LMFDB - Embedded newform 2151.4.a.g.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2151,4,Mod(1,2151)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2151, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2151.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2151 = 3^{2} \cdot 239 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2151.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:	$126.913108422$
Analytic rank:	$1$
Dimension:	$59$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Character	$\chi$	$=$	2151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.50845 q^{2} +22.3430 q^{4} +10.5610 q^{5} +7.83920 q^{7} -79.0077 q^{8} +O(q^{10})$	\(q-5.50845 q^{2} +22.3430 q^{4} +10.5610 q^{5} +7.83920 q^{7} -79.0077 q^{8} -58.1747 q^{10} +1.99748 q^{11} +45.5687 q^{13} -43.1818 q^{14} +256.466 q^{16} -69.6442 q^{17} -75.6903 q^{19} +235.964 q^{20} -11.0030 q^{22} -192.471 q^{23} -13.4654 q^{25} -251.013 q^{26} +175.151 q^{28} +241.246 q^{29} +208.009 q^{31} -780.667 q^{32} +383.631 q^{34} +82.7898 q^{35} +17.1277 q^{37} +416.936 q^{38} -834.400 q^{40} -279.675 q^{41} -186.934 q^{43} +44.6297 q^{44} +1060.21 q^{46} +526.711 q^{47} -281.547 q^{49} +74.1737 q^{50} +1018.14 q^{52} +646.542 q^{53} +21.0954 q^{55} -619.357 q^{56} -1328.89 q^{58} -867.869 q^{59} -41.7980 q^{61} -1145.81 q^{62} +2248.54 q^{64} +481.251 q^{65} +920.188 q^{67} -1556.06 q^{68} -456.043 q^{70} -1071.49 q^{71} +604.373 q^{73} -94.3472 q^{74} -1691.15 q^{76} +15.6587 q^{77} +1122.98 q^{79} +2708.53 q^{80} +1540.57 q^{82} -58.6057 q^{83} -735.512 q^{85} +1029.72 q^{86} -157.816 q^{88} -1275.90 q^{89} +357.222 q^{91} -4300.37 q^{92} -2901.36 q^{94} -799.364 q^{95} -1175.49 q^{97} +1550.89 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) $ 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	$ 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) $ 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * q^98

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 $

$2$	−5.50845	−1.94753	−0.973765	−	0.227555i	$-0.926927\pi$
$2$	−5.50845	−1.94753	−0.973765	+	0.227555i	$0.926927\pi$
$3$	0	0
$3$	0	0
$4$	22.3430	2.79288
$5$	10.5610	0.944604	0.472302	−	0.881437i	$-0.343423\pi$
$5$	10.5610	0.944604	0.472302	+	0.881437i	$0.343423\pi$
$6$	0	0
$7$	7.83920	0.423277	0.211639	−	0.977348i	$-0.432120\pi$
$7$	7.83920	0.423277	0.211639	+	0.977348i	$0.432120\pi$
$8$	−79.0077	−3.49168
$9$	0	0
$10$	−58.1747	−1.83964
$11$	1.99748	0.0547512	0.0273756	−	0.999625i	$-0.491285\pi$
$11$	1.99748	0.0547512	0.0273756	+	0.999625i	$0.491285\pi$
$12$	0	0
$13$	45.5687	0.972191	0.486096	−	0.873906i	$-0.338421\pi$
$13$	45.5687	0.972191	0.486096	+	0.873906i	$0.338421\pi$
$14$	−43.1818	−0.824345
$15$	0	0
$16$	256.466	4.00728
$17$	−69.6442	−0.993600	−0.496800	−	0.867865i	$-0.665492\pi$
$17$	−69.6442	−0.993600	−0.496800	+	0.867865i	$0.665492\pi$
$18$	0	0
$19$	−75.6903	−0.913923	−0.456961	−	0.889486i	$-0.651062\pi$
$19$	−75.6903	−0.913923	−0.456961	+	0.889486i	$0.651062\pi$
$20$	235.964	2.63816
$21$	0	0
$22$	−11.0030	−0.106630
$23$	−192.471	−1.74491	−0.872454	−	0.488696i	$-0.837473\pi$
$23$	−192.471	−1.74491	−0.872454	+	0.488696i	$0.837473\pi$
$24$	0	0
$25$	−13.4654	−0.107724
$26$	−251.013	−1.89337
$27$	0	0
$28$	175.151	1.18216
$29$	241.246	1.54477	0.772385	−	0.635155i	$-0.219064\pi$
$29$	241.246	1.54477	0.772385	+	0.635155i	$0.219064\pi$
$30$	0	0
$31$	208.009	1.20515	0.602573	−	0.798064i	$-0.294142\pi$
$31$	208.009	1.20515	0.602573	+	0.798064i	$0.294142\pi$
$32$	−780.667	−4.31261
$33$	0	0
$34$	383.631	1.93507
$35$	82.7898	0.399829
$36$	0	0
$37$	17.1277	0.0761022	0.0380511	−	0.999276i	$-0.487885\pi$
$37$	17.1277	0.0761022	0.0380511	+	0.999276i	$0.487885\pi$
$38$	416.936	1.77989
$39$	0	0
$40$	−834.400	−3.29825
$41$	−279.675	−1.06531	−0.532657	−	0.846331i	$-0.678806\pi$
$41$	−279.675	−1.06531	−0.532657	+	0.846331i	$0.678806\pi$
$42$	0	0
$43$	−186.934	−0.662957	−0.331479	−	0.943463i	$-0.607547\pi$
$43$	−186.934	−0.662957	−0.331479	+	0.943463i	$0.607547\pi$
$44$	44.6297	0.152913
$45$	0	0
$46$	1060.21	3.39826
$47$	526.711	1.63465	0.817327	−	0.576175i	$-0.195455\pi$
$47$	526.711	1.63465	0.817327	+	0.576175i	$0.195455\pi$
$48$	0	0
$49$	−281.547	−0.820836
$50$	74.1737	0.209795
$51$	0	0
$52$	1018.14	2.71521
$53$	646.542	1.67565	0.837825	−	0.545939i	$-0.183827\pi$
$53$	646.542	1.67565	0.837825	+	0.545939i	$0.183827\pi$
$54$	0	0
$55$	21.0954	0.0517182
$56$	−619.357	−1.47795
$57$	0	0
$58$	−1328.89	−3.00849
$59$	−867.869	−1.91503	−0.957516	−	0.288379i	$-0.906884\pi$
$59$	−867.869	−1.91503	−0.957516	+	0.288379i	$0.906884\pi$
$60$	0	0
$61$	−41.7980	−0.0877325	−0.0438662	−	0.999037i	$-0.513968\pi$
$61$	−41.7980	−0.0877325	−0.0438662	+	0.999037i	$0.513968\pi$
$62$	−1145.81	−2.34706
$63$	0	0
$64$	2248.54	4.39167
$65$	481.251	0.918335
$66$	0	0
$67$	920.188	1.67789	0.838947	−	0.544213i	$-0.183172\pi$
$67$	920.188	1.67789	0.838947	+	0.544213i	$0.183172\pi$
$68$	−1556.06	−2.77500
$69$	0	0
$70$	−456.043	−0.778680
$71$	−1071.49	−1.79101	−0.895506	−	0.445049i	$-0.853186\pi$
