LMFDB - Embedded newform 2124.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2124,2,Mod(1,2124)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2124, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2124.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2124 = 2^{2} \cdot 3^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2124.a (trivial)

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:	yes
Analytic conductor:	$16.9602253893$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.22896.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 9x^{2} - 4x + 3 $ x^4 - 9x^2 - 4x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.15690$ of defining polynomial
Character	$\chi$	$=$	2124.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.57516 q^{5} +0.184277 q^{7} +O(q^{10})$	$q-2.57516 q^{5} +0.184277 q^{7} -2.76602 q^{11} +2.34118 q^{13} +5.36351 q^{17} +3.96604 q^{19} -6.15690 q^{23} +1.63145 q^{25} +1.15690 q^{29} -3.11379 q^{31} -0.474542 q^{35} +10.7047 q^{37} -7.25631 q^{41} -10.7544 q^{43} -1.07707 q^{47} -6.96604 q^{49} -10.4210 q^{53} +7.12295 q^{55} +1.00000 q^{59} -11.5138 q^{61} -6.02891 q^{65} +5.97263 q^{67} -0.261352 q^{71} -12.6773 q^{73} -0.509714 q^{77} +10.4773 q^{79} -14.0732 q^{83} -13.8119 q^{85} -1.75944 q^{89} +0.431425 q^{91} -10.2132 q^{95} -11.1098 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{7}+O(q^{10}) $ 4 * q + 2 * q^7	$ 4 q + 2 q^{7} - 10 q^{11} - 2 q^{13} - 4 q^{17} - 6 q^{19} - 12 q^{23} + 4 q^{25} - 8 q^{29} - 8 q^{31} - 12 q^{35} + 6 q^{37} - 16 q^{41} - 6 q^{43} - 18 q^{47} - 6 q^{49} - 4 q^{53} - 6 q^{55} + 4 q^{59} - 18 q^{65} + 10 q^{67} - 16 q^{71} - 14 q^{77} + 12 q^{79} - 22 q^{83} - 6 q^{85} + 2 q^{89} + 20 q^{91} - 36 q^{95} + 6 q^{97}+O(q^{100}) $ 4 * q + 2 * q^7 - 10 * q^11 - 2 * q^13 - 4 * q^17 - 6 * q^19 - 12 * q^23 + 4 * q^25 - 8 * q^29 - 8 * q^31 - 12 * q^35 + 6 * q^37 - 16 * q^41 - 6 * q^43 - 18 * q^47 - 6 * q^49 - 4 * q^53 - 6 * q^55 + 4 * q^59 - 18 * q^65 + 10 * q^67 - 16 * q^71 - 14 * q^77 + 12 * q^79 - 22 * q^83 - 6 * q^85 + 2 * q^89 + 20 * q^91 - 36 * q^95 + 6 * q^97

Display 100 coefficients 10 coefficients

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	0
$3$	0	0
$4$	0	0
$5$	−2.57516	−1.15165	−0.575823	−	0.817574i	$-0.695318\pi$
$5$	−2.57516	−1.15165	−0.575823	+	0.817574i	$0.695318\pi$
$6$	0	0
$7$	0.184277	0.0696501	0.0348251	−	0.999393i	$-0.488913\pi$
$7$	0.184277	0.0696501	0.0348251	+	0.999393i	$0.488913\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.76602	−0.833987	−0.416993	−	0.908910i	$-0.636916\pi$
$11$	−2.76602	−0.833987	−0.416993	+	0.908910i	$0.636916\pi$
$12$	0	0
$13$	2.34118	0.649327	0.324663	−	0.945830i	$-0.394749\pi$
$13$	2.34118	0.649327	0.324663	+	0.945830i	$0.394749\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.36351	1.30084	0.650421	−	0.759574i	$-0.274592\pi$
$17$	5.36351	1.30084	0.650421	+	0.759574i	$0.274592\pi$
$18$	0	0
$19$	3.96604	0.909872	0.454936	−	0.890524i	$-0.349662\pi$
$19$	3.96604	0.909872	0.454936	+	0.890524i	$0.349662\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.15690	−1.28380	−0.641902	−	0.766787i	$-0.721854\pi$
$23$	−6.15690	−1.28380	−0.641902	+	0.766787i	$0.721854\pi$
$24$	0	0
$25$	1.63145	0.326289
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.15690	0.214832	0.107416	−	0.994214i	$-0.465742\pi$
$29$	1.15690	0.214832	0.107416	+	0.994214i	$0.465742\pi$
$30$	0	0
$31$	−3.11379	−0.559253	−0.279626	−	0.960109i	$-0.590211\pi$
$31$	−3.11379	−0.559253	−0.279626	+	0.960109i	$0.590211\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.474542	−0.0802123
$36$	0	0
$37$	10.7047	1.75984	0.879920	−	0.475122i	$-0.157596\pi$
$37$	10.7047	1.75984	0.879920	+	0.475122i	$0.157596\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.25631	−1.13324	−0.566622	−	0.823978i	$-0.691750\pi$
$41$	−7.25631	−1.13324	−0.566622	+	0.823978i	$0.691750\pi$
$42$	0	0
$43$	−10.7544	−1.64003	−0.820015	−	0.572342i	$-0.806035\pi$
$43$	−10.7544	−1.64003	−0.820015	+	0.572342i	$0.806035\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.07707	−0.157108	−0.0785538	−	0.996910i	$-0.525030\pi$
$47$	−1.07707	−0.157108	−0.0785538	+	0.996910i	$0.525030\pi$
$48$	0	0
$49$	−6.96604	−0.995149
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.4210	−1.43144	−0.715718	−	0.698390i	$-0.753900\pi$
$53$	−10.4210	−1.43144	−0.715718	+	0.698390i	$0.753900\pi$
$54$	0	0
$55$	7.12295	0.960458
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−11.5138	−1.47419	−0.737097	−	0.675787i	$-0.763804\pi$
