Properties

Label 1148.2.d.a.1065.1
Level $1148$
Weight $2$
Character 1148.1065
Analytic conductor $9.167$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1148.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.16682615204\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Defining polynomial: \(x^{20} + 39 x^{18} + 531 x^{16} + 3488 x^{14} + 12661 x^{12} + 27027 x^{10} + 34540 x^{8} + 25909 x^{6} + 10677 x^{4} + 2092 x^{2} + 144\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1065.1
Root \(0.544821i\) of defining polynomial
Character \(\chi\) \(=\) 1148.1065
Dual form 1148.2.d.a.1065.20

$q$-expansion

\(f(q)\) \(=\) \(q-3.32626i q^{3} -0.885775 q^{5} +1.00000i q^{7} -8.06402 q^{9} +O(q^{10})\) \(q-3.32626i q^{3} -0.885775 q^{5} +1.00000i q^{7} -8.06402 q^{9} -4.99006i q^{11} -4.26351i q^{13} +2.94632i q^{15} +5.28751i q^{17} -0.754013i q^{19} +3.32626 q^{21} -2.59135 q^{23} -4.21540 q^{25} +16.8442i q^{27} +2.99919i q^{29} +2.96961 q^{31} -16.5983 q^{33} -0.885775i q^{35} -2.09849 q^{37} -14.1816 q^{39} +(-5.15019 + 3.80468i) q^{41} +5.25668 q^{43} +7.14291 q^{45} -2.19765i q^{47} -1.00000 q^{49} +17.5876 q^{51} -12.9516i q^{53} +4.42007i q^{55} -2.50804 q^{57} -0.407891 q^{59} -3.50613 q^{61} -8.06402i q^{63} +3.77651i q^{65} +7.13251i q^{67} +8.61949i q^{69} +13.4730i q^{71} -1.38016 q^{73} +14.0215i q^{75} +4.99006 q^{77} -10.5799i q^{79} +31.8363 q^{81} -5.94978 q^{83} -4.68354i q^{85} +9.97610 q^{87} +7.56362i q^{89} +4.26351 q^{91} -9.87769i q^{93} +0.667886i q^{95} -18.7175i q^{97} +40.2400i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 4q^{5} - 20q^{9} + O(q^{10}) \) \( 20q + 4q^{5} - 20q^{9} + 4q^{21} + 8q^{31} + 20q^{37} + 4q^{39} - 16q^{41} + 20q^{43} - 4q^{45} - 20q^{49} + 52q^{51} - 36q^{57} + 20q^{59} - 4q^{61} - 12q^{73} + 8q^{77} + 20q^{81} - 48q^{83} + 44q^{87} - 4q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1148\mathbb{Z}\right)^\times\).

\(n\) \(493\) \(575\) \(785\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(3\) 3.32626i 1.92042i −0.279282 0.960209i \(-0.590097\pi\)
0.279282 0.960209i \(-0.409903\pi\)
\(4\) 0 0
\(5\) −0.885775 −0.396131 −0.198065 0.980189i \(-0.563466\pi\)
−0.198065 + 0.980189i \(0.563466\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) −8.06402 −2.68801
\(10\) 0 0
\(11\) 4.99006i 1.50456i −0.658843 0.752280i \(-0.728954\pi\)
0.658843 0.752280i \(-0.271046\pi\)
\(12\) 0 0
\(13\) 4.26351i 1.18248i −0.806494 0.591242i \(-0.798638\pi\)
0.806494 0.591242i \(-0.201362\pi\)
\(14\) 0 0
\(15\) 2.94632i 0.760737i
\(16\) 0 0
\(17\) 5.28751i 1.28241i 0.767370 + 0.641204i \(0.221565\pi\)
−0.767370 + 0.641204i \(0.778435\pi\)
\(18\) 0 0
\(19\) 0.754013i 0.172982i −0.996253 0.0864912i \(-0.972435\pi\)
0.996253 0.0864912i \(-0.0275654\pi\)
\(20\) 0 0
\(21\) 3.32626 0.725850
\(22\) 0 0
\(23\) −2.59135 −0.540333 −0.270166 0.962814i \(-0.587079\pi\)
−0.270166 + 0.962814i \(0.587079\pi\)
\(24\) 0 0
\(25\) −4.21540 −0.843080
\(26\) 0 0
\(27\) 16.8442i 3.24168i
\(28\) 0 0
\(29\) 2.99919i 0.556936i 0.960445 + 0.278468i \(0.0898267\pi\)
−0.960445 + 0.278468i \(0.910173\pi\)
\(30\) 0 0
\(31\) 2.96961 0.533357 0.266679 0.963786i \(-0.414074\pi\)
0.266679 + 0.963786i \(0.414074\pi\)
\(32\) 0 0
\(33\) −16.5983 −2.88939
\(34\) 0 0
\(35\) 0.885775i 0.149723i
\(36\) 0 0
\(37\) −2.09849 −0.344990 −0.172495 0.985010i \(-0.555183\pi\)
−0.172495 + 0.985010i \(0.555183\pi\)
\(38\) 0 0
\(39\) −14.1816 −2.27087
\(40\) 0 0