$71$	−1071.49	−1.79101	−0.895506	+	0.445049i	$0.853186\pi$
$72$	0	0
$73$	604.373	0.968993	0.484497	−	0.874793i	$-0.339003\pi$
$73$	604.373	0.968993	0.484497	+	0.874793i	$0.339003\pi$
$74$	−94.3472	−0.148211
$75$	0	0
$76$	−1691.15	−2.55247
$77$	15.6587	0.0231749
$78$	0	0
$79$	1122.98	1.59931	0.799655	−	0.600460i	$-0.205016\pi$
$79$	1122.98	1.59931	0.799655	+	0.600460i	$0.205016\pi$
$80$	2708.53	3.78529
$81$	0	0
$82$	1540.57	2.07473
$83$	−58.6057	−0.0775038	−0.0387519	−	0.999249i	$-0.512338\pi$
$83$	−58.6057	−0.0775038	−0.0387519	+	0.999249i	$0.512338\pi$
$84$	0	0
$85$	−735.512	−0.938558
$86$	1029.72	1.29113
$87$	0	0
$88$	−157.816	−0.191174
$89$	−1275.90	−1.51961	−0.759803	−	0.650153i	$-0.774705\pi$
$89$	−1275.90	−1.51961	−0.759803	+	0.650153i	$0.774705\pi$
$90$	0	0
$91$	357.222	0.411506
$92$	−4300.37	−4.87331
$93$	0	0
$94$	−2901.36	−3.18354
$95$	−799.364	−0.863295
$96$	0	0
$97$	−1175.49	−1.23044	−0.615219	−	0.788357i	$-0.710932\pi$
$97$	−1175.49	−1.23044	−0.615219	+	0.788357i	$0.710932\pi$
$98$	1550.89	1.59860
$99$	0	0
$100$	−300.858	−0.300858
$101$	−837.268	−0.824864	−0.412432	−	0.910988i	$-0.635321\pi$
$101$	−837.268	−0.824864	−0.412432	+	0.910988i	$0.635321\pi$
$102$	0	0
$103$	−166.747	−0.159515	−0.0797577	−	0.996814i	$-0.525415\pi$
$103$	−166.747	−0.159515	−0.0797577	+	0.996814i	$0.525415\pi$
$104$	−3600.28	−3.39458
$105$	0	0
$106$	−3561.44	−3.26338
$107$	−2112.88	−1.90897	−0.954484	−	0.298261i	$-0.903593\pi$
$107$	−2112.88	−1.90897	−0.954484	+	0.298261i	$0.903593\pi$
$108$	0	0
$109$	−1941.99	−1.70651	−0.853253	−	0.521497i	$-0.825374\pi$
$109$	−1941.99	−1.70651	−0.853253	+	0.521497i	$0.825374\pi$
$110$	−116.203	−0.100723
$111$	0	0
$112$	2010.49	1.69619
$113$	−100.874	−0.0839769	−0.0419884	−	0.999118i	$-0.513369\pi$
$113$	−100.874	−0.0839769	−0.0419884	+	0.999118i	$0.513369\pi$
$114$	0	0
$115$	−2032.68	−1.64825
$116$	5390.17	4.31435
$117$	0	0
$118$	4780.61	3.72958
$119$	−545.955	−0.420568
$120$	0	0
$121$	−1327.01	−0.997002
$122$	230.242	0.170862
$123$	0	0
$124$	4647.55	3.36582
$125$	−1462.33	−1.04636
$126$	0	0
$127$	−274.960	−0.192116	−0.0960582	−	0.995376i	$-0.530623\pi$
$127$	−274.960	−0.192116	−0.0960582	+	0.995376i	$0.530623\pi$
$128$	−6140.61	−4.24030
$129$	0	0
$130$	−2650.94	−1.78849
$131$	1277.00	0.851698	0.425849	−	0.904794i	$-0.359976\pi$
$131$	1277.00	0.851698	0.425849	+	0.904794i	$0.359976\pi$
$132$	0	0
$133$	−593.351	−0.386843
$134$	−5068.81	−3.26775
$135$	0	0
$136$	5502.43	3.46933
$137$	−968.683	−0.604089	−0.302044	−	0.953294i	$-0.597669\pi$
$137$	−968.683	−0.604089	−0.302044	+	0.953294i	$0.597669\pi$
$138$	0	0
$139$	1187.10	0.724376	0.362188	−	0.932105i	$-0.382030\pi$
$139$	1187.10	0.724376	0.362188	+	0.932105i	$0.382030\pi$
$140$	1849.77	1.11667
$141$	0	0
$142$	5902.22	3.48805
$143$	91.0227	0.0532287
$144$	0	0
$145$	2547.80	1.45920
$146$	−3329.16	−1.88714
$147$	0	0
$148$	382.685	0.212544
$149$	134.127	0.0737456	0.0368728	−	0.999320i	$-0.488260\pi$
$149$	134.127	0.0737456	0.0368728	+	0.999320i	$0.488260\pi$
$150$	0	0
$151$	−367.497	−0.198056	−0.0990282	−	0.995085i	$-0.531573\pi$
$151$	−367.497	−0.198056	−0.0990282	+	0.995085i	$0.531573\pi$
$152$	5980.11	3.19113
$153$	0	0
$154$	−86.2550	−0.0451339
$155$	2196.78	1.13839
$156$	0	0
$157$	1056.28	0.536944	0.268472	−	0.963287i	$-0.413481\pi$
$157$	1056.28	0.536944	0.268472	+	0.963287i	$0.413481\pi$
$158$	−6185.89	−3.11470
$159$	0	0
$160$	−8244.61	−4.07371
$161$	−1508.82	−0.738580
$162$	0	0
$163$	−1070.05	−0.514188	−0.257094	−	0.966386i	$-0.582765\pi$
$163$	−1070.05	−0.514188	−0.257094	+	0.966386i	$0.582765\pi$
$164$	−6248.78	−2.97529
$165$	0	0
$166$	322.827	0.150941
$167$	3582.55	1.66004	0.830019	−	0.557735i	$-0.188329\pi$
$167$	3582.55	1.66004	0.830019	+	0.557735i	$0.188329\pi$
$168$	0	0
$169$	−120.493	−0.0548444
$170$	4051.53	1.82787
$171$	0	0
$172$	−4176.66	−1.85156
$173$	3357.68	1.47561	0.737803	−	0.675016i	$-0.235863\pi$
$173$	3357.68	1.47561	0.737803	+	0.675016i	$0.235863\pi$
$174$	0	0
$175$	−105.558	−0.0455969
$176$	512.286	0.219403
$177$	0	0
$178$	7028.22	2.95948
$179$	937.441	0.391439	0.195720	−	0.980660i	$-0.437296\pi$
$179$	937.441	0.391439	0.195720	+	0.980660i	$0.437296\pi$
$180$	0	0
$181$	−394.462	−0.161990	−0.0809948	−	0.996715i	$-0.525810\pi$
$181$	−394.462	−0.161990	−0.0809948	+	0.996715i	$0.525810\pi$
$182$	−1967.74	−0.801421
$183$	0	0
$184$	15206.7	6.09266
$185$	180.886	0.0718865
$186$	0	0
$187$	−139.113	−0.0544008
$188$	11768.3	4.56538
$189$	0	0
$190$	4403.26	1.68129
$191$	−2099.68	−0.795430	−0.397715	−	0.917509i	$-0.630197\pi$
$191$	−2099.68	−0.795430	−0.397715	+	0.917509i	$0.630197\pi$
$192$	0	0
$193$	24.6580	0.00919649	0.00459824	−	0.999989i	$-0.498536\pi$
$193$	24.6580	0.00919649	0.00459824	+	0.999989i	$0.498536\pi$
$194$	6475.10	2.39631
$195$	0	0
$196$	−6290.60	−2.29249
$197$	552.688	0.199885	0.0999426	−	0.994993i	$-0.468134\pi$
$197$	552.688	0.199885	0.0999426	+	0.994993i	$0.468134\pi$
$198$	0	0
$199$	3371.51	1.20101	0.600503	−	0.799622i	$-0.294967\pi$
$199$	3371.51	1.20101	0.600503	+	0.799622i	$0.294967\pi$
$200$	1063.87	0.376136
$201$	0	0
$202$	4612.05	1.60645
