Properties

Label 8004.2.a.i.1.3
Level $8004$
Weight $2$
Character 8004.1
Self dual yes
Analytic conductor $63.912$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 5 x^{15} - 43 x^{14} + 234 x^{13} + 634 x^{12} - 4048 x^{11} - 3483 x^{10} + 32512 x^{9} - 137 x^{8} - 121665 x^{7} + 60168 x^{6} + 172218 x^{5} - 138024 x^{4} - 17844 x^{3} + 14352 x^{2} + 72 x - 208\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-2.66915\) of defining polynomial
Character \(\chi\) \(=\) 8004.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.66915 q^{5} -2.40707 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.66915 q^{5} -2.40707 q^{7} +1.00000 q^{9} -0.449906 q^{11} -7.00023 q^{13} +2.66915 q^{15} -4.21903 q^{17} +5.92988 q^{19} +2.40707 q^{21} +1.00000 q^{23} +2.12436 q^{25} -1.00000 q^{27} +1.00000 q^{29} +1.72098 q^{31} +0.449906 q^{33} +6.42483 q^{35} -10.3678 q^{37} +7.00023 q^{39} -2.78180 q^{41} -3.55621 q^{43} -2.66915 q^{45} -10.5877 q^{47} -1.20602 q^{49} +4.21903 q^{51} -14.2130 q^{53} +1.20087 q^{55} -5.92988 q^{57} -2.05076 q^{59} -14.9792 q^{61} -2.40707 q^{63} +18.6847 q^{65} -7.71769 q^{67} -1.00000 q^{69} -10.9512 q^{71} +5.88086 q^{73} -2.12436 q^{75} +1.08296 q^{77} -15.0372 q^{79} +1.00000 q^{81} +11.2646 q^{83} +11.2612 q^{85} -1.00000 q^{87} -6.99588 q^{89} +16.8501 q^{91} -1.72098 q^{93} -15.8277 q^{95} -13.7094 q^{97} -0.449906 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + O(q^{10}) \) \( 16q - 16q^{3} + 5q^{5} - 4q^{7} + 16q^{9} + 5q^{11} + 8q^{13} - 5q^{15} + 7q^{17} + q^{19} + 4q^{21} + 16q^{23} + 31q^{25} - 16q^{27} + 16q^{29} - 2q^{31} - 5q^{33} + 5q^{35} + 14q^{37} - 8q^{39} - q^{41} - 13q^{43} + 5q^{45} - 4q^{47} + 30q^{49} - 7q^{51} + 19q^{53} - 37q^{55} - q^{57} + 12q^{59} + 21q^{61} - 4q^{63} + 26q^{65} - 11q^{67} - 16q^{69} + 7q^{71} - 13q^{73} - 31q^{75} + 4q^{77} - 18q^{79} + 16q^{81} + 25q^{83} + 48q^{85} - 16q^{87} + 12q^{89} - 11q^{91} + 2q^{93} - q^{95} + 5q^{97} + 5q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −2.66915 −1.19368 −0.596840 0.802360i \(-0.703577\pi\)
−0.596840 + 0.802360i \(0.703577\pi\)
\(6\) 0 0
\(7\) −2.40707 −0.909787 −0.454893 0.890546i \(-0.650323\pi\)
−0.454893 + 0.890546i \(0.650323\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.449906 −0.135652 −0.0678259 0.997697i \(-0.521606\pi\)
−0.0678259 + 0.997697i \(0.521606\pi\)
\(12\) 0 0
\(13\) −7.00023 −1.94152 −0.970758 0.240061i \(-0.922833\pi\)
−0.970758 + 0.240061i \(0.922833\pi\)
\(14\) 0 0
\(15\) 2.66915 0.689171
\(16\) 0 0
\(17\) −4.21903 −1.02327 −0.511633 0.859204i \(-0.670959\pi\)
−0.511633 + 0.859204i \(0.670959\pi\)
\(18\) 0 0
\(19\) 5.92988 1.36041 0.680204 0.733023i \(-0.261891\pi\)
0.680204 + 0.733023i \(0.261891\pi\)
\(20\) 0 0
\(21\) 2.40707 0.525266
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 2.12436 0.424872
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 1.72098 0.309098 0.154549 0.987985i \(-0.450608\pi\)
0.154549 + 0.987985i \(0.450608\pi\)
\(32\) 0 0
\(33\) 0.449906 0.0783186
\(34\) 0 0
\(35\) 6.42483 1.08599
\(36\) 0 0
\(37\) −10.3678 −1.70445 −0.852226 0.523174i \(-0.824748\pi\)
−0.852226 + 0.523174i \(0.824748\pi\)
\(38\) 0 0
\(39\) 7.00023 1.12093
\(40\) 0 0
\(41\) −2.78180 −0.434445 −0.217222 0.976122i \(-0.569700\pi\)
−0.217222 + 0.976122i \(0.569700\pi\)
\(42\) 0 0
\(43\) −3.55621 −0.542317 −0.271159 0.962535i \(-0.587407\pi\)
−0.271159 + 0.962535i \(0.587407\pi\)
\(44\) 0 0
\(45\) −2.66915 −0.397893
\(46\) 0 0
\(47\) −10.5877 −1.54438 −0.772190 0.635392i \(-0.780838\pi\)
−0.772190 + 0.635392i \(0.780838\pi\)
\(48\) 0 0
\(49\) −1.20602 −0.172288
\(50\) 0 0
\(51\) 4.21903 0.590783
\(52\) 0 0
\(53\) −14.2130 −1.95230 −0.976150 0.217098i \(-0.930341\pi\)
−0.976150 + 0.217098i \(0.930341\pi\)
\(54\) 0 0
\(55\) 1.20087 0.161925
\(56\) 0 0
\(57\) −5.92988 −0.785432
\(58\) 0 0
\(59\) −2.05076 −0.266986 −0.133493 0.991050i \(-0.542619\pi\)
−0.133493 + 0.991050i \(0.542619\pi\)
\(60\) 0 0
\(61\) −14.9792 −1.91789 −0.958943 0.283600i \(-0.908471\pi\)
−0.958943 + 0.283600i \(0.908471\pi\)
\(62\) 0 0
\(63\) −2.40707 −0.303262
\(64\) 0 0
\(65\) 18.6847 2.31755
\(66\) 0 0
