Properties

Label 350.2.c.a.99.1
Level $350$
Weight $2$
Character 350.99
Analytic conductor $2.795$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 350.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.79476407074\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 350.99
Dual form 350.2.c.a.99.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} -3.00000i q^{3} -1.00000 q^{4} -3.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -6.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} -3.00000i q^{3} -1.00000 q^{4} -3.00000 q^{6} +1.00000i q^{7} +1.00000i q^{8} -6.00000 q^{9} -5.00000 q^{11} +3.00000i q^{12} -6.00000i q^{13} +1.00000 q^{14} +1.00000 q^{16} +1.00000i q^{17} +6.00000i q^{18} +3.00000 q^{19} +3.00000 q^{21} +5.00000i q^{22} +3.00000 q^{24} -6.00000 q^{26} +9.00000i q^{27} -1.00000i q^{28} +6.00000 q^{29} -4.00000 q^{31} -1.00000i q^{32} +15.0000i q^{33} +1.00000 q^{34} +6.00000 q^{36} -8.00000i q^{37} -3.00000i q^{38} -18.0000 q^{39} +11.0000 q^{41} -3.00000i q^{42} -8.00000i q^{43} +5.00000 q^{44} -2.00000i q^{47} -3.00000i q^{48} -1.00000 q^{49} +3.00000 q^{51} +6.00000i q^{52} +4.00000i q^{53} +9.00000 q^{54} -1.00000 q^{56} -9.00000i q^{57} -6.00000i q^{58} -4.00000 q^{59} -2.00000 q^{61} +4.00000i q^{62} -6.00000i q^{63} -1.00000 q^{64} +15.0000 q^{66} -9.00000i q^{67} -1.00000i q^{68} -10.0000 q^{71} -6.00000i q^{72} -7.00000i q^{73} -8.00000 q^{74} -3.00000 q^{76} -5.00000i q^{77} +18.0000i q^{78} +2.00000 q^{79} +9.00000 q^{81} -11.0000i q^{82} +11.0000i q^{83} -3.00000 q^{84} -8.00000 q^{86} -18.0000i q^{87} -5.00000i q^{88} +11.0000 q^{89} +6.00000 q^{91} +12.0000i q^{93} -2.00000 q^{94} -3.00000 q^{96} +10.0000i q^{97} +1.00000i q^{98} +30.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} - 6q^{6} - 12q^{9} + O(q^{10}) \) \( 2q - 2q^{4} - 6q^{6} - 12q^{9} - 10q^{11} + 2q^{14} + 2q^{16} + 6q^{19} + 6q^{21} + 6q^{24} - 12q^{26} + 12q^{29} - 8q^{31} + 2q^{34} + 12q^{36} - 36q^{39} + 22q^{41} + 10q^{44} - 2q^{49} + 6q^{51} + 18q^{54} - 2q^{56} - 8q^{59} - 4q^{61} - 2q^{64} + 30q^{66} - 20q^{71} - 16q^{74} - 6q^{76} + 4q^{79} + 18q^{81} - 6q^{84} - 16q^{86} + 22q^{89} + 12q^{91} - 4q^{94} - 6q^{96} + 60q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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
\(3\) − 3.00000i − 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −3.00000 −1.22474
\(7\) 1.00000i 0.377964i
\(8\) 1.00000i 0.353553i
\(9\) −6.00000 −2.00000
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) 3.00000i 0.866025i
\(13\) − 6.00000i − 1.66410i −0.554700 0.832050i \(-0.687167\pi\)
0.554700 0.832050i \(-0.312833\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.00000i 0.242536i 0.992620 + 0.121268i \(0.0386960\pi\)
−0.992620 + 0.121268i \(0.961304\pi\)
\(18\) 6.00000i 1.41421i
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 0 0
\(21\) 3.00000 0.654654
\(22\) 5.00000i 1.06600i
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 3.00000 0.612372
\(25\) 0 0
\(26\) −6.00000 −1.17670
\(27\) 9.00000i 1.73205i
\(28\) − 1.00000i − 0.188982i
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 15.0000i 2.61116i
\(34\) 1.00000 0.171499
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) − 8.00000i − 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) − 3.00000i − 0.486664i
\(39\) −18.0000 −2.88231
\(40\) 0 0
\(41\) 11.0000 1.71791 0.858956 0.512050i \(-0.171114\pi\)
0.858956 + 0.512050i \(0.171114\pi\)
\(42\) − 3.00000i − 0.462910i
\(43\) − 8.00000i − 1.21999i −0.792406 0.609994i \(-0.791172\pi\)
0.792406 0.609994i \(-0.208828\pi\)
\(44\) 5.00000 0.753778
\(45\) 0 0
\(46\) 0 0
\(47\) − 2.00000i − 0.291730i −0.989305 0.145865i \(-0.953403\pi\)
0.989305 0.145865i \(-0.0465965\pi\)
\(48\) − 3.00000i − 0.433013i
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 6.00000i 0.832050i
\(53\) 4.00000i 0.549442i 0.961524 + 0.274721i \(0.0885855\pi\)
−0.961524 + 0.274721i \(0.911414\pi\)
\(54\) 9.00000 1.22474
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) − 9.00000i − 1.19208i
\(58\) − 6.00000i − 0.787839i
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 4.00000i 0.508001i
\(63\) − 6.00000i − 0.755929i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 15.0000 1.84637
\(67\) − 9.00000i − 1.09952i −0.835321 0.549762i \(-0.814718\pi\)
0.835321 0.549762i \(-0.185282\pi\)
\(68\) − 1.00000i − 0.121268i
\(69\) 0 0
\(70\) 0 0
\(71\) −10.0000 −1.18678 −0.593391 0.804914i \(-0.702211\pi\)
−0.593391 + 0.804914i \(0.702211\pi\)
\(72\) − 6.00000i − 0.707107i
\(73\) − 7.00000i − 0.819288i −0.912245 0.409644i \(-0.865653\pi\)
0.912245 0.409644i \(-0.134347\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) −3.00000 −0.344124
