LMFDB - Embedded newform 8012.2.a.b.1.18

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			1.18
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)

$f(q)$	$=$	$q-1.98040 q^{3} +1.50325 q^{5} +0.615748 q^{7} +0.921997 q^{9} +O(q^{10})$	$q-1.98040 q^{3} +1.50325 q^{5} +0.615748 q^{7} +0.921997 q^{9} +2.22960 q^{11} +3.28817 q^{13} -2.97704 q^{15} +1.86319 q^{17} +3.99531 q^{19} -1.21943 q^{21} +5.43947 q^{23} -2.74024 q^{25} +4.11528 q^{27} +3.51905 q^{29} +5.06970 q^{31} -4.41551 q^{33} +0.925623 q^{35} +3.55705 q^{37} -6.51190 q^{39} +2.07409 q^{41} +1.17632 q^{43} +1.38599 q^{45} -9.98065 q^{47} -6.62085 q^{49} -3.68986 q^{51} +12.0805 q^{53} +3.35164 q^{55} -7.91232 q^{57} +2.81692 q^{59} +7.18059 q^{61} +0.567718 q^{63} +4.94293 q^{65} +2.24027 q^{67} -10.7723 q^{69} +1.53784 q^{71} -12.1137 q^{73} +5.42679 q^{75} +1.37287 q^{77} -4.02911 q^{79} -10.9159 q^{81} -0.251453 q^{83} +2.80083 q^{85} -6.96913 q^{87} +11.4533 q^{89} +2.02468 q^{91} -10.0400 q^{93} +6.00594 q^{95} -3.17662 q^{97} +2.05569 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.98040	−1.14339	−0.571693	−	0.820467i	$-0.693713\pi$
$3$	−1.98040	−1.14339	−0.571693	+	0.820467i	$0.693713\pi$
$4$	0	0
$5$	1.50325	0.672273	0.336137	−	0.941813i	$-0.390880\pi$
$5$	1.50325	0.672273	0.336137	+	0.941813i	$0.390880\pi$
$6$	0	0
$7$	0.615748	0.232731	0.116366	−	0.993206i	$-0.462876\pi$
$7$	0.615748	0.232731	0.116366	+	0.993206i	$0.462876\pi$
$8$	0	0
$9$	0.921997	0.307332
$10$	0	0
$11$	2.22960	0.672250	0.336125	−	0.941817i	$-0.390884\pi$
$11$	2.22960	0.672250	0.336125	+	0.941817i	$0.390884\pi$
$12$	0	0
$13$	3.28817	0.911974	0.455987	−	0.889987i	$-0.349286\pi$
$13$	3.28817	0.911974	0.455987	+	0.889987i	$0.349286\pi$
$14$	0	0
$15$	−2.97704	−0.768668
$16$	0	0
$17$	1.86319	0.451889	0.225945	−	0.974140i	$-0.427453\pi$
$17$	1.86319	0.451889	0.225945	+	0.974140i	$0.427453\pi$
$18$	0	0
$19$	3.99531	0.916586	0.458293	−	0.888801i	$-0.348461\pi$
$19$	3.99531	0.916586	0.458293	+	0.888801i	$0.348461\pi$
$20$	0	0
$21$	−1.21943	−0.266102
$22$	0	0
$23$	5.43947	1.13421	0.567104	−	0.823646i	$-0.308064\pi$
$23$	5.43947	1.13421	0.567104	+	0.823646i	$0.308064\pi$
$24$	0	0
$25$	−2.74024	−0.548049
$26$	0	0
$27$	4.11528	0.791987
$28$	0	0
$29$	3.51905	0.653470	0.326735	−	0.945116i	$-0.394051\pi$
$29$	3.51905	0.653470	0.326735	+	0.945116i	$0.394051\pi$
$30$	0	0
$31$	5.06970	0.910544	0.455272	−	0.890352i	$-0.349542\pi$
$31$	5.06970	0.910544	0.455272	+	0.890352i	$0.349542\pi$
$32$	0	0
$33$	−4.41551	−0.768641
$34$	0	0
$35$	0.925623	0.156459
$36$	0	0
$37$	3.55705	0.584776	0.292388	−	0.956300i	$-0.405550\pi$
$37$	3.55705	0.584776	0.292388	+	0.956300i	$0.405550\pi$
$38$	0	0
$39$	−6.51190	−1.04274
$40$	0	0
$41$	2.07409	0.323918	0.161959	−	0.986797i	$-0.448219\pi$
$41$	2.07409	0.323918	0.161959	+	0.986797i	$0.448219\pi$
$42$	0	0
$43$	1.17632	0.179387	0.0896935	−	0.995969i	$-0.471411\pi$
$43$	1.17632	0.179387	0.0896935	+	0.995969i	$0.471411\pi$
$44$	0	0
$45$	1.38599	0.206611
$46$	0	0
$47$	−9.98065	−1.45583	−0.727913	−	0.685669i	$-0.759510\pi$
$47$	−9.98065	−1.45583	−0.727913	+	0.685669i	$0.759510\pi$
$48$	0	0
$49$	−6.62085	−0.945836
$50$	0	0
$51$	−3.68986	−0.516684
$52$	0	0
$53$	12.0805	1.65939	0.829694	−	0.558219i	$-0.188515\pi$
$53$	12.0805	1.65939	0.829694	+	0.558219i	$0.188515\pi$
$54$	0	0
$55$	3.35164	0.451936
$56$	0	0
$57$	−7.91232	−1.04801
$58$	0	0
$59$	2.81692	0.366731	0.183366	−	0.983045i	$-0.441301\pi$
$59$	2.81692	0.366731	0.183366	+	0.983045i	$0.441301\pi$
$60$	0	0
$61$	7.18059	0.919380	0.459690	−	0.888080i	$-0.347961\pi$
$61$	7.18059	0.919380	0.459690	+	0.888080i	$0.347961\pi$
$62$	0	0
$63$	0.567718	0.0715258
$64$	0	0
$65$	4.94293	0.613095
$66$	0	0
$67$	2.24027	0.273693	0.136846	−	0.990592i	$-0.456303\pi$
$67$	2.24027	0.273693	0.136846	+	0.990592i	$0.456303\pi$
$68$	0	0
$69$	−10.7723	−1.29684
$70$	0	0
$71$	1.53784	0.182508	0.0912538	−	0.995828i	$-0.470913\pi$
$71$	1.53784	0.182508	0.0912538	+	0.995828i	$0.470913\pi$
$72$	0	0
$73$	−12.1137	−1.41780	−0.708900	−	0.705309i	$-0.750808\pi$
$73$	−12.1137	−1.41780	−0.708900	+	0.705309i	$0.750808\pi$
$74$	0	0
$75$	5.42679	0.626631
$76$	0	0
$77$	1.37287	0.156453
$78$	0	0
$79$	−4.02911	−0.453310	−0.226655	−	0.973975i	$-0.572779\pi$
$79$	−4.02911	−0.453310	−0.226655	+	0.973975i	$0.572779\pi$
$80$	0	0
$81$	−10.9159	−1.21288
$82$	0	0
$83$	−0.251453	−0.0276006	−0.0138003	−	0.999905i	$-0.504393\pi$
$83$	−0.251453	−0.0276006	−0.0138003	+	0.999905i	$0.504393\pi$
