LMFDB - Embedded newform 640.2.o.a.63.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [640,2,Mod(63,640)] 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("640.63"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(640, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 2, 3])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 640 = 2^{7} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	640.o (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-4,0,-4,0,-4,0,0,0,-8,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$5.11042572936$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			63.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	640.63
Dual form			640.2.o.a.447.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-2.00000 - 2.00000i) q^{3} +(-2.00000 - 1.00000i) q^{5} +(-2.00000 - 2.00000i) q^{7} +5.00000i q^{9} -4.00000 q^{11} +(3.00000 - 3.00000i) q^{13} +(2.00000 + 6.00000i) q^{15} +(-3.00000 + 3.00000i) q^{17} +8.00000i q^{21} +(6.00000 - 6.00000i) q^{23} +(3.00000 + 4.00000i) q^{25} +(4.00000 - 4.00000i) q^{27} -2.00000 q^{29} +4.00000i q^{31} +(8.00000 + 8.00000i) q^{33} +(2.00000 + 6.00000i) q^{35} +(3.00000 + 3.00000i) q^{37} -12.0000 q^{39} +(6.00000 + 6.00000i) q^{43} +(5.00000 - 10.0000i) q^{45} +(-6.00000 - 6.00000i) q^{47} +1.00000i q^{49} +12.0000 q^{51} +(-3.00000 + 3.00000i) q^{53} +(8.00000 + 4.00000i) q^{55} +8.00000i q^{59} -6.00000i q^{61} +(10.0000 - 10.0000i) q^{63} +(-9.00000 + 3.00000i) q^{65} +(-6.00000 + 6.00000i) q^{67} -24.0000 q^{69} +12.0000i q^{71} +(-5.00000 - 5.00000i) q^{73} +(2.00000 - 14.0000i) q^{75} +(8.00000 + 8.00000i) q^{77} -8.00000 q^{79} -1.00000 q^{81} +(-6.00000 - 6.00000i) q^{83} +(9.00000 - 3.00000i) q^{85} +(4.00000 + 4.00000i) q^{87} -12.0000 q^{91} +(8.00000 - 8.00000i) q^{93} +(11.0000 - 11.0000i) q^{97} -20.0000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} - 4 q^{5} - 4 q^{7} - 8 q^{11} + 6 q^{13} + 4 q^{15} - 6 q^{17} + 12 q^{23} + 6 q^{25} + 8 q^{27} - 4 q^{29} + 16 q^{33} + 4 q^{35} + 6 q^{37} - 24 q^{39} + 12 q^{43} + 10 q^{45} - 12 q^{47}+ \cdots + 22 q^{97}+O(q^{100}) $ 2 * q - 4 * q^3 - 4 * q^5 - 4 * q^7 - 8 * q^11 + 6 * q^13 + 4 * q^15 - 6 * q^17 + 12 * q^23 + 6 * q^25 + 8 * q^27 - 4 * q^29 + 16 * q^33 + 4 * q^35 + 6 * q^37 - 24 * q^39 + 12 * q^43 + 10 * q^45 - 12 * q^47 + 24 * q^51 - 6 * q^53 + 16 * q^55 + 20 * q^63 - 18 * q^65 - 12 * q^67 - 48 * q^69 - 10 * q^73 + 4 * q^75 + 16 * q^77 - 16 * q^79 - 2 * q^81 - 12 * q^83 + 18 * q^85 + 8 * q^87 - 24 * q^91 + 16 * q^93 + 22 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/640\mathbb{Z}\right)^\times$.

