Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [65,4,Mod(1,65)]
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, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("65.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 65 = 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 65.a (trivial) |
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: | yes |
| Analytic conductor: | \(3.83512415037\) |
| Analytic rank: | \(1\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{6}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - 6 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−3.44949 | 0.449490 | 3.89898 | 5.00000 | −1.55051 | −5.10102 | 14.1464 | −26.7980 | −17.2474 | ||||||||||||||||||||||||
| 1.2 | 1.44949 | −4.44949 | −5.89898 | 5.00000 | −6.44949 | −14.8990 | −20.1464 | −7.20204 | 7.24745 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \( -1 \) |
| \(13\) | \( +1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 65.4.a.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 585.4.a.i | 2 | ||
| 4.b | odd | 2 | 1 | 1040.4.a.m | 2 | ||
| 5.b | even | 2 | 1 | 325.4.a.h | 2 | ||
| 5.c | odd | 4 | 2 | 325.4.b.d | 4 | ||
| 13.b | even | 2 | 1 | 845.4.a.e | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 65.4.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 325.4.a.h | 2 | 5.b | even | 2 | 1 | ||
| 325.4.b.d | 4 | 5.c | odd | 4 | 2 | ||
| 585.4.a.i | 2 | 3.b | odd | 2 | 1 | ||
| 845.4.a.e | 2 | 13.b | even | 2 | 1 | ||
| 1040.4.a.m | 2 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + 2T_{2} - 5 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(65))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 2T - 5 \)
|
| $3$ |
\( T^{2} + 4T - 2 \)
|
| $5$ |
\( (T - 5)^{2} \)
|
| $7$ |
\( T^{2} + 20T + 76 \)
|
| $11$ |
\( T^{2} + 104T + 2698 \)
|
| $13$ |
\( (T + 13)^{2} \)
|
| $17$ |
\( T^{2} - 52T - 9908 \)
|
| $19$ |
\( T^{2} - 9126 \)
|
| $23$ |
\( T^{2} + 12T - 23778 \)
|
| $29$ |
\( T^{2} - 16T - 86336 \)
|
| $31$ |
\( T^{2} + 464T + 52810 \)
|
| $37$ |
\( T^{2} + 88T - 3464 \)
|
| $41$ |
\( T^{2} + 180T + 4044 \)
|
| $43$ |
\( T^{2} + 364T + 33070 \)
|
| $47$ |
\( T^{2} - 308T - 29300 \)
|
| $53$ |
\( T^{2} - 748T + 56332 \)
|
| $59$ |
\( T^{2} + 480T + 36714 \)
|
| $61$ |
\( T^{2} - 16T - 483872 \)
|
| $67$ |
\( T^{2} + 116T - 18236 \)
|
| $71$ |
\( T^{2} + 480T + 20154 \)
|
| $73$ |
\( T^{2} - 744T + 3384 \)
|
| $79$ |
\( T^{2} - 456T - 129672 \)
|
| $83$ |
\( T^{2} + 500T - 21044 \)
|
| $89$ |
\( T^{2} - 268T - 37340 \)
|
| $97$ |
\( T^{2} + 724T + 130180 \)
|
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