LMFDB - Embedded newform 4620.2.m.a.1121.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4620,2,Mod(1121,4620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 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("4620.1121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4620.m (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:	$36.8908857338$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1121.8
Character	$\chi$	$=$	4620.1121
Dual form			4620.2.m.a.1121.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.63283 + 0.577800i) q^{3} +1.00000i q^{5} +1.00000i q^{7} +(2.33229 - 1.88690i) q^{9} +O(q^{10})$	\(q+(-1.63283 + 0.577800i) q^{3} +1.00000i q^{5} +1.00000i q^{7} +(2.33229 - 1.88690i) q^{9} +(-0.787530 - 3.22177i) q^{11} -4.21336i q^{13} +(-0.577800 - 1.63283i) q^{15} -4.67066 q^{17} +3.45326i q^{19} +(-0.577800 - 1.63283i) q^{21} -7.19644i q^{23} -1.00000 q^{25} +(-2.71799 + 4.42860i) q^{27} -0.509605 q^{29} -3.13027 q^{31} +(3.14745 + 4.80558i) q^{33} -1.00000 q^{35} +3.27384 q^{37} +(2.43448 + 6.87972i) q^{39} +2.80280 q^{41} +2.06166i q^{43} +(1.88690 + 2.33229i) q^{45} +6.20047i q^{47} -1.00000 q^{49} +(7.62642 - 2.69871i) q^{51} -1.89015i q^{53} +(3.22177 - 0.787530i) q^{55} +(-1.99529 - 5.63860i) q^{57} -0.185289i q^{59} +14.7666i q^{61} +(1.88690 + 2.33229i) q^{63} +4.21336 q^{65} -13.2509 q^{67} +(4.15811 + 11.7506i) q^{69} +5.66861i q^{71} -4.59324i q^{73} +(1.63283 - 0.577800i) q^{75} +(3.22177 - 0.787530i) q^{77} +8.18804i q^{79} +(1.87918 - 8.80163i) q^{81} +11.1934 q^{83} -4.67066i q^{85} +(0.832100 - 0.294450i) q^{87} +12.1510i q^{89} +4.21336 q^{91} +(5.11122 - 1.80867i) q^{93} -3.45326 q^{95} +10.6957 q^{97} +(-7.91592 - 6.02812i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 4 q^{3} + 6 q^{9}+O(q^{10}) $ 48 * q - 4 * q^3 + 6 * q^9	$ 48 q - 4 q^{3} + 6 q^{9} - 6 q^{11} + 2 q^{15} + 4 q^{17} + 2 q^{21} - 48 q^{25} - 16 q^{27} + 36 q^{29} - 16 q^{31} + 16 q^{33} - 48 q^{35} - 8 q^{37} - 18 q^{39} - 48 q^{49} - 30 q^{51} + 4 q^{55} + 16 q^{57} - 12 q^{65} + 24 q^{67} + 4 q^{69} + 4 q^{75} + 4 q^{77} - 22 q^{81} + 20 q^{83} + 44 q^{87} - 12 q^{91} - 28 q^{93} + 8 q^{95} - 56 q^{97} + 26 q^{99}+O(q^{100}) $ 48 * q - 4 * q^3 + 6 * q^9 - 6 * q^11 + 2 * q^15 + 4 * q^17 + 2 * q^21 - 48 * q^25 - 16 * q^27 + 36 * q^29 - 16 * q^31 + 16 * q^33 - 48 * q^35 - 8 * q^37 - 18 * q^39 - 48 * q^49 - 30 * q^51 + 4 * q^55 + 16 * q^57 - 12 * q^65 + 24 * q^67 + 4 * q^69 + 4 * q^75 + 4 * q^77 - 22 * q^81 + 20 * q^83 + 44 * q^87 - 12 * q^91 - 28 * q^93 + 8 * q^95 - 56 * q^97 + 26 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4620\mathbb{Z}\right)^\times$.

$n$	$661$	$1541$	$2311$	$2521$	$3697$
$\chi(n)$	$1$	$-1$	$1$	$-1$	$1$

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
$3$	−1.63283	+	0.577800i	−0.942717	+	0.333593i
$3$	−1.63283	+	0.577800i	−0.942717	+	0.333593i
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	2.33229	−	1.88690i	0.777431	−	0.628968i
$10$		0			0
$11$	−0.787530	−	3.22177i	−0.237449	−	0.971400i
$11$	−0.787530	−	3.22177i	−0.237449	−	0.971400i
$12$		0			0
$13$		−	4.21336i		−	1.16858i	−0.811547	−	0.584288i	$-0.801374\pi$
$13$		−	4.21336i		−	1.16858i	0.811547	−	0.584288i	$-0.198626\pi$
$14$		0			0
$15$	−0.577800	−	1.63283i	−0.149187	−	0.421596i
$16$		0			0
$17$	−4.67066			−1.13280			−0.566401	−	0.824130i	$-0.691665\pi$
$17$	−4.67066			−1.13280			−0.566401	+	0.824130i	$0.691665\pi$
$18$		0			0
$19$			3.45326i			0.792232i	0.918201	+	0.396116i	$0.129642\pi$
$19$			3.45326i			0.792232i	−0.918201	+	0.396116i	$0.870358\pi$
$20$		0			0
$21$	−0.577800	−	1.63283i	−0.126086	−	0.356314i
$22$		0			0
$23$		−	7.19644i		−	1.50056i	−0.661119	−	0.750281i	$-0.729918\pi$
$23$		−	7.19644i		−	1.50056i	0.661119	−	0.750281i	$-0.270082\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	−2.71799	+	4.42860i	−0.523078	+	0.852285i
$28$		0			0
$29$	−0.509605			−0.0946312			−0.0473156	−	0.998880i	$-0.515067\pi$
$29$	−0.509605			−0.0946312			−0.0473156	+	0.998880i	$0.515067\pi$
$30$		0			0
$31$	−3.13027			−0.562214			−0.281107	−	0.959677i	$-0.590702\pi$
$31$	−3.13027			−0.562214			−0.281107	+	0.959677i	$0.590702\pi$
$32$		0			0
$33$	3.14745	+	4.80558i	0.547900	+	0.836544i
$34$		0			0
$35$	−1.00000			−0.169031
$36$		0			0
$37$	3.27384			0.538216			0.269108	−	0.963110i	$-0.413271\pi$
$37$	3.27384			0.538216			0.269108	+	0.963110i	$0.413271\pi$
$38$		0			0
$39$	2.43448	+	6.87972i	0.389829	+	1.10164i
$40$		0			0
$41$	2.80280			0.437723			0.218862	−	0.975756i	$-0.429766\pi$
$41$	2.80280			0.437723			0.218862	+	0.975756i	$0.429766\pi$
$42$		0			0
$43$			2.06166i			0.314400i	0.987567	+	0.157200i	$0.0502467\pi$
$43$			2.06166i			0.314400i	−0.987567	+	0.157200i	$0.949753\pi$
$44$		0			0
$45$	1.88690	+	2.33229i	0.281283	+	0.347678i
$46$		0			0
$47$			6.20047i			0.904432i	0.891908	+	0.452216i	$0.149366\pi$
$47$			6.20047i			0.904432i	−0.891908	+	0.452216i	$0.850634\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	7.62642	−	2.69871i	1.06791	−	0.377895i
$52$		0			0
$53$		−	1.89015i		−	0.259632i	−0.991538	−	0.129816i	$-0.958561\pi$
$53$		−	1.89015i		−	0.259632i	0.991538	−	0.129816i	$-0.0414387\pi$
$54$		0			0
$55$	3.22177	−	0.787530i	0.434423	−	0.106191i
$56$		0			0
$57$	−1.99529	−	5.63860i	−0.264283	−	0.746850i
$58$		0			0
$59$		−	0.185289i		−	0.0241226i	−0.999927	−	0.0120613i	$-0.996161\pi$
$59$		−	0.185289i		−	0.0241226i	0.999927	−	0.0120613i	$-0.00383933\pi$
$60$		0			0
$61$			14.7666i			1.89067i	0.326102	+	0.945335i	$0.394265\pi$
$61$			14.7666i			1.89067i	−0.326102	+	0.945335i	$0.605735\pi$
$62$		0			0
$63$	1.88690	+	2.33229i	0.237728	+	0.293841i
$64$		0			0
$65$	4.21336			0.522603
$66$		0			0
$67$	−13.2509			−1.61886			−0.809428	−	0.587219i	$-0.800223\pi$
$67$	−13.2509			−1.61886			−0.809428	+	0.587219i	$0.800223\pi$
$68$		0			0
