LMFDB - Embedded newform 2151.4.a.e.1.12

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:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 717)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
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)

$f(q)$	$=$	$q-2.35723 q^{2} -2.44345 q^{4} +8.17039 q^{5} +34.4525 q^{7} +24.6177 q^{8} +O(q^{10})$	\(q-2.35723 q^{2} -2.44345 q^{4} +8.17039 q^{5} +34.4525 q^{7} +24.6177 q^{8} -19.2595 q^{10} +20.0959 q^{11} +4.61504 q^{13} -81.2125 q^{14} -38.4819 q^{16} +39.0786 q^{17} +72.4577 q^{19} -19.9640 q^{20} -47.3706 q^{22} +183.476 q^{23} -58.2447 q^{25} -10.8787 q^{26} -84.1829 q^{28} +67.6196 q^{29} +261.423 q^{31} -106.230 q^{32} -92.1174 q^{34} +281.490 q^{35} +260.381 q^{37} -170.800 q^{38} +201.136 q^{40} -90.3254 q^{41} +181.936 q^{43} -49.1033 q^{44} -432.496 q^{46} -612.406 q^{47} +843.972 q^{49} +137.296 q^{50} -11.2766 q^{52} -260.323 q^{53} +164.191 q^{55} +848.139 q^{56} -159.395 q^{58} +592.573 q^{59} +641.983 q^{61} -616.234 q^{62} +558.265 q^{64} +37.7067 q^{65} +76.0336 q^{67} -95.4868 q^{68} -663.538 q^{70} +781.662 q^{71} -100.795 q^{73} -613.779 q^{74} -177.047 q^{76} +692.352 q^{77} +1141.46 q^{79} -314.412 q^{80} +212.918 q^{82} -900.743 q^{83} +319.288 q^{85} -428.867 q^{86} +494.713 q^{88} -651.019 q^{89} +159.000 q^{91} -448.315 q^{92} +1443.58 q^{94} +592.008 q^{95} +856.008 q^{97} -1989.44 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8}+O(q^{10}) $ 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8	$ 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8} + 32 q^{10} - 154 q^{11} + 100 q^{13} - 42 q^{14} + 719 q^{16} - 32 q^{17} + 202 q^{19} + 132 q^{20} + 265 q^{22} - 552 q^{23} + 1086 q^{25} + 280 q^{26} + 390 q^{28} + 154 q^{29} + 560 q^{31} - 444 q^{32} + 156 q^{34} - 394 q^{35} + 914 q^{37} - 111 q^{38} + 257 q^{40} + 914 q^{41} + 1722 q^{43} - 1243 q^{44} + 584 q^{46} - 380 q^{47} + 2446 q^{49} + 454 q^{50} + 1552 q^{52} - 370 q^{53} + 918 q^{55} + 499 q^{56} + 2446 q^{58} - 492 q^{59} + 668 q^{61} - 578 q^{62} + 6475 q^{64} - 736 q^{65} + 4548 q^{67} - 5253 q^{68} + 7793 q^{70} - 258 q^{71} + 3096 q^{73} - 449 q^{74} + 6814 q^{76} - 3804 q^{77} + 2864 q^{79} + 1052 q^{80} + 14145 q^{82} - 2364 q^{83} + 3088 q^{85} - 2811 q^{86} + 8329 q^{88} + 4172 q^{89} + 7350 q^{91} - 13644 q^{92} + 6122 q^{94} - 3336 q^{95} + 6370 q^{97} - 1572 q^{98}+O(q^{100}) $ 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8 + 32 * q^10 - 154 * q^11 + 100 * q^13 - 42 * q^14 + 719 * q^16 - 32 * q^17 + 202 * q^19 + 132 * q^20 + 265 * q^22 - 552 * q^23 + 1086 * q^25 + 280 * q^26 + 390 * q^28 + 154 * q^29 + 560 * q^31 - 444 * q^32 + 156 * q^34 - 394 * q^35 + 914 * q^37 - 111 * q^38 + 257 * q^40 + 914 * q^41 + 1722 * q^43 - 1243 * q^44 + 584 * q^46 - 380 * q^47 + 2446 * q^49 + 454 * q^50 + 1552 * q^52 - 370 * q^53 + 918 * q^55 + 499 * q^56 + 2446 * q^58 - 492 * q^59 + 668 * q^61 - 578 * q^62 + 6475 * q^64 - 736 * q^65 + 4548 * q^67 - 5253 * q^68 + 7793 * q^70 - 258 * q^71 + 3096 * q^73 - 449 * q^74 + 6814 * q^76 - 3804 * q^77 + 2864 * q^79 + 1052 * q^80 + 14145 * q^82 - 2364 * q^83 + 3088 * q^85 - 2811 * q^86 + 8329 * q^88 + 4172 * q^89 + 7350 * q^91 - 13644 * q^92 + 6122 * q^94 - 3336 * q^95 + 6370 * q^97 - 1572 * 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 $

$2$	−2.35723	−0.833408	−0.416704	−	0.909042i	$-0.636815\pi$
$2$	−2.35723	−0.833408	−0.416704	+	0.909042i	$0.636815\pi$
$3$	0	0
$3$	0	0
$4$	−2.44345	−0.305432
$5$	8.17039	0.730782	0.365391	−	0.930854i	$-0.380935\pi$
$5$	8.17039	0.730782	0.365391	+	0.930854i	$0.380935\pi$
$6$	0	0
$7$	34.4525	1.86026	0.930129	−	0.367233i	$-0.119695\pi$
$7$	34.4525	1.86026	0.930129	+	0.367233i	$0.119695\pi$
$8$	24.6177	1.08796
$9$	0	0
$10$	−19.2595	−0.609039
$11$	20.0959	0.550830	0.275415	−	0.961325i	$-0.411185\pi$
$11$	20.0959	0.550830	0.275415	+	0.961325i	$0.411185\pi$
$12$	0	0
$13$	4.61504	0.0984602	0.0492301	−	0.998787i	$-0.484323\pi$
$13$	4.61504	0.0984602	0.0492301	+	0.998787i	$0.484323\pi$
$14$	−81.2125	−1.55035
$15$	0	0
$16$	−38.4819	−0.601280
$17$	39.0786	0.557527	0.278764	−	0.960360i	$-0.410075\pi$
$17$	39.0786	0.557527	0.278764	+	0.960360i	$0.410075\pi$
$18$	0	0
$19$	72.4577	0.874891	0.437446	−	0.899245i	$-0.355883\pi$
$19$	72.4577	0.874891	0.437446	+	0.899245i	$0.355883\pi$
$20$	−19.9640	−0.223204
$21$	0	0
$22$	−47.3706	−0.459066
$23$	183.476	1.66336	0.831682	−	0.555252i	$-0.187378\pi$
$23$	183.476	1.66336	0.831682	+	0.555252i	$0.187378\pi$
$24$	0	0
$25$	−58.2447	−0.465958
$26$	−10.8787	−0.0820575
$27$	0	0
$28$	−84.1829	−0.568182
$29$	67.6196	0.432988	0.216494	−	0.976284i	$-0.430538\pi$
$29$	67.6196	0.432988	0.216494	+	0.976284i	$0.430538\pi$
$30$	0	0
$31$	261.423	1.51461	0.757305	−	0.653062i	$-0.226516\pi$
$31$	261.423	1.51461	0.757305	+	0.653062i	$0.226516\pi$
$32$	−106.230	−0.586845
$33$	0	0
$34$	−92.1174	−0.464647
$35$	281.490	1.35944
$36$	0	0
$37$	260.381	1.15693	0.578465	−	0.815707i	$-0.303652\pi$
$37$	260.381	1.15693	0.578465	+	0.815707i	$0.303652\pi$
$38$	−170.800	−0.729141
$39$	0	0
$40$	201.136	0.795059
$41$	−90.3254	−0.344060	−0.172030	−	0.985092i	$-0.555033\pi$
$41$	−90.3254	−0.344060	−0.172030	+	0.985092i	$0.555033\pi$
$42$	0	0
$43$	181.936	0.645234	0.322617	−	0.946530i	$-0.395437\pi$
