Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [6069,2,Mod(1,6069)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(6069, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 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("6069.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 6069 = 3 \cdot 7 \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 6069.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: | \(48.4612089867\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.1502576.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 9x^{3} + 18x - 2 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 357) |
| 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 a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 9x^{3} + 18x - 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 5\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 6\nu^{2} - \nu + 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 6\beta_{2} + \beta _1 + 20 \)
|
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 |
|
−2.38496 | −1.00000 | 3.68803 | 4.32895 | 2.38496 | 1.00000 | −4.02588 | 1.00000 | −10.3244 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.84908 | −1.00000 | 1.41908 | −2.50415 | 1.84908 | 1.00000 | 1.07416 | 1.00000 | 4.63037 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.111809 | −1.00000 | −1.98750 | −2.42985 | −0.111809 | 1.00000 | −0.445838 | 1.00000 | −0.271679 | ||||||||||||||||||||||||||||||||||
| 1.4 | 1.62293 | −1.00000 | 0.633897 | 3.47391 | −1.62293 | 1.00000 | −2.21709 | 1.00000 | 5.63791 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.49930 | −1.00000 | 4.24649 | 0.131141 | −2.49930 | 1.00000 | 5.61465 | 1.00000 | 0.327761 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(-1\) |
| \(17\) | \(1\) |
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 | 6069.2.a.v | 5 | |
| 17.b | even | 2 | 1 | 6069.2.a.w | 5 | ||
| 17.c | even | 4 | 2 | 357.2.f.b | ✓ | 10 | |
| 51.f | odd | 4 | 2 | 1071.2.f.b | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 357.2.f.b | ✓ | 10 | 17.c | even | 4 | 2 | |
| 1071.2.f.b | 10 | 51.f | odd | 4 | 2 | ||
| 6069.2.a.v | 5 | 1.a | even | 1 | 1 | trivial | |
| 6069.2.a.w | 5 | 17.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6069))\):
|
\( T_{2}^{5} - 9T_{2}^{3} + 18T_{2} - 2 \)
|
|
\( T_{5}^{5} - 3T_{5}^{4} - 17T_{5}^{3} + 29T_{5}^{2} + 88T_{5} - 12 \)
|
|
\( T_{11}^{5} - 3T_{11}^{4} - 9T_{11}^{3} + 27T_{11}^{2} - 4 \)
|
|
\( T_{23}^{5} - 7T_{23}^{4} - 45T_{23}^{3} + 343T_{23}^{2} - 368T_{23} - 52 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 9 T^{3} + \cdots - 2 \)
|
| $3$ |
\( (T + 1)^{5} \)
|
| $5$ |
\( T^{5} - 3 T^{4} + \cdots - 12 \)
|
| $7$ |
\( (T - 1)^{5} \)
|
| $11$ |
\( T^{5} - 3 T^{4} + \cdots - 4 \)
|
| $13$ |
\( T^{5} - 3 T^{4} + \cdots - 332 \)
|
| $17$ |
\( T^{5} \)
|
| $19$ |
\( T^{5} - 17 T^{4} + \cdots + 2612 \)
|
| $23$ |
\( T^{5} - 7 T^{4} + \cdots - 52 \)
|
| $29$ |
\( T^{5} - 10 T^{4} + \cdots - 1392 \)
|
| $31$ |
\( T^{5} + 14 T^{4} + \cdots - 256 \)
|
| $37$ |
\( T^{5} + 4 T^{4} + \cdots - 224 \)
|
| $41$ |
\( T^{5} + T^{4} + \cdots - 3812 \)
|
| $43$ |
\( T^{5} + T^{4} + \cdots + 20224 \)
|
| $47$ |
\( T^{5} - 6 T^{4} + \cdots - 32 \)
|
| $53$ |
\( T^{5} - 6 T^{4} + \cdots + 6408 \)
|
| $59$ |
\( T^{5} - 26 T^{4} + \cdots - 384 \)
|
| $61$ |
\( T^{5} - 14 T^{4} + \cdots - 2248 \)
|
| $67$ |
\( T^{5} + 4 T^{4} + \cdots - 4608 \)
|
| $71$ |
\( T^{5} - 16 T^{4} + \cdots - 3792 \)
|
| $73$ |
\( T^{5} + 14 T^{4} + \cdots + 672 \)
|
| $79$ |
\( T^{5} + 10 T^{4} + \cdots - 128 \)
|
| $83$ |
\( T^{5} - 8 T^{4} + \cdots + 12288 \)
|
| $89$ |
\( T^{5} - 24 T^{4} + \cdots + 3328 \)
|
| $97$ |
\( T^{5} + 14 T^{4} + \cdots - 2528 \)
|
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