LMFDB - Newform orbit 141.2.c.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [141,2,Mod(140,141)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(141, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("141.140");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 141 = 3 \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	141.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$1.12589066850$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	10.0.2239697333984375.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 9x^{5} + 32 $ x^10 - 9*x^5 + 32
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-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 $1,\beta_1,\ldots,\beta_{9}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_{3} q^{2} - \beta_1 q^{3} + ( - \beta_{5} - 2) q^{4} - \beta_{6} q^{6} + ( - \beta_{9} + \beta_{6}) q^{7} + ( - 2 \beta_{3} - \beta_{2} + \beta_1) q^{8} + ( - \beta_{4} + \beta_{3}) q^{9}+O(q^{10}) $ q + b3 * q^2 - b1 * q^3 + (-b5 - 2) * q^4 - b6 * q^6 + (-b9 + b6) * q^7 + (-2b3 - b2 + b1) q^8 + (-b4 + b3) * q^9	$ q + \beta_{3} q^{2} - \beta_1 q^{3} + ( - \beta_{5} - 2) q^{4} - \beta_{6} q^{6} + ( - \beta_{9} + \beta_{6}) q^{7} + ( - 2 \beta_{3} - \beta_{2} + \beta_1) q^{8} + ( - \beta_{4} + \beta_{3}) q^{9} + (\beta_{8} - \beta_{7} + \cdots + 2 \beta_1) q^{12}+ \cdots + (\beta_{9} + 3 \beta_{6} + \cdots + \beta_1) q^{98}+O(q^{100}) $ q + b3 * q^2 - b1 * q^3 + (-b5 - 2) * q^4 - b6 * q^6 + (-b9 + b6) * q^7 + (-2b3 - b2 + b1) q^8 + (-b4 + b3) * q^9 + (b8 - b7 - b3 + 2b1) q^12 + (2b7 + b5 - 2b1 + 1) * q^14 + (b8 + b6 + 2b5 + 2b4 - b3 + 4) * q^16 + (b9 - b8 - b5 + b2 + b1) * q^17 + (-b7 - b5 - b2 - 3) * q^18 + (b9 - b6 + b5 + b4 + b3 + b2) * q^21 + (-b9 - b7 + 2b6 + b4 + 3) q^24 - 5 * q^25 + (-b8 - b6 - b4 - b3) * q^27 + (2b9 - 4b6 - 2b4 + b3 + b2 + 3b1) * q^28 + (b9 - b8 + 2b7 + 4b3 + 2b2 - 2b1 + 1) * q^32 + (-b9 - 2b8 + b6 - 2b5 - 2b4 + b3 - b2 - 3b1) * q^34 + (-2b9 + b8 + 2b6 + 2b5 + b4 - 3b3 - b2 - b1) * q^36 + (b9 - b8 + b5 - b2 - 3b1) q^37 + (-b8 - b7 - 4b5 - 2b4 + 3b3 + 2b2 + b1 - 6) * q^42 + (-b9 + b8 - 2b7 - 1) q^47 + (-2b9 + b8 + 2b7 + 2b6 + b5 + b4 + 3b3 + b2 - 4b1) q^48 + (-b9 + b8 + 3b5 + b2 + 3b1 + 7) * q^49 - 5b3 q^50 + (2b9 + 2b7 + b4 - 3b3 + 3) q^51 + (b9 - 2b8 - 3b6 + 2b4) q^53 + (-b9 + b8 - b7 + 2b5 - b2 + 2b1 + 6) * q^54 + (b8 - 4b7 + 3b6 - 3b5 - 2b4 - b3 - 2b2 + 8b1 - 2) * q^56 + (-b9 + 2b8 + 3b6 - 2b4 + 2b3) * q^59 + (b9 + 2b8 + b6 + 4b4 - 2b3) q^61 + (2b7 - b5 - b2 - 3) q^63 + (3b9 - b8 - 4b6 - 4b5 - 2b4 + b3 - 8) * q^64 + (-b8 + 2b7 - 3b6 + 2b5 + 2b4 - 3b3 - 2b2 - 2b1 + 1) q^68 + (-b9 + b8 + b5 - 3b2 + b1) q^71 + (b9 + b8 + 2b7 - b6 + 4b5 + 2b4 + 2b3 + b2 - 4b1 + 6) q^72 + (-b9 - 3b6 + 2b5 + 2b4 + 3b3 + b2 - 5b1) q^74 + 5b1 q^75 + (-b9 + b8 - 3b5 + b2 + 3b1) * q^79 + (2b9 - 2b8 - b5 - b2) * q^81 + (-2b9 + 2b8 - 4b7 - 2) q^83 + (-b9 - 2b8 - b7 + b6 - 4b5 - b4 - 8b3 - 4b2 - 9) * q^84 + (-b9 + 2b8 + 3b6 - 2b4 - 4b3) * q^89 + (-3b9 + b8 + 4b6 + 2b4 - b3) q^94 + (3b9 - b8 + 2b7 - 4b6 - 4b5 - 2b4 + 2b2 + b1 - 6) * q^96 + (-3b9 - 2b8 + b6 - 4b4 + 2b3) * q^97 + (b9 + 3b6 - 2b5 - 2b4 + 12b3 + 3b2 + b1) q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 20 q^{4}+O(q^{10}) $ 10 * q - 20 * q^4	$ 10 q - 20 q^{4} + 5 q^{12} + 40 q^{16} - 25 q^{18} + 35 q^{24} - 50 q^{25} - 55 q^{42} - 10 q^{48} + 70 q^{49} + 20 q^{51} + 65 q^{54} - 40 q^{63} - 80 q^{64} + 50 q^{72} - 85 q^{84} - 70 q^{96}+O(q^{100}) $ 10 * q - 20 * q^4 + 5 * q^12 + 40 * q^16 - 25 * q^18 + 35 * q^24 - 50 * q^25 - 55 * q^42 - 10 * q^48 + 70 * q^49 + 20 * q^51 + 65 * q^54 - 40 * q^63 - 80 * q^64 + 50 * q^72 - 85 * q^84 - 70 * q^96

