Properties

Label 8281.2.a.cp.1.11
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(66.1241179138\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 16 x^{10} + 88 x^{8} - 197 x^{6} + 172 x^{4} - 36 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(2.30327\) of defining polynomial
Character \(\chi\) \(=\) 8281.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.30327 q^{2} -1.47336 q^{3} +3.30504 q^{4} +0.847292 q^{5} -3.39354 q^{6} +3.00585 q^{8} -0.829208 q^{9} +O(q^{10})\) \(q+2.30327 q^{2} -1.47336 q^{3} +3.30504 q^{4} +0.847292 q^{5} -3.39354 q^{6} +3.00585 q^{8} -0.829208 q^{9} +1.95154 q^{10} -1.50340 q^{11} -4.86951 q^{12} -1.24837 q^{15} +0.313194 q^{16} +2.07140 q^{17} -1.90989 q^{18} -0.0474272 q^{19} +2.80033 q^{20} -3.46274 q^{22} -7.81870 q^{23} -4.42870 q^{24} -4.28210 q^{25} +5.64180 q^{27} +1.35971 q^{29} -2.87532 q^{30} +7.86105 q^{31} -5.29033 q^{32} +2.21505 q^{33} +4.77099 q^{34} -2.74056 q^{36} +6.70219 q^{37} -0.109237 q^{38} +2.54683 q^{40} +10.0184 q^{41} +9.26566 q^{43} -4.96880 q^{44} -0.702581 q^{45} -18.0086 q^{46} +0.360014 q^{47} -0.461448 q^{48} -9.86281 q^{50} -3.05192 q^{51} +2.71181 q^{53} +12.9946 q^{54} -1.27382 q^{55} +0.0698773 q^{57} +3.13177 q^{58} +1.64120 q^{59} -4.12590 q^{60} +4.52194 q^{61} +18.1061 q^{62} -12.8114 q^{64} +5.10186 q^{66} -2.04266 q^{67} +6.84606 q^{68} +11.5198 q^{69} +14.2139 q^{71} -2.49247 q^{72} +6.76150 q^{73} +15.4369 q^{74} +6.30907 q^{75} -0.156749 q^{76} +11.6590 q^{79} +0.265367 q^{80} -5.82479 q^{81} +23.0751 q^{82} +11.5362 q^{83} +1.75508 q^{85} +21.3413 q^{86} -2.00334 q^{87} -4.51900 q^{88} -17.5112 q^{89} -1.61823 q^{90} -25.8411 q^{92} -11.5822 q^{93} +0.829208 q^{94} -0.0401846 q^{95} +7.79456 q^{96} +0.426229 q^{97} +1.24663 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 6q^{3} + 8q^{4} + 2q^{9} + O(q^{10}) \) \( 12q + 6q^{3} + 8q^{4} + 2q^{9} + 24q^{10} - 2q^{12} + 16q^{16} + 34q^{17} + 30q^{22} + 6q^{23} - 10q^{25} + 12q^{27} + 2q^{29} + 22q^{30} - 26q^{36} + 38q^{38} + 2q^{40} + 22q^{43} - 38q^{48} + 8q^{51} + 16q^{53} + 30q^{55} - 10q^{61} + 82q^{62} - 2q^{64} + 68q^{66} + 22q^{68} + 14q^{69} + 66q^{74} + 2q^{75} + 70q^{79} - 28q^{81} + 10q^{82} - 20q^{87} - 28q^{88} - 66q^{92} - 2q^{94} + 4q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.30327 1.62866 0.814328 0.580405i \(-0.197106\pi\)
0.814328 + 0.580405i \(0.197106\pi\)
\(3\) −1.47336 −0.850645 −0.425323 0.905042i \(-0.639839\pi\)
−0.425323 + 0.905042i \(0.639839\pi\)
\(4\) 3.30504 1.65252
\(5\) 0.847292 0.378920 0.189460 0.981888i \(-0.439326\pi\)
0.189460 + 0.981888i \(0.439326\pi\)
\(6\) −3.39354 −1.38541
\(7\) 0 0
\(8\) 3.00585 1.06273
\(9\) −0.829208 −0.276403
\(10\) 1.95154 0.617131
\(11\) −1.50340 −0.453293 −0.226646 0.973977i \(-0.572776\pi\)
−0.226646 + 0.973977i \(0.572776\pi\)
\(12\) −4.86951 −1.40571
\(13\) 0 0
\(14\) 0 0
\(15\) −1.24837 −0.322327
\(16\) 0.313194 0.0782985
\(17\) 2.07140 0.502389 0.251194 0.967937i \(-0.419177\pi\)
0.251194 + 0.967937i \(0.419177\pi\)
\(18\) −1.90989 −0.450164
\(19\) −0.0474272 −0.0108805 −0.00544027 0.999985i \(-0.501732\pi\)
−0.00544027 + 0.999985i \(0.501732\pi\)
\(20\) 2.80033 0.626173
\(21\) 0 0
\(22\) −3.46274 −0.738258
\(23\) −7.81870 −1.63031 −0.815156 0.579241i \(-0.803349\pi\)
−0.815156 + 0.579241i \(0.803349\pi\)
\(24\) −4.42870 −0.904004
\(25\) −4.28210 −0.856419
\(26\) 0 0
\(27\) 5.64180 1.08577
\(28\) 0 0
\(29\) 1.35971 0.252491 0.126246 0.991999i \(-0.459707\pi\)
0.126246 + 0.991999i \(0.459707\pi\)
\(30\) −2.87532 −0.524959
\(31\) 7.86105 1.41189 0.705943 0.708269i \(-0.250523\pi\)
0.705943 + 0.708269i \(0.250523\pi\)
\(32\) −5.29033 −0.935206
\(33\) 2.21505 0.385591
\(34\) 4.77099 0.818218
\(35\) 0 0
\(36\) −2.74056 −0.456760
\(37\) 6.70219 1.10183 0.550917 0.834560i \(-0.314278\pi\)
0.550917 + 0.834560i \(0.314278\pi\)
\(38\) −0.109237 −0.0177206
\(39\) 0 0
\(40\) 2.54683 0.402689
\(41\) 10.0184 1.56462 0.782309 0.622891i \(-0.214042\pi\)
0.782309 + 0.622891i \(0.214042\pi\)
\(42\) 0 0
\(43\) 9.26566 1.41300 0.706500 0.707713i \(-0.250273\pi\)
0.706500 + 0.707713i \(0.250273\pi\)
\(44\) −4.96880 −0.749075
\(45\) −0.702581 −0.104735
\(46\) −18.0086 −2.65522
\(47\) 0.360014 0.0525134 0.0262567 0.999655i \(-0.491641\pi\)
0.0262567 + 0.999655i \(0.491641\pi\)
\(48\) −0.461448 −0.0666042
\(49\) 0 0
\(50\) −9.86281 −1.39481
\(51\) −3.05192 −0.427355
\(52\) 0 0
\(53\) 2.71181 0.372496 0.186248 0.982503i \(-0.440367\pi\)
0.186248 + 0.982503i \(0.440367\pi\)
\(54\) 12.9946 1.76834
\(55\) −1.27382 −0.171762
\(56\) 0 0
\(57\) 0.0698773 0.00925548
\(58\) 3.13177 0.411222
\(59\) 1.64120 0.213666 0.106833 0.994277i \(-0.465929\pi\)
0.106833 + 0.994277i \(0.465929\pi\)
\(60\) −4.12590 −0.532651
\(61\) 4.52194 0.578975 0.289488 0.957182i \(-0.406515\pi\)
0.289488 + 0.957182i \(0.406515\pi\)
\(62\) 18.1061 2.29948
\(63\) 0 0
\(64\) −12.8114 −1.60143
\(65\) 0 0
\(66\) 5.10186 0.627995
