LMFDB - Embedded newform 8020.2.a.c.1.17

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			1.17
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)

$f(q)$	$=$	$q+0.910150 q^{3} -1.00000 q^{5} +1.40834 q^{7} -2.17163 q^{9} +O(q^{10})$	$q+0.910150 q^{3} -1.00000 q^{5} +1.40834 q^{7} -2.17163 q^{9} -2.72560 q^{11} +4.96710 q^{13} -0.910150 q^{15} +1.31929 q^{17} -1.68185 q^{19} +1.28180 q^{21} +4.30276 q^{23} +1.00000 q^{25} -4.70696 q^{27} -5.98564 q^{29} +0.173134 q^{31} -2.48070 q^{33} -1.40834 q^{35} -2.08751 q^{37} +4.52081 q^{39} -1.57854 q^{41} -10.3754 q^{43} +2.17163 q^{45} -2.61807 q^{47} -5.01659 q^{49} +1.20075 q^{51} +2.01074 q^{53} +2.72560 q^{55} -1.53074 q^{57} +3.37023 q^{59} +3.01769 q^{61} -3.05838 q^{63} -4.96710 q^{65} -1.07701 q^{67} +3.91616 q^{69} +0.238719 q^{71} -11.9587 q^{73} +0.910150 q^{75} -3.83856 q^{77} +4.59577 q^{79} +2.23084 q^{81} +1.18072 q^{83} -1.31929 q^{85} -5.44784 q^{87} +1.48814 q^{89} +6.99536 q^{91} +0.157578 q^{93} +1.68185 q^{95} -14.7986 q^{97} +5.91898 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.910150	0.525476	0.262738	−	0.964867i	$-0.415375\pi$
$3$	0.910150	0.525476	0.262738	+	0.964867i	$0.415375\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.40834	0.532302	0.266151	−	0.963931i	$-0.414248\pi$
$7$	1.40834	0.532302	0.266151	+	0.963931i	$0.414248\pi$
$8$	0	0
$9$	−2.17163	−0.723875
$10$	0	0
$11$	−2.72560	−0.821798	−0.410899	−	0.911681i	$-0.634785\pi$
$11$	−2.72560	−0.821798	−0.410899	+	0.911681i	$0.634785\pi$
$12$	0	0
$13$	4.96710	1.37763	0.688813	−	0.724939i	$-0.258132\pi$
$13$	4.96710	1.37763	0.688813	+	0.724939i	$0.258132\pi$
$14$	0	0
$15$	−0.910150	−0.235000
$16$	0	0
$17$	1.31929	0.319976	0.159988	−	0.987119i	$-0.448855\pi$
$17$	1.31929	0.319976	0.159988	+	0.987119i	$0.448855\pi$
$18$	0	0
$19$	−1.68185	−0.385844	−0.192922	−	0.981214i	$-0.561796\pi$
$19$	−1.68185	−0.385844	−0.192922	+	0.981214i	$0.561796\pi$
$20$	0	0
$21$	1.28180	0.279711
$22$	0	0
$23$	4.30276	0.897188	0.448594	−	0.893736i	$-0.351925\pi$
$23$	4.30276	0.897188	0.448594	+	0.893736i	$0.351925\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.70696	−0.905854
$28$	0	0
$29$	−5.98564	−1.11151	−0.555753	−	0.831347i	$-0.687570\pi$
$29$	−5.98564	−1.11151	−0.555753	+	0.831347i	$0.687570\pi$
$30$	0	0
$31$	0.173134	0.0310957	0.0155479	−	0.999879i	$-0.495051\pi$
$31$	0.173134	0.0310957	0.0155479	+	0.999879i	$0.495051\pi$
$32$	0	0
$33$	−2.48070	−0.431835
$34$	0	0
$35$	−1.40834	−0.238052
$36$	0	0
$37$	−2.08751	−0.343185	−0.171592	−	0.985168i	$-0.554891\pi$
$37$	−2.08751	−0.343185	−0.171592	+	0.985168i	$0.554891\pi$
$38$	0	0
$39$	4.52081	0.723909
$40$	0	0
$41$	−1.57854	−0.246526	−0.123263	−	0.992374i	$-0.539336\pi$
$41$	−1.57854	−0.246526	−0.123263	+	0.992374i	$0.539336\pi$
$42$	0	0
$43$	−10.3754	−1.58223	−0.791116	−	0.611666i	$-0.790500\pi$
$43$	−10.3754	−1.58223	−0.791116	+	0.611666i	$0.790500\pi$
$44$	0	0
$45$	2.17163	0.323727
$46$	0	0
$47$	−2.61807	−0.381885	−0.190942	−	0.981601i	$-0.561154\pi$
$47$	−2.61807	−0.381885	−0.190942	+	0.981601i	$0.561154\pi$
$48$	0	0
$49$	−5.01659	−0.716655
$50$	0	0
$51$	1.20075	0.168139
$52$	0	0
$53$	2.01074	0.276196	0.138098	−	0.990419i	$-0.455901\pi$
$53$	2.01074	0.276196	0.138098	+	0.990419i	$0.455901\pi$
$54$	0	0
$55$	2.72560	0.367519
$56$	0	0
$57$	−1.53074	−0.202751
$58$	0	0
$59$	3.37023	0.438767	0.219383	−	0.975639i	$-0.429595\pi$
$59$	3.37023	0.438767	0.219383	+	0.975639i	$0.429595\pi$
$60$	0	0
$61$	3.01769	0.386375	0.193188	−	0.981162i	$-0.438117\pi$
$61$	3.01769	0.386375	0.193188	+	0.981162i	$0.438117\pi$
$62$	0	0
$63$	−3.05838	−0.385320
$64$	0	0
$65$	−4.96710	−0.616093
$66$	0	0
$67$	−1.07701	−0.131578	−0.0657891	−	0.997834i	$-0.520956\pi$
$67$	−1.07701	−0.131578	−0.0657891	+	0.997834i	$0.520956\pi$
$68$	0	0
$69$	3.91616	0.471450
$70$	0	0
$71$	0.238719	0.0283307	0.0141654	−	0.999900i	$-0.495491\pi$
$71$	0.238719	0.0283307	0.0141654	+	0.999900i	$0.495491\pi$
$72$	0	0
$73$	−11.9587	−1.39967	−0.699833	−	0.714307i	$-0.746742\pi$
$73$	−11.9587	−1.39967	−0.699833	+	0.714307i	$0.746742\pi$
$74$	0	0
$75$	0.910150	0.105095
$76$	0	0
$77$	−3.83856	−0.437444
$78$	0	0
$79$	4.59577	0.517064	0.258532	−	0.966003i	$-0.416761\pi$
$79$	4.59577	0.517064	0.258532	+	0.966003i	$0.416761\pi$
$80$	0	0
$81$	2.23084	0.247871
$82$	0	0
$83$	1.18072	0.129601	0.0648004	−	0.997898i	$-0.479359\pi$
$83$	1.18072	0.129601	0.0648004	+	0.997898i	$0.479359\pi$
$84$	0	0
