Properties

Label 441.4.a.n.1.1
Level $441$
Weight $4$
Character 441.1
Self dual yes
Analytic conductor $26.020$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.0198423125\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 147)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.41421 q^{2} -2.17157 q^{4} -19.8995 q^{5} +24.5563 q^{8} +O(q^{10})\) \(q-2.41421 q^{2} -2.17157 q^{4} -19.8995 q^{5} +24.5563 q^{8} +48.0416 q^{10} -23.9411 q^{11} +87.3553 q^{13} -41.9117 q^{16} +5.63961 q^{17} +64.8873 q^{19} +43.2132 q^{20} +57.7990 q^{22} +25.5980 q^{23} +270.990 q^{25} -210.894 q^{26} -60.3188 q^{29} -122.711 q^{31} -95.2670 q^{32} -13.6152 q^{34} -56.1177 q^{37} -156.652 q^{38} -488.659 q^{40} +299.713 q^{41} -501.421 q^{43} +51.9899 q^{44} -61.7990 q^{46} -305.553 q^{47} -654.227 q^{50} -189.698 q^{52} +375.117 q^{53} +476.416 q^{55} +145.622 q^{58} +627.612 q^{59} +3.75736 q^{61} +296.250 q^{62} +565.288 q^{64} -1738.33 q^{65} -813.048 q^{67} -12.2468 q^{68} -165.902 q^{71} +619.100 q^{73} +135.480 q^{74} -140.908 q^{76} -138.246 q^{79} +834.021 q^{80} -723.571 q^{82} -621.137 q^{83} -112.225 q^{85} +1210.54 q^{86} -587.907 q^{88} -285.418 q^{89} -55.5879 q^{92} +737.671 q^{94} -1291.22 q^{95} -603.114 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 10q^{4} - 20q^{5} + 18q^{8} + O(q^{10}) \) \( 2q - 2q^{2} - 10q^{4} - 20q^{5} + 18q^{8} + 48q^{10} + 20q^{11} + 104q^{13} + 18q^{16} - 116q^{17} + 192q^{19} + 44q^{20} + 76q^{22} - 28q^{23} + 146q^{25} - 204q^{26} - 296q^{29} - 104q^{31} - 18q^{32} - 64q^{34} - 248q^{37} - 104q^{38} - 488q^{40} - 20q^{41} - 720q^{43} - 292q^{44} - 84q^{46} + 96q^{47} - 706q^{50} - 320q^{52} - 268q^{53} + 472q^{55} + 48q^{58} + 616q^{59} + 16q^{61} + 304q^{62} + 118q^{64} - 1740q^{65} - 144q^{67} + 940q^{68} - 988q^{71} + 104q^{73} + 56q^{74} - 1136q^{76} - 944q^{79} + 828q^{80} - 856q^{82} - 1016q^{83} - 100q^{85} + 1120q^{86} - 876q^{88} + 388q^{89} + 364q^{92} + 904q^{94} - 1304q^{95} + 488q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.41421 −0.853553 −0.426777 0.904357i \(-0.640351\pi\)
−0.426777 + 0.904357i \(0.640351\pi\)
\(3\) 0 0
\(4\) −2.17157 −0.271447
\(5\) −19.8995 −1.77986 −0.889932 0.456092i \(-0.849249\pi\)
−0.889932 + 0.456092i \(0.849249\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 24.5563 1.08525
\(9\) 0 0
\(10\) 48.0416 1.51921
\(11\) −23.9411 −0.656229 −0.328115 0.944638i \(-0.606413\pi\)
−0.328115 + 0.944638i \(0.606413\pi\)
\(12\) 0 0
\(13\) 87.3553 1.86369 0.931847 0.362852i \(-0.118197\pi\)
0.931847 + 0.362852i \(0.118197\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −41.9117 −0.654870
\(17\) 5.63961 0.0804592 0.0402296 0.999190i \(-0.487191\pi\)
0.0402296 + 0.999190i \(0.487191\pi\)
\(18\) 0 0
\(19\) 64.8873 0.783483 0.391741 0.920075i \(-0.371873\pi\)
0.391741 + 0.920075i \(0.371873\pi\)
\(20\) 43.2132 0.483138
\(21\) 0 0
\(22\) 57.7990 0.560127
\(23\) 25.5980 0.232067 0.116034 0.993245i \(-0.462982\pi\)
0.116034 + 0.993245i \(0.462982\pi\)
\(24\) 0 0
\(25\) 270.990 2.16792
\(26\) −210.894 −1.59076
\(27\) 0 0
\(28\) 0 0
\(29\) −60.3188 −0.386238 −0.193119 0.981175i \(-0.561860\pi\)
−0.193119 + 0.981175i \(0.561860\pi\)
\(30\) 0 0
\(31\) −122.711 −0.710951 −0.355476 0.934686i \(-0.615681\pi\)
−0.355476 + 0.934686i \(0.615681\pi\)
\(32\) −95.2670 −0.526281
\(33\) 0 0
\(34\) −13.6152 −0.0686762
\(35\) 0 0
\(36\) 0 0
\(37\) −56.1177 −0.249343 −0.124672 0.992198i \(-0.539788\pi\)
−0.124672 + 0.992198i \(0.539788\pi\)
\(38\) −156.652 −0.668744
\(39\) 0 0
\(40\) −488.659 −1.93159
\(41\) 299.713 1.14164 0.570820 0.821075i \(-0.306625\pi\)
0.570820 + 0.821075i \(0.306625\pi\)
\(42\) 0 0
\(43\) −501.421 −1.77828 −0.889140 0.457635i \(-0.848697\pi\)
−0.889140 + 0.457635i \(0.848697\pi\)
\(44\) 51.9899 0.178131
\(45\) 0 0
\(46\) −61.7990 −0.198082
\(47\) −305.553 −0.948288 −0.474144 0.880447i \(-0.657242\pi\)
−0.474144 + 0.880447i \(0.657242\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −654.227 −1.85043
\(51\) 0 0
\(52\) −189.698 −0.505893
\(53\) 375.117 0.972194 0.486097 0.873905i \(-0.338420\pi\)
0.486097 + 0.873905i \(0.338420\pi\)
\(54\) 0 0
\(55\) 476.416 1.16800
\(56\) 0 0
\(57\) 0 0
\(58\) 145.622 0.329675
\(59\) 627.612 1.38488 0.692442 0.721474i \(-0.256535\pi\)
0.692442 + 0.721474i \(0.256535\pi\)
\(60\) 0 0
\(61\) 3.75736 0.00788657 0.00394328 0.999992i \(-0.498745\pi\)
0.00394328 + 0.999992i \(0.498745\pi\)
\(62\) 296.250 0.606835
\(63\) 0 0
\(64\) 565.288 1.10408
