Properties

Label 1014.2.b.c.337.1
Level $1014$
Weight $2$
Character 1014.337
Analytic conductor $8.097$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1014.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.09683076496\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1014.337
Dual form 1014.2.b.c.337.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +3.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000 q^{3} -1.00000 q^{4} +3.00000i q^{5} -1.00000i q^{6} -2.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +3.00000 q^{10} -6.00000i q^{11} -1.00000 q^{12} -2.00000 q^{14} +3.00000i q^{15} +1.00000 q^{16} +3.00000 q^{17} -1.00000i q^{18} +2.00000i q^{19} -3.00000i q^{20} -2.00000i q^{21} -6.00000 q^{22} +6.00000 q^{23} +1.00000i q^{24} -4.00000 q^{25} +1.00000 q^{27} +2.00000i q^{28} +3.00000 q^{29} +3.00000 q^{30} -4.00000i q^{31} -1.00000i q^{32} -6.00000i q^{33} -3.00000i q^{34} +6.00000 q^{35} -1.00000 q^{36} +7.00000i q^{37} +2.00000 q^{38} -3.00000 q^{40} -3.00000i q^{41} -2.00000 q^{42} +10.0000 q^{43} +6.00000i q^{44} +3.00000i q^{45} -6.00000i q^{46} -6.00000i q^{47} +1.00000 q^{48} +3.00000 q^{49} +4.00000i q^{50} +3.00000 q^{51} +3.00000 q^{53} -1.00000i q^{54} +18.0000 q^{55} +2.00000 q^{56} +2.00000i q^{57} -3.00000i q^{58} -3.00000i q^{60} -7.00000 q^{61} -4.00000 q^{62} -2.00000i q^{63} -1.00000 q^{64} -6.00000 q^{66} -10.0000i q^{67} -3.00000 q^{68} +6.00000 q^{69} -6.00000i q^{70} +6.00000i q^{71} +1.00000i q^{72} +13.0000i q^{73} +7.00000 q^{74} -4.00000 q^{75} -2.00000i q^{76} -12.0000 q^{77} -4.00000 q^{79} +3.00000i q^{80} +1.00000 q^{81} -3.00000 q^{82} -6.00000i q^{83} +2.00000i q^{84} +9.00000i q^{85} -10.0000i q^{86} +3.00000 q^{87} +6.00000 q^{88} -18.0000i q^{89} +3.00000 q^{90} -6.00000 q^{92} -4.00000i q^{93} -6.00000 q^{94} -6.00000 q^{95} -1.00000i q^{96} +14.0000i q^{97} -3.00000i q^{98} -6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} + 6 q^{10} - 2 q^{12} - 4 q^{14} + 2 q^{16} + 6 q^{17} - 12 q^{22} + 12 q^{23} - 8 q^{25} + 2 q^{27} + 6 q^{29} + 6 q^{30} + 12 q^{35} - 2 q^{36} + 4 q^{38} - 6 q^{40} - 4 q^{42} + 20 q^{43} + 2 q^{48} + 6 q^{49} + 6 q^{51} + 6 q^{53} + 36 q^{55} + 4 q^{56} - 14 q^{61} - 8 q^{62} - 2 q^{64} - 12 q^{66} - 6 q^{68} + 12 q^{69} + 14 q^{74} - 8 q^{75} - 24 q^{77} - 8 q^{79} + 2 q^{81} - 6 q^{82} + 6 q^{87} + 12 q^{88} + 6 q^{90} - 12 q^{92} - 12 q^{94} - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) 3.00000i 1.34164i 0.741620 + 0.670820i \(0.234058\pi\)
−0.741620 + 0.670820i \(0.765942\pi\)
\(6\) − 1.00000i − 0.408248i
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) 3.00000 0.948683
\(11\) − 6.00000i − 1.80907i −0.426401 0.904534i \(-0.640219\pi\)
0.426401 0.904534i \(-0.359781\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −2.00000 −0.534522
\(15\) 3.00000i 0.774597i
\(16\) 1.00000 0.250000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 2.00000i 0.458831i 0.973329 + 0.229416i \(0.0736815\pi\)
−0.973329 + 0.229416i \(0.926318\pi\)
\(20\) − 3.00000i − 0.670820i
\(21\) − 2.00000i − 0.436436i
\(22\) −6.00000 −1.27920
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −4.00000 −0.800000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 2.00000i 0.377964i
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 3.00000 0.547723
\(31\) − 4.00000i − 0.718421i −0.933257 0.359211i \(-0.883046\pi\)
0.933257 0.359211i \(-0.116954\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 6.00000i − 1.04447i
\(34\) − 3.00000i − 0.514496i
\(35\) 6.00000 1.01419
\(36\) −1.00000 −0.166667
\(37\) 7.00000i 1.15079i 0.817875 + 0.575396i \(0.195152\pi\)
−0.817875 + 0.575396i \(0.804848\pi\)
\(38\) 2.00000 0.324443
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) − 3.00000i − 0.468521i −0.972174 0.234261i \(-0.924733\pi\)
0.972174 0.234261i \(-0.0752669\pi\)
\(42\) −2.00000 −0.308607
\(43\) 10.0000 1.52499 0.762493 0.646997i \(-0.223975\pi\)
0.762493 + 0.646997i \(0.223975\pi\)
\(44\) 6.00000i 0.904534i
\(45\) 3.00000i 0.447214i
\(46\) − 6.00000i − 0.884652i
\(47\) − 6.00000i − 0.875190i −0.899172 0.437595i \(-0.855830\pi\)
0.899172 0.437595i \(-0.144170\pi\)
\(48\) 1.00000 0.144338
\(49\) 3.00000 0.428571
\(50\) 4.00000i 0.565685i
\(51\) 3.00000 0.420084
\(52\) 0 0
\(53\) 3.00000 0.412082 0.206041 0.978543i \(-0.433942\pi\)
0.206041 + 0.978543i \(0.433942\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) 18.0000 2.42712
\(56\) 2.00000 0.267261
\(57\) 2.00000i 0.264906i
\(58\) − 3.00000i − 0.393919i
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) − 3.00000i − 0.387298i
\(61\) −7.00000 −0.896258 −0.448129 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) −4.00000 −0.508001
\(63\) − 2.00000i − 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) − 10.0000i − 1.22169i −0.791748 0.610847i \(-0.790829\pi\)
0.791748 0.610847i \(-0.209171\pi\)
\(68\) −3.00000 −0.363803
\(69\) 6.00000 0.722315
\(70\) − 6.00000i − 0.717137i
\(71\) 6.00000i 0.712069i 0.934473 + 0.356034i \(0.115871\pi\)
