Properties

Label 4650.2.d.o.3349.2
Level $4650$
Weight $2$
Character 4650.3349
Analytic conductor $37.130$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(37.1304369399\)
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 930)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3349.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 4650.3349
Dual form 4650.2.d.o.3349.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} -4.00000 q^{11} +1.00000i q^{12} +6.00000i q^{13} +1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} -4.00000i q^{22} -4.00000i q^{23} -1.00000 q^{24} -6.00000 q^{26} +1.00000i q^{27} -2.00000 q^{29} -1.00000 q^{31} +1.00000i q^{32} +4.00000i q^{33} +2.00000 q^{34} +1.00000 q^{36} +2.00000i q^{37} +4.00000i q^{38} +6.00000 q^{39} -6.00000 q^{41} -4.00000i q^{43} +4.00000 q^{44} +4.00000 q^{46} -1.00000i q^{48} +7.00000 q^{49} -2.00000 q^{51} -6.00000i q^{52} +2.00000i q^{53} -1.00000 q^{54} -4.00000i q^{57} -2.00000i q^{58} +4.00000 q^{59} -6.00000 q^{61} -1.00000i q^{62} -1.00000 q^{64} -4.00000 q^{66} -16.0000i q^{67} +2.00000i q^{68} -4.00000 q^{69} -12.0000 q^{71} +1.00000i q^{72} -6.00000i q^{73} -2.00000 q^{74} -4.00000 q^{76} +6.00000i q^{78} +16.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -12.0000i q^{83} +4.00000 q^{86} +2.00000i q^{87} +4.00000i q^{88} +18.0000 q^{89} +4.00000i q^{92} +1.00000i q^{93} +1.00000 q^{96} +14.0000i q^{97} +7.00000i q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{4} + 2 q^{6} - 2 q^{9} - 8 q^{11} + 2 q^{16} + 8 q^{19} - 2 q^{24} - 12 q^{26} - 4 q^{29} - 2 q^{31} + 4 q^{34} + 2 q^{36} + 12 q^{39} - 12 q^{41} + 8 q^{44} + 8 q^{46} + 14 q^{49} - 4 q^{51} - 2 q^{54} + 8 q^{59} - 12 q^{61} - 2 q^{64} - 8 q^{66} - 8 q^{69} - 24 q^{71} - 4 q^{74} - 8 q^{76} + 32 q^{79} + 2 q^{81} + 8 q^{86} + 36 q^{89} + 2 q^{96} + 8 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times\).

\(n\) \(1801\) \(2977\) \(3101\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 6.00000i 1.66410i 0.554700 + 0.832050i \(0.312833\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 2.00000i − 0.485071i −0.970143 0.242536i \(-0.922021\pi\)
0.970143 0.242536i \(-0.0779791\pi\)
\(18\) − 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
\(21\) 0 0
\(22\) − 4.00000i − 0.852803i
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −6.00000 −1.17670
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 1.00000i 0.176777i
\(33\) 4.00000i 0.696311i
\(34\) 2.00000 0.342997
\(35\) 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
\(39\) 6.00000 0.960769
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 4.00000 0.603023
\(45\) 0 0
\(46\) 4.00000 0.589768
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) − 6.00000i − 0.832050i
\(53\) 2.00000i 0.274721i 0.990521 + 0.137361i \(0.0438619\pi\)
−0.990521 + 0.137361i \(0.956138\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) − 4.00000i − 0.529813i
\(58\) − 2.00000i − 0.262613i
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −6.00000 −0.768221 −0.384111 0.923287i \(-0.625492\pi\)
−0.384111 + 0.923287i \(0.625492\pi\)
\(62\) − 1.00000i − 0.127000i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) − 16.0000i − 1.95471i −0.211604 0.977356i \(-0.567869\pi\)
0.211604 0.977356i \(-0.432131\pi\)
\(68\) 2.00000i 0.242536i
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) −12.0000 −1.42414 −0.712069 0.702109i \(-0.752242\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 6.00000i − 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) −2.00000 −0.232495
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 6.00000i 0.679366i
\(79\) 16.0000 1.80014 0.900070 0.435745i \(-0.143515\pi\)
0.900070 + 0.435745i \(0.143515\pi\)
\(80\) 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 \(-0.228845\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 2.00000i 0.214423i
\(88\) 4.00000i 0.426401i
\(89\) 18.0000 1.90800 0.953998 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.00000i 0.417029i
\(93\) 1.00000i 0.103695i
\(94\) 0 0
