Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1014,3,Mod(577,1014)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1014, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 3]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1014.577");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1014 = 2 \cdot 3 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1014.f (of order \(4\), degree \(2\), 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: | \(27.6294988061\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 78) |
| 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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( \zeta_{12}^{2} + \zeta_{12} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{12}^{3} \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{12}^{2} + \zeta_{12} \)
|
| \(\zeta_{12}\) | \(=\) |
\( ( \beta_{3} + \beta_1 ) / 2 \)
|
| \(\zeta_{12}^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta_1 ) / 2 \)
|
| \(\zeta_{12}^{3}\) | \(=\) |
\( \beta_{2} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).
| \(n\) | \(677\) | \(847\) |
| \(\chi(n)\) | \(1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 577.1 |
|
1.00000 | − | 1.00000i | −1.73205 | − | 2.00000i | −0.633975 | + | 0.633975i | −1.73205 | + | 1.73205i | −8.92820 | − | 8.92820i | −2.00000 | − | 2.00000i | 3.00000 | 1.26795i | |||||||||||||||||||
| 577.2 | 1.00000 | − | 1.00000i | 1.73205 | − | 2.00000i | −2.36603 | + | 2.36603i | 1.73205 | − | 1.73205i | 4.92820 | + | 4.92820i | −2.00000 | − | 2.00000i | 3.00000 | 4.73205i | ||||||||||||||||||||
| 775.1 | 1.00000 | + | 1.00000i | −1.73205 | 2.00000i | −0.633975 | − | 0.633975i | −1.73205 | − | 1.73205i | −8.92820 | + | 8.92820i | −2.00000 | + | 2.00000i | 3.00000 | − | 1.26795i | ||||||||||||||||||||
| 775.2 | 1.00000 | + | 1.00000i | 1.73205 | 2.00000i | −2.36603 | − | 2.36603i | 1.73205 | + | 1.73205i | 4.92820 | − | 4.92820i | −2.00000 | + | 2.00000i | 3.00000 | − | 4.73205i | ||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.d | odd | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1014.3.f.g | 4 | |
| 13.b | even | 2 | 1 | 1014.3.f.b | 4 | ||
| 13.d | odd | 4 | 1 | 1014.3.f.b | 4 | ||
| 13.d | odd | 4 | 1 | inner | 1014.3.f.g | 4 | |
| 13.e | even | 6 | 1 | 78.3.l.b | ✓ | 4 | |
| 13.f | odd | 12 | 1 | 78.3.l.b | ✓ | 4 | |
| 39.h | odd | 6 | 1 | 234.3.bb.a | 4 | ||
| 39.k | even | 12 | 1 | 234.3.bb.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 78.3.l.b | ✓ | 4 | 13.e | even | 6 | 1 | |
| 78.3.l.b | ✓ | 4 | 13.f | odd | 12 | 1 | |
| 234.3.bb.a | 4 | 39.h | odd | 6 | 1 | ||
| 234.3.bb.a | 4 | 39.k | even | 12 | 1 | ||
| 1014.3.f.b | 4 | 13.b | even | 2 | 1 | ||
| 1014.3.f.b | 4 | 13.d | odd | 4 | 1 | ||
| 1014.3.f.g | 4 | 1.a | even | 1 | 1 | trivial | |
| 1014.3.f.g | 4 | 13.d | odd | 4 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1014, [\chi])\):
|
\( T_{5}^{4} + 6T_{5}^{3} + 18T_{5}^{2} + 18T_{5} + 9 \)
|
|
\( T_{7}^{4} + 8T_{7}^{3} + 32T_{7}^{2} - 704T_{7} + 7744 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 2)^{2} \)
|
| $3$ |
\( (T^{2} - 3)^{2} \)
|
| $5$ |
\( T^{4} + 6 T^{3} + \cdots + 9 \)
|
| $7$ |
\( T^{4} + 8 T^{3} + \cdots + 7744 \)
|
| $11$ |
\( T^{4} + 24 T^{3} + \cdots + 576 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} + 906 T^{2} + 184041 \)
|
| $19$ |
\( T^{4} + 8 T^{3} + \cdots + 141376 \)
|
| $23$ |
\( T^{4} + 672 T^{2} + 97344 \)
|
| $29$ |
\( (T^{2} - 60 T + 825)^{2} \)
|
| $31$ |
\( T^{4} + 88 T^{3} + \cdots + 891136 \)
|
| $37$ |
\( T^{4} - 26 T^{3} + \cdots + 2209 \)
|
| $41$ |
\( T^{4} + 90 T^{3} + \cdots + 1022121 \)
|
| $43$ |
\( T^{4} + 4704 T^{2} + 1382976 \)
|
| $47$ |
\( (T^{2} - 60 T + 1800)^{2} \)
|
| $53$ |
\( (T^{2} - 30 T - 1227)^{2} \)
|
| $59$ |
\( T^{4} - 120 T^{3} + \cdots + 5089536 \)
|
| $61$ |
\( (T^{2} + 126 T + 3537)^{2} \)
|
| $67$ |
\( T^{4} + 32 T^{3} + \cdots + 3297856 \)
|
| $71$ |
\( T^{4} + 1382976 \)
|
| $73$ |
\( T^{4} - 178 T^{3} + \cdots + 5414929 \)
|
| $79$ |
\( (T^{2} + 96 T + 576)^{2} \)
|
| $83$ |
\( T^{4} + 168 T^{3} + \cdots + 6843456 \)
|
| $89$ |
\( T^{4} - 156 T^{3} + \cdots + 19044 \)
|
| $97$ |
\( T^{4} - 188 T^{3} + \cdots + 88472836 \)
|
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