Properties

Label 1734.2.b.b.577.1
Level $1734$
Weight $2$
Character 1734.577
Analytic conductor $13.846$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1734 = 2 \cdot 3 \cdot 17^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1734.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(13.8460597105\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 102)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 577.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 1734.577
Dual form 1734.2.b.b.577.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} +2.00000i q^{5} +1.00000i q^{6} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000i q^{3} +1.00000 q^{4} +2.00000i q^{5} +1.00000i q^{6} -1.00000 q^{8} -1.00000 q^{9} -2.00000i q^{10} -4.00000i q^{11} -1.00000i q^{12} -2.00000 q^{13} +2.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} -4.00000 q^{19} +2.00000i q^{20} +4.00000i q^{22} +1.00000i q^{24} +1.00000 q^{25} +2.00000 q^{26} +1.00000i q^{27} +10.0000i q^{29} -2.00000 q^{30} -8.00000i q^{31} -1.00000 q^{32} -4.00000 q^{33} -1.00000 q^{36} +2.00000i q^{37} +4.00000 q^{38} +2.00000i q^{39} -2.00000i q^{40} +10.0000i q^{41} -12.0000 q^{43} -4.00000i q^{44} -2.00000i q^{45} -1.00000i q^{48} +7.00000 q^{49} -1.00000 q^{50} -2.00000 q^{52} -6.00000 q^{53} -1.00000i q^{54} +8.00000 q^{55} +4.00000i q^{57} -10.0000i q^{58} -12.0000 q^{59} +2.00000 q^{60} -10.0000i q^{61} +8.00000i q^{62} +1.00000 q^{64} -4.00000i q^{65} +4.00000 q^{66} -12.0000 q^{67} +1.00000 q^{72} -10.0000i q^{73} -2.00000i q^{74} -1.00000i q^{75} -4.00000 q^{76} -2.00000i q^{78} -8.00000i q^{79} +2.00000i q^{80} +1.00000 q^{81} -10.0000i q^{82} -4.00000 q^{83} +12.0000 q^{86} +10.0000 q^{87} +4.00000i q^{88} -6.00000 q^{89} +2.00000i q^{90} -8.00000 q^{93} -8.00000i q^{95} +1.00000i q^{96} +14.0000i q^{97} -7.00000 q^{98} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{8} - 2 q^{9} - 4 q^{13} + 4 q^{15} + 2 q^{16} + 2 q^{18} - 8 q^{19} + 2 q^{25} + 4 q^{26} - 4 q^{30} - 2 q^{32} - 8 q^{33} - 2 q^{36} + 8 q^{38} - 24 q^{43} + 14 q^{49} - 2 q^{50} - 4 q^{52} - 12 q^{53} + 16 q^{55} - 24 q^{59} + 4 q^{60} + 2 q^{64} + 8 q^{66} - 24 q^{67} + 2 q^{72} - 8 q^{76} + 2 q^{81} - 8 q^{83} + 24 q^{86} + 20 q^{87} - 12 q^{89} - 16 q^{93} - 14 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1734\mathbb{Z}\right)^\times\).

\(n\) \(1157\) \(1159\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) − 1.00000i − 0.577350i
\(4\) 1.00000 0.500000
\(5\) 2.00000i 0.894427i 0.894427 + 0.447214i \(0.147584\pi\)
−0.894427 + 0.447214i \(0.852416\pi\)
\(6\) 1.00000i 0.408248i
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −1.00000 −0.353553
\(9\) −1.00000 −0.333333
\(10\) − 2.00000i − 0.632456i
\(11\) − 4.00000i − 1.20605i −0.797724 0.603023i \(-0.793963\pi\)
0.797724 0.603023i \(-0.206037\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 1.00000 0.250000
\(17\) 0 0
\(18\) 1.00000 0.235702
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 2.00000i 0.447214i
\(21\) 0 0
\(22\) 4.00000i 0.852803i
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 1.00000i 0.204124i
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 10.0000i 1.85695i 0.371391 + 0.928477i \(0.378881\pi\)
−0.371391 + 0.928477i \(0.621119\pi\)
\(30\) −2.00000 −0.365148
\(31\) − 8.00000i − 1.43684i −0.695608 0.718421i \(-0.744865\pi\)
0.695608 0.718421i \(-0.255135\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.00000 −0.696311
\(34\) 0 0
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 4.00000 0.648886
\(39\) 2.00000i 0.320256i
\(40\) − 2.00000i − 0.316228i
\(41\) 10.0000i 1.56174i 0.624695 + 0.780869i \(0.285223\pi\)
−0.624695 + 0.780869i \(0.714777\pi\)
\(42\) 0 0
\(43\) −12.0000 −1.82998 −0.914991 0.403473i \(-0.867803\pi\)
−0.914991 + 0.403473i \(0.867803\pi\)
\(44\) − 4.00000i − 0.603023i
\(45\) − 2.00000i − 0.298142i
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 7.00000 1.00000
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) 8.00000 1.07872
\(56\) 0 0
\(57\) 4.00000i 0.529813i
\(58\) − 10.0000i − 1.31306i
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 2.00000 0.258199
\(61\) − 10.0000i − 1.28037i −0.768221 0.640184i \(-0.778858\pi\)
0.768221 0.640184i \(-0.221142\pi\)
\(62\) 8.00000i 1.01600i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) − 4.00000i − 0.496139i
\(66\) 4.00000 0.492366
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 1.00000 0.117851
\(73\) − 10.0000i − 1.17041i −0.810885 0.585206i \(-0.801014\pi\)
