Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [800,2,Mod(101,800)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(800, base_ring=CyclotomicField(8))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("800.101");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 800 = 2^{5} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 800.y (of order \(8\), degree \(4\), 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: | \(6.38803216170\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 32) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{8}]$ |
$q$-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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/800\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(351\) | \(577\) |
| \(\chi(n)\) | \(\zeta_{8}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 101.1 |
|
1.41421i | −0.707107 | − | 1.70711i | −2.00000 | 0 | 2.41421 | − | 1.00000i | −1.00000 | − | 1.00000i | − | 2.82843i | −0.292893 | + | 0.292893i | 0 | |||||||||||||||||||||
| 301.1 | − | 1.41421i | −0.707107 | + | 1.70711i | −2.00000 | 0 | 2.41421 | + | 1.00000i | −1.00000 | + | 1.00000i | 2.82843i | −0.292893 | − | 0.292893i | 0 | ||||||||||||||||||||||
| 501.1 | − | 1.41421i | 0.707107 | − | 0.292893i | −2.00000 | 0 | −0.414214 | − | 1.00000i | −1.00000 | − | 1.00000i | 2.82843i | −1.70711 | + | 1.70711i | 0 | ||||||||||||||||||||||
| 701.1 | 1.41421i | 0.707107 | + | 0.292893i | −2.00000 | 0 | −0.414214 | + | 1.00000i | −1.00000 | + | 1.00000i | − | 2.82843i | −1.70711 | − | 1.70711i | 0 | ||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 32.g | even | 8 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 800.2.y.a | 4 | |
| 5.b | even | 2 | 1 | 32.2.g.a | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 800.2.ba.a | 4 | ||
| 5.c | odd | 4 | 1 | 800.2.ba.b | 4 | ||
| 15.d | odd | 2 | 1 | 288.2.v.a | 4 | ||
| 20.d | odd | 2 | 1 | 128.2.g.a | 4 | ||
| 32.g | even | 8 | 1 | inner | 800.2.y.a | 4 | |
| 40.e | odd | 2 | 1 | 256.2.g.a | 4 | ||
| 40.f | even | 2 | 1 | 256.2.g.b | 4 | ||
| 60.h | even | 2 | 1 | 1152.2.v.a | 4 | ||
| 80.k | odd | 4 | 1 | 512.2.g.b | 4 | ||
| 80.k | odd | 4 | 1 | 512.2.g.c | 4 | ||
| 80.q | even | 4 | 1 | 512.2.g.a | 4 | ||
| 80.q | even | 4 | 1 | 512.2.g.d | 4 | ||
| 160.v | odd | 8 | 1 | 800.2.ba.a | 4 | ||
| 160.y | odd | 8 | 1 | 128.2.g.a | 4 | ||
| 160.y | odd | 8 | 1 | 256.2.g.a | 4 | ||
| 160.y | odd | 8 | 1 | 512.2.g.b | 4 | ||
| 160.y | odd | 8 | 1 | 512.2.g.c | 4 | ||
| 160.z | even | 8 | 1 | 32.2.g.a | ✓ | 4 | |
| 160.z | even | 8 | 1 | 256.2.g.b | 4 | ||
| 160.z | even | 8 | 1 | 512.2.g.a | 4 | ||
| 160.z | even | 8 | 1 | 512.2.g.d | 4 | ||
| 160.bb | odd | 8 | 1 | 800.2.ba.b | 4 | ||
