LMFDB - Embedded newform 8020.2.a.e.1.20

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:	$0$
Dimension:	$35$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
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)

$f(q)$	$=$	$q+0.309116 q^{3} -1.00000 q^{5} -4.23374 q^{7} -2.90445 q^{9} +O(q^{10})$	$q+0.309116 q^{3} -1.00000 q^{5} -4.23374 q^{7} -2.90445 q^{9} -2.06262 q^{11} -4.51628 q^{13} -0.309116 q^{15} +4.22032 q^{17} -8.26598 q^{19} -1.30872 q^{21} -2.74441 q^{23} +1.00000 q^{25} -1.82516 q^{27} +0.483520 q^{29} -1.67771 q^{31} -0.637589 q^{33} +4.23374 q^{35} +1.15744 q^{37} -1.39606 q^{39} +5.46044 q^{41} -8.29850 q^{43} +2.90445 q^{45} -10.6395 q^{47} +10.9246 q^{49} +1.30457 q^{51} -12.1735 q^{53} +2.06262 q^{55} -2.55515 q^{57} -7.15113 q^{59} -4.76744 q^{61} +12.2967 q^{63} +4.51628 q^{65} -12.5426 q^{67} -0.848343 q^{69} -2.96217 q^{71} -8.35721 q^{73} +0.309116 q^{75} +8.73259 q^{77} -3.90790 q^{79} +8.14915 q^{81} -5.38578 q^{83} -4.22032 q^{85} +0.149464 q^{87} -0.998379 q^{89} +19.1208 q^{91} -0.518608 q^{93} +8.26598 q^{95} +14.3241 q^{97} +5.99077 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9}+O(q^{10}) $ 35 * q - q^3 - 35 * q^5 + 6 * q^7 + 52 * q^9	$ 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9} - 2 q^{11} - q^{13} + q^{15} + 18 q^{17} + 2 q^{19} + 12 q^{21} + 13 q^{23} + 35 q^{25} - 7 q^{27} + 25 q^{29} + 13 q^{31} + 14 q^{33} - 6 q^{35} - 19 q^{37} - 3 q^{39} + 24 q^{41} - 5 q^{43} - 52 q^{45} + 19 q^{47} + 55 q^{49} + 41 q^{53} + 2 q^{55} + 14 q^{57} + 3 q^{59} + 13 q^{61} + 70 q^{63} + q^{65} - 17 q^{67} + 64 q^{69} + 17 q^{71} - 63 q^{73} - q^{75} + 54 q^{77} + 11 q^{79} + 107 q^{81} - 8 q^{83} - 18 q^{85} + 36 q^{87} + 38 q^{89} - 27 q^{91} + q^{93} - 2 q^{95} - 54 q^{97} - 51 q^{99}+O(q^{100}) $ 35 * q - q^3 - 35 * q^5 + 6 * q^7 + 52 * q^9 - 2 * q^11 - q^13 + q^15 + 18 * q^17 + 2 * q^19 + 12 * q^21 + 13 * q^23 + 35 * q^25 - 7 * q^27 + 25 * q^29 + 13 * q^31 + 14 * q^33 - 6 * q^35 - 19 * q^37 - 3 * q^39 + 24 * q^41 - 5 * q^43 - 52 * q^45 + 19 * q^47 + 55 * q^49 + 41 * q^53 + 2 * q^55 + 14 * q^57 + 3 * q^59 + 13 * q^61 + 70 * q^63 + q^65 - 17 * q^67 + 64 * q^69 + 17 * q^71 - 63 * q^73 - q^75 + 54 * q^77 + 11 * q^79 + 107 * q^81 - 8 * q^83 - 18 * q^85 + 36 * q^87 + 38 * q^89 - 27 * q^91 + q^93 - 2 * q^95 - 54 * q^97 - 51 * 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
$3$	0.309116	0.178468	0.0892342	−	0.996011i	$-0.471558\pi$
$3$	0.309116	0.178468	0.0892342	+	0.996011i	$0.471558\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.23374	−1.60020	−0.800102	−	0.599864i	$-0.795221\pi$
$7$	−4.23374	−1.60020	−0.800102	+	0.599864i	$0.795221\pi$
$8$	0	0
$9$	−2.90445	−0.968149
$10$	0	0
$11$	−2.06262	−0.621903	−0.310952	−	0.950426i	$-0.600648\pi$
$11$	−2.06262	−0.621903	−0.310952	+	0.950426i	$0.600648\pi$
$12$	0	0
$13$	−4.51628	−1.25259	−0.626295	−	0.779586i	$-0.715430\pi$
$13$	−4.51628	−1.25259	−0.626295	+	0.779586i	$0.715430\pi$
$14$	0	0
$15$	−0.309116	−0.0798135
$16$	0	0
$17$	4.22032	1.02358	0.511789	−	0.859111i	$-0.328983\pi$
$17$	4.22032	1.02358	0.511789	+	0.859111i	$0.328983\pi$
$18$	0	0
$19$	−8.26598	−1.89635	−0.948173	−	0.317754i	$-0.897072\pi$
$19$	−8.26598	−1.89635	−0.948173	+	0.317754i	$0.897072\pi$
$20$	0	0
$21$	−1.30872	−0.285586
$22$	0	0
$23$	−2.74441	−0.572250	−0.286125	−	0.958192i	$-0.592367\pi$
$23$	−2.74441	−0.572250	−0.286125	+	0.958192i	$0.592367\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.82516	−0.351252
$28$	0	0
$29$	0.483520	0.0897874	0.0448937	−	0.998992i	$-0.485705\pi$
$29$	0.483520	0.0897874	0.0448937	+	0.998992i	$0.485705\pi$
$30$	0	0
$31$	−1.67771	−0.301326	−0.150663	−	0.988585i	$-0.548141\pi$
$31$	−1.67771	−0.301326	−0.150663	+	0.988585i	$0.548141\pi$
$32$	0	0
$33$	−0.637589	−0.110990
$34$	0	0
$35$	4.23374	0.715633
$36$	0	0
$37$	1.15744	0.190282	0.0951412	−	0.995464i	$-0.469670\pi$
$37$	1.15744	0.190282	0.0951412	+	0.995464i	$0.469670\pi$
$38$	0	0
$39$	−1.39606	−0.223548
$40$	0	0
$41$	5.46044	0.852778	0.426389	−	0.904540i	$-0.359785\pi$
$41$	5.46044	0.852778	0.426389	+	0.904540i	$0.359785\pi$
$42$	0	0
$43$	−8.29850	−1.26551	−0.632755	−	0.774352i	$-0.718076\pi$
$43$	−8.29850	−1.26551	−0.632755	+	0.774352i	$0.718076\pi$
$44$	0	0
$45$	2.90445	0.432969
$46$	0	0
$47$	−10.6395	−1.55193	−0.775966	−	0.630775i	$-0.782737\pi$
$47$	−10.6395	−1.55193	−0.775966	+	0.630775i	$0.782737\pi$
$48$	0	0
$49$	10.9246	1.56065
$50$	0	0
$51$	1.30457	0.182676
$52$	0	0
$53$	−12.1735	−1.67215	−0.836076	−	0.548614i	$-0.815156\pi$
$53$	−12.1735	−1.67215	−0.836076	+	0.548614i	$0.815156\pi$
$54$	0	0
$55$	2.06262	0.278123
$56$	0	0
$57$	−2.55515	−0.338438
$58$	0	0
$59$	−7.15113	−0.930997	−0.465499	−	0.885049i	$-0.654125\pi$
