Properties

Label 1150.2.b.f.599.4
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 599.4
Root \(2.30278i\) of defining polynomial
Character \(\chi\) \(=\) 1150.599
Dual form 1150.2.b.f.599.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +3.30278i q^{3} -1.00000 q^{4} -3.30278 q^{6} +0.302776i q^{7} -1.00000i q^{8} -7.90833 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +3.30278i q^{3} -1.00000 q^{4} -3.30278 q^{6} +0.302776i q^{7} -1.00000i q^{8} -7.90833 q^{9} -5.30278 q^{11} -3.30278i q^{12} -0.302776i q^{13} -0.302776 q^{14} +1.00000 q^{16} +3.90833i q^{17} -7.90833i q^{18} +4.90833 q^{19} -1.00000 q^{21} -5.30278i q^{22} -1.00000i q^{23} +3.30278 q^{24} +0.302776 q^{26} -16.2111i q^{27} -0.302776i q^{28} -4.60555 q^{29} +2.90833 q^{31} +1.00000i q^{32} -17.5139i q^{33} -3.90833 q^{34} +7.90833 q^{36} -8.00000i q^{37} +4.90833i q^{38} +1.00000 q^{39} -9.90833 q^{41} -1.00000i q^{42} +5.21110i q^{43} +5.30278 q^{44} +1.00000 q^{46} -4.60555i q^{47} +3.30278i q^{48} +6.90833 q^{49} -12.9083 q^{51} +0.302776i q^{52} +3.21110i q^{53} +16.2111 q^{54} +0.302776 q^{56} +16.2111i q^{57} -4.60555i q^{58} +10.6056 q^{59} -6.51388 q^{61} +2.90833i q^{62} -2.39445i q^{63} -1.00000 q^{64} +17.5139 q^{66} +4.00000i q^{67} -3.90833i q^{68} +3.30278 q^{69} -12.6972 q^{71} +7.90833i q^{72} +15.8167i q^{73} +8.00000 q^{74} -4.90833 q^{76} -1.60555i q^{77} +1.00000i q^{78} -14.4222 q^{79} +29.8167 q^{81} -9.90833i q^{82} -3.21110i q^{83} +1.00000 q^{84} -5.21110 q^{86} -15.2111i q^{87} +5.30278i q^{88} +0.0916731 q^{91} +1.00000i q^{92} +9.60555i q^{93} +4.60555 q^{94} -3.30278 q^{96} -2.69722i q^{97} +6.90833i q^{98} +41.9361 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} - 6q^{6} - 10q^{9} + O(q^{10}) \) \( 4q - 4q^{4} - 6q^{6} - 10q^{9} - 14q^{11} + 6q^{14} + 4q^{16} - 2q^{19} - 4q^{21} + 6q^{24} - 6q^{26} - 4q^{29} - 10q^{31} + 6q^{34} + 10q^{36} + 4q^{39} - 18q^{41} + 14q^{44} + 4q^{46} + 6q^{49} - 30q^{51} + 36q^{54} - 6q^{56} + 28q^{59} + 10q^{61} - 4q^{64} + 34q^{66} + 6q^{69} - 58q^{71} + 32q^{74} + 2q^{76} + 76q^{81} + 4q^{84} + 8q^{86} + 22q^{91} + 4q^{94} - 6q^{96} + 74q^{99} + O(q^{100}) \)

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
\(3\) 3.30278i 1.90686i 0.301617 + 0.953429i \(0.402474\pi\)
−0.301617 + 0.953429i \(0.597526\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −3.30278 −1.34835
\(7\) 0.302776i 0.114438i 0.998362 + 0.0572192i \(0.0182234\pi\)
−0.998362 + 0.0572192i \(0.981777\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −7.90833 −2.63611
\(10\) 0 0
\(11\) −5.30278 −1.59885 −0.799424 0.600768i \(-0.794862\pi\)
−0.799424 + 0.600768i \(0.794862\pi\)
\(12\) − 3.30278i − 0.953429i
\(13\) − 0.302776i − 0.0839749i −0.999118 0.0419874i \(-0.986631\pi\)
0.999118 0.0419874i \(-0.0133689\pi\)
\(14\) −0.302776 −0.0809202
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.90833i 0.947909i 0.880549 + 0.473954i \(0.157174\pi\)
−0.880549 + 0.473954i \(0.842826\pi\)
\(18\) − 7.90833i − 1.86401i
\(19\) 4.90833 1.12605 0.563024 0.826441i \(-0.309638\pi\)
0.563024 + 0.826441i \(0.309638\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) − 5.30278i − 1.13056i
\(23\) − 1.00000i − 0.208514i
\(24\) 3.30278 0.674176
\(25\) 0 0
\(26\) 0.302776 0.0593792
\(27\) − 16.2111i − 3.11983i
\(28\) − 0.302776i − 0.0572192i
\(29\) −4.60555 −0.855229 −0.427615 0.903961i \(-0.640646\pi\)
−0.427615 + 0.903961i \(0.640646\pi\)
\(30\) 0 0
\(31\) 2.90833 0.522351 0.261175 0.965291i \(-0.415890\pi\)
0.261175 + 0.965291i \(0.415890\pi\)
\(32\) 1.00000i 0.176777i
\(33\) − 17.5139i − 3.04877i
\(34\) −3.90833 −0.670273
\(35\) 0 0
\(36\) 7.90833 1.31805
\(37\) − 8.00000i − 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) 4.90833i 0.796236i
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −9.90833 −1.54742 −0.773710 0.633540i \(-0.781601\pi\)
−0.773710 + 0.633540i \(0.781601\pi\)
\(42\) − 1.00000i − 0.154303i
\(43\) 5.21110i 0.794686i 0.917670 + 0.397343i \(0.130068\pi\)
−0.917670 + 0.397343i \(0.869932\pi\)
\(44\) 5.30278 0.799424
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) − 4.60555i − 0.671789i −0.941900 0.335894i \(-0.890961\pi\)
0.941900 0.335894i \(-0.109039\pi\)
\(48\) 3.30278i 0.476715i
\(49\) 6.90833 0.986904
\(50\) 0 0
\(51\) −12.9083 −1.80753
\(52\) 0.302776i 0.0419874i
\(53\) 3.21110i 0.441079i 0.975378 + 0.220539i \(0.0707818\pi\)
−0.975378 + 0.220539i \(0.929218\pi\)
\(54\) 16.2111 2.20605
\(55\) 0 0
\(56\) 0.302776 0.0404601
\(57\) 16.2111i 2.14721i
\(58\) − 4.60555i − 0.604739i
\(59\) 10.6056 1.38073 0.690363 0.723464i \(-0.257451\pi\)
0.690363 + 0.723464i \(0.257451\pi\)
\(60\) 0 0
\(61\) −6.51388 −0.834017 −0.417008 0.908903i \(-0.636921\pi\)
−0.417008 + 0.908903i \(0.636921\pi\)
\(62\) 2.90833i 0.369358i
\(63\) − 2.39445i − 0.301672i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 17.5139 2.15581
