Properties

Label 3450.2.d.l.2899.1
Level $3450$
Weight $2$
Character 3450.2899
Analytic conductor $27.548$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(27.5483886973\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2899.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 3450.2899
Dual form 3450.2.d.l.2899.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -5.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.00000 q^{6} -5.00000i q^{7} +1.00000i q^{8} -1.00000 q^{9} -3.00000 q^{11} -1.00000i q^{12} +5.00000i q^{13} -5.00000 q^{14} +1.00000 q^{16} -6.00000i q^{17} +1.00000i q^{18} +1.00000 q^{19} +5.00000 q^{21} +3.00000i q^{22} +1.00000i q^{23} -1.00000 q^{24} +5.00000 q^{26} -1.00000i q^{27} +5.00000i q^{28} +5.00000 q^{29} -8.00000 q^{31} -1.00000i q^{32} -3.00000i q^{33} -6.00000 q^{34} +1.00000 q^{36} -4.00000i q^{37} -1.00000i q^{38} -5.00000 q^{39} -7.00000 q^{41} -5.00000i q^{42} -7.00000i q^{43} +3.00000 q^{44} +1.00000 q^{46} +6.00000i q^{47} +1.00000i q^{48} -18.0000 q^{49} +6.00000 q^{51} -5.00000i q^{52} +8.00000i q^{53} -1.00000 q^{54} +5.00000 q^{56} +1.00000i q^{57} -5.00000i q^{58} +10.0000 q^{59} -12.0000 q^{61} +8.00000i q^{62} +5.00000i q^{63} -1.00000 q^{64} -3.00000 q^{66} +12.0000i q^{67} +6.00000i q^{68} -1.00000 q^{69} +10.0000 q^{71} -1.00000i q^{72} +15.0000i q^{73} -4.00000 q^{74} -1.00000 q^{76} +15.0000i q^{77} +5.00000i q^{78} +5.00000 q^{79} +1.00000 q^{81} +7.00000i q^{82} -9.00000i q^{83} -5.00000 q^{84} -7.00000 q^{86} +5.00000i q^{87} -3.00000i q^{88} -14.0000 q^{89} +25.0000 q^{91} -1.00000i q^{92} -8.00000i q^{93} +6.00000 q^{94} +1.00000 q^{96} -16.0000i q^{97} +18.0000i q^{98} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 2q^{6} - 2q^{9} - 6q^{11} - 10q^{14} + 2q^{16} + 2q^{19} + 10q^{21} - 2q^{24} + 10q^{26} + 10q^{29} - 16q^{31} - 12q^{34} + 2q^{36} - 10q^{39} - 14q^{41} + 6q^{44} + 2q^{46} - 36q^{49} + 12q^{51} - 2q^{54} + 10q^{56} + 20q^{59} - 24q^{61} - 2q^{64} - 6q^{66} - 2q^{69} + 20q^{71} - 8q^{74} - 2q^{76} + 10q^{79} + 2q^{81} - 10q^{84} - 14q^{86} - 28q^{89} + 50q^{91} + 12q^{94} + 2q^{96} + 6q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3450\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 1.00000i 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) − 5.00000i − 1.88982i −0.327327 0.944911i \(-0.606148\pi\)
0.327327 0.944911i \(-0.393852\pi\)
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −3.00000 −0.904534 −0.452267 0.891883i \(-0.649385\pi\)
−0.452267 + 0.891883i \(0.649385\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) 5.00000i 1.38675i 0.720577 + 0.693375i \(0.243877\pi\)
−0.720577 + 0.693375i \(0.756123\pi\)
\(14\) −5.00000 −1.33631
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) − 6.00000i − 1.45521i −0.685994 0.727607i \(-0.740633\pi\)
0.685994 0.727607i \(-0.259367\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 5.00000 1.09109
\(22\) 3.00000i 0.639602i
\(23\) 1.00000i 0.208514i
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 5.00000 0.980581
\(27\) − 1.00000i − 0.192450i
\(28\) 5.00000i 0.944911i
\(29\) 5.00000 0.928477 0.464238 0.885710i \(-0.346328\pi\)
0.464238 + 0.885710i \(0.346328\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 3.00000i − 0.522233i
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 4.00000i − 0.657596i −0.944400 0.328798i \(-0.893356\pi\)
0.944400 0.328798i \(-0.106644\pi\)
\(38\) − 1.00000i − 0.162221i
\(39\) −5.00000 −0.800641
\(40\) 0 0
\(41\) −7.00000 −1.09322 −0.546608 0.837389i \(-0.684081\pi\)
−0.546608 + 0.837389i \(0.684081\pi\)
\(42\) − 5.00000i − 0.771517i
\(43\) − 7.00000i − 1.06749i −0.845645 0.533745i \(-0.820784\pi\)
0.845645 0.533745i \(-0.179216\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 6.00000i 0.875190i 0.899172 + 0.437595i \(0.144170\pi\)
−0.899172 + 0.437595i \(0.855830\pi\)
\(48\) 1.00000i 0.144338i
\(49\) −18.0000 −2.57143
\(50\) 0 0
\(51\) 6.00000 0.840168
\(52\) − 5.00000i − 0.693375i
\(53\) 8.00000i 1.09888i 0.835532 + 0.549442i \(0.185160\pi\)
−0.835532 + 0.549442i \(0.814840\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 5.00000 0.668153
\(57\) 1.00000i 0.132453i
\(58\) − 5.00000i − 0.656532i
\(59\) 10.0000 1.30189 0.650945 0.759125i \(-0.274373\pi\)
0.650945 + 0.759125i \(0.274373\pi\)
