Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [33,2,Mod(2,33)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(33, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([5, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("33.2");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 33 = 3 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 33.f (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: | \(0.263506326670\) |
| 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 \(\left(\mathbb{Z}/33\mathbb{Z}\right)^\times\).
| \(n\) | \(13\) | \(23\) |
| \(\chi(n)\) | \(\zeta_{20}^{2}\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2.1 |
|
−0.587785 | + | 1.80902i | −1.67229 | + | 0.451057i | −1.30902 | − | 0.951057i | 2.48990 | − | 0.809017i | 0.166977 | − | 3.29032i | 0.427051 | − | 0.587785i | −0.587785 | + | 0.427051i | 2.59310 | − | 1.50859i | 4.97980i | ||||||||||||||||||||||||||
| 2.2 | 0.587785 | − | 1.80902i | −0.945746 | + | 1.45106i | −1.30902 | − | 0.951057i | −2.48990 | + | 0.809017i | 2.06909 | + | 2.56378i | 0.427051 | − | 0.587785i | 0.587785 | − | 0.427051i | −1.21113 | − | 2.74466i | 4.97980i | |||||||||||||||||||||||||||
| 8.1 | −0.951057 | + | 0.690983i | 1.34786 | − | 1.08779i | −0.190983 | + | 0.587785i | −0.224514 | + | 0.309017i | −0.530249 | + | 1.96589i | −2.92705 | − | 0.951057i | −0.951057 | − | 2.92705i | 0.633446 | − | 2.93236i | − | 0.449028i | ||||||||||||||||||||||||||
| 8.2 | 0.951057 | − | 0.690983i | −1.72982 | − | 0.0877853i | −0.190983 | + | 0.587785i | 0.224514 | − | 0.309017i | −1.70582 | + | 1.11179i | −2.92705 | − | 0.951057i | 0.951057 | + | 2.92705i | 2.98459 | + | 0.303706i | − | 0.449028i | ||||||||||||||||||||||||||
| 17.1 | −0.587785 | − | 1.80902i | −1.67229 | − | 0.451057i | −1.30902 | + | 0.951057i | 2.48990 | + | 0.809017i | 0.166977 | + | 3.29032i | 0.427051 | + | 0.587785i | −0.587785 | − | 0.427051i | 2.59310 | + | 1.50859i | − | 4.97980i | ||||||||||||||||||||||||||
| 17.2 | 0.587785 | + | 1.80902i | −0.945746 | − | 1.45106i | −1.30902 | + | 0.951057i | −2.48990 | − | 0.809017i | 2.06909 | − | 2.56378i | 0.427051 | + | 0.587785i | 0.587785 | + | 0.427051i | −1.21113 | + | 2.74466i | − | 4.97980i | ||||||||||||||||||||||||||
| 29.1 | −0.951057 | − | 0.690983i | 1.34786 | + | 1.08779i | −0.190983 | − | 0.587785i | −0.224514 | − | 0.309017i | −0.530249 | − | 1.96589i | −2.92705 | + | 0.951057i | −0.951057 | + | 2.92705i | 0.633446 | + | 2.93236i | 0.449028i | |||||||||||||||||||||||||||
| 29.2 | 0.951057 | + | 0.690983i | −1.72982 | + | 0.0877853i | −0.190983 | − | 0.587785i | 0.224514 | + | 0.309017i | −1.70582 | − | 1.11179i | −2.92705 | + | 0.951057i | 0.951057 | − | 2.92705i | 2.98459 | − | 0.303706i | 0.449028i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 11.d | odd | 10 | 1 | inner |
| 33.f | even | 10 | 1 | inner |
Twists
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(33, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + 5 T^{6} + \cdots + 25 \)
|
| $3$ |
\( T^{8} + 6 T^{7} + \cdots + 81 \)
|
| $5$ |
\( T^{8} - 11 T^{6} + \cdots + 1 \)
|
| $7$ |
\( (T^{4} + 5 T^{3} + 5 T^{2} + \cdots + 5)^{2} \)
|
| $11$ |
\( T^{8} + 19 T^{6} + \cdots + 14641 \)
|
| $13$ |
\( (T^{4} + 5 T^{3} + 5 T^{2} + \cdots + 5)^{2} \)
|
| $17$ |
\( T^{8} + 250 T^{4} + \cdots + 15625 \)
|
| $19$ |
\( (T^{4} - 10 T^{3} + \cdots + 125)^{2} \)
|
| $23$ |
\( (T^{4} + 42 T^{2} + 121)^{2} \)
|
| $29$ |
\( T^{8} + 160 T^{4} + \cdots + 6400 \)
|
| $31$ |
\( (T^{4} + 10 T^{3} + \cdots + 25)^{2} \)
|
| $37$ |
\( (T^{4} + 3 T^{3} + 9 T^{2} + \cdots + 81)^{2} \)
|
| $41$ |
\( T^{8} - 5 T^{6} + \cdots + 25 \)
|
| $43$ |
\( (T^{4} + 50 T^{2} + 125)^{2} \)
|
| $47$ |
\( T^{8} - 79 T^{6} + \cdots + 13845841 \)
|
| $53$ |
\( T^{8} - 36 T^{6} + \cdots + 6561 \)
|
| $59$ |
\( T^{8} + 4 T^{6} + \cdots + 1 \)
|
| $61$ |
\( (T^{4} + 5 T^{3} + 125)^{2} \)
|
| $67$ |
\( (T^{2} + T - 61)^{4} \)
|
| $71$ |
\( T^{8} - 155 T^{6} + \cdots + 9150625 \)
|
| $73$ |
\( (T^{4} + 2560 T + 20480)^{2} \)
|
| $79$ |
\( (T^{4} - 25 T^{3} + \cdots + 1805)^{2} \)
|
| $83$ |
\( T^{8} + 315 T^{6} + \cdots + 70644025 \)
|
| $89$ |
\( (T^{4} + 90 T^{2} + 25)^{2} \)
|
| $97$ |
\( (T^{4} - 3 T^{3} + \cdots + 841)^{2} \)
|
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