LMFDB - Newform orbit 176.2.e.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [176,2,Mod(175,176)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(176, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("176.175");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 176 = 2^{4} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	176.e (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:	$1.40536707557$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{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 $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta_1 q^{3} + (\beta_{3} - 2) q^{5} + (\beta_{3} - 1) q^{9}+O(q^{10}) $ q + b1 * q^3 + (b3 - 2) * q^5 + (b3 - 1) * q^9	$ q + \beta_1 q^{3} + (\beta_{3} - 2) q^{5} + (\beta_{3} - 1) q^{9} + ( - \beta_{2} + 2 \beta_1) q^{11} + ( - 2 \beta_{2} - 3 \beta_1) q^{15} + (4 \beta_{2} + \beta_1) q^{23} + ( - 3 \beta_{3} + 7) q^{25} + ( - 2 \beta_{2} + \beta_1) q^{27} + (4 \beta_{2} - 3 \beta_1) q^{31} + (\beta_{3} - 6) q^{33} + ( - 3 \beta_{3} - 2) q^{37} + ( - 2 \beta_{3} + 10) q^{45} + (2 \beta_{2} - 4 \beta_1) q^{47} - 7 q^{49} + 6 q^{53} + ( - 4 \beta_{2} - 3 \beta_1) q^{55} + ( - 8 \beta_{2} + \beta_1) q^{59} + (8 \beta_{2} - 3 \beta_1) q^{67} + (5 \beta_{3} - 12) q^{69} + ( - 4 \beta_{2} + 5 \beta_1) q^{71} + (6 \beta_{2} + 10 \beta_1) q^{75} + (2 \beta_{3} - 3) q^{81} + (5 \beta_{3} + 2) q^{89} + (\beta_{3} + 4) q^{93} + ( - 3 \beta_{3} + 10) q^{97} + ( - 5 \beta_{2} - \beta_1) q^{99}+O(q^{100}) $ q + b1 * q^3 + (b3 - 2) * q^5 + (b3 - 1) * q^9 + (-b2 + 2b1) q^11 + (-2b2 - 3b1) * q^15 + (4b2 + b1) q^23 + (-3b3 + 7) q^25 + (-2b2 + b1) q^27 + (4b2 - 3b1) * q^31 + (b3 - 6) * q^33 + (-3b3 - 2) q^37 + (-2b3 + 10) q^45 + (2b2 - 4b1) * q^47 - 7 * q^49 + 6 * q^53 + (-4b2 - 3b1) * q^55 + (-8b2 + b1) q^59 + (8b2 - 3b1) * q^67 + (5b3 - 12) q^69 + (-4b2 + 5b1) * q^71 + (6b2 + 10b1) * q^75 + (2b3 - 3) q^81 + (5b3 + 2) q^89 + (b3 + 4) * q^93 + (-3b3 + 10) q^97 + (-5b2 - b1) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{5} - 2 q^{9}+O(q^{10}) $ 4 * q - 6 * q^5 - 2 * q^9	$ 4 q - 6 q^{5} - 2 q^{9} + 22 q^{25} - 22 q^{33} - 14 q^{37} + 36 q^{45} - 28 q^{49} + 24 q^{53} - 38 q^{69} - 8 q^{81} + 18 q^{89} + 18 q^{93} + 34 q^{97}+O(q^{100}) $ 4 * q - 6 * q^5 - 2 * q^9 + 22 * q^25 - 22 * q^33 - 14 * q^37 + 36 * q^45 - 28 * q^49 + 24 * q^53 - 38 * q^69 - 8 * q^81 + 18 * q^89 + 18 * q^93 + 34 * q^97

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2*x^2 - 3*x + 9 :

$\beta_{1}$	$=$	$ ( -\nu^{3} + \nu^{2} - \nu + 3 ) / 3 $ (-v^3 + v^2 - v + 3) / 3
$\beta_{2}$	$=$	$ ( \nu^{3} + 2\nu^{2} - 2\nu - 6 ) / 3 $ (v^3 + 2v^2 - 2v - 6) / 3
$\beta_{3}$	$=$	$ ( -\nu^{3} + \nu^{2} + 5\nu + 3 ) / 3 $ (-v^3 + v^2 + 5*v + 3) / 3

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

$\nu$	$=$	$ ( \beta_{3} - \beta_1 ) / 2 $ (b3 - b1) / 2
$\nu^{2}$	$=$	$ ( \beta_{3} + 2\beta_{2} + \beta _1 + 2 ) / 2 $ (b3 + 2*b2 + b1 + 2) / 2
$\nu^{3}$	$=$	$ \beta_{2} - 2\beta _1 + 4 $ b2 - 2*b1 + 4

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/176\mathbb{Z}\right)^\times$.

