Properties

Label 7623.2.a.bz.1.1
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 847)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.76156\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.76156 q^{2} +5.62620 q^{4} +2.62620 q^{5} +1.00000 q^{7} -10.0140 q^{8} +O(q^{10})\) \(q-2.76156 q^{2} +5.62620 q^{4} +2.62620 q^{5} +1.00000 q^{7} -10.0140 q^{8} -7.25240 q^{10} -2.38776 q^{13} -2.76156 q^{14} +16.4017 q^{16} +2.38776 q^{17} -1.72928 q^{19} +14.7755 q^{20} +0.626198 q^{23} +1.89692 q^{25} +6.59392 q^{26} +5.62620 q^{28} -1.72928 q^{29} -2.23844 q^{31} -25.2663 q^{32} -6.59392 q^{34} +2.62620 q^{35} -6.89692 q^{37} +4.77551 q^{38} -26.2986 q^{40} -10.3878 q^{41} +7.25240 q^{43} -1.72928 q^{46} -6.38776 q^{47} +1.00000 q^{49} -5.23844 q^{50} -13.4340 q^{52} +9.25240 q^{53} -10.0140 q^{56} +4.77551 q^{58} +1.76156 q^{59} -10.3878 q^{61} +6.18159 q^{62} +36.9711 q^{64} -6.27072 q^{65} -6.42003 q^{67} +13.4340 q^{68} -7.25240 q^{70} -8.08476 q^{71} +10.3878 q^{73} +19.0462 q^{74} -9.72928 q^{76} +15.2524 q^{79} +43.0741 q^{80} +28.6864 q^{82} -12.7755 q^{83} +6.27072 q^{85} -20.0279 q^{86} +14.1493 q^{89} -2.38776 q^{91} +3.52311 q^{92} +17.6402 q^{94} -4.54144 q^{95} -8.35548 q^{97} -2.76156 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 2q^{2} + 8q^{4} - q^{5} + 3q^{7} - 6q^{8} + O(q^{10}) \) \( 3q - 2q^{2} + 8q^{4} - q^{5} + 3q^{7} - 6q^{8} - 4q^{10} + 8q^{13} - 2q^{14} + 10q^{16} - 8q^{17} + 14q^{20} - 7q^{23} + 2q^{25} + 12q^{26} + 8q^{28} - 13q^{31} - 34q^{32} - 12q^{34} - q^{35} - 17q^{37} - 16q^{38} - 36q^{40} - 16q^{41} + 4q^{43} - 4q^{47} + 3q^{49} - 22q^{50} + 10q^{53} - 6q^{56} - 16q^{58} - q^{59} - 16q^{61} - 4q^{62} + 34q^{64} - 24q^{65} - 3q^{67} - 4q^{70} - 5q^{71} + 16q^{73} + 32q^{74} - 24q^{76} + 28q^{79} + 56q^{80} + 28q^{82} - 8q^{83} + 24q^{85} - 12q^{86} + 21q^{89} + 8q^{91} - 2q^{92} + 20q^{94} - 24q^{95} - 11q^{97} - 2q^{98} + O(q^{100}) \)

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 \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.76156 −1.95272 −0.976358 0.216160i \(-0.930647\pi\)
−0.976358 + 0.216160i \(0.930647\pi\)
\(3\) 0 0
\(4\) 5.62620 2.81310
\(5\) 2.62620 1.17447 0.587236 0.809416i \(-0.300216\pi\)
0.587236 + 0.809416i \(0.300216\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −10.0140 −3.54047
\(9\) 0 0
\(10\) −7.25240 −2.29341
\(11\) 0 0
\(12\) 0 0
\(13\) −2.38776 −0.662244 −0.331122 0.943588i \(-0.607427\pi\)
−0.331122 + 0.943588i \(0.607427\pi\)
\(14\) −2.76156 −0.738057
\(15\) 0 0
\(16\) 16.4017 4.10043
\(17\) 2.38776 0.579116 0.289558 0.957161i \(-0.406492\pi\)
0.289558 + 0.957161i \(0.406492\pi\)
\(18\) 0 0
\(19\) −1.72928 −0.396724 −0.198362 0.980129i \(-0.563562\pi\)
−0.198362 + 0.980129i \(0.563562\pi\)
\(20\) 14.7755 3.30390
\(21\) 0 0
\(22\) 0 0
\(23\) 0.626198 0.130571 0.0652857 0.997867i \(-0.479204\pi\)
0.0652857 + 0.997867i \(0.479204\pi\)
\(24\) 0 0
\(25\) 1.89692 0.379383
\(26\) 6.59392 1.29317
\(27\) 0 0
\(28\) 5.62620 1.06325
\(29\) −1.72928 −0.321120 −0.160560 0.987026i \(-0.551330\pi\)
−0.160560 + 0.987026i \(0.551330\pi\)
\(30\) 0 0
\(31\) −2.23844 −0.402036 −0.201018 0.979588i \(-0.564425\pi\)
−0.201018 + 0.979588i \(0.564425\pi\)
\(32\) −25.2663 −4.46650
\(33\) 0 0
\(34\) −6.59392 −1.13085
\(35\) 2.62620 0.443908
\(36\) 0 0
\(37\) −6.89692 −1.13385 −0.566923 0.823771i \(-0.691866\pi\)
−0.566923 + 0.823771i \(0.691866\pi\)
\(38\) 4.77551 0.774690
\(39\) 0 0
\(40\) −26.2986 −4.15818
\(41\) −10.3878 −1.62229 −0.811147 0.584842i \(-0.801157\pi\)
−0.811147 + 0.584842i \(0.801157\pi\)
\(42\) 0 0
\(43\) 7.25240 1.10598 0.552990 0.833188i \(-0.313487\pi\)
0.552990 + 0.833188i \(0.313487\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −1.72928 −0.254969
\(47\) −6.38776 −0.931750 −0.465875 0.884851i \(-0.654260\pi\)
−0.465875 + 0.884851i \(0.654260\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −5.23844 −0.740828
\(51\) 0 0
\(52\) −13.4340 −1.86296
\(53\) 9.25240 1.27091 0.635457 0.772136i \(-0.280812\pi\)
0.635457 + 0.772136i \(0.280812\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −10.0140 −1.33817
\(57\) 0 0
\(58\) 4.77551 0.627055
\(59\) 1.76156 0.229335 0.114668 0.993404i \(-0.463420\pi\)
0.114668 + 0.993404i \(0.463420\pi\)
\(60\) 0 0
\(61\) −10.3878 −1.33002 −0.665008 0.746836i \(-0.731572\pi\)
−0.665008 + 0.746836i \(0.731572\pi\)
\(62\) 6.18159 0.785062
\(63\) 0 0
\(64\) 36.9711 4.62138
\(65\) −6.27072 −0.777787
\(66\) 0 0
