Properties

Label 1150.2.b.f.599.2
Level $1150$
Weight $2$
Character 1150.599
Analytic conductor $9.183$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1150.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.18279623245\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.2
Root \(1.30278i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.2.b.f.599.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +0.302776i q^{3} -1.00000 q^{4} +0.302776 q^{6} +3.30278i q^{7} +1.00000i q^{8} +2.90833 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +0.302776i q^{3} -1.00000 q^{4} +0.302776 q^{6} +3.30278i q^{7} +1.00000i q^{8} +2.90833 q^{9} -1.69722 q^{11} -0.302776i q^{12} -3.30278i q^{13} +3.30278 q^{14} +1.00000 q^{16} +6.90833i q^{17} -2.90833i q^{18} -5.90833 q^{19} -1.00000 q^{21} +1.69722i q^{22} +1.00000i q^{23} -0.302776 q^{24} -3.30278 q^{26} +1.78890i q^{27} -3.30278i q^{28} +2.60555 q^{29} -7.90833 q^{31} -1.00000i q^{32} -0.513878i q^{33} +6.90833 q^{34} -2.90833 q^{36} +8.00000i q^{37} +5.90833i q^{38} +1.00000 q^{39} +0.908327 q^{41} +1.00000i q^{42} +9.21110i q^{43} +1.69722 q^{44} +1.00000 q^{46} -2.60555i q^{47} +0.302776i q^{48} -3.90833 q^{49} -2.09167 q^{51} +3.30278i q^{52} +11.2111i q^{53} +1.78890 q^{54} -3.30278 q^{56} -1.78890i q^{57} -2.60555i q^{58} +3.39445 q^{59} +11.5139 q^{61} +7.90833i q^{62} +9.60555i q^{63} -1.00000 q^{64} -0.513878 q^{66} -4.00000i q^{67} -6.90833i q^{68} -0.302776 q^{69} -16.3028 q^{71} +2.90833i q^{72} +5.81665i q^{73} +8.00000 q^{74} +5.90833 q^{76} -5.60555i q^{77} -1.00000i q^{78} +14.4222 q^{79} +8.18335 q^{81} -0.908327i q^{82} -11.2111i q^{83} +1.00000 q^{84} +9.21110 q^{86} +0.788897i q^{87} -1.69722i q^{88} +10.9083 q^{91} -1.00000i q^{92} -2.39445i q^{93} -2.60555 q^{94} +0.302776 q^{96} +6.30278i q^{97} +3.90833i q^{98} -4.93608 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} - 6q^{6} - 10q^{9} + O(q^{10}) \) \( 4q - 4q^{4} - 6q^{6} - 10q^{9} - 14q^{11} + 6q^{14} + 4q^{16} - 2q^{19} - 4q^{21} + 6q^{24} - 6q^{26} - 4q^{29} - 10q^{31} + 6q^{34} + 10q^{36} + 4q^{39} - 18q^{41} + 14q^{44} + 4q^{46} + 6q^{49} - 30q^{51} + 36q^{54} - 6q^{56} + 28q^{59} + 10q^{61} - 4q^{64} + 34q^{66} + 6q^{69} - 58q^{71} + 32q^{74} + 2q^{76} + 76q^{81} + 4q^{84} + 8q^{86} + 22q^{91} + 4q^{94} - 6q^{96} + 74q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 0.302776i 0.174808i 0.996173 + 0.0874038i \(0.0278570\pi\)
−0.996173 + 0.0874038i \(0.972143\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0.302776 0.123608
\(7\) 3.30278i 1.24833i 0.781292 + 0.624166i \(0.214561\pi\)
−0.781292 + 0.624166i \(0.785439\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 2.90833 0.969442
\(10\) 0 0
\(11\) −1.69722 −0.511732 −0.255866 0.966712i \(-0.582361\pi\)
−0.255866 + 0.966712i \(0.582361\pi\)
\(12\) − 0.302776i − 0.0874038i
\(13\) − 3.30278i − 0.916025i −0.888946 0.458013i \(-0.848561\pi\)
0.888946 0.458013i \(-0.151439\pi\)
\(14\) 3.30278 0.882704
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.90833i 1.67552i 0.546042 + 0.837758i \(0.316134\pi\)
−0.546042 + 0.837758i \(0.683866\pi\)
\(18\) − 2.90833i − 0.685499i
\(19\) −5.90833 −1.35546 −0.677732 0.735309i \(-0.737037\pi\)
−0.677732 + 0.735309i \(0.737037\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 1.69722i 0.361849i
\(23\) 1.00000i 0.208514i
\(24\) −0.302776 −0.0618038
\(25\) 0 0
\(26\) −3.30278 −0.647728
\(27\) 1.78890i 0.344273i
\(28\) − 3.30278i − 0.624166i
\(29\) 2.60555 0.483839 0.241919 0.970296i \(-0.422223\pi\)
0.241919 + 0.970296i \(0.422223\pi\)
\(30\) 0 0
\(31\) −7.90833 −1.42038 −0.710189 0.704011i \(-0.751390\pi\)
−0.710189 + 0.704011i \(0.751390\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 0.513878i − 0.0894547i
\(34\) 6.90833 1.18477
\(35\) 0 0
\(36\) −2.90833 −0.484721
\(37\) 8.00000i 1.31519i 0.753371 + 0.657596i \(0.228427\pi\)
−0.753371 + 0.657596i \(0.771573\pi\)
\(38\) 5.90833i 0.958457i
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 0.908327 0.141857 0.0709284 0.997481i \(-0.477404\pi\)
0.0709284 + 0.997481i \(0.477404\pi\)
\(42\) 1.00000i 0.154303i
\(43\) 9.21110i 1.40468i 0.711842 + 0.702340i \(0.247861\pi\)
−0.711842 + 0.702340i \(0.752139\pi\)
\(44\) 1.69722 0.255866
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) − 2.60555i − 0.380059i −0.981778 0.190029i \(-0.939142\pi\)
0.981778 0.190029i \(-0.0608583\pi\)
\(48\) 0.302776i 0.0437019i
\(49\) −3.90833 −0.558332
\(50\) 0 0
\(51\) −2.09167 −0.292893
\(52\) 3.30278i 0.458013i
\(53\) 11.2111i 1.53996i 0.638066 + 0.769982i \(0.279735\pi\)
−0.638066 + 0.769982i \(0.720265\pi\)
\(54\) 1.78890 0.243438
\(55\) 0 0
\(56\) −3.30278 −0.441352
\(57\) − 1.78890i − 0.236945i
\(58\) − 2.60555i − 0.342126i
\(59\) 3.39445 0.441920 0.220960 0.975283i \(-0.429081\pi\)
0.220960 + 0.975283i \(0.429081\pi\)
\(60\) 0 0
\(61\) 11.5139 1.47420 0.737101 0.675783i \(-0.236194\pi\)
0.737101 + 0.675783i \(0.236194\pi\)
