LMFDB - Embedded Newform 4004.2.a.j.1.7

Newspace parameters

Level:	$ N $	=	$ 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	=	$ 2 $
Character orbit:	$[\chi]$	=	4004.a (trivial)

Newform invariants

Self dual:	Yes
Analytic conductor:	$31.9721009693$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.497051$
Character	$\chi$	=	4004.1

$q$-expansion

$f(q)$	$=$	$q+0.497051 q^{3} -2.93955 q^{5} -1.00000 q^{7} -2.75294 q^{9} +O(q^{10})$	$q+0.497051 q^{3} -2.93955 q^{5} -1.00000 q^{7} -2.75294 q^{9} -1.00000 q^{11} -1.00000 q^{13} -1.46111 q^{15} +6.64567 q^{17} -8.43503 q^{19} -0.497051 q^{21} -7.56554 q^{23} +3.64094 q^{25} -2.85951 q^{27} -0.0222906 q^{29} +1.14381 q^{31} -0.497051 q^{33} +2.93955 q^{35} -0.840405 q^{37} -0.497051 q^{39} +4.48124 q^{41} +1.41468 q^{43} +8.09240 q^{45} +6.97254 q^{47} +1.00000 q^{49} +3.30324 q^{51} +11.8143 q^{53} +2.93955 q^{55} -4.19264 q^{57} -0.878482 q^{59} +1.01213 q^{61} +2.75294 q^{63} +2.93955 q^{65} +8.34149 q^{67} -3.76046 q^{69} -13.3351 q^{71} -8.97242 q^{73} +1.80973 q^{75} +1.00000 q^{77} -12.6865 q^{79} +6.83750 q^{81} +2.63360 q^{83} -19.5353 q^{85} -0.0110796 q^{87} -14.9420 q^{89} +1.00000 q^{91} +0.568531 q^{93} +24.7952 q^{95} +19.2853 q^{97} +2.75294 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10q - 2q^{3} - 2q^{5} - 10q^{7} + 10q^{9} + O(q^{10}) $	$ 10q - 2q^{3} - 2q^{5} - 10q^{7} + 10q^{9} - 10q^{11} - 10q^{13} - 4q^{15} + 9q^{17} + 3q^{19} + 2q^{21} - 4q^{23} + 22q^{25} - 8q^{27} + 13q^{29} - 17q^{31} + 2q^{33} + 2q^{35} + 11q^{37} + 2q^{39} + 8q^{41} + 21q^{43} - 15q^{45} + q^{47} + 10q^{49} + 19q^{51} + 22q^{53} + 2q^{55} + 8q^{57} - 24q^{59} + 16q^{61} - 10q^{63} + 2q^{65} + 19q^{67} + 51q^{69} - 25q^{71} + 22q^{73} + 28q^{75} + 10q^{77} + 41q^{79} + 54q^{81} + 8q^{83} + 9q^{85} - 5q^{87} + 36q^{89} + 10q^{91} + 12q^{93} + 13q^{95} - 5q^{97} - 10q^{99} + O(q^{100}) $

