Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [810,3,Mod(269,810)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(810, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([1, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("810.269");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 810 = 2 \cdot 3^{4} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 810.j (of order \(6\), 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: | \(22.0709014132\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\zeta_{24})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 270) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/810\mathbb{Z}\right)^\times\).
| \(n\) | \(487\) | \(731\) |
| \(\chi(n)\) | \(-1\) | \(1 - \zeta_{24}^{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 269.1 |
|
−0.707107 | + | 1.22474i | 0 | −1.00000 | − | 1.73205i | −4.82963 | − | 1.29410i | 0 | 4.33013 | + | 2.50000i | 2.82843 | 0 | 5.00000 | − | 5.00000i | ||||||||||||||||||||||||||||||||
| 269.2 | −0.707107 | + | 1.22474i | 0 | −1.00000 | − | 1.73205i | 1.29410 | − | 4.82963i | 0 | −4.33013 | − | 2.50000i | 2.82843 | 0 | 5.00000 | + | 5.00000i | |||||||||||||||||||||||||||||||||
| 269.3 | 0.707107 | − | 1.22474i | 0 | −1.00000 | − | 1.73205i | −1.29410 | + | 4.82963i | 0 | −4.33013 | − | 2.50000i | −2.82843 | 0 | 5.00000 | + | 5.00000i | |||||||||||||||||||||||||||||||||
| 269.4 | 0.707107 | − | 1.22474i | 0 | −1.00000 | − | 1.73205i | 4.82963 | + | 1.29410i | 0 | 4.33013 | + | 2.50000i | −2.82843 | 0 | 5.00000 | − | 5.00000i | |||||||||||||||||||||||||||||||||
| 539.1 | −0.707107 | − | 1.22474i | 0 | −1.00000 | + | 1.73205i | −4.82963 | + | 1.29410i | 0 | 4.33013 | − | 2.50000i | 2.82843 | 0 | 5.00000 | + | 5.00000i | |||||||||||||||||||||||||||||||||
| 539.2 | −0.707107 | − | 1.22474i | 0 | −1.00000 | + | 1.73205i | 1.29410 | + | 4.82963i | 0 | −4.33013 | + | 2.50000i | 2.82843 | 0 | 5.00000 | − | 5.00000i | |||||||||||||||||||||||||||||||||
| 539.3 | 0.707107 | + | 1.22474i | 0 | −1.00000 | + | 1.73205i | −1.29410 | − | 4.82963i | 0 | −4.33013 | + | 2.50000i | −2.82843 | 0 | 5.00000 | − | 5.00000i | |||||||||||||||||||||||||||||||||
| 539.4 | 0.707107 | + | 1.22474i | 0 | −1.00000 | + | 1.73205i | 4.82963 | − | 1.29410i | 0 | 4.33013 | − | 2.50000i | −2.82843 | 0 | 5.00000 | + | 5.00000i | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
| 9.c | even | 3 | 1 | inner |
| 9.d | odd | 6 | 1 | inner |
| 15.d | odd | 2 | 1 | inner |
| 45.h | odd | 6 | 1 | inner |
| 45.j | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 810.3.j.e | 8 | |
| 3.b | odd | 2 | 1 | inner | 810.3.j.e | 8 | |
| 5.b | even | 2 | 1 | inner | 810.3.j.e | 8 | |
| 9.c | even | 3 | 1 | 270.3.b.c | ✓ | 4 | |
| 9.c | even | 3 | 1 | inner | 810.3.j.e | 8 | |
| 9.d | odd | 6 | 1 | 270.3.b.c | ✓ | 4 | |
| 9.d | odd | 6 | 1 | inner | 810.3.j.e | 8 | |
