LMFDB - Embedded newform 8012.2.a.b.1.19

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:	$0$
Dimension:	$88$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
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)

$f(q)$	$=$	$q-1.83871 q^{3} -1.10593 q^{5} +4.51431 q^{7} +0.380865 q^{9} +O(q^{10})$	$q-1.83871 q^{3} -1.10593 q^{5} +4.51431 q^{7} +0.380865 q^{9} +4.69997 q^{11} -2.55862 q^{13} +2.03349 q^{15} +7.60170 q^{17} +5.26385 q^{19} -8.30052 q^{21} +2.14897 q^{23} -3.77691 q^{25} +4.81584 q^{27} +8.15716 q^{29} -3.55033 q^{31} -8.64189 q^{33} -4.99252 q^{35} +10.6827 q^{37} +4.70457 q^{39} +9.16633 q^{41} -2.70941 q^{43} -0.421211 q^{45} -0.286749 q^{47} +13.3790 q^{49} -13.9773 q^{51} +4.04037 q^{53} -5.19785 q^{55} -9.67871 q^{57} -4.75788 q^{59} -5.24030 q^{61} +1.71934 q^{63} +2.82967 q^{65} -9.59281 q^{67} -3.95134 q^{69} -0.983246 q^{71} +12.9185 q^{73} +6.94466 q^{75} +21.2171 q^{77} +15.4081 q^{79} -9.99754 q^{81} +10.9900 q^{83} -8.40697 q^{85} -14.9987 q^{87} -13.2022 q^{89} -11.5504 q^{91} +6.52804 q^{93} -5.82146 q^{95} -2.27798 q^{97} +1.79005 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 88 q + 19 q^{3} + 44 q^{7} + 99 q^{9}+O(q^{10}) $ 88 * q + 19 * q^3 + 44 * q^7 + 99 * q^9	$ 88 q + 19 q^{3} + 44 q^{7} + 99 q^{9} + 16 q^{11} - 3 q^{13} + 21 q^{15} + 32 q^{17} + 49 q^{19} + 7 q^{21} + 24 q^{23} + 114 q^{25} + 82 q^{27} + 2 q^{29} + 40 q^{31} + 31 q^{33} + 26 q^{35} + 37 q^{39} + 22 q^{41} + 98 q^{43} + 12 q^{45} + 48 q^{47} + 132 q^{49} + 55 q^{51} - q^{53} + 116 q^{55} + 62 q^{57} + 34 q^{59} + 132 q^{63} + 39 q^{65} + 75 q^{67} + 15 q^{69} + 24 q^{71} + 104 q^{73} + 87 q^{75} + 4 q^{77} + 111 q^{79} + 128 q^{81} + 64 q^{83} + 7 q^{85} + 115 q^{87} + 14 q^{89} + 73 q^{91} - 7 q^{93} + 51 q^{95} + 117 q^{97} + 72 q^{99}+O(q^{100}) $ 88 * q + 19 * q^3 + 44 * q^7 + 99 * q^9 + 16 * q^11 - 3 * q^13 + 21 * q^15 + 32 * q^17 + 49 * q^19 + 7 * q^21 + 24 * q^23 + 114 * q^25 + 82 * q^27 + 2 * q^29 + 40 * q^31 + 31 * q^33 + 26 * q^35 + 37 * q^39 + 22 * q^41 + 98 * q^43 + 12 * q^45 + 48 * q^47 + 132 * q^49 + 55 * q^51 - q^53 + 116 * q^55 + 62 * q^57 + 34 * q^59 + 132 * q^63 + 39 * q^65 + 75 * q^67 + 15 * q^69 + 24 * q^71 + 104 * q^73 + 87 * q^75 + 4 * q^77 + 111 * q^79 + 128 * q^81 + 64 * q^83 + 7 * q^85 + 115 * q^87 + 14 * q^89 + 73 * q^91 - 7 * q^93 + 51 * q^95 + 117 * q^97 + 72 * 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
$3$	−1.83871	−1.06158	−0.530791	−	0.847503i	$-0.678105\pi$
$3$	−1.83871	−1.06158	−0.530791	+	0.847503i	$0.678105\pi$
$4$	0	0
$5$	−1.10593	−0.494588	−0.247294	−	0.968940i	$-0.579541\pi$
$5$	−1.10593	−0.494588	−0.247294	+	0.968940i	$0.579541\pi$
$6$	0	0
$7$	4.51431	1.70625	0.853124	−	0.521708i	$-0.174705\pi$
$7$	4.51431	1.70625	0.853124	+	0.521708i	$0.174705\pi$
$8$	0	0
$9$	0.380865	0.126955
$10$	0	0
$11$	4.69997	1.41709	0.708546	−	0.705664i	$-0.249351\pi$
$11$	4.69997	1.41709	0.708546	+	0.705664i	$0.249351\pi$
$12$	0	0
$13$	−2.55862	−0.709634	−0.354817	−	0.934936i	$-0.615457\pi$
$13$	−2.55862	−0.709634	−0.354817	+	0.934936i	$0.615457\pi$
$14$	0	0
$15$	2.03349	0.525046
$16$	0	0
$17$	7.60170	1.84368	0.921841	−	0.387568i	$-0.126685\pi$
$17$	7.60170	1.84368	0.921841	+	0.387568i	$0.126685\pi$
$18$	0	0
$19$	5.26385	1.20761	0.603805	−	0.797132i	$-0.293651\pi$
$19$	5.26385	1.20761	0.603805	+	0.797132i	$0.293651\pi$
$20$	0	0
$21$	−8.30052	−1.81132
$22$	0	0
$23$	2.14897	0.448092	0.224046	−	0.974579i	$-0.428073\pi$
$23$	2.14897	0.448092	0.224046	+	0.974579i	$0.428073\pi$
$24$	0	0
$25$	−3.77691	−0.755382
$26$	0	0
$27$	4.81584	0.926808
$28$	0	0
$29$	8.15716	1.51475	0.757374	−	0.652982i	$-0.226482\pi$
$29$	8.15716	1.51475	0.757374	+	0.652982i	$0.226482\pi$
$30$	0	0
$31$	−3.55033	−0.637658	−0.318829	−	0.947812i	$-0.603290\pi$
$31$	−3.55033	−0.637658	−0.318829	+	0.947812i	$0.603290\pi$
$32$	0	0
$33$	−8.64189	−1.50436
$34$	0	0
$35$	−4.99252	−0.843891
$36$	0	0
$37$	10.6827	1.75622	0.878112	−	0.478456i	$-0.158803\pi$
$37$	10.6827	1.75622	0.878112	+	0.478456i	$0.158803\pi$
$38$	0	0
$39$	4.70457	0.753335
$40$	0	0
$41$	9.16633	1.43154	0.715770	−	0.698336i	$-0.246076\pi$
$41$	9.16633	1.43154	0.715770	+	0.698336i	$0.246076\pi$
$42$	0	0
$43$	−2.70941	−0.413181	−0.206590	−	0.978428i	$-0.566237\pi$
$43$	−2.70941	−0.413181	−0.206590	+	0.978428i	$0.566237\pi$
$44$	0	0
$45$	−0.421211	−0.0627904
$46$	0	0
$47$	−0.286749	−0.0418267	−0.0209133	−	0.999781i	$-0.506657\pi$
$47$	−0.286749	−0.0418267	−0.0209133	+	0.999781i	$0.506657\pi$
$48$	0	0
$49$	13.3790	1.91128
$50$	0	0
$51$	−13.9773	−1.95722
$52$	0	0
$53$	4.04037	0.554988	0.277494	−	0.960727i	$-0.410496\pi$
$53$	4.04037	0.554988	0.277494	+	0.960727i	$0.410496\pi$
$54$	0	0
$55$	−5.19785	−0.700878
$56$	0	0
$57$	−9.67871	−1.28198
$58$	0	0
