LMFDB - Embedded newform 1920.2.h.d.1151.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1920,2,Mod(1151,1920)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1920, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1920.1151"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$1920 = 2^{7} \cdot 3 \cdot 5$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	1920.h (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,-2,0,0,0,0,0,0,0,-16,0,4] 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:	$15.3312771881$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} + 3x^{2} + 1$ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$2$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.3
Root			$1.61803i$ of defining polynomial
Character	$\chi$	$=$	1920.1151
Dual form			1920.2.h.d.1151.4

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.618034 - 1.61803i) q^{3} -1.00000i q^{5} +4.47214i q^{7} +(-2.23607 - 2.00000i) q^{9} -4.00000 q^{11} -1.23607 q^{13} +(-1.61803 - 0.618034i) q^{15} +3.23607i q^{17} -2.76393i q^{19} +(7.23607 + 2.76393i) q^{21} +6.00000 q^{23} -1.00000 q^{25} +(-4.61803 + 2.38197i) q^{27} -4.47214i q^{29} +9.70820i q^{31} +(-2.47214 + 6.47214i) q^{33} +4.47214 q^{35} -2.76393 q^{37} +(-0.763932 + 2.00000i) q^{39} +6.47214i q^{41} +11.2361i q^{43} +(-2.00000 + 2.23607i) q^{45} +6.94427 q^{47} -13.0000 q^{49} +(5.23607 + 2.00000i) q^{51} +12.4721i q^{53} +4.00000i q^{55} +(-4.47214 - 1.70820i) q^{57} +8.00000 q^{59} -0.472136 q^{61} +(8.94427 - 10.0000i) q^{63} +1.23607i q^{65} +9.70820i q^{67} +(3.70820 - 9.70820i) q^{69} -4.00000 q^{71} -13.4164 q^{73} +(-0.618034 + 1.61803i) q^{75} -17.8885i q^{77} +3.23607i q^{79} +(1.00000 + 8.94427i) q^{81} -4.29180 q^{83} +3.23607 q^{85} +(-7.23607 - 2.76393i) q^{87} -14.4721i q^{89} -5.52786i q^{91} +(15.7082 + 6.00000i) q^{93} -2.76393 q^{95} +0.472136 q^{97} +(8.94427 + 8.00000i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 2 q^{3} - 16 q^{11} + 4 q^{13} - 2 q^{15} + 20 q^{21} + 24 q^{23} - 4 q^{25} - 14 q^{27} + 8 q^{33} - 20 q^{37} - 12 q^{39} - 8 q^{45} - 8 q^{47} - 52 q^{49} + 12 q^{51} + 32 q^{59} + 16 q^{61} - 12 q^{69}+ \cdots - 16 q^{97}+O(q^{100})$ 4 * q - 2 * q^3 - 16 * q^11 + 4 * q^13 - 2 * q^15 + 20 * q^21 + 24 * q^23 - 4 * q^25 - 14 * q^27 + 8 * q^33 - 20 * q^37 - 12 * q^39 - 8 * q^45 - 8 * q^47 - 52 * q^49 + 12 * q^51 + 32 * q^59 + 16 * q^61 - 12 * q^69 - 16 * q^71 + 2 * q^75 + 4 * q^81 - 44 * q^83 + 4 * q^85 - 20 * q^87 + 36 * q^93 - 20 * q^95 - 16 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1920\mathbb{Z}\right)^\times$ .

$n$	$511$	$641$	$901$	$1537$
$\chi(n)$	$-1$	$-1$	$1$	$1$

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See 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
$3$	0.618034	−	1.61803i	0.356822	−	0.934172i
$3$	0.618034	−	1.61803i	0.356822	−	0.934172i
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$			4.47214i			1.69031i	0.534522	+	0.845154i	$0.320491\pi$
$7$			4.47214i			1.69031i	−0.534522	+	0.845154i	$0.679509\pi$
$8$		0			0
$9$	−2.23607	−	2.00000i	−0.745356	−	0.666667i
$10$		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
$13$	−1.23607			−0.342824			−0.171412	−	0.985199i	$-0.554833\pi$
$13$	−1.23607			−0.342824			−0.171412	+	0.985199i	$0.554833\pi$
$14$		0			0
$15$	−1.61803	−	0.618034i	−0.417775	−	0.159576i
$16$		0			0
$17$			3.23607i			0.784862i	0.919781	+	0.392431i	$0.128366\pi$
$17$			3.23607i			0.784862i	−0.919781	+	0.392431i	$0.871634\pi$
$18$		0			0
$19$		−	2.76393i		−	0.634089i	−0.948411	−	0.317045i	$-0.897309\pi$
$19$		−	2.76393i		−	0.634089i	0.948411	−	0.317045i	$-0.102691\pi$
$20$		0			0
$21$	7.23607	+	2.76393i	1.57904	+	0.603139i
$22$		0			0
$23$	6.00000			1.25109			0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000			1.25109			0.625543	+	0.780189i	$0.284877\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−4.61803	+	2.38197i	−0.888741	+	0.458410i
$28$		0			0
$29$		−	4.47214i		−	0.830455i	−0.909718	−	0.415227i	$-0.863702\pi$
$29$		−	4.47214i		−	0.830455i	0.909718	−	0.415227i	$-0.136298\pi$
$30$		0			0
$31$			9.70820i			1.74364i	0.489822	+	0.871822i	$0.337062\pi$
$31$			9.70820i			1.74364i	−0.489822	+	0.871822i	$0.662938\pi$
$32$		0			0
$33$	−2.47214	+	6.47214i	−0.430344	+	1.12665i
$34$		0			0
$35$	4.47214			0.755929
$36$		0			0
$37$	−2.76393			−0.454388			−0.227194	−	0.973850i	$-0.572955\pi$
$37$	−2.76393			−0.454388			−0.227194	+	0.973850i	$0.572955\pi$
$38$		0			0
$39$	−0.763932	+	2.00000i	−0.122327	+	0.320256i
$40$		0			0
$41$			6.47214i			1.01078i	0.862892	+	0.505389i	$0.168651\pi$
$41$			6.47214i			1.01078i	−0.862892	+	0.505389i	$0.831349\pi$
$42$		0			0
$43$			11.2361i			1.71348i	0.515745	+	0.856742i	$0.327515\pi$
$43$			11.2361i			1.71348i	−0.515745	+	0.856742i	$0.672485\pi$
$44$		0			0
$45$	−2.00000	+	2.23607i	−0.298142	+	0.333333i
$46$		0			0
$47$	6.94427			1.01293			0.506463	−	0.862262i	$-0.330953\pi$
$47$	6.94427			1.01293			0.506463	+	0.862262i	$0.330953\pi$
$48$		0			0
$49$	−13.0000			−1.85714
$50$		0			0
$51$	5.23607	+	2.00000i	0.733196	+	0.280056i
$52$		0			0
$53$			12.4721i			1.71318i	0.515998	+	0.856590i	$0.327421\pi$
$53$			12.4721i			1.71318i	−0.515998	+	0.856590i	$0.672579\pi$
$54$		0			0
$55$			4.00000i			0.539360i
$56$		0			0
$57$	−4.47214	−	1.70820i	−0.592349	−	0.226257i
$58$		0			0
$59$	8.00000			1.04151			0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000			1.04151			0.520756	+	0.853706i	$0.325650\pi$
$60$		0			0
$61$	−0.472136			−0.0604508			−0.0302254	−	0.999543i	$-0.509623\pi$