$85$	−1.31929	−0.143097
$86$	0	0
$87$	−5.44784	−0.584069
$88$	0	0
$89$	1.48814	0.157743	0.0788715	−	0.996885i	$-0.474868\pi$
$89$	1.48814	0.157743	0.0788715	+	0.996885i	$0.474868\pi$
$90$	0	0
$91$	6.99536	0.733313
$92$	0	0
$93$	0.157578	0.0163400
$94$	0	0
$95$	1.68185	0.172555
$96$	0	0
$97$	−14.7986	−1.50257	−0.751287	−	0.659976i	$-0.770567\pi$
$97$	−14.7986	−1.50257	−0.751287	+	0.659976i	$0.770567\pi$
$98$	0	0
$99$	5.91898	0.594880
$100$	0	0
$101$	6.09561	0.606536	0.303268	−	0.952905i	$-0.401922\pi$
$101$	6.09561	0.606536	0.303268	+	0.952905i	$0.401922\pi$
$102$	0	0
$103$	19.0756	1.87957	0.939785	−	0.341766i	$-0.111025\pi$
$103$	19.0756	1.87957	0.939785	+	0.341766i	$0.111025\pi$
$104$	0	0
$105$	−1.28180	−0.125091
$106$	0	0
$107$	−3.44412	−0.332956	−0.166478	−	0.986045i	$-0.553239\pi$
$107$	−3.44412	−0.332956	−0.166478	+	0.986045i	$0.553239\pi$
$108$	0	0
$109$	17.5293	1.67900	0.839499	−	0.543361i	$-0.182848\pi$
$109$	17.5293	1.67900	0.839499	+	0.543361i	$0.182848\pi$
$110$	0	0
$111$	−1.89995	−0.180335
$112$	0	0
$113$	−0.663647	−0.0624306	−0.0312153	−	0.999513i	$-0.509938\pi$
$113$	−0.663647	−0.0624306	−0.0312153	+	0.999513i	$0.509938\pi$
$114$	0	0
$115$	−4.30276	−0.401235
$116$	0	0
$117$	−10.7867	−0.997230
$118$	0	0
$119$	1.85801	0.170323
$120$	0	0
$121$	−3.57112	−0.324648
$122$	0	0
$123$	−1.43670	−0.129543
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−14.2684	−1.26612	−0.633058	−	0.774104i	$-0.718201\pi$
$127$	−14.2684	−1.26612	−0.633058	+	0.774104i	$0.718201\pi$
$128$	0	0
$129$	−9.44317	−0.831425
$130$	0	0
$131$	−5.18673	−0.453166	−0.226583	−	0.973992i	$-0.572756\pi$
$131$	−5.18673	−0.453166	−0.226583	+	0.973992i	$0.572756\pi$
$132$	0	0
$133$	−2.36862	−0.205385
$134$	0	0
$135$	4.70696	0.405110
$136$	0	0
$137$	−2.96049	−0.252932	−0.126466	−	0.991971i	$-0.540363\pi$
$137$	−2.96049	−0.252932	−0.126466	+	0.991971i	$0.540363\pi$
$138$	0	0
$139$	−13.0029	−1.10289	−0.551444	−	0.834212i	$-0.685923\pi$
$139$	−13.0029	−1.10289	−0.551444	+	0.834212i	$0.685923\pi$
$140$	0	0
$141$	−2.38284	−0.200671
$142$	0	0
$143$	−13.5383	−1.13213
$144$	0	0
$145$	5.98564	0.497081
$146$	0	0
$147$	−4.56585	−0.376585
$148$	0	0
$149$	−8.13362	−0.666332	−0.333166	−	0.942868i	$-0.608117\pi$
$149$	−8.13362	−0.666332	−0.333166	+	0.942868i	$0.608117\pi$
$150$	0	0
$151$	5.42500	0.441481	0.220740	−	0.975333i	$-0.429153\pi$
$151$	5.42500	0.441481	0.220740	+	0.975333i	$0.429153\pi$
$152$	0	0
$153$	−2.86501	−0.231622
$154$	0	0
$155$	−0.173134	−0.0139064
$156$	0	0
$157$	12.6459	1.00925	0.504627	−	0.863338i	$-0.331630\pi$
$157$	12.6459	1.00925	0.504627	+	0.863338i	$0.331630\pi$
$158$	0	0
$159$	1.83007	0.145134
$160$	0	0
$161$	6.05974	0.477575
$162$	0	0
$163$	9.36001	0.733133	0.366566	−	0.930392i	$-0.380533\pi$
$163$	9.36001	0.733133	0.366566	+	0.930392i	$0.380533\pi$
$164$	0	0
$165$	2.48070	0.193122
$166$	0	0
$167$	−6.29438	−0.487073	−0.243537	−	0.969892i	$-0.578308\pi$
$167$	−6.29438	−0.487073	−0.243537	+	0.969892i	$0.578308\pi$
$168$	0	0
$169$	11.6721	0.897855
$170$	0	0
$171$	3.65236	0.279303
$172$	0	0
$173$	−16.9019	−1.28503	−0.642513	−	0.766275i	$-0.722108\pi$
$173$	−16.9019	−1.28503	−0.642513	+	0.766275i	$0.722108\pi$
$174$	0	0
$175$	1.40834	0.106460
$176$	0	0
$177$	3.06742	0.230561
$178$	0	0
$179$	−3.58148	−0.267692	−0.133846	−	0.991002i	$-0.542733\pi$
$179$	−3.58148	−0.267692	−0.133846	+	0.991002i	$0.542733\pi$
$180$	0	0
$181$	13.1076	0.974280	0.487140	−	0.873324i	$-0.338040\pi$
$181$	13.1076	0.974280	0.487140	+	0.873324i	$0.338040\pi$
$182$	0	0
$183$	2.74655	0.203031
$184$	0	0
$185$	2.08751	0.153477
$186$	0	0
$187$	−3.59586	−0.262955
$188$	0	0
$189$	−6.62898	−0.482188
$190$	0	0
$191$	13.5106	0.977590	0.488795	−	0.872399i	$-0.337437\pi$
$191$	13.5106	0.977590	0.488795	+	0.872399i	$0.337437\pi$
$192$	0	0
$193$	−24.0813	−1.73341	−0.866704	−	0.498824i	$-0.833766\pi$
$193$	−24.0813	−1.73341	−0.866704	+	0.498824i	$0.833766\pi$
$194$	0	0
$195$	−4.52081	−0.323742
$196$	0	0
$197$	−20.0530	−1.42872	−0.714359	−	0.699780i	$-0.753281\pi$
$197$	−20.0530	−1.42872	−0.714359	+	0.699780i	$0.753281\pi$
$198$	0	0
$199$	−19.9413	−1.41360	−0.706801	−	0.707413i	$-0.749862\pi$
$199$	−19.9413	−1.41360	−0.706801	+	0.707413i	$0.749862\pi$
$200$	0	0
$201$	−0.980245	−0.0691411
$202$	0	0
$203$	−8.42981	−0.591656
$204$	0	0
$205$	1.57854	0.110250
$206$	0	0
$207$	−9.34399	−0.649452
$208$	0	0
$209$	4.58405	0.317086
$210$	0	0
$211$	−10.1730	−0.700337	−0.350169	−	0.936687i	$-0.613876\pi$