$69$	4.15811	+	11.7506i	0.500577	+	1.41461i
$70$		0			0
$71$			5.66861i			0.672741i	0.941730	+	0.336370i	$0.109199\pi$
$71$			5.66861i			0.672741i	−0.941730	+	0.336370i	$0.890801\pi$
$72$		0			0
$73$		−	4.59324i		−	0.537599i	−0.963196	−	0.268799i	$-0.913373\pi$
$73$		−	4.59324i		−	0.537599i	0.963196	−	0.268799i	$-0.0866268\pi$
$74$		0			0
$75$	1.63283	−	0.577800i	0.188543	−	0.0667186i
$76$		0			0
$77$	3.22177	−	0.787530i	0.367155	−	0.0897474i
$78$		0			0
$79$			8.18804i			0.921227i	0.887601	+	0.460613i	$0.152370\pi$
$79$			8.18804i			0.921227i	−0.887601	+	0.460613i	$0.847630\pi$
$80$		0			0
$81$	1.87918	−	8.80163i	0.208798	−	0.977959i
$82$		0			0
$83$	11.1934			1.22864			0.614319	−	0.789058i	$-0.289431\pi$
$83$	11.1934			1.22864			0.614319	+	0.789058i	$0.289431\pi$
$84$		0			0
$85$		−	4.67066i		−	0.506605i
$86$		0			0
$87$	0.832100	−	0.294450i	0.0892104	−	0.0315683i
$88$		0			0
$89$			12.1510i			1.28800i	0.765024	+	0.644002i	$0.222727\pi$
$89$			12.1510i			1.28800i	−0.765024	+	0.644002i	$0.777273\pi$
$90$		0			0
$91$	4.21336			0.441680
$92$		0			0
$93$	5.11122	−	1.80867i	0.530008	−	0.187551i
$94$		0			0
$95$	−3.45326			−0.354297
$96$		0			0
$97$	10.6957			1.08598			0.542990	−	0.839739i	$-0.317292\pi$
$97$	10.6957			1.08598			0.542990	+	0.839739i	$0.317292\pi$
$98$		0			0
$99$	−7.91592	−	6.02812i	−0.795580	−	0.605848i
$100$		0			0
$101$	13.1014			1.30363			0.651817	−	0.758376i	$-0.274007\pi$
$101$	13.1014			1.30363			0.651817	+	0.758376i	$0.274007\pi$
$102$		0			0
$103$	−1.26589			−0.124732			−0.0623661	−	0.998053i	$-0.519865\pi$
$103$	−1.26589			−0.124732			−0.0623661	+	0.998053i	$0.519865\pi$
$104$		0			0
$105$	1.63283	−	0.577800i	0.159348	−	0.0563875i
$106$		0			0
$107$	−2.86924			−0.277380			−0.138690	−	0.990336i	$-0.544289\pi$
$107$	−2.86924			−0.277380			−0.138690	+	0.990336i	$0.544289\pi$
$108$		0			0
$109$		−	8.71393i		−	0.834643i	−0.908759	−	0.417321i	$-0.862969\pi$
$109$		−	8.71393i		−	0.834643i	0.908759	−	0.417321i	$-0.137031\pi$
$110$		0			0
$111$	−5.34564	+	1.89163i	−0.507385	+	0.179545i
$112$		0			0
$113$		−	4.89549i		−	0.460529i	−0.973128	−	0.230264i	$-0.926041\pi$
$113$		−	4.89549i		−	0.460529i	0.973128	−	0.230264i	$-0.0739591\pi$
$114$		0			0
$115$	7.19644			0.671072
$116$		0			0
$117$	−7.95021	−	9.82679i	−0.734997	−	0.908487i
$118$		0			0
$119$		−	4.67066i		−	0.428159i
$120$		0			0
$121$	−9.75959	+	5.07448i	−0.887236	+	0.461317i
$122$		0			0
$123$	−4.57650	+	1.61946i	−0.412649	+	0.146022i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			8.36390i			0.742176i	0.928598	+	0.371088i	$0.121015\pi$
$127$			8.36390i			0.742176i	−0.928598	+	0.371088i	$0.878985\pi$
$128$		0			0
$129$	−1.19123	−	3.36635i	−0.104882	−	0.296390i
$130$		0			0
$131$	4.84037			0.422905			0.211452	−	0.977388i	$-0.432181\pi$
$131$	4.84037			0.422905			0.211452	+	0.977388i	$0.432181\pi$
$132$		0			0
$133$	−3.45326			−0.299435
$134$		0			0
$135$	−4.42860	−	2.71799i	−0.381153	−	0.233928i
$136$		0			0
$137$		−	4.32317i		−	0.369354i	−0.982799	−	0.184677i	$-0.940876\pi$
$137$		−	4.32317i		−	0.369354i	0.982799	−	0.184677i	$-0.0591238\pi$
$138$		0			0
$139$			4.63233i			0.392909i	0.980513	+	0.196455i	$0.0629428\pi$
$139$			4.63233i			0.392909i	−0.980513	+	0.196455i	$0.937057\pi$
$140$		0			0
$141$	−3.58264	−	10.1243i	−0.301712	−	0.852624i
$142$		0			0
$143$	−13.5745	+	3.31815i	−1.13515	+	0.277477i
$144$		0			0
$145$		−	0.509605i		−	0.0423204i
$146$		0			0
$147$	1.63283	−	0.577800i	0.134674	−	0.0476562i
$148$		0			0
$149$	5.33914			0.437399			0.218700	−	0.975792i	$-0.429818\pi$
$149$	5.33914			0.437399			0.218700	+	0.975792i	$0.429818\pi$
$150$		0			0
$151$			22.7168i			1.84867i	0.381581	+	0.924335i	$0.375380\pi$
$151$			22.7168i			1.84867i	−0.381581	+	0.924335i	$0.624620\pi$
$152$		0			0
$153$	−10.8934	+	8.81310i	−0.880676	+	0.712497i
$154$		0			0
$155$		−	3.13027i		−	0.251430i
$156$		0			0
$157$	−2.44648			−0.195250			−0.0976252	−	0.995223i	$-0.531125\pi$
$157$	−2.44648			−0.195250			−0.0976252	+	0.995223i	$0.531125\pi$
$158$		0			0
$159$	1.09213	+	3.08630i	0.0866116	+	0.244760i
$160$		0			0
$161$	7.19644			0.567159
$162$		0			0
$163$	5.74398			0.449903			0.224951	−	0.974370i	$-0.427778\pi$
$163$	5.74398			0.449903			0.224951	+	0.974370i	$0.427778\pi$
$164$		0			0
$165$	−4.80558	+	3.14745i	−0.374114	+	0.245028i
$166$		0			0
$167$	−23.3103			−1.80380			−0.901902	−	0.431941i	$-0.857829\pi$
$167$	−23.3103			−1.80380			−0.901902	+	0.431941i	$0.857829\pi$
$168$		0			0
$169$	−4.75240			−0.365569
$170$		0			0
$171$	6.51597	+	8.05401i	0.498289	+	0.615906i
$172$		0			0
$173$	−6.27951			−0.477422			−0.238711	−	0.971091i	$-0.576725\pi$
$173$	−6.27951			−0.477422			−0.238711	+	0.971091i	$0.576725\pi$
$174$		0			0
$175$		−	1.00000i		−	0.0755929i
$176$		0			0
$177$	0.107060	+	0.302547i	0.00804714	+	0.0227408i
$178$		0			0
$179$			4.55257i			0.340275i	0.985420	+	0.170137i	$0.0544212\pi$
$179$			4.55257i			0.340275i	−0.985420	+	0.170137i	$0.945579\pi$
$180$		0			0
$181$	−16.2027			−1.20434			−0.602170	−	0.798368i	$-0.705697\pi$
$181$	−16.2027			−1.20434			−0.602170	+	0.798368i	$0.705697\pi$
$182$		0			0
$183$	−8.53215	−	24.1114i	−0.630714	−	1.78237i
$184$		0			0
$185$			3.27384i			0.240698i
$186$		0			0
$187$	3.67829	+	15.0478i	0.268983	+	1.10040i
$188$		0			0
$189$	−4.42860	−	2.71799i	−0.322133	−	0.197705i
$190$		0			0
$191$		−	7.37437i		−	0.533591i	−0.963753	−	0.266795i	$-0.914035\pi$
$191$		−	7.37437i		−	0.533591i	0.963753	−	0.266795i	$-0.0859647\pi$
$192$		0			0
$193$			17.4347i			1.25497i	0.778627	+	0.627487i	$0.215916\pi$
$193$			17.4347i			1.25497i	−0.778627	+	0.627487i	$0.784084\pi$
$194$		0			0
$195$	−6.87972	+	2.43448i	−0.492667	+	0.174337i
$196$		0			0
$197$	19.4096			1.38288			0.691439	−	0.722435i	$-0.256977\pi$
$197$	19.4096			1.38288			0.691439	+	0.722435i	$0.256977\pi$
$198$		0			0