\(60\) 0 0
\(61\) −12.0000 −1.53644 −0.768221 0.640184i \(-0.778858\pi\)
−0.768221 + 0.640184i \(0.778858\pi\)
\(62\) 8.00000i 1.01600i
\(63\) 5.00000i 0.629941i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −3.00000 −0.369274
\(67\) 12.0000i 1.46603i 0.680211 + 0.733017i \(0.261888\pi\)
−0.680211 + 0.733017i \(0.738112\pi\)
\(68\) 6.00000i 0.727607i
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) − 1.00000i − 0.117851i
\(73\) 15.0000i 1.75562i 0.479012 + 0.877809i \(0.340995\pi\)
−0.479012 + 0.877809i \(0.659005\pi\)
\(74\) −4.00000 −0.464991
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) 15.0000i 1.70941i
\(78\) 5.00000i 0.566139i
\(79\) 5.00000 0.562544 0.281272 0.959628i \(-0.409244\pi\)
0.281272 + 0.959628i \(0.409244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 7.00000i 0.773021i
\(83\) − 9.00000i − 0.987878i −0.869496 0.493939i \(-0.835557\pi\)
0.869496 0.493939i \(-0.164443\pi\)
\(84\) −5.00000 −0.545545
\(85\) 0 0
\(86\) −7.00000 −0.754829
\(87\) 5.00000i 0.536056i
\(88\) − 3.00000i − 0.319801i
\(89\) −14.0000 −1.48400 −0.741999 0.670402i \(-0.766122\pi\)
−0.741999 + 0.670402i \(0.766122\pi\)
\(90\) 0 0
\(91\) 25.0000 2.62071
\(92\) − 1.00000i − 0.104257i
\(93\) − 8.00000i − 0.829561i
\(94\) 6.00000 0.618853
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) − 16.0000i − 1.62455i −0.583272 0.812277i \(-0.698228\pi\)
0.583272 0.812277i \(-0.301772\pi\)
\(98\) 18.0000i 1.81827i
\(99\) 3.00000 0.301511
\(100\) 0 0
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) − 6.00000i − 0.594089i
\(103\) 19.0000i 1.87213i 0.351833 + 0.936063i \(0.385559\pi\)
−0.351833 + 0.936063i \(0.614441\pi\)
\(104\) −5.00000 −0.490290
\(105\) 0 0
\(106\) 8.00000 0.777029
\(107\) 12.0000i 1.16008i 0.814587 + 0.580042i \(0.196964\pi\)
−0.814587 + 0.580042i \(0.803036\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) −12.0000 −1.14939 −0.574696 0.818367i \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) − 5.00000i − 0.472456i
\(113\) − 10.0000i − 0.940721i −0.882474 0.470360i \(-0.844124\pi\)
0.882474 0.470360i \(-0.155876\pi\)
\(114\) 1.00000 0.0936586
\(115\) 0 0
\(116\) −5.00000 −0.464238
\(117\) − 5.00000i − 0.462250i
\(118\) − 10.0000i − 0.920575i
\(119\) −30.0000 −2.75010
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) 12.0000i 1.08643i
\(123\) − 7.00000i − 0.631169i
\(124\) 8.00000 0.718421
\(125\) 0 0
\(126\) 5.00000 0.445435
\(127\) − 6.00000i − 0.532414i −0.963916 0.266207i \(-0.914230\pi\)
0.963916 0.266207i \(-0.0857705\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 7.00000 0.616316
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 3.00000i 0.261116i
\(133\) − 5.00000i − 0.433555i
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 8.00000i 0.683486i 0.939793 + 0.341743i \(0.111017\pi\)
−0.939793 + 0.341743i \(0.888983\pi\)
\(138\) 1.00000i 0.0851257i
\(139\) −2.00000 −0.169638 −0.0848189 0.996396i \(-0.527031\pi\)
−0.0848189 + 0.996396i \(0.527031\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) − 10.0000i − 0.839181i
\(143\) − 15.0000i − 1.25436i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 15.0000 1.24141
\(147\) − 18.0000i − 1.48461i
\(148\) 4.00000i 0.328798i
\(149\) 20.0000 1.63846 0.819232 0.573462i \(-0.194400\pi\)
0.819232 + 0.573462i \(0.194400\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 1.00000i 0.0811107i
\(153\) 6.00000i 0.485071i
\(154\) 15.0000 1.20873
\(155\) 0 0
\(156\) 5.00000 0.400320
\(157\) − 4.00000i − 0.319235i −0.987179 0.159617i \(-0.948974\pi\)
0.987179 0.159617i \(-0.0510260\pi\)
\(158\) − 5.00000i − 0.397779i
\(159\) −8.00000 −0.634441
\(160\) 0 0
\(161\) 5.00000 0.394055
\(162\) − 1.00000i − 0.0785674i
\(163\) 12.0000i 0.939913i 0.882690 + 0.469956i \(0.155730\pi\)
−0.882690 + 0.469956i \(0.844270\pi\)
\(164\) 7.00000 0.546608
\(165\) 0 0
\(166\) −9.00000 −0.698535
\(167\) 12.0000i 0.928588i 0.885681 + 0.464294i \(0.153692\pi\)
−0.885681 + 0.464294i \(0.846308\pi\)
\(168\) 5.00000i 0.385758i
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 7.00000i 0.533745i
\(173\) 21.0000i 1.59660i 0.602260 + 0.798300i \(0.294267\pi\)
−0.602260 + 0.798300i \(0.705733\pi\)
\(174\) 5.00000 0.379049
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 10.0000i 0.751646i
\(178\) 14.0000i 1.04934i
