Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1600,3,Mod(1599,1600)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1600, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1600.1599");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1600 = 2^{6} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1600.h (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: | \(43.5968422976\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{11})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 5x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 800) |
| 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
\( x^{4} - 5x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 2\nu ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 8\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} - 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 5 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{2} + 4\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times\).
| \(n\) | \(577\) | \(901\) | \(1151\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1599.1 |
|
0 | −1.31662 | 0 | 0 | 0 | 2.63325 | 0 | −7.26650 | 0 | ||||||||||||||||||||||||||||||
| 1599.2 | 0 | −1.31662 | 0 | 0 | 0 | 2.63325 | 0 | −7.26650 | 0 | |||||||||||||||||||||||||||||||
| 1599.3 | 0 | 5.31662 | 0 | 0 | 0 | −10.6332 | 0 | 19.2665 | 0 | |||||||||||||||||||||||||||||||
| 1599.4 | 0 | 5.31662 | 0 | 0 | 0 | −10.6332 | 0 | 19.2665 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 20.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1600.3.h.l | 4 | |
| 4.b | odd | 2 | 1 | 1600.3.h.e | 4 | ||
| 5.b | even | 2 | 1 | 1600.3.h.e | 4 | ||
| 5.c | odd | 4 | 1 | 1600.3.b.l | 4 | ||
| 5.c | odd | 4 | 1 | 1600.3.b.m | 4 | ||
| 8.b | even | 2 | 1 | 800.3.h.d | 4 | ||
| 8.d | odd | 2 | 1 | 800.3.h.i | 4 | ||
| 20.d | odd | 2 | 1 | inner | 1600.3.h.l | 4 | |
| 20.e | even | 4 | 1 | 1600.3.b.l | 4 | ||
| 20.e | even | 4 | 1 | 1600.3.b.m | 4 | ||
| 40.e | odd | 2 | 1 | 800.3.h.d | 4 | ||
| 40.f | even | 2 | 1 | 800.3.h.i | 4 | ||
| 40.i | odd | 4 | 1 | 800.3.b.b | ✓ | 4 | |
| 40.i | odd | 4 | 1 | 800.3.b.c | yes | 4 | |
| 40.k | even | 4 | 1 | 800.3.b.b | ✓ | 4 | |
| 40.k | even | 4 | 1 | 800.3.b.c | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 800.3.b.b | ✓ | 4 | 40.i | odd | 4 | 1 | |
| 800.3.b.b | ✓ | 4 | 40.k | even | 4 | 1 | |
| 800.3.b.c | yes | 4 | 40.i | odd | 4 | 1 | |
| 800.3.b.c | yes | 4 | 40.k | even | 4 | 1 | |
| 800.3.h.d | 4 | 8.b | even | 2 | 1 | ||
| 800.3.h.d | 4 | 40.e | odd | 2 | 1 | ||
| 800.3.h.i | 4 | 8.d | odd | 2 | 1 | ||
| 800.3.h.i | 4 | 40.f | even | 2 | 1 | ||
| 1600.3.b.l | 4 | 5.c | odd | 4 | 1 | ||
| 1600.3.b.l | 4 | 20.e | even | 4 | 1 | ||
| 1600.3.b.m | 4 | 5.c | odd | 4 | 1 | ||
| 1600.3.b.m | 4 | 20.e | even | 4 | 1 | ||
| 1600.3.h.e | 4 | 4.b | odd | 2 | 1 | ||
| 1600.3.h.e | 4 | 5.b | even | 2 | 1 | ||
| 1600.3.h.l | 4 | 1.a | even | 1 | 1 | trivial | |
| 1600.3.h.l | 4 | 20.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 \(S_{3}^{\mathrm{new}}(1600, [\chi])\):
|
\( T_{3}^{2} - 4T_{3} - 7 \)
|
|
\( T_{7}^{2} + 8T_{7} - 28 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} - 4 T - 7)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 8 T - 28)^{2} \)
|
| $11$ |
\( T^{4} + 206T^{2} + 9025 \)
|
| $13$ |
\( (T^{2} + 16)^{2} \)
|
| $17$ |
\( T^{4} + 354 T^{2} + 30625 \)
|
| $19$ |
\( T^{4} + 1198 T^{2} + 2401 \)
|
| $23$ |
\( (T^{2} + 8 T - 380)^{2} \)
|
| $29$ |
\( (T^{2} - 48 T - 128)^{2} \)
|
| $31$ |
\( T^{4} + 2680 T^{2} + 1567504 \)
|
| $37$ |
\( T^{4} + 5704 T^{2} + 7728400 \)
|
| $41$ |
\( (T^{2} - 66 T + 385)^{2} \)
|
| $43$ |
\( (T^{2} - 8 T - 688)^{2} \)
|
| $47$ |
\( (T^{2} - 176)^{2} \)
|
| $53$ |
\( T^{4} + 2760T^{2} + 784 \)
|
| $59$ |
\( T^{4} + 10368 T^{2} + 13075456 \)
|
| $61$ |
\( (T^{2} + 108 T + 2212)^{2} \)
|
| $67$ |
\( (T^{2} - 204 T + 9865)^{2} \)
|
| $71$ |
\( T^{4} + 25440 T^{2} + 152967424 \)
|
| $73$ |
\( T^{4} + 4050 T^{2} + 2798929 \)
|
| $79$ |
\( T^{4} + 6200 T^{2} + 5326864 \)
|
| $83$ |
\( (T^{2} - 52 T - 655)^{2} \)
|
| $89$ |
\( (T^{2} + 110 T - 1375)^{2} \)
|
| $97$ |
\( (T^{2} + 2116)^{2} \)
|
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