Properties

Label 8281.2.a.cp.1.2
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

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 1.2
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 \(-0.802894\pi\)
−0.814328 + 0.580405i \(0.802894\pi\)
\(3\) −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 \(-0.560674\pi\)
−0.189460 + 0.981888i \(0.560674\pi\)
\(6\) 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 \(-0.427224\pi\)
0.226646 + 0.973977i \(0.427224\pi\)
\(12\) −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 \(-0.498268\pi\)
0.00544027 + 0.999985i \(0.498268\pi\)
\(20\) −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 \(-0.749477\pi\)
−0.705943 + 0.708269i \(0.749477\pi\)
\(32\) 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 \(-0.685722\pi\)
−0.550917 + 0.834560i \(0.685722\pi\)
\(38\) −0.109237 −0.0177206
\(39\) 0 0
\(40\) 2.54683 0.402689
\(41\) −10.0184 −1.56462 −0.782309 0.622891i \(-0.785958\pi\)
−0.782309 + 0.622891i \(0.785958\pi\)
\(42\) 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 \(-0.508359\pi\)
−0.0262567 + 0.999655i \(0.508359\pi\)
\(48\) −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 \(-0.534071\pi\)
−0.106833 + 0.994277i \(0.534071\pi\)
\(60\) 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 \(-0.460179\pi\)
0.124775 + 0.992185i \(0.460179\pi\)
\(68\) 6.84606 0.830206
\(69\) 11.5198 1.38682
\(70\) 0 0
\(71\) −14.2139 −1.68688 −0.843442 0.537220i \(-0.819474\pi\)
−0.843442 + 0.537220i \(0.819474\pi\)
\(72\) 2.49247 0.293741
\(73\) −6.76150 −0.791373 −0.395687 0.918386i \(-0.629493\pi\)
−0.395687 + 0.918386i \(0.629493\pi\)
\(74\) 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 \(-0.718232\pi\)
−0.633133 + 0.774043i \(0.718232\pi\)
\(84\) 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 \(-0.121448\pi\)
0.928093 + 0.372350i \(0.121448\pi\)
\(90\) −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 \(-0.506888\pi\)
−0.0216385 + 0.999766i \(0.506888\pi\)
\(98\) 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 \(-0.687589\pi\)
−0.555803 + 0.831314i \(0.687589\pi\)
\(110\) 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 \(-0.474617\pi\)
0.0796571 + 0.996822i \(0.474617\pi\)
\(138\) −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 \(-0.416058\pi\)
0.260664 + 0.965429i \(0.416058\pi\)
\(150\) −14.5315 −1.18649
\(151\) −0.664094 −0.0540432 −0.0270216 0.999635i \(-0.508602\pi\)
−0.0270216 + 0.999635i \(0.508602\pi\)
\(152\) −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 \(-0.615340\pi\)
−0.354475 + 0.935065i \(0.615340\pi\)
\(164\) −33.1113 −2.58556
\(165\) 1.87680 0.146108
\(166\) 26.5710 2.06231
\(167\) 2.65761 0.205652 0.102826 0.994699i \(-0.467212\pi\)
0.102826 + 0.994699i \(0.467212\pi\)
\(168\) 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 \(-0.580522\pi\)
−0.250278 + 0.968174i \(0.580522\pi\)
\(194\) 0.981719 0.0704834
\(195\) 0 0
\(196\) 0 0
\(197\) −15.4772 −1.10271 −0.551353 0.834272i \(-0.685888\pi\)
−0.551353 + 0.834272i \(0.685888\pi\)
\(198\) 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 \(-0.319307\pi\)
0.537664 + 0.843159i \(0.319307\pi\)
\(224\) 0 0
\(225\) 3.55075 0.236716
\(226\) 8.00979 0.532804
\(227\) −1.29581 −0.0860057 −0.0430029 0.999075i \(-0.513692\pi\)
−0.0430029 + 0.999075i \(0.513692\pi\)
\(228\) −0.230947 −0.0152948
\(229\) 20.8175 1.37566 0.687831 0.725871i \(-0.258563\pi\)
0.687831 + 0.725871i \(0.258563\pi\)
\(230\) −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 \(-0.642285\pi\)
−0.432263 + 0.901748i \(0.642285\pi\)
\(240\) 0.390981 0.0252377
\(241\) 0.834153 0.0537325 0.0268663 0.999639i \(-0.491447\pi\)
0.0268663 + 0.999639i \(0.491447\pi\)
\(242\) 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 \(-0.585717\pi\)
−0.266046 + 0.963960i \(0.585717\pi\)
\(272\) 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 \(-0.637368\pi\)
−0.418283 + 0.908317i \(0.637368\pi\)
\(282\) −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 \(-0.501852\pi\)
−0.00581972 + 0.999983i \(0.501852\pi\)
\(294\) 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 \(-0.782857\pi\)
−0.776204 + 0.630482i \(0.782857\pi\)