\(62\) 7.90833i 1.00436i
\(63\) 9.60555i 1.21019i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −0.513878 −0.0632540
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) − 6.90833i − 0.837758i
\(69\) −0.302776 −0.0364499
\(70\) 0 0
\(71\) −16.3028 −1.93478 −0.967392 0.253285i \(-0.918489\pi\)
−0.967392 + 0.253285i \(0.918489\pi\)
\(72\) 2.90833i 0.342750i
\(73\) 5.81665i 0.680788i 0.940283 + 0.340394i \(0.110560\pi\)
−0.940283 + 0.340394i \(0.889440\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) 5.90833 0.677732
\(77\) − 5.60555i − 0.638812i
\(78\) − 1.00000i − 0.113228i
\(79\) 14.4222 1.62262 0.811312 0.584613i \(-0.198754\pi\)
0.811312 + 0.584613i \(0.198754\pi\)
\(80\) 0 0
\(81\) 8.18335 0.909261
\(82\) − 0.908327i − 0.100308i
\(83\) − 11.2111i − 1.23058i −0.788301 0.615289i \(-0.789039\pi\)
0.788301 0.615289i \(-0.210961\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) 9.21110 0.993259
\(87\) 0.788897i 0.0845787i
\(88\) − 1.69722i − 0.180925i
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 10.9083 1.14350
\(92\) − 1.00000i − 0.104257i
\(93\) − 2.39445i − 0.248293i
\(94\) −2.60555 −0.268742
\(95\) 0 0
\(96\) 0.302776 0.0309019
\(97\) 6.30278i 0.639950i 0.947426 + 0.319975i \(0.103675\pi\)
−0.947426 + 0.319975i \(0.896325\pi\)
\(98\) 3.90833i 0.394801i
\(99\) −4.93608 −0.496095
\(100\) 0 0
\(101\) 2.60555 0.259262 0.129631 0.991562i \(-0.458621\pi\)
0.129631 + 0.991562i \(0.458621\pi\)
\(102\) 2.09167i 0.207106i
\(103\) − 8.11943i − 0.800031i −0.916508 0.400016i \(-0.869005\pi\)
0.916508 0.400016i \(-0.130995\pi\)
\(104\) 3.30278 0.323864
\(105\) 0 0
\(106\) 11.2111 1.08892
\(107\) − 2.60555i − 0.251888i −0.992037 0.125944i \(-0.959804\pi\)
0.992037 0.125944i \(-0.0401960\pi\)
\(108\) − 1.78890i − 0.172137i
\(109\) −1.48612 −0.142345 −0.0711723 0.997464i \(-0.522674\pi\)
−0.0711723 + 0.997464i \(0.522674\pi\)
\(110\) 0 0
\(111\) −2.42221 −0.229906
\(112\) 3.30278i 0.312083i
\(113\) 16.4222i 1.54487i 0.635093 + 0.772436i \(0.280962\pi\)
−0.635093 + 0.772436i \(0.719038\pi\)
\(114\) −1.78890 −0.167546
\(115\) 0 0
\(116\) −2.60555 −0.241919
\(117\) − 9.60555i − 0.888034i
\(118\) − 3.39445i − 0.312484i
\(119\) −22.8167 −2.09160
\(120\) 0 0
\(121\) −8.11943 −0.738130
\(122\) − 11.5139i − 1.04242i
\(123\) 0.275019i 0.0247977i
\(124\) 7.90833 0.710189
\(125\) 0 0
\(126\) 9.60555 0.855731
\(127\) 9.81665i 0.871087i 0.900168 + 0.435544i \(0.143444\pi\)
−0.900168 + 0.435544i \(0.856556\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −2.78890 −0.245549
\(130\) 0 0
\(131\) −11.2111 −0.979519 −0.489759 0.871858i \(-0.662915\pi\)
−0.489759 + 0.871858i \(0.662915\pi\)
\(132\) 0.513878i 0.0447274i
\(133\) − 19.5139i − 1.69207i
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −6.90833 −0.592384
\(137\) 3.90833i 0.333911i 0.985964 + 0.166955i \(0.0533936\pi\)
−0.985964 + 0.166955i \(0.946606\pi\)
\(138\) 0.302776i 0.0257740i
\(139\) 12.6056 1.06919 0.534594 0.845109i \(-0.320464\pi\)
0.534594 + 0.845109i \(0.320464\pi\)
\(140\) 0 0
\(141\) 0.788897 0.0664372
\(142\) 16.3028i 1.36810i
\(143\) 5.60555i 0.468760i
\(144\) 2.90833 0.242361
\(145\) 0 0
\(146\) 5.81665 0.481390
\(147\) − 1.18335i − 0.0976007i
\(148\) − 8.00000i − 0.657596i
\(149\) −13.3028 −1.08981 −0.544903 0.838499i \(-0.683433\pi\)
−0.544903 + 0.838499i \(0.683433\pi\)
\(150\) 0 0
\(151\) 8.90833 0.724949 0.362475 0.931994i \(-0.381932\pi\)
0.362475 + 0.931994i \(0.381932\pi\)
\(152\) − 5.90833i − 0.479229i
\(153\) 20.0917i 1.62432i
\(154\) −5.60555 −0.451708
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) − 18.6056i − 1.48488i −0.669910 0.742442i \(-0.733667\pi\)
0.669910 0.742442i \(-0.266333\pi\)
\(158\) − 14.4222i − 1.14737i
\(159\) −3.39445 −0.269197
\(160\) 0 0
\(161\) −3.30278 −0.260295
\(162\) − 8.18335i − 0.642944i
\(163\) − 9.30278i − 0.728650i −0.931272 0.364325i \(-0.881300\pi\)
0.931272 0.364325i \(-0.118700\pi\)
\(164\) −0.908327 −0.0709284
\(165\) 0 0
\(166\) −11.2111 −0.870150
\(167\) 6.78890i 0.525341i 0.964886 + 0.262670i \(0.0846032\pi\)
−0.964886 + 0.262670i \(0.915397\pi\)
\(168\) − 1.00000i − 0.0771517i
\(169\) 2.09167 0.160898
\(170\) 0 0
\(171\) −17.1833 −1.31404
\(172\) − 9.21110i − 0.702340i
\(173\) − 19.6972i − 1.49755i −0.662823 0.748776i \(-0.730642\pi\)
0.662823 0.748776i \(-0.269358\pi\)
\(174\) 0.788897 0.0598062
\(175\) 0 0
\(176\) −1.69722 −0.127933
\(177\) 1.02776i 0.0772509i
\(178\) 0 0
\(179\) −9.39445 −0.702174 −0.351087 0.936343i \(-0.614188\pi\)
−0.351087 + 0.936343i \(0.614188\pi\)
\(180\) 0 0
\(181\) 17.1194 1.27248 0.636239 0.771492i \(-0.280489\pi\)
0.636239 + 0.771492i \(0.280489\pi\)
\(182\) − 10.9083i − 0.808579i
\(183\) 3.48612i 0.257702i
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) −2.39445 −0.175569
\(187\) − 11.7250i − 0.857416i
\(188\) 2.60555i 0.190029i
\(189\) −5.90833 −0.429768
\(190\) 0 0
\(191\) −8.60555 −0.622676 −0.311338 0.950299i \(-0.600777\pi\)
