Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [855,2,Mod(514,855)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(855, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("855.514");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 855 = 3^{2} \cdot 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 855.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: | \(6.82720937282\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.16516096.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 9x^{4} + 13x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 95) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 9x^{4} + 13x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} + 8\nu^{2} + 5 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} + 10\nu^{3} + 6\nu^{2} + 19\nu - 1 ) / 4 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - \nu^{4} + 10\nu^{3} - 6\nu^{2} + 19\nu + 1 ) / 4 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} - 8\nu^{3} - 7\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta_{2} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + \beta_{4} + \beta_{3} - 6\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( -8\beta_{4} + 8\beta_{3} - 6\beta_{2} + 19 \)
|
| \(\nu^{5}\) | \(=\) |
\( -10\beta_{5} - 8\beta_{4} - 8\beta_{3} + 41\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/855\mathbb{Z}\right)^\times\).
| \(n\) | \(172\) | \(191\) | \(496\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 514.1 |
|
− | 2.41987i | 0 | −3.85577 | 2.07772 | + | 0.826491i | 0 | 3.18676i | 4.49073i | 0 | 2.00000 | − | 5.02781i | |||||||||||||||||||||||||||||||
| 514.2 | − | 1.82254i | 0 | −1.32164 | −1.94827 | + | 1.09737i | 0 | 1.45033i | − | 1.23634i | 0 | 2.00000 | + | 3.55080i | |||||||||||||||||||||||||||||||
| 514.3 | − | 0.906968i | 0 | 1.17741 | 0.370556 | + | 2.20515i | 0 | − | 2.59637i | − | 2.88181i | 0 | 2.00000 | − | 0.336083i | ||||||||||||||||||||||||||||||
| 514.4 | 0.906968i | 0 | 1.17741 | 0.370556 | − | 2.20515i | 0 | 2.59637i | 2.88181i | 0 | 2.00000 | + | 0.336083i | |||||||||||||||||||||||||||||||||
| 514.5 | 1.82254i | 0 | −1.32164 | −1.94827 | − | 1.09737i | 0 | − | 1.45033i | 1.23634i | 0 | 2.00000 | − | 3.55080i | ||||||||||||||||||||||||||||||||
| 514.6 | 2.41987i | 0 | −3.85577 | 2.07772 | − | 0.826491i | 0 | − | 3.18676i | − | 4.49073i | 0 | 2.00000 | + | 5.02781i | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 855.2.c.d | 6 | |
| 3.b | odd | 2 | 1 | 95.2.b.b | ✓ | 6 | |
| 5.b | even | 2 | 1 | inner | 855.2.c.d | 6 | |
| 5.c | odd | 4 | 2 | 4275.2.a.br | 6 | ||
| 12.b | even | 2 | 1 | 1520.2.d.h | 6 | ||
| 15.d | odd | 2 | 1 | 95.2.b.b | ✓ | 6 | |
| 15.e | even | 4 | 2 | 475.2.a.j | 6 | ||
| 57.d | even | 2 | 1 | 1805.2.b.e | 6 | ||
| 60.h | even | 2 | 1 | 1520.2.d.h | 6 | ||
| 60.l | odd | 4 | 2 | 7600.2.a.ck | 6 | ||
| 285.b | even | 2 | 1 | 1805.2.b.e | 6 | ||
| 285.j | odd | 4 | 2 | 9025.2.a.bx | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 95.2.b.b | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 95.2.b.b | ✓ | 6 | 15.d | odd | 2 | 1 | |
| 475.2.a.j | 6 | 15.e | even | 4 | 2 | ||
| 855.2.c.d | 6 | 1.a | even | 1 | 1 | trivial | |
| 855.2.c.d | 6 | 5.b | even | 2 | 1 | inner | |
| 1520.2.d.h | 6 | 12.b | even | 2 | 1 | ||
| 1520.2.d.h | 6 | 60.h | even | 2 | 1 | ||
| 1805.2.b.e | 6 | 57.d | even | 2 | 1 | ||
| 1805.2.b.e | 6 | 285.b | even | 2 | 1 | ||
| 4275.2.a.br | 6 | 5.c | odd | 4 | 2 | ||
| 7600.2.a.ck | 6 | 60.l | odd | 4 | 2 | ||
| 9025.2.a.bx | 6 | 285.j | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(855, [\chi])\):
|
\( T_{2}^{6} + 10T_{2}^{4} + 27T_{2}^{2} + 16 \)
|
|
\( T_{11}^{3} + T_{11}^{2} - 16T_{11} - 12 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 10 T^{4} + \cdots + 16 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - T^{5} + \cdots + 125 \)
|
| $7$ |
\( T^{6} + 19 T^{4} + \cdots + 144 \)
|
| $11$ |
\( (T^{3} + T^{2} - 16 T - 12)^{2} \)
|
| $13$ |
\( T^{6} + 28 T^{4} + \cdots + 576 \)
|
| $17$ |
\( T^{6} + 59 T^{4} + \cdots + 5184 \)
|
| $19$ |
\( (T - 1)^{6} \)
|
| $23$ |
\( T^{6} + 36 T^{4} + \cdots + 64 \)
|
| $29$ |
\( (T - 6)^{6} \)
|
| $31$ |
\( (T^{3} - 56 T + 128)^{2} \)
|
| $37$ |
\( T^{6} + 56 T^{4} + \cdots + 1296 \)
|
| $41$ |
\( (T^{3} + 6 T^{2} - 44 T + 24)^{2} \)
|
| $43$ |
\( T^{6} + 19 T^{4} + \cdots + 144 \)
|
| $47$ |
\( T^{6} + 187 T^{4} + \cdots + 85264 \)
|
| $53$ |
\( T^{6} + 156 T^{4} + \cdots + 64 \)
|
| $59$ |
\( (T^{3} - 10 T^{2} + \cdots + 48)^{2} \)
|
| $61$ |
\( (T^{3} + 7 T^{2} + \cdots - 776)^{2} \)
|
| $67$ |
\( T^{6} + 340 T^{4} + \cdots + 484416 \)
|
| $71$ |
\( (T^{3} + 26 T^{2} + \cdots + 432)^{2} \)
|
| $73$ |
\( T^{6} + 131 T^{4} + \cdots + 5184 \)
|
| $79$ |
\( (T^{3} - 12 T^{2} + \cdots + 32)^{2} \)
|
| $83$ |
\( T^{6} + 228 T^{4} + \cdots + 141376 \)
|
| $89$ |
\( (T^{3} - 12 T^{2} + \cdots + 3456)^{2} \)
|
| $97$ |
\( T^{6} + 28 T^{4} + \cdots + 576 \)
|
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