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) − 3.90833i − 0.473954i
\(69\) 3.30278 0.397607
\(70\) 0 0
\(71\) −12.6972 −1.50688 −0.753442 0.657515i \(-0.771608\pi\)
−0.753442 + 0.657515i \(0.771608\pi\)
\(72\) 7.90833i 0.932005i
\(73\) 15.8167i 1.85120i 0.378504 + 0.925600i \(0.376439\pi\)
−0.378504 + 0.925600i \(0.623561\pi\)
\(74\) 8.00000 0.929981
\(75\) 0 0
\(76\) −4.90833 −0.563024
\(77\) − 1.60555i − 0.182970i
\(78\) 1.00000i 0.113228i
\(79\) −14.4222 −1.62262 −0.811312 0.584613i \(-0.801246\pi\)
−0.811312 + 0.584613i \(0.801246\pi\)
\(80\) 0 0
\(81\) 29.8167 3.31296
\(82\) − 9.90833i − 1.09419i
\(83\) − 3.21110i − 0.352464i −0.984349 0.176232i \(-0.943609\pi\)
0.984349 0.176232i \(-0.0563909\pi\)
\(84\) 1.00000 0.109109
\(85\) 0 0
\(86\) −5.21110 −0.561928
\(87\) − 15.2111i − 1.63080i
\(88\) 5.30278i 0.565278i
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0.0916731 0.00960995
\(92\) 1.00000i 0.104257i
\(93\) 9.60555i 0.996049i
\(94\) 4.60555 0.475026
\(95\) 0 0
\(96\) −3.30278 −0.337088
\(97\) − 2.69722i − 0.273862i −0.990581 0.136931i \(-0.956276\pi\)
0.990581 0.136931i \(-0.0437238\pi\)
\(98\) 6.90833i 0.697846i
\(99\) 41.9361 4.21473
\(100\) 0 0
\(101\) −4.60555 −0.458269 −0.229135 0.973395i \(-0.573590\pi\)
−0.229135 + 0.973395i \(0.573590\pi\)
\(102\) − 12.9083i − 1.27811i
\(103\) − 17.1194i − 1.68683i −0.537265 0.843414i \(-0.680542\pi\)
0.537265 0.843414i \(-0.319458\pi\)
\(104\) −0.302776 −0.0296896
\(105\) 0 0
\(106\) −3.21110 −0.311890
\(107\) − 4.60555i − 0.445235i −0.974906 0.222618i \(-0.928540\pi\)
0.974906 0.222618i \(-0.0714602\pi\)
\(108\) 16.2111i 1.55991i
\(109\) −19.5139 −1.86909 −0.934545 0.355844i \(-0.884193\pi\)
−0.934545 + 0.355844i \(0.884193\pi\)
\(110\) 0 0
\(111\) 26.4222 2.50788
\(112\) 0.302776i 0.0286096i
\(113\) 12.4222i 1.16858i 0.811544 + 0.584291i \(0.198627\pi\)
−0.811544 + 0.584291i \(0.801373\pi\)
\(114\) −16.2111 −1.51831
\(115\) 0 0
\(116\) 4.60555 0.427615
\(117\) 2.39445i 0.221367i
\(118\) 10.6056i 0.976320i
\(119\) −1.18335 −0.108477
\(120\) 0 0
\(121\) 17.1194 1.55631
\(122\) − 6.51388i − 0.589739i
\(123\) − 32.7250i − 2.95071i
\(124\) −2.90833 −0.261175
\(125\) 0 0
\(126\) 2.39445 0.213314
\(127\) 11.8167i 1.04856i 0.851546 + 0.524279i \(0.175665\pi\)
−0.851546 + 0.524279i \(0.824335\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −17.2111 −1.51535
\(130\) 0 0
\(131\) 3.21110 0.280555 0.140278 0.990112i \(-0.455200\pi\)
0.140278 + 0.990112i \(0.455200\pi\)
\(132\) 17.5139i 1.52439i
\(133\) 1.48612i 0.128863i
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 3.90833 0.335136
\(137\) 6.90833i 0.590218i 0.955464 + 0.295109i \(0.0953560\pi\)
−0.955464 + 0.295109i \(0.904644\pi\)
\(138\) 3.30278i 0.281151i
\(139\) 5.39445 0.457551 0.228776 0.973479i \(-0.426528\pi\)
0.228776 + 0.973479i \(0.426528\pi\)
\(140\) 0 0
\(141\) 15.2111 1.28101
\(142\) − 12.6972i − 1.06553i
\(143\) 1.60555i 0.134263i
\(144\) −7.90833 −0.659027
\(145\) 0 0
\(146\) −15.8167 −1.30900
\(147\) 22.8167i 1.88189i
\(148\) 8.00000i 0.657596i
\(149\) −9.69722 −0.794428 −0.397214 0.917726i \(-0.630023\pi\)
−0.397214 + 0.917726i \(0.630023\pi\)
\(150\) 0 0
\(151\) −1.90833 −0.155297 −0.0776487 0.996981i \(-0.524741\pi\)
−0.0776487 + 0.996981i \(0.524741\pi\)
\(152\) − 4.90833i − 0.398118i
\(153\) − 30.9083i − 2.49879i
\(154\) 1.60555 0.129379
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) 11.3944i 0.909376i 0.890651 + 0.454688i \(0.150249\pi\)
−0.890651 + 0.454688i \(0.849751\pi\)
\(158\) − 14.4222i − 1.14737i
\(159\) −10.6056 −0.841075
\(160\) 0 0
\(161\) 0.302776 0.0238621
\(162\) 29.8167i 2.34262i
\(163\) 5.69722i 0.446241i 0.974791 + 0.223121i \(0.0716243\pi\)
−0.974791 + 0.223121i \(0.928376\pi\)
\(164\) 9.90833 0.773710
\(165\) 0 0
\(166\) 3.21110 0.249230
\(167\) − 21.2111i − 1.64136i −0.571385 0.820682i \(-0.693594\pi\)
0.571385 0.820682i \(-0.306406\pi\)
\(168\) 1.00000i 0.0771517i
\(169\) 12.9083 0.992948
\(170\) 0 0
\(171\) −38.8167 −2.96838
\(172\) − 5.21110i − 0.397343i
\(173\) 23.3028i 1.77168i 0.463993 + 0.885839i \(0.346416\pi\)
−0.463993 + 0.885839i \(0.653584\pi\)
\(174\) 15.2111 1.15315
\(175\) 0 0
\(176\) −5.30278 −0.399712
\(177\) 35.0278i 2.63285i
\(178\) 0 0
\(179\) −16.6056 −1.24116 −0.620579 0.784144i \(-0.713102\pi\)
−0.620579 + 0.784144i \(0.713102\pi\)
\(180\) 0 0
\(181\) −8.11943 −0.603512 −0.301756 0.953385i \(-0.597573\pi\)
−0.301756 + 0.953385i \(0.597573\pi\)
\(182\) 0.0916731i 0.00679526i
\(183\) − 21.5139i − 1.59035i
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) −9.60555 −0.704313
\(187\) − 20.7250i − 1.51556i
\(188\) 4.60555i 0.335894i
\(189\) 4.90833 0.357028
\(190\) 0 0
\(191\) −1.39445 −0.100899 −0.0504494 0.998727i \(-0.516065\pi\)
−0.0504494 + 0.998727i \(0.516065\pi\)
\(192\) − 3.30278i − 0.238357i
\(193\) 3.81665i 0.274729i 0.990521 + 0.137364i \(0.0438631\pi\)
−0.990521 + 0.137364i \(0.956137\pi\)
\(194\) 2.69722 0.193649