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 14.0000i 1.42148i 0.703452 + 0.710742i \(0.251641\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) 7.00000i 0.707107i
\(99\) 4.00000 0.402015
\(100\) 0 0
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) − 2.00000i − 0.198030i
\(103\) 8.00000i 0.788263i 0.919054 + 0.394132i \(0.128955\pi\)
−0.919054 + 0.394132i \(0.871045\pi\)
\(104\) 6.00000 0.588348
\(105\) 0 0
\(106\) −2.00000 −0.194257
\(107\) − 4.00000i − 0.386695i −0.981130 0.193347i \(-0.938066\pi\)
0.981130 0.193347i \(-0.0619344\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) −6.00000 −0.574696 −0.287348 0.957826i \(-0.592774\pi\)
−0.287348 + 0.957826i \(0.592774\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) − 10.0000i − 0.940721i −0.882474 0.470360i \(-0.844124\pi\)
0.882474 0.470360i \(-0.155876\pi\)
\(114\) 4.00000 0.374634
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) − 6.00000i − 0.554700i
\(118\) 4.00000i 0.368230i
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) − 6.00000i − 0.543214i
\(123\) 6.00000i 0.541002i
\(124\) 1.00000 0.0898027
\(125\) 0 0
\(126\) 0 0
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) − 4.00000i − 0.348155i
\(133\) 0 0
\(134\) 16.0000 1.38219
\(135\) 0 0
\(136\) −2.00000 −0.171499
\(137\) − 18.0000i − 1.53784i −0.639343 0.768922i \(-0.720793\pi\)
0.639343 0.768922i \(-0.279207\pi\)
\(138\) − 4.00000i − 0.340503i
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 12.0000i − 1.00702i
\(143\) − 24.0000i − 2.00698i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 6.00000 0.496564
\(147\) − 7.00000i − 0.577350i
\(148\) − 2.00000i − 0.164399i
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) − 4.00000i − 0.324443i
\(153\) 2.00000i 0.161690i
\(154\) 0 0
\(155\) 0 0
\(156\) −6.00000 −0.480384
\(157\) − 18.0000i − 1.43656i −0.695756 0.718278i \(-0.744931\pi\)
0.695756 0.718278i \(-0.255069\pi\)
\(158\) 16.0000i 1.27289i
\(159\) 2.00000 0.158610
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000i 0.0785674i
\(163\) − 24.0000i − 1.87983i −0.341415 0.939913i \(-0.610906\pi\)
0.341415 0.939913i \(-0.389094\pi\)
\(164\) 6.00000 0.468521
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) − 12.0000i − 0.928588i −0.885681 0.464294i \(-0.846308\pi\)
0.885681 0.464294i \(-0.153692\pi\)
\(168\) 0 0
\(169\) −23.0000 −1.76923
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 4.00000i 0.304997i
\(173\) − 14.0000i − 1.06440i −0.846619 0.532200i \(-0.821365\pi\)
0.846619 0.532200i \(-0.178635\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) − 4.00000i − 0.300658i
\(178\) 18.0000i 1.34916i
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 6.00000i 0.443533i
\(184\) −4.00000 −0.294884
\(185\) 0 0
\(186\) −1.00000 −0.0733236
\(187\) 8.00000i 0.585018i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4.00000 0.289430 0.144715 0.989473i \(-0.453773\pi\)
0.144715 + 0.989473i \(0.453773\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) − 22.0000i − 1.58359i −0.610784 0.791797i \(-0.709146\pi\)
0.610784 0.791797i \(-0.290854\pi\)
\(194\) −14.0000 −1.00514
\(195\) 0 0
\(196\) −7.00000 −0.500000
\(197\) − 2.00000i − 0.142494i −0.997459 0.0712470i \(-0.977302\pi\)
0.997459 0.0712470i \(-0.0226979\pi\)
\(198\) 4.00000i 0.284268i
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) −16.0000 −1.12855
\(202\) 2.00000i 0.140720i
\(203\) 0 0
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 4.00000i 0.278019i
\(208\) 6.00000i 0.416025i
\(209\) −16.0000 −1.10674
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) − 2.00000i − 0.137361i
\(213\) 12.0000i 0.822226i
\(214\) 4.00000 0.273434
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) − 6.00000i − 0.406371i
\(219\) −6.00000 −0.405442
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 2.00000i 0.134231i
\(223\) − 8.00000i − 0.535720i −0.963458 0.267860i \(-0.913684\pi\)
0.963458 0.267860i \(-0.0863164\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 10.0000 0.665190
\(227\) − 20.0000i − 1.32745i −0.747978 0.663723i \(-0.768975\pi\)
0.747978 0.663723i \(-0.231025\pi\)
\(228\) 4.00000i 0.264906i
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.00000i 0.131306i
