Properties

Label 225.4.a.n.1.1
Level $225$
Weight $4$
Character 225.1
Self dual yes
Analytic conductor $13.275$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.2754297513\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
Defining polynomial: \(x^{2} - 19\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.35890\) of defining polynomial
Character \(\chi\) \(=\) 225.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.35890 q^{2} +3.28220 q^{4} +30.4356 q^{7} +15.8466 q^{8} +O(q^{10})\) \(q-3.35890 q^{2} +3.28220 q^{4} +30.4356 q^{7} +15.8466 q^{8} -31.4356 q^{11} -60.7424 q^{13} -102.230 q^{14} -79.4848 q^{16} +121.178 q^{17} -14.4356 q^{19} +105.589 q^{22} -13.6932 q^{23} +204.028 q^{26} +99.8958 q^{28} +76.0492 q^{29} +183.049 q^{31} +140.208 q^{32} -407.025 q^{34} -37.3864 q^{37} +48.4877 q^{38} +30.6627 q^{41} +327.564 q^{43} -103.178 q^{44} +45.9941 q^{46} +449.485 q^{47} +583.325 q^{49} -199.369 q^{52} +301.951 q^{53} +482.301 q^{56} -255.441 q^{58} -340.970 q^{59} +619.098 q^{61} -614.844 q^{62} +164.932 q^{64} +256.890 q^{67} +397.731 q^{68} -499.178 q^{71} -19.1288 q^{73} +125.577 q^{74} -47.3805 q^{76} -956.761 q^{77} +257.424 q^{79} -102.993 q^{82} -914.909 q^{83} -1100.26 q^{86} -498.148 q^{88} +1059.68 q^{89} -1848.73 q^{91} -44.9439 q^{92} -1509.77 q^{94} +521.000 q^{97} -1959.33 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 24 q^{4} + 26 q^{7} + 84 q^{8} + O(q^{10}) \) \( 2 q + 2 q^{2} + 24 q^{4} + 26 q^{7} + 84 q^{8} - 28 q^{11} + 18 q^{13} - 126 q^{14} + 120 q^{16} + 68 q^{17} + 6 q^{19} + 124 q^{22} - 132 q^{23} + 626 q^{26} + 8 q^{28} - 92 q^{29} + 122 q^{31} + 664 q^{32} - 692 q^{34} - 284 q^{37} + 158 q^{38} - 392 q^{41} + 690 q^{43} - 32 q^{44} - 588 q^{46} + 620 q^{47} + 260 q^{49} + 1432 q^{52} + 848 q^{53} + 180 q^{56} - 1156 q^{58} - 124 q^{59} + 750 q^{61} - 942 q^{62} + 1376 q^{64} - 358 q^{67} - 704 q^{68} - 824 q^{71} - 108 q^{73} - 1196 q^{74} + 376 q^{76} - 972 q^{77} - 880 q^{79} - 2368 q^{82} - 156 q^{83} + 842 q^{86} - 264 q^{88} + 864 q^{89} - 2198 q^{91} - 2496 q^{92} - 596 q^{94} + 1042 q^{97} - 3692 q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.35890 −1.18755 −0.593775 0.804631i \(-0.702363\pi\)
−0.593775 + 0.804631i \(0.702363\pi\)
\(3\) 0 0
\(4\) 3.28220 0.410275
\(5\) 0 0
\(6\) 0 0
\(7\) 30.4356 1.64337 0.821684 0.569944i \(-0.193035\pi\)
0.821684 + 0.569944i \(0.193035\pi\)
\(8\) 15.8466 0.700328
\(9\) 0 0
\(10\) 0 0
\(11\) −31.4356 −0.861654 −0.430827 0.902435i \(-0.641778\pi\)
−0.430827 + 0.902435i \(0.641778\pi\)
\(12\) 0 0
\(13\) −60.7424 −1.29592 −0.647958 0.761676i \(-0.724377\pi\)
−0.647958 + 0.761676i \(0.724377\pi\)
\(14\) −102.230 −1.95158
\(15\) 0 0
\(16\) −79.4848 −1.24195
\(17\) 121.178 1.72882 0.864411 0.502786i \(-0.167692\pi\)
0.864411 + 0.502786i \(0.167692\pi\)
\(18\) 0 0
\(19\) −14.4356 −0.174303 −0.0871514 0.996195i \(-0.527776\pi\)
−0.0871514 + 0.996195i \(0.527776\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 105.589 1.02326
\(23\) −13.6932 −0.124141 −0.0620703 0.998072i \(-0.519770\pi\)
−0.0620703 + 0.998072i \(0.519770\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 204.028 1.53896
\(27\) 0 0
\(28\) 99.8958 0.674233
\(29\) 76.0492 0.486965 0.243482 0.969905i \(-0.421710\pi\)
0.243482 + 0.969905i \(0.421710\pi\)
\(30\) 0 0
\(31\) 183.049 1.06054 0.530268 0.847830i \(-0.322091\pi\)
0.530268 + 0.847830i \(0.322091\pi\)
\(32\) 140.208 0.774550
\(33\) 0 0
\(34\) −407.025 −2.05306
\(35\) 0 0
\(36\) 0 0
\(37\) −37.3864 −0.166116 −0.0830580 0.996545i \(-0.526469\pi\)
−0.0830580 + 0.996545i \(0.526469\pi\)
\(38\) 48.4877 0.206993
\(39\) 0 0
\(40\) 0 0
\(41\) 30.6627 0.116798 0.0583990 0.998293i \(-0.481400\pi\)
0.0583990 + 0.998293i \(0.481400\pi\)
\(42\) 0 0
\(43\) 327.564 1.16170 0.580850 0.814011i \(-0.302720\pi\)
0.580850 + 0.814011i \(0.302720\pi\)
\(44\) −103.178 −0.353515
\(45\) 0 0
\(46\) 45.9941 0.147423
\(47\) 449.485 1.39498 0.697490 0.716594i \(-0.254300\pi\)
0.697490 + 0.716594i \(0.254300\pi\)
\(48\) 0 0
\(49\) 583.325 1.70066
\(50\) 0 0
\(51\) 0 0
\(52\) −199.369 −0.531682
\(53\) 301.951 0.782569 0.391284 0.920270i \(-0.372031\pi\)
0.391284 + 0.920270i \(0.372031\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 482.301 1.15090
\(57\) 0 0
\(58\) −255.441 −0.578295
\(59\) −340.970 −0.752381 −0.376190 0.926542i \(-0.622766\pi\)
−0.376190 + 0.926542i \(0.622766\pi\)
\(60\) 0 0
