LMFDB - Newform orbit 648.3.e.b

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [648,3,Mod(161,648)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(648, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("648.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	648.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:	$17.6567211305$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-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_{2} + \beta_1) q^{5} + (\beta_{3} + 3) q^{7}+O(q^{10}) $ q + (-b2 + b1) * q^5 + (b3 + 3) * q^7	$ q + ( - \beta_{2} + \beta_1) q^{5} + (\beta_{3} + 3) q^{7} + ( - 3 \beta_{2} - 2 \beta_1) q^{11} + ( - \beta_{3} + 7) q^{13} + ( - 8 \beta_{2} + 2 \beta_1) q^{17} + (2 \beta_{3} + 2) q^{19} + ( - 5 \beta_{2} + \beta_1) q^{23} + (2 \beta_{3} - 10) q^{25} + (\beta_{2} - 3 \beta_1) q^{29} + (\beta_{3} - 37) q^{31} + 29 \beta_{2} q^{35} + (2 \beta_{3} - 30) q^{37} + 23 \beta_{2} q^{41} + (4 \beta_{3} - 5) q^{43} + (29 \beta_{2} + 3 \beta_1) q^{47} + (6 \beta_{3} + 56) q^{49} + ( - 32 \beta_{2} - 8 \beta_1) q^{53} + (\beta_{3} + 55) q^{55} + (3 \beta_{2} - 2 \beta_1) q^{59} + (\beta_{3} + 31) q^{61} + ( - 39 \beta_{2} + 10 \beta_1) q^{65} + (10 \beta_{3} + 11) q^{67} + (24 \beta_{2} + 2 \beta_1) q^{71} + (6 \beta_{3} + 10) q^{73} + ( - 73 \beta_{2} - 15 \beta_1) q^{77} + ( - 7 \beta_{3} + 43) q^{79} + ( - 11 \beta_{2} + 14 \beta_1) q^{83} + (10 \beta_{3} - 88) q^{85} + ( - 24 \beta_{2} + 6 \beta_1) q^{89} + (4 \beta_{3} - 75) q^{91} + (62 \beta_{2} - 4 \beta_1) q^{95} + (2 \beta_{3} - 121) q^{97}+O(q^{100}) $ q + (-b2 + b1) * q^5 + (b3 + 3) * q^7 + (-3b2 - 2b1) * q^11 + (-b3 + 7) * q^13 + (-8b2 + 2b1) * q^17 + (2b3 + 2) q^19 + (-5b2 + b1) q^23 + (2b3 - 10) q^25 + (b2 - 3b1) q^29 + (b3 - 37) * q^31 + 29b2 q^35 + (2b3 - 30) q^37 + 23b2 q^41 + (4b3 - 5) q^43 + (29b2 + 3b1) * q^47 + (6b3 + 56) q^49 + (-32b2 - 8b1) * q^53 + (b3 + 55) * q^55 + (3b2 - 2b1) * q^59 + (b3 + 31) * q^61 + (-39b2 + 10b1) * q^65 + (10b3 + 11) q^67 + (24b2 + 2b1) * q^71 + (6b3 + 10) q^73 + (-73b2 - 15b1) * q^77 + (-7b3 + 43) q^79 + (-11b2 + 14b1) * q^83 + (10b3 - 88) q^85 + (-24b2 + 6b1) * q^89 + (4b3 - 75) q^91 + (62b2 - 4b1) * q^95 + (2b3 - 121) q^97
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{7}+O(q^{10}) $ 4 * q + 12 * q^7	$ 4 q + 12 q^{7} + 28 q^{13} + 8 q^{19} - 40 q^{25} - 148 q^{31} - 120 q^{37} - 20 q^{43} + 224 q^{49} + 220 q^{55} + 124 q^{61} + 44 q^{67} + 40 q^{73} + 172 q^{79} - 352 q^{85} - 300 q^{91} - 484 q^{97}+O(q^{100}) $ 4 * q + 12 * q^7 + 28 * q^13 + 8 * q^19 - 40 * q^25 - 148 * q^31 - 120 * q^37 - 20 * q^43 + 224 * q^49 + 220 * q^55 + 124 * q^61 + 44 * q^67 + 40 * q^73 + 172 * q^79 - 352 * q^85 - 300 * q^91 - 484 * q^97

Display 100 coefficients 10 coefficients

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

$\beta_{1}$	$=$	$ 2\nu^{3} $ 2*v^3
$\beta_{2}$	$=$	$ \nu^{2} - 1 $ v^2 - 1
$\beta_{3}$	$=$	$ -2\nu^{3} + 8\nu $ -2v^3 + 8v

