LMFDB - Embedded newform 2151.4.a.g.1.8

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			1.8
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)

$f(q)$	$=$	$q-4.59357 q^{2} +13.1009 q^{4} +7.15514 q^{5} +33.8683 q^{7} -23.4314 q^{8} +O(q^{10})$	\(q-4.59357 q^{2} +13.1009 q^{4} +7.15514 q^{5} +33.8683 q^{7} -23.4314 q^{8} -32.8677 q^{10} -63.8154 q^{11} +54.1173 q^{13} -155.576 q^{14} +2.82656 q^{16} -7.31099 q^{17} +84.0618 q^{19} +93.7388 q^{20} +293.141 q^{22} -79.5642 q^{23} -73.8040 q^{25} -248.592 q^{26} +443.705 q^{28} +116.594 q^{29} -301.586 q^{31} +174.467 q^{32} +33.5835 q^{34} +242.332 q^{35} +69.9127 q^{37} -386.144 q^{38} -167.655 q^{40} -114.218 q^{41} +91.0380 q^{43} -836.040 q^{44} +365.484 q^{46} +238.651 q^{47} +804.060 q^{49} +339.024 q^{50} +708.986 q^{52} -588.700 q^{53} -456.608 q^{55} -793.581 q^{56} -535.585 q^{58} -70.1289 q^{59} -333.530 q^{61} +1385.36 q^{62} -824.040 q^{64} +387.217 q^{65} -885.306 q^{67} -95.7806 q^{68} -1113.17 q^{70} -29.9178 q^{71} -715.131 q^{73} -321.149 q^{74} +1101.29 q^{76} -2161.32 q^{77} -522.551 q^{79} +20.2244 q^{80} +524.670 q^{82} -1222.84 q^{83} -52.3111 q^{85} -418.190 q^{86} +1495.28 q^{88} -1086.73 q^{89} +1832.86 q^{91} -1042.36 q^{92} -1096.26 q^{94} +601.474 q^{95} -1304.84 q^{97} -3693.51 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−4.59357	−1.62407	−0.812037	−	0.583607i	$-0.801641\pi$
$2$	−4.59357	−1.62407	−0.812037	+	0.583607i	$0.801641\pi$
$3$	0	0
$3$	0	0
$4$	13.1009	1.63761
$5$	7.15514	0.639975	0.319988	−	0.947422i	$-0.396321\pi$
$5$	7.15514	0.639975	0.319988	+	0.947422i	$0.396321\pi$
$6$	0	0
$7$	33.8683	1.82871	0.914357	−	0.404908i	$-0.132696\pi$
$7$	33.8683	1.82871	0.914357	+	0.404908i	$0.132696\pi$
$8$	−23.4314	−1.03553
$9$	0	0
$10$	−32.8677	−1.03937
$11$	−63.8154	−1.74919	−0.874594	−	0.484855i	$-0.838872\pi$
$11$	−63.8154	−1.74919	−0.874594	+	0.484855i	$0.838872\pi$
$12$	0	0
$13$	54.1173	1.15457	0.577286	−	0.816542i	$-0.304112\pi$
$13$	54.1173	1.15457	0.577286	+	0.816542i	$0.304112\pi$
$14$	−155.576	−2.96997
$15$	0	0
$16$	2.82656	0.0441650
$17$	−7.31099	−0.104304	−0.0521522	−	0.998639i	$-0.516608\pi$
$17$	−7.31099	−0.104304	−0.0521522	+	0.998639i	$0.516608\pi$
$18$	0	0
$19$	84.0618	1.01501	0.507503	−	0.861650i	$-0.330569\pi$
$19$	84.0618	1.01501	0.507503	+	0.861650i	$0.330569\pi$
$20$	93.7388	1.04803
$21$	0	0
$22$	293.141	2.84081
$23$	−79.5642	−0.721316	−0.360658	−	0.932698i	$-0.617448\pi$
$23$	−79.5642	−0.721316	−0.360658	+	0.932698i	$0.617448\pi$
$24$	0	0
$25$	−73.8040	−0.590432
$26$	−248.592	−1.87511
$27$	0	0
$28$	443.705	2.99473
$29$	116.594	0.746588	0.373294	−	0.927713i	$-0.378228\pi$
$29$	116.594	0.746588	0.373294	+	0.927713i	$0.378228\pi$
$30$	0	0
$31$	−301.586	−1.74731	−0.873653	−	0.486549i	$-0.838256\pi$
$31$	−301.586	−1.74731	−0.873653	+	0.486549i	$0.838256\pi$
$32$	174.467	0.963804
$33$	0	0
$34$	33.5835	0.169398
$35$	242.332	1.17033
$36$	0	0
$37$	69.9127	0.310637	0.155319	−	0.987864i	$-0.450360\pi$
$37$	69.9127	0.310637	0.155319	+	0.987864i	$0.450360\pi$
$38$	−386.144	−1.64844
$39$	0	0
$40$	−167.655	−0.662714
$41$	−114.218	−0.435071	−0.217535	−	0.976052i	$-0.569802\pi$
$41$	−114.218	−0.435071	−0.217535	+	0.976052i	$0.569802\pi$
$42$	0	0
$43$	91.0380	0.322864	0.161432	−	0.986884i	$-0.448389\pi$
$43$	91.0380	0.322864	0.161432	+	0.986884i	$0.448389\pi$
$44$	−836.040	−2.86450
$45$	0	0
$46$	365.484	1.17147
$47$	238.651	0.740656	0.370328	−	0.928901i	$-0.379245\pi$
$47$	238.651	0.740656	0.370328	+	0.928901i	$0.379245\pi$
$48$	0	0
$49$	804.060	2.34420
$50$	339.024	0.958905
$51$	0	0
$52$	708.986	1.89074
$53$	−588.700	−1.52574	−0.762869	−	0.646553i	$-0.776210\pi$
$53$	−588.700	−1.52574	−0.762869	+	0.646553i	$0.776210\pi$
$54$	0	0
$55$	−456.608	−1.11944
$56$	−793.581	−1.89369
$57$	0	0
$58$	−535.585	−1.21251
$59$	−70.1289	−0.154746	−0.0773729	−	0.997002i	$-0.524653\pi$
$59$	−70.1289	−0.154746	−0.0773729	+	0.997002i	$0.524653\pi$
$60$	0	0
$61$	−333.530	−0.700068	−0.350034	−	0.936737i	$-0.613830\pi$
$61$	−333.530	−0.700068	−0.350034	+	0.936737i	$0.613830\pi$
$62$	1385.36	2.83775
$63$	0	0
$64$	−824.040	−1.60945
$65$	387.217	0.738898
$66$	0	0
$67$	−885.306	−1.61429	−0.807144	−	0.590354i	$-0.798988\pi$
$67$	−885.306	−1.61429	−0.807144	+	0.590354i	$0.798988\pi$
$68$	−95.7806	−0.170810
$69$	0	0
$70$	−1113.17	−1.90070
$71$	−29.9178	−0.0500082	−0.0250041	−	0.999687i	$-0.507960\pi$
$71$	−29.9178	−0.0500082	−0.0250041	+	0.999687i	$0.507960\pi$
$72$	0	0
$73$	−715.131	−1.14657	−0.573286	−	0.819356i	$-0.694331\pi$
$73$	−715.131	−1.14657	−0.573286	+	0.819356i	$0.694331\pi$
$74$	−321.149	−0.504498
$75$	0	0
$76$	1101.29	1.66219
$77$	−2161.32	−3.19877
$78$	0	0
$79$	−522.551	−0.744197	−0.372099	−	0.928193i	$-0.621362\pi$
$79$	−522.551	−0.744197	−0.372099	+	0.928193i	$0.621362\pi$