\(233\) 22.0000i 1.44127i 0.693316 + 0.720634i \(0.256149\pi\)
−0.693316 + 0.720634i \(0.743851\pi\)
\(234\) 6.00000 0.392232
\(235\) 0 0
\(236\) −4.00000 −0.260378
\(237\) − 16.0000i − 1.03931i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −14.0000 −0.901819 −0.450910 0.892570i \(-0.648900\pi\)
−0.450910 + 0.892570i \(0.648900\pi\)
\(242\) 5.00000i 0.321412i
\(243\) − 1.00000i − 0.0641500i
\(244\) 6.00000 0.384111
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) 24.0000i 1.52708i
\(248\) 1.00000i 0.0635001i
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) 20.0000 1.26239 0.631194 0.775625i \(-0.282565\pi\)
0.631194 + 0.775625i \(0.282565\pi\)
\(252\) 0 0
\(253\) 16.0000i 1.00591i
\(254\) −8.00000 −0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 22.0000i − 1.37232i −0.727450 0.686161i \(-0.759294\pi\)
0.727450 0.686161i \(-0.240706\pi\)
\(258\) − 4.00000i − 0.249029i
\(259\) 0 0
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) − 4.00000i − 0.247121i
\(263\) − 12.0000i − 0.739952i −0.929041 0.369976i \(-0.879366\pi\)
0.929041 0.369976i \(-0.120634\pi\)
\(264\) 4.00000 0.246183
\(265\) 0 0
\(266\) 0 0
\(267\) − 18.0000i − 1.10158i
\(268\) 16.0000i 0.977356i
\(269\) −18.0000 −1.09748 −0.548740 0.835993i \(-0.684892\pi\)
−0.548740 + 0.835993i \(0.684892\pi\)
\(270\) 0 0
\(271\) −8.00000 −0.485965 −0.242983 0.970031i \(-0.578126\pi\)
−0.242983 + 0.970031i \(0.578126\pi\)
\(272\) − 2.00000i − 0.121268i
\(273\) 0 0
\(274\) 18.0000 1.08742
\(275\) 0 0
\(276\) 4.00000 0.240772
\(277\) − 6.00000i − 0.360505i −0.983620 0.180253i \(-0.942309\pi\)
0.983620 0.180253i \(-0.0576915\pi\)
\(278\) − 8.00000i − 0.479808i
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) 10.0000 0.596550 0.298275 0.954480i \(-0.403589\pi\)
0.298275 + 0.954480i \(0.403589\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 12.0000 0.712069
\(285\) 0 0
\(286\) 24.0000 1.41915
\(287\) 0 0
\(288\) − 1.00000i − 0.0589256i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) 6.00000i 0.351123i
\(293\) − 6.00000i − 0.350524i −0.984522 0.175262i \(-0.943923\pi\)
0.984522 0.175262i \(-0.0560772\pi\)
\(294\) 7.00000 0.408248
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) − 4.00000i − 0.232104i
\(298\) − 10.0000i − 0.579284i
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) 0 0
\(302\) 16.0000i 0.920697i
\(303\) − 2.00000i − 0.114897i
\(304\) 4.00000 0.229416
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) 16.0000i 0.913168i 0.889680 + 0.456584i \(0.150927\pi\)
−0.889680 + 0.456584i \(0.849073\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −28.0000 −1.58773 −0.793867 0.608091i \(-0.791935\pi\)
−0.793867 + 0.608091i \(0.791935\pi\)
\(312\) − 6.00000i − 0.339683i
\(313\) 26.0000i 1.46961i 0.678280 + 0.734803i \(0.262726\pi\)
−0.678280 + 0.734803i \(0.737274\pi\)
\(314\) 18.0000 1.01580
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) − 2.00000i − 0.112331i −0.998421 0.0561656i \(-0.982113\pi\)
0.998421 0.0561656i \(-0.0178875\pi\)
\(318\) 2.00000i 0.112154i
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) −4.00000 −0.223258
\(322\) 0 0
\(323\) − 8.00000i − 0.445132i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 24.0000 1.32924
\(327\) 6.00000i 0.331801i
\(328\) 6.00000i 0.331295i
\(329\) 0 0
\(330\) 0 0
\(331\) 24.0000 1.31916 0.659580 0.751635i \(-0.270734\pi\)
0.659580 + 0.751635i \(0.270734\pi\)
\(332\) 12.0000i 0.658586i
\(333\) − 2.00000i − 0.109599i
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) − 2.00000i − 0.108947i −0.998515 0.0544735i \(-0.982652\pi\)
0.998515 0.0544735i \(-0.0173480\pi\)
\(338\) − 23.0000i − 1.25104i
\(339\) −10.0000 −0.543125
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) − 4.00000i − 0.216295i
\(343\) 0 0
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 14.0000 0.752645
\(347\) 4.00000i 0.214731i 0.994220 + 0.107366i \(0.0342415\pi\)
−0.994220 + 0.107366i \(0.965758\pi\)
\(348\) − 2.00000i − 0.107211i
\(349\) 2.00000 0.107058 0.0535288 0.998566i \(-0.482953\pi\)
0.0535288 + 0.998566i \(0.482953\pi\)
\(350\) 0 0
\(351\) −6.00000 −0.320256