$211$	−10.1730	−0.700337	−0.350169	+	0.936687i	$0.613876\pi$
$212$	0	0
$213$	0.217270	0.0148871
$214$	0	0
$215$	10.3754	0.707596
$216$	0	0
$217$	0.243830	0.0165523
$218$	0	0
$219$	−10.8843	−0.735490
$220$	0	0
$221$	6.55306	0.440807
$222$	0	0
$223$	5.14739	0.344695	0.172347	−	0.985036i	$-0.444865\pi$
$223$	5.14739	0.344695	0.172347	+	0.985036i	$0.444865\pi$
$224$	0	0
$225$	−2.17163	−0.144775
$226$	0	0
$227$	−22.8204	−1.51465	−0.757323	−	0.653041i	$-0.773493\pi$
$227$	−22.8204	−1.51465	−0.757323	+	0.653041i	$0.773493\pi$
$228$	0	0
$229$	−4.05583	−0.268017	−0.134008	−	0.990980i	$-0.542785\pi$
$229$	−4.05583	−0.268017	−0.134008	+	0.990980i	$0.542785\pi$
$230$	0	0
$231$	−3.49367	−0.229866
$232$	0	0
$233$	−25.6247	−1.67873	−0.839364	−	0.543570i	$-0.817072\pi$
$233$	−25.6247	−1.67873	−0.839364	+	0.543570i	$0.817072\pi$
$234$	0	0
$235$	2.61807	0.170784
$236$	0	0
$237$	4.18284	0.271705
$238$	0	0
$239$	−23.0528	−1.49116	−0.745582	−	0.666414i	$-0.767828\pi$
$239$	−23.0528	−1.49116	−0.745582	+	0.666414i	$0.767828\pi$
$240$	0	0
$241$	1.59104	0.102488	0.0512440	−	0.998686i	$-0.483681\pi$
$241$	1.59104	0.102488	0.0512440	+	0.998686i	$0.483681\pi$
$242$	0	0
$243$	16.1513	1.03610
$244$	0	0
$245$	5.01659	0.320498
$246$	0	0
$247$	−8.35394	−0.531548
$248$	0	0
$249$	1.07463	0.0681020
$250$	0	0
$251$	−17.9822	−1.13502	−0.567512	−	0.823365i	$-0.692094\pi$
$251$	−17.9822	−1.13502	−0.567512	+	0.823365i	$0.692094\pi$
$252$	0	0
$253$	−11.7276	−0.737308
$254$	0	0
$255$	−1.20075	−0.0751942
$256$	0	0
$257$	−27.1656	−1.69454	−0.847272	−	0.531159i	$-0.821757\pi$
$257$	−27.1656	−1.69454	−0.847272	+	0.531159i	$0.821757\pi$
$258$	0	0
$259$	−2.93992	−0.182678
$260$	0	0
$261$	12.9986	0.804592
$262$	0	0
$263$	−9.50076	−0.585842	−0.292921	−	0.956137i	$-0.594627\pi$
$263$	−9.50076	−0.585842	−0.292921	+	0.956137i	$0.594627\pi$
$264$	0	0
$265$	−2.01074	−0.123519
$266$	0	0
$267$	1.35444	0.0828901
$268$	0	0
$269$	2.25679	0.137599	0.0687995	−	0.997631i	$-0.478083\pi$
$269$	2.25679	0.137599	0.0687995	+	0.997631i	$0.478083\pi$
$270$	0	0
$271$	−3.67960	−0.223520	−0.111760	−	0.993735i	$-0.535649\pi$
$271$	−3.67960	−0.223520	−0.111760	+	0.993735i	$0.535649\pi$
$272$	0	0
$273$	6.36683	0.385338
$274$	0	0
$275$	−2.72560	−0.164360
$276$	0	0
$277$	19.2379	1.15589	0.577947	−	0.816075i	$-0.303854\pi$
$277$	19.2379	1.15589	0.577947	+	0.816075i	$0.303854\pi$
$278$	0	0
$279$	−0.375981	−0.0225094
$280$	0	0
$281$	−19.1217	−1.14071	−0.570353	−	0.821400i	$-0.693193\pi$
$281$	−19.1217	−1.14071	−0.570353	+	0.821400i	$0.693193\pi$
$282$	0	0
$283$	19.3929	1.15279	0.576394	−	0.817172i	$-0.304459\pi$
$283$	19.3929	1.15279	0.576394	+	0.817172i	$0.304459\pi$
$284$	0	0
$285$	1.53074	0.0906732
$286$	0	0
$287$	−2.22311	−0.131226
$288$	0	0
$289$	−15.2595	−0.897616
$290$	0	0
$291$	−13.4690	−0.789566
$292$	0	0
$293$	8.20722	0.479471	0.239735	−	0.970838i	$-0.422939\pi$
$293$	8.20722	0.479471	0.239735	+	0.970838i	$0.422939\pi$
$294$	0	0
$295$	−3.37023	−0.196222
$296$	0	0
$297$	12.8293	0.744430
$298$	0	0
$299$	21.3723	1.23599
$300$	0	0
$301$	−14.6121	−0.842225
$302$	0	0
$303$	5.54792	0.318720
$304$	0	0
$305$	−3.01769	−0.172792
$306$	0	0
$307$	17.1911	0.981150	0.490575	−	0.871399i	$-0.336787\pi$
$307$	17.1911	0.981150	0.490575	+	0.871399i	$0.336787\pi$
$308$	0	0
$309$	17.3616	0.987668
$310$	0	0
$311$	−4.02356	−0.228155	−0.114077	−	0.993472i	$-0.536391\pi$
$311$	−4.02356	−0.228155	−0.114077	+	0.993472i	$0.536391\pi$
$312$	0	0
$313$	29.9299	1.69174	0.845869	−	0.533390i	$-0.179082\pi$
$313$	29.9299	1.69174	0.845869	+	0.533390i	$0.179082\pi$
$314$	0	0
$315$	3.05838	0.172320
$316$	0	0
$317$	3.09734	0.173964	0.0869821	−	0.996210i	$-0.472278\pi$
$317$	3.09734	0.173964	0.0869821	+	0.996210i	$0.472278\pi$
$318$	0	0
$319$	16.3144	0.913434
$320$	0	0
$321$	−3.13467	−0.174960
$322$	0	0
$323$	−2.21886	−0.123461
$324$	0	0
$325$	4.96710	0.275525
$326$	0	0
$327$	15.9543	0.882273
$328$	0	0
$329$	−3.68712	−0.203278
$330$	0	0
$331$	−8.27948	−0.455082	−0.227541	−	0.973769i	$-0.573069\pi$
$331$	−8.27948	−0.455082	−0.227541	+	0.973769i	$0.573069\pi$
$332$	0	0
$333$	4.53329	0.248423
$334$	0	0
$335$	1.07701	0.0588436
$336$	0	0
$337$	17.2680	0.940650	0.470325	−	0.882493i	$-0.344137\pi$
$337$	17.2680	0.940650	0.470325	+	0.882493i	$0.344137\pi$
$338$	0	0
$339$	−0.604018	−0.0328058
$340$	0	0
$341$	−0.471892	−0.0255544
$342$	0	0
$343$	−16.9234	−0.913778
$344$	0	0
$345$	−3.91616	−0.210839