$59$	−4.75788	−0.619423	−0.309711	−	0.950831i	$-0.600232\pi$
$59$	−4.75788	−0.619423	−0.309711	+	0.950831i	$0.600232\pi$
$60$	0	0
$61$	−5.24030	−0.670951	−0.335476	−	0.942049i	$-0.608897\pi$
$61$	−5.24030	−0.670951	−0.335476	+	0.942049i	$0.608897\pi$
$62$	0	0
$63$	1.71934	0.216617
$64$	0	0
$65$	2.82967	0.350977
$66$	0	0
$67$	−9.59281	−1.17195	−0.585974	−	0.810330i	$-0.699288\pi$
$67$	−9.59281	−1.17195	−0.585974	+	0.810330i	$0.699288\pi$
$68$	0	0
$69$	−3.95134	−0.475686
$70$	0	0
$71$	−0.983246	−0.116690	−0.0583449	−	0.998296i	$-0.518582\pi$
$71$	−0.983246	−0.116690	−0.0583449	+	0.998296i	$0.518582\pi$
$72$	0	0
$73$	12.9185	1.51199	0.755995	−	0.654577i	$-0.227153\pi$
$73$	12.9185	1.51199	0.755995	+	0.654577i	$0.227153\pi$
$74$	0	0
$75$	6.94466	0.801900
$76$	0	0
$77$	21.2171	2.41791
$78$	0	0
$79$	15.4081	1.73355	0.866774	−	0.498700i	$-0.166189\pi$
$79$	15.4081	1.73355	0.866774	+	0.498700i	$0.166189\pi$
$80$	0	0
$81$	−9.99754	−1.11084
$82$	0	0
$83$	10.9900	1.20631	0.603153	−	0.797625i	$-0.293911\pi$
$83$	10.9900	1.20631	0.603153	+	0.797625i	$0.293911\pi$
$84$	0	0
$85$	−8.40697	−0.911864
$86$	0	0
$87$	−14.9987	−1.60803
$88$	0	0
$89$	−13.2022	−1.39943	−0.699715	−	0.714422i	$-0.746690\pi$
$89$	−13.2022	−1.39943	−0.699715	+	0.714422i	$0.746690\pi$
$90$	0	0
$91$	−11.5504	−1.21081
$92$	0	0
$93$	6.52804	0.676926
$94$	0	0
$95$	−5.82146	−0.597270
$96$	0	0
$97$	−2.27798	−0.231294	−0.115647	−	0.993290i	$-0.536894\pi$
$97$	−2.27798	−0.231294	−0.115647	+	0.993290i	$0.536894\pi$
$98$	0	0
$99$	1.79005	0.179907
$100$	0	0
$101$	4.72947	0.470600	0.235300	−	0.971923i	$-0.424393\pi$
$101$	4.72947	0.470600	0.235300	+	0.971923i	$0.424393\pi$
$102$	0	0
$103$	−13.9717	−1.37667	−0.688336	−	0.725392i	$-0.741659\pi$
$103$	−13.9717	−1.37667	−0.688336	+	0.725392i	$0.741659\pi$
$104$	0	0
$105$	9.17982	0.895859
$106$	0	0
$107$	−5.47720	−0.529501	−0.264751	−	0.964317i	$-0.585290\pi$
$107$	−5.47720	−0.529501	−0.264751	+	0.964317i	$0.585290\pi$
$108$	0	0
$109$	0.344645	0.0330110	0.0165055	−	0.999864i	$-0.494746\pi$
$109$	0.344645	0.0330110	0.0165055	+	0.999864i	$0.494746\pi$
$110$	0	0
$111$	−19.6424	−1.86437
$112$	0	0
$113$	−21.0306	−1.97839	−0.989196	−	0.146600i	$-0.953167\pi$
$113$	−21.0306	−1.97839	−0.989196	+	0.146600i	$0.953167\pi$
$114$	0	0
$115$	−2.37662	−0.221621
$116$	0	0
$117$	−0.974489	−0.0900916
$118$	0	0
$119$	34.3164	3.14578
$120$	0	0
$121$	11.0897	1.00815
$122$	0	0
$123$	−16.8542	−1.51970
$124$	0	0
$125$	9.70668	0.868192
$126$	0	0
$127$	3.35636	0.297828	0.148914	−	0.988850i	$-0.452422\pi$
$127$	3.35636	0.297828	0.148914	+	0.988850i	$0.452422\pi$
$128$	0	0
$129$	4.98182	0.438625
$130$	0	0
$131$	1.48497	0.129743	0.0648713	−	0.997894i	$-0.479336\pi$
$131$	1.48497	0.129743	0.0648713	+	0.997894i	$0.479336\pi$
$132$	0	0
$133$	23.7626	2.06048
$134$	0	0
$135$	−5.32599	−0.458389
$136$	0	0
$137$	0.712616	0.0608829	0.0304415	−	0.999537i	$-0.490309\pi$
$137$	0.712616	0.0608829	0.0304415	+	0.999537i	$0.490309\pi$
$138$	0	0
$139$	5.79644	0.491648	0.245824	−	0.969315i	$-0.420942\pi$
$139$	5.79644	0.491648	0.245824	+	0.969315i	$0.420942\pi$
$140$	0	0
$141$	0.527249	0.0444024
$142$	0	0
$143$	−12.0254	−1.00562
$144$	0	0
$145$	−9.02128	−0.749176
$146$	0	0
$147$	−24.6001	−2.02898
$148$	0	0
$149$	−12.4719	−1.02174	−0.510870	−	0.859658i	$-0.670677\pi$
$149$	−12.4719	−1.02174	−0.510870	+	0.859658i	$0.670677\pi$
$150$	0	0
$151$	7.60546	0.618924	0.309462	−	0.950912i	$-0.399851\pi$
$151$	7.60546	0.618924	0.309462	+	0.950912i	$0.399851\pi$
$152$	0	0
$153$	2.89522	0.234064
$154$	0	0
$155$	3.92643	0.315378
$156$	0	0
$157$	−3.65427	−0.291643	−0.145821	−	0.989311i	$-0.546582\pi$
$157$	−3.65427	−0.291643	−0.145821	+	0.989311i	$0.546582\pi$
$158$	0	0
$159$	−7.42909	−0.589165
$160$	0	0
$161$	9.70113	0.764556
$162$	0	0
$163$	−18.3951	−1.44082	−0.720409	−	0.693549i	$-0.756046\pi$
$163$	−18.3951	−1.44082	−0.720409	+	0.693549i	$0.756046\pi$
$164$	0	0
$165$	9.55735	0.744039
$166$	0	0
$167$	−6.48726	−0.501999	−0.251000	−	0.967987i	$-0.580759\pi$
$167$	−6.48726	−0.501999	−0.251000	+	0.967987i	$0.580759\pi$
$168$	0	0
$169$	−6.45345	−0.496419
$170$	0	0
$171$	2.00481	0.153312
$172$	0	0
$173$	−1.88232	−0.143110	−0.0715552	−	0.997437i	$-0.522796\pi$
$173$	−1.88232	−0.143110	−0.0715552	+	0.997437i	$0.522796\pi$
$174$	0	0
$175$	−17.0502	−1.28887
$176$	0	0
$177$	8.74837	0.657568
$178$	0	0
$179$	−17.6451	−1.31886	−0.659430	−	0.751766i	$-0.729202\pi$
$179$	−17.6451	−1.31886	−0.659430	+	0.751766i	$0.729202\pi$
$180$	0	0
$181$	−24.7200	−1.83742	−0.918712	−	0.394929i	$-0.870769\pi$
$181$	−24.7200	−1.83742	−0.918712	+	0.394929i	$0.870769\pi$
$182$	0	0
$183$	9.63540	0.712269
$184$	0	0
$185$	−11.8143	−0.868608
$186$	0	0
$187$	35.7277	2.61267
$188$	0	0
$189$	21.7402	1.58137
$190$	0	0
