Properties

Label 1150.2.b.j.599.6
Level $1150$
Weight $2$
Character 1150.599
Analytic conductor $9.183$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.77580864.1
Defining polynomial: \(x^{6} + 19 x^{4} + 105 x^{2} + 144\)
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 599.6
Root \(3.11903i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.2.b.j.599.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +3.11903i q^{3} -1.00000 q^{4} -3.11903 q^{6} +4.50973i q^{7} -1.00000i q^{8} -6.72833 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +3.11903i q^{3} -1.00000 q^{4} -3.11903 q^{6} +4.50973i q^{7} -1.00000i q^{8} -6.72833 q^{9} +4.33763 q^{11} -3.11903i q^{12} +3.72833i q^{13} -4.50973 q^{14} +1.00000 q^{16} +1.11903i q^{17} -6.72833i q^{18} -4.50973 q^{19} -14.0660 q^{21} +4.33763i q^{22} +1.00000i q^{23} +3.11903 q^{24} -3.72833 q^{26} -11.6288i q^{27} -4.50973i q^{28} +8.23805 q^{29} +1.72833 q^{31} +1.00000i q^{32} +13.5292i q^{33} -1.11903 q^{34} +6.72833 q^{36} -0.781399i q^{37} -4.50973i q^{38} -11.6288 q^{39} +3.90043 q^{41} -14.0660i q^{42} -8.00000i q^{43} -4.33763 q^{44} -1.00000 q^{46} -11.4567i q^{47} +3.11903i q^{48} -13.3376 q^{49} -3.49027 q^{51} -3.72833i q^{52} +6.00000i q^{53} +11.6288 q^{54} +4.50973 q^{56} -14.0660i q^{57} +8.23805i q^{58} +2.23805 q^{59} +3.55623 q^{61} +1.72833i q^{62} -30.3429i q^{63} -1.00000 q^{64} -13.5292 q^{66} +2.43720i q^{67} -1.11903i q^{68} -3.11903 q^{69} +7.11903 q^{71} +6.72833i q^{72} +9.45665i q^{73} +0.781399 q^{74} +4.50973 q^{76} +19.5615i q^{77} -11.6288i q^{78} +14.9133 q^{79} +16.0854 q^{81} +3.90043i q^{82} -2.78140i q^{83} +14.0660 q^{84} +8.00000 q^{86} +25.6947i q^{87} -4.33763i q^{88} +7.69471 q^{89} -16.8137 q^{91} -1.00000i q^{92} +5.39070i q^{93} +11.4567 q^{94} -3.11903 q^{96} -0.642920i q^{97} -13.3376i q^{98} -29.1850 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{4} + 2q^{6} - 20q^{9} + O(q^{10}) \) \( 6q - 6q^{4} + 2q^{6} - 20q^{9} + 6q^{11} - 6q^{14} + 6q^{16} - 6q^{19} - 44q^{21} - 2q^{24} - 2q^{26} + 8q^{29} - 10q^{31} + 14q^{34} + 20q^{36} - 28q^{39} + 2q^{41} - 6q^{44} - 6q^{46} - 60q^{49} - 42q^{51} + 28q^{54} + 6q^{56} - 28q^{59} + 2q^{61} - 6q^{64} - 18q^{66} + 2q^{69} + 22q^{71} + 4q^{74} + 6q^{76} + 8q^{79} + 14q^{81} + 44q^{84} + 48q^{86} - 36q^{89} + 2q^{91} + 28q^{94} + 2q^{96} - 114q^{99} + O(q^{100}) \)

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
\(3\) 3.11903i 1.80077i 0.435093 + 0.900385i \(0.356715\pi\)
−0.435093 + 0.900385i \(0.643285\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −3.11903 −1.27334
\(7\) 4.50973i 1.70452i 0.523122 + 0.852258i \(0.324767\pi\)
−0.523122 + 0.852258i \(0.675233\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −6.72833 −2.24278
\(10\) 0 0
\(11\) 4.33763 1.30784 0.653922 0.756562i \(-0.273122\pi\)
0.653922 + 0.756562i \(0.273122\pi\)
\(12\) − 3.11903i − 0.900385i
\(13\) 3.72833i 1.03405i 0.855970 + 0.517026i \(0.172961\pi\)
−0.855970 + 0.517026i \(0.827039\pi\)
\(14\) −4.50973 −1.20527
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.11903i 0.271404i 0.990750 + 0.135702i \(0.0433289\pi\)
−0.990750 + 0.135702i \(0.956671\pi\)
\(18\) − 6.72833i − 1.58588i
\(19\) −4.50973 −1.03460 −0.517301 0.855803i \(-0.673063\pi\)
−0.517301 + 0.855803i \(0.673063\pi\)
\(20\) 0 0
\(21\) −14.0660 −3.06944
\(22\) 4.33763i 0.924785i
\(23\) 1.00000i 0.208514i
\(24\) 3.11903 0.636669
\(25\) 0 0
\(26\) −3.72833 −0.731185
\(27\) − 11.6288i − 2.23795i
\(28\) − 4.50973i − 0.852258i
\(29\) 8.23805 1.52977 0.764884 0.644168i \(-0.222796\pi\)
0.764884 + 0.644168i \(0.222796\pi\)
\(30\) 0 0
\(31\) 1.72833 0.310417 0.155208 0.987882i \(-0.450395\pi\)
0.155208 + 0.987882i \(0.450395\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 13.5292i 2.35513i
\(34\) −1.11903 −0.191911
\(35\) 0 0
\(36\) 6.72833 1.12139
\(37\) − 0.781399i − 0.128461i −0.997935 0.0642306i \(-0.979541\pi\)
0.997935 0.0642306i \(-0.0204593\pi\)
\(38\) − 4.50973i − 0.731574i
\(39\) −11.6288 −1.86209
\(40\) 0 0
\(41\) 3.90043 0.609144 0.304572 0.952489i \(-0.401487\pi\)
0.304572 + 0.952489i \(0.401487\pi\)
\(42\) − 14.0660i − 2.17042i
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) −4.33763 −0.653922
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) − 11.4567i − 1.67112i −0.549396 0.835562i \(-0.685142\pi\)
0.549396 0.835562i \(-0.314858\pi\)
\(48\) 3.11903i 0.450193i
\(49\) −13.3376 −1.90538
\(50\) 0 0
\(51\) −3.49027 −0.488736
\(52\) − 3.72833i − 0.517026i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 11.6288 1.58247
\(55\) 0 0
\(56\) 4.50973 0.602637
\(57\) − 14.0660i − 1.86308i
\(58\) 8.23805i 1.08171i
\(59\) 2.23805 0.291370 0.145685 0.989331i \(-0.453461\pi\)
0.145685 + 0.989331i \(0.453461\pi\)
\(60\) 0 0
\(61\) 3.55623 0.455329 0.227664 0.973740i \(-0.426891\pi\)
0.227664 + 0.973740i \(0.426891\pi\)
\(62\) 1.72833i 0.219498i
\(63\) − 30.3429i − 3.82285i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −13.5292 −1.66533
