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: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 148556x^{3} - 20997404x^{2} - 256427072x + 44264019648 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 148556x^{3} - 20997404x^{2} - 256427072x + 44264019648 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -67\nu^{4} + 4229\nu^{3} + 9590106\nu^{2} + 834162828\nu - 17355924152 ) / 58323928 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 57\nu^{4} - 7461\nu^{3} - 7789168\nu^{2} - 221698300\nu + 41732843296 ) / 8972912 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 917\nu^{4} - 74621\nu^{3} - 132645092\nu^{2} - 7627377340\nu + 531906698624 ) / 116647856 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 2039\nu^{4} - 212403\nu^{3} - 280855608\nu^{2} - 13688812132\nu + 934627614848 ) / 116647856 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{3} - 2\beta_{2} + 11\beta _1 + 3 ) / 24 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 70\beta_{4} - 47\beta_{3} - 101\beta_{2} + 185\beta _1 + 178251 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 44971\beta_{4} + 22585\beta_{3} - 96416\beta_{2} + 305687\beta _1 + 76083603 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 12995059\beta_{4} - 4458338\beta_{3} - 20612135\beta_{2} + 50635028\beta _1 + 27142832862 ) / 3 \)
|
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 | −340.441 | −216.000 | −343.000 | −512.000 | 729.000 | 2723.53 | |||||||||||||||||||||||||||||||||
| 1.2 | −8.00000 | 27.0000 | 64.0000 | −78.1553 | −216.000 | −343.000 | −512.000 | 729.000 | 625.242 | ||||||||||||||||||||||||||||||||||
| 1.3 | −8.00000 | 27.0000 | 64.0000 | 186.173 | −216.000 | −343.000 | −512.000 | 729.000 | −1489.38 | ||||||||||||||||||||||||||||||||||
| 1.4 | −8.00000 | 27.0000 | 64.0000 | 242.602 | −216.000 | −343.000 | −512.000 | 729.000 | −1940.82 | ||||||||||||||||||||||||||||||||||
| 1.5 | −8.00000 | 27.0000 | 64.0000 | 498.821 | −216.000 | −343.000 | −512.000 | 729.000 | −3990.57 | ||||||||||||||||||||||||||||||||||
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.k | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 546.8.a.k | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{5} - 509T_{5}^{4} - 102633T_{5}^{3} + 61226049T_{5}^{2} - 2538310720T_{5} - 599455015900 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(546))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 8)^{5} \)
|
| $3$ |
\( (T - 27)^{5} \)
|
| $5$ |
\( T^{5} + \cdots - 599455015900 \)
|
| $7$ |
\( (T + 343)^{5} \)
|
| $11$ |
\( T^{5} + \cdots + 35\!\cdots\!72 \)
|
| $13$ |
\( (T - 2197)^{5} \)
|
| $17$ |
\( T^{5} + \cdots - 57\!\cdots\!68 \)
|
| $19$ |
\( T^{5} + \cdots - 39\!\cdots\!04 \)
|
| $23$ |
\( T^{5} + \cdots + 88\!\cdots\!16 \)
|
| $29$ |
\( T^{5} + \cdots - 19\!\cdots\!76 \)
|
| $31$ |
\( T^{5} + \cdots - 18\!\cdots\!08 \)
|
| $37$ |
\( T^{5} + \cdots + 49\!\cdots\!32 \)
|
| $41$ |
\( T^{5} + \cdots + 81\!\cdots\!16 \)
|
| $43$ |
\( T^{5} + \cdots - 72\!\cdots\!00 \)
|
| $47$ |
\( T^{5} + \cdots + 91\!\cdots\!12 \)
|
| $53$ |
\( T^{5} + \cdots + 28\!\cdots\!52 \)
|
| $59$ |
\( T^{5} + \cdots - 20\!\cdots\!48 \)
|
| $61$ |
\( T^{5} + \cdots - 55\!\cdots\!12 \)
|
| $67$ |
\( T^{5} + \cdots - 25\!\cdots\!56 \)
|
| $71$ |
\( T^{5} + \cdots + 76\!\cdots\!40 \)
|
| $73$ |
\( T^{5} + \cdots - 24\!\cdots\!76 \)
|
| $79$ |
\( T^{5} + \cdots + 21\!\cdots\!00 \)
|
| $83$ |
\( T^{5} + \cdots + 87\!\cdots\!68 \)
|
| $89$ |
\( T^{5} + \cdots + 67\!\cdots\!64 \)
|
| $97$ |
\( T^{5} + \cdots + 11\!\cdots\!64 \)
|
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