Properties

Label 7623.2.a.cg.1.2
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 0
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: \(0\)
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 1.2
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 \(-0.655434\pi\)
−0.469133 + 0.883128i \(0.655434\pi\)
\(3\) 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 \(-0.502603\pi\)
−0.00817619 + 0.999967i \(0.502603\pi\)
\(14\) 1.32691 0.354631
\(15\) 0 0
\(16\) −3.46410 −0.866025
\(17\) −6.28375 −1.52403 −0.762016 0.647558i \(-0.775790\pi\)
−0.762016 + 0.647558i \(0.775790\pi\)
\(18\) 0 0
\(19\) 2.43378 0.558347 0.279173 0.960241i \(-0.409940\pi\)
0.279173 + 0.960241i \(0.409940\pi\)
\(20\) 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 \(-0.839547\pi\)
−0.875620 + 0.483000i \(0.839547\pi\)
\(30\) 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 \(-0.498055\pi\)
0.00610902 + 0.999981i \(0.498055\pi\)
\(42\) 0 0
\(43\) 4.92177 0.750562 0.375281 0.926911i \(-0.377546\pi\)
0.375281 + 0.926911i \(0.377546\pi\)
\(44\) 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 \(-0.212050\pi\)
0.786190 + 0.617984i \(0.212050\pi\)
\(62\) −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 \(-0.314477\pi\)
0.550394 + 0.834905i \(0.314477\pi\)
\(74\) 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 \(-0.762978\pi\)
−0.735340 + 0.677698i \(0.762978\pi\)
\(80\) 1.13244 0.126611
\(81\) 0 0
\(82\) −0.103809 −0.0114638
\(83\) 7.49443 0.822620 0.411310 0.911496i \(-0.365071\pi\)
0.411310 + 0.911496i \(0.365071\pi\)
\(84\) 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 \(-0.638082\pi\)
−0.420319 + 0.907376i \(0.638082\pi\)
\(102\) 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 \(-0.419595\pi\)
0.249923 + 0.968266i \(0.419595\pi\)
\(108\) 0 0
\(109\) 3.02444 0.289689 0.144844 0.989454i \(-0.453732\pi\)
0.144844 + 0.989454i \(0.453732\pi\)
\(110\) 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 \(-0.393511\pi\)
0.328339 + 0.944560i \(0.393511\pi\)
\(128\) −8.87093 −0.784087
\(129\) 0 0
\(130\) −0.0255753 −0.00224310
\(131\) −10.9479 −0.956522 −0.478261 0.878218i \(-0.658733\pi\)
−0.478261 + 0.878218i \(0.658733\pi\)
\(132\) 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 \(-0.584630\pi\)
−0.262752 + 0.964863i \(0.584630\pi\)
\(140\) −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 \(-0.187101\pi\)
0.832166 + 0.554527i \(0.187101\pi\)
\(150\) 0 0
\(151\) 9.49959 0.773066 0.386533 0.922276i \(-0.373673\pi\)
0.386533 + 0.922276i \(0.373673\pi\)
\(152\) 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 \(-0.441273\pi\)
0.183451 + 0.983029i \(0.441273\pi\)
\(168\) 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 \(-0.401031\pi\)
0.305936 + 0.952052i \(0.401031\pi\)
\(174\) 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 \(-0.367982\pi\)
0.402958 + 0.915218i \(0.367982\pi\)
\(194\) 8.90724 0.639503
\(195\) 0 0
\(196\) −0.239314 −0.0170938
\(197\) −5.32512 −0.379399 −0.189700 0.981842i \(-0.560751\pi\)
−0.189700 + 0.981842i \(0.560751\pi\)
\(198\) 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 \(-0.0923710\pi\)
0.958189 + 0.286136i \(0.0923710\pi\)
\(212\) −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 \(-0.513904\pi\)
−0.0436683 + 0.999046i \(0.513904\pi\)
\(228\) 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 \(-0.444480\pi\)
0.173537 + 0.984827i \(0.444480\pi\)
\(234\) 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 \(-0.788298\pi\)
−0.786866 + 0.617123i \(0.788298\pi\)
\(240\) 0 0