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) − 25.0000i − 1.85312i
\(183\) − 12.0000i − 0.887066i
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) −8.00000 −0.586588
\(187\) 18.0000i 1.31629i
\(188\) − 6.00000i − 0.437595i
\(189\) −5.00000 −0.363696
\(190\) 0 0
\(191\) −13.0000 −0.940647 −0.470323 0.882494i \(-0.655863\pi\)
−0.470323 + 0.882494i \(0.655863\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) 6.00000i 0.431889i 0.976406 + 0.215945i \(0.0692831\pi\)
−0.976406 + 0.215945i \(0.930717\pi\)
\(194\) −16.0000 −1.14873
\(195\) 0 0
\(196\) 18.0000 1.28571
\(197\) 11.0000i 0.783718i 0.920025 + 0.391859i \(0.128168\pi\)
−0.920025 + 0.391859i \(0.871832\pi\)
\(198\) − 3.00000i − 0.213201i
\(199\) 1.00000 0.0708881 0.0354441 0.999372i \(-0.488715\pi\)
0.0354441 + 0.999372i \(0.488715\pi\)
\(200\) 0 0
\(201\) −12.0000 −0.846415
\(202\) 10.0000i 0.703598i
\(203\) − 25.0000i − 1.75466i
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) 19.0000 1.32379
\(207\) − 1.00000i − 0.0695048i
\(208\) 5.00000i 0.346688i
\(209\) −3.00000 −0.207514
\(210\) 0 0
\(211\) 10.0000 0.688428 0.344214 0.938891i \(-0.388145\pi\)
0.344214 + 0.938891i \(0.388145\pi\)
\(212\) − 8.00000i − 0.549442i
\(213\) 10.0000i 0.685189i
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 40.0000i 2.71538i
\(218\) 12.0000i 0.812743i
\(219\) −15.0000 −1.01361
\(220\) 0 0
\(221\) 30.0000 2.01802
\(222\) − 4.00000i − 0.268462i
\(223\) − 14.0000i − 0.937509i −0.883328 0.468755i \(-0.844703\pi\)
0.883328 0.468755i \(-0.155297\pi\)
\(224\) −5.00000 −0.334077
\(225\) 0 0
\(226\) −10.0000 −0.665190
\(227\) − 8.00000i − 0.530979i −0.964114 0.265489i \(-0.914466\pi\)
0.964114 0.265489i \(-0.0855335\pi\)
\(228\) − 1.00000i − 0.0662266i
\(229\) −16.0000 −1.05731 −0.528655 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) 0 0
\(231\) −15.0000 −0.986928
\(232\) 5.00000i 0.328266i
\(233\) 7.00000i 0.458585i 0.973358 + 0.229293i \(0.0736413\pi\)
−0.973358 + 0.229293i \(0.926359\pi\)
\(234\) −5.00000 −0.326860
\(235\) 0 0
\(236\) −10.0000 −0.650945
\(237\) 5.00000i 0.324785i
\(238\) 30.0000i 1.94461i
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) −18.0000 −1.15948 −0.579741 0.814801i \(-0.696846\pi\)
−0.579741 + 0.814801i \(0.696846\pi\)
\(242\) 2.00000i 0.128565i
\(243\) 1.00000i 0.0641500i
\(244\) 12.0000 0.768221
\(245\) 0 0
\(246\) −7.00000 −0.446304
\(247\) 5.00000i 0.318142i
\(248\) − 8.00000i − 0.508001i
\(249\) 9.00000 0.570352
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) − 5.00000i − 0.314970i
\(253\) − 3.00000i − 0.188608i
\(254\) −6.00000 −0.376473
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) − 6.00000i − 0.374270i −0.982334 0.187135i \(-0.940080\pi\)
0.982334 0.187135i \(-0.0599201\pi\)
\(258\) − 7.00000i − 0.435801i
\(259\) −20.0000 −1.24274
\(260\) 0 0
\(261\) −5.00000 −0.309492
\(262\) 4.00000i 0.247121i
\(263\) − 4.00000i − 0.246651i −0.992366 0.123325i \(-0.960644\pi\)
0.992366 0.123325i \(-0.0393559\pi\)
\(264\) 3.00000 0.184637
\(265\) 0 0
\(266\) −5.00000 −0.306570
\(267\) − 14.0000i − 0.856786i
\(268\) − 12.0000i − 0.733017i
\(269\) −3.00000 −0.182913 −0.0914566 0.995809i \(-0.529152\pi\)
−0.0914566 + 0.995809i \(0.529152\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) − 6.00000i − 0.363803i
\(273\) 25.0000i 1.51307i
\(274\) 8.00000 0.483298
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) 1.00000i 0.0600842i 0.999549 + 0.0300421i \(0.00956413\pi\)
−0.999549 + 0.0300421i \(0.990436\pi\)
\(278\) 2.00000i 0.119952i
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 6.00000i 0.357295i
\(283\) 20.0000i 1.18888i 0.804141 + 0.594438i \(0.202626\pi\)
−0.804141 + 0.594438i \(0.797374\pi\)
\(284\) −10.0000 −0.593391
\(285\) 0 0
\(286\) −15.0000 −0.886969
\(287\) 35.0000i 2.06598i
\(288\) 1.00000i 0.0589256i
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) 16.0000 0.937937
\(292\) − 15.0000i − 0.877809i
\(293\) − 28.0000i − 1.63578i −0.575376 0.817889i \(-0.695144\pi\)
0.575376 0.817889i \(-0.304856\pi\)
\(294\) −18.0000 −1.04978
\(295\) 0 0
\(296\) 4.00000 0.232495
\(297\) 3.00000i 0.174078i
\(298\) − 20.0000i − 1.15857i
\(299\) −5.00000 −0.289157
\(300\) 0 0
\(301\) −35.0000 −2.01737
\(302\) 10.0000i 0.575435i
\(303\) − 10.0000i − 0.574485i
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) 6.00000 0.342997
