Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [361,2,Mod(68,361)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(361, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("361.68");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 361 = 19^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 361.c (of order \(3\), degree \(2\), 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: | \(2.88259951297\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| 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: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 3 \) |
| Twist minimal: | no (minimal twist has level 19) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( \zeta_{18}^{3} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{18}^{5} + \zeta_{18} \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} \)
|
| \(\beta_{4}\) | \(=\) |
\( -\zeta_{18}^{5} + \zeta_{18}^{4} \)
|
| \(\beta_{5}\) | \(=\) |
\( -\zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18} \)
|
| \(\zeta_{18}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} + 2\beta_{2} ) / 3 \)
|
| \(\zeta_{18}^{2}\) | \(=\) |
\( ( -2\beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} ) / 3 \)
|
| \(\zeta_{18}^{3}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{18}^{4}\) | \(=\) |
\( ( -\beta_{5} + 2\beta_{4} + \beta_{2} ) / 3 \)
|
| \(\zeta_{18}^{5}\) | \(=\) |
\( ( -\beta_{5} - \beta_{4} + \beta_{2} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/361\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1 + \beta_{1}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 68.1 |
|
−0.439693 | + | 0.761570i | −0.266044 | + | 0.460802i | 0.613341 | + | 1.06234i | 1.26604 | − | 2.19285i | −0.233956 | − | 0.405223i | −1.87939 | −2.83750 | 1.35844 | + | 2.35289i | 1.11334 | + | 1.92836i | ||||||||||||||||||||||
| 68.2 | 0.673648 | − | 1.16679i | 1.43969 | − | 2.49362i | 0.0923963 | + | 0.160035i | −0.439693 | + | 0.761570i | −1.93969 | − | 3.35965i | 0.347296 | 2.94356 | −2.64543 | − | 4.58202i | 0.592396 | + | 1.02606i | |||||||||||||||||||||||
| 68.3 | 1.26604 | − | 2.19285i | 0.326352 | − | 0.565258i | −2.20574 | − | 3.82045i | 0.673648 | − | 1.16679i | −0.826352 | − | 1.43128i | 1.53209 | −6.10607 | 1.28699 | + | 2.22913i | −1.70574 | − | 2.95442i | |||||||||||||||||||||||
| 292.1 | −0.439693 | − | 0.761570i | −0.266044 | − | 0.460802i | 0.613341 | − | 1.06234i | 1.26604 | + | 2.19285i | −0.233956 | + | 0.405223i | −1.87939 | −2.83750 | 1.35844 | − | 2.35289i | 1.11334 | − | 1.92836i | |||||||||||||||||||||||
| 292.2 | 0.673648 | + | 1.16679i | 1.43969 | + | 2.49362i | 0.0923963 | − | 0.160035i | −0.439693 | − | 0.761570i | −1.93969 | + | 3.35965i | 0.347296 | 2.94356 | −2.64543 | + | 4.58202i | 0.592396 | − | 1.02606i | |||||||||||||||||||||||
| 292.3 | 1.26604 | + | 2.19285i | 0.326352 | + | 0.565258i | −2.20574 | + | 3.82045i | 0.673648 | + | 1.16679i | −0.826352 | + | 1.43128i | 1.53209 | −6.10607 | 1.28699 | − | 2.22913i | −1.70574 | + | 2.95442i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 361.2.c.i | 6 | |
| 19.b | odd | 2 | 1 | 361.2.c.h | 6 | ||
| 19.c | even | 3 | 1 | 361.2.a.g | 3 | ||
| 19.c | even | 3 | 1 | inner | 361.2.c.i | 6 | |
| 19.d | odd | 6 | 1 | 361.2.a.h | 3 | ||
| 19.d | odd | 6 | 1 | 361.2.c.h | 6 | ||
| 19.e | even | 9 | 2 | 19.2.e.a | ✓ | 6 | |
| 19.e | even | 9 | 2 | 361.2.e.f | 6 | ||
| 19.e | even | 9 | 2 | 361.2.e.g | 6 | ||
| 19.f | odd | 18 | 2 | 361.2.e.a | 6 | ||
| 19.f | odd | 18 | 2 | 361.2.e.b | 6 | ||
| 19.f | odd | 18 | 2 | 361.2.e.h | 6 | ||
| 57.f | even | 6 | 1 | 3249.2.a.s | 3 | ||
