Properties

Label 7623.2.a.cz.1.2
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 12
CM no
Inner twists 2

Related objects

Downloads

Learn more about

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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.41714\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.41714 q^{2} +3.84255 q^{4} +4.01054 q^{5} -1.00000 q^{7} -4.45369 q^{8} +O(q^{10})\) \(q-2.41714 q^{2} +3.84255 q^{4} +4.01054 q^{5} -1.00000 q^{7} -4.45369 q^{8} -9.69401 q^{10} +0.690498 q^{13} +2.41714 q^{14} +3.08008 q^{16} -5.78000 q^{17} +1.80100 q^{19} +15.4107 q^{20} +2.03655 q^{23} +11.0844 q^{25} -1.66903 q^{26} -3.84255 q^{28} -7.98110 q^{29} -10.4676 q^{31} +1.46241 q^{32} +13.9711 q^{34} -4.01054 q^{35} -7.44454 q^{37} -4.35325 q^{38} -17.8617 q^{40} +5.66440 q^{41} -3.44102 q^{43} -4.92263 q^{46} +7.32704 q^{47} +1.00000 q^{49} -26.7925 q^{50} +2.65327 q^{52} -1.79792 q^{53} +4.45369 q^{56} +19.2914 q^{58} +10.9949 q^{59} +6.95019 q^{61} +25.3017 q^{62} -9.69500 q^{64} +2.76927 q^{65} -11.8438 q^{67} -22.2099 q^{68} +9.69401 q^{70} -13.5523 q^{71} -7.20252 q^{73} +17.9945 q^{74} +6.92041 q^{76} +5.33847 q^{79} +12.3528 q^{80} -13.6916 q^{82} +8.59090 q^{83} -23.1809 q^{85} +8.31742 q^{86} +13.0692 q^{89} -0.690498 q^{91} +7.82555 q^{92} -17.7104 q^{94} +7.22296 q^{95} -9.50344 q^{97} -2.41714 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 20q^{4} - 12q^{7} + O(q^{10}) \) \( 12q + 20q^{4} - 12q^{7} - 20q^{10} - 20q^{13} + 28q^{16} - 12q^{19} + 32q^{25} - 20q^{28} - 16q^{31} - 24q^{34} + 4q^{37} - 48q^{40} - 16q^{43} - 24q^{46} + 12q^{49} - 96q^{52} + 20q^{58} - 44q^{61} + 76q^{64} + 20q^{70} - 52q^{73} + 8q^{79} - 68q^{82} - 72q^{85} + 20q^{91} - 20q^{94} - 32q^{97} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.41714 −1.70917 −0.854587 0.519309i \(-0.826189\pi\)
−0.854587 + 0.519309i \(0.826189\pi\)
\(3\) 0 0
\(4\) 3.84255 1.92127
\(5\) 4.01054 1.79357 0.896783 0.442470i \(-0.145898\pi\)
0.896783 + 0.442470i \(0.145898\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −4.45369 −1.57462
\(9\) 0 0
\(10\) −9.69401 −3.06552
\(11\) 0 0
\(12\) 0 0
\(13\) 0.690498 0.191510 0.0957549 0.995405i \(-0.469474\pi\)
0.0957549 + 0.995405i \(0.469474\pi\)
\(14\) 2.41714 0.646007
\(15\) 0 0
\(16\) 3.08008 0.770020
\(17\) −5.78000 −1.40186 −0.700928 0.713232i \(-0.747231\pi\)
−0.700928 + 0.713232i \(0.747231\pi\)
\(18\) 0 0
\(19\) 1.80100 0.413177 0.206588 0.978428i \(-0.433764\pi\)
0.206588 + 0.978428i \(0.433764\pi\)
\(20\) 15.4107 3.44593
\(21\) 0 0
\(22\) 0 0
\(23\) 2.03655 0.424651 0.212325 0.977199i \(-0.431896\pi\)
0.212325 + 0.977199i \(0.431896\pi\)
\(24\) 0 0
\(25\) 11.0844 2.21688
\(26\) −1.66903 −0.327323
\(27\) 0 0
\(28\) −3.84255 −0.726173
\(29\) −7.98110 −1.48205 −0.741027 0.671475i \(-0.765661\pi\)
−0.741027 + 0.671475i \(0.765661\pi\)
\(30\) 0 0
\(31\) −10.4676 −1.88004 −0.940020 0.341120i \(-0.889194\pi\)
−0.940020 + 0.341120i \(0.889194\pi\)
\(32\) 1.46241 0.258520
\(33\) 0 0
\(34\) 13.9711 2.39602
\(35\) −4.01054 −0.677904
\(36\) 0 0
\(37\) −7.44454 −1.22387 −0.611937 0.790906i \(-0.709610\pi\)
−0.611937 + 0.790906i \(0.709610\pi\)
\(38\) −4.35325 −0.706191
\(39\) 0 0
\(40\) −17.8617 −2.82418
\(41\) 5.66440 0.884631 0.442316 0.896859i \(-0.354157\pi\)
0.442316 + 0.896859i \(0.354157\pi\)
\(42\) 0 0
\(43\) −3.44102 −0.524751 −0.262376 0.964966i \(-0.584506\pi\)
−0.262376 + 0.964966i \(0.584506\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −4.92263 −0.725802
\(47\) 7.32704 1.06876 0.534379 0.845245i \(-0.320545\pi\)
0.534379 + 0.845245i \(0.320545\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −26.7925 −3.78903
\(51\) 0 0
\(52\) 2.65327 0.367943
\(53\) −1.79792 −0.246963 −0.123481 0.992347i \(-0.539406\pi\)
−0.123481 + 0.992347i \(0.539406\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.45369 0.595149
\(57\) 0 0
\(58\) 19.2914 2.53309
\(59\) 10.9949 1.43141 0.715704 0.698403i \(-0.246106\pi\)
0.715704 + 0.698403i \(0.246106\pi\)
\(60\) 0 0
\(61\) 6.95019 0.889881 0.444940 0.895560i \(-0.353225\pi\)
0.444940 + 0.895560i \(0.353225\pi\)
\(62\) 25.3017 3.21331
\(63\) 0 0
\(64\) −9.69500 −1.21187
\(65\) 2.76927 0.343485
\(66\) 0 0
\(67\) −11.8438 −1.44696 −0.723478 0.690347i \(-0.757458\pi\)
−0.723478 + 0.690347i \(0.757458\pi\)
\(68\) −22.2099 −2.69335
\(69\) 0 0
\(70\) 9.69401 1.15866