$84$	0	0
$85$	2.80083	0.303793
$86$	0	0
$87$	−6.96913	−0.747169
$88$	0	0
$89$	11.4533	1.21404	0.607022	−	0.794685i	$-0.292364\pi$
$89$	11.4533	1.21404	0.607022	+	0.794685i	$0.292364\pi$
$90$	0	0
$91$	2.02468	0.212245
$92$	0	0
$93$	−10.0400	−1.04110
$94$	0	0
$95$	6.00594	0.616196
$96$	0	0
$97$	−3.17662	−0.322537	−0.161268	−	0.986911i	$-0.551558\pi$
$97$	−3.17662	−0.322537	−0.161268	+	0.986911i	$0.551558\pi$
$98$	0	0
$99$	2.05569	0.206604
$100$	0	0
$101$	−15.9485	−1.58694	−0.793470	−	0.608610i	$-0.791728\pi$
$101$	−15.9485	−1.58694	−0.793470	+	0.608610i	$0.791728\pi$
$102$	0	0
$103$	4.84089	0.476987	0.238493	−	0.971144i	$-0.423346\pi$
$103$	4.84089	0.476987	0.238493	+	0.971144i	$0.423346\pi$
$104$	0	0
$105$	−1.83311	−0.178893
$106$	0	0
$107$	12.3688	1.19574	0.597871	−	0.801592i	$-0.296014\pi$
$107$	12.3688	1.19574	0.597871	+	0.801592i	$0.296014\pi$
$108$	0	0
$109$	10.0137	0.959137	0.479569	−	0.877504i	$-0.340793\pi$
$109$	10.0137	0.959137	0.479569	+	0.877504i	$0.340793\pi$
$110$	0	0
$111$	−7.04440	−0.668625
$112$	0	0
$113$	3.35223	0.315351	0.157676	−	0.987491i	$-0.449600\pi$
$113$	3.35223	0.315351	0.157676	+	0.987491i	$0.449600\pi$
$114$	0	0
$115$	8.17688	0.762498
$116$	0	0
$117$	3.03168	0.280279
$118$	0	0
$119$	1.14725	0.105169
$120$	0	0
$121$	−6.02888	−0.548080
$122$	0	0
$123$	−4.10753	−0.370364
$124$	0	0
$125$	−11.6355	−1.04071
$126$	0	0
$127$	−14.5365	−1.28991	−0.644954	−	0.764222i	$-0.723123\pi$
$127$	−14.5365	−1.28991	−0.644954	+	0.764222i	$0.723123\pi$
$128$	0	0
$129$	−2.32959	−0.205109
$130$	0	0
$131$	−9.67181	−0.845030	−0.422515	−	0.906356i	$-0.638853\pi$
$131$	−9.67181	−0.845030	−0.422515	+	0.906356i	$0.638853\pi$
$132$	0	0
$133$	2.46010	0.213318
$134$	0	0
$135$	6.18630	0.532432
$136$	0	0
$137$	−15.7112	−1.34230	−0.671150	−	0.741321i	$-0.734199\pi$
$137$	−15.7112	−1.34230	−0.671150	+	0.741321i	$0.734199\pi$
$138$	0	0
$139$	−0.539830	−0.0457878	−0.0228939	−	0.999738i	$-0.507288\pi$
$139$	−0.539830	−0.0457878	−0.0228939	+	0.999738i	$0.507288\pi$
$140$	0	0
$141$	19.7657	1.66457
$142$	0	0
$143$	7.33130	0.613074
$144$	0	0
$145$	5.29000	0.439311
$146$	0	0
$147$	13.1120	1.08146
$148$	0	0
$149$	−9.65610	−0.791059	−0.395529	−	0.918453i	$-0.629439\pi$
$149$	−9.65610	−0.791059	−0.395529	+	0.918453i	$0.629439\pi$
$150$	0	0
$151$	11.3219	0.921362	0.460681	−	0.887566i	$-0.347605\pi$
$151$	11.3219	0.921362	0.460681	+	0.887566i	$0.347605\pi$
$152$	0	0
$153$	1.71785	0.138880
$154$	0	0
$155$	7.62101	0.612135
$156$	0	0
$157$	23.5770	1.88165	0.940826	−	0.338890i	$-0.110052\pi$
$157$	23.5770	1.88165	0.940826	+	0.338890i	$0.110052\pi$
$158$	0	0
$159$	−23.9243	−1.89732
$160$	0	0
$161$	3.34935	0.263965
$162$	0	0
$163$	−19.6253	−1.53717	−0.768585	−	0.639747i	$-0.779039\pi$
$163$	−19.6253	−1.53717	−0.768585	+	0.639747i	$0.779039\pi$
$164$	0	0
$165$	−6.63761	−0.516737
$166$	0	0
$167$	11.6897	0.904580	0.452290	−	0.891871i	$-0.350607\pi$
$167$	11.6897	0.904580	0.452290	+	0.891871i	$0.350607\pi$
$168$	0	0
$169$	−2.18795	−0.168304
$170$	0	0
$171$	3.68366	0.281697
$172$	0	0
$173$	8.19932	0.623383	0.311692	−	0.950183i	$-0.399104\pi$
$173$	8.19932	0.623383	0.311692	+	0.950183i	$0.399104\pi$
$174$	0	0
$175$	−1.68730	−0.127548
$176$	0	0
$177$	−5.57863	−0.419315
$178$	0	0
$179$	0.0614968	0.00459649	0.00229824	−	0.999997i	$-0.499268\pi$
$179$	0.0614968	0.00459649	0.00229824	+	0.999997i	$0.499268\pi$
$180$	0	0
$181$	−0.0912624	−0.00678348	−0.00339174	−	0.999994i	$-0.501080\pi$
$181$	−0.0912624	−0.00678348	−0.00339174	+	0.999994i	$0.501080\pi$
$182$	0	0
$183$	−14.2205	−1.05121
$184$	0	0
$185$	5.34714	0.393129
$186$	0	0
$187$	4.15416	0.303782
$188$	0	0
$189$	2.53398	0.184320
$190$	0	0
$191$	17.2984	1.25167	0.625836	−	0.779955i	$-0.284758\pi$
$191$	17.2984	1.25167	0.625836	+	0.779955i	$0.284758\pi$
$192$	0	0
$193$	−8.52089	−0.613347	−0.306674	−	0.951815i	$-0.599216\pi$
$193$	−8.52089	−0.613347	−0.306674	+	0.951815i	$0.599216\pi$
$194$	0	0
$195$	−9.78900	−0.701005
$196$	0	0
$197$	−16.3665	−1.16607	−0.583034	−	0.812448i	$-0.698135\pi$
$197$	−16.3665	−1.16607	−0.583034	+	0.812448i	$0.698135\pi$
$198$	0	0
$199$	16.5474	1.17302	0.586508	−	0.809944i	$-0.300502\pi$
$199$	16.5474	1.17302	0.586508	+	0.809944i	$0.300502\pi$
$200$	0	0
$201$	−4.43664	−0.312937
$202$	0	0
$203$	2.16685	0.152083
$204$	0	0
$205$	3.11787	0.217762
$206$	0	0
$207$	5.01518	0.348579
$208$	0	0
$209$	8.90794	0.616175
$210$	0	0
$211$	8.26324	0.568865	0.284432	−	0.958696i	$-0.408195\pi$
