LMFDB - Embedded newform 950.6.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [950,6,Mod(1,950)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(950, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("950.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 950 = 2 \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	950.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:	$152.364628822$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{163}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 163 $ x^2 - 163
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 190)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$12.7671$ of defining polynomial
Character	$\chi$	$=$	950.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.00000 q^{2} +13.7671 q^{3} +16.0000 q^{4} +55.0686 q^{6} -26.9314 q^{7} +64.0000 q^{8} -53.4657 q^{9} +24.8772 q^{11} +220.274 q^{12} +681.301 q^{13} -107.726 q^{14} +256.000 q^{16} -2301.87 q^{17} -213.863 q^{18} +361.000 q^{19} -370.769 q^{21} +99.5088 q^{22} -1211.62 q^{23} +881.097 q^{24} +2725.21 q^{26} -4081.49 q^{27} -430.903 q^{28} -1513.81 q^{29} -5524.64 q^{31} +1024.00 q^{32} +342.488 q^{33} -9207.46 q^{34} -855.451 q^{36} -3737.03 q^{37} +1444.00 q^{38} +9379.58 q^{39} -6130.29 q^{41} -1483.08 q^{42} +174.491 q^{43} +398.035 q^{44} -4846.48 q^{46} +13027.8 q^{47} +3524.39 q^{48} -16081.7 q^{49} -31690.1 q^{51} +10900.8 q^{52} +13845.4 q^{53} -16325.9 q^{54} -1723.61 q^{56} +4969.94 q^{57} -6055.23 q^{58} -26866.2 q^{59} +14931.9 q^{61} -22098.6 q^{62} +1439.91 q^{63} +4096.00 q^{64} +1369.95 q^{66} +1693.58 q^{67} -36829.9 q^{68} -16680.6 q^{69} -16542.8 q^{71} -3421.81 q^{72} +44435.8 q^{73} -14948.1 q^{74} +5776.00 q^{76} -669.978 q^{77} +37518.3 q^{78} +4217.16 q^{79} -43198.3 q^{81} -24521.1 q^{82} +31576.6 q^{83} -5932.30 q^{84} +697.965 q^{86} -20840.8 q^{87} +1592.14 q^{88} -134369. q^{89} -18348.4 q^{91} -19385.9 q^{92} -76058.5 q^{93} +52111.3 q^{94} +14097.6 q^{96} +22484.7 q^{97} -64326.8 q^{98} -1330.08 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} + 2 q^{3} + 32 q^{4} + 8 q^{6} - 156 q^{7} + 128 q^{8} - 158 q^{9} - 512 q^{11} + 32 q^{12} + 1286 q^{13} - 624 q^{14} + 512 q^{16} - 1080 q^{17} - 632 q^{18} + 722 q^{19} + 1148 q^{21} - 2048 q^{22}+ \cdots + 54792 q^{99}+O(q^{100}) $ 2 * q + 8 * q^2 + 2 * q^3 + 32 * q^4 + 8 * q^6 - 156 * q^7 + 128 * q^8 - 158 * q^9 - 512 * q^11 + 32 * q^12 + 1286 * q^13 - 624 * q^14 + 512 * q^16 - 1080 * q^17 - 632 * q^18 + 722 * q^19 + 1148 * q^21 - 2048 * q^22 + 2224 * q^23 + 128 * q^24 + 5144 * q^26 + 8 * q^27 - 2496 * q^28 - 2568 * q^29 - 1908 * q^31 + 2048 * q^32 + 6660 * q^33 - 4320 * q^34 - 2528 * q^36 - 7806 * q^37 + 2888 * q^38 + 2264 * q^39 + 3060 * q^41 + 4592 * q^42 + 2596 * q^43 - 8192 * q^44 + 8896 * q^46 + 14412 * q^47 + 512 * q^48 - 16230 * q^49 - 46068 * q^51 + 20576 * q^52 + 9638 * q^53 + 32 * q^54 - 9984 * q^56 + 722 * q^57 - 10272 * q^58 - 43008 * q^59 - 25852 * q^61 - 7632 * q^62 + 14932 * q^63 + 8192 * q^64 + 26640 * q^66 + 27466 * q^67 - 17280 * q^68 - 57108 * q^69 - 87780 * q^71 - 10112 * q^72 + 25904 * q^73 - 31224 * q^74 + 11552 * q^76 + 68624 * q^77 + 9056 * q^78 - 3720 * q^79 - 65918 * q^81 + 12240 * q^82 - 38320 * q^83 + 18368 * q^84 + 10384 * q^86 - 8436 * q^87 - 32768 * q^88 - 167264 * q^89 - 96396 * q^91 + 35584 * q^92 - 118616 * q^93 + 57648 * q^94 + 2048 * q^96 + 4038 * q^97 - 64920 * q^98 + 54792 * q^99

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
$3$	13.7671	0.883163	0.441581	−	0.897221i	$-0.354418\pi$
$3$	13.7671	0.883163	0.441581	+	0.897221i	$0.354418\pi$
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	55.0686	0.624490
$7$	−26.9314	−0.207737	−0.103869	−	0.994591i	$-0.533122\pi$
$7$	−26.9314	−0.207737	−0.103869	+	0.994591i	$0.533122\pi$
$8$	64.0000	0.353553
$9$	−53.4657	−0.220023
$10$	0	0
$11$	24.8772	0.0619897	0.0309949	−	0.999520i	$-0.490132\pi$
$11$	24.8772	0.0619897	0.0309949	+	0.999520i	$0.490132\pi$
$12$	220.274	0.441581
$13$	681.301	1.11810	0.559050	−	0.829134i	$-0.311166\pi$
$13$	681.301	1.11810	0.559050	+	0.829134i	$0.311166\pi$
$14$	−107.726	−0.146892
$15$	0	0
$16$	256.000	0.250000
$17$	−2301.87	−1.93178	−0.965890	−	0.258952i	$-0.916623\pi$
$17$	−2301.87	−1.93178	−0.965890	+	0.258952i	$0.916623\pi$
$18$	−213.863	−0.155580
$19$	361.000	0.229416
$19$	361.000	0.229416
$20$	0	0
$21$	−370.769	−0.183466
$22$	99.5088	0.0438334
$23$	−1211.62	−0.477581	−0.238790	−	0.971071i	$-0.576751\pi$
$23$	−1211.62	−0.477581	−0.238790	+	0.971071i	$0.576751\pi$
$24$	881.097	0.312245
$25$	0	0
$26$	2725.21	0.790617
$27$	−4081.49	−1.07748
$28$	−430.903	−0.103869
$29$	−1513.81	−0.334254	−0.167127	−	0.985935i	$-0.553449\pi$
$29$	−1513.81	−0.334254	−0.167127	+	0.985935i	$0.553449\pi$
$30$	0	0
$31$	−5524.64	−1.03252	−0.516261	−	0.856431i	$-0.672677\pi$
$31$	−5524.64	−1.03252	−0.516261	+	0.856431i	$0.672677\pi$
$32$	1024.00	0.176777
$33$	342.488	0.0547470
$34$	−9207.46	−1.36597
$35$	0	0
$36$	−855.451	−0.110012
$37$	−3737.03	−0.448768	−0.224384	−	0.974501i	$-0.572037\pi$
$37$	−3737.03	−0.448768	−0.224384	+	0.974501i	$0.572037\pi$
$38$	1444.00	0.162221
$39$	9379.58	0.987465
$40$	0	0
$41$	−6130.29	−0.569536	−0.284768	−	0.958596i	$-0.591917\pi$
$41$	−6130.29	−0.569536	−0.284768	+	0.958596i	$0.591917\pi$
$42$	−1483.08	−0.129730
$43$	174.491	0.0143914	0.00719569	−	0.999974i	$-0.497710\pi$
$43$	174.491	0.0143914	0.00719569	+	0.999974i	$0.497710\pi$
$44$	398.035	0.0309949
$45$	0	0
$46$	−4846.48	−0.337701
$47$	13027.8	0.860255	0.430127	−	0.902768i	$-0.358469\pi$
$47$	13027.8	0.860255	0.430127	+	0.902768i	$0.358469\pi$
$48$	3524.39	0.220791
