LMFDB - Embedded newform 1840.4.a.j.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,4,Mod(1,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1840.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1840.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:	$108.563514411$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.318165.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 45x + 60 $ x^3 - x^2 - 45*x + 60
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$6.50182$ of defining polynomial
Character	$\chi$	$=$	1840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.50182 q^{3} +5.00000 q^{5} -2.73001 q^{7} +15.2736 q^{9} +O(q^{10})$	$q+6.50182 q^{3} +5.00000 q^{5} -2.73001 q^{7} +15.2736 q^{9} +58.0600 q^{11} +60.8209 q^{13} +32.5091 q^{15} -25.5200 q^{17} +135.292 q^{19} -17.7500 q^{21} -23.0000 q^{23} +25.0000 q^{25} -76.2427 q^{27} +76.0691 q^{29} -146.654 q^{31} +377.495 q^{33} -13.6500 q^{35} +411.565 q^{37} +395.446 q^{39} +279.340 q^{41} -444.971 q^{43} +76.3681 q^{45} -60.4237 q^{47} -335.547 q^{49} -165.926 q^{51} -417.414 q^{53} +290.300 q^{55} +879.643 q^{57} -474.367 q^{59} +430.793 q^{61} -41.6971 q^{63} +304.104 q^{65} -444.731 q^{67} -149.542 q^{69} +425.418 q^{71} -835.514 q^{73} +162.545 q^{75} -158.504 q^{77} +169.644 q^{79} -908.104 q^{81} -623.154 q^{83} -127.600 q^{85} +494.587 q^{87} +1674.53 q^{89} -166.041 q^{91} -953.515 q^{93} +676.459 q^{95} +1010.47 q^{97} +886.787 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + 15 q^{5} - 7 q^{7} + 10 q^{9}+O(q^{10}) $ 3 * q + q^3 + 15 * q^5 - 7 * q^7 + 10 * q^9	$ 3 q + q^{3} + 15 q^{5} - 7 q^{7} + 10 q^{9} - 27 q^{11} + 75 q^{13} + 5 q^{15} + 127 q^{17} + 185 q^{19} - 258 q^{21} - 69 q^{23} + 75 q^{25} - 98 q^{27} + 344 q^{29} - 397 q^{31} + 477 q^{33} - 35 q^{35} + 978 q^{37} - 198 q^{39} - 575 q^{41} - 812 q^{43} + 50 q^{45} + 270 q^{47} + 878 q^{49} - 955 q^{51} + 510 q^{53} - 135 q^{55} + 894 q^{57} - 142 q^{59} - 49 q^{61} + 1356 q^{63} + 375 q^{65} - 1616 q^{67} - 23 q^{69} + 471 q^{71} - 780 q^{73} + 25 q^{75} + 1032 q^{77} + 860 q^{79} - 1193 q^{81} + 288 q^{83} + 635 q^{85} - 1452 q^{87} - 90 q^{89} + 3963 q^{91} - 1592 q^{93} + 925 q^{95} + 1321 q^{97} + 1851 q^{99}+O(q^{100}) $ 3 * q + q^3 + 15 * q^5 - 7 * q^7 + 10 * q^9 - 27 * q^11 + 75 * q^13 + 5 * q^15 + 127 * q^17 + 185 * q^19 - 258 * q^21 - 69 * q^23 + 75 * q^25 - 98 * q^27 + 344 * q^29 - 397 * q^31 + 477 * q^33 - 35 * q^35 + 978 * q^37 - 198 * q^39 - 575 * q^41 - 812 * q^43 + 50 * q^45 + 270 * q^47 + 878 * q^49 - 955 * q^51 + 510 * q^53 - 135 * q^55 + 894 * q^57 - 142 * q^59 - 49 * q^61 + 1356 * q^63 + 375 * q^65 - 1616 * q^67 - 23 * q^69 + 471 * q^71 - 780 * q^73 + 25 * q^75 + 1032 * q^77 + 860 * q^79 - 1193 * q^81 + 288 * q^83 + 635 * q^85 - 1452 * q^87 - 90 * q^89 + 3963 * q^91 - 1592 * q^93 + 925 * q^95 + 1321 * q^97 + 1851 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	6.50182	1.25128	0.625638	−	0.780114i	$-0.284839\pi$
$3$	6.50182	1.25128	0.625638	+	0.780114i	$0.284839\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−2.73001	−0.147406	−0.0737032	−	0.997280i	$-0.523482\pi$
$7$	−2.73001	−0.147406	−0.0737032	+	0.997280i	$0.523482\pi$
$8$	0	0
$9$	15.2736	0.565690
$10$	0	0
$11$	58.0600	1.59143	0.795716	−	0.605671i	$-0.207095\pi$
$11$	58.0600	1.59143	0.795716	+	0.605671i	$0.207095\pi$
$12$	0	0
$13$	60.8209	1.29759	0.648795	−	0.760963i	$-0.275273\pi$
$13$	60.8209	1.29759	0.648795	+	0.760963i	$0.275273\pi$
$14$	0	0
$15$	32.5091	0.559587
$16$	0	0
$17$	−25.5200	−0.364089	−0.182044	−	0.983290i	$-0.558271\pi$
$17$	−25.5200	−0.364089	−0.182044	+	0.983290i	$0.558271\pi$
$18$	0	0
$19$	135.292	1.63358	0.816791	−	0.576933i	$-0.195751\pi$
$19$	135.292	1.63358	0.816791	+	0.576933i	$0.195751\pi$
$20$	0	0
$21$	−17.7500	−0.184446
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−76.2427	−0.543441
$28$	0	0
$29$	76.0691	0.487092	0.243546	−	0.969889i	$-0.421689\pi$
$29$	76.0691	0.487092	0.243546	+	0.969889i	$0.421689\pi$
$30$	0	0
$31$	−146.654	−0.849670	−0.424835	−	0.905271i	$-0.639668\pi$
$31$	−146.654	−0.849670	−0.424835	+	0.905271i	$0.639668\pi$
$32$	0	0
$33$	377.495	1.99132
$34$	0	0
$35$	−13.6500	−0.0659222
$36$	0	0
$37$	411.565	1.82867	0.914337	−	0.404954i	$-0.132712\pi$
$37$	411.565	1.82867	0.914337	+	0.404954i	$0.132712\pi$
$38$	0	0
$39$	395.446	1.62364
$40$	0	0
$41$	279.340	1.06404	0.532019	−	0.846732i	$-0.321433\pi$
$41$	279.340	1.06404	0.532019	+	0.846732i	$0.321433\pi$
$42$	0	0
$43$	−444.971	−1.57808	−0.789040	−	0.614342i	$-0.789422\pi$
$43$	−444.971	−1.57808	−0.789040	+	0.614342i	$0.789422\pi$
$44$	0	0
$45$	76.3681	0.252984
$46$	0	0
$47$	−60.4237	−0.187525	−0.0937627	−	0.995595i	$-0.529890\pi$
$47$	−60.4237	−0.187525	−0.0937627	+	0.995595i	$0.529890\pi$
$48$	0	0
$49$	−335.547	−0.978271
$50$	0	0
$51$	−165.926	−0.455575
$52$	0	0