$n$	$111$	$133$	$145$
$\chi(n)$	$-1$	$1$	$-1$

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} $

175.1

−1.18614	+	1.26217i
1.68614	+	0.396143i
1.68614	−	0.396143i
−1.18614	−	1.26217i

−

2.52434i

−4.37228

−3.37228

175.2

−

0.792287i

1.37228

2.37228

175.3

0.792287i

1.37228

2.37228

175.4

2.52434i

−4.37228

−3.37228

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	CM by $\Q(\sqrt{-11}) $
4.b	odd	2	1	inner
44.c	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	176.2.e.b	✓	4
3.b	odd	2	1		1584.2.o.e		4
4.b	odd	2	1	inner	176.2.e.b	✓	4
8.b	even	2	1		704.2.e.c		4
8.d	odd	2	1		704.2.e.c		4
11.b	odd	2	1	CM	176.2.e.b	✓	4
12.b	even	2	1		1584.2.o.e		4
16.e	even	4	2		2816.2.g.c		8
16.f	odd	4	2		2816.2.g.c		8
33.d	even	2	1		1584.2.o.e		4
44.c	even	2	1	inner	176.2.e.b	✓	4
88.b	odd	2	1		704.2.e.c		4
88.g	even	2	1		704.2.e.c		4
132.d	odd	2	1		1584.2.o.e		4
176.i	even	4	2		2816.2.g.c		8
176.l	odd	4	2		2816.2.g.c		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
176.2.e.b	✓	4	1.a	even	1	1	trivial
176.2.e.b	✓	4	4.b	odd	2	1	inner
176.2.e.b	✓	4	11.b	odd	2	1	CM
176.2.e.b	✓	4	44.c	even	2	1	inner
704.2.e.c		4	8.b	even	2	1
704.2.e.c		4	8.d	odd	2	1
704.2.e.c		4	88.b	odd	2	1
704.2.e.c		4	88.g	even	2	1
1584.2.o.e		4	3.b	odd	2	1
1584.2.o.e		4	12.b	even	2	1
1584.2.o.e		4	33.d	even	2	1
1584.2.o.e		4	132.d	odd	2	1
2816.2.g.c		8	16.e	even	4	2
2816.2.g.c		8	16.f	odd	4	2
2816.2.g.c		8	176.i	even	4	2
2816.2.g.c		8	176.l	odd	4	2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{4} + 7T_{3}^{2} + 4 $ T3^4 + 7*T3^2 + 4 acting on $S_{2}^{\mathrm{new}}(176, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} + 7T^{2} + 4 $ T^4 + 7*T^2 + 4
$5$	$ (T^{2} + 3 T - 6)^{2} $ (T^2 + 3*T - 6)^2
$7$	$ T^{4} $ T^4
$11$	$ (T^{2} + 11)^{2} $ (T^2 + 11)^2
$13$	$ T^{4} $ T^4
$17$	$ T^{4} $ T^4
$19$	$ T^{4} $ T^4
$23$	$ T^{4} + 127T^{2} + 3364 $ T^4 + 127*T^2 + 3364
$29$	$ T^{4} $ T^4
$31$	$ T^{4} + 87T^{2} + 36 $ T^4 + 87*T^2 + 36
$37$	$ (T^{2} + 7 T - 62)^{2} $ (T^2 + 7*T - 62)^2
$41$	$ T^{4} $ T^4
$43$	$ T^{4} $ T^4
$47$	$ (T^{2} + 44)^{2} $ (T^2 + 44)^2
$53$	$ (T - 6)^{4} $ (T - 6)^4
$59$	$ T^{4} + 343 T^{2} + 27556 $ T^4 + 343*T^2 + 27556
$61$	$ T^{4} $ T^4
$67$	$ T^{4} + 303 T^{2} + 10404 $ T^4 + 303*T^2 + 10404