\(307\) − 12.0000i − 0.684876i −0.939540 0.342438i \(-0.888747\pi\)
0.939540 0.342438i \(-0.111253\pi\)
\(308\) − 15.0000i − 0.854704i
\(309\) −19.0000 −1.08087
\(310\) 0 0
\(311\) 28.0000 1.58773 0.793867 0.608091i \(-0.208065\pi\)
0.793867 + 0.608091i \(0.208065\pi\)
\(312\) − 5.00000i − 0.283069i
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) −4.00000 −0.225733
\(315\) 0 0
\(316\) −5.00000 −0.281272
\(317\) − 9.00000i − 0.505490i −0.967533 0.252745i \(-0.918667\pi\)
0.967533 0.252745i \(-0.0813334\pi\)
\(318\) 8.00000i 0.448618i
\(319\) −15.0000 −0.839839
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) − 5.00000i − 0.278639i
\(323\) − 6.00000i − 0.333849i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 12.0000 0.664619
\(327\) − 12.0000i − 0.663602i
\(328\) − 7.00000i − 0.386510i
\(329\) 30.0000 1.65395
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 9.00000i 0.493939i
\(333\) 4.00000i 0.219199i
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) 5.00000 0.272772
\(337\) − 8.00000i − 0.435788i −0.975972 0.217894i \(-0.930081\pi\)
0.975972 0.217894i \(-0.0699187\pi\)
\(338\) 12.0000i 0.652714i
\(339\) 10.0000 0.543125
\(340\) 0 0
\(341\) 24.0000 1.29967
\(342\) 1.00000i 0.0540738i
\(343\) 55.0000i 2.96972i
\(344\) 7.00000 0.377415
\(345\) 0 0
\(346\) 21.0000 1.12897
\(347\) 4.00000i 0.214731i 0.994220 + 0.107366i \(0.0342415\pi\)
−0.994220 + 0.107366i \(0.965758\pi\)
\(348\) − 5.00000i − 0.268028i
\(349\) −1.00000 −0.0535288 −0.0267644 0.999642i \(-0.508520\pi\)
−0.0267644 + 0.999642i \(0.508520\pi\)
\(350\) 0 0
\(351\) 5.00000 0.266880
\(352\) 3.00000i 0.159901i
\(353\) 9.00000i 0.479022i 0.970894 + 0.239511i \(0.0769871\pi\)
−0.970894 + 0.239511i \(0.923013\pi\)
\(354\) 10.0000 0.531494
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) − 30.0000i − 1.58777i
\(358\) 6.00000i 0.317110i
\(359\) 7.00000 0.369446 0.184723 0.982791i \(-0.440861\pi\)
0.184723 + 0.982791i \(0.440861\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 14.0000i 0.735824i
\(363\) − 2.00000i − 0.104973i
\(364\) −25.0000 −1.31036
\(365\) 0 0
\(366\) −12.0000 −0.627250
\(367\) − 1.00000i − 0.0521996i −0.999659 0.0260998i \(-0.991691\pi\)
0.999659 0.0260998i \(-0.00830876\pi\)
\(368\) 1.00000i 0.0521286i
\(369\) 7.00000 0.364405
\(370\) 0 0
\(371\) 40.0000 2.07670
\(372\) 8.00000i 0.414781i
\(373\) − 26.0000i − 1.34623i −0.739538 0.673114i \(-0.764956\pi\)
0.739538 0.673114i \(-0.235044\pi\)
\(374\) 18.0000 0.930758
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) 25.0000i 1.28757i
\(378\) 5.00000i 0.257172i
\(379\) −12.0000 −0.616399 −0.308199 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) 0 0
\(381\) 6.00000 0.307389
\(382\) 13.0000i 0.665138i
\(383\) − 9.00000i − 0.459879i −0.973205 0.229939i \(-0.926147\pi\)
0.973205 0.229939i \(-0.0738528\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 6.00000 0.305392
\(387\) 7.00000i 0.355830i
\(388\) 16.0000i 0.812277i
\(389\) 38.0000 1.92668 0.963338 0.268290i \(-0.0864585\pi\)
0.963338 + 0.268290i \(0.0864585\pi\)
\(390\) 0 0
\(391\) 6.00000 0.303433
\(392\) − 18.0000i − 0.909137i
\(393\) − 4.00000i − 0.201773i
\(394\) 11.0000 0.554172
\(395\) 0 0
\(396\) −3.00000 −0.150756
\(397\) 2.00000i 0.100377i 0.998740 + 0.0501886i \(0.0159822\pi\)
−0.998740 + 0.0501886i \(0.984018\pi\)
\(398\) − 1.00000i − 0.0501255i
\(399\) 5.00000 0.250313
\(400\) 0 0
\(401\) 20.0000 0.998752 0.499376 0.866385i \(-0.333563\pi\)
0.499376 + 0.866385i \(0.333563\pi\)
\(402\) 12.0000i 0.598506i
\(403\) − 40.0000i − 1.99254i
\(404\) 10.0000 0.497519
\(405\) 0 0
\(406\) −25.0000 −1.24073
\(407\) 12.0000i 0.594818i
\(408\) 6.00000i 0.297044i
\(409\) −17.0000 −0.840596 −0.420298 0.907386i \(-0.638074\pi\)
−0.420298 + 0.907386i \(0.638074\pi\)
\(410\) 0 0
\(411\) −8.00000 −0.394611
\(412\) − 19.0000i − 0.936063i
\(413\) − 50.0000i − 2.46034i
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) 5.00000 0.245145
\(417\) − 2.00000i − 0.0979404i
\(418\) 3.00000i 0.146735i
\(419\) 1.00000 0.0488532 0.0244266 0.999702i \(-0.492224\pi\)
0.0244266 + 0.999702i \(0.492224\pi\)
\(420\) 0 0
\(421\) −18.0000 −0.877266 −0.438633 0.898666i \(-0.644537\pi\)
−0.438633 + 0.898666i \(0.644537\pi\)
\(422\) − 10.0000i − 0.486792i
\(423\) − 6.00000i − 0.291730i
