Properties

Label 3856.2.a.j.1.7
Level $3856$
Weight $2$
Character 3856.1
Self dual yes
Analytic conductor $30.790$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3856 = 2^{4} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.7903150194\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: 7.7.31056073.1
Defining polynomial: \(x^{7} - 3 x^{6} - 3 x^{5} + 11 x^{4} + x^{3} - 9 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 241)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-0.356270\) of defining polynomial
Character \(\chi\) \(=\) 3856.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.45059 q^{3} +2.74184 q^{5} +0.283608 q^{7} +3.00540 q^{9} +O(q^{10})\) \(q+2.45059 q^{3} +2.74184 q^{5} +0.283608 q^{7} +3.00540 q^{9} +4.12582 q^{11} +0.0271909 q^{13} +6.71913 q^{15} -1.28740 q^{17} +5.72717 q^{19} +0.695007 q^{21} +5.97702 q^{23} +2.51768 q^{25} +0.0132278 q^{27} -2.55610 q^{29} +2.02967 q^{31} +10.1107 q^{33} +0.777607 q^{35} +2.42844 q^{37} +0.0666338 q^{39} -11.0324 q^{41} -10.4984 q^{43} +8.24032 q^{45} -4.54349 q^{47} -6.91957 q^{49} -3.15490 q^{51} -9.30751 q^{53} +11.3123 q^{55} +14.0350 q^{57} +9.94762 q^{59} +8.17350 q^{61} +0.852354 q^{63} +0.0745531 q^{65} -4.40964 q^{67} +14.6472 q^{69} +3.80954 q^{71} -15.6571 q^{73} +6.16980 q^{75} +1.17011 q^{77} -6.69229 q^{79} -8.98378 q^{81} +4.32880 q^{83} -3.52985 q^{85} -6.26396 q^{87} +0.746861 q^{89} +0.00771155 q^{91} +4.97390 q^{93} +15.7030 q^{95} +11.9245 q^{97} +12.3997 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 3q^{3} - 8q^{5} + 7q^{7} - 2q^{9} + O(q^{10}) \) \( 7q + 3q^{3} - 8q^{5} + 7q^{7} - 2q^{9} + 18q^{11} - q^{13} + 11q^{15} - 2q^{17} + 6q^{19} - 2q^{21} + 22q^{23} + 5q^{25} - 3q^{27} - 16q^{29} + 18q^{31} + 4q^{33} - 7q^{35} + 8q^{37} + 9q^{39} - 15q^{41} - 14q^{43} + 3q^{45} + 10q^{47} + 6q^{49} - 13q^{51} + 15q^{53} - 29q^{55} + 14q^{57} + 18q^{59} + 4q^{61} + 16q^{63} - 7q^{65} - 18q^{67} + 26q^{69} + 50q^{71} - 16q^{75} + 17q^{77} + 15q^{79} - 9q^{81} + 24q^{83} - 2q^{85} - 12q^{87} - 13q^{89} + 12q^{91} + 14q^{93} + 41q^{95} + q^{97} + 20q^{99} + O(q^{100}) \)

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
\(3\) 2.45059 1.41485 0.707425 0.706789i \(-0.249857\pi\)
0.707425 + 0.706789i \(0.249857\pi\)
\(4\) 0 0
\(5\) 2.74184 1.22619 0.613094 0.790010i \(-0.289925\pi\)
0.613094 + 0.790010i \(0.289925\pi\)
\(6\) 0 0
\(7\) 0.283608 0.107194 0.0535968 0.998563i \(-0.482931\pi\)
0.0535968 + 0.998563i \(0.482931\pi\)
\(8\) 0 0
\(9\) 3.00540 1.00180
\(10\) 0 0
\(11\) 4.12582 1.24398 0.621990 0.783025i \(-0.286324\pi\)
0.621990 + 0.783025i \(0.286324\pi\)
\(12\) 0 0
\(13\) 0.0271909 0.00754140 0.00377070 0.999993i \(-0.498800\pi\)
0.00377070 + 0.999993i \(0.498800\pi\)
\(14\) 0 0
\(15\) 6.71913 1.73487
\(16\) 0 0
\(17\) −1.28740 −0.312241 −0.156121 0.987738i \(-0.549899\pi\)
−0.156121 + 0.987738i \(0.549899\pi\)
\(18\) 0 0
\(19\) 5.72717 1.31390 0.656952 0.753933i \(-0.271846\pi\)
0.656952 + 0.753933i \(0.271846\pi\)
\(20\) 0 0
\(21\) 0.695007 0.151663
\(22\) 0 0
\(23\) 5.97702 1.24629 0.623147 0.782104i \(-0.285854\pi\)
0.623147 + 0.782104i \(0.285854\pi\)
\(24\) 0 0
\(25\) 2.51768 0.503536
\(26\) 0 0
\(27\) 0.0132278 0.00254570
\(28\) 0 0
\(29\) −2.55610 −0.474656 −0.237328 0.971430i \(-0.576272\pi\)
−0.237328 + 0.971430i \(0.576272\pi\)
\(30\) 0 0
\(31\) 2.02967 0.364540 0.182270 0.983248i \(-0.441655\pi\)
0.182270 + 0.983248i \(0.441655\pi\)
\(32\) 0 0
\(33\) 10.1107 1.76005
\(34\) 0 0
\(35\) 0.777607 0.131440
\(36\) 0 0
\(37\) 2.42844 0.399233 0.199617 0.979874i \(-0.436030\pi\)
0.199617 + 0.979874i \(0.436030\pi\)
\(38\) 0 0
\(39\) 0.0666338 0.0106700
\(40\) 0 0
\(41\) −11.0324 −1.72297 −0.861483 0.507786i \(-0.830464\pi\)
−0.861483 + 0.507786i \(0.830464\pi\)
\(42\) 0 0
\(43\) −10.4984 −1.60099 −0.800493 0.599342i \(-0.795429\pi\)
−0.800493 + 0.599342i \(0.795429\pi\)
\(44\) 0 0
\(45\) 8.24032 1.22839
\(46\) 0 0
\(47\) −4.54349 −0.662736 −0.331368 0.943502i \(-0.607510\pi\)
−0.331368 + 0.943502i \(0.607510\pi\)
\(48\) 0 0
\(49\) −6.91957 −0.988510
\(50\) 0 0
\(51\) −3.15490 −0.441774
\(52\) 0 0
\(53\) −9.30751 −1.27848 −0.639242 0.769005i \(-0.720752\pi\)
−0.639242 + 0.769005i \(0.720752\pi\)
\(54\) 0 0
\(55\) 11.3123 1.52535
