Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4598,2,Mod(1,4598)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4598, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("4598.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4598 = 2 \cdot 11^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4598.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: | \(36.7152148494\) |
| Analytic rank: | \(1\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 2x^{7} - 10x^{6} + 16x^{5} + 26x^{4} - 32x^{3} - 16x^{2} + 20x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 418) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 2x^{7} - 10x^{6} + 16x^{5} + 26x^{4} - 32x^{3} - 16x^{2} + 20x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 8\nu^{3} + 12\nu^{2} + 12\nu - 10 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 3\nu^{5} - 7\nu^{4} + 22\nu^{3} + 6\nu^{2} - 28\nu + 4 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{6} - 3\nu^{5} - 7\nu^{4} + 22\nu^{3} + 4\nu^{2} - 28\nu + 10 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{7} - 3\nu^{6} - 7\nu^{5} + 22\nu^{4} + 6\nu^{3} - 30\nu^{2} + 2\nu + 4 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 12\nu^{5} + 7\nu^{4} + 40\nu^{3} - 14\nu^{2} - 34\nu + 12 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -2\nu^{7} + 3\nu^{6} + 21\nu^{5} - 20\nu^{4} - 58\nu^{3} + 24\nu^{2} + 38\nu - 8 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{3} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + 2\beta_{3} + \beta_{2} + 5\beta _1 + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{7} + 4\beta_{6} - 10\beta_{4} + 8\beta_{3} + 2\beta _1 + 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( 12\beta_{7} + 16\beta_{6} + 8\beta_{5} - 16\beta_{4} + 20\beta_{3} + 10\beta_{2} + 32\beta _1 + 26 \)
|
| \(\nu^{6}\) | \(=\) |
\( 28\beta_{7} + 54\beta_{6} + 2\beta_{5} - 90\beta_{4} + 68\beta_{3} + 8\beta_{2} + 28\beta _1 + 138 \)
|
| \(\nu^{7}\) | \(=\) |
\( 118\beta_{7} + 180\beta_{6} + 58\beta_{5} - 186\beta_{4} + 186\beta_{3} + 88\beta_{2} + 232\beta _1 + 274 \)
|
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 |
|
−1.00000 | −3.37266 | 1.00000 | −0.689994 | 3.37266 | −0.481411 | −1.00000 | 8.37481 | 0.689994 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −1.00000 | −2.90436 | 1.00000 | −2.98197 | 2.90436 | 2.79511 | −1.00000 | 5.43531 | 2.98197 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.00000 | −2.16972 | 1.00000 | 1.79068 | 2.16972 | −4.71351 | −1.00000 | 1.70770 | −1.79068 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.00000 | 0.158731 | 1.00000 | 3.07692 | −0.158731 | −1.35199 | −1.00000 | −2.97480 | −3.07692 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −1.00000 | 0.912987 | 1.00000 | −1.77724 | −0.912987 | 5.04753 | −1.00000 | −2.16645 | 1.77724 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −1.00000 | 2.05958 | 1.00000 | 1.96511 | −2.05958 | −4.12502 | −1.00000 | 1.24187 | −1.96511 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | −1.00000 | 2.30094 | 1.00000 | 2.62639 | −2.30094 | −2.74200 | −1.00000 | 2.29431 | −2.62639 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | −1.00000 | 3.01451 | 1.00000 | −2.00989 | −3.01451 | −2.42872 | −1.00000 | 6.08725 | 2.00989 | |||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(11\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 4598.2.a.bx | 8 | |
| 11.b | odd | 2 | 1 | 4598.2.a.ca | 8 | ||
| 11.d | odd | 10 | 2 | 418.2.f.f | ✓ | 16 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 418.2.f.f | ✓ | 16 | 11.d | odd | 10 | 2 | |
| 4598.2.a.bx | 8 | 1.a | even | 1 | 1 | trivial | |
| 4598.2.a.ca | 8 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):
|
\( T_{3}^{8} - 22T_{3}^{6} + 8T_{3}^{5} + 154T_{3}^{4} - 108T_{3}^{3} - 332T_{3}^{2} + 332T_{3} - 44 \)
|
|
\( T_{5}^{8} - 2T_{5}^{7} - 18T_{5}^{6} + 32T_{5}^{5} + 107T_{5}^{4} - 152T_{5}^{3} - 256T_{5}^{2} + 224T_{5} + 209 \)
|
|
\( T_{7}^{8} + 8T_{7}^{7} - 12T_{7}^{6} - 246T_{7}^{5} - 501T_{7}^{4} + 902T_{7}^{3} + 4018T_{7}^{2} + 4152T_{7} + 1189 \)
|
|
\( T_{13}^{8} + 18T_{13}^{7} + 68T_{13}^{6} - 472T_{13}^{5} - 3496T_{13}^{4} - 608T_{13}^{3} + 34048T_{13}^{2} + 46080T_{13} - 33344 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 1)^{8} \)
|
| $3$ |
\( T^{8} - 22 T^{6} + \cdots - 44 \)
|
| $5$ |
\( T^{8} - 2 T^{7} + \cdots + 209 \)
|
| $7$ |
\( T^{8} + 8 T^{7} + \cdots + 1189 \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} + 18 T^{7} + \cdots - 33344 \)
|
| $17$ |
\( T^{8} + 4 T^{7} + \cdots - 4259 \)
|
| $19$ |
\( (T - 1)^{8} \)
|
| $23$ |
\( T^{8} - 12 T^{7} + \cdots - 96331 \)
|
| $29$ |
\( T^{8} + 14 T^{7} + \cdots + 1900 \)
|
| $31$ |
\( T^{8} + 2 T^{7} + \cdots - 5036 \)
|
| $37$ |
\( T^{8} + 22 T^{7} + \cdots - 810644 \)
|
| $41$ |
\( T^{8} + 8 T^{7} + \cdots + 2695396 \)
|
| $43$ |
\( T^{8} + 28 T^{7} + \cdots + 51869 \)
|
| $47$ |
\( T^{8} - 6 T^{7} + \cdots + 59921 \)
|
| $53$ |
\( T^{8} + 24 T^{7} + \cdots + 815956 \)
|
| $59$ |
\( T^{8} - 46 T^{7} + \cdots - 3488320 \)
|
| $61$ |
\( T^{8} - 24 T^{7} + \cdots - 4721299 \)
|
| $67$ |
\( T^{8} + 22 T^{7} + \cdots + 10564 \)
|
| $71$ |
\( T^{8} - 8 T^{7} + \cdots + 1691036 \)
|
| $73$ |
\( T^{8} + 16 T^{7} + \cdots + 1356976 \)
|
| $79$ |
\( T^{8} + 4 T^{7} + \cdots - 7215500 \)
|
| $83$ |
\( T^{8} + 12 T^{7} + \cdots + 47849 \)
|
| $89$ |
\( T^{8} + 28 T^{7} + \cdots - 31782080 \)
|
| $97$ |
\( T^{8} + 22 T^{7} + \cdots + 19644956 \)
|
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