0.810885 0.585206i \(-0.198986\pi\)
\(74\) − 2.00000i − 0.232495i
\(75\) − 1.00000i − 0.115470i
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) − 2.00000i − 0.226455i
\(79\) − 8.00000i − 0.900070i −0.893011 0.450035i \(-0.851411\pi\)
0.893011 0.450035i \(-0.148589\pi\)
\(80\) 2.00000i 0.223607i
\(81\) 1.00000 0.111111
\(82\) − 10.0000i − 1.10432i
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 10.0000 1.07211
\(88\) 4.00000i 0.426401i
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 2.00000i 0.210819i
\(91\) 0 0
\(92\) 0 0
\(93\) −8.00000 −0.829561
\(94\) 0 0
\(95\) − 8.00000i − 0.820783i
\(96\) 1.00000i 0.102062i
\(97\) 14.0000i 1.42148i 0.703452 + 0.710742i \(0.251641\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) −7.00000 −0.707107
\(99\) 4.00000i 0.402015i
\(100\) 1.00000 0.100000
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 4.00000i 0.386695i 0.981130 + 0.193347i \(0.0619344\pi\)
−0.981130 + 0.193347i \(0.938066\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) − 10.0000i − 0.957826i −0.877862 0.478913i \(-0.841031\pi\)
0.877862 0.478913i \(-0.158969\pi\)
\(110\) −8.00000 −0.762770
\(111\) 2.00000 0.189832
\(112\) 0 0
\(113\) 2.00000i 0.188144i 0.995565 + 0.0940721i \(0.0299884\pi\)
−0.995565 + 0.0940721i \(0.970012\pi\)
\(114\) − 4.00000i − 0.374634i
\(115\) 0 0
\(116\) 10.0000i 0.928477i
\(117\) 2.00000 0.184900
\(118\) 12.0000 1.10469
\(119\) 0 0
\(120\) −2.00000 −0.182574
\(121\) −5.00000 −0.454545
\(122\) 10.0000i 0.905357i
\(123\) 10.0000 0.901670
\(124\) − 8.00000i − 0.718421i
\(125\) 12.0000i 1.07331i
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 12.0000i 1.05654i
\(130\) 4.00000i 0.350823i
\(131\) 12.0000i 1.04844i 0.851581 + 0.524222i \(0.175644\pi\)
−0.851581 + 0.524222i \(0.824356\pi\)
\(132\) −4.00000 −0.348155
\(133\) 0 0
\(134\) 12.0000 1.03664
\(135\) −2.00000 −0.172133
\(136\) 0 0
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) 4.00000i 0.339276i 0.985506 + 0.169638i \(0.0542598\pi\)
−0.985506 + 0.169638i \(0.945740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 8.00000i 0.668994i
\(144\) −1.00000 −0.0833333
\(145\) −20.0000 −1.66091
\(146\) 10.0000i 0.827606i
\(147\) − 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 \(0.634337\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) −24.0000 −1.95309 −0.976546 0.215308i \(-0.930924\pi\)
−0.976546 + 0.215308i \(0.930924\pi\)
\(152\) 4.00000 0.324443
\(153\) 0 0
\(154\) 0 0
\(155\) 16.0000 1.28515
\(156\) 2.00000i 0.160128i
\(157\) −2.00000 −0.159617 −0.0798087 0.996810i \(-0.525431\pi\)
−0.0798087 + 0.996810i \(0.525431\pi\)
\(158\) 8.00000i 0.636446i
\(159\) 6.00000i 0.475831i
\(160\) − 2.00000i − 0.158114i
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 4.00000i 0.313304i 0.987654 + 0.156652i \(0.0500701\pi\)
−0.987654 + 0.156652i \(0.949930\pi\)
\(164\) 10.0000i 0.780869i
\(165\) − 8.00000i − 0.622799i
\(166\) 4.00000 0.310460
\(167\) − 16.0000i − 1.23812i −0.785345 0.619059i \(-0.787514\pi\)
0.785345 0.619059i \(-0.212486\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 4.00000 0.305888
\(172\) −12.0000 −0.914991
\(173\) − 6.00000i − 0.456172i −0.973641 0.228086i \(-0.926753\pi\)
0.973641 0.228086i \(-0.0732467\pi\)
\(174\) −10.0000 −0.758098
\(175\) 0 0
\(176\) − 4.00000i − 0.301511i
\(177\) 12.0000i 0.901975i
\(178\) 6.00000 0.449719
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) − 2.00000i − 0.149071i
\(181\) 14.0000i 1.04061i 0.853980 + 0.520306i \(0.174182\pi\)
−0.853980 + 0.520306i \(0.825818\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 8.00000 0.586588
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 8.00000i 0.580381i
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) 18.0000i 1.29567i 0.761781 + 0.647834i \(0.224325\pi\)
−0.761781 + 0.647834i \(0.775675\pi\)
\(194\) − 14.0000i − 1.00514i
\(195\) −4.00000 −0.286446
\(196\) 7.00000 0.500000
\(197\) 14.0000i 0.997459i 0.866758 + 0.498729i \(0.166200\pi\)
−0.866758 + 0.498729i \(0.833800\pi\)
\(198\) − 4.00000i − 0.284268i
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 12.0000i 0.846415i
\(202\) 10.0000 0.703598
\(203\) 0 0
\(204\) 0 0
\(205\) −20.0000 −1.39686
\(206\) 8.00000 0.557386
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 16.0000i 1.10674i
\(210\) 0 0
\(211\) − 28.0000i − 1.92760i −0.266627 0.963800i \(-0.585909\pi\)
0.266627 0.963800i \(-0.414091\pi\)
\(212\) −6.00000 −0.412082
\(213\) 0 0
