Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2156,2,Mod(177,2156)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2156, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 2, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2156.177");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2156 = 2^{2} \cdot 7^{2} \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2156.i (of order \(3\), degree \(2\), not 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: | \(17.2157466758\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.27870912.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 7x^{4} + 2x^{3} + 38x^{2} - 12x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 308) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{6} - x^{5} + 7x^{4} + 2x^{3} + 38x^{2} - 12x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 7\nu^{4} + 49\nu^{3} - 38\nu^{2} + 12\nu - 84 ) / 254 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -7\nu^{5} + 49\nu^{4} - 89\nu^{3} + 266\nu^{2} - 84\nu + 1096 ) / 254 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 21\nu^{5} - 20\nu^{4} + 140\nu^{3} + 91\nu^{2} + 760\nu + 14 ) / 254 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -85\nu^{5} + 87\nu^{4} - 609\nu^{3} - 72\nu^{2} - 3306\nu + 1044 ) / 254 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} + 4\beta_{4} + \beta_{2} + \beta _1 - 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 7\beta_{2} - 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( -7\beta_{5} - 26\beta_{4} + 7\beta_{3} - 11\beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( -11\beta_{5} - 30\beta_{4} - 51\beta_{2} - 51\beta _1 + 30 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2156\mathbb{Z}\right)^\times\).
| \(n\) | \(981\) | \(1079\) | \(1277\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-\beta_{4}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 177.1 |
|
0 | −1.08870 | − | 1.88569i | 0 | 0.370556 | − | 0.641823i | 0 | 0 | 0 | −0.870556 | + | 1.50785i | 0 | ||||||||||||||||||||||||||||||
| 177.2 | 0 | 0.160819 | + | 0.278546i | 0 | −1.94827 | + | 3.37451i | 0 | 0 | 0 | 1.44827 | − | 2.50849i | 0 | |||||||||||||||||||||||||||||||
| 177.3 | 0 | 1.42789 | + | 2.47317i | 0 | 2.07772 | − | 3.59871i | 0 | 0 | 0 | −2.57772 | + | 4.46474i | 0 | |||||||||||||||||||||||||||||||
| 1145.1 | 0 | −1.08870 | + | 1.88569i | 0 | 0.370556 | + | 0.641823i | 0 | 0 | 0 | −0.870556 | − | 1.50785i | 0 | |||||||||||||||||||||||||||||||
| 1145.2 | 0 | 0.160819 | − | 0.278546i | 0 | −1.94827 | − | 3.37451i | 0 | 0 | 0 | 1.44827 | + | 2.50849i | 0 | |||||||||||||||||||||||||||||||
| 1145.3 | 0 | 1.42789 | − | 2.47317i | 0 | 2.07772 | + | 3.59871i | 0 | 0 | 0 | −2.57772 | − | 4.46474i | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2156.2.i.m | 6 | |