$59$	−7.15113	−0.930997	−0.465499	+	0.885049i	$0.654125\pi$
$60$	0	0
$61$	−4.76744	−0.610408	−0.305204	−	0.952287i	$-0.598725\pi$
$61$	−4.76744	−0.610408	−0.305204	+	0.952287i	$0.598725\pi$
$62$	0	0
$63$	12.2967	1.54924
$64$	0	0
$65$	4.51628	0.560176
$66$	0	0
$67$	−12.5426	−1.53232	−0.766162	−	0.642648i	$-0.777836\pi$
$67$	−12.5426	−1.53232	−0.766162	+	0.642648i	$0.777836\pi$
$68$	0	0
$69$	−0.848343	−0.102128
$70$	0	0
$71$	−2.96217	−0.351545	−0.175773	−	0.984431i	$-0.556242\pi$
$71$	−2.96217	−0.351545	−0.175773	+	0.984431i	$0.556242\pi$
$72$	0	0
$73$	−8.35721	−0.978137	−0.489069	−	0.872245i	$-0.662663\pi$
$73$	−8.35721	−0.978137	−0.489069	+	0.872245i	$0.662663\pi$
$74$	0	0
$75$	0.309116	0.0356937
$76$	0	0
$77$	8.73259	0.995171
$78$	0	0
$79$	−3.90790	−0.439673	−0.219837	−	0.975537i	$-0.570552\pi$
$79$	−3.90790	−0.439673	−0.219837	+	0.975537i	$0.570552\pi$
$80$	0	0
$81$	8.14915	0.905462
$82$	0	0
$83$	−5.38578	−0.591166	−0.295583	−	0.955317i	$-0.595514\pi$
$83$	−5.38578	−0.591166	−0.295583	+	0.955317i	$0.595514\pi$
$84$	0	0
$85$	−4.22032	−0.457758
$86$	0	0
$87$	0.149464	0.0160242
$88$	0	0
$89$	−0.998379	−0.105828	−0.0529140	−	0.998599i	$-0.516851\pi$
$89$	−0.998379	−0.105828	−0.0529140	+	0.998599i	$0.516851\pi$
$90$	0	0
$91$	19.1208	2.00440
$92$	0	0
$93$	−0.518608	−0.0537772
$94$	0	0
$95$	8.26598	0.848072
$96$	0	0
$97$	14.3241	1.45439	0.727195	−	0.686431i	$-0.240823\pi$
$97$	14.3241	1.45439	0.727195	+	0.686431i	$0.240823\pi$
$98$	0	0
$99$	5.99077	0.602095
$100$	0	0
$101$	−9.24151	−0.919564	−0.459782	−	0.888032i	$-0.652073\pi$
$101$	−9.24151	−0.919564	−0.459782	+	0.888032i	$0.652073\pi$
$102$	0	0
$103$	18.1150	1.78492	0.892461	−	0.451125i	$-0.148977\pi$
$103$	18.1150	1.78492	0.892461	+	0.451125i	$0.148977\pi$
$104$	0	0
$105$	1.30872	0.127718
$106$	0	0
$107$	−8.11012	−0.784035	−0.392017	−	0.919958i	$-0.628223\pi$
$107$	−8.11012	−0.784035	−0.392017	+	0.919958i	$0.628223\pi$
$108$	0	0
$109$	16.8089	1.61000	0.805002	−	0.593272i	$-0.202164\pi$
$109$	16.8089	1.61000	0.805002	+	0.593272i	$0.202164\pi$
$110$	0	0
$111$	0.357784	0.0339594
$112$	0	0
$113$	−11.2230	−1.05577	−0.527886	−	0.849316i	$-0.677015\pi$
$113$	−11.2230	−1.05577	−0.527886	+	0.849316i	$0.677015\pi$
$114$	0	0
$115$	2.74441	0.255918
$116$	0	0
$117$	13.1173	1.21269
$118$	0	0
$119$	−17.8677	−1.63793
$120$	0	0
$121$	−6.74560	−0.613237
$122$	0	0
$123$	1.68791	0.152194
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	16.9340	1.50265	0.751325	−	0.659932i	$-0.229415\pi$
$127$	16.9340	1.50265	0.751325	+	0.659932i	$0.229415\pi$
$128$	0	0
$129$	−2.56520	−0.225853
$130$	0	0
$131$	−7.09553	−0.619939	−0.309970	−	0.950746i	$-0.600319\pi$
$131$	−7.09553	−0.619939	−0.309970	+	0.950746i	$0.600319\pi$
$132$	0	0
$133$	34.9960	3.03454
$134$	0	0
$135$	1.82516	0.157085
$136$	0	0
$137$	9.66662	0.825875	0.412938	−	0.910759i	$-0.364503\pi$
$137$	9.66662	0.825875	0.412938	+	0.910759i	$0.364503\pi$
$138$	0	0
$139$	−5.69948	−0.483423	−0.241712	−	0.970348i	$-0.577709\pi$
$139$	−5.69948	−0.483423	−0.241712	+	0.970348i	$0.577709\pi$
$140$	0	0
$141$	−3.28884	−0.276971
$142$	0	0
$143$	9.31536	0.778990
$144$	0	0
$145$	−0.483520	−0.0401541
$146$	0	0
$147$	3.37696	0.278527
$148$	0	0
$149$	−6.86580	−0.562468	−0.281234	−	0.959639i	$-0.590744\pi$
$149$	−6.86580	−0.562468	−0.281234	+	0.959639i	$0.590744\pi$
$150$	0	0
$151$	17.7703	1.44613	0.723064	−	0.690781i	$-0.242733\pi$
$151$	17.7703	1.44613	0.723064	+	0.690781i	$0.242733\pi$
$152$	0	0
$153$	−12.2577	−0.990976
$154$	0	0
$155$	1.67771	0.134757
$156$	0	0
$157$	23.3517	1.86367	0.931834	−	0.362885i	$-0.118208\pi$
$157$	23.3517	1.86367	0.931834	+	0.362885i	$0.118208\pi$
$158$	0	0
$159$	−3.76301	−0.298426
$160$	0	0
$161$	11.6191	0.915716
$162$	0	0
$163$	−6.92809	−0.542650	−0.271325	−	0.962488i	$-0.587462\pi$
$163$	−6.92809	−0.542650	−0.271325	+	0.962488i	$0.587462\pi$
$164$	0	0
$165$	0.637589	0.0496362
$166$	0	0
$167$	13.8213	1.06953	0.534763	−	0.845002i	$-0.320401\pi$
$167$	13.8213	1.06953	0.534763	+	0.845002i	$0.320401\pi$
$168$	0	0
$169$	7.39678	0.568983
$170$	0	0
$171$	24.0081	1.83595
$172$	0	0
$173$	7.71437	0.586513	0.293256	−	0.956034i	$-0.405261\pi$
$173$	7.71437	0.586513	0.293256	+	0.956034i	$0.405261\pi$
$174$	0	0
$175$	−4.23374	−0.320041
$176$	0	0
$177$	−2.21053	−0.166154
$178$	0	0
$179$	−16.2736	−1.21635	−0.608175	−	0.793803i	$-0.708098\pi$
$179$	−16.2736	−1.21635	−0.608175	+	0.793803i	$0.708098\pi$
$180$	0	0
$181$	−5.61918	−0.417670	−0.208835	−	0.977951i	$-0.566967\pi$
$181$	−5.61918	−0.417670	−0.208835	+	0.977951i	$0.566967\pi$
$182$	0	0
$183$	−1.47369	−0.108938
$184$	0	0
$185$	−1.15744	−0.0850969
$186$	0	0
$187$	−8.70491	−0.636566
$188$	0	0
$189$	7.72725	0.562075
$190$	0	0
$191$	−7.58739	−0.549004	−0.274502	−	0.961587i	$-0.588513\pi$
