Properties

Label 975.2.c.e.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: \(\Z[a_1, a_2]\)
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.e.274.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000i q^{3} +1.00000 q^{4} -1.00000 q^{6} +3.00000i q^{8} -1.00000 q^{9} +4.00000 q^{11} +1.00000i q^{12} +1.00000i q^{13} -1.00000 q^{16} -2.00000i q^{17} -1.00000i q^{18} +4.00000 q^{19} +4.00000i q^{22} +8.00000i q^{23} -3.00000 q^{24} -1.00000 q^{26} -1.00000i q^{27} +2.00000 q^{29} -8.00000 q^{31} +5.00000i q^{32} +4.00000i q^{33} +2.00000 q^{34} -1.00000 q^{36} -6.00000i q^{37} +4.00000i q^{38} -1.00000 q^{39} -6.00000 q^{41} -4.00000i q^{43} +4.00000 q^{44} -8.00000 q^{46} +8.00000i q^{47} -1.00000i q^{48} +7.00000 q^{49} +2.00000 q^{51} +1.00000i q^{52} +6.00000i q^{53} +1.00000 q^{54} +4.00000i q^{57} +2.00000i q^{58} +12.0000 q^{59} -2.00000 q^{61} -8.00000i q^{62} -7.00000 q^{64} -4.00000 q^{66} +4.00000i q^{67} -2.00000i q^{68} -8.00000 q^{69} -3.00000i q^{72} -6.00000i q^{73} +6.00000 q^{74} +4.00000 q^{76} -1.00000i q^{78} -16.0000 q^{79} +1.00000 q^{81} -6.00000i q^{82} -4.00000i q^{83} +4.00000 q^{86} +2.00000i q^{87} +12.0000i q^{88} -10.0000 q^{89} +8.00000i q^{92} -8.00000i q^{93} -8.00000 q^{94} -5.00000 q^{96} -18.0000i q^{97} +7.00000i q^{98} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} - 2 q^{6} - 2 q^{9} + 8 q^{11} - 2 q^{16} + 8 q^{19} - 6 q^{24} - 2 q^{26} + 4 q^{29} - 16 q^{31} + 4 q^{34} - 2 q^{36} - 2 q^{39} - 12 q^{41} + 8 q^{44} - 16 q^{46} + 14 q^{49} + 4 q^{51} + 2 q^{54} + 24 q^{59} - 4 q^{61} - 14 q^{64} - 8 q^{66} - 16 q^{69} + 12 q^{74} + 8 q^{76} - 32 q^{79} + 2 q^{81} + 8 q^{86} - 20 q^{89} - 16 q^{94} - 10 q^{96} - 8 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i 0.935414 + 0.353553i \(0.115027\pi\)
−0.935414 + 0.353553i \(0.884973\pi\)
\(3\) 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 \(\pi\)
\(8\) 3.00000i 1.06066i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000i 0.288675i
\(13\) 1.00000i 0.277350i
\(14\) 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
\(23\) 8.00000i 1.66812i 0.551677 + 0.834058i \(0.313988\pi\)
−0.551677 + 0.834058i \(0.686012\pi\)
\(24\) −3.00000 −0.612372
\(25\) 0 0
\(26\) −1.00000 −0.196116
\(27\) − 1.00000i − 0.192450i
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 5.00000i 0.883883i
\(33\) 4.00000i 0.696311i
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) −1.00000 −0.166667
\(37\) − 6.00000i − 0.986394i −0.869918 0.493197i \(-0.835828\pi\)
0.869918 0.493197i \(-0.164172\pi\)
\(38\) 4.00000i 0.648886i
\(39\) −1.00000 −0.160128
\(40\) 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
\(46\) −8.00000 −1.17954
\(47\) 8.00000i 1.16692i 0.812142 + 0.583460i \(0.198301\pi\)
−0.812142 + 0.583460i \(0.801699\pi\)
\(48\) − 1.00000i − 0.144338i
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 1.00000i 0.138675i
\(53\) 6.00000i 0.824163i 0.911147 + 0.412082i \(0.135198\pi\)
−0.911147 + 0.412082i \(0.864802\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) 0 0
\(57\) 4.00000i 0.529813i
\(58\) 2.00000i 0.262613i
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) − 8.00000i − 1.01600i
\(63\) 0 0
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) − 2.00000i − 0.242536i
\(69\) −8.00000 −0.963087
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) − 3.00000i − 0.353553i
\(73\) − 6.00000i − 0.702247i −0.936329 0.351123i \(-0.885800\pi\)
0.936329 0.351123i \(-0.114200\pi\)
\(74\) 6.00000 0.697486
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) 0 0
\(78\) − 1.00000i − 0.113228i
\(79\) −16.0000 −1.80014 −0.900070 0.435745i \(-0.856485\pi\)
−0.900070 + 0.435745i \(0.856485\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 6.00000i − 0.662589i
\(83\) − 4.00000i − 0.439057i −0.975606 0.219529i \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 4.00000 0.431331
\(87\) 2.00000i 0.214423i
\(88\) 12.0000i 1.27920i
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 8.00000i 0.834058i
\(93\) − 8.00000i − 0.829561i
\(94\) −8.00000 −0.825137
\(95\) 0 0
\(96\) −5.00000 −0.510310