\(214\) − 4.00000i − 0.273434i
\(215\) − 24.0000i − 1.63679i
\(216\) − 1.00000i − 0.0680414i
\(217\) 0 0
\(218\) 10.0000i 0.677285i
\(219\) −10.0000 −0.675737
\(220\) 8.00000 0.539360
\(221\) 0 0
\(222\) −2.00000 −0.134231
\(223\) −16.0000 −1.07144 −0.535720 0.844396i \(-0.679960\pi\)
−0.535720 + 0.844396i \(0.679960\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) − 2.00000i − 0.133038i
\(227\) 4.00000i 0.265489i 0.991150 + 0.132745i \(0.0423790\pi\)
−0.991150 + 0.132745i \(0.957621\pi\)
\(228\) 4.00000i 0.264906i
\(229\) 26.0000 1.71813 0.859064 0.511868i \(-0.171046\pi\)
0.859064 + 0.511868i \(0.171046\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 10.0000i − 0.656532i
\(233\) − 26.0000i − 1.70332i −0.524097 0.851658i \(-0.675597\pi\)
0.524097 0.851658i \(-0.324403\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) −12.0000 −0.781133
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 2.00000 0.129099
\(241\) − 2.00000i − 0.128831i −0.997923 0.0644157i \(-0.979482\pi\)
0.997923 0.0644157i \(-0.0205183\pi\)
\(242\) 5.00000 0.321412
\(243\) − 1.00000i − 0.0641500i
\(244\) − 10.0000i − 0.640184i
\(245\) 14.0000i 0.894427i
\(246\) −10.0000 −0.637577
\(247\) 8.00000 0.509028
\(248\) 8.00000i 0.508001i
\(249\) 4.00000i 0.253490i
\(250\) − 12.0000i − 0.758947i
\(251\) 28.0000 1.76734 0.883672 0.468106i \(-0.155064\pi\)
0.883672 + 0.468106i \(0.155064\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −2.00000 −0.124757 −0.0623783 0.998053i \(-0.519869\pi\)
−0.0623783 + 0.998053i \(0.519869\pi\)
\(258\) − 12.0000i − 0.747087i
\(259\) 0 0
\(260\) − 4.00000i − 0.248069i
\(261\) − 10.0000i − 0.618984i
\(262\) − 12.0000i − 0.741362i
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 4.00000 0.246183
\(265\) − 12.0000i − 0.737154i
\(266\) 0 0
\(267\) 6.00000i 0.367194i
\(268\) −12.0000 −0.733017
\(269\) − 6.00000i − 0.365826i −0.983129 0.182913i \(-0.941447\pi\)
0.983129 0.182913i \(-0.0585527\pi\)
\(270\) 2.00000 0.121716
\(271\) 16.0000 0.971931 0.485965 0.873978i \(-0.338468\pi\)
0.485965 + 0.873978i \(0.338468\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −10.0000 −0.604122
\(275\) − 4.00000i − 0.241209i
\(276\) 0 0
\(277\) − 30.0000i − 1.80253i −0.433273 0.901263i \(-0.642641\pi\)
0.433273 0.901263i \(-0.357359\pi\)
\(278\) − 4.00000i − 0.239904i
\(279\) 8.00000i 0.478947i
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 12.0000i 0.713326i 0.934233 + 0.356663i \(0.116086\pi\)
−0.934233 + 0.356663i \(0.883914\pi\)
\(284\) 0 0
\(285\) −8.00000 −0.473879
\(286\) − 8.00000i − 0.473050i
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 0 0
\(290\) 20.0000 1.17444
\(291\) 14.0000 0.820695
\(292\) − 10.0000i − 0.585206i
\(293\) −26.0000 −1.51894 −0.759468 0.650545i \(-0.774541\pi\)
−0.759468 + 0.650545i \(0.774541\pi\)
\(294\) 7.00000i 0.408248i
\(295\) − 24.0000i − 1.39733i
\(296\) − 2.00000i − 0.116248i
\(297\) 4.00000 0.232104
\(298\) 10.0000 0.579284
\(299\) 0 0
\(300\) − 1.00000i − 0.0577350i
\(301\) 0 0
\(302\) 24.0000 1.38104
\(303\) 10.0000i 0.574485i
\(304\) −4.00000 −0.229416
\(305\) 20.0000 1.14520
\(306\) 0 0
\(307\) −12.0000 −0.684876 −0.342438 0.939540i \(-0.611253\pi\)
−0.342438 + 0.939540i \(0.611253\pi\)
\(308\) 0 0
\(309\) 8.00000i 0.455104i
\(310\) −16.0000 −0.908739
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) − 2.00000i − 0.113228i
\(313\) 10.0000i 0.565233i 0.959233 + 0.282617i \(0.0912024\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) 2.00000 0.112867
\(315\) 0 0
\(316\) − 8.00000i − 0.450035i
\(317\) 6.00000i 0.336994i 0.985702 + 0.168497i \(0.0538913\pi\)
−0.985702 + 0.168497i \(0.946109\pi\)
\(318\) − 6.00000i − 0.336463i
\(319\) 40.0000 2.23957
\(320\) 2.00000i 0.111803i
\(321\) 4.00000 0.223258
\(322\) 0 0
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) − 4.00000i − 0.221540i
\(327\) −10.0000 −0.553001
\(328\) − 10.0000i − 0.552158i
\(329\) 0 0
\(330\) 8.00000i 0.440386i
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) −4.00000 −0.219529
\(333\) − 2.00000i − 0.109599i
\(334\) 16.0000i 0.875481i
\(335\) − 24.0000i − 1.31126i
\(336\) 0 0
\(337\) 14.0000i 0.762629i 0.924445 + 0.381314i \(0.124528\pi\)
−0.924445 + 0.381314i \(0.875472\pi\)
\(338\) 9.00000 0.489535
\(339\) 2.00000 0.108625
\(340\) 0 0
\(341\) −32.0000 −1.73290
\(342\) −4.00000 −0.216295
\(343\) 0 0
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 6.00000i 0.322562i
\(347\) 28.0000i 1.50312i 0.659665 + 0.751559i \(0.270698\pi\)
−0.659665 + 0.751559i \(0.729302\pi\)
