Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [324,6,Mod(109,324)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(324, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("324.109");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 324 = 2^{2} \cdot 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 324.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: | \(51.9643576194\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 135x^{4} - 1720x^{3} + 6375x^{2} - 110642x + 644836 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{5} \) |
| Twist minimal: | yes |
| 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
\( x^{6} - 2x^{5} + 135x^{4} - 1720x^{3} + 6375x^{2} - 110642x + 644836 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 9\nu^{5} + 264\nu^{4} - 1122\nu^{3} + 2409\nu^{2} - 174804\nu + 677496 ) / 10609 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -6\nu^{5} + 30\nu^{4} - 900\nu^{3} + 13020\nu^{2} - 13656\nu + 662384 ) / 10609 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 100\nu^{5} + 530\nu^{4} + 17369\nu^{3} - 48389\nu^{2} + 287443\nu - 9147280 ) / 63654 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -6\nu^{4} + 48\nu^{3} - 426\nu^{2} + 7500\nu - 33684 ) / 103 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -4171\nu^{5} - 21272\nu^{4} - 712994\nu^{3} + 2049851\nu^{2} - 11873938\nu + 372753988 ) / 31827 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + 3\beta_{2} + 2\beta _1 + 12 ) / 36 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{5} - \beta_{4} - 160\beta_{3} + 13\beta_{2} - 4\beta _1 - 452 ) / 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 27\beta_{5} - 20\beta_{4} + 2268\beta_{3} - 66\beta_{2} - 73\beta _1 + 11940 ) / 18 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 286\beta_{5} + 175\beta_{4} + 23456\beta_{3} - 25\beta_{2} + 728\beta _1 + 33404 ) / 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -16830\beta_{5} - 161\beta_{4} - 1370160\beta_{3} + 33573\beta_{2} + 2228\beta _1 - 2076468 ) / 36 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).
| \(n\) | \(163\) | \(245\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{3}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 109.1 |
|
0 | 0 | 0 | −35.3933 | + | 61.3030i | 0 | 5.13203 | + | 8.88894i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||
| 109.2 | 0 | 0 | 0 | 20.7695 | − | 35.9737i | 0 | −100.152 | − | 173.468i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
| 109.3 | 0 | 0 | 0 | 31.1239 | − | 53.9081i | 0 | 110.020 | + | 190.560i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
| 217.1 | 0 | 0 | 0 | −35.3933 | − | 61.3030i | 0 | 5.13203 | − | 8.88894i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
| 217.2 | 0 | 0 | 0 | 20.7695 | + | 35.9737i | 0 | −100.152 | + | 173.468i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
| 217.3 | 0 | 0 | 0 | 31.1239 | + | 53.9081i | 0 | 110.020 | − | 190.560i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
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 | 324.6.e.i | 6 | |
| 3.b | odd | 2 | 1 | 324.6.e.h | 6 | ||
| 9.c | even | 3 | 1 | 324.6.a.a | ✓ | 3 | |
| 9.c | even | 3 | 1 | inner | 324.6.e.i | 6 | |
| 9.d | odd | 6 | 1 | 324.6.a.b | yes | 3 | |
| 9.d | odd | 6 | 1 | 324.6.e.h | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 324.6.a.a | ✓ | 3 | 9.c | even | 3 | 1 | |
| 324.6.a.b | yes | 3 | 9.d | odd | 6 | 1 | |
| 324.6.e.h | 6 | 3.b | odd | 2 | 1 | ||
| 324.6.e.h | 6 | 9.d | odd | 6 | 1 | ||
| 324.6.e.i | 6 | 1.a | even | 1 | 1 | trivial | |
| 324.6.e.i | 6 | 9.c | even | 3 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(324, [\chi])\):
|
\( T_{5}^{6} - 33T_{5}^{5} + 5850T_{5}^{4} - 208953T_{5}^{3} + 28707210T_{5}^{2} - 871420113T_{5} + 33501079089 \)
|
|
\( T_{7}^{6} - 30T_{7}^{5} + 44772T_{7}^{4} + 411392T_{7}^{3} + 1938323904T_{7}^{2} - 19846990848T_{7} + 204651283456 \)
|
|
\( T_{11}^{6} + 30 T_{11}^{5} + 428292 T_{11}^{4} + 205708032 T_{11}^{3} + 185941868544 T_{11}^{2} + \cdots + 11\!\cdots\!16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 33501079089 \)
|
| $7$ |
\( T^{6} + \cdots + 204651283456 \)
|
| $11$ |
\( T^{6} + \cdots + 11\!\cdots\!16 \)
|
| $13$ |
\( T^{6} + \cdots + 11\!\cdots\!25 \)
|
| $17$ |
\( (T^{3} + 543 T^{2} + \cdots + 1357241805)^{2} \)
|
| $19$ |
\( (T^{3} - 1410 T^{2} + \cdots + 8206069600)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 27\!\cdots\!96 \)
|
| $29$ |
\( T^{6} + \cdots + 25\!\cdots\!41 \)
|
| $31$ |
\( T^{6} + \cdots + 31\!\cdots\!00 \)
|
| $37$ |
\( (T^{3} - 7143 T^{2} + \cdots + 679812536575)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 79\!\cdots\!44 \)
|
| $43$ |
\( T^{6} + \cdots + 92\!\cdots\!00 \)
|
| $47$ |
\( T^{6} + \cdots + 17\!\cdots\!00 \)
|
| $53$ |
\( (T^{3} + \cdots - 2134967305752)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 30\!\cdots\!36 \)
|
| $61$ |
\( T^{6} + \cdots + 38\!\cdots\!25 \)
|
| $67$ |
\( T^{6} + \cdots + 99\!\cdots\!44 \)
|
| $71$ |
\( (T^{3} - 31866 T^{2} + \cdots - 16003712160)^{2} \)
|
| $73$ |
\( (T^{3} + \cdots - 27185744955815)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 19\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 43\!\cdots\!00 \)
|
| $89$ |
\( (T^{3} + \cdots + 13397848224165)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 43\!\cdots\!00 \)
|
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