Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [69,2,Mod(1,69)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(69, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("69.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 69 = 3 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 69.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: | \(0.550967773947\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{5}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(\beta = \sqrt{5}\). 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 |
|
−2.23607 | −1.00000 | 3.00000 | 1.23607 | 2.23607 | 3.23607 | −2.23607 | 1.00000 | −2.76393 | ||||||||||||||||||||||||
| 1.2 | 2.23607 | −1.00000 | 3.00000 | −3.23607 | −2.23607 | −1.23607 | 2.23607 | 1.00000 | −7.23607 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(23\) | \(-1\) |
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 | 69.2.a.b | ✓ | 2 |
| 3.b | odd | 2 | 1 | 207.2.a.c | 2 | ||
| 4.b | odd | 2 | 1 | 1104.2.a.m | 2 | ||
| 5.b | even | 2 | 1 | 1725.2.a.ba | 2 | ||
| 5.c | odd | 4 | 2 | 1725.2.b.o | 4 | ||
| 7.b | odd | 2 | 1 | 3381.2.a.t | 2 | ||
| 8.b | even | 2 | 1 | 4416.2.a.bm | 2 | ||
| 8.d | odd | 2 | 1 | 4416.2.a.bg | 2 | ||
| 11.b | odd | 2 | 1 | 8349.2.a.i | 2 | ||
| 12.b | even | 2 | 1 | 3312.2.a.bb | 2 | ||
| 15.d | odd | 2 | 1 | 5175.2.a.bk | 2 | ||
| 23.b | odd | 2 | 1 | 1587.2.a.i | 2 | ||
| 69.c | even | 2 | 1 | 4761.2.a.v | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 69.2.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 207.2.a.c | 2 | 3.b | odd | 2 | 1 | ||
| 1104.2.a.m | 2 | 4.b | odd | 2 | 1 | ||
| 1587.2.a.i | 2 | 23.b | odd | 2 | 1 | ||
| 1725.2.a.ba | 2 | 5.b | even | 2 | 1 | ||
| 1725.2.b.o | 4 | 5.c | odd | 4 | 2 | ||
| 3312.2.a.bb | 2 | 12.b | even | 2 | 1 | ||
| 3381.2.a.t | 2 | 7.b | odd | 2 | 1 | ||
| 4416.2.a.bg | 2 | 8.d | odd | 2 | 1 | ||
| 4416.2.a.bm | 2 | 8.b | even | 2 | 1 | ||
| 4761.2.a.v | 2 | 69.c | even | 2 | 1 | ||
| 5175.2.a.bk | 2 | 15.d | odd | 2 | 1 | ||
| 8349.2.a.i | 2 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - 5 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(69))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 5 \)
|
| $3$ |
\( (T + 1)^{2} \)
|
| $5$ |
\( T^{2} + 2T - 4 \)
|
| $7$ |
\( T^{2} - 2T - 4 \)
|
| $11$ |
\( (T - 4)^{2} \)
|
| $13$ |
\( T^{2} - 20 \)
|
| $17$ |
\( T^{2} + 10T + 20 \)
|
| $19$ |
\( T^{2} - 10T + 20 \)
|
| $23$ |
\( (T - 1)^{2} \)
|
| $29$ |
\( T^{2} - 20 \)
|
| $31$ |
\( T^{2} + 4T - 16 \)
|
| $37$ |
\( T^{2} - 20 \)
|
| $41$ |
\( T^{2} + 4T - 76 \)
|
| $43$ |
\( T^{2} - 2T - 44 \)
|
| $47$ |
\( (T + 4)^{2} \)
|
| $53$ |
\( T^{2} + 6T + 4 \)
|
| $59$ |
\( T^{2} - 8T - 64 \)
|
| $61$ |
\( T^{2} - 20 \)
|
| $67$ |
\( T^{2} - 6T + 4 \)
|
| $71$ |
\( (T + 8)^{2} \)
|
| $73$ |
\( T^{2} + 4T - 76 \)
|
| $79$ |
\( T^{2} - 6T - 36 \)
|
| $83$ |
\( (T - 4)^{2} \)
|
| $89$ |
\( T^{2} - 2T - 4 \)
|
| $97$ |
\( T^{2} - 8T - 4 \)
|
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