Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [650,2,Mod(7,650)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(650, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([3, 11]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("650.7");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 650 = 2 \cdot 5^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 650.t (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: | \(5.19027613138\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(301\) |
| \(\chi(n)\) | \(-\zeta_{12}^{3}\) | \(\zeta_{12}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 7.1 |
|
0.866025 | + | 0.500000i | 0.633975 | + | 2.36603i | 0.500000 | + | 0.866025i | 0 | −0.633975 | + | 2.36603i | 0.133975 | + | 0.232051i | 1.00000i | −2.59808 | + | 1.50000i | 0 | ||||||||||||||||||
| 93.1 | 0.866025 | − | 0.500000i | 0.633975 | − | 2.36603i | 0.500000 | − | 0.866025i | 0 | −0.633975 | − | 2.36603i | 0.133975 | − | 0.232051i | − | 1.00000i | −2.59808 | − | 1.50000i | 0 | ||||||||||||||||||
| 557.1 | −0.866025 | + | 0.500000i | 2.36603 | + | 0.633975i | 0.500000 | − | 0.866025i | 0 | −2.36603 | + | 0.633975i | 1.86603 | − | 3.23205i | 1.00000i | 2.59808 | + | 1.50000i | 0 | |||||||||||||||||||
| 643.1 | −0.866025 | − | 0.500000i | 2.36603 | − | 0.633975i | 0.500000 | + | 0.866025i | 0 | −2.36603 | − | 0.633975i | 1.86603 | + | 3.23205i | − | 1.00000i | 2.59808 | − | 1.50000i | 0 | ||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 65.t | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 650.2.t.b | yes | 4 |
| 5.b | even | 2 | 1 | 650.2.t.a | ✓ | 4 | |
| 5.c | odd | 4 | 1 | 650.2.w.a | yes | 4 | |
| 5.c | odd | 4 | 1 | 650.2.w.b | yes | 4 | |
| 13.f | odd | 12 | 1 | 650.2.w.b | yes | 4 | |
| 65.o | even | 12 | 1 | 650.2.t.a | ✓ | 4 | |
| 65.s | odd | 12 | 1 | 650.2.w.a | yes | 4 | |
| 65.t | even | 12 | 1 | inner | 650.2.t.b | yes | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 650.2.t.a | ✓ | 4 | 5.b | even | 2 | 1 | |
| 650.2.t.a | ✓ | 4 | 65.o | even | 12 | 1 | |
| 650.2.t.b | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 650.2.t.b | yes | 4 | 65.t | even | 12 | 1 | inner |
| 650.2.w.a | yes | 4 | 5.c | odd | 4 | 1 | |
| 650.2.w.a | yes | 4 | 65.s | odd | 12 | 1 | |
| 650.2.w.b | yes | 4 | 5.c | odd | 4 | 1 | |
| 650.2.w.b | yes | 4 | 13.f | odd | 12 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 6T_{3}^{3} + 18T_{3}^{2} - 36T_{3} + 36 \)
acting on \(S_{2}^{\mathrm{new}}(650, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - T^{2} + 1 \)
|
| $3$ |
\( T^{4} - 6 T^{3} + \cdots + 36 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} - 4 T^{3} + \cdots + 1 \)
|
| $11$ |
\( T^{4} - 4 T^{3} + \cdots + 1 \)
|
| $13$ |
\( T^{4} + 23T^{2} + 169 \)
|
| $17$ |
\( T^{4} + 2 T^{3} + \cdots + 4 \)
|
| $19$ |
\( T^{4} + 12 T^{3} + \cdots + 1521 \)
|
| $23$ |
\( T^{4} - 14 T^{3} + \cdots + 484 \)
|
| $29$ |
\( T^{4} - 12 T^{3} + \cdots + 576 \)
|
| $31$ |
\( (T^{2} - 6 T + 18)^{2} \)
|
| $37$ |
\( T^{4} - 4 T^{3} + \cdots + 1 \)
|
| $41$ |
\( T^{4} - 10 T^{3} + \cdots + 4 \)
|
| $43$ |
\( T^{4} + 12 T^{3} + \cdots + 5184 \)
|
| $47$ |
\( (T + 7)^{4} \)
|
| $53$ |
\( T^{4} - 22 T^{3} + \cdots + 3481 \)
|
| $59$ |
\( T^{4} + 6 T^{3} + \cdots + 36 \)
|
| $61$ |
\( T^{4} + 22 T^{3} + \cdots + 13924 \)
|
| $67$ |
\( T^{4} - 24 T^{3} + \cdots + 144 \)
|
| $71$ |
\( T^{4} + 28 T^{3} + \cdots + 21904 \)
|
| $73$ |
\( T^{4} + 24T^{2} + 36 \)
|
| $79$ |
\( T^{4} + 296 T^{2} + 21316 \)
|
| $83$ |
\( (T^{2} - 2 T - 2)^{2} \)
|
| $89$ |
\( T^{4} - 26 T^{3} + \cdots + 69169 \)
|
| $97$ |
\( T^{4} + 18 T^{3} + \cdots + 8836 \)
|
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