−0.934473 + 0.356034i \(0.884129\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 13.0000i 1.52153i 0.649025 + 0.760767i \(0.275177\pi\)
−0.649025 + 0.760767i \(0.724823\pi\)
\(74\) 7.00000 0.813733
\(75\) −4.00000 −0.461880
\(76\) − 2.00000i − 0.229416i
\(77\) −12.0000 −1.36753
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 3.00000i 0.335410i
\(81\) 1.00000 0.111111
\(82\) −3.00000 −0.331295
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 2.00000i 0.218218i
\(85\) 9.00000i 0.976187i
\(86\) − 10.0000i − 1.07833i
\(87\) 3.00000 0.321634
\(88\) 6.00000 0.639602
\(89\) − 18.0000i − 1.90800i −0.299813 0.953998i \(-0.596924\pi\)
0.299813 0.953998i \(-0.403076\pi\)
\(90\) 3.00000 0.316228
\(91\) 0 0
\(92\) −6.00000 −0.625543
\(93\) − 4.00000i − 0.414781i
\(94\) −6.00000 −0.618853
\(95\) −6.00000 −0.615587
\(96\) − 1.00000i − 0.102062i
\(97\) 14.0000i 1.42148i 0.703452 + 0.710742i \(0.251641\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) − 3.00000i − 0.303046i
\(99\) − 6.00000i − 0.603023i
\(100\) 4.00000 0.400000
\(101\) −15.0000 −1.49256 −0.746278 0.665635i \(-0.768161\pi\)
−0.746278 + 0.665635i \(0.768161\pi\)
\(102\) − 3.00000i − 0.297044i
\(103\) −14.0000 −1.37946 −0.689730 0.724066i \(-0.742271\pi\)
−0.689730 + 0.724066i \(0.742271\pi\)
\(104\) 0 0
\(105\) 6.00000 0.585540
\(106\) − 3.00000i − 0.291386i
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 14.0000i 1.34096i 0.741929 + 0.670478i \(0.233911\pi\)
−0.741929 + 0.670478i \(0.766089\pi\)
\(110\) − 18.0000i − 1.71623i
\(111\) 7.00000i 0.664411i
\(112\) − 2.00000i − 0.188982i
\(113\) −3.00000 −0.282216 −0.141108 0.989994i \(-0.545067\pi\)
−0.141108 + 0.989994i \(0.545067\pi\)
\(114\) 2.00000 0.187317
\(115\) 18.0000i 1.67851i
\(116\) −3.00000 −0.278543
\(117\) 0 0
\(118\) 0 0
\(119\) − 6.00000i − 0.550019i
\(120\) −3.00000 −0.273861
\(121\) −25.0000 −2.27273
\(122\) 7.00000i 0.633750i
\(123\) − 3.00000i − 0.270501i
\(124\) 4.00000i 0.359211i
\(125\) 3.00000i 0.268328i
\(126\) −2.00000 −0.178174
\(127\) 4.00000 0.354943 0.177471 0.984126i \(-0.443208\pi\)
0.177471 + 0.984126i \(0.443208\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 10.0000 0.880451
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 6.00000i 0.522233i
\(133\) 4.00000 0.346844
\(134\) −10.0000 −0.863868
\(135\) 3.00000i 0.258199i
\(136\) 3.00000i 0.257248i
\(137\) − 9.00000i − 0.768922i −0.923141 0.384461i \(-0.874387\pi\)
0.923141 0.384461i \(-0.125613\pi\)
\(138\) − 6.00000i − 0.510754i
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −6.00000 −0.507093
\(141\) − 6.00000i − 0.505291i
\(142\) 6.00000 0.503509
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 9.00000i 0.747409i
\(146\) 13.0000 1.07589
\(147\) 3.00000 0.247436
\(148\) − 7.00000i − 0.575396i
\(149\) − 9.00000i − 0.737309i −0.929567 0.368654i \(-0.879819\pi\)
0.929567 0.368654i \(-0.120181\pi\)
\(150\) 4.00000i 0.326599i
\(151\) 10.0000i 0.813788i 0.913475 + 0.406894i \(0.133388\pi\)
−0.913475 + 0.406894i \(0.866612\pi\)
\(152\) −2.00000 −0.162221
\(153\) 3.00000 0.242536
\(154\) 12.0000i 0.966988i
\(155\) 12.0000 0.963863
\(156\) 0 0
\(157\) 5.00000 0.399043 0.199522 0.979893i \(-0.436061\pi\)
0.199522 + 0.979893i \(0.436061\pi\)
\(158\) 4.00000i 0.318223i
\(159\) 3.00000 0.237915
\(160\) 3.00000 0.237171
\(161\) − 12.0000i − 0.945732i
\(162\) − 1.00000i − 0.0785674i
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) 3.00000i 0.234261i
\(165\) 18.0000 1.40130
\(166\) −6.00000 −0.465690
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 2.00000 0.154303
\(169\) 0 0
\(170\) 9.00000 0.690268
\(171\) 2.00000i 0.152944i
\(172\) −10.0000 −0.762493
\(173\) 6.00000 0.456172 0.228086 0.973641i \(-0.426753\pi\)
0.228086 + 0.973641i \(0.426753\pi\)
\(174\) − 3.00000i − 0.227429i
\(175\) 8.00000i 0.604743i
\(176\) − 6.00000i − 0.452267i
\(177\) 0 0
\(178\) −18.0000 −1.34916
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) − 3.00000i − 0.223607i
\(181\) 7.00000 0.520306 0.260153 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 0 0
\(183\) −7.00000 −0.517455
\(184\) 6.00000i 0.442326i
\(185\) −21.0000 −1.54395
\(186\) −4.00000 −0.293294
\(187\) − 18.0000i − 1.31629i
\(188\) 6.00000i 0.437595i
\(189\) − 2.00000i − 0.145479i
\(190\) 6.00000i 0.435286i
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) −1.00000 −0.0721688
\(193\) − 23.0000i − 1.65558i −0.561041 0.827788i \(-0.689599\pi\)
0.561041 0.827788i \(-0.310401\pi\)
\(194\) 14.0000 1.00514
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) −6.00000 −0.426401
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) − 4.00000i − 0.282843i
\(201\) − 10.0000i − 0.705346i
\(202\) 15.0000i 1.05540i
\(203\) − 6.00000i − 0.421117i
\(204\) −3.00000 −0.210042
\(205\) 9.00000 0.628587
\(206\) 14.0000i 0.975426i
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) − 6.00000i − 0.414039i
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) −3.00000 −0.206041
\(213\) 6.00000i 0.411113i
\(214\) 6.00000i 0.410152i
\(215\) 30.0000i 2.04598i
\(216\) 1.00000i 0.0680414i
\(217\) −8.00000 −0.543075
