LMFDB - Embedded newform 8012.2.a.a.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8012,2,Mod(1,8012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8012, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8012 = 2^{2} \cdot 2003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8012.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$63.9761420994$
Analytic rank:	$1$
Dimension:	$79$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.46774 q^{3} -2.83803 q^{5} +1.20715 q^{7} +3.08972 q^{9} +O(q^{10})$	$q-2.46774 q^{3} -2.83803 q^{5} +1.20715 q^{7} +3.08972 q^{9} +3.32223 q^{11} -6.01937 q^{13} +7.00350 q^{15} -6.27576 q^{17} +1.50849 q^{19} -2.97894 q^{21} +2.42727 q^{23} +3.05439 q^{25} -0.221415 q^{27} +8.28895 q^{29} +2.31403 q^{31} -8.19838 q^{33} -3.42593 q^{35} -8.10081 q^{37} +14.8542 q^{39} -1.92992 q^{41} +1.90640 q^{43} -8.76872 q^{45} -3.28024 q^{47} -5.54278 q^{49} +15.4869 q^{51} -5.57942 q^{53} -9.42857 q^{55} -3.72255 q^{57} +0.131214 q^{59} +10.6435 q^{61} +3.72977 q^{63} +17.0831 q^{65} +5.04586 q^{67} -5.98987 q^{69} +6.82341 q^{71} -11.9018 q^{73} -7.53743 q^{75} +4.01044 q^{77} +13.3912 q^{79} -8.72278 q^{81} +12.9089 q^{83} +17.8108 q^{85} -20.4550 q^{87} -7.46598 q^{89} -7.26630 q^{91} -5.71042 q^{93} -4.28113 q^{95} +14.4885 q^{97} +10.2648 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 79 q - 19 q^{3} - 40 q^{7} + 72 q^{9}+O(q^{10}) $ 79 * q - 19 * q^3 - 40 * q^7 + 72 * q^9	$ 79 q - 19 q^{3} - 40 q^{7} + 72 q^{9} - 14 q^{11} - 3 q^{13} - 19 q^{15} - 30 q^{17} - 49 q^{19} + 7 q^{21} - 40 q^{23} + 51 q^{25} - 70 q^{27} + 2 q^{29} - 48 q^{31} - 25 q^{33} - 34 q^{35} - 35 q^{39} - 20 q^{41} - 104 q^{43} + 12 q^{45} - 38 q^{47} + 51 q^{49} - 41 q^{51} - q^{53} - 112 q^{55} - 34 q^{57} - 24 q^{59} - 120 q^{63} - 21 q^{65} - 67 q^{67} + 15 q^{69} - 28 q^{71} - 88 q^{73} - 103 q^{75} + 4 q^{77} - 99 q^{79} + 47 q^{81} - 70 q^{83} + 7 q^{85} - 109 q^{87} - 50 q^{89} - 83 q^{91} - 7 q^{93} - 61 q^{95} - 93 q^{97} - 78 q^{99}+O(q^{100}) $ 79 * q - 19 * q^3 - 40 * q^7 + 72 * q^9 - 14 * q^11 - 3 * q^13 - 19 * q^15 - 30 * q^17 - 49 * q^19 + 7 * q^21 - 40 * q^23 + 51 * q^25 - 70 * q^27 + 2 * q^29 - 48 * q^31 - 25 * q^33 - 34 * q^35 - 35 * q^39 - 20 * q^41 - 104 * q^43 + 12 * q^45 - 38 * q^47 + 51 * q^49 - 41 * q^51 - q^53 - 112 * q^55 - 34 * q^57 - 24 * q^59 - 120 * q^63 - 21 * q^65 - 67 * q^67 + 15 * q^69 - 28 * q^71 - 88 * q^73 - 103 * q^75 + 4 * q^77 - 99 * q^79 + 47 * q^81 - 70 * q^83 + 7 * q^85 - 109 * q^87 - 50 * q^89 - 83 * q^91 - 7 * q^93 - 61 * q^95 - 93 * q^97 - 78 * q^99

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
$3$	−2.46774	−1.42475	−0.712374	−	0.701800i	$-0.752380\pi$
$3$	−2.46774	−1.42475	−0.712374	+	0.701800i	$0.752380\pi$
$4$	0	0
$5$	−2.83803	−1.26920	−0.634602	−	0.772839i	$-0.718836\pi$
$5$	−2.83803	−1.26920	−0.634602	+	0.772839i	$0.718836\pi$
$6$	0	0
$7$	1.20715	0.456261	0.228130	−	0.973631i	$-0.426739\pi$
$7$	1.20715	0.456261	0.228130	+	0.973631i	$0.426739\pi$
$8$	0	0
$9$	3.08972	1.02991
$10$	0	0
$11$	3.32223	1.00169	0.500845	−	0.865537i	$-0.333023\pi$
$11$	3.32223	1.00169	0.500845	+	0.865537i	$0.333023\pi$
$12$	0	0
$13$	−6.01937	−1.66947	−0.834736	−	0.550651i	$-0.814380\pi$
$13$	−6.01937	−1.66947	−0.834736	+	0.550651i	$0.814380\pi$
$14$	0	0
$15$	7.00350	1.80830
$16$	0	0
$17$	−6.27576	−1.52210	−0.761048	−	0.648696i	$-0.775315\pi$
$17$	−6.27576	−1.52210	−0.761048	+	0.648696i	$0.775315\pi$
$18$	0	0
$19$	1.50849	0.346071	0.173035	−	0.984916i	$-0.444643\pi$
$19$	1.50849	0.346071	0.173035	+	0.984916i	$0.444643\pi$
$20$	0	0
$21$	−2.97894	−0.650057
$22$	0	0
$23$	2.42727	0.506121	0.253061	−	0.967450i	$-0.418563\pi$
$23$	2.42727	0.506121	0.253061	+	0.967450i	$0.418563\pi$
$24$	0	0
$25$	3.05439	0.610878
$26$	0	0
$27$	−0.221415	−0.0426113
$28$	0	0
$29$	8.28895	1.53922	0.769610	−	0.638514i	$-0.220451\pi$
$29$	8.28895	1.53922	0.769610	+	0.638514i	$0.220451\pi$
$30$	0	0
$31$	2.31403	0.415612	0.207806	−	0.978170i	$-0.433368\pi$
$31$	2.31403	0.415612	0.207806	+	0.978170i	$0.433368\pi$
$32$	0	0
$33$	−8.19838	−1.42715
$34$	0	0
$35$	−3.42593	−0.579088
$36$	0	0
$37$	−8.10081	−1.33176	−0.665882	−	0.746057i	$-0.731945\pi$
$37$	−8.10081	−1.33176	−0.665882	+	0.746057i	$0.731945\pi$
$38$	0	0
$39$	14.8542	2.37858
$40$	0	0
$41$	−1.92992	−0.301402	−0.150701	−	0.988579i	$-0.548153\pi$
$41$	−1.92992	−0.301402	−0.150701	+	0.988579i	$0.548153\pi$
$42$	0	0
$43$	1.90640	0.290724	0.145362	−	0.989379i	$-0.453565\pi$
$43$	1.90640	0.290724	0.145362	+	0.989379i	$0.453565\pi$
$44$	0	0
$45$	−8.76872	−1.30716
$46$	0	0
$47$	−3.28024	−0.478473	−0.239236	−	0.970961i	$-0.576897\pi$
$47$	−3.28024	−0.478473	−0.239236	+	0.970961i	$0.576897\pi$
$48$	0	0
$49$	−5.54278	−0.791826
$50$	0	0
$51$	15.4869	2.16860
$52$	0	0