| 57.h | odd | 6 | 1 | 3249.2.a.z | 3 | ||
| 57.l | odd | 18 | 2 | 171.2.u.c | 6 | ||
| 76.f | even | 6 | 1 | 5776.2.a.bi | 3 | ||
| 76.g | odd | 6 | 1 | 5776.2.a.br | 3 | ||
| 76.l | odd | 18 | 2 | 304.2.u.b | 6 | ||
| 95.h | odd | 6 | 1 | 9025.2.a.x | 3 | ||
| 95.i | even | 6 | 1 | 9025.2.a.bd | 3 | ||
| 95.p | even | 18 | 2 | 475.2.l.a | 6 | ||
| 95.q | odd | 36 | 4 | 475.2.u.a | 12 | ||
| 133.u | even | 9 | 2 | 931.2.v.b | 6 | ||
| 133.w | even | 9 | 2 | 931.2.x.a | 6 | ||
| 133.x | odd | 18 | 2 | 931.2.v.a | 6 | ||
| 133.y | odd | 18 | 2 | 931.2.w.a | 6 | ||
| 133.z | odd | 18 | 2 | 931.2.x.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 19.2.e.a | ✓ | 6 | 19.e | even | 9 | 2 | |
| 171.2.u.c | 6 | 57.l | odd | 18 | 2 | ||
| 304.2.u.b | 6 | 76.l | odd | 18 | 2 | ||
| 361.2.a.g | 3 | 19.c | even | 3 | 1 | ||
| 361.2.a.h | 3 | 19.d | odd | 6 | 1 | ||
| 361.2.c.h | 6 | 19.b | odd | 2 | 1 | ||
| 361.2.c.h | 6 | 19.d | odd | 6 | 1 | ||
| 361.2.c.i | 6 | 1.a | even | 1 | 1 | trivial | |
| 361.2.c.i | 6 | 19.c | even | 3 | 1 | inner | |
| 361.2.e.a | 6 | 19.f | odd | 18 | 2 | ||
| 361.2.e.b | 6 | 19.f | odd | 18 | 2 | ||
| 361.2.e.f | 6 | 19.e | even | 9 | 2 | ||
| 361.2.e.g | 6 | 19.e | even | 9 | 2 | ||
| 361.2.e.h | 6 | 19.f | odd | 18 | 2 | ||
| 475.2.l.a | 6 | 95.p | even | 18 | 2 | ||
| 475.2.u.a | 12 | 95.q | odd | 36 | 4 | ||
| 931.2.v.a | 6 | 133.x | odd | 18 | 2 | ||
| 931.2.v.b | 6 | 133.u | even | 9 | 2 | ||
| 931.2.w.a | 6 | 133.y | odd | 18 | 2 | ||
| 931.2.x.a | 6 | 133.w | even | 9 | 2 | ||
| 931.2.x.b | 6 | 133.z | odd | 18 | 2 | ||
| 3249.2.a.s | 3 | 57.f | even | 6 | 1 | ||
| 3249.2.a.z | 3 | 57.h | odd | 6 | 1 | ||
| 5776.2.a.bi | 3 | 76.f | even | 6 | 1 | ||
| 5776.2.a.br | 3 | 76.g | odd | 6 | 1 | ||
| 9025.2.a.x | 3 | 95.h | odd | 6 | 1 | ||
| 9025.2.a.bd | 3 | 95.i | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(361, [\chi])\):
|
\( T_{2}^{6} - 3T_{2}^{5} + 9T_{2}^{4} - 6T_{2}^{3} + 9T_{2}^{2} + 9 \)
|
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\( T_{3}^{6} - 3T_{3}^{5} + 9T_{3}^{4} - 2T_{3}^{3} + 3T_{3}^{2} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 3 T^{5} + \cdots + 9 \)
|
| $3$ |
\( T^{6} - 3 T^{5} + \cdots + 1 \)
|
| $5$ |
\( T^{6} - 3 T^{5} + \cdots + 9 \)
|
| $7$ |
\( (T^{3} - 3 T + 1)^{2} \)
|
| $11$ |
\( (T^{3} - 9 T - 9)^{2} \)
|
| $13$ |
\( T^{6} + 21 T^{4} + \cdots + 1369 \)
|
| $17$ |
\( T^{6} + 6 T^{5} + \cdots + 9 \)
|
| $19$ |
\( T^{6} \)
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| $23$ |
\( T^{6} - 6 T^{5} + \cdots + 576 \)
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| $29$ |
\( T^{6} - 15 T^{5} + \cdots + 12321 \)
|
| $31$ |
\( (T^{3} + 9 T^{2} + 6 T - 53)^{2} \)
|
| $37$ |
\( (T^{3} - 21 T - 17)^{2} \)
|
| $41$ |
\( T^{6} - 12 T^{5} + \cdots + 12321 \)
|
| $43$ |
\( T^{6} + 57 T^{4} + \cdots + 26569 \)
|
| $47$ |
\( T^{6} - 6 T^{5} + \cdots + 9 \)
|
| $53$ |
\( T^{6} - 6 T^{5} + \cdots + 2601 \)
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| $59$ |
\( T^{6} - 21 T^{5} + \cdots + 71289 \)
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| $61$ |
\( T^{6} + 9 T^{5} + \cdots + 32761 \)
|
| $67$ |
\( T^{6} + 18 T^{5} + \cdots + 179776 \)
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| $71$ |
\( T^{6} - 30 T^{5} + \cdots + 788544 \)
|
| $73$ |
\( T^{6} + 48 T^{4} + \cdots + 4096 \)
|
| $79$ |
\( T^{6} - 9 T^{5} + \cdots + 654481 \)
|
| $83$ |
\( (T^{3} - 189 T + 459)^{2} \)
|
| $89$ |
\( T^{6} - 15 T^{5} + \cdots + 3249 \)
|
| $97$ |
\( T^{6} + 15 T^{5} + \cdots + 16129 \)
|
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