Properties

Label 150.2.c.a.49.1
Level $150$
Weight $2$
Character 150.49
Analytic conductor $1.198$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 150.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.19775603032\)
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: no (minimal twist has level 30)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 150.49
Dual form 150.2.c.a.49.2

$q$-expansion

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

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

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
\(3\) − 1.00000i − 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) − 4.00000i − 1.51186i −0.654654 0.755929i \(-0.727186\pi\)
0.654654 0.755929i \(-0.272814\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 1.00000i 0.288675i
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) −4.00000 −1.06904
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.00000i 1.45521i 0.685994 + 0.727607i \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 1.00000i 0.192450i
\(28\) 4.00000i 0.755929i
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) − 4.00000i − 0.648886i
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 4.00000i 0.617213i
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) −9.00000 −1.28571
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) 2.00000i 0.277350i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 4.00000 0.534522
\(57\) − 4.00000i − 0.529813i
\(58\) − 6.00000i − 0.787839i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) − 8.00000i − 1.01600i
\(63\) 4.00000i 0.503953i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) − 6.00000i − 0.727607i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) − 2.00000i − 0.234082i −0.993127 0.117041i \(-0.962659\pi\)
0.993127 0.117041i \(-0.0373409\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 2.00000i 0.226455i
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000i 0.662589i
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 4.00000 0.436436
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) − 6.00000i − 0.643268i
\(88\) 0 0
\(89\) −18.0000 −1.90800 −0.953998 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) 0 0
\(93\) − 8.00000i − 0.829561i
\(94\) 0 0
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 2.00000i 0.203069i 0.994832 + 0.101535i \(0.0323753\pi\)
−0.994832 + 0.101535i \(0.967625\pi\)
\(98\) 9.00000i 0.909137i
\(99\) 0 0
\(100\) 0 0
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) − 6.00000i − 0.594089i
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) − 4.00000i − 0.377964i
\(113\) 18.0000i 1.69330i 0.532152 + 0.846649i \(0.321383\pi\)
−0.532152 + 0.846649i \(0.678617\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 2.00000i 0.184900i
\(118\) 0 0
\(119\) 24.0000 2.20008
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 10.0000i 0.905357i
\(123\) 6.00000i 0.541002i
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 4.00000 0.356348
\(127\) 20.0000i 1.77471i 0.461084 + 0.887357i \(0.347461\pi\)
−0.461084 + 0.887357i \(0.652539\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) − 16.0000i − 1.38738i
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) 9.00000i 0.742307i
\(148\) − 2.00000i − 0.164399i
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 4.00000i 0.324443i
\(153\) − 6.00000i − 0.485071i
\(154\) 0 0
\(155\) 0 0
\(156\) 2.00000 0.160128
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) 8.00000i 0.636446i
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) − 1.00000i − 0.0785674i
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) − 4.00000i − 0.308607i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) − 4.00000i − 0.304997i