\(97\) − 18.0000i − 1.82762i −0.406138 0.913812i \(-0.633125\pi\)
0.406138 0.913812i \(-0.366875\pi\)
\(98\) 7.00000i 0.707107i
\(99\) −4.00000 −0.402015
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 2.00000i 0.198030i
\(103\) − 8.00000i − 0.788263i −0.919054 0.394132i \(-0.871045\pi\)
0.919054 0.394132i \(-0.128955\pi\)
\(104\) −3.00000 −0.294174
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) − 12.0000i − 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) − 1.00000i − 0.0962250i
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 6.00000 0.569495
\(112\) 0 0
\(113\) 2.00000i 0.188144i 0.995565 + 0.0940721i \(0.0299884\pi\)
−0.995565 + 0.0940721i \(0.970012\pi\)
\(114\) −4.00000 −0.374634
\(115\) 0 0
\(116\) 2.00000 0.185695
\(117\) − 1.00000i − 0.0924500i
\(118\) 12.0000i 1.10469i
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) − 2.00000i − 0.181071i
\(123\) − 6.00000i − 0.541002i
\(124\) −8.00000 −0.718421
\(125\) 0 0
\(126\) 0 0
\(127\) 16.0000i 1.41977i 0.704317 + 0.709885i \(0.251253\pi\)
−0.704317 + 0.709885i \(0.748747\pi\)
\(128\) 3.00000i 0.265165i
\(129\) 4.00000 0.352180
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 4.00000i 0.348155i
\(133\) 0 0
\(134\) −4.00000 −0.345547
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) − 8.00000i − 0.681005i
\(139\) 20.0000 1.69638 0.848189 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 4.00000i 0.334497i
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 6.00000 0.496564
\(147\) 7.00000i 0.577350i
\(148\) − 6.00000i − 0.493197i
\(149\) 10.0000 0.819232 0.409616 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 12.0000i 0.973329i
\(153\) 2.00000i 0.161690i
\(154\) 0 0
\(155\) 0 0
\(156\) −1.00000 −0.0800641
\(157\) 2.00000i 0.159617i 0.996810 + 0.0798087i \(0.0254309\pi\)
−0.996810 + 0.0798087i \(0.974569\pi\)
\(158\) − 16.0000i − 1.27289i
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000i 0.0785674i
\(163\) − 20.0000i − 1.56652i −0.621694 0.783260i \(-0.713555\pi\)
0.621694 0.783260i \(-0.286445\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 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
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) − 4.00000i − 0.304997i
\(173\) − 2.00000i − 0.152057i −0.997106 0.0760286i \(-0.975776\pi\)
0.997106 0.0760286i \(-0.0242240\pi\)
\(174\) −2.00000 −0.151620
\(175\) 0 0
\(176\) −4.00000 −0.301511
\(177\) 12.0000i 0.901975i
\(178\) − 10.0000i − 0.749532i
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) − 2.00000i − 0.147844i
\(184\) −24.0000 −1.76930
\(185\) 0 0
\(186\) 8.00000 0.586588
\(187\) − 8.00000i − 0.585018i
\(188\) 8.00000i 0.583460i
\(189\) 0 0
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) − 7.00000i − 0.505181i
\(193\) − 14.0000i − 1.00774i −0.863779 0.503871i \(-0.831909\pi\)
0.863779 0.503871i \(-0.168091\pi\)
\(194\) 18.0000 1.29232
\(195\) 0 0
\(196\) 7.00000 0.500000
\(197\) − 6.00000i − 0.427482i −0.976890 0.213741i \(-0.931435\pi\)
0.976890 0.213741i \(-0.0685649\pi\)
\(198\) − 4.00000i − 0.284268i
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 6.00000i 0.422159i
\(203\) 0 0
\(204\) 2.00000 0.140028
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) − 8.00000i − 0.556038i
\(208\) − 1.00000i − 0.0693375i
\(209\) 16.0000 1.10674
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) 6.00000i 0.412082i
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 3.00000 0.204124
\(217\) 0 0
\(218\) 2.00000i 0.135457i
\(219\) 6.00000 0.405442
\(220\) 0 0
\(221\) 2.00000 0.134535
\(222\) 6.00000i 0.402694i
\(223\) − 24.0000i − 1.60716i −0.595198 0.803579i \(-0.702926\pi\)
0.595198 0.803579i \(-0.297074\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −2.00000 −0.133038
\(227\) − 12.0000i − 0.796468i −0.917284 0.398234i \(-0.869623\pi\)
0.917284 0.398234i \(-0.130377\pi\)
\(228\) 4.00000i 0.264906i
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000i 0.393919i
\(233\) 26.0000i 1.70332i 0.524097 + 0.851658i \(0.324403\pi\)
−0.524097 + 0.851658i \(0.675597\pi\)
\(234\) 1.00000 0.0653720
\(235\) 0 0
\(236\) 12.0000 0.781133
