Properties

Label 585.2.a.n.1.3
Level $585$
Weight $2$
Character 585.1
Self dual yes
Analytic conductor $4.671$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 195)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.34292\) of defining polynomial
Character \(\chi\) \(=\) 585.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.48929 q^{2} +4.19656 q^{4} +1.00000 q^{5} -1.19656 q^{7} +5.46787 q^{8} +O(q^{10})\) \(q+2.48929 q^{2} +4.19656 q^{4} +1.00000 q^{5} -1.19656 q^{7} +5.46787 q^{8} +2.48929 q^{10} +1.19656 q^{11} +1.00000 q^{13} -2.97858 q^{14} +5.21798 q^{16} -6.17513 q^{17} +6.97858 q^{19} +4.19656 q^{20} +2.97858 q^{22} -4.17513 q^{23} +1.00000 q^{25} +2.48929 q^{26} -5.02142 q^{28} -6.00000 q^{29} -2.97858 q^{31} +2.05333 q^{32} -15.3717 q^{34} -1.19656 q^{35} +7.78202 q^{37} +17.3717 q^{38} +5.46787 q^{40} +6.17513 q^{41} -9.95715 q^{43} +5.02142 q^{44} -10.3931 q^{46} +1.02142 q^{47} -5.56825 q^{49} +2.48929 q^{50} +4.19656 q^{52} -10.1751 q^{53} +1.19656 q^{55} -6.54262 q^{56} -14.9357 q^{58} -5.37169 q^{59} +12.5682 q^{61} -7.41454 q^{62} -5.32464 q^{64} +1.00000 q^{65} +9.37169 q^{67} -25.9143 q^{68} -2.97858 q^{70} +5.19656 q^{71} -11.9572 q^{73} +19.3717 q^{74} +29.2860 q^{76} -1.43175 q^{77} -1.78202 q^{79} +5.21798 q^{80} +15.3717 q^{82} +5.37169 q^{83} -6.17513 q^{85} -24.7862 q^{86} +6.54262 q^{88} -10.1751 q^{89} -1.19656 q^{91} -17.5212 q^{92} +2.54262 q^{94} +6.97858 q^{95} -1.82487 q^{97} -13.8610 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 8 q^{4} + 3 q^{5} + q^{7} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 8 q^{4} + 3 q^{5} + q^{7} - 6 q^{8} - q^{11} + 3 q^{13} + 6 q^{14} + 26 q^{16} + q^{17} + 6 q^{19} + 8 q^{20} - 6 q^{22} + 7 q^{23} + 3 q^{25} - 30 q^{28} - 18 q^{29} + 6 q^{31} - 22 q^{32} - 22 q^{34} + q^{35} + 13 q^{37} + 28 q^{38} - 6 q^{40} - q^{41} + 30 q^{44} - 22 q^{46} + 18 q^{47} + 12 q^{49} + 8 q^{52} - 11 q^{53} - q^{55} + 16 q^{56} + 8 q^{59} + 9 q^{61} - 28 q^{62} + 30 q^{64} + 3 q^{65} + 4 q^{67} - 18 q^{68} + 6 q^{70} + 11 q^{71} - 6 q^{73} + 34 q^{74} + 4 q^{76} - 33 q^{77} + 5 q^{79} + 26 q^{80} + 22 q^{82} - 8 q^{83} + q^{85} - 56 q^{86} - 16 q^{88} - 11 q^{89} + q^{91} - 2 q^{92} - 28 q^{94} + 6 q^{95} - 25 q^{97} - 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.48929 1.76019 0.880096 0.474795i \(-0.157478\pi\)
0.880096 + 0.474795i \(0.157478\pi\)
\(3\) 0 0
\(4\) 4.19656 2.09828
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.19656 −0.452256 −0.226128 0.974098i \(-0.572607\pi\)
−0.226128 + 0.974098i \(0.572607\pi\)
\(8\) 5.46787 1.93318
\(9\) 0 0
\(10\) 2.48929 0.787182
\(11\) 1.19656 0.360776 0.180388 0.983596i \(-0.442265\pi\)
0.180388 + 0.983596i \(0.442265\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) −2.97858 −0.796058
\(15\) 0 0
\(16\) 5.21798 1.30450
\(17\) −6.17513 −1.49769 −0.748845 0.662745i \(-0.769391\pi\)
−0.748845 + 0.662745i \(0.769391\pi\)
\(18\) 0 0
\(19\) 6.97858 1.60100 0.800498 0.599336i \(-0.204569\pi\)
0.800498 + 0.599336i \(0.204569\pi\)
\(20\) 4.19656 0.938379
\(21\) 0 0
\(22\) 2.97858 0.635035
\(23\) −4.17513 −0.870576 −0.435288 0.900291i \(-0.643353\pi\)
−0.435288 + 0.900291i \(0.643353\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.48929 0.488190
\(27\) 0 0
\(28\) −5.02142 −0.948960
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −2.97858 −0.534968 −0.267484 0.963562i \(-0.586192\pi\)
−0.267484 + 0.963562i \(0.586192\pi\)
\(32\) 2.05333 0.362980
\(33\) 0 0
\(34\) −15.3717 −2.63622
\(35\) −1.19656 −0.202255
\(36\) 0 0
\(37\) 7.78202 1.27936 0.639678 0.768643i \(-0.279068\pi\)
0.639678 + 0.768643i \(0.279068\pi\)
\(38\) 17.3717 2.81806
\(39\) 0 0
\(40\) 5.46787 0.864545
\(41\) 6.17513 0.964394 0.482197 0.876063i \(-0.339839\pi\)
0.482197 + 0.876063i \(0.339839\pi\)
\(42\) 0 0
\(43\) −9.95715 −1.51845 −0.759226 0.650827i \(-0.774422\pi\)
−0.759226 + 0.650827i \(0.774422\pi\)
\(44\) 5.02142 0.757008
\(45\) 0 0
\(46\) −10.3931 −1.53238
\(47\) 1.02142 0.148990 0.0744949 0.997221i \(-0.476266\pi\)
0.0744949 + 0.997221i \(0.476266\pi\)
\(48\) 0 0
\(49\) −5.56825 −0.795464
\(50\) 2.48929 0.352039
\(51\) 0 0
\(52\) 4.19656 0.581958
\(53\) −10.1751 −1.39766 −0.698831 0.715287i \(-0.746296\pi\)
−0.698831 + 0.715287i \(0.746296\pi\)
\(54\) 0 0
\(55\) 1.19656 0.161344
\(56\) −6.54262 −0.874294
\(57\) 0 0
\(58\) −14.9357 −1.96116
\(59\) −5.37169 −0.699335 −0.349667 0.936874i \(-0.613705\pi\)
−0.349667 + 0.936874i \(0.613705\pi\)
\(60\) 0 0
\(61\) 12.5682 1.60920 0.804600 0.593818i \(-0.202380\pi\)