−0.311338 + 0.950299i \(0.600777\pi\)
\(192\) − 0.302776i − 0.0218509i
\(193\) 17.8167i 1.28247i 0.767344 + 0.641235i \(0.221578\pi\)
−0.767344 + 0.641235i \(0.778422\pi\)
\(194\) 6.30278 0.452513
\(195\) 0 0
\(196\) 3.90833 0.279166
\(197\) 4.30278i 0.306560i 0.988183 + 0.153280i \(0.0489837\pi\)
−0.988183 + 0.153280i \(0.951016\pi\)
\(198\) 4.93608i 0.350792i
\(199\) 20.4222 1.44769 0.723846 0.689962i \(-0.242373\pi\)
0.723846 + 0.689962i \(0.242373\pi\)
\(200\) 0 0
\(201\) 1.21110 0.0854246
\(202\) − 2.60555i − 0.183326i
\(203\) 8.60555i 0.603991i
\(204\) 2.09167 0.146446
\(205\) 0 0
\(206\) −8.11943 −0.565707
\(207\) 2.90833i 0.202143i
\(208\) − 3.30278i − 0.229006i
\(209\) 10.0278 0.693634
\(210\) 0 0
\(211\) 7.21110 0.496433 0.248216 0.968705i \(-0.420156\pi\)
0.248216 + 0.968705i \(0.420156\pi\)
\(212\) − 11.2111i − 0.769982i
\(213\) − 4.93608i − 0.338215i
\(214\) −2.60555 −0.178112
\(215\) 0 0
\(216\) −1.78890 −0.121719
\(217\) − 26.1194i − 1.77310i
\(218\) 1.48612i 0.100653i
\(219\) −1.76114 −0.119007
\(220\) 0 0
\(221\) 22.8167 1.53481
\(222\) 2.42221i 0.162568i
\(223\) 4.00000i 0.267860i 0.990991 + 0.133930i \(0.0427597\pi\)
−0.990991 + 0.133930i \(0.957240\pi\)
\(224\) 3.30278 0.220676
\(225\) 0 0
\(226\) 16.4222 1.09239
\(227\) − 14.6056i − 0.969404i −0.874679 0.484702i \(-0.838928\pi\)
0.874679 0.484702i \(-0.161072\pi\)
\(228\) 1.78890i 0.118473i
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 1.69722 0.111669
\(232\) 2.60555i 0.171063i
\(233\) − 25.8167i − 1.69131i −0.533734 0.845653i \(-0.679212\pi\)
0.533734 0.845653i \(-0.320788\pi\)
\(234\) −9.60555 −0.627935
\(235\) 0 0
\(236\) −3.39445 −0.220960
\(237\) 4.36669i 0.283647i
\(238\) 22.8167i 1.47898i
\(239\) −5.21110 −0.337078 −0.168539 0.985695i \(-0.553905\pi\)
−0.168539 + 0.985695i \(0.553905\pi\)
\(240\) 0 0
\(241\) −14.4222 −0.929016 −0.464508 0.885569i \(-0.653769\pi\)
−0.464508 + 0.885569i \(0.653769\pi\)
\(242\) 8.11943i 0.521937i
\(243\) 7.84441i 0.503219i
\(244\) −11.5139 −0.737101
\(245\) 0 0
\(246\) 0.275019 0.0175346
\(247\) 19.5139i 1.24164i
\(248\) − 7.90833i − 0.502179i
\(249\) 3.39445 0.215114
\(250\) 0 0
\(251\) 12.5139 0.789869 0.394934 0.918709i \(-0.370767\pi\)
0.394934 + 0.918709i \(0.370767\pi\)
\(252\) − 9.60555i − 0.605093i
\(253\) − 1.69722i − 0.106704i
\(254\) 9.81665 0.615952
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 1.81665i − 0.113320i −0.998394 0.0566599i \(-0.981955\pi\)
0.998394 0.0566599i \(-0.0180451\pi\)
\(258\) 2.78890i 0.173629i
\(259\) −26.4222 −1.64180
\(260\) 0 0
\(261\) 7.57779 0.469054
\(262\) 11.2111i 0.692624i
\(263\) − 3.51388i − 0.216675i −0.994114 0.108338i \(-0.965447\pi\)
0.994114 0.108338i \(-0.0345527\pi\)
\(264\) 0.513878 0.0316270
\(265\) 0 0
\(266\) −19.5139 −1.19647
\(267\) 0 0
\(268\) 4.00000i 0.244339i
\(269\) −4.18335 −0.255063 −0.127532 0.991835i \(-0.540705\pi\)
−0.127532 + 0.991835i \(0.540705\pi\)
\(270\) 0 0
\(271\) −2.69722 −0.163845 −0.0819224 0.996639i \(-0.526106\pi\)
−0.0819224 + 0.996639i \(0.526106\pi\)
\(272\) 6.90833i 0.418879i
\(273\) 3.30278i 0.199893i
\(274\) 3.90833 0.236111
\(275\) 0 0
\(276\) 0.302776 0.0182250
\(277\) − 27.2111i − 1.63496i −0.575959 0.817478i \(-0.695371\pi\)
0.575959 0.817478i \(-0.304629\pi\)
\(278\) − 12.6056i − 0.756031i
\(279\) −23.0000 −1.37697
\(280\) 0 0
\(281\) −26.6056 −1.58715 −0.793577 0.608470i \(-0.791784\pi\)
−0.793577 + 0.608470i \(0.791784\pi\)
\(282\) − 0.788897i − 0.0469782i
\(283\) − 2.00000i − 0.118888i −0.998232 0.0594438i \(-0.981067\pi\)
0.998232 0.0594438i \(-0.0189327\pi\)
\(284\) 16.3028 0.967392
\(285\) 0 0
\(286\) 5.60555 0.331463
\(287\) 3.00000i 0.177084i
\(288\) − 2.90833i − 0.171375i
\(289\) −30.7250 −1.80735
\(290\) 0 0
\(291\) −1.90833 −0.111868
\(292\) − 5.81665i − 0.340394i
\(293\) − 23.2111i − 1.35601i −0.735059 0.678004i \(-0.762845\pi\)
0.735059 0.678004i \(-0.237155\pi\)
\(294\) −1.18335 −0.0690142
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) − 3.03616i − 0.176176i
\(298\) 13.3028i 0.770609i
\(299\) 3.30278 0.191004
\(300\) 0 0
\(301\) −30.4222 −1.75351
\(302\) − 8.90833i − 0.512617i
\(303\) 0.788897i 0.0453210i
\(304\) −5.90833 −0.338866
\(305\) 0 0
\(306\) 20.0917 1.14856
\(307\) − 11.6972i − 0.667596i −0.942645 0.333798i \(-0.891670\pi\)
0.942645 0.333798i \(-0.108330\pi\)
\(308\) 5.60555i 0.319406i
\(309\) 2.45837 0.139852
\(310\) 0 0
\(311\) 22.4222 1.27145 0.635723 0.771917i \(-0.280702\pi\)
0.635723 + 0.771917i \(0.280702\pi\)
\(312\) 1.00000i 0.0566139i
\(313\) − 19.7250i − 1.11492i −0.830203 0.557461i \(-0.811776\pi\)
0.830203 0.557461i \(-0.188224\pi\)
\(314\) −18.6056 −1.04997
\(315\) 0 0
\(316\) −14.4222 −0.811312
\(317\) 17.7250i 0.995534i 0.867311 + 0.497767i \(0.165847\pi\)
−0.867311 + 0.497767i \(0.834153\pi\)
\(318\) 3.39445i 0.190351i
\(319\) −4.42221 −0.247596
\(320\) 0 0
\(321\) 0.788897 0.0440320
\(322\) 3.30278i 0.184056i
\(323\) − 40.8167i − 2.27110i
\(324\) −8.18335 −0.454630
