Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [132,2,Mod(47,132)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(132, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 5, 8]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("132.47");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 132 = 2^{2} \cdot 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 132.o (of order \(10\), 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: | \(1.05402530668\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{10})\) |
| Coefficient field: | \(\Q(\zeta_{20})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{6} + x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{10}]$ |
$q$-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 \(\zeta_{20}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/132\mathbb{Z}\right)^\times\).
| \(n\) | \(13\) | \(67\) | \(89\) |
| \(\chi(n)\) | \(-\zeta_{20}^{6}\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 47.1 |
|
−1.39680 | − | 0.221232i | 1.08779 | + | 1.34786i | 1.90211 | + | 0.618034i | 0.951057 | − | 1.30902i | −1.22123 | − | 2.12334i | 0.224514 | + | 0.0729490i | −2.52015 | − | 1.28408i | −0.633446 | + | 2.93236i | −1.61803 | + | 1.61803i | ||||||||||||||||||||||||
| 47.2 | −0.221232 | + | 1.39680i | −0.0877853 | + | 1.72982i | −1.90211 | − | 0.618034i | −0.951057 | + | 1.30902i | −2.39680 | − | 0.505311i | −0.224514 | − | 0.0729490i | 1.28408 | − | 2.52015i | −2.98459 | − | 0.303706i | −1.61803 | − | 1.61803i | |||||||||||||||||||||||||
| 59.1 | −1.39680 | + | 0.221232i | 1.08779 | − | 1.34786i | 1.90211 | − | 0.618034i | 0.951057 | + | 1.30902i | −1.22123 | + | 2.12334i | 0.224514 | − | 0.0729490i | −2.52015 | + | 1.28408i | −0.633446 | − | 2.93236i | −1.61803 | − | 1.61803i | |||||||||||||||||||||||||
| 59.2 | −0.221232 | − | 1.39680i | −0.0877853 | − | 1.72982i | −1.90211 | + | 0.618034i | −0.951057 | − | 1.30902i | −2.39680 | + | 0.505311i | −0.224514 | + | 0.0729490i | 1.28408 | + | 2.52015i | −2.98459 | + | 0.303706i | −1.61803 | + | 1.61803i | |||||||||||||||||||||||||
| 71.1 | −0.642040 | + | 1.26007i | 1.45106 | − | 0.945746i | −1.17557 | − | 1.61803i | −0.587785 | − | 0.190983i | 0.260074 | + | 2.43564i | 2.48990 | + | 3.42705i | 2.79360 | − | 0.442463i | 1.21113 | − | 2.74466i | 0.618034 | − | 0.618034i | |||||||||||||||||||||||||
| 71.2 | 1.26007 | + | 0.642040i | −0.451057 | + | 1.67229i | 1.17557 | + | 1.61803i | 0.587785 | + | 0.190983i | −1.64204 | + | 1.81761i | −2.48990 | − | 3.42705i | 0.442463 | + | 2.79360i | −2.59310 | − | 1.50859i | 0.618034 | + | 0.618034i | |||||||||||||||||||||||||
| 119.1 | −0.642040 | − | 1.26007i | 1.45106 | + | 0.945746i | −1.17557 | + | 1.61803i | −0.587785 | + | 0.190983i | 0.260074 | − | 2.43564i | 2.48990 | − | 3.42705i | 2.79360 | + | 0.442463i | 1.21113 | + | 2.74466i | 0.618034 | + | 0.618034i | |||||||||||||||||||||||||
