Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [98,10,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, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("98.67");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| 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: | \(50.4735119441\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{43})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 43x^{2} + 1849 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 2^{6} \) |
| Twist minimal: | yes |
| 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
\( x^{4} + 43x^{2} + 1849 \)
:
| \(\beta_{1}\) | \(=\) |
\( 8\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 43 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 8\nu^{3} ) / 43 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 8 \)
|
| \(\nu^{2}\) | \(=\) |
\( 43\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 43\beta_{3} ) / 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/98\mathbb{Z}\right)^\times\).
| \(n\) | \(3\) |
| \(\chi(n)\) | \(\beta_{2}\) |
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 |
|
8.00000 | + | 13.8564i | −26.2298 | + | 45.4313i | −128.000 | + | 221.703i | −918.041 | − | 1590.09i | −839.352 | 0 | −4096.00 | 8465.50 | + | 14662.7i | 14688.7 | − | 25441.5i | ||||||||||||||||||
| 67.2 | 8.00000 | + | 13.8564i | 26.2298 | − | 45.4313i | −128.000 | + | 221.703i | 918.041 | + | 1590.09i | 839.352 | 0 | −4096.00 | 8465.50 | + | 14662.7i | −14688.7 | + | 25441.5i | |||||||||||||||||||
| 79.1 | 8.00000 | − | 13.8564i | −26.2298 | − | 45.4313i | −128.000 | − | 221.703i | −918.041 | + | 1590.09i | −839.352 | 0 | −4096.00 | 8465.50 | − | 14662.7i | 14688.7 | + | 25441.5i | |||||||||||||||||||
| 79.2 | 8.00000 | − | 13.8564i | 26.2298 | + | 45.4313i | −128.000 | − | 221.703i | 918.041 | − | 1590.09i | 839.352 | 0 | −4096.00 | 8465.50 | − | 14662.7i | −14688.7 | − | 25441.5i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 98.10.c.i | 4 | |
| 7.b | odd | 2 | 1 | inner | 98.10.c.i | 4 | |
| 7.c | even | 3 | 1 | 98.10.a.d | ✓ | 2 | |
| 7.c | even | 3 | 1 | inner | 98.10.c.i | 4 | |
| 7.d | odd | 6 | 1 | 98.10.a.d | ✓ | 2 | |
| 7.d | odd | 6 | 1 | inner | 98.10.c.i | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 98.10.a.d | ✓ | 2 | 7.c | even | 3 | 1 | |
| 98.10.a.d | ✓ | 2 | 7.d | odd | 6 | 1 | |
| 98.10.c.i | 4 | 1.a | even | 1 | 1 | trivial | |
| 98.10.c.i | 4 | 7.b | odd | 2 | 1 | inner | |
| 98.10.c.i | 4 | 7.c | even | 3 | 1 | inner | |
| 98.10.c.i | 4 | 7.d | odd | 6 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 2752T_{3}^{2} + 7573504 \)
acting on \(S_{10}^{\mathrm{new}}(98, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 16 T + 256)^{2} \)
|
| $3$ |
\( T^{4} + 2752 T^{2} + 7573504 \)
|
| $5$ |
\( T^{4} + \cdots + 11364989440000 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} + 9380 T + 87984400)^{2} \)
|
| $13$ |
\( (T^{2} - 32056861888)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 91\!\cdots\!44 \)
|
| $19$ |
\( T^{4} + \cdots + 10\!\cdots\!64 \)
|
| $23$ |
\( (T^{2} - 978936 T + 958315692096)^{2} \)
|
| $29$ |
\( (T + 4317214)^{4} \)
|
| $31$ |
\( T^{4} + \cdots + 40\!\cdots\!64 \)
|
| $37$ |
\( (T^{2} + \cdots + 7367934787236)^{2} \)
|
| $41$ |
\( (T^{2} - 81042600137472)^{2} \)
|
| $43$ |
\( (T + 34755692)^{4} \)
|
| $47$ |
\( T^{4} + \cdots + 31\!\cdots\!64 \)
|
| $53$ |
\( (T^{2} + \cdots + 46\!\cdots\!76)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 13\!\cdots\!44 \)
|
| $61$ |
\( T^{4} + \cdots + 75\!\cdots\!24 \)
|
| $67$ |
\( (T^{2} + \cdots + 59\!\cdots\!00)^{2} \)
|
| $71$ |
\( (T + 94292464)^{4} \)
|
| $73$ |
\( T^{4} + \cdots + 31\!\cdots\!64 \)
|
| $79$ |
\( (T^{2} + \cdots + 45\!\cdots\!00)^{2} \)
|
| $83$ |
\( (T^{2} - 19\!\cdots\!12)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 95\!\cdots\!00 \)
|
| $97$ |
\( (T^{2} - 16\!\cdots\!00)^{2} \)
|
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