Properties

Label 7623.2.a.cn.1.3
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 4
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: \(4\)
Coefficient field: 4.4.7488.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2541)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.05896\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.32691 q^{2} -0.239314 q^{4} -0.326909 q^{5} +1.00000 q^{7} -2.97136 q^{8} +O(q^{10})\) \(q+1.32691 q^{2} -0.239314 q^{4} -0.326909 q^{5} +1.00000 q^{7} -2.97136 q^{8} -0.433778 q^{10} +0.0589594 q^{13} +1.32691 q^{14} -3.46410 q^{16} +6.28375 q^{17} -2.43378 q^{19} +0.0782337 q^{20} -5.93756 q^{23} -4.89313 q^{25} +0.0782337 q^{26} -0.239314 q^{28} +9.43072 q^{29} +1.98589 q^{31} +1.34618 q^{32} +8.33796 q^{34} -0.326909 q^{35} -6.53242 q^{37} -3.22940 q^{38} +0.971364 q^{40} -0.0782337 q^{41} -4.92177 q^{43} -7.87861 q^{46} -6.01621 q^{47} +1.00000 q^{49} -6.49274 q^{50} -0.0141098 q^{52} +5.94273 q^{53} -2.97136 q^{56} +12.5137 q^{58} -1.13719 q^{59} -12.2807 q^{61} +2.63509 q^{62} +8.71446 q^{64} -0.0192743 q^{65} +8.75721 q^{67} -1.50379 q^{68} -0.433778 q^{70} +1.65857 q^{71} -9.40514 q^{73} -8.66793 q^{74} +0.582436 q^{76} +13.0717 q^{79} +1.13244 q^{80} -0.103809 q^{82} -7.49443 q^{83} -2.05421 q^{85} -6.53073 q^{86} -10.6013 q^{89} +0.0589594 q^{91} +1.42094 q^{92} -7.98297 q^{94} +0.795623 q^{95} -6.71278 q^{97} +1.32691 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} + 4q^{4} + 2q^{5} + 4q^{7} + O(q^{10}) \) \( 4q + 2q^{2} + 4q^{4} + 2q^{5} + 4q^{7} - 10q^{10} - 10q^{13} + 2q^{14} + 6q^{17} - 18q^{19} + 2q^{23} - 8q^{25} + 4q^{28} + 6q^{29} + 12q^{32} - 2q^{34} + 2q^{35} - 4q^{37} - 8q^{40} - 20q^{43} - 16q^{46} + 6q^{47} + 4q^{49} - 24q^{50} - 8q^{52} + 24q^{58} + 6q^{59} + 10q^{61} - 16q^{64} - 10q^{65} + 4q^{67} + 28q^{68} - 10q^{70} + 6q^{71} - 34q^{73} - 36q^{74} - 36q^{76} - 24q^{79} - 12q^{80} + 28q^{82} + 6q^{83} + 8q^{85} - 38q^{86} - 18q^{89} - 10q^{91} - 24q^{92} + 6q^{94} - 18q^{95} - 10q^{97} + 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.32691 0.938266 0.469133 0.883128i \(-0.344566\pi\)
0.469133 + 0.883128i \(0.344566\pi\)
\(3\) 0 0
\(4\) −0.239314 −0.119657
\(5\) −0.326909 −0.146198 −0.0730990 0.997325i \(-0.523289\pi\)
−0.0730990 + 0.997325i \(0.523289\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −2.97136 −1.05054
\(9\) 0 0
\(10\) −0.433778 −0.137173
\(11\) 0 0
\(12\) 0 0
\(13\) 0.0589594 0.0163524 0.00817619 0.999967i \(-0.497397\pi\)
0.00817619 + 0.999967i \(0.497397\pi\)
\(14\) 1.32691 0.354631
\(15\) 0 0
\(16\) −3.46410 −0.866025
\(17\) 6.28375 1.52403 0.762016 0.647558i \(-0.224210\pi\)
0.762016 + 0.647558i \(0.224210\pi\)
\(18\) 0 0
\(19\) −2.43378 −0.558347 −0.279173 0.960241i \(-0.590060\pi\)
−0.279173 + 0.960241i \(0.590060\pi\)
\(20\) 0.0782337 0.0174936
\(21\) 0 0
\(22\) 0 0
\(23\) −5.93756 −1.23807 −0.619034 0.785364i \(-0.712476\pi\)
−0.619034 + 0.785364i \(0.712476\pi\)
\(24\) 0 0
\(25\) −4.89313 −0.978626
\(26\) 0.0782337 0.0153429
\(27\) 0 0
\(28\) −0.239314 −0.0452260
\(29\) 9.43072 1.75124 0.875620 0.483000i \(-0.160453\pi\)
0.875620 + 0.483000i \(0.160453\pi\)
\(30\) 0 0
\(31\) 1.98589 0.356676 0.178338 0.983969i \(-0.442928\pi\)
0.178338 + 0.983969i \(0.442928\pi\)
\(32\) 1.34618 0.237974
\(33\) 0 0
\(34\) 8.33796 1.42995
\(35\) −0.326909 −0.0552576
\(36\) 0 0
\(37\) −6.53242 −1.07392 −0.536962 0.843607i \(-0.680428\pi\)
−0.536962 + 0.843607i \(0.680428\pi\)
\(38\) −3.22940 −0.523878
\(39\) 0 0
\(40\) 0.971364 0.153586
\(41\) −0.0782337 −0.0122180 −0.00610902 0.999981i \(-0.501945\pi\)
−0.00610902 + 0.999981i \(0.501945\pi\)
\(42\) 0 0
\(43\) −4.92177 −0.750562 −0.375281 0.926911i \(-0.622454\pi\)
−0.375281 + 0.926911i \(0.622454\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −7.87861 −1.16164
\(47\) −6.01621 −0.877555 −0.438778 0.898596i \(-0.644588\pi\)
−0.438778 + 0.898596i \(0.644588\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −6.49274 −0.918212
\(51\) 0 0
\(52\) −0.0141098 −0.00195667
\(53\) 5.94273 0.816297 0.408148 0.912916i \(-0.366175\pi\)
0.408148 + 0.912916i \(0.366175\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −2.97136 −0.397065
\(57\) 0 0
\(58\) 12.5137 1.64313
\(59\) −1.13719 −0.148050 −0.0740250 0.997256i \(-0.523584\pi\)
−0.0740250 + 0.997256i \(0.523584\pi\)
\(60\) 0 0
\(61\) −12.2807 −1.57238 −0.786190 0.617984i \(-0.787950\pi\)
−0.786190 + 0.617984i \(0.787950\pi\)
\(62\) 2.63509 0.334657
\(63\) 0 0