$80$	20.2244	0.0282645
$81$	0	0
$82$	524.670	0.706586
$83$	−1222.84	−1.61716	−0.808581	−	0.588384i	$-0.799764\pi$
$83$	−1222.84	−1.61716	−0.808581	+	0.588384i	$0.799764\pi$
$84$	0	0
$85$	−52.3111	−0.0667522
$86$	−418.190	−0.524355
$87$	0	0
$88$	1495.28	1.81134
$89$	−1086.73	−1.29430	−0.647150	−	0.762362i	$-0.724039\pi$
$89$	−1086.73	−1.29430	−0.647150	+	0.762362i	$0.724039\pi$
$90$	0	0
$91$	1832.86	2.11138
$92$	−1042.36	−1.18124
$93$	0	0
$94$	−1096.26	−1.20288
$95$	601.474	0.649578
$96$	0	0
$97$	−1304.84	−1.36584	−0.682921	−	0.730492i	$-0.739291\pi$
$97$	−1304.84	−1.36584	−0.682921	+	0.730492i	$0.739291\pi$
$98$	−3693.51	−3.80715
$99$	0	0
$100$	−966.899	−0.966899
$101$	170.901	0.168369	0.0841846	−	0.996450i	$-0.473171\pi$
$101$	170.901	0.168369	0.0841846	+	0.996450i	$0.473171\pi$
$102$	0	0
$103$	647.150	0.619083	0.309541	−	0.950886i	$-0.399824\pi$
$103$	647.150	0.619083	0.309541	+	0.950886i	$0.399824\pi$
$104$	−1268.04	−1.19560
$105$	0	0
$106$	2704.24	2.47791
$107$	−800.769	−0.723488	−0.361744	−	0.932277i	$-0.617819\pi$
$107$	−800.769	−0.723488	−0.361744	+	0.932277i	$0.617819\pi$
$108$	0	0
$109$	293.322	0.257754	0.128877	−	0.991661i	$-0.458863\pi$
$109$	293.322	0.257754	0.128877	+	0.991661i	$0.458863\pi$
$110$	2097.46	1.81805
$111$	0	0
$112$	95.7306	0.0807651
$113$	1218.50	1.01440	0.507199	−	0.861829i	$-0.330681\pi$
$113$	1218.50	1.01440	0.507199	+	0.861829i	$0.330681\pi$
$114$	0	0
$115$	−569.293	−0.461624
$116$	1527.49	1.22262
$117$	0	0
$118$	322.142	0.251319
$119$	−247.610	−0.190743
$120$	0	0
$121$	2741.41	2.05966
$122$	1532.09	1.13696
$123$	0	0
$124$	−3951.06	−2.86141
$125$	−1422.47	−1.01784
$126$	0	0
$127$	1932.54	1.35028	0.675138	−	0.737692i	$-0.264084\pi$
$127$	1932.54	1.35028	0.675138	+	0.737692i	$0.264084\pi$
$128$	2389.55	1.65007
$129$	0	0
$130$	−1778.71	−1.20002
$131$	−203.894	−0.135987	−0.0679935	−	0.997686i	$-0.521660\pi$
$131$	−203.894	−0.135987	−0.0679935	+	0.997686i	$0.521660\pi$
$132$	0	0
$133$	2847.03	1.85615
$134$	4066.72	2.62172
$135$	0	0
$136$	171.307	0.108010
$137$	−220.148	−0.137288	−0.0686442	−	0.997641i	$-0.521867\pi$
$137$	−220.148	−0.137288	−0.0686442	+	0.997641i	$0.521867\pi$
$138$	0	0
$139$	−989.877	−0.604031	−0.302015	−	0.953303i	$-0.597659\pi$
$139$	−989.877	−0.604031	−0.302015	+	0.953303i	$0.597659\pi$
$140$	3174.77	1.91655
$141$	0	0
$142$	137.429	0.0812170
$143$	−3453.52	−2.01957
$144$	0	0
$145$	834.250	0.477798
$146$	3285.00	1.86212
$147$	0	0
$148$	915.921	0.508704
$149$	−2285.17	−1.25643	−0.628216	−	0.778039i	$-0.716215\pi$
$149$	−2285.17	−1.25643	−0.628216	+	0.778039i	$0.716215\pi$
$150$	0	0
$151$	−396.925	−0.213916	−0.106958	−	0.994264i	$-0.534111\pi$
$151$	−396.925	−0.213916	−0.106958	+	0.994264i	$0.534111\pi$
$152$	−1969.69	−1.05107
$153$	0	0
$154$	9928.17	5.19503
$155$	−2157.89	−1.11823
$156$	0	0
$157$	−1676.43	−0.852187	−0.426093	−	0.904679i	$-0.640111\pi$
$157$	−1676.43	−0.852187	−0.426093	+	0.904679i	$0.640111\pi$
$158$	2400.38	1.20863
$159$	0	0
$160$	1248.34	0.616811
$161$	−2694.70	−1.31908
$162$	0	0
$163$	−342.528	−0.164594	−0.0822971	−	0.996608i	$-0.526226\pi$
$163$	−342.528	−0.164594	−0.0822971	+	0.996608i	$0.526226\pi$
$164$	−1496.36	−0.712477
$165$	0	0
$166$	5617.22	2.62639
$167$	−562.445	−0.260619	−0.130309	−	0.991473i	$-0.541597\pi$
$167$	−562.445	−0.260619	−0.130309	+	0.991473i	$0.541597\pi$
$168$	0	0
$169$	731.683	0.333037
$170$	240.295	0.108410
$171$	0	0
$172$	1192.68	0.528727
$173$	−3896.49	−1.71240	−0.856199	−	0.516647i	$-0.827180\pi$
$173$	−3896.49	−1.71240	−0.856199	+	0.516647i	$0.827180\pi$
$174$	0	0
$175$	−2499.61	−1.07973
$176$	−180.378	−0.0772529
$177$	0	0
$178$	4991.96	2.10204
$179$	2642.79	1.10353	0.551764	−	0.834000i	$-0.313955\pi$
$179$	2642.79	1.10353	0.551764	+	0.834000i	$0.313955\pi$
$180$	0	0
$181$	2310.24	0.948722	0.474361	−	0.880330i	$-0.342679\pi$
$181$	2310.24	0.948722	0.474361	+	0.880330i	$0.342679\pi$
$182$	−8419.37	−3.42904
$183$	0	0
$184$	1864.30	0.746946
$185$	500.235	0.198800
$186$	0	0
$187$	466.554	0.182448
$188$	3126.55	1.21291
$189$	0	0
$190$	−2762.91	−1.05496
$191$	5147.86	1.95019	0.975095	−	0.221788i	$-0.0711894\pi$
$191$	5147.86	1.95019	0.975095	+	0.221788i	$0.0711894\pi$
$192$	0	0
$193$	−4976.01	−1.85586	−0.927929	−	0.372756i	$-0.878413\pi$
$193$	−4976.01	−1.85586	−0.927929	+	0.372756i	$0.878413\pi$
$194$	5993.89	2.21823
$195$	0	0
$196$	10533.9	3.83889
$197$	4981.42	1.80158	0.900791	−	0.434253i	$-0.142988\pi$
$197$	4981.42	1.80158	0.900791	+	0.434253i	$0.142988\pi$
$198$	0	0
$199$	2586.81	0.921479	0.460739	−	0.887536i	$-0.347584\pi$
$199$	2586.81	0.921479	0.460739	+	0.887536i	$0.347584\pi$
$200$	1729.33	0.611411
$201$	0	0
$202$	−785.047	−0.273444
$203$	3948.85	1.36530
$204$	0	0
$205$	−817.247	−0.278434
$206$	−2972.73	−1.00544
$207$	0	0