\(67\) −6.42003 −0.784332 −0.392166 0.919895i \(-0.628274\pi\)
−0.392166 + 0.919895i \(0.628274\pi\)
\(68\) 13.4340 1.62911
\(69\) 0 0
\(70\) −7.25240 −0.866827
\(71\) −8.08476 −0.959485 −0.479742 0.877409i \(-0.659270\pi\)
−0.479742 + 0.877409i \(0.659270\pi\)
\(72\) 0 0
\(73\) 10.3878 1.21579 0.607897 0.794016i \(-0.292013\pi\)
0.607897 + 0.794016i \(0.292013\pi\)
\(74\) 19.0462 2.21408
\(75\) 0 0
\(76\) −9.72928 −1.11603
\(77\) 0 0
\(78\) 0 0
\(79\) 15.2524 1.71603 0.858014 0.513626i \(-0.171698\pi\)
0.858014 + 0.513626i \(0.171698\pi\)
\(80\) 43.0741 4.81583
\(81\) 0 0
\(82\) 28.6864 3.16788
\(83\) −12.7755 −1.40229 −0.701147 0.713017i \(-0.747328\pi\)
−0.701147 + 0.713017i \(0.747328\pi\)
\(84\) 0 0
\(85\) 6.27072 0.680155
\(86\) −20.0279 −2.15966
\(87\) 0 0
\(88\) 0 0
\(89\) 14.1493 1.49982 0.749912 0.661538i \(-0.230096\pi\)
0.749912 + 0.661538i \(0.230096\pi\)
\(90\) 0 0
\(91\) −2.38776 −0.250305
\(92\) 3.52311 0.367310
\(93\) 0 0
\(94\) 17.6402 1.81944
\(95\) −4.54144 −0.465942
\(96\) 0 0
\(97\) −8.35548 −0.848370 −0.424185 0.905575i \(-0.639439\pi\)
−0.424185 + 0.905575i \(0.639439\pi\)
\(98\) −2.76156 −0.278959
\(99\) 0 0
\(100\) 10.6724 1.06724
\(101\) 13.8463 1.37776 0.688880 0.724875i \(-0.258103\pi\)
0.688880 + 0.724875i \(0.258103\pi\)
\(102\) 0 0
\(103\) −2.92919 −0.288622 −0.144311 0.989532i \(-0.546097\pi\)
−0.144311 + 0.989532i \(0.546097\pi\)
\(104\) 23.9109 2.34465
\(105\) 0 0
\(106\) −25.5510 −2.48173
\(107\) −3.45856 −0.334352 −0.167176 0.985927i \(-0.553465\pi\)
−0.167176 + 0.985927i \(0.553465\pi\)
\(108\) 0 0
\(109\) −16.2341 −1.55494 −0.777471 0.628919i \(-0.783498\pi\)
−0.777471 + 0.628919i \(0.783498\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 16.4017 1.54982
\(113\) −7.10308 −0.668202 −0.334101 0.942537i \(-0.608433\pi\)
−0.334101 + 0.942537i \(0.608433\pi\)
\(114\) 0 0
\(115\) 1.64452 0.153352
\(116\) −9.72928 −0.903341
\(117\) 0 0
\(118\) −4.86464 −0.447826
\(119\) 2.38776 0.218885
\(120\) 0 0
\(121\) 0 0
\(122\) 28.6864 2.59714
\(123\) 0 0
\(124\) −12.5939 −1.13097
\(125\) −8.14931 −0.728897
\(126\) 0 0
\(127\) −4.54144 −0.402987 −0.201494 0.979490i \(-0.564580\pi\)
−0.201494 + 0.979490i \(0.564580\pi\)
\(128\) −51.5650 −4.55774
\(129\) 0 0
\(130\) 17.3169 1.51880
\(131\) 6.27072 0.547875 0.273938 0.961747i \(-0.411674\pi\)
0.273938 + 0.961747i \(0.411674\pi\)
\(132\) 0 0
\(133\) −1.72928 −0.149948
\(134\) 17.7293 1.53158
\(135\) 0 0
\(136\) −23.9109 −2.05034
\(137\) −9.40171 −0.803242 −0.401621 0.915806i \(-0.631553\pi\)
−0.401621 + 0.915806i \(0.631553\pi\)
\(138\) 0 0
\(139\) 15.8217 1.34198 0.670991 0.741465i \(-0.265869\pi\)
0.670991 + 0.741465i \(0.265869\pi\)
\(140\) 14.7755 1.24876
\(141\) 0 0
\(142\) 22.3265 1.87360
\(143\) 0 0
\(144\) 0 0
\(145\) −4.54144 −0.377146
\(146\) −28.6864 −2.37410
\(147\) 0 0
\(148\) −38.8034 −3.18962
\(149\) −3.45856 −0.283337 −0.141668 0.989914i \(-0.545247\pi\)
−0.141668 + 0.989914i \(0.545247\pi\)
\(150\) 0 0
\(151\) −10.2986 −0.838090 −0.419045 0.907965i \(-0.637635\pi\)
−0.419045 + 0.907965i \(0.637635\pi\)
\(152\) 17.3169 1.40459
\(153\) 0 0
\(154\) 0 0
\(155\) −5.87859 −0.472180
\(156\) 0 0
\(157\) 11.3372 0.904804 0.452402 0.891814i \(-0.350567\pi\)
0.452402 + 0.891814i \(0.350567\pi\)
\(158\) −42.1204 −3.35092
\(159\) 0 0
\(160\) −66.3544 −5.24578
\(161\) 0.626198 0.0493513
\(162\) 0 0
\(163\) −9.72928 −0.762056 −0.381028 0.924563i \(-0.624430\pi\)
−0.381028 + 0.924563i \(0.624430\pi\)
\(164\) −58.4436 −4.56368
\(165\) 0 0
\(166\) 35.2803 2.73828
\(167\) −22.5048 −1.74147 −0.870737 0.491750i \(-0.836357\pi\)
−0.870737 + 0.491750i \(0.836357\pi\)
\(168\) 0 0
\(169\) −7.29862 −0.561433
\(170\) −17.3169 −1.32815
\(171\) 0 0
\(172\) 40.8034 3.11123
\(173\) −6.92919 −0.526817 −0.263408 0.964684i \(-0.584847\pi\)
−0.263408 + 0.964684i \(0.584847\pi\)
\(174\) 0 0
\(175\) 1.89692 0.143393
\(176\) 0 0
\(177\) 0 0
\(178\) −39.0741 −2.92873
\(179\) −13.9431 −1.04216 −0.521080 0.853508i \(-0.674471\pi\)
−0.521080 + 0.853508i \(0.674471\pi\)
\(180\) 0 0
\(181\) 3.16763 0.235448 0.117724 0.993046i \(-0.462440\pi\)
0.117724 + 0.993046i \(0.462440\pi\)
\(182\) 6.59392 0.488774
\(183\) 0 0
\(184\) −6.27072 −0.462283
\(185\) −18.1127 −1.33167
\(186\) 0 0
\(187\) 0 0
\(188\) −35.9388 −2.62110
\(189\) 0 0
\(190\) 12.5414 0.909851
\(191\) −18.3555 −1.32816 −0.664078 0.747663i \(-0.731176\pi\)
−0.664078 + 0.747663i \(0.731176\pi\)
\(192\) 0 0
\(193\) −3.04623 −0.219272 −0.109636 0.993972i \(-0.534969\pi\)