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
$3$	0.497051	0.286973	0.143486	−	0.989652i	$-0.454169\pi$
$3$	0.497051	0.286973	0.143486	+	0.989652i	$0.454169\pi$
$4$	0	0
$5$	−2.93955	−1.31461	−0.657303	−	0.753626i	$-0.728303\pi$
$5$	−2.93955	−1.31461	−0.657303	+	0.753626i	$0.728303\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.75294	−0.917647
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.46111	−0.377256
$16$	0	0
$17$	6.64567	1.61181	0.805905	−	0.592044i	$-0.201679\pi$
$17$	6.64567	1.61181	0.805905	+	0.592044i	$0.201679\pi$
$18$	0	0
$19$	−8.43503	−1.93513	−0.967564	−	0.252625i	$-0.918706\pi$
$19$	−8.43503	−1.93513	−0.967564	+	0.252625i	$0.918706\pi$
$20$	0	0
$21$	−0.497051	−0.108465
$22$	0	0
$23$	−7.56554	−1.57752	−0.788762	−	0.614699i	$-0.789278\pi$
$23$	−7.56554	−1.57752	−0.788762	+	0.614699i	$0.789278\pi$
$24$	0	0
$25$	3.64094	0.728189
$26$	0	0
$27$	−2.85951	−0.550312
$28$	0	0
$29$	−0.0222906	−0.00413926	−0.00206963	−	0.999998i	$-0.500659\pi$
$29$	−0.0222906	−0.00413926	−0.00206963	+	0.999998i	$0.500659\pi$
$30$	0	0
$31$	1.14381	0.205434	0.102717	−	0.994711i	$-0.467246\pi$
$31$	1.14381	0.205434	0.102717	+	0.994711i	$0.467246\pi$
$32$	0	0
$33$	−0.497051	−0.0865255
$34$	0	0
$35$	2.93955	0.496874
$36$	0	0
$37$	−0.840405	−0.138162	−0.0690809	−	0.997611i	$-0.522007\pi$
$37$	−0.840405	−0.138162	−0.0690809	+	0.997611i	$0.522007\pi$
$38$	0	0
$39$	−0.497051	−0.0795919
$40$	0	0
$41$	4.48124	0.699852	0.349926	−	0.936777i	$-0.386207\pi$
$41$	4.48124	0.699852	0.349926	+	0.936777i	$0.386207\pi$
$42$	0	0
$43$	1.41468	0.215737	0.107869	−	0.994165i	$-0.465597\pi$
$43$	1.41468	0.215737	0.107869	+	0.994165i	$0.465597\pi$
$44$	0	0
$45$	8.09240	1.20634
$46$	0	0
$47$	6.97254	1.01705	0.508524	−	0.861048i	$-0.330191\pi$
$47$	6.97254	1.01705	0.508524	+	0.861048i	$0.330191\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	3.30324	0.462546
$52$	0	0
$53$	11.8143	1.62282	0.811412	−	0.584475i	$-0.198700\pi$
$53$	11.8143	1.62282	0.811412	+	0.584475i	$0.198700\pi$
$54$	0	0
$55$	2.93955	0.396369
$56$	0	0
$57$	−4.19264	−0.555329
$58$	0	0
$59$	−0.878482	−0.114369	−0.0571843	−	0.998364i	$-0.518212\pi$
$59$	−0.878482	−0.114369	−0.0571843	+	0.998364i	$0.518212\pi$
$60$	0	0
$61$	1.01213	0.129590	0.0647950	−	0.997899i	$-0.479361\pi$
$61$	1.01213	0.129590	0.0647950	+	0.997899i	$0.479361\pi$
$62$	0	0
$63$	2.75294	0.346838
$64$	0	0
$65$	2.93955	0.364606
$66$	0	0
$67$	8.34149	1.01907	0.509537	−	0.860449i	$-0.329817\pi$
$67$	8.34149	1.01907	0.509537	+	0.860449i	$0.329817\pi$
$68$	0	0
$69$	−3.76046	−0.452706
$70$	0	0
$71$	−13.3351	−1.58258	−0.791292	−	0.611439i	$-0.790591\pi$
$71$	−13.3351	−1.58258	−0.791292	+	0.611439i	$0.790591\pi$
$72$	0	0
$73$	−8.97242	−1.05014	−0.525071	−	0.851058i	$-0.675961\pi$
$73$	−8.97242	−1.05014	−0.525071	+	0.851058i	$0.675961\pi$
$74$	0	0
$75$	1.80973	0.208970
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−12.6865	−1.42735	−0.713674	−	0.700478i	$-0.752970\pi$
$79$	−12.6865	−1.42735	−0.713674	+	0.700478i	$0.752970\pi$
$80$	0	0
$81$	6.83750	0.759722
$82$	0	0
$83$	2.63360	0.289076	0.144538	−	0.989499i	$-0.453830\pi$
$83$	2.63360	0.289076	0.144538	+	0.989499i	$0.453830\pi$
$84$	0	0
$85$	−19.5353	−2.11890
$86$	0	0
$87$	−0.0110796	−0.00118786
$88$	0	0
$89$	−14.9420	−1.58385	−0.791923	−	0.610621i	$-0.790920\pi$
$89$	−14.9420	−1.58385	−0.791923	+	0.610621i	$0.790920\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	0.568531	0.0589539
$94$	0	0
$95$	24.7952	2.54393
$96$	0	0
$97$	19.2853	1.95813	0.979064	−	0.203551i	$-0.0652483\pi$
$97$	19.2853	1.95813	0.979064	+	0.203551i	$0.0652483\pi$
$98$	0	0
$99$	2.75294	0.276681
$100$	0	0
$101$	−3.05893	−0.304375	−0.152188	−	0.988352i	$-0.548632\pi$
$101$	−3.05893	−0.304375	−0.152188	+	0.988352i	$0.548632\pi$
$102$	0	0
$103$	2.52685	0.248978	0.124489	−	0.992221i	$-0.460271\pi$
$103$	2.52685	0.248978	0.124489	+	0.992221i	$0.460271\pi$
$104$	0	0
$105$	1.46111	0.142589
$106$	0	0
$107$	16.4039	1.58583	0.792913	−	0.609335i	$-0.208564\pi$
$107$	16.4039	1.58583	0.792913	+	0.609335i	$0.208564\pi$
$108$	0	0
$109$	15.6346	1.49752	0.748762	−	0.662839i	$-0.230649\pi$
$109$	15.6346	1.49752	0.748762	+	0.662839i	$0.230649\pi$
$110$	0	0
$111$	−0.417724	−0.0396486
$112$	0	0
$113$	5.09219	0.479033	0.239516	−	0.970892i	$-0.423011\pi$
$113$	5.09219	0.479033	0.239516	+	0.970892i	$0.423011\pi$
$114$	0	0
$115$	22.2393	2.07382
$116$	0	0
$117$	2.75294	0.254509
$118$	0	0
$119$	−6.64567	−0.609207
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	2.22740	0.200838
$124$	0	0
$125$	3.99501	0.357325
$126$	0	0
$127$	7.32853	0.650302	0.325151	−	0.945662i	$-0.394585\pi$
$127$	7.32853	0.650302	0.325151	+	0.945662i	$0.394585\pi$
$128$	0	0
$129$	0.703170	0.0619106
$130$	0	0
$131$	0.588280	0.0513983	0.0256991	−	0.999670i	$-0.491819\pi$
$131$	0.588280	0.0513983	0.0256991	+	0.999670i	$0.491819\pi$
$132$	0	0
$133$	8.43503	0.731410
$134$	0	0
$135$	8.40565	0.723443
$136$	0	0
$137$	−18.3928	−1.57141	−0.785703	−	0.618604i	$-0.787698\pi$
$137$	−18.3928	−1.57141	−0.785703	+	0.618604i	$0.787698\pi$
$138$	0	0
$139$	2.14980	0.182344	0.0911719	−	0.995835i	$-0.470939\pi$
$139$	2.14980	0.182344	0.0911719	+	0.995835i	$0.470939\pi$
$140$	0	0
$141$	3.46571	0.291865
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	0.0655243	0.00544150
$146$	0	0
$147$	0.497051	0.0409961
$148$	0	0
$149$	−6.33684	−0.519134	−0.259567	−	0.965725i	$-0.583580\pi$
$149$	−6.33684	−0.519134	−0.259567	+	0.965725i	$0.583580\pi$
$150$	0	0
$151$	7.78880	0.633844	0.316922	−	0.948452i	$-0.397351\pi$