\(237\) − 16.0000i − 1.03931i
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 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
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 6.00000 0.382546
\(247\) 4.00000i 0.254514i
\(248\) − 24.0000i − 1.52400i
\(249\) 4.00000 0.253490
\(250\) 0 0
\(251\) −4.00000 −0.252478 −0.126239 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 0 0
\(253\) 32.0000i 2.01182i
\(254\) −16.0000 −1.00393
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 4.00000i 0.249029i
\(259\) 0 0
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) − 12.0000i − 0.741362i
\(263\) − 24.0000i − 1.47990i −0.672660 0.739952i \(-0.734848\pi\)
0.672660 0.739952i \(-0.265152\pi\)
\(264\) −12.0000 −0.738549
\(265\) 0 0
\(266\) 0 0
\(267\) − 10.0000i − 0.611990i
\(268\) 4.00000i 0.244339i
\(269\) −14.0000 −0.853595 −0.426798 0.904347i \(-0.640358\pi\)
−0.426798 + 0.904347i \(0.640358\pi\)
\(270\) 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
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) −8.00000 −0.481543
\(277\) 10.0000i 0.600842i 0.953807 + 0.300421i \(0.0971271\pi\)
−0.953807 + 0.300421i \(0.902873\pi\)
\(278\) 20.0000i 1.19952i
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) − 8.00000i − 0.476393i
\(283\) − 20.0000i − 1.18888i −0.804141 0.594438i \(-0.797374\pi\)
0.804141 0.594438i \(-0.202626\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −4.00000 −0.236525
\(287\) 0 0
\(288\) − 5.00000i − 0.294628i
\(289\) 13.0000 0.764706
\(290\) 0 0
\(291\) 18.0000 1.05518
\(292\) − 6.00000i − 0.351123i
\(293\) 6.00000i 0.350524i 0.984522 + 0.175262i \(0.0560772\pi\)
−0.984522 + 0.175262i \(0.943923\pi\)
\(294\) −7.00000 −0.408248
\(295\) 0 0
\(296\) 18.0000 1.04623
\(297\) − 4.00000i − 0.232104i
\(298\) 10.0000i 0.579284i
\(299\) −8.00000 −0.462652
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 6.00000i 0.344691i
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) −2.00000 −0.114332
\(307\) 4.00000i 0.228292i 0.993464 + 0.114146i \(0.0364132\pi\)
−0.993464 + 0.114146i \(0.963587\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) − 3.00000i − 0.169842i
\(313\) 26.0000i 1.46961i 0.678280 + 0.734803i \(0.262726\pi\)
−0.678280 + 0.734803i \(0.737274\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) −16.0000 −0.900070
\(317\) − 30.0000i − 1.68497i −0.538721 0.842484i \(-0.681092\pi\)
0.538721 0.842484i \(-0.318908\pi\)
\(318\) − 6.00000i − 0.336463i
\(319\) 8.00000 0.447914
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) − 8.00000i − 0.445132i
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 20.0000 1.10770
\(327\) 2.00000i 0.110600i
\(328\) − 18.0000i − 0.993884i
\(329\) 0 0
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) − 4.00000i − 0.219529i
\(333\) 6.00000i 0.328798i
\(334\) 16.0000 0.875481
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000i 0.762629i 0.924445 + 0.381314i \(0.124528\pi\)
−0.924445 + 0.381314i \(0.875472\pi\)
\(338\) − 1.00000i − 0.0543928i
\(339\) −2.00000 −0.108625
\(340\) 0 0
\(341\) −32.0000 −1.73290
\(342\) − 4.00000i − 0.216295i
\(343\) 0 0
\(344\) 12.0000 0.646997
\(345\) 0 0
\(346\) 2.00000 0.107521
\(347\) 4.00000i 0.214731i 0.994220 + 0.107366i \(0.0342415\pi\)
−0.994220 + 0.107366i \(0.965758\pi\)
\(348\) 2.00000i 0.107211i
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 20.0000i 1.06600i
\(353\) − 14.0000i − 0.745145i −0.928003 0.372572i \(-0.878476\pi\)
0.928003 0.372572i \(-0.121524\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) −10.0000 −0.529999
\(357\) 0 0
\(358\) 12.0000i 0.634220i
\(359\) −16.0000 −0.844448 −0.422224 0.906492i \(-0.638750\pi\)
−0.422224 + 0.906492i \(0.638750\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 22.0000i 1.15629i
\(363\) 5.00000i 0.262432i
\(364\) 0 0
\(365\) 0 0
\(366\) 2.00000 0.104542
\(367\) 16.0000i 0.835193i 0.908633 + 0.417597i \(0.137127\pi\)
−0.908633 + 0.417597i \(0.862873\pi\)
\(368\) − 8.00000i − 0.417029i
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 0 0
\(372\) − 8.00000i − 0.414781i
\(373\) 22.0000i 1.13912i 0.821951 + 0.569558i \(0.192886\pi\)
