LMFDB - Embedded Newform 8008.2.a.p.1.1

Newspace parameters

Level:	$ N $	=	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	=	$ 2 $
Character orbit:	$[\chi]$	=	8008.a (trivial)

Newform invariants

Self dual:	Yes
Analytic conductor:	$63.9442019386$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.82075$
Character	$\chi$	=	8008.1

$q$-expansion

$f(q)$	$=$	$q-2.82075 q^{3} -1.92109 q^{5} -1.00000 q^{7} +4.95665 q^{9} +O(q^{10})$	$q-2.82075 q^{3} -1.92109 q^{5} -1.00000 q^{7} +4.95665 q^{9} +1.00000 q^{11} +1.00000 q^{13} +5.41891 q^{15} -0.965104 q^{17} +3.09750 q^{19} +2.82075 q^{21} -3.72460 q^{23} -1.30943 q^{25} -5.51924 q^{27} -0.786974 q^{29} +2.78741 q^{31} -2.82075 q^{33} +1.92109 q^{35} -10.8226 q^{37} -2.82075 q^{39} -4.59093 q^{41} -4.94708 q^{43} -9.52216 q^{45} -0.799126 q^{47} +1.00000 q^{49} +2.72232 q^{51} -1.50368 q^{53} -1.92109 q^{55} -8.73730 q^{57} +3.65311 q^{59} +14.3415 q^{61} -4.95665 q^{63} -1.92109 q^{65} +12.8578 q^{67} +10.5062 q^{69} +4.44026 q^{71} -1.16794 q^{73} +3.69358 q^{75} -1.00000 q^{77} +9.23904 q^{79} +0.698463 q^{81} +6.28506 q^{83} +1.85405 q^{85} +2.21986 q^{87} +4.46976 q^{89} -1.00000 q^{91} -7.86259 q^{93} -5.95057 q^{95} +7.50530 q^{97} +4.95665 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9q + q^{3} - 4q^{5} - 9q^{7} + 4q^{9} + O(q^{10}) $	$ 9q + q^{3} - 4q^{5} - 9q^{7} + 4q^{9} + 9q^{11} + 9q^{13} - 9q^{15} - 11q^{17} + 10q^{19} - q^{21} - 14q^{23} - q^{25} - 5q^{27} - 10q^{29} + 5q^{31} + q^{33} + 4q^{35} - 16q^{37} + q^{39} + 2q^{41} + 4q^{43} - 30q^{45} + 9q^{49} + 3q^{51} - 23q^{53} - 4q^{55} + 14q^{57} + 9q^{59} - 14q^{61} - 4q^{63} - 4q^{65} + 8q^{67} - 26q^{69} - 20q^{71} - 23q^{73} + 32q^{75} - 9q^{77} + 2q^{79} - 11q^{81} - 9q^{83} - 3q^{85} - 7q^{87} - 6q^{89} - 9q^{91} - 19q^{93} - 4q^{95} - 3q^{97} + 4q^{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
$3$	−2.82075	−1.62856	−0.814282	−	0.580470i	$-0.802869\pi$
$3$	−2.82075	−1.62856	−0.814282	+	0.580470i	$0.802869\pi$
$4$	0	0
$5$	−1.92109	−0.859136	−0.429568	−	0.903035i	$-0.641334\pi$
$5$	−1.92109	−0.859136	−0.429568	+	0.903035i	$0.641334\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	4.95665	1.65222
$10$	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
$15$	5.41891	1.39916
$16$	0	0
$17$	−0.965104	−0.234072	−0.117036	−	0.993128i	$-0.537339\pi$
$17$	−0.965104	−0.234072	−0.117036	+	0.993128i	$0.537339\pi$
$18$	0	0
$19$	3.09750	0.710616	0.355308	−	0.934749i	$-0.384376\pi$
$19$	3.09750	0.710616	0.355308	+	0.934749i	$0.384376\pi$
$20$	0	0
$21$	2.82075	0.615539
$22$	0	0
$23$	−3.72460	−0.776632	−0.388316	−	0.921526i	$-0.626943\pi$
$23$	−3.72460	−0.776632	−0.388316	+	0.921526i	$0.626943\pi$
$24$	0	0
$25$	−1.30943	−0.261886
$26$	0	0
$27$	−5.51924	−1.06218
$28$	0	0
$29$	−0.786974	−0.146137	−0.0730687	−	0.997327i	$-0.523279\pi$
$29$	−0.786974	−0.146137	−0.0730687	+	0.997327i	$0.523279\pi$
$30$	0	0
$31$	2.78741	0.500633	0.250317	−	0.968164i	$-0.419465\pi$
$31$	2.78741	0.500633	0.250317	+	0.968164i	$0.419465\pi$
$32$	0	0
$33$	−2.82075	−0.491030
$34$	0	0
$35$	1.92109	0.324723
$36$	0	0
$37$	−10.8226	−1.77922	−0.889612	−	0.456716i	$-0.849025\pi$
$37$	−10.8226	−1.77922	−0.889612	+	0.456716i	$0.849025\pi$
$38$	0	0
$39$	−2.82075	−0.451682
$40$	0	0
$41$	−4.59093	−0.716983	−0.358492	−	0.933533i	$-0.616709\pi$
$41$	−4.59093	−0.716983	−0.358492	+	0.933533i	$0.616709\pi$
$42$	0	0
$43$	−4.94708	−0.754423	−0.377212	−	0.926127i	$-0.623117\pi$
$43$	−4.94708	−0.754423	−0.377212	+	0.926127i	$0.623117\pi$
$44$	0	0
$45$	−9.52216	−1.41948
$46$	0	0
$47$	−0.799126	−0.116564	−0.0582822	−	0.998300i	$-0.518562\pi$
$47$	−0.799126	−0.116564	−0.0582822	+	0.998300i	$0.518562\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.72232	0.381201
$52$	0	0
$53$	−1.50368	−0.206547	−0.103273	−	0.994653i	$-0.532932\pi$
$53$	−1.50368	−0.206547	−0.103273	+	0.994653i	$0.532932\pi$
$54$	0	0
$55$	−1.92109	−0.259039
$56$	0	0
$57$	−8.73730	−1.15728
$58$	0	0
$59$	3.65311	0.475595	0.237798	−	0.971315i	$-0.423575\pi$
$59$	3.65311	0.475595	0.237798	+	0.971315i	$0.423575\pi$
$60$	0	0
$61$	14.3415	1.83624	0.918122	−	0.396298i	$-0.129705\pi$
$61$	14.3415	1.83624	0.918122	+	0.396298i	$0.129705\pi$
$62$	0	0
$63$	−4.95665	−0.624480
$64$	0	0
$65$	−1.92109	−0.238281
$66$	0	0
$67$	12.8578	1.57083	0.785413	−	0.618972i	$-0.212451\pi$
$67$	12.8578	1.57083	0.785413	+	0.618972i	$0.212451\pi$
$68$	0	0
$69$	10.5062	1.26480
$70$	0	0
$71$	4.44026	0.526962	0.263481	−	0.964665i	$-0.415129\pi$
$71$	4.44026	0.526962	0.263481	+	0.964665i	$0.415129\pi$
$72$	0	0
$73$	−1.16794	−0.136697	−0.0683484	−	0.997662i	$-0.521773\pi$
$73$	−1.16794	−0.136697	−0.0683484	+	0.997662i	$0.521773\pi$
$74$	0	0
$75$	3.69358	0.426498
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	9.23904	1.03947	0.519737	−	0.854327i	$-0.326030\pi$
$79$	9.23904	1.03947	0.519737	+	0.854327i	$0.326030\pi$
$80$	0	0
$81$	0.698463	0.0776070
$82$	0	0
$83$	6.28506	0.689875	0.344938	−	0.938626i	$-0.387900\pi$
