Properties

Label 225.4.a.o.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: \(\Q(\sqrt{41}) \)
Defining polynomial: \(x^{2} - x - 10\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.70156\) of defining polynomial
Character \(\chi\) \(=\) 225.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.70156 q^{2} -5.10469 q^{4} -22.2094 q^{7} +22.2984 q^{8} +O(q^{10})\) \(q-1.70156 q^{2} -5.10469 q^{4} -22.2094 q^{7} +22.2984 q^{8} +1.79063 q^{11} -58.2094 q^{13} +37.7906 q^{14} +2.89531 q^{16} +18.9844 q^{17} +104.837 q^{19} -3.04686 q^{22} -49.6125 q^{23} +99.0469 q^{26} +113.372 q^{28} +293.466 q^{29} +64.4187 q^{31} -183.314 q^{32} -32.3031 q^{34} +19.8844 q^{37} -178.388 q^{38} +165.581 q^{41} +247.350 q^{43} -9.14059 q^{44} +84.4187 q^{46} +384.544 q^{47} +150.256 q^{49} +297.141 q^{52} +463.528 q^{53} -495.234 q^{56} -499.350 q^{58} +73.7906 q^{59} -137.350 q^{61} -109.612 q^{62} +288.758 q^{64} -173.906 q^{67} -96.9093 q^{68} +594.281 q^{71} -320.231 q^{73} -33.8345 q^{74} -535.163 q^{76} -39.7687 q^{77} -770.469 q^{79} -281.747 q^{82} +173.925 q^{83} -420.881 q^{86} +39.9282 q^{88} -1019.02 q^{89} +1292.79 q^{91} +253.256 q^{92} -654.325 q^{94} -384.375 q^{97} -255.670 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 9 q^{4} - 6 q^{7} + 51 q^{8} + O(q^{10}) \) \( 2 q + 3 q^{2} + 9 q^{4} - 6 q^{7} + 51 q^{8} + 42 q^{11} - 78 q^{13} + 114 q^{14} + 25 q^{16} + 102 q^{17} + 56 q^{19} + 186 q^{22} - 48 q^{23} + 6 q^{26} + 342 q^{28} + 318 q^{29} + 52 q^{31} - 309 q^{32} + 358 q^{34} - 306 q^{37} - 408 q^{38} + 408 q^{41} - 120 q^{43} + 558 q^{44} + 92 q^{46} + 180 q^{47} + 70 q^{49} + 18 q^{52} + 402 q^{53} - 30 q^{56} - 384 q^{58} + 186 q^{59} + 340 q^{61} - 168 q^{62} - 479 q^{64} - 732 q^{67} + 1074 q^{68} + 36 q^{71} - 1332 q^{73} - 1566 q^{74} - 1224 q^{76} + 612 q^{77} + 380 q^{79} + 858 q^{82} - 984 q^{83} - 2148 q^{86} + 1194 q^{88} - 1116 q^{89} + 972 q^{91} + 276 q^{92} - 1616 q^{94} + 768 q^{97} - 633 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.70156 −0.601593 −0.300797 0.953688i \(-0.597253\pi\)
−0.300797 + 0.953688i \(0.597253\pi\)
\(3\) 0 0
\(4\) −5.10469 −0.638086
\(5\) 0 0
\(6\) 0 0
\(7\) −22.2094 −1.19919 −0.599597 0.800302i \(-0.704672\pi\)
−0.599597 + 0.800302i \(0.704672\pi\)
\(8\) 22.2984 0.985461
\(9\) 0 0
\(10\) 0 0
\(11\) 1.79063 0.0490813 0.0245407 0.999699i \(-0.492188\pi\)
0.0245407 + 0.999699i \(0.492188\pi\)
\(12\) 0 0
\(13\) −58.2094 −1.24188 −0.620938 0.783860i \(-0.713248\pi\)
−0.620938 + 0.783860i \(0.713248\pi\)
\(14\) 37.7906 0.721426
\(15\) 0 0
\(16\) 2.89531 0.0452393
\(17\) 18.9844 0.270846 0.135423 0.990788i \(-0.456761\pi\)
0.135423 + 0.990788i \(0.456761\pi\)
\(18\) 0 0
\(19\) 104.837 1.26586 0.632931 0.774208i \(-0.281852\pi\)
0.632931 + 0.774208i \(0.281852\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −3.04686 −0.0295270
\(23\) −49.6125 −0.449779 −0.224890 0.974384i \(-0.572202\pi\)
−0.224890 + 0.974384i \(0.572202\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 99.0469 0.747103
\(27\) 0 0
\(28\) 113.372 0.765188
\(29\) 293.466 1.87914 0.939572 0.342350i \(-0.111223\pi\)
0.939572 + 0.342350i \(0.111223\pi\)
\(30\) 0 0
\(31\) 64.4187 0.373224 0.186612 0.982434i \(-0.440249\pi\)
0.186612 + 0.982434i \(0.440249\pi\)
\(32\) −183.314 −1.01268
\(33\) 0 0
\(34\) −32.3031 −0.162939
\(35\) 0 0
\(36\) 0 0
\(37\) 19.8844 0.0883505 0.0441752 0.999024i \(-0.485934\pi\)
0.0441752 + 0.999024i \(0.485934\pi\)
\(38\) −178.388 −0.761534
\(39\) 0 0
\(40\) 0 0
\(41\) 165.581 0.630718 0.315359 0.948972i \(-0.397875\pi\)
0.315359 + 0.948972i \(0.397875\pi\)
\(42\) 0 0
\(43\) 247.350 0.877221 0.438611 0.898677i \(-0.355471\pi\)
0.438611 + 0.898677i \(0.355471\pi\)
\(44\) −9.14059 −0.0313181
\(45\) 0 0
\(46\) 84.4187 0.270584
\(47\) 384.544 1.19344 0.596718 0.802451i \(-0.296471\pi\)
0.596718 + 0.802451i \(0.296471\pi\)
\(48\) 0 0
\(49\) 150.256 0.438065
\(50\) 0 0
\(51\) 0 0
\(52\) 297.141 0.792423
\(53\) 463.528 1.20133 0.600665 0.799501i \(-0.294903\pi\)
0.600665 + 0.799501i \(0.294903\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −495.234 −1.18176
\(57\) 0 0
\(58\) −499.350 −1.13048
\(59\) 73.7906 0.162826 0.0814129 0.996680i \(-0.474057\pi\)
0.0814129 + 0.996680i \(0.474057\pi\)
\(60\) 0 0
\(61\) −137.350 −0.288293 −0.144146 0.989556i \(-0.546044\pi\)
−0.144146 + 0.989556i \(0.546044\pi\)
\(62\) −109.612 −0.224529
\(63\) 0 0
\(64\) 288.758 0.563980
\(65\) 0 0
\(66\) 0 0