$61$	−0.472136			−0.0604508			−0.0302254	+	0.999543i	$0.509623\pi$
$62$		0			0
$63$	8.94427	−	10.0000i	1.12687	−	1.25988i
$64$		0			0
$65$			1.23607i			0.153315i
$66$		0			0
$67$			9.70820i			1.18605i	0.805186	+	0.593023i	$0.202066\pi$
$67$			9.70820i			1.18605i	−0.805186	+	0.593023i	$0.797934\pi$
$68$		0			0
$69$	3.70820	−	9.70820i	0.446415	−	1.16873i
$70$		0			0
$71$	−4.00000			−0.474713			−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000			−0.474713			−0.237356	+	0.971423i	$0.576281\pi$
$72$		0			0
$73$	−13.4164			−1.57027			−0.785136	−	0.619324i	$-0.787407\pi$
$73$	−13.4164			−1.57027			−0.785136	+	0.619324i	$0.787407\pi$
$74$		0			0
$75$	−0.618034	+	1.61803i	−0.0713644	+	0.186834i
$76$		0			0
$77$		−	17.8885i		−	2.03859i
$78$		0			0
$79$			3.23607i			0.364086i	0.983291	+	0.182043i	$0.0582710\pi$
$79$			3.23607i			0.364086i	−0.983291	+	0.182043i	$0.941729\pi$
$80$		0			0
$81$	1.00000	+	8.94427i	0.111111	+	0.993808i
$82$		0			0
$83$	−4.29180			−0.471086			−0.235543	−	0.971864i	$-0.575687\pi$
$83$	−4.29180			−0.471086			−0.235543	+	0.971864i	$0.575687\pi$
$84$		0			0
$85$	3.23607			0.351001
$86$		0			0
$87$	−7.23607	−	2.76393i	−0.775788	−	0.296325i
$88$		0			0
$89$		−	14.4721i		−	1.53404i	−0.641621	−	0.767022i	$-0.721738\pi$
$89$		−	14.4721i		−	1.53404i	0.641621	−	0.767022i	$-0.278262\pi$
$90$		0			0
$91$		−	5.52786i		−	0.579478i
$92$		0			0
$93$	15.7082	+	6.00000i	1.62886	+	0.622171i
$94$		0			0
$95$	−2.76393			−0.283573
$96$		0			0
$97$	0.472136			0.0479381			0.0239691	−	0.999713i	$-0.492370\pi$
$97$	0.472136			0.0479381			0.0239691	+	0.999713i	$0.492370\pi$
$98$		0			0
$99$	8.94427	+	8.00000i	0.898933	+	0.804030i
$100$		0			0
$101$			0.472136i			0.0469793i	0.999724	+	0.0234896i	$0.00747767\pi$
$101$			0.472136i			0.0469793i	−0.999724	+	0.0234896i	$0.992522\pi$
$102$		0			0
$103$			10.0000i			0.985329i	0.870219	+	0.492665i	$0.163977\pi$
$103$			10.0000i			0.985329i	−0.870219	+	0.492665i	$0.836023\pi$
$104$		0			0
$105$	2.76393	−	7.23607i	0.269732	−	0.706168i
$106$		0			0
$107$	−2.76393			−0.267199			−0.133600	−	0.991035i	$-0.542654\pi$
$107$	−2.76393			−0.267199			−0.133600	+	0.991035i	$0.542654\pi$
$108$		0			0
$109$	2.94427			0.282010			0.141005	−	0.990009i	$-0.454967\pi$
$109$	2.94427			0.282010			0.141005	+	0.990009i	$0.454967\pi$
$110$		0			0
$111$	−1.70820	+	4.47214i	−0.162136	+	0.424476i
$112$		0			0
$113$		−	13.7082i		−	1.28956i	−0.764368	−	0.644780i	$-0.776949\pi$
$113$		−	13.7082i		−	1.28956i	0.764368	−	0.644780i	$-0.223051\pi$
$114$		0			0
$115$		−	6.00000i		−	0.559503i
$116$		0			0
$117$	2.76393	+	2.47214i	0.255526	+	0.228549i
$118$		0			0
$119$	−14.4721			−1.32666
$120$		0			0
$121$	5.00000			0.454545
$122$		0			0
$123$	10.4721	+	4.00000i	0.944241	+	0.360668i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$		−	0.472136i		−	0.0418953i	−0.999781	−	0.0209476i	$-0.993332\pi$
$127$		−	0.472136i		−	0.0418953i	0.999781	−	0.0209476i	$-0.00666833\pi$
$128$		0			0
$129$	18.1803	+	6.94427i	1.60069	+	0.611409i
$130$		0			0
$131$	2.47214			0.215992			0.107996	−	0.994151i	$-0.465557\pi$
$131$	2.47214			0.215992			0.107996	+	0.994151i	$0.465557\pi$
$132$		0			0
$133$	12.3607			1.07181
$134$		0			0
$135$	2.38197	+	4.61803i	0.205007	+	0.397457i
$136$		0			0
$137$			0.180340i			0.0154075i	0.999970	+	0.00770374i	$0.00245220\pi$
$137$			0.180340i			0.0154075i	−0.999970	+	0.00770374i	$0.997548\pi$
$138$		0			0
$139$			6.18034i			0.524210i	0.965039	+	0.262105i	$0.0844166\pi$
$139$			6.18034i			0.524210i	−0.965039	+	0.262105i	$0.915583\pi$
$140$		0			0
$141$	4.29180	−	11.2361i	0.361434	−	0.946248i
$142$		0			0
$143$	4.94427			0.413461
$144$		0			0
$145$	−4.47214			−0.371391
$146$		0			0
$147$	−8.03444	+	21.0344i	−0.662670	+	1.73489i
$148$		0			0
$149$			10.9443i			0.896590i	0.893886	+	0.448295i	$0.147969\pi$
$149$			10.9443i			0.896590i	−0.893886	+	0.448295i	$0.852031\pi$
$150$		0			0
$151$			4.76393i			0.387683i	0.981033	+	0.193842i	$0.0620948\pi$
$151$			4.76393i			0.387683i	−0.981033	+	0.193842i	$0.937905\pi$
$152$		0			0
$153$	6.47214	−	7.23607i	0.523241	−	0.585001i
$154$		0			0
$155$	9.70820			0.779782
$156$		0			0
$157$	−1.23607			−0.0986490			−0.0493245	−	0.998783i	$-0.515707\pi$
$157$	−1.23607			−0.0986490			−0.0493245	+	0.998783i	$0.515707\pi$
$158$		0			0
$159$	20.1803	+	7.70820i	1.60041	+	0.611300i
$160$		0			0
$161$			26.8328i			2.11472i
$162$		0			0
$163$		−	22.6525i		−	1.77428i	−0.461502	−	0.887139i	$-0.652689\pi$
$163$		−	22.6525i		−	1.77428i	0.461502	−	0.887139i	$-0.347311\pi$
$164$		0			0
$165$	6.47214	+	2.47214i	0.503855	+	0.192456i
$166$		0			0
$167$	21.4164			1.65725			0.828626	−	0.559803i	$-0.189123\pi$
$167$	21.4164			1.65725			0.828626	+	0.559803i	$0.189123\pi$
$168$		0			0
$169$	−11.4721			−0.882472
$170$		0			0
$171$	−5.52786	+	6.18034i	−0.422726	+	0.472622i
$172$		0			0
$173$		−	16.4721i		−	1.25235i	−0.779681	−	0.626177i	$-0.784619\pi$
$173$		−	16.4721i		−	1.25235i	0.779681	−	0.626177i	$-0.215381\pi$
$174$		0			0
$175$		−	4.47214i		−	0.338062i
$176$		0			0
$177$	4.94427	−	12.9443i	0.371634	−	0.972951i
$178$		0			0
$179$	−25.8885			−1.93500			−0.967500	−	0.252870i	$-0.918626\pi$
$179$	−25.8885			−1.93500			−0.967500	+	0.252870i	$0.918626\pi$
$180$		0			0
$181$	13.4164			0.997234			0.498617	−	0.866822i	$-0.333841\pi$
$181$	13.4164			0.997234			0.498617	+	0.866822i	$0.333841\pi$
$182$		0			0
$183$	−0.291796	+	0.763932i	−0.0215702	+	0.0564715i
$184$		0			0
$185$			2.76393i			0.203208i
$186$		0			0
$187$		−	12.9443i		−	0.946579i
$188$		0			0