\(67\) −7.71769 −0.942866 −0.471433 0.881902i \(-0.656263\pi\)
−0.471433 + 0.881902i \(0.656263\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −10.9512 −1.29967 −0.649833 0.760077i \(-0.725161\pi\)
−0.649833 + 0.760077i \(0.725161\pi\)
\(72\) 0 0
\(73\) 5.88086 0.688302 0.344151 0.938914i \(-0.388167\pi\)
0.344151 + 0.938914i \(0.388167\pi\)
\(74\) 0 0
\(75\) −2.12436 −0.245300
\(76\) 0 0
\(77\) 1.08296 0.123414
\(78\) 0 0
\(79\) −15.0372 −1.69182 −0.845909 0.533328i \(-0.820941\pi\)
−0.845909 + 0.533328i \(0.820941\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 11.2646 1.23645 0.618224 0.786002i \(-0.287852\pi\)
0.618224 + 0.786002i \(0.287852\pi\)
\(84\) 0 0
\(85\) 11.2612 1.22145
\(86\) 0 0
\(87\) −1.00000 −0.107211
\(88\) 0 0
\(89\) −6.99588 −0.741561 −0.370781 0.928720i \(-0.620910\pi\)
−0.370781 + 0.928720i \(0.620910\pi\)
\(90\) 0 0
\(91\) 16.8501 1.76637
\(92\) 0 0
\(93\) −1.72098 −0.178458
\(94\) 0 0
\(95\) −15.8277 −1.62389
\(96\) 0 0
\(97\) −13.7094 −1.39197 −0.695987 0.718054i \(-0.745033\pi\)
−0.695987 + 0.718054i \(0.745033\pi\)
\(98\) 0 0
\(99\) −0.449906 −0.0452173
\(100\) 0 0
\(101\) −9.74398 −0.969562 −0.484781 0.874635i \(-0.661101\pi\)
−0.484781 + 0.874635i \(0.661101\pi\)
\(102\) 0 0
\(103\) −1.67993 −0.165529 −0.0827643 0.996569i \(-0.526375\pi\)
−0.0827643 + 0.996569i \(0.526375\pi\)
\(104\) 0 0
\(105\) −6.42483 −0.626999
\(106\) 0 0
\(107\) 7.89307 0.763052 0.381526 0.924358i \(-0.375399\pi\)
0.381526 + 0.924358i \(0.375399\pi\)
\(108\) 0 0
\(109\) 14.1653 1.35679 0.678397 0.734696i \(-0.262675\pi\)
0.678397 + 0.734696i \(0.262675\pi\)
\(110\) 0 0
\(111\) 10.3678 0.984066
\(112\) 0 0
\(113\) −3.79851 −0.357334 −0.178667 0.983910i \(-0.557178\pi\)
−0.178667 + 0.983910i \(0.557178\pi\)
\(114\) 0 0
\(115\) −2.66915 −0.248899
\(116\) 0 0
\(117\) −7.00023 −0.647172
\(118\) 0 0
\(119\) 10.1555 0.930954
\(120\) 0 0
\(121\) −10.7976 −0.981599
\(122\) 0 0
\(123\) 2.78180 0.250827
\(124\) 0 0
\(125\) 7.67552 0.686519
\(126\) 0 0
\(127\) 15.8892 1.40994 0.704970 0.709238i \(-0.250961\pi\)
0.704970 + 0.709238i \(0.250961\pi\)
\(128\) 0 0
\(129\) 3.55621 0.313107
\(130\) 0 0
\(131\) 15.5379 1.35755 0.678775 0.734346i \(-0.262511\pi\)
0.678775 + 0.734346i \(0.262511\pi\)
\(132\) 0 0
\(133\) −14.2736 −1.23768
\(134\) 0 0
\(135\) 2.66915 0.229724
\(136\) 0 0
\(137\) 13.8302 1.18159 0.590797 0.806820i \(-0.298813\pi\)
0.590797 + 0.806820i \(0.298813\pi\)
\(138\) 0 0
\(139\) 0.638550 0.0541611 0.0270806 0.999633i \(-0.491379\pi\)
0.0270806 + 0.999633i \(0.491379\pi\)
\(140\) 0 0
\(141\) 10.5877 0.891648
\(142\) 0 0
\(143\) 3.14945 0.263370
\(144\) 0 0
\(145\) −2.66915 −0.221661
\(146\) 0 0
\(147\) 1.20602 0.0994705
\(148\) 0 0
\(149\) 19.8241 1.62405 0.812027 0.583620i \(-0.198364\pi\)
0.812027 + 0.583620i \(0.198364\pi\)
\(150\) 0 0
\(151\) −13.9692 −1.13680 −0.568400 0.822753i \(-0.692437\pi\)
−0.568400 + 0.822753i \(0.692437\pi\)
\(152\) 0 0
\(153\) −4.21903 −0.341089
\(154\) 0 0
\(155\) −4.59357 −0.368964
\(156\) 0 0
\(157\) −0.437155 −0.0348887 −0.0174444 0.999848i \(-0.505553\pi\)
−0.0174444 + 0.999848i \(0.505553\pi\)
\(158\) 0 0
\(159\) 14.2130 1.12716
\(160\) 0 0
\(161\) −2.40707 −0.189704
\(162\) 0 0
\(163\) 20.0126 1.56751 0.783753 0.621073i \(-0.213303\pi\)
0.783753 + 0.621073i \(0.213303\pi\)
\(164\) 0 0
\(165\) −1.20087 −0.0934874
\(166\) 0 0
\(167\) −2.73360 −0.211532 −0.105766 0.994391i \(-0.533729\pi\)
−0.105766 + 0.994391i \(0.533729\pi\)
\(168\) 0 0
\(169\) 36.0033 2.76948
\(170\) 0 0
\(171\) 5.92988 0.453470
\(172\) 0 0
\(173\) −9.25088 −0.703332 −0.351666 0.936126i \(-0.614385\pi\)
−0.351666 + 0.936126i \(0.614385\pi\)
\(174\) 0 0
\(175\) −5.11348 −0.386543
\(176\) 0 0
\(177\) 2.05076 0.154144
\(178\) 0 0
\(179\) 13.7996 1.03143 0.515716 0.856760i \(-0.327526\pi\)
0.515716 + 0.856760i \(0.327526\pi\)
\(180\) 0 0
\(181\) 0.740084 0.0550100 0.0275050 0.999622i \(-0.491244\pi\)
0.0275050 + 0.999622i \(0.491244\pi\)
\(182\) 0 0
\(183\) 14.9792 1.10729
\(184\) 0 0
\(185\) 27.6731 2.03457
\(186\) 0 0
\(187\) 1.89817 0.138808
\(188\) 0 0
\(189\) 2.40707 0.175089
\(190\) 0 0
\(191\) −25.6565 −1.85644 −0.928219 0.372034i \(-0.878660\pi\)