$49$	−16081.7	−0.956845
$50$	0	0
$51$	−31690.1	−1.70608
$52$	10900.8	0.559050
$53$	13845.4	0.677041	0.338520	−	0.940959i	$-0.390074\pi$
$53$	13845.4	0.677041	0.338520	+	0.940959i	$0.390074\pi$
$54$	−16325.9	−0.761893
$55$	0	0
$56$	−1723.61	−0.0734462
$57$	4969.94	0.202611
$58$	−6055.23	−0.236353
$59$	−26866.2	−1.00479	−0.502396	−	0.864638i	$-0.667548\pi$
$59$	−26866.2	−1.00479	−0.502396	+	0.864638i	$0.667548\pi$
$60$	0	0
$61$	14931.9	0.513796	0.256898	−	0.966439i	$-0.417300\pi$
$61$	14931.9	0.513796	0.256898	+	0.966439i	$0.417300\pi$
$62$	−22098.6	−0.730104
$63$	1439.91	0.0457070
$64$	4096.00	0.125000
$65$	0	0
$66$	1369.95	0.0387120
$67$	1693.58	0.0460913	0.0230457	−	0.999734i	$-0.492664\pi$
$67$	1693.58	0.0460913	0.0230457	+	0.999734i	$0.492664\pi$
$68$	−36829.9	−0.965890
$69$	−16680.6	−0.421782
$70$	0	0
$71$	−16542.8	−0.389460	−0.194730	−	0.980857i	$-0.562383\pi$
$71$	−16542.8	−0.389460	−0.194730	+	0.980857i	$0.562383\pi$
$72$	−3421.81	−0.0777901
$73$	44435.8	0.975946	0.487973	−	0.872859i	$-0.337736\pi$
$73$	44435.8	0.975946	0.487973	+	0.872859i	$0.337736\pi$
$74$	−14948.1	−0.317327
$75$	0	0
$76$	5776.00	0.114708
$77$	−669.978	−0.0128776
$78$	37518.3	0.698243
$79$	4217.16	0.0760243	0.0380122	−	0.999277i	$-0.487897\pi$
$79$	4217.16	0.0760243	0.0380122	+	0.999277i	$0.487897\pi$
$80$	0	0
$81$	−43198.3	−0.731566
$82$	−24521.1	−0.402723
$83$	31576.6	0.503119	0.251559	−	0.967842i	$-0.419057\pi$
$83$	31576.6	0.503119	0.251559	+	0.967842i	$0.419057\pi$
$84$	−5932.30	−0.0917329
$85$	0	0
$86$	697.965	0.0101762
$87$	−20840.8	−0.295200
$88$	1592.14	0.0219167
$89$	−134369.	−1.79814	−0.899069	−	0.437808i	$-0.855755\pi$
$89$	−134369.	−1.79814	−0.899069	+	0.437808i	$0.855755\pi$
$90$	0	0
$91$	−18348.4	−0.232271
$92$	−19385.9	−0.238790
$93$	−76058.5	−0.911886
$94$	52111.3	0.608292
$95$	0	0
$96$	14097.6	0.156123
$97$	22484.7	0.242638	0.121319	−	0.992614i	$-0.461288\pi$
$97$	22484.7	0.242638	0.121319	+	0.992614i	$0.461288\pi$
$98$	−64326.8	−0.676592
$99$	−1330.08	−0.0136392
$100$	0	0
$101$	−105949.	−1.03346	−0.516732	−	0.856147i	$-0.672852\pi$
$101$	−105949.	−1.03346	−0.516732	+	0.856147i	$0.672852\pi$
$102$	−126760.	−1.20638
$103$	−90421.6	−0.839806	−0.419903	−	0.907569i	$-0.637936\pi$
$103$	−90421.6	−0.839806	−0.419903	+	0.907569i	$0.637936\pi$
$104$	43603.3	0.395308
$105$	0	0
$106$	55381.5	0.478740
$107$	−1105.67	−0.00933613	−0.00466806	−	0.999989i	$-0.501486\pi$
$107$	−1105.67	−0.00933613	−0.00466806	+	0.999989i	$0.501486\pi$
$108$	−65303.8	−0.538740
$109$	−157244.	−1.26768	−0.633839	−	0.773465i	$-0.718522\pi$
$109$	−157244.	−1.26768	−0.633839	+	0.773465i	$0.718522\pi$
$110$	0	0
$111$	−51448.2	−0.396335
$112$	−6894.44	−0.0519343
$113$	−5592.71	−0.0412027	−0.0206014	−	0.999788i	$-0.506558\pi$
$113$	−5592.71	−0.0412027	−0.0206014	+	0.999788i	$0.506558\pi$
$114$	19879.8	0.143268
$115$	0	0
$116$	−24220.9	−0.167127
$117$	−36426.3	−0.246008
$118$	−107465.	−0.710495
$119$	61992.5	0.401303
$120$	0	0
$121$	−160432.	−0.996157
$122$	59727.6	0.363309
$123$	−84396.6	−0.502993
$124$	−88394.2	−0.516261
$125$	0	0
$126$	5759.63	0.0323198
$127$	−205906.	−1.13282	−0.566408	−	0.824125i	$-0.691667\pi$
$127$	−205906.	−1.13282	−0.566408	+	0.824125i	$0.691667\pi$
$128$	16384.0	0.0883883
$129$	2402.25	0.0127099
$130$	0	0
$131$	132812.	0.676176	0.338088	−	0.941115i	$-0.390220\pi$
$131$	132812.	0.676176	0.338088	+	0.941115i	$0.390220\pi$
$132$	5479.81	0.0273735
$133$	−9722.24	−0.0476582
$134$	6774.33	0.0325915
$135$	0	0
$136$	−147319.	−0.682987
$137$	386282.	1.75834	0.879171	−	0.476506i	$-0.158097\pi$
$137$	386282.	1.75834	0.879171	+	0.476506i	$0.158097\pi$
$138$	−66722.2	−0.298245
$139$	269554.	1.18334	0.591670	−	0.806180i	$-0.298469\pi$
$139$	269554.	1.18334	0.591670	+	0.806180i	$0.298469\pi$
$140$	0	0
$141$	179356.	0.759745
$142$	−66171.1	−0.275390
$143$	16948.9	0.0693108
$144$	−13687.2	−0.0550059
$145$	0	0
$146$	177743.	0.690098
$147$	−221399.	−0.845050
$148$	−59792.4	−0.224384
$149$	254035.	0.937407	0.468704	−	0.883355i	$-0.344721\pi$
$149$	254035.	0.937407	0.468704	+	0.883355i	$0.344721\pi$
$150$	0	0
$151$	−510194.	−1.82093	−0.910465	−	0.413586i	$-0.864276\pi$
$151$	−510194.	−1.82093	−0.910465	+	0.413586i	$0.864276\pi$
$152$	23104.0	0.0811107
$153$	123071.	0.425037
$154$	−2679.91	−0.00910582
$155$	0	0
$156$	150073.	0.493732
$157$	−279654.	−0.905465	−0.452733	−	0.891646i	$-0.649551\pi$
$157$	−279654.	−0.905465	−0.452733	+	0.891646i	$0.649551\pi$
$158$	16868.6	0.0537573
$159$	190611.	0.597937
$160$	0	0
$161$	32630.7	0.0992113
$162$	−172793.	−0.517295
$163$	120901.	0.356419	0.178210	−	0.983993i	$-0.442970\pi$
$163$	120901.	0.356419	0.178210	+	0.983993i	$0.442970\pi$
$164$	−98084.6	−0.284768
$165$	0	0
$166$	126307.	0.355759
$167$	−282045.	−0.782576	−0.391288	−	0.920268i	$-0.627970\pi$
$167$	−282045.	−0.782576	−0.391288	+	0.920268i	$0.627970\pi$
$168$	−23729.2	−0.0648649
$169$	92878.6	0.250149
$170$	0	0
$171$	−19301.1	−0.0504769
$172$	2791.86	0.00719569
$173$	−71570.3	−0.181810	−0.0909050	−	0.995860i	$-0.528976\pi$
$173$	−71570.3	−0.181810	−0.0909050	+	0.995860i	$0.528976\pi$
$174$	−83363.3	−0.208738
$175$	0	0
$176$	6368.56	0.0154974
$177$	−369871.	−0.887395
$178$	−537475.	−1.27148
$179$	447833.	1.04468	0.522340	−	0.852737i	$-0.325059\pi$
$179$	447833.	1.04468	0.522340	+	0.852737i	$0.325059\pi$
$180$	0	0
$181$	−386611.	−0.877157	−0.438578	−	0.898693i	$-0.644518\pi$
$181$	−386611.	−0.877157	−0.438578	+	0.898693i	$0.644518\pi$
$182$	−73393.7	−0.164240
$183$	205570.	0.453766
$184$	−77543.7	−0.168850
$185$	0	0
$186$	−304234.	−0.644800
$187$	−57264.0	−0.119751
$188$	208445.	0.430127
$189$	109920.	0.223832