$203$	1891.18	0.653866
$204$	0	0
$205$	−2953.64	−1.00630
$206$	918.518	0.310661
$207$	0	0
$208$	11686.8	3.89584
$209$	−151.190	−0.0500384
$210$	0	0
$211$	−5795.89	−1.89102	−0.945511	−	0.325591i	$-0.894437\pi$
$211$	−5795.89	−1.89102	−0.945511	+	0.325591i	$0.894437\pi$
$212$	14445.7	4.67988
$213$	0	0
$214$	11638.7	3.71777
$215$	−1974.21	−0.626232
$216$	0	0
$217$	1630.63	0.510111
$218$	10697.4	3.32347
$219$	0	0
$220$	471.334	0.144443
$221$	−3173.60	−0.965969
$222$	0	0
$223$	−5070.39	−1.52259	−0.761297	−	0.648403i	$-0.775437\pi$
$223$	−5070.39	−1.52259	−0.761297	+	0.648403i	$0.775437\pi$
$224$	−6119.80	−1.82543
$225$	0	0
$226$	555.657	0.163548
$227$	5498.54	1.60771	0.803857	−	0.594823i	$-0.202778\pi$
$227$	5498.54	1.60771	0.803857	+	0.594823i	$0.202778\pi$
$228$	0	0
$229$	−2538.05	−0.732396	−0.366198	−	0.930537i	$-0.619341\pi$
$229$	−2538.05	−0.732396	−0.366198	+	0.930537i	$0.619341\pi$
$230$	11196.9	3.21001
$231$	0	0
$232$	−19060.3	−5.39384
$233$	−493.643	−0.138797	−0.0693984	−	0.997589i	$-0.522108\pi$
$233$	−493.643	−0.138797	−0.0693984	+	0.997589i	$0.522108\pi$
$234$	0	0
$235$	5562.59	1.54410
$236$	−19390.8	−5.34845
$237$	0	0
$238$	3007.37	0.819069
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	950.309	0.254003	0.127002	−	0.991903i	$-0.459465\pi$
$241$	950.309	0.254003	0.127002	+	0.991903i	$0.459465\pi$
$242$	7309.77	1.94169
$243$	0	0
$244$	−933.892	−0.245026
$245$	−2973.41	−0.775365
$246$	0	0
$247$	−3449.11	−0.888508
$248$	−16434.3	−4.20798
$249$	0	0
$250$	8055.18	2.03782
$251$	3074.35	0.773112	0.386556	−	0.922266i	$-0.373665\pi$
$251$	3074.35	0.773112	0.386556	+	0.922266i	$0.373665\pi$
$252$	0	0
$253$	−384.456	−0.0955359
$254$	1514.60	0.374152
$255$	0	0
$256$	15836.9	3.86644
$257$	−1860.18	−0.451497	−0.225748	−	0.974186i	$-0.572483\pi$
$257$	−1860.18	−0.451497	−0.225748	+	0.974186i	$0.572483\pi$
$258$	0	0
$259$	134.268	0.0322123
$260$	10752.6	2.56480
$261$	0	0
$262$	−7034.31	−1.65871
$263$	−165.740	−0.0388592	−0.0194296	−	0.999811i	$-0.506185\pi$
$263$	−165.740	−0.0388592	−0.0194296	+	0.999811i	$0.506185\pi$
$264$	0	0
$265$	6828.13	1.58282
$266$	3268.45	0.753388
$267$	0	0
$268$	20559.8	4.68615
$269$	142.980	0.0324076	0.0162038	−	0.999869i	$-0.494842\pi$
$269$	142.980	0.0324076	0.0162038	+	0.999869i	$0.494842\pi$
$270$	0	0
$271$	−962.279	−0.215698	−0.107849	−	0.994167i	$-0.534396\pi$
$271$	−962.279	−0.215698	−0.107849	+	0.994167i	$0.534396\pi$
$272$	−17861.3	−3.98163
$273$	0	0
$274$	5335.94	1.17648
$275$	−26.8970	−0.00589799
$276$	0	0
$277$	−8015.42	−1.73863	−0.869314	−	0.494260i	$-0.835439\pi$
$277$	−8015.42	−1.73863	−0.869314	+	0.494260i	$0.835439\pi$
$278$	−6539.06	−1.41074
$279$	0	0
$280$	−6541.03	−1.39608
$281$	4986.31	1.05857	0.529285	−	0.848444i	$-0.322460\pi$
$281$	4986.31	1.05857	0.529285	+	0.848444i	$0.322460\pi$
$282$	0	0
$283$	−402.830	−0.0846141	−0.0423070	−	0.999105i	$-0.513471\pi$
$283$	−402.830	−0.0846141	−0.0423070	+	0.999105i	$0.513471\pi$
$284$	−23940.2	−5.00207
$285$	0	0
$286$	−501.394	−0.103664
$287$	−2192.43	−0.450923
$288$	0	0
$289$	−62.6859	−0.0127592
$290$	−14034.4	−2.84183
$291$	0	0
$292$	13503.5	2.70628
$293$	−5930.25	−1.18242	−0.591210	−	0.806518i	$-0.701349\pi$
$293$	−5930.25	−1.18242	−0.591210	+	0.806518i	$0.701349\pi$
$294$	0	0
$295$	−9165.56	−1.80895
$296$	−1353.22	−0.265725
$297$	0	0
$298$	−738.830	−0.143622
$299$	−8770.63	−1.69638
$300$	0	0
$301$	−1465.41	−0.280615
$302$	2024.34	0.385721
$303$	0	0
$304$	−19412.0	−3.66234
$305$	−441.428	−0.0828724
$306$	0	0
$307$	9974.53	1.85432	0.927161	−	0.374664i	$-0.122242\pi$
$307$	9974.53	1.85432	0.927161	+	0.374664i	$0.122242\pi$
$308$	349.862	0.0647247
$309$	0	0
$310$	−12100.9	−2.21704
$311$	−5615.57	−1.02389	−0.511945	−	0.859019i	$-0.671075\pi$
$311$	−5615.57	−1.02389	−0.511945	+	0.859019i	$0.671075\pi$
$312$	0	0
$313$	−10825.1	−1.95486	−0.977431	−	0.211253i	$-0.932246\pi$
$313$	−10825.1	−1.95486	−0.977431	+	0.211253i	$0.932246\pi$
$314$	−5818.46	−1.04572
$315$	0	0
$316$	25090.8	4.46667
$317$	6372.42	1.12906	0.564528	−	0.825414i	$-0.309058\pi$
$317$	6372.42	1.12906	0.564528	+	0.825414i	$0.309058\pi$
$318$	0	0
$319$	481.885	0.0845781
$320$	23746.8	4.14839
$321$	0	0
$322$	8311.23	1.43841
$323$	5271.39	0.908074
$324$	0	0
$325$	−613.603	−0.104728
$326$	5894.30	1.00140
$327$	0	0
$328$	22096.5	3.71974
$329$	4129.00	0.691911
$330$	0	0
$331$	−1856.48	−0.308281	−0.154141	−	0.988049i	$-0.549261\pi$
$331$	−1856.48	−0.308281	−0.154141	+	0.988049i	$0.549261\pi$
$332$	−1309.43	−0.216458
$333$	0	0
$334$	−19734.3	−3.23298
$335$	9718.10	1.58494
$336$	0	0
$337$	1658.71	0.268117	0.134059	−	0.990973i	$-0.457199\pi$
$337$	1658.71	0.268117	0.134059	+	0.990973i	$0.457199\pi$
$338$	663.731	0.106811
$339$	0	0
$340$	−16433.5	−2.62128
$341$	415.494	0.0659832
$342$	0	0
$343$	−4895.95	−0.770719
$344$	14769.2	2.31483
$345$	0	0
$346$	−18495.6	−2.87379
$347$	37.1295	0.00574414	0.00287207	−	0.999996i	$-0.499086\pi$
$347$	37.1295	0.00574414	0.00287207	+	0.999996i	$0.499086\pi$
$348$	0	0
$349$	−1906.17	−0.292364	−0.146182	−	0.989258i	$-0.546698\pi$
$349$	−1906.17	−0.292364	−0.146182	+	0.989258i	$0.546698\pi$