\(61\) 619.098 1.29947 0.649733 0.760163i \(-0.274881\pi\)
0.649733 + 0.760163i \(0.274881\pi\)
\(62\) −614.844 −1.25944
\(63\) 0 0
\(64\) 164.932 0.322133
\(65\) 0 0
\(66\) 0 0
\(67\) 256.890 0.468419 0.234210 0.972186i \(-0.424750\pi\)
0.234210 + 0.972186i \(0.424750\pi\)
\(68\) 397.731 0.709293
\(69\) 0 0
\(70\) 0 0
\(71\) −499.178 −0.834388 −0.417194 0.908818i \(-0.636986\pi\)
−0.417194 + 0.908818i \(0.636986\pi\)
\(72\) 0 0
\(73\) −19.1288 −0.0306693 −0.0153346 0.999882i \(-0.504881\pi\)
−0.0153346 + 0.999882i \(0.504881\pi\)
\(74\) 125.577 0.197271
\(75\) 0 0
\(76\) −47.3805 −0.0715121
\(77\) −956.761 −1.41601
\(78\) 0 0
\(79\) 257.424 0.366613 0.183307 0.983056i \(-0.441320\pi\)
0.183307 + 0.983056i \(0.441320\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −102.993 −0.138703
\(83\) −914.909 −1.20993 −0.604965 0.796252i \(-0.706813\pi\)
−0.604965 + 0.796252i \(0.706813\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1100.26 −1.37958
\(87\) 0 0
\(88\) −498.148 −0.603440
\(89\) 1059.68 1.26209 0.631045 0.775746i \(-0.282626\pi\)
0.631045 + 0.775746i \(0.282626\pi\)
\(90\) 0 0
\(91\) −1848.73 −2.12967
\(92\) −44.9439 −0.0509318
\(93\) 0 0
\(94\) −1509.77 −1.65661
\(95\) 0 0
\(96\) 0 0
\(97\) 521.000 0.545356 0.272678 0.962105i \(-0.412091\pi\)
0.272678 + 0.962105i \(0.412091\pi\)
\(98\) −1959.33 −2.01962
\(99\) 0 0
\(100\) 0 0
\(101\) −347.080 −0.341938 −0.170969 0.985276i \(-0.554690\pi\)
−0.170969 + 0.985276i \(0.554690\pi\)
\(102\) 0 0
\(103\) −770.749 −0.737322 −0.368661 0.929564i \(-0.620184\pi\)
−0.368661 + 0.929564i \(0.620184\pi\)
\(104\) −962.561 −0.907566
\(105\) 0 0
\(106\) −1014.22 −0.929339
\(107\) 1415.37 1.27878 0.639390 0.768883i \(-0.279187\pi\)
0.639390 + 0.768883i \(0.279187\pi\)
\(108\) 0 0
\(109\) 908.386 0.798235 0.399118 0.916900i \(-0.369317\pi\)
0.399118 + 0.916900i \(0.369317\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2419.17 −2.04098
\(113\) 2049.94 1.70657 0.853283 0.521447i \(-0.174608\pi\)
0.853283 + 0.521447i \(0.174608\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 249.609 0.199790
\(117\) 0 0
\(118\) 1145.28 0.893490
\(119\) 3688.12 2.84109
\(120\) 0 0
\(121\) −342.803 −0.257553
\(122\) −2079.49 −1.54318
\(123\) 0 0
\(124\) 600.804 0.435111
\(125\) 0 0
\(126\) 0 0
\(127\) 281.644 0.196786 0.0983932 0.995148i \(-0.468630\pi\)
0.0983932 + 0.995148i \(0.468630\pi\)
\(128\) −1675.66 −1.15710
\(129\) 0 0
\(130\) 0 0
\(131\) −243.056 −0.162106 −0.0810531 0.996710i \(-0.525828\pi\)
−0.0810531 + 0.996710i \(0.525828\pi\)
\(132\) 0 0
\(133\) −439.356 −0.286444
\(134\) −862.867 −0.556271
\(135\) 0 0
\(136\) 1920.26 1.21074
\(137\) −909.386 −0.567110 −0.283555 0.958956i \(-0.591514\pi\)
−0.283555 + 0.958956i \(0.591514\pi\)
\(138\) 0 0
\(139\) −2049.52 −1.25063 −0.625317 0.780371i \(-0.715030\pi\)
−0.625317 + 0.780371i \(0.715030\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1676.69 0.990877
\(143\) 1909.47 1.11663
\(144\) 0 0
\(145\) 0 0
\(146\) 64.2517 0.0364213
\(147\) 0 0
\(148\) −122.710 −0.0681533
\(149\) −3601.14 −1.97998 −0.989990 0.141136i \(-0.954925\pi\)
−0.989990 + 0.141136i \(0.954925\pi\)
\(150\) 0 0
\(151\) 1383.38 0.745550 0.372775 0.927922i \(-0.378406\pi\)
0.372775 + 0.927922i \(0.378406\pi\)
\(152\) −228.755 −0.122069
\(153\) 0 0
\(154\) 3213.66 1.68159
\(155\) 0 0
\(156\) 0 0
\(157\) −131.749 −0.0669729 −0.0334864 0.999439i \(-0.510661\pi\)
−0.0334864 + 0.999439i \(0.510661\pi\)
\(158\) −864.661 −0.435372
\(159\) 0 0
\(160\) 0 0
\(161\) −416.761 −0.204009
\(162\) 0 0
\(163\) 2897.74 1.39244 0.696222 0.717827i \(-0.254863\pi\)
0.696222 + 0.717827i \(0.254863\pi\)
\(164\) 100.641 0.0479193
\(165\) 0 0
\(166\) 3073.09 1.43685
\(167\) −260.283 −0.120607 −0.0603034 0.998180i \(-0.519207\pi\)
−0.0603034 + 0.998180i \(0.519207\pi\)
\(168\) 0 0
\(169\) 1492.64 0.679398
\(170\) 0 0
\(171\) 0 0
\(172\) 1075.13 0.476617
\(173\) 1935.83 0.850742 0.425371 0.905019i \(-0.360144\pi\)
0.425371 + 0.905019i \(0.360144\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2498.65 1.07013
\(177\) 0 0
\(178\) −3559.36 −1.49880
\(179\) −576.627 −0.240777 −0.120389 0.992727i \(-0.538414\pi\)
−0.120389 + 0.992727i \(0.538414\pi\)
\(180\) 0 0
\(181\) −1962.04 −0.805733 −0.402866 0.915259i \(-0.631986\pi\)
−0.402866 + 0.915259i \(0.631986\pi\)
\(182\) 6209.70 2.52909
\(183\) 0 0
\(184\) −216.991 −0.0869390
\(185\) 0 0
\(186\) 0 0
\(187\) −3809.30 −1.48965
\(188\) 1475.30 0.572326