$53$	−417.414	−1.08182	−0.540908	−	0.841082i	$-0.681919\pi$
$53$	−417.414	−1.08182	−0.540908	+	0.841082i	$0.681919\pi$
$54$	0	0
$55$	290.300	0.711710
$56$	0	0
$57$	879.643	2.04406
$58$	0	0
$59$	−474.367	−1.04673	−0.523367	−	0.852107i	$-0.675324\pi$
$59$	−474.367	−1.04673	−0.523367	+	0.852107i	$0.675324\pi$
$60$	0	0
$61$	430.793	0.904219	0.452109	−	0.891963i	$-0.350672\pi$
$61$	430.793	0.904219	0.452109	+	0.891963i	$0.350672\pi$
$62$	0	0
$63$	−41.6971	−0.0833864
$64$	0	0
$65$	304.104	0.580300
$66$	0	0
$67$	−444.731	−0.810933	−0.405467	−	0.914110i	$-0.632891\pi$
$67$	−444.731	−0.810933	−0.405467	+	0.914110i	$0.632891\pi$
$68$	0	0
$69$	−149.542	−0.260909
$70$	0	0
$71$	425.418	0.711096	0.355548	−	0.934658i	$-0.384294\pi$
$71$	425.418	0.711096	0.355548	+	0.934658i	$0.384294\pi$
$72$	0	0
$73$	−835.514	−1.33958	−0.669791	−	0.742549i	$-0.733616\pi$
$73$	−835.514	−1.33958	−0.669791	+	0.742549i	$0.733616\pi$
$74$	0	0
$75$	162.545	0.250255
$76$	0	0
$77$	−158.504	−0.234587
$78$	0	0
$79$	169.644	0.241600	0.120800	−	0.992677i	$-0.461454\pi$
$79$	169.644	0.241600	0.120800	+	0.992677i	$0.461454\pi$
$80$	0	0
$81$	−908.104	−1.24568
$82$	0	0
$83$	−623.154	−0.824097	−0.412049	−	0.911162i	$-0.635187\pi$
$83$	−623.154	−0.824097	−0.412049	+	0.911162i	$0.635187\pi$
$84$	0	0
$85$	−127.600	−0.162825
$86$	0	0
$87$	494.587	0.609486
$88$	0	0
$89$	1674.53	1.99437	0.997187	−	0.0749529i	$-0.0238807\pi$
$89$	1674.53	1.99437	0.997187	+	0.0749529i	$0.0238807\pi$
$90$	0	0
$91$	−166.041	−0.191273
$92$	0	0
$93$	−953.515	−1.06317
$94$	0	0
$95$	676.459	0.730560
$96$	0	0
$97$	1010.47	1.05771	0.528855	−	0.848712i	$-0.322621\pi$
$97$	1010.47	1.05771	0.528855	+	0.848712i	$0.322621\pi$
$98$	0	0
$99$	886.787	0.900257
$100$	0	0
$101$	793.592	0.781836	0.390918	−	0.920426i	$-0.372158\pi$
$101$	793.592	0.781836	0.390918	+	0.920426i	$0.372158\pi$
$102$	0	0
$103$	675.460	0.646165	0.323083	−	0.946371i	$-0.395281\pi$
$103$	675.460	0.646165	0.323083	+	0.946371i	$0.395281\pi$
$104$	0	0
$105$	−88.7500	−0.0824868
$106$	0	0
$107$	779.649	0.704407	0.352203	−	0.935924i	$-0.385433\pi$
$107$	779.649	0.704407	0.352203	+	0.935924i	$0.385433\pi$
$108$	0	0
$109$	985.830	0.866288	0.433144	−	0.901325i	$-0.357404\pi$
$109$	985.830	0.866288	0.433144	+	0.901325i	$0.357404\pi$
$110$	0	0
$111$	2675.92	2.28817
$112$	0	0
$113$	213.916	0.178084	0.0890422	−	0.996028i	$-0.471619\pi$
$113$	213.916	0.178084	0.0890422	+	0.996028i	$0.471619\pi$
$114$	0	0
$115$	−115.000	−0.0932505
$116$	0	0
$117$	928.956	0.734034
$118$	0	0
$119$	69.6697	0.0536690
$120$	0	0
$121$	2039.96	1.53265
$122$	0	0
$123$	1816.22	1.33140
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−1990.90	−1.39105	−0.695527	−	0.718500i	$-0.744829\pi$
$127$	−1990.90	−1.39105	−0.695527	+	0.718500i	$0.744829\pi$
$128$	0	0
$129$	−2893.12	−1.97461
$130$	0	0
$131$	1128.87	0.752897	0.376449	−	0.926437i	$-0.377145\pi$
$131$	1128.87	0.752897	0.376449	+	0.926437i	$0.377145\pi$
$132$	0	0
$133$	−369.347	−0.240801
$134$	0	0
$135$	−381.214	−0.243034
$136$	0	0
$137$	651.068	0.406018	0.203009	−	0.979177i	$-0.434928\pi$
$137$	651.068	0.406018	0.203009	+	0.979177i	$0.434928\pi$
$138$	0	0
$139$	−227.553	−0.138855	−0.0694273	−	0.997587i	$-0.522117\pi$
$139$	−227.553	−0.138855	−0.0694273	+	0.997587i	$0.522117\pi$
$140$	0	0
$141$	−392.864	−0.234646
$142$	0	0
$143$	3531.26	2.06503
$144$	0	0
$145$	380.345	0.217834
$146$	0	0
$147$	−2181.67	−1.22409
$148$	0	0
$149$	−3190.52	−1.75421	−0.877107	−	0.480295i	$-0.840530\pi$
$149$	−3190.52	−1.75421	−0.877107	+	0.480295i	$0.840530\pi$
$150$	0	0
$151$	954.094	0.514193	0.257096	−	0.966386i	$-0.417234\pi$
$151$	954.094	0.514193	0.257096	+	0.966386i	$0.417234\pi$
$152$	0	0
$153$	−389.783	−0.205961
$154$	0	0
$155$	−733.268	−0.379984
$156$	0	0
$157$	2828.19	1.43767	0.718834	−	0.695181i	$-0.244676\pi$
$157$	2828.19	1.43767	0.718834	+	0.695181i	$0.244676\pi$
$158$	0	0
$159$	−2713.95	−1.35365
$160$	0	0
$161$	62.7901	0.0307364
$162$	0	0
$163$	−457.019	−0.219610	−0.109805	−	0.993953i	$-0.535023\pi$
$163$	−457.019	−0.219610	−0.109805	+	0.993953i	$0.535023\pi$
$164$	0	0
$165$	1887.48	0.890545
$166$	0	0
$167$	1914.18	0.886969	0.443484	−	0.896282i	$-0.353742\pi$
$167$	1914.18	0.886969	0.443484	+	0.896282i	$0.353742\pi$
$168$	0	0
$169$	1502.18	0.683741
$170$	0	0
$171$	2066.40	0.924101
$172$	0	0
$173$	−3334.32	−1.46534	−0.732670	−	0.680584i	$-0.761726\pi$
$173$	−3334.32	−1.46534	−0.732670	+	0.680584i	$0.761726\pi$
$174$	0	0
$175$	−68.2502	−0.0294813
$176$	0	0
$177$	−3084.25	−1.30975
$178$	0	0
$179$	4516.88	1.88607	0.943037	−	0.332688i	$-0.107956\pi$
$179$	4516.88	1.88607	0.943037	+	0.332688i	$0.107956\pi$
$180$	0	0
$181$	4463.52	1.83299	0.916494	−	0.400048i	$-0.131007\pi$
$181$	4463.52	1.83299	0.916494	+	0.400048i	$0.131007\pi$
$182$	0	0
$183$	2800.93	1.13143
$184$	0	0
$185$	2057.83	0.817808
$186$	0	0
$187$	−1481.69	−0.579422
$188$	0	0
