Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [56,3,Mod(43,56)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(56, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("56.43");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 56 = 2^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 56.g (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: | no |
| Analytic conductor: | \(1.52588948042\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.292213762624.3 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} - 2x^{6} - 2x^{5} + 24x^{4} - 8x^{3} - 32x^{2} - 64x + 256 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{5} \) |
| Twist minimal: | yes |
| 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 in terms of a root \(\nu\) of
\( x^{8} - x^{7} - 2x^{6} - 2x^{5} + 24x^{4} - 8x^{3} - 32x^{2} - 64x + 256 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{7} + 3\nu^{6} + 12\nu^{5} - 6\nu^{4} - 12\nu^{3} + 96\nu ) / 128 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} + \nu^{6} + 8\nu^{5} - 10\nu^{4} + 36\nu^{3} + 112\nu^{2} + 416\nu - 256 ) / 128 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{7} - 3\nu^{6} + 4\nu^{5} + 22\nu^{4} + 12\nu^{3} - 32\nu^{2} - 32\nu + 128 ) / 128 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} + 5\nu^{6} - 4\nu^{5} + 6\nu^{4} - 4\nu^{3} + 32\nu^{2} + 32\nu ) / 128 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} + \nu^{6} + 2\nu^{5} + 2\nu^{4} - 24\nu^{3} + 8\nu^{2} + 32\nu + 64 ) / 64 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -\nu^{7} - 3\nu^{6} + 6\nu^{5} + 10\nu^{4} - 16\nu^{3} - 88\nu^{2} + 64\nu + 128 ) / 64 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -5\nu^{7} + 3\nu^{6} + 18\nu^{4} - 4\nu^{3} + 48\nu^{2} - 32\nu + 128 ) / 128 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{6} + \beta_{4} - \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{6} + 4\beta_{5} - 3\beta_{4} + 3\beta_{3} + \beta_{2} - \beta _1 + 1 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4\beta_{7} + \beta_{6} - 12\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + 3\beta _1 + 9 ) / 4 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 4\beta_{7} + \beta_{6} + 4\beta_{5} + 9\beta_{4} + 15\beta_{3} + \beta_{2} - 5\beta _1 - 23 ) / 4 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -4\beta_{7} - 11\beta_{6} + 4\beta_{5} - 19\beta_{4} + 27\beta_{3} - 3\beta_{2} + 39\beta _1 - 11 ) / 4 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 12\beta_{7} + 9\beta_{6} - 44\beta_{5} + 81\beta_{4} - 25\beta_{3} - 15\beta_{2} + 51\beta _1 + 9 ) / 4 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -84\beta_{7} - 27\beta_{6} + 36\beta_{5} + 45\beta_{4} + 75\beta_{3} - 3\beta_{2} + 7\beta _1 + 21 ) / 4 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/56\mathbb{Z}\right)^\times\).
| \(n\) | \(15\) | \(17\) | \(29\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 43.1 |
|
−1.67467 | − | 1.09337i | 4.56747 | 1.60906 | + | 3.66209i | 5.73252i | −7.64902 | − | 4.99396i | − | 2.64575i | 1.30939 | − | 7.89212i | 11.8618 | 6.26779 | − | 9.60010i | |||||||||||||||||||||||||||||||
| 43.2 | −1.67467 | + | 1.09337i | 4.56747 | 1.60906 | − | 3.66209i | − | 5.73252i | −7.64902 | + | 4.99396i | 2.64575i | 1.30939 | + | 7.89212i | 11.8618 | 6.26779 | + | 9.60010i | ||||||||||||||||||||||||||||||||
| 43.3 | −1.05468 | − | 1.69931i | −3.44128 | −1.77532 | + | 3.58445i | 4.88287i | 3.62943 | + | 5.84780i | 2.64575i | 7.96347 | − | 0.763618i | 2.84239 | 8.29751 | − | 5.14984i | |||||||||||||||||||||||||||||||||
| 43.4 | −1.05468 | + | 1.69931i | −3.44128 | −1.77532 | − | 3.58445i | − | 4.88287i | 3.62943 | − | 5.84780i | − | 2.64575i | 7.96347 | + | 0.763618i | 2.84239 | 8.29751 | + | 5.14984i | |||||||||||||||||||||||||||||||
| 43.5 | 1.37098 | − | 1.45617i | −5.22363 | −0.240837 | − | 3.99274i | − | 6.26788i | −7.16148 | + | 7.60647i | 2.64575i | −6.14428 | − | 5.12327i | 18.2863 | −9.12707 | − | 8.59313i | ||||||||||||||||||||||||||||||||
