LMFDB - Embedded newform 164.5.d.b.163.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [164,5,Mod(163,164)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("164.163"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(164, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 164 = 2^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	164.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,-8,0,32,28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$16.9526739458$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4}\cdot 3\cdot 5 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			163.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	164.163
Dual form			164.5.d.b.163.2

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000 q^{2} +16.0000 q^{4} +14.0000 q^{5} -64.0000 q^{8} -81.0000 q^{9} -56.0000 q^{10} -240.000i q^{13} +256.000 q^{16} +480.000i q^{17} +324.000 q^{18} +224.000 q^{20} -429.000 q^{25} +960.000i q^{26} -1680.00i q^{29} -1024.00 q^{32} -1920.00i q^{34} -1296.00 q^{36} -2162.00 q^{37} -896.000 q^{40} +(-1519.00 + 720.000i) q^{41} -1134.00 q^{45} -2401.00 q^{49} +1716.00 q^{50} -3840.00i q^{52} -5040.00i q^{53} +6720.00i q^{58} -6958.00 q^{61} +4096.00 q^{64} -3360.00i q^{65} +7680.00i q^{68} +5184.00 q^{72} -1442.00 q^{73} +8648.00 q^{74} +3584.00 q^{80} +6561.00 q^{81} +(6076.00 - 2880.00i) q^{82} +6720.00i q^{85} +12480.0i q^{89} +4536.00 q^{90} -18720.0i q^{97} +9604.00 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} + 32 q^{4} + 28 q^{5} - 128 q^{8} - 162 q^{9} - 112 q^{10} + 512 q^{16} + 648 q^{18} + 448 q^{20} - 858 q^{25} - 2048 q^{32} - 2592 q^{36} - 4324 q^{37} - 1792 q^{40} - 3038 q^{41} - 2268 q^{45}+ \cdots + 19208 q^{98}+O(q^{100}) $ 2 * q - 8 * q^2 + 32 * q^4 + 28 * q^5 - 128 * q^8 - 162 * q^9 - 112 * q^10 + 512 * q^16 + 648 * q^18 + 448 * q^20 - 858 * q^25 - 2048 * q^32 - 2592 * q^36 - 4324 * q^37 - 1792 * q^40 - 3038 * q^41 - 2268 * q^45 - 4802 * q^49 + 3432 * q^50 - 13916 * q^61 + 8192 * q^64 + 10368 * q^72 - 2884 * q^73 + 17296 * q^74 + 7168 * q^80 + 13122 * q^81 + 12152 * q^82 + 9072 * q^90 + 19208 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/164\mathbb{Z}\right)^\times$.

$n$	$83$	$129$
$\chi(n)$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$. You can download additional coefficients here.

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−4.00000			−1.00000
$2$	−4.00000			−1.00000
$3$		0			0			−	1.00000i	$-0.5\pi$
$3$		0			0				1.00000i	$0.5\pi$
$4$	16.0000			1.00000
$5$	14.0000			0.560000			0.280000	−	0.960000i	$-0.409666\pi$
$5$	14.0000			0.560000			0.280000	+	0.960000i	$0.409666\pi$
$6$		0			0
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$	−64.0000			−1.00000
$9$	−81.0000			−1.00000
$10$	−56.0000			−0.560000
$11$		0			0			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$		−	240.000i		−	1.42012i	−0.704142	−	0.710059i	$-0.748668\pi$
$13$		−	240.000i		−	1.42012i	0.704142	−	0.710059i	$-0.251332\pi$
$14$		0			0
$15$		0			0
$16$	256.000			1.00000
$17$			480.000i			1.66090i	0.557093	+	0.830450i	$0.311917\pi$
$17$			480.000i			1.66090i	−0.557093	+	0.830450i	$0.688083\pi$
$18$	324.000			1.00000
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$	224.000			0.560000
$21$		0			0
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−429.000			−0.686400
$26$			960.000i			1.42012i
$27$		0			0
$28$		0			0
$29$		−	1680.00i		−	1.99762i	−0.0487515	−	0.998811i	$-0.515524\pi$
$29$		−	1680.00i		−	1.99762i	0.0487515	−	0.998811i	$-0.484476\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$	−1024.00			−1.00000
$33$		0			0
$34$		−	1920.00i		−	1.66090i
$35$		0			0
$36$	−1296.00			−1.00000
$37$	−2162.00			−1.57925			−0.789627	−	0.613587i	$-0.789726\pi$
$37$	−2162.00			−1.57925			−0.789627	+	0.613587i	$0.789726\pi$
$38$		0			0
$39$		0			0
$40$	−896.000			−0.560000
$41$	−1519.00	+	720.000i	−0.903629	+	0.428316i
$41$	−1519.00	+	720.000i	−0.903629	+	0.428316i
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	−1134.00			−0.560000
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$	−2401.00			−1.00000
$50$	1716.00			0.686400
$51$		0			0
$52$		−	3840.00i		−	1.42012i
$53$		−	5040.00i		−	1.79423i	−0.441794	−	0.897116i	$-0.645658\pi$
$53$		−	5040.00i		−	1.79423i	0.441794	−	0.897116i	$-0.354342\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$			6720.00i			1.99762i
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	−6958.00			−1.86993			−0.934964	−	0.354743i	$-0.884568\pi$
$61$	−6958.00			−1.86993			−0.934964	+	0.354743i	$0.884568\pi$
$62$		0			0
$63$		0			0
$64$	4096.00			1.00000
$65$		−	3360.00i		−	0.795266i
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$			7680.00i			1.66090i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	5184.00			1.00000
$73$	−1442.00			−0.270595			−0.135297	−	0.990805i	$-0.543199\pi$
$73$	−1442.00			−0.270595			−0.135297	+	0.990805i	$0.543199\pi$
$74$	8648.00			1.57925
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$	3584.00			0.560000
$81$	6561.00			1.00000
$82$	6076.00	−	2880.00i	0.903629	−	0.428316i
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$		0			0
$85$			6720.00i			0.930104i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			12480.0i			1.57556i	0.615958	+	0.787779i	$0.288769\pi$
$89$			12480.0i			1.57556i	−0.615958	+	0.787779i	$0.711231\pi$
$90$	4536.00			0.560000
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	18720.0i		−	1.98958i	−0.101924	−	0.994792i	$-0.532500\pi$
$97$		−	18720.0i		−	1.98958i	0.101924	−	0.994792i	$-0.467500\pi$
$98$	9604.00			1.00000
$99$		0			0