−0.109636 + 0.993972i \(0.534969\pi\)
\(194\) 23.0741 1.65663
\(195\) 0 0
\(196\) 5.62620 0.401871
\(197\) −4.95377 −0.352942 −0.176471 0.984306i \(-0.556468\pi\)
−0.176471 + 0.984306i \(0.556468\pi\)
\(198\) 0 0
\(199\) 15.7047 1.11328 0.556638 0.830755i \(-0.312091\pi\)
0.556638 + 0.830755i \(0.312091\pi\)
\(200\) −18.9956 −1.34319
\(201\) 0 0
\(202\) −38.2374 −2.69037
\(203\) −1.72928 −0.121372
\(204\) 0 0
\(205\) −27.2803 −1.90534
\(206\) 8.08913 0.563596
\(207\) 0 0
\(208\) −39.1633 −2.71548
\(209\) 0 0
\(210\) 0 0
\(211\) 17.5510 1.20826 0.604131 0.796885i \(-0.293520\pi\)
0.604131 + 0.796885i \(0.293520\pi\)
\(212\) 52.0558 3.57521
\(213\) 0 0
\(214\) 9.55102 0.652894
\(215\) 19.0462 1.29894
\(216\) 0 0
\(217\) −2.23844 −0.151955
\(218\) 44.8313 3.03636
\(219\) 0 0
\(220\) 0 0
\(221\) −5.70138 −0.383516
\(222\) 0 0
\(223\) −22.2943 −1.49293 −0.746467 0.665423i \(-0.768251\pi\)
−0.746467 + 0.665423i \(0.768251\pi\)
\(224\) −25.2663 −1.68818
\(225\) 0 0
\(226\) 19.6156 1.30481
\(227\) −14.2707 −0.947181 −0.473590 0.880745i \(-0.657042\pi\)
−0.473590 + 0.880745i \(0.657042\pi\)
\(228\) 0 0
\(229\) −3.87859 −0.256305 −0.128152 0.991754i \(-0.540905\pi\)
−0.128152 + 0.991754i \(0.540905\pi\)
\(230\) −4.54144 −0.299453
\(231\) 0 0
\(232\) 17.3169 1.13691
\(233\) −6.68305 −0.437821 −0.218911 0.975745i \(-0.570250\pi\)
−0.218911 + 0.975745i \(0.570250\pi\)
\(234\) 0 0
\(235\) −16.7755 −1.09431
\(236\) 9.91087 0.645143
\(237\) 0 0
\(238\) −6.59392 −0.427421
\(239\) 4.54144 0.293761 0.146881 0.989154i \(-0.453077\pi\)
0.146881 + 0.989154i \(0.453077\pi\)
\(240\) 0 0
\(241\) −3.70470 −0.238641 −0.119320 0.992856i \(-0.538072\pi\)
−0.119320 + 0.992856i \(0.538072\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −58.4436 −3.74147
\(245\) 2.62620 0.167782
\(246\) 0 0
\(247\) 4.12910 0.262728
\(248\) 22.4157 1.42340
\(249\) 0 0
\(250\) 22.5048 1.42333
\(251\) −8.50916 −0.537093 −0.268547 0.963267i \(-0.586543\pi\)
−0.268547 + 0.963267i \(0.586543\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 12.5414 0.786920
\(255\) 0 0
\(256\) 68.4575 4.27860
\(257\) 2.95377 0.184251 0.0921256 0.995747i \(-0.470634\pi\)
0.0921256 + 0.995747i \(0.470634\pi\)
\(258\) 0 0
\(259\) −6.89692 −0.428554
\(260\) −35.2803 −2.18799
\(261\) 0 0
\(262\) −17.3169 −1.06984
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) 24.2986 1.49265
\(266\) 4.77551 0.292805
\(267\) 0 0
\(268\) −36.1204 −2.20640
\(269\) 6.77551 0.413110 0.206555 0.978435i \(-0.433775\pi\)
0.206555 + 0.978435i \(0.433775\pi\)
\(270\) 0 0
\(271\) 3.45856 0.210093 0.105046 0.994467i \(-0.466501\pi\)
0.105046 + 0.994467i \(0.466501\pi\)
\(272\) 39.1633 2.37462
\(273\) 0 0
\(274\) 25.9634 1.56850
\(275\) 0 0
\(276\) 0 0
\(277\) −11.4586 −0.688478 −0.344239 0.938882i \(-0.611863\pi\)
−0.344239 + 0.938882i \(0.611863\pi\)
\(278\) −43.6926 −2.62051
\(279\) 0 0
\(280\) −26.2986 −1.57164
\(281\) 22.3265 1.33189 0.665945 0.746001i \(-0.268029\pi\)
0.665945 + 0.746001i \(0.268029\pi\)
\(282\) 0 0
\(283\) −12.3632 −0.734915 −0.367457 0.930040i \(-0.619772\pi\)
−0.367457 + 0.930040i \(0.619772\pi\)
\(284\) −45.4865 −2.69913
\(285\) 0 0
\(286\) 0 0
\(287\) −10.3878 −0.613170
\(288\) 0 0
\(289\) −11.2986 −0.664625
\(290\) 12.5414 0.736459
\(291\) 0 0
\(292\) 58.4436 3.42015
\(293\) −2.15368 −0.125819 −0.0629097 0.998019i \(-0.520038\pi\)
−0.0629097 + 0.998019i \(0.520038\pi\)
\(294\) 0 0
\(295\) 4.62620 0.269348
\(296\) 69.0654 4.01434
\(297\) 0 0
\(298\) 9.55102 0.553276
\(299\) −1.49521 −0.0864701
\(300\) 0 0
\(301\) 7.25240 0.418021
\(302\) 28.4402 1.63655
\(303\) 0 0
\(304\) −28.3632 −1.62674
\(305\) −27.2803 −1.56207
\(306\) 0 0
\(307\) −31.8217 −1.81616 −0.908081 0.418794i \(-0.862453\pi\)
−0.908081 + 0.418794i \(0.862453\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 16.2341 0.922033
\(311\) 10.9292 0.619738 0.309869 0.950779i \(-0.399715\pi\)
0.309869 + 0.950779i \(0.399715\pi\)
\(312\) 0 0
\(313\) 7.40171 0.418369 0.209185 0.977876i \(-0.432919\pi\)
0.209185 + 0.977876i \(0.432919\pi\)
\(314\) −31.3082 −1.76682
\(315\) 0 0
\(316\) 85.8130 4.82736
\(317\) 8.89692 0.499701 0.249850 0.968284i \(-0.419619\pi\)
0.249850 + 0.968284i \(0.419619\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 97.0933 5.42768
\(321\) 0 0
\(322\) −1.72928 −0.0963691
\(323\) −4.12910 −0.229749
\(324\) 0 0
\(325\) −4.52937 −0.251244
\(326\) 26.8680 1.48808
\(327\) 0 0
\(328\) 104.022 5.74368
\(329\) −6.38776 −0.352168
\(330\) 0 0
