Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [287,2,Mod(204,287)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(287, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("287.204");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 287 = 7 \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 287.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: | \(2.29170653801\) |
| Analytic rank: | \(0\) |
| Dimension: | \(12\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{12} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{12} + 18x^{10} + 113x^{8} + 290x^{6} + 258x^{4} + 49x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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 \(1,\beta_1,\ldots,\beta_{11}\) 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
\( x^{12} + 18x^{10} + 113x^{8} + 290x^{6} + 258x^{4} + 49x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{10} - 20\nu^{8} - 13\nu^{6} + 213\nu^{4} + 218\nu^{2} + 13 ) / 53 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{10} + 20\nu^{8} + 13\nu^{6} - 213\nu^{4} - 165\nu^{2} + 146 ) / 53 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -10\nu^{10} - 153\nu^{8} - 754\nu^{6} - 1320\nu^{4} - 447\nu^{2} + 171 ) / 53 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -8\nu^{10} - 133\nu^{8} - 741\nu^{6} - 1586\nu^{4} - 1036\nu^{2} - 107 ) / 53 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -15\nu^{11} - 256\nu^{9} - 1502\nu^{7} - 3570\nu^{5} - 2976\nu^{3} - 618\nu ) / 53 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 13\nu^{10} + 236\nu^{8} + 1489\nu^{6} + 3783\nu^{4} + 3141\nu^{2} + 366 ) / 53 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 13\nu^{11} + 236\nu^{9} + 1489\nu^{7} + 3783\nu^{5} + 3141\nu^{3} + 419\nu ) / 53 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -41\nu^{11} - 728\nu^{9} - 4480\nu^{7} - 11136\nu^{5} - 9205\nu^{3} - 1191\nu ) / 53 \)
|
| \(\beta_{10}\) | \(=\) |
\( ( 40\nu^{11} + 718\nu^{9} + 4500\nu^{7} + 11587\nu^{5} + 10533\nu^{3} + 2178\nu ) / 53 \)
|
| \(\beta_{11}\) | \(=\) |
\( ( 48\nu^{11} + 851\nu^{9} + 5241\nu^{7} + 13120\nu^{5} + 11092\nu^{3} + 1437\nu ) / 53 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta_{2} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} + 2\beta_{8} - \beta_{6} - 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{5} + \beta_{4} - 7\beta_{3} - 8\beta_{2} + 16 \)
|
| \(\nu^{5}\) | \(=\) |
\( -2\beta_{11} + \beta_{10} - 11\beta_{9} - 20\beta_{8} + 9\beta_{6} + 29\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{7} + 15\beta_{5} - 11\beta_{4} + 49\beta_{3} + 57\beta_{2} - 97 \)
|
| \(\nu^{7}\) | \(=\) |
\( 26\beta_{11} - 11\beta_{10} + 98\beta_{9} + 165\beta_{8} - 71\beta_{6} - 183\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( -26\beta_{7} - 150\beta_{5} + 97\beta_{4} - 351\beta_{3} - 405\beta_{2} + 630 \)
|
| \(\nu^{9}\) | \(=\) |
\( -247\beta_{11} + 97\beta_{10} - 802\beta_{9} - 1287\beta_{8} + 545\beta_{6} + 1218\beta_1 \)
|
| \(\nu^{10}\) | \(=\) |
\( 247\beta_{7} + 1296\beta_{5} - 792\beta_{4} + 2555\beta_{3} + 2910\beta_{2} - 4286 \)
|
| \(\nu^{11}\) | \(=\) |
\( 2088\beta_{11} - 792\beta_{10} + 6294\beta_{9} + 9806\beta_{8} - 4139\beta_{6} - 8414\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/287\mathbb{Z}\right)^\times\).
