Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2166,4,Mod(1,2166)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2166, 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("2166.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2166 = 2 \cdot 3 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2166.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: | \(127.798137072\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{6} - 129x^{4} - 10x^{3} + 4680x^{2} + 1440x - 45999 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 114) |
| 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
\( x^{6} - 129x^{4} - 10x^{3} + 4680x^{2} + 1440x - 45999 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -1000\nu^{5} + 2799\nu^{4} + 101421\nu^{3} - 342782\nu^{2} - 1995117\nu + 7314348 ) / 805902 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2759\nu^{5} + 11349\nu^{4} + 284253\nu^{3} - 1172194\nu^{2} - 5724942\nu + 21094179 ) / 805902 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5791\nu^{5} - 32730\nu^{4} - 562749\nu^{3} + 3330101\nu^{2} + 9781947\nu - 60623964 ) / 805902 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 3783\nu^{5} - 23886\nu^{4} - 399928\nu^{3} + 2395726\nu^{2} + 8355713\nu - 42572381 ) / 268634 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 11\beta_{3} - 19\beta_{2} + 43 \)
|
| \(\nu^{3}\) | \(=\) |
\( -7\beta_{5} + 18\beta_{4} + 5\beta_{3} + 11\beta_{2} + 62\beta _1 + 14 \)
|
| \(\nu^{4}\) | \(=\) |
\( 71\beta_{5} - 22\beta_{4} + 903\beta_{3} - 1813\beta_{2} - 15\beta _1 + 2416 \)
|
| \(\nu^{5}\) | \(=\) |
\( -854\beta_{5} + 1764\beta_{4} - 736\beta_{3} + 1748\beta_{2} + 4251\beta _1 + 757 \)
|
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 | 3.00000 | 4.00000 | −14.8013 | 6.00000 | 24.6767 | 8.00000 | 9.00000 | −29.6025 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | 3.00000 | 4.00000 | −4.48228 | 6.00000 | 20.7901 | 8.00000 | 9.00000 | −8.96455 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 3.00000 | 4.00000 | −3.21746 | 6.00000 | −12.7567 | 8.00000 | 9.00000 | −6.43491 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 3.00000 | 4.00000 | 4.44414 | 6.00000 | −4.60340 | 8.00000 | 9.00000 | 8.88827 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 2.00000 | 3.00000 | 4.00000 | 4.66707 | 6.00000 | −31.8989 | 8.00000 | 9.00000 | 9.33414 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 2.00000 | 3.00000 | 4.00000 | 10.3898 | 6.00000 | −8.20787 | 8.00000 | 9.00000 | 20.7796 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(19\) | \(-1\) |
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 | 2166.4.a.bg | 6 | |
| 19.b | odd | 2 | 1 | 2166.4.a.bb | 6 | ||
| 19.f | odd | 18 | 2 | 114.4.i.b | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 114.4.i.b | ✓ | 12 | 19.f | odd | 18 | 2 | |
| 2166.4.a.bb | 6 | 19.b | odd | 2 | 1 | ||
| 2166.4.a.bg | 6 | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2166))\):
|
\( T_{5}^{6} + 3T_{5}^{5} - 195T_{5}^{4} + 91T_{5}^{3} + 5805T_{5}^{2} - 3033T_{5} - 45999 \)
|
|
\( T_{13}^{6} - 15T_{13}^{5} - 4332T_{13}^{4} + 89359T_{13}^{3} + 4371288T_{13}^{2} - 121406136T_{13} + 488321299 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{6} \)
|
| $3$ |
\( (T - 3)^{6} \)
|
| $5$ |
\( T^{6} + 3 T^{5} + \cdots - 45999 \)
|
| $7$ |
\( T^{6} + 12 T^{5} + \cdots + 7887993 \)
|
| $11$ |
\( T^{6} + 93 T^{5} + \cdots + 364890573 \)
|
| $13$ |
\( T^{6} - 15 T^{5} + \cdots + 488321299 \)
|
| $17$ |
\( T^{6} + \cdots + 941310795141 \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( T^{6} + \cdots + 14501187639 \)
|
| $29$ |
\( T^{6} + \cdots - 4297518078597 \)
|
| $31$ |
\( T^{6} + \cdots + 67023429443 \)
|
| $37$ |
\( T^{6} + \cdots + 28266080172299 \)
|
| $41$ |
\( T^{6} + \cdots + 16231227964857 \)
|
| $43$ |
\( T^{6} + \cdots - 10\!\cdots\!29 \)
|
| $47$ |
\( T^{6} + \cdots - 247197577449897 \)
|
| $53$ |
\( T^{6} + \cdots - 91487074606803 \)
|
| $59$ |
\( T^{6} + \cdots - 44\!\cdots\!43 \)
|
| $61$ |
\( T^{6} + \cdots - 46\!\cdots\!69 \)
|
| $67$ |
\( T^{6} + \cdots + 629472526097112 \)
|
| $71$ |
\( T^{6} + \cdots - 646008367547997 \)
|
| $73$ |
\( T^{6} + \cdots + 51\!\cdots\!73 \)
|
| $79$ |
\( T^{6} + \cdots + 18\!\cdots\!83 \)
|
| $83$ |
\( T^{6} + \cdots - 15\!\cdots\!51 \)
|
| $89$ |
\( T^{6} + \cdots + 98\!\cdots\!07 \)
|
| $97$ |
\( T^{6} + \cdots - 43\!\cdots\!91 \)
|
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