Properties

Label 975.2.c.c.274.2
Level $975$
Weight $2$
Character 975.274
Analytic conductor $7.785$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 975.274
Dual form 975.2.c.c.274.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000i q^{2} -1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} -1.00000i q^{7} -1.00000 q^{9} +O(q^{10})\) \(q+2.00000i q^{2} -1.00000i q^{3} -2.00000 q^{4} +2.00000 q^{6} -1.00000i q^{7} -1.00000 q^{9} +5.00000 q^{11} +2.00000i q^{12} +1.00000i q^{13} +2.00000 q^{14} -4.00000 q^{16} -7.00000i q^{17} -2.00000i q^{18} +6.00000 q^{19} -1.00000 q^{21} +10.0000i q^{22} -3.00000i q^{23} -2.00000 q^{26} +1.00000i q^{27} +2.00000i q^{28} -2.00000 q^{29} +2.00000 q^{31} -8.00000i q^{32} -5.00000i q^{33} +14.0000 q^{34} +2.00000 q^{36} +7.00000i q^{37} +12.0000i q^{38} +1.00000 q^{39} +9.00000 q^{41} -2.00000i q^{42} +8.00000i q^{43} -10.0000 q^{44} +6.00000 q^{46} +10.0000i q^{47} +4.00000i q^{48} +6.00000 q^{49} -7.00000 q^{51} -2.00000i q^{52} -5.00000i q^{53} -2.00000 q^{54} -6.00000i q^{57} -4.00000i q^{58} +5.00000 q^{61} +4.00000i q^{62} +1.00000i q^{63} +8.00000 q^{64} +10.0000 q^{66} -4.00000i q^{67} +14.0000i q^{68} -3.00000 q^{69} +9.00000 q^{71} +6.00000i q^{73} -14.0000 q^{74} -12.0000 q^{76} -5.00000i q^{77} +2.00000i q^{78} +3.00000 q^{79} +1.00000 q^{81} +18.0000i q^{82} +4.00000i q^{83} +2.00000 q^{84} -16.0000 q^{86} +2.00000i q^{87} -11.0000 q^{89} +1.00000 q^{91} +6.00000i q^{92} -2.00000i q^{93} -20.0000 q^{94} -8.00000 q^{96} -11.0000i q^{97} +12.0000i q^{98} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{4} + 4 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{4} + 4 q^{6} - 2 q^{9} + 10 q^{11} + 4 q^{14} - 8 q^{16} + 12 q^{19} - 2 q^{21} - 4 q^{26} - 4 q^{29} + 4 q^{31} + 28 q^{34} + 4 q^{36} + 2 q^{39} + 18 q^{41} - 20 q^{44} + 12 q^{46} + 12 q^{49} - 14 q^{51} - 4 q^{54} + 10 q^{61} + 16 q^{64} + 20 q^{66} - 6 q^{69} + 18 q^{71} - 28 q^{74} - 24 q^{76} + 6 q^{79} + 2 q^{81} + 4 q^{84} - 32 q^{86} - 22 q^{89} + 2 q^{91} - 40 q^{94} - 16 q^{96} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) −2.00000 −1.00000
\(5\) 0 0
\(6\) 2.00000 0.816497
\(7\) − 1.00000i − 0.377964i −0.981981 0.188982i \(-0.939481\pi\)
0.981981 0.188982i \(-0.0605189\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 2.00000i 0.577350i
\(13\) 1.00000i 0.277350i
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) − 7.00000i − 1.69775i −0.528594 0.848875i \(-0.677281\pi\)
0.528594 0.848875i \(-0.322719\pi\)
\(18\) − 2.00000i − 0.471405i
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 10.0000i 2.13201i
\(23\) − 3.00000i − 0.625543i −0.949828 0.312772i \(-0.898743\pi\)
0.949828 0.312772i \(-0.101257\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 1.00000i 0.192450i
\(28\) 2.00000i 0.377964i
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 2.00000 0.359211 0.179605 0.983739i \(-0.442518\pi\)
0.179605 + 0.983739i \(0.442518\pi\)
\(32\) − 8.00000i − 1.41421i
\(33\) − 5.00000i − 0.870388i
\(34\) 14.0000 2.40098
\(35\) 0 0
\(36\) 2.00000 0.333333
\(37\) 7.00000i 1.15079i 0.817875 + 0.575396i \(0.195152\pi\)
−0.817875 + 0.575396i \(0.804848\pi\)
\(38\) 12.0000i 1.94666i
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 9.00000 1.40556 0.702782 0.711405i \(-0.251941\pi\)
0.702782 + 0.711405i \(0.251941\pi\)
\(42\) − 2.00000i − 0.308607i
\(43\) 8.00000i 1.21999i 0.792406 + 0.609994i \(0.208828\pi\)
−0.792406 + 0.609994i \(0.791172\pi\)
\(44\) −10.0000 −1.50756
\(45\) 0 0
\(46\) 6.00000 0.884652
\(47\) 10.0000i 1.45865i 0.684167 + 0.729325i \(0.260166\pi\)
−0.684167 + 0.729325i \(0.739834\pi\)
\(48\) 4.00000i 0.577350i
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) −7.00000 −0.980196
\(52\) − 2.00000i − 0.277350i
\(53\) − 5.00000i − 0.686803i −0.939189 0.343401i \(-0.888421\pi\)
0.939189 0.343401i \(-0.111579\pi\)
\(54\) −2.00000 −0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) − 6.00000i − 0.794719i
\(58\) − 4.00000i − 0.525226i
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 4.00000i 0.508001i
\(63\) 1.00000i 0.125988i
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 10.0000 1.23091
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) 14.0000i 1.69775i
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) 9.00000 1.06810 0.534052 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) 0 0
\(73\) 6.00000i 0.702247i 0.936329 + 0.351123i \(0.114200\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) −14.0000 −1.62747
\(75\) 0 0
\(76\) −12.0000 −1.37649
