Properties

Label 378.5.b.a.323.2
Level $378$
Weight $5$
Character 378.323
Analytic conductor $39.074$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 378.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(39.0738460457\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.5443747577856.29
Defining polynomial: \( x^{8} + 24x^{6} + 180x^{4} + 488x^{2} + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{4}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 323.2
Root \(2.14697i\) of defining polynomial
Character \(\chi\) \(=\) 378.323
Dual form 378.5.b.a.323.7

$q$-expansion

\(f(q)\) \(=\) \(q-2.82843i q^{2} -8.00000 q^{4} -9.19100i q^{5} +18.5203 q^{7} +22.6274i q^{8} +O(q^{10})\) \(q-2.82843i q^{2} -8.00000 q^{4} -9.19100i q^{5} +18.5203 q^{7} +22.6274i q^{8} -25.9961 q^{10} -68.1448i q^{11} +212.413 q^{13} -52.3832i q^{14} +64.0000 q^{16} +340.636i q^{17} -176.519 q^{19} +73.5280i q^{20} -192.743 q^{22} +241.733i q^{23} +540.526 q^{25} -600.795i q^{26} -148.162 q^{28} -549.225i q^{29} +980.834 q^{31} -181.019i q^{32} +963.463 q^{34} -170.220i q^{35} +743.228 q^{37} +499.271i q^{38} +207.968 q^{40} -1964.59i q^{41} -1353.97 q^{43} +545.158i q^{44} +683.725 q^{46} +1321.73i q^{47} +343.000 q^{49} -1528.84i q^{50} -1699.30 q^{52} -4062.59i q^{53} -626.318 q^{55} +419.066i q^{56} -1553.44 q^{58} -3523.36i q^{59} +5207.28 q^{61} -2774.22i q^{62} -512.000 q^{64} -1952.29i q^{65} +2249.24 q^{67} -2725.08i q^{68} -481.454 q^{70} -7437.67i q^{71} -1051.46 q^{73} -2102.17i q^{74} +1412.15 q^{76} -1262.06i q^{77} +3149.66 q^{79} -588.224i q^{80} -5556.71 q^{82} +467.561i q^{83} +3130.78 q^{85} +3829.62i q^{86} +1541.94 q^{88} +2865.60i q^{89} +3933.94 q^{91} -1933.87i q^{92} +3738.42 q^{94} +1622.38i q^{95} -15884.4 q^{97} -970.151i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 64 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 64 q^{4} - 224 q^{10} - 376 q^{13} + 512 q^{16} + 1120 q^{19} - 1120 q^{22} + 792 q^{25} - 880 q^{31} - 1792 q^{34} + 1576 q^{37} + 1792 q^{40} - 5768 q^{43} - 160 q^{46} + 2744 q^{49} + 3008 q^{52} + 488 q^{55} + 7552 q^{58} - 2560 q^{61} - 4096 q^{64} + 23784 q^{67} + 1568 q^{70} - 13176 q^{73} - 8960 q^{76} - 9592 q^{79} - 3360 q^{82} - 39880 q^{85} + 8960 q^{88} + 5096 q^{91} + 4608 q^{94} - 14016 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.82843i − 0.707107i
\(3\) 0 0
\(4\) −8.00000 −0.500000
\(5\) − 9.19100i − 0.367640i −0.982960 0.183820i \(-0.941154\pi\)
0.982960 0.183820i \(-0.0588463\pi\)
\(6\) 0 0
\(7\) 18.5203 0.377964
\(8\) 22.6274i 0.353553i
\(9\) 0 0
\(10\) −25.9961 −0.259961
\(11\) − 68.1448i − 0.563180i −0.959535 0.281590i \(-0.909138\pi\)
0.959535 0.281590i \(-0.0908618\pi\)
\(12\) 0 0
\(13\) 212.413 1.25688 0.628441 0.777857i \(-0.283693\pi\)
0.628441 + 0.777857i \(0.283693\pi\)
\(14\) − 52.3832i − 0.267261i
\(15\) 0 0
\(16\) 64.0000 0.250000
\(17\) 340.636i 1.17867i 0.807889 + 0.589335i \(0.200610\pi\)
−0.807889 + 0.589335i \(0.799390\pi\)
\(18\) 0 0
\(19\) −176.519 −0.488972 −0.244486 0.969653i \(-0.578619\pi\)
−0.244486 + 0.969653i \(0.578619\pi\)
\(20\) 73.5280i 0.183820i
\(21\) 0 0
\(22\) −192.743 −0.398228
\(23\) 241.733i 0.456963i 0.973548 + 0.228481i \(0.0733760\pi\)
−0.973548 + 0.228481i \(0.926624\pi\)
\(24\) 0 0
\(25\) 540.526 0.864841
\(26\) − 600.795i − 0.888750i
\(27\) 0 0
\(28\) −148.162 −0.188982
\(29\) − 549.225i − 0.653062i −0.945186 0.326531i \(-0.894120\pi\)
0.945186 0.326531i \(-0.105880\pi\)
\(30\) 0 0
\(31\) 980.834 1.02064 0.510319 0.859985i \(-0.329527\pi\)
0.510319 + 0.859985i \(0.329527\pi\)
\(32\) − 181.019i − 0.176777i
\(33\) 0 0
\(34\) 963.463 0.833445
\(35\) − 170.220i − 0.138955i
\(36\) 0 0
\(37\) 743.228 0.542899 0.271449 0.962453i \(-0.412497\pi\)
0.271449 + 0.962453i \(0.412497\pi\)
\(38\) 499.271i 0.345755i
\(39\) 0 0
\(40\) 207.968 0.129980
\(41\) − 1964.59i − 1.16871i −0.811500 0.584353i \(-0.801348\pi\)
0.811500 0.584353i \(-0.198652\pi\)
\(42\) 0 0
\(43\) −1353.97 −0.732274 −0.366137 0.930561i \(-0.619320\pi\)
−0.366137 + 0.930561i \(0.619320\pi\)
\(44\) 545.158i 0.281590i
\(45\) 0 0
\(46\) 683.725 0.323121
\(47\) 1321.73i 0.598339i 0.954200 + 0.299169i \(0.0967095\pi\)
−0.954200 + 0.299169i \(0.903290\pi\)
\(48\) 0 0
\(49\) 343.000 0.142857
\(50\) − 1528.84i − 0.611535i
\(51\) 0 0
\(52\) −1699.30 −0.628441
\(53\) − 4062.59i − 1.44628i −0.690703 0.723139i \(-0.742699\pi\)
0.690703 0.723139i \(-0.257301\pi\)
\(54\) 0 0
\(55\) −626.318 −0.207047
\(56\) 419.066i 0.133631i
\(57\) 0 0
\(58\) −1553.44 −0.461785
\(59\) − 3523.36i − 1.01217i −0.862484 0.506085i \(-0.831092\pi\)
0.862484 0.506085i \(-0.168908\pi\)
\(60\) 0 0