$53$	−5.57942	−0.766392	−0.383196	−	0.923667i	$-0.625177\pi$
$53$	−5.57942	−0.766392	−0.383196	+	0.923667i	$0.625177\pi$
$54$	0	0
$55$	−9.42857	−1.27135
$56$	0	0
$57$	−3.72255	−0.493064
$58$	0	0
$59$	0.131214	0.0170827	0.00854133	−	0.999964i	$-0.497281\pi$
$59$	0.131214	0.0170827	0.00854133	+	0.999964i	$0.497281\pi$
$60$	0	0
$61$	10.6435	1.36276	0.681382	−	0.731928i	$-0.261379\pi$
$61$	10.6435	1.36276	0.681382	+	0.731928i	$0.261379\pi$
$62$	0	0
$63$	3.72977	0.469907
$64$	0	0
$65$	17.0831	2.11890
$66$	0	0
$67$	5.04586	0.616450	0.308225	−	0.951314i	$-0.400265\pi$
$67$	5.04586	0.616450	0.308225	+	0.951314i	$0.400265\pi$
$68$	0	0
$69$	−5.98987	−0.721096
$70$	0	0
$71$	6.82341	0.809790	0.404895	−	0.914363i	$-0.367308\pi$
$71$	6.82341	0.809790	0.404895	+	0.914363i	$0.367308\pi$
$72$	0	0
$73$	−11.9018	−1.39300	−0.696502	−	0.717554i	$-0.745261\pi$
$73$	−11.9018	−1.39300	−0.696502	+	0.717554i	$0.745261\pi$
$74$	0	0
$75$	−7.53743	−0.870348
$76$	0	0
$77$	4.01044	0.457032
$78$	0	0
$79$	13.3912	1.50662	0.753311	−	0.657664i	$-0.228455\pi$
$79$	13.3912	1.50662	0.753311	+	0.657664i	$0.228455\pi$
$80$	0	0
$81$	−8.72278	−0.969198
$82$	0	0
$83$	12.9089	1.41694	0.708469	−	0.705741i	$-0.249386\pi$
$83$	12.9089	1.41694	0.708469	+	0.705741i	$0.249386\pi$
$84$	0	0
$85$	17.8108	1.93185
$86$	0	0
$87$	−20.4550	−2.19300
$88$	0	0
$89$	−7.46598	−0.791393	−0.395696	−	0.918381i	$-0.629497\pi$
$89$	−7.46598	−0.791393	−0.395696	+	0.918381i	$0.629497\pi$
$90$	0	0
$91$	−7.26630	−0.761715
$92$	0	0
$93$	−5.71042	−0.592143
$94$	0	0
$95$	−4.28113	−0.439234
$96$	0	0
$97$	14.4885	1.47109	0.735543	−	0.677478i	$-0.236927\pi$
$97$	14.4885	1.47109	0.735543	+	0.677478i	$0.236927\pi$
$98$	0	0
$99$	10.2648	1.03165
$100$	0	0
$101$	−3.75438	−0.373575	−0.186787	−	0.982400i	$-0.559808\pi$
$101$	−3.75438	−0.373575	−0.186787	+	0.982400i	$0.559808\pi$
$102$	0	0
$103$	12.6397	1.24543	0.622715	−	0.782449i	$-0.286030\pi$
$103$	12.6397	1.24543	0.622715	+	0.782449i	$0.286030\pi$
$104$	0	0
$105$	8.45430	0.825055
$106$	0	0
$107$	−3.38423	−0.327166	−0.163583	−	0.986530i	$-0.552305\pi$
$107$	−3.38423	−0.327166	−0.163583	+	0.986530i	$0.552305\pi$
$108$	0	0
$109$	12.7132	1.21770	0.608851	−	0.793285i	$-0.291631\pi$
$109$	12.7132	1.21770	0.608851	+	0.793285i	$0.291631\pi$
$110$	0	0
$111$	19.9907	1.89743
$112$	0	0
$113$	−0.493940	−0.0464659	−0.0232330	−	0.999730i	$-0.507396\pi$
$113$	−0.493940	−0.0464659	−0.0232330	+	0.999730i	$0.507396\pi$
$114$	0	0
$115$	−6.88866	−0.642371
$116$	0	0
$117$	−18.5982	−1.71940
$118$	0	0
$119$	−7.57580	−0.694473
$120$	0	0
$121$	0.0371905	0.00338095
$122$	0	0
$123$	4.76252	0.429422
$124$	0	0
$125$	5.52169	0.493875
$126$	0	0
$127$	−16.8443	−1.49469	−0.747345	−	0.664437i	$-0.768672\pi$
$127$	−16.8443	−1.49469	−0.747345	+	0.664437i	$0.768672\pi$
$128$	0	0
$129$	−4.70450	−0.414208
$130$	0	0
$131$	21.6183	1.88880	0.944402	−	0.328794i	$-0.106642\pi$
$131$	21.6183	1.88880	0.944402	+	0.328794i	$0.106642\pi$
$132$	0	0
$133$	1.82097	0.157899
$134$	0	0
$135$	0.628380	0.0540824
$136$	0	0
$137$	5.06801	0.432989	0.216495	−	0.976284i	$-0.430538\pi$
$137$	5.06801	0.432989	0.216495	+	0.976284i	$0.430538\pi$
$138$	0	0
$139$	5.55068	0.470802	0.235401	−	0.971898i	$-0.424360\pi$
$139$	5.55068	0.470802	0.235401	+	0.971898i	$0.424360\pi$
$140$	0	0
$141$	8.09478	0.681703
$142$	0	0
$143$	−19.9977	−1.67229
$144$	0	0
$145$	−23.5243	−1.95358
$146$	0	0
$147$	13.6781	1.12815
$148$	0	0
$149$	−14.5682	−1.19348	−0.596738	−	0.802436i	$-0.703537\pi$
$149$	−14.5682	−1.19348	−0.596738	+	0.802436i	$0.703537\pi$
$150$	0	0
$151$	−8.22635	−0.669451	−0.334726	−	0.942316i	$-0.608644\pi$
$151$	−8.22635	−0.669451	−0.334726	+	0.942316i	$0.608644\pi$
$152$	0	0
$153$	−19.3904	−1.56762
$154$	0	0
$155$	−6.56728	−0.527497
$156$	0	0
$157$	22.9289	1.82993	0.914965	−	0.403534i	$-0.132218\pi$
$157$	22.9289	1.82993	0.914965	+	0.403534i	$0.132218\pi$
$158$	0	0
$159$	13.7685	1.09192
$160$	0	0
$161$	2.93009	0.230923
$162$	0	0
$163$	18.9607	1.48512	0.742558	−	0.669781i	$-0.233612\pi$
$163$	18.9607	1.48512	0.742558	+	0.669781i	$0.233612\pi$
$164$	0	0
$165$	23.2672	1.81135
$166$	0	0
$167$	−14.8371	−1.14813	−0.574065	−	0.818810i	$-0.694634\pi$
$167$	−14.8371	−1.14813	−0.574065	+	0.818810i	$0.694634\pi$
$168$	0	0
$169$	23.2328	1.78714
$170$	0	0
$171$	4.66081	0.356421
$172$	0	0
$173$	−20.8043	−1.58172	−0.790862	−	0.611994i	$-0.790367\pi$
$173$	−20.8043	−1.58172	−0.790862	+	0.611994i	$0.790367\pi$
$174$	0	0
$175$	3.68712	0.278720
$176$	0	0
$177$	−0.323803	−0.0243385
$178$	0	0
$179$	−5.63113	−0.420890	−0.210445	−	0.977606i	$-0.567491\pi$
$179$	−5.63113	−0.420890	−0.210445	+	0.977606i	$0.567491\pi$
$180$	0	0
$181$	−7.69111	−0.571676	−0.285838	−	0.958278i	$-0.592272\pi$
$181$	−7.69111	−0.571676	−0.285838	+	0.958278i	$0.592272\pi$
$182$	0	0
$183$	−26.2654	−1.94159
$184$	0	0
$185$	22.9903	1.69028