$208$	152.966	0.0509916
$209$	−5364.44	−1.77544
$210$	0	0
$211$	5336.91	1.74127	0.870635	−	0.491929i	$-0.163708\pi$
$211$	5336.91	1.74127	0.870635	+	0.491929i	$0.163708\pi$
$212$	−7712.50	−2.49857
$213$	0	0
$214$	3678.39	1.17500
$215$	651.390	0.206625
$216$	0	0
$217$	−10214.2	−3.19533
$218$	−1347.40	−0.418611
$219$	0	0
$220$	−5981.98	−1.83321
$221$	−395.651	−0.120427
$222$	0	0
$223$	1531.01	0.459749	0.229874	−	0.973220i	$-0.426168\pi$
$223$	1531.01	0.459749	0.229874	+	0.973220i	$0.426168\pi$
$224$	5908.90	1.76252
$225$	0	0
$226$	−5597.27	−1.64746
$227$	−4486.94	−1.31193	−0.655966	−	0.754790i	$-0.727739\pi$
$227$	−4486.94	−1.31193	−0.655966	+	0.754790i	$0.727739\pi$
$228$	0	0
$229$	2640.88	0.762072	0.381036	−	0.924560i	$-0.375567\pi$
$229$	2640.88	0.762072	0.381036	+	0.924560i	$0.375567\pi$
$230$	2615.09	0.749712
$231$	0	0
$232$	−2731.97	−0.773115
$233$	4682.96	1.31670	0.658350	−	0.752712i	$-0.271255\pi$
$233$	4682.96	1.31670	0.658350	+	0.752712i	$0.271255\pi$
$234$	0	0
$235$	1707.58	0.474002
$236$	−918.752	−0.253414
$237$	0	0
$238$	1137.42	0.309781
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	−3304.99	−0.883375	−0.441688	−	0.897169i	$-0.645620\pi$
$241$	−3304.99	−0.883375	−0.441688	+	0.897169i	$0.645620\pi$
$242$	−12592.9	−3.34504
$243$	0	0
$244$	−4369.55	−1.14644
$245$	5753.16	1.50023
$246$	0	0
$247$	4549.20	1.17190
$248$	7066.59	1.80939
$249$	0	0
$250$	6534.22	1.65304
$251$	−2958.33	−0.743937	−0.371969	−	0.928245i	$-0.621317\pi$
$251$	−2958.33	−0.743937	−0.371969	+	0.928245i	$0.621317\pi$
$252$	0	0
$253$	5077.42	1.26172
$254$	−8877.25	−2.19295
$255$	0	0
$256$	−4384.25	−1.07037
$257$	−4320.44	−1.04864	−0.524322	−	0.851520i	$-0.675681\pi$
$257$	−4320.44	−1.04864	−0.524322	+	0.851520i	$0.675681\pi$
$258$	0	0
$259$	2367.82	0.568067
$260$	5072.89	1.21003
$261$	0	0
$262$	936.602	0.220853
$263$	−1985.21	−0.465450	−0.232725	−	0.972543i	$-0.574764\pi$
$263$	−1985.21	−0.465450	−0.232725	+	0.972543i	$0.574764\pi$
$264$	0	0
$265$	−4212.23	−0.976435
$266$	−13078.0	−3.01453
$267$	0	0
$268$	−11598.3	−2.64358
$269$	5838.30	1.32330	0.661650	−	0.749813i	$-0.269857\pi$
$269$	5838.30	1.32330	0.661650	+	0.749813i	$0.269857\pi$
$270$	0	0
$271$	−6813.05	−1.52717	−0.763585	−	0.645707i	$-0.776563\pi$
$271$	−6813.05	−1.52717	−0.763585	+	0.645707i	$0.776563\pi$
$272$	−20.6649	−0.00460660
$273$	0	0
$274$	1011.27	0.222966
$275$	4709.83	1.03278
$276$	0	0
$277$	−3602.86	−0.781498	−0.390749	−	0.920497i	$-0.627784\pi$
$277$	−3602.86	−0.781498	−0.390749	+	0.920497i	$0.627784\pi$
$278$	4547.07	0.980990
$279$	0	0
$280$	−5678.18	−1.21192
$281$	−1484.39	−0.315130	−0.157565	−	0.987509i	$-0.550364\pi$
$281$	−1484.39	−0.315130	−0.157565	+	0.987509i	$0.550364\pi$
$282$	0	0
$283$	1013.40	0.212863	0.106431	−	0.994320i	$-0.466058\pi$
$283$	1013.40	0.212863	0.106431	+	0.994320i	$0.466058\pi$
$284$	−391.950	−0.0818942
$285$	0	0
$286$	15864.0	3.27992
$287$	−3868.37	−0.795620
$288$	0	0
$289$	−4859.55	−0.989121
$290$	−3832.19	−0.775978
$291$	0	0
$292$	−9368.86	−1.87764
$293$	4928.25	0.982633	0.491317	−	0.870981i	$-0.336516\pi$
$293$	4928.25	0.982633	0.491317	+	0.870981i	$0.336516\pi$
$294$	0	0
$295$	−501.782	−0.0990335
$296$	−1638.15	−0.321675
$297$	0	0
$298$	10497.1	2.04054
$299$	−4305.80	−0.832812
$300$	0	0
$301$	3083.30	0.590427
$302$	1823.30	0.347415
$303$	0	0
$304$	237.605	0.0448277
$305$	−2386.45	−0.448026
$306$	0	0
$307$	9433.94	1.75382	0.876911	−	0.480653i	$-0.159601\pi$
$307$	9433.94	1.75382	0.876911	+	0.480653i	$0.159601\pi$
$308$	−28315.2	−5.23835
$309$	0	0
$310$	9912.44	1.81609
$311$	5864.84	1.06934	0.534670	−	0.845061i	$-0.320436\pi$
$311$	5864.84	1.06934	0.534670	+	0.845061i	$0.320436\pi$
$312$	0	0
$313$	−6104.05	−1.10231	−0.551153	−	0.834405i	$-0.685812\pi$
$313$	−6104.05	−1.10231	−0.551153	+	0.834405i	$0.685812\pi$
$314$	7700.78	1.38401
$315$	0	0
$316$	−6845.89	−1.21871
$317$	4913.64	0.870591	0.435295	−	0.900288i	$-0.356644\pi$
$317$	4913.64	0.870591	0.435295	+	0.900288i	$0.356644\pi$
$318$	0	0
$319$	−7440.53	−1.30592
$320$	−5896.12	−1.03001
$321$	0	0
$322$	12378.3	2.14229
$323$	−614.574	−0.105869
$324$	0	0
$325$	−3994.07	−0.681696
$326$	1573.43	0.267313
$327$	0	0
$328$	2676.29	0.450529
$329$	8082.70	1.35445
$330$	0	0
$331$	4981.03	0.827137	0.413568	−	0.910473i	$-0.364282\pi$
$331$	4981.03	0.827137	0.413568	+	0.910473i	$0.364282\pi$
$332$	−16020.4	−2.64829
$333$	0	0
$334$	2583.63	0.423264
$335$	−6334.49	−1.03310
$336$	0	0
$337$	8158.65	1.31878	0.659392	−	0.751799i	$-0.270814\pi$
$337$	8158.65	1.31878	0.659392	+	0.751799i	$0.270814\pi$
$338$	−3361.04	−0.540877
$339$	0	0
$340$	−685.323	−0.109314
$341$	19245.9	3.05637
$342$	0	0
$343$	15615.3	2.45816
$344$	−2133.15	−0.334336
$345$	0	0
$346$	17898.8	2.78106
$347$	−5704.38	−0.882500	−0.441250	−	0.897384i	$-0.645465\pi$