$191$	−7.58739	−0.549004	−0.274502	+	0.961587i	$0.588513\pi$
$192$	0	0
$193$	−22.8008	−1.64124	−0.820619	−	0.571476i	$-0.806371\pi$
$193$	−22.8008	−1.64124	−0.820619	+	0.571476i	$0.806371\pi$
$194$	0	0
$195$	1.39606	0.0999736
$196$	0	0
$197$	−18.0268	−1.28436	−0.642178	−	0.766556i	$-0.721969\pi$
$197$	−18.0268	−1.28436	−0.642178	+	0.766556i	$0.721969\pi$
$198$	0	0
$199$	15.2806	1.08321	0.541606	−	0.840633i	$-0.317817\pi$
$199$	15.2806	1.08321	0.541606	+	0.840633i	$0.317817\pi$
$200$	0	0
$201$	−3.87712	−0.273471
$202$	0	0
$203$	−2.04710	−0.143678
$204$	0	0
$205$	−5.46044	−0.381374
$206$	0	0
$207$	7.97101	0.554023
$208$	0	0
$209$	17.0496	1.17934
$210$	0	0
$211$	−1.86062	−0.128091	−0.0640453	−	0.997947i	$-0.520400\pi$
$211$	−1.86062	−0.128091	−0.0640453	+	0.997947i	$0.520400\pi$
$212$	0	0
$213$	−0.915656	−0.0627397
$214$	0	0
$215$	8.29850	0.565953
$216$	0	0
$217$	7.10300	0.482183
$218$	0	0
$219$	−2.58335	−0.174566
$220$	0	0
$221$	−19.0601	−1.28212
$222$	0	0
$223$	7.99133	0.535139	0.267570	−	0.963539i	$-0.413780\pi$
$223$	7.99133	0.535139	0.267570	+	0.963539i	$0.413780\pi$
$224$	0	0
$225$	−2.90445	−0.193630
$226$	0	0
$227$	18.6562	1.23825	0.619127	−	0.785291i	$-0.287487\pi$
$227$	18.6562	1.23825	0.619127	+	0.785291i	$0.287487\pi$
$228$	0	0
$229$	−5.38017	−0.355532	−0.177766	−	0.984073i	$-0.556887\pi$
$229$	−5.38017	−0.355532	−0.177766	+	0.984073i	$0.556887\pi$
$230$	0	0
$231$	2.69939	0.177607
$232$	0	0
$233$	22.8183	1.49488	0.747438	−	0.664332i	$-0.231284\pi$
$233$	22.8183	1.49488	0.747438	+	0.664332i	$0.231284\pi$
$234$	0	0
$235$	10.6395	0.694045
$236$	0	0
$237$	−1.20800	−0.0784678
$238$	0	0
$239$	19.1026	1.23564	0.617821	−	0.786318i	$-0.288015\pi$
$239$	19.1026	1.23564	0.617821	+	0.786318i	$0.288015\pi$
$240$	0	0
$241$	3.14679	0.202702	0.101351	−	0.994851i	$-0.467683\pi$
$241$	3.14679	0.202702	0.101351	+	0.994851i	$0.467683\pi$
$242$	0	0
$243$	7.99452	0.512848
$244$	0	0
$245$	−10.9246	−0.697944
$246$	0	0
$247$	37.3315	2.37535
$248$	0	0
$249$	−1.66483	−0.105504
$250$	0	0
$251$	−3.16724	−0.199914	−0.0999572	−	0.994992i	$-0.531871\pi$
$251$	−3.16724	−0.199914	−0.0999572	+	0.994992i	$0.531871\pi$
$252$	0	0
$253$	5.66068	0.355884
$254$	0	0
$255$	−1.30457	−0.0816953
$256$	0	0
$257$	−11.6608	−0.727380	−0.363690	−	0.931520i	$-0.618483\pi$
$257$	−11.6608	−0.727380	−0.363690	+	0.931520i	$0.618483\pi$
$258$	0	0
$259$	−4.90031	−0.304491
$260$	0	0
$261$	−1.40436	−0.0869276
$262$	0	0
$263$	14.5425	0.896730	0.448365	−	0.893851i	$-0.352006\pi$
$263$	14.5425	0.896730	0.448365	+	0.893851i	$0.352006\pi$
$264$	0	0
$265$	12.1735	0.747809
$266$	0	0
$267$	−0.308615	−0.0188869
$268$	0	0
$269$	30.5743	1.86415	0.932074	−	0.362267i	$-0.117997\pi$
$269$	30.5743	1.86415	0.932074	+	0.362267i	$0.117997\pi$
$270$	0	0
$271$	16.1224	0.979367	0.489683	−	0.871900i	$-0.337112\pi$
$271$	16.1224	0.979367	0.489683	+	0.871900i	$0.337112\pi$
$272$	0	0
$273$	5.91053	0.357722
$274$	0	0
$275$	−2.06262	−0.124381
$276$	0	0
$277$	3.14558	0.188999	0.0944997	−	0.995525i	$-0.469875\pi$
$277$	3.14558	0.188999	0.0944997	+	0.995525i	$0.469875\pi$
$278$	0	0
$279$	4.87283	0.291729
$280$	0	0
$281$	1.17376	0.0700206	0.0350103	−	0.999387i	$-0.488854\pi$
$281$	1.17376	0.0700206	0.0350103	+	0.999387i	$0.488854\pi$
$282$	0	0
$283$	−25.5428	−1.51836	−0.759182	−	0.650879i	$-0.774401\pi$
$283$	−25.5428	−1.51836	−0.759182	+	0.650879i	$0.774401\pi$
$284$	0	0
$285$	2.55515	0.151354
$286$	0	0
$287$	−23.1181	−1.36462
$288$	0	0
$289$	0.811085	0.0477109
$290$	0	0
$291$	4.42781	0.259563
$292$	0	0
$293$	−19.3298	−1.12926	−0.564631	−	0.825344i	$-0.690981\pi$
$293$	−19.3298	−1.12926	−0.564631	+	0.825344i	$0.690981\pi$
$294$	0	0
$295$	7.15113	0.416355
$296$	0	0
$297$	3.76461	0.218445
$298$	0	0
$299$	12.3945	0.716795
$300$	0	0
$301$	35.1337	2.02507
$302$	0	0
$303$	−2.85670	−0.164113
$304$	0	0
$305$	4.76744	0.272983
$306$	0	0
$307$	−18.6901	−1.06670	−0.533349	−	0.845895i	$-0.679067\pi$
$307$	−18.6901	−1.06670	−0.533349	+	0.845895i	$0.679067\pi$
$308$	0	0
$309$	5.59963	0.318552
$310$	0	0
$311$	−24.6532	−1.39795	−0.698977	−	0.715144i	$-0.746361\pi$
$311$	−24.6532	−1.39795	−0.698977	+	0.715144i	$0.746361\pi$
$312$	0	0
$313$	8.80006	0.497409	0.248705	−	0.968579i	$-0.419995\pi$
$313$	8.80006	0.497409	0.248705	+	0.968579i	$0.419995\pi$
$314$	0	0
$315$	−12.2967	−0.692839
$316$	0	0
$317$	27.3281	1.53490	0.767449	−	0.641111i	$-0.221526\pi$
$317$	27.3281	1.53490	0.767449	+	0.641111i	$0.221526\pi$
$318$	0	0
$319$	−0.997317	−0.0558390
$320$	0	0
$321$	−2.50697	−0.139925
$322$	0	0
$323$	−34.8851	−1.94106
$324$	0	0
$325$	−4.51628	−0.250518
$326$	0	0
$327$	5.19591	0.287335
$328$	0	0
$329$	45.0449	2.48341
$330$	0	0
$331$	−24.9420	−1.37094	−0.685468	−	0.728102i	$-0.740403\pi$