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

$\nu$	$=$	$ ( \beta_{3} + \beta_1 ) / 8 $ (b3 + b1) / 8
$\nu^{2}$	$=$	$ \beta_{2} + 1 $ b2 + 1
$\nu^{3}$	$=$	$ ( \beta_1 ) / 2 $ (b1) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$1$	$1$	$-1$

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} $

161.1

−1.22474	−	0.707107i
1.22474	−	0.707107i
1.22474	+	0.707107i
−1.22474	+	0.707107i

−

7.38891i

−6.79796

161.2

−

3.92480i

12.7980

161.3

3.92480i

12.7980

161.4

7.38891i

−6.79796

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	648.3.e.b		4
3.b	odd	2	1	inner	648.3.e.b		4
4.b	odd	2	1		1296.3.e.c		4
9.c	even	3	1		72.3.m.a	✓	4
9.c	even	3	1		216.3.m.a		4
9.d	odd	6	1		72.3.m.a	✓	4
9.d	odd	6	1		216.3.m.a		4
12.b	even	2	1		1296.3.e.c		4
36.f	odd	6	1		144.3.q.d		4
36.f	odd	6	1		432.3.q.c		4
36.h	even	6	1		144.3.q.d		4
36.h	even	6	1		432.3.q.c		4
72.j	odd	6	1		576.3.q.h		4
72.j	odd	6	1		1728.3.q.e		4
72.l	even	6	1		576.3.q.c		4
72.l	even	6	1		1728.3.q.f		4
72.n	even	6	1		576.3.q.h		4
72.n	even	6	1		1728.3.q.e		4
72.p	odd	6	1		576.3.q.c		4
72.p	odd	6	1		1728.3.q.f		4