\(173\) − 18.0000i − 1.36851i −0.729241 0.684257i \(-0.760127\pi\)
0.729241 0.684257i \(-0.239873\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 18.0000i 1.34916i
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 8.00000i 0.592999i
\(183\) 10.0000i 0.739221i
\(184\) 0 0
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 0 0
\(188\) 0 0
\(189\) 4.00000 0.290957
\(190\) 0 0
\(191\) −24.0000 −1.73658 −0.868290 0.496058i \(-0.834780\pi\)
−0.868290 + 0.496058i \(0.834780\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 22.0000i 1.58359i 0.610784 + 0.791797i \(0.290854\pi\)
−0.610784 + 0.791797i \(0.709146\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) 9.00000 0.642857
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) − 18.0000i − 1.26648i
\(203\) − 24.0000i − 1.68447i
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) 4.00000 0.278693
\(207\) 0 0
\(208\) − 2.00000i − 0.138675i
\(209\) 0 0
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) − 6.00000i − 0.412082i
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) − 32.0000i − 2.17230i
\(218\) − 10.0000i − 0.677285i
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) − 2.00000i − 0.134231i
\(223\) − 20.0000i − 1.33930i −0.742677 0.669650i \(-0.766444\pi\)
0.742677 0.669650i \(-0.233556\pi\)
\(224\) −4.00000 −0.267261
\(225\) 0 0
\(226\) 18.0000 1.19734
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 4.00000i 0.264906i
\(229\) 10.0000 0.660819 0.330409 0.943838i \(-0.392813\pi\)
0.330409 + 0.943838i \(0.392813\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000i 0.393919i
\(233\) 18.0000i 1.17922i 0.807688 + 0.589610i \(0.200718\pi\)
−0.807688 + 0.589610i \(0.799282\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 0 0
\(237\) 8.00000i 0.519656i
\(238\) − 24.0000i − 1.55569i
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 11.0000i 0.707107i
\(243\) − 1.00000i − 0.0641500i
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) − 8.00000i − 0.509028i
\(248\) 8.00000i 0.508001i
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) − 4.00000i − 0.251976i
\(253\) 0 0
\(254\) 20.0000 1.25491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) − 4.00000i − 0.249029i
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −16.0000 −0.981023
\(267\) 18.0000i 1.10158i
\(268\) 4.00000i 0.244339i
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 6.00000i 0.363803i
\(273\) 8.00000i 0.484182i
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000i 0.120168i 0.998193 + 0.0600842i \(0.0191369\pi\)
−0.998193 + 0.0600842i \(0.980863\pi\)
\(278\) − 4.00000i − 0.239904i
\(279\) −8.00000 −0.478947
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) 28.0000i 1.66443i 0.554455 + 0.832214i \(0.312927\pi\)
−0.554455 + 0.832214i \(0.687073\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 24.0000i 1.41668i
\(288\) 1.00000i 0.0589256i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) 2.00000i 0.117041i
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) 9.00000 0.524891
\(295\) 0 0
\(296\) −2.00000 −0.116248
\(297\) 0 0
\(298\) − 6.00000i − 0.347571i
\(299\) 0 0
\(300\) 0 0
\(301\) 16.0000 0.922225
\(302\) − 8.00000i − 0.460348i
\(303\) − 18.0000i − 1.03407i
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) −6.00000 −0.342997
\(307\) 20.0000i 1.14146i 0.821138 + 0.570730i \(0.193340\pi\)
−0.821138 + 0.570730i \(0.806660\pi\)
\(308\) 0 0
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) − 2.00000i − 0.113228i
\(313\) − 2.00000i − 0.113047i −0.998401 0.0565233i \(-0.981998\pi\)
0.998401 0.0565233i \(-0.0180015\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) − 6.00000i − 0.336463i
\(319\) 0 0
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) 24.0000i 1.33540i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) − 10.0000i − 0.553001i
