Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1369,2,Mod(1368,1369)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1369, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1369.1368");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1369 = 37^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1369.b (of order \(2\), degree \(1\), not 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: | \(10.9315200367\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.419904.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 6x^{4} + 9x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 37) |
| 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
\( x^{6} + 6x^{4} + 9x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} + 4\nu^{2} + 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} + 5\nu^{3} + 5\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} - 4\beta_{2} + 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} - 5\beta_{3} + 10\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1369\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1368.1 |
|
− | 1.87939i | 1.34730 | −1.53209 | 3.06418i | − | 2.53209i | −3.53209 | − | 0.879385i | −1.18479 | 5.75877 | |||||||||||||||||||||||||||||||||
| 1368.2 | − | 1.53209i | −0.879385 | −0.347296 | − | 0.694593i | 1.34730i | −2.34730 | − | 2.53209i | −2.22668 | −1.06418 | ||||||||||||||||||||||||||||||||||
| 1368.3 | − | 0.347296i | 2.53209 | 1.87939 | 3.75877i | − | 0.879385i | −0.120615 | − | 1.34730i | 3.41147 | 1.30541 | ||||||||||||||||||||||||||||||||||
| 1368.4 | 0.347296i | 2.53209 | 1.87939 | − | 3.75877i | 0.879385i | −0.120615 | 1.34730i | 3.41147 | 1.30541 | ||||||||||||||||||||||||||||||||||||
| 1368.5 | 1.53209i | −0.879385 | −0.347296 | 0.694593i | − | 1.34730i | −2.34730 | 2.53209i | −2.22668 | −1.06418 | ||||||||||||||||||||||||||||||||||||
| 1368.6 | 1.87939i | 1.34730 | −1.53209 | − | 3.06418i | 2.53209i | −3.53209 | 0.879385i | −1.18479 | 5.75877 | ||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 37.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1369.2.b.f | 6 | |
| 37.b | even | 2 | 1 | inner | 1369.2.b.f | 6 | |
| 37.d | odd | 4 | 1 | 1369.2.a.j | 3 | ||
| 37.d | odd | 4 | 1 | 1369.2.a.k | 3 | ||
| 37.i | odd | 36 | 2 | 37.2.f.a | ✓ | 6 | |
| 111.q | even | 36 | 2 | 333.2.x.c | 6 | ||
| 148.q | even | 36 | 2 | 592.2.bc.a | 6 | ||
| 185.z | even | 36 | 2 | 925.2.bc.a | 12 | ||
| 185.ba | odd | 36 | 2 | 925.2.p.b | 6 | ||
| 185.bc | even | 36 | 2 | 925.2.bc.a | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 37.2.f.a | ✓ | 6 | 37.i | odd | 36 | 2 | |
| 333.2.x.c | 6 | 111.q | even | 36 | 2 | ||
| 592.2.bc.a | 6 | 148.q | even | 36 | 2 | ||
| 925.2.p.b | 6 | 185.ba | odd | 36 | 2 | ||
| 925.2.bc.a | 12 | 185.z | even | 36 | 2 | ||
| 925.2.bc.a | 12 | 185.bc | even | 36 | 2 | ||
| 1369.2.a.j | 3 | 37.d | odd | 4 | 1 | ||
| 1369.2.a.k | 3 | 37.d | odd | 4 | 1 | ||
| 1369.2.b.f | 6 | 1.a | even | 1 | 1 | trivial | |
| 1369.2.b.f | 6 | 37.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1369, [\chi])\):
|
\( T_{2}^{6} + 6T_{2}^{4} + 9T_{2}^{2} + 1 \)
|
|
\( T_{3}^{3} - 3T_{3}^{2} + 3 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 6 T^{4} + \cdots + 1 \)
|
| $3$ |
\( (T^{3} - 3 T^{2} + 3)^{2} \)
|
| $5$ |
\( T^{6} + 24 T^{4} + \cdots + 64 \)
|
| $7$ |
\( (T^{3} + 6 T^{2} + 9 T + 1)^{2} \)
|
| $11$ |
\( (T^{3} - 3 T + 1)^{2} \)
|
| $13$ |
\( T^{6} + 21 T^{4} + \cdots + 1 \)
|
| $17$ |
\( T^{6} + 117 T^{4} + \cdots + 47961 \)
|
| $19$ |
\( T^{6} + 90 T^{4} + \cdots + 3249 \)
|
| $23$ |
\( T^{6} + 9 T^{4} + \cdots + 1 \)
|
| $29$ |
\( T^{6} + 99 T^{4} + \cdots + 3249 \)
|
| $31$ |
\( T^{6} + 27 T^{4} + \cdots + 361 \)
|
| $37$ |
\( T^{6} \)
|
| $41$ |
\( (T^{3} - 9 T - 9)^{2} \)
|
| $43$ |
\( T^{6} + 129 T^{4} + \cdots + 5041 \)
|
| $47$ |
\( (T^{3} + 15 T^{2} + \cdots + 57)^{2} \)
|
| $53$ |
\( (T^{3} + 3 T^{2} - 45 T + 17)^{2} \)
|
| $59$ |
\( T^{6} + 105 T^{4} + \cdots + 32041 \)
|
| $61$ |
\( T^{6} + 165 T^{4} + \cdots + 104329 \)
|
| $67$ |
\( (T^{3} - 3 T^{2} - 24 T + 53)^{2} \)
|
| $71$ |
\( (T^{3} + 15 T^{2} + \cdots - 753)^{2} \)
|
| $73$ |
\( (T^{3} + 6 T^{2} + \cdots - 109)^{2} \)
|
| $79$ |
\( T^{6} + 297 T^{4} + \cdots + 687241 \)
|
| $83$ |
\( (T^{3} + 21 T^{2} + \cdots + 51)^{2} \)
|
| $89$ |
\( T^{6} + 150 T^{4} + \cdots + 5329 \)
|
| $97$ |
\( T^{6} + 186 T^{4} + \cdots + 5041 \)
|
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