\(67\) −2.04266 −0.249551 −0.124775 0.992185i \(-0.539821\pi\)
−0.124775 + 0.992185i \(0.539821\pi\)
\(68\) 6.84606 0.830206
\(69\) 11.5198 1.38682
\(70\) 0 0
\(71\) 14.2139 1.68688 0.843442 0.537220i \(-0.180526\pi\)
0.843442 + 0.537220i \(0.180526\pi\)
\(72\) −2.49247 −0.293741
\(73\) 6.76150 0.791373 0.395687 0.918386i \(-0.370507\pi\)
0.395687 + 0.918386i \(0.370507\pi\)
\(74\) 15.4369 1.79451
\(75\) 6.30907 0.728509
\(76\) −0.156749 −0.0179803
\(77\) 0 0
\(78\) 0 0
\(79\) 11.6590 1.31175 0.655873 0.754871i \(-0.272301\pi\)
0.655873 + 0.754871i \(0.272301\pi\)
\(80\) 0.265367 0.0296689
\(81\) −5.82479 −0.647199
\(82\) 23.0751 2.54822
\(83\) 11.5362 1.26627 0.633133 0.774043i \(-0.281768\pi\)
0.633133 + 0.774043i \(0.281768\pi\)
\(84\) 0 0
\(85\) 1.75508 0.190365
\(86\) 21.3413 2.30129
\(87\) −2.00334 −0.214781
\(88\) −4.51900 −0.481727
\(89\) −17.5112 −1.85619 −0.928093 0.372350i \(-0.878552\pi\)
−0.928093 + 0.372350i \(0.878552\pi\)
\(90\) −1.61823 −0.170576
\(91\) 0 0
\(92\) −25.8411 −2.69412
\(93\) −11.5822 −1.20101
\(94\) 0.829208 0.0855262
\(95\) −0.0401846 −0.00412286
\(96\) 7.79456 0.795529
\(97\) 0.426229 0.0432770 0.0216385 0.999766i \(-0.493112\pi\)
0.0216385 + 0.999766i \(0.493112\pi\)
\(98\) 0 0
\(99\) 1.24663 0.125291
\(100\) −14.1525 −1.41525
\(101\) 9.66997 0.962198 0.481099 0.876666i \(-0.340238\pi\)
0.481099 + 0.876666i \(0.340238\pi\)
\(102\) −7.02939 −0.696013
\(103\) −9.97823 −0.983185 −0.491592 0.870825i \(-0.663585\pi\)
−0.491592 + 0.870825i \(0.663585\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 6.24603 0.606668
\(107\) 9.86223 0.953417 0.476709 0.879061i \(-0.341830\pi\)
0.476709 + 0.879061i \(0.341830\pi\)
\(108\) 18.6464 1.79425
\(109\) 11.6055 1.11161 0.555803 0.831314i \(-0.312411\pi\)
0.555803 + 0.831314i \(0.312411\pi\)
\(110\) −2.93395 −0.279741
\(111\) −9.87475 −0.937269
\(112\) 0 0
\(113\) −3.47758 −0.327143 −0.163572 0.986531i \(-0.552301\pi\)
−0.163572 + 0.986531i \(0.552301\pi\)
\(114\) 0.160946 0.0150740
\(115\) −6.62472 −0.617758
\(116\) 4.49388 0.417247
\(117\) 0 0
\(118\) 3.78011 0.347987
\(119\) 0 0
\(120\) −3.75240 −0.342546
\(121\) −8.73978 −0.794526
\(122\) 10.4152 0.942951
\(123\) −14.7608 −1.33093
\(124\) 25.9811 2.33317
\(125\) −7.86464 −0.703435
\(126\) 0 0
\(127\) −15.6998 −1.39313 −0.696567 0.717491i \(-0.745290\pi\)
−0.696567 + 0.717491i \(0.745290\pi\)
\(128\) −18.9275 −1.67297
\(129\) −13.6517 −1.20196
\(130\) 0 0
\(131\) −2.54517 −0.222373 −0.111186 0.993800i \(-0.535465\pi\)
−0.111186 + 0.993800i \(0.535465\pi\)
\(132\) 7.32083 0.637197
\(133\) 0 0
\(134\) −4.70479 −0.406432
\(135\) 4.78025 0.411419
\(136\) 6.22632 0.533902
\(137\) −1.86472 −0.159314 −0.0796571 0.996822i \(-0.525383\pi\)
−0.0796571 + 0.996822i \(0.525383\pi\)
\(138\) 26.5331 2.25865
\(139\) 15.6092 1.32396 0.661979 0.749522i \(-0.269717\pi\)
0.661979 + 0.749522i \(0.269717\pi\)
\(140\) 0 0
\(141\) −0.530430 −0.0446703
\(142\) 32.7385 2.74735
\(143\) 0 0
\(144\) −0.259703 −0.0216419
\(145\) 1.15207 0.0956741
\(146\) 15.5735 1.28887
\(147\) 0 0
\(148\) 22.1510 1.82080
\(149\) −6.36363 −0.521329 −0.260664 0.965429i \(-0.583942\pi\)
−0.260664 + 0.965429i \(0.583942\pi\)
\(150\) 14.5315 1.18649
\(151\) 0.664094 0.0540432 0.0270216 0.999635i \(-0.491398\pi\)
0.0270216 + 0.999635i \(0.491398\pi\)
\(152\) −0.142559 −0.0115630
\(153\) −1.71762 −0.138861
\(154\) 0 0
\(155\) 6.66060 0.534992
\(156\) 0 0
\(157\) −16.5760 −1.32291 −0.661453 0.749986i \(-0.730060\pi\)
−0.661453 + 0.749986i \(0.730060\pi\)
\(158\) 26.8539 2.13638
\(159\) −3.99548 −0.316862
\(160\) −4.48245 −0.354369
\(161\) 0 0
\(162\) −13.4160 −1.05406
\(163\) 9.05127 0.708950 0.354475 0.935065i \(-0.384660\pi\)
0.354475 + 0.935065i \(0.384660\pi\)
\(164\) 33.1113 2.58556
\(165\) 1.87680 0.146108
\(166\) 26.5710 2.06231
\(167\) −2.65761 −0.205652 −0.102826 0.994699i \(-0.532788\pi\)
−0.102826 + 0.994699i \(0.532788\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 4.04242 0.310039
\(171\) 0.0393270 0.00300741
\(172\) 30.6234 2.33501
\(173\) 19.5870 1.48918 0.744588 0.667525i \(-0.232646\pi\)
0.744588 + 0.667525i \(0.232646\pi\)
\(174\) −4.61423 −0.349804
\(175\) 0 0
\(176\) −0.470856 −0.0354921
\(177\) −2.41807 −0.181754
\(178\) −40.3330 −3.02309
\(179\) −2.89332 −0.216257 −0.108129 0.994137i \(-0.534486\pi\)
−0.108129 + 0.994137i \(0.534486\pi\)
\(180\) −2.32205 −0.173076
\(181\) 1.36804 0.101686 0.0508429 0.998707i \(-0.483809\pi\)
0.0508429 + 0.998707i \(0.483809\pi\)
\(182\) 0 0
\(183\) −6.66245 −0.492503
\(184\) −23.5018 −1.73258
\(185\) 5.67871 0.417507
\(186\) −26.6768 −1.95604
\(187\) −3.11415 −0.227729
\(188\) 1.18986 0.0867794
\(189\) 0 0
\(190\) −0.0925559 −0.00671471
\(191\) −1.51325 −0.109495 −0.0547475 0.998500i \(-0.517435\pi\)
−0.0547475 + 0.998500i \(0.517435\pi\)
\(192\) 18.8758 1.36225
\(193\) 6.95394 0.500556 0.250278 0.968174i \(-0.419478\pi\)
0.250278 + 0.968174i \(0.419478\pi\)
\(194\) 0.981719 0.0704834
\(195\) 0 0
\(196\) 0 0
