Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [867,2,Mod(616,867)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(867, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("867.616");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 867 = 3 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 867.e (of order \(4\), 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: | \(6.92302985525\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 51) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/867\mathbb{Z}\right)^\times\).
| \(n\) | \(290\) | \(292\) |
| \(\chi(n)\) | \(1\) | \(\zeta_{8}^{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 616.1 |
|
0 | −0.707107 | + | 0.707107i | 2.00000 | 2.12132 | − | 2.12132i | 0 | −2.82843 | − | 2.82843i | 0 | − | 1.00000i | 0 | |||||||||||||||||||||||
| 616.2 | 0 | 0.707107 | − | 0.707107i | 2.00000 | −2.12132 | + | 2.12132i | 0 | 2.82843 | + | 2.82843i | 0 | − | 1.00000i | 0 | ||||||||||||||||||||||||
| 829.1 | 0 | −0.707107 | − | 0.707107i | 2.00000 | 2.12132 | + | 2.12132i | 0 | −2.82843 | + | 2.82843i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||
| 829.2 | 0 | 0.707107 | + | 0.707107i | 2.00000 | −2.12132 | − | 2.12132i | 0 | 2.82843 | − | 2.82843i | 0 | 1.00000i | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
| 17.c | even | 4 | 2 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 867.2.e.e | 4 | |
| 17.b | even | 2 | 1 | inner | 867.2.e.e | 4 | |
| 17.c | even | 4 | 2 | inner | 867.2.e.e | 4 | |
| 17.d | even | 8 | 1 | 51.2.a.a | ✓ | 1 | |
| 17.d | even | 8 | 1 | 867.2.a.c | 1 | ||
| 17.d | even | 8 | 2 | 867.2.d.a | 2 | ||
| 17.e | odd | 16 | 8 | 867.2.h.c | 8 | ||
| 51.g | odd | 8 | 1 | 153.2.a.b | 1 | ||
| 51.g | odd | 8 | 1 | 2601.2.a.f | 1 | ||
| 68.g | odd | 8 | 1 | 816.2.a.g | 1 | ||
| 85.k | odd | 8 | 1 | 1275.2.b.b | 2 | ||
| 85.m | even | 8 | 1 | 1275.2.a.d | 1 | ||
| 85.n | odd | 8 | 1 | 1275.2.b.b | 2 | ||
| 119.l | odd | 8 | 1 | 2499.2.a.d | 1 | ||
| 136.o | even | 8 | 1 | 3264.2.a.a | 1 | ||
| 136.p | odd | 8 | 1 | 3264.2.a.r | 1 | ||
| 187.i | odd | 8 | 1 | 6171.2.a.e | 1 | ||
| 204.p | even | 8 | 1 | 2448.2.a.c | 1 | ||
| 221.p | even | 8 | 1 | 8619.2.a.g | 1 | ||
| 255.y | odd | 8 | 1 | 3825.2.a.i | 1 | ||
| 357.w | even | 8 | 1 | 7497.2.a.j | 1 | ||
| 408.bd | even | 8 | 1 | 9792.2.a.cd | 1 | ||
| 408.be | odd | 8 | 1 | 9792.2.a.by | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 51.2.a.a | ✓ | 1 | 17.d | even | 8 | 1 | |
| 153.2.a.b | 1 | 51.g | odd | 8 | 1 | ||
| 816.2.a.g | 1 | 68.g | odd | 8 | 1 | ||
| 867.2.a.c | 1 | 17.d | even | 8 | 1 | ||
| 867.2.d.a | 2 | 17.d | even | 8 | 2 | ||
| 867.2.e.e | 4 | 1.a | even | 1 | 1 | trivial | |
| 867.2.e.e | 4 | 17.b | even | 2 | 1 | inner | |
| 867.2.e.e | 4 | 17.c | even | 4 | 2 | inner | |
| 867.2.h.c | 8 | 17.e | odd | 16 | 8 | ||
| 1275.2.a.d | 1 | 85.m | even | 8 | 1 | ||
| 1275.2.b.b | 2 | 85.k | odd | 8 | 1 | ||
| 1275.2.b.b | 2 | 85.n | odd | 8 | 1 | ||
| 2448.2.a.c | 1 | 204.p | even | 8 | 1 | ||
| 2499.2.a.d | 1 | 119.l | odd | 8 | 1 | ||
| 2601.2.a.f | 1 | 51.g | odd | 8 | 1 | ||
| 3264.2.a.a | 1 | 136.o | even | 8 | 1 | ||
| 3264.2.a.r | 1 | 136.p | odd | 8 | 1 | ||
| 3825.2.a.i | 1 | 255.y | odd | 8 | 1 | ||
| 6171.2.a.e | 1 | 187.i | odd | 8 | 1 | ||
| 7497.2.a.j | 1 | 357.w | even | 8 | 1 | ||
| 8619.2.a.g | 1 | 221.p | even | 8 | 1 | ||
| 9792.2.a.by | 1 | 408.be | odd | 8 | 1 | ||
| 9792.2.a.cd | 1 | 408.bd | even | 8 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(867, [\chi])\):
|
\( T_{2} \)
|
|
\( T_{5}^{4} + 81 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} + 1 \)
|
| $5$ |
\( T^{4} + 81 \)
|
| $7$ |
\( T^{4} + 256 \)
|
| $11$ |
\( T^{4} + 81 \)
|
| $13$ |
\( (T - 1)^{4} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( (T^{2} + 1)^{2} \)
|
| $23$ |
\( T^{4} + 6561 \)
|
| $29$ |
\( T^{4} + 1296 \)
|
| $31$ |
\( T^{4} + 16 \)
|
| $37$ |
\( T^{4} + 256 \)
|
| $41$ |
\( T^{4} + 81 \)
|
| $43$ |
\( (T^{2} + 49)^{2} \)
|
| $47$ |
\( (T - 6)^{4} \)
|
| $53$ |
\( (T^{2} + 36)^{2} \)
|
| $59$ |
\( (T^{2} + 36)^{2} \)
|
| $61$ |
\( T^{4} + 4096 \)
|
| $67$ |
\( (T + 4)^{4} \)
|
| $71$ |
\( T^{4} + 20736 \)
|
| $73$ |
\( T^{4} + 16 \)
|
| $79$ |
\( T^{4} + 10000 \)
|
| $83$ |
\( (T^{2} + 36)^{2} \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( T^{4} + 65536 \)
|
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