$211$	8.26324	0.568865	0.284432	+	0.958696i	$0.408195\pi$
$212$	0	0
$213$	−3.04554	−0.208677
$214$	0	0
$215$	1.76830	0.120597
$216$	0	0
$217$	3.12166	0.211912
$218$	0	0
$219$	23.9900	1.62109
$220$	0	0
$221$	6.12647	0.412111
$222$	0	0
$223$	−5.05787	−0.338700	−0.169350	−	0.985556i	$-0.554167\pi$
$223$	−5.05787	−0.338700	−0.169350	+	0.985556i	$0.554167\pi$
$224$	0	0
$225$	−2.52650	−0.168433
$226$	0	0
$227$	10.9479	0.726635	0.363317	−	0.931665i	$-0.381644\pi$
$227$	10.9479	0.726635	0.363317	+	0.931665i	$0.381644\pi$
$228$	0	0
$229$	−6.19733	−0.409531	−0.204766	−	0.978811i	$-0.565643\pi$
$229$	−6.19733	−0.409531	−0.204766	+	0.978811i	$0.565643\pi$
$230$	0	0
$231$	−2.71884	−0.178887
$232$	0	0
$233$	−5.60756	−0.367364	−0.183682	−	0.982986i	$-0.558802\pi$
$233$	−5.60756	−0.367364	−0.183682	+	0.982986i	$0.558802\pi$
$234$	0	0
$235$	−15.0034	−0.978713
$236$	0	0
$237$	7.97926	0.518309
$238$	0	0
$239$	−11.2265	−0.726182	−0.363091	−	0.931754i	$-0.618279\pi$
$239$	−11.2265	−0.726182	−0.363091	+	0.931754i	$0.618279\pi$
$240$	0	0
$241$	−28.3611	−1.82690	−0.913449	−	0.406953i	$-0.866591\pi$
$241$	−28.3611	−1.82690	−0.913449	+	0.406953i	$0.866591\pi$
$242$	0	0
$243$	9.27206	0.594803
$244$	0	0
$245$	−9.95279	−0.635860
$246$	0	0
$247$	13.1372	0.835902
$248$	0	0
$249$	0.497979	0.0315581
$250$	0	0
$251$	8.07878	0.509928	0.254964	−	0.966951i	$-0.417936\pi$
$251$	8.07878	0.509928	0.254964	+	0.966951i	$0.417936\pi$
$252$	0	0
$253$	12.1278	0.762471
$254$	0	0
$255$	−5.54678	−0.347353
$256$	0	0
$257$	−11.9215	−0.743645	−0.371822	−	0.928304i	$-0.621267\pi$
$257$	−11.9215	−0.743645	−0.371822	+	0.928304i	$0.621267\pi$
$258$	0	0
$259$	2.19025	0.136096
$260$	0	0
$261$	3.24455	0.200833
$262$	0	0
$263$	−0.00124181	−7.65734e−5 0	−3.82867e−5	−	1.00000i	$-0.500012\pi$
$263$	−0.00124181	−7.65734e−5 0	−3.82867e−5		1.00000i	$0.500012\pi$
$264$	0	0
$265$	18.1600	1.11556
$266$	0	0
$267$	−22.6821	−1.38812
$268$	0	0
$269$	−17.1690	−1.04681	−0.523405	−	0.852084i	$-0.675338\pi$
$269$	−17.1690	−1.04681	−0.523405	+	0.852084i	$0.675338\pi$
$270$	0	0
$271$	24.7569	1.50387	0.751937	−	0.659235i	$-0.229120\pi$
$271$	24.7569	1.50387	0.751937	+	0.659235i	$0.229120\pi$
$272$	0	0
$273$	−4.00969	−0.242678
$274$	0	0
$275$	−6.10965	−0.368426
$276$	0	0
$277$	12.0386	0.723330	0.361665	−	0.932308i	$-0.382208\pi$
$277$	12.0386	0.723330	0.361665	+	0.932308i	$0.382208\pi$
$278$	0	0
$279$	4.67424	0.279840
$280$	0	0
$281$	−1.52055	−0.0907086	−0.0453543	−	0.998971i	$-0.514442\pi$
$281$	−1.52055	−0.0907086	−0.0453543	+	0.998971i	$0.514442\pi$
$282$	0	0
$283$	4.84795	0.288181	0.144090	−	0.989565i	$-0.453974\pi$
$283$	4.84795	0.288181	0.144090	+	0.989565i	$0.453974\pi$
$284$	0	0
$285$	−11.8942	−0.704551
$286$	0	0
$287$	1.27712	0.0753859
$288$	0	0
$289$	−13.5285	−0.795796
$290$	0	0
$291$	6.29098	0.368784
$292$	0	0
$293$	−16.7457	−0.978294	−0.489147	−	0.872201i	$-0.662692\pi$
$293$	−16.7457	−0.978294	−0.489147	+	0.872201i	$0.662692\pi$
$294$	0	0
$295$	4.23453	0.246544
$296$	0	0
$297$	9.17544	0.532413
$298$	0	0
$299$	17.8859	1.03437
$300$	0	0
$301$	0.724316	0.0417489
$302$	0	0
$303$	31.5846	1.81449
$304$	0	0
$305$	10.7942	0.618074
$306$	0	0
$307$	−10.2074	−0.582564	−0.291282	−	0.956637i	$-0.594082\pi$
$307$	−10.2074	−0.582564	−0.291282	+	0.956637i	$0.594082\pi$
$308$	0	0
$309$	−9.58691	−0.545380
$310$	0	0
$311$	10.4479	0.592446	0.296223	−	0.955119i	$-0.404273\pi$
$311$	10.4479	0.592446	0.296223	+	0.955119i	$0.404273\pi$
$312$	0	0
$313$	−19.1666	−1.08336	−0.541680	−	0.840585i	$-0.682212\pi$
$313$	−19.1666	−1.08336	−0.541680	+	0.840585i	$0.682212\pi$
$314$	0	0
$315$	0.853422	0.0480849
$316$	0	0
$317$	−30.4980	−1.71294	−0.856468	−	0.516199i	$-0.827346\pi$
$317$	−30.4980	−1.71294	−0.856468	+	0.516199i	$0.827346\pi$
$318$	0	0
$319$	7.84607	0.439295
$320$	0	0
$321$	−24.4953	−1.36719
$322$	0	0
$323$	7.44400	0.414196
$324$	0	0
$325$	−9.01038	−0.499806
$326$	0	0
$327$	−19.8311	−1.09666
$328$	0	0
$329$	−6.14557	−0.338816
$330$	0	0
$331$	−6.46283	−0.355229	−0.177615	−	0.984100i	$-0.556838\pi$
$331$	−6.46283	−0.355229	−0.177615	+	0.984100i	$0.556838\pi$
$332$	0	0
$333$	3.27959	0.179721
$334$	0	0
$335$	3.36769	0.183996
$336$	0	0
$337$	31.8722	1.73619	0.868096	−	0.496397i	$-0.165344\pi$
$337$	31.8722	1.73619	0.868096	+	0.496397i	$0.165344\pi$
$338$	0	0
$339$	−6.63876	−0.360568
$340$	0	0
$341$	11.3034	0.612113
$342$	0	0
$343$	−8.38702	−0.452856
$344$	0	0
$345$	−16.1935	−0.871830