\(71\) −13.5523 −1.60836 −0.804181 0.594385i \(-0.797396\pi\)
−0.804181 + 0.594385i \(0.797396\pi\)
\(72\) 0 0
\(73\) −7.20252 −0.842991 −0.421496 0.906830i \(-0.638495\pi\)
−0.421496 + 0.906830i \(0.638495\pi\)
\(74\) 17.9945 2.09181
\(75\) 0 0
\(76\) 6.92041 0.793826
\(77\) 0 0
\(78\) 0 0
\(79\) 5.33847 0.600625 0.300313 0.953841i \(-0.402909\pi\)
0.300313 + 0.953841i \(0.402909\pi\)
\(80\) 12.3528 1.38108
\(81\) 0 0
\(82\) −13.6916 −1.51199
\(83\) 8.59090 0.942974 0.471487 0.881873i \(-0.343717\pi\)
0.471487 + 0.881873i \(0.343717\pi\)
\(84\) 0 0
\(85\) −23.1809 −2.51432
\(86\) 8.31742 0.896891
\(87\) 0 0
\(88\) 0 0
\(89\) 13.0692 1.38533 0.692665 0.721260i \(-0.256437\pi\)
0.692665 + 0.721260i \(0.256437\pi\)
\(90\) 0 0
\(91\) −0.690498 −0.0723839
\(92\) 7.82555 0.815870
\(93\) 0 0
\(94\) −17.7104 −1.82669
\(95\) 7.22296 0.741060
\(96\) 0 0
\(97\) −9.50344 −0.964928 −0.482464 0.875916i \(-0.660258\pi\)
−0.482464 + 0.875916i \(0.660258\pi\)
\(98\) −2.41714 −0.244168
\(99\) 0 0
\(100\) 42.5924 4.25924
\(101\) 9.70420 0.965604 0.482802 0.875729i \(-0.339619\pi\)
0.482802 + 0.875729i \(0.339619\pi\)
\(102\) 0 0
\(103\) 7.32767 0.722017 0.361009 0.932563i \(-0.382433\pi\)
0.361009 + 0.932563i \(0.382433\pi\)
\(104\) −3.07526 −0.301554
\(105\) 0 0
\(106\) 4.34581 0.422103
\(107\) −12.4348 −1.20212 −0.601058 0.799205i \(-0.705254\pi\)
−0.601058 + 0.799205i \(0.705254\pi\)
\(108\) 0 0
\(109\) −12.5955 −1.20643 −0.603213 0.797580i \(-0.706113\pi\)
−0.603213 + 0.797580i \(0.706113\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.08008 −0.291040
\(113\) 11.8186 1.11180 0.555900 0.831249i \(-0.312374\pi\)
0.555900 + 0.831249i \(0.312374\pi\)
\(114\) 0 0
\(115\) 8.16767 0.761639
\(116\) −30.6678 −2.84743
\(117\) 0 0
\(118\) −26.5761 −2.44653
\(119\) 5.78000 0.529852
\(120\) 0 0
\(121\) 0 0
\(122\) −16.7996 −1.52096
\(123\) 0 0
\(124\) −40.2223 −3.61207
\(125\) 24.4017 2.18256
\(126\) 0 0
\(127\) 9.59200 0.851152 0.425576 0.904923i \(-0.360072\pi\)
0.425576 + 0.904923i \(0.360072\pi\)
\(128\) 20.5093 1.81278
\(129\) 0 0
\(130\) −6.69370 −0.587076
\(131\) 1.48293 0.129564 0.0647819 0.997899i \(-0.479365\pi\)
0.0647819 + 0.997899i \(0.479365\pi\)
\(132\) 0 0
\(133\) −1.80100 −0.156166
\(134\) 28.6282 2.47310
\(135\) 0 0
\(136\) 25.7423 2.20739
\(137\) −17.6212 −1.50548 −0.752740 0.658318i \(-0.771268\pi\)
−0.752740 + 0.658318i \(0.771268\pi\)
\(138\) 0 0
\(139\) 2.36252 0.200387 0.100193 0.994968i \(-0.468054\pi\)
0.100193 + 0.994968i \(0.468054\pi\)
\(140\) −15.4107 −1.30244
\(141\) 0 0
\(142\) 32.7577 2.74897
\(143\) 0 0
\(144\) 0 0
\(145\) −32.0085 −2.65816
\(146\) 17.4095 1.44082
\(147\) 0 0
\(148\) −28.6060 −2.35140
\(149\) −18.4024 −1.50758 −0.753791 0.657114i \(-0.771777\pi\)
−0.753791 + 0.657114i \(0.771777\pi\)
\(150\) 0 0
\(151\) −5.35792 −0.436021 −0.218011 0.975946i \(-0.569957\pi\)
−0.218011 + 0.975946i \(0.569957\pi\)
\(152\) −8.02107 −0.650595
\(153\) 0 0
\(154\) 0 0
\(155\) −41.9808 −3.37198
\(156\) 0 0
\(157\) −5.46584 −0.436222 −0.218111 0.975924i \(-0.569989\pi\)
−0.218111 + 0.975924i \(0.569989\pi\)
\(158\) −12.9038 −1.02657
\(159\) 0 0
\(160\) 5.86505 0.463673
\(161\) −2.03655 −0.160503
\(162\) 0 0
\(163\) −17.4528 −1.36701 −0.683505 0.729946i \(-0.739545\pi\)
−0.683505 + 0.729946i \(0.739545\pi\)
\(164\) 21.7657 1.69962
\(165\) 0 0
\(166\) −20.7654 −1.61171
\(167\) −11.8534 −0.917247 −0.458623 0.888631i \(-0.651657\pi\)
−0.458623 + 0.888631i \(0.651657\pi\)
\(168\) 0 0
\(169\) −12.5232 −0.963324
\(170\) 56.0314 4.29741
\(171\) 0 0
\(172\) −13.2223 −1.00819
\(173\) −15.1122 −1.14896 −0.574481 0.818518i \(-0.694796\pi\)
−0.574481 + 0.818518i \(0.694796\pi\)
\(174\) 0 0
\(175\) −11.0844 −0.837902
\(176\) 0 0
\(177\) 0 0
\(178\) −31.5900 −2.36777
\(179\) 20.3747 1.52287 0.761436 0.648240i \(-0.224494\pi\)
0.761436 + 0.648240i \(0.224494\pi\)
\(180\) 0 0
\(181\) −25.0574 −1.86250 −0.931251 0.364379i \(-0.881281\pi\)
−0.931251 + 0.364379i \(0.881281\pi\)
\(182\) 1.66903 0.123717
\(183\) 0 0
\(184\) −9.07017 −0.668662
\(185\) −29.8566 −2.19510
\(186\) 0 0
\(187\) 0 0
\(188\) 28.1545 2.05338
\(189\) 0 0
\(190\) −17.4589 −1.26660
\(191\) −7.44391 −0.538622 −0.269311 0.963053i \(-0.586796\pi\)
−0.269311 + 0.963053i \(0.586796\pi\)
\(192\) 0 0
\(193\) −23.0191 −1.65695 −0.828477 0.560023i \(-0.810792\pi\)
−0.828477 + 0.560023i \(0.810792\pi\)
\(194\) 22.9711 1.64923
\(195\) 0 0
\(196\) 3.84255 0.274468
