LMFDB - Embedded Newform 4028.2.a.d.1.5

Newspace parameters

Level:	$ N $	=	$ 4028 = 2^{2} \cdot 19 \cdot 53 $
Weight:	$ k $	=	$ 2 $
Character orbit:	$[\chi]$	=	4028.a (trivial)

Newform invariants

Self dual:	Yes
Analytic conductor:	$32.1637419342$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
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			1.5
Root			$2.05170$
Character	$\chi$	=	4028.1

$q$-expansion

$f(q)$	$=$	$q-2.05170 q^{3} +2.67326 q^{5} +0.688434 q^{7} +1.20949 q^{9} +O(q^{10})$	$q-2.05170 q^{3} +2.67326 q^{5} +0.688434 q^{7} +1.20949 q^{9} -5.36452 q^{11} +1.75191 q^{13} -5.48473 q^{15} +3.58294 q^{17} -1.00000 q^{19} -1.41246 q^{21} +1.71126 q^{23} +2.14631 q^{25} +3.67360 q^{27} -10.2791 q^{29} -1.27276 q^{31} +11.0064 q^{33} +1.84036 q^{35} +1.26348 q^{37} -3.59440 q^{39} +9.70817 q^{41} +2.83145 q^{43} +3.23327 q^{45} -7.30266 q^{47} -6.52606 q^{49} -7.35114 q^{51} +1.00000 q^{53} -14.3408 q^{55} +2.05170 q^{57} -7.26302 q^{59} -14.4852 q^{61} +0.832651 q^{63} +4.68331 q^{65} -4.38693 q^{67} -3.51100 q^{69} -0.131451 q^{71} +9.46729 q^{73} -4.40359 q^{75} -3.69312 q^{77} -7.19961 q^{79} -11.1656 q^{81} -6.22376 q^{83} +9.57814 q^{85} +21.0898 q^{87} +6.04098 q^{89} +1.20607 q^{91} +2.61132 q^{93} -2.67326 q^{95} +10.7584 q^{97} -6.48832 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19q - 4q^{3} - 2q^{5} - 11q^{7} + 19q^{9} + O(q^{10}) $	$ 19q - 4q^{3} - 2q^{5} - 11q^{7} + 19q^{9} - 11q^{11} + q^{13} - 20q^{15} - q^{17} - 19q^{19} - 10q^{21} - 16q^{23} + 21q^{25} - 4q^{27} - 9q^{31} + 7q^{33} - 25q^{37} - 25q^{39} + q^{41} - 41q^{43} - 27q^{45} - 29q^{47} + 14q^{49} - 24q^{51} + 19q^{53} - 28q^{55} + 4q^{57} - 42q^{59} + q^{61} - 41q^{63} + 2q^{65} - 41q^{67} - 25q^{69} - 20q^{73} + 11q^{75} - 19q^{77} - 38q^{79} + 23q^{81} - 36q^{83} - 58q^{85} - 30q^{87} - 25q^{89} - 55q^{91} - 38q^{93} + 2q^{95} - 13q^{97} + q^{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
$3$	−2.05170	−1.18455	−0.592276	−	0.805735i	$-0.701770\pi$
$3$	−2.05170	−1.18455	−0.592276	+	0.805735i	$0.701770\pi$
$4$	0	0
$5$	2.67326	1.19552	0.597759	−	0.801676i	$-0.296058\pi$
$5$	2.67326	1.19552	0.597759	+	0.801676i	$0.296058\pi$
$6$	0	0
$7$	0.688434	0.260203	0.130102	−	0.991501i	$-0.458470\pi$
$7$	0.688434	0.260203	0.130102	+	0.991501i	$0.458470\pi$
$8$	0	0
$9$	1.20949	0.403162
$10$	0	0
$11$	−5.36452	−1.61746	−0.808732	−	0.588177i	$-0.799846\pi$
$11$	−5.36452	−1.61746	−0.808732	+	0.588177i	$0.799846\pi$
$12$	0	0
$13$	1.75191	0.485892	0.242946	−	0.970040i	$-0.421886\pi$
$13$	1.75191	0.485892	0.242946	+	0.970040i	$0.421886\pi$
$14$	0	0
$15$	−5.48473	−1.41615
$16$	0	0
$17$	3.58294	0.868992	0.434496	−	0.900674i	$-0.356927\pi$
$17$	3.58294	0.868992	0.434496	+	0.900674i	$0.356927\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.41246	−0.308224
$22$	0	0
$23$	1.71126	0.356823	0.178411	−	0.983956i	$-0.442904\pi$
$23$	1.71126	0.356823	0.178411	+	0.983956i	$0.442904\pi$
$24$	0	0
$25$	2.14631	0.429262
$26$	0	0
$27$	3.67360	0.706985
$28$	0	0
$29$	−10.2791	−1.90879	−0.954395	−	0.298547i	$-0.903498\pi$
$29$	−10.2791	−1.90879	−0.954395	+	0.298547i	$0.903498\pi$
$30$	0	0
$31$	−1.27276	−0.228594	−0.114297	−	0.993447i	$-0.536462\pi$
$31$	−1.27276	−0.228594	−0.114297	+	0.993447i	$0.536462\pi$
$32$	0	0
$33$	11.0064	1.91597
$34$	0	0
$35$	1.84036	0.311078
$36$	0	0
$37$	1.26348	0.207715	0.103857	−	0.994592i	$-0.466881\pi$
$37$	1.26348	0.207715	0.103857	+	0.994592i	$0.466881\pi$
$38$	0	0
$39$	−3.59440	−0.575564
$40$	0	0
$41$	9.70817	1.51616	0.758081	−	0.652161i	$-0.226137\pi$
$41$	9.70817	1.51616	0.758081	+	0.652161i	$0.226137\pi$
$42$	0	0
$43$	2.83145	0.431793	0.215896	−	0.976416i	$-0.430733\pi$
$43$	2.83145	0.431793	0.215896	+	0.976416i	$0.430733\pi$
$44$	0	0
$45$	3.23327	0.481987
$46$	0	0
$47$	−7.30266	−1.06520	−0.532601	−	0.846366i	$-0.678785\pi$
$47$	−7.30266	−1.06520	−0.532601	+	0.846366i	$0.678785\pi$
$48$	0	0
$49$	−6.52606	−0.932294
$50$	0	0
$51$	−7.35114	−1.02937
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	−14.3408	−1.93371
$56$	0	0
$57$	2.05170	0.271755
$58$	0	0
$59$	−7.26302	−0.945565	−0.472782	−	0.881179i	$-0.656750\pi$
$59$	−7.26302	−0.945565	−0.472782	+	0.881179i	$0.656750\pi$
$60$	0	0
$61$	−14.4852	−1.85464	−0.927318	−	0.374274i	$-0.877892\pi$
$61$	−14.4852	−1.85464	−0.927318	+	0.374274i	$0.877892\pi$
$62$	0	0
$63$	0.832651	0.104904
$64$	0	0
$65$	4.68331	0.580893
$66$	0	0
$67$	−4.38693	−0.535949	−0.267974	−	0.963426i	$-0.586354\pi$
$67$	−4.38693	−0.535949	−0.267974	+	0.963426i	$0.586354\pi$
$68$	0	0
$69$	−3.51100	−0.422675
$70$	0	0
$71$	−0.131451	−0.0156003	−0.00780015	−	0.999970i	$-0.502483\pi$
$71$	−0.131451	−0.0156003	−0.00780015	+	0.999970i	$0.502483\pi$
$72$	0	0
$73$	9.46729	1.10806	0.554031	−	0.832496i	$-0.313089\pi$
$73$	9.46729	1.10806	0.554031	+	0.832496i	$0.313089\pi$
$74$	0	0
$75$	−4.40359	−0.508483
$76$	0	0
$77$	−3.69312	−0.420870
$78$	0	0
$79$	−7.19961	−0.810020	−0.405010	−	0.914312i	$-0.632732\pi$