$346$	0	0
$347$	15.8394	0.850304	0.425152	−	0.905122i	$-0.360221\pi$
$347$	15.8394	0.850304	0.425152	+	0.905122i	$0.360221\pi$
$348$	0	0
$349$	−15.0303	−0.804556	−0.402278	−	0.915518i	$-0.631781\pi$
$349$	−15.0303	−0.804556	−0.402278	+	0.915518i	$0.631781\pi$
$350$	0	0
$351$	13.5317	0.722271
$352$	0	0
$353$	−7.70498	−0.410095	−0.205047	−	0.978752i	$-0.565735\pi$
$353$	−7.70498	−0.410095	−0.205047	+	0.978752i	$0.565735\pi$
$354$	0	0
$355$	2.31175	0.122695
$356$	0	0
$357$	−2.27203	−0.120248
$358$	0	0
$359$	25.9144	1.36771	0.683854	−	0.729618i	$-0.260302\pi$
$359$	25.9144	1.36771	0.683854	+	0.729618i	$0.260302\pi$
$360$	0	0
$361$	−3.03753	−0.159870
$362$	0	0
$363$	11.9396	0.626668
$364$	0	0
$365$	−18.2099	−0.953149
$366$	0	0
$367$	17.4140	0.909003	0.454501	−	0.890746i	$-0.349817\pi$
$367$	17.4140	0.909003	0.454501	+	0.890746i	$0.349817\pi$
$368$	0	0
$369$	1.91230	0.0995506
$370$	0	0
$371$	7.43856	0.386191
$372$	0	0
$373$	30.2497	1.56627	0.783135	−	0.621852i	$-0.213619\pi$
$373$	30.2497	1.56627	0.783135	+	0.621852i	$0.213619\pi$
$374$	0	0
$375$	23.0430	1.18994
$376$	0	0
$377$	11.5712	0.595948
$378$	0	0
$379$	24.3083	1.24864	0.624318	−	0.781170i	$-0.285377\pi$
$379$	24.3083	1.24864	0.624318	+	0.781170i	$0.285377\pi$
$380$	0	0
$381$	28.7882	1.47486
$382$	0	0
$383$	27.5395	1.40720	0.703601	−	0.710595i	$-0.251574\pi$
$383$	27.5395	1.40720	0.703601	+	0.710595i	$0.251574\pi$
$384$	0	0
$385$	2.06377	0.105179
$386$	0	0
$387$	1.08456	0.0551314
$388$	0	0
$389$	−1.10301	−0.0559247	−0.0279624	−	0.999609i	$-0.508902\pi$
$389$	−1.10301	−0.0559247	−0.0279624	+	0.999609i	$0.508902\pi$
$390$	0	0
$391$	10.1348	0.512536
$392$	0	0
$393$	19.1541	0.966196
$394$	0	0
$395$	−6.05676	−0.304748
$396$	0	0
$397$	5.18377	0.260166	0.130083	−	0.991503i	$-0.458476\pi$
$397$	5.18377	0.260166	0.130083	+	0.991503i	$0.458476\pi$
$398$	0	0
$399$	−4.87200	−0.243905
$400$	0	0
$401$	3.62397	0.180972	0.0904861	−	0.995898i	$-0.471158\pi$
$401$	3.62397	0.180972	0.0904861	+	0.995898i	$0.471158\pi$
$402$	0	0
$403$	16.6700	0.830392
$404$	0	0
$405$	−16.4093	−0.815386
$406$	0	0
$407$	7.93081	0.393116
$408$	0	0
$409$	0.127928	0.00632562	0.00316281	−	0.999995i	$-0.498993\pi$
$409$	0.127928	0.00632562	0.00316281	+	0.999995i	$0.498993\pi$
$410$	0	0
$411$	31.1146	1.53477
$412$	0	0
$413$	1.73451	0.0853497
$414$	0	0
$415$	−0.377997	−0.0185551
$416$	0	0
$417$	1.06908	0.0523531
$418$	0	0
$419$	−7.27462	−0.355389	−0.177694	−	0.984086i	$-0.556864\pi$
$419$	−7.27462	−0.355389	−0.177694	+	0.984086i	$0.556864\pi$
$420$	0	0
$421$	18.1709	0.885595	0.442797	−	0.896622i	$-0.353986\pi$
$421$	18.1709	0.885595	0.442797	+	0.896622i	$0.353986\pi$
$422$	0	0
$423$	−9.20213	−0.447423
$424$	0	0
$425$	−5.10559	−0.247657
$426$	0	0
$427$	4.42143	0.213968
$428$	0	0
$429$	−14.5189	−0.700980
$430$	0	0
$431$	−20.2826	−0.976979	−0.488489	−	0.872570i	$-0.662452\pi$
$431$	−20.2826	−0.976979	−0.488489	+	0.872570i	$0.662452\pi$
$432$	0	0
$433$	−0.396050	−0.0190329	−0.00951647	−	0.999955i	$-0.503029\pi$
$433$	−0.396050	−0.0190329	−0.00951647	+	0.999955i	$0.503029\pi$
$434$	0	0
$435$	−10.4763	−0.502302
$436$	0	0
$437$	21.7324	1.03960
$438$	0	0
$439$	−12.8114	−0.611457	−0.305728	−	0.952119i	$-0.598900\pi$
$439$	−12.8114	−0.611457	−0.305728	+	0.952119i	$0.598900\pi$
$440$	0	0
$441$	−6.10441	−0.290686
$442$	0	0
$443$	34.1973	1.62476	0.812382	−	0.583126i	$-0.198171\pi$
$443$	34.1973	1.62476	0.812382	+	0.583126i	$0.198171\pi$
$444$	0	0
$445$	17.2171	0.816169
$446$	0	0
$447$	19.1230	0.904486
$448$	0	0
$449$	−6.74408	−0.318273	−0.159137	−	0.987257i	$-0.550871\pi$
$449$	−6.74408	−0.318273	−0.159137	+	0.987257i	$0.550871\pi$
$450$	0	0
$451$	4.62439	0.217754
$452$	0	0
$453$	−22.4219	−1.05347
$454$	0	0
$455$	3.04360	0.142686
$456$	0	0
$457$	−0.714091	−0.0334038	−0.0167019	−	0.999861i	$-0.505317\pi$
$457$	−0.714091	−0.0334038	−0.0167019	+	0.999861i	$0.505317\pi$
$458$	0	0
$459$	7.66754	0.357890
$460$	0	0
$461$	16.6156	0.773868	0.386934	−	0.922107i	$-0.373534\pi$
$461$	16.6156	0.773868	0.386934	+	0.922107i	$0.373534\pi$
$462$	0	0
$463$	−23.9896	−1.11489	−0.557447	−	0.830213i	$-0.688219\pi$
$463$	−23.9896	−1.11489	−0.557447	+	0.830213i	$0.688219\pi$
$464$	0	0
$465$	−15.0927	−0.699906
$466$	0	0
$467$	−4.92240	−0.227782	−0.113891	−	0.993493i	$-0.536331\pi$
$467$	−4.92240	−0.227782	−0.113891	+	0.993493i	$0.536331\pi$
$468$	0	0
$469$	1.37944	0.0636968
$470$	0	0
$471$	−46.6920	−2.15146
$472$	0	0
$473$	2.62272	0.120593
$474$	0	0