0.804600 + 0.593818i \(0.202380\pi\)
\(62\) −7.41454 −0.941647
\(63\) 0 0
\(64\) −5.32464 −0.665579
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) 9.37169 1.14493 0.572467 0.819928i \(-0.305986\pi\)
0.572467 + 0.819928i \(0.305986\pi\)
\(68\) −25.9143 −3.14257
\(69\) 0 0
\(70\) −2.97858 −0.356008
\(71\) 5.19656 0.616718 0.308359 0.951270i \(-0.400220\pi\)
0.308359 + 0.951270i \(0.400220\pi\)
\(72\) 0 0
\(73\) −11.9572 −1.39948 −0.699740 0.714398i \(-0.746701\pi\)
−0.699740 + 0.714398i \(0.746701\pi\)
\(74\) 19.3717 2.25191
\(75\) 0 0
\(76\) 29.2860 3.35933
\(77\) −1.43175 −0.163163
\(78\) 0 0
\(79\) −1.78202 −0.200493 −0.100246 0.994963i \(-0.531963\pi\)
−0.100246 + 0.994963i \(0.531963\pi\)
\(80\) 5.21798 0.583388
\(81\) 0 0
\(82\) 15.3717 1.69752
\(83\) 5.37169 0.589620 0.294810 0.955556i \(-0.404744\pi\)
0.294810 + 0.955556i \(0.404744\pi\)
\(84\) 0 0
\(85\) −6.17513 −0.669787
\(86\) −24.7862 −2.67277
\(87\) 0 0
\(88\) 6.54262 0.697445
\(89\) −10.1751 −1.07856 −0.539281 0.842126i \(-0.681304\pi\)
−0.539281 + 0.842126i \(0.681304\pi\)
\(90\) 0 0
\(91\) −1.19656 −0.125433
\(92\) −17.5212 −1.82671
\(93\) 0 0
\(94\) 2.54262 0.262251
\(95\) 6.97858 0.715987
\(96\) 0 0
\(97\) −1.82487 −0.185287 −0.0926435 0.995699i \(-0.529532\pi\)
−0.0926435 + 0.995699i \(0.529532\pi\)
\(98\) −13.8610 −1.40017
\(99\) 0 0
\(100\) 4.19656 0.419656
\(101\) 10.3503 1.02989 0.514945 0.857223i \(-0.327812\pi\)
0.514945 + 0.857223i \(0.327812\pi\)
\(102\) 0 0
\(103\) 18.7434 1.84684 0.923420 0.383790i \(-0.125381\pi\)
0.923420 + 0.383790i \(0.125381\pi\)
\(104\) 5.46787 0.536168
\(105\) 0 0
\(106\) −25.3288 −2.46016
\(107\) 18.5682 1.79506 0.897530 0.440953i \(-0.145359\pi\)
0.897530 + 0.440953i \(0.145359\pi\)
\(108\) 0 0
\(109\) 8.39312 0.803915 0.401957 0.915658i \(-0.368330\pi\)
0.401957 + 0.915658i \(0.368330\pi\)
\(110\) 2.97858 0.283996
\(111\) 0 0
\(112\) −6.24361 −0.589966
\(113\) −7.95715 −0.748546 −0.374273 0.927319i \(-0.622108\pi\)
−0.374273 + 0.927319i \(0.622108\pi\)
\(114\) 0 0
\(115\) −4.17513 −0.389333
\(116\) −25.1793 −2.33784
\(117\) 0 0
\(118\) −13.3717 −1.23096
\(119\) 7.38890 0.677340
\(120\) 0 0
\(121\) −9.56825 −0.869841
\(122\) 31.2860 2.83250
\(123\) 0 0
\(124\) −12.4998 −1.12251
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −10.3931 −0.922240 −0.461120 0.887338i \(-0.652552\pi\)
−0.461120 + 0.887338i \(0.652552\pi\)
\(128\) −17.3612 −1.53453
\(129\) 0 0
\(130\) 2.48929 0.218325
\(131\) 6.39312 0.558569 0.279285 0.960208i \(-0.409903\pi\)
0.279285 + 0.960208i \(0.409903\pi\)
\(132\) 0 0
\(133\) −8.35027 −0.724060
\(134\) 23.3288 2.01531
\(135\) 0 0
\(136\) −33.7648 −2.89531
\(137\) −16.7434 −1.43048 −0.715242 0.698877i \(-0.753684\pi\)
−0.715242 + 0.698877i \(0.753684\pi\)
\(138\) 0 0
\(139\) 5.78202 0.490424 0.245212 0.969469i \(-0.421142\pi\)
0.245212 + 0.969469i \(0.421142\pi\)
\(140\) −5.02142 −0.424388
\(141\) 0 0
\(142\) 12.9357 1.08554
\(143\) 1.19656 0.100061
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) −29.7648 −2.46335
\(147\) 0 0
\(148\) 32.6577 2.68445
\(149\) −15.3461 −1.25720 −0.628599 0.777730i \(-0.716371\pi\)
−0.628599 + 0.777730i \(0.716371\pi\)
\(150\) 0 0
\(151\) −8.58546 −0.698675 −0.349337 0.936997i \(-0.613593\pi\)
−0.349337 + 0.936997i \(0.613593\pi\)
\(152\) 38.1579 3.09502
\(153\) 0 0
\(154\) −3.56404 −0.287198
\(155\) −2.97858 −0.239245
\(156\) 0 0
\(157\) 2.78623 0.222365 0.111183 0.993800i \(-0.464536\pi\)
0.111183 + 0.993800i \(0.464536\pi\)
\(158\) −4.43596 −0.352906
\(159\) 0 0
\(160\) 2.05333 0.162330
\(161\) 4.99579 0.393723
\(162\) 0 0
\(163\) 8.76060 0.686183 0.343091 0.939302i \(-0.388526\pi\)
0.343091 + 0.939302i \(0.388526\pi\)
\(164\) 25.9143 2.02357
\(165\) 0 0
\(166\) 13.3717 1.03784
\(167\) 17.3717 1.34426 0.672131 0.740432i \(-0.265379\pi\)
0.672131 + 0.740432i \(0.265379\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −15.3717 −1.17895
\(171\) 0 0
\(172\) −41.7858 −3.18614
\(173\) 7.95715 0.604971 0.302486 0.953154i \(-0.402184\pi\)
0.302486 + 0.953154i \(0.402184\pi\)
\(174\) 0 0
\(175\) −1.19656 −0.0904513
\(176\) 6.24361 0.470630
\(177\) 0 0
\(178\) −25.3288 −1.89848
\(179\) 15.5640 1.16331 0.581655 0.813435i \(-0.302405\pi\)
0.581655 + 0.813435i \(0.302405\pi\)
\(180\) 0 0
\(181\) 15.7820 1.17307 0.586534 0.809925i \(-0.300492\pi\)
0.586534 + 0.809925i \(0.300492\pi\)
\(182\) −2.97858 −0.220787
\(183\) 0 0
\(184\) −22.8291 −1.68298
\(185\) 7.78202 0.572145
\(186\) 0 0
\(187\) −7.38890 −0.540330
\(188\) 4.28646 0.312622
\(189\) 0 0
\(190\) 17.3717 1.26028
