Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [618,2,Mod(355,618)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(618, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("618.355");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 618 = 2 \cdot 3 \cdot 103 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 618.e (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: | \(4.93475484492\) |
| 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: |
\( x^{8} - x^{7} + 7x^{6} + 4x^{5} + 33x^{4} + 2x^{3} + 25x^{2} + 4x + 16 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{8} - x^{7} + 7x^{6} + 4x^{5} + 33x^{4} + 2x^{3} + 25x^{2} + 4x + 16 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 57\nu^{7} + 2871\nu^{6} - 2213\nu^{5} + 18284\nu^{4} + 18961\nu^{3} + 74646\nu^{2} + 15577\nu + 9988 ) / 20516 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -277\nu^{7} + 1165\nu^{6} - 2563\nu^{5} + 4368\nu^{4} - 3961\nu^{3} + 30290\nu^{2} - 2633\nu + 22368 ) / 20516 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 567\nu^{7} - 2755\nu^{6} + 6061\nu^{5} - 14644\nu^{4} + 9367\nu^{3} - 71630\nu^{2} - 13497\nu - 52896 ) / 20516 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 185\nu^{7} - 130\nu^{6} + 286\nu^{5} + 1934\nu^{4} + 442\nu^{3} - 3380\nu^{2} - 21609\nu - 2496 ) / 5129 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 222\nu^{7} - 156\nu^{6} + 1369\nu^{5} + 1295\nu^{4} + 7711\nu^{3} + 1073\nu^{2} + 740\nu + 1108 ) / 5129 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -1176\nu^{7} + 965\nu^{6} - 7252\nu^{5} - 6860\nu^{4} - 34055\nu^{3} - 5684\nu^{2} - 3920\nu - 7117 ) / 5129 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{6} + 3\beta_{3} - \beta_{2} + \beta _1 - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + 6\beta_{6} - \beta_{4} - \beta_{2} - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -\beta_{5} - 7\beta_{4} - 17\beta_{3} - 11\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( -7\beta_{7} - 42\beta_{6} - 7\beta_{5} - 32\beta_{3} + 12\beta_{2} - 42\beta _1 + 25 \)
|
| \(\nu^{6}\) | \(=\) |
\( -12\beta_{7} - 98\beta_{6} + 49\beta_{4} + 49\beta_{2} + 107 \)
|
| \(\nu^{7}\) | \(=\) |
\( 49\beta_{5} + 110\beta_{4} + 282\beta_{3} + 315\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/618\mathbb{Z}\right)^\times\).
| \(n\) | \(211\) | \(413\) |
| \(\chi(n)\) | \(-\beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 355.1 |
|
−0.500000 | − | 0.866025i | −1.00000 | −0.500000 | + | 0.866025i | −0.411201 | + | 0.712222i | 0.500000 | + | 0.866025i | 0.542009 | − | 0.938788i | 1.00000 | 1.00000 | 0.822403 | ||||||||||||||||||||||||||||||||
| 355.2 | −0.500000 | − | 0.866025i | −1.00000 | −0.500000 | + | 0.866025i | 0.551769 | − | 0.955691i | 0.500000 | + | 0.866025i | −0.288120 | + | 0.499038i | 1.00000 | 1.00000 | −1.10354 | |||||||||||||||||||||||||||||||||
| 355.3 | −0.500000 | − | 0.866025i | −1.00000 | −0.500000 | + | 0.866025i | 1.42129 | − | 2.46174i | 0.500000 | + | 0.866025i | 1.75484 | − | 3.03948i | 1.00000 | 1.00000 | −2.84258 | |||||||||||||||||||||||||||||||||