\(61\) 5207.28 1.39943 0.699715 0.714423i \(-0.253311\pi\)
0.699715 + 0.714423i \(0.253311\pi\)
\(62\) − 2774.22i − 0.721700i
\(63\) 0 0
\(64\) −512.000 −0.125000
\(65\) − 1952.29i − 0.462080i
\(66\) 0 0
\(67\) 2249.24 0.501056 0.250528 0.968109i \(-0.419396\pi\)
0.250528 + 0.968109i \(0.419396\pi\)
\(68\) − 2725.08i − 0.589335i
\(69\) 0 0
\(70\) −481.454 −0.0982559
\(71\) − 7437.67i − 1.47544i −0.675109 0.737718i \(-0.735903\pi\)
0.675109 0.737718i \(-0.264097\pi\)
\(72\) 0 0
\(73\) −1051.46 −0.197308 −0.0986542 0.995122i \(-0.531454\pi\)
−0.0986542 + 0.995122i \(0.531454\pi\)
\(74\) − 2102.17i − 0.383887i
\(75\) 0 0
\(76\) 1412.15 0.244486
\(77\) − 1262.06i − 0.212862i
\(78\) 0 0
\(79\) 3149.66 0.504673 0.252336 0.967640i \(-0.418801\pi\)
0.252336 + 0.967640i \(0.418801\pi\)
\(80\) − 588.224i − 0.0919100i
\(81\) 0 0
\(82\) −5556.71 −0.826400
\(83\) 467.561i 0.0678707i 0.999424 + 0.0339354i \(0.0108040\pi\)
−0.999424 + 0.0339354i \(0.989196\pi\)
\(84\) 0 0
\(85\) 3130.78 0.433326
\(86\) 3829.62i 0.517796i
\(87\) 0 0
\(88\) 1541.94 0.199114
\(89\) 2865.60i 0.361773i 0.983504 + 0.180886i \(0.0578966\pi\)
−0.983504 + 0.180886i \(0.942103\pi\)
\(90\) 0 0
\(91\) 3933.94 0.475057
\(92\) − 1933.87i − 0.228481i
\(93\) 0 0
\(94\) 3738.42 0.423089
\(95\) 1622.38i 0.179766i
\(96\) 0 0
\(97\) −15884.4 −1.68821 −0.844107 0.536175i \(-0.819869\pi\)
−0.844107 + 0.536175i \(0.819869\pi\)
\(98\) − 970.151i − 0.101015i
\(99\) 0 0
\(100\) −4324.20 −0.432420
\(101\) − 8321.92i − 0.815795i −0.913028 0.407897i \(-0.866262\pi\)
0.913028 0.407897i \(-0.133738\pi\)
\(102\) 0 0
\(103\) −2251.19 −0.212196 −0.106098 0.994356i \(-0.533836\pi\)
−0.106098 + 0.994356i \(0.533836\pi\)
\(104\) 4806.36i 0.444375i
\(105\) 0 0
\(106\) −11490.8 −1.02267
\(107\) − 21923.5i − 1.91488i −0.288629 0.957441i \(-0.593200\pi\)
0.288629 0.957441i \(-0.406800\pi\)
\(108\) 0 0
\(109\) −3095.26 −0.260522 −0.130261 0.991480i \(-0.541582\pi\)
−0.130261 + 0.991480i \(0.541582\pi\)
\(110\) 1771.50i 0.146405i
\(111\) 0 0
\(112\) 1185.30 0.0944911
\(113\) 4295.01i 0.336362i 0.985756 + 0.168181i \(0.0537893\pi\)
−0.985756 + 0.168181i \(0.946211\pi\)
\(114\) 0 0
\(115\) 2221.77 0.167998
\(116\) 4393.80i 0.326531i
\(117\) 0 0
\(118\) −9965.57 −0.715712
\(119\) 6308.66i 0.445495i
\(120\) 0 0
\(121\) 9997.29 0.682828
\(122\) − 14728.4i − 0.989546i
\(123\) 0 0
\(124\) −7846.67 −0.510319
\(125\) − 10712.3i − 0.685590i
\(126\) 0 0
\(127\) 3845.70 0.238434 0.119217 0.992868i \(-0.461962\pi\)
0.119217 + 0.992868i \(0.461962\pi\)
\(128\) 1448.15i 0.0883883i
\(129\) 0 0
\(130\) −5521.90 −0.326740
\(131\) − 12992.3i − 0.757083i −0.925584 0.378542i \(-0.876426\pi\)
0.925584 0.378542i \(-0.123574\pi\)
\(132\) 0 0
\(133\) −3269.17 −0.184814
\(134\) − 6361.81i − 0.354300i
\(135\) 0 0
\(136\) −7707.70 −0.416723
\(137\) 7048.28i 0.375528i 0.982214 + 0.187764i \(0.0601240\pi\)
−0.982214 + 0.187764i \(0.939876\pi\)
\(138\) 0 0
\(139\) 13159.3 0.681089 0.340544 0.940228i \(-0.389389\pi\)
0.340544 + 0.940228i \(0.389389\pi\)
\(140\) 1361.76i 0.0694774i
\(141\) 0 0
\(142\) −21036.9 −1.04329
\(143\) − 14474.8i − 0.707851i
\(144\) 0 0
\(145\) −5047.93 −0.240092
\(146\) 2973.97i 0.139518i
\(147\) 0 0
\(148\) −5945.83 −0.271449
\(149\) 26816.8i 1.20791i 0.797018 + 0.603955i \(0.206409\pi\)
−0.797018 + 0.603955i \(0.793591\pi\)
\(150\) 0 0
\(151\) 24578.6 1.07796 0.538980 0.842319i \(-0.318810\pi\)
0.538980 + 0.842319i \(0.318810\pi\)
\(152\) − 3994.17i − 0.172878i
\(153\) 0 0
\(154\) −3569.64 −0.150516
\(155\) − 9014.84i − 0.375227i
\(156\) 0 0
\(157\) 31474.8 1.27692 0.638460 0.769655i \(-0.279572\pi\)
0.638460 + 0.769655i \(0.279572\pi\)
\(158\) − 8908.59i − 0.356858i
\(159\) 0 0
\(160\) −1663.75 −0.0649902
\(161\) 4476.96i 0.172716i
\(162\) 0 0
\(163\) −32656.3 −1.22911 −0.614556 0.788873i \(-0.710665\pi\)
−0.614556 + 0.788873i \(0.710665\pi\)
\(164\) 15716.8i 0.584353i
\(165\) 0 0
\(166\) 1322.46 0.0479918
\(167\) 30734.2i 1.10202i 0.834499 + 0.551010i \(0.185757\pi\)
−0.834499 + 0.551010i \(0.814243\pi\)
\(168\) 0 0
\(169\) 16558.3 0.579752
\(170\) − 8855.18i − 0.306408i
\(171\) 0 0
\(172\) 10831.8 0.366137
\(173\) 32068.3i 1.07148i 0.844383 + 0.535740i \(0.179967\pi\)
−0.844383 + 0.535740i \(0.820033\pi\)
\(174\) 0 0
\(175\) 10010.7 0.326879
\(176\) − 4361.27i − 0.140795i
\(177\) 0 0
\(178\) 8105.14 0.255812
\(179\) 18927.5i 0.590729i 0.955385 + 0.295364i \(0.0954410\pi\)
−0.955385 + 0.295364i \(0.904559\pi\)
\(180\) 0 0
\(181\) 16502.4 0.503720 0.251860 0.967764i \(-0.418958\pi\)
0.251860 + 0.967764i \(0.418958\pi\)
\(182\) − 11126.9i − 0.335916i
\(183\) 0 0
\(184\) −5469.80 −0.161561
\(185\) − 6831.01i − 0.199591i
\(186\) 0 0
\(187\) 23212.5 0.663803
