Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [465,2,Mod(211,465)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(465, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("465.211");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 465 = 3 \cdot 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 465.i (of order \(3\), degree \(2\), 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: | \(3.71304369399\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Relative dimension: | \(5\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} - x^{9} + 9x^{8} - 4x^{7} + 56x^{6} - 31x^{5} + 137x^{4} - 92x^{3} + 262x^{2} - 154x + 121 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{10} - x^{9} + 9x^{8} - 4x^{7} + 56x^{6} - 31x^{5} + 137x^{4} - 92x^{3} + 262x^{2} - 154x + 121 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( - 1819 \nu^{9} - 40340 \nu^{8} + 114969 \nu^{7} - 367043 \nu^{6} + 590981 \nu^{5} + \cdots - 8605025 ) / 10775564 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 25627 \nu^{9} + 35180 \nu^{8} - 100263 \nu^{7} + 571185 \nu^{6} - 515387 \nu^{5} + 1628834 \nu^{4} + \cdots + 5239167 ) / 5387782 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 782275 \nu^{9} - 802284 \nu^{8} + 6596735 \nu^{7} - 1864441 \nu^{6} + 39769927 \nu^{5} + \cdots + 28079865 ) / 118531204 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 902085 \nu^{9} - 1018344 \nu^{8} + 9906397 \nu^{7} - 2776531 \nu^{6} + 45629097 \nu^{5} + \cdots + 30547055 ) / 118531204 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 45797 \nu^{9} - 50660 \nu^{8} + 144381 \nu^{7} + 41241 \nu^{6} + 742169 \nu^{5} - 2345558 \nu^{4} + \cdots - 15336741 ) / 5387782 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 155023 \nu^{9} + 590300 \nu^{8} - 1682355 \nu^{7} + 4671133 \nu^{6} - 8647895 \nu^{5} + \cdots + 61617563 ) / 10775564 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( - 2785161 \nu^{9} + 1980456 \nu^{8} - 25579093 \nu^{7} + 748559 \nu^{6} - 143773437 \nu^{5} + \cdots - 117207739 ) / 118531204 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 3815125 \nu^{9} - 1780048 \nu^{8} + 32550825 \nu^{7} - 10172395 \nu^{6} + 219499077 \nu^{5} + \cdots - 466753705 ) / 118531204 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{8} - \beta_{6} + \beta_{5} + 3\beta_{4} + \beta_{3} + \beta_{2} + \beta _1 - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} + 2\beta_{3} + 6\beta_{2} \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{9} - 7\beta_{8} - \beta_{7} - 8\beta_{5} - 13\beta_{4} - 10\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{9} - 9\beta_{8} + 9\beta_{6} - 18\beta_{5} - 6\beta_{4} - 18\beta_{3} - 38\beta_{2} - 38\beta _1 + 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( 9\beta_{7} + 45\beta_{6} - 56\beta_{3} - 79\beta_{2} + 68 \)
|
| \(\nu^{7}\) | \(=\) |
\( 11\beta_{9} + 72\beta_{8} + 11\beta_{7} + 135\beta_{5} + 71\beta_{4} + 250\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 63 \beta_{9} + 291 \beta_{8} - 291 \beta_{6} + 385 \beta_{5} + 397 \beta_{4} + 385 \beta_{3} + \cdots - 397 \)
|
| \(\nu^{9}\) | \(=\) |
\( -94\beta_{7} - 548\beta_{6} + 965\beta_{3} + 1684\beta_{2} - 616 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).
| \(n\) | \(187\) | \(311\) | \(406\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1 + \beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 211.1 |
|
−2.09156 | −0.500000 | − | 0.866025i | 2.37462 | 0.500000 | − | 0.866025i | 1.04578 | + | 1.81134i | 1.26639 | + | 2.19346i | −0.783545 | −0.500000 | + | 0.866025i | −1.04578 | + | 1.81134i | ||||||||||||||||||||||||||||||||||||
| 211.2 | −1.75347 | −0.500000 | − | 0.866025i | 1.07467 | 0.500000 | − | 0.866025i | 0.876737 | + | 1.51855i | −1.23659 | − | 2.14184i | 1.62254 | −0.500000 | + | 0.866025i | −0.876737 | + | 1.51855i | |||||||||||||||||||||||||||||||||||||