$190$	0	0
$191$	−874928.	−1.73536	−0.867679	−	0.497125i	$-0.834389\pi$
$191$	−874928.	−1.73536	−0.867679	+	0.497125i	$0.834389\pi$
$192$	56390.2	0.110395
$193$	−458388.	−0.885809	−0.442904	−	0.896569i	$-0.646052\pi$
$193$	−458388.	−0.885809	−0.442904	+	0.896569i	$0.646052\pi$
$194$	89938.9	0.171571
$195$	0	0
$196$	−257307.	−0.478423
$197$	707081.	1.29809	0.649043	−	0.760752i	$-0.275170\pi$
$197$	707081.	1.29809	0.649043	+	0.760752i	$0.275170\pi$
$198$	−5320.31	−0.00964437
$199$	468851.	0.839271	0.419635	−	0.907693i	$-0.362158\pi$
$199$	468851.	0.839271	0.419635	+	0.907693i	$0.362158\pi$
$200$	0	0
$201$	23315.8	0.0407061
$202$	−423798.	−0.730770
$203$	40769.0	0.0694369
$204$	−507042.	−0.853038
$205$	0	0
$206$	−361686.	−0.593832
$207$	64780.1	0.105079
$208$	174413.	0.279525
$209$	8980.67	0.0142214
$210$	0	0
$211$	−296627.	−0.458674	−0.229337	−	0.973347i	$-0.573656\pi$
$211$	−296627.	−0.458674	−0.229337	+	0.973347i	$0.573656\pi$
$212$	221526.	0.338520
$213$	−227747.	−0.343956
$214$	−4422.69	−0.00660164
$215$	0	0
$216$	−261215.	−0.380946
$217$	148786.	0.214493
$218$	−628978.	−0.896384
$219$	611754.	0.861919
$220$	0	0
$221$	−1.56826e6	−2.15992
$222$	−205793.	−0.280251
$223$	−702180.	−0.945553	−0.472777	−	0.881182i	$-0.656748\pi$
$223$	−702180.	−0.945553	−0.472777	+	0.881182i	$0.656748\pi$
$224$	−27577.8	−0.0367231
$225$	0	0
$226$	−22370.8	−0.0291347
$227$	1.14364e6	1.47307	0.736535	−	0.676399i	$-0.236461\pi$
$227$	1.14364e6	1.47307	0.736535	+	0.676399i	$0.236461\pi$
$228$	79519.0	0.101306
$229$	−1.11140e6	−1.40050	−0.700248	−	0.713899i	$-0.746927\pi$
$229$	−1.11140e6	−1.40050	−0.700248	+	0.713899i	$0.746927\pi$
$230$	0	0
$231$	−9223.69	−0.0113730
$232$	−96883.8	−0.118176
$233$	1.08911e6	1.31426	0.657131	−	0.753776i	$-0.271770\pi$
$233$	1.08911e6	1.31426	0.657131	+	0.753776i	$0.271770\pi$
$234$	−145705.	−0.173954
$235$	0	0
$236$	−429859.	−0.502396
$237$	58058.3	0.0671418
$238$	247970.	0.283764
$239$	966815.	1.09483	0.547417	−	0.836860i	$-0.315611\pi$
$239$	966815.	1.09483	0.547417	+	0.836860i	$0.315611\pi$
$240$	0	0
$241$	−50065.8	−0.0555263	−0.0277631	−	0.999615i	$-0.508838\pi$
$241$	−50065.8	−0.0555263	−0.0277631	+	0.999615i	$0.508838\pi$
$242$	−641729.	−0.704390
$243$	397085.	0.431387
$244$	238911.	0.256898
$245$	0	0
$246$	−337586.	−0.355670
$247$	245950.	0.256510
$248$	−353577.	−0.365052
$249$	434720.	0.444336
$250$	0	0
$251$	717366.	0.718715	0.359357	−	0.933200i	$-0.382996\pi$
$251$	717366.	0.718715	0.359357	+	0.933200i	$0.382996\pi$
$252$	23038.5	0.0228535
$253$	−30141.7	−0.0296051
$254$	−823623.	−0.801022
$255$	0	0
$256$	65536.0	0.0625000
$257$	579824.	0.547600	0.273800	−	0.961787i	$-0.411719\pi$
$257$	579824.	0.547600	0.273800	+	0.961787i	$0.411719\pi$
$258$	9608.98	0.00898728
$259$	100643.	0.0932258
$260$	0	0
$261$	80936.9	0.0735436
$262$	531249.	0.478129
$263$	−42940.0	−0.0382801	−0.0191400	−	0.999817i	$-0.506093\pi$
$263$	−42940.0	−0.0382801	−0.0191400	+	0.999817i	$0.506093\pi$
$264$	21919.2	0.0193560
$265$	0	0
$266$	−38889.0	−0.0336994
$267$	−1.84987e6	−1.58805
$268$	27097.3	0.0230457
$269$	545240.	0.459417	0.229709	−	0.973259i	$-0.426223\pi$
$269$	545240.	0.459417	0.229709	+	0.973259i	$0.426223\pi$
$270$	0	0
$271$	−91509.4	−0.0756907	−0.0378454	−	0.999284i	$-0.512049\pi$
$271$	−91509.4	−0.0756907	−0.0378454	+	0.999284i	$0.512049\pi$
$272$	−589278.	−0.482945
$273$	−252605.	−0.205133
$274$	1.54513e6	1.24334
$275$	0	0
$276$	−266889.	−0.210891
$277$	−1.55374e6	−1.21669	−0.608343	−	0.793674i	$-0.708166\pi$
$277$	−1.55374e6	−1.21669	−0.608343	+	0.793674i	$0.708166\pi$
$278$	1.07822e6	0.836748
$279$	295379.	0.227179
$280$	0	0
$281$	−1.45769e6	−1.10129	−0.550643	−	0.834741i	$-0.685618\pi$
$281$	−1.45769e6	−1.10129	−0.550643	+	0.834741i	$0.685618\pi$
$282$	717423.	0.537221
$283$	−937012.	−0.695471	−0.347736	−	0.937593i	$-0.613049\pi$
$283$	−937012.	−0.695471	−0.347736	+	0.937593i	$0.613049\pi$
$284$	−264684.	−0.194730
$285$	0	0
$286$	67795.5	0.0490101
$287$	165097.	0.118314
$288$	−54748.9	−0.0388950
$289$	3.87873e6	2.73178
$290$	0	0
$291$	309551.	0.214289
$292$	710972.	0.487973
$293$	1.28030e6	0.871247	0.435624	−	0.900129i	$-0.356528\pi$
$293$	1.28030e6	0.871247	0.435624	+	0.900129i	$0.356528\pi$
$294$	−885596.	−0.597541
$295$	0	0
$296$	−239170.	−0.158663
$297$	−101536.	−0.0667927
$298$	1.01614e6	0.662847
$299$	−825479.	−0.533984
$300$	0	0
$301$	−4699.30	−0.00298962
$302$	−2.04078e6	−1.28759
$303$	−1.45862e6	−0.912717
$304$	92416.0	0.0573539
$305$	0	0
$306$	492284.	0.300547
$307$	1.32295e6	0.801120	0.400560	−	0.916271i	$-0.368816\pi$
$307$	1.32295e6	0.801120	0.400560	+	0.916271i	$0.368816\pi$
$308$	−10719.7	−0.00643878
$309$	−1.24485e6	−0.741685
$310$	0	0
$311$	−33244.6	−0.0194904	−0.00974520	−	0.999953i	$-0.503102\pi$
$311$	−33244.6	−0.0194904	−0.00974520	+	0.999953i	$0.503102\pi$
$312$	600293.	0.349122
$313$	−2.43889e6	−1.40712	−0.703561	−	0.710635i	$-0.748408\pi$
$313$	−2.43889e6	−1.40712	−0.703561	+	0.710635i	$0.748408\pi$
$314$	−1.11862e6	−0.640261
$315$	0	0
$316$	67474.6	0.0380122
$317$	582943.	0.325820	0.162910	−	0.986641i	$-0.447912\pi$
$317$	582943.	0.325820	0.162910	+	0.986641i	$0.447912\pi$
$318$	762445.	0.422806
$319$	−37659.3	−0.0207203
$320$	0	0
$321$	−15221.9	−0.00824532
$322$	130523.	0.0701530
$323$	−830974.	−0.443181
$324$	−691172.	−0.365783
$325$	0	0
$326$	483604.	0.252026
$327$	−2.16481e6	−1.11957
$328$	−392338.	−0.201361
$329$	−350858.	−0.178707
$330$	0	0
$331$	1.40961e6	0.707177	0.353588	−	0.935401i	$-0.384961\pi$
$331$	1.40961e6	0.707177	0.353588	+	0.935401i	$0.384961\pi$
$332$	505226.	0.251559
$333$	199803.	0.0987395
$334$	−1.12818e6	−0.553365