$61$	−11.5138	−1.47419	−0.737097	+	0.675787i	$0.763804\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−6.02891	−0.747795
$66$	0	0
$67$	5.97263	0.729673	0.364836	−	0.931072i	$-0.381125\pi$
$67$	5.97263	0.729673	0.364836	+	0.931072i	$0.381125\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.261352	−0.0310167	−0.0155084	−	0.999880i	$-0.504937\pi$
$71$	−0.261352	−0.0310167	−0.0155084	+	0.999880i	$0.504937\pi$
$72$	0	0
$73$	−12.6773	−1.48377	−0.741884	−	0.670528i	$-0.766067\pi$
$73$	−12.6773	−1.48377	−0.741884	+	0.670528i	$0.766067\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.509714	−0.0580873
$78$	0	0
$79$	10.4773	1.17879	0.589394	−	0.807846i	$-0.299367\pi$
$79$	10.4773	1.17879	0.589394	+	0.807846i	$0.299367\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−14.0732	−1.54474	−0.772370	−	0.635173i	$-0.780929\pi$
$83$	−14.0732	−1.54474	−0.772370	+	0.635173i	$0.780929\pi$
$84$	0	0
$85$	−13.8119	−1.49811
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−1.75944	−0.186500	−0.0932499	−	0.995643i	$-0.529726\pi$
$89$	−1.75944	−0.186500	−0.0932499	+	0.995643i	$0.529726\pi$
$90$	0	0
$91$	0.431425	0.0452257
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−10.2132	−1.04785
$96$	0	0
$97$	−11.1098	−1.12803	−0.564013	−	0.825766i	$-0.690743\pi$
$97$	−11.1098	−1.12803	−0.564013	+	0.825766i	$0.690743\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.69553	0.268215	0.134108	−	0.990967i	$-0.457183\pi$
$101$	2.69553	0.268215	0.134108	+	0.990967i	$0.457183\pi$
$102$	0	0
$103$	−5.53204	−0.545088	−0.272544	−	0.962143i	$-0.587865\pi$
$103$	−5.53204	−0.545088	−0.272544	+	0.962143i	$0.587865\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−19.3088	−1.86665	−0.933324	−	0.359034i	$-0.883106\pi$
$107$	−19.3088	−1.86665	−0.933324	+	0.359034i	$0.883106\pi$
$108$	0	0
$109$	16.6499	1.59478	0.797388	−	0.603467i	$-0.206215\pi$
$109$	16.6499	1.59478	0.797388	+	0.603467i	$0.206215\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	19.3257	1.81801	0.909005	−	0.416784i	$-0.136843\pi$
$113$	19.3257	1.81801	0.909005	+	0.416784i	$0.136843\pi$
$114$	0	0
$115$	15.8550	1.47849
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.988371	0.0906038
$120$	0	0
$121$	−3.34913	−0.304466
$122$	0	0
$123$	0	0
$124$	0	0
$125$	8.67456	0.775877
$126$	0	0
$127$	−2.73085	−0.242324	−0.121162	−	0.992633i	$-0.538662\pi$
$127$	−2.73085	−0.242324	−0.121162	+	0.992633i	$0.538662\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−12.8812	−1.12543	−0.562716	−	0.826650i	$-0.690244\pi$
$131$	−12.8812	−1.12543	−0.562716	+	0.826650i	$0.690244\pi$
$132$	0	0
$133$	0.730850	0.0633727
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.47576	−0.553261	−0.276631	−	0.960976i	$-0.589218\pi$
$137$	−6.47576	−0.553261	−0.276631	+	0.960976i	$0.589218\pi$
$138$	0	0
$139$	3.46796	0.294148	0.147074	−	0.989125i	$-0.453014\pi$
$139$	3.46796	0.294148	0.147074	+	0.989125i	$0.453014\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−6.47576	−0.541530
$144$	0	0
$145$	−2.97921	−0.247410
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−3.88775	−0.318497	−0.159249	−	0.987239i	$-0.550907\pi$
$149$	−3.88775	−0.318497	−0.159249	+	0.987239i	$0.550907\pi$
$150$	0	0
$151$	0.530829	0.0431983	0.0215991	−	0.999767i	$-0.493124\pi$
$151$	0.530829	0.0431983	0.0215991	+	0.999767i	$0.493124\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	8.01850	0.644061
$156$	0	0
$157$	12.3727	0.987446	0.493723	−	0.869619i	$-0.335636\pi$
$157$	12.3727	0.987446	0.493723	+	0.869619i	$0.335636\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1.13458	−0.0894170
$162$	0	0
$163$	−3.34777	−0.262217	−0.131109	−	0.991368i	$-0.541854\pi$
$163$	−3.34777	−0.262217	−0.131109	+	0.991368i	$0.541854\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.74369	−0.212313	−0.106157	−	0.994349i	$-0.533855\pi$
$167$	−2.74369	−0.212313	−0.106157	+	0.994349i	$0.533855\pi$
$168$	0	0
$169$	−7.51887	−0.578375
$170$	0	0
$171$	0	0
$172$	0	0
$173$	3.09024	0.234947	0.117473	−	0.993076i	$-0.462520\pi$
$173$	3.09024	0.234947	0.117473	+	0.993076i	$0.462520\pi$
$174$	0	0
$175$	0.300638	0.0227261
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−12.4290	−0.928984	−0.464492	−	0.885577i	$-0.653763\pi$
$179$	−12.4290	−0.928984	−0.464492	+	0.885577i	$0.653763\pi$
$180$	0	0
$181$	−1.82889	−0.135941	−0.0679703	−	0.997687i	$-0.521652\pi$
$181$	−1.82889	−0.135941	−0.0679703	+	0.997687i	$0.521652\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−27.5663	−2.02671
$186$	0	0
$187$	−14.8356	−1.08489
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.75593	0.488842	0.244421	−	0.969669i	$-0.421402\pi$
