LMFDB - Embedded newform 950.6.a.d.1.1

Newspace parameters

Level:	$ N $	$=$	$ 950 = 2 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	950.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	$152.364628822$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1441}) $
Defining polynomial:	$x^{2} - x - 360$
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 38)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$19.4803$ of defining polynomial
Character	$\chi$	$=$	950.1

$q$-expansion

$f(q)$	$=$	$q+4.00000 q^{2} -20.4803 q^{3} +16.0000 q^{4} -81.9210 q^{6} +18.9210 q^{7} +64.0000 q^{8} +176.441 q^{9} +O(q^{10})$	\(q+4.00000 q^{2} -20.4803 q^{3} +16.0000 q^{4} -81.9210 q^{6} +18.9210 q^{7} +64.0000 q^{8} +176.441 q^{9} +349.480 q^{11} -327.684 q^{12} -711.599 q^{13} +75.6840 q^{14} +256.000 q^{16} -221.803 q^{17} +705.763 q^{18} +361.000 q^{19} -387.507 q^{21} +1397.92 q^{22} +662.468 q^{23} -1310.74 q^{24} -2846.39 q^{26} +1363.15 q^{27} +302.736 q^{28} -7219.28 q^{29} +5407.76 q^{31} +1024.00 q^{32} -7157.44 q^{33} -887.210 q^{34} +2823.05 q^{36} -1979.40 q^{37} +1444.00 q^{38} +14573.7 q^{39} -3111.11 q^{41} -1550.03 q^{42} -318.049 q^{43} +5591.68 q^{44} +2649.87 q^{46} +27240.5 q^{47} -5242.94 q^{48} -16449.0 q^{49} +4542.57 q^{51} -11385.6 q^{52} +1114.63 q^{53} +5452.60 q^{54} +1210.94 q^{56} -7393.37 q^{57} -28877.1 q^{58} -37904.9 q^{59} +37469.2 q^{61} +21631.0 q^{62} +3338.44 q^{63} +4096.00 q^{64} -28629.8 q^{66} +54955.3 q^{67} -3548.84 q^{68} -13567.5 q^{69} -7177.04 q^{71} +11292.2 q^{72} -64746.1 q^{73} -7917.59 q^{74} +5776.00 q^{76} +6612.52 q^{77} +58294.9 q^{78} +36104.4 q^{79} -70792.8 q^{81} -12444.4 q^{82} +51782.7 q^{83} -6200.11 q^{84} -1272.20 q^{86} +147853. q^{87} +22366.7 q^{88} +145254. q^{89} -13464.2 q^{91} +10599.5 q^{92} -110752. q^{93} +108962. q^{94} -20971.8 q^{96} -39512.8 q^{97} -65796.0 q^{98} +61662.6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2q + 8q^{2} - 3q^{3} + 32q^{4} - 12q^{6} - 114q^{7} + 128q^{8} + 239q^{9} + O(q^{10}) $	\( 2q + 8q^{2} - 3q^{3} + 32q^{4} - 12q^{6} - 114q^{7} + 128q^{8} + 239q^{9} + 661q^{11} - 48q^{12} - 1613q^{13} - 456q^{14} + 512q^{16} - 64q^{17} + 956q^{18} + 722q^{19} - 2711q^{21} + 2644q^{22} + 3185q^{23} - 192q^{24} - 6452q^{26} - 1791q^{27} - 1824q^{28} - 2481q^{29} - 1180q^{31} + 2048q^{32} - 1712q^{33} - 256q^{34} + 3824q^{36} - 10488q^{37} + 2888q^{38} - 1183q^{39} + 16630q^{41} - 10844q^{42} - 11303q^{43} + 10576q^{44} + 12740q^{46} + 12155q^{47} - 768q^{48} - 15588q^{49} + 7301q^{51} - 25808q^{52} - 20585q^{53} - 7164q^{54} - 7296q^{56} - 1083q^{57} - 9924q^{58} - 78581q^{59} + 43621q^{61} - 4720q^{62} - 4977q^{63} + 8192q^{64} - 6848q^{66} - 7805q^{67} - 1024q^{68} + 30527q^{69} - 62488q^{71} + 15296q^{72} - 16218q^{73} - 41952q^{74} + 11552q^{76} - 34795q^{77} - 4732q^{78} + 67122q^{79} - 141130q^{81} + 66520q^{82} + 10714q^{83} - 43376q^{84} - 45212q^{86} + 230679q^{87} + 42304q^{88} + 128188q^{89} + 106351q^{91} + 50960q^{92} - 225908q^{93} + 48620q^{94} - 3072q^{96} - 178558q^{97} - 62352q^{98} + 81151q^{99} + O(q^{100}) \)

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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	−20.4803	−1.31381	−0.656904	−	0.753974i	$-0.728135\pi$
$3$	−20.4803	−1.31381	−0.656904	+	0.753974i	$0.728135\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	−81.9210	−0.929003
$7$	18.9210	0.145948	0.0729742	−	0.997334i	$-0.476751\pi$
$7$	18.9210	0.145948	0.0729742	+	0.997334i	$0.476751\pi$
$8$	64.0000	0.353553
$9$	176.441	0.726094
$10$	0	0
$11$	349.480	0.870845	0.435423	−	0.900226i	$-0.356599\pi$
$11$	349.480	0.870845	0.435423	+	0.900226i	$0.356599\pi$
$12$	−327.684	−0.656904
$13$	−711.599	−1.16782	−0.583911	−	0.811818i	$-0.698478\pi$
$13$	−711.599	−1.16782	−0.583911	+	0.811818i	$0.698478\pi$
$14$	75.6840	0.103201
$15$	0	0
$16$	256.000	0.250000
$17$	−221.803	−0.186142	−0.0930710	−	0.995659i	$-0.529668\pi$
$17$	−221.803	−0.186142	−0.0930710	+	0.995659i	$0.529668\pi$
$18$	705.763	0.513426
$19$	361.000	0.229416
$19$	361.000	0.229416
$20$	0	0
$21$	−387.507	−0.191748
$22$	1397.92	0.615780
$23$	662.468	0.261123	0.130561	−	0.991440i	$-0.458322\pi$
$23$	662.468	0.261123	0.130561	+	0.991440i	$0.458322\pi$
$24$	−1310.74	−0.464502
$25$	0	0
$26$	−2846.39	−0.825775
$27$	1363.15	0.359861
$28$	302.736	0.0729742
$29$	−7219.28	−1.59404	−0.797019	−	0.603954i	$-0.793591\pi$
$29$	−7219.28	−1.59404	−0.797019	+	0.603954i	$0.793591\pi$
$30$	0	0
$31$	5407.76	1.01068	0.505339	−	0.862921i	$-0.331367\pi$
$31$	5407.76	1.01068	0.505339	+	0.862921i	$0.331367\pi$
$32$	1024.00	0.176777
$33$	−7157.44	−1.14412
$34$	−887.210	−0.131622
$35$	0	0
$36$	2823.05	0.363047
$37$	−1979.40	−0.237700	−0.118850	−	0.992912i	$-0.537921\pi$
$37$	−1979.40	−0.237700	−0.118850	+	0.992912i	$0.537921\pi$
$38$	1444.00	0.162221
$39$	14573.7	1.53430
$40$	0	0
$41$	−3111.11	−0.289039	−0.144519	−	0.989502i	$-0.546164\pi$
$41$	−3111.11	−0.289039	−0.144519	+	0.989502i	$0.546164\pi$
$42$	−1550.03	−0.135586
$43$	−318.049	−0.0262315	−0.0131157	−	0.999914i	$-0.504175\pi$
$43$	−318.049	−0.0262315	−0.0131157	+	0.999914i	$0.504175\pi$
$44$	5591.68	0.435423
$45$	0	0
$46$	2649.87	0.184642
$47$	27240.5	1.79875	0.899374	−	0.437181i	$-0.144023\pi$
$47$	27240.5	1.79875	0.899374	+	0.437181i	$0.144023\pi$
$48$	−5242.94	−0.328452
$49$	−16449.0	−0.978699
$50$	0	0
$51$	4542.57	0.244555
$52$	−11385.6	−0.583911
$53$	1114.63	0.0545057	0.0272528	−	0.999629i	$-0.491324\pi$
$53$	1114.63	0.0545057	0.0272528	+	0.999629i	$0.491324\pi$
$54$	5452.60	0.254460
$55$	0	0
$56$	1210.94	0.0516005
