Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [98,10,Mod(1,98)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(98, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("98.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 98.a (trivial) |
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: | yes |
| Analytic conductor: | \(50.4735119441\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{106}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 106 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\beta = 4\sqrt{106}\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
16.0000 | −123.548 | 256.000 | 1276.66 | −1976.76 | 0 | 4096.00 | −4419.00 | 20426.5 | ||||||||||||||||||||||||
| 1.2 | 16.0000 | 123.548 | 256.000 | −1276.66 | 1976.76 | 0 | 4096.00 | −4419.00 | −20426.5 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 98.10.a.f | ✓ | 2 |
| 7.b | odd | 2 | 1 | inner | 98.10.a.f | ✓ | 2 |
| 7.c | even | 3 | 2 | 98.10.c.g | 4 | ||
| 7.d | odd | 6 | 2 | 98.10.c.g | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 98.10.a.f | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 98.10.a.f | ✓ | 2 | 7.b | odd | 2 | 1 | inner |
| 98.10.c.g | 4 | 7.c | even | 3 | 2 | ||
| 98.10.c.g | 4 | 7.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} - 15264 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(98))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 16)^{2} \)
|
| $3$ |
\( T^{2} - 15264 \)
|
| $5$ |
\( T^{2} - 1629856 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( (T + 27644)^{2} \)
|
| $13$ |
\( T^{2} - 6940929184 \)
|
| $17$ |
\( T^{2} - 307175727744 \)
|
| $19$ |
\( T^{2} - 117994588576 \)
|
| $23$ |
\( (T + 535704)^{2} \)
|
| $29$ |
\( (T + 2591006)^{2} \)
|
| $31$ |
\( T^{2} - 37176297809536 \)
|
| $37$ |
\( (T + 17807910)^{2} \)
|
| $41$ |
\( T^{2} - 932312422855296 \)
|
| $43$ |
\( (T + 8759756)^{2} \)
|
| $47$ |
\( T^{2} - 743814320619136 \)
|
| $53$ |
\( (T + 83488490)^{2} \)
|
| $59$ |
\( T^{2} - 66\!\cdots\!96 \)
|
| $61$ |
\( T^{2} - 24\!\cdots\!00 \)
|
| $67$ |
\( (T + 185492924)^{2} \)
|
| $71$ |
\( (T - 370270672)^{2} \)
|
| $73$ |
\( T^{2} - 60\!\cdots\!16 \)
|
| $79$ |
\( (T + 519907256)^{2} \)
|
| $83$ |
\( T^{2} - 89\!\cdots\!24 \)
|
| $89$ |
\( T^{2} - 88\!\cdots\!44 \)
|
| $97$ |
\( T^{2} - 60\!\cdots\!00 \)
|
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