Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,6,Mod(1,441)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 441.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: | \(70.7292645375\) |
| 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: |
\( x^{6} - 2x^{5} - 59x^{4} + 122x^{3} + 941x^{2} - 1856x - 2338 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 7^{3} \) |
| Twist minimal: | no (minimal twist has level 147) |
| 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
\( x^{6} - 2x^{5} - 59x^{4} + 122x^{3} + 941x^{2} - 1856x - 2338 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 28\nu^{5} - 35\nu^{4} - 546\nu^{3} + 742\nu^{2} - 7063\nu + 11802 ) / 1941 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -87\nu^{5} - 53\nu^{4} + 3961\nu^{3} + 2547\nu^{2} - 42269\nu - 27289 ) / 1941 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 158\nu^{5} - 521\nu^{4} - 6316\nu^{3} + 21656\nu^{2} + 33579\nu - 143678 ) / 1941 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 108\nu^{5} + 512\nu^{4} - 5988\nu^{3} - 17195\nu^{2} + 81453\nu + 71402 ) / 647 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -774\nu^{5} - 1297\nu^{4} + 39032\nu^{3} + 55188\nu^{2} - 447553\nu - 433643 ) / 1941 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{5} - \beta_{3} - 12\beta_{2} - 4\beta _1 + 5 ) / 28 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{5} + 4\beta_{4} + 5\beta_{3} - 4\beta_{2} - 4\beta _1 + 567 ) / 28 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 29\beta_{5} + 4\beta_{4} - 28\beta_{3} - 302\beta_{2} - 25\beta _1 - 129 ) / 28 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 99\beta_{5} + 176\beta_{4} + 155\beta_{3} + 80\beta_{2} + 74\beta _1 + 14843 ) / 28 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 862\beta_{5} + 192\beta_{4} - 737\beta_{3} - 8710\beta_{2} + 643\beta _1 - 9528 ) / 28 \)
|
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 |
|
−10.0089 | 0 | 68.1779 | 70.3512 | 0 | 0 | −362.101 | 0 | −704.138 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −8.20863 | 0 | 35.3816 | 29.2259 | 0 | 0 | −27.7583 | 0 | −239.905 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −3.38033 | 0 | −20.5734 | −54.5253 | 0 | 0 | 177.715 | 0 | 184.313 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 3.09163 | 0 | −22.4418 | 13.7926 | 0 | 0 | −168.314 | 0 | 42.6416 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 5.31815 | 0 | −3.71724 | 103.471 | 0 | 0 | −189.950 | 0 | 550.272 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 11.1881 | 0 | 93.1729 | −62.3150 | 0 | 0 | 684.407 | 0 | −697.185 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(1\) |
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 | 441.6.a.bb | 6 | |
| 3.b | odd | 2 | 1 | 147.6.a.n | ✓ | 6 | |
| 7.b | odd | 2 | 1 | 441.6.a.ba | 6 | ||
| 21.c | even | 2 | 1 | 147.6.a.o | yes | 6 | |
| 21.g | even | 6 | 2 | 147.6.e.p | 12 | ||
| 21.h | odd | 6 | 2 | 147.6.e.q | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 147.6.a.n | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 147.6.a.o | yes | 6 | 21.c | even | 2 | 1 | |
| 147.6.e.p | 12 | 21.g | even | 6 | 2 | ||
| 147.6.e.q | 12 | 21.h | odd | 6 | 2 | ||
| 441.6.a.ba | 6 | 7.b | odd | 2 | 1 | ||
| 441.6.a.bb | 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 \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):
|
\( T_{2}^{6} + 2T_{2}^{5} - 169T_{2}^{4} - 336T_{2}^{3} + 6472T_{2}^{2} + 4256T_{2} - 51088 \)
|
|
\( T_{5}^{6} - 100T_{5}^{5} - 6778T_{5}^{4} + 651312T_{5}^{3} + 9669292T_{5}^{2} - 959211664T_{5} + 9969962312 \)
|
|
\( T_{13}^{6} + 1352 T_{13}^{5} - 126598 T_{13}^{4} - 808285728 T_{13}^{3} - 312398022260 T_{13}^{2} - 5552830868704 T_{13} + 88\!\cdots\!04 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 2 T^{5} - 169 T^{4} + \cdots - 51088 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - 100 T^{5} + \cdots + 9969962312 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} + 604 T^{5} + \cdots - 55273527989696 \)
|
| $13$ |
\( T^{6} + 1352 T^{5} + \cdots + 88\!\cdots\!04 \)
|
| $17$ |
\( T^{6} - 3028 T^{5} + \cdots - 77\!\cdots\!12 \)
|
| $19$ |
\( T^{6} + 1728 T^{5} + \cdots - 37\!\cdots\!68 \)
|
| $23$ |
\( T^{6} - 4484 T^{5} + \cdots - 10\!\cdots\!48 \)
|
| $29$ |
\( T^{6} - 5320 T^{5} + \cdots - 45\!\cdots\!84 \)
|
| $31$ |
\( T^{6} + 3976 T^{5} + \cdots - 32\!\cdots\!04 \)
|
| $37$ |
\( T^{6} - 22680 T^{5} + \cdots - 79\!\cdots\!96 \)
|
| $41$ |
\( T^{6} - 28756 T^{5} + \cdots + 18\!\cdots\!56 \)
|
| $43$ |
\( T^{6} + 6768 T^{5} + \cdots - 28\!\cdots\!68 \)
|
| $47$ |
\( T^{6} - 51552 T^{5} + \cdots + 13\!\cdots\!28 \)
|
| $53$ |
\( T^{6} + 80884 T^{5} + \cdots + 93\!\cdots\!88 \)
|
| $59$ |
\( T^{6} - 8872 T^{5} + \cdots - 34\!\cdots\!52 \)
|
| $61$ |
\( T^{6} + 50896 T^{5} + \cdots - 18\!\cdots\!36 \)
|
| $67$ |
\( T^{6} - 6480 T^{5} + \cdots - 24\!\cdots\!32 \)
|
| $71$ |
\( T^{6} - 110852 T^{5} + \cdots - 29\!\cdots\!48 \)
|
| $73$ |
\( T^{6} + 64232 T^{5} + \cdots + 32\!\cdots\!84 \)
|
| $79$ |
\( T^{6} - 111696 T^{5} + \cdots - 10\!\cdots\!68 \)
|
| $83$ |
\( T^{6} - 101128 T^{5} + \cdots - 19\!\cdots\!68 \)
|
| $89$ |
\( T^{6} + 35012 T^{5} + \cdots + 48\!\cdots\!72 \)
|
| $97$ |
\( T^{6} + 70952 T^{5} + \cdots - 13\!\cdots\!88 \)
|
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