\(41\) −5.15019 + 3.80468i −0.804324 + 0.594191i
\(42\) 0 0
\(43\) 5.25668 0.801636 0.400818 0.916158i \(-0.368726\pi\)
0.400818 + 0.916158i \(0.368726\pi\)
\(44\) 0 0
\(45\) 7.14291 1.06480
\(46\) 0 0
\(47\) 2.19765i 0.320560i −0.987072 0.160280i \(-0.948760\pi\)
0.987072 0.160280i \(-0.0512397\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 17.5876 2.46276
\(52\) 0 0
\(53\) 12.9516i 1.77904i −0.456899 0.889518i \(-0.651040\pi\)
0.456899 0.889518i \(-0.348960\pi\)
\(54\) 0 0
\(55\) 4.42007i 0.596003i
\(56\) 0 0
\(57\) −2.50804 −0.332199
\(58\) 0 0
\(59\) −0.407891 −0.0531029 −0.0265514 0.999647i \(-0.508453\pi\)
−0.0265514 + 0.999647i \(0.508453\pi\)
\(60\) 0 0
\(61\) −3.50613 −0.448914 −0.224457 0.974484i \(-0.572061\pi\)
−0.224457 + 0.974484i \(0.572061\pi\)
\(62\) 0 0
\(63\) 8.06402i 1.01597i
\(64\) 0 0
\(65\) 3.77651i 0.468419i
\(66\) 0 0
\(67\) 7.13251i 0.871375i 0.900098 + 0.435687i \(0.143495\pi\)
−0.900098 + 0.435687i \(0.856505\pi\)
\(68\) 0 0
\(69\) 8.61949i 1.03766i
\(70\) 0 0
\(71\) 13.4730i 1.59895i 0.600700 + 0.799475i \(0.294889\pi\)
−0.600700 + 0.799475i \(0.705111\pi\)
\(72\) 0 0
\(73\) −1.38016 −0.161536 −0.0807678 0.996733i \(-0.525737\pi\)
−0.0807678 + 0.996733i \(0.525737\pi\)
\(74\) 0 0
\(75\) 14.0215i 1.61907i
\(76\) 0 0
\(77\) 4.99006 0.568671
\(78\) 0 0
\(79\) 10.5799i 1.19033i −0.803603 0.595165i \(-0.797086\pi\)
0.803603 0.595165i \(-0.202914\pi\)
\(80\) 0 0
\(81\) 31.8363 3.53737
\(82\) 0 0
\(83\) −5.94978 −0.653073 −0.326537 0.945185i \(-0.605882\pi\)
−0.326537 + 0.945185i \(0.605882\pi\)
\(84\) 0 0
\(85\) 4.68354i 0.508001i
\(86\) 0 0
\(87\) 9.97610 1.06955
\(88\) 0 0
\(89\) 7.56362i 0.801742i 0.916135 + 0.400871i \(0.131292\pi\)
−0.916135 + 0.400871i \(0.868708\pi\)
\(90\) 0 0
\(91\) 4.26351 0.446937
\(92\) 0 0
\(93\) 9.87769i 1.02427i
\(94\) 0 0
\(95\) 0.667886i 0.0685236i
\(96\) 0 0
\(97\) 18.7175i 1.90048i −0.311523 0.950239i \(-0.600839\pi\)
0.311523 0.950239i \(-0.399161\pi\)
\(98\) 0 0
\(99\) 40.2400i 4.04427i
\(100\) 0 0
\(101\) 8.49673i 0.845456i 0.906257 + 0.422728i \(0.138927\pi\)
−0.906257 + 0.422728i \(0.861073\pi\)
\(102\) 0 0
\(103\) −11.8406 −1.16669 −0.583345 0.812225i \(-0.698256\pi\)
−0.583345 + 0.812225i \(0.698256\pi\)
\(104\) 0 0
\(105\) −2.94632 −0.287531
\(106\) 0 0
\(107\) 1.21695 0.117647 0.0588236 0.998268i \(-0.481265\pi\)
0.0588236 + 0.998268i \(0.481265\pi\)
\(108\) 0 0
\(109\) 8.90196i 0.852653i −0.904569 0.426326i \(-0.859807\pi\)
0.904569 0.426326i \(-0.140193\pi\)
\(110\) 0 0
\(111\) 6.98014i 0.662526i
\(112\) 0 0
\(113\) 13.9211 1.30959 0.654794 0.755808i \(-0.272755\pi\)
0.654794 + 0.755808i \(0.272755\pi\)
\(114\) 0 0
\(115\) 2.29535 0.214042
\(116\) 0 0
\(117\) 34.3810i 3.17853i
\(118\) 0 0
\(119\) −5.28751 −0.484705
\(120\) 0 0
\(121\) −13.9007 −1.26370
\(122\) 0 0
\(123\) 12.6554 + 17.1309i 1.14110 + 1.54464i
\(124\) 0 0
\(125\) 8.16277 0.730101
\(126\) 0 0
\(127\) 0.872741 0.0774433 0.0387216 0.999250i \(-0.487671\pi\)
0.0387216 + 0.999250i \(0.487671\pi\)
\(128\) 0 0
\(129\) 17.4851i 1.53948i
\(130\) 0 0
\(131\) −7.00013 −0.611605 −0.305802 0.952095i \(-0.598925\pi\)
−0.305802 + 0.952095i \(0.598925\pi\)
\(132\) 0 0
\(133\) 0.754013 0.0653812
\(134\) 0 0
\(135\) 14.9202i 1.28413i
\(136\) 0 0
\(137\) 8.07841i 0.690185i −0.938569 0.345092i \(-0.887848\pi\)
0.938569 0.345092i \(-0.112152\pi\)
\(138\) 0 0
\(139\) −7.36668 −0.624833 −0.312417 0.949945i \(-0.601139\pi\)
−0.312417 + 0.949945i \(0.601139\pi\)
\(140\) 0 0
