Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1734,2,Mod(577,1734)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1734, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1734.577");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1734 = 2 \cdot 3 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1734.b (of order \(2\), degree \(1\), 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: | \(13.8460597105\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\Q(\zeta_{16})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 102) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( \zeta_{16}^{4} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{16}^{5} + \zeta_{16}^{3} \)
|
| \(\beta_{3}\) | \(=\) |
\( \zeta_{16}^{6} + \zeta_{16}^{2} \)
|
| \(\beta_{4}\) | \(=\) |
\( \zeta_{16}^{7} + \zeta_{16} \)
|
| \(\beta_{5}\) | \(=\) |
\( -\zeta_{16}^{7} + \zeta_{16} \)
|
| \(\beta_{6}\) | \(=\) |
\( -\zeta_{16}^{6} + \zeta_{16}^{2} \)
|
| \(\beta_{7}\) | \(=\) |
\( -\zeta_{16}^{5} + \zeta_{16}^{3} \)
|
| \(\zeta_{16}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} ) / 2 \)
|
| \(\zeta_{16}^{2}\) | \(=\) |
\( ( \beta_{6} + \beta_{3} ) / 2 \)
|
| \(\zeta_{16}^{3}\) | \(=\) |
\( ( \beta_{7} + \beta_{2} ) / 2 \)
|
| \(\zeta_{16}^{4}\) | \(=\) |
\( \beta_1 \)
|
| \(\zeta_{16}^{5}\) | \(=\) |
\( ( -\beta_{7} + \beta_{2} ) / 2 \)
|
| \(\zeta_{16}^{6}\) | \(=\) |
\( ( -\beta_{6} + \beta_{3} ) / 2 \)
|
| \(\zeta_{16}^{7}\) | \(=\) |
\( ( -\beta_{5} + \beta_{4} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1734\mathbb{Z}\right)^\times\).
| \(n\) | \(1157\) | \(1159\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 577.1 |
|
1.00000 | − | 1.00000i | 1.00000 | − | 1.26197i | − | 1.00000i | − | 2.94495i | 1.00000 | −1.00000 | − | 1.26197i | |||||||||||||||||||||||||||||||||||||
| 577.2 | 1.00000 | − | 1.00000i | 1.00000 | 2.43355i | − | 1.00000i | 0.116520i | 1.00000 | −1.00000 | 2.43355i | |||||||||||||||||||||||||||||||||||||||||
| 577.3 | 1.00000 | − | 1.00000i | 1.00000 | 2.64885i | − | 1.00000i | 5.10973i | 1.00000 | −1.00000 | 2.64885i | |||||||||||||||||||||||||||||||||||||||||
| 577.4 | 1.00000 | − | 1.00000i | 1.00000 | 4.17958i | − | 1.00000i | − | 2.28130i | 1.00000 | −1.00000 | 4.17958i | ||||||||||||||||||||||||||||||||||||||||
| 577.5 | 1.00000 | 1.00000i | 1.00000 | − | 4.17958i | 1.00000i | 2.28130i | 1.00000 | −1.00000 | − | 4.17958i | |||||||||||||||||||||||||||||||||||||||||
| 577.6 | 1.00000 | 1.00000i | 1.00000 | − | 2.64885i | 1.00000i | − | 5.10973i | 1.00000 | −1.00000 | − | 2.64885i | ||||||||||||||||||||||||||||||||||||||||
| 577.7 | 1.00000 | 1.00000i | 1.00000 | − | 2.43355i | 1.00000i | − | 0.116520i | 1.00000 | −1.00000 | − | 2.43355i | ||||||||||||||||||||||||||||||||||||||||
| 577.8 | 1.00000 | 1.00000i | 1.00000 | 1.26197i | 1.00000i | 2.94495i | 1.00000 | −1.00000 | 1.26197i | |||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1734.2.b.l | 8 | |