\(56\) 0 0
\(57\) 14.0350 1.85898
\(58\) 0 0
\(59\) 9.94762 1.29507 0.647535 0.762036i \(-0.275800\pi\)
0.647535 + 0.762036i \(0.275800\pi\)
\(60\) 0 0
\(61\) 8.17350 1.04651 0.523255 0.852176i \(-0.324718\pi\)
0.523255 + 0.852176i \(0.324718\pi\)
\(62\) 0 0
\(63\) 0.852354 0.107387
\(64\) 0 0
\(65\) 0.0745531 0.00924717
\(66\) 0 0
\(67\) −4.40964 −0.538723 −0.269361 0.963039i \(-0.586813\pi\)
−0.269361 + 0.963039i \(0.586813\pi\)
\(68\) 0 0
\(69\) 14.6472 1.76332
\(70\) 0 0
\(71\) 3.80954 0.452109 0.226054 0.974115i \(-0.427417\pi\)
0.226054 + 0.974115i \(0.427417\pi\)
\(72\) 0 0
\(73\) −15.6571 −1.83252 −0.916261 0.400583i \(-0.868808\pi\)
−0.916261 + 0.400583i \(0.868808\pi\)
\(74\) 0 0
\(75\) 6.16980 0.712427
\(76\) 0 0
\(77\) 1.17011 0.133347
\(78\) 0 0
\(79\) −6.69229 −0.752942 −0.376471 0.926428i \(-0.622862\pi\)
−0.376471 + 0.926428i \(0.622862\pi\)
\(80\) 0 0
\(81\) −8.98378 −0.998197
\(82\) 0 0
\(83\) 4.32880 0.475147 0.237574 0.971370i \(-0.423648\pi\)
0.237574 + 0.971370i \(0.423648\pi\)
\(84\) 0 0
\(85\) −3.52985 −0.382866
\(86\) 0 0
\(87\) −6.26396 −0.671567
\(88\) 0 0
\(89\) 0.746861 0.0791671 0.0395835 0.999216i \(-0.487397\pi\)
0.0395835 + 0.999216i \(0.487397\pi\)
\(90\) 0 0
\(91\) 0.00771155 0.000808391 0
\(92\) 0 0
\(93\) 4.97390 0.515770
\(94\) 0 0
\(95\) 15.7030 1.61109
\(96\) 0 0
\(97\) 11.9245 1.21075 0.605373 0.795942i \(-0.293024\pi\)
0.605373 + 0.795942i \(0.293024\pi\)
\(98\) 0 0
\(99\) 12.3997 1.24622
\(100\) 0 0
\(101\) 17.7456 1.76576 0.882878 0.469603i \(-0.155603\pi\)
0.882878 + 0.469603i \(0.155603\pi\)
\(102\) 0 0
\(103\) −5.10848 −0.503354 −0.251677 0.967811i \(-0.580982\pi\)
−0.251677 + 0.967811i \(0.580982\pi\)
\(104\) 0 0
\(105\) 1.90560 0.185967
\(106\) 0 0
\(107\) 0.619213 0.0598615 0.0299308 0.999552i \(-0.490471\pi\)
0.0299308 + 0.999552i \(0.490471\pi\)
\(108\) 0 0
\(109\) 3.34125 0.320034 0.160017 0.987114i \(-0.448845\pi\)
0.160017 + 0.987114i \(0.448845\pi\)
\(110\) 0 0
\(111\) 5.95112 0.564855
\(112\) 0 0
\(113\) −3.42957 −0.322627 −0.161313 0.986903i \(-0.551573\pi\)
−0.161313 + 0.986903i \(0.551573\pi\)
\(114\) 0 0
\(115\) 16.3880 1.52819
\(116\) 0 0
\(117\) 0.0817195 0.00755497
\(118\) 0 0
\(119\) −0.365118 −0.0334703
\(120\) 0 0
\(121\) 6.02237 0.547488
\(122\) 0 0
\(123\) −27.0358 −2.43774
\(124\) 0 0
\(125\) −6.80613 −0.608758
\(126\) 0 0
\(127\) −8.11791 −0.720348 −0.360174 0.932885i \(-0.617283\pi\)
−0.360174 + 0.932885i \(0.617283\pi\)
\(128\) 0 0
\(129\) −25.7272 −2.26516
\(130\) 0 0
\(131\) −2.73258 −0.238747 −0.119373 0.992849i \(-0.538089\pi\)
−0.119373 + 0.992849i \(0.538089\pi\)
\(132\) 0 0
\(133\) 1.62427 0.140842
\(134\) 0 0
\(135\) 0.0362686 0.00312151
\(136\) 0 0
\(137\) 5.47355 0.467637 0.233819 0.972280i \(-0.424878\pi\)
0.233819 + 0.972280i \(0.424878\pi\)
\(138\) 0 0
\(139\) 16.3761 1.38901 0.694503 0.719490i \(-0.255624\pi\)
0.694503 + 0.719490i \(0.255624\pi\)
\(140\) 0 0
\(141\) −11.1342 −0.937671
\(142\) 0 0
\(143\) 0.112185 0.00938136
\(144\) 0 0
\(145\) −7.00842 −0.582017
\(146\) 0 0
\(147\) −16.9570 −1.39859
\(148\) 0 0
\(149\) −15.2495 −1.24929 −0.624645 0.780909i \(-0.714756\pi\)
−0.624645 + 0.780909i \(0.714756\pi\)
\(150\) 0 0
\(151\) 3.19011 0.259607 0.129804 0.991540i \(-0.458565\pi\)
0.129804 + 0.991540i \(0.458565\pi\)
\(152\) 0 0
\(153\) −3.86916 −0.312803
\(154\) 0 0
\(155\) 5.56504 0.446995
\(156\) 0 0
\(157\) 12.7755 1.01959 0.509797 0.860295i \(-0.329721\pi\)
0.509797 + 0.860295i \(0.329721\pi\)
\(158\) 0 0
\(159\) −22.8089 −1.80886
\(160\) 0 0
\(161\) 1.69513 0.133595
\(162\) 0 0
\(163\) 14.1420 1.10769 0.553844 0.832620i \(-0.313160\pi\)
0.553844 + 0.832620i \(0.313160\pi\)
\(164\) 0 0
\(165\) 27.7219 2.15815
\(166\) 0 0
\(167\) −9.14541 −0.707693 −0.353846 0.935304i \(-0.615126\pi\)
−0.353846 + 0.935304i \(0.615126\pi\)
\(168\) 0 0
\(169\) −12.9993 −0.999943
\(170\) 0 0
\(171\) 17.2124 1.31627
\(172\) 0 0
\(173\) 14.0992 1.07194 0.535970 0.844237i \(-0.319946\pi\)
0.535970 + 0.844237i \(0.319946\pi\)
\(174\) 0 0
\(175\) 0.714033 0.0539758
\(176\) 0 0
\(177\) 24.3776 1.83233
\(178\) 0 0
\(179\) −23.3458 −1.74495 −0.872474 0.488660i \(-0.837486\pi\)
−0.872474 + 0.488660i \(0.837486\pi\)
\(180\) 0 0
\(181\) 19.8207 1.47326 0.736631 0.676295i \(-0.236415\pi\)