$186$	0	0
$187$	−20.8495	−1.52467
$188$	0	0
$189$	−0.267281	−0.0194419
$190$	0	0
$191$	19.8357	1.43526	0.717631	−	0.696423i	$-0.245226\pi$
$191$	19.8357	1.43526	0.717631	+	0.696423i	$0.245226\pi$
$192$	0	0
$193$	1.53579	0.110548	0.0552742	−	0.998471i	$-0.482397\pi$
$193$	1.53579	0.110548	0.0552742	+	0.998471i	$0.482397\pi$
$194$	0	0
$195$	−42.1566	−3.01890
$196$	0	0
$197$	3.54828	0.252804	0.126402	−	0.991979i	$-0.459657\pi$
$197$	3.54828	0.252804	0.126402	+	0.991979i	$0.459657\pi$
$198$	0	0
$199$	−11.3509	−0.804646	−0.402323	−	0.915498i	$-0.631797\pi$
$199$	−11.3509	−0.804646	−0.402323	+	0.915498i	$0.631797\pi$
$200$	0	0
$201$	−12.4518	−0.878285
$202$	0	0
$203$	10.0060	0.702286
$204$	0	0
$205$	5.47715	0.382541
$206$	0	0
$207$	7.49960	0.521258
$208$	0	0
$209$	5.01154	0.346655
$210$	0	0
$211$	−1.12076	−0.0771562	−0.0385781	−	0.999256i	$-0.512283\pi$
$211$	−1.12076	−0.0771562	−0.0385781	+	0.999256i	$0.512283\pi$
$212$	0	0
$213$	−16.8384	−1.15375
$214$	0	0
$215$	−5.41042	−0.368987
$216$	0	0
$217$	2.79339	0.189628
$218$	0	0
$219$	29.3706	1.98468
$220$	0	0
$221$	37.7761	2.54110
$222$	0	0
$223$	−4.86576	−0.325835	−0.162918	−	0.986640i	$-0.552090\pi$
$223$	−4.86576	−0.325835	−0.162918	+	0.986640i	$0.552090\pi$
$224$	0	0
$225$	9.43722	0.629148
$226$	0	0
$227$	−6.50490	−0.431745	−0.215873	−	0.976422i	$-0.569260\pi$
$227$	−6.50490	−0.431745	−0.215873	+	0.976422i	$0.569260\pi$
$228$	0	0
$229$	3.07753	0.203369	0.101684	−	0.994817i	$-0.467577\pi$
$229$	3.07753	0.203369	0.101684	+	0.994817i	$0.467577\pi$
$230$	0	0
$231$	−9.89670	−0.651155
$232$	0	0
$233$	−5.90603	−0.386917	−0.193459	−	0.981108i	$-0.561971\pi$
$233$	−5.90603	−0.386917	−0.193459	+	0.981108i	$0.561971\pi$
$234$	0	0
$235$	9.30941	0.607279
$236$	0	0
$237$	−33.0458	−2.14656
$238$	0	0
$239$	11.7050	0.757133	0.378567	−	0.925574i	$-0.376417\pi$
$239$	11.7050	0.757133	0.378567	+	0.925574i	$0.376417\pi$
$240$	0	0
$241$	−22.8888	−1.47439	−0.737197	−	0.675678i	$-0.763851\pi$
$241$	−22.8888	−1.47439	−0.737197	+	0.675678i	$0.763851\pi$
$242$	0	0
$243$	22.1898	1.42347
$244$	0	0
$245$	15.7306	1.00499
$246$	0	0
$247$	−9.08014	−0.577755
$248$	0	0
$249$	−31.8558	−2.01878
$250$	0	0
$251$	−4.96998	−0.313703	−0.156851	−	0.987622i	$-0.550134\pi$
$251$	−4.96998	−0.313703	−0.156851	+	0.987622i	$0.550134\pi$
$252$	0	0
$253$	8.06395	0.506976
$254$	0	0
$255$	−43.9523	−2.75240
$256$	0	0
$257$	−9.03181	−0.563389	−0.281694	−	0.959504i	$-0.590896\pi$
$257$	−9.03181	−0.563389	−0.281694	+	0.959504i	$0.590896\pi$
$258$	0	0
$259$	−9.77891	−0.607632
$260$	0	0
$261$	25.6106	1.58525
$262$	0	0
$263$	−5.53000	−0.340994	−0.170497	−	0.985358i	$-0.554537\pi$
$263$	−5.53000	−0.340994	−0.170497	+	0.985358i	$0.554537\pi$
$264$	0	0
$265$	15.8345	0.972707
$266$	0	0
$267$	18.4241	1.12754
$268$	0	0
$269$	25.9114	1.57984	0.789922	−	0.613207i	$-0.210121\pi$
$269$	25.9114	1.57984	0.789922	+	0.613207i	$0.210121\pi$
$270$	0	0
$271$	−12.8890	−0.782952	−0.391476	−	0.920188i	$-0.628035\pi$
$271$	−12.8890	−0.782952	−0.391476	+	0.920188i	$0.628035\pi$
$272$	0	0
$273$	17.9313	1.08525
$274$	0	0
$275$	10.1474	0.611910
$276$	0	0
$277$	−11.3081	−0.679436	−0.339718	−	0.940527i	$-0.610332\pi$
$277$	−11.3081	−0.679436	−0.339718	+	0.940527i	$0.610332\pi$
$278$	0	0
$279$	7.14972	0.428043
$280$	0	0
$281$	15.8225	0.943891	0.471946	−	0.881628i	$-0.343552\pi$
$281$	15.8225	0.943891	0.471946	+	0.881628i	$0.343552\pi$
$282$	0	0
$283$	−7.75262	−0.460845	−0.230423	−	0.973091i	$-0.574011\pi$
$283$	−7.75262	−0.460845	−0.230423	+	0.973091i	$0.574011\pi$
$284$	0	0
$285$	10.5647	0.625798
$286$	0	0
$287$	−2.32970	−0.137518
$288$	0	0
$289$	22.3852	1.31678
$290$	0	0
$291$	−35.7539	−2.09593
$292$	0	0
$293$	7.96476	0.465306	0.232653	−	0.972560i	$-0.425259\pi$
$293$	7.96476	0.465306	0.232653	+	0.972560i	$0.425259\pi$
$294$	0	0
$295$	−0.372390	−0.0216814
$296$	0	0
$297$	−0.735590	−0.0426832
$298$	0	0
$299$	−14.6106	−0.844955
$300$	0	0
$301$	2.30132	0.132646
$302$	0	0
$303$	9.26482	0.532250
$304$	0	0
$305$	−30.2066	−1.72962
$306$	0	0
$307$	14.0455	0.801619	0.400809	−	0.916161i	$-0.368729\pi$
$307$	14.0455	0.801619	0.400809	+	0.916161i	$0.368729\pi$
$308$	0	0
$309$	−31.1915	−1.77442
$310$	0	0
$311$	29.1823	1.65478	0.827389	−	0.561629i	$-0.189825\pi$
$311$	29.1823	1.65478	0.827389	+	0.561629i	$0.189825\pi$
$312$	0	0
$313$	−7.44829	−0.421002	−0.210501	−	0.977594i	$-0.567510\pi$
$313$	−7.44829	−0.421002	−0.210501	+	0.977594i	$0.567510\pi$
$314$	0	0
$315$	−10.5852	−0.596407
$316$	0	0
$317$	7.16057	0.402177	0.201089	−	0.979573i	$-0.435552\pi$
$317$	7.16057	0.402177	0.201089	+	0.979573i	$0.435552\pi$
$318$	0	0
$319$	27.5378	1.54182
$320$	0	0
$321$	8.35139	0.466129
$322$	0	0
$323$	−9.46690	−0.526753
$324$	0	0
$325$	−18.3855	−1.01984
$326$	0	0
$327$	−31.3728	−1.73492
$328$	0	0
$329$	−3.95976	−0.218308
$330$	0	0