−0.928219 + 0.372034i \(0.878660\pi\)
\(192\) 0 0
\(193\) −0.108371 −0.00780073 −0.00390037 0.999992i \(-0.501242\pi\)
−0.00390037 + 0.999992i \(0.501242\pi\)
\(194\) 0 0
\(195\) −18.6847 −1.33804
\(196\) 0 0
\(197\) −6.44256 −0.459014 −0.229507 0.973307i \(-0.573711\pi\)
−0.229507 + 0.973307i \(0.573711\pi\)
\(198\) 0 0
\(199\) −23.6398 −1.67578 −0.837892 0.545836i \(-0.816212\pi\)
−0.837892 + 0.545836i \(0.816212\pi\)
\(200\) 0 0
\(201\) 7.71769 0.544364
\(202\) 0 0
\(203\) −2.40707 −0.168943
\(204\) 0 0
\(205\) 7.42505 0.518588
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −2.66789 −0.184542
\(210\) 0 0
\(211\) −0.615086 −0.0423443 −0.0211721 0.999776i \(-0.506740\pi\)
−0.0211721 + 0.999776i \(0.506740\pi\)
\(212\) 0 0
\(213\) 10.9512 0.750363
\(214\) 0 0
\(215\) 9.49206 0.647353
\(216\) 0 0
\(217\) −4.14253 −0.281213
\(218\) 0 0
\(219\) −5.88086 −0.397392
\(220\) 0 0
\(221\) 29.5342 1.98669
\(222\) 0 0
\(223\) −21.9551 −1.47022 −0.735110 0.677948i \(-0.762870\pi\)
−0.735110 + 0.677948i \(0.762870\pi\)
\(224\) 0 0
\(225\) 2.12436 0.141624
\(226\) 0 0
\(227\) 2.38076 0.158017 0.0790084 0.996874i \(-0.474825\pi\)
0.0790084 + 0.996874i \(0.474825\pi\)
\(228\) 0 0
\(229\) 17.7836 1.17517 0.587586 0.809161i \(-0.300078\pi\)
0.587586 + 0.809161i \(0.300078\pi\)
\(230\) 0 0
\(231\) −1.08296 −0.0712532
\(232\) 0 0
\(233\) −25.3471 −1.66054 −0.830271 0.557359i \(-0.811815\pi\)
−0.830271 + 0.557359i \(0.811815\pi\)
\(234\) 0 0
\(235\) 28.2602 1.84349
\(236\) 0 0
\(237\) 15.0372 0.976771
\(238\) 0 0
\(239\) 6.87771 0.444882 0.222441 0.974946i \(-0.428597\pi\)
0.222441 + 0.974946i \(0.428597\pi\)
\(240\) 0 0
\(241\) 1.68799 0.108733 0.0543663 0.998521i \(-0.482686\pi\)
0.0543663 + 0.998521i \(0.482686\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 3.21903 0.205657
\(246\) 0 0
\(247\) −41.5106 −2.64125
\(248\) 0 0
\(249\) −11.2646 −0.713864
\(250\) 0 0
\(251\) −18.9014 −1.19305 −0.596524 0.802595i \(-0.703452\pi\)
−0.596524 + 0.802595i \(0.703452\pi\)
\(252\) 0 0
\(253\) −0.449906 −0.0282854
\(254\) 0 0
\(255\) −11.2612 −0.705206
\(256\) 0 0
\(257\) 2.03066 0.126669 0.0633345 0.997992i \(-0.479826\pi\)
0.0633345 + 0.997992i \(0.479826\pi\)
\(258\) 0 0
\(259\) 24.9560 1.55069
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) −2.51086 −0.154826 −0.0774130 0.996999i \(-0.524666\pi\)
−0.0774130 + 0.996999i \(0.524666\pi\)
\(264\) 0 0
\(265\) 37.9365 2.33042
\(266\) 0 0
\(267\) 6.99588 0.428141
\(268\) 0 0
\(269\) −15.1102 −0.921287 −0.460643 0.887585i \(-0.652381\pi\)
−0.460643 + 0.887585i \(0.652381\pi\)
\(270\) 0 0
\(271\) −2.81007 −0.170700 −0.0853499 0.996351i \(-0.527201\pi\)
−0.0853499 + 0.996351i \(0.527201\pi\)
\(272\) 0 0
\(273\) −16.8501 −1.01981
\(274\) 0 0
\(275\) −0.955762 −0.0576346
\(276\) 0 0
\(277\) 10.3519 0.621983 0.310992 0.950413i \(-0.399339\pi\)
0.310992 + 0.950413i \(0.399339\pi\)
\(278\) 0 0
\(279\) 1.72098 0.103033
\(280\) 0 0
\(281\) 11.0253 0.657716 0.328858 0.944379i \(-0.393336\pi\)
0.328858 + 0.944379i \(0.393336\pi\)
\(282\) 0 0
\(283\) −22.1196 −1.31487 −0.657437 0.753510i \(-0.728359\pi\)
−0.657437 + 0.753510i \(0.728359\pi\)
\(284\) 0 0
\(285\) 15.8277 0.937555
\(286\) 0 0
\(287\) 6.69599 0.395252
\(288\) 0 0
\(289\) 0.800250 0.0470735
\(290\) 0 0
\(291\) 13.7094 0.803657
\(292\) 0 0
\(293\) 2.71146 0.158405 0.0792026 0.996859i \(-0.474763\pi\)
0.0792026 + 0.996859i \(0.474763\pi\)
\(294\) 0 0
\(295\) 5.47378 0.318696
\(296\) 0 0
\(297\) 0.449906 0.0261062
\(298\) 0 0
\(299\) −7.00023 −0.404834
\(300\) 0 0
\(301\) 8.56005 0.493393
\(302\) 0 0
\(303\) 9.74398 0.559777
\(304\) 0 0
\(305\) 39.9816 2.28934
\(306\) 0 0
\(307\) −9.99953 −0.570703 −0.285352 0.958423i \(-0.592110\pi\)
−0.285352 + 0.958423i \(0.592110\pi\)
\(308\) 0 0
\(309\) 1.67993 0.0955680
\(310\) 0 0
\(311\) 28.8982 1.63867 0.819334 0.573317i \(-0.194344\pi\)
0.819334 + 0.573317i \(0.194344\pi\)
\(312\) 0 0
\(313\) −23.7841 −1.34436 −0.672178 0.740389i \(-0.734641\pi\)
−0.672178 + 0.740389i \(0.734641\pi\)
\(314\) 0 0
\(315\) 6.42483 0.361998
\(316\) 0 0
\(317\) 10.7561 0.604124 0.302062 0.953288i \(-0.402325\pi\)
0.302062 + 0.953288i \(0.402325\pi\)