\(352\) − 4.00000i − 0.213201i
\(353\) − 6.00000i − 0.319348i −0.987170 0.159674i \(-0.948956\pi\)
0.987170 0.159674i \(-0.0510443\pi\)
\(354\) 4.00000 0.212598
\(355\) 0 0
\(356\) −18.0000 −0.953998
\(357\) 0 0
\(358\) 4.00000i 0.211407i
\(359\) 20.0000 1.05556 0.527780 0.849381i \(-0.323025\pi\)
0.527780 + 0.849381i \(0.323025\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) − 6.00000i − 0.315353i
\(363\) − 5.00000i − 0.262432i
\(364\) 0 0
\(365\) 0 0
\(366\) −6.00000 −0.313625
\(367\) − 16.0000i − 0.835193i −0.908633 0.417597i \(-0.862873\pi\)
0.908633 0.417597i \(-0.137127\pi\)
\(368\) − 4.00000i − 0.208514i
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) − 1.00000i − 0.0518476i
\(373\) − 38.0000i − 1.96757i −0.179364 0.983783i \(-0.557404\pi\)
0.179364 0.983783i \(-0.442596\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) 0 0
\(377\) − 12.0000i − 0.618031i
\(378\) 0 0
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) 8.00000 0.409852
\(382\) 4.00000i 0.204658i
\(383\) 4.00000i 0.204390i 0.994764 + 0.102195i \(0.0325866\pi\)
−0.994764 + 0.102195i \(0.967413\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 22.0000 1.11977
\(387\) 4.00000i 0.203331i
\(388\) − 14.0000i − 0.710742i
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) − 7.00000i − 0.353553i
\(393\) 4.00000i 0.201773i
\(394\) 2.00000 0.100759
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) 14.0000i 0.702640i 0.936255 + 0.351320i \(0.114267\pi\)
−0.936255 + 0.351320i \(0.885733\pi\)
\(398\) − 16.0000i − 0.802008i
\(399\) 0 0
\(400\) 0 0
\(401\) 14.0000 0.699127 0.349563 0.936913i \(-0.386330\pi\)
0.349563 + 0.936913i \(0.386330\pi\)
\(402\) − 16.0000i − 0.798007i
\(403\) − 6.00000i − 0.298881i
\(404\) −2.00000 −0.0995037
\(405\) 0 0
\(406\) 0 0
\(407\) − 8.00000i − 0.396545i
\(408\) 2.00000i 0.0990148i
\(409\) 6.00000 0.296681 0.148340 0.988936i \(-0.452607\pi\)
0.148340 + 0.988936i \(0.452607\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) − 8.00000i − 0.394132i
\(413\) 0 0
\(414\) −4.00000 −0.196589
\(415\) 0 0
\(416\) −6.00000 −0.294174
\(417\) 8.00000i 0.391762i
\(418\) − 16.0000i − 0.782586i
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 14.0000 0.682318 0.341159 0.940006i \(-0.389181\pi\)
0.341159 + 0.940006i \(0.389181\pi\)
\(422\) − 4.00000i − 0.194717i
\(423\) 0 0
\(424\) 2.00000 0.0971286
\(425\) 0 0
\(426\) −12.0000 −0.581402
\(427\) 0 0
\(428\) 4.00000i 0.193347i
\(429\) −24.0000 −1.15873
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 2.00000i 0.0961139i 0.998845 + 0.0480569i \(0.0153029\pi\)
−0.998845 + 0.0480569i \(0.984697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 6.00000 0.287348
\(437\) − 16.0000i − 0.765384i
\(438\) − 6.00000i − 0.286691i
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 12.0000i 0.570782i
\(443\) − 4.00000i − 0.190046i −0.995475 0.0950229i \(-0.969708\pi\)
0.995475 0.0950229i \(-0.0302924\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 0 0
\(446\) 8.00000 0.378811
\(447\) 10.0000i 0.472984i
\(448\) 0 0
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) 10.0000i 0.470360i
\(453\) − 16.0000i − 0.751746i
\(454\) 20.0000 0.938647
\(455\) 0 0
\(456\) −4.00000 −0.187317
\(457\) 22.0000i 1.02912i 0.857455 + 0.514558i \(0.172044\pi\)
−0.857455 + 0.514558i \(0.827956\pi\)
\(458\) − 10.0000i − 0.467269i
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) − 24.0000i − 1.11537i −0.830051 0.557687i \(-0.811689\pi\)
0.830051 0.557687i \(-0.188311\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −22.0000 −1.01913
\(467\) 4.00000i 0.185098i 0.995708 + 0.0925490i \(0.0295015\pi\)
−0.995708 + 0.0925490i \(0.970499\pi\)
\(468\) 6.00000i 0.277350i
\(469\) 0 0
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) − 4.00000i − 0.184115i
\(473\) 16.0000i 0.735681i
\(474\) 16.0000 0.734904
\(475\) 0 0
\(476\) 0 0
\(477\) − 2.00000i − 0.0915737i
\(478\) 0 0
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) −12.0000 −0.547153
\(482\) − 14.0000i − 0.637683i
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 40.0000i 1.81257i 0.422664 + 0.906287i \(0.361095\pi\)