$151$	7.78880	0.633844	0.316922	+	0.948452i	$0.397351\pi$
$152$	0	0
$153$	−18.2951	−1.47907
$154$	0	0
$155$	−3.36228	−0.270065
$156$	0	0
$157$	9.41916	0.751731	0.375865	−	0.926674i	$-0.377346\pi$
$157$	9.41916	0.751731	0.375865	+	0.926674i	$0.377346\pi$
$158$	0	0
$159$	5.87233	0.465706
$160$	0	0
$161$	7.56554	0.596248
$162$	0	0
$163$	22.0204	1.72477	0.862386	−	0.506251i	$-0.168969\pi$
$163$	22.0204	1.72477	0.862386	+	0.506251i	$0.168969\pi$
$164$	0	0
$165$	1.46111	0.113747
$166$	0	0
$167$	−19.5196	−1.51047	−0.755236	−	0.655453i	$-0.772478\pi$
$167$	−19.5196	−1.51047	−0.755236	+	0.655453i	$0.772478\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	23.2211	1.77576
$172$	0	0
$173$	−6.93854	−0.527527	−0.263764	−	0.964587i	$-0.584964\pi$
$173$	−6.93854	−0.527527	−0.263764	+	0.964587i	$0.584964\pi$
$174$	0	0
$175$	−3.64094	−0.275229
$176$	0	0
$177$	−0.436651	−0.0328207
$178$	0	0
$179$	6.01410	0.449515	0.224757	−	0.974415i	$-0.427841\pi$
$179$	6.01410	0.449515	0.224757	+	0.974415i	$0.427841\pi$
$180$	0	0
$181$	10.4942	0.780029	0.390014	−	0.920809i	$-0.372470\pi$
$181$	10.4942	0.780029	0.390014	+	0.920809i	$0.372470\pi$
$182$	0	0
$183$	0.503081	0.0371888
$184$	0	0
$185$	2.47041	0.181628
$186$	0	0
$187$	−6.64567	−0.485979
$188$	0	0
$189$	2.85951	0.207998
$190$	0	0
$191$	8.16705	0.590947	0.295473	−	0.955351i	$-0.404523\pi$
$191$	8.16705	0.590947	0.295473	+	0.955351i	$0.404523\pi$
$192$	0	0
$193$	−2.47887	−0.178433	−0.0892164	−	0.996012i	$-0.528436\pi$
$193$	−2.47887	−0.178433	−0.0892164	+	0.996012i	$0.528436\pi$
$194$	0	0
$195$	1.46111	0.104632
$196$	0	0
$197$	13.2215	0.941992	0.470996	−	0.882135i	$-0.343895\pi$
$197$	13.2215	0.941992	0.470996	+	0.882135i	$0.343895\pi$
$198$	0	0
$199$	0.643373	0.0456075	0.0228037	−	0.999740i	$-0.492741\pi$
$199$	0.643373	0.0456075	0.0228037	+	0.999740i	$0.492741\pi$
$200$	0	0
$201$	4.14614	0.292447
$202$	0	0
$203$	0.0222906	0.00156449
$204$	0	0
$205$	−13.1728	−0.920030
$206$	0	0
$207$	20.8275	1.44761
$208$	0	0
$209$	8.43503	0.583463
$210$	0	0
$211$	8.75163	0.602487	0.301244	−	0.953547i	$-0.402598\pi$
$211$	8.75163	0.602487	0.301244	+	0.953547i	$0.402598\pi$
$212$	0	0
$213$	−6.62822	−0.454158
$214$	0	0
$215$	−4.15853	−0.283609
$216$	0	0
$217$	−1.14381	−0.0776468
$218$	0	0
$219$	−4.45975	−0.301362
$220$	0	0
$221$	−6.64567	−0.447036
$222$	0	0
$223$	−1.34705	−0.0902048	−0.0451024	−	0.998982i	$-0.514361\pi$
$223$	−1.34705	−0.0902048	−0.0451024	+	0.998982i	$0.514361\pi$
$224$	0	0
$225$	−10.0233	−0.668220
$226$	0	0
$227$	−0.959150	−0.0636610	−0.0318305	−	0.999493i	$-0.510134\pi$
$227$	−0.959150	−0.0636610	−0.0318305	+	0.999493i	$0.510134\pi$
$228$	0	0
$229$	−10.3814	−0.686023	−0.343011	−	0.939331i	$-0.611447\pi$
$229$	−10.3814	−0.686023	−0.343011	+	0.939331i	$0.611447\pi$
$230$	0	0
$231$	0.497051	0.0327036
$232$	0	0
$233$	6.48083	0.424573	0.212287	−	0.977207i	$-0.431909\pi$
$233$	6.48083	0.424573	0.212287	+	0.977207i	$0.431909\pi$
$234$	0	0
$235$	−20.4961	−1.33702
$236$	0	0
$237$	−6.30586	−0.409610
$238$	0	0
$239$	−1.47533	−0.0954314	−0.0477157	−	0.998861i	$-0.515194\pi$
$239$	−1.47533	−0.0954314	−0.0477157	+	0.998861i	$0.515194\pi$
$240$	0	0
$241$	−9.57997	−0.617100	−0.308550	−	0.951208i	$-0.599844\pi$
$241$	−9.57997	−0.617100	−0.308550	+	0.951208i	$0.599844\pi$
$242$	0	0
$243$	11.9771	0.768331
$244$	0	0
$245$	−2.93955	−0.187801
$246$	0	0
$247$	8.43503	0.536708
$248$	0	0
$249$	1.30904	0.0829568
$250$	0	0
$251$	20.8727	1.31747	0.658737	−	0.752374i	$-0.271091\pi$
$251$	20.8727	1.31747	0.658737	+	0.752374i	$0.271091\pi$
$252$	0	0
$253$	7.56554	0.475641
$254$	0	0
$255$	−9.71002	−0.608065
$256$	0	0
$257$	26.6086	1.65980	0.829900	−	0.557912i	$-0.188397\pi$
$257$	26.6086	1.65980	0.829900	+	0.557912i	$0.188397\pi$
$258$	0	0
$259$	0.840405	0.0522202
$260$	0	0
$261$	0.0613647	0.00379838
$262$	0	0
$263$	−5.05996	−0.312011	−0.156005	−	0.987756i	$-0.549862\pi$
$263$	−5.05996	−0.312011	−0.156005	+	0.987756i	$0.549862\pi$
$264$	0	0
$265$	−34.7288	−2.13337
$266$	0	0
$267$	−7.42693	−0.454520
$268$	0	0
$269$	−11.6505	−0.710341	−0.355170	−	0.934802i	$-0.615577\pi$
$269$	−11.6505	−0.710341	−0.355170	+	0.934802i	$0.615577\pi$
$270$	0	0
$271$	−24.5537	−1.49153	−0.745765	−	0.666210i	$-0.767916\pi$
$271$	−24.5537	−1.49153	−0.745765	+	0.666210i	$0.767916\pi$
$272$	0	0
$273$	0.497051	0.0300829
$274$	0	0
$275$	−3.64094	−0.219557
$276$	0	0
$277$	8.80267	0.528901	0.264451	−	0.964399i	$-0.414809\pi$
$277$	8.80267	0.528901	0.264451	+	0.964399i	$0.414809\pi$
$278$	0	0
$279$	−3.14884	−0.188516
$280$	0	0
$281$	2.58544	0.154234	0.0771172	−	0.997022i	$-0.475428\pi$
$281$	2.58544	0.154234	0.0771172	+	0.997022i	$0.475428\pi$
$282$	0	0
$283$	−15.0212	−0.892920	−0.446460	−	0.894804i	$-0.647315\pi$
$283$	−15.0212	−0.892920	−0.446460	+	0.894804i	$0.647315\pi$
$284$	0	0
$285$	12.3245	0.730039
$286$	0	0
$287$	−4.48124	−0.264519
$288$	0	0
$289$	27.1649	1.59793
$290$	0	0
$291$	9.58579	0.561929
$292$	0	0
$293$	5.57879	0.325917	0.162958	−	0.986633i	$-0.447896\pi$
$293$	5.57879	0.325917	0.162958	+	0.986633i	$0.447896\pi$
$294$	0	0
$295$	2.58234	0.150350
$296$	0	0
$297$	2.85951	0.165925
$298$	0	0
$299$	7.56554	0.437526
$300$	0	0
$301$	−1.41468	−0.0815410
$302$	0	0
$303$	−1.52045	−0.0873474
$304$	0	0
$305$	−2.97521	−0.170360
$306$	0	0
$307$	30.8736	1.76205	0.881024	−	0.473072i	$-0.156855\pi$
$307$	30.8736	1.76205	0.881024	+	0.473072i	$0.156855\pi$
$308$	0	0
$309$	1.25597	0.0714498
$310$	0	0
$311$	−17.5943	−0.997683	−0.498842	−	0.866693i	$-0.666241\pi$