\(218\) 14.0000 0.948200
\(219\) 13.0000i 0.878459i
\(220\) −18.0000 −1.21356
\(221\) 0 0
\(222\) 7.00000 0.469809
\(223\) 8.00000i 0.535720i 0.963458 + 0.267860i \(0.0863164\pi\)
−0.963458 + 0.267860i \(0.913684\pi\)
\(224\) −2.00000 −0.133631
\(225\) −4.00000 −0.266667
\(226\) 3.00000i 0.199557i
\(227\) 18.0000i 1.19470i 0.801980 + 0.597351i \(0.203780\pi\)
−0.801980 + 0.597351i \(0.796220\pi\)
\(228\) − 2.00000i − 0.132453i
\(229\) 22.0000i 1.45380i 0.686743 + 0.726900i \(0.259040\pi\)
−0.686743 + 0.726900i \(0.740960\pi\)
\(230\) 18.0000 1.18688
\(231\) −12.0000 −0.789542
\(232\) 3.00000i 0.196960i
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 18.0000 1.17419
\(236\) 0 0
\(237\) −4.00000 −0.259828
\(238\) −6.00000 −0.388922
\(239\) 6.00000i 0.388108i 0.980991 + 0.194054i \(0.0621637\pi\)
−0.980991 + 0.194054i \(0.937836\pi\)
\(240\) 3.00000i 0.193649i
\(241\) 1.00000i 0.0644157i 0.999481 + 0.0322078i \(0.0102538\pi\)
−0.999481 + 0.0322078i \(0.989746\pi\)
\(242\) 25.0000i 1.60706i
\(243\) 1.00000 0.0641500
\(244\) 7.00000 0.448129
\(245\) 9.00000i 0.574989i
\(246\) −3.00000 −0.191273
\(247\) 0 0
\(248\) 4.00000 0.254000
\(249\) − 6.00000i − 0.380235i
\(250\) 3.00000 0.189737
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 2.00000i 0.125988i
\(253\) − 36.0000i − 2.26330i
\(254\) − 4.00000i − 0.250982i
\(255\) 9.00000i 0.563602i
\(256\) 1.00000 0.0625000
\(257\) 3.00000 0.187135 0.0935674 0.995613i \(-0.470173\pi\)
0.0935674 + 0.995613i \(0.470173\pi\)
\(258\) − 10.0000i − 0.622573i
\(259\) 14.0000 0.869918
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) 0 0
\(263\) −6.00000 −0.369976 −0.184988 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) 6.00000 0.369274
\(265\) 9.00000i 0.552866i
\(266\) − 4.00000i − 0.245256i
\(267\) − 18.0000i − 1.10158i
\(268\) 10.0000i 0.610847i
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 3.00000 0.182574
\(271\) 16.0000i 0.971931i 0.873978 + 0.485965i \(0.161532\pi\)
−0.873978 + 0.485965i \(0.838468\pi\)
\(272\) 3.00000 0.181902
\(273\) 0 0
\(274\) −9.00000 −0.543710
\(275\) 24.0000i 1.44725i
\(276\) −6.00000 −0.361158
\(277\) −17.0000 −1.02143 −0.510716 0.859750i \(-0.670619\pi\)
−0.510716 + 0.859750i \(0.670619\pi\)
\(278\) 4.00000i 0.239904i
\(279\) − 4.00000i − 0.239474i
\(280\) 6.00000i 0.358569i
\(281\) − 9.00000i − 0.536895i −0.963294 0.268447i \(-0.913489\pi\)
0.963294 0.268447i \(-0.0865106\pi\)
\(282\) −6.00000 −0.357295
\(283\) −14.0000 −0.832214 −0.416107 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) − 6.00000i − 0.356034i
\(285\) −6.00000 −0.355409
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) − 1.00000i − 0.0589256i
\(289\) −8.00000 −0.470588
\(290\) 9.00000 0.528498
\(291\) 14.0000i 0.820695i
\(292\) − 13.0000i − 0.760767i
\(293\) 21.0000i 1.22683i 0.789760 + 0.613417i \(0.210205\pi\)
−0.789760 + 0.613417i \(0.789795\pi\)
\(294\) − 3.00000i − 0.174964i
\(295\) 0 0
\(296\) −7.00000 −0.406867
\(297\) − 6.00000i − 0.348155i
\(298\) −9.00000 −0.521356
\(299\) 0 0
\(300\) 4.00000 0.230940
\(301\) − 20.0000i − 1.15278i
\(302\) 10.0000 0.575435
\(303\) −15.0000 −0.861727
\(304\) 2.00000i 0.114708i
\(305\) − 21.0000i − 1.20246i
\(306\) − 3.00000i − 0.171499i
\(307\) 10.0000i 0.570730i 0.958419 + 0.285365i \(0.0921148\pi\)
−0.958419 + 0.285365i \(0.907885\pi\)
\(308\) 12.0000 0.683763
\(309\) −14.0000 −0.796432
\(310\) − 12.0000i − 0.681554i
\(311\) 30.0000 1.70114 0.850572 0.525859i \(-0.176256\pi\)
0.850572 + 0.525859i \(0.176256\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) − 5.00000i − 0.282166i
\(315\) 6.00000 0.338062
\(316\) 4.00000 0.225018
\(317\) 3.00000i 0.168497i 0.996445 + 0.0842484i \(0.0268489\pi\)
−0.996445 + 0.0842484i \(0.973151\pi\)
\(318\) − 3.00000i − 0.168232i
\(319\) − 18.0000i − 1.00781i
\(320\) − 3.00000i − 0.167705i
\(321\) −6.00000 −0.334887
\(322\) −12.0000 −0.668734
\(323\) 6.00000i 0.333849i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 4.00000 0.221540
\(327\) 14.0000i 0.774202i
\(328\) 3.00000 0.165647
\(329\) −12.0000 −0.661581
\(330\) − 18.0000i − 0.990867i
\(331\) − 4.00000i − 0.219860i −0.993939 0.109930i \(-0.964937\pi\)
0.993939 0.109930i \(-0.0350627\pi\)
\(332\) 6.00000i 0.329293i
\(333\) 7.00000i 0.383598i
\(334\) 0 0
\(335\) 30.0000 1.63908
\(336\) − 2.00000i − 0.109109i
\(337\) −23.0000 −1.25289 −0.626445 0.779466i \(-0.715491\pi\)
−0.626445 + 0.779466i \(0.715491\pi\)
\(338\) 0 0
\(339\) −3.00000 −0.162938
\(340\) − 9.00000i − 0.488094i
\(341\) −24.0000 −1.29967
\(342\) 2.00000 0.108148
\(343\) − 20.0000i − 1.07990i
\(344\) 10.0000i 0.539164i
\(345\) 18.0000i 0.969087i
\(346\) − 6.00000i − 0.322562i
\(347\) −30.0000 −1.61048 −0.805242 0.592946i \(-0.797965\pi\)
−0.805242 + 0.592946i \(0.797965\pi\)
\(348\) −3.00000 −0.160817
\(349\) 10.0000i 0.535288i 0.963518 + 0.267644i \(0.0862451\pi\)
−0.963518 + 0.267644i \(0.913755\pi\)
\(350\) 8.00000 0.427618
\(351\) 0 0
\(352\) −6.00000 −0.319801
\(353\) − 15.0000i − 0.798369i −0.916871 0.399185i \(-0.869293\pi\)
0.916871 0.399185i \(-0.130707\pi\)