\(67\) −173.906 −0.317105 −0.158552 0.987351i \(-0.550683\pi\)
−0.158552 + 0.987351i \(0.550683\pi\)
\(68\) −96.9093 −0.172823
\(69\) 0 0
\(70\) 0 0
\(71\) 594.281 0.993355 0.496677 0.867935i \(-0.334553\pi\)
0.496677 + 0.867935i \(0.334553\pi\)
\(72\) 0 0
\(73\) −320.231 −0.513428 −0.256714 0.966487i \(-0.582640\pi\)
−0.256714 + 0.966487i \(0.582640\pi\)
\(74\) −33.8345 −0.0531510
\(75\) 0 0
\(76\) −535.163 −0.807728
\(77\) −39.7687 −0.0588580
\(78\) 0 0
\(79\) −770.469 −1.09727 −0.548636 0.836061i \(-0.684853\pi\)
−0.548636 + 0.836061i \(0.684853\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −281.747 −0.379436
\(83\) 173.925 0.230009 0.115004 0.993365i \(-0.463312\pi\)
0.115004 + 0.993365i \(0.463312\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −420.881 −0.527730
\(87\) 0 0
\(88\) 39.9282 0.0483677
\(89\) −1019.02 −1.21367 −0.606834 0.794829i \(-0.707561\pi\)
−0.606834 + 0.794829i \(0.707561\pi\)
\(90\) 0 0
\(91\) 1292.79 1.48925
\(92\) 253.256 0.286998
\(93\) 0 0
\(94\) −654.325 −0.717962
\(95\) 0 0
\(96\) 0 0
\(97\) −384.375 −0.402344 −0.201172 0.979556i \(-0.564475\pi\)
−0.201172 + 0.979556i \(0.564475\pi\)
\(98\) −255.670 −0.263537
\(99\) 0 0
\(100\) 0 0
\(101\) −34.4906 −0.0339796 −0.0169898 0.999856i \(-0.505408\pi\)
−0.0169898 + 0.999856i \(0.505408\pi\)
\(102\) 0 0
\(103\) 1756.30 1.68013 0.840066 0.542484i \(-0.182516\pi\)
0.840066 + 0.542484i \(0.182516\pi\)
\(104\) −1297.98 −1.22382
\(105\) 0 0
\(106\) −788.722 −0.722712
\(107\) −1361.74 −1.23032 −0.615159 0.788403i \(-0.710908\pi\)
−0.615159 + 0.788403i \(0.710908\pi\)
\(108\) 0 0
\(109\) 321.119 0.282180 0.141090 0.989997i \(-0.454939\pi\)
0.141090 + 0.989997i \(0.454939\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −64.3031 −0.0542506
\(113\) −1582.25 −1.31721 −0.658607 0.752487i \(-0.728854\pi\)
−0.658607 + 0.752487i \(0.728854\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1498.05 −1.19906
\(117\) 0 0
\(118\) −125.559 −0.0979549
\(119\) −421.631 −0.324797
\(120\) 0 0
\(121\) −1327.79 −0.997591
\(122\) 233.709 0.173435
\(123\) 0 0
\(124\) −328.837 −0.238149
\(125\) 0 0
\(126\) 0 0
\(127\) 1197.14 0.836449 0.418225 0.908344i \(-0.362652\pi\)
0.418225 + 0.908344i \(0.362652\pi\)
\(128\) 975.173 0.673390
\(129\) 0 0
\(130\) 0 0
\(131\) 321.647 0.214522 0.107261 0.994231i \(-0.465792\pi\)
0.107261 + 0.994231i \(0.465792\pi\)
\(132\) 0 0
\(133\) −2328.37 −1.51801
\(134\) 295.912 0.190768
\(135\) 0 0
\(136\) 423.322 0.266909
\(137\) 354.291 0.220942 0.110471 0.993879i \(-0.464764\pi\)
0.110471 + 0.993879i \(0.464764\pi\)
\(138\) 0 0
\(139\) 77.2562 0.0471424 0.0235712 0.999722i \(-0.492496\pi\)
0.0235712 + 0.999722i \(0.492496\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1011.21 −0.597595
\(143\) −104.231 −0.0609529
\(144\) 0 0
\(145\) 0 0
\(146\) 544.893 0.308875
\(147\) 0 0
\(148\) −101.503 −0.0563752
\(149\) 1705.38 0.937651 0.468826 0.883291i \(-0.344677\pi\)
0.468826 + 0.883291i \(0.344677\pi\)
\(150\) 0 0
\(151\) 758.281 0.408663 0.204331 0.978902i \(-0.434498\pi\)
0.204331 + 0.978902i \(0.434498\pi\)
\(152\) 2337.71 1.24746
\(153\) 0 0
\(154\) 67.6689 0.0354086
\(155\) 0 0
\(156\) 0 0
\(157\) 1769.05 0.899273 0.449636 0.893212i \(-0.351554\pi\)
0.449636 + 0.893212i \(0.351554\pi\)
\(158\) 1311.00 0.660111
\(159\) 0 0
\(160\) 0 0
\(161\) 1101.86 0.539372
\(162\) 0 0
\(163\) 881.719 0.423690 0.211845 0.977303i \(-0.432053\pi\)
0.211845 + 0.977303i \(0.432053\pi\)
\(164\) −845.240 −0.402452
\(165\) 0 0
\(166\) −295.944 −0.138372
\(167\) −216.900 −0.100504 −0.0502522 0.998737i \(-0.516003\pi\)
−0.0502522 + 0.998737i \(0.516003\pi\)
\(168\) 0 0
\(169\) 1191.33 0.542254
\(170\) 0 0
\(171\) 0 0
\(172\) −1262.64 −0.559742
\(173\) 4125.91 1.81322 0.906610 0.421970i \(-0.138661\pi\)
0.906610 + 0.421970i \(0.138661\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 5.18443 0.00222040
\(177\) 0 0
\(178\) 1733.93 0.730134
\(179\) 3213.14 1.34168 0.670842 0.741600i \(-0.265933\pi\)
0.670842 + 0.741600i \(0.265933\pi\)
\(180\) 0 0
\(181\) 3394.42 1.39395 0.696976 0.717095i \(-0.254529\pi\)
0.696976 + 0.717095i \(0.254529\pi\)
\(182\) −2199.77 −0.895921
\(183\) 0 0
\(184\) −1106.28 −0.443240
\(185\) 0 0
\(186\) 0 0
\(187\) 33.9939 0.0132935
\(188\) −1962.98 −0.761514
\(189\) 0 0
\(190\) 0 0
\(191\) 3467.49 1.31361 0.656804 0.754062i \(-0.271908\pi\)
0.656804 + 0.754062i \(0.271908\pi\)
\(192\) 0 0
\(193\) −1792.14 −0.668401 −0.334200 0.942502i \(-0.608466\pi\)