$311$	−17.5943	−0.997683	−0.498842	+	0.866693i	$0.666241\pi$
$312$	0	0
$313$	5.67492	0.320765	0.160383	−	0.987055i	$-0.448727\pi$
$313$	5.67492	0.320765	0.160383	+	0.987055i	$0.448727\pi$
$314$	0	0
$315$	−8.09240	−0.455955
$316$	0	0
$317$	−28.7277	−1.61351	−0.806754	−	0.590888i	$-0.798778\pi$
$317$	−28.7277	−1.61351	−0.806754	+	0.590888i	$0.798778\pi$
$318$	0	0
$319$	0.0222906	0.00124803
$320$	0	0
$321$	8.15358	0.455089
$322$	0	0
$323$	−56.0564	−3.11906
$324$	0	0
$325$	−3.64094	−0.201963
$326$	0	0
$327$	7.77120	0.429748
$328$	0	0
$329$	−6.97254	−0.384408
$330$	0	0
$331$	22.6487	1.24488	0.622442	−	0.782666i	$-0.286141\pi$
$331$	22.6487	1.24488	0.622442	+	0.782666i	$0.286141\pi$
$332$	0	0
$333$	2.31359	0.126784
$334$	0	0
$335$	−24.5202	−1.33968
$336$	0	0
$337$	3.33158	0.181483	0.0907414	−	0.995874i	$-0.471076\pi$
$337$	3.33158	0.181483	0.0907414	+	0.995874i	$0.471076\pi$
$338$	0	0
$339$	2.53108	0.137469
$340$	0	0
$341$	−1.14381	−0.0619407
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	11.0541	0.595130
$346$	0	0
$347$	−12.8753	−0.691183	−0.345591	−	0.938385i	$-0.612322\pi$
$347$	−12.8753	−0.691183	−0.345591	+	0.938385i	$0.612322\pi$
$348$	0	0
$349$	7.23667	0.387370	0.193685	−	0.981064i	$-0.437956\pi$
$349$	7.23667	0.387370	0.193685	+	0.981064i	$0.437956\pi$
$350$	0	0
$351$	2.85951	0.152629
$352$	0	0
$353$	−31.3422	−1.66817	−0.834087	−	0.551632i	$-0.814005\pi$
$353$	−31.3422	−1.66817	−0.834087	+	0.551632i	$0.814005\pi$
$354$	0	0
$355$	39.1991	2.08047
$356$	0	0
$357$	−3.30324	−0.174826
$358$	0	0
$359$	−8.63676	−0.455831	−0.227915	−	0.973681i	$-0.573191\pi$
$359$	−8.63676	−0.455831	−0.227915	+	0.973681i	$0.573191\pi$
$360$	0	0
$361$	52.1497	2.74472
$362$	0	0
$363$	0.497051	0.0260884
$364$	0	0
$365$	26.3748	1.38052
$366$	0	0
$367$	−23.7636	−1.24045	−0.620225	−	0.784424i	$-0.712959\pi$
$367$	−23.7636	−1.24045	−0.620225	+	0.784424i	$0.712959\pi$
$368$	0	0
$369$	−12.3366	−0.642217
$370$	0	0
$371$	−11.8143	−0.613369
$372$	0	0
$373$	30.2981	1.56878	0.784389	−	0.620269i	$-0.212977\pi$
$373$	30.2981	1.56878	0.784389	+	0.620269i	$0.212977\pi$
$374$	0	0
$375$	1.98573	0.102542
$376$	0	0
$377$	0.0222906	0.00114803
$378$	0	0
$379$	−22.8240	−1.17239	−0.586196	−	0.810170i	$-0.699375\pi$
$379$	−22.8240	−1.17239	−0.586196	+	0.810170i	$0.699375\pi$
$380$	0	0
$381$	3.64265	0.186619
$382$	0	0
$383$	−9.44337	−0.482534	−0.241267	−	0.970459i	$-0.577563\pi$
$383$	−9.44337	−0.482534	−0.241267	+	0.970459i	$0.577563\pi$
$384$	0	0
$385$	−2.93955	−0.149813
$386$	0	0
$387$	−3.89454	−0.197970
$388$	0	0
$389$	−0.167140	−0.00847433	−0.00423716	−	0.999991i	$-0.501349\pi$
$389$	−0.167140	−0.00847433	−0.00423716	+	0.999991i	$0.501349\pi$
$390$	0	0
$391$	−50.2781	−2.54267
$392$	0	0
$393$	0.292405	0.0147499
$394$	0	0
$395$	37.2927	1.87640
$396$	0	0
$397$	−6.43898	−0.323163	−0.161581	−	0.986859i	$-0.551659\pi$
$397$	−6.43898	−0.323163	−0.161581	+	0.986859i	$0.551659\pi$
$398$	0	0
$399$	4.19264	0.209895
$400$	0	0
$401$	22.4519	1.12119	0.560597	−	0.828089i	$-0.310572\pi$
$401$	22.4519	1.12119	0.560597	+	0.828089i	$0.310572\pi$
$402$	0	0
$403$	−1.14381	−0.0569771
$404$	0	0
$405$	−20.0992	−0.998735
$406$	0	0
$407$	0.840405	0.0416573
$408$	0	0
$409$	31.2869	1.54704	0.773518	−	0.633774i	$-0.218495\pi$
$409$	31.2869	1.54704	0.773518	+	0.633774i	$0.218495\pi$
$410$	0	0
$411$	−9.14218	−0.450950
$412$	0	0
$413$	0.878482	0.0432273
$414$	0	0
$415$	−7.74161	−0.380021
$416$	0	0
$417$	1.06856	0.0523277
$418$	0	0
$419$	5.99916	0.293078	0.146539	−	0.989205i	$-0.453187\pi$
$419$	5.99916	0.293078	0.146539	+	0.989205i	$0.453187\pi$
$420$	0	0
$421$	10.4215	0.507911	0.253956	−	0.967216i	$-0.418268\pi$
$421$	10.4215	0.507911	0.253956	+	0.967216i	$0.418268\pi$
$422$	0	0
$423$	−19.1950	−0.933292
$424$	0	0
$425$	24.1965	1.17370
$426$	0	0
$427$	−1.01213	−0.0489804
$428$	0	0
$429$	0.497051	0.0239979
$430$	0	0
$431$	−18.2146	−0.877367	−0.438684	−	0.898642i	$-0.644555\pi$
$431$	−18.2146	−0.877367	−0.438684	+	0.898642i	$0.644555\pi$
$432$	0	0
$433$	−4.08666	−0.196392	−0.0981962	−	0.995167i	$-0.531307\pi$
$433$	−4.08666	−0.196392	−0.0981962	+	0.995167i	$0.531307\pi$
$434$	0	0
$435$	0.0325689	0.00156156
$436$	0	0
$437$	63.8156	3.05271
$438$	0	0
$439$	−0.274007	−0.0130777	−0.00653883	−	0.999979i	$-0.502081\pi$
$439$	−0.274007	−0.0130777	−0.00653883	+	0.999979i	$0.502081\pi$
$440$	0	0
$441$	−2.75294	−0.131092
$442$	0	0
$443$	21.2676	1.01045	0.505226	−	0.862987i	$-0.331409\pi$
$443$	21.2676	1.01045	0.505226	+	0.862987i	$0.331409\pi$
$444$	0	0
$445$	43.9227	2.08213
$446$	0	0
$447$	−3.14973	−0.148977
$448$	0	0
$449$	−12.2838	−0.579710	−0.289855	−	0.957071i	$-0.593607\pi$
$449$	−12.2838	−0.579710	−0.289855	+	0.957071i	$0.593607\pi$
$450$	0	0
$451$	−4.48124	−0.211013
$452$	0	0
$453$	3.87143	0.181896
$454$	0	0
$455$	−2.93955	−0.137808
$456$	0	0
$457$	−11.7234	−0.548399	−0.274199	−	0.961673i	$-0.588413\pi$
$457$	−11.7234	−0.548399	−0.274199	+	0.961673i	$0.588413\pi$
$458$	0	0
$459$	−19.0033	−0.886999
$460$	0	0
$461$	3.19309	0.148717	0.0743585	−	0.997232i	$-0.476309\pi$
$461$	3.19309	0.148717	0.0743585	+	0.997232i	$0.476309\pi$
$462$	0	0
$463$	19.9860	0.928830	0.464415	−	0.885618i	$-0.346265\pi$
$463$	19.9860	0.928830	0.464415	+	0.885618i	$0.346265\pi$
$464$	0	0
$465$	−1.67122	−0.0775012
$466$	0	0
$467$	−38.4088	−1.77735	−0.888673	−	0.458541i	$-0.848372\pi$
$467$	−38.4088	−1.77735	−0.888673	+	0.458541i	$0.848372\pi$
$468$	0	0
$469$	−8.34149	−0.385174
$470$	0	0
$471$	4.68180	0.215726
$472$	0	0
$473$	−1.41468	−0.0650472
$474$	0	0
$475$	−30.7115	−1.40914