\(328\) − 6.00000i − 0.331295i
\(329\) 0 0
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 12.0000i 0.658586i
\(333\) − 2.00000i − 0.109599i
\(334\) 0 0
\(335\) 0 0
\(336\) −4.00000 −0.218218
\(337\) 26.0000i 1.41631i 0.706057 + 0.708155i \(0.250472\pi\)
−0.706057 + 0.708155i \(0.749528\pi\)
\(338\) − 9.00000i − 0.489535i
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) 0 0
\(342\) 4.00000i 0.216295i
\(343\) 8.00000i 0.431959i
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) 12.0000i 0.644194i 0.946707 + 0.322097i \(0.104388\pi\)
−0.946707 + 0.322097i \(0.895612\pi\)
\(348\) 6.00000i 0.321634i
\(349\) 10.0000 0.535288 0.267644 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 2.00000 0.106752
\(352\) 0 0
\(353\) − 6.00000i − 0.319348i −0.987170 0.159674i \(-0.948956\pi\)
0.987170 0.159674i \(-0.0510443\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 18.0000 0.953998
\(357\) − 24.0000i − 1.27021i
\(358\) 24.0000i 1.26844i
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) − 14.0000i − 0.735824i
\(363\) 11.0000i 0.577350i
\(364\) 8.00000 0.419314
\(365\) 0 0
\(366\) 10.0000 0.522708
\(367\) − 28.0000i − 1.46159i −0.682598 0.730794i \(-0.739150\pi\)
0.682598 0.730794i \(-0.260850\pi\)
\(368\) 0 0
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 24.0000 1.24602
\(372\) 8.00000i 0.414781i
\(373\) − 26.0000i − 1.34623i −0.739538 0.673114i \(-0.764956\pi\)
0.739538 0.673114i \(-0.235044\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 12.0000i − 0.618031i
\(378\) − 4.00000i − 0.205738i
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) 20.0000 1.02463
\(382\) 24.0000i 1.22795i
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 22.0000 1.11977
\(387\) − 4.00000i − 0.203331i
\(388\) − 2.00000i − 0.101535i
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) − 9.00000i − 0.454569i
\(393\) 0 0
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) 0 0
\(397\) − 22.0000i − 1.10415i −0.833795 0.552074i \(-0.813837\pi\)
0.833795 0.552074i \(-0.186163\pi\)
\(398\) 8.00000i 0.401004i
\(399\) −16.0000 −0.801002
\(400\) 0 0
\(401\) −6.00000 −0.299626 −0.149813 0.988714i \(-0.547867\pi\)
−0.149813 + 0.988714i \(0.547867\pi\)
\(402\) 4.00000i 0.199502i
\(403\) − 16.0000i − 0.797017i
\(404\) −18.0000 −0.895533
\(405\) 0 0
\(406\) −24.0000 −1.19110
\(407\) 0 0
\(408\) 6.00000i 0.297044i
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) 6.00000 0.295958
\(412\) − 4.00000i − 0.197066i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) − 4.00000i − 0.195881i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) − 20.0000i − 0.973585i
\(423\) 0 0
\(424\) −6.00000 −0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) 40.0000i 1.93574i
\(428\) 12.0000i 0.580042i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) − 26.0000i − 1.24948i −0.780833 0.624740i \(-0.785205\pi\)
0.780833 0.624740i \(-0.214795\pi\)
\(434\) −32.0000 −1.53605
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 0 0
\(438\) 2.00000i 0.0955637i
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) 9.00000 0.428571
\(442\) − 12.0000i − 0.570782i
\(443\) − 12.0000i − 0.570137i −0.958507 0.285069i \(-0.907984\pi\)
0.958507 0.285069i \(-0.0920164\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) −20.0000 −0.947027
\(447\) − 6.00000i − 0.283790i
\(448\) 4.00000i 0.188982i
\(449\) 6.00000 0.283158 0.141579 0.989927i \(-0.454782\pi\)
0.141579 + 0.989927i \(0.454782\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) − 18.0000i − 0.846649i
\(453\) − 8.00000i − 0.375873i
\(454\) −12.0000 −0.563188
\(455\) 0 0
\(456\) 4.00000 0.187317
\(457\) 26.0000i 1.21623i 0.793849 + 0.608114i \(0.208074\pi\)
−0.793849 + 0.608114i \(0.791926\pi\)
\(458\) − 10.0000i − 0.467269i