\(65\) −1738.33 −3.31712
\(66\) 0 0
\(67\) −813.048 −1.48253 −0.741266 0.671212i \(-0.765774\pi\)
−0.741266 + 0.671212i \(0.765774\pi\)
\(68\) −12.2468 −0.0218404
\(69\) 0 0
\(70\) 0 0
\(71\) −165.902 −0.277310 −0.138655 0.990341i \(-0.544278\pi\)
−0.138655 + 0.990341i \(0.544278\pi\)
\(72\) 0 0
\(73\) 619.100 0.992605 0.496302 0.868150i \(-0.334691\pi\)
0.496302 + 0.868150i \(0.334691\pi\)
\(74\) 135.480 0.212828
\(75\) 0 0
\(76\) −140.908 −0.212674
\(77\) 0 0
\(78\) 0 0
\(79\) −138.246 −0.196884 −0.0984421 0.995143i \(-0.531386\pi\)
−0.0984421 + 0.995143i \(0.531386\pi\)
\(80\) 834.021 1.16558
\(81\) 0 0
\(82\) −723.571 −0.974451
\(83\) −621.137 −0.821430 −0.410715 0.911764i \(-0.634721\pi\)
−0.410715 + 0.911764i \(0.634721\pi\)
\(84\) 0 0
\(85\) −112.225 −0.143207
\(86\) 1210.54 1.51786
\(87\) 0 0
\(88\) −587.907 −0.712171
\(89\) −285.418 −0.339936 −0.169968 0.985450i \(-0.554366\pi\)
−0.169968 + 0.985450i \(0.554366\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −55.5879 −0.0629939
\(93\) 0 0
\(94\) 737.671 0.809415
\(95\) −1291.22 −1.39449
\(96\) 0 0
\(97\) −603.114 −0.631309 −0.315654 0.948874i \(-0.602224\pi\)
−0.315654 + 0.948874i \(0.602224\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −588.474 −0.588474
\(101\) −457.209 −0.450436 −0.225218 0.974308i \(-0.572309\pi\)
−0.225218 + 0.974308i \(0.572309\pi\)
\(102\) 0 0
\(103\) 786.045 0.751954 0.375977 0.926629i \(-0.377307\pi\)
0.375977 + 0.926629i \(0.377307\pi\)
\(104\) 2145.13 2.02257
\(105\) 0 0
\(106\) −905.612 −0.829819
\(107\) 196.461 0.177501 0.0887504 0.996054i \(-0.471713\pi\)
0.0887504 + 0.996054i \(0.471713\pi\)
\(108\) 0 0
\(109\) −306.343 −0.269196 −0.134598 0.990900i \(-0.542974\pi\)
−0.134598 + 0.990900i \(0.542974\pi\)
\(110\) −1150.17 −0.996950
\(111\) 0 0
\(112\) 0 0
\(113\) −1997.63 −1.66302 −0.831508 0.555512i \(-0.812522\pi\)
−0.831508 + 0.555512i \(0.812522\pi\)
\(114\) 0 0
\(115\) −509.387 −0.413048
\(116\) 130.987 0.104843
\(117\) 0 0
\(118\) −1515.19 −1.18207
\(119\) 0 0
\(120\) 0 0
\(121\) −757.823 −0.569363
\(122\) −9.07107 −0.00673161
\(123\) 0 0
\(124\) 266.475 0.192985
\(125\) −2905.13 −2.07874
\(126\) 0 0
\(127\) −2311.40 −1.61499 −0.807494 0.589875i \(-0.799177\pi\)
−0.807494 + 0.589875i \(0.799177\pi\)
\(128\) −602.591 −0.416109
\(129\) 0 0
\(130\) 4196.69 2.83134
\(131\) 155.018 0.103389 0.0516945 0.998663i \(-0.483538\pi\)
0.0516945 + 0.998663i \(0.483538\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1962.87 1.26542
\(135\) 0 0
\(136\) 138.488 0.0873182
\(137\) −516.936 −0.322371 −0.161186 0.986924i \(-0.551532\pi\)
−0.161186 + 0.986924i \(0.551532\pi\)
\(138\) 0 0
\(139\) 958.067 0.584620 0.292310 0.956324i \(-0.405576\pi\)
0.292310 + 0.956324i \(0.405576\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 400.524 0.236699
\(143\) −2091.39 −1.22301
\(144\) 0 0
\(145\) 1200.31 0.687452
\(146\) −1494.64 −0.847241
\(147\) 0 0
\(148\) 121.864 0.0676834
\(149\) 1770.63 0.973526 0.486763 0.873534i \(-0.338178\pi\)
0.486763 + 0.873534i \(0.338178\pi\)
\(150\) 0 0
\(151\) −2540.24 −1.36902 −0.684508 0.729005i \(-0.739983\pi\)
−0.684508 + 0.729005i \(0.739983\pi\)
\(152\) 1593.40 0.850272
\(153\) 0 0
\(154\) 0 0
\(155\) 2441.88 1.26540
\(156\) 0 0
\(157\) −1083.34 −0.550702 −0.275351 0.961344i \(-0.588794\pi\)
−0.275351 + 0.961344i \(0.588794\pi\)
\(158\) 333.754 0.168051
\(159\) 0 0
\(160\) 1895.77 0.936709
\(161\) 0 0
\(162\) 0 0
\(163\) 2968.72 1.42655 0.713277 0.700882i \(-0.247210\pi\)
0.713277 + 0.700882i \(0.247210\pi\)
\(164\) −650.848 −0.309895
\(165\) 0 0
\(166\) 1499.56 0.701134
\(167\) 2091.53 0.969149 0.484574 0.874750i \(-0.338975\pi\)
0.484574 + 0.874750i \(0.338975\pi\)
\(168\) 0 0
\(169\) 5433.96 2.47335
\(170\) 270.936 0.122234
\(171\) 0 0
\(172\) 1088.87 0.482708
\(173\) 470.148 0.206617 0.103308 0.994649i \(-0.467057\pi\)
0.103308 + 0.994649i \(0.467057\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1003.41 0.429745
\(177\) 0 0
\(178\) 689.061 0.290153
\(179\) −1056.46 −0.441137 −0.220569 0.975371i \(-0.570791\pi\)
−0.220569 + 0.975371i \(0.570791\pi\)
\(180\) 0 0
\(181\) −406.470 −0.166921 −0.0834605 0.996511i \(-0.526597\pi\)
−0.0834605 + 0.996511i \(0.526597\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 628.593 0.251850
\(185\) 1116.71 0.443797
\(186\) 0 0
\(187\) −135.019 −0.0527997
\(188\) 663.531 0.257410
\(189\) 0 0
\(190\) 3117.29 1.19027
\(191\) −179.410 −0.0679666 −0.0339833 0.999422i \(-0.510819\pi\)