\(64\) 8.71446 1.08931
\(65\) −0.0192743 −0.00239069
\(66\) 0 0
\(67\) 8.75721 1.06986 0.534932 0.844895i \(-0.320337\pi\)
0.534932 + 0.844895i \(0.320337\pi\)
\(68\) −1.50379 −0.182361
\(69\) 0 0
\(70\) −0.433778 −0.0518464
\(71\) 1.65857 0.196836 0.0984178 0.995145i \(-0.468622\pi\)
0.0984178 + 0.995145i \(0.468622\pi\)
\(72\) 0 0
\(73\) −9.40514 −1.10079 −0.550394 0.834905i \(-0.685523\pi\)
−0.550394 + 0.834905i \(0.685523\pi\)
\(74\) −8.66793 −1.00763
\(75\) 0 0
\(76\) 0.582436 0.0668100
\(77\) 0 0
\(78\) 0 0
\(79\) 13.0717 1.47068 0.735340 0.677698i \(-0.237022\pi\)
0.735340 + 0.677698i \(0.237022\pi\)
\(80\) 1.13244 0.126611
\(81\) 0 0
\(82\) −0.103809 −0.0114638
\(83\) −7.49443 −0.822620 −0.411310 0.911496i \(-0.634929\pi\)
−0.411310 + 0.911496i \(0.634929\pi\)
\(84\) 0 0
\(85\) −2.05421 −0.222810
\(86\) −6.53073 −0.704227
\(87\) 0 0
\(88\) 0 0
\(89\) −10.6013 −1.12373 −0.561867 0.827227i \(-0.689917\pi\)
−0.561867 + 0.827227i \(0.689917\pi\)
\(90\) 0 0
\(91\) 0.0589594 0.00618062
\(92\) 1.42094 0.148143
\(93\) 0 0
\(94\) −7.98297 −0.823380
\(95\) 0.795623 0.0816292
\(96\) 0 0
\(97\) −6.71278 −0.681579 −0.340790 0.940140i \(-0.610694\pi\)
−0.340790 + 0.940140i \(0.610694\pi\)
\(98\) 1.32691 0.134038
\(99\) 0 0
\(100\) 1.17099 0.117099
\(101\) 8.44830 0.840638 0.420319 0.907376i \(-0.361918\pi\)
0.420319 + 0.907376i \(0.361918\pi\)
\(102\) 0 0
\(103\) −15.9631 −1.57289 −0.786447 0.617657i \(-0.788082\pi\)
−0.786447 + 0.617657i \(0.788082\pi\)
\(104\) −0.175190 −0.0171788
\(105\) 0 0
\(106\) 7.88546 0.765903
\(107\) −5.17044 −0.499845 −0.249923 0.968266i \(-0.580405\pi\)
−0.249923 + 0.968266i \(0.580405\pi\)
\(108\) 0 0
\(109\) −3.02444 −0.289689 −0.144844 0.989454i \(-0.546268\pi\)
−0.144844 + 0.989454i \(0.546268\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.46410 −0.327327
\(113\) 6.23584 0.586618 0.293309 0.956018i \(-0.405243\pi\)
0.293309 + 0.956018i \(0.405243\pi\)
\(114\) 0 0
\(115\) 1.94104 0.181003
\(116\) −2.25690 −0.209548
\(117\) 0 0
\(118\) −1.50895 −0.138910
\(119\) 6.28375 0.576030
\(120\) 0 0
\(121\) 0 0
\(122\) −16.2953 −1.47531
\(123\) 0 0
\(124\) −0.475251 −0.0426788
\(125\) 3.23415 0.289271
\(126\) 0 0
\(127\) −7.40039 −0.656679 −0.328339 0.944560i \(-0.606489\pi\)
−0.328339 + 0.944560i \(0.606489\pi\)
\(128\) 8.87093 0.784087
\(129\) 0 0
\(130\) −0.0255753 −0.00224310
\(131\) 10.9479 0.956522 0.478261 0.878218i \(-0.341267\pi\)
0.478261 + 0.878218i \(0.341267\pi\)
\(132\) 0 0
\(133\) −2.43378 −0.211035
\(134\) 11.6200 1.00382
\(135\) 0 0
\(136\) −18.6713 −1.60105
\(137\) −4.96620 −0.424291 −0.212146 0.977238i \(-0.568045\pi\)
−0.212146 + 0.977238i \(0.568045\pi\)
\(138\) 0 0
\(139\) 6.19560 0.525504 0.262752 0.964863i \(-0.415370\pi\)
0.262752 + 0.964863i \(0.415370\pi\)
\(140\) 0.0782337 0.00661195
\(141\) 0 0
\(142\) 2.20077 0.184684
\(143\) 0 0
\(144\) 0 0
\(145\) −3.08298 −0.256028
\(146\) −12.4798 −1.03283
\(147\) 0 0
\(148\) 1.56330 0.128502
\(149\) −20.3158 −1.66433 −0.832166 0.554527i \(-0.812899\pi\)
−0.832166 + 0.554527i \(0.812899\pi\)
\(150\) 0 0
\(151\) −9.49959 −0.773066 −0.386533 0.922276i \(-0.626327\pi\)
−0.386533 + 0.922276i \(0.626327\pi\)
\(152\) 7.23164 0.586564
\(153\) 0 0
\(154\) 0 0
\(155\) −0.649205 −0.0521454
\(156\) 0 0
\(157\) −17.4500 −1.39266 −0.696330 0.717721i \(-0.745185\pi\)
−0.696330 + 0.717721i \(0.745185\pi\)
\(158\) 17.3449 1.37989
\(159\) 0 0
\(160\) −0.440079 −0.0347913
\(161\) −5.93756 −0.467946
\(162\) 0 0
\(163\) −7.35304 −0.575934 −0.287967 0.957640i \(-0.592979\pi\)
−0.287967 + 0.957640i \(0.592979\pi\)
\(164\) 0.0187224 0.00146197
\(165\) 0 0
\(166\) −9.94442 −0.771836
\(167\) −4.74141 −0.366901 −0.183451 0.983029i \(-0.558727\pi\)
−0.183451 + 0.983029i \(0.558727\pi\)
\(168\) 0 0
\(169\) −12.9965 −0.999733
\(170\) −2.72575 −0.209055
\(171\) 0 0
\(172\) 1.17785 0.0898099
\(173\) −8.04791 −0.611871 −0.305936 0.952052i \(-0.598969\pi\)
−0.305936 + 0.952052i \(0.598969\pi\)
\(174\) 0 0
\(175\) −4.89313 −0.369886
\(176\) 0 0
\(177\) 0 0
\(178\) −14.0669 −1.05436
\(179\) 10.5393 0.787742 0.393871 0.919166i \(-0.371136\pi\)
0.393871 + 0.919166i \(0.371136\pi\)
\(180\) 0 0
\(181\) −23.5389 −1.74963 −0.874815 0.484457i \(-0.839017\pi\)
−0.874815 + 0.484457i \(0.839017\pi\)
\(182\) 0.0782337 0.00579907
\(183\) 0 0
\(184\) 17.6427 1.30063
\(185\) 2.13550 0.157005
\(186\) 0 0
\(187\) 0 0
\(188\) 1.43976 0.105005
\(189\) 0 0
\(190\) 1.05572 0.0765899
\(191\) −17.9188 −1.29656 −0.648281 0.761401i \(-0.724512\pi\)
