Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1428,1,Mod(1427,1428)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1428, 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("1428.1427");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1428 = 2^{2} \cdot 3 \cdot 7 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1428.b (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: | \(0.712664838040\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\Q(\zeta_{20})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{6} + x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{10}\) |
| Projective field: | Galois closure of 10.0.349294796411904.1 |
$q$-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 \(\left(\mathbb{Z}/1428\mathbb{Z}\right)^\times\).
| \(n\) | \(409\) | \(715\) | \(953\) | \(1261\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1427.1 |
|
−0.809017 | − | 0.587785i | −0.587785 | − | 0.809017i | 0.309017 | + | 0.951057i | 0.618034i | 1.00000i | − | 1.00000i | 0.309017 | − | 0.951057i | −0.309017 | + | 0.951057i | 0.363271 | − | 0.500000i | |||||||||||||||||||||||||||||
| 1427.2 | −0.809017 | − | 0.587785i | 0.587785 | + | 0.809017i | 0.309017 | + | 0.951057i | − | 0.618034i | − | 1.00000i | 1.00000i | 0.309017 | − | 0.951057i | −0.309017 | + | 0.951057i | −0.363271 | + | 0.500000i | |||||||||||||||||||||||||||||
| 1427.3 | −0.809017 | + | 0.587785i | −0.587785 | + | 0.809017i | 0.309017 | − | 0.951057i | − | 0.618034i | − | 1.00000i | 1.00000i | 0.309017 | + | 0.951057i | −0.309017 | − | 0.951057i | 0.363271 | + | 0.500000i | |||||||||||||||||||||||||||||
| 1427.4 | −0.809017 | + | 0.587785i | 0.587785 | − | 0.809017i | 0.309017 | − | 0.951057i | 0.618034i | 1.00000i | − | 1.00000i | 0.309017 | + | 0.951057i | −0.309017 | − | 0.951057i | −0.363271 | − | 0.500000i | ||||||||||||||||||||||||||||||
| 1427.5 | 0.309017 | − | 0.951057i | −0.951057 | + | 0.309017i | −0.809017 | − | 0.587785i | − | 1.61803i | 1.00000i | − | 1.00000i | −0.809017 | + | 0.587785i | 0.809017 | − | 0.587785i | −1.53884 | − | 0.500000i | |||||||||||||||||||||||||||||
| 1427.6 | 0.309017 | − | 0.951057i | 0.951057 | − | 0.309017i | −0.809017 | − | 0.587785i | 1.61803i | − | 1.00000i | 1.00000i | −0.809017 | + | 0.587785i | 0.809017 | − | 0.587785i | 1.53884 | + | 0.500000i | ||||||||||||||||||||||||||||||
| 1427.7 | 0.309017 | + | 0.951057i | −0.951057 | − | 0.309017i | −0.809017 | + | 0.587785i | 1.61803i | − | 1.00000i | 1.00000i | −0.809017 | − | 0.587785i | 0.809017 | + | 0.587785i | −1.53884 | + | 0.500000i | ||||||||||||||||||||||||||||||
| 1427.8 | 0.309017 | + | 0.951057i | 0.951057 | + | 0.309017i | −0.809017 | + | 0.587785i | − | 1.61803i | 1.00000i | − | 1.00000i | −0.809017 | − | 0.587785i | 0.809017 | + | 0.587785i | 1.53884 | − | 0.500000i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 119.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-119}) \) |