$331$	10.5245	0.578478	0.289239	−	0.957257i	$-0.406598\pi$
$331$	10.5245	0.578478	0.289239	+	0.957257i	$0.406598\pi$
$332$	0	0
$333$	−25.0293	−1.37159
$334$	0	0
$335$	−14.3203	−0.782400
$336$	0	0
$337$	−16.1765	−0.881188	−0.440594	−	0.897707i	$-0.645232\pi$
$337$	−16.1765	−0.881188	−0.440594	+	0.897707i	$0.645232\pi$
$338$	0	0
$339$	1.21891	0.0662022
$340$	0	0
$341$	7.68774	0.416314
$342$	0	0
$343$	−15.1411	−0.817540
$344$	0	0
$345$	16.9994	0.915217
$346$	0	0
$347$	16.2125	0.870332	0.435166	−	0.900350i	$-0.356690\pi$
$347$	16.2125	0.870332	0.435166	+	0.900350i	$0.356690\pi$
$348$	0	0
$349$	−31.8852	−1.70678	−0.853389	−	0.521275i	$-0.825456\pi$
$349$	−31.8852	−1.70678	−0.853389	+	0.521275i	$0.825456\pi$
$350$	0	0
$351$	1.33278	0.0711383
$352$	0	0
$353$	−19.5396	−1.03999	−0.519995	−	0.854169i	$-0.674066\pi$
$353$	−19.5396	−1.03999	−0.519995	+	0.854169i	$0.674066\pi$
$354$	0	0
$355$	−19.3650	−1.02779
$356$	0	0
$357$	18.6951	0.989449
$358$	0	0
$359$	−27.1446	−1.43263	−0.716317	−	0.697775i	$-0.754174\pi$
$359$	−27.1446	−1.43263	−0.716317	+	0.697775i	$0.754174\pi$
$360$	0	0
$361$	−16.7245	−0.880235
$362$	0	0
$363$	−0.0917762	−0.00481700
$364$	0	0
$365$	33.7777	1.76801
$366$	0	0
$367$	−27.2382	−1.42182	−0.710911	−	0.703282i	$-0.751717\pi$
$367$	−27.2382	−1.42182	−0.710911	+	0.703282i	$0.751717\pi$
$368$	0	0
$369$	−5.96291	−0.310416
$370$	0	0
$371$	−6.73521	−0.349675
$372$	0	0
$373$	−10.5394	−0.545710	−0.272855	−	0.962055i	$-0.587968\pi$
$373$	−10.5394	−0.545710	−0.272855	+	0.962055i	$0.587968\pi$
$374$	0	0
$375$	−13.6261	−0.703647
$376$	0	0
$377$	−49.8942	−2.56968
$378$	0	0
$379$	−12.4414	−0.639072	−0.319536	−	0.947574i	$-0.603527\pi$
$379$	−12.4414	−0.639072	−0.319536	+	0.947574i	$0.603527\pi$
$380$	0	0
$381$	41.5673	2.12956
$382$	0	0
$383$	0.806037	0.0411866	0.0205933	−	0.999788i	$-0.493444\pi$
$383$	0.806037	0.0411866	0.0205933	+	0.999788i	$0.493444\pi$
$384$	0	0
$385$	−11.3817	−0.580066
$386$	0	0
$387$	5.89025	0.299418
$388$	0	0
$389$	−25.7835	−1.30728	−0.653639	−	0.756807i	$-0.726758\pi$
$389$	−25.7835	−1.30728	−0.653639	+	0.756807i	$0.726758\pi$
$390$	0	0
$391$	−15.2330	−0.770365
$392$	0	0
$393$	−53.3484	−2.69107
$394$	0	0
$395$	−38.0044	−1.91221
$396$	0	0
$397$	−4.19088	−0.210334	−0.105167	−	0.994455i	$-0.533538\pi$
$397$	−4.19088	−0.210334	−0.105167	+	0.994455i	$0.533538\pi$
$398$	0	0
$399$	−4.49369	−0.224966
$400$	0	0
$401$	34.2199	1.70886	0.854430	−	0.519567i	$-0.173907\pi$
$401$	34.2199	1.70886	0.854430	+	0.519567i	$0.173907\pi$
$402$	0	0
$403$	−13.9290	−0.693853
$404$	0	0
$405$	24.7555	1.23011
$406$	0	0
$407$	−26.9127	−1.33401
$408$	0	0
$409$	−15.8967	−0.786041	−0.393020	−	0.919530i	$-0.628570\pi$
$409$	−15.8967	−0.786041	−0.393020	+	0.919530i	$0.628570\pi$
$410$	0	0
$411$	−12.5065	−0.616901
$412$	0	0
$413$	0.158396	0.00779415
$414$	0	0
$415$	−36.6359	−1.79838
$416$	0	0
$417$	−13.6976	−0.670775
$418$	0	0
$419$	−7.28835	−0.356059	−0.178030	−	0.984025i	$-0.556972\pi$
$419$	−7.28835	−0.356059	−0.178030	+	0.984025i	$0.556972\pi$
$420$	0	0
$421$	−35.1736	−1.71426	−0.857128	−	0.515103i	$-0.827754\pi$
$421$	−35.1736	−1.71426	−0.857128	+	0.515103i	$0.827754\pi$
$422$	0	0
$423$	−10.1350	−0.492783
$424$	0	0
$425$	−19.1686	−0.929815
$426$	0	0
$427$	12.8484	0.621776
$428$	0	0
$429$	49.3491	2.38259
$430$	0	0
$431$	2.33907	0.112669	0.0563345	−	0.998412i	$-0.482059\pi$
$431$	2.33907	0.112669	0.0563345	+	0.998412i	$0.482059\pi$
$432$	0	0
$433$	−3.18429	−0.153027	−0.0765136	−	0.997069i	$-0.524379\pi$
$433$	−3.18429	−0.153027	−0.0765136	+	0.997069i	$0.524379\pi$
$434$	0	0
$435$	58.0517	2.78337
$436$	0	0
$437$	3.66151	0.175154
$438$	0	0
$439$	−6.24647	−0.298128	−0.149064	−	0.988828i	$-0.547626\pi$
$439$	−6.24647	−0.298128	−0.149064	+	0.988828i	$0.547626\pi$
$440$	0	0
$441$	−17.1257	−0.815508
$442$	0	0
$443$	29.6274	1.40764	0.703821	−	0.710378i	$-0.251476\pi$
$443$	29.6274	1.40764	0.703821	+	0.710378i	$0.251476\pi$
$444$	0	0
$445$	21.1887	1.00444
$446$	0	0
$447$	35.9505	1.70040
$448$	0	0
$449$	18.1994	0.858884	0.429442	−	0.903094i	$-0.358710\pi$
$449$	18.1994	0.858884	0.429442	+	0.903094i	$0.358710\pi$
$450$	0	0
$451$	−6.41162	−0.301911
$452$	0	0
$453$	20.3005	0.953799
$454$	0	0
$455$	20.6219	0.966771
$456$	0	0
$457$	−9.97359	−0.466545	−0.233272	−	0.972411i	$-0.574943\pi$
$457$	−9.97359	−0.466545	−0.233272	+	0.972411i	$0.574943\pi$
$458$	0	0
$459$	1.38955	0.0648584
$460$	0	0
$461$	−7.23992	−0.337196	−0.168598	−	0.985685i	$-0.553924\pi$
$461$	−7.23992	−0.337196	−0.168598	+	0.985685i	$0.553924\pi$
$462$	0	0
$463$	−22.5684	−1.04884	−0.524421	−	0.851459i	$-0.675718\pi$
$463$	−22.5684	−1.04884	−0.524421	+	0.851459i	$0.675718\pi$
$464$	0	0
$465$	16.2063	0.751550
$466$	0	0
$467$	−31.9623	−1.47904	−0.739518	−	0.673136i	$-0.764947\pi$
$467$	−31.9623	−1.47904	−0.739518	+	0.673136i	$0.764947\pi$