\(348\) 10.0000 0.536056
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) − 2.00000i − 0.106752i
\(352\) 4.00000i 0.213201i
\(353\) −30.0000 −1.59674 −0.798369 0.602168i \(-0.794304\pi\)
−0.798369 + 0.602168i \(0.794304\pi\)
\(354\) − 12.0000i − 0.637793i
\(355\) 0 0
\(356\) −6.00000 −0.317999
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 2.00000i 0.105409i
\(361\) −3.00000 −0.157895
\(362\) − 14.0000i − 0.735824i
\(363\) 5.00000i 0.262432i
\(364\) 0 0
\(365\) 20.0000 1.04685
\(366\) 10.0000 0.522708
\(367\) 24.0000i 1.25279i 0.779506 + 0.626395i \(0.215470\pi\)
−0.779506 + 0.626395i \(0.784530\pi\)
\(368\) 0 0
\(369\) − 10.0000i − 0.520579i
\(370\) 4.00000 0.207950
\(371\) 0 0
\(372\) −8.00000 −0.414781
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 0 0
\(375\) 12.0000 0.619677
\(376\) 0 0
\(377\) − 20.0000i − 1.03005i
\(378\) 0 0
\(379\) 4.00000i 0.205466i 0.994709 + 0.102733i \(0.0327588\pi\)
−0.994709 + 0.102733i \(0.967241\pi\)
\(380\) − 8.00000i − 0.410391i
\(381\) 0 0
\(382\) 16.0000 0.818631
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) 0 0
\(386\) − 18.0000i − 0.916176i
\(387\) 12.0000 0.609994
\(388\) 14.0000i 0.710742i
\(389\) 26.0000 1.31825 0.659126 0.752032i \(-0.270926\pi\)
0.659126 + 0.752032i \(0.270926\pi\)
\(390\) 4.00000 0.202548
\(391\) 0 0
\(392\) −7.00000 −0.353553
\(393\) 12.0000 0.605320
\(394\) − 14.0000i − 0.705310i
\(395\) 16.0000 0.805047
\(396\) 4.00000i 0.201008i
\(397\) − 26.0000i − 1.30490i −0.757831 0.652451i \(-0.773741\pi\)
0.757831 0.652451i \(-0.226259\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) − 14.0000i − 0.699127i −0.936913 0.349563i \(-0.886330\pi\)
0.936913 0.349563i \(-0.113670\pi\)
\(402\) − 12.0000i − 0.598506i
\(403\) 16.0000i 0.797017i
\(404\) −10.0000 −0.497519
\(405\) 2.00000i 0.0993808i
\(406\) 0 0
\(407\) 8.00000 0.396545
\(408\) 0 0
\(409\) 26.0000 1.28562 0.642809 0.766027i \(-0.277769\pi\)
0.642809 + 0.766027i \(0.277769\pi\)
\(410\) 20.0000 0.987730
\(411\) − 10.0000i − 0.493264i
\(412\) −8.00000 −0.394132
\(413\) 0 0
\(414\) 0 0
\(415\) − 8.00000i − 0.392705i
\(416\) 2.00000 0.0980581
\(417\) 4.00000 0.195881
\(418\) − 16.0000i − 0.782586i
\(419\) 4.00000i 0.195413i 0.995215 + 0.0977064i \(0.0311506\pi\)
−0.995215 + 0.0977064i \(0.968849\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 28.0000i 1.36302i
\(423\) 0 0
\(424\) 6.00000 0.291386
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 4.00000i 0.193347i
\(429\) 8.00000 0.386244
\(430\) 24.0000i 1.15738i
\(431\) − 8.00000i − 0.385346i −0.981263 0.192673i \(-0.938284\pi\)
0.981263 0.192673i \(-0.0617157\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 20.0000i 0.958927i
\(436\) − 10.0000i − 0.478913i
\(437\) 0 0
\(438\) 10.0000 0.477818
\(439\) − 16.0000i − 0.763638i −0.924237 0.381819i \(-0.875298\pi\)
0.924237 0.381819i \(-0.124702\pi\)
\(440\) −8.00000 −0.381385
\(441\) −7.00000 −0.333333
\(442\) 0 0
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 2.00000 0.0949158
\(445\) − 12.0000i − 0.568855i
\(446\) 16.0000 0.757622
\(447\) 10.0000i 0.472984i
\(448\) 0 0
\(449\) 2.00000i 0.0943858i 0.998886 + 0.0471929i \(0.0150276\pi\)
−0.998886 + 0.0471929i \(0.984972\pi\)
\(450\) 1.00000 0.0471405
\(451\) 40.0000 1.88353
\(452\) 2.00000i 0.0940721i
\(453\) 24.0000i 1.12762i
\(454\) − 4.00000i − 0.187729i
\(455\) 0 0
\(456\) − 4.00000i − 0.187317i
\(457\) −10.0000 −0.467780 −0.233890 0.972263i \(-0.575146\pi\)
−0.233890 + 0.972263i \(0.575146\pi\)
\(458\) −26.0000 −1.21490
\(459\) 0 0
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) 32.0000 1.48717 0.743583 0.668644i \(-0.233125\pi\)
0.743583 + 0.668644i \(0.233125\pi\)
\(464\) 10.0000i 0.464238i
\(465\) − 16.0000i − 0.741982i
\(466\) 26.0000i 1.20443i
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 2.00000 0.0924500
\(469\) 0 0
\(470\) 0 0
\(471\) 2.00000i 0.0921551i
\(472\) 12.0000 0.552345
\(473\) 48.0000i 2.20704i
\(474\) 8.00000 0.367452
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) − 24.0000i − 1.09659i −0.836286 0.548294i \(-0.815277\pi\)
0.836286 0.548294i \(-0.184723\pi\)
\(480\) −2.00000 −0.0912871
\(481\) − 4.00000i − 0.182384i
\(482\) 2.00000i 0.0910975i
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) −28.0000 −1.27141
\(486\) 1.00000i 0.0453609i
\(487\) − 16.0000i − 0.725029i −0.931978 0.362515i \(-0.881918\pi\)
0.931978 0.362515i \(-0.118082\pi\)
\(488\) 10.0000i 0.452679i
\(489\) 4.00000 0.180886
\(490\) − 14.0000i − 0.632456i
