Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [775,2,Mod(249,775)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(775, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("775.249");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 775 = 5^{2} \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 775.b (of order \(2\), degree \(1\), not 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: | \(6.18840615665\) |
| 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: |
\( x^{10} + 13x^{8} + 56x^{6} + 99x^{4} + 67x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{10} + 13x^{8} + 56x^{6} + 99x^{4} + 67x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{9} + 56\nu^{7} + 175\nu^{5} + 138\nu^{3} - 31\nu ) / 21 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{9} - 14\nu^{7} + 14\nu^{5} + 159\nu^{3} + 151\nu ) / 21 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 4\nu^{9} + 49\nu^{7} + 182\nu^{5} + 207\nu^{3} - 8\nu ) / 21 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{8} - 14\nu^{6} - 63\nu^{4} - 99\nu^{2} - 33 ) / 7 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{8} - 35\nu^{6} - 119\nu^{4} - 122\nu^{2} - 15 ) / 7 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 5\nu^{8} + 56\nu^{6} + 182\nu^{4} + 194\nu^{2} + 46 ) / 7 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 6\nu^{9} + 70\nu^{7} + 245\nu^{5} + 293\nu^{3} + 86\nu ) / 7 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -8\nu^{8} - 91\nu^{6} - 301\nu^{4} - 309\nu^{2} - 40 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{9} + \beta_{7} - \beta_{6} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{4} + \beta_{3} + 2\beta_{2} - 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -7\beta_{9} - 6\beta_{7} + 9\beta_{6} - \beta_{5} + 14 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{8} + 16\beta_{4} - 9\beta_{3} - 20\beta_{2} + 29\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 45\beta_{9} + 35\beta_{7} - 64\beta_{6} + 7\beta_{5} - 77 \)
|
| \(\nu^{7}\) | \(=\) |
\( -10\beta_{8} - 109\beta_{4} + 67\beta_{3} + 150\beta_{2} - 179\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( -288\beta_{9} - 211\beta_{7} + 428\beta_{6} - 42\beta_{5} + 460 \)
|
| \(\nu^{9}\) | \(=\) |
\( 77\beta_{8} + 716\beta_{4} - 463\beta_{3} - 1031\beta_{2} + 1134\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/775\mathbb{Z}\right)^\times\).
| \(n\) | \(251\) | \(652\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 249.1 |
|
− | 2.77799i | − | 2.60920i | −5.71723 | 0 | −7.24833 | − | 3.19118i | 10.3264i | −3.80792 | 0 | |||||||||||||||||||||||||||||||||||||||||||||
| 249.2 | − | 2.35347i | 1.91723i | −3.53883 | 0 | 4.51216 | 0.845802i | 3.62160i | −0.675783 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 249.3 | − | 1.54180i | 0.124960i | −0.377151 | 0 | 0.192664 | 4.01817i | − | 2.50211i | 2.98438 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 249.4 | − | 0.581067i | 2.46572i | 1.66236 | 0 | 1.43275 | − | 1.67419i | − | 2.12808i | −3.07975 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 249.5 | − | 0.170728i | 0.648789i | 1.97085 | 0 | 0.110767 | − | 2.03774i | − | 0.677937i | 2.57907 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 249.6 | 0.170728i | − | 0.648789i | 1.97085 | 0 | 0.110767 | 2.03774i | 0.677937i | 2.57907 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 249.7 | 0.581067i | − | 2.46572i | 1.66236 | 0 | 1.43275 | 1.67419i | 2.12808i | −3.07975 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 249.8 | 1.54180i | − | 0.124960i | −0.377151 | 0 | 0.192664 | − | 4.01817i | 2.50211i | 2.98438 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 249.9 | 2.35347i | − | 1.91723i | −3.53883 | 0 | 4.51216 | − | 0.845802i | − | 3.62160i | −0.675783 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||
| 249.10 | 2.77799i | 2.60920i | −5.71723 | 0 | −7.24833 | 3.19118i | − | 10.3264i | −3.80792 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
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 | 775.2.b.h | 10 | |
| 5.b | even | 2 | 1 | inner | 775.2.b.h | 10 | |
| 5.c | odd | 4 | 1 | 775.2.a.i | ✓ | 5 | |
| 5.c | odd | 4 | 1 | 775.2.a.j | yes | 5 | |
| 15.e | even | 4 | 1 | 6975.2.a.bq | 5 | ||
| 15.e | even | 4 | 1 | 6975.2.a.bx | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 775.2.a.i | ✓ | 5 | 5.c | odd | 4 | 1 | |
| 775.2.a.j | yes | 5 | 5.c | odd | 4 | 1 | |
| 775.2.b.h | 10 | 1.a | even | 1 | 1 | trivial | |
| 775.2.b.h | 10 | 5.b | even | 2 | 1 | inner | |
| 6975.2.a.bq | 5 | 15.e | even | 4 | 1 | ||
| 6975.2.a.bx | 5 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 16T_{2}^{8} + 80T_{2}^{6} + 129T_{2}^{4} + 38T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(775, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 16 T^{8} + \cdots + 1 \)
|
| $3$ |
\( T^{10} + 17 T^{8} + \cdots + 1 \)
|
| $5$ |
\( T^{10} \)
|
| $7$ |
\( T^{10} + 34 T^{8} + \cdots + 1369 \)
|
| $11$ |
\( (T^{5} - 23 T^{3} + \cdots + 37)^{2} \)
|
| $13$ |
\( T^{10} + 30 T^{8} + \cdots + 81 \)
|
| $17$ |
\( T^{10} + 95 T^{8} + \cdots + 3481 \)
|
| $19$ |
\( (T^{5} - 4 T^{4} + \cdots - 135)^{2} \)
|
| $23$ |
\( T^{10} + 82 T^{8} + \cdots + 27889 \)
|
| $29$ |
\( (T^{5} - 6 T^{4} + \cdots - 135)^{2} \)
|
| $31$ |
\( (T - 1)^{10} \)
|
| $37$ |
\( T^{10} + \cdots + 758396521 \)
|
| $41$ |
\( (T^{5} + 2 T^{4} - 67 T^{3} + \cdots - 27)^{2} \)
|
| $43$ |
\( T^{10} + \cdots + 137522529 \)
|
| $47$ |
\( T^{10} + \cdots + 2527977841 \)
|
| $53$ |
\( T^{10} + 229 T^{8} + \cdots + 2954961 \)
|
| $59$ |
\( (T^{5} + 4 T^{4} + \cdots - 4595)^{2} \)
|
| $61$ |
\( (T^{5} + 17 T^{4} + \cdots - 20609)^{2} \)
|
| $67$ |
\( T^{10} + 229 T^{8} + \cdots + 113569 \)
|
| $71$ |
\( (T^{5} + 6 T^{4} + \cdots - 549)^{2} \)
|
| $73$ |
\( T^{10} + 167 T^{8} + \cdots + 30327049 \)
|
| $79$ |
\( (T^{5} + 12 T^{4} + \cdots + 14985)^{2} \)
|
| $83$ |
\( T^{10} + \cdots + 504406681 \)
|
| $89$ |
\( (T^{5} - 3 T^{4} + \cdots + 39965)^{2} \)
|
| $97$ |
\( T^{10} + \cdots + 28668246489 \)
|
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