$199$	−12.5558			−0.890059			−0.445030	−	0.895516i	$-0.646807\pi$
$199$	−12.5558			−0.890059			−0.445030	+	0.895516i	$0.646807\pi$
$200$		0			0
$201$	21.6365	−	7.65638i	1.52612	−	0.540039i
$202$		0			0
$203$		−	0.509605i		−	0.0357672i
$204$		0			0
$205$			2.80280i			0.195756i
$206$		0			0
$207$	−13.5790	−	16.7842i	−0.943805	−	1.16658i
$208$		0			0
$209$	11.1256	−	2.71955i	0.769574	−	0.188115i
$210$		0			0
$211$			15.3161i			1.05441i	0.849739	+	0.527203i	$0.176759\pi$
$211$			15.3161i			1.05441i	−0.849739	+	0.527203i	$0.823241\pi$
$212$		0			0
$213$	−3.27533	−	9.25591i	−0.224422	−	0.634204i
$214$		0			0
$215$	−2.06166			−0.140604
$216$		0			0
$217$		−	3.13027i		−	0.212497i
$218$		0			0
$219$	2.65398	+	7.50001i	0.179339	+	0.506803i
$220$		0			0
$221$			19.6792i			1.32377i
$222$		0			0
$223$	−1.74763			−0.117030			−0.0585149	−	0.998287i	$-0.518637\pi$
$223$	−1.74763			−0.117030			−0.0585149	+	0.998287i	$0.518637\pi$
$224$		0			0
$225$	−2.33229	+	1.88690i	−0.155486	+	0.125794i
$226$		0			0
$227$	−24.4951			−1.62580			−0.812900	−	0.582404i	$-0.802112\pi$
$227$	−24.4951			−1.62580			−0.812900	+	0.582404i	$0.802112\pi$
$228$		0			0
$229$	−3.04916			−0.201494			−0.100747	−	0.994912i	$-0.532123\pi$
$229$	−3.04916			−0.201494			−0.100747	+	0.994912i	$0.532123\pi$
$230$		0			0
$231$	−4.80558	+	3.14745i	−0.316184	+	0.207087i
$232$		0			0
$233$	−17.9918			−1.17868			−0.589339	−	0.807886i	$-0.700612\pi$
$233$	−17.9918			−1.17868			−0.589339	+	0.807886i	$0.700612\pi$
$234$		0			0
$235$	−6.20047			−0.404474
$236$		0			0
$237$	−4.73105	−	13.3697i	−0.307315	−	0.868456i
$238$		0			0
$239$	−14.6396			−0.946957			−0.473479	−	0.880805i	$-0.657002\pi$
$239$	−14.6396			−0.946957			−0.473479	+	0.880805i	$0.657002\pi$
$240$		0			0
$241$			13.8263i			0.890628i	0.895374	+	0.445314i	$0.146908\pi$
$241$			13.8263i			0.890628i	−0.895374	+	0.445314i	$0.853092\pi$
$242$		0			0
$243$	2.01719	+	15.4574i	0.129403	+	0.991592i
$244$		0			0
$245$		−	1.00000i		−	0.0638877i
$246$		0			0
$247$	14.5498			0.925783
$248$		0			0
$249$	−18.2770	+	6.46757i	−1.15826	+	0.409865i
$250$		0			0
$251$			8.57520i			0.541262i	0.962683	+	0.270631i	$0.0872323\pi$
$251$			8.57520i			0.541262i	−0.962683	+	0.270631i	$0.912768\pi$
$252$		0			0
$253$	−23.1853	+	5.66742i	−1.45765	+	0.356307i
$254$		0			0
$255$	2.69871	+	7.62642i	0.169000	+	0.477585i
$256$		0			0
$257$		−	10.3427i		−	0.645162i	−0.946542	−	0.322581i	$-0.895450\pi$
$257$		−	10.3427i		−	0.645162i	0.946542	−	0.322581i	$-0.104550\pi$
$258$		0			0
$259$			3.27384i			0.203427i
$260$		0			0
$261$	−1.18855	+	0.961575i	−0.0735692	+	0.0595200i
$262$		0			0
$263$	−2.69783			−0.166355			−0.0831776	−	0.996535i	$-0.526507\pi$
$263$	−2.69783			−0.166355			−0.0831776	+	0.996535i	$0.526507\pi$
$264$		0			0
$265$	1.89015			0.116111
$266$		0			0
$267$	−7.02085	−	19.8406i	−0.429669	−	1.21422i
$268$		0			0
$269$		−	26.4670i		−	1.61372i	−0.590740	−	0.806862i	$-0.701164\pi$
$269$		−	26.4670i		−	1.61372i	0.590740	−	0.806862i	$-0.298836\pi$
$270$		0			0
$271$			31.6260i			1.92114i	0.278037	+	0.960570i	$0.410316\pi$
$271$			31.6260i			1.92114i	−0.278037	+	0.960570i	$0.589684\pi$
$272$		0			0
$273$	−6.87972	+	2.43448i	−0.416379	+	0.147341i
$274$		0			0
$275$	0.787530	+	3.22177i	0.0474899	+	0.194280i
$276$		0			0
$277$		−	30.3858i		−	1.82571i	−0.408286	−	0.912854i	$-0.633873\pi$
$277$		−	30.3858i		−	1.82571i	0.408286	−	0.912854i	$-0.366127\pi$
$278$		0			0
$279$	−7.30071	+	5.90652i	−0.437082	+	0.353614i
$280$		0			0
$281$	3.73984			0.223100			0.111550	−	0.993759i	$-0.464418\pi$
$281$	3.73984			0.223100			0.111550	+	0.993759i	$0.464418\pi$
$282$		0			0
$283$			22.0886i			1.31303i	0.754313	+	0.656515i	$0.227970\pi$
$283$			22.0886i			1.31303i	−0.754313	+	0.656515i	$0.772030\pi$
$284$		0			0
$285$	5.63860	−	1.99529i	0.334002	−	0.118191i
$286$		0			0
$287$			2.80280i			0.165444i
$288$		0			0
$289$	4.81510			0.283241
$290$		0			0
$291$	−17.4642	+	6.17995i	−1.02377	+	0.362275i
$292$		0			0
$293$	−0.307411			−0.0179591			−0.00897956	−	0.999960i	$-0.502858\pi$
$293$	−0.307411			−0.0179591			−0.00897956	+	0.999960i	$0.502858\pi$
$294$		0			0
$295$	0.185289			0.0107880
$296$		0			0
$297$	16.4084	+	5.26909i	0.952114	+	0.305744i
$298$		0			0
$299$	−30.3212			−1.75352
$300$		0			0
$301$	−2.06166			−0.118832
$302$		0			0
$303$	−21.3923	+	7.56997i	−1.22896	+	0.434883i
$304$		0			0
$305$	−14.7666			−0.845533
$306$		0			0
$307$			14.4174i			0.822845i	0.911445	+	0.411423i	$0.134968\pi$
$307$			14.4174i			0.822845i	−0.911445	+	0.411423i	$0.865032\pi$
$308$		0			0
$309$	2.06699	−	0.731434i	0.117587	−	0.0416098i
$310$		0			0
$311$			27.0040i			1.53126i	0.643282	+	0.765629i	$0.277572\pi$
$311$			27.0040i			1.53126i	−0.643282	+	0.765629i	$0.722428\pi$
$312$		0			0
$313$	25.2479			1.42709			0.713547	−	0.700607i	$-0.247088\pi$
$313$	25.2479			1.42709			0.713547	+	0.700607i	$0.247088\pi$
$314$		0			0
$315$	−2.33229	+	1.88690i	−0.131410	+	0.106315i
$316$		0			0
$317$			7.86544i			0.441767i	0.975300	+	0.220884i	$0.0708941\pi$
$317$			7.86544i			0.441767i	−0.975300	+	0.220884i	$0.929106\pi$
$318$		0			0
$319$	0.401329	+	1.64183i	0.0224701	+	0.0919247i
$320$		0			0
$321$	4.68499	−	1.65785i	0.261490	−	0.0925320i
$322$		0			0
$323$		−	16.1290i		−	0.897442i
$324$		0			0
$325$			4.21336i			0.233715i
$326$		0			0
$327$	5.03491	+	14.2284i	0.278431	+	0.786832i
$328$		0			0
$329$	−6.20047			−0.341843
$330$		0			0
$331$	20.8237			1.14458			0.572288	−	0.820053i	$-0.306056\pi$
$331$	20.8237			1.14458			0.572288	+	0.820053i	$0.306056\pi$
$332$		0			0
$333$	7.63556	−	6.17742i	0.418426	−	0.338521i
$334$		0			0
$335$		−	13.2509i		−	0.723974i
$336$		0			0
$337$			24.8024i			1.35107i	0.737326	+	0.675537i	$0.236088\pi$
$337$			24.8024i			1.35107i	−0.737326	+	0.675537i	$0.763912\pi$
$338$		0			0
$339$	2.82861	+	7.99352i	0.153629	+	0.434148i
