Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [570,2,Mod(83,570)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(570, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 9, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("570.83");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 570.v (of order \(12\), degree \(4\), 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: | \(4.55147291521\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{12})\) |
| 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: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 \(\left(\mathbb{Z}/570\mathbb{Z}\right)^\times\).
| \(n\) | \(191\) | \(211\) | \(457\) |
| \(\chi(n)\) | \(-1\) | \(-1 + \zeta_{24}^{4}\) | \(-\zeta_{24}^{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 83.1 |
|
−0.258819 | + | 0.965926i | 1.72474 | + | 0.158919i | −0.866025 | − | 0.500000i | −1.67303 | − | 1.48356i | −0.599900 | + | 1.62484i | −2.12132 | + | 2.12132i | 0.707107 | − | 0.707107i | 2.94949 | + | 0.548188i | 1.86603 | − | 1.23205i | ||||||||||||||||||||||||
| 83.2 | 0.258819 | − | 0.965926i | −0.724745 | + | 1.57313i | −0.866025 | − | 0.500000i | 1.67303 | + | 1.48356i | 1.33195 | + | 1.10721i | 2.12132 | − | 2.12132i | −0.707107 | + | 0.707107i | −1.94949 | − | 2.28024i | 1.86603 | − | 1.23205i | |||||||||||||||||||||||||
| 197.1 | −0.965926 | − | 0.258819i | 1.72474 | + | 0.158919i | 0.866025 | + | 0.500000i | 0.448288 | + | 2.19067i | −1.62484 | − | 0.599900i | 2.12132 | + | 2.12132i | −0.707107 | − | 0.707107i | 2.94949 | + | 0.548188i | 0.133975 | − | 2.23205i | |||||||||||||||||||||||||
| 197.2 | 0.965926 | + | 0.258819i | −0.724745 | + | 1.57313i | 0.866025 | + | 0.500000i | −0.448288 | − | 2.19067i | −1.10721 | + | 1.33195i | −2.12132 | − | 2.12132i | 0.707107 | + | 0.707107i | −1.94949 | − | 2.28024i | 0.133975 | − | 2.23205i | |||||||||||||||||||||||||
| 353.1 | −0.965926 | + | 0.258819i | 1.72474 | − | 0.158919i | 0.866025 | − | 0.500000i | 0.448288 | − | 2.19067i | −1.62484 | + | 0.599900i | 2.12132 | − | 2.12132i | −0.707107 | + | 0.707107i | 2.94949 | − | 0.548188i | 0.133975 | + | 2.23205i | |||||||||||||||||||||||||
| 353.2 | 0.965926 | − | 0.258819i | −0.724745 | − | 1.57313i | 0.866025 | − | 0.500000i | −0.448288 | + | 2.19067i | −1.10721 | − | 1.33195i | −2.12132 | + | 2.12132i | 0.707107 | − | 0.707107i | −1.94949 | + | 2.28024i | 0.133975 | + | 2.23205i | |||||||||||||||||||||||||
| 467.1 | −0.258819 | − | 0.965926i | 1.72474 | − | 0.158919i | −0.866025 | + | 0.500000i | −1.67303 | + | 1.48356i | −0.599900 | − | 1.62484i | −2.12132 | − | 2.12132i | 0.707107 | + | 0.707107i | 2.94949 | − | 0.548188i | 1.86603 | + | 1.23205i | |||||||||||||||||||||||||