$191$	6.75593	0.488842	0.244421	+	0.969669i	$0.421402\pi$
$192$	0	0
$193$	−11.9453	−0.859838	−0.429919	−	0.902867i	$-0.641458\pi$
$193$	−11.9453	−0.859838	−0.429919	+	0.902867i	$0.641458\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.21823	0.229290	0.114645	−	0.993407i	$-0.463427\pi$
$197$	3.21823	0.229290	0.114645	+	0.993407i	$0.463427\pi$
$198$	0	0
$199$	−12.3439	−0.875038	−0.437519	−	0.899209i	$-0.644143\pi$
$199$	−12.3439	−0.875038	−0.437519	+	0.899209i	$0.644143\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.213191	0.0149630
$204$	0	0
$205$	18.6861	1.30510
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−10.9702	−0.758822
$210$	0	0
$211$	23.2329	1.59942	0.799709	−	0.600387i	$-0.204987\pi$
$211$	23.2329	1.59942	0.799709	+	0.600387i	$0.204987\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	27.6943	1.88873
$216$	0	0
$217$	−0.573799	−0.0389520
$218$	0	0
$219$	0	0
$220$	0	0
$221$	12.5569	0.844672
$222$	0	0
$223$	−17.7779	−1.19050	−0.595249	−	0.803541i	$-0.702947\pi$
$223$	−17.7779	−1.19050	−0.595249	+	0.803541i	$0.702947\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.243138	0.0161376	0.00806880	−	0.999967i	$-0.497432\pi$
$227$	0.243138	0.0161376	0.00806880	+	0.999967i	$0.497432\pi$
$228$	0	0
$229$	4.41826	0.291967	0.145983	−	0.989287i	$-0.453365\pi$
$229$	4.41826	0.291967	0.145983	+	0.989287i	$0.453365\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	26.2591	1.72029	0.860144	−	0.510051i	$-0.170373\pi$
$233$	26.2591	1.72029	0.860144	+	0.510051i	$0.170373\pi$
$234$	0	0
$235$	2.77364	0.180932
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−26.3794	−1.70634	−0.853172	−	0.521630i	$-0.825324\pi$
$239$	−26.3794	−1.70634	−0.853172	+	0.521630i	$0.825324\pi$
$240$	0	0
$241$	10.2591	0.660844	0.330422	−	0.943833i	$-0.392809\pi$
$241$	10.2591	0.660844	0.330422	+	0.943833i	$0.392809\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	17.9387	1.14606
$246$	0	0
$247$	9.28522	0.590804
$248$	0	0
$249$	0	0
$250$	0	0
$251$	25.2343	1.59277	0.796386	−	0.604788i	$-0.206742\pi$
$251$	25.2343	1.59277	0.796386	+	0.604788i	$0.206742\pi$
$252$	0	0
$253$	17.0301	1.07067
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.89434	−0.305300	−0.152650	−	0.988280i	$-0.548781\pi$
$257$	−4.89434	−0.305300	−0.152650	+	0.988280i	$0.548781\pi$
$258$	0	0
$259$	1.97263	0.122573
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−26.0852	−1.60848	−0.804240	−	0.594305i	$-0.797427\pi$
$263$	−26.0852	−1.60848	−0.804240	+	0.594305i	$0.797427\pi$
$264$	0	0
$265$	26.8358	1.64851
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−10.9572	−0.668070	−0.334035	−	0.942561i	$-0.608410\pi$
$269$	−10.9572	−0.668070	−0.334035	+	0.942561i	$0.608410\pi$
$270$	0	0
$271$	4.41947	0.268464	0.134232	−	0.990950i	$-0.457143\pi$
$271$	4.41947	0.268464	0.134232	+	0.990950i	$0.457143\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.51262	−0.272121
$276$	0	0
$277$	2.93208	0.176172	0.0880859	−	0.996113i	$-0.471925\pi$
$277$	2.93208	0.176172	0.0880859	+	0.996113i	$0.471925\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−15.4357	−0.920818	−0.460409	−	0.887707i	$-0.652297\pi$
$281$	−15.4357	−0.920818	−0.460409	+	0.887707i	$0.652297\pi$
$282$	0	0
$283$	18.9638	1.12728	0.563639	−	0.826021i	$-0.309401\pi$
$283$	18.9638	1.12728	0.563639	+	0.826021i	$0.309401\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.33717	−0.0789306
$288$	0	0
$289$	11.7672	0.692190
$290$	0	0
$291$	0	0
$292$	0	0
$293$	22.7830	1.33100	0.665498	−	0.746400i	$-0.268219\pi$
$293$	22.7830	1.33100	0.665498	+	0.746400i	$0.268219\pi$
$294$	0	0
$295$	−2.57516	−0.149932
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−14.4144	−0.833608
$300$	0	0
$301$	−1.98179	−0.114228
$302$	0	0
$303$	0	0
$304$	0	0
$305$	29.6499	1.69775
$306$	0	0
$307$	−12.7007	−0.724866	−0.362433	−	0.932010i	$-0.618054\pi$
$307$	−12.7007	−0.724866	−0.362433	+	0.932010i	$0.618054\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−17.2509	−0.978211	−0.489105	−	0.872225i	$-0.662677\pi$
$311$	−17.2509	−0.978211	−0.489105	+	0.872225i	$0.662677\pi$
$312$	0	0
$313$	−12.1373	−0.686042	−0.343021	−	0.939328i	$-0.611450\pi$
$313$	−12.1373	−0.686042	−0.343021	+	0.939328i	$0.611450\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	20.8626	1.17176	0.585881	−	0.810397i	$-0.300749\pi$
$317$	20.8626	1.17176	0.585881	+	0.810397i	$0.300749\pi$
$318$	0	0
$319$	−3.20002	−0.179167
$320$	0	0
$321$	0	0
$322$	0	0
$323$	21.2719	1.18360
$324$	0	0
$325$	3.81951	0.211868
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−0.198480	−0.0109426
$330$	0	0
