Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2415,1,Mod(137,2415)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2415, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 3, 8, 6]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2415.137");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2415 = 3 \cdot 5 \cdot 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2415.bq (of order \(12\), 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: | \(1.20524200551\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | \(\Q(\zeta_{24})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(S_{4}\) |
| Projective field: | Galois closure of 4.2.1267875.1 |
$q$-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 \(\left(\mathbb{Z}/2415\mathbb{Z}\right)^\times\).
| \(n\) | \(346\) | \(806\) | \(967\) | \(1891\) |
| \(\chi(n)\) | \(-\zeta_{24}^{4}\) | \(-1\) | \(-\zeta_{24}^{6}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 137.1 |
|
−1.36603 | − | 0.366025i | 0.866025 | + | 0.500000i | 0.866025 | + | 0.500000i | −0.258819 | + | 0.965926i | −1.00000 | − | 1.00000i | 0.965926 | + | 0.258819i | 0 | 0.500000 | + | 0.866025i | 0.707107 | − | 1.22474i | ||||||||||||||||||||||||||
| 137.2 | −1.36603 | − | 0.366025i | 0.866025 | + | 0.500000i | 0.866025 | + | 0.500000i | 0.258819 | − | 0.965926i | −1.00000 | − | 1.00000i | −0.965926 | − | 0.258819i | 0 | 0.500000 | + | 0.866025i | −0.707107 | + | 1.22474i | |||||||||||||||||||||||||||
| 758.1 | −1.36603 | + | 0.366025i | 0.866025 | − | 0.500000i | 0.866025 | − | 0.500000i | −0.258819 | − | 0.965926i | −1.00000 | + | 1.00000i | 0.965926 | − | 0.258819i | 0 | 0.500000 | − | 0.866025i | 0.707107 | + | 1.22474i | |||||||||||||||||||||||||||
| 758.2 | −1.36603 | + | 0.366025i | 0.866025 | − | 0.500000i | 0.866025 | − | 0.500000i | 0.258819 | + | 0.965926i | −1.00000 | + | 1.00000i | −0.965926 | + | 0.258819i | 0 | 0.500000 | − | 0.866025i | −0.707107 | − | 1.22474i | |||||||||||||||||||||||||||
| 1103.1 | 0.366025 | − | 1.36603i | −0.866025 | − | 0.500000i | −0.866025 | − | 0.500000i | −0.965926 | − | 0.258819i | −1.00000 | + | 1.00000i | 0.258819 | − | 0.965926i | 0 | 0.500000 | + | 0.866025i | −0.707107 | + | 1.22474i | |||||||||||||||||||||||||||
| 1103.2 | 0.366025 | − | 1.36603i | −0.866025 | − | 0.500000i | −0.866025 | − | 0.500000i | 0.965926 | + | 0.258819i | −1.00000 | + | 1.00000i | −0.258819 | + | 0.965926i | 0 | 0.500000 | + | 0.866025i | 0.707107 | − | 1.22474i | |||||||||||||||||||||||||||
| 2207.1 | 0.366025 | + | 1.36603i | −0.866025 | + | 0.500000i | −0.866025 | + | 0.500000i | −0.965926 | + | 0.258819i | −1.00000 | − | 1.00000i | 0.258819 | + | 0.965926i | 0 | 0.500000 | − | 0.866025i | −0.707107 | − | 1.22474i | |||||||||||||||||||||||||||
| 2207.2 | 0.366025 | + | 1.36603i | −0.866025 | + | 0.500000i | −0.866025 | + | 0.500000i | 0.965926 | − | 0.258819i | −1.00000 | − | 1.00000i | −0.258819 | − | 0.965926i | 0 | 0.500000 | − | 0.866025i | 0.707107 | + | 1.22474i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