$476$	0	0
$477$	−32.5241	−1.48918
$478$	0	0
$479$	−40.6119	−1.85561	−0.927803	−	0.373070i	$-0.878305\pi$
$479$	−40.6119	−1.85561	−0.927803	+	0.373070i	$0.878305\pi$
$480$	0	0
$481$	0.840405	0.0383192
$482$	0	0
$483$	3.76046	0.171107
$484$	0	0
$485$	−56.6902	−2.57417
$486$	0	0
$487$	29.8727	1.35366	0.676831	−	0.736139i	$-0.263353\pi$
$487$	29.8727	1.35366	0.676831	+	0.736139i	$0.263353\pi$
$488$	0	0
$489$	10.9453	0.494962
$490$	0	0
$491$	−5.02149	−0.226617	−0.113308	−	0.993560i	$-0.536145\pi$
$491$	−5.02149	−0.226617	−0.113308	+	0.993560i	$0.536145\pi$
$492$	0	0
$493$	−0.148136	−0.00667171
$494$	0	0
$495$	−8.09240	−0.363726
$496$	0	0
$497$	13.3351	0.598160
$498$	0	0
$499$	−31.2389	−1.39844	−0.699222	−	0.714905i	$-0.746470\pi$
$499$	−31.2389	−1.39844	−0.699222	+	0.714905i	$0.746470\pi$
$500$	0	0
$501$	−9.70223	−0.433464
$502$	0	0
$503$	19.6425	0.875818	0.437909	−	0.899019i	$-0.355719\pi$
$503$	19.6425	0.875818	0.437909	+	0.899019i	$0.355719\pi$
$504$	0	0
$505$	8.99188	0.400133
$506$	0	0
$507$	0.497051	0.0220748
$508$	0	0
$509$	−25.1361	−1.11414	−0.557068	−	0.830467i	$-0.688074\pi$
$509$	−25.1361	−1.11414	−0.557068	+	0.830467i	$0.688074\pi$
$510$	0	0
$511$	8.97242	0.396916
$512$	0	0
$513$	24.1200	1.06492
$514$	0	0
$515$	−7.42779	−0.327308
$516$	0	0
$517$	−6.97254	−0.306652
$518$	0	0
$519$	−3.44881	−0.151386
$520$	0	0
$521$	−11.1585	−0.488862	−0.244431	−	0.969667i	$-0.578601\pi$
$521$	−11.1585	−0.488862	−0.244431	+	0.969667i	$0.578601\pi$
$522$	0	0
$523$	−6.91985	−0.302584	−0.151292	−	0.988489i	$-0.548343\pi$
$523$	−6.91985	−0.302584	−0.151292	+	0.988489i	$0.548343\pi$
$524$	0	0
$525$	−1.80973	−0.0789833
$526$	0	0
$527$	7.60137	0.331121
$528$	0	0
$529$	34.2374	1.48858
$530$	0	0
$531$	2.41841	0.104950
$532$	0	0
$533$	−4.48124	−0.194104
$534$	0	0
$535$	−48.2201	−2.08474
$536$	0	0
$537$	2.98931	0.128998
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	39.5333	1.69967	0.849836	−	0.527048i	$-0.176701\pi$
$541$	39.5333	1.69967	0.849836	+	0.527048i	$0.176701\pi$
$542$	0	0
$543$	5.21616	0.223847
$544$	0	0
$545$	−45.9587	−1.96865
$546$	0	0
$547$	44.8339	1.91696	0.958479	−	0.285162i	$-0.0920474\pi$
$547$	44.8339	1.91696	0.958479	+	0.285162i	$0.0920474\pi$
$548$	0	0
$549$	−2.78634	−0.118918
$550$	0	0
$551$	0.188022	0.00801001
$552$	0	0
$553$	12.6865	0.539487
$554$	0	0
$555$	1.22792	0.0521223
$556$	0	0
$557$	−19.8445	−0.840839	−0.420419	−	0.907330i	$-0.638117\pi$
$557$	−19.8445	−0.840839	−0.420419	+	0.907330i	$0.638117\pi$
$558$	0	0
$559$	−1.41468	−0.0598347
$560$	0	0
$561$	−3.30324	−0.139463
$562$	0	0
$563$	−19.3194	−0.814216	−0.407108	−	0.913380i	$-0.633463\pi$
$563$	−19.3194	−0.814216	−0.407108	+	0.913380i	$0.633463\pi$
$564$	0	0
$565$	−14.9687	−0.629739
$566$	0	0
$567$	−6.83750	−0.287148
$568$	0	0
$569$	−29.3941	−1.23227	−0.616133	−	0.787642i	$-0.711302\pi$
$569$	−29.3941	−1.23227	−0.616133	+	0.787642i	$0.711302\pi$
$570$	0	0
$571$	−19.6340	−0.821659	−0.410829	−	0.911712i	$-0.634761\pi$
$571$	−19.6340	−0.821659	−0.410829	+	0.911712i	$0.634761\pi$
$572$	0	0
$573$	4.05944	0.169585
$574$	0	0
$575$	−27.5457	−1.14874
$576$	0	0
$577$	4.99071	0.207766	0.103883	−	0.994590i	$-0.466873\pi$
$577$	4.99071	0.207766	0.103883	+	0.994590i	$0.466873\pi$
$578$	0	0
$579$	−1.23212	−0.0512053
$580$	0	0
$581$	−2.63360	−0.109260
$582$	0	0
$583$	−11.8143	−0.489300
$584$	0	0
$585$	−8.09240	−0.334580
$586$	0	0
$587$	−7.15675	−0.295391	−0.147695	−	0.989033i	$-0.547186\pi$
$587$	−7.15675	−0.295391	−0.147695	+	0.989033i	$0.547186\pi$
$588$	0	0
$589$	−9.64806	−0.397541
$590$	0	0
$591$	6.57176	0.270326
$592$	0	0
$593$	46.1999	1.89720	0.948601	−	0.316474i	$-0.102499\pi$
$593$	46.1999	1.89720	0.948601	+	0.316474i	$0.102499\pi$
$594$	0	0
$595$	19.5353	0.800867
$596$	0	0
$597$	0.319789	0.0130881
$598$	0	0
$599$	12.3134	0.503113	0.251556	−	0.967843i	$-0.419058\pi$
$599$	12.3134	0.503113	0.251556	+	0.967843i	$0.419058\pi$
$600$	0	0
$601$	−4.36559	−0.178076	−0.0890380	−	0.996028i	$-0.528379\pi$
$601$	−4.36559	−0.178076	−0.0890380	+	0.996028i	$0.528379\pi$
$602$	0	0
$603$	−22.9636	−0.935151
$604$	0	0
$605$	−2.93955	−0.119510
$606$	0	0
$607$	−7.34490	−0.298120	−0.149060	−	0.988828i	$-0.547625\pi$
$607$	−7.34490	−0.298120	−0.149060	+	0.988828i	$0.547625\pi$
$608$	0	0
$609$	0.0110796	0.000448967 0
$610$	0	0
$611$	−6.97254	−0.282079
$612$	0	0
$613$	40.0829	1.61893	0.809466	−	0.587167i	$-0.199757\pi$
$613$	40.0829	1.61893	0.809466	+	0.587167i	$0.199757\pi$
$614$	0	0
$615$	−6.54756	−0.264023
$616$	0	0
$617$	36.6470	1.47535	0.737677	−	0.675154i	$-0.235923\pi$
$617$	36.6470	1.47535	0.737677	+	0.675154i	$0.235923\pi$
$618$	0	0
$619$	2.87216	0.115442	0.0577209	−	0.998333i	$-0.481617\pi$
$619$	2.87216	0.115442	0.0577209	+	0.998333i	$0.481617\pi$
$620$	0	0
$621$	21.6337	0.868131
$622$	0	0
$623$	14.9420	0.598638
$624$	0	0
$625$	−29.9482	−1.19793
$626$	0	0
$627$	4.19264	0.167438
$628$	0	0
$629$	−5.58505	−0.222691
$630$	0	0
$631$	−18.0530	−0.718678	−0.359339	−	0.933207i	$-0.616998\pi$
$631$	−18.0530	−0.718678	−0.359339	+	0.933207i	$0.616998\pi$
$632$	0	0
$633$	4.35001	0.172897
$634$	0	0
$635$	−21.5426	−0.854891
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	36.7107	1.45225
$640$	0	0
$641$	−21.7333	−0.858412	−0.429206	−	0.903207i	$-0.641207\pi$
$641$	−21.7333	−0.858412	−0.429206	+	0.903207i	$0.641207\pi$
$642$	0	0
$643$	−36.3416	−1.43317	−0.716586	−	0.697499i	$-0.754296\pi$
$643$	−36.3416	−1.43317	−0.716586	+	0.697499i	$0.754296\pi$
$644$	0	0
$645$	−2.06700	−0.0813881
$646$	0	0
$647$	45.7454	1.79844	0.899219	−	0.437499i	$-0.144136\pi$