\(354\) 0 0
\(355\) −18.0000 −0.955341
\(356\) 18.0000i 0.953998i
\(357\) − 6.00000i − 0.317554i
\(358\) 6.00000i 0.317110i
\(359\) − 6.00000i − 0.316668i −0.987386 0.158334i \(-0.949388\pi\)
0.987386 0.158334i \(-0.0506123\pi\)
\(360\) −3.00000 −0.158114
\(361\) 15.0000 0.789474
\(362\) − 7.00000i − 0.367912i
\(363\) −25.0000 −1.31216
\(364\) 0 0
\(365\) −39.0000 −2.04135
\(366\) 7.00000i 0.365896i
\(367\) 2.00000 0.104399 0.0521996 0.998637i \(-0.483377\pi\)
0.0521996 + 0.998637i \(0.483377\pi\)
\(368\) 6.00000 0.312772
\(369\) − 3.00000i − 0.156174i
\(370\) 21.0000i 1.09174i
\(371\) − 6.00000i − 0.311504i
\(372\) 4.00000i 0.207390i
\(373\) 29.0000 1.50156 0.750782 0.660551i \(-0.229677\pi\)
0.750782 + 0.660551i \(0.229677\pi\)
\(374\) −18.0000 −0.930758
\(375\) 3.00000i 0.154919i
\(376\) 6.00000 0.309426
\(377\) 0 0
\(378\) −2.00000 −0.102869
\(379\) 20.0000i 1.02733i 0.857991 + 0.513665i \(0.171713\pi\)
−0.857991 + 0.513665i \(0.828287\pi\)
\(380\) 6.00000 0.307794
\(381\) 4.00000 0.204926
\(382\) 12.0000i 0.613973i
\(383\) 24.0000i 1.22634i 0.789950 + 0.613171i \(0.210106\pi\)
−0.789950 + 0.613171i \(0.789894\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) − 36.0000i − 1.83473i
\(386\) −23.0000 −1.17067
\(387\) 10.0000 0.508329
\(388\) − 14.0000i − 0.710742i
\(389\) −39.0000 −1.97738 −0.988689 0.149979i \(-0.952080\pi\)
−0.988689 + 0.149979i \(0.952080\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) 3.00000i 0.151523i
\(393\) 0 0
\(394\) 6.00000 0.302276
\(395\) − 12.0000i − 0.603786i
\(396\) 6.00000i 0.301511i
\(397\) − 14.0000i − 0.702640i −0.936255 0.351320i \(-0.885733\pi\)
0.936255 0.351320i \(-0.114267\pi\)
\(398\) − 10.0000i − 0.501255i
\(399\) 4.00000 0.200250
\(400\) −4.00000 −0.200000
\(401\) 3.00000i 0.149813i 0.997191 + 0.0749064i \(0.0238658\pi\)
−0.997191 + 0.0749064i \(0.976134\pi\)
\(402\) −10.0000 −0.498755
\(403\) 0 0
\(404\) 15.0000 0.746278
\(405\) 3.00000i 0.149071i
\(406\) −6.00000 −0.297775
\(407\) 42.0000 2.08186
\(408\) 3.00000i 0.148522i
\(409\) − 1.00000i − 0.0494468i −0.999694 0.0247234i \(-0.992129\pi\)
0.999694 0.0247234i \(-0.00787051\pi\)
\(410\) − 9.00000i − 0.444478i
\(411\) − 9.00000i − 0.443937i
\(412\) 14.0000 0.689730
\(413\) 0 0
\(414\) − 6.00000i − 0.294884i
\(415\) 18.0000 0.883585
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) − 12.0000i − 0.586939i
\(419\) 24.0000 1.17248 0.586238 0.810139i \(-0.300608\pi\)
0.586238 + 0.810139i \(0.300608\pi\)
\(420\) −6.00000 −0.292770
\(421\) 29.0000i 1.41337i 0.707527 + 0.706687i \(0.249811\pi\)
−0.707527 + 0.706687i \(0.750189\pi\)
\(422\) 16.0000i 0.778868i
\(423\) − 6.00000i − 0.291730i
\(424\) 3.00000i 0.145693i
\(425\) −12.0000 −0.582086
\(426\) 6.00000 0.290701
\(427\) 14.0000i 0.677507i
\(428\) 6.00000 0.290021
\(429\) 0 0
\(430\) 30.0000 1.44673
\(431\) − 6.00000i − 0.289010i −0.989504 0.144505i \(-0.953841\pi\)
0.989504 0.144505i \(-0.0461589\pi\)
\(432\) 1.00000 0.0481125
\(433\) 13.0000 0.624740 0.312370 0.949960i \(-0.398877\pi\)
0.312370 + 0.949960i \(0.398877\pi\)
\(434\) 8.00000i 0.384012i
\(435\) 9.00000i 0.431517i
\(436\) − 14.0000i − 0.670478i
\(437\) 12.0000i 0.574038i
\(438\) 13.0000 0.621164
\(439\) −14.0000 −0.668184 −0.334092 0.942541i \(-0.608430\pi\)
−0.334092 + 0.942541i \(0.608430\pi\)
\(440\) 18.0000i 0.858116i
\(441\) 3.00000 0.142857
\(442\) 0 0
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) − 7.00000i − 0.332205i
\(445\) 54.0000 2.55985
\(446\) 8.00000 0.378811
\(447\) − 9.00000i − 0.425685i
\(448\) 2.00000i 0.0944911i
\(449\) − 18.0000i − 0.849473i −0.905317 0.424736i \(-0.860367\pi\)
0.905317 0.424736i \(-0.139633\pi\)
\(450\) 4.00000i 0.188562i
\(451\) −18.0000 −0.847587
\(452\) 3.00000 0.141108
\(453\) 10.0000i 0.469841i
\(454\) 18.0000 0.844782
\(455\) 0 0
\(456\) −2.00000 −0.0936586
\(457\) 11.0000i 0.514558i 0.966337 + 0.257279i \(0.0828260\pi\)
−0.966337 + 0.257279i \(0.917174\pi\)
\(458\) 22.0000 1.02799
\(459\) 3.00000 0.140028
\(460\) − 18.0000i − 0.839254i
\(461\) 15.0000i 0.698620i 0.937007 + 0.349310i \(0.113584\pi\)
−0.937007 + 0.349310i \(0.886416\pi\)
\(462\) 12.0000i 0.558291i
\(463\) − 38.0000i − 1.76601i −0.469364 0.883005i \(-0.655517\pi\)
0.469364 0.883005i \(-0.344483\pi\)
\(464\) 3.00000 0.139272
\(465\) 12.0000 0.556487
\(466\) − 6.00000i − 0.277945i
\(467\) 18.0000 0.832941 0.416470 0.909149i \(-0.363267\pi\)
0.416470 + 0.909149i \(0.363267\pi\)
\(468\) 0 0
\(469\) −20.0000 −0.923514
\(470\) − 18.0000i − 0.830278i
\(471\) 5.00000 0.230388
\(472\) 0 0
\(473\) − 60.0000i − 2.75880i
\(474\) 4.00000i 0.183726i
\(475\) − 8.00000i − 0.367065i
\(476\) 6.00000i 0.275010i
\(477\) 3.00000 0.137361
\(478\) 6.00000 0.274434
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 3.00000 0.136931
\(481\) 0 0
\(482\) 1.00000 0.0455488
\(483\) − 12.0000i − 0.546019i
\(484\) 25.0000 1.13636
\(485\) −42.0000 −1.90712
\(486\) − 1.00000i − 0.0453609i
\(487\) 2.00000i 0.0906287i 0.998973 + 0.0453143i \(0.0144289\pi\)
−0.998973 + 0.0453143i \(0.985571\pi\)
\(488\) − 7.00000i − 0.316875i