\(77\) − 5.00000i − 0.569803i
\(78\) 2.00000i 0.226455i
\(79\) 3.00000 0.337526 0.168763 0.985657i \(-0.446023\pi\)
0.168763 + 0.985657i \(0.446023\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 18.0000i 1.98777i
\(83\) 4.00000i 0.439057i 0.975606 + 0.219529i \(0.0704519\pi\)
−0.975606 + 0.219529i \(0.929548\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −16.0000 −1.72532
\(87\) 2.00000i 0.214423i
\(88\) 0 0
\(89\) −11.0000 −1.16600 −0.582999 0.812473i \(-0.698121\pi\)
−0.582999 + 0.812473i \(0.698121\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 6.00000i 0.625543i
\(93\) − 2.00000i − 0.207390i
\(94\) −20.0000 −2.06284
\(95\) 0 0
\(96\) −8.00000 −0.816497
\(97\) − 11.0000i − 1.11688i −0.829545 0.558440i \(-0.811400\pi\)
0.829545 0.558440i \(-0.188600\pi\)
\(98\) 12.0000i 1.21218i
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) − 14.0000i − 1.38621i
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) − 17.0000i − 1.64345i −0.569883 0.821726i \(-0.693011\pi\)
0.569883 0.821726i \(-0.306989\pi\)
\(108\) − 2.00000i − 0.192450i
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 7.00000 0.664411
\(112\) 4.00000i 0.377964i
\(113\) − 10.0000i − 0.940721i −0.882474 0.470360i \(-0.844124\pi\)
0.882474 0.470360i \(-0.155876\pi\)
\(114\) 12.0000 1.12390
\(115\) 0 0
\(116\) 4.00000 0.371391
\(117\) − 1.00000i − 0.0924500i
\(118\) 0 0
\(119\) −7.00000 −0.641689
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 10.0000i 0.905357i
\(123\) − 9.00000i − 0.811503i
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) − 2.00000i − 0.177471i −0.996055 0.0887357i \(-0.971717\pi\)
0.996055 0.0887357i \(-0.0282826\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) −22.0000 −1.92215 −0.961074 0.276289i \(-0.910895\pi\)
−0.961074 + 0.276289i \(0.910895\pi\)
\(132\) 10.0000i 0.870388i
\(133\) − 6.00000i − 0.520266i
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 0 0
\(137\) − 14.0000i − 1.19610i −0.801459 0.598050i \(-0.795942\pi\)
0.801459 0.598050i \(-0.204058\pi\)
\(138\) − 6.00000i − 0.510754i
\(139\) −15.0000 −1.27228 −0.636142 0.771572i \(-0.719471\pi\)
−0.636142 + 0.771572i \(0.719471\pi\)
\(140\) 0 0
\(141\) 10.0000 0.842152
\(142\) 18.0000i 1.51053i
\(143\) 5.00000i 0.418121i
\(144\) 4.00000 0.333333
\(145\) 0 0
\(146\) −12.0000 −0.993127
\(147\) − 6.00000i − 0.494872i
\(148\) − 14.0000i − 1.15079i
\(149\) −15.0000 −1.22885 −0.614424 0.788976i \(-0.710612\pi\)
−0.614424 + 0.788976i \(0.710612\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 7.00000i 0.565916i
\(154\) 10.0000 0.805823
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) 18.0000i 1.43656i 0.695756 + 0.718278i \(0.255069\pi\)
−0.695756 + 0.718278i \(0.744931\pi\)
\(158\) 6.00000i 0.477334i
\(159\) −5.00000 −0.396526
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 2.00000i 0.157135i
\(163\) − 15.0000i − 1.17489i −0.809264 0.587445i \(-0.800134\pi\)
0.809264 0.587445i \(-0.199866\pi\)
\(164\) −18.0000 −1.40556
\(165\) 0 0
\(166\) −8.00000 −0.620920
\(167\) − 24.0000i − 1.85718i −0.371113 0.928588i \(-0.621024\pi\)
0.371113 0.928588i \(-0.378976\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) − 16.0000i − 1.21999i
\(173\) 18.0000i 1.36851i 0.729241 + 0.684257i \(0.239873\pi\)
−0.729241 + 0.684257i \(0.760127\pi\)
\(174\) −4.00000 −0.303239
\(175\) 0 0
\(176\) −20.0000 −1.50756
\(177\) 0 0
\(178\) − 22.0000i − 1.64897i
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 2.00000i 0.148250i
\(183\) − 5.00000i − 0.369611i
\(184\) 0 0
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) − 35.0000i − 2.55945i
\(188\) − 20.0000i − 1.45865i
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) − 8.00000i − 0.577350i
\(193\) 17.0000i 1.22369i 0.790979 + 0.611843i \(0.209572\pi\)
−0.790979 + 0.611843i \(0.790428\pi\)
\(194\) 22.0000 1.57951
\(195\) 0 0
\(196\) −12.0000 −0.857143
\(197\) 24.0000i 1.70993i 0.518686 + 0.854965i \(0.326421\pi\)
−0.518686 + 0.854965i \(0.673579\pi\)
\(198\) − 10.0000i − 0.710669i
\(199\) 28.0000 1.98487 0.992434 0.122782i \(-0.0391815\pi\)
0.992434 + 0.122782i \(0.0391815\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) 2.00000i 0.140372i
\(204\) 14.0000 0.980196
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 3.00000i 0.208514i
\(208\) − 4.00000i − 0.277350i
\(209\) 30.0000 2.07514
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 10.0000i 0.686803i
\(213\) − 9.00000i − 0.616670i
\(214\) 34.0000 2.32419
\(215\) 0 0
\(216\) 0 0
\(217\) − 2.00000i − 0.135769i
\(218\) − 8.00000i − 0.541828i