$475$	−10.9481	−0.502334
$476$	0	0
$477$	11.1382	0.509984
$478$	0	0
$479$	4.05892	0.185457	0.0927285	−	0.995691i	$-0.470441\pi$
$479$	4.05892	0.185457	0.0927285	+	0.995691i	$0.470441\pi$
$480$	0	0
$481$	11.6962	0.533300
$482$	0	0
$483$	−6.63305	−0.301814
$484$	0	0
$485$	−4.77525	−0.216833
$486$	0	0
$487$	1.25221	0.0567432	0.0283716	−	0.999597i	$-0.490968\pi$
$487$	1.25221	0.0567432	0.0283716	+	0.999597i	$0.490968\pi$
$488$	0	0
$489$	38.8660	1.75758
$490$	0	0
$491$	10.9907	0.496005	0.248002	−	0.968759i	$-0.420226\pi$
$491$	10.9907	0.496005	0.248002	+	0.968759i	$0.420226\pi$
$492$	0	0
$493$	6.55664	0.295296
$494$	0	0
$495$	3.09021	0.138894
$496$	0	0
$497$	0.946921	0.0424752
$498$	0	0
$499$	−0.743405	−0.0332794	−0.0166397	−	0.999862i	$-0.505297\pi$
$499$	−0.743405	−0.0332794	−0.0166397	+	0.999862i	$0.505297\pi$
$500$	0	0
$501$	−23.1504	−1.03428
$502$	0	0
$503$	31.4573	1.40261	0.701305	−	0.712861i	$-0.252601\pi$
$503$	31.4573	1.40261	0.701305	+	0.712861i	$0.252601\pi$
$504$	0	0
$505$	−23.9746	−1.06686
$506$	0	0
$507$	4.33303	0.192437
$508$	0	0
$509$	7.37474	0.326880	0.163440	−	0.986553i	$-0.447741\pi$
$509$	7.37474	0.326880	0.163440	+	0.986553i	$0.447741\pi$
$510$	0	0
$511$	−7.45899	−0.329966
$512$	0	0
$513$	16.4418	0.725924
$514$	0	0
$515$	7.27706	0.320665
$516$	0	0
$517$	−22.2528	−0.978679
$518$	0	0
$519$	−16.2380	−0.712768
$520$	0	0
$521$	9.01996	0.395172	0.197586	−	0.980286i	$-0.436690\pi$
$521$	9.01996	0.395172	0.197586	+	0.980286i	$0.436690\pi$
$522$	0	0
$523$	−0.913607	−0.0399492	−0.0199746	−	0.999800i	$-0.506359\pi$
$523$	−0.913607	−0.0399492	−0.0199746	+	0.999800i	$0.506359\pi$
$524$	0	0
$525$	3.34154	0.145837
$526$	0	0
$527$	9.44579	0.411465
$528$	0	0
$529$	6.58784	0.286428
$530$	0	0
$531$	2.59719	0.112708
$532$	0	0
$533$	6.81995	0.295405
$534$	0	0
$535$	18.5935	0.803865
$536$	0	0
$537$	−0.121788	−0.00525556
$538$	0	0
$539$	−14.7619	−0.635838
$540$	0	0
$541$	−11.0814	−0.476427	−0.238213	−	0.971213i	$-0.576562\pi$
$541$	−11.0814	−0.476427	−0.238213	+	0.971213i	$0.576562\pi$
$542$	0	0
$543$	0.180736	0.00775614
$544$	0	0
$545$	15.0531	0.644802
$546$	0	0
$547$	22.8294	0.976114	0.488057	−	0.872812i	$-0.337706\pi$
$547$	22.8294	0.976114	0.488057	+	0.872812i	$0.337706\pi$
$548$	0	0
$549$	6.62048	0.282555
$550$	0	0
$551$	14.0597	0.598962
$552$	0	0
$553$	−2.48092	−0.105499
$554$	0	0
$555$	−10.5895	−0.449499
$556$	0	0
$557$	−13.9272	−0.590114	−0.295057	−	0.955480i	$-0.595339\pi$
$557$	−13.9272	−0.590114	−0.295057	+	0.955480i	$0.595339\pi$
$558$	0	0
$559$	3.86793	0.163596
$560$	0	0
$561$	−8.22692	−0.347341
$562$	0	0
$563$	44.6494	1.88175	0.940874	−	0.338758i	$-0.110007\pi$
$563$	44.6494	1.88175	0.940874	+	0.338758i	$0.110007\pi$
$564$	0	0
$565$	5.03923	0.212002
$566$	0	0
$567$	−6.72146	−0.282275
$568$	0	0
$569$	32.2186	1.35067	0.675337	−	0.737509i	$-0.263998\pi$
$569$	32.2186	1.35067	0.675337	+	0.737509i	$0.263998\pi$
$570$	0	0
$571$	−26.5581	−1.11142	−0.555711	−	0.831376i	$-0.687554\pi$
$571$	−26.5581	−1.11142	−0.555711	+	0.831376i	$0.687554\pi$
$572$	0	0
$573$	−34.2579	−1.43114
$574$	0	0
$575$	−14.9055	−0.621601
$576$	0	0
$577$	28.6793	1.19394	0.596968	−	0.802265i	$-0.296372\pi$
$577$	28.6793	1.19394	0.596968	+	0.802265i	$0.296372\pi$
$578$	0	0
$579$	16.8748	0.701293
$580$	0	0
$581$	−0.154832	−0.00642351
$582$	0	0
$583$	26.9347	1.11552
$584$	0	0
$585$	4.55737	0.188424
$586$	0	0
$587$	−40.4466	−1.66941	−0.834704	−	0.550699i	$-0.814361\pi$
$587$	−40.4466	−1.66941	−0.834704	+	0.550699i	$0.814361\pi$
$588$	0	0
$589$	20.2550	0.834592
$590$	0	0
$591$	32.4124	1.33327
$592$	0	0
$593$	13.5470	0.556307	0.278154	−	0.960537i	$-0.410278\pi$
$593$	13.5470	0.556307	0.278154	+	0.960537i	$0.410278\pi$
$594$	0	0
$595$	1.72461	0.0707021
$596$	0	0
$597$	−32.7706	−1.34121
$598$	0	0
$599$	9.76398	0.398946	0.199473	−	0.979903i	$-0.436077\pi$
$599$	9.76398	0.398946	0.199473	+	0.979903i	$0.436077\pi$
$600$	0	0
$601$	0.853350	0.0348089	0.0174044	−	0.999849i	$-0.494460\pi$
$601$	0.853350	0.0348089	0.0174044	+	0.999849i	$0.494460\pi$
$602$	0	0
$603$	2.06552	0.0841147
$604$	0	0
$605$	−9.06291	−0.368460
$606$	0	0
$607$	26.3452	1.06932	0.534659	−	0.845068i	$-0.320440\pi$
$607$	26.3452	1.06932	0.534659	+	0.845068i	$0.320440\pi$
$608$	0	0
$609$	−4.29123	−0.173889
$610$	0	0
$611$	−32.8180	−1.32768
$612$	0	0
$613$	40.5379	1.63731	0.818656	−	0.574285i	$-0.194720\pi$
$613$	40.5379	1.63731	0.818656	+	0.574285i	$0.194720\pi$
$614$	0	0
$615$	−6.17465	−0.248986