$191$	−2.19343	−0.158711	−0.0793556	−	0.996846i	$-0.525286\pi$
$191$	−2.19343	−0.158711	−0.0793556	+	0.996846i	$0.525286\pi$
$192$	0	0
$193$	22.6288	1.62886	0.814428	−	0.580264i	$-0.197051\pi$
$193$	22.6288	1.62886	0.814428	+	0.580264i	$0.197051\pi$
$194$	0	0
$195$	−5.20294	−0.372590
$196$	0	0
$197$	−10.4814	−0.746769	−0.373385	−	0.927677i	$-0.621803\pi$
$197$	−10.4814	−0.746769	−0.373385	+	0.927677i	$0.621803\pi$
$198$	0	0
$199$	14.9661	1.06092	0.530458	−	0.847711i	$-0.322020\pi$
$199$	14.9661	1.06092	0.530458	+	0.847711i	$0.322020\pi$
$200$	0	0
$201$	17.6384	1.24412
$202$	0	0
$203$	36.8240	2.58454
$204$	0	0
$205$	−10.1373	−0.708023
$206$	0	0
$207$	0.818468	0.0568874
$208$	0	0
$209$	24.7399	1.71130
$210$	0	0
$211$	−17.3372	−1.19354	−0.596770	−	0.802412i	$-0.703550\pi$
$211$	−17.3372	−1.19354	−0.596770	+	0.802412i	$0.703550\pi$
$212$	0	0
$213$	1.80791	0.123876
$214$	0	0
$215$	2.99642	0.204354
$216$	0	0
$217$	−16.0273	−1.08800
$218$	0	0
$219$	−23.7533	−1.60510
$220$	0	0
$221$	−19.4499	−1.30834
$222$	0	0
$223$	−2.24433	−0.150291	−0.0751457	−	0.997173i	$-0.523942\pi$
$223$	−2.24433	−0.150291	−0.0751457	+	0.997173i	$0.523942\pi$
$224$	0	0
$225$	−1.43849	−0.0958995
$226$	0	0
$227$	−13.6517	−0.906096	−0.453048	−	0.891486i	$-0.649663\pi$
$227$	−13.6517	−0.906096	−0.453048	+	0.891486i	$0.649663\pi$
$228$	0	0
$229$	−20.8130	−1.37536	−0.687679	−	0.726014i	$-0.741371\pi$
$229$	−20.8130	−1.37536	−0.687679	+	0.726014i	$0.741371\pi$
$230$	0	0
$231$	−39.0122	−2.56681
$232$	0	0
$233$	20.3606	1.33386	0.666932	−	0.745118i	$-0.267607\pi$
$233$	20.3606	1.33386	0.666932	+	0.745118i	$0.267607\pi$
$234$	0	0
$235$	0.317125	0.0206870
$236$	0	0
$237$	−28.3311	−1.84030
$238$	0	0
$239$	11.4133	0.738264	0.369132	−	0.929377i	$-0.379655\pi$
$239$	11.4133	0.738264	0.369132	+	0.929377i	$0.379655\pi$
$240$	0	0
$241$	13.4369	0.865548	0.432774	−	0.901503i	$-0.357535\pi$
$241$	13.4369	0.865548	0.432774	+	0.901503i	$0.357535\pi$
$242$	0	0
$243$	3.93508	0.252436
$244$	0	0
$245$	−14.7963	−0.945299
$246$	0	0
$247$	−13.4682	−0.856961
$248$	0	0
$249$	−20.2074	−1.28059
$250$	0	0
$251$	−24.3737	−1.53846	−0.769229	−	0.638974i	$-0.779359\pi$
$251$	−24.3737	−1.53846	−0.769229	+	0.638974i	$0.779359\pi$
$252$	0	0
$253$	10.1001	0.634988
$254$	0	0
$255$	15.4580	0.968017
$256$	0	0
$257$	−1.98520	−0.123834	−0.0619168	−	0.998081i	$-0.519721\pi$
$257$	−1.98520	−0.123834	−0.0619168	+	0.998081i	$0.519721\pi$
$258$	0	0
$259$	48.2250	2.99655
$260$	0	0
$261$	3.10678	0.192305
$262$	0	0
$263$	−1.15577	−0.0712680	−0.0356340	−	0.999365i	$-0.511345\pi$
$263$	−1.15577	−0.0712680	−0.0356340	+	0.999365i	$0.511345\pi$
$264$	0	0
$265$	−4.46838	−0.274491
$266$	0	0
$267$	24.2751	1.48561
$268$	0	0
$269$	7.84545	0.478346	0.239173	−	0.970977i	$-0.423124\pi$
$269$	7.84545	0.478346	0.239173	+	0.970977i	$0.423124\pi$
$270$	0	0
$271$	8.42636	0.511865	0.255932	−	0.966695i	$-0.417618\pi$
$271$	8.42636	0.511865	0.255932	+	0.966695i	$0.417618\pi$
$272$	0	0
$273$	21.2379	1.28538
$274$	0	0
$275$	−17.7514	−1.07045
$276$	0	0
$277$	−20.4489	−1.22866	−0.614329	−	0.789050i	$-0.710573\pi$
$277$	−20.4489	−1.22866	−0.614329	+	0.789050i	$0.710573\pi$
$278$	0	0
$279$	−1.35220	−0.0809538
$280$	0	0
$281$	−4.61778	−0.275473	−0.137737	−	0.990469i	$-0.543983\pi$
$281$	−4.61778	−0.275473	−0.137737	+	0.990469i	$0.543983\pi$
$282$	0	0
$283$	11.3247	0.673184	0.336592	−	0.941651i	$-0.390726\pi$
$283$	11.3247	0.673184	0.336592	+	0.941651i	$0.390726\pi$
$284$	0	0
$285$	10.7040	0.634050
$286$	0	0
$287$	41.3796	2.44256
$288$	0	0
$289$	40.7858	2.39916
$290$	0	0
$291$	4.18856	0.245537
$292$	0	0
$293$	2.94249	0.171902	0.0859511	−	0.996299i	$-0.472607\pi$
$293$	2.94249	0.171902	0.0859511	+	0.996299i	$0.472607\pi$
$294$	0	0
$295$	5.26189	0.306359
$296$	0	0
$297$	22.6343	1.31337
$298$	0	0
$299$	−5.49841	−0.317981
$300$	0	0
$301$	−12.2311	−0.704989
$302$	0	0
$303$	−8.69613	−0.499580
$304$	0	0
$305$	5.79542	0.331845
$306$	0	0
$307$	18.3183	1.04548	0.522740	−	0.852492i	$-0.324910\pi$
$307$	18.3183	1.04548	0.522740	+	0.852492i	$0.324910\pi$
$308$	0	0
$309$	25.6900	1.46145
$310$	0	0
$311$	23.9244	1.35663	0.678313	−	0.734773i	$-0.262711\pi$
$311$	23.9244	1.35663	0.678313	+	0.734773i	$0.262711\pi$
$312$	0	0
$313$	−5.23881	−0.296115	−0.148058	−	0.988979i	$-0.547302\pi$
$313$	−5.23881	−0.296115	−0.148058	+	0.988979i	$0.547302\pi$
$314$	0	0
$315$	−1.90148	−0.107136
$316$	0	0
$317$	11.3928	0.639885	0.319943	−	0.947437i	$-0.396336\pi$
$317$	11.3928	0.639885	0.319943	+	0.947437i	$0.396336\pi$
$318$	0	0
$319$	38.3384	2.14654
$320$	0	0
$321$	10.0710	0.562109
$322$	0	0
$323$	40.0142	2.22645
$324$	0	0
$325$	9.66369	0.536045
$326$	0	0
$327$	−0.633703	−0.0350438
$328$	0	0
$329$	−1.29447	−0.0713667
$330$	0	0
$331$	0.192437	0.0105773	0.00528865	−	0.999986i	$-0.498317\pi$