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.5.d.b.163.1	✓	2
4.3	odd	2	CM	164.5.d.b.163.1	✓	2
41.40	even	2	inner	164.5.d.b.163.2	yes	2
164.163	odd	2	inner	164.5.d.b.163.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.5.d.b.163.1	✓	2	1.1	even	1	trivial
164.5.d.b.163.1	✓	2	4.3	odd	2	CM
164.5.d.b.163.2	yes	2	41.40	even	2	inner
164.5.d.b.163.2	yes	2	164.163	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 164 = 2^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	164.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.00000 q^{2} +16.0000 q^{4} +14.0000 q^{5} -64.0000 q^{8} -81.0000 q^{9} -56.0000 q^{10} -240.000i q^{13} +256.000 q^{16} +480.000i q^{17} +324.000 q^{18} +224.000 q^{20} -429.000 q^{25} +960.000i q^{26} -1680.00i q^{29} -1024.00 q^{32} -1920.00i q^{34} -1296.00 q^{36} -2162.00 q^{37} -896.000 q^{40} +(-1519.00 + 720.000i) q^{41} -1134.00 q^{45} -2401.00 q^{49} +1716.00 q^{50} -3840.00i q^{52} -5040.00i q^{53} +6720.00i q^{58} -6958.00 q^{61} +4096.00 q^{64} -3360.00i q^{65} +7680.00i q^{68} +5184.00 q^{72} -1442.00 q^{73} +8648.00 q^{74} +3584.00 q^{80} +6561.00 q^{81} +(6076.00 - 2880.00i) q^{82} +6720.00i q^{85} +12480.0i q^{89} +4536.00 q^{90} -18720.0i q^{97} +9604.00 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} + 32 q^{4} + 28 q^{5} - 128 q^{8} - 162 q^{9} - 112 q^{10} + 512 q^{16} + 648 q^{18} + 448 q^{20} - 858 q^{25} - 2048 q^{32} - 2592 q^{36} - 4324 q^{37} - 1792 q^{40} - 3038 q^{41} - 2268 q^{45}+ \cdots + 19208 q^{98}+O(q^{100}) \) 2 * q - 8 * q^2 + 32 * q^4 + 28 * q^5 - 128 * q^8 - 162 * q^9 - 112 * q^10 + 512 * q^16 + 648 * q^18 + 448 * q^20 - 858 * q^25 - 2048 * q^32 - 2592 * q^36 - 4324 * q^37 - 1792 * q^40 - 3038 * q^41 - 2268 * q^45 - 4802 * q^49 + 3432 * q^50 - 13916 * q^61 + 8192 * q^64 + 10368 * q^72 - 2884 * q^73 + 17296 * q^74 + 7168 * q^80 + 13122 * q^81 + 12152 * q^82 + 9072 * q^90 + 19208 * q^98

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