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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{21}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 5 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.
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 | −4.79129 | 0 | 18.9564 | 0 | −8.95644 | 0 | −4.04356 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | −0.208712 | 0 | −3.95644 | 0 | 13.9564 | 0 | −26.9564 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(17\) | \(1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1156.4.a.b | ✓ | 2 |
| 17.b | even | 2 | 1 | 1156.4.a.f | yes | 2 | |
| 17.c | even | 4 | 2 | 1156.4.b.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1156.4.a.b | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 1156.4.a.f | yes | 2 | 17.b | even | 2 | 1 | |
| 1156.4.b.d | 4 | 17.c | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 5T_{3} + 1 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1156))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 5T + 1 \)
|
| $5$ |
\( T^{2} - 15T - 75 \)
|
| $7$ |
\( T^{2} - 5T - 125 \)
|
| $11$ |
\( T^{2} + 18T - 2019 \)
|
| $13$ |
\( T^{2} - 85T - 89 \)
|
| $17$ |
\( T^{2} \)
|
| $19$ |
\( T^{2} + 59T - 5561 \)
|
| $23$ |
\( T^{2} - 15T - 1839 \)
|
| $29$ |
\( T^{2} + 249T - 6681 \)
|
| $31$ |
\( T^{2} - 260T + 12175 \)
|
| $37$ |
\( T^{2} + 130T - 101636 \)
|
| $41$ |
\( (T + 366)^{2} \)
|
| $43$ |
\( T^{2} - 160T + 1675 \)
|
| $47$ |
\( T^{2} + 150T - 230331 \)
|
| $53$ |
\( T^{2} + 345T - 11829 \)
|
| $59$ |
\( T^{2} - 756T + 134484 \)
|
| $61$ |
\( T^{2} + 532T - 17969 \)
|
| $67$ |
\( T^{2} - 850T + 102484 \)
|
| $71$ |
\( T^{2} - 192T - 916884 \)
|
| $73$ |
\( T^{2} - 320T - 133349 \)
|
| $79$ |
\( T^{2} + 1864 T + 849724 \)
|
| $83$ |
\( T^{2} + 585T - 4539 \)
|
| $89$ |
\( T^{2} - 1416 T + 146364 \)
|
| $97$ |
\( T^{2} + 2005 T + 957625 \)
|
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