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 10.0000 0.450835
\(493\) 0 0
\(494\) −8.00000 −0.359937
\(495\) −8.00000 −0.359573
\(496\) − 8.00000i − 0.359211i
\(497\) 0 0
\(498\) − 4.00000i − 0.179244i
\(499\) 4.00000i 0.179065i 0.995984 + 0.0895323i \(0.0285372\pi\)
−0.995984 + 0.0895323i \(0.971463\pi\)
\(500\) 12.0000i 0.536656i
\(501\) −16.0000 −0.714827
\(502\) −28.0000 −1.24970
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) − 20.0000i − 0.889988i
\(506\) 0 0
\(507\) 9.00000i 0.399704i
\(508\) 0 0
\(509\) 30.0000 1.32973 0.664863 0.746965i \(-0.268490\pi\)
0.664863 + 0.746965i \(0.268490\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) − 4.00000i − 0.176604i
\(514\) 2.00000 0.0882162
\(515\) − 16.0000i − 0.705044i
\(516\) 12.0000i 0.528271i
\(517\) 0 0
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 4.00000i 0.175412i
\(521\) − 22.0000i − 0.963837i −0.876216 0.481919i \(-0.839940\pi\)
0.876216 0.481919i \(-0.160060\pi\)
\(522\) 10.0000i 0.437688i
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 12.0000i 0.524222i
\(525\) 0 0
\(526\) −8.00000 −0.348817
\(527\) 0 0
\(528\) −4.00000 −0.174078
\(529\) 23.0000 1.00000
\(530\) 12.0000i 0.521247i
\(531\) 12.0000 0.520756
\(532\) 0 0
\(533\) − 20.0000i − 0.866296i
\(534\) − 6.00000i − 0.259645i
\(535\) −8.00000 −0.345870
\(536\) 12.0000 0.518321
\(537\) − 12.0000i − 0.517838i
\(538\) 6.00000i 0.258678i
\(539\) − 28.0000i − 1.20605i
\(540\) −2.00000 −0.0860663
\(541\) 10.0000i 0.429934i 0.976621 + 0.214967i \(0.0689643\pi\)
−0.976621 + 0.214967i \(0.931036\pi\)
\(542\) −16.0000 −0.687259
\(543\) 14.0000 0.600798
\(544\) 0 0
\(545\) 20.0000 0.856706
\(546\) 0 0
\(547\) 12.0000i 0.513083i 0.966533 + 0.256541i \(0.0825830\pi\)
−0.966533 + 0.256541i \(0.917417\pi\)
\(548\) 10.0000 0.427179
\(549\) 10.0000i 0.426790i
\(550\) 4.00000i 0.170561i
\(551\) − 40.0000i − 1.70406i
\(552\) 0 0
\(553\) 0 0
\(554\) 30.0000i 1.27458i
\(555\) 4.00000i 0.169791i
\(556\) 4.00000i 0.169638i
\(557\) −34.0000 −1.44063 −0.720313 0.693649i \(-0.756002\pi\)
−0.720313 + 0.693649i \(0.756002\pi\)
\(558\) − 8.00000i − 0.338667i
\(559\) 24.0000 1.01509
\(560\) 0 0
\(561\) 0 0
\(562\) −6.00000 −0.253095
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 0 0
\(565\) −4.00000 −0.168281
\(566\) − 12.0000i − 0.504398i
\(567\) 0 0
\(568\) 0 0
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 8.00000 0.335083
\(571\) 28.0000i 1.17176i 0.810397 + 0.585882i \(0.199252\pi\)
−0.810397 + 0.585882i \(0.800748\pi\)
\(572\) 8.00000i 0.334497i
\(573\) 16.0000i 0.668410i
\(574\) 0 0
\(575\) 0 0
\(576\) −1.00000 −0.0416667
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) 18.0000 0.748054
\(580\) −20.0000 −0.830455
\(581\) 0 0
\(582\) −14.0000 −0.580319
\(583\) 24.0000i 0.993978i
\(584\) 10.0000i 0.413803i
\(585\) 4.00000i 0.165380i
\(586\) 26.0000 1.07405
\(587\) 20.0000 0.825488 0.412744 0.910847i \(-0.364570\pi\)
0.412744 + 0.910847i \(0.364570\pi\)
\(588\) − 7.00000i − 0.288675i
\(589\) 32.0000i 1.31854i
\(590\) 24.0000i 0.988064i
\(591\) 14.0000 0.575883
\(592\) 2.00000i 0.0821995i
\(593\) 14.0000 0.574911 0.287456 0.957794i \(-0.407191\pi\)
0.287456 + 0.957794i \(0.407191\pi\)
\(594\) −4.00000 −0.164122
\(595\) 0 0
\(596\) −10.0000 −0.409616
\(597\) 0 0
\(598\) 0 0
\(599\) −8.00000 −0.326871 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) 1.00000i 0.0408248i
\(601\) − 6.00000i − 0.244745i −0.992484 0.122373i \(-0.960950\pi\)
0.992484 0.122373i \(-0.0390503\pi\)
\(602\) 0 0
\(603\) 12.0000 0.488678
\(604\) −24.0000 −0.976546
\(605\) − 10.0000i − 0.406558i
\(606\) − 10.0000i − 0.406222i
\(607\) 24.0000i 0.974130i 0.873366 + 0.487065i \(0.161933\pi\)
−0.873366 + 0.487065i \(0.838067\pi\)
\(608\) 4.00000 0.162221
\(609\) 0 0
\(610\) −20.0000 −0.809776
\(611\) 0 0
\(612\) 0 0
\(613\) 38.0000 1.53481 0.767403 0.641165i \(-0.221549\pi\)
0.767403 + 0.641165i \(0.221549\pi\)
\(614\) 12.0000 0.484281
\(615\) 20.0000i 0.806478i
\(616\) 0 0
\(617\) 6.00000i 0.241551i 0.992680 + 0.120775i \(0.0385381\pi\)
−0.992680 + 0.120775i \(0.961462\pi\)
\(618\) − 8.00000i − 0.321807i
\(619\) − 20.0000i − 0.803868i −0.915669 0.401934i \(-0.868338\pi\)
0.915669 0.401934i \(-0.131662\pi\)
\(620\) 16.0000 0.642575
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 2.00000i 0.0800641i
\(625\) −19.0000 −0.760000
\(626\) − 10.0000i − 0.399680i
\(627\) 16.0000 0.638978
\(628\) −2.00000 −0.0798087
\(629\) 0 0
\(630\) 0 0
\(631\) 24.0000 0.955425 0.477712 0.878516i \(-0.341466\pi\)