\(331\) 8.56165 0.470591 0.235295 0.971924i \(-0.424394\pi\)
0.235295 + 0.971924i \(0.424394\pi\)
\(332\) −71.8776 −3.94479
\(333\) 0 0
\(334\) 62.1483 3.40060
\(335\) −16.8603 −0.921175
\(336\) 0 0
\(337\) −9.72928 −0.529988 −0.264994 0.964250i \(-0.585370\pi\)
−0.264994 + 0.964250i \(0.585370\pi\)
\(338\) 20.1556 1.09632
\(339\) 0 0
\(340\) 35.2803 1.91334
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −72.6252 −3.91569
\(345\) 0 0
\(346\) 19.1354 1.02872
\(347\) −14.8401 −0.796656 −0.398328 0.917243i \(-0.630409\pi\)
−0.398328 + 0.917243i \(0.630409\pi\)
\(348\) 0 0
\(349\) 16.2461 0.869636 0.434818 0.900518i \(-0.356813\pi\)
0.434818 + 0.900518i \(0.356813\pi\)
\(350\) −5.23844 −0.280007
\(351\) 0 0
\(352\) 0 0
\(353\) −30.8603 −1.64253 −0.821263 0.570549i \(-0.806730\pi\)
−0.821263 + 0.570549i \(0.806730\pi\)
\(354\) 0 0
\(355\) −21.2322 −1.12689
\(356\) 79.6068 4.21915
\(357\) 0 0
\(358\) 38.5048 2.03504
\(359\) 11.7938 0.622455 0.311227 0.950335i \(-0.399260\pi\)
0.311227 + 0.950335i \(0.399260\pi\)
\(360\) 0 0
\(361\) −16.0096 −0.842610
\(362\) −8.74760 −0.459764
\(363\) 0 0
\(364\) −13.4340 −0.704132
\(365\) 27.2803 1.42792
\(366\) 0 0
\(367\) −3.55539 −0.185590 −0.0927949 0.995685i \(-0.529580\pi\)
−0.0927949 + 0.995685i \(0.529580\pi\)
\(368\) 10.2707 0.535398
\(369\) 0 0
\(370\) 50.0192 2.60037
\(371\) 9.25240 0.480360
\(372\) 0 0
\(373\) 22.5048 1.16525 0.582627 0.812740i \(-0.302025\pi\)
0.582627 + 0.812740i \(0.302025\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 63.9667 3.29883
\(377\) 4.12910 0.212660
\(378\) 0 0
\(379\) 23.4296 1.20350 0.601749 0.798685i \(-0.294471\pi\)
0.601749 + 0.798685i \(0.294471\pi\)
\(380\) −25.5510 −1.31074
\(381\) 0 0
\(382\) 50.6897 2.59351
\(383\) 3.42629 0.175075 0.0875376 0.996161i \(-0.472100\pi\)
0.0875376 + 0.996161i \(0.472100\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 8.41233 0.428177
\(387\) 0 0
\(388\) −47.0096 −2.38655
\(389\) −8.05685 −0.408499 −0.204249 0.978919i \(-0.565475\pi\)
−0.204249 + 0.978919i \(0.565475\pi\)
\(390\) 0 0
\(391\) 1.49521 0.0756159
\(392\) −10.0140 −0.505781
\(393\) 0 0
\(394\) 13.6801 0.689195
\(395\) 40.0558 2.01543
\(396\) 0 0
\(397\) −26.3632 −1.32313 −0.661565 0.749888i \(-0.730107\pi\)
−0.661565 + 0.749888i \(0.730107\pi\)
\(398\) −43.3694 −2.17391
\(399\) 0 0
\(400\) 31.1127 1.55563
\(401\) 22.5972 1.12845 0.564226 0.825620i \(-0.309175\pi\)
0.564226 + 0.825620i \(0.309175\pi\)
\(402\) 0 0
\(403\) 5.34485 0.266246
\(404\) 77.9021 3.87578
\(405\) 0 0
\(406\) 4.77551 0.237005
\(407\) 0 0
\(408\) 0 0
\(409\) 9.30488 0.460097 0.230048 0.973179i \(-0.426112\pi\)
0.230048 + 0.973179i \(0.426112\pi\)
\(410\) 75.3361 3.72059
\(411\) 0 0
\(412\) −16.4802 −0.811922
\(413\) 1.76156 0.0866806
\(414\) 0 0
\(415\) −33.5510 −1.64695
\(416\) 60.3299 2.95791
\(417\) 0 0
\(418\) 0 0
\(419\) 13.8463 0.676437 0.338218 0.941068i \(-0.390176\pi\)
0.338218 + 0.941068i \(0.390176\pi\)
\(420\) 0 0
\(421\) −35.7572 −1.74270 −0.871349 0.490663i \(-0.836755\pi\)
−0.871349 + 0.490663i \(0.836755\pi\)
\(422\) −48.4681 −2.35939
\(423\) 0 0
\(424\) −92.6531 −4.49963
\(425\) 4.52937 0.219707
\(426\) 0 0
\(427\) −10.3878 −0.502699
\(428\) −19.4586 −0.940565
\(429\) 0 0
\(430\) −52.5972 −2.53646
\(431\) 31.3082 1.50806 0.754032 0.656838i \(-0.228106\pi\)
0.754032 + 0.656838i \(0.228106\pi\)
\(432\) 0 0
\(433\) 21.6079 1.03841 0.519204 0.854650i \(-0.326228\pi\)
0.519204 + 0.854650i \(0.326228\pi\)
\(434\) 6.18159 0.296726
\(435\) 0 0
\(436\) −91.3361 −4.37421
\(437\) −1.08287 −0.0518008
\(438\) 0 0
\(439\) 15.5877 0.743959 0.371979 0.928241i \(-0.378679\pi\)
0.371979 + 0.928241i \(0.378679\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 15.7447 0.748898
\(443\) −23.4942 −1.11624 −0.558121 0.829760i \(-0.688477\pi\)
−0.558121 + 0.829760i \(0.688477\pi\)
\(444\) 0 0
\(445\) 37.1589 1.76150
\(446\) 61.5669 2.91528
\(447\) 0 0
\(448\) 36.9711 1.74672
\(449\) −22.7832 −1.07521 −0.537603 0.843198i \(-0.680670\pi\)
−0.537603 + 0.843198i \(0.680670\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −39.9634 −1.87972
\(453\) 0 0
\(454\) 39.4094 1.84957
\(455\) −6.27072 −0.293976
\(456\) 0 0
\(457\) −14.5048 −0.678506 −0.339253 0.940695i \(-0.610174\pi\)
−0.339253 + 0.940695i \(0.610174\pi\)
\(458\) 10.7110 0.500490
\(459\) 0 0
\(460\) 9.25240 0.431395
\(461\) 31.1633 1.45142 0.725709 0.688002i \(-0.241512\pi\)
0.725709 + 0.688002i \(0.241512\pi\)
\(462\) 0 0
\(463\) −5.87859 −0.273201 −0.136601 0.990626i \(-0.543618\pi\)