\(318\) 0 0
\(319\) −0.449906 −0.0251899
\(320\) 0 0
\(321\) −7.89307 −0.440548
\(322\) 0 0
\(323\) −25.0184 −1.39206
\(324\) 0 0
\(325\) −14.8710 −0.824895
\(326\) 0 0
\(327\) −14.1653 −0.783345
\(328\) 0 0
\(329\) 25.4854 1.40506
\(330\) 0 0
\(331\) −4.47707 −0.246082 −0.123041 0.992402i \(-0.539265\pi\)
−0.123041 + 0.992402i \(0.539265\pi\)
\(332\) 0 0
\(333\) −10.3678 −0.568151
\(334\) 0 0
\(335\) 20.5997 1.12548
\(336\) 0 0
\(337\) −6.61471 −0.360326 −0.180163 0.983637i \(-0.557663\pi\)
−0.180163 + 0.983637i \(0.557663\pi\)
\(338\) 0 0
\(339\) 3.79851 0.206307
\(340\) 0 0
\(341\) −0.774282 −0.0419297
\(342\) 0 0
\(343\) 19.7525 1.06653
\(344\) 0 0
\(345\) 2.66915 0.143702
\(346\) 0 0
\(347\) −22.2158 −1.19261 −0.596303 0.802760i \(-0.703364\pi\)
−0.596303 + 0.802760i \(0.703364\pi\)
\(348\) 0 0
\(349\) 27.2496 1.45864 0.729318 0.684175i \(-0.239838\pi\)
0.729318 + 0.684175i \(0.239838\pi\)
\(350\) 0 0
\(351\) 7.00023 0.373645
\(352\) 0 0
\(353\) 2.64067 0.140549 0.0702744 0.997528i \(-0.477613\pi\)
0.0702744 + 0.997528i \(0.477613\pi\)
\(354\) 0 0
\(355\) 29.2303 1.55139
\(356\) 0 0
\(357\) −10.1555 −0.537487
\(358\) 0 0
\(359\) 2.65624 0.140191 0.0700954 0.997540i \(-0.477670\pi\)
0.0700954 + 0.997540i \(0.477670\pi\)
\(360\) 0 0
\(361\) 16.1635 0.850712
\(362\) 0 0
\(363\) 10.7976 0.566726
\(364\) 0 0
\(365\) −15.6969 −0.821613
\(366\) 0 0
\(367\) −19.6512 −1.02578 −0.512892 0.858453i \(-0.671426\pi\)
−0.512892 + 0.858453i \(0.671426\pi\)
\(368\) 0 0
\(369\) −2.78180 −0.144815
\(370\) 0 0
\(371\) 34.2116 1.77618
\(372\) 0 0
\(373\) 22.7065 1.17570 0.587850 0.808970i \(-0.299974\pi\)
0.587850 + 0.808970i \(0.299974\pi\)
\(374\) 0 0
\(375\) −7.67552 −0.396362
\(376\) 0 0
\(377\) −7.00023 −0.360530
\(378\) 0 0
\(379\) −18.4758 −0.949040 −0.474520 0.880245i \(-0.657378\pi\)
−0.474520 + 0.880245i \(0.657378\pi\)
\(380\) 0 0
\(381\) −15.8892 −0.814029
\(382\) 0 0
\(383\) 14.3973 0.735668 0.367834 0.929891i \(-0.380099\pi\)
0.367834 + 0.929891i \(0.380099\pi\)
\(384\) 0 0
\(385\) −2.89057 −0.147317
\(386\) 0 0
\(387\) −3.55621 −0.180772
\(388\) 0 0
\(389\) 18.5518 0.940614 0.470307 0.882503i \(-0.344143\pi\)
0.470307 + 0.882503i \(0.344143\pi\)
\(390\) 0 0
\(391\) −4.21903 −0.213366
\(392\) 0 0
\(393\) −15.5379 −0.783782
\(394\) 0 0
\(395\) 40.1365 2.01949
\(396\) 0 0
\(397\) −16.6792 −0.837104 −0.418552 0.908193i \(-0.637462\pi\)
−0.418552 + 0.908193i \(0.637462\pi\)
\(398\) 0 0
\(399\) 14.2736 0.714576
\(400\) 0 0
\(401\) 23.2765 1.16237 0.581186 0.813771i \(-0.302589\pi\)
0.581186 + 0.813771i \(0.302589\pi\)
\(402\) 0 0
\(403\) −12.0473 −0.600119
\(404\) 0 0
\(405\) −2.66915 −0.132631
\(406\) 0 0
\(407\) 4.66453 0.231212
\(408\) 0 0
\(409\) −29.3692 −1.45221 −0.726107 0.687582i \(-0.758672\pi\)
−0.726107 + 0.687582i \(0.758672\pi\)
\(410\) 0 0
\(411\) −13.8302 −0.682194
\(412\) 0 0
\(413\) 4.93632 0.242900
\(414\) 0 0
\(415\) −30.0668 −1.47592
\(416\) 0 0
\(417\) −0.638550 −0.0312699
\(418\) 0 0
\(419\) −6.70884 −0.327748 −0.163874 0.986481i \(-0.552399\pi\)
−0.163874 + 0.986481i \(0.552399\pi\)
\(420\) 0 0
\(421\) −11.5441 −0.562623 −0.281311 0.959617i \(-0.590769\pi\)
−0.281311 + 0.959617i \(0.590769\pi\)
\(422\) 0 0
\(423\) −10.5877 −0.514793
\(424\) 0 0
\(425\) −8.96274 −0.434757
\(426\) 0 0
\(427\) 36.0559 1.74487
\(428\) 0 0
\(429\) −3.14945 −0.152057
\(430\) 0 0
\(431\) 3.32052 0.159944 0.0799720 0.996797i \(-0.474517\pi\)
0.0799720 + 0.996797i \(0.474517\pi\)
\(432\) 0 0
\(433\) −41.0947 −1.97489 −0.987443 0.157979i \(-0.949502\pi\)
−0.987443 + 0.157979i \(0.949502\pi\)
\(434\) 0 0
\(435\) 2.66915 0.127976
\(436\) 0 0
\(437\) 5.92988 0.283665
\(438\) 0 0
\(439\) 38.7312 1.84854 0.924270 0.381740i \(-0.124675\pi\)
0.924270 + 0.381740i \(0.124675\pi\)
\(440\) 0 0
\(441\) −1.20602 −0.0574293
\(442\) 0 0
\(443\) 29.2691 1.39062 0.695308 0.718712i \(-0.255268\pi\)
0.695308 + 0.718712i \(0.255268\pi\)
\(444\) 0 0
\(445\) 18.6730 0.885187
\(446\) 0 0
\(447\) −19.8241 −0.937648
\(448\) 0 0
\(449\) 20.8790 0.985339 0.492669 0.870217i \(-0.336021\pi\)
0.492669 + 0.870217i \(0.336021\pi\)
\(450\) 0 0
\(451\) 1.25155 0.0589332
\(452\) 0 0
\(453\) 13.9692 0.656331