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) 7.00000 0.470871
\(222\) 14.0000i 0.939618i
\(223\) 8.00000i 0.535720i 0.963458 + 0.267860i \(0.0863164\pi\)
−0.963458 + 0.267860i \(0.913684\pi\)
\(224\) −8.00000 −0.534522
\(225\) 0 0
\(226\) 20.0000 1.33038
\(227\) − 2.00000i − 0.132745i −0.997795 0.0663723i \(-0.978857\pi\)
0.997795 0.0663723i \(-0.0211425\pi\)
\(228\) 12.0000i 0.794719i
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) −5.00000 −0.328976
\(232\) 0 0
\(233\) − 19.0000i − 1.24473i −0.782727 0.622366i \(-0.786172\pi\)
0.782727 0.622366i \(-0.213828\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) 0 0
\(237\) − 3.00000i − 0.194871i
\(238\) − 14.0000i − 0.907485i
\(239\) −9.00000 −0.582162 −0.291081 0.956698i \(-0.594015\pi\)
−0.291081 + 0.956698i \(0.594015\pi\)
\(240\) 0 0
\(241\) 22.0000 1.41714 0.708572 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) 28.0000i 1.79991i
\(243\) − 1.00000i − 0.0641500i
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) 18.0000 1.14764
\(247\) 6.00000i 0.381771i
\(248\) 0 0
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) − 2.00000i − 0.125988i
\(253\) − 15.0000i − 0.943042i
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 2.00000i 0.124757i 0.998053 + 0.0623783i \(0.0198685\pi\)
−0.998053 + 0.0623783i \(0.980131\pi\)
\(258\) 16.0000i 0.996116i
\(259\) 7.00000 0.434959
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) − 44.0000i − 2.71833i
\(263\) 16.0000i 0.986602i 0.869859 + 0.493301i \(0.164210\pi\)
−0.869859 + 0.493301i \(0.835790\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 12.0000 0.735767
\(267\) 11.0000i 0.673189i
\(268\) 8.00000i 0.488678i
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) −22.0000 −1.33640 −0.668202 0.743980i \(-0.732936\pi\)
−0.668202 + 0.743980i \(0.732936\pi\)
\(272\) 28.0000i 1.69775i
\(273\) − 1.00000i − 0.0605228i
\(274\) 28.0000 1.69154
\(275\) 0 0
\(276\) 6.00000 0.361158
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) − 30.0000i − 1.79928i
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 20.0000i 1.19098i
\(283\) − 20.0000i − 1.18888i −0.804141 0.594438i \(-0.797374\pi\)
0.804141 0.594438i \(-0.202626\pi\)
\(284\) −18.0000 −1.06810
\(285\) 0 0
\(286\) −10.0000 −0.591312
\(287\) − 9.00000i − 0.531253i
\(288\) 8.00000i 0.471405i
\(289\) −32.0000 −1.88235
\(290\) 0 0
\(291\) −11.0000 −0.644831
\(292\) − 12.0000i − 0.702247i
\(293\) − 4.00000i − 0.233682i −0.993151 0.116841i \(-0.962723\pi\)
0.993151 0.116841i \(-0.0372769\pi\)
\(294\) 12.0000 0.699854
\(295\) 0 0
\(296\) 0 0
\(297\) 5.00000i 0.290129i
\(298\) − 30.0000i − 1.73785i
\(299\) 3.00000 0.173494
\(300\) 0 0
\(301\) 8.00000 0.461112
\(302\) − 16.0000i − 0.920697i
\(303\) 0 0
\(304\) −24.0000 −1.37649
\(305\) 0 0
\(306\) −14.0000 −0.800327
\(307\) 23.0000i 1.31268i 0.754466 + 0.656340i \(0.227896\pi\)
−0.754466 + 0.656340i \(0.772104\pi\)
\(308\) 10.0000i 0.569803i
\(309\) 4.00000 0.227552
\(310\) 0 0
\(311\) 20.0000 1.13410 0.567048 0.823685i \(-0.308085\pi\)
0.567048 + 0.823685i \(0.308085\pi\)
\(312\) 0 0
\(313\) 22.0000i 1.24351i 0.783210 + 0.621757i \(0.213581\pi\)
−0.783210 + 0.621757i \(0.786419\pi\)
\(314\) −36.0000 −2.03160
\(315\) 0 0
\(316\) −6.00000 −0.337526
\(317\) − 24.0000i − 1.34797i −0.738743 0.673987i \(-0.764580\pi\)
0.738743 0.673987i \(-0.235420\pi\)
\(318\) − 10.0000i − 0.560772i
\(319\) −10.0000 −0.559893
\(320\) 0 0
\(321\) −17.0000 −0.948847
\(322\) − 6.00000i − 0.334367i
\(323\) − 42.0000i − 2.33694i
\(324\) −2.00000 −0.111111
\(325\) 0 0
\(326\) 30.0000 1.66155
\(327\) 4.00000i 0.221201i
\(328\) 0 0
\(329\) 10.0000 0.551318
\(330\) 0 0
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) − 8.00000i − 0.439057i
\(333\) − 7.00000i − 0.383598i
\(334\) 48.0000 2.62644
\(335\) 0 0
\(336\) 4.00000 0.218218
\(337\) 8.00000i 0.435788i 0.975972 + 0.217894i \(0.0699187\pi\)
−0.975972 + 0.217894i \(0.930081\pi\)
\(338\) − 2.00000i − 0.108786i
\(339\) −10.0000 −0.543125
\(340\) 0 0
\(341\) 10.0000 0.541530
\(342\) − 12.0000i − 0.648886i
\(343\) − 13.0000i − 0.701934i
\(344\) 0 0
\(345\) 0 0
\(346\) −36.0000 −1.93537
\(347\) 11.0000i 0.590511i 0.955418 + 0.295255i \(0.0954048\pi\)
−0.955418 + 0.295255i \(0.904595\pi\)
\(348\) − 4.00000i − 0.214423i
\(349\) −24.0000 −1.28469 −0.642345 0.766415i \(-0.722038\pi\)
−0.642345 + 0.766415i \(0.722038\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) − 40.0000i − 2.13201i
\(353\) 6.00000i 0.319348i 0.987170 + 0.159674i \(0.0510443\pi\)
−0.987170 + 0.159674i \(0.948956\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 22.0000 1.16600