\(195\) 0 0
\(196\) −6.90833 −0.493452
\(197\) − 0.697224i − 0.0496752i −0.999691 0.0248376i \(-0.992093\pi\)
0.999691 0.0248376i \(-0.00790686\pi\)
\(198\) 41.9361i 2.98027i
\(199\) −8.42221 −0.597034 −0.298517 0.954404i \(-0.596492\pi\)
−0.298517 + 0.954404i \(0.596492\pi\)
\(200\) 0 0
\(201\) −13.2111 −0.931839
\(202\) − 4.60555i − 0.324045i
\(203\) − 1.39445i − 0.0978711i
\(204\) 12.9083 0.903764
\(205\) 0 0
\(206\) 17.1194 1.19277
\(207\) 7.90833i 0.549667i
\(208\) − 0.302776i − 0.0209937i
\(209\) −26.0278 −1.80038
\(210\) 0 0
\(211\) −7.21110 −0.496433 −0.248216 0.968705i \(-0.579844\pi\)
−0.248216 + 0.968705i \(0.579844\pi\)
\(212\) − 3.21110i − 0.220539i
\(213\) − 41.9361i − 2.87341i
\(214\) 4.60555 0.314829
\(215\) 0 0
\(216\) −16.2111 −1.10303
\(217\) 0.880571i 0.0597770i
\(218\) − 19.5139i − 1.32165i
\(219\) −52.2389 −3.52997
\(220\) 0 0
\(221\) 1.18335 0.0796005
\(222\) 26.4222i 1.77334i
\(223\) − 4.00000i − 0.267860i −0.990991 0.133930i \(-0.957240\pi\)
0.990991 0.133930i \(-0.0427597\pi\)
\(224\) −0.302776 −0.0202300
\(225\) 0 0
\(226\) −12.4222 −0.826313
\(227\) 7.39445i 0.490787i 0.969424 + 0.245393i \(0.0789171\pi\)
−0.969424 + 0.245393i \(0.921083\pi\)
\(228\) − 16.2111i − 1.07361i
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 5.30278 0.348897
\(232\) 4.60555i 0.302369i
\(233\) 4.18335i 0.274060i 0.990567 + 0.137030i \(0.0437557\pi\)
−0.990567 + 0.137030i \(0.956244\pi\)
\(234\) −2.39445 −0.156530
\(235\) 0 0
\(236\) −10.6056 −0.690363
\(237\) − 47.6333i − 3.09412i
\(238\) − 1.18335i − 0.0767049i
\(239\) 9.21110 0.595817 0.297908 0.954594i \(-0.403711\pi\)
0.297908 + 0.954594i \(0.403711\pi\)
\(240\) 0 0
\(241\) 14.4222 0.929016 0.464508 0.885569i \(-0.346231\pi\)
0.464508 + 0.885569i \(0.346231\pi\)
\(242\) 17.1194i 1.10048i
\(243\) 49.8444i 3.19752i
\(244\) 6.51388 0.417008
\(245\) 0 0
\(246\) 32.7250 2.08647
\(247\) − 1.48612i − 0.0945597i
\(248\) − 2.90833i − 0.184679i
\(249\) 10.6056 0.672100
\(250\) 0 0
\(251\) −5.51388 −0.348033 −0.174016 0.984743i \(-0.555675\pi\)
−0.174016 + 0.984743i \(0.555675\pi\)
\(252\) 2.39445i 0.150836i
\(253\) 5.30278i 0.333383i
\(254\) −11.8167 −0.741443
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 19.8167i − 1.23613i −0.786127 0.618064i \(-0.787917\pi\)
0.786127 0.618064i \(-0.212083\pi\)
\(258\) − 17.2111i − 1.07152i
\(259\) 2.42221 0.150509
\(260\) 0 0
\(261\) 36.4222 2.25448
\(262\) 3.21110i 0.198383i
\(263\) − 14.5139i − 0.894964i −0.894293 0.447482i \(-0.852321\pi\)
0.894293 0.447482i \(-0.147679\pi\)
\(264\) −17.5139 −1.07790
\(265\) 0 0
\(266\) −1.48612 −0.0911200
\(267\) 0 0
\(268\) − 4.00000i − 0.244339i
\(269\) −25.8167 −1.57407 −0.787035 0.616909i \(-0.788385\pi\)
−0.787035 + 0.616909i \(0.788385\pi\)
\(270\) 0 0
\(271\) −6.30278 −0.382866 −0.191433 0.981506i \(-0.561314\pi\)
−0.191433 + 0.981506i \(0.561314\pi\)
\(272\) 3.90833i 0.236977i
\(273\) 0.302776i 0.0183248i
\(274\) −6.90833 −0.417347
\(275\) 0 0
\(276\) −3.30278 −0.198804
\(277\) 12.7889i 0.768410i 0.923248 + 0.384205i \(0.125524\pi\)
−0.923248 + 0.384205i \(0.874476\pi\)
\(278\) 5.39445i 0.323538i
\(279\) −23.0000 −1.37697
\(280\) 0 0
\(281\) −19.3944 −1.15698 −0.578488 0.815691i \(-0.696357\pi\)
−0.578488 + 0.815691i \(0.696357\pi\)
\(282\) 15.2111i 0.905808i
\(283\) 2.00000i 0.118888i 0.998232 + 0.0594438i \(0.0189327\pi\)
−0.998232 + 0.0594438i \(0.981067\pi\)
\(284\) 12.6972 0.753442
\(285\) 0 0
\(286\) −1.60555 −0.0949382
\(287\) − 3.00000i − 0.177084i
\(288\) − 7.90833i − 0.466003i
\(289\) 1.72498 0.101469
\(290\) 0 0
\(291\) 8.90833 0.522215
\(292\) − 15.8167i − 0.925600i
\(293\) 8.78890i 0.513453i 0.966484 + 0.256726i \(0.0826439\pi\)
−0.966484 + 0.256726i \(0.917356\pi\)
\(294\) −22.8167 −1.33069
\(295\) 0 0
\(296\) −8.00000 −0.464991
\(297\) 85.9638i 4.98813i
\(298\) − 9.69722i − 0.561745i
\(299\) −0.302776 −0.0175100
\(300\) 0 0
\(301\) −1.57779 −0.0909426
\(302\) − 1.90833i − 0.109812i
\(303\) − 15.2111i − 0.873855i
\(304\) 4.90833 0.281512
\(305\) 0 0
\(306\) 30.9083 1.76691
\(307\) 15.3028i 0.873376i 0.899613 + 0.436688i \(0.143849\pi\)
−0.899613 + 0.436688i \(0.856151\pi\)
\(308\) 1.60555i 0.0914848i
\(309\) 56.5416 3.21654
\(310\) 0 0
\(311\) −6.42221 −0.364170 −0.182085 0.983283i \(-0.558285\pi\)
−0.182085 + 0.983283i \(0.558285\pi\)
\(312\) − 1.00000i − 0.0566139i
\(313\) − 12.7250i − 0.719258i −0.933095 0.359629i \(-0.882903\pi\)
0.933095 0.359629i \(-0.117097\pi\)
\(314\) −11.3944 −0.643026
\(315\) 0 0
\(316\) 14.4222 0.811312
\(317\) 14.7250i 0.827037i 0.910496 + 0.413519i \(0.135700\pi\)
−0.910496 + 0.413519i \(0.864300\pi\)
\(318\) − 10.6056i − 0.594730i
\(319\) 24.4222 1.36738
\(320\) 0 0
\(321\) 15.2111 0.849001
\(322\) 0.302776i 0.0168730i
\(323\) 19.1833i 1.06739i
\(324\) −29.8167 −1.65648
\(325\) 0 0
\(326\) −5.69722 −0.315540
\(327\) − 64.4500i − 3.56409i
\(328\) 9.90833i 0.547096i
\(329\) 1.39445 0.0768784
