Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [800,4,Mod(1,800)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(800, 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("800.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 800 = 2^{5} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 800.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: | \(47.2015280046\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.5685.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 11x + 6 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | yes |
| 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 a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 11x + 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( 4\nu^{2} - 2\nu - 30 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + \beta _1 + 31 ) / 4 \)
|
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 | −5.25747 | 0 | 0 | 0 | −9.15590 | 0 | 0.640965 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 2.06677 | 0 | 0 | 0 | 28.8620 | 0 | −22.7285 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 8.19070 | 0 | 0 | 0 | −21.7061 | 0 | 40.0875 | 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 | 800.4.a.w | yes | 3 |
| 4.b | odd | 2 | 1 | 800.4.a.v | yes | 3 | |
| 5.b | even | 2 | 1 | 800.4.a.u | ✓ | 3 | |
| 5.c | odd | 4 | 2 | 800.4.c.m | 6 | ||
| 8.b | even | 2 | 1 | 1600.4.a.cr | 3 | ||
| 8.d | odd | 2 | 1 | 1600.4.a.cs | 3 | ||
| 20.d | odd | 2 | 1 | 800.4.a.x | yes | 3 | |
| 20.e | even | 4 | 2 | 800.4.c.n | 6 | ||
| 40.e | odd | 2 | 1 | 1600.4.a.cq | 3 | ||
| 40.f | even | 2 | 1 | 1600.4.a.ct | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 800.4.a.u | ✓ | 3 | 5.b | even | 2 | 1 | |
| 800.4.a.v | yes | 3 | 4.b | odd | 2 | 1 | |
| 800.4.a.w | yes | 3 | 1.a | even | 1 | 1 | trivial |
| 800.4.a.x | yes | 3 | 20.d | odd | 2 | 1 | |
| 800.4.c.m | 6 | 5.c | odd | 4 | 2 | ||
| 800.4.c.n | 6 | 20.e | even | 4 | 2 | ||
| 1600.4.a.cq | 3 | 40.e | odd | 2 | 1 | ||
| 1600.4.a.cr | 3 | 8.b | even | 2 | 1 | ||
| 1600.4.a.cs | 3 | 8.d | odd | 2 | 1 | ||
| 1600.4.a.ct | 3 | 40.f | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(800))\):
|
\( T_{3}^{3} - 5T_{3}^{2} - 37T_{3} + 89 \)
|
|
\( T_{11}^{3} + 31T_{11}^{2} - 485T_{11} - 8475 \)
|
|
\( T_{13}^{3} - 8T_{13}^{2} - 6000T_{13} + 192000 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} - 5 T^{2} + \cdots + 89 \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( T^{3} + 2 T^{2} + \cdots - 5736 \)
|
| $11$ |
\( T^{3} + 31 T^{2} + \cdots - 8475 \)
|
| $13$ |
\( T^{3} - 8 T^{2} + \cdots + 192000 \)
|
| $17$ |
\( T^{3} + 89 T^{2} + \cdots - 490725 \)
|
| $19$ |
\( T^{3} + 87 T^{2} + \cdots + 7925 \)
|
| $23$ |
\( T^{3} + 122 T^{2} + \cdots - 68808 \)
|
| $29$ |
\( T^{3} - 84 T^{2} + \cdots + 278528 \)
|
| $31$ |
\( T^{3} + 294 T^{2} + \cdots - 1628600 \)
|
| $37$ |
\( T^{3} + 94 T^{2} + \cdots - 10497400 \)
|
| $41$ |
\( T^{3} + 345 T^{2} + \cdots - 14242325 \)
|
| $43$ |
\( T^{3} + 412 T^{2} + \cdots - 9200896 \)
|
| $47$ |
\( T^{3} - 824 T^{2} + \cdots + 11410304 \)
|
| $53$ |
\( T^{3} - 74 T^{2} + \cdots - 76600 \)
|
| $59$ |
\( T^{3} + 1040 T^{2} + \cdots - 140160000 \)
|
| $61$ |
\( T^{3} - 482 T^{2} + \cdots + 80633480 \)
|
| $67$ |
\( T^{3} - 735 T^{2} + \cdots + 240845067 \)
|
| $71$ |
\( T^{3} + 2048 T^{2} + \cdots + 220992000 \)
|
| $73$ |
\( T^{3} + 15 T^{2} + \cdots - 18845875 \)
|
| $79$ |
\( T^{3} + 998 T^{2} + \cdots - 361942200 \)
|
| $83$ |
\( T^{3} + \cdots - 1162222465 \)
|
| $89$ |
\( T^{3} - 1189 T^{2} + \cdots + 641970729 \)
|
| $97$ |
\( T^{3} + \cdots - 3811625000 \)
|
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