$331$	−21.2968	−1.17058	−0.585289	−	0.810825i	$-0.699019\pi$
$331$	−21.2968	−1.17058	−0.585289	+	0.810825i	$0.699019\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−15.3805	−0.840325
$336$	0	0
$337$	−19.6329	−1.06947	−0.534737	−	0.845018i	$-0.679589\pi$
$337$	−19.6329	−1.06947	−0.534737	+	0.845018i	$0.679589\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	8.61280	0.466409
$342$	0	0
$343$	−2.57362	−0.138962
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.0763	1.07775	0.538877	−	0.842385i	$-0.318849\pi$
$347$	20.0763	1.07775	0.538877	+	0.842385i	$0.318849\pi$
$348$	0	0
$349$	−9.60407	−0.514094	−0.257047	−	0.966399i	$-0.582750\pi$
$349$	−9.60407	−0.514094	−0.257047	+	0.966399i	$0.582750\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	30.7845	1.63849	0.819247	−	0.573440i	$-0.194392\pi$
$353$	30.7845	1.63849	0.819247	+	0.573440i	$0.194392\pi$
$354$	0	0
$355$	0.673022	0.0357203
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10.2560	0.541290	0.270645	−	0.962679i	$-0.412763\pi$
$359$	10.2560	0.541290	0.270645	+	0.962679i	$0.412763\pi$
$360$	0	0
$361$	−3.27051	−0.172132
$362$	0	0
$363$	0	0
$364$	0	0
$365$	32.6461	1.70878
$366$	0	0
$367$	−23.8631	−1.24565	−0.622823	−	0.782363i	$-0.714014\pi$
$367$	−23.8631	−1.24565	−0.622823	+	0.782363i	$0.714014\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.92035	−0.0996997
$372$	0	0
$373$	3.96604	0.205354	0.102677	−	0.994715i	$-0.467259\pi$
$373$	3.96604	0.205354	0.102677	+	0.994715i	$0.467259\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.70852	0.139496
$378$	0	0
$379$	−34.8798	−1.79165	−0.895827	−	0.444404i	$-0.853416\pi$
$379$	−34.8798	−1.79165	−0.895827	+	0.444404i	$0.853416\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5.20035	0.265725	0.132863	−	0.991134i	$-0.457583\pi$
$383$	5.20035	0.265725	0.132863	+	0.991134i	$0.457583\pi$
$384$	0	0
$385$	1.31259	0.0668960
$386$	0	0
$387$	0	0
$388$	0	0
$389$	28.6692	1.45359	0.726793	−	0.686857i	$-0.241010\pi$
$389$	28.6692	1.45359	0.726793	+	0.686857i	$0.241010\pi$
$390$	0	0
$391$	−33.0226	−1.67003
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−26.9807	−1.35755
$396$	0	0
$397$	11.6065	0.582514	0.291257	−	0.956645i	$-0.405927\pi$
$397$	11.6065	0.582514	0.291257	+	0.956645i	$0.405927\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−2.89401	−0.144520	−0.0722600	−	0.997386i	$-0.523021\pi$
$401$	−2.89401	−0.144520	−0.0722600	+	0.997386i	$0.523021\pi$
$402$	0	0
$403$	−7.28994	−0.363138
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−29.6094	−1.46768
$408$	0	0
$409$	−13.6314	−0.674032	−0.337016	−	0.941499i	$-0.609418\pi$
$409$	−13.6314	−0.674032	−0.337016	+	0.941499i	$0.609418\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0.184277	0.00906767
$414$	0	0
$415$	36.2408	1.77899
$416$	0	0
$417$	0	0
$418$	0	0
$419$	16.7007	0.815882	0.407941	−	0.913008i	$-0.366247\pi$
$419$	16.7007	0.815882	0.407941	+	0.913008i	$0.366247\pi$
$420$	0	0
$421$	−27.3141	−1.33121	−0.665604	−	0.746305i	$-0.731826\pi$
$421$	−27.3141	−1.33121	−0.665604	+	0.746305i	$0.731826\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	8.75028	0.424451
$426$	0	0
$427$	−2.12173	−0.102678
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13.9922	0.673979	0.336989	−	0.941508i	$-0.390591\pi$
$431$	13.9922	0.673979	0.336989	+	0.941508i	$0.390591\pi$
$432$	0	0
$433$	−10.7478	−0.516507	−0.258253	−	0.966077i	$-0.583147\pi$
$433$	−10.7478	−0.516507	−0.258253	+	0.966077i	$0.583147\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−24.4185	−1.16810
$438$	0	0
$439$	20.7509	0.990386	0.495193	−	0.868783i	$-0.335097\pi$
$439$	20.7509	0.990386	0.495193	+	0.868783i	$0.335097\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9.33871	−0.443696	−0.221848	−	0.975081i	$-0.571209\pi$
$443$	−9.33871	−0.443696	−0.221848	+	0.975081i	$0.571209\pi$
$444$	0	0
$445$	4.53083	0.214782
$446$	0	0
$447$	0	0
$448$	0	0
$449$	4.34759	0.205175	0.102588	−	0.994724i	$-0.467288\pi$
$449$	4.34759	0.205175	0.102588	+	0.994724i	$0.467288\pi$
$450$	0	0
$451$	20.0711	0.945111
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.11099	−0.0520840
$456$	0	0
$457$	18.8214	0.880427	0.440213	−	0.897893i	$-0.354903\pi$
$457$	18.8214	0.880427	0.440213	+	0.897893i	$0.354903\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−2.78298	−0.129616	−0.0648081	−	0.997898i	$-0.520644\pi$
$461$	−2.78298	−0.129616	−0.0648081	+	0.997898i	$0.520644\pi$
$462$	0	0
$463$	−20.7047	−0.962229	−0.481114	−	0.876658i	$-0.659768\pi$
$463$	−20.7047	−0.962229	−0.481114	+	0.876658i	$0.659768\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	23.7492	1.09898	0.549490	−	0.835500i	$-0.314822\pi$