−0.821951 + 0.569558i \(0.807114\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) −24.0000 −1.23771
\(377\) 2.00000i 0.103005i
\(378\) 0 0
\(379\) −36.0000 −1.84920 −0.924598 0.380945i \(-0.875599\pi\)
−0.924598 + 0.380945i \(0.875599\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 16.0000i 0.818631i
\(383\) 24.0000i 1.22634i 0.789950 + 0.613171i \(0.210106\pi\)
−0.789950 + 0.613171i \(0.789894\pi\)
\(384\) −3.00000 −0.153093
\(385\) 0 0
\(386\) 14.0000 0.712581
\(387\) 4.00000i 0.203331i
\(388\) − 18.0000i − 0.913812i
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 21.0000i 1.06066i
\(393\) − 12.0000i − 0.605320i
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) −4.00000 −0.201008
\(397\) − 14.0000i − 0.702640i −0.936255 0.351320i \(-0.885733\pi\)
0.936255 0.351320i \(-0.114267\pi\)
\(398\) 8.00000i 0.401004i
\(399\) 0 0
\(400\) 0 0
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) − 4.00000i − 0.199502i
\(403\) − 8.00000i − 0.398508i
\(404\) 6.00000 0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) − 24.0000i − 1.18964i
\(408\) 6.00000i 0.297044i
\(409\) 38.0000 1.87898 0.939490 0.342578i \(-0.111300\pi\)
0.939490 + 0.342578i \(0.111300\pi\)
\(410\) 0 0
\(411\) −6.00000 −0.295958
\(412\) − 8.00000i − 0.394132i
\(413\) 0 0
\(414\) 8.00000 0.393179
\(415\) 0 0
\(416\) −5.00000 −0.245145
\(417\) 20.0000i 0.979404i
\(418\) 16.0000i 0.782586i
\(419\) −20.0000 −0.977064 −0.488532 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 20.0000i 0.973585i
\(423\) − 8.00000i − 0.388973i
\(424\) −18.0000 −0.874157
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) − 12.0000i − 0.580042i
\(429\) −4.00000 −0.193122
\(430\) 0 0
\(431\) 8.00000 0.385346 0.192673 0.981263i \(-0.438284\pi\)
0.192673 + 0.981263i \(0.438284\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 18.0000i 0.865025i 0.901628 + 0.432512i \(0.142373\pi\)
−0.901628 + 0.432512i \(0.857627\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 32.0000i 1.53077i
\(438\) 6.00000i 0.286691i
\(439\) 24.0000 1.14546 0.572729 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 2.00000i 0.0951303i
\(443\) − 4.00000i − 0.190046i −0.995475 0.0950229i \(-0.969708\pi\)
0.995475 0.0950229i \(-0.0302924\pi\)
\(444\) 6.00000 0.284747
\(445\) 0 0
\(446\) 24.0000 1.13643
\(447\) 10.0000i 0.472984i
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) 2.00000i 0.0940721i
\(453\) 0 0
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) −12.0000 −0.561951
\(457\) − 26.0000i − 1.21623i −0.793849 0.608114i \(-0.791926\pi\)
0.793849 0.608114i \(-0.208074\pi\)
\(458\) − 22.0000i − 1.02799i
\(459\) −2.00000 −0.0933520
\(460\) 0 0
\(461\) −18.0000 −0.838344 −0.419172 0.907907i \(-0.637680\pi\)
−0.419172 + 0.907907i \(0.637680\pi\)
\(462\) 0 0
\(463\) 8.00000i 0.371792i 0.982569 + 0.185896i \(0.0595187\pi\)
−0.982569 + 0.185896i \(0.940481\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) −26.0000 −1.20443
\(467\) − 4.00000i − 0.185098i −0.995708 0.0925490i \(-0.970499\pi\)
0.995708 0.0925490i \(-0.0295015\pi\)
\(468\) − 1.00000i − 0.0462250i
\(469\) 0 0
\(470\) 0 0
\(471\) −2.00000 −0.0921551
\(472\) 36.0000i 1.65703i
\(473\) − 16.0000i − 0.735681i
\(474\) 16.0000 0.734904
\(475\) 0 0
\(476\) 0 0
\(477\) − 6.00000i − 0.274721i
\(478\) 24.0000i 1.09773i
\(479\) −8.00000 −0.365529 −0.182765 0.983157i \(-0.558505\pi\)
−0.182765 + 0.983157i \(0.558505\pi\)
\(480\) 0 0
\(481\) 6.00000 0.273576
\(482\) − 14.0000i − 0.637683i
\(483\) 0 0
\(484\) 5.00000 0.227273
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) − 16.0000i − 0.725029i −0.931978 0.362515i \(-0.881918\pi\)
0.931978 0.362515i \(-0.118082\pi\)
\(488\) − 6.00000i − 0.271607i
\(489\) 20.0000 0.904431
\(490\) 0 0
\(491\) −20.0000 −0.902587 −0.451294 0.892375i \(-0.649037\pi\)
−0.451294 + 0.892375i \(0.649037\pi\)
\(492\) − 6.00000i − 0.270501i
\(493\) − 4.00000i − 0.180151i
\(494\) −4.00000 −0.179969
\(495\) 0 0
\(496\) 8.00000 0.359211
\(497\) 0 0
\(498\) 4.00000i 0.179244i
\(499\) −28.0000 −1.25345 −0.626726 0.779240i \(-0.715605\pi\)
−0.626726 + 0.779240i \(0.715605\pi\)
