Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [510,2,Mod(169,510)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(510, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("510.169");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 510 = 2 \cdot 3 \cdot 5 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 510.f (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: | \(4.07237050309\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.350464.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -3\nu^{5} + \nu^{4} + 11\nu^{3} - 26\nu^{2} + 6\nu - 1 ) / 23 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} + 8\nu - 9 ) / 23 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 6\nu^{5} - 2\nu^{4} + \nu^{3} + 6\nu^{2} + 80\nu + 2 ) / 23 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7\nu^{5} - 10\nu^{4} + 5\nu^{3} + 30\nu^{2} + 32\nu - 13 ) / 23 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -16\nu^{5} + 36\nu^{4} - 41\nu^{3} - 16\nu^{2} - 60\nu + 56 ) / 23 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} + 4\beta_{4} - \beta_{3} + 2\beta_1 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} - 2\beta_{2} + 2\beta _1 - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} + 2\beta_{3} - 5\beta_{2} - 7 \)
|
| \(\nu^{5}\) | \(=\) |
\( -9\beta_{4} + 5\beta_{3} - 8\beta_{2} - 8\beta _1 - 9 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/510\mathbb{Z}\right)^\times\).
| \(n\) | \(241\) | \(307\) | \(341\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 169.1 |
|
− | 1.00000i | 1.00000 | −1.00000 | −1.48119 | + | 1.67513i | − | 1.00000i | −4.15633 | 1.00000i | 1.00000 | 1.67513 | + | 1.48119i | ||||||||||||||||||||||||||||||
| 169.2 | − | 1.00000i | 1.00000 | −1.00000 | 0.311108 | − | 2.21432i | − | 1.00000i | 1.52543 | 1.00000i | 1.00000 | −2.21432 | − | 0.311108i | |||||||||||||||||||||||||||||||
| 169.3 | − | 1.00000i | 1.00000 | −1.00000 | 2.17009 | + | 0.539189i | − | 1.00000i | 0.630898 | 1.00000i | 1.00000 | 0.539189 | − | 2.17009i | |||||||||||||||||||||||||||||||
| 169.4 | 1.00000i | 1.00000 | −1.00000 | −1.48119 | − | 1.67513i | 1.00000i | −4.15633 | − | 1.00000i | 1.00000 | 1.67513 | − | 1.48119i | ||||||||||||||||||||||||||||||||
| 169.5 | 1.00000i | 1.00000 | −1.00000 | 0.311108 | + | 2.21432i | 1.00000i | 1.52543 | − | 1.00000i | 1.00000 | −2.21432 | + | 0.311108i | ||||||||||||||||||||||||||||||||
| 169.6 | 1.00000i | 1.00000 | −1.00000 | 2.17009 | − | 0.539189i | 1.00000i | 0.630898 | − | 1.00000i | 1.00000 | 0.539189 | + | 2.17009i | ||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 85.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 510.2.f.d | yes | 6 |
| 3.b | odd | 2 | 1 | 1530.2.f.j | 6 | ||
| 5.b | even | 2 | 1 | 510.2.f.c | ✓ | 6 | |
| 5.c | odd | 4 | 1 | 2550.2.c.r | 6 | ||
| 5.c | odd | 4 | 1 | 2550.2.c.s | 6 | ||
| 15.d | odd | 2 | 1 | 1530.2.f.k | 6 | ||
| 17.b | even | 2 | 1 | 510.2.f.c | ✓ | 6 | |
| 51.c | odd | 2 | 1 | 1530.2.f.k | 6 | ||
| 85.c | even | 2 | 1 | inner | 510.2.f.d | yes | 6 |
| 85.g | odd | 4 | 1 | 2550.2.c.r | 6 | ||
| 85.g | odd | 4 | 1 | 2550.2.c.s | 6 | ||
| 255.h | odd | 2 | 1 | 1530.2.f.j | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 510.2.f.c | ✓ | 6 | 5.b | even | 2 | 1 | |
| 510.2.f.c | ✓ | 6 | 17.b | even | 2 | 1 | |
| 510.2.f.d | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 510.2.f.d | yes | 6 | 85.c | even | 2 | 1 | inner |
| 1530.2.f.j | 6 | 3.b | odd | 2 | 1 | ||
| 1530.2.f.j | 6 | 255.h | odd | 2 | 1 | ||
| 1530.2.f.k | 6 | 15.d | odd | 2 | 1 | ||
| 1530.2.f.k | 6 | 51.c | odd | 2 | 1 | ||
| 2550.2.c.r | 6 | 5.c | odd | 4 | 1 | ||
| 2550.2.c.r | 6 | 85.g | odd | 4 | 1 | ||
| 2550.2.c.s | 6 | 5.c | odd | 4 | 1 | ||
| 2550.2.c.s | 6 | 85.g | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{3} + 2T_{7}^{2} - 8T_{7} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(510, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{3} \)
|
| $3$ |
\( (T - 1)^{6} \)
|
| $5$ |
\( T^{6} - 2 T^{5} + \cdots + 125 \)
|
| $7$ |
\( (T^{3} + 2 T^{2} - 8 T + 4)^{2} \)
|
| $11$ |
\( T^{6} + 44 T^{4} + \cdots + 1600 \)
|
| $13$ |
\( T^{6} + 32 T^{4} + \cdots + 16 \)
|
| $17$ |
\( T^{6} + 12 T^{5} + \cdots + 4913 \)
|
| $19$ |
\( (T^{3} - 8 T^{2} + \cdots + 160)^{2} \)
|
| $23$ |
\( (T^{3} + 14 T^{2} + \cdots - 296)^{2} \)
|
| $29$ |
\( T^{6} + 32 T^{4} + \cdots + 256 \)
|
| $31$ |
\( T^{6} + 76 T^{4} + \cdots + 1600 \)
|
| $37$ |
\( (T^{3} - 6 T^{2} + \cdots + 632)^{2} \)
|
| $41$ |
\( T^{6} + 188 T^{4} + \cdots + 10000 \)
|
| $43$ |
\( T^{6} + 188 T^{4} + \cdots + 150544 \)
|
| $47$ |
\( T^{6} + 80 T^{4} + \cdots + 256 \)
|
| $53$ |
\( T^{6} + 172 T^{4} + \cdots + 10816 \)
|
| $59$ |
\( (T^{3} - 48 T + 20)^{2} \)
|
| $61$ |
\( T^{6} + 332 T^{4} + \cdots + 846400 \)
|
| $67$ |
\( T^{6} + 220 T^{4} + \cdots + 8464 \)
|
| $71$ |
\( T^{6} + 96 T^{4} + \cdots + 400 \)
|
| $73$ |
\( (T^{3} - 8 T^{2} + 12 T - 4)^{2} \)
|
| $79$ |
\( T^{6} + 140 T^{4} + \cdots + 64 \)
|
| $83$ |
\( T^{6} + 80 T^{4} + \cdots + 1024 \)
|
| $89$ |
\( (T^{3} + 14 T^{2} + \cdots - 4280)^{2} \)
|
| $97$ |
\( (T^{3} + 4 T^{2} + \cdots - 1108)^{2} \)
|
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