$468$	0	0
$469$	6.09112	0.281262
$470$	0	0
$471$	−56.5826	−2.60719
$472$	0	0
$473$	6.33350	0.291215
$474$	0	0
$475$	4.60751	0.211407
$476$	0	0
$477$	−17.2389	−0.789313
$478$	0	0
$479$	−16.8734	−0.770966	−0.385483	−	0.922715i	$-0.625965\pi$
$479$	−16.8734	−0.770966	−0.385483	+	0.922715i	$0.625965\pi$
$480$	0	0
$481$	48.7617	2.22334
$482$	0	0
$483$	−7.23069	−0.329008
$484$	0	0
$485$	−41.1188	−1.86711
$486$	0	0
$487$	−1.19546	−0.0541713	−0.0270857	−	0.999633i	$-0.508623\pi$
$487$	−1.19546	−0.0541713	−0.0270857	+	0.999633i	$0.508623\pi$
$488$	0	0
$489$	−46.7900	−2.11592
$490$	0	0
$491$	−34.8373	−1.57218	−0.786092	−	0.618110i	$-0.787899\pi$
$491$	−34.8373	−1.57218	−0.786092	+	0.618110i	$0.787899\pi$
$492$	0	0
$493$	−52.0195	−2.34284
$494$	0	0
$495$	−29.1317	−1.30937
$496$	0	0
$497$	8.23690	0.369475
$498$	0	0
$499$	3.27147	0.146451	0.0732255	−	0.997315i	$-0.476671\pi$
$499$	3.27147	0.146451	0.0732255	+	0.997315i	$0.476671\pi$
$500$	0	0
$501$	36.6141	1.63580
$502$	0	0
$503$	−6.72097	−0.299673	−0.149837	−	0.988711i	$-0.547875\pi$
$503$	−6.72097	−0.299673	−0.149837	+	0.988711i	$0.547875\pi$
$504$	0	0
$505$	10.6550	0.474143
$506$	0	0
$507$	−57.3323	−2.54622
$508$	0	0
$509$	−9.99263	−0.442915	−0.221458	−	0.975170i	$-0.571081\pi$
$509$	−9.99263	−0.442915	−0.221458	+	0.975170i	$0.571081\pi$
$510$	0	0
$511$	−14.3673	−0.635574
$512$	0	0
$513$	−0.334001	−0.0147465
$514$	0	0
$515$	−35.8719	−1.58070
$516$	0	0
$517$	−10.8977	−0.479281
$518$	0	0
$519$	51.3396	2.25356
$520$	0	0
$521$	40.5925	1.77839	0.889194	−	0.457530i	$-0.151266\pi$
$521$	40.5925	1.77839	0.889194	+	0.457530i	$0.151266\pi$
$522$	0	0
$523$	−39.2064	−1.71438	−0.857189	−	0.515002i	$-0.827791\pi$
$523$	−39.2064	−1.71438	−0.857189	+	0.515002i	$0.827791\pi$
$524$	0	0
$525$	−9.09883	−0.397106
$526$	0	0
$527$	−14.5223	−0.632602
$528$	0	0
$529$	−17.1083	−0.743841
$530$	0	0
$531$	0.405416	0.0175936
$532$	0	0
$533$	11.6169	0.503182
$534$	0	0
$535$	9.60454	0.415240
$536$	0	0
$537$	13.8961	0.599663
$538$	0	0
$539$	−18.4144	−0.793163
$540$	0	0
$541$	17.8267	0.766429	0.383214	−	0.923659i	$-0.374817\pi$
$541$	17.8267	0.766429	0.383214	+	0.923659i	$0.374817\pi$
$542$	0	0
$543$	18.9796	0.814494
$544$	0	0
$545$	−36.0803	−1.54551
$546$	0	0
$547$	9.70215	0.414834	0.207417	−	0.978253i	$-0.433494\pi$
$547$	9.70215	0.414834	0.207417	+	0.978253i	$0.433494\pi$
$548$	0	0
$549$	32.8855	1.40352
$550$	0	0
$551$	12.5038	0.532679
$552$	0	0
$553$	16.1652	0.687413
$554$	0	0
$555$	−56.7340	−2.40822
$556$	0	0
$557$	13.1860	0.558707	0.279353	−	0.960188i	$-0.409880\pi$
$557$	13.1860	0.558707	0.279353	+	0.960188i	$0.409880\pi$
$558$	0	0
$559$	−11.4753	−0.485355
$560$	0	0
$561$	51.4511	2.17227
$562$	0	0
$563$	−30.6826	−1.29312	−0.646559	−	0.762864i	$-0.723793\pi$
$563$	−30.6826	−1.29312	−0.646559	+	0.762864i	$0.723793\pi$
$564$	0	0
$565$	1.40181	0.0589747
$566$	0	0
$567$	−10.5297	−0.442207
$568$	0	0
$569$	−34.5430	−1.44812	−0.724058	−	0.689739i	$-0.757725\pi$
$569$	−34.5430	−1.44812	−0.724058	+	0.689739i	$0.757725\pi$
$570$	0	0
$571$	5.39999	0.225983	0.112991	−	0.993596i	$-0.463957\pi$
$571$	5.39999	0.225983	0.112991	+	0.993596i	$0.463957\pi$
$572$	0	0
$573$	−48.9493	−2.04489
$574$	0	0
$575$	7.41384	0.309179
$576$	0	0
$577$	26.6177	1.10811	0.554055	−	0.832480i	$-0.313080\pi$
$577$	26.6177	1.10811	0.554055	+	0.832480i	$0.313080\pi$
$578$	0	0
$579$	−3.78992	−0.157504
$580$	0	0
$581$	15.5831	0.646494
$582$	0	0
$583$	−18.5361	−0.767686
$584$	0	0
$585$	52.7821	2.18227
$586$	0	0
$587$	−8.42185	−0.347607	−0.173803	−	0.984780i	$-0.555606\pi$
$587$	−8.42185	−0.347607	−0.173803	+	0.984780i	$0.555606\pi$
$588$	0	0
$589$	3.49069	0.143831
$590$	0	0
$591$	−8.75621	−0.360183
$592$	0	0
$593$	−21.8039	−0.895379	−0.447690	−	0.894189i	$-0.647753\pi$
$593$	−21.8039	−0.895379	−0.447690	+	0.894189i	$0.647753\pi$
$594$	0	0
$595$	21.5003	0.881427
$596$	0	0
$597$	28.0111	1.14642
$598$	0	0
$599$	−31.6332	−1.29250	−0.646249	−	0.763126i	$-0.723663\pi$
$599$	−31.6332	−1.29250	−0.646249	+	0.763126i	$0.723663\pi$
$600$	0	0
$601$	−7.95703	−0.324574	−0.162287	−	0.986744i	$-0.551887\pi$
$601$	−7.95703	−0.324574	−0.162287	+	0.986744i	$0.551887\pi$
$602$	0	0
$603$	15.5903	0.634886
$604$	0	0
$605$	−0.105547	−0.00429111
$606$	0	0
$607$	−26.4167	−1.07222	−0.536110	−	0.844148i	$-0.680107\pi$
$607$	−26.4167	−1.07222	−0.536110	+	0.844148i	$0.680107\pi$
$608$	0	0
$609$	−24.6923	−1.00058
$610$	0	0
$611$	19.7450	0.798796
$612$	0	0
$613$	−2.03338	−0.0821274	−0.0410637	−	0.999157i	$-0.513075\pi$
$613$	−2.03338	−0.0821274	−0.0410637	+	0.999157i	$0.513075\pi$
$614$	0	0
$615$	−13.5162	−0.545024
$616$	0	0
$617$	−14.0533	−0.565764	−0.282882	−	0.959155i	$-0.591290\pi$
$617$	−14.0533	−0.565764	−0.282882	+	0.959155i	$0.591290\pi$
$618$	0	0
$619$	−19.2425	−0.773421	−0.386711	−	0.922201i	$-0.626389\pi$
