Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [85,2,Mod(84,85)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(85, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("85.84");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 85 = 5 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 85.c (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: | \(0.678728417181\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.10070523904.11 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 10x^{4} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2\cdot 3^{2} \) |
| 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
\( x^{8} - 10x^{4} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{6} + 9\nu^{4} + \nu^{2} - 45 ) / 36 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - 9\nu^{5} - 37\nu^{3} + 9\nu ) / 108 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{6} + 19\nu^{2} ) / 18 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{7} - 3\nu^{5} - \nu^{3} + 3\nu ) / 36 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{7} - 3\nu^{5} + \nu^{3} + 39\nu ) / 36 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 5\nu^{6} + 9\nu^{4} - 5\nu^{2} - 45 ) / 36 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 7\nu^{7} + 9\nu^{5} - 43\nu^{3} + 99\nu ) / 108 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + 2\beta_{5} - \beta_{2} ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} + 3\beta_{3} - \beta_1 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{7} + \beta_{5} + 6\beta_{4} - 8\beta_{2} ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 2\beta_{6} + 10\beta _1 + 15 ) / 3 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 7\beta_{7} - 4\beta_{5} - 18\beta_{4} - 7\beta_{2} ) / 3 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 19\beta_{6} + 3\beta_{3} - 19\beta_1 ) / 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 17\beta_{7} - 17\beta_{5} + 60\beta_{4} - 26\beta_{2} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/85\mathbb{Z}\right)^\times\).
| \(n\) | \(52\) | \(71\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 84.1 |
|
− | 2.57794i | −2.37608 | −4.64575 | 1.95522 | − | 1.08495i | 6.12538i | −1.53436 | 6.82058i | 2.64575 | −2.79694 | − | 5.04042i | |||||||||||||||||||||||||||||||||||||
| 84.2 | − | 2.57794i | 2.37608 | −4.64575 | −1.95522 | + | 1.08495i | − | 6.12538i | 1.53436 | 6.82058i | 2.64575 | 2.79694 | + | 5.04042i | |||||||||||||||||||||||||||||||||||||
| 84.3 | − | 1.16372i | −0.595188 | 0.645751 | −1.08495 | − | 1.95522i | 0.692633i | 2.76510 | − | 3.07892i | −2.64575 | −2.27533 | + | 1.26258i | |||||||||||||||||||||||||||||||||||||
| 84.4 | − | 1.16372i | 0.595188 | 0.645751 | 1.08495 | + | 1.95522i | − | 0.692633i | −2.76510 | − | 3.07892i | −2.64575 | 2.27533 | − | 1.26258i | ||||||||||||||||||||||||||||||||||||
| 84.5 | 1.16372i | −0.595188 | 0.645751 | −1.08495 | + | 1.95522i | − | 0.692633i | 2.76510 | 3.07892i | −2.64575 | −2.27533 | − | 1.26258i | ||||||||||||||||||||||||||||||||||||||
| 84.6 | 1.16372i | 0.595188 | 0.645751 | 1.08495 | − | 1.95522i | 0.692633i | −2.76510 | 3.07892i | −2.64575 | 2.27533 | + | 1.26258i | |||||||||||||||||||||||||||||||||||||||