\(189\) 0 0
\(190\) 0 0
\(191\) 4318.75 1.63609 0.818047 0.575152i \(-0.195057\pi\)
0.818047 + 0.575152i \(0.195057\pi\)
\(192\) 0 0
\(193\) 2.97647 0.00111011 0.000555054 1.00000i \(-0.499823\pi\)
0.000555054 1.00000i \(0.499823\pi\)
\(194\) −1749.99 −0.647638
\(195\) 0 0
\(196\) 1914.59 0.697738
\(197\) −569.705 −0.206040 −0.103020 0.994679i \(-0.532851\pi\)
−0.103020 + 0.994679i \(0.532851\pi\)
\(198\) 0 0
\(199\) −3050.73 −1.08674 −0.543368 0.839494i \(-0.682851\pi\)
−0.543368 + 0.839494i \(0.682851\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 1165.81 0.406068
\(203\) 2314.60 0.800262
\(204\) 0 0
\(205\) 0 0
\(206\) 2588.87 0.875607
\(207\) 0 0
\(208\) 4828.09 1.60946
\(209\) 453.792 0.150189
\(210\) 0 0
\(211\) −50.5104 −0.0164800 −0.00824000 0.999966i \(-0.502623\pi\)
−0.00824000 + 0.999966i \(0.502623\pi\)
\(212\) 991.064 0.321069
\(213\) 0 0
\(214\) −4754.10 −1.51862
\(215\) 0 0
\(216\) 0 0
\(217\) 5571.21 1.74285
\(218\) −3051.18 −0.947944
\(219\) 0 0
\(220\) 0 0
\(221\) −7360.64 −2.24041
\(222\) 0 0
\(223\) 5453.55 1.63765 0.818827 0.574040i \(-0.194625\pi\)
0.818827 + 0.574040i \(0.194625\pi\)
\(224\) 4267.33 1.27287
\(225\) 0 0
\(226\) −6885.54 −2.02663
\(227\) 4777.14 1.39678 0.698392 0.715715i \(-0.253899\pi\)
0.698392 + 0.715715i \(0.253899\pi\)
\(228\) 0 0
\(229\) −2085.51 −0.601808 −0.300904 0.953654i \(-0.597288\pi\)
−0.300904 + 0.953654i \(0.597288\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1205.12 0.341035
\(233\) −6484.53 −1.82324 −0.911622 0.411030i \(-0.865169\pi\)
−0.911622 + 0.411030i \(0.865169\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1119.13 −0.308683
\(237\) 0 0
\(238\) −12388.0 −3.37394
\(239\) −2234.62 −0.604792 −0.302396 0.953182i \(-0.597786\pi\)
−0.302396 + 0.953182i \(0.597786\pi\)
\(240\) 0 0
\(241\) −2393.01 −0.639616 −0.319808 0.947482i \(-0.603618\pi\)
−0.319808 + 0.947482i \(0.603618\pi\)
\(242\) 1151.44 0.305857
\(243\) 0 0
\(244\) 2032.01 0.533139
\(245\) 0 0
\(246\) 0 0
\(247\) 876.852 0.225882
\(248\) 2900.71 0.742722
\(249\) 0 0
\(250\) 0 0
\(251\) 612.661 0.154067 0.0770335 0.997029i \(-0.475455\pi\)
0.0770335 + 0.997029i \(0.475455\pi\)
\(252\) 0 0
\(253\) 430.454 0.106966
\(254\) −946.014 −0.233694
\(255\) 0 0
\(256\) 4308.91 1.05198
\(257\) 306.112 0.0742987 0.0371493 0.999310i \(-0.488172\pi\)
0.0371493 + 0.999310i \(0.488172\pi\)
\(258\) 0 0
\(259\) −1137.88 −0.272990
\(260\) 0 0
\(261\) 0 0
\(262\) 816.401 0.192509
\(263\) −283.839 −0.0665484 −0.0332742 0.999446i \(-0.510593\pi\)
−0.0332742 + 0.999446i \(0.510593\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 1475.75 0.340166
\(267\) 0 0
\(268\) 843.165 0.192181
\(269\) −2426.21 −0.549920 −0.274960 0.961456i \(-0.588665\pi\)
−0.274960 + 0.961456i \(0.588665\pi\)
\(270\) 0 0
\(271\) −174.946 −0.0392148 −0.0196074 0.999808i \(-0.506242\pi\)
−0.0196074 + 0.999808i \(0.506242\pi\)
\(272\) −9631.80 −2.14711
\(273\) 0 0
\(274\) 3054.54 0.673472
\(275\) 0 0
\(276\) 0 0
\(277\) −7807.07 −1.69344 −0.846718 0.532042i \(-0.821425\pi\)
−0.846718 + 0.532042i \(0.821425\pi\)
\(278\) 6884.14 1.48519
\(279\) 0 0
\(280\) 0 0
\(281\) −584.171 −0.124017 −0.0620084 0.998076i \(-0.519751\pi\)
−0.0620084 + 0.998076i \(0.519751\pi\)
\(282\) 0 0
\(283\) −5897.31 −1.23872 −0.619362 0.785106i \(-0.712609\pi\)
−0.619362 + 0.785106i \(0.712609\pi\)
\(284\) −1638.40 −0.342329
\(285\) 0 0
\(286\) −6413.73 −1.32605
\(287\) 933.239 0.191942
\(288\) 0 0
\(289\) 9771.10 1.98883
\(290\) 0 0
\(291\) 0 0
\(292\) −62.7846 −0.0125828
\(293\) −1609.73 −0.320960 −0.160480 0.987039i \(-0.551304\pi\)
−0.160480 + 0.987039i \(0.551304\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −592.448 −0.116336
\(297\) 0 0
\(298\) 12095.9 2.35133
\(299\) 831.758 0.160876
\(300\) 0 0
\(301\) 9969.62 1.90910
\(302\) −4646.64 −0.885378
\(303\) 0 0
\(304\) 1147.41 0.216475
\(305\) 0 0
\(306\) 0 0
\(307\) 234.473 0.0435898 0.0217949 0.999762i \(-0.493062\pi\)
0.0217949 + 0.999762i \(0.493062\pi\)
\(308\) −3140.28 −0.580955
\(309\) 0 0
\(310\) 0 0
\(311\) 1795.25 0.327329 0.163665 0.986516i \(-0.447669\pi\)
0.163665 + 0.986516i \(0.447669\pi\)
\(312\) 0 0
\(313\) −8440.61 −1.52425 −0.762127 0.647427i \(-0.775845\pi\)
−0.762127 + 0.647427i \(0.775845\pi\)
\(314\) 442.533 0.0795336
\(315\) 0 0
\(316\) 844.917 0.150412
\(317\) −10551.7 −1.86953 −0.934766 0.355264i \(-0.884391\pi\)
−0.934766 + 0.355264i \(0.884391\pi\)
