LMFDB - Embedded newform 8020.2.a.c.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8020,2,Mod(1,8020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8020, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8020.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8020 = 2^{2} \cdot 5 \cdot 401 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8020.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:	$64.0400224211$
Analytic rank:	$1$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	8020.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.62538 q^{3} -1.00000 q^{5} +4.42157 q^{7} -0.358130 q^{9} +O(q^{10})$	$q-1.62538 q^{3} -1.00000 q^{5} +4.42157 q^{7} -0.358130 q^{9} -1.81850 q^{11} -1.06376 q^{13} +1.62538 q^{15} -1.23283 q^{17} +4.12363 q^{19} -7.18675 q^{21} -4.61525 q^{23} +1.00000 q^{25} +5.45825 q^{27} +0.847315 q^{29} -0.697606 q^{31} +2.95575 q^{33} -4.42157 q^{35} +3.45352 q^{37} +1.72901 q^{39} -3.20797 q^{41} -10.5027 q^{43} +0.358130 q^{45} +4.77104 q^{47} +12.5503 q^{49} +2.00381 q^{51} +3.05860 q^{53} +1.81850 q^{55} -6.70247 q^{57} -10.5261 q^{59} +9.43455 q^{61} -1.58350 q^{63} +1.06376 q^{65} +4.76942 q^{67} +7.50154 q^{69} +0.533751 q^{71} -7.64125 q^{73} -1.62538 q^{75} -8.04062 q^{77} -9.62077 q^{79} -7.79735 q^{81} -6.01040 q^{83} +1.23283 q^{85} -1.37721 q^{87} -6.21176 q^{89} -4.70348 q^{91} +1.13388 q^{93} -4.12363 q^{95} +12.6850 q^{97} +0.651259 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	$ 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) $ 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * 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
$3$	−1.62538	−0.938415	−0.469208	−	0.883088i	$-0.655460\pi$
$3$	−1.62538	−0.938415	−0.469208	+	0.883088i	$0.655460\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.42157	1.67120	0.835599	−	0.549340i	$-0.185121\pi$
$7$	4.42157	1.67120	0.835599	+	0.549340i	$0.185121\pi$
$8$	0	0
$9$	−0.358130	−0.119377
$10$	0	0
$11$	−1.81850	−0.548297	−0.274149	−	0.961687i	$-0.588396\pi$
$11$	−1.81850	−0.548297	−0.274149	+	0.961687i	$0.588396\pi$
$12$	0	0
$13$	−1.06376	−0.295033	−0.147517	−	0.989060i	$-0.547128\pi$
$13$	−1.06376	−0.295033	−0.147517	+	0.989060i	$0.547128\pi$
$14$	0	0
$15$	1.62538	0.419672
$16$	0	0
$17$	−1.23283	−0.299004	−0.149502	−	0.988761i	$-0.547767\pi$
$17$	−1.23283	−0.299004	−0.149502	+	0.988761i	$0.547767\pi$
$18$	0	0
$19$	4.12363	0.946024	0.473012	−	0.881056i	$-0.343167\pi$
$19$	4.12363	0.946024	0.473012	+	0.881056i	$0.343167\pi$
$20$	0	0
$21$	−7.18675	−1.56828
$22$	0	0
$23$	−4.61525	−0.962345	−0.481173	−	0.876626i	$-0.659789\pi$
$23$	−4.61525	−0.962345	−0.481173	+	0.876626i	$0.659789\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.45825	1.05044
$28$	0	0
$29$	0.847315	0.157342	0.0786712	−	0.996901i	$-0.474932\pi$
$29$	0.847315	0.157342	0.0786712	+	0.996901i	$0.474932\pi$
$30$	0	0
$31$	−0.697606	−0.125294	−0.0626469	−	0.998036i	$-0.519954\pi$
$31$	−0.697606	−0.125294	−0.0626469	+	0.998036i	$0.519954\pi$
$32$	0	0
$33$	2.95575	0.514531
$34$	0	0
$35$	−4.42157	−0.747382
$36$	0	0
$37$	3.45352	0.567755	0.283877	−	0.958861i	$-0.408379\pi$
$37$	3.45352	0.567755	0.283877	+	0.958861i	$0.408379\pi$
$38$	0	0
$39$	1.72901	0.276864
$40$	0	0
$41$	−3.20797	−0.501001	−0.250500	−	0.968116i	$-0.580595\pi$
$41$	−3.20797	−0.501001	−0.250500	+	0.968116i	$0.580595\pi$
$42$	0	0
$43$	−10.5027	−1.60164	−0.800820	−	0.598905i	$-0.795603\pi$
$43$	−10.5027	−1.60164	−0.800820	+	0.598905i	$0.795603\pi$
$44$	0	0
$45$	0.358130	0.0533869
$46$	0	0
$47$	4.77104	0.695928	0.347964	−	0.937508i	$-0.386873\pi$
$47$	4.77104	0.695928	0.347964	+	0.937508i	$0.386873\pi$
$48$	0	0
$49$	12.5503	1.79290
$50$	0	0
$51$	2.00381	0.280590
$52$	0	0
$53$	3.05860	0.420131	0.210066	−	0.977687i	$-0.432632\pi$
$53$	3.05860	0.420131	0.210066	+	0.977687i	$0.432632\pi$
$54$	0	0
$55$	1.81850	0.245206
$56$	0	0
$57$	−6.70247	−0.887764
$58$	0	0
$59$	−10.5261	−1.37038	−0.685191	−	0.728364i	$-0.740281\pi$
$59$	−10.5261	−1.37038	−0.685191	+	0.728364i	$0.740281\pi$
$60$	0	0
$61$	9.43455	1.20797	0.603985	−	0.796996i	$-0.293579\pi$
$61$	9.43455	1.20797	0.603985	+	0.796996i	$0.293579\pi$
$62$	0	0
$63$	−1.58350	−0.199502
$64$	0	0
$65$	1.06376	0.131943
$66$	0	0
$67$	4.76942	0.582677	0.291339	−	0.956620i	$-0.405899\pi$
$67$	4.76942	0.582677	0.291339	+	0.956620i	$0.405899\pi$
$68$	0	0
$69$	7.50154	0.903080
$70$	0	0
$71$	0.533751	0.0633446	0.0316723	−	0.999498i	$-0.489917\pi$
$71$	0.533751	0.0633446	0.0316723	+	0.999498i	$0.489917\pi$
$72$	0	0
$73$	−7.64125	−0.894341	−0.447171	−	0.894449i	$-0.647568\pi$
$73$	−7.64125	−0.894341	−0.447171	+	0.894449i	$0.647568\pi$