\(191\) −10.7434 −0.777364 −0.388682 0.921372i \(-0.627070\pi\)
−0.388682 + 0.921372i \(0.627070\pi\)
\(192\) 0 0
\(193\) 9.73917 0.701041 0.350521 0.936555i \(-0.386005\pi\)
0.350521 + 0.936555i \(0.386005\pi\)
\(194\) −4.54262 −0.326141
\(195\) 0 0
\(196\) −23.3675 −1.66911
\(197\) 9.56404 0.681410 0.340705 0.940170i \(-0.389334\pi\)
0.340705 + 0.940170i \(0.389334\pi\)
\(198\) 0 0
\(199\) 5.95715 0.422291 0.211146 0.977455i \(-0.432281\pi\)
0.211146 + 0.977455i \(0.432281\pi\)
\(200\) 5.46787 0.386636
\(201\) 0 0
\(202\) 25.7648 1.81281
\(203\) 7.17935 0.503891
\(204\) 0 0
\(205\) 6.17513 0.431290
\(206\) 46.6577 3.25080
\(207\) 0 0
\(208\) 5.21798 0.361802
\(209\) 8.35027 0.577600
\(210\) 0 0
\(211\) 23.9143 1.64633 0.823164 0.567803i \(-0.192206\pi\)
0.823164 + 0.567803i \(0.192206\pi\)
\(212\) −42.7005 −2.93269
\(213\) 0 0
\(214\) 46.2217 3.15965
\(215\) −9.95715 −0.679072
\(216\) 0 0
\(217\) 3.56404 0.241943
\(218\) 20.8929 1.41504
\(219\) 0 0
\(220\) 5.02142 0.338544
\(221\) −6.17513 −0.415385
\(222\) 0 0
\(223\) 2.62831 0.176004 0.0880022 0.996120i \(-0.471952\pi\)
0.0880022 + 0.996120i \(0.471952\pi\)
\(224\) −2.45692 −0.164160
\(225\) 0 0
\(226\) −19.8077 −1.31759
\(227\) −15.7648 −1.04635 −0.523174 0.852226i \(-0.675252\pi\)
−0.523174 + 0.852226i \(0.675252\pi\)
\(228\) 0 0
\(229\) 8.74338 0.577779 0.288890 0.957362i \(-0.406714\pi\)
0.288890 + 0.957362i \(0.406714\pi\)
\(230\) −10.3931 −0.685302
\(231\) 0 0
\(232\) −32.8072 −2.15390
\(233\) 2.17513 0.142498 0.0712489 0.997459i \(-0.477302\pi\)
0.0712489 + 0.997459i \(0.477302\pi\)
\(234\) 0 0
\(235\) 1.02142 0.0666303
\(236\) −22.5426 −1.46740
\(237\) 0 0
\(238\) 18.3931 1.19225
\(239\) −2.80344 −0.181340 −0.0906698 0.995881i \(-0.528901\pi\)
−0.0906698 + 0.995881i \(0.528901\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) −23.8181 −1.53109
\(243\) 0 0
\(244\) 52.7434 3.37655
\(245\) −5.56825 −0.355742
\(246\) 0 0
\(247\) 6.97858 0.444036
\(248\) −16.2865 −1.03419
\(249\) 0 0
\(250\) 2.48929 0.157436
\(251\) 23.9143 1.50946 0.754729 0.656037i \(-0.227768\pi\)
0.754729 + 0.656037i \(0.227768\pi\)
\(252\) 0 0
\(253\) −4.99579 −0.314083
\(254\) −25.8715 −1.62332
\(255\) 0 0
\(256\) −32.5678 −2.03549
\(257\) 19.9572 1.24489 0.622447 0.782662i \(-0.286139\pi\)
0.622447 + 0.782662i \(0.286139\pi\)
\(258\) 0 0
\(259\) −9.31163 −0.578597
\(260\) 4.19656 0.260259
\(261\) 0 0
\(262\) 15.9143 0.983189
\(263\) −8.00000 −0.493301 −0.246651 0.969104i \(-0.579330\pi\)
−0.246651 + 0.969104i \(0.579330\pi\)
\(264\) 0 0
\(265\) −10.1751 −0.625054
\(266\) −20.7862 −1.27449
\(267\) 0 0
\(268\) 39.3288 2.40239
\(269\) 2.35027 0.143298 0.0716492 0.997430i \(-0.477174\pi\)
0.0716492 + 0.997430i \(0.477174\pi\)
\(270\) 0 0
\(271\) 10.9786 0.666901 0.333451 0.942768i \(-0.391787\pi\)
0.333451 + 0.942768i \(0.391787\pi\)
\(272\) −32.2217 −1.95373
\(273\) 0 0
\(274\) −41.6791 −2.51793
\(275\) 1.19656 0.0721551
\(276\) 0 0
\(277\) 1.21377 0.0729283 0.0364642 0.999335i \(-0.488391\pi\)
0.0364642 + 0.999335i \(0.488391\pi\)
\(278\) 14.3931 0.863242
\(279\) 0 0
\(280\) −6.54262 −0.390996
\(281\) −11.9572 −0.713304 −0.356652 0.934237i \(-0.616082\pi\)
−0.356652 + 0.934237i \(0.616082\pi\)
\(282\) 0 0
\(283\) −29.8715 −1.77567 −0.887837 0.460158i \(-0.847793\pi\)
−0.887837 + 0.460158i \(0.847793\pi\)
\(284\) 21.8077 1.29405
\(285\) 0 0
\(286\) 2.97858 0.176127
\(287\) −7.38890 −0.436153
\(288\) 0 0
\(289\) 21.1323 1.24308
\(290\) −14.9357 −0.877056
\(291\) 0 0
\(292\) −50.1789 −2.93650
\(293\) −0.777809 −0.0454401 −0.0227200 0.999742i \(-0.507233\pi\)
−0.0227200 + 0.999742i \(0.507233\pi\)
\(294\) 0 0
\(295\) −5.37169 −0.312752
\(296\) 42.5510 2.47323
\(297\) 0 0
\(298\) −38.2008 −2.21291
\(299\) −4.17513 −0.241454
\(300\) 0 0
\(301\) 11.9143 0.686729
\(302\) −21.3717 −1.22980
\(303\) 0 0
\(304\) 36.4141 2.08849
\(305\) 12.5682 0.719656
\(306\) 0 0
\(307\) 0.760597 0.0434095 0.0217048 0.999764i \(-0.493091\pi\)
0.0217048 + 0.999764i \(0.493091\pi\)
\(308\) −6.00842 −0.342362
\(309\) 0 0
\(310\) −7.41454 −0.421117
\(311\) 23.1281 1.31147 0.655736 0.754990i \(-0.272358\pi\)
0.655736 + 0.754990i \(0.272358\pi\)
\(312\) 0 0
\(313\) −33.9143 −1.91695 −0.958475 0.285176i \(-0.907948\pi\)
−0.958475 + 0.285176i \(0.907948\pi\)
\(314\) 6.93573 0.391406
\(315\) 0 0
\(316\) −7.47835 −0.420690
\(317\) −9.64973 −0.541983 −0.270991 0.962582i \(-0.587352\pi\)
−0.270991 + 0.962582i \(0.587352\pi\)
\(318\) 0 0
\(319\) −7.17935 −0.401966
\(320\) −5.32464 −0.297656
\(321\) 0 0
\(322\) 12.4360 0.693029
\(323\) −43.0937 −2.39780