\(67\) 2.43720i 0.297752i 0.988856 + 0.148876i \(0.0475655\pi\)
−0.988856 + 0.148876i \(0.952435\pi\)
\(68\) − 1.11903i − 0.135702i
\(69\) −3.11903 −0.375487
\(70\) 0 0
\(71\) 7.11903 0.844873 0.422437 0.906393i \(-0.361175\pi\)
0.422437 + 0.906393i \(0.361175\pi\)
\(72\) 6.72833i 0.792941i
\(73\) 9.45665i 1.10682i 0.832910 + 0.553409i \(0.186673\pi\)
−0.832910 + 0.553409i \(0.813327\pi\)
\(74\) 0.781399 0.0908357
\(75\) 0 0
\(76\) 4.50973 0.517301
\(77\) 19.5615i 2.22924i
\(78\) − 11.6288i − 1.31670i
\(79\) 14.9133 1.67788 0.838939 0.544225i \(-0.183176\pi\)
0.838939 + 0.544225i \(0.183176\pi\)
\(80\) 0 0
\(81\) 16.0854 1.78727
\(82\) 3.90043i 0.430730i
\(83\) − 2.78140i − 0.305298i −0.988280 0.152649i \(-0.951220\pi\)
0.988280 0.152649i \(-0.0487804\pi\)
\(84\) 14.0660 1.53472
\(85\) 0 0
\(86\) 8.00000 0.862662
\(87\) 25.6947i 2.75476i
\(88\) − 4.33763i − 0.462393i
\(89\) 7.69471 0.815637 0.407819 0.913063i \(-0.366290\pi\)
0.407819 + 0.913063i \(0.366290\pi\)
\(90\) 0 0
\(91\) −16.8137 −1.76256
\(92\) − 1.00000i − 0.104257i
\(93\) 5.39070i 0.558989i
\(94\) 11.4567 1.18166
\(95\) 0 0
\(96\) −3.11903 −0.318334
\(97\) − 0.642920i − 0.0652786i −0.999467 0.0326393i \(-0.989609\pi\)
0.999467 0.0326393i \(-0.0103913\pi\)
\(98\) − 13.3376i − 1.34730i
\(99\) −29.1850 −2.93320
\(100\) 0 0
\(101\) −8.23805 −0.819717 −0.409858 0.912149i \(-0.634422\pi\)
−0.409858 + 0.912149i \(0.634422\pi\)
\(102\) − 3.49027i − 0.345589i
\(103\) − 12.3376i − 1.21566i −0.794066 0.607831i \(-0.792040\pi\)
0.794066 0.607831i \(-0.207960\pi\)
\(104\) 3.72833 0.365593
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) − 15.9328i − 1.54028i −0.637876 0.770139i \(-0.720187\pi\)
0.637876 0.770139i \(-0.279813\pi\)
\(108\) 11.6288i 1.11898i
\(109\) 1.49027 0.142742 0.0713712 0.997450i \(-0.477263\pi\)
0.0713712 + 0.997450i \(0.477263\pi\)
\(110\) 0 0
\(111\) 2.43720 0.231329
\(112\) 4.50973i 0.426129i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 14.0660 1.31740
\(115\) 0 0
\(116\) −8.23805 −0.764884
\(117\) − 25.0854i − 2.31915i
\(118\) 2.23805i 0.206030i
\(119\) −5.04650 −0.462612
\(120\) 0 0
\(121\) 7.81502 0.710456
\(122\) 3.55623i 0.321966i
\(123\) 12.1655i 1.09693i
\(124\) −1.72833 −0.155208
\(125\) 0 0
\(126\) 30.3429 2.70316
\(127\) − 0.675256i − 0.0599193i −0.999551 0.0299597i \(-0.990462\pi\)
0.999551 0.0299597i \(-0.00953788\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) 24.9522 2.19692
\(130\) 0 0
\(131\) −13.6947 −1.19651 −0.598256 0.801305i \(-0.704139\pi\)
−0.598256 + 0.801305i \(0.704139\pi\)
\(132\) − 13.5292i − 1.17756i
\(133\) − 20.3376i − 1.76350i
\(134\) −2.43720 −0.210542
\(135\) 0 0
\(136\) 1.11903 0.0959557
\(137\) 7.52918i 0.643261i 0.946865 + 0.321631i \(0.104231\pi\)
−0.946865 + 0.321631i \(0.895769\pi\)
\(138\) − 3.11903i − 0.265509i
\(139\) −4.67526 −0.396550 −0.198275 0.980146i \(-0.563534\pi\)
−0.198275 + 0.980146i \(0.563534\pi\)
\(140\) 0 0
\(141\) 35.7336 3.00931
\(142\) 7.11903i 0.597415i
\(143\) 16.1721i 1.35238i
\(144\) −6.72833 −0.560694
\(145\) 0 0
\(146\) −9.45665 −0.782638
\(147\) − 41.6004i − 3.43114i
\(148\) 0.781399i 0.0642306i
\(149\) −7.52918 −0.616814 −0.308407 0.951254i \(-0.599796\pi\)
−0.308407 + 0.951254i \(0.599796\pi\)
\(150\) 0 0
\(151\) −13.3571 −1.08698 −0.543492 0.839414i \(-0.682898\pi\)
−0.543492 + 0.839414i \(0.682898\pi\)
\(152\) 4.50973i 0.365787i
\(153\) − 7.52918i − 0.608698i
\(154\) −19.5615 −1.57631
\(155\) 0 0
\(156\) 11.6288 0.931045
\(157\) 16.2381i 1.29594i 0.761667 + 0.647969i \(0.224381\pi\)
−0.761667 + 0.647969i \(0.775619\pi\)
\(158\) 14.9133i 1.18644i
\(159\) −18.7142 −1.48413
\(160\) 0 0
\(161\) −4.50973 −0.355416
\(162\) 16.0854i 1.26379i
\(163\) 3.29112i 0.257781i 0.991659 + 0.128890i \(0.0411415\pi\)
−0.991659 + 0.128890i \(0.958858\pi\)
\(164\) −3.90043 −0.304572
\(165\) 0 0
\(166\) 2.78140 0.215878
\(167\) 22.9133i 1.77309i 0.462647 + 0.886543i \(0.346900\pi\)
−0.462647 + 0.886543i \(0.653100\pi\)
\(168\) 14.0660i 1.08521i
\(169\) −0.900425 −0.0692635
\(170\) 0 0
\(171\) 30.3429 2.32038
\(172\) 8.00000i 0.609994i
\(173\) − 0.575681i − 0.0437683i −0.999761 0.0218841i \(-0.993034\pi\)
0.999761 0.0218841i \(-0.00696649\pi\)
\(174\) −25.6947 −1.94791
\(175\) 0 0
\(176\) 4.33763 0.326961
\(177\) 6.98055i 0.524690i
\(178\) 7.69471i 0.576743i
\(179\) −5.01945 −0.375171 −0.187586 0.982248i \(-0.560066\pi\)
−0.187586 + 0.982248i \(0.560066\pi\)
\(180\) 0 0
\(181\) −11.5292 −0.856957 −0.428479 0.903552i \(-0.640950\pi\)
−0.428479 + 0.903552i \(0.640950\pi\)
\(182\) − 16.8137i − 1.24632i
\(183\) 11.0920i 0.819942i
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) −5.39070 −0.395265
\(187\) 4.85392i 0.354954i
\(188\) 11.4567i 0.835562i
\(189\) 52.4425 3.81463
\(190\) 0 0
\(191\) −18.7142 −1.35411 −0.677055 0.735933i \(-0.736744\pi\)
−0.677055 + 0.735933i \(0.736744\pi\)
\(192\) − 3.11903i − 0.225096i