$347$	−5704.38	−0.882500	−0.441250	+	0.897384i	$0.645465\pi$
$348$	0	0
$349$	−3602.61	−0.552560	−0.276280	−	0.961077i	$-0.589102\pi$
$349$	−3602.61	−0.552560	−0.276280	+	0.961077i	$0.589102\pi$
$350$	11482.2	1.75356
$351$	0	0
$352$	−11133.7	−1.68588
$353$	2803.75	0.422744	0.211372	−	0.977406i	$-0.432207\pi$
$353$	2803.75	0.422744	0.211372	+	0.977406i	$0.432207\pi$
$354$	0	0
$355$	−214.066	−0.0320040
$356$	−14237.1	−2.11956
$357$	0	0
$358$	−12139.9	−1.79221
$359$	−355.464	−0.0522582	−0.0261291	−	0.999659i	$-0.508318\pi$
$359$	−355.464	−0.0522582	−0.0261291	+	0.999659i	$0.508318\pi$
$360$	0	0
$361$	207.384	0.0302353
$362$	−10612.2	−1.54079
$363$	0	0
$364$	24012.1	3.45763
$365$	−5116.86	−0.733777
$366$	0	0
$367$	−7157.52	−1.01804	−0.509018	−	0.860756i	$-0.669991\pi$
$367$	−7157.52	−1.01804	−0.509018	+	0.860756i	$0.669991\pi$
$368$	−224.893	−0.0318569
$369$	0	0
$370$	−2297.87	−0.322866
$371$	−19938.2	−2.79014
$372$	0	0
$373$	−7324.99	−1.01682	−0.508410	−	0.861115i	$-0.669766\pi$
$373$	−7324.99	−1.01682	−0.508410	+	0.861115i	$0.669766\pi$
$374$	−2143.15	−0.296309
$375$	0	0
$376$	−5591.93	−0.766973
$377$	6309.78	0.861990
$378$	0	0
$379$	−8650.46	−1.17241	−0.586206	−	0.810162i	$-0.699379\pi$
$379$	−8650.46	−1.17241	−0.586206	+	0.810162i	$0.699379\pi$
$380$	7879.85	1.06376
$381$	0	0
$382$	−23647.1	−3.16725
$383$	−6852.10	−0.914167	−0.457084	−	0.889424i	$-0.651106\pi$
$383$	−6852.10	−0.914167	−0.457084	+	0.889424i	$0.651106\pi$
$384$	0	0
$385$	−15464.5	−2.04713
$386$	22857.6	3.01405
$387$	0	0
$388$	−17094.6	−2.23672
$389$	−9408.56	−1.22631	−0.613153	−	0.789964i	$-0.710099\pi$
$389$	−9408.56	−1.22631	−0.613153	+	0.789964i	$0.710099\pi$
$390$	0	0
$391$	581.692	0.0752364
$392$	−18840.2	−2.42749
$393$	0	0
$394$	−22882.5	−2.92590
$395$	−3738.92	−0.476268
$396$	0	0
$397$	−6882.28	−0.870055	−0.435028	−	0.900417i	$-0.643261\pi$
$397$	−6882.28	−0.870055	−0.435028	+	0.900417i	$0.643261\pi$
$398$	−11882.7	−1.49655
$399$	0	0
$400$	−208.611	−0.0260764
$401$	5316.52	0.662081	0.331040	−	0.943617i	$-0.392600\pi$
$401$	5316.52	0.662081	0.331040	+	0.943617i	$0.392600\pi$
$402$	0	0
$403$	−16321.0	−2.01739
$404$	2238.96	0.275724
$405$	0	0
$406$	−18139.3	−2.21734
$407$	−4461.51	−0.543363
$408$	0	0
$409$	−7886.23	−0.953421	−0.476711	−	0.879060i	$-0.658171\pi$
$409$	−7886.23	−0.953421	−0.476711	+	0.879060i	$0.658171\pi$
$410$	3754.08	0.452198
$411$	0	0
$412$	8478.25	1.01382
$413$	−2375.14	−0.282986
$414$	0	0
$415$	−8749.62	−1.03494
$416$	9441.69	1.11278
$417$	0	0
$418$	24641.9	2.88344
$419$	−3196.80	−0.372730	−0.186365	−	0.982481i	$-0.559671\pi$
$419$	−3196.80	−0.372730	−0.186365	+	0.982481i	$0.559671\pi$
$420$	0	0
$421$	6687.59	0.774188	0.387094	−	0.922040i	$-0.373479\pi$
$421$	6687.59	0.774188	0.387094	+	0.922040i	$0.373479\pi$
$422$	−24515.5	−2.82795
$423$	0	0
$424$	13794.1	1.57995
$425$	539.580	0.0615846
$426$	0	0
$427$	−11296.1	−1.28022
$428$	−10490.8	−1.18479
$429$	0	0
$430$	−2992.21	−0.335574
$431$	13231.6	1.47876	0.739380	−	0.673289i	$-0.235119\pi$
$431$	13231.6	1.47876	0.739380	+	0.673289i	$0.235119\pi$
$432$	0	0
$433$	1504.88	0.167021	0.0835104	−	0.996507i	$-0.473387\pi$
$433$	1504.88	0.167021	0.0835104	+	0.996507i	$0.473387\pi$
$434$	46919.7	5.18944
$435$	0	0
$436$	3842.79	0.422101
$437$	−6688.31	−0.732140
$438$	0	0
$439$	−7916.92	−0.860715	−0.430357	−	0.902659i	$-0.641612\pi$
$439$	−7916.92	−0.860715	−0.430357	+	0.902659i	$0.641612\pi$
$440$	10699.0	1.15921
$441$	0	0
$442$	1817.45	0.195582
$443$	−5699.27	−0.611243	−0.305621	−	0.952153i	$-0.598864\pi$
$443$	−5699.27	−0.611243	−0.305621	+	0.952153i	$0.598864\pi$
$444$	0	0
$445$	−7775.68	−0.828320
$446$	−7032.80	−0.746666
$447$	0	0
$448$	−27908.8	−2.94323
$449$	2780.79	0.292279	0.146140	−	0.989264i	$-0.453315\pi$
$449$	2780.79	0.292279	0.146140	+	0.989264i	$0.453315\pi$
$450$	0	0
$451$	7288.89	0.761020
$452$	15963.5	1.66119
$453$	0	0
$454$	20611.1	2.13067
$455$	13114.4	1.35123
$456$	0	0
$457$	10301.9	1.05449	0.527246	−	0.849713i	$-0.323225\pi$
$457$	10301.9	1.05449	0.527246	+	0.849713i	$0.323225\pi$
$458$	−12131.1	−1.23766
$459$	0	0
$460$	−7458.25	−0.755963
$461$	−10080.3	−1.01841	−0.509203	−	0.860647i	$-0.670060\pi$
$461$	−10080.3	−1.01841	−0.509203	+	0.860647i	$0.670060\pi$
$462$	0	0
$463$	17730.1	1.77967	0.889837	−	0.456278i	$-0.150818\pi$
$463$	17730.1	1.77967	0.889837	+	0.456278i	$0.150818\pi$
$464$	329.561	0.0329730
$465$	0	0
$466$	−21511.5	−2.13842
$467$	8715.56	0.863614	0.431807	−	0.901966i	$-0.357876\pi$
$467$	8715.56	0.863614	0.431807	+	0.901966i	$0.357876\pi$
$468$	0	0
$469$	−29983.8	−2.95207
$470$	−7843.90	−0.769813
$471$	0	0
$472$	1643.22	0.160244
$473$	−5809.63	−0.564751
$474$	0	0
$475$	−6204.09	−0.599291
$476$	−3243.92	−0.312363
$477$	0	0
$478$	−1097.86	−0.105053
$479$	7158.26	0.682817	0.341409	−	0.939915i	$-0.389096\pi$