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
72.3.m.a	✓	4	9.c	even	3	1
72.3.m.a	✓	4	9.d	odd	6	1
144.3.q.d		4	36.f	odd	6	1
144.3.q.d		4	36.h	even	6	1
216.3.m.a		4	9.c	even	3	1
216.3.m.a		4	9.d	odd	6	1
432.3.q.c		4	36.f	odd	6	1
432.3.q.c		4	36.h	even	6	1
576.3.q.c		4	72.l	even	6	1
576.3.q.c		4	72.p	odd	6	1
576.3.q.h		4	72.j	odd	6	1
576.3.q.h		4	72.n	even	6	1
648.3.e.b		4	1.a	even	1	1	trivial
648.3.e.b		4	3.b	odd	2	1	inner
1296.3.e.c		4	4.b	odd	2	1
1296.3.e.c		4	12.b	even	2	1
1728.3.q.e		4	72.j	odd	6	1
1728.3.q.e		4	72.n	even	6	1
1728.3.q.f		4	72.l	even	6	1
1728.3.q.f		4	72.p	odd	6	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{4} + 70T_{5}^{2} + 841 $ T5^4 + 70*T5^2 + 841 acting on $S_{3}^{\mathrm{new}}(648, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{4} $ T^4
$3$	$ T^{4} $ T^4
$5$	$ T^{4} + 70T^{2} + 841 $ T^4 + 70*T^2 + 841
$7$	$ (T^{2} - 6 T - 87)^{2} $ (T^2 - 6*T - 87)^2
$11$	$ T^{4} + 310 T^{2} + 10201 $ T^4 + 310*T^2 + 10201
$13$	$ (T^{2} - 14 T - 47)^{2} $ (T^2 - 14*T - 47)^2
$17$	$ T^{4} + 640T^{2} + 4096 $ T^4 + 640*T^2 + 4096
$19$	$ (T^{2} - 4 T - 380)^{2} $ (T^2 - 4*T - 380)^2
$23$	$ T^{4} + 214T^{2} + 1849 $ T^4 + 214*T^2 + 1849
$29$	$ T^{4} + 582 T^{2} + 81225 $ T^4 + 582*T^2 + 81225
$31$	$ (T^{2} + 74 T + 1273)^{2} $ (T^2 + 74*T + 1273)^2
$37$	$ (T^{2} + 60 T + 516)^{2} $ (T^2 + 60*T + 516)^2
$41$	$ (T^{2} + 1587)^{2} $ (T^2 + 1587)^2
$43$	$ (T^{2} + 10 T - 1511)^{2} $ (T^2 + 10*T - 1511)^2
$47$	$ T^{4} + 5622 T^{2} + 4995225 $ T^4 + 5622*T^2 + 4995225
$53$	$ T^{4} + 10240 T^{2} + 1048576 $ T^4 + 10240*T^2 + 1048576
$59$	$ T^{4} + 310 T^{2} + 10201 $ T^4 + 310*T^2 + 10201
$61$	$ (T^{2} - 62 T + 865)^{2} $ (T^2 - 62*T + 865)^2
$67$	$ (T^{2} - 22 T - 9479)^{2} $ (T^2 - 22*T - 9479)^2
$71$	$ T^{4} + 3712 T^{2} + 2560000 $ T^4 + 3712*T^2 + 2560000
$73$	$ (T^{2} - 20 T - 3356)^{2} $ (T^2 - 20*T - 3356)^2
$79$	$ (T^{2} - 86 T - 2855)^{2} $ (T^2 - 86*T - 2855)^2
$83$	$ T^{4} + 13270 T^{2} + 34916281 $ T^4 + 13270*T^2 + 34916281
$89$	$ T^{4} + 5760 T^{2} + 331776 $ T^4 + 5760*T^2 + 331776
$97$	$ (T^{2} + 242 T + 14257)^{2} $ (T^2 + 242*T + 14257)^2
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Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	648.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{2} + \beta_1) q^{5} + (\beta_{3} + 3) q^{7}+O(q^{10}) \) q + (-b2 + b1) * q^5 + (b3 + 3) * q^7	\( q + ( - \beta_{2} + \beta_1) q^{5} + (\beta_{3} + 3) q^{7} + ( - 3 \beta_{2} - 2 \beta_1) q^{11} + ( - \beta_{3} + 7) q^{13} + ( - 8 \beta_{2} + 2 \beta_1) q^{17} + (2 \beta_{3} + 2) q^{19} + ( - 5 \beta_{2} + \beta_1) q^{23} + (2 \beta_{3} - 10) q^{25} + (\beta_{2} - 3 \beta_1) q^{29} + (\beta_{3} - 37) q^{31} + 29 \beta_{2} q^{35} + (2 \beta_{3} - 30) q^{37} + 23 \beta_{2} q^{41} + (4 \beta_{3} - 5) q^{43} + (29 \beta_{2} + 3 \beta_1) q^{47} + (6 \beta_{3} + 56) q^{49} + ( - 32 \beta_{2} - 8 \beta_1) q^{53} + (\beta_{3} + 55) q^{55} + (3 \beta_{2} - 2 \beta_1) q^{59} + (\beta_{3} + 31) q^{61} + ( - 39 \beta_{2} + 10 \beta_1) q^{65} + (10 \beta_{3} + 11) q^{67} + (24 \beta_{2} + 2 \beta_1) q^{71} + (6 \beta_{3} + 10) q^{73} + ( - 73 \beta_{2} - 15 \beta_1) q^{77} + ( - 7 \beta_{3} + 43) q^{79} + ( - 11 \beta_{2} + 14 \beta_1) q^{83} + (10 \beta_{3} - 88) q^{85} + ( - 24 \beta_{2} + 6 \beta_1) q^{89} + (4 \beta_{3} - 75) q^{91} + (62 \beta_{2} - 4 \beta_1) q^{95} + (2 \beta_{3} - 121) q^{97}+O(q^{100}) \) q + (-b2 + b1) * q^5 + (b3 + 3) * q^7 + (-3b2 - 2b1) * q^11 + (-b3 + 7) * q^13 + (-8b2 + 2b1) * q^17 + (2b3 + 2) q^19 + (-5b2 + b1) q^23 + (2b3 - 10) q^25 + (b2 - 3b1) q^29 + (b3 - 37) * q^31 + 29b2 q^35 + (2b3 - 30) q^37 + 23b2 q^41 + (4b3 - 5) q^43 + (29b2 + 3b1) * q^47 + (6b3 + 56) q^49 + (-32b2 - 8b1) * q^53 + (b3 + 55) * q^55 + (3b2 - 2b1) * q^59 + (b3 + 31) * q^61 + (-39b2 + 10b1) * q^65 + (10b3 + 11) q^67 + (24b2 + 2b1) * q^71 + (6b3 + 10) q^73 + (-73b2 - 15b1) * q^77 + (-7b3 + 43) q^79 + (-11b2 + 14b1) * q^83 + (10b3 - 88) q^85 + (-24b2 + 6b1) * q^89 + (4b3 - 75) q^91 + (62b2 - 4b1) * q^95 + (2b3 - 121) q^97
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{7}+O(q^{10}) \) 4 * q + 12 * q^7	\( 4 q + 12 q^{7} + 28 q^{13} + 8 q^{19} - 40 q^{25} - 148 q^{31} - 120 q^{37} - 20 q^{43} + 224 q^{49} + 220 q^{55} + 124 q^{61} + 44 q^{67} + 40 q^{73} + 172 q^{79} - 352 q^{85} - 300 q^{91} - 484 q^{97}+O(q^{100}) \) 4 * q + 12 * q^7 + 28 * q^13 + 8 * q^19 - 40 * q^25 - 148 * q^31 - 120 * q^37 - 20 * q^43 + 224 * q^49 + 220 * q^55 + 124 * q^61 + 44 * q^67 + 40 * q^73 + 172 * q^79 - 352 * q^85 - 300 * q^91 - 484 * q^97

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