\(197\) 15.4772 1.10271 0.551353 0.834272i \(-0.314112\pi\)
0.551353 + 0.834272i \(0.314112\pi\)
\(198\) 2.87133 0.204056
\(199\) 6.61529 0.468945 0.234473 0.972123i \(-0.424664\pi\)
0.234473 + 0.972123i \(0.424664\pi\)
\(200\) −12.8713 −0.910140
\(201\) 3.00958 0.212279
\(202\) 22.2725 1.56709
\(203\) 0 0
\(204\) −10.0867 −0.706211
\(205\) 8.48854 0.592865
\(206\) −22.9825 −1.60127
\(207\) 6.48333 0.450622
\(208\) 0 0
\(209\) 0.0713021 0.00493207
\(210\) 0 0
\(211\) −8.09428 −0.557234 −0.278617 0.960402i \(-0.589876\pi\)
−0.278617 + 0.960402i \(0.589876\pi\)
\(212\) 8.96264 0.615557
\(213\) −20.9423 −1.43494
\(214\) 22.7153 1.55279
\(215\) 7.85072 0.535415
\(216\) 16.9584 1.15387
\(217\) 0 0
\(218\) 26.7306 1.81042
\(219\) −9.96212 −0.673178
\(220\) −4.21002 −0.283840
\(221\) 0 0
\(222\) −22.7442 −1.52649
\(223\) −16.0581 −1.07533 −0.537664 0.843159i \(-0.680693\pi\)
−0.537664 + 0.843159i \(0.680693\pi\)
\(224\) 0 0
\(225\) 3.55075 0.236716
\(226\) −8.00979 −0.532804
\(227\) 1.29581 0.0860057 0.0430029 0.999075i \(-0.486308\pi\)
0.0430029 + 0.999075i \(0.486308\pi\)
\(228\) 0.230947 0.0152948
\(229\) −20.8175 −1.37566 −0.687831 0.725871i \(-0.741437\pi\)
−0.687831 + 0.725871i \(0.741437\pi\)
\(230\) −15.2585 −1.00612
\(231\) 0 0
\(232\) 4.08707 0.268330
\(233\) −13.3043 −0.871591 −0.435796 0.900046i \(-0.643533\pi\)
−0.435796 + 0.900046i \(0.643533\pi\)
\(234\) 0 0
\(235\) 0.305037 0.0198984
\(236\) 5.42421 0.353086
\(237\) −17.1780 −1.11583
\(238\) 0 0
\(239\) 13.3652 0.864525 0.432263 0.901748i \(-0.357715\pi\)
0.432263 + 0.901748i \(0.357715\pi\)
\(240\) −0.390981 −0.0252377
\(241\) −0.834153 −0.0537325 −0.0268663 0.999639i \(-0.508553\pi\)
−0.0268663 + 0.999639i \(0.508553\pi\)
\(242\) −20.1300 −1.29401
\(243\) −8.34339 −0.535229
\(244\) 14.9452 0.956767
\(245\) 0 0
\(246\) −33.9980 −2.16763
\(247\) 0 0
\(248\) 23.6291 1.50045
\(249\) −16.9970 −1.07714
\(250\) −18.1144 −1.14565
\(251\) 27.2721 1.72140 0.860699 0.509114i \(-0.170027\pi\)
0.860699 + 0.509114i \(0.170027\pi\)
\(252\) 0 0
\(253\) 11.7547 0.739009
\(254\) −36.1609 −2.26894
\(255\) −2.58587 −0.161933
\(256\) −17.9721 −1.12326
\(257\) 6.55188 0.408695 0.204348 0.978898i \(-0.434493\pi\)
0.204348 + 0.978898i \(0.434493\pi\)
\(258\) −31.4434 −1.95758
\(259\) 0 0
\(260\) 0 0
\(261\) −1.12748 −0.0697893
\(262\) −5.86221 −0.362168
\(263\) −22.5891 −1.39290 −0.696450 0.717605i \(-0.745238\pi\)
−0.696450 + 0.717605i \(0.745238\pi\)
\(264\) 6.65811 0.409779
\(265\) 2.29770 0.141146
\(266\) 0 0
\(267\) 25.8003 1.57896
\(268\) −6.75107 −0.412387
\(269\) 16.0013 0.975617 0.487808 0.872951i \(-0.337797\pi\)
0.487808 + 0.872951i \(0.337797\pi\)
\(270\) 11.0102 0.670059
\(271\) 8.75935 0.532093 0.266046 0.963960i \(-0.414283\pi\)
0.266046 + 0.963960i \(0.414283\pi\)
\(272\) 0.648750 0.0393363
\(273\) 0 0
\(274\) −4.29496 −0.259468
\(275\) 6.43771 0.388209
\(276\) 38.0733 2.29174
\(277\) −19.9183 −1.19677 −0.598387 0.801208i \(-0.704191\pi\)
−0.598387 + 0.801208i \(0.704191\pi\)
\(278\) 35.9522 2.15627
\(279\) −6.51844 −0.390249
\(280\) 0 0
\(281\) 14.0234 0.836566 0.418283 0.908317i \(-0.362632\pi\)
0.418283 + 0.908317i \(0.362632\pi\)
\(282\) −1.22172 −0.0727525
\(283\) 1.01259 0.0601922 0.0300961 0.999547i \(-0.490419\pi\)
0.0300961 + 0.999547i \(0.490419\pi\)
\(284\) 46.9776 2.78761
\(285\) 0.0592065 0.00350709
\(286\) 0 0
\(287\) 0 0
\(288\) 4.38678 0.258493
\(289\) −12.7093 −0.747606
\(290\) 2.65352 0.155820
\(291\) −0.627989 −0.0368134
\(292\) 22.3470 1.30776
\(293\) 0.199235 0.0116394 0.00581972 0.999983i \(-0.498148\pi\)
0.00581972 + 0.999983i \(0.498148\pi\)
\(294\) 0 0
\(295\) 1.39057 0.0809622
\(296\) 20.1458 1.17095
\(297\) −8.48190 −0.492170
\(298\) −14.6571 −0.849065
\(299\) 0 0
\(300\) 20.8517 1.20387
\(301\) 0 0
\(302\) 1.52958 0.0880177
\(303\) −14.2474 −0.818489
\(304\) −0.0148539 −0.000851930 0
\(305\) 3.83140 0.219386
\(306\) −3.95614 −0.226157
\(307\) 27.2004 1.55241 0.776204 0.630482i \(-0.217143\pi\)
0.776204 + 0.630482i \(0.217143\pi\)
\(308\) 0 0
\(309\) 14.7015 0.836341
\(310\) 15.3411 0.871318
\(311\) 27.1009 1.53675 0.768376 0.639999i \(-0.221065\pi\)
0.768376 + 0.639999i \(0.221065\pi\)
\(312\) 0 0
\(313\) −22.0785 −1.24795 −0.623975 0.781445i \(-0.714483\pi\)
−0.623975 + 0.781445i \(0.714483\pi\)
\(314\) −38.1789 −2.15456
\(315\) 0 0
\(316\) 38.5336 2.16768
\(317\) −7.06823 −0.396991 −0.198496 0.980102i \(-0.563606\pi\)
−0.198496 + 0.980102i \(0.563606\pi\)
\(318\) −9.20265 −0.516059
\(319\) −2.04419 −0.114453
\(320\) −10.8550 −0.606813
\(321\) −14.5306 −0.811020
\(322\) 0 0
\(323\) −0.0982407 −0.00546626
\(324\) −19.2512 −1.06951
\(325\) 0 0
\(326\) 20.8475 1.15464
\(327\) −17.0991 −0.945582
\(328\) 30.1139 1.66276
\(329\) 0 0
\(330\) 4.32276 0.237960
\(331\) −6.58858 −0.362141 −0.181071 0.983470i \(-0.557956\pi\)
−0.181071 + 0.983470i \(0.557956\pi\)
\(332\) 38.1277 2.09253
\(333\) −5.55751 −0.304550
\(334\) −6.12118 −0.334936
\(335\) −1.73073 −0.0945599