$74$	0	0
$75$	−1.62538	−0.187683
$76$	0	0
$77$	−8.04062	−0.916313
$78$	0	0
$79$	−9.62077	−1.08242	−0.541210	−	0.840887i	$-0.682034\pi$
$79$	−9.62077	−1.08242	−0.541210	+	0.840887i	$0.682034\pi$
$80$	0	0
$81$	−7.79735	−0.866372
$82$	0	0
$83$	−6.01040	−0.659727	−0.329864	−	0.944029i	$-0.607003\pi$
$83$	−6.01040	−0.659727	−0.329864	+	0.944029i	$0.607003\pi$
$84$	0	0
$85$	1.23283	0.133719
$86$	0	0
$87$	−1.37721	−0.147653
$88$	0	0
$89$	−6.21176	−0.658445	−0.329222	−	0.944252i	$-0.606787\pi$
$89$	−6.21176	−0.658445	−0.329222	+	0.944252i	$0.606787\pi$
$90$	0	0
$91$	−4.70348	−0.493059
$92$	0	0
$93$	1.13388	0.117578
$94$	0	0
$95$	−4.12363	−0.423075
$96$	0	0
$97$	12.6850	1.28797	0.643984	−	0.765039i	$-0.277280\pi$
$97$	12.6850	1.28797	0.643984	+	0.765039i	$0.277280\pi$
$98$	0	0
$99$	0.651259	0.0654540
$100$	0	0
$101$	−9.56613	−0.951865	−0.475933	−	0.879482i	$-0.657889\pi$
$101$	−9.56613	−0.951865	−0.475933	+	0.879482i	$0.657889\pi$
$102$	0	0
$103$	−9.56470	−0.942438	−0.471219	−	0.882016i	$-0.656186\pi$
$103$	−9.56470	−0.942438	−0.471219	+	0.882016i	$0.656186\pi$
$104$	0	0
$105$	7.18675	0.701355
$106$	0	0
$107$	6.77911	0.655361	0.327681	−	0.944789i	$-0.393733\pi$
$107$	6.77911	0.655361	0.327681	+	0.944789i	$0.393733\pi$
$108$	0	0
$109$	5.53790	0.530435	0.265217	−	0.964189i	$-0.414556\pi$
$109$	5.53790	0.530435	0.265217	+	0.964189i	$0.414556\pi$
$110$	0	0
$111$	−5.61329	−0.532790
$112$	0	0
$113$	−16.2741	−1.53094	−0.765471	−	0.643470i	$-0.777494\pi$
$113$	−16.2741	−1.53094	−0.765471	+	0.643470i	$0.777494\pi$
$114$	0	0
$115$	4.61525	0.430374
$116$	0	0
$117$	0.380964	0.0352201
$118$	0	0
$119$	−5.45103	−0.499695
$120$	0	0
$121$	−7.69307	−0.699370
$122$	0	0
$123$	5.21418	0.470147
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−1.53521	−0.136228	−0.0681140	−	0.997678i	$-0.521698\pi$
$127$	−1.53521	−0.136228	−0.0681140	+	0.997678i	$0.521698\pi$
$128$	0	0
$129$	17.0708	1.50300
$130$	0	0
$131$	17.1128	1.49515	0.747576	−	0.664176i	$-0.231218\pi$
$131$	17.1128	1.49515	0.747576	+	0.664176i	$0.231218\pi$
$132$	0	0
$133$	18.2329	1.58099
$134$	0	0
$135$	−5.45825	−0.469771
$136$	0	0
$137$	20.8091	1.77784	0.888920	−	0.458062i	$-0.151456\pi$
$137$	20.8091	1.77784	0.888920	+	0.458062i	$0.151456\pi$
$138$	0	0
$139$	1.51691	0.128662	0.0643312	−	0.997929i	$-0.479509\pi$
$139$	1.51691	0.128662	0.0643312	+	0.997929i	$0.479509\pi$
$140$	0	0
$141$	−7.75477	−0.653069
$142$	0	0
$143$	1.93444	0.161766
$144$	0	0
$145$	−0.847315	−0.0703657
$146$	0	0
$147$	−20.3991	−1.68249
$148$	0	0
$149$	22.4601	1.84000	0.920000	−	0.391919i	$-0.128189\pi$
$149$	22.4601	1.84000	0.920000	+	0.391919i	$0.128189\pi$
$150$	0	0
$151$	19.6913	1.60245	0.801227	−	0.598360i	$-0.204181\pi$
$151$	19.6913	1.60245	0.801227	+	0.598360i	$0.204181\pi$
$152$	0	0
$153$	0.441512	0.0356942
$154$	0	0
$155$	0.697606	0.0560331
$156$	0	0
$157$	2.04837	0.163478	0.0817388	−	0.996654i	$-0.473953\pi$
$157$	2.04837	0.163478	0.0817388	+	0.996654i	$0.473953\pi$
$158$	0	0
$159$	−4.97140	−0.394258
$160$	0	0
$161$	−20.4067	−1.60827
$162$	0	0
$163$	−3.99410	−0.312842	−0.156421	−	0.987690i	$-0.549996\pi$
$163$	−3.99410	−0.312842	−0.156421	+	0.987690i	$0.549996\pi$
$164$	0	0
$165$	−2.95575	−0.230105
$166$	0	0
$167$	−15.3495	−1.18778	−0.593889	−	0.804547i	$-0.702408\pi$
$167$	−15.3495	−1.18778	−0.593889	+	0.804547i	$0.702408\pi$
$168$	0	0
$169$	−11.8684	−0.912955
$170$	0	0
$171$	−1.47680	−0.112933
$172$	0	0
$173$	−24.3080	−1.84810	−0.924051	−	0.382270i	$-0.875143\pi$
$173$	−24.3080	−1.84810	−0.924051	+	0.382270i	$0.875143\pi$
$174$	0	0
$175$	4.42157	0.334240
$176$	0	0
$177$	17.1089	1.28599
$178$	0	0
$179$	5.21931	0.390110	0.195055	−	0.980792i	$-0.437512\pi$
$179$	5.21931	0.390110	0.195055	+	0.980792i	$0.437512\pi$
$180$	0	0
$181$	7.44881	0.553666	0.276833	−	0.960918i	$-0.410715\pi$
$181$	7.44881	0.553666	0.276833	+	0.960918i	$0.410715\pi$
$182$	0	0
$183$	−15.3347	−1.13358
$184$	0	0
$185$	−3.45352	−0.253908
$186$	0	0
$187$	2.24189	0.163943
$188$	0	0
$189$	24.1340	1.75549
$190$	0	0
$191$	−10.4691	−0.757515	−0.378757	−	0.925496i	$-0.623649\pi$
$191$	−10.4691	−0.757515	−0.378757	+	0.925496i	$0.623649\pi$
$192$	0	0
$193$	2.75123	0.198038	0.0990189	−	0.995086i	$-0.468430\pi$
$193$	2.75123	0.198038	0.0990189	+	0.995086i	$0.468430\pi$
$194$	0	0
$195$	−1.72901	−0.123817
$196$	0	0
$197$	1.62752	0.115956	0.0579781	−	0.998318i	$-0.481535\pi$
$197$	1.62752	0.115956	0.0579781	+	0.998318i	$0.481535\pi$
$198$	0	0
$199$	−17.7008	−1.25477	−0.627387	−	0.778707i	$-0.715876\pi$
$199$	−17.7008	−1.25477	−0.627387	+	0.778707i	$0.715876\pi$
$200$	0	0
$201$	−7.75213	−0.546793
$202$	0	0
$203$	3.74647	0.262950
$204$	0	0
