Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [722,2,Mod(99,722)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(722, base_ring=CyclotomicField(18))
chi = DirichletCharacter(H, H._module([2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("722.99");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 722 = 2 \cdot 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 722.e (of order \(9\), degree \(6\), 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: | \(5.76519902594\) |
| 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: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 38) |
| 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 \(\left(\mathbb{Z}/722\mathbb{Z}\right)^\times\).
| \(n\) | \(363\) |
| \(\chi(n)\) | \(-\zeta_{18}^{5}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 99.1 |
|
−0.939693 | − | 0.342020i | −0.173648 | − | 0.984808i | 0.766044 | + | 0.642788i | −3.06418 | + | 2.57115i | −0.173648 | + | 0.984808i | −1.50000 | + | 2.59808i | −0.500000 | − | 0.866025i | 1.87939 | − | 0.684040i | 3.75877 | − | 1.36808i | ||||||||||||||||||
| 245.1 | 0.173648 | − | 0.984808i | −0.766044 | − | 0.642788i | −0.939693 | − | 0.342020i | 3.75877 | − | 1.36808i | −0.766044 | + | 0.642788i | −1.50000 | − | 2.59808i | −0.500000 | + | 0.866025i | −0.347296 | − | 1.96962i | −0.694593 | − | 3.93923i | |||||||||||||||||||
| 389.1 | 0.173648 | + | 0.984808i | −0.766044 | + | 0.642788i | −0.939693 | + | 0.342020i | 3.75877 | + | 1.36808i | −0.766044 | − | 0.642788i | −1.50000 | + | 2.59808i | −0.500000 | − | 0.866025i | −0.347296 | + | 1.96962i | −0.694593 | + | 3.93923i | |||||||||||||||||||
| 415.1 | 0.766044 | + | 0.642788i | 0.939693 | − | 0.342020i | 0.173648 | + | 0.984808i | −0.694593 | + | 3.93923i | 0.939693 | + | 0.342020i | −1.50000 | − | 2.59808i | −0.500000 | + | 0.866025i | −1.53209 | + | 1.28558i | −3.06418 | + | 2.57115i | |||||||||||||||||||
| 423.1 | −0.939693 | + | 0.342020i | −0.173648 | + | 0.984808i | 0.766044 | − | 0.642788i | −3.06418 | − | 2.57115i | −0.173648 | − | 0.984808i | −1.50000 | − | 2.59808i | −0.500000 | + | 0.866025i | 1.87939 | + | 0.684040i | 3.75877 | + | 1.36808i | |||||||||||||||||||
| 595.1 | 0.766044 | − | 0.642788i | 0.939693 | + | 0.342020i | 0.173648 | − | 0.984808i | −0.694593 | − | 3.93923i | 0.939693 | − | 0.342020i | −1.50000 | + | 2.59808i | −0.500000 | − | 0.866025i | −1.53209 | − | 1.28558i | −3.06418 | − | 2.57115i | |||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 2 | inner |
| 19.e | even | 9 | 3 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 722.2.e.c | 6 | |
| 19.b | odd | 2 | 1 | 722.2.e.d | 6 | ||
| 19.c | even | 3 | 2 | inner | 722.2.e.c | 6 | |
| 19.d | odd | 6 | 2 | 722.2.e.d | 6 | ||
| 19.e | even | 9 | 1 | 38.2.a.b | ✓ | 1 | |
| 19.e | even | 9 | 2 | 722.2.c.d | 2 | ||
| 19.e | even | 9 | 3 | inner | 722.2.e.c | 6 | |
| 19.f | odd | 18 | 1 | 722.2.a.b | 1 | ||
| 19.f | odd | 18 | 2 | 722.2.c.f | 2 | ||
| 19.f | odd | 18 | 3 | 722.2.e.d | 6 | ||