\(188\) − 10573.8i − 0.299169i
\(189\) 0 0
\(190\) 4588.79 0.127113
\(191\) 33728.4i 0.924548i 0.886737 + 0.462274i \(0.152966\pi\)
−0.886737 + 0.462274i \(0.847034\pi\)
\(192\) 0 0
\(193\) 29108.7 0.781462 0.390731 0.920505i \(-0.372222\pi\)
0.390731 + 0.920505i \(0.372222\pi\)
\(194\) 44927.9i 1.19375i
\(195\) 0 0
\(196\) −2744.00 −0.0714286
\(197\) − 61088.1i − 1.57407i −0.616908 0.787035i \(-0.711615\pi\)
0.616908 0.787035i \(-0.288385\pi\)
\(198\) 0 0
\(199\) 22664.7 0.572328 0.286164 0.958181i \(-0.407620\pi\)
0.286164 + 0.958181i \(0.407620\pi\)
\(200\) 12230.7i 0.305767i
\(201\) 0 0
\(202\) −23538.0 −0.576854
\(203\) − 10171.8i − 0.246834i
\(204\) 0 0
\(205\) −18056.6 −0.429663
\(206\) 6367.33i 0.150045i
\(207\) 0 0
\(208\) 13594.4 0.314220
\(209\) 12028.8i 0.275379i
\(210\) 0 0
\(211\) −47754.5 −1.07263 −0.536314 0.844018i \(-0.680184\pi\)
−0.536314 + 0.844018i \(0.680184\pi\)
\(212\) 32500.8i 0.723139i
\(213\) 0 0
\(214\) −62009.0 −1.35403
\(215\) 12444.4i 0.269213i
\(216\) 0 0
\(217\) 18165.3 0.385765
\(218\) 8754.73i 0.184217i
\(219\) 0 0
\(220\) 5010.55 0.103524
\(221\) 72355.4i 1.48145i
\(222\) 0 0
\(223\) 40317.3 0.810741 0.405370 0.914153i \(-0.367143\pi\)
0.405370 + 0.914153i \(0.367143\pi\)
\(224\) − 3352.53i − 0.0668153i
\(225\) 0 0
\(226\) 12148.1 0.237844
\(227\) 69798.9i 1.35455i 0.735728 + 0.677277i \(0.236840\pi\)
−0.735728 + 0.677277i \(0.763160\pi\)
\(228\) 0 0
\(229\) 58230.3 1.11040 0.555199 0.831718i \(-0.312642\pi\)
0.555199 + 0.831718i \(0.312642\pi\)
\(230\) − 6284.11i − 0.118792i
\(231\) 0 0
\(232\) 12427.5 0.230892
\(233\) 36975.2i 0.681081i 0.940230 + 0.340540i \(0.110610\pi\)
−0.940230 + 0.340540i \(0.889390\pi\)
\(234\) 0 0
\(235\) 12148.0 0.219973
\(236\) 28186.9i 0.506085i
\(237\) 0 0
\(238\) 17843.6 0.315013
\(239\) 44554.3i 0.779999i 0.920815 + 0.390000i \(0.127525\pi\)
−0.920815 + 0.390000i \(0.872475\pi\)
\(240\) 0 0
\(241\) 79999.2 1.37737 0.688686 0.725060i \(-0.258188\pi\)
0.688686 + 0.725060i \(0.258188\pi\)
\(242\) − 28276.6i − 0.482833i
\(243\) 0 0
\(244\) −41658.2 −0.699715
\(245\) − 3152.51i − 0.0525200i
\(246\) 0 0
\(247\) −37494.9 −0.614580
\(248\) 22193.7i 0.360850i
\(249\) 0 0
\(250\) −30299.1 −0.484785
\(251\) − 47272.6i − 0.750346i −0.926955 0.375173i \(-0.877583\pi\)
0.926955 0.375173i \(-0.122417\pi\)
\(252\) 0 0
\(253\) 16472.9 0.257352
\(254\) − 10877.3i − 0.168598i
\(255\) 0 0
\(256\) 4096.00 0.0625000
\(257\) 53489.4i 0.809845i 0.914351 + 0.404922i \(0.132701\pi\)
−0.914351 + 0.404922i \(0.867299\pi\)
\(258\) 0 0
\(259\) 13764.8 0.205196
\(260\) 15618.3i 0.231040i
\(261\) 0 0
\(262\) −36747.8 −0.535339
\(263\) 59690.1i 0.862961i 0.902122 + 0.431480i \(0.142009\pi\)
−0.902122 + 0.431480i \(0.857991\pi\)
\(264\) 0 0
\(265\) −37339.3 −0.531709
\(266\) 9246.62i 0.130683i
\(267\) 0 0
\(268\) −17993.9 −0.250528
\(269\) − 76612.4i − 1.05875i −0.848387 0.529376i \(-0.822426\pi\)
0.848387 0.529376i \(-0.177574\pi\)
\(270\) 0 0
\(271\) 13224.9 0.180076 0.0900379 0.995938i \(-0.471301\pi\)
0.0900379 + 0.995938i \(0.471301\pi\)
\(272\) 21800.7i 0.294667i
\(273\) 0 0
\(274\) 19935.6 0.265538
\(275\) − 36834.0i − 0.487061i
\(276\) 0 0
\(277\) −2728.19 −0.0355562 −0.0177781 0.999842i \(-0.505659\pi\)
−0.0177781 + 0.999842i \(0.505659\pi\)
\(278\) − 37220.2i − 0.481602i
\(279\) 0 0
\(280\) 3851.63 0.0491279
\(281\) 92335.3i 1.16938i 0.811258 + 0.584689i \(0.198783\pi\)
−0.811258 + 0.584689i \(0.801217\pi\)
\(282\) 0 0
\(283\) −117175. −1.46306 −0.731528 0.681812i \(-0.761192\pi\)
−0.731528 + 0.681812i \(0.761192\pi\)
\(284\) 59501.4i 0.737718i
\(285\) 0 0
\(286\) −40941.0 −0.500526
\(287\) − 36384.8i − 0.441729i
\(288\) 0 0
\(289\) −32511.6 −0.389262
\(290\) 14277.7i 0.169770i
\(291\) 0 0
\(292\) 8411.65 0.0986542
\(293\) − 67218.3i − 0.782982i −0.920182 0.391491i \(-0.871959\pi\)
0.920182 0.391491i \(-0.128041\pi\)
\(294\) 0 0
\(295\) −32383.2 −0.372114
\(296\) 16817.3i 0.191944i
\(297\) 0 0
\(298\) 75849.4 0.854122
\(299\) 51347.3i 0.574348i
\(300\) 0 0
\(301\) −25075.9 −0.276773
\(302\) − 69518.6i − 0.762232i
\(303\) 0 0
\(304\) −11297.2 −0.122243
\(305\) − 47860.0i − 0.514486i
\(306\) 0 0
\(307\) −108355. −1.14966 −0.574832 0.818271i \(-0.694933\pi\)
−0.574832 + 0.818271i \(0.694933\pi\)
\(308\) 10096.5i 0.106431i
\(309\) 0 0
\(310\) −25497.8 −0.265326
\(311\) − 95709.1i − 0.989538i −0.869024 0.494769i \(-0.835253\pi\)
0.869024 0.494769i \(-0.164747\pi\)
\(312\) 0 0
\(313\) 158112. 1.61390 0.806951 0.590619i \(-0.201116\pi\)
0.806951 + 0.590619i \(0.201116\pi\)
\(314\) − 89024.2i − 0.902919i
\(315\) 0 0
\(316\) −25197.3 −0.252336
\(317\) 91133.8i 0.906903i 0.891281 + 0.453451i \(0.149807\pi\)