$205$	3.20797	0.224054
$206$	0	0
$207$	1.65286	0.114882
$208$	0	0
$209$	−7.49880	−0.518703
$210$	0	0
$211$	8.62214	0.593573	0.296786	−	0.954944i	$-0.404085\pi$
$211$	8.62214	0.593573	0.296786	+	0.954944i	$0.404085\pi$
$212$	0	0
$213$	−0.867550	−0.0594435
$214$	0	0
$215$	10.5027	0.716276
$216$	0	0
$217$	−3.08452	−0.209391
$218$	0	0
$219$	12.4200	0.839263
$220$	0	0
$221$	1.31143	0.0882162
$222$	0	0
$223$	12.0986	0.810185	0.405092	−	0.914276i	$-0.367239\pi$
$223$	12.0986	0.810185	0.405092	+	0.914276i	$0.367239\pi$
$224$	0	0
$225$	−0.358130	−0.0238754
$226$	0	0
$227$	−26.8451	−1.78177	−0.890887	−	0.454226i	$-0.849916\pi$
$227$	−26.8451	−1.78177	−0.890887	+	0.454226i	$0.849916\pi$
$228$	0	0
$229$	−7.38780	−0.488200	−0.244100	−	0.969750i	$-0.578492\pi$
$229$	−7.38780	−0.488200	−0.244100	+	0.969750i	$0.578492\pi$
$230$	0	0
$231$	13.0691	0.859882
$232$	0	0
$233$	−21.1908	−1.38825	−0.694127	−	0.719853i	$-0.744209\pi$
$233$	−21.1908	−1.38825	−0.694127	+	0.719853i	$0.744209\pi$
$234$	0	0
$235$	−4.77104	−0.311228
$236$	0	0
$237$	15.6374	1.01576
$238$	0	0
$239$	5.29768	0.342678	0.171339	−	0.985212i	$-0.445191\pi$
$239$	5.29768	0.342678	0.171339	+	0.985212i	$0.445191\pi$
$240$	0	0
$241$	5.28096	0.340177	0.170088	−	0.985429i	$-0.445595\pi$
$241$	5.28096	0.340177	0.170088	+	0.985429i	$0.445595\pi$
$242$	0	0
$243$	−3.70106	−0.237423
$244$	0	0
$245$	−12.5503	−0.801810
$246$	0	0
$247$	−4.38654	−0.279109
$248$	0	0
$249$	9.76920	0.619098
$250$	0	0
$251$	5.21532	0.329188	0.164594	−	0.986361i	$-0.447369\pi$
$251$	5.21532	0.329188	0.164594	+	0.986361i	$0.447369\pi$
$252$	0	0
$253$	8.39281	0.527651
$254$	0	0
$255$	−2.00381	−0.125484
$256$	0	0
$257$	−8.33349	−0.519829	−0.259914	−	0.965632i	$-0.583694\pi$
$257$	−8.33349	−0.519829	−0.259914	+	0.965632i	$0.583694\pi$
$258$	0	0
$259$	15.2700	0.948831
$260$	0	0
$261$	−0.303449	−0.0187830
$262$	0	0
$263$	15.9773	0.985203	0.492601	−	0.870255i	$-0.336046\pi$
$263$	15.9773	0.985203	0.492601	+	0.870255i	$0.336046\pi$
$264$	0	0
$265$	−3.05860	−0.187888
$266$	0	0
$267$	10.0965	0.617895
$268$	0	0
$269$	3.74626	0.228413	0.114207	−	0.993457i	$-0.463567\pi$
$269$	3.74626	0.228413	0.114207	+	0.993457i	$0.463567\pi$
$270$	0	0
$271$	−13.8456	−0.841060	−0.420530	−	0.907279i	$-0.638156\pi$
$271$	−13.8456	−0.841060	−0.420530	+	0.907279i	$0.638156\pi$
$272$	0	0
$273$	7.64496	0.462694
$274$	0	0
$275$	−1.81850	−0.109659
$276$	0	0
$277$	1.46676	0.0881289	0.0440645	−	0.999029i	$-0.485969\pi$
$277$	1.46676	0.0881289	0.0440645	+	0.999029i	$0.485969\pi$
$278$	0	0
$279$	0.249834	0.0149572
$280$	0	0
$281$	−29.0767	−1.73457	−0.867284	−	0.497813i	$-0.834136\pi$
$281$	−29.0767	−1.73457	−0.867284	+	0.497813i	$0.834136\pi$
$282$	0	0
$283$	25.6843	1.52678	0.763388	−	0.645941i	$-0.223535\pi$
$283$	25.6843	1.52678	0.763388	+	0.645941i	$0.223535\pi$
$284$	0	0
$285$	6.70247	0.397020
$286$	0	0
$287$	−14.1843	−0.837272
$288$	0	0
$289$	−15.4801	−0.910597
$290$	0	0
$291$	−20.6180	−1.20865
$292$	0	0
$293$	5.96190	0.348298	0.174149	−	0.984719i	$-0.444283\pi$
$293$	5.96190	0.348298	0.174149	+	0.984719i	$0.444283\pi$
$294$	0	0
$295$	10.5261	0.612853
$296$	0	0
$297$	−9.92581	−0.575954
$298$	0	0
$299$	4.90950	0.283924
$300$	0	0
$301$	−46.4383	−2.67666
$302$	0	0
$303$	15.5486	0.893245
$304$	0	0
$305$	−9.43455	−0.540220
$306$	0	0
$307$	−5.14528	−0.293656	−0.146828	−	0.989162i	$-0.546906\pi$
$307$	−5.14528	−0.293656	−0.146828	+	0.989162i	$0.546906\pi$
$308$	0	0
$309$	15.5463	0.884399
$310$	0	0
$311$	1.26295	0.0716154	0.0358077	−	0.999359i	$-0.488600\pi$
$311$	1.26295	0.0716154	0.0358077	+	0.999359i	$0.488600\pi$
$312$	0	0
$313$	13.8392	0.782239	0.391120	−	0.920340i	$-0.372088\pi$
$313$	13.8392	0.782239	0.391120	+	0.920340i	$0.372088\pi$
$314$	0	0
$315$	1.58350	0.0892201
$316$	0	0
$317$	4.93169	0.276991	0.138496	−	0.990363i	$-0.455773\pi$
$317$	4.93169	0.276991	0.138496	+	0.990363i	$0.455773\pi$
$318$	0	0
$319$	−1.54084	−0.0862704
$320$	0	0
$321$	−11.0186	−0.615001
$322$	0	0
$323$	−5.08371	−0.282865
$324$	0	0
$325$	−1.06376	−0.0590067
$326$	0	0
$327$	−9.00121	−0.497768
$328$	0	0
$329$	21.0955	1.16303
$330$	0	0
$331$	−10.4049	−0.571908	−0.285954	−	0.958243i	$-0.592310\pi$
$331$	−10.4049	−0.571908	−0.285954	+	0.958243i	$0.592310\pi$
$332$	0	0
$333$	−1.23681	−0.0677768
$334$	0	0
$335$	−4.76942	−0.260581
$336$	0	0
$337$	−25.0621	−1.36522	−0.682610	−	0.730783i	$-0.739155\pi$
$337$	−25.0621	−1.36522	−0.682610	+	0.730783i	$0.739155\pi$
$338$	0	0
$339$	26.4517	1.43666
$340$	0	0
$341$	1.26859	0.0686983
$342$	0	0
$343$	24.5411	1.32510
$344$	0	0
$345$	−7.50154	−0.403870
$346$	0	0
$347$	13.8943	0.745887	0.372943	−	0.927854i	$-0.378349\pi$