\(357\) 7.00000i 0.370479i
\(358\) − 12.0000i − 0.634220i
\(359\) 16.0000 0.844448 0.422224 0.906492i \(-0.361250\pi\)
0.422224 + 0.906492i \(0.361250\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) − 14.0000i − 0.735824i
\(363\) − 14.0000i − 0.734809i
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) 10.0000 0.522708
\(367\) 4.00000i 0.208798i 0.994535 + 0.104399i \(0.0332919\pi\)
−0.994535 + 0.104399i \(0.966708\pi\)
\(368\) 12.0000i 0.625543i
\(369\) −9.00000 −0.468521
\(370\) 0 0
\(371\) −5.00000 −0.259587
\(372\) 4.00000i 0.207390i
\(373\) 32.0000i 1.65690i 0.560065 + 0.828449i \(0.310776\pi\)
−0.560065 + 0.828449i \(0.689224\pi\)
\(374\) 70.0000 3.61961
\(375\) 0 0
\(376\) 0 0
\(377\) − 2.00000i − 0.103005i
\(378\) 2.00000i 0.102869i
\(379\) −10.0000 −0.513665 −0.256833 0.966456i \(-0.582679\pi\)
−0.256833 + 0.966456i \(0.582679\pi\)
\(380\) 0 0
\(381\) −2.00000 −0.102463
\(382\) − 24.0000i − 1.22795i
\(383\) 6.00000i 0.306586i 0.988181 + 0.153293i \(0.0489878\pi\)
−0.988181 + 0.153293i \(0.951012\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −34.0000 −1.73055
\(387\) − 8.00000i − 0.406663i
\(388\) 22.0000i 1.11688i
\(389\) 4.00000 0.202808 0.101404 0.994845i \(-0.467667\pi\)
0.101404 + 0.994845i \(0.467667\pi\)
\(390\) 0 0
\(391\) −21.0000 −1.06202
\(392\) 0 0
\(393\) 22.0000i 1.10975i
\(394\) −48.0000 −2.41821
\(395\) 0 0
\(396\) 10.0000 0.502519
\(397\) 15.0000i 0.752828i 0.926451 + 0.376414i \(0.122843\pi\)
−0.926451 + 0.376414i \(0.877157\pi\)
\(398\) 56.0000i 2.80703i
\(399\) −6.00000 −0.300376
\(400\) 0 0
\(401\) −14.0000 −0.699127 −0.349563 0.936913i \(-0.613670\pi\)
−0.349563 + 0.936913i \(0.613670\pi\)
\(402\) − 8.00000i − 0.399004i
\(403\) 2.00000i 0.0996271i
\(404\) 0 0
\(405\) 0 0
\(406\) −4.00000 −0.198517
\(407\) 35.0000i 1.73489i
\(408\) 0 0
\(409\) −30.0000 −1.48340 −0.741702 0.670729i \(-0.765981\pi\)
−0.741702 + 0.670729i \(0.765981\pi\)
\(410\) 0 0
\(411\) −14.0000 −0.690569
\(412\) − 8.00000i − 0.394132i
\(413\) 0 0
\(414\) −6.00000 −0.294884
\(415\) 0 0
\(416\) 8.00000 0.392232
\(417\) 15.0000i 0.734553i
\(418\) 60.0000i 2.93470i
\(419\) −26.0000 −1.27018 −0.635092 0.772437i \(-0.719038\pi\)
−0.635092 + 0.772437i \(0.719038\pi\)
\(420\) 0 0
\(421\) −20.0000 −0.974740 −0.487370 0.873195i \(-0.662044\pi\)
−0.487370 + 0.873195i \(0.662044\pi\)
\(422\) − 24.0000i − 1.16830i
\(423\) − 10.0000i − 0.486217i
\(424\) 0 0
\(425\) 0 0
\(426\) 18.0000 0.872103
\(427\) − 5.00000i − 0.241967i
\(428\) 34.0000i 1.64345i
\(429\) 5.00000 0.241402
\(430\) 0 0
\(431\) −40.0000 −1.92673 −0.963366 0.268190i \(-0.913575\pi\)
−0.963366 + 0.268190i \(0.913575\pi\)
\(432\) − 4.00000i − 0.192450i
\(433\) 20.0000i 0.961139i 0.876957 + 0.480569i \(0.159570\pi\)
−0.876957 + 0.480569i \(0.840430\pi\)
\(434\) 4.00000 0.192006
\(435\) 0 0
\(436\) 8.00000 0.383131
\(437\) − 18.0000i − 0.861057i
\(438\) 12.0000i 0.573382i
\(439\) −15.0000 −0.715911 −0.357955 0.933739i \(-0.616526\pi\)
−0.357955 + 0.933739i \(0.616526\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 14.0000i 0.665912i
\(443\) 1.00000i 0.0475114i 0.999718 + 0.0237557i \(0.00756239\pi\)
−0.999718 + 0.0237557i \(0.992438\pi\)
\(444\) −14.0000 −0.664411
\(445\) 0 0
\(446\) −16.0000 −0.757622
\(447\) 15.0000i 0.709476i
\(448\) − 8.00000i − 0.377964i
\(449\) −9.00000 −0.424736 −0.212368 0.977190i \(-0.568118\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) 0 0
\(451\) 45.0000 2.11897
\(452\) 20.0000i 0.940721i
\(453\) 8.00000i 0.375873i
\(454\) 4.00000 0.187729
\(455\) 0 0
\(456\) 0 0
\(457\) − 7.00000i − 0.327446i −0.986506 0.163723i \(-0.947650\pi\)
0.986506 0.163723i \(-0.0523504\pi\)
\(458\) − 28.0000i − 1.30835i
\(459\) 7.00000 0.326732
\(460\) 0 0
\(461\) 37.0000 1.72326 0.861631 0.507535i \(-0.169443\pi\)
0.861631 + 0.507535i \(0.169443\pi\)
\(462\) − 10.0000i − 0.465242i
\(463\) − 15.0000i − 0.697109i −0.937288 0.348555i \(-0.886673\pi\)
0.937288 0.348555i \(-0.113327\pi\)
\(464\) 8.00000 0.371391
\(465\) 0 0
\(466\) 38.0000 1.76032
\(467\) − 1.00000i − 0.0462745i −0.999732 0.0231372i \(-0.992635\pi\)
0.999732 0.0231372i \(-0.00736547\pi\)
\(468\) 2.00000i 0.0924500i
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 18.0000 0.829396
\(472\) 0 0
\(473\) 40.0000i 1.83920i
\(474\) 6.00000 0.275589
\(475\) 0 0
\(476\) 14.0000 0.641689
\(477\) 5.00000i 0.228934i
\(478\) − 18.0000i − 0.823301i
\(479\) −3.00000 −0.137073 −0.0685367 0.997649i \(-0.521833\pi\)
−0.0685367 + 0.997649i \(0.521833\pi\)
\(480\) 0 0