−0.334200 + 0.942502i \(0.608466\pi\)
\(194\) 654.038 0.242047
\(195\) 0 0
\(196\) −767.011 −0.279523
\(197\) 1678.19 0.606935 0.303467 0.952842i \(-0.401856\pi\)
0.303467 + 0.952842i \(0.401856\pi\)
\(198\) 0 0
\(199\) 3108.23 1.10722 0.553610 0.832776i \(-0.313250\pi\)
0.553610 + 0.832776i \(0.313250\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 58.6878 0.0204419
\(203\) −6517.69 −2.25346
\(204\) 0 0
\(205\) 0 0
\(206\) −2988.46 −1.01076
\(207\) 0 0
\(208\) −168.534 −0.0561815
\(209\) 187.725 0.0621301
\(210\) 0 0
\(211\) −4473.27 −1.45949 −0.729745 0.683719i \(-0.760361\pi\)
−0.729745 + 0.683719i \(0.760361\pi\)
\(212\) −2366.17 −0.766551
\(213\) 0 0
\(214\) 2317.08 0.740151
\(215\) 0 0
\(216\) 0 0
\(217\) −1430.70 −0.447568
\(218\) −546.403 −0.169757
\(219\) 0 0
\(220\) 0 0
\(221\) −1105.07 −0.336357
\(222\) 0 0
\(223\) 1753.42 0.526535 0.263268 0.964723i \(-0.415200\pi\)
0.263268 + 0.964723i \(0.415200\pi\)
\(224\) 4071.29 1.21440
\(225\) 0 0
\(226\) 2692.29 0.792427
\(227\) −936.900 −0.273939 −0.136970 0.990575i \(-0.543736\pi\)
−0.136970 + 0.990575i \(0.543736\pi\)
\(228\) 0 0
\(229\) −2582.06 −0.745096 −0.372548 0.928013i \(-0.621516\pi\)
−0.372548 + 0.928013i \(0.621516\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6543.82 1.85182
\(233\) 2295.01 0.645284 0.322642 0.946521i \(-0.395429\pi\)
0.322642 + 0.946521i \(0.395429\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −376.678 −0.103897
\(237\) 0 0
\(238\) 717.432 0.195396
\(239\) 2294.01 0.620866 0.310433 0.950595i \(-0.399526\pi\)
0.310433 + 0.950595i \(0.399526\pi\)
\(240\) 0 0
\(241\) 382.287 0.102180 0.0510898 0.998694i \(-0.483731\pi\)
0.0510898 + 0.998694i \(0.483731\pi\)
\(242\) 2259.32 0.600144
\(243\) 0 0
\(244\) 701.128 0.183956
\(245\) 0 0
\(246\) 0 0
\(247\) −6102.52 −1.57204
\(248\) 1436.44 0.367798
\(249\) 0 0
\(250\) 0 0
\(251\) 2259.98 0.568322 0.284161 0.958777i \(-0.408285\pi\)
0.284161 + 0.958777i \(0.408285\pi\)
\(252\) 0 0
\(253\) −88.8375 −0.0220758
\(254\) −2037.01 −0.503202
\(255\) 0 0
\(256\) −3969.38 −0.969087
\(257\) −92.7843 −0.0225203 −0.0112602 0.999937i \(-0.503584\pi\)
−0.0112602 + 0.999937i \(0.503584\pi\)
\(258\) 0 0
\(259\) −441.619 −0.105949
\(260\) 0 0
\(261\) 0 0
\(262\) −547.302 −0.129055
\(263\) −568.312 −0.133246 −0.0666229 0.997778i \(-0.521222\pi\)
−0.0666229 + 0.997778i \(0.521222\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 3961.87 0.913226
\(267\) 0 0
\(268\) 887.737 0.202340
\(269\) −7582.41 −1.71862 −0.859309 0.511458i \(-0.829106\pi\)
−0.859309 + 0.511458i \(0.829106\pi\)
\(270\) 0 0
\(271\) 7943.69 1.78061 0.890304 0.455366i \(-0.150492\pi\)
0.890304 + 0.455366i \(0.150492\pi\)
\(272\) 54.9657 0.0122529
\(273\) 0 0
\(274\) −602.847 −0.132917
\(275\) 0 0
\(276\) 0 0
\(277\) −6823.00 −1.47998 −0.739990 0.672618i \(-0.765170\pi\)
−0.739990 + 0.672618i \(0.765170\pi\)
\(278\) −131.456 −0.0283605
\(279\) 0 0
\(280\) 0 0
\(281\) −3315.86 −0.703942 −0.351971 0.936011i \(-0.614488\pi\)
−0.351971 + 0.936011i \(0.614488\pi\)
\(282\) 0 0
\(283\) −6602.76 −1.38690 −0.693451 0.720504i \(-0.743910\pi\)
−0.693451 + 0.720504i \(0.743910\pi\)
\(284\) −3033.62 −0.633846
\(285\) 0 0
\(286\) 177.356 0.0366688
\(287\) −3677.46 −0.756353
\(288\) 0 0
\(289\) −4552.59 −0.926642
\(290\) 0 0
\(291\) 0 0
\(292\) 1634.68 0.327611
\(293\) 5814.14 1.15927 0.579634 0.814877i \(-0.303195\pi\)
0.579634 + 0.814877i \(0.303195\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 443.390 0.0870660
\(297\) 0 0
\(298\) −2901.81 −0.564084
\(299\) 2887.91 0.558570
\(300\) 0 0
\(301\) −5493.49 −1.05196
\(302\) −1290.26 −0.245849
\(303\) 0 0
\(304\) 303.537 0.0572667
\(305\) 0 0
\(306\) 0 0
\(307\) 8124.86 1.51046 0.755229 0.655462i \(-0.227526\pi\)
0.755229 + 0.655462i \(0.227526\pi\)
\(308\) 203.007 0.0375564
\(309\) 0 0
\(310\) 0 0
\(311\) −7336.26 −1.33762 −0.668812 0.743432i \(-0.733197\pi\)
−0.668812 + 0.743432i \(0.733197\pi\)
\(312\) 0 0
\(313\) 2202.66 0.397768 0.198884 0.980023i \(-0.436268\pi\)
0.198884 + 0.980023i \(0.436268\pi\)
\(314\) −3010.15 −0.540996
\(315\) 0 0
\(316\) 3933.00 0.700154
\(317\) 10008.9 1.77336 0.886679 0.462386i \(-0.153007\pi\)
0.886679 + 0.462386i \(0.153007\pi\)
\(318\) 0 0
\(319\) 525.488 0.0922309
\(320\) 0 0
\(321\) 0 0
\(322\) −1874.89 −0.324483
\(323\) 1990.27 0.342854
\(324\) 0 0
\(325\) 0 0
\(326\) −1500.30 −0.254889
\(327\) 0 0
\(328\) 3692.20 0.621548
\(329\) −8540.47 −1.43116
\(330\) 0 0