| 211.3 | 0.817167 | −0.500000 | − | 0.866025i | −1.33224 | 0.500000 | − | 0.866025i | −0.408584 | − | 0.707687i | −1.47074 | − | 2.54740i | −2.72300 | −0.500000 | + | 0.866025i | 0.408584 | − | 0.707687i | |||||||||||||||||||||||||||||||||||||
| 211.4 | 1.39301 | −0.500000 | − | 0.866025i | −0.0595120 | 0.500000 | − | 0.866025i | −0.696507 | − | 1.20639i | −0.125047 | − | 0.216589i | −2.86893 | −0.500000 | + | 0.866025i | 0.696507 | − | 1.20639i | |||||||||||||||||||||||||||||||||||||
| 211.5 | 2.63485 | −0.500000 | − | 0.866025i | 4.94245 | 0.500000 | − | 0.866025i | −1.31743 | − | 2.28185i | −0.434013 | − | 0.751733i | 7.75293 | −0.500000 | + | 0.866025i | 1.31743 | − | 2.28185i | |||||||||||||||||||||||||||||||||||||
| 346.1 | −2.09156 | −0.500000 | + | 0.866025i | 2.37462 | 0.500000 | + | 0.866025i | 1.04578 | − | 1.81134i | 1.26639 | − | 2.19346i | −0.783545 | −0.500000 | − | 0.866025i | −1.04578 | − | 1.81134i | |||||||||||||||||||||||||||||||||||||
| 346.2 | −1.75347 | −0.500000 | + | 0.866025i | 1.07467 | 0.500000 | + | 0.866025i | 0.876737 | − | 1.51855i | −1.23659 | + | 2.14184i | 1.62254 | −0.500000 | − | 0.866025i | −0.876737 | − | 1.51855i | |||||||||||||||||||||||||||||||||||||
| 346.3 | 0.817167 | −0.500000 | + | 0.866025i | −1.33224 | 0.500000 | + | 0.866025i | −0.408584 | + | 0.707687i | −1.47074 | + | 2.54740i | −2.72300 | −0.500000 | − | 0.866025i | 0.408584 | + | 0.707687i | |||||||||||||||||||||||||||||||||||||
| 346.4 | 1.39301 | −0.500000 | + | 0.866025i | −0.0595120 | 0.500000 | + | 0.866025i | −0.696507 | + | 1.20639i | −0.125047 | + | 0.216589i | −2.86893 | −0.500000 | − | 0.866025i | 0.696507 | + | 1.20639i | |||||||||||||||||||||||||||||||||||||
| 346.5 | 2.63485 | −0.500000 | + | 0.866025i | 4.94245 | 0.500000 | + | 0.866025i | −1.31743 | + | 2.28185i | −0.434013 | + | 0.751733i | 7.75293 | −0.500000 | − | 0.866025i | 1.31743 | + | 2.28185i | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 31.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 465.2.i.e | ✓ | 10 |
| 31.c | even | 3 | 1 | inner | 465.2.i.e | ✓ | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 465.2.i.e | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 465.2.i.e | ✓ | 10 | 31.c | even | 3 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(465, [\chi])\):
|
\( T_{2}^{5} - T_{2}^{4} - 8T_{2}^{3} + 6T_{2}^{2} + 14T_{2} - 11 \)
|
|
\( T_{7}^{10} + 4 T_{7}^{9} + 19 T_{7}^{8} + 38 T_{7}^{7} + 131 T_{7}^{6} + 247 T_{7}^{5} + 575 T_{7}^{4} + \cdots + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{5} - T^{4} - 8 T^{3} + \cdots - 11)^{2} \)
|
| $3$ |
\( (T^{2} + T + 1)^{5} \)
|
| $5$ |
\( (T^{2} - T + 1)^{5} \)
|
| $7$ |
\( T^{10} + 4 T^{9} + \cdots + 16 \)
|
| $11$ |
\( T^{10} - T^{9} + \cdots + 15376 \)
|
| $13$ |
\( T^{10} + T^{9} + \cdots + 3936256 \)
|
| $17$ |
\( T^{10} - T^{9} + \cdots + 5929 \)
|
| $19$ |
\( T^{10} + 7 T^{9} + \cdots + 10201 \)
|
| $23$ |
\( (T^{5} + 13 T^{4} + \cdots + 144)^{2} \)
|
| $29$ |
\( (T^{5} - 3 T^{4} - 34 T^{3} + \cdots - 4)^{2} \)
|
| $31$ |
\( T^{10} + 8 T^{9} + \cdots + 28629151 \)
|
| $37$ |
\( T^{10} - 2 T^{9} + \cdots + 1106704 \)
|
| $41$ |
\( T^{10} + 14 T^{9} + \cdots + 13075456 \)
|
| $43$ |
\( T^{10} - 8 T^{9} + \cdots + 394384 \)
|
| $47$ |
\( (T^{5} - 7 T^{4} + \cdots + 163)^{2} \)
|
| $53$ |
\( T^{10} + \cdots + 266211856 \)
|
| $59$ |
\( T^{10} + 3 T^{9} + \cdots + 795664 \)
|
| $61$ |
\( (T^{5} + T^{4} + \cdots - 11873)^{2} \)
|
| $67$ |
\( T^{10} - 18 T^{9} + \cdots + 35664784 \)
|
| $71$ |
\( T^{10} + \cdots + 361456144 \)
|
| $73$ |
\( T^{10} + \cdots + 33192467344 \)
|
| $79$ |
\( T^{10} + \cdots + 434847609 \)
|
| $83$ |
\( T^{10} + \cdots + 201554809 \)
|
| $89$ |
\( (T^{5} + 19 T^{4} + \cdots + 67196)^{2} \)
|
| $97$ |
\( (T^{5} - 19 T^{4} + \cdots + 292)^{2} \)
|
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