| 320.bf | even | 16 | 2 | 4096.2.a.e | 4 | ||
| 320.bh | odd | 16 | 2 | 4096.2.a.f | 4 | ||
| 480.bs | even | 8 | 1 | 1152.2.v.a | 4 | ||
| 480.bu | odd | 8 | 1 | 288.2.v.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 32.2.g.a | ✓ | 4 | 5.b | even | 2 | 1 | |
| 32.2.g.a | ✓ | 4 | 160.z | even | 8 | 1 | |
| 128.2.g.a | 4 | 20.d | odd | 2 | 1 | ||
| 128.2.g.a | 4 | 160.y | odd | 8 | 1 | ||
| 256.2.g.a | 4 | 40.e | odd | 2 | 1 | ||
| 256.2.g.a | 4 | 160.y | odd | 8 | 1 | ||
| 256.2.g.b | 4 | 40.f | even | 2 | 1 | ||
| 256.2.g.b | 4 | 160.z | even | 8 | 1 | ||
| 288.2.v.a | 4 | 15.d | odd | 2 | 1 | ||
| 288.2.v.a | 4 | 480.bu | odd | 8 | 1 | ||
| 512.2.g.a | 4 | 80.q | even | 4 | 1 | ||
| 512.2.g.a | 4 | 160.z | even | 8 | 1 | ||
| 512.2.g.b | 4 | 80.k | odd | 4 | 1 | ||
| 512.2.g.b | 4 | 160.y | odd | 8 | 1 | ||
| 512.2.g.c | 4 | 80.k | odd | 4 | 1 | ||
| 512.2.g.c | 4 | 160.y | odd | 8 | 1 | ||
| 512.2.g.d | 4 | 80.q | even | 4 | 1 | ||
| 512.2.g.d | 4 | 160.z | even | 8 | 1 | ||
| 800.2.y.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 800.2.y.a | 4 | 32.g | even | 8 | 1 | inner | |
| 800.2.ba.a | 4 | 5.c | odd | 4 | 1 | ||
| 800.2.ba.a | 4 | 160.v | odd | 8 | 1 | ||
| 800.2.ba.b | 4 | 5.c | odd | 4 | 1 | ||
| 800.2.ba.b | 4 | 160.bb | odd | 8 | 1 | ||
| 1152.2.v.a | 4 | 60.h | even | 2 | 1 | ||
| 1152.2.v.a | 4 | 480.bs | even | 8 | 1 | ||
| 4096.2.a.e | 4 | 320.bf | even | 16 | 2 | ||
| 4096.2.a.f | 4 | 320.bh | odd | 16 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 2T_{3}^{2} - 4T_{3} + 2 \)
acting on \(S_{2}^{\mathrm{new}}(800, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2)^{2} \)
|
| $3$ |
\( T^{4} + 2 T^{2} + \cdots + 2 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T^{2} + 2 T + 2)^{2} \)
|
| $11$ |
\( T^{4} + 8 T^{3} + \cdots + 2 \)
|
| $13$ |
\( T^{4} + 4 T^{3} + \cdots + 2 \)
|
| $17$ |
\( (T^{2} + 8)^{2} \)
|
| $19$ |
\( T^{4} + 8 T^{3} + \cdots + 578 \)
|
| $23$ |
\( T^{4} + 12 T^{3} + \cdots + 4 \)
|
| $29$ |
\( T^{4} + 4 T^{3} + \cdots + 98 \)
|
| $31$ |
\( (T + 4)^{4} \)
|
| $37$ |
\( T^{4} + 4 T^{3} + \cdots + 2 \)
|
| $41$ |
\( T^{4} + 12 T^{3} + \cdots + 4 \)
|
| $43$ |
\( T^{4} + 16 T^{3} + \cdots + 1922 \)
|
| $47$ |
\( T^{4} + 136T^{2} + 16 \)
|
| $53$ |
\( T^{4} + 4 T^{3} + \cdots + 98 \)
|
| $59$ |
\( T^{4} + 16 T^{3} + \cdots + 1058 \)
|
| $61$ |
\( T^{4} - 4 T^{3} + \cdots + 2 \)
|
| $67$ |
\( T^{4} - 8 T^{3} + \cdots + 578 \)
|
| $71$ |
\( T^{4} + 12 T^{3} + \cdots + 4 \)
|
| $73$ |
\( (T^{2} + 14 T + 98)^{2} \)
|
| $79$ |
\( (T^{2} + 36)^{2} \)
|
| $83$ |
\( T^{4} + 16 T^{3} + \cdots + 1058 \)
|
| $89$ |
\( T^{4} - 12 T^{3} + \cdots + 2116 \)
|
| $97$ |
\( (T^{2} - 20 T + 28)^{2} \)
|
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