Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1600,4,Mod(1,1600)]
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([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("1600.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1600 = 2^{6} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1600.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: | \(94.4030560092\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{6}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 6 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 40) |
| 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 \(\beta = 2\sqrt{6}\). 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 | −2.89898 | 0 | 0 | 0 | −16.6969 | 0 | −18.5959 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 6.89898 | 0 | 0 | 0 | 12.6969 | 0 | 20.5959 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(5\) | \(-1\) |
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 | 1600.4.a.cl | 2 | |
| 4.b | odd | 2 | 1 | 1600.4.a.cf | 2 | ||
| 5.b | even | 2 | 1 | 1600.4.a.ce | 2 | ||
| 5.c | odd | 4 | 2 | 320.4.c.g | 4 | ||
| 8.b | even | 2 | 1 | 200.4.a.k | 2 | ||
| 8.d | odd | 2 | 1 | 400.4.a.x | 2 | ||
| 20.d | odd | 2 | 1 | 1600.4.a.cm | 2 | ||
| 20.e | even | 4 | 2 | 320.4.c.h | 4 | ||
| 24.h | odd | 2 | 1 | 1800.4.a.bk | 2 | ||
| 40.e | odd | 2 | 1 | 400.4.a.v | 2 | ||
| 40.f | even | 2 | 1 | 200.4.a.l | 2 | ||
| 40.i | odd | 4 | 2 | 40.4.c.a | ✓ | 4 | |
| 40.k | even | 4 | 2 | 80.4.c.c | 4 | ||
| 120.i | odd | 2 | 1 | 1800.4.a.bp | 2 | ||
| 120.q | odd | 4 | 2 | 720.4.f.m | 4 | ||
| 120.w | even | 4 | 2 | 360.4.f.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 40.4.c.a | ✓ | 4 | 40.i | odd | 4 | 2 | |
| 80.4.c.c | 4 | 40.k | even | 4 | 2 | ||
| 200.4.a.k | 2 | 8.b | even | 2 | 1 | ||
| 200.4.a.l | 2 | 40.f | even | 2 | 1 | ||
| 320.4.c.g | 4 | 5.c | odd | 4 | 2 | ||
| 320.4.c.h | 4 | 20.e | even | 4 | 2 | ||
| 360.4.f.e | 4 | 120.w | even | 4 | 2 | ||
| 400.4.a.v | 2 | 40.e | odd | 2 | 1 | ||
| 400.4.a.x | 2 | 8.d | odd | 2 | 1 | ||
| 720.4.f.m | 4 | 120.q | odd | 4 | 2 | ||
| 1600.4.a.ce | 2 | 5.b | even | 2 | 1 | ||
| 1600.4.a.cf | 2 | 4.b | odd | 2 | 1 | ||
| 1600.4.a.cl | 2 | 1.a | even | 1 | 1 | trivial | |
| 1600.4.a.cm | 2 | 20.d | odd | 2 | 1 | ||
| 1800.4.a.bk | 2 | 24.h | odd | 2 | 1 | ||
| 1800.4.a.bp | 2 | 120.i | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1600))\):
|
\( T_{3}^{2} - 4T_{3} - 20 \)
|
|
\( T_{7}^{2} + 4T_{7} - 212 \)
|
|
\( T_{11}^{2} + 40T_{11} - 1136 \)
|
|
\( T_{13}^{2} - 104T_{13} + 2608 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} - 4T - 20 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} + 4T - 212 \)
|
| $11$ |
\( T^{2} + 40T - 1136 \)
|
| $13$ |
\( T^{2} - 104T + 2608 \)
|
| $17$ |
\( T^{2} + 96T - 3840 \)
|
| $19$ |
\( T^{2} + 40T - 1136 \)
|
| $23$ |
\( T^{2} + 284T + 16108 \)
|
| $29$ |
\( T^{2} - 140T - 1244 \)
|
| $31$ |
\( T^{2} - 192T - 46080 \)
|
| $37$ |
\( T^{2} - 200T - 17744 \)
|
| $41$ |
\( T^{2} + 524T + 65188 \)
|
| $43$ |
\( T^{2} - 372T + 24012 \)
|
| $47$ |
\( T^{2} + 84T - 99636 \)
|
| $53$ |
\( T^{2} + 296T - 268496 \)
|
| $59$ |
\( T^{2} + 696T - 64752 \)
|
| $61$ |
\( T^{2} - 692T + 100900 \)
|
| $67$ |
\( T^{2} - 316T + 23788 \)
|
| $71$ |
\( T^{2} - 688T + 93760 \)
|
| $73$ |
\( T^{2} - 656T - 95552 \)
|
| $79$ |
\( T^{2} + 736T + 129280 \)
|
| $83$ |
\( T^{2} + 1628 T + 604972 \)
|
| $89$ |
\( T^{2} + 660T + 53604 \)
|
| $97$ |
\( T^{2} + 896T + 162304 \)
|
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