\(331\) −8695.94 −1.44402 −0.722012 0.691881i \(-0.756782\pi\)
−0.722012 + 0.691881i \(0.756782\pi\)
\(332\) −887.832 −0.146765
\(333\) 0 0
\(334\) 369.069 0.0604627
\(335\) 0 0
\(336\) 0 0
\(337\) 7400.61 1.19625 0.598126 0.801402i \(-0.295912\pi\)
0.598126 + 0.801402i \(0.295912\pi\)
\(338\) −2027.12 −0.326216
\(339\) 0 0
\(340\) 0 0
\(341\) 115.350 0.0183183
\(342\) 0 0
\(343\) 4280.72 0.673869
\(344\) 5515.52 0.864467
\(345\) 0 0
\(346\) −7020.49 −1.09082
\(347\) 7841.44 1.21311 0.606557 0.795040i \(-0.292550\pi\)
0.606557 + 0.795040i \(0.292550\pi\)
\(348\) 0 0
\(349\) −4961.26 −0.760946 −0.380473 0.924792i \(-0.624239\pi\)
−0.380473 + 0.924792i \(0.624239\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −328.247 −0.0497035
\(353\) −12163.0 −1.83392 −0.916959 0.398981i \(-0.869364\pi\)
−0.916959 + 0.398981i \(0.869364\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 5201.80 0.774424
\(357\) 0 0
\(358\) −5467.36 −0.807148
\(359\) −5193.79 −0.763559 −0.381779 0.924253i \(-0.624689\pi\)
−0.381779 + 0.924253i \(0.624689\pi\)
\(360\) 0 0
\(361\) 4131.90 0.602406
\(362\) −5775.81 −0.838591
\(363\) 0 0
\(364\) −6599.31 −0.950268
\(365\) 0 0
\(366\) 0 0
\(367\) 6086.09 0.865644 0.432822 0.901479i \(-0.357518\pi\)
0.432822 + 0.901479i \(0.357518\pi\)
\(368\) −143.644 −0.0203477
\(369\) 0 0
\(370\) 0 0
\(371\) −10294.7 −1.44063
\(372\) 0 0
\(373\) 10581.9 1.46893 0.734466 0.678646i \(-0.237433\pi\)
0.734466 + 0.678646i \(0.237433\pi\)
\(374\) −57.8428 −0.00799727
\(375\) 0 0
\(376\) 8574.72 1.17608
\(377\) −17082.4 −2.33366
\(378\) 0 0
\(379\) 11655.2 1.57964 0.789822 0.613336i \(-0.210173\pi\)
0.789822 + 0.613336i \(0.210173\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −5900.16 −0.790257
\(383\) 6364.97 0.849177 0.424588 0.905387i \(-0.360419\pi\)
0.424588 + 0.905387i \(0.360419\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3049.44 0.402105
\(387\) 0 0
\(388\) 1962.11 0.256730
\(389\) −6134.33 −0.799545 −0.399773 0.916614i \(-0.630911\pi\)
−0.399773 + 0.916614i \(0.630911\pi\)
\(390\) 0 0
\(391\) −941.862 −0.121821
\(392\) 3350.48 0.431696
\(393\) 0 0
\(394\) −2855.55 −0.365128
\(395\) 0 0
\(396\) 0 0
\(397\) −9746.46 −1.23214 −0.616072 0.787690i \(-0.711277\pi\)
−0.616072 + 0.787690i \(0.711277\pi\)
\(398\) −5288.85 −0.666096
\(399\) 0 0
\(400\) 0 0
\(401\) 1306.44 0.162695 0.0813474 0.996686i \(-0.474078\pi\)
0.0813474 + 0.996686i \(0.474078\pi\)
\(402\) 0 0
\(403\) −3749.77 −0.463498
\(404\) 176.063 0.0216819
\(405\) 0 0
\(406\) 11090.2 1.35566
\(407\) 35.6055 0.00433636
\(408\) 0 0
\(409\) −3876.93 −0.468709 −0.234354 0.972151i \(-0.575298\pi\)
−0.234354 + 0.972151i \(0.575298\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −8965.38 −1.07207
\(413\) −1638.84 −0.195260
\(414\) 0 0
\(415\) 0 0
\(416\) 10670.6 1.25762
\(417\) 0 0
\(418\) −319.426 −0.0373771
\(419\) 16022.5 1.86814 0.934071 0.357088i \(-0.116230\pi\)
0.934071 + 0.357088i \(0.116230\pi\)
\(420\) 0 0
\(421\) −8119.73 −0.939980 −0.469990 0.882672i \(-0.655742\pi\)
−0.469990 + 0.882672i \(0.655742\pi\)
\(422\) 7611.54 0.878019
\(423\) 0 0
\(424\) 10336.0 1.18386
\(425\) 0 0
\(426\) 0 0
\(427\) 3050.46 0.345719
\(428\) 6951.24 0.785049
\(429\) 0 0
\(430\) 0 0
\(431\) 5713.99 0.638592 0.319296 0.947655i \(-0.396554\pi\)
0.319296 + 0.947655i \(0.396554\pi\)
\(432\) 0 0
\(433\) −6251.34 −0.693811 −0.346906 0.937900i \(-0.612768\pi\)
−0.346906 + 0.937900i \(0.612768\pi\)
\(434\) 2434.42 0.269254
\(435\) 0 0
\(436\) −1639.21 −0.180055
\(437\) −5201.25 −0.569358
\(438\) 0 0
\(439\) −4230.97 −0.459984 −0.229992 0.973192i \(-0.573870\pi\)
−0.229992 + 0.973192i \(0.573870\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1880.34 0.202350
\(443\) 6314.29 0.677203 0.338601 0.940930i \(-0.390046\pi\)
0.338601 + 0.940930i \(0.390046\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −2983.55 −0.316760
\(447\) 0 0
\(448\) −6413.13 −0.676321
\(449\) 9349.71 0.982717 0.491358 0.870957i \(-0.336501\pi\)
0.491358 + 0.870957i \(0.336501\pi\)
\(450\) 0 0
\(451\) 296.494 0.0309565
\(452\) 8076.87 0.840496
\(453\) 0 0
\(454\) 1594.19 0.164800
\(455\) 0 0
\(456\) 0 0
\(457\) 9547.46 0.977268 0.488634 0.872489i \(-0.337495\pi\)
0.488634 + 0.872489i \(0.337495\pi\)
\(458\) 4393.53 0.448245
\(459\) 0 0
\(460\) 0 0
\(461\) −6237.23 −0.630145 −0.315073 0.949068i \(-0.602029\pi\)
−0.315073 + 0.949068i \(0.602029\pi\)
\(462\) 0 0
\(463\) −6469.98 −0.649428 −0.324714 0.945812i \(-0.605268\pi\)