\(330\) 0 0
\(331\) 9.39445 0.516366 0.258183 0.966096i \(-0.416876\pi\)
0.258183 + 0.966096i \(0.416876\pi\)
\(332\) 3.21110i 0.176232i
\(333\) 63.2666i 3.46699i
\(334\) 21.2111 1.16062
\(335\) 0 0
\(336\) −1.00000 −0.0545545
\(337\) 4.48612i 0.244375i 0.992507 + 0.122187i \(0.0389909\pi\)
−0.992507 + 0.122187i \(0.961009\pi\)
\(338\) 12.9083i 0.702120i
\(339\) −41.0278 −2.22832
\(340\) 0 0
\(341\) −15.4222 −0.835159
\(342\) − 38.8167i − 2.09896i
\(343\) 4.21110i 0.227378i
\(344\) 5.21110 0.280964
\(345\) 0 0
\(346\) −23.3028 −1.25276
\(347\) 25.5416i 1.37115i 0.728004 + 0.685573i \(0.240448\pi\)
−0.728004 + 0.685573i \(0.759552\pi\)
\(348\) 15.2111i 0.815401i
\(349\) 12.7889 0.684574 0.342287 0.939595i \(-0.388798\pi\)
0.342287 + 0.939595i \(0.388798\pi\)
\(350\) 0 0
\(351\) −4.90833 −0.261987
\(352\) − 5.30278i − 0.282639i
\(353\) 18.4222i 0.980515i 0.871578 + 0.490258i \(0.163097\pi\)
−0.871578 + 0.490258i \(0.836903\pi\)
\(354\) −35.0278 −1.86170
\(355\) 0 0
\(356\) 0 0
\(357\) − 3.90833i − 0.206851i
\(358\) − 16.6056i − 0.877631i
\(359\) 3.21110 0.169476 0.0847378 0.996403i \(-0.472995\pi\)
0.0847378 + 0.996403i \(0.472995\pi\)
\(360\) 0 0
\(361\) 5.09167 0.267983
\(362\) − 8.11943i − 0.426748i
\(363\) 56.5416i 2.96767i
\(364\) −0.0916731 −0.00480498
\(365\) 0 0
\(366\) 21.5139 1.12455
\(367\) − 29.2111i − 1.52481i −0.647102 0.762404i \(-0.724019\pi\)
0.647102 0.762404i \(-0.275981\pi\)
\(368\) − 1.00000i − 0.0521286i
\(369\) 78.3583 4.07917
\(370\) 0 0
\(371\) −0.972244 −0.0504764
\(372\) − 9.60555i − 0.498025i
\(373\) − 2.60555i − 0.134910i −0.997722 0.0674552i \(-0.978512\pi\)
0.997722 0.0674552i \(-0.0214880\pi\)
\(374\) 20.7250 1.07166
\(375\) 0 0
\(376\) −4.60555 −0.237513
\(377\) 1.39445i 0.0718178i
\(378\) 4.90833i 0.252457i
\(379\) −4.09167 −0.210175 −0.105088 0.994463i \(-0.533512\pi\)
−0.105088 + 0.994463i \(0.533512\pi\)
\(380\) 0 0
\(381\) −39.0278 −1.99945
\(382\) − 1.39445i − 0.0713462i
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 3.30278 0.168544
\(385\) 0 0
\(386\) −3.81665 −0.194263
\(387\) − 41.2111i − 2.09488i
\(388\) 2.69722i 0.136931i
\(389\) −20.9361 −1.06150 −0.530751 0.847528i \(-0.678090\pi\)
−0.530751 + 0.847528i \(0.678090\pi\)
\(390\) 0 0
\(391\) 3.90833 0.197653
\(392\) − 6.90833i − 0.348923i
\(393\) 10.6056i 0.534979i
\(394\) 0.697224 0.0351257
\(395\) 0 0
\(396\) −41.9361 −2.10737
\(397\) 21.7250i 1.09035i 0.838324 + 0.545173i \(0.183536\pi\)
−0.838324 + 0.545173i \(0.816464\pi\)
\(398\) − 8.42221i − 0.422167i
\(399\) −4.90833 −0.245724
\(400\) 0 0
\(401\) −1.39445 −0.0696354 −0.0348177 0.999394i \(-0.511085\pi\)
−0.0348177 + 0.999394i \(0.511085\pi\)
\(402\) − 13.2111i − 0.658910i
\(403\) − 0.880571i − 0.0438643i
\(404\) 4.60555 0.229135
\(405\) 0 0
\(406\) 1.39445 0.0692053
\(407\) 42.4222i 2.10279i
\(408\) 12.9083i 0.639057i
\(409\) 15.0917 0.746235 0.373118 0.927784i \(-0.378289\pi\)
0.373118 + 0.927784i \(0.378289\pi\)
\(410\) 0 0
\(411\) −22.8167 −1.12546
\(412\) 17.1194i 0.843414i
\(413\) 3.21110i 0.158008i
\(414\) −7.90833 −0.388673
\(415\) 0 0
\(416\) 0.302776 0.0148448
\(417\) 17.8167i 0.872485i
\(418\) − 26.0278i − 1.27306i
\(419\) 39.6333 1.93621 0.968107 0.250538i \(-0.0806073\pi\)
0.968107 + 0.250538i \(0.0806073\pi\)
\(420\) 0 0
\(421\) 34.3028 1.67181 0.835907 0.548870i \(-0.184942\pi\)
0.835907 + 0.548870i \(0.184942\pi\)
\(422\) − 7.21110i − 0.351031i
\(423\) 36.4222i 1.77091i
\(424\) 3.21110 0.155945
\(425\) 0 0
\(426\) 41.9361 2.03181
\(427\) − 1.97224i − 0.0954436i
\(428\) 4.60555i 0.222618i
\(429\) −5.30278 −0.256020
\(430\) 0 0
\(431\) 20.2389 0.974872 0.487436 0.873159i \(-0.337932\pi\)
0.487436 + 0.873159i \(0.337932\pi\)
\(432\) − 16.2111i − 0.779957i
\(433\) − 34.9083i − 1.67759i −0.544450 0.838794i \(-0.683261\pi\)
0.544450 0.838794i \(-0.316739\pi\)
\(434\) −0.880571 −0.0422687
\(435\) 0 0
\(436\) 19.5139 0.934545
\(437\) − 4.90833i − 0.234797i
\(438\) − 52.2389i − 2.49607i
\(439\) 18.3028 0.873544 0.436772 0.899572i \(-0.356122\pi\)
0.436772 + 0.899572i \(0.356122\pi\)
\(440\) 0 0
\(441\) −54.6333 −2.60159
\(442\) 1.18335i 0.0562860i
\(443\) 35.5139i 1.68732i 0.536882 + 0.843658i \(0.319602\pi\)
−0.536882 + 0.843658i \(0.680398\pi\)
\(444\) −26.4222 −1.25394
\(445\) 0 0
\(446\) 4.00000 0.189405
\(447\) − 32.0278i − 1.51486i
\(448\) − 0.302776i − 0.0143048i
\(449\) 12.9083 0.609182 0.304591 0.952483i \(-0.401480\pi\)
0.304591 + 0.952483i \(0.401480\pi\)
\(450\) 0 0
\(451\) 52.5416 2.47409
\(452\) − 12.4222i − 0.584291i
\(453\) − 6.30278i − 0.296130i
\(454\) −7.39445 −0.347039
\(455\) 0 0
\(456\) 16.2111 0.759154
\(457\) 3.57779i 0.167362i 0.996493 + 0.0836811i \(0.0266677\pi\)
−0.996493 + 0.0836811i \(0.973332\pi\)
\(458\) − 2.00000i − 0.0934539i
\(459\) 63.3583 2.95731
\(460\) 0 0
\(461\) 31.8167 1.48185 0.740925 0.671588i \(-0.234388\pi\)
0.740925 + 0.671588i \(0.234388\pi\)
\(462\) 5.30278i 0.246707i