\(318\) 0 0
\(319\) −2390.65 −0.419595
\(320\) 0 0
\(321\) 0 0
\(322\) 1399.86 0.242270
\(323\) −1749.28 −0.301339
\(324\) 0 0
\(325\) 0 0
\(326\) −9733.21 −1.65360
\(327\) 0 0
\(328\) 485.900 0.0817968
\(329\) 13680.3 2.29247
\(330\) 0 0
\(331\) 6743.17 1.11975 0.559876 0.828576i \(-0.310849\pi\)
0.559876 + 0.828576i \(0.310849\pi\)
\(332\) −3002.91 −0.496405
\(333\) 0 0
\(334\) 874.265 0.143227
\(335\) 0 0
\(336\) 0 0
\(337\) −8437.26 −1.36382 −0.681909 0.731437i \(-0.738850\pi\)
−0.681909 + 0.731437i \(0.738850\pi\)
\(338\) −5013.62 −0.806819
\(339\) 0 0
\(340\) 0 0
\(341\) −5754.26 −0.913814
\(342\) 0 0
\(343\) 7314.45 1.15144
\(344\) 5190.78 0.813571
\(345\) 0 0
\(346\) −6502.25 −1.01030
\(347\) 1848.85 0.286027 0.143013 0.989721i \(-0.454321\pi\)
0.143013 + 0.989721i \(0.454321\pi\)
\(348\) 0 0
\(349\) −1148.38 −0.176136 −0.0880678 0.996114i \(-0.528069\pi\)
−0.0880678 + 0.996114i \(0.528069\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −4407.54 −0.667393
\(353\) 5753.60 0.867516 0.433758 0.901029i \(-0.357187\pi\)
0.433758 + 0.901029i \(0.357187\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 3478.09 0.517804
\(357\) 0 0
\(358\) 1936.83 0.285935
\(359\) −5452.01 −0.801521 −0.400761 0.916183i \(-0.631254\pi\)
−0.400761 + 0.916183i \(0.631254\pi\)
\(360\) 0 0
\(361\) −6650.61 −0.969619
\(362\) 6590.31 0.956848
\(363\) 0 0
\(364\) −6067.91 −0.873749
\(365\) 0 0
\(366\) 0 0
\(367\) 8385.93 1.19276 0.596379 0.802703i \(-0.296606\pi\)
0.596379 + 0.802703i \(0.296606\pi\)
\(368\) 1088.40 0.154176
\(369\) 0 0
\(370\) 0 0
\(371\) 9190.05 1.28605
\(372\) 0 0
\(373\) 2728.30 0.378730 0.189365 0.981907i \(-0.439357\pi\)
0.189365 + 0.981907i \(0.439357\pi\)
\(374\) 12795.1 1.76903
\(375\) 0 0
\(376\) 7122.81 0.976944
\(377\) −4619.41 −0.631065
\(378\) 0 0
\(379\) 3348.99 0.453895 0.226947 0.973907i \(-0.427125\pi\)
0.226947 + 0.973907i \(0.427125\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −14506.2 −1.94294
\(383\) −10430.3 −1.39155 −0.695774 0.718261i \(-0.744938\pi\)
−0.695774 + 0.718261i \(0.744938\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −9.99766 −0.00131831
\(387\) 0 0
\(388\) 1710.03 0.223746
\(389\) −9827.23 −1.28088 −0.640438 0.768010i \(-0.721247\pi\)
−0.640438 + 0.768010i \(0.721247\pi\)
\(390\) 0 0
\(391\) −1659.32 −0.214617
\(392\) 9243.73 1.19102
\(393\) 0 0
\(394\) 1913.58 0.244682
\(395\) 0 0
\(396\) 0 0
\(397\) −436.382 −0.0551672 −0.0275836 0.999619i \(-0.508781\pi\)
−0.0275836 + 0.999619i \(0.508781\pi\)
\(398\) 10247.1 1.29055
\(399\) 0 0
\(400\) 0 0
\(401\) 14501.5 1.80591 0.902955 0.429736i \(-0.141393\pi\)
0.902955 + 0.429736i \(0.141393\pi\)
\(402\) 0 0
\(403\) −11118.8 −1.37436
\(404\) −1139.19 −0.140289
\(405\) 0 0
\(406\) −7774.51 −0.950351
\(407\) 1175.26 0.143134
\(408\) 0 0
\(409\) −12058.4 −1.45782 −0.728911 0.684609i \(-0.759973\pi\)
−0.728911 + 0.684609i \(0.759973\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −2529.76 −0.302505
\(413\) −10377.6 −1.23644
\(414\) 0 0
\(415\) 0 0
\(416\) −8516.60 −1.00375
\(417\) 0 0
\(418\) −1524.24 −0.178356
\(419\) −6042.95 −0.704577 −0.352288 0.935892i \(-0.614596\pi\)
−0.352288 + 0.935892i \(0.614596\pi\)
\(420\) 0 0
\(421\) −9994.67 −1.15703 −0.578516 0.815671i \(-0.696368\pi\)
−0.578516 + 0.815671i \(0.696368\pi\)
\(422\) 169.659 0.0195708
\(423\) 0 0
\(424\) 4784.90 0.548054
\(425\) 0 0
\(426\) 0 0
\(427\) 18842.6 2.13550
\(428\) 4645.55 0.524652
\(429\) 0 0
\(430\) 0 0
\(431\) −9327.32 −1.04242 −0.521208 0.853430i \(-0.674518\pi\)
−0.521208 + 0.853430i \(0.674518\pi\)
\(432\) 0 0
\(433\) −7861.22 −0.872485 −0.436243 0.899829i \(-0.643691\pi\)
−0.436243 + 0.899829i \(0.643691\pi\)
\(434\) −18713.1 −2.06972
\(435\) 0 0
\(436\) 2981.51 0.327496
\(437\) 197.670 0.0216380
\(438\) 0 0
\(439\) 7412.06 0.805828 0.402914 0.915238i \(-0.367997\pi\)
0.402914 + 0.915238i \(0.367997\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 24723.6 2.66060
\(443\) 3043.66 0.326430 0.163215 0.986591i \(-0.447814\pi\)
0.163215 + 0.986591i \(0.447814\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −18317.9 −1.94480
\(447\) 0 0
\(448\) 5019.81 0.529383
\(449\) 9547.87 1.00355 0.501773 0.864999i \(-0.332681\pi\)
0.501773 + 0.864999i \(0.332681\pi\)
\(450\) 0 0
\(451\) −963.902 −0.100639
\(452\) 6728.31 0.700162
\(453\) 0 0
\(454\) −16045.9 −1.65875
\(455\) 0 0
\(456\) 0 0
\(457\) 13401.9 1.37180 0.685901 0.727695i \(-0.259408\pi\)