$189$	208.143	0.0801068
$190$	0	0
$191$	−1766.54	−0.669227	−0.334614	−	0.942355i	$-0.608606\pi$
$191$	−1766.54	−0.669227	−0.334614	+	0.942355i	$0.608606\pi$
$192$	0	0
$193$	598.155	0.223089	0.111544	−	0.993759i	$-0.464420\pi$
$193$	598.155	0.223089	0.111544	+	0.993759i	$0.464420\pi$
$194$	0	0
$195$	1977.23	0.726115
$196$	0	0
$197$	2354.76	0.851623	0.425812	−	0.904812i	$-0.359989\pi$
$197$	2354.76	0.851623	0.425812	+	0.904812i	$0.359989\pi$
$198$	0	0
$199$	−2728.83	−0.972070	−0.486035	−	0.873939i	$-0.661557\pi$
$199$	−2728.83	−0.972070	−0.486035	+	0.873939i	$0.661557\pi$
$200$	0	0
$201$	−2891.56	−1.01470
$202$	0	0
$203$	−207.669	−0.0718005
$204$	0	0
$205$	1396.70	0.475852
$206$	0	0
$207$	−351.293	−0.117955
$208$	0	0
$209$	7855.04	2.59973
$210$	0	0
$211$	−3649.23	−1.19063	−0.595315	−	0.803492i	$-0.702973\pi$
$211$	−3649.23	−1.19063	−0.595315	+	0.803492i	$0.702973\pi$
$212$	0	0
$213$	2765.99	0.889777
$214$	0	0
$215$	−2224.85	−0.705739
$216$	0	0
$217$	400.365	0.125247
$218$	0	0
$219$	−5432.36	−1.67619
$220$	0	0
$221$	−1552.15	−0.472438
$222$	0	0
$223$	825.309	0.247833	0.123916	−	0.992293i	$-0.460455\pi$
$223$	825.309	0.247833	0.123916	+	0.992293i	$0.460455\pi$
$224$	0	0
$225$	381.841	0.113138
$226$	0	0
$227$	−6047.64	−1.76826	−0.884132	−	0.467238i	$-0.845249\pi$
$227$	−6047.64	−1.76826	−0.884132	+	0.467238i	$0.845249\pi$
$228$	0	0
$229$	−4557.36	−1.31510	−0.657551	−	0.753410i	$-0.728408\pi$
$229$	−4557.36	−1.31510	−0.657551	+	0.753410i	$0.728408\pi$
$230$	0	0
$231$	−1030.56	−0.293533
$232$	0	0
$233$	−4755.29	−1.33704	−0.668518	−	0.743696i	$-0.733071\pi$
$233$	−4755.29	−1.33704	−0.668518	+	0.743696i	$0.733071\pi$
$234$	0	0
$235$	−302.118	−0.0838639
$236$	0	0
$237$	1102.99	0.302308
$238$	0	0
$239$	4935.94	1.33590	0.667948	−	0.744208i	$-0.267173\pi$
$239$	4935.94	1.33590	0.667948	+	0.744208i	$0.267173\pi$
$240$	0	0
$241$	−2511.38	−0.671253	−0.335626	−	0.941995i	$-0.608948\pi$
$241$	−2511.38	−0.671253	−0.335626	+	0.941995i	$0.608948\pi$
$242$	0	0
$243$	−3845.77	−1.01525
$244$	0	0
$245$	−1677.74	−0.437496
$246$	0	0
$247$	8228.57	2.11972
$248$	0	0
$249$	−4051.64	−1.03117
$250$	0	0
$251$	5.50802	0.00138511	0.000692556	−	1.00000i	$-0.499780\pi$
$251$	5.50802	0.00138511	0.000692556		1.00000i	$0.499780\pi$
$252$	0	0
$253$	−1335.38	−0.331836
$254$	0	0
$255$	−829.632	−0.203739
$256$	0	0
$257$	1516.34	0.368041	0.184021	−	0.982922i	$-0.441089\pi$
$257$	1516.34	0.368041	0.184021	+	0.982922i	$0.441089\pi$
$258$	0	0
$259$	−1123.58	−0.269558
$260$	0	0
$261$	1161.85	0.275543
$262$	0	0
$263$	−1712.56	−0.401524	−0.200762	−	0.979640i	$-0.564342\pi$
$263$	−1712.56	−0.401524	−0.200762	+	0.979640i	$0.564342\pi$
$264$	0	0
$265$	−2087.07	−0.483803
$266$	0	0
$267$	10887.5	2.49551
$268$	0	0
$269$	−4081.95	−0.925208	−0.462604	−	0.886565i	$-0.653085\pi$
$269$	−4081.95	−0.925208	−0.462604	+	0.886565i	$0.653085\pi$
$270$	0	0
$271$	1313.70	0.294471	0.147235	−	0.989101i	$-0.452963\pi$
$271$	1313.70	0.294471	0.147235	+	0.989101i	$0.452963\pi$
$272$	0	0
$273$	−1079.57	−0.239336
$274$	0	0
$275$	1451.50	0.318286
$276$	0	0
$277$	2787.83	0.604709	0.302355	−	0.953196i	$-0.402227\pi$
$277$	2787.83	0.604709	0.302355	+	0.953196i	$0.402227\pi$
$278$	0	0
$279$	−2239.93	−0.480650
$280$	0	0
$281$	−2017.23	−0.428249	−0.214124	−	0.976806i	$-0.568690\pi$
$281$	−2017.23	−0.428249	−0.214124	+	0.976806i	$0.568690\pi$
$282$	0	0
$283$	325.902	0.0684554	0.0342277	−	0.999414i	$-0.489103\pi$
$283$	325.902	0.0684554	0.0342277	+	0.999414i	$0.489103\pi$
$284$	0	0
$285$	4398.21	0.914132
$286$	0	0
$287$	−762.600	−0.156846
$288$	0	0
$289$	−4261.73	−0.867439
$290$	0	0
$291$	6569.91	1.32349
$292$	0	0
$293$	2696.09	0.537568	0.268784	−	0.963200i	$-0.413378\pi$
$293$	2696.09	0.537568	0.268784	+	0.963200i	$0.413378\pi$
$294$	0	0
$295$	−2371.84	−0.468114
$296$	0	0
$297$	−4426.65	−0.864850
$298$	0	0
$299$	−1398.88	−0.270566
$300$	0	0
$301$	1214.77	0.232619
$302$	0	0
$303$	5159.79	0.978292
$304$	0	0
$305$	2153.96	0.404379
$306$	0	0
$307$	2411.07	0.448232	0.224116	−	0.974562i	$-0.428051\pi$
$307$	2411.07	0.448232	0.224116	+	0.974562i	$0.428051\pi$
$308$	0	0
$309$	4391.72	0.808531
$310$	0	0
$311$	7951.50	1.44980	0.724901	−	0.688853i	$-0.241886\pi$
$311$	7951.50	1.44980	0.724901	+	0.688853i	$0.241886\pi$
$312$	0	0
$313$	−989.412	−0.178674	−0.0893368	−	0.996001i	$-0.528475\pi$
$313$	−989.412	−0.178674	−0.0893368	+	0.996001i	$0.528475\pi$
$314$	0	0
$315$	−208.486	−0.0372915
$316$	0	0
$317$	−2862.59	−0.507189	−0.253594	−	0.967311i	$-0.581613\pi$
$317$	−2862.59	−0.507189	−0.253594	+	0.967311i	$0.581613\pi$
$318$	0	0
$319$	4416.57	0.775174
$320$	0	0
$321$	5069.14	0.881407
$322$	0	0
$323$	−3452.65	−0.594769
$324$	0	0
$325$	1520.52	0.259518
$326$	0	0
$327$	6409.69	1.08396
$328$	0	0
$329$	164.957	0.0276425
$330$	0	0
$331$	−6982.52	−1.15950	−0.579749	−	0.814795i	$-0.696849\pi$