\(141\) −7.30995 −0.615609
\(142\) 0 0
\(143\) −21.2752 −1.77912
\(144\) 0 0
\(145\) 2.65661i 0.220620i
\(146\) 0 0
\(147\) 3.32626i 0.274345i
\(148\) 0 0
\(149\) 10.8414i 0.888165i −0.895986 0.444082i \(-0.853530\pi\)
0.895986 0.444082i \(-0.146470\pi\)
\(150\) 0 0
\(151\) 7.14528i 0.581475i −0.956803 0.290737i \(-0.906099\pi\)
0.956803 0.290737i \(-0.0939006\pi\)
\(152\) 0 0
\(153\) 42.6385i 3.44712i
\(154\) 0 0
\(155\) −2.63040 −0.211279
\(156\) 0 0
\(157\) 21.7642i 1.73698i −0.495710 0.868488i \(-0.665092\pi\)
0.495710 0.868488i \(-0.334908\pi\)
\(158\) 0 0
\(159\) −43.0804 −3.41649
\(160\) 0 0
\(161\) 2.59135i 0.204227i
\(162\) 0 0
\(163\) 20.7447 1.62485 0.812425 0.583066i \(-0.198147\pi\)
0.812425 + 0.583066i \(0.198147\pi\)
\(164\) 0 0
\(165\) 14.7023 1.14457
\(166\) 0 0
\(167\) 0.148320i 0.0114774i −0.999984 0.00573868i \(-0.998173\pi\)
0.999984 0.00573868i \(-0.00182669\pi\)
\(168\) 0 0
\(169\) −5.17752 −0.398271
\(170\) 0 0
\(171\) 6.08037i 0.464978i
\(172\) 0 0
\(173\) −22.0166 −1.67389 −0.836946 0.547286i \(-0.815661\pi\)
−0.836946 + 0.547286i \(0.815661\pi\)
\(174\) 0 0
\(175\) 4.21540i 0.318654i
\(176\) 0 0
\(177\) 1.35675i 0.101980i
\(178\) 0 0
\(179\) 19.2143i 1.43614i −0.695970 0.718071i \(-0.745025\pi\)
0.695970 0.718071i \(-0.254975\pi\)
\(180\) 0 0
\(181\) 22.2636i 1.65484i −0.561582 0.827421i \(-0.689807\pi\)
0.561582 0.827421i \(-0.310193\pi\)
\(182\) 0 0
\(183\) 11.6623i 0.862103i
\(184\) 0 0
\(185\) 1.85879 0.136661
\(186\) 0 0
\(187\) 26.3850 1.92946
\(188\) 0 0
\(189\) −16.8442 −1.22524
\(190\) 0 0
\(191\) 10.7178i 0.775515i −0.921761 0.387758i \(-0.873250\pi\)
0.921761 0.387758i \(-0.126750\pi\)
\(192\) 0 0
\(193\) 14.8149i 1.06640i 0.845990 + 0.533199i \(0.179010\pi\)
−0.845990 + 0.533199i \(0.820990\pi\)
\(194\) 0 0
\(195\) 12.5617 0.899559
\(196\) 0 0
\(197\) −19.4190 −1.38355 −0.691775 0.722113i \(-0.743171\pi\)
−0.691775 + 0.722113i \(0.743171\pi\)
\(198\) 0 0
\(199\) 3.45691i 0.245054i 0.992465 + 0.122527i \(0.0390998\pi\)
−0.992465 + 0.122527i \(0.960900\pi\)
\(200\) 0 0
\(201\) 23.7246 1.67340
\(202\) 0 0
\(203\) −2.99919 −0.210502
\(204\) 0 0
\(205\) 4.56191 3.37009i 0.318617 0.235377i
\(206\) 0 0
\(207\) 20.8967 1.45242
\(208\) 0 0
\(209\) −3.76257 −0.260263
\(210\) 0 0
\(211\) 25.3680i 1.74640i −0.487358 0.873202i \(-0.662039\pi\)
0.487358 0.873202i \(-0.337961\pi\)
\(212\) 0 0
\(213\) 44.8147 3.07065
\(214\) 0 0
\(215\) −4.65624 −0.317553
\(216\) 0 0
\(217\) 2.96961i 0.201590i
\(218\) 0 0
\(219\) 4.59078i 0.310216i
\(220\) 0 0
\(221\) 22.5433 1.51643
\(222\) 0 0
\(223\) −19.7290 −1.32115 −0.660574 0.750761i \(-0.729687\pi\)
−0.660574 + 0.750761i \(0.729687\pi\)
\(224\) 0 0
\(225\) 33.9931 2.26621
\(226\) 0 0
\(227\) 7.52428i 0.499404i 0.968323 + 0.249702i \(0.0803327\pi\)
−0.968323 + 0.249702i \(0.919667\pi\)
\(228\) 0 0
\(229\) 14.0607i 0.929155i 0.885533 + 0.464577i \(0.153794\pi\)
−0.885533 + 0.464577i \(0.846206\pi\)
\(230\) 0 0
\(231\) 16.5983i 1.09209i
\(232\) 0 0
\(233\) 13.1209i 0.859581i 0.902929 + 0.429790i \(0.141413\pi\)
−0.902929 + 0.429790i \(0.858587\pi\)
\(234\) 0 0
\(235\) 1.94662i 0.126984i
\(236\) 0 0
\(237\) −35.1915 −2.28593
\(238\) 0 0
\(239\) 27.8883i 1.80394i −0.431795 0.901972i \(-0.642120\pi\)
0.431795 0.901972i \(-0.357880\pi\)
\(240\) 0 0
\(241\) −25.3127 −1.63053 −0.815266 0.579086i \(-0.803409\pi\)
−0.815266 + 0.579086i \(0.803409\pi\)
\(242\) 0 0
\(243\) 55.3632i 3.55155i
\(244\) 0 0
\(245\) 0.885775 0.0565901
\(246\) 0 0