| 15.e | even | 4 | 1 | inner |
| 23.b | odd | 2 | 1 | inner |
| 105.x | even | 12 | 1 | inner |
| 161.f | odd | 6 | 1 | inner |
| 345.l | odd | 4 | 1 | inner |
| 2415.bq | odd | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2415.1.bq.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 2415.1.bq.b | yes | 8 | |
| 5.c | odd | 4 | 1 | 2415.1.bq.b | yes | 8 | |
| 7.c | even | 3 | 1 | inner | 2415.1.bq.a | ✓ | 8 |
| 15.e | even | 4 | 1 | inner | 2415.1.bq.a | ✓ | 8 |
| 21.h | odd | 6 | 1 | 2415.1.bq.b | yes | 8 | |
| 23.b | odd | 2 | 1 | inner | 2415.1.bq.a | ✓ | 8 |
| 35.l | odd | 12 | 1 | 2415.1.bq.b | yes | 8 | |
| 69.c | even | 2 | 1 | 2415.1.bq.b | yes | 8 | |
| 105.x | even | 12 | 1 | inner | 2415.1.bq.a | ✓ | 8 |
| 115.e | even | 4 | 1 | 2415.1.bq.b | yes | 8 | |
| 161.f | odd | 6 | 1 | inner | 2415.1.bq.a | ✓ | 8 |
| 345.l | odd | 4 | 1 | inner | 2415.1.bq.a | ✓ | 8 |
| 483.m | even | 6 | 1 | 2415.1.bq.b | yes | 8 | |
| 805.y | even | 12 | 1 | 2415.1.bq.b | yes | 8 | |
| 2415.bq | odd | 12 | 1 | inner | 2415.1.bq.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2415.1.bq.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 2415.1.bq.a | ✓ | 8 | 7.c | even | 3 | 1 | inner |
| 2415.1.bq.a | ✓ | 8 | 15.e | even | 4 | 1 | inner |
| 2415.1.bq.a | ✓ | 8 | 23.b | odd | 2 | 1 | inner |
| 2415.1.bq.a | ✓ | 8 | 105.x | even | 12 | 1 | inner |
| 2415.1.bq.a | ✓ | 8 | 161.f | odd | 6 | 1 | inner |
| 2415.1.bq.a | ✓ | 8 | 345.l | odd | 4 | 1 | inner |
| 2415.1.bq.a | ✓ | 8 | 2415.bq | odd | 12 | 1 | inner |
| 2415.1.bq.b | yes | 8 | 3.b | odd | 2 | 1 | |
| 2415.1.bq.b | yes | 8 | 5.c | odd | 4 | 1 | |
| 2415.1.bq.b | yes | 8 | 21.h | odd | 6 | 1 | |
| 2415.1.bq.b | yes | 8 | 35.l | odd | 12 | 1 | |
| 2415.1.bq.b | yes | 8 | 69.c | even | 2 | 1 | |
| 2415.1.bq.b | yes | 8 | 115.e | even | 4 | 1 | |
| 2415.1.bq.b | yes | 8 | 483.m | even | 6 | 1 | |
| 2415.1.bq.b | yes | 8 | 805.y | even | 12 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 2T_{2}^{3} + 2T_{2}^{2} + 4T_{2} + 4 \)
acting on \(S_{1}^{\mathrm{new}}(2415, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \)
|
| $3$ |
\( (T^{4} - T^{2} + 1)^{2} \)
|
| $5$ |
\( T^{8} - T^{4} + 1 \)
|
| $7$ |
\( T^{8} - T^{4} + 1 \)
|
| $11$ |
\( (T^{4} + 2 T^{2} + 4)^{2} \)
|
| $13$ |
\( T^{8} \)
|
| $17$ |
\( T^{8} - T^{4} + 1 \)
|
| $19$ |
\( T^{8} \)
|
| $23$ |
\( T^{8} - T^{4} + 1 \)
|
| $29$ |
\( (T - 1)^{8} \)
|
| $31$ |
\( (T^{2} + T + 1)^{4} \)
|
| $37$ |
\( T^{8} - T^{4} + 1 \)
|
| $41$ |
\( (T^{2} + 4)^{4} \)
|
| $43$ |
\( (T^{4} + 1)^{2} \)
|
| $47$ |
\( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \)
|
| $53$ |
\( T^{8} - T^{4} + 1 \)
|
| $59$ |
\( (T^{2} - T + 1)^{4} \)
|
| $61$ |
\( T^{8} \)
|
| $67$ |
\( T^{8} \)
|
| $71$ |
\( (T^{2} + 1)^{4} \)
|
| $73$ |
\( T^{8} \)
|
| $79$ |
\( T^{8} \)
|
| $83$ |
\( (T^{4} + 1)^{2} \)
|
| $89$ |
\( (T^{4} - 2 T^{2} + 4)^{2} \)
|
| $97$ |
\( (T^{4} + 1)^{2} \)
|
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