\(77\) − 5.00000i − 0.569803i
\(78\) 18.0000i 2.03810i
\(79\) 2.00000 0.225018 0.112509 0.993651i \(-0.464111\pi\)
0.112509 + 0.993651i \(0.464111\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) − 11.0000i − 1.21475i
\(83\) 11.0000i 1.20741i 0.797209 + 0.603703i \(0.206309\pi\)
−0.797209 + 0.603703i \(0.793691\pi\)
\(84\) −3.00000 −0.327327
\(85\) 0 0
\(86\) −8.00000 −0.862662
\(87\) − 18.0000i − 1.92980i
\(88\) − 5.00000i − 0.533002i
\(89\) 11.0000 1.16600 0.582999 0.812473i \(-0.301879\pi\)
0.582999 + 0.812473i \(0.301879\pi\)
\(90\) 0 0
\(91\) 6.00000 0.628971
\(92\) 0 0
\(93\) 12.0000i 1.24434i
\(94\) −2.00000 −0.206284
\(95\) 0 0
\(96\) −3.00000 −0.306186
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 1.00000i 0.101015i
\(99\) 30.0000 3.01511
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) − 3.00000i − 0.297044i
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) 4.00000 0.388514
\(107\) − 3.00000i − 0.290021i −0.989430 0.145010i \(-0.953678\pi\)
0.989430 0.145010i \(-0.0463216\pi\)
\(108\) − 9.00000i − 0.866025i
\(109\) 18.0000 1.72409 0.862044 0.506834i \(-0.169184\pi\)
0.862044 + 0.506834i \(0.169184\pi\)
\(110\) 0 0
\(111\) −24.0000 −2.27798
\(112\) 1.00000i 0.0944911i
\(113\) − 1.00000i − 0.0940721i −0.998893 0.0470360i \(-0.985022\pi\)
0.998893 0.0470360i \(-0.0149776\pi\)
\(114\) −9.00000 −0.842927
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 36.0000i 3.32820i
\(118\) 4.00000i 0.368230i
\(119\) −1.00000 −0.0916698
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 2.00000i 0.181071i
\(123\) − 33.0000i − 2.97551i
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) −6.00000 −0.534522
\(127\) 14.0000i 1.24230i 0.783692 + 0.621150i \(0.213334\pi\)
−0.783692 + 0.621150i \(0.786666\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −24.0000 −2.11308
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) − 15.0000i − 1.30558i
\(133\) 3.00000i 0.260133i
\(134\) −9.00000 −0.777482
\(135\) 0 0
\(136\) −1.00000 −0.0857493
\(137\) 3.00000i 0.256307i 0.991754 + 0.128154i \(0.0409051\pi\)
−0.991754 + 0.128154i \(0.959095\pi\)
\(138\) 0 0
\(139\) 11.0000 0.933008 0.466504 0.884519i \(-0.345513\pi\)
0.466504 + 0.884519i \(0.345513\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 10.0000i 0.839181i
\(143\) 30.0000i 2.50873i
\(144\) −6.00000 −0.500000
\(145\) 0 0
\(146\) −7.00000 −0.579324
\(147\) 3.00000i 0.247436i
\(148\) 8.00000i 0.657596i
\(149\) −12.0000 −0.983078 −0.491539 0.870855i \(-0.663566\pi\)
−0.491539 + 0.870855i \(0.663566\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 3.00000i 0.243332i
\(153\) − 6.00000i − 0.485071i
\(154\) −5.00000 −0.402911
\(155\) 0 0
\(156\) 18.0000 1.44115
\(157\) − 4.00000i − 0.319235i −0.987179 0.159617i \(-0.948974\pi\)
0.987179 0.159617i \(-0.0510260\pi\)
\(158\) − 2.00000i − 0.159111i
\(159\) 12.0000 0.951662
\(160\) 0 0
\(161\) 0 0
\(162\) − 9.00000i − 0.707107i
\(163\) − 19.0000i − 1.48819i −0.668071 0.744097i \(-0.732880\pi\)
0.668071 0.744097i \(-0.267120\pi\)
\(164\) −11.0000 −0.858956
\(165\) 0 0
\(166\) 11.0000 0.853766
\(167\) − 12.0000i − 0.928588i −0.885681 0.464294i \(-0.846308\pi\)
0.885681 0.464294i \(-0.153692\pi\)
\(168\) 3.00000i 0.231455i
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) −18.0000 −1.37649
\(172\) 8.00000i 0.609994i
\(173\) 2.00000i 0.152057i 0.997106 + 0.0760286i \(0.0242240\pi\)
−0.997106 + 0.0760286i \(0.975776\pi\)
\(174\) −18.0000 −1.36458
\(175\) 0 0
\(176\) −5.00000 −0.376889
\(177\) 12.0000i 0.901975i
\(178\) − 11.0000i − 0.824485i
\(179\) −3.00000 −0.224231 −0.112115 0.993695i \(-0.535763\pi\)
−0.112115 + 0.993695i \(0.535763\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) − 6.00000i − 0.444750i
\(183\) 6.00000i 0.443533i
\(184\) 0 0
\(185\) 0 0
\(186\) 12.0000 0.879883
\(187\) − 5.00000i − 0.365636i
\(188\) 2.00000i 0.145865i
\(189\) −9.00000 −0.654654
\(190\) 0 0
\(191\) −6.00000 −0.434145 −0.217072 0.976156i \(-0.569651\pi\)
−0.217072 + 0.976156i \(0.569651\pi\)
\(192\) 3.00000i 0.216506i
\(193\) − 19.0000i − 1.36765i −0.729646 0.683825i \(-0.760315\pi\)
0.729646 0.683825i \(-0.239685\pi\)
\(194\) 10.0000 0.717958
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 22.0000i 1.56744i 0.621117 + 0.783718i \(0.286679\pi\)
−0.621117 + 0.783718i \(0.713321\pi\)
\(198\) − 30.0000i − 2.13201i
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) −27.0000 −1.90443
\(202\) 0 0
\(203\) 6.00000i 0.421117i
\(204\) −3.00000 −0.210042
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) − 6.00000i − 0.416025i
\(209\) −15.0000 −1.03757
\(210\) 0 0
\(211\) 1.00000 0.0688428 0.0344214 0.999407i \(-0.489041\pi\)