$43$	181.936	0.645234	0.322617	+	0.946530i	$0.395437\pi$
$44$	−49.1033	−0.168241
$45$	0	0
$46$	−432.496	−1.38626
$47$	−612.406	−1.90061	−0.950304	−	0.311325i	$-0.899227\pi$
$47$	−612.406	−1.90061	−0.950304	+	0.311325i	$0.899227\pi$
$48$	0	0
$49$	843.972	2.46056
$50$	137.296	0.388333
$51$	0	0
$52$	−11.2766	−0.0300728
$53$	−260.323	−0.674683	−0.337341	−	0.941382i	$-0.609528\pi$
$53$	−260.323	−0.674683	−0.337341	+	0.941382i	$0.609528\pi$
$54$	0	0
$55$	164.191	0.402537
$56$	848.139	2.02388
$57$	0	0
$58$	−159.395	−0.360855
$59$	592.573	1.30757	0.653783	−	0.756682i	$-0.273181\pi$
$59$	592.573	1.30757	0.653783	+	0.756682i	$0.273181\pi$
$60$	0	0
$61$	641.983	1.34750	0.673750	−	0.738959i	$-0.264682\pi$
$61$	641.983	1.34750	0.673750	+	0.738959i	$0.264682\pi$
$62$	−616.234	−1.26229
$63$	0	0
$64$	558.265	1.09036
$65$	37.7067	0.0719529
$66$	0	0
$67$	76.0336	0.138642	0.0693208	−	0.997594i	$-0.477917\pi$
$67$	76.0336	0.138642	0.0693208	+	0.997594i	$0.477917\pi$
$68$	−95.4868	−0.170286
$69$	0	0
$70$	−663.538	−1.13297
$71$	781.662	1.30657	0.653283	−	0.757114i	$-0.273391\pi$
$71$	781.662	1.30657	0.653283	+	0.757114i	$0.273391\pi$
$72$	0	0
$73$	−100.795	−0.161604	−0.0808021	−	0.996730i	$-0.525748\pi$
$73$	−100.795	−0.161604	−0.0808021	+	0.996730i	$0.525748\pi$
$74$	−613.779	−0.964194
$75$	0	0
$76$	−177.047	−0.267219
$77$	692.352	1.02469
$78$	0	0
$79$	1141.46	1.62562	0.812812	−	0.582526i	$-0.197936\pi$
$79$	1141.46	1.62562	0.812812	+	0.582526i	$0.197936\pi$
$80$	−314.412	−0.439405
$81$	0	0
$82$	212.918	0.286742
$83$	−900.743	−1.19120	−0.595599	−	0.803282i	$-0.703085\pi$
$83$	−900.743	−1.19120	−0.595599	+	0.803282i	$0.703085\pi$
$84$	0	0
$85$	319.288	0.407431
$86$	−428.867	−0.537743
$87$	0	0
$88$	494.713	0.599279
$89$	−651.019	−0.775369	−0.387685	−	0.921792i	$-0.626725\pi$
$89$	−651.019	−0.775369	−0.387685	+	0.921792i	$0.626725\pi$
$90$	0	0
$91$	159.000	0.183161
$92$	−448.315	−0.508044
$93$	0	0
$94$	1443.58	1.58398
$95$	592.008	0.639355
$96$	0	0
$97$	856.008	0.896026	0.448013	−	0.894027i	$-0.352132\pi$
$97$	856.008	0.896026	0.448013	+	0.894027i	$0.352132\pi$
$98$	−1989.44	−2.05065
$99$	0	0
$100$	142.318	0.142318
$101$	−1622.93	−1.59889	−0.799445	−	0.600740i	$-0.794873\pi$
$101$	−1622.93	−1.59889	−0.799445	+	0.600740i	$0.794873\pi$
$102$	0	0
$103$	−1800.13	−1.72206	−0.861028	−	0.508557i	$-0.830179\pi$
$103$	−1800.13	−1.72206	−0.861028	+	0.508557i	$0.830179\pi$
$104$	113.611	0.107120
$105$	0	0
$106$	613.643	0.562286
$107$	−576.698	−0.521042	−0.260521	−	0.965468i	$-0.583894\pi$
$107$	−576.698	−0.521042	−0.260521	+	0.965468i	$0.583894\pi$
$108$	0	0
$109$	−805.934	−0.708206	−0.354103	−	0.935206i	$-0.615214\pi$
$109$	−805.934	−0.708206	−0.354103	+	0.935206i	$0.615214\pi$
$110$	−387.037	−0.335477
$111$	0	0
$112$	−1325.80	−1.11854
$113$	148.948	0.123999	0.0619994	−	0.998076i	$-0.480252\pi$
$113$	148.948	0.123999	0.0619994	+	0.998076i	$0.480252\pi$
$114$	0	0
$115$	1499.07	1.21556
$116$	−165.225	−0.132248
$117$	0	0
$118$	−1396.83	−1.08974
$119$	1346.36	1.03714
$120$	0	0
$121$	−927.156	−0.696586
$122$	−1513.30	−1.12302
$123$	0	0
$124$	−638.774	−0.462609
$125$	−1497.18	−1.07130
$126$	0	0
$127$	−290.448	−0.202938	−0.101469	−	0.994839i	$-0.532354\pi$
$127$	−290.448	−0.202938	−0.101469	+	0.994839i	$0.532354\pi$
$128$	−466.118	−0.321870
$129$	0	0
$130$	−88.8835	−0.0599661
$131$	663.457	0.442492	0.221246	−	0.975218i	$-0.428988\pi$
$131$	663.457	0.442492	0.221246	+	0.975218i	$0.428988\pi$
$132$	0	0
$133$	2496.35	1.62752
$134$	−179.229	−0.115545
$135$	0	0
$136$	962.024	0.606565
$137$	255.785	0.159512	0.0797561	−	0.996814i	$-0.474586\pi$
$137$	255.785	0.159512	0.0797561	+	0.996814i	$0.474586\pi$
$138$	0	0
$139$	−1307.81	−0.798034	−0.399017	−	0.916944i	$-0.630649\pi$
$139$	−1307.81	−0.798034	−0.399017	+	0.916944i	$0.630649\pi$
$140$	−687.808	−0.415217
$141$	0	0
$142$	−1842.56	−1.08890
$143$	92.7433	0.0542348
$144$	0	0
$145$	552.478	0.316420
$146$	237.596	0.134682
$147$	0	0
$148$	−636.229	−0.353363
$149$	−2970.56	−1.63327	−0.816637	−	0.577152i	$-0.804164\pi$
$149$	−2970.56	−1.63327	−0.816637	+	0.577152i	$0.804164\pi$
$150$	0	0
$151$	−1701.11	−0.916786	−0.458393	−	0.888750i	$-0.651575\pi$
$151$	−1701.11	−0.916786	−0.458393	+	0.888750i	$0.651575\pi$
$152$	1783.74	0.951844
$153$	0	0
$154$	−1632.03	−0.853981
$155$	2135.92	1.10685
$156$	0	0
$157$	−710.749	−0.361299	−0.180649	−	0.983548i	$-0.557820\pi$
$157$	−710.749	−0.361299	−0.180649	+	0.983548i	$0.557820\pi$
$158$	−2690.69	−1.35481
$159$	0	0
$160$	−867.944	−0.428856
$161$	6321.20	3.09429
$162$	0	0
$163$	193.312	0.0928917	0.0464458	−	0.998921i	$-0.485211\pi$
$163$	193.312	0.0928917	0.0464458	+	0.998921i	$0.485211\pi$
$164$	220.706	0.105087
$165$	0	0
$166$	2123.26	0.992753
$167$	−1976.82	−0.915991	−0.457996	−	0.888954i	$-0.651432\pi$
$167$	−1976.82	−0.915991	−0.457996	+	0.888954i	$0.651432\pi$
$168$	0	0
$169$	−2175.70	−0.990306
$170$	−752.636	−0.339556
$171$	0	0
$172$	−444.553	−0.197075
$173$	−4034.98	−1.77326	−0.886630	−	0.462479i	$-0.846960\pi$
$173$	−4034.98	−1.77326	−0.886630	+	0.462479i	$0.846960\pi$
$174$	0	0
$175$	−2006.67	−0.866801
$176$	−773.327	−0.331203
$177$	0	0
$178$	1534.60	0.646199
$179$	−3411.01	−1.42431	−0.712154	−	0.702023i	$-0.752280\pi$
$179$	−3411.01	−1.42431	−0.712154	+	0.702023i	$0.752280\pi$
$180$	0	0