\(247\) −3.21474 −0.204549
\(248\) 0 0
\(249\) 19.7905i 1.25417i
\(250\) 0 0
\(251\) −24.8073 −1.56582 −0.782912 0.622132i \(-0.786267\pi\)
−0.782912 + 0.622132i \(0.786267\pi\)
\(252\) 0 0
\(253\) 12.9310i 0.812964i
\(254\) 0 0
\(255\) −15.5787 −0.975575
\(256\) 0 0
\(257\) 13.4127i 0.836659i 0.908295 + 0.418329i \(0.137384\pi\)
−0.908295 + 0.418329i \(0.862616\pi\)
\(258\) 0 0
\(259\) 2.09849i 0.130394i
\(260\) 0 0
\(261\) 24.1855i 1.49705i
\(262\) 0 0
\(263\) 11.8133i 0.728442i 0.931313 + 0.364221i \(0.118665\pi\)
−0.931313 + 0.364221i \(0.881335\pi\)
\(264\) 0 0
\(265\) 11.4722i 0.704731i
\(266\) 0 0
\(267\) 25.1586 1.53968
\(268\) 0 0
\(269\) −19.3704 −1.18103 −0.590516 0.807026i \(-0.701076\pi\)
−0.590516 + 0.807026i \(0.701076\pi\)
\(270\) 0 0
\(271\) 20.3389 1.23550 0.617750 0.786375i \(-0.288044\pi\)
0.617750 + 0.786375i \(0.288044\pi\)
\(272\) 0 0
\(273\) 14.1816i 0.858306i
\(274\) 0 0
\(275\) 21.0351i 1.26847i
\(276\) 0 0
\(277\) −4.06079 −0.243989 −0.121995 0.992531i \(-0.538929\pi\)
−0.121995 + 0.992531i \(0.538929\pi\)
\(278\) 0 0
\(279\) −23.9470 −1.43367
\(280\) 0 0
\(281\) 16.2106i 0.967042i −0.875333 0.483521i \(-0.839358\pi\)
0.875333 0.483521i \(-0.160642\pi\)
\(282\) 0 0
\(283\) −12.2524 −0.728328 −0.364164 0.931335i \(-0.618645\pi\)
−0.364164 + 0.931335i \(0.618645\pi\)
\(284\) 0 0
\(285\) 2.22156 0.131594
\(286\) 0 0
\(287\) −3.80468 5.15019i −0.224583 0.304006i
\(288\) 0 0
\(289\) −10.9577 −0.644572
\(290\) 0 0
\(291\) −62.2594 −3.64971
\(292\) 0 0
\(293\) 10.2231i 0.597237i 0.954373 + 0.298619i \(0.0965258\pi\)
−0.954373 + 0.298619i \(0.903474\pi\)
\(294\) 0 0
\(295\) 0.361300 0.0210357
\(296\) 0 0
\(297\) 84.0539 4.87730
\(298\) 0 0
\(299\) 11.0482i 0.638935i
\(300\) 0 0
\(301\) 5.25668i 0.302990i
\(302\) 0 0
\(303\) 28.2623 1.62363
\(304\) 0 0
\(305\) 3.10564 0.177829
\(306\) 0 0
\(307\) 25.1727 1.43668 0.718341 0.695691i \(-0.244902\pi\)
0.718341 + 0.695691i \(0.244902\pi\)
\(308\) 0 0
\(309\) 39.3849i 2.24053i
\(310\) 0 0
\(311\) 33.4518i 1.89688i 0.316962 + 0.948438i \(0.397337\pi\)
−0.316962 + 0.948438i \(0.602663\pi\)
\(312\) 0 0
\(313\) 33.5808i 1.89810i −0.315127 0.949050i \(-0.602047\pi\)
0.315127 0.949050i \(-0.397953\pi\)
\(314\) 0 0
\(315\) 7.14291i 0.402457i
\(316\) 0 0
\(317\) 26.1878i 1.47086i −0.677603 0.735428i \(-0.736981\pi\)
0.677603 0.735428i \(-0.263019\pi\)
\(318\) 0 0
\(319\) 14.9662 0.837945
\(320\) 0 0
\(321\) 4.04790i 0.225932i
\(322\) 0 0
\(323\) 3.98685 0.221834
\(324\) 0 0
\(325\) 17.9724i 0.996930i
\(326\) 0 0
\(327\) −29.6102 −1.63745
\(328\) 0 0
\(329\) 2.19765 0.121160
\(330\) 0 0
\(331\) 7.39984i 0.406732i 0.979103 + 0.203366i \(0.0651881\pi\)
−0.979103 + 0.203366i \(0.934812\pi\)
\(332\) 0 0
\(333\) 16.9223 0.927336
\(334\) 0 0
\(335\) 6.31780i 0.345178i
\(336\) 0 0
\(337\) 18.2525 0.994278 0.497139 0.867671i \(-0.334384\pi\)
0.497139 + 0.867671i \(0.334384\pi\)
\(338\) 0 0
\(339\) 46.3052i 2.51496i
\(340\) 0 0
\(341\) 14.8185i 0.802468i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 7.63493i 0.411051i
\(346\) 0 0
\(347\) 5.41714i 0.290807i 0.989372 + 0.145404i \(0.0464481\pi\)
−0.989372 + 0.145404i \(0.953552\pi\)
\(348\) 0 0
\(349\) 15.9953 0.856211 0.428105 0.903729i \(-0.359181\pi\)
0.428105 + 0.903729i \(0.359181\pi\)
\(350\) 0 0
\(351\) 71.8156 3.83323
\(352\) 0 0
\(353\) −14.7025 −0.782534 −0.391267 0.920277i \(-0.627963\pi\)
−0.391267 + 0.920277i \(0.627963\pi\)
\(354\) 0 0
\(355\) 11.9340i 0.633393i
\(356\) 0 0