$340$		0			0
$341$	2.46518	+	10.0850i	0.133497	+	0.546134i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	−11.7506	+	4.15811i	−0.632631	+	0.223865i
$346$		0			0
$347$	8.70935			0.467543			0.233771	−	0.972292i	$-0.424893\pi$
$347$	8.70935			0.467543			0.233771	+	0.972292i	$0.424893\pi$
$348$		0			0
$349$			19.0874i			1.02172i	0.859663	+	0.510862i	$0.170674\pi$
$349$			19.0874i			1.02172i	−0.859663	+	0.510862i	$0.829326\pi$
$350$		0			0
$351$	18.6593	+	11.4519i	0.995959	+	0.611256i
$352$		0			0
$353$		−	15.9608i		−	0.849505i	−0.905309	−	0.424753i	$-0.860361\pi$
$353$		−	15.9608i		−	0.849505i	0.905309	−	0.424753i	$-0.139639\pi$
$354$		0			0
$355$	−5.66861			−0.300859
$356$		0			0
$357$	2.69871	+	7.62642i	0.142831	+	0.403633i
$358$		0			0
$359$	−15.3704			−0.811218			−0.405609	−	0.914047i	$-0.632941\pi$
$359$	−15.3704			−0.811218			−0.405609	+	0.914047i	$0.632941\pi$
$360$		0			0
$361$	7.07501			0.372369
$362$		0			0
$363$	13.0038	−	13.9249i	0.682520	−	0.730867i
$364$		0			0
$365$	4.59324			0.240421
$366$		0			0
$367$	18.4329			0.962192			0.481096	−	0.876668i	$-0.340239\pi$
$367$	18.4329			0.962192			0.481096	+	0.876668i	$0.340239\pi$
$368$		0			0
$369$	6.53695	−	5.28861i	0.340300	−	0.275314i
$370$		0			0
$371$	1.89015			0.0981318
$372$		0			0
$373$			13.6831i			0.708483i	0.935154	+	0.354242i	$0.115261\pi$
$373$			13.6831i			0.708483i	−0.935154	+	0.354242i	$0.884739\pi$
$374$		0			0
$375$	0.577800	+	1.63283i	0.0298375	+	0.0843192i
$376$		0			0
$377$			2.14715i			0.110584i
$378$		0			0
$379$	−6.42225			−0.329889			−0.164944	−	0.986303i	$-0.552744\pi$
$379$	−6.42225			−0.329889			−0.164944	+	0.986303i	$0.552744\pi$
$380$		0			0
$381$	−4.83267	−	13.6569i	−0.247585	−	0.699662i
$382$		0			0
$383$		−	0.339305i		−	0.0173377i	−0.999962	−	0.00866884i	$-0.997241\pi$
$383$		−	0.339305i		−	0.0173377i	0.999962	−	0.00866884i	$-0.00275941\pi$
$384$		0			0
$385$	0.787530	+	3.22177i	0.0401363	+	0.164197i
$386$		0			0
$387$	3.89015	+	4.80839i	0.197748	+	0.244424i
$388$		0			0
$389$			18.0879i			0.917095i	0.888670	+	0.458547i	$0.151630\pi$
$389$			18.0879i			0.917095i	−0.888670	+	0.458547i	$0.848370\pi$
$390$		0			0
$391$			33.6122i			1.69984i
$392$		0			0
$393$	−7.90351	+	2.79677i	−0.398680	+	0.141078i
$394$		0			0
$395$	−8.18804			−0.411985
$396$		0			0
$397$	4.49417			0.225556			0.112778	−	0.993620i	$-0.464025\pi$
$397$	4.49417			0.225556			0.112778	+	0.993620i	$0.464025\pi$
$398$		0			0
$399$	5.63860	−	1.99529i	0.282283	−	0.0998897i
$400$		0			0
$401$		−	2.89361i		−	0.144500i	−0.997387	−	0.0722500i	$-0.976982\pi$
$401$		−	2.89361i		−	0.144500i	0.997387	−	0.0722500i	$-0.0230180\pi$
$402$		0			0
$403$			13.1890i			0.656989i
$404$		0			0
$405$	8.80163	+	1.87918i	0.437356	+	0.0933774i
$406$		0			0
$407$	−2.57825	−	10.5476i	−0.127799	−	0.522823i
$408$		0			0
$409$		−	4.14181i		−	0.204799i	−0.994743	−	0.102400i	$-0.967348\pi$
$409$		−	4.14181i		−	0.204799i	0.994743	−	0.102400i	$-0.0326520\pi$
$410$		0			0
$411$	2.49793	+	7.05903i	0.123214	+	0.348196i
$412$		0			0
$413$	0.185289			0.00911749
$414$		0			0
$415$			11.1934i			0.549464i
$416$		0			0
$417$	−2.67656	−	7.56383i	−0.131072	−	0.370402i
$418$		0			0
$419$		−	31.6620i		−	1.54679i	−0.633924	−	0.773396i	$-0.718557\pi$
$419$		−	31.6620i		−	1.54679i	0.633924	−	0.773396i	$-0.281443\pi$
$420$		0			0
$421$	−39.8837			−1.94381			−0.971906	−	0.235369i	$-0.924370\pi$
$421$	−39.8837			−1.94381			−0.971906	+	0.235369i	$0.924370\pi$
$422$		0			0
$423$	11.6997	+	14.4613i	0.568859	+	0.703134i
$424$		0			0
$425$	4.67066			0.226560
$426$		0			0
$427$	−14.7666			−0.714606
$428$		0			0
$429$	20.2476	−	13.2613i	0.977565	−	0.640263i
$430$		0			0
$431$	26.7421			1.28812			0.644060	−	0.764975i	$-0.277249\pi$
$431$	26.7421			1.28812			0.644060	+	0.764975i	$0.277249\pi$
$432$		0			0
$433$	6.50444			0.312584			0.156292	−	0.987711i	$-0.450046\pi$
$433$	6.50444			0.312584			0.156292	+	0.987711i	$0.450046\pi$
$434$		0			0
$435$	0.294450	+	0.832100i	0.0141178	+	0.0398961i
$436$		0			0
$437$	24.8512			1.18879
$438$		0			0
$439$			8.19776i			0.391258i	0.980678	+	0.195629i	$0.0626748\pi$
$439$			8.19776i			0.391258i	−0.980678	+	0.195629i	$0.937325\pi$
$440$		0			0
$441$	−2.33229	+	1.88690i	−0.111062	+	0.0898526i
$442$		0			0
$443$			2.25694i			0.107230i	0.998562	+	0.0536152i	$0.0170744\pi$
$443$			2.25694i			0.107230i	−0.998562	+	0.0536152i	$0.982926\pi$
$444$		0			0
$445$	−12.1510			−0.576013
$446$		0			0
$447$	−8.71793	+	3.08496i	−0.412344	+	0.145913i
$448$		0			0
$449$		−	16.9844i		−	0.801545i	−0.916178	−	0.400772i	$-0.868742\pi$
$449$		−	16.9844i		−	0.801545i	0.916178	−	0.400772i	$-0.131258\pi$
$450$		0			0
$451$	−2.20729	−	9.02997i	−0.103937	−	0.425205i
$452$		0			0
$453$	−13.1258	−	37.0928i	−0.616704	−	1.74277i
$454$		0			0
$455$			4.21336i			0.197525i
$456$		0			0
$457$			28.1915i			1.31874i	0.751817	+	0.659371i	$0.229178\pi$
$457$			28.1915i			1.31874i	−0.751817	+	0.659371i	$0.770822\pi$
$458$		0			0
$459$	12.6948	−	20.6845i	0.592544	−	0.965470i
$460$		0			0
$461$	15.1023			0.703383			0.351691	−	0.936116i	$-0.385607\pi$
$461$	15.1023			0.703383			0.351691	+	0.936116i	$0.385607\pi$
$462$		0			0
$463$	−33.9407			−1.57736			−0.788679	−	0.614805i	$-0.789235\pi$
$463$	−33.9407			−1.57736			−0.788679	+	0.614805i	$0.789235\pi$
$464$		0			0
$465$	1.80867	+	5.11122i	0.0838752	+	0.237027i
$466$		0			0
$467$			7.93772i			0.367314i	0.982990	+	0.183657i	$0.0587936\pi$
$467$			7.93772i			0.367314i	−0.982990	+	0.183657i	$0.941206\pi$
$468$		0			0
$469$		−	13.2509i		−	0.611870i
$470$		0			0
$471$	3.99470	−	1.41358i	0.184066	−	0.0651342i
$472$		0			0
$473$	6.64219	−	1.62362i	0.305408	−	0.0746541i
$474$		0			0
$475$		−	3.45326i		−	0.158446i
$476$		0			0
$477$	−3.56654	−	4.40839i	−0.163300	−	0.201846i
$478$		0			0
$479$	35.7229			1.63222			0.816111	−	0.577895i	$-0.196126\pi$