\(241\) −26.5165 −1.70808 −0.854040 0.520208i \(-0.825855\pi\)
−0.854040 + 0.520208i \(0.825855\pi\)
\(242\) 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 \(-0.310338\pi\)
0.561206 + 0.827676i \(0.310338\pi\)
\(264\) 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 \(-0.645437\pi\)
−0.441170 + 0.897423i \(0.645437\pi\)
\(272\) 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 \(-0.420134\pi\)
0.248282 + 0.968688i \(0.420134\pi\)
\(278\) 8.22100 0.493063
\(279\) 0 0
\(280\) 0.971364 0.0580501
\(281\) 24.0080 1.43220 0.716098 0.697999i \(-0.245926\pi\)
0.716098 + 0.697999i \(0.245926\pi\)
\(282\) 0 0
\(283\) 16.0512 0.954142 0.477071 0.878865i \(-0.341698\pi\)
0.477071 + 0.878865i \(0.341698\pi\)
\(284\) −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 \(-0.571266\pi\)
−0.222024 + 0.975041i \(0.571266\pi\)
\(294\) 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 \(-0.309588\pi\)
0.563153 + 0.826353i \(0.309588\pi\)
\(308\) 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 \(-0.229624\pi\)
0.750891 + 0.660426i \(0.229624\pi\)
\(338\) 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 \(-0.162108\pi\)
0.873097 + 0.487547i \(0.162108\pi\)
\(348\) 0 0
\(349\) 28.7099 1.53680 0.768402 0.639968i \(-0.221052\pi\)
0.768402 + 0.639968i \(0.221052\pi\)
\(350\) −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 \(-0.569199\pi\)
−0.215688 + 0.976462i \(0.569199\pi\)
\(360\) 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 \(-0.370404\pi\)
0.395982 + 0.918258i \(0.370404\pi\)
\(374\) 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 \(-0.859616\pi\)
−0.904312 + 0.426871i \(0.859616\pi\)
\(410\) 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 \(-0.472282\pi\)
0.0869696 + 0.996211i \(0.472282\pi\)
\(432\) 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 \(-0.711945\pi\)
−0.617723 + 0.786396i \(0.711945\pi\)
\(440\) 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 \(-0.685337\pi\)
−0.549908 + 0.835226i \(0.685337\pi\)
\(458\) 0.583944 0.0272859
\(459\) 0 0
\(460\) −0.464518 −0.0216582
\(461\) 40.9211 1.90589 0.952943 0.303149i \(-0.0980380\pi\)
0.952943 + 0.303149i \(0.0980380\pi\)
\(462\) 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 \(-0.754622\pi\)
−0.717300 + 0.696764i \(0.754622\pi\)
\(480\) 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 \(-0.584389\pi\)
−0.262022 + 0.965062i \(0.584389\pi\)
\(492\) 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 \(-0.241303\pi\)
0.726160 + 0.687526i \(0.241303\pi\)
\(504\) 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 \(-0.441885\pi\)
0.181561 + 0.983380i \(0.441885\pi\)
\(524\) 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 \(-0.667733\pi\)
−0.502899 + 0.864345i \(0.667733\pi\)
\(542\) 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 \(-0.634340\pi\)
−0.409623 + 0.912255i \(0.634340\pi\)
\(548\) 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 \(-0.684935\pi\)
−0.548852 + 0.835920i \(0.684935\pi\)
\(558\) 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 \(-0.482660\pi\)
0.0544473 + 0.998517i \(0.482660\pi\)
\(564\) 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 \(-0.680901\pi\)
−0.538215 + 0.842807i \(0.680901\pi\)
\(570\) 0 0
\(571\) 1.12907 0.0472500 0.0236250 0.999721i \(-0.492479\pi\)
0.0236250 + 0.999721i \(0.492479\pi\)
\(572\) 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 \(-0.281152\pi\)
0.634630 + 0.772816i \(0.281152\pi\)
\(594\) 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 \(-0.310111\pi\)
0.561795 + 0.827276i \(0.310111\pi\)