$616$	0	0
$617$	15.7649	0.634670	0.317335	−	0.948314i	$-0.397212\pi$
$617$	15.7649	0.634670	0.317335	+	0.948314i	$0.397212\pi$
$618$	0	0
$619$	−38.2761	−1.53845	−0.769223	−	0.638981i	$-0.779356\pi$
$619$	−38.2761	−1.53845	−0.769223	+	0.638981i	$0.779356\pi$
$620$	0	0
$621$	22.3850	0.898278
$622$	0	0
$623$	7.05233	0.282546
$624$	0	0
$625$	−3.78985	−0.151594
$626$	0	0
$627$	−17.6413	−0.704526
$628$	0	0
$629$	6.62746	0.264254
$630$	0	0
$631$	32.8492	1.30770	0.653852	−	0.756622i	$-0.273152\pi$
$631$	32.8492	1.30770	0.653852	+	0.756622i	$0.273152\pi$
$632$	0	0
$633$	−16.3645	−0.650432
$634$	0	0
$635$	−21.8520	−0.867170
$636$	0	0
$637$	−21.7705	−0.862578
$638$	0	0
$639$	1.41788	0.0560905
$640$	0	0
$641$	9.46242	0.373743	0.186872	−	0.982384i	$-0.440165\pi$
$641$	9.46242	0.373743	0.186872	+	0.982384i	$0.440165\pi$
$642$	0	0
$643$	−10.4020	−0.410215	−0.205107	−	0.978739i	$-0.565754\pi$
$643$	−10.4020	−0.410215	−0.205107	+	0.978739i	$0.565754\pi$
$644$	0	0
$645$	−3.50195	−0.137889
$646$	0	0
$647$	11.3401	0.445824	0.222912	−	0.974839i	$-0.428444\pi$
$647$	11.3401	0.445824	0.222912	+	0.974839i	$0.428444\pi$
$648$	0	0
$649$	6.28060	0.246535
$650$	0	0
$651$	−6.18214	−0.242297
$652$	0	0
$653$	−12.1395	−0.475057	−0.237528	−	0.971381i	$-0.576337\pi$
$653$	−12.1395	−0.475057	−0.237528	+	0.971381i	$0.576337\pi$
$654$	0	0
$655$	−14.5391	−0.568091
$656$	0	0
$657$	−11.1688	−0.435736
$658$	0	0
$659$	20.0777	0.782118	0.391059	−	0.920366i	$-0.372109\pi$
$659$	20.0777	0.782118	0.391059	+	0.920366i	$0.372109\pi$
$660$	0	0
$661$	7.28394	0.283313	0.141656	−	0.989916i	$-0.454757\pi$
$661$	7.28394	0.283313	0.141656	+	0.989916i	$0.454757\pi$
$662$	0	0
$663$	−12.1329	−0.471202
$664$	0	0
$665$	3.69815	0.143408
$666$	0	0
$667$	19.1417	0.741171
$668$	0	0
$669$	10.0166	0.387265
$670$	0	0
$671$	16.0098	0.618053
$672$	0	0
$673$	21.3613	0.823417	0.411708	−	0.911316i	$-0.364932\pi$
$673$	21.3613	0.823417	0.411708	+	0.911316i	$0.364932\pi$
$674$	0	0
$675$	−11.2769	−0.434047
$676$	0	0
$677$	−26.4545	−1.01673	−0.508365	−	0.861142i	$-0.669750\pi$
$677$	−26.4545	−1.01673	−0.508365	+	0.861142i	$0.669750\pi$
$678$	0	0
$679$	−1.95600	−0.0750643
$680$	0	0
$681$	−21.6812	−0.830825
$682$	0	0
$683$	5.64177	0.215876	0.107938	−	0.994158i	$-0.465575\pi$
$683$	5.64177	0.215876	0.107938	+	0.994158i	$0.465575\pi$
$684$	0	0
$685$	−23.6179	−0.902393
$686$	0	0
$687$	12.2732	0.468252
$688$	0	0
$689$	39.7228	1.51332
$690$	0	0
$691$	12.3640	0.470348	0.235174	−	0.971953i	$-0.424434\pi$
$691$	12.3640	0.470348	0.235174	+	0.971953i	$0.424434\pi$
$692$	0	0
$693$	1.26578	0.0480832
$694$	0	0
$695$	−0.811499	−0.0307819
$696$	0	0
$697$	3.86442	0.146375
$698$	0	0
$699$	11.1052	0.420039
$700$	0	0
$701$	7.76431	0.293254	0.146627	−	0.989192i	$-0.453158\pi$
$701$	7.76431	0.293254	0.146627	+	0.989192i	$0.453158\pi$
$702$	0	0
$703$	14.2115	0.535998
$704$	0	0
$705$	29.7128	1.11905
$706$	0	0
$707$	−9.82029	−0.369330
$708$	0	0
$709$	−13.5796	−0.509993	−0.254997	−	0.966942i	$-0.582074\pi$
$709$	−13.5796	−0.509993	−0.254997	+	0.966942i	$0.582074\pi$
$710$	0	0
$711$	−3.71483	−0.139317
$712$	0	0
$713$	27.5765	1.03275
$714$	0	0
$715$	11.0208	0.412153
$716$	0	0
$717$	22.2330	0.830307
$718$	0	0
$719$	−4.76710	−0.177783	−0.0888914	−	0.996041i	$-0.528332\pi$
$719$	−4.76710	−0.177783	−0.0888914	+	0.996041i	$0.528332\pi$
$720$	0	0
$721$	2.98077	0.111010
$722$	0	0
$723$	56.1664	2.08885
$724$	0	0
$725$	−9.64304	−0.358134
$726$	0	0
$727$	−49.1407	−1.82253	−0.911264	−	0.411822i	$-0.864892\pi$
$727$	−49.1407	−1.82253	−0.911264	+	0.411822i	$0.864892\pi$
$728$	0	0
$729$	14.3853	0.532790
$730$	0	0
$731$	2.19170	0.0810630
$732$	0	0
$733$	31.3447	1.15774	0.578871	−	0.815419i	$-0.303494\pi$
$733$	31.3447	1.15774	0.578871	+	0.815419i	$0.303494\pi$
$734$	0	0
$735$	19.7105	0.727034
$736$	0	0
$737$	4.99491	0.183990
$738$	0	0
$739$	−44.7428	−1.64589	−0.822946	−	0.568120i	$-0.807671\pi$
$739$	−44.7428	−1.64589	−0.822946	+	0.568120i	$0.807671\pi$
$740$	0	0
$741$	−26.0170	−0.955759
$742$	0	0
$743$	−26.9238	−0.987737	−0.493869	−	0.869537i	$-0.664418\pi$
$743$	−26.9238	−0.987737	−0.493869	+	0.869537i	$0.664418\pi$
$744$	0	0
$745$	−14.5155	−0.531808
$746$	0	0
$747$	−0.231839	−0.00848255
$748$	0	0
$749$	7.61610	0.278286
$750$	0	0
$751$	17.6246	0.643132	0.321566	−	0.946887i	$-0.395791\pi$
$751$	17.6246	0.643132	0.321566	+	0.946887i	$0.395791\pi$
$752$	0	0
$753$	−15.9992	−0.583045
$754$	0	0
$755$	17.0196	0.619407
$756$	0	0