$79$	−7.19961	−0.810020	−0.405010	+	0.914312i	$0.632732\pi$
$80$	0	0
$81$	−11.1656	−1.24062
$82$	0	0
$83$	−6.22376	−0.683146	−0.341573	−	0.939855i	$-0.610960\pi$
$83$	−6.22376	−0.683146	−0.341573	+	0.939855i	$0.610960\pi$
$84$	0	0
$85$	9.57814	1.03889
$86$	0	0
$87$	21.0898	2.26106
$88$	0	0
$89$	6.04098	0.640343	0.320171	−	0.947360i	$-0.396260\pi$
$89$	6.04098	0.640343	0.320171	+	0.947360i	$0.396260\pi$
$90$	0	0
$91$	1.20607	0.126431
$92$	0	0
$93$	2.61132	0.270781
$94$	0	0
$95$	−2.67326	−0.274271
$96$	0	0
$97$	10.7584	1.09235	0.546176	−	0.837670i	$-0.316083\pi$
$97$	10.7584	1.09235	0.546176	+	0.837670i	$0.316083\pi$
$98$	0	0
$99$	−6.48832	−0.652100
$100$	0	0
$101$	1.43012	0.142302	0.0711512	−	0.997466i	$-0.477333\pi$
$101$	1.43012	0.142302	0.0711512	+	0.997466i	$0.477333\pi$
$102$	0	0
$103$	−4.52410	−0.445773	−0.222886	−	0.974844i	$-0.571548\pi$
$103$	−4.52410	−0.445773	−0.222886	+	0.974844i	$0.571548\pi$
$104$	0	0
$105$	−3.77588	−0.368488
$106$	0	0
$107$	9.61719	0.929728	0.464864	−	0.885382i	$-0.346103\pi$
$107$	9.61719	0.929728	0.464864	+	0.885382i	$0.346103\pi$
$108$	0	0
$109$	2.61052	0.250043	0.125021	−	0.992154i	$-0.460100\pi$
$109$	2.61052	0.250043	0.125021	+	0.992154i	$0.460100\pi$
$110$	0	0
$111$	−2.59229	−0.246049
$112$	0	0
$113$	16.2523	1.52889	0.764444	−	0.644690i	$-0.223013\pi$
$113$	16.2523	1.52889	0.764444	+	0.644690i	$0.223013\pi$
$114$	0	0
$115$	4.57464	0.426588
$116$	0	0
$117$	2.11891	0.195893
$118$	0	0
$119$	2.46662	0.226115
$120$	0	0
$121$	17.7781	1.61619
$122$	0	0
$123$	−19.9183	−1.79597
$124$	0	0
$125$	−7.62865	−0.682327
$126$	0	0
$127$	−22.3178	−1.98038	−0.990192	−	0.139711i	$-0.955383\pi$
$127$	−22.3178	−1.98038	−0.990192	+	0.139711i	$0.955383\pi$
$128$	0	0
$129$	−5.80930	−0.511481
$130$	0	0
$131$	−9.70655	−0.848065	−0.424033	−	0.905647i	$-0.639386\pi$
$131$	−9.70655	−0.848065	−0.424033	+	0.905647i	$0.639386\pi$
$132$	0	0
$133$	−0.688434	−0.0596948
$134$	0	0
$135$	9.82049	0.845213
$136$	0	0
$137$	11.6636	0.996488	0.498244	−	0.867037i	$-0.333978\pi$
$137$	11.6636	0.996488	0.498244	+	0.867037i	$0.333978\pi$
$138$	0	0
$139$	3.49402	0.296359	0.148179	−	0.988961i	$-0.452659\pi$
$139$	3.49402	0.296359	0.148179	+	0.988961i	$0.452659\pi$
$140$	0	0
$141$	14.9829	1.26179
$142$	0	0
$143$	−9.39816	−0.785913
$144$	0	0
$145$	−27.4788	−2.28199
$146$	0	0
$147$	13.3895	1.10435
$148$	0	0
$149$	1.16712	0.0956141	0.0478070	−	0.998857i	$-0.484777\pi$
$149$	1.16712	0.0956141	0.0478070	+	0.998857i	$0.484777\pi$
$150$	0	0
$151$	−5.68464	−0.462610	−0.231305	−	0.972881i	$-0.574299\pi$
$151$	−5.68464	−0.462610	−0.231305	+	0.972881i	$0.574299\pi$
$152$	0	0
$153$	4.33352	0.350345
$154$	0	0
$155$	−3.40241	−0.273288
$156$	0	0
$157$	−10.7537	−0.858241	−0.429120	−	0.903247i	$-0.641176\pi$
$157$	−10.7537	−0.858241	−0.429120	+	0.903247i	$0.641176\pi$
$158$	0	0
$159$	−2.05170	−0.162711
$160$	0	0
$161$	1.17809	0.0928465
$162$	0	0
$163$	3.77174	0.295426	0.147713	−	0.989030i	$-0.452809\pi$
$163$	3.77174	0.295426	0.147713	+	0.989030i	$0.452809\pi$
$164$	0	0
$165$	29.4230	2.29058
$166$	0	0
$167$	−10.0827	−0.780221	−0.390110	−	0.920768i	$-0.627563\pi$
$167$	−10.0827	−0.780221	−0.390110	+	0.920768i	$0.627563\pi$
$168$	0	0
$169$	−9.93081	−0.763909
$170$	0	0
$171$	−1.20949	−0.0924917
$172$	0	0
$173$	−13.8105	−1.05000	−0.524998	−	0.851104i	$-0.675934\pi$
$173$	−13.8105	−1.05000	−0.524998	+	0.851104i	$0.675934\pi$
$174$	0	0
$175$	1.47759	0.111696
$176$	0	0
$177$	14.9016	1.12007
$178$	0	0
$179$	−1.58868	−0.118744	−0.0593718	−	0.998236i	$-0.518910\pi$
$179$	−1.58868	−0.118744	−0.0593718	+	0.998236i	$0.518910\pi$
$180$	0	0
$181$	−19.3651	−1.43939	−0.719697	−	0.694289i	$-0.755719\pi$
$181$	−19.3651	−1.43939	−0.719697	+	0.694289i	$0.755719\pi$
$182$	0	0
$183$	29.7193	2.19691
$184$	0	0
$185$	3.37761	0.248327
$186$	0	0
$187$	−19.2208	−1.40556
$188$	0	0
$189$	2.52903	0.183960
$190$	0	0
$191$	−20.9567	−1.51637	−0.758187	−	0.652037i	$-0.773914\pi$
$191$	−20.9567	−1.51637	−0.758187	+	0.652037i	$0.773914\pi$
$192$	0	0
$193$	−6.33053	−0.455682	−0.227841	−	0.973698i	$-0.573167\pi$
$193$	−6.33053	−0.455682	−0.227841	+	0.973698i	$0.573167\pi$
$194$	0	0
$195$	−9.60876	−0.688097
$196$	0	0
$197$	14.9006	1.06163	0.530813	−	0.847489i	$-0.321887\pi$
$197$	14.9006	1.06163	0.530813	+	0.847489i	$0.321887\pi$
$198$	0	0
$199$	−21.0567	−1.49267	−0.746335	−	0.665570i	$-0.768188\pi$
$199$	−21.0567	−1.49267	−0.746335	+	0.665570i	$0.768188\pi$
$200$	0	0
$201$	9.00068	0.634859
$202$	0	0
$203$	−7.07651	−0.496674
$204$	0	0
$205$	25.9524	1.81260
$206$	0	0
$207$	2.06975	0.143857
$208$	0	0
$209$	5.36452	0.371072
$210$	0	0
$211$	3.83490	0.264006	0.132003	−	0.991249i	$-0.457859\pi$
$211$	3.83490	0.264006	0.132003	+	0.991249i	$0.457859\pi$
$212$	0	0
$213$	0.269698	0.0184794
$214$	0	0
$215$	7.56921	0.516216
$216$	0	0
$217$	−0.876209	−0.0594809
$218$	0	0
$219$	−19.4241	−1.31256
$220$	0	0
$221$	6.27699	0.422236
$222$	0	0
$223$	26.2227	1.75600	0.878001	−	0.478658i	$-0.158877\pi$
$223$	26.2227	1.75600	0.878001	+	0.478658i	$0.158877\pi$