$346$	0	0
$347$	−23.5763	−1.26564	−0.632820	−	0.774299i	$-0.718103\pi$
$347$	−23.5763	−1.26564	−0.632820	+	0.774299i	$0.718103\pi$
$348$	0	0
$349$	−29.7277	−1.59129	−0.795644	−	0.605765i	$-0.792867\pi$
$349$	−29.7277	−1.59129	−0.795644	+	0.605765i	$0.792867\pi$
$350$	0	0
$351$	−23.3799	−1.24793
$352$	0	0
$353$	−32.9290	−1.75264	−0.876318	−	0.481734i	$-0.840007\pi$
$353$	−32.9290	−1.75264	−0.876318	+	0.481734i	$0.840007\pi$
$354$	0	0
$355$	−0.238719	−0.0126699
$356$	0	0
$357$	1.69107	0.0895008
$358$	0	0
$359$	17.3352	0.914915	0.457458	−	0.889231i	$-0.348760\pi$
$359$	17.3352	0.914915	0.457458	+	0.889231i	$0.348760\pi$
$360$	0	0
$361$	−16.1714	−0.851125
$362$	0	0
$363$	−3.25026	−0.170594
$364$	0	0
$365$	11.9587	0.625949
$366$	0	0
$367$	−30.1011	−1.57127	−0.785634	−	0.618692i	$-0.787663\pi$
$367$	−30.1011	−1.57127	−0.785634	+	0.618692i	$0.787663\pi$
$368$	0	0
$369$	3.42799	0.178454
$370$	0	0
$371$	2.83180	0.147020
$372$	0	0
$373$	−13.5408	−0.701115	−0.350558	−	0.936541i	$-0.614008\pi$
$373$	−13.5408	−0.701115	−0.350558	+	0.936541i	$0.614008\pi$
$374$	0	0
$375$	−0.910150	−0.0470000
$376$	0	0
$377$	−29.7313	−1.53124
$378$	0	0
$379$	28.6588	1.47210	0.736052	−	0.676925i	$-0.236688\pi$
$379$	28.6588	1.47210	0.736052	+	0.676925i	$0.236688\pi$
$380$	0	0
$381$	−12.9864	−0.665313
$382$	0	0
$383$	−10.9371	−0.558861	−0.279431	−	0.960166i	$-0.590146\pi$
$383$	−10.9371	−0.558861	−0.279431	+	0.960166i	$0.590146\pi$
$384$	0	0
$385$	3.83856	0.195631
$386$	0	0
$387$	22.5315	1.14534
$388$	0	0
$389$	12.4862	0.633076	0.316538	−	0.948580i	$-0.397479\pi$
$389$	12.4862	0.633076	0.316538	+	0.948580i	$0.397479\pi$
$390$	0	0
$391$	5.67660	0.287078
$392$	0	0
$393$	−4.72070	−0.238128
$394$	0	0
$395$	−4.59577	−0.231238
$396$	0	0
$397$	21.1365	1.06081	0.530406	−	0.847744i	$-0.322039\pi$
$397$	21.1365	1.06081	0.530406	+	0.847744i	$0.322039\pi$
$398$	0	0
$399$	−2.15580	−0.107925
$400$	0	0
$401$	1.00000	0.0499376
$401$	1.00000	0.0499376
$402$	0	0
$403$	0.859972	0.0428383
$404$	0	0
$405$	−2.23084	−0.110851
$406$	0	0
$407$	5.68971	0.282029
$408$	0	0
$409$	16.1784	0.799973	0.399986	−	0.916521i	$-0.369015\pi$
$409$	16.1784	0.799973	0.399986	+	0.916521i	$0.369015\pi$
$410$	0	0
$411$	−2.69449	−0.132910
$412$	0	0
$413$	4.74642	0.233556
$414$	0	0
$415$	−1.18072	−0.0579592
$416$	0	0
$417$	−11.8346	−0.579540
$418$	0	0
$419$	2.50376	0.122317	0.0611583	−	0.998128i	$-0.480521\pi$
$419$	2.50376	0.122317	0.0611583	+	0.998128i	$0.480521\pi$
$420$	0	0
$421$	21.1000	1.02835	0.514175	−	0.857685i	$-0.328098\pi$
$421$	21.1000	1.02835	0.514175	+	0.857685i	$0.328098\pi$
$422$	0	0
$423$	5.68547	0.276437
$424$	0	0
$425$	1.31929	0.0639951
$426$	0	0
$427$	4.24992	0.205668
$428$	0	0
$429$	−12.3219	−0.594907
$430$	0	0
$431$	−8.25772	−0.397760	−0.198880	−	0.980024i	$-0.563730\pi$
$431$	−8.25772	−0.397760	−0.198880	+	0.980024i	$0.563730\pi$
$432$	0	0
$433$	26.1255	1.25551	0.627756	−	0.778410i	$-0.283973\pi$
$433$	26.1255	1.25551	0.627756	+	0.778410i	$0.283973\pi$
$434$	0	0
$435$	5.44784	0.261204
$436$	0	0
$437$	−7.23662	−0.346174
$438$	0	0
$439$	−19.5302	−0.932124	−0.466062	−	0.884752i	$-0.654328\pi$
$439$	−19.5302	−0.932124	−0.466062	+	0.884752i	$0.654328\pi$
$440$	0	0
$441$	10.8941	0.518769
$442$	0	0
$443$	−33.8710	−1.60926	−0.804629	−	0.593778i	$-0.797636\pi$
$443$	−33.8710	−1.60926	−0.804629	+	0.593778i	$0.797636\pi$
$444$	0	0
$445$	−1.48814	−0.0705448
$446$	0	0
$447$	−7.40282	−0.350141
$448$	0	0
$449$	10.4004	0.490827	0.245414	−	0.969418i	$-0.421076\pi$
$449$	10.4004	0.490827	0.245414	+	0.969418i	$0.421076\pi$
$450$	0	0
$451$	4.30245	0.202595
$452$	0	0
$453$	4.93757	0.231987
$454$	0	0
$455$	−6.99536	−0.327947
$456$	0	0
$457$	9.74839	0.456011	0.228005	−	0.973660i	$-0.426780\pi$
$457$	9.74839	0.456011	0.228005	+	0.973660i	$0.426780\pi$
$458$	0	0
$459$	−6.20986	−0.289851
$460$	0	0
$461$	−2.69522	−0.125529	−0.0627644	−	0.998028i	$-0.519992\pi$
$461$	−2.69522	−0.125529	−0.0627644	+	0.998028i	$0.519992\pi$
$462$	0	0
$463$	2.80445	0.130334	0.0651669	−	0.997874i	$-0.479242\pi$
$463$	2.80445	0.130334	0.0651669	+	0.997874i	$0.479242\pi$
$464$	0	0
$465$	−0.157578	−0.00730748
$466$	0	0
$467$	14.1427	0.654448	0.327224	−	0.944947i	$-0.393887\pi$
$467$	14.1427	0.654448	0.327224	+	0.944947i	$0.393887\pi$
$468$	0	0
$469$	−1.51680	−0.0700393
$470$	0	0
$471$	11.5097	0.530338
$472$	0	0
$473$	28.2791	1.30028
$474$	0	0
$475$	−1.68185	−0.0771687
$476$	0	0
$477$	−4.36657	−0.199931
$478$	0	0