| 57.j | even | 18 | 1 | 6498.2.a.y | 1 | ||
| 57.l | odd | 18 | 1 | 342.2.a.d | 1 | ||
| 76.k | even | 18 | 1 | 5776.2.a.d | 1 | ||
| 76.l | odd | 18 | 1 | 304.2.a.d | 1 | ||
| 95.p | even | 18 | 1 | 950.2.a.b | 1 | ||
| 95.q | odd | 36 | 2 | 950.2.b.c | 2 | ||
| 133.y | odd | 18 | 1 | 1862.2.a.f | 1 | ||
| 152.t | even | 18 | 1 | 1216.2.a.n | 1 | ||
| 152.u | odd | 18 | 1 | 1216.2.a.g | 1 | ||
| 209.q | odd | 18 | 1 | 4598.2.a.a | 1 | ||
| 228.v | even | 18 | 1 | 2736.2.a.w | 1 | ||
| 247.bn | even | 18 | 1 | 6422.2.a.b | 1 | ||
| 285.bd | odd | 18 | 1 | 8550.2.a.u | 1 | ||
| 380.ba | odd | 18 | 1 | 7600.2.a.h | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 38.2.a.b | ✓ | 1 | 19.e | even | 9 | 1 | |
| 304.2.a.d | 1 | 76.l | odd | 18 | 1 | ||
| 342.2.a.d | 1 | 57.l | odd | 18 | 1 | ||
| 722.2.a.b | 1 | 19.f | odd | 18 | 1 | ||
| 722.2.c.d | 2 | 19.e | even | 9 | 2 | ||
| 722.2.c.f | 2 | 19.f | odd | 18 | 2 | ||
| 722.2.e.c | 6 | 1.a | even | 1 | 1 | trivial | |
| 722.2.e.c | 6 | 19.c | even | 3 | 2 | inner | |
| 722.2.e.c | 6 | 19.e | even | 9 | 3 | inner | |
| 722.2.e.d | 6 | 19.b | odd | 2 | 1 | ||
| 722.2.e.d | 6 | 19.d | odd | 6 | 2 | ||
| 722.2.e.d | 6 | 19.f | odd | 18 | 3 | ||
| 950.2.a.b | 1 | 95.p | even | 18 | 1 | ||
| 950.2.b.c | 2 | 95.q | odd | 36 | 2 | ||
| 1216.2.a.g | 1 | 152.u | odd | 18 | 1 | ||
| 1216.2.a.n | 1 | 152.t | even | 18 | 1 | ||
| 1862.2.a.f | 1 | 133.y | odd | 18 | 1 | ||
| 2736.2.a.w | 1 | 228.v | even | 18 | 1 | ||
| 4598.2.a.a | 1 | 209.q | odd | 18 | 1 | ||
| 5776.2.a.d | 1 | 76.k | even | 18 | 1 | ||
| 6422.2.a.b | 1 | 247.bn | even | 18 | 1 | ||
| 6498.2.a.y | 1 | 57.j | even | 18 | 1 | ||
| 7600.2.a.h | 1 | 380.ba | odd | 18 | 1 | ||
| 8550.2.a.u | 1 | 285.bd | odd | 18 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(722, [\chi])\):
|
\( T_{3}^{6} - T_{3}^{3} + 1 \)
|
|
\( T_{5}^{6} - 64T_{5}^{3} + 4096 \)
|
|
\( T_{7}^{2} + 3T_{7} + 9 \)
|
|
\( T_{13}^{6} - T_{13}^{3} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{3} + 1 \)
|
| $3$ |
\( T^{6} - T^{3} + 1 \)
|
| $5$ |
\( T^{6} - 64T^{3} + 4096 \)
|
| $7$ |
\( (T^{2} + 3 T + 9)^{3} \)
|
| $11$ |
\( (T^{2} + 2 T + 4)^{3} \)
|
| $13$ |
\( T^{6} - T^{3} + 1 \)
|
| $17$ |
\( T^{6} + 27T^{3} + 729 \)
|
| $19$ |
\( T^{6} \)
|
| $23$ |
\( T^{6} - T^{3} + 1 \)
|
| $29$ |
\( T^{6} - 125 T^{3} + 15625 \)
|
| $31$ |
\( (T^{2} - 8 T + 64)^{3} \)
|
| $37$ |
\( (T + 2)^{6} \)
|
| $41$ |
\( T^{6} - 512 T^{3} + 262144 \)
|
| $43$ |
\( T^{6} + 64T^{3} + 4096 \)
|
| $47$ |
\( T^{6} + 512 T^{3} + 262144 \)
|
| $53$ |
\( T^{6} - T^{3} + 1 \)
|
| $59$ |
\( T^{6} + 3375 T^{3} + 11390625 \)
|
| $61$ |
\( T^{6} + 8T^{3} + 64 \)
|
| $67$ |
\( T^{6} + 27T^{3} + 729 \)
|
| $71$ |
\( T^{6} + 8T^{3} + 64 \)
|
| $73$ |
\( T^{6} + 729 T^{3} + 531441 \)
|
| $79$ |
\( T^{6} - 1000 T^{3} + 1000000 \)
|
| $83$ |
\( (T^{2} - 6 T + 36)^{3} \)
|
| $89$ |
\( T^{6} \)
|
| $97$ |
\( T^{6} - 8T^{3} + 64 \)
|
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