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) 4.00000i 0.185896i 0.995671 + 0.0929479i \(0.0296290\pi\)
−0.995671 + 0.0929479i \(0.970371\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 18.0000 0.833834
\(467\) − 36.0000i − 1.66588i −0.553362 0.832941i \(-0.686655\pi\)
0.553362 0.832941i \(-0.313345\pi\)
\(468\) − 2.00000i − 0.0924500i
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) 0 0
\(474\) 8.00000 0.367452
\(475\) 0 0
\(476\) −24.0000 −1.10004
\(477\) − 6.00000i − 0.274721i
\(478\) 24.0000i 1.09773i
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) − 2.00000i − 0.0910975i
\(483\) 0 0
\(484\) 11.0000 0.500000
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) − 28.0000i − 1.26880i −0.773004 0.634401i \(-0.781247\pi\)
0.773004 0.634401i \(-0.218753\pi\)
\(488\) − 10.0000i − 0.452679i
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) 24.0000 1.08310 0.541552 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) − 6.00000i − 0.270501i
\(493\) 36.0000i 1.62136i
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 12.0000i 0.537733i
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 24.0000i 1.07117i
\(503\) − 24.0000i − 1.07011i −0.844818 0.535054i \(-0.820291\pi\)
0.844818 0.535054i \(-0.179709\pi\)
\(504\) −4.00000 −0.178174
\(505\) 0 0
\(506\) 0 0
\(507\) − 9.00000i − 0.399704i
\(508\) − 20.0000i − 0.887357i
\(509\) 6.00000 0.265945 0.132973 0.991120i \(-0.457548\pi\)
0.132973 + 0.991120i \(0.457548\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) − 1.00000i − 0.0441942i
\(513\) 4.00000i 0.176604i
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) 0 0
\(518\) − 8.00000i − 0.351500i
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) 6.00000i 0.262613i
\(523\) − 20.0000i − 0.874539i −0.899331 0.437269i \(-0.855946\pi\)
0.899331 0.437269i \(-0.144054\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 48.0000i 2.09091i
\(528\) 0 0
\(529\) 23.0000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 16.0000i 0.693688i
\(533\) 12.0000i 0.519778i
\(534\) 18.0000 0.778936
\(535\) 0 0
\(536\) 4.00000 0.172774
\(537\) 24.0000i 1.03568i
\(538\) − 6.00000i − 0.258678i
\(539\) 0 0
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 16.0000i 0.687259i
\(543\) − 14.0000i − 0.600798i
\(544\) 6.00000 0.257248
\(545\) 0 0
\(546\) 8.00000 0.342368
\(547\) − 28.0000i − 1.19719i −0.801050 0.598597i \(-0.795725\pi\)
0.801050 0.598597i \(-0.204275\pi\)
\(548\) − 6.00000i − 0.256307i
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) 24.0000 1.02243
\(552\) 0 0
\(553\) 32.0000i 1.36078i
\(554\) 2.00000 0.0849719
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 18.0000i 0.762684i 0.924434 + 0.381342i \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) 8.00000i 0.338667i
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) − 18.0000i − 0.759284i
\(563\) 12.0000i 0.505740i 0.967500 + 0.252870i \(0.0813744\pi\)
−0.967500 + 0.252870i \(0.918626\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 28.0000 1.17693
\(567\) − 4.00000i − 0.167984i
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) 24.0000i 1.00261i
\(574\) 24.0000 1.00174
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 2.00000i 0.0832611i 0.999133 + 0.0416305i \(0.0132552\pi\)
−0.999133 + 0.0416305i \(0.986745\pi\)
\(578\) 19.0000i 0.790296i
\(579\) 22.0000 0.914289
\(580\) 0 0
\(581\) −48.0000 −1.99138
\(582\) − 2.00000i − 0.0829027i
\(583\) 0 0
\(584\) 2.00000 0.0827606
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) − 12.0000i − 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) − 9.00000i − 0.371154i
\(589\) 32.0000 1.31854
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 2.00000i 0.0821995i
\(593\) − 30.0000i − 1.23195i −0.787765 0.615976i \(-0.788762\pi\)