0.0344214 + 0.999407i \(0.489041\pi\)
\(212\) − 4.00000i − 0.274721i
\(213\) 30.0000i 2.05557i
\(214\) −3.00000 −0.205076
\(215\) 0 0
\(216\) −9.00000 −0.612372
\(217\) − 4.00000i − 0.271538i
\(218\) − 18.0000i − 1.21911i
\(219\) −21.0000 −1.41905
\(220\) 0 0
\(221\) 6.00000 0.403604
\(222\) 24.0000i 1.61077i
\(223\) 22.0000i 1.47323i 0.676313 + 0.736614i \(0.263577\pi\)
−0.676313 + 0.736614i \(0.736423\pi\)
\(224\) 1.00000 0.0668153
\(225\) 0 0
\(226\) −1.00000 −0.0665190
\(227\) 28.0000i 1.85843i 0.369546 + 0.929213i \(0.379513\pi\)
−0.369546 + 0.929213i \(0.620487\pi\)
\(228\) 9.00000i 0.596040i
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) −15.0000 −0.986928
\(232\) 6.00000i 0.393919i
\(233\) 6.00000i 0.393073i 0.980497 + 0.196537i \(0.0629694\pi\)
−0.980497 + 0.196537i \(0.937031\pi\)
\(234\) 36.0000 2.35339
\(235\) 0 0
\(236\) 4.00000 0.260378
\(237\) − 6.00000i − 0.389742i
\(238\) 1.00000i 0.0648204i
\(239\) −4.00000 −0.258738 −0.129369 0.991596i \(-0.541295\pi\)
−0.129369 + 0.991596i \(0.541295\pi\)
\(240\) 0 0
\(241\) −5.00000 −0.322078 −0.161039 0.986948i \(-0.551485\pi\)
−0.161039 + 0.986948i \(0.551485\pi\)
\(242\) − 14.0000i − 0.899954i
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 0 0
\(246\) −33.0000 −2.10400
\(247\) − 18.0000i − 1.14531i
\(248\) − 4.00000i − 0.254000i
\(249\) 33.0000 2.09129
\(250\) 0 0
\(251\) 27.0000 1.70422 0.852112 0.523359i \(-0.175321\pi\)
0.852112 + 0.523359i \(0.175321\pi\)
\(252\) 6.00000i 0.377964i
\(253\) 0 0
\(254\) 14.0000 0.878438
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 2.00000i − 0.124757i −0.998053 0.0623783i \(-0.980131\pi\)
0.998053 0.0623783i \(-0.0198685\pi\)
\(258\) 24.0000i 1.49417i
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) −36.0000 −2.22834
\(262\) − 8.00000i − 0.494242i
\(263\) − 10.0000i − 0.616626i −0.951285 0.308313i \(-0.900236\pi\)
0.951285 0.308313i \(-0.0997645\pi\)
\(264\) −15.0000 −0.923186
\(265\) 0 0
\(266\) 3.00000 0.183942
\(267\) − 33.0000i − 2.01957i
\(268\) 9.00000i 0.549762i
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) −6.00000 −0.364474 −0.182237 0.983255i \(-0.558334\pi\)
−0.182237 + 0.983255i \(0.558334\pi\)
\(272\) 1.00000i 0.0606339i
\(273\) − 18.0000i − 1.08941i
\(274\) 3.00000 0.181237
\(275\) 0 0
\(276\) 0 0
\(277\) 30.0000i 1.80253i 0.433273 + 0.901263i \(0.357359\pi\)
−0.433273 + 0.901263i \(0.642641\pi\)
\(278\) − 11.0000i − 0.659736i
\(279\) 24.0000 1.43684
\(280\) 0 0
\(281\) 14.0000 0.835170 0.417585 0.908638i \(-0.362877\pi\)
0.417585 + 0.908638i \(0.362877\pi\)
\(282\) 6.00000i 0.357295i
\(283\) − 13.0000i − 0.772770i −0.922338 0.386385i \(-0.873724\pi\)
0.922338 0.386385i \(-0.126276\pi\)
\(284\) 10.0000 0.593391
\(285\) 0 0
\(286\) 30.0000 1.77394
\(287\) 11.0000i 0.649309i
\(288\) 6.00000i 0.353553i
\(289\) 16.0000 0.941176
\(290\) 0 0
\(291\) 30.0000 1.75863
\(292\) 7.00000i 0.409644i
\(293\) − 14.0000i − 0.817889i −0.912559 0.408944i \(-0.865897\pi\)
0.912559 0.408944i \(-0.134103\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) 8.00000 0.464991
\(297\) − 45.0000i − 2.61116i
\(298\) 12.0000i 0.695141i
\(299\) 0 0
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) − 8.00000i − 0.460348i
\(303\) 0 0
\(304\) 3.00000 0.172062
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) − 13.0000i − 0.741949i −0.928643 0.370975i \(-0.879024\pi\)
0.928643 0.370975i \(-0.120976\pi\)
\(308\) 5.00000i 0.284901i
\(309\) 12.0000 0.682656
\(310\) 0 0
\(311\) 6.00000 0.340229 0.170114 0.985424i \(-0.445586\pi\)
0.170114 + 0.985424i \(0.445586\pi\)
\(312\) − 18.0000i − 1.01905i
\(313\) − 10.0000i − 0.565233i −0.959233 0.282617i \(-0.908798\pi\)
0.959233 0.282617i \(-0.0912024\pi\)
\(314\) −4.00000 −0.225733
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) − 4.00000i − 0.224662i −0.993671 0.112331i \(-0.964168\pi\)
0.993671 0.112331i \(-0.0358318\pi\)
\(318\) − 12.0000i − 0.672927i
\(319\) −30.0000 −1.67968
\(320\) 0 0
\(321\) −9.00000 −0.502331
\(322\) 0 0
\(323\) 3.00000i 0.166924i
\(324\) −9.00000 −0.500000
\(325\) 0 0
\(326\) −19.0000 −1.05231
\(327\) − 54.0000i − 2.98621i
\(328\) 11.0000i 0.607373i
\(329\) 2.00000 0.110264
\(330\) 0 0
\(331\) −17.0000 −0.934405 −0.467202 0.884150i \(-0.654738\pi\)
−0.467202 + 0.884150i \(0.654738\pi\)
\(332\) − 11.0000i − 0.603703i
\(333\) 48.0000i 2.63038i
\(334\) −12.0000 −0.656611
\(335\) 0 0
\(336\) 3.00000 0.163663
\(337\) 29.0000i 1.57973i 0.613280 + 0.789865i \(0.289850\pi\)
−0.613280 + 0.789865i \(0.710150\pi\)
\(338\) 23.0000i 1.25104i
\(339\) −3.00000 −0.162938
\(340\) 0 0
\(341\) 20.0000 1.08306
\(342\) 18.0000i 0.973329i
\(343\) − 1.00000i − 0.0539949i
