Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1210,4,Mod(1,1210)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1210, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1210.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1210 = 2 \cdot 5 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1210.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: | \(71.3923111069\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} - 112x^{4} + 199x^{3} + 3005x^{2} - 4454x + 1444 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 11 \) |
| Twist minimal: | no (minimal twist has level 110) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} - 112x^{4} + 199x^{3} + 3005x^{2} - 4454x + 1444 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 37 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 61\nu + 43 ) / 9 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 112\nu^{3} + 41\nu^{2} + 3021\nu - 2168 ) / 54 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 2\nu^{4} - 114\nu^{3} + 167\nu^{2} + 3161\nu - 2538 ) / 54 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} + 37 \)
|
| \(\nu^{3}\) | \(=\) |
\( 9\beta_{3} + 3\beta_{2} + 61\beta _1 - 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( -27\beta_{5} + 27\beta_{4} - 9\beta_{3} + 186\beta_{2} + 9\beta _1 + 2152 \)
|
| \(\nu^{5}\) | \(=\) |
\( 54\beta_{4} + 1008\beta_{3} + 213\beta_{2} + 3811\beta _1 - 21 \)
|
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 |
|
−2.00000 | −8.13548 | 4.00000 | −5.00000 | 16.2710 | 24.7864 | −8.00000 | 39.1860 | 10.0000 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | −6.58257 | 4.00000 | −5.00000 | 13.1651 | −9.19445 | −8.00000 | 16.3303 | 10.0000 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 0.489967 | 4.00000 | −5.00000 | −0.979934 | −23.6928 | −8.00000 | −26.7599 | 10.0000 | |||||||||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 0.947647 | 4.00000 | −5.00000 | −1.89529 | 8.84290 | −8.00000 | −26.1020 | 10.0000 | |||||||||||||||||||||||||||||||||||||
| 1.5 | −2.00000 | 7.09261 | 4.00000 | −5.00000 | −14.1852 | 11.4708 | −8.00000 | 23.3051 | 10.0000 | |||||||||||||||||||||||||||||||||||||
| 1.6 | −2.00000 | 8.18783 | 4.00000 | −5.00000 | −16.3757 | −28.2129 | −8.00000 | 40.0406 | 10.0000 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(5\) | \( +1 \) |
| \(11\) | \( +1 \) |
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 | 1210.4.a.bd | 6 | |
| 11.b | odd | 2 | 1 | 1210.4.a.bg | 6 | ||
| 11.c | even | 5 | 2 | 110.4.g.c | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 110.4.g.c | ✓ | 12 | 11.c | even | 5 | 2 | |
| 1210.4.a.bd | 6 | 1.a | even | 1 | 1 | trivial | |
| 1210.4.a.bg | 6 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1210))\):
|
\( T_{3}^{6} - 2T_{3}^{5} - 112T_{3}^{4} + 199T_{3}^{3} + 3005T_{3}^{2} - 4454T_{3} + 1444 \)
|
|
\( T_{7}^{6} + 16T_{7}^{5} - 1005T_{7}^{4} - 11077T_{7}^{3} + 262269T_{7}^{2} + 837430T_{7} - 15452305 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{6} \)
|
| $3$ |
\( T^{6} - 2 T^{5} + \cdots + 1444 \)
|
| $5$ |
\( (T + 5)^{6} \)
|
| $7$ |
\( T^{6} + 16 T^{5} + \cdots - 15452305 \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( T^{6} + 74 T^{5} + \cdots - 286114419 \)
|
| $17$ |
\( T^{6} + \cdots - 1015234516 \)
|
| $19$ |
\( T^{6} + \cdots + 4068392031 \)
|
| $23$ |
\( T^{6} + \cdots + 35478920461 \)
|
| $29$ |
\( T^{6} + \cdots - 6227920756420 \)
|
| $31$ |
\( T^{6} + \cdots + 48878946172996 \)
|
| $37$ |
\( T^{6} + \cdots + 316174202431 \)
|
| $41$ |
\( T^{6} + \cdots - 3945511103249 \)
|
| $43$ |
\( T^{6} + \cdots + 78741334432420 \)
|
| $47$ |
\( T^{6} + \cdots - 82641459711689 \)
|
| $53$ |
\( T^{6} + \cdots + 12\!\cdots\!95 \)
|
| $59$ |
\( T^{6} + \cdots - 313956112801111 \)
|
| $61$ |
\( T^{6} + \cdots + 7628316133680 \)
|
| $67$ |
\( T^{6} + \cdots + 13\!\cdots\!44 \)
|
| $71$ |
\( T^{6} + \cdots - 35\!\cdots\!80 \)
|
| $73$ |
\( T^{6} + \cdots - 14\!\cdots\!76 \)
|
| $79$ |
\( T^{6} + \cdots - 41\!\cdots\!76 \)
|
| $83$ |
\( T^{6} + \cdots + 22\!\cdots\!84 \)
|
| $89$ |
\( T^{6} + \cdots + 27\!\cdots\!19 \)
|
| $97$ |
\( T^{6} + \cdots + 15\!\cdots\!56 \)
|
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