\(481\) −7.00000 −0.319173
\(482\) 44.0000i 2.00415i
\(483\) 3.00000i 0.136505i
\(484\) −28.0000 −1.27273
\(485\) 0 0
\(486\) 2.00000 0.0907218
\(487\) 5.00000i 0.226572i 0.993562 + 0.113286i \(0.0361376\pi\)
−0.993562 + 0.113286i \(0.963862\pi\)
\(488\) 0 0
\(489\) −15.0000 −0.678323
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 18.0000i 0.811503i
\(493\) 14.0000i 0.630528i
\(494\) −12.0000 −0.539906
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) − 9.00000i − 0.403705i
\(498\) 8.00000i 0.358489i
\(499\) 14.0000 0.626726 0.313363 0.949633i \(-0.398544\pi\)
0.313363 + 0.949633i \(0.398544\pi\)
\(500\) 0 0
\(501\) −24.0000 −1.07224
\(502\) 0 0
\(503\) − 36.0000i − 1.60516i −0.596544 0.802580i \(-0.703460\pi\)
0.596544 0.802580i \(-0.296540\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 30.0000 1.33366
\(507\) 1.00000i 0.0444116i
\(508\) 4.00000i 0.177471i
\(509\) −27.0000 −1.19675 −0.598377 0.801215i \(-0.704187\pi\)
−0.598377 + 0.801215i \(0.704187\pi\)
\(510\) 0 0
\(511\) 6.00000 0.265424
\(512\) 32.0000i 1.41421i
\(513\) 6.00000i 0.264906i
\(514\) −4.00000 −0.176432
\(515\) 0 0
\(516\) −16.0000 −0.704361
\(517\) 50.0000i 2.19900i
\(518\) 14.0000i 0.615125i
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 4.00000i 0.175075i
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 44.0000 1.92215
\(525\) 0 0
\(526\) −32.0000 −1.39527
\(527\) − 14.0000i − 0.609850i
\(528\) 20.0000i 0.870388i
\(529\) 14.0000 0.608696
\(530\) 0 0
\(531\) 0 0
\(532\) 12.0000i 0.520266i
\(533\) 9.00000i 0.389833i
\(534\) −22.0000 −0.952033
\(535\) 0 0
\(536\) 0 0
\(537\) 6.00000i 0.258919i
\(538\) 48.0000i 2.06943i
\(539\) 30.0000 1.29219
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) − 44.0000i − 1.88996i
\(543\) 7.00000i 0.300399i
\(544\) −56.0000 −2.40098
\(545\) 0 0
\(546\) 2.00000 0.0855921
\(547\) 36.0000i 1.53925i 0.638497 + 0.769624i \(0.279557\pi\)
−0.638497 + 0.769624i \(0.720443\pi\)
\(548\) 28.0000i 1.19610i
\(549\) −5.00000 −0.213395
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) − 3.00000i − 0.127573i
\(554\) 4.00000 0.169944
\(555\) 0 0
\(556\) 30.0000 1.27228
\(557\) − 2.00000i − 0.0847427i −0.999102 0.0423714i \(-0.986509\pi\)
0.999102 0.0423714i \(-0.0134913\pi\)
\(558\) − 4.00000i − 0.169334i
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) −35.0000 −1.47770
\(562\) 36.0000i 1.51857i
\(563\) − 11.0000i − 0.463595i −0.972764 0.231797i \(-0.925539\pi\)
0.972764 0.231797i \(-0.0744606\pi\)
\(564\) −20.0000 −0.842152
\(565\) 0 0
\(566\) 40.0000 1.68133
\(567\) − 1.00000i − 0.0419961i
\(568\) 0 0
\(569\) 20.0000 0.838444 0.419222 0.907884i \(-0.362303\pi\)
0.419222 + 0.907884i \(0.362303\pi\)
\(570\) 0 0
\(571\) −39.0000 −1.63210 −0.816050 0.577982i \(-0.803840\pi\)
−0.816050 + 0.577982i \(0.803840\pi\)
\(572\) − 10.0000i − 0.418121i
\(573\) 12.0000i 0.501307i
\(574\) 18.0000 0.751305
\(575\) 0 0
\(576\) −8.00000 −0.333333
\(577\) − 7.00000i − 0.291414i −0.989328 0.145707i \(-0.953454\pi\)
0.989328 0.145707i \(-0.0465456\pi\)
\(578\) − 64.0000i − 2.66205i
\(579\) 17.0000 0.706496
\(580\) 0 0
\(581\) 4.00000 0.165948
\(582\) − 22.0000i − 0.911929i
\(583\) − 25.0000i − 1.03539i
\(584\) 0 0
\(585\) 0 0
\(586\) 8.00000 0.330477
\(587\) − 2.00000i − 0.0825488i −0.999148 0.0412744i \(-0.986858\pi\)
0.999148 0.0412744i \(-0.0131418\pi\)
\(588\) 12.0000i 0.494872i
\(589\) 12.0000 0.494451
\(590\) 0 0
\(591\) 24.0000 0.987228
\(592\) − 28.0000i − 1.15079i
\(593\) 16.0000i 0.657041i 0.944497 + 0.328521i \(0.106550\pi\)
−0.944497 + 0.328521i \(0.893450\pi\)
\(594\) −10.0000 −0.410305
\(595\) 0 0
\(596\) 30.0000 1.22885
\(597\) − 28.0000i − 1.14596i
\(598\) 6.00000i 0.245358i
\(599\) −20.0000 −0.817178 −0.408589 0.912719i \(-0.633979\pi\)
−0.408589 + 0.912719i \(0.633979\pi\)
\(600\) 0 0
\(601\) −5.00000 −0.203954 −0.101977 0.994787i \(-0.532517\pi\)
−0.101977 + 0.994787i \(0.532517\pi\)
\(602\) 16.0000i 0.652111i
\(603\) 4.00000i 0.162893i
\(604\) 16.0000 0.651031
\(605\) 0 0
\(606\) 0 0
\(607\) − 32.0000i − 1.29884i −0.760430 0.649420i \(-0.775012\pi\)
0.760430 0.649420i \(-0.224988\pi\)
\(608\) − 48.0000i − 1.94666i
\(609\) 2.00000 0.0810441
\(610\) 0 0
\(611\) −10.0000 −0.404557
\(612\) − 14.0000i − 0.565916i
\(613\) − 9.00000i − 0.363507i −0.983344 0.181753i \(-0.941823\pi\)
0.983344 0.181753i \(-0.0581772\pi\)
\(614\) −46.0000 −1.85641
\(615\) 0 0
\(616\) 0 0
\(617\) 18.0000i 0.724653i 0.932051 + 0.362326i \(0.118017\pi\)
−0.932051 + 0.362326i \(0.881983\pi\)
\(618\) 8.00000i 0.321807i
\(619\) 38.0000 1.52735 0.763674 0.645601i \(-0.223393\pi\)