\(463\) − 25.6333i − 1.19128i −0.803251 0.595640i \(-0.796898\pi\)
0.803251 0.595640i \(-0.203102\pi\)
\(464\) −4.60555 −0.213807
\(465\) 0 0
\(466\) −4.18335 −0.193790
\(467\) − 19.8167i − 0.917005i −0.888693 0.458503i \(-0.848386\pi\)
0.888693 0.458503i \(-0.151614\pi\)
\(468\) − 2.39445i − 0.110683i
\(469\) −1.21110 −0.0559235
\(470\) 0 0
\(471\) −37.6333 −1.73405
\(472\) − 10.6056i − 0.488160i
\(473\) − 27.6333i − 1.27058i
\(474\) 47.6333 2.18787
\(475\) 0 0
\(476\) 1.18335 0.0542386
\(477\) − 25.3944i − 1.16273i
\(478\) 9.21110i 0.421306i
\(479\) 30.0000 1.37073 0.685367 0.728197i \(-0.259642\pi\)
0.685367 + 0.728197i \(0.259642\pi\)
\(480\) 0 0
\(481\) −2.42221 −0.110443
\(482\) 14.4222i 0.656913i
\(483\) 1.00000i 0.0455016i
\(484\) −17.1194 −0.778156
\(485\) 0 0
\(486\) −49.8444 −2.26099
\(487\) 11.8167i 0.535464i 0.963493 + 0.267732i \(0.0862741\pi\)
−0.963493 + 0.267732i \(0.913726\pi\)
\(488\) 6.51388i 0.294869i
\(489\) −18.8167 −0.850918
\(490\) 0 0
\(491\) −25.8167 −1.16509 −0.582545 0.812799i \(-0.697943\pi\)
−0.582545 + 0.812799i \(0.697943\pi\)
\(492\) 32.7250i 1.47536i
\(493\) − 18.0000i − 0.810679i
\(494\) 1.48612 0.0668638
\(495\) 0 0
\(496\) 2.90833 0.130588
\(497\) − 3.84441i − 0.172445i
\(498\) 10.6056i 0.475246i
\(499\) −11.6333 −0.520778 −0.260389 0.965504i \(-0.583851\pi\)
−0.260389 + 0.965504i \(0.583851\pi\)
\(500\) 0 0
\(501\) 70.0555 3.12985
\(502\) − 5.51388i − 0.246096i
\(503\) − 2.72498i − 0.121501i −0.998153 0.0607504i \(-0.980651\pi\)
0.998153 0.0607504i \(-0.0193494\pi\)
\(504\) −2.39445 −0.106657
\(505\) 0 0
\(506\) −5.30278 −0.235737
\(507\) 42.6333i 1.89341i
\(508\) − 11.8167i − 0.524279i
\(509\) −29.4500 −1.30535 −0.652673 0.757639i \(-0.726353\pi\)
−0.652673 + 0.757639i \(0.726353\pi\)
\(510\) 0 0
\(511\) −4.78890 −0.211848
\(512\) 1.00000i 0.0441942i
\(513\) − 79.5694i − 3.51307i
\(514\) 19.8167 0.874075
\(515\) 0 0
\(516\) 17.2111 0.757677
\(517\) 24.4222i 1.07409i
\(518\) 2.42221i 0.106426i
\(519\) −76.9638 −3.37834
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 36.4222i 1.59416i
\(523\) 8.42221i 0.368277i 0.982900 + 0.184139i \(0.0589495\pi\)
−0.982900 + 0.184139i \(0.941050\pi\)
\(524\) −3.21110 −0.140278
\(525\) 0 0
\(526\) 14.5139 0.632835
\(527\) 11.3667i 0.495141i
\(528\) − 17.5139i − 0.762194i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −83.8722 −3.63974
\(532\) − 1.48612i − 0.0644316i
\(533\) 3.00000i 0.129944i
\(534\) 0 0
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) − 54.8444i − 2.36671i
\(538\) − 25.8167i − 1.11303i
\(539\) −36.6333 −1.57791
\(540\) 0 0
\(541\) −28.8444 −1.24012 −0.620059 0.784555i \(-0.712891\pi\)
−0.620059 + 0.784555i \(0.712891\pi\)
\(542\) − 6.30278i − 0.270727i
\(543\) − 26.8167i − 1.15081i
\(544\) −3.90833 −0.167568
\(545\) 0 0
\(546\) −0.302776 −0.0129576
\(547\) − 7.51388i − 0.321270i −0.987014 0.160635i \(-0.948646\pi\)
0.987014 0.160635i \(-0.0513542\pi\)
\(548\) − 6.90833i − 0.295109i
\(549\) 51.5139 2.19856
\(550\) 0 0
\(551\) −22.6056 −0.963029
\(552\) − 3.30278i − 0.140575i
\(553\) − 4.36669i − 0.185691i
\(554\) −12.7889 −0.543348
\(555\) 0 0
\(556\) −5.39445 −0.228776
\(557\) 6.42221i 0.272118i 0.990701 + 0.136059i \(0.0434436\pi\)
−0.990701 + 0.136059i \(0.956556\pi\)
\(558\) − 23.0000i − 0.973668i
\(559\) 1.57779 0.0667336
\(560\) 0 0
\(561\) 68.4500 2.88996
\(562\) − 19.3944i − 0.818105i
\(563\) 39.6333i 1.67034i 0.549988 + 0.835172i \(0.314632\pi\)
−0.549988 + 0.835172i \(0.685368\pi\)
\(564\) −15.2111 −0.640503
\(565\) 0 0
\(566\) −2.00000 −0.0840663
\(567\) 9.02776i 0.379130i
\(568\) 12.6972i 0.532764i
\(569\) 0.422205 0.0176998 0.00884988 0.999961i \(-0.497183\pi\)
0.00884988 + 0.999961i \(0.497183\pi\)
\(570\) 0 0
\(571\) 9.11943 0.381636 0.190818 0.981625i \(-0.438886\pi\)
0.190818 + 0.981625i \(0.438886\pi\)
\(572\) − 1.60555i − 0.0671315i
\(573\) − 4.60555i − 0.192400i
\(574\) 3.00000 0.125218
\(575\) 0 0
\(576\) 7.90833 0.329514
\(577\) − 2.00000i − 0.0832611i −0.999133 0.0416305i \(-0.986745\pi\)
0.999133 0.0416305i \(-0.0132552\pi\)
\(578\) 1.72498i 0.0717497i
\(579\) −12.6056 −0.523869
\(580\) 0 0
\(581\) 0.972244 0.0403355
\(582\) 8.90833i 0.369262i
\(583\) − 17.0278i − 0.705218i
\(584\) 15.8167 0.654498
\(585\) 0 0
\(586\) −8.78890 −0.363066
\(587\) 37.5416i 1.54951i 0.632262 + 0.774755i \(0.282127\pi\)
−0.632262 + 0.774755i \(0.717873\pi\)
\(588\) − 22.8167i − 0.940943i
\(589\) 14.2750 0.588192
\(590\) 0 0
\(591\) 2.30278 0.0947235
\(592\) − 8.00000i − 0.328798i
\(593\) 19.8167i 0.813772i 0.913479 + 0.406886i \(0.133385\pi\)
−0.913479 + 0.406886i \(0.866615\pi\)
\(594\) −85.9638 −3.52714
\(595\) 0 0
\(596\) 9.69722 0.397214
\(597\) − 27.8167i − 1.13846i
\(598\) − 0.302776i − 0.0123814i
\(599\) −4.33053 −0.176941 −0.0884704 0.996079i \(-0.528198\pi\)
−0.0884704 + 0.996079i \(0.528198\pi\)
\(600\) 0 0
\(601\) −3.93608 −0.160556 −0.0802781 0.996773i \(-0.525581\pi\)