\(500\) 0 0
\(501\) 16.0000 0.714827
\(502\) − 4.00000i − 0.178529i
\(503\) 24.0000i 1.07011i 0.844818 + 0.535054i \(0.179709\pi\)
−0.844818 + 0.535054i \(0.820291\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −32.0000 −1.42257
\(507\) − 1.00000i − 0.0444116i
\(508\) 16.0000i 0.709885i
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 11.0000i − 0.486136i
\(513\) − 4.00000i − 0.176604i
\(514\) 18.0000 0.793946
\(515\) 0 0
\(516\) 4.00000 0.176090
\(517\) 32.0000i 1.40736i
\(518\) 0 0
\(519\) 2.00000 0.0877903
\(520\) 0 0
\(521\) −22.0000 −0.963837 −0.481919 0.876216i \(-0.660060\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) − 2.00000i − 0.0875376i
\(523\) − 20.0000i − 0.874539i −0.899331 0.437269i \(-0.855946\pi\)
0.899331 0.437269i \(-0.144054\pi\)
\(524\) −12.0000 −0.524222
\(525\) 0 0
\(526\) 24.0000 1.04645
\(527\) 16.0000i 0.696971i
\(528\) − 4.00000i − 0.174078i
\(529\) −41.0000 −1.78261
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) − 6.00000i − 0.259889i
\(534\) 10.0000 0.432742
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) 12.0000i 0.517838i
\(538\) − 14.0000i − 0.603583i
\(539\) 28.0000 1.20605
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) − 8.00000i − 0.343629i
\(543\) 22.0000i 0.944110i
\(544\) 10.0000 0.428746
\(545\) 0 0
\(546\) 0 0
\(547\) − 36.0000i − 1.53925i −0.638497 0.769624i \(-0.720443\pi\)
0.638497 0.769624i \(-0.279557\pi\)
\(548\) 6.00000i 0.256307i
\(549\) 2.00000 0.0853579
\(550\) 0 0
\(551\) 8.00000 0.340811
\(552\) − 24.0000i − 1.02151i
\(553\) 0 0
\(554\) −10.0000 −0.424859
\(555\) 0 0
\(556\) 20.0000 0.848189
\(557\) 18.0000i 0.762684i 0.924434 + 0.381342i \(0.124538\pi\)
−0.924434 + 0.381342i \(0.875462\pi\)
\(558\) 8.00000i 0.338667i
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) − 22.0000i − 0.928014i
\(563\) 4.00000i 0.168580i 0.996441 + 0.0842900i \(0.0268622\pi\)
−0.996441 + 0.0842900i \(0.973138\pi\)
\(564\) −8.00000 −0.336861
\(565\) 0 0
\(566\) 20.0000 0.840663
\(567\) 0 0
\(568\) 0 0
\(569\) 38.0000 1.59304 0.796521 0.604610i \(-0.206671\pi\)
0.796521 + 0.604610i \(0.206671\pi\)
\(570\) 0 0
\(571\) 44.0000 1.84134 0.920671 0.390339i \(-0.127642\pi\)
0.920671 + 0.390339i \(0.127642\pi\)
\(572\) 4.00000i 0.167248i
\(573\) 16.0000i 0.668410i
\(574\) 0 0
\(575\) 0 0
\(576\) 7.00000 0.291667
\(577\) − 18.0000i − 0.749350i −0.927156 0.374675i \(-0.877754\pi\)
0.927156 0.374675i \(-0.122246\pi\)
\(578\) 13.0000i 0.540729i
\(579\) 14.0000 0.581820
\(580\) 0 0
\(581\) 0 0
\(582\) 18.0000i 0.746124i
\(583\) 24.0000i 0.993978i
\(584\) 18.0000 0.744845
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) 12.0000i 0.495293i 0.968850 + 0.247647i \(0.0796572\pi\)
−0.968850 + 0.247647i \(0.920343\pi\)
\(588\) 7.00000i 0.288675i
\(589\) −32.0000 −1.31854
\(590\) 0 0
\(591\) 6.00000 0.246807
\(592\) 6.00000i 0.246598i
\(593\) − 30.0000i − 1.23195i −0.787765 0.615976i \(-0.788762\pi\)
0.787765 0.615976i \(-0.211238\pi\)
\(594\) 4.00000 0.164122
\(595\) 0 0
\(596\) 10.0000 0.409616
\(597\) 8.00000i 0.327418i
\(598\) − 8.00000i − 0.327144i
\(599\) −40.0000 −1.63436 −0.817178 0.576386i \(-0.804463\pi\)
−0.817178 + 0.576386i \(0.804463\pi\)
\(600\) 0 0
\(601\) −38.0000 −1.55005 −0.775026 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) 0 0
\(603\) − 4.00000i − 0.162893i
\(604\) 0 0
\(605\) 0 0
\(606\) −6.00000 −0.243733
\(607\) − 32.0000i − 1.29884i −0.760430 0.649420i \(-0.775012\pi\)
0.760430 0.649420i \(-0.224988\pi\)
\(608\) 20.0000i 0.811107i
\(609\) 0 0
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) 2.00000i 0.0808452i
\(613\) 6.00000i 0.242338i 0.992632 + 0.121169i \(0.0386643\pi\)
−0.992632 + 0.121169i \(0.961336\pi\)
\(614\) −4.00000 −0.161427
\(615\) 0 0
\(616\) 0 0
\(617\) 38.0000i 1.52982i 0.644136 + 0.764911i \(0.277217\pi\)
−0.644136 + 0.764911i \(0.722783\pi\)
\(618\) 8.00000i 0.321807i
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) 8.00000 0.321029
\(622\) 24.0000i 0.962312i
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) 0 0
\(626\) −26.0000 −1.03917
\(627\) 16.0000i 0.638978i