| 467.2 | 0.258819 | + | 0.965926i | −0.724745 | − | 1.57313i | −0.866025 | + | 0.500000i | 1.67303 | − | 1.48356i | 1.33195 | − | 1.10721i | 2.12132 | + | 2.12132i | −0.707107 | − | 0.707107i | −1.94949 | + | 2.28024i | 1.86603 | + | 1.23205i | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 15.e | even | 4 | 1 | inner |
| 19.c | even | 3 | 1 | inner |
| 285.v | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 570.2.v.b | yes | 8 |
| 3.b | odd | 2 | 1 | 570.2.v.a | ✓ | 8 | |
| 5.c | odd | 4 | 1 | 570.2.v.a | ✓ | 8 | |
| 15.e | even | 4 | 1 | inner | 570.2.v.b | yes | 8 |
| 19.c | even | 3 | 1 | inner | 570.2.v.b | yes | 8 |
| 57.h | odd | 6 | 1 | 570.2.v.a | ✓ | 8 | |
| 95.m | odd | 12 | 1 | 570.2.v.a | ✓ | 8 | |
| 285.v | even | 12 | 1 | inner | 570.2.v.b | yes | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 570.2.v.a | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 570.2.v.a | ✓ | 8 | 5.c | odd | 4 | 1 | |
| 570.2.v.a | ✓ | 8 | 57.h | odd | 6 | 1 | |
| 570.2.v.a | ✓ | 8 | 95.m | odd | 12 | 1 | |
| 570.2.v.b | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 570.2.v.b | yes | 8 | 15.e | even | 4 | 1 | inner |
| 570.2.v.b | yes | 8 | 19.c | even | 3 | 1 | inner |
| 570.2.v.b | yes | 8 | 285.v | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(570, [\chi])\):
|
\( T_{7}^{4} + 81 \)
|
|
\( T_{17}^{8} - 12T_{17}^{7} + 72T_{17}^{6} - 528T_{17}^{5} + 2972T_{17}^{4} - 7392T_{17}^{3} + 14112T_{17}^{2} - 32928T_{17} + 38416 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - T^{4} + 1 \)
|
| $3$ |
\( (T^{4} - 2 T^{3} + T^{2} + \cdots + 9)^{2} \)
|
| $5$ |
\( T^{8} + 8 T^{6} + \cdots + 625 \)
|
| $7$ |
\( (T^{4} + 81)^{2} \)
|
| $11$ |
\( (T^{4} + 34 T^{2} + 1)^{2} \)
|
| $13$ |
\( (T^{4} - 4 T^{3} + 8 T^{2} + \cdots + 64)^{2} \)
|
| $17$ |
\( T^{8} - 12 T^{7} + \cdots + 38416 \)
|
| $19$ |
\( (T^{4} - 37 T^{2} + 361)^{2} \)
|
| $23$ |
\( T^{8} - 81T^{4} + 6561 \)
|
| $29$ |
\( (T^{4} + 12 T^{3} + \cdots + 784)^{2} \)
|
| $31$ |
\( (T^{2} - 4 T - 14)^{4} \)
|
| $37$ |
\( (T^{4} + 81)^{2} \)
|
| $41$ |
\( (T^{4} - 9 T^{2} + 81)^{2} \)
|
| $43$ |
\( (T^{4} + 6 T^{3} + \cdots + 324)^{2} \)
|
| $47$ |
\( T^{8} - 24 T^{7} + \cdots + 21381376 \)
|
| $53$ |
\( T^{8} - 24 T^{7} + \cdots + 25411681 \)
|
| $59$ |
\( (T^{4} + 32 T^{2} + 1024)^{2} \)
|
| $61$ |
\( (T^{4} - 4 T^{3} + \cdots + 196)^{2} \)
|
| $67$ |
\( (T^{4} - 8 T^{3} + \cdots + 1024)^{2} \)
|
| $71$ |
\( T^{8} - 292 T^{6} + \cdots + 406586896 \)
|
| $73$ |
\( T^{8} + 12 T^{7} + \cdots + 252047376 \)
|
| $79$ |
\( T^{8} - 152 T^{6} + \cdots + 21381376 \)
|
| $83$ |
\( (T^{2} - 6 T + 18)^{4} \)
|
| $89$ |
\( (T^{4} + 18 T^{3} + \cdots + 5329)^{2} \)
|
| $97$ |
\( T^{8} + 24 T^{7} + \cdots + 1679616 \)
|
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