$467$	23.7492	1.09898	0.549490	+	0.835500i	$0.314822\pi$
$468$	0	0
$469$	1.10062	0.0508218
$470$	0	0
$471$	0	0
$472$	0	0
$473$	29.7469	1.36776
$474$	0	0
$475$	6.47038	0.296882
$476$	0	0
$477$	0	0
$478$	0	0
$479$	38.6343	1.76525	0.882624	−	0.470080i	$-0.155775\pi$
$479$	38.6343	1.76525	0.882624	+	0.470080i	$0.155775\pi$
$480$	0	0
$481$	25.0616	1.14271
$482$	0	0
$483$	0	0
$484$	0	0
$485$	28.6094	1.29909
$486$	0	0
$487$	−40.9805	−1.85701	−0.928503	−	0.371326i	$-0.878903\pi$
$487$	−40.9805	−1.85701	−0.928503	+	0.371326i	$0.878903\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−21.1569	−0.954798	−0.477399	−	0.878687i	$-0.658420\pi$
$491$	−21.1569	−0.954798	−0.477399	+	0.878687i	$0.658420\pi$
$492$	0	0
$493$	6.20506	0.279462
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.0481611	−0.00216032
$498$	0	0
$499$	6.43264	0.287964	0.143982	−	0.989580i	$-0.454009\pi$
$499$	6.43264	0.287964	0.143982	+	0.989580i	$0.454009\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	21.0771	0.939780	0.469890	−	0.882725i	$-0.344294\pi$
$503$	21.0771	0.939780	0.469890	+	0.882725i	$0.344294\pi$
$504$	0	0
$505$	−6.94142	−0.308889
$506$	0	0
$507$	0	0
$508$	0	0
$509$	3.65761	0.162121	0.0810603	−	0.996709i	$-0.474169\pi$
$509$	3.65761	0.162121	0.0810603	+	0.996709i	$0.474169\pi$
$510$	0	0
$511$	−2.33614	−0.103345
$512$	0	0
$513$	0	0
$514$	0	0
$515$	14.2459	0.627749
$516$	0	0
$517$	2.97921	0.131026
$518$	0	0
$519$	0	0
$520$	0	0
$521$	3.55180	0.155607	0.0778035	−	0.996969i	$-0.475209\pi$
$521$	3.55180	0.155607	0.0778035	+	0.996969i	$0.475209\pi$
$522$	0	0
$523$	−22.2653	−0.973594	−0.486797	−	0.873515i	$-0.661835\pi$
$523$	−22.2653	−0.973594	−0.486797	+	0.873515i	$0.661835\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−16.7008	−0.727499
$528$	0	0
$529$	14.9075	0.648151
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−16.9883	−0.735846
$534$	0	0
$535$	49.7231	2.14972
$536$	0	0
$537$	0	0
$538$	0	0
$539$	19.2682	0.829941
$540$	0	0
$541$	33.6224	1.44554	0.722770	−	0.691089i	$-0.242869\pi$
$541$	33.6224	1.44554	0.722770	+	0.691089i	$0.242869\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−42.8763	−1.83662
$546$	0	0
$547$	29.8749	1.27736	0.638679	−	0.769474i	$-0.279481\pi$
$547$	29.8749	1.27736	0.638679	+	0.769474i	$0.279481\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.58833	0.195469
$552$	0	0
$553$	1.93072	0.0821027
$554$	0	0
$555$	0	0
$556$	0	0
$557$	32.9962	1.39809	0.699046	−	0.715076i	$-0.253608\pi$
$557$	32.9962	1.39809	0.699046	+	0.715076i	$0.253608\pi$
$558$	0	0
$559$	−25.1780	−1.06491
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5.94783	−0.250671	−0.125336	−	0.992114i	$-0.540001\pi$
$563$	−5.94783	−0.250671	−0.125336	+	0.992114i	$0.540001\pi$
$564$	0	0
$565$	−49.7668	−2.09371
$566$	0	0
$567$	0	0
$568$	0	0
$569$	25.6236	1.07420	0.537099	−	0.843519i	$-0.319520\pi$
$569$	25.6236	1.07420	0.537099	+	0.843519i	$0.319520\pi$
$570$	0	0
$571$	−2.31484	−0.0968731	−0.0484365	−	0.998826i	$-0.515424\pi$
$571$	−2.31484	−0.0968731	−0.0484365	+	0.998826i	$0.515424\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−10.0447	−0.418891
$576$	0	0
$577$	22.1112	0.920500	0.460250	−	0.887789i	$-0.347760\pi$
$577$	22.1112	0.920500	0.460250	+	0.887789i	$0.347760\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−2.59337	−0.107591
$582$	0	0
$583$	28.8247	1.19380
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−21.0829	−0.870185	−0.435092	−	0.900386i	$-0.643284\pi$
$587$	−21.0829	−0.870185	−0.435092	+	0.900386i	$0.643284\pi$
$588$	0	0
$589$	−12.3494	−0.508849
$590$	0	0
$591$	0	0
$592$	0	0
$593$	16.7042	0.685959	0.342979	−	0.939343i	$-0.388564\pi$
$593$	16.7042	0.685959	0.342979	+	0.939343i	$0.388564\pi$
$594$	0	0
$595$	−2.54521	−0.104344
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−38.6175	−1.57787	−0.788935	−	0.614477i	$-0.789367\pi$
$599$	−38.6175	−1.57787	−0.788935	+	0.614477i	$0.789367\pi$
$600$	0	0
$601$	14.6915	0.599280	0.299640	−	0.954052i	$-0.403133\pi$
$601$	14.6915	0.599280	0.299640	+	0.954052i	$0.403133\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	8.62453	0.350637
$606$	0	0
$607$	7.33979	0.297913	0.148956	−	0.988844i	$-0.452409\pi$
$607$	7.33979	0.297913	0.148956	+	0.988844i	$0.452409\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.52163	−0.102014
$612$	0	0
$613$	21.9767	0.887632	0.443816	−	0.896118i	$-0.353624\pi$
$613$	21.9767	0.887632	0.443816	+	0.896118i	$0.353624\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−41.4522	−1.66880	−0.834401	−	0.551158i	$-0.814186\pi$