−0.891281 + 0.453451i \(0.850193\pi\)
\(318\) 0 0
\(319\) −37426.8 −0.367791
\(320\) 4705.79i 0.0459550i
\(321\) 0 0
\(322\) 12662.8 0.122128
\(323\) − 60128.6i − 0.576336i
\(324\) 0 0
\(325\) 114815. 1.08700
\(326\) 92365.9i 0.869113i
\(327\) 0 0
\(328\) 44453.7 0.413200
\(329\) 24478.8i 0.226151i
\(330\) 0 0
\(331\) −129014. −1.17755 −0.588777 0.808296i \(-0.700390\pi\)
−0.588777 + 0.808296i \(0.700390\pi\)
\(332\) − 3740.49i − 0.0339354i
\(333\) 0 0
\(334\) 86929.6 0.779246
\(335\) − 20672.8i − 0.184208i
\(336\) 0 0
\(337\) −166293. −1.46425 −0.732124 0.681172i \(-0.761471\pi\)
−0.732124 + 0.681172i \(0.761471\pi\)
\(338\) − 46833.9i − 0.409947i
\(339\) 0 0
\(340\) −25046.2 −0.216663
\(341\) − 66838.7i − 0.574803i
\(342\) 0 0
\(343\) 6352.45 0.0539949
\(344\) − 30636.9i − 0.258898i
\(345\) 0 0
\(346\) 90702.9 0.757650
\(347\) 229085.i 1.90256i 0.308335 + 0.951278i \(0.400228\pi\)
−0.308335 + 0.951278i \(0.599772\pi\)
\(348\) 0 0
\(349\) 97978.4 0.804413 0.402207 0.915549i \(-0.368243\pi\)
0.402207 + 0.915549i \(0.368243\pi\)
\(350\) − 28314.5i − 0.231138i
\(351\) 0 0
\(352\) −12335.5 −0.0995571
\(353\) − 212359.i − 1.70420i −0.523380 0.852100i \(-0.675329\pi\)
0.523380 0.852100i \(-0.324671\pi\)
\(354\) 0 0
\(355\) −68359.6 −0.542429
\(356\) − 22924.8i − 0.180886i
\(357\) 0 0
\(358\) 53535.2 0.417708
\(359\) − 72001.6i − 0.558668i −0.960194 0.279334i \(-0.909886\pi\)
0.960194 0.279334i \(-0.0901136\pi\)
\(360\) 0 0
\(361\) −99162.1 −0.760907
\(362\) − 46675.8i − 0.356184i
\(363\) 0 0
\(364\) −31471.6 −0.237528
\(365\) 9663.93i 0.0725384i
\(366\) 0 0
\(367\) −75888.0 −0.563432 −0.281716 0.959498i \(-0.590904\pi\)
−0.281716 + 0.959498i \(0.590904\pi\)
\(368\) 15470.9i 0.114241i
\(369\) 0 0
\(370\) −19321.0 −0.141132
\(371\) − 75240.3i − 0.546642i
\(372\) 0 0
\(373\) −141578. −1.01760 −0.508800 0.860885i \(-0.669911\pi\)
−0.508800 + 0.860885i \(0.669911\pi\)
\(374\) − 65654.9i − 0.469380i
\(375\) 0 0
\(376\) −29907.3 −0.211545
\(377\) − 116663.i − 0.820822i
\(378\) 0 0
\(379\) 81013.3 0.563999 0.281999 0.959415i \(-0.409002\pi\)
0.281999 + 0.959415i \(0.409002\pi\)
\(380\) − 12979.1i − 0.0898828i
\(381\) 0 0
\(382\) 95398.4 0.653754
\(383\) − 81113.4i − 0.552961i −0.961019 0.276481i \(-0.910832\pi\)
0.961019 0.276481i \(-0.0891682\pi\)
\(384\) 0 0
\(385\) −11599.6 −0.0782565
\(386\) − 82331.8i − 0.552577i
\(387\) 0 0
\(388\) 127075. 0.844107
\(389\) 77440.9i 0.511766i 0.966708 + 0.255883i \(0.0823662\pi\)
−0.966708 + 0.255883i \(0.917634\pi\)
\(390\) 0 0
\(391\) −82342.9 −0.538608
\(392\) 7761.20i 0.0505076i
\(393\) 0 0
\(394\) −172783. −1.11304
\(395\) − 28948.5i − 0.185538i
\(396\) 0 0
\(397\) 18460.7 0.117130 0.0585650 0.998284i \(-0.481348\pi\)
0.0585650 + 0.998284i \(0.481348\pi\)
\(398\) − 64105.6i − 0.404697i
\(399\) 0 0
\(400\) 34593.6 0.216210
\(401\) − 201249.i − 1.25154i −0.780008 0.625769i \(-0.784785\pi\)
0.780008 0.625769i \(-0.215215\pi\)
\(402\) 0 0
\(403\) 208342. 1.28282
\(404\) 66575.4i 0.407897i
\(405\) 0 0
\(406\) −28770.2 −0.174538
\(407\) − 50647.1i − 0.305750i
\(408\) 0 0
\(409\) 5492.43 0.0328335 0.0164168 0.999865i \(-0.494774\pi\)
0.0164168 + 0.999865i \(0.494774\pi\)
\(410\) 51071.7i 0.303818i
\(411\) 0 0
\(412\) 18009.5 0.106098
\(413\) − 65253.6i − 0.382564i
\(414\) 0 0
\(415\) 4297.36 0.0249520
\(416\) − 38450.9i − 0.222187i
\(417\) 0 0
\(418\) 34022.7 0.194722
\(419\) − 84105.3i − 0.479066i −0.970888 0.239533i \(-0.923006\pi\)
0.970888 0.239533i \(-0.0769943\pi\)
\(420\) 0 0
\(421\) −49470.9 −0.279117 −0.139558 0.990214i \(-0.544568\pi\)
−0.139558 + 0.990214i \(0.544568\pi\)
\(422\) 135070.i 0.758463i
\(423\) 0 0
\(424\) 91926.0 0.511336
\(425\) 184122.i 1.01936i
\(426\) 0 0
\(427\) 96440.1 0.528934
\(428\) 175388.i 0.957441i
\(429\) 0 0
\(430\) 35198.0 0.190362
\(431\) 193441.i 1.04135i 0.853756 + 0.520673i \(0.174319\pi\)
−0.853756 + 0.520673i \(0.825681\pi\)
\(432\) 0 0
\(433\) −153071. −0.816425 −0.408213 0.912887i \(-0.633848\pi\)
−0.408213 + 0.912887i \(0.633848\pi\)
\(434\) − 51379.2i − 0.272777i
\(435\) 0 0
\(436\) 24762.1 0.130261
\(437\) − 42670.5i − 0.223442i
\(438\) 0 0
\(439\) −138512. −0.718717 −0.359359 0.933200i \(-0.617005\pi\)
−0.359359 + 0.933200i \(0.617005\pi\)
\(440\) − 14172.0i − 0.0732023i
\(441\) 0 0
\(442\) 204652. 1.04754
\(443\) 32183.4i 0.163993i 0.996633 + 0.0819963i \(0.0261296\pi\)
−0.996633 + 0.0819963i \(0.973870\pi\)
\(444\) 0 0
\(445\) 26337.7 0.133002
\(446\) − 114035.i − 0.573280i
\(447\) 0 0
\(448\) −9482.37 −0.0472456
\(449\) − 100743.i − 0.499713i −0.968283 0.249857i \(-0.919617\pi\)
0.968283 0.249857i \(-0.0803835\pi\)
\(450\) 0 0
\(451\) −133877. −0.658192
