Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1156,4,Mod(1,1156)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1156, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1156.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1156 = 2^{2} \cdot 17^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1156.a (trivial) |
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: | \(68.2062079666\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.5999648.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 21x^{2} + 2 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | no (minimal twist has level 68) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{4} - 21x^{2} + 2 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{3} - 42\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( 4\nu^{2} - 42 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 42 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{2} + 21\beta_1 ) / 2 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | −9.14425 | 0 | 0.874867 | 0 | 8.26938 | 0 | 56.6173 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.618624 | 0 | 12.9319 | 0 | −12.3133 | 0 | −26.6173 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 0.618624 | 0 | −12.9319 | 0 | 12.3133 | 0 | −26.6173 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 9.14425 | 0 | −0.874867 | 0 | −8.26938 | 0 | 56.6173 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(17\) | \(1\) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 17.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1156.4.a.h | 4 | |
| 17.b | even | 2 | 1 | inner | 1156.4.a.h | 4 | |
| 17.c | even | 4 | 2 | 68.4.b.a | ✓ | 4 | |
| 51.f | odd | 4 | 2 | 612.4.b.b | 4 | ||
| 68.f | odd | 4 | 2 | 272.4.b.c | 4 | ||
| 85.f | odd | 4 | 2 | 1700.4.g.a | 8 | ||
| 85.i | odd | 4 | 2 | 1700.4.g.a | 8 | ||
| 85.j | even | 4 | 2 | 1700.4.c.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 68.4.b.a | ✓ | 4 | 17.c | even | 4 | 2 | |
| 272.4.b.c | 4 | 68.f | odd | 4 | 2 | ||
| 612.4.b.b | 4 | 51.f | odd | 4 | 2 | ||
| 1156.4.a.h | 4 | 1.a | even | 1 | 1 | trivial | |
| 1156.4.a.h | 4 | 17.b | even | 2 | 1 | inner | |
| 1700.4.c.a | 4 | 85.j | even | 4 | 2 | ||
| 1700.4.g.a | 8 | 85.f | odd | 4 | 2 | ||
| 1700.4.g.a | 8 | 85.i | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 84T_{3}^{2} + 32 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1156))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 84T^{2} + 32 \)
|
| $5$ |
\( T^{4} - 168T^{2} + 128 \)
|
| $7$ |
\( T^{4} - 220 T^{2} + 10368 \)
|
| $11$ |
\( T^{4} - 692 T^{2} + 34848 \)
|
| $13$ |
\( (T^{2} + 16 T - 1668)^{2} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( (T^{2} + 80 T - 5328)^{2} \)
|
| $23$ |
\( T^{4} - 47996 T^{2} + 526436352 \)
|
| $29$ |
\( T^{4} - 41000 T^{2} + 208080000 \)
|
| $31$ |
\( T^{4} + \cdots + 1351168128 \)
|
| $37$ |
\( T^{4} + \cdots + 4128133248 \)
|
| $41$ |
\( T^{4} + \cdots + 3053242368 \)
|
| $43$ |
\( (T^{2} - 72 T - 26416)^{2} \)
|
| $47$ |
\( (T^{2} - 560 T + 50688)^{2} \)
|
| $53$ |
\( (T^{2} + 12 T - 249372)^{2} \)
|
| $59$ |
\( (T^{2} - 168 T - 242352)^{2} \)
|
| $61$ |
\( T^{4} + \cdots + 107699974272 \)
|
| $67$ |
\( (T^{2} - 160 T - 55952)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 1666260992 \)
|
| $73$ |
\( T^{4} + \cdots + 27614380032 \)
|
| $79$ |
\( T^{4} + \cdots + 26013892608 \)
|
| $83$ |
\( (T^{2} + 1112 T + 198288)^{2} \)
|
| $89$ |
\( (T^{2} + 1568 T + 571356)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 1258180190208 \)
|
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