−0.422664 + 0.906287i \(0.638905\pi\)
\(488\) 6.00000i 0.271607i
\(489\) −24.0000 −1.08532
\(490\) 0 0
\(491\) 20.0000 0.902587 0.451294 0.892375i \(-0.350963\pi\)
0.451294 + 0.892375i \(0.350963\pi\)
\(492\) − 6.00000i − 0.270501i
\(493\) 4.00000i 0.180151i
\(494\) −24.0000 −1.07981
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) 0 0
\(498\) − 12.0000i − 0.537733i
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) 20.0000i 0.892644i
\(503\) 32.0000i 1.42681i 0.700752 + 0.713405i \(0.252848\pi\)
−0.700752 + 0.713405i \(0.747152\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −16.0000 −0.711287
\(507\) 23.0000i 1.02147i
\(508\) − 8.00000i − 0.354943i
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000i 0.0441942i
\(513\) 4.00000i 0.176604i
\(514\) 22.0000 0.970378
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 0 0
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) −14.0000 −0.613351 −0.306676 0.951814i \(-0.599217\pi\)
−0.306676 + 0.951814i \(0.599217\pi\)
\(522\) 2.00000i 0.0875376i
\(523\) 20.0000i 0.874539i 0.899331 + 0.437269i \(0.144054\pi\)
−0.899331 + 0.437269i \(0.855946\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) 12.0000 0.523225
\(527\) 2.00000i 0.0871214i
\(528\) 4.00000i 0.174078i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 0 0
\(533\) − 36.0000i − 1.55933i
\(534\) 18.0000 0.778936
\(535\) 0 0
\(536\) −16.0000 −0.691095
\(537\) − 4.00000i − 0.172613i
\(538\) − 18.0000i − 0.776035i
\(539\) −28.0000 −1.20605
\(540\) 0 0
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) − 8.00000i − 0.343629i
\(543\) 6.00000i 0.257485i
\(544\) 2.00000 0.0857493
\(545\) 0 0
\(546\) 0 0
\(547\) − 16.0000i − 0.684111i −0.939680 0.342055i \(-0.888877\pi\)
0.939680 0.342055i \(-0.111123\pi\)
\(548\) 18.0000i 0.768922i
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) −8.00000 −0.340811
\(552\) 4.00000i 0.170251i
\(553\) 0 0
\(554\) 6.00000 0.254916
\(555\) 0 0
\(556\) 8.00000 0.339276
\(557\) 30.0000i 1.27114i 0.772043 + 0.635570i \(0.219235\pi\)
−0.772043 + 0.635570i \(0.780765\pi\)
\(558\) 1.00000i 0.0423334i
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 10.0000i 0.421825i
\(563\) − 12.0000i − 0.505740i −0.967500 0.252870i \(-0.918626\pi\)
0.967500 0.252870i \(-0.0813744\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 12.0000i 0.503509i
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 40.0000 1.67395 0.836974 0.547243i \(-0.184323\pi\)
0.836974 + 0.547243i \(0.184323\pi\)
\(572\) 24.0000i 1.00349i
\(573\) − 4.00000i − 0.167102i
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 30.0000i 1.24892i 0.781058 + 0.624458i \(0.214680\pi\)
−0.781058 + 0.624458i \(0.785320\pi\)
\(578\) 13.0000i 0.540729i
\(579\) −22.0000 −0.914289
\(580\) 0 0
\(581\) 0 0
\(582\) 14.0000i 0.580319i
\(583\) − 8.00000i − 0.331326i
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) 6.00000 0.247858
\(587\) − 4.00000i − 0.165098i −0.996587 0.0825488i \(-0.973694\pi\)
0.996587 0.0825488i \(-0.0263060\pi\)
\(588\) 7.00000i 0.288675i
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) −2.00000 −0.0822690
\(592\) 2.00000i 0.0821995i
\(593\) 14.0000i 0.574911i 0.957794 + 0.287456i \(0.0928094\pi\)
−0.957794 + 0.287456i \(0.907191\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 16.0000i 0.654836i
\(598\) 24.0000i 0.981433i
\(599\) 4.00000 0.163436 0.0817178 0.996656i \(-0.473959\pi\)
0.0817178 + 0.996656i \(0.473959\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) 16.0000i 0.651570i
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) 2.00000 0.0812444
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 4.00000i 0.162221i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) − 2.00000i − 0.0808452i
\(613\) − 26.0000i − 1.05013i −0.851062 0.525065i \(-0.824041\pi\)
0.851062 0.525065i \(-0.175959\pi\)
\(614\) −16.0000 −0.645707
\(615\) 0 0
\(616\) 0 0
\(617\) − 46.0000i − 1.85189i −0.377658 0.925945i \(-0.623271\pi\)
0.377658 0.925945i \(-0.376729\pi\)
\(618\) 8.00000i 0.321807i
\(619\) −32.0000 −1.28619 −0.643094 0.765787i \(-0.722350\pi\)