$335$	0	0
$336$	−94916.8	−0.0458664
$337$	−3.44544e6	−1.65261	−0.826304	−	0.563225i	$-0.809560\pi$
$337$	−3.44544e6	−1.65261	−0.826304	+	0.563225i	$0.809560\pi$
$338$	371515.	0.176882
$339$	−76995.6	−0.0363887
$340$	0	0
$341$	−137438.	−0.0640058
$342$	−77204.5	−0.0356925
$343$	885739.	0.406509
$344$	11167.4	0.00508812
$345$	0	0
$346$	−286281.	−0.128559
$347$	−4.33057e6	−1.93073	−0.965365	−	0.260901i	$-0.915980\pi$
$347$	−4.33057e6	−1.93073	−0.965365	+	0.260901i	$0.915980\pi$
$348$	−333453.	−0.147600
$349$	2.92386e6	1.28497	0.642485	−	0.766298i	$-0.277903\pi$
$349$	2.92386e6	1.28497	0.642485	+	0.766298i	$0.277903\pi$
$350$	0	0
$351$	−2.78072e6	−1.20473
$352$	25474.3	0.0109583
$353$	2.91262e6	1.24407	0.622037	−	0.782988i	$-0.286305\pi$
$353$	2.91262e6	1.24407	0.622037	+	0.782988i	$0.286305\pi$
$354$	−1.47948e6	−0.627483
$355$	0	0
$356$	−2.14990e6	−0.899069
$357$	853460.	0.354415
$358$	1.79133e6	0.738700
$359$	−1.69303e6	−0.693311	−0.346655	−	0.937993i	$-0.612683\pi$
$359$	−1.69303e6	−0.693311	−0.346655	+	0.937993i	$0.612683\pi$
$360$	0	0
$361$	130321.	0.0526316
$362$	−1.54644e6	−0.620244
$363$	−2.20869e6	−0.879769
$364$	−293575.	−0.116136
$365$	0	0
$366$	822279.	0.320861
$367$	1.79156e6	0.694330	0.347165	−	0.937804i	$-0.387144\pi$
$367$	1.79156e6	0.694330	0.347165	+	0.937804i	$0.387144\pi$
$368$	−310175.	−0.119395
$369$	327760.	0.125311
$370$	0	0
$371$	−372876.	−0.140647
$372$	−1.21694e6	−0.455943
$373$	3.22590e6	1.20055	0.600273	−	0.799795i	$-0.295059\pi$
$373$	3.22590e6	1.20055	0.600273	+	0.799795i	$0.295059\pi$
$374$	−229056.	−0.0846764
$375$	0	0
$376$	833780.	0.304146
$377$	−1.03136e6	−0.373729
$378$	439681.	0.158273
$379$	−2.37691e6	−0.849991	−0.424995	−	0.905195i	$-0.639724\pi$
$379$	−2.37691e6	−0.849991	−0.424995	+	0.905195i	$0.639724\pi$
$380$	0	0
$381$	−2.83474e6	−1.00046
$382$	−3.49971e6	−1.22708
$383$	−627187.	−0.218474	−0.109237	−	0.994016i	$-0.534841\pi$
$383$	−627187.	−0.218474	−0.109237	+	0.994016i	$0.534841\pi$
$384$	225561.	0.0780613
$385$	0	0
$386$	−1.83355e6	−0.626362
$387$	−9329.30	−0.00316644
$388$	359756.	0.121319
$389$	4.19629e6	1.40602	0.703010	−	0.711180i	$-0.251839\pi$
$389$	4.19629e6	1.40602	0.703010	+	0.711180i	$0.251839\pi$
$390$	0	0
$391$	2.78899e6	0.922582
$392$	−1.02923e6	−0.338296
$393$	1.82844e6	0.597173
$394$	2.82832e6	0.917885
$395$	0	0
$396$	−21281.2	−0.00681960
$397$	−4.76904e6	−1.51864	−0.759320	−	0.650718i	$-0.774468\pi$
$397$	−4.76904e6	−1.51864	−0.759320	+	0.650718i	$0.774468\pi$
$398$	1.87540e6	0.593454
$399$	−133848.	−0.0420899
$400$	0	0
$401$	1.58560e6	0.492416	0.246208	−	0.969217i	$-0.420815\pi$
$401$	1.58560e6	0.492416	0.246208	+	0.969217i	$0.420815\pi$
$402$	93263.2	0.0287836
$403$	−3.76394e6	−1.15446
$404$	−1.69519e6	−0.516732
$405$	0	0
$406$	163076.	0.0490993
$407$	−92966.8	−0.0278190
$408$	−2.02817e6	−0.603189
$409$	1.19815e6	0.354163	0.177081	−	0.984196i	$-0.443334\pi$
$409$	1.19815e6	0.354163	0.177081	+	0.984196i	$0.443334\pi$
$410$	0	0
$411$	5.31800e6	1.55290
$412$	−1.44675e6	−0.419903
$413$	723545.	0.208733
$414$	259121.	0.0743021
$415$	0	0
$416$	697653.	0.197654
$417$	3.71100e6	1.04508
$418$	35922.7	0.0100561
$419$	−2.21392e6	−0.616066	−0.308033	−	0.951376i	$-0.599671\pi$
$419$	−2.21392e6	−0.616066	−0.308033	+	0.951376i	$0.599671\pi$
$420$	0	0
$421$	−3.93507e6	−1.08205	−0.541025	−	0.841007i	$-0.681964\pi$
$421$	−3.93507e6	−1.08205	−0.541025	+	0.841007i	$0.681964\pi$
$422$	−1.18651e6	−0.324331
$423$	−696542.	−0.189276
$424$	886104.	0.239370
$425$	0	0
$426$	−910987.	−0.243214
$427$	−402138.	−0.106735
$428$	−17690.7	−0.00466806
$429$	233338.	0.0612127
$430$	0	0
$431$	5.66370e6	1.46861	0.734307	−	0.678818i	$-0.237507\pi$
$431$	5.66370e6	1.46861	0.734307	+	0.678818i	$0.237507\pi$
$432$	−1.04486e6	−0.269370
$433$	3.10634e6	0.796213	0.398106	−	0.917339i	$-0.369667\pi$
$433$	3.10634e6	0.796213	0.398106	+	0.917339i	$0.369667\pi$
$434$	595145.	0.151670
$435$	0	0
$436$	−2.51591e6	−0.633839
$437$	−437395.	−0.109565
$438$	2.44702e6	0.609469
$439$	6.46172e6	1.60025	0.800123	−	0.599836i	$-0.204768\pi$
$439$	6.46172e6	1.60025	0.800123	+	0.599836i	$0.204768\pi$
$440$	0	0
$441$	859819.	0.210528
$442$	−6.27306e6	−1.52730
$443$	−59065.0	−0.0142995	−0.00714975	−	0.999974i	$-0.502276\pi$
$443$	−59065.0	−0.0142995	−0.00714975	+	0.999974i	$0.502276\pi$
$444$	−823171.	−0.198168
$445$	0	0
$446$	−2.80872e6	−0.668607
$447$	3.49734e6	0.827883
$448$	−110311.	−0.0259671
$449$	−8.12621e6	−1.90227	−0.951135	−	0.308776i	$-0.900081\pi$
$449$	−8.12621e6	−1.90227	−0.951135	+	0.308776i	$0.900081\pi$
$450$	0	0
$451$	−152504.	−0.0353054
$452$	−89483.3	−0.0206014
$453$	−7.02392e6	−1.60818
$454$	4.57455e6	1.04162
$455$	0	0
$456$	318076.	0.0716340
$457$	3.23112e6	0.723707	0.361853	−	0.932235i	$-0.382144\pi$
$457$	3.23112e6	0.723707	0.361853	+	0.932235i	$0.382144\pi$
$458$	−4.44560e6	−0.990301
$459$	9.39504e6	2.08145
$460$	0	0
$461$	−5.92201e6	−1.29783	−0.648913	−	0.760862i	$-0.724776\pi$
$461$	−5.92201e6	−1.29783	−0.648913	+	0.760862i	$0.724776\pi$
$462$	−36894.7	−0.00804192
$463$	3.20431e6	0.694675	0.347337	−	0.937740i	$-0.387086\pi$
$463$	3.20431e6	0.694675	0.347337	+	0.937740i	$0.387086\pi$
$464$	−387535.	−0.0835634
$465$	0	0
$466$	4.35644e6	0.929324
$467$	−3.05936e6	−0.649140	−0.324570	−	0.945862i	$-0.605220\pi$
$467$	−3.05936e6	−0.649140	−0.324570	+	0.945862i	$0.605220\pi$
$468$	−582820.	−0.123004
$469$	−45610.6	−0.00957488
$470$	0	0
$471$	−3.85004e6	−0.799673
$472$	−1.71944e6	−0.355248
$473$	4340.85	0.000892118 0
$474$	232233.	0.0474764
$475$	0	0
$476$	991880.	0.200651
$477$	−740253.	−0.148965
$478$	3.86726e6	0.774165
