ |
\( (T^{2} + \cdots - 10\!\cdots\!08)^{2} \)
|
| $43$ |
\( (T^{2} + \cdots - 36\!\cdots\!32)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 33\!\cdots\!76 \)
|
| $53$ |
\( T^{4} + \cdots + 20\!\cdots\!64 \)
|
| $59$ |
\( T^{4} + \cdots + 13\!\cdots\!24 \)
|
| $61$ |
\( T^{4} + \cdots + 30\!\cdots\!44 \)
|
| $67$ |
\( T^{4} + \cdots + 20\!\cdots\!64 \)
|
| $71$ |
\( (T^{2} + \cdots - 12\!\cdots\!60)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 58\!\cdots\!56 \)
|
| $79$ |
\( T^{4} + \cdots + 12\!\cdots\!16 \)
|
| $83$ |
\( (T^{2} + \cdots - 26\!\cdots\!32)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 45\!\cdots\!00 \)
|
| $97$ |
\( (T^{2} + \cdots + 68\!\cdots\!60)^{2} \)
|
| show more | |
| show less |