| 119.2 | 1.26007 | − | 0.642040i | −0.451057 | − | 1.67229i | 1.17557 | − | 1.61803i | 0.587785 | − | 0.190983i | −1.64204 | − | 1.81761i | −2.48990 | + | 3.42705i | 0.442463 | − | 2.79360i | −2.59310 | + | 1.50859i | 0.618034 | − | 0.618034i | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 1 | inner |
| 12.b | even | 2 | 1 | inner |
| 132.o | even | 10 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 132.2.o.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 132.2.o.b | yes | 8 | |
| 4.b | odd | 2 | 1 | 132.2.o.b | yes | 8 | |
| 11.c | even | 5 | 1 | inner | 132.2.o.a | ✓ | 8 |
| 12.b | even | 2 | 1 | inner | 132.2.o.a | ✓ | 8 |
| 33.h | odd | 10 | 1 | 132.2.o.b | yes | 8 | |
| 44.h | odd | 10 | 1 | 132.2.o.b | yes | 8 | |
| 132.o | even | 10 | 1 | inner | 132.2.o.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 132.2.o.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 132.2.o.a | ✓ | 8 | 11.c | even | 5 | 1 | inner |
| 132.2.o.a | ✓ | 8 | 12.b | even | 2 | 1 | inner |
| 132.2.o.a | ✓ | 8 | 132.o | even | 10 | 1 | inner |
| 132.2.o.b | yes | 8 | 3.b | odd | 2 | 1 | |
| 132.2.o.b | yes | 8 | 4.b | odd | 2 | 1 | |
| 132.2.o.b | yes | 8 | 33.h | odd | 10 | 1 | |
| 132.2.o.b | yes | 8 | 44.h | odd | 10 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(132, [\chi])\):
|
\( T_{5}^{8} + T_{5}^{6} + 6T_{5}^{4} - 4T_{5}^{2} + 1 \)
|
|
\( T_{23}^{2} + 8T_{23} + 11 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 2 T^{7} + \cdots + 16 \)
|
| $3$ |
\( T^{8} - 4 T^{7} + \cdots + 81 \)
|
| $5$ |
\( T^{8} + T^{6} + 6 T^{4} + \cdots + 1 \)
|
| $7$ |
\( T^{8} + 11 T^{6} + \cdots + 1 \)
|
| $11$ |
\( (T^{4} - 11 T^{3} + \cdots + 121)^{2} \)
|
| $13$ |
\( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \)
|
| $17$ |
\( T^{8} - 44 T^{6} + \cdots + 923521 \)
|
| $19$ |
\( T^{8} - 36 T^{6} + \cdots + 6561 \)
|
| $23$ |
\( (T^{2} + 8 T + 11)^{4} \)
|
| $29$ |
\( T^{8} - 16 T^{6} + \cdots + 65536 \)
|
| $31$ |
\( T^{8} + 20 T^{6} + \cdots + 625 \)
|
| $37$ |
\( (T^{4} + 3 T^{3} + \cdots + 3481)^{2} \)
|
| $41$ |
\( T^{8} - 145 T^{6} + \cdots + 625 \)
|
| $43$ |
\( (T^{4} + 42 T^{2} + 121)^{2} \)
|
| $47$ |
\( (T^{4} - 3 T^{3} + \cdots + 841)^{2} \)
|
| $53$ |
\( T^{8} + 44 T^{6} + \cdots + 13845841 \)
|
| $59$ |
\( (T^{4} - 10 T^{3} + \cdots + 25)^{2} \)
|
| $61$ |
\( (T^{4} - 13 T^{3} + \cdots + 961)^{2} \)
|
| $67$ |
\( (T^{4} + 23 T^{2} + 121)^{2} \)
|
| $71$ |
\( (T^{4} + 13 T^{3} + \cdots + 961)^{2} \)
|
| $73$ |
\( (T^{4} + 14 T^{3} + \cdots + 1936)^{2} \)
|
| $79$ |
\( T^{8} + 139 T^{6} + \cdots + 62742241 \)
|
| $83$ |
\( (T^{4} + T^{3} + 141 T^{2} + \cdots + 3721)^{2} \)
|
| $89$ |
\( (T^{4} + 282 T^{2} + 11881)^{2} \)
|
| $97$ |
\( (T^{4} + 13 T^{3} + \cdots + 1681)^{2} \)
|
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