$479$	−22.4348	−1.02507	−0.512536	−	0.858666i	$-0.671294\pi$
$479$	−22.4348	−1.02507	−0.512536	+	0.858666i	$0.671294\pi$
$480$	0	0
$481$	−10.3689	−0.472780
$482$	0	0
$483$	5.51528	0.250954
$484$	0	0
$485$	14.7986	0.671971
$486$	0	0
$487$	−9.55046	−0.432773	−0.216386	−	0.976308i	$-0.569427\pi$
$487$	−9.55046	−0.432773	−0.216386	+	0.976308i	$0.569427\pi$
$488$	0	0
$489$	8.51902	0.385243
$490$	0	0
$491$	−22.4807	−1.01454	−0.507270	−	0.861787i	$-0.669345\pi$
$491$	−22.4807	−1.01454	−0.507270	+	0.861787i	$0.669345\pi$
$492$	0	0
$493$	−7.89682	−0.355655
$494$	0	0
$495$	−5.91898	−0.266038
$496$	0	0
$497$	0.336197	0.0150805
$498$	0	0
$499$	31.2833	1.40043	0.700217	−	0.713930i	$-0.253086\pi$
$499$	31.2833	1.40043	0.700217	+	0.713930i	$0.253086\pi$
$500$	0	0
$501$	−5.72883	−0.255945
$502$	0	0
$503$	34.9055	1.55636	0.778180	−	0.628042i	$-0.216143\pi$
$503$	34.9055	1.55636	0.778180	+	0.628042i	$0.216143\pi$
$504$	0	0
$505$	−6.09561	−0.271251
$506$	0	0
$507$	10.6234	0.471801
$508$	0	0
$509$	−5.70394	−0.252823	−0.126411	−	0.991978i	$-0.540346\pi$
$509$	−5.70394	−0.252823	−0.126411	+	0.991978i	$0.540346\pi$
$510$	0	0
$511$	−16.8420	−0.745044
$512$	0	0
$513$	7.91641	0.349518
$514$	0	0
$515$	−19.0756	−0.840569
$516$	0	0
$517$	7.13580	0.313832
$518$	0	0
$519$	−15.3832	−0.675249
$520$	0	0
$521$	−18.4893	−0.810032	−0.405016	−	0.914310i	$-0.632734\pi$
$521$	−18.4893	−0.810032	−0.405016	+	0.914310i	$0.632734\pi$
$522$	0	0
$523$	0.185716	0.00812078	0.00406039	−	0.999992i	$-0.498708\pi$
$523$	0.185716	0.00812078	0.00406039	+	0.999992i	$0.498708\pi$
$524$	0	0
$525$	1.28180	0.0559423
$526$	0	0
$527$	0.228414	0.00994986
$528$	0	0
$529$	−4.48623	−0.195054
$530$	0	0
$531$	−7.31888	−0.317612
$532$	0	0
$533$	−7.84075	−0.339621
$534$	0	0
$535$	3.44412	0.148902
$536$	0	0
$537$	−3.25969	−0.140666
$538$	0	0
$539$	13.6732	0.588946
$540$	0	0
$541$	32.6216	1.40251	0.701256	−	0.712909i	$-0.252623\pi$
$541$	32.6216	1.40251	0.701256	+	0.712909i	$0.252623\pi$
$542$	0	0
$543$	11.9299	0.511960
$544$	0	0
$545$	−17.5293	−0.750871
$546$	0	0
$547$	−23.7955	−1.01742	−0.508710	−	0.860938i	$-0.669878\pi$
$547$	−23.7955	−1.01742	−0.508710	+	0.860938i	$0.669878\pi$
$548$	0	0
$549$	−6.55328	−0.279687
$550$	0	0
$551$	10.0670	0.428868
$552$	0	0
$553$	6.47239	0.275234
$554$	0	0
$555$	1.89995	0.0806483
$556$	0	0
$557$	23.7350	1.00568	0.502842	−	0.864378i	$-0.332288\pi$
$557$	23.7350	1.00568	0.502842	+	0.864378i	$0.332288\pi$
$558$	0	0
$559$	−51.5356	−2.17973
$560$	0	0
$561$	−3.27277	−0.138177
$562$	0	0
$563$	15.3811	0.648235	0.324117	−	0.946017i	$-0.394933\pi$
$563$	15.3811	0.648235	0.324117	+	0.946017i	$0.394933\pi$
$564$	0	0
$565$	0.663647	0.0279198
$566$	0	0
$567$	3.14178	0.131942
$568$	0	0
$569$	−41.8330	−1.75373	−0.876865	−	0.480737i	$-0.840369\pi$
$569$	−41.8330	−1.75373	−0.876865	+	0.480737i	$0.840369\pi$
$570$	0	0
$571$	3.80060	0.159050	0.0795251	−	0.996833i	$-0.474660\pi$
$571$	3.80060	0.159050	0.0795251	+	0.996833i	$0.474660\pi$
$572$	0	0
$573$	12.2966	0.513700
$574$	0	0
$575$	4.30276	0.179438
$576$	0	0
$577$	27.5594	1.14731	0.573657	−	0.819095i	$-0.305524\pi$
$577$	27.5594	1.14731	0.573657	+	0.819095i	$0.305524\pi$
$578$	0	0
$579$	−21.9176	−0.910863
$580$	0	0
$581$	1.66285	0.0689867
$582$	0	0
$583$	−5.48046	−0.226977
$584$	0	0
$585$	10.7867	0.445975
$586$	0	0
$587$	−31.4787	−1.29926	−0.649632	−	0.760249i	$-0.725077\pi$
$587$	−31.4787	−1.29926	−0.649632	+	0.760249i	$0.725077\pi$
$588$	0	0
$589$	−0.291185	−0.0119981
$590$	0	0
$591$	−18.2512	−0.750756
$592$	0	0
$593$	4.54584	0.186675	0.0933376	−	0.995635i	$-0.470246\pi$
$593$	4.54584	0.186675	0.0933376	+	0.995635i	$0.470246\pi$
$594$	0	0
$595$	−1.85801	−0.0761710
$596$	0	0
$597$	−18.1496	−0.742813
$598$	0	0
$599$	32.3791	1.32297	0.661486	−	0.749957i	$-0.269926\pi$
$599$	32.3791	1.32297	0.661486	+	0.749957i	$0.269926\pi$
$600$	0	0
$601$	3.98405	0.162513	0.0812563	−	0.996693i	$-0.474107\pi$
$601$	3.98405	0.162513	0.0812563	+	0.996693i	$0.474107\pi$
$602$	0	0
$603$	2.33887	0.0952462
$604$	0	0
$605$	3.57112	0.145187
$606$	0	0
$607$	−0.788641	−0.0320100	−0.0160050	−	0.999872i	$-0.505095\pi$
$607$	−0.788641	−0.0320100	−0.0160050	+	0.999872i	$0.505095\pi$
$608$	0	0
$609$	−7.67239	−0.310901
$610$	0	0
$611$	−13.0042	−0.526094
$612$	0	0
$613$	22.3260	0.901740	0.450870	−	0.892590i	$-0.351114\pi$
$613$	22.3260	0.901740	0.450870	+	0.892590i	$0.351114\pi$
$614$	0	0
$615$	1.43670	0.0579335
$616$	0	0
