Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [65,2,Mod(51,65)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(65, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("65.51");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 65 = 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 65.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: | \(0.519027613138\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.5089536.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 24\nu^{4} - 6\nu^{3} - \nu^{2} + 6\nu + 285 ) / 131 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{5} - 11\nu^{4} + 101\nu^{3} - 136\nu^{2} + 292\nu - 147 ) / 393 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 6\nu^{5} - 13\nu^{4} + 36\nu^{3} + 6\nu^{2} - 36\nu - 7 ) / 131 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 23\nu^{5} - 28\nu^{4} + 7\nu^{3} + 154\nu^{2} + 386\nu - 267 ) / 393 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -23\nu^{5} + 28\nu^{4} - 7\nu^{3} - 23\nu^{2} - 386\nu + 267 ) / 131 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} - \beta_{3} + \beta_{2} + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 3\beta_{4} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{5} + 2\beta_{4} + 2\beta_{3} + 2\beta_{2} + \beta _1 - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{3} + 6\beta _1 - 13 \)
|
| \(\nu^{5}\) | \(=\) |
\( -7\beta_{5} - 12\beta_{4} + 9\beta_{3} - 9\beta_{2} + 7\beta _1 - 12 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/65\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(41\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 51.1 |
|
− | 2.51414i | 1.51414 | −4.32088 | 1.00000i | − | 3.80675i | 3.32088i | 5.83502i | −0.707389 | 2.51414 | ||||||||||||||||||||||||||||||||||
| 51.2 | − | 2.08613i | −3.08613 | −2.35194 | − | 1.00000i | 6.43807i | − | 1.35194i | 0.734191i | 6.52420 | −2.08613 | ||||||||||||||||||||||||||||||||||
| 51.3 | − | 0.571993i | −0.428007 | 1.67282 | 1.00000i | 0.244817i | − | 2.67282i | − | 2.10083i | −2.81681 | 0.571993 | ||||||||||||||||||||||||||||||||||
| 51.4 | 0.571993i | −0.428007 | 1.67282 | − | 1.00000i | − | 0.244817i | 2.67282i | 2.10083i | −2.81681 | 0.571993 | |||||||||||||||||||||||||||||||||||
| 51.5 | 2.08613i | −3.08613 | −2.35194 | 1.00000i | − | 6.43807i | 1.35194i | − | 0.734191i | 6.52420 | −2.08613 | |||||||||||||||||||||||||||||||||||
| 51.6 | 2.51414i | 1.51414 | −4.32088 | − | 1.00000i | 3.80675i | − | 3.32088i | − | 5.83502i | −0.707389 | 2.51414 | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 65.2.c.a | ✓ | 6 |
| 3.b | odd | 2 | 1 | 585.2.b.g | 6 | ||
| 4.b | odd | 2 | 1 | 1040.2.k.d | 6 | ||
| 5.b | even | 2 | 1 | 325.2.c.g | 6 | ||
| 5.c | odd | 4 | 1 | 325.2.d.e | 6 | ||
| 5.c | odd | 4 | 1 | 325.2.d.f | 6 | ||
| 13.b | even | 2 | 1 | inner | 65.2.c.a | ✓ | 6 |
| 13.c | even | 3 | 2 | 845.2.m.h | 12 | ||
| 13.d | odd | 4 | 1 | 845.2.a.i | 3 | ||