\(197\) 6.34515 0.452073 0.226037 0.974119i \(-0.427423\pi\)
0.226037 + 0.974119i \(0.427423\pi\)
\(198\) 0 0
\(199\) −21.3877 −1.51614 −0.758069 0.652175i \(-0.773857\pi\)
−0.758069 + 0.652175i \(0.773857\pi\)
\(200\) −49.3665 −3.49074
\(201\) 0 0
\(202\) −23.4564 −1.65039
\(203\) 7.98110 0.560164
\(204\) 0 0
\(205\) 22.7173 1.58665
\(206\) −17.7120 −1.23405
\(207\) 0 0
\(208\) 2.12679 0.147466
\(209\) 0 0
\(210\) 0 0
\(211\) 3.37623 0.232429 0.116215 0.993224i \(-0.462924\pi\)
0.116215 + 0.993224i \(0.462924\pi\)
\(212\) −6.90858 −0.474484
\(213\) 0 0
\(214\) 30.0566 2.05463
\(215\) −13.8003 −0.941176
\(216\) 0 0
\(217\) 10.4676 0.710588
\(218\) 30.4449 2.06199
\(219\) 0 0
\(220\) 0 0
\(221\) −3.99108 −0.268469
\(222\) 0 0
\(223\) −1.94696 −0.130378 −0.0651889 0.997873i \(-0.520765\pi\)
−0.0651889 + 0.997873i \(0.520765\pi\)
\(224\) −1.46241 −0.0977113
\(225\) 0 0
\(226\) −28.5672 −1.90026
\(227\) −4.59774 −0.305163 −0.152581 0.988291i \(-0.548759\pi\)
−0.152581 + 0.988291i \(0.548759\pi\)
\(228\) 0 0
\(229\) 1.42275 0.0940178 0.0470089 0.998894i \(-0.485031\pi\)
0.0470089 + 0.998894i \(0.485031\pi\)
\(230\) −19.7424 −1.30177
\(231\) 0 0
\(232\) 35.5454 2.33367
\(233\) 9.75278 0.638926 0.319463 0.947599i \(-0.396497\pi\)
0.319463 + 0.947599i \(0.396497\pi\)
\(234\) 0 0
\(235\) 29.3854 1.91689
\(236\) 42.2483 2.75013
\(237\) 0 0
\(238\) −13.9711 −0.905609
\(239\) −0.842040 −0.0544670 −0.0272335 0.999629i \(-0.508670\pi\)
−0.0272335 + 0.999629i \(0.508670\pi\)
\(240\) 0 0
\(241\) −26.8301 −1.72828 −0.864141 0.503250i \(-0.832137\pi\)
−0.864141 + 0.503250i \(0.832137\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 26.7064 1.70971
\(245\) 4.01054 0.256224
\(246\) 0 0
\(247\) 1.24358 0.0791274
\(248\) 46.6195 2.96034
\(249\) 0 0
\(250\) −58.9823 −3.73037
\(251\) −13.8130 −0.871867 −0.435933 0.899979i \(-0.643582\pi\)
−0.435933 + 0.899979i \(0.643582\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −23.1852 −1.45477
\(255\) 0 0
\(256\) −30.1838 −1.88649
\(257\) −4.15222 −0.259008 −0.129504 0.991579i \(-0.541339\pi\)
−0.129504 + 0.991579i \(0.541339\pi\)
\(258\) 0 0
\(259\) 7.44454 0.462581
\(260\) 10.6410 0.659930
\(261\) 0 0
\(262\) −3.58443 −0.221447
\(263\) 6.34086 0.390994 0.195497 0.980704i \(-0.437368\pi\)
0.195497 + 0.980704i \(0.437368\pi\)
\(264\) 0 0
\(265\) −7.21061 −0.442945
\(266\) 4.35325 0.266915
\(267\) 0 0
\(268\) −45.5106 −2.78000
\(269\) 7.00736 0.427246 0.213623 0.976916i \(-0.431474\pi\)
0.213623 + 0.976916i \(0.431474\pi\)
\(270\) 0 0
\(271\) 28.1196 1.70815 0.854073 0.520153i \(-0.174125\pi\)
0.854073 + 0.520153i \(0.174125\pi\)
\(272\) −17.8029 −1.07946
\(273\) 0 0
\(274\) 42.5928 2.57313
\(275\) 0 0
\(276\) 0 0
\(277\) 16.1096 0.967932 0.483966 0.875087i \(-0.339196\pi\)
0.483966 + 0.875087i \(0.339196\pi\)
\(278\) −5.71054 −0.342495
\(279\) 0 0
\(280\) 17.8617 1.06744
\(281\) −9.29098 −0.554254 −0.277127 0.960833i \(-0.589382\pi\)
−0.277127 + 0.960833i \(0.589382\pi\)
\(282\) 0 0
\(283\) 15.9612 0.948796 0.474398 0.880310i \(-0.342666\pi\)
0.474398 + 0.880310i \(0.342666\pi\)
\(284\) −52.0753 −3.09010
\(285\) 0 0
\(286\) 0 0
\(287\) −5.66440 −0.334359
\(288\) 0 0
\(289\) 16.4084 0.965202
\(290\) 77.3689 4.54326
\(291\) 0 0
\(292\) −27.6760 −1.61962
\(293\) 7.55261 0.441228 0.220614 0.975361i \(-0.429194\pi\)
0.220614 + 0.975361i \(0.429194\pi\)
\(294\) 0 0
\(295\) 44.0953 2.56733
\(296\) 33.1557 1.92713
\(297\) 0 0
\(298\) 44.4811 2.57672
\(299\) 1.40624 0.0813247
\(300\) 0 0
\(301\) 3.44102 0.198337
\(302\) 12.9508 0.745236
\(303\) 0 0
\(304\) 5.54721 0.318154
\(305\) 27.8740 1.59606
\(306\) 0 0
\(307\) 0.903714 0.0515777 0.0257888 0.999667i \(-0.491790\pi\)
0.0257888 + 0.999667i \(0.491790\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 101.473 5.76329
\(311\) 6.62277 0.375543 0.187771 0.982213i \(-0.439874\pi\)
0.187771 + 0.982213i \(0.439874\pi\)
\(312\) 0 0
\(313\) −9.09175 −0.513896 −0.256948 0.966425i \(-0.582717\pi\)
−0.256948 + 0.966425i \(0.582717\pi\)
\(314\) 13.2117 0.745578
\(315\) 0 0
\(316\) 20.5133 1.15397
\(317\) −20.9949 −1.17919 −0.589596 0.807698i \(-0.700713\pi\)
−0.589596 + 0.807698i \(0.700713\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −38.8822 −2.17358
\(321\) 0 0
\(322\) 4.92263 0.274327
\(323\) −10.4098 −0.579214
\(324\) 0 0
\(325\) 7.65376 0.424554
\(326\) 42.1858 2.33646
\(327\) 0 0