\(336\) 0 0
\(337\) 4.22290 0.230036 0.115018 0.993363i \(-0.463307\pi\)
0.115018 + 0.993363i \(0.463307\pi\)
\(338\) 0 0
\(339\) 5.12373 0.278283
\(340\) 5.80061 0.314582
\(341\) −11.8183 −0.639998
\(342\) 0.0905805 0.00489803
\(343\) 0 0
\(344\) 27.8512 1.50163
\(345\) 9.76060 0.525493
\(346\) 45.1142 2.42535
\(347\) −9.09478 −0.488233 −0.244117 0.969746i \(-0.578498\pi\)
−0.244117 + 0.969746i \(0.578498\pi\)
\(348\) −6.62111 −0.354929
\(349\) 9.22053 0.493564 0.246782 0.969071i \(-0.420627\pi\)
0.246782 + 0.969071i \(0.420627\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 7.95349 0.423922
\(353\) 2.15449 0.114672 0.0573359 0.998355i \(-0.481739\pi\)
0.0573359 + 0.998355i \(0.481739\pi\)
\(354\) −5.56947 −0.296014
\(355\) 12.0433 0.639195
\(356\) −57.8752 −3.06738
\(357\) 0 0
\(358\) −6.66410 −0.352208
\(359\) 8.55756 0.451651 0.225825 0.974168i \(-0.427492\pi\)
0.225825 + 0.974168i \(0.427492\pi\)
\(360\) −2.11185 −0.111304
\(361\) −18.9978 −0.999882
\(362\) 3.15096 0.165611
\(363\) 12.8769 0.675860
\(364\) 0 0
\(365\) 5.72896 0.299867
\(366\) −15.3454 −0.802117
\(367\) 2.29823 0.119967 0.0599833 0.998199i \(-0.480895\pi\)
0.0599833 + 0.998199i \(0.480895\pi\)
\(368\) −2.44877 −0.127651
\(369\) −8.30736 −0.432464
\(370\) 13.0796 0.679975
\(371\) 0 0
\(372\) −38.2795 −1.98470
\(373\) 11.7684 0.609343 0.304672 0.952457i \(-0.401453\pi\)
0.304672 + 0.952457i \(0.401453\pi\)
\(374\) −7.17271 −0.370892
\(375\) 11.5875 0.598374
\(376\) 1.08215 0.0558074
\(377\) 0 0
\(378\) 0 0
\(379\) 7.99093 0.410466 0.205233 0.978713i \(-0.434205\pi\)
0.205233 + 0.978713i \(0.434205\pi\)
\(380\) −0.132812 −0.00681310
\(381\) 23.1315 1.18506
\(382\) −3.48542 −0.178329
\(383\) 28.2446 1.44323 0.721616 0.692294i \(-0.243400\pi\)
0.721616 + 0.692294i \(0.243400\pi\)
\(384\) 27.8870 1.42310
\(385\) 0 0
\(386\) 16.0168 0.815233
\(387\) −7.68316 −0.390557
\(388\) 1.40870 0.0715161
\(389\) 7.68086 0.389435 0.194717 0.980859i \(-0.437621\pi\)
0.194717 + 0.980859i \(0.437621\pi\)
\(390\) 0 0
\(391\) −16.1957 −0.819050
\(392\) 0 0
\(393\) 3.74996 0.189160
\(394\) 35.6481 1.79593
\(395\) 9.87862 0.497047
\(396\) 4.12017 0.207046
\(397\) 7.45281 0.374046 0.187023 0.982356i \(-0.440116\pi\)
0.187023 + 0.982356i \(0.440116\pi\)
\(398\) 15.2368 0.763750
\(399\) 0 0
\(400\) −1.34113 −0.0670563
\(401\) 18.1982 0.908777 0.454389 0.890804i \(-0.349858\pi\)
0.454389 + 0.890804i \(0.349858\pi\)
\(402\) 6.93186 0.345730
\(403\) 0 0
\(404\) 31.9596 1.59005
\(405\) −4.93530 −0.245237
\(406\) 0 0
\(407\) −10.0761 −0.499453
\(408\) −9.17361 −0.454161
\(409\) 29.2825 1.44793 0.723964 0.689838i \(-0.242318\pi\)
0.723964 + 0.689838i \(0.242318\pi\)
\(410\) 19.5514 0.965573
\(411\) 2.74741 0.135520
\(412\) −32.9784 −1.62473
\(413\) 0 0
\(414\) 14.9328 0.733909
\(415\) 9.77456 0.479814
\(416\) 0 0
\(417\) −22.9980 −1.12622
\(418\) 0.164228 0.00803264
\(419\) −20.7393 −1.01318 −0.506591 0.862187i \(-0.669095\pi\)
−0.506591 + 0.862187i \(0.669095\pi\)
\(420\) 0 0
\(421\) −24.8696 −1.21207 −0.606036 0.795437i \(-0.707241\pi\)
−0.606036 + 0.795437i \(0.707241\pi\)
\(422\) −18.6433 −0.907541
\(423\) −0.298526 −0.0145148
\(424\) 8.15130 0.395862
\(425\) −8.86994 −0.430255
\(426\) −48.2356 −2.33702
\(427\) 0 0
\(428\) 32.5950 1.57554
\(429\) 0 0
\(430\) 18.0823 0.872006
\(431\) −21.1688 −1.01966 −0.509832 0.860274i \(-0.670292\pi\)
−0.509832 + 0.860274i \(0.670292\pi\)
\(432\) 1.76698 0.0850138
\(433\) −23.4296 −1.12595 −0.562977 0.826472i \(-0.690344\pi\)
−0.562977 + 0.826472i \(0.690344\pi\)
\(434\) 0 0
\(435\) −1.69741 −0.0813848
\(436\) 38.3566 1.83695
\(437\) 0.370819 0.0177387
\(438\) −22.9454 −1.09637
\(439\) −12.0384 −0.574561 −0.287280 0.957847i \(-0.592751\pi\)
−0.287280 + 0.957847i \(0.592751\pi\)
\(440\) −3.82891 −0.182536
\(441\) 0 0
\(442\) 0 0
\(443\) 15.7331 0.747503 0.373752 0.927529i \(-0.378071\pi\)
0.373752 + 0.927529i \(0.378071\pi\)
\(444\) −32.6364 −1.54885
\(445\) −14.8371 −0.703346
\(446\) −36.9860 −1.75134
\(447\) 9.37592 0.443466
\(448\) 0 0
\(449\) −26.0012 −1.22707 −0.613536 0.789667i \(-0.710253\pi\)
−0.613536 + 0.789667i \(0.710253\pi\)
\(450\) 8.17832 0.385530
\(451\) −15.0617 −0.709230
\(452\) −11.4935 −0.540610
\(453\) −0.978449 −0.0459716
\(454\) 2.98459 0.140074
\(455\) 0 0
\(456\) 0.210041 0.00983605
\(457\) 30.7958 1.44057 0.720284 0.693679i \(-0.244011\pi\)
0.720284 + 0.693679i \(0.244011\pi\)
\(458\) −47.9483 −2.24048
\(459\) 11.6864 0.545476
\(460\) −21.8949 −1.02086
\(461\) −34.0958 −1.58800 −0.794000 0.607918i \(-0.792005\pi\)
−0.794000 + 0.607918i \(0.792005\pi\)
\(462\) 0 0
\(463\) −1.69184 −0.0786263 −0.0393131 0.999227i \(-0.512517\pi\)
−0.0393131 + 0.999227i \(0.512517\pi\)
\(464\) 0.425852 0.0197697
\(465\) −9.81347 −0.455089
\(466\) −30.6433 −1.41952
\(467\) 28.3524 1.31199 0.655996 0.754764i \(-0.272249\pi\)
0.655996 + 0.754764i \(0.272249\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0.702581 0.0324076
