Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [912,4,Mod(607,912)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(912, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("912.607");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 912 = 2^{4} \cdot 3 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 912.k (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: | \(53.8097419252\) |
| 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: |
\( x^{10} - 4 x^{9} + 228 x^{8} + 702 x^{7} + 60343 x^{6} - 96364 x^{5} + 8487621 x^{4} + 8543982 x^{3} + \cdots + 889372467 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{19}]\) |
| Coefficient ring index: | \( 2^{13} \) |
| 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
\( x^{10} - 4 x^{9} + 228 x^{8} + 702 x^{7} + 60343 x^{6} - 96364 x^{5} + 8487621 x^{4} + 8543982 x^{3} + \cdots + 889372467 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( - 36\!\cdots\!52 \nu^{9} + \cdots + 98\!\cdots\!85 ) / 49\!\cdots\!43 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 36\!\cdots\!52 \nu^{9} + \cdots - 14\!\cdots\!28 ) / 49\!\cdots\!43 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( - 69\!\cdots\!48 \nu^{9} + \cdots - 13\!\cdots\!08 ) / 49\!\cdots\!43 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 79\!\cdots\!34 \nu^{9} + \cdots + 41\!\cdots\!91 ) / 49\!\cdots\!43 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 67\!\cdots\!35 \nu^{9} + \cdots + 40\!\cdots\!33 ) / 26\!\cdots\!72 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 16\!\cdots\!41 \nu^{9} + \cdots + 12\!\cdots\!39 ) / 39\!\cdots\!80 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 20\!\cdots\!39 \nu^{9} + \cdots - 13\!\cdots\!56 ) / 19\!\cdots\!90 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( 59\!\cdots\!11 \nu^{9} + \cdots + 54\!\cdots\!14 ) / 36\!\cdots\!80 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( 30\!\cdots\!34 \nu^{9} + \cdots + 21\!\cdots\!56 ) / 76\!\cdots\!65 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{7} - \beta_{6} - 7\beta_{5} + \beta_{4} - 3\beta_{2} + \beta _1 - 89 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( - 9 \beta_{9} - 24 \beta_{8} - 74 \beta_{7} - 6 \beta_{6} + 18 \beta_{5} + 6 \beta_{4} - 3 \beta_{3} + \cdots - 2014 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( - 11 \beta_{9} - 36 \beta_{8} + 234 \beta_{7} + 581 \beta_{6} + 415 \beta_{5} - 263 \beta_{4} + \cdots - 32487 ) / 2 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( - 2691 \beta_{9} + 10128 \beta_{8} + 13618 \beta_{7} + 13764 \beta_{6} + 10800 \beta_{5} + \cdots + 390056 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( - 16309 \beta_{9} + 45780 \beta_{8} + 21020 \beta_{7} - 83188 \beta_{6} + 198000 \beta_{5} + \cdots + 4175068 ) / 2 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 659516 \beta_{9} - 353280 \beta_{8} + 731740 \beta_{7} - 1913528 \beta_{6} - 2136592 \beta_{5} + \cdots + 26099305 ) / 2 \)
|
| \(\nu^{8}\) | \(=\) |
\( ( 6054103 \beta_{9} - 11423076 \beta_{8} - 6385158 \beta_{7} - 18022597 \beta_{6} - 20555759 \beta_{5} + \cdots + 497305865 ) / 2 \)
|
| \(\nu^{9}\) | \(=\) |
\( ( 13124341 \beta_{9} - 402846504 \beta_{8} - 531499654 \beta_{7} + 198551714 \beta_{6} + \cdots - 31378624738 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).