$83$	6.28506	0.689875	0.344938	+	0.938626i	$0.387900\pi$
$84$	0	0
$85$	1.85405	0.201100
$86$	0	0
$87$	2.21986	0.237994
$88$	0	0
$89$	4.46976	0.473793	0.236897	−	0.971535i	$-0.423870\pi$
$89$	4.46976	0.473793	0.236897	+	0.971535i	$0.423870\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−7.86259	−0.815313
$94$	0	0
$95$	−5.95057	−0.610516
$96$	0	0
$97$	7.50530	0.762048	0.381024	−	0.924565i	$-0.375572\pi$
$97$	7.50530	0.762048	0.381024	+	0.924565i	$0.375572\pi$
$98$	0	0
$99$	4.95665	0.498163
$100$	0	0
$101$	1.12014	0.111459	0.0557293	−	0.998446i	$-0.482252\pi$
$101$	1.12014	0.111459	0.0557293	+	0.998446i	$0.482252\pi$
$102$	0	0
$103$	0.999512	0.0984849	0.0492424	−	0.998787i	$-0.484319\pi$
$103$	0.999512	0.0984849	0.0492424	+	0.998787i	$0.484319\pi$
$104$	0	0
$105$	−5.41891	−0.528832
$106$	0	0
$107$	5.56947	0.538421	0.269210	−	0.963081i	$-0.413237\pi$
$107$	5.56947	0.538421	0.269210	+	0.963081i	$0.413237\pi$
$108$	0	0
$109$	1.41533	0.135564	0.0677821	−	0.997700i	$-0.478408\pi$
$109$	1.41533	0.135564	0.0677821	+	0.997700i	$0.478408\pi$
$110$	0	0
$111$	30.5279	2.89758
$112$	0	0
$113$	0.238232	0.0224110	0.0112055	−	0.999937i	$-0.496433\pi$
$113$	0.238232	0.0224110	0.0112055	+	0.999937i	$0.496433\pi$
$114$	0	0
$115$	7.15527	0.667233
$116$	0	0
$117$	4.95665	0.458243
$118$	0	0
$119$	0.965104	0.0884709
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	12.9499	1.16765
$124$	0	0
$125$	12.1210	1.08413
$126$	0	0
$127$	−2.54582	−0.225905	−0.112952	−	0.993600i	$-0.536031\pi$
$127$	−2.54582	−0.225905	−0.112952	+	0.993600i	$0.536031\pi$
$128$	0	0
$129$	13.9545	1.22863
$130$	0	0
$131$	7.05388	0.616300	0.308150	−	0.951338i	$-0.400290\pi$
$131$	7.05388	0.616300	0.308150	+	0.951338i	$0.400290\pi$
$132$	0	0
$133$	−3.09750	−0.268588
$134$	0	0
$135$	10.6029	0.912556
$136$	0	0
$137$	−19.1999	−1.64036	−0.820180	−	0.572106i	$-0.806127\pi$
$137$	−19.1999	−1.64036	−0.820180	+	0.572106i	$0.806127\pi$
$138$	0	0
$139$	14.8024	1.25552	0.627762	−	0.778406i	$-0.283971\pi$
$139$	14.8024	1.25552	0.627762	+	0.778406i	$0.283971\pi$
$140$	0	0
$141$	2.25414	0.189833
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	1.51185	0.125552
$146$	0	0
$147$	−2.82075	−0.232652
$148$	0	0
$149$	−11.2743	−0.923623	−0.461811	−	0.886978i	$-0.652800\pi$
$149$	−11.2743	−0.923623	−0.461811	+	0.886978i	$0.652800\pi$
$150$	0	0
$151$	4.09117	0.332934	0.166467	−	0.986047i	$-0.446764\pi$
$151$	4.09117	0.332934	0.166467	+	0.986047i	$0.446764\pi$
$152$	0	0
$153$	−4.78369	−0.386738
$154$	0	0
$155$	−5.35485	−0.430112
$156$	0	0
$157$	−12.6226	−1.00739	−0.503696	−	0.863881i	$-0.668027\pi$
$157$	−12.6226	−1.00739	−0.503696	+	0.863881i	$0.668027\pi$
$158$	0	0
$159$	4.24152	0.336374
$160$	0	0
$161$	3.72460	0.293539
$162$	0	0
$163$	−1.58008	−0.123762	−0.0618808	−	0.998084i	$-0.519710\pi$
$163$	−1.58008	−0.123762	−0.0618808	+	0.998084i	$0.519710\pi$
$164$	0	0
$165$	5.41891	0.421862
$166$	0	0
$167$	−10.4625	−0.809611	−0.404805	−	0.914403i	$-0.632661\pi$
$167$	−10.4625	−0.809611	−0.404805	+	0.914403i	$0.632661\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	15.3533	1.17409
$172$	0	0
$173$	8.90837	0.677291	0.338645	−	0.940914i	$-0.390031\pi$
$173$	8.90837	0.677291	0.338645	+	0.940914i	$0.390031\pi$
$174$	0	0
$175$	1.30943	0.0989836
$176$	0	0
$177$	−10.3045	−0.774537
$178$	0	0
$179$	−17.8195	−1.33189	−0.665947	−	0.745999i	$-0.731972\pi$
$179$	−17.8195	−1.33189	−0.665947	+	0.745999i	$0.731972\pi$
$180$	0	0
$181$	−15.2593	−1.13421	−0.567106	−	0.823645i	$-0.691937\pi$
$181$	−15.2593	−1.13421	−0.567106	+	0.823645i	$0.691937\pi$
$182$	0	0
$183$	−40.4539	−2.99044
$184$	0	0
$185$	20.7911	1.52860
$186$	0	0
$187$	−0.965104	−0.0705754
$188$	0	0
$189$	5.51924	0.401466
$190$	0	0
$191$	12.6559	0.915751	0.457876	−	0.889016i	$-0.348611\pi$
$191$	12.6559	0.915751	0.457876	+	0.889016i	$0.348611\pi$
$192$	0	0
$193$	−16.9963	−1.22342	−0.611711	−	0.791081i	$-0.709518\pi$
$193$	−16.9963	−1.22342	−0.611711	+	0.791081i	$0.709518\pi$
$194$	0	0
$195$	5.41891	0.388056
$196$	0	0
$197$	9.70263	0.691284	0.345642	−	0.938367i	$-0.387661\pi$
$197$	9.70263	0.691284	0.345642	+	0.938367i	$0.387661\pi$
$198$	0	0
$199$	−18.9379	−1.34247	−0.671235	−	0.741245i	$-0.734236\pi$
$199$	−18.9379	−1.34247	−0.671235	+	0.741245i	$0.734236\pi$
$200$	0	0
$201$	−36.2686	−2.55819
$202$	0	0
$203$	0.786974	0.0552348
$204$	0	0
$205$	8.81958	0.615986
$206$	0	0
$207$	−18.4615	−1.28317
$208$	0	0
$209$	3.09750	0.214259
$210$	0	0
$211$	11.8938	0.818804	0.409402	−	0.912354i	$-0.365737\pi$
$211$	11.8938	0.818804	0.409402	+	0.912354i	$0.365737\pi$
$212$	0	0
$213$	−12.5249	−0.858192
$214$	0	0
$215$	9.50377	0.648152
$216$	0	0
$217$	−2.78741	−0.189222
$218$	0	0
$219$	3.29447	0.222619
$220$	0	0
$221$	−0.965104	−0.0649199
$222$	0	0
$223$	0.973431	0.0651857	0.0325929	−	0.999469i	$-0.489624\pi$
$223$	0.973431	0.0651857	0.0325929	+	0.999469i	$0.489624\pi$
$224$	0	0
$225$	−6.49039	−0.432693
$226$	0	0
$227$	−2.50622	−0.166344	−0.0831719	−	0.996535i	$-0.526505\pi$