\(328\) −25.2275 −1.39296
\(329\) −7.32704 −0.403953
\(330\) 0 0
\(331\) 34.8916 1.91782 0.958909 0.283715i \(-0.0915670\pi\)
0.958909 + 0.283715i \(0.0915670\pi\)
\(332\) 33.0110 1.81171
\(333\) 0 0
\(334\) 28.6514 1.56773
\(335\) −47.5002 −2.59521
\(336\) 0 0
\(337\) −22.6494 −1.23379 −0.616897 0.787044i \(-0.711610\pi\)
−0.616897 + 0.787044i \(0.711610\pi\)
\(338\) 30.2703 1.64649
\(339\) 0 0
\(340\) −89.0738 −4.83070
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 15.3252 0.826282
\(345\) 0 0
\(346\) 36.5283 1.96377
\(347\) −15.5034 −0.832268 −0.416134 0.909303i \(-0.636615\pi\)
−0.416134 + 0.909303i \(0.636615\pi\)
\(348\) 0 0
\(349\) 16.3823 0.876927 0.438463 0.898749i \(-0.355523\pi\)
0.438463 + 0.898749i \(0.355523\pi\)
\(350\) 26.7925 1.43212
\(351\) 0 0
\(352\) 0 0
\(353\) 19.9280 1.06066 0.530330 0.847792i \(-0.322068\pi\)
0.530330 + 0.847792i \(0.322068\pi\)
\(354\) 0 0
\(355\) −54.3520 −2.88470
\(356\) 50.2189 2.66160
\(357\) 0 0
\(358\) −49.2483 −2.60285
\(359\) −21.2044 −1.11913 −0.559563 0.828788i \(-0.689031\pi\)
−0.559563 + 0.828788i \(0.689031\pi\)
\(360\) 0 0
\(361\) −15.7564 −0.829285
\(362\) 60.5671 3.18334
\(363\) 0 0
\(364\) −2.65327 −0.139069
\(365\) −28.8860 −1.51196
\(366\) 0 0
\(367\) −3.65357 −0.190715 −0.0953575 0.995443i \(-0.530399\pi\)
−0.0953575 + 0.995443i \(0.530399\pi\)
\(368\) 6.27274 0.326989
\(369\) 0 0
\(370\) 72.1675 3.75181
\(371\) 1.79792 0.0933432
\(372\) 0 0
\(373\) −19.1169 −0.989837 −0.494919 0.868939i \(-0.664802\pi\)
−0.494919 + 0.868939i \(0.664802\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −32.6323 −1.68288
\(377\) −5.51094 −0.283828
\(378\) 0 0
\(379\) 33.3753 1.71438 0.857188 0.515003i \(-0.172209\pi\)
0.857188 + 0.515003i \(0.172209\pi\)
\(380\) 27.7546 1.42378
\(381\) 0 0
\(382\) 17.9929 0.920599
\(383\) −34.3561 −1.75552 −0.877758 0.479105i \(-0.840961\pi\)
−0.877758 + 0.479105i \(0.840961\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 55.6404 2.83202
\(387\) 0 0
\(388\) −36.5174 −1.85389
\(389\) 9.01029 0.456840 0.228420 0.973563i \(-0.426644\pi\)
0.228420 + 0.973563i \(0.426644\pi\)
\(390\) 0 0
\(391\) −11.7713 −0.595299
\(392\) −4.45369 −0.224945
\(393\) 0 0
\(394\) −15.3371 −0.772672
\(395\) 21.4101 1.07726
\(396\) 0 0
\(397\) −4.43406 −0.222539 −0.111269 0.993790i \(-0.535492\pi\)
−0.111269 + 0.993790i \(0.535492\pi\)
\(398\) 51.6971 2.59134
\(399\) 0 0
\(400\) 34.1408 1.70704
\(401\) −6.23579 −0.311401 −0.155700 0.987804i \(-0.549763\pi\)
−0.155700 + 0.987804i \(0.549763\pi\)
\(402\) 0 0
\(403\) −7.22787 −0.360046
\(404\) 37.2889 1.85519
\(405\) 0 0
\(406\) −19.2914 −0.957417
\(407\) 0 0
\(408\) 0 0
\(409\) 13.1954 0.652471 0.326236 0.945289i \(-0.394220\pi\)
0.326236 + 0.945289i \(0.394220\pi\)
\(410\) −54.9108 −2.71185
\(411\) 0 0
\(412\) 28.1569 1.38719
\(413\) −10.9949 −0.541022
\(414\) 0 0
\(415\) 34.4541 1.69129
\(416\) 1.00979 0.0495091
\(417\) 0 0
\(418\) 0 0
\(419\) −0.552355 −0.0269843 −0.0134922 0.999909i \(-0.504295\pi\)
−0.0134922 + 0.999909i \(0.504295\pi\)
\(420\) 0 0
\(421\) −6.61758 −0.322521 −0.161260 0.986912i \(-0.551556\pi\)
−0.161260 + 0.986912i \(0.551556\pi\)
\(422\) −8.16081 −0.397262
\(423\) 0 0
\(424\) 8.00737 0.388872
\(425\) −64.0679 −3.10775
\(426\) 0 0
\(427\) −6.95019 −0.336343
\(428\) −47.7813 −2.30960
\(429\) 0 0
\(430\) 33.3573 1.60863
\(431\) 0.520604 0.0250766 0.0125383 0.999921i \(-0.496009\pi\)
0.0125383 + 0.999921i \(0.496009\pi\)
\(432\) 0 0
\(433\) 0.949704 0.0456398 0.0228199 0.999740i \(-0.492736\pi\)
0.0228199 + 0.999740i \(0.492736\pi\)
\(434\) −25.3017 −1.21452
\(435\) 0 0
\(436\) −48.3987 −2.31788
\(437\) 3.66782 0.175456
\(438\) 0 0
\(439\) 36.6885 1.75105 0.875524 0.483174i \(-0.160516\pi\)
0.875524 + 0.483174i \(0.160516\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 9.64699 0.458860
\(443\) −10.2133 −0.485249 −0.242624 0.970120i \(-0.578008\pi\)
−0.242624 + 0.970120i \(0.578008\pi\)
\(444\) 0 0
\(445\) 52.4144 2.48468
\(446\) 4.70606 0.222838
\(447\) 0 0
\(448\) 9.69500 0.458046
\(449\) 20.0890 0.948059 0.474030 0.880509i \(-0.342799\pi\)
0.474030 + 0.880509i \(0.342799\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 45.4135 2.13607
\(453\) 0 0
\(454\) 11.1134 0.521576
\(455\) −2.76927 −0.129825
\(456\) 0 0
\(457\) 0.566344 0.0264925 0.0132462 0.999912i \(-0.495783\pi\)
0.0132462 + 0.999912i \(0.495783\pi\)
\(458\) −3.43897 −0.160693
