Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1232,3,Mod(1231,1232)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1232, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 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("1232.1231");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1232 = 2^{4} \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1232.m (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: | \(33.5695685692\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{7}, \sqrt{11})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 9x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{6} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 9x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 4\nu^{2} - 18 \)
|
| \(\beta_{3}\) | \(=\) |
\( 8\nu^{3} - 70\nu \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 18 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} + 35\beta_1 ) / 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1232\mathbb{Z}\right)^\times\).
| \(n\) | \(309\) | \(353\) | \(463\) | \(673\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1231.1 |
|
0 | −5.96238 | 0 | 0 | 0 | −7.00000 | 0 | 26.5499 | 0 | ||||||||||||||||||||||||||||||
| 1231.2 | 0 | −0.670873 | 0 | 0 | 0 | −7.00000 | 0 | −8.54993 | 0 | |||||||||||||||||||||||||||||||
| 1231.3 | 0 | 0.670873 | 0 | 0 | 0 | −7.00000 | 0 | −8.54993 | 0 | |||||||||||||||||||||||||||||||
| 1231.4 | 0 | 5.96238 | 0 | 0 | 0 | −7.00000 | 0 | 26.5499 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 308.g | odd | 2 | 1 | CM by \(\Q(\sqrt{-77}) \) |
| 7.b | odd | 2 | 1 | inner |
| 44.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1232.3.m.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | 1232.3.m.b | yes | 4 | |
| 7.b | odd | 2 | 1 | inner | 1232.3.m.a | ✓ | 4 |
| 11.b | odd | 2 | 1 | 1232.3.m.b | yes | 4 | |
| 28.d | even | 2 | 1 | 1232.3.m.b | yes | 4 | |
| 44.c | even | 2 | 1 | inner | 1232.3.m.a | ✓ | 4 |
| 77.b | even | 2 | 1 | 1232.3.m.b | yes | 4 | |
| 308.g | odd | 2 | 1 | CM | 1232.3.m.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1232.3.m.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1232.3.m.a | ✓ | 4 | 7.b | odd | 2 | 1 | inner |
| 1232.3.m.a | ✓ | 4 | 44.c | even | 2 | 1 | inner |
| 1232.3.m.a | ✓ | 4 | 308.g | odd | 2 | 1 | CM |
| 1232.3.m.b | yes | 4 | 4.b | odd | 2 | 1 | |
| 1232.3.m.b | yes | 4 | 11.b | odd | 2 | 1 | |
| 1232.3.m.b | yes | 4 | 28.d | even | 2 | 1 | |
| 1232.3.m.b | yes | 4 | 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 \(S_{3}^{\mathrm{new}}(1232, [\chi])\):
|
\( T_{3}^{4} - 36T_{3}^{2} + 16 \)
|
|
\( T_{107} + 38 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 36T^{2} + 16 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( (T + 7)^{4} \)
|
| $11$ |
\( (T + 11)^{4} \)
|
| $13$ |
\( T^{4} - 676 T^{2} + 44944 \)
|
| $17$ |
\( T^{4} - 1156 T^{2} + 309136 \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( T^{4} - 3844 T^{2} + 3225616 \)
|
| $37$ |
\( (T^{2} - 2772)^{2} \)
|
| $41$ |
\( T^{4} - 6724 T^{2} + 7907344 \)
|
| $43$ |
\( (T^{2} - 2772)^{2} \)
|
| $47$ |
\( T^{4} - 8836 T^{2} + 3083536 \)
|
| $53$ |
\( (T^{2} - 2772)^{2} \)
|
| $59$ |
\( T^{4} - 13924 T^{2} + 10523536 \)
|
| $61$ |
\( T^{4} - 14884 T^{2} + 39790864 \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( T^{4} - 21316 T^{2} + 113124496 \)
|
| $79$ |
\( (T^{2} - 24948)^{2} \)
|
| $83$ |
\( T^{4} \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( T^{4} \)
|
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