\(454\) 0 0
\(455\) −44.9753 −2.10847
\(456\) 0 0
\(457\) 7.93241 0.371063 0.185531 0.982638i \(-0.440599\pi\)
0.185531 + 0.982638i \(0.440599\pi\)
\(458\) 0 0
\(459\) 4.21903 0.196928
\(460\) 0 0
\(461\) −5.72475 −0.266628 −0.133314 0.991074i \(-0.542562\pi\)
−0.133314 + 0.991074i \(0.542562\pi\)
\(462\) 0 0
\(463\) −10.2304 −0.475447 −0.237724 0.971333i \(-0.576401\pi\)
−0.237724 + 0.971333i \(0.576401\pi\)
\(464\) 0 0
\(465\) 4.59357 0.213021
\(466\) 0 0
\(467\) 14.3498 0.664031 0.332015 0.943274i \(-0.392271\pi\)
0.332015 + 0.943274i \(0.392271\pi\)
\(468\) 0 0
\(469\) 18.5770 0.857807
\(470\) 0 0
\(471\) 0.437155 0.0201430
\(472\) 0 0
\(473\) 1.59996 0.0735663
\(474\) 0 0
\(475\) 12.5972 0.577999
\(476\) 0 0
\(477\) −14.2130 −0.650767
\(478\) 0 0
\(479\) 19.2456 0.879354 0.439677 0.898156i \(-0.355093\pi\)
0.439677 + 0.898156i \(0.355093\pi\)
\(480\) 0 0
\(481\) 72.5769 3.30922
\(482\) 0 0
\(483\) 2.40707 0.109525
\(484\) 0 0
\(485\) 36.5923 1.66157
\(486\) 0 0
\(487\) −3.45404 −0.156517 −0.0782587 0.996933i \(-0.524936\pi\)
−0.0782587 + 0.996933i \(0.524936\pi\)
\(488\) 0 0
\(489\) −20.0126 −0.905000
\(490\) 0 0
\(491\) −6.13006 −0.276646 −0.138323 0.990387i \(-0.544171\pi\)
−0.138323 + 0.990387i \(0.544171\pi\)
\(492\) 0 0
\(493\) −4.21903 −0.190016
\(494\) 0 0
\(495\) 1.20087 0.0539750
\(496\) 0 0
\(497\) 26.3603 1.18242
\(498\) 0 0
\(499\) −0.886059 −0.0396654 −0.0198327 0.999803i \(-0.506313\pi\)
−0.0198327 + 0.999803i \(0.506313\pi\)
\(500\) 0 0
\(501\) 2.73360 0.122128
\(502\) 0 0
\(503\) 33.9578 1.51410 0.757051 0.653356i \(-0.226639\pi\)
0.757051 + 0.653356i \(0.226639\pi\)
\(504\) 0 0
\(505\) 26.0081 1.15735
\(506\) 0 0
\(507\) −36.0033 −1.59896
\(508\) 0 0
\(509\) 7.31256 0.324123 0.162062 0.986781i \(-0.448186\pi\)
0.162062 + 0.986781i \(0.448186\pi\)
\(510\) 0 0
\(511\) −14.1556 −0.626209
\(512\) 0 0
\(513\) −5.92988 −0.261811
\(514\) 0 0
\(515\) 4.48399 0.197588
\(516\) 0 0
\(517\) 4.76349 0.209498
\(518\) 0 0
\(519\) 9.25088 0.406069
\(520\) 0 0
\(521\) 33.9412 1.48699 0.743495 0.668742i \(-0.233167\pi\)
0.743495 + 0.668742i \(0.233167\pi\)
\(522\) 0 0
\(523\) −22.0212 −0.962919 −0.481459 0.876468i \(-0.659893\pi\)
−0.481459 + 0.876468i \(0.659893\pi\)
\(524\) 0 0
\(525\) 5.11348 0.223171
\(526\) 0 0
\(527\) −7.26089 −0.316289
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −2.05076 −0.0889954
\(532\) 0 0
\(533\) 19.4733 0.843481
\(534\) 0 0
\(535\) −21.0678 −0.910840
\(536\) 0 0
\(537\) −13.7996 −0.595497
\(538\) 0 0
\(539\) 0.542594 0.0233712
\(540\) 0 0
\(541\) 28.3084 1.21707 0.608537 0.793525i \(-0.291757\pi\)
0.608537 + 0.793525i \(0.291757\pi\)
\(542\) 0 0
\(543\) −0.740084 −0.0317600
\(544\) 0 0
\(545\) −37.8094 −1.61958
\(546\) 0 0
\(547\) −22.6366 −0.967870 −0.483935 0.875104i \(-0.660793\pi\)
−0.483935 + 0.875104i \(0.660793\pi\)
\(548\) 0 0
\(549\) −14.9792 −0.639295
\(550\) 0 0
\(551\) 5.92988 0.252622
\(552\) 0 0
\(553\) 36.1956 1.53919
\(554\) 0 0
\(555\) −27.6731 −1.17466
\(556\) 0 0
\(557\) −42.6407 −1.80674 −0.903372 0.428857i \(-0.858916\pi\)
−0.903372 + 0.428857i \(0.858916\pi\)
\(558\) 0 0
\(559\) 24.8943 1.05292
\(560\) 0 0
\(561\) −1.89817 −0.0801408
\(562\) 0 0
\(563\) 24.7479 1.04300 0.521500 0.853251i \(-0.325372\pi\)
0.521500 + 0.853251i \(0.325372\pi\)
\(564\) 0 0
\(565\) 10.1388 0.426542
\(566\) 0 0
\(567\) −2.40707 −0.101087
\(568\) 0 0
\(569\) −38.6922 −1.62206 −0.811030 0.585004i \(-0.801093\pi\)
−0.811030 + 0.585004i \(0.801093\pi\)
\(570\) 0 0
\(571\) −14.6552 −0.613299 −0.306650 0.951822i \(-0.599208\pi\)
−0.306650 + 0.951822i \(0.599208\pi\)
\(572\) 0 0
\(573\) 25.6565 1.07181
\(574\) 0 0
\(575\) 2.12436 0.0885919
\(576\) 0 0
\(577\) −11.8436 −0.493054 −0.246527 0.969136i \(-0.579289\pi\)
−0.246527 + 0.969136i \(0.579289\pi\)
\(578\) 0 0
\(579\) 0.108371 0.00450376
\(580\) 0 0
\(581\) −27.1146 −1.12490
\(582\) 0 0
\(583\) 6.39450 0.264833
\(584\) 0 0
\(585\) 18.6847 0.772516
\(586\) 0 0
\(587\) −33.9403 −1.40086 −0.700432 0.713719i \(-0.747009\pi\)
−0.700432 + 0.713719i \(0.747009\pi\)
\(588\) 0 0
\(589\) 10.2052 0.420500
\(590\) 0 0
\(591\) 6.44256 0.265012
\(592\) 0 0