$647$	45.7454	1.79844	0.899219	+	0.437499i	$0.144136\pi$
$648$	0	0
$649$	0.878482	0.0344835
$650$	0	0
$651$	−0.568531	−0.0222825
$652$	0	0
$653$	0.230521	0.00902100	0.00451050	−	0.999990i	$-0.498564\pi$
$653$	0.230521	0.00902100	0.00451050	+	0.999990i	$0.498564\pi$
$654$	0	0
$655$	−1.72928	−0.0675685
$656$	0	0
$657$	24.7005	0.963659
$658$	0	0
$659$	−19.7196	−0.768167	−0.384083	−	0.923298i	$-0.625482\pi$
$659$	−19.7196	−0.768167	−0.384083	+	0.923298i	$0.625482\pi$
$660$	0	0
$661$	−37.0997	−1.44301	−0.721505	−	0.692409i	$-0.756549\pi$
$661$	−37.0997	−1.44301	−0.721505	+	0.692409i	$0.756549\pi$
$662$	0	0
$663$	−3.30324	−0.128287
$664$	0	0
$665$	−24.7952	−0.961516
$666$	0	0
$667$	0.168641	0.00652979
$668$	0	0
$669$	−0.669550	−0.0258863
$670$	0	0
$671$	−1.01213	−0.0390729
$672$	0	0
$673$	23.2123	0.894768	0.447384	−	0.894342i	$-0.352356\pi$
$673$	23.2123	0.894768	0.447384	+	0.894342i	$0.352356\pi$
$674$	0	0
$675$	−10.4113	−0.400731
$676$	0	0
$677$	21.3929	0.822198	0.411099	−	0.911591i	$-0.365145\pi$
$677$	21.3929	0.822198	0.411099	+	0.911591i	$0.365145\pi$
$678$	0	0
$679$	−19.2853	−0.740103
$680$	0	0
$681$	−0.476747	−0.0182690
$682$	0	0
$683$	26.8470	1.02727	0.513636	−	0.858008i	$-0.328298\pi$
$683$	26.8470	1.02727	0.513636	+	0.858008i	$0.328298\pi$
$684$	0	0
$685$	54.0666	2.06578
$686$	0	0
$687$	−5.16009	−0.196870
$688$	0	0
$689$	−11.8143	−0.450090
$690$	0	0
$691$	7.08393	0.269485	0.134743	−	0.990881i	$-0.456979\pi$
$691$	7.08393	0.269485	0.134743	+	0.990881i	$0.456979\pi$
$692$	0	0
$693$	−2.75294	−0.104576
$694$	0	0
$695$	−6.31945	−0.239710
$696$	0	0
$697$	29.7808	1.12803
$698$	0	0
$699$	3.22130	0.121841
$700$	0	0
$701$	−34.7313	−1.31178	−0.655892	−	0.754855i	$-0.727707\pi$
$701$	−34.7313	−1.31178	−0.655892	+	0.754855i	$0.727707\pi$
$702$	0	0
$703$	7.08884	0.267361
$704$	0	0
$705$	−10.1876	−0.383688
$706$	0	0
$707$	3.05893	0.115043
$708$	0	0
$709$	17.0404	0.639966	0.319983	−	0.947423i	$-0.396323\pi$
$709$	17.0404	0.639966	0.319983	+	0.947423i	$0.396323\pi$
$710$	0	0
$711$	34.9253	1.30980
$712$	0	0
$713$	−8.65353	−0.324077
$714$	0	0
$715$	−2.93955	−0.109933
$716$	0	0
$717$	−0.733316	−0.0273862
$718$	0	0
$719$	−3.21732	−0.119986	−0.0599929	−	0.998199i	$-0.519108\pi$
$719$	−3.21732	−0.119986	−0.0599929	+	0.998199i	$0.519108\pi$
$720$	0	0
$721$	−2.52685	−0.0941048
$722$	0	0
$723$	−4.76173	−0.177091
$724$	0	0
$725$	−0.0811589	−0.00301416
$726$	0	0
$727$	−9.07075	−0.336416	−0.168208	−	0.985752i	$-0.553798\pi$
$727$	−9.07075	−0.336416	−0.168208	+	0.985752i	$0.553798\pi$
$728$	0	0
$729$	−14.5593	−0.539232
$730$	0	0
$731$	9.40151	0.347727
$732$	0	0
$733$	−30.8602	−1.13985	−0.569923	−	0.821698i	$-0.693027\pi$
$733$	−30.8602	−1.13985	−0.569923	+	0.821698i	$0.693027\pi$
$734$	0	0
$735$	−1.46111	−0.0538937
$736$	0	0
$737$	−8.34149	−0.307263
$738$	0	0
$739$	30.3010	1.11464	0.557320	−	0.830298i	$-0.311830\pi$
$739$	30.3010	1.11464	0.557320	+	0.830298i	$0.311830\pi$
$740$	0	0
$741$	4.19264	0.154021
$742$	0	0
$743$	35.3660	1.29745	0.648726	−	0.761022i	$-0.275302\pi$
$743$	35.3660	1.29745	0.648726	+	0.761022i	$0.275302\pi$
$744$	0	0
$745$	18.6275	0.682457
$746$	0	0
$747$	−7.25016	−0.265269
$748$	0	0
$749$	−16.4039	−0.599386
$750$	0	0
$751$	22.9692	0.838158	0.419079	−	0.907950i	$-0.362353\pi$
$751$	22.9692	0.838158	0.419079	+	0.907950i	$0.362353\pi$
$752$	0	0
$753$	10.3748	0.378079
$754$	0	0
$755$	−22.8956	−0.833255
$756$	0	0
$757$	10.9833	0.399196	0.199598	−	0.979878i	$-0.436036\pi$
$757$	10.9833	0.399196	0.199598	+	0.979878i	$0.436036\pi$
$758$	0	0
$759$	3.76046	0.136496
$760$	0	0
$761$	−32.4052	−1.17469	−0.587344	−	0.809337i	$-0.699827\pi$
$761$	−32.4052	−1.17469	−0.587344	+	0.809337i	$0.699827\pi$
$762$	0	0
$763$	−15.6346	−0.566011
$764$	0	0
$765$	53.7794	1.94440
$766$	0	0
$767$	0.878482	0.0317202
$768$	0	0
$769$	−3.19787	−0.115318	−0.0576590	−	0.998336i	$-0.518364\pi$
$769$	−3.19787	−0.115318	−0.0576590	+	0.998336i	$0.518364\pi$
$770$	0	0
$771$	13.2258	0.476317
$772$	0	0
$773$	−42.5718	−1.53120	−0.765601	−	0.643315i	$-0.777558\pi$
$773$	−42.5718	−1.53120	−0.765601	+	0.643315i	$0.777558\pi$
$774$	0	0
$775$	4.16454	0.149595
$776$	0	0
$777$	0.417724	0.0149858
$778$	0	0
$779$	−37.7994	−1.35430
$780$	0	0
$781$	13.3351	0.477167
$782$	0	0
$783$	0.0637401	0.00227789
$784$	0	0
$785$	−27.6881	−0.988230
$786$	0	0
$787$	−25.4656	−0.907752	−0.453876	−	0.891065i	$-0.649959\pi$
$787$	−25.4656	−0.907752	−0.453876	+	0.891065i	$0.649959\pi$
$788$	0	0
$789$	−2.51506	−0.0895385
$790$	0	0
$791$	−5.09219	−0.181057
$792$	0	0
$793$	−1.01213	−0.0359418
$794$	0	0
$795$	−17.2620	−0.612219
$796$	0	0
$797$	1.19398	0.0422928	0.0211464	−	0.999776i	$-0.493268\pi$
$797$	1.19398	0.0422928	0.0211464	+	0.999776i	$0.493268\pi$
$798$	0	0
$799$	46.3372	1.63929
$800$	0	0
$801$	41.1344	1.45341
$802$	0	0
$803$	8.97242	0.316630
$804$	0	0
$805$	−22.2393	−0.783831
$806$	0	0
$807$	−5.79087	−0.203848
$808$	0	0
$809$	46.3350	1.62905	0.814526	−	0.580128i	$-0.196997\pi$
$809$	46.3350	1.62905	0.814526	+	0.580128i	$0.196997\pi$
$810$	0	0
$811$	6.62835	0.232753	0.116376	−	0.993205i	$-0.462872\pi$
$811$	6.62835	0.232753	0.116376	+	0.993205i	$0.462872\pi$
$812$	0	0
$813$	−12.2044	−0.428028
$814$	0	0
$815$	−64.7301	−2.26740
$816$	0	0
$817$	−11.9329	−0.417479
$818$	0	0
$819$	−2.75294	−0.0961955
$820$	0	0
$821$	−37.1054	−1.29499	−0.647493	−	0.762071i	$-0.724183\pi$
$821$	−37.1054	−1.29499	−0.647493	+	0.762071i	$0.724183\pi$
$822$	0	0
$823$	−18.0648	−0.629701	−0.314850	−	0.949141i	$-0.601954\pi$