\(471\) 24.4224 1.12532
\(472\) 4.93318 0.227068
\(473\) −13.9300 −0.640503
\(474\) −39.5655 −1.81730
\(475\) 0.203088 0.00931830
\(476\) 0 0
\(477\) −2.24866 −0.102959
\(478\) 30.7837 1.40801
\(479\) −6.28246 −0.287053 −0.143526 0.989646i \(-0.545844\pi\)
−0.143526 + 0.989646i \(0.545844\pi\)
\(480\) 6.60426 0.301442
\(481\) 0 0
\(482\) −1.92128 −0.0875117
\(483\) 0 0
\(484\) −28.8853 −1.31297
\(485\) 0.361140 0.0163985
\(486\) −19.2171 −0.871703
\(487\) 13.0176 0.589883 0.294942 0.955515i \(-0.404700\pi\)
0.294942 + 0.955515i \(0.404700\pi\)
\(488\) 13.5923 0.615293
\(489\) −13.3358 −0.603065
\(490\) 0 0
\(491\) −12.3523 −0.557453 −0.278726 0.960371i \(-0.589912\pi\)
−0.278726 + 0.960371i \(0.589912\pi\)
\(492\) −48.7849 −2.19939
\(493\) 2.81650 0.126849
\(494\) 0 0
\(495\) 1.05626 0.0474754
\(496\) 2.46203 0.110549
\(497\) 0 0
\(498\) −39.1487 −1.75430
\(499\) 9.15340 0.409763 0.204881 0.978787i \(-0.434319\pi\)
0.204881 + 0.978787i \(0.434319\pi\)
\(500\) −25.9929 −1.16244
\(501\) 3.91562 0.174937
\(502\) 62.8149 2.80357
\(503\) 22.5037 1.00339 0.501696 0.865044i \(-0.332710\pi\)
0.501696 + 0.865044i \(0.332710\pi\)
\(504\) 0 0
\(505\) 8.19329 0.364596
\(506\) 27.0741 1.20359
\(507\) 0 0
\(508\) −51.8885 −2.30218
\(509\) −38.6606 −1.71360 −0.856800 0.515649i \(-0.827551\pi\)
−0.856800 + 0.515649i \(0.827551\pi\)
\(510\) −5.95594 −0.263734
\(511\) 0 0
\(512\) −3.53972 −0.156435
\(513\) −0.267575 −0.0118137
\(514\) 15.0907 0.665623
\(515\) −8.45447 −0.372549
\(516\) −45.1193 −1.98626
\(517\) −0.541245 −0.0238039
\(518\) 0 0
\(519\) −28.8588 −1.26676
\(520\) 0 0
\(521\) 40.2351 1.76273 0.881366 0.472434i \(-0.156625\pi\)
0.881366 + 0.472434i \(0.156625\pi\)
\(522\) −2.59689 −0.113663
\(523\) −0.732146 −0.0320145 −0.0160073 0.999872i \(-0.505095\pi\)
−0.0160073 + 0.999872i \(0.505095\pi\)
\(524\) −8.41189 −0.367475
\(525\) 0 0
\(526\) −52.0286 −2.26856
\(527\) 16.2834 0.709316
\(528\) 0.693741 0.0301912
\(529\) 38.1321 1.65792
\(530\) 5.29221 0.229879
\(531\) −1.36089 −0.0590577
\(532\) 0 0
\(533\) 0 0
\(534\) 59.4251 2.57157
\(535\) 8.35618 0.361269
\(536\) −6.13993 −0.265204
\(537\) 4.26291 0.183958
\(538\) 36.8553 1.58894
\(539\) 0 0
\(540\) 15.7989 0.679877
\(541\) 23.6537 1.01695 0.508476 0.861076i \(-0.330209\pi\)
0.508476 + 0.861076i \(0.330209\pi\)
\(542\) 20.1751 0.866595
\(543\) −2.01562 −0.0864985
\(544\) −10.9584 −0.469837
\(545\) 9.83325 0.421210
\(546\) 0 0
\(547\) −12.9472 −0.553582 −0.276791 0.960930i \(-0.589271\pi\)
−0.276791 + 0.960930i \(0.589271\pi\)
\(548\) −6.16298 −0.263270
\(549\) −3.74963 −0.160030
\(550\) 14.8278 0.632258
\(551\) −0.0644871 −0.00274724
\(552\) 34.6267 1.47381
\(553\) 0 0
\(554\) −45.8771 −1.94913
\(555\) −8.36679 −0.355150
\(556\) 51.5891 2.18787
\(557\) −6.40680 −0.271465 −0.135732 0.990746i \(-0.543339\pi\)
−0.135732 + 0.990746i \(0.543339\pi\)
\(558\) −15.0137 −0.635581
\(559\) 0 0
\(560\) 0 0
\(561\) 4.58827 0.193717
\(562\) 32.2996 1.36248
\(563\) 7.32084 0.308537 0.154268 0.988029i \(-0.450698\pi\)
0.154268 + 0.988029i \(0.450698\pi\)
\(564\) −1.75309 −0.0738185
\(565\) −2.94652 −0.123961
\(566\) 2.33226 0.0980324
\(567\) 0 0
\(568\) 42.7249 1.79270
\(569\) −4.31743 −0.180996 −0.0904981 0.995897i \(-0.528846\pi\)
−0.0904981 + 0.995897i \(0.528846\pi\)
\(570\) 0.136368 0.00571184
\(571\) 34.1695 1.42995 0.714974 0.699152i \(-0.246439\pi\)
0.714974 + 0.699152i \(0.246439\pi\)
\(572\) 0 0
\(573\) 2.22956 0.0931413
\(574\) 0 0
\(575\) 33.4804 1.39623
\(576\) 10.6233 0.442639
\(577\) 6.35656 0.264627 0.132314 0.991208i \(-0.457759\pi\)
0.132314 + 0.991208i \(0.457759\pi\)
\(578\) −29.2729 −1.21759
\(579\) −10.2457 −0.425795
\(580\) 3.80763 0.158103
\(581\) 0 0
\(582\) −1.44643 −0.0599563
\(583\) −4.07695 −0.168850
\(584\) 20.3240 0.841014
\(585\) 0 0
\(586\) 0.458892 0.0189566
\(587\) −31.4120 −1.29651 −0.648256 0.761422i \(-0.724501\pi\)
−0.648256 + 0.761422i \(0.724501\pi\)
\(588\) 0 0
\(589\) −0.372827 −0.0153621
\(590\) 3.20286 0.131860
\(591\) −22.8035 −0.938011
\(592\) 2.09909 0.0862719
\(593\) −0.473013 −0.0194243 −0.00971215 0.999953i \(-0.503092\pi\)
−0.00971215 + 0.999953i \(0.503092\pi\)
\(594\) −19.5361 −0.801575
\(595\) 0 0
\(596\) −21.0320 −0.861505
\(597\) −9.74670 −0.398906
\(598\) 0 0
\(599\) −9.62695 −0.393347 −0.196673 0.980469i \(-0.563014\pi\)
−0.196673 + 0.980469i \(0.563014\pi\)
\(600\) 18.9641 0.774207
\(601\) −41.0799 −1.67568 −0.837842 0.545914i \(-0.816183\pi\)
−0.837842 + 0.545914i \(0.816183\pi\)
\(602\) 0 0
\(603\) 1.69379 0.0689765
\(604\) 2.19485 0.0893073
\(605\) −7.40514 −0.301062
\(606\) −32.8155 −1.33304
\(607\) 19.0858 0.774668 0.387334 0.921939i \(-0.373396\pi\)
0.387334 + 0.921939i \(0.373396\pi\)
\(608\) 0.250905 0.0101755
\(609\) 0 0
\(610\) 8.82474 0.357303
\(611\) 0 0
\(612\) −5.67680 −0.229471
\(613\) −38.0048 −1.53500 −0.767500 0.641049i \(-0.778499\pi\)
−0.767500 + 0.641049i \(0.778499\pi\)
\(614\) 62.6498 2.52834
\(615\) −12.5067 −0.504318