$224$	0	0
$225$	2.59593	0.173062
$226$	0	0
$227$	6.75562	0.448386	0.224193	−	0.974545i	$-0.428025\pi$
$227$	6.75562	0.448386	0.224193	+	0.974545i	$0.428025\pi$
$228$	0	0
$229$	−7.52706	−0.497402	−0.248701	−	0.968580i	$-0.580004\pi$
$229$	−7.52706	−0.497402	−0.248701	+	0.968580i	$0.580004\pi$
$230$	0	0
$231$	7.57718	0.498542
$232$	0	0
$233$	−19.6746	−1.28893	−0.644464	−	0.764634i	$-0.722920\pi$
$233$	−19.6746	−1.28893	−0.644464	+	0.764634i	$0.722920\pi$
$234$	0	0
$235$	−19.5219	−1.27347
$236$	0	0
$237$	14.7715	0.959510
$238$	0	0
$239$	2.86141	0.185089	0.0925445	−	0.995709i	$-0.470500\pi$
$239$	2.86141	0.185089	0.0925445	+	0.995709i	$0.470500\pi$
$240$	0	0
$241$	−14.2261	−0.916386	−0.458193	−	0.888853i	$-0.651503\pi$
$241$	−14.2261	−0.916386	−0.458193	+	0.888853i	$0.651503\pi$
$242$	0	0
$243$	11.8877	0.762596
$244$	0	0
$245$	−17.4458	−1.11457
$246$	0	0
$247$	−1.75191	−0.111471
$248$	0	0
$249$	12.7693	0.809222
$250$	0	0
$251$	−1.21246	−0.0765297	−0.0382648	−	0.999268i	$-0.512183\pi$
$251$	−1.21246	−0.0765297	−0.0382648	+	0.999268i	$0.512183\pi$
$252$	0	0
$253$	−9.18010	−0.577148
$254$	0	0
$255$	−19.6515	−1.23062
$256$	0	0
$257$	−15.9057	−0.992173	−0.496086	−	0.868273i	$-0.665230\pi$
$257$	−15.9057	−0.992173	−0.496086	+	0.868273i	$0.665230\pi$
$258$	0	0
$259$	0.869822	0.0540481
$260$	0	0
$261$	−12.4325	−0.769552
$262$	0	0
$263$	21.2856	1.31253	0.656263	−	0.754532i	$-0.272136\pi$
$263$	21.2856	1.31253	0.656263	+	0.754532i	$0.272136\pi$
$264$	0	0
$265$	2.67326	0.164217
$266$	0	0
$267$	−12.3943	−0.758519
$268$	0	0
$269$	−22.0518	−1.34452	−0.672261	−	0.740315i	$-0.734677\pi$
$269$	−22.0518	−1.34452	−0.672261	+	0.740315i	$0.734677\pi$
$270$	0	0
$271$	−27.2142	−1.65314	−0.826572	−	0.562831i	$-0.809712\pi$
$271$	−27.2142	−1.65314	−0.826572	+	0.562831i	$0.809712\pi$
$272$	0	0
$273$	−2.47450	−0.149764
$274$	0	0
$275$	−11.5139	−0.694316
$276$	0	0
$277$	0.665156	0.0399654	0.0199827	−	0.999800i	$-0.493639\pi$
$277$	0.665156	0.0399654	0.0199827	+	0.999800i	$0.493639\pi$
$278$	0	0
$279$	−1.53938	−0.0921604
$280$	0	0
$281$	13.9904	0.834600	0.417300	−	0.908769i	$-0.362976\pi$
$281$	13.9904	0.834600	0.417300	+	0.908769i	$0.362976\pi$
$282$	0	0
$283$	5.64144	0.335349	0.167674	−	0.985842i	$-0.446374\pi$
$283$	5.64144	0.335349	0.167674	+	0.985842i	$0.446374\pi$
$284$	0	0
$285$	5.48473	0.324888
$286$	0	0
$287$	6.68343	0.394510
$288$	0	0
$289$	−4.16251	−0.244853
$290$	0	0
$291$	−22.0731	−1.29395
$292$	0	0
$293$	−30.8081	−1.79983	−0.899915	−	0.436065i	$-0.856372\pi$
$293$	−30.8081	−1.79983	−0.899915	+	0.436065i	$0.856372\pi$
$294$	0	0
$295$	−19.4159	−1.13044
$296$	0	0
$297$	−19.7071	−1.14352
$298$	0	0
$299$	2.99797	0.173377
$300$	0	0
$301$	1.94927	0.112354
$302$	0	0
$303$	−2.93419	−0.168565
$304$	0	0
$305$	−38.7226	−2.21725
$306$	0	0
$307$	20.1847	1.15200	0.576000	−	0.817450i	$-0.304613\pi$
$307$	20.1847	1.15200	0.576000	+	0.817450i	$0.304613\pi$
$308$	0	0
$309$	9.28211	0.528041
$310$	0	0
$311$	−15.8353	−0.897936	−0.448968	−	0.893548i	$-0.648208\pi$
$311$	−15.8353	−0.897936	−0.448968	+	0.893548i	$0.648208\pi$
$312$	0	0
$313$	−30.1257	−1.70280	−0.851402	−	0.524514i	$-0.824247\pi$
$313$	−30.1257	−1.70280	−0.851402	+	0.524514i	$0.824247\pi$
$314$	0	0
$315$	2.22589	0.125415
$316$	0	0
$317$	−9.37205	−0.526387	−0.263193	−	0.964743i	$-0.584776\pi$
$317$	−9.37205	−0.526387	−0.263193	+	0.964743i	$0.584776\pi$
$318$	0	0
$319$	55.1427	3.08740
$320$	0	0
$321$	−19.7316	−1.10131
$322$	0	0
$323$	−3.58294	−0.199360
$324$	0	0
$325$	3.76014	0.208575
$326$	0	0
$327$	−5.35602	−0.296189
$328$	0	0
$329$	−5.02740	−0.277169
$330$	0	0
$331$	13.8940	0.763682	0.381841	−	0.924228i	$-0.375290\pi$
$331$	13.8940	0.763682	0.381841	+	0.924228i	$0.375290\pi$
$332$	0	0
$333$	1.52816	0.0837427
$334$	0	0
$335$	−11.7274	−0.640736
$336$	0	0
$337$	−16.7493	−0.912392	−0.456196	−	0.889879i	$-0.650788\pi$
$337$	−16.7493	−0.912392	−0.456196	+	0.889879i	$0.650788\pi$
$338$	0	0
$339$	−33.3449	−1.81105
$340$	0	0
$341$	6.82774	0.369743
$342$	0	0
$343$	−9.31179	−0.502790
$344$	0	0
$345$	−9.38581	−0.505315
$346$	0	0
$347$	−18.4496	−0.990427	−0.495214	−	0.868771i	$-0.664910\pi$
$347$	−18.4496	−0.990427	−0.495214	+	0.868771i	$0.664910\pi$
$348$	0	0
$349$	−8.47158	−0.453473	−0.226737	−	0.973956i	$-0.572806\pi$
$349$	−8.47158	−0.453473	−0.226737	+	0.973956i	$0.572806\pi$
$350$	0	0
$351$	6.43582	0.343519
$352$	0	0
$353$	21.5749	1.14832	0.574158	−	0.818744i	$-0.305329\pi$
$353$	21.5749	1.14832	0.574158	+	0.818744i	$0.305329\pi$
$354$	0	0
$355$	−0.351401	−0.0186504
$356$	0	0
$357$	−5.06077	−0.267844
$358$	0	0
$359$	5.12932	0.270715	0.135358	−	0.990797i	$-0.456782\pi$
$359$	5.12932	0.270715	0.135358	+	0.990797i	$0.456782\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−36.4754	−1.91446
$364$	0	0
$365$	25.3085	1.32471
$366$	0	0
$367$	−1.04598	−0.0545998	−0.0272999	−	0.999627i	$-0.508691\pi$
$367$	−1.04598	−0.0545998	−0.0272999	+	0.999627i	$0.508691\pi$
$368$	0	0
$369$	11.7419	0.611259
$370$	0	0
$371$	0.688434	0.0357417