$57$	−7393.37	−0.301408
$58$	−28877.1	−1.12716
$59$	−37904.9	−1.41764	−0.708820	−	0.705390i	$-0.750772\pi$
$59$	−37904.9	−1.41764	−0.708820	+	0.705390i	$0.750772\pi$
$60$	0	0
$61$	37469.2	1.28929	0.644644	−	0.764483i	$-0.277006\pi$
$61$	37469.2	1.28929	0.644644	+	0.764483i	$0.277006\pi$
$62$	21631.0	0.714658
$63$	3338.44	0.105972
$64$	4096.00	0.125000
$65$	0	0
$66$	−28629.8	−0.809018
$67$	54955.3	1.49562	0.747812	−	0.663911i	$-0.231105\pi$
$67$	54955.3	1.49562	0.747812	+	0.663911i	$0.231105\pi$
$68$	−3548.84	−0.0930710
$69$	−13567.5	−0.343066
$70$	0	0
$71$	−7177.04	−0.168966	−0.0844830	−	0.996425i	$-0.526924\pi$
$71$	−7177.04	−0.168966	−0.0844830	+	0.996425i	$0.526924\pi$
$72$	11292.2	0.256713
$73$	−64746.1	−1.42202	−0.711011	−	0.703181i	$-0.751762\pi$
$73$	−64746.1	−1.42202	−0.711011	+	0.703181i	$0.751762\pi$
$74$	−7917.59	−0.168079
$75$	0	0
$76$	5776.00	0.114708
$77$	6612.52	0.127098
$78$	58294.9	1.08491
$79$	36104.4	0.650866	0.325433	−	0.945565i	$-0.394490\pi$
$79$	36104.4	0.650866	0.325433	+	0.945565i	$0.394490\pi$
$80$	0	0
$81$	−70792.8	−1.19888
$82$	−12444.4	−0.204381
$83$	51782.7	0.825067	0.412534	−	0.910942i	$-0.364644\pi$
$83$	51782.7	0.825067	0.412534	+	0.910942i	$0.364644\pi$
$84$	−6200.11	−0.0958741
$85$	0	0
$86$	−1272.20	−0.0185485
$87$	147853.	2.09426
$88$	22366.7	0.307890
$89$	145254.	1.94380	0.971900	−	0.235392i	$-0.0756375\pi$
$89$	145254.	1.94380	0.971900	+	0.235392i	$0.0756375\pi$
$90$	0	0
$91$	−13464.2	−0.170442
$92$	10599.5	0.130561
$93$	−110752.	−1.32784
$94$	108962.	1.27191
$95$	0	0
$96$	−20971.8	−0.232251
$97$	−39512.8	−0.426391	−0.213196	−	0.977010i	$-0.568387\pi$
$97$	−39512.8	−0.426391	−0.213196	+	0.977010i	$0.568387\pi$
$98$	−65796.0	−0.692045
$99$	61662.6	0.632315
$100$	0	0
$101$	−16722.9	−0.163120	−0.0815602	−	0.996668i	$-0.525990\pi$
$101$	−16722.9	−0.163120	−0.0815602	+	0.996668i	$0.525990\pi$
$102$	18170.3	0.172926
$103$	−92664.4	−0.860636	−0.430318	−	0.902677i	$-0.641599\pi$
$103$	−92664.4	−0.860636	−0.430318	+	0.902677i	$0.641599\pi$
$104$	−45542.3	−0.412888
$105$	0	0
$106$	4458.53	0.0385413
$107$	47156.3	0.398180	0.199090	−	0.979981i	$-0.436201\pi$
$107$	47156.3	0.398180	0.199090	+	0.979981i	$0.436201\pi$
$108$	21810.4	0.179930
$109$	52450.1	0.422844	0.211422	−	0.977395i	$-0.432191\pi$
$109$	52450.1	0.422844	0.211422	+	0.977395i	$0.432191\pi$
$110$	0	0
$111$	40538.5	0.312292
$112$	4843.78	0.0364871
$113$	−150596.	−1.10947	−0.554737	−	0.832026i	$-0.687181\pi$
$113$	−150596.	−1.10947	−0.554737	+	0.832026i	$0.687181\pi$
$114$	−29573.5	−0.213128
$115$	0	0
$116$	−115508.	−0.797019
$117$	−125555.	−0.847948
$118$	−151620.	−1.00242
$119$	−4196.73	−0.0271671
$120$	0	0
$121$	−38914.6	−0.241629
$122$	149877.	0.911664
$123$	63716.4	0.379742
$124$	86524.2	0.505339
$125$	0	0
$126$	13353.8	0.0749337
$127$	−352240.	−1.93789	−0.968945	−	0.247276i	$-0.920465\pi$
$127$	−352240.	−1.93789	−0.968945	+	0.247276i	$0.920465\pi$
$128$	16384.0	0.0883883
$129$	6513.72	0.0344632
$130$	0	0
$131$	19070.2	0.0970907	0.0485453	−	0.998821i	$-0.484541\pi$
$131$	19070.2	0.0970907	0.0485453	+	0.998821i	$0.484541\pi$
$132$	−114519.	−0.572062
$133$	6830.49	0.0334829
$134$	219821.	1.05757
$135$	0	0
$136$	−14195.4	−0.0658111
$137$	266677.	1.21390	0.606952	−	0.794738i	$-0.292392\pi$
$137$	266677.	1.21390	0.606952	+	0.794738i	$0.292392\pi$
$138$	−54270.0	−0.242584
$139$	−294888.	−1.29456	−0.647278	−	0.762254i	$-0.724093\pi$
$139$	−294888.	−1.29456	−0.647278	+	0.762254i	$0.724093\pi$
$140$	0	0
$141$	−557892.	−2.36321
$142$	−28708.2	−0.119477
$143$	−248690.	−1.01699
$144$	45168.8	0.181523
$145$	0	0
$146$	−258984.	−1.00552
$147$	336880.	1.28582
$148$	−31670.3	−0.118850
$149$	−103990.	−0.383729	−0.191864	−	0.981421i	$-0.561453\pi$
$149$	−103990.	−0.383729	−0.191864	+	0.981421i	$0.561453\pi$
$150$	0	0
$151$	−477343.	−1.70368	−0.851840	−	0.523802i	$-0.824513\pi$
$151$	−477343.	−1.70368	−0.851840	+	0.523802i	$0.824513\pi$
$152$	23104.0	0.0811107
$153$	−39135.0	−0.135156
$154$	26450.1	0.0898722
$155$	0	0
$156$	233180.	0.767148
$157$	−434955.	−1.40830	−0.704151	−	0.710051i	$-0.748672\pi$
$157$	−434955.	−1.40830	−0.704151	+	0.710051i	$0.748672\pi$
$158$	144417.	0.460232
$159$	−22827.9	−0.0716101
$160$	0	0
$161$	12534.6	0.0381105
$162$	−283171.	−0.847737
$163$	255083.	0.751990	0.375995	−	0.926622i	$-0.377301\pi$
$163$	255083.	0.751990	0.375995	+	0.926622i	$0.377301\pi$
$164$	−49777.8	−0.144519
$165$	0	0
$166$	207131.	0.583411
$167$	−510501.	−1.41646	−0.708232	−	0.705980i	$-0.750507\pi$
$167$	−510501.	−1.41646	−0.708232	+	0.705980i	$0.750507\pi$
$168$	−24800.5	−0.0677932
$169$	135080.	0.363809
$170$	0	0
$171$	63695.1	0.166577
$172$	−5088.78	−0.0131157
$173$	−773453.	−1.96480	−0.982401	−	0.186783i	$-0.940194\pi$
$173$	−773453.	−1.96480	−0.982401	+	0.186783i	$0.940194\pi$
$174$	591411.	1.48087
$175$	0	0
$176$	89466.9	0.217711
$177$	776303.	1.86251
$178$	581014.	1.37447
$179$	−477664.	−1.11427	−0.557135	−	0.830422i	$-0.688099\pi$
$179$	−477664.	−1.11427	−0.557135	+	0.830422i	$0.688099\pi$
$180$	0	0
$181$	729114.	1.65424	0.827121	−	0.562024i	$-0.189977\pi$
$181$	729114.	1.65424	0.827121	+	0.562024i	$0.189977\pi$
$182$	−53856.7	−0.120521
$183$	−767379.	−1.69388
$184$	42397.9	0.0923209
$185$	0	0
$186$	−443009.	−0.938924
$187$	−77515.6	−0.162101
$188$	435848.	0.899374
$189$	25792.2	0.0525211
$190$	0	0
$191$	−285584.	−0.566435	−0.283218	−	0.959056i	$-0.591402\pi$
$191$	−285584.	−0.566435	−0.283218	+	0.959056i	$0.591402\pi$
$192$	−83887.1	−0.164226
$193$	53690.2	0.103753	0.0518766	−	0.998654i	$-0.483480\pi$
$193$	53690.2	0.103753	0.0518766	+	0.998654i	$0.483480\pi$
$194$	−158051.	−0.301504