0.685901 + 0.727695i \(0.259408\pi\)
\(458\) 7005.00 0.714677
\(459\) 0 0
\(460\) 0 0
\(461\) 4137.03 0.417962 0.208981 0.977920i \(-0.432985\pi\)
0.208981 + 0.977920i \(0.432985\pi\)
\(462\) 0 0
\(463\) −13976.3 −1.40288 −0.701439 0.712729i \(-0.747459\pi\)
−0.701439 + 0.712729i \(0.747459\pi\)
\(464\) −6044.75 −0.604786
\(465\) 0 0
\(466\) 21780.9 2.16519
\(467\) 10796.5 1.06982 0.534908 0.844910i \(-0.320346\pi\)
0.534908 + 0.844910i \(0.320346\pi\)
\(468\) 0 0
\(469\) 7818.60 0.769785
\(470\) 0 0
\(471\) 0 0
\(472\) −5403.21 −0.526913
\(473\) −10297.2 −1.00098
\(474\) 0 0
\(475\) 0 0
\(476\) 12105.2 1.16563
\(477\) 0 0
\(478\) 7505.85 0.718221
\(479\) −14568.4 −1.38966 −0.694830 0.719174i \(-0.744521\pi\)
−0.694830 + 0.719174i \(0.744521\pi\)
\(480\) 0 0
\(481\) 2270.94 0.215272
\(482\) 8037.89 0.759577
\(483\) 0 0
\(484\) −1125.15 −0.105668
\(485\) 0 0
\(486\) 0 0
\(487\) −11456.6 −1.06601 −0.533004 0.846113i \(-0.678937\pi\)
−0.533004 + 0.846113i \(0.678937\pi\)
\(488\) 9810.61 0.910052
\(489\) 0 0
\(490\) 0 0
\(491\) 19666.5 1.80761 0.903804 0.427948i \(-0.140763\pi\)
0.903804 + 0.427948i \(0.140763\pi\)
\(492\) 0 0
\(493\) 9215.48 0.841875
\(494\) −2945.26 −0.268246
\(495\) 0 0
\(496\) −14549.6 −1.31713
\(497\) −15192.8 −1.37121
\(498\) 0 0
\(499\) 8379.31 0.751722 0.375861 0.926676i \(-0.377347\pi\)
0.375861 + 0.926676i \(0.377347\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −2057.87 −0.182962
\(503\) 15678.1 1.38976 0.694881 0.719124i \(-0.255457\pi\)
0.694881 + 0.719124i \(0.255457\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −1445.85 −0.127028
\(507\) 0 0
\(508\) 924.413 0.0807366
\(509\) 17037.3 1.48363 0.741813 0.670606i \(-0.233966\pi\)
0.741813 + 0.670606i \(0.233966\pi\)
\(510\) 0 0
\(511\) −582.197 −0.0504009
\(512\) −1067.93 −0.0921798
\(513\) 0 0
\(514\) −1028.20 −0.0882334
\(515\) 0 0
\(516\) 0 0
\(517\) −14129.8 −1.20199
\(518\) 3822.02 0.324189
\(519\) 0 0
\(520\) 0 0
\(521\) 8776.12 0.737982 0.368991 0.929433i \(-0.379703\pi\)
0.368991 + 0.929433i \(0.379703\pi\)
\(522\) 0 0
\(523\) 11120.4 0.929753 0.464877 0.885375i \(-0.346099\pi\)
0.464877 + 0.885375i \(0.346099\pi\)
\(524\) −797.759 −0.0665082
\(525\) 0 0
\(526\) 953.385 0.0790296
\(527\) 22181.5 1.83348
\(528\) 0 0
\(529\) −11979.5 −0.984589
\(530\) 0 0
\(531\) 0 0
\(532\) −1442.06 −0.117521
\(533\) −1862.53 −0.151360
\(534\) 0 0
\(535\) 0 0
\(536\) 4070.83 0.328047
\(537\) 0 0
\(538\) 8149.39 0.653058
\(539\) −18337.2 −1.46538
\(540\) 0 0
\(541\) 21730.6 1.72693 0.863467 0.504405i \(-0.168288\pi\)
0.863467 + 0.504405i \(0.168288\pi\)
\(542\) 587.626 0.0465695
\(543\) 0 0
\(544\) 16990.2 1.33906
\(545\) 0 0
\(546\) 0 0
\(547\) −6926.17 −0.541392 −0.270696 0.962665i \(-0.587254\pi\)
−0.270696 + 0.962665i \(0.587254\pi\)
\(548\) −2984.79 −0.232671
\(549\) 0 0
\(550\) 0 0
\(551\) −1097.82 −0.0848793
\(552\) 0 0
\(553\) 7834.85 0.602480
\(554\) 26223.2 2.01104
\(555\) 0 0
\(556\) −6726.95 −0.513104
\(557\) 6589.22 0.501246 0.250623 0.968085i \(-0.419364\pi\)
0.250623 + 0.968085i \(0.419364\pi\)
\(558\) 0 0
\(559\) −19897.0 −1.50547
\(560\) 0 0
\(561\) 0 0
\(562\) 1962.17 0.147276
\(563\) −3839.63 −0.287426 −0.143713 0.989619i \(-0.545904\pi\)
−0.143713 + 0.989619i \(0.545904\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 19808.5 1.47105
\(567\) 0 0
\(568\) −7910.28 −0.584345
\(569\) 20874.0 1.53794 0.768968 0.639287i \(-0.220771\pi\)
0.768968 + 0.639287i \(0.220771\pi\)
\(570\) 0 0
\(571\) 21175.9 1.55199 0.775994 0.630740i \(-0.217248\pi\)
0.775994 + 0.630740i \(0.217248\pi\)
\(572\) 6267.28 0.458126
\(573\) 0 0
\(574\) −3134.66 −0.227941
\(575\) 0 0
\(576\) 0 0
\(577\) −14924.2 −1.07678 −0.538391 0.842695i \(-0.680968\pi\)
−0.538391 + 0.842695i \(0.680968\pi\)
\(578\) −32820.1 −2.36183
\(579\) 0 0
\(580\) 0 0
\(581\) −27845.8 −1.98836
\(582\) 0 0
\(583\) −9492.00 −0.674303
\(584\) −303.127 −0.0214785
\(585\) 0 0
\(586\) 5406.92 0.381156
\(587\) −25218.0 −1.77318 −0.886592 0.462552i \(-0.846934\pi\)
−0.886592 + 0.462552i \(0.846934\pi\)
\(588\) 0 0
\(589\) −2642.42 −0.184854
\(590\) 0 0
\(591\) 0 0
\(592\) 2971.65 0.206308
\(593\) −5011.77 −0.347063 −0.173532 0.984828i \(-0.555518\pi\)
−0.173532 + 0.984828i \(0.555518\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −11819.7 −0.812337
\(597\) 0 0
\(598\) −2793.79 −0.191048
\(599\) −4943.41 −0.337199 −0.168600 0.985685i \(-0.553925\pi\)