\(357\) 17.5876i 0.930836i
\(358\) 0 0
\(359\) −14.4172 −0.760912 −0.380456 0.924799i \(-0.624233\pi\)
−0.380456 + 0.924799i \(0.624233\pi\)
\(360\) 0 0
\(361\) 18.4315 0.970077
\(362\) 0 0
\(363\) 46.2375i 2.42684i
\(364\) 0 0
\(365\) 1.22251 0.0639892
\(366\) 0 0
\(367\) 30.6227 1.59849 0.799245 0.601005i \(-0.205233\pi\)
0.799245 + 0.601005i \(0.205233\pi\)
\(368\) 0 0
\(369\) 41.5312 30.6810i 2.16203 1.59719i
\(370\) 0 0
\(371\) 12.9516 0.672413
\(372\) 0 0
\(373\) −9.39430 −0.486418 −0.243209 0.969974i \(-0.578200\pi\)
−0.243209 + 0.969974i \(0.578200\pi\)
\(374\) 0 0
\(375\) 27.1515i 1.40210i
\(376\) 0 0
\(377\) 12.7871 0.658569
\(378\) 0 0
\(379\) 4.49107 0.230691 0.115346 0.993325i \(-0.463202\pi\)
0.115346 + 0.993325i \(0.463202\pi\)
\(380\) 0 0
\(381\) 2.90297i 0.148723i
\(382\) 0 0
\(383\) 11.9010i 0.608111i 0.952654 + 0.304055i \(0.0983408\pi\)
−0.952654 + 0.304055i \(0.901659\pi\)
\(384\) 0 0
\(385\) −4.42007 −0.225268
\(386\) 0 0
\(387\) −42.3900 −2.15480
\(388\) 0 0
\(389\) −3.77845 −0.191575 −0.0957874 0.995402i \(-0.530537\pi\)
−0.0957874 + 0.995402i \(0.530537\pi\)
\(390\) 0 0
\(391\) 13.7018i 0.692927i
\(392\) 0 0
\(393\) 23.2843i 1.17454i
\(394\) 0 0
\(395\) 9.37141i 0.471527i
\(396\) 0 0
\(397\) 1.90061i 0.0953891i 0.998862 + 0.0476945i \(0.0151874\pi\)
−0.998862 + 0.0476945i \(0.984813\pi\)
\(398\) 0 0
\(399\) 2.50804i 0.125559i
\(400\) 0 0
\(401\) −13.4642 −0.672370 −0.336185 0.941796i \(-0.609137\pi\)
−0.336185 + 0.941796i \(0.609137\pi\)
\(402\) 0 0
\(403\) 12.6609i 0.630687i
\(404\) 0 0
\(405\) −28.1998 −1.40126
\(406\) 0 0
\(407\) 10.4716i 0.519059i
\(408\) 0 0
\(409\) 15.4442 0.763665 0.381832 0.924232i \(-0.375293\pi\)
0.381832 + 0.924232i \(0.375293\pi\)
\(410\) 0 0
\(411\) −26.8709 −1.32544
\(412\) 0 0
\(413\) 0.407891i 0.0200710i
\(414\) 0 0
\(415\) 5.27017 0.258702
\(416\) 0 0
\(417\) 24.5035i 1.19994i
\(418\) 0 0
\(419\) 31.9780 1.56223 0.781113 0.624390i \(-0.214652\pi\)
0.781113 + 0.624390i \(0.214652\pi\)
\(420\) 0 0
\(421\) 32.8813i 1.60254i −0.598306 0.801268i \(-0.704159\pi\)
0.598306 0.801268i \(-0.295841\pi\)
\(422\) 0 0
\(423\) 17.7219i 0.861666i
\(424\) 0 0
\(425\) 22.2890i 1.08117i
\(426\) 0 0
\(427\) 3.50613i 0.169674i
\(428\) 0 0
\(429\) 70.7668i 3.41666i
\(430\) 0 0
\(431\) 11.7455 0.565760 0.282880 0.959155i \(-0.408710\pi\)
0.282880 + 0.959155i \(0.408710\pi\)
\(432\) 0 0
\(433\) 40.7925 1.96036 0.980181 0.198103i \(-0.0634782\pi\)
0.980181 + 0.198103i \(0.0634782\pi\)
\(434\) 0 0
\(435\) −8.83658 −0.423682
\(436\) 0 0
\(437\) 1.95391i 0.0934681i
\(438\) 0 0
\(439\) 2.01206i 0.0960302i 0.998847 + 0.0480151i \(0.0152896\pi\)
−0.998847 + 0.0480151i \(0.984710\pi\)
\(440\) 0 0
\(441\) 8.06402 0.384001
\(442\) 0 0
\(443\) 30.7331 1.46017 0.730087 0.683354i \(-0.239480\pi\)
0.730087 + 0.683354i \(0.239480\pi\)
\(444\) 0 0
\(445\) 6.69966i 0.317595i
\(446\) 0 0
\(447\) −36.0614 −1.70565
\(448\) 0 0
\(449\) −5.35997 −0.252952 −0.126476 0.991970i \(-0.540367\pi\)
−0.126476 + 0.991970i \(0.540367\pi\)
\(450\) 0 0
\(451\) 18.9856 + 25.6998i 0.893997 + 1.21015i
\(452\) 0 0
\(453\) −23.7671 −1.11667
\(454\) 0 0
\(455\) −3.77651 −0.177046
\(456\) 0 0
\(457\) 27.9170i 1.30590i 0.757400 + 0.652951i \(0.226469\pi\)
−0.757400 + 0.652951i \(0.773531\pi\)
\(458\) 0 0
\(459\) −89.0641 −4.15715
\(460\) 0 0
\(461\) 25.5388 1.18946 0.594731 0.803925i \(-0.297258\pi\)
0.594731 + 0.803925i \(0.297258\pi\)
\(462\) 0 0
\(463\) 16.5418i 0.768761i −0.923175 0.384381i \(-0.874415\pi\)