$619$	−19.2425	−0.773421	−0.386711	+	0.922201i	$0.626389\pi$
$620$	0	0
$621$	−0.537434	−0.0215665
$622$	0	0
$623$	−9.01258	−0.361082
$624$	0	0
$625$	−30.9427	−1.23771
$626$	0	0
$627$	−12.3672	−0.493896
$628$	0	0
$629$	50.8387	2.02707
$630$	0	0
$631$	10.8099	0.430337	0.215168	−	0.976577i	$-0.430970\pi$
$631$	10.8099	0.430337	0.215168	+	0.976577i	$0.430970\pi$
$632$	0	0
$633$	2.76574	0.109928
$634$	0	0
$635$	47.8045	1.89707
$636$	0	0
$637$	33.3640	1.32193
$638$	0	0
$639$	21.0825	0.834009
$640$	0	0
$641$	10.9006	0.430546	0.215273	−	0.976554i	$-0.430936\pi$
$641$	10.9006	0.430546	0.215273	+	0.976554i	$0.430936\pi$
$642$	0	0
$643$	−11.7276	−0.462491	−0.231245	−	0.972895i	$-0.574280\pi$
$643$	−11.7276	−0.462491	−0.231245	+	0.972895i	$0.574280\pi$
$644$	0	0
$645$	13.3515	0.525714
$646$	0	0
$647$	−22.0657	−0.867491	−0.433746	−	0.901035i	$-0.642808\pi$
$647$	−22.0657	−0.867491	−0.433746	+	0.901035i	$0.642808\pi$
$648$	0	0
$649$	0.435924	0.0171115
$650$	0	0
$651$	−6.89335	−0.270172
$652$	0	0
$653$	38.5946	1.51033	0.755163	−	0.655537i	$-0.227558\pi$
$653$	38.5946	1.51033	0.755163	+	0.655537i	$0.227558\pi$
$654$	0	0
$655$	−61.3534	−2.39728
$656$	0	0
$657$	−36.7734	−1.43467
$658$	0	0
$659$	8.42173	0.328064	0.164032	−	0.986455i	$-0.447550\pi$
$659$	8.42173	0.328064	0.164032	+	0.986455i	$0.447550\pi$
$660$	0	0
$661$	−38.1634	−1.48438	−0.742192	−	0.670187i	$-0.766214\pi$
$661$	−38.1634	−1.48438	−0.742192	+	0.670187i	$0.766214\pi$
$662$	0	0
$663$	−93.2215	−3.62042
$664$	0	0
$665$	−5.16797	−0.200405
$666$	0	0
$667$	20.1196	0.779032
$668$	0	0
$669$	12.0074	0.464233
$670$	0	0
$671$	35.3602	1.36506
$672$	0	0
$673$	−33.6274	−1.29624	−0.648121	−	0.761537i	$-0.724445\pi$
$673$	−33.6274	−1.29624	−0.648121	+	0.761537i	$0.724445\pi$
$674$	0	0
$675$	−0.676287	−0.0260303
$676$	0	0
$677$	39.6002	1.52196	0.760980	−	0.648776i	$-0.224719\pi$
$677$	39.6002	1.52196	0.760980	+	0.648776i	$0.224719\pi$
$678$	0	0
$679$	17.4899	0.671199
$680$	0	0
$681$	16.0524	0.615129
$682$	0	0
$683$	24.3056	0.930028	0.465014	−	0.885303i	$-0.346049\pi$
$683$	24.3056	0.930028	0.465014	+	0.885303i	$0.346049\pi$
$684$	0	0
$685$	−14.3831	−0.549551
$686$	0	0
$687$	−7.59452	−0.289749
$688$	0	0
$689$	33.5845	1.27947
$690$	0	0
$691$	−30.1408	−1.14661	−0.573306	−	0.819342i	$-0.694339\pi$
$691$	−30.1408	−1.14661	−0.573306	+	0.819342i	$0.694339\pi$
$692$	0	0
$693$	12.3911	0.470700
$694$	0	0
$695$	−15.7530	−0.597544
$696$	0	0
$697$	12.1117	0.458763
$698$	0	0
$699$	14.5745	0.551260
$700$	0	0
$701$	7.42015	0.280255	0.140128	−	0.990133i	$-0.455249\pi$
$701$	7.42015	0.280255	0.140128	+	0.990133i	$0.455249\pi$
$702$	0	0
$703$	−12.2200	−0.460885
$704$	0	0
$705$	−22.9732	−0.865220
$706$	0	0
$707$	−4.53211	−0.170448
$708$	0	0
$709$	−0.188564	−0.00708168	−0.00354084	−	0.999994i	$-0.501127\pi$
$709$	−0.188564	−0.00708168	−0.00354084	+	0.999994i	$0.501127\pi$
$710$	0	0
$711$	41.3750	1.55168
$712$	0	0
$713$	5.61679	0.210350
$714$	0	0
$715$	56.7540	2.12248
$716$	0	0
$717$	−28.8848	−1.07872
$718$	0	0
$719$	19.0552	0.710638	0.355319	−	0.934745i	$-0.384372\pi$
$719$	19.0552	0.710638	0.355319	+	0.934745i	$0.384372\pi$
$720$	0	0
$721$	15.2581	0.568241
$722$	0	0
$723$	56.4834	2.10064
$724$	0	0
$725$	25.3177	0.940276
$726$	0	0
$727$	20.9946	0.778648	0.389324	−	0.921101i	$-0.372709\pi$
$727$	20.9946	0.778648	0.389324	+	0.921101i	$0.372709\pi$
$728$	0	0
$729$	−28.5902	−1.05889
$730$	0	0
$731$	−11.9641	−0.442509
$732$	0	0
$733$	23.4219	0.865107	0.432553	−	0.901608i	$-0.357613\pi$
$733$	23.4219	0.865107	0.432553	+	0.901608i	$0.357613\pi$
$734$	0	0
$735$	−38.8189	−1.43186
$736$	0	0
$737$	16.7635	0.617491
$738$	0	0
$739$	−47.0720	−1.73157	−0.865785	−	0.500416i	$-0.833180\pi$
$739$	−47.0720	−1.73157	−0.865785	+	0.500416i	$0.833180\pi$
$740$	0	0
$741$	22.4074	0.823156
$742$	0	0
$743$	47.0903	1.72758	0.863788	−	0.503855i	$-0.168085\pi$
$743$	47.0903	1.72758	0.863788	+	0.503855i	$0.168085\pi$
$744$	0	0
$745$	41.3450	1.51476
$746$	0	0
$747$	39.8850	1.45932
$748$	0	0
$749$	−4.08529	−0.149273
$750$	0	0
$751$	34.3163	1.25222	0.626111	−	0.779734i	$-0.284646\pi$
$751$	34.3163	1.25222	0.626111	+	0.779734i	$0.284646\pi$
$752$	0	0
$753$	12.2646	0.446947
$754$	0	0
$755$	23.3466	0.849670
$756$	0	0
$757$	14.5990	0.530611	0.265306	−	0.964164i	$-0.414527\pi$
$757$	14.5990	0.530611	0.265306	+	0.964164i	$0.414527\pi$
$758$	0	0
$759$	−19.8997	−0.722314
$760$	0	0
$761$	−39.1562	−1.41941	−0.709706	−	0.704498i	$-0.751172\pi$
$761$	−39.1562	−1.41941	−0.709706	+	0.704498i	$0.751172\pi$
$762$	0	0
$763$	15.3468	0.555590
$764$	0	0
$765$	55.0304	1.98963
$766$	0	0
$767$	−0.789827	−0.0285190
$768$	0	0
$769$	−29.5117	−1.06422	−0.532109	−	0.846676i	$-0.678600\pi$
$769$	−29.5117	−1.06422	−0.532109	+	0.846676i	$0.678600\pi$
$770$	0	0
$771$	22.2881	0.802687
$772$	0	0
