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: | \(1\) |
| 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: |
\( x^{6} - 187x^{4} + 9570x^{2} - 135576 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2} \) |
| Twist minimal: | no (minimal twist has level 63) |
| 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
\( x^{6} - 187x^{4} + 9570x^{2} - 135576 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 139\nu^{3} + 3318\nu ) / 84 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 62 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + 181\nu^{3} - 6888\nu ) / 42 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{4} - 139\nu^{2} + 3388 ) / 14 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 62 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} + 2\beta_{2} + 85\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 14\beta_{5} + 139\beta_{3} + 5230 \)
|
| \(\nu^{5}\) | \(=\) |
\( 139\beta_{4} + 362\beta_{2} + 8497\beta_1 \)
|
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 |
|
−10.6223 | 0 | 80.8340 | 67.4751 | 0 | 0 | −518.731 | 0 | −716.743 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −7.08933 | 0 | 18.2586 | −82.2041 | 0 | 0 | 97.4172 | 0 | 582.772 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −4.88952 | 0 | −8.09260 | 42.7504 | 0 | 0 | 196.034 | 0 | −209.029 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 4.88952 | 0 | −8.09260 | −42.7504 | 0 | 0 | −196.034 | 0 | −209.029 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 7.08933 | 0 | 18.2586 | 82.2041 | 0 | 0 | −97.4172 | 0 | 582.772 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 10.6223 | 0 | 80.8340 | −67.4751 | 0 | 0 | 518.731 | 0 | −716.743 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 441.6.a.bc | 6 | |
| 3.b | odd | 2 | 1 | inner | 441.6.a.bc | 6 | |
| 7.b | odd | 2 | 1 | 441.6.a.bd | 6 | ||
| 7.c | even | 3 | 2 | 63.6.e.f | ✓ | 12 | |
| 21.c | even | 2 | 1 | 441.6.a.bd | 6 | ||
| 21.h | odd | 6 | 2 | 63.6.e.f | ✓ | 12 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 63.6.e.f | ✓ | 12 | 7.c | even | 3 | 2 | |
| 63.6.e.f | ✓ | 12 | 21.h | odd | 6 | 2 | |
| 441.6.a.bc | 6 | 1.a | even | 1 | 1 | trivial | |
| 441.6.a.bc | 6 | 3.b | odd | 2 | 1 | inner | |
| 441.6.a.bd | 6 | 7.b | odd | 2 | 1 | ||
| 441.6.a.bd | 6 | 21.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\):
|
\( T_{2}^{6} - 187T_{2}^{4} + 9570T_{2}^{2} - 135576 \)
|
|
\( T_{5}^{6} - 13138T_{5}^{4} + 51437073T_{5}^{2} - 56228247936 \)
|
|
\( T_{13}^{3} - 77T_{13}^{2} - 783976T_{13} + 60080636 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 187 T^{4} + \cdots - 135576 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots - 56228247936 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( T^{6} + \cdots - 17\!\cdots\!76 \)
|
| $13$ |
\( (T^{3} - 77 T^{2} + \cdots + 60080636)^{2} \)
|
| $17$ |
\( T^{6} + \cdots - 14\!\cdots\!76 \)
|
| $19$ |
\( (T^{3} + 4711 T^{2} + \cdots + 3713875508)^{2} \)
|
| $23$ |
\( T^{6} + \cdots - 11\!\cdots\!84 \)
|
| $29$ |
\( T^{6} + \cdots - 75\!\cdots\!56 \)
|
| $31$ |
\( (T^{3} + 11711 T^{2} + \cdots - 85806884751)^{2} \)
|
| $37$ |
\( (T^{3} - 9091 T^{2} + \cdots + 32349623604)^{2} \)
|
| $41$ |
\( T^{6} + \cdots - 13\!\cdots\!64 \)
|
| $43$ |
\( (T^{3} + \cdots - 1643621929904)^{2} \)
|
| $47$ |
\( T^{6} + \cdots - 19\!\cdots\!64 \)
|
| $53$ |
\( T^{6} + \cdots - 16\!\cdots\!64 \)
|
| $59$ |
\( T^{6} + \cdots - 31\!\cdots\!04 \)
|
| $61$ |
\( (T^{3} + \cdots + 5346133312352)^{2} \)
|
| $67$ |
\( (T^{3} + \cdots + 7112548657724)^{2} \)
|
| $71$ |
\( T^{6} + \cdots - 30\!\cdots\!44 \)
|
| $73$ |
\( (T^{3} - 50029 T^{2} + \cdots - 937824660612)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots - 111559545344717)^{2} \)
|
| $83$ |
\( T^{6} + \cdots - 11\!\cdots\!64 \)
|
| $89$ |
\( T^{6} + \cdots - 60\!\cdots\!96 \)
|
| $97$ |
\( (T^{3} + \cdots - 546802680855102)^{2} \)
|
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