Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [242,8,Mod(1,242)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(242, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("242.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 242 = 2 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 242.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: | \(75.5971761672\) |
| 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: |
\( x^{6} - x^{5} - 1169x^{4} + 9903x^{3} + 334564x^{2} - 5434366x + 21800869 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 11^{3} \) |
| Twist minimal: | no (minimal twist has level 22) |
| 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
\( x^{6} - x^{5} - 1169x^{4} + 9903x^{3} + 334564x^{2} - 5434366x + 21800869 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -30701\nu^{5} - 289817\nu^{4} + 33074778\nu^{3} + 30085506\nu^{2} - 10075302941\nu + 69133433878 ) / 53535015 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 41257\nu^{5} + 302084\nu^{4} - 45764076\nu^{3} + 14364708\nu^{2} + 13912205332\nu - 100241769171 ) / 29741675 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -62692\nu^{5} - 561974\nu^{4} + 65957906\nu^{3} + 56652612\nu^{2} - 19371394892\nu + 129112518856 ) / 29741675 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 285872 \nu^{5} - 2302214 \nu^{4} + 322655496 \nu^{3} + 112080657 \nu^{2} - 99944100797 \nu + 684279028966 ) / 89225025 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 1019902 \nu^{5} + 7792849 \nu^{4} - 1126419486 \nu^{3} + 151231338 \nu^{2} + 344495777227 \nu - 2450833547006 ) / 89225025 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - 7\beta_{2} + 3\beta _1 + 1 ) / 22 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -9\beta_{5} + 14\beta_{4} + 20\beta_{3} + 91\beta_{2} - 111\beta _1 + 8647 ) / 22 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 593\beta_{5} + 96\beta_{4} - 185\beta_{3} - 4069\beta_{2} + 2121\beta _1 - 97809 ) / 22 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -10317\beta_{5} + 7332\beta_{4} + 15330\beta_{3} + 81053\beta_{2} - 106893\beta _1 + 5053369 ) / 22 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 199624\beta_{5} + 23964\beta_{4} - 162210\beta_{3} - 1381175\beta_{2} + 1081200\beta _1 - 47694759 ) / 11 \)
|
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 | −64.3514 | 64.0000 | 46.6686 | −514.811 | −662.930 | 512.000 | 1954.10 | 373.349 | ||||||||||||||||||||||||||||||||||||
| 1.2 | 8.00000 | −62.4780 | 64.0000 | −259.683 | −499.824 | −756.554 | 512.000 | 1716.50 | −2077.46 | |||||||||||||||||||||||||||||||||||||
| 1.3 | 8.00000 | 6.83217 | 64.0000 | −47.4682 | 54.6573 | 1463.04 | 512.000 | −2140.32 | −379.746 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 8.00000 | 11.2980 | 64.0000 | 185.480 | 90.3843 | 152.908 | 512.000 | −2059.35 | 1483.84 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 8.00000 | 32.3753 | 64.0000 | 130.567 | 259.003 | −1201.79 | 512.000 | −1138.84 | 1044.54 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 8.00000 | 63.3239 | 64.0000 | −435.565 | 506.591 | 77.3271 | 512.000 | 1822.91 | −3484.52 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(11\) | \(1\) |
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 | 242.8.a.q | 6 | |
| 11.b | odd | 2 | 1 | 242.8.a.o | 6 | ||
| 11.c | even | 5 | 2 | 22.8.c.a | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 22.8.c.a | ✓ | 12 | 11.c | even | 5 | 2 | |
| 242.8.a.o | 6 | 11.b | odd | 2 | 1 | ||
| 242.8.a.q | 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 \(S_{8}^{\mathrm{new}}(\Gamma_0(242))\):
|
\( T_{3}^{6} + 13T_{3}^{5} - 6554T_{3}^{4} - 12351T_{3}^{3} + 10036026T_{3}^{2} - 159070095T_{3} + 636250869 \)
|
|
\( T_{7}^{6} + 928 T_{7}^{5} - 1882403 T_{7}^{4} - 2238439176 T_{7}^{3} - 296294672471 T_{7}^{2} + \cdots - 10\!\cdots\!80 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 8)^{6} \)
|
| $3$ |
\( T^{6} + 13 T^{5} + \cdots + 636250869 \)
|
| $5$ |
\( T^{6} + \cdots - 6068136136100 \)
|
| $7$ |
\( T^{6} + \cdots - 10\!\cdots\!80 \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( T^{6} + \cdots - 40\!\cdots\!20 \)
|
| $17$ |
\( T^{6} + \cdots + 16\!\cdots\!55 \)
|
| $19$ |
\( T^{6} + \cdots + 12\!\cdots\!95 \)
|
| $23$ |
\( T^{6} + \cdots - 12\!\cdots\!16 \)
|
| $29$ |
\( T^{6} + \cdots - 12\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots + 63\!\cdots\!80 \)
|
| $37$ |
\( T^{6} + \cdots + 87\!\cdots\!04 \)
|
| $41$ |
\( T^{6} + \cdots - 12\!\cdots\!01 \)
|
| $43$ |
\( T^{6} + \cdots + 13\!\cdots\!00 \)
|
| $47$ |
\( T^{6} + \cdots + 34\!\cdots\!20 \)
|
| $53$ |
\( T^{6} + \cdots - 13\!\cdots\!80 \)
|
| $59$ |
\( T^{6} + \cdots - 34\!\cdots\!05 \)
|
| $61$ |
\( T^{6} + \cdots - 21\!\cdots\!00 \)
|
| $67$ |
\( T^{6} + \cdots - 17\!\cdots\!84 \)
|
| $71$ |
\( T^{6} + \cdots + 39\!\cdots\!80 \)
|
| $73$ |
\( T^{6} + \cdots + 62\!\cdots\!11 \)
|
| $79$ |
\( T^{6} + \cdots - 30\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots - 12\!\cdots\!95 \)
|
| $89$ |
\( T^{6} + \cdots - 34\!\cdots\!20 \)
|
| $97$ |
\( T^{6} + \cdots + 67\!\cdots\!51 \)
|
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