0.477712 + 0.878516i \(0.341466\pi\)
\(632\) 8.00000i 0.318223i
\(633\) −28.0000 −1.11290
\(634\) − 6.00000i − 0.238290i
\(635\) 0 0
\(636\) 6.00000i 0.237915i
\(637\) −14.0000 −0.554700
\(638\) −40.0000 −1.58362
\(639\) 0 0
\(640\) − 2.00000i − 0.0790569i
\(641\) − 2.00000i − 0.0789953i −0.999220 0.0394976i \(-0.987424\pi\)
0.999220 0.0394976i \(-0.0125758\pi\)
\(642\) −4.00000 −0.157867
\(643\) 28.0000i 1.10421i 0.833774 + 0.552106i \(0.186176\pi\)
−0.833774 + 0.552106i \(0.813824\pi\)
\(644\) 0 0
\(645\) −24.0000 −0.944999
\(646\) 0 0
\(647\) −40.0000 −1.57256 −0.786281 0.617869i \(-0.787996\pi\)
−0.786281 + 0.617869i \(0.787996\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 48.0000i 1.88416i
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) 4.00000i 0.156652i
\(653\) 6.00000i 0.234798i 0.993085 + 0.117399i \(0.0374557\pi\)
−0.993085 + 0.117399i \(0.962544\pi\)
\(654\) 10.0000 0.391031
\(655\) −24.0000 −0.937758
\(656\) 10.0000i 0.390434i
\(657\) 10.0000i 0.390137i
\(658\) 0 0
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) − 8.00000i − 0.311400i
\(661\) −6.00000 −0.233373 −0.116686 0.993169i \(-0.537227\pi\)
−0.116686 + 0.993169i \(0.537227\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) 4.00000 0.155230
\(665\) 0 0
\(666\) 2.00000i 0.0774984i
\(667\) 0 0
\(668\) − 16.0000i − 0.619059i
\(669\) 16.0000i 0.618596i
\(670\) 24.0000i 0.927201i
\(671\) −40.0000 −1.54418
\(672\) 0 0
\(673\) − 46.0000i − 1.77317i −0.462566 0.886585i \(-0.653071\pi\)
0.462566 0.886585i \(-0.346929\pi\)
\(674\) − 14.0000i − 0.539260i
\(675\) 1.00000i 0.0384900i
\(676\) −9.00000 −0.346154
\(677\) − 46.0000i − 1.76792i −0.467559 0.883962i \(-0.654866\pi\)
0.467559 0.883962i \(-0.345134\pi\)
\(678\) −2.00000 −0.0768095
\(679\) 0 0
\(680\) 0 0
\(681\) 4.00000 0.153280
\(682\) 32.0000 1.22534
\(683\) 20.0000i 0.765279i 0.923898 + 0.382639i \(0.124985\pi\)
−0.923898 + 0.382639i \(0.875015\pi\)
\(684\) 4.00000 0.152944
\(685\) 20.0000i 0.764161i
\(686\) 0 0
\(687\) − 26.0000i − 0.991962i
\(688\) −12.0000 −0.457496
\(689\) 12.0000 0.457164
\(690\) 0 0
\(691\) − 28.0000i − 1.06517i −0.846376 0.532585i \(-0.821221\pi\)
0.846376 0.532585i \(-0.178779\pi\)
\(692\) − 6.00000i − 0.228086i
\(693\) 0 0
\(694\) − 28.0000i − 1.06287i
\(695\) −8.00000 −0.303457
\(696\) −10.0000 −0.379049
\(697\) 0 0
\(698\) 14.0000 0.529908
\(699\) −26.0000 −0.983410
\(700\) 0 0
\(701\) 46.0000 1.73740 0.868698 0.495342i \(-0.164957\pi\)
0.868698 + 0.495342i \(0.164957\pi\)
\(702\) 2.00000i 0.0754851i
\(703\) − 8.00000i − 0.301726i
\(704\) − 4.00000i − 0.150756i
\(705\) 0 0
\(706\) 30.0000 1.12906
\(707\) 0 0
\(708\) 12.0000i 0.450988i
\(709\) − 46.0000i − 1.72757i −0.503864 0.863783i \(-0.668089\pi\)
0.503864 0.863783i \(-0.331911\pi\)
\(710\) 0 0
\(711\) 8.00000i 0.300023i
\(712\) 6.00000 0.224860
\(713\) 0 0
\(714\) 0 0
\(715\) −16.0000 −0.598366
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) −24.0000 −0.895672
\(719\) − 40.0000i − 1.49175i −0.666087 0.745874i \(-0.732032\pi\)
0.666087 0.745874i \(-0.267968\pi\)
\(720\) − 2.00000i − 0.0745356i
\(721\) 0 0
\(722\) 3.00000 0.111648
\(723\) −2.00000 −0.0743808
\(724\) 14.0000i 0.520306i
\(725\) 10.0000i 0.371391i
\(726\) − 5.00000i − 0.185567i
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) −20.0000 −0.740233
\(731\) 0 0
\(732\) −10.0000 −0.369611
\(733\) −46.0000 −1.69905 −0.849524 0.527549i \(-0.823111\pi\)
−0.849524 + 0.527549i \(0.823111\pi\)
\(734\) − 24.0000i − 0.885856i
\(735\) 14.0000 0.516398
\(736\) 0 0
\(737\) 48.0000i 1.76810i
\(738\) 10.0000i 0.368105i
\(739\) −52.0000 −1.91285 −0.956425 0.291977i \(-0.905687\pi\)
−0.956425 + 0.291977i \(0.905687\pi\)
\(740\) −4.00000 −0.147043
\(741\) − 8.00000i − 0.293887i
\(742\) 0 0
\(743\) 16.0000i 0.586983i 0.955962 + 0.293492i \(0.0948173\pi\)
−0.955962 + 0.293492i \(0.905183\pi\)
\(744\) 8.00000 0.293294
\(745\) − 20.0000i − 0.732743i
\(746\) −6.00000 −0.219676
\(747\) 4.00000 0.146352
\(748\) 0 0
\(749\) 0 0
\(750\) −12.0000 −0.438178
\(751\) − 8.00000i − 0.291924i −0.989290 0.145962i \(-0.953372\pi\)
0.989290 0.145962i \(-0.0466277\pi\)
\(752\) 0 0
\(753\) − 28.0000i − 1.02038i
\(754\) 20.0000i 0.728357i
\(755\) − 48.0000i − 1.74690i
\(756\) 0 0
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) − 4.00000i − 0.145287i
\(759\) 0 0
\(760\) 8.00000i 0.290191i
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −16.0000 −0.578860
\(765\) 0 0
\(766\) −16.0000 −0.578103