$347$	13.8943	0.745887	0.372943	+	0.927854i	$0.378349\pi$
$348$	0	0
$349$	−11.8128	−0.632327	−0.316163	−	0.948705i	$-0.602395\pi$
$349$	−11.8128	−0.632327	−0.316163	+	0.948705i	$0.602395\pi$
$350$	0	0
$351$	−5.80625	−0.309915
$352$	0	0
$353$	−18.2525	−0.971485	−0.485743	−	0.874102i	$-0.661451\pi$
$353$	−18.2525	−0.971485	−0.485743	+	0.874102i	$0.661451\pi$
$354$	0	0
$355$	−0.533751	−0.0283286
$356$	0	0
$357$	8.86001	0.468921
$358$	0	0
$359$	−28.0793	−1.48197	−0.740986	−	0.671521i	$-0.765641\pi$
$359$	−28.0793	−1.48197	−0.740986	+	0.671521i	$0.765641\pi$
$360$	0	0
$361$	−1.99572	−0.105038
$362$	0	0
$363$	12.5042	0.656299
$364$	0	0
$365$	7.64125	0.399961
$366$	0	0
$367$	0.686369	0.0358282	0.0179141	−	0.999840i	$-0.494297\pi$
$367$	0.686369	0.0358282	0.0179141	+	0.999840i	$0.494297\pi$
$368$	0	0
$369$	1.14887	0.0598079
$370$	0	0
$371$	13.5238	0.702123
$372$	0	0
$373$	−5.58581	−0.289222	−0.144611	−	0.989489i	$-0.546193\pi$
$373$	−5.58581	−0.289222	−0.144611	+	0.989489i	$0.546193\pi$
$374$	0	0
$375$	1.62538	0.0839344
$376$	0	0
$377$	−0.901338	−0.0464213
$378$	0	0
$379$	−9.47553	−0.486726	−0.243363	−	0.969935i	$-0.578251\pi$
$379$	−9.47553	−0.486726	−0.243363	+	0.969935i	$0.578251\pi$
$380$	0	0
$381$	2.49531	0.127838
$382$	0	0
$383$	3.02327	0.154482	0.0772409	−	0.997012i	$-0.475389\pi$
$383$	3.02327	0.154482	0.0772409	+	0.997012i	$0.475389\pi$
$384$	0	0
$385$	8.04062	0.409788
$386$	0	0
$387$	3.76132	0.191199
$388$	0	0
$389$	−19.8063	−1.00422	−0.502109	−	0.864804i	$-0.667442\pi$
$389$	−19.8063	−1.00422	−0.502109	+	0.864804i	$0.667442\pi$
$390$	0	0
$391$	5.68979	0.287745
$392$	0	0
$393$	−27.8148	−1.40307
$394$	0	0
$395$	9.62077	0.484073
$396$	0	0
$397$	−23.0475	−1.15672	−0.578362	−	0.815781i	$-0.696308\pi$
$397$	−23.0475	−1.15672	−0.578362	+	0.815781i	$0.696308\pi$
$398$	0	0
$399$	−29.6355	−1.48363
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	0.742084	0.0369658
$404$	0	0
$405$	7.79735	0.387454
$406$	0	0
$407$	−6.28021	−0.311299
$408$	0	0
$409$	−29.1460	−1.44118	−0.720588	−	0.693363i	$-0.756128\pi$
$409$	−29.1460	−1.44118	−0.720588	+	0.693363i	$0.756128\pi$
$410$	0	0
$411$	−33.8227	−1.66835
$412$	0	0
$413$	−46.5419	−2.29018
$414$	0	0
$415$	6.01040	0.295039
$416$	0	0
$417$	−2.46556	−0.120739
$418$	0	0
$419$	−10.2572	−0.501096	−0.250548	−	0.968104i	$-0.580611\pi$
$419$	−10.2572	−0.501096	−0.250548	+	0.968104i	$0.580611\pi$
$420$	0	0
$421$	−1.67406	−0.0815889	−0.0407945	−	0.999168i	$-0.512989\pi$
$421$	−1.67406	−0.0815889	−0.0407945	+	0.999168i	$0.512989\pi$
$422$	0	0
$423$	−1.70865	−0.0830776
$424$	0	0
$425$	−1.23283	−0.0598008
$426$	0	0
$427$	41.7155	2.01876
$428$	0	0
$429$	−3.14421	−0.151804
$430$	0	0
$431$	−24.7069	−1.19009	−0.595045	−	0.803692i	$-0.702866\pi$
$431$	−24.7069	−1.19009	−0.595045	+	0.803692i	$0.702866\pi$
$432$	0	0
$433$	31.4314	1.51049	0.755247	−	0.655440i	$-0.227517\pi$
$433$	31.4314	1.51049	0.755247	+	0.655440i	$0.227517\pi$
$434$	0	0
$435$	1.37721	0.0660322
$436$	0	0
$437$	−19.0315	−0.910402
$438$	0	0
$439$	−14.9372	−0.712914	−0.356457	−	0.934312i	$-0.616015\pi$
$439$	−14.9372	−0.712914	−0.356457	+	0.934312i	$0.616015\pi$
$440$	0	0
$441$	−4.49465	−0.214031
$442$	0	0
$443$	0.673491	0.0319985	0.0159993	−	0.999872i	$-0.494907\pi$
$443$	0.673491	0.0319985	0.0159993	+	0.999872i	$0.494907\pi$
$444$	0	0
$445$	6.21176	0.294466
$446$	0	0
$447$	−36.5062	−1.72668
$448$	0	0
$449$	−8.62968	−0.407260	−0.203630	−	0.979048i	$-0.565274\pi$
$449$	−8.62968	−0.407260	−0.203630	+	0.979048i	$0.565274\pi$
$450$	0	0
$451$	5.83368	0.274697
$452$	0	0
$453$	−32.0059	−1.50377
$454$	0	0
$455$	4.70348	0.220503
$456$	0	0
$457$	−30.5412	−1.42866	−0.714329	−	0.699810i	$-0.753268\pi$
$457$	−30.5412	−1.42866	−0.714329	+	0.699810i	$0.753268\pi$
$458$	0	0
$459$	−6.72907	−0.314086
$460$	0	0
$461$	−20.8363	−0.970444	−0.485222	−	0.874391i	$-0.661261\pi$
$461$	−20.8363	−0.970444	−0.485222	+	0.874391i	$0.661261\pi$
$462$	0	0
$463$	11.9447	0.555118	0.277559	−	0.960709i	$-0.410475\pi$
$463$	11.9447	0.555118	0.277559	+	0.960709i	$0.410475\pi$
$464$	0	0
$465$	−1.13388	−0.0525823
$466$	0	0
$467$	−20.9025	−0.967253	−0.483627	−	0.875274i	$-0.660681\pi$
$467$	−20.9025	−0.967253	−0.483627	+	0.875274i	$0.660681\pi$
$468$	0	0
$469$	21.0883	0.973769
$470$	0	0
$471$	−3.32938	−0.153410
$472$	0	0
$473$	19.0991	0.878175
$474$	0	0
$475$	4.12363	0.189205
$476$	0	0
$477$	−1.09538	−0.0501539
$478$	0	0
$479$	28.5391	1.30398	0.651992	−	0.758226i	$-0.273934\pi$
$479$	28.5391	1.30398	0.651992	+	0.758226i	$0.273934\pi$
$480$	0	0
$481$	−3.67371	−0.167507
$482$	0	0
$483$	33.1686	1.50922
$484$	0	0
$485$	−12.6850	−0.575997
$486$	0	0
$487$	25.1344	1.13895	0.569473	−	0.822010i	$-0.307147\pi$