\(344\) 8.00000 0.431331
\(345\) 0 0
\(346\) 2.00000 0.107521
\(347\) − 19.0000i − 1.01997i −0.860182 0.509987i \(-0.829650\pi\)
0.860182 0.509987i \(-0.170350\pi\)
\(348\) 18.0000i 0.964901i
\(349\) 8.00000 0.428230 0.214115 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) 0 0
\(351\) 54.0000 2.88231
\(352\) 5.00000i 0.266501i
\(353\) − 18.0000i − 0.958043i −0.877803 0.479022i \(-0.840992\pi\)
0.877803 0.479022i \(-0.159008\pi\)
\(354\) 12.0000 0.637793
\(355\) 0 0
\(356\) −11.0000 −0.582999
\(357\) 3.00000i 0.158777i
\(358\) 3.00000i 0.158555i
\(359\) −26.0000 −1.37223 −0.686114 0.727494i \(-0.740685\pi\)
−0.686114 + 0.727494i \(0.740685\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) − 10.0000i − 0.525588i
\(363\) − 42.0000i − 2.20443i
\(364\) −6.00000 −0.314485
\(365\) 0 0
\(366\) 6.00000 0.313625
\(367\) 8.00000i 0.417597i 0.977959 + 0.208798i \(0.0669552\pi\)
−0.977959 + 0.208798i \(0.933045\pi\)
\(368\) 0 0
\(369\) −66.0000 −3.43582
\(370\) 0 0
\(371\) −4.00000 −0.207670
\(372\) − 12.0000i − 0.622171i
\(373\) − 4.00000i − 0.207112i −0.994624 0.103556i \(-0.966978\pi\)
0.994624 0.103556i \(-0.0330221\pi\)
\(374\) −5.00000 −0.258544
\(375\) 0 0
\(376\) 2.00000 0.103142
\(377\) − 36.0000i − 1.85409i
\(378\) 9.00000i 0.462910i
\(379\) −9.00000 −0.462299 −0.231149 0.972918i \(-0.574249\pi\)
−0.231149 + 0.972918i \(0.574249\pi\)
\(380\) 0 0
\(381\) 42.0000 2.15173
\(382\) 6.00000i 0.306987i
\(383\) − 6.00000i − 0.306586i −0.988181 0.153293i \(-0.951012\pi\)
0.988181 0.153293i \(-0.0489878\pi\)
\(384\) 3.00000 0.153093
\(385\) 0 0
\(386\) −19.0000 −0.967075
\(387\) 48.0000i 2.43998i
\(388\) − 10.0000i − 0.507673i
\(389\) −8.00000 −0.405616 −0.202808 0.979219i \(-0.565007\pi\)
−0.202808 + 0.979219i \(0.565007\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) − 1.00000i − 0.0505076i
\(393\) − 24.0000i − 1.21064i
\(394\) 22.0000 1.10834
\(395\) 0 0
\(396\) −30.0000 −1.50756
\(397\) − 10.0000i − 0.501886i −0.968002 0.250943i \(-0.919259\pi\)
0.968002 0.250943i \(-0.0807406\pi\)
\(398\) − 10.0000i − 0.501255i
\(399\) 9.00000 0.450564
\(400\) 0 0
\(401\) 37.0000 1.84769 0.923846 0.382765i \(-0.125028\pi\)
0.923846 + 0.382765i \(0.125028\pi\)
\(402\) 27.0000i 1.34664i
\(403\) 24.0000i 1.19553i
\(404\) 0 0
\(405\) 0 0
\(406\) 6.00000 0.297775
\(407\) 40.0000i 1.98273i
\(408\) 3.00000i 0.148522i
\(409\) 21.0000 1.03838 0.519192 0.854658i \(-0.326233\pi\)
0.519192 + 0.854658i \(0.326233\pi\)
\(410\) 0 0
\(411\) 9.00000 0.443937
\(412\) − 4.00000i − 0.197066i
\(413\) − 4.00000i − 0.196827i
\(414\) 0 0
\(415\) 0 0
\(416\) −6.00000 −0.294174
\(417\) − 33.0000i − 1.61602i
\(418\) 15.0000i 0.733674i
\(419\) 39.0000 1.90527 0.952637 0.304109i \(-0.0983586\pi\)
0.952637 + 0.304109i \(0.0983586\pi\)
\(420\) 0 0
\(421\) 20.0000 0.974740 0.487370 0.873195i \(-0.337956\pi\)
0.487370 + 0.873195i \(0.337956\pi\)
\(422\) − 1.00000i − 0.0486792i
\(423\) 12.0000i 0.583460i
\(424\) −4.00000 −0.194257
\(425\) 0 0
\(426\) 30.0000 1.45350
\(427\) − 2.00000i − 0.0967868i
\(428\) 3.00000i 0.145010i
\(429\) 90.0000 4.34524
\(430\) 0 0
\(431\) −36.0000 −1.73406 −0.867029 0.498257i \(-0.833974\pi\)
−0.867029 + 0.498257i \(0.833974\pi\)
\(432\) 9.00000i 0.433013i
\(433\) − 1.00000i − 0.0480569i −0.999711 0.0240285i \(-0.992351\pi\)
0.999711 0.0240285i \(-0.00764923\pi\)
\(434\) −4.00000 −0.192006
\(435\) 0 0
\(436\) −18.0000 −0.862044
\(437\) 0 0
\(438\) 21.0000i 1.00342i
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 6.00000 0.285714
\(442\) − 6.00000i − 0.285391i
\(443\) 37.0000i 1.75792i 0.476893 + 0.878962i \(0.341763\pi\)
−0.476893 + 0.878962i \(0.658237\pi\)
\(444\) 24.0000 1.13899
\(445\) 0 0
\(446\) 22.0000 1.04173
\(447\) 36.0000i 1.70274i
\(448\) − 1.00000i − 0.0472456i
\(449\) −33.0000 −1.55737 −0.778683 0.627417i \(-0.784112\pi\)
−0.778683 + 0.627417i \(0.784112\pi\)
\(450\) 0 0
\(451\) −55.0000 −2.58985
\(452\) 1.00000i 0.0470360i
\(453\) − 24.0000i − 1.12762i
\(454\) 28.0000 1.31411
\(455\) 0 0
\(456\) 9.00000 0.421464
\(457\) − 25.0000i − 1.16945i −0.811231 0.584725i \(-0.801202\pi\)
0.811231 0.584725i \(-0.198798\pi\)
\(458\) − 14.0000i − 0.654177i
\(459\) −9.00000 −0.420084
\(460\) 0 0
\(461\) −38.0000 −1.76984 −0.884918 0.465746i \(-0.845786\pi\)
−0.884918 + 0.465746i \(0.845786\pi\)
\(462\) 15.0000i 0.697863i
\(463\) 8.00000i 0.371792i 0.982569 + 0.185896i \(0.0595187\pi\)
−0.982569 + 0.185896i \(0.940481\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) 4.00000i 0.185098i 0.995708 + 0.0925490i \(0.0295015\pi\)
−0.995708 + 0.0925490i \(0.970499\pi\)
\(468\) − 36.0000i − 1.66410i
\(469\) 9.00000 0.415581
\(470\) 0 0
\(471\) −12.0000 −0.552931