$350$	581.463	0.0888014
$351$	0	0
$352$	−1559.37	−0.236121
$353$	−4773.80	−0.719784	−0.359892	−	0.932994i	$-0.617187\pi$
$353$	−4773.80	−0.719784	−0.359892	+	0.932994i	$0.617187\pi$
$354$	0	0
$355$	−11315.9	−1.69180
$356$	−28507.4	−4.24407
$357$	0	0
$358$	−5163.84	−0.762340
$359$	3863.87	0.568042	0.284021	−	0.958818i	$-0.408331\pi$
$359$	3863.87	0.568042	0.284021	+	0.958818i	$0.408331\pi$
$360$	0	0
$361$	−1129.98	−0.164745
$362$	2172.87	0.315480
$363$	0	0
$364$	7981.42	1.14929
$365$	6382.78	0.915315
$366$	0	0
$367$	3339.79	0.475030	0.237515	−	0.971384i	$-0.423667\pi$
$367$	3339.79	0.475030	0.237515	+	0.971384i	$0.423667\pi$
$368$	−49362.1	−6.99233
$369$	0	0
$370$	−996.400	−0.140001
$371$	5068.38	0.709264
$372$	0	0
$373$	−1324.25	−0.183826	−0.0919132	−	0.995767i	$-0.529298\pi$
$373$	−1324.25	−0.183826	−0.0919132	+	0.995767i	$0.529298\pi$
$374$	766.297	0.105947
$375$	0	0
$376$	−41614.2	−5.70768
$377$	10993.3	1.50181
$378$	0	0
$379$	−2138.72	−0.289864	−0.144932	−	0.989442i	$-0.546296\pi$
$379$	−2138.72	−0.289864	−0.144932	+	0.989442i	$0.546296\pi$
$380$	−17860.2	−2.41108
$381$	0	0
$382$	11566.0	1.54912
$383$	−1766.82	−0.235719	−0.117859	−	0.993030i	$-0.537603\pi$
$383$	−1766.82	−0.235719	−0.117859	+	0.993030i	$0.537603\pi$
$384$	0	0
$385$	165.371	0.0218911
$386$	−135.827	−0.0179104
$387$	0	0
$388$	−26263.9	−3.43646
$389$	−9388.56	−1.22370	−0.611850	−	0.790974i	$-0.709574\pi$
$389$	−9388.56	−1.22370	−0.611850	+	0.790974i	$0.709574\pi$
$390$	0	0
$391$	13404.5	1.73374
$392$	22244.4	2.86610
$393$	0	0
$394$	−3044.45	−0.389283
$395$	11859.8	1.51071
$396$	0	0
$397$	−11015.9	−1.39263	−0.696314	−	0.717738i	$-0.745178\pi$
$397$	−11015.9	−1.39263	−0.696314	+	0.717738i	$0.745178\pi$
$398$	−18571.8	−2.33900
$399$	0	0
$400$	−3453.42	−0.431678
$401$	10180.2	1.26777	0.633886	−	0.773426i	$-0.281459\pi$
$401$	10180.2	1.26777	0.633886	+	0.773426i	$0.281459\pi$
$402$	0	0
$403$	9478.70	1.17163
$404$	−18707.1	−2.30374
$405$	0	0
$406$	−10417.5	−1.27342
$407$	34.2123	0.00416669
$408$	0	0
$409$	−7790.94	−0.941901	−0.470950	−	0.882160i	$-0.656089\pi$
$409$	−7790.94	−0.941901	−0.470950	+	0.882160i	$0.656089\pi$
$410$	16270.0	1.95980
$411$	0	0
$412$	−3725.63	−0.445507
$413$	−6803.40	−0.810590
$414$	0	0
$415$	−618.934	−0.0732104
$416$	−35574.0	−4.19269
$417$	0	0
$418$	832.822	0.0974513
$419$	−7927.41	−0.924294	−0.462147	−	0.886803i	$-0.652921\pi$
$419$	−7927.41	−0.924294	−0.462147	+	0.886803i	$0.652921\pi$
$420$	0	0
$421$	2106.86	0.243900	0.121950	−	0.992536i	$-0.461085\pi$
$421$	2106.86	0.243900	0.121950	+	0.992536i	$0.461085\pi$
$422$	31926.4	3.68282
$423$	0	0
$424$	−51081.8	−5.85083
$425$	937.790	0.107034
$426$	0	0
$427$	−327.663	−0.0371352
$428$	−47208.0	−5.33151
$429$	0	0
$430$	10874.8	1.21961
$431$	10875.8	1.21548	0.607738	−	0.794138i	$-0.292077\pi$
$431$	10875.8	1.21548	0.607738	+	0.794138i	$0.292077\pi$
$432$	0	0
$433$	8497.98	0.943157	0.471579	−	0.881824i	$-0.343684\pi$
$433$	8497.98	0.943157	0.471579	+	0.881824i	$0.343684\pi$
$434$	−8982.21	−0.993456
$435$	0	0
$436$	−43390.0	−4.76606
$437$	14568.1	1.59471
$438$	0	0
$439$	2339.92	0.254393	0.127196	−	0.991878i	$-0.459402\pi$
$439$	2339.92	0.254393	0.127196	+	0.991878i	$0.459402\pi$
$440$	−1666.70	−0.180583
$441$	0	0
$442$	17481.6	1.88125
$443$	17940.9	1.92415	0.962076	−	0.272782i	$-0.0879439\pi$
$443$	17940.9	1.92415	0.962076	+	0.272782i	$0.0879439\pi$
$444$	0	0
$445$	−13474.8	−1.43543
$446$	27930.0	2.96530
$447$	0	0
$448$	17626.7	1.85889
$449$	−8422.31	−0.885242	−0.442621	−	0.896709i	$-0.645951\pi$
$449$	−8422.31	−0.885242	−0.442621	+	0.896709i	$0.645951\pi$
$450$	0	0
$451$	−558.646	−0.0583273
$452$	−2253.82	−0.234537
$453$	0	0
$454$	−30288.4	−3.13107
$455$	3772.62	0.388711
$456$	0	0
$457$	−1158.06	−0.118538	−0.0592688	−	0.998242i	$-0.518877\pi$
$457$	−1158.06	−0.118538	−0.0592688	+	0.998242i	$0.518877\pi$
$458$	13980.7	1.42636
$459$	0	0
$460$	−45416.2	−4.60335
$461$	11152.7	1.12676	0.563378	−	0.826199i	$-0.309501\pi$
$461$	11152.7	1.12676	0.563378	+	0.826199i	$0.309501\pi$
$462$	0	0
$463$	−5448.05	−0.546852	−0.273426	−	0.961893i	$-0.588157\pi$
$463$	−5448.05	−0.546852	−0.273426	+	0.961893i	$0.588157\pi$
$464$	61871.4	6.19032
$465$	0	0
$466$	2719.21	0.270311
$467$	−17824.4	−1.76620	−0.883098	−	0.469188i	$-0.844547\pi$
$467$	−17824.4	−1.76620	−0.883098	+	0.469188i	$0.844547\pi$
$468$	0	0
$469$	7213.54	0.710214
$470$	−30641.2	−3.00718
$471$	0	0
$472$	68568.3	6.68668
$473$	−373.397	−0.0362977
$474$	0	0
$475$	1019.20	0.0984510
$476$	−12198.3	−1.17459
$477$	0	0
$478$	−1316.52	−0.125975
$479$	−7010.70	−0.668741	−0.334370	−	0.942442i	$-0.608524\pi$
$479$	−7010.70	−0.668741	−0.334370	+	0.942442i	$0.608524\pi$
$480$	0	0
$481$	780.489	0.0739859
$482$	−5234.73	−0.494679
$483$	0	0
$484$	−29649.4	−2.78450
$485$	−12414.3	−1.16228
$486$	0	0
$487$	6282.23	0.584549	0.292274	−	0.956335i	$-0.405588\pi$
$487$	6282.23	0.584549	0.292274	+	0.956335i	$0.405588\pi$
$488$	3302.36	0.306334
$489$	0	0
$490$	16378.9	1.51005
$491$	−5133.86	−0.471869	−0.235935	−	0.971769i	$-0.575815\pi$
$491$	−5133.86	−0.471869	−0.235935	+	0.971769i	$0.575815\pi$