−0.643094 + 0.765787i \(0.722350\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) − 28.0000i − 1.12270i
\(623\) 0 0
\(624\) 6.00000 0.240192
\(625\) 0 0
\(626\) −26.0000 −1.03917
\(627\) 16.0000i 0.638978i
\(628\) 18.0000i 0.718278i
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) − 16.0000i − 0.636446i
\(633\) 4.00000i 0.158986i
\(634\) 2.00000 0.0794301
\(635\) 0 0
\(636\) −2.00000 −0.0793052
\(637\) 42.0000i 1.66410i
\(638\) 8.00000i 0.316723i
\(639\) 12.0000 0.474713
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) − 4.00000i − 0.157867i
\(643\) − 20.0000i − 0.788723i −0.918955 0.394362i \(-0.870966\pi\)
0.918955 0.394362i \(-0.129034\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) 12.0000i 0.471769i 0.971781 + 0.235884i \(0.0757987\pi\)
−0.971781 + 0.235884i \(0.924201\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 24.0000i 0.939913i
\(653\) 18.0000i 0.704394i 0.935926 + 0.352197i \(0.114565\pi\)
−0.935926 + 0.352197i \(0.885435\pi\)
\(654\) −6.00000 −0.234619
\(655\) 0 0
\(656\) −6.00000 −0.234261
\(657\) 6.00000i 0.234082i
\(658\) 0 0
\(659\) −36.0000 −1.40236 −0.701180 0.712984i \(-0.747343\pi\)
−0.701180 + 0.712984i \(0.747343\pi\)
\(660\) 0 0
\(661\) −34.0000 −1.32245 −0.661223 0.750189i \(-0.729962\pi\)
−0.661223 + 0.750189i \(0.729962\pi\)
\(662\) 24.0000i 0.932786i
\(663\) − 12.0000i − 0.466041i
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) 8.00000i 0.309761i
\(668\) 12.0000i 0.464294i
\(669\) −8.00000 −0.309298
\(670\) 0 0
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) 34.0000i 1.31060i 0.755367 + 0.655302i \(0.227459\pi\)
−0.755367 + 0.655302i \(0.772541\pi\)
\(674\) 2.00000 0.0770371
\(675\) 0 0
\(676\) 23.0000 0.884615
\(677\) − 42.0000i − 1.61419i −0.590421 0.807096i \(-0.701038\pi\)
0.590421 0.807096i \(-0.298962\pi\)
\(678\) − 10.0000i − 0.384048i
\(679\) 0 0
\(680\) 0 0
\(681\) −20.0000 −0.766402
\(682\) 4.00000i 0.153168i
\(683\) − 36.0000i − 1.37750i −0.724998 0.688751i \(-0.758159\pi\)
0.724998 0.688751i \(-0.241841\pi\)
\(684\) 4.00000 0.152944
\(685\) 0 0
\(686\) 0 0
\(687\) 10.0000i 0.381524i
\(688\) − 4.00000i − 0.152499i
\(689\) −12.0000 −0.457164
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) 14.0000i 0.532200i
\(693\) 0 0
\(694\) −4.00000 −0.151838
\(695\) 0 0
\(696\) 2.00000 0.0758098
\(697\) 12.0000i 0.454532i
\(698\) 2.00000i 0.0757011i
\(699\) 22.0000 0.832116
\(700\) 0 0
\(701\) −38.0000 −1.43524 −0.717620 0.696435i \(-0.754769\pi\)
−0.717620 + 0.696435i \(0.754769\pi\)
\(702\) − 6.00000i − 0.226455i
\(703\) 8.00000i 0.301726i
\(704\) 4.00000 0.150756
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 0 0
\(708\) 4.00000i 0.150329i
\(709\) 46.0000 1.72757 0.863783 0.503864i \(-0.168089\pi\)
0.863783 + 0.503864i \(0.168089\pi\)
\(710\) 0 0
\(711\) −16.0000 −0.600047
\(712\) − 18.0000i − 0.674579i
\(713\) 4.00000i 0.149801i
\(714\) 0 0
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) 20.0000i 0.746393i
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 3.00000i − 0.111648i
\(723\) 14.0000i 0.520666i
\(724\) 6.00000 0.222988
\(725\) 0 0
\(726\) 5.00000 0.185567
\(727\) − 16.0000i − 0.593407i −0.954970 0.296704i \(-0.904113\pi\)
0.954970 0.296704i \(-0.0958873\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) − 6.00000i − 0.221766i
\(733\) 2.00000i 0.0738717i 0.999318 + 0.0369358i \(0.0117597\pi\)
−0.999318 + 0.0369358i \(0.988240\pi\)
\(734\) 16.0000 0.590571
\(735\) 0 0
\(736\) 4.00000 0.147442
\(737\) 64.0000i 2.35747i
\(738\) 6.00000i 0.220863i
\(739\) 16.0000 0.588570 0.294285 0.955718i \(-0.404919\pi\)
0.294285 + 0.955718i \(0.404919\pi\)
\(740\) 0 0
\(741\) 24.0000 0.881662
\(742\) 0 0
\(743\) 12.0000i 0.440237i 0.975473 + 0.220119i \(0.0706445\pi\)
−0.975473 + 0.220119i \(0.929356\pi\)
\(744\) 1.00000 0.0366618
\(745\) 0 0
\(746\) 38.0000 1.39128
\(747\) 12.0000i 0.439057i
\(748\) − 8.00000i − 0.292509i
\(749\) 0 0
\(750\) 0 0
\(751\) −24.0000 −0.875772 −0.437886 0.899030i \(-0.644273\pi\)
−0.437886 + 0.899030i \(0.644273\pi\)
\(752\) 0 0
\(753\) − 20.0000i − 0.728841i
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) 18.0000i 0.654221i 0.944986 + 0.327111i \(0.106075\pi\)