\(452\) − 34360.1i − 0.168181i
\(453\) 0 0
\(454\) 197421. 0.957815
\(455\) − 36156.9i − 0.174650i
\(456\) 0 0
\(457\) 249340. 1.19387 0.596937 0.802288i \(-0.296384\pi\)
0.596937 + 0.802288i \(0.296384\pi\)
\(458\) − 164700.i − 0.785169i
\(459\) 0 0
\(460\) −17774.2 −0.0839989
\(461\) − 340783.i − 1.60353i −0.597640 0.801764i \(-0.703895\pi\)
0.597640 0.801764i \(-0.296105\pi\)
\(462\) 0 0
\(463\) −86247.9 −0.402334 −0.201167 0.979557i \(-0.564473\pi\)
−0.201167 + 0.979557i \(0.564473\pi\)
\(464\) − 35150.4i − 0.163266i
\(465\) 0 0
\(466\) 104582. 0.481597
\(467\) − 21803.1i − 0.0999734i −0.998750 0.0499867i \(-0.984082\pi\)
0.998750 0.0499867i \(-0.0159179\pi\)
\(468\) 0 0
\(469\) 41656.5 0.189381
\(470\) − 34359.8i − 0.155544i
\(471\) 0 0
\(472\) 79724.5 0.357856
\(473\) 92266.2i 0.412402i
\(474\) 0 0
\(475\) −95412.9 −0.422883
\(476\) − 50469.3i − 0.222748i
\(477\) 0 0
\(478\) 126019. 0.551543
\(479\) − 71881.6i − 0.313290i −0.987655 0.156645i \(-0.949932\pi\)
0.987655 0.156645i \(-0.0500679\pi\)
\(480\) 0 0
\(481\) 157871. 0.682359
\(482\) − 226272.i − 0.973949i
\(483\) 0 0
\(484\) −79978.3 −0.341414
\(485\) 145994.i 0.620655i
\(486\) 0 0
\(487\) −319320. −1.34638 −0.673191 0.739469i \(-0.735077\pi\)
−0.673191 + 0.739469i \(0.735077\pi\)
\(488\) 117827.i 0.494773i
\(489\) 0 0
\(490\) −8916.65 −0.0371372
\(491\) − 250141.i − 1.03758i −0.854902 0.518790i \(-0.826383\pi\)
0.854902 0.518790i \(-0.173617\pi\)
\(492\) 0 0
\(493\) 187086. 0.769744
\(494\) 106052.i 0.434574i
\(495\) 0 0
\(496\) 62773.4 0.255160
\(497\) − 137748.i − 0.557662i
\(498\) 0 0
\(499\) −154356. −0.619899 −0.309950 0.950753i \(-0.600312\pi\)
−0.309950 + 0.950753i \(0.600312\pi\)
\(500\) 85698.7i 0.342795i
\(501\) 0 0
\(502\) −133707. −0.530575
\(503\) − 344575.i − 1.36191i −0.732325 0.680955i \(-0.761565\pi\)
0.732325 0.680955i \(-0.238435\pi\)
\(504\) 0 0
\(505\) −76486.8 −0.299919
\(506\) − 46592.3i − 0.181976i
\(507\) 0 0
\(508\) −30765.6 −0.119217
\(509\) 127372.i 0.491632i 0.969317 + 0.245816i \(0.0790559\pi\)
−0.969317 + 0.245816i \(0.920944\pi\)
\(510\) 0 0
\(511\) −19473.3 −0.0745756
\(512\) − 11585.2i − 0.0441942i
\(513\) 0 0
\(514\) 151291. 0.572647
\(515\) 20690.7i 0.0780118i
\(516\) 0 0
\(517\) 90069.0 0.336972
\(518\) − 38932.7i − 0.145096i
\(519\) 0 0
\(520\) 44175.2 0.163370
\(521\) 310653.i 1.14446i 0.820094 + 0.572229i \(0.193921\pi\)
−0.820094 + 0.572229i \(0.806079\pi\)
\(522\) 0 0
\(523\) 200802. 0.734116 0.367058 0.930198i \(-0.380365\pi\)
0.367058 + 0.930198i \(0.380365\pi\)
\(524\) 103938.i 0.378542i
\(525\) 0 0
\(526\) 168829. 0.610205
\(527\) 334107.i 1.20300i
\(528\) 0 0
\(529\) 221406. 0.791185
\(530\) 105611.i 0.375975i
\(531\) 0 0
\(532\) 26153.4 0.0924070
\(533\) − 417306.i − 1.46893i
\(534\) 0 0
\(535\) −201499. −0.703987
\(536\) 50894.5i 0.177150i
\(537\) 0 0
\(538\) −216693. −0.748651
\(539\) − 23373.7i − 0.0804543i
\(540\) 0 0
\(541\) 286252. 0.978033 0.489016 0.872275i \(-0.337356\pi\)
0.489016 + 0.872275i \(0.337356\pi\)
\(542\) − 37405.8i − 0.127333i
\(543\) 0 0
\(544\) 61661.6 0.208361
\(545\) 28448.6i 0.0957783i
\(546\) 0 0
\(547\) 78649.5 0.262858 0.131429 0.991326i \(-0.458043\pi\)
0.131429 + 0.991326i \(0.458043\pi\)
\(548\) − 56386.3i − 0.187764i
\(549\) 0 0
\(550\) −104182. −0.344404
\(551\) 96948.6i 0.319329i
\(552\) 0 0
\(553\) 58332.6 0.190748
\(554\) 7716.48i 0.0251420i
\(555\) 0 0
\(556\) −105275. −0.340544
\(557\) 580730.i 1.87182i 0.352241 + 0.935909i \(0.385420\pi\)
−0.352241 + 0.935909i \(0.614580\pi\)
\(558\) 0 0
\(559\) −287602. −0.920381
\(560\) − 10894.1i − 0.0347387i
\(561\) 0 0
\(562\) 261164. 0.826875
\(563\) − 293483.i − 0.925906i −0.886383 0.462953i \(-0.846790\pi\)
0.886383 0.462953i \(-0.153210\pi\)
\(564\) 0 0
\(565\) 39475.4 0.123660
\(566\) 331420.i 1.03454i
\(567\) 0 0
\(568\) 168295. 0.521645
\(569\) − 149286.i − 0.461100i −0.973060 0.230550i \(-0.925947\pi\)
0.973060 0.230550i \(-0.0740526\pi\)
\(570\) 0 0
\(571\) −522954. −1.60395 −0.801976 0.597356i \(-0.796218\pi\)
−0.801976 + 0.597356i \(0.796218\pi\)
\(572\) 115799.i 0.353925i
\(573\) 0 0
\(574\) −102912. −0.312350
\(575\) 130663.i 0.395200i
\(576\) 0 0
\(577\) −351985. −1.05724 −0.528619 0.848859i \(-0.677290\pi\)
−0.528619 + 0.848859i \(0.677290\pi\)
\(578\) 91956.6i 0.275250i
\(579\) 0 0
\(580\) 40383.4 0.120046
\(581\) 8659.36i 0.0256527i
\(582\) 0 0
\(583\) −276845. −0.814515
\(584\) − 23791.7i − 0.0697591i
\(585\) 0 0
\(586\) −190122. −0.553652
\(587\) − 256546.i − 0.744541i −0.928124 0.372270i \(-0.878579\pi\)
0.928124 0.372270i \(-0.121421\pi\)
\(588\) 0 0
\(589\) −173136. −0.499064
\(590\) 91593.5i 0.263124i
\(591\) 0 0
\(592\) 47566.6 0.135725
