Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [616,2,Mod(177,616)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(616, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("616.177");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 616 = 2^{3} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 616.q (of order \(3\), 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: | \(4.91878476451\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 9x^{6} - 2x^{5} + 66x^{4} - 9x^{3} + 136x^{2} + 15x + 225 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{8} + 9x^{6} - 2x^{5} + 66x^{4} - 9x^{3} + 136x^{2} + 15x + 225 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 27\nu^{7} - 330\nu^{6} + 198\nu^{5} - 2474\nu^{4} + 2112\nu^{3} - 21978\nu^{2} + 3122\nu - 44550 ) / 36705 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 9\nu^{7} - 110\nu^{6} + 66\nu^{5} - 9\nu^{4} + 704\nu^{3} + 15\nu^{2} + 225\nu + 14514 ) / 7341 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 18\nu^{7} - 220\nu^{6} + 132\nu^{5} - 2465\nu^{4} + 1408\nu^{3} - 14652\nu^{2} + 2897\nu - 29700 ) / 7341 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 424 \nu^{7} - 1615 \nu^{6} - 11266 \nu^{5} - 6917 \nu^{4} - 63074 \nu^{3} - 21914 \nu^{2} + \cdots + 112320 ) / 110115 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 66\nu^{7} + 9\nu^{6} + 484\nu^{5} - 66\nu^{4} + 4347\nu^{3} + 110\nu^{2} + 1650\nu + 1215 ) / 7341 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( - 1993 \nu^{7} - 2830 \nu^{6} - 10537 \nu^{5} - 12689 \nu^{4} - 55298 \nu^{3} - 102833 \nu^{2} + \cdots - 51705 ) / 110115 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} - 5\beta_{2} - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{7} + 5\beta_{6} + \beta_{5} - \beta_{4} - 2\beta_{3} - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -9\beta_{4} + 30\beta_{2} + \beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( -9\beta_{7} - 30\beta_{6} - 18\beta_{5} + 10\beta_{4} + 19\beta_{3} - 14\beta_{2} - 30\beta _1 + 5 \)
|
| \(\nu^{6}\) | \(=\) |
\( 2\beta_{7} + 14\beta_{6} + \beta_{5} - \beta_{4} - 68\beta_{3} + 128 \)
|
| \(\nu^{7}\) | \(=\) |
\( -66\beta_{7} + 66\beta_{5} - 18\beta_{4} + 141\beta_{2} + 196\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/616\mathbb{Z}\right)^\times\).
| \(n\) | \(57\) | \(309\) | \(353\) | \(463\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1 - \beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 177.1 |
|
0 | −1.03812 | − | 1.79807i | 0 | −0.193499 | + | 0.335150i | 0 | 1.09525 | + | 2.40841i | 0 | −0.655380 | + | 1.13515i | 0 | ||||||||||||||||||||||||||||||||||
| 177.2 | 0 | −0.703938 | − | 1.21926i | 0 | 1.30500 | − | 2.26033i | 0 | −2.64229 | − | 0.135337i | 0 | 0.508942 | − | 0.881513i | 0 | |||||||||||||||||||||||||||||||||||
| 177.3 | 0 | 1.08177 | + | 1.87368i | 0 | 1.74132 | − | 3.01606i | 0 | 2.52609 | + | 0.786673i | 0 | −0.840445 | + | 1.45569i | 0 | |||||||||||||||||||||||||||||||||||