$372$	0	0
$373$	−4.18140	−0.216505	−0.108252	−	0.994123i	$-0.534525\pi$
$373$	−4.18140	−0.216505	−0.108252	+	0.994123i	$0.534525\pi$
$374$	0	0
$375$	15.6517	0.808252
$376$	0	0
$377$	−18.0081	−0.927466
$378$	0	0
$379$	1.97671	0.101537	0.0507683	−	0.998710i	$-0.483833\pi$
$379$	1.97671	0.101537	0.0507683	+	0.998710i	$0.483833\pi$
$380$	0	0
$381$	45.7895	2.34587
$382$	0	0
$383$	38.6302	1.97391	0.986955	−	0.160999i	$-0.0514716\pi$
$383$	38.6302	1.97391	0.986955	+	0.160999i	$0.0514716\pi$
$384$	0	0
$385$	−9.87266	−0.503157
$386$	0	0
$387$	3.42460	0.174082
$388$	0	0
$389$	−7.88051	−0.399558	−0.199779	−	0.979841i	$-0.564022\pi$
$389$	−7.88051	−0.399558	−0.199779	+	0.979841i	$0.564022\pi$
$390$	0	0
$391$	6.13135	0.310076
$392$	0	0
$393$	19.9150	1.00458
$394$	0	0
$395$	−19.2464	−0.968393
$396$	0	0
$397$	−15.9614	−0.801080	−0.400540	−	0.916279i	$-0.631177\pi$
$397$	−15.9614	−0.801080	−0.400540	+	0.916279i	$0.631177\pi$
$398$	0	0
$399$	1.41246	0.0707115
$400$	0	0
$401$	6.22351	0.310787	0.155394	−	0.987853i	$-0.450335\pi$
$401$	6.22351	0.310787	0.155394	+	0.987853i	$0.450335\pi$
$402$	0	0
$403$	−2.22976	−0.111072
$404$	0	0
$405$	−29.8485	−1.48319
$406$	0	0
$407$	−6.77796	−0.335971
$408$	0	0
$409$	20.9180	1.03433	0.517164	−	0.855886i	$-0.326988\pi$
$409$	20.9180	1.03433	0.517164	+	0.855886i	$0.326988\pi$
$410$	0	0
$411$	−23.9302	−1.18039
$412$	0	0
$413$	−5.00011	−0.246039
$414$	0	0
$415$	−16.6377	−0.816714
$416$	0	0
$417$	−7.16869	−0.351052
$418$	0	0
$419$	−18.3679	−0.897331	−0.448665	−	0.893700i	$-0.648100\pi$
$419$	−18.3679	−0.897331	−0.448665	+	0.893700i	$0.648100\pi$
$420$	0	0
$421$	20.3859	0.993548	0.496774	−	0.867880i	$-0.334518\pi$
$421$	20.3859	0.993548	0.496774	+	0.867880i	$0.334518\pi$
$422$	0	0
$423$	−8.83246	−0.429449
$424$	0	0
$425$	7.69011	0.373025
$426$	0	0
$427$	−9.97208	−0.482583
$428$	0	0
$429$	19.2822	0.930955
$430$	0	0
$431$	−10.6564	−0.513301	−0.256651	−	0.966504i	$-0.582619\pi$
$431$	−10.6564	−0.513301	−0.256651	+	0.966504i	$0.582619\pi$
$432$	0	0
$433$	23.6612	1.13708	0.568542	−	0.822654i	$-0.307508\pi$
$433$	23.6612	1.13708	0.568542	+	0.822654i	$0.307508\pi$
$434$	0	0
$435$	56.3784	2.70314
$436$	0	0
$437$	−1.71126	−0.0818607
$438$	0	0
$439$	33.3773	1.59301	0.796506	−	0.604630i	$-0.206679\pi$
$439$	33.3773	1.59301	0.796506	+	0.604630i	$0.206679\pi$
$440$	0	0
$441$	−7.89318	−0.375866
$442$	0	0
$443$	−29.2843	−1.39134	−0.695670	−	0.718361i	$-0.744893\pi$
$443$	−29.2843	−1.39134	−0.695670	+	0.718361i	$0.744893\pi$
$444$	0	0
$445$	16.1491	0.765541
$446$	0	0
$447$	−2.39458	−0.113260
$448$	0	0
$449$	24.9640	1.17812	0.589061	−	0.808089i	$-0.299498\pi$
$449$	24.9640	1.17812	0.589061	+	0.808089i	$0.299498\pi$
$450$	0	0
$451$	−52.0797	−2.45234
$452$	0	0
$453$	11.6632	0.547985
$454$	0	0
$455$	3.22415	0.151150
$456$	0	0
$457$	−19.0056	−0.889044	−0.444522	−	0.895768i	$-0.646626\pi$
$457$	−19.0056	−0.889044	−0.444522	+	0.895768i	$0.646626\pi$
$458$	0	0
$459$	13.1623	0.614364
$460$	0	0
$461$	−20.4148	−0.950811	−0.475405	−	0.879767i	$-0.657699\pi$
$461$	−20.4148	−0.950811	−0.475405	+	0.879767i	$0.657699\pi$
$462$	0	0
$463$	24.6655	1.14630	0.573151	−	0.819450i	$-0.305721\pi$
$463$	24.6655	1.14630	0.573151	+	0.819450i	$0.305721\pi$
$464$	0	0
$465$	6.98073	0.323724
$466$	0	0
$467$	−37.8873	−1.75322	−0.876608	−	0.481206i	$-0.840199\pi$
$467$	−37.8873	−1.75322	−0.876608	+	0.481206i	$0.840199\pi$
$468$	0	0
$469$	−3.02011	−0.139456
$470$	0	0
$471$	22.0635	1.01663
$472$	0	0
$473$	−15.1894	−0.698409
$474$	0	0
$475$	−2.14631	−0.0984795
$476$	0	0
$477$	1.20949	0.0553786
$478$	0	0
$479$	9.44817	0.431698	0.215849	−	0.976427i	$-0.430748\pi$
$479$	9.44817	0.431698	0.215849	+	0.976427i	$0.430748\pi$
$480$	0	0
$481$	2.21350	0.100927
$482$	0	0
$483$	−2.41709	−0.109981
$484$	0	0
$485$	28.7601	1.30593
$486$	0	0
$487$	−37.7835	−1.71213	−0.856067	−	0.516865i	$-0.827099\pi$
$487$	−37.7835	−1.71213	−0.856067	+	0.516865i	$0.827099\pi$
$488$	0	0
$489$	−7.73850	−0.349947
$490$	0	0
$491$	−11.6092	−0.523914	−0.261957	−	0.965079i	$-0.584368\pi$
$491$	−11.6092	−0.523914	−0.261957	+	0.965079i	$0.584368\pi$
$492$	0	0
$493$	−36.8296	−1.65872
$494$	0	0
$495$	−17.3449	−0.779597
$496$	0	0
$497$	−0.0904950	−0.00405925
$498$	0	0
$499$	−7.23875	−0.324051	−0.162026	−	0.986787i	$-0.551803\pi$
$499$	−7.23875	−0.324051	−0.162026	+	0.986787i	$0.551803\pi$
$500$	0	0
$501$	20.6867	0.924212
$502$	0	0
$503$	−11.7813	−0.525302	−0.262651	−	0.964891i	$-0.584597\pi$
$503$	−11.7813	−0.525302	−0.262651	+	0.964891i	$0.584597\pi$
$504$	0	0
$505$	3.82309	0.170125
$506$	0	0
$507$	20.3751	0.904889
$508$	0	0
$509$	40.8441	1.81038	0.905191	−	0.425005i	$-0.139728\pi$
$509$	40.8441	1.81038	0.905191	+	0.425005i	$0.139728\pi$
$510$	0	0
$511$	6.51760	0.288322
$512$	0	0
$513$	−3.67360	−0.162194
$514$	0	0
$515$	−12.0941	−0.532929
$516$	0	0
$517$	39.1753	1.72293
$518$	0	0
$519$	28.3351	1.24377
$520$	0	0
$521$	−20.2565	−0.887453	−0.443726	−	0.896162i	$-0.646344\pi$
$521$	−20.2565	−0.887453	−0.443726	+	0.896162i	$0.646344\pi$