\(472\) − 4.00000i − 0.184115i
\(473\) 40.0000i 1.83920i
\(474\) −6.00000 −0.275589
\(475\) 0 0
\(476\) 1.00000 0.0458349
\(477\) − 24.0000i − 1.09888i
\(478\) 4.00000i 0.182956i
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) −48.0000 −2.18861
\(482\) 5.00000i 0.227744i
\(483\) 0 0
\(484\) −14.0000 −0.636364
\(485\) 0 0
\(486\) 0 0
\(487\) 34.0000i 1.54069i 0.637629 + 0.770344i \(0.279915\pi\)
−0.637629 + 0.770344i \(0.720085\pi\)
\(488\) − 2.00000i − 0.0905357i
\(489\) −57.0000 −2.57763
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 33.0000i 1.48775i
\(493\) 6.00000i 0.270226i
\(494\) −18.0000 −0.809858
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) − 10.0000i − 0.448561i
\(498\) − 33.0000i − 1.47877i
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) 0 0
\(501\) −36.0000 −1.60836
\(502\) − 27.0000i − 1.20507i
\(503\) − 30.0000i − 1.33763i −0.743427 0.668817i \(-0.766801\pi\)
0.743427 0.668817i \(-0.233199\pi\)
\(504\) 6.00000 0.267261
\(505\) 0 0
\(506\) 0 0
\(507\) 69.0000i 3.06440i
\(508\) − 14.0000i − 0.621150i
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) 7.00000 0.309662
\(512\) − 1.00000i − 0.0441942i
\(513\) 27.0000i 1.19208i
\(514\) −2.00000 −0.0882162
\(515\) 0 0
\(516\) 24.0000 1.05654
\(517\) 10.0000i 0.439799i
\(518\) − 8.00000i − 0.351500i
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 11.0000 0.481919 0.240959 0.970535i \(-0.422538\pi\)
0.240959 + 0.970535i \(0.422538\pi\)
\(522\) 36.0000i 1.57568i
\(523\) 13.0000i 0.568450i 0.958758 + 0.284225i \(0.0917363\pi\)
−0.958758 + 0.284225i \(0.908264\pi\)
\(524\) −8.00000 −0.349482
\(525\) 0 0
\(526\) −10.0000 −0.436021
\(527\) − 4.00000i − 0.174243i
\(528\) 15.0000i 0.652791i
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 24.0000 1.04151
\(532\) − 3.00000i − 0.130066i
\(533\) − 66.0000i − 2.85878i
\(534\) −33.0000 −1.42805
\(535\) 0 0
\(536\) 9.00000 0.388741
\(537\) 9.00000i 0.388379i
\(538\) 18.0000i 0.776035i
\(539\) 5.00000 0.215365
\(540\) 0 0
\(541\) −42.0000 −1.80572 −0.902861 0.429934i \(-0.858537\pi\)
−0.902861 + 0.429934i \(0.858537\pi\)
\(542\) 6.00000i 0.257722i
\(543\) − 30.0000i − 1.28742i
\(544\) 1.00000 0.0428746
\(545\) 0 0
\(546\) −18.0000 −0.770329
\(547\) − 27.0000i − 1.15444i −0.816590 0.577218i \(-0.804138\pi\)
0.816590 0.577218i \(-0.195862\pi\)
\(548\) − 3.00000i − 0.128154i
\(549\) 12.0000 0.512148
\(550\) 0 0
\(551\) 18.0000 0.766826
\(552\) 0 0
\(553\) 2.00000i 0.0850487i
\(554\) 30.0000 1.27458
\(555\) 0 0
\(556\) −11.0000 −0.466504
\(557\) − 4.00000i − 0.169485i −0.996403 0.0847427i \(-0.972993\pi\)
0.996403 0.0847427i \(-0.0270068\pi\)
\(558\) − 24.0000i − 1.01600i
\(559\) −48.0000 −2.03018
\(560\) 0 0
\(561\) −15.0000 −0.633300
\(562\) − 14.0000i − 0.590554i
\(563\) 20.0000i 0.842900i 0.906852 + 0.421450i \(0.138479\pi\)
−0.906852 + 0.421450i \(0.861521\pi\)
\(564\) 6.00000 0.252646
\(565\) 0 0
\(566\) −13.0000 −0.546431
\(567\) 9.00000i 0.377964i
\(568\) − 10.0000i − 0.419591i
\(569\) −21.0000 −0.880366 −0.440183 0.897908i \(-0.645086\pi\)
−0.440183 + 0.897908i \(0.645086\pi\)
\(570\) 0 0
\(571\) 12.0000 0.502184 0.251092 0.967963i \(-0.419210\pi\)
0.251092 + 0.967963i \(0.419210\pi\)
\(572\) − 30.0000i − 1.25436i
\(573\) 18.0000i 0.751961i
\(574\) 11.0000 0.459131
\(575\) 0 0
\(576\) 6.00000 0.250000
\(577\) − 13.0000i − 0.541197i −0.962692 0.270599i \(-0.912778\pi\)
0.962692 0.270599i \(-0.0872216\pi\)
\(578\) − 16.0000i − 0.665512i
\(579\) −57.0000 −2.36884
\(580\) 0 0
\(581\) −11.0000 −0.456357
\(582\) − 30.0000i − 1.24354i
\(583\) − 20.0000i − 0.828315i
\(584\) 7.00000 0.289662
\(585\) 0 0
\(586\) −14.0000 −0.578335
\(587\) 13.0000i 0.536567i 0.963340 + 0.268284i \(0.0864565\pi\)
−0.963340 + 0.268284i \(0.913544\pi\)
\(588\) − 3.00000i − 0.123718i
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) 66.0000 2.71488
\(592\) − 8.00000i − 0.328798i
\(593\) − 39.0000i − 1.60154i −0.598973 0.800769i \(-0.704424\pi\)
0.598973 0.800769i \(-0.295576\pi\)
\(594\) −45.0000 −1.84637
\(595\) 0 0
\(596\) 12.0000 0.491539
\(597\) − 30.0000i − 1.22782i
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) 21.0000 0.856608 0.428304 0.903635i \(-0.359111\pi\)
0.428304 + 0.903635i \(0.359111\pi\)
\(602\) − 8.00000i − 0.326056i
\(603\) 54.0000i 2.19905i
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 0 0
\(607\) − 28.0000i − 1.13648i −0.822861 0.568242i \(-0.807624\pi\)
0.822861 0.568242i \(-0.192376\pi\)
\(608\) − 3.00000i − 0.121666i
\(609\) 18.0000 0.729397
\(610\) 0 0
\(611\) −12.0000 −0.485468
\(612\) 6.00000i 0.242536i
\(613\) 18.0000i 0.727013i 0.931592 + 0.363507i \(0.118421\pi\)
−0.931592 + 0.363507i \(0.881579\pi\)