−0.136601 + 0.990626i \(0.543618\pi\)
\(464\) −28.3632 −1.31673
\(465\) 0 0
\(466\) 18.4556 0.854941
\(467\) 23.7249 1.09786 0.548929 0.835869i \(-0.315036\pi\)
0.548929 + 0.835869i \(0.315036\pi\)
\(468\) 0 0
\(469\) −6.42003 −0.296449
\(470\) 46.3265 2.13688
\(471\) 0 0
\(472\) −17.6402 −0.811954
\(473\) 0 0
\(474\) 0 0
\(475\) −3.28030 −0.150511
\(476\) 13.4340 0.615746
\(477\) 0 0
\(478\) −12.5414 −0.573632
\(479\) −8.41233 −0.384369 −0.192185 0.981359i \(-0.561557\pi\)
−0.192185 + 0.981359i \(0.561557\pi\)
\(480\) 0 0
\(481\) 16.4681 0.750883
\(482\) 10.2307 0.465998
\(483\) 0 0
\(484\) 0 0
\(485\) −21.9431 −0.996387
\(486\) 0 0
\(487\) 24.7957 1.12360 0.561801 0.827273i \(-0.310109\pi\)
0.561801 + 0.827273i \(0.310109\pi\)
\(488\) 104.022 4.70888
\(489\) 0 0
\(490\) −7.25240 −0.327630
\(491\) −12.2062 −0.550857 −0.275428 0.961322i \(-0.588820\pi\)
−0.275428 + 0.961322i \(0.588820\pi\)
\(492\) 0 0
\(493\) −4.12910 −0.185965
\(494\) −11.4028 −0.513034
\(495\) 0 0
\(496\) −36.7143 −1.64852
\(497\) −8.08476 −0.362651
\(498\) 0 0
\(499\) 22.3265 0.999473 0.499736 0.866178i \(-0.333430\pi\)
0.499736 + 0.866178i \(0.333430\pi\)
\(500\) −45.8496 −2.05046
\(501\) 0 0
\(502\) 23.4985 1.04879
\(503\) 4.54144 0.202493 0.101246 0.994861i \(-0.467717\pi\)
0.101246 + 0.994861i \(0.467717\pi\)
\(504\) 0 0
\(505\) 36.3632 1.61814
\(506\) 0 0
\(507\) 0 0
\(508\) −25.5510 −1.13364
\(509\) −4.12141 −0.182678 −0.0913391 0.995820i \(-0.529115\pi\)
−0.0913391 + 0.995820i \(0.529115\pi\)
\(510\) 0 0
\(511\) 10.3878 0.459527
\(512\) −85.9194 −3.79714
\(513\) 0 0
\(514\) −8.15701 −0.359790
\(515\) −7.69264 −0.338978
\(516\) 0 0
\(517\) 0 0
\(518\) 19.0462 0.836843
\(519\) 0 0
\(520\) 62.7947 2.75373
\(521\) −39.2880 −1.72124 −0.860619 0.509249i \(-0.829923\pi\)
−0.860619 + 0.509249i \(0.829923\pi\)
\(522\) 0 0
\(523\) −2.14162 −0.0936464 −0.0468232 0.998903i \(-0.514910\pi\)
−0.0468232 + 0.998903i \(0.514910\pi\)
\(524\) 35.2803 1.54123
\(525\) 0 0
\(526\) −44.1849 −1.92655
\(527\) −5.34485 −0.232825
\(528\) 0 0
\(529\) −22.6079 −0.982951
\(530\) −67.1020 −2.91473
\(531\) 0 0
\(532\) −9.72928 −0.421818
\(533\) 24.8034 1.07436
\(534\) 0 0
\(535\) −9.08287 −0.392687
\(536\) 64.2899 2.77690
\(537\) 0 0
\(538\) −18.7110 −0.806687
\(539\) 0 0
\(540\) 0 0
\(541\) −43.0462 −1.85070 −0.925351 0.379112i \(-0.876230\pi\)
−0.925351 + 0.379112i \(0.876230\pi\)
\(542\) −9.55102 −0.410251
\(543\) 0 0
\(544\) −60.3299 −2.58662
\(545\) −42.6339 −1.82624
\(546\) 0 0
\(547\) −29.0096 −1.24036 −0.620180 0.784459i \(-0.712940\pi\)
−0.620180 + 0.784459i \(0.712940\pi\)
\(548\) −52.8959 −2.25960
\(549\) 0 0
\(550\) 0 0
\(551\) 2.99042 0.127396
\(552\) 0 0
\(553\) 15.2524 0.648598
\(554\) 31.6435 1.34440
\(555\) 0 0
\(556\) 89.0162 3.77513
\(557\) 1.49521 0.0633540 0.0316770 0.999498i \(-0.489915\pi\)
0.0316770 + 0.999498i \(0.489915\pi\)
\(558\) 0 0
\(559\) −17.3169 −0.732429
\(560\) 43.0741 1.82021
\(561\) 0 0
\(562\) −61.6560 −2.60080
\(563\) −6.27072 −0.264279 −0.132140 0.991231i \(-0.542185\pi\)
−0.132140 + 0.991231i \(0.542185\pi\)
\(564\) 0 0
\(565\) −18.6541 −0.784784
\(566\) 34.1416 1.43508
\(567\) 0 0
\(568\) 80.9604 3.39702
\(569\) −7.82174 −0.327904 −0.163952 0.986468i \(-0.552424\pi\)
−0.163952 + 0.986468i \(0.552424\pi\)
\(570\) 0 0
\(571\) −15.1753 −0.635068 −0.317534 0.948247i \(-0.602855\pi\)
−0.317534 + 0.948247i \(0.602855\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 28.6864 1.19735
\(575\) 1.18785 0.0495366
\(576\) 0 0
\(577\) 23.6445 0.984334 0.492167 0.870501i \(-0.336205\pi\)
0.492167 + 0.870501i \(0.336205\pi\)
\(578\) 31.2018 1.29782
\(579\) 0 0
\(580\) −25.5510 −1.06095
\(581\) −12.7755 −0.530017
\(582\) 0 0
\(583\) 0 0
\(584\) −104.022 −4.30448
\(585\) 0 0
\(586\) 5.94751 0.245690
\(587\) 10.1537 0.419087 0.209544 0.977799i \(-0.432802\pi\)
0.209544 + 0.977799i \(0.432802\pi\)
\(588\) 0 0
\(589\) 3.87090 0.159498
\(590\) −12.7755 −0.525959
\(591\) 0 0
\(592\) −113.121 −4.64925
\(593\) −33.7972 −1.38788 −0.693942 0.720031i \(-0.744127\pi\)
−0.693942 + 0.720031i \(0.744127\pi\)
\(594\) 0 0
\(595\) 6.27072 0.257074
\(596\) −19.4586 −0.797054
\(597\) 0 0
\(598\) 4.12910 0.168852
\(599\) 18.3265 0.748802 0.374401 0.927267i \(-0.377848\pi\)
0.374401 + 0.927267i \(0.377848\pi\)
\(600\) 0 0
\(601\) 48.4802 1.97755 0.988775 0.149415i \(-0.0477391\pi\)
0.988775 + 0.149415i \(0.0477391\pi\)
\(602\) −20.0279 −0.816277
\(603\) 0 0
\(604\) −57.9421 −2.35763
\(605\) 0 0
\(606\) 0 0
