Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3420,2,Mod(1369,3420)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3420, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("3420.1369");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3420.f (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: | \(27.3088374913\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.14077504.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 9x^{4} + 14x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 380) |
| 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
\( x^{6} + 9x^{4} + 14x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} + 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + 9\nu^{3} + 14\nu ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{5} - \nu^{4} - 8\nu^{3} - 8\nu^{2} - 8\nu - 6 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} - 8\nu^{3} + 8\nu^{2} - 4\nu + 6 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + \nu^{4} - 8\nu^{3} + 8\nu^{2} - 8\nu + 6 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{5} + \beta_{4} ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta _1 - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( 4\beta_{5} - 3\beta_{4} + \beta_{3} + 2\beta_{2} \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{5} - \beta_{3} - 8\beta _1 + 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( -29\beta_{5} + 20\beta_{4} - 9\beta_{3} - 16\beta_{2} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times\).
| \(n\) | \(1711\) | \(1901\) | \(2737\) | \(3061\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1369.1 |
|
0 | 0 | 0 | −1.58777 | − | 1.57448i | 0 | 1.93210i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1369.2 | 0 | 0 | 0 | −1.58777 | + | 1.57448i | 0 | − | 1.93210i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1369.3 | 0 | 0 | 0 | 0.163664 | − | 2.23007i | 0 | − | 0.872810i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
| 1369.4 | 0 | 0 | 0 | 0.163664 | + | 2.23007i | 0 | 0.872810i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||
| 1369.5 | 0 | 0 | 0 | 1.92411 | − | 1.13921i | 0 | 4.74397i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||||||
| 1369.6 | 0 | 0 | 0 | 1.92411 | + | 1.13921i | 0 | − | 4.74397i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||||||
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 | 3420.2.f.c | 6 | |
| 3.b | odd | 2 | 1 | 380.2.c.b | ✓ | 6 | |
| 5.b | even | 2 | 1 | inner | 3420.2.f.c | 6 | |
| 12.b | even | 2 | 1 | 1520.2.d.i | 6 | ||
| 15.d | odd | 2 | 1 | 380.2.c.b | ✓ | 6 | |
| 15.e | even | 4 | 2 | 1900.2.a.k | 6 | ||
| 60.h | even | 2 | 1 | 1520.2.d.i | 6 | ||
| 60.l | odd | 4 | 2 | 7600.2.a.cj | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 380.2.c.b | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 380.2.c.b | ✓ | 6 | 15.d | odd | 2 | 1 | |
| 1520.2.d.i | 6 | 12.b | even | 2 | 1 | ||
| 1520.2.d.i | 6 | 60.h | even | 2 | 1 | ||
| 1900.2.a.k | 6 | 15.e | even | 4 | 2 | ||
| 3420.2.f.c | 6 | 1.a | even | 1 | 1 | trivial | |
| 3420.2.f.c | 6 | 5.b | even | 2 | 1 | inner | |
| 7600.2.a.cj | 6 | 60.l | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{6} + 27T_{7}^{4} + 104T_{7}^{2} + 64 \)
acting on \(S_{2}^{\mathrm{new}}(3420, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} - T^{5} + \cdots + 125 \)
|
| $7$ |
\( T^{6} + 27 T^{4} + \cdots + 64 \)
|
| $11$ |
\( (T^{3} + 9 T^{2} + 14 T - 28)^{2} \)
|
| $13$ |
\( T^{6} + 26 T^{4} + \cdots + 64 \)
|
| $17$ |
\( T^{6} + 51 T^{4} + \cdots + 3136 \)
|
| $19$ |
\( (T + 1)^{6} \)
|
| $23$ |
\( T^{6} + 44 T^{4} + \cdots + 256 \)
|
| $29$ |
\( (T^{3} + 2 T^{2} - 84 T + 88)^{2} \)
|
| $31$ |
\( (T^{3} - 4 T^{2} + \cdots + 256)^{2} \)
|
| $37$ |
\( T^{6} + 14 T^{4} + \cdots + 16 \)
|
| $41$ |
\( (T^{3} + 2 T^{2} + \cdots - 488)^{2} \)
|
| $43$ |
\( T^{6} + 227 T^{4} + \cdots + 3136 \)
|
| $47$ |
\( T^{6} + 243 T^{4} + \cdots + 118336 \)
|
| $53$ |
\( T^{6} + 114 T^{4} + \cdots + 23104 \)
|
| $59$ |
\( (T^{3} - 14 T^{2} + \cdots + 128)^{2} \)
|
| $61$ |
\( (T^{3} - 15 T^{2} + \cdots - 44)^{2} \)
|
| $67$ |
\( T^{6} + 378 T^{4} + \cdots + 222784 \)
|
| $71$ |
\( (T^{3} + 22 T^{2} + \cdots + 32)^{2} \)
|
| $73$ |
\( T^{6} + 251 T^{4} + \cdots + 118336 \)
|
| $79$ |
\( (T + 4)^{6} \)
|
| $83$ |
\( T^{6} + 44 T^{4} + \cdots + 256 \)
|
| $89$ |
\( (T^{3} - 4 T^{2} - 44 T + 64)^{2} \)
|
| $97$ |
\( T^{6} + 114 T^{4} + \cdots + 23104 \)
|
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