\(325\) 0 0
\(326\) −9.30278 −0.515233
\(327\) − 0.449961i − 0.0248829i
\(328\) 0.908327i 0.0501540i
\(329\) 8.60555 0.474439
\(330\) 0 0
\(331\) 16.6056 0.912724 0.456362 0.889794i \(-0.349152\pi\)
0.456362 + 0.889794i \(0.349152\pi\)
\(332\) 11.2111i 0.615289i
\(333\) 23.2666i 1.27500i
\(334\) 6.78890 0.371472
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) − 22.5139i − 1.22641i −0.789924 0.613205i \(-0.789880\pi\)
0.789924 0.613205i \(-0.210120\pi\)
\(338\) − 2.09167i − 0.113772i
\(339\) −4.97224 −0.270055
\(340\) 0 0
\(341\) 13.4222 0.726853
\(342\) 17.1833i 0.929169i
\(343\) 10.2111i 0.551348i
\(344\) −9.21110 −0.496629
\(345\) 0 0
\(346\) −19.6972 −1.05893
\(347\) 28.5416i 1.53220i 0.642724 + 0.766098i \(0.277804\pi\)
−0.642724 + 0.766098i \(0.722196\pi\)
\(348\) − 0.788897i − 0.0422893i
\(349\) 27.2111 1.45658 0.728288 0.685271i \(-0.240316\pi\)
0.728288 + 0.685271i \(0.240316\pi\)
\(350\) 0 0
\(351\) 5.90833 0.315363
\(352\) 1.69722i 0.0904624i
\(353\) 10.4222i 0.554718i 0.960766 + 0.277359i \(0.0894591\pi\)
−0.960766 + 0.277359i \(0.910541\pi\)
\(354\) 1.02776 0.0546246
\(355\) 0 0
\(356\) 0 0
\(357\) − 6.90833i − 0.365627i
\(358\) 9.39445i 0.496512i
\(359\) −11.2111 −0.591699 −0.295850 0.955235i \(-0.595603\pi\)
−0.295850 + 0.955235i \(0.595603\pi\)
\(360\) 0 0
\(361\) 15.9083 0.837280
\(362\) − 17.1194i − 0.899777i
\(363\) − 2.45837i − 0.129031i
\(364\) −10.9083 −0.571752
\(365\) 0 0
\(366\) 3.48612 0.182223
\(367\) 14.7889i 0.771974i 0.922504 + 0.385987i \(0.126139\pi\)
−0.922504 + 0.385987i \(0.873861\pi\)
\(368\) 1.00000i 0.0521286i
\(369\) 2.64171 0.137522
\(370\) 0 0
\(371\) −37.0278 −1.92239
\(372\) 2.39445i 0.124146i
\(373\) − 4.60555i − 0.238466i −0.992866 0.119233i \(-0.961956\pi\)
0.992866 0.119233i \(-0.0380436\pi\)
\(374\) −11.7250 −0.606284
\(375\) 0 0
\(376\) 2.60555 0.134371
\(377\) − 8.60555i − 0.443208i
\(378\) 5.90833i 0.303892i
\(379\) −14.9083 −0.765789 −0.382895 0.923792i \(-0.625073\pi\)
−0.382895 + 0.923792i \(0.625073\pi\)
\(380\) 0 0
\(381\) −2.97224 −0.152273
\(382\) 8.60555i 0.440298i
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) −0.302776 −0.0154510
\(385\) 0 0
\(386\) 17.8167 0.906844
\(387\) 26.7889i 1.36176i
\(388\) − 6.30278i − 0.319975i
\(389\) 25.9361 1.31501 0.657506 0.753449i \(-0.271612\pi\)
0.657506 + 0.753449i \(0.271612\pi\)
\(390\) 0 0
\(391\) −6.90833 −0.349369
\(392\) − 3.90833i − 0.197400i
\(393\) − 3.39445i − 0.171227i
\(394\) 4.30278 0.216771
\(395\) 0 0
\(396\) 4.93608 0.248048
\(397\) 10.7250i 0.538271i 0.963102 + 0.269136i \(0.0867380\pi\)
−0.963102 + 0.269136i \(0.913262\pi\)
\(398\) − 20.4222i − 1.02367i
\(399\) 5.90833 0.295786
\(400\) 0 0
\(401\) −8.60555 −0.429741 −0.214870 0.976643i \(-0.568933\pi\)
−0.214870 + 0.976643i \(0.568933\pi\)
\(402\) − 1.21110i − 0.0604043i
\(403\) 26.1194i 1.30110i
\(404\) −2.60555 −0.129631
\(405\) 0 0
\(406\) 8.60555 0.427086
\(407\) − 13.5778i − 0.673026i
\(408\) − 2.09167i − 0.103553i
\(409\) 25.9083 1.28108 0.640542 0.767923i \(-0.278710\pi\)
0.640542 + 0.767923i \(0.278710\pi\)
\(410\) 0 0
\(411\) −1.18335 −0.0583702
\(412\) 8.11943i 0.400016i
\(413\) 11.2111i 0.551662i
\(414\) 2.90833 0.142936
\(415\) 0 0
\(416\) −3.30278 −0.161932
\(417\) 3.81665i 0.186902i
\(418\) − 10.0278i − 0.490474i
\(419\) −3.63331 −0.177499 −0.0887493 0.996054i \(-0.528287\pi\)
−0.0887493 + 0.996054i \(0.528287\pi\)
\(420\) 0 0
\(421\) 30.6972 1.49609 0.748046 0.663647i \(-0.230992\pi\)
0.748046 + 0.663647i \(0.230992\pi\)
\(422\) − 7.21110i − 0.351031i
\(423\) − 7.57779i − 0.368445i
\(424\) −11.2111 −0.544459
\(425\) 0 0
\(426\) −4.93608 −0.239154
\(427\) 38.0278i 1.84029i
\(428\) 2.60555i 0.125944i
\(429\) −1.69722 −0.0819428
\(430\) 0 0
\(431\) −30.2389 −1.45655 −0.728277 0.685283i \(-0.759679\pi\)
−0.728277 + 0.685283i \(0.759679\pi\)
\(432\) 1.78890i 0.0860684i
\(433\) 24.0917i 1.15777i 0.815409 + 0.578886i \(0.196512\pi\)
−0.815409 + 0.578886i \(0.803488\pi\)
\(434\) −26.1194 −1.25377
\(435\) 0 0
\(436\) 1.48612 0.0711723
\(437\) − 5.90833i − 0.282634i
\(438\) 1.76114i 0.0841506i
\(439\) 14.6972 0.701460 0.350730 0.936477i \(-0.385933\pi\)
0.350730 + 0.936477i \(0.385933\pi\)
\(440\) 0 0
\(441\) −11.3667 −0.541271
\(442\) − 22.8167i − 1.08528i
\(443\) − 17.4861i − 0.830791i −0.909641 0.415395i \(-0.863643\pi\)
0.909641 0.415395i \(-0.136357\pi\)
\(444\) 2.42221 0.114953
\(445\) 0 0
\(446\) 4.00000 0.189405
\(447\) − 4.02776i − 0.190506i
\(448\) − 3.30278i − 0.156041i
\(449\) 2.09167 0.0987122 0.0493561 0.998781i \(-0.484283\pi\)
0.0493561 + 0.998781i \(0.484283\pi\)
\(450\) 0 0
\(451\) −1.54163 −0.0725927
\(452\) − 16.4222i − 0.772436i
\(453\) 2.69722i 0.126727i
\(454\) −14.6056 −0.685472
\(455\) 0 0
\(456\) 1.78890 0.0837728
\(457\) − 32.4222i − 1.51665i −0.651879 0.758323i \(-0.726019\pi\)
0.651879 0.758323i \(-0.273981\pi\)
\(458\) 2.00000i 0.0934539i
\(459\) −12.3583 −0.576836
\(460\) 0 0