\(767\) 24.0000 0.866590
\(768\) − 1.00000i − 0.0360844i
\(769\) −30.0000 −1.08183 −0.540914 0.841078i \(-0.681921\pi\)
−0.540914 + 0.841078i \(0.681921\pi\)
\(770\) 0 0
\(771\) 2.00000i 0.0720282i
\(772\) 18.0000i 0.647834i
\(773\) −38.0000 −1.36677 −0.683383 0.730061i \(-0.739492\pi\)
−0.683383 + 0.730061i \(0.739492\pi\)
\(774\) −12.0000 −0.431331
\(775\) − 8.00000i − 0.287368i
\(776\) − 14.0000i − 0.502571i
\(777\) 0 0
\(778\) −26.0000 −0.932145
\(779\) − 40.0000i − 1.43315i
\(780\) −4.00000 −0.143223
\(781\) 0 0
\(782\) 0 0
\(783\) −10.0000 −0.357371
\(784\) 7.00000 0.250000
\(785\) − 4.00000i − 0.142766i
\(786\) −12.0000 −0.428026
\(787\) − 4.00000i − 0.142585i −0.997455 0.0712923i \(-0.977288\pi\)
0.997455 0.0712923i \(-0.0227123\pi\)
\(788\) 14.0000i 0.498729i
\(789\) − 8.00000i − 0.284808i
\(790\) −16.0000 −0.569254
\(791\) 0 0
\(792\) − 4.00000i − 0.142134i
\(793\) 20.0000i 0.710221i
\(794\) 26.0000i 0.922705i
\(795\) −12.0000 −0.425596
\(796\) 0 0
\(797\) −14.0000 −0.495905 −0.247953 0.968772i \(-0.579758\pi\)
−0.247953 + 0.968772i \(0.579758\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.00000 −0.0353553
\(801\) 6.00000 0.212000
\(802\) 14.0000i 0.494357i
\(803\) −40.0000 −1.41157
\(804\) 12.0000i 0.423207i
\(805\) 0 0
\(806\) − 16.0000i − 0.563576i
\(807\) −6.00000 −0.211210
\(808\) 10.0000 0.351799
\(809\) − 22.0000i − 0.773479i −0.922189 0.386739i \(-0.873601\pi\)
0.922189 0.386739i \(-0.126399\pi\)
\(810\) − 2.00000i − 0.0702728i
\(811\) 20.0000i 0.702295i 0.936320 + 0.351147i \(0.114208\pi\)
−0.936320 + 0.351147i \(0.885792\pi\)
\(812\) 0 0
\(813\) − 16.0000i − 0.561144i
\(814\) −8.00000 −0.280400
\(815\) −8.00000 −0.280228
\(816\) 0 0
\(817\) 48.0000 1.67931
\(818\) −26.0000 −0.909069
\(819\) 0 0
\(820\) −20.0000 −0.698430
\(821\) 34.0000i 1.18661i 0.804978 + 0.593304i \(0.202177\pi\)
−0.804978 + 0.593304i \(0.797823\pi\)
\(822\) 10.0000i 0.348790i
\(823\) 32.0000i 1.11545i 0.830026 + 0.557725i \(0.188326\pi\)
−0.830026 + 0.557725i \(0.811674\pi\)
\(824\) 8.00000 0.278693
\(825\) −4.00000 −0.139262
\(826\) 0 0
\(827\) − 20.0000i − 0.695468i −0.937593 0.347734i \(-0.886951\pi\)
0.937593 0.347734i \(-0.113049\pi\)
\(828\) 0 0
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 8.00000i 0.277684i
\(831\) −30.0000 −1.04069
\(832\) −2.00000 −0.0693375
\(833\) 0 0
\(834\) −4.00000 −0.138509
\(835\) 32.0000 1.10741
\(836\) 16.0000i 0.553372i
\(837\) 8.00000 0.276520
\(838\) − 4.00000i − 0.138178i
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) −71.0000 −2.44828
\(842\) −22.0000 −0.758170
\(843\) − 6.00000i − 0.206651i
\(844\) − 28.0000i − 0.963800i
\(845\) − 18.0000i − 0.619219i
\(846\) 0 0
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) 12.0000 0.411839
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 2.00000i 0.0684787i 0.999414 + 0.0342393i \(0.0109009\pi\)
−0.999414 + 0.0342393i \(0.989099\pi\)
\(854\) 0 0
\(855\) 8.00000i 0.273594i
\(856\) − 4.00000i − 0.136717i
\(857\) − 6.00000i − 0.204956i −0.994735 0.102478i \(-0.967323\pi\)
0.994735 0.102478i \(-0.0326771\pi\)
\(858\) −8.00000 −0.273115
\(859\) 36.0000 1.22830 0.614152 0.789188i \(-0.289498\pi\)
0.614152 + 0.789188i \(0.289498\pi\)
\(860\) − 24.0000i − 0.818393i
\(861\) 0 0
\(862\) 8.00000i 0.272481i
\(863\) −32.0000 −1.08929 −0.544646 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) − 1.00000i − 0.0340207i
\(865\) 12.0000 0.408012
\(866\) −14.0000 −0.475739
\(867\) 0 0
\(868\) 0 0
\(869\) −32.0000 −1.08553
\(870\) − 20.0000i − 0.678064i
\(871\) 24.0000 0.813209
\(872\) 10.0000i 0.338643i
\(873\) − 14.0000i − 0.473828i
\(874\) 0 0
\(875\) 0 0
\(876\) −10.0000 −0.337869
\(877\) 22.0000i 0.742887i 0.928456 + 0.371444i \(0.121137\pi\)
−0.928456 + 0.371444i \(0.878863\pi\)
\(878\) 16.0000i 0.539974i
\(879\) 26.0000i 0.876958i
\(880\) 8.00000 0.269680
\(881\) − 2.00000i − 0.0673817i −0.999432 0.0336909i \(-0.989274\pi\)
0.999432 0.0336909i \(-0.0107262\pi\)
\(882\) 7.00000 0.235702
\(883\) 36.0000 1.21150 0.605748 0.795656i \(-0.292874\pi\)
0.605748 + 0.795656i \(0.292874\pi\)
\(884\) 0 0
\(885\) −24.0000 −0.806751
\(886\) 4.00000 0.134383
\(887\) 48.0000i 1.61168i 0.592132 + 0.805841i \(0.298286\pi\)
−0.592132 + 0.805841i \(0.701714\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 0 0
\(890\) 12.0000i 0.402241i
\(891\) − 4.00000i − 0.134005i
\(892\) −16.0000 −0.535720
\(893\) 0 0
\(894\) − 10.0000i − 0.334450i
\(895\) 24.0000i 0.802232i
\(896\) 0 0
\(897\) 0 0
\(898\) − 2.00000i − 0.0667409i