$227$	−2.50622	−0.166344	−0.0831719	+	0.996535i	$0.526505\pi$
$228$	0	0
$229$	28.9136	1.91066	0.955332	−	0.295534i	$-0.0954974\pi$
$229$	28.9136	1.91066	0.955332	+	0.295534i	$0.0954974\pi$
$230$	0	0
$231$	2.82075	0.185592
$232$	0	0
$233$	7.98505	0.523118	0.261559	−	0.965187i	$-0.415763\pi$
$233$	7.98505	0.523118	0.261559	+	0.965187i	$0.415763\pi$
$234$	0	0
$235$	1.53519	0.100145
$236$	0	0
$237$	−26.0611	−1.69285
$238$	0	0
$239$	6.12982	0.396505	0.198253	−	0.980151i	$-0.436473\pi$
$239$	6.12982	0.396505	0.198253	+	0.980151i	$0.436473\pi$
$240$	0	0
$241$	−6.12777	−0.394725	−0.197362	−	0.980331i	$-0.563238\pi$
$241$	−6.12777	−0.394725	−0.197362	+	0.980331i	$0.563238\pi$
$242$	0	0
$243$	14.5875	0.935791
$244$	0	0
$245$	−1.92109	−0.122734
$246$	0	0
$247$	3.09750	0.197089
$248$	0	0
$249$	−17.7286	−1.12351
$250$	0	0
$251$	16.1518	1.01949	0.509747	−	0.860324i	$-0.329739\pi$
$251$	16.1518	1.01949	0.509747	+	0.860324i	$0.329739\pi$
$252$	0	0
$253$	−3.72460	−0.234163
$254$	0	0
$255$	−5.22981	−0.327504
$256$	0	0
$257$	−7.69320	−0.479889	−0.239944	−	0.970787i	$-0.577129\pi$
$257$	−7.69320	−0.479889	−0.239944	+	0.970787i	$0.577129\pi$
$258$	0	0
$259$	10.8226	0.672484
$260$	0	0
$261$	−3.90076	−0.241451
$262$	0	0
$263$	−13.1633	−0.811681	−0.405841	−	0.913944i	$-0.633021\pi$
$263$	−13.1633	−0.811681	−0.405841	+	0.913944i	$0.633021\pi$
$264$	0	0
$265$	2.88870	0.177452
$266$	0	0
$267$	−12.6081	−0.771602
$268$	0	0
$269$	−1.62237	−0.0989175	−0.0494587	−	0.998776i	$-0.515750\pi$
$269$	−1.62237	−0.0989175	−0.0494587	+	0.998776i	$0.515750\pi$
$270$	0	0
$271$	−3.46096	−0.210239	−0.105119	−	0.994460i	$-0.533522\pi$
$271$	−3.46096	−0.210239	−0.105119	+	0.994460i	$0.533522\pi$
$272$	0	0
$273$	2.82075	0.170720
$274$	0	0
$275$	−1.30943	−0.0789616
$276$	0	0
$277$	−1.41294	−0.0848954	−0.0424477	−	0.999099i	$-0.513516\pi$
$277$	−1.41294	−0.0848954	−0.0424477	+	0.999099i	$0.513516\pi$
$278$	0	0
$279$	13.8162	0.827155
$280$	0	0
$281$	−7.83859	−0.467611	−0.233806	−	0.972283i	$-0.575118\pi$
$281$	−7.83859	−0.467611	−0.233806	+	0.972283i	$0.575118\pi$
$282$	0	0
$283$	11.6854	0.694627	0.347313	−	0.937749i	$-0.387094\pi$
$283$	11.6854	0.694627	0.347313	+	0.937749i	$0.387094\pi$
$284$	0	0
$285$	16.7851	0.994263
$286$	0	0
$287$	4.59093	0.270994
$288$	0	0
$289$	−16.0686	−0.945210
$290$	0	0
$291$	−21.1706	−1.24104
$292$	0	0
$293$	−27.1820	−1.58799	−0.793996	−	0.607923i	$-0.792003\pi$
$293$	−27.1820	−1.58799	−0.793996	+	0.607923i	$0.792003\pi$
$294$	0	0
$295$	−7.01795	−0.408601
$296$	0	0
$297$	−5.51924	−0.320259
$298$	0	0
$299$	−3.72460	−0.215399
$300$	0	0
$301$	4.94708	0.285145
$302$	0	0
$303$	−3.15965	−0.181517
$304$	0	0
$305$	−27.5513	−1.57758
$306$	0	0
$307$	24.7444	1.41224	0.706118	−	0.708094i	$-0.250445\pi$
$307$	24.7444	1.41224	0.706118	+	0.708094i	$0.250445\pi$
$308$	0	0
$309$	−2.81938	−0.160389
$310$	0	0
$311$	−19.2913	−1.09391	−0.546956	−	0.837161i	$-0.684214\pi$
$311$	−19.2913	−1.09391	−0.546956	+	0.837161i	$0.684214\pi$
$312$	0	0
$313$	11.9577	0.675892	0.337946	−	0.941166i	$-0.390268\pi$
$313$	11.9577	0.675892	0.337946	+	0.941166i	$0.390268\pi$
$314$	0	0
$315$	9.52216	0.536513
$316$	0	0
$317$	15.9623	0.896531	0.448266	−	0.893900i	$-0.352042\pi$
$317$	15.9623	0.896531	0.448266	+	0.893900i	$0.352042\pi$
$318$	0	0
$319$	−0.786974	−0.0440621
$320$	0	0
$321$	−15.7101	−0.876852
$322$	0	0
$323$	−2.98941	−0.166335
$324$	0	0
$325$	−1.30943	−0.0726341
$326$	0	0
$327$	−3.99230	−0.220775
$328$	0	0
$329$	0.799126	0.0440572
$330$	0	0
$331$	−34.4160	−1.89168	−0.945838	−	0.324640i	$-0.894757\pi$
$331$	−34.4160	−1.89168	−0.945838	+	0.324640i	$0.894757\pi$
$332$	0	0
$333$	−53.6439	−2.93967
$334$	0	0
$335$	−24.7009	−1.34955
$336$	0	0
$337$	7.86366	0.428361	0.214180	−	0.976794i	$-0.431292\pi$
$337$	7.86366	0.428361	0.214180	+	0.976794i	$0.431292\pi$
$338$	0	0
$339$	−0.671995	−0.0364978
$340$	0	0
$341$	2.78741	0.150947
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−20.1833	−1.08663
$346$	0	0
$347$	9.82358	0.527357	0.263679	−	0.964611i	$-0.415064\pi$
$347$	9.82358	0.527357	0.263679	+	0.964611i	$0.415064\pi$
$348$	0	0
$349$	−23.4893	−1.25735	−0.628676	−	0.777667i	$-0.716403\pi$
$349$	−23.4893	−1.25735	−0.628676	+	0.777667i	$0.716403\pi$
$350$	0	0
$351$	−5.51924	−0.294595
$352$	0	0
$353$	−12.6559	−0.673607	−0.336804	−	0.941575i	$-0.609346\pi$
$353$	−12.6559	−0.673607	−0.336804	+	0.941575i	$0.609346\pi$
$354$	0	0
$355$	−8.53013	−0.452732
$356$	0	0
$357$	−2.72232	−0.144081
$358$	0	0
$359$	−15.4424	−0.815019	−0.407509	−	0.913201i	$-0.633603\pi$
$359$	−15.4424	−0.815019	−0.407509	+	0.913201i	$0.633603\pi$
$360$	0	0
$361$	−9.40547	−0.495025
$362$	0	0
$363$	−2.82075	−0.148051
$364$	0	0
$365$	2.24371	0.117441
$366$	0	0
$367$	10.5254	0.549423	0.274712	−	0.961527i	$-0.411418\pi$
$367$	10.5254	0.549423	0.274712	+	0.961527i	$0.411418\pi$
$368$	0	0
$369$	−22.7557	−1.18461
$370$	0	0
$371$	1.50368	0.0780673
$372$	0	0
$373$	31.1512	1.61295	0.806474	−	0.591270i	$-0.201373\pi$
$373$	31.1512	1.61295	0.806474	+	0.591270i	$0.201373\pi$