−0.324714 + 0.945812i \(0.605268\pi\)
\(464\) 849.675 0.0850111
\(465\) 0 0
\(466\) −3905.10 −0.388198
\(467\) 7206.64 0.714097 0.357049 0.934086i \(-0.383783\pi\)
0.357049 + 0.934086i \(0.383783\pi\)
\(468\) 0 0
\(469\) 3862.35 0.380270
\(470\) 0 0
\(471\) 0 0
\(472\) 1645.42 0.160458
\(473\) 442.912 0.0430552
\(474\) 0 0
\(475\) 0 0
\(476\) 2152.29 0.207248
\(477\) 0 0
\(478\) −3903.39 −0.373509
\(479\) 10851.8 1.03514 0.517571 0.855640i \(-0.326836\pi\)
0.517571 + 0.855640i \(0.326836\pi\)
\(480\) 0 0
\(481\) −1157.46 −0.109720
\(482\) −650.485 −0.0614705
\(483\) 0 0
\(484\) 6777.97 0.636549
\(485\) 0 0
\(486\) 0 0
\(487\) −12757.1 −1.18702 −0.593510 0.804827i \(-0.702258\pi\)
−0.593510 + 0.804827i \(0.702258\pi\)
\(488\) −3062.69 −0.284101
\(489\) 0 0
\(490\) 0 0
\(491\) 7016.52 0.644911 0.322455 0.946585i \(-0.395492\pi\)
0.322455 + 0.946585i \(0.395492\pi\)
\(492\) 0 0
\(493\) 5571.26 0.508960
\(494\) 10383.8 0.945729
\(495\) 0 0
\(496\) 186.512 0.0168844
\(497\) −13198.6 −1.19122
\(498\) 0 0
\(499\) 11372.3 1.02023 0.510113 0.860107i \(-0.329604\pi\)
0.510113 + 0.860107i \(0.329604\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −3845.50 −0.341899
\(503\) −5587.37 −0.495285 −0.247643 0.968851i \(-0.579656\pi\)
−0.247643 + 0.968851i \(0.579656\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 151.163 0.0132806
\(507\) 0 0
\(508\) −6111.03 −0.533726
\(509\) −16256.7 −1.41565 −0.707825 0.706388i \(-0.750324\pi\)
−0.707825 + 0.706388i \(0.750324\pi\)
\(510\) 0 0
\(511\) 7112.14 0.615699
\(512\) −1047.24 −0.0903943
\(513\) 0 0
\(514\) 157.878 0.0135481
\(515\) 0 0
\(516\) 0 0
\(517\) 688.574 0.0585754
\(518\) 751.442 0.0637384
\(519\) 0 0
\(520\) 0 0
\(521\) −19748.4 −1.66064 −0.830320 0.557286i \(-0.811843\pi\)
−0.830320 + 0.557286i \(0.811843\pi\)
\(522\) 0 0
\(523\) 7843.44 0.655774 0.327887 0.944717i \(-0.393663\pi\)
0.327887 + 0.944717i \(0.393663\pi\)
\(524\) −1641.91 −0.136884
\(525\) 0 0
\(526\) 967.019 0.0801597
\(527\) 1222.95 0.101086
\(528\) 0 0
\(529\) −9705.60 −0.797699
\(530\) 0 0
\(531\) 0 0
\(532\) 11885.6 0.968622
\(533\) −9638.38 −0.783273
\(534\) 0 0
\(535\) 0 0
\(536\) −3877.84 −0.312495
\(537\) 0 0
\(538\) 12902.0 1.03391
\(539\) 269.053 0.0215008
\(540\) 0 0
\(541\) 7383.29 0.586751 0.293376 0.955997i \(-0.405221\pi\)
0.293376 + 0.955997i \(0.405221\pi\)
\(542\) −13516.7 −1.07120
\(543\) 0 0
\(544\) −3480.10 −0.274280
\(545\) 0 0
\(546\) 0 0
\(547\) 3354.90 0.262240 0.131120 0.991367i \(-0.458143\pi\)
0.131120 + 0.991367i \(0.458143\pi\)
\(548\) −1808.54 −0.140980
\(549\) 0 0
\(550\) 0 0
\(551\) 30766.2 2.37874
\(552\) 0 0
\(553\) 17111.6 1.31584
\(554\) 11609.8 0.890346
\(555\) 0 0
\(556\) −394.369 −0.0300809
\(557\) 20771.8 1.58012 0.790061 0.613028i \(-0.210049\pi\)
0.790061 + 0.613028i \(0.210049\pi\)
\(558\) 0 0
\(559\) −14398.1 −1.08940
\(560\) 0 0
\(561\) 0 0
\(562\) 5642.15 0.423487
\(563\) 7194.86 0.538592 0.269296 0.963057i \(-0.413209\pi\)
0.269296 + 0.963057i \(0.413209\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 11235.0 0.834350
\(567\) 0 0
\(568\) 13251.5 0.978913
\(569\) −11549.5 −0.850931 −0.425466 0.904975i \(-0.639890\pi\)
−0.425466 + 0.904975i \(0.639890\pi\)
\(570\) 0 0
\(571\) 1482.54 0.108655 0.0543277 0.998523i \(-0.482698\pi\)
0.0543277 + 0.998523i \(0.482698\pi\)
\(572\) 532.068 0.0388932
\(573\) 0 0
\(574\) 6257.42 0.455017
\(575\) 0 0
\(576\) 0 0
\(577\) 15264.0 1.10130 0.550649 0.834737i \(-0.314380\pi\)
0.550649 + 0.834737i \(0.314380\pi\)
\(578\) 7746.52 0.557462
\(579\) 0 0
\(580\) 0 0
\(581\) −3862.76 −0.275825
\(582\) 0 0
\(583\) 830.006 0.0589628
\(584\) −7140.66 −0.505963
\(585\) 0 0
\(586\) −9893.12 −0.697408
\(587\) 1736.89 0.122128 0.0610639 0.998134i \(-0.480551\pi\)
0.0610639 + 0.998134i \(0.480551\pi\)
\(588\) 0 0
\(589\) 6753.50 0.472450
\(590\) 0 0
\(591\) 0 0
\(592\) 57.5714 0.00399691
\(593\) −11764.8 −0.814707 −0.407353 0.913271i \(-0.633548\pi\)
−0.407353 + 0.913271i \(0.633548\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −8705.42 −0.598302
\(597\) 0 0
\(598\) −4913.96 −0.336032
\(599\) 9451.99 0.644737 0.322369 0.946614i \(-0.395521\pi\)
0.322369 + 0.946614i \(0.395521\pi\)
\(600\) 0 0
\(601\) −3131.93 −0.212569 −0.106285 0.994336i \(-0.533895\pi\)
−0.106285 + 0.994336i \(0.533895\pi\)
\(602\) 9347.51 0.632851
\(603\) 0 0
\(604\) −3870.79 −0.260762
\(605\) 0 0
\(606\) 0 0
\(607\) −22700.8 −1.51795 −0.758975 0.651120i \(-0.774300\pi\)