−0.0802781 + 0.996773i \(0.525581\pi\)
\(602\) − 1.57779i − 0.0643061i
\(603\) − 31.6333i − 1.28821i
\(604\) 1.90833 0.0776487
\(605\) 0 0
\(606\) 15.2111 0.617909
\(607\) 26.0555i 1.05756i 0.848759 + 0.528780i \(0.177350\pi\)
−0.848759 + 0.528780i \(0.822650\pi\)
\(608\) 4.90833i 0.199059i
\(609\) 4.60555 0.186626
\(610\) 0 0
\(611\) −1.39445 −0.0564134
\(612\) 30.9083i 1.24940i
\(613\) 32.4222i 1.30952i 0.755837 + 0.654760i \(0.227230\pi\)
−0.755837 + 0.654760i \(0.772770\pi\)
\(614\) −15.3028 −0.617570
\(615\) 0 0
\(616\) −1.60555 −0.0646895
\(617\) − 8.09167i − 0.325758i −0.986646 0.162879i \(-0.947922\pi\)
0.986646 0.162879i \(-0.0520781\pi\)
\(618\) 56.5416i 2.27444i
\(619\) −27.3305 −1.09851 −0.549253 0.835656i \(-0.685088\pi\)
−0.549253 + 0.835656i \(0.685088\pi\)
\(620\) 0 0
\(621\) −16.2111 −0.650529
\(622\) − 6.42221i − 0.257507i
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) 0 0
\(626\) 12.7250 0.508593
\(627\) − 85.9638i − 3.43307i
\(628\) − 11.3944i − 0.454688i
\(629\) 31.2666 1.24668
\(630\) 0 0
\(631\) 30.6056 1.21839 0.609194 0.793021i \(-0.291493\pi\)
0.609194 + 0.793021i \(0.291493\pi\)
\(632\) 14.4222i 0.573685i
\(633\) − 23.8167i − 0.946627i
\(634\) −14.7250 −0.584804
\(635\) 0 0
\(636\) 10.6056 0.420537
\(637\) − 2.09167i − 0.0828751i
\(638\) 24.4222i 0.966884i
\(639\) 100.414 3.97231
\(640\) 0 0
\(641\) −36.0000 −1.42191 −0.710957 0.703235i \(-0.751738\pi\)
−0.710957 + 0.703235i \(0.751738\pi\)
\(642\) 15.2111i 0.600334i
\(643\) 16.2389i 0.640398i 0.947350 + 0.320199i \(0.103750\pi\)
−0.947350 + 0.320199i \(0.896250\pi\)
\(644\) −0.302776 −0.0119310
\(645\) 0 0
\(646\) −19.1833 −0.754759
\(647\) 30.8444i 1.21262i 0.795229 + 0.606309i \(0.207351\pi\)
−0.795229 + 0.606309i \(0.792649\pi\)
\(648\) − 29.8167i − 1.17131i
\(649\) −56.2389 −2.20757
\(650\) 0 0
\(651\) −2.90833 −0.113986
\(652\) − 5.69722i − 0.223121i
\(653\) 9.27502i 0.362960i 0.983395 + 0.181480i \(0.0580887\pi\)
−0.983395 + 0.181480i \(0.941911\pi\)
\(654\) 64.4500 2.52019
\(655\) 0 0
\(656\) −9.90833 −0.386855
\(657\) − 125.083i − 4.87996i
\(658\) 1.39445i 0.0543613i
\(659\) 27.6333 1.07644 0.538220 0.842804i \(-0.319097\pi\)
0.538220 + 0.842804i \(0.319097\pi\)
\(660\) 0 0
\(661\) −24.0917 −0.937057 −0.468529 0.883448i \(-0.655216\pi\)
−0.468529 + 0.883448i \(0.655216\pi\)
\(662\) 9.39445i 0.365126i
\(663\) 3.90833i 0.151787i
\(664\) −3.21110 −0.124615
\(665\) 0 0
\(666\) −63.2666 −2.45153
\(667\) 4.60555i 0.178328i
\(668\) 21.2111i 0.820682i
\(669\) 13.2111 0.510771
\(670\) 0 0
\(671\) 34.5416 1.33347
\(672\) − 1.00000i − 0.0385758i
\(673\) 5.63331i 0.217148i 0.994088 + 0.108574i \(0.0346284\pi\)
−0.994088 + 0.108574i \(0.965372\pi\)
\(674\) −4.48612 −0.172799
\(675\) 0 0
\(676\) −12.9083 −0.496474
\(677\) 12.4222i 0.477424i 0.971090 + 0.238712i \(0.0767252\pi\)
−0.971090 + 0.238712i \(0.923275\pi\)
\(678\) − 41.0278i − 1.57566i
\(679\) 0.816654 0.0313403
\(680\) 0 0
\(681\) −24.4222 −0.935861
\(682\) − 15.4222i − 0.590547i
\(683\) − 32.7250i − 1.25219i −0.779748 0.626093i \(-0.784653\pi\)
0.779748 0.626093i \(-0.215347\pi\)
\(684\) 38.8167 1.48419
\(685\) 0 0
\(686\) −4.21110 −0.160781
\(687\) − 6.60555i − 0.252018i
\(688\) 5.21110i 0.198671i
\(689\) 0.972244 0.0370395
\(690\) 0 0
\(691\) 30.1833 1.14823 0.574114 0.818775i \(-0.305347\pi\)
0.574114 + 0.818775i \(0.305347\pi\)
\(692\) − 23.3028i − 0.885839i
\(693\) 12.6972i 0.482328i
\(694\) −25.5416 −0.969547
\(695\) 0 0
\(696\) −15.2111 −0.576575
\(697\) − 38.7250i − 1.46681i
\(698\) 12.7889i 0.484067i
\(699\) −13.8167 −0.522594
\(700\) 0 0
\(701\) 42.9083 1.62063 0.810313 0.585998i \(-0.199297\pi\)
0.810313 + 0.585998i \(0.199297\pi\)
\(702\) − 4.90833i − 0.185253i
\(703\) − 39.2666i − 1.48097i
\(704\) 5.30278 0.199856
\(705\) 0 0
\(706\) −18.4222 −0.693329
\(707\) − 1.39445i − 0.0524436i
\(708\) − 35.0278i − 1.31642i
\(709\) 41.1194 1.54427 0.772136 0.635457i \(-0.219188\pi\)
0.772136 + 0.635457i \(0.219188\pi\)
\(710\) 0 0
\(711\) 114.056 4.27742
\(712\) 0 0
\(713\) − 2.90833i − 0.108918i
\(714\) 3.90833 0.146265
\(715\) 0 0
\(716\) 16.6056 0.620579
\(717\) 30.4222i 1.13614i
\(718\) 3.21110i 0.119837i
\(719\) −14.3028 −0.533404 −0.266702 0.963779i \(-0.585934\pi\)
−0.266702 + 0.963779i \(0.585934\pi\)
\(720\) 0 0
\(721\) 5.18335 0.193038
\(722\) 5.09167i 0.189492i
\(723\) 47.6333i 1.77150i
\(724\) 8.11943 0.301756
\(725\) 0 0
\(726\) −56.5416 −2.09846
\(727\) 7.90833i 0.293304i 0.989188 + 0.146652i \(0.0468497\pi\)
−0.989188 + 0.146652i \(0.953150\pi\)
\(728\) − 0.0916731i − 0.00339763i
\(729\) −75.1749 −2.78426
\(730\) 0 0
\(731\) −20.3667 −0.753289
\(732\) 21.5139i 0.795176i
\(733\) − 13.6333i − 0.503558i −0.967785 0.251779i \(-0.918984\pi\)
0.967785 0.251779i \(-0.0810156\pi\)
\(734\) 29.2111 1.07820
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) − 21.2111i − 0.781321i
\(738\) 78.3583i 2.88441i