| 17.b | even | 2 | 1 | inner | 1734.2.b.l | 8 | |
| 17.c | even | 4 | 1 | 1734.2.a.t | 4 | ||
| 17.c | even | 4 | 1 | 1734.2.a.u | 4 | ||
| 17.d | even | 8 | 2 | 1734.2.f.k | 8 | ||
| 17.d | even | 8 | 2 | 1734.2.f.l | 8 | ||
| 17.e | odd | 16 | 2 | 102.2.h.b | ✓ | 8 | |
| 51.f | odd | 4 | 1 | 5202.2.a.bu | 4 | ||
| 51.f | odd | 4 | 1 | 5202.2.a.bx | 4 | ||
| 51.i | even | 16 | 2 | 306.2.l.e | 8 | ||
| 68.i | even | 16 | 2 | 816.2.bq.c | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 102.2.h.b | ✓ | 8 | 17.e | odd | 16 | 2 | |
| 306.2.l.e | 8 | 51.i | even | 16 | 2 | ||
| 816.2.bq.c | 8 | 68.i | even | 16 | 2 | ||
| 1734.2.a.t | 4 | 17.c | even | 4 | 1 | ||
| 1734.2.a.u | 4 | 17.c | even | 4 | 1 | ||
| 1734.2.b.l | 8 | 1.a | even | 1 | 1 | trivial | |
| 1734.2.b.l | 8 | 17.b | even | 2 | 1 | inner | |
| 1734.2.f.k | 8 | 17.d | even | 8 | 2 | ||
| 1734.2.f.l | 8 | 17.d | even | 8 | 2 | ||
| 5202.2.a.bu | 4 | 51.f | odd | 4 | 1 | ||
| 5202.2.a.bx | 4 | 51.f | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1734, [\chi])\):
|
\( T_{5}^{8} + 32T_{5}^{6} + 316T_{5}^{4} + 1152T_{5}^{2} + 1156 \)
|
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\( T_{7}^{8} + 40T_{7}^{6} + 408T_{7}^{4} + 1184T_{7}^{2} + 16 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{8} \)
|
| $3$ |
\( (T^{2} + 1)^{4} \)
|
| $5$ |
\( T^{8} + 32 T^{6} + \cdots + 1156 \)
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| $7$ |
\( T^{8} + 40 T^{6} + \cdots + 16 \)
|
| $11$ |
\( T^{8} + 80 T^{6} + \cdots + 73984 \)
|
| $13$ |
\( (T^{4} - 44 T^{2} + \cdots + 316)^{2} \)
|
| $17$ |
\( T^{8} \)
|
| $19$ |
\( (T^{4} + 8 T^{3} - 32 T - 8)^{2} \)
|
| $23$ |
\( (T^{2} + 2)^{4} \)
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| $29$ |
\( T^{8} + 144 T^{6} + \cdots + 1156 \)
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| $31$ |
\( T^{8} + 56 T^{6} + \cdots + 16 \)
|
| $37$ |
\( T^{8} + 112 T^{6} + \cdots + 9604 \)
|
| $41$ |
\( T^{8} + 56 T^{6} + \cdots + 3844 \)
|
| $43$ |
\( (T^{4} - 32 T^{2} + \cdots - 32)^{2} \)
|
| $47$ |
\( (T^{4} + 16 T^{3} + \cdots - 376)^{2} \)
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| $53$ |
\( (T^{4} - 16 T^{3} + \cdots + 316)^{2} \)
|
| $59$ |
\( (T^{4} - 24 T^{3} + \cdots - 392)^{2} \)
|
| $61$ |
\( T^{8} + 128 T^{6} + \cdots + 8836 \)
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| $67$ |
\( (T^{4} - 8 T^{3} + \cdots - 136)^{2} \)
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| $71$ |
\( T^{8} + 520 T^{6} + \cdots + 107412496 \)
|
| $73$ |
\( T^{8} + 264 T^{6} + \cdots + 8836 \)
|
| $79$ |
\( T^{8} + 296 T^{6} + \cdots + 6574096 \)
|
| $83$ |
\( (T^{4} - 8 T^{3} + \cdots - 2696)^{2} \)
|
| $89$ |
\( (T^{4} + 16 T^{3} + \cdots + 6596)^{2} \)
|
| $97$ |
\( T^{8} + 456 T^{6} + \cdots + 81468676 \)
|
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