$331$	0.192437	0.0105773	0.00528865	+	0.999986i	$0.498317\pi$
$332$	0	0
$333$	4.06866	0.222961
$334$	0	0
$335$	10.6090	0.579632
$336$	0	0
$337$	−17.3027	−0.942536	−0.471268	−	0.881990i	$-0.656204\pi$
$337$	−17.3027	−0.942536	−0.471268	+	0.881990i	$0.656204\pi$
$338$	0	0
$339$	38.6692	2.10022
$340$	0	0
$341$	−16.6864	−0.903621
$342$	0	0
$343$	28.7967	1.55488
$344$	0	0
$345$	4.36992	0.235269
$346$	0	0
$347$	−14.0215	−0.752715	−0.376357	−	0.926475i	$-0.622824\pi$
$347$	−14.0215	−0.752715	−0.376357	+	0.926475i	$0.622824\pi$
$348$	0	0
$349$	−24.7040	−1.32238	−0.661188	−	0.750220i	$-0.729947\pi$
$349$	−24.7040	−1.32238	−0.661188	+	0.750220i	$0.729947\pi$
$350$	0	0
$351$	−12.3219	−0.657695
$352$	0	0
$353$	−9.69603	−0.516068	−0.258034	−	0.966136i	$-0.583075\pi$
$353$	−9.69603	−0.516068	−0.258034	+	0.966136i	$0.583075\pi$
$354$	0	0
$355$	1.08740	0.0577134
$356$	0	0
$357$	−63.0980	−3.33950
$358$	0	0
$359$	21.8583	1.15364	0.576819	−	0.816872i	$-0.304294\pi$
$359$	21.8583	1.15364	0.576819	+	0.816872i	$0.304294\pi$
$360$	0	0
$361$	8.70811	0.458321
$362$	0	0
$363$	−20.3907	−1.07024
$364$	0	0
$365$	−14.2869	−0.747813
$366$	0	0
$367$	−14.5772	−0.760925	−0.380462	−	0.924796i	$-0.624235\pi$
$367$	−14.5772	−0.760925	−0.380462	+	0.924796i	$0.624235\pi$
$368$	0	0
$369$	3.49113	0.181741
$370$	0	0
$371$	18.2395	0.946948
$372$	0	0
$373$	−27.0898	−1.40266	−0.701328	−	0.712839i	$-0.747409\pi$
$373$	−27.0898	−1.40266	−0.701328	+	0.712839i	$0.747409\pi$
$374$	0	0
$375$	−17.8478	−0.921656
$376$	0	0
$377$	−20.8711	−1.07492
$378$	0	0
$379$	23.7551	1.22022	0.610110	−	0.792317i	$-0.291125\pi$
$379$	23.7551	1.22022	0.610110	+	0.792317i	$0.291125\pi$
$380$	0	0
$381$	−6.17137	−0.316169
$382$	0	0
$383$	13.2137	0.675190	0.337595	−	0.941291i	$-0.390387\pi$
$383$	13.2137	0.675190	0.337595	+	0.941291i	$0.390387\pi$
$384$	0	0
$385$	−23.4647	−1.19587
$386$	0	0
$387$	−1.03192	−0.0524553
$388$	0	0
$389$	−19.4983	−0.988603	−0.494301	−	0.869291i	$-0.664576\pi$
$389$	−19.4983	−0.988603	−0.494301	+	0.869291i	$0.664576\pi$
$390$	0	0
$391$	16.3358	0.826139
$392$	0	0
$393$	−2.73044	−0.137732
$394$	0	0
$395$	−17.0403	−0.857393
$396$	0	0
$397$	29.3407	1.47257	0.736283	−	0.676673i	$-0.236579\pi$
$397$	29.3407	1.47257	0.736283	+	0.676673i	$0.236579\pi$
$398$	0	0
$399$	−43.6927	−2.18737
$400$	0	0
$401$	4.67120	0.233269	0.116634	−	0.993175i	$-0.462789\pi$
$401$	4.67120	0.233269	0.116634	+	0.993175i	$0.462789\pi$
$402$	0	0
$403$	9.08396	0.452504
$404$	0	0
$405$	11.0566	0.549407
$406$	0	0
$407$	50.2083	2.48873
$408$	0	0
$409$	−7.67264	−0.379388	−0.189694	−	0.981843i	$-0.560750\pi$
$409$	−7.67264	−0.379388	−0.189694	+	0.981843i	$0.560750\pi$
$410$	0	0
$411$	−1.31030	−0.0646322
$412$	0	0
$413$	−21.4785	−1.05689
$414$	0	0
$415$	−12.1542	−0.596625
$416$	0	0
$417$	−10.6580	−0.521924
$418$	0	0
$419$	25.4348	1.24257	0.621285	−	0.783585i	$-0.286611\pi$
$419$	25.4348	1.24257	0.621285	+	0.783585i	$0.286611\pi$
$420$	0	0
$421$	14.1522	0.689735	0.344868	−	0.938651i	$-0.387924\pi$
$421$	14.1522	0.689735	0.344868	+	0.938651i	$0.387924\pi$
$422$	0	0
$423$	−0.109213	−0.00531010
$424$	0	0
$425$	−28.7109	−1.39268
$426$	0	0
$427$	−23.6563	−1.14481
$428$	0	0
$429$	22.1113	1.06755
$430$	0	0
$431$	−31.7503	−1.52936	−0.764679	−	0.644411i	$-0.777103\pi$
$431$	−31.7503	−1.52936	−0.764679	+	0.644411i	$0.777103\pi$
$432$	0	0
$433$	24.4130	1.17321	0.586607	−	0.809872i	$-0.300463\pi$
$433$	24.4130	1.17321	0.586607	+	0.809872i	$0.300463\pi$
$434$	0	0
$435$	16.5875	0.795312
$436$	0	0
$437$	11.3119	0.541120
$438$	0	0
$439$	−1.05533	−0.0503680	−0.0251840	−	0.999683i	$-0.508017\pi$
$439$	−1.05533	−0.0503680	−0.0251840	+	0.999683i	$0.508017\pi$
$440$	0	0
$441$	5.09559	0.242647
$442$	0	0
$443$	−10.4076	−0.494481	−0.247241	−	0.968954i	$-0.579524\pi$
$443$	−10.4076	−0.494481	−0.247241	+	0.968954i	$0.579524\pi$
$444$	0	0
$445$	14.6008	0.692142
$446$	0	0
$447$	22.9323	1.08466
$448$	0	0
$449$	32.6668	1.54164	0.770820	−	0.637053i	$-0.219847\pi$
$449$	32.6668	1.54164	0.770820	+	0.637053i	$0.219847\pi$
$450$	0	0
$451$	43.0814	2.02862
$452$	0	0
$453$	−13.9843	−0.657038
$454$	0	0
$455$	12.7740	0.598854
$456$	0	0
$457$	36.8367	1.72315	0.861574	−	0.507631i	$-0.169479\pi$
$457$	36.8367	1.72315	0.861574	+	0.507631i	$0.169479\pi$
$458$	0	0
$459$	36.6085	1.70874
$460$	0	0
$461$	−28.5032	−1.32753	−0.663763	−	0.747943i	$-0.731042\pi$
$461$	−28.5032	−1.32753	−0.663763	+	0.747943i	$0.731042\pi$
$462$	0	0
$463$	−13.3251	−0.619271	−0.309636	−	0.950855i	$-0.600207\pi$
$463$	−13.3251	−0.619271	−0.309636	+	0.950855i	$0.600207\pi$
$464$	0	0
$465$	−7.21957	−0.334800
$466$	0	0
$467$	−9.06828	−0.419630	−0.209815	−	0.977741i	$-0.567286\pi$
$467$	−9.06828	−0.419630	−0.209815	+	0.977741i	$0.567286\pi$
$468$	0	0