0.787765 0.615976i \(-0.211238\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 8.00000i 0.327418i
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) − 16.0000i − 0.652111i
\(603\) 4.00000i 0.162893i
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) −18.0000 −0.731200
\(607\) − 4.00000i − 0.162355i −0.996700 0.0811775i \(-0.974132\pi\)
0.996700 0.0811775i \(-0.0258681\pi\)
\(608\) − 4.00000i − 0.162221i
\(609\) −24.0000 −0.972529
\(610\) 0 0
\(611\) 0 0
\(612\) 6.00000i 0.242536i
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) 20.0000 0.807134
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000i 1.20775i 0.797077 + 0.603877i \(0.206378\pi\)
−0.797077 + 0.603877i \(0.793622\pi\)
\(618\) − 4.00000i − 0.160904i
\(619\) −44.0000 −1.76851 −0.884255 0.467005i \(-0.845333\pi\)
−0.884255 + 0.467005i \(0.845333\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 72.0000i 2.88462i
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) −2.00000 −0.0799361
\(627\) 0 0
\(628\) − 2.00000i − 0.0798087i
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 32.0000 1.27390 0.636950 0.770905i \(-0.280196\pi\)
0.636950 + 0.770905i \(0.280196\pi\)
\(632\) − 8.00000i − 0.318223i
\(633\) − 20.0000i − 0.794929i
\(634\) 18.0000 0.714871
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 18.0000i 0.713186i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 12.0000i 0.473602i
\(643\) 4.00000i 0.157745i 0.996885 + 0.0788723i \(0.0251319\pi\)
−0.996885 + 0.0788723i \(0.974868\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) 24.0000i 0.943537i 0.881722 + 0.471769i \(0.156384\pi\)
−0.881722 + 0.471769i \(0.843616\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) −32.0000 −1.25418
\(652\) − 4.00000i − 0.156652i
\(653\) − 18.0000i − 0.704394i −0.935926 0.352197i \(-0.885435\pi\)
0.935926 0.352197i \(-0.114565\pi\)
\(654\) −10.0000 −0.391031
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 2.00000i 0.0780274i
\(658\) 0 0
\(659\) −48.0000 −1.86981 −0.934907 0.354892i \(-0.884518\pi\)
−0.934907 + 0.354892i \(0.884518\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 28.0000i 1.08825i
\(663\) − 12.0000i − 0.466041i
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) −2.00000 −0.0774984
\(667\) 0 0
\(668\) 0 0
\(669\) −20.0000 −0.773245
\(670\) 0 0
\(671\) 0 0
\(672\) 4.00000i 0.154303i
\(673\) − 26.0000i − 1.00223i −0.865382 0.501113i \(-0.832924\pi\)
0.865382 0.501113i \(-0.167076\pi\)
\(674\) 26.0000 1.00148
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) − 6.00000i − 0.230599i −0.993331 0.115299i \(-0.963217\pi\)
0.993331 0.115299i \(-0.0367827\pi\)
\(678\) − 18.0000i − 0.691286i
\(679\) 8.00000 0.307012
\(680\) 0 0
\(681\) −12.0000 −0.459841
\(682\) 0 0
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) 8.00000 0.305441
\(687\) − 10.0000i − 0.381524i
\(688\) 4.00000i 0.152499i
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) 44.0000 1.67384 0.836919 0.547326i \(-0.184354\pi\)
0.836919 + 0.547326i \(0.184354\pi\)
\(692\) 18.0000i 0.684257i
\(693\) 0 0
\(694\) 12.0000 0.455514
\(695\) 0 0
\(696\) 6.00000 0.227429
\(697\) − 36.0000i − 1.36360i
\(698\) − 10.0000i − 0.378506i
\(699\) 18.0000 0.680823
\(700\) 0 0
\(701\) −6.00000 −0.226617 −0.113308 0.993560i \(-0.536145\pi\)
−0.113308 + 0.993560i \(0.536145\pi\)
\(702\) − 2.00000i − 0.0754851i
\(703\) 8.00000i 0.301726i
\(704\) 0 0
\(705\) 0 0
\(706\) −6.00000 −0.225813
\(707\) − 72.0000i − 2.70784i
\(708\) 0 0
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) − 18.0000i − 0.674579i
\(713\) 0 0
\(714\) −24.0000 −0.898177
\(715\) 0 0
\(716\) 24.0000 0.896922
\(717\) 24.0000i 0.896296i
\(718\) − 24.0000i − 0.895672i
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 3.00000i 0.111648i