−0.648281 + 0.761401i \(0.724512\pi\)
\(192\) 0 0
\(193\) −11.1962 −0.805917 −0.402958 0.915218i \(-0.632018\pi\)
−0.402958 + 0.915218i \(0.632018\pi\)
\(194\) −8.90724 −0.639503
\(195\) 0 0
\(196\) −0.239314 −0.0170938
\(197\) 5.32512 0.379399 0.189700 0.981842i \(-0.439249\pi\)
0.189700 + 0.981842i \(0.439249\pi\)
\(198\) 0 0
\(199\) 9.05938 0.642202 0.321101 0.947045i \(-0.395947\pi\)
0.321101 + 0.947045i \(0.395947\pi\)
\(200\) 14.5393 1.02808
\(201\) 0 0
\(202\) 11.2101 0.788742
\(203\) 9.43072 0.661907
\(204\) 0 0
\(205\) 0.0255753 0.00178625
\(206\) −21.1816 −1.47579
\(207\) 0 0
\(208\) −0.204241 −0.0141616
\(209\) 0 0
\(210\) 0 0
\(211\) −27.8370 −1.91638 −0.958189 0.286136i \(-0.907629\pi\)
−0.958189 + 0.286136i \(0.907629\pi\)
\(212\) −1.42218 −0.0976755
\(213\) 0 0
\(214\) −6.86070 −0.468988
\(215\) 1.60897 0.109731
\(216\) 0 0
\(217\) 1.98589 0.134811
\(218\) −4.01315 −0.271805
\(219\) 0 0
\(220\) 0 0
\(221\) 0.370486 0.0249216
\(222\) 0 0
\(223\) 19.3064 1.29285 0.646426 0.762977i \(-0.276263\pi\)
0.646426 + 0.762977i \(0.276263\pi\)
\(224\) 1.34618 0.0899456
\(225\) 0 0
\(226\) 8.27439 0.550404
\(227\) 1.31586 0.0873366 0.0436683 0.999046i \(-0.486096\pi\)
0.0436683 + 0.999046i \(0.486096\pi\)
\(228\) 0 0
\(229\) −0.440079 −0.0290812 −0.0145406 0.999894i \(-0.504629\pi\)
−0.0145406 + 0.999894i \(0.504629\pi\)
\(230\) 2.57558 0.169829
\(231\) 0 0
\(232\) −28.0221 −1.83974
\(233\) −5.29786 −0.347074 −0.173537 0.984827i \(-0.555520\pi\)
−0.173537 + 0.984827i \(0.555520\pi\)
\(234\) 0 0
\(235\) 1.96675 0.128297
\(236\) 0.272146 0.0177152
\(237\) 0 0
\(238\) 8.33796 0.540470
\(239\) 24.3293 1.57373 0.786866 0.617123i \(-0.211702\pi\)
0.786866 + 0.617123i \(0.211702\pi\)
\(240\) 0 0
\(241\) 26.5165 1.70808 0.854040 0.520208i \(-0.174145\pi\)
0.854040 + 0.520208i \(0.174145\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 2.93894 0.188146
\(245\) −0.326909 −0.0208854
\(246\) 0 0
\(247\) −0.143494 −0.00913030
\(248\) −5.90080 −0.374701
\(249\) 0 0
\(250\) 4.29142 0.271413
\(251\) 19.1828 1.21081 0.605403 0.795919i \(-0.293012\pi\)
0.605403 + 0.795919i \(0.293012\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −9.81965 −0.616139
\(255\) 0 0
\(256\) −5.65801 −0.353626
\(257\) −12.2417 −0.763618 −0.381809 0.924241i \(-0.624699\pi\)
−0.381809 + 0.924241i \(0.624699\pi\)
\(258\) 0 0
\(259\) −6.53242 −0.405905
\(260\) 0.00461261 0.000286062 0
\(261\) 0 0
\(262\) 14.5269 0.897472
\(263\) −18.2025 −1.12241 −0.561206 0.827676i \(-0.689662\pi\)
−0.561206 + 0.827676i \(0.689662\pi\)
\(264\) 0 0
\(265\) −1.94273 −0.119341
\(266\) −3.22940 −0.198007
\(267\) 0 0
\(268\) −2.09572 −0.128016
\(269\) −9.40683 −0.573545 −0.286772 0.957999i \(-0.592582\pi\)
−0.286772 + 0.957999i \(0.592582\pi\)
\(270\) 0 0
\(271\) 14.5252 0.882341 0.441170 0.897423i \(-0.354563\pi\)
0.441170 + 0.897423i \(0.354563\pi\)
\(272\) −21.7675 −1.31985
\(273\) 0 0
\(274\) −6.58969 −0.398098
\(275\) 0 0
\(276\) 0 0
\(277\) −8.26447 −0.496564 −0.248282 0.968688i \(-0.579866\pi\)
−0.248282 + 0.968688i \(0.579866\pi\)
\(278\) 8.22100 0.493063
\(279\) 0 0
\(280\) 0.971364 0.0580501
\(281\) −24.0080 −1.43220 −0.716098 0.697999i \(-0.754074\pi\)
−0.716098 + 0.697999i \(0.754074\pi\)
\(282\) 0 0
\(283\) −16.0512 −0.954142 −0.477071 0.878865i \(-0.658302\pi\)
−0.477071 + 0.878865i \(0.658302\pi\)
\(284\) −0.396917 −0.0235527
\(285\) 0 0
\(286\) 0 0
\(287\) −0.0782337 −0.00461799
\(288\) 0 0
\(289\) 22.4855 1.32268
\(290\) −4.09084 −0.240222
\(291\) 0 0
\(292\) 2.25078 0.131717
\(293\) 7.60088 0.444048 0.222024 0.975041i \(-0.428734\pi\)
0.222024 + 0.975041i \(0.428734\pi\)
\(294\) 0 0
\(295\) 0.371758 0.0216446
\(296\) 19.4102 1.12820
\(297\) 0 0
\(298\) −26.9572 −1.56159
\(299\) −0.350075 −0.0202454
\(300\) 0 0
\(301\) −4.92177 −0.283686
\(302\) −12.6051 −0.725341
\(303\) 0 0
\(304\) 8.43085 0.483543
\(305\) 4.01466 0.229879
\(306\) 0 0
\(307\) −19.7345 −1.12631 −0.563153 0.826353i \(-0.690412\pi\)
−0.563153 + 0.826353i \(0.690412\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −0.861435 −0.0489262
\(311\) 30.3127 1.71888 0.859438 0.511240i \(-0.170814\pi\)
0.859438 + 0.511240i \(0.170814\pi\)
\(312\) 0 0
\(313\) −19.7850 −1.11832 −0.559158 0.829061i \(-0.688875\pi\)
−0.559158 + 0.829061i \(0.688875\pi\)
\(314\) −23.1545 −1.30669
\(315\) 0 0
\(316\) −3.12824 −0.175977
\(317\) 18.9639 1.06512 0.532558 0.846393i \(-0.321231\pi\)