$181$	−4564.67	−1.87452	−0.937262	−	0.348625i	$-0.886649\pi$
$181$	−4564.67	−1.87452	−0.937262	+	0.348625i	$0.886649\pi$
$182$	−374.799	−0.152648
$183$	0	0
$184$	4516.75	1.80967
$185$	2127.42	0.845463
$186$	0	0
$187$	785.319	0.307103
$188$	1496.38	0.580505
$189$	0	0
$190$	−1395.50	−0.532843
$191$	3550.82	1.34517	0.672587	−	0.740018i	$-0.265183\pi$
$191$	3550.82	1.34517	0.672587	+	0.740018i	$0.265183\pi$
$192$	0	0
$193$	−4420.52	−1.64868	−0.824342	−	0.566091i	$-0.808455\pi$
$193$	−4420.52	−1.64868	−0.824342	+	0.566091i	$0.808455\pi$
$194$	−2017.81	−0.746755
$195$	0	0
$196$	−2062.21	−0.751533
$197$	−3983.56	−1.44070	−0.720348	−	0.693613i	$-0.756018\pi$
$197$	−3983.56	−1.44070	−0.720348	+	0.693613i	$0.756018\pi$
$198$	0	0
$199$	5575.10	1.98597	0.992986	−	0.118228i	$-0.0377215\pi$
$199$	5575.10	1.98597	0.992986	+	0.118228i	$0.0377215\pi$
$200$	−1433.85	−0.506942
$201$	0	0
$202$	3825.63	1.33253
$203$	2329.66	0.805469
$204$	0	0
$205$	−737.994	−0.251433
$206$	4243.32	1.43517
$207$	0	0
$208$	−177.596	−0.0592021
$209$	1456.10	0.481916
$210$	0	0
$211$	3418.58	1.11538	0.557689	−	0.830050i	$-0.311688\pi$
$211$	3418.58	1.11538	0.557689	+	0.830050i	$0.311688\pi$
$212$	636.088	0.206069
$213$	0	0
$214$	1359.41	0.434241
$215$	1486.49	0.471525
$216$	0	0
$217$	9006.65	2.81756
$218$	1899.77	0.590225
$219$	0	0
$220$	−401.193	−0.122947
$221$	180.350	0.0548942
$222$	0	0
$223$	6297.69	1.89114	0.945571	−	0.325417i	$-0.105505\pi$
$223$	6297.69	1.89114	0.945571	+	0.325417i	$0.105505\pi$
$224$	−3659.90	−1.09168
$225$	0	0
$226$	−351.105	−0.103341
$227$	1431.58	0.418579	0.209290	−	0.977854i	$-0.432885\pi$
$227$	1431.58	0.418579	0.209290	+	0.977854i	$0.432885\pi$
$228$	0	0
$229$	4419.80	1.27541	0.637704	−	0.770281i	$-0.279884\pi$
$229$	4419.80	1.27541	0.637704	+	0.770281i	$0.279884\pi$
$230$	−3533.66	−1.01305
$231$	0	0
$232$	1664.64	0.471072
$233$	−26.5441	−0.00746335	−0.00373167	−	0.999993i	$-0.501188\pi$
$233$	−26.5441	−0.00746335	−0.00373167	+	0.999993i	$0.501188\pi$
$234$	0	0
$235$	−5003.59	−1.38893
$236$	−1447.92	−0.399372
$237$	0	0
$238$	−3173.67	−0.864364
$239$	239.000	0.0646846
$239$	239.000	0.0646846
$240$	0	0
$241$	−536.528	−0.143406	−0.0717029	−	0.997426i	$-0.522843\pi$
$241$	−536.528	−0.143406	−0.0717029	+	0.997426i	$0.522843\pi$
$242$	2185.52	0.580540
$243$	0	0
$244$	−1568.65	−0.411569
$245$	6895.58	1.79813
$246$	0	0
$247$	334.395	0.0861419
$248$	6435.61	1.64783
$249$	0	0
$250$	3529.20	0.892826
$251$	7396.22	1.85994	0.929971	−	0.367633i	$-0.119832\pi$
$251$	7396.22	1.85994	0.929971	+	0.367633i	$0.119832\pi$
$252$	0	0
$253$	3687.11	0.916231
$254$	684.654	0.169130
$255$	0	0
$256$	−3367.37	−0.822112
$257$	1240.11	0.300996	0.150498	−	0.988610i	$-0.451912\pi$
$257$	1240.11	0.300996	0.150498	+	0.988610i	$0.451912\pi$
$258$	0	0
$259$	8970.77	2.15219
$260$	−92.1345	−0.0219767
$261$	0	0
$262$	−1563.92	−0.368776
$263$	6053.27	1.41924	0.709621	−	0.704584i	$-0.248866\pi$
$263$	6053.27	1.41924	0.709621	+	0.704584i	$0.248866\pi$
$264$	0	0
$265$	−2126.94	−0.493046
$266$	−5884.47	−1.35639
$267$	0	0
$268$	−185.785	−0.0423455
$269$	−45.4646	−0.0103049	−0.00515247	−	0.999987i	$-0.501640\pi$
$269$	−45.4646	−0.0103049	−0.00515247	+	0.999987i	$0.501640\pi$
$270$	0	0
$271$	−5415.06	−1.21381	−0.606903	−	0.794776i	$-0.707588\pi$
$271$	−5415.06	−1.21381	−0.606903	+	0.794776i	$0.707588\pi$
$272$	−1503.82	−0.335230
$273$	0	0
$274$	−602.944	−0.132939
$275$	−1170.48	−0.256663
$276$	0	0
$277$	−2752.10	−0.596959	−0.298479	−	0.954416i	$-0.596479\pi$
$277$	−2752.10	−0.596959	−0.298479	+	0.954416i	$0.596479\pi$
$278$	3082.81	0.665088
$279$	0	0
$280$	6929.62	1.47902
$281$	−8393.45	−1.78189	−0.890946	−	0.454110i	$-0.849957\pi$
$281$	−8393.45	−1.78189	−0.890946	+	0.454110i	$0.849957\pi$
$282$	0	0
$283$	−2985.38	−0.627075	−0.313538	−	0.949576i	$-0.601514\pi$
$283$	−2985.38	−0.627075	−0.313538	+	0.949576i	$0.601514\pi$
$284$	−1909.95	−0.399066
$285$	0	0
$286$	−218.617	−0.0451997
$287$	−3111.93	−0.640040
$288$	0	0
$289$	−3385.86	−0.689164
$290$	−1302.32	−0.263707
$291$	0	0
$292$	246.287	0.0493591
$293$	−517.361	−0.103155	−0.0515777	−	0.998669i	$-0.516425\pi$
$293$	−517.361	−0.103155	−0.0515777	+	0.998669i	$0.516425\pi$
$294$	0	0
$295$	4841.55	0.955546
$296$	6409.97	1.25869
$297$	0	0
$298$	7002.30	1.36118
$299$	846.749	0.163775
$300$	0	0
$301$	6268.16	1.20030
$302$	4009.92	0.764057
$303$	0	0
$304$	−2788.31	−0.526055
$305$	5245.25	0.984729
$306$	0	0
$307$	1336.43	0.248451	0.124225	−	0.992254i	$-0.460355\pi$
$307$	1336.43	0.248451	0.124225	+	0.992254i	$0.460355\pi$
$308$	−1691.73	−0.312971
$309$	0	0
$310$	−5034.87	−0.922457
$311$	−2188.00	−0.398939	−0.199469	−	0.979904i	$-0.563922\pi$
$311$	−2188.00	−0.398939	−0.199469	+	0.979904i	$0.563922\pi$
$312$	0	0
$313$	4558.49	0.823198	0.411599	−	0.911365i	$-0.364970\pi$
$313$	4558.49	0.823198	0.411599	+	0.911365i	$0.364970\pi$
$314$	1675.40	0.301109
$315$	0	0
$316$	−2789.10	−0.496517
$317$	3629.26	0.643027	0.321514	−	0.946905i	$-0.395808\pi$
$317$	3629.26	0.643027	0.321514	+	0.946905i	$0.395808\pi$
$318$	0	0
$319$	1358.87	0.238503
$320$	4561.24	0.796817
$321$	0	0
$322$	−14900.5	−2.57880
$323$	2831.55	0.487776
$324$	0	0
$325$	−268.802	−0.0458783
$326$	−455.681	−0.0774166
$327$	0	0
$328$	−2223.60	−0.374322
$329$	−21098.9	−3.53562
$330$	0	0