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) 21.8077 1.20781
\(327\) 0 0
\(328\) 33.7648 1.86435
\(329\) −1.22219 −0.0673816
\(330\) 0 0
\(331\) 15.3288 0.842550 0.421275 0.906933i \(-0.361583\pi\)
0.421275 + 0.906933i \(0.361583\pi\)
\(332\) 22.5426 1.23719
\(333\) 0 0
\(334\) 43.2432 2.36616
\(335\) 9.37169 0.512030
\(336\) 0 0
\(337\) −22.3503 −1.21750 −0.608748 0.793363i \(-0.708328\pi\)
−0.608748 + 0.793363i \(0.708328\pi\)
\(338\) 2.48929 0.135399
\(339\) 0 0
\(340\) −25.9143 −1.40540
\(341\) −3.56404 −0.193004
\(342\) 0 0
\(343\) 15.0386 0.812010
\(344\) −54.4444 −2.93544
\(345\) 0 0
\(346\) 19.8077 1.06487
\(347\) 5.78202 0.310395 0.155198 0.987883i \(-0.450399\pi\)
0.155198 + 0.987883i \(0.450399\pi\)
\(348\) 0 0
\(349\) −27.5212 −1.47318 −0.736588 0.676342i \(-0.763564\pi\)
−0.736588 + 0.676342i \(0.763564\pi\)
\(350\) −2.97858 −0.159212
\(351\) 0 0
\(352\) 2.45692 0.130955
\(353\) 28.7434 1.52986 0.764928 0.644116i \(-0.222775\pi\)
0.764928 + 0.644116i \(0.222775\pi\)
\(354\) 0 0
\(355\) 5.19656 0.275805
\(356\) −42.7005 −2.26312
\(357\) 0 0
\(358\) 38.7434 2.04765
\(359\) 12.5855 0.664235 0.332118 0.943238i \(-0.392237\pi\)
0.332118 + 0.943238i \(0.392237\pi\)
\(360\) 0 0
\(361\) 29.7005 1.56319
\(362\) 39.2860 2.06483
\(363\) 0 0
\(364\) −5.02142 −0.263194
\(365\) −11.9572 −0.625866
\(366\) 0 0
\(367\) −27.9143 −1.45712 −0.728558 0.684985i \(-0.759809\pi\)
−0.728558 + 0.684985i \(0.759809\pi\)
\(368\) −21.7858 −1.13566
\(369\) 0 0
\(370\) 19.3717 1.00709
\(371\) 12.1751 0.632102
\(372\) 0 0
\(373\) −2.35027 −0.121692 −0.0608462 0.998147i \(-0.519380\pi\)
−0.0608462 + 0.998147i \(0.519380\pi\)
\(374\) −18.3931 −0.951085
\(375\) 0 0
\(376\) 5.58500 0.288025
\(377\) −6.00000 −0.309016
\(378\) 0 0
\(379\) 24.5510 1.26110 0.630551 0.776148i \(-0.282829\pi\)
0.630551 + 0.776148i \(0.282829\pi\)
\(380\) 29.2860 1.50234
\(381\) 0 0
\(382\) −26.7434 −1.36831
\(383\) −5.80765 −0.296757 −0.148379 0.988931i \(-0.547405\pi\)
−0.148379 + 0.988931i \(0.547405\pi\)
\(384\) 0 0
\(385\) −1.43175 −0.0729687
\(386\) 24.2436 1.23397
\(387\) 0 0
\(388\) −7.65815 −0.388784
\(389\) −13.6497 −0.692069 −0.346034 0.938222i \(-0.612472\pi\)
−0.346034 + 0.938222i \(0.612472\pi\)
\(390\) 0 0
\(391\) 25.7820 1.30385
\(392\) −30.4464 −1.53778
\(393\) 0 0
\(394\) 23.8077 1.19941
\(395\) −1.78202 −0.0896631
\(396\) 0 0
\(397\) −12.1323 −0.608902 −0.304451 0.952528i \(-0.598473\pi\)
−0.304451 + 0.952528i \(0.598473\pi\)
\(398\) 14.8291 0.743314
\(399\) 0 0
\(400\) 5.21798 0.260899
\(401\) 37.4439 1.86986 0.934930 0.354832i \(-0.115462\pi\)
0.934930 + 0.354832i \(0.115462\pi\)
\(402\) 0 0
\(403\) −2.97858 −0.148373
\(404\) 43.4355 2.16100
\(405\) 0 0
\(406\) 17.8715 0.886946
\(407\) 9.31163 0.461561
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 15.3717 0.759154
\(411\) 0 0
\(412\) 78.6577 3.87519
\(413\) 6.42754 0.316279
\(414\) 0 0
\(415\) 5.37169 0.263686
\(416\) 2.05333 0.100673
\(417\) 0 0
\(418\) 20.7862 1.01669
\(419\) −3.17935 −0.155321 −0.0776606 0.996980i \(-0.524745\pi\)
−0.0776606 + 0.996980i \(0.524745\pi\)
\(420\) 0 0
\(421\) −16.3074 −0.794775 −0.397388 0.917651i \(-0.630083\pi\)
−0.397388 + 0.917651i \(0.630083\pi\)
\(422\) 59.5296 2.89786
\(423\) 0 0
\(424\) −55.6363 −2.70194
\(425\) −6.17513 −0.299538
\(426\) 0 0
\(427\) −15.0386 −0.727771
\(428\) 77.9227 3.76654
\(429\) 0 0
\(430\) −24.7862 −1.19530
\(431\) −4.58546 −0.220874 −0.110437 0.993883i \(-0.535225\pi\)
−0.110437 + 0.993883i \(0.535225\pi\)
\(432\) 0 0
\(433\) −38.3503 −1.84300 −0.921498 0.388383i \(-0.873034\pi\)
−0.921498 + 0.388383i \(0.873034\pi\)
\(434\) 8.87192 0.425866
\(435\) 0 0
\(436\) 35.2222 1.68684
\(437\) −29.1365 −1.39379
\(438\) 0 0
\(439\) 7.73917 0.369371 0.184685 0.982798i \(-0.440873\pi\)
0.184685 + 0.982798i \(0.440873\pi\)
\(440\) 6.54262 0.311907
\(441\) 0 0
\(442\) −15.3717 −0.731157
\(443\) −34.9185 −1.65903 −0.829514 0.558485i \(-0.811383\pi\)
−0.829514 + 0.558485i \(0.811383\pi\)
\(444\) 0 0
\(445\) −10.1751 −0.482348
\(446\) 6.54262 0.309802
\(447\) 0 0
\(448\) 6.37123 0.301012
\(449\) 6.17513 0.291423 0.145711 0.989327i \(-0.453453\pi\)
0.145711 + 0.989327i \(0.453453\pi\)
\(450\) 0 0
\(451\) 7.38890 0.347930
\(452\) −33.3927 −1.57066
\(453\) 0 0
\(454\) −39.2432 −1.84177
\(455\) −1.19656 −0.0560955
\(456\) 0 0
\(457\) 1.38890 0.0649702 0.0324851 0.999472i \(-0.489658\pi\)
0.0324851 + 0.999472i \(0.489658\pi\)
\(458\) 21.7648 1.01700
\(459\) 0 0
\(460\) −17.5212 −0.816930
\(461\) −28.4826 −1.32657 −0.663283 0.748369i \(-0.730837\pi\)