$479$	7158.26	0.682817	0.341409	+	0.939915i	$0.389096\pi$
$480$	0	0
$481$	3783.49	0.358653
$482$	15181.7	1.43467
$483$	0	0
$484$	35915.0	3.37293
$485$	−9336.33	−0.874105
$486$	0	0
$487$	372.979	0.0347049	0.0173525	−	0.999849i	$-0.494476\pi$
$487$	372.979	0.0347049	0.0173525	+	0.999849i	$0.494476\pi$
$488$	7815.07	0.724942
$489$	0	0
$490$	−26427.6	−2.43648
$491$	7266.72	0.667907	0.333954	−	0.942589i	$-0.391617\pi$
$491$	7266.72	0.667907	0.333954	+	0.942589i	$0.391617\pi$
$492$	0	0
$493$	−852.420	−0.0778724
$494$	−20897.1	−1.90325
$495$	0	0
$496$	−852.451	−0.0771697
$497$	−1013.26	−0.0914508
$498$	0	0
$499$	3812.20	0.341999	0.171000	−	0.985271i	$-0.445300\pi$
$499$	3812.20	0.341999	0.171000	+	0.985271i	$0.445300\pi$
$500$	−18635.7	−1.66682
$501$	0	0
$502$	13589.3	1.20821
$503$	6948.38	0.615930	0.307965	−	0.951398i	$-0.400352\pi$
$503$	6948.38	0.615930	0.307965	+	0.951398i	$0.400352\pi$
$504$	0	0
$505$	1222.82	0.107752
$506$	−23323.5	−2.04912
$507$	0	0
$508$	25318.0	2.21123
$509$	−774.325	−0.0674289	−0.0337145	−	0.999432i	$-0.510734\pi$
$509$	−774.325	−0.0674289	−0.0337145	+	0.999432i	$0.510734\pi$
$510$	0	0
$511$	−24220.2	−2.09675
$512$	1022.98	0.0883006
$513$	0	0
$514$	19846.3	1.70308
$515$	4630.44	0.396198
$516$	0	0
$517$	−15229.6	−1.29555
$518$	−10876.8	−0.922583
$519$	0	0
$520$	−9073.03	−0.765152
$521$	142.642	0.0119948	0.00599739	−	0.999982i	$-0.498091\pi$
$521$	142.642	0.0119948	0.00599739	+	0.999982i	$0.498091\pi$
$522$	0	0
$523$	−20177.4	−1.68699	−0.843494	−	0.537138i	$-0.819505\pi$
$523$	−20177.4	−1.68699	−0.843494	+	0.537138i	$0.819505\pi$
$524$	−2671.20	−0.222694
$525$	0	0
$526$	9119.21	0.755925
$527$	2204.89	0.182252
$528$	0	0
$529$	−5836.54	−0.479703
$530$	19349.2	1.58580
$531$	0	0
$532$	37298.7	3.03966
$533$	−6181.18	−0.502320
$534$	0	0
$535$	−5729.61	−0.463014
$536$	20744.0	1.67165
$537$	0	0
$538$	−26818.7	−2.14913
$539$	−51311.4	−4.10045
$540$	0	0
$541$	−24399.9	−1.93907	−0.969533	−	0.244960i	$-0.921225\pi$
$541$	−24399.9	−1.93907	−0.969533	+	0.244960i	$0.921225\pi$
$542$	31296.2	2.48024
$543$	0	0
$544$	−1275.53	−0.100529
$545$	2098.76	0.164956
$546$	0	0
$547$	17403.3	1.36035	0.680173	−	0.733051i	$-0.261905\pi$
$547$	17403.3	1.36035	0.680173	+	0.733051i	$0.261905\pi$
$548$	−2884.14	−0.224825
$549$	0	0
$550$	−21635.0	−1.67731
$551$	9801.14	0.757791
$552$	0	0
$553$	−17697.9	−1.36092
$554$	16550.0	1.26921
$555$	0	0
$556$	−12968.3	−0.989169
$557$	14768.4	1.12344	0.561719	−	0.827328i	$-0.310140\pi$
$557$	14768.4	1.12344	0.561719	+	0.827328i	$0.310140\pi$
$558$	0	0
$559$	4926.73	0.372770
$560$	684.966	0.0516877
$561$	0	0
$562$	6818.67	0.511794
$563$	−17890.7	−1.33926	−0.669630	−	0.742695i	$-0.733547\pi$
$563$	−17890.7	−1.33926	−0.669630	+	0.742695i	$0.733547\pi$
$564$	0	0
$565$	8718.54	0.649189
$566$	−4655.10	−0.345704
$567$	0	0
$568$	701.015	0.0517851
$569$	−16858.7	−1.24210	−0.621049	−	0.783772i	$-0.713293\pi$
$569$	−16858.7	−1.24210	−0.621049	+	0.783772i	$0.713293\pi$
$570$	0	0
$571$	23454.2	1.71896	0.859482	−	0.511166i	$-0.170786\pi$
$571$	23454.2	1.71896	0.859482	+	0.511166i	$0.170786\pi$
$572$	−45244.2	−3.30727
$573$	0	0
$574$	17769.7	1.29214
$575$	5872.15	0.425888
$576$	0	0
$577$	−96.7723	−0.00698212	−0.00349106	−	0.999994i	$-0.501111\pi$
$577$	−96.7723	−0.00698212	−0.00349106	+	0.999994i	$0.501111\pi$
$578$	22322.7	1.60640
$579$	0	0
$580$	10929.4	0.782448
$581$	−41415.6	−2.95733
$582$	0	0
$583$	37568.1	2.66880
$584$	16756.5	1.18731
$585$	0	0
$586$	−22638.3	−1.59587
$587$	7330.94	0.515469	0.257734	−	0.966216i	$-0.417024\pi$
$587$	7330.94	0.515469	0.257734	+	0.966216i	$0.417024\pi$
$588$	0	0
$589$	−25351.9	−1.77353
$590$	2304.97	0.160838
$591$	0	0
$592$	197.612	0.0137193
$593$	−18885.7	−1.30783	−0.653916	−	0.756567i	$-0.726875\pi$
$593$	−18885.7	−1.30783	−0.653916	+	0.756567i	$0.726875\pi$
$594$	0	0
$595$	−1771.69	−0.122071
$596$	−29937.8	−2.05755
$597$	0	0
$598$	19779.0	1.35255
$599$	17752.6	1.21094	0.605468	−	0.795870i	$-0.292986\pi$
$599$	17752.6	1.21094	0.605468	+	0.795870i	$0.292986\pi$
$600$	0	0
$601$	2004.86	0.136073	0.0680364	−	0.997683i	$-0.478327\pi$
$601$	2004.86	0.136073	0.0680364	+	0.997683i	$0.478327\pi$
$602$	−14163.4	−0.958896
$603$	0	0
$604$	−5200.07	−0.350311
$605$	19615.2	1.31813
$606$	0	0
$607$	−10759.4	−0.719457	−0.359728	−	0.933057i	$-0.617131\pi$
$607$	−10759.4	−0.719457	−0.359728	+	0.933057i	$0.617131\pi$
$608$	14666.0	0.978266
$609$	0	0
$610$	10962.3	0.727627
$611$	12915.2	0.855141
$612$	0	0
$613$	−1411.80	−0.0930212	−0.0465106	−	0.998918i	$-0.514810\pi$
$613$	−1411.80	−0.0930212	−0.0465106	+	0.998918i	$0.514810\pi$
$614$	−43335.5	−2.84833
$615$	0	0
$616$	50642.7	3.31242
$617$	18223.9	1.18909	0.594543	−	0.804064i	$-0.297333\pi$
$617$	18223.9	1.18909	0.594543	+	0.804064i	$0.297333\pi$
$618$	0	0