$469$	−43.3049	−1.99963
$470$	0	0
$471$	6.71916	0.309603
$472$	0	0
$473$	−12.7341	−0.585515
$474$	0	0
$475$	−19.8811	−0.912207
$476$	0	0
$477$	1.53884	0.0704585
$478$	0	0
$479$	28.6687	1.30991	0.654954	−	0.755669i	$-0.272688\pi$
$479$	28.6687	1.30991	0.654954	+	0.755669i	$0.272688\pi$
$480$	0	0
$481$	−27.3330	−1.24628
$482$	0	0
$483$	−17.8376	−0.811638
$484$	0	0
$485$	2.51930	0.114395
$486$	0	0
$487$	−25.4300	−1.15235	−0.576173	−	0.817328i	$-0.695454\pi$
$487$	−25.4300	−1.15235	−0.576173	+	0.817328i	$0.695454\pi$
$488$	0	0
$489$	33.8234	1.52955
$490$	0	0
$491$	−4.58084	−0.206730	−0.103365	−	0.994643i	$-0.532961\pi$
$491$	−4.58084	−0.206730	−0.103365	+	0.994643i	$0.532961\pi$
$492$	0	0
$493$	62.0083	2.79271
$494$	0	0
$495$	−1.97968	−0.0889798
$496$	0	0
$497$	−4.43868	−0.199102
$498$	0	0
$499$	−18.6496	−0.834869	−0.417435	−	0.908707i	$-0.637071\pi$
$499$	−18.6496	−0.834869	−0.417435	+	0.908707i	$0.637071\pi$
$500$	0	0
$501$	11.9282	0.532913
$502$	0	0
$503$	28.2577	1.25995	0.629975	−	0.776615i	$-0.283065\pi$
$503$	28.2577	1.25995	0.629975	+	0.776615i	$0.283065\pi$
$504$	0	0
$505$	−5.23047	−0.232753
$506$	0	0
$507$	11.8660	0.526989
$508$	0	0
$509$	−6.44018	−0.285456	−0.142728	−	0.989762i	$-0.545587\pi$
$509$	−6.44018	−0.285456	−0.142728	+	0.989762i	$0.545587\pi$
$510$	0	0
$511$	58.3179	2.57983
$512$	0	0
$513$	25.3498	1.11922
$514$	0	0
$515$	15.4518	0.680886
$516$	0	0
$517$	−1.34771	−0.0592723
$518$	0	0
$519$	3.46105	0.151923
$520$	0	0
$521$	−41.7922	−1.83095	−0.915476	−	0.402374i	$-0.868185\pi$
$521$	−41.7922	−1.83095	−0.915476	+	0.402374i	$0.868185\pi$
$522$	0	0
$523$	20.4560	0.894477	0.447239	−	0.894415i	$-0.352408\pi$
$523$	20.4560	0.894477	0.447239	+	0.894415i	$0.352408\pi$
$524$	0	0
$525$	31.3503	1.36824
$526$	0	0
$527$	−26.9885	−1.17564
$528$	0	0
$529$	−18.3819	−0.799214
$530$	0	0
$531$	−1.81211	−0.0786388
$532$	0	0
$533$	−23.4532	−1.01587
$534$	0	0
$535$	6.05742	0.261885
$536$	0	0
$537$	32.4443	1.40008
$538$	0	0
$539$	62.8808	2.70847
$540$	0	0
$541$	19.6481	0.844739	0.422369	−	0.906424i	$-0.361199\pi$
$541$	19.6481	0.844739	0.422369	+	0.906424i	$0.361199\pi$
$542$	0	0
$543$	45.4530	1.95057
$544$	0	0
$545$	−0.381154	−0.0163268
$546$	0	0
$547$	−13.9528	−0.596578	−0.298289	−	0.954476i	$-0.596416\pi$
$547$	−13.9528	−0.596578	−0.298289	+	0.954476i	$0.596416\pi$
$548$	0	0
$549$	−1.99584	−0.0851805
$550$	0	0
$551$	42.9381	1.82922
$552$	0	0
$553$	69.5570	2.95787
$554$	0	0
$555$	21.7232	0.922098
$556$	0	0
$557$	−33.8759	−1.43537	−0.717685	−	0.696368i	$-0.754798\pi$
$557$	−33.8759	−1.43537	−0.717685	+	0.696368i	$0.754798\pi$
$558$	0	0
$559$	6.93235	0.293207
$560$	0	0
$561$	−65.6930	−2.77356
$562$	0	0
$563$	14.4550	0.609207	0.304604	−	0.952479i	$-0.401476\pi$
$563$	14.4550	0.609207	0.304604	+	0.952479i	$0.401476\pi$
$564$	0	0
$565$	23.2584	0.978489
$566$	0	0
$567$	−45.1320	−1.89536
$568$	0	0
$569$	−2.80371	−0.117538	−0.0587689	−	0.998272i	$-0.518718\pi$
$569$	−2.80371	−0.117538	−0.0587689	+	0.998272i	$0.518718\pi$
$570$	0	0
$571$	34.9657	1.46327	0.731634	−	0.681697i	$-0.238758\pi$
$571$	34.9657	1.46327	0.731634	+	0.681697i	$0.238758\pi$
$572$	0	0
$573$	4.03309	0.168485
$574$	0	0
$575$	−8.11648	−0.338481
$576$	0	0
$577$	40.9367	1.70422	0.852108	−	0.523366i	$-0.175324\pi$
$577$	40.9367	1.70422	0.852108	+	0.523366i	$0.175324\pi$
$578$	0	0
$579$	−41.6079	−1.72916
$580$	0	0
$581$	49.6122	2.05826
$582$	0	0
$583$	18.9896	0.786470
$584$	0	0
$585$	1.07772	0.0445582
$586$	0	0
$587$	39.6680	1.63727	0.818636	−	0.574313i	$-0.194731\pi$
$587$	39.6680	1.63727	0.818636	+	0.574313i	$0.194731\pi$
$588$	0	0
$589$	−18.6884	−0.770042
$590$	0	0
$591$	19.2723	0.792756
$592$	0	0
$593$	26.3439	1.08182	0.540908	−	0.841082i	$-0.318081\pi$
$593$	26.3439	1.08182	0.540908	+	0.841082i	$0.318081\pi$
$594$	0	0
$595$	−37.9516	−1.55587
$596$	0	0
$597$	−27.5183	−1.12625
$598$	0	0
$599$	20.5188	0.838374	0.419187	−	0.907900i	$-0.362315\pi$
$599$	20.5188	0.838374	0.419187	+	0.907900i	$0.362315\pi$
$600$	0	0
$601$	33.5639	1.36910	0.684550	−	0.728966i	$-0.259999\pi$
$601$	33.5639	1.36910	0.684550	+	0.728966i	$0.259999\pi$
$602$	0	0
$603$	−3.65356	−0.148785
$604$	0	0
$605$	−12.2644	−0.498620
$606$	0	0
$607$	−19.1829	−0.778609	−0.389305	−	0.921109i	$-0.627285\pi$
$607$	−19.1829	−0.778609	−0.389305	+	0.921109i	$0.627285\pi$
$608$	0	0
$609$	−67.7087	−2.74369
$610$	0	0
$611$	0.733683	0.0296816
$612$	0	0
$613$	5.70029	0.230232	0.115116	−	0.993352i	$-0.463276\pi$
$613$	5.70029	0.230232	0.115116	+	0.993352i	$0.463276\pi$
$614$	0	0
$615$	18.6397	0.751624
$616$	0	0
$617$	−41.5169	−1.67141	−0.835704	−	0.549180i	$-0.814940\pi$
$617$	−41.5169	−1.67141	−0.835704	+	0.549180i	$0.814940\pi$
$618$	0	0
$619$	−40.2650	−1.61839	−0.809194	−	0.587542i	$-0.800096\pi$