−0.663283 + 0.748369i \(0.730837\pi\)
\(462\) 0 0
\(463\) 16.3759 0.761053 0.380526 0.924770i \(-0.375743\pi\)
0.380526 + 0.924770i \(0.375743\pi\)
\(464\) −31.3079 −1.45343
\(465\) 0 0
\(466\) 5.41454 0.250824
\(467\) −25.7476 −1.19146 −0.595728 0.803186i \(-0.703136\pi\)
−0.595728 + 0.803186i \(0.703136\pi\)
\(468\) 0 0
\(469\) −11.2138 −0.517804
\(470\) 2.54262 0.117282
\(471\) 0 0
\(472\) −29.3717 −1.35194
\(473\) −11.9143 −0.547820
\(474\) 0 0
\(475\) 6.97858 0.320199
\(476\) 31.0080 1.42125
\(477\) 0 0
\(478\) −6.97858 −0.319193
\(479\) 7.58967 0.346781 0.173391 0.984853i \(-0.444528\pi\)
0.173391 + 0.984853i \(0.444528\pi\)
\(480\) 0 0
\(481\) 7.78202 0.354830
\(482\) −14.9357 −0.680304
\(483\) 0 0
\(484\) −40.1537 −1.82517
\(485\) −1.82487 −0.0828629
\(486\) 0 0
\(487\) 4.41033 0.199851 0.0999255 0.994995i \(-0.468140\pi\)
0.0999255 + 0.994995i \(0.468140\pi\)
\(488\) 68.7215 3.11088
\(489\) 0 0
\(490\) −13.8610 −0.626175
\(491\) 0.0856914 0.00386720 0.00193360 0.999998i \(-0.499385\pi\)
0.00193360 + 0.999998i \(0.499385\pi\)
\(492\) 0 0
\(493\) 37.0508 1.66868
\(494\) 17.3717 0.781589
\(495\) 0 0
\(496\) −15.5422 −0.697863
\(497\) −6.21798 −0.278915
\(498\) 0 0
\(499\) −17.7220 −0.793344 −0.396672 0.917960i \(-0.629835\pi\)
−0.396672 + 0.917960i \(0.629835\pi\)
\(500\) 4.19656 0.187676
\(501\) 0 0
\(502\) 59.5296 2.65694
\(503\) 8.70054 0.387938 0.193969 0.981008i \(-0.437864\pi\)
0.193969 + 0.981008i \(0.437864\pi\)
\(504\) 0 0
\(505\) 10.3503 0.460581
\(506\) −12.4360 −0.552846
\(507\) 0 0
\(508\) −43.6153 −1.93512
\(509\) 33.3545 1.47841 0.739206 0.673480i \(-0.235201\pi\)
0.739206 + 0.673480i \(0.235201\pi\)
\(510\) 0 0
\(511\) 14.3074 0.632923
\(512\) −46.3482 −2.04832
\(513\) 0 0
\(514\) 49.6791 2.19125
\(515\) 18.7434 0.825932
\(516\) 0 0
\(517\) 1.22219 0.0537519
\(518\) −23.1793 −1.01844
\(519\) 0 0
\(520\) 5.46787 0.239782
\(521\) −18.7005 −0.819285 −0.409643 0.912246i \(-0.634347\pi\)
−0.409643 + 0.912246i \(0.634347\pi\)
\(522\) 0 0
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 26.8291 1.17203
\(525\) 0 0
\(526\) −19.9143 −0.868305
\(527\) 18.3931 0.801217
\(528\) 0 0
\(529\) −5.56825 −0.242098
\(530\) −25.3288 −1.10021
\(531\) 0 0
\(532\) −35.0424 −1.51928
\(533\) 6.17513 0.267475
\(534\) 0 0
\(535\) 18.5682 0.802775
\(536\) 51.2432 2.21337
\(537\) 0 0
\(538\) 5.85050 0.252233
\(539\) −6.66273 −0.286984
\(540\) 0 0
\(541\) −41.5296 −1.78550 −0.892749 0.450555i \(-0.851226\pi\)
−0.892749 + 0.450555i \(0.851226\pi\)
\(542\) 27.3288 1.17387
\(543\) 0 0
\(544\) −12.6796 −0.543632
\(545\) 8.39312 0.359522
\(546\) 0 0
\(547\) −7.91431 −0.338391 −0.169196 0.985582i \(-0.554117\pi\)
−0.169196 + 0.985582i \(0.554117\pi\)
\(548\) −70.2646 −3.00155
\(549\) 0 0
\(550\) 2.97858 0.127007
\(551\) −41.8715 −1.78378
\(552\) 0 0
\(553\) 2.13229 0.0906742
\(554\) 3.02142 0.128368
\(555\) 0 0
\(556\) 24.2646 1.02905
\(557\) −42.7005 −1.80928 −0.904640 0.426177i \(-0.859860\pi\)
−0.904640 + 0.426177i \(0.859860\pi\)
\(558\) 0 0
\(559\) −9.95715 −0.421143
\(560\) −6.24361 −0.263841
\(561\) 0 0
\(562\) −29.7648 −1.25555
\(563\) −1.04706 −0.0441282 −0.0220641 0.999757i \(-0.507024\pi\)
−0.0220641 + 0.999757i \(0.507024\pi\)
\(564\) 0 0
\(565\) −7.95715 −0.334760
\(566\) −74.3587 −3.12553
\(567\) 0 0
\(568\) 28.4141 1.19223
\(569\) −16.7778 −0.703362 −0.351681 0.936120i \(-0.614390\pi\)
−0.351681 + 0.936120i \(0.614390\pi\)
\(570\) 0 0
\(571\) −20.6111 −0.862548 −0.431274 0.902221i \(-0.641936\pi\)
−0.431274 + 0.902221i \(0.641936\pi\)
\(572\) 5.02142 0.209956
\(573\) 0 0
\(574\) −18.3931 −0.767714
\(575\) −4.17513 −0.174115
\(576\) 0 0
\(577\) 1.38890 0.0578208 0.0289104 0.999582i \(-0.490796\pi\)
0.0289104 + 0.999582i \(0.490796\pi\)
\(578\) 52.6044 2.18805
\(579\) 0 0
\(580\) −25.1793 −1.04552
\(581\) −6.42754 −0.266659
\(582\) 0 0
\(583\) −12.1751 −0.504243
\(584\) −65.3801 −2.70545
\(585\) 0 0
\(586\) −1.93619 −0.0799833
\(587\) 0.935731 0.0386218 0.0193109 0.999814i \(-0.493853\pi\)
0.0193109 + 0.999814i \(0.493853\pi\)
\(588\) 0 0
\(589\) −20.7862 −0.856482
\(590\) −13.3717 −0.550504
\(591\) 0 0
\(592\) 40.6064 1.66891
\(593\) 0.478807 0.0196622 0.00983112 0.999952i \(-0.496871\pi\)
0.00983112 + 0.999952i \(0.496871\pi\)
\(594\) 0 0
\(595\) 7.38890 0.302916
\(596\) −64.4006 −2.63795
\(597\) 0 0
\(598\) −10.3931 −0.425006
\(599\) −29.0852 −1.18839 −0.594195 0.804321i \(-0.702529\pi\)
−0.594195 + 0.804321i \(0.702529\pi\)
\(600\) 0 0
\(601\) 11.4318 0.466311 0.233155 0.972439i \(-0.425095\pi\)
0.233155 + 0.972439i \(0.425095\pi\)