0.923175 0.384381i \(-0.125585\pi\)
\(464\) 0 0
\(465\) 8.74941i 0.405744i
\(466\) 0 0
\(467\) 16.8249 0.778565 0.389282 0.921118i \(-0.372723\pi\)
0.389282 + 0.921118i \(0.372723\pi\)
\(468\) 0 0
\(469\) −7.13251 −0.329349
\(470\) 0 0
\(471\) −72.3936 −3.33572
\(472\) 0 0
\(473\) 26.2312i 1.20611i
\(474\) 0 0
\(475\) 3.17847i 0.145838i
\(476\) 0 0
\(477\) 104.442i 4.78206i
\(478\) 0 0
\(479\) 27.4094i 1.25237i −0.779675 0.626184i \(-0.784616\pi\)
0.779675 0.626184i \(-0.215384\pi\)
\(480\) 0 0
\(481\) 8.94695i 0.407946i
\(482\) 0 0
\(483\) −8.61949 −0.392200
\(484\) 0 0
\(485\) 16.5795i 0.752837i
\(486\) 0 0
\(487\) −10.6649 −0.483273 −0.241637 0.970367i \(-0.577684\pi\)
−0.241637 + 0.970367i \(0.577684\pi\)
\(488\) 0 0
\(489\) 69.0023i 3.12039i
\(490\) 0 0
\(491\) −30.6296 −1.38229 −0.691147 0.722714i \(-0.742894\pi\)
−0.691147 + 0.722714i \(0.742894\pi\)
\(492\) 0 0
\(493\) −15.8583 −0.714220
\(494\) 0 0
\(495\) 35.6436i 1.60206i
\(496\) 0 0
\(497\) −13.4730 −0.604346
\(498\) 0 0
\(499\) 31.5750i 1.41349i −0.707467 0.706746i \(-0.750162\pi\)
0.707467 0.706746i \(-0.249838\pi\)
\(500\) 0 0
\(501\) −0.493352 −0.0220413
\(502\) 0 0
\(503\) 15.3179i 0.682994i 0.939883 + 0.341497i \(0.110934\pi\)
−0.939883 + 0.341497i \(0.889066\pi\)
\(504\) 0 0
\(505\) 7.52619i 0.334911i
\(506\) 0 0
\(507\) 17.2218i 0.764846i
\(508\) 0 0
\(509\) 28.4719i 1.26199i −0.775785 0.630997i \(-0.782646\pi\)
0.775785 0.630997i \(-0.217354\pi\)
\(510\) 0 0
\(511\) 1.38016i 0.0610547i
\(512\) 0 0
\(513\) 12.7008 0.560753
\(514\) 0 0
\(515\) 10.4881 0.462161
\(516\) 0 0
\(517\) −10.9664 −0.482301
\(518\) 0 0
\(519\) 73.2330i 3.21457i
\(520\) 0 0
\(521\) 18.0313i 0.789966i 0.918688 + 0.394983i \(0.129249\pi\)
−0.918688 + 0.394983i \(0.870751\pi\)
\(522\) 0 0
\(523\) 11.3389 0.495815 0.247907 0.968784i \(-0.420257\pi\)
0.247907 + 0.968784i \(0.420257\pi\)
\(524\) 0 0
\(525\) −14.0215 −0.611950
\(526\) 0 0
\(527\) 15.7018i 0.683982i
\(528\) 0 0
\(529\) −16.2849 −0.708040
\(530\) 0 0
\(531\) 3.28924 0.142741
\(532\) 0 0
\(533\) 16.2213 + 21.9579i 0.702622 + 0.951101i
\(534\) 0 0
\(535\) −1.07795 −0.0466037
\(536\) 0 0
\(537\) −63.9117 −2.75799
\(538\) 0 0
\(539\) 4.99006i 0.214937i
\(540\) 0 0
\(541\) −0.655488 −0.0281816 −0.0140908 0.999901i \(-0.504485\pi\)
−0.0140908 + 0.999901i \(0.504485\pi\)
\(542\) 0 0
\(543\) −74.0547 −3.17799
\(544\) 0 0
\(545\) 7.88513i 0.337762i
\(546\) 0 0
\(547\) 31.9749i 1.36715i −0.729881 0.683574i \(-0.760425\pi\)
0.729881 0.683574i \(-0.239575\pi\)
\(548\) 0 0
\(549\) 28.2735 1.20668
\(550\) 0 0
\(551\) 2.26143 0.0963402
\(552\) 0 0
\(553\) 10.5799 0.449903
\(554\) 0 0
\(555\) 6.18284i 0.262447i
\(556\) 0 0
\(557\) 33.6085i 1.42404i 0.702159 + 0.712020i \(0.252219\pi\)
−0.702159 + 0.712020i \(0.747781\pi\)
\(558\) 0 0
\(559\) 22.4119i 0.947923i
\(560\) 0 0
\(561\) 87.7634i 3.70537i
\(562\) 0 0
\(563\) 41.0263i 1.72905i 0.502588 + 0.864526i \(0.332381\pi\)
−0.502588 + 0.864526i \(0.667619\pi\)
\(564\) 0 0
\(565\) −12.3310 −0.518768
\(566\) 0 0
\(567\) 31.8363i 1.33700i
\(568\) 0 0
\(569\) −26.9544 −1.12999 −0.564993 0.825095i \(-0.691121\pi\)
−0.564993 + 0.825095i \(0.691121\pi\)
\(570\) 0 0
\(571\) 35.8336i 1.49959i 0.661670 + 0.749795i \(0.269848\pi\)
−0.661670 + 0.749795i \(0.730152\pi\)
\(572\) 0 0
\(573\) −35.6503 −1.48931
\(574\) 0 0
\(575\) 10.9236 0.455544
\(576\) 0 0
\(577\) 20.1767i 0.839966i 0.907532 + 0.419983i \(0.137964\pi\)
−0.907532 + 0.419983i \(0.862036\pi\)