0.763674 + 0.645601i \(0.223393\pi\)
\(620\) 0 0
\(621\) 3.00000 0.120386
\(622\) 40.0000i 1.60385i
\(623\) 11.0000i 0.440706i
\(624\) −4.00000 −0.160128
\(625\) 0 0
\(626\) −44.0000 −1.75859
\(627\) − 30.0000i − 1.19808i
\(628\) − 36.0000i − 1.43656i
\(629\) 49.0000 1.95376
\(630\) 0 0
\(631\) −20.0000 −0.796187 −0.398094 0.917345i \(-0.630328\pi\)
−0.398094 + 0.917345i \(0.630328\pi\)
\(632\) 0 0
\(633\) 12.0000i 0.476957i
\(634\) 48.0000 1.90632
\(635\) 0 0
\(636\) 10.0000 0.396526
\(637\) 6.00000i 0.237729i
\(638\) − 20.0000i − 0.791808i
\(639\) −9.00000 −0.356034
\(640\) 0 0
\(641\) 20.0000 0.789953 0.394976 0.918691i \(-0.370753\pi\)
0.394976 + 0.918691i \(0.370753\pi\)
\(642\) − 34.0000i − 1.34187i
\(643\) 37.0000i 1.45914i 0.683907 + 0.729569i \(0.260279\pi\)
−0.683907 + 0.729569i \(0.739721\pi\)
\(644\) 6.00000 0.236433
\(645\) 0 0
\(646\) 84.0000 3.30494
\(647\) 17.0000i 0.668339i 0.942513 + 0.334169i \(0.108456\pi\)
−0.942513 + 0.334169i \(0.891544\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −2.00000 −0.0783862
\(652\) 30.0000i 1.17489i
\(653\) − 18.0000i − 0.704394i −0.935926 0.352197i \(-0.885435\pi\)
0.935926 0.352197i \(-0.114565\pi\)
\(654\) −8.00000 −0.312825
\(655\) 0 0
\(656\) −36.0000 −1.40556
\(657\) − 6.00000i − 0.234082i
\(658\) 20.0000i 0.779681i
\(659\) −24.0000 −0.934907 −0.467454 0.884018i \(-0.654829\pi\)
−0.467454 + 0.884018i \(0.654829\pi\)
\(660\) 0 0
\(661\) −40.0000 −1.55582 −0.777910 0.628376i \(-0.783720\pi\)
−0.777910 + 0.628376i \(0.783720\pi\)
\(662\) 0 0
\(663\) − 7.00000i − 0.271857i
\(664\) 0 0
\(665\) 0 0
\(666\) 14.0000 0.542489
\(667\) 6.00000i 0.232321i
\(668\) 48.0000i 1.85718i
\(669\) 8.00000 0.309298
\(670\) 0 0
\(671\) 25.0000 0.965114
\(672\) 8.00000i 0.308607i
\(673\) − 42.0000i − 1.61898i −0.587133 0.809491i \(-0.699743\pi\)
0.587133 0.809491i \(-0.300257\pi\)
\(674\) −16.0000 −0.616297
\(675\) 0 0
\(676\) 2.00000 0.0769231
\(677\) 21.0000i 0.807096i 0.914959 + 0.403548i \(0.132223\pi\)
−0.914959 + 0.403548i \(0.867777\pi\)
\(678\) − 20.0000i − 0.768095i
\(679\) −11.0000 −0.422141
\(680\) 0 0
\(681\) −2.00000 −0.0766402
\(682\) 20.0000i 0.765840i
\(683\) − 16.0000i − 0.612223i −0.951996 0.306111i \(-0.900972\pi\)
0.951996 0.306111i \(-0.0990280\pi\)
\(684\) 12.0000 0.458831
\(685\) 0 0
\(686\) 26.0000 0.992685
\(687\) 14.0000i 0.534133i
\(688\) − 32.0000i − 1.21999i
\(689\) 5.00000 0.190485
\(690\) 0 0
\(691\) 10.0000 0.380418 0.190209 0.981744i \(-0.439083\pi\)
0.190209 + 0.981744i \(0.439083\pi\)
\(692\) − 36.0000i − 1.36851i
\(693\) 5.00000i 0.189934i
\(694\) −22.0000 −0.835109
\(695\) 0 0
\(696\) 0 0
\(697\) − 63.0000i − 2.38630i
\(698\) − 48.0000i − 1.81683i
\(699\) −19.0000 −0.718646
\(700\) 0 0
\(701\) 28.0000 1.05755 0.528773 0.848763i \(-0.322652\pi\)
0.528773 + 0.848763i \(0.322652\pi\)
\(702\) − 2.00000i − 0.0754851i
\(703\) 42.0000i 1.58406i
\(704\) 40.0000 1.50756
\(705\) 0 0
\(706\) −12.0000 −0.451626
\(707\) 0 0
\(708\) 0 0
\(709\) 44.0000 1.65245 0.826227 0.563337i \(-0.190483\pi\)
0.826227 + 0.563337i \(0.190483\pi\)
\(710\) 0 0
\(711\) −3.00000 −0.112509
\(712\) 0 0
\(713\) − 6.00000i − 0.224702i
\(714\) −14.0000 −0.523937
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 9.00000i 0.336111i
\(718\) 32.0000i 1.19423i
\(719\) −16.0000 −0.596699 −0.298350 0.954457i \(-0.596436\pi\)
−0.298350 + 0.954457i \(0.596436\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 34.0000i 1.26535i
\(723\) − 22.0000i − 0.818189i
\(724\) 14.0000 0.520306
\(725\) 0 0
\(726\) 28.0000 1.03918
\(727\) − 18.0000i − 0.667583i −0.942647 0.333792i \(-0.891672\pi\)
0.942647 0.333792i \(-0.108328\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 56.0000 2.07123
\(732\) 10.0000i 0.369611i
\(733\) − 43.0000i − 1.58824i −0.607760 0.794121i \(-0.707932\pi\)
0.607760 0.794121i \(-0.292068\pi\)
\(734\) −8.00000 −0.295285
\(735\) 0 0
\(736\) −24.0000 −0.884652
\(737\) − 20.0000i − 0.736709i
\(738\) − 18.0000i − 0.662589i
\(739\) 6.00000 0.220714 0.110357 0.993892i \(-0.464801\pi\)
0.110357 + 0.993892i \(0.464801\pi\)
\(740\) 0 0
\(741\) 6.00000 0.220416
\(742\) − 10.0000i − 0.367112i
\(743\) − 30.0000i − 1.10059i −0.834969 0.550297i \(-0.814515\pi\)
0.834969 0.550297i \(-0.185485\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −64.0000 −2.34321
\(747\) − 4.00000i − 0.146352i
\(748\) 70.0000i 2.55945i
\(749\) −17.0000 −0.621166
\(750\) 0 0
\(751\) −13.0000 −0.474377 −0.237188 0.971464i \(-0.576226\pi\)
−0.237188 + 0.971464i \(0.576226\pi\)
\(752\) − 40.0000i − 1.45865i
\(753\) 0 0