$487$	25.1344	1.13895	0.569473	+	0.822010i	$0.307147\pi$
$488$	0	0
$489$	6.49194	0.293576
$490$	0	0
$491$	−43.2133	−1.95019	−0.975093	−	0.221794i	$-0.928809\pi$
$491$	−43.2133	−1.95019	−0.975093	+	0.221794i	$0.928809\pi$
$492$	0	0
$493$	−1.04459	−0.0470460
$494$	0	0
$495$	−0.651259	−0.0292719
$496$	0	0
$497$	2.36002	0.105861
$498$	0	0
$499$	30.2184	1.35276	0.676381	−	0.736552i	$-0.263547\pi$
$499$	30.2184	1.35276	0.676381	+	0.736552i	$0.263547\pi$
$500$	0	0
$501$	24.9488	1.11463
$502$	0	0
$503$	−24.5662	−1.09535	−0.547676	−	0.836690i	$-0.684488\pi$
$503$	−24.5662	−1.09535	−0.547676	+	0.836690i	$0.684488\pi$
$504$	0	0
$505$	9.56613	0.425687
$506$	0	0
$507$	19.2907	0.856731
$508$	0	0
$509$	−14.2824	−0.633057	−0.316529	−	0.948583i	$-0.602517\pi$
$509$	−14.2824	−0.633057	−0.316529	+	0.948583i	$0.602517\pi$
$510$	0	0
$511$	−33.7864	−1.49462
$512$	0	0
$513$	22.5078	0.993742
$514$	0	0
$515$	9.56470	0.421471
$516$	0	0
$517$	−8.67612	−0.381575
$518$	0	0
$519$	39.5098	1.73429
$520$	0	0
$521$	−28.8306	−1.26309	−0.631545	−	0.775339i	$-0.717579\pi$
$521$	−28.8306	−1.26309	−0.631545	+	0.775339i	$0.717579\pi$
$522$	0	0
$523$	25.7614	1.12647	0.563234	−	0.826297i	$-0.309557\pi$
$523$	25.7614	1.12647	0.563234	+	0.826297i	$0.309557\pi$
$524$	0	0
$525$	−7.18675	−0.313656
$526$	0	0
$527$	0.860027	0.0374634
$528$	0	0
$529$	−1.69950	−0.0738912
$530$	0	0
$531$	3.76972	0.163592
$532$	0	0
$533$	3.41250	0.147812
$534$	0	0
$535$	−6.77911	−0.293086
$536$	0	0
$537$	−8.48338	−0.366085
$538$	0	0
$539$	−22.8227	−0.983044
$540$	0	0
$541$	−29.2832	−1.25898	−0.629492	−	0.777007i	$-0.716737\pi$
$541$	−29.2832	−1.25898	−0.629492	+	0.777007i	$0.716737\pi$
$542$	0	0
$543$	−12.1072	−0.519568
$544$	0	0
$545$	−5.53790	−0.237218
$546$	0	0
$547$	−19.2524	−0.823175	−0.411587	−	0.911370i	$-0.635025\pi$
$547$	−19.2524	−0.823175	−0.411587	+	0.911370i	$0.635025\pi$
$548$	0	0
$549$	−3.37880	−0.144204
$550$	0	0
$551$	3.49401	0.148850
$552$	0	0
$553$	−42.5389	−1.80894
$554$	0	0
$555$	5.61329	0.238271
$556$	0	0
$557$	14.8252	0.628162	0.314081	−	0.949396i	$-0.398304\pi$
$557$	14.8252	0.628162	0.314081	+	0.949396i	$0.398304\pi$
$558$	0	0
$559$	11.1723	0.472537
$560$	0	0
$561$	−3.64393	−0.153847
$562$	0	0
$563$	−13.0922	−0.551770	−0.275885	−	0.961191i	$-0.588971\pi$
$563$	−13.0922	−0.551770	−0.275885	+	0.961191i	$0.588971\pi$
$564$	0	0
$565$	16.2741	0.684658
$566$	0	0
$567$	−34.4766	−1.44788
$568$	0	0
$569$	21.4256	0.898207	0.449104	−	0.893480i	$-0.351743\pi$
$569$	21.4256	0.898207	0.449104	+	0.893480i	$0.351743\pi$
$570$	0	0
$571$	14.6522	0.613175	0.306587	−	0.951843i	$-0.400813\pi$
$571$	14.6522	0.613175	0.306587	+	0.951843i	$0.400813\pi$
$572$	0	0
$573$	17.0162	0.710863
$574$	0	0
$575$	−4.61525	−0.192469
$576$	0	0
$577$	−5.21697	−0.217185	−0.108593	−	0.994086i	$-0.534634\pi$
$577$	−5.21697	−0.217185	−0.108593	+	0.994086i	$0.534634\pi$
$578$	0	0
$579$	−4.47180	−0.185842
$580$	0	0
$581$	−26.5754	−1.10253
$582$	0	0
$583$	−5.56206	−0.230357
$584$	0	0
$585$	−0.380964	−0.0157509
$586$	0	0
$587$	36.5655	1.50922	0.754611	−	0.656173i	$-0.227826\pi$
$587$	36.5655	1.50922	0.754611	+	0.656173i	$0.227826\pi$
$588$	0	0
$589$	−2.87667	−0.118531
$590$	0	0
$591$	−2.64535	−0.108815
$592$	0	0
$593$	−33.1765	−1.36239	−0.681197	−	0.732100i	$-0.738540\pi$
$593$	−33.1765	−1.36239	−0.681197	+	0.732100i	$0.738540\pi$
$594$	0	0
$595$	5.45103	0.223470
$596$	0	0
$597$	28.7705	1.17750
$598$	0	0
$599$	−9.54284	−0.389910	−0.194955	−	0.980812i	$-0.562456\pi$
$599$	−9.54284	−0.389910	−0.194955	+	0.980812i	$0.562456\pi$
$600$	0	0
$601$	−36.3137	−1.48127	−0.740634	−	0.671909i	$-0.765475\pi$
$601$	−36.3137	−1.48127	−0.740634	+	0.671909i	$0.765475\pi$
$602$	0	0
$603$	−1.70807	−0.0695581
$604$	0	0
$605$	7.69307	0.312768
$606$	0	0
$607$	−22.7975	−0.925321	−0.462660	−	0.886536i	$-0.653105\pi$
$607$	−22.7975	−0.925321	−0.462660	+	0.886536i	$0.653105\pi$
$608$	0	0
$609$	−6.08944	−0.246757
$610$	0	0
$611$	−5.07523	−0.205322
$612$	0	0
$613$	−14.7378	−0.595254	−0.297627	−	0.954682i	$-0.596195\pi$
$613$	−14.7378	−0.595254	−0.297627	+	0.954682i	$0.596195\pi$
$614$	0	0
$615$	−5.21418	−0.210256
$616$	0	0
$617$	10.0465	0.404456	0.202228	−	0.979338i	$-0.435182\pi$
$617$	10.0465	0.404456	0.202228	+	0.979338i	$0.435182\pi$
$618$	0	0
$619$	−41.3237	−1.66094	−0.830469	−	0.557065i	$-0.811928\pi$
$619$	−41.3237	−1.66094	−0.830469	+	0.557065i	$0.811928\pi$
$620$	0	0
$621$	−25.1912	−1.01089
$622$	0	0
$623$	−27.4657	−1.10039
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	12.1884	0.486759
$628$	0	0
$629$	−4.25759	−0.169761
$630$	0	0
$631$	32.4599	1.29221	0.646105	−	0.763249i	$-0.276397\pi$
