Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1872,3,Mod(1169,1872)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1872, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1872.1169");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1872 = 2^{4} \cdot 3^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1872.l (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: | \(51.0083054882\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.138169810944.5 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 2x^{6} - 3x^{4} + 18x^{2} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{17}]\) |
| Coefficient ring index: | \( 2^{7}\cdot 3^{3} \) |
| Twist minimal: | no (minimal twist has level 117) |
| 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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} + 2x^{6} - 3x^{4} + 18x^{2} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{7} - \nu^{5} + 21\nu^{3} - 81\nu ) / 27 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{6} - 2\nu^{4} + 12\nu^{2} - 9 ) / 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{7} - 5\nu^{5} + 3\nu^{3} + 63\nu ) / 27 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -2\nu^{7} - 4\nu^{5} + 6\nu^{3} + 18\nu ) / 27 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{6} + 4\nu^{4} + 6\nu^{2} - 27 ) / 9 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 2\nu^{7} + 10\nu^{5} + 24\nu^{3} + 108\nu ) / 27 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 8\nu^{6} + 4\nu^{4} + 24\nu^{2} + 162 ) / 9 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + 3\beta_{4} - 2\beta_1 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} + 2\beta_{5} + 6\beta_{2} - 6 ) / 12 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2\beta_{6} + 3\beta_{3} + 5\beta_1 ) / 6 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( \beta_{7} + 20\beta_{5} - 12\beta_{2} + 30 ) / 12 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 13\beta_{6} - 9\beta_{4} - 30\beta_{3} - 8\beta_1 ) / 12 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 5\beta_{7} - 8\beta_{5} - 6\beta_{2} - 120 ) / 6 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -5\beta_{6} - 117\beta_{4} + 78\beta_{3} + 28\beta_1 ) / 12 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1872\mathbb{Z}\right)^\times\).
| \(n\) | \(145\) | \(209\) | \(469\) | \(703\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1169.1 |
|
0 | 0 | 0 | −4.83537 | 0 | − | 7.62512i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||
| 1169.2 | 0 | 0 | 0 | −4.83537 | 0 | 7.62512i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1169.3 | 0 | 0 | 0 | −2.14923 | 0 | − | 1.36290i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1169.4 | 0 | 0 | 0 | −2.14923 | 0 | 1.36290i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1169.5 | 0 | 0 | 0 | 2.14923 | 0 | − | 1.36290i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1169.6 | 0 | 0 | 0 | 2.14923 | 0 | 1.36290i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
| 1169.7 | 0 | 0 | 0 | 4.83537 | 0 | − | 7.62512i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||||||||||
| 1169.8 | 0 | 0 | 0 | 4.83537 | 0 | 7.62512i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 13.b | even | 2 | 1 | inner |
| 39.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1872.3.l.d | 8 | |
| 3.b | odd | 2 | 1 | inner | 1872.3.l.d | 8 | |
| 4.b | odd | 2 | 1 | 117.3.d.a | ✓ | 8 | |
| 12.b | even | 2 | 1 | 117.3.d.a | ✓ | 8 | |
| 13.b | even | 2 | 1 | inner | 1872.3.l.d | 8 | |
| 39.d | odd | 2 | 1 | inner | 1872.3.l.d | 8 | |
| 52.b | odd | 2 | 1 | 117.3.d.a | ✓ | 8 | |
| 156.h | even | 2 | 1 | 117.3.d.a | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 117.3.d.a | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 117.3.d.a | ✓ | 8 | 12.b | even | 2 | 1 | |
| 117.3.d.a | ✓ | 8 | 52.b | odd | 2 | 1 | |
| 117.3.d.a | ✓ | 8 | 156.h | even | 2 | 1 | |
| 1872.3.l.d | 8 | 1.a | even | 1 | 1 | trivial | |
| 1872.3.l.d | 8 | 3.b | odd | 2 | 1 | inner | |
| 1872.3.l.d | 8 | 13.b | even | 2 | 1 | inner | |
| 1872.3.l.d | 8 | 39.d | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1872, [\chi])\):
|
\( T_{5}^{4} - 28T_{5}^{2} + 108 \)
|
|
\( T_{7}^{4} + 60T_{7}^{2} + 108 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{4} - 28 T^{2} + 108)^{2} \)
|
| $7$ |
\( (T^{4} + 60 T^{2} + 108)^{2} \)
|
| $11$ |
\( (T^{4} - 244 T^{2} + 12)^{2} \)
|
| $13$ |
\( (T^{4} + 12 T^{3} + \cdots + 28561)^{2} \)
|
| $17$ |
\( (T^{4} + 828 T^{2} + 142884)^{2} \)
|
| $19$ |
\( (T^{4} + 924 T^{2} + 117612)^{2} \)
|
| $23$ |
\( (T^{4} + 936 T^{2} + 104976)^{2} \)
|
| $29$ |
\( (T^{4} + 3492 T^{2} + 2022084)^{2} \)
|
| $31$ |
\( (T^{4} + 1428 T^{2} + 375948)^{2} \)
|
| $37$ |
\( (T^{4} + 2040 T^{2} + 657072)^{2} \)
|
| $41$ |
\( (T^{4} - 220 T^{2} + 1452)^{2} \)
|
| $43$ |
\( (T^{2} + 4 T - 788)^{4} \)
|
| $47$ |
\( (T^{4} - 6580 T^{2} + 2312652)^{2} \)
|
| $53$ |
\( (T^{4} + 5148 T^{2} + 3175524)^{2} \)
|
| $59$ |
\( (T^{4} - 3484 T^{2} + 2027052)^{2} \)
|
| $61$ |
\( (T^{2} + 60 T + 548)^{4} \)
|
| $67$ |
\( (T^{4} + 11412 T^{2} + 32551308)^{2} \)
|
| $71$ |
\( (T^{4} - 23452 T^{2} + 100433388)^{2} \)
|
| $73$ |
\( (T^{4} + 17064 T^{2} + 29428272)^{2} \)
|
| $79$ |
\( (T^{2} + 144 T + 5096)^{4} \)
|
| $83$ |
\( (T^{4} - 17596 T^{2} + 77358252)^{2} \)
|
| $89$ |
\( (T^{4} - 11068 T^{2} + 4869228)^{2} \)
|
| $97$ |
\( (T^{4} + 28968 T^{2} + 197998128)^{2} \)
|
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