\(628\) 2.00000i 0.0798087i
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) − 48.0000i − 1.90934i
\(633\) 20.0000i 0.794929i
\(634\) 30.0000 1.19145
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 7.00000i 0.277350i
\(638\) 8.00000i 0.316723i
\(639\) 0 0
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 12.0000i 0.473602i
\(643\) 28.0000i 1.10421i 0.833774 + 0.552106i \(0.186176\pi\)
−0.833774 + 0.552106i \(0.813824\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 8.00000 0.314756
\(647\) − 24.0000i − 0.943537i −0.881722 0.471769i \(-0.843616\pi\)
0.881722 0.471769i \(-0.156384\pi\)
\(648\) 3.00000i 0.117851i
\(649\) 48.0000 1.88416
\(650\) 0 0
\(651\) 0 0
\(652\) − 20.0000i − 0.783260i
\(653\) 30.0000i 1.17399i 0.809590 + 0.586995i \(0.199689\pi\)
−0.809590 + 0.586995i \(0.800311\pi\)
\(654\) −2.00000 −0.0782062
\(655\) 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
\(661\) 38.0000 1.47803 0.739014 0.673690i \(-0.235292\pi\)
0.739014 + 0.673690i \(0.235292\pi\)
\(662\) 20.0000i 0.777322i
\(663\) 2.00000i 0.0776736i
\(664\) 12.0000 0.465690
\(665\) 0 0
\(666\) −6.00000 −0.232495
\(667\) 16.0000i 0.619522i
\(668\) − 16.0000i − 0.619059i
\(669\) 24.0000 0.927894
\(670\) 0 0
\(671\) −8.00000 −0.308837
\(672\) 0 0
\(673\) 34.0000i 1.31060i 0.755367 + 0.655302i \(0.227459\pi\)
−0.755367 + 0.655302i \(0.772541\pi\)
\(674\) −14.0000 −0.539260
\(675\) 0 0
\(676\) −1.00000 −0.0384615
\(677\) 10.0000i 0.384331i 0.981363 + 0.192166i \(0.0615511\pi\)
−0.981363 + 0.192166i \(0.938449\pi\)
\(678\) − 2.00000i − 0.0768095i
\(679\) 0 0
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) − 32.0000i − 1.22534i
\(683\) 4.00000i 0.153056i 0.997067 + 0.0765279i \(0.0243834\pi\)
−0.997067 + 0.0765279i \(0.975617\pi\)
\(684\) −4.00000 −0.152944
\(685\) 0 0
\(686\) 0 0
\(687\) − 22.0000i − 0.839352i
\(688\) 4.00000i 0.152499i
\(689\) −6.00000 −0.228582
\(690\) 0 0
\(691\) −36.0000 −1.36950 −0.684752 0.728776i \(-0.740090\pi\)
−0.684752 + 0.728776i \(0.740090\pi\)
\(692\) − 2.00000i − 0.0760286i
\(693\) 0 0
\(694\) −4.00000 −0.151838
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) 12.0000i 0.454532i
\(698\) − 14.0000i − 0.529908i
\(699\) −26.0000 −0.983410
\(700\) 0 0
\(701\) −34.0000 −1.28416 −0.642081 0.766637i \(-0.721929\pi\)
−0.642081 + 0.766637i \(0.721929\pi\)
\(702\) 1.00000i 0.0377426i
\(703\) − 24.0000i − 0.905177i
\(704\) −28.0000 −1.05529
\(705\) 0 0
\(706\) 14.0000 0.526897
\(707\) 0 0
\(708\) 12.0000i 0.450988i
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 16.0000 0.600047
\(712\) − 30.0000i − 1.12430i
\(713\) − 64.0000i − 2.39682i
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 24.0000i 0.896296i
\(718\) − 16.0000i − 0.597115i
\(719\) 48.0000 1.79010 0.895049 0.445968i \(-0.147140\pi\)
0.895049 + 0.445968i \(0.147140\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 3.00000i − 0.111648i
\(723\) − 14.0000i − 0.520666i
\(724\) 22.0000 0.817624
\(725\) 0 0
\(726\) −5.00000 −0.185567
\(727\) 24.0000i 0.890111i 0.895503 + 0.445055i \(0.146816\pi\)
−0.895503 + 0.445055i \(0.853184\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) − 2.00000i − 0.0739221i
\(733\) 30.0000i 1.10808i 0.832492 + 0.554038i \(0.186914\pi\)
−0.832492 + 0.554038i \(0.813086\pi\)
\(734\) −16.0000 −0.590571
\(735\) 0 0
\(736\) −40.0000 −1.47442
\(737\) 16.0000i 0.589368i
\(738\) 6.00000i 0.220863i
\(739\) 4.00000 0.147142 0.0735712 0.997290i \(-0.476560\pi\)
0.0735712 + 0.997290i \(0.476560\pi\)
\(740\) 0 0
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 24.0000 0.879883
\(745\) 0 0
\(746\) −22.0000 −0.805477
\(747\) 4.00000i 0.146352i
\(748\) − 8.00000i − 0.292509i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) − 8.00000i − 0.291730i
\(753\) − 4.00000i − 0.145768i
\(754\) −2.00000 −0.0728357
\(755\) 0 0
\(756\) 0 0
\(757\) 42.0000i 1.52652i 0.646094 + 0.763258i \(0.276401\pi\)
−0.646094 + 0.763258i \(0.723599\pi\)
\(758\) − 36.0000i − 1.30758i
\(759\) −32.0000 −1.16153
\(760\) 0 0
\(761\) 10.0000 0.362500 0.181250 0.983437i \(-0.441986\pi\)
0.181250 + 0.983437i \(0.441986\pi\)
\(762\) − 16.0000i − 0.579619i
