Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1764,4,Mod(361,1764)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 4]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1764.361");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1764.k (of order \(3\), degree \(2\), 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: | \(104.079369250\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{25}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 28) |
| 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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).
| \(n\) | \(785\) | \(883\) | \(1081\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 361.1 |
|
0 | 0 | 0 | −4.00000 | − | 6.92820i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||||||||
| 1549.1 | 0 | 0 | 0 | −4.00000 | + | 6.92820i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1764.4.k.d | 2 | |
| 3.b | odd | 2 | 1 | 196.4.e.f | 2 | ||
| 7.b | odd | 2 | 1 | 1764.4.k.m | 2 | ||
| 7.c | even | 3 | 1 | 252.4.a.d | 1 | ||
| 7.c | even | 3 | 1 | inner | 1764.4.k.d | 2 | |
| 7.d | odd | 6 | 1 | 1764.4.a.c | 1 | ||
| 7.d | odd | 6 | 1 | 1764.4.k.m | 2 | ||
| 21.c | even | 2 | 1 | 196.4.e.a | 2 | ||
| 21.g | even | 6 | 1 | 196.4.a.d | 1 | ||
| 21.g | even | 6 | 1 | 196.4.e.a | 2 | ||
| 21.h | odd | 6 | 1 | 28.4.a.a | ✓ | 1 | |
| 21.h | odd | 6 | 1 | 196.4.e.f | 2 | ||
| 28.g | odd | 6 | 1 | 1008.4.a.o | 1 | ||
| 84.j | odd | 6 | 1 | 784.4.a.a | 1 | ||
| 84.n | even | 6 | 1 | 112.4.a.g | 1 | ||
| 105.o | odd | 6 | 1 | 700.4.a.n | 1 | ||
| 105.x | even | 12 | 2 | 700.4.e.a | 2 | ||
| 168.s | odd | 6 | 1 | 448.4.a.p | 1 | ||
| 168.v | even | 6 | 1 | 448.4.a.a | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 28.4.a.a | ✓ | 1 | 21.h | odd | 6 | 1 | |
| 112.4.a.g | 1 | 84.n | even | 6 | 1 | ||
| 196.4.a.d | 1 | 21.g | even | 6 | 1 | ||
| 196.4.e.a | 2 | 21.c | even | 2 | 1 | ||
| 196.4.e.a | 2 | 21.g | even | 6 | 1 | ||
| 196.4.e.f | 2 | 3.b | odd | 2 | 1 | ||
| 196.4.e.f | 2 | 21.h | odd | 6 | 1 | ||
| 252.4.a.d | 1 | 7.c | even | 3 | 1 | ||
| 448.4.a.a | 1 | 168.v | even | 6 | 1 | ||
| 448.4.a.p | 1 | 168.s | odd | 6 | 1 | ||
| 700.4.a.n | 1 | 105.o | odd | 6 | 1 | ||
| 700.4.e.a | 2 | 105.x | even | 12 | 2 | ||
| 784.4.a.a | 1 | 84.j | odd | 6 | 1 | ||
| 1008.4.a.o | 1 | 28.g | odd | 6 | 1 | ||
| 1764.4.a.c | 1 | 7.d | odd | 6 | 1 | ||
| 1764.4.k.d | 2 | 1.a | even | 1 | 1 | trivial | |
| 1764.4.k.d | 2 | 7.c | even | 3 | 1 | inner | |
| 1764.4.k.m | 2 | 7.b | odd | 2 | 1 | ||
| 1764.4.k.m | 2 | 7.d | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(1764, [\chi])\):
|
\( T_{5}^{2} + 8T_{5} + 64 \)
|
|
\( T_{11}^{2} + 40T_{11} + 1600 \)
|
|
\( T_{13} + 12 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} + 8T + 64 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} + 40T + 1600 \)
|
| $13$ |
\( (T + 12)^{2} \)
|
| $17$ |
\( T^{2} + 58T + 3364 \)
|
| $19$ |
\( T^{2} + 26T + 676 \)
|
| $23$ |
\( T^{2} + 64T + 4096 \)
|
| $29$ |
\( (T - 62)^{2} \)
|
| $31$ |
\( T^{2} + 252T + 63504 \)
|
| $37$ |
\( T^{2} + 26T + 676 \)
|
| $41$ |
\( (T + 6)^{2} \)
|
| $43$ |
\( (T - 416)^{2} \)
|
| $47$ |
\( T^{2} + 396T + 156816 \)
|
| $53$ |
\( T^{2} + 450T + 202500 \)
|
| $59$ |
\( T^{2} - 274T + 75076 \)
|
| $61$ |
\( T^{2} - 576T + 331776 \)
|
| $67$ |
\( T^{2} - 476T + 226576 \)
|
| $71$ |
\( (T - 448)^{2} \)
|
| $73$ |
\( T^{2} - 158T + 24964 \)
|
| $79$ |
\( T^{2} - 936T + 876096 \)
|
| $83$ |
\( (T + 530)^{2} \)
|
| $89$ |
\( T^{2} + 390T + 152100 \)
|
| $97$ |
\( (T - 214)^{2} \)
|
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