$331$	10627.0	1.76469	0.882344	−	0.470606i	$-0.155965\pi$
$331$	10627.0	1.76469	0.882344	+	0.470606i	$0.155965\pi$
$332$	2200.92	0.363829
$333$	0	0
$334$	4659.81	0.763394
$335$	621.225	0.101317
$336$	0	0
$337$	−8882.96	−1.43586	−0.717931	−	0.696114i	$-0.754911\pi$
$337$	−8882.96	−1.43586	−0.717931	+	0.696114i	$0.754911\pi$
$338$	5128.64	0.825328
$339$	0	0
$340$	−780.164	−0.124442
$341$	5253.51	0.834292
$342$	0	0
$343$	17259.7	2.71702
$344$	4478.85	0.701986
$345$	0	0
$346$	9511.39	1.47785
$347$	−2862.14	−0.442789	−0.221394	−	0.975184i	$-0.571061\pi$
$347$	−2862.14	−0.442789	−0.221394	+	0.975184i	$0.571061\pi$
$348$	0	0
$349$	2052.27	0.314773	0.157386	−	0.987537i	$-0.449693\pi$
$349$	2052.27	0.314773	0.157386	+	0.987537i	$0.449693\pi$
$350$	4730.20	0.722399
$351$	0	0
$352$	−2134.79	−0.323252
$353$	−4490.41	−0.677055	−0.338528	−	0.940956i	$-0.609929\pi$
$353$	−4490.41	−0.677055	−0.338528	+	0.940956i	$0.609929\pi$
$354$	0	0
$355$	6386.48	0.954815
$356$	1590.73	0.236822
$357$	0	0
$358$	8040.55	1.18703
$359$	−920.320	−0.135300	−0.0676499	−	0.997709i	$-0.521550\pi$
$359$	−920.320	−0.135300	−0.0676499	+	0.997709i	$0.521550\pi$
$360$	0	0
$361$	−1608.88	−0.234565
$362$	10760.0	1.56224
$363$	0	0
$364$	−388.508	−0.0559433
$365$	−823.531	−0.118098
$366$	0	0
$367$	−11519.2	−1.63841	−0.819205	−	0.573501i	$-0.805585\pi$
$367$	−11519.2	−1.63841	−0.819205	+	0.573501i	$0.805585\pi$
$368$	−7060.51	−1.00015
$369$	0	0
$370$	−5014.81	−0.704616
$371$	−8968.78	−1.25508
$372$	0	0
$373$	6428.88	0.892426	0.446213	−	0.894927i	$-0.352772\pi$
$373$	6428.88	0.892426	0.446213	+	0.894927i	$0.352772\pi$
$374$	−1851.18	−0.255942
$375$	0	0
$376$	−15076.0	−2.06778
$377$	312.067	0.0426320
$378$	0	0
$379$	−1954.14	−0.264848	−0.132424	−	0.991193i	$-0.542276\pi$
$379$	−1954.14	−0.264848	−0.132424	+	0.991193i	$0.542276\pi$
$380$	−1446.54	−0.195279
$381$	0	0
$382$	−8370.11	−1.12108
$383$	8373.50	1.11714	0.558571	−	0.829456i	$-0.311350\pi$
$383$	8373.50	1.11714	0.558571	+	0.829456i	$0.311350\pi$
$384$	0	0
$385$	5656.79	0.748822
$386$	10420.2	1.37403
$387$	0	0
$388$	−2091.62	−0.273674
$389$	−3503.79	−0.456681	−0.228341	−	0.973581i	$-0.573330\pi$
$389$	−3503.79	−0.456681	−0.228341	+	0.973581i	$0.573330\pi$
$390$	0	0
$391$	7169.99	0.927371
$392$	20776.6	2.67698
$393$	0	0
$394$	9390.19	1.20069
$395$	9326.17	1.18798
$396$	0	0
$397$	−675.958	−0.0854543	−0.0427271	−	0.999087i	$-0.513605\pi$
$397$	−675.958	−0.0854543	−0.0427271	+	0.999087i	$0.513605\pi$
$398$	−13141.8	−1.65513
$399$	0	0
$400$	2241.37	0.280171
$401$	−15382.9	−1.91568	−0.957838	−	0.287308i	$-0.907240\pi$
$401$	−15382.9	−1.91568	−0.957838	+	0.287308i	$0.907240\pi$
$402$	0	0
$403$	1206.48	0.149129
$404$	3965.56	0.488351
$405$	0	0
$406$	−5491.55	−0.671284
$407$	5232.58	0.637272
$408$	0	0
$409$	−1384.62	−0.167397	−0.0836983	−	0.996491i	$-0.526673\pi$
$409$	−1384.62	−0.167397	−0.0836983	+	0.996491i	$0.526673\pi$
$410$	1739.62	0.209546
$411$	0	0
$412$	4398.52	0.525970
$413$	20415.6	2.43241
$414$	0	0
$415$	−7359.42	−0.870506
$416$	−490.258	−0.0577809
$417$	0	0
$418$	−3432.37	−0.401633
$419$	−8637.00	−1.00703	−0.503515	−	0.863987i	$-0.667960\pi$
$419$	−8637.00	−1.00703	−0.503515	+	0.863987i	$0.667960\pi$
$420$	0	0
$421$	1523.46	0.176364	0.0881818	−	0.996104i	$-0.471894\pi$
$421$	1523.46	0.176364	0.0881818	+	0.996104i	$0.471894\pi$
$422$	−8058.38	−0.929564
$423$	0	0
$424$	−6408.55	−0.734026
$425$	−2276.12	−0.259784
$426$	0	0
$427$	22117.9	2.50670
$428$	1409.13	0.159143
$429$	0	0
$430$	−3504.01	−0.392973
$431$	−3321.50	−0.371209	−0.185604	−	0.982625i	$-0.559424\pi$
$431$	−3321.50	−0.371209	−0.185604	+	0.982625i	$0.559424\pi$
$432$	0	0
$433$	−10953.8	−1.21572	−0.607862	−	0.794043i	$-0.707972\pi$
$433$	−10953.8	−1.21572	−0.607862	+	0.794043i	$0.707972\pi$
$434$	−21230.8	−2.34818
$435$	0	0
$436$	1969.26	0.216309
$437$	13294.2	1.45526
$438$	0	0
$439$	−312.062	−0.0339269	−0.0169634	−	0.999856i	$-0.505400\pi$
$439$	−312.062	−0.0339269	−0.0169634	+	0.999856i	$0.505400\pi$
$440$	4042.00	0.437943
$441$	0	0
$442$	−425.126	−0.0457493
$443$	−812.555	−0.0871459	−0.0435730	−	0.999050i	$-0.513874\pi$
$443$	−812.555	−0.0871459	−0.0435730	+	0.999050i	$0.513874\pi$
$444$	0	0
$445$	−5319.08	−0.566626
$446$	−14845.1	−1.57609
$447$	0	0
$448$	19233.6	2.02835
$449$	13700.7	1.44004	0.720020	−	0.693953i	$-0.244133\pi$
$449$	13700.7	1.44004	0.720020	+	0.693953i	$0.244133\pi$
$450$	0	0
$451$	−1815.17	−0.189519
$452$	−363.948	−0.0378731
$453$	0	0
$454$	−3374.57	−0.348847
$455$	1299.09	0.133851
$456$	0	0
$457$	11685.9	1.19615	0.598077	−	0.801439i	$-0.295932\pi$
$457$	11685.9	1.19615	0.598077	+	0.801439i	$0.295932\pi$
$458$	−10418.5	−1.06294
$459$	0	0
$460$	−3662.91	−0.371269
$461$	−1677.31	−0.169458	−0.0847288	−	0.996404i	$-0.527002\pi$
$461$	−1677.31	−0.169458	−0.0847288	+	0.996404i	$0.527002\pi$
$462$	0	0
$463$	4782.05	0.480002	0.240001	−	0.970773i	$-0.422852\pi$
$463$	4782.05	0.480002	0.240001	+	0.970773i	$0.422852\pi$
$464$	−2602.13	−0.260347
$465$	0	0
$466$	62.5705	0.00622001
$467$	7587.16	0.751803	0.375901	−	0.926660i	$-0.377333\pi$
$467$	7587.16	0.751803	0.375901	+	0.926660i	$0.377333\pi$
$468$	0	0
$469$	2619.55	0.257909
$470$	11794.6	1.15754
$471$	0	0
$472$	14587.8	1.42258
$473$	3656.17	0.355414
$474$	0	0
$475$	−4220.28	−0.407662
$476$	−3289.75	−0.316777
$477$	0	0