$617$	−41.4522	−1.66880	−0.834401	+	0.551158i	$0.814186\pi$
$618$	0	0
$619$	−32.2999	−1.29824	−0.649121	−	0.760685i	$-0.724863\pi$
$619$	−32.2999	−1.29824	−0.649121	+	0.760685i	$0.724863\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−0.324223	−0.0129897
$624$	0	0
$625$	−30.4956	−1.21982
$626$	0	0
$627$	0	0
$628$	0	0
$629$	57.4147	2.28927
$630$	0	0
$631$	14.4804	0.576455	0.288227	−	0.957562i	$-0.406934\pi$
$631$	14.4804	0.576455	0.288227	+	0.957562i	$0.406934\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	7.03237	0.279071
$636$	0	0
$637$	−16.3088	−0.646177
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−22.2828	−0.880116	−0.440058	−	0.897969i	$-0.645042\pi$
$641$	−22.2828	−0.880116	−0.440058	+	0.897969i	$0.645042\pi$
$642$	0	0
$643$	−14.6647	−0.578318	−0.289159	−	0.957281i	$-0.593376\pi$
$643$	−14.6647	−0.578318	−0.289159	+	0.957281i	$0.593376\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21.8063	0.857296	0.428648	−	0.903472i	$-0.358990\pi$
$647$	21.8063	0.857296	0.428648	+	0.903472i	$0.358990\pi$
$648$	0	0
$649$	−2.76602	−0.108576
$650$	0	0
$651$	0	0
$652$	0	0
$653$	40.4121	1.58145	0.790725	−	0.612172i	$-0.209704\pi$
$653$	40.4121	1.58145	0.790725	+	0.612172i	$0.209704\pi$
$654$	0	0
$655$	33.1711	1.29610
$656$	0	0
$657$	0	0
$658$	0	0
$659$	48.1416	1.87533	0.937665	−	0.347541i	$-0.112983\pi$
$659$	48.1416	1.87533	0.937665	+	0.347541i	$0.112983\pi$
$660$	0	0
$661$	1.46549	0.0570008	0.0285004	−	0.999594i	$-0.490927\pi$
$661$	1.46549	0.0570008	0.0285004	+	0.999594i	$0.490927\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.88205	−0.0729830
$666$	0	0
$667$	−7.12295	−0.275802
$668$	0	0
$669$	0	0
$670$	0	0
$671$	31.8475	1.22946
$672$	0	0
$673$	40.3640	1.55592	0.777958	−	0.628316i	$-0.216255\pi$
$673$	40.3640	1.55592	0.777958	+	0.628316i	$0.216255\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21.2944	0.818410	0.409205	−	0.912443i	$-0.365806\pi$
$677$	21.2944	0.818410	0.409205	+	0.912443i	$0.365806\pi$
$678$	0	0
$679$	−2.04727	−0.0785672
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−21.3988	−0.818802	−0.409401	−	0.912355i	$-0.634262\pi$
$683$	−21.3988	−0.818802	−0.409401	+	0.912355i	$0.634262\pi$
$684$	0	0
$685$	16.6761	0.637161
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−24.3975	−0.929469
$690$	0	0
$691$	36.0654	1.37199	0.685996	−	0.727605i	$-0.259367\pi$
$691$	36.0654	1.37199	0.685996	+	0.727605i	$0.259367\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−8.93054	−0.338755
$696$	0	0
$697$	−38.9193	−1.47417
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−18.8865	−0.713333	−0.356667	−	0.934232i	$-0.616087\pi$
$701$	−18.8865	−0.713333	−0.356667	+	0.934232i	$0.616087\pi$
$702$	0	0
$703$	42.4553	1.60123
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0.496724	0.0186812
$708$	0	0
$709$	1.21823	0.0457518	0.0228759	−	0.999738i	$-0.492718\pi$
$709$	1.21823	0.0457518	0.0228759	+	0.999738i	$0.492718\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	19.1713	0.717970
$714$	0	0
$715$	16.6761	0.623651
$716$	0	0
$717$	0	0
$718$	0	0
$719$	19.5500	0.729093	0.364547	−	0.931185i	$-0.381224\pi$
$719$	19.5500	0.729093	0.364547	+	0.931185i	$0.381224\pi$
$720$	0	0
$721$	−1.01943	−0.0379655
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.88743	0.0700973
$726$	0	0
$727$	22.6986	0.841845	0.420922	−	0.907097i	$-0.361706\pi$
$727$	22.6986	0.841845	0.420922	+	0.907097i	$0.361706\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−57.6813	−2.13342
$732$	0	0
$733$	−16.7617	−0.619107	−0.309553	−	0.950882i	$-0.600180\pi$
$733$	−16.7617	−0.619107	−0.309553	+	0.950882i	$0.600180\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16.5204	−0.608537
$738$	0	0
$739$	−39.1078	−1.43860	−0.719302	−	0.694697i	$-0.755538\pi$
$739$	−39.1078	−1.43860	−0.719302	+	0.694697i	$0.755538\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	29.6693	1.08846	0.544230	−	0.838936i	$-0.316822\pi$
$743$	29.6693	1.08846	0.544230	+	0.838936i	$0.316822\pi$
$744$	0	0
$745$	10.0116	0.366796
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.55816	−0.130012
$750$	0	0
$751$	6.59885	0.240795	0.120398	−	0.992726i	$-0.461583\pi$
$751$	6.59885	0.240795	0.120398	+	0.992726i	$0.461583\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.36697	−0.0497491
$756$	0	0
$757$	3.11944	0.113378	0.0566890	−	0.998392i	$-0.481946\pi$
$757$	3.11944	0.113378	0.0566890	+	0.998392i	$0.481946\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−39.3569	−1.42669	−0.713344	−	0.700814i	$-0.752820\pi$
