Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1305,2,Mod(811,1305)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1305, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1305.811");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1305 = 3^{2} \cdot 5 \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1305.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: | \(10.4204774638\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} + 13x^{8} + 56x^{6} + 89x^{4} + 41x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} + 13x^{8} + 56x^{6} + 89x^{4} + 41x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 4\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} + 6\nu^{2} + 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{6} + 9\nu^{4} + 20\nu^{2} + 5 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{9} - 12\nu^{7} - 46\nu^{5} - 59\nu^{3} - 12\nu ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{8} + 12\nu^{6} + 46\nu^{4} + 59\nu^{2} + 12 ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( -\nu^{9} - 13\nu^{7} - 54\nu^{5} - 73\nu^{3} - 13\nu ) / 2 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( \nu^{9} + 13\nu^{7} + 56\nu^{5} + 87\nu^{3} + 31\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} - 6\beta_{2} + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{9} + \beta_{8} - 7\beta_{3} + 19\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( \beta_{5} - 9\beta_{4} + 34\beta_{2} - 71 \)
|
| \(\nu^{7}\) | \(=\) |
\( -8\beta_{9} - 10\beta_{8} + 2\beta_{6} + 42\beta_{3} - 97\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 2\beta_{7} - 12\beta_{5} + 62\beta_{4} - 191\beta_{2} + 373 \)
|
| \(\nu^{9}\) | \(=\) |
\( 50\beta_{9} + 74\beta_{8} - 26\beta_{6} - 241\beta_{3} + 514\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times\).
| \(n\) | \(146\) | \(262\) | \(901\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 811.1 |
|
− | 2.38254i | 0 | −3.67649 | 1.00000 | 0 | 1.34246 | 3.99431i | 0 | − | 2.38254i | ||||||||||||||||||||||||||||||||||||||||||||||
| 811.2 | − | 2.09920i | 0 | −2.40666 | 1.00000 | 0 | −2.25380 | 0.853652i | 0 | − | 2.09920i | |||||||||||||||||||||||||||||||||||||||||||||||
| 811.3 | − | 1.49936i | 0 | −0.248070 | 1.00000 | 0 | 0.522278 | − | 2.62677i | 0 | − | 1.49936i | ||||||||||||||||||||||||||||||||||||||||||||||
| 811.4 | − | 0.732188i | 0 | 1.46390 | 1.00000 | 0 | −4.08783 | − | 2.53623i | 0 | − | 0.732188i | ||||||||||||||||||||||||||||||||||||||||||||||
| 811.5 | − | 0.364257i | 0 | 1.86732 | 1.00000 | 0 | 2.47690 | − | 1.40870i | 0 | − | 0.364257i | ||||||||||||||||||||||||||||||||||||||||||||||
| 811.6 | 0.364257i | 0 | 1.86732 | 1.00000 | 0 | 2.47690 | 1.40870i | 0 | 0.364257i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 811.7 | 0.732188i | 0 | 1.46390 | 1.00000 | 0 | −4.08783 | 2.53623i | 0 | 0.732188i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 811.8 | 1.49936i | 0 | −0.248070 | 1.00000 | 0 | 0.522278 | 2.62677i | 0 | 1.49936i | |||||||||||||||||||||||||||||||||||||||||||||||||
| 811.9 | 2.09920i | 0 | −2.40666 | 1.00000 | 0 | −2.25380 | − | 0.853652i | 0 | 2.09920i | ||||||||||||||||||||||||||||||||||||||||||||||||
| 811.10 | 2.38254i | 0 | −3.67649 | 1.00000 | 0 | 1.34246 | − | 3.99431i | 0 | 2.38254i | ||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 29.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1305.2.d.f | yes | 10 |
| 3.b | odd | 2 | 1 | 1305.2.d.e | ✓ | 10 | |
| 29.b | even | 2 | 1 | inner | 1305.2.d.f | yes | 10 |
| 87.d | odd | 2 | 1 | 1305.2.d.e | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1305.2.d.e | ✓ | 10 | 3.b | odd | 2 | 1 | |
| 1305.2.d.e | ✓ | 10 | 87.d | odd | 2 | 1 | |
| 1305.2.d.f | yes | 10 | 1.a | even | 1 | 1 | trivial |
| 1305.2.d.f | yes | 10 | 29.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1305, [\chi])\):
|
\( T_{2}^{10} + 13T_{2}^{8} + 56T_{2}^{6} + 89T_{2}^{4} + 41T_{2}^{2} + 4 \)
|
|
\( T_{23}^{5} - 5T_{23}^{4} - 86T_{23}^{3} + 270T_{23}^{2} + 1728T_{23} + 648 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 13 T^{8} + \cdots + 4 \)
|
| $3$ |
\( T^{10} \)
|
| $5$ |
\( (T - 1)^{10} \)
|
| $7$ |
\( (T^{5} + 2 T^{4} - 13 T^{3} + \cdots - 16)^{2} \)
|
| $11$ |
\( T^{10} + 47 T^{8} + \cdots + 6724 \)
|
| $13$ |
\( (T^{5} - 2 T^{4} - 25 T^{3} + \cdots + 8)^{2} \)
|
| $17$ |
\( T^{10} + 62 T^{8} + \cdots + 4096 \)
|
| $19$ |
\( T^{10} + 140 T^{8} + \cdots + 186624 \)
|
| $23$ |
\( (T^{5} - 5 T^{4} + \cdots + 648)^{2} \)
|
| $29$ |
\( T^{10} - 43 T^{8} + \cdots + 20511149 \)
|
| $31$ |
\( T^{10} + 212 T^{8} + \cdots + 6718464 \)
|
| $37$ |
\( T^{10} + 229 T^{8} + \cdots + 186624 \)
|
| $41$ |
\( T^{10} + 197 T^{8} + \cdots + 1024 \)
|
| $43$ |
\( T^{10} + \cdots + 215737344 \)
|
| $47$ |
\( T^{10} + 314 T^{8} + \cdots + 63744256 \)
|
| $53$ |
\( (T^{5} + 7 T^{4} + \cdots + 6480)^{2} \)
|
| $59$ |
\( (T^{5} - 16 T^{4} + \cdots - 46656)^{2} \)
|
| $61$ |
\( T^{10} + 512 T^{8} + \cdots + 74649600 \)
|
| $67$ |
\( (T^{5} + 4 T^{4} + \cdots + 2836)^{2} \)
|
| $71$ |
\( (T^{5} + 4 T^{4} + \cdots + 3456)^{2} \)
|
| $73$ |
\( T^{10} + \cdots + 967458816 \)
|
| $79$ |
\( T^{10} + 140 T^{8} + \cdots + 186624 \)
|
| $83$ |
\( (T^{5} - 9 T^{4} + \cdots - 38664)^{2} \)
|
| $89$ |
\( T^{10} + \cdots + 132342016 \)
|
| $97$ |
\( T^{10} + \cdots + 191102976 \)
|
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