$195$	0	0
$196$	−263184.	−0.489350
$197$	−14538.9	−0.0266910	−0.0133455	−	0.999911i	$-0.504248\pi$
$197$	−14538.9	−0.0266910	−0.0133455	+	0.999911i	$0.504248\pi$
$198$	246650.	0.447114
$199$	−143700.	−0.257232	−0.128616	−	0.991695i	$-0.541053\pi$
$199$	−143700.	−0.257232	−0.128616	+	0.991695i	$0.541053\pi$
$200$	0	0
$201$	−1.12550e6	−1.96496
$202$	−66891.6	−0.115343
$203$	−136596.	−0.232647
$204$	72681.2	0.122277
$205$	0	0
$206$	−370658.	−0.608562
$207$	116886.	0.189600
$208$	−182169.	−0.291956
$209$	126162.	0.199786
$210$	0	0
$211$	83653.6	0.129354	0.0646768	−	0.997906i	$-0.479398\pi$
$211$	83653.6	0.129354	0.0646768	+	0.997906i	$0.479398\pi$
$212$	17834.1	0.0272528
$213$	146988.	0.221989
$214$	188625.	0.281556
$215$	0	0
$216$	87241.6	0.127230
$217$	102320.	0.147507
$218$	209800.	0.298996
$219$	1.32602e6	1.86826
$220$	0	0
$221$	157834.	0.217381
$222$	162154.	0.220824
$223$	665414.	0.896045	0.448022	−	0.894022i	$-0.352128\pi$
$223$	665414.	0.896045	0.448022	+	0.894022i	$0.352128\pi$
$224$	19375.1	0.0258003
$225$	0	0
$226$	−602384.	−0.784517
$227$	−1.15382e6	−1.48619	−0.743096	−	0.669185i	$-0.766643\pi$
$227$	−1.15382e6	−1.48619	−0.743096	+	0.669185i	$0.766643\pi$
$228$	−118294.	−0.150704
$229$	−433691.	−0.546502	−0.273251	−	0.961943i	$-0.588099\pi$
$229$	−433691.	−0.546502	−0.273251	+	0.961943i	$0.588099\pi$
$230$	0	0
$231$	−135426.	−0.166983
$232$	−462034.	−0.563578
$233$	−387173.	−0.467214	−0.233607	−	0.972331i	$-0.575053\pi$
$233$	−387173.	−0.467214	−0.233607	+	0.972331i	$0.575053\pi$
$234$	−502220.	−0.599590
$235$	0	0
$236$	−606479.	−0.708820
$237$	−739426.	−0.855114
$238$	−16786.9	−0.0192100
$239$	−622463.	−0.704886	−0.352443	−	0.935833i	$-0.614649\pi$
$239$	−622463.	−0.704886	−0.352443	+	0.935833i	$0.614649\pi$
$240$	0	0
$241$	−371454.	−0.411967	−0.205984	−	0.978555i	$-0.566039\pi$
$241$	−371454.	−0.411967	−0.205984	+	0.978555i	$0.566039\pi$
$242$	−155658.	−0.170857
$243$	1.11861e6	1.21524
$244$	599507.	0.644644
$245$	0	0
$246$	254865.	0.268518
$247$	−256887.	−0.267917
$248$	346097.	0.357329
$249$	−1.06052e6	−1.08398
$250$	0	0
$251$	376098.	0.376805	0.188403	−	0.982092i	$-0.439669\pi$
$251$	376098.	0.376805	0.188403	+	0.982092i	$0.439669\pi$
$252$	53415.0	0.0529861
$253$	231519.	0.227398
$254$	−1.40896e6	−1.37030
$255$	0	0
$256$	65536.0	0.0625000
$257$	−1.64382e6	−1.55247	−0.776233	−	0.630446i	$-0.782872\pi$
$257$	−1.64382e6	−1.55247	−0.776233	+	0.630446i	$0.782872\pi$
$258$	26054.9	0.0243691
$259$	−37452.2	−0.0346919
$260$	0	0
$261$	−1.27378e6	−1.15742
$262$	76280.9	0.0686535
$263$	−775299.	−0.691162	−0.345581	−	0.938389i	$-0.612318\pi$
$263$	−775299.	−0.691162	−0.345581	+	0.938389i	$0.612318\pi$
$264$	−458076.	−0.404509
$265$	0	0
$266$	27321.9	0.0236760
$267$	−2.97483e6	−2.55378
$268$	879284.	0.747812
$269$	334851.	0.282144	0.141072	−	0.989999i	$-0.454945\pi$
$269$	334851.	0.282144	0.141072	+	0.989999i	$0.454945\pi$
$270$	0	0
$271$	893377.	0.738944	0.369472	−	0.929242i	$-0.379539\pi$
$271$	893377.	0.738944	0.369472	+	0.929242i	$0.379539\pi$
$272$	−56781.4	−0.0465355
$273$	275750.	0.223928
$274$	1.06671e6	0.858360
$275$	0	0
$276$	−217080.	−0.171533
$277$	1.13462e6	0.888483	0.444242	−	0.895907i	$-0.353473\pi$
$277$	1.13462e6	0.888483	0.444242	+	0.895907i	$0.353473\pi$
$278$	−1.17955e6	−0.915389
$279$	954149.	0.733847
$280$	0	0
$281$	−1.98111e6	−1.49672	−0.748362	−	0.663290i	$-0.769159\pi$
$281$	−1.98111e6	−1.49672	−0.748362	+	0.663290i	$0.769159\pi$
$282$	−2.23157e6	−1.67104
$283$	−1.32192e6	−0.981161	−0.490580	−	0.871396i	$-0.663215\pi$
$283$	−1.32192e6	−0.981161	−0.490580	+	0.871396i	$0.663215\pi$
$284$	−114833.	−0.0844830
$285$	0	0
$286$	−994759.	−0.719122
$287$	−58865.4	−0.0421847
$288$	180675.	0.128356
$289$	−1.37066e6	−0.965351
$290$	0	0
$291$	809232.	0.560197
$292$	−1.03594e6	−0.711011
$293$	−1.35808e6	−0.924179	−0.462089	−	0.886833i	$-0.652900\pi$
$293$	−1.35808e6	−0.924179	−0.462089	+	0.886833i	$0.652900\pi$
$294$	1.34752e6	0.909215
$295$	0	0
$296$	−126681.	−0.0840395
$297$	476394.	0.313383
$298$	−415959.	−0.271337
$299$	−471411.	−0.304945
$300$	0	0
$301$	−6017.81	−0.00382844
$302$	−1.90937e6	−1.20468
$303$	342489.	0.214309
$304$	92416.0	0.0573539
$305$	0	0
$306$	−156540.	−0.0955701
$307$	1.23842e6	0.749930	0.374965	−	0.927039i	$-0.377655\pi$
$307$	1.23842e6	0.749930	0.374965	+	0.927039i	$0.377655\pi$
$308$	105800.	0.0635492
$309$	1.89779e6	1.13071
$310$	0	0
$311$	−1.78565e6	−1.04688	−0.523439	−	0.852063i	$-0.675351\pi$
$311$	−1.78565e6	−1.04688	−0.523439	+	0.852063i	$0.675351\pi$
$312$	932718.	0.542455
$313$	3.01608e6	1.74013	0.870066	−	0.492935i	$-0.164076\pi$
$313$	3.01608e6	1.74013	0.870066	+	0.492935i	$0.164076\pi$
$314$	−1.73982e6	−0.995819
$315$	0	0
$316$	577670.	0.325433
$317$	−515839.	−0.288314	−0.144157	−	0.989555i	$-0.546047\pi$
$317$	−515839.	−0.288314	−0.144157	+	0.989555i	$0.546047\pi$
$318$	−91311.8	−0.0506360
$319$	−2.52300e6	−1.38816
$320$	0	0
$321$	−965772.	−0.523133
$322$	50138.2	0.0269482
$323$	−80070.7	−0.0427039
$324$	−1.13268e6	−0.599441
$325$	0	0
$326$	1.02033e6	0.531737
$327$	−1.07419e6	−0.555536
$328$	−199111.	−0.102191
$329$	515417.	0.262524
$330$	0	0
$331$	−2.25449e6	−1.13104	−0.565521	−	0.824734i	$-0.691325\pi$
$331$	−2.25449e6	−1.13104	−0.565521	+	0.824734i	$0.691325\pi$
$332$	828523.	0.412534
$333$	−349246.	−0.172592
$334$	−2.04200e6	−1.00159
$335$	0	0
$336$	−99201.8	−0.0479371
$337$	−2.40261e6	−1.15242	−0.576208	−	0.817303i	$-0.695468\pi$
$337$	−2.40261e6	−1.15242	−0.576208	+	0.817303i	$0.695468\pi$
$338$	540319.	0.257252
$339$	3.08424e6	1.45764
$340$	0	0
$341$	1.88991e6	0.880145