$823$	−18.0648	−0.629701	−0.314850	+	0.949141i	$0.601954\pi$
$824$	0	0
$825$	−1.80973	−0.0630069
$826$	0	0
$827$	38.9556	1.35462	0.677310	−	0.735698i	$-0.263146\pi$
$827$	38.9556	1.35462	0.677310	+	0.735698i	$0.263146\pi$
$828$	0	0
$829$	25.6819	0.891971	0.445985	−	0.895040i	$-0.352853\pi$
$829$	25.6819	0.891971	0.445985	+	0.895040i	$0.352853\pi$
$830$	0	0
$831$	4.37538	0.151780
$832$	0	0
$833$	6.64567	0.230259
$834$	0	0
$835$	57.3788	1.98567
$836$	0	0
$837$	−3.27073	−0.113053
$838$	0	0
$839$	46.6244	1.60965	0.804827	−	0.593509i	$-0.202258\pi$
$839$	46.6244	1.60965	0.804827	+	0.593509i	$0.202258\pi$
$840$	0	0
$841$	−28.9995	−0.999983
$842$	0	0
$843$	1.28510	0.0442610
$844$	0	0
$845$	−2.93955	−0.101124
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−7.46632	−0.256243
$850$	0	0
$851$	6.35812	0.217953
$852$	0	0
$853$	−5.03415	−0.172366	−0.0861830	−	0.996279i	$-0.527467\pi$
$853$	−5.03415	−0.172366	−0.0861830	+	0.996279i	$0.527467\pi$
$854$	0	0
$855$	−68.2596	−2.33443
$856$	0	0
$857$	18.5698	0.634333	0.317166	−	0.948370i	$-0.397269\pi$
$857$	18.5698	0.634333	0.317166	+	0.948370i	$0.397269\pi$
$858$	0	0
$859$	29.0060	0.989673	0.494837	−	0.868986i	$-0.335228\pi$
$859$	29.0060	0.989673	0.494837	+	0.868986i	$0.335228\pi$
$860$	0	0
$861$	−2.22740	−0.0759098
$862$	0	0
$863$	−13.2955	−0.452583	−0.226292	−	0.974060i	$-0.572660\pi$
$863$	−13.2955	−0.452583	−0.226292	+	0.974060i	$0.572660\pi$
$864$	0	0
$865$	20.3962	0.693490
$866$	0	0
$867$	13.5023	0.458563
$868$	0	0
$869$	12.6865	0.430362
$870$	0	0
$871$	−8.34149	−0.282640
$872$	0	0
$873$	−53.0914	−1.79687
$874$	0	0
$875$	−3.99501	−0.135056
$876$	0	0
$877$	−24.5596	−0.829320	−0.414660	−	0.909976i	$-0.636099\pi$
$877$	−24.5596	−0.829320	−0.414660	+	0.909976i	$0.636099\pi$
$878$	0	0
$879$	2.77295	0.0935291
$880$	0	0
$881$	−8.92309	−0.300627	−0.150313	−	0.988638i	$-0.548028\pi$
$881$	−8.92309	−0.300627	−0.150313	+	0.988638i	$0.548028\pi$
$882$	0	0
$883$	−7.12193	−0.239672	−0.119836	−	0.992794i	$-0.538237\pi$
$883$	−7.12193	−0.239672	−0.119836	+	0.992794i	$0.538237\pi$
$884$	0	0
$885$	1.28356	0.0431462
$886$	0	0
$887$	−32.3572	−1.08645	−0.543224	−	0.839588i	$-0.682796\pi$
$887$	−32.3572	−1.08645	−0.543224	+	0.839588i	$0.682796\pi$
$888$	0	0
$889$	−7.32853	−0.245791
$890$	0	0
$891$	−6.83750	−0.229065
$892$	0	0
$893$	−58.8136	−1.96812
$894$	0	0
$895$	−17.6787	−0.590935
$896$	0	0
$897$	3.76046	0.125558
$898$	0	0
$899$	−0.0254962	−0.000850345 0
$900$	0	0
$901$	78.5141	2.61568
$902$	0	0
$903$	−0.703170	−0.0234000
$904$	0	0
$905$	−30.8482	−1.02543
$906$	0	0
$907$	−40.6121	−1.34850	−0.674251	−	0.738502i	$-0.735534\pi$
$907$	−40.6121	−1.34850	−0.674251	+	0.738502i	$0.735534\pi$
$908$	0	0
$909$	8.42106	0.279309
$910$	0	0
$911$	−4.77972	−0.158359	−0.0791796	−	0.996860i	$-0.525230\pi$
$911$	−4.77972	−0.158359	−0.0791796	+	0.996860i	$0.525230\pi$
$912$	0	0
$913$	−2.63360	−0.0871596
$914$	0	0
$915$	−1.47883	−0.0488886
$916$	0	0
$917$	−0.588280	−0.0194267
$918$	0	0
$919$	36.2542	1.19591	0.597957	−	0.801528i	$-0.295979\pi$
$919$	36.2542	1.19591	0.597957	+	0.801528i	$0.295979\pi$
$920$	0	0
$921$	15.3457	0.505659
$922$	0	0
$923$	13.3351	0.438930
$924$	0	0
$925$	−3.05987	−0.100608
$926$	0	0
$927$	−6.95626	−0.228474
$928$	0	0
$929$	21.3237	0.699607	0.349803	−	0.936823i	$-0.386248\pi$
$929$	21.3237	0.699607	0.349803	+	0.936823i	$0.386248\pi$
$930$	0	0
$931$	−8.43503	−0.276447
$932$	0	0
$933$	−8.74528	−0.286308
$934$	0	0
$935$	19.5353	0.638871
$936$	0	0
$937$	21.4550	0.700904	0.350452	−	0.936581i	$-0.386028\pi$
$937$	21.4550	0.700904	0.350452	+	0.936581i	$0.386028\pi$
$938$	0	0
$939$	2.82073	0.0920509
$940$	0	0
$941$	36.4897	1.18953	0.594766	−	0.803899i	$-0.297245\pi$
$941$	36.4897	1.18953	0.594766	+	0.803899i	$0.297245\pi$
$942$	0	0
$943$	−33.9030	−1.10403
$944$	0	0
$945$	−8.40565	−0.273436
$946$	0	0
$947$	51.1608	1.66250	0.831251	−	0.555897i	$-0.187625\pi$
$947$	51.1608	1.66250	0.831251	+	0.555897i	$0.187625\pi$
$948$	0	0
$949$	8.97242	0.291257
$950$	0	0
$951$	−14.2791	−0.463032
$952$	0	0
$953$	−21.1360	−0.684661	−0.342331	−	0.939580i	$-0.611216\pi$
$953$	−21.1360	−0.684661	−0.342331	+	0.939580i	$0.611216\pi$
$954$	0	0
$955$	−24.0074	−0.776862
$956$	0	0
$957$	0.0110796	0.000358152 0
$958$	0	0
$959$	18.3928	0.593935
$960$	0	0
$961$	−29.6917	−0.957797
$962$	0	0
$963$	−45.1590	−1.45523
$964$	0	0
$965$	7.28675	0.234569
$966$	0	0
$967$	−53.5874	−1.72325	−0.861627	−	0.507542i	$-0.830554\pi$
$967$	−53.5874	−1.72325	−0.861627	+	0.507542i	$0.830554\pi$
$968$	0	0
$969$	−27.8629	−0.895085
$970$	0	0
$971$	40.8287	1.31026	0.655128	−	0.755518i	$-0.272615\pi$
$971$	40.8287	1.31026	0.655128	+	0.755518i	$0.272615\pi$
$972$	0	0
$973$	−2.14980	−0.0689195
$974$	0	0
$975$	−1.80973	−0.0579579
$976$	0	0
$977$	59.8042	1.91330	0.956652	−	0.291232i	$-0.0940651\pi$
$977$	59.8042	1.91330	0.956652	+	0.291232i	$0.0940651\pi$
$978$	0	0
$979$	14.9420	0.477548
$980$	0	0
$981$	−43.0411	−1.37420
$982$	0	0
$983$	−55.4918	−1.76991	−0.884957	−	0.465673i	$-0.845812\pi$
$983$	−55.4918	−1.76991	−0.884957	+	0.465673i	$0.845812\pi$
$984$	0	0
$985$	−38.8652	−1.23835
$986$	0	0
$987$	−3.46571	−0.110315
$988$	0	0
$989$	−10.7028	−0.340330
$990$	0	0
$991$	−53.4449	−1.69773	−0.848866	−	0.528608i	$-0.822714\pi$
$991$	−53.4449	−1.69773	−0.848866	+	0.528608i	$0.822714\pi$
$992$	0	0
$993$	11.2576	0.357248
$994$	0	0
$995$	−1.89122	−0.0599559
$996$	0	0
$997$	44.0110	1.39384	0.696922	−	0.717147i	$-0.254553\pi$
$997$	44.0110	1.39384	0.696922	+	0.717147i	$0.254553\pi$
$998$	0	0
$999$	2.40314	0.0760321