$478$	−563.379	−0.0539087
$479$	−238.788	−0.0227776	−0.0113888	−	0.999935i	$-0.503625\pi$
$479$	−238.788	−0.0227776	−0.0113888	+	0.999935i	$0.503625\pi$
$480$	0	0
$481$	1201.67	0.113911
$482$	1264.72	0.119516
$483$	0	0
$484$	2265.46	0.212759
$485$	6993.92	0.654799
$486$	0	0
$487$	3867.13	0.359828	0.179914	−	0.983682i	$-0.442418\pi$
$487$	3867.13	0.359828	0.179914	+	0.983682i	$0.442418\pi$
$488$	15804.1	1.46602
$489$	0	0
$490$	−16254.5	−1.49858
$491$	14575.1	1.33965	0.669824	−	0.742520i	$-0.266370\pi$
$491$	14575.1	1.33965	0.669824	+	0.742520i	$0.266370\pi$
$492$	0	0
$493$	2642.48	0.241402
$494$	−788.248	−0.0717914
$495$	0	0
$496$	−10060.0	−0.910704
$497$	26930.2	2.43055
$498$	0	0
$499$	−18560.5	−1.66510	−0.832548	−	0.553953i	$-0.813119\pi$
$499$	−18560.5	−1.66510	−0.832548	+	0.553953i	$0.813119\pi$
$500$	3658.29	0.327207
$501$	0	0
$502$	−17434.6	−1.55009
$503$	−852.272	−0.0755486	−0.0377743	−	0.999286i	$-0.512027\pi$
$503$	−852.272	−0.0755486	−0.0377743	+	0.999286i	$0.512027\pi$
$504$	0	0
$505$	−13260.0	−1.16844
$506$	−8691.37	−0.763594
$507$	0	0
$508$	709.697	0.0619836
$509$	7352.32	0.640248	0.320124	−	0.947376i	$-0.396275\pi$
$509$	7352.32	0.640248	0.320124	+	0.947376i	$0.396275\pi$
$510$	0	0
$511$	−3472.62	−0.300626
$512$	11666.6	1.00703
$513$	0	0
$514$	−2923.23	−0.250852
$515$	−14707.7	−1.25845
$516$	0	0
$517$	−12306.8	−1.04691
$518$	−21146.2	−1.79365
$519$	0	0
$520$	928.250	0.0782817
$521$	−13892.7	−1.16823	−0.584115	−	0.811671i	$-0.698558\pi$
$521$	−13892.7	−1.16823	−0.584115	+	0.811671i	$0.698558\pi$
$522$	0	0
$523$	−20748.1	−1.73470	−0.867352	−	0.497695i	$-0.834180\pi$
$523$	−20748.1	−1.73470	−0.867352	+	0.497695i	$0.834180\pi$
$524$	−1621.12	−0.135151
$525$	0	0
$526$	−14269.0	−1.18281
$527$	10216.0	0.844436
$528$	0	0
$529$	21496.4	1.76678
$530$	5013.70	0.410908
$531$	0	0
$532$	−6099.70	−0.497097
$533$	−416.855	−0.0338762
$534$	0	0
$535$	−4711.85	−0.380768
$536$	1871.77	0.150836
$537$	0	0
$538$	107.171	0.00858822
$539$	16960.3	1.35535
$540$	0	0
$541$	10691.4	0.849643	0.424822	−	0.905277i	$-0.360337\pi$
$541$	10691.4	0.849643	0.424822	+	0.905277i	$0.360337\pi$
$542$	12764.5	1.01159
$543$	0	0
$544$	−4151.34	−0.327182
$545$	−6584.80	−0.517544
$546$	0	0
$547$	−6570.97	−0.513628	−0.256814	−	0.966461i	$-0.582673\pi$
$547$	−6570.97	−0.513628	−0.256814	+	0.966461i	$0.582673\pi$
$548$	−624.998	−0.0487200
$549$	0	0
$550$	2759.09	0.213905
$551$	4899.56	0.378817
$552$	0	0
$553$	39326.1	3.02408
$554$	6487.34	0.497510
$555$	0	0
$556$	3195.56	0.243745
$557$	−12342.5	−0.938905	−0.469453	−	0.882958i	$-0.655549\pi$
$557$	−12342.5	−0.938905	−0.469453	+	0.882958i	$0.655549\pi$
$558$	0	0
$559$	839.644	0.0635298
$560$	−10832.3	−0.817406
$561$	0	0
$562$	19785.3	1.48504
$563$	7206.53	0.539466	0.269733	−	0.962935i	$-0.413065\pi$
$563$	7206.53	0.539466	0.269733	+	0.962935i	$0.413065\pi$
$564$	0	0
$565$	1216.96	0.0906160
$566$	7037.23	0.522609
$567$	0	0
$568$	19242.7	1.42149
$569$	21324.5	1.57112	0.785562	−	0.618783i	$-0.212374\pi$
$569$	21324.5	1.57112	0.785562	+	0.618783i	$0.212374\pi$
$570$	0	0
$571$	7463.09	0.546971	0.273486	−	0.961876i	$-0.411823\pi$
$571$	7463.09	0.546971	0.273486	+	0.961876i	$0.411823\pi$
$572$	−226.614	−0.0165650
$573$	0	0
$574$	7335.55	0.533414
$575$	−10686.5	−0.775057
$576$	0	0
$577$	−22634.4	−1.63307	−0.816537	−	0.577293i	$-0.804109\pi$
$577$	−22634.4	−1.63307	−0.816537	+	0.577293i	$0.804109\pi$
$578$	7981.26	0.574354
$579$	0	0
$580$	−1349.95	−0.0966445
$581$	−31032.8	−2.21593
$582$	0	0
$583$	−5231.43	−0.371636
$584$	−2481.33	−0.175818
$585$	0	0
$586$	1219.54	0.0859706
$587$	−394.214	−0.0277188	−0.0138594	−	0.999904i	$-0.504412\pi$
$587$	−394.214	−0.0277188	−0.0138594	+	0.999904i	$0.504412\pi$
$588$	0	0
$589$	18942.1	1.32512
$590$	−11412.7	−0.796360
$591$	0	0
$592$	−10020.0	−0.695639
$593$	703.737	0.0487336	0.0243668	−	0.999703i	$-0.492243\pi$
$593$	703.737	0.0487336	0.0243668	+	0.999703i	$0.492243\pi$
$594$	0	0
$595$	11000.2	0.757926
$596$	7258.42	0.498853
$597$	0	0
$598$	−1995.99	−0.136491
$599$	−16450.7	−1.12213	−0.561066	−	0.827771i	$-0.689608\pi$
$599$	−16450.7	−1.12213	−0.561066	+	0.827771i	$0.689608\pi$
$600$	0	0
$601$	−8308.16	−0.563888	−0.281944	−	0.959431i	$-0.590979\pi$
$601$	−8308.16	−0.563888	−0.281944	+	0.959431i	$0.590979\pi$
$602$	−14775.5	−1.00034
$603$	0	0
$604$	4156.59	0.280015
$605$	−7575.23	−0.509053
$606$	0	0
$607$	16830.2	1.12540	0.562701	−	0.826661i	$-0.309762\pi$
$607$	16830.2	1.12540	0.562701	+	0.826661i	$0.309762\pi$
$608$	−7697.21	−0.513426
$609$	0	0
$610$	−12364.3	−0.820681
$611$	−2826.28	−0.187134
$612$	0	0
$613$	−8865.84	−0.584157	−0.292078	−	0.956394i	$-0.594347\pi$
$613$	−8865.84	−0.584157	−0.292078	+	0.956394i	$0.594347\pi$
$614$	−3150.29	−0.207061
$615$	0	0
$616$	17044.1	1.11481
$617$	15857.3	1.03467	0.517335	−	0.855783i	$-0.326924\pi$
$617$	15857.3	1.03467	0.517335	+	0.855783i	$0.326924\pi$
$618$	0	0
$619$	−4929.79	−0.320105	−0.160053	−	0.987108i	$-0.551166\pi$
$619$	−4929.79	−0.320105	−0.160053	+	0.987108i	$0.551166\pi$
$620$	−5219.03	−0.338067
$621$	0	0
$622$	5157.62	0.332479
$623$	−22429.2	−1.44239
$624$	0	0
$625$	−4951.97	−0.316926
$626$	−10745.4	−0.686060
$627$	0	0
$628$	1736.68	0.110352
$629$	10175.3	0.645019
$630$	0	0
$631$	21300.5	1.34384	0.671918	−	0.740626i	$-0.265471\pi$