$342$	254780.	0.117788
$343$	−629237.	−0.288788
$344$	−20355.1	−0.00927423
$345$	0	0
$346$	−3.09381e6	−1.38933
$347$	684693.	0.305262	0.152631	−	0.988283i	$-0.451225\pi$
$347$	684693.	0.305262	0.152631	+	0.988283i	$0.451225\pi$
$348$	2.36564e6	1.04713
$349$	2.19857e6	0.966220	0.483110	−	0.875560i	$-0.339507\pi$
$349$	2.19857e6	0.966220	0.483110	+	0.875560i	$0.339507\pi$
$350$	0	0
$351$	−970016.	−0.420253
$352$	357868.	0.153945
$353$	−2.03446e6	−0.868987	−0.434493	−	0.900675i	$-0.643073\pi$
$353$	−2.03446e6	−0.868987	−0.434493	+	0.900675i	$0.643073\pi$
$354$	3.10521e6	1.31699
$355$	0	0
$356$	2.32406e6	0.971900
$357$	85950.1	0.0356924
$358$	−1.91066e6	−0.787907
$359$	2.30592e6	0.944298	0.472149	−	0.881519i	$-0.343478\pi$
$359$	2.30592e6	0.944298	0.472149	+	0.881519i	$0.343478\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	2.91646e6	1.16973
$363$	796980.	0.317454
$364$	−215427.	−0.0852209
$365$	0	0
$366$	−3.06952e6	−1.19775
$367$	−2.64475e6	−1.02499	−0.512495	−	0.858690i	$-0.671279\pi$
$367$	−2.64475e6	−1.02499	−0.512495	+	0.858690i	$0.671279\pi$
$368$	169592.	0.0652807
$369$	−548927.	−0.209869
$370$	0	0
$371$	21090.0	0.00795502
$372$	−1.77204e6	−0.663919
$373$	−173405.	−0.0645342	−0.0322671	−	0.999479i	$-0.510273\pi$
$373$	−173405.	−0.0645342	−0.0322671	+	0.999479i	$0.510273\pi$
$374$	−310062.	−0.114623
$375$	0	0
$376$	1.74339e6	0.635953
$377$	5.13723e6	1.86155
$378$	103169.	0.0371380
$379$	2.23408e6	0.798915	0.399458	−	0.916752i	$-0.369198\pi$
$379$	2.23408e6	0.798915	0.399458	+	0.916752i	$0.369198\pi$
$380$	0	0
$381$	7.21396e6	2.54602
$382$	−1.14234e6	−0.400530
$383$	4.44976e6	1.55003	0.775014	−	0.631944i	$-0.217743\pi$
$383$	4.44976e6	1.55003	0.775014	+	0.631944i	$0.217743\pi$
$384$	−335548.	−0.116125
$385$	0	0
$386$	214761.	0.0733646
$387$	−56116.8	−0.0190465
$388$	−632204.	−0.213196
$389$	2.43083e6	0.814480	0.407240	−	0.913321i	$-0.366491\pi$
$389$	2.43083e6	0.814480	0.407240	+	0.913321i	$0.366491\pi$
$390$	0	0
$391$	−146937.	−0.0486059
$392$	−1.05274e6	−0.346022
$393$	−390563.	−0.127559
$394$	−58155.5	−0.0188734
$395$	0	0
$396$	986601.	0.316158
$397$	−2.61205e6	−0.831775	−0.415887	−	0.909416i	$-0.636529\pi$
$397$	−2.61205e6	−0.831775	−0.415887	+	0.909416i	$0.636529\pi$
$398$	−574800.	−0.181890
$399$	−139890.	−0.0439901
$400$	0	0
$401$	1.42660e6	0.443038	0.221519	−	0.975156i	$-0.428899\pi$
$401$	1.42660e6	0.443038	0.221519	+	0.975156i	$0.428899\pi$
$402$	−4.50199e6	−1.38944
$403$	−3.84816e6	−1.18029
$404$	−267566.	−0.0815602
$405$	0	0
$406$	−546384.	−0.164507
$407$	−691760.	−0.207000
$408$	290725.	0.0864632
$409$	4.73321e6	1.39910	0.699548	−	0.714586i	$-0.253385\pi$
$409$	4.73321e6	1.39910	0.699548	+	0.714586i	$0.253385\pi$
$410$	0	0
$411$	−5.46162e6	−1.59484
$412$	−1.48263e6	−0.430318
$413$	−717200.	−0.206902
$414$	467545.	0.134067
$415$	0	0
$416$	−728677.	−0.206444
$417$	6.03939e6	1.70080
$418$	504649.	0.141270
$419$	−357759.	−0.0995531	−0.0497766	−	0.998760i	$-0.515851\pi$
$419$	−357759.	−0.0995531	−0.0497766	+	0.998760i	$0.515851\pi$
$420$	0	0
$421$	652504.	0.179423	0.0897115	−	0.995968i	$-0.471405\pi$
$421$	652504.	0.179423	0.0897115	+	0.995968i	$0.471405\pi$
$422$	334614.	0.0914668
$423$	4.80633e6	1.30606
$424$	71336.4	0.0192707
$425$	0	0
$426$	587950.	0.156970
$427$	708955.	0.188169
$428$	754500.	0.199090
$429$	5.09323e6	1.33613
$430$	0	0
$431$	−6.25148e6	−1.62103	−0.810513	−	0.585721i	$-0.800812\pi$
$431$	−6.25148e6	−1.62103	−0.810513	+	0.585721i	$0.800812\pi$
$432$	348966.	0.0899651
$433$	−4.45832e6	−1.14275	−0.571375	−	0.820689i	$-0.693590\pi$
$433$	−4.45832e6	−1.14275	−0.571375	+	0.820689i	$0.693590\pi$
$434$	409281.	0.104303
$435$	0	0
$436$	839201.	0.211422
$437$	239151.	0.0599057
$438$	5.30406e6	1.32106
$439$	5.00652e6	1.23986	0.619932	−	0.784655i	$-0.287160\pi$
$439$	5.00652e6	1.23986	0.619932	+	0.784655i	$0.287160\pi$
$440$	0	0
$441$	−2.90227e6	−0.710627
$442$	631338.	0.153711
$443$	−715908.	−0.173320	−0.0866599	−	0.996238i	$-0.527619\pi$
$443$	−715908.	−0.173320	−0.0866599	+	0.996238i	$0.527619\pi$
$444$	648617.	0.156146
$445$	0	0
$446$	2.66166e6	0.633599
$447$	2.12974e6	0.504147
$448$	77500.5	0.0182435
$449$	−4.83183e6	−1.13109	−0.565544	−	0.824718i	$-0.691334\pi$
$449$	−4.83183e6	−1.13109	−0.565544	+	0.824718i	$0.691334\pi$
$450$	0	0
$451$	−1.08727e6	−0.251708
$452$	−2.40954e6	−0.554737
$453$	9.77610e6	2.23831
$454$	−4.61530e6	−1.05090
$455$	0	0
$456$	−473176.	−0.106564
$457$	6.44410e6	1.44335	0.721675	−	0.692232i	$-0.243372\pi$
$457$	6.44410e6	1.44335	0.721675	+	0.692232i	$0.243372\pi$
$458$	−1.73477e6	−0.386436
$459$	−302350.	−0.0669851
$460$	0	0
$461$	−4.10108e6	−0.898764	−0.449382	−	0.893340i	$-0.648356\pi$
$461$	−4.10108e6	−0.898764	−0.449382	+	0.893340i	$0.648356\pi$
$462$	−541704.	−0.118075
$463$	−8.98683e6	−1.94829	−0.974146	−	0.225919i	$-0.927461\pi$
$463$	−8.98683e6	−1.94829	−0.974146	+	0.225919i	$0.927461\pi$
$464$	−1.84814e6	−0.398510
$465$	0	0
$466$	−1.54869e6	−0.330370
$467$	8.84409e6	1.87655	0.938276	−	0.345886i	$-0.112422\pi$
$467$	8.84409e6	1.87655	0.938276	+	0.345886i	$0.112422\pi$
$468$	−2.00888e6	−0.423974
$469$	1.03981e6	0.218284
$470$	0	0
$471$	8.90800e6	1.85024
$472$	−2.42592e6	−0.501211
$473$	−111152.	−0.0228436
$474$	−2.95771e6	−0.604657
$475$	0	0
$476$	−67147.7	−0.0135836
$477$	196667.	0.0395762
$478$	−2.48985e6	−0.498430
$479$	2.89320e6	0.576155	0.288077	−	0.957607i	$-0.406984\pi$
$479$	2.89320e6	0.576155	0.288077	+	0.957607i	$0.406984\pi$
$480$	0	0
$481$	1.40854e6	0.277591
$482$	−1.48582e6	−0.291305
$483$	−256711.	−0.0500699
$484$	−622633.	−0.120814
$485$	0	0
$486$	4.47443e6	0.859305
