Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [98,14,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, 14, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("98.1");
S:= CuspForms(chi, 14);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 98 = 2 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| 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: | \(105.086310373\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 692090x^{2} - 221993874x - 1534236795 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3\cdot 7^{2} \) |
| Twist minimal: | no (minimal twist has level 14) |
| 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 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
\( x^{4} - 2x^{3} - 692090x^{2} - 221993874x - 1534236795 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 37\nu^{3} - 15203\nu^{2} - 18352979\nu - 928867275 ) / 480060 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -223\nu^{3} + 152177\nu^{2} + 63870761\nu - 15332437575 ) / 480060 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -33\nu^{3} + 20287\nu^{2} + 15655751\nu - 1499456175 ) / 13335 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - 4\beta_{2} + 8\beta _1 + 170 ) / 336 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 193\beta_{3} - 106\beta_{2} + 5558\beta _1 + 29070590 ) / 84 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 813235\beta_{3} - 2158324\beta_{2} + 17462624\beta _1 + 56298922430 ) / 336 \)
|
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 |
|
64.0000 | −1711.34 | 4096.00 | 25091.5 | −109526. | 0 | 262144. | 1.33435e6 | 1.60586e6 | ||||||||||||||||||||||||||||||
| 1.2 | 64.0000 | −1359.31 | 4096.00 | −58022.5 | −86996.0 | 0 | 262144. | 253406. | −3.71344e6 | |||||||||||||||||||||||||||||||
| 1.3 | 64.0000 | 603.940 | 4096.00 | 5043.92 | 38652.2 | 0 | 262144. | −1.22958e6 | 322811. | |||||||||||||||||||||||||||||||
| 1.4 | 64.0000 | 2284.71 | 4096.00 | −36512.9 | 146221. | 0 | 262144. | 3.62557e6 | −2.33683e6 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(7\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 98.14.a.k | 4 | |
| 7.b | odd | 2 | 1 | 98.14.a.l | 4 | ||
| 7.c | even | 3 | 2 | 98.14.c.n | 8 | ||
| 7.d | odd | 6 | 2 | 14.14.c.a | ✓ | 8 | |
| 21.g | even | 6 | 2 | 126.14.g.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 14.14.c.a | ✓ | 8 | 7.d | odd | 6 | 2 | |
| 98.14.a.k | 4 | 1.a | even | 1 | 1 | trivial | |
| 98.14.a.l | 4 | 7.b | odd | 2 | 1 | ||
| 98.14.c.n | 8 | 7.c | even | 3 | 2 | ||
| 126.14.g.d | 8 | 21.g | even | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 182T_{3}^{3} - 5163960T_{3}^{2} - 2482729326T_{3} + 3209810951295 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(98))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 64)^{4} \)
|
| $3$ |
\( T^{4} + \cdots + 3209810951295 \)
|
| $5$ |
\( T^{4} + \cdots + 26\!\cdots\!25 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots - 54\!\cdots\!45 \)
|
| $13$ |
\( T^{4} + \cdots - 94\!\cdots\!80 \)
|
| $17$ |
\( T^{4} + \cdots + 13\!\cdots\!05 \)
|
| $19$ |
\( T^{4} + \cdots - 36\!\cdots\!65 \)
|
| $23$ |
\( T^{4} + \cdots + 42\!\cdots\!91 \)
|
| $29$ |
\( T^{4} + \cdots + 47\!\cdots\!28 \)
|
| $31$ |
\( T^{4} + \cdots - 58\!\cdots\!05 \)
|
| $37$ |
\( T^{4} + \cdots - 23\!\cdots\!19 \)
|
| $41$ |
\( T^{4} + \cdots - 10\!\cdots\!08 \)
|
| $43$ |
\( T^{4} + \cdots - 18\!\cdots\!00 \)
|
| $47$ |
\( T^{4} + \cdots - 28\!\cdots\!45 \)
|
| $53$ |
\( T^{4} + \cdots - 25\!\cdots\!79 \)
|
| $59$ |
\( T^{4} + \cdots + 38\!\cdots\!75 \)
|
| $61$ |
\( T^{4} + \cdots + 11\!\cdots\!25 \)
|
| $67$ |
\( T^{4} + \cdots - 31\!\cdots\!85 \)
|
| $71$ |
\( T^{4} + \cdots - 79\!\cdots\!40 \)
|
| $73$ |
\( T^{4} + \cdots - 45\!\cdots\!95 \)
|
| $79$ |
\( T^{4} + \cdots + 58\!\cdots\!75 \)
|
| $83$ |
\( T^{4} + \cdots + 13\!\cdots\!00 \)
|
| $89$ |
\( T^{4} + \cdots - 79\!\cdots\!55 \)
|
| $97$ |
\( T^{4} + \cdots - 27\!\cdots\!00 \)
|
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