\(616\) 0 0
\(617\) 8.31519 0.334757 0.167378 0.985893i \(-0.446470\pi\)
0.167378 + 0.985893i \(0.446470\pi\)
\(618\) 33.8616 1.36211
\(619\) −44.4728 −1.78751 −0.893756 0.448553i \(-0.851940\pi\)
−0.893756 + 0.448553i \(0.851940\pi\)
\(620\) 22.0135 0.884085
\(621\) −44.1116 −1.77014
\(622\) 62.4206 2.50284
\(623\) 0 0
\(624\) 0 0
\(625\) 14.7468 0.589874
\(626\) −50.8526 −2.03248
\(627\) −0.105054 −0.00419544
\(628\) −54.7842 −2.18613
\(629\) 13.8829 0.553549
\(630\) 0 0
\(631\) 11.7524 0.467858 0.233929 0.972254i \(-0.424842\pi\)
0.233929 + 0.972254i \(0.424842\pi\)
\(632\) 35.0453 1.39403
\(633\) 11.9258 0.474008
\(634\) −16.2800 −0.646562
\(635\) −13.3023 −0.527887
\(636\) −13.2052 −0.523620
\(637\) 0 0
\(638\) −4.70831 −0.186404
\(639\) −11.7863 −0.466259
\(640\) −16.0371 −0.633921
\(641\) 10.4868 0.414205 0.207102 0.978319i \(-0.433597\pi\)
0.207102 + 0.978319i \(0.433597\pi\)
\(642\) −33.4679 −1.32087
\(643\) 31.2822 1.23365 0.616825 0.787101i \(-0.288419\pi\)
0.616825 + 0.787101i \(0.288419\pi\)
\(644\) 0 0
\(645\) −11.5669 −0.455448
\(646\) −0.226275 −0.00890265
\(647\) −26.8675 −1.05627 −0.528135 0.849160i \(-0.677109\pi\)
−0.528135 + 0.849160i \(0.677109\pi\)
\(648\) −17.5084 −0.687796
\(649\) −2.46738 −0.0968530
\(650\) 0 0
\(651\) 0 0
\(652\) 29.9148 1.17155
\(653\) −4.14161 −0.162074 −0.0810369 0.996711i \(-0.525823\pi\)
−0.0810369 + 0.996711i \(0.525823\pi\)
\(654\) −39.3838 −1.54003
\(655\) −2.15650 −0.0842615
\(656\) 3.13771 0.122507
\(657\) −5.60668 −0.218738
\(658\) 0 0
\(659\) 21.4551 0.835773 0.417887 0.908499i \(-0.362771\pi\)
0.417887 + 0.908499i \(0.362771\pi\)
\(660\) 6.20288 0.241447
\(661\) 42.3872 1.64867 0.824335 0.566102i \(-0.191549\pi\)
0.824335 + 0.566102i \(0.191549\pi\)
\(662\) −15.1753 −0.589803
\(663\) 0 0
\(664\) 34.6762 1.34570
\(665\) 0 0
\(666\) −12.8004 −0.496006
\(667\) −10.6312 −0.411640
\(668\) −8.78349 −0.339844
\(669\) 23.6593 0.914722
\(670\) −3.98633 −0.154005
\(671\) −6.79830 −0.262445
\(672\) 0 0
\(673\) −29.5856 −1.14044 −0.570220 0.821492i \(-0.693142\pi\)
−0.570220 + 0.821492i \(0.693142\pi\)
\(674\) 9.72645 0.374649
\(675\) −24.1588 −0.929871
\(676\) 0 0
\(677\) 32.1659 1.23624 0.618118 0.786085i \(-0.287895\pi\)
0.618118 + 0.786085i \(0.287895\pi\)
\(678\) 11.8013 0.453227
\(679\) 0 0
\(680\) 5.27551 0.202306
\(681\) −1.90919 −0.0731604
\(682\) −27.2207 −1.04234
\(683\) −8.60236 −0.329160 −0.164580 0.986364i \(-0.552627\pi\)
−0.164580 + 0.986364i \(0.552627\pi\)
\(684\) 0.129977 0.00496980
\(685\) −1.57997 −0.0603674
\(686\) 0 0
\(687\) 30.6717 1.17020
\(688\) 2.90195 0.110636
\(689\) 0 0
\(690\) 22.4813 0.855847
\(691\) 20.4420 0.777651 0.388826 0.921311i \(-0.372881\pi\)
0.388826 + 0.921311i \(0.372881\pi\)
\(692\) 64.7359 2.46089
\(693\) 0 0
\(694\) −20.9477 −0.795164
\(695\) 13.2256 0.501675
\(696\) −6.02174 −0.228253
\(697\) 20.7522 0.786046
\(698\) 21.2373 0.803845
\(699\) 19.6020 0.741415
\(700\) 0 0
\(701\) 25.1373 0.949422 0.474711 0.880142i \(-0.342553\pi\)
0.474711 + 0.880142i \(0.342553\pi\)
\(702\) 0 0
\(703\) −0.317866 −0.0119885
\(704\) 19.2607 0.725915
\(705\) −0.449429 −0.0169265
\(706\) 4.96236 0.186761
\(707\) 0 0
\(708\) −7.99182 −0.300351
\(709\) −29.4929 −1.10763 −0.553814 0.832640i \(-0.686828\pi\)
−0.553814 + 0.832640i \(0.686828\pi\)
\(710\) 27.7390 1.04103
\(711\) −9.66777 −0.362570
\(712\) −52.6360 −1.97262
\(713\) −61.4632 −2.30182
\(714\) 0 0
\(715\) 0 0
\(716\) −9.56254 −0.357369
\(717\) −19.6918 −0.735404
\(718\) 19.7104 0.735584
\(719\) −8.33153 −0.310713 −0.155357 0.987858i \(-0.549653\pi\)
−0.155357 + 0.987858i \(0.549653\pi\)
\(720\) −0.220044 −0.00820055
\(721\) 0 0
\(722\) −43.7569 −1.62846
\(723\) 1.22901 0.0457073
\(724\) 4.52143 0.168038
\(725\) −5.82240 −0.216239
\(726\) 29.6588 1.10074
\(727\) −9.66141 −0.358322 −0.179161 0.983820i \(-0.557338\pi\)
−0.179161 + 0.983820i \(0.557338\pi\)
\(728\) 0 0
\(729\) 29.7672 1.10249
\(730\) 13.1953 0.488381
\(731\) 19.1929 0.709875
\(732\) −22.0197 −0.813870
\(733\) −14.0179 −0.517762 −0.258881 0.965909i \(-0.583354\pi\)
−0.258881 + 0.965909i \(0.583354\pi\)
\(734\) 5.29344 0.195384
\(735\) 0 0
\(736\) 41.3635 1.52468
\(737\) 3.07094 0.113120
\(738\) −19.1341 −0.704335
\(739\) −38.8147 −1.42782 −0.713910 0.700237i \(-0.753077\pi\)
−0.713910 + 0.700237i \(0.753077\pi\)
\(740\) 18.7683 0.689938
\(741\) 0 0
\(742\) 0 0
\(743\) −34.3942 −1.26180 −0.630901 0.775863i \(-0.717315\pi\)
−0.630901 + 0.775863i \(0.717315\pi\)
\(744\) −34.8142 −1.27635
\(745\) −5.39185 −0.197542
\(746\) 27.1057 0.992410
\(747\) −9.56594 −0.349999
\(748\) −10.2924 −0.376327
\(749\) 0 0
\(750\) 26.6890 0.974545
\(751\) 48.1470 1.75691 0.878454 0.477827i \(-0.158575\pi\)
0.878454 + 0.477827i \(0.158575\pi\)
\(752\) 0.112754 0.00411172
\(753\) −40.1816 −1.46430
\(754\) 0 0
\(755\) 0.562681 0.0204781
\(756\) 0 0
\(757\) −6.90638 −0.251016 −0.125508 0.992093i \(-0.540056\pi\)
−0.125508 + 0.992093i \(0.540056\pi\)
