Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [270,4,Mod(109,270)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(270, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("270.109");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 270 = 2 \cdot 3^{3} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 270.c (of order \(2\), degree \(1\), 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: | \(15.9305157015\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.34768823296.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 48x^{3} + 361x^{2} + 190x + 50 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{6} - 2x^{5} + 2x^{4} + 48x^{3} + 361x^{2} + 190x + 50 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 17\nu^{5} + 76\nu^{4} - 96\nu^{3} + 861\nu^{2} + 15572\nu + 29245 ) / 2265 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 25\nu^{5} - 57\nu^{4} + 72\nu^{3} + 1053\nu^{2} + 8857\nu + 2415 ) / 2265 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 25\nu^{5} - 57\nu^{4} + 72\nu^{3} + 1053\nu^{2} + 15652\nu + 150 ) / 2265 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 29\nu^{5} - 199\nu^{4} + 609\nu^{3} + 696\nu^{2} + 290\nu - 20815 ) / 2265 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 254\nu^{5} - 561\nu^{4} + 351\nu^{3} + 13344\nu^{2} + 83627\nu + 22815 ) / 2265 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_{2} + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta_{3} - 13\beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{5} + 22\beta_{4} - 73\beta_{2} + 25\beta _1 - 73 ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( 8\beta_{4} - 29\beta_{3} + 29\beta _1 - 299 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -135\beta_{5} - 135\beta_{4} - 679\beta_{3} + 2479\beta_{2} - 2479 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/270\mathbb{Z}\right)^\times\).
| \(n\) | \(191\) | \(217\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 109.1 |
|
− | 2.00000i | 0 | −4.00000 | −7.41822 | + | 8.36481i | 0 | − | 6.05342i | 8.00000i | 0 | 16.7296 | + | 14.8364i | ||||||||||||||||||||||||||||||
| 109.2 | − | 2.00000i | 0 | −4.00000 | −4.61308 | − | 10.1843i | 0 | − | 21.7974i | 8.00000i | 0 | −20.3686 | + | 9.22615i | |||||||||||||||||||||||||||||||
| 109.3 | − | 2.00000i | 0 | −4.00000 | 11.0313 | + | 1.81947i | 0 | 5.85077i | 8.00000i | 0 | 3.63895 | − | 22.0626i | ||||||||||||||||||||||||||||||||
| 109.4 | 2.00000i | 0 | −4.00000 | −7.41822 | − | 8.36481i | 0 | 6.05342i | − | 8.00000i | 0 | 16.7296 | − | 14.8364i | ||||||||||||||||||||||||||||||||
| 109.5 | 2.00000i | 0 | −4.00000 | −4.61308 | + | 10.1843i | 0 | 21.7974i | − | 8.00000i | 0 | −20.3686 | − | 9.22615i | ||||||||||||||||||||||||||||||||
| 109.6 | 2.00000i | 0 | −4.00000 | 11.0313 | − | 1.81947i | 0 | − | 5.85077i | − | 8.00000i | 0 | 3.63895 | + | 22.0626i | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 270.4.c.c | ✓ | 6 |
| 3.b | odd | 2 | 1 | 270.4.c.d | yes | 6 | |
| 5.b | even | 2 | 1 | inner | 270.4.c.c | ✓ | 6 |
| 5.c | odd | 4 | 1 | 1350.4.a.bq | 3 | ||
| 5.c | odd | 4 | 1 | 1350.4.a.bt | 3 | ||
| 15.d | odd | 2 | 1 | 270.4.c.d | yes | 6 | |
| 15.e | even | 4 | 1 | 1350.4.a.br | 3 | ||
| 15.e | even | 4 | 1 | 1350.4.a.bs | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 270.4.c.c | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 270.4.c.c | ✓ | 6 | 5.b | even | 2 | 1 | inner |
| 270.4.c.d | yes | 6 | 3.b | odd | 2 | 1 | |
| 270.4.c.d | yes | 6 | 15.d | odd | 2 | 1 | |
| 1350.4.a.bq | 3 | 5.c | odd | 4 | 1 | ||
| 1350.4.a.br | 3 | 15.e | even | 4 | 1 | ||
| 1350.4.a.bs | 3 | 15.e | even | 4 | 1 | ||
| 1350.4.a.bt | 3 | 5.c | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(270, [\chi])\):
|
\( T_{7}^{6} + 546T_{7}^{4} + 34929T_{7}^{2} + 595984 \)
|
|
\( T_{11}^{3} - 31T_{11}^{2} - 1966T_{11} + 17350 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4)^{3} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + 2 T^{5} + \cdots + 1953125 \)
|
| $7$ |
\( T^{6} + 546 T^{4} + \cdots + 595984 \)
|
| $11$ |
\( (T^{3} - 31 T^{2} + \cdots + 17350)^{2} \)
|
| $13$ |
\( T^{6} + 3915 T^{4} + \cdots + 102515625 \)
|
| $17$ |
\( T^{6} + \cdots + 215303424064 \)
|
| $19$ |
\( (T^{3} + 12 T^{2} + \cdots + 869238)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 2578060541956 \)
|
| $29$ |
\( (T^{3} + 153 T^{2} + \cdots - 1255338)^{2} \)
|
| $31$ |
\( (T^{3} + 147 T^{2} + \cdots - 11395300)^{2} \)
|
| $37$ |
\( T^{6} + \cdots + 14\!\cdots\!00 \)
|
| $41$ |
\( (T^{3} - 550 T^{2} + \cdots - 3400460)^{2} \)
|
| $43$ |
\( T^{6} + \cdots + 184761928256656 \)
|
| $47$ |
\( T^{6} + \cdots + 328017361984 \)
|
| $53$ |
\( T^{6} + \cdots + 548903174400 \)
|
| $59$ |
\( (T^{3} + 1364 T^{2} + \cdots + 28685344)^{2} \)
|
| $61$ |
\( (T^{3} + 330 T^{2} + \cdots + 154840)^{2} \)
|
| $67$ |
\( T^{6} + \cdots + 17\!\cdots\!84 \)
|
| $71$ |
\( (T^{3} - 1446 T^{2} + \cdots + 56345220)^{2} \)
|
| $73$ |
\( T^{6} + \cdots + 25\!\cdots\!00 \)
|
| $79$ |
\( (T^{3} + 45 T^{2} + \cdots + 1769283)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 2387816105536 \)
|
| $89$ |
\( (T^{3} + 2566 T^{2} + \cdots + 298865120)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 689711027522500 \)
|
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