$189$	−10.6525	−	20.6525i	−0.774854	−	1.50225i
$190$		0			0
$191$	−24.3607			−1.76268			−0.881338	−	0.472485i	$-0.843357\pi$
$191$	−24.3607			−1.76268			−0.881338	+	0.472485i	$0.843357\pi$
$192$		0			0
$193$	10.9443			0.787786			0.393893	−	0.919156i	$-0.371128\pi$
$193$	10.9443			0.787786			0.393893	+	0.919156i	$0.371128\pi$
$194$		0			0
$195$	2.00000	+	0.763932i	0.143223	+	0.0547063i
$196$		0			0
$197$		−	6.94427i		−	0.494759i	−0.968919	−	0.247379i	$-0.920431\pi$
$197$		−	6.94427i		−	0.494759i	0.968919	−	0.247379i	$-0.0795694\pi$
$198$		0			0
$199$		−	9.70820i		−	0.688196i	−0.938934	−	0.344098i	$-0.888185\pi$
$199$		−	9.70820i		−	0.688196i	0.938934	−	0.344098i	$-0.111815\pi$
$200$		0			0
$201$	15.7082	+	6.00000i	1.10797	+	0.423207i
$202$		0			0
$203$	20.0000			1.40372
$204$		0			0
$205$	6.47214			0.452034
$206$		0			0
$207$	−13.4164	−	12.0000i	−0.932505	−	0.834058i
$208$		0			0
$209$			11.0557i			0.764741i
$210$		0			0
$211$			15.1246i			1.04122i	0.853794	+	0.520611i	$0.174296\pi$
$211$			15.1246i			1.04122i	−0.853794	+	0.520611i	$0.825704\pi$
$212$		0			0
$213$	−2.47214	+	6.47214i	−0.169388	+	0.443463i
$214$		0			0
$215$	11.2361			0.766293
$216$		0			0
$217$	−43.4164			−2.94730
$218$		0			0
$219$	−8.29180	+	21.7082i	−0.560308	+	1.46690i
$220$		0			0
$221$		−	4.00000i		−	0.269069i
$222$		0			0
$223$			2.58359i			0.173010i	0.996251	+	0.0865051i	$0.0275699\pi$
$223$			2.58359i			0.173010i	−0.996251	+	0.0865051i	$0.972430\pi$
$224$		0			0
$225$	2.23607	+	2.00000i	0.149071	+	0.133333i
$226$		0			0
$227$	9.23607			0.613019			0.306510	−	0.951868i	$-0.400839\pi$
$227$	9.23607			0.613019			0.306510	+	0.951868i	$0.400839\pi$
$228$		0			0
$229$	−15.8885			−1.04994			−0.524972	−	0.851119i	$-0.675924\pi$
$229$	−15.8885			−1.04994			−0.524972	+	0.851119i	$0.675924\pi$
$230$		0			0
$231$	−28.9443	−	11.0557i	−1.90439	−	0.727414i
$232$		0			0
$233$		−	23.5967i		−	1.54587i	−0.634483	−	0.772937i	$-0.718787\pi$
$233$		−	23.5967i		−	1.54587i	0.634483	−	0.772937i	$-0.281213\pi$
$234$		0			0
$235$		−	6.94427i		−	0.452994i
$236$		0			0
$237$	5.23607	+	2.00000i	0.340119	+	0.129914i
$238$		0			0
$239$	−10.4721			−0.677386			−0.338693	−	0.940897i	$-0.609985\pi$
$239$	−10.4721			−0.677386			−0.338693	+	0.940897i	$0.609985\pi$
$240$		0			0
$241$	−9.05573			−0.583331			−0.291665	−	0.956520i	$-0.594209\pi$
$241$	−9.05573			−0.583331			−0.291665	+	0.956520i	$0.594209\pi$
$242$		0			0
$243$	15.0902	+	3.90983i	0.968035	+	0.250816i
$244$		0			0
$245$			13.0000i			0.830540i
$246$		0			0
$247$			3.41641i			0.217381i
$248$		0			0
$249$	−2.65248	+	6.94427i	−0.168094	+	0.440075i
$250$		0			0
$251$	4.00000			0.252478			0.126239	−	0.992000i	$-0.459709\pi$
$251$	4.00000			0.252478			0.126239	+	0.992000i	$0.459709\pi$
$252$		0			0
$253$	−24.0000			−1.50887
$254$		0			0
$255$	2.00000	−	5.23607i	0.125245	−	0.327895i
$256$		0			0
$257$			21.7082i			1.35412i	0.735928	+	0.677060i	$0.236746\pi$
$257$			21.7082i			1.35412i	−0.735928	+	0.677060i	$0.763254\pi$
$258$		0			0
$259$		−	12.3607i		−	0.768055i
$260$		0			0
$261$	−8.94427	+	10.0000i	−0.553637	+	0.618984i
$262$		0			0
$263$	25.4164			1.56724			0.783621	−	0.621239i	$-0.213370\pi$
$263$	25.4164			1.56724			0.783621	+	0.621239i	$0.213370\pi$
$264$		0			0
$265$	12.4721			0.766157
$266$		0			0
$267$	−23.4164	−	8.94427i	−1.43306	−	0.547381i
$268$		0			0
$269$			22.9443i			1.39894i	0.714663	+	0.699468i	$0.246580\pi$
$269$			22.9443i			1.39894i	−0.714663	+	0.699468i	$0.753420\pi$
$270$		0			0
$271$			21.7082i			1.31868i	0.751845	+	0.659340i	$0.229164\pi$
$271$			21.7082i			1.31868i	−0.751845	+	0.659340i	$0.770836\pi$
$272$		0			0
$273$	−8.94427	−	3.41641i	−0.541332	−	0.206770i
$274$		0			0
$275$	4.00000			0.241209
$276$		0			0
$277$	−23.7082			−1.42449			−0.712244	−	0.701932i	$-0.752321\pi$
$277$	−23.7082			−1.42449			−0.712244	+	0.701932i	$0.752321\pi$
$278$		0			0
$279$	19.4164	−	21.7082i	1.16243	−	1.29964i
$280$		0			0
$281$		−	23.4164i		−	1.39691i	−0.715656	−	0.698453i	$-0.753872\pi$
$281$		−	23.4164i		−	1.39691i	0.715656	−	0.698453i	$-0.246128\pi$
$282$		0			0
$283$		−	16.1803i		−	0.961821i	−0.876770	−	0.480911i	$-0.840306\pi$
$283$		−	16.1803i		−	0.961821i	0.876770	−	0.480911i	$-0.159694\pi$
$284$		0			0
$285$	−1.70820	+	4.47214i	−0.101185	+	0.264906i
$286$		0			0
$287$	−28.9443			−1.70853
$288$		0			0
$289$	6.52786			0.383992
$290$		0			0
$291$	0.291796	−	0.763932i	0.0171054	−	0.0447825i
$292$		0			0
$293$			18.3607i			1.07264i	0.844014	+	0.536321i	$0.180186\pi$
$293$			18.3607i			1.07264i	−0.844014	+	0.536321i	$0.819814\pi$
$294$		0			0
$295$		−	8.00000i		−	0.465778i
$296$		0			0
$297$	18.4721	−	9.52786i	1.07186	−	0.552863i
$298$		0			0
$299$	−7.41641			−0.428902
$300$		0			0
$301$	−50.2492			−2.89632
$302$		0			0
$303$	0.763932	+	0.291796i	0.0438867	+	0.0167632i
$304$		0			0
$305$			0.472136i			0.0270344i
$306$		0			0
$307$		−	2.29180i		−	0.130800i	−0.997859	−	0.0653999i	$-0.979168\pi$
$307$		−	2.29180i		−	0.130800i	0.997859	−	0.0653999i	$-0.0208323\pi$
$308$		0			0
$309$	16.1803	+	6.18034i	0.920467	+	0.351587i
$310$		0			0
$311$	−5.88854			−0.333909			−0.166954	−	0.985965i	$-0.553393\pi$
$311$	−5.88854			−0.333909			−0.166954	+	0.985965i	$0.553393\pi$
$312$		0			0
$313$	−11.8885			−0.671980			−0.335990	−	0.941866i	$-0.609071\pi$
$313$	−11.8885			−0.671980			−0.335990	+	0.941866i	$0.609071\pi$
$314$		0			0
$315$	−10.0000	−	8.94427i	−0.563436	−	0.503953i
$316$		0			0
$317$		−	10.9443i		−	0.614692i	−0.951598	−	0.307346i	$-0.900559\pi$
$317$		−	10.9443i		−	0.614692i	0.951598	−	0.307346i	$-0.0994408\pi$