$374$	0	0
$375$	−34.1902	−1.76558
$376$	0	0
$377$	−0.786974	−0.0405312
$378$	0	0
$379$	17.2227	0.884669	0.442334	−	0.896850i	$-0.354150\pi$
$379$	17.2227	0.884669	0.442334	+	0.896850i	$0.354150\pi$
$380$	0	0
$381$	7.18113	0.367900
$382$	0	0
$383$	13.2561	0.677353	0.338677	−	0.940903i	$-0.390021\pi$
$383$	13.2561	0.677353	0.338677	+	0.940903i	$0.390021\pi$
$384$	0	0
$385$	1.92109	0.0979076
$386$	0	0
$387$	−24.5210	−1.24647
$388$	0	0
$389$	−25.5281	−1.29433	−0.647163	−	0.762352i	$-0.724045\pi$
$389$	−25.5281	−1.29433	−0.647163	+	0.762352i	$0.724045\pi$
$390$	0	0
$391$	3.59462	0.181788
$392$	0	0
$393$	−19.8973	−1.00368
$394$	0	0
$395$	−17.7490	−0.893049
$396$	0	0
$397$	−3.14865	−0.158026	−0.0790131	−	0.996874i	$-0.525177\pi$
$397$	−3.14865	−0.158026	−0.0790131	+	0.996874i	$0.525177\pi$
$398$	0	0
$399$	8.73730	0.437412
$400$	0	0
$401$	−17.3655	−0.867192	−0.433596	−	0.901107i	$-0.642756\pi$
$401$	−17.3655	−0.867192	−0.433596	+	0.901107i	$0.642756\pi$
$402$	0	0
$403$	2.78741	0.138851
$404$	0	0
$405$	−1.34181	−0.0666749
$406$	0	0
$407$	−10.8226	−0.536456
$408$	0	0
$409$	11.4713	0.567220	0.283610	−	0.958940i	$-0.408468\pi$
$409$	11.4713	0.567220	0.283610	+	0.958940i	$0.408468\pi$
$410$	0	0
$411$	54.1582	2.67143
$412$	0	0
$413$	−3.65311	−0.179758
$414$	0	0
$415$	−12.0741	−0.592696
$416$	0	0
$417$	−41.7539	−2.04470
$418$	0	0
$419$	30.9086	1.50998	0.754992	−	0.655734i	$-0.227641\pi$
$419$	30.9086	1.50998	0.754992	+	0.655734i	$0.227641\pi$
$420$	0	0
$421$	11.0837	0.540186	0.270093	−	0.962834i	$-0.412945\pi$
$421$	11.0837	0.540186	0.270093	+	0.962834i	$0.412945\pi$
$422$	0	0
$423$	−3.96099	−0.192590
$424$	0	0
$425$	1.26374	0.0613002
$426$	0	0
$427$	−14.3415	−0.694035
$428$	0	0
$429$	−2.82075	−0.136187
$430$	0	0
$431$	−0.506075	−0.0243768	−0.0121884	−	0.999926i	$-0.503880\pi$
$431$	−0.506075	−0.0243768	−0.0121884	+	0.999926i	$0.503880\pi$
$432$	0	0
$433$	16.2174	0.779360	0.389680	−	0.920950i	$-0.372586\pi$
$433$	16.2174	0.779360	0.389680	+	0.920950i	$0.372586\pi$
$434$	0	0
$435$	−4.26454	−0.204469
$436$	0	0
$437$	−11.5370	−0.551888
$438$	0	0
$439$	6.42312	0.306559	0.153279	−	0.988183i	$-0.451017\pi$
$439$	6.42312	0.306559	0.153279	+	0.988183i	$0.451017\pi$
$440$	0	0
$441$	4.95665	0.236031
$442$	0	0
$443$	−16.8382	−0.800009	−0.400005	−	0.916513i	$-0.630991\pi$
$443$	−16.8382	−0.800009	−0.400005	+	0.916513i	$0.630991\pi$
$444$	0	0
$445$	−8.58678	−0.407053
$446$	0	0
$447$	31.8019	1.50418
$448$	0	0
$449$	34.9320	1.64854	0.824272	−	0.566194i	$-0.191585\pi$
$449$	34.9320	1.64854	0.824272	+	0.566194i	$0.191585\pi$
$450$	0	0
$451$	−4.59093	−0.216179
$452$	0	0
$453$	−11.5402	−0.542205
$454$	0	0
$455$	1.92109	0.0900619
$456$	0	0
$457$	−27.7876	−1.29985	−0.649925	−	0.759998i	$-0.725200\pi$
$457$	−27.7876	−1.29985	−0.649925	+	0.759998i	$0.725200\pi$
$458$	0	0
$459$	5.32664	0.248626
$460$	0	0
$461$	−21.2125	−0.987963	−0.493981	−	0.869472i	$-0.664459\pi$
$461$	−21.2125	−0.987963	−0.493981	+	0.869472i	$0.664459\pi$
$462$	0	0
$463$	−32.8563	−1.52696	−0.763481	−	0.645830i	$-0.776511\pi$
$463$	−32.8563	−1.52696	−0.763481	+	0.645830i	$0.776511\pi$
$464$	0	0
$465$	15.1047	0.700464
$466$	0	0
$467$	−25.1183	−1.16233	−0.581167	−	0.813784i	$-0.697404\pi$
$467$	−25.1183	−1.16233	−0.581167	+	0.813784i	$0.697404\pi$
$468$	0	0
$469$	−12.8578	−0.593716
$470$	0	0
$471$	35.6052	1.64060
$472$	0	0
$473$	−4.94708	−0.227467
$474$	0	0
$475$	−4.05596	−0.186100
$476$	0	0
$477$	−7.45323	−0.341260
$478$	0	0
$479$	−18.8332	−0.860512	−0.430256	−	0.902707i	$-0.641577\pi$
$479$	−18.8332	−0.860512	−0.430256	+	0.902707i	$0.641577\pi$
$480$	0	0
$481$	−10.8226	−0.493468
$482$	0	0
$483$	−10.5062	−0.478048
$484$	0	0
$485$	−14.4183	−0.654702
$486$	0	0
$487$	−23.9582	−1.08565	−0.542824	−	0.839847i	$-0.682645\pi$
$487$	−23.9582	−1.08565	−0.542824	+	0.839847i	$0.682645\pi$
$488$	0	0
$489$	4.45703	0.201554
$490$	0	0
$491$	−6.97873	−0.314946	−0.157473	−	0.987523i	$-0.550335\pi$
$491$	−6.97873	−0.314946	−0.157473	+	0.987523i	$0.550335\pi$
$492$	0	0
$493$	0.759512	0.0342067
$494$	0	0
$495$	−9.52216	−0.427989
$496$	0	0
$497$	−4.44026	−0.199173
$498$	0	0
$499$	12.2887	0.550116	0.275058	−	0.961428i	$-0.411303\pi$
$499$	12.2887	0.550116	0.275058	+	0.961428i	$0.411303\pi$
$500$	0	0
$501$	29.5121	1.31850
$502$	0	0
$503$	−1.30018	−0.0579722	−0.0289861	−	0.999580i	$-0.509228\pi$
$503$	−1.30018	−0.0579722	−0.0289861	+	0.999580i	$0.509228\pi$
$504$	0	0
$505$	−2.15189	−0.0957581
$506$	0	0
$507$	−2.82075	−0.125274
$508$	0	0
$509$	−24.7209	−1.09574	−0.547868	−	0.836565i	$-0.684560\pi$
$509$	−24.7209	−1.09574	−0.547868	+	0.836565i	$0.684560\pi$
$510$	0	0
$511$	1.16794	0.0516665
$512$	0	0
$513$	−17.0959	−0.754801
$514$	0	0
$515$	−1.92015	−0.0846119
$516$	0	0
$517$	−0.799126	−0.0351455
$518$	0	0
$519$	−25.1283	−1.10301
$520$	0	0
$521$	6.00539	0.263101	0.131550	−	0.991309i	$-0.458004\pi$
$521$	6.00539	0.263101	0.131550	+	0.991309i	$0.458004\pi$
$522$	0	0
$523$	−11.7372	−0.513230	−0.256615	−	0.966514i	$-0.582607\pi$
