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: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 21x^{4} + 111x^{2} + 47 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 3 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 21x^{4} + 111x^{2} + 47 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} + 16\nu^{3} + 10\nu^{2} + 55\nu + 1 ) / 12 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + \nu^{2} - 11\nu + 7 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} + \nu^{4} + 22\nu^{3} + 16\nu^{2} + 109\nu + 43 ) / 12 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} + 5\nu^{4} - 16\nu^{3} + 62\nu^{2} - 55\nu + 65 ) / 12 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 - 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - \beta_{3} - \beta_{2} - 10\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{5} - 12\beta_{4} - 12\beta_{3} + 14\beta_{2} - 12\beta _1 + 73 \)
|
| \(\nu^{5}\) | \(=\) |
\( -2\beta_{5} - 14\beta_{4} + 18\beta_{3} + 24\beta_{2} + 107\beta _1 - 4 \)
|
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 |
|
− | 3.50091i | −1.38788 | + | 2.65966i | −8.25635 | − | 7.89925i | 9.31122 | + | 4.85883i | −1.81458 | 14.9011i | −5.14758 | − | 7.38257i | −27.6545 | ||||||||||||||||||||||||||||
| 122.2 | − | 2.87759i | 2.10316 | − | 2.13932i | −4.28055 | 4.94788i | −6.15610 | − | 6.05206i | 11.1801 | 0.807314i | −0.153394 | − | 8.99869i | 14.2380 | ||||||||||||||||||||||||||||||
| 122.3 | − | 0.680517i | 1.28471 | + | 2.71100i | 3.53690 | 6.49002i | 1.84488 | − | 0.874269i | −9.36551 | − | 5.12898i | −5.69902 | + | 6.96571i | 4.41657 | |||||||||||||||||||||||||||||
| 122.4 | 0.680517i | 1.28471 | − | 2.71100i | 3.53690 | − | 6.49002i | 1.84488 | + | 0.874269i | −9.36551 | 5.12898i | −5.69902 | − | 6.96571i | 4.41657 | ||||||||||||||||||||||||||||||
| 122.5 | 2.87759i | 2.10316 | + | 2.13932i | −4.28055 | − | 4.94788i | −6.15610 | + | 6.05206i | 11.1801 | − | 0.807314i | −0.153394 | + | 8.99869i | 14.2380 | |||||||||||||||||||||||||||||
| 122.6 | 3.50091i | −1.38788 | − | 2.65966i | −8.25635 | 7.89925i | 9.31122 | − | 4.85883i | −1.81458 | − | 14.9011i | −5.14758 | + | 7.38257i | −27.6545 | ||||||||||||||||||||||||||||||
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.k | yes | 6 |
| 3.b | odd | 2 | 1 | inner | 363.3.b.k | yes | 6 |
| 11.b | odd | 2 | 1 | 363.3.b.j | ✓ | 6 | |
| 11.c | even | 5 | 4 | 363.3.h.p | 24 | ||
| 11.d | odd | 10 | 4 | 363.3.h.q | 24 | ||
| 33.d | even | 2 | 1 | 363.3.b.j | ✓ | 6 | |
| 33.f | even | 10 | 4 | 363.3.h.q | 24 | ||
| 33.h | odd | 10 | 4 | 363.3.h.p | 24 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 363.3.b.j | ✓ | 6 | 11.b | odd | 2 | 1 | |
| 363.3.b.j | ✓ | 6 | 33.d | even | 2 | 1 | |
| 363.3.b.k | yes | 6 | 1.a | even | 1 | 1 | trivial |
| 363.3.b.k | yes | 6 | 3.b | odd | 2 | 1 | inner |
| 363.3.h.p | 24 | 11.c | even | 5 | 4 | ||
| 363.3.h.p | 24 | 33.h | odd | 10 | 4 | ||
| 363.3.h.q | 24 | 11.d | odd | 10 | 4 | ||
| 363.3.h.q | 24 | 33.f | even | 10 | 4 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(363, [\chi])\):
|
\( T_{2}^{6} + 21T_{2}^{4} + 111T_{2}^{2} + 47 \)
|
|
\( T_{5}^{6} + 129T_{5}^{4} + 5187T_{5}^{2} + 64343 \)
|
|
\( T_{7}^{3} - 108T_{7} - 190 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 21 T^{4} + \cdots + 47 \)
|
| $3$ |
\( T^{6} - 4 T^{5} + \cdots + 729 \)
|
| $5$ |
\( T^{6} + 129 T^{4} + \cdots + 64343 \)
|
| $7$ |
\( (T^{3} - 108 T - 190)^{2} \)
|
| $11$ |
\( T^{6} \)
|
| $13$ |
\( (T^{3} - 21 T^{2} + 69 T + 5)^{2} \)
|
| $17$ |
\( T^{6} + 909 T^{4} + \cdots + 47 \)
|
| $19$ |
\( (T^{3} + 42 T^{2} + \cdots + 1458)^{2} \)
|
| $23$ |
\( T^{6} + 900 T^{4} + \cdots + 2752508 \)
|
| $29$ |
\( T^{6} + 3669 T^{4} + \cdots + 593987175 \)
|
| $31$ |
\( (T^{3} - 1350 T - 18334)^{2} \)
|
| $37$ |
\( (T^{3} + 21 T^{2} + \cdots + 2115)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 11934438575 \)
|
| $43$ |
\( (T^{3} - 78 T^{2} + \cdots + 180)^{2} \)
|
| $47$ |
\( T^{6} + 1080 T^{4} + \cdots + 18183548 \)
|
| $53$ |
\( T^{6} + \cdots + 16518150423 \)
|
| $59$ |
\( T^{6} + \cdots + 2211801200 \)
|
| $61$ |
\( (T^{3} + 132 T^{2} + \cdots + 54576)^{2} \)
|
| $67$ |
\( (T^{3} - 12 T^{2} + \cdots + 259810)^{2} \)
|
| $71$ |
\( T^{6} + 2868 T^{4} + \cdots + 1522800 \)
|
| $73$ |
\( (T^{3} + 36 T^{2} + \cdots - 150820)^{2} \)
|
| $79$ |
\( (T^{3} - 9630 T - 356894)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 280649740572 \)
|
| $89$ |
\( T^{6} + \cdots + 1393457175 \)
|
| $97$ |
\( (T^{3} - 63 T^{2} + \cdots - 563825)^{2} \)
|
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