0.736631 + 0.676295i \(0.236415\pi\)
\(182\) 0 0
\(183\) 20.0299 1.48065
\(184\) 0 0
\(185\) 6.65839 0.489535
\(186\) 0 0
\(187\) −5.31159 −0.388422
\(188\) 0 0
\(189\) 0.00375152 0.000272883 0
\(190\) 0 0
\(191\) −1.89196 −0.136898 −0.0684488 0.997655i \(-0.521805\pi\)
−0.0684488 + 0.997655i \(0.521805\pi\)
\(192\) 0 0
\(193\) 17.2319 1.24038 0.620189 0.784452i \(-0.287056\pi\)
0.620189 + 0.784452i \(0.287056\pi\)
\(194\) 0 0
\(195\) 0.182699 0.0130834
\(196\) 0 0
\(197\) 4.62238 0.329331 0.164665 0.986349i \(-0.447346\pi\)
0.164665 + 0.986349i \(0.447346\pi\)
\(198\) 0 0
\(199\) 17.7065 1.25518 0.627589 0.778545i \(-0.284042\pi\)
0.627589 + 0.778545i \(0.284042\pi\)
\(200\) 0 0
\(201\) −10.8062 −0.762212
\(202\) 0 0
\(203\) −0.724930 −0.0508801
\(204\) 0 0
\(205\) −30.2490 −2.11268
\(206\) 0 0
\(207\) 17.9633 1.24854
\(208\) 0 0
\(209\) 23.6293 1.63447
\(210\) 0 0
\(211\) 11.1970 0.770834 0.385417 0.922742i \(-0.374058\pi\)
0.385417 + 0.922742i \(0.374058\pi\)
\(212\) 0 0
\(213\) 9.33561 0.639666
\(214\) 0 0
\(215\) −28.7848 −1.96311
\(216\) 0 0
\(217\) 0.575631 0.0390764
\(218\) 0 0
\(219\) −38.3691 −2.59274
\(220\) 0 0
\(221\) −0.0350057 −0.00235474
\(222\) 0 0
\(223\) 2.90292 0.194394 0.0971969 0.995265i \(-0.469012\pi\)
0.0971969 + 0.995265i \(0.469012\pi\)
\(224\) 0 0
\(225\) 7.56662 0.504442
\(226\) 0 0
\(227\) −21.4022 −1.42051 −0.710255 0.703944i \(-0.751421\pi\)
−0.710255 + 0.703944i \(0.751421\pi\)
\(228\) 0 0
\(229\) 17.5833 1.16194 0.580968 0.813926i \(-0.302674\pi\)
0.580968 + 0.813926i \(0.302674\pi\)
\(230\) 0 0
\(231\) 2.86747 0.188666
\(232\) 0 0
\(233\) −28.8991 −1.89324 −0.946621 0.322348i \(-0.895528\pi\)
−0.946621 + 0.322348i \(0.895528\pi\)
\(234\) 0 0
\(235\) −12.4575 −0.812638
\(236\) 0 0
\(237\) −16.4001 −1.06530
\(238\) 0 0
\(239\) −25.6630 −1.66000 −0.830000 0.557763i \(-0.811660\pi\)
−0.830000 + 0.557763i \(0.811660\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157
\(242\) 0 0
\(243\) −22.0553 −1.41484
\(244\) 0 0
\(245\) −18.9723 −1.21210
\(246\) 0 0
\(247\) 0.155727 0.00990868
\(248\) 0 0
\(249\) 10.6081 0.672262
\(250\) 0 0
\(251\) 28.3138 1.78715 0.893576 0.448913i \(-0.148189\pi\)
0.893576 + 0.448913i \(0.148189\pi\)
\(252\) 0 0
\(253\) 24.6601 1.55037
\(254\) 0 0
\(255\) −8.65022 −0.541698
\(256\) 0 0
\(257\) −17.0359 −1.06267 −0.531336 0.847161i \(-0.678310\pi\)
−0.531336 + 0.847161i \(0.678310\pi\)
\(258\) 0 0
\(259\) 0.688725 0.0427953
\(260\) 0 0
\(261\) −7.68210 −0.475510
\(262\) 0 0
\(263\) 21.9895 1.35593 0.677966 0.735094i \(-0.262862\pi\)
0.677966 + 0.735094i \(0.262862\pi\)
\(264\) 0 0
\(265\) −25.5197 −1.56766
\(266\) 0 0
\(267\) 1.83025 0.112009
\(268\) 0 0
\(269\) 8.91005 0.543255 0.271628 0.962402i \(-0.412438\pi\)
0.271628 + 0.962402i \(0.412438\pi\)
\(270\) 0 0
\(271\) −29.9968 −1.82218 −0.911088 0.412213i \(-0.864756\pi\)
−0.911088 + 0.412213i \(0.864756\pi\)
\(272\) 0 0
\(273\) 0.0188979 0.00114375
\(274\) 0 0
\(275\) 10.3875 0.626389
\(276\) 0 0
\(277\) −14.9125 −0.896007 −0.448003 0.894032i \(-0.647865\pi\)
−0.448003 + 0.894032i \(0.647865\pi\)
\(278\) 0 0
\(279\) 6.09998 0.365196
\(280\) 0 0
\(281\) −28.8871 −1.72326 −0.861631 0.507535i \(-0.830557\pi\)
−0.861631 + 0.507535i \(0.830557\pi\)
\(282\) 0 0
\(283\) 16.5841 0.985820 0.492910 0.870080i \(-0.335933\pi\)
0.492910 + 0.870080i \(0.335933\pi\)
\(284\) 0 0
\(285\) 38.4816 2.27945
\(286\) 0 0
\(287\) −3.12886 −0.184691
\(288\) 0 0
\(289\) −15.3426 −0.902505
\(290\) 0 0
\(291\) 29.2220 1.71302
\(292\) 0 0
\(293\) −11.2756 −0.658729 −0.329364 0.944203i \(-0.606834\pi\)
−0.329364 + 0.944203i \(0.606834\pi\)
\(294\) 0 0
\(295\) 27.2748 1.58800
\(296\) 0 0
\(297\) 0.0545757 0.00316680
\(298\) 0 0
\(299\) 0.162521 0.00939881
\(300\) 0 0
\(301\) −2.97742 −0.171616
\(302\) 0 0
\(303\) 43.4873 2.49828
\(304\) 0 0
\(305\) 22.4104 1.28322
\(306\) 0 0
\(307\) 1.24678 0.0711575 0.0355787 0.999367i \(-0.488673\pi\)
0.0355787 + 0.999367i \(0.488673\pi\)
\(308\) 0 0
\(309\) −12.5188 −0.712170
\(310\) 0 0
\(311\) 17.8941 1.01468 0.507340 0.861746i \(-0.330629\pi\)
0.507340 + 0.861746i \(0.330629\pi\)
\(312\) 0 0
\(313\) −30.0820 −1.70033 −0.850166 0.526514i \(-0.823499\pi\)
−0.850166 + 0.526514i \(0.823499\pi\)
