Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [66,4,Mod(25,66)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(66, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 8]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("66.25");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 66 = 2 \cdot 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 66.e (of order \(5\), 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: | \(3.89412606038\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{10})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + x^{2} - x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{5}]$ |
$q$-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 \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/66\mathbb{Z}\right)^\times\).
| \(n\) | \(13\) | \(23\) |
| \(\chi(n)\) | \(\zeta_{10}^{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 25.1 |
|
−1.61803 | + | 1.17557i | −0.927051 | − | 2.85317i | 1.23607 | − | 3.80423i | −4.30902 | − | 3.13068i | 4.85410 | + | 3.52671i | −10.5451 | + | 32.4544i | 2.47214 | + | 7.60845i | −7.28115 | + | 5.29007i | 10.6525 | ||||||||||||||
| 31.1 | 0.618034 | − | 1.90211i | 2.42705 | − | 1.76336i | −3.23607 | − | 2.35114i | −3.19098 | − | 9.82084i | −1.85410 | − | 5.70634i | −4.95492 | − | 3.59996i | −6.47214 | + | 4.70228i | 2.78115 | − | 8.55951i | −20.6525 | |||||||||||||||
| 37.1 | −1.61803 | − | 1.17557i | −0.927051 | + | 2.85317i | 1.23607 | + | 3.80423i | −4.30902 | + | 3.13068i | 4.85410 | − | 3.52671i | −10.5451 | − | 32.4544i | 2.47214 | − | 7.60845i | −7.28115 | − | 5.29007i | 10.6525 | |||||||||||||||
| 49.1 | 0.618034 | + | 1.90211i | 2.42705 | + | 1.76336i | −3.23607 | + | 2.35114i | −3.19098 | + | 9.82084i | −1.85410 | + | 5.70634i | −4.95492 | + | 3.59996i | −6.47214 | − | 4.70228i | 2.78115 | + | 8.55951i | −20.6525 | |||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.c | even | 5 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 66.4.e.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 198.4.f.c | 4 | ||
| 11.c | even | 5 | 1 | inner | 66.4.e.a | ✓ | 4 |
| 11.c | even | 5 | 1 | 726.4.a.q | 2 | ||
| 11.d | odd | 10 | 1 | 726.4.a.l | 2 | ||
| 33.f | even | 10 | 1 | 2178.4.a.bk | 2 | ||
| 33.h | odd | 10 | 1 | 198.4.f.c | 4 | ||
| 33.h | odd | 10 | 1 | 2178.4.a.ba | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 66.4.e.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 66.4.e.a | ✓ | 4 | 11.c | even | 5 | 1 | inner |
| 198.4.f.c | 4 | 3.b | odd | 2 | 1 | ||
| 198.4.f.c | 4 | 33.h | odd | 10 | 1 | ||
| 726.4.a.l | 2 | 11.d | odd | 10 | 1 | ||
| 726.4.a.q | 2 | 11.c | even | 5 | 1 | ||
| 2178.4.a.ba | 2 | 33.h | odd | 10 | 1 | ||
| 2178.4.a.bk | 2 | 33.f | even | 10 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 15T_{5}^{3} + 190T_{5}^{2} + 1100T_{5} + 3025 \)
acting on \(S_{4}^{\mathrm{new}}(66, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 2 T^{3} + \cdots + 16 \)
|
| $3$ |
\( T^{4} - 3 T^{3} + \cdots + 81 \)
|
| $5$ |
\( T^{4} + 15 T^{3} + \cdots + 3025 \)
|
| $7$ |
\( T^{4} + 31 T^{3} + \cdots + 43681 \)
|
| $11$ |
\( T^{4} + 11 T^{3} + \cdots + 1771561 \)
|
| $13$ |
\( T^{4} + 85 T^{3} + \cdots + 198025 \)
|
| $17$ |
\( T^{4} + 50 T^{3} + \cdots + 9025 \)
|
| $19$ |
\( T^{4} - 4 T^{3} + \cdots + 9740641 \)
|
| $23$ |
\( (T^{2} - 152 T - 3469)^{2} \)
|
| $29$ |
\( T^{4} - 422 T^{3} + \cdots + 690848656 \)
|
| $31$ |
\( T^{4} + \cdots + 1657385521 \)
|
| $37$ |
\( T^{4} + 171 T^{3} + \cdots + 41873841 \)
|
| $41$ |
\( T^{4} + \cdots + 1304726641 \)
|
| $43$ |
\( (T^{2} + 306 T + 8829)^{2} \)
|
| $47$ |
\( T^{4} + 79 T^{3} + \cdots + 39601 \)
|
| $53$ |
\( T^{4} + 74 T^{3} + \cdots + 19456921 \)
|
| $59$ |
\( T^{4} + \cdots + 288851427601 \)
|
| $61$ |
\( T^{4} + \cdots + 1311091681 \)
|
| $67$ |
\( (T^{2} + 553 T - 18079)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 97120112881 \)
|
| $73$ |
\( T^{4} + \cdots + 69065942416 \)
|
| $79$ |
\( T^{4} + \cdots + 79400531961 \)
|
| $83$ |
\( T^{4} + \cdots + 27439591201 \)
|
| $89$ |
\( (T^{2} + 274 T - 193411)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 327219464961 \)
|
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