\(614\) −13.0000 −0.524637
\(615\) 0 0
\(616\) 5.00000 0.201456
\(617\) − 14.0000i − 0.563619i −0.959470 0.281809i \(-0.909065\pi\)
0.959470 0.281809i \(-0.0909346\pi\)
\(618\) − 12.0000i − 0.482711i
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 6.00000i − 0.240578i
\(623\) 11.0000i 0.440706i
\(624\) −18.0000 −0.720577
\(625\) 0 0
\(626\) −10.0000 −0.399680
\(627\) 45.0000i 1.79713i
\(628\) 4.00000i 0.159617i
\(629\) 8.00000 0.318981
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 2.00000i 0.0795557i
\(633\) − 3.00000i − 0.119239i
\(634\) −4.00000 −0.158860
\(635\) 0 0
\(636\) −12.0000 −0.475831
\(637\) 6.00000i 0.237729i
\(638\) 30.0000i 1.18771i
\(639\) 60.0000 2.37356
\(640\) 0 0
\(641\) 2.00000 0.0789953 0.0394976 0.999220i \(-0.487424\pi\)
0.0394976 + 0.999220i \(0.487424\pi\)
\(642\) 9.00000i 0.355202i
\(643\) 16.0000i 0.630978i 0.948929 + 0.315489i \(0.102169\pi\)
−0.948929 + 0.315489i \(0.897831\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 3.00000 0.118033
\(647\) 8.00000i 0.314512i 0.987558 + 0.157256i \(0.0502649\pi\)
−0.987558 + 0.157256i \(0.949735\pi\)
\(648\) 9.00000i 0.353553i
\(649\) 20.0000 0.785069
\(650\) 0 0
\(651\) −12.0000 −0.470317
\(652\) 19.0000i 0.744097i
\(653\) − 28.0000i − 1.09572i −0.836569 0.547862i \(-0.815442\pi\)
0.836569 0.547862i \(-0.184558\pi\)
\(654\) −54.0000 −2.11157
\(655\) 0 0
\(656\) 11.0000 0.429478
\(657\) 42.0000i 1.63858i
\(658\) − 2.00000i − 0.0779681i
\(659\) −1.00000 −0.0389545 −0.0194772 0.999810i \(-0.506200\pi\)
−0.0194772 + 0.999810i \(0.506200\pi\)
\(660\) 0 0
\(661\) −50.0000 −1.94477 −0.972387 0.233373i \(-0.925024\pi\)
−0.972387 + 0.233373i \(0.925024\pi\)
\(662\) 17.0000i 0.660724i
\(663\) − 18.0000i − 0.699062i
\(664\) −11.0000 −0.426883
\(665\) 0 0
\(666\) 48.0000 1.85996
\(667\) 0 0
\(668\) 12.0000i 0.464294i
\(669\) 66.0000 2.55171
\(670\) 0 0
\(671\) 10.0000 0.386046
\(672\) − 3.00000i − 0.115728i
\(673\) 34.0000i 1.31060i 0.755367 + 0.655302i \(0.227459\pi\)
−0.755367 + 0.655302i \(0.772541\pi\)
\(674\) 29.0000 1.11704
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) 48.0000i 1.84479i 0.386248 + 0.922395i \(0.373771\pi\)
−0.386248 + 0.922395i \(0.626229\pi\)
\(678\) 3.00000i 0.115214i
\(679\) −10.0000 −0.383765
\(680\) 0 0
\(681\) 84.0000 3.21889
\(682\) − 20.0000i − 0.765840i
\(683\) − 13.0000i − 0.497431i −0.968577 0.248716i \(-0.919992\pi\)
0.968577 0.248716i \(-0.0800084\pi\)
\(684\) 18.0000 0.688247
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) − 42.0000i − 1.60240i
\(688\) − 8.00000i − 0.304997i
\(689\) 24.0000 0.914327
\(690\) 0 0
\(691\) 49.0000 1.86405 0.932024 0.362397i \(-0.118041\pi\)
0.932024 + 0.362397i \(0.118041\pi\)
\(692\) − 2.00000i − 0.0760286i
\(693\) 30.0000i 1.13961i
\(694\) −19.0000 −0.721230
\(695\) 0 0
\(696\) 18.0000 0.682288
\(697\) 11.0000i 0.416655i
\(698\) − 8.00000i − 0.302804i
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) −32.0000 −1.20862 −0.604312 0.796748i \(-0.706552\pi\)
−0.604312 + 0.796748i \(0.706552\pi\)
\(702\) − 54.0000i − 2.03810i
\(703\) − 24.0000i − 0.905177i
\(704\) 5.00000 0.188445
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 0 0
\(708\) − 12.0000i − 0.450988i
\(709\) −4.00000 −0.150223 −0.0751116 0.997175i \(-0.523931\pi\)
−0.0751116 + 0.997175i \(0.523931\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) 11.0000i 0.412242i
\(713\) 0 0
\(714\) 3.00000 0.112272
\(715\) 0 0
\(716\) 3.00000 0.112115
\(717\) 12.0000i 0.448148i
\(718\) 26.0000i 0.970311i
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) 10.0000i 0.372161i
\(723\) 15.0000i 0.557856i
\(724\) −10.0000 −0.371647
\(725\) 0 0
\(726\) −42.0000 −1.55877
\(727\) − 6.00000i − 0.222528i −0.993791 0.111264i \(-0.964510\pi\)
0.993791 0.111264i \(-0.0354899\pi\)
\(728\) 6.00000i 0.222375i
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 8.00000 0.295891
\(732\) − 6.00000i − 0.221766i
\(733\) 40.0000i 1.47743i 0.674016 + 0.738717i \(0.264568\pi\)
−0.674016 + 0.738717i \(0.735432\pi\)
\(734\) 8.00000 0.295285
\(735\) 0 0
\(736\) 0 0
\(737\) 45.0000i 1.65760i
\(738\) 66.0000i 2.42949i
\(739\) −4.00000 −0.147142 −0.0735712 0.997290i \(-0.523440\pi\)
−0.0735712 + 0.997290i \(0.523440\pi\)
\(740\) 0 0
\(741\) −54.0000 −1.98374
\(742\) 4.00000i 0.146845i
\(743\) − 24.0000i − 0.880475i −0.897881 0.440237i \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) −12.0000 −0.439941
\(745\) 0 0
\(746\) −4.00000 −0.146450
\(747\) − 66.0000i − 2.41481i
\(748\) 5.00000i 0.182818i
\(749\) 3.00000 0.109618
\(750\) 0 0
\(751\) 50.0000 1.82453 0.912263 0.409605i \(-0.134333\pi\)
0.912263 + 0.409605i \(0.134333\pi\)
\(752\) − 2.00000i − 0.0729325i