$631$	21300.5	1.34384	0.671918	+	0.740626i	$0.265471\pi$
$632$	28100.1	1.76861
$633$	0	0
$634$	−8555.01	−0.535904
$635$	−2373.08	−0.148303
$636$	0	0
$637$	3894.97	0.242267
$638$	−3203.18	−0.198770
$639$	0	0
$640$	−3808.37	−0.235217
$641$	−2762.44	−0.170218	−0.0851090	−	0.996372i	$-0.527124\pi$
$641$	−2762.44	−0.170218	−0.0851090	+	0.996372i	$0.527124\pi$
$642$	0	0
$643$	31108.5	1.90793	0.953965	−	0.299919i	$-0.0969597\pi$
$643$	31108.5	1.90793	0.953965	+	0.299919i	$0.0969597\pi$
$644$	−15445.5	−0.945093
$645$	0	0
$646$	−6674.62	−0.406516
$647$	−17242.4	−1.04771	−0.523855	−	0.851808i	$-0.675506\pi$
$647$	−17242.4	−1.04771	−0.523855	+	0.851808i	$0.675506\pi$
$648$	0	0
$649$	11908.3	0.720247
$650$	633.628	0.0382353
$651$	0	0
$652$	−472.348	−0.0283720
$653$	−7624.58	−0.456926	−0.228463	−	0.973553i	$-0.573370\pi$
$653$	−7624.58	−0.456926	−0.228463	+	0.973553i	$0.573370\pi$
$654$	0	0
$655$	5420.70	0.323365
$656$	3475.89	0.206876
$657$	0	0
$658$	49735.0	2.94661
$659$	−8713.42	−0.515063	−0.257532	−	0.966270i	$-0.582909\pi$
$659$	−8713.42	−0.515063	−0.257532	+	0.966270i	$0.582909\pi$
$660$	0	0
$661$	22069.2	1.29863	0.649314	−	0.760521i	$-0.275056\pi$
$661$	22069.2	1.29863	0.649314	+	0.760521i	$0.275056\pi$
$662$	−25050.3	−1.47070
$663$	0	0
$664$	−22174.2	−1.29597
$665$	20396.1	1.18936
$666$	0	0
$667$	12406.6	0.720216
$668$	4830.25	0.279773
$669$	0	0
$670$	−1464.37	−0.0844382
$671$	12901.2	0.742244
$672$	0	0
$673$	6557.61	0.375598	0.187799	−	0.982208i	$-0.439865\pi$
$673$	6557.61	0.375598	0.187799	+	0.982208i	$0.439865\pi$
$674$	20939.2	1.19666
$675$	0	0
$676$	5316.22	0.302471
$677$	−6131.50	−0.348083	−0.174042	−	0.984738i	$-0.555683\pi$
$677$	−6131.50	−0.348083	−0.174042	+	0.984738i	$0.555683\pi$
$678$	0	0
$679$	29491.6	1.66684
$680$	7860.11	0.443267
$681$	0	0
$682$	−12383.8	−0.695306
$683$	4988.14	0.279452	0.139726	−	0.990190i	$-0.455378\pi$
$683$	4988.14	0.279452	0.139726	+	0.990190i	$0.455378\pi$
$684$	0	0
$685$	2089.86	0.116569
$686$	−40685.2	−2.26438
$687$	0	0
$688$	−7001.26	−0.387966
$689$	−1201.40	−0.0664294
$690$	0	0
$691$	14759.5	0.812556	0.406278	−	0.913749i	$-0.366826\pi$
$691$	14759.5	0.812556	0.406278	+	0.913749i	$0.366826\pi$
$692$	9859.29	0.541610
$693$	0	0
$694$	6746.73	0.369024
$695$	−10685.3	−0.583189
$696$	0	0
$697$	−3529.79	−0.191823
$698$	−4837.69	−0.262334
$699$	0	0
$700$	4903.21	0.264749
$701$	−18101.0	−0.975272	−0.487636	−	0.873047i	$-0.662141\pi$
$701$	−18101.0	−0.975272	−0.487636	+	0.873047i	$0.662141\pi$
$702$	0	0
$703$	18866.6	1.01219
$704$	11218.8	0.600604
$705$	0	0
$706$	10584.9	0.564263
$707$	−55914.0	−2.97435
$708$	0	0
$709$	−4875.32	−0.258246	−0.129123	−	0.991629i	$-0.541216\pi$
$709$	−4875.32	−0.258246	−0.129123	+	0.991629i	$0.541216\pi$
$710$	−15054.4	−0.795750
$711$	0	0
$712$	−16026.6	−0.843568
$713$	47964.8	2.51935
$714$	0	0
$715$	757.749	0.0396338
$716$	8334.65	0.435029
$717$	0	0
$718$	2169.41	0.112760
$719$	−9678.95	−0.502036	−0.251018	−	0.967982i	$-0.580765\pi$
$719$	−9678.95	−0.502036	−0.251018	+	0.967982i	$0.580765\pi$
$720$	0	0
$721$	−62018.8	−3.20347
$722$	3792.51	0.195489
$723$	0	0
$724$	11153.5	0.572539
$725$	−3938.48	−0.201754
$726$	0	0
$727$	2276.10	0.116115	0.0580577	−	0.998313i	$-0.481509\pi$
$727$	2276.10	0.116115	0.0580577	+	0.998313i	$0.481509\pi$
$728$	3914.20	0.199272
$729$	0	0
$730$	1941.25	0.0984234
$731$	7109.83	0.359735
$732$	0	0
$733$	−18220.1	−0.918108	−0.459054	−	0.888408i	$-0.651812\pi$
$733$	−18220.1	−0.918108	−0.459054	+	0.888408i	$0.651812\pi$
$734$	27153.4	1.36546
$735$	0	0
$736$	−19490.7	−0.976138
$737$	1527.96	0.0763680
$738$	0	0
$739$	18202.5	0.906075	0.453037	−	0.891492i	$-0.350340\pi$
$739$	18202.5	0.906075	0.453037	+	0.891492i	$0.350340\pi$
$740$	−5198.24	−0.258231
$741$	0	0
$742$	21141.5	1.04600
$743$	−24540.7	−1.21172	−0.605861	−	0.795570i	$-0.707171\pi$
$743$	−24540.7	−1.21172	−0.605861	+	0.795570i	$0.707171\pi$
$744$	0	0
$745$	−24270.6	−1.19357
$746$	−15154.4	−0.743754
$747$	0	0
$748$	−1918.89	−0.0937989
$749$	−19868.7	−0.969273
$750$	0	0
$751$	−8767.16	−0.425989	−0.212995	−	0.977053i	$-0.568322\pi$
$751$	−8767.16	−0.425989	−0.212995	+	0.977053i	$0.568322\pi$
$752$	23566.5	1.14280
$753$	0	0
$754$	−735.615	−0.0355299
$755$	−13898.8	−0.669971
$756$	0	0
$757$	−16022.1	−0.769266	−0.384633	−	0.923069i	$-0.625672\pi$
$757$	−16022.1	−0.769266	−0.384633	+	0.923069i	$0.625672\pi$
$758$	4606.36	0.220726
$759$	0	0
$760$	14573.8	0.695590
$761$	21923.6	1.04432	0.522162	−	0.852846i	$-0.325126\pi$
$761$	21923.6	1.04432	0.522162	+	0.852846i	$0.325126\pi$
$762$	0	0
$763$	−27766.4	−1.31745
$764$	−8676.26	−0.410859
$765$	0	0
$766$	−19738.3	−0.931035
$767$	2734.75	0.128743
$768$	0	0
$769$	−8282.63	−0.388400	−0.194200	−	0.980962i	$-0.562211\pi$
$769$	−8282.63	−0.388400	−0.194200	+	0.980962i	$0.562211\pi$
$770$	−13334.4	−0.624074
$771$	0	0
$772$	10801.3	0.503560
$773$	25704.7	1.19603	0.598016	−	0.801484i	$-0.295956\pi$
$773$	25704.7	1.19603	0.598016	+	0.801484i	$0.295956\pi$
$774$	0	0
$775$	−15226.5	−0.705744
$776$	21072.9	0.974837
$777$	0	0
$778$	8259.24	0.380602
$779$	−6544.77	−0.301015
$780$	0	0
$781$	15708.2	0.719696
$782$	−16901.3	−0.772878
$783$	0	0
$784$	−32477.7	−1.47949
$785$	−5807.10	−0.264031
$786$	0	0
$787$	−185.901	−0.00842015	−0.00421007	−	0.999991i	$-0.501340\pi$