\(607\) −20.5972 −0.836017 −0.418008 0.908443i \(-0.637272\pi\)
−0.418008 + 0.908443i \(0.637272\pi\)
\(608\) 43.6926 1.77197
\(609\) 0 0
\(610\) 75.3361 3.05027
\(611\) 15.2524 0.617046
\(612\) 0 0
\(613\) 18.8122 0.759816 0.379908 0.925024i \(-0.375956\pi\)
0.379908 + 0.925024i \(0.375956\pi\)
\(614\) 87.8776 3.54645
\(615\) 0 0
\(616\) 0 0
\(617\) 14.2062 0.571919 0.285959 0.958242i \(-0.407688\pi\)
0.285959 + 0.958242i \(0.407688\pi\)
\(618\) 0 0
\(619\) −16.0235 −0.644040 −0.322020 0.946733i \(-0.604362\pi\)
−0.322020 + 0.946733i \(0.604362\pi\)
\(620\) −33.0741 −1.32829
\(621\) 0 0
\(622\) −30.1816 −1.21017
\(623\) 14.1493 0.566880
\(624\) 0 0
\(625\) −30.8863 −1.23545
\(626\) −20.4402 −0.816956
\(627\) 0 0
\(628\) 63.7851 2.54530
\(629\) −16.4681 −0.656628
\(630\) 0 0
\(631\) 15.1955 0.604925 0.302462 0.953161i \(-0.402191\pi\)
0.302462 + 0.953161i \(0.402191\pi\)
\(632\) −152.737 −6.07554
\(633\) 0 0
\(634\) −24.5693 −0.975773
\(635\) −11.9267 −0.473297
\(636\) 0 0
\(637\) −2.38776 −0.0946063
\(638\) 0 0
\(639\) 0 0
\(640\) −135.420 −5.35294
\(641\) 10.2264 0.403918 0.201959 0.979394i \(-0.435269\pi\)
0.201959 + 0.979394i \(0.435269\pi\)
\(642\) 0 0
\(643\) 20.5650 0.811003 0.405502 0.914094i \(-0.367097\pi\)
0.405502 + 0.914094i \(0.367097\pi\)
\(644\) 3.52311 0.138830
\(645\) 0 0
\(646\) 11.4028 0.448635
\(647\) 25.4542 1.00071 0.500354 0.865821i \(-0.333203\pi\)
0.500354 + 0.865821i \(0.333203\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 12.5081 0.490609
\(651\) 0 0
\(652\) −54.7389 −2.14374
\(653\) 45.2514 1.77082 0.885411 0.464809i \(-0.153877\pi\)
0.885411 + 0.464809i \(0.153877\pi\)
\(654\) 0 0
\(655\) 16.4681 0.643464
\(656\) −170.377 −6.65210
\(657\) 0 0
\(658\) 17.6402 0.687685
\(659\) 50.3544 1.96153 0.980765 0.195191i \(-0.0625327\pi\)
0.980765 + 0.195191i \(0.0625327\pi\)
\(660\) 0 0
\(661\) −23.2234 −0.903287 −0.451644 0.892198i \(-0.649162\pi\)
−0.451644 + 0.892198i \(0.649162\pi\)
\(662\) −23.6435 −0.918930
\(663\) 0 0
\(664\) 127.933 4.96478
\(665\) −4.54144 −0.176109
\(666\) 0 0
\(667\) −1.08287 −0.0419290
\(668\) −126.616 −4.89894
\(669\) 0 0
\(670\) 46.5606 1.79879
\(671\) 0 0
\(672\) 0 0
\(673\) −3.22449 −0.124295 −0.0621475 0.998067i \(-0.519795\pi\)
−0.0621475 + 0.998067i \(0.519795\pi\)
\(674\) 26.8680 1.03492
\(675\) 0 0
\(676\) −41.0635 −1.57937
\(677\) −42.6218 −1.63809 −0.819045 0.573729i \(-0.805496\pi\)
−0.819045 + 0.573729i \(0.805496\pi\)
\(678\) 0 0
\(679\) −8.35548 −0.320654
\(680\) −62.7947 −2.40807
\(681\) 0 0
\(682\) 0 0
\(683\) −15.5877 −0.596445 −0.298223 0.954496i \(-0.596394\pi\)
−0.298223 + 0.954496i \(0.596394\pi\)
\(684\) 0 0
\(685\) −24.6907 −0.943385
\(686\) −2.76156 −0.105437
\(687\) 0 0
\(688\) 118.952 4.53499
\(689\) −22.0925 −0.841656
\(690\) 0 0
\(691\) 41.9311 1.59513 0.797567 0.603231i \(-0.206120\pi\)
0.797567 + 0.603231i \(0.206120\pi\)
\(692\) −38.9850 −1.48199
\(693\) 0 0
\(694\) 40.9817 1.55564
\(695\) 41.5510 1.57612
\(696\) 0 0
\(697\) −24.8034 −0.939496
\(698\) −44.8646 −1.69815
\(699\) 0 0
\(700\) 10.6724 0.403380
\(701\) 15.4094 0.582005 0.291003 0.956722i \(-0.406011\pi\)
0.291003 + 0.956722i \(0.406011\pi\)
\(702\) 0 0
\(703\) 11.9267 0.449824
\(704\) 0 0
\(705\) 0 0
\(706\) 85.2224 3.20739
\(707\) 13.8463 0.520744
\(708\) 0 0
\(709\) 2.86027 0.107420 0.0537099 0.998557i \(-0.482895\pi\)
0.0537099 + 0.998557i \(0.482895\pi\)
\(710\) 58.6339 2.20049
\(711\) 0 0
\(712\) −141.691 −5.31008
\(713\) −1.40171 −0.0524944
\(714\) 0 0
\(715\) 0 0
\(716\) −78.4469 −2.93170
\(717\) 0 0
\(718\) −32.5693 −1.21548
\(719\) 34.4725 1.28561 0.642804 0.766031i \(-0.277771\pi\)
0.642804 + 0.766031i \(0.277771\pi\)
\(720\) 0 0
\(721\) −2.92919 −0.109089
\(722\) 44.2114 1.64538
\(723\) 0 0
\(724\) 17.8217 0.662340
\(725\) −3.28030 −0.121827
\(726\) 0 0
\(727\) −40.3309 −1.49579 −0.747895 0.663817i \(-0.768935\pi\)
−0.747895 + 0.663817i \(0.768935\pi\)
\(728\) 23.9109 0.886196
\(729\) 0 0
\(730\) −75.3361 −2.78831
\(731\) 17.3169 0.640490
\(732\) 0 0
\(733\) −28.7634 −1.06240 −0.531201 0.847246i \(-0.678259\pi\)
−0.531201 + 0.847246i \(0.678259\pi\)
\(734\) 9.81841 0.362404
\(735\) 0 0
\(736\) −15.8217 −0.583197
\(737\) 0 0
\(738\) 0 0
\(739\) 35.1020 1.29125 0.645625 0.763655i \(-0.276597\pi\)
0.645625 + 0.763655i \(0.276597\pi\)
\(740\) −101.905 −3.74612
\(741\) 0 0
\(742\) −25.5510 −0.938007
\(743\) 32.4681 1.19114 0.595570 0.803303i \(-0.296926\pi\)
0.595570 + 0.803303i \(0.296926\pi\)
\(744\) 0 0