\(489\) 4.00000i 0.180886i
\(490\) 9.00000 0.406579
\(491\) 18.0000 0.812329 0.406164 0.913800i \(-0.366866\pi\)
0.406164 + 0.913800i \(0.366866\pi\)
\(492\) 3.00000i 0.135250i
\(493\) 9.00000 0.405340
\(494\) 0 0
\(495\) 18.0000 0.809040
\(496\) − 4.00000i − 0.179605i
\(497\) 12.0000 0.538274
\(498\) −6.00000 −0.268866
\(499\) 32.0000i 1.43252i 0.697835 + 0.716258i \(0.254147\pi\)
−0.697835 + 0.716258i \(0.745853\pi\)
\(500\) − 3.00000i − 0.134164i
\(501\) 0 0
\(502\) − 12.0000i − 0.535586i
\(503\) 6.00000 0.267527 0.133763 0.991013i \(-0.457294\pi\)
0.133763 + 0.991013i \(0.457294\pi\)
\(504\) 2.00000 0.0890871
\(505\) − 45.0000i − 2.00247i
\(506\) −36.0000 −1.60040
\(507\) 0 0
\(508\) −4.00000 −0.177471
\(509\) 3.00000i 0.132973i 0.997787 + 0.0664863i \(0.0211789\pi\)
−0.997787 + 0.0664863i \(0.978821\pi\)
\(510\) 9.00000 0.398527
\(511\) 26.0000 1.15017
\(512\) − 1.00000i − 0.0441942i
\(513\) 2.00000i 0.0883022i
\(514\) − 3.00000i − 0.132324i
\(515\) − 42.0000i − 1.85074i
\(516\) −10.0000 −0.440225
\(517\) −36.0000 −1.58328
\(518\) − 14.0000i − 0.615125i
\(519\) 6.00000 0.263371
\(520\) 0 0
\(521\) 33.0000 1.44576 0.722878 0.690976i \(-0.242819\pi\)
0.722878 + 0.690976i \(0.242819\pi\)
\(522\) − 3.00000i − 0.131306i
\(523\) −34.0000 −1.48672 −0.743358 0.668894i \(-0.766768\pi\)
−0.743358 + 0.668894i \(0.766768\pi\)
\(524\) 0 0
\(525\) 8.00000i 0.349149i
\(526\) 6.00000i 0.261612i
\(527\) − 12.0000i − 0.522728i
\(528\) − 6.00000i − 0.261116i
\(529\) 13.0000 0.565217
\(530\) 9.00000 0.390935
\(531\) 0 0
\(532\) −4.00000 −0.173422
\(533\) 0 0
\(534\) −18.0000 −0.778936
\(535\) − 18.0000i − 0.778208i
\(536\) 10.0000 0.431934
\(537\) −6.00000 −0.258919
\(538\) − 18.0000i − 0.776035i
\(539\) − 18.0000i − 0.775315i
\(540\) − 3.00000i − 0.129099i
\(541\) − 29.0000i − 1.24681i −0.781900 0.623404i \(-0.785749\pi\)
0.781900 0.623404i \(-0.214251\pi\)
\(542\) 16.0000 0.687259
\(543\) 7.00000 0.300399
\(544\) − 3.00000i − 0.128624i
\(545\) −42.0000 −1.79908
\(546\) 0 0
\(547\) −34.0000 −1.45374 −0.726868 0.686778i \(-0.759025\pi\)
−0.726868 + 0.686778i \(0.759025\pi\)
\(548\) 9.00000i 0.384461i
\(549\) −7.00000 −0.298753
\(550\) 24.0000 1.02336
\(551\) 6.00000i 0.255609i
\(552\) 6.00000i 0.255377i
\(553\) 8.00000i 0.340195i
\(554\) 17.0000i 0.722261i
\(555\) −21.0000 −0.891400
\(556\) 4.00000 0.169638
\(557\) − 3.00000i − 0.127114i −0.997978 0.0635570i \(-0.979756\pi\)
0.997978 0.0635570i \(-0.0202445\pi\)
\(558\) −4.00000 −0.169334
\(559\) 0 0
\(560\) 6.00000 0.253546
\(561\) − 18.0000i − 0.759961i
\(562\) −9.00000 −0.379642
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 6.00000i 0.252646i
\(565\) − 9.00000i − 0.378633i
\(566\) 14.0000i 0.588464i
\(567\) − 2.00000i − 0.0839921i
\(568\) −6.00000 −0.251754
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 6.00000i 0.251312i
\(571\) 22.0000 0.920671 0.460336 0.887745i \(-0.347729\pi\)
0.460336 + 0.887745i \(0.347729\pi\)
\(572\) 0 0
\(573\) −12.0000 −0.501307
\(574\) 6.00000i 0.250435i
\(575\) −24.0000 −1.00087
\(576\) −1.00000 −0.0416667
\(577\) 11.0000i 0.457936i 0.973434 + 0.228968i \(0.0735351\pi\)
−0.973434 + 0.228968i \(0.926465\pi\)
\(578\) 8.00000i 0.332756i
\(579\) − 23.0000i − 0.955847i
\(580\) − 9.00000i − 0.373705i
\(581\) −12.0000 −0.497844
\(582\) 14.0000 0.580319
\(583\) − 18.0000i − 0.745484i
\(584\) −13.0000 −0.537944
\(585\) 0 0
\(586\) 21.0000 0.867502
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) −3.00000 −0.123718
\(589\) 8.00000 0.329634
\(590\) 0 0
\(591\) 6.00000i 0.246807i
\(592\) 7.00000i 0.287698i
\(593\) − 9.00000i − 0.369586i −0.982777 0.184793i \(-0.940839\pi\)
0.982777 0.184793i \(-0.0591614\pi\)
\(594\) −6.00000 −0.246183
\(595\) 18.0000 0.737928
\(596\) 9.00000i 0.368654i
\(597\) 10.0000 0.409273
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) − 4.00000i − 0.163299i
\(601\) −37.0000 −1.50926 −0.754631 0.656150i \(-0.772184\pi\)
−0.754631 + 0.656150i \(0.772184\pi\)
\(602\) −20.0000 −0.815139
\(603\) − 10.0000i − 0.407231i
\(604\) − 10.0000i − 0.406894i
\(605\) − 75.0000i − 3.04918i
\(606\) 15.0000i 0.609333i
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) 2.00000 0.0811107
\(609\) − 6.00000i − 0.243132i
\(610\) −21.0000 −0.850265
\(611\) 0 0
\(612\) −3.00000 −0.121268
\(613\) − 31.0000i − 1.25208i −0.779792 0.626039i \(-0.784675\pi\)
0.779792 0.626039i \(-0.215325\pi\)
\(614\) 10.0000 0.403567
\(615\) 9.00000 0.362915
\(616\) − 12.0000i − 0.483494i
\(617\) − 15.0000i − 0.603877i −0.953327 0.301939i \(-0.902366\pi\)
0.953327 0.301939i \(-0.0976338\pi\)
\(618\) 14.0000i 0.563163i
\(619\) − 8.00000i − 0.321547i −0.986991 0.160774i \(-0.948601\pi\)
0.986991 0.160774i \(-0.0513989\pi\)
\(620\) −12.0000 −0.481932
\(621\) 6.00000 0.240772
\(622\) − 30.0000i − 1.20289i
\(623\) −36.0000 −1.44231
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 10.0000i 0.399680i
\(627\) 12.0000 0.479234
\(628\) −5.00000 −0.199522
\(629\) 21.0000i 0.837325i
\(630\) − 6.00000i − 0.239046i
\(631\) − 20.0000i − 0.796187i −0.917345 0.398094i \(-0.869672\pi\)
