Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [4851,2,Mod(1,4851)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(4851, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("4851.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 4851.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: | \(38.7354300205\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - 20x^{8} + 75x^{6} - 68x^{4} + 21x^{2} - 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| 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 a basis \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} - 20x^{8} + 75x^{6} - 68x^{4} + 21x^{2} - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{6} - 19\nu^{4} + 57\nu^{2} - 18 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{8} + 58\nu^{6} - 186\nu^{4} + 75\nu^{2} - 2 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 6\nu^{9} - 117\nu^{7} + 391\nu^{5} - 203\nu^{3} - 2\nu ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -3\nu^{8} + 59\nu^{6} - 205\nu^{4} + 130\nu^{2} - 14 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 7\nu^{9} - 140\nu^{7} + 524\nu^{5} - 457\nu^{3} + 90\nu ) / 4 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5\nu^{9} - 98\nu^{7} + 336\nu^{5} - 209\nu^{3} + 28\nu ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 10\nu^{8} - 197\nu^{6} + 691\nu^{4} - 475\nu^{2} + 74 ) / 4 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 10\nu^{9} - 197\nu^{7} + 691\nu^{5} - 475\nu^{3} + 78\nu ) / 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{5} + 2\beta_{3} + 2\beta_{2} + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -5\beta_{9} + 3\beta_{7} + 2\beta_{6} + \beta_{4} + 11\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 3\beta_{8} - 12\beta_{5} + 34\beta_{3} + 35\beta_{2} + 35 \)
|
| \(\nu^{5}\) | \(=\) |
\( -88\beta_{9} + 57\beta_{7} + 34\beta_{6} + 12\beta_{4} + 159\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 57\beta_{8} - 171\beta_{5} + 532\beta_{3} + 555\beta_{2} + 512 \)
|
| \(\nu^{7}\) | \(=\) |
\( -1391\beta_{9} + 916\beta_{7} + 532\beta_{6} + 171\beta_{4} + 2416\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 916\beta_{8} - 2587\beta_{5} + 8226\beta_{3} + 8610\beta_{2} + 7803 \)
|
| \(\nu^{9}\) | \(=\) |
\( -21559\beta_{9} + 14249\beta_{7} + 8226\beta_{6} + 2587\beta_{4} + 37123\beta_1 \)
|
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 |
|
−2.06220 | 0 | 2.25267 | −0.107797 | 0 | 0 | −0.521060 | 0 | 0.222298 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 1.2 | −2.06220 | 0 | 2.25267 | 0.107797 | 0 | 0 | −0.521060 | 0 | −0.222298 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.3 | −1.50315 | 0 | 0.259455 | −3.22776 | 0 | 0 | 2.61630 | 0 | 4.85180 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.4 | −1.50315 | 0 | 0.259455 | 3.22776 | 0 | 0 | 2.61630 | 0 | −4.85180 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.5 | −0.374626 | 0 | −1.85966 | −2.19650 | 0 | 0 | 1.44593 | 0 | 0.822868 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.6 | −0.374626 | 0 | −1.85966 | 2.19650 | 0 | 0 | 1.44593 | 0 | −0.822868 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.7 | 2.19994 | 0 | 2.83974 | −3.29701 | 0 | 0 | 1.84739 | 0 | −7.25322 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.8 | 2.19994 | 0 | 2.83974 | 3.29701 | 0 | 0 | 1.84739 | 0 | 7.25322 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.9 | 2.74003 | 0 | 5.50778 | −2.80625 | 0 | 0 | 9.61145 | 0 | −7.68923 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 1.10 | 2.74003 | 0 | 5.50778 | 2.80625 | 0 | 0 | 9.61145 | 0 | 7.68923 | |||||||||||||||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(1\) |
| \(11\) | \(-1\) |
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 | 4851.2.a.ch | yes | 10 |
| 3.b | odd | 2 | 1 | 4851.2.a.cf | ✓ | 10 | |
| 7.b | odd | 2 | 1 | inner | 4851.2.a.ch | yes | 10 |
| 21.c | even | 2 | 1 | 4851.2.a.cf | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 4851.2.a.cf | ✓ | 10 | 3.b | odd | 2 | 1 | |
| 4851.2.a.cf | ✓ | 10 | 21.c | even | 2 | 1 | |
| 4851.2.a.ch | yes | 10 | 1.a | even | 1 | 1 | trivial |
| 4851.2.a.ch | yes | 10 | 7.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4851))\):
|
\( T_{2}^{5} - T_{2}^{4} - 9T_{2}^{3} + 3T_{2}^{2} + 21T_{2} + 7 \)
|
|
\( T_{5}^{10} - 34T_{5}^{8} + 422T_{5}^{6} - 2252T_{5}^{4} + 4329T_{5}^{2} - 50 \)
|
|
\( T_{13}^{10} - 78T_{13}^{8} + 1918T_{13}^{6} - 19768T_{13}^{4} + 88233T_{13}^{2} - 138338 \)
|
|
\( T_{17}^{10} - 70T_{17}^{8} + 1764T_{17}^{6} - 19816T_{17}^{4} + 98672T_{17}^{2} - 170528 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{5} - T^{4} - 9 T^{3} + \cdots + 7)^{2} \)
|
| $3$ |
\( T^{10} \)
|
| $5$ |
\( T^{10} - 34 T^{8} + \cdots - 50 \)
|
| $7$ |
\( T^{10} \)
|
| $11$ |
\( (T - 1)^{10} \)
|
| $13$ |
\( T^{10} - 78 T^{8} + \cdots - 138338 \)
|
| $17$ |
\( T^{10} - 70 T^{8} + \cdots - 170528 \)
|
| $19$ |
\( T^{10} - 84 T^{8} + \cdots - 968 \)
|
| $23$ |
\( (T^{5} - 8 T^{4} + \cdots - 880)^{2} \)
|
| $29$ |
\( (T^{5} - 18 T^{4} + \cdots + 866)^{2} \)
|
| $31$ |
\( T^{10} - 268 T^{8} + \cdots - 22471808 \)
|
| $37$ |
\( (T^{5} + 4 T^{4} - 82 T^{3} + \cdots + 56)^{2} \)
|
| $41$ |
\( T^{10} - 162 T^{8} + \cdots - 301088 \)
|
| $43$ |
\( (T^{5} - 8 T^{4} + \cdots - 2336)^{2} \)
|
| $47$ |
\( T^{10} - 184 T^{8} + \cdots - 968 \)
|
| $53$ |
\( (T^{5} - 20 T^{4} + \cdots - 68288)^{2} \)
|
| $59$ |
\( T^{10} - 192 T^{8} + \cdots - 2508800 \)
|
| $61$ |
\( T^{10} - 210 T^{8} + \cdots - 19870208 \)
|
| $67$ |
\( (T^{5} - 4 T^{4} + \cdots - 63272)^{2} \)
|
| $71$ |
\( (T^{5} - 24 T^{4} + \cdots + 352)^{2} \)
|
| $73$ |
\( T^{10} + \cdots - 512960450 \)
|
| $79$ |
\( (T^{5} + 20 T^{4} + \cdots - 1088)^{2} \)
|
| $83$ |
\( T^{10} + \cdots - 145726592 \)
|
| $89$ |
\( T^{10} + \cdots - 907039232 \)
|
| $97$ |
\( T^{10} + \cdots - 1106757152 \)
|
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