\(193\) − 23.4956i − 1.69125i −0.533780 0.845624i \(-0.679229\pi\)
0.533780 0.845624i \(-0.320771\pi\)
\(194\) 0.642920 0.0461590
\(195\) 0 0
\(196\) 13.3376 0.952688
\(197\) − 18.1385i − 1.29231i −0.763205 0.646157i \(-0.776375\pi\)
0.763205 0.646157i \(-0.223625\pi\)
\(198\) − 29.1850i − 2.07409i
\(199\) 23.2575 1.64868 0.824340 0.566094i \(-0.191546\pi\)
0.824340 + 0.566094i \(0.191546\pi\)
\(200\) 0 0
\(201\) −7.60170 −0.536183
\(202\) − 8.23805i − 0.579627i
\(203\) 37.1514i 2.60751i
\(204\) 3.49027 0.244368
\(205\) 0 0
\(206\) 12.3376 0.859603
\(207\) − 6.72833i − 0.467651i
\(208\) 3.72833i 0.258513i
\(209\) −19.5615 −1.35310
\(210\) 0 0
\(211\) 4.34420 0.299067 0.149533 0.988757i \(-0.452223\pi\)
0.149533 + 0.988757i \(0.452223\pi\)
\(212\) − 6.00000i − 0.412082i
\(213\) 22.2044i 1.52142i
\(214\) 15.9328 1.08914
\(215\) 0 0
\(216\) −11.6288 −0.791236
\(217\) 7.79428i 0.529110i
\(218\) 1.49027i 0.100934i
\(219\) −29.4956 −1.99313
\(220\) 0 0
\(221\) −4.17210 −0.280646
\(222\) 2.43720i 0.163574i
\(223\) − 12.4761i − 0.835462i −0.908571 0.417731i \(-0.862825\pi\)
0.908571 0.417731i \(-0.137175\pi\)
\(224\) −4.50973 −0.301319
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 15.9328i 1.05749i 0.848779 + 0.528747i \(0.177338\pi\)
−0.848779 + 0.528747i \(0.822662\pi\)
\(228\) 14.0660i 0.931541i
\(229\) 3.56280 0.235436 0.117718 0.993047i \(-0.462442\pi\)
0.117718 + 0.993047i \(0.462442\pi\)
\(230\) 0 0
\(231\) −61.0129 −4.01435
\(232\) − 8.23805i − 0.540855i
\(233\) 27.4956i 1.80129i 0.434552 + 0.900647i \(0.356907\pi\)
−0.434552 + 0.900647i \(0.643093\pi\)
\(234\) 25.0854 1.63988
\(235\) 0 0
\(236\) −2.23805 −0.145685
\(237\) 46.5150i 3.02147i
\(238\) − 5.04650i − 0.327116i
\(239\) −10.0389 −0.649363 −0.324681 0.945823i \(-0.605257\pi\)
−0.324681 + 0.945823i \(0.605257\pi\)
\(240\) 0 0
\(241\) −23.6947 −1.52631 −0.763155 0.646215i \(-0.776351\pi\)
−0.763155 + 0.646215i \(0.776351\pi\)
\(242\) 7.81502i 0.502368i
\(243\) 15.2846i 0.980505i
\(244\) −3.55623 −0.227664
\(245\) 0 0
\(246\) −12.1655 −0.775646
\(247\) − 16.8137i − 1.06983i
\(248\) − 1.72833i − 0.109749i
\(249\) 8.67526 0.549772
\(250\) 0 0
\(251\) 12.4425 0.785363 0.392681 0.919675i \(-0.371548\pi\)
0.392681 + 0.919675i \(0.371548\pi\)
\(252\) 30.3429i 1.91142i
\(253\) 4.33763i 0.272704i
\(254\) 0.675256 0.0423693
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 5.45665i 0.340377i 0.985412 + 0.170188i \(0.0544376\pi\)
−0.985412 + 0.170188i \(0.945562\pi\)
\(258\) 24.9522i 1.55346i
\(259\) 3.52389 0.218964
\(260\) 0 0
\(261\) −55.4283 −3.43093
\(262\) − 13.6947i − 0.846062i
\(263\) − 0.138479i − 0.00853895i −0.999991 0.00426948i \(-0.998641\pi\)
0.999991 0.00426948i \(-0.00135902\pi\)
\(264\) 13.5292 0.832663
\(265\) 0 0
\(266\) 20.3376 1.24698
\(267\) 24.0000i 1.46878i
\(268\) − 2.43720i − 0.148876i
\(269\) −14.6753 −0.894766 −0.447383 0.894342i \(-0.647644\pi\)
−0.447383 + 0.894342i \(0.647644\pi\)
\(270\) 0 0
\(271\) −8.31058 −0.504832 −0.252416 0.967619i \(-0.581225\pi\)
−0.252416 + 0.967619i \(0.581225\pi\)
\(272\) 1.11903i 0.0678510i
\(273\) − 52.4425i − 3.17396i
\(274\) −7.52918 −0.454854
\(275\) 0 0
\(276\) 3.11903 0.187743
\(277\) 12.9133i 0.775886i 0.921683 + 0.387943i \(0.126814\pi\)
−0.921683 + 0.387943i \(0.873186\pi\)
\(278\) − 4.67526i − 0.280403i
\(279\) −11.6288 −0.696195
\(280\) 0 0
\(281\) 2.67526 0.159592 0.0797962 0.996811i \(-0.474573\pi\)
0.0797962 + 0.996811i \(0.474573\pi\)
\(282\) 35.7336i 2.12791i
\(283\) − 0.742495i − 0.0441367i −0.999756 0.0220684i \(-0.992975\pi\)
0.999756 0.0220684i \(-0.00702515\pi\)
\(284\) −7.11903 −0.422437
\(285\) 0 0
\(286\) −16.1721 −0.956276
\(287\) 17.5898i 1.03830i
\(288\) − 6.72833i − 0.396470i
\(289\) 15.7478 0.926340
\(290\) 0 0
\(291\) 2.00528 0.117552
\(292\) − 9.45665i − 0.553409i
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 41.6004 2.42619
\(295\) 0 0
\(296\) −0.781399 −0.0454179
\(297\) − 50.4412i − 2.92690i
\(298\) − 7.52918i − 0.436154i
\(299\) −3.72833 −0.215615
\(300\) 0 0
\(301\) 36.0778 2.07949
\(302\) − 13.3571i − 0.768614i
\(303\) − 25.6947i − 1.47612i
\(304\) −4.50973 −0.258651
\(305\) 0 0
\(306\) 7.52918 0.430414
\(307\) 30.5084i 1.74121i 0.491984 + 0.870604i \(0.336272\pi\)
−0.491984 + 0.870604i \(0.663728\pi\)
\(308\) − 19.5615i − 1.11462i
\(309\) 38.4814 2.18913
\(310\) 0 0
\(311\) 5.56280 0.315437 0.157719 0.987484i \(-0.449586\pi\)
0.157719 + 0.987484i \(0.449586\pi\)
\(312\) 11.6288i 0.658348i
\(313\) − 4.07252i − 0.230193i −0.993354 0.115096i \(-0.963282\pi\)
0.993354 0.115096i \(-0.0367177\pi\)
\(314\) −16.2381 −0.916366
\(315\) 0 0
\(316\) −14.9133 −0.838939
\(317\) − 6.16553i − 0.346291i −0.984896 0.173145i \(-0.944607\pi\)
0.984896 0.173145i \(-0.0553930\pi\)
\(318\) − 18.7142i − 1.04944i
\(319\) 35.7336 2.00070
\(320\) 0 0
\(321\) 49.6947 2.77369
\(322\) − 4.50973i − 0.251317i
\(323\) − 5.04650i − 0.280795i
\(324\) −16.0854 −0.893634
\(325\) 0 0