\(593\) −19.8322 −0.814411 −0.407205 0.913337i \(-0.633497\pi\)
−0.407205 + 0.913337i \(0.633497\pi\)
\(594\) 0 0
\(595\) −27.1066 −1.11126
\(596\) 0 0
\(597\) 23.6398 0.967515
\(598\) 0 0
\(599\) 1.02753 0.0419839 0.0209919 0.999780i \(-0.493318\pi\)
0.0209919 + 0.999780i \(0.493318\pi\)
\(600\) 0 0
\(601\) 15.5376 0.633791 0.316896 0.948460i \(-0.397359\pi\)
0.316896 + 0.948460i \(0.397359\pi\)
\(602\) 0 0
\(603\) −7.71769 −0.314289
\(604\) 0 0
\(605\) 28.8204 1.17171
\(606\) 0 0
\(607\) 25.2532 1.02500 0.512498 0.858688i \(-0.328720\pi\)
0.512498 + 0.858688i \(0.328720\pi\)
\(608\) 0 0
\(609\) 2.40707 0.0975394
\(610\) 0 0
\(611\) 74.1166 2.99844
\(612\) 0 0
\(613\) 11.5036 0.464628 0.232314 0.972641i \(-0.425370\pi\)
0.232314 + 0.972641i \(0.425370\pi\)
\(614\) 0 0
\(615\) −7.42505 −0.299407
\(616\) 0 0
\(617\) 48.6255 1.95759 0.978795 0.204841i \(-0.0656676\pi\)
0.978795 + 0.204841i \(0.0656676\pi\)
\(618\) 0 0
\(619\) 3.75055 0.150748 0.0753738 0.997155i \(-0.475985\pi\)
0.0753738 + 0.997155i \(0.475985\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 16.8396 0.674663
\(624\) 0 0
\(625\) −31.1089 −1.24436
\(626\) 0 0
\(627\) 2.66789 0.106545
\(628\) 0 0
\(629\) 43.7420 1.74411
\(630\) 0 0
\(631\) −13.5526 −0.539521 −0.269760 0.962927i \(-0.586944\pi\)
−0.269760 + 0.962927i \(0.586944\pi\)
\(632\) 0 0
\(633\) 0.615086 0.0244475
\(634\) 0 0
\(635\) −42.4107 −1.68302
\(636\) 0 0
\(637\) 8.44239 0.334500
\(638\) 0 0
\(639\) −10.9512 −0.433222
\(640\) 0 0
\(641\) −5.80626 −0.229334 −0.114667 0.993404i \(-0.536580\pi\)
−0.114667 + 0.993404i \(0.536580\pi\)
\(642\) 0 0
\(643\) −24.8634 −0.980516 −0.490258 0.871577i \(-0.663097\pi\)
−0.490258 + 0.871577i \(0.663097\pi\)
\(644\) 0 0
\(645\) −9.49206 −0.373749
\(646\) 0 0
\(647\) −0.929782 −0.0365535 −0.0182768 0.999833i \(-0.505818\pi\)
−0.0182768 + 0.999833i \(0.505818\pi\)
\(648\) 0 0
\(649\) 0.922649 0.0362172
\(650\) 0 0
\(651\) 4.14253 0.162359
\(652\) 0 0
\(653\) 37.2741 1.45865 0.729324 0.684169i \(-0.239835\pi\)
0.729324 + 0.684169i \(0.239835\pi\)
\(654\) 0 0
\(655\) −41.4729 −1.62048
\(656\) 0 0
\(657\) 5.88086 0.229434
\(658\) 0 0
\(659\) −0.651527 −0.0253799 −0.0126899 0.999919i \(-0.504039\pi\)
−0.0126899 + 0.999919i \(0.504039\pi\)
\(660\) 0 0
\(661\) 29.7004 1.15521 0.577605 0.816316i \(-0.303987\pi\)
0.577605 + 0.816316i \(0.303987\pi\)
\(662\) 0 0
\(663\) −29.5342 −1.14701
\(664\) 0 0
\(665\) 38.0985 1.47740
\(666\) 0 0
\(667\) 1.00000 0.0387202
\(668\) 0 0
\(669\) 21.9551 0.848832
\(670\) 0 0
\(671\) 6.73922 0.260165
\(672\) 0 0
\(673\) −23.1497 −0.892357 −0.446179 0.894944i \(-0.647215\pi\)
−0.446179 + 0.894944i \(0.647215\pi\)
\(674\) 0 0
\(675\) −2.12436 −0.0817666
\(676\) 0 0
\(677\) −4.30960 −0.165631 −0.0828157 0.996565i \(-0.526391\pi\)
−0.0828157 + 0.996565i \(0.526391\pi\)
\(678\) 0 0
\(679\) 32.9994 1.26640
\(680\) 0 0
\(681\) −2.38076 −0.0912310
\(682\) 0 0
\(683\) 18.5128 0.708373 0.354186 0.935175i \(-0.384758\pi\)
0.354186 + 0.935175i \(0.384758\pi\)
\(684\) 0 0
\(685\) −36.9149 −1.41045
\(686\) 0 0
\(687\) −17.7836 −0.678486
\(688\) 0 0
\(689\) 99.4940 3.79042
\(690\) 0 0
\(691\) −36.3777 −1.38387 −0.691936 0.721959i \(-0.743242\pi\)
−0.691936 + 0.721959i \(0.743242\pi\)
\(692\) 0 0
\(693\) 1.08296 0.0411381
\(694\) 0 0
\(695\) −1.70439 −0.0646511
\(696\) 0 0
\(697\) 11.7365 0.444552
\(698\) 0 0
\(699\) 25.3471 0.958715
\(700\) 0 0
\(701\) −10.5888 −0.399935 −0.199968 0.979803i \(-0.564084\pi\)
−0.199968 + 0.979803i \(0.564084\pi\)
\(702\) 0 0
\(703\) −61.4797 −2.31875
\(704\) 0 0
\(705\) −28.2602 −1.06434
\(706\) 0 0
\(707\) 23.4544 0.882095
\(708\) 0 0
\(709\) −29.6458 −1.11337 −0.556686 0.830723i \(-0.687927\pi\)
−0.556686 + 0.830723i \(0.687927\pi\)
\(710\) 0 0
\(711\) −15.0372 −0.563939
\(712\) 0 0
\(713\) 1.72098 0.0644514
\(714\) 0 0
\(715\) −8.40635 −0.314380
\(716\) 0 0
\(717\) −6.87771 −0.256853
\(718\) 0 0
\(719\) −16.8976 −0.630175 −0.315087 0.949063i \(-0.602034\pi\)
−0.315087 + 0.949063i \(0.602034\pi\)
\(720\) 0 0
\(721\) 4.04371 0.150596
\(722\) 0 0
\(723\) −1.68799 −0.0627768
\(724\) 0 0
\(725\) 2.12436 0.0788967
\(726\) 0 0
\(727\) −36.6130 −1.35790 −0.678950 0.734184i \(-0.737565\pi\)