\(308\) 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 \(-0.436394\pi\)
0.198496 + 0.980102i \(0.436394\pi\)
\(318\) 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 \(-0.442044\pi\)
0.181071 + 0.983470i \(0.442044\pi\)
\(332\) −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 \(-0.579373\pi\)
−0.246782 + 0.969071i \(0.579373\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 7.95349 0.423922
\(353\) −2.15449 −0.114672 −0.0573359 0.998355i \(-0.518261\pi\)
−0.0573359 + 0.998355i \(0.518261\pi\)
\(354\) −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 \(-0.572508\pi\)
−0.225825 + 0.974168i \(0.572508\pi\)
\(360\) −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 \(-0.565795\pi\)
−0.205233 + 0.978713i \(0.565795\pi\)
\(380\) −0.132812 −0.00681310
\(381\) 23.1315 1.18506
\(382\) 3.48542 0.178329
\(383\) −28.2446 −1.44323 −0.721616 0.692294i \(-0.756600\pi\)
−0.721616 + 0.692294i \(0.756600\pi\)
\(384\) −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 \(-0.559884\pi\)
−0.187023 + 0.982356i \(0.559884\pi\)
\(398\) −15.2368 −0.763750
\(399\) 0 0
\(400\) −1.34113 −0.0670563
\(401\) −18.1982 −0.908777 −0.454389 0.890804i \(-0.650142\pi\)
−0.454389 + 0.890804i \(0.650142\pi\)
\(402\) 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 \(-0.757682\pi\)
−0.723964 + 0.689838i \(0.757682\pi\)
\(410\) −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 \(-0.292759\pi\)
0.606036 + 0.795437i \(0.292759\pi\)
\(422\) 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 \(-0.329708\pi\)
0.509832 + 0.860274i \(0.329708\pi\)
\(432\) 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 \(-0.289747\pi\)
0.613536 + 0.789667i \(0.289747\pi\)
\(450\) −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 \(-0.755989\pi\)
−0.720284 + 0.693679i \(0.755989\pi\)
\(458\) −47.9483 −2.24048
\(459\) 11.6864 0.545476
\(460\) 21.8949 1.02086
\(461\) 34.0958 1.58800 0.794000 0.607918i \(-0.207995\pi\)
0.794000 + 0.607918i \(0.207995\pi\)
\(462\) 0 0
\(463\) 1.69184 0.0786263 0.0393131 0.999227i \(-0.487483\pi\)
0.0393131 + 0.999227i \(0.487483\pi\)
\(464\) 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 \(-0.454156\pi\)
0.143526 + 0.989646i \(0.454156\pi\)
\(480\) 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 \(-0.595300\pi\)
−0.294942 + 0.955515i \(0.595300\pi\)
\(488\) −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 \(-0.565681\pi\)
−0.204881 + 0.978787i \(0.565681\pi\)
\(500\) 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 \(-0.172449\pi\)
0.856800 + 0.515649i \(0.172449\pi\)
\(510\) −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 \(-0.669791\pi\)
−0.508476 + 0.861076i \(0.669791\pi\)
\(542\) 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 \(-0.456661\pi\)
0.135732 + 0.990746i \(0.456661\pi\)
\(558\) −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 \(-0.542241\pi\)
−0.132314 + 0.991208i \(0.542241\pi\)
\(578\) 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 \(-0.275499\pi\)
0.648256 + 0.761422i \(0.275499\pi\)
\(588\) 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 \(-0.496908\pi\)
0.00971215 + 0.999953i \(0.496908\pi\)
\(594\) −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 \(-0.221501\pi\)
0.767500 + 0.641049i \(0.221501\pi\)
\(614\) 62.6498 2.52834
\(615\) −12.5067 −0.504318
\(616\) 0 0
\(617\) −8.31519 −0.334757 −0.167378 0.985893i \(-0.553530\pi\)
−0.167378 + 0.985893i \(0.553530\pi\)
\(618\) −33.8616 −1.36211
\(619\) 44.4728 1.78751 0.893756 0.448553i \(-0.148060\pi\)
0.893756 + 0.448553i \(0.148060\pi\)
\(620\) 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 \(-0.575158\pi\)
−0.233929 + 0.972254i \(0.575158\pi\)
\(632\) −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 \(-0.711581\pi\)
−0.616825 + 0.787101i \(0.711581\pi\)
\(644\) 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 \(-0.808451\pi\)
−0.824335 + 0.566102i \(0.808451\pi\)
\(662\) −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 \(-0.447373\pi\)
0.164580 + 0.986364i \(0.447373\pi\)
\(684\) −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 \(-0.627119\pi\)
−0.388826 + 0.921311i \(0.627119\pi\)