$617$	42.9781	1.73023	0.865116	−	0.501572i	$-0.167245\pi$
$617$	42.9781	1.73023	0.865116	+	0.501572i	$0.167245\pi$
$618$	0	0
$619$	31.8889	1.28172	0.640861	−	0.767657i	$-0.278577\pi$
$619$	31.8889	1.28172	0.640861	+	0.767657i	$0.278577\pi$
$620$	0	0
$621$	−20.2529	−0.812722
$622$	0	0
$623$	2.09581	0.0839668
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	4.17218	0.166621
$628$	0	0
$629$	−2.75404	−0.109811
$630$	0	0
$631$	46.1653	1.83781	0.918905	−	0.394479i	$-0.129075\pi$
$631$	46.1653	1.83781	0.918905	+	0.394479i	$0.129075\pi$
$632$	0	0
$633$	−9.25895	−0.368010
$634$	0	0
$635$	14.2684	0.566224
$636$	0	0
$637$	−24.9179	−0.987283
$638$	0	0
$639$	−0.518408	−0.0205079
$640$	0	0
$641$	−38.8169	−1.53318	−0.766588	−	0.642140i	$-0.778047\pi$
$641$	−38.8169	−1.53318	−0.766588	+	0.642140i	$0.778047\pi$
$642$	0	0
$643$	−22.2086	−0.875820	−0.437910	−	0.899019i	$-0.644281\pi$
$643$	−22.2086	−0.875820	−0.437910	+	0.899019i	$0.644281\pi$
$644$	0	0
$645$	9.44317	0.371824
$646$	0	0
$647$	−0.631871	−0.0248414	−0.0124207	−	0.999923i	$-0.503954\pi$
$647$	−0.631871	−0.0248414	−0.0124207	+	0.999923i	$0.503954\pi$
$648$	0	0
$649$	−9.18589	−0.360578
$650$	0	0
$651$	0.221922	0.00869782
$652$	0	0
$653$	−16.2224	−0.634830	−0.317415	−	0.948287i	$-0.602815\pi$
$653$	−16.2224	−0.634830	−0.317415	+	0.948287i	$0.602815\pi$
$654$	0	0
$655$	5.18673	0.202662
$656$	0	0
$657$	25.9699	1.01318
$658$	0	0
$659$	−45.6183	−1.77704	−0.888518	−	0.458841i	$-0.848265\pi$
$659$	−45.6183	−1.77704	−0.888518	+	0.458841i	$0.848265\pi$
$660$	0	0
$661$	−29.6362	−1.15272	−0.576358	−	0.817197i	$-0.695527\pi$
$661$	−29.6362	−1.15272	−0.576358	+	0.817197i	$0.695527\pi$
$662$	0	0
$663$	5.96427	0.231633
$664$	0	0
$665$	2.36862	0.0918510
$666$	0	0
$667$	−25.7548	−0.997230
$668$	0	0
$669$	4.68490	0.181129
$670$	0	0
$671$	−8.22499	−0.317522
$672$	0	0
$673$	28.6613	1.10481	0.552407	−	0.833575i	$-0.313710\pi$
$673$	28.6613	1.10481	0.552407	+	0.833575i	$0.313710\pi$
$674$	0	0
$675$	−4.70696	−0.181171
$676$	0	0
$677$	−45.6801	−1.75563	−0.877815	−	0.479000i	$-0.840999\pi$
$677$	−45.6801	−1.75563	−0.877815	+	0.479000i	$0.840999\pi$
$678$	0	0
$679$	−20.8415	−0.799822
$680$	0	0
$681$	−20.7700	−0.795909
$682$	0	0
$683$	20.0350	0.766616	0.383308	−	0.923621i	$-0.374785\pi$
$683$	20.0350	0.766616	0.383308	+	0.923621i	$0.374785\pi$
$684$	0	0
$685$	2.96049	0.113115
$686$	0	0
$687$	−3.69141	−0.140836
$688$	0	0
$689$	9.98754	0.380495
$690$	0	0
$691$	1.54348	0.0587166	0.0293583	−	0.999569i	$-0.490654\pi$
$691$	1.54348	0.0587166	0.0293583	+	0.999569i	$0.490654\pi$
$692$	0	0
$693$	8.33592	0.316655
$694$	0	0
$695$	13.0029	0.493226
$696$	0	0
$697$	−2.08255	−0.0788822
$698$	0	0
$699$	−23.3223	−0.882130
$700$	0	0
$701$	10.4916	0.396264	0.198132	−	0.980175i	$-0.436513\pi$
$701$	10.4916	0.396264	0.198132	+	0.980175i	$0.436513\pi$
$702$	0	0
$703$	3.51089	0.132416
$704$	0	0
$705$	2.38284	0.0897428
$706$	0	0
$707$	8.58467	0.322860
$708$	0	0
$709$	−3.08650	−0.115916	−0.0579579	−	0.998319i	$-0.518459\pi$
$709$	−3.08650	−0.115916	−0.0579579	+	0.998319i	$0.518459\pi$
$710$	0	0
$711$	−9.98029	−0.374290
$712$	0	0
$713$	0.744953	0.0278987
$714$	0	0
$715$	13.5383	0.506304
$716$	0	0
$717$	−20.9815	−0.783570
$718$	0	0
$719$	43.5852	1.62545	0.812727	−	0.582645i	$-0.197982\pi$
$719$	43.5852	1.62545	0.812727	+	0.582645i	$0.197982\pi$
$720$	0	0
$721$	26.8648	1.00050
$722$	0	0
$723$	1.44809	0.0538550
$724$	0	0
$725$	−5.98564	−0.222301
$726$	0	0
$727$	14.9249	0.553532	0.276766	−	0.960937i	$-0.410737\pi$
$727$	14.9249	0.553532	0.276766	+	0.960937i	$0.410737\pi$
$728$	0	0
$729$	8.00757	0.296577
$730$	0	0
$731$	−13.6882	−0.506276
$732$	0	0
$733$	−20.0259	−0.739673	−0.369837	−	0.929097i	$-0.620586\pi$
$733$	−20.0259	−0.739673	−0.369837	+	0.929097i	$0.620586\pi$
$734$	0	0
$735$	4.56585	0.168414
$736$	0	0
$737$	2.93551	0.108131
$738$	0	0
$739$	22.2677	0.819130	0.409565	−	0.912281i	$-0.365680\pi$
$739$	22.2677	0.819130	0.409565	+	0.912281i	$0.365680\pi$
$740$	0	0
$741$	−7.60334	−0.279316
$742$	0	0
$743$	35.6037	1.30617	0.653086	−	0.757284i	$-0.273474\pi$
$743$	35.6037	1.30617	0.653086	+	0.757284i	$0.273474\pi$
$744$	0	0
$745$	8.13362	0.297993
$746$	0	0
$747$	−2.56408	−0.0938148
$748$	0	0
$749$	−4.85049	−0.177233
$750$	0	0
$751$	−3.72066	−0.135769	−0.0678844	−	0.997693i	$-0.521625\pi$
$751$	−3.72066	−0.135769	−0.0678844	+	0.997693i	$0.521625\pi$
$752$	0	0
$753$	−16.3665	−0.596427
$754$	0	0
$755$	−5.42500	−0.197436
$756$	0	0