$479$	−74969.9	−0.0149296	−0.00746480	−	0.999972i	$-0.502376\pi$
$479$	−74969.9	−0.0149296	−0.00746480	+	0.999972i	$0.502376\pi$
$480$	0	0
$481$	−2.54604e6	−0.501768
$482$	−200263.	−0.0392630
$483$	449231.	0.0876197
$484$	−2.56691e6	−0.498079
$485$	0	0
$486$	1.58834e6	0.305037
$487$	−1.37282e6	−0.262297	−0.131148	−	0.991363i	$-0.541866\pi$
$487$	−1.37282e6	−0.262297	−0.131148	+	0.991363i	$0.541866\pi$
$488$	955642.	0.181654
$489$	1.66446e6	0.314776
$490$	0	0
$491$	−2.24035e6	−0.419384	−0.209692	−	0.977767i	$-0.567246\pi$
$491$	−2.24035e6	−0.419384	−0.209692	+	0.977767i	$0.567246\pi$
$492$	−1.35034e6	−0.251496
$493$	3.48458e6	0.645704
$494$	983799.	0.181380
$495$	0	0
$496$	−1.41431e6	−0.258131
$497$	445520.	0.0809052
$498$	1.73888e6	0.314193
$499$	−6.25455e6	−1.12446	−0.562231	−	0.826980i	$-0.690057\pi$
$499$	−6.25455e6	−1.12446	−0.562231	+	0.826980i	$0.690057\pi$
$500$	0	0
$501$	−3.88295e6	−0.691142
$502$	2.86947e6	0.508208
$503$	7.83485e6	1.38074	0.690368	−	0.723459i	$-0.257449\pi$
$503$	7.83485e6	1.38074	0.690368	+	0.723459i	$0.257449\pi$
$504$	92154.1	0.0161599
$505$	0	0
$506$	−120567.	−0.0209340
$507$	1.27867e6	0.220922
$508$	−3.29449e6	−0.566408
$509$	−5.46950e6	−0.935736	−0.467868	−	0.883798i	$-0.654978\pi$
$509$	−5.46950e6	−0.935736	−0.467868	+	0.883798i	$0.654978\pi$
$510$	0	0
$511$	−1.19672e6	−0.202740
$512$	262144.	0.0441942
$513$	−1.47342e6	−0.247191
$514$	2.31929e6	0.387211
$515$	0	0
$516$	38435.9	0.00635497
$517$	324096.	0.0533270
$518$	402574.	0.0659206
$519$	−985319.	−0.160568
$520$	0	0
$521$	−3.58095e6	−0.577968	−0.288984	−	0.957334i	$-0.593317\pi$
$521$	−3.58095e6	−0.577968	−0.288984	+	0.957334i	$0.593317\pi$
$522$	323747.	0.0520032
$523$	−2.71881e6	−0.434634	−0.217317	−	0.976101i	$-0.569731\pi$
$523$	−2.71881e6	−0.434634	−0.217317	+	0.976101i	$0.569731\pi$
$524$	2.12499e6	0.338088
$525$	0	0
$526$	−171760.	−0.0270681
$527$	1.27170e7	1.99461
$528$	87676.9	0.0136868
$529$	−4.96832e6	−0.771916
$530$	0	0
$531$	1.43642e6	0.221078
$532$	−155556.	−0.0238291
$533$	−4.17657e6	−0.636799
$534$	−7.39949e6	−1.12292
$535$	0	0
$536$	108389.	0.0162957
$537$	6.16538e6	0.922623
$538$	2.18096e6	0.324857
$539$	−400068.	−0.0593146
$540$	0	0
$541$	2.46153e6	0.361587	0.180793	−	0.983521i	$-0.442133\pi$
$541$	2.46153e6	0.361587	0.180793	+	0.983521i	$0.442133\pi$
$542$	−366038.	−0.0535214
$543$	−5.32252e6	−0.774672
$544$	−2.35711e6	−0.341494
$545$	0	0
$546$	−1.01042e6	−0.145051
$547$	1.04612e7	1.49490	0.747449	−	0.664319i	$-0.231278\pi$
$547$	1.04612e7	1.49490	0.747449	+	0.664319i	$0.231278\pi$
$548$	6.18052e6	0.879171
$549$	−798345.	−0.113047
$550$	0	0
$551$	−546485.	−0.0766830
$552$	−1.06756e6	−0.149122
$553$	−113574.	−0.0157931
$554$	−6.21496e6	−0.860327
$555$	0	0
$556$	4.31287e6	0.591670
$557$	498171.	0.0680363	0.0340181	−	0.999421i	$-0.489170\pi$
$557$	498171.	0.0680363	0.0340181	+	0.999421i	$0.489170\pi$
$558$	1.18151e6	0.160640
$559$	118881.	0.0160910
$560$	0	0
$561$	−788361.	−0.105759
$562$	−5.83077e6	−0.778727
$563$	5.45739e6	0.725628	0.362814	−	0.931862i	$-0.381816\pi$
$563$	5.45739e6	0.725628	0.362814	+	0.931862i	$0.381816\pi$
$564$	2.86969e6	0.379872
$565$	0	0
$566$	−3.74805e6	−0.491772
$567$	1.16339e6	0.151973
$568$	−1.05874e6	−0.137695
$569$	−3.58354e6	−0.464015	−0.232007	−	0.972714i	$-0.574529\pi$
$569$	−3.58354e6	−0.464015	−0.232007	+	0.972714i	$0.574529\pi$
$570$	0	0
$571$	−353458.	−0.0453678	−0.0226839	−	0.999743i	$-0.507221\pi$
$571$	−353458.	−0.0453678	−0.0226839	+	0.999743i	$0.507221\pi$
$572$	271182.	0.0346554
$573$	−1.20453e7	−1.53260
$574$	660389.	0.0836605
$575$	0	0
$576$	−218996.	−0.0275029
$577$	8.54711e6	1.06876	0.534379	−	0.845245i	$-0.320545\pi$
$577$	8.54711e6	1.06876	0.534379	+	0.845245i	$0.320545\pi$
$578$	1.55149e7	1.93166
$579$	−6.31070e6	−0.782314
$580$	0	0
$581$	−850404.	−0.104516
$582$	1.23820e6	0.151525
$583$	344434.	0.0419696
$584$	2.84389e6	0.345049
$585$	0	0
$586$	5.12119e6	0.616065
$587$	7.33730e6	0.878903	0.439452	−	0.898266i	$-0.355173\pi$
$587$	7.33730e6	0.878903	0.439452	+	0.898266i	$0.355173\pi$
$588$	−3.54239e6	−0.422525
$589$	−1.99439e6	−0.236877
$590$	0	0
$591$	9.73448e6	1.14642
$592$	−956679.	−0.112192
$593$	636290.	0.0743050	0.0371525	−	0.999310i	$-0.488171\pi$
$593$	636290.	0.0743050	0.0371525	+	0.999310i	$0.488171\pi$
$594$	−406144.	−0.0472295
$595$	0	0
$596$	4.06456e6	0.468704
$597$	6.45474e6	0.741213
$598$	−3.30192e6	−0.377583
$599$	5.64075e6	0.642347	0.321174	−	0.947020i	$-0.395923\pi$
$599$	5.64075e6	0.642347	0.321174	+	0.947020i	$0.395923\pi$
$600$	0	0
$601$	−1.73832e6	−0.196310	−0.0981551	−	0.995171i	$-0.531294\pi$
$601$	−1.73832e6	−0.196310	−0.0981551	+	0.995171i	$0.531294\pi$
$602$	−18797.2	−0.00211398
$603$	−90548.6	−0.0101412
$604$	−8.16311e6	−0.910465
$605$	0	0
$606$	−5.83449e6	−0.645389
$607$	140440.	0.0154710	0.00773550	−	0.999970i	$-0.497538\pi$
$607$	140440.	0.0154710	0.00773550	+	0.999970i	$0.497538\pi$
$608$	369664.	0.0405554
$609$	561273.	0.0613241
$610$	0	0
$611$	8.87587e6	0.961851
$612$	1.96913e6	0.212519
$613$	889627.	0.0956217	0.0478109	−	0.998856i	$-0.484776\pi$
$613$	889627.	0.0956217	0.0478109	+	0.998856i	$0.484776\pi$
$614$	5.29180e6	0.566477
$615$	0	0
$616$	−42878.6	−0.00455291
$617$	1.53129e7	1.61937	0.809684	−	0.586866i	$-0.199638\pi$
$617$	1.53129e7	1.61937	0.809684	+	0.586866i	$0.199638\pi$
$618$	−4.97939e6	−0.524451
$619$	1.49975e7	1.57323	0.786614	−	0.617445i	$-0.211832\pi$
$619$	1.49975e7	1.57323	0.786614	+	0.617445i	$0.211832\pi$
$620$	0	0
$621$	4.94521e6	0.514584
$622$	−132979.	−0.0137818
$623$	3.61874e6	0.373540
$624$	2.40117e6	0.246866
$625$	0	0
$626$	−9.75556e6	−0.994985
$627$	123638.	0.0125598