\(739\) 7.63331 0.280796 0.140398 0.990095i \(-0.455162\pi\)
0.140398 + 0.990095i \(0.455162\pi\)
\(740\) 0 0
\(741\) 4.90833 0.180312
\(742\) − 0.972244i − 0.0356922i
\(743\) 7.33053i 0.268931i 0.990918 + 0.134466i \(0.0429318\pi\)
−0.990918 + 0.134466i \(0.957068\pi\)
\(744\) 9.60555 0.352157
\(745\) 0 0
\(746\) 2.60555 0.0953960
\(747\) 25.3944i 0.929134i
\(748\) 20.7250i 0.757780i
\(749\) 1.39445 0.0509520
\(750\) 0 0
\(751\) 0.183346 0.00669040 0.00334520 0.999994i \(-0.498935\pi\)
0.00334520 + 0.999994i \(0.498935\pi\)
\(752\) − 4.60555i − 0.167947i
\(753\) − 18.2111i − 0.663649i
\(754\) −1.39445 −0.0507828
\(755\) 0 0
\(756\) −4.90833 −0.178514
\(757\) 1.21110i 0.0440183i 0.999758 + 0.0220091i \(0.00700629\pi\)
−0.999758 + 0.0220091i \(0.992994\pi\)
\(758\) − 4.09167i − 0.148616i
\(759\) −17.5139 −0.635714
\(760\) 0 0
\(761\) 4.54163 0.164634 0.0823171 0.996606i \(-0.473768\pi\)
0.0823171 + 0.996606i \(0.473768\pi\)
\(762\) − 39.0278i − 1.41383i
\(763\) − 5.90833i − 0.213896i
\(764\) 1.39445 0.0504494
\(765\) 0 0
\(766\) 0 0
\(767\) − 3.21110i − 0.115946i
\(768\) 3.30278i 0.119179i
\(769\) 41.2666 1.48811 0.744056 0.668117i \(-0.232900\pi\)
0.744056 + 0.668117i \(0.232900\pi\)
\(770\) 0 0
\(771\) 65.4500 2.35712
\(772\) − 3.81665i − 0.137364i
\(773\) 12.0000i 0.431610i 0.976436 + 0.215805i \(0.0692376\pi\)
−0.976436 + 0.215805i \(0.930762\pi\)
\(774\) 41.2111 1.48130
\(775\) 0 0
\(776\) −2.69722 −0.0968247
\(777\) 8.00000i 0.286998i
\(778\) − 20.9361i − 0.750595i
\(779\) −48.6333 −1.74247
\(780\) 0 0
\(781\) 67.3305 2.40928
\(782\) 3.90833i 0.139761i
\(783\) 74.6611i 2.66817i
\(784\) 6.90833 0.246726
\(785\) 0 0
\(786\) −10.6056 −0.378287
\(787\) 27.4500i 0.978485i 0.872148 + 0.489243i \(0.162727\pi\)
−0.872148 + 0.489243i \(0.837273\pi\)
\(788\) 0.697224i 0.0248376i
\(789\) 47.9361 1.70657
\(790\) 0 0
\(791\) −3.76114 −0.133731
\(792\) − 41.9361i − 1.49013i
\(793\) 1.97224i 0.0700364i
\(794\) −21.7250 −0.770991
\(795\) 0 0
\(796\) 8.42221 0.298517
\(797\) 19.8167i 0.701942i 0.936386 + 0.350971i \(0.114148\pi\)
−0.936386 + 0.350971i \(0.885852\pi\)
\(798\) − 4.90833i − 0.173753i
\(799\) 18.0000 0.636794
\(800\) 0 0
\(801\) 0 0
\(802\) − 1.39445i − 0.0492397i
\(803\) − 83.8722i − 2.95978i
\(804\) 13.2111 0.465920
\(805\) 0 0
\(806\) 0.880571 0.0310168
\(807\) − 85.2666i − 3.00153i
\(808\) 4.60555i 0.162023i
\(809\) −18.2750 −0.642515 −0.321258 0.946992i \(-0.604106\pi\)
−0.321258 + 0.946992i \(0.604106\pi\)
\(810\) 0 0
\(811\) −4.97224 −0.174599 −0.0872995 0.996182i \(-0.527824\pi\)
−0.0872995 + 0.996182i \(0.527824\pi\)
\(812\) 1.39445i 0.0489356i
\(813\) − 20.8167i − 0.730072i
\(814\) −42.4222 −1.48690
\(815\) 0 0
\(816\) −12.9083 −0.451882
\(817\) 25.5778i 0.894854i
\(818\) 15.0917i 0.527668i
\(819\) −0.724981 −0.0253329
\(820\) 0 0
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) − 22.8167i − 0.795822i
\(823\) − 0.788897i − 0.0274992i −0.999905 0.0137496i \(-0.995623\pi\)
0.999905 0.0137496i \(-0.00437678\pi\)
\(824\) −17.1194 −0.596384
\(825\) 0 0
\(826\) −3.21110 −0.111729
\(827\) − 35.4500i − 1.23272i −0.787466 0.616358i \(-0.788607\pi\)
0.787466 0.616358i \(-0.211393\pi\)
\(828\) − 7.90833i − 0.274833i
\(829\) −16.7889 −0.583103 −0.291551 0.956555i \(-0.594171\pi\)
−0.291551 + 0.956555i \(0.594171\pi\)
\(830\) 0 0
\(831\) −42.2389 −1.46525
\(832\) 0.302776i 0.0104969i
\(833\) 27.0000i 0.935495i
\(834\) −17.8167 −0.616940
\(835\) 0 0
\(836\) 26.0278 0.900189
\(837\) − 47.1472i − 1.62965i
\(838\) 39.6333i 1.36911i
\(839\) 22.1833 0.765854 0.382927 0.923779i \(-0.374916\pi\)
0.382927 + 0.923779i \(0.374916\pi\)
\(840\) 0 0
\(841\) −7.78890 −0.268583
\(842\) 34.3028i 1.18215i
\(843\) − 64.0555i − 2.20619i
\(844\) 7.21110 0.248216
\(845\) 0 0
\(846\) −36.4222 −1.25222
\(847\) 5.18335i 0.178102i
\(848\) 3.21110i 0.110270i
\(849\) −6.60555 −0.226702
\(850\) 0 0
\(851\) −8.00000 −0.274236
\(852\) 41.9361i 1.43671i
\(853\) 10.7250i 0.367216i 0.983000 + 0.183608i \(0.0587778\pi\)
−0.983000 + 0.183608i \(0.941222\pi\)
\(854\) 1.97224 0.0674888
\(855\) 0 0
\(856\) −4.60555 −0.157415
\(857\) 33.6333i 1.14889i 0.818543 + 0.574446i \(0.194782\pi\)
−0.818543 + 0.574446i \(0.805218\pi\)
\(858\) − 5.30278i − 0.181034i
\(859\) 14.1833 0.483930 0.241965 0.970285i \(-0.422208\pi\)
0.241965 + 0.970285i \(0.422208\pi\)
\(860\) 0 0
\(861\) 9.90833 0.337675
\(862\) 20.2389i 0.689338i
\(863\) 23.4500i 0.798246i 0.916897 + 0.399123i \(0.130685\pi\)
−0.916897 + 0.399123i \(0.869315\pi\)
\(864\) 16.2111 0.551513
\(865\) 0 0
\(866\) 34.9083 1.18623
\(867\) 5.69722i 0.193488i
\(868\) − 0.880571i − 0.0298885i
\(869\) 76.4777 2.59433
\(870\) 0 0
\(871\) 1.21110 0.0410366
\(872\) 19.5139i 0.660823i
\(873\) 21.3305i 0.721929i
\(874\) 4.90833 0.166027
\(875\) 0 0
\(876\) 52.2389 1.76499
\(877\) − 49.1749i − 1.66052i −0.557376 0.830260i \(-0.688192\pi\)
0.557376 0.830260i \(-0.311808\pi\)
\(878\) 18.3028i 0.617689i