$523$	−11.7372	−0.513230	−0.256615	+	0.966514i	$0.582607\pi$
$524$	0	0
$525$	−3.69358	−0.161201
$526$	0	0
$527$	−2.69014	−0.117184
$528$	0	0
$529$	−9.12737	−0.396842
$530$	0	0
$531$	18.1072	0.785787
$532$	0	0
$533$	−4.59093	−0.198855
$534$	0	0
$535$	−10.6994	−0.462576
$536$	0	0
$537$	50.2645	2.16907
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−36.1370	−1.55365	−0.776826	−	0.629715i	$-0.783172\pi$
$541$	−36.1370	−1.55365	−0.776826	+	0.629715i	$0.783172\pi$
$542$	0	0
$543$	43.0426	1.84714
$544$	0	0
$545$	−2.71897	−0.116468
$546$	0	0
$547$	6.13682	0.262392	0.131196	−	0.991356i	$-0.458118\pi$
$547$	6.13682	0.262392	0.131196	+	0.991356i	$0.458118\pi$
$548$	0	0
$549$	71.0860	3.03388
$550$	0	0
$551$	−2.43766	−0.103848
$552$	0	0
$553$	−9.23904	−0.392884
$554$	0	0
$555$	−58.6467	−2.48941
$556$	0	0
$557$	2.07144	0.0877696	0.0438848	−	0.999037i	$-0.486027\pi$
$557$	2.07144	0.0877696	0.0438848	+	0.999037i	$0.486027\pi$
$558$	0	0
$559$	−4.94708	−0.209239
$560$	0	0
$561$	2.72232	0.114937
$562$	0	0
$563$	−12.3135	−0.518953	−0.259476	−	0.965749i	$-0.583550\pi$
$563$	−12.3135	−0.518953	−0.259476	+	0.965749i	$0.583550\pi$
$564$	0	0
$565$	−0.457665	−0.0192541
$566$	0	0
$567$	−0.698463	−0.0293327
$568$	0	0
$569$	3.27822	0.137430	0.0687150	−	0.997636i	$-0.478110\pi$
$569$	3.27822	0.137430	0.0687150	+	0.997636i	$0.478110\pi$
$570$	0	0
$571$	−12.5889	−0.526828	−0.263414	−	0.964683i	$-0.584849\pi$
$571$	−12.5889	−0.526828	−0.263414	+	0.964683i	$0.584849\pi$
$572$	0	0
$573$	−35.6993	−1.49136
$574$	0	0
$575$	4.87710	0.203389
$576$	0	0
$577$	30.3850	1.26494	0.632471	−	0.774584i	$-0.282041\pi$
$577$	30.3850	1.26494	0.632471	+	0.774584i	$0.282041\pi$
$578$	0	0
$579$	47.9424	1.99242
$580$	0	0
$581$	−6.28506	−0.260748
$582$	0	0
$583$	−1.50368	−0.0622762
$584$	0	0
$585$	−9.52216	−0.393693
$586$	0	0
$587$	−23.1269	−0.954549	−0.477275	−	0.878754i	$-0.658375\pi$
$587$	−23.1269	−0.954549	−0.477275	+	0.878754i	$0.658375\pi$
$588$	0	0
$589$	8.63400	0.355758
$590$	0	0
$591$	−27.3687	−1.12580
$592$	0	0
$593$	−15.0928	−0.619785	−0.309893	−	0.950772i	$-0.600293\pi$
$593$	−15.0928	−0.619785	−0.309893	+	0.950772i	$0.600293\pi$
$594$	0	0
$595$	−1.85405	−0.0760085
$596$	0	0
$597$	53.4190	2.18630
$598$	0	0
$599$	−45.8369	−1.87285	−0.936423	−	0.350873i	$-0.885885\pi$
$599$	−45.8369	−1.87285	−0.936423	+	0.350873i	$0.885885\pi$
$600$	0	0
$601$	−22.0710	−0.900295	−0.450147	−	0.892954i	$-0.648629\pi$
$601$	−22.0710	−0.900295	−0.450147	+	0.892954i	$0.648629\pi$
$602$	0	0
$603$	63.7315	2.59535
$604$	0	0
$605$	−1.92109	−0.0781032
$606$	0	0
$607$	0.770767	0.0312845	0.0156422	−	0.999878i	$-0.495021\pi$
$607$	0.770767	0.0312845	0.0156422	+	0.999878i	$0.495021\pi$
$608$	0	0
$609$	−2.21986	−0.0899533
$610$	0	0
$611$	−0.799126	−0.0323292
$612$	0	0
$613$	−22.3061	−0.900935	−0.450467	−	0.892793i	$-0.648743\pi$
$613$	−22.3061	−0.900935	−0.450467	+	0.892793i	$0.648743\pi$
$614$	0	0
$615$	−24.8779	−1.00317
$616$	0	0
$617$	8.72611	0.351300	0.175650	−	0.984453i	$-0.443797\pi$
$617$	8.72611	0.351300	0.175650	+	0.984453i	$0.443797\pi$
$618$	0	0
$619$	3.10198	0.124679	0.0623396	−	0.998055i	$-0.480144\pi$
$619$	3.10198	0.124679	0.0623396	+	0.998055i	$0.480144\pi$
$620$	0	0
$621$	20.5570	0.824922
$622$	0	0
$623$	−4.46976	−0.179077
$624$	0	0
$625$	−16.7382	−0.669530
$626$	0	0
$627$	−8.73730	−0.348934
$628$	0	0
$629$	10.4449	0.416467
$630$	0	0
$631$	21.7298	0.865048	0.432524	−	0.901622i	$-0.357623\pi$
$631$	21.7298	0.865048	0.432524	+	0.901622i	$0.357623\pi$
$632$	0	0
$633$	−33.5495	−1.33347
$634$	0	0
$635$	4.89074	0.194083
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	22.0089	0.870657
$640$	0	0
$641$	−4.59909	−0.181653	−0.0908265	−	0.995867i	$-0.528951\pi$
$641$	−4.59909	−0.181653	−0.0908265	+	0.995867i	$0.528951\pi$
$642$	0	0
$643$	24.7546	0.976226	0.488113	−	0.872780i	$-0.337685\pi$
$643$	24.7546	0.976226	0.488113	+	0.872780i	$0.337685\pi$
$644$	0	0
$645$	−26.8078	−1.05556
$646$	0	0
$647$	−27.5816	−1.08434	−0.542172	−	0.840268i	$-0.682398\pi$
$647$	−27.5816	−1.08434	−0.542172	+	0.840268i	$0.682398\pi$
$648$	0	0
$649$	3.65311	0.143397
$650$	0	0
$651$	7.86259	0.308159
$652$	0	0
$653$	−13.5587	−0.530595	−0.265297	−	0.964167i	$-0.585470\pi$
$653$	−13.5587	−0.530595	−0.265297	+	0.964167i	$0.585470\pi$
$654$	0	0
$655$	−13.5511	−0.529485
$656$	0	0
$657$	−5.78906	−0.225853
$658$	0	0
$659$	27.9805	1.08997	0.544983	−	0.838447i	$-0.316536\pi$
$659$	27.9805	1.08997	0.544983	+	0.838447i	$0.316536\pi$
$660$	0	0
$661$	−2.77272	−0.107846	−0.0539232	−	0.998545i	$-0.517173\pi$
$661$	−2.77272	−0.107846	−0.0539232	+	0.998545i	$0.517173\pi$
$662$	0	0
$663$	2.72232	0.105726
$664$	0	0
$665$	5.95057	0.230753
$666$	0	0
$667$	2.93116	0.113495
$668$	0	0
$669$	−2.74581	−0.106159
$670$	0	0
$671$	14.3415	0.553648
$672$	0	0
$673$	−11.0314	−0.425229	−0.212614	−	0.977136i	$-0.568198\pi$
$673$	−11.0314	−0.425229	−0.212614	+	0.977136i	$0.568198\pi$
$674$	0	0
$675$	7.22706	0.278170
$676$	0	0
$677$	−4.07374	−0.156567	−0.0782834	−	0.996931i	$-0.524944\pi$