\(326\) −3.29112 −0.182279
\(327\) 4.64820i 0.257046i
\(328\) − 3.90043i − 0.215365i
\(329\) 51.6664 2.84846
\(330\) 0 0
\(331\) 27.5886 1.51640 0.758202 0.652019i \(-0.226078\pi\)
0.758202 + 0.652019i \(0.226078\pi\)
\(332\) 2.78140i 0.152649i
\(333\) 5.25751i 0.288110i
\(334\) −22.9133 −1.25376
\(335\) 0 0
\(336\) −14.0660 −0.767361
\(337\) − 17.4230i − 0.949093i −0.880230 0.474547i \(-0.842612\pi\)
0.880230 0.474547i \(-0.157388\pi\)
\(338\) − 0.900425i − 0.0489767i
\(339\) −18.7142 −1.01641
\(340\) 0 0
\(341\) 7.49684 0.405977
\(342\) 30.3429i 1.64076i
\(343\) − 28.5810i − 1.54323i
\(344\) −8.00000 −0.431331
\(345\) 0 0
\(346\) 0.575681 0.0309488
\(347\) 4.88097i 0.262024i 0.991381 + 0.131012i \(0.0418227\pi\)
−0.991381 + 0.131012i \(0.958177\pi\)
\(348\) − 25.6947i − 1.37738i
\(349\) −24.0389 −1.28677 −0.643387 0.765542i \(-0.722471\pi\)
−0.643387 + 0.765542i \(0.722471\pi\)
\(350\) 0 0
\(351\) 43.3558 2.31416
\(352\) 4.33763i 0.231196i
\(353\) 14.3442i 0.763464i 0.924273 + 0.381732i \(0.124672\pi\)
−0.924273 + 0.381732i \(0.875328\pi\)
\(354\) −6.98055 −0.371012
\(355\) 0 0
\(356\) −7.69471 −0.407819
\(357\) − 15.7402i − 0.833059i
\(358\) − 5.01945i − 0.265286i
\(359\) −26.7814 −1.41347 −0.706734 0.707479i \(-0.749832\pi\)
−0.706734 + 0.707479i \(0.749832\pi\)
\(360\) 0 0
\(361\) 1.33763 0.0704015
\(362\) − 11.5292i − 0.605960i
\(363\) 24.3752i 1.27937i
\(364\) 16.8137 0.881279
\(365\) 0 0
\(366\) −11.0920 −0.579787
\(367\) − 20.4761i − 1.06884i −0.845218 0.534422i \(-0.820529\pi\)
0.845218 0.534422i \(-0.179471\pi\)
\(368\) 1.00000i 0.0521286i
\(369\) −26.2433 −1.36617
\(370\) 0 0
\(371\) −27.0584 −1.40480
\(372\) − 5.39070i − 0.279495i
\(373\) 3.89386i 0.201616i 0.994906 + 0.100808i \(0.0321428\pi\)
−0.994906 + 0.100808i \(0.967857\pi\)
\(374\) −4.85392 −0.250990
\(375\) 0 0
\(376\) −11.4567 −0.590832
\(377\) 30.7142i 1.58186i
\(378\) 52.4425i 2.69735i
\(379\) 30.3765 1.56034 0.780169 0.625569i \(-0.215133\pi\)
0.780169 + 0.625569i \(0.215133\pi\)
\(380\) 0 0
\(381\) 2.10614 0.107901
\(382\) − 18.7142i − 0.957500i
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 3.11903 0.159167
\(385\) 0 0
\(386\) 23.4956 1.19589
\(387\) 53.8266i 2.73616i
\(388\) 0.642920i 0.0326393i
\(389\) 18.6818 0.947206 0.473603 0.880738i \(-0.342953\pi\)
0.473603 + 0.880738i \(0.342953\pi\)
\(390\) 0 0
\(391\) −1.11903 −0.0565916
\(392\) 13.3376i 0.673652i
\(393\) − 42.7142i − 2.15464i
\(394\) 18.1385 0.913803
\(395\) 0 0
\(396\) 29.1850 1.46660
\(397\) − 28.5757i − 1.43417i −0.696985 0.717086i \(-0.745475\pi\)
0.696985 0.717086i \(-0.254525\pi\)
\(398\) 23.2575i 1.16579i
\(399\) 63.4336 3.17565
\(400\) 0 0
\(401\) 12.1061 0.604552 0.302276 0.953220i \(-0.402254\pi\)
0.302276 + 0.953220i \(0.402254\pi\)
\(402\) − 7.60170i − 0.379138i
\(403\) 6.44377i 0.320987i
\(404\) 8.23805 0.409858
\(405\) 0 0
\(406\) −37.1514 −1.84379
\(407\) − 3.38942i − 0.168007i
\(408\) 3.49027i 0.172794i
\(409\) −25.2911 −1.25057 −0.625283 0.780398i \(-0.715016\pi\)
−0.625283 + 0.780398i \(0.715016\pi\)
\(410\) 0 0
\(411\) −23.4837 −1.15837
\(412\) 12.3376i 0.607831i
\(413\) 10.0930i 0.496644i
\(414\) 6.72833 0.330679
\(415\) 0 0
\(416\) −3.72833 −0.182796
\(417\) − 14.5822i − 0.714096i
\(418\) − 19.5615i − 0.956785i
\(419\) 17.3505 0.847628 0.423814 0.905749i \(-0.360691\pi\)
0.423814 + 0.905749i \(0.360691\pi\)
\(420\) 0 0
\(421\) 21.4230 1.04409 0.522047 0.852916i \(-0.325168\pi\)
0.522047 + 0.852916i \(0.325168\pi\)
\(422\) 4.34420i 0.211472i
\(423\) 77.0841i 3.74796i
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) −22.2044 −1.07581
\(427\) 16.0376i 0.776115i
\(428\) 15.9328i 0.770139i
\(429\) −50.4412 −2.43532
\(430\) 0 0
\(431\) 22.5822 1.08775 0.543874 0.839167i \(-0.316957\pi\)
0.543874 + 0.839167i \(0.316957\pi\)
\(432\) − 11.6288i − 0.559489i
\(433\) − 1.01417i − 0.0487378i −0.999703 0.0243689i \(-0.992242\pi\)
0.999703 0.0243689i \(-0.00775763\pi\)
\(434\) −7.79428 −0.374138
\(435\) 0 0
\(436\) −1.49027 −0.0713712
\(437\) − 4.50973i − 0.215729i
\(438\) − 29.4956i − 1.40935i
\(439\) 26.7478 1.27660 0.638301 0.769787i \(-0.279638\pi\)
0.638301 + 0.769787i \(0.279638\pi\)
\(440\) 0 0
\(441\) 89.7399 4.27333
\(442\) − 4.17210i − 0.198446i
\(443\) − 10.2044i − 0.484827i −0.970173 0.242414i \(-0.922061\pi\)
0.970173 0.242414i \(-0.0779391\pi\)
\(444\) −2.43720 −0.115665
\(445\) 0 0
\(446\) 12.4761 0.590761
\(447\) − 23.4837i − 1.11074i
\(448\) − 4.50973i − 0.213065i
\(449\) −38.7867 −1.83046 −0.915228 0.402936i \(-0.867990\pi\)
−0.915228 + 0.402936i \(0.867990\pi\)
\(450\) 0 0
\(451\) 16.9186 0.796665
\(452\) − 6.00000i − 0.282216i
\(453\) − 41.6611i − 1.95741i
\(454\) −15.9328 −0.747762
\(455\) 0 0
\(456\) −14.0660 −0.658699
\(457\) 34.9522i 1.63500i 0.575932 + 0.817498i \(0.304639\pi\)
−0.575932 + 0.817498i \(0.695361\pi\)
\(458\) 3.56280i 0.166479i
\(459\) 13.0129 0.607389
\(460\) 0 0
\(461\) 16.3700 0.762425 0.381213 0.924487i \(-0.375507\pi\)