\(745\) −9.08287 −0.332771
\(746\) −62.1483 −2.27541
\(747\) 0 0
\(748\) 0 0
\(749\) −3.45856 −0.126373
\(750\) 0 0
\(751\) −28.3834 −1.03572 −0.517862 0.855464i \(-0.673272\pi\)
−0.517862 + 0.855464i \(0.673272\pi\)
\(752\) −104.770 −3.82057
\(753\) 0 0
\(754\) −11.4028 −0.415264
\(755\) −27.0462 −0.984313
\(756\) 0 0
\(757\) −17.0462 −0.619556 −0.309778 0.950809i \(-0.600255\pi\)
−0.309778 + 0.950809i \(0.600255\pi\)
\(758\) −64.7022 −2.35009
\(759\) 0 0
\(760\) 45.4777 1.64965
\(761\) −27.9388 −1.01278 −0.506390 0.862305i \(-0.669020\pi\)
−0.506390 + 0.862305i \(0.669020\pi\)
\(762\) 0 0
\(763\) −16.2341 −0.587713
\(764\) −103.272 −3.73623
\(765\) 0 0
\(766\) −9.46189 −0.341872
\(767\) −4.20617 −0.151876
\(768\) 0 0
\(769\) 6.69512 0.241432 0.120716 0.992687i \(-0.461481\pi\)
0.120716 + 0.992687i \(0.461481\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −17.1387 −0.616835
\(773\) −27.9142 −1.00400 −0.502002 0.864866i \(-0.667403\pi\)
−0.502002 + 0.864866i \(0.667403\pi\)
\(774\) 0 0
\(775\) −4.24614 −0.152526
\(776\) 83.6714 3.00363
\(777\) 0 0
\(778\) 22.2495 0.797682
\(779\) 17.9634 0.643604
\(780\) 0 0
\(781\) 0 0
\(782\) −4.12910 −0.147656
\(783\) 0 0
\(784\) 16.4017 0.585775
\(785\) 29.7736 1.06267
\(786\) 0 0
\(787\) 21.8584 0.779167 0.389584 0.920991i \(-0.372619\pi\)
0.389584 + 0.920991i \(0.372619\pi\)
\(788\) −27.8709 −0.992860
\(789\) 0 0
\(790\) −110.616 −3.93556
\(791\) −7.10308 −0.252557
\(792\) 0 0
\(793\) 24.8034 0.880795
\(794\) 72.8034 2.58370
\(795\) 0 0
\(796\) 88.3578 3.13176
\(797\) 34.6262 1.22652 0.613261 0.789880i \(-0.289857\pi\)
0.613261 + 0.789880i \(0.289857\pi\)
\(798\) 0 0
\(799\) −15.2524 −0.539591
\(800\) −47.9282 −1.69452
\(801\) 0 0
\(802\) −62.4036 −2.20355
\(803\) 0 0
\(804\) 0 0
\(805\) 1.64452 0.0579617
\(806\) −14.7601 −0.519903
\(807\) 0 0
\(808\) −138.656 −4.87791
\(809\) −35.0462 −1.23216 −0.616080 0.787684i \(-0.711280\pi\)
−0.616080 + 0.787684i \(0.711280\pi\)
\(810\) 0 0
\(811\) −18.1416 −0.637038 −0.318519 0.947916i \(-0.603186\pi\)
−0.318519 + 0.947916i \(0.603186\pi\)
\(812\) −9.72928 −0.341431
\(813\) 0 0
\(814\) 0 0
\(815\) −25.5510 −0.895013
\(816\) 0 0
\(817\) −12.5414 −0.438769
\(818\) −25.6960 −0.898438
\(819\) 0 0
\(820\) −153.484 −5.35991
\(821\) 3.45856 0.120705 0.0603524 0.998177i \(-0.480778\pi\)
0.0603524 + 0.998177i \(0.480778\pi\)
\(822\) 0 0
\(823\) −33.5308 −1.16881 −0.584405 0.811462i \(-0.698672\pi\)
−0.584405 + 0.811462i \(0.698672\pi\)
\(824\) 29.3328 1.02186
\(825\) 0 0
\(826\) −4.86464 −0.169263
\(827\) 26.2986 0.914493 0.457246 0.889340i \(-0.348836\pi\)
0.457246 + 0.889340i \(0.348836\pi\)
\(828\) 0 0
\(829\) −29.0019 −1.00728 −0.503639 0.863914i \(-0.668006\pi\)
−0.503639 + 0.863914i \(0.668006\pi\)
\(830\) 92.6531 3.21603
\(831\) 0 0
\(832\) −88.2778 −3.06048
\(833\) 2.38776 0.0827308
\(834\) 0 0
\(835\) −59.1020 −2.04531
\(836\) 0 0
\(837\) 0 0
\(838\) −38.2374 −1.32089
\(839\) −29.1710 −1.00709 −0.503547 0.863968i \(-0.667972\pi\)
−0.503547 + 0.863968i \(0.667972\pi\)
\(840\) 0 0
\(841\) −26.0096 −0.896882
\(842\) 98.7455 3.40300
\(843\) 0 0
\(844\) 98.7455 3.39896
\(845\) −19.1676 −0.659387
\(846\) 0 0
\(847\) 0 0
\(848\) 151.755 5.21129
\(849\) 0 0
\(850\) −12.5081 −0.429025
\(851\) −4.31884 −0.148048
\(852\) 0 0
\(853\) 10.6218 0.363685 0.181842 0.983328i \(-0.441794\pi\)
0.181842 + 0.983328i \(0.441794\pi\)
\(854\) 28.6864 0.981628
\(855\) 0 0
\(856\) 34.6339 1.18376
\(857\) −37.8463 −1.29281 −0.646403 0.762996i \(-0.723727\pi\)
−0.646403 + 0.762996i \(0.723727\pi\)
\(858\) 0 0
\(859\) −19.0785 −0.650950 −0.325475 0.945551i \(-0.605524\pi\)
−0.325475 + 0.945551i \(0.605524\pi\)
\(860\) 107.158 3.65405
\(861\) 0 0
\(862\) −86.4594 −2.94482
\(863\) −11.1753 −0.380413 −0.190206 0.981744i \(-0.560916\pi\)
−0.190206 + 0.981744i \(0.560916\pi\)
\(864\) 0 0
\(865\) −18.1974 −0.618731
\(866\) −59.6714 −2.02772
\(867\) 0 0
\(868\) −12.5939 −0.427466
\(869\) 0 0
\(870\) 0 0
\(871\) 15.3295 0.519419
\(872\) 162.567 5.50522
\(873\) 0 0
\(874\) 2.99042 0.101152
\(875\) −8.14931 −0.275497
\(876\) 0 0
\(877\) −5.85838 −0.197824 −0.0989118 0.995096i \(-0.531536\pi\)
−0.0989118 + 0.995096i \(0.531536\pi\)
\(878\) −43.0462 −1.45274
\(879\) 0 0
\(880\) 0 0
\(881\) 25.9065 0.872812 0.436406 0.899750i \(-0.356251\pi\)
0.436406 + 0.899750i \(0.356251\pi\)
\(882\) 0 0
\(883\) 13.4219 0.451684 0.225842 0.974164i \(-0.427487\pi\)
0.225842 + 0.974164i \(0.427487\pi\)
\(884\) −32.0771 −1.07887
\(885\) 0 0