$757$	4.80917	0.174792	0.0873962	−	0.996174i	$-0.472145\pi$
$757$	4.80917	0.174792	0.0873962	+	0.996174i	$0.472145\pi$
$758$	0	0
$759$	−10.6739	−0.387437
$760$	0	0
$761$	−41.8969	−1.51876	−0.759381	−	0.650647i	$-0.774498\pi$
$761$	−41.8969	−1.51876	−0.759381	+	0.650647i	$0.774498\pi$
$762$	0	0
$763$	24.6871	0.893734
$764$	0	0
$765$	2.86501	0.103585
$766$	0	0
$767$	16.7403	0.604456
$768$	0	0
$769$	−12.5554	−0.452761	−0.226381	−	0.974039i	$-0.572689\pi$
$769$	−12.5554	−0.452761	−0.226381	+	0.974039i	$0.572689\pi$
$770$	0	0
$771$	−24.7248	−0.890442
$772$	0	0
$773$	4.05283	0.145770	0.0728850	−	0.997340i	$-0.476779\pi$
$773$	4.05283	0.145770	0.0728850	+	0.997340i	$0.476779\pi$
$774$	0	0
$775$	0.173134	0.00621914
$776$	0	0
$777$	−2.67577	−0.0959927
$778$	0	0
$779$	2.65487	0.0951204
$780$	0	0
$781$	−0.650651	−0.0232821
$782$	0	0
$783$	28.1742	1.00686
$784$	0	0
$785$	−12.6459	−0.451352
$786$	0	0
$787$	−8.18545	−0.291780	−0.145890	−	0.989301i	$-0.546605\pi$
$787$	−8.18545	−0.291780	−0.145890	+	0.989301i	$0.546605\pi$
$788$	0	0
$789$	−8.64712	−0.307846
$790$	0	0
$791$	−0.934638	−0.0332319
$792$	0	0
$793$	14.9892	0.532280
$794$	0	0
$795$	−1.83007	−0.0649060
$796$	0	0
$797$	−17.0042	−0.602320	−0.301160	−	0.953574i	$-0.597374\pi$
$797$	−17.0042	−0.602320	−0.301160	+	0.953574i	$0.597374\pi$
$798$	0	0
$799$	−3.45400	−0.122194
$800$	0	0
$801$	−3.23169	−0.114186
$802$	0	0
$803$	32.5947	1.15024
$804$	0	0
$805$	−6.05974	−0.213578
$806$	0	0
$807$	2.05402	0.0723049
$808$	0	0
$809$	−13.6213	−0.478900	−0.239450	−	0.970909i	$-0.576967\pi$
$809$	−13.6213	−0.478900	−0.239450	+	0.970909i	$0.576967\pi$
$810$	0	0
$811$	−9.23919	−0.324432	−0.162216	−	0.986755i	$-0.551864\pi$
$811$	−9.23919	−0.324432	−0.162216	+	0.986755i	$0.551864\pi$
$812$	0	0
$813$	−3.34899	−0.117454
$814$	0	0
$815$	−9.36001	−0.327867
$816$	0	0
$817$	17.4499	0.610495
$818$	0	0
$819$	−15.1913	−0.530827
$820$	0	0
$821$	21.2747	0.742493	0.371246	−	0.928534i	$-0.378931\pi$
$821$	21.2747	0.742493	0.371246	+	0.928534i	$0.378931\pi$
$822$	0	0
$823$	−12.1633	−0.423984	−0.211992	−	0.977271i	$-0.567995\pi$
$823$	−12.1633	−0.423984	−0.211992	+	0.977271i	$0.567995\pi$
$824$	0	0
$825$	−2.48070	−0.0863670
$826$	0	0
$827$	8.71185	0.302941	0.151470	−	0.988462i	$-0.451599\pi$
$827$	8.71185	0.302941	0.151470	+	0.988462i	$0.451599\pi$
$828$	0	0
$829$	19.2545	0.668736	0.334368	−	0.942443i	$-0.391477\pi$
$829$	19.2545	0.668736	0.334368	+	0.942443i	$0.391477\pi$
$830$	0	0
$831$	17.5094	0.607394
$832$	0	0
$833$	−6.61835	−0.229312
$834$	0	0
$835$	6.29438	0.217826
$836$	0	0
$837$	−0.814932	−0.0281682
$838$	0	0
$839$	24.7907	0.855869	0.427934	−	0.903810i	$-0.359241\pi$
$839$	24.7907	0.855869	0.427934	+	0.903810i	$0.359241\pi$
$840$	0	0
$841$	6.82793	0.235446
$842$	0	0
$843$	−17.4036	−0.599413
$844$	0	0
$845$	−11.6721	−0.401533
$846$	0	0
$847$	−5.02935	−0.172810
$848$	0	0
$849$	17.6505	0.605762
$850$	0	0
$851$	−8.98206	−0.307901
$852$	0	0
$853$	0.640816	0.0219411	0.0109706	−	0.999940i	$-0.496508\pi$
$853$	0.640816	0.0219411	0.0109706	+	0.999940i	$0.496508\pi$
$854$	0	0
$855$	−3.65236	−0.124908
$856$	0	0
$857$	−38.2067	−1.30512	−0.652558	−	0.757738i	$-0.726304\pi$
$857$	−38.2067	−1.30512	−0.652558	+	0.757738i	$0.726304\pi$
$858$	0	0
$859$	8.38812	0.286199	0.143099	−	0.989708i	$-0.454293\pi$
$859$	8.38812	0.286199	0.143099	+	0.989708i	$0.454293\pi$
$860$	0	0
$861$	−2.02337	−0.0689561
$862$	0	0
$863$	1.70332	0.0579818	0.0289909	−	0.999580i	$-0.490771\pi$
$863$	1.70332	0.0579818	0.0289909	+	0.999580i	$0.490771\pi$
$864$	0	0
$865$	16.9019	0.574681
$866$	0	0
$867$	−13.8884	−0.471675
$868$	0	0
$869$	−12.5262	−0.424923
$870$	0	0
$871$	−5.34964	−0.181266
$872$	0	0
$873$	32.1371	1.08768
$874$	0	0
$875$	−1.40834	−0.0476105
$876$	0	0
$877$	52.0454	1.75745	0.878725	−	0.477329i	$-0.158395\pi$
$877$	52.0454	1.75745	0.878725	+	0.477329i	$0.158395\pi$
$878$	0	0
$879$	7.46980	0.251950
$880$	0	0
$881$	−7.56874	−0.254997	−0.127499	−	0.991839i	$-0.540695\pi$
$881$	−7.56874	−0.254997	−0.127499	+	0.991839i	$0.540695\pi$
$882$	0	0
$883$	59.0344	1.98666	0.993332	−	0.115288i	$-0.0367792\pi$
$883$	59.0344	1.98666	0.993332	+	0.115288i	$0.0367792\pi$
$884$	0	0
$885$	−3.06742	−0.103110
$886$	0	0
$887$	−0.169272	−0.00568361	−0.00284180	−	0.999996i	$-0.500905\pi$
$887$	−0.169272	−0.00568361	−0.00284180	+	0.999996i	$0.500905\pi$
$888$	0	0
$889$	−20.0947	−0.673956
$890$	0	0
$891$	−6.08037	−0.203700
$892$	0	0
$893$	4.40321	0.147348