\(753\) − 81.0000i − 2.95180i
\(754\) −36.0000 −1.31104
\(755\) 0 0
\(756\) 9.00000 0.327327
\(757\) 2.00000i 0.0726912i 0.999339 + 0.0363456i \(0.0115717\pi\)
−0.999339 + 0.0363456i \(0.988428\pi\)
\(758\) 9.00000i 0.326895i
\(759\) 0 0
\(760\) 0 0
\(761\) 27.0000 0.978749 0.489375 0.872074i \(-0.337225\pi\)
0.489375 + 0.872074i \(0.337225\pi\)
\(762\) − 42.0000i − 1.52150i
\(763\) 18.0000i 0.651644i
\(764\) 6.00000 0.217072
\(765\) 0 0
\(766\) −6.00000 −0.216789
\(767\) 24.0000i 0.866590i
\(768\) − 3.00000i − 0.108253i
\(769\) −19.0000 −0.685158 −0.342579 0.939489i \(-0.611300\pi\)
−0.342579 + 0.939489i \(0.611300\pi\)
\(770\) 0 0
\(771\) −6.00000 −0.216085
\(772\) 19.0000i 0.683825i
\(773\) 36.0000i 1.29483i 0.762138 + 0.647415i \(0.224150\pi\)
−0.762138 + 0.647415i \(0.775850\pi\)
\(774\) 48.0000 1.72532
\(775\) 0 0
\(776\) −10.0000 −0.358979
\(777\) − 24.0000i − 0.860995i
\(778\) 8.00000i 0.286814i
\(779\) 33.0000 1.18235
\(780\) 0 0
\(781\) 50.0000 1.78914
\(782\) 0 0
\(783\) 54.0000i 1.92980i
\(784\) −1.00000 −0.0357143
\(785\) 0 0
\(786\) −24.0000 −0.856052
\(787\) − 52.0000i − 1.85360i −0.375555 0.926800i \(-0.622548\pi\)
0.375555 0.926800i \(-0.377452\pi\)
\(788\) − 22.0000i − 0.783718i
\(789\) −30.0000 −1.06803
\(790\) 0 0
\(791\) 1.00000 0.0355559
\(792\) 30.0000i 1.06600i
\(793\) 12.0000i 0.426132i
\(794\) −10.0000 −0.354887
\(795\) 0 0
\(796\) −10.0000 −0.354441
\(797\) − 42.0000i − 1.48772i −0.668338 0.743858i \(-0.732994\pi\)
0.668338 0.743858i \(-0.267006\pi\)
\(798\) − 9.00000i − 0.318597i
\(799\) 2.00000 0.0707549
\(800\) 0 0
\(801\) −66.0000 −2.33200
\(802\) − 37.0000i − 1.30652i
\(803\) 35.0000i 1.23512i
\(804\) 27.0000 0.952217
\(805\) 0 0
\(806\) 24.0000 0.845364
\(807\) 54.0000i 1.90089i
\(808\) 0 0
\(809\) −6.00000 −0.210949 −0.105474 0.994422i \(-0.533636\pi\)
−0.105474 + 0.994422i \(0.533636\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) − 6.00000i − 0.210559i
\(813\) 18.0000i 0.631288i
\(814\) 40.0000 1.40200
\(815\) 0 0
\(816\) 3.00000 0.105021
\(817\) − 24.0000i − 0.839654i
\(818\) − 21.0000i − 0.734248i
\(819\) −36.0000 −1.25794
\(820\) 0 0
\(821\) −24.0000 −0.837606 −0.418803 0.908077i \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) − 9.00000i − 0.313911i
\(823\) 10.0000i 0.348578i 0.984695 + 0.174289i \(0.0557627\pi\)
−0.984695 + 0.174289i \(0.944237\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) −4.00000 −0.139178
\(827\) − 41.0000i − 1.42571i −0.701312 0.712855i \(-0.747402\pi\)
0.701312 0.712855i \(-0.252598\pi\)
\(828\) 0 0
\(829\) 4.00000 0.138926 0.0694629 0.997585i \(-0.477871\pi\)
0.0694629 + 0.997585i \(0.477871\pi\)
\(830\) 0 0
\(831\) 90.0000 3.12207
\(832\) 6.00000i 0.208013i
\(833\) − 1.00000i − 0.0346479i
\(834\) −33.0000 −1.14270
\(835\) 0 0
\(836\) 15.0000 0.518786
\(837\) − 36.0000i − 1.24434i
\(838\) − 39.0000i − 1.34723i
\(839\) −2.00000 −0.0690477 −0.0345238 0.999404i \(-0.510991\pi\)
−0.0345238 + 0.999404i \(0.510991\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) − 20.0000i − 0.689246i
\(843\) − 42.0000i − 1.44656i
\(844\) −1.00000 −0.0344214
\(845\) 0 0
\(846\) 12.0000 0.412568
\(847\) 14.0000i 0.481046i
\(848\) 4.00000i 0.137361i
\(849\) −39.0000 −1.33848
\(850\) 0 0
\(851\) 0 0
\(852\) − 30.0000i − 1.02778i
\(853\) − 34.0000i − 1.16414i −0.813139 0.582069i \(-0.802243\pi\)
0.813139 0.582069i \(-0.197757\pi\)
\(854\) −2.00000 −0.0684386
\(855\) 0 0
\(856\) 3.00000 0.102538
\(857\) − 3.00000i − 0.102478i −0.998686 0.0512390i \(-0.983683\pi\)
0.998686 0.0512390i \(-0.0163170\pi\)
\(858\) − 90.0000i − 3.07255i
\(859\) 51.0000 1.74010 0.870049 0.492966i \(-0.164087\pi\)
0.870049 + 0.492966i \(0.164087\pi\)
\(860\) 0 0
\(861\) 33.0000 1.12464
\(862\) 36.0000i 1.22616i
\(863\) 4.00000i 0.136162i 0.997680 + 0.0680808i \(0.0216876\pi\)
−0.997680 + 0.0680808i \(0.978312\pi\)
\(864\) 9.00000 0.306186
\(865\) 0 0
\(866\) −1.00000 −0.0339814
\(867\) − 48.0000i − 1.63017i
\(868\) 4.00000i 0.135769i
\(869\) −10.0000 −0.339227
\(870\) 0 0
\(871\) −54.0000 −1.82972
\(872\) 18.0000i 0.609557i
\(873\) − 60.0000i − 2.03069i
\(874\) 0 0
\(875\) 0 0
\(876\) 21.0000 0.709524
\(877\) − 32.0000i − 1.08056i −0.841484 0.540282i \(-0.818318\pi\)
0.841484 0.540282i \(-0.181682\pi\)
\(878\) 28.0000i 0.944954i
\(879\) −42.0000 −1.41662
\(880\) 0 0
\(881\) 26.0000 0.875962 0.437981 0.898984i \(-0.355694\pi\)
0.437981 + 0.898984i \(0.355694\pi\)
\(882\) − 6.00000i − 0.202031i
\(883\) 15.0000i 0.504790i 0.967624 + 0.252395i \(0.0812183\pi\)
−0.967624 + 0.252395i \(0.918782\pi\)
\(884\) −6.00000 −0.201802
\(885\) 0 0
\(886\) 37.0000 1.24304
\(887\) 34.0000i 1.14161i 0.821086 + 0.570804i \(0.193368\pi\)