$631$	32.4599	1.29221	0.646105	+	0.763249i	$0.276397\pi$
$632$	0	0
$633$	−14.0143	−0.557018
$634$	0	0
$635$	1.53521	0.0609230
$636$	0	0
$637$	−13.3505	−0.528966
$638$	0	0
$639$	−0.191152	−0.00756187
$640$	0	0
$641$	−3.29789	−0.130259	−0.0651293	−	0.997877i	$-0.520746\pi$
$641$	−3.29789	−0.130259	−0.0651293	+	0.997877i	$0.520746\pi$
$642$	0	0
$643$	−6.89098	−0.271754	−0.135877	−	0.990726i	$-0.543385\pi$
$643$	−6.89098	−0.271754	−0.135877	+	0.990726i	$0.543385\pi$
$644$	0	0
$645$	−17.0708	−0.672164
$646$	0	0
$647$	−8.35604	−0.328510	−0.164255	−	0.986418i	$-0.552522\pi$
$647$	−8.35604	−0.328510	−0.164255	+	0.986418i	$0.552522\pi$
$648$	0	0
$649$	19.1417	0.751376
$650$	0	0
$651$	5.01352	0.196495
$652$	0	0
$653$	34.1612	1.33683	0.668416	−	0.743788i	$-0.266973\pi$
$653$	34.1612	1.33683	0.668416	+	0.743788i	$0.266973\pi$
$654$	0	0
$655$	−17.1128	−0.668652
$656$	0	0
$657$	2.73656	0.106764
$658$	0	0
$659$	19.9226	0.776074	0.388037	−	0.921644i	$-0.373153\pi$
$659$	19.9226	0.776074	0.388037	+	0.921644i	$0.373153\pi$
$660$	0	0
$661$	41.8320	1.62708	0.813539	−	0.581511i	$-0.197538\pi$
$661$	41.8320	1.62708	0.813539	+	0.581511i	$0.197538\pi$
$662$	0	0
$663$	−2.13157	−0.0827834
$664$	0	0
$665$	−18.2329	−0.707042
$666$	0	0
$667$	−3.91057	−0.151418
$668$	0	0
$669$	−19.6649	−0.760290
$670$	0	0
$671$	−17.1567	−0.662327
$672$	0	0
$673$	−5.82606	−0.224578	−0.112289	−	0.993676i	$-0.535818\pi$
$673$	−5.82606	−0.224578	−0.112289	+	0.993676i	$0.535818\pi$
$674$	0	0
$675$	5.45825	0.210088
$676$	0	0
$677$	−13.8212	−0.531193	−0.265596	−	0.964084i	$-0.585569\pi$
$677$	−13.8212	−0.531193	−0.265596	+	0.964084i	$0.585569\pi$
$678$	0	0
$679$	56.0877	2.15245
$680$	0	0
$681$	43.6336	1.67204
$682$	0	0
$683$	−5.57951	−0.213494	−0.106747	−	0.994286i	$-0.534043\pi$
$683$	−5.57951	−0.213494	−0.106747	+	0.994286i	$0.534043\pi$
$684$	0	0
$685$	−20.8091	−0.795075
$686$	0	0
$687$	12.0080	0.458134
$688$	0	0
$689$	−3.25361	−0.123953
$690$	0	0
$691$	−40.4727	−1.53965	−0.769826	−	0.638254i	$-0.779657\pi$
$691$	−40.4727	−1.53965	−0.769826	+	0.638254i	$0.779657\pi$
$692$	0	0
$693$	2.87959	0.109387
$694$	0	0
$695$	−1.51691	−0.0575396
$696$	0	0
$697$	3.95487	0.149801
$698$	0	0
$699$	34.4431	1.30276
$700$	0	0
$701$	−19.7507	−0.745975	−0.372987	−	0.927836i	$-0.621666\pi$
$701$	−19.7507	−0.745975	−0.372987	+	0.927836i	$0.621666\pi$
$702$	0	0
$703$	14.2410	0.537110
$704$	0	0
$705$	7.75477	0.292061
$706$	0	0
$707$	−42.2973	−1.59076
$708$	0	0
$709$	−25.3283	−0.951222	−0.475611	−	0.879656i	$-0.657773\pi$
$709$	−25.3283	−0.951222	−0.475611	+	0.879656i	$0.657773\pi$
$710$	0	0
$711$	3.44549	0.129216
$712$	0	0
$713$	3.21963	0.120576
$714$	0	0
$715$	−1.93444	−0.0723440
$716$	0	0
$717$	−8.61075	−0.321574
$718$	0	0
$719$	−7.17722	−0.267665	−0.133833	−	0.991004i	$-0.542728\pi$
$719$	−7.17722	−0.267665	−0.133833	+	0.991004i	$0.542728\pi$
$720$	0	0
$721$	−42.2910	−1.57500
$722$	0	0
$723$	−8.58359	−0.319227
$724$	0	0
$725$	0.847315	0.0314685
$726$	0	0
$727$	27.7846	1.03047	0.515237	−	0.857048i	$-0.327704\pi$
$727$	27.7846	1.03047	0.515237	+	0.857048i	$0.327704\pi$
$728$	0	0
$729$	29.4077	1.08917
$730$	0	0
$731$	12.9479	0.478897
$732$	0	0
$733$	−23.5720	−0.870653	−0.435327	−	0.900273i	$-0.643367\pi$
$733$	−23.5720	−0.870653	−0.435327	+	0.900273i	$0.643367\pi$
$734$	0	0
$735$	20.3991	0.752431
$736$	0	0
$737$	−8.67317	−0.319480
$738$	0	0
$739$	−48.1819	−1.77240	−0.886200	−	0.463303i	$-0.846664\pi$
$739$	−48.1819	−1.77240	−0.886200	+	0.463303i	$0.846664\pi$
$740$	0	0
$741$	7.12980	0.261920
$742$	0	0
$743$	24.9637	0.915830	0.457915	−	0.888996i	$-0.348596\pi$
$743$	24.9637	0.915830	0.457915	+	0.888996i	$0.348596\pi$
$744$	0	0
$745$	−22.4601	−0.822873
$746$	0	0
$747$	2.15251	0.0787561
$748$	0	0
$749$	29.9743	1.09524
$750$	0	0
$751$	−3.31934	−0.121125	−0.0605623	−	0.998164i	$-0.519289\pi$
$751$	−3.31934	−0.121125	−0.0605623	+	0.998164i	$0.519289\pi$
$752$	0	0
$753$	−8.47690	−0.308915
$754$	0	0
$755$	−19.6913	−0.716640
$756$	0	0
$757$	29.4811	1.07151	0.535754	−	0.844374i	$-0.320028\pi$
$757$	29.4811	1.07151	0.535754	+	0.844374i	$0.320028\pi$
$758$	0	0
$759$	−13.6415	−0.495156
$760$	0	0
$761$	−17.4559	−0.632777	−0.316388	−	0.948630i	$-0.602470\pi$
$761$	−17.4559	−0.632777	−0.316388	+	0.948630i	$0.602470\pi$
$762$	0	0
$763$	24.4863	0.886462
$764$	0	0
$765$	−0.441512	−0.0159629
$766$	0	0
$767$	11.1972	0.404308
$768$	0	0
$769$	18.1727	0.655325	0.327662	−	0.944795i	$-0.393739\pi$
$769$	18.1727	0.655325	0.327662	+	0.944795i	$0.393739\pi$
$770$	0	0
$771$	13.5451	0.487815
$772$	0	0
$773$	33.8354	1.21697	0.608487	−	0.793564i	$-0.291777\pi$
$773$	33.8354	1.21697	0.608487	+	0.793564i	$0.291777\pi$
$774$	0	0