$331$	−24.9420	−1.37094	−0.685468	+	0.728102i	$0.740403\pi$
$332$	0	0
$333$	−3.36173	−0.184222
$334$	0	0
$335$	12.5426	0.685276
$336$	0	0
$337$	−17.5644	−0.956792	−0.478396	−	0.878144i	$-0.658782\pi$
$337$	−17.5644	−0.956792	−0.478396	+	0.878144i	$0.658782\pi$
$338$	0	0
$339$	−3.46921	−0.188422
$340$	0	0
$341$	3.46048	0.187396
$342$	0	0
$343$	−16.6155	−0.897155
$344$	0	0
$345$	0.848343	0.0456732
$346$	0	0
$347$	11.0901	0.595348	0.297674	−	0.954668i	$-0.403789\pi$
$347$	11.0901	0.595348	0.297674	+	0.954668i	$0.403789\pi$
$348$	0	0
$349$	17.7307	0.949103	0.474552	−	0.880228i	$-0.342610\pi$
$349$	17.7307	0.949103	0.474552	+	0.880228i	$0.342610\pi$
$350$	0	0
$351$	8.24293	0.439975
$352$	0	0
$353$	−26.2416	−1.39670	−0.698351	−	0.715756i	$-0.746082\pi$
$353$	−26.2416	−1.39670	−0.698351	+	0.715756i	$0.746082\pi$
$354$	0	0
$355$	2.96217	0.157216
$356$	0	0
$357$	−5.52320	−0.292319
$358$	0	0
$359$	14.5447	0.767639	0.383819	−	0.923408i	$-0.374608\pi$
$359$	14.5447	0.767639	0.383819	+	0.923408i	$0.374608\pi$
$360$	0	0
$361$	49.3265	2.59613
$362$	0	0
$363$	−2.08518	−0.109443
$364$	0	0
$365$	8.35721	0.437436
$366$	0	0
$367$	−5.27479	−0.275342	−0.137671	−	0.990478i	$-0.543962\pi$
$367$	−5.27479	−0.275342	−0.137671	+	0.990478i	$0.543962\pi$
$368$	0	0
$369$	−15.8596	−0.825616
$370$	0	0
$371$	51.5392	2.67578
$372$	0	0
$373$	−11.6994	−0.605770	−0.302885	−	0.953027i	$-0.597950\pi$
$373$	−11.6994	−0.605770	−0.302885	+	0.953027i	$0.597950\pi$
$374$	0	0
$375$	−0.309116	−0.0159627
$376$	0	0
$377$	−2.18371	−0.112467
$378$	0	0
$379$	8.22949	0.422720	0.211360	−	0.977408i	$-0.432211\pi$
$379$	8.22949	0.422720	0.211360	+	0.977408i	$0.432211\pi$
$380$	0	0
$381$	5.23458	0.268175
$382$	0	0
$383$	−22.0455	−1.12647	−0.563237	−	0.826295i	$-0.690444\pi$
$383$	−22.0455	−1.12647	−0.563237	+	0.826295i	$0.690444\pi$
$384$	0	0
$385$	−8.73259	−0.445054
$386$	0	0
$387$	24.1026	1.22520
$388$	0	0
$389$	−15.1660	−0.768944	−0.384472	−	0.923137i	$-0.625617\pi$
$389$	−15.1660	−0.768944	−0.384472	+	0.923137i	$0.625617\pi$
$390$	0	0
$391$	−11.5823	−0.585742
$392$	0	0
$393$	−2.19334	−0.110639
$394$	0	0
$395$	3.90790	0.196628
$396$	0	0
$397$	−34.6742	−1.74025	−0.870124	−	0.492834i	$-0.835961\pi$
$397$	−34.6742	−1.74025	−0.870124	+	0.492834i	$0.835961\pi$
$398$	0	0
$399$	10.8178	0.541569
$400$	0	0
$401$	−1.00000	−0.0499376
$401$	−1.00000	−0.0499376
$402$	0	0
$403$	7.57702	0.377438
$404$	0	0
$405$	−8.14915	−0.404935
$406$	0	0
$407$	−2.38736	−0.118337
$408$	0	0
$409$	−9.84717	−0.486911	−0.243456	−	0.969912i	$-0.578281\pi$
$409$	−9.84717	−0.486911	−0.243456	+	0.969912i	$0.578281\pi$
$410$	0	0
$411$	2.98811	0.147393
$412$	0	0
$413$	30.2760	1.48979
$414$	0	0
$415$	5.38578	0.264377
$416$	0	0
$417$	−1.76180	−0.0862758
$418$	0	0
$419$	29.3420	1.43345	0.716725	−	0.697356i	$-0.245640\pi$
$419$	29.3420	1.43345	0.716725	+	0.697356i	$0.245640\pi$
$420$	0	0
$421$	−22.9165	−1.11688	−0.558440	−	0.829545i	$-0.688600\pi$
$421$	−22.9165	−1.11688	−0.558440	+	0.829545i	$0.688600\pi$
$422$	0	0
$423$	30.9019	1.50250
$424$	0	0
$425$	4.22032	0.204716
$426$	0	0
$427$	20.1841	0.976777
$428$	0	0
$429$	2.87953	0.139025
$430$	0	0
$431$	−21.7290	−1.04665	−0.523325	−	0.852133i	$-0.675309\pi$
$431$	−21.7290	−1.04665	−0.523325	+	0.852133i	$0.675309\pi$
$432$	0	0
$433$	−30.6405	−1.47249	−0.736245	−	0.676715i	$-0.763403\pi$
$433$	−30.6405	−1.47249	−0.736245	+	0.676715i	$0.763403\pi$
$434$	0	0
$435$	−0.149464	−0.00716624
$436$	0	0
$437$	22.6853	1.08518
$438$	0	0
$439$	−2.19603	−0.104811	−0.0524054	−	0.998626i	$-0.516689\pi$
$439$	−2.19603	−0.104811	−0.0524054	+	0.998626i	$0.516689\pi$
$440$	0	0
$441$	−31.7298	−1.51094
$442$	0	0
$443$	24.5468	1.16625	0.583127	−	0.812381i	$-0.301829\pi$
$443$	24.5468	1.16625	0.583127	+	0.812381i	$0.301829\pi$
$444$	0	0
$445$	0.998379	0.0473277
$446$	0	0
$447$	−2.12233	−0.100383
$448$	0	0
$449$	10.0957	0.476445	0.238223	−	0.971211i	$-0.423435\pi$
$449$	10.0957	0.476445	0.238223	+	0.971211i	$0.423435\pi$
$450$	0	0
$451$	−11.2628	−0.530345
$452$	0	0
$453$	5.49309	0.258088
$454$	0	0
$455$	−19.1208	−0.896395
$456$	0	0
$457$	−13.2033	−0.617625	−0.308813	−	0.951123i	$-0.599932\pi$
$457$	−13.2033	−0.617625	−0.308813	+	0.951123i	$0.599932\pi$
$458$	0	0
$459$	−7.70276	−0.359534
$460$	0	0
$461$	−25.5041	−1.18784	−0.593921	−	0.804523i	$-0.702421\pi$
$461$	−25.5041	−1.18784	−0.593921	+	0.804523i	$0.702421\pi$
$462$	0	0
$463$	−41.0312	−1.90688	−0.953440	−	0.301582i	$-0.902485\pi$
$463$	−41.0312	−1.90688	−0.953440	+	0.301582i	$0.902485\pi$
$464$	0	0
$465$	0.518608	0.0240499
$466$	0	0
$467$	−33.2023	−1.53642	−0.768210	−	0.640198i	$-0.778852\pi$
$467$	−33.2023	−1.53642	−0.768210	+	0.640198i	$0.778852\pi$
$468$	0	0
$469$	53.1022	2.45203
$470$	0	0
$471$	7.21838	0.332606
$472$	0	0