\(461\) 10.1833 0.474286 0.237143 0.971475i \(-0.423789\pi\)
0.237143 + 0.971475i \(0.423789\pi\)
\(462\) − 1.69722i − 0.0789620i
\(463\) − 17.6333i − 0.819489i −0.912200 0.409745i \(-0.865618\pi\)
0.912200 0.409745i \(-0.134382\pi\)
\(464\) 2.60555 0.120960
\(465\) 0 0
\(466\) −25.8167 −1.19593
\(467\) − 1.81665i − 0.0840647i −0.999116 0.0420324i \(-0.986617\pi\)
0.999116 0.0420324i \(-0.0133833\pi\)
\(468\) 9.60555i 0.444017i
\(469\) 13.2111 0.610032
\(470\) 0 0
\(471\) 5.63331 0.259569
\(472\) 3.39445i 0.156242i
\(473\) − 15.6333i − 0.718820i
\(474\) 4.36669 0.200569
\(475\) 0 0
\(476\) 22.8167 1.04580
\(477\) 32.6056i 1.49291i
\(478\) 5.21110i 0.238350i
\(479\) 30.0000 1.37073 0.685367 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) 0 0
\(481\) 26.4222 1.20475
\(482\) 14.4222i 0.656913i
\(483\) − 1.00000i − 0.0455016i
\(484\) 8.11943 0.369065
\(485\) 0 0
\(486\) 7.84441 0.355830
\(487\) 9.81665i 0.444835i 0.974952 + 0.222418i \(0.0713948\pi\)
−0.974952 + 0.222418i \(0.928605\pi\)
\(488\) 11.5139i 0.521209i
\(489\) 2.81665 0.127373
\(490\) 0 0
\(491\) −4.18335 −0.188792 −0.0943959 0.995535i \(-0.530092\pi\)
−0.0943959 + 0.995535i \(0.530092\pi\)
\(492\) − 0.275019i − 0.0123988i
\(493\) 18.0000i 0.810679i
\(494\) 19.5139 0.877971
\(495\) 0 0
\(496\) −7.90833 −0.355094
\(497\) − 53.8444i − 2.41525i
\(498\) − 3.39445i − 0.152109i
\(499\) 31.6333 1.41610 0.708051 0.706162i \(-0.249575\pi\)
0.708051 + 0.706162i \(0.249575\pi\)
\(500\) 0 0
\(501\) −2.05551 −0.0918335
\(502\) − 12.5139i − 0.558522i
\(503\) − 29.7250i − 1.32537i −0.748898 0.662686i \(-0.769417\pi\)
0.748898 0.662686i \(-0.230583\pi\)
\(504\) −9.60555 −0.427865
\(505\) 0 0
\(506\) −1.69722 −0.0754508
\(507\) 0.633308i 0.0281262i
\(508\) − 9.81665i − 0.435544i
\(509\) 35.4500 1.57129 0.785646 0.618676i \(-0.212331\pi\)
0.785646 + 0.618676i \(0.212331\pi\)
\(510\) 0 0
\(511\) −19.2111 −0.849849
\(512\) − 1.00000i − 0.0441942i
\(513\) − 10.5694i − 0.466650i
\(514\) −1.81665 −0.0801292
\(515\) 0 0
\(516\) 2.78890 0.122774
\(517\) 4.42221i 0.194488i
\(518\) 26.4222i 1.16093i
\(519\) 5.96384 0.261784
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) − 7.57779i − 0.331671i
\(523\) 20.4222i 0.893001i 0.894783 + 0.446500i \(0.147330\pi\)
−0.894783 + 0.446500i \(0.852670\pi\)
\(524\) 11.2111 0.489759
\(525\) 0 0
\(526\) −3.51388 −0.153212
\(527\) − 54.6333i − 2.37986i
\(528\) − 0.513878i − 0.0223637i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 9.87217 0.428416
\(532\) 19.5139i 0.846034i
\(533\) − 3.00000i − 0.129944i
\(534\) 0 0
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) − 2.84441i − 0.122745i
\(538\) 4.18335i 0.180357i
\(539\) 6.63331 0.285717
\(540\) 0 0
\(541\) 28.8444 1.24012 0.620059 0.784555i \(-0.287109\pi\)
0.620059 + 0.784555i \(0.287109\pi\)
\(542\) 2.69722i 0.115856i
\(543\) 5.18335i 0.222439i
\(544\) 6.90833 0.296192
\(545\) 0 0
\(546\) 3.30278 0.141346
\(547\) − 10.5139i − 0.449541i −0.974412 0.224770i \(-0.927837\pi\)
0.974412 0.224770i \(-0.0721632\pi\)
\(548\) − 3.90833i − 0.166955i
\(549\) 33.4861 1.42915
\(550\) 0 0
\(551\) −15.3944 −0.655826
\(552\) − 0.302776i − 0.0128870i
\(553\) 47.6333i 2.02557i
\(554\) −27.2111 −1.15609
\(555\) 0 0
\(556\) −12.6056 −0.534594
\(557\) 22.4222i 0.950059i 0.879970 + 0.475030i \(0.157563\pi\)
−0.879970 + 0.475030i \(0.842437\pi\)
\(558\) 23.0000i 0.973668i
\(559\) 30.4222 1.28672
\(560\) 0 0
\(561\) 3.55004 0.149883
\(562\) 26.6056i 1.12229i
\(563\) 3.63331i 0.153126i 0.997065 + 0.0765628i \(0.0243946\pi\)
−0.997065 + 0.0765628i \(0.975605\pi\)
\(564\) −0.788897 −0.0332186
\(565\) 0 0
\(566\) −2.00000 −0.0840663
\(567\) 27.0278i 1.13506i
\(568\) − 16.3028i − 0.684049i
\(569\) −28.4222 −1.19152 −0.595760 0.803162i \(-0.703149\pi\)
−0.595760 + 0.803162i \(0.703149\pi\)
\(570\) 0 0
\(571\) −16.1194 −0.674577 −0.337289 0.941401i \(-0.609510\pi\)
−0.337289 + 0.941401i \(0.609510\pi\)
\(572\) − 5.60555i − 0.234380i
\(573\) − 2.60555i − 0.108848i
\(574\) 3.00000 0.125218
\(575\) 0 0
\(576\) −2.90833 −0.121180
\(577\) 2.00000i 0.0832611i 0.999133 + 0.0416305i \(0.0132552\pi\)
−0.999133 + 0.0416305i \(0.986745\pi\)
\(578\) 30.7250i 1.27799i
\(579\) −5.39445 −0.224186
\(580\) 0 0
\(581\) 37.0278 1.53617
\(582\) 1.90833i 0.0791027i
\(583\) − 19.0278i − 0.788049i
\(584\) −5.81665 −0.240695
\(585\) 0 0
\(586\) −23.2111 −0.958842
\(587\) 16.5416i 0.682746i 0.939928 + 0.341373i \(0.110892\pi\)
−0.939928 + 0.341373i \(0.889108\pi\)
\(588\) 1.18335i 0.0488004i
\(589\) 46.7250 1.92527
\(590\) 0 0
\(591\) −1.30278 −0.0535890
\(592\) 8.00000i 0.328798i
\(593\) 1.81665i 0.0746010i 0.999304 + 0.0373005i \(0.0118759\pi\)
−0.999304 + 0.0373005i \(0.988124\pi\)
\(594\) −3.03616 −0.124575
\(595\) 0 0
\(596\) 13.3028 0.544903
\(597\) 6.18335i 0.253068i
\(598\) − 3.30278i − 0.135061i
\(599\) 35.3305 1.44357 0.721783 0.692119i \(-0.243323\pi\)
0.721783 + 0.692119i \(0.243323\pi\)
\(600\) 0 0
\(601\) 42.9361 1.75140 0.875700 0.482856i \(-0.160401\pi\)