\(758\) 18.4052 0.668508
\(759\) −17.3188 −0.628634
\(760\) −0.120789 −0.00438147
\(761\) −31.9730 −1.15902 −0.579511 0.814965i \(-0.696756\pi\)
−0.579511 + 0.814965i \(0.696756\pi\)
\(762\) 53.2781 1.93006
\(763\) 0 0
\(764\) −5.00135 −0.180942
\(765\) −1.45533 −0.0526174
\(766\) 65.0548 2.35053
\(767\) 0 0
\(768\) 26.4795 0.955495
\(769\) −14.3950 −0.519099 −0.259549 0.965730i \(-0.583574\pi\)
−0.259549 + 0.965730i \(0.583574\pi\)
\(770\) 0 0
\(771\) −9.65328 −0.347654
\(772\) 22.9830 0.827177
\(773\) 37.2771 1.34076 0.670382 0.742016i \(-0.266130\pi\)
0.670382 + 0.742016i \(0.266130\pi\)
\(774\) −17.6964 −0.636083
\(775\) −33.6618 −1.20917
\(776\) 1.28118 0.0459917
\(777\) 0 0
\(778\) 17.6911 0.634255
\(779\) −0.475146 −0.0170239
\(780\) 0 0
\(781\) −21.3693 −0.764652
\(782\) −37.3029 −1.33395
\(783\) 7.67121 0.274147
\(784\) 0 0
\(785\) −14.0447 −0.501276
\(786\) 8.63715 0.308077
\(787\) −14.3486 −0.511472 −0.255736 0.966747i \(-0.582318\pi\)
−0.255736 + 0.966747i \(0.582318\pi\)
\(788\) 51.1527 1.82224
\(789\) 33.2818 1.18486
\(790\) 22.7531 0.809518
\(791\) 0 0
\(792\) 3.74719 0.133150
\(793\) 0 0
\(794\) 17.1658 0.609192
\(795\) −3.38534 −0.120066
\(796\) 21.8638 0.774940
\(797\) 11.0844 0.392629 0.196314 0.980541i \(-0.437103\pi\)
0.196314 + 0.980541i \(0.437103\pi\)
\(798\) 0 0
\(799\) 0.745733 0.0263821
\(800\) 22.6537 0.800929
\(801\) 14.5204 0.513054
\(802\) 41.9154 1.48008
\(803\) −10.1652 −0.358724
\(804\) 9.94676 0.350795
\(805\) 0 0
\(806\) 0 0
\(807\) −23.5757 −0.829904
\(808\) 29.0665 1.02255
\(809\) −42.5536 −1.49610 −0.748052 0.663640i \(-0.769011\pi\)
−0.748052 + 0.663640i \(0.769011\pi\)
\(810\) −11.3673 −0.399406
\(811\) −16.3622 −0.574554 −0.287277 0.957848i \(-0.592750\pi\)
−0.287277 + 0.957848i \(0.592750\pi\)
\(812\) 0 0
\(813\) −12.9057 −0.452622
\(814\) −23.2079 −0.813437
\(815\) 7.66906 0.268636
\(816\) −0.955843 −0.0334612
\(817\) −0.439444 −0.0153742
\(818\) 67.4455 2.35818
\(819\) 0 0
\(820\) 28.0549 0.979721
\(821\) 3.10550 0.108383 0.0541913 0.998531i \(-0.482742\pi\)
0.0541913 + 0.998531i \(0.482742\pi\)
\(822\) 6.32802 0.220715
\(823\) 49.0164 1.70860 0.854301 0.519778i \(-0.173985\pi\)
0.854301 + 0.519778i \(0.173985\pi\)
\(824\) −29.9930 −1.04486
\(825\) −9.48507 −0.330228
\(826\) 0 0
\(827\) −13.0887 −0.455140 −0.227570 0.973762i \(-0.573078\pi\)
−0.227570 + 0.973762i \(0.573078\pi\)
\(828\) 21.4276 0.744662
\(829\) 49.2565 1.71075 0.855374 0.518010i \(-0.173327\pi\)
0.855374 + 0.518010i \(0.173327\pi\)
\(830\) 22.5134 0.781452
\(831\) 29.3468 1.01803
\(832\) 0 0
\(833\) 0 0
\(834\) −52.9706 −1.83422
\(835\) −2.25177 −0.0779257
\(836\) 0.235656 0.00815034
\(837\) 44.3505 1.53298
\(838\) −47.7682 −1.65012
\(839\) −17.2636 −0.596007 −0.298004 0.954565i \(-0.596321\pi\)
−0.298004 + 0.954565i \(0.596321\pi\)
\(840\) 0 0
\(841\) −27.1512 −0.936248
\(842\) −57.2814 −1.97405
\(843\) −20.6615 −0.711621
\(844\) −26.7519 −0.920839
\(845\) 0 0
\(846\) −0.687585 −0.0236397
\(847\) 0 0
\(848\) 0.849323 0.0291659
\(849\) −1.49191 −0.0512022
\(850\) −20.4298 −0.700738
\(851\) −52.4024 −1.79633
\(852\) −69.2149 −2.37126
\(853\) −52.4163 −1.79470 −0.897350 0.441319i \(-0.854511\pi\)
−0.897350 + 0.441319i \(0.854511\pi\)
\(854\) 0 0
\(855\) 0.0333214 0.00113957
\(856\) 29.6443 1.01322
\(857\) 10.1271 0.345935 0.172967 0.984928i \(-0.444664\pi\)
0.172967 + 0.984928i \(0.444664\pi\)
\(858\) 0 0
\(859\) −0.510237 −0.0174090 −0.00870452 0.999962i \(-0.502771\pi\)
−0.00870452 + 0.999962i \(0.502771\pi\)
\(860\) 25.9469 0.884782
\(861\) 0 0
\(862\) −48.7573 −1.66068
\(863\) −20.4991 −0.697797 −0.348898 0.937161i \(-0.613444\pi\)
−0.348898 + 0.937161i \(0.613444\pi\)
\(864\) −29.8470 −1.01541
\(865\) 16.5959 0.564279
\(866\) −53.9646 −1.83379
\(867\) 18.7254 0.635947
\(868\) 0 0
\(869\) −17.5282 −0.594605
\(870\) −3.90960 −0.132548
\(871\) 0 0
\(872\) 34.8844 1.18133
\(873\) −0.353433 −0.0119619
\(874\) 0.854095 0.0288902
\(875\) 0 0
\(876\) −32.9252 −1.11244
\(877\) 11.2906 0.381256 0.190628 0.981662i \(-0.438948\pi\)
0.190628 + 0.981662i \(0.438948\pi\)
\(878\) −27.7276 −0.935762
\(879\) −0.293545 −0.00990103
\(880\) −0.398953 −0.0134487
\(881\) 22.5268 0.758947 0.379474 0.925203i \(-0.376105\pi\)
0.379474 + 0.925203i \(0.376105\pi\)
\(882\) 0 0
\(883\) 28.0268 0.943178 0.471589 0.881819i \(-0.343681\pi\)
0.471589 + 0.881819i \(0.343681\pi\)
\(884\) 0 0
\(885\) −2.04881 −0.0688701
\(886\) 36.2376 1.21743
\(887\) −20.6235 −0.692470 −0.346235 0.938148i \(-0.612540\pi\)
−0.346235 + 0.938148i \(0.612540\pi\)
\(888\) −29.6820 −0.996062
\(889\) 0 0
\(890\) −34.1738 −1.14551
\(891\) 8.75700 0.293371
\(892\) −53.0725 −1.77700
\(893\) −0.0170744 −0.000571374 0
\(894\) 21.5952 0.722253
\(895\) −2.45149 −0.0819443
\(896\) 0 0
\(897\) 0 0
\(898\) −59.8876 −1.99848
\(899\) 10.6887 0.356489
\(900\) 11.7353 0.391178
\(901\) 5.61725 0.187138
\(902\) −34.6912 −1.15509
\(903\) 0 0
\(904\) −10.4531 −0.347664