\(314\) 0 0
\(315\) 2.33702 0.131676
\(316\) 0 0
\(317\) −8.17835 −0.459342 −0.229671 0.973268i \(-0.573765\pi\)
−0.229671 + 0.973268i \(0.573765\pi\)
\(318\) 0 0
\(319\) −10.5460 −0.590463
\(320\) 0 0
\(321\) 1.51744 0.0846951
\(322\) 0 0
\(323\) −7.37318 −0.410255
\(324\) 0 0
\(325\) 0.0684580 0.00379736
\(326\) 0 0
\(327\) 8.18805 0.452800
\(328\) 0 0
\(329\) −1.28857 −0.0710410
\(330\) 0 0
\(331\) −13.4643 −0.740067 −0.370034 0.929018i \(-0.620654\pi\)
−0.370034 + 0.929018i \(0.620654\pi\)
\(332\) 0 0
\(333\) 7.29843 0.399952
\(334\) 0 0
\(335\) −12.0905 −0.660575
\(336\) 0 0
\(337\) −6.06342 −0.330295 −0.165148 0.986269i \(-0.552810\pi\)
−0.165148 + 0.986269i \(0.552810\pi\)
\(338\) 0 0
\(339\) −8.40448 −0.456469
\(340\) 0 0
\(341\) 8.37407 0.453481
\(342\) 0 0
\(343\) −3.94770 −0.213156
\(344\) 0 0
\(345\) 40.1603 2.16216
\(346\) 0 0
\(347\) −23.7297 −1.27388 −0.636938 0.770915i \(-0.719799\pi\)
−0.636938 + 0.770915i \(0.719799\pi\)
\(348\) 0 0
\(349\) 8.46091 0.452902 0.226451 0.974023i \(-0.427288\pi\)
0.226451 + 0.974023i \(0.427288\pi\)
\(350\) 0 0
\(351\) 0.000359677 0 1.91982e−5 0
\(352\) 0 0
\(353\) 6.66847 0.354927 0.177463 0.984127i \(-0.443211\pi\)
0.177463 + 0.984127i \(0.443211\pi\)
\(354\) 0 0
\(355\) 10.4451 0.554370
\(356\) 0 0
\(357\) −0.894754 −0.0473554
\(358\) 0 0
\(359\) 28.3212 1.49474 0.747369 0.664409i \(-0.231317\pi\)
0.747369 + 0.664409i \(0.231317\pi\)
\(360\) 0 0
\(361\) 13.8005 0.726342
\(362\) 0 0
\(363\) 14.7584 0.774614
\(364\) 0 0
\(365\) −42.9292 −2.24701
\(366\) 0 0
\(367\) −7.48573 −0.390752 −0.195376 0.980728i \(-0.562593\pi\)
−0.195376 + 0.980728i \(0.562593\pi\)
\(368\) 0 0
\(369\) −33.1566 −1.72607
\(370\) 0 0
\(371\) −2.63968 −0.137045
\(372\) 0 0
\(373\) 17.6125 0.911943 0.455972 0.889994i \(-0.349292\pi\)
0.455972 + 0.889994i \(0.349292\pi\)
\(374\) 0 0
\(375\) −16.6790 −0.861302
\(376\) 0 0
\(377\) −0.0695028 −0.00357957
\(378\) 0 0
\(379\) −7.61632 −0.391224 −0.195612 0.980681i \(-0.562669\pi\)
−0.195612 + 0.980681i \(0.562669\pi\)
\(380\) 0 0
\(381\) −19.8937 −1.01918
\(382\) 0 0
\(383\) 27.2242 1.39109 0.695546 0.718482i \(-0.255163\pi\)
0.695546 + 0.718482i \(0.255163\pi\)
\(384\) 0 0
\(385\) 3.20826 0.163508
\(386\) 0 0
\(387\) −31.5518 −1.60387
\(388\) 0 0
\(389\) 23.0380 1.16807 0.584036 0.811728i \(-0.301473\pi\)
0.584036 + 0.811728i \(0.301473\pi\)
\(390\) 0 0
\(391\) −7.69483 −0.389145
\(392\) 0 0
\(393\) −6.69643 −0.337790
\(394\) 0 0
\(395\) −18.3492 −0.923248
\(396\) 0 0
\(397\) −17.0475 −0.855587 −0.427794 0.903876i \(-0.640709\pi\)
−0.427794 + 0.903876i \(0.640709\pi\)
\(398\) 0 0
\(399\) 3.98042 0.199270
\(400\) 0 0
\(401\) 6.19034 0.309131 0.154565 0.987983i \(-0.450602\pi\)
0.154565 + 0.987983i \(0.450602\pi\)
\(402\) 0 0
\(403\) 0.0551887 0.00274914
\(404\) 0 0
\(405\) −24.6321 −1.22398
\(406\) 0 0
\(407\) 10.0193 0.496638
\(408\) 0 0
\(409\) 11.3675 0.562088 0.281044 0.959695i \(-0.409319\pi\)
0.281044 + 0.959695i \(0.409319\pi\)
\(410\) 0 0
\(411\) 13.4134 0.661636
\(412\) 0 0
\(413\) 2.82122 0.138823
\(414\) 0 0
\(415\) 11.8689 0.582620
\(416\) 0 0
\(417\) 40.1312 1.96523
\(418\) 0 0
\(419\) −4.14954 −0.202718 −0.101359 0.994850i \(-0.532319\pi\)
−0.101359 + 0.994850i \(0.532319\pi\)
\(420\) 0 0
\(421\) −1.44805 −0.0705734 −0.0352867 0.999377i \(-0.511234\pi\)
−0.0352867 + 0.999377i \(0.511234\pi\)
\(422\) 0 0
\(423\) −13.6550 −0.663928
\(424\) 0 0
\(425\) −3.24127 −0.157225
\(426\) 0 0
\(427\) 2.31807 0.112179
\(428\) 0 0
\(429\) 0.274919 0.0132732
\(430\) 0 0
\(431\) 22.1450 1.06669 0.533344 0.845898i \(-0.320935\pi\)
0.533344 + 0.845898i \(0.320935\pi\)
\(432\) 0 0
\(433\) 10.3894 0.499284 0.249642 0.968338i \(-0.419687\pi\)
0.249642 + 0.968338i \(0.419687\pi\)
\(434\) 0 0
\(435\) −17.1748 −0.823467
\(436\) 0 0
\(437\) 34.2314 1.63751
\(438\) 0 0
\(439\) −39.5672 −1.88844 −0.944220 0.329316i \(-0.893182\pi\)
−0.944220 + 0.329316i \(0.893182\pi\)
\(440\) 0 0
\(441\) −20.7961 −0.990288
\(442\) 0 0
\(443\) 2.60234 0.123641 0.0618203 0.998087i \(-0.480309\pi\)
0.0618203 + 0.998087i \(0.480309\pi\)
\(444\) 0 0
\(445\) 2.04777 0.0970737
\(446\) 0 0
\(447\) −37.3703 −1.76756
\(448\) 0 0
\(449\) −29.9486 −1.41336 −0.706681 0.707533i \(-0.749808\pi\)
−0.706681 + 0.707533i \(0.749808\pi\)