\(754\) 4.00000 0.145671
\(755\) 0 0
\(756\) −2.00000 −0.0727393
\(757\) 20.0000i 0.726912i 0.931611 + 0.363456i \(0.118403\pi\)
−0.931611 + 0.363456i \(0.881597\pi\)
\(758\) − 20.0000i − 0.726433i
\(759\) −15.0000 −0.544466
\(760\) 0 0
\(761\) −42.0000 −1.52250 −0.761249 0.648459i \(-0.775414\pi\)
−0.761249 + 0.648459i \(0.775414\pi\)
\(762\) − 4.00000i − 0.144905i
\(763\) 4.00000i 0.144810i
\(764\) 24.0000 0.868290
\(765\) 0 0
\(766\) −12.0000 −0.433578
\(767\) 0 0
\(768\) − 16.0000i − 0.577350i
\(769\) 28.0000 1.00971 0.504853 0.863205i \(-0.331547\pi\)
0.504853 + 0.863205i \(0.331547\pi\)
\(770\) 0 0
\(771\) 2.00000 0.0720282
\(772\) − 34.0000i − 1.22369i
\(773\) 20.0000i 0.719350i 0.933078 + 0.359675i \(0.117112\pi\)
−0.933078 + 0.359675i \(0.882888\pi\)
\(774\) 16.0000 0.575108
\(775\) 0 0
\(776\) 0 0
\(777\) − 7.00000i − 0.251124i
\(778\) 8.00000i 0.286814i
\(779\) 54.0000 1.93475
\(780\) 0 0
\(781\) 45.0000 1.61023
\(782\) − 42.0000i − 1.50192i
\(783\) − 2.00000i − 0.0714742i
\(784\) −24.0000 −0.857143
\(785\) 0 0
\(786\) −44.0000 −1.56943
\(787\) 44.0000i 1.56843i 0.620489 + 0.784215i \(0.286934\pi\)
−0.620489 + 0.784215i \(0.713066\pi\)
\(788\) − 48.0000i − 1.70993i
\(789\) 16.0000 0.569615
\(790\) 0 0
\(791\) −10.0000 −0.355559
\(792\) 0 0
\(793\) 5.00000i 0.177555i
\(794\) −30.0000 −1.06466
\(795\) 0 0
\(796\) −56.0000 −1.98487
\(797\) 39.0000i 1.38145i 0.723117 + 0.690725i \(0.242709\pi\)
−0.723117 + 0.690725i \(0.757291\pi\)
\(798\) − 12.0000i − 0.424795i
\(799\) 70.0000 2.47642
\(800\) 0 0
\(801\) 11.0000 0.388666
\(802\) − 28.0000i − 0.988714i
\(803\) 30.0000i 1.05868i
\(804\) 8.00000 0.282138
\(805\) 0 0
\(806\) −4.00000 −0.140894
\(807\) − 24.0000i − 0.844840i
\(808\) 0 0
\(809\) 50.0000 1.75791 0.878953 0.476908i \(-0.158243\pi\)
0.878953 + 0.476908i \(0.158243\pi\)
\(810\) 0 0
\(811\) 24.0000 0.842754 0.421377 0.906886i \(-0.361547\pi\)
0.421377 + 0.906886i \(0.361547\pi\)
\(812\) − 4.00000i − 0.140372i
\(813\) 22.0000i 0.771574i
\(814\) −70.0000 −2.45350
\(815\) 0 0
\(816\) 28.0000 0.980196
\(817\) 48.0000i 1.67931i
\(818\) − 60.0000i − 2.09785i
\(819\) −1.00000 −0.0349428
\(820\) 0 0
\(821\) 17.0000 0.593304 0.296652 0.954986i \(-0.404130\pi\)
0.296652 + 0.954986i \(0.404130\pi\)
\(822\) − 28.0000i − 0.976612i
\(823\) 4.00000i 0.139431i 0.997567 + 0.0697156i \(0.0222092\pi\)
−0.997567 + 0.0697156i \(0.977791\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 26.0000i − 0.904109i −0.891990 0.452054i \(-0.850691\pi\)
0.891990 0.452054i \(-0.149309\pi\)
\(828\) − 6.00000i − 0.208514i
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) −2.00000 −0.0693792
\(832\) 8.00000i 0.277350i
\(833\) − 42.0000i − 1.45521i
\(834\) −30.0000 −1.03882
\(835\) 0 0
\(836\) −60.0000 −2.07514
\(837\) 2.00000i 0.0691301i
\(838\) − 52.0000i − 1.79631i
\(839\) −17.0000 −0.586905 −0.293453 0.955974i \(-0.594804\pi\)
−0.293453 + 0.955974i \(0.594804\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) − 40.0000i − 1.37849i
\(843\) − 18.0000i − 0.619953i
\(844\) 24.0000 0.826114
\(845\) 0 0
\(846\) 20.0000 0.687614
\(847\) − 14.0000i − 0.481046i
\(848\) 20.0000i 0.686803i
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 21.0000 0.719871
\(852\) 18.0000i 0.616670i
\(853\) 39.0000i 1.33533i 0.744460 + 0.667667i \(0.232707\pi\)
−0.744460 + 0.667667i \(0.767293\pi\)
\(854\) 10.0000 0.342193
\(855\) 0 0
\(856\) 0 0
\(857\) − 45.0000i − 1.53717i −0.639747 0.768585i \(-0.720961\pi\)
0.639747 0.768585i \(-0.279039\pi\)
\(858\) 10.0000i 0.341394i
\(859\) −19.0000 −0.648272 −0.324136 0.946011i \(-0.605073\pi\)
−0.324136 + 0.946011i \(0.605073\pi\)
\(860\) 0 0
\(861\) −9.00000 −0.306719
\(862\) − 80.0000i − 2.72481i
\(863\) − 6.00000i − 0.204242i −0.994772 0.102121i \(-0.967437\pi\)
0.994772 0.102121i \(-0.0325630\pi\)
\(864\) 8.00000 0.272166
\(865\) 0 0
\(866\) −40.0000 −1.35926
\(867\) 32.0000i 1.08678i
\(868\) 4.00000i 0.135769i
\(869\) 15.0000 0.508840
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 0 0
\(873\) 11.0000i 0.372294i
\(874\) 36.0000 1.21772
\(875\) 0 0
\(876\) −12.0000 −0.405442
\(877\) 18.0000i 0.607817i 0.952701 + 0.303908i \(0.0982917\pi\)
−0.952701 + 0.303908i \(0.901708\pi\)
\(878\) − 30.0000i − 1.01245i
\(879\) −4.00000 −0.134917
\(880\) 0 0
\(881\) 14.0000 0.471672 0.235836 0.971793i \(-0.424217\pi\)
0.235836 + 0.971793i \(0.424217\pi\)
\(882\) − 12.0000i − 0.404061i
\(883\) − 44.0000i − 1.48072i −0.672212 0.740359i \(-0.734656\pi\)
0.672212 0.740359i \(-0.265344\pi\)
\(884\) −14.0000 −0.470871
\(885\) 0 0