−0.0339833 + 0.999422i \(0.510819\pi\)
\(192\) 0 0
\(193\) −2388.37 −0.890769 −0.445385 0.895339i \(-0.646933\pi\)
−0.445385 + 0.895339i \(0.646933\pi\)
\(194\) 1456.05 0.538856
\(195\) 0 0
\(196\) 0 0
\(197\) 2665.35 0.963952 0.481976 0.876184i \(-0.339919\pi\)
0.481976 + 0.876184i \(0.339919\pi\)
\(198\) 0 0
\(199\) −1342.31 −0.478159 −0.239079 0.971000i \(-0.576846\pi\)
−0.239079 + 0.971000i \(0.576846\pi\)
\(200\) 6654.52 2.35273
\(201\) 0 0
\(202\) 1103.80 0.384471
\(203\) 0 0
\(204\) 0 0
\(205\) −5964.13 −2.03197
\(206\) −1897.68 −0.641833
\(207\) 0 0
\(208\) −3661.21 −1.22048
\(209\) −1553.48 −0.514144
\(210\) 0 0
\(211\) 628.442 0.205042 0.102521 0.994731i \(-0.467309\pi\)
0.102521 + 0.994731i \(0.467309\pi\)
\(212\) −814.594 −0.263899
\(213\) 0 0
\(214\) −474.299 −0.151506
\(215\) 9978.03 3.16510
\(216\) 0 0
\(217\) 0 0
\(218\) 739.578 0.229773
\(219\) 0 0
\(220\) −1034.57 −0.317049
\(221\) 492.650 0.149951
\(222\) 0 0
\(223\) −969.970 −0.291273 −0.145637 0.989338i \(-0.546523\pi\)
−0.145637 + 0.989338i \(0.546523\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 4822.70 1.41947
\(227\) 4748.64 1.38845 0.694225 0.719758i \(-0.255747\pi\)
0.694225 + 0.719758i \(0.255747\pi\)
\(228\) 0 0
\(229\) −4801.99 −1.38570 −0.692848 0.721083i \(-0.743644\pi\)
−0.692848 + 0.721083i \(0.743644\pi\)
\(230\) 1229.77 0.352559
\(231\) 0 0
\(232\) −1481.21 −0.419164
\(233\) 3155.29 0.887166 0.443583 0.896233i \(-0.353707\pi\)
0.443583 + 0.896233i \(0.353707\pi\)
\(234\) 0 0
\(235\) 6080.36 1.68782
\(236\) −1362.91 −0.375922
\(237\) 0 0
\(238\) 0 0
\(239\) −4241.93 −1.14806 −0.574032 0.818833i \(-0.694622\pi\)
−0.574032 + 0.818833i \(0.694622\pi\)
\(240\) 0 0
\(241\) −4342.99 −1.16081 −0.580407 0.814326i \(-0.697107\pi\)
−0.580407 + 0.814326i \(0.697107\pi\)
\(242\) 1829.55 0.485982
\(243\) 0 0
\(244\) −8.15938 −0.00214078
\(245\) 0 0
\(246\) 0 0
\(247\) 5668.25 1.46017
\(248\) −3013.33 −0.771558
\(249\) 0 0
\(250\) 7013.59 1.77431
\(251\) −3003.01 −0.755172 −0.377586 0.925974i \(-0.623246\pi\)
−0.377586 + 0.925974i \(0.623246\pi\)
\(252\) 0 0
\(253\) −612.844 −0.152289
\(254\) 5580.21 1.37848
\(255\) 0 0
\(256\) −3067.52 −0.748907
\(257\) −4468.84 −1.08466 −0.542332 0.840164i \(-0.682459\pi\)
−0.542332 + 0.840164i \(0.682459\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 3774.90 0.900422
\(261\) 0 0
\(262\) −374.246 −0.0882480
\(263\) −6160.13 −1.44430 −0.722148 0.691738i \(-0.756845\pi\)
−0.722148 + 0.691738i \(0.756845\pi\)
\(264\) 0 0
\(265\) −7464.64 −1.73037
\(266\) 0 0
\(267\) 0 0
\(268\) 1765.59 0.402428
\(269\) −4988.90 −1.13078 −0.565388 0.824825i \(-0.691274\pi\)
−0.565388 + 0.824825i \(0.691274\pi\)
\(270\) 0 0
\(271\) 4433.73 0.993837 0.496918 0.867797i \(-0.334465\pi\)
0.496918 + 0.867797i \(0.334465\pi\)
\(272\) −236.366 −0.0526903
\(273\) 0 0
\(274\) 1247.99 0.275161
\(275\) −6487.80 −1.42265
\(276\) 0 0
\(277\) 1112.37 0.241284 0.120642 0.992696i \(-0.461505\pi\)
0.120642 + 0.992696i \(0.461505\pi\)
\(278\) −2312.98 −0.499005
\(279\) 0 0
\(280\) 0 0
\(281\) −2813.22 −0.597233 −0.298616 0.954373i \(-0.596525\pi\)
−0.298616 + 0.954373i \(0.596525\pi\)
\(282\) 0 0
\(283\) 3147.54 0.661137 0.330569 0.943782i \(-0.392759\pi\)
0.330569 + 0.943782i \(0.392759\pi\)
\(284\) 360.269 0.0752748
\(285\) 0 0
\(286\) 5049.05 1.04390
\(287\) 0 0
\(288\) 0 0
\(289\) −4881.19 −0.993526
\(290\) −2897.81 −0.586777
\(291\) 0 0
\(292\) −1344.42 −0.269439
\(293\) −9143.04 −1.82301 −0.911505 0.411289i \(-0.865079\pi\)
−0.911505 + 0.411289i \(0.865079\pi\)
\(294\) 0 0
\(295\) −12489.2 −2.46491
\(296\) −1378.05 −0.270599
\(297\) 0 0
\(298\) −4274.67 −0.830956
\(299\) 2236.12 0.432502
\(300\) 0 0
\(301\) 0 0
\(302\) 6132.67 1.16853
\(303\) 0 0
\(304\) −2719.54 −0.513079
\(305\) −74.7696 −0.0140370
\(306\) 0 0
\(307\) −4648.90 −0.864257 −0.432129 0.901812i \(-0.642237\pi\)
−0.432129 + 0.901812i \(0.642237\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −5895.22 −1.08008
\(311\) 6417.18 1.17005 0.585024 0.811016i \(-0.301085\pi\)
0.585024 + 0.811016i \(0.301085\pi\)
\(312\) 0 0
\(313\) 5868.22 1.05972 0.529858 0.848086i \(-0.322245\pi\)
0.529858 + 0.848086i \(0.322245\pi\)
\(314\) 2615.42 0.470054
\(315\) 0 0
\(316\) 300.210 0.0534435
\(317\) 3974.52 0.704200 0.352100 0.935962i \(-0.385468\pi\)
0.352100 + 0.935962i \(0.385468\pi\)
\(318\) 0 0
\(319\) 1444.10 0.253461
\(320\) −11249.0 −1.96511
\(321\) 0 0
\(322\) 0 0
\(323\) 365.939 0.0630384
\(324\) 0 0