\(723\) − 2.00000i − 0.0743808i
\(724\) −14.0000 −0.520306
\(725\) 0 0
\(726\) 11.0000 0.408248
\(727\) − 28.0000i − 1.03846i −0.854634 0.519231i \(-0.826218\pi\)
0.854634 0.519231i \(-0.173782\pi\)
\(728\) − 8.00000i − 0.296500i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −24.0000 −0.887672
\(732\) − 10.0000i − 0.369611i
\(733\) 22.0000i 0.812589i 0.913742 + 0.406294i \(0.133179\pi\)
−0.913742 + 0.406294i \(0.866821\pi\)
\(734\) −28.0000 −1.03350
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) − 6.00000i − 0.220863i
\(739\) 52.0000 1.91285 0.956425 0.291977i \(-0.0943129\pi\)
0.956425 + 0.291977i \(0.0943129\pi\)
\(740\) 0 0
\(741\) −8.00000 −0.293887
\(742\) − 24.0000i − 0.881068i
\(743\) 24.0000i 0.880475i 0.897881 + 0.440237i \(0.145106\pi\)
−0.897881 + 0.440237i \(0.854894\pi\)
\(744\) 8.00000 0.293294
\(745\) 0 0
\(746\) −26.0000 −0.951928
\(747\) 12.0000i 0.439057i
\(748\) 0 0
\(749\) −48.0000 −1.75388
\(750\) 0 0
\(751\) −40.0000 −1.45962 −0.729810 0.683650i \(-0.760392\pi\)
−0.729810 + 0.683650i \(0.760392\pi\)
\(752\) 0 0
\(753\) 24.0000i 0.874609i
\(754\) −12.0000 −0.437014
\(755\) 0 0
\(756\) −4.00000 −0.145479
\(757\) 2.00000i 0.0726912i 0.999339 + 0.0363456i \(0.0115717\pi\)
−0.999339 + 0.0363456i \(0.988428\pi\)
\(758\) − 4.00000i − 0.145287i
\(759\) 0 0
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) − 20.0000i − 0.724524i
\(763\) − 40.0000i − 1.44810i
\(764\) 24.0000 0.868290
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) − 1.00000i − 0.0360844i
\(769\) −2.00000 −0.0721218 −0.0360609 0.999350i \(-0.511481\pi\)
−0.0360609 + 0.999350i \(0.511481\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) − 22.0000i − 0.791797i
\(773\) − 42.0000i − 1.51064i −0.655359 0.755318i \(-0.727483\pi\)
0.655359 0.755318i \(-0.272517\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) −2.00000 −0.0717958
\(777\) − 8.00000i − 0.286998i
\(778\) − 6.00000i − 0.215110i
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 6.00000i 0.214423i
\(784\) −9.00000 −0.321429
\(785\) 0 0
\(786\) 0 0
\(787\) − 4.00000i − 0.142585i −0.997455 0.0712923i \(-0.977288\pi\)
0.997455 0.0712923i \(-0.0227123\pi\)
\(788\) 6.00000i 0.213741i
\(789\) 0 0
\(790\) 0 0
\(791\) 72.0000 2.56003
\(792\) 0 0
\(793\) 20.0000i 0.710221i
\(794\) −22.0000 −0.780751
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) − 30.0000i − 1.06265i −0.847167 0.531327i \(-0.821693\pi\)
0.847167 0.531327i \(-0.178307\pi\)
\(798\) 16.0000i 0.566394i
\(799\) 0 0
\(800\) 0 0
\(801\) 18.0000 0.635999
\(802\) 6.00000i 0.211867i
\(803\) 0 0
\(804\) 4.00000 0.141069
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) − 6.00000i − 0.211210i
\(808\) 18.0000i 0.633238i
\(809\) 54.0000 1.89854 0.949269 0.314464i \(-0.101825\pi\)
0.949269 + 0.314464i \(0.101825\pi\)
\(810\) 0 0
\(811\) −4.00000 −0.140459 −0.0702295 0.997531i \(-0.522373\pi\)
−0.0702295 + 0.997531i \(0.522373\pi\)
\(812\) 24.0000i 0.842235i
\(813\) 16.0000i 0.561144i
\(814\) 0 0
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) 16.0000i 0.559769i
\(818\) 26.0000i 0.909069i
\(819\) 8.00000 0.279543
\(820\) 0 0
\(821\) 18.0000 0.628204 0.314102 0.949389i \(-0.398297\pi\)
0.314102 + 0.949389i \(0.398297\pi\)
\(822\) − 6.00000i − 0.209274i
\(823\) − 20.0000i − 0.697156i −0.937280 0.348578i \(-0.886665\pi\)
0.937280 0.348578i \(-0.113335\pi\)
\(824\) −4.00000 −0.139347
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 0 0
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 2.00000i 0.0693375i
\(833\) − 54.0000i − 1.87099i
\(834\) −4.00000 −0.138509
\(835\) 0 0
\(836\) 0 0
\(837\) 8.00000i 0.276520i
\(838\) 0 0
\(839\) −24.0000 −0.828572 −0.414286 0.910147i \(-0.635969\pi\)
−0.414286 + 0.910147i \(0.635969\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 10.0000i 0.344623i