| 15.d | odd | 2 | 1 | inner | 810.3.j.e | 8 | |
| 36.f | odd | 6 | 1 | 2160.3.c.i | 4 | ||
| 36.h | even | 6 | 1 | 2160.3.c.i | 4 | ||
| 45.h | odd | 6 | 1 | 270.3.b.c | ✓ | 4 | |
| 45.h | odd | 6 | 1 | inner | 810.3.j.e | 8 | |
| 45.j | even | 6 | 1 | 270.3.b.c | ✓ | 4 | |
| 45.j | even | 6 | 1 | inner | 810.3.j.e | 8 | |
| 45.k | odd | 12 | 1 | 1350.3.d.f | 2 | ||
| 45.k | odd | 12 | 1 | 1350.3.d.g | 2 | ||
| 45.l | even | 12 | 1 | 1350.3.d.f | 2 | ||
| 45.l | even | 12 | 1 | 1350.3.d.g | 2 | ||
| 180.n | even | 6 | 1 | 2160.3.c.i | 4 | ||
| 180.p | odd | 6 | 1 | 2160.3.c.i | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 270.3.b.c | ✓ | 4 | 9.c | even | 3 | 1 | |
| 270.3.b.c | ✓ | 4 | 9.d | odd | 6 | 1 | |
| 270.3.b.c | ✓ | 4 | 45.h | odd | 6 | 1 | |
| 270.3.b.c | ✓ | 4 | 45.j | even | 6 | 1 | |
| 810.3.j.e | 8 | 1.a | even | 1 | 1 | trivial | |
| 810.3.j.e | 8 | 3.b | odd | 2 | 1 | inner | |
| 810.3.j.e | 8 | 5.b | even | 2 | 1 | inner | |
| 810.3.j.e | 8 | 9.c | even | 3 | 1 | inner | |
| 810.3.j.e | 8 | 9.d | odd | 6 | 1 | inner | |
| 810.3.j.e | 8 | 15.d | odd | 2 | 1 | inner | |
| 810.3.j.e | 8 | 45.h | odd | 6 | 1 | inner | |
| 810.3.j.e | 8 | 45.j | even | 6 | 1 | inner | |
| 1350.3.d.f | 2 | 45.k | odd | 12 | 1 | ||
| 1350.3.d.f | 2 | 45.l | even | 12 | 1 | ||
| 1350.3.d.g | 2 | 45.k | odd | 12 | 1 | ||
| 1350.3.d.g | 2 | 45.l | even | 12 | 1 | ||
| 2160.3.c.i | 4 | 36.f | odd | 6 | 1 | ||
| 2160.3.c.i | 4 | 36.h | even | 6 | 1 | ||
| 2160.3.c.i | 4 | 180.n | even | 6 | 1 | ||
| 2160.3.c.i | 4 | 180.p | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(810, [\chi])\):
|
\( T_{7}^{4} - 25T_{7}^{2} + 625 \)
|
|
\( T_{17}^{2} - 128 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 2 T^{2} + 4)^{2} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} - 625 T^{4} + 390625 \)
|
| $7$ |
\( (T^{4} - 25 T^{2} + 625)^{2} \)
|
| $11$ |
\( (T^{4} - 2 T^{2} + 4)^{2} \)
|
| $13$ |
\( (T^{4} - 81 T^{2} + 6561)^{2} \)
|
| $17$ |
\( (T^{2} - 128)^{4} \)
|
| $19$ |
\( (T - 21)^{8} \)
|
| $23$ |
\( (T^{4} + 2 T^{2} + 4)^{2} \)
|
| $29$ |
\( (T^{4} - 1458 T^{2} + 2125764)^{2} \)
|
| $31$ |
\( (T^{2} + 40 T + 1600)^{4} \)
|
| $37$ |
\( (T^{2} + 625)^{4} \)
|
| $41$ |
\( (T^{4} - 2738 T^{2} + 7496644)^{2} \)
|
| $43$ |
\( (T^{4} - 4096 T^{2} + 16777216)^{2} \)
|
| $47$ |
\( (T^{4} + 512 T^{2} + 262144)^{2} \)
|
| $53$ |
\( (T^{2} - 5202)^{4} \)
|
| $59$ |
\( (T^{4} - 8192 T^{2} + 67108864)^{2} \)
|
| $61$ |
\( (T^{2} - 97 T + 9409)^{4} \)
|
| $67$ |
\( (T^{4} - 17161 T^{2} + 294499921)^{2} \)
|
| $71$ |
\( (T^{2} + 7938)^{4} \)
|
| $73$ |
\( (T^{2} + 289)^{4} \)
|
| $79$ |
\( (T^{2} - 117 T + 13689)^{4} \)
|
| $83$ |
\( (T^{4} + 3362 T^{2} + 11303044)^{2} \)
|
| $89$ |
\( (T^{2} + 21632)^{4} \)
|
| $97$ |
\( (T^{4} - 1681 T^{2} + 2825761)^{2} \)
|
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