$757$	−3.76980	−0.137016	−0.0685078	−	0.997651i	$-0.521824\pi$
$757$	−3.76980	−0.137016	−0.0685078	+	0.997651i	$0.521824\pi$
$758$	0	0
$759$	−24.0180	−0.871799
$760$	0	0
$761$	−41.9979	−1.52242	−0.761212	−	0.648504i	$-0.775395\pi$
$761$	−41.9979	−1.52242	−0.761212	+	0.648504i	$0.775395\pi$
$762$	0	0
$763$	6.16591	0.223221
$764$	0	0
$765$	2.58236	0.0933655
$766$	0	0
$767$	9.26249	0.334449
$768$	0	0
$769$	22.3837	0.807178	0.403589	−	0.914940i	$-0.367763\pi$
$769$	22.3837	0.807178	0.403589	+	0.914940i	$0.367763\pi$
$770$	0	0
$771$	23.6094	0.850273
$772$	0	0
$773$	8.68465	0.312365	0.156183	−	0.987728i	$-0.450081\pi$
$773$	8.68465	0.312365	0.156183	+	0.987728i	$0.450081\pi$
$774$	0	0
$775$	−13.8922	−0.499022
$776$	0	0
$777$	−4.33758	−0.155610
$778$	0	0
$779$	8.28662	0.296899
$780$	0	0
$781$	3.42876	0.122691
$782$	0	0
$783$	14.4819	0.517540
$784$	0	0
$785$	35.4421	1.26498
$786$	0	0
$787$	31.0984	1.10854	0.554269	−	0.832338i	$-0.312998\pi$
$787$	31.0984	1.10854	0.554269	+	0.832338i	$0.312998\pi$
$788$	0	0
$789$	0.00245929	8.75530e−5 0
$790$	0	0
$791$	2.06413	0.0733920
$792$	0	0
$793$	23.6110	0.838450
$794$	0	0
$795$	−35.9642	−1.27552
$796$	0	0
$797$	−10.6107	−0.375851	−0.187925	−	0.982183i	$-0.560176\pi$
$797$	−10.6107	−0.375851	−0.187925	+	0.982183i	$0.560176\pi$
$798$	0	0
$799$	−18.5958	−0.657873
$800$	0	0
$801$	10.5599	0.373115
$802$	0	0
$803$	−27.0087	−0.953116
$804$	0	0
$805$	5.03490	0.177457
$806$	0	0
$807$	34.0015	1.19691
$808$	0	0
$809$	−16.8966	−0.594051	−0.297026	−	0.954870i	$-0.595995\pi$
$809$	−16.8966	−0.594051	−0.297026	+	0.954870i	$0.595995\pi$
$810$	0	0
$811$	5.76212	0.202336	0.101168	−	0.994869i	$-0.467742\pi$
$811$	5.76212	0.202336	0.101168	+	0.994869i	$0.467742\pi$
$812$	0	0
$813$	−49.0286	−1.71951
$814$	0	0
$815$	−29.5017	−1.03340
$816$	0	0
$817$	4.69975	0.164424
$818$	0	0
$819$	1.86675	0.0652296
$820$	0	0
$821$	−19.0621	−0.665273	−0.332636	−	0.943055i	$-0.607938\pi$
$821$	−19.0621	−0.665273	−0.332636	+	0.943055i	$0.607938\pi$
$822$	0	0
$823$	−9.76291	−0.340314	−0.170157	−	0.985417i	$-0.554427\pi$
$823$	−9.76291	−0.340314	−0.170157	+	0.985417i	$0.554427\pi$
$824$	0	0
$825$	12.0996	0.421253
$826$	0	0
$827$	−36.5908	−1.27239	−0.636194	−	0.771529i	$-0.719492\pi$
$827$	−36.5908	−1.27239	−0.636194	+	0.771529i	$0.719492\pi$
$828$	0	0
$829$	36.6882	1.27424	0.637118	−	0.770766i	$-0.280126\pi$
$829$	36.6882	1.27424	0.637118	+	0.770766i	$0.280126\pi$
$830$	0	0
$831$	−23.8413	−0.827045
$832$	0	0
$833$	−12.3359	−0.427413
$834$	0	0
$835$	17.5726	0.608125
$836$	0	0
$837$	20.8632	0.721139
$838$	0	0
$839$	−28.9491	−0.999433	−0.499716	−	0.866189i	$-0.666562\pi$
$839$	−28.9491	−0.999433	−0.499716	+	0.866189i	$0.666562\pi$
$840$	0	0
$841$	−16.6163	−0.572976
$842$	0	0
$843$	3.01131	0.103715
$844$	0	0
$845$	−3.28904	−0.113146
$846$	0	0
$847$	−3.71228	−0.127555
$848$	0	0
$849$	−9.60090	−0.329502
$850$	0	0
$851$	19.3485	0.663258
$852$	0	0
$853$	−0.509191	−0.0174344	−0.00871718	−	0.999962i	$-0.502775\pi$
$853$	−0.509191	−0.0174344	−0.00871718	+	0.999962i	$0.502775\pi$
$854$	0	0
$855$	5.53746	0.189377
$856$	0	0
$857$	−16.5787	−0.566317	−0.283159	−	0.959073i	$-0.591382\pi$
$857$	−16.5787	−0.566317	−0.283159	+	0.959073i	$0.591382\pi$
$858$	0	0
$859$	39.9083	1.36166	0.680828	−	0.732444i	$-0.261620\pi$
$859$	39.9083	1.36166	0.680828	+	0.732444i	$0.261620\pi$
$860$	0	0
$861$	−2.52921	−0.0861952
$862$	0	0
$863$	−19.1407	−0.651556	−0.325778	−	0.945446i	$-0.605626\pi$
$863$	−19.1407	−0.651556	−0.325778	+	0.945446i	$0.605626\pi$
$864$	0	0
$865$	12.3256	0.419084
$866$	0	0
$867$	26.7920	0.909902
$868$	0	0
$869$	−8.98331	−0.304738
$870$	0	0
$871$	7.36639	0.249601
$872$	0	0
$873$	−2.92883	−0.0991260
$874$	0	0
$875$	−7.16455	−0.242206
$876$	0	0
$877$	8.97461	0.303051	0.151526	−	0.988453i	$-0.451581\pi$
$877$	8.97461	0.303051	0.151526	+	0.988453i	$0.451581\pi$
$878$	0	0
$879$	33.1632	1.11857
$880$	0	0
$881$	−56.7354	−1.91146	−0.955732	−	0.294237i	$-0.904935\pi$
$881$	−56.7354	−1.91146	−0.955732	+	0.294237i	$0.904935\pi$
$882$	0	0
$883$	1.16895	0.0393382	0.0196691	−	0.999807i	$-0.493739\pi$
$883$	1.16895	0.0393382	0.0196691	+	0.999807i	$0.493739\pi$
$884$	0	0
$885$	−8.38607	−0.281895
$886$	0	0
$887$	−41.7627	−1.40225	−0.701127	−	0.713037i	$-0.747319\pi$
$887$	−41.7627	−1.40225	−0.701127	+	0.713037i	$0.747319\pi$
$888$	0	0
$889$	−8.95084	−0.300202
$890$	0	0
$891$	−24.3381	−0.815358
$892$	0	0
$893$	−39.8757	−1.33439
$894$	0	0
$895$	0.0924450	0.00309010