$331$	−6982.52	−1.15950	−0.579749	+	0.814795i	$0.696849\pi$
$332$	0	0
$333$	6286.10	1.03446
$334$	0	0
$335$	−2223.66	−0.362660
$336$	0	0
$337$	5037.32	0.814244	0.407122	−	0.913374i	$-0.366532\pi$
$337$	5037.32	0.814244	0.407122	+	0.913374i	$0.366532\pi$
$338$	0	0
$339$	1390.84	0.222833
$340$	0	0
$341$	−8514.71	−1.35219
$342$	0	0
$343$	1852.44	0.291610
$344$	0	0
$345$	−747.709	−0.116682
$346$	0	0
$347$	−9612.35	−1.48708	−0.743542	−	0.668690i	$-0.766855\pi$
$347$	−9612.35	−1.48708	−0.743542	+	0.668690i	$0.766855\pi$
$348$	0	0
$349$	−841.492	−0.129066	−0.0645330	−	0.997916i	$-0.520556\pi$
$349$	−841.492	−0.129066	−0.0645330	+	0.997916i	$0.520556\pi$
$350$	0	0
$351$	−4637.15	−0.705165
$352$	0	0
$353$	−3687.33	−0.555968	−0.277984	−	0.960586i	$-0.589666\pi$
$353$	−3687.33	−0.555968	−0.277984	+	0.960586i	$0.589666\pi$
$354$	0	0
$355$	2127.09	0.318012
$356$	0	0
$357$	452.980	0.0671547
$358$	0	0
$359$	−3947.28	−0.580305	−0.290153	−	0.956980i	$-0.593706\pi$
$359$	−3947.28	−0.580305	−0.290153	+	0.956980i	$0.593706\pi$
$360$	0	0
$361$	11444.9	1.66859
$362$	0	0
$363$	13263.5	1.91777
$364$	0	0
$365$	−4177.57	−0.599080
$366$	0	0
$367$	7067.93	1.00529	0.502647	−	0.864492i	$-0.332360\pi$
$367$	7067.93	1.00529	0.502647	+	0.864492i	$0.332360\pi$
$368$	0	0
$369$	4266.53	0.601916
$370$	0	0
$371$	1139.54	0.159467
$372$	0	0
$373$	3798.67	0.527313	0.263657	−	0.964617i	$-0.415071\pi$
$373$	3798.67	0.527313	0.263657	+	0.964617i	$0.415071\pi$
$374$	0	0
$375$	812.727	0.111917
$376$	0	0
$377$	4626.59	0.632046
$378$	0	0
$379$	5405.21	0.732577	0.366289	−	0.930501i	$-0.380628\pi$
$379$	5405.21	0.732577	0.366289	+	0.930501i	$0.380628\pi$
$380$	0	0
$381$	−12944.5	−1.74059
$382$	0	0
$383$	−8357.40	−1.11500	−0.557498	−	0.830178i	$-0.688239\pi$
$383$	−8357.40	−1.11500	−0.557498	+	0.830178i	$0.688239\pi$
$384$	0	0
$385$	−792.521	−0.104911
$386$	0	0
$387$	−6796.32	−0.892704
$388$	0	0
$389$	8070.10	1.05185	0.525926	−	0.850530i	$-0.323719\pi$
$389$	8070.10	1.05185	0.525926	+	0.850530i	$0.323719\pi$
$390$	0	0
$391$	586.960	0.0759177
$392$	0	0
$393$	7339.69	0.942082
$394$	0	0
$395$	848.219	0.108047
$396$	0	0
$397$	7274.45	0.919633	0.459816	−	0.888014i	$-0.347915\pi$
$397$	7274.45	0.919633	0.459816	+	0.888014i	$0.347915\pi$
$398$	0	0
$399$	−2401.43	−0.301308
$400$	0	0
$401$	13501.5	1.68138	0.840692	−	0.541514i	$-0.182149\pi$
$401$	13501.5	1.68138	0.840692	+	0.541514i	$0.182149\pi$
$402$	0	0
$403$	−8919.61	−1.10252
$404$	0	0
$405$	−4540.52	−0.557087
$406$	0	0
$407$	23895.5	2.91021
$408$	0	0
$409$	−3887.99	−0.470046	−0.235023	−	0.971990i	$-0.575517\pi$
$409$	−3887.99	−0.470046	−0.235023	+	0.971990i	$0.575517\pi$
$410$	0	0
$411$	4233.13	0.508041
$412$	0	0
$413$	1295.03	0.154295
$414$	0	0
$415$	−3115.77	−0.368547
$416$	0	0
$417$	−1479.51	−0.173745
$418$	0	0
$419$	−4416.78	−0.514973	−0.257486	−	0.966282i	$-0.582894\pi$
$419$	−4416.78	−0.514973	−0.257486	+	0.966282i	$0.582894\pi$
$420$	0	0
$421$	−14065.5	−1.62829	−0.814145	−	0.580661i	$-0.802794\pi$
$421$	−14065.5	−1.62829	−0.814145	+	0.580661i	$0.802794\pi$
$422$	0	0
$423$	−922.888	−0.106081
$424$	0	0
$425$	−638.000	−0.0728177
$426$	0	0
$427$	−1176.07	−0.133288
$428$	0	0
$429$	22959.6	2.58392
$430$	0	0
$431$	−12344.3	−1.37959	−0.689794	−	0.724006i	$-0.742299\pi$
$431$	−12344.3	−1.37959	−0.689794	+	0.724006i	$0.742299\pi$
$432$	0	0
$433$	5439.20	0.603675	0.301837	−	0.953359i	$-0.402400\pi$
$433$	5439.20	0.603675	0.301837	+	0.953359i	$0.402400\pi$
$434$	0	0
$435$	2472.94	0.272571
$436$	0	0
$437$	−3111.71	−0.340626
$438$	0	0
$439$	−1216.82	−0.132290	−0.0661452	−	0.997810i	$-0.521070\pi$
$439$	−1216.82	−0.132290	−0.0661452	+	0.997810i	$0.521070\pi$
$440$	0	0
$441$	−5125.02	−0.553398
$442$	0	0
$443$	−14338.4	−1.53779	−0.768893	−	0.639378i	$-0.779192\pi$
$443$	−14338.4	−1.53779	−0.768893	+	0.639378i	$0.779192\pi$
$444$	0	0
$445$	8372.63	0.891911
$446$	0	0
$447$	−20744.2	−2.19500
$448$	0	0
$449$	−3502.78	−0.368166	−0.184083	−	0.982911i	$-0.558932\pi$
$449$	−3502.78	−0.368166	−0.184083	+	0.982911i	$0.558932\pi$
$450$	0	0
$451$	16218.5	1.69334
$452$	0	0
$453$	6203.35	0.643397
$454$	0	0
$455$	−830.207	−0.0855400
$456$	0	0
$457$	−4207.17	−0.430641	−0.215321	−	0.976543i	$-0.569080\pi$
$457$	−4207.17	−0.430641	−0.215321	+	0.976543i	$0.569080\pi$
$458$	0	0
$459$	1945.71	0.197861
$460$	0	0
$461$	−468.801	−0.0473628	−0.0236814	−	0.999720i	$-0.507539\pi$
$461$	−468.801	−0.0473628	−0.0236814	+	0.999720i	$0.507539\pi$
$462$	0	0
$463$	3612.40	0.362598	0.181299	−	0.983428i	$-0.441970\pi$
$463$	3612.40	0.362598	0.181299	+	0.983428i	$0.441970\pi$
$464$	0	0
$465$	−4767.58	−0.475465
$466$	0	0
$467$	−4794.07	−0.475038	−0.237519	−	0.971383i	$-0.576334\pi$
$467$	−4794.07	−0.475038	−0.237519	+	0.971383i	$0.576334\pi$
$468$	0	0
$469$	1214.12	0.119537
$470$	0	0
$471$	18388.4	1.79892
$472$	0	0
$473$	−25835.0	−2.51141
$474$	0	0
$475$	3382.30	0.326717
$476$	0	0