−0.944986 + 0.327111i \(0.893925\pi\)
\(758\) 20.0000i 0.726433i
\(759\) 16.0000 0.580763
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 8.00000i 0.289809i
\(763\) 0 0
\(764\) −4.00000 −0.144715
\(765\) 0 0
\(766\) −4.00000 −0.144526
\(767\) 24.0000i 0.866590i
\(768\) − 1.00000i − 0.0360844i
\(769\) 14.0000 0.504853 0.252426 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) −22.0000 −0.792311
\(772\) 22.0000i 0.791797i
\(773\) 50.0000i 1.79838i 0.437564 + 0.899188i \(0.355842\pi\)
−0.437564 + 0.899188i \(0.644158\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 14.0000 0.502571
\(777\) 0 0
\(778\) 6.00000i 0.215110i
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 48.0000 1.71758
\(782\) − 8.00000i − 0.286079i
\(783\) − 2.00000i − 0.0714742i
\(784\) 7.00000 0.250000
\(785\) 0 0
\(786\) −4.00000 −0.142675
\(787\) − 28.0000i − 0.998092i −0.866575 0.499046i \(-0.833684\pi\)
0.866575 0.499046i \(-0.166316\pi\)
\(788\) 2.00000i 0.0712470i
\(789\) −12.0000 −0.427211
\(790\) 0 0
\(791\) 0 0
\(792\) − 4.00000i − 0.142134i
\(793\) − 36.0000i − 1.27840i
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) 16.0000 0.567105
\(797\) − 18.0000i − 0.637593i −0.947823 0.318796i \(-0.896721\pi\)
0.947823 0.318796i \(-0.103279\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −18.0000 −0.635999
\(802\) 14.0000i 0.494357i
\(803\) 24.0000i 0.846942i
\(804\) 16.0000 0.564276
\(805\) 0 0
\(806\) 6.00000 0.211341
\(807\) 18.0000i 0.633630i
\(808\) − 2.00000i − 0.0703598i
\(809\) 10.0000 0.351581 0.175791 0.984428i \(-0.443752\pi\)
0.175791 + 0.984428i \(0.443752\pi\)
\(810\) 0 0
\(811\) 12.0000 0.421377 0.210688 0.977553i \(-0.432429\pi\)
0.210688 + 0.977553i \(0.432429\pi\)
\(812\) 0 0
\(813\) 8.00000i 0.280572i
\(814\) 8.00000 0.280400
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) − 16.0000i − 0.559769i
\(818\) 6.00000i 0.209785i
\(819\) 0 0
\(820\) 0 0
\(821\) −22.0000 −0.767805 −0.383903 0.923374i \(-0.625420\pi\)
−0.383903 + 0.923374i \(0.625420\pi\)
\(822\) − 18.0000i − 0.627822i
\(823\) 40.0000i 1.39431i 0.716919 + 0.697156i \(0.245552\pi\)
−0.716919 + 0.697156i \(0.754448\pi\)
\(824\) 8.00000 0.278693
\(825\) 0 0
\(826\) 0 0
\(827\) − 52.0000i − 1.80822i −0.427303 0.904109i \(-0.640536\pi\)
0.427303 0.904109i \(-0.359464\pi\)
\(828\) − 4.00000i − 0.139010i
\(829\) −50.0000 −1.73657 −0.868286 0.496064i \(-0.834778\pi\)
−0.868286 + 0.496064i \(0.834778\pi\)
\(830\) 0 0
\(831\) −6.00000 −0.208138
\(832\) − 6.00000i − 0.208013i
\(833\) − 14.0000i − 0.485071i
\(834\) −8.00000 −0.277017
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) − 1.00000i − 0.0345651i
\(838\) 12.0000i 0.414533i
\(839\) −28.0000 −0.966667 −0.483334 0.875436i \(-0.660574\pi\)
−0.483334 + 0.875436i \(0.660574\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 14.0000i 0.482472i
\(843\) − 10.0000i − 0.344418i
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 2.00000i 0.0686803i
\(849\) 0 0
\(850\) 0 0
\(851\) 8.00000 0.274236
\(852\) − 12.0000i − 0.411113i
\(853\) 58.0000i 1.98588i 0.118609 + 0.992941i \(0.462157\pi\)
−0.118609 + 0.992941i \(0.537843\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −4.00000 −0.136717
\(857\) − 54.0000i − 1.84460i −0.386469 0.922302i \(-0.626305\pi\)
0.386469 0.922302i \(-0.373695\pi\)
\(858\) − 24.0000i − 0.819346i
\(859\) −48.0000 −1.63774 −0.818869 0.573980i \(-0.805399\pi\)
−0.818869 + 0.573980i \(0.805399\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 12.0000i − 0.408722i
\(863\) 36.0000i 1.22545i 0.790295 + 0.612727i \(0.209928\pi\)
−0.790295 + 0.612727i \(0.790072\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −2.00000 −0.0679628
\(867\) − 13.0000i − 0.441503i
\(868\) 0 0
\(869\) −64.0000 −2.17105
\(870\) 0 0
\(871\) 96.0000 3.25284
\(872\) 6.00000i 0.203186i
\(873\) − 14.0000i − 0.473828i
\(874\) 16.0000 0.541208
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) 38.0000i 1.28317i 0.767052 + 0.641584i \(0.221723\pi\)
−0.767052 + 0.641584i \(0.778277\pi\)
\(878\) 32.0000i 1.07995i