$896$	0	0
$897$	−35.4213	−1.18268
$898$	0	0
$899$	17.8405	0.595014
$900$	0	0
$901$	22.5083	0.749860
$902$	0	0
$903$	−1.43444	−0.0477351
$904$	0	0
$905$	−0.137190	−0.00456035
$906$	0	0
$907$	25.6006	0.850054	0.425027	−	0.905181i	$-0.360265\pi$
$907$	25.6006	0.850054	0.425027	+	0.905181i	$0.360265\pi$
$908$	0	0
$909$	−14.7045	−0.487718
$910$	0	0
$911$	8.04440	0.266523	0.133261	−	0.991081i	$-0.457455\pi$
$911$	8.04440	0.266523	0.133261	+	0.991081i	$0.457455\pi$
$912$	0	0
$913$	−0.560640	−0.0185545
$914$	0	0
$915$	−21.3769	−0.706698
$916$	0	0
$917$	−5.95540	−0.196665
$918$	0	0
$919$	18.6460	0.615074	0.307537	−	0.951536i	$-0.400495\pi$
$919$	18.6460	0.615074	0.307537	+	0.951536i	$0.400495\pi$
$920$	0	0
$921$	20.2147	0.666096
$922$	0	0
$923$	5.05666	0.166442
$924$	0	0
$925$	−9.74719	−0.320486
$926$	0	0
$927$	4.46328	0.146593
$928$	0	0
$929$	−35.2253	−1.15571	−0.577853	−	0.816141i	$-0.696109\pi$
$929$	−35.2253	−1.15571	−0.577853	+	0.816141i	$0.696109\pi$
$930$	0	0
$931$	−26.4523	−0.866940
$932$	0	0
$933$	−20.6911	−0.677395
$934$	0	0
$935$	6.24474	0.204225
$936$	0	0
$937$	−37.3199	−1.21919	−0.609595	−	0.792713i	$-0.708668\pi$
$937$	−37.3199	−1.21919	−0.609595	+	0.792713i	$0.708668\pi$
$938$	0	0
$939$	37.9576	1.23870
$940$	0	0
$941$	42.7831	1.39469	0.697344	−	0.716737i	$-0.254365\pi$
$941$	42.7831	1.39469	0.697344	+	0.716737i	$0.254365\pi$
$942$	0	0
$943$	11.2819	0.367391
$944$	0	0
$945$	3.80920	0.123913
$946$	0	0
$947$	1.97041	0.0640298	0.0320149	−	0.999487i	$-0.489808\pi$
$947$	1.97041	0.0640298	0.0320149	+	0.999487i	$0.489808\pi$
$948$	0	0
$949$	−39.8318	−1.29300
$950$	0	0
$951$	60.3983	1.95855
$952$	0	0
$953$	59.3778	1.92343	0.961717	−	0.274044i	$-0.0883613\pi$
$953$	59.3778	1.92343	0.961717	+	0.274044i	$0.0883613\pi$
$954$	0	0
$955$	26.0039	0.841465
$956$	0	0
$957$	−15.5384	−0.502284
$958$	0	0
$959$	−9.67416	−0.312395
$960$	0	0
$961$	−5.29819	−0.170909
$962$	0	0
$963$	11.4040	0.367490
$964$	0	0
$965$	−12.8090	−0.412337
$966$	0	0
$967$	55.0530	1.77038	0.885192	−	0.465226i	$-0.154027\pi$
$967$	55.0530	1.77038	0.885192	+	0.465226i	$0.154027\pi$
$968$	0	0
$969$	−14.7421	−0.473586
$970$	0	0
$971$	−42.4415	−1.36201	−0.681006	−	0.732278i	$-0.738457\pi$
$971$	−42.4415	−1.36201	−0.681006	+	0.732278i	$0.738457\pi$
$972$	0	0
$973$	−0.332399	−0.0106562
$974$	0	0
$975$	17.8442	0.571471
$976$	0	0
$977$	38.9746	1.24691	0.623455	−	0.781859i	$-0.285729\pi$
$977$	38.9746	1.24691	0.623455	+	0.781859i	$0.285729\pi$
$978$	0	0
$979$	25.5362	0.816140
$980$	0	0
$981$	9.23259	0.294774
$982$	0	0
$983$	14.5307	0.463457	0.231729	−	0.972780i	$-0.425562\pi$
$983$	14.5307	0.463457	0.231729	+	0.972780i	$0.425562\pi$
$984$	0	0
$985$	−24.6030	−0.783916
$986$	0	0
$987$	12.1707	0.387398
$988$	0	0
$989$	6.39855	0.203462
$990$	0	0
$991$	−4.27445	−0.135782	−0.0678912	−	0.997693i	$-0.521627\pi$
$991$	−4.27445	−0.135782	−0.0678912	+	0.997693i	$0.521627\pi$
$992$	0	0
$993$	12.7990	0.406164
$994$	0	0
$995$	24.8749	0.788587
$996$	0	0
$997$	3.31025	0.104837	0.0524184	−	0.998625i	$-0.483307\pi$
$997$	3.31025	0.104837	0.0524184	+	0.998625i	$0.483307\pi$
$998$	0	0
$999$	14.6383	0.463135

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8012.2.a.b.1.18	✓	88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8012.2.a.b.1.18	✓	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)

\(f(q)\)	\(=\)	\(q-1.98040 q^{3} +1.50325 q^{5} +0.615748 q^{7} +0.921997 q^{9} +O(q^{10})\)	\(q-1.98040 q^{3} +1.50325 q^{5} +0.615748 q^{7} +0.921997 q^{9} +2.22960 q^{11} +3.28817 q^{13} -2.97704 q^{15} +1.86319 q^{17} +3.99531 q^{19} -1.21943 q^{21} +5.43947 q^{23} -2.74024 q^{25} +4.11528 q^{27} +3.51905 q^{29} +5.06970 q^{31} -4.41551 q^{33} +0.925623 q^{35} +3.55705 q^{37} -6.51190 q^{39} +2.07409 q^{41} +1.17632 q^{43} +1.38599 q^{45} -9.98065 q^{47} -6.62085 q^{49} -3.68986 q^{51} +12.0805 q^{53} +3.35164 q^{55} -7.91232 q^{57} +2.81692 q^{59} +7.18059 q^{61} +0.567718 q^{63} +4.94293 q^{65} +2.24027 q^{67} -10.7723 q^{69} +1.53784 q^{71} -12.1137 q^{73} +5.42679 q^{75} +1.37287 q^{77} -4.02911 q^{79} -10.9159 q^{81} -0.251453 q^{83} +2.80083 q^{85} -6.96913 q^{87} +11.4533 q^{89} +2.02468 q^{91} -10.0400 q^{93} +6.00594 q^{95} -3.17662 q^{97} +2.05569 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q + 19 q^{3} + 44 q^{7} + 99 q^{9}+O(q^{10}) \) 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

Groups

Database

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