Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1764,4,Mod(1,1764)]
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, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1764.1");
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.a (trivial) |
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: | yes |
| Analytic conductor: | \(104.079369250\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{57}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 14 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 84) |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\beta = \frac{1}{2}(-1 + 3\sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | 0 | 0 | −9.82475 | 0 | 0 | 0 | 0 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | 12.8248 | 0 | 0 | 0 | 0 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(7\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1764.4.a.x | 2 | |
| 3.b | odd | 2 | 1 | 588.4.a.g | 2 | ||
| 7.b | odd | 2 | 1 | 1764.4.a.p | 2 | ||
| 7.c | even | 3 | 2 | 252.4.k.d | 4 | ||
| 7.d | odd | 6 | 2 | 1764.4.k.z | 4 | ||
| 12.b | even | 2 | 1 | 2352.4.a.cb | 2 | ||
| 21.c | even | 2 | 1 | 588.4.a.h | 2 | ||
| 21.g | even | 6 | 2 | 588.4.i.i | 4 | ||
| 21.h | odd | 6 | 2 | 84.4.i.b | ✓ | 4 | |
| 84.h | odd | 2 | 1 | 2352.4.a.bp | 2 | ||
| 84.n | even | 6 | 2 | 336.4.q.h | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 84.4.i.b | ✓ | 4 | 21.h | odd | 6 | 2 | |
| 252.4.k.d | 4 | 7.c | even | 3 | 2 | ||
| 336.4.q.h | 4 | 84.n | even | 6 | 2 | ||
| 588.4.a.g | 2 | 3.b | odd | 2 | 1 | ||
| 588.4.a.h | 2 | 21.c | even | 2 | 1 | ||
| 588.4.i.i | 4 | 21.g | even | 6 | 2 | ||
| 1764.4.a.p | 2 | 7.b | odd | 2 | 1 | ||
| 1764.4.a.x | 2 | 1.a | even | 1 | 1 | trivial | |
| 1764.4.k.z | 4 | 7.d | odd | 6 | 2 | ||
| 2352.4.a.bp | 2 | 84.h | odd | 2 | 1 | ||
| 2352.4.a.cb | 2 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1764))\):
|
\( T_{5}^{2} - 3T_{5} - 126 \)
|
|
\( T_{11}^{2} - 51T_{11} + 522 \)
|
|
\( T_{13}^{2} - 61T_{13} - 2276 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} \)
|
| $5$ |
\( T^{2} - 3T - 126 \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} - 51T + 522 \)
|
| $13$ |
\( T^{2} - 61T - 2276 \)
|
| $17$ |
\( T^{2} + 24T - 8064 \)
|
| $19$ |
\( T^{2} - 169T + 7012 \)
|
| $23$ |
\( (T - 96)^{2} \)
|
| $29$ |
\( T^{2} - 39T - 36684 \)
|
| $31$ |
\( T^{2} + 92T - 10709 \)
|
| $37$ |
\( T^{2} + 173T - 38816 \)
|
| $41$ |
\( T^{2} + 174T - 140688 \)
|
| $43$ |
\( T^{2} + 497T + 15454 \)
|
| $47$ |
\( T^{2} + 180T - 24732 \)
|
| $53$ |
\( T^{2} + 285T - 73188 \)
|
| $59$ |
\( T^{2} - 1269 T + 401436 \)
|
| $61$ |
\( T^{2} - 328T - 139316 \)
|
| $67$ |
\( T^{2} + 875T + 181018 \)
|
| $71$ |
\( T^{2} - 1404 T + 361476 \)
|
| $73$ |
\( T^{2} + 1361 T + 406522 \)
|
| $79$ |
\( T^{2} + 182T + 6229 \)
|
| $83$ |
\( T^{2} - 399T - 197334 \)
|
| $89$ |
\( T^{2} + 822T + 106848 \)
|
| $97$ |
\( T^{2} - 841T + 120262 \)
|
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