\(325\) 23672.4 4.04034
\(326\) −7167.13 −1.21764
\(327\) 0 0
\(328\) 7359.85 1.23896
\(329\) 0 0
\(330\) 0 0
\(331\) −8912.82 −1.48004 −0.740020 0.672585i \(-0.765184\pi\)
−0.740020 + 0.672585i \(0.765184\pi\)
\(332\) 1348.84 0.222974
\(333\) 0 0
\(334\) −5049.41 −0.827220
\(335\) 16179.2 2.63871
\(336\) 0 0
\(337\) 3977.06 0.642862 0.321431 0.946933i \(-0.395836\pi\)
0.321431 + 0.946933i \(0.395836\pi\)
\(338\) −13118.7 −2.11114
\(339\) 0 0
\(340\) 243.706 0.0388729
\(341\) 2937.83 0.466547
\(342\) 0 0
\(343\) 0 0
\(344\) −12313.1 −1.92987
\(345\) 0 0
\(346\) −1135.04 −0.176358
\(347\) −6826.43 −1.05609 −0.528043 0.849218i \(-0.677074\pi\)
−0.528043 + 0.849218i \(0.677074\pi\)
\(348\) 0 0
\(349\) −807.342 −0.123828 −0.0619141 0.998081i \(-0.519720\pi\)
−0.0619141 + 0.998081i \(0.519720\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2280.80 0.345361
\(353\) 7919.20 1.19404 0.597020 0.802226i \(-0.296351\pi\)
0.597020 + 0.802226i \(0.296351\pi\)
\(354\) 0 0
\(355\) 3301.38 0.493574
\(356\) 619.807 0.0922744
\(357\) 0 0
\(358\) 2550.52 0.376534
\(359\) 8819.21 1.29655 0.648273 0.761408i \(-0.275491\pi\)
0.648273 + 0.761408i \(0.275491\pi\)
\(360\) 0 0
\(361\) −2648.64 −0.386155
\(362\) 981.307 0.142476
\(363\) 0 0
\(364\) 0 0
\(365\) −12319.8 −1.76670
\(366\) 0 0
\(367\) −11161.8 −1.58758 −0.793788 0.608194i \(-0.791894\pi\)
−0.793788 + 0.608194i \(0.791894\pi\)
\(368\) −1072.85 −0.151974
\(369\) 0 0
\(370\) −2695.99 −0.378805
\(371\) 0 0
\(372\) 0 0
\(373\) 2727.86 0.378668 0.189334 0.981913i \(-0.439367\pi\)
0.189334 + 0.981913i \(0.439367\pi\)
\(374\) 325.964 0.0450673
\(375\) 0 0
\(376\) −7503.28 −1.02913
\(377\) −5269.17 −0.719830
\(378\) 0 0
\(379\) 4086.49 0.553849 0.276924 0.960892i \(-0.410685\pi\)
0.276924 + 0.960892i \(0.410685\pi\)
\(380\) 2803.99 0.378530
\(381\) 0 0
\(382\) 433.133 0.0580131
\(383\) −13032.9 −1.73878 −0.869389 0.494129i \(-0.835487\pi\)
−0.869389 + 0.494129i \(0.835487\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 5766.03 0.760319
\(387\) 0 0
\(388\) 1309.71 0.171367
\(389\) 194.991 0.0254151 0.0127075 0.999919i \(-0.495955\pi\)
0.0127075 + 0.999919i \(0.495955\pi\)
\(390\) 0 0
\(391\) 144.363 0.0186719
\(392\) 0 0
\(393\) 0 0
\(394\) −6434.73 −0.822785
\(395\) 2751.02 0.350427
\(396\) 0 0
\(397\) −14183.0 −1.79300 −0.896501 0.443042i \(-0.853899\pi\)
−0.896501 + 0.443042i \(0.853899\pi\)
\(398\) 3240.61 0.408134
\(399\) 0 0
\(400\) −11357.6 −1.41971
\(401\) −10005.0 −1.24596 −0.622978 0.782239i \(-0.714077\pi\)
−0.622978 + 0.782239i \(0.714077\pi\)
\(402\) 0 0
\(403\) −10719.4 −1.32500
\(404\) 992.863 0.122269
\(405\) 0 0
\(406\) 0 0
\(407\) 1343.52 0.163626
\(408\) 0 0
\(409\) 4634.93 0.560349 0.280174 0.959949i \(-0.409608\pi\)
0.280174 + 0.959949i \(0.409608\pi\)
\(410\) 14398.7 1.73439
\(411\) 0 0
\(412\) −1706.95 −0.204115
\(413\) 0 0
\(414\) 0 0
\(415\) 12360.3 1.46203
\(416\) −8322.08 −0.980826
\(417\) 0 0
\(418\) 3750.42 0.438849
\(419\) −4998.31 −0.582777 −0.291388 0.956605i \(-0.594117\pi\)
−0.291388 + 0.956605i \(0.594117\pi\)
\(420\) 0 0
\(421\) −704.160 −0.0815170 −0.0407585 0.999169i \(-0.512977\pi\)
−0.0407585 + 0.999169i \(0.512977\pi\)
\(422\) −1517.19 −0.175014
\(423\) 0 0
\(424\) 9211.50 1.05507
\(425\) 1528.28 0.174429
\(426\) 0 0
\(427\) 0 0
\(428\) −426.629 −0.0481820
\(429\) 0 0
\(430\) −24089.1 −2.70158
\(431\) 10332.8 1.15479 0.577393 0.816466i \(-0.304070\pi\)
0.577393 + 0.816466i \(0.304070\pi\)
\(432\) 0 0
\(433\) 11106.8 1.23270 0.616348 0.787474i \(-0.288611\pi\)
0.616348 + 0.787474i \(0.288611\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 665.246 0.0730723
\(437\) 1660.98 0.181821
\(438\) 0 0
\(439\) −7299.28 −0.793566 −0.396783 0.917912i \(-0.629874\pi\)
−0.396783 + 0.917912i \(0.629874\pi\)
\(440\) 11699.0 1.26757
\(441\) 0 0
\(442\) −1189.36 −0.127991
\(443\) −16089.7 −1.72560 −0.862802 0.505542i \(-0.831293\pi\)
−0.862802 + 0.505542i \(0.831293\pi\)
\(444\) 0 0
\(445\) 5679.68 0.605040
\(446\) 2341.71 0.248617
\(447\) 0 0
\(448\) 0 0
\(449\) −13561.7 −1.42543 −0.712715 0.701454i \(-0.752534\pi\)
−0.712715 + 0.701454i \(0.752534\pi\)
\(450\) 0 0
\(451\) −7175.46 −0.749178
\(452\) 4337.99 0.451420
\(453\) 0 0
\(454\) −11464.2 −1.18512
\(455\) 0 0
\(456\) 0 0
\(457\) 10848.6 1.11045 0.555224 0.831701i \(-0.312633\pi\)
0.555224 + 0.831701i \(0.312633\pi\)
\(458\) 11593.0 1.18277
\(459\) 0 0
\(460\) 1106.17 0.112121
\(461\) 1758.69 0.177679 0.0888397 0.996046i \(-0.471684\pi\)