\(450\) 0 0
\(451\) −45.5175 −2.14334
\(452\) 0 0
\(453\) 7.81765 0.367305
\(454\) 0 0
\(455\) 0.0211438 0.000991238 0
\(456\) 0 0
\(457\) −14.5052 −0.678526 −0.339263 0.940691i \(-0.610178\pi\)
−0.339263 + 0.940691i \(0.610178\pi\)
\(458\) 0 0
\(459\) −0.0170296 −0.000794872 0
\(460\) 0 0
\(461\) 38.4334 1.79002 0.895011 0.446044i \(-0.147168\pi\)
0.895011 + 0.446044i \(0.147168\pi\)
\(462\) 0 0
\(463\) −36.2326 −1.68387 −0.841936 0.539578i \(-0.818584\pi\)
−0.841936 + 0.539578i \(0.818584\pi\)
\(464\) 0 0
\(465\) 13.6376 0.632430
\(466\) 0 0
\(467\) 25.1111 1.16200 0.581001 0.813903i \(-0.302661\pi\)
0.581001 + 0.813903i \(0.302661\pi\)
\(468\) 0 0
\(469\) −1.25061 −0.0577477
\(470\) 0 0
\(471\) 31.3074 1.44257
\(472\) 0 0
\(473\) −43.3144 −1.99160
\(474\) 0 0
\(475\) 14.4192 0.661597
\(476\) 0 0
\(477\) −27.9728 −1.28079
\(478\) 0 0
\(479\) 22.0372 1.00691 0.503454 0.864022i \(-0.332062\pi\)
0.503454 + 0.864022i \(0.332062\pi\)
\(480\) 0 0
\(481\) 0.0660315 0.00301078
\(482\) 0 0
\(483\) 4.15407 0.189017
\(484\) 0 0
\(485\) 32.6950 1.48460
\(486\) 0 0
\(487\) 15.1309 0.685649 0.342824 0.939400i \(-0.388616\pi\)
0.342824 + 0.939400i \(0.388616\pi\)
\(488\) 0 0
\(489\) 34.6563 1.56721
\(490\) 0 0
\(491\) 40.4874 1.82717 0.913585 0.406649i \(-0.133303\pi\)
0.913585 + 0.406649i \(0.133303\pi\)
\(492\) 0 0
\(493\) 3.29073 0.148207
\(494\) 0 0
\(495\) 33.9980 1.52810
\(496\) 0 0
\(497\) 1.08041 0.0484632
\(498\) 0 0
\(499\) 22.5598 1.00992 0.504959 0.863144i \(-0.331508\pi\)
0.504959 + 0.863144i \(0.331508\pi\)
\(500\) 0 0
\(501\) −22.4117 −1.00128
\(502\) 0 0
\(503\) −25.5277 −1.13823 −0.569113 0.822259i \(-0.692713\pi\)
−0.569113 + 0.822259i \(0.692713\pi\)
\(504\) 0 0
\(505\) 48.6556 2.16515
\(506\) 0 0
\(507\) −31.8559 −1.41477
\(508\) 0 0
\(509\) −16.9047 −0.749287 −0.374643 0.927169i \(-0.622235\pi\)
−0.374643 + 0.927169i \(0.622235\pi\)
\(510\) 0 0
\(511\) −4.44047 −0.196435
\(512\) 0 0
\(513\) 0.0757581 0.00334480
\(514\) 0 0
\(515\) −14.0066 −0.617206
\(516\) 0 0
\(517\) −18.7456 −0.824430
\(518\) 0 0
\(519\) 34.5513 1.51663
\(520\) 0 0
\(521\) 28.9409 1.26792 0.633961 0.773365i \(-0.281428\pi\)
0.633961 + 0.773365i \(0.281428\pi\)
\(522\) 0 0
\(523\) −15.3129 −0.669586 −0.334793 0.942292i \(-0.608666\pi\)
−0.334793 + 0.942292i \(0.608666\pi\)
\(524\) 0 0
\(525\) 1.74980 0.0763677
\(526\) 0 0
\(527\) −2.61301 −0.113824
\(528\) 0 0
\(529\) 12.7248 0.553250
\(530\) 0 0
\(531\) 29.8966 1.29740
\(532\) 0 0
\(533\) −0.299980 −0.0129936
\(534\) 0 0
\(535\) 1.69778 0.0734015
\(536\) 0 0
\(537\) −57.2111 −2.46884
\(538\) 0 0
\(539\) −28.5489 −1.22969
\(540\) 0 0
\(541\) 11.0774 0.476253 0.238127 0.971234i \(-0.423467\pi\)
0.238127 + 0.971234i \(0.423467\pi\)
\(542\) 0 0
\(543\) 48.5725 2.08444
\(544\) 0 0
\(545\) 9.16118 0.392422
\(546\) 0 0
\(547\) −4.93535 −0.211020 −0.105510 0.994418i \(-0.533648\pi\)
−0.105510 + 0.994418i \(0.533648\pi\)
\(548\) 0 0
\(549\) 24.5646 1.04839
\(550\) 0 0
\(551\) −14.6392 −0.623652
\(552\) 0 0
\(553\) −1.89799 −0.0807106
\(554\) 0 0
\(555\) 16.3170 0.692618
\(556\) 0 0
\(557\) −14.0586 −0.595681 −0.297840 0.954616i \(-0.596266\pi\)
−0.297840 + 0.954616i \(0.596266\pi\)
\(558\) 0 0
\(559\) −0.285460 −0.0120737
\(560\) 0 0
\(561\) −13.0165 −0.549559
\(562\) 0 0
\(563\) −44.3164 −1.86771 −0.933856 0.357650i \(-0.883578\pi\)
−0.933856 + 0.357650i \(0.883578\pi\)
\(564\) 0 0
\(565\) −9.40333 −0.395601
\(566\) 0 0
\(567\) −2.54787 −0.107000
\(568\) 0 0
\(569\) −6.64972 −0.278771 −0.139385 0.990238i \(-0.544513\pi\)
−0.139385 + 0.990238i \(0.544513\pi\)
\(570\) 0 0
\(571\) 22.3573 0.935625 0.467812 0.883828i \(-0.345042\pi\)
0.467812 + 0.883828i \(0.345042\pi\)
\(572\) 0 0
\(573\) −4.63642 −0.193689
\(574\) 0 0
\(575\) 15.0482 0.627554
\(576\) 0 0
\(577\) 33.2047 1.38233 0.691165 0.722697i \(-0.257098\pi\)
0.691165 + 0.722697i \(0.257098\pi\)
\(578\) 0 0
\(579\) 42.2283 1.75495
\(580\) 0 0
\(581\) 1.22768 0.0509328
\(582\) 0 0
\(583\) −38.4011 −1.59041
\(584\) 0 0
\(585\) 0.224062 0.00926381
\(586\) 0 0
\(587\) −28.0737 −1.15873 −0.579363 0.815069i \(-0.696699\pi\)
−0.579363 + 0.815069i \(0.696699\pi\)
\(588\) 0 0
\(589\) 11.6243 0.478971
\(590\) 0 0
\(591\) 11.3276 0.465954
\(592\) 0 0