−0.678950 + 0.734184i \(0.737565\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 15.0038 0.554935
\(732\) 0 0
\(733\) −4.31448 −0.159359 −0.0796795 0.996821i \(-0.525390\pi\)
−0.0796795 + 0.996821i \(0.525390\pi\)
\(734\) 0 0
\(735\) −3.21903 −0.118736
\(736\) 0 0
\(737\) 3.47224 0.127902
\(738\) 0 0
\(739\) −41.0753 −1.51098 −0.755490 0.655160i \(-0.772601\pi\)
−0.755490 + 0.655160i \(0.772601\pi\)
\(740\) 0 0
\(741\) 41.5106 1.52493
\(742\) 0 0
\(743\) 37.6566 1.38149 0.690743 0.723100i \(-0.257284\pi\)
0.690743 + 0.723100i \(0.257284\pi\)
\(744\) 0 0
\(745\) −52.9135 −1.93860
\(746\) 0 0
\(747\) 11.2646 0.412149
\(748\) 0 0
\(749\) −18.9992 −0.694215
\(750\) 0 0
\(751\) 11.9634 0.436551 0.218275 0.975887i \(-0.429957\pi\)
0.218275 + 0.975887i \(0.429957\pi\)
\(752\) 0 0
\(753\) 18.9014 0.688806
\(754\) 0 0
\(755\) 37.2860 1.35697
\(756\) 0 0
\(757\) −32.3223 −1.17477 −0.587387 0.809306i \(-0.699843\pi\)
−0.587387 + 0.809306i \(0.699843\pi\)
\(758\) 0 0
\(759\) 0.449906 0.0163306
\(760\) 0 0
\(761\) −27.1869 −0.985524 −0.492762 0.870164i \(-0.664013\pi\)
−0.492762 + 0.870164i \(0.664013\pi\)
\(762\) 0 0
\(763\) −34.0970 −1.23439
\(764\) 0 0
\(765\) 11.2612 0.407151
\(766\) 0 0
\(767\) 14.3558 0.518358
\(768\) 0 0
\(769\) −38.1911 −1.37721 −0.688603 0.725138i \(-0.741776\pi\)
−0.688603 + 0.725138i \(0.741776\pi\)
\(770\) 0 0
\(771\) −2.03066 −0.0731324
\(772\) 0 0
\(773\) −11.1603 −0.401409 −0.200704 0.979652i \(-0.564323\pi\)
−0.200704 + 0.979652i \(0.564323\pi\)
\(774\) 0 0
\(775\) 3.65599 0.131327
\(776\) 0 0
\(777\) −24.9560 −0.895290
\(778\) 0 0
\(779\) −16.4958 −0.591022
\(780\) 0 0
\(781\) 4.92701 0.176302
\(782\) 0 0
\(783\) −1.00000 −0.0357371
\(784\) 0 0
\(785\) 1.16683 0.0416460
\(786\) 0 0
\(787\) −17.0856 −0.609035 −0.304518 0.952507i \(-0.598495\pi\)
−0.304518 + 0.952507i \(0.598495\pi\)
\(788\) 0 0
\(789\) 2.51086 0.0893889
\(790\) 0 0
\(791\) 9.14327 0.325097
\(792\) 0 0
\(793\) 104.858 3.72360
\(794\) 0 0
\(795\) −37.9365 −1.34547
\(796\) 0 0
\(797\) −24.7388 −0.876294 −0.438147 0.898903i \(-0.644365\pi\)
−0.438147 + 0.898903i \(0.644365\pi\)
\(798\) 0 0
\(799\) 44.6700 1.58031
\(800\) 0 0
\(801\) −6.99588 −0.247187
\(802\) 0 0
\(803\) −2.64583 −0.0933695
\(804\) 0 0
\(805\) 6.42483 0.226445
\(806\) 0 0
\(807\) 15.1102 0.531905
\(808\) 0 0
\(809\) 14.3789 0.505533 0.252767 0.967527i \(-0.418659\pi\)
0.252767 + 0.967527i \(0.418659\pi\)
\(810\) 0 0
\(811\) 12.6400 0.443852 0.221926 0.975064i \(-0.428766\pi\)
0.221926 + 0.975064i \(0.428766\pi\)
\(812\) 0 0
\(813\) 2.81007 0.0985535
\(814\) 0 0
\(815\) −53.4165 −1.87110
\(816\) 0 0
\(817\) −21.0879 −0.737773
\(818\) 0 0
\(819\) 16.8501 0.588788
\(820\) 0 0
\(821\) −41.4996 −1.44835 −0.724173 0.689618i \(-0.757778\pi\)
−0.724173 + 0.689618i \(0.757778\pi\)
\(822\) 0 0
\(823\) −5.67046 −0.197660 −0.0988299 0.995104i \(-0.531510\pi\)
−0.0988299 + 0.995104i \(0.531510\pi\)
\(824\) 0 0
\(825\) 0.955762 0.0332754
\(826\) 0 0
\(827\) 12.9498 0.450307 0.225154 0.974323i \(-0.427712\pi\)
0.225154 + 0.974323i \(0.427712\pi\)
\(828\) 0 0
\(829\) −42.4241 −1.47345 −0.736725 0.676192i \(-0.763629\pi\)
−0.736725 + 0.676192i \(0.763629\pi\)
\(830\) 0 0
\(831\) −10.3519 −0.359102
\(832\) 0 0
\(833\) 5.08822 0.176296
\(834\) 0 0
\(835\) 7.29638 0.252502
\(836\) 0 0
\(837\) −1.72098 −0.0594859
\(838\) 0 0
\(839\) −53.2397 −1.83804 −0.919019 0.394213i \(-0.871017\pi\)
−0.919019 + 0.394213i \(0.871017\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −11.0253 −0.379733
\(844\) 0 0
\(845\) −96.0981 −3.30588
\(846\) 0 0
\(847\) 25.9905 0.893045
\(848\) 0 0
\(849\) 22.1196 0.759142
\(850\) 0 0
\(851\) −10.3678 −0.355403
\(852\) 0 0
\(853\) 11.8955 0.407296 0.203648 0.979044i \(-0.434720\pi\)
0.203648 + 0.979044i \(0.434720\pi\)
\(854\) 0 0
\(855\) −15.8277 −0.541298
\(856\) 0 0
\(857\) −16.4990 −0.563595 −0.281797 0.959474i \(-0.590931\pi\)
−0.281797 + 0.959474i \(0.590931\pi\)
\(858\) 0 0
\(859\) 20.9828 0.715923 0.357962 0.933736i \(-0.383472\pi\)
0.357962 + 0.933736i \(0.383472\pi\)
\(860\) 0 0
\(861\) −6.69599 −0.228199
\(862\) 0 0
\(863\) −19.3952 −0.660220 −0.330110 0.943943i \(-0.607086\pi\)