| 7.b | odd | 2 | 1 | inner |
| 12.b | even | 2 | 1 | inner |
| 17.b | even | 2 | 1 | inner |
| 84.h | odd | 2 | 1 | inner |
| 204.h | even | 2 | 1 | inner |
| 1428.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1428.1.b.e | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1428.1.b.f | yes | 8 | |
| 4.b | odd | 2 | 1 | 1428.1.b.f | yes | 8 | |
| 7.b | odd | 2 | 1 | inner | 1428.1.b.e | ✓ | 8 |
| 12.b | even | 2 | 1 | inner | 1428.1.b.e | ✓ | 8 |
| 17.b | even | 2 | 1 | inner | 1428.1.b.e | ✓ | 8 |
| 21.c | even | 2 | 1 | 1428.1.b.f | yes | 8 | |
| 28.d | even | 2 | 1 | 1428.1.b.f | yes | 8 | |
| 51.c | odd | 2 | 1 | 1428.1.b.f | yes | 8 | |
| 68.d | odd | 2 | 1 | 1428.1.b.f | yes | 8 | |
| 84.h | odd | 2 | 1 | inner | 1428.1.b.e | ✓ | 8 |
| 119.d | odd | 2 | 1 | CM | 1428.1.b.e | ✓ | 8 |
| 204.h | even | 2 | 1 | inner | 1428.1.b.e | ✓ | 8 |
| 357.c | even | 2 | 1 | 1428.1.b.f | yes | 8 | |
| 476.e | even | 2 | 1 | 1428.1.b.f | yes | 8 | |
| 1428.b | odd | 2 | 1 | inner | 1428.1.b.e | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1428.1.b.e | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 1428.1.b.e | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 1428.1.b.e | ✓ | 8 | 12.b | even | 2 | 1 | inner |
| 1428.1.b.e | ✓ | 8 | 17.b | even | 2 | 1 | inner |
| 1428.1.b.e | ✓ | 8 | 84.h | odd | 2 | 1 | inner |
| 1428.1.b.e | ✓ | 8 | 119.d | odd | 2 | 1 | CM |
| 1428.1.b.e | ✓ | 8 | 204.h | even | 2 | 1 | inner |
| 1428.1.b.e | ✓ | 8 | 1428.b | odd | 2 | 1 | inner |
| 1428.1.b.f | yes | 8 | 3.b | odd | 2 | 1 | |
| 1428.1.b.f | yes | 8 | 4.b | odd | 2 | 1 | |
| 1428.1.b.f | yes | 8 | 21.c | even | 2 | 1 | |
| 1428.1.b.f | yes | 8 | 28.d | even | 2 | 1 | |
| 1428.1.b.f | yes | 8 | 51.c | odd | 2 | 1 | |
| 1428.1.b.f | yes | 8 | 68.d | odd | 2 | 1 | |
| 1428.1.b.f | yes | 8 | 357.c | even | 2 | 1 | |
| 1428.1.b.f | yes | 8 | 476.e | even | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1428, [\chi])\):
|
\( T_{5}^{4} + 3T_{5}^{2} + 1 \)
|
|
\( T_{179}^{2} - T_{179} - 1 \)
|
|
\( T_{293} \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $3$ |
\( T^{8} - T^{6} + T^{4} + \cdots + 1 \)
|
| $5$ |
\( (T^{4} + 3 T^{2} + 1)^{2} \)
|
| $7$ |
\( (T^{2} + 1)^{4} \)
|
| $11$ |
\( T^{8} \)
|
| $13$ |
\( T^{8} \)
|
| $17$ |
\( (T^{2} + 1)^{4} \)
|
| $19$ |
\( T^{8} \)
|
| $23$ |
\( T^{8} \)
|
| $29$ |
\( T^{8} \)
|
| $31$ |
\( (T^{4} + 3 T^{2} + 1)^{2} \)
|
| $37$ |
\( T^{8} \)
|
| $41$ |
\( (T^{4} + 3 T^{2} + 1)^{2} \)
|
| $43$ |
\( (T^{4} + 5 T^{2} + 5)^{2} \)
|
| $47$ |
\( T^{8} \)
|
| $53$ |
\( (T^{4} + 5 T^{2} + 5)^{2} \)
|
| $59$ |
\( T^{8} \)
|
| $61$ |
\( (T^{4} - 5 T^{2} + 5)^{2} \)
|
| $67$ |
\( (T^{4} + 5 T^{2} + 5)^{2} \)
|
| $71$ |
\( T^{8} \)
|
| $73$ |
\( (T^{4} - 5 T^{2} + 5)^{2} \)
|
| $79$ |
\( T^{8} \)
|
| $83$ |
\( T^{8} \)
|
| $89$ |
\( T^{8} \)
|
| $97$ |
\( (T^{4} - 5 T^{2} + 5)^{2} \)
|
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