−0.821086 + 0.570804i \(0.806632\pi\)
\(888\) − 24.0000i − 0.805387i
\(889\) −14.0000 −0.469545
\(890\) 0 0
\(891\) −45.0000 −1.50756
\(892\) − 22.0000i − 0.736614i
\(893\) − 6.00000i − 0.200782i
\(894\) 36.0000 1.20402
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 33.0000i 1.10122i
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) −4.00000 −0.133259
\(902\) 55.0000i 1.83130i
\(903\) − 24.0000i − 0.798670i
\(904\) 1.00000 0.0332595
\(905\) 0 0
\(906\) −24.0000 −0.797347
\(907\) 4.00000i 0.132818i 0.997792 + 0.0664089i \(0.0211542\pi\)
−0.997792 + 0.0664089i \(0.978846\pi\)
\(908\) − 28.0000i − 0.929213i
\(909\) 0 0
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) − 9.00000i − 0.298020i
\(913\) − 55.0000i − 1.82023i
\(914\) −25.0000 −0.826927
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) 8.00000i 0.264183i
\(918\) 9.00000i 0.297044i
\(919\) 34.0000 1.12156 0.560778 0.827966i \(-0.310502\pi\)
0.560778 + 0.827966i \(0.310502\pi\)
\(920\) 0 0
\(921\) −39.0000 −1.28509
\(922\) 38.0000i 1.25146i
\(923\) 60.0000i 1.97492i
\(924\) 15.0000 0.493464
\(925\) 0 0
\(926\) 8.00000 0.262896
\(927\) − 24.0000i − 0.788263i
\(928\) − 6.00000i − 0.196960i
\(929\) −46.0000 −1.50921 −0.754606 0.656179i \(-0.772172\pi\)
−0.754606 + 0.656179i \(0.772172\pi\)
\(930\) 0 0
\(931\) −3.00000 −0.0983210
\(932\) − 6.00000i − 0.196537i
\(933\) − 18.0000i − 0.589294i
\(934\) 4.00000 0.130884
\(935\) 0 0
\(936\) −36.0000 −1.17670
\(937\) 7.00000i 0.228680i 0.993442 + 0.114340i \(0.0364753\pi\)
−0.993442 + 0.114340i \(0.963525\pi\)
\(938\) − 9.00000i − 0.293860i
\(939\) −30.0000 −0.979013
\(940\) 0 0
\(941\) 56.0000 1.82555 0.912774 0.408465i \(-0.133936\pi\)
0.912774 + 0.408465i \(0.133936\pi\)
\(942\) 12.0000i 0.390981i
\(943\) 0 0
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 40.0000 1.30051
\(947\) 4.00000i 0.129983i 0.997886 + 0.0649913i \(0.0207020\pi\)
−0.997886 + 0.0649913i \(0.979298\pi\)
\(948\) 6.00000i 0.194871i
\(949\) −42.0000 −1.36338
\(950\) 0 0
\(951\) −12.0000 −0.389127
\(952\) − 1.00000i − 0.0324102i
\(953\) 9.00000i 0.291539i 0.989319 + 0.145769i \(0.0465657\pi\)
−0.989319 + 0.145769i \(0.953434\pi\)
\(954\) −24.0000 −0.777029
\(955\) 0 0
\(956\) 4.00000 0.129369
\(957\) 90.0000i 2.90929i
\(958\) − 6.00000i − 0.193851i
\(959\) −3.00000 −0.0968751
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 48.0000i 1.54758i
\(963\) 18.0000i 0.580042i
\(964\) 5.00000 0.161039
\(965\) 0 0
\(966\) 0 0
\(967\) 2.00000i 0.0643157i 0.999483 + 0.0321578i \(0.0102379\pi\)
−0.999483 + 0.0321578i \(0.989762\pi\)
\(968\) 14.0000i 0.449977i
\(969\) 9.00000 0.289122
\(970\) 0 0
\(971\) −51.0000 −1.63667 −0.818334 0.574743i \(-0.805102\pi\)
−0.818334 + 0.574743i \(0.805102\pi\)
\(972\) 0 0
\(973\) 11.0000i 0.352644i
\(974\) 34.0000 1.08943
\(975\) 0 0
\(976\) −2.00000 −0.0640184
\(977\) − 21.0000i − 0.671850i −0.941889 0.335925i \(-0.890951\pi\)
0.941889 0.335925i \(-0.109049\pi\)
\(978\) 57.0000i 1.82266i
\(979\) −55.0000 −1.75781
\(980\) 0 0
\(981\) −108.000 −3.44817
\(982\) 12.0000i 0.382935i
\(983\) − 4.00000i − 0.127580i −0.997963 0.0637901i \(-0.979681\pi\)
0.997963 0.0637901i \(-0.0203188\pi\)
\(984\) 33.0000 1.05200
\(985\) 0 0
\(986\) 6.00000 0.191079
\(987\) − 6.00000i − 0.190982i
\(988\) 18.0000i 0.572656i
\(989\) 0 0
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 4.00000i 0.127000i
\(993\) 51.0000i 1.61844i
\(994\) −10.0000 −0.317181
\(995\) 0 0
\(996\) −33.0000 −1.04565
\(997\) 58.0000i 1.83688i 0.395562 + 0.918439i \(0.370550\pi\)
−0.395562 + 0.918439i \(0.629450\pi\)
\(998\) 36.0000i 1.13956i
\(999\) 72.0000 2.27798
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 350.2.c.a.99.1 2
3.2 odd 2 3150.2.g.v.2899.2 2
4.3 odd 2 2800.2.g.a.449.2 2
5.2 odd 4 350.2.a.d.1.1 yes 1
5.3 odd 4 350.2.a.c.1.1 1
5.4 even 2 inner 350.2.c.a.99.2 2
7.6 odd 2 2450.2.c.r.99.1 2
15.2 even 4 3150.2.a.j.1.1 1
15.8 even 4 3150.2.a.bq.1.1 1
15.14 odd 2 3150.2.g.v.2899.1 2
20.3 even 4 2800.2.a.b.1.1 1
20.7 even 4 2800.2.a.bg.1.1 1
20.19 odd 2 2800.2.g.a.449.1 2
35.13 even 4 2450.2.a.a.1.1 1
35.27 even 4 2450.2.a.bg.1.1 1
35.34 odd 2 2450.2.c.r.99.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
350.2.a.c.1.1 1 5.3 odd 4
350.2.a.d.1.1 yes 1 5.2 odd 4
350.2.c.a.99.1 2 1.1 even 1 trivial
350.2.c.a.99.2 2 5.4 even 2 inner
2450.2.a.a.1.1 1 35.13 even 4
2450.2.a.bg.1.1 1 35.27 even 4
2450.2.c.r.99.1 2 7.6 odd 2
2450.2.c.r.99.2 2 35.34 odd 2
2800.2.a.b.1.1 1 20.3 even 4
2800.2.a.bg.1.1 1 20.7 even 4
2800.2.g.a.449.1 2 20.19 odd 2
2800.2.g.a.449.2 2 4.3 odd 2
3150.2.a.j.1.1 1 15.2 even 4
3150.2.a.bq.1.1 1 15.8 even 4
3150.2.g.v.2899.1 2 15.14 odd 2
3150.2.g.v.2899.2 2 3.2 odd 2