0.381213 + 0.924487i \(0.375507\pi\)
\(462\) − 61.0129i − 2.83858i
\(463\) − 29.2186i − 1.35790i −0.734183 0.678952i \(-0.762435\pi\)
0.734183 0.678952i \(-0.237565\pi\)
\(464\) 8.23805 0.382442
\(465\) 0 0
\(466\) −27.4956 −1.27371
\(467\) 24.2770i 1.12340i 0.827340 + 0.561702i \(0.189853\pi\)
−0.827340 + 0.561702i \(0.810147\pi\)
\(468\) 25.0854i 1.15957i
\(469\) −10.9911 −0.507523
\(470\) 0 0
\(471\) −50.6469 −2.33369
\(472\) − 2.23805i − 0.103015i
\(473\) − 34.7010i − 1.59555i
\(474\) −46.5150 −2.13651
\(475\) 0 0
\(476\) 5.04650 0.231306
\(477\) − 40.3700i − 1.84841i
\(478\) − 10.0389i − 0.459169i
\(479\) −24.6080 −1.12437 −0.562185 0.827012i \(-0.690039\pi\)
−0.562185 + 0.827012i \(0.690039\pi\)
\(480\) 0 0
\(481\) 2.91331 0.132835
\(482\) − 23.6947i − 1.07926i
\(483\) − 14.0660i − 0.640023i
\(484\) −7.81502 −0.355228
\(485\) 0 0
\(486\) −15.2846 −0.693322
\(487\) − 30.2381i − 1.37022i −0.728441 0.685108i \(-0.759755\pi\)
0.728441 0.685108i \(-0.240245\pi\)
\(488\) − 3.55623i − 0.160983i
\(489\) −10.2651 −0.464204
\(490\) 0 0
\(491\) 12.3311 0.556493 0.278246 0.960510i \(-0.410247\pi\)
0.278246 + 0.960510i \(0.410247\pi\)
\(492\) − 12.1655i − 0.548464i
\(493\) 9.21860i 0.415185i
\(494\) 16.8137 0.756486
\(495\) 0 0
\(496\) 1.72833 0.0776042
\(497\) 32.1049i 1.44010i
\(498\) 8.67526i 0.388748i
\(499\) 26.9133 1.20481 0.602403 0.798192i \(-0.294210\pi\)
0.602403 + 0.798192i \(0.294210\pi\)
\(500\) 0 0
\(501\) −71.4672 −3.19292
\(502\) 12.4425i 0.555335i
\(503\) − 20.5097i − 0.914483i −0.889342 0.457242i \(-0.848837\pi\)
0.889342 0.457242i \(-0.151163\pi\)
\(504\) −30.3429 −1.35158
\(505\) 0 0
\(506\) −4.33763 −0.192831
\(507\) − 2.80845i − 0.124728i
\(508\) 0.675256i 0.0299597i
\(509\) 36.7142 1.62733 0.813663 0.581336i \(-0.197470\pi\)
0.813663 + 0.581336i \(0.197470\pi\)
\(510\) 0 0
\(511\) −42.6469 −1.88659
\(512\) 1.00000i 0.0441942i
\(513\) 52.4425i 2.31539i
\(514\) −5.45665 −0.240683
\(515\) 0 0
\(516\) −24.9522 −1.09846
\(517\) − 49.6947i − 2.18557i
\(518\) 3.52389i 0.154831i
\(519\) 1.79557 0.0788166
\(520\) 0 0
\(521\) −4.91331 −0.215256 −0.107628 0.994191i \(-0.534326\pi\)
−0.107628 + 0.994191i \(0.534326\pi\)
\(522\) − 55.4283i − 2.42603i
\(523\) 0.344196i 0.0150506i 0.999972 + 0.00752531i \(0.00239540\pi\)
−0.999972 + 0.00752531i \(0.997605\pi\)
\(524\) 13.6947 0.598256
\(525\) 0 0
\(526\) 0.138479 0.00603795
\(527\) 1.93404i 0.0842483i
\(528\) 13.5292i 0.588782i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −15.0584 −0.653477
\(532\) 20.3376i 0.881748i
\(533\) 14.5421i 0.629887i
\(534\) −24.0000 −1.03858
\(535\) 0 0
\(536\) 2.43720 0.105271
\(537\) − 15.6558i − 0.675598i
\(538\) − 14.6753i − 0.632695i
\(539\) −57.8537 −2.49193
\(540\) 0 0
\(541\) −6.13191 −0.263631 −0.131816 0.991274i \(-0.542081\pi\)
−0.131816 + 0.991274i \(0.542081\pi\)
\(542\) − 8.31058i − 0.356970i
\(543\) − 35.9598i − 1.54318i
\(544\) −1.11903 −0.0479779
\(545\) 0 0
\(546\) 52.4425 2.24433
\(547\) − 9.18498i − 0.392721i −0.980532 0.196361i \(-0.937088\pi\)
0.980532 0.196361i \(-0.0629124\pi\)
\(548\) − 7.52918i − 0.321631i
\(549\) −23.9275 −1.02120
\(550\) 0 0
\(551\) −37.1514 −1.58270
\(552\) 3.11903i 0.132755i
\(553\) 67.2549i 2.85997i
\(554\) −12.9133 −0.548634
\(555\) 0 0
\(556\) 4.67526 0.198275
\(557\) − 4.30529i − 0.182421i −0.995832 0.0912105i \(-0.970926\pi\)
0.995832 0.0912105i \(-0.0290736\pi\)
\(558\) − 11.6288i − 0.492284i
\(559\) 29.8266 1.26153
\(560\) 0 0
\(561\) −15.1395 −0.639191
\(562\) 2.67526i 0.112849i
\(563\) − 11.1256i − 0.468888i −0.972130 0.234444i \(-0.924673\pi\)
0.972130 0.234444i \(-0.0753269\pi\)
\(564\) −35.7336 −1.50466
\(565\) 0 0
\(566\) 0.742495 0.0312094
\(567\) 72.5408i 3.04643i
\(568\) − 7.11903i − 0.298708i
\(569\) 16.0389 0.672386 0.336193 0.941793i \(-0.390861\pi\)
0.336193 + 0.941793i \(0.390861\pi\)
\(570\) 0 0
\(571\) 17.9004 0.749109 0.374555 0.927205i \(-0.377796\pi\)
0.374555 + 0.927205i \(0.377796\pi\)
\(572\) − 16.1721i − 0.676189i
\(573\) − 58.3700i − 2.43844i
\(574\) −17.5898 −0.734186
\(575\) 0 0
\(576\) 6.72833 0.280347
\(577\) − 9.12559i − 0.379903i −0.981793 0.189952i \(-0.939167\pi\)
0.981793 0.189952i \(-0.0608332\pi\)
\(578\) 15.7478i 0.655021i
\(579\) 73.2833 3.04555
\(580\) 0 0
\(581\) 12.5433 0.520386
\(582\) 2.00528i 0.0831217i
\(583\) 26.0258i 1.07788i
\(584\) 9.45665 0.391319
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) − 33.6340i − 1.38823i −0.719866 0.694113i \(-0.755797\pi\)
0.719866 0.694113i \(-0.244203\pi\)
\(588\) 41.6004i 1.71557i
\(589\) −7.79428 −0.321158
\(590\) 0 0
\(591\) 56.5744 2.32716
\(592\) − 0.781399i − 0.0321153i
\(593\) 17.4567i 0.716859i 0.933557 + 0.358429i \(0.116688\pi\)
−0.933557 + 0.358429i \(0.883312\pi\)
\(594\) 50.4412 2.06963
\(595\) 0 0
\(596\) 7.52918 0.308407
\(597\) 72.5408i 2.96890i
\(598\) − 3.72833i − 0.152463i
\(599\) 11.5951 0.473764 0.236882 0.971538i \(-0.423874\pi\)
0.236882 + 0.971538i \(0.423874\pi\)