| \(n\) | \(97\) | \(229\) | \(305\) | \(799\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 607.1 |
|
0 | 3.00000 | 0 | −17.3568 | 0 | − | 20.6610i | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||
| 607.2 | 0 | 3.00000 | 0 | −17.3568 | 0 | 20.6610i | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 607.3 | 0 | 3.00000 | 0 | −13.6039 | 0 | − | 24.7646i | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 607.4 | 0 | 3.00000 | 0 | −13.6039 | 0 | 24.7646i | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 607.5 | 0 | 3.00000 | 0 | 1.97267 | 0 | − | 1.05996i | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 607.6 | 0 | 3.00000 | 0 | 1.97267 | 0 | 1.05996i | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 607.7 | 0 | 3.00000 | 0 | 4.89205 | 0 | − | 19.2993i | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 607.8 | 0 | 3.00000 | 0 | 4.89205 | 0 | 19.2993i | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
| 607.9 | 0 | 3.00000 | 0 | 20.0960 | 0 | − | 19.6024i | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 607.10 | 0 | 3.00000 | 0 | 20.0960 | 0 | 19.6024i | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 76.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 912.4.k.b | yes | 10 |
| 4.b | odd | 2 | 1 | 912.4.k.a | ✓ | 10 | |
| 19.b | odd | 2 | 1 | 912.4.k.a | ✓ | 10 | |
| 76.d | even | 2 | 1 | inner | 912.4.k.b | yes | 10 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 912.4.k.a | ✓ | 10 | 4.b | odd | 2 | 1 | |
| 912.4.k.a | ✓ | 10 | 19.b | odd | 2 | 1 | |
| 912.4.k.b | yes | 10 | 1.a | even | 1 | 1 | trivial |
| 912.4.k.b | yes | 10 | 76.d | 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 \(S_{4}^{\mathrm{new}}(912, [\chi])\):
|
\( T_{5}^{5} + 4T_{5}^{4} - 451T_{5}^{3} - 1990T_{5}^{2} + 28848T_{5} - 45792 \)
|
|
\( T_{31}^{5} + 6T_{31}^{4} - 64132T_{31}^{3} - 9301176T_{31}^{2} - 451413504T_{31} - 6894129024 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} \)
|
| $3$ |
\( (T - 3)^{10} \)
|
| $5$ |
\( (T^{5} + 4 T^{4} + \cdots - 45792)^{2} \)
|
| $7$ |
\( T^{10} + \cdots + 42096412800 \)
|
| $11$ |
\( T^{10} + \cdots + 14281775991072 \)
|
| $13$ |
\( T^{10} + \cdots + 14\!\cdots\!68 \)
|
| $17$ |
\( (T^{5} + 64 T^{4} + \cdots - 77802552)^{2} \)
|
| $19$ |
\( T^{10} + \cdots + 15\!\cdots\!99 \)
|
| $23$ |
\( T^{10} + \cdots + 71\!\cdots\!00 \)
|
| $29$ |
\( T^{10} + \cdots + 39\!\cdots\!92 \)
|
| $31$ |
\( (T^{5} + 6 T^{4} + \cdots - 6894129024)^{2} \)
|
| $37$ |
\( T^{10} + \cdots + 14\!\cdots\!12 \)
|
| $41$ |
\( T^{10} + \cdots + 53\!\cdots\!92 \)
|
| $43$ |
\( T^{10} + \cdots + 25\!\cdots\!00 \)
|
| $47$ |
\( T^{10} + \cdots + 75\!\cdots\!68 \)
|
| $53$ |
\( T^{10} + \cdots + 15\!\cdots\!68 \)
|
| $59$ |
\( (T^{5} + \cdots - 5564557357056)^{2} \)
|
| $61$ |
\( (T^{5} + \cdots + 5188288025952)^{2} \)
|
| $67$ |
\( (T^{5} + \cdots - 10302645090816)^{2} \)
|
| $71$ |
\( (T^{5} + \cdots - 6622497460224)^{2} \)
|
| $73$ |
\( (T^{5} + \cdots + 6232066147968)^{2} \)
|
| $79$ |
\( (T^{5} + \cdots - 10864846075392)^{2} \)
|
| $83$ |
\( T^{10} + \cdots + 74\!\cdots\!52 \)
|
| $89$ |
\( T^{10} + \cdots + 15\!\cdots\!88 \)
|
| $97$ |
\( T^{10} + \cdots + 71\!\cdots\!92 \)
|
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