Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [405,4,Mod(136,405)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(405, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("405.136");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 405 = 3^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 405.e (of order \(3\), degree \(2\), not minimal) |
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: | no |
| Analytic conductor: | \(23.8957735523\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.95327307.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} + 20x^{4} - 35x^{3} + 85x^{2} - 68x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3^{3} \) |
| Twist minimal: | no (minimal twist has level 135) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{6} - 3x^{5} + 20x^{4} - 35x^{3} + 85x^{2} - 68x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} + 14\nu^{2} - 13\nu + 24 ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{4} + 2\nu^{3} - 8\nu^{2} + 7\nu + 12 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{5} - 5\nu^{4} + 38\nu^{3} - 52\nu^{2} + 149\nu - 64 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 11\nu^{5} - 28\nu^{4} + 206\nu^{3} - 287\nu^{2} + 786\nu - 356 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -7\nu^{5} + 18\nu^{4} - 132\nu^{3} + 183\nu^{2} - 514\nu + 220 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{5} - 2\beta_{4} - 3\beta_{3} - \beta_{2} - \beta _1 + 6 ) / 9 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{5} - 2\beta_{4} - 3\beta_{3} + 2\beta_{2} + 5\beta _1 - 48 ) / 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 17\beta_{5} + 8\beta_{4} + 75\beta_{3} + 13\beta_{2} + 13\beta _1 - 114 ) / 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 12\beta_{5} + 6\beta_{4} + 51\beta_{3} - 5\beta_{2} - 7\beta _1 + 102 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -136\beta_{5} - 10\beta_{4} - 879\beta_{3} - 158\beta_{2} - 95\beta _1 + 1524 ) / 9 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/405\mathbb{Z}\right)^\times\).
| \(n\) | \(82\) | \(326\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 136.1 |
|
−2.60034 | + | 4.50391i | 0 | −9.52349 | − | 16.4952i | 2.50000 | + | 4.33013i | 0 | −12.2007 | + | 21.1322i | 57.4517 | 0 | −26.0034 | ||||||||||||||||||||||||||||
| 136.2 | −0.129356 | + | 0.224051i | 0 | 3.96653 | + | 6.87024i | 2.50000 | + | 4.33013i | 0 | −7.25871 | + | 12.5725i | −4.12208 | 0 | −1.29356 | |||||||||||||||||||||||||||||
| 136.3 | 2.22969 | − | 3.86194i | 0 | −5.94305 | − | 10.2937i | 2.50000 | + | 4.33013i | 0 | −2.54062 | + | 4.40048i | −17.3296 | 0 | 22.2969 | |||||||||||||||||||||||||||||
| 271.1 | −2.60034 | − | 4.50391i | 0 | −9.52349 | + | 16.4952i | 2.50000 | − | 4.33013i | 0 | −12.2007 | − | 21.1322i | 57.4517 | 0 | −26.0034 | |||||||||||||||||||||||||||||
| 271.2 | −0.129356 | − | 0.224051i | 0 | 3.96653 | − | 6.87024i | 2.50000 | − | 4.33013i | 0 | −7.25871 | − | 12.5725i | −4.12208 | 0 | −1.29356 | |||||||||||||||||||||||||||||
| 271.3 | 2.22969 | + | 3.86194i | 0 | −5.94305 | + | 10.2937i | 2.50000 | − | 4.33013i | 0 | −2.54062 | − | 4.40048i | −17.3296 | 0 | 22.2969 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 405.4.e.r | 6 | |
| 3.b | odd | 2 | 1 | 405.4.e.t | 6 | ||
| 9.c | even | 3 | 1 | 135.4.a.g | yes | 3 | |
| 9.c | even | 3 | 1 | inner | 405.4.e.r | 6 | |
| 9.d | odd | 6 | 1 | 135.4.a.f | ✓ | 3 | |
| 9.d | odd | 6 | 1 | 405.4.e.t | 6 | ||
| 36.f | odd | 6 | 1 | 2160.4.a.be | 3 | ||
| 36.h | even | 6 | 1 | 2160.4.a.bm | 3 | ||
| 45.h | odd | 6 | 1 | 675.4.a.r | 3 | ||
| 45.j | even | 6 | 1 | 675.4.a.q | 3 | ||
| 45.k | odd | 12 | 2 | 675.4.b.k | 6 | ||
| 45.l | even | 12 | 2 | 675.4.b.l | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 135.4.a.f | ✓ | 3 | 9.d | odd | 6 | 1 | |
| 135.4.a.g | yes | 3 | 9.c | even | 3 | 1 | |
| 405.4.e.r | 6 | 1.a | even | 1 | 1 | trivial | |
| 405.4.e.r | 6 | 9.c | even | 3 | 1 | inner | |
| 405.4.e.t | 6 | 3.b | odd | 2 | 1 | ||
| 405.4.e.t | 6 | 9.d | odd | 6 | 1 | ||
| 675.4.a.q | 3 | 45.j | even | 6 | 1 | ||
| 675.4.a.r | 3 | 45.h | odd | 6 | 1 | ||
| 675.4.b.k | 6 | 45.k | odd | 12 | 2 | ||
| 675.4.b.l | 6 | 45.l | even | 12 | 2 | ||
| 2160.4.a.be | 3 | 36.f | odd | 6 | 1 | ||
| 2160.4.a.bm | 3 | 36.h | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(405, [\chi])\):
|
\( T_{2}^{6} + T_{2}^{5} + 24T_{2}^{4} - 11T_{2}^{3} + 535T_{2}^{2} + 138T_{2} + 36 \)
|
|
\( T_{7}^{6} + 44T_{7}^{5} + 1384T_{7}^{4} + 20688T_{7}^{3} + 225504T_{7}^{2} + 993600T_{7} + 3240000 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} + \cdots + 36 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T^{2} - 5 T + 25)^{3} \)
|
| $7$ |
\( T^{6} + 44 T^{5} + \cdots + 3240000 \)
|
| $11$ |
\( T^{6} + \cdots + 6935558400 \)
|
| $13$ |
\( T^{6} + \cdots + 10024014400 \)
|
| $17$ |
\( (T^{3} - 19 T^{2} + \cdots + 553887)^{2} \)
|
| $19$ |
\( (T^{3} - 187 T^{2} + \cdots + 525871)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 4177858328361 \)
|
| $29$ |
\( T^{6} + \cdots + 62295660417600 \)
|
| $31$ |
\( T^{6} + \cdots + 60674035041 \)
|
| $37$ |
\( (T^{3} - 78 T^{2} + \cdots + 13637080)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 146812964889600 \)
|
| $43$ |
\( T^{6} + \cdots + 340939975993600 \)
|
| $47$ |
\( T^{6} + \cdots + 80202240000 \)
|
| $53$ |
\( (T^{3} + 521 T^{2} + \cdots - 939789)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 11\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots + 3177687716449 \)
|
| $67$ |
\( T^{6} + \cdots + 127577025000000 \)
|
| $71$ |
\( (T^{3} - 602 T^{2} + \cdots + 280550880)^{2} \)
|
| $73$ |
\( (T^{3} - 1294 T^{2} + \cdots + 404091280)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 4041318151809 \)
|
| $83$ |
\( T^{6} + \cdots + 11\!\cdots\!21 \)
|
| $89$ |
\( (T^{3} + 2154 T^{2} + \cdots + 74325600)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 40\!\cdots\!00 \)
|
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