$761$	−39.3569	−1.42669	−0.713344	+	0.700814i	$0.752820\pi$
$762$	0	0
$763$	3.06820	0.111076
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.34118	0.0845351
$768$	0	0
$769$	−1.61154	−0.0581138	−0.0290569	−	0.999578i	$-0.509250\pi$
$769$	−1.61154	−0.0581138	−0.0290569	+	0.999578i	$0.509250\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	25.2102	0.906748	0.453374	−	0.891320i	$-0.350220\pi$
$773$	25.2102	0.906748	0.453374	+	0.891320i	$0.350220\pi$
$774$	0	0
$775$	−5.07998	−0.182478
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−28.7788	−1.03111
$780$	0	0
$781$	0.722904	0.0258676
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−31.8616	−1.13719
$786$	0	0
$787$	−18.1860	−0.648261	−0.324130	−	0.946012i	$-0.605072\pi$
$787$	−18.1860	−0.648261	−0.324130	+	0.946012i	$0.605072\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	3.56128	0.126625
$792$	0	0
$793$	−26.9560	−0.957234
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−6.49636	−0.230113	−0.115057	−	0.993359i	$-0.536705\pi$
$797$	−6.49636	−0.230113	−0.115057	+	0.993359i	$0.536705\pi$
$798$	0	0
$799$	−5.77690	−0.204372
$800$	0	0
$801$	0	0
$802$	0	0
$803$	35.0657	1.23744
$804$	0	0
$805$	2.92171	0.102977
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−26.5621	−0.933873	−0.466937	−	0.884291i	$-0.654642\pi$
$809$	−26.5621	−0.933873	−0.466937	+	0.884291i	$0.654642\pi$
$810$	0	0
$811$	−3.45862	−0.121449	−0.0607243	−	0.998155i	$-0.519341\pi$
$811$	−3.45862	−0.121449	−0.0607243	+	0.998155i	$0.519341\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	8.62103	0.301982
$816$	0	0
$817$	−42.6524	−1.49222
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.53429	0.193148	0.0965740	−	0.995326i	$-0.469212\pi$
$821$	5.53429	0.193148	0.0965740	+	0.995326i	$0.469212\pi$
$822$	0	0
$823$	−56.8700	−1.98236	−0.991181	−	0.132514i	$-0.957695\pi$
$823$	−56.8700	−1.98236	−0.991181	+	0.132514i	$0.957695\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−25.9274	−0.901583	−0.450791	−	0.892629i	$-0.648858\pi$
$827$	−25.9274	−0.901583	−0.450791	+	0.892629i	$0.648858\pi$
$828$	0	0
$829$	−31.2670	−1.08595	−0.542975	−	0.839749i	$-0.682702\pi$
$829$	−31.2670	−1.08595	−0.542975	+	0.839749i	$0.682702\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−37.3624	−1.29453
$834$	0	0
$835$	7.06545	0.244510
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−48.0396	−1.65851	−0.829255	−	0.558870i	$-0.811235\pi$
$839$	−48.0396	−1.65851	−0.829255	+	0.558870i	$0.811235\pi$
$840$	0	0
$841$	−27.6616	−0.953847
$842$	0	0
$843$	0	0
$844$	0	0
$845$	19.3623	0.666083
$846$	0	0
$847$	−0.617167	−0.0212061
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−65.9077	−2.25929
$852$	0	0
$853$	44.1288	1.51094	0.755470	−	0.655183i	$-0.227408\pi$
$853$	44.1288	1.51094	0.755470	+	0.655183i	$0.227408\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.15597	0.107806	0.0539030	−	0.998546i	$-0.482834\pi$
$857$	3.15597	0.107806	0.0539030	+	0.998546i	$0.482834\pi$
$858$	0	0
$859$	41.3546	1.41100	0.705501	−	0.708709i	$-0.250722\pi$
$859$	41.3546	1.41100	0.705501	+	0.708709i	$0.250722\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−6.65363	−0.226492	−0.113246	−	0.993567i	$-0.536125\pi$
$863$	−6.65363	−0.226492	−0.113246	+	0.993567i	$0.536125\pi$
$864$	0	0
$865$	−7.95787	−0.270576
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−28.9804	−0.983094
$870$	0	0
$871$	13.9830	0.473796
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.59852	0.0540399
$876$	0	0
$877$	−5.04541	−0.170371	−0.0851856	−	0.996365i	$-0.527148\pi$
$877$	−5.04541	−0.170371	−0.0851856	+	0.996365i	$0.527148\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−26.5150	−0.893314	−0.446657	−	0.894705i	$-0.647386\pi$
$881$	−26.5150	−0.893314	−0.446657	+	0.894705i	$0.647386\pi$
$882$	0	0
$883$	−25.0221	−0.842062	−0.421031	−	0.907046i	$-0.638332\pi$
$883$	−25.0221	−0.842062	−0.421031	+	0.907046i	$0.638332\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	13.9194	0.467368	0.233684	−	0.972313i	$-0.424922\pi$
$887$	13.9194	0.467368	0.233684	+	0.972313i	$0.424922\pi$
$888$	0	0
$889$	−0.503233	−0.0168779
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−4.27172	−0.142948
$894$	0	0
$895$	32.0065	1.06986
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3.60235	−0.120145
$900$	0	0
$901$	−55.8932	−1.86207
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.70969	0.156555
$906$	0	0
$907$	27.2175	0.903742	0.451871	−	0.892083i	$-0.350757\pi$
$907$	27.2175	0.903742	0.451871	+	0.892083i	$0.350757\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−17.7790	−0.589045	−0.294522	−	0.955645i	$-0.595161\pi$