0.917345 0.398094i \(-0.130328\pi\)
\(632\) − 4.00000i − 0.159111i
\(633\) −16.0000 −0.635943
\(634\) 3.00000 0.119145
\(635\) 12.0000i 0.476205i
\(636\) −3.00000 −0.118958
\(637\) 0 0
\(638\) −18.0000 −0.712627
\(639\) 6.00000i 0.237356i
\(640\) −3.00000 −0.118585
\(641\) 3.00000 0.118493 0.0592464 0.998243i \(-0.481130\pi\)
0.0592464 + 0.998243i \(0.481130\pi\)
\(642\) 6.00000i 0.236801i
\(643\) − 16.0000i − 0.630978i −0.948929 0.315489i \(-0.897831\pi\)
0.948929 0.315489i \(-0.102169\pi\)
\(644\) 12.0000i 0.472866i
\(645\) 30.0000i 1.18125i
\(646\) 6.00000 0.236067
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 0 0
\(650\) 0 0
\(651\) −8.00000 −0.313545
\(652\) − 4.00000i − 0.156652i
\(653\) 42.0000 1.64359 0.821794 0.569785i \(-0.192974\pi\)
0.821794 + 0.569785i \(0.192974\pi\)
\(654\) 14.0000 0.547443
\(655\) 0 0
\(656\) − 3.00000i − 0.117130i
\(657\) 13.0000i 0.507178i
\(658\) 12.0000i 0.467809i
\(659\) 24.0000 0.934907 0.467454 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) −18.0000 −0.700649
\(661\) − 5.00000i − 0.194477i −0.995261 0.0972387i \(-0.968999\pi\)
0.995261 0.0972387i \(-0.0310010\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) 12.0000i 0.465340i
\(666\) 7.00000 0.271244
\(667\) 18.0000 0.696963
\(668\) 0 0
\(669\) 8.00000i 0.309298i
\(670\) − 30.0000i − 1.15900i
\(671\) 42.0000i 1.62139i
\(672\) −2.00000 −0.0771517
\(673\) 13.0000 0.501113 0.250557 0.968102i \(-0.419386\pi\)
0.250557 + 0.968102i \(0.419386\pi\)
\(674\) 23.0000i 0.885927i
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) 18.0000 0.691796 0.345898 0.938272i \(-0.387574\pi\)
0.345898 + 0.938272i \(0.387574\pi\)
\(678\) 3.00000i 0.115214i
\(679\) 28.0000 1.07454
\(680\) −9.00000 −0.345134
\(681\) 18.0000i 0.689761i
\(682\) 24.0000i 0.919007i
\(683\) 48.0000i 1.83667i 0.395805 + 0.918334i \(0.370466\pi\)
−0.395805 + 0.918334i \(0.629534\pi\)
\(684\) − 2.00000i − 0.0764719i
\(685\) 27.0000 1.03162
\(686\) −20.0000 −0.763604
\(687\) 22.0000i 0.839352i
\(688\) 10.0000 0.381246
\(689\) 0 0
\(690\) 18.0000 0.685248
\(691\) 26.0000i 0.989087i 0.869153 + 0.494543i \(0.164665\pi\)
−0.869153 + 0.494543i \(0.835335\pi\)
\(692\) −6.00000 −0.228086
\(693\) −12.0000 −0.455842
\(694\) 30.0000i 1.13878i
\(695\) − 12.0000i − 0.455186i
\(696\) 3.00000i 0.113715i
\(697\) − 9.00000i − 0.340899i
\(698\) 10.0000 0.378506
\(699\) 6.00000 0.226941
\(700\) − 8.00000i − 0.302372i
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) −14.0000 −0.528020
\(704\) 6.00000i 0.226134i
\(705\) 18.0000 0.677919
\(706\) −15.0000 −0.564532
\(707\) 30.0000i 1.12827i
\(708\) 0 0
\(709\) − 5.00000i − 0.187779i −0.995583 0.0938895i \(-0.970070\pi\)
0.995583 0.0938895i \(-0.0299300\pi\)
\(710\) 18.0000i 0.675528i
\(711\) −4.00000 −0.150012
\(712\) 18.0000 0.674579
\(713\) − 24.0000i − 0.898807i
\(714\) −6.00000 −0.224544
\(715\) 0 0
\(716\) 6.00000 0.224231
\(717\) 6.00000i 0.224074i
\(718\) −6.00000 −0.223918
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 3.00000i 0.111803i
\(721\) 28.0000i 1.04277i
\(722\) − 15.0000i − 0.558242i
\(723\) 1.00000i 0.0371904i
\(724\) −7.00000 −0.260153
\(725\) −12.0000 −0.445669
\(726\) 25.0000i 0.927837i
\(727\) −14.0000 −0.519231 −0.259616 0.965712i \(-0.583596\pi\)
−0.259616 + 0.965712i \(0.583596\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 39.0000i 1.44345i
\(731\) 30.0000 1.10959
\(732\) 7.00000 0.258727
\(733\) − 31.0000i − 1.14501i −0.819901 0.572506i \(-0.805971\pi\)
0.819901 0.572506i \(-0.194029\pi\)
\(734\) − 2.00000i − 0.0738213i
\(735\) 9.00000i 0.331970i
\(736\) − 6.00000i − 0.221163i
\(737\) −60.0000 −2.21013
\(738\) −3.00000 −0.110432
\(739\) 16.0000i 0.588570i 0.955718 + 0.294285i \(0.0950814\pi\)
−0.955718 + 0.294285i \(0.904919\pi\)
\(740\) 21.0000 0.771975
\(741\) 0 0
\(742\) −6.00000 −0.220267
\(743\) − 36.0000i − 1.32071i −0.750953 0.660356i \(-0.770405\pi\)
0.750953 0.660356i \(-0.229595\pi\)
\(744\) 4.00000 0.146647
\(745\) 27.0000 0.989203
\(746\) − 29.0000i − 1.06177i
\(747\) − 6.00000i − 0.219529i
\(748\) 18.0000i 0.658145i
\(749\) 12.0000i 0.438470i
\(750\) 3.00000 0.109545
\(751\) −14.0000 −0.510867 −0.255434 0.966827i \(-0.582218\pi\)
−0.255434 + 0.966827i \(0.582218\pi\)
\(752\) − 6.00000i − 0.218797i
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) −30.0000 −1.09181
\(756\) 2.00000i 0.0727393i
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 20.0000 0.726433
\(759\) − 36.0000i − 1.30672i
\(760\) − 6.00000i − 0.217643i
\(761\) 30.0000i 1.08750i 0.839248 + 0.543750i \(0.182996\pi\)
−0.839248 + 0.543750i \(0.817004\pi\)
\(762\) − 4.00000i − 0.144905i
\(763\) 28.0000 1.01367
\(764\) 12.0000 0.434145
\(765\) 9.00000i 0.325396i
\(766\) 24.0000 0.867155
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) 14.0000i 0.504853i 0.967616 + 0.252426i \(0.0812286\pi\)
−0.967616 + 0.252426i \(0.918771\pi\)
\(770\) −36.0000 −1.29735
\(771\) 3.00000 0.108042
\(772\) 23.0000i 0.827788i
\(773\) − 30.0000i − 1.07903i −0.841978 0.539513i \(-0.818609\pi\)
0.841978 0.539513i \(-0.181391\pi\)
\(774\) − 10.0000i − 0.359443i