\(602\) 29.6582 1.20878
\(603\) 0 0
\(604\) −36.0294 −1.46601
\(605\) −9.56825 −0.389005
\(606\) 0 0
\(607\) 27.9143 1.13301 0.566503 0.824059i \(-0.308296\pi\)
0.566503 + 0.824059i \(0.308296\pi\)
\(608\) 14.3293 0.581130
\(609\) 0 0
\(610\) 31.2860 1.26673
\(611\) 1.02142 0.0413223
\(612\) 0 0
\(613\) −4.65394 −0.187971 −0.0939855 0.995574i \(-0.529961\pi\)
−0.0939855 + 0.995574i \(0.529961\pi\)
\(614\) 1.89334 0.0764092
\(615\) 0 0
\(616\) −7.82862 −0.315424
\(617\) 15.9572 0.642411 0.321205 0.947010i \(-0.395912\pi\)
0.321205 + 0.947010i \(0.395912\pi\)
\(618\) 0 0
\(619\) 1.02142 0.0410545 0.0205272 0.999789i \(-0.493466\pi\)
0.0205272 + 0.999789i \(0.493466\pi\)
\(620\) −12.4998 −0.502003
\(621\) 0 0
\(622\) 57.5725 2.30845
\(623\) 12.1751 0.487786
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −84.4225 −3.37420
\(627\) 0 0
\(628\) 11.6926 0.466585
\(629\) −48.0550 −1.91608
\(630\) 0 0
\(631\) −20.4998 −0.816083 −0.408041 0.912963i \(-0.633788\pi\)
−0.408041 + 0.912963i \(0.633788\pi\)
\(632\) −9.74384 −0.387589
\(633\) 0 0
\(634\) −24.0210 −0.953994
\(635\) −10.3931 −0.412438
\(636\) 0 0
\(637\) −5.56825 −0.220622
\(638\) −17.8715 −0.707538
\(639\) 0 0
\(640\) −17.3612 −0.686262
\(641\) −38.2646 −1.51136 −0.755680 0.654941i \(-0.772693\pi\)
−0.755680 + 0.654941i \(0.772693\pi\)
\(642\) 0 0
\(643\) −33.1109 −1.30577 −0.652883 0.757459i \(-0.726440\pi\)
−0.652883 + 0.757459i \(0.726440\pi\)
\(644\) 20.9651 0.826141
\(645\) 0 0
\(646\) −107.273 −4.22058
\(647\) 16.9614 0.666820 0.333410 0.942782i \(-0.391801\pi\)
0.333410 + 0.942782i \(0.391801\pi\)
\(648\) 0 0
\(649\) −6.42754 −0.252303
\(650\) 2.48929 0.0976379
\(651\) 0 0
\(652\) 36.7643 1.43980
\(653\) 19.1709 0.750216 0.375108 0.926981i \(-0.377606\pi\)
0.375108 + 0.926981i \(0.377606\pi\)
\(654\) 0 0
\(655\) 6.39312 0.249800
\(656\) 32.2217 1.25805
\(657\) 0 0
\(658\) −3.04239 −0.118605
\(659\) −37.8715 −1.47526 −0.737631 0.675204i \(-0.764056\pi\)
−0.737631 + 0.675204i \(0.764056\pi\)
\(660\) 0 0
\(661\) 24.3931 0.948782 0.474391 0.880314i \(-0.342668\pi\)
0.474391 + 0.880314i \(0.342668\pi\)
\(662\) 38.1579 1.48305
\(663\) 0 0
\(664\) 29.3717 1.13984
\(665\) −8.35027 −0.323810
\(666\) 0 0
\(667\) 25.0508 0.969971
\(668\) 72.9013 2.82064
\(669\) 0 0
\(670\) 23.3288 0.901272
\(671\) 15.0386 0.580560
\(672\) 0 0
\(673\) −21.1281 −0.814428 −0.407214 0.913333i \(-0.633500\pi\)
−0.407214 + 0.913333i \(0.633500\pi\)
\(674\) −55.6363 −2.14303
\(675\) 0 0
\(676\) 4.19656 0.161406
\(677\) 15.3973 0.591767 0.295884 0.955224i \(-0.404386\pi\)
0.295884 + 0.955224i \(0.404386\pi\)
\(678\) 0 0
\(679\) 2.18356 0.0837972
\(680\) −33.7648 −1.29482
\(681\) 0 0
\(682\) −8.87192 −0.339723
\(683\) −30.0722 −1.15068 −0.575341 0.817914i \(-0.695131\pi\)
−0.575341 + 0.817914i \(0.695131\pi\)
\(684\) 0 0
\(685\) −16.7434 −0.639732
\(686\) 37.4355 1.42929
\(687\) 0 0
\(688\) −51.9562 −1.98081
\(689\) −10.1751 −0.387642
\(690\) 0 0
\(691\) −8.14950 −0.310022 −0.155011 0.987913i \(-0.549541\pi\)
−0.155011 + 0.987913i \(0.549541\pi\)
\(692\) 33.3927 1.26940
\(693\) 0 0
\(694\) 14.3931 0.546355
\(695\) 5.78202 0.219325
\(696\) 0 0
\(697\) −38.1323 −1.44436
\(698\) −68.5082 −2.59307
\(699\) 0 0
\(700\) −5.02142 −0.189792
\(701\) 28.6921 1.08369 0.541843 0.840480i \(-0.317727\pi\)
0.541843 + 0.840480i \(0.317727\pi\)
\(702\) 0 0
\(703\) 54.3074 2.04824
\(704\) −6.37123 −0.240125
\(705\) 0 0
\(706\) 71.5506 2.69284
\(707\) −12.3847 −0.465774
\(708\) 0 0
\(709\) 12.3074 0.462215 0.231108 0.972928i \(-0.425765\pi\)
0.231108 + 0.972928i \(0.425765\pi\)
\(710\) 12.9357 0.485469
\(711\) 0 0
\(712\) −55.6363 −2.08506
\(713\) 12.4360 0.465730
\(714\) 0 0
\(715\) 1.19656 0.0447487
\(716\) 65.3154 2.44095
\(717\) 0 0
\(718\) 31.3288 1.16918
\(719\) 28.7862 1.07355 0.536773 0.843727i \(-0.319643\pi\)
0.536773 + 0.843727i \(0.319643\pi\)
\(720\) 0 0
\(721\) −22.4275 −0.835245
\(722\) 73.9332 2.75151
\(723\) 0 0
\(724\) 66.2302 2.46142
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 34.3931 1.27557 0.637785 0.770214i \(-0.279851\pi\)
0.637785 + 0.770214i \(0.279851\pi\)
\(728\) −6.54262 −0.242485
\(729\) 0 0
\(730\) −29.7648 −1.10164
\(731\) 61.4868 2.27417
\(732\) 0 0
\(733\) −29.0042 −1.07129 −0.535647 0.844442i \(-0.679932\pi\)
−0.535647 + 0.844442i \(0.679932\pi\)
\(734\) −69.4868 −2.56480
\(735\) 0 0
\(736\) −8.57292 −0.316002
\(737\) 11.2138 0.413065
\(738\) 0 0
\(739\) −6.27804 −0.230941 −0.115471 0.993311i \(-0.536838\pi\)
−0.115471 + 0.993311i \(0.536838\pi\)
\(740\) 32.6577 1.20052
\(741\) 0 0