\(424\) −8.00000 −0.388514
\(425\) 0 0
\(426\) 10.0000 0.484502
\(427\) 60.0000i 2.90360i
\(428\) − 12.0000i − 0.580042i
\(429\) 15.0000 0.724207
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) − 28.0000i − 1.34559i −0.739827 0.672797i \(-0.765093\pi\)
0.739827 0.672797i \(-0.234907\pi\)
\(434\) 40.0000 1.92006
\(435\) 0 0
\(436\) 12.0000 0.574696
\(437\) 1.00000i 0.0478365i
\(438\) 15.0000i 0.716728i
\(439\) 12.0000 0.572729 0.286364 0.958121i \(-0.407553\pi\)
0.286364 + 0.958121i \(0.407553\pi\)
\(440\) 0 0
\(441\) 18.0000 0.857143
\(442\) − 30.0000i − 1.42695i
\(443\) − 4.00000i − 0.190046i −0.995475 0.0950229i \(-0.969708\pi\)
0.995475 0.0950229i \(-0.0302924\pi\)
\(444\) −4.00000 −0.189832
\(445\) 0 0
\(446\) −14.0000 −0.662919
\(447\) 20.0000i 0.945968i
\(448\) 5.00000i 0.236228i
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 0 0
\(451\) 21.0000 0.988851
\(452\) 10.0000i 0.470360i
\(453\) − 10.0000i − 0.469841i
\(454\) −8.00000 −0.375459
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) 18.0000i 0.842004i 0.907060 + 0.421002i \(0.138322\pi\)
−0.907060 + 0.421002i \(0.861678\pi\)
\(458\) 16.0000i 0.747631i
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) 7.00000 0.326023 0.163011 0.986624i \(-0.447879\pi\)
0.163011 + 0.986624i \(0.447879\pi\)
\(462\) 15.0000i 0.697863i
\(463\) 4.00000i 0.185896i 0.995671 + 0.0929479i \(0.0296290\pi\)
−0.995671 + 0.0929479i \(0.970371\pi\)
\(464\) 5.00000 0.232119
\(465\) 0 0
\(466\) 7.00000 0.324269
\(467\) − 21.0000i − 0.971764i −0.874024 0.485882i \(-0.838498\pi\)
0.874024 0.485882i \(-0.161502\pi\)
\(468\) 5.00000i 0.231125i
\(469\) 60.0000 2.77054
\(470\) 0 0
\(471\) 4.00000 0.184310
\(472\) 10.0000i 0.460287i
\(473\) 21.0000i 0.965581i
\(474\) 5.00000 0.229658
\(475\) 0 0
\(476\) 30.0000 1.37505
\(477\) − 8.00000i − 0.366295i
\(478\) 24.0000i 1.09773i
\(479\) 25.0000 1.14228 0.571140 0.820853i \(-0.306501\pi\)
0.571140 + 0.820853i \(0.306501\pi\)
\(480\) 0 0
\(481\) 20.0000 0.911922
\(482\) 18.0000i 0.819878i
\(483\) 5.00000i 0.227508i
\(484\) 2.00000 0.0909091
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) − 2.00000i − 0.0906287i −0.998973 0.0453143i \(-0.985571\pi\)
0.998973 0.0453143i \(-0.0144289\pi\)
\(488\) − 12.0000i − 0.543214i
\(489\) −12.0000 −0.542659
\(490\) 0 0
\(491\) −32.0000 −1.44414 −0.722070 0.691820i \(-0.756809\pi\)
−0.722070 + 0.691820i \(0.756809\pi\)
\(492\) 7.00000i 0.315584i
\(493\) − 30.0000i − 1.35113i
\(494\) 5.00000 0.224961
\(495\) 0 0
\(496\) −8.00000 −0.359211
\(497\) − 50.0000i − 2.24281i
\(498\) − 9.00000i − 0.403300i
\(499\) 6.00000 0.268597 0.134298 0.990941i \(-0.457122\pi\)
0.134298 + 0.990941i \(0.457122\pi\)
\(500\) 0 0
\(501\) −12.0000 −0.536120
\(502\) − 12.0000i − 0.535586i
\(503\) − 19.0000i − 0.847168i −0.905857 0.423584i \(-0.860772\pi\)
0.905857 0.423584i \(-0.139228\pi\)
\(504\) −5.00000 −0.222718
\(505\) 0 0
\(506\) −3.00000 −0.133366
\(507\) − 12.0000i − 0.532939i
\(508\) 6.00000i 0.266207i
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) 75.0000 3.31780
\(512\) − 1.00000i − 0.0441942i
\(513\) − 1.00000i − 0.0441511i
\(514\) −6.00000 −0.264649
\(515\) 0 0
\(516\) −7.00000 −0.308158
\(517\) − 18.0000i − 0.791639i
\(518\) 20.0000i 0.878750i
\(519\) −21.0000 −0.921798
\(520\) 0 0
\(521\) −4.00000 −0.175243 −0.0876216 0.996154i \(-0.527927\pi\)
−0.0876216 + 0.996154i \(0.527927\pi\)
\(522\) 5.00000i 0.218844i
\(523\) − 1.00000i − 0.0437269i −0.999761 0.0218635i \(-0.993040\pi\)
0.999761 0.0218635i \(-0.00695991\pi\)
\(524\) 4.00000 0.174741
\(525\) 0 0
\(526\) −4.00000 −0.174408
\(527\) 48.0000i 2.09091i
\(528\) − 3.00000i − 0.130558i
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −10.0000 −0.433963
\(532\) 5.00000i 0.216777i
\(533\) − 35.0000i − 1.51602i
\(534\) −14.0000 −0.605839
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) − 6.00000i − 0.258919i
\(538\) 3.00000i 0.129339i
\(539\) 54.0000 2.32594
\(540\) 0 0
\(541\) −39.0000 −1.67674 −0.838370 0.545101i \(-0.816491\pi\)
−0.838370 + 0.545101i \(0.816491\pi\)
\(542\) 28.0000i 1.20270i
\(543\) − 14.0000i − 0.600798i
\(544\) −6.00000 −0.257248
\(545\) 0 0
\(546\) 25.0000 1.06990
\(547\) − 30.0000i − 1.28271i −0.767245 0.641354i \(-0.778373\pi\)
0.767245 0.641354i \(-0.221627\pi\)
\(548\) − 8.00000i − 0.341743i