| 177.4 | 0 | 1.66029 | + | 2.87570i | 0 | −0.852828 | + | 1.47714i | 0 | −1.47905 | − | 2.19372i | 0 | −4.01312 | + | 6.95092i | 0 | |||||||||||||||||||||||||||||||||||
| 529.1 | 0 | −1.03812 | + | 1.79807i | 0 | −0.193499 | − | 0.335150i | 0 | 1.09525 | − | 2.40841i | 0 | −0.655380 | − | 1.13515i | 0 | |||||||||||||||||||||||||||||||||||
| 529.2 | 0 | −0.703938 | + | 1.21926i | 0 | 1.30500 | + | 2.26033i | 0 | −2.64229 | + | 0.135337i | 0 | 0.508942 | + | 0.881513i | 0 | |||||||||||||||||||||||||||||||||||
| 529.3 | 0 | 1.08177 | − | 1.87368i | 0 | 1.74132 | + | 3.01606i | 0 | 2.52609 | − | 0.786673i | 0 | −0.840445 | − | 1.45569i | 0 | |||||||||||||||||||||||||||||||||||
| 529.4 | 0 | 1.66029 | − | 2.87570i | 0 | −0.852828 | − | 1.47714i | 0 | −1.47905 | + | 2.19372i | 0 | −4.01312 | − | 6.95092i | 0 | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 616.2.q.d | ✓ | 8 |
| 4.b | odd | 2 | 1 | 1232.2.q.n | 8 | ||
| 7.c | even | 3 | 1 | inner | 616.2.q.d | ✓ | 8 |
| 7.c | even | 3 | 1 | 4312.2.a.y | 4 | ||
| 7.d | odd | 6 | 1 | 4312.2.a.bd | 4 | ||
| 28.f | even | 6 | 1 | 8624.2.a.cr | 4 | ||
| 28.g | odd | 6 | 1 | 1232.2.q.n | 8 | ||
| 28.g | odd | 6 | 1 | 8624.2.a.cz | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 616.2.q.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 616.2.q.d | ✓ | 8 | 7.c | even | 3 | 1 | inner |
| 1232.2.q.n | 8 | 4.b | odd | 2 | 1 | ||
| 1232.2.q.n | 8 | 28.g | odd | 6 | 1 | ||
| 4312.2.a.y | 4 | 7.c | even | 3 | 1 | ||
| 4312.2.a.bd | 4 | 7.d | odd | 6 | 1 | ||
| 8624.2.a.cr | 4 | 28.f | even | 6 | 1 | ||
| 8624.2.a.cz | 4 | 28.g | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{8} - 2T_{3}^{7} + 13T_{3}^{6} + 78T_{3}^{4} + 3T_{3}^{3} + 270T_{3}^{2} + 189T_{3} + 441 \)
acting on \(S_{2}^{\mathrm{new}}(616, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - 2 T^{7} + \cdots + 441 \)
|
| $5$ |
\( T^{8} - 4 T^{7} + \cdots + 36 \)
|
| $7$ |
\( T^{8} + T^{7} + \cdots + 2401 \)
|
| $11$ |
\( (T^{2} - T + 1)^{4} \)
|
| $13$ |
\( (T^{4} + 2 T^{3} - 9 T^{2} + \cdots + 21)^{2} \)
|
| $17$ |
\( T^{8} + 3 T^{7} + \cdots + 8100 \)
|
| $19$ |
\( T^{8} - 13 T^{7} + \cdots + 256 \)
|
| $23$ |
\( T^{8} + 10 T^{7} + \cdots + 3136 \)
|
| $29$ |
\( (T^{4} - 4 T^{3} + \cdots + 211)^{2} \)
|
| $31$ |
\( T^{8} - T^{7} + \cdots + 25600 \)
|
| $37$ |
\( T^{8} - 8 T^{7} + \cdots + 16384 \)
|
| $41$ |
\( (T^{4} - 9 T^{3} + \cdots + 492)^{2} \)
|
| $43$ |
\( (T^{4} + 14 T^{3} + \cdots + 48)^{2} \)
|
| $47$ |
\( T^{8} - 11 T^{7} + \cdots + 11222500 \)
|
| $53$ |
\( T^{8} - T^{7} + \cdots + 62500 \)
|
| $59$ |
\( T^{8} - 33 T^{7} + \cdots + 164025 \)
|
| $61$ |
\( T^{8} - 9 T^{7} + \cdots + 34574400 \)
|
| $67$ |
\( T^{8} + 25 T^{7} + \cdots + 26460736 \)
|
| $71$ |
\( (T^{4} - 3 T^{3} - 48 T^{2} + \cdots - 90)^{2} \)
|
| $73$ |
\( T^{8} + 213 T^{6} + \cdots + 19289664 \)
|
| $79$ |
\( T^{8} + 28 T^{7} + \cdots + 12201049 \)
|
| $83$ |
\( (T^{4} - 5 T^{3} + \cdots + 3242)^{2} \)
|
| $89$ |
\( T^{8} + 3 T^{7} + \cdots + 10381284 \)
|
| $97$ |
\( (T^{4} - 20 T^{3} + \cdots + 1687)^{2} \)
|
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