\(593\) 469980.i 1.33650i 0.743935 + 0.668252i \(0.232957\pi\)
−0.743935 + 0.668252i \(0.767043\pi\)
\(594\) 0 0
\(595\) 57982.9 0.163782
\(596\) − 214535.i − 0.603955i
\(597\) 0 0
\(598\) 145232. 0.406126
\(599\) 667523.i 1.86043i 0.367021 + 0.930213i \(0.380378\pi\)
−0.367021 + 0.930213i \(0.619622\pi\)
\(600\) 0 0
\(601\) 578943. 1.60283 0.801414 0.598109i \(-0.204081\pi\)
0.801414 + 0.598109i \(0.204081\pi\)
\(602\) 70925.5i 0.195708i
\(603\) 0 0
\(604\) −196628. −0.538980
\(605\) − 91885.1i − 0.251035i
\(606\) 0 0
\(607\) −393712. −1.06857 −0.534283 0.845306i \(-0.679418\pi\)
−0.534283 + 0.845306i \(0.679418\pi\)
\(608\) 31953.3i 0.0864388i
\(609\) 0 0
\(610\) −135369. −0.363796
\(611\) 280753.i 0.752041i
\(612\) 0 0
\(613\) −112288. −0.298823 −0.149412 0.988775i \(-0.547738\pi\)
−0.149412 + 0.988775i \(0.547738\pi\)
\(614\) 306473.i 0.812935i
\(615\) 0 0
\(616\) 28557.1 0.0752581
\(617\) − 106565.i − 0.279927i −0.990157 0.139964i \(-0.955301\pi\)
0.990157 0.139964i \(-0.0446986\pi\)
\(618\) 0 0
\(619\) −115118. −0.300444 −0.150222 0.988652i \(-0.547999\pi\)
−0.150222 + 0.988652i \(0.547999\pi\)
\(620\) 72118.7i 0.187614i
\(621\) 0 0
\(622\) −270706. −0.699709
\(623\) 53071.7i 0.136737i
\(624\) 0 0
\(625\) 239371. 0.612791
\(626\) − 447209.i − 1.14120i
\(627\) 0 0
\(628\) −251798. −0.638460
\(629\) 253170.i 0.639898i
\(630\) 0 0
\(631\) −776896. −1.95121 −0.975606 0.219530i \(-0.929547\pi\)
−0.975606 + 0.219530i \(0.929547\pi\)
\(632\) 71268.7i 0.178429i
\(633\) 0 0
\(634\) 257765. 0.641277
\(635\) − 35345.8i − 0.0876577i
\(636\) 0 0
\(637\) 72857.7 0.179555
\(638\) 105859.i 0.260068i
\(639\) 0 0
\(640\) 13310.0 0.0324951
\(641\) 574512.i 1.39824i 0.715002 + 0.699122i \(0.246426\pi\)
−0.715002 + 0.699122i \(0.753574\pi\)
\(642\) 0 0
\(643\) 702571. 1.69929 0.849647 0.527352i \(-0.176815\pi\)
0.849647 + 0.527352i \(0.176815\pi\)
\(644\) − 35815.7i − 0.0863578i
\(645\) 0 0
\(646\) −170069. −0.407531
\(647\) 51929.4i 0.124052i 0.998075 + 0.0620262i \(0.0197562\pi\)
−0.998075 + 0.0620262i \(0.980244\pi\)
\(648\) 0 0
\(649\) −240099. −0.570033
\(650\) − 324745.i − 0.768627i
\(651\) 0 0
\(652\) 261250. 0.614556
\(653\) 608223.i 1.42638i 0.700969 + 0.713192i \(0.252751\pi\)
−0.700969 + 0.713192i \(0.747249\pi\)
\(654\) 0 0
\(655\) −119412. −0.278334
\(656\) − 125734.i − 0.292176i
\(657\) 0 0
\(658\) 69236.4 0.159913
\(659\) 556649.i 1.28177i 0.767636 + 0.640886i \(0.221433\pi\)
−0.767636 + 0.640886i \(0.778567\pi\)
\(660\) 0 0
\(661\) 211725. 0.484584 0.242292 0.970203i \(-0.422101\pi\)
0.242292 + 0.970203i \(0.422101\pi\)
\(662\) 364907.i 0.832656i
\(663\) 0 0
\(664\) −10579.7 −0.0239959
\(665\) 30047.0i 0.0679450i
\(666\) 0 0
\(667\) 132766. 0.298425
\(668\) − 245874.i − 0.551010i
\(669\) 0 0
\(670\) −58471.4 −0.130255
\(671\) − 354849.i − 0.788130i
\(672\) 0 0
\(673\) −476473. −1.05198 −0.525991 0.850490i \(-0.676306\pi\)
−0.525991 + 0.850490i \(0.676306\pi\)
\(674\) 470348.i 1.03538i
\(675\) 0 0
\(676\) −132466. −0.289876
\(677\) 582092.i 1.27003i 0.772499 + 0.635016i \(0.219006\pi\)
−0.772499 + 0.635016i \(0.780994\pi\)
\(678\) 0 0
\(679\) −294183. −0.638085
\(680\) 70841.5i 0.153204i
\(681\) 0 0
\(682\) −189048. −0.406447
\(683\) 685947.i 1.47045i 0.677825 + 0.735223i \(0.262923\pi\)
−0.677825 + 0.735223i \(0.737077\pi\)
\(684\) 0 0
\(685\) 64780.8 0.138059
\(686\) − 17967.4i − 0.0381802i
\(687\) 0 0
\(688\) −86654.3 −0.183068
\(689\) − 862948.i − 1.81780i
\(690\) 0 0
\(691\) −244679. −0.512437 −0.256219 0.966619i \(-0.582477\pi\)
−0.256219 + 0.966619i \(0.582477\pi\)
\(692\) − 256546.i − 0.535740i
\(693\) 0 0
\(694\) 647950. 1.34531
\(695\) − 120947.i − 0.250395i
\(696\) 0 0
\(697\) 669211. 1.37752
\(698\) − 277125.i − 0.568806i
\(699\) 0 0
\(700\) −80085.4 −0.163440
\(701\) − 275489.i − 0.560619i −0.959910 0.280310i \(-0.909563\pi\)
0.959910 0.280310i \(-0.0904371\pi\)
\(702\) 0 0
\(703\) −131194. −0.265462
\(704\) 34890.1i 0.0703975i
\(705\) 0 0
\(706\) −600641. −1.20505
\(707\) − 154124.i − 0.308341i
\(708\) 0 0
\(709\) 732370. 1.45693 0.728464 0.685084i \(-0.240234\pi\)
0.728464 + 0.685084i \(0.240234\pi\)
\(710\) 193350.i 0.383555i
\(711\) 0 0
\(712\) −64841.2 −0.127906
\(713\) 237100.i 0.466394i
\(714\) 0 0
\(715\) −133038. −0.260234
\(716\) − 151420.i − 0.295364i
\(717\) 0 0
\(718\) −203651. −0.395038
\(719\) 900565.i 1.74204i 0.491251 + 0.871018i \(0.336540\pi\)
−0.491251 + 0.871018i \(0.663460\pi\)
\(720\) 0 0
\(721\) −41692.6 −0.0802026
\(722\) 280473.i 0.538042i
\(723\) 0 0
\(724\) −132019. −0.251860
\(725\) − 296870.i − 0.564795i
\(726\) 0 0
\(727\) 681602. 1.28962 0.644811 0.764342i \(-0.276936\pi\)
0.644811 + 0.764342i \(0.276936\pi\)