$318$		0			0
$319$			17.8885i			1.00157i
$320$		0			0
$321$	−1.70820	+	4.47214i	−0.0953426	+	0.249610i
$322$		0			0
$323$	8.94427			0.497673
$324$		0			0
$325$	1.23607			0.0685647
$326$		0			0
$327$	1.81966	−	4.76393i	0.100627	−	0.263446i
$328$		0			0
$329$			31.0557i			1.71216i
$330$		0			0
$331$		−	23.7082i		−	1.30312i	−0.758597	−	0.651560i	$-0.774115\pi$
$331$		−	23.7082i		−	1.30312i	0.758597	−	0.651560i	$-0.225885\pi$
$332$		0			0
$333$	6.18034	+	5.52786i	0.338681	+	0.302925i
$334$		0			0
$335$	9.70820			0.530416
$336$		0			0
$337$	22.3607			1.21806			0.609032	−	0.793146i	$-0.291558\pi$
$337$	22.3607			1.21806			0.609032	+	0.793146i	$0.291558\pi$
$338$		0			0
$339$	−22.1803	−	8.47214i	−1.20467	−	0.460143i
$340$		0			0
$341$		−	38.8328i		−	2.10291i
$342$		0			0
$343$		−	26.8328i		−	1.44884i
$344$		0			0
$345$	−9.70820	−	3.70820i	−0.522672	−	0.199643i
$346$		0			0
$347$	−5.23607			−0.281087			−0.140543	−	0.990075i	$-0.544885\pi$
$347$	−5.23607			−0.281087			−0.140543	+	0.990075i	$0.544885\pi$
$348$		0			0
$349$	10.0000			0.535288			0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000			0.535288			0.267644	+	0.963518i	$0.413755\pi$
$350$		0			0
$351$	5.70820	−	2.94427i	0.304681	−	0.157154i
$352$		0			0
$353$		−	29.7082i		−	1.58121i	−0.612328	−	0.790604i	$-0.709767\pi$
$353$		−	29.7082i		−	1.58121i	0.612328	−	0.790604i	$-0.290233\pi$
$354$		0			0
$355$			4.00000i			0.212298i
$356$		0			0
$357$	−8.94427	+	23.4164i	−0.473381	+	1.23933i
$358$		0			0
$359$	5.52786			0.291750			0.145875	−	0.989303i	$-0.453400\pi$
$359$	5.52786			0.291750			0.145875	+	0.989303i	$0.453400\pi$
$360$		0			0
$361$	11.3607			0.597931
$362$		0			0
$363$	3.09017	−	8.09017i	0.162192	−	0.424624i
$364$		0			0
$365$			13.4164i			0.702247i
$366$		0			0
$367$		−	10.9443i		−	0.571286i	−0.958336	−	0.285643i	$-0.907793\pi$
$367$		−	10.9443i		−	0.571286i	0.958336	−	0.285643i	$-0.0922072\pi$
$368$		0			0
$369$	12.9443	−	14.4721i	0.673852	−	0.753389i
$370$		0			0
$371$	−55.7771			−2.89580
$372$		0			0
$373$	−17.2361			−0.892450			−0.446225	−	0.894921i	$-0.647232\pi$
$373$	−17.2361			−0.892450			−0.446225	+	0.894921i	$0.647232\pi$
$374$		0			0
$375$	1.61803	+	0.618034i	0.0835549	+	0.0319151i
$376$		0			0
$377$			5.52786i			0.284699i
$378$		0			0
$379$		−	15.7082i		−	0.806876i	−0.915007	−	0.403438i	$-0.867815\pi$
$379$		−	15.7082i		−	0.806876i	0.915007	−	0.403438i	$-0.132185\pi$
$380$		0			0
$381$	−0.763932	−	0.291796i	−0.0391374	−	0.0149492i
$382$		0			0
$383$	2.58359			0.132015			0.0660077	−	0.997819i	$-0.478974\pi$
$383$	2.58359			0.132015			0.0660077	+	0.997819i	$0.478974\pi$
$384$		0			0
$385$	−17.8885			−0.911685
$386$		0			0
$387$	22.4721	−	25.1246i	1.14232	−	1.27716i
$388$		0			0
$389$			22.9443i			1.16332i	0.813432	+	0.581660i	$0.197597\pi$
$389$			22.9443i			1.16332i	−0.813432	+	0.581660i	$0.802403\pi$
$390$		0			0
$391$			19.4164i			0.981930i
$392$		0			0
$393$	1.52786	−	4.00000i	0.0770705	−	0.201773i
$394$		0			0
$395$	3.23607			0.162824
$396$		0			0
$397$	0.291796			0.0146448			0.00732241	−	0.999973i	$-0.497669\pi$
$397$	0.291796			0.0146448			0.00732241	+	0.999973i	$0.497669\pi$
$398$		0			0
$399$	7.63932	−	20.0000i	0.382444	−	1.00125i
$400$		0			0
$401$			16.9443i			0.846157i	0.906093	+	0.423078i	$0.139050\pi$
$401$			16.9443i			0.846157i	−0.906093	+	0.423078i	$0.860950\pi$
$402$		0			0
$403$		−	12.0000i		−	0.597763i
$404$		0			0
$405$	8.94427	−	1.00000i	0.444444	−	0.0496904i
$406$		0			0
$407$	11.0557			0.548012
$408$		0			0
$409$	39.3050			1.94350			0.971752	−	0.236003i	$-0.0758374\pi$
$409$	39.3050			1.94350			0.971752	+	0.236003i	$0.0758374\pi$
$410$		0			0
$411$	0.291796	+	0.111456i	0.0143932	+	0.00549773i
$412$		0			0
$413$			35.7771i			1.76048i
$414$		0			0
$415$			4.29180i			0.210676i
$416$		0			0
$417$	10.0000	+	3.81966i	0.489702	+	0.187050i
$418$		0			0
$419$	3.41641			0.166902			0.0834512	−	0.996512i	$-0.473406\pi$
$419$	3.41641			0.166902			0.0834512	+	0.996512i	$0.473406\pi$
$420$		0			0
$421$	15.5279			0.756782			0.378391	−	0.925646i	$-0.376478\pi$
$421$	15.5279			0.756782			0.378391	+	0.925646i	$0.376478\pi$
$422$		0			0
$423$	−15.5279	−	13.8885i	−0.754991	−	0.675284i
$424$		0			0
$425$		−	3.23607i		−	0.156972i
$426$		0			0
$427$		−	2.11146i		−	0.102181i
$428$		0			0
$429$	3.05573	−	8.00000i	0.147532	−	0.386244i
$430$		0			0
$431$	0.944272			0.0454840			0.0227420	−	0.999741i	$-0.492760\pi$
$431$	0.944272			0.0454840			0.0227420	+	0.999741i	$0.492760\pi$
$432$		0			0
$433$	−20.4721			−0.983828			−0.491914	−	0.870644i	$-0.663703\pi$
$433$	−20.4721			−0.983828			−0.491914	+	0.870644i	$0.663703\pi$
$434$		0			0
$435$	−2.76393	+	7.23607i	−0.132520	+	0.346943i
$436$		0			0
$437$		−	16.5836i		−	0.793301i
$438$		0			0
$439$			11.2361i			0.536268i	0.963382	+	0.268134i	$0.0864070\pi$
$439$			11.2361i			0.536268i	−0.963382	+	0.268134i	$0.913593\pi$
$440$		0			0
$441$	29.0689	+	26.0000i	1.38423	+	1.23810i
$442$		0			0
$443$	13.5967			0.646001			0.323000	−	0.946399i	$-0.395308\pi$
$443$	13.5967			0.646001			0.323000	+	0.946399i	$0.395308\pi$
$444$		0			0
$445$	−14.4721			−0.686045
$446$		0			0
$447$	17.7082	+	6.76393i	0.837569	+	0.319923i
$448$		0			0
$449$			4.94427i			0.233335i	0.993171	+	0.116667i	$0.0372211\pi$
$449$			4.94427i			0.233335i	−0.993171	+	0.116667i	$0.962779\pi$
$450$		0			0
$451$		−	25.8885i		−	1.21904i
$452$		0			0
$453$	7.70820	+	2.94427i	0.362163	+	0.138334i
$454$		0			0
$455$	−5.52786			−0.259150
$456$		0			0
$457$	40.4721			1.89321			0.946603	−	0.322401i	$-0.104490\pi$