| 43.6 | 1.37098 | + | 1.45617i | −5.22363 | −0.240837 | + | 3.99274i | 6.26788i | −7.16148 | − | 7.60647i | − | 2.64575i | −6.14428 | + | 5.12327i | 18.2863 | −9.12707 | + | 8.59313i | ||||||||||||||||||||||||||||||||
| 43.7 | 1.85837 | − | 0.739226i | 0.0974366 | 2.90709 | − | 2.74751i | 3.46547i | 0.181073 | − | 0.0720276i | − | 2.64575i | 3.37142 | − | 7.25490i | −8.99051 | 2.56177 | + | 6.44013i | ||||||||||||||||||||||||||||||||
| 43.8 | 1.85837 | + | 0.739226i | 0.0974366 | 2.90709 | + | 2.74751i | − | 3.46547i | 0.181073 | + | 0.0720276i | 2.64575i | 3.37142 | + | 7.25490i | −8.99051 | 2.56177 | − | 6.44013i | ||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 56.3.g.b | ✓ | 8 |
| 3.b | odd | 2 | 1 | 504.3.g.b | 8 | ||
| 4.b | odd | 2 | 1 | 224.3.g.b | 8 | ||
| 7.b | odd | 2 | 1 | 392.3.g.m | 8 | ||
| 7.c | even | 3 | 2 | 392.3.k.o | 16 | ||
| 7.d | odd | 6 | 2 | 392.3.k.n | 16 | ||
| 8.b | even | 2 | 1 | 224.3.g.b | 8 | ||
| 8.d | odd | 2 | 1 | inner | 56.3.g.b | ✓ | 8 |
| 12.b | even | 2 | 1 | 2016.3.g.b | 8 | ||
| 16.e | even | 4 | 2 | 1792.3.d.j | 16 | ||
| 16.f | odd | 4 | 2 | 1792.3.d.j | 16 | ||
| 24.f | even | 2 | 1 | 504.3.g.b | 8 | ||
| 24.h | odd | 2 | 1 | 2016.3.g.b | 8 | ||
| 28.d | even | 2 | 1 | 1568.3.g.m | 8 | ||
| 56.e | even | 2 | 1 | 392.3.g.m | 8 | ||
| 56.h | odd | 2 | 1 | 1568.3.g.m | 8 | ||
| 56.k | odd | 6 | 2 | 392.3.k.o | 16 | ||
| 56.m | even | 6 | 2 | 392.3.k.n | 16 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 56.3.g.b | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 56.3.g.b | ✓ | 8 | 8.d | odd | 2 | 1 | inner |
| 224.3.g.b | 8 | 4.b | odd | 2 | 1 | ||
| 224.3.g.b | 8 | 8.b | even | 2 | 1 | ||
| 392.3.g.m | 8 | 7.b | odd | 2 | 1 | ||
| 392.3.g.m | 8 | 56.e | even | 2 | 1 | ||
| 392.3.k.n | 16 | 7.d | odd | 6 | 2 | ||
| 392.3.k.n | 16 | 56.m | even | 6 | 2 | ||
| 392.3.k.o | 16 | 7.c | even | 3 | 2 | ||
| 392.3.k.o | 16 | 56.k | odd | 6 | 2 | ||
| 504.3.g.b | 8 | 3.b | odd | 2 | 1 | ||
| 504.3.g.b | 8 | 24.f | even | 2 | 1 | ||
| 1568.3.g.m | 8 | 28.d | even | 2 | 1 | ||
| 1568.3.g.m | 8 | 56.h | odd | 2 | 1 | ||
| 1792.3.d.j | 16 | 16.e | even | 4 | 2 | ||
| 1792.3.d.j | 16 | 16.f | odd | 4 | 2 | ||
| 2016.3.g.b | 8 | 12.b | even | 2 | 1 | ||
| 2016.3.g.b | 8 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} + 4T_{3}^{3} - 22T_{3}^{2} - 80T_{3} + 8 \)
acting on \(S_{3}^{\mathrm{new}}(56, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - T^{7} + \cdots + 256 \)
|
| $3$ |
\( (T^{4} + 4 T^{3} - 22 T^{2} + \cdots + 8)^{2} \)
|
| $5$ |
\( T^{8} + 108 T^{6} + \cdots + 369664 \)
|
| $7$ |
\( (T^{2} + 7)^{4} \)
|
| $11$ |
\( (T^{4} + 16 T^{3} + \cdots - 864)^{2} \)
|
| $13$ |
\( T^{8} + 908 T^{6} + \cdots + 133448704 \)
|
| $17$ |
\( (T^{4} + 40 T^{3} + \cdots - 752)^{2} \)
|
| $19$ |
\( (T^{4} - 28 T^{3} + \cdots + 1288)^{2} \)
|
| $23$ |
\( T^{8} + \cdots + 15607005184 \)
|
| $29$ |
\( T^{8} + \cdots + 13389266944 \)
|
| $31$ |
\( T^{8} + \cdots + 61917364224 \)
|
| $37$ |
\( T^{8} + 7440 T^{6} + \cdots + 554696704 \)
|
| $41$ |
\( (T^{4} - 64 T^{3} + \cdots + 837776)^{2} \)
|
| $43$ |
\( (T^{4} - 2716 T^{2} + \cdots - 217952)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 7542537191424 \)
|
| $53$ |
\( T^{8} + 3552 T^{6} + \cdots + 16777216 \)
|
| $59$ |
\( (T^{4} - 52 T^{3} + \cdots - 11252856)^{2} \)
|
| $61$ |
\( T^{8} + \cdots + 3223777158144 \)
|
| $67$ |
\( (T^{4} - 152 T^{3} + \cdots - 791808)^{2} \)
|
| $71$ |
\( T^{8} + \cdots + 221437256269824 \)
|
| $73$ |
\( (T^{4} + 56 T^{3} + \cdots - 1726704)^{2} \)
|
| $79$ |
\( T^{8} + \cdots + 89369947930624 \)
|
| $83$ |
\( (T^{4} - 36 T^{3} + \cdots + 3180872)^{2} \)
|
| $89$ |
\( (T^{4} + 256 T^{3} + \cdots - 618736)^{2} \)
|
| $97$ |
\( (T^{4} - 32 T^{3} + \cdots + 9539216)^{2} \)
|
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