Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [156,2,Mod(7,156)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(156, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 0, 11]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("156.7");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 156 = 2^{2} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 156.w (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: | \(1.24566627153\) |
| 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: | \(\Z[a_1, a_2, a_3]\) |
| 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 \(\left(\mathbb{Z}/156\mathbb{Z}\right)^\times\).
| \(n\) | \(53\) | \(79\) | \(145\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(\zeta_{12}\) |
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 |
|
1.36603 | + | 0.366025i | −0.866025 | − | 0.500000i | 1.73205 | + | 1.00000i | 0.732051 | + | 0.732051i | −1.00000 | − | 1.00000i | 0.232051 | − | 0.866025i | 2.00000 | + | 2.00000i | 0.500000 | + | 0.866025i | 0.732051 | + | 1.26795i | ||||||||||||
| 19.1 | −0.366025 | + | 1.36603i | 0.866025 | + | 0.500000i | −1.73205 | − | 1.00000i | −2.73205 | + | 2.73205i | −1.00000 | + | 1.00000i | −3.23205 | − | 0.866025i | 2.00000 | − | 2.00000i | 0.500000 | + | 0.866025i | −2.73205 | − | 4.73205i | |||||||||||||
| 67.1 | 1.36603 | − | 0.366025i | −0.866025 | + | 0.500000i | 1.73205 | − | 1.00000i | 0.732051 | − | 0.732051i | −1.00000 | + | 1.00000i | 0.232051 | + | 0.866025i | 2.00000 | − | 2.00000i | 0.500000 | − | 0.866025i | 0.732051 | − | 1.26795i | |||||||||||||
| 115.1 | −0.366025 | − | 1.36603i | 0.866025 | − | 0.500000i | −1.73205 | + | 1.00000i | −2.73205 | − | 2.73205i | −1.00000 | − | 1.00000i | −3.23205 | + | 0.866025i | 2.00000 | + | 2.00000i | 0.500000 | − | 0.866025i | −2.73205 | + | 4.73205i | |||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 52.l | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 156.2.w.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 468.2.cb.c | 4 | ||
| 4.b | odd | 2 | 1 | 156.2.w.b | yes | 4 | |
| 12.b | even | 2 | 1 | 468.2.cb.a | 4 | ||
| 13.f | odd | 12 | 1 | 156.2.w.b | yes | 4 | |
| 39.k | even | 12 | 1 | 468.2.cb.a | 4 | ||
| 52.l | even | 12 | 1 | inner | 156.2.w.a | ✓ | 4 |
| 156.v | odd | 12 | 1 | 468.2.cb.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 156.2.w.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 156.2.w.a | ✓ | 4 | 52.l | even | 12 | 1 | inner |
| 156.2.w.b | yes | 4 | 4.b | odd | 2 | 1 | |
| 156.2.w.b | yes | 4 | 13.f | odd | 12 | 1 | |
| 468.2.cb.a | 4 | 12.b | even | 2 | 1 | ||
| 468.2.cb.a | 4 | 39.k | even | 12 | 1 | ||
| 468.2.cb.c | 4 | 3.b | odd | 2 | 1 | ||
| 468.2.cb.c | 4 | 156.v | odd | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(156, [\chi])\):
|
\( T_{5}^{4} + 4T_{5}^{3} + 8T_{5}^{2} - 16T_{5} + 16 \)
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\( T_{7}^{4} + 6T_{7}^{3} + 9T_{7}^{2} + 9 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 2 T^{3} + \cdots + 4 \)
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| $3$ |
\( T^{4} - T^{2} + 1 \)
|
| $5$ |
\( T^{4} + 4 T^{3} + \cdots + 16 \)
|
| $7$ |
\( T^{4} + 6 T^{3} + \cdots + 9 \)
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| $11$ |
\( T^{4} - 4 T^{3} + \cdots + 16 \)
|
| $13$ |
\( T^{4} - T^{2} + 169 \)
|
| $17$ |
\( T^{4} - 16T^{2} + 256 \)
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| $19$ |
\( T^{4} + 10 T^{3} + \cdots + 4 \)
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| $23$ |
\( T^{4} + 12T^{2} + 144 \)
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| $29$ |
\( T^{4} + 8 T^{3} + \cdots + 16 \)
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| $31$ |
\( T^{4} + 6 T^{3} + \cdots + 9 \)
|
| $37$ |
\( T^{4} - 6 T^{3} + \cdots + 324 \)
|
| $41$ |
\( T^{4} - 8 T^{3} + \cdots + 256 \)
|
| $43$ |
\( T^{4} + 2 T^{3} + \cdots + 2209 \)
|
| $47$ |
\( T^{4} - 20 T^{3} + \cdots + 1936 \)
|
| $53$ |
\( (T^{2} - 12 T + 24)^{2} \)
|
| $59$ |
\( T^{4} + 28 T^{3} + \cdots + 16 \)
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| $61$ |
\( T^{4} + 8 T^{3} + \cdots + 169 \)
|
| $67$ |
\( T^{4} - 32 T^{3} + \cdots + 11881 \)
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| $71$ |
\( T^{4} - 12 T^{3} + \cdots + 144 \)
|
| $73$ |
\( T^{4} - 2 T^{3} + \cdots + 5329 \)
|
| $79$ |
\( T^{4} + 98T^{2} + 2209 \)
|
| $83$ |
\( T^{4} + 16 T^{3} + \cdots + 4096 \)
|
| $89$ |
\( T^{4} + 16 T^{3} + \cdots + 256 \)
|
| $97$ |
\( T^{4} - 14 T^{3} + \cdots + 1 \)
|
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