Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [111,2,Mod(73,111)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(111, 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("111.73");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 111 = 3 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 111.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: | \(0.886339462436\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.27648.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 6x^{2} + 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 6x^{2} + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 5\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 5\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/111\mathbb{Z}\right)^\times\).
| \(n\) | \(38\) | \(76\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 73.1 |
|
− | 2.33441i | 1.00000 | −3.44949 | − | 1.04930i | − | 2.33441i | −2.00000 | 3.38371i | 1.00000 | −2.44949 | |||||||||||||||||||||||||||
| 73.2 | − | 0.741964i | 1.00000 | 1.44949 | 3.30136i | − | 0.741964i | −2.00000 | − | 2.55940i | 1.00000 | 2.44949 | ||||||||||||||||||||||||||||
| 73.3 | 0.741964i | 1.00000 | 1.44949 | − | 3.30136i | 0.741964i | −2.00000 | 2.55940i | 1.00000 | 2.44949 | ||||||||||||||||||||||||||||||
| 73.4 | 2.33441i | 1.00000 | −3.44949 | 1.04930i | 2.33441i | −2.00000 | − | 3.38371i | 1.00000 | −2.44949 | ||||||||||||||||||||||||||||||
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 | 111.2.c.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 333.2.c.c | 4 | ||
| 4.b | odd | 2 | 1 | 1776.2.h.e | 4 | ||
| 12.b | even | 2 | 1 | 5328.2.h.l | 4 | ||
| 37.b | even | 2 | 1 | inner | 111.2.c.b | ✓ | 4 |
| 37.d | odd | 4 | 2 | 4107.2.a.g | 4 | ||
| 111.d | odd | 2 | 1 | 333.2.c.c | 4 | ||
| 148.b | odd | 2 | 1 | 1776.2.h.e | 4 | ||
| 444.g | even | 2 | 1 | 5328.2.h.l | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 111.2.c.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 111.2.c.b | ✓ | 4 | 37.b | even | 2 | 1 | inner |
| 333.2.c.c | 4 | 3.b | odd | 2 | 1 | ||
| 333.2.c.c | 4 | 111.d | odd | 2 | 1 | ||
| 1776.2.h.e | 4 | 4.b | odd | 2 | 1 | ||
| 1776.2.h.e | 4 | 148.b | odd | 2 | 1 | ||
| 4107.2.a.g | 4 | 37.d | odd | 4 | 2 | ||
| 5328.2.h.l | 4 | 12.b | even | 2 | 1 | ||
| 5328.2.h.l | 4 | 444.g | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 6T_{2}^{2} + 3 \)
acting on \(S_{2}^{\mathrm{new}}(111, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 6T^{2} + 3 \)
|
| $3$ |
\( (T - 1)^{4} \)
|
| $5$ |
\( T^{4} + 12T^{2} + 12 \)
|
| $7$ |
\( (T + 2)^{4} \)
|
| $11$ |
\( (T^{2} - 24)^{2} \)
|
| $13$ |
\( T^{4} + 48T^{2} + 192 \)
|
| $17$ |
\( T^{4} + 12T^{2} + 12 \)
|
| $19$ |
\( T^{4} + 24T^{2} + 48 \)
|
| $23$ |
\( T^{4} + 36T^{2} + 300 \)
|
| $29$ |
\( T^{4} + 108T^{2} + 12 \)
|
| $31$ |
\( T^{4} + 72T^{2} + 1200 \)
|
| $37$ |
\( T^{4} - 4 T^{3} + \cdots + 1369 \)
|
| $41$ |
\( (T^{2} + 12 T + 12)^{2} \)
|
| $43$ |
\( T^{4} + 264 T^{2} + 17328 \)
|
| $47$ |
\( (T^{2} - 24)^{2} \)
|
| $53$ |
\( (T^{2} - 12 T + 12)^{2} \)
|
| $59$ |
\( T^{4} + 36T^{2} + 300 \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( (T + 2)^{4} \)
|
| $71$ |
\( (T^{2} - 24)^{2} \)
|
| $73$ |
\( (T^{2} + 16 T + 40)^{2} \)
|
| $79$ |
\( T^{4} + 24T^{2} + 48 \)
|
| $83$ |
\( (T^{2} - 96)^{2} \)
|
| $89$ |
\( T^{4} + 108T^{2} + 12 \)
|
| $97$ |
\( T^{4} + 144T^{2} + 4800 \)
|
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