Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [147,3,Mod(116,147)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(147, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 4]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("147.116");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 147 = 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 147.h (of order \(6\), degree \(2\), not 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: | \(4.00545988610\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 5x^{2} + 25 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 21) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - 5x^{2} + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 5\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/147\mathbb{Z}\right)^\times\).
| \(n\) | \(50\) | \(52\) |
| \(\chi(n)\) | \(-1\) | \(-1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 116.1 |
|
−1.93649 | + | 1.11803i | −0.936492 | − | 2.85008i | 0.500000 | − | 0.866025i | −1.93649 | + | 1.11803i | 5.00000 | + | 4.47214i | 0 | − | 6.70820i | −7.24597 | + | 5.33816i | 2.50000 | − | 4.33013i | |||||||||||||||
| 116.2 | 1.93649 | − | 1.11803i | 2.93649 | − | 0.614017i | 0.500000 | − | 0.866025i | 1.93649 | − | 1.11803i | 5.00000 | − | 4.47214i | 0 | 6.70820i | 8.24597 | − | 3.60611i | 2.50000 | − | 4.33013i | |||||||||||||||||
| 128.1 | −1.93649 | − | 1.11803i | −0.936492 | + | 2.85008i | 0.500000 | + | 0.866025i | −1.93649 | − | 1.11803i | 5.00000 | − | 4.47214i | 0 | 6.70820i | −7.24597 | − | 5.33816i | 2.50000 | + | 4.33013i | |||||||||||||||||
| 128.2 | 1.93649 | + | 1.11803i | 2.93649 | + | 0.614017i | 0.500000 | + | 0.866025i | 1.93649 | + | 1.11803i | 5.00000 | + | 4.47214i | 0 | − | 6.70820i | 8.24597 | + | 3.60611i | 2.50000 | + | 4.33013i | ||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 21.h | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 147.3.h.b | 4 | |
| 3.b | odd | 2 | 1 | inner | 147.3.h.b | 4 | |
| 7.b | odd | 2 | 1 | 21.3.h.b | ✓ | 4 | |
| 7.c | even | 3 | 1 | 147.3.b.c | 2 | ||
| 7.c | even | 3 | 1 | inner | 147.3.h.b | 4 | |
| 7.d | odd | 6 | 1 | 21.3.h.b | ✓ | 4 | |
| 7.d | odd | 6 | 1 | 147.3.b.d | 2 | ||
| 21.c | even | 2 | 1 | 21.3.h.b | ✓ | 4 | |
| 21.g | even | 6 | 1 | 21.3.h.b | ✓ | 4 | |
| 21.g | even | 6 | 1 | 147.3.b.d | 2 | ||
| 21.h | odd | 6 | 1 | 147.3.b.c | 2 | ||
| 21.h | odd | 6 | 1 | inner | 147.3.h.b | 4 | |
| 28.d | even | 2 | 1 | 336.3.bn.f | 4 | ||
| 28.f | even | 6 | 1 | 336.3.bn.f | 4 | ||
| 84.h | odd | 2 | 1 | 336.3.bn.f | 4 | ||
| 84.j | odd | 6 | 1 | 336.3.bn.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 21.3.h.b | ✓ | 4 | 7.b | odd | 2 | 1 | |
| 21.3.h.b | ✓ | 4 | 7.d | odd | 6 | 1 | |
| 21.3.h.b | ✓ | 4 | 21.c | even | 2 | 1 | |
| 21.3.h.b | ✓ | 4 | 21.g | even | 6 | 1 | |
| 147.3.b.c | 2 | 7.c | even | 3 | 1 | ||
| 147.3.b.c | 2 | 21.h | odd | 6 | 1 | ||
| 147.3.b.d | 2 | 7.d | odd | 6 | 1 | ||
| 147.3.b.d | 2 | 21.g | even | 6 | 1 | ||
| 147.3.h.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 147.3.h.b | 4 | 3.b | odd | 2 | 1 | inner | |
| 147.3.h.b | 4 | 7.c | even | 3 | 1 | inner | |
| 147.3.h.b | 4 | 21.h | odd | 6 | 1 | inner | |
| 336.3.bn.f | 4 | 28.d | even | 2 | 1 | ||
| 336.3.bn.f | 4 | 28.f | even | 6 | 1 | ||
| 336.3.bn.f | 4 | 84.h | odd | 2 | 1 | ||
| 336.3.bn.f | 4 | 84.j | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(147, [\chi])\):
|
\( T_{2}^{4} - 5T_{2}^{2} + 25 \)
|
|
\( T_{5}^{4} - 5T_{5}^{2} + 25 \)
|
|
\( T_{13} - 2 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 5T^{2} + 25 \)
|
| $3$ |
\( T^{4} - 4 T^{3} + \cdots + 81 \)
|
| $5$ |
\( T^{4} - 5T^{2} + 25 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} - 125 T^{2} + 15625 \)
|
| $13$ |
\( (T - 2)^{4} \)
|
| $17$ |
\( T^{4} - 720 T^{2} + 518400 \)
|
| $19$ |
\( (T^{2} - 16 T + 256)^{2} \)
|
| $23$ |
\( T^{4} - 180 T^{2} + 32400 \)
|
| $29$ |
\( (T^{2} + 245)^{2} \)
|
| $31$ |
\( (T^{2} + 3 T + 9)^{2} \)
|
| $37$ |
\( (T^{2} + 12 T + 144)^{2} \)
|
| $41$ |
\( (T^{2} + 980)^{2} \)
|
| $43$ |
\( (T - 44)^{4} \)
|
| $47$ |
\( T^{4} - 180 T^{2} + 32400 \)
|
| $53$ |
\( T^{4} - 405 T^{2} + 164025 \)
|
| $59$ |
\( T^{4} - 405 T^{2} + 164025 \)
|
| $61$ |
\( (T^{2} + 26 T + 676)^{2} \)
|
| $67$ |
\( (T^{2} + 52 T + 2704)^{2} \)
|
| $71$ |
\( (T^{2} + 8820)^{2} \)
|
| $73$ |
\( (T^{2} - 18 T + 324)^{2} \)
|
| $79$ |
\( (T^{2} - 79 T + 6241)^{2} \)
|
| $83$ |
\( (T^{2} + 19845)^{2} \)
|
| $89$ |
\( T^{4} - 2420 T^{2} + 5856400 \)
|
| $97$ |
\( (T - 93)^{4} \)
|
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