$487$	3.18166e6	0.607900	0.303950	−	0.952688i	$-0.401694\pi$
$487$	3.18166e6	0.607900	0.303950	+	0.952688i	$0.401694\pi$
$488$	2.39803e6	0.455832
$489$	−5.22416e6	−0.987971
$490$	0	0
$491$	−4.45509e6	−0.833975	−0.416987	−	0.908912i	$-0.636914\pi$
$491$	−4.45509e6	−0.833975	−0.416987	+	0.908912i	$0.636914\pi$
$492$	1.01946e6	0.189871
$493$	1.60125e6	0.296717
$494$	−1.02755e6	−0.189446
$495$	0	0
$496$	1.38439e6	0.252670
$497$	−135797.	−0.0246603
$498$	−4.24209e6	−0.766490
$499$	9.31472e6	1.67463	0.837315	−	0.546721i	$-0.184124\pi$
$499$	9.31472e6	1.67463	0.837315	+	0.546721i	$0.184124\pi$
$500$	0	0
$501$	1.04552e7	1.86096
$502$	1.50439e6	0.266442
$503$	2.04835e6	0.360980	0.180490	−	0.983577i	$-0.442232\pi$
$503$	2.04835e6	0.360980	0.180490	+	0.983577i	$0.442232\pi$
$504$	213660.	0.0374668
$505$	0	0
$506$	926077.	0.160794
$507$	−2.76647e6	−0.477976
$508$	−5.63584e6	−0.968945
$509$	−166211.	−0.0284358	−0.0142179	−	0.999899i	$-0.504526\pi$
$509$	−166211.	−0.0284358	−0.0142179	+	0.999899i	$0.504526\pi$
$510$	0	0
$511$	−1.22506e6	−0.207542
$512$	262144.	0.0441942
$513$	492097.	0.0825577
$514$	−6.57529e6	−1.09776
$515$	0	0
$516$	104220.	0.0172316
$517$	9.52001e6	1.56643
$518$	−149809.	−0.0245309
$519$	1.58405e7	2.58137
$520$	0	0
$521$	2.64602e6	0.427069	0.213535	−	0.976935i	$-0.431502\pi$
$521$	2.64602e6	0.427069	0.213535	+	0.976935i	$0.431502\pi$
$522$	−5.09510e6	−0.818421
$523$	7.02287e6	1.12269	0.561346	−	0.827581i	$-0.310284\pi$
$523$	7.02287e6	1.12269	0.561346	+	0.827581i	$0.310284\pi$
$524$	305124.	0.0485453
$525$	0	0
$526$	−3.10120e6	−0.488726
$527$	−1.19945e6	−0.188130
$528$	−1.83231e6	−0.286031
$529$	−5.99748e6	−0.931815
$530$	0	0
$531$	−6.68798e6	−1.02934
$532$	109288.	0.0167414
$533$	2.21386e6	0.337546
$534$	−1.18993e7	−1.80580
$535$	0	0
$536$	3.51714e6	0.528783
$537$	9.78268e6	1.46394
$538$	1.33940e6	0.199506
$539$	−5.74860e6	−0.852295
$540$	0	0
$541$	4.35066e6	0.639090	0.319545	−	0.947571i	$-0.396470\pi$
$541$	4.35066e6	0.639090	0.319545	+	0.947571i	$0.396470\pi$
$542$	3.57351e6	0.522512
$543$	−1.49324e7	−2.17336
$544$	−227126.	−0.0329056
$545$	0	0
$546$	1.10300e6	0.158341
$547$	−9.77794e6	−1.39727	−0.698633	−	0.715480i	$-0.746208\pi$
$547$	−9.77794e6	−1.39727	−0.698633	+	0.715480i	$0.746208\pi$
$548$	4.26684e6	0.606952
$549$	6.61110e6	0.936144
$550$	0	0
$551$	−2.60616e6	−0.365698
$552$	−868320.	−0.121292
$553$	683131.	0.0949929
$554$	4.53846e6	0.628252
$555$	0	0
$556$	−4.71822e6	−0.647278
$557$	−6.69323e6	−0.914108	−0.457054	−	0.889439i	$-0.651095\pi$
$557$	−6.69323e6	−0.914108	−0.457054	+	0.889439i	$0.651095\pi$
$558$	3.81660e6	0.518909
$559$	226323.	0.0306337
$560$	0	0
$561$	1.58754e6	0.212969
$562$	−7.92442e6	−1.05834
$563$	−7.54034e6	−1.00258	−0.501291	−	0.865279i	$-0.667141\pi$
$563$	−7.54034e6	−1.00258	−0.501291	+	0.865279i	$0.667141\pi$
$564$	−8.92627e6	−1.18161
$565$	0	0
$566$	−5.28769e6	−0.693785
$567$	−1.33947e6	−0.174975
$568$	−459331.	−0.0597385
$569$	1.34726e6	0.174450	0.0872252	−	0.996189i	$-0.472200\pi$
$569$	1.34726e6	0.174450	0.0872252	+	0.996189i	$0.472200\pi$
$570$	0	0
$571$	2.05762e6	0.264104	0.132052	−	0.991243i	$-0.457843\pi$
$571$	2.05762e6	0.264104	0.132052	+	0.991243i	$0.457843\pi$
$572$	−3.97904e6	−0.508496
$573$	5.84883e6	0.744188
$574$	−235462.	−0.0298291
$575$	0	0
$576$	722701.	0.0907617
$577$	−7.88561e6	−0.986043	−0.493021	−	0.870017i	$-0.664108\pi$
$577$	−7.88561e6	−0.986043	−0.493021	+	0.870017i	$0.664108\pi$
$578$	−5.48264e6	−0.682606
$579$	−1.09959e6	−0.136312
$580$	0	0
$581$	979781.	0.120417
$582$	3.23693e6	0.396119
$583$	389542.	0.0474660
$584$	−4.14375e6	−0.502761
$585$	0	0
$586$	−5.43232e6	−0.653493
$587$	−1.90439e6	−0.228119	−0.114059	−	0.993474i	$-0.536385\pi$
$587$	−1.90439e6	−0.228119	−0.114059	+	0.993474i	$0.536385\pi$
$588$	5.39007e6	0.642912
$589$	1.95220e6	0.231866
$590$	0	0
$591$	297760.	0.0350669
$592$	−506726.	−0.0594249
$593$	1.35835e7	1.58626	0.793132	−	0.609050i	$-0.208449\pi$
$593$	1.35835e7	1.58626	0.793132	+	0.609050i	$0.208449\pi$
$594$	1.90558e6	0.221595
$595$	0	0
$596$	−1.66384e6	−0.191864
$597$	2.94301e6	0.337953
$598$	−1.88564e6	−0.215629
$599$	−1.04361e7	−1.18842	−0.594212	−	0.804309i	$-0.702536\pi$
$599$	−1.04361e7	−1.18842	−0.594212	+	0.804309i	$0.702536\pi$
$600$	0	0
$601$	−1.17196e7	−1.32351	−0.661753	−	0.749722i	$-0.730187\pi$
$601$	−1.17196e7	−1.32351	−0.661753	+	0.749722i	$0.730187\pi$
$602$	−24071.2	−0.00270712
$603$	9.69635e6	1.08596
$604$	−7.63749e6	−0.851840
$605$	0	0
$606$	1.36996e6	0.151539
$607$	7.53524e6	0.830091	0.415045	−	0.909801i	$-0.363766\pi$
$607$	7.53524e6	0.830091	0.415045	+	0.909801i	$0.363766\pi$
$608$	369664.	0.0405554
$609$	2.79752e6	0.305654
$610$	0	0
$611$	−1.93843e7	−2.10062
$612$	−626160.	−0.0675782
$613$	4.37292e6	0.470025	0.235012	−	0.971992i	$-0.424487\pi$
$613$	4.37292e6	0.470025	0.235012	+	0.971992i	$0.424487\pi$
$614$	4.95366e6	0.530280
$615$	0	0
$616$	423201.	0.0449361
$617$	7.50595e6	0.793767	0.396883	−	0.917869i	$-0.370092\pi$
$617$	7.50595e6	0.793767	0.396883	+	0.917869i	$0.370092\pi$
$618$	7.59116e6	0.799534
$619$	−1.30877e7	−1.37289	−0.686447	−	0.727180i	$-0.740831\pi$
$619$	−1.30877e7	−1.37289	−0.686447	+	0.727180i	$0.740831\pi$
$620$	0	0
$621$	903043.	0.0939679
$622$	−7.14261e6	−0.740254
$623$	2.74834e6	0.283695
$624$	3.73087e6	0.383574
$625$	0	0
$626$	1.20643e7	1.23046
$627$	−2.58384e6	−0.262480
$628$	−6.95929e6	−0.704151
$629$	439035.	0.0442459
$630$	0	0
$631$	8.92096e6	0.891945	0.445972	−	0.895047i	$-0.352858\pi$
$631$	8.92096e6	0.891945	0.445972	+	0.895047i	$0.352858\pi$
$632$	2.31068e6	0.230116
$633$	−1.71325e6	−0.169946
$634$	−2.06336e6	−0.203869