$492$	0	0
$493$	−16801.4	−1.53488
$494$	18999.2	1.73040
$495$	0	0
$496$	53347.2	4.82935
$497$	−8399.59	−0.758095
$498$	0	0
$499$	8319.22	0.746332	0.373166	−	0.927765i	$-0.378272\pi$
$499$	8319.22	0.746332	0.373166	+	0.927765i	$0.378272\pi$
$500$	−32672.9	−2.92235
$501$	0	0
$502$	−16934.9	−1.50566
$503$	5639.36	0.499894	0.249947	−	0.968259i	$-0.419587\pi$
$503$	5639.36	0.499894	0.249947	+	0.968259i	$0.419587\pi$
$504$	0	0
$505$	−8842.38	−0.779170
$506$	2117.76	0.186059
$507$	0	0
$508$	−6143.44	−0.536557
$509$	−7593.79	−0.661275	−0.330637	−	0.943758i	$-0.607264\pi$
$509$	−7593.79	−0.661275	−0.330637	+	0.943758i	$0.607264\pi$
$510$	0	0
$511$	4737.80	0.410153
$512$	−38112.1	−3.28971
$513$	0	0
$514$	10246.7	0.879304
$515$	−1761.02	−0.150679
$516$	0	0
$517$	1052.10	0.0894993
$518$	−739.607	−0.0627345
$519$	0	0
$520$	−38022.5	−3.20653
$521$	−22251.0	−1.87108	−0.935540	−	0.353220i	$-0.885087\pi$
$521$	−22251.0	−1.87108	−0.935540	+	0.353220i	$0.885087\pi$
$522$	0	0
$523$	−10911.0	−0.912250	−0.456125	−	0.889916i	$-0.650763\pi$
$523$	−10911.0	−0.912250	−0.456125	+	0.889916i	$0.650763\pi$
$524$	28532.1	2.37868
$525$	0	0
$526$	912.972	0.0756796
$527$	−14486.6	−1.19743
$528$	0	0
$529$	24877.9	2.04470
$530$	−37612.4	−3.08260
$531$	0	0
$532$	−13257.3	−1.08040
$533$	−12744.4	−1.03569
$534$	0	0
$535$	−22314.1	−1.80322
$536$	−72701.9	−5.85867
$537$	0	0
$538$	−787.598	−0.0631148
$539$	−562.385	−0.0449418
$540$	0	0
$541$	−8012.61	−0.636764	−0.318382	−	0.947963i	$-0.603139\pi$
$541$	−8012.61	−0.636764	−0.318382	+	0.947963i	$0.603139\pi$
$542$	5300.66	0.420079
$543$	0	0
$544$	54368.9	4.28501
$545$	−20509.4	−1.61197
$546$	0	0
$547$	−18981.4	−1.48371	−0.741853	−	0.670563i	$-0.766053\pi$
$547$	−18981.4	−1.48371	−0.741853	+	0.670563i	$0.766053\pi$
$548$	−21643.3	−1.68714
$549$	0	0
$550$	148.161	0.0114865
$551$	−18260.0	−1.41180
$552$	0	0
$553$	8803.29	0.676951
$554$	44152.5	3.38603
$555$	0	0
$556$	26523.3	2.02309
$557$	−15478.2	−1.17743	−0.588717	−	0.808339i	$-0.700367\pi$
$557$	−15478.2	−1.17743	−0.588717	+	0.808339i	$0.700367\pi$
$558$	0	0
$559$	−8518.34	−0.644521
$560$	21232.7	1.60223
$561$	0	0
$562$	−27466.8	−2.06160
$563$	−2074.89	−0.155322	−0.0776609	−	0.996980i	$-0.524745\pi$
$563$	−2074.89	−0.155322	−0.0776609	+	0.996980i	$0.524745\pi$
$564$	0	0
$565$	−1065.33	−0.0793249
$566$	2218.97	0.164788
$567$	0	0
$568$	84655.6	6.25364
$569$	17795.6	1.31113	0.655563	−	0.755140i	$-0.272431\pi$
$569$	17795.6	1.31113	0.655563	+	0.755140i	$0.272431\pi$
$570$	0	0
$571$	−8545.21	−0.626280	−0.313140	−	0.949707i	$-0.601381\pi$
$571$	−8545.21	−0.626280	−0.313140	+	0.949707i	$0.601381\pi$
$572$	2033.72	0.148661
$573$	0	0
$574$	12076.9	0.878187
$575$	2591.70	0.187968
$576$	0	0
$577$	20661.6	1.49074	0.745368	−	0.666653i	$-0.232274\pi$
$577$	20661.6	1.49074	0.745368	+	0.666653i	$0.232274\pi$
$578$	345.302	0.0248489
$579$	0	0
$580$	56925.5	4.07535
$581$	−459.422	−0.0328056
$582$	0	0
$583$	1291.46	0.0917439
$584$	−47750.1	−3.38341
$585$	0	0
$586$	32666.5	2.30280
$587$	17607.6	1.23806	0.619031	−	0.785366i	$-0.287525\pi$
$587$	17607.6	1.23806	0.619031	+	0.785366i	$0.287525\pi$
$588$	0	0
$589$	−15744.3	−1.10141
$590$	50488.0	3.52298
$591$	0	0
$592$	4392.68	0.304963
$593$	−2180.35	−0.150989	−0.0754943	−	0.997146i	$-0.524053\pi$
$593$	−2180.35	−0.150989	−0.0754943	+	0.997146i	$0.524053\pi$
$594$	0	0
$595$	−5765.83	−0.397270
$596$	2996.79	0.205962
$597$	0	0
$598$	48312.6	3.30376
$599$	10967.9	0.748141	0.374070	−	0.927400i	$-0.377962\pi$
$599$	10967.9	0.748141	0.374070	+	0.927400i	$0.377962\pi$
$600$	0	0
$601$	−13808.1	−0.937178	−0.468589	−	0.883416i	$-0.655237\pi$
$601$	−13808.1	−0.937178	−0.468589	+	0.883416i	$0.655237\pi$
$602$	8072.15	0.546506
$603$	0	0
$604$	−8211.00	−0.553147
$605$	−14014.5	−0.941772
$606$	0	0
$607$	21992.4	1.47058	0.735292	−	0.677751i	$-0.237045\pi$
$607$	21992.4	1.47058	0.735292	+	0.677751i	$0.237045\pi$
$608$	59088.9	3.94140
$609$	0	0
$610$	2431.58	0.161397
$611$	24001.5	1.58920
$612$	0	0
$613$	−7358.60	−0.484847	−0.242424	−	0.970171i	$-0.577942\pi$
$613$	−7358.60	−0.484847	−0.242424	+	0.970171i	$0.577942\pi$
$614$	−54944.2	−3.61135
$615$	0	0
$616$	−1237.16	−0.0809195
$617$	16620.9	1.08450	0.542248	−	0.840218i	$-0.317573\pi$
$617$	16620.9	1.08450	0.542248	+	0.840218i	$0.317573\pi$
$618$	0	0
$619$	−590.078	−0.0383154	−0.0191577	−	0.999816i	$-0.506098\pi$
$619$	−590.078	−0.0383154	−0.0191577	+	0.999816i	$0.506098\pi$
$620$	49082.7	3.17937
$621$	0	0
$622$	30933.1	1.99406
$623$	−10002.0	−0.643215
$624$	0	0
$625$	−13760.5	−0.880672
$626$	59629.6	3.80716
$627$	0	0
$628$	23600.4	1.49962
$629$	−1192.85	−0.0756152
$630$	0	0
$631$	−23313.2	−1.47081	−0.735407	−	0.677625i	$-0.763009\pi$
$631$	−23313.2	−1.47081	−0.735407	+	0.677625i	$0.763009\pi$
$632$	−88724.3	−5.58427
$633$	0	0
$634$	−35102.1	−2.19887
$635$	−2903.85	−0.181474
$636$	0	0
$637$	−12829.7	−0.798010
$638$	−2654.44	−0.164718
$639$	0	0
$640$	−64850.9	−4.00540
$641$	12309.3	0.758486	0.379243	−	0.925297i	$-0.376184\pi$
$641$	12309.3	0.758486	0.379243	+	0.925297i	$0.376184\pi$
$642$	0	0