$628$	−4.47446e6	−0.452733
$629$	8.60214e6	0.866921
$630$	0	0
$631$	1.56254e7	1.56228	0.781138	−	0.624358i	$-0.214639\pi$
$631$	1.56254e7	1.56228	0.781138	+	0.624358i	$0.214639\pi$
$632$	269898.	0.0268787
$633$	−4.08370e6	−0.405084
$634$	2.33177e6	0.230390
$635$	0	0
$636$	3.04978e6	0.298969
$637$	−1.09565e7	−1.06985
$638$	−150637.	−0.0146515
$639$	884471.	0.0856903
$640$	0	0
$641$	8.70192e6	0.836508	0.418254	−	0.908330i	$-0.362642\pi$
$641$	8.70192e6	0.836508	0.418254	+	0.908330i	$0.362642\pi$
$642$	−60887.8	−0.00583032
$643$	−1.65258e7	−1.57628	−0.788141	−	0.615495i	$-0.788956\pi$
$643$	−1.65258e7	−1.57628	−0.788141	+	0.615495i	$0.788956\pi$
$644$	522091.	0.0496056
$645$	0	0
$646$	−3.32389e6	−0.313376
$647$	1.50588e7	1.41426	0.707129	−	0.707085i	$-0.249990\pi$
$647$	1.50588e7	1.41426	0.707129	+	0.707085i	$0.249990\pi$
$648$	−2.76469e6	−0.258648
$649$	−668356.	−0.0622868
$650$	0	0
$651$	2.04836e6	0.189432
$652$	1.93442e6	0.178210
$653$	−7.93953e6	−0.728638	−0.364319	−	0.931274i	$-0.618698\pi$
$653$	−7.93953e6	−0.728638	−0.364319	+	0.931274i	$0.618698\pi$
$654$	−8.65923e6	−0.791653
$655$	0	0
$656$	−1.56935e6	−0.142384
$657$	−2.37579e6	−0.214731
$658$	−1.40343e6	−0.126365
$659$	3.69910e6	0.331805	0.165902	−	0.986142i	$-0.446946\pi$
$659$	3.69910e6	0.331805	0.165902	+	0.986142i	$0.446946\pi$
$660$	0	0
$661$	−1.18580e7	−1.05562	−0.527809	−	0.849363i	$-0.676986\pi$
$661$	−1.18580e7	−1.05562	−0.527809	+	0.849363i	$0.676986\pi$
$662$	5.63843e6	0.500050
$663$	−2.15905e7	−1.90757
$664$	2.02090e6	0.177879
$665$	0	0
$666$	799211.	0.0698194
$667$	1.83416e6	0.159633
$668$	−4.51271e6	−0.391288
$669$	−9.66701e6	−0.835078
$670$	0	0
$671$	371464.	0.0318501
$672$	−379667.	−0.0324325
$673$	5.02558e6	0.427709	0.213855	−	0.976865i	$-0.431398\pi$
$673$	5.02558e6	0.427709	0.213855	+	0.976865i	$0.431398\pi$
$674$	−1.37818e7	−1.16857
$675$	0	0
$676$	1.48606e6	0.125075
$677$	1.10904e7	0.929981	0.464990	−	0.885316i	$-0.346058\pi$
$677$	1.10904e7	0.929981	0.464990	+	0.885316i	$0.346058\pi$
$678$	−307983.	−0.0257307
$679$	−605546.	−0.0504049
$680$	0	0
$681$	1.57446e7	1.30096
$682$	−549750.	−0.0452589
$683$	−2.15629e7	−1.76871	−0.884355	−	0.466816i	$-0.845401\pi$
$683$	−2.15629e7	−1.76871	−0.884355	+	0.466816i	$0.845401\pi$
$684$	−308818.	−0.0252384
$685$	0	0
$686$	3.54296e6	0.287446
$687$	−1.53008e7	−1.23687
$688$	44669.7	0.00359784
$689$	9.43287e6	0.757000
$690$	0	0
$691$	1.80076e6	0.143470	0.0717351	−	0.997424i	$-0.477146\pi$
$691$	1.80076e6	0.143470	0.0717351	+	0.997424i	$0.477146\pi$
$692$	−1.14512e6	−0.0909050
$693$	35820.9	0.00283337
$694$	−1.73223e7	−1.36523
$695$	0	0
$696$	−1.33381e6	−0.104369
$697$	1.41111e7	1.10022
$698$	1.16954e7	0.908611
$699$	1.49939e7	1.16071
$700$	0	0
$701$	1.83428e7	1.40984	0.704922	−	0.709285i	$-0.250982\pi$
$701$	1.83428e7	1.40984	0.704922	+	0.709285i	$0.250982\pi$
$702$	−1.11229e7	−0.851873
$703$	−1.34907e6	−0.102954
$704$	101897.	0.00774872
$705$	0	0
$706$	1.16505e7	0.879694
$707$	2.85337e6	0.214689
$708$	−5.91793e6	−0.443698
$709$	−637036.	−0.0475935	−0.0237968	−	0.999717i	$-0.507575\pi$
$709$	−637036.	−0.0475935	−0.0237968	+	0.999717i	$0.507575\pi$
$710$	0	0
$711$	−225474.	−0.0167271
$712$	−8.59959e6	−0.635738
$713$	6.69376e6	0.493113
$714$	3.41384e6	0.250610
$715$	0	0
$716$	7.16533e6	0.522340
$717$	1.33103e7	0.966917
$718$	−6.77211e6	−0.490245
$719$	−4.55484e6	−0.328587	−0.164294	−	0.986411i	$-0.552534\pi$
$719$	−4.55484e6	−0.328587	−0.164294	+	0.986411i	$0.552534\pi$
$720$	0	0
$721$	2.43518e6	0.174459
$722$	521284.	0.0372161
$723$	−689263.	−0.0490387
$724$	−6.18577e6	−0.438578
$725$	0	0
$726$	−8.83477e6	−0.622091
$727$	2.66974e7	1.87341	0.936704	−	0.350121i	$-0.113860\pi$
$727$	2.66974e7	1.87341	0.936704	+	0.350121i	$0.113860\pi$
$728$	−1.17430e6	−0.0821202
$729$	1.59639e7	1.11255
$730$	0	0
$731$	−401655.	−0.0278010
$732$	3.28912e6	0.226883
$733$	1.38386e7	0.951332	0.475666	−	0.879626i	$-0.342207\pi$
$733$	1.38386e7	0.951332	0.475666	+	0.879626i	$0.342207\pi$
$734$	7.16624e6	0.490965
$735$	0	0
$736$	−1.24070e6	−0.0844252
$737$	42131.6	0.00285719
$738$	1.31104e6	0.0886085
$739$	2.18127e7	1.46926	0.734629	−	0.678469i	$-0.237356\pi$
$739$	2.18127e7	1.46926	0.734629	+	0.678469i	$0.237356\pi$
$740$	0	0
$741$	3.38603e6	0.226540
$742$	−1.49150e6	−0.0994521
$743$	1.18744e7	0.789112	0.394556	−	0.918872i	$-0.370898\pi$
$743$	1.18744e7	0.789112	0.394556	+	0.918872i	$0.370898\pi$
$744$	−4.86774e6	−0.322400
$745$	0	0
$746$	1.29036e7	0.848914
$747$	−1.68827e6	−0.110698
$748$	−916224.	−0.0598753
$749$	29777.3	0.00193946
$750$	0	0
$751$	−4.30891e6	−0.278784	−0.139392	−	0.990237i	$-0.544515\pi$
$751$	−4.30891e6	−0.278784	−0.139392	+	0.990237i	$0.544515\pi$
$752$	3.33512e6	0.215064
$753$	9.87609e6	0.634742
$754$	−4.12544e6	−0.264266
$755$	0	0
$756$	1.75872e6	0.111916
$757$	−2.74658e7	−1.74202	−0.871008	−	0.491269i	$-0.836533\pi$
$757$	−2.74658e7	−1.74202	−0.871008	+	0.491269i	$0.836533\pi$
$758$	−9.50763e6	−0.601034
$759$	−414965.	−0.0261461
$760$	0	0
$761$	−6.72569e6	−0.420993	−0.210497	−	0.977595i	$-0.567508\pi$
$761$	−6.72569e6	−0.420993	−0.210497	+	0.977595i	$0.567508\pi$
$762$	−1.13389e7	−0.707433
$763$	4.23482e6	0.263344
$764$	−1.39989e7	−0.867679
$765$	0	0
$766$	−2.50875e6	−0.154485
$767$	−1.83040e7	−1.12346
$768$	902244.	0.0551977
$769$	9.60352e6	0.585618	0.292809	−	0.956171i	$-0.405410\pi$
$769$	9.60352e6	0.585618	0.292809	+	0.956171i	$0.405410\pi$
$770$	0	0
$771$	7.98252e6	0.483619
$772$	−7.33421e6	−0.442904
$773$	−1.43535e7	−0.863991	−0.431995	−	0.901876i	$-0.642190\pi$
$773$	−1.43535e7	−0.863991	−0.431995	+	0.901876i	$0.642190\pi$
$774$	−37317.2	−0.00223901
$775$	0	0