0.0888397 + 0.996046i \(0.471684\pi\)
\(462\) 0 0
\(463\) −5411.95 −0.543228 −0.271614 0.962406i \(-0.587557\pi\)
−0.271614 + 0.962406i \(0.587557\pi\)
\(464\) 2528.06 0.252936
\(465\) 0 0
\(466\) −7617.54 −0.757244
\(467\) −8111.34 −0.803744 −0.401872 0.915696i \(-0.631640\pi\)
−0.401872 + 0.915696i \(0.631640\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −14679.3 −1.44065
\(471\) 0 0
\(472\) 15411.9 1.50294
\(473\) 12004.6 1.16696
\(474\) 0 0
\(475\) 17583.8 1.69853
\(476\) 0 0
\(477\) 0 0
\(478\) 10240.9 0.979935
\(479\) −2095.76 −0.199912 −0.0999559 0.994992i \(-0.531870\pi\)
−0.0999559 + 0.994992i \(0.531870\pi\)
\(480\) 0 0
\(481\) −4902.18 −0.464699
\(482\) 10484.9 0.990817
\(483\) 0 0
\(484\) 1645.67 0.154552
\(485\) 12001.7 1.12364
\(486\) 0 0
\(487\) 9610.08 0.894197 0.447099 0.894485i \(-0.352457\pi\)
0.447099 + 0.894485i \(0.352457\pi\)
\(488\) 92.2670 0.00855888
\(489\) 0 0
\(490\) 0 0
\(491\) 11717.3 1.07698 0.538488 0.842633i \(-0.318996\pi\)
0.538488 + 0.842633i \(0.318996\pi\)
\(492\) 0 0
\(493\) −340.174 −0.0310764
\(494\) −13684.4 −1.24633
\(495\) 0 0
\(496\) 5143.01 0.465581
\(497\) 0 0
\(498\) 0 0
\(499\) −9195.19 −0.824916 −0.412458 0.910977i \(-0.635330\pi\)
−0.412458 + 0.910977i \(0.635330\pi\)
\(500\) 6308.69 0.564266
\(501\) 0 0
\(502\) 7249.90 0.644580
\(503\) −16118.8 −1.42883 −0.714414 0.699724i \(-0.753307\pi\)
−0.714414 + 0.699724i \(0.753307\pi\)
\(504\) 0 0
\(505\) 9098.23 0.801715
\(506\) 1479.54 0.129987
\(507\) 0 0
\(508\) 5019.37 0.438383
\(509\) −4918.78 −0.428333 −0.214166 0.976797i \(-0.568703\pi\)
−0.214166 + 0.976797i \(0.568703\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 12226.4 1.05534
\(513\) 0 0
\(514\) 10788.7 0.925819
\(515\) −15641.9 −1.33838
\(516\) 0 0
\(517\) 7315.29 0.622294
\(518\) 0 0
\(519\) 0 0
\(520\) −42687.0 −3.59990
\(521\) −13963.4 −1.17418 −0.587089 0.809522i \(-0.699726\pi\)
−0.587089 + 0.809522i \(0.699726\pi\)
\(522\) 0 0
\(523\) −13755.3 −1.15005 −0.575024 0.818136i \(-0.695007\pi\)
−0.575024 + 0.818136i \(0.695007\pi\)
\(524\) −336.632 −0.0280646
\(525\) 0 0
\(526\) 14871.9 1.23278
\(527\) −692.040 −0.0572026
\(528\) 0 0
\(529\) −11511.7 −0.946145
\(530\) 18021.2 1.47697
\(531\) 0 0
\(532\) 0 0
\(533\) 26181.5 2.12767
\(534\) 0 0
\(535\) −3909.47 −0.315928
\(536\) −19965.5 −1.60891
\(537\) 0 0
\(538\) 12044.3 0.965178
\(539\) 0 0
\(540\) 0 0
\(541\) 14462.7 1.14935 0.574676 0.818381i \(-0.305128\pi\)
0.574676 + 0.818381i \(0.305128\pi\)
\(542\) −10704.0 −0.848293
\(543\) 0 0
\(544\) −537.269 −0.0423441
\(545\) 6096.07 0.479132
\(546\) 0 0
\(547\) 13682.5 1.06951 0.534755 0.845007i \(-0.320404\pi\)
0.534755 + 0.845007i \(0.320404\pi\)
\(548\) 1122.56 0.0875065
\(549\) 0 0
\(550\) 15662.9 1.21431
\(551\) −3913.92 −0.302611
\(552\) 0 0
\(553\) 0 0
\(554\) −2685.49 −0.205949
\(555\) 0 0
\(556\) −2080.51 −0.158693
\(557\) 7663.13 0.582939 0.291470 0.956580i \(-0.405856\pi\)
0.291470 + 0.956580i \(0.405856\pi\)
\(558\) 0 0
\(559\) −43801.8 −3.31417
\(560\) 0 0
\(561\) 0 0
\(562\) 6791.71 0.509770
\(563\) −17470.5 −1.30780 −0.653902 0.756580i \(-0.726869\pi\)
−0.653902 + 0.756580i \(0.726869\pi\)
\(564\) 0 0
\(565\) 39751.8 2.95995
\(566\) −7598.83 −0.564316
\(567\) 0 0
\(568\) −4073.96 −0.300950
\(569\) 13873.9 1.02219 0.511094 0.859525i \(-0.329240\pi\)
0.511094 + 0.859525i \(0.329240\pi\)
\(570\) 0 0
\(571\) −3777.52 −0.276855 −0.138428 0.990373i \(-0.544205\pi\)
−0.138428 + 0.990373i \(0.544205\pi\)
\(572\) 4541.60 0.331982
\(573\) 0 0
\(574\) 0 0
\(575\) 6936.79 0.503103
\(576\) 0 0
\(577\) 13880.5 1.00148 0.500738 0.865599i \(-0.333062\pi\)
0.500738 + 0.865599i \(0.333062\pi\)
\(578\) 11784.2 0.848028
\(579\) 0 0
\(580\) −2606.57 −0.186607
\(581\) 0 0
\(582\) 0 0
\(583\) −8980.72 −0.637982
\(584\) 15202.8 1.07722
\(585\) 0 0
\(586\) 22073.2 1.55604
\(587\) 2395.61 0.168445 0.0842227 0.996447i \(-0.473159\pi\)
0.0842227 + 0.996447i \(0.473159\pi\)
\(588\) 0 0
\(589\) −7962.36 −0.557018
\(590\) 30151.5 2.10393
\(591\) 0 0
\(592\) 2351.99 0.163287
\(593\) −6603.50 −0.457290 −0.228645 0.973510i \(-0.573430\pi\)
−0.228645 + 0.973510i \(0.573430\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −3845.04 −0.264260
\(597\) 0 0
\(598\) −5398.47 −0.369164
\(599\) −17252.1 −1.17680 −0.588399 0.808571i \(-0.700241\pi\)
−0.588399 + 0.808571i \(0.700241\pi\)
\(600\) 0 0
\(601\) 12833.1 0.871005 0.435503 0.900187i \(-0.356571\pi\)
0.435503 + 0.900187i \(0.356571\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 5516.31 0.371615