| 7.b | odd | 2 | 1 | 2156.2.i.k | 6 | ||
| 7.c | even | 3 | 1 | 308.2.a.c | ✓ | 3 | |
| 7.c | even | 3 | 1 | inner | 2156.2.i.m | 6 | |
| 7.d | odd | 6 | 1 | 2156.2.a.j | 3 | ||
| 7.d | odd | 6 | 1 | 2156.2.i.k | 6 | ||
| 21.h | odd | 6 | 1 | 2772.2.a.s | 3 | ||
| 28.f | even | 6 | 1 | 8624.2.a.cj | 3 | ||
| 28.g | odd | 6 | 1 | 1232.2.a.r | 3 | ||
| 35.j | even | 6 | 1 | 7700.2.a.y | 3 | ||
| 35.l | odd | 12 | 2 | 7700.2.e.p | 6 | ||
| 56.k | odd | 6 | 1 | 4928.2.a.bx | 3 | ||
| 56.p | even | 6 | 1 | 4928.2.a.ca | 3 | ||
| 77.h | odd | 6 | 1 | 3388.2.a.o | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 308.2.a.c | ✓ | 3 | 7.c | even | 3 | 1 | |
| 1232.2.a.r | 3 | 28.g | odd | 6 | 1 | ||
| 2156.2.a.j | 3 | 7.d | odd | 6 | 1 | ||
| 2156.2.i.k | 6 | 7.b | odd | 2 | 1 | ||
| 2156.2.i.k | 6 | 7.d | odd | 6 | 1 | ||
| 2156.2.i.m | 6 | 1.a | even | 1 | 1 | trivial | |
| 2156.2.i.m | 6 | 7.c | even | 3 | 1 | inner | |
| 2772.2.a.s | 3 | 21.h | odd | 6 | 1 | ||
| 3388.2.a.o | 3 | 77.h | odd | 6 | 1 | ||
| 4928.2.a.bx | 3 | 56.k | odd | 6 | 1 | ||
| 4928.2.a.ca | 3 | 56.p | even | 6 | 1 | ||
| 7700.2.a.y | 3 | 35.j | even | 6 | 1 | ||
| 7700.2.e.p | 6 | 35.l | odd | 12 | 2 | ||
| 8624.2.a.cj | 3 | 28.f | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2156, [\chi])\):
|
\( T_{3}^{6} - T_{3}^{5} + 7T_{3}^{4} + 2T_{3}^{3} + 38T_{3}^{2} - 12T_{3} + 4 \)
|
|
\( T_{5}^{6} - T_{5}^{5} + 17T_{5}^{4} - 8T_{5}^{3} + 268T_{5}^{2} - 192T_{5} + 144 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
|
| $3$ |
\( T^{6} - T^{5} + 7 T^{4} + \cdots + 4 \)
|
| $5$ |
\( T^{6} - T^{5} + \cdots + 144 \)
|
| $7$ |
\( T^{6} \)
|
| $11$ |
\( (T^{2} + T + 1)^{3} \)
|
| $13$ |
\( (T^{3} - 12 T^{2} + 34 T + 8)^{2} \)
|
| $17$ |
\( T^{6} + 2 T^{5} + \cdots + 24336 \)
|
| $19$ |
\( T^{6} - 2 T^{5} + \cdots + 256 \)
|
| $23$ |
\( T^{6} - 7 T^{5} + \cdots + 5184 \)
|
| $29$ |
\( (T^{3} - 6 T^{2} - 44 T - 24)^{2} \)
|
| $31$ |
\( T^{6} - 9 T^{5} + \cdots + 21316 \)
|
| $37$ |
\( T^{6} + 7 T^{5} + \cdots + 1024 \)
|
| $41$ |
\( (T^{3} + 2 T^{2} + \cdots - 156)^{2} \)
|
| $43$ |
\( (T^{3} + 4 T^{2} - 20 T - 32)^{2} \)
|
| $47$ |
\( T^{6} - 4 T^{5} + \cdots + 9216 \)
|
| $53$ |
\( T^{6} + 2 T^{5} + \cdots + 9216 \)
|
| $59$ |
\( T^{6} - 23 T^{5} + \cdots + 161604 \)
|
| $61$ |
\( T^{6} + 14 T^{5} + \cdots + 839056 \)
|
| $67$ |
\( T^{6} - 5 T^{5} + \cdots + 64 \)
|
| $71$ |
\( (T^{3} + 5 T^{2} + \cdots + 312)^{2} \)
|
| $73$ |
\( T^{6} - 14 T^{5} + \cdots + 59536 \)
|
| $79$ |
\( T^{6} + 14 T^{5} + \cdots + 876096 \)
|
| $83$ |
\( (T^{3} - 4 T^{2} + \cdots + 1248)^{2} \)
|
| $89$ |
\( T^{6} - 19 T^{5} + \cdots + 1648656 \)
|
| $97$ |
\( (T^{3} - 15 T^{2} + 16 T - 4)^{2} \)
|
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