$522$	0	0
$523$	16.0528	0.701941	0.350970	−	0.936387i	$-0.385852\pi$
$523$	16.0528	0.701941	0.350970	+	0.936387i	$0.385852\pi$
$524$	0	0
$525$	−3.03158	−0.132309
$526$	0	0
$527$	−4.56022	−0.198646
$528$	0	0
$529$	−20.0716	−0.872678
$530$	0	0
$531$	−8.78452	−0.381216
$532$	0	0
$533$	17.0078	0.736691
$534$	0	0
$535$	25.7092	1.11151
$536$	0	0
$537$	3.25950	0.140658
$538$	0	0
$539$	35.0092	1.50795
$540$	0	0
$541$	−33.1284	−1.42430	−0.712151	−	0.702026i	$-0.752279\pi$
$541$	−33.1284	−1.42430	−0.712151	+	0.702026i	$0.752279\pi$
$542$	0	0
$543$	39.7314	1.70504
$544$	0	0
$545$	6.97860	0.298931
$546$	0	0
$547$	−9.67593	−0.413713	−0.206856	−	0.978371i	$-0.566323\pi$
$547$	−9.67593	−0.413713	−0.206856	+	0.978371i	$0.566323\pi$
$548$	0	0
$549$	−17.5196	−0.747719
$550$	0	0
$551$	10.2791	0.437906
$552$	0	0
$553$	−4.95646	−0.210770
$554$	0	0
$555$	−6.92985	−0.294156
$556$	0	0
$557$	13.9545	0.591270	0.295635	−	0.955301i	$-0.404469\pi$
$557$	13.9545	0.591270	0.295635	+	0.955301i	$0.404469\pi$
$558$	0	0
$559$	4.96045	0.209805
$560$	0	0
$561$	39.4353	1.66496
$562$	0	0
$563$	−12.5614	−0.529402	−0.264701	−	0.964331i	$-0.585273\pi$
$563$	−12.5614	−0.529402	−0.264701	+	0.964331i	$0.585273\pi$
$564$	0	0
$565$	43.4466	1.82781
$566$	0	0
$567$	−7.68678	−0.322814
$568$	0	0
$569$	30.0275	1.25882	0.629410	−	0.777074i	$-0.283297\pi$
$569$	30.0275	1.25882	0.629410	+	0.777074i	$0.283297\pi$
$570$	0	0
$571$	−29.6364	−1.24024	−0.620122	−	0.784505i	$-0.712917\pi$
$571$	−29.6364	−1.24024	−0.620122	+	0.784505i	$0.712917\pi$
$572$	0	0
$573$	42.9969	1.79622
$574$	0	0
$575$	3.67290	0.153170
$576$	0	0
$577$	−14.4863	−0.603072	−0.301536	−	0.953455i	$-0.597499\pi$
$577$	−14.4863	−0.603072	−0.301536	+	0.953455i	$0.597499\pi$
$578$	0	0
$579$	12.9884	0.539779
$580$	0	0
$581$	−4.28465	−0.177757
$582$	0	0
$583$	−5.36452	−0.222176
$584$	0	0
$585$	5.66439	0.234194
$586$	0	0
$587$	22.2906	0.920030	0.460015	−	0.887911i	$-0.347844\pi$
$587$	22.2906	0.920030	0.460015	+	0.887911i	$0.347844\pi$
$588$	0	0
$589$	1.27276	0.0524431
$590$	0	0
$591$	−30.5717	−1.25755
$592$	0	0
$593$	25.1368	1.03224	0.516121	−	0.856515i	$-0.327375\pi$
$593$	25.1368	1.03224	0.516121	+	0.856515i	$0.327375\pi$
$594$	0	0
$595$	6.59391	0.270324
$596$	0	0
$597$	43.2021	1.76814
$598$	0	0
$599$	1.40362	0.0573504	0.0286752	−	0.999589i	$-0.490871\pi$
$599$	1.40362	0.0573504	0.0286752	+	0.999589i	$0.490871\pi$
$600$	0	0
$601$	−12.1969	−0.497523	−0.248761	−	0.968565i	$-0.580023\pi$
$601$	−12.1969	−0.497523	−0.248761	+	0.968565i	$0.580023\pi$
$602$	0	0
$603$	−5.30593	−0.216074
$604$	0	0
$605$	47.5254	1.93218
$606$	0	0
$607$	−12.5491	−0.509352	−0.254676	−	0.967026i	$-0.581969\pi$
$607$	−12.5491	−0.509352	−0.254676	+	0.967026i	$0.581969\pi$
$608$	0	0
$609$	14.5189	0.588336
$610$	0	0
$611$	−12.7936	−0.517573
$612$	0	0
$613$	13.9971	0.565337	0.282669	−	0.959218i	$-0.408780\pi$
$613$	13.9971	0.565337	0.282669	+	0.959218i	$0.408780\pi$
$614$	0	0
$615$	−53.2467	−2.14711
$616$	0	0
$617$	18.9172	0.761576	0.380788	−	0.924662i	$-0.375653\pi$
$617$	18.9172	0.761576	0.380788	+	0.924662i	$0.375653\pi$
$618$	0	0
$619$	3.15793	0.126928	0.0634640	−	0.997984i	$-0.479785\pi$
$619$	3.15793	0.126928	0.0634640	+	0.997984i	$0.479785\pi$
$620$	0	0
$621$	6.28649	0.252268
$622$	0	0
$623$	4.15881	0.166619
$624$	0	0
$625$	−31.1249	−1.24500
$626$	0	0
$627$	−11.0064	−0.439554
$628$	0	0
$629$	4.52698	0.180502
$630$	0	0
$631$	−22.4939	−0.895469	−0.447734	−	0.894167i	$-0.647769\pi$
$631$	−22.4939	−0.895469	−0.447734	+	0.894167i	$0.647769\pi$
$632$	0	0
$633$	−7.86808	−0.312728
$634$	0	0
$635$	−59.6613	−2.36758
$636$	0	0
$637$	−11.4331	−0.452995
$638$	0	0
$639$	−0.158988	−0.00628945
$640$	0	0
$641$	−10.6681	−0.421366	−0.210683	−	0.977554i	$-0.567569\pi$
$641$	−10.6681	−0.421366	−0.210683	+	0.977554i	$0.567569\pi$
$642$	0	0
$643$	41.6557	1.64274	0.821370	−	0.570396i	$-0.193210\pi$
$643$	41.6557	1.64274	0.821370	+	0.570396i	$0.193210\pi$
$644$	0	0
$645$	−15.5298	−0.611484
$646$	0	0
$647$	2.12688	0.0836162	0.0418081	−	0.999126i	$-0.486688\pi$
$647$	2.12688	0.0836162	0.0418081	+	0.999126i	$0.486688\pi$
$648$	0	0
$649$	38.9626	1.52942
$650$	0	0
$651$	1.79772	0.0704582
$652$	0	0
$653$	7.11558	0.278454	0.139227	−	0.990260i	$-0.455538\pi$
$653$	7.11558	0.278454	0.139227	+	0.990260i	$0.455538\pi$
$654$	0	0
$655$	−25.9481	−1.01388
$656$	0	0
$657$	11.4506	0.446729
$658$	0	0
$659$	34.9080	1.35982	0.679911	−	0.733295i	$-0.262018\pi$
$659$	34.9080	1.35982	0.679911	+	0.733295i	$0.262018\pi$
$660$	0	0
$661$	35.6020	1.38476	0.692379	−	0.721534i	$-0.256563\pi$
$661$	35.6020	1.38476	0.692379	+	0.721534i	$0.256563\pi$
$662$	0	0
$663$	−12.8785	−0.500161
$664$	0	0
$665$	−1.84036	−0.0713661
$666$	0	0
$667$	−17.5903	−0.681099
$668$	0	0
$669$	−53.8012	−2.08008
$670$	0	0
$671$	77.7060	2.99981
$672$	0	0
$673$	12.1762	0.469360	0.234680	−	0.972073i	$-0.424596\pi$
$673$	12.1762	0.469360	0.234680	+	0.972073i	$0.424596\pi$
$674$	0	0
$675$	7.88469	0.303482
$676$	0	0