$457$	40.4721			1.89321			0.946603	+	0.322401i	$0.104490\pi$
$458$		0			0
$459$	−7.70820	−	14.9443i	−0.359788	−	0.697539i
$460$		0			0
$461$			5.41641i			0.252267i	0.992013	+	0.126134i	$0.0402568\pi$
$461$			5.41641i			0.252267i	−0.992013	+	0.126134i	$0.959743\pi$
$462$		0			0
$463$			14.9443i			0.694519i	0.937769	+	0.347260i	$0.112888\pi$
$463$			14.9443i			0.694519i	−0.937769	+	0.347260i	$0.887112\pi$
$464$		0			0
$465$	6.00000	−	15.7082i	0.278243	−	0.728451i
$466$		0			0
$467$	−26.1803			−1.21148			−0.605741	−	0.795662i	$-0.707123\pi$
$467$	−26.1803			−1.21148			−0.605741	+	0.795662i	$0.707123\pi$
$468$		0			0
$469$	−43.4164			−2.00478
$470$		0			0
$471$	−0.763932	+	2.00000i	−0.0352001	+	0.0921551i
$472$		0			0
$473$		−	44.9443i		−	2.06654i
$474$		0			0
$475$			2.76393i			0.126818i
$476$		0			0
$477$	24.9443	−	27.8885i	1.14212	−	1.27693i
$478$		0			0
$479$	−10.4721			−0.478484			−0.239242	−	0.970960i	$-0.576899\pi$
$479$	−10.4721			−0.478484			−0.239242	+	0.970960i	$0.576899\pi$
$480$		0			0
$481$	3.41641			0.155775
$482$		0			0
$483$	43.4164	+	16.5836i	1.97551	+	0.754580i
$484$		0			0
$485$		−	0.472136i		−	0.0214386i
$486$		0			0
$487$			34.0000i			1.54069i	0.637629	+	0.770344i	$0.279915\pi$
$487$			34.0000i			1.54069i	−0.637629	+	0.770344i	$0.720085\pi$
$488$		0			0
$489$	−36.6525	−	14.0000i	−1.65748	−	0.633102i
$490$		0			0
$491$	−4.36068			−0.196795			−0.0983974	−	0.995147i	$-0.531372\pi$
$491$	−4.36068			−0.196795			−0.0983974	+	0.995147i	$0.531372\pi$
$492$		0			0
$493$	14.4721			0.651792
$494$		0			0
$495$	8.00000	−	8.94427i	0.359573	−	0.402015i
$496$		0			0
$497$		−	17.8885i		−	0.802411i
$498$		0			0
$499$			6.18034i			0.276670i	0.990385	+	0.138335i	$0.0441751\pi$
$499$			6.18034i			0.276670i	−0.990385	+	0.138335i	$0.955825\pi$
$500$		0			0
$501$	13.2361	−	34.6525i	0.591344	−	1.54816i
$502$		0			0
$503$	−0.472136			−0.0210515			−0.0105258	−	0.999945i	$-0.503351\pi$
$503$	−0.472136			−0.0210515			−0.0105258	+	0.999945i	$0.503351\pi$
$504$		0			0
$505$	0.472136			0.0210098
$506$		0			0
$507$	−7.09017	+	18.5623i	−0.314886	+	0.824381i
$508$		0			0
$509$		−	13.4164i		−	0.594672i	−0.954773	−	0.297336i	$-0.903902\pi$
$509$		−	13.4164i		−	0.594672i	0.954773	−	0.297336i	$-0.0960981\pi$
$510$		0			0
$511$		−	60.0000i		−	2.65424i
$512$		0			0
$513$	6.58359	+	12.7639i	0.290673	+	0.563541i
$514$		0			0
$515$	10.0000			0.440653
$516$		0			0
$517$	−27.7771			−1.22163
$518$		0			0
$519$	−26.6525	−	10.1803i	−1.16991	−	0.446867i
$520$		0			0
$521$		−	5.88854i		−	0.257982i	−0.991646	−	0.128991i	$-0.958826\pi$
$521$		−	5.88854i		−	0.257982i	0.991646	−	0.128991i	$-0.0411738\pi$
$522$		0			0
$523$		−	13.7082i		−	0.599418i	−0.954031	−	0.299709i	$-0.903110\pi$
$523$		−	13.7082i		−	0.599418i	0.954031	−	0.299709i	$-0.0968896\pi$
$524$		0			0
$525$	−7.23607	−	2.76393i	−0.315808	−	0.120628i
$526$		0			0
$527$	−31.4164			−1.36852
$528$		0			0
$529$	13.0000			0.565217
$530$		0			0
$531$	−17.8885	−	16.0000i	−0.776297	−	0.694341i
$532$		0			0
$533$		−	8.00000i		−	0.346518i
$534$		0			0
$535$			2.76393i			0.119495i
$536$		0			0
$537$	−16.0000	+	41.8885i	−0.690451	+	1.80762i
$538$		0			0
$539$	52.0000			2.23980
$540$		0			0
$541$	−33.4164			−1.43668			−0.718342	−	0.695690i	$-0.755099\pi$
$541$	−33.4164			−1.43668			−0.718342	+	0.695690i	$0.755099\pi$
$542$		0			0
$543$	8.29180	−	21.7082i	0.355835	−	0.931588i
$544$		0			0
$545$		−	2.94427i		−	0.126119i
$546$		0			0
$547$			11.2361i			0.480420i	0.970721	+	0.240210i	$0.0772162\pi$
$547$			11.2361i			0.480420i	−0.970721	+	0.240210i	$0.922784\pi$
$548$		0			0
$549$	1.05573	+	0.944272i	0.0450574	+	0.0403005i
$550$		0			0
$551$	−12.3607			−0.526583
$552$		0			0
$553$	−14.4721			−0.615418
$554$		0			0
$555$	4.47214	+	1.70820i	0.189832	+	0.0725092i
$556$		0			0
$557$			12.4721i			0.528461i	0.964460	+	0.264231i	$0.0851180\pi$
$557$			12.4721i			0.528461i	−0.964460	+	0.264231i	$0.914882\pi$
$558$		0			0
$559$		−	13.8885i		−	0.587423i
$560$		0			0
$561$	−20.9443	−	8.00000i	−0.884268	−	0.337760i
$562$		0			0
$563$	−33.5967			−1.41593			−0.707967	−	0.706245i	$-0.750387\pi$
$563$	−33.5967			−1.41593			−0.707967	+	0.706245i	$0.750387\pi$
$564$		0			0
$565$	−13.7082			−0.576708
$566$		0			0
$567$	−40.0000	+	4.47214i	−1.67984	+	0.187812i
$568$		0			0
$569$			6.83282i			0.286447i	0.989690	+	0.143223i	$0.0457467\pi$
$569$			6.83282i			0.286447i	−0.989690	+	0.143223i	$0.954253\pi$
$570$		0			0
$571$			44.0689i			1.84423i	0.386921	+	0.922113i	$0.373538\pi$
$571$			44.0689i			1.84423i	−0.386921	+	0.922113i	$0.626462\pi$
$572$		0			0
$573$	−15.0557	+	39.4164i	−0.628962	+	1.64664i
$574$		0			0
$575$	−6.00000			−0.250217
$576$		0			0
$577$	14.3607			0.597843			0.298921	−	0.954278i	$-0.403373\pi$
$577$	14.3607			0.597843			0.298921	+	0.954278i	$0.403373\pi$
$578$		0			0
$579$	6.76393	−	17.7082i	0.281099	−	0.735928i
$580$		0			0
$581$		−	19.1935i		−	0.796280i
$582$		0			0
$583$		−	49.8885i		−	2.06617i
$584$		0			0
$585$	2.47214	−	2.76393i	0.102210	−	0.114275i
$586$		0			0
$587$	−12.6525			−0.522224			−0.261112	−	0.965309i	$-0.584089\pi$
$587$	−12.6525			−0.522224			−0.261112	+	0.965309i	$0.584089\pi$
$588$		0			0
$589$	26.8328			1.10563
$590$		0			0
$591$	−11.2361	−	4.29180i	−0.462190	−	0.176541i
$592$		0			0
$593$			2.29180i			0.0941128i	0.998892	+	0.0470564i	$0.0149840\pi$
$593$			2.29180i			0.0941128i	−0.998892	+	0.0470564i	$0.985016\pi$
$594$		0			0
$595$			14.4721i			0.593300i
$596$		0			0
$597$	−15.7082	−	6.00000i	−0.642894	−	0.245564i
$598$		0			0