\(549\) 12.0000 0.512148
\(550\) 0 0
\(551\) 5.00000 0.213007
\(552\) − 1.00000i − 0.0425628i
\(553\) − 25.0000i − 1.06311i
\(554\) 1.00000 0.0424859
\(555\) 0 0
\(556\) 2.00000 0.0848189
\(557\) 10.0000i 0.423714i 0.977301 + 0.211857i \(0.0679510\pi\)
−0.977301 + 0.211857i \(0.932049\pi\)
\(558\) − 8.00000i − 0.338667i
\(559\) 35.0000 1.48034
\(560\) 0 0
\(561\) −18.0000 −0.759961
\(562\) − 2.00000i − 0.0843649i
\(563\) − 21.0000i − 0.885044i −0.896758 0.442522i \(-0.854084\pi\)
0.896758 0.442522i \(-0.145916\pi\)
\(564\) 6.00000 0.252646
\(565\) 0 0
\(566\) 20.0000 0.840663
\(567\) − 5.00000i − 0.209980i
\(568\) 10.0000i 0.419591i
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 15.0000i 0.627182i
\(573\) − 13.0000i − 0.543083i
\(574\) 35.0000 1.46087
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 23.0000i − 0.957503i −0.877951 0.478751i \(-0.841090\pi\)
0.877951 0.478751i \(-0.158910\pi\)
\(578\) 19.0000i 0.790296i
\(579\) −6.00000 −0.249351
\(580\) 0 0
\(581\) −45.0000 −1.86691
\(582\) − 16.0000i − 0.663221i
\(583\) − 24.0000i − 0.993978i
\(584\) −15.0000 −0.620704
\(585\) 0 0
\(586\) −28.0000 −1.15667
\(587\) − 42.0000i − 1.73353i −0.498721 0.866763i \(-0.666197\pi\)
0.498721 0.866763i \(-0.333803\pi\)
\(588\) 18.0000i 0.742307i
\(589\) −8.00000 −0.329634
\(590\) 0 0
\(591\) −11.0000 −0.452480
\(592\) − 4.00000i − 0.164399i
\(593\) 3.00000i 0.123195i 0.998101 + 0.0615976i \(0.0196196\pi\)
−0.998101 + 0.0615976i \(0.980380\pi\)
\(594\) 3.00000 0.123091
\(595\) 0 0
\(596\) −20.0000 −0.819232
\(597\) 1.00000i 0.0409273i
\(598\) 5.00000i 0.204465i
\(599\) 44.0000 1.79779 0.898896 0.438163i \(-0.144371\pi\)
0.898896 + 0.438163i \(0.144371\pi\)
\(600\) 0 0
\(601\) −38.0000 −1.55005 −0.775026 0.631929i \(-0.782263\pi\)
−0.775026 + 0.631929i \(0.782263\pi\)
\(602\) 35.0000i 1.42649i
\(603\) − 12.0000i − 0.488678i
\(604\) 10.0000 0.406894
\(605\) 0 0
\(606\) −10.0000 −0.406222
\(607\) 22.0000i 0.892952i 0.894795 + 0.446476i \(0.147321\pi\)
−0.894795 + 0.446476i \(0.852679\pi\)
\(608\) − 1.00000i − 0.0405554i
\(609\) 25.0000 1.01305
\(610\) 0 0
\(611\) −30.0000 −1.21367
\(612\) − 6.00000i − 0.242536i
\(613\) 10.0000i 0.403896i 0.979396 + 0.201948i \(0.0647272\pi\)
−0.979396 + 0.201948i \(0.935273\pi\)
\(614\) −12.0000 −0.484281
\(615\) 0 0
\(616\) −15.0000 −0.604367
\(617\) 2.00000i 0.0805170i 0.999189 + 0.0402585i \(0.0128181\pi\)
−0.999189 + 0.0402585i \(0.987182\pi\)
\(618\) 19.0000i 0.764292i
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) − 28.0000i − 1.12270i
\(623\) 70.0000i 2.80449i
\(624\) −5.00000 −0.200160
\(625\) 0 0
\(626\) 0 0
\(627\) − 3.00000i − 0.119808i
\(628\) 4.00000i 0.159617i
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) 29.0000 1.15447 0.577236 0.816577i \(-0.304131\pi\)
0.577236 + 0.816577i \(0.304131\pi\)
\(632\) 5.00000i 0.198889i
\(633\) 10.0000i 0.397464i
\(634\) −9.00000 −0.357436
\(635\) 0 0
\(636\) 8.00000 0.317221
\(637\) − 90.0000i − 3.56593i
\(638\) 15.0000i 0.593856i
\(639\) −10.0000 −0.395594
\(640\) 0 0
\(641\) −36.0000 −1.42191 −0.710957 0.703235i \(-0.751738\pi\)
−0.710957 + 0.703235i \(0.751738\pi\)
\(642\) 12.0000i 0.473602i
\(643\) 1.00000i 0.0394362i 0.999806 + 0.0197181i \(0.00627687\pi\)
−0.999806 + 0.0197181i \(0.993723\pi\)
\(644\) −5.00000 −0.197028
\(645\) 0 0
\(646\) −6.00000 −0.236067
\(647\) − 24.0000i − 0.943537i −0.881722 0.471769i \(-0.843616\pi\)
0.881722 0.471769i \(-0.156384\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) −30.0000 −1.17760
\(650\) 0 0
\(651\) −40.0000 −1.56772
\(652\) − 12.0000i − 0.469956i
\(653\) 11.0000i 0.430463i 0.976563 + 0.215232i \(0.0690506\pi\)
−0.976563 + 0.215232i \(0.930949\pi\)
\(654\) −12.0000 −0.469237
\(655\) 0 0
\(656\) −7.00000 −0.273304
\(657\) − 15.0000i − 0.585206i
\(658\) − 30.0000i − 1.16952i
\(659\) −3.00000 −0.116863 −0.0584317 0.998291i \(-0.518610\pi\)
−0.0584317 + 0.998291i \(0.518610\pi\)
\(660\) 0 0
\(661\) 40.0000 1.55582 0.777910 0.628376i \(-0.216280\pi\)
0.777910 + 0.628376i \(0.216280\pi\)
\(662\) − 4.00000i − 0.155464i
\(663\) 30.0000i 1.16510i
\(664\) 9.00000 0.349268
\(665\) 0 0
\(666\) 4.00000 0.154997
\(667\) 5.00000i 0.193601i
\(668\) − 12.0000i − 0.464294i
\(669\) 14.0000 0.541271
\(670\) 0 0
\(671\) 36.0000 1.38976
\(672\) − 5.00000i − 0.192879i