$775$	−0.697606	−0.0250588
$776$	0	0
$777$	−24.8196	−0.890397
$778$	0	0
$779$	−13.2285	−0.473959
$780$	0	0
$781$	−0.970624	−0.0347317
$782$	0	0
$783$	4.62485	0.165279
$784$	0	0
$785$	−2.04837	−0.0731094
$786$	0	0
$787$	9.92229	0.353692	0.176846	−	0.984239i	$-0.443411\pi$
$787$	9.92229	0.353692	0.176846	+	0.984239i	$0.443411\pi$
$788$	0	0
$789$	−25.9692	−0.924529
$790$	0	0
$791$	−71.9573	−2.55851
$792$	0	0
$793$	−10.0361	−0.356391
$794$	0	0
$795$	4.97140	0.176317
$796$	0	0
$797$	−28.2666	−1.00126	−0.500628	−	0.865663i	$-0.666897\pi$
$797$	−28.2666	−1.00126	−0.500628	+	0.865663i	$0.666897\pi$
$798$	0	0
$799$	−5.88186	−0.208085
$800$	0	0
$801$	2.22462	0.0786030
$802$	0	0
$803$	13.8956	0.490365
$804$	0	0
$805$	20.4067	0.719240
$806$	0	0
$807$	−6.08910	−0.214346
$808$	0	0
$809$	−39.7494	−1.39752	−0.698758	−	0.715358i	$-0.746264\pi$
$809$	−39.7494	−1.39752	−0.698758	+	0.715358i	$0.746264\pi$
$810$	0	0
$811$	5.60060	0.196664	0.0983319	−	0.995154i	$-0.468649\pi$
$811$	5.60060	0.196664	0.0983319	+	0.995154i	$0.468649\pi$
$812$	0	0
$813$	22.5044	0.789264
$814$	0	0
$815$	3.99410	0.139907
$816$	0	0
$817$	−43.3090	−1.51519
$818$	0	0
$819$	1.68446	0.0588598
$820$	0	0
$821$	−29.6542	−1.03494	−0.517469	−	0.855702i	$-0.673126\pi$
$821$	−29.6542	−1.03494	−0.517469	+	0.855702i	$0.673126\pi$
$822$	0	0
$823$	−12.4945	−0.435531	−0.217765	−	0.976001i	$-0.569877\pi$
$823$	−12.4945	−0.435531	−0.217765	+	0.976001i	$0.569877\pi$
$824$	0	0
$825$	2.95575	0.102906
$826$	0	0
$827$	−33.2840	−1.15740	−0.578700	−	0.815541i	$-0.696440\pi$
$827$	−33.2840	−1.15740	−0.578700	+	0.815541i	$0.696440\pi$
$828$	0	0
$829$	−33.0337	−1.14731	−0.573654	−	0.819098i	$-0.694474\pi$
$829$	−33.0337	−1.14731	−0.573654	+	0.819098i	$0.694474\pi$
$830$	0	0
$831$	−2.38404	−0.0827015
$832$	0	0
$833$	−15.4723	−0.536085
$834$	0	0
$835$	15.3495	0.531191
$836$	0	0
$837$	−3.80771	−0.131614
$838$	0	0
$839$	−23.3786	−0.807120	−0.403560	−	0.914953i	$-0.632227\pi$
$839$	−23.3786	−0.807120	−0.403560	+	0.914953i	$0.632227\pi$
$840$	0	0
$841$	−28.2821	−0.975243
$842$	0	0
$843$	47.2607	1.62775
$844$	0	0
$845$	11.8684	0.408286
$846$	0	0
$847$	−34.0155	−1.16879
$848$	0	0
$849$	−41.7469	−1.43275
$850$	0	0
$851$	−15.9388	−0.546376
$852$	0	0
$853$	32.9431	1.12795	0.563975	−	0.825792i	$-0.309271\pi$
$853$	32.9431	1.12795	0.563975	+	0.825792i	$0.309271\pi$
$854$	0	0
$855$	1.47680	0.0505053
$856$	0	0
$857$	7.72706	0.263951	0.131976	−	0.991253i	$-0.457868\pi$
$857$	7.72706	0.263951	0.131976	+	0.991253i	$0.457868\pi$
$858$	0	0
$859$	−21.8310	−0.744863	−0.372431	−	0.928060i	$-0.621476\pi$
$859$	−21.8310	−0.744863	−0.372431	+	0.928060i	$0.621476\pi$
$860$	0	0
$861$	23.0549	0.785709
$862$	0	0
$863$	−24.3905	−0.830263	−0.415131	−	0.909761i	$-0.636264\pi$
$863$	−24.3905	−0.830263	−0.415131	+	0.909761i	$0.636264\pi$
$864$	0	0
$865$	24.3080	0.826496
$866$	0	0
$867$	25.1612	0.854518
$868$	0	0
$869$	17.4953	0.593489
$870$	0	0
$871$	−5.07350	−0.171909
$872$	0	0
$873$	−4.54289	−0.153753
$874$	0	0
$875$	−4.42157	−0.149476
$876$	0	0
$877$	−5.89927	−0.199204	−0.0996021	−	0.995027i	$-0.531757\pi$
$877$	−5.89927	−0.199204	−0.0996021	+	0.995027i	$0.531757\pi$
$878$	0	0
$879$	−9.69038	−0.326848
$880$	0	0
$881$	16.9087	0.569668	0.284834	−	0.958577i	$-0.408061\pi$
$881$	16.9087	0.569668	0.284834	+	0.958577i	$0.408061\pi$
$882$	0	0
$883$	26.9248	0.906091	0.453046	−	0.891487i	$-0.350337\pi$
$883$	26.9248	0.906091	0.453046	+	0.891487i	$0.350337\pi$
$884$	0	0
$885$	−17.1089	−0.575111
$886$	0	0
$887$	35.5378	1.19324	0.596621	−	0.802523i	$-0.296510\pi$
$887$	35.5378	1.19324	0.596621	+	0.802523i	$0.296510\pi$
$888$	0	0
$889$	−6.78805	−0.227664
$890$	0	0
$891$	14.1795	0.475030
$892$	0	0
$893$	19.6740	0.658365
$894$	0	0
$895$	−5.21931	−0.174462
$896$	0	0
$897$	−7.97982	−0.266439
$898$	0	0
$899$	−0.591092	−0.0197140
$900$	0	0
$901$	−3.77072	−0.125621
$902$	0	0
$903$	75.4800	2.51182
$904$	0	0
$905$	−7.44881	−0.247607
$906$	0	0
$907$	1.02937	0.0341796	0.0170898	−	0.999854i	$-0.494560\pi$
$907$	1.02937	0.0341796	0.0170898	+	0.999854i	$0.494560\pi$
$908$	0	0
$909$	3.42592	0.113631
$910$	0	0
$911$	14.9038	0.493785	0.246893	−	0.969043i	$-0.420591\pi$
$911$	14.9038	0.493785	0.246893	+	0.969043i	$0.420591\pi$
$912$	0	0
$913$	10.9299	0.361727
$914$	0	0
$915$	15.3347	0.506951
$916$	0	0
$917$	75.6655	2.49869
$918$	0	0
$919$	−8.36916	−0.276073	−0.138037	−	0.990427i	$-0.544079\pi$
$919$	−8.36916	−0.276073	−0.138037	+	0.990427i	$0.544079\pi$
$920$	0	0
$921$	8.36305	0.275572
$922$	0	0
$923$	−0.567782	−0.0186888
$924$	0	0
$925$	3.45352	0.113551
$926$	0	0
$927$	3.42541	0.112505
$928$	0	0
$929$	−43.9155	−1.44082	−0.720410	−	0.693548i	$-0.756046\pi$