\(775\) 16.0000i 0.574737i
\(776\) −14.0000 −0.502571
\(777\) 14.0000 0.502247
\(778\) 39.0000i 1.39822i
\(779\) 6.00000 0.214972
\(780\) 0 0
\(781\) 36.0000 1.28818
\(782\) − 18.0000i − 0.643679i
\(783\) 3.00000 0.107211
\(784\) 3.00000 0.107143
\(785\) 15.0000i 0.535373i
\(786\) 0 0
\(787\) 28.0000i 0.998092i 0.866575 + 0.499046i \(0.166316\pi\)
−0.866575 + 0.499046i \(0.833684\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) −6.00000 −0.213606
\(790\) −12.0000 −0.426941
\(791\) 6.00000i 0.213335i
\(792\) 6.00000 0.213201
\(793\) 0 0
\(794\) −14.0000 −0.496841
\(795\) 9.00000i 0.319197i
\(796\) −10.0000 −0.354441
\(797\) −30.0000 −1.06265 −0.531327 0.847167i \(-0.678307\pi\)
−0.531327 + 0.847167i \(0.678307\pi\)
\(798\) − 4.00000i − 0.141598i
\(799\) − 18.0000i − 0.636794i
\(800\) 4.00000i 0.141421i
\(801\) − 18.0000i − 0.635999i
\(802\) 3.00000 0.105934
\(803\) 78.0000 2.75256
\(804\) 10.0000i 0.352673i
\(805\) 36.0000 1.26883
\(806\) 0 0
\(807\) 18.0000 0.633630
\(808\) − 15.0000i − 0.527698i
\(809\) −51.0000 −1.79306 −0.896532 0.442978i \(-0.853922\pi\)
−0.896532 + 0.442978i \(0.853922\pi\)
\(810\) 3.00000 0.105409
\(811\) − 4.00000i − 0.140459i −0.997531 0.0702295i \(-0.977627\pi\)
0.997531 0.0702295i \(-0.0223732\pi\)
\(812\) 6.00000i 0.210559i
\(813\) 16.0000i 0.561144i
\(814\) − 42.0000i − 1.47210i
\(815\) −12.0000 −0.420342
\(816\) 3.00000 0.105021
\(817\) 20.0000i 0.699711i
\(818\) −1.00000 −0.0349642
\(819\) 0 0
\(820\) −9.00000 −0.314294
\(821\) 18.0000i 0.628204i 0.949389 + 0.314102i \(0.101703\pi\)
−0.949389 + 0.314102i \(0.898297\pi\)
\(822\) −9.00000 −0.313911
\(823\) 40.0000 1.39431 0.697156 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) − 14.0000i − 0.487713i
\(825\) 24.0000i 0.835573i
\(826\) 0 0
\(827\) 48.0000i 1.66912i 0.550914 + 0.834562i \(0.314279\pi\)
−0.550914 + 0.834562i \(0.685721\pi\)
\(828\) −6.00000 −0.208514
\(829\) −17.0000 −0.590434 −0.295217 0.955430i \(-0.595392\pi\)
−0.295217 + 0.955430i \(0.595392\pi\)
\(830\) − 18.0000i − 0.624789i
\(831\) −17.0000 −0.589723
\(832\) 0 0
\(833\) 9.00000 0.311832
\(834\) 4.00000i 0.138509i
\(835\) 0 0
\(836\) −12.0000 −0.415029
\(837\) − 4.00000i − 0.138260i
\(838\) − 24.0000i − 0.829066i
\(839\) 12.0000i 0.414286i 0.978311 + 0.207143i \(0.0664165\pi\)
−0.978311 + 0.207143i \(0.933583\pi\)
\(840\) 6.00000i 0.207020i
\(841\) −20.0000 −0.689655
\(842\) 29.0000 0.999406
\(843\) − 9.00000i − 0.309976i
\(844\) 16.0000 0.550743
\(845\) 0 0
\(846\) −6.00000 −0.206284
\(847\) 50.0000i 1.71802i
\(848\) 3.00000 0.103020
\(849\) −14.0000 −0.480479
\(850\) 12.0000i 0.411597i
\(851\) 42.0000i 1.43974i
\(852\) − 6.00000i − 0.205557i
\(853\) 19.0000i 0.650548i 0.945620 + 0.325274i \(0.105456\pi\)
−0.945620 + 0.325274i \(0.894544\pi\)
\(854\) 14.0000 0.479070
\(855\) −6.00000 −0.205196
\(856\) − 6.00000i − 0.205076i
\(857\) −21.0000 −0.717346 −0.358673 0.933463i \(-0.616771\pi\)
−0.358673 + 0.933463i \(0.616771\pi\)
\(858\) 0 0
\(859\) 26.0000 0.887109 0.443554 0.896248i \(-0.353717\pi\)
0.443554 + 0.896248i \(0.353717\pi\)
\(860\) − 30.0000i − 1.02299i
\(861\) −6.00000 −0.204479
\(862\) −6.00000 −0.204361
\(863\) 18.0000i 0.612727i 0.951915 + 0.306364i \(0.0991123\pi\)
−0.951915 + 0.306364i \(0.900888\pi\)
\(864\) − 1.00000i − 0.0340207i
\(865\) 18.0000i 0.612018i
\(866\) − 13.0000i − 0.441758i
\(867\) −8.00000 −0.271694
\(868\) 8.00000 0.271538
\(869\) 24.0000i 0.814144i
\(870\) 9.00000 0.305129
\(871\) 0 0
\(872\) −14.0000 −0.474100
\(873\) 14.0000i 0.473828i
\(874\) 12.0000 0.405906
\(875\) 6.00000 0.202837
\(876\) − 13.0000i − 0.439229i
\(877\) 41.0000i 1.38447i 0.721671 + 0.692236i \(0.243374\pi\)
−0.721671 + 0.692236i \(0.756626\pi\)
\(878\) 14.0000i 0.472477i
\(879\) 21.0000i 0.708312i
\(880\) 18.0000 0.606780
\(881\) −33.0000 −1.11180 −0.555899 0.831250i \(-0.687626\pi\)
−0.555899 + 0.831250i \(0.687626\pi\)
\(882\) − 3.00000i − 0.101015i
\(883\) −8.00000 −0.269221 −0.134611 0.990899i \(-0.542978\pi\)
−0.134611 + 0.990899i \(0.542978\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 36.0000i 1.20944i
\(887\) −48.0000 −1.61168 −0.805841 0.592132i \(-0.798286\pi\)
−0.805841 + 0.592132i \(0.798286\pi\)
\(888\) −7.00000 −0.234905
\(889\) − 8.00000i − 0.268311i
\(890\) − 54.0000i − 1.81008i
\(891\) − 6.00000i − 0.201008i
\(892\) − 8.00000i − 0.267860i
\(893\) 12.0000 0.401565
\(894\) −9.00000 −0.301005
\(895\) − 18.0000i − 0.601674i
\(896\) 2.00000 0.0668153
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) − 12.0000i − 0.400222i
\(900\) 4.00000 0.133333
\(901\) 9.00000 0.299833
\(902\) 18.0000i 0.599334i
\(903\) − 20.0000i − 0.665558i
\(904\) − 3.00000i − 0.0997785i
\(905\) 21.0000i 0.698064i
\(906\) 10.0000 0.332228
\(907\) −44.0000 −1.46100 −0.730498 0.682915i \(-0.760712\pi\)
−0.730498 + 0.682915i \(0.760712\pi\)
\(908\) − 18.0000i − 0.597351i
\(909\) −15.0000 −0.497519
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 2.00000i 0.0662266i
\(913\) −36.0000 −1.19143
\(914\) 11.0000 0.363848
\(915\) − 21.0000i − 0.694239i