$911$	−17.7790	−0.589045	−0.294522	+	0.955645i	$0.595161\pi$
$912$	0	0
$913$	38.9269	1.28829
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−2.37370	−0.0783865
$918$	0	0
$919$	−29.8905	−0.985997	−0.492998	−	0.870030i	$-0.664099\pi$
$919$	−29.8905	−0.985997	−0.492998	+	0.870030i	$0.664099\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−0.611872	−0.0201400
$924$	0	0
$925$	17.4641	0.574217
$926$	0	0
$927$	0	0
$928$	0	0
$929$	42.2408	1.38588	0.692938	−	0.720997i	$-0.256316\pi$
$929$	42.2408	1.38588	0.692938	+	0.720997i	$0.256316\pi$
$930$	0	0
$931$	−27.6276	−0.905459
$932$	0	0
$933$	0	0
$934$	0	0
$935$	38.2040	1.24940
$936$	0	0
$937$	−17.8084	−0.581775	−0.290887	−	0.956757i	$-0.593950\pi$
$937$	−17.8084	−0.581775	−0.290887	+	0.956757i	$0.593950\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	27.3312	0.890973	0.445486	−	0.895289i	$-0.353031\pi$
$941$	27.3312	0.890973	0.445486	+	0.895289i	$0.353031\pi$
$942$	0	0
$943$	44.6764	1.45486
$944$	0	0
$945$	0	0
$946$	0	0
$947$	33.6386	1.09311	0.546554	−	0.837424i	$-0.315939\pi$
$947$	33.6386	1.09311	0.546554	+	0.837424i	$0.315939\pi$
$948$	0	0
$949$	−29.6799	−0.963450
$950$	0	0
$951$	0	0
$952$	0	0
$953$	5.87734	0.190386	0.0951928	−	0.995459i	$-0.469653\pi$
$953$	5.87734	0.190386	0.0951928	+	0.995459i	$0.469653\pi$
$954$	0	0
$955$	−17.3976	−0.562973
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.19333	−0.0385347
$960$	0	0
$961$	−21.3043	−0.687236
$962$	0	0
$963$	0	0
$964$	0	0
$965$	30.7609	0.990229
$966$	0	0
$967$	−0.960671	−0.0308931	−0.0154465	−	0.999881i	$-0.504917\pi$
$967$	−0.960671	−0.0308931	−0.0154465	+	0.999881i	$0.504917\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−35.1567	−1.12823	−0.564116	−	0.825696i	$-0.690783\pi$
$971$	−35.1567	−1.12823	−0.564116	+	0.825696i	$0.690783\pi$
$972$	0	0
$973$	0.639064	0.0204875
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−50.1598	−1.60475	−0.802377	−	0.596817i	$-0.796432\pi$
$977$	−50.1598	−1.60475	−0.802377	+	0.596817i	$0.796432\pi$
$978$	0	0
$979$	4.86664	0.155538
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−23.9533	−0.763993	−0.381996	−	0.924164i	$-0.624763\pi$
$983$	−23.9533	−0.763993	−0.381996	+	0.924164i	$0.624763\pi$
$984$	0	0
$985$	−8.28747	−0.264061
$986$	0	0
$987$	0	0
$988$	0	0
$989$	66.2138	2.10548
$990$	0	0
$991$	−18.6625	−0.592833	−0.296416	−	0.955059i	$-0.595792\pi$
$991$	−18.6625	−0.592833	−0.296416	+	0.955059i	$0.595792\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	31.7876	1.00773
$996$	0	0
$997$	34.9006	1.10531	0.552656	−	0.833410i	$-0.313614\pi$
$997$	34.9006	1.10531	0.552656	+	0.833410i	$0.313614\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2124.2.a.i.1.1	✓	4
3.2	odd	2		2124.2.a.j.1.4	yes	4
4.3	odd	2		8496.2.a.bs.1.1		4
12.11	even	2		8496.2.a.bp.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2124.2.a.i.1.1	✓	4	1.1	even	1	trivial
2124.2.a.j.1.4	yes	4	3.2	odd	2
8496.2.a.bp.1.4		4	12.11	even	2
8496.2.a.bs.1.1		4	4.3	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:	\( N \)	\(=\)	\( 2124 = 2^{2} \cdot 3^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2124.a (trivial)

\(f(q)\)	\(=\)	\(q-2.57516 q^{5} +0.184277 q^{7} +O(q^{10})\)	\(q-2.57516 q^{5} +0.184277 q^{7} -2.76602 q^{11} +2.34118 q^{13} +5.36351 q^{17} +3.96604 q^{19} -6.15690 q^{23} +1.63145 q^{25} +1.15690 q^{29} -3.11379 q^{31} -0.474542 q^{35} +10.7047 q^{37} -7.25631 q^{41} -10.7544 q^{43} -1.07707 q^{47} -6.96604 q^{49} -10.4210 q^{53} +7.12295 q^{55} +1.00000 q^{59} -11.5138 q^{61} -6.02891 q^{65} +5.97263 q^{67} -0.261352 q^{71} -12.6773 q^{73} -0.509714 q^{77} +10.4773 q^{79} -14.0732 q^{83} -13.8119 q^{85} -1.75944 q^{89} +0.431425 q^{91} -10.2132 q^{95} -11.1098 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{7}+O(q^{10}) \) 4 * q + 2 * q^7	\( 4 q + 2 q^{7} - 10 q^{11} - 2 q^{13} - 4 q^{17} - 6 q^{19} - 12 q^{23} + 4 q^{25} - 8 q^{29} - 8 q^{31} - 12 q^{35} + 6 q^{37} - 16 q^{41} - 6 q^{43} - 18 q^{47} - 6 q^{49} - 4 q^{53} - 6 q^{55} + 4 q^{59} - 18 q^{65} + 10 q^{67} - 16 q^{71} - 14 q^{77} + 12 q^{79} - 22 q^{83} - 6 q^{85} + 2 q^{89} + 20 q^{91} - 36 q^{95} + 6 q^{97}+O(q^{100}) \) 4 * q + 2 * q^7 - 10 * q^11 - 2 * q^13 - 4 * q^17 - 6 * q^19 - 12 * q^23 + 4 * q^25 - 8 * q^29 - 8 * q^31 - 12 * q^35 + 6 * q^37 - 16 * q^41 - 6 * q^43 - 18 * q^47 - 6 * q^49 - 4 * q^53 - 6 * q^55 + 4 * q^59 - 18 * q^65 + 10 * q^67 - 16 * q^71 - 14 * q^77 + 12 * q^79 - 22 * q^83 - 6 * q^85 + 2 * q^89 + 20 * q^91 - 36 * q^95 + 6 * q^97

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