Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [684,2,Mod(73,684)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(684, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([0, 0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("684.73");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 684.bo (of order \(9\), degree \(6\), 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: | \(5.46176749826\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\Q(\zeta_{18})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{3} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 228) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).
| \(n\) | \(325\) | \(343\) | \(533\) |
| \(\chi(n)\) | \(\zeta_{18}^{2}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 73.1 |
|
0 | 0 | 0 | −0.233956 | + | 1.32683i | 0 | −1.20574 | − | 2.08840i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||
| 253.1 | 0 | 0 | 0 | −0.233956 | − | 1.32683i | 0 | −1.20574 | + | 2.08840i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
| 289.1 | 0 | 0 | 0 | −1.93969 | + | 1.62760i | 0 | 1.61334 | − | 2.79439i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
| 397.1 | 0 | 0 | 0 | −0.826352 | + | 0.300767i | 0 | 1.09240 | + | 1.89209i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
| 541.1 | 0 | 0 | 0 | −0.826352 | − | 0.300767i | 0 | 1.09240 | − | 1.89209i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
| 613.1 | 0 | 0 | 0 | −1.93969 | − | 1.62760i | 0 | 1.61334 | + | 2.79439i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.e | even | 9 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 684.2.bo.a | 6 | |
| 3.b | odd | 2 | 1 | 228.2.q.a | ✓ | 6 | |
| 12.b | even | 2 | 1 | 912.2.bo.e | 6 | ||
| 19.e | even | 9 | 1 | inner | 684.2.bo.a | 6 | |
| 57.j | even | 18 | 1 | 4332.2.a.n | 3 | ||
| 57.l | odd | 18 | 1 | 228.2.q.a | ✓ | 6 | |
| 57.l | odd | 18 | 1 | 4332.2.a.o | 3 | ||
| 228.v | even | 18 | 1 | 912.2.bo.e | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 228.2.q.a | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 228.2.q.a | ✓ | 6 | 57.l | odd | 18 | 1 | |
| 684.2.bo.a | 6 | 1.a | even | 1 | 1 | trivial | |
| 684.2.bo.a | 6 | 19.e | even | 9 | 1 | inner | |
| 912.2.bo.e | 6 | 12.b | even | 2 | 1 | ||
| 912.2.bo.e | 6 | 228.v | even | 18 | 1 | ||
| 4332.2.a.n | 3 | 57.j | even | 18 | 1 | ||
| 4332.2.a.o | 3 | 57.l | odd | 18 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{6} + 6T_{5}^{5} + 18T_{5}^{4} + 30T_{5}^{3} + 36T_{5}^{2} + 27T_{5} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(684, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + 6 T^{5} + \cdots + 9 \)
|
| $7$ |
\( T^{6} - 3 T^{5} + \cdots + 289 \)
|
| $11$ |
\( T^{6} + 3 T^{5} + \cdots + 9 \)
|
| $13$ |
\( T^{6} - 6 T^{5} + \cdots + 289 \)
|
| $17$ |
\( T^{6} + 12 T^{5} + \cdots + 9 \)
|
| $19$ |
\( T^{6} - 6 T^{5} + \cdots + 6859 \)
|
| $23$ |
\( T^{6} - 15 T^{5} + \cdots + 3249 \)
|
| $29$ |
\( T^{6} - 12 T^{5} + \cdots + 12321 \)
|
| $31$ |
\( T^{6} + 6 T^{5} + \cdots + 361 \)
|
| $37$ |
\( (T^{3} + 6 T^{2} - 9 T - 17)^{2} \)
|
| $41$ |
\( T^{6} - 18 T^{5} + \cdots + 729 \)
|
| $43$ |
\( T^{6} + 18 T^{5} + \cdots + 361 \)
|
| $47$ |
\( T^{6} - 3 T^{5} + \cdots + 12321 \)
|
| $53$ |
\( T^{6} - 24 T^{5} + \cdots + 45369 \)
|
| $59$ |
\( T^{6} + 18 T^{5} + \cdots + 81 \)
|
| $61$ |
\( T^{6} + 9 T^{5} + \cdots + 5041 \)
|
| $67$ |
\( T^{6} + 6 T^{5} + \cdots + 64 \)
|
| $71$ |
\( T^{6} - 18 T^{5} + \cdots + 263169 \)
|
| $73$ |
\( T^{6} - 21 T^{5} + \cdots + 361 \)
|
| $79$ |
\( T^{6} - 6 T^{5} + \cdots + 375769 \)
|
| $83$ |
\( T^{6} + 15 T^{5} + \cdots + 9 \)
|
| $89$ |
\( T^{6} + 15 T^{5} + \cdots + 145161 \)
|
| $97$ |
\( T^{6} - 9 T^{5} + \cdots + 143641 \)
|
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