$473$	17.1166	0.787024
$474$	0	0
$475$	−8.26598	−0.379269
$476$	0	0
$477$	35.3571	1.61889
$478$	0	0
$479$	−26.0473	−1.19013	−0.595066	−	0.803677i	$-0.702874\pi$
$479$	−26.0473	−1.19013	−0.595066	+	0.803677i	$0.702874\pi$
$480$	0	0
$481$	−5.22734	−0.238346
$482$	0	0
$483$	3.59166	0.163426
$484$	0	0
$485$	−14.3241	−0.650423
$486$	0	0
$487$	−10.8837	−0.493187	−0.246593	−	0.969119i	$-0.579311\pi$
$487$	−10.8837	−0.493187	−0.246593	+	0.969119i	$0.579311\pi$
$488$	0	0
$489$	−2.14158	−0.0968458
$490$	0	0
$491$	−43.3226	−1.95512	−0.977560	−	0.210658i	$-0.932439\pi$
$491$	−43.3226	−1.95512	−0.977560	+	0.210658i	$0.932439\pi$
$492$	0	0
$493$	2.04061	0.0919043
$494$	0	0
$495$	−5.99077	−0.269265
$496$	0	0
$497$	12.5411	0.562544
$498$	0	0
$499$	28.7932	1.28896	0.644480	−	0.764621i	$-0.277074\pi$
$499$	28.7932	1.28896	0.644480	+	0.764621i	$0.277074\pi$
$500$	0	0
$501$	4.27239	0.190876
$502$	0	0
$503$	−38.8083	−1.73038	−0.865188	−	0.501448i	$-0.832801\pi$
$503$	−38.8083	−1.73038	−0.865188	+	0.501448i	$0.832801\pi$
$504$	0	0
$505$	9.24151	0.411242
$506$	0	0
$507$	2.28647	0.101546
$508$	0	0
$509$	−0.102805	−0.00455676	−0.00227838	−	0.999997i	$-0.500725\pi$
$509$	−0.102805	−0.00455676	−0.00227838	+	0.999997i	$0.500725\pi$
$510$	0	0
$511$	35.3822	1.56522
$512$	0	0
$513$	15.0867	0.666096
$514$	0	0
$515$	−18.1150	−0.798241
$516$	0	0
$517$	21.9452	0.965151
$518$	0	0
$519$	2.38464	0.104674
$520$	0	0
$521$	10.7339	0.470262	0.235131	−	0.971964i	$-0.424448\pi$
$521$	10.7339	0.470262	0.235131	+	0.971964i	$0.424448\pi$
$522$	0	0
$523$	−15.8280	−0.692109	−0.346055	−	0.938214i	$-0.612479\pi$
$523$	−15.8280	−0.692109	−0.346055	+	0.938214i	$0.612479\pi$
$524$	0	0
$525$	−1.30872	−0.0571171
$526$	0	0
$527$	−7.08048	−0.308431
$528$	0	0
$529$	−15.4682	−0.672530
$530$	0	0
$531$	20.7701	0.901344
$532$	0	0
$533$	−24.6609	−1.06818
$534$	0	0
$535$	8.11012	0.350631
$536$	0	0
$537$	−5.03045	−0.217080
$538$	0	0
$539$	−22.5332	−0.970573
$540$	0	0
$541$	−37.2788	−1.60274	−0.801371	−	0.598168i	$-0.795896\pi$
$541$	−37.2788	−1.60274	−0.801371	+	0.598168i	$0.795896\pi$
$542$	0	0
$543$	−1.73698	−0.0745409
$544$	0	0
$545$	−16.8089	−0.720016
$546$	0	0
$547$	−9.17836	−0.392438	−0.196219	−	0.980560i	$-0.562866\pi$
$547$	−9.17836	−0.392438	−0.196219	+	0.980560i	$0.562866\pi$
$548$	0	0
$549$	13.8468	0.590966
$550$	0	0
$551$	−3.99677	−0.170268
$552$	0	0
$553$	16.5450	0.703567
$554$	0	0
$555$	−0.357784	−0.0151871
$556$	0	0
$557$	−20.1147	−0.852289	−0.426144	−	0.904655i	$-0.640128\pi$
$557$	−20.1147	−0.852289	−0.426144	+	0.904655i	$0.640128\pi$
$558$	0	0
$559$	37.4784	1.58517
$560$	0	0
$561$	−2.69083	−0.113607
$562$	0	0
$563$	−24.4648	−1.03107	−0.515535	−	0.856869i	$-0.672407\pi$
$563$	−24.4648	−1.03107	−0.515535	+	0.856869i	$0.672407\pi$
$564$	0	0
$565$	11.2230	0.472155
$566$	0	0
$567$	−34.5014	−1.44892
$568$	0	0
$569$	−29.6100	−1.24132	−0.620658	−	0.784081i	$-0.713134\pi$
$569$	−29.6100	−1.24132	−0.620658	+	0.784081i	$0.713134\pi$
$570$	0	0
$571$	−3.39946	−0.142263	−0.0711315	−	0.997467i	$-0.522661\pi$
$571$	−3.39946	−0.142263	−0.0711315	+	0.997467i	$0.522661\pi$
$572$	0	0
$573$	−2.34538	−0.0979798
$574$	0	0
$575$	−2.74441	−0.114450
$576$	0	0
$577$	−34.9222	−1.45383	−0.726914	−	0.686728i	$-0.759046\pi$
$577$	−34.9222	−1.45383	−0.726914	+	0.686728i	$0.759046\pi$
$578$	0	0
$579$	−7.04810	−0.292909
$580$	0	0
$581$	22.8020	0.945986
$582$	0	0
$583$	25.1092	1.03992
$584$	0	0
$585$	−13.1173	−0.542333
$586$	0	0
$587$	10.2591	0.423439	0.211720	−	0.977330i	$-0.432094\pi$
$587$	10.2591	0.423439	0.211720	+	0.977330i	$0.432094\pi$
$588$	0	0
$589$	13.8679	0.571419
$590$	0	0
$591$	−5.57237	−0.229217
$592$	0	0
$593$	−36.2894	−1.49023	−0.745114	−	0.666937i	$-0.767605\pi$
$593$	−36.2894	−1.49023	−0.745114	+	0.666937i	$0.767605\pi$
$594$	0	0
$595$	17.8677	0.732506
$596$	0	0
$597$	4.72348	0.193319
$598$	0	0
$599$	27.1930	1.11108	0.555538	−	0.831491i	$-0.312512\pi$
$599$	27.1930	1.11108	0.555538	+	0.831491i	$0.312512\pi$
$600$	0	0
$601$	29.1306	1.18826	0.594130	−	0.804369i	$-0.297496\pi$
$601$	29.1306	1.18826	0.594130	+	0.804369i	$0.297496\pi$
$602$	0	0
$603$	36.4294	1.48352
$604$	0	0
$605$	6.74560	0.274248
$606$	0	0
$607$	−23.0113	−0.934002	−0.467001	−	0.884257i	$-0.654665\pi$
$607$	−23.0113	−0.934002	−0.467001	+	0.884257i	$0.654665\pi$
$608$	0	0
$609$	−0.632791	−0.0256420
$610$	0	0
$611$	48.0510	1.94393
$612$	0	0
$613$	17.5839	0.710206	0.355103	−	0.934827i	$-0.384446\pi$
$613$	17.5839	0.710206	0.355103	+	0.934827i	$0.384446\pi$
$614$	0	0
$615$	−1.68791	−0.0680632
$616$	0	0
$617$	9.69794	0.390424	0.195212	−	0.980761i	$-0.437460\pi$
$617$	9.69794	0.390424	0.195212	+	0.980761i	$0.437460\pi$
$618$	0	0
$619$	15.4633	0.621524	0.310762	−	0.950488i	$-0.399416\pi$