$619$	−13680.6	−0.888318	−0.444159	−	0.895948i	$-0.646497\pi$
$619$	−13680.6	−0.888318	−0.444159	+	0.895948i	$0.646497\pi$
$620$	−28270.4	−1.83123
$621$	0	0
$622$	−26940.6	−1.73669
$623$	−36805.5	−2.36691
$624$	0	0
$625$	−952.474	−0.0609583
$626$	28039.4	1.79022
$627$	0	0
$628$	−21962.7	−1.39555
$629$	−511.131	−0.0324008
$630$	0	0
$631$	10884.1	0.686670	0.343335	−	0.939213i	$-0.388443\pi$
$631$	10884.1	0.686670	0.343335	+	0.939213i	$0.388443\pi$
$632$	12244.1	0.770640
$633$	0	0
$634$	−22571.1	−1.41390
$635$	13827.6	0.864142
$636$	0	0
$637$	43513.6	2.70655
$638$	34178.6	2.12092
$639$	0	0
$640$	17097.6	1.05600
$641$	−1889.40	−0.116423	−0.0582113	−	0.998304i	$-0.518540\pi$
$641$	−1889.40	−0.116423	−0.0582113	+	0.998304i	$0.518540\pi$
$642$	0	0
$643$	19403.7	1.19006	0.595029	−	0.803705i	$-0.297141\pi$
$643$	19403.7	1.19006	0.595029	+	0.803705i	$0.297141\pi$
$644$	−35303.0	−2.16015
$645$	0	0
$646$	2823.09	0.171940
$647$	−29568.5	−1.79669	−0.898344	−	0.439293i	$-0.855229\pi$
$647$	−29568.5	−1.79669	−0.898344	+	0.439293i	$0.855229\pi$
$648$	0	0
$649$	4475.31	0.270680
$650$	18347.1	1.10712
$651$	0	0
$652$	−4487.43	−0.269542
$653$	16106.0	0.965200	0.482600	−	0.875841i	$-0.339692\pi$
$653$	16106.0	0.965200	0.482600	+	0.875841i	$0.339692\pi$
$654$	0	0
$655$	−1458.89	−0.0870283
$656$	−322.844	−0.0192149
$657$	0	0
$658$	−37128.5	−2.19972
$659$	21.8800	0.00129336	0.000646681	−	1.00000i	$-0.499794\pi$
$659$	21.8800	0.00129336	0.000646681		1.00000i	$0.499794\pi$
$660$	0	0
$661$	−9693.77	−0.570415	−0.285207	−	0.958466i	$-0.592062\pi$
$661$	−9693.77	−0.570415	−0.285207	+	0.958466i	$0.592062\pi$
$662$	−22880.7	−1.34333
$663$	0	0
$664$	28652.9	1.67462
$665$	20370.9	1.18789
$666$	0	0
$667$	−9276.74	−0.538526
$668$	−7368.54	−0.426792
$669$	0	0
$670$	29097.9	1.67784
$671$	21284.4	1.22455
$672$	0	0
$673$	−1668.31	−0.0955552	−0.0477776	−	0.998858i	$-0.515214\pi$
$673$	−1668.31	−0.0955552	−0.0477776	+	0.998858i	$0.515214\pi$
$674$	−37477.4	−2.14180
$675$	0	0
$676$	9585.71	0.545386
$677$	6747.46	0.383052	0.191526	−	0.981488i	$-0.438656\pi$
$677$	6747.46	0.383052	0.191526	+	0.981488i	$0.438656\pi$
$678$	0	0
$679$	−44192.8	−2.49774
$680$	1225.72	0.0691240
$681$	0	0
$682$	−88407.3	−4.96377
$683$	544.019	0.0304778	0.0152389	−	0.999884i	$-0.495149\pi$
$683$	544.019	0.0304778	0.0152389	+	0.999884i	$0.495149\pi$
$684$	0	0
$685$	−1575.19	−0.0878612
$686$	−71730.0	−3.99222
$687$	0	0
$688$	257.324	0.0142593
$689$	−31858.9	−1.76158
$690$	0	0
$691$	−27800.4	−1.53050	−0.765251	−	0.643732i	$-0.777385\pi$
$691$	−27800.4	−1.53050	−0.765251	+	0.643732i	$0.777385\pi$
$692$	−51047.6	−2.80425
$693$	0	0
$694$	26203.5	1.43324
$695$	−7082.71	−0.386565
$696$	0	0
$697$	835.048	0.0453798
$698$	16548.9	0.897398
$699$	0	0
$700$	−32747.2	−1.76818
$701$	29517.3	1.59037	0.795187	−	0.606365i	$-0.207373\pi$
$701$	29517.3	1.59037	0.795187	+	0.606365i	$0.207373\pi$
$702$	0	0
$703$	5876.99	0.315299
$704$	52586.5	2.81524
$705$	0	0
$706$	−12879.2	−0.686567
$707$	5788.13	0.307899
$708$	0	0
$709$	24796.0	1.31345	0.656724	−	0.754131i	$-0.271942\pi$
$709$	24796.0	1.31345	0.656724	+	0.754131i	$0.271942\pi$
$710$	983.327	0.0519769
$711$	0	0
$712$	25463.5	1.34029
$713$	23995.5	1.26036
$714$	0	0
$715$	−24710.4	−1.29247
$716$	34623.0	1.80715
$717$	0	0
$718$	1632.85	0.0848711
$719$	−26709.8	−1.38540	−0.692702	−	0.721224i	$-0.743580\pi$
$719$	−26709.8	−1.38540	−0.692702	+	0.721224i	$0.743580\pi$
$720$	0	0
$721$	21917.8	1.13213
$722$	−952.634	−0.0491044
$723$	0	0
$724$	30266.2	1.55364
$725$	−8605.14	−0.440809
$726$	0	0
$727$	−5438.42	−0.277441	−0.138721	−	0.990332i	$-0.544299\pi$
$727$	−5438.42	−0.277441	−0.138721	+	0.990332i	$0.544299\pi$
$728$	−42946.5	−2.18640
$729$	0	0
$730$	23504.7	1.19171
$731$	−665.578	−0.0336762
$732$	0	0
$733$	−14528.5	−0.732091	−0.366045	−	0.930597i	$-0.619288\pi$
$733$	−14528.5	−0.732091	−0.366045	+	0.930597i	$0.619288\pi$
$734$	32878.6	1.65337
$735$	0	0
$736$	−13881.3	−0.695208
$737$	56496.2	2.82370
$738$	0	0
$739$	−13795.9	−0.686724	−0.343362	−	0.939203i	$-0.611566\pi$
$739$	−13795.9	−0.686724	−0.343362	+	0.939203i	$0.611566\pi$
$740$	6553.54	0.325558
$741$	0	0
$742$	91587.8	4.53139
$743$	22223.4	1.09730	0.548652	−	0.836051i	$-0.315141\pi$
$743$	22223.4	1.09730	0.548652	+	0.836051i	$0.315141\pi$
$744$	0	0
$745$	−16350.7	−0.804086
$746$	33647.9	1.65139
$747$	0	0
$748$	6112.28	0.298779
$749$	−27120.6	−1.32305
$750$	0	0
$751$	28755.8	1.39722	0.698612	−	0.715501i	$-0.253801\pi$
$751$	28755.8	1.39722	0.698612	+	0.715501i	$0.253801\pi$
$752$	674.561	0.0327111
$753$	0	0
$754$	−28984.4	−1.39993
$755$	−2840.05	−0.136901
$756$	0	0
$757$	−4735.85	−0.227381	−0.113691	−	0.993516i	$-0.536267\pi$
$757$	−4735.85	−0.227381	−0.113691	+	0.993516i	$0.536267\pi$
$758$	39736.5	1.90408
$759$	0	0
$760$	−14093.4	−0.672658