\(742\) 30.3074 1.11262
\(743\) −12.2352 −0.448866 −0.224433 0.974490i \(-0.572053\pi\)
−0.224433 + 0.974490i \(0.572053\pi\)
\(744\) 0 0
\(745\) −15.3461 −0.562236
\(746\) −5.85050 −0.214202
\(747\) 0 0
\(748\) −31.0080 −1.13376
\(749\) −22.2180 −0.811827
\(750\) 0 0
\(751\) −28.8757 −1.05369 −0.526844 0.849962i \(-0.676625\pi\)
−0.526844 + 0.849962i \(0.676625\pi\)
\(752\) 5.32976 0.194357
\(753\) 0 0
\(754\) −14.9357 −0.543927
\(755\) −8.58546 −0.312457
\(756\) 0 0
\(757\) 30.3503 1.10310 0.551550 0.834142i \(-0.314037\pi\)
0.551550 + 0.834142i \(0.314037\pi\)
\(758\) 61.1146 2.21978
\(759\) 0 0
\(760\) 38.1579 1.38413
\(761\) 27.1709 0.984945 0.492473 0.870328i \(-0.336093\pi\)
0.492473 + 0.870328i \(0.336093\pi\)
\(762\) 0 0
\(763\) −10.0428 −0.363575
\(764\) −45.0852 −1.63113
\(765\) 0 0
\(766\) −14.4569 −0.522350
\(767\) −5.37169 −0.193961
\(768\) 0 0
\(769\) −38.3503 −1.38295 −0.691473 0.722402i \(-0.743038\pi\)
−0.691473 + 0.722402i \(0.743038\pi\)
\(770\) −3.56404 −0.128439
\(771\) 0 0
\(772\) 40.8710 1.47098
\(773\) 50.2646 1.80789 0.903946 0.427647i \(-0.140658\pi\)
0.903946 + 0.427647i \(0.140658\pi\)
\(774\) 0 0
\(775\) −2.97858 −0.106994
\(776\) −9.97812 −0.358194
\(777\) 0 0
\(778\) −33.9781 −1.21817
\(779\) 43.0937 1.54399
\(780\) 0 0
\(781\) 6.21798 0.222497
\(782\) 64.1789 2.29503
\(783\) 0 0
\(784\) −29.0550 −1.03768
\(785\) 2.78623 0.0994448
\(786\) 0 0
\(787\) −13.2860 −0.473595 −0.236797 0.971559i \(-0.576098\pi\)
−0.236797 + 0.971559i \(0.576098\pi\)
\(788\) 40.1360 1.42979
\(789\) 0 0
\(790\) −4.43596 −0.157824
\(791\) 9.52119 0.338535
\(792\) 0 0
\(793\) 12.5682 0.446312
\(794\) −30.2008 −1.07179
\(795\) 0 0
\(796\) 24.9995 0.886085
\(797\) −9.82487 −0.348015 −0.174007 0.984744i \(-0.555672\pi\)
−0.174007 + 0.984744i \(0.555672\pi\)
\(798\) 0 0
\(799\) −6.30742 −0.223141
\(800\) 2.05333 0.0725961
\(801\) 0 0
\(802\) 93.2087 3.29131
\(803\) −14.3074 −0.504898
\(804\) 0 0
\(805\) 4.99579 0.176078
\(806\) −7.41454 −0.261166
\(807\) 0 0
\(808\) 56.5939 1.99097
\(809\) 9.91431 0.348569 0.174284 0.984695i \(-0.444239\pi\)
0.174284 + 0.984695i \(0.444239\pi\)
\(810\) 0 0
\(811\) −36.5855 −1.28469 −0.642345 0.766416i \(-0.722038\pi\)
−0.642345 + 0.766416i \(0.722038\pi\)
\(812\) 30.1285 1.05730
\(813\) 0 0
\(814\) 23.1793 0.812436
\(815\) 8.76060 0.306870
\(816\) 0 0
\(817\) −69.4868 −2.43103
\(818\) −34.8500 −1.21850
\(819\) 0 0
\(820\) 25.9143 0.904967
\(821\) 34.4741 1.20316 0.601578 0.798814i \(-0.294539\pi\)
0.601578 + 0.798814i \(0.294539\pi\)
\(822\) 0 0
\(823\) 13.2566 0.462097 0.231048 0.972942i \(-0.425784\pi\)
0.231048 + 0.972942i \(0.425784\pi\)
\(824\) 102.486 3.57028
\(825\) 0 0
\(826\) 16.0000 0.556711
\(827\) −28.1495 −0.978854 −0.489427 0.872044i \(-0.662794\pi\)
−0.489427 + 0.872044i \(0.662794\pi\)
\(828\) 0 0
\(829\) 16.3418 0.567576 0.283788 0.958887i \(-0.408409\pi\)
0.283788 + 0.958887i \(0.408409\pi\)
\(830\) 13.3717 0.464138
\(831\) 0 0
\(832\) −5.32464 −0.184599
\(833\) 34.3847 1.19136
\(834\) 0 0
\(835\) 17.3717 0.601172
\(836\) 35.0424 1.21197
\(837\) 0 0
\(838\) −7.91431 −0.273395
\(839\) 30.3675 1.04840 0.524201 0.851595i \(-0.324364\pi\)
0.524201 + 0.851595i \(0.324364\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −40.5939 −1.39896
\(843\) 0 0
\(844\) 100.358 3.45446
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) 11.4490 0.393391
\(848\) −53.0937 −1.82324
\(849\) 0 0
\(850\) −15.3717 −0.527245
\(851\) −32.4910 −1.11378
\(852\) 0 0
\(853\) −42.1407 −1.44287 −0.721435 0.692482i \(-0.756517\pi\)
−0.721435 + 0.692482i \(0.756517\pi\)
\(854\) −37.4355 −1.28102
\(855\) 0 0
\(856\) 101.529 3.47018
\(857\) 2.17513 0.0743012 0.0371506 0.999310i \(-0.488172\pi\)
0.0371506 + 0.999310i \(0.488172\pi\)
\(858\) 0 0
\(859\) 18.5682 0.633541 0.316770 0.948502i \(-0.397402\pi\)
0.316770 + 0.948502i \(0.397402\pi\)
\(860\) −41.7858 −1.42488
\(861\) 0 0
\(862\) −11.4145 −0.388781
\(863\) −33.7220 −1.14791 −0.573954 0.818887i \(-0.694591\pi\)
−0.573954 + 0.818887i \(0.694591\pi\)
\(864\) 0 0
\(865\) 7.95715 0.270551
\(866\) −95.4649 −3.24403
\(867\) 0 0
\(868\) 14.9567 0.507663
\(869\) −2.13229 −0.0723330
\(870\) 0 0
\(871\) 9.37169 0.317548
\(872\) 45.8924 1.55411
\(873\) 0 0
\(874\) −72.5292 −2.45334
\(875\) −1.19656 −0.0404510
\(876\) 0 0
\(877\) 43.4868 1.46844 0.734222 0.678910i \(-0.237547\pi\)
0.734222 + 0.678910i \(0.237547\pi\)
\(878\) 19.2650 0.650164
\(879\) 0 0
\(880\) 6.24361 0.210472
\(881\) −26.7005 −0.899564 −0.449782 0.893138i \(-0.648498\pi\)
−0.449782 + 0.893138i \(0.648498\pi\)