$479$	35.7229			1.63222			0.816111	+	0.577895i	$0.196126\pi$
$480$		0			0
$481$		−	13.7939i		−	0.628946i
$482$		0			0
$483$	−11.7506	+	4.15811i	−0.534670	+	0.189200i
$484$		0			0
$485$			10.6957i			0.485665i
$486$		0			0
$487$	−41.0282			−1.85917			−0.929584	−	0.368611i	$-0.879833\pi$
$487$	−41.0282			−1.85917			−0.929584	+	0.368611i	$0.879833\pi$
$488$		0			0
$489$	−9.37896	+	3.31887i	−0.424131	+	0.150085i
$490$		0			0
$491$	34.8627			1.57333			0.786665	−	0.617380i	$-0.211806\pi$
$491$	34.8627			1.57333			0.786665	+	0.617380i	$0.211806\pi$
$492$		0			0
$493$	2.38019			0.107198
$494$		0			0
$495$	6.02812	−	7.91592i	0.270944	−	0.355794i
$496$		0			0
$497$	−5.66861			−0.254272
$498$		0			0
$499$	25.5214			1.14250			0.571248	−	0.820778i	$-0.306460\pi$
$499$	25.5214			1.14250			0.571248	+	0.820778i	$0.306460\pi$
$500$		0			0
$501$	38.0618	−	13.4687i	1.70048	−	0.601737i
$502$		0			0
$503$	−24.7201			−1.10222			−0.551108	−	0.834434i	$-0.685795\pi$
$503$	−24.7201			−1.10222			−0.551108	+	0.834434i	$0.685795\pi$
$504$		0			0
$505$			13.1014i			0.583003i
$506$		0			0
$507$	7.75987	−	2.74594i	0.344628	−	0.121951i
$508$		0			0
$509$			30.7977i			1.36508i	0.730846	+	0.682542i	$0.239126\pi$
$509$			30.7977i			1.36508i	−0.730846	+	0.682542i	$0.760874\pi$
$510$		0			0
$511$	4.59324			0.203193
$512$		0			0
$513$	−15.2931	−	9.38593i	−0.675207	−	0.414399i
$514$		0			0
$515$		−	1.26589i		−	0.0557819i
$516$		0			0
$517$	19.9765	−	4.88306i	0.878565	−	0.214757i
$518$		0			0
$519$	10.2534	−	3.62830i	0.450074	−	0.159265i
$520$		0			0
$521$			7.62883i			0.334225i	0.985938	+	0.167113i	$0.0534443\pi$
$521$			7.62883i			0.334225i	−0.985938	+	0.167113i	$0.946556\pi$
$522$		0			0
$523$		−	1.67837i		−	0.0733900i	−0.999327	−	0.0366950i	$-0.988317\pi$
$523$		−	1.67837i		−	0.0733900i	0.999327	−	0.0366950i	$-0.0116830\pi$
$524$		0			0
$525$	0.577800	+	1.63283i	0.0252173	+	0.0712627i
$526$		0			0
$527$	14.6205			0.636877
$528$		0			0
$529$	−28.7888			−1.25169
$530$		0			0
$531$	−0.349623	−	0.432149i	−0.0151724	−	0.0187537i
$532$		0			0
$533$		−	11.8092i		−	0.511513i
$534$		0			0
$535$		−	2.86924i		−	0.124048i
$536$		0			0
$537$	−2.63047	−	7.43358i	−0.113513	−	0.320783i
$538$		0			0
$539$	0.787530	+	3.22177i	0.0339213	+	0.138771i
$540$		0			0
$541$		−	9.90310i		−	0.425767i	−0.977078	−	0.212884i	$-0.931714\pi$
$541$		−	9.90310i		−	0.425767i	0.977078	−	0.212884i	$-0.0682856\pi$
$542$		0			0
$543$	26.4564	−	9.36195i	1.13535	−	0.401760i
$544$		0			0
$545$	8.71393			0.373264
$546$		0			0
$547$		−	15.4767i		−	0.661734i	−0.943677	−	0.330867i	$-0.892659\pi$
$547$		−	15.4767i		−	0.661734i	0.943677	−	0.330867i	$-0.107341\pi$
$548$		0			0
$549$	27.8632	+	34.4400i	1.18917	+	1.46987i
$550$		0			0
$551$		−	1.75980i		−	0.0749698i
$552$		0			0
$553$	−8.18804			−0.348191
$554$		0			0
$555$	−1.89163	−	5.34564i	−0.0802951	−	0.226910i
$556$		0			0
$557$	−13.7039			−0.580652			−0.290326	−	0.956928i	$-0.593764\pi$
$557$	−13.7039			−0.580652			−0.290326	+	0.956928i	$0.593764\pi$
$558$		0			0
$559$	8.68651			0.367400
$560$		0			0
$561$	−14.7007	−	22.4452i	−0.620662	−	0.947639i
$562$		0			0
$563$	−30.0472			−1.26634			−0.633170	−	0.774013i	$-0.718247\pi$
$563$	−30.0472			−1.26634			−0.633170	+	0.774013i	$0.718247\pi$
$564$		0			0
$565$	4.89549			0.205955
$566$		0			0
$567$	8.80163	+	1.87918i	0.369634	+	0.0789183i
$568$		0			0
$569$	−42.4022			−1.77759			−0.888796	−	0.458303i	$-0.848457\pi$
$569$	−42.4022			−1.77759			−0.888796	+	0.458303i	$0.848457\pi$
$570$		0			0
$571$		−	34.6612i		−	1.45052i	−0.688473	−	0.725262i	$-0.741718\pi$
$571$		−	34.6612i		−	1.45052i	0.688473	−	0.725262i	$-0.258282\pi$
$572$		0			0
$573$	4.26091	+	12.0411i	0.178002	+	0.503025i
$574$		0			0
$575$			7.19644i			0.300112i
$576$		0			0
$577$	−24.7332			−1.02966			−0.514828	−	0.857294i	$-0.672144\pi$
$577$	−24.7332			−1.02966			−0.514828	+	0.857294i	$0.672144\pi$
$578$		0			0
$579$	−10.0738	−	28.4679i	−0.418651	−	1.18309i
$580$		0			0
$581$			11.1934i			0.464382i
$582$		0			0
$583$	−6.08963	+	1.48855i	−0.252207	+	0.0616495i
$584$		0			0
$585$	9.82679	−	7.95021i	0.406288	−	0.328701i
$586$		0			0
$587$		−	6.91292i		−	0.285327i	−0.989771	−	0.142663i	$-0.954433\pi$
$587$		−	6.91292i		−	0.285327i	0.989771	−	0.142663i	$-0.0455667\pi$
$588$		0			0
$589$		−	10.8096i		−	0.445403i
$590$		0			0
$591$	−31.6927	+	11.2149i	−1.30366	+	0.461319i
$592$		0			0
$593$	−25.2787			−1.03807			−0.519037	−	0.854752i	$-0.673709\pi$
$593$	−25.2787			−1.03807			−0.519037	+	0.854752i	$0.673709\pi$
$594$		0			0
$595$	4.67066			0.191479
$596$		0			0
$597$	20.5016	−	7.25476i	0.839074	−	0.296918i
$598$		0			0
$599$		−	38.5694i		−	1.57590i	−0.615738	−	0.787951i	$-0.711142\pi$
$599$		−	38.5694i		−	1.57590i	0.615738	−	0.787951i	$-0.288858\pi$
$600$		0			0
$601$		−	1.33246i		−	0.0543523i	−0.999631	−	0.0271762i	$-0.991348\pi$
$601$		−	1.33246i		−	0.0543523i	0.999631	−	0.0271762i	$-0.00865151\pi$
$602$		0			0
$603$	−30.9050	+	25.0032i	−1.25855	+	1.01821i
$604$		0			0
$605$	−5.07448	−	9.75959i	−0.206307	−	0.396784i
$606$		0			0
$607$			10.0389i			0.407466i	0.979027	+	0.203733i	$0.0653074\pi$
$607$			10.0389i			0.407466i	−0.979027	+	0.203733i	$0.934693\pi$
$608$		0			0
$609$	0.294450	+	0.832100i	0.0119317	+	0.0337184i
$610$		0			0
$611$	26.1248			1.05690
$612$		0			0
$613$		−	19.9779i		−	0.806898i	−0.915002	−	0.403449i	$-0.867811\pi$
$613$		−	19.9779i		−	0.806898i	0.915002	−	0.403449i	$-0.132189\pi$
$614$		0			0
$615$	−1.61946	−	4.57650i	−0.0653028	−	0.184542i
$616$		0			0
$617$		−	7.26545i		−	0.292496i	−0.989248	−	0.146248i	$-0.953280\pi$
$617$		−	7.26545i		−	0.292496i	0.989248	−	0.146248i	$-0.0467197\pi$
$618$		0			0
$619$	15.2376			0.612449			0.306225	−	0.951959i	$-0.400934\pi$
$619$	15.2376			0.612449			0.306225	+	0.951959i	$0.400934\pi$
$620$		0			0