$635$	0	0
$636$	−365247.	−0.0358050
$637$	1.17051e7	1.14295
$638$	−1.00920e7	−0.981578
$639$	−1.26632e6	−0.122685
$640$	0	0
$641$	1.45438e7	1.39808	0.699041	−	0.715082i	$-0.253611\pi$
$641$	1.45438e7	1.39808	0.699041	+	0.715082i	$0.253611\pi$
$642$	−3.86309e6	−0.369911
$643$	−1.45757e7	−1.39028	−0.695140	−	0.718874i	$-0.744658\pi$
$643$	−1.45757e7	−1.39028	−0.695140	+	0.718874i	$0.744658\pi$
$644$	200553.	0.0190552
$645$	0	0
$646$	−320283.	−0.0301962
$647$	2.24522e6	0.210862	0.105431	−	0.994427i	$-0.466378\pi$
$647$	2.24522e6	0.210862	0.105431	+	0.994427i	$0.466378\pi$
$648$	−4.53074e6	−0.423869
$649$	−1.32470e7	−1.23454
$650$	0	0
$651$	−2.09555e6	−0.193796
$652$	4.08132e6	0.375995
$653$	−1.54461e7	−1.41754	−0.708770	−	0.705440i	$-0.750749\pi$
$653$	−1.54461e7	−1.41754	−0.708770	+	0.705440i	$0.750749\pi$
$654$	−4.29676e6	−0.392823
$655$	0	0
$656$	−796445.	−0.0722597
$657$	−1.14238e7	−1.03252
$658$	2.06167e6	0.185633
$659$	8.68940e6	0.779429	0.389714	−	0.920936i	$-0.372574\pi$
$659$	8.68940e6	0.779429	0.389714	+	0.920936i	$0.372574\pi$
$660$	0	0
$661$	−2.03442e7	−1.81108	−0.905538	−	0.424266i	$-0.860532\pi$
$661$	−2.03442e7	−1.81108	−0.905538	+	0.424266i	$0.860532\pi$
$662$	−9.01797e6	−0.799767
$663$	−3.23249e6	−0.285597
$664$	3.31409e6	0.291705
$665$	0	0
$666$	−1.39698e6	−0.122041
$667$	−4.78254e6	−0.416240
$668$	−8.16802e6	−0.708232
$669$	−1.36278e7	−1.17723
$670$	0	0
$671$	1.30947e7	1.12277
$672$	−396807.	−0.0338966
$673$	−1.71139e7	−1.45650	−0.728251	−	0.685310i	$-0.759667\pi$
$673$	−1.71139e7	−1.45650	−0.728251	+	0.685310i	$0.759667\pi$
$674$	−9.61045e6	−0.814881
$675$	0	0
$676$	2.16128e6	0.181905
$677$	−2.25299e6	−0.188924	−0.0944621	−	0.995528i	$-0.530113\pi$
$677$	−2.25299e6	−0.188924	−0.0944621	+	0.995528i	$0.530113\pi$
$678$	1.23370e7	1.03071
$679$	−747622.	−0.0622311
$680$	0	0
$681$	2.36306e7	1.95257
$682$	7.55962e6	0.622356
$683$	−5.33481e6	−0.437590	−0.218795	−	0.975771i	$-0.570213\pi$
$683$	−5.33481e6	−0.437590	−0.218795	+	0.975771i	$0.570213\pi$
$684$	1.01912e6	0.0832887
$685$	0	0
$686$	−2.51695e6	−0.204204
$687$	8.88211e6	0.718000
$688$	−81420.5	−0.00655787
$689$	−793171.	−0.0636530
$690$	0	0
$691$	8.08495e6	0.644143	0.322071	−	0.946715i	$-0.395621\pi$
$691$	8.08495e6	0.644143	0.322071	+	0.946715i	$0.395621\pi$
$692$	−1.23753e7	−0.982401
$693$	1.16672e6	0.0922854
$694$	2.73877e6	0.215853
$695$	0	0
$696$	9.46257e6	0.740433
$697$	690053.	0.0538022
$698$	8.79427e6	0.683221
$699$	7.92941e6	0.613829
$700$	0	0
$701$	7.88116e6	0.605752	0.302876	−	0.953030i	$-0.402053\pi$
$701$	7.88116e6	0.605752	0.302876	+	0.953030i	$0.402053\pi$
$702$	−3.88006e6	−0.297164
$703$	−714562.	−0.0545320
$704$	1.43147e6	0.108856
$705$	0	0
$706$	−8.13786e6	−0.614467
$707$	−316414.	−0.0238071
$708$	1.24208e7	0.931254
$709$	−2.07701e7	−1.55175	−0.775876	−	0.630885i	$-0.782692\pi$
$709$	−2.07701e7	−1.55175	−0.775876	+	0.630885i	$0.782692\pi$
$710$	0	0
$711$	6.37028e6	0.472590
$712$	9.29623e6	0.687237
$713$	3.58247e6	0.263911
$714$	343800.	0.0252383
$715$	0	0
$716$	−7.64263e6	−0.557135
$717$	1.27482e7	0.926086
$718$	9.22370e6	0.667719
$719$	1.03826e7	0.749004	0.374502	−	0.927226i	$-0.377814\pi$
$719$	1.03826e7	0.749004	0.374502	+	0.927226i	$0.377814\pi$
$720$	0	0
$721$	−1.75330e6	−0.125608
$722$	521284.	0.0372161
$723$	7.60748e6	0.541246
$724$	1.16658e7	0.827121
$725$	0	0
$726$	3.18792e6	0.224474
$727$	6.52486e6	0.457863	0.228931	−	0.973443i	$-0.426477\pi$
$727$	6.52486e6	0.457863	0.228931	+	0.973443i	$0.426477\pi$
$728$	−861707.	−0.0602603
$729$	−5.70674e6	−0.397712
$730$	0	0
$731$	70544.1	0.00488278
$732$	−1.22781e7	−0.846939
$733$	2.46101e6	0.169182	0.0845908	−	0.996416i	$-0.473042\pi$
$733$	2.46101e6	0.169182	0.0845908	+	0.996416i	$0.473042\pi$
$734$	−1.05790e7	−0.724778
$735$	0	0
$736$	678367.	0.0461605
$737$	1.92058e7	1.30246
$738$	−2.19571e6	−0.148400
$739$	−7.33586e6	−0.494128	−0.247064	−	0.968999i	$-0.579466\pi$
$739$	−7.33586e6	−0.494128	−0.247064	+	0.968999i	$0.579466\pi$
$740$	0	0
$741$	5.26111e6	0.351991
$742$	84359.9	0.00562505
$743$	−2.44394e7	−1.62412	−0.812060	−	0.583574i	$-0.801654\pi$
$743$	−2.44394e7	−1.62412	−0.812060	+	0.583574i	$0.801654\pi$
$744$	−7.08815e6	−0.469462
$745$	0	0
$746$	−693621.	−0.0456326
$747$	9.13658e6	0.599076
$748$	−1.24025e6	−0.0810504
$749$	892244.	0.0581138
$750$	0	0
$751$	−2.11169e6	−0.136625	−0.0683126	−	0.997664i	$-0.521762\pi$
$751$	−2.11169e6	−0.136625	−0.0683126	+	0.997664i	$0.521762\pi$
$752$	6.97356e6	0.449687
$753$	−7.70259e6	−0.495050
$754$	2.05489e7	1.31632
$755$	0	0
$756$	412675.	0.0262605
$757$	1.47627e7	0.936325	0.468162	−	0.883642i	$-0.344916\pi$
$757$	1.47627e7	0.936325	0.468162	+	0.883642i	$0.344916\pi$
$758$	8.93632e6	0.564919
$759$	−4.74157e6	−0.298757
$760$	0	0
$761$	2.67074e7	1.67174	0.835871	−	0.548926i	$-0.184963\pi$
$761$	2.67074e7	1.67174	0.835871	+	0.548926i	$0.184963\pi$
$762$	2.88558e7	1.80031
$763$	992408.	0.0617133
$764$	−4.56934e6	−0.283218
$765$	0	0
$766$	1.77990e7	1.09604
$767$	2.69731e7	1.65555
$768$	−1.34219e6	−0.0821131
$769$	3.05104e6	0.186051	0.0930256	−	0.995664i	$-0.470346\pi$
$769$	3.05104e6	0.186051	0.0930256	+	0.995664i	$0.470346\pi$
$770$	0	0
$771$	3.36659e7	2.03964
$772$	859043.	0.0518766
$773$	2.18723e7	1.31657	0.658287	−	0.752767i	$-0.271281\pi$
$773$	2.18723e7	1.31657	0.658287	+	0.752767i	$0.271281\pi$
$774$	−224467.	−0.0134679
$775$	0	0
$776$	−2.52882e6	−0.150752
$777$	767030.	0.0455785
$778$	9.72331e6	0.575924
$779$	−1.12311e6	−0.0663100
$780$	0	0
$781$	−2.50823e6	−0.147143
$782$	−587748.	−0.0343696
$783$	−9.84096e6	−0.573632
$784$	−4.21094e6	−0.244675
$785$	0	0