\(605\) 15080.3 1.01339
\(606\) 0 0
\(607\) 8620.16 0.576411 0.288206 0.957569i \(-0.406941\pi\)
0.288206 + 0.957569i \(0.406941\pi\)
\(608\) −6181.62 −0.412332
\(609\) 0 0
\(610\) 180.510 0.0119813
\(611\) −26691.7 −1.76732
\(612\) 0 0
\(613\) 5136.73 0.338451 0.169226 0.985577i \(-0.445873\pi\)
0.169226 + 0.985577i \(0.445873\pi\)
\(614\) 11223.4 0.737690
\(615\) 0 0
\(616\) 0 0
\(617\) 1759.82 0.114826 0.0574131 0.998351i \(-0.481715\pi\)
0.0574131 + 0.998351i \(0.481715\pi\)
\(618\) 0 0
\(619\) −3560.24 −0.231176 −0.115588 0.993297i \(-0.536875\pi\)
−0.115588 + 0.993297i \(0.536875\pi\)
\(620\) −5302.72 −0.343488
\(621\) 0 0
\(622\) −15492.4 −0.998698
\(623\) 0 0
\(624\) 0 0
\(625\) 23936.8 1.53195
\(626\) −14167.1 −0.904524
\(627\) 0 0
\(628\) 2352.56 0.149486
\(629\) −316.482 −0.0200620
\(630\) 0 0
\(631\) −27321.4 −1.72369 −0.861845 0.507172i \(-0.830691\pi\)
−0.861845 + 0.507172i \(0.830691\pi\)
\(632\) −3394.81 −0.213668
\(633\) 0 0
\(634\) −9595.34 −0.601072
\(635\) 45995.7 2.87446
\(636\) 0 0
\(637\) 0 0
\(638\) −3486.36 −0.216342
\(639\) 0 0
\(640\) 11991.3 0.740619
\(641\) 21927.0 1.35112 0.675558 0.737307i \(-0.263903\pi\)
0.675558 + 0.737307i \(0.263903\pi\)
\(642\) 0 0
\(643\) −5826.04 −0.357320 −0.178660 0.983911i \(-0.557176\pi\)
−0.178660 + 0.983911i \(0.557176\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −883.455 −0.0538066
\(647\) −24210.7 −1.47113 −0.735565 0.677454i \(-0.763083\pi\)
−0.735565 + 0.677454i \(0.763083\pi\)
\(648\) 0 0
\(649\) −15025.7 −0.908801
\(650\) −57150.3 −3.44864
\(651\) 0 0
\(652\) −6446.80 −0.387233
\(653\) 25623.3 1.53556 0.767778 0.640716i \(-0.221363\pi\)
0.767778 + 0.640716i \(0.221363\pi\)
\(654\) 0 0
\(655\) −3084.77 −0.184018
\(656\) −12561.5 −0.747626
\(657\) 0 0
\(658\) 0 0
\(659\) 23273.7 1.37574 0.687871 0.725833i \(-0.258545\pi\)
0.687871 + 0.725833i \(0.258545\pi\)
\(660\) 0 0
\(661\) −20036.5 −1.17902 −0.589508 0.807763i \(-0.700678\pi\)
−0.589508 + 0.807763i \(0.700678\pi\)
\(662\) 21517.5 1.26329
\(663\) 0 0
\(664\) −15252.9 −0.891454
\(665\) 0 0
\(666\) 0 0
\(667\) −1544.04 −0.0896333
\(668\) −4541.92 −0.263072
\(669\) 0 0
\(670\) −39060.1 −2.25228
\(671\) −89.9554 −0.00517540
\(672\) 0 0
\(673\) −18127.8 −1.03830 −0.519149 0.854684i \(-0.673751\pi\)
−0.519149 + 0.854684i \(0.673751\pi\)
\(674\) −9601.48 −0.548717
\(675\) 0 0
\(676\) −11800.2 −0.671383
\(677\) −13815.5 −0.784301 −0.392150 0.919901i \(-0.628269\pi\)
−0.392150 + 0.919901i \(0.628269\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −2755.85 −0.155415
\(681\) 0 0
\(682\) −7092.55 −0.398223
\(683\) 4604.23 0.257944 0.128972 0.991648i \(-0.458832\pi\)
0.128972 + 0.991648i \(0.458832\pi\)
\(684\) 0 0
\(685\) 10286.8 0.573777
\(686\) 0 0
\(687\) 0 0
\(688\) 21015.4 1.16454
\(689\) 32768.5 1.81187
\(690\) 0 0
\(691\) 17913.7 0.986205 0.493103 0.869971i \(-0.335863\pi\)
0.493103 + 0.869971i \(0.335863\pi\)
\(692\) −1020.96 −0.0560854
\(693\) 0 0
\(694\) 16480.5 0.901426
\(695\) −19065.1 −1.04054
\(696\) 0 0
\(697\) 1690.26 0.0918555
\(698\) 1949.10 0.105694
\(699\) 0 0
\(700\) 0 0
\(701\) −11303.7 −0.609035 −0.304518 0.952507i \(-0.598495\pi\)
−0.304518 + 0.952507i \(0.598495\pi\)
\(702\) 0 0
\(703\) −3641.33 −0.195356
\(704\) −13533.6 −0.724529
\(705\) 0 0
\(706\) −19118.6 −1.01918
\(707\) 0 0
\(708\) 0 0
\(709\) −16046.3 −0.849973 −0.424987 0.905200i \(-0.639721\pi\)
−0.424987 + 0.905200i \(0.639721\pi\)
\(710\) −7970.22 −0.421292
\(711\) 0 0
\(712\) −7008.83 −0.368915
\(713\) −3141.15 −0.164989
\(714\) 0 0
\(715\) 41617.5 2.17679
\(716\) 2294.18 0.119745
\(717\) 0 0
\(718\) −21291.5 −1.10667
\(719\) 25190.5 1.30660 0.653300 0.757099i \(-0.273384\pi\)
0.653300 + 0.757099i \(0.273384\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 6394.38 0.329604
\(723\) 0 0
\(724\) 882.680 0.0453102
\(725\) −16345.8 −0.837334
\(726\) 0 0
\(727\) 11277.2 0.575307 0.287653 0.957735i \(-0.407125\pi\)
0.287653 + 0.957735i \(0.407125\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 29742.6 1.50797
\(731\) −2827.82 −0.143079
\(732\) 0 0
\(733\) 23720.0 1.19525 0.597626 0.801775i \(-0.296111\pi\)
0.597626 + 0.801775i \(0.296111\pi\)
\(734\) 26946.9 1.35508
\(735\) 0 0
\(736\) −2438.64 −0.122133
\(737\) 19465.3 0.972880
\(738\) 0 0
\(739\) 8124.72 0.404428 0.202214 0.979341i \(-0.435186\pi\)
0.202214 + 0.979341i \(0.435186\pi\)
\(740\) −2425.03 −0.120467
\(741\) 0 0
\(742\) 0 0