| 84.7 | 2.57794i | −2.37608 | −4.64575 | 1.95522 | + | 1.08495i | − | 6.12538i | −1.53436 | − | 6.82058i | 2.64575 | −2.79694 | + | 5.04042i | |||||||||||||||||||||||||||||||||||||
| 84.8 | 2.57794i | 2.37608 | −4.64575 | −1.95522 | − | 1.08495i | 6.12538i | 1.53436 | − | 6.82058i | 2.64575 | 2.79694 | − | 5.04042i | ||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 17.b | even | 2 | 1 | inner |
| 85.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 85.2.c.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 765.2.d.c | 8 | ||
| 4.b | odd | 2 | 1 | 1360.2.o.f | 8 | ||
| 5.b | even | 2 | 1 | inner | 85.2.c.a | ✓ | 8 |
| 5.c | odd | 4 | 2 | 425.2.d.d | 8 | ||
| 15.d | odd | 2 | 1 | 765.2.d.c | 8 | ||
| 17.b | even | 2 | 1 | inner | 85.2.c.a | ✓ | 8 |
| 17.c | even | 4 | 2 | 1445.2.b.b | 8 | ||
| 20.d | odd | 2 | 1 | 1360.2.o.f | 8 | ||
| 51.c | odd | 2 | 1 | 765.2.d.c | 8 | ||
| 68.d | odd | 2 | 1 | 1360.2.o.f | 8 | ||
| 85.c | even | 2 | 1 | inner | 85.2.c.a | ✓ | 8 |
| 85.f | odd | 4 | 2 | 7225.2.a.bl | 8 | ||
| 85.g | odd | 4 | 2 | 425.2.d.d | 8 | ||
| 85.i | odd | 4 | 2 | 7225.2.a.bl | 8 | ||
| 85.j | even | 4 | 2 | 1445.2.b.b | 8 | ||
| 255.h | odd | 2 | 1 | 765.2.d.c | 8 | ||
| 340.d | odd | 2 | 1 | 1360.2.o.f | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 85.2.c.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 85.2.c.a | ✓ | 8 | 5.b | even | 2 | 1 | inner |
| 85.2.c.a | ✓ | 8 | 17.b | even | 2 | 1 | inner |
| 85.2.c.a | ✓ | 8 | 85.c | even | 2 | 1 | inner |
| 425.2.d.d | 8 | 5.c | odd | 4 | 2 | ||
| 425.2.d.d | 8 | 85.g | odd | 4 | 2 | ||
| 765.2.d.c | 8 | 3.b | odd | 2 | 1 | ||
| 765.2.d.c | 8 | 15.d | odd | 2 | 1 | ||
| 765.2.d.c | 8 | 51.c | odd | 2 | 1 | ||
| 765.2.d.c | 8 | 255.h | odd | 2 | 1 | ||
| 1360.2.o.f | 8 | 4.b | odd | 2 | 1 | ||
| 1360.2.o.f | 8 | 20.d | odd | 2 | 1 | ||
| 1360.2.o.f | 8 | 68.d | odd | 2 | 1 | ||
| 1360.2.o.f | 8 | 340.d | odd | 2 | 1 | ||
| 1445.2.b.b | 8 | 17.c | even | 4 | 2 | ||
| 1445.2.b.b | 8 | 85.j | even | 4 | 2 | ||
| 7225.2.a.bl | 8 | 85.f | odd | 4 | 2 | ||
| 7225.2.a.bl | 8 | 85.i | odd | 4 | 2 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(85, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 8 T^{2} + 9)^{2} \)
|
| $3$ |
\( (T^{4} - 6 T^{2} + 2)^{2} \)
|
| $5$ |
\( T^{8} + 22T^{4} + 625 \)
|
| $7$ |
\( (T^{4} - 10 T^{2} + 18)^{2} \)
|
| $11$ |
\( (T^{4} + 26 T^{2} + 162)^{2} \)
|
| $13$ |
\( (T^{2} + 18)^{4} \)
|
| $17$ |
\( T^{8} - 36 T^{6} + \cdots + 83521 \)
|
| $19$ |
\( (T - 2)^{8} \)
|
| $23$ |
\( (T^{4} - 38 T^{2} + 18)^{2} \)
|
| $29$ |
\( (T^{4} + 80 T^{2} + 1152)^{2} \)
|
| $31$ |
\( (T^{4} + 54 T^{2} + 162)^{2} \)
|
| $37$ |
\( (T^{4} - 40 T^{2} + 288)^{2} \)
|
| $41$ |
\( (T^{4} + 20 T^{2} + 72)^{2} \)
|
| $43$ |
\( (T^{2} + 18)^{4} \)
|
| $47$ |
\( (T^{4} + 116 T^{2} + 2916)^{2} \)
|
| $53$ |
\( (T^{4} + 32 T^{2} + 144)^{2} \)
|
| $59$ |
\( (T - 6)^{8} \)
|
| $61$ |
\( (T^{4} + 108 T^{2} + 648)^{2} \)
|
| $67$ |
\( (T^{2} + 18)^{4} \)
|
| $71$ |
\( (T^{4} + 266 T^{2} + 882)^{2} \)
|
| $73$ |
\( (T^{4} - 76 T^{2} + 72)^{2} \)
|
| $79$ |
\( (T^{4} + 54 T^{2} + 162)^{2} \)
|
| $83$ |
\( (T^{4} + 44 T^{2} + 36)^{2} \)
|
| $89$ |
\( (T^{2} + 6 T - 54)^{4} \)
|
| $97$ |
\( (T^{4} - 208 T^{2} + 10368)^{2} \)
|
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