Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3136,1,Mod(99,3136)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3136, base_ring=CyclotomicField(16))
chi = DirichletCharacter(H, H._module([8, 11, 0]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3136.99");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3136 = 2^{6} \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3136.bm (of order \(16\), degree \(8\), 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: | \(1.56506787962\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\Q(\zeta_{16})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Projective image: | \(D_{16}\) |
| Projective field: | Galois closure of \(\mathbb{Q}[x]/(x^{16} + \cdots)\) |
$q$-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 \(\left(\mathbb{Z}/3136\mathbb{Z}\right)^\times\).
| \(n\) | \(197\) | \(1471\) | \(1473\) |
| \(\chi(n)\) | \(\zeta_{16}^{3}\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 99.1 |
|
−0.382683 | − | 0.923880i | 0 | −0.707107 | + | 0.707107i | 0 | 0 | 0 | 0.923880 | + | 0.382683i | 0.923880 | + | 0.382683i | 0 | ||||||||||||||||||||||||||||||||||
| 491.1 | 0.382683 | − | 0.923880i | 0 | −0.707107 | − | 0.707107i | 0 | 0 | 0 | −0.923880 | + | 0.382683i | −0.923880 | + | 0.382683i | 0 | |||||||||||||||||||||||||||||||||||
| 883.1 | 0.923880 | − | 0.382683i | 0 | 0.707107 | − | 0.707107i | 0 | 0 | 0 | 0.382683 | − | 0.923880i | 0.382683 | − | 0.923880i | 0 | |||||||||||||||||||||||||||||||||||
| 1275.1 | 0.923880 | + | 0.382683i | 0 | 0.707107 | + | 0.707107i | 0 | 0 | 0 | 0.382683 | + | 0.923880i | 0.382683 | + | 0.923880i | 0 | |||||||||||||||||||||||||||||||||||
| 1667.1 | 0.382683 | + | 0.923880i | 0 | −0.707107 | + | 0.707107i | 0 | 0 | 0 | −0.923880 | − | 0.382683i | −0.923880 | − | 0.382683i | 0 | |||||||||||||||||||||||||||||||||||
| 2059.1 | −0.382683 | + | 0.923880i | 0 | −0.707107 | − | 0.707107i | 0 | 0 | 0 | 0.923880 | − | 0.382683i | 0.923880 | − | 0.382683i | 0 | |||||||||||||||||||||||||||||||||||
| 2451.1 | −0.923880 | + | 0.382683i | 0 | 0.707107 | − | 0.707107i | 0 | 0 | 0 | −0.382683 | + | 0.923880i | −0.382683 | + | 0.923880i | 0 | |||||||||||||||||||||||||||||||||||
| 2843.1 | −0.923880 | − | 0.382683i | 0 | 0.707107 | + | 0.707107i | 0 | 0 | 0 | −0.382683 | − | 0.923880i | −0.382683 | − | 0.923880i | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-7}) \) |
| 64.j | odd | 16 | 1 | inner |
| 448.bd | even | 16 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 3136.1.bm.a | ✓ | 8 |
| 7.b | odd | 2 | 1 | CM | 3136.1.bm.a | ✓ | 8 |
| 7.c | even | 3 | 2 | 3136.1.cf.a | 16 | ||
| 7.d | odd | 6 | 2 | 3136.1.cf.a | 16 | ||
| 64.j | odd | 16 | 1 | inner | 3136.1.bm.a | ✓ | 8 |
| 448.bd | even | 16 | 1 | inner | 3136.1.bm.a | ✓ | 8 |
| 448.bl | odd | 48 | 2 | 3136.1.cf.a | 16 | ||
| 448.bm | even | 48 | 2 | 3136.1.cf.a | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3136.1.bm.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 3136.1.bm.a | ✓ | 8 | 7.b | odd | 2 | 1 | CM |
| 3136.1.bm.a | ✓ | 8 | 64.j | odd | 16 | 1 | inner |
| 3136.1.bm.a | ✓ | 8 | 448.bd | even | 16 | 1 | inner |
| 3136.1.cf.a | 16 | 7.c | even | 3 | 2 | ||
| 3136.1.cf.a | 16 | 7.d | odd | 6 | 2 | ||
| 3136.1.cf.a | 16 | 448.bl | odd | 48 | 2 | ||
| 3136.1.cf.a | 16 | 448.bm | even | 48 | 2 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(3136, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 1 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} \)
|
| $7$ |
\( T^{8} \)
|
| $11$ |
\( T^{8} + 2 T^{4} + \cdots + 2 \)
|
| $13$ |
\( T^{8} \)
|
| $17$ |
\( T^{8} \)
|
| $19$ |
\( T^{8} \)
|
| $23$ |
\( (T^{4} + 2 T^{2} - 4 T + 2)^{2} \)
|
| $29$ |
\( T^{8} - 8 T^{5} + \cdots + 2 \)
|
| $31$ |
\( T^{8} \)
|
| $37$ |
\( T^{8} + 8 T^{5} + \cdots + 2 \)
|
| $41$ |
\( T^{8} \)
|
| $43$ |
\( T^{8} + 4 T^{6} + \cdots + 2 \)
|
| $47$ |
\( T^{8} \)
|
| $53$ |
\( T^{8} + 4 T^{6} + \cdots + 2 \)
|
| $59$ |
\( T^{8} \)
|
| $61$ |
\( T^{8} \)
|
| $67$ |
\( T^{8} + 8 T^{7} + \cdots + 2 \)
|
| $71$ |
\( T^{8} + 16 \)
|
| $73$ |
\( T^{8} \)
|
| $79$ |
\( T^{8} + 12T^{4} + 4 \)
|
| $83$ |
\( T^{8} \)
|
| $89$ |
\( T^{8} \)
|
| $97$ |
\( T^{8} \)
|
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