$619$	−40.2650	−1.61839	−0.809194	+	0.587542i	$0.800096\pi$
$620$	0	0
$621$	10.3491	0.415295
$622$	0	0
$623$	−59.5988	−2.38778
$624$	0	0
$625$	8.14963	0.325985
$626$	0	0
$627$	−45.4896	−1.81668
$628$	0	0
$629$	81.2066	3.23792
$630$	0	0
$631$	−6.72929	−0.267889	−0.133944	−	0.990989i	$-0.542764\pi$
$631$	−6.72929	−0.267889	−0.133944	+	0.990989i	$0.542764\pi$
$632$	0	0
$633$	31.8781	1.26704
$634$	0	0
$635$	−3.71190	−0.147302
$636$	0	0
$637$	−34.2318	−1.35631
$638$	0	0
$639$	−0.374484	−0.0148143
$640$	0	0
$641$	6.61742	0.261372	0.130686	−	0.991424i	$-0.458282\pi$
$641$	6.61742	0.261372	0.130686	+	0.991424i	$0.458282\pi$
$642$	0	0
$643$	8.39790	0.331181	0.165590	−	0.986195i	$-0.447047\pi$
$643$	8.39790	0.331181	0.165590	+	0.986195i	$0.447047\pi$
$644$	0	0
$645$	−5.50956	−0.216939
$646$	0	0
$647$	−16.2911	−0.640469	−0.320235	−	0.947338i	$-0.603762\pi$
$647$	−16.2911	−0.640469	−0.320235	+	0.947338i	$0.603762\pi$
$648$	0	0
$649$	−22.3619	−0.877780
$650$	0	0
$651$	29.4696	1.15500
$652$	0	0
$653$	−9.76537	−0.382149	−0.191074	−	0.981576i	$-0.561197\pi$
$653$	−9.76537	−0.382149	−0.191074	+	0.981576i	$0.561197\pi$
$654$	0	0
$655$	−1.64228	−0.0641692
$656$	0	0
$657$	4.92018	0.191955
$658$	0	0
$659$	38.1082	1.48448	0.742242	−	0.670132i	$-0.233762\pi$
$659$	38.1082	1.48448	0.742242	+	0.670132i	$0.233762\pi$
$660$	0	0
$661$	−28.1671	−1.09557	−0.547786	−	0.836618i	$-0.684529\pi$
$661$	−28.1671	−1.09557	−0.547786	+	0.836618i	$0.684529\pi$
$662$	0	0
$663$	35.7627	1.38891
$664$	0	0
$665$	−26.2799	−1.01909
$666$	0	0
$667$	17.5295	0.678746
$668$	0	0
$669$	4.12668	0.159546
$670$	0	0
$671$	−24.6292	−0.950800
$672$	0	0
$673$	33.9208	1.30755	0.653775	−	0.756689i	$-0.273184\pi$
$673$	33.9208	1.30755	0.653775	+	0.756689i	$0.273184\pi$
$674$	0	0
$675$	−18.1890	−0.700095
$676$	0	0
$677$	−17.4894	−0.672173	−0.336086	−	0.941831i	$-0.609103\pi$
$677$	−17.4894	−0.672173	−0.336086	+	0.941831i	$0.609103\pi$
$678$	0	0
$679$	−10.2835	−0.394645
$680$	0	0
$681$	25.1016	0.961894
$682$	0	0
$683$	−16.1735	−0.618860	−0.309430	−	0.950922i	$-0.600138\pi$
$683$	−16.1735	−0.618860	−0.309430	+	0.950922i	$0.600138\pi$
$684$	0	0
$685$	−0.788106	−0.0301120
$686$	0	0
$687$	38.2691	1.46006
$688$	0	0
$689$	−10.3378	−0.393839
$690$	0	0
$691$	3.36243	0.127913	0.0639564	−	0.997953i	$-0.479628\pi$
$691$	3.36243	0.127913	0.0639564	+	0.997953i	$0.479628\pi$
$692$	0	0
$693$	8.08084	0.306966
$694$	0	0
$695$	−6.41048	−0.243163
$696$	0	0
$697$	69.6796	2.63930
$698$	0	0
$699$	−37.4372	−1.41601
$700$	0	0
$701$	−29.9852	−1.13253	−0.566263	−	0.824225i	$-0.691611\pi$
$701$	−29.9852	−1.13253	−0.566263	+	0.824225i	$0.691611\pi$
$702$	0	0
$703$	56.2321	2.12083
$704$	0	0
$705$	−0.583103	−0.0219609
$706$	0	0
$707$	21.3503	0.802960
$708$	0	0
$709$	43.1690	1.62125	0.810623	−	0.585569i	$-0.199129\pi$
$709$	43.1690	1.62125	0.810623	+	0.585569i	$0.199129\pi$
$710$	0	0
$711$	5.86841	0.220083
$712$	0	0
$713$	−7.62956	−0.285729
$714$	0	0
$715$	13.2993	0.497367
$716$	0	0
$717$	−20.9857	−0.783727
$718$	0	0
$719$	−33.4338	−1.24687	−0.623434	−	0.781876i	$-0.714263\pi$
$719$	−33.4338	−1.24687	−0.623434	+	0.781876i	$0.714263\pi$
$720$	0	0
$721$	−63.0726	−2.34895
$722$	0	0
$723$	−24.7066	−0.918849
$724$	0	0
$725$	−30.8089	−1.14421
$726$	0	0
$727$	48.7808	1.80918	0.904590	−	0.426283i	$-0.140177\pi$
$727$	48.7808	1.80918	0.904590	+	0.426283i	$0.140177\pi$
$728$	0	0
$729$	22.7571	0.842856
$730$	0	0
$731$	−20.5961	−0.761774
$732$	0	0
$733$	48.4413	1.78922	0.894610	−	0.446847i	$-0.147453\pi$
$733$	48.4413	1.78922	0.894610	+	0.446847i	$0.147453\pi$
$734$	0	0
$735$	27.2061	1.00351
$736$	0	0
$737$	−45.0859	−1.66076
$738$	0	0
$739$	2.93103	0.107820	0.0539098	−	0.998546i	$-0.482832\pi$
$739$	2.93103	0.107820	0.0539098	+	0.998546i	$0.482832\pi$
$740$	0	0
$741$	24.7642	0.909734
$742$	0	0
$743$	−3.92192	−0.143881	−0.0719406	−	0.997409i	$-0.522919\pi$
$743$	−3.92192	−0.143881	−0.0719406	+	0.997409i	$0.522919\pi$
$744$	0	0
$745$	13.7931	0.505340
$746$	0	0
$747$	4.18569	0.153147
$748$	0	0
$749$	−24.7258	−0.903461
$750$	0	0
$751$	−44.7024	−1.63121	−0.815606	−	0.578607i	$-0.803596\pi$
$751$	−44.7024	−1.63121	−0.815606	+	0.578607i	$0.803596\pi$
$752$	0	0
$753$	44.8163	1.63320
$754$	0	0
$755$	−8.41113	−0.306112
$756$	0	0
$757$	−1.70313	−0.0619014	−0.0309507	−	0.999521i	$-0.509853\pi$
$757$	−1.70313	−0.0619014	−0.0309507	+	0.999521i	$0.509853\pi$
$758$	0	0
$759$	−18.5712	−0.674091
$760$	0	0
$761$	−30.9671	−1.12255	−0.561277	−	0.827628i	$-0.689690\pi$
$761$	−30.9671	−1.12255	−0.561277	+	0.827628i	$0.689690\pi$
$762$	0	0
$763$	1.55583	0.0563249
$764$	0	0
$765$	−3.20192	−0.115766
$766$	0	0
$767$	12.1736	0.439564
$768$	0	0
$769$	50.7674	1.83072	0.915360	−	0.402637i	$-0.131906\pi$
$769$	50.7674	1.83072	0.915360	+	0.402637i	$0.131906\pi$
$770$	0	0