$621$	31.8702	+	19.5599i	1.27891	+	0.784911i
$622$		0			0
$623$	−12.1510			−0.486820
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	−16.5949	+	10.8689i	−0.662737	+	0.434064i
$628$		0			0
$629$	−15.2910			−0.609692
$630$		0			0
$631$	−49.7521			−1.98060			−0.990299	−	0.138953i	$-0.955626\pi$
$631$	−49.7521			−1.98060			−0.990299	+	0.138953i	$0.955626\pi$
$632$		0			0
$633$	−8.84967	−	25.0087i	−0.351743	−	0.994006i
$634$		0			0
$635$	−8.36390			−0.331911
$636$		0			0
$637$			4.21336i			0.166939i
$638$		0			0
$639$	10.6961	+	13.2209i	0.423132	+	0.523010i
$640$		0			0
$641$			3.85467i			0.152250i	0.997098	+	0.0761252i	$0.0242549\pi$
$641$			3.85467i			0.152250i	−0.997098	+	0.0761252i	$0.975745\pi$
$642$		0			0
$643$	−46.3113			−1.82634			−0.913169	−	0.407581i	$-0.866373\pi$
$643$	−46.3113			−1.82634			−0.913169	+	0.407581i	$0.866373\pi$
$644$		0			0
$645$	3.36635	−	1.19123i	0.132550	−	0.0469045i
$646$		0			0
$647$			16.9609i			0.666800i	0.942785	+	0.333400i	$0.108196\pi$
$647$			16.9609i			0.666800i	−0.942785	+	0.333400i	$0.891804\pi$
$648$		0			0
$649$	−0.596959	+	0.145921i	−0.0234327	+	0.00572790i
$650$		0			0
$651$	1.80867	+	5.11122i	0.0708875	+	0.200324i
$652$		0			0
$653$			6.03560i			0.236191i	0.993002	+	0.118096i	$0.0376790\pi$
$653$			6.03560i			0.236191i	−0.993002	+	0.118096i	$0.962321\pi$
$654$		0			0
$655$			4.84037i			0.189129i
$656$		0			0
$657$	−8.66701	−	10.7128i	−0.338132	−	0.417946i
$658$		0			0
$659$	−43.6261			−1.69943			−0.849715	−	0.527243i	$-0.823226\pi$
$659$	−43.6261			−1.69943			−0.849715	+	0.527243i	$0.823226\pi$
$660$		0			0
$661$	44.1175			1.71597			0.857986	−	0.513674i	$-0.171716\pi$
$661$	44.1175			1.71597			0.857986	+	0.513674i	$0.171716\pi$
$662$		0			0
$663$	−11.3706	−	32.1328i	−0.441599	−	1.24794i
$664$		0			0
$665$		−	3.45326i		−	0.133912i
$666$		0			0
$667$			3.66734i			0.142000i
$668$		0			0
$669$	2.85358	−	1.00978i	0.110326	−	0.0390403i
$670$		0			0
$671$	47.5746	−	11.6291i	1.83660	−	0.448938i
$672$		0			0
$673$		−	3.86261i		−	0.148893i	−0.997225	−	0.0744463i	$-0.976281\pi$
$673$		−	3.86261i		−	0.148893i	0.997225	−	0.0744463i	$-0.0237189\pi$
$674$		0			0
$675$	2.71799	−	4.42860i	0.104616	−	0.170457i
$676$		0			0
$677$	42.9897			1.65223			0.826114	−	0.563503i	$-0.190547\pi$
$677$	42.9897			1.65223			0.826114	+	0.563503i	$0.190547\pi$
$678$		0			0
$679$			10.6957i			0.410462i
$680$		0			0
$681$	39.9965	−	14.1533i	1.53267	−	0.542356i
$682$		0			0
$683$		−	35.1641i		−	1.34552i	−0.739862	−	0.672759i	$-0.765109\pi$
$683$		−	35.1641i		−	1.34552i	0.739862	−	0.672759i	$-0.234891\pi$
$684$		0			0
$685$	4.32317			0.165180
$686$		0			0
$687$	4.97878	−	1.76181i	0.189952	−	0.0672171i
$688$		0			0
$689$	−7.96389			−0.303400
$690$		0			0
$691$	−6.14925			−0.233928			−0.116964	−	0.993136i	$-0.537316\pi$
$691$	−6.14925			−0.233928			−0.116964	+	0.993136i	$0.537316\pi$
$692$		0			0
$693$	6.02812	−	7.91592i	0.228989	−	0.300701i
$694$		0			0
$695$	−4.63233			−0.175714
$696$		0			0
$697$	−13.0909			−0.495854
$698$		0			0
$699$	29.3775	−	10.3956i	1.11116	−	0.393199i
$700$		0			0
$701$	−15.8074			−0.597037			−0.298519	−	0.954404i	$-0.596493\pi$
$701$	−15.8074			−0.597037			−0.298519	+	0.954404i	$0.596493\pi$
$702$		0			0
$703$			11.3054i			0.426392i
$704$		0			0
$705$	10.1243	−	3.58264i	0.381305	−	0.134930i
$706$		0			0
$707$			13.1014i			0.492727i
$708$		0			0
$709$	39.4336			1.48096			0.740480	−	0.672078i	$-0.234598\pi$
$709$	39.4336			1.48096			0.740480	+	0.672078i	$0.234598\pi$
$710$		0			0
$711$	15.4500	+	19.0969i	0.579422	+	0.716190i
$712$		0			0
$713$			22.5268i			0.843636i
$714$		0			0
$715$	−3.31815	−	13.5745i	−0.124092	−	0.507656i
$716$		0			0
$717$	23.9040	−	8.45877i	0.892713	−	0.315899i
$718$		0			0
$719$			19.4081i			0.723800i	0.932217	+	0.361900i	$0.117872\pi$
$719$			19.4081i			0.723800i	−0.932217	+	0.361900i	$0.882128\pi$
$720$		0			0
$721$		−	1.26589i		−	0.0471443i
$722$		0			0
$723$	−7.98882	−	22.5760i	−0.297108	−	0.839610i
$724$		0			0
$725$	0.509605			0.0189262
$726$		0			0
$727$	9.03203			0.334979			0.167490	−	0.985874i	$-0.446434\pi$
$727$	9.03203			0.334979			0.167490	+	0.985874i	$0.446434\pi$
$728$		0			0
$729$	−12.2250	−	24.0738i	−0.452779	−	0.891623i
$730$		0			0
$731$		−	9.62931i		−	0.356153i
$732$		0			0
$733$		−	22.4049i		−	0.827545i	−0.910380	−	0.413773i	$-0.864211\pi$
$733$		−	22.4049i		−	0.827545i	0.910380	−	0.413773i	$-0.135789\pi$
$734$		0			0
$735$	0.577800	+	1.63283i	0.0213125	+	0.0602280i
$736$		0			0
$737$	10.4355	+	42.6914i	0.384396	+	1.57256i
$738$		0			0
$739$		−	13.4927i		−	0.496335i	−0.968717	−	0.248168i	$-0.920172\pi$
$739$		−	13.4927i		−	0.496335i	0.968717	−	0.248168i	$-0.0798284\pi$
$740$		0			0
$741$	−23.7574	+	8.40689i	−0.872751	+	0.308835i
$742$		0			0
$743$	18.0037			0.660493			0.330247	−	0.943895i	$-0.392868\pi$
$743$	18.0037			0.660493			0.330247	+	0.943895i	$0.392868\pi$
$744$		0			0
$745$			5.33914i			0.195611i
$746$		0			0
$747$	26.1064	−	21.1209i	0.955182	−	0.772774i
$748$		0			0
$749$		−	2.86924i		−	0.104840i
$750$		0			0
$751$	21.5209			0.785308			0.392654	−	0.919686i	$-0.371557\pi$
$751$	21.5209			0.785308			0.392654	+	0.919686i	$0.371557\pi$
$752$		0			0
$753$	−4.95475	−	14.0019i	−0.180561	−	0.510257i
$754$		0			0
$755$	−22.7168			−0.826751
$756$		0			0
$757$	−11.8880			−0.432078			−0.216039	−	0.976385i	$-0.569314\pi$
$757$	−11.8880			−0.432078			−0.216039	+	0.976385i	$0.569314\pi$
$758$		0			0
$759$	34.5831	−	22.6504i	1.25529	−	0.822158i
$760$		0			0
$761$	19.6237			0.711357			0.355679	−	0.934608i	$-0.384250\pi$
$761$	19.6237			0.711357			0.355679	+	0.934608i	$0.384250\pi$
$762$		0			0
$763$	8.71393			0.315465
$764$		0			0
$765$	−8.81310	−	10.8934i	−0.318638	−	0.393850i
$766$		0			0
$767$	−0.780690			−0.0281891
$768$		0			0
$769$			5.52466i			0.199224i	0.995026	+	0.0996122i	$0.0317602\pi$
