Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [161,2,Mod(1,161)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(161, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("161.1");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 161 = 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 161.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: | \(1.28559147254\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.2147108.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 2x^{4} - 9x^{3} + 17x^{2} + 16x - 27 \)
|
| 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 a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 2x^{4} - 9x^{3} + 17x^{2} + 16x - 27 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 8\nu^{2} + 5\nu + 11 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} + \nu^{3} - 10\nu^{2} - 5\nu + 19 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{4} - \beta_{3} + \beta_{2} + 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + \beta_{3} + 9\beta_{2} + 21 \)
|
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 |
|
−2.54577 | 2.46268 | 4.48096 | 2.78847 | −6.26943 | 1.00000 | −6.31597 | 3.06481 | −7.09882 | |||||||||||||||||||||||||||||||||
| 1.2 | −1.50216 | −3.04067 | 0.256481 | −3.82405 | 4.56757 | 1.00000 | 2.61904 | 6.24568 | 5.74433 | ||||||||||||||||||||||||||||||||||
| 1.3 | 1.23828 | 2.68857 | −0.466664 | −1.86253 | 3.32920 | 1.00000 | −3.05442 | 4.22838 | −2.30633 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.11948 | −1.84074 | 2.49221 | 2.40920 | −3.90141 | 1.00000 | 1.04322 | 0.388311 | 5.10626 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.69017 | −0.269842 | 5.23702 | −3.51109 | −0.725921 | 1.00000 | 8.70812 | −2.92719 | −9.44544 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(7\) | \(-1\) |
| \(23\) | \(1\) |
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 | 161.2.a.d | ✓ | 5 |
| 3.b | odd | 2 | 1 | 1449.2.a.r | 5 | ||
| 4.b | odd | 2 | 1 | 2576.2.a.bd | 5 | ||
| 5.b | even | 2 | 1 | 4025.2.a.p | 5 | ||
| 7.b | odd | 2 | 1 | 1127.2.a.h | 5 | ||
| 23.b | odd | 2 | 1 | 3703.2.a.j | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 161.2.a.d | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 1127.2.a.h | 5 | 7.b | odd | 2 | 1 | ||
| 1449.2.a.r | 5 | 3.b | odd | 2 | 1 | ||
| 2576.2.a.bd | 5 | 4.b | odd | 2 | 1 | ||
| 3703.2.a.j | 5 | 23.b | odd | 2 | 1 | ||
| 4025.2.a.p | 5 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 2T_{2}^{4} - 9T_{2}^{3} + 17T_{2}^{2} + 16T_{2} - 27 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(161))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 2 T^{4} + \cdots - 27 \)
|
| $3$ |
\( T^{5} - 13 T^{3} + \cdots + 10 \)
|
| $5$ |
\( T^{5} + 4 T^{4} + \cdots + 168 \)
|
| $7$ |
\( (T - 1)^{5} \)
|
| $11$ |
\( T^{5} + 4 T^{4} + \cdots - 48 \)
|
| $13$ |
\( T^{5} + 6 T^{4} + \cdots + 56 \)
|
| $17$ |
\( T^{5} + 12 T^{4} + \cdots - 1536 \)
|
| $19$ |
\( T^{5} - 6 T^{4} + \cdots + 128 \)
|
| $23$ |
\( (T + 1)^{5} \)
|
| $29$ |
\( T^{5} + 4 T^{4} + \cdots - 1452 \)
|
| $31$ |
\( T^{5} - 30 T^{4} + \cdots - 5206 \)
|
| $37$ |
\( T^{5} - 4 T^{4} + \cdots - 32 \)
|
| $41$ |
\( T^{5} - 6 T^{4} + \cdots - 456 \)
|
| $43$ |
\( T^{5} + 12 T^{4} + \cdots + 256 \)
|
| $47$ |
\( T^{5} - 10 T^{4} + \cdots + 11142 \)
|
| $53$ |
\( T^{5} - 16 T^{4} + \cdots - 480 \)
|
| $59$ |
\( T^{5} - 22 T^{4} + \cdots + 1440 \)
|
| $61$ |
\( T^{5} + 18 T^{4} + \cdots + 56 \)
|
| $67$ |
\( T^{5} + 2 T^{4} + \cdots + 61936 \)
|
| $71$ |
\( T^{5} - 4 T^{4} + \cdots - 5184 \)
|
| $73$ |
\( T^{5} + 2 T^{4} + \cdots - 27656 \)
|
| $79$ |
\( T^{5} - 30 T^{4} + \cdots + 1936 \)
|
| $83$ |
\( T^{5} - 8 T^{4} + \cdots - 5376 \)
|
| $89$ |
\( T^{5} + 20 T^{4} + \cdots - 4704 \)
|
| $97$ |
\( T^{5} + 12 T^{4} + \cdots + 4120 \)
|
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