$776$	1.43902e6	0.0857854
$777$	1.38557e6	0.0823335
$778$	1.67852e7	0.994206
$779$	−2.21303e6	−0.130661
$780$	0	0
$781$	−411538.	−0.0241425
$782$	1.11560e7	0.652364
$783$	6.17859e6	0.360151
$784$	−4.11691e6	−0.239211
$785$	0	0
$786$	7.31378e6	0.422265
$787$	7.07627e6	0.407256	0.203628	−	0.979048i	$-0.434727\pi$
$787$	7.07627e6	0.407256	0.203628	+	0.979048i	$0.434727\pi$
$788$	1.13133e7	0.649043
$789$	−591162.	−0.0338076
$790$	0	0
$791$	150620.	0.00855934
$792$	−85124.9	−0.00482218
$793$	1.01731e7	0.574476
$794$	−1.90762e7	−1.07384
$795$	0	0
$796$	7.50162e6	0.419635
$797$	−1.09539e7	−0.610836	−0.305418	−	0.952218i	$-0.598796\pi$
$797$	−1.09539e7	−0.610836	−0.305418	+	0.952218i	$0.598796\pi$
$798$	−535390.	−0.0297621
$799$	−2.99883e7	−1.66182
$800$	0	0
$801$	7.18411e6	0.395632
$802$	6.34239e6	0.348190
$803$	1.10544e6	0.0604986
$804$	373053.	0.0203531
$805$	0	0
$806$	−1.50558e7	−0.816330
$807$	7.50640e6	0.405740
$808$	−6.78077e6	−0.365385
$809$	1.06616e7	0.572733	0.286366	−	0.958120i	$-0.407553\pi$
$809$	1.06616e7	0.572733	0.286366	+	0.958120i	$0.407553\pi$
$810$	0	0
$811$	2.38087e7	1.27111	0.635556	−	0.772055i	$-0.280771\pi$
$811$	2.38087e7	1.27111	0.635556	+	0.772055i	$0.280771\pi$
$812$	652304.	0.0347184
$813$	−1.25982e6	−0.0668472
$814$	−371867.	−0.0196710
$815$	0	0
$816$	−8.11267e6	−0.426519
$817$	62991.3	0.00330161
$818$	4.79260e6	0.250431
$819$	981011.	0.0511051
$820$	0	0
$821$	−1.11417e7	−0.576889	−0.288444	−	0.957497i	$-0.593138\pi$
$821$	−1.11417e7	−0.576889	−0.288444	+	0.957497i	$0.593138\pi$
$822$	2.12720e7	1.09807
$823$	5.60804e6	0.288610	0.144305	−	0.989533i	$-0.453905\pi$
$823$	5.60804e6	0.288610	0.144305	+	0.989533i	$0.453905\pi$
$824$	−5.78698e6	−0.296916
$825$	0	0
$826$	2.89418e6	0.147596
$827$	−2.38056e7	−1.21036	−0.605181	−	0.796088i	$-0.706899\pi$
$827$	−2.38056e7	−1.21036	−0.605181	+	0.796088i	$0.706899\pi$
$828$	1.03648e6	0.0525395
$829$	6.44338e6	0.325632	0.162816	−	0.986656i	$-0.447942\pi$
$829$	6.44338e6	0.325632	0.162816	+	0.986656i	$0.447942\pi$
$830$	0	0
$831$	−2.13906e7	−1.07453
$832$	2.79061e6	0.139763
$833$	3.70179e7	1.84841
$834$	1.48440e7	0.738985
$835$	0	0
$836$	143691.	0.00711071
$837$	2.25487e7	1.11252
$838$	−8.85569e6	−0.435624
$839$	−2.81950e7	−1.38283	−0.691413	−	0.722460i	$-0.743012\pi$
$839$	−2.81950e7	−1.38283	−0.691413	+	0.722460i	$0.743012\pi$
$840$	0	0
$841$	−1.82195e7	−0.888275
$842$	−1.57403e7	−0.765125
$843$	−2.00683e7	−0.972615
$844$	−4.74603e6	−0.229337
$845$	0	0
$846$	−2.78617e6	−0.133839
$847$	4.32066e6	0.206939
$848$	3.54442e6	0.169260
$849$	−1.29000e7	−0.614214
$850$	0	0
$851$	4.52786e6	0.214323
$852$	−3.64395e6	−0.171978
$853$	−2.45452e7	−1.15503	−0.577515	−	0.816380i	$-0.695978\pi$
$853$	−2.45452e7	−1.15503	−0.577515	+	0.816380i	$0.695978\pi$
$854$	−1.60855e6	−0.0754727
$855$	0	0
$856$	−70763.0	−0.00330082
$857$	−1.96052e6	−0.0911841	−0.0455920	−	0.998960i	$-0.514517\pi$
$857$	−1.96052e6	−0.0911841	−0.0455920	+	0.998960i	$0.514517\pi$
$858$	933350.	0.0432839
$859$	−1.69840e7	−0.785338	−0.392669	−	0.919680i	$-0.628448\pi$
$859$	−1.69840e7	−0.785338	−0.392669	+	0.919680i	$0.628448\pi$
$860$	0	0
$861$	2.27292e6	0.104490
$862$	2.26548e7	1.03847
$863$	3.88177e6	0.177420	0.0887101	−	0.996057i	$-0.471726\pi$
$863$	3.88177e6	0.177420	0.0887101	+	0.996057i	$0.471726\pi$
$864$	−4.17944e6	−0.190473
$865$	0	0
$866$	1.24254e7	0.563007
$867$	5.33990e7	2.41260
$868$	2.38058e6	0.107247
$869$	104911.	0.00471273
$870$	0	0
$871$	1.15384e6	0.0515347
$872$	−1.00636e7	−0.448192
$873$	−1.20216e6	−0.0533860
$874$	−1.74958e6	−0.0774739
$875$	0	0
$876$	9.78806e6	0.430959
$877$	−4.17538e6	−0.183315	−0.0916573	−	0.995791i	$-0.529216\pi$
$877$	−4.17538e6	−0.183315	−0.0916573	+	0.995791i	$0.529216\pi$
$878$	2.58469e7	1.13154
$879$	1.76260e7	0.769453
$880$	0	0
$881$	1.53863e7	0.667873	0.333937	−	0.942595i	$-0.391623\pi$
$881$	1.53863e7	0.667873	0.333937	+	0.942595i	$0.391623\pi$
$882$	3.43928e6	0.148866
$883$	3.56568e7	1.53901	0.769504	−	0.638642i	$-0.220504\pi$
$883$	3.56568e7	1.53901	0.769504	+	0.638642i	$0.220504\pi$
$884$	−2.50922e7	−1.07996
$885$	0	0
$886$	−236260.	−0.0101113
$887$	−1.00835e7	−0.430329	−0.215164	−	0.976578i	$-0.569029\pi$
$887$	−1.00835e7	−0.430329	−0.215164	+	0.976578i	$0.569029\pi$
$888$	−3.29268e6	−0.140126
$889$	5.54534e6	0.235328
$890$	0	0
$891$	−1.07465e6	−0.0453496
$892$	−1.12349e7	−0.472777
$893$	4.70304e6	0.197356
$894$	1.39894e7	0.585402
$895$	0	0
$896$	−441244.	−0.0183615
$897$	−1.13645e7	−0.471594
$898$	−3.25048e7	−1.34511
$899$	8.36324e6	0.345124
$900$	0	0
$901$	−3.18702e7	−1.30789
$902$	−610017.	−0.0249647
$903$	−64695.9	−0.00264032
$904$	−357933.	−0.0145674
$905$	0	0
$906$	−2.80957e7	−1.13715
$907$	7.77219e6	0.313708	0.156854	−	0.987622i	$-0.449865\pi$
$907$	7.77219e6	0.313708	0.156854	+	0.987622i	$0.449865\pi$
$908$	1.82982e7	0.736535
$909$	5.66467e6	0.227386
$910$	0	0
$911$	4.23937e7	1.69241	0.846204	−	0.532859i	$-0.178882\pi$
$911$	4.23937e7	1.69241	0.846204	+	0.532859i	$0.178882\pi$
$912$	1.27230e6	0.0506529
$913$	785538.	0.0311882
$914$	1.29245e7	0.511738
$915$	0	0
$916$	−1.77824e7	−0.700248
$917$	−3.57682e6	−0.140467
$918$	3.75801e7	1.47181
$919$	6.83543e6	0.266979	0.133490	−	0.991050i	$-0.457382\pi$
$919$	6.83543e6	0.266979	0.133490	+	0.991050i	$0.457382\pi$
$920$	0	0
$921$	1.82133e7	0.707519
$922$	−2.36880e7	−0.917702
$923$	−1.12706e7	−0.435455
$924$	−147579.	−0.00568649
$925$	0	0
$926$	1.28172e7	0.491209
$927$	4.83445e6	0.184777
$928$	−1.55014e6	−0.0590882
$929$	4.98907e6	0.189662	0.0948310	−	0.995493i	$-0.469769\pi$
$929$	4.98907e6	0.189662	0.0948310	+	0.995493i	$0.469769\pi$
$930$	0	0
