Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [810,3,Mod(161,810)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(810, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("810.161");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 810 = 2 \cdot 3^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 810.d (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: | \(22.0709014132\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.3317760000.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 8x^{6} + 13x^{4} + 12x^{2} + 36 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 8x^{6} + 13x^{4} + 12x^{2} + 36 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{7} + 14\nu^{5} - 25\nu^{3} - 78\nu ) / 72 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{6} + 6\nu^{4} - 9\nu^{2} + 2 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 14\nu^{5} + 61\nu^{3} - 30\nu ) / 36 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5\nu^{7} - 22\nu^{5} + 96\nu^{4} + 5\nu^{3} - 384\nu^{2} + 30\nu - 144 ) / 144 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -7\nu^{7} + 6\nu^{6} + 50\nu^{5} - 36\nu^{4} - 55\nu^{3} - 42\nu^{2} + 102\nu + 180 ) / 144 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -7\nu^{7} - 12\nu^{6} + 50\nu^{5} + 72\nu^{4} - 55\nu^{3} + 84\nu^{2} + 102\nu - 360 ) / 144 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 17\nu^{7} - 94\nu^{5} + 65\nu^{3} + 246\nu ) / 144 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + 2\beta_{5} - 2\beta_1 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} - \beta_{5} - \beta_{2} + 4 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + 2\beta_{5} + 2\beta_{3} + 2\beta_1 ) / 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -\beta_{7} + 4\beta_{6} - 4\beta_{5} + 3\beta_{4} - 4\beta_{2} - \beta _1 + 19 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 7\beta_{7} + 5\beta_{6} + 10\beta_{5} + 5\beta_{3} + 17\beta_1 ) / 2 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( -6\beta_{7} + 15\beta_{6} - 15\beta_{5} + 18\beta_{4} - 31\beta_{2} - 6\beta _1 + 82 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 47\beta_{7} + 19\beta_{6} + 38\beta_{5} + 20\beta_{3} + 96\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).
| \(n\) | \(487\) | \(731\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 161.1 |
|
− | 1.41421i | 0 | −2.00000 | − | 2.23607i | 0 | 3.01793 | 2.82843i | 0 | −3.16228 | ||||||||||||||||||||||||||||||||||||||||
| 161.2 | − | 1.41421i | 0 | −2.00000 | − | 2.23607i | 0 | 4.46891 | 2.82843i | 0 | −3.16228 | |||||||||||||||||||||||||||||||||||||||||
| 161.3 | − | 1.41421i | 0 | −2.00000 | 2.23607i | 0 | −11.9461 | 2.82843i | 0 | 3.16228 | ||||||||||||||||||||||||||||||||||||||||||
| 161.4 | − | 1.41421i | 0 | −2.00000 | 2.23607i | 0 | 0.459298 | 2.82843i | 0 | 3.16228 | ||||||||||||||||||||||||||||||||||||||||||
| 161.5 | 1.41421i | 0 | −2.00000 | − | 2.23607i | 0 | −11.9461 | − | 2.82843i | 0 | 3.16228 | |||||||||||||||||||||||||||||||||||||||||
| 161.6 | 1.41421i | 0 | −2.00000 | − | 2.23607i | 0 | 0.459298 | − | 2.82843i | 0 | 3.16228 | |||||||||||||||||||||||||||||||||||||||||
| 161.7 | 1.41421i | 0 | −2.00000 | 2.23607i | 0 | 3.01793 | − | 2.82843i | 0 | −3.16228 | ||||||||||||||||||||||||||||||||||||||||||
| 161.8 | 1.41421i | 0 | −2.00000 | 2.23607i | 0 | 4.46891 | − | 2.82843i | 0 | −3.16228 | ||||||||||||||||||||||||||||||||||||||||||
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 | 810.3.d.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 810.3.d.a | ✓ | 8 |
| 9.c | even | 3 | 2 | 810.3.h.e | 16 | ||
| 9.d | odd | 6 | 2 | 810.3.h.e | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 810.3.d.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 810.3.d.a | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 810.3.h.e | 16 | 9.c | even | 3 | 2 | ||
| 810.3.h.e | 16 | 9.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + 4T_{7}^{3} - 78T_{7}^{2} + 196T_{7} - 74 \)
acting on \(S_{3}^{\mathrm{new}}(810, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2)^{4} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{2} + 5)^{4} \)
|
| $7$ |
\( (T^{4} + 4 T^{3} - 78 T^{2} + \cdots - 74)^{2} \)
|
| $11$ |
\( T^{8} + 408 T^{6} + \cdots + 33074001 \)
|
| $13$ |
\( (T^{4} + 4 T^{3} + \cdots + 664)^{2} \)
|
| $17$ |
\( T^{8} + 84 T^{6} + \cdots + 2916 \)
|
| $19$ |
\( (T^{4} + 16 T^{3} + \cdots - 71)^{2} \)
|
| $23$ |
\( T^{8} + 2280 T^{6} + \cdots + 29160000 \)
|
| $29$ |
\( T^{8} + \cdots + 61877060001 \)
|
| $31$ |
\( (T^{4} - 44 T^{3} + \cdots + 374881)^{2} \)
|
| $37$ |
\( (T^{4} - 20 T^{3} + \cdots - 59306)^{2} \)
|
| $41$ |
\( T^{8} + \cdots + 14419950564321 \)
|
| $43$ |
\( (T^{4} - 80 T^{3} + \cdots - 3049466)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 39511103076 \)
|
| $53$ |
\( T^{8} + \cdots + 428980461156 \)
|
| $59$ |
\( T^{8} + \cdots + 257020887760641 \)
|
| $61$ |
\( (T^{4} + 124 T^{3} + \cdots - 651164)^{2} \)
|
| $67$ |
\( (T^{4} + 52 T^{3} + \cdots + 7024216)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 15946385841 \)
|
| $73$ |
\( (T^{4} + 196 T^{3} + \cdots - 230954)^{2} \)
|
| $79$ |
\( (T^{4} - 44 T^{3} + \cdots + 87417124)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 49250836195236 \)
|
| $89$ |
\( T^{8} + \cdots + 72157396691601 \)
|
| $97$ |
\( (T^{4} - 128 T^{3} + \cdots + 5964886)^{2} \)
|
| show more | |
| show less |