−0.758975 + 0.651120i \(0.774300\pi\)
\(608\) −19218.2 −1.28191
\(609\) 0 0
\(610\) 0 0
\(611\) −22384.0 −1.48210
\(612\) 0 0
\(613\) −28911.6 −1.90494 −0.952471 0.304629i \(-0.901468\pi\)
−0.952471 + 0.304629i \(0.901468\pi\)
\(614\) −13825.0 −0.908681
\(615\) 0 0
\(616\) −886.780 −0.0580023
\(617\) −5566.87 −0.363231 −0.181616 0.983370i \(-0.558133\pi\)
−0.181616 + 0.983370i \(0.558133\pi\)
\(618\) 0 0
\(619\) 4150.32 0.269492 0.134746 0.990880i \(-0.456978\pi\)
0.134746 + 0.990880i \(0.456978\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 12483.1 0.804705
\(623\) 22631.9 1.45542
\(624\) 0 0
\(625\) 0 0
\(626\) −3747.96 −0.239295
\(627\) 0 0
\(628\) −9030.46 −0.573813
\(629\) 377.492 0.0239294
\(630\) 0 0
\(631\) −4090.09 −0.258041 −0.129021 0.991642i \(-0.541183\pi\)
−0.129021 + 0.991642i \(0.541183\pi\)
\(632\) −17180.2 −1.08132
\(633\) 0 0
\(634\) −17030.7 −1.06684
\(635\) 0 0
\(636\) 0 0
\(637\) −8746.32 −0.544022
\(638\) −894.150 −0.0554855
\(639\) 0 0
\(640\) 0 0
\(641\) −3909.35 −0.240890 −0.120445 0.992720i \(-0.538432\pi\)
−0.120445 + 0.992720i \(0.538432\pi\)
\(642\) 0 0
\(643\) −30539.5 −1.87303 −0.936516 0.350624i \(-0.885969\pi\)
−0.936516 + 0.350624i \(0.885969\pi\)
\(644\) −5624.66 −0.344166
\(645\) 0 0
\(646\) −3386.58 −0.206259
\(647\) −12707.7 −0.772167 −0.386083 0.922464i \(-0.626172\pi\)
−0.386083 + 0.922464i \(0.626172\pi\)
\(648\) 0 0
\(649\) 132.132 0.00799170
\(650\) 0 0
\(651\) 0 0
\(652\) −4500.90 −0.270351
\(653\) 12777.6 0.765737 0.382869 0.923803i \(-0.374936\pi\)
0.382869 + 0.923803i \(0.374936\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 479.410 0.0285332
\(657\) 0 0
\(658\) 14532.1 0.860976
\(659\) −23563.5 −1.39287 −0.696435 0.717620i \(-0.745232\pi\)
−0.696435 + 0.717620i \(0.745232\pi\)
\(660\) 0 0
\(661\) −4361.31 −0.256634 −0.128317 0.991733i \(-0.540958\pi\)
−0.128317 + 0.991733i \(0.540958\pi\)
\(662\) 14796.7 0.868715
\(663\) 0 0
\(664\) 3878.25 0.226665
\(665\) 0 0
\(666\) 0 0
\(667\) −14559.6 −0.845200
\(668\) 1107.21 0.0641304
\(669\) 0 0
\(670\) 0 0
\(671\) −245.943 −0.0141498
\(672\) 0 0
\(673\) 8203.52 0.469870 0.234935 0.972011i \(-0.424512\pi\)
0.234935 + 0.972011i \(0.424512\pi\)
\(674\) −12592.6 −0.719657
\(675\) 0 0
\(676\) −6081.37 −0.346004
\(677\) 28057.1 1.59279 0.796397 0.604774i \(-0.206737\pi\)
0.796397 + 0.604774i \(0.206737\pi\)
\(678\) 0 0
\(679\) 8536.73 0.482488
\(680\) 0 0
\(681\) 0 0
\(682\) −196.275 −0.0110202
\(683\) −3344.62 −0.187377 −0.0936885 0.995602i \(-0.529866\pi\)
−0.0936885 + 0.995602i \(0.529866\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −7283.91 −0.405395
\(687\) 0 0
\(688\) 716.156 0.0396849
\(689\) −26981.7 −1.49190
\(690\) 0 0
\(691\) 12964.8 0.713757 0.356879 0.934151i \(-0.383841\pi\)
0.356879 + 0.934151i \(0.383841\pi\)
\(692\) −21061.5 −1.15699
\(693\) 0 0
\(694\) −13342.7 −0.729801
\(695\) 0 0
\(696\) 0 0
\(697\) 3143.46 0.170828
\(698\) 8441.90 0.457780
\(699\) 0 0
\(700\) 0 0
\(701\) 16162.1 0.870806 0.435403 0.900236i \(-0.356606\pi\)
0.435403 + 0.900236i \(0.356606\pi\)
\(702\) 0 0
\(703\) 2084.63 0.111839
\(704\) 517.058 0.0276809
\(705\) 0 0
\(706\) 20696.2 1.10327
\(707\) 766.014 0.0407481
\(708\) 0 0
\(709\) −14244.4 −0.754529 −0.377265 0.926105i \(-0.623135\pi\)
−0.377265 + 0.926105i \(0.623135\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −22722.7 −1.19602
\(713\) −3195.97 −0.167868
\(714\) 0 0
\(715\) 0 0
\(716\) −16402.1 −0.856109
\(717\) 0 0
\(718\) 8837.55 0.459352
\(719\) 27638.5 1.43358 0.716790 0.697289i \(-0.245611\pi\)
0.716790 + 0.697289i \(0.245611\pi\)
\(720\) 0 0
\(721\) −39006.4 −2.01480
\(722\) −7030.68 −0.362403
\(723\) 0 0
\(724\) −17327.4 −0.889460
\(725\) 0 0
\(726\) 0 0
\(727\) 2525.52 0.128840 0.0644199 0.997923i \(-0.479480\pi\)
0.0644199 + 0.997923i \(0.479480\pi\)
\(728\) 28827.3 1.46760
\(729\) 0 0
\(730\) 0 0
\(731\) 4695.79 0.237592
\(732\) 0 0
\(733\) 8400.27 0.423289 0.211645 0.977347i \(-0.432118\pi\)
0.211645 + 0.977347i \(0.432118\pi\)
\(734\) −10355.9 −0.520765
\(735\) 0 0
\(736\) 9094.67 0.455481
\(737\) −311.401 −0.0155639
\(738\) 0 0
\(739\) −19689.1 −0.980074 −0.490037 0.871702i \(-0.663017\pi\)
−0.490037 + 0.871702i \(0.663017\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 17517.0 0.866671
\(743\) −22526.6 −1.11227 −0.556137 0.831091i \(-0.687717\pi\)
−0.556137 + 0.831091i \(0.687717\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −18005.8 −0.883699