$619$	15.4633	0.621524	0.310762	+	0.950488i	$0.399416\pi$
$620$	0	0
$621$	5.00900	0.201004
$622$	0	0
$623$	4.22688	0.169346
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	5.27030	0.210475
$628$	0	0
$629$	4.88478	0.194769
$630$	0	0
$631$	−17.7521	−0.706699	−0.353350	−	0.935491i	$-0.614957\pi$
$631$	−17.7521	−0.706699	−0.353350	+	0.935491i	$0.614957\pi$
$632$	0	0
$633$	−0.575148	−0.0228601
$634$	0	0
$635$	−16.9340	−0.672006
$636$	0	0
$637$	−49.3383	−1.95486
$638$	0	0
$639$	8.60348	0.340348
$640$	0	0
$641$	34.9559	1.38068	0.690338	−	0.723487i	$-0.257462\pi$
$641$	34.9559	1.38068	0.690338	+	0.723487i	$0.257462\pi$
$642$	0	0
$643$	−25.7046	−1.01369	−0.506846	−	0.862037i	$-0.669189\pi$
$643$	−25.7046	−1.01369	−0.506846	+	0.862037i	$0.669189\pi$
$644$	0	0
$645$	2.56520	0.101005
$646$	0	0
$647$	1.44793	0.0569242	0.0284621	−	0.999595i	$-0.490939\pi$
$647$	1.44793	0.0569242	0.0284621	+	0.999595i	$0.490939\pi$
$648$	0	0
$649$	14.7500	0.578990
$650$	0	0
$651$	2.19565	0.0860544
$652$	0	0
$653$	−18.2786	−0.715298	−0.357649	−	0.933856i	$-0.616422\pi$
$653$	−18.2786	−0.715298	−0.357649	+	0.933856i	$0.616422\pi$
$654$	0	0
$655$	7.09553	0.277245
$656$	0	0
$657$	24.2731	0.946982
$658$	0	0
$659$	21.9140	0.853648	0.426824	−	0.904335i	$-0.359632\pi$
$659$	21.9140	0.853648	0.426824	+	0.904335i	$0.359632\pi$
$660$	0	0
$661$	−18.7423	−0.728991	−0.364495	−	0.931205i	$-0.618759\pi$
$661$	−18.7423	−0.728991	−0.364495	+	0.931205i	$0.618759\pi$
$662$	0	0
$663$	−5.89180	−0.228818
$664$	0	0
$665$	−34.9960	−1.35709
$666$	0	0
$667$	−1.32698	−0.0513808
$668$	0	0
$669$	2.47025	0.0955054
$670$	0	0
$671$	9.83341	0.379615
$672$	0	0
$673$	24.8480	0.957819	0.478910	−	0.877864i	$-0.341032\pi$
$673$	24.8480	0.957819	0.478910	+	0.877864i	$0.341032\pi$
$674$	0	0
$675$	−1.82516	−0.0702504
$676$	0	0
$677$	27.7843	1.06784	0.533918	−	0.845536i	$-0.320719\pi$
$677$	27.7843	1.06784	0.533918	+	0.845536i	$0.320719\pi$
$678$	0	0
$679$	−60.6445	−2.32732
$680$	0	0
$681$	5.76692	0.220989
$682$	0	0
$683$	−32.4623	−1.24214	−0.621068	−	0.783757i	$-0.713301\pi$
$683$	−32.4623	−1.24214	−0.621068	+	0.783757i	$0.713301\pi$
$684$	0	0
$685$	−9.66662	−0.369343
$686$	0	0
$687$	−1.66310	−0.0634511
$688$	0	0
$689$	54.9787	2.09452
$690$	0	0
$691$	20.9032	0.795194	0.397597	−	0.917560i	$-0.369844\pi$
$691$	20.9032	0.795194	0.397597	+	0.917560i	$0.369844\pi$
$692$	0	0
$693$	−25.3634	−0.963474
$694$	0	0
$695$	5.69948	0.216193
$696$	0	0
$697$	23.0448	0.872884
$698$	0	0
$699$	7.05350	0.266788
$700$	0	0
$701$	48.0977	1.81663	0.908313	−	0.418290i	$-0.137371\pi$
$701$	48.0977	1.81663	0.908313	+	0.418290i	$0.137371\pi$
$702$	0	0
$703$	−9.56741	−0.360842
$704$	0	0
$705$	3.28884	0.123865
$706$	0	0
$707$	39.1261	1.47149
$708$	0	0
$709$	−4.06952	−0.152834	−0.0764171	−	0.997076i	$-0.524348\pi$
$709$	−4.06952	−0.152834	−0.0764171	+	0.997076i	$0.524348\pi$
$710$	0	0
$711$	11.3503	0.425669
$712$	0	0
$713$	4.60434	0.172434
$714$	0	0
$715$	−9.31536	−0.348375
$716$	0	0
$717$	5.90491	0.220523
$718$	0	0
$719$	−38.5861	−1.43902	−0.719509	−	0.694483i	$-0.755633\pi$
$719$	−38.5861	−1.43902	−0.719509	+	0.694483i	$0.755633\pi$
$720$	0	0
$721$	−76.6941	−2.85624
$722$	0	0
$723$	0.972722	0.0361759
$724$	0	0
$725$	0.483520	0.0179575
$726$	0	0
$727$	−15.4258	−0.572111	−0.286055	−	0.958213i	$-0.592344\pi$
$727$	−15.4258	−0.572111	−0.286055	+	0.958213i	$0.592344\pi$
$728$	0	0
$729$	−21.9762	−0.813934
$730$	0	0
$731$	−35.0223	−1.29535
$732$	0	0
$733$	4.30290	0.158931	0.0794655	−	0.996838i	$-0.474679\pi$
$733$	4.30290	0.158931	0.0794655	+	0.996838i	$0.474679\pi$
$734$	0	0
$735$	−3.37696	−0.124561
$736$	0	0
$737$	25.8706	0.952957
$738$	0	0
$739$	−17.8776	−0.657638	−0.328819	−	0.944393i	$-0.606651\pi$
$739$	−17.8776	−0.657638	−0.328819	+	0.944393i	$0.606651\pi$
$740$	0	0
$741$	11.5398	0.423924
$742$	0	0
$743$	−0.384453	−0.0141042	−0.00705211	−	0.999975i	$-0.502245\pi$
$743$	−0.384453	−0.0141042	−0.00705211	+	0.999975i	$0.502245\pi$
$744$	0	0
$745$	6.86580	0.251544
$746$	0	0
$747$	15.6427	0.572337
$748$	0	0
$749$	34.3361	1.25462
$750$	0	0
$751$	28.2942	1.03247	0.516234	−	0.856447i	$-0.327333\pi$
$751$	28.2942	1.03247	0.516234	+	0.856447i	$0.327333\pi$
$752$	0	0
$753$	−0.979045	−0.0356784
$754$	0	0
$755$	−17.7703	−0.646728
$756$	0	0
$757$	−30.4246	−1.10580	−0.552900	−	0.833248i	$-0.686479\pi$
$757$	−30.4246	−1.10580	−0.552900	+	0.833248i	$0.686479\pi$
$758$	0	0
$759$	1.74981	0.0635140
$760$	0	0
$761$	8.24405	0.298847	0.149423	−	0.988773i	$-0.452258\pi$
$761$	8.24405	0.298847	0.149423	+	0.988773i	$0.452258\pi$
$762$	0	0
$763$	−71.1647	−2.57633
$764$	0	0
$765$	12.2577	0.443178
$766$	0	0
$767$	32.2965	1.16616
$768$	0	0
$769$	20.9736	0.756329	0.378164	−	0.925738i	$-0.376555\pi$
$769$	20.9736	0.756329	0.378164	+	0.925738i	$0.376555\pi$