$786$	−1.56225e6	−0.0901975
$787$	1.26837e7	0.729975	0.364987	−	0.931013i	$-0.381073\pi$
$787$	1.26837e7	0.729975	0.364987	+	0.931013i	$0.381073\pi$
$788$	−232622.	−0.0133455
$789$	1.58783e7	0.908055
$790$	0	0
$791$	−2.84943e6	−0.161926
$792$	3.94640e6	0.223557
$793$	−2.66630e7	−1.50566
$794$	−1.04482e7	−0.588154
$795$	0	0
$796$	−2.29920e6	−0.128616
$797$	−7.28989e6	−0.406514	−0.203257	−	0.979125i	$-0.565153\pi$
$797$	−7.28989e6	−0.406514	−0.203257	+	0.979125i	$0.565153\pi$
$798$	−559560.	−0.0311057
$799$	−6.04201e6	−0.334822
$800$	0	0
$801$	2.56286e7	1.41138
$802$	5.70640e6	0.313275
$803$	−2.26275e7	−1.23836
$804$	−1.80080e7	−0.982482
$805$	0	0
$806$	−1.53926e7	−0.834593
$807$	−6.85783e6	−0.370683
$808$	−1.07027e6	−0.0576717
$809$	−2.72506e7	−1.46388	−0.731939	−	0.681370i	$-0.761385\pi$
$809$	−2.72506e7	−1.46388	−0.731939	+	0.681370i	$0.761385\pi$
$810$	0	0
$811$	1.45954e7	0.779229	0.389615	−	0.920978i	$-0.372608\pi$
$811$	1.45954e7	0.779229	0.389615	+	0.920978i	$0.372608\pi$
$812$	−2.18554e6	−0.116324
$813$	−1.82966e7	−0.970831
$814$	−2.76704e6	−0.146371
$815$	0	0
$816$	1.16290e6	0.0611387
$817$	−114816.	−0.00601791
$818$	1.89328e7	0.989310
$819$	−2.37563e6	−0.123757
$820$	0	0
$821$	4.16570e6	0.215690	0.107845	−	0.994168i	$-0.465605\pi$
$821$	4.16570e6	0.215690	0.107845	+	0.994168i	$0.465605\pi$
$822$	−2.18465e7	−1.12772
$823$	9.74487e6	0.501506	0.250753	−	0.968051i	$-0.419322\pi$
$823$	9.74487e6	0.501506	0.250753	+	0.968051i	$0.419322\pi$
$824$	−5.93052e6	−0.304281
$825$	0	0
$826$	−2.86880e6	−0.146302
$827$	−1.04970e6	−0.0533707	−0.0266854	−	0.999644i	$-0.508495\pi$
$827$	−1.04970e6	−0.0533707	−0.0266854	+	0.999644i	$0.508495\pi$
$828$	1.87018e6	0.0947999
$829$	1.87001e7	0.945056	0.472528	−	0.881316i	$-0.343342\pi$
$829$	1.87001e7	0.945056	0.472528	+	0.881316i	$0.343342\pi$
$830$	0	0
$831$	−2.32372e7	−1.16730
$832$	−2.91471e6	−0.145978
$833$	3.64843e6	0.182177
$834$	2.41576e7	1.20265
$835$	0	0
$836$	2.01860e6	0.0998928
$837$	7.37159e6	0.363703
$838$	−1.43103e6	−0.0703947
$839$	1.96094e7	0.961742	0.480871	−	0.876791i	$-0.340320\pi$
$839$	1.96094e7	0.961742	0.480871	+	0.876791i	$0.340320\pi$
$840$	0	0
$841$	3.16068e7	1.54096
$842$	2.61002e6	0.126871
$843$	4.05735e7	1.96641
$844$	1.33846e6	0.0646768
$845$	0	0
$846$	1.92253e7	0.923523
$847$	−736303.	−0.0352653
$848$	285346.	0.0136264
$849$	2.70733e7	1.28906
$850$	0	0
$851$	−1.31129e6	−0.0620688
$852$	2.35180e6	0.110995
$853$	5.00288e6	0.235422	0.117711	−	0.993048i	$-0.462444\pi$
$853$	5.00288e6	0.235422	0.117711	+	0.993048i	$0.462444\pi$
$854$	2.83582e6	0.133056
$855$	0	0
$856$	3.01800e6	0.140778
$857$	−1.36218e7	−0.633553	−0.316776	−	0.948500i	$-0.602600\pi$
$857$	−1.36218e7	−0.633553	−0.316776	+	0.948500i	$0.602600\pi$
$858$	2.03729e7	0.944789
$859$	1.88600e7	0.872084	0.436042	−	0.899926i	$-0.356380\pi$
$859$	1.88600e7	0.872084	0.436042	+	0.899926i	$0.356380\pi$
$860$	0	0
$861$	1.20558e6	0.0554227
$862$	−2.50059e7	−1.14624
$863$	3.62744e7	1.65796	0.828979	−	0.559280i	$-0.188922\pi$
$863$	3.62744e7	1.65796	0.828979	+	0.559280i	$0.188922\pi$
$864$	1.39587e6	0.0636150
$865$	0	0
$866$	−1.78333e7	−0.808047
$867$	2.80715e7	1.26829
$868$	1.63712e6	0.0737535
$869$	1.26178e7	0.566804
$870$	0	0
$871$	−3.91061e7	−1.74662
$872$	3.35680e6	0.149498
$873$	−6.97166e6	−0.309600
$874$	956603.	0.0423597
$875$	0	0
$876$	2.12163e7	0.934132
$877$	1.47531e7	0.647715	0.323857	−	0.946106i	$-0.395020\pi$
$877$	1.47531e7	0.647715	0.323857	+	0.946106i	$0.395020\pi$
$878$	2.00261e7	0.876717
$879$	2.78138e7	1.21419
$880$	0	0
$881$	1.41746e7	0.615277	0.307638	−	0.951503i	$-0.400461\pi$
$881$	1.41746e7	0.615277	0.307638	+	0.951503i	$0.400461\pi$
$882$	−1.16091e7	−0.502489
$883$	−2.05570e7	−0.887276	−0.443638	−	0.896206i	$-0.646312\pi$
$883$	−2.05570e7	−0.887276	−0.443638	+	0.896206i	$0.646312\pi$
$884$	2.52535e6	0.108690
$885$	0	0
$886$	−2.86363e6	−0.122556
$887$	9.48962e6	0.404986	0.202493	−	0.979284i	$-0.435096\pi$
$887$	9.48962e6	0.404986	0.202493	+	0.979284i	$0.435096\pi$
$888$	2.59447e6	0.110412
$889$	−6.66473e6	−0.282832
$890$	0	0
$891$	−2.47407e7	−1.04404
$892$	1.06466e7	0.448022
$893$	9.83381e6	0.412661
$894$	8.51894e6	0.356485
$895$	0	0
$896$	310002.	0.0129001
$897$	9.65462e6	0.400640
$898$	−1.93273e7	−0.799800
$899$	−3.90401e7	−1.61106
$900$	0	0
$901$	−247228.	−0.0101458
$902$	−4.34909e6	−0.177984
$903$	123246.	0.00502984
$904$	−9.63814e6	−0.392258
$905$	0	0
$906$	3.91044e7	1.58272
$907$	−1.30672e7	−0.527428	−0.263714	−	0.964601i	$-0.584947\pi$
$907$	−1.30672e7	−0.527428	−0.263714	+	0.964601i	$0.584947\pi$
$908$	−1.84612e7	−0.743096
$909$	−2.95060e6	−0.118441
$910$	0	0
$911$	−4.60549e7	−1.83857	−0.919284	−	0.393594i	$-0.871232\pi$
$911$	−4.60549e7	−1.83857	−0.919284	+	0.393594i	$0.871232\pi$
$912$	−1.89270e6	−0.0753521
$913$	1.80970e7	0.718506
$914$	2.57764e7	1.02060
$915$	0	0
$916$	−6.93906e6	−0.273251
$917$	360828.	0.0141702
$918$	−1.20940e6	−0.0473656
$919$	−6.67836e6	−0.260844	−0.130422	−	0.991459i	$-0.541633\pi$
$919$	−6.67836e6	−0.260844	−0.130422	+	0.991459i	$0.541633\pi$
$920$	0	0
$921$	−2.53631e7	−0.985264
$922$	−1.64043e7	−0.635522
$923$	5.10717e6	0.197322
$924$	−2.16682e6	−0.0834915
$925$	0	0
$926$	−3.59473e7	−1.37765
$927$	−1.63498e7	−0.624903
$928$	−7.39254e6	−0.281789
$929$	−84737.5	−0.00322134	−0.00161067	−	0.999999i	$-0.500513\pi$
$929$	−84737.5	−0.00322134	−0.00161067	+	0.999999i	$0.500513\pi$
$930$	0	0
$931$	−5.93809e6	−0.224529
$932$	−6.19477e6	−0.233607
$933$	3.65706e7	1.37540
$934$	3.53763e7	1.32692
$935$	0	0
$936$	−8.03552e6	−0.299795
$937$	3.12820e7	1.16398	0.581989	−	0.813196i	$-0.302275\pi$