$677$	−7.14162	−0.274475	−0.137237	−	0.990538i	$-0.543822\pi$
$677$	−7.14162	−0.274475	−0.137237	+	0.990538i	$0.543822\pi$
$678$	0	0
$679$	7.40646	0.284234
$680$	0	0
$681$	−13.8605	−0.531137
$682$	0	0
$683$	39.6862	1.51855	0.759276	−	0.650769i	$-0.225553\pi$
$683$	39.6862	1.51855	0.759276	+	0.650769i	$0.225553\pi$
$684$	0	0
$685$	31.1798	1.19132
$686$	0	0
$687$	15.4433	0.589199
$688$	0	0
$689$	1.75191	0.0667424
$690$	0	0
$691$	33.7169	1.28265	0.641325	−	0.767269i	$-0.278385\pi$
$691$	33.7169	1.28265	0.641325	+	0.767269i	$0.278385\pi$
$692$	0	0
$693$	−4.46677	−0.169679
$694$	0	0
$695$	9.34041	0.354302
$696$	0	0
$697$	34.7838	1.31753
$698$	0	0
$699$	40.3665	1.52680
$700$	0	0
$701$	7.55443	0.285327	0.142664	−	0.989771i	$-0.454433\pi$
$701$	7.55443	0.285327	0.142664	+	0.989771i	$0.454433\pi$
$702$	0	0
$703$	−1.26348	−0.0476530
$704$	0	0
$705$	40.0531	1.50849
$706$	0	0
$707$	0.984544	0.0370276
$708$	0	0
$709$	49.9410	1.87558	0.937788	−	0.347209i	$-0.112871\pi$
$709$	49.9410	1.87558	0.937788	+	0.347209i	$0.112871\pi$
$710$	0	0
$711$	−8.70783	−0.326569
$712$	0	0
$713$	−2.17802	−0.0815675
$714$	0	0
$715$	−25.1237	−0.939573
$716$	0	0
$717$	−5.87076	−0.219247
$718$	0	0
$719$	−8.33614	−0.310885	−0.155443	−	0.987845i	$-0.549680\pi$
$719$	−8.33614	−0.310885	−0.155443	+	0.987845i	$0.549680\pi$
$720$	0	0
$721$	−3.11454	−0.115992
$722$	0	0
$723$	29.1878	1.08551
$724$	0	0
$725$	−22.0622	−0.819371
$726$	0	0
$727$	16.8955	0.626621	0.313311	−	0.949651i	$-0.398562\pi$
$727$	16.8955	0.626621	0.313311	+	0.949651i	$0.398562\pi$
$728$	0	0
$729$	9.10679	0.337288
$730$	0	0
$731$	10.1449	0.375224
$732$	0	0
$733$	−30.5227	−1.12738	−0.563691	−	0.825986i	$-0.690619\pi$
$733$	−30.5227	−1.12738	−0.563691	+	0.825986i	$0.690619\pi$
$734$	0	0
$735$	35.7937	1.32027
$736$	0	0
$737$	23.5338	0.866878
$738$	0	0
$739$	26.5270	0.975812	0.487906	−	0.872896i	$-0.337761\pi$
$739$	26.5270	0.975812	0.487906	+	0.872896i	$0.337761\pi$
$740$	0	0
$741$	3.59440	0.132044
$742$	0	0
$743$	−22.9543	−0.842113	−0.421057	−	0.907034i	$-0.638341\pi$
$743$	−22.9543	−0.842113	−0.421057	+	0.907034i	$0.638341\pi$
$744$	0	0
$745$	3.12001	0.114308
$746$	0	0
$747$	−7.52755	−0.275419
$748$	0	0
$749$	6.62079	0.241919
$750$	0	0
$751$	−21.0347	−0.767567	−0.383783	−	0.923423i	$-0.625379\pi$
$751$	−21.0347	−0.767567	−0.383783	+	0.923423i	$0.625379\pi$
$752$	0	0
$753$	2.48760	0.0906533
$754$	0	0
$755$	−15.1965	−0.553058
$756$	0	0
$757$	−36.2915	−1.31904	−0.659519	−	0.751688i	$-0.729240\pi$
$757$	−36.2915	−1.31904	−0.659519	+	0.751688i	$0.729240\pi$
$758$	0	0
$759$	18.8348	0.683661
$760$	0	0
$761$	4.90249	0.177715	0.0888575	−	0.996044i	$-0.471678\pi$
$761$	4.90249	0.177715	0.0888575	+	0.996044i	$0.471678\pi$
$762$	0	0
$763$	1.79717	0.0650620
$764$	0	0
$765$	11.5846	0.418843
$766$	0	0
$767$	−12.7242	−0.459443
$768$	0	0
$769$	−1.30765	−0.0471550	−0.0235775	−	0.999722i	$-0.507506\pi$
$769$	−1.30765	−0.0471550	−0.0235775	+	0.999722i	$0.507506\pi$
$770$	0	0
$771$	32.6338	1.17528
$772$	0	0
$773$	16.3103	0.586641	0.293321	−	0.956014i	$-0.405240\pi$
$773$	16.3103	0.586641	0.293321	+	0.956014i	$0.405240\pi$
$774$	0	0
$775$	−2.73173	−0.0981267
$776$	0	0
$777$	−1.78462	−0.0640228
$778$	0	0
$779$	−9.70817	−0.347831
$780$	0	0
$781$	0.705169	0.0252329
$782$	0	0
$783$	−37.7615	−1.34949
$784$	0	0
$785$	−28.7475	−1.02604
$786$	0	0
$787$	45.5598	1.62403	0.812015	−	0.583636i	$-0.198371\pi$
$787$	45.5598	1.62403	0.812015	+	0.583636i	$0.198371\pi$
$788$	0	0
$789$	−43.6717	−1.55475
$790$	0	0
$791$	11.1886	0.397822
$792$	0	0
$793$	−25.3767	−0.901154
$794$	0	0
$795$	−5.48473	−0.194523
$796$	0	0
$797$	−8.76534	−0.310484	−0.155242	−	0.987876i	$-0.549616\pi$
$797$	−8.76534	−0.310484	−0.155242	+	0.987876i	$0.549616\pi$
$798$	0	0
$799$	−26.1650	−0.925652
$800$	0	0
$801$	7.30648	0.258162
$802$	0	0
$803$	−50.7875	−1.79225
$804$	0	0
$805$	3.14934	0.111000
$806$	0	0
$807$	45.2437	1.59265
$808$	0	0
$809$	12.8952	0.453373	0.226686	−	0.973968i	$-0.427211\pi$
$809$	12.8952	0.453373	0.226686	+	0.973968i	$0.427211\pi$
$810$	0	0
$811$	−27.8727	−0.978742	−0.489371	−	0.872076i	$-0.662774\pi$
$811$	−27.8727	−0.978742	−0.489371	+	0.872076i	$0.662774\pi$
$812$	0	0
$813$	55.8354	1.95823
$814$	0	0
$815$	10.0828	0.353187
$816$	0	0
$817$	−2.83145	−0.0990600
$818$	0	0
$819$	1.45873	0.0509721
$820$	0	0
$821$	−28.0674	−0.979559	−0.489780	−	0.871846i	$-0.662923\pi$
$821$	−28.0674	−0.979559	−0.489780	+	0.871846i	$0.662923\pi$
$822$	0	0
$823$	21.1117	0.735909	0.367954	−	0.929844i	$-0.380058\pi$
$823$	21.1117	0.735909	0.367954	+	0.929844i	$0.380058\pi$
$824$	0	0
$825$	23.6232	0.822453
$826$	0	0
$827$	20.9063	0.726983	0.363491	−	0.931598i	$-0.381585\pi$
$827$	20.9063	0.726983	0.363491	+	0.931598i	$0.381585\pi$
$828$	0	0
$829$	−40.8405	−1.41845	−0.709225	−	0.704982i	$-0.750955\pi$
$829$	−40.8405	−1.41845	−0.709225	+	0.704982i	$0.750955\pi$
$830$	0	0
$831$	−1.36470	−0.0473410
$832$	0	0
$833$	−23.3825	−0.810156
$834$	0	0
$835$	−26.9536	−0.932768
$836$	0	0
$837$	−4.67561	−0.161613
$838$	0	0