$787$	−185.901	−0.00842015	−0.00421007	+	0.999991i	$0.501340\pi$
$788$	9733.65	0.440034
$789$	0	0
$790$	−21984.0	−0.990069
$791$	5131.63	0.230670
$792$	0	0
$793$	2962.78	0.132675
$794$	1593.39	0.0712182
$795$	0	0
$796$	−13622.5	−0.606579
$797$	5591.12	0.248492	0.124246	−	0.992251i	$-0.460349\pi$
$797$	5591.12	0.248492	0.124246	+	0.992251i	$0.460349\pi$
$798$	0	0
$799$	−23932.0	−1.05964
$800$	6187.36	0.273445
$801$	0	0
$802$	36261.1	1.59654
$803$	−2025.55	−0.0890165
$804$	0	0
$805$	51646.7	2.26125
$806$	−2843.95	−0.124285
$807$	0	0
$808$	−39952.8	−1.73952
$809$	−22428.5	−0.974715	−0.487357	−	0.873203i	$-0.662039\pi$
$809$	−22428.5	−0.974715	−0.487357	+	0.873203i	$0.662039\pi$
$810$	0	0
$811$	17778.8	0.769788	0.384894	−	0.922961i	$-0.374238\pi$
$811$	17778.8	0.769788	0.384894	+	0.922961i	$0.374238\pi$
$812$	−5692.42	−0.246016
$813$	0	0
$814$	−12334.4	−0.531107
$815$	1579.43	0.0678836
$816$	0	0
$817$	13182.7	0.564509
$818$	3263.88	0.139510
$819$	0	0
$820$	1803.25	0.0767955
$821$	−11923.1	−0.506843	−0.253422	−	0.967356i	$-0.581556\pi$
$821$	−11923.1	−0.506843	−0.253422	+	0.967356i	$0.581556\pi$
$822$	0	0
$823$	30363.2	1.28602	0.643010	−	0.765857i	$-0.277685\pi$
$823$	30363.2	1.28602	0.643010	+	0.765857i	$0.277685\pi$
$824$	−44314.9	−1.87352
$825$	0	0
$826$	−48124.3	−2.02719
$827$	−34954.4	−1.46975	−0.734875	−	0.678202i	$-0.762759\pi$
$827$	−34954.4	−1.46975	−0.734875	+	0.678202i	$0.762759\pi$
$828$	0	0
$829$	−1152.53	−0.0482857	−0.0241429	−	0.999709i	$-0.507686\pi$
$829$	−1152.53	−0.0482857	−0.0241429	+	0.999709i	$0.507686\pi$
$830$	17347.9	0.725486
$831$	0	0
$832$	2576.42	0.107357
$833$	32981.3	1.37183
$834$	0	0
$835$	−16151.4	−0.669390
$836$	−3557.91	−0.147192
$837$	0	0
$838$	20359.4	0.839266
$839$	28808.9	1.18545	0.592726	−	0.805404i	$-0.298052\pi$
$839$	28808.9	1.18545	0.592726	+	0.805404i	$0.298052\pi$
$840$	0	0
$841$	−19816.6	−0.812522
$842$	−3591.16	−0.146983
$843$	0	0
$844$	−8353.13	−0.340671
$845$	−17776.3	−0.723698
$846$	0	0
$847$	−31942.8	−1.29583
$848$	10017.7	0.405673
$849$	0	0
$850$	5365.35	0.216506
$851$	47773.7	1.92440
$852$	0	0
$853$	11629.6	0.466811	0.233406	−	0.972379i	$-0.425013\pi$
$853$	11629.6	0.466811	0.233406	+	0.972379i	$0.425013\pi$
$854$	−52137.0	−2.08910
$855$	0	0
$856$	−14197.0	−0.566871
$857$	8245.78	0.328670	0.164335	−	0.986405i	$-0.447452\pi$
$857$	8245.78	0.328670	0.164335	+	0.986405i	$0.447452\pi$
$858$	0	0
$859$	38129.6	1.51451	0.757255	−	0.653119i	$-0.226540\pi$
$859$	38129.6	1.51451	0.757255	+	0.653119i	$0.226540\pi$
$860$	−3632.17	−0.144019
$861$	0	0
$862$	7829.55	0.309368
$863$	−18680.4	−0.736835	−0.368417	−	0.929661i	$-0.620100\pi$
$863$	−18680.4	−0.736835	−0.368417	+	0.929661i	$0.620100\pi$
$864$	0	0
$865$	−32967.4	−1.29587
$866$	25820.8	1.01319
$867$	0	0
$868$	−22007.3	−0.860573
$869$	22938.6	0.895442
$870$	0	0
$871$	350.898	0.0136507
$872$	−19840.2	−0.770498
$873$	0	0
$874$	−31337.6	−1.21283
$875$	−51581.6	−1.99289
$876$	0	0
$877$	−19632.4	−0.755916	−0.377958	−	0.925823i	$-0.623374\pi$
$877$	−19632.4	−0.755916	−0.377958	+	0.925823i	$0.623374\pi$
$878$	735.602	0.0282749
$879$	0	0
$880$	−6318.39	−0.242037
$881$	39980.7	1.52893	0.764464	−	0.644666i	$-0.223004\pi$
$881$	39980.7	1.52893	0.764464	+	0.644666i	$0.223004\pi$
$882$	0	0
$883$	−40009.1	−1.52482	−0.762409	−	0.647096i	$-0.775983\pi$
$883$	−40009.1	−1.52482	−0.762409	+	0.647096i	$0.775983\pi$
$884$	−440.676	−0.0167664
$885$	0	0
$886$	1915.38	0.0726281
$887$	−47588.7	−1.80143	−0.900717	−	0.434406i	$-0.856958\pi$
$887$	−47588.7	−1.80143	−0.900717	+	0.434406i	$0.856958\pi$
$888$	0	0
$889$	−10006.7	−0.377517
$890$	12538.3	0.472231
$891$	0	0
$892$	−15388.1	−0.577614
$893$	−44373.5	−1.66282
$894$	0	0
$895$	−27869.3	−1.04086
$896$	−16058.9	−0.598762
$897$	0	0
$898$	−32295.8	−1.20014
$899$	17677.3	0.655807
$900$	0	0
$901$	−10173.1	−0.376154
$902$	4278.77	0.157946
$903$	0	0
$904$	3666.75	0.134905
$905$	−37295.1	−1.36987
$906$	0	0
$907$	21421.5	0.784220	0.392110	−	0.919918i	$-0.371745\pi$
$907$	21421.5	0.784220	0.392110	+	0.919918i	$0.371745\pi$
$908$	−3498.00	−0.127847
$909$	0	0
$910$	−3062.25	−0.111552
$911$	−27736.1	−1.00871	−0.504356	−	0.863496i	$-0.668270\pi$
$911$	−27736.1	−1.00871	−0.504356	+	0.863496i	$0.668270\pi$
$912$	0	0
$913$	−18101.2	−0.656147
$914$	−27546.3	−0.996884
$915$	0	0
$916$	−10799.6	−0.389550
$917$	22857.7	0.823150
$918$	0	0
$919$	870.448	0.0312442	0.0156221	−	0.999878i	$-0.495027\pi$
$919$	870.448	0.0312442	0.0156221	+	0.999878i	$0.495027\pi$
$920$	36903.6	1.32247
$921$	0	0
$922$	3953.80	0.141227
$923$	3607.40	0.128645
$924$	0	0
$925$	−15165.8	−0.539080
$926$	−11272.4	−0.400037
$927$	0	0
$928$	−7183.25	−0.254097
$929$	23166.7	0.818165	0.409082	−	0.912497i	$-0.365849\pi$
$929$	23166.7	0.818165	0.409082	+	0.912497i	$0.365849\pi$
$930$	0	0
$931$	61152.3	2.15272
$932$	64.8592	0.00227954
$933$	0	0
$934$	−17884.7	−0.626558
$935$	6416.36	0.224425
$936$	0	0
$937$	−19336.1	−0.674155	−0.337078	−	0.941477i	$-0.609439\pi$
$937$	−19336.1	−0.674155	−0.337078	+	0.941477i	$0.609439\pi$
$938$	−6174.88	−0.214943
$939$	0	0
$940$	12226.0	0.424223
$941$	−24224.6	−0.839213	−0.419607	−	0.907706i	$-0.637832\pi$
$941$	−24224.6	−0.839213	−0.419607	+	0.907706i	$0.637832\pi$
$942$	0	0
$943$	−16572.5	−0.572297
$944$	−22803.3	−0.786214
$945$	0	0
$946$	−8618.44	−0.296205