\(692\) 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 \(-0.313172\pi\)
0.553814 + 0.832640i \(0.313172\pi\)
\(710\) −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 \(-0.416646\pi\)
0.258881 + 0.965909i \(0.416646\pi\)
\(734\) −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 \(-0.246923\pi\)
0.713910 + 0.700237i \(0.246923\pi\)
\(740\) 18.7683 0.689938
\(741\) 0 0
\(742\) 0 0
\(743\) 34.3942 1.26180 0.630901 0.775863i \(-0.282685\pi\)
0.630901 + 0.775863i \(0.282685\pi\)
\(744\) −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 \(-0.303244\pi\)
0.579511 + 0.814965i \(0.303244\pi\)
\(762\) −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 \(-0.416426\pi\)
0.259549 + 0.965730i \(0.416426\pi\)
\(770\) 0 0
\(771\) −9.65328 −0.347654
\(772\) −22.9830 −0.827177
\(773\) −37.2771 −1.34076 −0.670382 0.742016i \(-0.733870\pi\)
−0.670382 + 0.742016i \(0.733870\pi\)
\(774\) 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 \(-0.417682\pi\)
0.255736 + 0.966747i \(0.417682\pi\)
\(788\) −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 \(-0.407250\pi\)
0.287277 + 0.957848i \(0.407250\pi\)
\(812\) 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 \(-0.517258\pi\)
−0.0541913 + 0.998531i \(0.517258\pi\)
\(822\) 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 \(-0.426922\pi\)
0.227570 + 0.973762i \(0.426922\pi\)
\(828\) 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 \(-0.403679\pi\)
0.298004 + 0.954565i \(0.403679\pi\)
\(840\) 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 \(-0.145489\pi\)
0.897350 + 0.441319i \(0.145489\pi\)
\(854\) 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 \(-0.386556\pi\)
0.348898 + 0.937161i \(0.386556\pi\)
\(864\) 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 \(-0.561052\pi\)
−0.190628 + 0.981662i \(0.561052\pi\)
\(878\) 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 \(-0.660094\pi\)
−0.482012 + 0.876165i \(0.660094\pi\)
\(930\) 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 \(-0.628518\pi\)
−0.392872 + 0.919593i \(0.628518\pi\)
\(942\) −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 \(-0.517285\pi\)
−0.0542751 + 0.998526i \(0.517285\pi\)
\(948\) −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 \(-0.223402\pi\)
0.763657 + 0.645623i \(0.223402\pi\)
\(968\) 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 \(-0.568494\pi\)
−0.213522 + 0.976938i \(0.568494\pi\)
\(978\) −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 \(-0.435941\pi\)
0.199893 + 0.979818i \(0.435941\pi\)
\(984\) −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.1.2 12
7.2 even 3 1183.2.e.j.508.11 24
7.4 even 3 1183.2.e.j.170.11 24
7.6 odd 2 8281.2.a.co.1.2 12
13.2 odd 12 637.2.q.g.589.1 12
13.7 odd 12 637.2.q.g.491.1 12
13.12 even 2 inner 8281.2.a.cp.1.11 12
91.2 odd 12 91.2.k.b.4.6 12
91.20 even 12 637.2.q.i.491.1 12
91.25 even 6 1183.2.e.j.170.2 24
91.33 even 12 637.2.u.g.361.6 12
91.41 even 12 637.2.q.i.589.1 12
91.46 odd 12 91.2.k.b.23.1 yes 12
91.51 even 6 1183.2.e.j.508.2 24
91.54 even 12 637.2.k.i.459.6 12
91.59 even 12 637.2.k.i.569.1 12
91.67 odd 12 91.2.u.b.30.6 yes 12
91.72 odd 12 91.2.u.b.88.6 yes 12
91.80 even 12 637.2.u.g.30.6 12
91.90 odd 2 8281.2.a.co.1.11 12
273.2 even 12 819.2.bm.f.550.1 12
273.137 even 12 819.2.bm.f.478.6 12
273.158 even 12 819.2.do.e.667.1 12
273.254 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 91.2 odd 12
91.2.k.b.23.1 yes 12 91.46 odd 12
91.2.u.b.30.6 yes 12 91.67 odd 12
91.2.u.b.88.6 yes 12 91.72 odd 12
637.2.k.i.459.6 12 91.54 even 12
637.2.k.i.569.1 12 91.59 even 12
637.2.q.g.491.1 12 13.7 odd 12
637.2.q.g.589.1 12 13.2 odd 12
637.2.q.i.491.1 12 91.20 even 12
637.2.q.i.589.1 12 91.41 even 12
637.2.u.g.30.6 12 91.80 even 12
637.2.u.g.361.6 12 91.33 even 12
819.2.bm.f.478.6 12 273.137 even 12
819.2.bm.f.550.1 12 273.2 even 12
819.2.do.e.361.1 12 273.254 even 12
819.2.do.e.667.1 12 273.158 even 12
1183.2.e.j.170.2 24 91.25 even 6
1183.2.e.j.170.11 24 7.4 even 3
1183.2.e.j.508.2 24 91.51 even 6
1183.2.e.j.508.11 24 7.2 even 3
8281.2.a.co.1.2 12 7.6 odd 2
8281.2.a.co.1.11 12 91.90 odd 2
8281.2.a.cp.1.2 12 1.1 even 1 trivial
8281.2.a.cp.1.11 12 13.12 even 2 inner