\(600\) 0 0
\(601\) −31.6611 −1.29148 −0.645741 0.763556i \(-0.723452\pi\)
−0.645741 + 0.763556i \(0.723452\pi\)
\(602\) 36.0778i 1.47042i
\(603\) − 16.3983i − 0.667790i
\(604\) 13.3571 0.543492
\(605\) 0 0
\(606\) 25.6947 1.04378
\(607\) − 36.0778i − 1.46435i −0.681115 0.732177i \(-0.738505\pi\)
0.681115 0.732177i \(-0.261495\pi\)
\(608\) − 4.50973i − 0.182894i
\(609\) −115.876 −4.69554
\(610\) 0 0
\(611\) 42.7142 1.72803
\(612\) 7.52918i 0.304349i
\(613\) 32.0389i 1.29404i 0.762473 + 0.647020i \(0.223985\pi\)
−0.762473 + 0.647020i \(0.776015\pi\)
\(614\) −30.5084 −1.23122
\(615\) 0 0
\(616\) 19.5615 0.788156
\(617\) − 13.3960i − 0.539302i −0.962958 0.269651i \(-0.913092\pi\)
0.962958 0.269651i \(-0.0869083\pi\)
\(618\) 38.4814i 1.54795i
\(619\) −37.1309 −1.49242 −0.746208 0.665713i \(-0.768128\pi\)
−0.746208 + 0.665713i \(0.768128\pi\)
\(620\) 0 0
\(621\) 11.6288 0.466646
\(622\) 5.56280i 0.223048i
\(623\) 34.7010i 1.39027i
\(624\) −11.6288 −0.465523
\(625\) 0 0
\(626\) 4.07252 0.162771
\(627\) − 61.0129i − 2.43662i
\(628\) − 16.2381i − 0.647969i
\(629\) 0.874406 0.0348648
\(630\) 0 0
\(631\) 11.1125 0.442380 0.221190 0.975231i \(-0.429006\pi\)
0.221190 + 0.975231i \(0.429006\pi\)
\(632\) − 14.9133i − 0.593220i
\(633\) 13.5497i 0.538551i
\(634\) 6.16553 0.244864
\(635\) 0 0
\(636\) 18.7142 0.742065
\(637\) − 49.7270i − 1.97026i
\(638\) 35.7336i 1.41471i
\(639\) −47.8991 −1.89486
\(640\) 0 0
\(641\) 12.3831 0.489103 0.244552 0.969636i \(-0.421359\pi\)
0.244552 + 0.969636i \(0.421359\pi\)
\(642\) 49.6947i 1.96129i
\(643\) 37.4956i 1.47868i 0.673332 + 0.739340i \(0.264862\pi\)
−0.673332 + 0.739340i \(0.735138\pi\)
\(644\) 4.50973 0.177708
\(645\) 0 0
\(646\) 5.04650 0.198552
\(647\) 14.5691i 0.572771i 0.958114 + 0.286385i \(0.0924538\pi\)
−0.958114 + 0.286385i \(0.907546\pi\)
\(648\) − 16.0854i − 0.631894i
\(649\) 9.70784 0.381066
\(650\) 0 0
\(651\) −24.3106 −0.952807
\(652\) − 3.29112i − 0.128890i
\(653\) 4.41672i 0.172840i 0.996259 + 0.0864198i \(0.0275426\pi\)
−0.996259 + 0.0864198i \(0.972457\pi\)
\(654\) −4.64820 −0.181759
\(655\) 0 0
\(656\) 3.90043 0.152286
\(657\) − 63.6275i − 2.48234i
\(658\) 51.6664i 2.01416i
\(659\) 31.8655 1.24130 0.620652 0.784086i \(-0.286868\pi\)
0.620652 + 0.784086i \(0.286868\pi\)
\(660\) 0 0
\(661\) 33.1190 1.28818 0.644090 0.764949i \(-0.277236\pi\)
0.644090 + 0.764949i \(0.277236\pi\)
\(662\) 27.5886i 1.07226i
\(663\) − 13.0129i − 0.505379i
\(664\) −2.78140 −0.107939
\(665\) 0 0
\(666\) −5.25751 −0.203724
\(667\) 8.23805i 0.318979i
\(668\) − 22.9133i − 0.886543i
\(669\) 38.9133 1.50448
\(670\) 0 0
\(671\) 15.4256 0.595499
\(672\) − 14.0660i − 0.542606i
\(673\) − 19.3505i − 0.745907i −0.927850 0.372954i \(-0.878345\pi\)
0.927850 0.372954i \(-0.121655\pi\)
\(674\) 17.4230 0.671110
\(675\) 0 0
\(676\) 0.900425 0.0346317
\(677\) − 6.00000i − 0.230599i −0.993331 0.115299i \(-0.963217\pi\)
0.993331 0.115299i \(-0.0367827\pi\)
\(678\) − 18.7142i − 0.718713i
\(679\) 2.89939 0.111268
\(680\) 0 0
\(681\) −49.6947 −1.90431
\(682\) 7.49684i 0.287069i
\(683\) − 38.3495i − 1.46740i −0.679472 0.733701i \(-0.737791\pi\)
0.679472 0.733701i \(-0.262209\pi\)
\(684\) −30.3429 −1.16019
\(685\) 0 0
\(686\) 28.5810 1.09123
\(687\) 11.1125i 0.423967i
\(688\) − 8.00000i − 0.304997i
\(689\) −22.3700 −0.852228
\(690\) 0 0
\(691\) −21.0195 −0.799618 −0.399809 0.916599i \(-0.630923\pi\)
−0.399809 + 0.916599i \(0.630923\pi\)
\(692\) 0.575681i 0.0218841i
\(693\) − 131.616i − 4.99969i
\(694\) −4.88097 −0.185279
\(695\) 0 0
\(696\) 25.6947 0.973955
\(697\) 4.36468i 0.165324i
\(698\) − 24.0389i − 0.909886i
\(699\) −85.7594 −3.24372
\(700\) 0 0
\(701\) 19.3169 0.729589 0.364794 0.931088i \(-0.381139\pi\)
0.364794 + 0.931088i \(0.381139\pi\)
\(702\) 43.3558i 1.63636i
\(703\) 3.52389i 0.132906i
\(704\) −4.33763 −0.163481
\(705\) 0 0
\(706\) −14.3442 −0.539851
\(707\) − 37.1514i − 1.39722i
\(708\) − 6.98055i − 0.262345i
\(709\) −12.2315 −0.459363 −0.229682 0.973266i \(-0.573768\pi\)
−0.229682 + 0.973266i \(0.573768\pi\)
\(710\) 0 0
\(711\) −100.342 −3.76311
\(712\) − 7.69471i − 0.288371i
\(713\) 1.72833i 0.0647264i
\(714\) 15.7402 0.589061
\(715\) 0 0
\(716\) 5.01945 0.187586
\(717\) − 31.3116i − 1.16935i
\(718\) − 26.7814i − 0.999473i
\(719\) 40.6416 1.51568 0.757839 0.652442i \(-0.226255\pi\)
0.757839 + 0.652442i \(0.226255\pi\)
\(720\) 0 0
\(721\) 55.6393 2.07212
\(722\) 1.33763i 0.0497814i
\(723\) − 73.9044i − 2.74854i
\(724\) 11.5292 0.428479
\(725\) 0 0
\(726\) −24.3752 −0.904650
\(727\) − 23.4501i − 0.869716i −0.900499 0.434858i \(-0.856799\pi\)
0.900499 0.434858i \(-0.143201\pi\)
\(728\) 16.8137i 0.623158i
\(729\) 0.583281 0.0216030
\(730\) 0 0
\(731\) 8.95221 0.331110
\(732\) − 11.0920i − 0.409971i
\(733\) 12.5150i 0.462252i 0.972924 + 0.231126i \(0.0742410\pi\)
−0.972924 + 0.231126i \(0.925759\pi\)
\(734\) 20.4761 0.755787
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) 10.5717i 0.389413i