\(602\) 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 \(-0.614867\pi\)
−0.353085 + 0.935591i \(0.614867\pi\)
\(608\) −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 \(-0.498854\pi\)
0.00359973 + 0.999994i \(0.498854\pi\)
\(614\) −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 \(-0.428118\pi\)
0.223910 + 0.974610i \(0.428118\pi\)
\(660\) 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 \(-0.542830\pi\)
−0.134148 + 0.990961i \(0.542830\pi\)
\(674\) −36.5816 −1.40907
\(675\) 0 0
\(676\) 3.11025 0.119625
\(677\) 24.6140 0.945993 0.472996 0.881064i \(-0.343172\pi\)
0.472996 + 0.881064i \(0.343172\pi\)
\(678\) 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 \(-0.314069\pi\)
0.551465 + 0.834198i \(0.314069\pi\)
\(702\) 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 \(-0.478905\pi\)
0.0662218 + 0.997805i \(0.478905\pi\)
\(734\) 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 \(-0.622565\pi\)
−0.375604 + 0.926780i \(0.622565\pi\)
\(740\) −0.511055 −0.0187868
\(741\) 0 0
\(742\) 7.88546 0.289484
\(743\) −26.7582 −0.981662 −0.490831 0.871255i \(-0.663307\pi\)
−0.490831 + 0.871255i \(0.663307\pi\)
\(744\) 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 \(-0.513752\pi\)
−0.0431908 + 0.999067i \(0.513752\pi\)
\(762\) 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 \(-0.363574\pi\)
0.415593 + 0.909551i \(0.363574\pi\)
\(770\) 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 \(-0.542170\pi\)
−0.132095 + 0.991237i \(0.542170\pi\)
\(788\) 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 \(-0.766319\pi\)
−0.742413 + 0.669942i \(0.766319\pi\)
\(810\) 0 0
\(811\) −10.1590 −0.356730 −0.178365 0.983964i \(-0.557081\pi\)
−0.178365 + 0.983964i \(0.557081\pi\)
\(812\) −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 \(-0.352276\pi\)
0.447607 + 0.894230i \(0.352276\pi\)
\(822\) 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 \(-0.425208\pi\)
0.232810 + 0.972522i \(0.425208\pi\)
\(828\) 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 \(-0.722846\pi\)
−0.644287 + 0.764784i \(0.722846\pi\)
\(854\) 16.2953 0.557615
\(855\) 0 0
\(856\) 15.3633 0.525106
\(857\) 12.9962 0.443943 0.221972 0.975053i \(-0.428751\pi\)
0.221972 + 0.975053i \(0.428751\pi\)
\(858\) 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 \(-0.626916\pi\)
−0.388236 + 0.921560i \(0.626916\pi\)
\(878\) 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 \(-0.218600\pi\)
0.773309 + 0.634029i \(0.218600\pi\)
\(888\) 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 \(-0.729978\pi\)
−0.661259 + 0.750158i \(0.729978\pi\)
\(920\) 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 \(-0.495305\pi\)
0.0147484 + 0.999891i \(0.495305\pi\)
\(938\) 11.6200 0.379407
\(939\) 0 0
\(940\) −0.470671 −0.0153516
\(941\) −34.8442 −1.13589 −0.567943 0.823068i \(-0.692261\pi\)
−0.567943 + 0.823068i \(0.692261\pi\)
\(942\) 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 \(-0.508746\pi\)
−0.0274720 + 0.999623i \(0.508746\pi\)
\(954\) 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 \(-0.368136\pi\)
0.402514 + 0.915414i \(0.368136\pi\)
\(968\) 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 \(-0.463889\pi\)
0.113204 + 0.993572i \(0.463889\pi\)
\(998\) −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.cg.1.2 4
3.2 odd 2 2541.2.a.bp.1.3 yes 4
11.10 odd 2 7623.2.a.cn.1.3 4
33.32 even 2 2541.2.a.bl.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2541.2.a.bl.1.2 4 33.32 even 2
2541.2.a.bp.1.3 yes 4 3.2 odd 2
7623.2.a.cg.1.2 4 1.1 even 1 trivial
7623.2.a.cn.1.3 4 11.10 odd 2