\(673\) 45.0000i 1.73462i 0.497766 + 0.867311i \(0.334154\pi\)
−0.497766 + 0.867311i \(0.665846\pi\)
\(674\) −8.00000 −0.308148
\(675\) 0 0
\(676\) 12.0000 0.461538
\(677\) − 24.0000i − 0.922395i −0.887298 0.461197i \(-0.847420\pi\)
0.887298 0.461197i \(-0.152580\pi\)
\(678\) − 10.0000i − 0.384048i
\(679\) −80.0000 −3.07012
\(680\) 0 0
\(681\) 8.00000 0.306561
\(682\) − 24.0000i − 0.919007i
\(683\) − 10.0000i − 0.382639i −0.981528 0.191320i \(-0.938723\pi\)
0.981528 0.191320i \(-0.0612767\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) 55.0000 2.09991
\(687\) − 16.0000i − 0.610438i
\(688\) − 7.00000i − 0.266872i
\(689\) −40.0000 −1.52388
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) − 21.0000i − 0.798300i
\(693\) − 15.0000i − 0.569803i
\(694\) 4.00000 0.151838
\(695\) 0 0
\(696\) −5.00000 −0.189525
\(697\) 42.0000i 1.59086i
\(698\) 1.00000i 0.0378506i
\(699\) −7.00000 −0.264764
\(700\) 0 0
\(701\) 12.0000 0.453234 0.226617 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) − 5.00000i − 0.188713i
\(703\) − 4.00000i − 0.150863i
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) 9.00000 0.338719
\(707\) 50.0000i 1.88044i
\(708\) − 10.0000i − 0.375823i
\(709\) −12.0000 −0.450669 −0.225335 0.974281i \(-0.572348\pi\)
−0.225335 + 0.974281i \(0.572348\pi\)
\(710\) 0 0
\(711\) −5.00000 −0.187515
\(712\) − 14.0000i − 0.524672i
\(713\) − 8.00000i − 0.299602i
\(714\) −30.0000 −1.12272
\(715\) 0 0
\(716\) 6.00000 0.224231
\(717\) − 24.0000i − 0.896296i
\(718\) − 7.00000i − 0.261238i
\(719\) 22.0000 0.820462 0.410231 0.911982i \(-0.365448\pi\)
0.410231 + 0.911982i \(0.365448\pi\)
\(720\) 0 0
\(721\) 95.0000 3.53798
\(722\) 18.0000i 0.669891i
\(723\) − 18.0000i − 0.669427i
\(724\) 14.0000 0.520306
\(725\) 0 0
\(726\) −2.00000 −0.0742270
\(727\) − 32.0000i − 1.18681i −0.804902 0.593407i \(-0.797782\pi\)
0.804902 0.593407i \(-0.202218\pi\)
\(728\) 25.0000i 0.926562i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −42.0000 −1.55343
\(732\) 12.0000i 0.443533i
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) −1.00000 −0.0369107
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) − 36.0000i − 1.32608i
\(738\) − 7.00000i − 0.257674i
\(739\) 24.0000 0.882854 0.441427 0.897297i \(-0.354472\pi\)
0.441427 + 0.897297i \(0.354472\pi\)
\(740\) 0 0
\(741\) −5.00000 −0.183680
\(742\) − 40.0000i − 1.46845i
\(743\) 11.0000i 0.403551i 0.979432 + 0.201775i \(0.0646711\pi\)
−0.979432 + 0.201775i \(0.935329\pi\)
\(744\) 8.00000 0.293294
\(745\) 0 0
\(746\) −26.0000 −0.951928
\(747\) 9.00000i 0.329293i
\(748\) − 18.0000i − 0.658145i
\(749\) 60.0000 2.19235
\(750\) 0 0
\(751\) −43.0000 −1.56909 −0.784546 0.620070i \(-0.787104\pi\)
−0.784546 + 0.620070i \(0.787104\pi\)
\(752\) 6.00000i 0.218797i
\(753\) 12.0000i 0.437304i
\(754\) 25.0000 0.910446
\(755\) 0 0
\(756\) 5.00000 0.181848
\(757\) − 28.0000i − 1.01768i −0.860862 0.508839i \(-0.830075\pi\)
0.860862 0.508839i \(-0.169925\pi\)
\(758\) 12.0000i 0.435860i
\(759\) 3.00000 0.108893
\(760\) 0 0
\(761\) 51.0000 1.84875 0.924374 0.381487i \(-0.124588\pi\)
0.924374 + 0.381487i \(0.124588\pi\)
\(762\) − 6.00000i − 0.217357i
\(763\) 60.0000i 2.17215i
\(764\) 13.0000 0.470323
\(765\) 0 0
\(766\) −9.00000 −0.325183
\(767\) 50.0000i 1.80540i
\(768\) 1.00000i 0.0360844i
\(769\) 2.00000 0.0721218 0.0360609 0.999350i \(-0.488519\pi\)
0.0360609 + 0.999350i \(0.488519\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) − 6.00000i − 0.215945i
\(773\) − 12.0000i − 0.431610i −0.976436 0.215805i \(-0.930762\pi\)
0.976436 0.215805i \(-0.0692376\pi\)
\(774\) 7.00000 0.251610
\(775\) 0 0
\(776\) 16.0000 0.574367
\(777\) − 20.0000i − 0.717496i
\(778\) − 38.0000i − 1.36237i
\(779\) −7.00000 −0.250801
\(780\) 0 0
\(781\) −30.0000 −1.07348
\(782\) − 6.00000i − 0.214560i
\(783\) − 5.00000i − 0.178685i
\(784\) −18.0000 −0.642857
\(785\) 0 0
\(786\) −4.00000 −0.142675
\(787\) 17.0000i 0.605985i 0.952993 + 0.302992i \(0.0979856\pi\)
−0.952993 + 0.302992i \(0.902014\pi\)
\(788\) − 11.0000i − 0.391859i
\(789\) 4.00000 0.142404
\(790\) 0 0
\(791\) −50.0000 −1.77780
\(792\) 3.00000i 0.106600i
\(793\) − 60.0000i − 2.13066i
\(794\) 2.00000 0.0709773
\(795\) 0 0
\(796\) −1.00000 −0.0354441
\(797\) − 42.0000i − 1.48772i −0.668338 0.743858i \(-0.732994\pi\)
0.668338 0.743858i \(-0.267006\pi\)