$769$			5.52466i			0.199224i	−0.995026	+	0.0996122i	$0.968240\pi$
$770$		0			0
$771$	5.97604	+	16.8880i	0.215222	+	0.608205i
$772$		0			0
$773$		−	26.1770i		−	0.941521i	−0.882261	−	0.470761i	$-0.843980\pi$
$773$		−	26.1770i		−	0.941521i	0.882261	−	0.470761i	$-0.156020\pi$
$774$		0			0
$775$	3.13027			0.112443
$776$		0			0
$777$	−1.89163	−	5.34564i	−0.0678617	−	0.191774i
$778$		0			0
$779$			9.67879i			0.346778i
$780$		0			0
$781$	18.2630	−	4.46421i	0.653500	−	0.159742i
$782$		0			0
$783$	1.38510	−	2.25684i	0.0494995	−	0.0806527i
$784$		0			0
$785$		−	2.44648i		−	0.0873186i
$786$		0			0
$787$			29.4615i			1.05019i	0.851044	+	0.525095i	$0.175970\pi$
$787$			29.4615i			1.05019i	−0.851044	+	0.525095i	$0.824030\pi$
$788$		0			0
$789$	4.40511	−	1.55881i	0.156826	−	0.0554950i
$790$		0			0
$791$	4.89549			0.174064
$792$		0			0
$793$	62.2170			2.20939
$794$		0			0
$795$	−3.08630	+	1.09213i	−0.109460	+	0.0387339i
$796$		0			0
$797$		−	0.382096i		−	0.0135345i	−0.999977	−	0.00676727i	$-0.997846\pi$
$797$		−	0.382096i		−	0.0135345i	0.999977	−	0.00676727i	$-0.00215410\pi$
$798$		0			0
$799$		−	28.9603i		−	1.02454i
$800$		0			0
$801$	22.9278	+	28.3397i	0.810113	+	1.00133i
$802$		0			0
$803$	−14.7984	+	3.61732i	−0.522223	+	0.127652i
$804$		0			0
$805$			7.19644i			0.253641i
$806$		0			0
$807$	15.2927	+	43.2163i	0.538327	+	1.52129i
$808$		0			0
$809$	9.03411			0.317622			0.158811	−	0.987309i	$-0.449234\pi$
$809$	9.03411			0.317622			0.158811	+	0.987309i	$0.449234\pi$
$810$		0			0
$811$		−	21.3483i		−	0.749640i	−0.927098	−	0.374820i	$-0.877704\pi$
$811$		−	21.3483i		−	0.749640i	0.927098	−	0.374820i	$-0.122296\pi$
$812$		0			0
$813$	−18.2735	−	51.6400i	−0.640880	−	1.81109i
$814$		0			0
$815$			5.74398i			0.201203i
$816$		0			0
$817$	−7.11944			−0.249078
$818$		0			0
$819$	9.82679	−	7.95021i	0.343376	−	0.277803i
$820$		0			0
$821$	5.04211			0.175971			0.0879854	−	0.996122i	$-0.471957\pi$
$821$	5.04211			0.175971			0.0879854	+	0.996122i	$0.471957\pi$
$822$		0			0
$823$	12.8659			0.448478			0.224239	−	0.974534i	$-0.428010\pi$
$823$	12.8659			0.448478			0.224239	+	0.974534i	$0.428010\pi$
$824$		0			0
$825$	−3.14745	−	4.80558i	−0.109580	−	0.167309i
$826$		0			0
$827$	13.7830			0.479280			0.239640	−	0.970862i	$-0.422971\pi$
$827$	13.7830			0.479280			0.239640	+	0.970862i	$0.422971\pi$
$828$		0			0
$829$	50.4433			1.75197			0.875983	−	0.482341i	$-0.160214\pi$
$829$	50.4433			1.75197			0.875983	+	0.482341i	$0.160214\pi$
$830$		0			0
$831$	17.5570	+	49.6150i	0.609044	+	1.72113i
$832$		0			0
$833$	4.67066			0.161829
$834$		0			0
$835$		−	23.3103i		−	0.806686i
$836$		0			0
$837$	8.50806	−	13.8627i	0.294082	−	0.479166i
$838$		0			0
$839$			0.484108i			0.0167133i	0.999965	+	0.00835664i	$0.00266003\pi$
$839$			0.484108i			0.0167133i	−0.999965	+	0.00835664i	$0.997340\pi$
$840$		0			0
$841$	−28.7403			−0.991045
$842$		0			0
$843$	−6.10653	+	2.16088i	−0.210320	+	0.0744246i
$844$		0			0
$845$		−	4.75240i		−	0.163487i
$846$		0			0
$847$	−5.07448	−	9.75959i	−0.174361	−	0.335344i
$848$		0			0
$849$	−12.7628	−	36.0670i	−0.438018	−	1.23782i
$850$		0			0
$851$		−	23.5600i		−	0.807626i
$852$		0			0
$853$		−	1.96937i		−	0.0674300i	−0.999431	−	0.0337150i	$-0.989266\pi$
$853$		−	1.96937i		−	0.0674300i	0.999431	−	0.0337150i	$-0.0107339\pi$
$854$		0			0
$855$	−8.05401	+	6.51597i	−0.275441	+	0.222841i
$856$		0			0
$857$	−11.3911			−0.389114			−0.194557	−	0.980891i	$-0.562327\pi$
$857$	−11.3911			−0.389114			−0.194557	+	0.980891i	$0.562327\pi$
$858$		0			0
$859$	−41.7042			−1.42293			−0.711464	−	0.702722i	$-0.751968\pi$
$859$	−41.7042			−1.42293			−0.711464	+	0.702722i	$0.751968\pi$
$860$		0			0
$861$	−1.61946	−	4.57650i	−0.0551910	−	0.155967i
$862$		0			0
$863$			32.8995i			1.11991i	0.828522	+	0.559956i	$0.189182\pi$
$863$			32.8995i			1.11991i	−0.828522	+	0.559956i	$0.810818\pi$
$864$		0			0
$865$		−	6.27951i		−	0.213510i
$866$		0			0
$867$	−7.86226	+	2.78217i	−0.267016	+	0.0944873i
$868$		0			0
$869$	26.3800	−	6.44833i	0.894879	−	0.218745i
$870$		0			0
$871$			55.8308i			1.89176i
$872$		0			0
$873$	24.9454	−	20.1817i	0.844274	−	0.683046i
$874$		0			0
$875$	1.00000			0.0338062
$876$		0			0
$877$		−	39.7438i		−	1.34205i	−0.741433	−	0.671026i	$-0.765854\pi$
$877$		−	39.7438i		−	1.34205i	0.741433	−	0.671026i	$-0.234146\pi$
$878$		0			0
$879$	0.501950	−	0.177622i	0.0169304	−	0.00599104i
$880$		0			0
$881$		−	38.2012i		−	1.28703i	−0.765434	−	0.643515i	$-0.777475\pi$
$881$		−	38.2012i		−	1.28703i	0.765434	−	0.643515i	$-0.222525\pi$
$882$		0			0
$883$	45.4506			1.52954			0.764768	−	0.644306i	$-0.222854\pi$
$883$	45.4506			1.52954			0.764768	+	0.644306i	$0.222854\pi$
$884$		0			0
$885$	−0.302547	+	0.107060i	−0.0101700	+	0.00359879i
$886$		0			0
$887$	−44.5954			−1.49737			−0.748683	−	0.662928i	$-0.769313\pi$
$887$	−44.5954			−1.49737			−0.748683	+	0.662928i	$0.769313\pi$
$888$		0			0
$889$	−8.36390			−0.280516
$890$		0			0
$891$	−29.8367	+	0.877252i	−0.999568	+	0.0293890i
$892$		0			0
$893$	−21.4118			−0.716520
$894$		0			0
$895$	−4.55257			−0.152176
$896$		0			0
$897$	49.5095	−	17.5196i	1.65307	−	0.584962i
$898$		0			0
$899$	1.59520			0.0532029
$900$		0			0
$901$			8.82827i			0.294112i
$902$		0			0
$903$	3.36635	−	1.19123i	0.112025	−	0.0396416i
$904$		0			0
$905$		−	16.2027i		−	0.538598i
$906$		0			0
$907$	−8.22201			−0.273007			−0.136504	−	0.990640i	$-0.543587\pi$
$907$	−8.22201			−0.273007			−0.136504	+	0.990640i	$0.543587\pi$
$908$		0			0
$909$	30.5562	−	24.7210i	1.01349	−	0.819944i
$910$		0			0
$911$			12.4065i			0.411046i	0.978652	+	0.205523i	$0.0658896\pi$
$911$			12.4065i			0.411046i	−0.978652	+	0.205523i	$0.934110\pi$
$912$		0			0
$913$	−8.81516	−	36.0626i	−0.291739	−	1.19350i
$914$		0			0
$915$	24.1114	−	8.53215i	0.797098	−	0.282064i
$916$		0			0
$917$			4.84037i			0.159843i
$918$		0			0