$771$	3.65022	0.131460
$772$	0	0
$773$	−45.2783	−1.62855	−0.814273	−	0.580482i	$-0.802864\pi$
$773$	−45.2783	−1.62855	−0.814273	+	0.580482i	$0.802864\pi$
$774$	0	0
$775$	13.4093	0.481676
$776$	0	0
$777$	−88.6719	−3.18109
$778$	0	0
$779$	48.2502	1.72874
$780$	0	0
$781$	−4.62122	−0.165360
$782$	0	0
$783$	39.2836	1.40388
$784$	0	0
$785$	4.04138	0.144243
$786$	0	0
$787$	−20.0929	−0.716234	−0.358117	−	0.933677i	$-0.616581\pi$
$787$	−20.0929	−0.716234	−0.358117	+	0.933677i	$0.616581\pi$
$788$	0	0
$789$	2.12513	0.0756568
$790$	0	0
$791$	−94.9386	−3.37563
$792$	0	0
$793$	13.4079	0.476130
$794$	0	0
$795$	8.21608	0.291394
$796$	0	0
$797$	11.5342	0.408562	0.204281	−	0.978912i	$-0.434514\pi$
$797$	11.5342	0.408562	0.204281	+	0.978912i	$0.434514\pi$
$798$	0	0
$799$	−2.17978	−0.0771151
$800$	0	0
$801$	−5.02825	−0.177665
$802$	0	0
$803$	60.7163	2.14263
$804$	0	0
$805$	−10.7288	−0.378140
$806$	0	0
$807$	−14.4255	−0.507803
$808$	0	0
$809$	1.79074	0.0629590	0.0314795	−	0.999504i	$-0.489978\pi$
$809$	1.79074	0.0629590	0.0314795	+	0.999504i	$0.489978\pi$
$810$	0	0
$811$	52.4331	1.84118	0.920588	−	0.390535i	$-0.127710\pi$
$811$	52.4331	1.84118	0.920588	+	0.390535i	$0.127710\pi$
$812$	0	0
$813$	−15.4936	−0.543386
$814$	0	0
$815$	20.3438	0.712612
$816$	0	0
$817$	−14.2619	−0.498961
$818$	0	0
$819$	−4.39915	−0.153719
$820$	0	0
$821$	−8.53712	−0.297947	−0.148974	−	0.988841i	$-0.547597\pi$
$821$	−8.53712	−0.297947	−0.148974	+	0.988841i	$0.547597\pi$
$822$	0	0
$823$	50.6211	1.76454	0.882270	−	0.470744i	$-0.156014\pi$
$823$	50.6211	1.76454	0.882270	+	0.470744i	$0.156014\pi$
$824$	0	0
$825$	32.6396	1.13637
$826$	0	0
$827$	−16.1033	−0.559966	−0.279983	−	0.960005i	$-0.590329\pi$
$827$	−16.1033	−0.559966	−0.279983	+	0.960005i	$0.590329\pi$
$828$	0	0
$829$	−39.4847	−1.37136	−0.685681	−	0.727903i	$-0.740495\pi$
$829$	−39.4847	−1.37136	−0.685681	+	0.727903i	$0.740495\pi$
$830$	0	0
$831$	37.5997	1.30432
$832$	0	0
$833$	101.703	3.52380
$834$	0	0
$835$	7.17447	0.248283
$836$	0	0
$837$	−17.0978	−0.590987
$838$	0	0
$839$	−43.9795	−1.51834	−0.759170	−	0.650893i	$-0.774395\pi$
$839$	−43.9795	−1.51834	−0.759170	+	0.650893i	$0.774395\pi$
$840$	0	0
$841$	37.5393	1.29446
$842$	0	0
$843$	8.49077	0.292437
$844$	0	0
$845$	7.13708	0.245523
$846$	0	0
$847$	50.0622	1.72016
$848$	0	0
$849$	−20.8229	−0.714639
$850$	0	0
$851$	22.9568	0.786949
$852$	0	0
$853$	−25.2672	−0.865133	−0.432567	−	0.901602i	$-0.642392\pi$
$853$	−25.2672	−0.865133	−0.432567	+	0.901602i	$0.642392\pi$
$854$	0	0
$855$	−2.21719	−0.0758263
$856$	0	0
$857$	−7.52530	−0.257059	−0.128530	−	0.991706i	$-0.541026\pi$
$857$	−7.52530	−0.257059	−0.128530	+	0.991706i	$0.541026\pi$
$858$	0	0
$859$	−31.7821	−1.08439	−0.542195	−	0.840252i	$-0.682407\pi$
$859$	−31.7821	−1.08439	−0.542195	+	0.840252i	$0.682407\pi$
$860$	0	0
$861$	−76.0853	−2.59298
$862$	0	0
$863$	−22.6079	−0.769582	−0.384791	−	0.923004i	$-0.625727\pi$
$863$	−22.6079	−0.769582	−0.384791	+	0.923004i	$0.625727\pi$
$864$	0	0
$865$	2.08172	0.0707807
$866$	0	0
$867$	−74.9933	−2.54691
$868$	0	0
$869$	72.4176	2.45660
$870$	0	0
$871$	24.5444	0.831654
$872$	0	0
$873$	−0.867603	−0.0293639
$874$	0	0
$875$	43.8189	1.48135
$876$	0	0
$877$	−26.6834	−0.901035	−0.450518	−	0.892768i	$-0.648761\pi$
$877$	−26.6834	−0.901035	−0.450518	+	0.892768i	$0.648761\pi$
$878$	0	0
$879$	−5.41040	−0.182488
$880$	0	0
$881$	−41.6488	−1.40318	−0.701592	−	0.712579i	$-0.747527\pi$
$881$	−41.6488	−1.40318	−0.701592	+	0.712579i	$0.747527\pi$
$882$	0	0
$883$	−40.7491	−1.37132	−0.685659	−	0.727923i	$-0.740486\pi$
$883$	−40.7491	−1.37132	−0.685659	+	0.727923i	$0.740486\pi$
$884$	0	0
$885$	−9.67511	−0.325225
$886$	0	0
$887$	14.2080	0.477058	0.238529	−	0.971135i	$-0.423335\pi$
$887$	14.2080	0.477058	0.238529	+	0.971135i	$0.423335\pi$
$888$	0	0
$889$	15.1516	0.508169
$890$	0	0
$891$	−46.9881	−1.57416
$892$	0	0
$893$	−1.50940	−0.0505103
$894$	0	0
$895$	19.5143	0.652293
$896$	0	0
$897$	10.1100	0.337563
$898$	0	0
$899$	−28.9606	−0.965891
$900$	0	0
$901$	30.7137	1.02322
$902$	0	0
$903$	22.4895	0.748403
$904$	0	0
$905$	27.3387	0.908768
$906$	0	0
$907$	42.4103	1.40821	0.704106	−	0.710095i	$-0.251348\pi$
$907$	42.4103	1.40821	0.704106	+	0.710095i	$0.251348\pi$
$908$	0	0
$909$	1.80129	0.0597449
$910$	0	0
$911$	−4.50287	−0.149187	−0.0745934	−	0.997214i	$-0.523766\pi$
$911$	−4.50287	−0.149187	−0.0745934	+	0.997214i	$0.523766\pi$
$912$	0	0
$913$	51.6525	1.70945
$914$	0	0
$915$	−10.6561	−0.352280
$916$	0	0
$917$	6.70363	0.221373
$918$	0	0
$919$	4.25838	0.140471	0.0702356	−	0.997530i	$-0.477625\pi$
$919$	4.25838	0.140471	0.0702356	+	0.997530i	$0.477625\pi$
$920$	0	0
$921$	−33.6821	−1.10986
$922$	0	0
$923$	2.51576	0.0828071
$924$	0	0
$925$	−40.3476	−1.32662
$926$	0	0
$927$	−5.32133	−0.174775
$928$	0	0