0.875700 + 0.482856i \(0.160401\pi\)
\(602\) 30.4222i 1.23992i
\(603\) − 11.6333i − 0.473745i
\(604\) −8.90833 −0.362475
\(605\) 0 0
\(606\) 0.788897 0.0320468
\(607\) 46.0555i 1.86934i 0.355522 + 0.934668i \(0.384303\pi\)
−0.355522 + 0.934668i \(0.615697\pi\)
\(608\) 5.90833i 0.239614i
\(609\) −2.60555 −0.105582
\(610\) 0 0
\(611\) −8.60555 −0.348143
\(612\) − 20.0917i − 0.812158i
\(613\) − 3.57779i − 0.144506i −0.997386 0.0722529i \(-0.976981\pi\)
0.997386 0.0722529i \(-0.0230189\pi\)
\(614\) −11.6972 −0.472062
\(615\) 0 0
\(616\) 5.60555 0.225854
\(617\) 18.9083i 0.761221i 0.924736 + 0.380610i \(0.124286\pi\)
−0.924736 + 0.380610i \(0.875714\pi\)
\(618\) − 2.45837i − 0.0988900i
\(619\) 12.3305 0.495606 0.247803 0.968810i \(-0.420291\pi\)
0.247803 + 0.968810i \(0.420291\pi\)
\(620\) 0 0
\(621\) −1.78890 −0.0717860
\(622\) − 22.4222i − 0.899049i
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) 0 0
\(626\) −19.7250 −0.788369
\(627\) 3.03616i 0.121253i
\(628\) 18.6056i 0.742442i
\(629\) −55.2666 −2.20362
\(630\) 0 0
\(631\) 23.3944 0.931318 0.465659 0.884964i \(-0.345817\pi\)
0.465659 + 0.884964i \(0.345817\pi\)
\(632\) 14.4222i 0.573685i
\(633\) 2.18335i 0.0867802i
\(634\) 17.7250 0.703949
\(635\) 0 0
\(636\) 3.39445 0.134599
\(637\) 12.9083i 0.511447i
\(638\) 4.42221i 0.175077i
\(639\) −47.4138 −1.87566
\(640\) 0 0
\(641\) −36.0000 −1.42191 −0.710957 0.703235i \(-0.751738\pi\)
−0.710957 + 0.703235i \(0.751738\pi\)
\(642\) − 0.788897i − 0.0311353i
\(643\) 34.2389i 1.35025i 0.737704 + 0.675124i \(0.235910\pi\)
−0.737704 + 0.675124i \(0.764090\pi\)
\(644\) 3.30278 0.130148
\(645\) 0 0
\(646\) −40.8167 −1.60591
\(647\) 26.8444i 1.05536i 0.849442 + 0.527681i \(0.176938\pi\)
−0.849442 + 0.527681i \(0.823062\pi\)
\(648\) 8.18335i 0.321472i
\(649\) −5.76114 −0.226145
\(650\) 0 0
\(651\) 7.90833 0.309952
\(652\) 9.30278i 0.364325i
\(653\) − 41.7250i − 1.63282i −0.577469 0.816412i \(-0.695960\pi\)
0.577469 0.816412i \(-0.304040\pi\)
\(654\) −0.449961 −0.0175949
\(655\) 0 0
\(656\) 0.908327 0.0354642
\(657\) 16.9167i 0.659985i
\(658\) − 8.60555i − 0.335479i
\(659\) −15.6333 −0.608987 −0.304494 0.952514i \(-0.598487\pi\)
−0.304494 + 0.952514i \(0.598487\pi\)
\(660\) 0 0
\(661\) −34.9083 −1.35778 −0.678888 0.734242i \(-0.737538\pi\)
−0.678888 + 0.734242i \(0.737538\pi\)
\(662\) − 16.6056i − 0.645393i
\(663\) 6.90833i 0.268297i
\(664\) 11.2111 0.435075
\(665\) 0 0
\(666\) 23.2666 0.901563
\(667\) 2.60555i 0.100887i
\(668\) − 6.78890i − 0.262670i
\(669\) −1.21110 −0.0468239
\(670\) 0 0
\(671\) −19.5416 −0.754396
\(672\) 1.00000i 0.0385758i
\(673\) 37.6333i 1.45066i 0.688403 + 0.725329i \(0.258312\pi\)
−0.688403 + 0.725329i \(0.741688\pi\)
\(674\) −22.5139 −0.867202
\(675\) 0 0
\(676\) −2.09167 −0.0804490
\(677\) 16.4222i 0.631157i 0.948900 + 0.315578i \(0.102198\pi\)
−0.948900 + 0.315578i \(0.897802\pi\)
\(678\) 4.97224i 0.190958i
\(679\) −20.8167 −0.798870
\(680\) 0 0
\(681\) 4.42221 0.169459
\(682\) − 13.4222i − 0.513963i
\(683\) 0.275019i 0.0105233i 0.999986 + 0.00526166i \(0.00167485\pi\)
−0.999986 + 0.00526166i \(0.998325\pi\)
\(684\) 17.1833 0.657022
\(685\) 0 0
\(686\) 10.2111 0.389862
\(687\) − 0.605551i − 0.0231032i
\(688\) 9.21110i 0.351170i
\(689\) 37.0278 1.41065
\(690\) 0 0
\(691\) 51.8167 1.97120 0.985599 0.169098i \(-0.0540855\pi\)
0.985599 + 0.169098i \(0.0540855\pi\)
\(692\) 19.6972i 0.748776i
\(693\) − 16.3028i − 0.619291i
\(694\) 28.5416 1.08343
\(695\) 0 0
\(696\) −0.788897 −0.0299031
\(697\) 6.27502i 0.237683i
\(698\) − 27.2111i − 1.02996i
\(699\) 7.81665 0.295653
\(700\) 0 0
\(701\) 32.0917 1.21209 0.606043 0.795432i \(-0.292756\pi\)
0.606043 + 0.795432i \(0.292756\pi\)
\(702\) − 5.90833i − 0.222995i
\(703\) − 47.2666i − 1.78269i
\(704\) 1.69722 0.0639665
\(705\) 0 0
\(706\) 10.4222 0.392245
\(707\) 8.60555i 0.323645i
\(708\) − 1.02776i − 0.0386254i
\(709\) 15.8806 0.596407 0.298204 0.954502i \(-0.403613\pi\)
0.298204 + 0.954502i \(0.403613\pi\)
\(710\) 0 0
\(711\) 41.9445 1.57304
\(712\) 0 0
\(713\) − 7.90833i − 0.296169i
\(714\) −6.90833 −0.258538
\(715\) 0 0
\(716\) 9.39445 0.351087
\(717\) − 1.57779i − 0.0589238i
\(718\) 11.2111i 0.418395i
\(719\) −10.6972 −0.398939 −0.199470 0.979904i \(-0.563922\pi\)
−0.199470 + 0.979904i \(0.563922\pi\)
\(720\) 0 0
\(721\) 26.8167 0.998704
\(722\) − 15.9083i − 0.592047i
\(723\) − 4.36669i − 0.162399i
\(724\) −17.1194 −0.636239
\(725\) 0 0
\(726\) −2.45837 −0.0912385
\(727\) 2.90833i 0.107864i 0.998545 + 0.0539319i \(0.0171754\pi\)
−0.998545 + 0.0539319i \(0.982825\pi\)
\(728\) 10.9083i 0.404289i
\(729\) 22.1749 0.821294
\(730\) 0 0
\(731\) −63.6333 −2.35356
\(732\) − 3.48612i − 0.128851i
\(733\) − 29.6333i − 1.09453i −0.836959 0.547266i \(-0.815669\pi\)
0.836959 0.547266i \(-0.184331\pi\)
\(734\) 14.7889 0.545868
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 6.78890i 0.250072i
\(738\) − 2.64171i − 0.0972427i
