Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [968,1,Mod(243,968)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(968, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("968.243");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 968 = 2^{3} \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 968.f (of order \(2\), degree \(1\), 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: | yes |
| Analytic conductor: | \(0.483094932229\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{5}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 88) |
| Projective image: | \(D_{5}\) |
| Projective field: | Galois closure of 5.1.937024.1 |
| Artin image: | $D_5$ |
| Artin field: | Galois closure of 5.1.937024.1 |
$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/968\mathbb{Z}\right)^\times\).
| \(n\) | \(485\) | \(727\) | \(849\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 243.1 |
|
1.00000 | −1.61803 | 1.00000 | 0 | −1.61803 | 0 | 1.00000 | 1.61803 | 0 | ||||||||||||||||||||||||
| 243.2 | 1.00000 | 0.618034 | 1.00000 | 0 | 0.618034 | 0 | 1.00000 | −0.618034 | 0 | |||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-2}) \) |
Twists
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 88.1.l.a | ✓ | 4 | 11.c | even | 5 | 2 | |
| 88.1.l.a | ✓ | 4 | 88.l | odd | 10 | 2 | |
| 352.1.t.a | 4 | 44.h | odd | 10 | 2 | ||
| 352.1.t.a | 4 | 88.o | even | 10 | 2 | ||
| 792.1.bu.a | 4 | 33.h | odd | 10 | 2 | ||
| 792.1.bu.a | 4 | 264.w | even | 10 | 2 | ||
| 968.1.f.a | 2 | 11.b | odd | 2 | 1 | ||
| 968.1.f.a | 2 | 88.g | even | 2 | 1 | ||
| 968.1.f.b | 2 | 1.a | even | 1 | 1 | trivial | |
| 968.1.f.b | 2 | 8.d | odd | 2 | 1 | CM | |
| 968.1.l.a | 4 | 11.c | even | 5 | 2 | ||
| 968.1.l.a | 4 | 88.l | odd | 10 | 2 | ||
| 968.1.l.b | 4 | 11.d | odd | 10 | 2 | ||
| 968.1.l.b | 4 | 88.k | even | 10 | 2 | ||
| 968.1.l.c | 4 | 11.d | odd | 10 | 2 | ||
| 968.1.l.c | 4 | 88.k | even | 10 | 2 | ||
| 2200.1.cl.a | 4 | 55.j | even | 10 | 2 | ||
| 2200.1.cl.a | 4 | 440.bh | odd | 10 | 2 | ||
| 2200.1.dd.a | 8 | 55.k | odd | 20 | 4 | ||
| 2200.1.dd.a | 8 | 440.bs | even | 20 | 4 | ||
| 2816.1.v.c | 8 | 176.v | odd | 20 | 4 | ||
| 2816.1.v.c | 8 | 176.w | even | 20 | 4 | ||
| 3168.1.ck.a | 4 | 132.o | even | 10 | 2 | ||
| 3168.1.ck.a | 4 | 264.t | odd | 10 | 2 | ||
| 3872.1.f.a | 2 | 4.b | odd | 2 | 1 | ||
| 3872.1.f.a | 2 | 8.b | even | 2 | 1 | ||
| 3872.1.f.b | 2 | 44.c | even | 2 | 1 | ||
| 3872.1.f.b | 2 | 88.b | odd | 2 | 1 | ||
| 3872.1.t.a | 4 | 44.g | even | 10 | 2 | ||
| 3872.1.t.a | 4 | 88.p | odd | 10 | 2 | ||
| 3872.1.t.b | 4 | 44.h | odd | 10 | 2 | ||
| 3872.1.t.b | 4 | 88.o | even | 10 | 2 | ||
| 3872.1.t.c | 4 | 44.g | even | 10 | 2 | ||
| 3872.1.t.c | 4 | 88.p | odd | 10 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{17}^{2} + T_{17} - 1 \)
acting on \(S_{1}^{\mathrm{new}}(968, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{2} \)
|
| $3$ |
\( T^{2} + T - 1 \)
|
| $5$ |
\( T^{2} \)
|
| $7$ |
\( T^{2} \)
|
| $11$ |
\( T^{2} \)
|
| $13$ |
\( T^{2} \)
|
| $17$ |
\( T^{2} + T - 1 \)
|
| $19$ |
\( T^{2} + T - 1 \)
|
| $23$ |
\( T^{2} \)
|
| $29$ |
\( T^{2} \)
|
| $31$ |
\( T^{2} \)
|
| $37$ |
\( T^{2} \)
|
| $41$ |
\( T^{2} + T - 1 \)
|
| $43$ |
\( T^{2} + T - 1 \)
|
| $47$ |
\( T^{2} \)
|
| $53$ |
\( T^{2} \)
|
| $59$ |
\( T^{2} + T - 1 \)
|
| $61$ |
\( T^{2} \)
|
| $67$ |
\( T^{2} + T - 1 \)
|
| $71$ |
\( T^{2} \)
|
| $73$ |
\( T^{2} + T - 1 \)
|
| $79$ |
\( T^{2} \)
|
| $83$ |
\( T^{2} + T - 1 \)
|
| $89$ |
\( T^{2} + T - 1 \)
|
| $97$ |
\( T^{2} + T - 1 \)
|
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