$929$	−44.6261	−1.46414	−0.732068	−	0.681231i	$-0.761445\pi$
$929$	−44.6261	−1.46414	−0.732068	+	0.681231i	$0.761445\pi$
$930$	0	0
$931$	70.4250	2.30809
$932$	0	0
$933$	−43.9900	−1.44017
$934$	0	0
$935$	−39.5125	−1.29220
$936$	0	0
$937$	41.6688	1.36126	0.680630	−	0.732627i	$-0.261706\pi$
$937$	41.6688	1.36126	0.680630	+	0.732627i	$0.261706\pi$
$938$	0	0
$939$	9.63267	0.314350
$940$	0	0
$941$	13.3449	0.435033	0.217516	−	0.976057i	$-0.430204\pi$
$941$	13.3449	0.435033	0.217516	+	0.976057i	$0.430204\pi$
$942$	0	0
$943$	19.6982	0.641461
$944$	0	0
$945$	−24.0432	−0.782125
$946$	0	0
$947$	2.68746	0.0873306	0.0436653	−	0.999046i	$-0.486096\pi$
$947$	2.68746	0.0873306	0.0436653	+	0.999046i	$0.486096\pi$
$948$	0	0
$949$	−33.0534	−1.07296
$950$	0	0
$951$	−20.9481	−0.679290
$952$	0	0
$953$	−6.13767	−0.198819	−0.0994093	−	0.995047i	$-0.531695\pi$
$953$	−6.13767	−0.198819	−0.0994093	+	0.995047i	$0.531695\pi$
$954$	0	0
$955$	2.42579	0.0784967
$956$	0	0
$957$	−70.4933	−2.27872
$958$	0	0
$959$	3.21697	0.103881
$960$	0	0
$961$	−18.3951	−0.593392
$962$	0	0
$963$	−2.08607	−0.0672228
$964$	0	0
$965$	−25.0259	−0.805613
$966$	0	0
$967$	−32.1730	−1.03461	−0.517307	−	0.855800i	$-0.673065\pi$
$967$	−32.1730	−1.03461	−0.517307	+	0.855800i	$0.673065\pi$
$968$	0	0
$969$	−73.5746	−2.36356
$970$	0	0
$971$	−6.09678	−0.195655	−0.0978275	−	0.995203i	$-0.531189\pi$
$971$	−6.09678	−0.195655	−0.0978275	+	0.995203i	$0.531189\pi$
$972$	0	0
$973$	26.1669	0.838873
$974$	0	0
$975$	−17.7688	−0.569056
$976$	0	0
$977$	−32.1479	−1.02850	−0.514251	−	0.857640i	$-0.671930\pi$
$977$	−32.1479	−1.02850	−0.514251	+	0.857640i	$0.671930\pi$
$978$	0	0
$979$	−62.0499	−1.98312
$980$	0	0
$981$	0.131263	0.00419091
$982$	0	0
$983$	−45.1305	−1.43944	−0.719720	−	0.694264i	$-0.755730\pi$
$983$	−45.1305	−1.43944	−0.719720	+	0.694264i	$0.755730\pi$
$984$	0	0
$985$	11.5917	0.369343
$986$	0	0
$987$	2.38017	0.0757616
$988$	0	0
$989$	−5.82244	−0.185143
$990$	0	0
$991$	−9.75111	−0.309754	−0.154877	−	0.987934i	$-0.549498\pi$
$991$	−9.75111	−0.309754	−0.154877	+	0.987934i	$0.549498\pi$
$992$	0	0
$993$	−0.353836	−0.0112287
$994$	0	0
$995$	−16.5515	−0.524717
$996$	0	0
$997$	−9.98456	−0.316214	−0.158107	−	0.987422i	$-0.550539\pi$
$997$	−9.98456	−0.316214	−0.158107	+	0.987422i	$0.550539\pi$
$998$	0	0
$999$	51.4461	1.62768

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8012.2.a.b.1.19	✓	88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8012.2.a.b.1.19	✓	88	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q-1.83871 q^{3} -1.10593 q^{5} +4.51431 q^{7} +0.380865 q^{9} +O(q^{10})\)	\(q-1.83871 q^{3} -1.10593 q^{5} +4.51431 q^{7} +0.380865 q^{9} +4.69997 q^{11} -2.55862 q^{13} +2.03349 q^{15} +7.60170 q^{17} +5.26385 q^{19} -8.30052 q^{21} +2.14897 q^{23} -3.77691 q^{25} +4.81584 q^{27} +8.15716 q^{29} -3.55033 q^{31} -8.64189 q^{33} -4.99252 q^{35} +10.6827 q^{37} +4.70457 q^{39} +9.16633 q^{41} -2.70941 q^{43} -0.421211 q^{45} -0.286749 q^{47} +13.3790 q^{49} -13.9773 q^{51} +4.04037 q^{53} -5.19785 q^{55} -9.67871 q^{57} -4.75788 q^{59} -5.24030 q^{61} +1.71934 q^{63} +2.82967 q^{65} -9.59281 q^{67} -3.95134 q^{69} -0.983246 q^{71} +12.9185 q^{73} +6.94466 q^{75} +21.2171 q^{77} +15.4081 q^{79} -9.99754 q^{81} +10.9900 q^{83} -8.40697 q^{85} -14.9987 q^{87} -13.2022 q^{89} -11.5504 q^{91} +6.52804 q^{93} -5.82146 q^{95} -2.27798 q^{97} +1.79005 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 19 q^{3} + 44 q^{7} + 99 q^{9}+O(q^{10}) \) 88 * q + 19 * q^3 + 44 * q^7 + 99 * q^9	\( 88 q + 19 q^{3} + 44 q^{7} + 99 q^{9} + 16 q^{11} - 3 q^{13} + 21 q^{15} + 32 q^{17} + 49 q^{19} + 7 q^{21} + 24 q^{23} + 114 q^{25} + 82 q^{27} + 2 q^{29} + 40 q^{31} + 31 q^{33} + 26 q^{35} + 37 q^{39} + 22 q^{41} + 98 q^{43} + 12 q^{45} + 48 q^{47} + 132 q^{49} + 55 q^{51} - q^{53} + 116 q^{55} + 62 q^{57} + 34 q^{59} + 132 q^{63} + 39 q^{65} + 75 q^{67} + 15 q^{69} + 24 q^{71} + 104 q^{73} + 87 q^{75} + 4 q^{77} + 111 q^{79} + 128 q^{81} + 64 q^{83} + 7 q^{85} + 115 q^{87} + 14 q^{89} + 73 q^{91} - 7 q^{93} + 51 q^{95} + 117 q^{97} + 72 q^{99}+O(q^{100}) \) 88 * q + 19 * q^3 + 44 * q^7 + 99 * q^9 + 16 * q^11 - 3 * q^13 + 21 * q^15 + 32 * q^17 + 49 * q^19 + 7 * q^21 + 24 * q^23 + 114 * q^25 + 82 * q^27 + 2 * q^29 + 40 * q^31 + 31 * q^33 + 26 * q^35 + 37 * q^39 + 22 * q^41 + 98 * q^43 + 12 * q^45 + 48 * q^47 + 132 * q^49 + 55 * q^51 - q^53 + 116 * q^55 + 62 * q^57 + 34 * q^59 + 132 * q^63 + 39 * q^65 + 75 * q^67 + 15 * q^69 + 24 * q^71 + 104 * q^73 + 87 * q^75 + 4 * q^77 + 111 * q^79 + 128 * q^81 + 64 * q^83 + 7 * q^85 + 115 * q^87 + 14 * q^89 + 73 * q^91 - 7 * q^93 + 51 * q^95 + 117 * q^97 + 72 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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