\(593\) −0.499968 −0.0205312 −0.0102656 0.999947i \(-0.503268\pi\)
−0.0102656 + 0.999947i \(0.503268\pi\)
\(594\) 0 0
\(595\) −1.00109 −0.0410408
\(596\) 0 0
\(597\) 43.3913 1.77589
\(598\) 0 0
\(599\) 5.66864 0.231614 0.115807 0.993272i \(-0.463055\pi\)
0.115807 + 0.993272i \(0.463055\pi\)
\(600\) 0 0
\(601\) 22.5848 0.921252 0.460626 0.887594i \(-0.347625\pi\)
0.460626 + 0.887594i \(0.347625\pi\)
\(602\) 0 0
\(603\) −13.2527 −0.539692
\(604\) 0 0
\(605\) 16.5124 0.671323
\(606\) 0 0
\(607\) 4.01260 0.162867 0.0814333 0.996679i \(-0.474050\pi\)
0.0814333 + 0.996679i \(0.474050\pi\)
\(608\) 0 0
\(609\) −1.77651 −0.0719877
\(610\) 0 0
\(611\) −0.123542 −0.00499796
\(612\) 0 0
\(613\) −31.5947 −1.27610 −0.638048 0.769996i \(-0.720258\pi\)
−0.638048 + 0.769996i \(0.720258\pi\)
\(614\) 0 0
\(615\) −74.1278 −2.98912
\(616\) 0 0
\(617\) −2.64125 −0.106333 −0.0531664 0.998586i \(-0.516931\pi\)
−0.0531664 + 0.998586i \(0.516931\pi\)
\(618\) 0 0
\(619\) −13.8609 −0.557115 −0.278558 0.960419i \(-0.589856\pi\)
−0.278558 + 0.960419i \(0.589856\pi\)
\(620\) 0 0
\(621\) 0.0790631 0.00317269
\(622\) 0 0
\(623\) 0.211815 0.00848621
\(624\) 0 0
\(625\) −31.2497 −1.24999
\(626\) 0 0
\(627\) 57.9057 2.31253
\(628\) 0 0
\(629\) −3.12638 −0.124657
\(630\) 0 0
\(631\) 28.9779 1.15359 0.576795 0.816889i \(-0.304303\pi\)
0.576795 + 0.816889i \(0.304303\pi\)
\(632\) 0 0
\(633\) 27.4393 1.09061
\(634\) 0 0
\(635\) −22.2580 −0.883281
\(636\) 0 0
\(637\) −0.188149 −0.00745475
\(638\) 0 0
\(639\) 11.4492 0.452922
\(640\) 0 0
\(641\) −36.8202 −1.45431 −0.727154 0.686474i \(-0.759158\pi\)
−0.727154 + 0.686474i \(0.759158\pi\)
\(642\) 0 0
\(643\) −14.9327 −0.588890 −0.294445 0.955668i \(-0.595135\pi\)
−0.294445 + 0.955668i \(0.595135\pi\)
\(644\) 0 0
\(645\) −70.5399 −2.77751
\(646\) 0 0
\(647\) −1.09720 −0.0431356 −0.0215678 0.999767i \(-0.506866\pi\)
−0.0215678 + 0.999767i \(0.506866\pi\)
\(648\) 0 0
\(649\) 41.0421 1.61104
\(650\) 0 0
\(651\) 1.41064 0.0552872
\(652\) 0 0
\(653\) 9.51269 0.372260 0.186130 0.982525i \(-0.440405\pi\)
0.186130 + 0.982525i \(0.440405\pi\)
\(654\) 0 0
\(655\) −7.49229 −0.292748
\(656\) 0 0
\(657\) −47.0557 −1.83582
\(658\) 0 0
\(659\) −3.98928 −0.155400 −0.0777001 0.996977i \(-0.524758\pi\)
−0.0777001 + 0.996977i \(0.524758\pi\)
\(660\) 0 0
\(661\) −25.6014 −0.995780 −0.497890 0.867240i \(-0.665892\pi\)
−0.497890 + 0.867240i \(0.665892\pi\)
\(662\) 0 0
\(663\) −0.0857846 −0.00333160
\(664\) 0 0
\(665\) 4.45349 0.172699
\(666\) 0 0
\(667\) −15.2779 −0.591561
\(668\) 0 0
\(669\) 7.11387 0.275038
\(670\) 0 0
\(671\) 33.7224 1.30184
\(672\) 0 0
\(673\) 0.273027 0.0105244 0.00526221 0.999986i \(-0.498325\pi\)
0.00526221 + 0.999986i \(0.498325\pi\)
\(674\) 0 0
\(675\) 0.0333035 0.00128185
\(676\) 0 0
\(677\) −5.73279 −0.220329 −0.110165 0.993913i \(-0.535138\pi\)
−0.110165 + 0.993913i \(0.535138\pi\)
\(678\) 0 0
\(679\) 3.38187 0.129784
\(680\) 0 0
\(681\) −52.4479 −2.00981
\(682\) 0 0
\(683\) 18.8046 0.719538 0.359769 0.933041i \(-0.382855\pi\)
0.359769 + 0.933041i \(0.382855\pi\)
\(684\) 0 0
\(685\) 15.0076 0.573411
\(686\) 0 0
\(687\) 43.0895 1.64397
\(688\) 0 0
\(689\) −0.253080 −0.00964157
\(690\) 0 0
\(691\) −15.8329 −0.602310 −0.301155 0.953575i \(-0.597372\pi\)
−0.301155 + 0.953575i \(0.597372\pi\)
\(692\) 0 0
\(693\) 3.51666 0.133587
\(694\) 0 0
\(695\) 44.9007 1.70318
\(696\) 0 0
\(697\) 14.2031 0.537981
\(698\) 0 0
\(699\) −70.8199 −2.67865
\(700\) 0 0
\(701\) 40.3523 1.52409 0.762043 0.647527i \(-0.224197\pi\)
0.762043 + 0.647527i \(0.224197\pi\)
\(702\) 0 0
\(703\) 13.9081 0.524554
\(704\) 0 0
\(705\) −30.5283 −1.14976
\(706\) 0 0
\(707\) 5.03280 0.189278
\(708\) 0 0
\(709\) −41.5401 −1.56007 −0.780036 0.625734i \(-0.784800\pi\)
−0.780036 + 0.625734i \(0.784800\pi\)
\(710\) 0 0
\(711\) −20.1130 −0.754296
\(712\) 0 0
\(713\) 12.1314 0.454325
\(714\) 0 0
\(715\) 0.307592 0.0115033
\(716\) 0 0
\(717\) −62.8895 −2.34865
\(718\) 0 0
\(719\) 10.2324 0.381603 0.190801 0.981629i \(-0.438891\pi\)
0.190801 + 0.981629i \(0.438891\pi\)
\(720\) 0 0
\(721\) −1.44881 −0.0539563
\(722\) 0 0
\(723\) −2.45059 −0.0911385
\(724\) 0 0
\(725\) −6.43544 −0.239006
\(726\) 0 0
\(727\) 45.5533 1.68948 0.844739 0.535178i \(-0.179756\pi\)
0.844739 + 0.535178i \(0.179756\pi\)