\(886\) 64.8805 2.17970
\(887\) 49.6068 1.66563 0.832817 0.553548i \(-0.186726\pi\)
0.832817 + 0.553548i \(0.186726\pi\)
\(888\) 0 0
\(889\) −4.54144 −0.152315
\(890\) −102.616 −3.43971
\(891\) 0 0
\(892\) −125.432 −4.19977
\(893\) 11.0462 0.369648
\(894\) 0 0
\(895\) −36.6175 −1.22399
\(896\) −51.5650 −1.72266
\(897\) 0 0
\(898\) 62.9171 2.09957
\(899\) 3.87090 0.129102
\(900\) 0 0
\(901\) 22.0925 0.736006
\(902\) 0 0
\(903\) 0 0
\(904\) 71.1299 2.36575
\(905\) 8.31884 0.276527
\(906\) 0 0
\(907\) −47.8776 −1.58975 −0.794874 0.606775i \(-0.792463\pi\)
−0.794874 + 0.606775i \(0.792463\pi\)
\(908\) −80.2899 −2.66451
\(909\) 0 0
\(910\) 17.3169 0.574051
\(911\) −22.8122 −0.755800 −0.377900 0.925846i \(-0.623354\pi\)
−0.377900 + 0.925846i \(0.623354\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 40.0558 1.32493
\(915\) 0 0
\(916\) −21.8217 −0.721011
\(917\) 6.27072 0.207077
\(918\) 0 0
\(919\) −40.0000 −1.31948 −0.659739 0.751495i \(-0.729333\pi\)
−0.659739 + 0.751495i \(0.729333\pi\)
\(920\) −16.4681 −0.542939
\(921\) 0 0
\(922\) −86.0591 −2.83421
\(923\) 19.3044 0.635413
\(924\) 0 0
\(925\) −13.0829 −0.430162
\(926\) 16.2341 0.533485
\(927\) 0 0
\(928\) 43.6926 1.43428
\(929\) 16.0366 0.526145 0.263073 0.964776i \(-0.415264\pi\)
0.263073 + 0.964776i \(0.415264\pi\)
\(930\) 0 0
\(931\) −1.72928 −0.0566749
\(932\) −37.6002 −1.23163
\(933\) 0 0
\(934\) −65.5177 −2.14380
\(935\) 0 0
\(936\) 0 0
\(937\) −25.3049 −0.826674 −0.413337 0.910578i \(-0.635637\pi\)
−0.413337 + 0.910578i \(0.635637\pi\)
\(938\) 17.7293 0.578882
\(939\) 0 0
\(940\) −94.3823 −3.07841
\(941\) −20.5294 −0.669238 −0.334619 0.942353i \(-0.608608\pi\)
−0.334619 + 0.942353i \(0.608608\pi\)
\(942\) 0 0
\(943\) −6.50479 −0.211825
\(944\) 28.8925 0.940372
\(945\) 0 0
\(946\) 0 0
\(947\) 16.6907 0.542376 0.271188 0.962526i \(-0.412583\pi\)
0.271188 + 0.962526i \(0.412583\pi\)
\(948\) 0 0
\(949\) −24.8034 −0.805153
\(950\) 9.05874 0.293904
\(951\) 0 0
\(952\) −23.9109 −0.774956
\(953\) −56.4681 −1.82918 −0.914591 0.404379i \(-0.867488\pi\)
−0.914591 + 0.404379i \(0.867488\pi\)
\(954\) 0 0
\(955\) −48.2051 −1.55988
\(956\) 25.5510 0.826379
\(957\) 0 0
\(958\) 23.2311 0.750564
\(959\) −9.40171 −0.303597
\(960\) 0 0
\(961\) −25.9894 −0.838367
\(962\) −45.4777 −1.46626
\(963\) 0 0
\(964\) −20.8434 −0.671320
\(965\) −8.00000 −0.257529
\(966\) 0 0
\(967\) 47.3082 1.52133 0.760665 0.649145i \(-0.224873\pi\)
0.760665 + 0.649145i \(0.224873\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 60.5972 1.94566
\(971\) 25.2355 0.809846 0.404923 0.914351i \(-0.367298\pi\)
0.404923 + 0.914351i \(0.367298\pi\)
\(972\) 0 0
\(973\) 15.8217 0.507222
\(974\) −68.4748 −2.19407
\(975\) 0 0
\(976\) −170.377 −5.45363
\(977\) −7.10308 −0.227248 −0.113624 0.993524i \(-0.536246\pi\)
−0.113624 + 0.993524i \(0.536246\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 14.7755 0.471986
\(981\) 0 0
\(982\) 33.7080 1.07567
\(983\) 14.1893 0.452568 0.226284 0.974061i \(-0.427342\pi\)
0.226284 + 0.974061i \(0.427342\pi\)
\(984\) 0 0
\(985\) −13.0096 −0.414520
\(986\) 11.4028 0.363138
\(987\) 0 0
\(988\) 23.2311 0.739081
\(989\) 4.54144 0.144409
\(990\) 0 0
\(991\) 4.23407 0.134500 0.0672499 0.997736i \(-0.478578\pi\)
0.0672499 + 0.997736i \(0.478578\pi\)
\(992\) 56.5573 1.79570
\(993\) 0 0
\(994\) 22.3265 0.708155
\(995\) 41.2437 1.30751
\(996\) 0 0
\(997\) −27.7047 −0.877417 −0.438708 0.898629i \(-0.644564\pi\)
−0.438708 + 0.898629i \(0.644564\pi\)
\(998\) −61.6560 −1.95169
\(999\) 0 0
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 7623.2.a.bz.1.1 3
3.2 odd 2 847.2.a.j.1.3 yes 3
11.10 odd 2 7623.2.a.ce.1.3 3
21.20 even 2 5929.2.a.y.1.3 3
33.2 even 10 847.2.f.u.323.3 12
33.5 odd 10 847.2.f.t.729.1 12
33.8 even 10 847.2.f.u.372.1 12
33.14 odd 10 847.2.f.t.372.3 12
33.17 even 10 847.2.f.u.729.3 12
33.20 odd 10 847.2.f.t.323.1 12
33.26 odd 10 847.2.f.t.148.3 12
33.29 even 10 847.2.f.u.148.1 12
33.32 even 2 847.2.a.i.1.1 3
231.230 odd 2 5929.2.a.t.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
847.2.a.i.1.1 3 33.32 even 2
847.2.a.j.1.3 yes 3 3.2 odd 2
847.2.f.t.148.3 12 33.26 odd 10
847.2.f.t.323.1 12 33.20 odd 10
847.2.f.t.372.3 12 33.14 odd 10
847.2.f.t.729.1 12 33.5 odd 10
847.2.f.u.148.1 12 33.29 even 10
847.2.f.u.323.3 12 33.2 even 10
847.2.f.u.372.1 12 33.8 even 10
847.2.f.u.729.3 12 33.17 even 10
5929.2.a.t.1.1 3 231.230 odd 2
5929.2.a.y.1.3 3 21.20 even 2
7623.2.a.bz.1.1 3 1.1 even 1 trivial
7623.2.a.ce.1.3 3 11.10 odd 2