Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [161,4,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, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("161.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 161 = 7 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| 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: | \(9.49930751092\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 18x^{3} - 4x^{2} + 44x + 24 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{5} - x^{4} - 18x^{3} - 4x^{2} + 44x + 24 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} - 14\nu - 4 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 15\nu^{2} + 10\nu + 24 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 7 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + \beta_{2} + 16\beta _1 + 11 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} + 4\beta_{3} + 17\beta_{2} + 52\beta _1 + 103 \)
|
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 |
|
−4.14816 | −4.24155 | 9.20723 | −7.96567 | 17.5946 | 7.00000 | −5.00776 | −9.00922 | 33.0429 | |||||||||||||||||||||||||||||||||
| 1.2 | −2.59849 | 5.80739 | −1.24784 | −7.55955 | −15.0904 | 7.00000 | 24.0304 | 6.72576 | 19.6434 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.59486 | −8.64488 | −5.45643 | 11.1013 | 13.7874 | 7.00000 | 21.4611 | 47.7340 | −17.7050 | ||||||||||||||||||||||||||||||||||
| 1.4 | 0.743749 | 0.765559 | −7.44684 | 3.68128 | 0.569384 | 7.00000 | −11.4886 | −26.4139 | 2.73795 | ||||||||||||||||||||||||||||||||||
| 1.5 | 3.59776 | −4.68651 | 4.94388 | −3.25737 | −16.8609 | 7.00000 | −10.9952 | −5.03662 | −11.7192 | ||||||||||||||||||||||||||||||||||
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 | 161.4.a.a | ✓ | 5 |
| 3.b | odd | 2 | 1 | 1449.4.a.e | 5 | ||
| 7.b | odd | 2 | 1 | 1127.4.a.d | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 161.4.a.a | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 1127.4.a.d | 5 | 7.b | odd | 2 | 1 | ||
| 1449.4.a.e | 5 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 4T_{2}^{4} - 12T_{2}^{3} - 54T_{2}^{2} - 17T_{2} + 46 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(161))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 4 T^{4} + \cdots + 46 \)
|
| $3$ |
\( T^{5} + 11 T^{4} + \cdots + 764 \)
|
| $5$ |
\( T^{5} + 4 T^{4} + \cdots + 8016 \)
|
| $7$ |
\( (T - 7)^{5} \)
|
| $11$ |
\( T^{5} + 36 T^{4} + \cdots - 2036752 \)
|
| $13$ |
\( T^{5} + 69 T^{4} + \cdots - 142624 \)
|
| $17$ |
\( T^{5} + 42 T^{4} + \cdots + 69249152 \)
|
| $19$ |
\( T^{5} + 140 T^{4} + \cdots + 5938176 \)
|
| $23$ |
\( (T + 23)^{5} \)
|
| $29$ |
\( T^{5} + \cdots - 4712427144 \)
|
| $31$ |
\( T^{5} + \cdots + 1002629384 \)
|
| $37$ |
\( T^{5} + \cdots + 433525518848 \)
|
| $41$ |
\( T^{5} + \cdots - 341026572752 \)
|
| $43$ |
\( T^{5} + \cdots + 215123486848 \)
|
| $47$ |
\( T^{5} + \cdots + 533217618272 \)
|
| $53$ |
\( T^{5} + \cdots - 22043277353472 \)
|
| $59$ |
\( T^{5} + \cdots + 3783080243952 \)
|
| $61$ |
\( T^{5} + \cdots + 16114431705048 \)
|
| $67$ |
\( T^{5} + \cdots + 91664243429376 \)
|
| $71$ |
\( T^{5} + \cdots + 109108067649024 \)
|
| $73$ |
\( T^{5} + \cdots + 32974565300272 \)
|
| $79$ |
\( T^{5} + \cdots + 26006513682944 \)
|
| $83$ |
\( T^{5} + \cdots - 3541881302016 \)
|
| $89$ |
\( T^{5} + \cdots - 7355431797264 \)
|
| $97$ |
\( T^{5} + \cdots - 80961807928968 \)
|
| show more | |
| show less |