$677$	−4.07374	−0.156567	−0.0782834	+	0.996931i	$0.524944\pi$
$678$	0	0
$679$	−7.50530	−0.288027
$680$	0	0
$681$	7.06944	0.270901
$682$	0	0
$683$	−17.4545	−0.667878	−0.333939	−	0.942595i	$-0.608378\pi$
$683$	−17.4545	−0.667878	−0.333939	+	0.942595i	$0.608378\pi$
$684$	0	0
$685$	36.8847	1.40929
$686$	0	0
$687$	−81.5582	−3.11164
$688$	0	0
$689$	−1.50368	−0.0572857
$690$	0	0
$691$	2.95460	0.112398	0.0561992	−	0.998420i	$-0.482102\pi$
$691$	2.95460	0.112398	0.0561992	+	0.998420i	$0.482102\pi$
$692$	0	0
$693$	−4.95665	−0.188288
$694$	0	0
$695$	−28.4367	−1.07866
$696$	0	0
$697$	4.43073	0.167826
$698$	0	0
$699$	−22.5239	−0.851931
$700$	0	0
$701$	33.6736	1.27184	0.635918	−	0.771757i	$-0.280622\pi$
$701$	33.6736	1.27184	0.635918	+	0.771757i	$0.280622\pi$
$702$	0	0
$703$	−33.5231	−1.26435
$704$	0	0
$705$	−4.33039	−0.163092
$706$	0	0
$707$	−1.12014	−0.0421274
$708$	0	0
$709$	4.54524	0.170700	0.0853500	−	0.996351i	$-0.472799\pi$
$709$	4.54524	0.170700	0.0853500	+	0.996351i	$0.472799\pi$
$710$	0	0
$711$	45.7947	1.71744
$712$	0	0
$713$	−10.3820	−0.388808
$714$	0	0
$715$	−1.92109	−0.0718445
$716$	0	0
$717$	−17.2907	−0.645734
$718$	0	0
$719$	4.41887	0.164796	0.0823979	−	0.996600i	$-0.473742\pi$
$719$	4.41887	0.164796	0.0823979	+	0.996600i	$0.473742\pi$
$720$	0	0
$721$	−0.999512	−0.0372238
$722$	0	0
$723$	17.2849	0.642834
$724$	0	0
$725$	1.03049	0.0382713
$726$	0	0
$727$	29.7898	1.10484	0.552422	−	0.833565i	$-0.313704\pi$
$727$	29.7898	1.10484	0.552422	+	0.833565i	$0.313704\pi$
$728$	0	0
$729$	−43.2432	−1.60160
$730$	0	0
$731$	4.77445	0.176589
$732$	0	0
$733$	15.3862	0.568301	0.284150	−	0.958780i	$-0.408288\pi$
$733$	15.3862	0.568301	0.284150	+	0.958780i	$0.408288\pi$
$734$	0	0
$735$	5.41891	0.199880
$736$	0	0
$737$	12.8578	0.473622
$738$	0	0
$739$	−2.58332	−0.0950291	−0.0475146	−	0.998871i	$-0.515130\pi$
$739$	−2.58332	−0.0950291	−0.0475146	+	0.998871i	$0.515130\pi$
$740$	0	0
$741$	−8.73730	−0.320973
$742$	0	0
$743$	1.52107	0.0558028	0.0279014	−	0.999611i	$-0.491118\pi$
$743$	1.52107	0.0558028	0.0279014	+	0.999611i	$0.491118\pi$
$744$	0	0
$745$	21.6588	0.793517
$746$	0	0
$747$	31.1529	1.13982
$748$	0	0
$749$	−5.56947	−0.203504
$750$	0	0
$751$	9.78675	0.357124	0.178562	−	0.983929i	$-0.442856\pi$
$751$	9.78675	0.357124	0.178562	+	0.983929i	$0.442856\pi$
$752$	0	0
$753$	−45.5603	−1.66031
$754$	0	0
$755$	−7.85948	−0.286036
$756$	0	0
$757$	−37.0214	−1.34557	−0.672784	−	0.739839i	$-0.734902\pi$
$757$	−37.0214	−1.34557	−0.672784	+	0.739839i	$0.734902\pi$
$758$	0	0
$759$	10.5062	0.381350
$760$	0	0
$761$	9.66051	0.350193	0.175097	−	0.984551i	$-0.443976\pi$
$761$	9.66051	0.350193	0.175097	+	0.984551i	$0.443976\pi$
$762$	0	0
$763$	−1.41533	−0.0512384
$764$	0	0
$765$	9.18987	0.332261
$766$	0	0
$767$	3.65311	0.131906
$768$	0	0
$769$	−22.6345	−0.816222	−0.408111	−	0.912932i	$-0.633812\pi$
$769$	−22.6345	−0.816222	−0.408111	+	0.912932i	$0.633812\pi$
$770$	0	0
$771$	21.7006	0.781529
$772$	0	0
$773$	−10.4568	−0.376107	−0.188053	−	0.982159i	$-0.560218\pi$
$773$	−10.4568	−0.376107	−0.188053	+	0.982159i	$0.560218\pi$
$774$	0	0
$775$	−3.64991	−0.131109
$776$	0	0
$777$	−30.5279	−1.09518
$778$	0	0
$779$	−14.2204	−0.509500
$780$	0	0
$781$	4.44026	0.158885
$782$	0	0
$783$	4.34350	0.155224
$784$	0	0
$785$	24.2491	0.865487
$786$	0	0
$787$	18.3322	0.653471	0.326735	−	0.945116i	$-0.394051\pi$
$787$	18.3322	0.653471	0.326735	+	0.945116i	$0.394051\pi$
$788$	0	0
$789$	37.1303	1.32187
$790$	0	0
$791$	−0.238232	−0.00847057
$792$	0	0
$793$	14.3415	0.509282
$794$	0	0
$795$	−8.14832	−0.288991
$796$	0	0
$797$	−41.3042	−1.46307	−0.731535	−	0.681804i	$-0.761196\pi$
$797$	−41.3042	−1.46307	−0.731535	+	0.681804i	$0.761196\pi$
$798$	0	0
$799$	0.771239	0.0272845
$800$	0	0
$801$	22.1550	0.782810
$802$	0	0
$803$	−1.16794	−0.0412156
$804$	0	0
$805$	−7.15527	−0.252190
$806$	0	0
$807$	4.57630	0.161093
$808$	0	0
$809$	7.79972	0.274224	0.137112	−	0.990556i	$-0.456218\pi$
$809$	7.79972	0.274224	0.137112	+	0.990556i	$0.456218\pi$
$810$	0	0
$811$	49.1702	1.72660	0.863300	−	0.504692i	$-0.168394\pi$
$811$	49.1702	1.72660	0.863300	+	0.504692i	$0.168394\pi$
$812$	0	0
$813$	9.76253	0.342387
$814$	0	0
$815$	3.03548	0.106328
$816$	0	0
$817$	−15.3236	−0.536105
$818$	0	0
$819$	−4.95665	−0.173200
$820$	0	0
$821$	22.3354	0.779511	0.389755	−	0.920918i	$-0.372560\pi$
$821$	22.3354	0.779511	0.389755	+	0.920918i	$0.372560\pi$
$822$	0	0
$823$	−43.1116	−1.50277	−0.751387	−	0.659861i	$-0.770615\pi$
$823$	−43.1116	−1.50277	−0.751387	+	0.659861i	$0.770615\pi$
$824$	0	0
$825$	3.69358	0.128594
$826$	0	0
$827$	−21.2067	−0.737431	−0.368715	−	0.929542i	$-0.620202\pi$
$827$	−21.2067	−0.737431	−0.368715	+	0.929542i	$0.620202\pi$
$828$	0	0
$829$	6.96521	0.241912	0.120956	−	0.992658i	$-0.461404\pi$
$829$	6.96521	0.241912	0.120956	+	0.992658i	$0.461404\pi$
$830$	0	0
$831$	3.98556	0.138258
$832$	0	0
$833$	−0.965104	−0.0334389
$834$	0	0
$835$	20.0993	0.695565
$836$	0	0
$837$	−15.3844	−0.531762
$838$	0	0
