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: | \(0\) |
| 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: |
\( x^{8} - 16x^{6} - 4x^{5} + 75x^{4} + 32x^{3} - 90x^{2} - 28x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 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
\( x^{8} - 16x^{6} - 4x^{5} + 75x^{4} + 32x^{3} - 90x^{2} - 28x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - \nu^{6} - 15\nu^{5} + 11\nu^{4} + 64\nu^{3} - 24\nu^{2} - 66\nu - 2 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} - 2\nu^{5} - 13\nu^{4} + 20\nu^{3} + 44\nu^{2} - 40\nu - 18 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} + \nu^{6} - 19\nu^{5} - 15\nu^{4} + 108\nu^{3} + 56\nu^{2} - 166\nu - 18 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3\nu^{7} - 2\nu^{6} - 45\nu^{5} + 14\nu^{4} + 196\nu^{3} + 8\nu^{2} - 218\nu - 32 ) / 4 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5\nu^{7} + 3\nu^{6} - 87\nu^{5} - 53\nu^{4} + 448\nu^{3} + 232\nu^{2} - 610\nu - 98 ) / 8 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 7\nu^{7} - 3\nu^{6} - 109\nu^{5} + 13\nu^{4} + 496\nu^{3} + 72\nu^{2} - 582\nu - 78 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{7} + \beta_{5} + \beta_{3} + \beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{7} + 2\beta_{5} + \beta_{4} + 5\beta _1 + 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -9\beta_{7} + \beta_{6} + 8\beta_{5} + 6\beta_{3} + 10\beta_{2} + 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( -25\beta_{7} + 3\beta_{6} + 24\beta_{5} + 8\beta_{4} - 2\beta_{3} + 30\beta _1 + 12 \)
|
| \(\nu^{6}\) | \(=\) |
\( -83\beta_{7} + 19\beta_{6} + 68\beta_{5} - 4\beta_{4} + 34\beta_{3} + 86\beta_{2} + 124 \)
|
| \(\nu^{7}\) | \(=\) |
\( -255\beta_{7} + 53\beta_{6} + 236\beta_{5} + 52\beta_{4} - 38\beta_{3} + 8\beta_{2} + 196\beta _1 + 116 \)
|
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 | −2.48823 | 1.00000 | 3.25915 | −2.48823 | 4.58972 | 1.00000 | 3.19127 | 3.25915 | ||||||||||||||||||||||||||||||||||||||||||
| 1.2 | 1.00000 | −2.11129 | 1.00000 | −2.75070 | −2.11129 | 3.66381 | 1.00000 | 1.45756 | −2.75070 | |||||||||||||||||||||||||||||||||||||||||||
| 1.3 | 1.00000 | −0.103249 | 1.00000 | −3.91798 | −0.103249 | −4.45623 | 1.00000 | −2.98934 | −3.91798 | |||||||||||||||||||||||||||||||||||||||||||
| 1.4 | 1.00000 | 1.55594 | 1.00000 | 2.24033 | 1.55594 | −2.27752 | 1.00000 | −0.579065 | 2.24033 | |||||||||||||||||||||||||||||||||||||||||||
| 1.5 | 1.00000 | 1.67352 | 1.00000 | −0.566489 | 1.67352 | 3.40033 | 1.00000 | −0.199325 | −0.566489 | |||||||||||||||||||||||||||||||||||||||||||
| 1.6 | 1.00000 | 2.79123 | 1.00000 | 4.02325 | 2.79123 | 0.274917 | 1.00000 | 4.79095 | 4.02325 | |||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 1.00000 | 3.30349 | 1.00000 | 1.88260 | 3.30349 | 2.41197 | 1.00000 | 7.91304 | 1.88260 | |||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 1.00000 | 3.37860 | 1.00000 | −4.17017 | 3.37860 | −3.60700 | 1.00000 | 8.41492 | −4.17017 | |||||||||||||||||||||||||||||||||||||||||||
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.cb | yes | 8 |
| 11.b | odd | 2 | 1 | 4598.2.a.by | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4598.2.a.by | ✓ | 8 | 11.b | odd | 2 | 1 | |
| 4598.2.a.cb | yes | 8 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4598))\):
|
\( T_{3}^{8} - 8T_{3}^{7} + 9T_{3}^{6} + 72T_{3}^{5} - 183T_{3}^{4} - 80T_{3}^{3} + 557T_{3}^{2} - 368T_{3} - 44 \)
|
|
\( T_{5}^{8} - 38T_{5}^{6} + 12T_{5}^{5} + 470T_{5}^{4} - 288T_{5}^{3} - 1984T_{5}^{2} + 1536T_{5} + 1408 \)
|
|
\( T_{7}^{8} - 4T_{7}^{7} - 37T_{7}^{6} + 152T_{7}^{5} + 387T_{7}^{4} - 1716T_{7}^{3} - 867T_{7}^{2} + 5408T_{7} - 1388 \)
|
|
\( T_{13}^{8} + 12T_{13}^{7} + 34T_{13}^{6} - 84T_{13}^{5} - 451T_{13}^{4} - 72T_{13}^{3} + 1088T_{13}^{2} + 96T_{13} - 368 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{8} \)
|
| $3$ |
\( T^{8} - 8 T^{7} + \cdots - 44 \)
|
| $5$ |
\( T^{8} - 38 T^{6} + \cdots + 1408 \)
|
| $7$ |
\( T^{8} - 4 T^{7} + \cdots - 1388 \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} + 12 T^{7} + \cdots - 368 \)
|
| $17$ |
\( T^{8} + 4 T^{7} + \cdots - 92 \)
|
| $19$ |
\( (T + 1)^{8} \)
|
| $23$ |
\( T^{8} - 14 T^{7} + \cdots + 35296 \)
|
| $29$ |
\( T^{8} + 2 T^{7} + \cdots + 964 \)
|
| $31$ |
\( T^{8} - 132 T^{6} + \cdots - 9344 \)
|
| $37$ |
\( T^{8} - 24 T^{7} + \cdots + 29569 \)
|
| $41$ |
\( T^{8} - 8 T^{7} + \cdots - 3872 \)
|
| $43$ |
\( T^{8} - 8 T^{7} + \cdots - 67328 \)
|
| $47$ |
\( T^{8} + 16 T^{7} + \cdots - 2659283 \)
|
| $53$ |
\( T^{8} - 36 T^{7} + \cdots + 2770816 \)
|
| $59$ |
\( T^{8} + 24 T^{7} + \cdots - 20032604 \)
|
| $61$ |
\( T^{8} - 12 T^{7} + \cdots + 16192 \)
|
| $67$ |
\( T^{8} - 16 T^{7} + \cdots + 457168 \)
|
| $71$ |
\( T^{8} - 4 T^{7} + \cdots + 3806464 \)
|
| $73$ |
\( T^{8} + 20 T^{7} + \cdots - 37382204 \)
|
| $79$ |
\( T^{8} + 12 T^{7} + \cdots + 51808 \)
|
| $83$ |
\( T^{8} - 20 T^{7} + \cdots - 251648 \)
|
| $89$ |
\( T^{8} - 8 T^{7} + \cdots - 320672 \)
|
| $97$ |
\( T^{8} - 4 T^{7} + \cdots - 1921664 \)
|
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