$773$	−5.60709	−0.201673	−0.100836	−	0.994903i	$-0.532152\pi$
$773$	−5.60709	−0.201673	−0.100836	+	0.994903i	$0.532152\pi$
$774$	0	0
$775$	7.06796	0.253889
$776$	0	0
$777$	24.1318	0.865723
$778$	0	0
$779$	−2.91125	−0.104306
$780$	0	0
$781$	22.6689	0.811158
$782$	0	0
$783$	−1.83530	−0.0655881
$784$	0	0
$785$	−65.0729	−2.32255
$786$	0	0
$787$	30.1637	1.07522	0.537609	−	0.843194i	$-0.319328\pi$
$787$	30.1637	1.07522	0.537609	+	0.843194i	$0.319328\pi$
$788$	0	0
$789$	13.6466	0.485831
$790$	0	0
$791$	−0.596261	−0.0212006
$792$	0	0
$793$	−64.0672	−2.27509
$794$	0	0
$795$	−39.0754	−1.38586
$796$	0	0
$797$	42.1719	1.49380	0.746902	−	0.664934i	$-0.231540\pi$
$797$	42.1719	1.49380	0.746902	+	0.664934i	$0.231540\pi$
$798$	0	0
$799$	20.5860	0.728281
$800$	0	0
$801$	−23.0678	−0.815062
$802$	0	0
$803$	−39.5406	−1.39536
$804$	0	0
$805$	−8.31567	−0.293089
$806$	0	0
$807$	−63.9424	−2.25088
$808$	0	0
$809$	36.1854	1.27221	0.636106	−	0.771602i	$-0.280544\pi$
$809$	36.1854	1.27221	0.636106	+	0.771602i	$0.280544\pi$
$810$	0	0
$811$	−40.9142	−1.43669	−0.718345	−	0.695687i	$-0.755100\pi$
$811$	−40.9142	−1.43669	−0.718345	+	0.695687i	$0.755100\pi$
$812$	0	0
$813$	31.8067	1.11551
$814$	0	0
$815$	−53.8110	−1.88492
$816$	0	0
$817$	2.87578	0.100611
$818$	0	0
$819$	−22.4508	−0.784496
$820$	0	0
$821$	−18.8310	−0.657207	−0.328604	−	0.944468i	$-0.606578\pi$
$821$	−18.8310	−0.657207	−0.328604	+	0.944468i	$0.606578\pi$
$822$	0	0
$823$	−29.0052	−1.01106	−0.505529	−	0.862810i	$-0.668703\pi$
$823$	−29.0052	−1.01106	−0.505529	+	0.862810i	$0.668703\pi$
$824$	0	0
$825$	−25.0411	−0.871818
$826$	0	0
$827$	6.71926	0.233651	0.116826	−	0.993152i	$-0.462728\pi$
$827$	6.71926	0.233651	0.116826	+	0.993152i	$0.462728\pi$
$828$	0	0
$829$	3.29359	0.114391	0.0571956	−	0.998363i	$-0.481784\pi$
$829$	3.29359	0.114391	0.0571956	+	0.998363i	$0.481784\pi$
$830$	0	0
$831$	27.9053	0.968026
$832$	0	0
$833$	34.7852	1.20523
$834$	0	0
$835$	42.1081	1.45721
$836$	0	0
$837$	−0.512361	−0.0177098
$838$	0	0
$839$	16.0913	0.555535	0.277767	−	0.960648i	$-0.410406\pi$
$839$	16.0913	0.555535	0.277767	+	0.960648i	$0.410406\pi$
$840$	0	0
$841$	39.7067	1.36920
$842$	0	0
$843$	−39.0458	−1.34481
$844$	0	0
$845$	−65.9352	−2.26824
$846$	0	0
$847$	0.0448946	0.00154260
$848$	0	0
$849$	19.1314	0.656589
$850$	0	0
$851$	−19.6629	−0.674035
$852$	0	0
$853$	−3.93970	−0.134893	−0.0674464	−	0.997723i	$-0.521485\pi$
$853$	−3.93970	−0.134893	−0.0674464	+	0.997723i	$0.521485\pi$
$854$	0	0
$855$	−13.2275	−0.452371
$856$	0	0
$857$	29.7266	1.01544	0.507721	−	0.861522i	$-0.330488\pi$
$857$	29.7266	1.01544	0.507721	+	0.861522i	$0.330488\pi$
$858$	0	0
$859$	0.673462	0.0229782	0.0114891	−	0.999934i	$-0.496343\pi$
$859$	0.673462	0.0229782	0.0114891	+	0.999934i	$0.496343\pi$
$860$	0	0
$861$	5.74909	0.195929
$862$	0	0
$863$	4.00145	0.136211	0.0681054	−	0.997678i	$-0.478305\pi$
$863$	4.00145	0.136211	0.0681054	+	0.997678i	$0.478305\pi$
$864$	0	0
$865$	59.0432	2.00753
$866$	0	0
$867$	−55.2407	−1.87607
$868$	0	0
$869$	44.4885	1.50917
$870$	0	0
$871$	−30.3729	−1.02915
$872$	0	0
$873$	44.7655	1.51508
$874$	0	0
$875$	6.66552	0.225336
$876$	0	0
$877$	16.6593	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$877$	16.6593	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$878$	0	0
$879$	−19.6549	−0.662945
$880$	0	0
$881$	−13.1999	−0.444717	−0.222358	−	0.974965i	$-0.571375\pi$
$881$	−13.1999	−0.444717	−0.222358	+	0.974965i	$0.571375\pi$
$882$	0	0
$883$	−29.0896	−0.978944	−0.489472	−	0.872019i	$-0.662810\pi$
$883$	−29.0896	−0.978944	−0.489472	+	0.872019i	$0.662810\pi$
$884$	0	0
$885$	0.918960	0.0308905
$886$	0	0
$887$	−6.61080	−0.221969	−0.110985	−	0.993822i	$-0.535400\pi$
$887$	−6.61080	−0.221969	−0.110985	+	0.993822i	$0.535400\pi$
$888$	0	0
$889$	−20.3336	−0.681968
$890$	0	0
$891$	−28.9790	−0.970835
$892$	0	0
$893$	−4.94820	−0.165585
$894$	0	0
$895$	15.9813	0.534195
$896$	0	0
$897$	36.0552	1.20385
$898$	0	0
$899$	19.1809	0.639719
$900$	0	0
$901$	35.0151	1.16652
$902$	0	0
$903$	−5.67905	−0.188987
$904$	0	0
$905$	21.8276	0.725573
$906$	0	0
$907$	18.3938	0.610755	0.305377	−	0.952231i	$-0.401217\pi$
$907$	18.3938	0.610755	0.305377	+	0.952231i	$0.401217\pi$
$908$	0	0
$909$	−11.6000	−0.384748
$910$	0	0
$911$	56.6153	1.87575	0.937875	−	0.346974i	$-0.112791\pi$
$911$	56.6153	1.87575	0.937875	+	0.346974i	$0.112791\pi$
$912$	0	0
$913$	42.8864	1.41933
$914$	0	0
$915$	74.5419	2.46428
$916$	0	0
$917$	26.0966	0.861787
$918$	0	0
$919$	−14.2054	−0.468592	−0.234296	−	0.972165i	$-0.575279\pi$
$919$	−14.2054	−0.468592	−0.234296	+	0.972165i	$0.575279\pi$
$920$	0	0
$921$	−34.6606	−1.14211
$922$	0	0
$923$	−41.0726	−1.35192
$924$	0	0
$925$	−24.7430	−0.813546
$926$	0	0
$927$	39.0533	1.28268
$928$	0	0
$929$	−29.2143	−0.958490	−0.479245	−	0.877681i	$-0.659089\pi$
$929$	−29.2143	−0.958490	−0.479245	+	0.877681i	$0.659089\pi$