$n$	$257$	$261$	$511$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$	−2.00000	−	2.00000i	−1.15470	−	1.15470i	−0.985599	−	0.169102i	$-0.945913\pi$
$3$	−2.00000	−	2.00000i	−1.15470	−	1.15470i	−0.169102	−	0.985599i	$-0.554087\pi$
$4$		0			0
$5$	−2.00000	−	1.00000i	−0.894427	−	0.447214i
$5$	−2.00000	−	1.00000i	−0.894427	−	0.447214i
$6$		0			0
$7$	−2.00000	−	2.00000i	−0.755929	−	0.755929i	0.219650	−	0.975579i	$-0.429509\pi$
$7$	−2.00000	−	2.00000i	−0.755929	−	0.755929i	−0.975579	+	0.219650i	$0.929509\pi$
$8$		0			0
$9$			5.00000i			1.66667i
$10$		0			0
$11$	−4.00000			−1.20605			−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000			−1.20605			−0.603023	+	0.797724i	$0.706037\pi$
$12$		0			0
$13$	3.00000	−	3.00000i	0.832050	−	0.832050i	−0.155747	−	0.987797i	$-0.549778\pi$
$13$	3.00000	−	3.00000i	0.832050	−	0.832050i	0.987797	+	0.155747i	$0.0497784\pi$
$14$		0			0
$15$	2.00000	+	6.00000i	0.516398	+	1.54919i
$16$		0			0
$17$	−3.00000	+	3.00000i	−0.727607	+	0.727607i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	−3.00000	+	3.00000i	−0.727607	+	0.727607i	0.242536	+	0.970143i	$0.422021\pi$
$18$		0			0
$19$		0			0		1.00000			$0$
$19$		0			0		−1.00000			$\pi$
$20$		0			0
$21$			8.00000i			1.74574i
$22$		0			0
$23$	6.00000	−	6.00000i	1.25109	−	1.25109i	0.295853	−	0.955233i	$-0.404396\pi$
$23$	6.00000	−	6.00000i	1.25109	−	1.25109i	0.955233	−	0.295853i	$-0.0956039\pi$
$24$		0			0
$25$	3.00000	+	4.00000i	0.600000	+	0.800000i
$26$		0			0
$27$	4.00000	−	4.00000i	0.769800	−	0.769800i
$28$		0			0
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$			4.00000i			0.718421i	0.933257	+	0.359211i	$0.116954\pi$
$31$			4.00000i			0.718421i	−0.933257	+	0.359211i	$0.883046\pi$
$32$		0			0
$33$	8.00000	+	8.00000i	1.39262	+	1.39262i
$34$		0			0
$35$	2.00000	+	6.00000i	0.338062	+	1.01419i
$36$		0			0
$37$	3.00000	+	3.00000i	0.493197	+	0.493197i	0.909312	−	0.416115i	$-0.136609\pi$
$37$	3.00000	+	3.00000i	0.493197	+	0.493197i	−0.416115	+	0.909312i	$0.636609\pi$
$38$		0			0
$39$	−12.0000			−1.92154
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	6.00000	+	6.00000i	0.914991	+	0.914991i	0.996660	−	0.0816682i	$-0.0260248\pi$
$43$	6.00000	+	6.00000i	0.914991	+	0.914991i	−0.0816682	+	0.996660i	$0.526025\pi$
$44$		0			0
$45$	5.00000	−	10.0000i	0.745356	−	1.49071i
$46$		0			0
$47$	−6.00000	−	6.00000i	−0.875190	−	0.875190i	0.117842	−	0.993032i	$-0.462402\pi$
$47$	−6.00000	−	6.00000i	−0.875190	−	0.875190i	−0.993032	+	0.117842i	$0.962402\pi$
$48$		0			0
$49$			1.00000i			0.142857i
$50$		0			0
$51$	12.0000			1.68034
$52$		0			0
$53$	−3.00000	+	3.00000i	−0.412082	+	0.412082i	−0.882463	−	0.470381i	$-0.844116\pi$