\(886\) −2.00000 −0.0671913
\(887\) 13.0000i 0.436497i 0.975893 + 0.218249i \(0.0700344\pi\)
−0.975893 + 0.218249i \(0.929966\pi\)
\(888\) 0 0
\(889\) −2.00000 −0.0670778
\(890\) 0 0
\(891\) 5.00000 0.167506
\(892\) − 16.0000i − 0.535720i
\(893\) 60.0000i 2.00782i
\(894\) −30.0000 −1.00335
\(895\) 0 0
\(896\) 0 0
\(897\) − 3.00000i − 0.100167i
\(898\) − 18.0000i − 0.600668i
\(899\) −4.00000 −0.133407
\(900\) 0 0
\(901\) −35.0000 −1.16602
\(902\) 90.0000i 2.99667i
\(903\) − 8.00000i − 0.266223i
\(904\) 0 0
\(905\) 0 0
\(906\) −16.0000 −0.531564
\(907\) 38.0000i 1.26177i 0.775877 + 0.630885i \(0.217308\pi\)
−0.775877 + 0.630885i \(0.782692\pi\)
\(908\) 4.00000i 0.132745i
\(909\) 0 0
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 24.0000i 0.794719i
\(913\) 20.0000i 0.661903i
\(914\) 14.0000 0.463079
\(915\) 0 0
\(916\) 28.0000 0.925146
\(917\) 22.0000i 0.726504i
\(918\) 14.0000i 0.462069i
\(919\) 19.0000 0.626752 0.313376 0.949629i \(-0.398540\pi\)
0.313376 + 0.949629i \(0.398540\pi\)
\(920\) 0 0
\(921\) 23.0000 0.757876
\(922\) 74.0000i 2.43706i
\(923\) 9.00000i 0.296239i
\(924\) 10.0000 0.328976
\(925\) 0 0
\(926\) 30.0000 0.985861
\(927\) − 4.00000i − 0.131377i
\(928\) 16.0000i 0.525226i
\(929\) 29.0000 0.951459 0.475730 0.879592i \(-0.342184\pi\)
0.475730 + 0.879592i \(0.342184\pi\)
\(930\) 0 0
\(931\) 36.0000 1.17985
\(932\) 38.0000i 1.24473i
\(933\) − 20.0000i − 0.654771i
\(934\) 2.00000 0.0654420
\(935\) 0 0
\(936\) 0 0
\(937\) − 6.00000i − 0.196011i −0.995186 0.0980057i \(-0.968754\pi\)
0.995186 0.0980057i \(-0.0312463\pi\)
\(938\) − 8.00000i − 0.261209i
\(939\) 22.0000 0.717943
\(940\) 0 0
\(941\) 15.0000 0.488986 0.244493 0.969651i \(-0.421378\pi\)
0.244493 + 0.969651i \(0.421378\pi\)
\(942\) 36.0000i 1.17294i
\(943\) − 27.0000i − 0.879241i
\(944\) 0 0
\(945\) 0 0
\(946\) −80.0000 −2.60102
\(947\) 12.0000i 0.389948i 0.980808 + 0.194974i \(0.0624622\pi\)
−0.980808 + 0.194974i \(0.937538\pi\)
\(948\) 6.00000i 0.194871i
\(949\) −6.00000 −0.194768
\(950\) 0 0
\(951\) −24.0000 −0.778253
\(952\) 0 0
\(953\) 15.0000i 0.485898i 0.970039 + 0.242949i \(0.0781147\pi\)
−0.970039 + 0.242949i \(0.921885\pi\)
\(954\) −10.0000 −0.323762
\(955\) 0 0
\(956\) 18.0000 0.582162
\(957\) 10.0000i 0.323254i
\(958\) − 6.00000i − 0.193851i
\(959\) −14.0000 −0.452084
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) − 14.0000i − 0.451378i
\(963\) 17.0000i 0.547817i
\(964\) −44.0000 −1.41714
\(965\) 0 0
\(966\) −6.00000 −0.193047
\(967\) − 8.00000i − 0.257263i −0.991692 0.128631i \(-0.958942\pi\)
0.991692 0.128631i \(-0.0410584\pi\)
\(968\) 0 0
\(969\) −42.0000 −1.34923
\(970\) 0 0
\(971\) −28.0000 −0.898563 −0.449281 0.893390i \(-0.648320\pi\)
−0.449281 + 0.893390i \(0.648320\pi\)
\(972\) 2.00000i 0.0641500i
\(973\) 15.0000i 0.480878i
\(974\) −10.0000 −0.320421
\(975\) 0 0
\(976\) −20.0000 −0.640184
\(977\) − 12.0000i − 0.383914i −0.981403 0.191957i \(-0.938517\pi\)
0.981403 0.191957i \(-0.0614834\pi\)
\(978\) − 30.0000i − 0.959294i
\(979\) −55.0000 −1.75781
\(980\) 0 0
\(981\) 4.00000 0.127710
\(982\) 0 0
\(983\) 4.00000i 0.127580i 0.997963 + 0.0637901i \(0.0203188\pi\)
−0.997963 + 0.0637901i \(0.979681\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −28.0000 −0.891702
\(987\) − 10.0000i − 0.318304i
\(988\) − 12.0000i − 0.381771i
\(989\) 24.0000 0.763156
\(990\) 0 0
\(991\) −57.0000 −1.81066 −0.905332 0.424704i \(-0.860378\pi\)
−0.905332 + 0.424704i \(0.860378\pi\)
\(992\) − 16.0000i − 0.508001i
\(993\) 0 0
\(994\) 18.0000 0.570925
\(995\) 0 0
\(996\) −8.00000 −0.253490
\(997\) 8.00000i 0.253363i 0.991943 + 0.126681i \(0.0404325\pi\)
−0.991943 + 0.126681i \(0.959567\pi\)
\(998\) 28.0000i 0.886325i
\(999\) −7.00000 −0.221470
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 975.2.c.c.274.2 2
3.2 odd 2 2925.2.c.a.2224.1 2
5.2 odd 4 975.2.a.a.1.1 1
5.3 odd 4 195.2.a.c.1.1 1
5.4 even 2 inner 975.2.c.c.274.1 2
15.2 even 4 2925.2.a.s.1.1 1
15.8 even 4 585.2.a.c.1.1 1
15.14 odd 2 2925.2.c.a.2224.2 2
20.3 even 4 3120.2.a.d.1.1 1
35.13 even 4 9555.2.a.u.1.1 1
60.23 odd 4 9360.2.a.bv.1.1 1
65.38 odd 4 2535.2.a.d.1.1 1
195.38 even 4 7605.2.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.c.1.1 1 5.3 odd 4
585.2.a.c.1.1 1 15.8 even 4
975.2.a.a.1.1 1 5.2 odd 4
975.2.c.c.274.1 2 5.4 even 2 inner
975.2.c.c.274.2 2 1.1 even 1 trivial
2535.2.a.d.1.1 1 65.38 odd 4
2925.2.a.s.1.1 1 15.2 even 4
2925.2.c.a.2224.1 2 3.2 odd 2
2925.2.c.a.2224.2 2 15.14 odd 2
3120.2.a.d.1.1 1 20.3 even 4
7605.2.a.t.1.1 1 195.38 even 4
9360.2.a.bv.1.1 1 60.23 odd 4
9555.2.a.u.1.1 1 35.13 even 4