$930$	0	0
$931$	−8.36121	−0.274028
$932$	0	0
$933$	−72.0143	−2.35764
$934$	0	0
$935$	59.1714	1.93511
$936$	0	0
$937$	−29.6437	−0.968418	−0.484209	−	0.874952i	$-0.660893\pi$
$937$	−29.6437	−0.968418	−0.484209	+	0.874952i	$0.660893\pi$
$938$	0	0
$939$	18.3804	0.599823
$940$	0	0
$941$	−32.9786	−1.07507	−0.537535	−	0.843241i	$-0.680645\pi$
$941$	−32.9786	−1.07507	−0.537535	+	0.843241i	$0.680645\pi$
$942$	0	0
$943$	−4.68443	−0.152546
$944$	0	0
$945$	0.758551	0.0246757
$946$	0	0
$947$	46.4508	1.50945	0.754724	−	0.656043i	$-0.227771\pi$
$947$	46.4508	1.50945	0.754724	+	0.656043i	$0.227771\pi$
$948$	0	0
$949$	71.6415	2.32558
$950$	0	0
$951$	−17.6704	−0.573002
$952$	0	0
$953$	20.6226	0.668033	0.334016	−	0.942567i	$-0.391596\pi$
$953$	20.6226	0.668033	0.334016	+	0.942567i	$0.391596\pi$
$954$	0	0
$955$	−56.2943	−1.82164
$956$	0	0
$957$	−67.9560	−2.19671
$958$	0	0
$959$	6.11786	0.197556
$960$	0	0
$961$	−25.6453	−0.827266
$962$	0	0
$963$	−10.4563	−0.336951
$964$	0	0
$965$	−4.35861	−0.140308
$966$	0	0
$967$	11.6391	0.374288	0.187144	−	0.982332i	$-0.440077\pi$
$967$	11.6391	0.374288	0.187144	+	0.982332i	$0.440077\pi$
$968$	0	0
$969$	23.3618	0.750490
$970$	0	0
$971$	10.9618	0.351782	0.175891	−	0.984410i	$-0.443719\pi$
$971$	10.9618	0.351782	0.175891	+	0.984410i	$0.443719\pi$
$972$	0	0
$973$	6.70052	0.214809
$974$	0	0
$975$	45.3706	1.45302
$976$	0	0
$977$	−22.3441	−0.714853	−0.357426	−	0.933941i	$-0.616346\pi$
$977$	−22.3441	−0.714853	−0.357426	+	0.933941i	$0.616346\pi$
$978$	0	0
$979$	−24.8037	−0.792729
$980$	0	0
$981$	39.2802	1.25412
$982$	0	0
$983$	39.7074	1.26647	0.633235	−	0.773960i	$-0.281727\pi$
$983$	39.7074	1.26647	0.633235	+	0.773960i	$0.281727\pi$
$984$	0	0
$985$	−10.0701	−0.320860
$986$	0	0
$987$	9.77163	0.311034
$988$	0	0
$989$	4.62736	0.147141
$990$	0	0
$991$	3.29881	0.104790	0.0523951	−	0.998626i	$-0.483314\pi$
$991$	3.29881	0.104790	0.0523951	+	0.998626i	$0.483314\pi$
$992$	0	0
$993$	−25.9716	−0.824185
$994$	0	0
$995$	32.2142	1.02126
$996$	0	0
$997$	8.48108	0.268598	0.134299	−	0.990941i	$-0.457122\pi$
$997$	8.48108	0.268598	0.134299	+	0.990941i	$0.457122\pi$
$998$	0	0
$999$	1.79364	0.0567482

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8012.2.a.a.1.14	✓	79

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8012.2.a.a.1.14	✓	79	1.1	even	1	trivial

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Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8012 = 2^{2} \cdot 2003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8012.a (trivial)

\(f(q)\)	\(=\)	\(q-2.46774 q^{3} -2.83803 q^{5} +1.20715 q^{7} +3.08972 q^{9} +O(q^{10})\)	\(q-2.46774 q^{3} -2.83803 q^{5} +1.20715 q^{7} +3.08972 q^{9} +3.32223 q^{11} -6.01937 q^{13} +7.00350 q^{15} -6.27576 q^{17} +1.50849 q^{19} -2.97894 q^{21} +2.42727 q^{23} +3.05439 q^{25} -0.221415 q^{27} +8.28895 q^{29} +2.31403 q^{31} -8.19838 q^{33} -3.42593 q^{35} -8.10081 q^{37} +14.8542 q^{39} -1.92992 q^{41} +1.90640 q^{43} -8.76872 q^{45} -3.28024 q^{47} -5.54278 q^{49} +15.4869 q^{51} -5.57942 q^{53} -9.42857 q^{55} -3.72255 q^{57} +0.131214 q^{59} +10.6435 q^{61} +3.72977 q^{63} +17.0831 q^{65} +5.04586 q^{67} -5.98987 q^{69} +6.82341 q^{71} -11.9018 q^{73} -7.53743 q^{75} +4.01044 q^{77} +13.3912 q^{79} -8.72278 q^{81} +12.9089 q^{83} +17.8108 q^{85} -20.4550 q^{87} -7.46598 q^{89} -7.26630 q^{91} -5.71042 q^{93} -4.28113 q^{95} +14.4885 q^{97} +10.2648 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 79 q - 19 q^{3} - 40 q^{7} + 72 q^{9}+O(q^{10}) \) 79 * q - 19 * q^3 - 40 * q^7 + 72 * q^9	\( 79 q - 19 q^{3} - 40 q^{7} + 72 q^{9} - 14 q^{11} - 3 q^{13} - 19 q^{15} - 30 q^{17} - 49 q^{19} + 7 q^{21} - 40 q^{23} + 51 q^{25} - 70 q^{27} + 2 q^{29} - 48 q^{31} - 25 q^{33} - 34 q^{35} - 35 q^{39} - 20 q^{41} - 104 q^{43} + 12 q^{45} - 38 q^{47} + 51 q^{49} - 41 q^{51} - q^{53} - 112 q^{55} - 34 q^{57} - 24 q^{59} - 120 q^{63} - 21 q^{65} - 67 q^{67} + 15 q^{69} - 28 q^{71} - 88 q^{73} - 103 q^{75} + 4 q^{77} - 99 q^{79} + 47 q^{81} - 70 q^{83} + 7 q^{85} - 109 q^{87} - 50 q^{89} - 83 q^{91} - 7 q^{93} - 61 q^{95} - 93 q^{97} - 78 q^{99}+O(q^{100}) \) 79 * q - 19 * q^3 - 40 * q^7 + 72 * q^9 - 14 * q^11 - 3 * q^13 - 19 * q^15 - 30 * q^17 - 49 * q^19 + 7 * q^21 - 40 * q^23 + 51 * q^25 - 70 * q^27 + 2 * q^29 - 48 * q^31 - 25 * q^33 - 34 * q^35 - 35 * q^39 - 20 * q^41 - 104 * q^43 + 12 * q^45 - 38 * q^47 + 51 * q^49 - 41 * q^51 - q^53 - 112 * q^55 - 34 * q^57 - 24 * q^59 - 120 * q^63 - 21 * q^65 - 67 * q^67 + 15 * q^69 - 28 * q^71 - 88 * q^73 - 103 * q^75 + 4 * q^77 - 99 * q^79 + 47 * q^81 - 70 * q^83 + 7 * q^85 - 109 * q^87 - 50 * q^89 - 83 * q^91 - 7 * q^93 - 61 * q^95 - 93 * q^97 - 78 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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