\(882\) 0 0
\(883\) 20.2990 0.683116 0.341558 0.939861i \(-0.389045\pi\)
0.341558 + 0.939861i \(0.389045\pi\)
\(884\) −25.9143 −0.871593
\(885\) 0 0
\(886\) −86.9223 −2.92021
\(887\) −36.0550 −1.21061 −0.605305 0.795994i \(-0.706949\pi\)
−0.605305 + 0.795994i \(0.706949\pi\)
\(888\) 0 0
\(889\) 12.4360 0.417089
\(890\) −25.3288 −0.849025
\(891\) 0 0
\(892\) 11.0298 0.369307
\(893\) 7.12808 0.238532
\(894\) 0 0
\(895\) 15.5640 0.520248
\(896\) 20.7737 0.694000
\(897\) 0 0
\(898\) 15.3717 0.512960
\(899\) 17.8715 0.596047
\(900\) 0 0
\(901\) 62.8328 2.09326
\(902\) 18.3931 0.612424
\(903\) 0 0
\(904\) −43.5087 −1.44708
\(905\) 15.7820 0.524612
\(906\) 0 0
\(907\) −7.26504 −0.241232 −0.120616 0.992699i \(-0.538487\pi\)
−0.120616 + 0.992699i \(0.538487\pi\)
\(908\) −66.1579 −2.19553
\(909\) 0 0
\(910\) −2.97858 −0.0987389
\(911\) 6.65769 0.220579 0.110290 0.993899i \(-0.464822\pi\)
0.110290 + 0.993899i \(0.464822\pi\)
\(912\) 0 0
\(913\) 6.42754 0.212721
\(914\) 3.45738 0.114360
\(915\) 0 0
\(916\) 36.6921 1.21234
\(917\) −7.64973 −0.252616
\(918\) 0 0
\(919\) −27.1831 −0.896688 −0.448344 0.893861i \(-0.647986\pi\)
−0.448344 + 0.893861i \(0.647986\pi\)
\(920\) −22.8291 −0.752652
\(921\) 0 0
\(922\) −70.9013 −2.33501
\(923\) 5.19656 0.171047
\(924\) 0 0
\(925\) 7.78202 0.255871
\(926\) 40.7643 1.33960
\(927\) 0 0
\(928\) −12.3200 −0.404423
\(929\) 15.3973 0.505170 0.252585 0.967575i \(-0.418719\pi\)
0.252585 + 0.967575i \(0.418719\pi\)
\(930\) 0 0
\(931\) −38.8585 −1.27353
\(932\) 9.12808 0.299000
\(933\) 0 0
\(934\) −64.0932 −2.09719
\(935\) −7.38890 −0.241643
\(936\) 0 0
\(937\) 1.12808 0.0368527 0.0184264 0.999830i \(-0.494134\pi\)
0.0184264 + 0.999830i \(0.494134\pi\)
\(938\) −27.9143 −0.911434
\(939\) 0 0
\(940\) 4.28646 0.139809
\(941\) 30.1407 0.982559 0.491280 0.871002i \(-0.336529\pi\)
0.491280 + 0.871002i \(0.336529\pi\)
\(942\) 0 0
\(943\) −25.7820 −0.839578
\(944\) −28.0294 −0.912279
\(945\) 0 0
\(946\) −29.6582 −0.964270
\(947\) 20.0294 0.650868 0.325434 0.945565i \(-0.394490\pi\)
0.325434 + 0.945565i \(0.394490\pi\)
\(948\) 0 0
\(949\) −11.9572 −0.388146
\(950\) 17.3717 0.563612
\(951\) 0 0
\(952\) 40.4015 1.30942
\(953\) 43.2259 1.40023 0.700113 0.714032i \(-0.253133\pi\)
0.700113 + 0.714032i \(0.253133\pi\)
\(954\) 0 0
\(955\) −10.7434 −0.347648
\(956\) −11.7648 −0.380501
\(957\) 0 0
\(958\) 18.8929 0.610401
\(959\) 20.0344 0.646945
\(960\) 0 0
\(961\) −22.1281 −0.713809
\(962\) 19.3717 0.624568
\(963\) 0 0
\(964\) −25.1793 −0.810972
\(965\) 9.73917 0.313515
\(966\) 0 0
\(967\) 57.6875 1.85511 0.927553 0.373691i \(-0.121908\pi\)
0.927553 + 0.373691i \(0.121908\pi\)
\(968\) −52.3179 −1.68156
\(969\) 0 0
\(970\) −4.54262 −0.145855
\(971\) 19.5296 0.626735 0.313368 0.949632i \(-0.398543\pi\)
0.313368 + 0.949632i \(0.398543\pi\)
\(972\) 0 0
\(973\) −6.91852 −0.221798
\(974\) 10.9786 0.351776
\(975\) 0 0
\(976\) 65.5809 2.09919
\(977\) 40.3074 1.28955 0.644774 0.764373i \(-0.276951\pi\)
0.644774 + 0.764373i \(0.276951\pi\)
\(978\) 0 0
\(979\) −12.1751 −0.389119
\(980\) −23.3675 −0.746447
\(981\) 0 0
\(982\) 0.213311 0.00680702
\(983\) −32.2008 −1.02705 −0.513523 0.858076i \(-0.671660\pi\)
−0.513523 + 0.858076i \(0.671660\pi\)
\(984\) 0 0
\(985\) 9.56404 0.304736
\(986\) 92.2302 2.93721
\(987\) 0 0
\(988\) 29.2860 0.931712
\(989\) 41.5725 1.32193
\(990\) 0 0
\(991\) 26.4826 0.841246 0.420623 0.907235i \(-0.361811\pi\)
0.420623 + 0.907235i \(0.361811\pi\)
\(992\) −6.11599 −0.194183
\(993\) 0 0
\(994\) −15.4783 −0.490943
\(995\) 5.95715 0.188854
\(996\) 0 0
\(997\) 35.1365 1.11278 0.556392 0.830920i \(-0.312185\pi\)
0.556392 + 0.830920i \(0.312185\pi\)
\(998\) −44.1151 −1.39644
\(999\) 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 585.2.a.n.1.3 3
3.2 odd 2 195.2.a.e.1.1 3
4.3 odd 2 9360.2.a.dd.1.2 3
5.2 odd 4 2925.2.c.w.2224.5 6
5.3 odd 4 2925.2.c.w.2224.2 6
5.4 even 2 2925.2.a.bh.1.1 3
12.11 even 2 3120.2.a.bj.1.2 3
13.12 even 2 7605.2.a.bx.1.1 3
15.2 even 4 975.2.c.i.274.2 6
15.8 even 4 975.2.c.i.274.5 6
15.14 odd 2 975.2.a.o.1.3 3
21.20 even 2 9555.2.a.bq.1.1 3
39.38 odd 2 2535.2.a.bc.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.e.1.1 3 3.2 odd 2
585.2.a.n.1.3 3 1.1 even 1 trivial
975.2.a.o.1.3 3 15.14 odd 2
975.2.c.i.274.2 6 15.2 even 4
975.2.c.i.274.5 6 15.8 even 4
2535.2.a.bc.1.3 3 39.38 odd 2
2925.2.a.bh.1.1 3 5.4 even 2
2925.2.c.w.2224.2 6 5.3 odd 4
2925.2.c.w.2224.5 6 5.2 odd 4
3120.2.a.bj.1.2 3 12.11 even 2
7605.2.a.bx.1.1 3 13.12 even 2
9360.2.a.dd.1.2 3 4.3 odd 2
9555.2.a.bq.1.1 3 21.20 even 2