$477$	−6375.43	−0.611973
$478$	0	0
$479$	−1544.58	−0.147335	−0.0736677	−	0.997283i	$-0.523470\pi$
$479$	−1544.58	−0.147335	−0.0736677	+	0.997283i	$0.523470\pi$
$480$	0	0
$481$	25031.8	2.37287
$482$	0	0
$483$	408.250	0.0384597
$484$	0	0
$485$	5052.36	0.473023
$486$	0	0
$487$	14437.6	1.34339	0.671695	−	0.740828i	$-0.265567\pi$
$487$	14437.6	1.34339	0.671695	+	0.740828i	$0.265567\pi$
$488$	0	0
$489$	−2971.46	−0.274793
$490$	0	0
$491$	−17252.2	−1.58570	−0.792851	−	0.609416i	$-0.791404\pi$
$491$	−17252.2	−1.58570	−0.792851	+	0.609416i	$0.791404\pi$
$492$	0	0
$493$	−1941.28	−0.177345
$494$	0	0
$495$	4433.93	0.402607
$496$	0	0
$497$	−1161.39	−0.104820
$498$	0	0
$499$	−20118.4	−1.80486	−0.902430	−	0.430836i	$-0.858219\pi$
$499$	−20118.4	−1.80486	−0.902430	+	0.430836i	$0.858219\pi$
$500$	0	0
$501$	12445.7	1.10984
$502$	0	0
$503$	1537.38	0.136279	0.0681397	−	0.997676i	$-0.478294\pi$
$503$	1537.38	0.136279	0.0681397	+	0.997676i	$0.478294\pi$
$504$	0	0
$505$	3967.96	0.349648
$506$	0	0
$507$	9766.90	0.855549
$508$	0	0
$509$	7757.27	0.675511	0.337755	−	0.941234i	$-0.390332\pi$
$509$	7757.27	0.675511	0.337755	+	0.941234i	$0.390332\pi$
$510$	0	0
$511$	2280.96	0.197463
$512$	0	0
$513$	−10315.0	−0.887756
$514$	0	0
$515$	3377.30	0.288974
$516$	0	0
$517$	−3508.20	−0.298434
$518$	0	0
$519$	−21679.2	−1.83354
$520$	0	0
$521$	−19463.5	−1.63668	−0.818342	−	0.574732i	$-0.805106\pi$
$521$	−19463.5	−1.63668	−0.818342	+	0.574732i	$0.805106\pi$
$522$	0	0
$523$	−8640.74	−0.722435	−0.361217	−	0.932482i	$-0.617639\pi$
$523$	−8640.74	−0.722435	−0.361217	+	0.932482i	$0.617639\pi$
$524$	0	0
$525$	−443.750	−0.0368892
$526$	0	0
$527$	3742.60	0.309355
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−7245.31	−0.592127
$532$	0	0
$533$	16989.7	1.38069
$534$	0	0
$535$	3898.25	0.315020
$536$	0	0
$537$	29367.9	2.36000
$538$	0	0
$539$	−19481.9	−1.55685
$540$	0	0
$541$	−3428.37	−0.272453	−0.136226	−	0.990678i	$-0.543497\pi$
$541$	−3428.37	−0.272453	−0.136226	+	0.990678i	$0.543497\pi$
$542$	0	0
$543$	29021.0	2.29357
$544$	0	0
$545$	4929.15	0.387416
$546$	0	0
$547$	−6246.03	−0.488228	−0.244114	−	0.969746i	$-0.578497\pi$
$547$	−6246.03	−0.488228	−0.244114	+	0.969746i	$0.578497\pi$
$548$	0	0
$549$	6579.77	0.511507
$550$	0	0
$551$	10291.5	0.795705
$552$	0	0
$553$	−463.129	−0.0356134
$554$	0	0
$555$	13379.6	1.02330
$556$	0	0
$557$	19065.0	1.45028	0.725142	−	0.688599i	$-0.241774\pi$
$557$	19065.0	1.45028	0.725142	+	0.688599i	$0.241774\pi$
$558$	0	0
$559$	−27063.5	−2.04770
$560$	0	0
$561$	−9633.68	−0.725016
$562$	0	0
$563$	14965.2	1.12027	0.560133	−	0.828403i	$-0.310750\pi$
$563$	14965.2	1.12027	0.560133	+	0.828403i	$0.310750\pi$
$564$	0	0
$565$	1069.58	0.0796417
$566$	0	0
$567$	2479.13	0.183622
$568$	0	0
$569$	12421.6	0.915185	0.457593	−	0.889162i	$-0.348712\pi$
$569$	12421.6	0.915185	0.457593	+	0.889162i	$0.348712\pi$
$570$	0	0
$571$	−4517.92	−0.331120	−0.165560	−	0.986200i	$-0.552943\pi$
$571$	−4517.92	−0.331120	−0.165560	+	0.986200i	$0.552943\pi$
$572$	0	0
$573$	−11485.7	−0.837388
$574$	0	0
$575$	−575.000	−0.0417029
$576$	0	0
$577$	−18827.0	−1.35837	−0.679184	−	0.733968i	$-0.737666\pi$
$577$	−18827.0	−1.35837	−0.679184	+	0.733968i	$0.737666\pi$
$578$	0	0
$579$	3889.10	0.279146
$580$	0	0
$581$	1701.22	0.121477
$582$	0	0
$583$	−24235.1	−1.72164
$584$	0	0
$585$	4644.78	0.328270
$586$	0	0
$587$	11036.1	0.775993	0.387996	−	0.921661i	$-0.373167\pi$
$587$	11036.1	0.775993	0.387996	+	0.921661i	$0.373167\pi$
$588$	0	0
$589$	−19841.0	−1.38801
$590$	0	0
$591$	15310.2	1.06562
$592$	0	0
$593$	−5459.17	−0.378046	−0.189023	−	0.981973i	$-0.560532\pi$
$593$	−5459.17	−0.378046	−0.189023	+	0.981973i	$0.560532\pi$
$594$	0	0
$595$	348.349	0.0240015
$596$	0	0
$597$	−17742.4	−1.21633
$598$	0	0
$599$	−4065.23	−0.277297	−0.138648	−	0.990342i	$-0.544276\pi$
$599$	−4065.23	−0.277297	−0.138648	+	0.990342i	$0.544276\pi$
$600$	0	0
$601$	13664.7	0.927443	0.463722	−	0.885981i	$-0.346514\pi$
$601$	13664.7	0.927443	0.463722	+	0.885981i	$0.346514\pi$
$602$	0	0
$603$	−6792.66	−0.458737
$604$	0	0
$605$	10199.8	0.685423
$606$	0	0
$607$	22214.5	1.48543	0.742716	−	0.669606i	$-0.233537\pi$
$607$	22214.5	1.48543	0.742716	+	0.669606i	$0.233537\pi$
$608$	0	0
$609$	−1350.23	−0.0898422
$610$	0	0
$611$	−3675.02	−0.243331
$612$	0	0
$613$	18941.2	1.24800	0.624002	−	0.781423i	$-0.285506\pi$
$613$	18941.2	1.24800	0.624002	+	0.781423i	$0.285506\pi$
$614$	0	0
$615$	9081.08	0.595422
$616$	0	0
$617$	13188.8	0.860550	0.430275	−	0.902698i	$-0.358417\pi$
$617$	13188.8	0.860550	0.430275	+	0.902698i	$0.358417\pi$
$618$	0	0
$619$	−15094.4	−0.980124	−0.490062	−	0.871688i	$-0.663026\pi$
$619$	−15094.4	−0.980124	−0.490062	+	0.871688i	$0.663026\pi$
$620$	0	0
$621$	1753.58	0.113315
$622$	0	0
$623$	−4571.46	−0.293984
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	51072.0	3.25298
$628$	0	0
$629$	−10503.1	−0.665799
$630$	0	0