$919$		−	7.26376i		−	0.239609i	−0.992797	−	0.119805i	$-0.961773\pi$
$919$		−	7.26376i		−	0.239609i	0.992797	−	0.119805i	$-0.0382268\pi$
$920$		0			0
$921$	−8.33039	−	23.5412i	−0.274496	−	0.775710i
$922$		0			0
$923$	23.8839			0.786148
$924$		0			0
$925$	−3.27384			−0.107643
$926$		0			0
$927$	−2.95244	+	2.38862i	−0.0969707	+	0.0784526i
$928$		0			0
$929$			20.0383i			0.657436i	0.944428	+	0.328718i	$0.106617\pi$
$929$			20.0383i			0.657436i	−0.944428	+	0.328718i	$0.893383\pi$
$930$		0			0
$931$		−	3.45326i		−	0.113176i
$932$		0			0
$933$	−15.6029	−	44.0931i	−0.510817	−	1.44354i
$934$		0			0
$935$	−15.0478	+	3.67829i	−0.492116	+	0.120293i
$936$		0			0
$937$			11.3461i			0.370662i	0.982676	+	0.185331i	$0.0593357\pi$
$937$			11.3461i			0.370662i	−0.982676	+	0.185331i	$0.940664\pi$
$938$		0			0
$939$	−41.2256	+	14.5882i	−1.34535	+	0.476069i
$940$		0			0
$941$	43.5347			1.41919			0.709596	−	0.704609i	$-0.248878\pi$
$941$	43.5347			1.41919			0.709596	+	0.704609i	$0.248878\pi$
$942$		0			0
$943$		−	20.1702i		−	0.656831i
$944$		0			0
$945$	2.71799	−	4.42860i	0.0884163	−	0.144062i
$946$		0			0
$947$			53.1278i			1.72642i	0.504843	+	0.863211i	$0.331550\pi$
$947$			53.1278i			1.72642i	−0.504843	+	0.863211i	$0.668450\pi$
$948$		0			0
$949$	−19.3530			−0.628225
$950$		0			0
$951$	−4.54465	−	12.8430i	−0.147371	−	0.416461i
$952$		0			0
$953$	9.33763			0.302475			0.151238	−	0.988497i	$-0.451674\pi$
$953$	9.33763			0.302475			0.151238	+	0.988497i	$0.451674\pi$
$954$		0			0
$955$	7.37437			0.238629
$956$		0			0
$957$	−1.60395	−	2.44894i	−0.0518484	−	0.0791631i
$958$		0			0
$959$	4.32317			0.139603
$960$		0			0
$961$	−21.2014			−0.683916
$962$		0			0
$963$	−6.69190	+	5.41398i	−0.215644	+	0.174463i
$964$		0			0
$965$	−17.4347			−0.561241
$966$		0			0
$967$		−	21.9886i		−	0.707105i	−0.935415	−	0.353553i	$-0.884974\pi$
$967$		−	21.9886i		−	0.707105i	0.935415	−	0.353553i	$-0.115026\pi$
$968$		0			0
$969$	9.31935	+	26.3360i	0.299381	+	0.846034i
$970$		0			0
$971$		−	29.5256i		−	0.947523i	−0.880653	−	0.473762i	$-0.842896\pi$
$971$		−	29.5256i		−	0.947523i	0.880653	−	0.473762i	$-0.157104\pi$
$972$		0			0
$973$	−4.63233			−0.148506
$974$		0			0
$975$	−2.43448	−	6.87972i	−0.0779658	−	0.220327i
$976$		0			0
$977$			43.1574i			1.38073i	0.723462	+	0.690364i	$0.242550\pi$
$977$			43.1574i			1.38073i	−0.723462	+	0.690364i	$0.757450\pi$
$978$		0			0
$979$	39.1477	−	9.56928i	1.25117	−	0.305836i
$980$		0			0
$981$	−16.4423	−	20.3234i	−0.524964	−	0.648877i
$982$		0			0
$983$			43.1994i			1.37785i	0.724835	+	0.688923i	$0.241916\pi$
$983$			43.1994i			1.37785i	−0.724835	+	0.688923i	$0.758084\pi$
$984$		0			0
$985$			19.4096i			0.618442i
$986$		0			0
$987$	10.1243	−	3.58264i	0.322261	−	0.114037i
$988$		0			0
$989$	14.8366			0.471777
$990$		0			0
$991$	21.6630			0.688149			0.344074	−	0.938942i	$-0.388193\pi$
$991$	21.6630			0.688149			0.344074	+	0.938942i	$0.388193\pi$
$992$		0			0
$993$	−34.0017	+	12.0320i	−1.07901	+	0.381822i
$994$		0			0
$995$		−	12.5558i		−	0.398047i
$996$		0			0
$997$		−	46.7740i		−	1.48135i	−0.671864	−	0.740675i	$-0.734506\pi$
$997$		−	46.7740i		−	1.48135i	0.671864	−	0.740675i	$-0.265494\pi$
$998$		0			0
$999$	−8.89828	+	14.4985i	−0.281529	+	0.458713i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4620.2.m.a.1121.8	yes	48
3.2	odd	2		4620.2.m.b.1121.7	yes	48
11.10	odd	2		4620.2.m.b.1121.8	yes	48
33.32	even	2	inner	4620.2.m.a.1121.7	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4620.2.m.a.1121.7	✓	48	33.32	even	2	inner
4620.2.m.a.1121.8	yes	48	1.1	even	1	trivial
4620.2.m.b.1121.7	yes	48	3.2	odd	2
4620.2.m.b.1121.8	yes	48	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4620.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.63283 + 0.577800i) q^{3} +1.00000i q^{5} +1.00000i q^{7} +(2.33229 - 1.88690i) q^{9} +O(q^{10})\)	\(q+(-1.63283 + 0.577800i) q^{3} +1.00000i q^{5} +1.00000i q^{7} +(2.33229 - 1.88690i) q^{9} +(-0.787530 - 3.22177i) q^{11} -4.21336i q^{13} +(-0.577800 - 1.63283i) q^{15} -4.67066 q^{17} +3.45326i q^{19} +(-0.577800 - 1.63283i) q^{21} -7.19644i q^{23} -1.00000 q^{25} +(-2.71799 + 4.42860i) q^{27} -0.509605 q^{29} -3.13027 q^{31} +(3.14745 + 4.80558i) q^{33} -1.00000 q^{35} +3.27384 q^{37} +(2.43448 + 6.87972i) q^{39} +2.80280 q^{41} +2.06166i q^{43} +(1.88690 + 2.33229i) q^{45} +6.20047i q^{47} -1.00000 q^{49} +(7.62642 - 2.69871i) q^{51} -1.89015i q^{53} +(3.22177 - 0.787530i) q^{55} +(-1.99529 - 5.63860i) q^{57} -0.185289i q^{59} +14.7666i q^{61} +(1.88690 + 2.33229i) q^{63} +4.21336 q^{65} -13.2509 q^{67} +(4.15811 + 11.7506i) q^{69} +5.66861i q^{71} -4.59324i q^{73} +(1.63283 - 0.577800i) q^{75} +(3.22177 - 0.787530i) q^{77} +8.18804i q^{79} +(1.87918 - 8.80163i) q^{81} +11.1934 q^{83} -4.67066i q^{85} +(0.832100 - 0.294450i) q^{87} +12.1510i q^{89} +4.21336 q^{91} +(5.11122 - 1.80867i) q^{93} -3.45326 q^{95} +10.6957 q^{97} +(-7.91592 - 6.02812i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 4 q^{3} + 6 q^{9}+O(q^{10}) \) 48 * q - 4 * q^3 + 6 * q^9	\( 48 q - 4 q^{3} + 6 q^{9} - 6 q^{11} + 2 q^{15} + 4 q^{17} + 2 q^{21} - 48 q^{25} - 16 q^{27} + 36 q^{29} - 16 q^{31} + 16 q^{33} - 48 q^{35} - 8 q^{37} - 18 q^{39} - 48 q^{49} - 30 q^{51} + 4 q^{55} + 16 q^{57} - 12 q^{65} + 24 q^{67} + 4 q^{69} + 4 q^{75} + 4 q^{77} - 22 q^{81} + 20 q^{83} + 44 q^{87} - 12 q^{91} - 28 q^{93} + 8 q^{95} - 56 q^{97} + 26 q^{99}+O(q^{100}) \) 48 * q - 4 * q^3 + 6 * q^9 - 6 * q^11 + 2 * q^15 + 4 * q^17 + 2 * q^21 - 48 * q^25 - 16 * q^27 + 36 * q^29 - 16 * q^31 + 16 * q^33 - 48 * q^35 - 8 * q^37 - 18 * q^39 - 48 * q^49 - 30 * q^51 + 4 * q^55 + 16 * q^57 - 12 * q^65 + 24 * q^67 + 4 * q^69 + 4 * q^75 + 4 * q^77 - 22 * q^81 + 20 * q^83 + 44 * q^87 - 12 * q^91 - 28 * q^93 + 8 * q^95 - 56 * q^97 + 26 * q^99

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