\(843\) − 18.0000i − 0.619953i
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) 0 0
\(847\) 44.0000i 1.51186i
\(848\) 6.00000i 0.206041i
\(849\) 28.0000 0.960958
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 46.0000i 1.57501i 0.616308 + 0.787505i \(0.288628\pi\)
−0.616308 + 0.787505i \(0.711372\pi\)
\(854\) 40.0000 1.36877
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) − 18.0000i − 0.614868i −0.951569 0.307434i \(-0.900530\pi\)
0.951569 0.307434i \(-0.0994704\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) 24.0000 0.817918
\(862\) 0 0
\(863\) − 24.0000i − 0.816970i −0.912765 0.408485i \(-0.866057\pi\)
0.912765 0.408485i \(-0.133943\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −26.0000 −0.883516
\(867\) 19.0000i 0.645274i
\(868\) 32.0000i 1.08615i
\(869\) 0 0
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 10.0000i 0.338643i
\(873\) − 2.00000i − 0.0676897i
\(874\) 0 0
\(875\) 0 0
\(876\) 2.00000 0.0675737
\(877\) 2.00000i 0.0675352i 0.999430 + 0.0337676i \(0.0107506\pi\)
−0.999430 + 0.0337676i \(0.989249\pi\)
\(878\) 8.00000i 0.269987i
\(879\) 6.00000 0.202375
\(880\) 0 0
\(881\) −54.0000 −1.81931 −0.909653 0.415369i \(-0.863653\pi\)
−0.909653 + 0.415369i \(0.863653\pi\)
\(882\) − 9.00000i − 0.303046i
\(883\) 4.00000i 0.134611i 0.997732 + 0.0673054i \(0.0214402\pi\)
−0.997732 + 0.0673054i \(0.978560\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) −12.0000 −0.403148
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 2.00000i 0.0671156i
\(889\) 80.0000 2.68311
\(890\) 0 0
\(891\) 0 0
\(892\) 20.0000i 0.669650i
\(893\) 0 0
\(894\) −6.00000 −0.200670
\(895\) 0 0
\(896\) 4.00000 0.133631
\(897\) 0 0
\(898\) − 6.00000i − 0.200223i
\(899\) 48.0000 1.60089
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) − 16.0000i − 0.532447i
\(904\) −18.0000 −0.598671
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) 44.0000i 1.46100i 0.682915 + 0.730498i \(0.260712\pi\)
−0.682915 + 0.730498i \(0.739288\pi\)
\(908\) 12.0000i 0.398234i
\(909\) −18.0000 −0.597022
\(910\) 0 0
\(911\) −48.0000 −1.59031 −0.795155 0.606406i \(-0.792611\pi\)
−0.795155 + 0.606406i \(0.792611\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) 0 0
\(914\) 26.0000 0.860004
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) 6.00000i 0.198030i
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 30.0000i 0.987997i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 4.00000 0.131448
\(927\) − 4.00000i − 0.131377i
\(928\) − 6.00000i − 0.196960i
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) −36.0000 −1.17985
\(932\) − 18.0000i − 0.589610i
\(933\) 0 0
\(934\) −36.0000 −1.17796
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 26.0000i 0.849383i 0.905338 + 0.424691i \(0.139617\pi\)
−0.905338 + 0.424691i \(0.860383\pi\)
\(938\) 16.0000i 0.522419i
\(939\) −2.00000 −0.0652675
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) − 2.00000i − 0.0651635i
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 36.0000i 1.16984i 0.811090 + 0.584921i \(0.198875\pi\)
−0.811090 + 0.584921i \(0.801125\pi\)
\(948\) − 8.00000i − 0.259828i
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 24.0000i 0.777844i
\(953\) − 6.00000i − 0.194359i −0.995267 0.0971795i \(-0.969018\pi\)
0.995267 0.0971795i \(-0.0309821\pi\)
\(954\) −6.00000 −0.194257
\(955\) 0 0
\(956\) 24.0000 0.776215
\(957\) 0 0
\(958\) − 24.0000i − 0.775405i
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) − 4.00000i − 0.128965i
\(963\) 12.0000i 0.386695i
\(964\) −2.00000 −0.0644157
\(965\) 0 0
\(966\) 0 0
\(967\) − 4.00000i − 0.128631i −0.997930 0.0643157i \(-0.979514\pi\)
0.997930 0.0643157i \(-0.0204865\pi\)
\(968\) − 11.0000i − 0.353553i