$631$	9183.83	0.579401	0.289701	−	0.957117i	$-0.406444\pi$
$631$	9183.83	0.579401	0.289701	+	0.957117i	$0.406444\pi$
$632$	0	0
$633$	−23726.6	−1.48981
$634$	0	0
$635$	−9954.51	−0.622098
$636$	0	0
$637$	−20408.3	−1.26940
$638$	0	0
$639$	6497.68	0.402260
$640$	0	0
$641$	5367.34	0.330729	0.165364	−	0.986233i	$-0.447120\pi$
$641$	5367.34	0.330729	0.165364	+	0.986233i	$0.447120\pi$
$642$	0	0
$643$	27594.2	1.69240	0.846198	−	0.532868i	$-0.178886\pi$
$643$	27594.2	1.69240	0.846198	+	0.532868i	$0.178886\pi$
$644$	0	0
$645$	−14465.6	−0.883073
$646$	0	0
$647$	4129.95	0.250951	0.125475	−	0.992097i	$-0.459954\pi$
$647$	4129.95	0.250951	0.125475	+	0.992097i	$0.459954\pi$
$648$	0	0
$649$	−27541.7	−1.66581
$650$	0	0
$651$	2603.10	0.156718
$652$	0	0
$653$	17419.5	1.04391	0.521957	−	0.852972i	$-0.325202\pi$
$653$	17419.5	1.04391	0.521957	+	0.852972i	$0.325202\pi$
$654$	0	0
$655$	5644.33	0.336706
$656$	0	0
$657$	−12761.3	−0.757788
$658$	0	0
$659$	−577.196	−0.0341189	−0.0170594	−	0.999854i	$-0.505430\pi$
$659$	−577.196	−0.0341189	−0.0170594	+	0.999854i	$0.505430\pi$
$660$	0	0
$661$	−3858.11	−0.227024	−0.113512	−	0.993537i	$-0.536210\pi$
$661$	−3858.11	−0.227024	−0.113512	+	0.993537i	$0.536210\pi$
$662$	0	0
$663$	−10091.8	−0.591150
$664$	0	0
$665$	−1846.74	−0.107689
$666$	0	0
$667$	−1749.59	−0.101566
$668$	0	0
$669$	5366.01	0.310107
$670$	0	0
$671$	25011.8	1.43900
$672$	0	0
$673$	−28439.6	−1.62892	−0.814461	−	0.580218i	$-0.802967\pi$
$673$	−28439.6	−1.62892	−0.814461	+	0.580218i	$0.802967\pi$
$674$	0	0
$675$	−1906.07	−0.108688
$676$	0	0
$677$	−13186.6	−0.748601	−0.374300	−	0.927308i	$-0.622117\pi$
$677$	−13186.6	−0.748601	−0.374300	+	0.927308i	$0.622117\pi$
$678$	0	0
$679$	−2758.60	−0.155913
$680$	0	0
$681$	−39320.6	−2.21258
$682$	0	0
$683$	−3508.39	−0.196552	−0.0982758	−	0.995159i	$-0.531333\pi$
$683$	−3508.39	−0.196552	−0.0982758	+	0.995159i	$0.531333\pi$
$684$	0	0
$685$	3255.34	0.181577
$686$	0	0
$687$	−29631.1	−1.64556
$688$	0	0
$689$	−25387.5	−1.40375
$690$	0	0
$691$	−22652.9	−1.24712	−0.623558	−	0.781777i	$-0.714313\pi$
$691$	−22652.9	−1.24712	−0.623558	+	0.781777i	$0.714313\pi$
$692$	0	0
$693$	−2420.93	−0.132704
$694$	0	0
$695$	−1137.76	−0.0620976
$696$	0	0
$697$	−7128.75	−0.387404
$698$	0	0
$699$	−30918.0	−1.67300
$700$	0	0
$701$	15393.8	0.829412	0.414706	−	0.909956i	$-0.363885\pi$
$701$	15393.8	0.829412	0.414706	+	0.909956i	$0.363885\pi$
$702$	0	0
$703$	55681.4	2.98729
$704$	0	0
$705$	−1964.32	−0.104937
$706$	0	0
$707$	−2166.51	−0.115248
$708$	0	0
$709$	−33826.6	−1.79180	−0.895898	−	0.444260i	$-0.853467\pi$
$709$	−33826.6	−1.79180	−0.895898	+	0.444260i	$0.853467\pi$
$710$	0	0
$711$	2591.08	0.136671
$712$	0	0
$713$	3373.03	0.177168
$714$	0	0
$715$	17656.3	0.923508
$716$	0	0
$717$	32092.6	1.67157
$718$	0	0
$719$	20975.5	1.08798	0.543988	−	0.839093i	$-0.316914\pi$
$719$	20975.5	1.08798	0.543988	+	0.839093i	$0.316914\pi$
$720$	0	0
$721$	−1844.01	−0.0952490
$722$	0	0
$723$	−16328.5	−0.839922
$724$	0	0
$725$	1901.73	0.0974184
$726$	0	0
$727$	−24780.4	−1.26417	−0.632086	−	0.774898i	$-0.717801\pi$
$727$	−24780.4	−1.26417	−0.632086	+	0.774898i	$0.717801\pi$
$728$	0	0
$729$	−485.707	−0.0246765
$730$	0	0
$731$	11355.7	0.574561
$732$	0	0
$733$	−16951.6	−0.854189	−0.427094	−	0.904207i	$-0.640463\pi$
$733$	−16951.6	−0.854189	−0.427094	+	0.904207i	$0.640463\pi$
$734$	0	0
$735$	−10908.3	−0.547428
$736$	0	0
$737$	−25821.1	−1.29054
$738$	0	0
$739$	9849.67	0.490292	0.245146	−	0.969486i	$-0.421164\pi$
$739$	9849.67	0.490292	0.245146	+	0.969486i	$0.421164\pi$
$740$	0	0
$741$	53500.6	2.65236
$742$	0	0
$743$	−37095.1	−1.83161	−0.915805	−	0.401623i	$-0.868446\pi$
$743$	−37095.1	−1.83161	−0.915805	+	0.401623i	$0.868446\pi$
$744$	0	0
$745$	−15952.6	−0.784508
$746$	0	0
$747$	−9517.83	−0.466184
$748$	0	0
$749$	−2128.45	−0.103834
$750$	0	0
$751$	−10943.1	−0.531718	−0.265859	−	0.964012i	$-0.585656\pi$
$751$	−10943.1	−0.531718	−0.265859	+	0.964012i	$0.585656\pi$
$752$	0	0
$753$	35.8121	0.00173316
$754$	0	0
$755$	4770.47	0.229954
$756$	0	0
$757$	36604.3	1.75747	0.878737	−	0.477307i	$-0.158387\pi$
$757$	36604.3	1.75747	0.878737	+	0.477307i	$0.158387\pi$
$758$	0	0
$759$	−8682.39	−0.415219
$760$	0	0
$761$	12046.3	0.573823	0.286912	−	0.957957i	$-0.407371\pi$
$761$	12046.3	0.573823	0.286912	+	0.957957i	$0.407371\pi$
$762$	0	0
$763$	−2691.32	−0.127696
$764$	0	0
$765$	−1948.91	−0.0921087
$766$	0	0
$767$	−28851.4	−1.35823
$768$	0	0
$769$	−27690.3	−1.29849	−0.649243	−	0.760581i	$-0.724914\pi$
$769$	−27690.3	−1.29849	−0.649243	+	0.760581i	$0.724914\pi$
$770$	0	0
$771$	9858.96	0.460521
$772$	0	0
$773$	−26922.2	−1.25268	−0.626341	−	0.779550i	$-0.715448\pi$
$773$	−26922.2	−1.25268	−0.626341	+	0.779550i	$0.715448\pi$
$774$	0	0
$775$	−3666.34	−0.169934
$776$	0	0
$777$	−7305.29	−0.337292
$778$	0	0
$779$	37792.4	1.73819
$780$	0	0
$781$	24699.8	1.13166
$782$	0	0