$839$	−18.8012	−0.649090	−0.324545	−	0.945870i	$-0.605211\pi$
$839$	−18.8012	−0.649090	−0.324545	+	0.945870i	$0.605211\pi$
$840$	0	0
$841$	−28.3807	−0.978644
$842$	0	0
$843$	22.1107	0.761535
$844$	0	0
$845$	−1.92109	−0.0660874
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−32.9617	−1.13124
$850$	0	0
$851$	40.3098	1.38180
$852$	0	0
$853$	21.1583	0.724448	0.362224	−	0.932091i	$-0.382018\pi$
$853$	21.1583	0.724448	0.362224	+	0.932091i	$0.382018\pi$
$854$	0	0
$855$	−29.4949	−1.00871
$856$	0	0
$857$	−21.4458	−0.732575	−0.366288	−	0.930502i	$-0.619371\pi$
$857$	−21.4458	−0.732575	−0.366288	+	0.930502i	$0.619371\pi$
$858$	0	0
$859$	−27.6487	−0.943360	−0.471680	−	0.881770i	$-0.656352\pi$
$859$	−27.6487	−0.943360	−0.471680	+	0.881770i	$0.656352\pi$
$860$	0	0
$861$	−12.9499	−0.441331
$862$	0	0
$863$	−7.47498	−0.254451	−0.127226	−	0.991874i	$-0.540607\pi$
$863$	−7.47498	−0.254451	−0.127226	+	0.991874i	$0.540607\pi$
$864$	0	0
$865$	−17.1137	−0.581885
$866$	0	0
$867$	45.3255	1.53933
$868$	0	0
$869$	9.23904	0.313413
$870$	0	0
$871$	12.8578	0.435669
$872$	0	0
$873$	37.2012	1.25907
$874$	0	0
$875$	−12.1210	−0.409763
$876$	0	0
$877$	−19.3194	−0.652368	−0.326184	−	0.945306i	$-0.605763\pi$
$877$	−19.3194	−0.652368	−0.326184	+	0.945306i	$0.605763\pi$
$878$	0	0
$879$	76.6739	2.58614
$880$	0	0
$881$	−17.5591	−0.591579	−0.295790	−	0.955253i	$-0.595583\pi$
$881$	−17.5591	−0.591579	−0.295790	+	0.955253i	$0.595583\pi$
$882$	0	0
$883$	48.9084	1.64590	0.822950	−	0.568114i	$-0.192327\pi$
$883$	48.9084	1.64590	0.822950	+	0.568114i	$0.192327\pi$
$884$	0	0
$885$	19.7959	0.665432
$886$	0	0
$887$	17.9645	0.603187	0.301594	−	0.953437i	$-0.402481\pi$
$887$	17.9645	0.603187	0.301594	+	0.953437i	$0.402481\pi$
$888$	0	0
$889$	2.54582	0.0853840
$890$	0	0
$891$	0.698463	0.0233994
$892$	0	0
$893$	−2.47529	−0.0828326
$894$	0	0
$895$	34.2328	1.14428
$896$	0	0
$897$	10.5062	0.350791
$898$	0	0
$899$	−2.19362	−0.0731612
$900$	0	0
$901$	1.45121	0.0483468
$902$	0	0
$903$	−13.9545	−0.464377
$904$	0	0
$905$	29.3144	0.974442
$906$	0	0
$907$	21.3944	0.710390	0.355195	−	0.934792i	$-0.384414\pi$
$907$	21.3944	0.710390	0.355195	+	0.934792i	$0.384414\pi$
$908$	0	0
$909$	5.55217	0.184154
$910$	0	0
$911$	−47.8635	−1.58579	−0.792894	−	0.609359i	$-0.791427\pi$
$911$	−47.8635	−1.58579	−0.792894	+	0.609359i	$0.791427\pi$
$912$	0	0
$913$	6.28506	0.208005
$914$	0	0
$915$	77.7154	2.56919
$916$	0	0
$917$	−7.05388	−0.232940
$918$	0	0
$919$	44.9869	1.48398	0.741991	−	0.670410i	$-0.233882\pi$
$919$	44.9869	1.48398	0.741991	+	0.670410i	$0.233882\pi$
$920$	0	0
$921$	−69.7978	−2.29992
$922$	0	0
$923$	4.44026	0.146153
$924$	0	0
$925$	14.1714	0.465954
$926$	0	0
$927$	4.95424	0.162719
$928$	0	0
$929$	21.9391	0.719800	0.359900	−	0.932991i	$-0.382811\pi$
$929$	21.9391	0.719800	0.359900	+	0.932991i	$0.382811\pi$
$930$	0	0
$931$	3.09750	0.101517
$932$	0	0
$933$	54.4162	1.78151
$934$	0	0
$935$	1.85405	0.0606338
$936$	0	0
$937$	−54.4149	−1.77766	−0.888829	−	0.458238i	$-0.848481\pi$
$937$	−54.4149	−1.77766	−0.888829	+	0.458238i	$0.848481\pi$
$938$	0	0
$939$	−33.7299	−1.10073
$940$	0	0
$941$	−20.7915	−0.677782	−0.338891	−	0.940826i	$-0.610052\pi$
$941$	−20.7915	−0.677782	−0.338891	+	0.940826i	$0.610052\pi$
$942$	0	0
$943$	17.0994	0.556833
$944$	0	0
$945$	−10.6029	−0.344914
$946$	0	0
$947$	47.0249	1.52810	0.764052	−	0.645154i	$-0.223207\pi$
$947$	47.0249	1.52810	0.764052	+	0.645154i	$0.223207\pi$
$948$	0	0
$949$	−1.16794	−0.0379129
$950$	0	0
$951$	−45.0257	−1.46006
$952$	0	0
$953$	−13.8050	−0.447187	−0.223594	−	0.974682i	$-0.571779\pi$
$953$	−13.8050	−0.447187	−0.223594	+	0.974682i	$0.571779\pi$
$954$	0	0
$955$	−24.3131	−0.786754
$956$	0	0
$957$	2.21986	0.0717579
$958$	0	0
$959$	19.1999	0.619998
$960$	0	0
$961$	−23.2304	−0.749367
$962$	0	0
$963$	27.6059	0.889588
$964$	0	0
$965$	32.6514	1.05109
$966$	0	0
$967$	26.2050	0.842697	0.421348	−	0.906899i	$-0.361557\pi$
$967$	26.2050	0.842697	0.421348	+	0.906899i	$0.361557\pi$
$968$	0	0
$969$	8.43240	0.270888
$970$	0	0
$971$	−11.9621	−0.383883	−0.191942	−	0.981406i	$-0.561478\pi$
$971$	−11.9621	−0.383883	−0.191942	+	0.981406i	$0.561478\pi$
$972$	0	0
$973$	−14.8024	−0.474543
$974$	0	0
$975$	3.69358	0.118289
$976$	0	0
$977$	19.7252	0.631064	0.315532	−	0.948915i	$-0.397817\pi$
$977$	19.7252	0.631064	0.315532	+	0.948915i	$0.397817\pi$
$978$	0	0
$979$	4.46976	0.142854
$980$	0	0
$981$	7.01531	0.223982
$982$	0	0
$983$	3.94017	0.125672	0.0628360	−	0.998024i	$-0.479985\pi$
$983$	3.94017	0.125672	0.0628360	+	0.998024i	$0.479985\pi$
$984$	0	0
$985$	−18.6396	−0.593907
$986$	0	0
$987$	−2.25414	−0.0717500
$988$	0	0
$989$	18.4259	0.585909
$990$	0	0
$991$	−34.6737	−1.10145	−0.550723	−	0.834688i	$-0.685648\pi$
$991$	−34.6737	−1.10145	−0.550723	+	0.834688i	$0.685648\pi$
$992$	0	0
$993$	97.0791	3.08071
$994$	0	0
$995$	36.3813	1.15336
$996$	0	0
$997$	−23.9427	−0.758274	−0.379137	−	0.925341i	$-0.623779\pi$
$997$	−23.9427	−0.758274	−0.379137	+	0.925341i	$0.623779\pi$
$998$	0	0
$999$	59.7326	1.88985