$937$	3.12820e7	1.16398	0.581989	+	0.813196i	$0.302275\pi$
$938$	4.15924e6	0.154350
$939$	−6.17701e7	−2.28620
$940$	0	0
$941$	2.88841e7	1.06337	0.531685	−	0.846942i	$-0.321559\pi$
$941$	2.88841e7	1.06337	0.531685	+	0.846942i	$0.321559\pi$
$942$	3.56320e7	1.30832
$943$	−2.06101e6	−0.0754746
$944$	−9.70367e6	−0.354410
$945$	0	0
$946$	−444607.	−0.0161528
$947$	−1.99925e7	−0.724422	−0.362211	−	0.932096i	$-0.617978\pi$
$947$	−1.99925e7	−0.724422	−0.362211	+	0.932096i	$0.617978\pi$
$948$	−1.18308e7	−0.427557
$949$	4.60732e7	1.66067
$950$	0	0
$951$	1.05645e7	0.378790
$952$	−268591.	−0.00960502
$953$	−5.22187e7	−1.86249	−0.931244	−	0.364396i	$-0.881276\pi$
$953$	−5.22187e7	−1.86249	−0.931244	+	0.364396i	$0.881276\pi$
$954$	786666.	0.0279846
$955$	0	0
$956$	−9.95941e6	−0.352443
$957$	5.16716e7	1.82378
$958$	1.15728e7	0.407403
$959$	5.04580e6	0.177167
$960$	0	0
$961$	614716.	0.0214717
$962$	5.63414e6	0.196286
$963$	8.32029e6	0.289116
$964$	−5.94327e6	−0.205984
$965$	0	0
$966$	−1.02684e6	−0.0354047
$967$	−2.66303e7	−0.915818	−0.457909	−	0.888999i	$-0.651402\pi$
$967$	−2.66303e7	−0.915818	−0.457909	+	0.888999i	$0.651402\pi$
$968$	−2.49053e6	−0.0854287
$969$	1.63987e6	0.0561047
$970$	0	0
$971$	1.04206e7	0.354686	0.177343	−	0.984149i	$-0.443250\pi$
$971$	1.04206e7	0.354686	0.177343	+	0.984149i	$0.443250\pi$
$972$	1.78977e7	0.607620
$973$	−5.57959e6	−0.188938
$974$	1.27267e7	0.429850
$975$	0	0
$976$	9.59212e6	0.322322
$977$	−1.82795e7	−0.612673	−0.306336	−	0.951923i	$-0.599103\pi$
$977$	−1.82795e7	−0.612673	−0.306336	+	0.951923i	$0.599103\pi$
$978$	−2.08966e7	−0.698601
$979$	5.07633e7	1.69275
$980$	0	0
$981$	9.25433e6	0.307024
$982$	−1.78204e7	−0.589709
$983$	−2.25159e7	−0.743199	−0.371600	−	0.928393i	$-0.621191\pi$
$983$	−2.25159e7	−0.743199	−0.371600	+	0.928393i	$0.621191\pi$
$984$	4.07785e6	0.134259
$985$	0	0
$986$	6.40502e6	0.209811
$987$	−1.05559e7	−0.344907
$988$	−4.11019e6	−0.133958
$989$	−210697.	−0.00684964
$990$	0	0
$991$	3.42068e7	1.10644	0.553221	−	0.833034i	$-0.313398\pi$
$991$	3.42068e7	1.10644	0.553221	+	0.833034i	$0.313398\pi$
$992$	5.53755e6	0.178664
$993$	4.61726e7	1.48597
$994$	−543187.	−0.0174375
$995$	0	0
$996$	−1.69684e7	−0.541990
$997$	1.12682e7	0.359018	0.179509	−	0.983756i	$-0.442549\pi$
$997$	1.12682e7	0.359018	0.179509	+	0.983756i	$0.442549\pi$
$998$	3.72589e7	1.18414
$999$	−2.69821e6	−0.0855387

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.6.a.d.1.1		2
5.4	even	2		38.6.a.c.1.2	✓	2
15.14	odd	2		342.6.a.i.1.1		2
20.19	odd	2		304.6.a.f.1.1		2
95.94	odd	2		722.6.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.6.a.c.1.2	✓	2	5.4	even	2
304.6.a.f.1.1		2	20.19	odd	2
342.6.a.i.1.1		2	15.14	odd	2
722.6.a.c.1.1		2	95.94	odd	2
950.6.a.d.1.1		2	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:	\( N \)	\(=\)	\( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	950.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} -20.4803 q^{3} +16.0000 q^{4} -81.9210 q^{6} +18.9210 q^{7} +64.0000 q^{8} +176.441 q^{9} +O(q^{10})\)	\(q+4.00000 q^{2} -20.4803 q^{3} +16.0000 q^{4} -81.9210 q^{6} +18.9210 q^{7} +64.0000 q^{8} +176.441 q^{9} +349.480 q^{11} -327.684 q^{12} -711.599 q^{13} +75.6840 q^{14} +256.000 q^{16} -221.803 q^{17} +705.763 q^{18} +361.000 q^{19} -387.507 q^{21} +1397.92 q^{22} +662.468 q^{23} -1310.74 q^{24} -2846.39 q^{26} +1363.15 q^{27} +302.736 q^{28} -7219.28 q^{29} +5407.76 q^{31} +1024.00 q^{32} -7157.44 q^{33} -887.210 q^{34} +2823.05 q^{36} -1979.40 q^{37} +1444.00 q^{38} +14573.7 q^{39} -3111.11 q^{41} -1550.03 q^{42} -318.049 q^{43} +5591.68 q^{44} +2649.87 q^{46} +27240.5 q^{47} -5242.94 q^{48} -16449.0 q^{49} +4542.57 q^{51} -11385.6 q^{52} +1114.63 q^{53} +5452.60 q^{54} +1210.94 q^{56} -7393.37 q^{57} -28877.1 q^{58} -37904.9 q^{59} +37469.2 q^{61} +21631.0 q^{62} +3338.44 q^{63} +4096.00 q^{64} -28629.8 q^{66} +54955.3 q^{67} -3548.84 q^{68} -13567.5 q^{69} -7177.04 q^{71} +11292.2 q^{72} -64746.1 q^{73} -7917.59 q^{74} +5776.00 q^{76} +6612.52 q^{77} +58294.9 q^{78} +36104.4 q^{79} -70792.8 q^{81} -12444.4 q^{82} +51782.7 q^{83} -6200.11 q^{84} -1272.20 q^{86} +147853. q^{87} +22366.7 q^{88} +145254. q^{89} -13464.2 q^{91} +10599.5 q^{92} -110752. q^{93} +108962. q^{94} -20971.8 q^{96} -39512.8 q^{97} -65796.0 q^{98} +61662.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2q + 8q^{2} - 3q^{3} + 32q^{4} - 12q^{6} - 114q^{7} + 128q^{8} + 239q^{9} + O(q^{10}) \)	\( 2q + 8q^{2} - 3q^{3} + 32q^{4} - 12q^{6} - 114q^{7} + 128q^{8} + 239q^{9} + 661q^{11} - 48q^{12} - 1613q^{13} - 456q^{14} + 512q^{16} - 64q^{17} + 956q^{18} + 722q^{19} - 2711q^{21} + 2644q^{22} + 3185q^{23} - 192q^{24} - 6452q^{26} - 1791q^{27} - 1824q^{28} - 2481q^{29} - 1180q^{31} + 2048q^{32} - 1712q^{33} - 256q^{34} + 3824q^{36} - 10488q^{37} + 2888q^{38} - 1183q^{39} + 16630q^{41} - 10844q^{42} - 11303q^{43} + 10576q^{44} + 12740q^{46} + 12155q^{47} - 768q^{48} - 15588q^{49} + 7301q^{51} - 25808q^{52} - 20585q^{53} - 7164q^{54} - 7296q^{56} - 1083q^{57} - 9924q^{58} - 78581q^{59} + 43621q^{61} - 4720q^{62} - 4977q^{63} + 8192q^{64} - 6848q^{66} - 7805q^{67} - 1024q^{68} + 30527q^{69} - 62488q^{71} + 15296q^{72} - 16218q^{73} - 41952q^{74} + 11552q^{76} - 34795q^{77} - 4732q^{78} + 67122q^{79} - 141130q^{81} + 66520q^{82} + 10714q^{83} - 43376q^{84} - 45212q^{86} + 230679q^{87} + 42304q^{88} + 128188q^{89} + 106351q^{91} + 50960q^{92} - 225908q^{93} + 48620q^{94} - 3072q^{96} - 178558q^{97} - 62352q^{98} + 81151q^{99} + O(q^{100}) \)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Related objects

Downloads

Learn more about