$783$	−5799.71	−0.264706
$784$	0	0
$785$	14140.9	0.642945
$786$	0	0
$787$	−36138.1	−1.63683	−0.818414	−	0.574629i	$-0.805147\pi$
$787$	−36138.1	−1.63683	−0.818414	+	0.574629i	$0.805147\pi$
$788$	0	0
$789$	−11134.7	−0.502417
$790$	0	0
$791$	−583.992	−0.0262508
$792$	0	0
$793$	26201.2	1.17331
$794$	0	0
$795$	−13569.8	−0.605371
$796$	0	0
$797$	12371.9	0.549855	0.274928	−	0.961465i	$-0.411346\pi$
$797$	12371.9	0.549855	0.274928	+	0.961465i	$0.411346\pi$
$798$	0	0
$799$	1542.01	0.0682759
$800$	0	0
$801$	25576.1	1.12820
$802$	0	0
$803$	−48509.9	−2.13185
$804$	0	0
$805$	313.951	0.0137457
$806$	0	0
$807$	−26540.1	−1.15769
$808$	0	0
$809$	−21068.0	−0.915588	−0.457794	−	0.889058i	$-0.651360\pi$
$809$	−21068.0	−0.915588	−0.457794	+	0.889058i	$0.651360\pi$
$810$	0	0
$811$	12595.5	0.545360	0.272680	−	0.962105i	$-0.412090\pi$
$811$	12595.5	0.545360	0.272680	+	0.962105i	$0.412090\pi$
$812$	0	0
$813$	8541.43	0.368464
$814$	0	0
$815$	−2285.10	−0.0982128
$816$	0	0
$817$	−60200.9	−2.57792
$818$	0	0
$819$	−2536.05	−0.108201
$820$	0	0
$821$	33947.7	1.44310	0.721549	−	0.692363i	$-0.243430\pi$
$821$	33947.7	1.44310	0.721549	+	0.692363i	$0.243430\pi$
$822$	0	0
$823$	−13026.1	−0.551715	−0.275857	−	0.961199i	$-0.588962\pi$
$823$	−13026.1	−0.551715	−0.275857	+	0.961199i	$0.588962\pi$
$824$	0	0
$825$	9437.38	0.398264
$826$	0	0
$827$	−6209.08	−0.261077	−0.130539	−	0.991443i	$-0.541671\pi$
$827$	−6209.08	−0.261077	−0.130539	+	0.991443i	$0.541671\pi$
$828$	0	0
$829$	7291.20	0.305469	0.152735	−	0.988267i	$-0.451192\pi$
$829$	7291.20	0.305469	0.152735	+	0.988267i	$0.451192\pi$
$830$	0	0
$831$	18126.0	0.756658
$832$	0	0
$833$	8563.16	0.356177
$834$	0	0
$835$	9570.91	0.396664
$836$	0	0
$837$	11181.3	0.461746
$838$	0	0
$839$	−764.741	−0.0314682	−0.0157341	−	0.999876i	$-0.505009\pi$
$839$	−764.741	−0.0314682	−0.0157341	+	0.999876i	$0.505009\pi$
$840$	0	0
$841$	−18602.5	−0.762741
$842$	0	0
$843$	−13115.7	−0.535857
$844$	0	0
$845$	7510.90	0.305778
$846$	0	0
$847$	−5569.11	−0.225923
$848$	0	0
$849$	2118.96	0.0856566
$850$	0	0
$851$	−9466.00	−0.381305
$852$	0	0
$853$	−2306.94	−0.0926003	−0.0463002	−	0.998928i	$-0.514743\pi$
$853$	−2306.94	−0.0926003	−0.0463002	+	0.998928i	$0.514743\pi$
$854$	0	0
$855$	10332.0	0.413271
$856$	0	0
$857$	−34838.6	−1.38864	−0.694320	−	0.719666i	$-0.744295\pi$
$857$	−34838.6	−1.38864	−0.694320	+	0.719666i	$0.744295\pi$
$858$	0	0
$859$	38882.0	1.54440	0.772199	−	0.635380i	$-0.219157\pi$
$859$	38882.0	1.54440	0.772199	+	0.635380i	$0.219157\pi$
$860$	0	0
$861$	−4958.28	−0.196258
$862$	0	0
$863$	−19124.0	−0.754334	−0.377167	−	0.926145i	$-0.623102\pi$
$863$	−19124.0	−0.754334	−0.377167	+	0.926145i	$0.623102\pi$
$864$	0	0
$865$	−16671.6	−0.655320
$866$	0	0
$867$	−27709.0	−1.08541
$868$	0	0
$869$	9849.51	0.384490
$870$	0	0
$871$	−27048.9	−1.05226
$872$	0	0
$873$	15433.6	0.598336
$874$	0	0
$875$	−341.251	−0.0131844
$876$	0	0
$877$	−38827.9	−1.49501	−0.747505	−	0.664256i	$-0.768749\pi$
$877$	−38827.9	−1.49501	−0.747505	+	0.664256i	$0.768749\pi$
$878$	0	0
$879$	17529.5	0.672646
$880$	0	0
$881$	30392.1	1.16224	0.581121	−	0.813817i	$-0.302614\pi$
$881$	30392.1	1.16224	0.581121	+	0.813817i	$0.302614\pi$
$882$	0	0
$883$	−2520.10	−0.0960453	−0.0480226	−	0.998846i	$-0.515292\pi$
$883$	−2520.10	−0.0960453	−0.0480226	+	0.998846i	$0.515292\pi$
$884$	0	0
$885$	−15421.2	−0.585739
$886$	0	0
$887$	−16141.7	−0.611032	−0.305516	−	0.952187i	$-0.598829\pi$
$887$	−16141.7	−0.611032	−0.305516	+	0.952187i	$0.598829\pi$
$888$	0	0
$889$	5435.17	0.205050
$890$	0	0
$891$	−52724.5	−1.98242
$892$	0	0
$893$	−8174.82	−0.306338
$894$	0	0
$895$	22584.4	0.843478
$896$	0	0
$897$	−9095.26	−0.338553
$898$	0	0
$899$	−11155.8	−0.413868
$900$	0	0
$901$	10652.4	0.393877
$902$	0	0
$903$	7898.24	0.291071
$904$	0	0
$905$	22317.6	0.819737
$906$	0	0
$907$	−54155.5	−1.98258	−0.991292	−	0.131684i	$-0.957962\pi$
$907$	−54155.5	−1.98258	−0.991292	+	0.131684i	$0.957962\pi$
$908$	0	0
$909$	12121.0	0.442277
$910$	0	0
$911$	20067.1	0.729805	0.364903	−	0.931046i	$-0.381102\pi$
$911$	20067.1	0.729805	0.364903	+	0.931046i	$0.381102\pi$
$912$	0	0
$913$	−36180.3	−1.31149
$914$	0	0
$915$	14004.7	0.505989
$916$	0	0
$917$	−3081.81	−0.110982
$918$	0	0
$919$	−34816.2	−1.24971	−0.624854	−	0.780742i	$-0.714841\pi$
$919$	−34816.2	−1.24971	−0.624854	+	0.780742i	$0.714841\pi$
$920$	0	0
$921$	15676.4	0.560862
$922$	0	0
$923$	25874.3	0.922712
$924$	0	0
$925$	10289.1	0.365735
$926$	0	0
$927$	10316.7	0.365529
$928$	0	0
$929$	10936.8	0.386250	0.193125	−	0.981174i	$-0.438138\pi$
$929$	10936.8	0.386250	0.193125	+	0.981174i	$0.438138\pi$
$930$	0	0
$931$	−45396.8	−1.59809
$932$	0	0
$933$	51699.2	1.81410
$934$	0	0
$935$	−7408.45	−0.259125
$936$	0	0
$937$	−41240.6	−1.43786	−0.718928	−	0.695085i	$-0.755367\pi$
$937$	−41240.6	−1.43786	−0.718928	+	0.695085i	$0.755367\pi$
$938$	0	0
$939$	−6432.97	−0.223570
$940$	0	0