$53$	−3.00000	+	3.00000i	−0.412082	+	0.412082i	0.470381	+	0.882463i	$0.344116\pi$
$54$		0			0
$55$	8.00000	+	4.00000i	1.07872	+	0.539360i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			8.00000i			1.04151i	0.853706	+	0.520756i	$0.174350\pi$
$59$			8.00000i			1.04151i	−0.853706	+	0.520756i	$0.825650\pi$
$60$		0			0
$61$		−	6.00000i		−	0.768221i	−0.923287	−	0.384111i	$-0.874508\pi$
$61$		−	6.00000i		−	0.768221i	0.923287	−	0.384111i	$-0.125492\pi$
$62$		0			0
$63$	10.0000	−	10.0000i	1.25988	−	1.25988i
$64$		0			0
$65$	−9.00000	+	3.00000i	−1.11631	+	0.372104i
$66$		0			0
$67$	−6.00000	+	6.00000i	−0.733017	+	0.733017i	−0.971216	−	0.238200i	$-0.923443\pi$
$67$	−6.00000	+	6.00000i	−0.733017	+	0.733017i	0.238200	+	0.971216i	$0.423443\pi$
$68$		0			0
$69$	−24.0000			−2.88926
$70$		0			0
$71$			12.0000i			1.42414i	0.702109	+	0.712069i	$0.252242\pi$
$71$			12.0000i			1.42414i	−0.702109	+	0.712069i	$0.747758\pi$
$72$		0			0
$73$	−5.00000	−	5.00000i	−0.585206	−	0.585206i	0.351123	−	0.936329i	$-0.385800\pi$
$73$	−5.00000	−	5.00000i	−0.585206	−	0.585206i	−0.936329	+	0.351123i	$0.885800\pi$
$74$		0			0
$75$	2.00000	−	14.0000i	0.230940	−	1.61658i
$76$		0			0
$77$	8.00000	+	8.00000i	0.911685	+	0.911685i
$78$		0			0
$79$	−8.00000			−0.900070			−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000			−0.900070			−0.450035	+	0.893011i	$0.648589\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$	−6.00000	−	6.00000i	−0.658586	−	0.658586i	0.296460	−	0.955045i	$-0.404194\pi$
$83$	−6.00000	−	6.00000i	−0.658586	−	0.658586i	−0.955045	+	0.296460i	$0.904194\pi$
$84$		0			0
$85$	9.00000	−	3.00000i	0.976187	−	0.325396i
$86$		0			0
$87$	4.00000	+	4.00000i	0.428845	+	0.428845i
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$	−12.0000			−1.25794
$92$		0			0
$93$	8.00000	−	8.00000i	0.829561	−	0.829561i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	11.0000	−	11.0000i	1.11688	−	1.11688i	0.124684	−	0.992196i	$-0.460208\pi$
$97$	11.0000	−	11.0000i	1.11688	−	1.11688i	0.992196	−	0.124684i	$-0.0397918\pi$
$98$		0			0
$99$		−	20.0000i		−	2.01008i

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	640.2.o.a.63.1	✓	2
4.3	odd	2		640.2.o.g.63.1	yes	2
5.2	odd	4		640.2.o.b.447.1	yes	2
8.3	odd	2		640.2.o.b.63.1	yes	2
8.5	even	2		640.2.o.h.63.1	yes	2
16.3	odd	4		1280.2.n.a.1023.1		2
16.5	even	4		1280.2.n.b.1023.1		2
16.11	odd	4		1280.2.n.l.1023.1		2
16.13	even	4		1280.2.n.k.1023.1		2
20.7	even	4		640.2.o.h.447.1	yes	2
40.27	even	4	inner	640.2.o.a.447.1	yes	2
40.37	odd	4		640.2.o.g.447.1	yes	2
80.27	even	4		1280.2.n.b.767.1		2
80.37	odd	4		1280.2.n.l.767.1		2
80.67	even	4		1280.2.n.k.767.1		2