$643$	−5054.50	−0.310000	−0.155000	−	0.987914i	$-0.549538\pi$
$643$	−5054.50	−0.310000	−0.155000	+	0.987914i	$0.549538\pi$
$644$	−33711.5	−2.06276
$645$	0	0
$646$	−29037.2	−1.76850
$647$	−20978.5	−1.27473	−0.637366	−	0.770561i	$-0.719976\pi$
$647$	−20978.5	−1.27473	−0.637366	+	0.770561i	$0.719976\pi$
$648$	0	0
$649$	−1733.55	−0.104850
$650$	3380.00	0.203961
$651$	0	0
$652$	−23908.1	−1.43606
$653$	24927.9	1.49388	0.746939	−	0.664892i	$-0.231523\pi$
$653$	24927.9	1.49388	0.746939	+	0.664892i	$0.231523\pi$
$654$	0	0
$655$	13486.4	0.804517
$656$	−71727.0	−4.26901
$657$	0	0
$658$	−22744.4	−1.34752
$659$	−16232.2	−0.959512	−0.479756	−	0.877402i	$-0.659275\pi$
$659$	−16232.2	−0.959512	−0.479756	+	0.877402i	$0.659275\pi$
$660$	0	0
$661$	−17837.7	−1.04963	−0.524814	−	0.851217i	$-0.675865\pi$
$661$	−17837.7	−1.04963	−0.524814	+	0.851217i	$0.675865\pi$
$662$	10226.3	0.600387
$663$	0	0
$664$	4630.30	0.270618
$665$	−6266.38	−0.365413
$666$	0	0
$667$	−46432.8	−2.69548
$668$	80045.0	4.63628
$669$	0	0
$670$	−53531.6	−3.08673
$671$	−83.4907	−0.00480346
$672$	0	0
$673$	3050.26	0.174709	0.0873544	−	0.996177i	$-0.472159\pi$
$673$	3050.26	0.174709	0.0873544	+	0.996177i	$0.472159\pi$
$674$	−9136.90	−0.522166
$675$	0	0
$676$	−2692.18	−0.153174
$677$	−13940.7	−0.791407	−0.395704	−	0.918378i	$-0.629499\pi$
$677$	−13940.7	−0.791407	−0.395704	+	0.918378i	$0.629499\pi$
$678$	0	0
$679$	−9214.87	−0.520816
$680$	58111.1	3.27714
$681$	0	0
$682$	−2288.73	−0.128504
$683$	3706.59	0.207655	0.103828	−	0.994595i	$-0.466891\pi$
$683$	3706.59	0.207655	0.103828	+	0.994595i	$0.466891\pi$
$684$	0	0
$685$	−10230.3	−0.570625
$686$	26969.1	1.50100
$687$	0	0
$688$	−47942.1	−2.65665
$689$	29462.1	1.62905
$690$	0	0
$691$	10501.8	0.578160	0.289080	−	0.957305i	$-0.406651\pi$
$691$	10501.8	0.578160	0.289080	+	0.957305i	$0.406651\pi$
$692$	75020.7	4.12118
$693$	0	0
$694$	−204.526	−0.0111869
$695$	12536.9	0.684248
$696$	0	0
$697$	19477.7	1.05850
$698$	10500.0	0.569387
$699$	0	0
$700$	−2358.49	−0.127346
$701$	20261.3	1.09167	0.545834	−	0.837893i	$-0.316213\pi$
$701$	20261.3	1.09167	0.545834	+	0.837893i	$0.316213\pi$
$702$	0	0
$703$	−1296.40	−0.0695516
$704$	4491.41	0.240449
$705$	0	0
$706$	26296.2	1.40180
$707$	−6563.51	−0.349146
$708$	0	0
$709$	−24165.1	−1.28003	−0.640013	−	0.768364i	$-0.721071\pi$
$709$	−24165.1	−1.28003	−0.640013	+	0.768364i	$0.721071\pi$
$710$	62333.3	3.29483
$711$	0	0
$712$	100806.	5.30598
$713$	−40035.6	−2.10287
$714$	0	0
$715$	961.290	0.0502800
$716$	20945.2	1.09324
$717$	0	0
$718$	−21283.9	−1.10628
$719$	−21667.4	−1.12386	−0.561930	−	0.827185i	$-0.689941\pi$
$719$	−21667.4	−1.12386	−0.561930	+	0.827185i	$0.689941\pi$
$720$	0	0
$721$	−1307.17	−0.0675192
$722$	6224.46	0.320846
$723$	0	0
$724$	−8813.47	−0.452417
$725$	−3248.49	−0.166408
$726$	0	0
$727$	−17390.3	−0.887167	−0.443583	−	0.896233i	$-0.646293\pi$
$727$	−17390.3	−0.887167	−0.443583	+	0.896233i	$0.646293\pi$
$728$	−28223.3	−1.43685
$729$	0	0
$730$	−35159.2	−1.78260
$731$	13018.9	0.658714
$732$	0	0
$733$	6148.73	0.309834	0.154917	−	0.987927i	$-0.450489\pi$
$733$	6148.73	0.309834	0.154917	+	0.987927i	$0.450489\pi$
$734$	−18397.1	−0.925135
$735$	0	0
$736$	150255.	7.52512
$737$	1838.06	0.0918667
$738$	0	0
$739$	1621.93	0.0807358	0.0403679	−	0.999185i	$-0.487147\pi$
$739$	1621.93	0.0807358	0.0403679	+	0.999185i	$0.487147\pi$
$740$	4041.53	0.200770
$741$	0	0
$742$	−27918.9	−1.38131
$743$	−29316.7	−1.44754	−0.723772	−	0.690040i	$-0.757593\pi$
$743$	−29316.7	−1.44754	−0.723772	+	0.690040i	$0.757593\pi$
$744$	0	0
$745$	1416.51	0.0696603
$746$	7294.58	0.358007
$747$	0	0
$748$	−3108.20	−0.151935
$749$	−16563.3	−0.808023
$750$	0	0
$751$	14045.6	0.682467	0.341234	−	0.939979i	$-0.389155\pi$
$751$	14045.6	0.682467	0.341234	+	0.939979i	$0.389155\pi$
$752$	135083.	6.55051
$753$	0	0
$754$	−60555.9	−2.92482
$755$	−3881.14	−0.187085
$756$	0	0
$757$	3030.98	0.145525	0.0727627	−	0.997349i	$-0.476818\pi$
$757$	3030.98	0.145525	0.0727627	+	0.997349i	$0.476818\pi$
$758$	11781.0	0.564519
$759$	0	0
$760$	63155.9	3.01435
$761$	7201.80	0.343055	0.171528	−	0.985179i	$-0.445130\pi$
$761$	7201.80	0.343055	0.171528	+	0.985179i	$0.445130\pi$
$762$	0	0
$763$	−15223.7	−0.722325
$764$	−46913.1	−2.22154
$765$	0	0
$766$	9732.44	0.459070
$767$	−39547.7	−1.86178
$768$	0	0
$769$	19617.1	0.919910	0.459955	−	0.887942i	$-0.347866\pi$
$769$	19617.1	0.919910	0.459955	+	0.887942i	$0.347866\pi$
$770$	−910.938	−0.0426337
$771$	0	0
$772$	550.934	0.0256846
$773$	−14853.4	−0.691123	−0.345562	−	0.938396i	$-0.612312\pi$
$773$	−14853.4	−0.691123	−0.345562	+	0.938396i	$0.612312\pi$
$774$	0	0
$775$	−2800.93	−0.129823
$776$	92872.3	4.29629
$777$	0	0
$778$	51716.4	2.38319
$779$	21168.7	0.973615
$780$	0	0
$781$	−2140.27	−0.0980601
$782$	−73837.8	−3.37651
$783$	0	0
$784$	−72207.1	−3.28932
$785$	11155.4	0.507200
$786$	0	0
$787$	13843.0	0.627001	0.313500	−	0.949588i	$-0.398498\pi$
$787$	13843.0	0.627001	0.313500	+	0.949588i	$0.398498\pi$
$788$	12348.7	0.558255
$789$	0	0
$790$	−65329.2	−2.94216
$791$	−790.769	−0.0355455
$792$	0	0
$793$	−1904.68	−0.0852927
$794$	60680.6	2.71218