$929$	−43.9155	−1.44082	−0.720410	+	0.693548i	$0.756046\pi$
$930$	0	0
$931$	51.7528	1.69613
$932$	0	0
$933$	−2.05278	−0.0672049
$934$	0	0
$935$	−2.24189	−0.0733176
$936$	0	0
$937$	42.5033	1.38852	0.694261	−	0.719724i	$-0.255731\pi$
$937$	42.5033	1.38852	0.694261	+	0.719724i	$0.255731\pi$
$938$	0	0
$939$	−22.4940	−0.734065
$940$	0	0
$941$	−19.9326	−0.649783	−0.324891	−	0.945751i	$-0.605328\pi$
$941$	−19.9326	−0.649783	−0.324891	+	0.945751i	$0.605328\pi$
$942$	0	0
$943$	14.8056	0.482136
$944$	0	0
$945$	−24.1340	−0.785081
$946$	0	0
$947$	−39.7373	−1.29129	−0.645644	−	0.763639i	$-0.723411\pi$
$947$	−39.7373	−1.29129	−0.645644	+	0.763639i	$0.723411\pi$
$948$	0	0
$949$	8.12844	0.263860
$950$	0	0
$951$	−8.01589	−0.259933
$952$	0	0
$953$	6.10032	0.197609	0.0988044	−	0.995107i	$-0.468498\pi$
$953$	6.10032	0.197609	0.0988044	+	0.995107i	$0.468498\pi$
$954$	0	0
$955$	10.4691	0.338771
$956$	0	0
$957$	2.50445	0.0809575
$958$	0	0
$959$	92.0089	2.97112
$960$	0	0
$961$	−30.5133	−0.984301
$962$	0	0
$963$	−2.42781	−0.0782349
$964$	0	0
$965$	−2.75123	−0.0885652
$966$	0	0
$967$	−21.4093	−0.688478	−0.344239	−	0.938882i	$-0.611863\pi$
$967$	−21.4093	−0.688478	−0.344239	+	0.938882i	$0.611863\pi$
$968$	0	0
$969$	8.26298	0.265445
$970$	0	0
$971$	19.5770	0.628256	0.314128	−	0.949381i	$-0.398288\pi$
$971$	19.5770	0.628256	0.314128	+	0.949381i	$0.398288\pi$
$972$	0	0
$973$	6.70712	0.215020
$974$	0	0
$975$	1.72901	0.0553728
$976$	0	0
$977$	21.0987	0.675009	0.337504	−	0.941324i	$-0.390417\pi$
$977$	21.0987	0.675009	0.337504	+	0.941324i	$0.390417\pi$
$978$	0	0
$979$	11.2961	0.361024
$980$	0	0
$981$	−1.98329	−0.0633216
$982$	0	0
$983$	22.5009	0.717667	0.358834	−	0.933402i	$-0.383175\pi$
$983$	22.5009	0.717667	0.358834	+	0.933402i	$0.383175\pi$
$984$	0	0
$985$	−1.62752	−0.0518572
$986$	0	0
$987$	−34.2883	−1.09141
$988$	0	0
$989$	48.4724	1.54133
$990$	0	0
$991$	25.6980	0.816322	0.408161	−	0.912910i	$-0.366170\pi$
$991$	25.6980	0.816322	0.408161	+	0.912910i	$0.366170\pi$
$992$	0	0
$993$	16.9120	0.536687
$994$	0	0
$995$	17.7008	0.561152
$996$	0	0
$997$	50.6604	1.60443	0.802216	−	0.597034i	$-0.203654\pi$
$997$	50.6604	1.60443	0.802216	+	0.597034i	$0.203654\pi$
$998$	0	0
$999$	18.8502	0.596393

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.c.1.6	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.c.1.6	✓	28	1.1	even	1	trivial

Overview	Random
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:	\( N \)	\(=\)	\( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8020.a (trivial)

\(f(q)\)	\(=\)	\(q-1.62538 q^{3} -1.00000 q^{5} +4.42157 q^{7} -0.358130 q^{9} +O(q^{10})\)	\(q-1.62538 q^{3} -1.00000 q^{5} +4.42157 q^{7} -0.358130 q^{9} -1.81850 q^{11} -1.06376 q^{13} +1.62538 q^{15} -1.23283 q^{17} +4.12363 q^{19} -7.18675 q^{21} -4.61525 q^{23} +1.00000 q^{25} +5.45825 q^{27} +0.847315 q^{29} -0.697606 q^{31} +2.95575 q^{33} -4.42157 q^{35} +3.45352 q^{37} +1.72901 q^{39} -3.20797 q^{41} -10.5027 q^{43} +0.358130 q^{45} +4.77104 q^{47} +12.5503 q^{49} +2.00381 q^{51} +3.05860 q^{53} +1.81850 q^{55} -6.70247 q^{57} -10.5261 q^{59} +9.43455 q^{61} -1.58350 q^{63} +1.06376 q^{65} +4.76942 q^{67} +7.50154 q^{69} +0.533751 q^{71} -7.64125 q^{73} -1.62538 q^{75} -8.04062 q^{77} -9.62077 q^{79} -7.79735 q^{81} -6.01040 q^{83} +1.23283 q^{85} -1.37721 q^{87} -6.21176 q^{89} -4.70348 q^{91} +1.13388 q^{93} -4.12363 q^{95} +12.6850 q^{97} +0.651259 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9} + 2 q^{11} + 3 q^{13} - 3 q^{15} - 10 q^{17} - 2 q^{19} - 12 q^{21} - 23 q^{23} + 28 q^{25} + 9 q^{27} - 37 q^{29} - 11 q^{31} + 2 q^{33} + 4 q^{35} - 3 q^{37} - 19 q^{39} - 30 q^{41} + 13 q^{43} - 17 q^{45} - 15 q^{47} + 12 q^{49} - 8 q^{51} - 35 q^{53} - 2 q^{55} - 22 q^{57} - q^{59} - 33 q^{61} - 20 q^{63} - 3 q^{65} + 19 q^{67} - 8 q^{69} - 31 q^{71} + 31 q^{73} + 3 q^{75} - 42 q^{77} - 29 q^{79} - 36 q^{81} + 14 q^{83} + 10 q^{85} - 32 q^{87} - 32 q^{89} - 7 q^{91} - 11 q^{93} + 2 q^{95} + 2 q^{97} - 39 q^{99}+O(q^{100}) \) 28 * q + 3 * q^3 - 28 * q^5 - 4 * q^7 + 17 * q^9 + 2 * q^11 + 3 * q^13 - 3 * q^15 - 10 * q^17 - 2 * q^19 - 12 * q^21 - 23 * q^23 + 28 * q^25 + 9 * q^27 - 37 * q^29 - 11 * q^31 + 2 * q^33 + 4 * q^35 - 3 * q^37 - 19 * q^39 - 30 * q^41 + 13 * q^43 - 17 * q^45 - 15 * q^47 + 12 * q^49 - 8 * q^51 - 35 * q^53 - 2 * q^55 - 22 * q^57 - q^59 - 33 * q^61 - 20 * q^63 - 3 * q^65 + 19 * q^67 - 8 * q^69 - 31 * q^71 + 31 * q^73 + 3 * q^75 - 42 * q^77 - 29 * q^79 - 36 * q^81 + 14 * q^83 + 10 * q^85 - 32 * q^87 - 32 * q^89 - 7 * q^91 - 11 * q^93 + 2 * q^95 + 2 * q^97 - 39 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more