\(728\) 89015.0i 0.167958i
\(729\) 0 0
\(730\) 27333.7 0.0512924
\(731\) − 461212.i − 0.863109i
\(732\) 0 0
\(733\) 369951. 0.688552 0.344276 0.938869i \(-0.388124\pi\)
0.344276 + 0.938869i \(0.388124\pi\)
\(734\) 214644.i 0.398406i
\(735\) 0 0
\(736\) 43758.4 0.0807804
\(737\) − 153274.i − 0.282185i
\(738\) 0 0
\(739\) 318069. 0.582416 0.291208 0.956660i \(-0.405943\pi\)
0.291208 + 0.956660i \(0.405943\pi\)
\(740\) 54648.1i 0.0997956i
\(741\) 0 0
\(742\) −212812. −0.386534
\(743\) − 531939.i − 0.963571i −0.876289 0.481786i \(-0.839988\pi\)
0.876289 0.481786i \(-0.160012\pi\)
\(744\) 0 0
\(745\) 246473. 0.444076
\(746\) 400442.i 0.719552i
\(747\) 0 0
\(748\) −185700. −0.331902
\(749\) − 406029.i − 0.723757i
\(750\) 0 0
\(751\) 411990. 0.730477 0.365239 0.930914i \(-0.380987\pi\)
0.365239 + 0.930914i \(0.380987\pi\)
\(752\) 84590.7i 0.149585i
\(753\) 0 0
\(754\) −329972. −0.580409
\(755\) − 225901.i − 0.396301i
\(756\) 0 0
\(757\) 1.11562e6 1.94681 0.973407 0.229081i \(-0.0735722\pi\)
0.973407 + 0.229081i \(0.0735722\pi\)
\(758\) − 229140.i − 0.398807i
\(759\) 0 0
\(760\) −36710.4 −0.0635567
\(761\) − 251236.i − 0.433822i −0.976191 0.216911i \(-0.930402\pi\)
0.976191 0.216911i \(-0.0695982\pi\)
\(762\) 0 0
\(763\) −57325.1 −0.0984681
\(764\) − 269827.i − 0.462274i
\(765\) 0 0
\(766\) −229423. −0.391003
\(767\) − 748408.i − 1.27218i
\(768\) 0 0
\(769\) 215136. 0.363799 0.181899 0.983317i \(-0.441776\pi\)
0.181899 + 0.983317i \(0.441776\pi\)
\(770\) 32808.6i 0.0553357i
\(771\) 0 0
\(772\) −232870. −0.390731
\(773\) 377669.i 0.632051i 0.948751 + 0.316026i \(0.102349\pi\)
−0.948751 + 0.316026i \(0.897651\pi\)
\(774\) 0 0
\(775\) 530166. 0.882690
\(776\) − 359423.i − 0.596874i
\(777\) 0 0
\(778\) 219036. 0.361873
\(779\) 346788.i 0.571464i
\(780\) 0 0
\(781\) −506838. −0.830936
\(782\) 232901.i 0.380853i
\(783\) 0 0
\(784\) 21952.0 0.0357143
\(785\) − 289285.i − 0.469447i
\(786\) 0 0
\(787\) −610887. −0.986306 −0.493153 0.869943i \(-0.664156\pi\)
−0.493153 + 0.869943i \(0.664156\pi\)
\(788\) 488705.i 0.787035i
\(789\) 0 0
\(790\) −81878.8 −0.131195
\(791\) 79544.7i 0.127133i
\(792\) 0 0
\(793\) 1.10609e6 1.75892
\(794\) − 52214.9i − 0.0828234i
\(795\) 0 0
\(796\) −181318. −0.286164
\(797\) 144095.i 0.226847i 0.993547 + 0.113423i \(0.0361817\pi\)
−0.993547 + 0.113423i \(0.963818\pi\)
\(798\) 0 0
\(799\) −450228. −0.705243
\(800\) − 97845.6i − 0.152884i
\(801\) 0 0
\(802\) −569217. −0.884971
\(803\) 71651.3i 0.111120i
\(804\) 0 0
\(805\) 41147.8 0.0634972
\(806\) − 589280.i − 0.907092i
\(807\) 0 0
\(808\) 188304. 0.288427
\(809\) 469089.i 0.716735i 0.933581 + 0.358367i \(0.116666\pi\)
−0.933581 + 0.358367i \(0.883334\pi\)
\(810\) 0 0
\(811\) −651439. −0.990449 −0.495224 0.868765i \(-0.664914\pi\)
−0.495224 + 0.868765i \(0.664914\pi\)
\(812\) 81374.3i 0.123417i
\(813\) 0 0
\(814\) −143252. −0.216198
\(815\) 300144.i 0.451870i
\(816\) 0 0
\(817\) 239002. 0.358061
\(818\) − 15534.9i − 0.0232168i
\(819\) 0 0
\(820\) 144453. 0.214831
\(821\) − 344638.i − 0.511301i −0.966769 0.255650i \(-0.917710\pi\)
0.966769 0.255650i \(-0.0822896\pi\)
\(822\) 0 0
\(823\) −77588.8 −0.114551 −0.0572755 0.998358i \(-0.518241\pi\)
−0.0572755 + 0.998358i \(0.518241\pi\)
\(824\) − 50938.6i − 0.0750227i
\(825\) 0 0
\(826\) −184565. −0.270514
\(827\) − 449172.i − 0.656752i −0.944547 0.328376i \(-0.893499\pi\)
0.944547 0.328376i \(-0.106501\pi\)
\(828\) 0 0
\(829\) −1.21343e6 −1.76565 −0.882824 0.469704i \(-0.844361\pi\)
−0.882824 + 0.469704i \(0.844361\pi\)
\(830\) − 12154.8i − 0.0176437i
\(831\) 0 0
\(832\) −108755. −0.157110
\(833\) 116838.i 0.168381i
\(834\) 0 0
\(835\) 282478. 0.405147
\(836\) − 96230.7i − 0.137690i
\(837\) 0 0
\(838\) −237886. −0.338751
\(839\) 290391.i 0.412534i 0.978496 + 0.206267i \(0.0661316\pi\)
−0.978496 + 0.206267i \(0.933868\pi\)
\(840\) 0 0
\(841\) 405633. 0.573510
\(842\) 139925.i 0.197365i
\(843\) 0 0
\(844\) 382036. 0.536314
\(845\) − 152187.i − 0.213140i
\(846\) 0 0
\(847\) 185152. 0.258085
\(848\) − 260006.i − 0.361569i
\(849\) 0 0
\(850\) 520776. 0.720798
\(851\) 179663.i 0.248084i
\(852\) 0 0
\(853\) 1.13718e6 1.56291 0.781453 0.623964i \(-0.214479\pi\)
0.781453 + 0.623964i \(0.214479\pi\)
\(854\) − 272774.i − 0.374013i
\(855\) 0 0
\(856\) 496072. 0.677013
\(857\) − 1.10331e6i − 1.50223i −0.660174 0.751113i \(-0.729517\pi\)
0.660174 0.751113i \(-0.270483\pi\)
\(858\) 0 0
\(859\) −57100.5 −0.0773844 −0.0386922 0.999251i \(-0.512319\pi\)
−0.0386922 + 0.999251i \(0.512319\pi\)
\(860\) − 99554.9i − 0.134606i
\(861\) 0 0
\(862\) 547135. 0.736342
\(863\) 787528.i 1.05741i 0.848805 + 0.528706i \(0.177323\pi\)
−0.848805 + 0.528706i \(0.822677\pi\)