\(763\) 0 0
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) 12.0000i 0.433295i
\(768\) − 17.0000i − 0.613435i
\(769\) 30.0000 1.08183 0.540914 0.841078i \(-0.318079\pi\)
0.540914 + 0.841078i \(0.318079\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) − 14.0000i − 0.503871i
\(773\) 6.00000i 0.215805i 0.994161 + 0.107903i \(0.0344134\pi\)
−0.994161 + 0.107903i \(0.965587\pi\)
\(774\) −4.00000 −0.143777
\(775\) 0 0
\(776\) 54.0000 1.93849
\(777\) 0 0
\(778\) − 6.00000i − 0.215110i
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 0 0
\(782\) 16.0000i 0.572159i
\(783\) − 2.00000i − 0.0714742i
\(784\) −7.00000 −0.250000
\(785\) 0 0
\(786\) 12.0000 0.428026
\(787\) 4.00000i 0.142585i 0.997455 + 0.0712923i \(0.0227123\pi\)
−0.997455 + 0.0712923i \(0.977288\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 0 0
\(792\) − 12.0000i − 0.426401i
\(793\) − 2.00000i − 0.0710221i
\(794\) 14.0000 0.496841
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) − 46.0000i − 1.62940i −0.579880 0.814702i \(-0.696901\pi\)
0.579880 0.814702i \(-0.303099\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) − 30.0000i − 1.05934i
\(803\) − 24.0000i − 0.846942i
\(804\) −4.00000 −0.141069
\(805\) 0 0
\(806\) 8.00000 0.281788
\(807\) − 14.0000i − 0.492823i
\(808\) 18.0000i 0.633238i
\(809\) −42.0000 −1.47664 −0.738321 0.674450i \(-0.764381\pi\)
−0.738321 + 0.674450i \(0.764381\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 0 0
\(813\) − 8.00000i − 0.280572i
\(814\) 24.0000 0.841200
\(815\) 0 0
\(816\) −2.00000 −0.0700140
\(817\) − 16.0000i − 0.559769i
\(818\) 38.0000i 1.32864i
\(819\) 0 0
\(820\) 0 0
\(821\) 22.0000 0.767805 0.383903 0.923374i \(-0.374580\pi\)
0.383903 + 0.923374i \(0.374580\pi\)
\(822\) − 6.00000i − 0.209274i
\(823\) − 24.0000i − 0.836587i −0.908312 0.418294i \(-0.862628\pi\)
0.908312 0.418294i \(-0.137372\pi\)
\(824\) 24.0000 0.836080
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) − 8.00000i − 0.278019i
\(829\) 34.0000 1.18087 0.590434 0.807086i \(-0.298956\pi\)
0.590434 + 0.807086i \(0.298956\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) − 7.00000i − 0.242681i
\(833\) − 14.0000i − 0.485071i
\(834\) −20.0000 −0.692543
\(835\) 0 0
\(836\) 16.0000 0.553372
\(837\) 8.00000i 0.276520i
\(838\) − 20.0000i − 0.690889i
\(839\) −16.0000 −0.552381 −0.276191 0.961103i \(-0.589072\pi\)
−0.276191 + 0.961103i \(0.589072\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) − 10.0000i − 0.344623i
\(843\) − 22.0000i − 0.757720i
\(844\) 20.0000 0.688428
\(845\) 0 0
\(846\) 8.00000 0.275046
\(847\) 0 0
\(848\) − 6.00000i − 0.206041i
\(849\) 20.0000 0.686398
\(850\) 0 0
\(851\) 48.0000 1.64542
\(852\) 0 0
\(853\) − 42.0000i − 1.43805i −0.694983 0.719026i \(-0.744588\pi\)
0.694983 0.719026i \(-0.255412\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 36.0000 1.23045
\(857\) − 10.0000i − 0.341593i −0.985306 0.170797i \(-0.945366\pi\)
0.985306 0.170797i \(-0.0546341\pi\)
\(858\) − 4.00000i − 0.136558i
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 8.00000i 0.272481i
\(863\) − 40.0000i − 1.36162i −0.732462 0.680808i \(-0.761629\pi\)
0.732462 0.680808i \(-0.238371\pi\)
\(864\) 5.00000 0.170103
\(865\) 0 0
\(866\) −18.0000 −0.611665
\(867\) 13.0000i 0.441503i
\(868\) 0 0
\(869\) −64.0000 −2.17105
\(870\) 0 0
\(871\) −4.00000 −0.135535
\(872\) 6.00000i 0.203186i
\(873\) 18.0000i 0.609208i
\(874\) −32.0000 −1.08242
\(875\) 0 0
\(876\) 6.00000 0.202721
\(877\) 50.0000i 1.68838i 0.536044 + 0.844190i \(0.319918\pi\)
−0.536044 + 0.844190i \(0.680082\pi\)
\(878\) 24.0000i 0.809961i
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) −14.0000 −0.471672 −0.235836 0.971793i \(-0.575783\pi\)
−0.235836 + 0.971793i \(0.575783\pi\)
\(882\) − 7.00000i − 0.235702i
\(883\) 36.0000i 1.21150i 0.795656 + 0.605748i \(0.207126\pi\)
−0.795656 + 0.605748i \(0.792874\pi\)
\(884\) 2.00000 0.0672673
\(885\) 0 0
\(886\) 4.00000 0.134383
\(887\) − 24.0000i − 0.805841i −0.915235 0.402921i \(-0.867995\pi\)
0.915235 0.402921i \(-0.132005\pi\)
\(888\) 18.0000i 0.604040i
\(889\) 0 0