\(916\) − 22.0000i − 0.726900i
\(917\) 0 0
\(918\) − 3.00000i − 0.0990148i
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) −18.0000 −0.593442
\(921\) 10.0000i 0.329511i
\(922\) 15.0000 0.493999
\(923\) 0 0
\(924\) 12.0000 0.394771
\(925\) − 28.0000i − 0.920634i
\(926\) −38.0000 −1.24876
\(927\) −14.0000 −0.459820
\(928\) − 3.00000i − 0.0984798i
\(929\) 33.0000i 1.08269i 0.840799 + 0.541347i \(0.182086\pi\)
−0.840799 + 0.541347i \(0.817914\pi\)
\(930\) − 12.0000i − 0.393496i
\(931\) 6.00000i 0.196642i
\(932\) −6.00000 −0.196537
\(933\) 30.0000 0.982156
\(934\) − 18.0000i − 0.588978i
\(935\) 54.0000 1.76599
\(936\) 0 0
\(937\) 47.0000 1.53542 0.767712 0.640796i \(-0.221395\pi\)
0.767712 + 0.640796i \(0.221395\pi\)
\(938\) 20.0000i 0.653023i
\(939\) −10.0000 −0.326338
\(940\) −18.0000 −0.587095
\(941\) 42.0000i 1.36916i 0.728937 + 0.684580i \(0.240015\pi\)
−0.728937 + 0.684580i \(0.759985\pi\)
\(942\) − 5.00000i − 0.162909i
\(943\) − 18.0000i − 0.586161i
\(944\) 0 0
\(945\) 6.00000 0.195180
\(946\) −60.0000 −1.95077
\(947\) − 24.0000i − 0.779895i −0.920837 0.389948i \(-0.872493\pi\)
0.920837 0.389948i \(-0.127507\pi\)
\(948\) 4.00000 0.129914
\(949\) 0 0
\(950\) −8.00000 −0.259554
\(951\) 3.00000i 0.0972817i
\(952\) 6.00000 0.194461
\(953\) −54.0000 −1.74923 −0.874616 0.484817i \(-0.838886\pi\)
−0.874616 + 0.484817i \(0.838886\pi\)
\(954\) − 3.00000i − 0.0971286i
\(955\) − 36.0000i − 1.16493i
\(956\) − 6.00000i − 0.194054i
\(957\) − 18.0000i − 0.581857i
\(958\) 0 0
\(959\) −18.0000 −0.581250
\(960\) − 3.00000i − 0.0968246i
\(961\) 15.0000 0.483871
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) − 1.00000i − 0.0322078i
\(965\) 69.0000 2.22119
\(966\) −12.0000 −0.386094
\(967\) − 22.0000i − 0.707472i −0.935345 0.353736i \(-0.884911\pi\)
0.935345 0.353736i \(-0.115089\pi\)
\(968\) − 25.0000i − 0.803530i
\(969\) 6.00000i 0.192748i
\(970\) 42.0000i 1.34854i
\(971\) 60.0000 1.92549 0.962746 0.270408i \(-0.0871586\pi\)
0.962746 + 0.270408i \(0.0871586\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 8.00000i 0.256468i
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) −7.00000 −0.224065
\(977\) − 3.00000i − 0.0959785i −0.998848 0.0479893i \(-0.984719\pi\)
0.998848 0.0479893i \(-0.0152813\pi\)
\(978\) 4.00000 0.127906
\(979\) −108.000 −3.45169
\(980\) − 9.00000i − 0.287494i
\(981\) 14.0000i 0.446986i
\(982\) − 18.0000i − 0.574403i
\(983\) 36.0000i 1.14822i 0.818778 + 0.574111i \(0.194652\pi\)
−0.818778 + 0.574111i \(0.805348\pi\)
\(984\) 3.00000 0.0956365
\(985\) −18.0000 −0.573528
\(986\) − 9.00000i − 0.286618i
\(987\) −12.0000 −0.381964
\(988\) 0 0
\(989\) 60.0000 1.90789
\(990\) − 18.0000i − 0.572078i
\(991\) 38.0000 1.20711 0.603555 0.797321i \(-0.293750\pi\)
0.603555 + 0.797321i \(0.293750\pi\)
\(992\) −4.00000 −0.127000
\(993\) − 4.00000i − 0.126936i
\(994\) − 12.0000i − 0.380617i
\(995\) 30.0000i 0.951064i
\(996\) 6.00000i 0.190117i
\(997\) 5.00000 0.158352 0.0791758 0.996861i \(-0.474771\pi\)
0.0791758 + 0.996861i \(0.474771\pi\)
\(998\) 32.0000 1.01294
\(999\) 7.00000i 0.221470i
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 1014.2.b.c.337.1 2
3.2 odd 2 3042.2.b.h.1351.2 2
13.2 odd 12 78.2.e.a.61.1 yes 2
13.3 even 3 1014.2.i.b.823.2 4
13.4 even 6 1014.2.i.b.361.2 4
13.5 odd 4 1014.2.a.c.1.1 1
13.6 odd 12 78.2.e.a.55.1 2
13.7 odd 12 1014.2.e.a.991.1 2
13.8 odd 4 1014.2.a.f.1.1 1
13.9 even 3 1014.2.i.b.361.1 4
13.10 even 6 1014.2.i.b.823.1 4
13.11 odd 12 1014.2.e.a.529.1 2
13.12 even 2 inner 1014.2.b.c.337.2 2
39.2 even 12 234.2.h.a.217.1 2
39.5 even 4 3042.2.a.i.1.1 1
39.8 even 4 3042.2.a.h.1.1 1
39.32 even 12 234.2.h.a.55.1 2
39.38 odd 2 3042.2.b.h.1351.1 2
52.15 even 12 624.2.q.g.529.1 2
52.19 even 12 624.2.q.g.289.1 2
52.31 even 4 8112.2.a.m.1.1 1
52.47 even 4 8112.2.a.c.1.1 1
65.2 even 12 1950.2.z.g.1699.2 4
65.19 odd 12 1950.2.i.m.601.1 2
65.28 even 12 1950.2.z.g.1699.1 4
65.32 even 12 1950.2.z.g.1849.1 4
65.54 odd 12 1950.2.i.m.451.1 2
65.58 even 12 1950.2.z.g.1849.2 4
156.71 odd 12 1872.2.t.c.289.1 2
156.119 odd 12 1872.2.t.c.1153.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
78.2.e.a.55.1 2 13.6 odd 12
78.2.e.a.61.1 yes 2 13.2 odd 12
234.2.h.a.55.1 2 39.32 even 12
234.2.h.a.217.1 2 39.2 even 12
624.2.q.g.289.1 2 52.19 even 12
624.2.q.g.529.1 2 52.15 even 12
1014.2.a.c.1.1 1 13.5 odd 4
1014.2.a.f.1.1 1 13.8 odd 4
1014.2.b.c.337.1 2 1.1 even 1 trivial
1014.2.b.c.337.2 2 13.12 even 2 inner
1014.2.e.a.529.1 2 13.11 odd 12
1014.2.e.a.991.1 2 13.7 odd 12
1014.2.i.b.361.1 4 13.9 even 3
1014.2.i.b.361.2 4 13.4 even 6
1014.2.i.b.823.1 4 13.10 even 6
1014.2.i.b.823.2 4 13.3 even 3
1872.2.t.c.289.1 2 156.71 odd 12
1872.2.t.c.1153.1 2 156.119 odd 12
1950.2.i.m.451.1 2 65.54 odd 12
1950.2.i.m.601.1 2 65.19 odd 12
1950.2.z.g.1699.1 4 65.28 even 12
1950.2.z.g.1699.2 4 65.2 even 12
1950.2.z.g.1849.1 4 65.32 even 12
1950.2.z.g.1849.2 4 65.58 even 12
3042.2.a.h.1.1 1 39.8 even 4
3042.2.a.i.1.1 1 39.5 even 4
3042.2.b.h.1351.1 2 39.38 odd 2
3042.2.b.h.1351.2 2 3.2 odd 2
8112.2.a.c.1.1 1 52.47 even 4
8112.2.a.m.1.1 1 52.31 even 4