$761$	−28657.3	−1.36508	−0.682541	−	0.730847i	$-0.739125\pi$
$761$	−28657.3	−1.36508	−0.682541	+	0.730847i	$0.739125\pi$
$762$	0	0
$763$	9934.31	0.471358
$764$	67441.7	3.19366
$765$	0	0
$766$	31475.6	1.48467
$767$	−3795.19	−0.178665
$768$	0	0
$769$	10064.5	0.471955	0.235978	−	0.971758i	$-0.424171\pi$
$769$	10064.5	0.471955	0.235978	+	0.971758i	$0.424171\pi$
$770$	71037.5	3.32469
$771$	0	0
$772$	−65190.2	−3.03918
$773$	8043.29	0.374252	0.187126	−	0.982336i	$-0.440083\pi$
$773$	8043.29	0.374252	0.187126	+	0.982336i	$0.440083\pi$
$774$	0	0
$775$	22258.3	1.03167
$776$	30574.3	1.41437
$777$	0	0
$778$	43218.9	1.99161
$779$	−9601.39	−0.441599
$780$	0	0
$781$	1909.21	0.0874739
$782$	−2672.05	−0.122189
$783$	0	0
$784$	2272.72	0.103531
$785$	−11995.1	−0.545378
$786$	0	0
$787$	5620.03	0.254552	0.127276	−	0.991867i	$-0.459377\pi$
$787$	5620.03	0.254552	0.127276	+	0.991867i	$0.459377\pi$
$788$	65261.2	2.95030
$789$	0	0
$790$	17175.0	0.773494
$791$	41268.5	1.85504
$792$	0	0
$793$	−18049.7	−0.808279
$794$	31614.3	1.41303
$795$	0	0
$796$	33889.6	1.50903
$797$	−8000.11	−0.355556	−0.177778	−	0.984071i	$-0.556891\pi$
$797$	−8000.11	−0.355556	−0.177778	+	0.984071i	$0.556891\pi$
$798$	0	0
$799$	−1744.77	−0.0772537
$800$	−12876.4	−0.569061
$801$	0	0
$802$	−24421.8	−1.07527
$803$	45636.4	2.00557
$804$	0	0
$805$	−19281.0	−0.844180
$806$	74971.9	3.27639
$807$	0	0
$808$	−4004.45	−0.174352
$809$	23973.0	1.04184	0.520919	−	0.853606i	$-0.325589\pi$
$809$	23973.0	1.04184	0.520919	+	0.853606i	$0.325589\pi$
$810$	0	0
$811$	−20319.9	−0.879814	−0.439907	−	0.898043i	$-0.644989\pi$
$811$	−20319.9	−0.879814	−0.439907	+	0.898043i	$0.644989\pi$
$812$	51733.6	2.23583
$813$	0	0
$814$	20494.3	0.882462
$815$	−2450.83	−0.105336
$816$	0	0
$817$	7652.82	0.327709
$818$	36226.0	1.54843
$819$	0	0
$820$	−10706.7	−0.455968
$821$	−5690.35	−0.241894	−0.120947	−	0.992659i	$-0.538593\pi$
$821$	−5690.35	−0.241894	−0.120947	+	0.992659i	$0.538593\pi$
$822$	0	0
$823$	−9583.95	−0.405924	−0.202962	−	0.979187i	$-0.565057\pi$
$823$	−9583.95	−0.405924	−0.202962	+	0.979187i	$0.565057\pi$
$824$	−15163.6	−0.641080
$825$	0	0
$826$	10910.4	0.459590
$827$	−9760.07	−0.410388	−0.205194	−	0.978721i	$-0.565783\pi$
$827$	−9760.07	−0.410388	−0.205194	+	0.978721i	$0.565783\pi$
$828$	0	0
$829$	3388.68	0.141971	0.0709854	−	0.997477i	$-0.477386\pi$
$829$	3388.68	0.141971	0.0709854	+	0.997477i	$0.477386\pi$
$830$	40192.0	1.68082
$831$	0	0
$832$	−44594.8	−1.85823
$833$	−5878.47	−0.244510
$834$	0	0
$835$	−4024.37	−0.166789
$836$	−70279.0	−2.90748
$837$	0	0
$838$	14684.7	0.605342
$839$	−38109.7	−1.56817	−0.784085	−	0.620654i	$-0.786867\pi$
$839$	−38109.7	−1.56817	−0.784085	+	0.620654i	$0.786867\pi$
$840$	0	0
$841$	−10794.7	−0.442606
$842$	−30719.9	−1.25734
$843$	0	0
$844$	69918.4	2.85153
$845$	5235.29	0.213136
$846$	0	0
$847$	92846.8	3.76653
$848$	−1663.99	−0.0673842
$849$	0	0
$850$	−2478.60	−0.100018
$851$	−5562.55	−0.224068
$852$	0	0
$853$	14268.6	0.572742	0.286371	−	0.958119i	$-0.407551\pi$
$853$	14268.6	0.572742	0.286371	+	0.958119i	$0.407551\pi$
$854$	51889.4	2.07918
$855$	0	0
$856$	18763.1	0.749195
$857$	−22417.8	−0.893557	−0.446778	−	0.894645i	$-0.647429\pi$
$857$	−22417.8	−0.893557	−0.446778	+	0.894645i	$0.647429\pi$
$858$	0	0
$859$	6514.22	0.258745	0.129373	−	0.991596i	$-0.458704\pi$
$859$	6514.22	0.258745	0.129373	+	0.991596i	$0.458704\pi$
$860$	8533.80	0.338372
$861$	0	0
$862$	−60780.5	−2.40161
$863$	−39574.1	−1.56097	−0.780486	−	0.625173i	$-0.785028\pi$
$863$	−39574.1	−1.56097	−0.780486	+	0.625173i	$0.785028\pi$
$864$	0	0
$865$	−27879.9	−1.09589
$866$	−6912.78	−0.271254
$867$	0	0
$868$	−133815.	−5.23271
$869$	33346.8	1.30174
$870$	0	0
$871$	−47910.4	−1.86381
$872$	−6872.95	−0.266912
$873$	0	0
$874$	30723.2	1.18905
$875$	−48176.6	−1.86133
$876$	0	0
$877$	−11286.1	−0.434553	−0.217277	−	0.976110i	$-0.569717\pi$
$877$	−11286.1	−0.434553	−0.217277	+	0.976110i	$0.569717\pi$
$878$	36366.9	1.39786
$879$	0	0
$880$	−1290.63	−0.0494399
$881$	18300.7	0.699850	0.349925	−	0.936778i	$-0.386207\pi$
$881$	18300.7	0.699850	0.349925	+	0.936778i	$0.386207\pi$
$882$	0	0
$883$	−43089.5	−1.64222	−0.821108	−	0.570773i	$-0.806644\pi$
$883$	−43089.5	−1.64222	−0.821108	+	0.570773i	$0.806644\pi$
$884$	−5183.39	−0.197213
$885$	0	0
$886$	26180.0	0.992703
$887$	6532.95	0.247300	0.123650	−	0.992326i	$-0.460540\pi$
$887$	6532.95	0.247300	0.123650	+	0.992326i	$0.460540\pi$
$888$	0	0
$889$	65451.7	2.46927
$890$	35718.1	1.34525
$891$	0	0
$892$	20057.6	0.752891
$893$	20061.4	0.751770
$894$	0	0
$895$	18909.6	0.706231
$896$	80930.0	3.01750
$897$	0	0
$898$	−12773.7	−0.474683
$899$	−35163.3	−1.30452
$900$	0	0
$901$	4303.98	0.159141
$902$	−33482.0	−1.23595
$903$	0	0
$904$	−28551.2	−1.05044
$905$	16530.1	0.607158
$906$	0	0