$795$	0	0
$796$	75329.8	3.35426
$797$	−20159.8	−0.895982	−0.447991	−	0.894038i	$-0.647860\pi$
$797$	−20159.8	−0.895982	−0.447991	+	0.894038i	$0.647860\pi$
$798$	0	0
$799$	−36682.4	−1.62419
$800$	10512.0	0.464570
$801$	0	0
$802$	−56077.3	−2.46902
$803$	1207.22	0.0530536
$804$	0	0
$805$	−15934.6	−0.697665
$806$	−52212.9	−2.28179
$807$	0	0
$808$	66150.6	2.88016
$809$	4023.03	0.174836	0.0874178	−	0.996172i	$-0.472138\pi$
$809$	4023.03	0.174836	0.0874178	+	0.996172i	$0.472138\pi$
$810$	0	0
$811$	−9998.22	−0.432904	−0.216452	−	0.976293i	$-0.569448\pi$
$811$	−9998.22	−0.432904	−0.216452	+	0.976293i	$0.569448\pi$
$812$	42254.6	1.82617
$813$	0	0
$814$	−188.457	−0.00811476
$815$	−11300.8	−0.485704
$816$	0	0
$817$	14149.1	0.605892
$818$	42916.0	1.83438
$819$	0	0
$820$	−65993.3	−2.81047
$821$	−32692.4	−1.38974	−0.694868	−	0.719138i	$-0.744537\pi$
$821$	−32692.4	−1.38974	−0.694868	+	0.719138i	$0.744537\pi$
$822$	0	0
$823$	14711.6	0.623102	0.311551	−	0.950229i	$-0.399152\pi$
$823$	14711.6	0.623102	0.311551	+	0.950229i	$0.399152\pi$
$824$	13174.3	0.556977
$825$	0	0
$826$	37476.2	1.57865
$827$	−1609.72	−0.0676848	−0.0338424	−	0.999427i	$-0.510774\pi$
$827$	−1609.72	−0.0676848	−0.0338424	+	0.999427i	$0.510774\pi$
$828$	0	0
$829$	35330.6	1.48020	0.740098	−	0.672499i	$-0.234779\pi$
$829$	35330.6	1.48020	0.740098	+	0.672499i	$0.234779\pi$
$830$	3409.37	0.142579
$831$	0	0
$832$	102463.	4.26954
$833$	19608.1	0.815583
$834$	0	0
$835$	37835.3	1.56808
$836$	−3378.04	−0.139751
$837$	0	0
$838$	43667.7	1.80009
$839$	−37947.8	−1.56151	−0.780753	−	0.624840i	$-0.785164\pi$
$839$	−37947.8	−1.56151	−0.780753	+	0.624840i	$0.785164\pi$
$840$	0	0
$841$	33810.8	1.38631
$842$	−11605.5	−0.475003
$843$	0	0
$844$	−129498.	−5.28139
$845$	−1272.53	−0.0518063
$846$	0	0
$847$	−10402.7	−0.422008
$848$	165816.	6.71479
$849$	0	0
$850$	−5165.77	−0.208452
$851$	−3296.58	−0.132791
$852$	0	0
$853$	25415.7	1.02019	0.510093	−	0.860119i	$-0.329611\pi$
$853$	25415.7	1.02019	0.510093	+	0.860119i	$0.329611\pi$
$854$	1804.91	0.0723219
$855$	0	0
$856$	166934.	6.66551
$857$	19890.4	0.792814	0.396407	−	0.918075i	$-0.370257\pi$
$857$	19890.4	0.792814	0.396407	+	0.918075i	$0.370257\pi$
$858$	0	0
$859$	−38133.3	−1.51466	−0.757328	−	0.653034i	$-0.773496\pi$
$859$	−38133.3	−1.51466	−0.757328	+	0.653034i	$0.773496\pi$
$860$	−44109.7	−1.74899
$861$	0	0
$862$	−59908.9	−2.36718
$863$	31421.2	1.23938	0.619692	−	0.784845i	$-0.287257\pi$
$863$	31421.2	1.23938	0.619692	+	0.784845i	$0.287257\pi$
$864$	0	0
$865$	35460.5	1.39386
$866$	−46810.7	−1.83683
$867$	0	0
$868$	36433.1	1.42468
$869$	2243.14	0.0875641
$870$	0	0
$871$	41931.8	1.63123
$872$	153432.	5.95857
$873$	0	0
$874$	−80247.9	−3.10575
$875$	−11463.5	−0.442900
$876$	0	0
$877$	−7517.74	−0.289460	−0.144730	−	0.989471i	$-0.546231\pi$
$877$	−7517.74	−0.289460	−0.144730	+	0.989471i	$0.546231\pi$
$878$	−12889.3	−0.495437
$879$	0	0
$880$	5410.24	0.207249
$881$	−5727.47	−0.219028	−0.109514	−	0.993985i	$-0.534929\pi$
$881$	−5727.47	−0.219028	−0.109514	+	0.993985i	$0.534929\pi$
$882$	0	0
$883$	−27334.5	−1.04177	−0.520883	−	0.853628i	$-0.674397\pi$
$883$	−27334.5	−1.04177	−0.520883	+	0.853628i	$0.674397\pi$
$884$	−70907.7	−2.69783
$885$	0	0
$886$	−98826.7	−3.74734
$887$	−8580.36	−0.324803	−0.162401	−	0.986725i	$-0.551924\pi$
$887$	−8580.36	−0.324803	−0.162401	+	0.986725i	$0.551924\pi$
$888$	0	0
$889$	−2155.47	−0.0813185
$890$	74225.0	2.79554
$891$	0	0
$892$	−113288.	−4.25241
$893$	−39866.9	−1.49395
$894$	0	0
$895$	9900.30	0.369755
$896$	−48137.5	−1.79482
$897$	0	0
$898$	46393.9	1.72404
$899$	50181.4	1.86167
$900$	0	0
$901$	−45027.9	−1.66492
$902$	3077.27	0.113594
$903$	0	0
$904$	7969.79	0.293220
$905$	−4165.91	−0.153016
$906$	0	0
$907$	−6405.19	−0.234488	−0.117244	−	0.993103i	$-0.537406\pi$
$907$	−6405.19	−0.234488	−0.117244	+	0.993103i	$0.537406\pi$
$908$	122854.	4.49014
$909$	0	0
$910$	−20781.3	−0.757026
$911$	−18005.8	−0.654839	−0.327419	−	0.944879i	$-0.606179\pi$
$911$	−18005.8	−0.654839	−0.327419	+	0.944879i	$0.606179\pi$
$912$	0	0
$913$	−117.064	−0.00424343
$914$	6379.11	0.230856
$915$	0	0
$916$	−56707.6	−2.04549
$917$	10010.7	0.360504
$918$	0	0
$919$	−4778.16	−0.171509	−0.0857546	−	0.996316i	$-0.527330\pi$
$919$	−4778.16	−0.171509	−0.0857546	+	0.996316i	$0.527330\pi$
$920$	160597.	5.75515
$921$	0	0
$922$	−61434.3	−2.19439
$923$	−48826.2	−1.74121
$924$	0	0
$925$	−230.632	−0.00819800
$926$	30010.3	1.06501
$927$	0	0
$928$	−188333.	−6.66200
$929$	47560.0	1.67965	0.839825	−	0.542858i	$-0.182658\pi$
$929$	47560.0	1.67965	0.839825	+	0.542858i	$0.182658\pi$
$930$	0	0
$931$	21310.4	0.750181
$932$	−11029.5	−0.387642
$933$	0	0
$934$	98184.7	3.43972
$935$	−1469.17	−0.0513872
$936$	0	0
$937$	−21536.2	−0.750863	−0.375431	−	0.926850i	$-0.622505\pi$
$937$	−21536.2	−0.750863	−0.375431	+	0.926850i	$0.622505\pi$
$938$	−39735.4	−1.38316
$939$	0	0
$940$	124285.	4.31248
$941$	37432.4	1.29677	0.648385	−	0.761313i	$-0.275445\pi$
$941$	37432.4	1.29677	0.648385	+	0.761313i	$0.275445\pi$
$942$	0	0
$943$	53829.2	1.85888
$944$	−222579.	−7.67407
$945$	0	0
$946$	2056.84	0.0706909
