Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [465,2,Mod(94,465)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(465, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("465.94");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 465 = 3 \cdot 5 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 465.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: | \(3.71304369399\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | 10.0.1016580161536.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} + 12x^{8} + 48x^{6} + 72x^{4} + 36x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{10} + 12x^{8} + 48x^{6} + 72x^{4} + 36x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + 6\nu^{3} + 6\nu ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{6} - 6\nu^{4} - 6\nu^{2} ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} + 8\nu^{4} + 16\nu^{2} + 4 ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{7} + 8\nu^{5} + 18\nu^{3} + 12\nu ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{8} + 11\nu^{6} + 38\nu^{4} + 42\nu^{2} + 10 ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{9} + 11\nu^{7} + 38\nu^{5} + 42\nu^{3} + 10\nu ) / 2 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -\nu^{9} - 12\nu^{7} - 47\nu^{5} - 64\nu^{3} - 20\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{9} + \beta_{8} + \beta_{6} + \beta_{3} - 4\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} + \beta_{4} - 5\beta_{2} + 8 \)
|
| \(\nu^{5}\) | \(=\) |
\( -6\beta_{9} - 6\beta_{8} - 6\beta_{6} - 4\beta_{3} + 18\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -6\beta_{5} - 8\beta_{4} + 24\beta_{2} - 36 \)
|
| \(\nu^{7}\) | \(=\) |
\( 30\beta_{9} + 30\beta_{8} + 32\beta_{6} + 14\beta_{3} - 84\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 2\beta_{7} + 28\beta_{5} + 50\beta_{4} - 116\beta_{2} + 166 \)
|
| \(\nu^{9}\) | \(=\) |
\( -144\beta_{9} - 142\beta_{8} - 166\beta_{6} - 44\beta_{3} + 398\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/465\mathbb{Z}\right)^\times\).
| \(n\) | \(187\) | \(311\) | \(406\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 94.1 |
|
− | 2.58691i | − | 1.00000i | −4.69208 | −2.10518 | − | 0.753811i | −2.58691 | − | 3.58691i | 6.96416i | −1.00000 | −1.95004 | + | 5.44589i | |||||||||||||||||||||||||||||||||||||||||
| 94.2 | − | 1.86307i | − | 1.00000i | −1.47102 | 0.392048 | + | 2.20143i | −1.86307 | − | 2.86307i | − | 0.985527i | −1.00000 | 4.10141 | − | 0.730412i | |||||||||||||||||||||||||||||||||||||||||
| 94.3 | − | 1.39276i | 1.00000i | 0.0602300 | −1.33253 | − | 1.79565i | 1.39276 | − | 0.392756i | − | 2.86940i | −1.00000 | −2.50091 | + | 1.85588i | ||||||||||||||||||||||||||||||||||||||||||
| 94.4 | − | 0.697747i | 1.00000i | 1.51315 | 0.815403 | + | 2.08209i | 0.697747 | 0.302253i | − | 2.45129i | −1.00000 | 1.45277 | − | 0.568945i | |||||||||||||||||||||||||||||||||||||||||||
| 94.5 | − | 0.640530i | − | 1.00000i | 1.58972 | 2.23025 | − | 0.161179i | −0.640530 | − | 1.64053i | − | 2.29932i | −1.00000 | −0.103240 | − | 1.42854i | |||||||||||||||||||||||||||||||||||||||||
