Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [363,3,Mod(122,363)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(363, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("363.122");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 363 = 3 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 363.b (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: | \(9.89103359628\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 3x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 33) |
| 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
\( x^{4} + 3x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 2\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 2\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).
| \(n\) | \(122\) | \(244\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 122.1 |
|
− | 1.00000i | −2.00000 | − | 2.23607i | 3.00000 | 4.38197i | −2.23607 | + | 2.00000i | 2.14590 | − | 7.00000i | −1.00000 | + | 8.94427i | 4.38197 | ||||||||||||||||||||||
| 122.2 | − | 1.00000i | −2.00000 | + | 2.23607i | 3.00000 | 6.61803i | 2.23607 | + | 2.00000i | 8.85410 | − | 7.00000i | −1.00000 | − | 8.94427i | 6.61803 | |||||||||||||||||||||||
| 122.3 | 1.00000i | −2.00000 | − | 2.23607i | 3.00000 | − | 6.61803i | 2.23607 | − | 2.00000i | 8.85410 | 7.00000i | −1.00000 | + | 8.94427i | 6.61803 | ||||||||||||||||||||||||
| 122.4 | 1.00000i | −2.00000 | + | 2.23607i | 3.00000 | − | 4.38197i | −2.23607 | − | 2.00000i | 2.14590 | 7.00000i | −1.00000 | − | 8.94427i | 4.38197 | ||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 363.3.b.g | 4 | |
| 3.b | odd | 2 | 1 | inner | 363.3.b.g | 4 | |
| 11.b | odd | 2 | 1 | 363.3.b.f | 4 | ||
| 11.c | even | 5 | 2 | 363.3.h.g | 8 | ||
| 11.c | even | 5 | 2 | 363.3.h.i | 8 | ||
| 11.d | odd | 10 | 2 | 33.3.h.a | ✓ | 8 | |
| 11.d | odd | 10 | 2 | 363.3.h.h | 8 | ||
| 33.d | even | 2 | 1 | 363.3.b.f | 4 | ||
| 33.f | even | 10 | 2 | 33.3.h.a | ✓ | 8 | |
| 33.f | even | 10 | 2 | 363.3.h.h | 8 | ||
| 33.h | odd | 10 | 2 | 363.3.h.g | 8 | ||
| 33.h | odd | 10 | 2 | 363.3.h.i | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 33.3.h.a | ✓ | 8 | 11.d | odd | 10 | 2 | |
| 33.3.h.a | ✓ | 8 | 33.f | even | 10 | 2 | |
| 363.3.b.f | 4 | 11.b | odd | 2 | 1 | ||
| 363.3.b.f | 4 | 33.d | even | 2 | 1 | ||
| 363.3.b.g | 4 | 1.a | even | 1 | 1 | trivial | |
| 363.3.b.g | 4 | 3.b | odd | 2 | 1 | inner | |
| 363.3.h.g | 8 | 11.c | even | 5 | 2 | ||
| 363.3.h.g | 8 | 33.h | odd | 10 | 2 | ||
| 363.3.h.h | 8 | 11.d | odd | 10 | 2 | ||
| 363.3.h.h | 8 | 33.f | even | 10 | 2 | ||
| 363.3.h.i | 8 | 11.c | even | 5 | 2 | ||
| 363.3.h.i | 8 | 33.h | odd | 10 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(363, [\chi])\):
|
\( T_{2}^{2} + 1 \)
|
|
\( T_{5}^{4} + 63T_{5}^{2} + 841 \)
|
|
\( T_{7}^{2} - 11T_{7} + 19 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{2} \)
|
| $3$ |
\( (T^{2} + 4 T + 9)^{2} \)
|
| $5$ |
\( T^{4} + 63T^{2} + 841 \)
|
| $7$ |
\( (T^{2} - 11 T + 19)^{2} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( (T^{2} - 22 T + 116)^{2} \)
|
| $17$ |
\( (T^{2} + 36)^{2} \)
|
| $19$ |
\( (T^{2} - 22 T - 284)^{2} \)
|
| $23$ |
\( T^{4} + 1008 T^{2} + 215296 \)
|
| $29$ |
\( T^{4} + 2943 T^{2} + 1565001 \)
|
| $31$ |
\( (T^{2} + 37 T + 191)^{2} \)
|
| $37$ |
\( (T^{2} - 10 T - 580)^{2} \)
|
| $41$ |
\( T^{4} + 1932 T^{2} + 59536 \)
|
| $43$ |
\( (T^{2} - 66 T + 1044)^{2} \)
|
| $47$ |
\( T^{4} + 332T^{2} + 5776 \)
|
| $53$ |
\( T^{4} + 623 T^{2} + 63001 \)
|
| $59$ |
\( T^{4} + 1575 T^{2} + 525625 \)
|
| $61$ |
\( (T^{2} - 66 T - 2556)^{2} \)
|
| $67$ |
\( (T^{2} + 6 T - 5436)^{2} \)
|
| $71$ |
\( T^{4} + 12108 T^{2} + 33686416 \)
|
| $73$ |
\( (T^{2} + 77 T + 931)^{2} \)
|
| $79$ |
\( (T^{2} + 55 T - 7445)^{2} \)
|
| $83$ |
\( T^{4} + 24503 T^{2} + 150087001 \)
|
| $89$ |
\( T^{4} + 9632 T^{2} + 891136 \)
|
| $97$ |
\( (T^{2} + 21 T - 12141)^{2} \)
|
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