−0.168600 + 0.985685i \(0.553925\pi\)
\(600\) 0 0
\(601\) −24334.8 −1.65164 −0.825821 0.563932i \(-0.809288\pi\)
−0.825821 + 0.563932i \(0.809288\pi\)
\(602\) −33486.9 −2.26715
\(603\) 0 0
\(604\) 4540.54 0.305881
\(605\) 0 0
\(606\) 0 0
\(607\) 28973.8 1.93741 0.968707 0.248207i \(-0.0798415\pi\)
0.968707 + 0.248207i \(0.0798415\pi\)
\(608\) −2023.99 −0.135006
\(609\) 0 0
\(610\) 0 0
\(611\) −27302.8 −1.80778
\(612\) 0 0
\(613\) 15139.1 0.997490 0.498745 0.866749i \(-0.333794\pi\)
0.498745 + 0.866749i \(0.333794\pi\)
\(614\) −787.571 −0.0517651
\(615\) 0 0
\(616\) −15161.4 −0.991673
\(617\) −13894.8 −0.906617 −0.453309 0.891354i \(-0.649756\pi\)
−0.453309 + 0.891354i \(0.649756\pi\)
\(618\) 0 0
\(619\) −4589.69 −0.298021 −0.149011 0.988836i \(-0.547609\pi\)
−0.149011 + 0.988836i \(0.547609\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −6030.07 −0.388720
\(623\) 32252.0 2.07408
\(624\) 0 0
\(625\) 0 0
\(626\) 28351.2 1.81013
\(627\) 0 0
\(628\) −432.428 −0.0274773
\(629\) −4530.41 −0.287185
\(630\) 0 0
\(631\) 3005.77 0.189632 0.0948160 0.995495i \(-0.469774\pi\)
0.0948160 + 0.995495i \(0.469774\pi\)
\(632\) 4079.29 0.256749
\(633\) 0 0
\(634\) 35442.0 2.22016
\(635\) 0 0
\(636\) 0 0
\(637\) −35432.6 −2.20391
\(638\) 8029.96 0.498290
\(639\) 0 0
\(640\) 0 0
\(641\) 5631.47 0.347004 0.173502 0.984834i \(-0.444492\pi\)
0.173502 + 0.984834i \(0.444492\pi\)
\(642\) 0 0
\(643\) 11305.1 0.693358 0.346679 0.937984i \(-0.387309\pi\)
0.346679 + 0.937984i \(0.387309\pi\)
\(644\) −1367.89 −0.0836997
\(645\) 0 0
\(646\) 5875.64 0.357855
\(647\) 8614.30 0.523436 0.261718 0.965144i \(-0.415711\pi\)
0.261718 + 0.965144i \(0.415711\pi\)
\(648\) 0 0
\(649\) 10718.6 0.648291
\(650\) 0 0
\(651\) 0 0
\(652\) 9510.96 0.571285
\(653\) −12639.8 −0.757479 −0.378739 0.925503i \(-0.623642\pi\)
−0.378739 + 0.925503i \(0.623642\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −2437.22 −0.145057
\(657\) 0 0
\(658\) −45950.9 −2.72242
\(659\) −13640.4 −0.806306 −0.403153 0.915133i \(-0.632086\pi\)
−0.403153 + 0.915133i \(0.632086\pi\)
\(660\) 0 0
\(661\) −17052.0 −1.00340 −0.501699 0.865042i \(-0.667291\pi\)
−0.501699 + 0.865042i \(0.667291\pi\)
\(662\) −22649.6 −1.32976
\(663\) 0 0
\(664\) −14498.2 −0.847348
\(665\) 0 0
\(666\) 0 0
\(667\) −1041.36 −0.0604521
\(668\) −854.302 −0.0494820
\(669\) 0 0
\(670\) 0 0
\(671\) −19461.7 −1.11969
\(672\) 0 0
\(673\) −16419.7 −0.940467 −0.470234 0.882542i \(-0.655830\pi\)
−0.470234 + 0.882542i \(0.655830\pi\)
\(674\) 28339.9 1.61960
\(675\) 0 0
\(676\) 4899.14 0.278740
\(677\) 8670.47 0.492221 0.246110 0.969242i \(-0.420847\pi\)
0.246110 + 0.969242i \(0.420847\pi\)
\(678\) 0 0
\(679\) 15856.9 0.896220
\(680\) 0 0
\(681\) 0 0
\(682\) 19328.0 1.08520
\(683\) −5973.36 −0.334647 −0.167324 0.985902i \(-0.553512\pi\)
−0.167324 + 0.985902i \(0.553512\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −24568.5 −1.36739
\(687\) 0 0
\(688\) −26036.4 −1.44277
\(689\) −18341.2 −1.01414
\(690\) 0 0
\(691\) −15316.3 −0.843212 −0.421606 0.906779i \(-0.638533\pi\)
−0.421606 + 0.906779i \(0.638533\pi\)
\(692\) 6353.78 0.349038
\(693\) 0 0
\(694\) −6210.09 −0.339671
\(695\) 0 0
\(696\) 0 0
\(697\) 3715.65 0.201923
\(698\) 3857.29 0.209170
\(699\) 0 0
\(700\) 0 0
\(701\) −34583.1 −1.86332 −0.931660 0.363333i \(-0.881639\pi\)
−0.931660 + 0.363333i \(0.881639\pi\)
\(702\) 0 0
\(703\) 539.695 0.0289545
\(704\) −5184.74 −0.277567
\(705\) 0 0
\(706\) −19325.8 −1.03022
\(707\) −10563.6 −0.561929
\(708\) 0 0
\(709\) 11194.1 0.592955 0.296477 0.955040i \(-0.404188\pi\)
0.296477 + 0.955040i \(0.404188\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 16792.4 0.883877
\(713\) −2506.53 −0.131655
\(714\) 0 0
\(715\) 0 0
\(716\) −1892.61 −0.0987850
\(717\) 0 0
\(718\) 18312.8 0.951847
\(719\) 15491.9 0.803549 0.401774 0.915739i \(-0.368394\pi\)
0.401774 + 0.915739i \(0.368394\pi\)
\(720\) 0 0
\(721\) −23458.2 −1.21169
\(722\) 22338.7 1.15147
\(723\) 0 0
\(724\) −6439.83 −0.330572
\(725\) 0 0
\(726\) 0 0
\(727\) −6272.72 −0.320003 −0.160002 0.987117i \(-0.551150\pi\)
−0.160002 + 0.987117i \(0.551150\pi\)
\(728\) −29296.1 −1.49146
\(729\) 0 0
\(730\) 0 0
\(731\) 39693.6 2.00837
\(732\) 0 0
\(733\) 24980.5 1.25877 0.629383 0.777095i \(-0.283308\pi\)
0.629383 + 0.777095i \(0.283308\pi\)
\(734\) −28167.5 −1.41646
\(735\) 0 0
\(736\) −1919.90 −0.0961530
\(737\) −8075.49 −0.403615
\(738\) 0 0
\(739\) −30660.7 −1.52621 −0.763107 0.646272i \(-0.776327\pi\)