$599$	−21.5279			−0.879605			−0.439802	−	0.898095i	$-0.644952\pi$
$599$	−21.5279			−0.879605			−0.439802	+	0.898095i	$0.644952\pi$
$600$		0			0
$601$	22.3607			0.912111			0.456056	−	0.889951i	$-0.349262\pi$
$601$	22.3607			0.912111			0.456056	+	0.889951i	$0.349262\pi$
$602$		0			0
$603$	19.4164	−	21.7082i	0.790697	−	0.884026i
$604$		0			0
$605$		−	5.00000i		−	0.203279i
$606$		0			0
$607$		−	22.0000i		−	0.892952i	−0.894795	−	0.446476i	$-0.852679\pi$
$607$		−	22.0000i		−	0.892952i	0.894795	−	0.446476i	$-0.147321\pi$
$608$		0			0
$609$	12.3607	−	32.3607i	0.500880	−	1.31132i
$610$		0			0
$611$	−8.58359			−0.347255
$612$		0			0
$613$	39.1246			1.58023			0.790114	−	0.612960i	$-0.210021\pi$
$613$	39.1246			1.58023			0.790114	+	0.612960i	$0.210021\pi$
$614$		0			0
$615$	4.00000	−	10.4721i	0.161296	−	0.422277i
$616$		0			0
$617$			20.5410i			0.826950i	0.910515	+	0.413475i	$0.135685\pi$
$617$			20.5410i			0.826950i	−0.910515	+	0.413475i	$0.864315\pi$
$618$		0			0
$619$			27.1246i			1.09023i	0.838361	+	0.545115i	$0.183514\pi$
$619$			27.1246i			1.09023i	−0.838361	+	0.545115i	$0.816486\pi$
$620$		0			0
$621$	−27.7082	+	14.2918i	−1.11189	+	0.573510i
$622$		0			0
$623$	64.7214			2.59301
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	17.8885	+	6.83282i	0.714400	+	0.272876i
$628$		0			0
$629$		−	8.94427i		−	0.356631i
$630$		0			0
$631$			18.6525i			0.742543i	0.928524	+	0.371272i	$0.121078\pi$
$631$			18.6525i			0.742543i	−0.928524	+	0.371272i	$0.878922\pi$
$632$		0			0
$633$	24.4721	+	9.34752i	0.972680	+	0.371531i
$634$		0			0
$635$	−0.472136			−0.0187361
$636$		0			0
$637$	16.0689			0.636672
$638$		0			0
$639$	8.94427	+	8.00000i	0.353830	+	0.316475i
$640$		0			0
$641$		−	27.4164i		−	1.08288i	−0.840738	−	0.541442i	$-0.817879\pi$
$641$		−	27.4164i		−	1.08288i	0.840738	−	0.541442i	$-0.182121\pi$
$642$		0			0
$643$		−	3.59675i		−	0.141842i	−0.997482	−	0.0709209i	$-0.977406\pi$
$643$		−	3.59675i		−	0.141842i	0.997482	−	0.0709209i	$-0.0225938\pi$
$644$		0			0
$645$	6.94427	−	18.1803i	0.273430	−	0.715850i
$646$		0			0
$647$	−27.3050			−1.07347			−0.536734	−	0.843751i	$-0.680342\pi$
$647$	−27.3050			−1.07347			−0.536734	+	0.843751i	$0.680342\pi$
$648$		0			0
$649$	−32.0000			−1.25611
$650$		0			0
$651$	−26.8328	+	70.2492i	−1.05166	+	2.75328i
$652$		0			0
$653$			13.4164i			0.525025i	0.964929	+	0.262512i	$0.0845510\pi$
$653$			13.4164i			0.525025i	−0.964929	+	0.262512i	$0.915449\pi$
$654$		0			0
$655$		−	2.47214i		−	0.0965943i
$656$		0			0
$657$	30.0000	+	26.8328i	1.17041	+	1.04685i
$658$		0			0
$659$	41.8885			1.63175			0.815873	−	0.578231i	$-0.196257\pi$
$659$	41.8885			1.63175			0.815873	+	0.578231i	$0.196257\pi$
$660$		0			0
$661$	14.3607			0.558566			0.279283	−	0.960209i	$-0.409903\pi$
$661$	14.3607			0.558566			0.279283	+	0.960209i	$0.409903\pi$
$662$		0			0
$663$	−6.47214	−	2.47214i	−0.251357	−	0.0960098i
$664$		0			0
$665$		−	12.3607i		−	0.479327i
$666$		0			0
$667$		−	26.8328i		−	1.03897i
$668$		0			0
$669$	4.18034	+	1.59675i	0.161621	+	0.0617338i
$670$		0			0
$671$	1.88854			0.0729064
$672$		0			0
$673$	11.5279			0.444367			0.222183	−	0.975005i	$-0.428682\pi$
$673$	11.5279			0.444367			0.222183	+	0.975005i	$0.428682\pi$
$674$		0			0
$675$	4.61803	−	2.38197i	0.177748	−	0.0916819i
$676$		0			0
$677$			32.2492i			1.23944i	0.784824	+	0.619719i	$0.212753\pi$
$677$			32.2492i			1.23944i	−0.784824	+	0.619719i	$0.787247\pi$
$678$		0			0
$679$			2.11146i			0.0810303i
$680$		0			0
$681$	5.70820	−	14.9443i	0.218739	−	0.572666i
$682$		0			0
$683$	−23.7082			−0.907169			−0.453585	−	0.891213i	$-0.649855\pi$
$683$	−23.7082			−0.907169			−0.453585	+	0.891213i	$0.649855\pi$
$684$		0			0
$685$	0.180340			0.00689043
$686$		0			0
$687$	−9.81966	+	25.7082i	−0.374643	+	0.980829i
$688$		0			0
$689$		−	15.4164i		−	0.587318i
$690$		0			0
$691$			19.1246i			0.727535i	0.931490	+	0.363767i	$0.118510\pi$
$691$			19.1246i			0.727535i	−0.931490	+	0.363767i	$0.881490\pi$
$692$		0			0
$693$	−35.7771	+	40.0000i	−1.35906	+	1.51947i
$694$		0			0
$695$	6.18034			0.234434
$696$		0			0
$697$	−20.9443			−0.793321
$698$		0			0
$699$	−38.1803	−	14.5836i	−1.44411	−	0.551602i
$700$		0			0
$701$		−	42.0000i		−	1.58632i	−0.609015	−	0.793159i	$-0.708435\pi$
$701$		−	42.0000i		−	1.58632i	0.609015	−	0.793159i	$-0.291565\pi$
$702$		0			0
$703$			7.63932i			0.288122i
$704$		0			0
$705$	−11.2361	−	4.29180i	−0.423175	−	0.161638i
$706$		0			0
$707$	−2.11146			−0.0794095
$708$		0			0
$709$	−10.9443			−0.411021			−0.205510	−	0.978655i	$-0.565885\pi$
$709$	−10.9443			−0.411021			−0.205510	+	0.978655i	$0.565885\pi$
$710$		0			0
$711$	6.47214	−	7.23607i	0.242724	−	0.271374i
$712$		0			0
$713$			58.2492i			2.18145i
$714$		0			0
$715$		−	4.94427i		−	0.184905i
$716$		0			0
$717$	−6.47214	+	16.9443i	−0.241706	+	0.632795i
$718$		0			0
$719$	−44.9443			−1.67614			−0.838069	−	0.545564i	$-0.816316\pi$
$719$	−44.9443			−1.67614			−0.838069	+	0.545564i	$0.816316\pi$
$720$		0			0
$721$	−44.7214			−1.66551
$722$		0			0
$723$	−5.59675	+	14.6525i	−0.208145	+	0.544931i
$724$		0			0
$725$			4.47214i			0.166091i
$726$		0			0
$727$			10.0000i			0.370879i	0.982656	+	0.185440i	$0.0593710\pi$
$727$			10.0000i			0.370879i	−0.982656	+	0.185440i	$0.940629\pi$
$728$		0			0
$729$	15.6525	−	22.0000i	0.579721	−	0.814815i
$730$		0			0
$731$	−36.3607			−1.34485
$732$		0			0
$733$	50.1803			1.85345			0.926727	−	0.375736i	$-0.122610\pi$
$733$	50.1803			1.85345			0.926727	+	0.375736i	$0.122610\pi$
$734$		0			0
$735$	21.0344	+	8.03444i	0.775867	+	0.296355i
$736$		0			0
