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: | \(8\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 24x^{6} + 164x^{4} - 111x^{2} + 100 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{4}\cdot 5^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 24x^{6} + 164x^{4} - 111x^{2} + 100 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 3\nu^{7} - 68\nu^{5} + 422\nu^{3} + 23\nu ) / 310 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{6} + 33\nu^{4} - 306\nu^{2} + 385 ) / 31 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -3\nu^{7} + 68\nu^{5} - 360\nu^{3} - 581\nu ) / 62 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -12\nu^{6} + 303\nu^{4} - 1905\nu^{2} - 309 ) / 31 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 8\nu^{6} - 171\nu^{4} + 1053\nu^{2} - 414 ) / 31 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -18\nu^{7} + 439\nu^{5} - 3152\nu^{3} + 4481\nu ) / 155 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 147\nu^{7} - 3642\nu^{5} + 25638\nu^{3} - 15303\nu ) / 310 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + 5\beta_{6} + 4\beta_{3} + 31\beta_1 ) / 60 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} - 4\beta_{2} + 73 ) / 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{7} + 15\beta_{6} + 32\beta_{3} + 193\beta_1 ) / 20 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 19\beta_{5} + 15\beta_{4} - 28\beta_{2} + 751 ) / 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 31\beta_{7} + 455\beta_{6} + 1324\beta_{3} + 10561\beta_1 ) / 60 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 107\beta_{5} + 63\beta_{4} - 24\beta_{2} + 2355 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -571\beta_{7} + 3945\beta_{6} + 16476\beta_{3} + 163899\beta_1 ) / 60 \)
|
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 | −7.52557 | − | 8.26836i | 0 | − | 26.3948i | 8.00000i | 0 | −16.5367 | + | 15.0511i | ||||||||||||||||||||||||||||||||||||
| 109.2 | − | 2.00000i | 0 | −4.00000 | −6.93656 | + | 8.76836i | 0 | 22.9416i | 8.00000i | 0 | 17.5367 | + | 13.8731i | ||||||||||||||||||||||||||||||||||||||
| 109.3 | − | 2.00000i | 0 | −4.00000 | 6.93656 | + | 8.76836i | 0 | − | 22.9416i | 8.00000i | 0 | 17.5367 | − | 13.8731i | |||||||||||||||||||||||||||||||||||||
| 109.4 | − | 2.00000i | 0 | −4.00000 | 7.52557 | − | 8.26836i | 0 | 26.3948i | 8.00000i | 0 | −16.5367 | − | 15.0511i | ||||||||||||||||||||||||||||||||||||||
| 109.5 | 2.00000i | 0 | −4.00000 | −7.52557 | + | 8.26836i | 0 | 26.3948i | − | 8.00000i | 0 | −16.5367 | − | 15.0511i | ||||||||||||||||||||||||||||||||||||||
| 109.6 | 2.00000i | 0 | −4.00000 | −6.93656 | − | 8.76836i | 0 | − | 22.9416i | − | 8.00000i | 0 | 17.5367 | − | 13.8731i | |||||||||||||||||||||||||||||||||||||
| 109.7 | 2.00000i | 0 | −4.00000 | 6.93656 | − | 8.76836i | 0 | 22.9416i | − | 8.00000i | 0 | 17.5367 | + | 13.8731i | ||||||||||||||||||||||||||||||||||||||
| 109.8 | 2.00000i | 0 | −4.00000 | 7.52557 | + | 8.26836i | 0 | − | 26.3948i | − | 8.00000i | 0 | −16.5367 | + | 15.0511i | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
| 15.d | odd | 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.e | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 270.4.c.e | ✓ | 8 |
| 5.b | even | 2 | 1 | inner | 270.4.c.e | ✓ | 8 |
| 5.c | odd | 4 | 1 | 1350.4.a.bu | 4 | ||
| 5.c | odd | 4 | 1 | 1350.4.a.bv | 4 | ||
| 15.d | odd | 2 | 1 | inner | 270.4.c.e | ✓ | 8 |
| 15.e | even | 4 | 1 | 1350.4.a.bu | 4 | ||
| 15.e | even | 4 | 1 | 1350.4.a.bv | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 270.4.c.e | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 270.4.c.e | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 270.4.c.e | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 270.4.c.e | ✓ | 8 | 15.d | odd | 2 | 1 | inner |
| 1350.4.a.bu | 4 | 5.c | odd | 4 | 1 | ||
| 1350.4.a.bu | 4 | 15.e | even | 4 | 1 | ||
| 1350.4.a.bv | 4 | 5.c | odd | 4 | 1 | ||
| 1350.4.a.bv | 4 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(270, [\chi])\):
|
\( T_{7}^{4} + 1223T_{7}^{2} + 366676 \)
|
|
\( T_{11}^{4} - 1223T_{11}^{2} + 366676 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4)^{4} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + 81 T^{6} + \cdots + 244140625 \)
|
| $7$ |
\( (T^{4} + 1223 T^{2} + 366676)^{2} \)
|
| $11$ |
\( (T^{4} - 1223 T^{2} + 366676)^{2} \)
|
| $13$ |
\( (T^{4} + 10908 T^{2} + 20342016)^{2} \)
|
| $17$ |
\( (T^{4} + 16025 T^{2} + 42250000)^{2} \)
|
| $19$ |
\( (T^{2} + 21 T - 7146)^{4} \)
|
| $23$ |
\( (T^{4} + 16373 T^{2} + 40018276)^{2} \)
|
| $29$ |
\( (T^{4} - 98172 T^{2} + 1647703296)^{2} \)
|
| $31$ |
\( (T^{2} - 200 T - 19025)^{4} \)
|
| $37$ |
\( (T^{4} + 132812 T^{2} + 262548736)^{2} \)
|
| $41$ |
\( (T^{4} - 227432 T^{2} + 9169688656)^{2} \)
|
| $43$ |
\( (T^{4} + 113000 T^{2} + 916690000)^{2} \)
|
| $47$ |
\( (T^{4} + 132548 T^{2} + 67634176)^{2} \)
|
| $53$ |
\( (T^{4} + 249498 T^{2} + 11545717401)^{2} \)
|
| $59$ |
\( (T^{4} - 566408 T^{2} + 67407499216)^{2} \)
|
| $61$ |
\( (T^{2} - 781 T - 28916)^{4} \)
|
| $67$ |
\( (T^{4} + 708428 T^{2} + 110836055296)^{2} \)
|
| $71$ |
\( (T^{4} - 272700 T^{2} + 12713760000)^{2} \)
|
| $73$ |
\( (T^{4} + 662447 T^{2} + 106363578196)^{2} \)
|
| $79$ |
\( (T^{2} + 1629 T + 598104)^{4} \)
|
| $83$ |
\( (T^{4} + 722162 T^{2} + 26039954161)^{2} \)
|
| $89$ |
\( (T^{4} - 116648 T^{2} + 1839447376)^{2} \)
|
| $97$ |
\( (T^{4} + 556883 T^{2} + 365828416)^{2} \)
|
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