$839$	22.1992	0.766401	0.383200	−	0.923665i	$-0.374822\pi$
$839$	22.1992	0.766401	0.383200	+	0.923665i	$0.374822\pi$
$840$	0	0
$841$	76.6609	2.64348
$842$	0	0
$843$	−28.7042	−0.988627
$844$	0	0
$845$	−26.5476	−0.913266
$846$	0	0
$847$	12.2390	0.420538
$848$	0	0
$849$	−11.5746	−0.397238
$850$	0	0
$851$	2.16214	0.0741173
$852$	0	0
$853$	48.3086	1.65406	0.827028	−	0.562160i	$-0.190030\pi$
$853$	48.3086	1.65406	0.827028	+	0.562160i	$0.190030\pi$
$854$	0	0
$855$	−3.23327	−0.110575
$856$	0	0
$857$	35.4816	1.21203	0.606014	−	0.795454i	$-0.292767\pi$
$857$	35.4816	1.21203	0.606014	+	0.795454i	$0.292767\pi$
$858$	0	0
$859$	−31.8670	−1.08729	−0.543644	−	0.839316i	$-0.682956\pi$
$859$	−31.8670	−1.08729	−0.543644	+	0.839316i	$0.682956\pi$
$860$	0	0
$861$	−13.7124	−0.467318
$862$	0	0
$863$	22.3988	0.762464	0.381232	−	0.924479i	$-0.375500\pi$
$863$	22.3988	0.762464	0.381232	+	0.924479i	$0.375500\pi$
$864$	0	0
$865$	−36.9191	−1.25529
$866$	0	0
$867$	8.54023	0.290041
$868$	0	0
$869$	38.6225	1.31018
$870$	0	0
$871$	−7.68550	−0.260413
$872$	0	0
$873$	13.0122	0.440395
$874$	0	0
$875$	−5.25182	−0.177544
$876$	0	0
$877$	−38.3390	−1.29462	−0.647308	−	0.762228i	$-0.724105\pi$
$877$	−38.3390	−1.29462	−0.647308	+	0.762228i	$0.724105\pi$
$878$	0	0
$879$	63.2091	2.13199
$880$	0	0
$881$	40.5900	1.36751	0.683755	−	0.729712i	$-0.260346\pi$
$881$	40.5900	1.36751	0.683755	+	0.729712i	$0.260346\pi$
$882$	0	0
$883$	−16.0431	−0.539892	−0.269946	−	0.962875i	$-0.587006\pi$
$883$	−16.0431	−0.539892	−0.269946	+	0.962875i	$0.587006\pi$
$884$	0	0
$885$	39.8357	1.33906
$886$	0	0
$887$	−3.37948	−0.113472	−0.0567359	−	0.998389i	$-0.518069\pi$
$887$	−3.37948	−0.113472	−0.0567359	+	0.998389i	$0.518069\pi$
$888$	0	0
$889$	−15.3643	−0.515303
$890$	0	0
$891$	59.8981	2.00666
$892$	0	0
$893$	7.30266	0.244374
$894$	0	0
$895$	−4.24695	−0.141960
$896$	0	0
$897$	−6.15095	−0.205374
$898$	0	0
$899$	13.0829	0.436338
$900$	0	0
$901$	3.58294	0.119365
$902$	0	0
$903$	−3.99932	−0.133089
$904$	0	0
$905$	−51.7678	−1.72082
$906$	0	0
$907$	−2.67056	−0.0886745	−0.0443372	−	0.999017i	$-0.514118\pi$
$907$	−2.67056	−0.0886745	−0.0443372	+	0.999017i	$0.514118\pi$
$908$	0	0
$909$	1.72971	0.0573710
$910$	0	0
$911$	18.7800	0.622209	0.311105	−	0.950376i	$-0.399301\pi$
$911$	18.7800	0.622209	0.311105	+	0.950376i	$0.399301\pi$
$912$	0	0
$913$	33.3875	1.10496
$914$	0	0
$915$	79.4473	2.62645
$916$	0	0
$917$	−6.68232	−0.220670
$918$	0	0
$919$	43.8555	1.44666	0.723330	−	0.690503i	$-0.242611\pi$
$919$	43.8555	1.44666	0.723330	+	0.690503i	$0.242611\pi$
$920$	0	0
$921$	−41.4130	−1.36460
$922$	0	0
$923$	−0.230289	−0.00758007
$924$	0	0
$925$	2.71182	0.0891641
$926$	0	0
$927$	−5.47184	−0.179719
$928$	0	0
$929$	−30.2236	−0.991604	−0.495802	−	0.868436i	$-0.665126\pi$
$929$	−30.2236	−0.991604	−0.495802	+	0.868436i	$0.665126\pi$
$930$	0	0
$931$	6.52606	0.213883
$932$	0	0
$933$	32.4893	1.06365
$934$	0	0
$935$	−51.3821	−1.68038
$936$	0	0
$937$	−5.88444	−0.192236	−0.0961182	−	0.995370i	$-0.530643\pi$
$937$	−5.88444	−0.192236	−0.0961182	+	0.995370i	$0.530643\pi$
$938$	0	0
$939$	61.8090	2.01706
$940$	0	0
$941$	−35.4073	−1.15424	−0.577122	−	0.816658i	$-0.695824\pi$
$941$	−35.4073	−1.15424	−0.577122	+	0.816658i	$0.695824\pi$
$942$	0	0
$943$	16.6132	0.541001
$944$	0	0
$945$	6.76076	0.219927
$946$	0	0
$947$	23.6233	0.767653	0.383827	−	0.923405i	$-0.374606\pi$
$947$	23.6233	0.767653	0.383827	+	0.923405i	$0.374606\pi$
$948$	0	0
$949$	16.5858	0.538399
$950$	0	0
$951$	19.2287	0.623532
$952$	0	0
$953$	8.57550	0.277788	0.138894	−	0.990307i	$-0.455645\pi$
$953$	8.57550	0.277788	0.138894	+	0.990307i	$0.455645\pi$
$954$	0	0
$955$	−56.0227	−1.81285
$956$	0	0
$957$	−113.136	−3.65718
$958$	0	0
$959$	8.02961	0.259290
$960$	0	0
$961$	−29.3801	−0.947745
$962$	0	0
$963$	11.6319	0.374831
$964$	0	0
$965$	−16.9232	−0.544776
$966$	0	0
$967$	52.5010	1.68832	0.844160	−	0.536092i	$-0.180100\pi$
$967$	52.5010	1.68832	0.844160	+	0.536092i	$0.180100\pi$
$968$	0	0
$969$	7.35114	0.236153
$970$	0	0
$971$	−17.5733	−0.563955	−0.281977	−	0.959421i	$-0.590990\pi$
$971$	−17.5733	−0.563955	−0.281977	+	0.959421i	$0.590990\pi$
$972$	0	0
$973$	2.40540	0.0771135
$974$	0	0
$975$	−7.71470	−0.247068
$976$	0	0
$977$	−22.1812	−0.709641	−0.354820	−	0.934935i	$-0.615458\pi$
$977$	−22.1812	−0.709641	−0.354820	+	0.934935i	$0.615458\pi$
$978$	0	0
$979$	−32.4070	−1.03573
$980$	0	0
$981$	3.15739	0.100808
$982$	0	0
$983$	−61.4344	−1.95945	−0.979727	−	0.200336i	$-0.935797\pi$
$983$	−61.4344	−1.95945	−0.979727	+	0.200336i	$0.935797\pi$
$984$	0	0
$985$	39.8333	1.26919
$986$	0	0
$987$	10.3147	0.328321
$988$	0	0
$989$	4.84536	0.154073
$990$	0	0
$991$	4.45220	0.141429	0.0707143	−	0.997497i	$-0.477472\pi$
$991$	4.45220	0.141429	0.0707143	+	0.997497i	$0.477472\pi$
$992$	0	0
$993$	−28.5063	−0.904620
$994$	0	0
$995$	−56.2900	−1.78451
$996$	0	0
$997$	−35.1944	−1.11462	−0.557308	−	0.830306i	$-0.688166\pi$
$997$	−35.1944	−1.11462	−0.557308	+	0.830306i	$0.688166\pi$
$998$	0	0
$999$	4.64152	0.146851