\(739\) −35.6333 −1.31079 −0.655396 0.755285i \(-0.727498\pi\)
−0.655396 + 0.755285i \(0.727498\pi\)
\(740\) 0 0
\(741\) −5.90833 −0.217048
\(742\) 37.0278i 1.35933i
\(743\) 32.3305i 1.18609i 0.805169 + 0.593046i \(0.202075\pi\)
−0.805169 + 0.593046i \(0.797925\pi\)
\(744\) 2.39445 0.0877847
\(745\) 0 0
\(746\) −4.60555 −0.168621
\(747\) − 32.6056i − 1.19297i
\(748\) 11.7250i 0.428708i
\(749\) 8.60555 0.314440
\(750\) 0 0
\(751\) 21.8167 0.796101 0.398051 0.917364i \(-0.369687\pi\)
0.398051 + 0.917364i \(0.369687\pi\)
\(752\) − 2.60555i − 0.0950147i
\(753\) 3.78890i 0.138075i
\(754\) −8.60555 −0.313396
\(755\) 0 0
\(756\) 5.90833 0.214884
\(757\) 13.2111i 0.480166i 0.970752 + 0.240083i \(0.0771746\pi\)
−0.970752 + 0.240083i \(0.922825\pi\)
\(758\) 14.9083i 0.541495i
\(759\) 0.513878 0.0186526
\(760\) 0 0
\(761\) −49.5416 −1.79588 −0.897941 0.440115i \(-0.854938\pi\)
−0.897941 + 0.440115i \(0.854938\pi\)
\(762\) 2.97224i 0.107673i
\(763\) − 4.90833i − 0.177693i
\(764\) 8.60555 0.311338
\(765\) 0 0
\(766\) 0 0
\(767\) − 11.2111i − 0.404809i
\(768\) 0.302776i 0.0109255i
\(769\) −45.2666 −1.63236 −0.816178 0.577801i \(-0.803911\pi\)
−0.816178 + 0.577801i \(0.803911\pi\)
\(770\) 0 0
\(771\) 0.550039 0.0198092
\(772\) − 17.8167i − 0.641235i
\(773\) − 12.0000i − 0.431610i −0.976436 0.215805i \(-0.930762\pi\)
0.976436 0.215805i \(-0.0692376\pi\)
\(774\) 26.7889 0.962907
\(775\) 0 0
\(776\) −6.30278 −0.226256
\(777\) − 8.00000i − 0.286998i
\(778\) − 25.9361i − 0.929854i
\(779\) −5.36669 −0.192282
\(780\) 0 0
\(781\) 27.6695 0.990091
\(782\) 6.90833i 0.247041i
\(783\) 4.66106i 0.166573i
\(784\) −3.90833 −0.139583
\(785\) 0 0
\(786\) −3.39445 −0.121076
\(787\) 37.4500i 1.33495i 0.744634 + 0.667473i \(0.232624\pi\)
−0.744634 + 0.667473i \(0.767376\pi\)
\(788\) − 4.30278i − 0.153280i
\(789\) 1.06392 0.0378764
\(790\) 0 0
\(791\) −54.2389 −1.92851
\(792\) − 4.93608i − 0.175396i
\(793\) − 38.0278i − 1.35041i
\(794\) 10.7250 0.380615
\(795\) 0 0
\(796\) −20.4222 −0.723846
\(797\) 1.81665i 0.0643492i 0.999482 + 0.0321746i \(0.0102433\pi\)
−0.999482 + 0.0321746i \(0.989757\pi\)
\(798\) − 5.90833i − 0.209153i
\(799\) 18.0000 0.636794
\(800\) 0 0
\(801\) 0 0
\(802\) 8.60555i 0.303873i
\(803\) − 9.87217i − 0.348381i
\(804\) −1.21110 −0.0427123
\(805\) 0 0
\(806\) 26.1194 0.920018
\(807\) − 1.26662i − 0.0445870i
\(808\) 2.60555i 0.0916630i
\(809\) −50.7250 −1.78340 −0.891698 0.452631i \(-0.850485\pi\)
−0.891698 + 0.452631i \(0.850485\pi\)
\(810\) 0 0
\(811\) −41.0278 −1.44068 −0.720340 0.693621i \(-0.756014\pi\)
−0.720340 + 0.693621i \(0.756014\pi\)
\(812\) − 8.60555i − 0.301996i
\(813\) − 0.816654i − 0.0286413i
\(814\) −13.5778 −0.475901
\(815\) 0 0
\(816\) −2.09167 −0.0732232
\(817\) − 54.4222i − 1.90399i
\(818\) − 25.9083i − 0.905863i
\(819\) 31.7250 1.10856
\(820\) 0 0
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) 1.18335i 0.0412739i
\(823\) 15.2111i 0.530226i 0.964217 + 0.265113i \(0.0854092\pi\)
−0.964217 + 0.265113i \(0.914591\pi\)
\(824\) 8.11943 0.282854
\(825\) 0 0
\(826\) 11.2111 0.390084
\(827\) − 29.4500i − 1.02408i −0.858963 0.512038i \(-0.828891\pi\)
0.858963 0.512038i \(-0.171109\pi\)
\(828\) − 2.90833i − 0.101071i
\(829\) −31.2111 −1.08401 −0.542003 0.840376i \(-0.682334\pi\)
−0.542003 + 0.840376i \(0.682334\pi\)
\(830\) 0 0
\(831\) 8.23886 0.285803
\(832\) 3.30278i 0.114503i
\(833\) − 27.0000i − 0.935495i
\(834\) 3.81665 0.132160
\(835\) 0 0
\(836\) −10.0278 −0.346817
\(837\) − 14.1472i − 0.488998i
\(838\) 3.63331i 0.125511i
\(839\) 43.8167 1.51272 0.756359 0.654156i \(-0.226976\pi\)
0.756359 + 0.654156i \(0.226976\pi\)
\(840\) 0 0
\(841\) −22.2111 −0.765900
\(842\) − 30.6972i − 1.05790i
\(843\) − 8.05551i − 0.277447i
\(844\) −7.21110 −0.248216
\(845\) 0 0
\(846\) −7.57779 −0.260530
\(847\) − 26.8167i − 0.921431i
\(848\) 11.2111i 0.384991i
\(849\) 0.605551 0.0207825
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 4.93608i 0.169107i
\(853\) 21.7250i 0.743849i 0.928263 + 0.371925i \(0.121302\pi\)
−0.928263 + 0.371925i \(0.878698\pi\)
\(854\) 38.0278 1.30128
\(855\) 0 0
\(856\) 2.60555 0.0890559
\(857\) 9.63331i 0.329068i 0.986371 + 0.164534i \(0.0526119\pi\)
−0.986371 + 0.164534i \(0.947388\pi\)
\(858\) 1.69722i 0.0579423i
\(859\) 35.8167 1.22205 0.611024 0.791612i \(-0.290758\pi\)
0.611024 + 0.791612i \(0.290758\pi\)
\(860\) 0 0
\(861\) −0.908327 −0.0309557
\(862\) 30.2389i 1.02994i
\(863\) 41.4500i 1.41097i 0.708723 + 0.705487i \(0.249271\pi\)
−0.708723 + 0.705487i \(0.750729\pi\)
\(864\) 1.78890 0.0608595
\(865\) 0 0
\(866\) 24.0917 0.818668
\(867\) − 9.30278i − 0.315939i
\(868\) 26.1194i 0.886551i
\(869\) −24.4777 −0.830350
\(870\) 0 0
\(871\) −13.2111 −0.447641
\(872\) − 1.48612i − 0.0503264i
\(873\) 18.3305i 0.620395i
\(874\) −5.90833 −0.199852
\(875\) 0 0
\(876\) 1.76114 0.0595034
\(877\) − 48.1749i − 1.62675i −0.581738 0.813376i \(-0.697627\pi\)
0.581738 0.813376i \(-0.302373\pi\)
\(878\) − 14.6972i − 0.496007i