0.532558 + 0.846393i \(0.321231\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −2.84883 −0.159255
\(321\) 0 0
\(322\) −7.87861 −0.439057
\(323\) −15.2932 −0.850939
\(324\) 0 0
\(325\) −0.288496 −0.0160029
\(326\) −9.75681 −0.540380
\(327\) 0 0
\(328\) 0.232461 0.0128355
\(329\) −6.01621 −0.331685
\(330\) 0 0
\(331\) −3.98895 −0.219253 −0.109626 0.993973i \(-0.534965\pi\)
−0.109626 + 0.993973i \(0.534965\pi\)
\(332\) 1.79352 0.0984321
\(333\) 0 0
\(334\) −6.29142 −0.344251
\(335\) −2.86281 −0.156412
\(336\) 0 0
\(337\) −27.5690 −1.50178 −0.750891 0.660426i \(-0.770376\pi\)
−0.750891 + 0.660426i \(0.770376\pi\)
\(338\) −17.2452 −0.938015
\(339\) 0 0
\(340\) 0.491601 0.0266608
\(341\) 0 0
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 14.6244 0.788493
\(345\) 0 0
\(346\) −10.6788 −0.574098
\(347\) −32.5280 −1.74619 −0.873097 0.487547i \(-0.837892\pi\)
−0.873097 + 0.487547i \(0.837892\pi\)
\(348\) 0 0
\(349\) −28.7099 −1.53680 −0.768402 0.639968i \(-0.778948\pi\)
−0.768402 + 0.639968i \(0.778948\pi\)
\(350\) −6.49274 −0.347051
\(351\) 0 0
\(352\) 0 0
\(353\) 0.421356 0.0224265 0.0112133 0.999937i \(-0.496431\pi\)
0.0112133 + 0.999937i \(0.496431\pi\)
\(354\) 0 0
\(355\) −0.542199 −0.0287770
\(356\) 2.53703 0.134463
\(357\) 0 0
\(358\) 13.9847 0.739112
\(359\) 8.17340 0.431376 0.215688 0.976462i \(-0.430801\pi\)
0.215688 + 0.976462i \(0.430801\pi\)
\(360\) 0 0
\(361\) −13.0767 −0.688249
\(362\) −31.2339 −1.64162
\(363\) 0 0
\(364\) −0.0141098 −0.000739554 0
\(365\) 3.07462 0.160933
\(366\) 0 0
\(367\) −10.9415 −0.571139 −0.285570 0.958358i \(-0.592183\pi\)
−0.285570 + 0.958358i \(0.592183\pi\)
\(368\) 20.5683 1.07220
\(369\) 0 0
\(370\) 2.83362 0.147313
\(371\) 5.94273 0.308531
\(372\) 0 0
\(373\) −15.2953 −0.791963 −0.395982 0.918258i \(-0.629596\pi\)
−0.395982 + 0.918258i \(0.629596\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 17.8764 0.921903
\(377\) 0.556029 0.0286370
\(378\) 0 0
\(379\) 32.1698 1.65245 0.826226 0.563339i \(-0.190484\pi\)
0.826226 + 0.563339i \(0.190484\pi\)
\(380\) −0.190403 −0.00976749
\(381\) 0 0
\(382\) −23.7767 −1.21652
\(383\) 20.5485 1.04998 0.524991 0.851108i \(-0.324069\pi\)
0.524991 + 0.851108i \(0.324069\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −14.8563 −0.756164
\(387\) 0 0
\(388\) 1.60646 0.0815556
\(389\) −16.9025 −0.856992 −0.428496 0.903544i \(-0.640956\pi\)
−0.428496 + 0.903544i \(0.640956\pi\)
\(390\) 0 0
\(391\) −37.3102 −1.88686
\(392\) −2.97136 −0.150077
\(393\) 0 0
\(394\) 7.06595 0.355977
\(395\) −4.27325 −0.215011
\(396\) 0 0
\(397\) 17.7606 0.891378 0.445689 0.895188i \(-0.352959\pi\)
0.445689 + 0.895188i \(0.352959\pi\)
\(398\) 12.0210 0.602556
\(399\) 0 0
\(400\) 16.9503 0.847515
\(401\) −8.19963 −0.409470 −0.204735 0.978817i \(-0.565633\pi\)
−0.204735 + 0.978817i \(0.565633\pi\)
\(402\) 0 0
\(403\) 0.117087 0.00583251
\(404\) −2.02179 −0.100588
\(405\) 0 0
\(406\) 12.5137 0.621044
\(407\) 0 0
\(408\) 0 0
\(409\) 36.5772 1.80862 0.904312 0.426871i \(-0.140384\pi\)
0.904312 + 0.426871i \(0.140384\pi\)
\(410\) 0.0339360 0.00167598
\(411\) 0 0
\(412\) 3.82020 0.188208
\(413\) −1.13719 −0.0559576
\(414\) 0 0
\(415\) 2.44999 0.120265
\(416\) 0.0793701 0.00389144
\(417\) 0 0
\(418\) 0 0
\(419\) 13.8296 0.675618 0.337809 0.941215i \(-0.390314\pi\)
0.337809 + 0.941215i \(0.390314\pi\)
\(420\) 0 0
\(421\) −30.8503 −1.50355 −0.751775 0.659419i \(-0.770802\pi\)
−0.751775 + 0.659419i \(0.770802\pi\)
\(422\) −36.9371 −1.79807
\(423\) 0 0
\(424\) −17.6580 −0.857549
\(425\) −30.7472 −1.49146
\(426\) 0 0
\(427\) −12.2807 −0.594304
\(428\) 1.23736 0.0598099
\(429\) 0 0
\(430\) 2.13495 0.102957
\(431\) −3.61107 −0.173939 −0.0869696 0.996211i \(-0.527718\pi\)
−0.0869696 + 0.996211i \(0.527718\pi\)
\(432\) 0 0
\(433\) 38.0051 1.82641 0.913203 0.407504i \(-0.133601\pi\)
0.913203 + 0.407504i \(0.133601\pi\)
\(434\) 2.63509 0.126489
\(435\) 0 0
\(436\) 0.723790 0.0346632
\(437\) 14.4507 0.691271
\(438\) 0 0
\(439\) 25.8855 1.23545 0.617723 0.786396i \(-0.288055\pi\)
0.617723 + 0.786396i \(0.288055\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0.491601 0.0233831
\(443\) −14.8247 −0.704342 −0.352171 0.935936i \(-0.614556\pi\)
−0.352171 + 0.935936i \(0.614556\pi\)
\(444\) 0 0
\(445\) 3.46565 0.164288
\(446\) 25.6178 1.21304
\(447\) 0 0
\(448\) 8.71446 0.411720
\(449\) −37.8071 −1.78423 −0.892114 0.451810i \(-0.850779\pi\)
−0.892114 + 0.451810i \(0.850779\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −1.49232 −0.0701929