| 13.d | odd | 4 | 1 | 845.2.a.k | 3 | ||
| 13.e | even | 6 | 2 | 845.2.m.h | 12 | ||
| 13.f | odd | 12 | 2 | 845.2.e.i | 6 | ||
| 13.f | odd | 12 | 2 | 845.2.e.k | 6 | ||
| 39.d | odd | 2 | 1 | 585.2.b.g | 6 | ||
| 39.f | even | 4 | 1 | 7605.2.a.bs | 3 | ||
| 39.f | even | 4 | 1 | 7605.2.a.cc | 3 | ||
| 52.b | odd | 2 | 1 | 1040.2.k.d | 6 | ||
| 65.d | even | 2 | 1 | 325.2.c.g | 6 | ||
| 65.g | odd | 4 | 1 | 4225.2.a.bc | 3 | ||
| 65.g | odd | 4 | 1 | 4225.2.a.be | 3 | ||
| 65.h | odd | 4 | 1 | 325.2.d.e | 6 | ||
| 65.h | odd | 4 | 1 | 325.2.d.f | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 65.2.c.a | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 65.2.c.a | ✓ | 6 | 13.b | even | 2 | 1 | inner |
| 325.2.c.g | 6 | 5.b | even | 2 | 1 | ||
| 325.2.c.g | 6 | 65.d | even | 2 | 1 | ||
| 325.2.d.e | 6 | 5.c | odd | 4 | 1 | ||
| 325.2.d.e | 6 | 65.h | odd | 4 | 1 | ||
| 325.2.d.f | 6 | 5.c | odd | 4 | 1 | ||
| 325.2.d.f | 6 | 65.h | odd | 4 | 1 | ||
| 585.2.b.g | 6 | 3.b | odd | 2 | 1 | ||
| 585.2.b.g | 6 | 39.d | odd | 2 | 1 | ||
| 845.2.a.i | 3 | 13.d | odd | 4 | 1 | ||
| 845.2.a.k | 3 | 13.d | odd | 4 | 1 | ||
| 845.2.e.i | 6 | 13.f | odd | 12 | 2 | ||
| 845.2.e.k | 6 | 13.f | odd | 12 | 2 | ||
| 845.2.m.h | 12 | 13.c | even | 3 | 2 | ||
| 845.2.m.h | 12 | 13.e | even | 6 | 2 | ||
| 1040.2.k.d | 6 | 4.b | odd | 2 | 1 | ||
| 1040.2.k.d | 6 | 52.b | odd | 2 | 1 | ||
| 4225.2.a.bc | 3 | 65.g | odd | 4 | 1 | ||
| 4225.2.a.be | 3 | 65.g | odd | 4 | 1 | ||
| 7605.2.a.bs | 3 | 39.f | even | 4 | 1 | ||
| 7605.2.a.cc | 3 | 39.f | even | 4 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(65, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 11 T^{4} + \cdots + 9 \)
|
| $3$ |
\( (T^{3} + 2 T^{2} - 4 T - 2)^{2} \)
|
| $5$ |
\( (T^{2} + 1)^{3} \)
|
| $7$ |
\( T^{6} + 20 T^{4} + \cdots + 144 \)
|
| $11$ |
\( T^{6} + 48 T^{4} + \cdots + 2916 \)
|
| $13$ |
\( T^{6} + 8 T^{5} + \cdots + 2197 \)
|
| $17$ |
\( (T^{3} - 4 T^{2} - 32 T + 96)^{2} \)
|
| $19$ |
\( T^{6} + 44 T^{4} + \cdots + 36 \)
|
| $23$ |
\( (T^{3} - 8 T^{2} + 16 T - 6)^{2} \)
|
| $29$ |
\( (T^{3} + 2 T^{2} - 8 T - 12)^{2} \)
|
| $31$ |
\( T^{6} + 56 T^{4} + \cdots + 36 \)
|
| $37$ |
\( T^{6} + 152 T^{4} + \cdots + 51984 \)
|
| $41$ |
\( T^{6} + 92 T^{4} + \cdots + 5184 \)
|
| $43$ |
\( (T^{3} - 12 T^{2} + 12 T - 2)^{2} \)
|
| $47$ |
\( T^{6} + 20 T^{4} + \cdots + 144 \)
|
| $53$ |
\( (T^{3} - 6 T^{2} - 60 T - 72)^{2} \)
|
| $59$ |
\( T^{6} + 84 T^{4} + \cdots + 324 \)
|
| $61$ |
\( (T^{3} + 6 T^{2} - 12 T - 76)^{2} \)
|
| $67$ |
\( T^{6} + 156 T^{4} + \cdots + 11664 \)
|
| $71$ |
\( T^{6} + 44 T^{4} + \cdots + 36 \)
|
| $73$ |
\( T^{6} + 296 T^{4} + \cdots + 266256 \)
|
| $79$ |
\( (T^{3} - 12 T^{2} + \cdots + 32)^{2} \)
|
| $83$ |
\( T^{6} + 156 T^{4} + \cdots + 1296 \)
|
| $89$ |
\( T^{6} + 192 T^{4} + \cdots + 82944 \)
|
| $97$ |
\( T^{6} + 140 T^{4} + \cdots + 576 \)
|
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