$947$	−12749.1	−0.437475	−0.218738	−	0.975784i	$-0.570194\pi$
$947$	−12749.1	−0.437475	−0.218738	+	0.975784i	$0.570194\pi$
$948$	0	0
$949$	−465.171	−0.0159116
$950$	9948.18	0.339749
$951$	0	0
$952$	33144.1	1.12837
$953$	−9344.42	−0.317624	−0.158812	−	0.987309i	$-0.550766\pi$
$953$	−9344.42	−0.317624	−0.158812	+	0.987309i	$0.550766\pi$
$954$	0	0
$955$	29011.6	0.983030
$956$	−583.985	−0.0197567
$957$	0	0
$958$	562.878	0.0189831
$959$	8812.41	0.296734
$960$	0	0
$961$	38550.8	1.29404
$962$	−2832.62	−0.0949347
$963$	0	0
$964$	1310.98	0.0438007
$965$	−36117.4	−1.20483
$966$	0	0
$967$	1682.05	0.0559370	0.0279685	−	0.999609i	$-0.491096\pi$
$967$	1682.05	0.0559370	0.0279685	+	0.999609i	$0.491096\pi$
$968$	−22824.4	−0.757856
$969$	0	0
$970$	−16486.3	−0.545715
$971$	−51783.5	−1.71144	−0.855722	−	0.517436i	$-0.826887\pi$
$971$	−51783.5	−1.71144	−0.855722	+	0.517436i	$0.826887\pi$
$972$	0	0
$973$	−45057.2	−1.48455
$974$	−9115.73	−0.299884
$975$	0	0
$976$	−24704.7	−0.810225
$977$	16169.2	0.529477	0.264738	−	0.964320i	$-0.414714\pi$
$977$	16169.2	0.529477	0.264738	+	0.964320i	$0.414714\pi$
$978$	0	0
$979$	−13082.8	−0.427097
$980$	−16849.0	−0.549207
$981$	0	0
$982$	−34357.0	−1.11647
$983$	57090.3	1.85239	0.926194	−	0.377047i	$-0.123060\pi$
$983$	57090.3	1.85239	0.926194	+	0.377047i	$0.123060\pi$
$984$	0	0
$985$	−32547.3	−1.05283
$986$	−6228.94	−0.201187
$987$	0	0
$988$	−817.079	−0.0263105
$989$	33381.0	1.07326
$990$	0	0
$991$	−12152.3	−0.389537	−0.194768	−	0.980849i	$-0.562395\pi$
$991$	−12152.3	−0.389537	−0.194768	+	0.980849i	$0.562395\pi$
$992$	−27771.0	−0.888841
$993$	0	0
$994$	−63480.7	−2.02564
$995$	45550.8	1.45131
$996$	0	0
$997$	−15398.0	−0.489126	−0.244563	−	0.969633i	$-0.578645\pi$
$997$	−15398.0	−0.489126	−0.244563	+	0.969633i	$0.578645\pi$
$998$	43751.5	1.38770
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2151.4.a.e.1.12		32
3.2	odd	2		717.4.a.c.1.21	✓	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
717.4.a.c.1.21	✓	32	3.2	odd	2
2151.4.a.e.1.12		32	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:	\( N \)	\(=\)	\( 2151 = 3^{2} \cdot 239 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2151.a (trivial)

\(f(q)\)	\(=\)	\(q-2.35723 q^{2} -2.44345 q^{4} +8.17039 q^{5} +34.4525 q^{7} +24.6177 q^{8} +O(q^{10})\)	\(q-2.35723 q^{2} -2.44345 q^{4} +8.17039 q^{5} +34.4525 q^{7} +24.6177 q^{8} -19.2595 q^{10} +20.0959 q^{11} +4.61504 q^{13} -81.2125 q^{14} -38.4819 q^{16} +39.0786 q^{17} +72.4577 q^{19} -19.9640 q^{20} -47.3706 q^{22} +183.476 q^{23} -58.2447 q^{25} -10.8787 q^{26} -84.1829 q^{28} +67.6196 q^{29} +261.423 q^{31} -106.230 q^{32} -92.1174 q^{34} +281.490 q^{35} +260.381 q^{37} -170.800 q^{38} +201.136 q^{40} -90.3254 q^{41} +181.936 q^{43} -49.1033 q^{44} -432.496 q^{46} -612.406 q^{47} +843.972 q^{49} +137.296 q^{50} -11.2766 q^{52} -260.323 q^{53} +164.191 q^{55} +848.139 q^{56} -159.395 q^{58} +592.573 q^{59} +641.983 q^{61} -616.234 q^{62} +558.265 q^{64} +37.7067 q^{65} +76.0336 q^{67} -95.4868 q^{68} -663.538 q^{70} +781.662 q^{71} -100.795 q^{73} -613.779 q^{74} -177.047 q^{76} +692.352 q^{77} +1141.46 q^{79} -314.412 q^{80} +212.918 q^{82} -900.743 q^{83} +319.288 q^{85} -428.867 q^{86} +494.713 q^{88} -651.019 q^{89} +159.000 q^{91} -448.315 q^{92} +1443.58 q^{94} +592.008 q^{95} +856.008 q^{97} -1989.44 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8}+O(q^{10}) \) 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8	\( 32 q - 3 q^{2} + 151 q^{4} + 14 q^{5} + 72 q^{7} - 57 q^{8} + 32 q^{10} - 154 q^{11} + 100 q^{13} - 42 q^{14} + 719 q^{16} - 32 q^{17} + 202 q^{19} + 132 q^{20} + 265 q^{22} - 552 q^{23} + 1086 q^{25} + 280 q^{26} + 390 q^{28} + 154 q^{29} + 560 q^{31} - 444 q^{32} + 156 q^{34} - 394 q^{35} + 914 q^{37} - 111 q^{38} + 257 q^{40} + 914 q^{41} + 1722 q^{43} - 1243 q^{44} + 584 q^{46} - 380 q^{47} + 2446 q^{49} + 454 q^{50} + 1552 q^{52} - 370 q^{53} + 918 q^{55} + 499 q^{56} + 2446 q^{58} - 492 q^{59} + 668 q^{61} - 578 q^{62} + 6475 q^{64} - 736 q^{65} + 4548 q^{67} - 5253 q^{68} + 7793 q^{70} - 258 q^{71} + 3096 q^{73} - 449 q^{74} + 6814 q^{76} - 3804 q^{77} + 2864 q^{79} + 1052 q^{80} + 14145 q^{82} - 2364 q^{83} + 3088 q^{85} - 2811 q^{86} + 8329 q^{88} + 4172 q^{89} + 7350 q^{91} - 13644 q^{92} + 6122 q^{94} - 3336 q^{95} + 6370 q^{97} - 1572 q^{98}+O(q^{100}) \) 32 * q - 3 * q^2 + 151 * q^4 + 14 * q^5 + 72 * q^7 - 57 * q^8 + 32 * q^10 - 154 * q^11 + 100 * q^13 - 42 * q^14 + 719 * q^16 - 32 * q^17 + 202 * q^19 + 132 * q^20 + 265 * q^22 - 552 * q^23 + 1086 * q^25 + 280 * q^26 + 390 * q^28 + 154 * q^29 + 560 * q^31 - 444 * q^32 + 156 * q^34 - 394 * q^35 + 914 * q^37 - 111 * q^38 + 257 * q^40 + 914 * q^41 + 1722 * q^43 - 1243 * q^44 + 584 * q^46 - 380 * q^47 + 2446 * q^49 + 454 * q^50 + 1552 * q^52 - 370 * q^53 + 918 * q^55 + 499 * q^56 + 2446 * q^58 - 492 * q^59 + 668 * q^61 - 578 * q^62 + 6475 * q^64 - 736 * q^65 + 4548 * q^67 - 5253 * q^68 + 7793 * q^70 - 258 * q^71 + 3096 * q^73 - 449 * q^74 + 6814 * q^76 - 3804 * q^77 + 2864 * q^79 + 1052 * q^80 + 14145 * q^82 - 2364 * q^83 + 3088 * q^85 - 2811 * q^86 + 8329 * q^88 + 4172 * q^89 + 7350 * q^91 - 13644 * q^92 + 6122 * q^94 - 3336 * q^95 + 6370 * q^97 - 1572 * q^98

Introduction

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