$737$		−	38.8328i		−	1.43043i
$738$		0			0
$739$		−	30.7639i		−	1.13167i	−0.824519	−	0.565835i	$-0.808554\pi$
$739$		−	30.7639i		−	1.13167i	0.824519	−	0.565835i	$-0.191446\pi$
$740$		0			0
$741$	5.52786	+	2.11146i	0.203071	+	0.0775663i
$742$		0			0
$743$	47.8885			1.75686			0.878430	−	0.477871i	$-0.158591\pi$
$743$	47.8885			1.75686			0.878430	+	0.477871i	$0.158591\pi$
$744$		0			0
$745$	10.9443			0.400967
$746$		0			0
$747$	9.59675	+	8.58359i	0.351127	+	0.314057i
$748$		0			0
$749$		−	12.3607i		−	0.451649i
$750$		0			0
$751$		−	27.2361i		−	0.993858i	−0.867791	−	0.496929i	$-0.834461\pi$
$751$		−	27.2361i		−	0.993858i	0.867791	−	0.496929i	$-0.165539\pi$
$752$		0			0
$753$	2.47214	−	6.47214i	0.0900896	−	0.235858i
$754$		0			0
$755$	4.76393			0.173377
$756$		0			0
$757$	37.2361			1.35337			0.676684	−	0.736274i	$-0.263416\pi$
$757$	37.2361			1.35337			0.676684	+	0.736274i	$0.263416\pi$
$758$		0			0
$759$	−14.8328	+	38.8328i	−0.538397	+	1.40954i
$760$		0			0
$761$			21.8885i			0.793459i	0.917936	+	0.396730i	$0.129855\pi$
$761$			21.8885i			0.793459i	−0.917936	+	0.396730i	$0.870145\pi$
$762$		0			0
$763$			13.1672i			0.476684i
$764$		0			0
$765$	−7.23607	−	6.47214i	−0.261621	−	0.234001i
$766$		0			0
$767$	−9.88854			−0.357055
$768$		0			0
$769$	−14.3607			−0.517859			−0.258930	−	0.965896i	$-0.583370\pi$
$769$	−14.3607			−0.517859			−0.258930	+	0.965896i	$0.583370\pi$
$770$		0			0
$771$	35.1246	+	13.4164i	1.26498	+	0.483180i
$772$		0			0
$773$		−	46.9443i		−	1.68847i	−0.535975	−	0.844234i	$-0.680056\pi$
$773$		−	46.9443i		−	1.68847i	0.535975	−	0.844234i	$-0.319944\pi$
$774$		0			0
$775$		−	9.70820i		−	0.348729i
$776$		0			0
$777$	−20.0000	−	7.63932i	−0.717496	−	0.274059i
$778$		0			0
$779$	17.8885			0.640924
$780$		0			0
$781$	16.0000			0.572525
$782$		0			0
$783$	10.6525	+	20.6525i	0.380688	+	0.738059i
$784$		0			0
$785$			1.23607i			0.0441172i
$786$		0			0
$787$		−	16.7639i		−	0.597570i	−0.954321	−	0.298785i	$-0.903419\pi$
$787$		−	16.7639i		−	0.597570i	0.954321	−	0.298785i	$-0.0965813\pi$
$788$		0			0
$789$	15.7082	−	41.1246i	0.559227	−	1.46407i
$790$		0			0
$791$	61.3050			2.17975
$792$		0			0
$793$	0.583592			0.0207240
$794$		0			0
$795$	7.70820	−	20.1803i	0.273382	−	0.715723i
$796$		0			0
$797$			26.0000i			0.920967i	0.887668	+	0.460484i	$0.152324\pi$
$797$			26.0000i			0.920967i	−0.887668	+	0.460484i	$0.847676\pi$
$798$		0			0
$799$			22.4721i			0.795007i
$800$		0			0
$801$	−28.9443	+	32.3607i	−1.02270	+	1.14341i
$802$		0			0
$803$	53.6656			1.89382
$804$		0			0
$805$	26.8328			0.945732
$806$		0			0
$807$	37.1246	+	14.1803i	1.30685	+	0.499172i
$808$		0			0
$809$			49.3050i			1.73347i	0.498769	+	0.866735i	$0.333786\pi$
$809$			49.3050i			1.73347i	−0.498769	+	0.866735i	$0.666214\pi$
$810$		0			0
$811$		−	11.7082i		−	0.411131i	−0.978643	−	0.205565i	$-0.934097\pi$
$811$		−	11.7082i		−	0.411131i	0.978643	−	0.205565i	$-0.0659033\pi$
$812$		0			0
$813$	35.1246	+	13.4164i	1.23187	+	0.470534i
$814$		0			0
$815$	−22.6525			−0.793482
$816$		0			0
$817$	31.0557			1.08650
$818$		0			0
$819$	−11.0557	+	12.3607i	−0.386318	+	0.431917i
$820$		0			0
$821$		−	21.0557i		−	0.734850i	−0.930053	−	0.367425i	$-0.880239\pi$
$821$		−	21.0557i		−	0.734850i	0.930053	−	0.367425i	$-0.119761\pi$
$822$		0			0
$823$		−	4.11146i		−	0.143316i	−0.997429	−	0.0716582i	$-0.977171\pi$
$823$		−	4.11146i		−	0.143316i	0.997429	−	0.0716582i	$-0.0228291\pi$
$824$		0			0
$825$	2.47214	−	6.47214i	0.0860687	−	0.225331i
$826$		0			0
$827$	40.6525			1.41363			0.706813	−	0.707401i	$-0.250132\pi$
$827$	40.6525			1.41363			0.706813	+	0.707401i	$0.250132\pi$
$828$		0			0
$829$	15.8885			0.551832			0.275916	−	0.961182i	$-0.411019\pi$
$829$	15.8885			0.551832			0.275916	+	0.961182i	$0.411019\pi$
$830$		0			0
$831$	−14.6525	+	38.3607i	−0.508289	+	1.33072i
$832$		0			0
$833$		−	42.0689i		−	1.45760i
$834$		0			0
$835$		−	21.4164i		−	0.741145i
$836$		0			0
$837$	−23.1246	−	44.8328i	−0.799304	−	1.54965i
$838$		0			0
$839$	5.52786			0.190843			0.0954215	−	0.995437i	$-0.469580\pi$
$839$	5.52786			0.190843			0.0954215	+	0.995437i	$0.469580\pi$
$840$		0			0
$841$	9.00000			0.310345
$842$		0			0
$843$	−37.8885	−	14.4721i	−1.30495	−	0.498447i
$844$		0			0
$845$			11.4721i			0.394653i
$846$		0			0
$847$			22.3607i			0.768322i
$848$		0			0
$849$	−26.1803	−	10.0000i	−0.898507	−	0.343199i
$850$		0			0
$851$	−16.5836			−0.568478
$852$		0			0
$853$	3.34752			0.114617			0.0573085	−	0.998357i	$-0.481748\pi$
$853$	3.34752			0.114617			0.0573085	+	0.998357i	$0.481748\pi$
$854$		0			0
$855$	6.18034	+	5.52786i	0.211363	+	0.189049i
$856$		0			0
$857$			42.0689i			1.43705i	0.695503	+	0.718523i	$0.255181\pi$
$857$			42.0689i			1.43705i	−0.695503	+	0.718523i	$0.744819\pi$
$858$		0			0
$859$		−	9.59675i		−	0.327437i	−0.986507	−	0.163718i	$-0.947651\pi$
$859$		−	9.59675i		−	0.327437i	0.986507	−	0.163718i	$-0.0523488\pi$
$860$		0			0
$861$	−17.8885	+	46.8328i	−0.609640	+	1.59606i
$862$		0			0
$863$	22.0000			0.748889			0.374444	−	0.927249i	$-0.377833\pi$
$863$	22.0000			0.748889			0.374444	+	0.927249i	$0.377833\pi$
$864$		0			0
$865$	−16.4721			−0.560069
$866$		0			0
$867$	4.03444	−	10.5623i	0.137017	−	0.358715i
$868$		0			0
$869$		−	12.9443i		−	0.439104i
$870$		0			0
$871$		−	12.0000i		−	0.406604i
$872$		0			0
$873$	−1.05573	−	0.944272i	−0.0357310	−	0.0319588i
$874$		0			0
$875$	−4.47214			−0.151186
$876$		0			0
$877$	58.5410			1.97679			0.988395	−	0.151906i	$-0.0485412\pi$
$877$	58.5410			1.97679			0.988395	+	0.151906i	$0.0485412\pi$
$878$		0			0
