Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [231,3,Mod(230,231)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(231, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("231.230");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 231 = 3 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 231.h (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: | yes |
| Analytic conductor: | \(6.29429410672\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.6237.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 12x - 5 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - 12x - 5 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/231\mathbb{Z}\right)^\times\).
| \(n\) | \(155\) | \(199\) | \(211\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 230.1 |
|
−3.65617 | −3.00000 | 9.36755 | 9.02372 | 10.9685 | 7.00000 | −19.6247 | 9.00000 | −32.9922 | |||||||||||||||||||||||||||
| 230.2 | 0.422973 | −3.00000 | −3.82109 | −8.24407 | −1.26892 | 7.00000 | −3.30811 | 9.00000 | −3.48702 | ||||||||||||||||||||||||||||
| 230.3 | 3.23319 | −3.00000 | 6.45354 | −0.779652 | −9.69958 | 7.00000 | 7.93277 | 9.00000 | −2.52077 | ||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 231.h | odd | 2 | 1 | CM by \(\Q(\sqrt{-231}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 231.3.h.b | yes | 3 |
| 3.b | odd | 2 | 1 | 231.3.h.d | yes | 3 | |
| 7.b | odd | 2 | 1 | 231.3.h.c | yes | 3 | |
| 11.b | odd | 2 | 1 | 231.3.h.a | ✓ | 3 | |
| 21.c | even | 2 | 1 | 231.3.h.a | ✓ | 3 | |
| 33.d | even | 2 | 1 | 231.3.h.c | yes | 3 | |
| 77.b | even | 2 | 1 | 231.3.h.d | yes | 3 | |
| 231.h | odd | 2 | 1 | CM | 231.3.h.b | yes | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 231.3.h.a | ✓ | 3 | 11.b | odd | 2 | 1 | |
| 231.3.h.a | ✓ | 3 | 21.c | even | 2 | 1 | |
| 231.3.h.b | yes | 3 | 1.a | even | 1 | 1 | trivial |
| 231.3.h.b | yes | 3 | 231.h | odd | 2 | 1 | CM |
| 231.3.h.c | yes | 3 | 7.b | odd | 2 | 1 | |
| 231.3.h.c | yes | 3 | 33.d | even | 2 | 1 | |
| 231.3.h.d | yes | 3 | 3.b | odd | 2 | 1 | |
| 231.3.h.d | yes | 3 | 77.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 \(S_{3}^{\mathrm{new}}(231, [\chi])\):
|
\( T_{2}^{3} - 12T_{2} + 5 \)
|
|
\( T_{5}^{3} - 75T_{5} - 58 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 12T + 5 \)
|
| $3$ |
\( (T + 3)^{3} \)
|
| $5$ |
\( T^{3} - 75T - 58 \)
|
| $7$ |
\( (T - 7)^{3} \)
|
| $11$ |
\( (T - 11)^{3} \)
|
| $13$ |
\( T^{3} - 507T - 1094 \)
|
| $17$ |
\( T^{3} \)
|
| $19$ |
\( T^{3} - 1083T - 13190 \)
|
| $23$ |
\( T^{3} \)
|
| $29$ |
\( T^{3} - 2523T + 3722 \)
|
| $31$ |
\( T^{3} \)
|
| $37$ |
\( T^{3} - 4107T - 97610 \)
|
| $41$ |
\( T^{3} \)
|
| $43$ |
\( T^{3} \)
|
| $47$ |
\( T^{3} - 6627 T + 176846 \)
|
| $53$ |
\( T^{3} \)
|
| $59$ |
\( T^{3} - 10443 T - 185530 \)
|
| $61$ |
\( (T + 10)^{3} \)
|
| $67$ |
\( T^{3} - 13467 T - 556250 \)
|
| $71$ |
\( T^{3} \)
|
| $73$ |
\( T^{3} - 15987 T + 733234 \)
|
| $79$ |
\( T^{3} \)
|
| $83$ |
\( T^{3} \)
|
| $89$ |
\( (T - 130)^{3} \)
|
| $97$ |
\( T^{3} \)
|
| show more | |
| show less |