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{10} - 9x^{5} + 32 $ x^10 - 9*x^5 + 32 :

$\beta_{1}$	$=$	$ ( \nu^{7} - 5\nu^{2} ) / 4 $ (v^7 - 5*v^2) / 4
$\beta_{2}$	$=$	$ -\nu^{3} + \nu^{2} $ -v^3 + v^2
$\beta_{3}$	$=$	$ ( \nu^{9} - 9\nu^{4} + 16\nu ) / 16 $ (v^9 - 9v^4 + 16v) / 16
$\beta_{4}$	$=$	$ ( \nu^{9} - \nu^{4} + 8\nu ) / 8 $ (v^9 - v^4 + 8*v) / 8
$\beta_{5}$	$=$	$ ( \nu^{8} - 9\nu^{3} - 8\nu^{2} ) / 8 $ (v^8 - 9v^3 - 8v^2) / 8
$\beta_{6}$	$=$	$ ( \nu^{8} - 2\nu^{6} - 5\nu^{3} + 10\nu ) / 4 $ (v^8 - 2v^6 - 5v^3 + 10*v) / 4
$\beta_{7}$	$=$	$ ( -\nu^{8} + 4\nu^{5} + 5\nu^{3} - 20 ) / 4 $ (-v^8 + 4v^5 + 5v^3 - 20) / 4
$\beta_{8}$	$=$	$ ( -3\nu^{9} - 4\nu^{8} + 11\nu^{4} + 20\nu^{3} + 16\nu ) / 16 $ (-3v^9 - 4v^8 + 11v^4 + 20v^3 + 16*v) / 16
$\beta_{9}$	$=$	$ ( -3\nu^{9} + 4\nu^{8} + 11\nu^{4} - 20\nu^{3} + 16\nu ) / 16 $ (-3v^9 + 4v^8 + 11v^4 - 20v^3 + 16*v) / 16

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_{9} + \beta_{8} + 2\beta_{4} + 2\beta_{3} ) / 6 $ (b9 + b8 + 2b4 + 2b3) / 6
$\nu^{2}$	$=$	$ ( \beta_{9} - \beta_{8} - 4\beta_{5} + 2\beta_{2} ) / 6 $ (b9 - b8 - 4b5 + 2b2) / 6
$\nu^{3}$	$=$	$ ( \beta_{9} - \beta_{8} - 4\beta_{5} - 4\beta_{2} ) / 6 $ (b9 - b8 - 4b5 - 4b2) / 6
$\nu^{4}$	$=$	$ ( \beta_{9} + \beta_{8} + 8\beta_{4} - 10\beta_{3} ) / 6 $ (b9 + b8 + 8b4 - 10b3) / 6
$\nu^{5}$	$=$	$ ( \beta_{9} - \beta_{8} + 2\beta_{7} + 10 ) / 2 $ (b9 - b8 + 2*b7 + 10) / 2
$\nu^{6}$	$=$	$ ( 11\beta_{9} - \beta_{8} - 12\beta_{6} + 10\beta_{4} + 10\beta_{3} ) / 6 $ (11b9 - b8 - 12b6 + 10b4 + 10b3) / 6
$\nu^{7}$	$=$	$ ( 5\beta_{9} - 5\beta_{8} - 20\beta_{5} + 10\beta_{2} + 24\beta_1 ) / 6 $ (5b9 - 5b8 - 20b5 + 10b2 + 24*b1) / 6
$\nu^{8}$	$=$	$ ( 17\beta_{9} - 17\beta_{8} - 20\beta_{5} - 20\beta_{2} ) / 6 $ (17b9 - 17b8 - 20b5 - 20b2) / 6
$\nu^{9}$	$=$	$ ( -7\beta_{9} - 7\beta_{8} + 40\beta_{4} - 26\beta_{3} ) / 6 $ (-7b9 - 7b8 + 40b4 - 26b3) / 6

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/141\mathbb{Z}\right)^\times$.

$n$	$52$	$95$
$\chi(n)$	$-1$	$-1$

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} $

140.1

0.258701	−	1.39035i
0.607934	−	1.27688i
−1.02652	−	0.972757i
−1.24236	−	0.675681i
1.40224	−	0.183603i
1.40224	+	0.183603i
−1.24236	+	0.675681i
−1.02652	+	0.972757i
0.607934	+	1.27688i
0.258701	+	1.39035i

−

2.78070i

0.383200

−

1.68913i

−5.73230

−4.69696

−

1.06556i

4.37783

10.3784i

−2.70632

−

1.29455i

140.2

−

2.55375i

−1.48804

0.886414i

−4.52166

2.26368

3.80010i

−5.28882

6.43971i

1.42854

−

2.63804i

140.3

−

1.94551i

1.72487

0.157525i

−1.78502

0.306467

−

3.35576i

−1.79466

−

0.418243i

2.95037

0.543422i

140.4

−

1.35136i

−1.30286

1.14130i

0.173822

1.54230

1.76064i

4.17966

−

2.93762i

0.394890

−

2.97390i

140.5

−

0.367206i

0.682830

1.59177i

1.86516

0.584509

−

0.250739i

−1.47401

−

1.41931i

−2.06749

2.17382i

140.6