\(578\) 0 0
\(579\) 49.2781 2.04793
\(580\) 0 0
\(581\) 5.94978i 0.246839i
\(582\) 0 0
\(583\) −64.6292 −2.67667
\(584\) 0 0
\(585\) 30.4539i 1.25911i
\(586\) 0 0
\(587\) 27.8661i 1.15016i 0.818098 + 0.575079i \(0.195029\pi\)
−0.818098 + 0.575079i \(0.804971\pi\)
\(588\) 0 0
\(589\) 2.23912i 0.0922614i
\(590\) 0 0
\(591\) 64.5928i 2.65699i
\(592\) 0 0
\(593\) 0.849655i 0.0348911i 0.999848 + 0.0174456i \(0.00555338\pi\)
−0.999848 + 0.0174456i \(0.994447\pi\)
\(594\) 0 0
\(595\) 4.68354 0.192006
\(596\) 0 0
\(597\) 11.4986 0.470606
\(598\) 0 0
\(599\) 27.3828 1.11883 0.559416 0.828887i \(-0.311025\pi\)
0.559416 + 0.828887i \(0.311025\pi\)
\(600\) 0 0
\(601\) 41.2015i 1.68064i −0.542088 0.840322i \(-0.682366\pi\)
0.542088 0.840322i \(-0.317634\pi\)
\(602\) 0 0
\(603\) 57.5167i 2.34226i
\(604\) 0 0
\(605\) 12.3129 0.500592
\(606\) 0 0
\(607\) 4.42813 0.179732 0.0898662 0.995954i \(-0.471356\pi\)
0.0898662 + 0.995954i \(0.471356\pi\)
\(608\) 0 0
\(609\) 9.97610i 0.404252i
\(610\) 0 0
\(611\) −9.36969 −0.379057
\(612\) 0 0
\(613\) −31.0549 −1.25430 −0.627148 0.778900i \(-0.715778\pi\)
−0.627148 + 0.778900i \(0.715778\pi\)
\(614\) 0 0
\(615\) −11.2098 15.1741i −0.452023 0.611879i
\(616\) 0 0
\(617\) 37.8646 1.52437 0.762187 0.647357i \(-0.224126\pi\)
0.762187 + 0.647357i \(0.224126\pi\)
\(618\) 0 0
\(619\) 23.9899 0.964235 0.482117 0.876107i \(-0.339868\pi\)
0.482117 + 0.876107i \(0.339868\pi\)
\(620\) 0 0
\(621\) 43.6493i 1.75158i
\(622\) 0 0
\(623\) −7.56362 −0.303030
\(624\) 0 0
\(625\) 13.8466 0.553865
\(626\) 0 0
\(627\) 12.5153i 0.499813i
\(628\) 0 0
\(629\) 11.0958i 0.442419i
\(630\) 0 0
\(631\) 24.2633 0.965907 0.482953 0.875646i \(-0.339564\pi\)
0.482953 + 0.875646i \(0.339564\pi\)
\(632\) 0 0
\(633\) −84.3806 −3.35383
\(634\) 0 0
\(635\) −0.773053 −0.0306777
\(636\) 0 0
\(637\) 4.26351i 0.168926i
\(638\) 0 0
\(639\) 108.646i 4.29799i
\(640\) 0 0
\(641\) 27.0510i 1.06845i −0.845342 0.534225i \(-0.820604\pi\)
0.845342 0.534225i \(-0.179396\pi\)
\(642\) 0 0
\(643\) 4.86194i 0.191736i 0.995394 + 0.0958680i \(0.0305627\pi\)
−0.995394 + 0.0958680i \(0.969437\pi\)
\(644\) 0 0
\(645\) 15.4879i 0.609834i
\(646\) 0 0
\(647\) 10.4608 0.411255 0.205627 0.978630i \(-0.434076\pi\)
0.205627 + 0.978630i \(0.434076\pi\)
\(648\) 0 0
\(649\) 2.03540i 0.0798965i
\(650\) 0 0
\(651\) 9.87769 0.387137
\(652\) 0 0
\(653\) 23.2678i 0.910539i 0.890354 + 0.455269i \(0.150457\pi\)
−0.890354 + 0.455269i \(0.849543\pi\)
\(654\) 0 0
\(655\) 6.20054 0.242275
\(656\) 0 0
\(657\) 11.1296 0.434209
\(658\) 0 0
\(659\) 6.49703i 0.253088i −0.991961 0.126544i \(-0.959611\pi\)
0.991961 0.126544i \(-0.0403885\pi\)
\(660\) 0 0
\(661\) −18.8388 −0.732746 −0.366373 0.930468i \(-0.619401\pi\)
−0.366373 + 0.930468i \(0.619401\pi\)
\(662\) 0 0
\(663\) 74.9850i 2.91218i
\(664\) 0 0
\(665\) −0.667886 −0.0258995
\(666\) 0 0
\(667\) 7.77195i 0.300931i
\(668\) 0 0
\(669\) 65.6237i 2.53716i
\(670\) 0 0
\(671\) 17.4958i 0.675419i
\(672\) 0 0
\(673\) 24.1914i 0.932510i 0.884650 + 0.466255i \(0.154397\pi\)
−0.884650 + 0.466255i \(0.845603\pi\)
\(674\) 0 0
\(675\) 71.0053i 2.73299i
\(676\) 0 0
\(677\) 18.8143 0.723094 0.361547 0.932354i \(-0.382249\pi\)
0.361547 + 0.932354i \(0.382249\pi\)
\(678\) 0 0
\(679\) 18.7175 0.718313
\(680\) 0 0
\(681\) 25.0277 0.959065
\(682\) 0 0
\(683\) 28.4360i 1.08807i 0.839061 + 0.544037i \(0.183105\pi\)
−0.839061 + 0.544037i \(0.816895\pi\)
\(684\) 0 0
\(685\) 7.15565i 0.273403i
\(686\) 0 0