$71$	$ T^{4} + 151T^{2} + 3844 $ T^4 + 151*T^2 + 3844
$73$	$ T^{4} $ T^4
$79$	$ T^{4} $ T^4
$83$	$ T^{4} $ T^4
$89$	$ (T^{2} - 9 T - 186)^{2} $ (T^2 - 9*T - 186)^2
$97$	$ (T^{2} - 17 T - 2)^{2} $ (T^2 - 17*T - 2)^2
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Level:	\( N \)	\(=\)	\( 176 = 2^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	176.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + \beta_1 q^{3} + (\beta_{3} - 2) q^{5} + (\beta_{3} - 1) q^{9}+O(q^{10}) \) q + b1 * q^3 + (b3 - 2) * q^5 + (b3 - 1) * q^9	\( q + \beta_1 q^{3} + (\beta_{3} - 2) q^{5} + (\beta_{3} - 1) q^{9} + ( - \beta_{2} + 2 \beta_1) q^{11} + ( - 2 \beta_{2} - 3 \beta_1) q^{15} + (4 \beta_{2} + \beta_1) q^{23} + ( - 3 \beta_{3} + 7) q^{25} + ( - 2 \beta_{2} + \beta_1) q^{27} + (4 \beta_{2} - 3 \beta_1) q^{31} + (\beta_{3} - 6) q^{33} + ( - 3 \beta_{3} - 2) q^{37} + ( - 2 \beta_{3} + 10) q^{45} + (2 \beta_{2} - 4 \beta_1) q^{47} - 7 q^{49} + 6 q^{53} + ( - 4 \beta_{2} - 3 \beta_1) q^{55} + ( - 8 \beta_{2} + \beta_1) q^{59} + (8 \beta_{2} - 3 \beta_1) q^{67} + (5 \beta_{3} - 12) q^{69} + ( - 4 \beta_{2} + 5 \beta_1) q^{71} + (6 \beta_{2} + 10 \beta_1) q^{75} + (2 \beta_{3} - 3) q^{81} + (5 \beta_{3} + 2) q^{89} + (\beta_{3} + 4) q^{93} + ( - 3 \beta_{3} + 10) q^{97} + ( - 5 \beta_{2} - \beta_1) q^{99}+O(q^{100}) \) q + b1 * q^3 + (b3 - 2) * q^5 + (b3 - 1) * q^9 + (-b2 + 2b1) q^11 + (-2b2 - 3b1) * q^15 + (4b2 + b1) q^23 + (-3b3 + 7) q^25 + (-2b2 + b1) q^27 + (4b2 - 3b1) * q^31 + (b3 - 6) * q^33 + (-3b3 - 2) q^37 + (-2b3 + 10) q^45 + (2b2 - 4b1) * q^47 - 7 * q^49 + 6 * q^53 + (-4b2 - 3b1) * q^55 + (-8b2 + b1) q^59 + (8b2 - 3b1) * q^67 + (5b3 - 12) q^69 + (-4b2 + 5b1) * q^71 + (6b2 + 10b1) * q^75 + (2b3 - 3) q^81 + (5b3 + 2) q^89 + (b3 + 4) * q^93 + (-3b3 + 10) q^97 + (-5b2 - b1) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{5} - 2 q^{9}+O(q^{10}) \) 4 * q - 6 * q^5 - 2 * q^9	\( 4 q - 6 q^{5} - 2 q^{9} + 22 q^{25} - 22 q^{33} - 14 q^{37} + 36 q^{45} - 28 q^{49} + 24 q^{53} - 38 q^{69} - 8 q^{81} + 18 q^{89} + 18 q^{93} + 34 q^{97}+O(q^{100}) \) 4 * q - 6 * q^5 - 2 * q^9 + 22 * q^25 - 22 * q^33 - 14 * q^37 + 36 * q^45 - 28 * q^49 + 24 * q^53 - 38 * q^69 - 8 * q^81 + 18 * q^89 + 18 * q^93 + 34 * q^97

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