−0.330110 + 0.943943i \(0.607086\pi\)
\(864\) 0 0
\(865\) 24.6920 0.839553
\(866\) 0 0
\(867\) −0.800250 −0.0271779
\(868\) 0 0
\(869\) 6.76533 0.229498
\(870\) 0 0
\(871\) 54.0256 1.83059
\(872\) 0 0
\(873\) −13.7094 −0.463991
\(874\) 0 0
\(875\) −18.4755 −0.624586
\(876\) 0 0
\(877\) 3.29776 0.111357 0.0556787 0.998449i \(-0.482268\pi\)
0.0556787 + 0.998449i \(0.482268\pi\)
\(878\) 0 0
\(879\) −2.71146 −0.0914552
\(880\) 0 0
\(881\) 51.5936 1.73823 0.869116 0.494609i \(-0.164689\pi\)
0.869116 + 0.494609i \(0.164689\pi\)
\(882\) 0 0
\(883\) −12.1958 −0.410422 −0.205211 0.978718i \(-0.565788\pi\)
−0.205211 + 0.978718i \(0.565788\pi\)
\(884\) 0 0
\(885\) −5.47378 −0.183999
\(886\) 0 0
\(887\) −18.3971 −0.617714 −0.308857 0.951108i \(-0.599946\pi\)
−0.308857 + 0.951108i \(0.599946\pi\)
\(888\) 0 0
\(889\) −38.2464 −1.28274
\(890\) 0 0
\(891\) −0.449906 −0.0150724
\(892\) 0 0
\(893\) −62.7840 −2.10099
\(894\) 0 0
\(895\) −36.8332 −1.23120
\(896\) 0 0
\(897\) 7.00023 0.233731
\(898\) 0 0
\(899\) 1.72098 0.0573981
\(900\) 0 0
\(901\) 59.9649 1.99772
\(902\) 0 0
\(903\) −8.56005 −0.284861
\(904\) 0 0
\(905\) −1.97539 −0.0656643
\(906\) 0 0
\(907\) −49.2532 −1.63543 −0.817713 0.575626i \(-0.804759\pi\)
−0.817713 + 0.575626i \(0.804759\pi\)
\(908\) 0 0
\(909\) −9.74398 −0.323187
\(910\) 0 0
\(911\) 26.5858 0.880828 0.440414 0.897795i \(-0.354832\pi\)
0.440414 + 0.897795i \(0.354832\pi\)
\(912\) 0 0
\(913\) −5.06800 −0.167726
\(914\) 0 0
\(915\) −39.9816 −1.32175
\(916\) 0 0
\(917\) −37.4008 −1.23508
\(918\) 0 0
\(919\) −21.8848 −0.721913 −0.360956 0.932583i \(-0.617550\pi\)
−0.360956 + 0.932583i \(0.617550\pi\)
\(920\) 0 0
\(921\) 9.99953 0.329496
\(922\) 0 0
\(923\) 76.6609 2.52332
\(924\) 0 0
\(925\) −22.0249 −0.724174
\(926\) 0 0
\(927\) −1.67993 −0.0551762
\(928\) 0 0
\(929\) −12.3640 −0.405648 −0.202824 0.979215i \(-0.565012\pi\)
−0.202824 + 0.979215i \(0.565012\pi\)
\(930\) 0 0
\(931\) −7.15153 −0.234382
\(932\) 0 0
\(933\) −28.8982 −0.946085
\(934\) 0 0
\(935\) −5.06650 −0.165692
\(936\) 0 0
\(937\) −35.8561 −1.17137 −0.585684 0.810539i \(-0.699174\pi\)
−0.585684 + 0.810539i \(0.699174\pi\)
\(938\) 0 0
\(939\) 23.7841 0.776165
\(940\) 0 0
\(941\) −36.6024 −1.19320 −0.596602 0.802537i \(-0.703483\pi\)
−0.596602 + 0.802537i \(0.703483\pi\)
\(942\) 0 0
\(943\) −2.78180 −0.0905880
\(944\) 0 0
\(945\) −6.42483 −0.209000
\(946\) 0 0
\(947\) 46.7935 1.52058 0.760292 0.649582i \(-0.225056\pi\)
0.760292 + 0.649582i \(0.225056\pi\)
\(948\) 0 0
\(949\) −41.1674 −1.33635
\(950\) 0 0
\(951\) −10.7561 −0.348791
\(952\) 0 0
\(953\) 8.94600 0.289789 0.144895 0.989447i \(-0.453716\pi\)
0.144895 + 0.989447i \(0.453716\pi\)
\(954\) 0 0
\(955\) 68.4810 2.21599
\(956\) 0 0
\(957\) 0.449906 0.0145434
\(958\) 0 0
\(959\) −33.2903 −1.07500
\(960\) 0 0
\(961\) −28.0382 −0.904458
\(962\) 0 0
\(963\) 7.89307 0.254351
\(964\) 0 0
\(965\) 0.289259 0.00931158
\(966\) 0 0
\(967\) 24.6114 0.791450 0.395725 0.918369i \(-0.370493\pi\)
0.395725 + 0.918369i \(0.370493\pi\)
\(968\) 0 0
\(969\) 25.0184 0.803706
\(970\) 0 0
\(971\) 12.2735 0.393876 0.196938 0.980416i \(-0.436900\pi\)
0.196938 + 0.980416i \(0.436900\pi\)
\(972\) 0 0
\(973\) −1.53704 −0.0492751
\(974\) 0 0
\(975\) 14.8710 0.476253
\(976\) 0 0
\(977\) 34.3612 1.09931 0.549655 0.835391i \(-0.314759\pi\)
0.549655 + 0.835391i \(0.314759\pi\)
\(978\) 0 0
\(979\) 3.14749 0.100594
\(980\) 0 0
\(981\) 14.1653 0.452265
\(982\) 0 0
\(983\) −27.1793 −0.866885 −0.433442 0.901181i \(-0.642701\pi\)
−0.433442 + 0.901181i \(0.642701\pi\)
\(984\) 0 0
\(985\) 17.1962 0.547915
\(986\) 0 0
\(987\) −25.4854 −0.811209
\(988\) 0 0
\(989\) −3.55621 −0.113081
\(990\) 0 0
\(991\) 3.74004 0.118806 0.0594032 0.998234i \(-0.481080\pi\)
0.0594032 + 0.998234i \(0.481080\pi\)
\(992\) 0 0
\(993\) 4.47707 0.142076
\(994\) 0 0
\(995\) 63.0983 2.00035
\(996\) 0 0
\(997\) 11.7393 0.371786 0.185893 0.982570i \(-0.440482\pi\)
0.185893 + 0.982570i \(0.440482\pi\)
\(998\) 0 0
\(999\) 10.3678 0.328022
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 8004.2.a.i.1.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8004.2.a.i.1.3 16 1.1 even 1 trivial