\(743\) −20955.3 −1.03469 −0.517346 0.855777i \(-0.673080\pi\)
−0.517346 + 0.855777i \(0.673080\pi\)
\(744\) 0 0
\(745\) −35234.6 −1.73274
\(746\) −6585.63 −0.323213
\(747\) 0 0
\(748\) 293.203 0.0143323
\(749\) 0 0
\(750\) 0 0
\(751\) −38208.1 −1.85650 −0.928251 0.371954i \(-0.878688\pi\)
−0.928251 + 0.371954i \(0.878688\pi\)
\(752\) 12806.3 0.621006
\(753\) 0 0
\(754\) 12720.9 0.614413
\(755\) 50549.4 2.43666
\(756\) 0 0
\(757\) 30958.1 1.48638 0.743191 0.669079i \(-0.233311\pi\)
0.743191 + 0.669079i \(0.233311\pi\)
\(758\) −9865.65 −0.472740
\(759\) 0 0
\(760\) −31707.8 −1.51337
\(761\) 40049.4 1.90774 0.953871 0.300218i \(-0.0970594\pi\)
0.953871 + 0.300218i \(0.0970594\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 389.601 0.0184493
\(765\) 0 0
\(766\) 31464.3 1.48414
\(767\) 54825.3 2.58100
\(768\) 0 0
\(769\) −8002.01 −0.375240 −0.187620 0.982242i \(-0.560077\pi\)
−0.187620 + 0.982242i \(0.560077\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 5186.52 0.241796
\(773\) 13933.4 0.648316 0.324158 0.946003i \(-0.394919\pi\)
0.324158 + 0.946003i \(0.394919\pi\)
\(774\) 0 0
\(775\) −33253.4 −1.54128
\(776\) −14810.3 −0.685126
\(777\) 0 0
\(778\) −470.751 −0.0216931
\(779\) 19447.6 0.894456
\(780\) 0 0
\(781\) 3971.89 0.181979
\(782\) −348.522 −0.0159375
\(783\) 0 0
\(784\) 0 0
\(785\) 21558.0 0.980175
\(786\) 0 0
\(787\) 37581.7 1.70222 0.851108 0.524991i \(-0.175931\pi\)
0.851108 + 0.524991i \(0.175931\pi\)
\(788\) −5788.01 −0.261662
\(789\) 0 0
\(790\) −6641.54 −0.299108
\(791\) 0 0
\(792\) 0 0
\(793\) 328.225 0.0146981
\(794\) 34240.7 1.53042
\(795\) 0 0
\(796\) 2914.92 0.129795
\(797\) 9458.78 0.420385 0.210193 0.977660i \(-0.432591\pi\)
0.210193 + 0.977660i \(0.432591\pi\)
\(798\) 0 0
\(799\) −1723.20 −0.0762985
\(800\) −25816.4 −1.14093
\(801\) 0 0
\(802\) 24154.3 1.06349
\(803\) −14821.9 −0.651376
\(804\) 0 0
\(805\) 0 0
\(806\) 25879.0 1.13095
\(807\) 0 0
\(808\) −11227.4 −0.488834
\(809\) 1909.94 0.0830034 0.0415017 0.999138i \(-0.486786\pi\)
0.0415017 + 0.999138i \(0.486786\pi\)
\(810\) 0 0
\(811\) 43110.6 1.86661 0.933303 0.359091i \(-0.116913\pi\)
0.933303 + 0.359091i \(0.116913\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −3243.55 −0.139664
\(815\) −59076.1 −2.53907
\(816\) 0 0
\(817\) −32535.9 −1.39325
\(818\) −11189.7 −0.478287
\(819\) 0 0
\(820\) 12951.5 0.551570
\(821\) −4026.56 −0.171167 −0.0855834 0.996331i \(-0.527275\pi\)
−0.0855834 + 0.996331i \(0.527275\pi\)
\(822\) 0 0
\(823\) 39668.1 1.68012 0.840062 0.542491i \(-0.182519\pi\)
0.840062 + 0.542491i \(0.182519\pi\)
\(824\) 19302.4 0.816056
\(825\) 0 0
\(826\) 0 0
\(827\) −30137.7 −1.26722 −0.633611 0.773652i \(-0.718428\pi\)
−0.633611 + 0.773652i \(0.718428\pi\)
\(828\) 0 0
\(829\) 23278.0 0.975245 0.487622 0.873055i \(-0.337864\pi\)
0.487622 + 0.873055i \(0.337864\pi\)
\(830\) −29840.4 −1.24792
\(831\) 0 0
\(832\) 49381.0 2.05766
\(833\) 0 0
\(834\) 0 0
\(835\) −41620.5 −1.72495
\(836\) 3373.48 0.139563
\(837\) 0 0
\(838\) 12067.0 0.497431
\(839\) 9494.43 0.390684 0.195342 0.980735i \(-0.437418\pi\)
0.195342 + 0.980735i \(0.437418\pi\)
\(840\) 0 0
\(841\) −20750.6 −0.850820
\(842\) 1699.99 0.0695791
\(843\) 0 0
\(844\) −1364.71 −0.0556578
\(845\) −108133. −4.40223
\(846\) 0 0
\(847\) 0 0
\(848\) −15721.8 −0.636661
\(849\) 0 0
\(850\) −3689.59 −0.148885
\(851\) −1436.50 −0.0578644
\(852\) 0 0
\(853\) −12692.4 −0.509471 −0.254736 0.967011i \(-0.581988\pi\)
−0.254736 + 0.967011i \(0.581988\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 4824.36 0.192632
\(857\) −22206.0 −0.885114 −0.442557 0.896740i \(-0.645929\pi\)
−0.442557 + 0.896740i \(0.645929\pi\)
\(858\) 0 0
\(859\) 19820.5 0.787271 0.393636 0.919267i \(-0.371217\pi\)
0.393636 + 0.919267i \(0.371217\pi\)
\(860\) −21668.0 −0.859155
\(861\) 0 0
\(862\) −24945.5 −0.985671
\(863\) −36413.8 −1.43631 −0.718157 0.695881i \(-0.755014\pi\)
−0.718157 + 0.695881i \(0.755014\pi\)
\(864\) 0 0
\(865\) −9355.70 −0.367749
\(866\) −26814.1 −1.05217
\(867\) 0 0
\(868\) 0 0
\(869\) 3309.76 0.129201
\(870\) 0 0
\(871\) −71024.1 −2.76298
\(872\) −7522.67 −0.292144
\(873\) 0 0
\(874\) −4009.97 −0.155194
\(875\) 0 0
\(876\) 0 0
\(877\) 19442.6 0.748609 0.374305 0.927306i \(-0.377881\pi\)
0.374305 + 0.927306i \(0.377881\pi\)
\(878\) 17622.0 0.677351
\(879\) 0 0
\(880\) −19967.4 −0.764888
\(881\) 25184.2 0.963082 0.481541 0.876423i \(-0.340077\pi\)
0.481541 + 0.876423i \(0.340077\pi\)
\(882\) 0 0
\(883\) −4050.03 −0.154354 −0.0771769 0.997017i \(-0.524591\pi\)