\(879\) −29.0278 −0.979082
\(880\) 0 0
\(881\) −31.2666 −1.05340 −0.526700 0.850052i \(-0.676571\pi\)
−0.526700 + 0.850052i \(0.676571\pi\)
\(882\) − 54.6333i − 1.83960i
\(883\) 40.7250i 1.37050i 0.728306 + 0.685252i \(0.240308\pi\)
−0.728306 + 0.685252i \(0.759692\pi\)
\(884\) −1.18335 −0.0398002
\(885\) 0 0
\(886\) −35.5139 −1.19311
\(887\) 15.6333i 0.524915i 0.964944 + 0.262458i \(0.0845330\pi\)
−0.964944 + 0.262458i \(0.915467\pi\)
\(888\) − 26.4222i − 0.886671i
\(889\) −3.57779 −0.119995
\(890\) 0 0
\(891\) −158.111 −5.29692
\(892\) 4.00000i 0.133930i
\(893\) − 22.6056i − 0.756466i
\(894\) 32.0278 1.07117
\(895\) 0 0
\(896\) 0.302776 0.0101150
\(897\) − 1.00000i − 0.0333890i
\(898\) 12.9083i 0.430756i
\(899\) −13.3944 −0.446730
\(900\) 0 0
\(901\) −12.5500 −0.418102
\(902\) 52.5416i 1.74945i
\(903\) − 5.21110i − 0.173415i
\(904\) 12.4222 0.413156
\(905\) 0 0
\(906\) 6.30278 0.209396
\(907\) 30.6611i 1.01808i 0.860742 + 0.509042i \(0.170000\pi\)
−0.860742 + 0.509042i \(0.830000\pi\)
\(908\) − 7.39445i − 0.245393i
\(909\) 36.4222 1.20805
\(910\) 0 0
\(911\) −25.8167 −0.855344 −0.427672 0.903934i \(-0.640666\pi\)
−0.427672 + 0.903934i \(0.640666\pi\)
\(912\) 16.2111i 0.536803i
\(913\) 17.0278i 0.563536i
\(914\) −3.57779 −0.118343
\(915\) 0 0
\(916\) 2.00000 0.0660819
\(917\) 0.972244i 0.0321063i
\(918\) 63.3583i 2.09114i
\(919\) −44.0000 −1.45143 −0.725713 0.687998i \(-0.758490\pi\)
−0.725713 + 0.687998i \(0.758490\pi\)
\(920\) 0 0
\(921\) −50.5416 −1.66540
\(922\) 31.8167i 1.04783i
\(923\) 3.84441i 0.126540i
\(924\) −5.30278 −0.174449
\(925\) 0 0
\(926\) 25.6333 0.842363
\(927\) 135.386i 4.44666i
\(928\) − 4.60555i − 0.151185i
\(929\) 57.6333 1.89089 0.945444 0.325785i \(-0.105629\pi\)
0.945444 + 0.325785i \(0.105629\pi\)
\(930\) 0 0
\(931\) 33.9083 1.11130
\(932\) − 4.18335i − 0.137030i
\(933\) − 21.2111i − 0.694420i
\(934\) 19.8167 0.648421
\(935\) 0 0
\(936\) 2.39445 0.0782650
\(937\) 44.9638i 1.46890i 0.678660 + 0.734452i \(0.262561\pi\)
−0.678660 + 0.734452i \(0.737439\pi\)
\(938\) − 1.21110i − 0.0395439i
\(939\) 42.0278 1.37152
\(940\) 0 0
\(941\) −20.9361 −0.682497 −0.341248 0.939973i \(-0.610850\pi\)
−0.341248 + 0.939973i \(0.610850\pi\)
\(942\) − 37.6333i − 1.22616i
\(943\) 9.90833i 0.322660i
\(944\) 10.6056 0.345181
\(945\) 0 0
\(946\) 27.6333 0.898436
\(947\) − 41.9361i − 1.36274i −0.731939 0.681370i \(-0.761385\pi\)
0.731939 0.681370i \(-0.238615\pi\)
\(948\) 47.6333i 1.54706i
\(949\) 4.78890 0.155454
\(950\) 0 0
\(951\) −48.6333 −1.57704
\(952\) 1.18335i 0.0383525i
\(953\) − 1.66947i − 0.0540794i −0.999634 0.0270397i \(-0.991392\pi\)
0.999634 0.0270397i \(-0.00860805\pi\)
\(954\) 25.3944 0.822176
\(955\) 0 0
\(956\) −9.21110 −0.297908
\(957\) 80.6611i 2.60740i
\(958\) 30.0000i 0.969256i
\(959\) −2.09167 −0.0675436
\(960\) 0 0
\(961\) −22.5416 −0.727150
\(962\) − 2.42221i − 0.0780950i
\(963\) 36.4222i 1.17369i
\(964\) −14.4222 −0.464508
\(965\) 0 0
\(966\) −1.00000 −0.0321745
\(967\) 5.39445i 0.173474i 0.996231 + 0.0867369i \(0.0276439\pi\)
−0.996231 + 0.0867369i \(0.972356\pi\)
\(968\) − 17.1194i − 0.550239i
\(969\) −63.3583 −2.03536
\(970\) 0 0
\(971\) −27.9083 −0.895621 −0.447810 0.894129i \(-0.647796\pi\)
−0.447810 + 0.894129i \(0.647796\pi\)
\(972\) − 49.8444i − 1.59876i
\(973\) 1.63331i 0.0523614i
\(974\) −11.8167 −0.378630
\(975\) 0 0
\(976\) −6.51388 −0.208504
\(977\) 11.5139i 0.368362i 0.982892 + 0.184181i \(0.0589632\pi\)
−0.982892 + 0.184181i \(0.941037\pi\)
\(978\) − 18.8167i − 0.601690i
\(979\) 0 0
\(980\) 0 0
\(981\) 154.322 4.92713
\(982\) − 25.8167i − 0.823843i
\(983\) − 19.5416i − 0.623281i −0.950200 0.311641i \(-0.899121\pi\)
0.950200 0.311641i \(-0.100879\pi\)
\(984\) −32.7250 −1.04323
\(985\) 0 0
\(986\) 18.0000 0.573237
\(987\) 4.60555i 0.146596i
\(988\) 1.48612i 0.0472798i
\(989\) 5.21110 0.165703
\(990\) 0 0
\(991\) 24.3305 0.772885 0.386442 0.922314i \(-0.373704\pi\)
0.386442 + 0.922314i \(0.373704\pi\)
\(992\) 2.90833i 0.0923395i
\(993\) 31.0278i 0.984636i
\(994\) 3.84441 0.121937
\(995\) 0 0
\(996\) −10.6056 −0.336050
\(997\) 31.2111i 0.988466i 0.869330 + 0.494233i \(0.164551\pi\)
−0.869330 + 0.494233i \(0.835449\pi\)
\(998\) − 11.6333i − 0.368246i
\(999\) −129.689 −4.10317
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.599.4 4
5.2 odd 4 230.2.a.b.1.2 2
5.3 odd 4 1150.2.a.m.1.1 2
5.4 even 2 inner 1150.2.b.f.599.1 4
15.2 even 4 2070.2.a.w.1.1 2
20.3 even 4 9200.2.a.ca.1.2 2
20.7 even 4 1840.2.a.j.1.1 2
40.27 even 4 7360.2.a.bu.1.2 2
40.37 odd 4 7360.2.a.bc.1.1 2
115.22 even 4 5290.2.a.j.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
230.2.a.b.1.2 2 5.2 odd 4
1150.2.a.m.1.1 2 5.3 odd 4
1150.2.b.f.599.1 4 5.4 even 2 inner
1150.2.b.f.599.4 4 1.1 even 1 trivial
1840.2.a.j.1.1 2 20.7 even 4
2070.2.a.w.1.1 2 15.2 even 4
5290.2.a.j.1.2 2 115.22 even 4
7360.2.a.bc.1.1 2 40.37 odd 4
7360.2.a.bu.1.2 2 40.27 even 4
9200.2.a.ca.1.2 2 20.3 even 4