\(899\) 80.0000 2.66815
\(900\) −1.00000 −0.0333333
\(901\) 0 0
\(902\) −40.0000 −1.33185
\(903\) 0 0
\(904\) − 2.00000i − 0.0665190i
\(905\) −28.0000 −0.930751
\(906\) − 24.0000i − 0.797347i
\(907\) 28.0000i 0.929725i 0.885383 + 0.464862i \(0.153896\pi\)
−0.885383 + 0.464862i \(0.846104\pi\)
\(908\) 4.00000i 0.132745i
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) 8.00000i 0.265052i 0.991180 + 0.132526i \(0.0423088\pi\)
−0.991180 + 0.132526i \(0.957691\pi\)
\(912\) 4.00000i 0.132453i
\(913\) 16.0000i 0.529523i
\(914\) 10.0000 0.330771
\(915\) − 20.0000i − 0.661180i
\(916\) 26.0000 0.859064
\(917\) 0 0
\(918\) 0 0
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) 0 0
\(921\) 12.0000i 0.395413i
\(922\) 30.0000 0.987997
\(923\) 0 0
\(924\) 0 0
\(925\) 2.00000i 0.0657596i
\(926\) −32.0000 −1.05159
\(927\) 8.00000 0.262754
\(928\) − 10.0000i − 0.328266i
\(929\) 2.00000i 0.0656179i 0.999462 + 0.0328089i \(0.0104453\pi\)
−0.999462 + 0.0328089i \(0.989555\pi\)
\(930\) 16.0000i 0.524661i
\(931\) −28.0000 −0.917663
\(932\) − 26.0000i − 0.851658i
\(933\) 0 0
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) − 10.0000i − 0.325991i −0.986627 0.162995i \(-0.947884\pi\)
0.986627 0.162995i \(-0.0521156\pi\)
\(942\) − 2.00000i − 0.0651635i
\(943\) 0 0
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) − 48.0000i − 1.56061i
\(947\) 28.0000i 0.909878i 0.890523 + 0.454939i \(0.150339\pi\)
−0.890523 + 0.454939i \(0.849661\pi\)
\(948\) −8.00000 −0.259828
\(949\) 20.0000i 0.649227i
\(950\) 4.00000 0.129777
\(951\) 6.00000 0.194563
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) −6.00000 −0.194257
\(955\) − 32.0000i − 1.03550i
\(956\) 0 0
\(957\) − 40.0000i − 1.29302i
\(958\) 24.0000i 0.775405i
\(959\) 0 0
\(960\) 2.00000 0.0645497
\(961\) −33.0000 −1.06452
\(962\) 4.00000i 0.128965i
\(963\) − 4.00000i − 0.128898i
\(964\) − 2.00000i − 0.0644157i
\(965\) −36.0000 −1.15888
\(966\) 0 0
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 5.00000 0.160706
\(969\) 0 0
\(970\) 28.0000 0.899026
\(971\) 4.00000 0.128366 0.0641831 0.997938i \(-0.479556\pi\)
0.0641831 + 0.997938i \(0.479556\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) 0 0
\(974\) 16.0000i 0.512673i
\(975\) 2.00000i 0.0640513i
\(976\) − 10.0000i − 0.320092i
\(977\) 46.0000 1.47167 0.735835 0.677161i \(-0.236790\pi\)
0.735835 + 0.677161i \(0.236790\pi\)
\(978\) −4.00000 −0.127906
\(979\) 24.0000i 0.767043i
\(980\) 14.0000i 0.447214i
\(981\) 10.0000i 0.319275i
\(982\) 12.0000 0.382935
\(983\) − 48.0000i − 1.53096i −0.643458 0.765481i \(-0.722501\pi\)
0.643458 0.765481i \(-0.277499\pi\)
\(984\) −10.0000 −0.318788
\(985\) −28.0000 −0.892154
\(986\) 0 0
\(987\) 0 0
\(988\) 8.00000 0.254514
\(989\) 0 0
\(990\) 8.00000 0.254257
\(991\) 40.0000i 1.27064i 0.772248 + 0.635321i \(0.219132\pi\)
−0.772248 + 0.635321i \(0.780868\pi\)
\(992\) 8.00000i 0.254000i
\(993\) − 20.0000i − 0.634681i
\(994\) 0 0
\(995\) 0 0
\(996\) 4.00000i 0.126745i
\(997\) − 2.00000i − 0.0633406i −0.999498 0.0316703i \(-0.989917\pi\)
0.999498 0.0316703i \(-0.0100827\pi\)
\(998\) − 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 1734.2.b.b.577.1 2
17.2 even 8 1734.2.f.e.829.1 4
17.4 even 4 1734.2.a.j.1.1 1
17.8 even 8 1734.2.f.e.1483.1 4
17.9 even 8 1734.2.f.e.1483.2 4
17.13 even 4 102.2.a.c.1.1 1
17.15 even 8 1734.2.f.e.829.2 4
17.16 even 2 inner 1734.2.b.b.577.2 2
51.38 odd 4 5202.2.a.c.1.1 1
51.47 odd 4 306.2.a.b.1.1 1
68.47 odd 4 816.2.a.b.1.1 1
85.13 odd 4 2550.2.d.m.2449.1 2
85.47 odd 4 2550.2.d.m.2449.2 2
85.64 even 4 2550.2.a.c.1.1 1
119.13 odd 4 4998.2.a.be.1.1 1
136.13 even 4 3264.2.a.m.1.1 1
136.115 odd 4 3264.2.a.bc.1.1 1
204.47 even 4 2448.2.a.p.1.1 1
255.149 odd 4 7650.2.a.ca.1.1 1
408.149 odd 4 9792.2.a.k.1.1 1
408.251 even 4 9792.2.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
102.2.a.c.1.1 1 17.13 even 4
306.2.a.b.1.1 1 51.47 odd 4
816.2.a.b.1.1 1 68.47 odd 4
1734.2.a.j.1.1 1 17.4 even 4
1734.2.b.b.577.1 2 1.1 even 1 trivial
1734.2.b.b.577.2 2 17.16 even 2 inner
1734.2.f.e.829.1 4 17.2 even 8
1734.2.f.e.829.2 4 17.15 even 8
1734.2.f.e.1483.1 4 17.8 even 8
1734.2.f.e.1483.2 4 17.9 even 8
2448.2.a.p.1.1 1 204.47 even 4
2550.2.a.c.1.1 1 85.64 even 4
2550.2.d.m.2449.1 2 85.13 odd 4
2550.2.d.m.2449.2 2 85.47 odd 4
3264.2.a.m.1.1 1 136.13 even 4
3264.2.a.bc.1.1 1 136.115 odd 4
4998.2.a.be.1.1 1 119.13 odd 4
5202.2.a.c.1.1 1 51.38 odd 4
7650.2.a.ca.1.1 1 255.149 odd 4
9792.2.a.k.1.1 1 408.149 odd 4
9792.2.a.l.1.1 1 408.251 even 4