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) − 16.0000i − 0.512936i
\(974\) −28.0000 −0.897178
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) − 42.0000i − 1.34370i −0.740688 0.671850i \(-0.765500\pi\)
0.740688 0.671850i \(-0.234500\pi\)
\(978\) − 4.00000i − 0.127906i
\(979\) 0 0
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) − 24.0000i − 0.765871i
\(983\) 24.0000i 0.765481i 0.923856 + 0.382741i \(0.125020\pi\)
−0.923856 + 0.382741i \(0.874980\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) 36.0000 1.14647
\(987\) 0 0
\(988\) 8.00000i 0.254514i
\(989\) 0 0
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) − 8.00000i − 0.254000i
\(993\) 28.0000i 0.888553i
\(994\) 0 0
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) 26.0000i 0.823428i 0.911313 + 0.411714i \(0.135070\pi\)
−0.911313 + 0.411714i \(0.864930\pi\)
\(998\) − 4.00000i − 0.126618i
\(999\) −2.00000 −0.0632772
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 150.2.c.a.49.1 2
3.2 odd 2 450.2.c.b.199.2 2
4.3 odd 2 1200.2.f.e.49.2 2
5.2 odd 4 150.2.a.b.1.1 1
5.3 odd 4 30.2.a.a.1.1 1
5.4 even 2 inner 150.2.c.a.49.2 2
8.3 odd 2 4800.2.f.w.3649.1 2
8.5 even 2 4800.2.f.p.3649.2 2
12.11 even 2 3600.2.f.i.2449.2 2
15.2 even 4 450.2.a.d.1.1 1
15.8 even 4 90.2.a.c.1.1 1
15.14 odd 2 450.2.c.b.199.1 2
20.3 even 4 240.2.a.b.1.1 1
20.7 even 4 1200.2.a.k.1.1 1
20.19 odd 2 1200.2.f.e.49.1 2
35.3 even 12 1470.2.i.q.961.1 2
35.13 even 4 1470.2.a.d.1.1 1
35.18 odd 12 1470.2.i.o.961.1 2
35.23 odd 12 1470.2.i.o.361.1 2
35.27 even 4 7350.2.a.ct.1.1 1
35.33 even 12 1470.2.i.q.361.1 2
40.3 even 4 960.2.a.p.1.1 1
40.13 odd 4 960.2.a.e.1.1 1
40.19 odd 2 4800.2.f.w.3649.2 2
40.27 even 4 4800.2.a.d.1.1 1
40.29 even 2 4800.2.f.p.3649.1 2
40.37 odd 4 4800.2.a.cq.1.1 1
45.13 odd 12 810.2.e.l.541.1 2
45.23 even 12 810.2.e.b.541.1 2
45.38 even 12 810.2.e.b.271.1 2
45.43 odd 12 810.2.e.l.271.1 2
55.43 even 4 3630.2.a.w.1.1 1
60.23 odd 4 720.2.a.j.1.1 1
60.47 odd 4 3600.2.a.f.1.1 1
60.59 even 2 3600.2.f.i.2449.1 2
65.8 even 4 5070.2.b.k.1351.1 2
65.18 even 4 5070.2.b.k.1351.2 2
65.38 odd 4 5070.2.a.w.1.1 1
80.3 even 4 3840.2.k.f.1921.1 2
80.13 odd 4 3840.2.k.y.1921.2 2
80.43 even 4 3840.2.k.f.1921.2 2
80.53 odd 4 3840.2.k.y.1921.1 2
85.33 odd 4 8670.2.a.g.1.1 1
105.83 odd 4 4410.2.a.z.1.1 1
120.53 even 4 2880.2.a.a.1.1 1
120.83 odd 4 2880.2.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.2.a.a.1.1 1 5.3 odd 4
90.2.a.c.1.1 1 15.8 even 4
150.2.a.b.1.1 1 5.2 odd 4
150.2.c.a.49.1 2 1.1 even 1 trivial
150.2.c.a.49.2 2 5.4 even 2 inner
240.2.a.b.1.1 1 20.3 even 4
450.2.a.d.1.1 1 15.2 even 4
450.2.c.b.199.1 2 15.14 odd 2
450.2.c.b.199.2 2 3.2 odd 2
720.2.a.j.1.1 1 60.23 odd 4
810.2.e.b.271.1 2 45.38 even 12
810.2.e.b.541.1 2 45.23 even 12
810.2.e.l.271.1 2 45.43 odd 12
810.2.e.l.541.1 2 45.13 odd 12
960.2.a.e.1.1 1 40.13 odd 4
960.2.a.p.1.1 1 40.3 even 4
1200.2.a.k.1.1 1 20.7 even 4
1200.2.f.e.49.1 2 20.19 odd 2
1200.2.f.e.49.2 2 4.3 odd 2
1470.2.a.d.1.1 1 35.13 even 4
1470.2.i.o.361.1 2 35.23 odd 12
1470.2.i.o.961.1 2 35.18 odd 12
1470.2.i.q.361.1 2 35.33 even 12
1470.2.i.q.961.1 2 35.3 even 12
2880.2.a.a.1.1 1 120.53 even 4
2880.2.a.q.1.1 1 120.83 odd 4
3600.2.a.f.1.1 1 60.47 odd 4
3600.2.f.i.2449.1 2 60.59 even 2
3600.2.f.i.2449.2 2 12.11 even 2
3630.2.a.w.1.1 1 55.43 even 4
3840.2.k.f.1921.1 2 80.3 even 4
3840.2.k.f.1921.2 2 80.43 even 4
3840.2.k.y.1921.1 2 80.53 odd 4
3840.2.k.y.1921.2 2 80.13 odd 4
4410.2.a.z.1.1 1 105.83 odd 4
4800.2.a.d.1.1 1 40.27 even 4
4800.2.a.cq.1.1 1 40.37 odd 4
4800.2.f.p.3649.1 2 40.29 even 2
4800.2.f.p.3649.2 2 8.5 even 2
4800.2.f.w.3649.1 2 8.3 odd 2
4800.2.f.w.3649.2 2 40.19 odd 2
5070.2.a.w.1.1 1 65.38 odd 4
5070.2.b.k.1351.1 2 65.8 even 4
5070.2.b.k.1351.2 2 65.18 even 4
7350.2.a.ct.1.1 1 35.27 even 4
8670.2.a.g.1.1 1 85.33 odd 4