$931$	−5.80549e6	−0.219515
$932$	1.74258e7	0.657131
$933$	−457684.	−0.0172132
$934$	−1.22374e7	−0.459011
$935$	0	0
$936$	−2.33128e6	−0.0869771
$937$	3.41535e6	0.127083	0.0635413	−	0.997979i	$-0.479761\pi$
$937$	3.41535e6	0.127083	0.0635413	+	0.997979i	$0.479761\pi$
$938$	−182442.	−0.00677046
$939$	−3.35766e7	−1.24272
$940$	0	0
$941$	2.36479e7	0.870600	0.435300	−	0.900285i	$-0.356642\pi$
$941$	2.36479e7	0.870600	0.435300	+	0.900285i	$0.356642\pi$
$942$	−1.54001e7	−0.565454
$943$	7.42758e6	0.272000
$944$	−6.87775e6	−0.251198
$945$	0	0
$946$	17363.4	0.000630822 0
$947$	−2.60917e7	−0.945426	−0.472713	−	0.881217i	$-0.656725\pi$
$947$	−2.60917e7	−0.945426	−0.472713	+	0.881217i	$0.656725\pi$
$948$	928932.	0.0335709
$949$	3.02742e7	1.09121
$950$	0	0
$951$	8.02547e6	0.287752
$952$	3.96752e6	0.141882
$953$	3.60456e7	1.28564	0.642821	−	0.766016i	$-0.277764\pi$
$953$	3.60456e7	1.28564	0.642821	+	0.766016i	$0.277764\pi$
$954$	−2.96101e6	−0.105334
$955$	0	0
$956$	1.54690e7	0.547417
$957$	−518461.	−0.0182994
$958$	−299880.	−0.0105568
$959$	−1.04031e7	−0.365273
$960$	0	0
$961$	1.89247e6	0.0661031
$962$	−1.01842e7	−0.354803
$963$	59115.5	0.00205417
$964$	−801053.	−0.0277631
$965$	0	0
$966$	1.79692e6	0.0619565
$967$	1.15017e7	0.395546	0.197773	−	0.980248i	$-0.436629\pi$
$967$	1.15017e7	0.395546	0.197773	+	0.980248i	$0.436629\pi$
$968$	−1.02677e7	−0.352195
$969$	−1.14401e7	−0.391401
$970$	0	0
$971$	2.44992e7	0.833879	0.416940	−	0.908934i	$-0.363103\pi$
$971$	2.44992e7	0.833879	0.416940	+	0.908934i	$0.363103\pi$
$972$	6.35335e6	0.215694
$973$	−7.25948e6	−0.245824
$974$	−5.49130e6	−0.185472
$975$	0	0
$976$	3.82257e6	0.128449
$977$	−3.27314e7	−1.09706	−0.548528	−	0.836132i	$-0.684812\pi$
$977$	−3.27314e7	−1.09706	−0.548528	+	0.836132i	$0.684812\pi$
$978$	6.65785e6	0.222580
$979$	−3.34272e6	−0.111466
$980$	0	0
$981$	8.40719e6	0.278919
$982$	−8.96140e6	−0.296549
$983$	4.67333e7	1.54256	0.771281	−	0.636494i	$-0.219616\pi$
$983$	4.67333e7	1.54256	0.771281	+	0.636494i	$0.219616\pi$
$984$	−5.40138e6	−0.177835
$985$	0	0
$986$	1.39383e7	0.456582
$987$	−4.83031e6	−0.157827
$988$	3.93520e6	0.128255
$989$	−211417.	−0.00687305
$990$	0	0
$991$	2.80933e7	0.908697	0.454348	−	0.890824i	$-0.349872\pi$
$991$	2.80933e7	0.908697	0.454348	+	0.890824i	$0.349872\pi$
$992$	−5.65723e6	−0.182526
$993$	1.94063e7	0.624552
$994$	1.78208e6	0.0572086
$995$	0	0
$996$	6.95552e6	0.222168
$997$	4.91873e6	0.156716	0.0783582	−	0.996925i	$-0.475032\pi$
$997$	4.91873e6	0.156716	0.0783582	+	0.996925i	$0.475032\pi$
$998$	−2.50182e7	−0.795114
$999$	1.52526e7	0.483538

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	950.6.a.e.1.2		2
5.4	even	2		190.6.a.b.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.6.a.b.1.1	✓	2	5.4	even	2
950.6.a.e.1.2		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)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} +13.7671 q^{3} +16.0000 q^{4} +55.0686 q^{6} -26.9314 q^{7} +64.0000 q^{8} -53.4657 q^{9} +24.8772 q^{11} +220.274 q^{12} +681.301 q^{13} -107.726 q^{14} +256.000 q^{16} -2301.87 q^{17} -213.863 q^{18} +361.000 q^{19} -370.769 q^{21} +99.5088 q^{22} -1211.62 q^{23} +881.097 q^{24} +2725.21 q^{26} -4081.49 q^{27} -430.903 q^{28} -1513.81 q^{29} -5524.64 q^{31} +1024.00 q^{32} +342.488 q^{33} -9207.46 q^{34} -855.451 q^{36} -3737.03 q^{37} +1444.00 q^{38} +9379.58 q^{39} -6130.29 q^{41} -1483.08 q^{42} +174.491 q^{43} +398.035 q^{44} -4846.48 q^{46} +13027.8 q^{47} +3524.39 q^{48} -16081.7 q^{49} -31690.1 q^{51} +10900.8 q^{52} +13845.4 q^{53} -16325.9 q^{54} -1723.61 q^{56} +4969.94 q^{57} -6055.23 q^{58} -26866.2 q^{59} +14931.9 q^{61} -22098.6 q^{62} +1439.91 q^{63} +4096.00 q^{64} +1369.95 q^{66} +1693.58 q^{67} -36829.9 q^{68} -16680.6 q^{69} -16542.8 q^{71} -3421.81 q^{72} +44435.8 q^{73} -14948.1 q^{74} +5776.00 q^{76} -669.978 q^{77} +37518.3 q^{78} +4217.16 q^{79} -43198.3 q^{81} -24521.1 q^{82} +31576.6 q^{83} -5932.30 q^{84} +697.965 q^{86} -20840.8 q^{87} +1592.14 q^{88} -134369. q^{89} -18348.4 q^{91} -19385.9 q^{92} -76058.5 q^{93} +52111.3 q^{94} +14097.6 q^{96} +22484.7 q^{97} -64326.8 q^{98} -1330.08 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} + 2 q^{3} + 32 q^{4} + 8 q^{6} - 156 q^{7} + 128 q^{8} - 158 q^{9} - 512 q^{11} + 32 q^{12} + 1286 q^{13} - 624 q^{14} + 512 q^{16} - 1080 q^{17} - 632 q^{18} + 722 q^{19} + 1148 q^{21} - 2048 q^{22}+ \cdots + 54792 q^{99}+O(q^{100}) \) 2 * q + 8 * q^2 + 2 * q^3 + 32 * q^4 + 8 * q^6 - 156 * q^7 + 128 * q^8 - 158 * q^9 - 512 * q^11 + 32 * q^12 + 1286 * q^13 - 624 * q^14 + 512 * q^16 - 1080 * q^17 - 632 * q^18 + 722 * q^19 + 1148 * q^21 - 2048 * q^22 + 2224 * q^23 + 128 * q^24 + 5144 * q^26 + 8 * q^27 - 2496 * q^28 - 2568 * q^29 - 1908 * q^31 + 2048 * q^32 + 6660 * q^33 - 4320 * q^34 - 2528 * q^36 - 7806 * q^37 + 2888 * q^38 + 2264 * q^39 + 3060 * q^41 + 4592 * q^42 + 2596 * q^43 - 8192 * q^44 + 8896 * q^46 + 14412 * q^47 + 512 * q^48 - 16230 * q^49 - 46068 * q^51 + 20576 * q^52 + 9638 * q^53 + 32 * q^54 - 9984 * q^56 + 722 * q^57 - 10272 * q^58 - 43008 * q^59 - 25852 * q^61 - 7632 * q^62 + 14932 * q^63 + 8192 * q^64 + 26640 * q^66 + 27466 * q^67 - 17280 * q^68 - 57108 * q^69 - 87780 * q^71 - 10112 * q^72 + 25904 * q^73 - 31224 * q^74 + 11552 * q^76 + 68624 * q^77 + 9056 * q^78 - 3720 * q^79 - 65918 * q^81 + 12240 * q^82 - 38320 * q^83 + 18368 * q^84 + 10384 * q^86 - 8436 * q^87 - 32768 * q^88 - 167264 * q^89 - 96396 * q^91 + 35584 * q^92 - 118616 * q^93 + 57648 * q^94 + 2048 * q^96 + 4038 * q^97 - 64920 * q^98 + 54792 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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