Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [360,2,Mod(77,360)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(360, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([0, 6, 10, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("360.77");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 360.br (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: | \(2.87461447277\) |
| Analytic rank: | \(1\) |
| 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: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(\left(\mathbb{Z}/360\mathbb{Z}\right)^\times\).
| \(n\) | \(181\) | \(217\) | \(271\) | \(281\) |
| \(\chi(n)\) | \(-1\) | \(\zeta_{12}^{3}\) | \(1\) | \(\zeta_{12}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 77.1 |
|
−1.00000 | − | 1.00000i | 0.866025 | − | 1.50000i | 2.00000i | −2.23205 | − | 0.133975i | −2.36603 | + | 0.633975i | −0.767949 | + | 2.86603i | 2.00000 | − | 2.00000i | −1.50000 | − | 2.59808i | 2.09808 | + | 2.36603i | ||||||||||||||
| 173.1 | −1.00000 | + | 1.00000i | 0.866025 | + | 1.50000i | − | 2.00000i | −2.23205 | + | 0.133975i | −2.36603 | − | 0.633975i | −0.767949 | − | 2.86603i | 2.00000 | + | 2.00000i | −1.50000 | + | 2.59808i | 2.09808 | − | 2.36603i | ||||||||||||||
| 293.1 | −1.00000 | + | 1.00000i | −0.866025 | + | 1.50000i | − | 2.00000i | 1.23205 | + | 1.86603i | −0.633975 | − | 2.36603i | −4.23205 | − | 1.13397i | 2.00000 | + | 2.00000i | −1.50000 | − | 2.59808i | −3.09808 | − | 0.633975i | ||||||||||||||
| 317.1 | −1.00000 | − | 1.00000i | −0.866025 | − | 1.50000i | 2.00000i | 1.23205 | − | 1.86603i | −0.633975 | + | 2.36603i | −4.23205 | + | 1.13397i | 2.00000 | − | 2.00000i | −1.50000 | + | 2.59808i | −3.09808 | + | 0.633975i | |||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 360.br | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 360.2.br.a | ✓ | 4 |
| 5.c | odd | 4 | 1 | 360.2.br.b | yes | 4 | |
| 8.b | even | 2 | 1 | 360.2.br.d | yes | 4 | |
| 9.d | odd | 6 | 1 | 360.2.br.c | yes | 4 | |
| 40.i | odd | 4 | 1 | 360.2.br.c | yes | 4 | |
| 45.l | even | 12 | 1 | 360.2.br.d | yes | 4 | |
| 72.j | odd | 6 | 1 | 360.2.br.b | yes | 4 | |
| 360.br | even | 12 | 1 | inner | 360.2.br.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 360.2.br.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 360.2.br.a | ✓ | 4 | 360.br | even | 12 | 1 | inner |
| 360.2.br.b | yes | 4 | 5.c | odd | 4 | 1 | |
| 360.2.br.b | yes | 4 | 72.j | odd | 6 | 1 | |
| 360.2.br.c | yes | 4 | 9.d | odd | 6 | 1 | |
| 360.2.br.c | yes | 4 | 40.i | odd | 4 | 1 | |
| 360.2.br.d | yes | 4 | 8.b | even | 2 | 1 | |
| 360.2.br.d | yes | 4 | 45.l | even | 12 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\):
|
\( T_{7}^{4} + 10T_{7}^{3} + 41T_{7}^{2} + 104T_{7} + 169 \)
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\( T_{11}^{2} + 2T_{11} + 4 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2 T + 2)^{2} \)
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| $3$ |
\( T^{4} + 3T^{2} + 9 \)
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| $5$ |
\( T^{4} + 2 T^{3} + \cdots + 25 \)
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| $7$ |
\( T^{4} + 10 T^{3} + \cdots + 169 \)
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| $11$ |
\( (T^{2} + 2 T + 4)^{2} \)
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| $13$ |
\( T^{4} + 12 T^{3} + \cdots + 144 \)
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| $17$ |
\( (T^{2} + 6 T + 18)^{2} \)
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| $19$ |
\( (T^{2} + 6 T + 6)^{2} \)
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| $23$ |
\( T^{4} + 6 T^{3} + \cdots + 9 \)
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| $29$ |
\( T^{4} - 18 T^{3} + \cdots + 529 \)
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| $31$ |
\( T^{4} - 6 T^{3} + \cdots + 324 \)
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| $37$ |
\( T^{4} - 12 T^{3} + \cdots + 36 \)
|
| $41$ |
\( T^{4} + 12 T^{3} + \cdots + 9 \)
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| $43$ |
\( T^{4} + 18 T^{3} + \cdots + 36 \)
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| $47$ |
\( T^{4} + 12 T^{3} + \cdots + 81 \)
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| $53$ |
\( T^{4} + 2916 \)
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| $59$ |
\( T^{4} + 18 T^{3} + \cdots + 484 \)
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| $61$ |
\( T^{4} + 30 T^{3} + \cdots + 1521 \)
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| $67$ |
\( T^{4} + 12 T^{3} + \cdots + 1521 \)
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| $71$ |
\( (T^{2} + 144)^{2} \)
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| $73$ |
\( T^{4} + 20 T^{3} + \cdots + 16 \)
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| $79$ |
\( T^{4} + 18 T^{3} + \cdots + 324 \)
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| $83$ |
\( T^{4} + 26 T^{3} + \cdots + 14641 \)
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| $89$ |
\( (T^{2} - 12 T - 39)^{2} \)
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| $97$ |
\( T^{4} + 10 T^{3} + \cdots + 2500 \)
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