\(453\) 0 0
\(454\) 1.74602 0.0819450
\(455\) −0.0192743 −0.000903594 0
\(456\) 0 0
\(457\) 23.5114 1.09982 0.549908 0.835226i \(-0.314663\pi\)
0.549908 + 0.835226i \(0.314663\pi\)
\(458\) −0.583944 −0.0272859
\(459\) 0 0
\(460\) −0.464518 −0.0216582
\(461\) −40.9211 −1.90589 −0.952943 0.303149i \(-0.901962\pi\)
−0.952943 + 0.303149i \(0.901962\pi\)
\(462\) 0 0
\(463\) 0.0607472 0.00282316 0.00141158 0.999999i \(-0.499551\pi\)
0.00141158 + 0.999999i \(0.499551\pi\)
\(464\) −32.6690 −1.51662
\(465\) 0 0
\(466\) −7.02977 −0.325648
\(467\) 37.4362 1.73234 0.866170 0.499749i \(-0.166575\pi\)
0.866170 + 0.499749i \(0.166575\pi\)
\(468\) 0 0
\(469\) 8.75721 0.404370
\(470\) 2.60970 0.120376
\(471\) 0 0
\(472\) 3.37902 0.155532
\(473\) 0 0
\(474\) 0 0
\(475\) 11.9088 0.546413
\(476\) −1.50379 −0.0689259
\(477\) 0 0
\(478\) 32.2828 1.47658
\(479\) 31.3978 1.43460 0.717300 0.696764i \(-0.245378\pi\)
0.717300 + 0.696764i \(0.245378\pi\)
\(480\) 0 0
\(481\) −0.385147 −0.0175612
\(482\) 35.1850 1.60263
\(483\) 0 0
\(484\) 0 0
\(485\) 2.19446 0.0996455
\(486\) 0 0
\(487\) −20.3881 −0.923873 −0.461937 0.886913i \(-0.652845\pi\)
−0.461937 + 0.886913i \(0.652845\pi\)
\(488\) 36.4904 1.65184
\(489\) 0 0
\(490\) −0.433778 −0.0195961
\(491\) 11.6120 0.524043 0.262022 0.965062i \(-0.415611\pi\)
0.262022 + 0.965062i \(0.415611\pi\)
\(492\) 0 0
\(493\) 59.2602 2.66895
\(494\) −0.190403 −0.00856665
\(495\) 0 0
\(496\) −6.87933 −0.308891
\(497\) 1.65857 0.0743968
\(498\) 0 0
\(499\) 38.1606 1.70830 0.854151 0.520025i \(-0.174078\pi\)
0.854151 + 0.520025i \(0.174078\pi\)
\(500\) −0.773976 −0.0346133
\(501\) 0 0
\(502\) 25.4538 1.13606
\(503\) −32.5721 −1.45232 −0.726160 0.687526i \(-0.758697\pi\)
−0.726160 + 0.687526i \(0.758697\pi\)
\(504\) 0 0
\(505\) −2.76182 −0.122899
\(506\) 0 0
\(507\) 0 0
\(508\) 1.77102 0.0785761
\(509\) 23.4898 1.04117 0.520584 0.853811i \(-0.325714\pi\)
0.520584 + 0.853811i \(0.325714\pi\)
\(510\) 0 0
\(511\) −9.40514 −0.416059
\(512\) −25.2495 −1.11588
\(513\) 0 0
\(514\) −16.2436 −0.716477
\(515\) 5.21849 0.229954
\(516\) 0 0
\(517\) 0 0
\(518\) −8.66793 −0.380847
\(519\) 0 0
\(520\) 0.0572710 0.00251150
\(521\) 34.5918 1.51549 0.757747 0.652548i \(-0.226300\pi\)
0.757747 + 0.652548i \(0.226300\pi\)
\(522\) 0 0
\(523\) −8.30429 −0.363121 −0.181561 0.983380i \(-0.558115\pi\)
−0.181561 + 0.983380i \(0.558115\pi\)
\(524\) −2.61998 −0.114454
\(525\) 0 0
\(526\) −24.1530 −1.05312
\(527\) 12.4788 0.543586
\(528\) 0 0
\(529\) 12.2547 0.532812
\(530\) −2.57782 −0.111974
\(531\) 0 0
\(532\) 0.582436 0.0252518
\(533\) −0.00461261 −0.000199794 0
\(534\) 0 0
\(535\) 1.69026 0.0730764
\(536\) −26.0209 −1.12393
\(537\) 0 0
\(538\) −12.4820 −0.538137
\(539\) 0 0
\(540\) 0 0
\(541\) 23.3943 1.00580 0.502899 0.864345i \(-0.332267\pi\)
0.502899 + 0.864345i \(0.332267\pi\)
\(542\) 19.2736 0.827871
\(543\) 0 0
\(544\) 8.45907 0.362680
\(545\) 0.988715 0.0423519
\(546\) 0 0
\(547\) 19.1606 0.819247 0.409623 0.912255i \(-0.365660\pi\)
0.409623 + 0.912255i \(0.365660\pi\)
\(548\) 1.18848 0.0507693
\(549\) 0 0
\(550\) 0 0
\(551\) −22.9523 −0.977800
\(552\) 0 0
\(553\) 13.0717 0.555865
\(554\) −10.9662 −0.465909
\(555\) 0 0
\(556\) −1.48269 −0.0628801
\(557\) 25.9067 1.09770 0.548852 0.835920i \(-0.315065\pi\)
0.548852 + 0.835920i \(0.315065\pi\)
\(558\) 0 0
\(559\) −0.290184 −0.0122735
\(560\) 1.13244 0.0478545
\(561\) 0 0
\(562\) −31.8564 −1.34378
\(563\) −2.58381 −0.108895 −0.0544473 0.998517i \(-0.517340\pi\)
−0.0544473 + 0.998517i \(0.517340\pi\)
\(564\) 0 0
\(565\) −2.03855 −0.0857624
\(566\) −21.2984 −0.895239
\(567\) 0 0
\(568\) −4.92820 −0.206783
\(569\) 25.6769 1.07643 0.538215 0.842807i \(-0.319099\pi\)
0.538215 + 0.842807i \(0.319099\pi\)
\(570\) 0 0
\(571\) −1.12907 −0.0472500 −0.0236250 0.999721i \(-0.507521\pi\)
−0.0236250 + 0.999721i \(0.507521\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −0.103809 −0.00433290
\(575\) 29.0533 1.21161
\(576\) 0 0
\(577\) −33.9141 −1.41186 −0.705932 0.708280i \(-0.749472\pi\)
−0.705932 + 0.708280i \(0.749472\pi\)
\(578\) 29.8362 1.24102
\(579\) 0 0
\(580\) 0.737800 0.0306355
\(581\) −7.49443 −0.310921
\(582\) 0 0
\(583\) 0 0
\(584\) 27.9461 1.15642
\(585\) 0 0
\(586\) 10.0857 0.416635
\(587\) −42.7729 −1.76543 −0.882713 0.469912i \(-0.844286\pi\)
−0.882713 + 0.469912i \(0.844286\pi\)
\(588\) 0 0
\(589\) −4.83322 −0.199149
\(590\) 0.493289 0.0203084
\(591\) 0 0
\(592\) 22.6290 0.930045