$770$	0	0
$771$	−3.60454	−0.129814
$772$	0	0
$773$	−49.8508	−1.79301	−0.896504	−	0.443035i	$-0.853902\pi$
$773$	−49.8508	−1.79301	−0.896504	+	0.443035i	$0.853902\pi$
$774$	0	0
$775$	−1.67771	−0.0602652
$776$	0	0
$777$	−1.51477	−0.0543419
$778$	0	0
$779$	−45.1359	−1.61716
$780$	0	0
$781$	6.10984	0.218627
$782$	0	0
$783$	−0.882501	−0.0315380
$784$	0	0
$785$	−23.3517	−0.833457
$786$	0	0
$787$	15.8326	0.564372	0.282186	−	0.959360i	$-0.408941\pi$
$787$	15.8326	0.564372	0.282186	+	0.959360i	$0.408941\pi$
$788$	0	0
$789$	4.49533	0.160038
$790$	0	0
$791$	47.5153	1.68945
$792$	0	0
$793$	21.5311	0.764591
$794$	0	0
$795$	3.76301	0.133460
$796$	0	0
$797$	7.74360	0.274292	0.137146	−	0.990551i	$-0.456207\pi$
$797$	7.74360	0.274292	0.137146	+	0.990551i	$0.456207\pi$
$798$	0	0
$799$	−44.9021	−1.58852
$800$	0	0
$801$	2.89974	0.102457
$802$	0	0
$803$	17.2377	0.608306
$804$	0	0
$805$	−11.6191	−0.409521
$806$	0	0
$807$	9.45102	0.332692
$808$	0	0
$809$	−5.97581	−0.210098	−0.105049	−	0.994467i	$-0.533500\pi$
$809$	−5.97581	−0.210098	−0.105049	+	0.994467i	$0.533500\pi$
$810$	0	0
$811$	11.8327	0.415504	0.207752	−	0.978182i	$-0.433385\pi$
$811$	11.8327	0.415504	0.207752	+	0.978182i	$0.433385\pi$
$812$	0	0
$813$	4.98370	0.174786
$814$	0	0
$815$	6.92809	0.242680
$816$	0	0
$817$	68.5953	2.39985
$818$	0	0
$819$	−55.5352	−1.94056
$820$	0	0
$821$	2.00901	0.0701149	0.0350575	−	0.999385i	$-0.488839\pi$
$821$	2.00901	0.0701149	0.0350575	+	0.999385i	$0.488839\pi$
$822$	0	0
$823$	5.13252	0.178908	0.0894542	−	0.995991i	$-0.471488\pi$
$823$	5.13252	0.178908	0.0894542	+	0.995991i	$0.471488\pi$
$824$	0	0
$825$	−0.637589	−0.0221980
$826$	0	0
$827$	23.1543	0.805153	0.402576	−	0.915386i	$-0.368115\pi$
$827$	23.1543	0.805153	0.402576	+	0.915386i	$0.368115\pi$
$828$	0	0
$829$	−13.3337	−0.463100	−0.231550	−	0.972823i	$-0.574380\pi$
$829$	−13.3337	−0.463100	−0.231550	+	0.972823i	$0.574380\pi$
$830$	0	0
$831$	0.972348	0.0337304
$832$	0	0
$833$	46.1051	1.59745
$834$	0	0
$835$	−13.8213	−0.478306
$836$	0	0
$837$	3.06209	0.105841
$838$	0	0
$839$	30.6651	1.05868	0.529338	−	0.848411i	$-0.322440\pi$
$839$	30.6651	1.05868	0.529338	+	0.848411i	$0.322440\pi$
$840$	0	0
$841$	−28.7662	−0.991938
$842$	0	0
$843$	0.362828	0.0124965
$844$	0	0
$845$	−7.39678	−0.254457
$846$	0	0
$847$	28.5591	0.981303
$848$	0	0
$849$	−7.89570	−0.270980
$850$	0	0
$851$	−3.17650	−0.108889
$852$	0	0
$853$	29.4801	1.00938	0.504689	−	0.863301i	$-0.331607\pi$
$853$	29.4801	1.00938	0.504689	+	0.863301i	$0.331607\pi$
$854$	0	0
$855$	−24.0081	−0.821060
$856$	0	0
$857$	43.0671	1.47114	0.735571	−	0.677447i	$-0.236914\pi$
$857$	43.0671	1.47114	0.735571	+	0.677447i	$0.236914\pi$
$858$	0	0
$859$	35.2299	1.20203	0.601015	−	0.799238i	$-0.294763\pi$
$859$	35.2299	1.20203	0.601015	+	0.799238i	$0.294763\pi$
$860$	0	0
$861$	−7.14618	−0.243541
$862$	0	0
$863$	47.8606	1.62919	0.814596	−	0.580028i	$-0.196959\pi$
$863$	47.8606	1.62919	0.814596	+	0.580028i	$0.196959\pi$
$864$	0	0
$865$	−7.71437	−0.262297
$866$	0	0
$867$	0.250720	0.00851489
$868$	0	0
$869$	8.06051	0.273434
$870$	0	0
$871$	56.6459	1.91937
$872$	0	0
$873$	−41.6036	−1.40807
$874$	0	0
$875$	4.23374	0.143127
$876$	0	0
$877$	12.9798	0.438298	0.219149	−	0.975691i	$-0.429672\pi$
$877$	12.9798	0.438298	0.219149	+	0.975691i	$0.429672\pi$
$878$	0	0
$879$	−5.97517	−0.201537
$880$	0	0
$881$	−3.13613	−0.105659	−0.0528295	−	0.998604i	$-0.516824\pi$
$881$	−3.13613	−0.105659	−0.0528295	+	0.998604i	$0.516824\pi$
$882$	0	0
$883$	−38.0676	−1.28108	−0.640538	−	0.767926i	$-0.721289\pi$
$883$	−38.0676	−1.28108	−0.640538	+	0.767926i	$0.721289\pi$
$884$	0	0
$885$	2.21053	0.0743061
$886$	0	0
$887$	−22.7164	−0.762743	−0.381372	−	0.924422i	$-0.624548\pi$
$887$	−22.7164	−0.762743	−0.381372	+	0.924422i	$0.624548\pi$
$888$	0	0
$889$	−71.6942	−2.40455
$890$	0	0
$891$	−16.8086	−0.563109
$892$	0	0
$893$	87.9460	2.94300
$894$	0	0
$895$	16.2736	0.543968
$896$	0	0
$897$	3.83135	0.127925
$898$	0	0
$899$	−0.811207	−0.0270553
$900$	0	0
$901$	−51.3758	−1.71158
$902$	0	0
$903$	10.8604	0.361411
$904$	0	0
$905$	5.61918	0.186788
$906$	0	0
$907$	−1.56449	−0.0519482	−0.0259741	−	0.999663i	$-0.508269\pi$
$907$	−1.56449	−0.0519482	−0.0259741	+	0.999663i	$0.508269\pi$
$908$	0	0
$909$	26.8415	0.890275
$910$	0	0
$911$	6.51833	0.215962	0.107981	−	0.994153i	$-0.465561\pi$
$911$	6.51833	0.215962	0.107981	+	0.994153i	$0.465561\pi$
$912$	0	0
$913$	11.1088	0.367648
$914$	0	0
$915$	1.47369	0.0487188
$916$	0	0
$917$	30.0406	0.992029
$918$	0	0
$919$	6.03420	0.199050	0.0995249	−	0.995035i	$-0.468268\pi$
$919$	6.03420	0.199050	0.0995249	+	0.995035i	$0.468268\pi$
$920$	0	0
$921$	−5.77740	−0.190372
$922$	0	0
$923$	13.3780	0.440342
$924$	0	0
$925$	1.15744	0.0380565
$926$	0	0
$927$	−52.6140	−1.72807