$879$	29.7082	+	11.3475i	1.00203	+	0.382742i
$880$		0			0
$881$			17.3050i			0.583019i	0.956568	+	0.291509i	$0.0941574\pi$
$881$			17.3050i			0.583019i	−0.956568	+	0.291509i	$0.905843\pi$
$882$		0			0
$883$			5.12461i			0.172457i	0.996275	+	0.0862285i	$0.0274815\pi$
$883$			5.12461i			0.172457i	−0.996275	+	0.0862285i	$0.972519\pi$
$884$		0			0
$885$	−12.9443	−	4.94427i	−0.435117	−	0.166200i
$886$		0			0
$887$	1.63932			0.0550430			0.0275215	−	0.999621i	$-0.491239\pi$
$887$	1.63932			0.0550430			0.0275215	+	0.999621i	$0.491239\pi$
$888$		0			0
$889$	2.11146			0.0708160
$890$		0			0
$891$	−4.00000	−	35.7771i	−0.134005	−	1.19858i
$892$		0			0
$893$		−	19.1935i		−	0.642286i
$894$		0			0
$895$			25.8885i			0.865359i
$896$		0			0
$897$	−4.58359	+	12.0000i	−0.153042	+	0.400668i
$898$		0			0
$899$	43.4164			1.44802
$900$		0			0
$901$	−40.3607			−1.34461
$902$		0			0
$903$	−31.0557	+	81.3050i	−1.03347	+	2.70566i
$904$		0			0
$905$		−	13.4164i		−	0.445976i
$906$		0			0
$907$			46.0689i			1.52969i	0.644213	+	0.764846i	$0.277185\pi$
$907$			46.0689i			1.52969i	−0.644213	+	0.764846i	$0.722815\pi$
$908$		0			0
$909$	0.944272	−	1.05573i	0.0313195	−	0.0350163i
$910$		0			0
$911$	4.00000			0.132526			0.0662630	−	0.997802i	$-0.478892\pi$
$911$	4.00000			0.132526			0.0662630	+	0.997802i	$0.478892\pi$
$912$		0			0
$913$	17.1672			0.568151
$914$		0			0
$915$	0.763932	+	0.291796i	0.0252548	+	0.00964648i
$916$		0			0
$917$			11.0557i			0.365092i
$918$		0			0
$919$			25.1246i			0.828784i	0.910098	+	0.414392i	$0.136006\pi$
$919$			25.1246i			0.828784i	−0.910098	+	0.414392i	$0.863994\pi$
$920$		0			0
$921$	−3.70820	−	1.41641i	−0.122189	−	0.0466722i
$922$		0			0
$923$	4.94427			0.162743
$924$		0			0
$925$	2.76393			0.0908775
$926$		0			0
$927$	20.0000	−	22.3607i	0.656886	−	0.734421i
$928$		0			0
$929$			6.11146i			0.200510i	0.994962	+	0.100255i	$0.0319659\pi$
$929$			6.11146i			0.200510i	−0.994962	+	0.100255i	$0.968034\pi$
$930$		0			0
$931$			35.9311i			1.17759i
$932$		0			0
$933$	−3.63932	+	9.52786i	−0.119146	+	0.311928i
$934$		0			0
$935$	−12.9443			−0.423323
$936$		0			0
$937$	−39.8885			−1.30310			−0.651551	−	0.758605i	$-0.725881\pi$
$937$	−39.8885			−1.30310			−0.651551	+	0.758605i	$0.725881\pi$
$938$		0			0
$939$	−7.34752	+	19.2361i	−0.239777	+	0.627745i
$940$		0			0
$941$		−	0.472136i		−	0.0153912i	−0.999970	−	0.00769560i	$-0.997550\pi$
$941$		−	0.472136i		−	0.0153912i	0.999970	−	0.00769560i	$-0.00244961\pi$
$942$		0			0
$943$			38.8328i			1.26457i
$944$		0			0
$945$	−20.6525	+	10.6525i	−0.671825	+	0.346525i
$946$		0			0
$947$	−51.4853			−1.67305			−0.836524	−	0.547931i	$-0.815416\pi$
$947$	−51.4853			−1.67305			−0.836524	+	0.547931i	$0.815416\pi$
$948$		0			0
$949$	16.5836			0.538326
$950$		0			0
$951$	−17.7082	−	6.76393i	−0.574228	−	0.219336i
$952$		0			0
$953$		−	25.3475i		−	0.821087i	−0.911841	−	0.410543i	$-0.865339\pi$
$953$		−	25.3475i		−	0.821087i	0.911841	−	0.410543i	$-0.134661\pi$
$954$		0			0
$955$			24.3607i			0.788293i
$956$		0			0
$957$	28.9443	+	11.0557i	0.935635	+	0.357381i
$958$		0			0
$959$	−0.806504			−0.0260434
$960$		0			0
$961$	−63.2492			−2.04030
$962$		0			0
$963$	6.18034	+	5.52786i	0.199159	+	0.178133i
$964$		0			0
$965$		−	10.9443i		−	0.352309i
$966$		0			0
$967$		−	22.0000i		−	0.707472i	−0.935345	−	0.353736i	$-0.884911\pi$
$967$		−	22.0000i		−	0.707472i	0.935345	−	0.353736i	$-0.115089\pi$
$968$		0			0
$969$	5.52786	−	14.4721i	0.177581	−	0.464912i
$970$		0			0
$971$	−15.0557			−0.483161			−0.241581	−	0.970381i	$-0.577666\pi$
$971$	−15.0557			−0.483161			−0.241581	+	0.970381i	$0.577666\pi$
$972$		0			0
$973$	−27.6393			−0.886076
$974$		0			0
$975$	0.763932	−	2.00000i	0.0244654	−	0.0640513i
$976$		0			0
$977$		−	17.7082i		−	0.566536i	−0.959041	−	0.283268i	$-0.908581\pi$
$977$		−	17.7082i		−	0.566536i	0.959041	−	0.283268i	$-0.0914185\pi$
$978$		0			0
$979$			57.8885i			1.85013i
$980$		0			0
$981$	−6.58359	−	5.88854i	−0.210198	−	0.188007i
$982$		0			0
$983$	33.7771			1.07732			0.538661	−	0.842523i	$-0.318930\pi$
$983$	33.7771			1.07732			0.538661	+	0.842523i	$0.318930\pi$
$984$		0			0
$985$	−6.94427			−0.221263
$986$		0			0
$987$	50.2492	+	19.1935i	1.59945	+	0.610936i
$988$		0			0
$989$			67.4164i			2.14372i
$990$		0			0
$991$			8.76393i			0.278395i	0.990265	+	0.139198i	$0.0444524\pi$
$991$			8.76393i			0.278395i	−0.990265	+	0.139198i	$0.955548\pi$
$992$		0			0
$993$	−38.3607	−	14.6525i	−1.21734	−	0.464982i
$994$		0			0
$995$	−9.70820			−0.307771
$996$		0			0
$997$	−24.0689			−0.762269			−0.381135	−	0.924520i	$-0.624467\pi$
$997$	−24.0689			−0.762269			−0.381135	+	0.924520i	$0.624467\pi$
$998$		0			0
$999$	12.7639	−	6.58359i	0.403833	−	0.208296i

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1920.2.h.d.1151.3	yes	4
3.2	odd	2		1920.2.h.n.1151.1	yes	4
4.3	odd	2		1920.2.h.n.1151.2	yes	4
8.3	odd	2		1920.2.h.c.1151.3	✓	4
8.5	even	2		1920.2.h.m.1151.2	yes	4
12.11	even	2	inner	1920.2.h.d.1151.4	yes	4
24.5	odd	2		1920.2.h.c.1151.4	yes	4
24.11	even	2		1920.2.h.m.1151.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.h.c.1151.3	✓	4	8.3	odd	2
1920.2.h.c.1151.4	yes	4	24.5	odd	2
1920.2.h.d.1151.3	yes	4	1.1	even	1	trivial
1920.2.h.d.1151.4	yes	4	12.11	even	2	inner
1920.2.h.m.1151.1	yes	4	24.11	even	2
1920.2.h.m.1151.2	yes	4	8.5	even	2
1920.2.h.n.1151.1	yes	4	3.2	odd	2
1920.2.h.n.1151.2	yes	4	4.3	odd	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1920.2.h.d.1151.3

Level	$1920$
Weight	$2$
Character	1920.1151
Analytic conductor	$15.331$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$