Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1764,3,Mod(197,1764)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1764, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1764.197");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1764.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: | \(48.0655186332\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-23})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} + 17x^{2} - 16x + 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 252) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} + 17x^{2} - 16x + 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{3} - 3\nu^{2} + 25\nu - 12 ) / 5 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu + 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{3} - 3\nu^{2} + 35\nu - 17 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} - \beta _1 - 15 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -11\beta_{3} + 3\beta_{2} + 16\beta _1 - 23 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).
| \(n\) | \(785\) | \(883\) | \(1081\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 197.1 |
|
0 | 0 | 0 | − | 7.62426i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||
| 197.2 | 0 | 0 | 0 | − | 1.96740i | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 197.3 | 0 | 0 | 0 | 1.96740i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 197.4 | 0 | 0 | 0 | 7.62426i | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1764.3.c.e | 4 | |
| 3.b | odd | 2 | 1 | inner | 1764.3.c.e | 4 | |
| 7.b | odd | 2 | 1 | 1764.3.c.g | 4 | ||
| 7.c | even | 3 | 2 | 252.3.bk.b | ✓ | 8 | |
| 7.d | odd | 6 | 2 | 1764.3.bk.f | 8 | ||
| 21.c | even | 2 | 1 | 1764.3.c.g | 4 | ||
| 21.g | even | 6 | 2 | 1764.3.bk.f | 8 | ||
| 21.h | odd | 6 | 2 | 252.3.bk.b | ✓ | 8 | |
| 28.g | odd | 6 | 2 | 1008.3.dc.d | 8 | ||
| 84.n | even | 6 | 2 | 1008.3.dc.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 252.3.bk.b | ✓ | 8 | 7.c | even | 3 | 2 | |
| 252.3.bk.b | ✓ | 8 | 21.h | odd | 6 | 2 | |
| 1008.3.dc.d | 8 | 28.g | odd | 6 | 2 | ||
| 1008.3.dc.d | 8 | 84.n | even | 6 | 2 | ||
| 1764.3.c.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 1764.3.c.e | 4 | 3.b | odd | 2 | 1 | inner | |
| 1764.3.c.g | 4 | 7.b | odd | 2 | 1 | ||
| 1764.3.c.g | 4 | 21.c | even | 2 | 1 | ||
| 1764.3.bk.f | 8 | 7.d | odd | 6 | 2 | ||
| 1764.3.bk.f | 8 | 21.g | even | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1764, [\chi])\):
|
\( T_{5}^{4} + 62T_{5}^{2} + 225 \)
|
|
\( T_{13}^{2} + 28T_{13} + 150 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 62T^{2} + 225 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + 242T^{2} + 5625 \)
|
| $13$ |
\( (T^{2} + 28 T + 150)^{2} \)
|
| $17$ |
\( T^{4} + 836 T^{2} + 101124 \)
|
| $19$ |
\( (T^{2} - 4 T - 410)^{2} \)
|
| $23$ |
\( T^{4} + 2948 T^{2} + 2160900 \)
|
| $29$ |
\( T^{4} + 62T^{2} + 225 \)
|
| $31$ |
\( (T^{2} + 10 T - 21)^{2} \)
|
| $37$ |
\( (T^{2} + 80 T + 1186)^{2} \)
|
| $41$ |
\( T^{4} + 2232 T^{2} + 291600 \)
|
| $43$ |
\( (T^{2} + 20 T - 314)^{2} \)
|
| $47$ |
\( (T^{2} + 1458)^{2} \)
|
| $53$ |
\( T^{4} + 558 T^{2} + 18225 \)
|
| $59$ |
\( T^{4} + 14354 T^{2} + 24235929 \)
|
| $61$ |
\( (T^{2} + 4 T - 2250)^{2} \)
|
| $67$ |
\( (T^{2} - 72 T + 560)^{2} \)
|
| $71$ |
\( T^{4} + 8132 T^{2} + 11088900 \)
|
| $73$ |
\( (T^{2} + 144 T + 584)^{2} \)
|
| $79$ |
\( (T^{2} + 70 T + 1179)^{2} \)
|
| $83$ |
\( T^{4} + 21698 T^{2} + 9455625 \)
|
| $89$ |
\( T^{4} + 18504 T^{2} + 31945104 \)
|
| $97$ |
\( (T^{2} + 138 T - 7015)^{2} \)
|
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