Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1369,4,Mod(1,1369)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1369, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1369.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1369 = 37^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1369.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: | \(80.7736147979\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{5} - x^{4} - 27x^{3} + 3x^{2} + 176x + 144 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 37) |
| 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
\( x^{5} - x^{4} - 27x^{3} + 3x^{2} + 176x + 144 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 15\nu^{2} + 3\nu + 12 ) / 8 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 23\nu^{2} + 11\nu + 92 ) / 8 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + 5\nu^{3} + 23\nu^{2} - 71\nu - 140 ) / 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{3} + \beta_{2} + \beta _1 + 10 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} + 2\beta_{3} + 15\beta _1 + 12 \)
|
| \(\nu^{4}\) | \(=\) |
\( 2\beta_{4} - 13\beta_{3} + 23\beta_{2} + 27\beta _1 + 150 \)
|
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 |
|
−4.79672 | −2.86359 | 15.0086 | −0.463335 | 13.7359 | 1.01485 | −33.6181 | −18.7998 | 2.22249 | |||||||||||||||||||||||||||||||||
| 1.2 | −3.40805 | 7.32702 | 3.61479 | 17.2892 | −24.9708 | −15.1635 | 14.9450 | 26.6852 | −58.9226 | ||||||||||||||||||||||||||||||||||
| 1.3 | −1.96027 | 3.37210 | −4.15734 | −14.4391 | −6.61023 | 7.73331 | 23.8317 | −15.6289 | 28.3045 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.56768 | 9.45274 | −1.40700 | −1.50802 | 24.2716 | 12.6891 | −24.1542 | 62.3543 | −3.87211 | ||||||||||||||||||||||||||||||||||
| 1.5 | 3.59736 | −4.28827 | 4.94099 | −11.8788 | −15.4264 | 17.7262 | −11.0044 | −8.61076 | −42.7323 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(37\) | \(1\) |
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 | 1369.4.a.d | 5 | |
| 37.b | even | 2 | 1 | 37.4.a.b | ✓ | 5 | |
| 111.d | odd | 2 | 1 | 333.4.a.f | 5 | ||
| 148.b | odd | 2 | 1 | 592.4.a.g | 5 | ||
| 185.d | even | 2 | 1 | 925.4.a.b | 5 | ||
| 259.b | odd | 2 | 1 | 1813.4.a.c | 5 | ||
| 296.e | even | 2 | 1 | 2368.4.a.m | 5 | ||
| 296.h | odd | 2 | 1 | 2368.4.a.r | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 37.4.a.b | ✓ | 5 | 37.b | even | 2 | 1 | |
| 333.4.a.f | 5 | 111.d | odd | 2 | 1 | ||
| 592.4.a.g | 5 | 148.b | odd | 2 | 1 | ||
| 925.4.a.b | 5 | 185.d | even | 2 | 1 | ||
| 1369.4.a.d | 5 | 1.a | even | 1 | 1 | trivial | |
| 1813.4.a.c | 5 | 259.b | odd | 2 | 1 | ||
| 2368.4.a.m | 5 | 296.e | even | 2 | 1 | ||
| 2368.4.a.r | 5 | 296.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 4T_{2}^{4} - 21T_{2}^{3} - 74T_{2}^{2} + 102T_{2} + 296 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1369))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 4 T^{4} + \cdots + 296 \)
|
| $3$ |
\( T^{5} - 13 T^{4} + \cdots - 2868 \)
|
| $5$ |
\( T^{5} + 11 T^{4} + \cdots - 2072 \)
|
| $7$ |
\( T^{5} - 24 T^{4} + \cdots + 26768 \)
|
| $11$ |
\( T^{5} - 61 T^{4} + \cdots + 5135996 \)
|
| $13$ |
\( T^{5} - 37 T^{4} + \cdots + 7098184 \)
|
| $17$ |
\( T^{5} + \cdots - 1123875456 \)
|
| $19$ |
\( T^{5} - 22 T^{4} + \cdots + 69604224 \)
|
| $23$ |
\( T^{5} + \cdots - 14836947832 \)
|
| $29$ |
\( T^{5} + \cdots + 45459799656 \)
|
| $31$ |
\( T^{5} + \cdots - 11228647136 \)
|
| $37$ |
\( T^{5} \)
|
| $41$ |
\( T^{5} + \cdots - 762162331986 \)
|
| $43$ |
\( T^{5} + \cdots + 965840235296 \)
|
| $47$ |
\( T^{5} + \cdots - 11399037456 \)
|
| $53$ |
\( T^{5} + \cdots - 12174094047032 \)
|
| $59$ |
\( T^{5} + \cdots - 199509626624 \)
|
| $61$ |
\( T^{5} + \cdots - 5262207537472 \)
|
| $67$ |
\( T^{5} + \cdots - 69671300293888 \)
|
| $71$ |
\( T^{5} + \cdots - 105494295472656 \)
|
| $73$ |
\( T^{5} + \cdots + 24654270137062 \)
|
| $79$ |
\( T^{5} + \cdots - 429381365248 \)
|
| $83$ |
\( T^{5} + \cdots + 8663698210944 \)
|
| $89$ |
\( T^{5} + \cdots + 2194912910336 \)
|
| $97$ |
\( T^{5} + \cdots - 108205552171136 \)
|
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