\(593\) −30.9085 −1.26926 −0.634630 0.772816i \(-0.718848\pi\)
−0.634630 + 0.772816i \(0.718848\pi\)
\(594\) 0 0
\(595\) −2.05421 −0.0842144
\(596\) 4.86184 0.199149
\(597\) 0 0
\(598\) −0.464518 −0.0189955
\(599\) −10.2428 −0.418509 −0.209255 0.977861i \(-0.567104\pi\)
−0.209255 + 0.977861i \(0.567104\pi\)
\(600\) 0 0
\(601\) −27.5452 −1.12359 −0.561795 0.827276i \(-0.689889\pi\)
−0.561795 + 0.827276i \(0.689889\pi\)
\(602\) −6.53073 −0.266173
\(603\) 0 0
\(604\) 2.27338 0.0925026
\(605\) 0 0
\(606\) 0 0
\(607\) 17.3982 0.706171 0.353085 0.935591i \(-0.385133\pi\)
0.353085 + 0.935591i \(0.385133\pi\)
\(608\) −3.27631 −0.132872
\(609\) 0 0
\(610\) 5.32709 0.215688
\(611\) −0.354712 −0.0143501
\(612\) 0 0
\(613\) −0.178250 −0.00719945 −0.00359973 0.999994i \(-0.501146\pi\)
−0.00359973 + 0.999994i \(0.501146\pi\)
\(614\) −26.1858 −1.05677
\(615\) 0 0
\(616\) 0 0
\(617\) 3.66139 0.147402 0.0737010 0.997280i \(-0.476519\pi\)
0.0737010 + 0.997280i \(0.476519\pi\)
\(618\) 0 0
\(619\) 23.7370 0.954070 0.477035 0.878884i \(-0.341712\pi\)
0.477035 + 0.878884i \(0.341712\pi\)
\(620\) 0.155364 0.00623955
\(621\) 0 0
\(622\) 40.2222 1.61276
\(623\) −10.6013 −0.424732
\(624\) 0 0
\(625\) 23.4084 0.936335
\(626\) −26.2529 −1.04928
\(627\) 0 0
\(628\) 4.17602 0.166641
\(629\) −41.0481 −1.63669
\(630\) 0 0
\(631\) 16.0214 0.637801 0.318901 0.947788i \(-0.396686\pi\)
0.318901 + 0.947788i \(0.396686\pi\)
\(632\) −38.8408 −1.54500
\(633\) 0 0
\(634\) 25.1633 0.999363
\(635\) 2.41925 0.0960051
\(636\) 0 0
\(637\) 0.0589594 0.00233606
\(638\) 0 0
\(639\) 0 0
\(640\) −2.89998 −0.114632
\(641\) 26.3387 1.04032 0.520158 0.854070i \(-0.325873\pi\)
0.520158 + 0.854070i \(0.325873\pi\)
\(642\) 0 0
\(643\) 32.9889 1.30095 0.650477 0.759526i \(-0.274569\pi\)
0.650477 + 0.759526i \(0.274569\pi\)
\(644\) 1.42094 0.0559929
\(645\) 0 0
\(646\) −20.2927 −0.798407
\(647\) 15.1898 0.597171 0.298585 0.954383i \(-0.403485\pi\)
0.298585 + 0.954383i \(0.403485\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −0.382808 −0.0150150
\(651\) 0 0
\(652\) 1.75968 0.0689145
\(653\) −34.1357 −1.33583 −0.667916 0.744236i \(-0.732814\pi\)
−0.667916 + 0.744236i \(0.732814\pi\)
\(654\) 0 0
\(655\) −3.57896 −0.139842
\(656\) 0.271009 0.0105811
\(657\) 0 0
\(658\) −7.98297 −0.311208
\(659\) −11.4960 −0.447820 −0.223910 0.974610i \(-0.571882\pi\)
−0.223910 + 0.974610i \(0.571882\pi\)
\(660\) 0 0
\(661\) 8.84312 0.343957 0.171979 0.985101i \(-0.444984\pi\)
0.171979 + 0.985101i \(0.444984\pi\)
\(662\) −5.29297 −0.205717
\(663\) 0 0
\(664\) 22.2687 0.864192
\(665\) 0.795623 0.0308529
\(666\) 0 0
\(667\) −55.9955 −2.16815
\(668\) 1.13468 0.0439023
\(669\) 0 0
\(670\) −3.79868 −0.146756
\(671\) 0 0
\(672\) 0 0
\(673\) 6.96022 0.268297 0.134148 0.990961i \(-0.457170\pi\)
0.134148 + 0.990961i \(0.457170\pi\)
\(674\) −36.5816 −1.40907
\(675\) 0 0
\(676\) 3.11025 0.119625
\(677\) −24.6140 −0.945993 −0.472996 0.881064i \(-0.656828\pi\)
−0.472996 + 0.881064i \(0.656828\pi\)
\(678\) 0 0
\(679\) −6.71278 −0.257613
\(680\) 6.10381 0.234070
\(681\) 0 0
\(682\) 0 0
\(683\) 2.65760 0.101690 0.0508451 0.998707i \(-0.483809\pi\)
0.0508451 + 0.998707i \(0.483809\pi\)
\(684\) 0 0
\(685\) 1.62349 0.0620305
\(686\) 1.32691 0.0506616
\(687\) 0 0
\(688\) 17.0495 0.650006
\(689\) 0.350380 0.0133484
\(690\) 0 0
\(691\) 6.06065 0.230558 0.115279 0.993333i \(-0.463224\pi\)
0.115279 + 0.993333i \(0.463224\pi\)
\(692\) 1.92597 0.0732146
\(693\) 0 0
\(694\) −43.1617 −1.63839
\(695\) −2.02539 −0.0768276
\(696\) 0 0
\(697\) −0.491601 −0.0186207
\(698\) −38.0953 −1.44193
\(699\) 0 0
\(700\) 1.17099 0.0442594
\(701\) −29.2016 −1.10293 −0.551465 0.834198i \(-0.685931\pi\)
−0.551465 + 0.834198i \(0.685931\pi\)
\(702\) 0 0
\(703\) 15.8985 0.599622
\(704\) 0 0
\(705\) 0 0
\(706\) 0.559101 0.0210421
\(707\) 8.44830 0.317731
\(708\) 0 0
\(709\) 5.34783 0.200842 0.100421 0.994945i \(-0.467981\pi\)
0.100421 + 0.994945i \(0.467981\pi\)
\(710\) −0.719449 −0.0270004
\(711\) 0 0
\(712\) 31.5003 1.18052
\(713\) −11.7914 −0.441590
\(714\) 0 0
\(715\) 0 0
\(716\) −2.52219 −0.0942588
\(717\) 0 0
\(718\) 10.8454 0.404745
\(719\) −39.3239 −1.46653 −0.733267 0.679941i \(-0.762005\pi\)
−0.733267 + 0.679941i \(0.762005\pi\)
\(720\) 0 0
\(721\) −15.9631 −0.594498
\(722\) −17.3516 −0.645760
\(723\) 0 0
\(724\) 5.63317 0.209355
\(725\) −46.1457 −1.71381
\(726\) 0 0
\(727\) −3.52389 −0.130694 −0.0653470 0.997863i \(-0.520815\pi\)
−0.0653470 + 0.997863i \(0.520815\pi\)
\(728\) −0.175190 −0.00649296