\(747\) 0 0
\(748\) −173.528 −0.00848239
\(749\) 30243.3 1.47539
\(750\) 0 0
\(751\) 34691.1 1.68562 0.842808 0.538215i \(-0.180901\pi\)
0.842808 + 0.538215i \(0.180901\pi\)
\(752\) 1113.37 0.0539902
\(753\) 0 0
\(754\) 29066.8 1.40392
\(755\) 0 0
\(756\) 0 0
\(757\) −6619.98 −0.317843 −0.158922 0.987291i \(-0.550802\pi\)
−0.158922 + 0.987291i \(0.550802\pi\)
\(758\) −19832.0 −0.950303
\(759\) 0 0
\(760\) 0 0
\(761\) 29368.7 1.39897 0.699483 0.714649i \(-0.253414\pi\)
0.699483 + 0.714649i \(0.253414\pi\)
\(762\) 0 0
\(763\) −7131.84 −0.338388
\(764\) −17700.5 −0.838194
\(765\) 0 0
\(766\) −10830.4 −0.510859
\(767\) −4295.31 −0.202209
\(768\) 0 0
\(769\) 32677.4 1.53235 0.766174 0.642633i \(-0.222158\pi\)
0.766174 + 0.642633i \(0.222158\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 9148.33 0.426497
\(773\) −28047.5 −1.30504 −0.652522 0.757770i \(-0.726289\pi\)
−0.652522 + 0.757770i \(0.726289\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −8570.96 −0.396494
\(777\) 0 0
\(778\) 10438.0 0.481001
\(779\) 17359.1 0.798402
\(780\) 0 0
\(781\) 1064.14 0.0487552
\(782\) 1602.64 0.0732867
\(783\) 0 0
\(784\) 435.039 0.0198177
\(785\) 0 0
\(786\) 0 0
\(787\) 22172.1 1.00426 0.502128 0.864793i \(-0.332551\pi\)
0.502128 + 0.864793i \(0.332551\pi\)
\(788\) −8566.64 −0.387276
\(789\) 0 0
\(790\) 0 0
\(791\) 35140.7 1.57960
\(792\) 0 0
\(793\) 7995.06 0.358024
\(794\) 16584.2 0.741249
\(795\) 0 0
\(796\) −15866.5 −0.706501
\(797\) −24170.3 −1.07422 −0.537112 0.843511i \(-0.680485\pi\)
−0.537112 + 0.843511i \(0.680485\pi\)
\(798\) 0 0
\(799\) 7300.32 0.323238
\(800\) 0 0
\(801\) 0 0
\(802\) −2222.99 −0.0978761
\(803\) −573.415 −0.0251997
\(804\) 0 0
\(805\) 0 0
\(806\) 6380.47 0.278837
\(807\) 0 0
\(808\) −769.085 −0.0334856
\(809\) −15304.2 −0.665102 −0.332551 0.943085i \(-0.607909\pi\)
−0.332551 + 0.943085i \(0.607909\pi\)
\(810\) 0 0
\(811\) −27002.2 −1.16914 −0.584572 0.811342i \(-0.698738\pi\)
−0.584572 + 0.811342i \(0.698738\pi\)
\(812\) 33270.7 1.43790
\(813\) 0 0
\(814\) −60.5849 −0.00260872
\(815\) 0 0
\(816\) 0 0
\(817\) 25931.5 1.11044
\(818\) 6596.84 0.281972
\(819\) 0 0
\(820\) 0 0
\(821\) −25061.4 −1.06535 −0.532673 0.846321i \(-0.678812\pi\)
−0.532673 + 0.846321i \(0.678812\pi\)
\(822\) 0 0
\(823\) 24896.4 1.05448 0.527238 0.849718i \(-0.323228\pi\)
0.527238 + 0.849718i \(0.323228\pi\)
\(824\) 39162.8 1.65571
\(825\) 0 0
\(826\) 2788.59 0.117467
\(827\) −20063.2 −0.843612 −0.421806 0.906686i \(-0.638604\pi\)
−0.421806 + 0.906686i \(0.638604\pi\)
\(828\) 0 0
\(829\) −13884.2 −0.581687 −0.290844 0.956771i \(-0.593936\pi\)
−0.290844 + 0.956771i \(0.593936\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −16808.4 −0.700393
\(833\) 2852.52 0.118648
\(834\) 0 0
\(835\) 0 0
\(836\) −958.277 −0.0396444
\(837\) 0 0
\(838\) −27263.3 −1.12386
\(839\) 13678.1 0.562838 0.281419 0.959585i \(-0.409195\pi\)
0.281419 + 0.959585i \(0.409195\pi\)
\(840\) 0 0
\(841\) 61733.1 2.53118
\(842\) 13816.2 0.565485
\(843\) 0 0
\(844\) 22834.6 0.931280
\(845\) 0 0
\(846\) 0 0
\(847\) 29489.5 1.19630
\(848\) 1342.06 0.0543473
\(849\) 0 0
\(850\) 0 0
\(851\) −986.512 −0.0397382
\(852\) 0 0
\(853\) −29802.9 −1.19629 −0.598143 0.801390i \(-0.704094\pi\)
−0.598143 + 0.801390i \(0.704094\pi\)
\(854\) −5190.54 −0.207982
\(855\) 0 0
\(856\) −30364.6 −1.21243
\(857\) 22045.2 0.878706 0.439353 0.898314i \(-0.355208\pi\)
0.439353 + 0.898314i \(0.355208\pi\)
\(858\) 0 0
\(859\) 33609.5 1.33497 0.667487 0.744622i \(-0.267370\pi\)
0.667487 + 0.744622i \(0.267370\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −9722.70 −0.384172
\(863\) 33775.6 1.33226 0.666128 0.745838i \(-0.267951\pi\)
0.666128 + 0.745838i \(0.267951\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 10637.0 0.417392
\(867\) 0 0
\(868\) 7303.27 0.285587
\(869\) −1379.62 −0.0538556
\(870\) 0 0
\(871\) 10123.0 0.393805
\(872\) 7160.44 0.278077
\(873\) 0 0
\(874\) 8850.25 0.342522
\(875\) 0 0
\(876\) 0 0
\(877\) −12637.0 −0.486570 −0.243285 0.969955i \(-0.578225\pi\)
−0.243285 + 0.969955i \(0.578225\pi\)
\(878\) 7199.25 0.276723
\(879\) 0 0
\(880\) 0 0
\(881\) 6579.45 0.251609 0.125804 0.992055i \(-0.459849\pi\)
0.125804 + 0.992055i \(0.459849\pi\)
\(882\) 0 0
\(883\) 50442.1 1.92244 0.961219 0.275786i \(-0.0889382\pi\)
0.961219 + 0.275786i \(0.0889382\pi\)
\(884\) 5641.03 0.214625
\(885\) 0 0
\(886\) −10744.2 −0.407401
\(887\) 984.823 0.0372797 0.0186399 0.999826i \(-0.494066\pi\)