\(459\) 0 0
\(460\) 31.3847 1.46332
\(461\) −12.2429 −0.570209 −0.285105 0.958496i \(-0.592028\pi\)
−0.285105 + 0.958496i \(0.592028\pi\)
\(462\) 0 0
\(463\) −23.5815 −1.09593 −0.547963 0.836502i \(-0.684597\pi\)
−0.547963 + 0.836502i \(0.684597\pi\)
\(464\) −24.5824 −1.14121
\(465\) 0 0
\(466\) −23.5738 −1.09204
\(467\) −14.4991 −0.670937 −0.335468 0.942051i \(-0.608895\pi\)
−0.335468 + 0.942051i \(0.608895\pi\)
\(468\) 0 0
\(469\) 11.8438 0.546898
\(470\) −71.0284 −3.27630
\(471\) 0 0
\(472\) −48.9677 −2.25392
\(473\) 0 0
\(474\) 0 0
\(475\) 19.9630 0.915964
\(476\) 22.2099 1.01799
\(477\) 0 0
\(478\) 2.03533 0.0930936
\(479\) 29.8824 1.36536 0.682681 0.730717i \(-0.260814\pi\)
0.682681 + 0.730717i \(0.260814\pi\)
\(480\) 0 0
\(481\) −5.14044 −0.234384
\(482\) 64.8521 2.95393
\(483\) 0 0
\(484\) 0 0
\(485\) −38.1139 −1.73066
\(486\) 0 0
\(487\) 23.4828 1.06411 0.532053 0.846711i \(-0.321421\pi\)
0.532053 + 0.846711i \(0.321421\pi\)
\(488\) −30.9540 −1.40122
\(489\) 0 0
\(490\) −9.69401 −0.437931
\(491\) −14.1496 −0.638561 −0.319281 0.947660i \(-0.603441\pi\)
−0.319281 + 0.947660i \(0.603441\pi\)
\(492\) 0 0
\(493\) 46.1308 2.07763
\(494\) −3.00591 −0.135242
\(495\) 0 0
\(496\) −32.2411 −1.44767
\(497\) 13.5523 0.607903
\(498\) 0 0
\(499\) 4.41009 0.197423 0.0987114 0.995116i \(-0.468528\pi\)
0.0987114 + 0.995116i \(0.468528\pi\)
\(500\) 93.7648 4.19329
\(501\) 0 0
\(502\) 33.3878 1.49017
\(503\) 13.9458 0.621812 0.310906 0.950441i \(-0.399368\pi\)
0.310906 + 0.950441i \(0.399368\pi\)
\(504\) 0 0
\(505\) 38.9191 1.73188
\(506\) 0 0
\(507\) 0 0
\(508\) 36.8577 1.63530
\(509\) 10.1933 0.451812 0.225906 0.974149i \(-0.427466\pi\)
0.225906 + 0.974149i \(0.427466\pi\)
\(510\) 0 0
\(511\) 7.20252 0.318621
\(512\) 31.9398 1.41155
\(513\) 0 0
\(514\) 10.0365 0.442690
\(515\) 29.3879 1.29499
\(516\) 0 0
\(517\) 0 0
\(518\) −17.9945 −0.790631
\(519\) 0 0
\(520\) −12.3335 −0.540858
\(521\) 0.311793 0.0136599 0.00682994 0.999977i \(-0.497826\pi\)
0.00682994 + 0.999977i \(0.497826\pi\)
\(522\) 0 0
\(523\) 1.83936 0.0804296 0.0402148 0.999191i \(-0.487196\pi\)
0.0402148 + 0.999191i \(0.487196\pi\)
\(524\) 5.69821 0.248928
\(525\) 0 0
\(526\) −15.3267 −0.668277
\(527\) 60.5029 2.63555
\(528\) 0 0
\(529\) −18.8525 −0.819672
\(530\) 17.4290 0.757069
\(531\) 0 0
\(532\) −6.92041 −0.300038
\(533\) 3.91126 0.169416
\(534\) 0 0
\(535\) −49.8702 −2.15608
\(536\) 52.7488 2.27840
\(537\) 0 0
\(538\) −16.9377 −0.730238
\(539\) 0 0
\(540\) 0 0
\(541\) −1.05220 −0.0452374 −0.0226187 0.999744i \(-0.507200\pi\)
−0.0226187 + 0.999744i \(0.507200\pi\)
\(542\) −67.9690 −2.91952
\(543\) 0 0
\(544\) −8.45273 −0.362408
\(545\) −50.5146 −2.16381
\(546\) 0 0
\(547\) −46.4795 −1.98732 −0.993659 0.112434i \(-0.964135\pi\)
−0.993659 + 0.112434i \(0.964135\pi\)
\(548\) −67.7103 −2.89244
\(549\) 0 0
\(550\) 0 0
\(551\) −14.3739 −0.612350
\(552\) 0 0
\(553\) −5.33847 −0.227015
\(554\) −38.9391 −1.65436
\(555\) 0 0
\(556\) 9.07811 0.384997
\(557\) −11.5830 −0.490789 −0.245395 0.969423i \(-0.578918\pi\)
−0.245395 + 0.969423i \(0.578918\pi\)
\(558\) 0 0
\(559\) −2.37602 −0.100495
\(560\) −12.3528 −0.522000
\(561\) 0 0
\(562\) 22.4576 0.947316
\(563\) 35.4172 1.49266 0.746328 0.665578i \(-0.231815\pi\)
0.746328 + 0.665578i \(0.231815\pi\)
\(564\) 0 0
\(565\) 47.3989 1.99409
\(566\) −38.5805 −1.62166
\(567\) 0 0
\(568\) 60.3577 2.53255
\(569\) −36.2759 −1.52077 −0.760383 0.649475i \(-0.774989\pi\)
−0.760383 + 0.649475i \(0.774989\pi\)
\(570\) 0 0
\(571\) 7.31960 0.306316 0.153158 0.988202i \(-0.451056\pi\)
0.153158 + 0.988202i \(0.451056\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 13.6916 0.571478
\(575\) 22.5740 0.941400
\(576\) 0 0
\(577\) 14.7700 0.614883 0.307441 0.951567i \(-0.400527\pi\)
0.307441 + 0.951567i \(0.400527\pi\)
\(578\) −39.6614 −1.64970
\(579\) 0 0
\(580\) −122.994 −5.10706
\(581\) −8.59090 −0.356411
\(582\) 0 0
\(583\) 0 0
\(584\) 32.0778 1.32739
\(585\) 0 0
\(586\) −18.2557 −0.754136
\(587\) −9.27769 −0.382931 −0.191466 0.981499i \(-0.561324\pi\)
−0.191466 + 0.981499i \(0.561324\pi\)
\(588\) 0 0
\(589\) −18.8521 −0.776788
\(590\) −106.584 −4.38801
\(591\) 0 0
\(592\) −22.9298 −0.942407
\(593\) −22.1174 −0.908252 −0.454126 0.890937i \(-0.650048\pi\)
−0.454126 + 0.890937i \(0.650048\pi\)
\(594\) 0 0
\(595\) 23.1809 0.950325
\(596\) −70.7120 −2.89648
\(597\) 0 0
\(598\) −3.39906 −0.138998