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) − 24.0000i − 0.803579i
\(893\) 32.0000i 1.07084i
\(894\) −10.0000 −0.334450
\(895\) 0 0
\(896\) 0 0
\(897\) − 8.00000i − 0.267112i
\(898\) − 18.0000i − 0.600668i
\(899\) −16.0000 −0.533630
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) − 24.0000i − 0.799113i
\(903\) 0 0
\(904\) −6.00000 −0.199557
\(905\) 0 0
\(906\) 0 0
\(907\) 4.00000i 0.132818i 0.997792 + 0.0664089i \(0.0211542\pi\)
−0.997792 + 0.0664089i \(0.978846\pi\)
\(908\) − 12.0000i − 0.398234i
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 32.0000 1.06021 0.530104 0.847933i \(-0.322153\pi\)
0.530104 + 0.847933i \(0.322153\pi\)
\(912\) − 4.00000i − 0.132453i
\(913\) − 16.0000i − 0.529523i
\(914\) 26.0000 0.860004
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) 0 0
\(918\) − 2.00000i − 0.0660098i
\(919\) 40.0000 1.31948 0.659739 0.751495i \(-0.270667\pi\)
0.659739 + 0.751495i \(0.270667\pi\)
\(920\) 0 0
\(921\) −4.00000 −0.131804
\(922\) − 18.0000i − 0.592798i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) −8.00000 −0.262896
\(927\) 8.00000i 0.262754i
\(928\) 10.0000i 0.328266i
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) 28.0000 0.917663
\(932\) 26.0000i 0.851658i
\(933\) 24.0000i 0.785725i
\(934\) 4.00000 0.130884
\(935\) 0 0
\(936\) 3.00000 0.0980581
\(937\) 22.0000i 0.718709i 0.933201 + 0.359354i \(0.117003\pi\)
−0.933201 + 0.359354i \(0.882997\pi\)
\(938\) 0 0
\(939\) −26.0000 −0.848478
\(940\) 0 0
\(941\) 46.0000 1.49956 0.749779 0.661689i \(-0.230160\pi\)
0.749779 + 0.661689i \(0.230160\pi\)
\(942\) − 2.00000i − 0.0651635i
\(943\) − 48.0000i − 1.56310i
\(944\) −12.0000 −0.390567
\(945\) 0 0
\(946\) 16.0000 0.520205
\(947\) − 12.0000i − 0.389948i −0.980808 0.194974i \(-0.937538\pi\)
0.980808 0.194974i \(-0.0624622\pi\)
\(948\) − 16.0000i − 0.519656i
\(949\) 6.00000 0.194768
\(950\) 0 0
\(951\) 30.0000 0.972817
\(952\) 0 0
\(953\) 10.0000i 0.323932i 0.986796 + 0.161966i \(0.0517835\pi\)
−0.986796 + 0.161966i \(0.948217\pi\)
\(954\) 6.00000 0.194257
\(955\) 0 0
\(956\) 24.0000 0.776215
\(957\) 8.00000i 0.258603i
\(958\) − 8.00000i − 0.258468i
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 6.00000i 0.193448i
\(963\) 12.0000i 0.386695i
\(964\) −14.0000 −0.450910
\(965\) 0 0
\(966\) 0 0
\(967\) 16.0000i 0.514525i 0.966342 + 0.257263i \(0.0828206\pi\)
−0.966342 + 0.257263i \(0.917179\pi\)
\(968\) 15.0000i 0.482118i
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 0 0
\(974\) 16.0000 0.512673
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) 30.0000i 0.959785i 0.877327 + 0.479893i \(0.159324\pi\)
−0.877327 + 0.479893i \(0.840676\pi\)
\(978\) 20.0000i 0.639529i
\(979\) −40.0000 −1.27841
\(980\) 0 0
\(981\) −2.00000 −0.0638551
\(982\) − 20.0000i − 0.638226i
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 18.0000 0.573819
\(985\) 0 0
\(986\) 4.00000 0.127386
\(987\) 0 0
\(988\) 4.00000i 0.127257i
\(989\) 32.0000 1.01754
\(990\) 0 0
\(991\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) − 40.0000i − 1.27000i
\(993\) 20.0000i 0.634681i
\(994\) 0 0
\(995\) 0 0
\(996\) 4.00000 0.126745
\(997\) − 38.0000i − 1.20347i −0.798695 0.601736i \(-0.794476\pi\)
0.798695 0.601736i \(-0.205524\pi\)
\(998\) − 28.0000i − 0.886325i
\(999\) −6.00000 −0.189832
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.e.274.2 2
3.2 odd 2 2925.2.c.f.2224.1 2
5.2 odd 4 195.2.a.a.1.1 1
5.3 odd 4 975.2.a.i.1.1 1
5.4 even 2 inner 975.2.c.e.274.1 2
15.2 even 4 585.2.a.g.1.1 1
15.8 even 4 2925.2.a.d.1.1 1
15.14 odd 2 2925.2.c.f.2224.2 2
20.7 even 4 3120.2.a.k.1.1 1
35.27 even 4 9555.2.a.b.1.1 1
60.47 odd 4 9360.2.a.o.1.1 1
65.12 odd 4 2535.2.a.k.1.1 1
195.77 even 4 7605.2.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.a.1.1 1 5.2 odd 4
585.2.a.g.1.1 1 15.2 even 4
975.2.a.i.1.1 1 5.3 odd 4
975.2.c.e.274.1 2 5.4 even 2 inner
975.2.c.e.274.2 2 1.1 even 1 trivial
2535.2.a.k.1.1 1 65.12 odd 4
2925.2.a.d.1.1 1 15.8 even 4
2925.2.c.f.2224.1 2 3.2 odd 2
2925.2.c.f.2224.2 2 15.14 odd 2
3120.2.a.k.1.1 1 20.7 even 4
7605.2.a.h.1.1 1 195.77 even 4
9360.2.a.o.1.1 1 60.47 odd 4
9555.2.a.b.1.1 1 35.27 even 4