80.77	odd	4		1280.2.n.a.767.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.2.o.a.63.1	✓	2	1.1	even	1	trivial
640.2.o.a.447.1	yes	2	40.27	even	4	inner
640.2.o.b.63.1	yes	2	8.3	odd	2
640.2.o.b.447.1	yes	2	5.2	odd	4
640.2.o.g.63.1	yes	2	4.3	odd	2
640.2.o.g.447.1	yes	2	40.37	odd	4
640.2.o.h.63.1	yes	2	8.5	even	2
640.2.o.h.447.1	yes	2	20.7	even	4
1280.2.n.a.767.1		2	80.77	odd	4
1280.2.n.a.1023.1		2	16.3	odd	4
1280.2.n.b.767.1		2	80.27	even	4
1280.2.n.b.1023.1		2	16.5	even	4
1280.2.n.k.767.1		2	80.67	even	4
1280.2.n.k.1023.1		2	16.13	even	4
1280.2.n.l.767.1		2	80.37	odd	4
1280.2.n.l.1023.1		2	16.11	odd	4

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 640 = 2^{7} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	640.o (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 - 2.00000i) q^{3} +(-2.00000 - 1.00000i) q^{5} +(-2.00000 - 2.00000i) q^{7} +5.00000i q^{9} -4.00000 q^{11} +(3.00000 - 3.00000i) q^{13} +(2.00000 + 6.00000i) q^{15} +(-3.00000 + 3.00000i) q^{17} +8.00000i q^{21} +(6.00000 - 6.00000i) q^{23} +(3.00000 + 4.00000i) q^{25} +(4.00000 - 4.00000i) q^{27} -2.00000 q^{29} +4.00000i q^{31} +(8.00000 + 8.00000i) q^{33} +(2.00000 + 6.00000i) q^{35} +(3.00000 + 3.00000i) q^{37} -12.0000 q^{39} +(6.00000 + 6.00000i) q^{43} +(5.00000 - 10.0000i) q^{45} +(-6.00000 - 6.00000i) q^{47} +1.00000i q^{49} +12.0000 q^{51} +(-3.00000 + 3.00000i) q^{53} +(8.00000 + 4.00000i) q^{55} +8.00000i q^{59} -6.00000i q^{61} +(10.0000 - 10.0000i) q^{63} +(-9.00000 + 3.00000i) q^{65} +(-6.00000 + 6.00000i) q^{67} -24.0000 q^{69} +12.0000i q^{71} +(-5.00000 - 5.00000i) q^{73} +(2.00000 - 14.0000i) q^{75} +(8.00000 + 8.00000i) q^{77} -8.00000 q^{79} -1.00000 q^{81} +(-6.00000 - 6.00000i) q^{83} +(9.00000 - 3.00000i) q^{85} +(4.00000 + 4.00000i) q^{87} -12.0000 q^{91} +(8.00000 - 8.00000i) q^{93} +(11.0000 - 11.0000i) q^{97} -20.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} - 4 q^{5} - 4 q^{7} - 8 q^{11} + 6 q^{13} + 4 q^{15} - 6 q^{17} + 12 q^{23} + 6 q^{25} + 8 q^{27} - 4 q^{29} + 16 q^{33} + 4 q^{35} + 6 q^{37} - 24 q^{39} + 12 q^{43} + 10 q^{45} - 12 q^{47}+ \cdots + 22 q^{97}+O(q^{100}) \) 2 * q - 4 * q^3 - 4 * q^5 - 4 * q^7 - 8 * q^11 + 6 * q^13 + 4 * q^15 - 6 * q^17 + 12 * q^23 + 6 * q^25 + 8 * q^27 - 4 * q^29 + 16 * q^33 + 4 * q^35 + 6 * q^37 - 24 * q^39 + 12 * q^43 + 10 * q^45 - 12 * q^47 + 24 * q^51 - 6 * q^53 + 16 * q^55 + 20 * q^63 - 18 * q^65 - 12 * q^67 - 48 * q^69 - 10 * q^73 + 4 * q^75 + 16 * q^77 - 16 * q^79 - 2 * q^81 - 12 * q^83 + 18 * q^85 + 8 * q^87 - 24 * q^91 + 16 * q^93 + 22 * q^97

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