\(729\) 0 0
\(730\) 4.07974 0.150998
\(731\) −30.9271 −1.14388
\(732\) 0 0
\(733\) −3.58578 −0.132444 −0.0662218 0.997805i \(-0.521095\pi\)
−0.0662218 + 0.997805i \(0.521095\pi\)
\(734\) −14.5183 −0.535881
\(735\) 0 0
\(736\) −7.99305 −0.294628
\(737\) 0 0
\(738\) 0 0
\(739\) 20.4213 0.751208 0.375604 0.926780i \(-0.377435\pi\)
0.375604 + 0.926780i \(0.377435\pi\)
\(740\) −0.511055 −0.0187868
\(741\) 0 0
\(742\) 7.88546 0.289484
\(743\) 26.7582 0.981662 0.490831 0.871255i \(-0.336693\pi\)
0.490831 + 0.871255i \(0.336693\pi\)
\(744\) 0 0
\(745\) 6.64140 0.243322
\(746\) −20.2955 −0.743072
\(747\) 0 0
\(748\) 0 0
\(749\) −5.17044 −0.188924
\(750\) 0 0
\(751\) 40.7545 1.48715 0.743576 0.668652i \(-0.233128\pi\)
0.743576 + 0.668652i \(0.233128\pi\)
\(752\) 20.8408 0.759985
\(753\) 0 0
\(754\) 0.737800 0.0268691
\(755\) 3.10550 0.113021
\(756\) 0 0
\(757\) 10.4923 0.381350 0.190675 0.981653i \(-0.438932\pi\)
0.190675 + 0.981653i \(0.438932\pi\)
\(758\) 42.6864 1.55044
\(759\) 0 0
\(760\) −2.36409 −0.0857544
\(761\) 2.38294 0.0863816 0.0431908 0.999067i \(-0.486248\pi\)
0.0431908 + 0.999067i \(0.486248\pi\)
\(762\) 0 0
\(763\) −3.02444 −0.109492
\(764\) 4.28822 0.155142
\(765\) 0 0
\(766\) 27.2660 0.985162
\(767\) −0.0670482 −0.00242097
\(768\) 0 0
\(769\) −23.0495 −0.831186 −0.415593 0.909551i \(-0.636426\pi\)
−0.415593 + 0.909551i \(0.636426\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2.67939 0.0964334
\(773\) 30.2708 1.08876 0.544382 0.838837i \(-0.316764\pi\)
0.544382 + 0.838837i \(0.316764\pi\)
\(774\) 0 0
\(775\) −9.71722 −0.349053
\(776\) 19.9461 0.716023
\(777\) 0 0
\(778\) −22.4281 −0.804087
\(779\) 0.190403 0.00682191
\(780\) 0 0
\(781\) 0 0
\(782\) −49.5072 −1.77037
\(783\) 0 0
\(784\) −3.46410 −0.123718
\(785\) 5.70455 0.203604
\(786\) 0 0
\(787\) 7.41144 0.264189 0.132095 0.991237i \(-0.457830\pi\)
0.132095 + 0.991237i \(0.457830\pi\)
\(788\) −1.27437 −0.0453977
\(789\) 0 0
\(790\) −5.67021 −0.201737
\(791\) 6.23584 0.221721
\(792\) 0 0
\(793\) −0.724062 −0.0257122
\(794\) 23.5667 0.836350
\(795\) 0 0
\(796\) −2.16803 −0.0768439
\(797\) 33.5597 1.18875 0.594373 0.804190i \(-0.297400\pi\)
0.594373 + 0.804190i \(0.297400\pi\)
\(798\) 0 0
\(799\) −37.8044 −1.33742
\(800\) −6.58705 −0.232887
\(801\) 0 0
\(802\) −10.8802 −0.384192
\(803\) 0 0
\(804\) 0 0
\(805\) 1.94104 0.0684127
\(806\) 0.155364 0.00547245
\(807\) 0 0
\(808\) −25.1030 −0.883120
\(809\) 42.2328 1.48483 0.742413 0.669942i \(-0.233681\pi\)
0.742413 + 0.669942i \(0.233681\pi\)
\(810\) 0 0
\(811\) 10.1590 0.356730 0.178365 0.983964i \(-0.442919\pi\)
0.178365 + 0.983964i \(0.442919\pi\)
\(812\) −2.25690 −0.0792017
\(813\) 0 0
\(814\) 0 0
\(815\) 2.40377 0.0842004
\(816\) 0 0
\(817\) 11.9785 0.419074
\(818\) 48.5346 1.69697
\(819\) 0 0
\(820\) −0.00612051 −0.000213737 0
\(821\) −25.6507 −0.895214 −0.447607 0.894230i \(-0.647724\pi\)
−0.447607 + 0.894230i \(0.647724\pi\)
\(822\) 0 0
\(823\) −47.8879 −1.66927 −0.834634 0.550805i \(-0.814321\pi\)
−0.834634 + 0.550805i \(0.814321\pi\)
\(824\) 47.4323 1.65238
\(825\) 0 0
\(826\) −1.50895 −0.0525031
\(827\) −13.3901 −0.465619 −0.232810 0.972522i \(-0.574792\pi\)
−0.232810 + 0.972522i \(0.574792\pi\)
\(828\) 0 0
\(829\) −19.7867 −0.687221 −0.343610 0.939112i \(-0.611650\pi\)
−0.343610 + 0.939112i \(0.611650\pi\)
\(830\) 3.25092 0.112841
\(831\) 0 0
\(832\) 0.513799 0.0178128
\(833\) 6.28375 0.217719
\(834\) 0 0
\(835\) 1.55001 0.0536402
\(836\) 0 0
\(837\) 0 0
\(838\) 18.3506 0.633910
\(839\) −46.1948 −1.59482 −0.797410 0.603437i \(-0.793797\pi\)
−0.797410 + 0.603437i \(0.793797\pi\)
\(840\) 0 0
\(841\) 59.9384 2.06684
\(842\) −40.9355 −1.41073
\(843\) 0 0
\(844\) 6.66177 0.229308
\(845\) 4.24867 0.146159
\(846\) 0 0
\(847\) 0 0
\(848\) −20.5862 −0.706934
\(849\) 0 0
\(850\) −40.7987 −1.39938
\(851\) 38.7867 1.32959
\(852\) 0 0
\(853\) 37.6343 1.28857 0.644287 0.764784i \(-0.277154\pi\)
0.644287 + 0.764784i \(0.277154\pi\)
\(854\) −16.2953 −0.557615
\(855\) 0 0
\(856\) 15.3633 0.525106
\(857\) −12.9962 −0.443943 −0.221972 0.975053i \(-0.571249\pi\)
−0.221972 + 0.975053i \(0.571249\pi\)
\(858\) 0 0
\(859\) −26.0560 −0.889020 −0.444510 0.895774i \(-0.646622\pi\)
−0.444510 + 0.895774i \(0.646622\pi\)
\(860\) −0.385048 −0.0131300
\(861\) 0 0
\(862\) −4.79156 −0.163201
\(863\) 19.3134 0.657434 0.328717 0.944428i \(-0.393384\pi\)
0.328717 + 0.944428i \(0.393384\pi\)
\(864\) 0 0
\(865\) 2.63093 0.0894543
\(866\) 50.4292 1.71366
\(867\) 0 0