$928$	0	0
$929$	−12.2622	−0.402311	−0.201155	−	0.979559i	$-0.564470\pi$
$929$	−12.2622	−0.402311	−0.201155	+	0.979559i	$0.564470\pi$
$930$	0	0
$931$	−90.3022	−2.95953
$932$	0	0
$933$	−7.62070	−0.249490
$934$	0	0
$935$	8.70491	0.284681
$936$	0	0
$937$	−54.1471	−1.76891	−0.884454	−	0.466627i	$-0.845469\pi$
$937$	−54.1471	−1.76891	−0.884454	+	0.466627i	$0.845469\pi$
$938$	0	0
$939$	2.72024	0.0887718
$940$	0	0
$941$	1.41222	0.0460371	0.0230186	−	0.999735i	$-0.492672\pi$
$941$	1.41222	0.0460371	0.0230186	+	0.999735i	$0.492672\pi$
$942$	0	0
$943$	−14.9857	−0.488002
$944$	0	0
$945$	−7.72725	−0.251368
$946$	0	0
$947$	−47.5098	−1.54386	−0.771930	−	0.635707i	$-0.780709\pi$
$947$	−47.5098	−1.54386	−0.771930	+	0.635707i	$0.780709\pi$
$948$	0	0
$949$	37.7435	1.22521
$950$	0	0
$951$	8.44755	0.273930
$952$	0	0
$953$	−3.53119	−0.114386	−0.0571932	−	0.998363i	$-0.518215\pi$
$953$	−3.53119	−0.114386	−0.0571932	+	0.998363i	$0.518215\pi$
$954$	0	0
$955$	7.58739	0.245522
$956$	0	0
$957$	−0.308287	−0.00996550
$958$	0	0
$959$	−40.9260	−1.32157
$960$	0	0
$961$	−28.1853	−0.909203
$962$	0	0
$963$	23.5554	0.759063
$964$	0	0
$965$	22.8008	0.733984
$966$	0	0
$967$	−53.0185	−1.70496	−0.852480	−	0.522760i	$-0.824903\pi$
$967$	−53.0185	−1.70496	−0.852480	+	0.522760i	$0.824903\pi$
$968$	0	0
$969$	−10.7835	−0.346417
$970$	0	0
$971$	26.9777	0.865756	0.432878	−	0.901453i	$-0.357498\pi$
$971$	26.9777	0.865756	0.432878	+	0.901453i	$0.357498\pi$
$972$	0	0
$973$	24.1301	0.773576
$974$	0	0
$975$	−1.39606	−0.0447095
$976$	0	0
$977$	37.5654	1.20182	0.600912	−	0.799315i	$-0.294804\pi$
$977$	37.5654	1.20182	0.600912	+	0.799315i	$0.294804\pi$
$978$	0	0
$979$	2.05928	0.0658147
$980$	0	0
$981$	−48.8207	−1.55872
$982$	0	0
$983$	15.0837	0.481095	0.240548	−	0.970637i	$-0.422673\pi$
$983$	15.0837	0.481095	0.240548	+	0.970637i	$0.422673\pi$
$984$	0	0
$985$	18.0268	0.574381
$986$	0	0
$987$	13.9241	0.443209
$988$	0	0
$989$	22.7745	0.724188
$990$	0	0
$991$	1.30347	0.0414061	0.0207031	−	0.999786i	$-0.493410\pi$
$991$	1.30347	0.0414061	0.0207031	+	0.999786i	$0.493410\pi$
$992$	0	0
$993$	−7.70998	−0.244669
$994$	0	0
$995$	−15.2806	−0.484427
$996$	0	0
$997$	−14.4320	−0.457067	−0.228533	−	0.973536i	$-0.573393\pi$
$997$	−14.4320	−0.457067	−0.228533	+	0.973536i	$0.573393\pi$
$998$	0	0
$999$	−2.11252	−0.0668371

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.e.1.20	✓	35

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.e.1.20	✓	35	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)

\(f(q)\)	\(=\)	\(q+0.309116 q^{3} -1.00000 q^{5} -4.23374 q^{7} -2.90445 q^{9} +O(q^{10})\)	\(q+0.309116 q^{3} -1.00000 q^{5} -4.23374 q^{7} -2.90445 q^{9} -2.06262 q^{11} -4.51628 q^{13} -0.309116 q^{15} +4.22032 q^{17} -8.26598 q^{19} -1.30872 q^{21} -2.74441 q^{23} +1.00000 q^{25} -1.82516 q^{27} +0.483520 q^{29} -1.67771 q^{31} -0.637589 q^{33} +4.23374 q^{35} +1.15744 q^{37} -1.39606 q^{39} +5.46044 q^{41} -8.29850 q^{43} +2.90445 q^{45} -10.6395 q^{47} +10.9246 q^{49} +1.30457 q^{51} -12.1735 q^{53} +2.06262 q^{55} -2.55515 q^{57} -7.15113 q^{59} -4.76744 q^{61} +12.2967 q^{63} +4.51628 q^{65} -12.5426 q^{67} -0.848343 q^{69} -2.96217 q^{71} -8.35721 q^{73} +0.309116 q^{75} +8.73259 q^{77} -3.90790 q^{79} +8.14915 q^{81} -5.38578 q^{83} -4.22032 q^{85} +0.149464 q^{87} -0.998379 q^{89} +19.1208 q^{91} -0.518608 q^{93} +8.26598 q^{95} +14.3241 q^{97} +5.99077 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9}+O(q^{10}) \) 35 * q - q^3 - 35 * q^5 + 6 * q^7 + 52 * q^9	\( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9} - 2 q^{11} - q^{13} + q^{15} + 18 q^{17} + 2 q^{19} + 12 q^{21} + 13 q^{23} + 35 q^{25} - 7 q^{27} + 25 q^{29} + 13 q^{31} + 14 q^{33} - 6 q^{35} - 19 q^{37} - 3 q^{39} + 24 q^{41} - 5 q^{43} - 52 q^{45} + 19 q^{47} + 55 q^{49} + 41 q^{53} + 2 q^{55} + 14 q^{57} + 3 q^{59} + 13 q^{61} + 70 q^{63} + q^{65} - 17 q^{67} + 64 q^{69} + 17 q^{71} - 63 q^{73} - q^{75} + 54 q^{77} + 11 q^{79} + 107 q^{81} - 8 q^{83} - 18 q^{85} + 36 q^{87} + 38 q^{89} - 27 q^{91} + q^{93} - 2 q^{95} - 54 q^{97} - 51 q^{99}+O(q^{100}) \) 35 * q - q^3 - 35 * q^5 + 6 * q^7 + 52 * q^9 - 2 * q^11 - q^13 + q^15 + 18 * q^17 + 2 * q^19 + 12 * q^21 + 13 * q^23 + 35 * q^25 - 7 * q^27 + 25 * q^29 + 13 * q^31 + 14 * q^33 - 6 * q^35 - 19 * q^37 - 3 * q^39 + 24 * q^41 - 5 * q^43 - 52 * q^45 + 19 * q^47 + 55 * q^49 + 41 * q^53 + 2 * q^55 + 14 * q^57 + 3 * q^59 + 13 * q^61 + 70 * q^63 + q^65 - 17 * q^67 + 64 * q^69 + 17 * q^71 - 63 * q^73 - q^75 + 54 * q^77 + 11 * q^79 + 107 * q^81 - 8 * q^83 - 18 * q^85 + 36 * q^87 + 38 * q^89 - 27 * q^91 + q^93 - 2 * q^95 - 54 * q^97 - 51 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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