\(599\) −5.17361 −0.211388 −0.105694 0.994399i \(-0.533706\pi\)
−0.105694 + 0.994399i \(0.533706\pi\)
\(600\) 0 0
\(601\) −9.56657 −0.390228 −0.195114 0.980781i \(-0.562508\pi\)
−0.195114 + 0.980781i \(0.562508\pi\)
\(602\) −8.31742 −0.338993
\(603\) 0 0
\(604\) −20.5881 −0.837716
\(605\) 0 0
\(606\) 0 0
\(607\) −24.6208 −0.999328 −0.499664 0.866219i \(-0.666543\pi\)
−0.499664 + 0.866219i \(0.666543\pi\)
\(608\) 2.63379 0.106814
\(609\) 0 0
\(610\) −67.3753 −2.72794
\(611\) 5.05931 0.204678
\(612\) 0 0
\(613\) 39.7281 1.60460 0.802301 0.596920i \(-0.203609\pi\)
0.802301 + 0.596920i \(0.203609\pi\)
\(614\) −2.18440 −0.0881552
\(615\) 0 0
\(616\) 0 0
\(617\) 35.9804 1.44852 0.724258 0.689529i \(-0.242183\pi\)
0.724258 + 0.689529i \(0.242183\pi\)
\(618\) 0 0
\(619\) −0.447180 −0.0179737 −0.00898683 0.999960i \(-0.502861\pi\)
−0.00898683 + 0.999960i \(0.502861\pi\)
\(620\) −161.313 −6.47849
\(621\) 0 0
\(622\) −16.0081 −0.641868
\(623\) −13.0692 −0.523605
\(624\) 0 0
\(625\) 42.4420 1.69768
\(626\) 21.9760 0.878337
\(627\) 0 0
\(628\) −21.0028 −0.838101
\(629\) 43.0295 1.71570
\(630\) 0 0
\(631\) −6.09051 −0.242459 −0.121230 0.992624i \(-0.538684\pi\)
−0.121230 + 0.992624i \(0.538684\pi\)
\(632\) −23.7759 −0.945754
\(633\) 0 0
\(634\) 50.7476 2.01544
\(635\) 38.4691 1.52660
\(636\) 0 0
\(637\) 0.690498 0.0273585
\(638\) 0 0
\(639\) 0 0
\(640\) 82.2534 3.25135
\(641\) −35.5955 −1.40594 −0.702969 0.711220i \(-0.748143\pi\)
−0.702969 + 0.711220i \(0.748143\pi\)
\(642\) 0 0
\(643\) −29.2126 −1.15203 −0.576017 0.817438i \(-0.695394\pi\)
−0.576017 + 0.817438i \(0.695394\pi\)
\(644\) −7.82555 −0.308370
\(645\) 0 0
\(646\) 25.1618 0.989978
\(647\) 9.44881 0.371471 0.185736 0.982600i \(-0.440533\pi\)
0.185736 + 0.982600i \(0.440533\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −18.5002 −0.725637
\(651\) 0 0
\(652\) −67.0633 −2.62640
\(653\) 24.2269 0.948073 0.474036 0.880505i \(-0.342796\pi\)
0.474036 + 0.880505i \(0.342796\pi\)
\(654\) 0 0
\(655\) 5.94733 0.232381
\(656\) 17.4468 0.681184
\(657\) 0 0
\(658\) 17.7104 0.690425
\(659\) −19.2004 −0.747943 −0.373971 0.927440i \(-0.622004\pi\)
−0.373971 + 0.927440i \(0.622004\pi\)
\(660\) 0 0
\(661\) 6.95611 0.270561 0.135281 0.990807i \(-0.456806\pi\)
0.135281 + 0.990807i \(0.456806\pi\)
\(662\) −84.3378 −3.27788
\(663\) 0 0
\(664\) −38.2612 −1.48482
\(665\) −7.22296 −0.280094
\(666\) 0 0
\(667\) −16.2539 −0.629355
\(668\) −45.5474 −1.76228
\(669\) 0 0
\(670\) 114.814 4.43567
\(671\) 0 0
\(672\) 0 0
\(673\) −49.2066 −1.89677 −0.948387 0.317114i \(-0.897286\pi\)
−0.948387 + 0.317114i \(0.897286\pi\)
\(674\) 54.7468 2.10877
\(675\) 0 0
\(676\) −48.1210 −1.85081
\(677\) −14.8594 −0.571093 −0.285546 0.958365i \(-0.592175\pi\)
−0.285546 + 0.958365i \(0.592175\pi\)
\(678\) 0 0
\(679\) 9.50344 0.364708
\(680\) 103.241 3.95910
\(681\) 0 0
\(682\) 0 0
\(683\) −22.6395 −0.866275 −0.433138 0.901328i \(-0.642594\pi\)
−0.433138 + 0.901328i \(0.642594\pi\)
\(684\) 0 0
\(685\) −70.6705 −2.70018
\(686\) 2.41714 0.0922867
\(687\) 0 0
\(688\) −10.5986 −0.404069
\(689\) −1.24146 −0.0472958
\(690\) 0 0
\(691\) 13.0444 0.496232 0.248116 0.968730i \(-0.420189\pi\)
0.248116 + 0.968730i \(0.420189\pi\)
\(692\) −58.0695 −2.20747
\(693\) 0 0
\(694\) 37.4739 1.42249
\(695\) 9.47498 0.359407
\(696\) 0 0
\(697\) −32.7403 −1.24013
\(698\) −39.5984 −1.49882
\(699\) 0 0
\(700\) −42.5924 −1.60984
\(701\) −0.471163 −0.0177956 −0.00889778 0.999960i \(-0.502832\pi\)
−0.00889778 + 0.999960i \(0.502832\pi\)
\(702\) 0 0
\(703\) −13.4076 −0.505676
\(704\) 0 0
\(705\) 0 0
\(706\) −48.1686 −1.81285
\(707\) −9.70420 −0.364964
\(708\) 0 0
\(709\) 32.8816 1.23490 0.617448 0.786612i \(-0.288167\pi\)
0.617448 + 0.786612i \(0.288167\pi\)
\(710\) 131.376 4.93046
\(711\) 0 0
\(712\) −58.2060 −2.18136
\(713\) −21.3179 −0.798360
\(714\) 0 0
\(715\) 0 0
\(716\) 78.2906 2.92586
\(717\) 0 0
\(718\) 51.2539 1.91278
\(719\) 16.6079 0.619372 0.309686 0.950839i \(-0.399776\pi\)
0.309686 + 0.950839i \(0.399776\pi\)
\(720\) 0 0
\(721\) −7.32767 −0.272897
\(722\) 38.0854 1.41739
\(723\) 0 0
\(724\) −96.2843 −3.57838
\(725\) −88.4658 −3.28554
\(726\) 0 0
\(727\) 8.21481 0.304670 0.152335 0.988329i \(-0.451321\pi\)
0.152335 + 0.988329i \(0.451321\pi\)
\(728\) 3.07526 0.113977
\(729\) 0 0
\(730\) 69.8213 2.58420
\(731\) 19.8891 0.735626
\(732\) 0 0
\(733\) 9.37274 0.346190 0.173095 0.984905i \(-0.444623\pi\)