\(879\) 7.02776 0.237040
\(880\) 0 0
\(881\) 55.2666 1.86198 0.930990 0.365045i \(-0.118946\pi\)
0.930990 + 0.365045i \(0.118946\pi\)
\(882\) 11.3667i 0.382736i
\(883\) − 8.27502i − 0.278477i −0.990259 0.139238i \(-0.955535\pi\)
0.990259 0.139238i \(-0.0444654\pi\)
\(884\) −22.8167 −0.767407
\(885\) 0 0
\(886\) −17.4861 −0.587458
\(887\) 27.6333i 0.927836i 0.885878 + 0.463918i \(0.153557\pi\)
−0.885878 + 0.463918i \(0.846443\pi\)
\(888\) − 2.42221i − 0.0812839i
\(889\) −32.4222 −1.08741
\(890\) 0 0
\(891\) −13.8890 −0.465298
\(892\) − 4.00000i − 0.133930i
\(893\) 15.3944i 0.515156i
\(894\) −4.02776 −0.134708
\(895\) 0 0
\(896\) −3.30278 −0.110338
\(897\) 1.00000i 0.0333890i
\(898\) − 2.09167i − 0.0698000i
\(899\) −20.6056 −0.687234
\(900\) 0 0
\(901\) −77.4500 −2.58023
\(902\) 1.54163i 0.0513308i
\(903\) − 9.21110i − 0.306526i
\(904\) −16.4222 −0.546194
\(905\) 0 0
\(906\) 2.69722 0.0896093
\(907\) 48.6611i 1.61576i 0.589344 + 0.807882i \(0.299386\pi\)
−0.589344 + 0.807882i \(0.700614\pi\)
\(908\) 14.6056i 0.484702i
\(909\) 7.57779 0.251340
\(910\) 0 0
\(911\) −4.18335 −0.138600 −0.0693002 0.997596i \(-0.522077\pi\)
−0.0693002 + 0.997596i \(0.522077\pi\)
\(912\) − 1.78890i − 0.0592363i
\(913\) 19.0278i 0.629727i
\(914\) −32.4222 −1.07243
\(915\) 0 0
\(916\) 2.00000 0.0660819
\(917\) − 37.0278i − 1.22276i
\(918\) 12.3583i 0.407884i
\(919\) −44.0000 −1.45143 −0.725713 0.687998i \(-0.758490\pi\)
−0.725713 + 0.687998i \(0.758490\pi\)
\(920\) 0 0
\(921\) 3.54163 0.116701
\(922\) − 10.1833i − 0.335371i
\(923\) 53.8444i 1.77231i
\(924\) −1.69722 −0.0558346
\(925\) 0 0
\(926\) −17.6333 −0.579466
\(927\) − 23.6140i − 0.775584i
\(928\) − 2.60555i − 0.0855314i
\(929\) 14.3667 0.471356 0.235678 0.971831i \(-0.424269\pi\)
0.235678 + 0.971831i \(0.424269\pi\)
\(930\) 0 0
\(931\) 23.0917 0.756799
\(932\) 25.8167i 0.845653i
\(933\) 6.78890i 0.222259i
\(934\) −1.81665 −0.0594427
\(935\) 0 0
\(936\) 9.60555 0.313967
\(937\) 37.9638i 1.24022i 0.784513 + 0.620112i \(0.212913\pi\)
−0.784513 + 0.620112i \(0.787087\pi\)
\(938\) − 13.2111i − 0.431358i
\(939\) 5.97224 0.194897
\(940\) 0 0
\(941\) 25.9361 0.845492 0.422746 0.906248i \(-0.361066\pi\)
0.422746 + 0.906248i \(0.361066\pi\)
\(942\) − 5.63331i − 0.183543i
\(943\) 0.908327i 0.0295792i
\(944\) 3.39445 0.110480
\(945\) 0 0
\(946\) −15.6333 −0.508283
\(947\) − 4.93608i − 0.160401i −0.996779 0.0802006i \(-0.974444\pi\)
0.996779 0.0802006i \(-0.0255561\pi\)
\(948\) − 4.36669i − 0.141824i
\(949\) 19.2111 0.623619
\(950\) 0 0
\(951\) −5.36669 −0.174027
\(952\) − 22.8167i − 0.739492i
\(953\) 41.3305i 1.33883i 0.742890 + 0.669414i \(0.233455\pi\)
−0.742890 + 0.669414i \(0.766545\pi\)
\(954\) 32.6056 1.05564
\(955\) 0 0
\(956\) 5.21110 0.168539
\(957\) − 1.33894i − 0.0432817i
\(958\) − 30.0000i − 0.969256i
\(959\) −12.9083 −0.416832
\(960\) 0 0
\(961\) 31.5416 1.01747
\(962\) − 26.4222i − 0.851886i
\(963\) − 7.57779i − 0.244191i
\(964\) 14.4222 0.464508
\(965\) 0 0
\(966\) −1.00000 −0.0321745
\(967\) − 12.6056i − 0.405367i −0.979244 0.202684i \(-0.935034\pi\)
0.979244 0.202684i \(-0.0649663\pi\)
\(968\) − 8.11943i − 0.260968i
\(969\) 12.3583 0.397005
\(970\) 0 0
\(971\) −17.0917 −0.548498 −0.274249 0.961659i \(-0.588429\pi\)
−0.274249 + 0.961659i \(0.588429\pi\)
\(972\) − 7.84441i − 0.251610i
\(973\) 41.6333i 1.33470i
\(974\) 9.81665 0.314546
\(975\) 0 0
\(976\) 11.5139 0.368550
\(977\) 6.51388i 0.208397i 0.994556 + 0.104199i \(0.0332278\pi\)
−0.994556 + 0.104199i \(0.966772\pi\)
\(978\) − 2.81665i − 0.0900667i
\(979\) 0 0
\(980\) 0 0
\(981\) −4.32213 −0.137995
\(982\) 4.18335i 0.133496i
\(983\) − 34.5416i − 1.10171i −0.834602 0.550854i \(-0.814302\pi\)
0.834602 0.550854i \(-0.185698\pi\)
\(984\) −0.275019 −0.00876729
\(985\) 0 0
\(986\) 18.0000 0.573237
\(987\) 2.60555i 0.0829356i
\(988\) − 19.5139i − 0.620819i
\(989\) −9.21110 −0.292896
\(990\) 0 0
\(991\) −15.3305 −0.486990 −0.243495 0.969902i \(-0.578294\pi\)
−0.243495 + 0.969902i \(0.578294\pi\)
\(992\) 7.90833i 0.251090i
\(993\) 5.02776i 0.159551i
\(994\) −53.8444 −1.70784
\(995\) 0 0
\(996\) −3.39445 −0.107557
\(997\) − 16.7889i − 0.531710i −0.964013 0.265855i \(-0.914346\pi\)
0.964013 0.265855i \(-0.0856542\pi\)
\(998\) − 31.6333i − 1.00133i
\(999\) −14.3112 −0.452786
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 1150.2.b.f.599.2 4
5.2 odd 4 1150.2.a.m.1.2 2
5.3 odd 4 230.2.a.b.1.1 2
5.4 even 2 inner 1150.2.b.f.599.3 4
15.8 even 4 2070.2.a.w.1.2 2
20.3 even 4 1840.2.a.j.1.2 2
20.7 even 4 9200.2.a.ca.1.1 2
40.3 even 4 7360.2.a.bu.1.1 2
40.13 odd 4 7360.2.a.bc.1.2 2
115.68 even 4 5290.2.a.j.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.b.1.1 2 5.3 odd 4
1150.2.a.m.1.2 2 5.2 odd 4
1150.2.b.f.599.2 4 1.1 even 1 trivial
1150.2.b.f.599.3 4 5.4 even 2 inner
1840.2.a.j.1.2 2 20.3 even 4
2070.2.a.w.1.2 2 15.8 even 4
5290.2.a.j.1.1 2 115.68 even 4
7360.2.a.bc.1.2 2 40.13 odd 4
7360.2.a.bu.1.1 2 40.3 even 4
9200.2.a.ca.1.1 2 20.7 even 4