Introduction	Features
Universe	News

Level:	\( N \)	=	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	=	\( 2 \)
Character orbit:	\([\chi]\)	=	8008.a (trivial)

\(f(q)\)	\(=\)	\(q-2.82075 q^{3} -1.92109 q^{5} -1.00000 q^{7} +4.95665 q^{9} +O(q^{10})\)	\(q-2.82075 q^{3} -1.92109 q^{5} -1.00000 q^{7} +4.95665 q^{9} +1.00000 q^{11} +1.00000 q^{13} +5.41891 q^{15} -0.965104 q^{17} +3.09750 q^{19} +2.82075 q^{21} -3.72460 q^{23} -1.30943 q^{25} -5.51924 q^{27} -0.786974 q^{29} +2.78741 q^{31} -2.82075 q^{33} +1.92109 q^{35} -10.8226 q^{37} -2.82075 q^{39} -4.59093 q^{41} -4.94708 q^{43} -9.52216 q^{45} -0.799126 q^{47} +1.00000 q^{49} +2.72232 q^{51} -1.50368 q^{53} -1.92109 q^{55} -8.73730 q^{57} +3.65311 q^{59} +14.3415 q^{61} -4.95665 q^{63} -1.92109 q^{65} +12.8578 q^{67} +10.5062 q^{69} +4.44026 q^{71} -1.16794 q^{73} +3.69358 q^{75} -1.00000 q^{77} +9.23904 q^{79} +0.698463 q^{81} +6.28506 q^{83} +1.85405 q^{85} +2.21986 q^{87} +4.46976 q^{89} -1.00000 q^{91} -7.86259 q^{93} -5.95057 q^{95} +7.50530 q^{97} +4.95665 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9q + q^{3} - 4q^{5} - 9q^{7} + 4q^{9} + O(q^{10}) \)	\( 9q + q^{3} - 4q^{5} - 9q^{7} + 4q^{9} + 9q^{11} + 9q^{13} - 9q^{15} - 11q^{17} + 10q^{19} - q^{21} - 14q^{23} - q^{25} - 5q^{27} - 10q^{29} + 5q^{31} + q^{33} + 4q^{35} - 16q^{37} + q^{39} + 2q^{41} + 4q^{43} - 30q^{45} + 9q^{49} + 3q^{51} - 23q^{53} - 4q^{55} + 14q^{57} + 9q^{59} - 14q^{61} - 4q^{63} - 4q^{65} + 8q^{67} - 26q^{69} - 20q^{71} - 23q^{73} + 32q^{75} - 9q^{77} + 2q^{79} - 11q^{81} - 9q^{83} - 3q^{85} - 7q^{87} - 6q^{89} - 9q^{91} - 19q^{93} - 4q^{95} - 3q^{97} + 4q^{99} + O(q^{100}) \)

Introduction and more

L-functions

Modular Forms

Varieties

Fields

Representations

Groups

Knowledge

Properties

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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	8008.2.a.p.1.1

Level	8008
Weight	2
Character	8008.1
Self dual	Yes
Analytic conductor	63.944
Analytic rank	1
Dimension	9
CM	No
Inner twists	1

Self dual:	Yes
Analytic conductor:	\(63.9442019386\)
Analytic rank:	\(1\)
Dimension:	\(9\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Coefficient ring:	\(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index:	\( 1 \)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$