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −42.0000 −1.41502 −0.707508 0.706705i \(-0.750181\pi\)
−0.707508 + 0.706705i \(0.750181\pi\)
\(882\) − 7.00000i − 0.235702i
\(883\) 52.0000i 1.74994i 0.484178 + 0.874970i \(0.339119\pi\)
−0.484178 + 0.874970i \(0.660881\pi\)
\(884\) −12.0000 −0.403604
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) 24.0000i 0.805841i 0.915235 + 0.402921i \(0.132005\pi\)
−0.915235 + 0.402921i \(0.867995\pi\)
\(888\) − 2.00000i − 0.0671156i
\(889\) 0 0
\(890\) 0 0
\(891\) −4.00000 −0.134005
\(892\) 8.00000i 0.267860i
\(893\) 0 0
\(894\) −10.0000 −0.334450
\(895\) 0 0
\(896\) 0 0
\(897\) − 24.0000i − 0.801337i
\(898\) 18.0000i 0.600668i
\(899\) 2.00000 0.0667037
\(900\) 0 0
\(901\) 4.00000 0.133259
\(902\) 24.0000i 0.799113i
\(903\) 0 0
\(904\) −10.0000 −0.332595
\(905\) 0 0
\(906\) 16.0000 0.531564
\(907\) 56.0000i 1.85945i 0.368255 + 0.929725i \(0.379955\pi\)
−0.368255 + 0.929725i \(0.620045\pi\)
\(908\) 20.0000i 0.663723i
\(909\) −2.00000 −0.0663358
\(910\) 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
\(913\) 48.0000i 1.58857i
\(914\) −22.0000 −0.727695
\(915\) 0 0
\(916\) 10.0000 0.330409
\(917\) 0 0
\(918\) 2.00000i 0.0660098i
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) 16.0000 0.527218
\(922\) − 14.0000i − 0.461065i
\(923\) − 72.0000i − 2.36991i
\(924\) 0 0
\(925\) 0 0
\(926\) 24.0000 0.788689
\(927\) − 8.00000i − 0.262754i
\(928\) − 2.00000i − 0.0656532i
\(929\) 42.0000 1.37798 0.688988 0.724773i \(-0.258055\pi\)
0.688988 + 0.724773i \(0.258055\pi\)
\(930\) 0 0
\(931\) 28.0000 0.917663
\(932\) − 22.0000i − 0.720634i
\(933\) 28.0000i 0.916679i
\(934\) −4.00000 −0.130884
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) − 10.0000i − 0.326686i −0.986569 0.163343i \(-0.947772\pi\)
0.986569 0.163343i \(-0.0522277\pi\)
\(938\) 0 0
\(939\) 26.0000 0.848478
\(940\) 0 0
\(941\) 18.0000 0.586783 0.293392 0.955992i \(-0.405216\pi\)
0.293392 + 0.955992i \(0.405216\pi\)
\(942\) − 18.0000i − 0.586472i
\(943\) 24.0000i 0.781548i
\(944\) 4.00000 0.130189
\(945\) 0 0
\(946\) −16.0000 −0.520205
\(947\) 44.0000i 1.42981i 0.699223 + 0.714904i \(0.253530\pi\)
−0.699223 + 0.714904i \(0.746470\pi\)
\(948\) 16.0000i 0.519656i
\(949\) 36.0000 1.16861
\(950\) 0 0
\(951\) −2.00000 −0.0648544
\(952\) 0 0
\(953\) − 30.0000i − 0.971795i −0.874016 0.485898i \(-0.838493\pi\)
0.874016 0.485898i \(-0.161507\pi\)
\(954\) 2.00000 0.0647524
\(955\) 0 0
\(956\) 0 0
\(957\) − 8.00000i − 0.258603i
\(958\) − 36.0000i − 1.16311i
\(959\) 0 0
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) − 12.0000i − 0.386896i
\(963\) 4.00000i 0.128898i
\(964\) 14.0000 0.450910
\(965\) 0 0
\(966\) 0 0
\(967\) 8.00000i 0.257263i 0.991692 + 0.128631i \(0.0410584\pi\)
−0.991692 + 0.128631i \(0.958942\pi\)
\(968\) − 5.00000i − 0.160706i
\(969\) −8.00000 −0.256997
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 0 0
\(974\) −40.0000 −1.28168
\(975\) 0 0
\(976\) −6.00000 −0.192055
\(977\) − 22.0000i − 0.703842i −0.936030 0.351921i \(-0.885529\pi\)
0.936030 0.351921i \(-0.114471\pi\)
\(978\) − 24.0000i − 0.767435i
\(979\) −72.0000 −2.30113
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) 20.0000i 0.638226i
\(983\) − 4.00000i − 0.127580i −0.997963 0.0637901i \(-0.979681\pi\)
0.997963 0.0637901i \(-0.0203188\pi\)
\(984\) 6.00000 0.191273
\(985\) 0 0
\(986\) −4.00000 −0.127386
\(987\) 0 0
\(988\) − 24.0000i − 0.763542i
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) − 1.00000i − 0.0317500i
\(993\) − 24.0000i − 0.761617i
\(994\) 0 0
\(995\) 0 0
\(996\) 12.0000 0.380235
\(997\) − 26.0000i − 0.823428i −0.911313 0.411714i \(-0.864930\pi\)
0.911313 0.411714i \(-0.135070\pi\)
\(998\) − 16.0000i − 0.506471i
\(999\) −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 4650.2.d.o.3349.2 2
5.2 odd 4 930.2.a.b.1.1 1
5.3 odd 4 4650.2.a.bp.1.1 1
5.4 even 2 inner 4650.2.d.o.3349.1 2
15.2 even 4 2790.2.a.ba.1.1 1
20.7 even 4 7440.2.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.b.1.1 1 5.2 odd 4
2790.2.a.ba.1.1 1 15.2 even 4
4650.2.a.bp.1.1 1 5.3 odd 4
4650.2.d.o.3349.1 2 5.4 even 2 inner
4650.2.d.o.3349.2 2 1.1 even 1 trivial
7440.2.a.q.1.1 1 20.7 even 4