$947$	−17622.9	−0.604717	−0.302358	−	0.953194i	$-0.597774\pi$
$947$	−17622.9	−0.604717	−0.302358	+	0.953194i	$0.597774\pi$
$948$	0	0
$949$	27540.5	0.942046
$950$	−5614.22	−0.191736
$951$	0	0
$952$	43134.6	1.46849
$953$	−10322.2	−0.350860	−0.175430	−	0.984492i	$-0.556132\pi$
$953$	−10322.2	−0.350860	−0.175430	+	0.984492i	$0.556132\pi$
$954$	0	0
$955$	−22174.7	−0.751367
$956$	5339.98	0.180656
$957$	0	0
$958$	38618.1	1.30239
$959$	−7593.70	−0.255697
$960$	0	0
$961$	13476.7	0.452377
$962$	−4299.28	−0.144090
$963$	0	0
$964$	21232.8	0.709399
$965$	260.413	0.00868704
$966$	0	0
$967$	16993.8	0.565133	0.282567	−	0.959248i	$-0.408814\pi$
$967$	16993.8	0.565133	0.282567	+	0.959248i	$0.408814\pi$
$968$	104844.	3.48121
$969$	0	0
$970$	68383.5	2.26357
$971$	−35368.3	−1.16892	−0.584460	−	0.811422i	$-0.698694\pi$
$971$	−35368.3	−1.16892	−0.584460	+	0.811422i	$0.698694\pi$
$972$	0	0
$973$	9305.89	0.306612
$974$	−34605.4	−1.13843
$975$	0	0
$976$	−10719.7	−0.351568
$977$	−58537.4	−1.91687	−0.958433	−	0.285319i	$-0.907900\pi$
$977$	−58537.4	−1.91687	−0.958433	+	0.285319i	$0.907900\pi$
$978$	0	0
$979$	−2548.58	−0.0832003
$980$	−66435.0	−2.16550
$981$	0	0
$982$	28279.6	0.918980
$983$	58363.6	1.89370	0.946851	−	0.321671i	$-0.104245\pi$
$983$	58363.6	1.89370	0.946851	+	0.321671i	$0.104245\pi$
$984$	0	0
$985$	5836.94	0.188812
$986$	92549.7	2.98923
$987$	0	0
$988$	−77063.4	−2.48149
$989$	35979.3	1.15680
$990$	0	0
$991$	−8406.46	−0.269465	−0.134733	−	0.990882i	$-0.543018\pi$
$991$	−8406.46	−0.269465	−0.134733	+	0.990882i	$0.543018\pi$
$992$	−162386.	−5.19733
$993$	0	0
$994$	46268.7	1.47641
$995$	35606.5	1.13448
$996$	0	0
$997$	−13058.3	−0.414804	−0.207402	−	0.978256i	$-0.566501\pi$
$997$	−13058.3	−0.414804	−0.207402	+	0.978256i	$0.566501\pi$
$998$	−45826.0	−1.45350
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.g.1.2	✓	59
3.2	odd	2		2151.4.a.h.1.58	yes	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.2	✓	59	1.1	even	1	trivial
2151.4.a.h.1.58	yes	59	3.2	odd	2

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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-5.50845 q^{2} +22.3430 q^{4} +10.5610 q^{5} +7.83920 q^{7} -79.0077 q^{8} +O(q^{10})\)	\(q-5.50845 q^{2} +22.3430 q^{4} +10.5610 q^{5} +7.83920 q^{7} -79.0077 q^{8} -58.1747 q^{10} +1.99748 q^{11} +45.5687 q^{13} -43.1818 q^{14} +256.466 q^{16} -69.6442 q^{17} -75.6903 q^{19} +235.964 q^{20} -11.0030 q^{22} -192.471 q^{23} -13.4654 q^{25} -251.013 q^{26} +175.151 q^{28} +241.246 q^{29} +208.009 q^{31} -780.667 q^{32} +383.631 q^{34} +82.7898 q^{35} +17.1277 q^{37} +416.936 q^{38} -834.400 q^{40} -279.675 q^{41} -186.934 q^{43} +44.6297 q^{44} +1060.21 q^{46} +526.711 q^{47} -281.547 q^{49} +74.1737 q^{50} +1018.14 q^{52} +646.542 q^{53} +21.0954 q^{55} -619.357 q^{56} -1328.89 q^{58} -867.869 q^{59} -41.7980 q^{61} -1145.81 q^{62} +2248.54 q^{64} +481.251 q^{65} +920.188 q^{67} -1556.06 q^{68} -456.043 q^{70} -1071.49 q^{71} +604.373 q^{73} -94.3472 q^{74} -1691.15 q^{76} +15.6587 q^{77} +1122.98 q^{79} +2708.53 q^{80} +1540.57 q^{82} -58.6057 q^{83} -735.512 q^{85} +1029.72 q^{86} -157.816 q^{88} -1275.90 q^{89} +357.222 q^{91} -4300.37 q^{92} -2901.36 q^{94} -799.364 q^{95} -1175.49 q^{97} +1550.89 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8} - 36 q^{10} - 132 q^{11} + 104 q^{13} - 280 q^{14} + 822 q^{16} - 408 q^{17} + 20 q^{19} - 800 q^{20} - 2 q^{22} - 276 q^{23} + 1477 q^{25} - 780 q^{26} + 224 q^{28} - 696 q^{29} - 380 q^{31} - 896 q^{32} - 72 q^{34} - 700 q^{35} + 224 q^{37} - 988 q^{38} - 258 q^{40} - 2706 q^{41} - 156 q^{43} - 1584 q^{44} + 428 q^{46} - 1316 q^{47} + 2135 q^{49} - 1400 q^{50} + 1092 q^{52} - 1484 q^{53} - 992 q^{55} - 3360 q^{56} - 120 q^{58} - 3186 q^{59} - 254 q^{61} - 1240 q^{62} + 3054 q^{64} - 5120 q^{65} + 288 q^{67} - 9420 q^{68} + 1108 q^{70} - 4468 q^{71} - 1770 q^{73} - 6214 q^{74} + 720 q^{76} - 6352 q^{77} - 746 q^{79} - 7040 q^{80} + 276 q^{82} - 5484 q^{83} + 588 q^{85} - 10152 q^{86} + 1186 q^{88} - 11570 q^{89} + 1768 q^{91} - 15366 q^{92} - 2142 q^{94} - 5736 q^{95} + 2390 q^{97} - 6912 q^{98}+O(q^{100}) \) 59 * q - 8 * q^2 + 238 * q^4 - 80 * q^5 - 10 * q^7 - 96 * q^8 - 36 * q^10 - 132 * q^11 + 104 * q^13 - 280 * q^14 + 822 * q^16 - 408 * q^17 + 20 * q^19 - 800 * q^20 - 2 * q^22 - 276 * q^23 + 1477 * q^25 - 780 * q^26 + 224 * q^28 - 696 * q^29 - 380 * q^31 - 896 * q^32 - 72 * q^34 - 700 * q^35 + 224 * q^37 - 988 * q^38 - 258 * q^40 - 2706 * q^41 - 156 * q^43 - 1584 * q^44 + 428 * q^46 - 1316 * q^47 + 2135 * q^49 - 1400 * q^50 + 1092 * q^52 - 1484 * q^53 - 992 * q^55 - 3360 * q^56 - 120 * q^58 - 3186 * q^59 - 254 * q^61 - 1240 * q^62 + 3054 * q^64 - 5120 * q^65 + 288 * q^67 - 9420 * q^68 + 1108 * q^70 - 4468 * q^71 - 1770 * q^73 - 6214 * q^74 + 720 * q^76 - 6352 * q^77 - 746 * q^79 - 7040 * q^80 + 276 * q^82 - 5484 * q^83 + 588 * q^85 - 10152 * q^86 + 1186 * q^88 - 11570 * q^89 + 1768 * q^91 - 15366 * q^92 - 2142 * q^94 - 5736 * q^95 + 2390 * q^97 - 6912 * q^98

Introduction

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