\(738\) − 26.2433i − 0.966031i
\(739\) 21.3505 0.785391 0.392696 0.919668i \(-0.371543\pi\)
0.392696 + 0.919668i \(0.371543\pi\)
\(740\) 0 0
\(741\) 52.4425 1.92652
\(742\) − 27.0584i − 0.993343i
\(743\) − 24.9858i − 0.916641i −0.888787 0.458321i \(-0.848451\pi\)
0.888787 0.458321i \(-0.151549\pi\)
\(744\) 5.39070 0.197633
\(745\) 0 0
\(746\) −3.89386 −0.142564
\(747\) 18.7142i 0.684715i
\(748\) − 4.85392i − 0.177477i
\(749\) 71.8524 2.62543
\(750\) 0 0
\(751\) −33.6275 −1.22708 −0.613542 0.789662i \(-0.710256\pi\)
−0.613542 + 0.789662i \(0.710256\pi\)
\(752\) − 11.4567i − 0.417781i
\(753\) 38.8085i 1.41426i
\(754\) −30.7142 −1.11854
\(755\) 0 0
\(756\) −52.4425 −1.90731
\(757\) − 37.1230i − 1.34926i −0.738156 0.674630i \(-0.764303\pi\)
0.738156 0.674630i \(-0.235697\pi\)
\(758\) 30.3765i 1.10333i
\(759\) −13.5292 −0.491078
\(760\) 0 0
\(761\) −3.87337 −0.140410 −0.0702048 0.997533i \(-0.522365\pi\)
−0.0702048 + 0.997533i \(0.522365\pi\)
\(762\) 2.10614i 0.0762975i
\(763\) 6.72073i 0.243307i
\(764\) 18.7142 0.677055
\(765\) 0 0
\(766\) 0 0
\(767\) 8.34420i 0.301291i
\(768\) 3.11903i 0.112548i
\(769\) −23.1645 −0.835333 −0.417667 0.908600i \(-0.637152\pi\)
−0.417667 + 0.908600i \(0.637152\pi\)
\(770\) 0 0
\(771\) −17.0195 −0.612941
\(772\) 23.4956i 0.845624i
\(773\) 8.78140i 0.315845i 0.987452 + 0.157922i \(0.0504796\pi\)
−0.987452 + 0.157922i \(0.949520\pi\)
\(774\) −53.8266 −1.93476
\(775\) 0 0
\(776\) −0.642920 −0.0230795
\(777\) 10.9911i 0.394304i
\(778\) 18.6818i 0.669776i
\(779\) −17.5898 −0.630222
\(780\) 0 0
\(781\) 30.8797 1.10496
\(782\) − 1.11903i − 0.0400163i
\(783\) − 95.7983i − 3.42355i
\(784\) −13.3376 −0.476344
\(785\) 0 0
\(786\) 42.7142 1.52356
\(787\) 49.6275i 1.76903i 0.466513 + 0.884514i \(0.345510\pi\)
−0.466513 + 0.884514i \(0.654490\pi\)
\(788\) 18.1385i 0.646157i
\(789\) 0.431918 0.0153767
\(790\) 0 0
\(791\) −27.0584 −0.962084
\(792\) 29.1850i 1.03704i
\(793\) 13.2588i 0.470833i
\(794\) 28.5757 1.01411
\(795\) 0 0
\(796\) −23.2575 −0.824340
\(797\) 18.3311i 0.649319i 0.945831 + 0.324660i \(0.105250\pi\)
−0.945831 + 0.324660i \(0.894750\pi\)
\(798\) 63.4336i 2.24553i
\(799\) 12.8203 0.453550
\(800\) 0 0
\(801\) −51.7725 −1.82929
\(802\) 12.1061i 0.427483i
\(803\) 41.0195i 1.44755i
\(804\) 7.60170 0.268091
\(805\) 0 0
\(806\) −6.44377 −0.226972
\(807\) − 45.7725i − 1.61127i
\(808\) 8.23805i 0.289814i
\(809\) 1.93933 0.0681832 0.0340916 0.999419i \(-0.489146\pi\)
0.0340916 + 0.999419i \(0.489146\pi\)
\(810\) 0 0
\(811\) −5.41775 −0.190243 −0.0951215 0.995466i \(-0.530324\pi\)
−0.0951215 + 0.995466i \(0.530324\pi\)
\(812\) − 37.1514i − 1.30376i
\(813\) − 25.9209i − 0.909086i
\(814\) 3.38942 0.118799
\(815\) 0 0
\(816\) −3.49027 −0.122184
\(817\) 36.0778i 1.26220i
\(818\) − 25.2911i − 0.884283i
\(819\) 113.128 3.95302
\(820\) 0 0
\(821\) 37.8655 1.32152 0.660758 0.750599i \(-0.270235\pi\)
0.660758 + 0.750599i \(0.270235\pi\)
\(822\) − 23.4837i − 0.819088i
\(823\) − 43.7336i − 1.52446i −0.647308 0.762229i \(-0.724105\pi\)
0.647308 0.762229i \(-0.275895\pi\)
\(824\) −12.3376 −0.429802
\(825\) 0 0
\(826\) −10.0930 −0.351181
\(827\) 28.1991i 0.980581i 0.871559 + 0.490290i \(0.163109\pi\)
−0.871559 + 0.490290i \(0.836891\pi\)
\(828\) 6.72833i 0.233826i
\(829\) −1.12559 −0.0390935 −0.0195468 0.999809i \(-0.506222\pi\)
−0.0195468 + 0.999809i \(0.506222\pi\)
\(830\) 0 0
\(831\) −40.2770 −1.39719
\(832\) − 3.72833i − 0.129256i
\(833\) − 14.9252i − 0.517126i
\(834\) 14.5822 0.504942
\(835\) 0 0
\(836\) 19.5615 0.676549
\(837\) − 20.0983i − 0.694699i
\(838\) 17.3505i 0.599364i
\(839\) −39.9328 −1.37863 −0.689316 0.724461i \(-0.742089\pi\)
−0.689316 + 0.724461i \(0.742089\pi\)
\(840\) 0 0
\(841\) 38.8655 1.34019
\(842\) 21.4230i 0.738287i
\(843\) 8.34420i 0.287389i
\(844\) −4.34420 −0.149533
\(845\) 0 0
\(846\) −77.0841 −2.65021
\(847\) 35.2436i 1.21098i
\(848\) 6.00000i 0.206041i
\(849\) 2.31586 0.0794801
\(850\) 0 0
\(851\) 0.781399 0.0267860
\(852\) − 22.2044i − 0.760711i
\(853\) − 47.9921i − 1.64322i −0.570050 0.821610i \(-0.693076\pi\)
0.570050 0.821610i \(-0.306924\pi\)
\(854\) −16.0376 −0.548796
\(855\) 0 0
\(856\) −15.9328 −0.544571
\(857\) − 43.4283i − 1.48348i −0.670686 0.741742i \(-0.734000\pi\)
0.670686 0.741742i \(-0.266000\pi\)
\(858\) − 50.4412i − 1.72203i
\(859\) −32.5433 −1.11036 −0.555182 0.831729i \(-0.687352\pi\)
−0.555182 + 0.831729i \(0.687352\pi\)
\(860\) 0 0
\(861\) −54.8632 −1.86973
\(862\) 22.5822i 0.769154i
\(863\) − 46.1036i − 1.56938i −0.619886 0.784692i \(-0.712821\pi\)
0.619886 0.784692i \(-0.287179\pi\)
\(864\) 11.6288 0.395618
\(865\) 0 0
\(866\) 1.01417 0.0344628
\(867\) 49.1177i 1.66813i
\(868\) − 7.79428i − 0.264555i
\(869\) 64.6884 2.19440
\(870\) 0 0
\(871\) −9.08669 −0.307891
\(872\) − 1.49027i − 0.0504670i
\(873\) 4.32578i 0.146405i
\(874\) 4.50973 0.152544
\(875\) 0 0
\(876\) 29.4956 0.996563
\(877\) − 24.0996i − 0.813785i −0.913476 0.406892i \(-0.866612\pi\)