Introduction	Features
Universe	News

Level:	\( N \)	=	\( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	=	\( 2 \)
Character orbit:	\([\chi]\)	=	4004.a (trivial)

\(f(q)\)	\(=\)	\(q+0.497051 q^{3} -2.93955 q^{5} -1.00000 q^{7} -2.75294 q^{9} +O(q^{10})\)	\(q+0.497051 q^{3} -2.93955 q^{5} -1.00000 q^{7} -2.75294 q^{9} -1.00000 q^{11} -1.00000 q^{13} -1.46111 q^{15} +6.64567 q^{17} -8.43503 q^{19} -0.497051 q^{21} -7.56554 q^{23} +3.64094 q^{25} -2.85951 q^{27} -0.0222906 q^{29} +1.14381 q^{31} -0.497051 q^{33} +2.93955 q^{35} -0.840405 q^{37} -0.497051 q^{39} +4.48124 q^{41} +1.41468 q^{43} +8.09240 q^{45} +6.97254 q^{47} +1.00000 q^{49} +3.30324 q^{51} +11.8143 q^{53} +2.93955 q^{55} -4.19264 q^{57} -0.878482 q^{59} +1.01213 q^{61} +2.75294 q^{63} +2.93955 q^{65} +8.34149 q^{67} -3.76046 q^{69} -13.3351 q^{71} -8.97242 q^{73} +1.80973 q^{75} +1.00000 q^{77} -12.6865 q^{79} +6.83750 q^{81} +2.63360 q^{83} -19.5353 q^{85} -0.0110796 q^{87} -14.9420 q^{89} +1.00000 q^{91} +0.568531 q^{93} +24.7952 q^{95} +19.2853 q^{97} +2.75294 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10q - 2q^{3} - 2q^{5} - 10q^{7} + 10q^{9} + O(q^{10}) \)	\( 10q - 2q^{3} - 2q^{5} - 10q^{7} + 10q^{9} - 10q^{11} - 10q^{13} - 4q^{15} + 9q^{17} + 3q^{19} + 2q^{21} - 4q^{23} + 22q^{25} - 8q^{27} + 13q^{29} - 17q^{31} + 2q^{33} + 2q^{35} + 11q^{37} + 2q^{39} + 8q^{41} + 21q^{43} - 15q^{45} + q^{47} + 10q^{49} + 19q^{51} + 22q^{53} + 2q^{55} + 8q^{57} - 24q^{59} + 16q^{61} - 10q^{63} + 2q^{65} + 19q^{67} + 51q^{69} - 25q^{71} + 22q^{73} + 28q^{75} + 10q^{77} + 41q^{79} + 54q^{81} + 8q^{83} + 9q^{85} - 5q^{87} + 36q^{89} + 10q^{91} + 12q^{93} + 13q^{95} - 5q^{97} - 10q^{99} + O(q^{100}) \)

Introduction and more

L-functions

Modular Forms

Varieties

Fields

Representations

Groups

Knowledge

Properties

Related objects

Downloads

Learn more about

Newspace parameters

Newform invariants

Embedding invariants

$q$-expansion

Coefficient data

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	4004.2.a.j.1.7

Level	4004
Weight	2
Character	4004.1
Self dual	Yes
Analytic conductor	31.972
Analytic rank	0
Dimension	10
CM	No
Inner twists	1

Self dual:	Yes
Analytic conductor:	\(31.9721009693\)
Analytic rank:	\(0\)
Dimension:	\(10\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Fricke sign:	\(-1\)
Sato-Tate group:	$\mathrm{SU}(2)$