\(728\) 0 0
\(729\) −27.0971 −1.00360
\(730\) 0 0
\(731\) 13.5156 0.499894
\(732\) 0 0
\(733\) −20.6241 −0.761770 −0.380885 0.924622i \(-0.624381\pi\)
−0.380885 + 0.924622i \(0.624381\pi\)
\(734\) 0 0
\(735\) −46.4934 −1.71494
\(736\) 0 0
\(737\) −18.1934 −0.670161
\(738\) 0 0
\(739\) 17.1975 0.632620 0.316310 0.948656i \(-0.397556\pi\)
0.316310 + 0.948656i \(0.397556\pi\)
\(740\) 0 0
\(741\) 0.381623 0.0140193
\(742\) 0 0
\(743\) 40.8356 1.49811 0.749056 0.662506i \(-0.230507\pi\)
0.749056 + 0.662506i \(0.230507\pi\)
\(744\) 0 0
\(745\) −41.8117 −1.53186
\(746\) 0 0
\(747\) 13.0098 0.476002
\(748\) 0 0
\(749\) 0.175613 0.00641678
\(750\) 0 0
\(751\) −20.2798 −0.740021 −0.370010 0.929028i \(-0.620646\pi\)
−0.370010 + 0.929028i \(0.620646\pi\)
\(752\) 0 0
\(753\) 69.3856 2.52855
\(754\) 0 0
\(755\) 8.74676 0.318327
\(756\) 0 0
\(757\) 13.7148 0.498473 0.249237 0.968443i \(-0.419820\pi\)
0.249237 + 0.968443i \(0.419820\pi\)
\(758\) 0 0
\(759\) 60.4318 2.19354
\(760\) 0 0
\(761\) −6.41706 −0.232618 −0.116309 0.993213i \(-0.537106\pi\)
−0.116309 + 0.993213i \(0.537106\pi\)
\(762\) 0 0
\(763\) 0.947605 0.0343056
\(764\) 0 0
\(765\) −10.6086 −0.383555
\(766\) 0 0
\(767\) 0.270485 0.00976665
\(768\) 0 0
\(769\) −27.8170 −1.00311 −0.501553 0.865127i \(-0.667238\pi\)
−0.501553 + 0.865127i \(0.667238\pi\)
\(770\) 0 0
\(771\) −41.7481 −1.50352
\(772\) 0 0
\(773\) 12.8524 0.462270 0.231135 0.972922i \(-0.425756\pi\)
0.231135 + 0.972922i \(0.425756\pi\)
\(774\) 0 0
\(775\) 5.11007 0.183559
\(776\) 0 0
\(777\) 1.68778 0.0605489
\(778\) 0 0
\(779\) −63.1843 −2.26381
\(780\) 0 0
\(781\) 15.7174 0.562414
\(782\) 0 0
\(783\) −0.0338117 −0.00120833
\(784\) 0 0
\(785\) 35.0283 1.25021
\(786\) 0 0
\(787\) −9.53803 −0.339994 −0.169997 0.985445i \(-0.554376\pi\)
−0.169997 + 0.985445i \(0.554376\pi\)
\(788\) 0 0
\(789\) 53.8873 1.91844
\(790\) 0 0
\(791\) −0.972653 −0.0345836
\(792\) 0 0
\(793\) 0.222245 0.00789215
\(794\) 0 0
\(795\) −62.5383 −2.21801
\(796\) 0 0
\(797\) −7.72801 −0.273740 −0.136870 0.990589i \(-0.543704\pi\)
−0.136870 + 0.990589i \(0.543704\pi\)
\(798\) 0 0
\(799\) 5.84930 0.206933
\(800\) 0 0
\(801\) 2.24461 0.0793095
\(802\) 0 0
\(803\) −64.5982 −2.27962
\(804\) 0 0
\(805\) 4.64777 0.163812
\(806\) 0 0
\(807\) 21.8349 0.768624
\(808\) 0 0
\(809\) −1.99109 −0.0700030 −0.0350015 0.999387i \(-0.511144\pi\)
−0.0350015 + 0.999387i \(0.511144\pi\)
\(810\) 0 0
\(811\) −28.5346 −1.00198 −0.500992 0.865452i \(-0.667031\pi\)
−0.500992 + 0.865452i \(0.667031\pi\)
\(812\) 0 0
\(813\) −73.5099 −2.57810
\(814\) 0 0
\(815\) 38.7751 1.35823
\(816\) 0 0
\(817\) −60.1260 −2.10354
\(818\) 0 0
\(819\) 0.0231763 0.000809845 0
\(820\) 0 0
\(821\) 48.3459 1.68728 0.843642 0.536907i \(-0.180407\pi\)
0.843642 + 0.536907i \(0.180407\pi\)
\(822\) 0 0
\(823\) 36.4809 1.27164 0.635821 0.771836i \(-0.280662\pi\)
0.635821 + 0.771836i \(0.280662\pi\)
\(824\) 0 0
\(825\) 25.4555 0.886246
\(826\) 0 0
\(827\) −23.2928 −0.809971 −0.404985 0.914323i \(-0.632723\pi\)
−0.404985 + 0.914323i \(0.632723\pi\)
\(828\) 0 0
\(829\) −29.8113 −1.03539 −0.517695 0.855565i \(-0.673210\pi\)
−0.517695 + 0.855565i \(0.673210\pi\)
\(830\) 0 0
\(831\) −36.5445 −1.26771
\(832\) 0 0
\(833\) 8.90827 0.308653
\(834\) 0 0
\(835\) −25.0752 −0.867764
\(836\) 0 0
\(837\) 0.0268482 0.000928010 0
\(838\) 0 0
\(839\) 48.3508 1.66925 0.834627 0.550815i \(-0.185683\pi\)
0.834627 + 0.550815i \(0.185683\pi\)
\(840\) 0 0
\(841\) −22.4663 −0.774702
\(842\) 0 0
\(843\) −70.7906 −2.43816
\(844\) 0 0
\(845\) −35.6419 −1.22612
\(846\) 0 0
\(847\) 1.70799 0.0586873
\(848\) 0 0
\(849\) 40.6408 1.39479
\(850\) 0 0
\(851\) 14.5148 0.497562
\(852\) 0 0
\(853\) 5.13297 0.175750 0.0878748 0.996132i \(-0.471992\pi\)
0.0878748 + 0.996132i \(0.471992\pi\)
\(854\) 0 0
\(855\) 47.1937 1.61399
\(856\) 0 0
\(857\) 20.8453 0.712062 0.356031 0.934474i \(-0.384130\pi\)
0.356031 + 0.934474i \(0.384130\pi\)
\(858\) 0 0
\(859\) −24.4260 −0.833403 −0.416701 0.909043i \(-0.636814\pi\)
−0.416701 + 0.909043i \(0.636814\pi\)
\(860\) 0 0
\(861\) −7.66757 −0.261310
\(862\) 0 0
\(863\) −25.5588 −0.870033 −0.435017 0.900422i \(-0.643257\pi\)
−0.435017 + 0.900422i \(0.643257\pi\)
\(864\) 0 0
\(865\) 38.6576 1.31440
\(866\) 0 0