\(687\) 46.7694 1.78437
\(688\) 0 0
\(689\) −55.2192 −2.10368
\(690\) 0 0
\(691\) 23.5898i 0.897397i 0.893683 + 0.448699i \(0.148112\pi\)
−0.893683 + 0.448699i \(0.851888\pi\)
\(692\) 0 0
\(693\) −40.2400 −1.52859
\(694\) 0 0
\(695\) 6.52522 0.247516
\(696\) 0 0
\(697\) −20.1173 27.2316i −0.761996 1.03147i
\(698\) 0 0
\(699\) 43.6436 1.65075
\(700\) 0 0
\(701\) −45.2228 −1.70804 −0.854021 0.520238i \(-0.825843\pi\)
−0.854021 + 0.520238i \(0.825843\pi\)
\(702\) 0 0
\(703\) 1.58229i 0.0596773i
\(704\) 0 0
\(705\) 6.47497 0.243861
\(706\) 0 0
\(707\) −8.49673 −0.319552
\(708\) 0 0
\(709\) 2.74052i 0.102923i 0.998675 + 0.0514613i \(0.0163879\pi\)
−0.998675 + 0.0514613i \(0.983612\pi\)
\(710\) 0 0
\(711\) 85.3164i 3.19962i
\(712\) 0 0
\(713\) −7.69528 −0.288190
\(714\) 0 0
\(715\) 18.8450 0.704764
\(716\) 0 0
\(717\) −92.7637 −3.46433
\(718\) 0 0
\(719\) 0.134357i 0.00501066i 0.999997 + 0.00250533i \(0.000797472\pi\)
−0.999997 + 0.00250533i \(0.999203\pi\)
\(720\) 0 0
\(721\) 11.8406i 0.440967i
\(722\) 0 0
\(723\) 84.1966i 3.13130i
\(724\) 0 0
\(725\) 12.6428i 0.469542i
\(726\) 0 0
\(727\) 1.46508i 0.0543369i 0.999631 + 0.0271684i \(0.00864905\pi\)
−0.999631 + 0.0271684i \(0.991351\pi\)
\(728\) 0 0
\(729\) −88.6435 −3.28309
\(730\) 0 0
\(731\) 27.7947i 1.02803i
\(732\) 0 0
\(733\) 8.35740 0.308688 0.154344 0.988017i \(-0.450674\pi\)
0.154344 + 0.988017i \(0.450674\pi\)
\(734\) 0 0
\(735\) 2.94632i 0.108677i
\(736\) 0 0
\(737\) 35.5917 1.31104
\(738\) 0 0
\(739\) 44.2671 1.62839 0.814197 0.580589i \(-0.197178\pi\)
0.814197 + 0.580589i \(0.197178\pi\)
\(740\) 0 0
\(741\) 10.6931i 0.392820i
\(742\) 0 0
\(743\) 46.5738 1.70863 0.854313 0.519758i \(-0.173978\pi\)
0.854313 + 0.519758i \(0.173978\pi\)
\(744\) 0 0
\(745\) 9.60307i 0.351829i
\(746\) 0 0
\(747\) 47.9791 1.75547
\(748\) 0 0
\(749\) 1.21695i 0.0444665i
\(750\) 0 0
\(751\) 15.9358i 0.581507i −0.956798 0.290753i \(-0.906094\pi\)
0.956798 0.290753i \(-0.0939060\pi\)
\(752\) 0 0
\(753\) 82.5156i 3.00704i
\(754\) 0 0
\(755\) 6.32911i 0.230340i
\(756\) 0 0
\(757\) 9.80863i 0.356501i 0.983985 + 0.178250i \(0.0570437\pi\)
−0.983985 + 0.178250i \(0.942956\pi\)
\(758\) 0 0
\(759\) 43.0118 1.56123
\(760\) 0 0
\(761\) −21.3088 −0.772443 −0.386221 0.922406i \(-0.626220\pi\)
−0.386221 + 0.922406i \(0.626220\pi\)
\(762\) 0 0
\(763\) 8.90196 0.322273
\(764\) 0 0
\(765\) 37.7682i 1.36551i
\(766\) 0 0
\(767\) 1.73905i 0.0627934i
\(768\) 0 0
\(769\) −9.24679 −0.333448 −0.166724 0.986004i \(-0.553319\pi\)
−0.166724 + 0.986004i \(0.553319\pi\)
\(770\) 0 0
\(771\) 44.6140 1.60673
\(772\) 0 0
\(773\) 34.0114i 1.22330i −0.791127 0.611652i \(-0.790505\pi\)
0.791127 0.611652i \(-0.209495\pi\)
\(774\) 0 0
\(775\) −12.5181 −0.449663
\(776\) 0 0
\(777\) −6.98014 −0.250411
\(778\) 0 0
\(779\) 2.86878 + 3.88331i 0.102785 + 0.139134i
\(780\) 0 0
\(781\) 67.2311 2.40572
\(782\) 0 0
\(783\) −50.5192 −1.80541
\(784\) 0 0
\(785\) 19.2782i 0.688070i
\(786\) 0 0
\(787\) 12.0908 0.430989 0.215494 0.976505i \(-0.430864\pi\)
0.215494 + 0.976505i \(0.430864\pi\)
\(788\) 0 0
\(789\) 39.2943 1.39891
\(790\) 0 0
\(791\) 13.9211i 0.494978i
\(792\) 0 0
\(793\) 14.9484i 0.530834i
\(794\) 0 0
\(795\) 38.1595 1.35338
\(796\) 0 0
\(797\) 2.80245 0.0992680 0.0496340 0.998767i \(-0.484195\pi\)
0.0496340 + 0.998767i \(0.484195\pi\)
\(798\) 0 0
\(799\) 11.6201 0.411088
\(800\) 0 0
\(801\) 60.9931i 2.15509i
\(802\) 0 0
\(803\) 6.88709i 0.243040i
\(804\) 0 0
\(805\) 2.29535i 0.0809004i
\(806\) 0 0