0.0186399 + 0.999826i \(0.494066\pi\)
\(888\) 0 0
\(889\) −26587.7 −1.00306
\(890\) 0 0
\(891\) 0 0
\(892\) −8950.64 −0.335975
\(893\) 40314.6 1.51072
\(894\) 0 0
\(895\) 0 0
\(896\) −21658.0 −0.807525
\(897\) 0 0
\(898\) −15909.1 −0.591196
\(899\) 18904.7 0.701342
\(900\) 0 0
\(901\) 8799.79 0.325376
\(902\) −504.503 −0.0186232
\(903\) 0 0
\(904\) −35281.6 −1.29806
\(905\) 0 0
\(906\) 0 0
\(907\) −43679.9 −1.59908 −0.799541 0.600612i \(-0.794924\pi\)
−0.799541 + 0.600612i \(0.794924\pi\)
\(908\) 4782.58 0.174797
\(909\) 0 0
\(910\) 0 0
\(911\) 10364.3 0.376930 0.188465 0.982080i \(-0.439649\pi\)
0.188465 + 0.982080i \(0.439649\pi\)
\(912\) 0 0
\(913\) 311.435 0.0112891
\(914\) −16245.6 −0.587918
\(915\) 0 0
\(916\) 13180.6 0.475435
\(917\) −7143.58 −0.257254
\(918\) 0 0
\(919\) 11451.9 0.411059 0.205530 0.978651i \(-0.434108\pi\)
0.205530 + 0.978651i \(0.434108\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 10613.0 0.379091
\(923\) −34592.7 −1.23362
\(924\) 0 0
\(925\) 0 0
\(926\) 11009.1 0.390692
\(927\) 0 0
\(928\) −53796.4 −1.90297
\(929\) 27701.8 0.978326 0.489163 0.872192i \(-0.337302\pi\)
0.489163 + 0.872192i \(0.337302\pi\)
\(930\) 0 0
\(931\) 15752.5 0.554529
\(932\) −11715.3 −0.411746
\(933\) 0 0
\(934\) −12262.5 −0.429596
\(935\) 0 0
\(936\) 0 0
\(937\) 5878.01 0.204937 0.102469 0.994736i \(-0.467326\pi\)
0.102469 + 0.994736i \(0.467326\pi\)
\(938\) −6572.03 −0.228768
\(939\) 0 0
\(940\) 0 0
\(941\) 28786.0 0.997234 0.498617 0.866823i \(-0.333841\pi\)
0.498617 + 0.866823i \(0.333841\pi\)
\(942\) 0 0
\(943\) −8214.90 −0.283684
\(944\) 213.647 0.00736612
\(945\) 0 0
\(946\) −753.642 −0.0259017
\(947\) −1695.04 −0.0581641 −0.0290821 0.999577i \(-0.509258\pi\)
−0.0290821 + 0.999577i \(0.509258\pi\)
\(948\) 0 0
\(949\) 18640.5 0.637613
\(950\) 0 0
\(951\) 0 0
\(952\) −9401.72 −0.320075
\(953\) 31929.4 1.08530 0.542651 0.839958i \(-0.317420\pi\)
0.542651 + 0.839958i \(0.317420\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −11710.2 −0.396166
\(957\) 0 0
\(958\) −18465.1 −0.622735
\(959\) −7868.57 −0.264952
\(960\) 0 0
\(961\) −25641.2 −0.860704
\(962\) 1969.48 0.0660069
\(963\) 0 0
\(964\) −1951.46 −0.0651994
\(965\) 0 0
\(966\) 0 0
\(967\) 10897.1 0.362385 0.181193 0.983448i \(-0.442004\pi\)
0.181193 + 0.983448i \(0.442004\pi\)
\(968\) −29607.7 −0.983087
\(969\) 0 0
\(970\) 0 0
\(971\) −7041.97 −0.232737 −0.116368 0.993206i \(-0.537125\pi\)
−0.116368 + 0.993206i \(0.537125\pi\)
\(972\) 0 0
\(973\) −1715.81 −0.0565328
\(974\) 21707.0 0.714103
\(975\) 0 0
\(976\) −397.671 −0.0130422
\(977\) −37607.6 −1.23150 −0.615749 0.787943i \(-0.711146\pi\)
−0.615749 + 0.787943i \(0.711146\pi\)
\(978\) 0 0
\(979\) −1824.69 −0.0595684
\(980\) 0 0
\(981\) 0 0
\(982\) −11939.0 −0.387974
\(983\) −25297.7 −0.820826 −0.410413 0.911900i \(-0.634615\pi\)
−0.410413 + 0.911900i \(0.634615\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −9479.85 −0.306187
\(987\) 0 0
\(988\) 31151.5 1.00310
\(989\) −12271.6 −0.394556
\(990\) 0 0
\(991\) −41686.5 −1.33624 −0.668120 0.744053i \(-0.732901\pi\)
−0.668120 + 0.744053i \(0.732901\pi\)
\(992\) −11808.9 −0.377955
\(993\) 0 0
\(994\) 22458.3 0.716633
\(995\) 0 0
\(996\) 0 0
\(997\) −25465.9 −0.808939 −0.404470 0.914551i \(-0.632544\pi\)
−0.404470 + 0.914551i \(0.632544\pi\)
\(998\) −19350.6 −0.613761
\(999\) 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.o.1.1 2
3.2 odd 2 75.4.a.c.1.2 2
5.2 odd 4 45.4.b.b.19.2 4
5.3 odd 4 45.4.b.b.19.3 4
5.4 even 2 225.4.a.i.1.2 2
12.11 even 2 1200.4.a.bt.1.2 2
15.2 even 4 15.4.b.a.4.3 yes 4
15.8 even 4 15.4.b.a.4.2 4
15.14 odd 2 75.4.a.f.1.1 2
20.3 even 4 720.4.f.j.289.3 4
20.7 even 4 720.4.f.j.289.4 4
60.23 odd 4 240.4.f.f.49.3 4
60.47 odd 4 240.4.f.f.49.1 4
60.59 even 2 1200.4.a.bn.1.1 2
120.53 even 4 960.4.f.q.769.4 4
120.77 even 4 960.4.f.q.769.2 4
120.83 odd 4 960.4.f.p.769.2 4
120.107 odd 4 960.4.f.p.769.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.b.a.4.2 4 15.8 even 4
15.4.b.a.4.3 yes 4 15.2 even 4
45.4.b.b.19.2 4 5.2 odd 4
45.4.b.b.19.3 4 5.3 odd 4
75.4.a.c.1.2 2 3.2 odd 2
75.4.a.f.1.1 2 15.14 odd 2
225.4.a.i.1.2 2 5.4 even 2
225.4.a.o.1.1 2 1.1 even 1 trivial
240.4.f.f.49.1 4 60.47 odd 4
240.4.f.f.49.3 4 60.23 odd 4
720.4.f.j.289.3 4 20.3 even 4
720.4.f.j.289.4 4 20.7 even 4
960.4.f.p.769.2 4 120.83 odd 4
960.4.f.p.769.4 4 120.107 odd 4
960.4.f.q.769.2 4 120.77 even 4
960.4.f.q.769.4 4 120.53 even 4
1200.4.a.bn.1.1 2 60.59 even 2
1200.4.a.bt.1.2 2 12.11 even 2