\(867\) −37.5984 −1.27691
\(868\) 0 0
\(869\) −27.6112 −0.936645
\(870\) 0 0
\(871\) −0.119902 −0.00406273
\(872\) 0 0
\(873\) 35.8378 1.21292
\(874\) 0 0
\(875\) −1.93027 −0.0652550
\(876\) 0 0
\(877\) −7.53456 −0.254424 −0.127212 0.991876i \(-0.540603\pi\)
−0.127212 + 0.991876i \(0.540603\pi\)
\(878\) 0 0
\(879\) −27.6319 −0.932002
\(880\) 0 0
\(881\) 21.7361 0.732308 0.366154 0.930554i \(-0.380674\pi\)
0.366154 + 0.930554i \(0.380674\pi\)
\(882\) 0 0
\(883\) 38.6202 1.29967 0.649837 0.760074i \(-0.274837\pi\)
0.649837 + 0.760074i \(0.274837\pi\)
\(884\) 0 0
\(885\) 66.8393 2.24678
\(886\) 0 0
\(887\) 14.3165 0.480702 0.240351 0.970686i \(-0.422738\pi\)
0.240351 + 0.970686i \(0.422738\pi\)
\(888\) 0 0
\(889\) −2.30230 −0.0772167
\(890\) 0 0
\(891\) −37.0654 −1.24174
\(892\) 0 0
\(893\) −26.0213 −0.870771
\(894\) 0 0
\(895\) −64.0105 −2.13963
\(896\) 0 0
\(897\) 0.398272 0.0132979
\(898\) 0 0
\(899\) −5.18805 −0.173031
\(900\) 0 0
\(901\) 11.9825 0.399196
\(902\) 0 0
\(903\) −7.29644 −0.242810
\(904\) 0 0
\(905\) 54.3452 1.80650
\(906\) 0 0
\(907\) 47.6042 1.58067 0.790336 0.612674i \(-0.209906\pi\)
0.790336 + 0.612674i \(0.209906\pi\)
\(908\) 0 0
\(909\) 53.3327 1.76893
\(910\) 0 0
\(911\) 40.1570 1.33046 0.665230 0.746639i \(-0.268334\pi\)
0.665230 + 0.746639i \(0.268334\pi\)
\(912\) 0 0
\(913\) 17.8598 0.591074
\(914\) 0 0
\(915\) 54.9188 1.81556
\(916\) 0 0
\(917\) −0.774981 −0.0255921
\(918\) 0 0
\(919\) 18.0383 0.595029 0.297514 0.954717i \(-0.403842\pi\)
0.297514 + 0.954717i \(0.403842\pi\)
\(920\) 0 0
\(921\) 3.05535 0.100677
\(922\) 0 0
\(923\) 0.103585 0.00340953
\(924\) 0 0
\(925\) 6.11403 0.201028
\(926\) 0 0
\(927\) −15.3530 −0.504260
\(928\) 0 0
\(929\) −0.679116 −0.0222811 −0.0111405 0.999938i \(-0.503546\pi\)
−0.0111405 + 0.999938i \(0.503546\pi\)
\(930\) 0 0
\(931\) −39.6295 −1.29881
\(932\) 0 0
\(933\) 43.8510 1.43562
\(934\) 0 0
\(935\) −14.5635 −0.476278
\(936\) 0 0
\(937\) −25.7202 −0.840243 −0.420122 0.907468i \(-0.638013\pi\)
−0.420122 + 0.907468i \(0.638013\pi\)
\(938\) 0 0
\(939\) −73.7186 −2.40571
\(940\) 0 0
\(941\) −41.9988 −1.36912 −0.684560 0.728956i \(-0.740006\pi\)
−0.684560 + 0.728956i \(0.740006\pi\)
\(942\) 0 0
\(943\) −65.9407 −2.14732
\(944\) 0 0
\(945\) 0.0102861 0.000334606 0
\(946\) 0 0
\(947\) 15.2367 0.495128 0.247564 0.968872i \(-0.420370\pi\)
0.247564 + 0.968872i \(0.420370\pi\)
\(948\) 0 0
\(949\) −0.425730 −0.0138198
\(950\) 0 0
\(951\) −20.0418 −0.649900
\(952\) 0 0
\(953\) 43.1360 1.39731 0.698656 0.715457i \(-0.253782\pi\)
0.698656 + 0.715457i \(0.253782\pi\)
\(954\) 0 0
\(955\) −5.18745 −0.167862
\(956\) 0 0
\(957\) −25.8440 −0.835417
\(958\) 0 0
\(959\) 1.55234 0.0501277
\(960\) 0 0
\(961\) −26.8804 −0.867110
\(962\) 0 0
\(963\) 1.86098 0.0599692
\(964\) 0 0
\(965\) 47.2471 1.52094
\(966\) 0 0
\(967\) −45.6699 −1.46864 −0.734322 0.678801i \(-0.762500\pi\)
−0.734322 + 0.678801i \(0.762500\pi\)
\(968\) 0 0
\(969\) −18.0687 −0.580449
\(970\) 0 0
\(971\) 2.32251 0.0745330 0.0372665 0.999305i \(-0.488135\pi\)
0.0372665 + 0.999305i \(0.488135\pi\)
\(972\) 0 0
\(973\) 4.64440 0.148893
\(974\) 0 0
\(975\) 0.167762 0.00537270
\(976\) 0 0
\(977\) −50.8644 −1.62730 −0.813649 0.581357i \(-0.802522\pi\)
−0.813649 + 0.581357i \(0.802522\pi\)
\(978\) 0 0
\(979\) 3.08141 0.0984823
\(980\) 0 0
\(981\) 10.0418 0.320610
\(982\) 0 0
\(983\) 21.2587 0.678047 0.339023 0.940778i \(-0.389903\pi\)
0.339023 + 0.940778i \(0.389903\pi\)
\(984\) 0 0
\(985\) 12.6738 0.403821
\(986\) 0 0
\(987\) −3.15775 −0.100512
\(988\) 0 0
\(989\) −62.7490 −1.99530
\(990\) 0 0
\(991\) −51.6637 −1.64115 −0.820575 0.571539i \(-0.806347\pi\)
−0.820575 + 0.571539i \(0.806347\pi\)
\(992\) 0 0
\(993\) −32.9956 −1.04708
\(994\) 0 0
\(995\) 48.5483 1.53908
\(996\) 0 0
\(997\) −32.2376 −1.02097 −0.510487 0.859885i \(-0.670535\pi\)
−0.510487 + 0.859885i \(0.670535\pi\)
\(998\) 0 0
\(999\) 0.0321230 0.00101633
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 3856.2.a.j.1.7 7
4.3 odd 2 241.2.a.a.1.3 7
12.11 even 2 2169.2.a.e.1.5 7
20.19 odd 2 6025.2.a.f.1.5 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
241.2.a.a.1.3 7 4.3 odd 2
2169.2.a.e.1.5 7 12.11 even 2
3856.2.a.j.1.7 7 1.1 even 1 trivial
6025.2.a.f.1.5 7 20.19 odd 2