−0.763107 + 0.646272i \(0.776327\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −30868.5 −1.52725
\(743\) 17205.7 0.849551 0.424776 0.905299i \(-0.360353\pi\)
0.424776 + 0.905299i \(0.360353\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −9164.09 −0.449760
\(747\) 0 0
\(748\) −12502.9 −0.611165
\(749\) 43077.8 2.10151
\(750\) 0 0
\(751\) 18397.1 0.893901 0.446950 0.894559i \(-0.352510\pi\)
0.446950 + 0.894559i \(0.352510\pi\)
\(752\) −35727.2 −1.73250
\(753\) 0 0
\(754\) 15516.1 0.749422
\(755\) 0 0
\(756\) 0 0
\(757\) 22305.1 1.07093 0.535465 0.844557i \(-0.320136\pi\)
0.535465 + 0.844557i \(0.320136\pi\)
\(758\) −11248.9 −0.539023
\(759\) 0 0
\(760\) 0 0
\(761\) −14458.5 −0.688727 −0.344364 0.938836i \(-0.611905\pi\)
−0.344364 + 0.938836i \(0.611905\pi\)
\(762\) 0 0
\(763\) 27647.3 1.31179
\(764\) 14175.0 0.671249
\(765\) 0 0
\(766\) 35034.3 1.65253
\(767\) 20711.3 0.975022
\(768\) 0 0
\(769\) −39897.6 −1.87093 −0.935463 0.353424i \(-0.885017\pi\)
−0.935463 + 0.353424i \(0.885017\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 9.76938 0.000455450 0
\(773\) −20070.2 −0.933863 −0.466931 0.884294i \(-0.654641\pi\)
−0.466931 + 0.884294i \(0.654641\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 8256.08 0.381928
\(777\) 0 0
\(778\) 33008.7 1.52110
\(779\) −442.635 −0.0203582
\(780\) 0 0
\(781\) 15692.0 0.718953
\(782\) 5573.47 0.254868
\(783\) 0 0
\(784\) −46365.5 −2.11213
\(785\) 0 0
\(786\) 0 0
\(787\) 10733.0 0.486137 0.243069 0.970009i \(-0.421846\pi\)
0.243069 + 0.970009i \(0.421846\pi\)
\(788\) −1869.89 −0.0845329
\(789\) 0 0
\(790\) 0 0
\(791\) 62391.1 2.80452
\(792\) 0 0
\(793\) −37605.5 −1.68400
\(794\) 1465.76 0.0655139
\(795\) 0 0
\(796\) −10013.1 −0.445861
\(797\) 14335.8 0.637140 0.318570 0.947899i \(-0.396797\pi\)
0.318570 + 0.947899i \(0.396797\pi\)
\(798\) 0 0
\(799\) 54467.7 2.41167
\(800\) 0 0
\(801\) 0 0
\(802\) −48709.0 −2.14461
\(803\) 601.325 0.0264263
\(804\) 0 0
\(805\) 0 0
\(806\) 37347.1 1.63213
\(807\) 0 0
\(808\) −5500.03 −0.239468
\(809\) −20920.7 −0.909187 −0.454593 0.890699i \(-0.650215\pi\)
−0.454593 + 0.890699i \(0.650215\pi\)
\(810\) 0 0
\(811\) 12816.5 0.554931 0.277465 0.960736i \(-0.410506\pi\)
0.277465 + 0.960736i \(0.410506\pi\)
\(812\) 7596.99 0.328328
\(813\) 0 0
\(814\) −3947.59 −0.169979
\(815\) 0 0
\(816\) 0 0
\(817\) −4728.59 −0.202488
\(818\) 40502.9 1.73124
\(819\) 0 0
\(820\) 0 0
\(821\) −7253.55 −0.308344 −0.154172 0.988044i \(-0.549271\pi\)
−0.154172 + 0.988044i \(0.549271\pi\)
\(822\) 0 0
\(823\) 35288.1 1.49461 0.747307 0.664479i \(-0.231346\pi\)
0.747307 + 0.664479i \(0.231346\pi\)
\(824\) −12213.8 −0.516367
\(825\) 0 0
\(826\) 34857.3 1.46833
\(827\) 32205.9 1.35418 0.677092 0.735899i \(-0.263240\pi\)
0.677092 + 0.735899i \(0.263240\pi\)
\(828\) 0 0
\(829\) 29993.3 1.25659 0.628294 0.777976i \(-0.283754\pi\)
0.628294 + 0.777976i \(0.283754\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −10018.4 −0.417457
\(833\) 70686.2 2.94013
\(834\) 0 0
\(835\) 0 0
\(836\) 1489.44 0.0616187
\(837\) 0 0
\(838\) 20297.7 0.836720
\(839\) 35608.7 1.46526 0.732628 0.680629i \(-0.238294\pi\)
0.732628 + 0.680629i \(0.238294\pi\)
\(840\) 0 0
\(841\) −18605.5 −0.762865
\(842\) 33571.1 1.37403
\(843\) 0 0
\(844\) −165.785 −0.00676134
\(845\) 0 0
\(846\) 0 0
\(847\) −10433.4 −0.423255
\(848\) −24000.5 −0.971911
\(849\) 0 0
\(850\) 0 0
\(851\) 511.940 0.0206217
\(852\) 0 0
\(853\) −11229.3 −0.450744 −0.225372 0.974273i \(-0.572360\pi\)
−0.225372 + 0.974273i \(0.572360\pi\)
\(854\) −63290.5 −2.53601
\(855\) 0 0
\(856\) 22428.9 0.895565
\(857\) −22136.1 −0.882327 −0.441164 0.897427i \(-0.645434\pi\)
−0.441164 + 0.897427i \(0.645434\pi\)
\(858\) 0 0
\(859\) −820.727 −0.0325994 −0.0162997 0.999867i \(-0.505189\pi\)
−0.0162997 + 0.999867i \(0.505189\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 31329.5 1.23792
\(863\) 245.223 0.00967264 0.00483632 0.999988i \(-0.498461\pi\)
0.00483632 + 0.999988i \(0.498461\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 26405.0 1.03612
\(867\) 0 0
\(868\) 18285.8 0.715048
\(869\) −8092.27 −0.315894
\(870\) 0 0
\(871\) −15604.1 −0.607032
\(872\) 14394.8 0.559026
\(873\) 0 0
\(874\) −663.952 −0.0256963
\(875\) 0 0
\(876\) 0 0
\(877\) 37727.8 1.45265 0.726326 0.687350i \(-0.241226\pi\)
0.726326 + 0.687350i \(0.241226\pi\)
\(878\) −24896.4 −0.956961
\(879\) 0 0
\(880\) 0 0