$907$	17353.6	0.635298	0.317649	−	0.948208i	$-0.397107\pi$
$907$	17353.6	0.635298	0.317649	+	0.948208i	$0.397107\pi$
$908$	−58783.0	−2.14844
$909$	0	0
$910$	−60241.8	−2.19450
$911$	37198.8	1.35285	0.676427	−	0.736509i	$-0.263527\pi$
$911$	37198.8	1.35285	0.676427	+	0.736509i	$0.263527\pi$
$912$	0	0
$913$	78036.3	2.82872
$914$	−47322.5	−1.71257
$915$	0	0
$916$	34598.0	1.24798
$917$	−6905.54	−0.248681
$918$	0	0
$919$	−329.251	−0.0118183	−0.00590914	−	0.999983i	$-0.501881\pi$
$919$	−329.251	−0.0118183	−0.00590914	+	0.999983i	$0.501881\pi$
$920$	13339.3	0.478027
$921$	0	0
$922$	46304.4	1.65396
$923$	−1619.07	−0.0577381
$924$	0	0
$925$	−5159.84	−0.183410
$926$	−81444.7	−2.89032
$927$	0	0
$928$	20341.9	0.719565
$929$	−51527.0	−1.81975	−0.909875	−	0.414883i	$-0.863822\pi$
$929$	−51527.0	−1.81975	−0.909875	+	0.414883i	$0.863822\pi$
$930$	0	0
$931$	67590.7	2.37937
$932$	61351.0	2.15624
$933$	0	0
$934$	−40035.5	−1.40257
$935$	3338.26	0.116762
$936$	0	0
$937$	−14507.4	−0.505801	−0.252901	−	0.967492i	$-0.581385\pi$
$937$	−14507.4	−0.505801	−0.252901	+	0.967492i	$0.581385\pi$
$938$	137733.	4.79438
$939$	0	0
$940$	22370.9	0.776232
$941$	39570.3	1.37083	0.685417	−	0.728151i	$-0.259620\pi$
$941$	39570.3	1.37083	0.685417	+	0.728151i	$0.259620\pi$
$942$	0	0
$943$	9087.68	0.313823
$944$	−198.223	−0.00683434
$945$	0	0
$946$	26687.0	0.917196
$947$	−4868.20	−0.167049	−0.0835245	−	0.996506i	$-0.526618\pi$
$947$	−4868.20	−0.167049	−0.0835245	+	0.996506i	$0.526618\pi$
$948$	0	0
$949$	−38700.9	−1.32380
$950$	28499.0	0.973293
$951$	0	0
$952$	5801.86	0.197520
$953$	57910.8	1.96843	0.984216	−	0.176971i	$-0.0566298\pi$
$953$	57910.8	1.96843	0.984216	+	0.176971i	$0.0566298\pi$
$954$	0	0
$955$	36833.7	1.24807
$956$	3131.12	0.105928
$957$	0	0
$958$	−32882.0	−1.10895
$959$	−7456.03	−0.251061
$960$	0	0
$961$	61163.3	2.05308
$962$	−17379.7	−0.582479
$963$	0	0
$964$	−43298.4	−1.44663
$965$	−35604.0	−1.18770
$966$	0	0
$967$	−3804.25	−0.126511	−0.0632556	−	0.997997i	$-0.520148\pi$
$967$	−3804.25	−0.126511	−0.0632556	+	0.997997i	$0.520148\pi$
$968$	−64235.1	−2.13284
$969$	0	0
$970$	42887.1	1.41961
$971$	−24345.3	−0.804612	−0.402306	−	0.915505i	$-0.631791\pi$
$971$	−24345.3	−0.804612	−0.402306	+	0.915505i	$0.631791\pi$
$972$	0	0
$973$	−33525.4	−1.10460
$974$	−1713.31	−0.0563634
$975$	0	0
$976$	−942.742	−0.0309185
$977$	15871.9	0.519741	0.259870	−	0.965644i	$-0.416320\pi$
$977$	15871.9	0.519741	0.259870	+	0.965644i	$0.416320\pi$
$978$	0	0
$979$	69349.9	2.26398
$980$	75371.6	2.45679
$981$	0	0
$982$	−33380.2	−1.08473
$983$	9897.19	0.321131	0.160565	−	0.987025i	$-0.448668\pi$
$983$	9897.19	0.321131	0.160565	+	0.987025i	$0.448668\pi$
$984$	0	0
$985$	35642.8	1.15297
$986$	3915.66	0.126470
$987$	0	0
$988$	59598.6	1.91911
$989$	−7243.36	−0.232887
$990$	0	0
$991$	−2580.10	−0.0827038	−0.0413519	−	0.999145i	$-0.513166\pi$
$991$	−2580.10	−0.0827038	−0.0413519	+	0.999145i	$0.513166\pi$
$992$	−52616.9	−1.68406
$993$	0	0
$994$	4654.50	0.148523
$995$	18509.0	0.589723
$996$	0	0
$997$	26895.5	0.854353	0.427176	−	0.904168i	$-0.359508\pi$
$997$	26895.5	0.854353	0.427176	+	0.904168i	$0.359508\pi$
$998$	−17511.6	−0.555432
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.g.1.8	✓	59
3.2	odd	2		2151.4.a.h.1.52	yes	59

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2151.4.a.g.1.8	✓	59	1.1	even	1	trivial
2151.4.a.h.1.52	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)

\(f(q)\)	\(=\)	\(q-4.59357 q^{2} +13.1009 q^{4} +7.15514 q^{5} +33.8683 q^{7} -23.4314 q^{8} +O(q^{10})\)	\(q-4.59357 q^{2} +13.1009 q^{4} +7.15514 q^{5} +33.8683 q^{7} -23.4314 q^{8} -32.8677 q^{10} -63.8154 q^{11} +54.1173 q^{13} -155.576 q^{14} +2.82656 q^{16} -7.31099 q^{17} +84.0618 q^{19} +93.7388 q^{20} +293.141 q^{22} -79.5642 q^{23} -73.8040 q^{25} -248.592 q^{26} +443.705 q^{28} +116.594 q^{29} -301.586 q^{31} +174.467 q^{32} +33.5835 q^{34} +242.332 q^{35} +69.9127 q^{37} -386.144 q^{38} -167.655 q^{40} -114.218 q^{41} +91.0380 q^{43} -836.040 q^{44} +365.484 q^{46} +238.651 q^{47} +804.060 q^{49} +339.024 q^{50} +708.986 q^{52} -588.700 q^{53} -456.608 q^{55} -793.581 q^{56} -535.585 q^{58} -70.1289 q^{59} -333.530 q^{61} +1385.36 q^{62} -824.040 q^{64} +387.217 q^{65} -885.306 q^{67} -95.7806 q^{68} -1113.17 q^{70} -29.9178 q^{71} -715.131 q^{73} -321.149 q^{74} +1101.29 q^{76} -2161.32 q^{77} -522.551 q^{79} +20.2244 q^{80} +524.670 q^{82} -1222.84 q^{83} -52.3111 q^{85} -418.190 q^{86} +1495.28 q^{88} -1086.73 q^{89} +1832.86 q^{91} -1042.36 q^{92} -1096.26 q^{94} +601.474 q^{95} -1304.84 q^{97} -3693.51 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 59 q - 8 q^{2} + 238 q^{4} - 80 q^{5} - 10 q^{7} - 96 q^{8}+O(q^{10}) \) 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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