0.173095 + 0.984905i \(0.444623\pi\)
\(734\) 8.83119 0.325965
\(735\) 0 0
\(736\) 2.97827 0.109781
\(737\) 0 0
\(738\) 0 0
\(739\) −39.0743 −1.43737 −0.718685 0.695336i \(-0.755256\pi\)
−0.718685 + 0.695336i \(0.755256\pi\)
\(740\) −114.725 −4.21739
\(741\) 0 0
\(742\) −4.34581 −0.159540
\(743\) 35.1057 1.28790 0.643952 0.765066i \(-0.277294\pi\)
0.643952 + 0.765066i \(0.277294\pi\)
\(744\) 0 0
\(745\) −73.8034 −2.70395
\(746\) 46.2082 1.69180
\(747\) 0 0
\(748\) 0 0
\(749\) 12.4348 0.454357
\(750\) 0 0
\(751\) −0.0765251 −0.00279244 −0.00139622 0.999999i \(-0.500444\pi\)
−0.00139622 + 0.999999i \(0.500444\pi\)
\(752\) 22.5678 0.822965
\(753\) 0 0
\(754\) 13.3207 0.485111
\(755\) −21.4881 −0.782033
\(756\) 0 0
\(757\) −9.60692 −0.349170 −0.174585 0.984642i \(-0.555858\pi\)
−0.174585 + 0.984642i \(0.555858\pi\)
\(758\) −80.6728 −2.93017
\(759\) 0 0
\(760\) −32.1688 −1.16689
\(761\) 31.3129 1.13509 0.567545 0.823342i \(-0.307893\pi\)
0.567545 + 0.823342i \(0.307893\pi\)
\(762\) 0 0
\(763\) 12.5955 0.455986
\(764\) −28.6036 −1.03484
\(765\) 0 0
\(766\) 83.0434 3.00048
\(767\) 7.59193 0.274129
\(768\) 0 0
\(769\) 9.43619 0.340278 0.170139 0.985420i \(-0.445578\pi\)
0.170139 + 0.985420i \(0.445578\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −88.4521 −3.18346
\(773\) 11.2386 0.404226 0.202113 0.979362i \(-0.435219\pi\)
0.202113 + 0.979362i \(0.435219\pi\)
\(774\) 0 0
\(775\) −116.027 −4.16782
\(776\) 42.3254 1.51939
\(777\) 0 0
\(778\) −21.7791 −0.780818
\(779\) 10.2016 0.365509
\(780\) 0 0
\(781\) 0 0
\(782\) 28.4528 1.01747
\(783\) 0 0
\(784\) 3.08008 0.110003
\(785\) −21.9210 −0.782392
\(786\) 0 0
\(787\) −26.8005 −0.955333 −0.477667 0.878541i \(-0.658517\pi\)
−0.477667 + 0.878541i \(0.658517\pi\)
\(788\) 24.3815 0.868557
\(789\) 0 0
\(790\) −51.7512 −1.84123
\(791\) −11.8186 −0.420221
\(792\) 0 0
\(793\) 4.79910 0.170421
\(794\) 10.7177 0.380358
\(795\) 0 0
\(796\) −82.1835 −2.91292
\(797\) −47.4878 −1.68211 −0.841053 0.540953i \(-0.818064\pi\)
−0.841053 + 0.540953i \(0.818064\pi\)
\(798\) 0 0
\(799\) −42.3503 −1.49825
\(800\) 16.2099 0.573108
\(801\) 0 0
\(802\) 15.0728 0.532238
\(803\) 0 0
\(804\) 0 0
\(805\) −8.16767 −0.287873
\(806\) 17.4707 0.615381
\(807\) 0 0
\(808\) −43.2195 −1.52046
\(809\) −23.5047 −0.826382 −0.413191 0.910644i \(-0.635586\pi\)
−0.413191 + 0.910644i \(0.635586\pi\)
\(810\) 0 0
\(811\) 49.9468 1.75387 0.876934 0.480610i \(-0.159585\pi\)
0.876934 + 0.480610i \(0.159585\pi\)
\(812\) 30.6678 1.07623
\(813\) 0 0
\(814\) 0 0
\(815\) −69.9952 −2.45182
\(816\) 0 0
\(817\) −6.19727 −0.216815
\(818\) −31.8951 −1.11519
\(819\) 0 0
\(820\) 87.2923 3.04838
\(821\) 44.3034 1.54620 0.773100 0.634284i \(-0.218705\pi\)
0.773100 + 0.634284i \(0.218705\pi\)
\(822\) 0 0
\(823\) 25.4653 0.887666 0.443833 0.896109i \(-0.353618\pi\)
0.443833 + 0.896109i \(0.353618\pi\)
\(824\) −32.6352 −1.13690
\(825\) 0 0
\(826\) 26.5761 0.924700
\(827\) 11.4958 0.399748 0.199874 0.979822i \(-0.435947\pi\)
0.199874 + 0.979822i \(0.435947\pi\)
\(828\) 0 0
\(829\) −1.58988 −0.0552187 −0.0276094 0.999619i \(-0.508789\pi\)
−0.0276094 + 0.999619i \(0.508789\pi\)
\(830\) −83.2803 −2.89070
\(831\) 0 0
\(832\) −6.69438 −0.232086
\(833\) −5.78000 −0.200265
\(834\) 0 0
\(835\) −47.5387 −1.64514
\(836\) 0 0
\(837\) 0 0
\(838\) 1.33512 0.0461209
\(839\) −5.98145 −0.206503 −0.103251 0.994655i \(-0.532925\pi\)
−0.103251 + 0.994655i \(0.532925\pi\)
\(840\) 0 0
\(841\) 34.6980 1.19648
\(842\) 15.9956 0.551244
\(843\) 0 0
\(844\) 12.9733 0.446560
\(845\) −50.2248 −1.72779
\(846\) 0 0
\(847\) 0 0
\(848\) −5.53773 −0.190166
\(849\) 0 0
\(850\) 154.861 5.31168
\(851\) −15.1612 −0.519719
\(852\) 0 0
\(853\) 8.89990 0.304727 0.152363 0.988325i \(-0.451312\pi\)
0.152363 + 0.988325i \(0.451312\pi\)
\(854\) 16.7996 0.574869
\(855\) 0 0
\(856\) 55.3807 1.89287
\(857\) 52.2243 1.78395 0.891973 0.452088i \(-0.149321\pi\)
0.891973 + 0.452088i \(0.149321\pi\)
\(858\) 0 0
\(859\) 5.76678 0.196760 0.0983800 0.995149i \(-0.468634\pi\)
0.0983800 + 0.995149i \(0.468634\pi\)
\(860\) −53.0285 −1.80826
\(861\) 0 0
\(862\) −1.25837 −0.0428603
\(863\) −26.9515 −0.917440 −0.458720 0.888581i \(-0.651692\pi\)
−0.458720 + 0.888581i \(0.651692\pi\)
\(864\) 0 0
\(865\) −60.6082 −2.06074
\(866\) −2.29556 −0.0780064
\(867\) 0 0
\(868\) 40.2223 1.36523
\(869\) 0 0
\(870\) 0 0
\(871\) −8.17816 −0.277106
\(872\) 56.0963 1.89966