| 94.6 | 0.640530i | 1.00000i | 1.58972 | 2.23025 | + | 0.161179i | −0.640530 | 1.64053i | 2.29932i | −1.00000 | −0.103240 | + | 1.42854i | |||||||||||||||||||||||||||||||||||||||||||||
| 94.7 | 0.697747i | − | 1.00000i | 1.51315 | 0.815403 | − | 2.08209i | 0.697747 | − | 0.302253i | 2.45129i | −1.00000 | 1.45277 | + | 0.568945i | |||||||||||||||||||||||||||||||||||||||||||
| 94.8 | 1.39276i | − | 1.00000i | 0.0602300 | −1.33253 | + | 1.79565i | 1.39276 | 0.392756i | 2.86940i | −1.00000 | −2.50091 | − | 1.85588i | ||||||||||||||||||||||||||||||||||||||||||||
| 94.9 | 1.86307i | 1.00000i | −1.47102 | 0.392048 | − | 2.20143i | −1.86307 | 2.86307i | 0.985527i | −1.00000 | 4.10141 | + | 0.730412i | |||||||||||||||||||||||||||||||||||||||||||||
| 94.10 | 2.58691i | 1.00000i | −4.69208 | −2.10518 | + | 0.753811i | −2.58691 | 3.58691i | − | 6.96416i | −1.00000 | −1.95004 | − | 5.44589i | ||||||||||||||||||||||||||||||||||||||||||||
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 | 465.2.c.a | ✓ | 10 |
| 3.b | odd | 2 | 1 | 1395.2.c.f | 10 | ||
| 5.b | even | 2 | 1 | inner | 465.2.c.a | ✓ | 10 |
| 5.c | odd | 4 | 1 | 2325.2.a.w | 5 | ||
| 5.c | odd | 4 | 1 | 2325.2.a.x | 5 | ||
| 15.d | odd | 2 | 1 | 1395.2.c.f | 10 | ||
| 15.e | even | 4 | 1 | 6975.2.a.bs | 5 | ||
| 15.e | even | 4 | 1 | 6975.2.a.bv | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 465.2.c.a | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 465.2.c.a | ✓ | 10 | 5.b | even | 2 | 1 | inner |
| 1395.2.c.f | 10 | 3.b | odd | 2 | 1 | ||
| 1395.2.c.f | 10 | 15.d | odd | 2 | 1 | ||
| 2325.2.a.w | 5 | 5.c | odd | 4 | 1 | ||
| 2325.2.a.x | 5 | 5.c | odd | 4 | 1 | ||
| 6975.2.a.bs | 5 | 15.e | even | 4 | 1 | ||
| 6975.2.a.bv | 5 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 13T_{2}^{8} + 54T_{2}^{6} + 86T_{2}^{4} + 49T_{2}^{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(465, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 13 T^{8} + \cdots + 9 \)
|
| $3$ |
\( (T^{2} + 1)^{5} \)
|
| $5$ |
\( T^{10} + T^{8} + \cdots + 3125 \)
|
| $7$ |
\( T^{10} + 24 T^{8} + \cdots + 4 \)
|
| $11$ |
\( (T^{5} - 12 T^{3} + \cdots - 24)^{2} \)
|
| $13$ |
\( T^{10} + 36 T^{8} + \cdots + 2116 \)
|
| $17$ |
\( T^{10} + 76 T^{8} + \cdots + 1296 \)
|
| $19$ |
\( (T^{5} - 4 T^{4} - 8 T^{3} + \cdots - 16)^{2} \)
|
| $23$ |
\( T^{10} + 140 T^{8} + \cdots + 4787344 \)
|
| $29$ |
\( (T^{5} + 4 T^{4} + \cdots + 326)^{2} \)
|
| $31$ |
\( (T + 1)^{10} \)
|
| $37$ |
\( T^{10} + 184 T^{8} + \cdots + 91204 \)
|
| $41$ |
\( (T^{5} - 120 T^{3} + \cdots + 232)^{2} \)
|
| $43$ |
\( T^{10} + 160 T^{8} + \cdots + 6031936 \)
|
| $47$ |
\( T^{10} + 320 T^{8} + \cdots + 80496784 \)
|
| $53$ |
\( T^{10} + \cdots + 280495504 \)
|
| $59$ |
\( (T^{5} + 10 T^{4} + \cdots + 474)^{2} \)
|
| $61$ |
\( (T^{5} + 14 T^{4} + \cdots - 848)^{2} \)
|
| $67$ |
\( T^{10} + 188 T^{8} + \cdots + 3452164 \)
|
| $71$ |
\( (T^{5} - 6 T^{4} + \cdots - 35762)^{2} \)
|
| $73$ |
\( T^{10} + 524 T^{8} + \cdots + 20016676 \)
|
| $79$ |
\( (T^{5} - 24 T^{4} + \cdots + 22404)^{2} \)
|
| $83$ |
\( T^{10} + \cdots + 3122127376 \)
|
| $89$ |
\( (T^{5} + 10 T^{4} + \cdots - 1186)^{2} \)
|
| $97$ |
\( T^{10} + 424 T^{8} + \cdots + 7011904 \)
|
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