\(905\) 1.15913 0.0385308
\(906\) −2.25363 −0.0748718
\(907\) 41.4631 1.37676 0.688379 0.725351i \(-0.258322\pi\)
0.688379 + 0.725351i \(0.258322\pi\)
\(908\) 4.28269 0.142126
\(909\) −8.01841 −0.265954
\(910\) 0 0
\(911\) 40.8187 1.35239 0.676193 0.736725i \(-0.263629\pi\)
0.676193 + 0.736725i \(0.263629\pi\)
\(912\) 0.0218852 0.000724690 0
\(913\) −17.3436 −0.573990
\(914\) 70.9310 2.34619
\(915\) −5.64504 −0.186619
\(916\) −68.8027 −2.27331
\(917\) 0 0
\(918\) 26.9170 0.888393
\(919\) −48.7678 −1.60870 −0.804350 0.594155i \(-0.797486\pi\)
−0.804350 + 0.594155i \(0.797486\pi\)
\(920\) −19.9129 −0.656509
\(921\) −40.0760 −1.32055
\(922\) −78.5317 −2.58630
\(923\) 0 0
\(924\) 0 0
\(925\) −28.6994 −0.943631
\(926\) −3.89675 −0.128055
\(927\) 8.27403 0.271755
\(928\) −7.19330 −0.236132
\(929\) 29.3829 0.964023 0.482012 0.876165i \(-0.339906\pi\)
0.482012 + 0.876165i \(0.339906\pi\)
\(930\) −22.6030 −0.741183
\(931\) 0 0
\(932\) −43.9711 −1.44032
\(933\) −39.9294 −1.30723
\(934\) 65.3031 2.13678
\(935\) −2.63859 −0.0862912
\(936\) 0 0
\(937\) −21.0196 −0.686681 −0.343340 0.939211i \(-0.611558\pi\)
−0.343340 + 0.939211i \(0.611558\pi\)
\(938\) 0 0
\(939\) 32.5296 1.06156
\(940\) 1.00816 0.0328825
\(941\) 24.1033 0.785744 0.392872 0.919593i \(-0.371482\pi\)
0.392872 + 0.919593i \(0.371482\pi\)
\(942\) 56.2513 1.83277
\(943\) −78.3312 −2.55081
\(944\) 0.514013 0.0167297
\(945\) 0 0
\(946\) −32.0845 −1.04316
\(947\) 3.34046 0.108550 0.0542751 0.998526i \(-0.482715\pi\)
0.0542751 + 0.998526i \(0.482715\pi\)
\(948\) −56.7739 −1.84393
\(949\) 0 0
\(950\) 0.467765 0.0151763
\(951\) 10.4141 0.337699
\(952\) 0 0
\(953\) −4.97124 −0.161034 −0.0805171 0.996753i \(-0.525657\pi\)
−0.0805171 + 0.996753i \(0.525657\pi\)
\(954\) −5.17925 −0.167685
\(955\) −1.28216 −0.0414898
\(956\) 44.1726 1.42864
\(957\) 3.01183 0.0973585
\(958\) −14.4702 −0.467510
\(959\) 0 0
\(960\) 15.9933 0.516183
\(961\) 30.7961 0.993423
\(962\) 0 0
\(963\) −8.17783 −0.263527
\(964\) −2.75691 −0.0887939
\(965\) 5.89202 0.189671
\(966\) 0 0
\(967\) −47.4943 −1.52731 −0.763657 0.645623i \(-0.776598\pi\)
−0.763657 + 0.645623i \(0.776598\pi\)
\(968\) −26.2705 −0.844364
\(969\) 0.144744 0.00464985
\(970\) 0.831803 0.0267076
\(971\) 34.4715 1.10624 0.553121 0.833101i \(-0.313437\pi\)
0.553121 + 0.833101i \(0.313437\pi\)
\(972\) −27.5752 −0.884476
\(973\) 0 0
\(974\) 29.9830 0.960717
\(975\) 0 0
\(976\) 1.41624 0.0453329
\(977\) 13.3481 0.427044 0.213522 0.976938i \(-0.431506\pi\)
0.213522 + 0.976938i \(0.431506\pi\)
\(978\) −30.7159 −0.982185
\(979\) 26.3264 0.841395
\(980\) 0 0
\(981\) −9.62337 −0.307251
\(982\) −28.4507 −0.907898
\(983\) −12.5344 −0.399785 −0.199893 0.979818i \(-0.564059\pi\)
−0.199893 + 0.979818i \(0.564059\pi\)
\(984\) −44.3686 −1.41442
\(985\) 13.1137 0.417838
\(986\) 6.48715 0.206593
\(987\) 0 0
\(988\) 0 0
\(989\) −72.4455 −2.30363
\(990\) 2.43285 0.0773211
\(991\) −10.4119 −0.330745 −0.165373 0.986231i \(-0.552883\pi\)
−0.165373 + 0.986231i \(0.552883\pi\)
\(992\) −41.5875 −1.32040
\(993\) 9.70736 0.308054
\(994\) 0 0
\(995\) 5.60508 0.177693
\(996\) −56.1759 −1.78000
\(997\) 5.75270 0.182190 0.0910949 0.995842i \(-0.470963\pi\)
0.0910949 + 0.995842i \(0.470963\pi\)
\(998\) 21.0827 0.667362
\(999\) 37.8125 1.19633
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 8281.2.a.cp.1.11 12
7.2 even 3 1183.2.e.j.508.2 24
7.4 even 3 1183.2.e.j.170.2 24
7.6 odd 2 8281.2.a.co.1.11 12
13.6 odd 12 637.2.q.g.491.1 12
13.11 odd 12 637.2.q.g.589.1 12
13.12 even 2 inner 8281.2.a.cp.1.2 12
91.6 even 12 637.2.q.i.491.1 12
91.11 odd 12 91.2.u.b.30.6 yes 12
91.19 even 12 637.2.u.g.361.6 12
91.24 even 12 637.2.u.g.30.6 12
91.25 even 6 1183.2.e.j.170.11 24
91.32 odd 12 91.2.k.b.23.1 yes 12
91.37 odd 12 91.2.k.b.4.6 12
91.45 even 12 637.2.k.i.569.1 12
91.51 even 6 1183.2.e.j.508.11 24
91.58 odd 12 91.2.u.b.88.6 yes 12
91.76 even 12 637.2.q.i.589.1 12
91.89 even 12 637.2.k.i.459.6 12
91.90 odd 2 8281.2.a.co.1.2 12
273.11 even 12 819.2.do.e.667.1 12
273.32 even 12 819.2.bm.f.478.6 12
273.128 even 12 819.2.bm.f.550.1 12
273.149 even 12 819.2.do.e.361.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.k.b.4.6 12 91.37 odd 12
91.2.k.b.23.1 yes 12 91.32 odd 12
91.2.u.b.30.6 yes 12 91.11 odd 12
91.2.u.b.88.6 yes 12 91.58 odd 12
637.2.k.i.459.6 12 91.89 even 12
637.2.k.i.569.1 12 91.45 even 12
637.2.q.g.491.1 12 13.6 odd 12
637.2.q.g.589.1 12 13.11 odd 12
637.2.q.i.491.1 12 91.6 even 12
637.2.q.i.589.1 12 91.76 even 12
637.2.u.g.30.6 12 91.24 even 12
637.2.u.g.361.6 12 91.19 even 12
819.2.bm.f.478.6 12 273.32 even 12
819.2.bm.f.550.1 12 273.128 even 12
819.2.do.e.361.1 12 273.149 even 12
819.2.do.e.667.1 12 273.11 even 12
1183.2.e.j.170.2 24 7.4 even 3
1183.2.e.j.170.11 24 91.25 even 6
1183.2.e.j.508.2 24 7.2 even 3
1183.2.e.j.508.11 24 91.51 even 6
8281.2.a.co.1.2 12 91.90 odd 2
8281.2.a.co.1.11 12 7.6 odd 2
8281.2.a.cp.1.2 12 13.12 even 2 inner
8281.2.a.cp.1.11 12 1.1 even 1 trivial