\(868\) −0.475251 −0.0161311
\(869\) 0 0
\(870\) 0 0
\(871\) 0.516320 0.0174948
\(872\) 8.98671 0.304328
\(873\) 0 0
\(874\) 19.1748 0.648596
\(875\) 3.23415 0.109334
\(876\) 0 0
\(877\) 22.9946 0.776472 0.388236 0.921560i \(-0.373084\pi\)
0.388236 + 0.921560i \(0.373084\pi\)
\(878\) 34.3476 1.15918
\(879\) 0 0
\(880\) 0 0
\(881\) 50.4759 1.70058 0.850288 0.526318i \(-0.176428\pi\)
0.850288 + 0.526318i \(0.176428\pi\)
\(882\) 0 0
\(883\) 14.8564 0.499958 0.249979 0.968251i \(-0.419576\pi\)
0.249979 + 0.968251i \(0.419576\pi\)
\(884\) −0.0886623 −0.00298204
\(885\) 0 0
\(886\) −19.6710 −0.660860
\(887\) −46.0622 −1.54662 −0.773309 0.634029i \(-0.781400\pi\)
−0.773309 + 0.634029i \(0.781400\pi\)
\(888\) 0 0
\(889\) −7.40039 −0.248201
\(890\) 4.59861 0.154146
\(891\) 0 0
\(892\) −4.62029 −0.154699
\(893\) 14.6421 0.489980
\(894\) 0 0
\(895\) −3.44538 −0.115166
\(896\) 8.87093 0.296357
\(897\) 0 0
\(898\) −50.1666 −1.67408
\(899\) 18.7284 0.624626
\(900\) 0 0
\(901\) 37.3426 1.24406
\(902\) 0 0
\(903\) 0 0
\(904\) −18.5289 −0.616264
\(905\) 7.69505 0.255792
\(906\) 0 0
\(907\) 28.5358 0.947516 0.473758 0.880655i \(-0.342897\pi\)
0.473758 + 0.880655i \(0.342897\pi\)
\(908\) −0.314903 −0.0104504
\(909\) 0 0
\(910\) −0.0255753 −0.000847812 0
\(911\) −31.5458 −1.04516 −0.522580 0.852590i \(-0.675030\pi\)
−0.522580 + 0.852590i \(0.675030\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 31.1974 1.03192
\(915\) 0 0
\(916\) 0.105317 0.00347977
\(917\) 10.9479 0.361531
\(918\) 0 0
\(919\) 40.0921 1.32252 0.661259 0.750158i \(-0.270022\pi\)
0.661259 + 0.750158i \(0.270022\pi\)
\(920\) −5.76754 −0.190150
\(921\) 0 0
\(922\) −54.2986 −1.78823
\(923\) 0.0977880 0.00321873
\(924\) 0 0
\(925\) 31.9640 1.05097
\(926\) 0.0806060 0.00264888
\(927\) 0 0
\(928\) 12.6955 0.416749
\(929\) −4.40597 −0.144555 −0.0722777 0.997385i \(-0.523027\pi\)
−0.0722777 + 0.997385i \(0.523027\pi\)
\(930\) 0 0
\(931\) −2.43378 −0.0797638
\(932\) 1.26785 0.0415298
\(933\) 0 0
\(934\) 49.6744 1.62540
\(935\) 0 0
\(936\) 0 0
\(937\) −0.902908 −0.0294967 −0.0147484 0.999891i \(-0.504695\pi\)
−0.0147484 + 0.999891i \(0.504695\pi\)
\(938\) 11.6200 0.379407
\(939\) 0 0
\(940\) −0.470671 −0.0153516
\(941\) 34.8442 1.13589 0.567943 0.823068i \(-0.307739\pi\)
0.567943 + 0.823068i \(0.307739\pi\)
\(942\) 0 0
\(943\) 0.464518 0.0151268
\(944\) 3.93935 0.128215
\(945\) 0 0
\(946\) 0 0
\(947\) −25.9556 −0.843444 −0.421722 0.906725i \(-0.638574\pi\)
−0.421722 + 0.906725i \(0.638574\pi\)
\(948\) 0 0
\(949\) −0.554521 −0.0180005
\(950\) 15.8019 0.512681
\(951\) 0 0
\(952\) −18.6713 −0.605140
\(953\) 1.69616 0.0549440 0.0274720 0.999623i \(-0.491254\pi\)
0.0274720 + 0.999623i \(0.491254\pi\)
\(954\) 0 0
\(955\) 5.85782 0.189555
\(956\) −5.82234 −0.188308
\(957\) 0 0
\(958\) 41.6620 1.34604
\(959\) −4.96620 −0.160367
\(960\) 0 0
\(961\) −27.0562 −0.872782
\(962\) −0.511055 −0.0164771
\(963\) 0 0
\(964\) −6.34577 −0.204383
\(965\) 3.66012 0.117823
\(966\) 0 0
\(967\) −25.0337 −0.805028 −0.402514 0.915414i \(-0.631864\pi\)
−0.402514 + 0.915414i \(0.631864\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 2.91185 0.0934940
\(971\) 57.3356 1.83999 0.919994 0.391933i \(-0.128193\pi\)
0.919994 + 0.391933i \(0.128193\pi\)
\(972\) 0 0
\(973\) 6.19560 0.198622
\(974\) −27.0532 −0.866839
\(975\) 0 0
\(976\) 42.5415 1.36172
\(977\) −19.6196 −0.627687 −0.313844 0.949475i \(-0.601617\pi\)
−0.313844 + 0.949475i \(0.601617\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0.0782337 0.00249908
\(981\) 0 0
\(982\) 15.4081 0.491692
\(983\) −32.4742 −1.03577 −0.517883 0.855451i \(-0.673280\pi\)
−0.517883 + 0.855451i \(0.673280\pi\)
\(984\) 0 0
\(985\) −1.74083 −0.0554674
\(986\) 78.6329 2.50418
\(987\) 0 0
\(988\) 0.0343401 0.00109250
\(989\) 29.2233 0.929247
\(990\) 0 0
\(991\) 21.7384 0.690543 0.345271 0.938503i \(-0.387787\pi\)
0.345271 + 0.938503i \(0.387787\pi\)
\(992\) 2.67337 0.0848796
\(993\) 0 0
\(994\) 2.20077 0.0698040
\(995\) −2.96159 −0.0938886
\(996\) 0 0
\(997\) −7.14889 −0.226408 −0.113204 0.993572i \(-0.536111\pi\)
−0.113204 + 0.993572i \(0.536111\pi\)
\(998\) 50.6356 1.60284
\(999\) 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.cn.1.3 4
3.2 odd 2 2541.2.a.bl.1.2 4
11.10 odd 2 7623.2.a.cg.1.2 4
33.32 even 2 2541.2.a.bp.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2541.2.a.bl.1.2 4 3.2 odd 2
2541.2.a.bp.1.3 yes 4 33.32 even 2
7623.2.a.cg.1.2 4 11.10 odd 2
7623.2.a.cn.1.3 4 1.1 even 1 trivial