Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [195,3,Mod(8,195)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(195, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("195.8");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 195 = 3 \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 195.j (of order \(4\), degree \(2\), 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: | \(5.31336515503\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| 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: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(\left(\mathbb{Z}/195\mathbb{Z}\right)^\times\).
| \(n\) | \(106\) | \(131\) | \(157\) |
| \(\chi(n)\) | \(\zeta_{8}^{2}\) | \(-1\) | \(-\zeta_{8}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 8.1 |
|
− | 2.82843i | 1.29289 | + | 2.70711i | −4.00000 | 4.94975 | + | 0.707107i | 7.65685 | − | 3.65685i | 5.00000i | 0 | −5.65685 | + | 7.00000i | 2.00000 | − | 14.0000i | |||||||||||||||||||
| 8.2 | 2.82843i | 2.70711 | + | 1.29289i | −4.00000 | −4.94975 | − | 0.707107i | −3.65685 | + | 7.65685i | 5.00000i | 0 | 5.65685 | + | 7.00000i | 2.00000 | − | 14.0000i | |||||||||||||||||||||
| 122.1 | − | 2.82843i | 2.70711 | − | 1.29289i | −4.00000 | −4.94975 | + | 0.707107i | −3.65685 | − | 7.65685i | − | 5.00000i | 0 | 5.65685 | − | 7.00000i | 2.00000 | + | 14.0000i | |||||||||||||||||||
| 122.2 | 2.82843i | 1.29289 | − | 2.70711i | −4.00000 | 4.94975 | − | 0.707107i | 7.65685 | + | 3.65685i | − | 5.00000i | 0 | −5.65685 | − | 7.00000i | 2.00000 | + | 14.0000i | ||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 65.k | even | 4 | 1 | inner |
| 195.j | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 195.3.j.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 195.3.j.c | ✓ | 4 |
| 5.c | odd | 4 | 1 | 195.3.u.c | yes | 4 | |
| 13.d | odd | 4 | 1 | 195.3.u.c | yes | 4 | |
| 15.e | even | 4 | 1 | 195.3.u.c | yes | 4 | |
| 39.f | even | 4 | 1 | 195.3.u.c | yes | 4 | |
| 65.k | even | 4 | 1 | inner | 195.3.j.c | ✓ | 4 |
| 195.j | odd | 4 | 1 | inner | 195.3.j.c | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 195.3.j.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 195.3.j.c | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 195.3.j.c | ✓ | 4 | 65.k | even | 4 | 1 | inner |
| 195.3.j.c | ✓ | 4 | 195.j | odd | 4 | 1 | inner |
| 195.3.u.c | yes | 4 | 5.c | odd | 4 | 1 | |
| 195.3.u.c | yes | 4 | 13.d | odd | 4 | 1 | |
| 195.3.u.c | yes | 4 | 15.e | even | 4 | 1 | |
| 195.3.u.c | yes | 4 | 39.f | even | 4 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(195, [\chi])\):
|
\( T_{2}^{2} + 8 \)
|
|
\( T_{11}^{4} + 28561 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 8)^{2} \)
|
| $3$ |
\( T^{4} - 8 T^{3} + \cdots + 81 \)
|
| $5$ |
\( T^{4} - 48T^{2} + 625 \)
|
| $7$ |
\( (T^{2} + 25)^{2} \)
|
| $11$ |
\( T^{4} + 28561 \)
|
| $13$ |
\( (T^{2} + 169)^{2} \)
|
| $17$ |
\( T^{4} + 130321 \)
|
| $19$ |
\( (T^{2} - 44 T + 968)^{2} \)
|
| $23$ |
\( T^{4} + 194481 \)
|
| $29$ |
\( (T^{2} - 1458)^{2} \)
|
| $31$ |
\( (T^{2} + 26 T + 338)^{2} \)
|
| $37$ |
\( (T^{2} + 225)^{2} \)
|
| $41$ |
\( T^{4} + 1500625 \)
|
| $43$ |
\( (T^{2} - 34 T + 578)^{2} \)
|
| $47$ |
\( (T^{2} - 5202)^{2} \)
|
| $53$ |
\( T^{4} + 28561 \)
|
| $59$ |
\( T^{4} + 21381376 \)
|
| $61$ |
\( (T - 67)^{4} \)
|
| $67$ |
\( (T + 100)^{4} \)
|
| $71$ |
\( T^{4} + 7890481 \)
|
| $73$ |
\( (T + 76)^{4} \)
|
| $79$ |
\( (T^{2} + 7921)^{2} \)
|
| $83$ |
\( (T^{2} - 8712)^{2} \)
|
| $89$ |
\( T^{4} + 2401 \)
|
| $97$ |
\( (T + 25)^{4} \)
|
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