$941$	−39978.4	−1.38497	−0.692486	−	0.721431i	$-0.743485\pi$
$941$	−39978.4	−1.38497	−0.692486	+	0.721431i	$0.743485\pi$
$942$	0	0
$943$	−6424.82	−0.221867
$944$	0	0
$945$	1040.72	0.0358249
$946$	0	0
$947$	23729.0	0.814242	0.407121	−	0.913374i	$-0.366533\pi$
$947$	23729.0	0.814242	0.407121	+	0.913374i	$0.366533\pi$
$948$	0	0
$949$	−50816.7	−1.73823
$950$	0	0
$951$	−18612.0	−0.634633
$952$	0	0
$953$	−1214.35	−0.0412767	−0.0206383	−	0.999787i	$-0.506570\pi$
$953$	−1214.35	−0.0412767	−0.0206383	+	0.999787i	$0.506570\pi$
$954$	0	0
$955$	−8832.70	−0.299288
$956$	0	0
$957$	28715.7	0.969956
$958$	0	0
$959$	−1777.42	−0.0598498
$960$	0	0
$961$	−8283.70	−0.278061
$962$	0	0
$963$	11908.1	0.398476
$964$	0	0
$965$	2990.78	0.0997684
$966$	0	0
$967$	−42431.8	−1.41108	−0.705540	−	0.708670i	$-0.749296\pi$
$967$	−42431.8	−1.41108	−0.705540	+	0.708670i	$0.749296\pi$
$968$	0	0
$969$	−22448.5	−0.744220
$970$	0	0
$971$	3273.42	0.108187	0.0540933	−	0.998536i	$-0.482773\pi$
$971$	3273.42	0.108187	0.0540933	+	0.998536i	$0.482773\pi$
$972$	0	0
$973$	621.221	0.0204681
$974$	0	0
$975$	9886.16	0.324729
$976$	0	0
$977$	−32057.9	−1.04977	−0.524884	−	0.851174i	$-0.675891\pi$
$977$	−32057.9	−1.04977	−0.524884	+	0.851174i	$0.675891\pi$
$978$	0	0
$979$	97222.9	3.17391
$980$	0	0
$981$	15057.2	0.490050
$982$	0	0
$983$	−21721.1	−0.704777	−0.352388	−	0.935854i	$-0.614630\pi$
$983$	−21721.1	−0.704777	−0.352388	+	0.935854i	$0.614630\pi$
$984$	0	0
$985$	11773.8	0.380858
$986$	0	0
$987$	1072.52	0.0345883
$988$	0	0
$989$	10234.3	0.329052
$990$	0	0
$991$	−5969.62	−0.191353	−0.0956767	−	0.995412i	$-0.530501\pi$
$991$	−5969.62	−0.191353	−0.0956767	+	0.995412i	$0.530501\pi$
$992$	0	0
$993$	−45399.0	−1.45085
$994$	0	0
$995$	−13644.2	−0.434723
$996$	0	0
$997$	35140.4	1.11626	0.558128	−	0.829755i	$-0.311520\pi$
$997$	35140.4	1.11626	0.558128	+	0.829755i	$0.311520\pi$
$998$	0	0
$999$	−31378.9	−0.993777

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.4.a.j.1.3		3
4.3	odd	2		230.4.a.g.1.1	✓	3
12.11	even	2		2070.4.a.ba.1.2		3
20.3	even	4		1150.4.b.l.599.4		6
20.7	even	4		1150.4.b.l.599.3		6
20.19	odd	2		1150.4.a.m.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.4.a.g.1.1	✓	3	4.3	odd	2
1150.4.a.m.1.3		3	20.19	odd	2
1150.4.b.l.599.3		6	20.7	even	4
1150.4.b.l.599.4		6	20.3	even	4
1840.4.a.j.1.3		3	1.1	even	1	trivial
2070.4.a.ba.1.2		3	12.11	even	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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1840.a (trivial)

\(f(q)\)	\(=\)	\(q+6.50182 q^{3} +5.00000 q^{5} -2.73001 q^{7} +15.2736 q^{9} +O(q^{10})\)	\(q+6.50182 q^{3} +5.00000 q^{5} -2.73001 q^{7} +15.2736 q^{9} +58.0600 q^{11} +60.8209 q^{13} +32.5091 q^{15} -25.5200 q^{17} +135.292 q^{19} -17.7500 q^{21} -23.0000 q^{23} +25.0000 q^{25} -76.2427 q^{27} +76.0691 q^{29} -146.654 q^{31} +377.495 q^{33} -13.6500 q^{35} +411.565 q^{37} +395.446 q^{39} +279.340 q^{41} -444.971 q^{43} +76.3681 q^{45} -60.4237 q^{47} -335.547 q^{49} -165.926 q^{51} -417.414 q^{53} +290.300 q^{55} +879.643 q^{57} -474.367 q^{59} +430.793 q^{61} -41.6971 q^{63} +304.104 q^{65} -444.731 q^{67} -149.542 q^{69} +425.418 q^{71} -835.514 q^{73} +162.545 q^{75} -158.504 q^{77} +169.644 q^{79} -908.104 q^{81} -623.154 q^{83} -127.600 q^{85} +494.587 q^{87} +1674.53 q^{89} -166.041 q^{91} -953.515 q^{93} +676.459 q^{95} +1010.47 q^{97} +886.787 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + 15 q^{5} - 7 q^{7} + 10 q^{9}+O(q^{10}) \) 3 * q + q^3 + 15 * q^5 - 7 * q^7 + 10 * q^9	\( 3 q + q^{3} + 15 q^{5} - 7 q^{7} + 10 q^{9} - 27 q^{11} + 75 q^{13} + 5 q^{15} + 127 q^{17} + 185 q^{19} - 258 q^{21} - 69 q^{23} + 75 q^{25} - 98 q^{27} + 344 q^{29} - 397 q^{31} + 477 q^{33} - 35 q^{35} + 978 q^{37} - 198 q^{39} - 575 q^{41} - 812 q^{43} + 50 q^{45} + 270 q^{47} + 878 q^{49} - 955 q^{51} + 510 q^{53} - 135 q^{55} + 894 q^{57} - 142 q^{59} - 49 q^{61} + 1356 q^{63} + 375 q^{65} - 1616 q^{67} - 23 q^{69} + 471 q^{71} - 780 q^{73} + 25 q^{75} + 1032 q^{77} + 860 q^{79} - 1193 q^{81} + 288 q^{83} + 635 q^{85} - 1452 q^{87} - 90 q^{89} + 3963 q^{91} - 1592 q^{93} + 925 q^{95} + 1321 q^{97} + 1851 q^{99}+O(q^{100}) \) 3 * q + q^3 + 15 * q^5 - 7 * q^7 + 10 * q^9 - 27 * q^11 + 75 * q^13 + 5 * q^15 + 127 * q^17 + 185 * q^19 - 258 * q^21 - 69 * q^23 + 75 * q^25 - 98 * q^27 + 344 * q^29 - 397 * q^31 + 477 * q^33 - 35 * q^35 + 978 * q^37 - 198 * q^39 - 575 * q^41 - 812 * q^43 + 50 * q^45 + 270 * q^47 + 878 * q^49 - 955 * q^51 + 510 * q^53 - 135 * q^55 + 894 * q^57 - 142 * q^59 - 49 * q^61 + 1356 * q^63 + 375 * q^65 - 1616 * q^67 - 23 * q^69 + 471 * q^71 - 780 * q^73 + 25 * q^75 + 1032 * q^77 + 860 * q^79 - 1193 * q^81 + 288 * q^83 + 635 * q^85 - 1452 * q^87 - 90 * q^89 + 3963 * q^91 - 1592 * q^93 + 925 * q^95 + 1321 * q^97 + 1851 * q^99

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