\(864\) 0 0
\(865\) 294740. 0.393919
\(866\) 432949.i 0.577300i
\(867\) 0 0
\(868\) −145322. −0.192883
\(869\) − 214633.i − 0.284222i
\(870\) 0 0
\(871\) 477768. 0.629768
\(872\) − 70037.8i − 0.0921085i
\(873\) 0 0
\(874\) −120690. −0.157997
\(875\) − 198395.i − 0.259129i
\(876\) 0 0
\(877\) −804114. −1.04549 −0.522743 0.852490i \(-0.675091\pi\)
−0.522743 + 0.852490i \(0.675091\pi\)
\(878\) 391771.i 0.508210i
\(879\) 0 0
\(880\) −40084.4 −0.0517618
\(881\) − 824622.i − 1.06244i −0.847235 0.531218i \(-0.821734\pi\)
0.847235 0.531218i \(-0.178266\pi\)
\(882\) 0 0
\(883\) −78126.7 −0.100202 −0.0501012 0.998744i \(-0.515954\pi\)
−0.0501012 + 0.998744i \(0.515954\pi\)
\(884\) − 578843.i − 0.740724i
\(885\) 0 0
\(886\) 91028.4 0.115960
\(887\) − 372894.i − 0.473956i −0.971515 0.236978i \(-0.923843\pi\)
0.971515 0.236978i \(-0.0761570\pi\)
\(888\) 0 0
\(889\) 71223.3 0.0901195
\(890\) − 74494.3i − 0.0940466i
\(891\) 0 0
\(892\) −322539. −0.405370
\(893\) − 233310.i − 0.292571i
\(894\) 0 0
\(895\) 173963. 0.217175
\(896\) 26820.2i 0.0334077i
\(897\) 0 0
\(898\) −284943. −0.353351
\(899\) − 538699.i − 0.666540i
\(900\) 0 0
\(901\) 1.38386e6 1.70468
\(902\) 378661.i 0.465412i
\(903\) 0 0
\(904\) −97184.9 −0.118922
\(905\) − 151673.i − 0.185188i
\(906\) 0 0
\(907\) −915350. −1.11269 −0.556343 0.830953i \(-0.687796\pi\)
−0.556343 + 0.830953i \(0.687796\pi\)
\(908\) − 558391.i − 0.677277i
\(909\) 0 0
\(910\) −102267. −0.123496
\(911\) − 1.09746e6i − 1.32237i −0.750224 0.661184i \(-0.770054\pi\)
0.750224 0.661184i \(-0.229946\pi\)
\(912\) 0 0
\(913\) 31861.9 0.0382234
\(914\) − 705239.i − 0.844197i
\(915\) 0 0
\(916\) −465843. −0.555199
\(917\) − 240621.i − 0.286151i
\(918\) 0 0
\(919\) 916021. 1.08461 0.542306 0.840181i \(-0.317551\pi\)
0.542306 + 0.840181i \(0.317551\pi\)
\(920\) 50272.9i 0.0593962i
\(921\) 0 0
\(922\) −963881. −1.13387
\(923\) − 1.57986e6i − 1.85445i
\(924\) 0 0
\(925\) 401734. 0.469521
\(926\) 243946.i 0.284493i
\(927\) 0 0
\(928\) −99420.4 −0.115446
\(929\) 977241.i 1.13232i 0.824294 + 0.566161i \(0.191572\pi\)
−0.824294 + 0.566161i \(0.808428\pi\)
\(930\) 0 0
\(931\) −60546.0 −0.0698531
\(932\) − 295802.i − 0.340540i
\(933\) 0 0
\(934\) −61668.5 −0.0706919
\(935\) − 213346.i − 0.244040i
\(936\) 0 0
\(937\) −211902. −0.241355 −0.120678 0.992692i \(-0.538507\pi\)
−0.120678 + 0.992692i \(0.538507\pi\)
\(938\) − 117822.i − 0.133913i
\(939\) 0 0
\(940\) −97184.1 −0.109987
\(941\) 886510.i 1.00116i 0.865690 + 0.500581i \(0.166880\pi\)
−0.865690 + 0.500581i \(0.833120\pi\)
\(942\) 0 0
\(943\) 474908. 0.534055
\(944\) − 225495.i − 0.253042i
\(945\) 0 0
\(946\) 260968. 0.291612
\(947\) − 391263.i − 0.436283i −0.975917 0.218142i \(-0.930001\pi\)
0.975917 0.218142i \(-0.0699995\pi\)
\(948\) 0 0
\(949\) −223343. −0.247993
\(950\) 269869.i 0.299023i
\(951\) 0 0
\(952\) −142749. −0.157506
\(953\) 445329.i 0.490337i 0.969480 + 0.245169i \(0.0788433\pi\)
−0.969480 + 0.245169i \(0.921157\pi\)
\(954\) 0 0
\(955\) 309998. 0.339901
\(956\) − 356435.i − 0.390000i
\(957\) 0 0
\(958\) −203312. −0.221530
\(959\) 130536.i 0.141936i
\(960\) 0 0
\(961\) 38513.8 0.0417032
\(962\) − 446528.i − 0.482501i
\(963\) 0 0
\(964\) −639993. −0.688686
\(965\) − 267538.i − 0.287297i
\(966\) 0 0
\(967\) −1.31391e6 −1.40511 −0.702557 0.711627i \(-0.747959\pi\)
−0.702557 + 0.711627i \(0.747959\pi\)
\(968\) 226213.i 0.241416i
\(969\) 0 0
\(970\) 412932. 0.438869
\(971\) − 1.12629e6i − 1.19457i −0.802029 0.597285i \(-0.796246\pi\)
0.802029 0.597285i \(-0.203754\pi\)
\(972\) 0 0
\(973\) 243714. 0.257427
\(974\) 903174.i 0.952036i
\(975\) 0 0
\(976\) 333266. 0.349857
\(977\) 1.70721e6i 1.78854i 0.447528 + 0.894270i \(0.352304\pi\)
−0.447528 + 0.894270i \(0.647696\pi\)
\(978\) 0 0
\(979\) 195276. 0.203743
\(980\) 25220.1i 0.0262600i
\(981\) 0 0
\(982\) −707505. −0.733680
\(983\) 438109.i 0.453394i 0.973965 + 0.226697i \(0.0727926\pi\)
−0.973965 + 0.226697i \(0.927207\pi\)
\(984\) 0 0
\(985\) −561460. −0.578691
\(986\) − 529158.i − 0.544292i
\(987\) 0 0
\(988\) 299959. 0.307290
\(989\) − 327301.i − 0.334622i
\(990\) 0 0
\(991\) −1.40134e6 −1.42690 −0.713452 0.700704i \(-0.752870\pi\)
−0.713452 + 0.700704i \(0.752870\pi\)
\(992\) − 177550.i − 0.180425i
\(993\) 0 0
\(994\) −389609. −0.394327
\(995\) − 208312.i − 0.210410i
\(996\) 0 0
\(997\) −971355. −0.977209 −0.488605 0.872505i \(-0.662494\pi\)
−0.488605 + 0.872505i \(0.662494\pi\)
\(998\) 436583.i 0.438335i
\(999\) 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 378.5.b.a.323.2 8
3.2 odd 2 inner 378.5.b.a.323.7 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.5.b.a.323.2 8 1.1 even 1 trivial
378.5.b.a.323.7 yes 8 3.2 odd 2 inner