| 355.4 | −0.500000 | − | 0.866025i | −1.00000 | −0.500000 | + | 0.866025i | 1.93814 | − | 3.35697i | 0.500000 | + | 0.866025i | −2.50873 | + | 4.34525i | 1.00000 | 1.00000 | −3.87629 | |||||||||||||||||||||||||||||||||
| 571.1 | −0.500000 | + | 0.866025i | −1.00000 | −0.500000 | − | 0.866025i | −0.411201 | − | 0.712222i | 0.500000 | − | 0.866025i | 0.542009 | + | 0.938788i | 1.00000 | 1.00000 | 0.822403 | |||||||||||||||||||||||||||||||||
| 571.2 | −0.500000 | + | 0.866025i | −1.00000 | −0.500000 | − | 0.866025i | 0.551769 | + | 0.955691i | 0.500000 | − | 0.866025i | −0.288120 | − | 0.499038i | 1.00000 | 1.00000 | −1.10354 | |||||||||||||||||||||||||||||||||
| 571.3 | −0.500000 | + | 0.866025i | −1.00000 | −0.500000 | − | 0.866025i | 1.42129 | + | 2.46174i | 0.500000 | − | 0.866025i | 1.75484 | + | 3.03948i | 1.00000 | 1.00000 | −2.84258 | |||||||||||||||||||||||||||||||||
| 571.4 | −0.500000 | + | 0.866025i | −1.00000 | −0.500000 | − | 0.866025i | 1.93814 | + | 3.35697i | 0.500000 | − | 0.866025i | −2.50873 | − | 4.34525i | 1.00000 | 1.00000 | −3.87629 | |||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 103.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 618.2.e.d | ✓ | 8 |
| 3.b | odd | 2 | 1 | 1854.2.f.i | 8 | ||
| 103.c | even | 3 | 1 | inner | 618.2.e.d | ✓ | 8 |
| 309.h | odd | 6 | 1 | 1854.2.f.i | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 618.2.e.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 618.2.e.d | ✓ | 8 | 103.c | even | 3 | 1 | inner |
| 1854.2.f.i | 8 | 3.b | odd | 2 | 1 | ||
| 1854.2.f.i | 8 | 309.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} - 7T_{5}^{7} + 37T_{5}^{6} - 90T_{5}^{5} + 175T_{5}^{4} - 104T_{5}^{3} + 129T_{5}^{2} - 30T_{5} + 100 \)
acting on \(S_{2}^{\mathrm{new}}(618, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + T + 1)^{4} \)
|
| $3$ |
\( (T + 1)^{8} \)
|
| $5$ |
\( T^{8} - 7 T^{7} + \cdots + 100 \)
|
| $7$ |
\( T^{8} + T^{7} + \cdots + 121 \)
|
| $11$ |
\( T^{8} + T^{7} + \cdots + 49 \)
|
| $13$ |
\( (T^{4} - T^{3} - 17 T^{2} + \cdots + 64)^{2} \)
|
| $17$ |
\( T^{8} - T^{7} + \cdots + 6724 \)
|
| $19$ |
\( T^{8} - 3 T^{7} + \cdots + 2116 \)
|
| $23$ |
\( (T^{4} - T^{3} - 96 T^{2} + \cdots + 1738)^{2} \)
|
| $29$ |
\( T^{8} - 7 T^{7} + \cdots + 61009 \)
|
| $31$ |
\( (T^{4} - 11 T^{3} + \cdots - 640)^{2} \)
|
| $37$ |
\( (T^{4} - 2 T^{3} + \cdots + 352)^{2} \)
|
| $41$ |
\( T^{8} + 15 T^{7} + \cdots + 10956100 \)
|
| $43$ |
\( T^{8} + T^{7} + \cdots + 80656 \)
|
| $47$ |
\( T^{8} + 9 T^{7} + \cdots + 62500 \)
|
| $53$ |
\( T^{8} + 13 T^{7} + \cdots + 256036 \)
|
| $59$ |
\( T^{8} + 13 T^{7} + \cdots + 10857025 \)
|
| $61$ |
\( (T^{4} - 15 T^{3} + \cdots - 6140)^{2} \)
|
| $67$ |
\( T^{8} - 8 T^{7} + \cdots + 529984 \)
|
| $71$ |
\( T^{8} - 7 T^{7} + \cdots + 206116 \)
|
| $73$ |
\( (T^{4} - 17 T^{3} + \cdots - 641)^{2} \)
|
| $79$ |
\( (T^{4} + 7 T^{3} + \cdots + 235)^{2} \)
|
| $83$ |
\( T^{8} + 2 T^{7} + \cdots + 279841 \)
|
| $89$ |
\( (T^{4} - 5 T^{3} + \cdots - 1720)^{2} \)
|
| $97$ |
\( T^{8} - 8 T^{7} + \cdots + 7579009 \)
|
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