$894$	0	0
$895$	3.58148	0.119716
$896$	0	0
$897$	19.4520	0.649483
$898$	0	0
$899$	−1.03632	−0.0345631
$900$	0	0
$901$	2.65275	0.0883760
$902$	0	0
$903$	−13.2992	−0.442569
$904$	0	0
$905$	−13.1076	−0.435711
$906$	0	0
$907$	20.2540	0.672523	0.336262	−	0.941769i	$-0.390837\pi$
$907$	20.2540	0.672523	0.336262	+	0.941769i	$0.390837\pi$
$908$	0	0
$909$	−13.2374	−0.439056
$910$	0	0
$911$	40.6549	1.34696	0.673478	−	0.739207i	$-0.264800\pi$
$911$	40.6549	1.34696	0.673478	+	0.739207i	$0.264800\pi$
$912$	0	0
$913$	−3.21816	−0.106506
$914$	0	0
$915$	−2.74655	−0.0907981
$916$	0	0
$917$	−7.30466	−0.241221
$918$	0	0
$919$	58.0652	1.91539	0.957696	−	0.287780i	$-0.0929173\pi$
$919$	58.0652	1.91539	0.957696	+	0.287780i	$0.0929173\pi$
$920$	0	0
$921$	15.6465	0.515571
$922$	0	0
$923$	1.18574	0.0390291
$924$	0	0
$925$	−2.08751	−0.0686369
$926$	0	0
$927$	−41.4250	−1.36057
$928$	0	0
$929$	37.8235	1.24095	0.620474	−	0.784227i	$-0.286940\pi$
$929$	37.8235	1.24095	0.620474	+	0.784227i	$0.286940\pi$
$930$	0	0
$931$	8.43716	0.276517
$932$	0	0
$933$	−3.66204	−0.119890
$934$	0	0
$935$	3.59586	0.117597
$936$	0	0
$937$	−18.6064	−0.607844	−0.303922	−	0.952697i	$-0.598296\pi$
$937$	−18.6064	−0.607844	−0.303922	+	0.952697i	$0.598296\pi$
$938$	0	0
$939$	27.2407	0.888967
$940$	0	0
$941$	−32.9383	−1.07376	−0.536879	−	0.843659i	$-0.680397\pi$
$941$	−32.9383	−1.07376	−0.536879	+	0.843659i	$0.680397\pi$
$942$	0	0
$943$	−6.79206	−0.221180
$944$	0	0
$945$	6.62898	0.215641
$946$	0	0
$947$	−21.4528	−0.697123	−0.348561	−	0.937286i	$-0.613330\pi$
$947$	−21.4528	−0.697123	−0.348561	+	0.937286i	$0.613330\pi$
$948$	0	0
$949$	−59.4003	−1.92822
$950$	0	0
$951$	2.81905	0.0914139
$952$	0	0
$953$	48.2736	1.56374	0.781868	−	0.623444i	$-0.214267\pi$
$953$	48.2736	1.56374	0.781868	+	0.623444i	$0.214267\pi$
$954$	0	0
$955$	−13.5106	−0.437192
$956$	0	0
$957$	14.8486	0.479987
$958$	0	0
$959$	−4.16937	−0.134636
$960$	0	0
$961$	−30.9700	−0.999033
$962$	0	0
$963$	7.47935	0.241019
$964$	0	0
$965$	24.0813	0.775203
$966$	0	0
$967$	−40.8247	−1.31283	−0.656416	−	0.754399i	$-0.727929\pi$
$967$	−40.8247	−1.31283	−0.656416	+	0.754399i	$0.727929\pi$
$968$	0	0
$969$	−2.01949	−0.0648755
$970$	0	0
$971$	54.4176	1.74634	0.873171	−	0.487413i	$-0.162060\pi$
$971$	54.4176	1.74634	0.873171	+	0.487413i	$0.162060\pi$
$972$	0	0
$973$	−18.3124	−0.587069
$974$	0	0
$975$	4.52081	0.144782
$976$	0	0
$977$	−5.82295	−0.186293	−0.0931464	−	0.995652i	$-0.529692\pi$
$977$	−5.82295	−0.186293	−0.0931464	+	0.995652i	$0.529692\pi$
$978$	0	0
$979$	−4.05608	−0.129633
$980$	0	0
$981$	−38.0670	−1.21539
$982$	0	0
$983$	−26.3549	−0.840590	−0.420295	−	0.907388i	$-0.638073\pi$
$983$	−26.3549	−0.840590	−0.420295	+	0.907388i	$0.638073\pi$
$984$	0	0
$985$	20.0530	0.638942
$986$	0	0
$987$	−3.35584	−0.106817
$988$	0	0
$989$	−44.6429	−1.41956
$990$	0	0
$991$	1.87927	0.0596969	0.0298485	−	0.999554i	$-0.490498\pi$
$991$	1.87927	0.0596969	0.0298485	+	0.999554i	$0.490498\pi$
$992$	0	0
$993$	−7.53558	−0.239134
$994$	0	0
$995$	19.9413	0.632182
$996$	0	0
$997$	−50.5797	−1.60188	−0.800938	−	0.598748i	$-0.795665\pi$
$997$	−50.5797	−1.60188	−0.800938	+	0.598748i	$0.795665\pi$
$998$	0	0
$999$	9.82582	0.310875

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8020.2.a.c.1.17	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8020.2.a.c.1.17	✓	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)

\(f(q)\)	\(=\)	\(q+0.910150 q^{3} -1.00000 q^{5} +1.40834 q^{7} -2.17163 q^{9} +O(q^{10})\)	\(q+0.910150 q^{3} -1.00000 q^{5} +1.40834 q^{7} -2.17163 q^{9} -2.72560 q^{11} +4.96710 q^{13} -0.910150 q^{15} +1.31929 q^{17} -1.68185 q^{19} +1.28180 q^{21} +4.30276 q^{23} +1.00000 q^{25} -4.70696 q^{27} -5.98564 q^{29} +0.173134 q^{31} -2.48070 q^{33} -1.40834 q^{35} -2.08751 q^{37} +4.52081 q^{39} -1.57854 q^{41} -10.3754 q^{43} +2.17163 q^{45} -2.61807 q^{47} -5.01659 q^{49} +1.20075 q^{51} +2.01074 q^{53} +2.72560 q^{55} -1.53074 q^{57} +3.37023 q^{59} +3.01769 q^{61} -3.05838 q^{63} -4.96710 q^{65} -1.07701 q^{67} +3.91616 q^{69} +0.238719 q^{71} -11.9587 q^{73} +0.910150 q^{75} -3.83856 q^{77} +4.59577 q^{79} +2.23084 q^{81} +1.18072 q^{83} -1.31929 q^{85} -5.44784 q^{87} +1.48814 q^{89} +6.99536 q^{91} +0.157578 q^{93} +1.68185 q^{95} -14.7986 q^{97} +5.91898 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 3 q^{3} - 28 q^{5} - 4 q^{7} + 17 q^{9}+O(q^{10}) \) 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