\(881\) −21738.8 −0.831326 −0.415663 0.909519i \(-0.636450\pi\)
−0.415663 + 0.909519i \(0.636450\pi\)
\(882\) 0 0
\(883\) −44340.4 −1.68989 −0.844946 0.534852i \(-0.820368\pi\)
−0.844946 + 0.534852i \(0.820368\pi\)
\(884\) −24159.1 −0.919184
\(885\) 0 0
\(886\) −10223.3 −0.387652
\(887\) 681.008 0.0257790 0.0128895 0.999917i \(-0.495897\pi\)
0.0128895 + 0.999917i \(0.495897\pi\)
\(888\) 0 0
\(889\) 8572.00 0.323392
\(890\) 0 0
\(891\) 0 0
\(892\) 17899.7 0.671889
\(893\) −6488.58 −0.243149
\(894\) 0 0
\(895\) 0 0
\(896\) −50999.6 −1.90154
\(897\) 0 0
\(898\) −32070.3 −1.19176
\(899\) 13920.7 0.516443
\(900\) 0 0
\(901\) 36589.8 1.35292
\(902\) 3237.65 0.119514
\(903\) 0 0
\(904\) 32484.6 1.19516
\(905\) 0 0
\(906\) 0 0
\(907\) 5348.01 0.195786 0.0978929 0.995197i \(-0.468790\pi\)
0.0978929 + 0.995197i \(0.468790\pi\)
\(908\) 15679.5 0.573066
\(909\) 0 0
\(910\) 0 0
\(911\) −14488.7 −0.526930 −0.263465 0.964669i \(-0.584865\pi\)
−0.263465 + 0.964669i \(0.584865\pi\)
\(912\) 0 0
\(913\) 28760.7 1.04254
\(914\) −45015.6 −1.62908
\(915\) 0 0
\(916\) −6845.05 −0.246907
\(917\) −7397.56 −0.266400
\(918\) 0 0
\(919\) −22546.9 −0.809308 −0.404654 0.914470i \(-0.632608\pi\)
−0.404654 + 0.914470i \(0.632608\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −13895.9 −0.496351
\(923\) 30321.3 1.08130
\(924\) 0 0
\(925\) 0 0
\(926\) 46944.9 1.66599
\(927\) 0 0
\(928\) 10662.7 0.377178
\(929\) 13980.7 0.493747 0.246874 0.969048i \(-0.420597\pi\)
0.246874 + 0.969048i \(0.420597\pi\)
\(930\) 0 0
\(931\) −8420.65 −0.296429
\(932\) −21283.5 −0.748032
\(933\) 0 0
\(934\) −36264.5 −1.27046
\(935\) 0 0
\(936\) 0 0
\(937\) 26362.6 0.919133 0.459567 0.888143i \(-0.348005\pi\)
0.459567 + 0.888143i \(0.348005\pi\)
\(938\) −26261.9 −0.914159
\(939\) 0 0
\(940\) 0 0
\(941\) 14715.6 0.509792 0.254896 0.966968i \(-0.417959\pi\)
0.254896 + 0.966968i \(0.417959\pi\)
\(942\) 0 0
\(943\) −419.871 −0.0144994
\(944\) 27101.9 0.934419
\(945\) 0 0
\(946\) 34587.2 1.18872
\(947\) 11818.2 0.405532 0.202766 0.979227i \(-0.435007\pi\)
0.202766 + 0.979227i \(0.435007\pi\)
\(948\) 0 0
\(949\) 1161.93 0.0397448
\(950\) 0 0
\(951\) 0 0
\(952\) 58444.2 1.98969
\(953\) 43832.3 1.48989 0.744945 0.667125i \(-0.232476\pi\)
0.744945 + 0.667125i \(0.232476\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −7334.46 −0.248131
\(957\) 0 0
\(958\) 48933.8 1.65029
\(959\) −27677.7 −0.931971
\(960\) 0 0
\(961\) 3716.00 0.124736
\(962\) −7627.86 −0.255647
\(963\) 0 0
\(964\) −7854.36 −0.262419
\(965\) 0 0
\(966\) 0 0
\(967\) 10696.2 0.355706 0.177853 0.984057i \(-0.443085\pi\)
0.177853 + 0.984057i \(0.443085\pi\)
\(968\) −5432.27 −0.180372
\(969\) 0 0
\(970\) 0 0
\(971\) −27933.0 −0.923187 −0.461593 0.887092i \(-0.652722\pi\)
−0.461593 + 0.887092i \(0.652722\pi\)
\(972\) 0 0
\(973\) −62378.4 −2.05525
\(974\) 38481.4 1.26594
\(975\) 0 0
\(976\) −49208.9 −1.61387
\(977\) −24341.7 −0.797094 −0.398547 0.917148i \(-0.630485\pi\)
−0.398547 + 0.917148i \(0.630485\pi\)
\(978\) 0 0
\(979\) −33311.7 −1.08748
\(980\) 0 0
\(981\) 0 0
\(982\) −66057.7 −2.14662
\(983\) −12553.0 −0.407301 −0.203651 0.979044i \(-0.565281\pi\)
−0.203651 + 0.979044i \(0.565281\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −30953.9 −0.999769
\(987\) 0 0
\(988\) 2878.01 0.0926737
\(989\) −4485.41 −0.144214
\(990\) 0 0
\(991\) −45631.8 −1.46271 −0.731353 0.681999i \(-0.761111\pi\)
−0.731353 + 0.681999i \(0.761111\pi\)
\(992\) 25665.0 0.821437
\(993\) 0 0
\(994\) 51031.0 1.62838
\(995\) 0 0
\(996\) 0 0
\(997\) −34499.0 −1.09588 −0.547941 0.836517i \(-0.684588\pi\)
−0.547941 + 0.836517i \(0.684588\pi\)
\(998\) −28145.3 −0.892708
\(999\) 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 225.4.a.n.1.1 2
3.2 odd 2 75.4.a.d.1.2 2
5.2 odd 4 225.4.b.h.199.2 4
5.3 odd 4 225.4.b.h.199.3 4
5.4 even 2 225.4.a.j.1.2 2
12.11 even 2 1200.4.a.bl.1.1 2
15.2 even 4 75.4.b.c.49.3 4
15.8 even 4 75.4.b.c.49.2 4
15.14 odd 2 75.4.a.e.1.1 yes 2
60.23 odd 4 1200.4.f.v.49.2 4
60.47 odd 4 1200.4.f.v.49.3 4
60.59 even 2 1200.4.a.bu.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.4.a.d.1.2 2 3.2 odd 2
75.4.a.e.1.1 yes 2 15.14 odd 2
75.4.b.c.49.2 4 15.8 even 4
75.4.b.c.49.3 4 15.2 even 4
225.4.a.j.1.2 2 5.4 even 2
225.4.a.n.1.1 2 1.1 even 1 trivial
225.4.b.h.199.2 4 5.2 odd 4
225.4.b.h.199.3 4 5.3 odd 4
1200.4.a.bl.1.1 2 12.11 even 2
1200.4.a.bu.1.2 2 60.59 even 2
1200.4.f.v.49.2 4 60.23 odd 4
1200.4.f.v.49.3 4 60.47 odd 4