0.913476 0.406892i \(-0.133388\pi\)
\(878\) 26.7478i 0.902694i
\(879\) −18.7142 −0.631213
\(880\) 0 0
\(881\) 2.34420 0.0789780 0.0394890 0.999220i \(-0.487427\pi\)
0.0394890 + 0.999220i \(0.487427\pi\)
\(882\) 89.7399i 3.02170i
\(883\) 41.0505i 1.38146i 0.723113 + 0.690730i \(0.242711\pi\)
−0.723113 + 0.690730i \(0.757289\pi\)
\(884\) 4.17210 0.140323
\(885\) 0 0
\(886\) 10.2044 0.342825
\(887\) 54.7788i 1.83929i 0.392747 + 0.919647i \(0.371525\pi\)
−0.392747 + 0.919647i \(0.628475\pi\)
\(888\) − 2.43720i − 0.0817872i
\(889\) 3.04522 0.102133
\(890\) 0 0
\(891\) 69.7725 2.33747
\(892\) 12.4761i 0.417731i
\(893\) 51.6664i 1.72895i
\(894\) 23.4837 0.785413
\(895\) 0 0
\(896\) 4.50973 0.150659
\(897\) − 11.6288i − 0.388273i
\(898\) − 38.7867i − 1.29433i
\(899\) 14.2381 0.474866
\(900\) 0 0
\(901\) −6.71416 −0.223681
\(902\) 16.9186i 0.563328i
\(903\) 112.528i 3.74469i
\(904\) 6.00000 0.199557
\(905\) 0 0
\(906\) 41.6611 1.38410
\(907\) − 10.1061i − 0.335569i −0.985824 0.167784i \(-0.946339\pi\)
0.985824 0.167784i \(-0.0536613\pi\)
\(908\) − 15.9328i − 0.528747i
\(909\) 55.4283 1.83844
\(910\) 0 0
\(911\) −25.4178 −0.842128 −0.421064 0.907031i \(-0.638343\pi\)
−0.421064 + 0.907031i \(0.638343\pi\)
\(912\) − 14.0660i − 0.465770i
\(913\) − 12.0647i − 0.399282i
\(914\) −34.9522 −1.15612
\(915\) 0 0
\(916\) −3.56280 −0.117718
\(917\) − 61.7594i − 2.03947i
\(918\) 13.0129i 0.429489i
\(919\) −23.6017 −0.778548 −0.389274 0.921122i \(-0.627274\pi\)
−0.389274 + 0.921122i \(0.627274\pi\)
\(920\) 0 0
\(921\) −95.1566 −3.13552
\(922\) 16.3700i 0.539116i
\(923\) 26.5421i 0.873643i
\(924\) 61.0129 2.00718
\(925\) 0 0
\(926\) 29.2186 0.960183
\(927\) 83.0116i 2.72646i
\(928\) 8.23805i 0.270427i
\(929\) 7.08669 0.232507 0.116253 0.993220i \(-0.462912\pi\)
0.116253 + 0.993220i \(0.462912\pi\)
\(930\) 0 0
\(931\) 60.1490 1.97131
\(932\) − 27.4956i − 0.900647i
\(933\) 17.3505i 0.568030i
\(934\) −24.2770 −0.794366
\(935\) 0 0
\(936\) −25.0854 −0.819942
\(937\) 27.3169i 0.892404i 0.894932 + 0.446202i \(0.147224\pi\)
−0.894932 + 0.446202i \(0.852776\pi\)
\(938\) − 10.9911i − 0.358873i
\(939\) 12.7023 0.414524
\(940\) 0 0
\(941\) 55.8979 1.82222 0.911109 0.412165i \(-0.135227\pi\)
0.911109 + 0.412165i \(0.135227\pi\)
\(942\) − 50.6469i − 1.65017i
\(943\) 3.90043i 0.127015i
\(944\) 2.23805 0.0728424
\(945\) 0 0
\(946\) 34.7010 1.12823
\(947\) 37.5939i 1.22164i 0.791771 + 0.610818i \(0.209159\pi\)
−0.791771 + 0.610818i \(0.790841\pi\)
\(948\) − 46.5150i − 1.51074i
\(949\) −35.2575 −1.14451
\(950\) 0 0
\(951\) 19.2305 0.623590
\(952\) 5.04650i 0.163558i
\(953\) 29.3828i 0.951804i 0.879498 + 0.475902i \(0.157878\pi\)
−0.879498 + 0.475902i \(0.842122\pi\)
\(954\) 40.3700 1.30703
\(955\) 0 0
\(956\) 10.0389 0.324681
\(957\) 111.454i 3.60280i
\(958\) − 24.6080i − 0.795049i
\(959\) −33.9545 −1.09645
\(960\) 0 0
\(961\) −28.0129 −0.903641
\(962\) 2.91331i 0.0939289i
\(963\) 107.201i 3.45450i
\(964\) 23.6947 0.763155
\(965\) 0 0
\(966\) 14.0660 0.452565
\(967\) − 49.2292i − 1.58310i −0.611102 0.791552i \(-0.709274\pi\)
0.611102 0.791552i \(-0.290726\pi\)
\(968\) − 7.81502i − 0.251184i
\(969\) 15.7402 0.505647
\(970\) 0 0
\(971\) 9.62347 0.308832 0.154416 0.988006i \(-0.450650\pi\)
0.154416 + 0.988006i \(0.450650\pi\)
\(972\) − 15.2846i − 0.490252i
\(973\) − 21.0841i − 0.675926i
\(974\) 30.2381 0.968890
\(975\) 0 0
\(976\) 3.55623 0.113832
\(977\) 18.9858i 0.607411i 0.952766 + 0.303705i \(0.0982238\pi\)
−0.952766 + 0.303705i \(0.901776\pi\)
\(978\) − 10.2651i − 0.328242i
\(979\) 33.3768 1.06673
\(980\) 0 0
\(981\) −10.0271 −0.320139
\(982\) 12.3311i 0.393500i
\(983\) − 4.33763i − 0.138349i −0.997605 0.0691744i \(-0.977963\pi\)
0.997605 0.0691744i \(-0.0220365\pi\)
\(984\) 12.1655 0.387823
\(985\) 0 0
\(986\) −9.21860 −0.293580
\(987\) 161.149i 5.12942i
\(988\) 16.8137i 0.534916i
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) 1.96766 0.0625049 0.0312524 0.999512i \(-0.490050\pi\)
0.0312524 + 0.999512i \(0.490050\pi\)
\(992\) 1.72833i 0.0548744i
\(993\) 86.0495i 2.73070i
\(994\) −32.1049 −1.01830
\(995\) 0 0
\(996\) −8.67526 −0.274886
\(997\) 3.96110i 0.125449i 0.998031 + 0.0627246i \(0.0199790\pi\)
−0.998031 + 0.0627246i \(0.980021\pi\)
\(998\) 26.9133i 0.851926i
\(999\) −9.08669 −0.287490
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.j.599.6 6
5.2 odd 4 1150.2.a.q.1.3 3
5.3 odd 4 230.2.a.d.1.1 3
5.4 even 2 inner 1150.2.b.j.599.1 6
15.8 even 4 2070.2.a.z.1.3 3
20.3 even 4 1840.2.a.r.1.3 3
20.7 even 4 9200.2.a.cf.1.1 3
40.3 even 4 7360.2.a.ce.1.1 3
40.13 odd 4 7360.2.a.bz.1.3 3
115.68 even 4 5290.2.a.r.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.d.1.1 3 5.3 odd 4
1150.2.a.q.1.3 3 5.2 odd 4
1150.2.b.j.599.1 6 5.4 even 2 inner
1150.2.b.j.599.6 6 1.1 even 1 trivial
1840.2.a.r.1.3 3 20.3 even 4
2070.2.a.z.1.3 3 15.8 even 4
5290.2.a.r.1.1 3 115.68 even 4
7360.2.a.bz.1.3 3 40.13 odd 4
7360.2.a.ce.1.1 3 40.3 even 4
9200.2.a.cf.1.1 3 20.7 even 4