−0.0771769 + 0.997017i \(0.524591\pi\)
\(884\) −1069.83 −0.0407038
\(885\) 0 0
\(886\) 38843.9 1.47290
\(887\) −41604.2 −1.57489 −0.787447 0.616383i \(-0.788598\pi\)
−0.787447 + 0.616383i \(0.788598\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −13712.0 −0.516434
\(891\) 0 0
\(892\) 2106.36 0.0790652
\(893\) −19826.5 −0.742967
\(894\) 0 0
\(895\) 21023.0 0.785165
\(896\) 0 0
\(897\) 0 0
\(898\) 32740.9 1.21668
\(899\) 7401.76 0.274597
\(900\) 0 0
\(901\) 2115.51 0.0782219
\(902\) 17323.1 0.639463
\(903\) 0 0
\(904\) −49054.4 −1.80478
\(905\) 8088.56 0.297097
\(906\) 0 0
\(907\) −3572.60 −0.130790 −0.0653949 0.997859i \(-0.520831\pi\)
−0.0653949 + 0.997859i \(0.520831\pi\)
\(908\) −10312.0 −0.376890
\(909\) 0 0
\(910\) 0 0
\(911\) −29457.7 −1.07133 −0.535663 0.844432i \(-0.679938\pi\)
−0.535663 + 0.844432i \(0.679938\pi\)
\(912\) 0 0
\(913\) 14870.7 0.539046
\(914\) −26190.8 −0.947827
\(915\) 0 0
\(916\) 10427.9 0.376143
\(917\) 0 0
\(918\) 0 0
\(919\) −3310.65 −0.118834 −0.0594168 0.998233i \(-0.518924\pi\)
−0.0594168 + 0.998233i \(0.518924\pi\)
\(920\) −12508.7 −0.448260
\(921\) 0 0
\(922\) −4245.85 −0.151659
\(923\) −14492.5 −0.516820
\(924\) 0 0
\(925\) −15207.3 −0.540556
\(926\) 13065.6 0.463674
\(927\) 0 0
\(928\) 5746.39 0.203270
\(929\) 31467.5 1.11132 0.555660 0.831410i \(-0.312466\pi\)
0.555660 + 0.831410i \(0.312466\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −6851.94 −0.240818
\(933\) 0 0
\(934\) 19582.5 0.686038
\(935\) 2686.80 0.0939763
\(936\) 0 0
\(937\) −17363.4 −0.605375 −0.302688 0.953090i \(-0.597884\pi\)
−0.302688 + 0.953090i \(0.597884\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −13203.9 −0.458154
\(941\) 5547.77 0.192192 0.0960958 0.995372i \(-0.469364\pi\)
0.0960958 + 0.995372i \(0.469364\pi\)
\(942\) 0 0
\(943\) 7672.04 0.264937
\(944\) −26304.3 −0.906919
\(945\) 0 0
\(946\) −28981.6 −0.996062
\(947\) 37960.3 1.30258 0.651290 0.758829i \(-0.274228\pi\)
0.651290 + 0.758829i \(0.274228\pi\)
\(948\) 0 0
\(949\) 54081.7 1.84991
\(950\) −42451.1 −1.44978
\(951\) 0 0
\(952\) 0 0
\(953\) 10019.3 0.340563 0.170282 0.985395i \(-0.445532\pi\)
0.170282 + 0.985395i \(0.445532\pi\)
\(954\) 0 0
\(955\) 3570.16 0.120971
\(956\) 9211.65 0.311638
\(957\) 0 0
\(958\) 5059.61 0.170635
\(959\) 0 0
\(960\) 0 0
\(961\) −14733.1 −0.494548
\(962\) 11834.9 0.396646
\(963\) 0 0
\(964\) 9431.11 0.315099
\(965\) 47527.3 1.58545
\(966\) 0 0
\(967\) 27834.4 0.925641 0.462820 0.886452i \(-0.346837\pi\)
0.462820 + 0.886452i \(0.346837\pi\)
\(968\) −18609.4 −0.617900
\(969\) 0 0
\(970\) −28974.6 −0.959090
\(971\) 18275.3 0.604000 0.302000 0.953308i \(-0.402346\pi\)
0.302000 + 0.953308i \(0.402346\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −23200.8 −0.763245
\(975\) 0 0
\(976\) −157.477 −0.00516468
\(977\) 43028.9 1.40903 0.704513 0.709691i \(-0.251166\pi\)
0.704513 + 0.709691i \(0.251166\pi\)
\(978\) 0 0
\(979\) 6833.24 0.223076
\(980\) 0 0
\(981\) 0 0
\(982\) −28288.1 −0.919256
\(983\) −30559.2 −0.991544 −0.495772 0.868453i \(-0.665115\pi\)
−0.495772 + 0.868453i \(0.665115\pi\)
\(984\) 0 0
\(985\) −53039.2 −1.71571
\(986\) 821.253 0.0265254
\(987\) 0 0
\(988\) −12309.0 −0.396358
\(989\) −12835.4 −0.412681
\(990\) 0 0
\(991\) −44945.9 −1.44072 −0.720360 0.693600i \(-0.756024\pi\)
−0.720360 + 0.693600i \(0.756024\pi\)
\(992\) 11690.3 0.374160
\(993\) 0 0
\(994\) 0 0
\(995\) 26711.2 0.851058
\(996\) 0 0
\(997\) 29006.1 0.921397 0.460698 0.887557i \(-0.347599\pi\)
0.460698 + 0.887557i \(0.347599\pi\)
\(998\) 22199.1 0.704110
\(999\) 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 441.4.a.n.1.1 2
3.2 odd 2 147.4.a.j.1.2 2
7.2 even 3 441.4.e.v.361.2 4
7.3 odd 6 441.4.e.u.226.2 4
7.4 even 3 441.4.e.v.226.2 4
7.5 odd 6 441.4.e.u.361.2 4
7.6 odd 2 441.4.a.o.1.1 2
12.11 even 2 2352.4.a.cf.1.2 2
21.2 odd 6 147.4.e.k.67.1 4
21.5 even 6 147.4.e.j.67.1 4
21.11 odd 6 147.4.e.k.79.1 4
21.17 even 6 147.4.e.j.79.1 4
21.20 even 2 147.4.a.k.1.2 yes 2
84.83 odd 2 2352.4.a.bl.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.4.a.j.1.2 2 3.2 odd 2
147.4.a.k.1.2 yes 2 21.20 even 2
147.4.e.j.67.1 4 21.5 even 6
147.4.e.j.79.1 4 21.17 even 6
147.4.e.k.67.1 4 21.2 odd 6
147.4.e.k.79.1 4 21.11 odd 6
441.4.a.n.1.1 2 1.1 even 1 trivial
441.4.a.o.1.1 2 7.6 odd 2
441.4.e.u.226.2 4 7.3 odd 6
441.4.e.u.361.2 4 7.5 odd 6
441.4.e.v.226.2 4 7.4 even 3
441.4.e.v.361.2 4 7.2 even 3
2352.4.a.bl.1.1 2 84.83 odd 2
2352.4.a.cf.1.2 2 12.11 even 2