\(873\) 0 0
\(874\) −8.86563 −0.299884
\(875\) −24.4017 −0.824929
\(876\) 0 0
\(877\) 23.0226 0.777419 0.388709 0.921360i \(-0.372921\pi\)
0.388709 + 0.921360i \(0.372921\pi\)
\(878\) −88.6812 −2.99285
\(879\) 0 0
\(880\) 0 0
\(881\) 9.80782 0.330434 0.165217 0.986257i \(-0.447168\pi\)
0.165217 + 0.986257i \(0.447168\pi\)
\(882\) 0 0
\(883\) −15.0392 −0.506109 −0.253054 0.967452i \(-0.581435\pi\)
−0.253054 + 0.967452i \(0.581435\pi\)
\(884\) −15.3359 −0.515803
\(885\) 0 0
\(886\) 24.6870 0.829374
\(887\) 11.0331 0.370455 0.185227 0.982696i \(-0.440698\pi\)
0.185227 + 0.982696i \(0.440698\pi\)
\(888\) 0 0
\(889\) −9.59200 −0.321705
\(890\) −126.693 −4.24675
\(891\) 0 0
\(892\) −7.48127 −0.250491
\(893\) 13.1960 0.441586
\(894\) 0 0
\(895\) 81.7133 2.73137
\(896\) −20.5093 −0.685168
\(897\) 0 0
\(898\) −48.5579 −1.62040
\(899\) 83.5431 2.78632
\(900\) 0 0
\(901\) 10.3920 0.346207
\(902\) 0 0
\(903\) 0 0
\(904\) −52.6364 −1.75066
\(905\) −100.494 −3.34052
\(906\) 0 0
\(907\) −14.5803 −0.484131 −0.242066 0.970260i \(-0.577825\pi\)
−0.242066 + 0.970260i \(0.577825\pi\)
\(908\) −17.6670 −0.586301
\(909\) 0 0
\(910\) 6.69370 0.221894
\(911\) 3.29983 0.109328 0.0546642 0.998505i \(-0.482591\pi\)
0.0546642 + 0.998505i \(0.482591\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −1.36893 −0.0452802
\(915\) 0 0
\(916\) 5.46697 0.180634
\(917\) −1.48293 −0.0489705
\(918\) 0 0
\(919\) 26.6221 0.878181 0.439091 0.898443i \(-0.355301\pi\)
0.439091 + 0.898443i \(0.355301\pi\)
\(920\) −36.3763 −1.19929
\(921\) 0 0
\(922\) 29.5928 0.974587
\(923\) −9.35784 −0.308017
\(924\) 0 0
\(925\) −82.5183 −2.71318
\(926\) 56.9998 1.87313
\(927\) 0 0
\(928\) −11.6716 −0.383140
\(929\) −4.34942 −0.142700 −0.0713500 0.997451i \(-0.522731\pi\)
−0.0713500 + 0.997451i \(0.522731\pi\)
\(930\) 0 0
\(931\) 1.80100 0.0590252
\(932\) 37.4755 1.22755
\(933\) 0 0
\(934\) 35.0462 1.14675
\(935\) 0 0
\(936\) 0 0
\(937\) −48.1141 −1.57182 −0.785909 0.618342i \(-0.787805\pi\)
−0.785909 + 0.618342i \(0.787805\pi\)
\(938\) −28.6282 −0.934744
\(939\) 0 0
\(940\) 112.915 3.68287
\(941\) 19.5441 0.637118 0.318559 0.947903i \(-0.396801\pi\)
0.318559 + 0.947903i \(0.396801\pi\)
\(942\) 0 0
\(943\) 11.5359 0.375659
\(944\) 33.8650 1.10221
\(945\) 0 0
\(946\) 0 0
\(947\) −54.2546 −1.76304 −0.881519 0.472149i \(-0.843478\pi\)
−0.881519 + 0.472149i \(0.843478\pi\)
\(948\) 0 0
\(949\) −4.97333 −0.161441
\(950\) −48.2532 −1.56554
\(951\) 0 0
\(952\) −25.7423 −0.834314
\(953\) −26.0971 −0.845368 −0.422684 0.906277i \(-0.638912\pi\)
−0.422684 + 0.906277i \(0.638912\pi\)
\(954\) 0 0
\(955\) −29.8541 −0.966055
\(956\) −3.23558 −0.104646
\(957\) 0 0
\(958\) −72.2298 −2.33364
\(959\) 17.6212 0.569018
\(960\) 0 0
\(961\) 78.5710 2.53455
\(962\) 12.4251 0.400603
\(963\) 0 0
\(964\) −103.096 −3.32050
\(965\) −92.3191 −2.97186
\(966\) 0 0
\(967\) −13.7935 −0.443567 −0.221784 0.975096i \(-0.571188\pi\)
−0.221784 + 0.975096i \(0.571188\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 92.1265 2.95800
\(971\) 15.2502 0.489403 0.244701 0.969599i \(-0.421310\pi\)
0.244701 + 0.969599i \(0.421310\pi\)
\(972\) 0 0
\(973\) −2.36252 −0.0757390
\(974\) −56.7611 −1.81874
\(975\) 0 0
\(976\) 21.4071 0.685226
\(977\) −12.2867 −0.393088 −0.196544 0.980495i \(-0.562972\pi\)
−0.196544 + 0.980495i \(0.562972\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 15.4107 0.492276
\(981\) 0 0
\(982\) 34.2014 1.09141
\(983\) −12.7524 −0.406737 −0.203369 0.979102i \(-0.565189\pi\)
−0.203369 + 0.979102i \(0.565189\pi\)
\(984\) 0 0
\(985\) 25.4475 0.810824
\(986\) −111.504 −3.55102
\(987\) 0 0
\(988\) 4.77853 0.152025
\(989\) −7.00783 −0.222836
\(990\) 0 0
\(991\) 49.0744 1.55890 0.779449 0.626465i \(-0.215499\pi\)
0.779449 + 0.626465i \(0.215499\pi\)
\(992\) −15.3079 −0.486028
\(993\) 0 0
\(994\) −32.7577 −1.03901
\(995\) −85.7764 −2.71929
\(996\) 0 0
\(997\) −26.9368 −0.853098 −0.426549 0.904464i \(-0.640271\pi\)
−0.426549 + 0.904464i \(0.640271\pi\)
\(998\) −10.6598 −0.337430
\(999\) 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.cz.1.2 12
3.2 odd 2 inner 7623.2.a.cz.1.11 yes 12
11.10 odd 2 7623.2.a.da.1.11 yes 12
33.32 even 2 7623.2.a.da.1.2 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7623.2.a.cz.1.2 12 1.1 even 1 trivial
7623.2.a.cz.1.11 yes 12 3.2 odd 2 inner
7623.2.a.da.1.2 yes 12 33.32 even 2
7623.2.a.da.1.11 yes 12 11.10 odd 2