Introduction	Features
Universe	News

Level:	\( N \)	=	\( 4028 = 2^{2} \cdot 19 \cdot 53 \)
Weight:	\( k \)	=	\( 2 \)
Character orbit:	\([\chi]\)	=	4028.a (trivial)

\(f(q)\)	\(=\)	\(q-2.05170 q^{3} +2.67326 q^{5} +0.688434 q^{7} +1.20949 q^{9} +O(q^{10})\)	\(q-2.05170 q^{3} +2.67326 q^{5} +0.688434 q^{7} +1.20949 q^{9} -5.36452 q^{11} +1.75191 q^{13} -5.48473 q^{15} +3.58294 q^{17} -1.00000 q^{19} -1.41246 q^{21} +1.71126 q^{23} +2.14631 q^{25} +3.67360 q^{27} -10.2791 q^{29} -1.27276 q^{31} +11.0064 q^{33} +1.84036 q^{35} +1.26348 q^{37} -3.59440 q^{39} +9.70817 q^{41} +2.83145 q^{43} +3.23327 q^{45} -7.30266 q^{47} -6.52606 q^{49} -7.35114 q^{51} +1.00000 q^{53} -14.3408 q^{55} +2.05170 q^{57} -7.26302 q^{59} -14.4852 q^{61} +0.832651 q^{63} +4.68331 q^{65} -4.38693 q^{67} -3.51100 q^{69} -0.131451 q^{71} +9.46729 q^{73} -4.40359 q^{75} -3.69312 q^{77} -7.19961 q^{79} -11.1656 q^{81} -6.22376 q^{83} +9.57814 q^{85} +21.0898 q^{87} +6.04098 q^{89} +1.20607 q^{91} +2.61132 q^{93} -2.67326 q^{95} +10.7584 q^{97} -6.48832 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19q - 4q^{3} - 2q^{5} - 11q^{7} + 19q^{9} + O(q^{10}) \)	\( 19q - 4q^{3} - 2q^{5} - 11q^{7} + 19q^{9} - 11q^{11} + q^{13} - 20q^{15} - q^{17} - 19q^{19} - 10q^{21} - 16q^{23} + 21q^{25} - 4q^{27} - 9q^{31} + 7q^{33} - 25q^{37} - 25q^{39} + q^{41} - 41q^{43} - 27q^{45} - 29q^{47} + 14q^{49} - 24q^{51} + 19q^{53} - 28q^{55} + 4q^{57} - 42q^{59} + q^{61} - 41q^{63} + 2q^{65} - 41q^{67} - 25q^{69} - 20q^{73} + 11q^{75} - 19q^{77} - 38q^{79} + 23q^{81} - 36q^{83} - 58q^{85} - 30q^{87} - 25q^{89} - 55q^{91} - 38q^{93} + 2q^{95} - 13q^{97} + q^{99} + O(q^{100}) \)

Introduction and more

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Modular Forms

Varieties

Fields

Representations

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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	4028.2.a.d.1.5

Level	4028
Weight	2
Character	4028.1
Self dual	Yes
Analytic conductor	32.164
Analytic rank	1
Dimension	19
CM	No
Inner twists	1

Self dual:	Yes
Analytic conductor:	\(32.1637419342\)
Analytic rank:	\(1\)
Dimension:	\(19\)
Coefficient field:	\(\mathbb{Q}[x]/(x^{19} - \cdots)\)
Coefficient ring:	\(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index:	\( 1 \)
Fricke sign:	\(1\)
Sato-Tate group:	$\mathrm{SU}(2)$