Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [546,8,Mod(1,546)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(546, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("546.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 546.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: | \(170.562223914\) |
| 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: |
\( x^{6} - x^{5} - 264981x^{4} + 17519669x^{3} + 15113237808x^{2} - 1787613752904x - 21984668630064 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 264981x^{4} + 17519669x^{3} + 15113237808x^{2} - 1787613752904x - 21984668630064 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 664523 \nu^{5} + 1394854154 \nu^{4} - 75579217837 \nu^{3} - 255406399982096 \nu^{2} + \cdots + 48\!\cdots\!48 ) / 13\!\cdots\!80 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 46974981 \nu^{5} + 15607782922 \nu^{4} + 22419068741319 \nu^{3} + \cdots + 12\!\cdots\!44 ) / 27\!\cdots\!00 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 1679580891 \nu^{5} - 237883755898 \nu^{4} + 392168012091729 \nu^{3} + \cdots + 62\!\cdots\!44 ) / 10\!\cdots\!00 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 1123997957 \nu^{5} - 205637389526 \nu^{4} + 256239695015823 \nu^{3} + \cdots - 17\!\cdots\!92 ) / 54\!\cdots\!00 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 12\beta_{5} - 16\beta_{4} + 2\beta_{3} + 9\beta_{2} - 97\beta _1 + 88343 \)
|
| \(\nu^{3}\) | \(=\) |
\( -880\beta_{5} + 800\beta_{4} + 2540\beta_{3} - 2959\beta_{2} + 151193\beta _1 - 8652389 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2326902\beta_{5} - 3046716\beta_{4} + 223442\beta_{3} + 2927757\beta_{2} - 26613237\beta _1 + 13322692095 \)
|
| \(\nu^{5}\) | \(=\) |
\( - 372179492 \beta_{5} + 336610536 \beta_{4} + 588565048 \beta_{3} - 983423979 \beta_{2} + \cdots - 2337559494441 \)
|
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 |
|
8.00000 | −27.0000 | 64.0000 | −356.179 | −216.000 | −343.000 | 512.000 | 729.000 | −2849.43 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 8.00000 | −27.0000 | 64.0000 | −178.575 | −216.000 | −343.000 | 512.000 | 729.000 | −1428.60 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 8.00000 | −27.0000 | 64.0000 | −136.698 | −216.000 | −343.000 | 512.000 | 729.000 | −1093.58 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 8.00000 | −27.0000 | 64.0000 | 45.2455 | −216.000 | −343.000 | 512.000 | 729.000 | 361.964 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 8.00000 | −27.0000 | 64.0000 | 332.282 | −216.000 | −343.000 | 512.000 | 729.000 | 2658.26 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 8.00000 | −27.0000 | 64.0000 | 496.924 | −216.000 | −343.000 | 512.000 | 729.000 | 3975.39 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(1\) |
| \(7\) | \(1\) |
| \(13\) | \(-1\) |
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 | 546.8.a.r | ✓ | 6 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 546.8.a.r | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} - 203T_{5}^{5} - 247811T_{5}^{4} + 17743227T_{5}^{3} + 15081987830T_{5}^{2} + 740548691900T_{5} - 64956644533000 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(546))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8)^{6} \)
|
| $3$ |
\( (T + 27)^{6} \)
|
| $5$ |
\( T^{6} + \cdots - 64956644533000 \)
|
| $7$ |
\( (T + 343)^{6} \)
|
| $11$ |
\( T^{6} + \cdots - 69\!\cdots\!52 \)
|
| $13$ |
\( (T - 2197)^{6} \)
|
| $17$ |
\( T^{6} + \cdots + 13\!\cdots\!12 \)
|
| $19$ |
\( T^{6} + \cdots - 14\!\cdots\!52 \)
|
| $23$ |
\( T^{6} + \cdots - 94\!\cdots\!84 \)
|
| $29$ |
\( T^{6} + \cdots - 20\!\cdots\!12 \)
|
| $31$ |
\( T^{6} + \cdots + 14\!\cdots\!16 \)
|
| $37$ |
\( T^{6} + \cdots + 15\!\cdots\!24 \)
|
| $41$ |
\( T^{6} + \cdots - 10\!\cdots\!08 \)
|
| $43$ |
\( T^{6} + \cdots + 36\!\cdots\!88 \)
|
| $47$ |
\( T^{6} + \cdots - 10\!\cdots\!92 \)
|
| $53$ |
\( T^{6} + \cdots + 21\!\cdots\!96 \)
|
| $59$ |
\( T^{6} + \cdots - 10\!\cdots\!56 \)
|
| $61$ |
\( T^{6} + \cdots - 16\!\cdots\!52 \)
|
| $67$ |
\( T^{6} + \cdots - 92\!\cdots\!28 \)
|
| $71$ |
\( T^{6} + \cdots + 10\!\cdots\!52 \)
|
| $73$ |
\( T^{6} + \cdots + 19\!\cdots\!92 \)
|
| $79$ |
\( T^{6} + \cdots + 76\!\cdots\!36 \)
|
| $83$ |
\( T^{6} + \cdots - 43\!\cdots\!60 \)
|
| $89$ |
\( T^{6} + \cdots + 67\!\cdots\!52 \)
|
| $97$ |
\( T^{6} + \cdots - 58\!\cdots\!28 \)
|
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