Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [420,1,Mod(419,420)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(420, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("420.419");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 420.o (of order \(2\), degree \(1\), 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: | \(0.209607305306\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(i)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{2}\) |
| Projective field: | Galois closure of \(\Q(\sqrt{-5}, \sqrt{-21})\) |
| Artin image: | $D_4:C_2$ |
| Artin field: | Galois closure of 8.0.70560000.1 |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/420\mathbb{Z}\right)^\times\).
| \(n\) | \(211\) | \(241\) | \(281\) | \(337\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 419.1 |
|
− | 1.00000i | − | 1.00000i | −1.00000 | −1.00000 | −1.00000 | − | 1.00000i | 1.00000i | −1.00000 | 1.00000i | |||||||||||||||||||||
| 419.2 | 1.00000i | 1.00000i | −1.00000 | −1.00000 | −1.00000 | 1.00000i | − | 1.00000i | −1.00000 | − | 1.00000i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 20.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-5}) \) |
| 84.h | odd | 2 | 1 | CM by \(\Q(\sqrt{-21}) \) |
| 105.g | even | 2 | 1 | RM by \(\Q(\sqrt{105}) \) |
| 4.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
| 21.c | even | 2 | 1 | inner |
| 420.o | odd | 2 | 1 | inner |
Twists
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{41} - 2 \)
acting on \(S_{1}^{\mathrm{new}}(420, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 1 \)
|
| $3$ |
\( T^{2} + 1 \)
|
| $5$ |
\( (T + 1)^{2} \)
|
| $7$ |
\( T^{2} + 1 \)
|
| $11$ |
\( T^{2} \)
|
| $13$ |
\( T^{2} \)
|
| $17$ |
\( T^{2} \)
|
| $19$ |
\( T^{2} \)
|
| $23$ |
\( T^{2} + 4 \)
|
| $29$ |
\( T^{2} \)
|
| $31$ |
\( T^{2} \)
|
| $37$ |
\( T^{2} \)
|
| $41$ |
\( (T - 2)^{2} \)
|
| $43$ |
\( T^{2} \)
|
| $47$ |
\( T^{2} \)
|
| $53$ |
\( T^{2} \)
|
| $59$ |
\( T^{2} \)
|
| $61$ |
\( T^{2} \)
|
| $67$ |
\( T^{2} \)
|
| $71$ |
\( T^{2} \)
|
| $73$ |
\( T^{2} \)
|
| $79$ |
\( T^{2} \)
|
| $83$ |
\( T^{2} \)
|
| $89$ |
\( (T - 2)^{2} \)
|
| $97$ |
\( T^{2} \)
|
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