| \(n\) | \(206\) | \(211\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 204.1 |
|
−2.72816 | − | 1.72816i | 5.44285 | −1.58817 | 4.71469i | − | 1.00000i | −9.39263 | 0.0134697 | 4.33279 | ||||||||||||||||||||||||||||||||||||||||||||||||||||
| 204.2 | −2.72816 | 1.72816i | 5.44285 | −1.58817 | − | 4.71469i | 1.00000i | −9.39263 | 0.0134697 | 4.33279 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 204.3 | −2.16363 | − | 1.16363i | 2.68131 | 2.90865 | 2.51768i | − | 1.00000i | −1.47412 | 1.64595 | −6.29325 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 204.4 | −2.16363 | 1.16363i | 2.68131 | 2.90865 | − | 2.51768i | 1.00000i | −1.47412 | 1.64595 | −6.29325 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 204.5 | −0.473876 | − | 0.526124i | −1.77544 | −1.62273 | 0.249318i | 1.00000i | 1.78909 | 2.72319 | 0.768972 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 204.6 | −0.473876 | 0.526124i | −1.77544 | −1.62273 | − | 0.249318i | − | 1.00000i | 1.78909 | 2.72319 | 0.768972 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 204.7 | 0.152182 | − | 1.15218i | −1.97684 | 3.83685 | − | 0.175342i | 1.00000i | −0.605204 | 1.67248 | 0.583900 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 204.8 | 0.152182 | 1.15218i | −1.97684 | 3.83685 | 0.175342i | − | 1.00000i | −0.605204 | 1.67248 | 0.583900 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 204.9 | 1.12463 | − | 2.12463i | −0.735213 | −4.04675 | − | 2.38941i | 1.00000i | −3.07610 | −1.51404 | −4.55109 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 204.10 | 1.12463 | 2.12463i | −0.735213 | −4.04675 | 2.38941i | − | 1.00000i | −3.07610 | −1.51404 | −4.55109 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
| 204.11 | 2.08886 | − | 3.08886i | 2.36333 | 1.51216 | − | 6.45219i | 1.00000i | 0.758953 | −6.54105 | 3.15868 | |||||||||||||||||||||||||||||||||||||||||||||||||||||
| 204.12 | 2.08886 | 3.08886i | 2.36333 | 1.51216 | 6.45219i | − | 1.00000i | 0.758953 | −6.54105 | 3.15868 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 41.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 287.2.c.b | ✓ | 12 |
| 41.b | even | 2 | 1 | inner | 287.2.c.b | ✓ | 12 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 287.2.c.b | ✓ | 12 | 1.a | even | 1 | 1 | trivial |
| 287.2.c.b | ✓ | 12 | 41.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 2T_{2}^{5} - 7T_{2}^{4} - 10T_{2}^{3} + 12T_{2}^{2} + 5T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(287, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{6} + 2 T^{5} - 7 T^{4} + \cdots - 1)^{2} \)
|
| $3$ |
\( T^{12} + 20 T^{10} + \cdots + 64 \)
|
| $5$ |
\( (T^{6} - T^{5} - 23 T^{4} + \cdots - 176)^{2} \)
|
| $7$ |
\( (T^{2} + 1)^{6} \)
|
| $11$ |
\( T^{12} + 106 T^{10} + \cdots + 6553600 \)
|
| $13$ |
\( T^{12} + 55 T^{10} + \cdots + 256 \)
|
| $17$ |
\( T^{12} + 107 T^{10} + \cdots + 1183744 \)
|
| $19$ |
\( T^{12} + 120 T^{10} + \cdots + 3356224 \)
|
| $23$ |
\( (T^{6} - 8 T^{5} + \cdots - 5488)^{2} \)
|
| $29$ |
\( T^{12} + \cdots + 307511296 \)
|
| $31$ |
\( (T^{6} + 3 T^{5} + \cdots + 1088)^{2} \)
|
| $37$ |
\( (T^{6} + 5 T^{5} + \cdots - 1384)^{2} \)
|
| $41$ |
\( T^{12} + \cdots + 4750104241 \)
|
| $43$ |
\( (T^{6} + 25 T^{5} + \cdots + 1396)^{2} \)
|
| $47$ |
\( T^{12} + \cdots + 167133184 \)
|
| $53$ |
\( T^{12} + 167 T^{10} + \cdots + 51380224 \)
|
| $59$ |
\( (T^{6} + 35 T^{5} + \cdots + 17440)^{2} \)
|
| $61$ |
\( (T^{6} - 26 T^{5} + \cdots + 164368)^{2} \)
|
| $67$ |
\( T^{12} + \cdots + 101052780544 \)
|
| $71$ |
\( T^{12} + \cdots + 1086361600 \)
|
| $73$ |
\( (T^{6} + 32 T^{5} + \cdots + 80752)^{2} \)
|
| $79$ |
\( T^{12} + 220 T^{10} + \cdots + 1048576 \)
|
| $83$ |
\( (T^{6} - 30 T^{5} + \cdots - 176224)^{2} \)
|
| $89$ |
\( T^{12} + \cdots + 52833780736 \)
|
| $97$ |
\( T^{12} + \cdots + 2422214656 \)
|
| show more | |
| show less |