\(798\) − 5.00000i − 0.176998i
\(799\) 36.0000 1.27359
\(800\) 0 0
\(801\) 14.0000 0.494666
\(802\) − 20.0000i − 0.706225i
\(803\) − 45.0000i − 1.58802i
\(804\) 12.0000 0.423207
\(805\) 0 0
\(806\) −40.0000 −1.40894
\(807\) − 3.00000i − 0.105605i
\(808\) − 10.0000i − 0.351799i
\(809\) −19.0000 −0.668004 −0.334002 0.942572i \(-0.608399\pi\)
−0.334002 + 0.942572i \(0.608399\pi\)
\(810\) 0 0
\(811\) −30.0000 −1.05344 −0.526721 0.850038i \(-0.676579\pi\)
−0.526721 + 0.850038i \(0.676579\pi\)
\(812\) 25.0000i 0.877328i
\(813\) − 28.0000i − 0.982003i
\(814\) 12.0000 0.420600
\(815\) 0 0
\(816\) 6.00000 0.210042
\(817\) − 7.00000i − 0.244899i
\(818\) 17.0000i 0.594391i
\(819\) −25.0000 −0.873571
\(820\) 0 0
\(821\) 9.00000 0.314102 0.157051 0.987590i \(-0.449801\pi\)
0.157051 + 0.987590i \(0.449801\pi\)
\(822\) 8.00000i 0.279032i
\(823\) 50.0000i 1.74289i 0.490493 + 0.871445i \(0.336817\pi\)
−0.490493 + 0.871445i \(0.663183\pi\)
\(824\) −19.0000 −0.661896
\(825\) 0 0
\(826\) −50.0000 −1.73972
\(827\) 27.0000i 0.938882i 0.882964 + 0.469441i \(0.155545\pi\)
−0.882964 + 0.469441i \(0.844455\pi\)
\(828\) 1.00000i 0.0347524i
\(829\) 31.0000 1.07667 0.538337 0.842729i \(-0.319053\pi\)
0.538337 + 0.842729i \(0.319053\pi\)
\(830\) 0 0
\(831\) −1.00000 −0.0346896
\(832\) − 5.00000i − 0.173344i
\(833\) 108.000i 3.74198i
\(834\) −2.00000 −0.0692543
\(835\) 0 0
\(836\) 3.00000 0.103757
\(837\) 8.00000i 0.276520i
\(838\) − 1.00000i − 0.0345444i
\(839\) −7.00000 −0.241667 −0.120833 0.992673i \(-0.538557\pi\)
−0.120833 + 0.992673i \(0.538557\pi\)
\(840\) 0 0
\(841\) −4.00000 −0.137931
\(842\) 18.0000i 0.620321i
\(843\) 2.00000i 0.0688837i
\(844\) −10.0000 −0.344214
\(845\) 0 0
\(846\) −6.00000 −0.206284
\(847\) 10.0000i 0.343604i
\(848\) 8.00000i 0.274721i
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 4.00000 0.137118
\(852\) − 10.0000i − 0.342594i
\(853\) 9.00000i 0.308154i 0.988059 + 0.154077i \(0.0492404\pi\)
−0.988059 + 0.154077i \(0.950760\pi\)
\(854\) 60.0000 2.05316
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) 22.0000i 0.751506i 0.926720 + 0.375753i \(0.122616\pi\)
−0.926720 + 0.375753i \(0.877384\pi\)
\(858\) − 15.0000i − 0.512092i
\(859\) 30.0000 1.02359 0.511793 0.859109i \(-0.328981\pi\)
0.511793 + 0.859109i \(0.328981\pi\)
\(860\) 0 0
\(861\) −35.0000 −1.19280
\(862\) 0 0
\(863\) 18.0000i 0.612727i 0.951915 + 0.306364i \(0.0991123\pi\)
−0.951915 + 0.306364i \(0.900888\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −28.0000 −0.951479
\(867\) − 19.0000i − 0.645274i
\(868\) − 40.0000i − 1.35769i
\(869\) −15.0000 −0.508840
\(870\) 0 0
\(871\) −60.0000 −2.03302
\(872\) − 12.0000i − 0.406371i
\(873\) 16.0000i 0.541518i
\(874\) 1.00000 0.0338255
\(875\) 0 0
\(876\) 15.0000 0.506803
\(877\) − 42.0000i − 1.41824i −0.705088 0.709120i \(-0.749093\pi\)
0.705088 0.709120i \(-0.250907\pi\)
\(878\) − 12.0000i − 0.404980i
\(879\) 28.0000 0.944417
\(880\) 0 0
\(881\) −24.0000 −0.808581 −0.404290 0.914631i \(-0.632481\pi\)
−0.404290 + 0.914631i \(0.632481\pi\)
\(882\) − 18.0000i − 0.606092i
\(883\) 8.00000i 0.269221i 0.990899 + 0.134611i \(0.0429784\pi\)
−0.990899 + 0.134611i \(0.957022\pi\)
\(884\) −30.0000 −1.00901
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) − 22.0000i − 0.738688i −0.929293 0.369344i \(-0.879582\pi\)
0.929293 0.369344i \(-0.120418\pi\)
\(888\) 4.00000i 0.134231i
\(889\) −30.0000 −1.00617
\(890\) 0 0
\(891\) −3.00000 −0.100504
\(892\) 14.0000i 0.468755i
\(893\) 6.00000i 0.200782i
\(894\) 20.0000 0.668900
\(895\) 0 0
\(896\) 5.00000 0.167038
\(897\) − 5.00000i − 0.166945i
\(898\) − 2.00000i − 0.0667409i
\(899\) −40.0000 −1.33407
\(900\) 0 0
\(901\) 48.0000 1.59911
\(902\) − 21.0000i − 0.699224i
\(903\) − 35.0000i − 1.16473i
\(904\) 10.0000 0.332595
\(905\) 0 0
\(906\) −10.0000 −0.332228
\(907\) 37.0000i 1.22856i 0.789086 + 0.614282i \(0.210554\pi\)
−0.789086 + 0.614282i \(0.789446\pi\)
\(908\) 8.00000i 0.265489i
\(909\) 10.0000 0.331679
\(910\) 0 0
\(911\) −15.0000 −0.496972 −0.248486 0.968635i \(-0.579933\pi\)
−0.248486 + 0.968635i \(0.579933\pi\)
\(912\) 1.00000i 0.0331133i
\(913\) 27.0000i 0.893570i
\(914\) 18.0000 0.595387
\(915\) 0 0
\(916\) 16.0000 0.528655
\(917\) 20.0000i 0.660458i
\(918\) 6.00000i 0.198030i
\(919\) 12.0000 0.395843 0.197922 0.980218i \(-0.436581\pi\)
0.197922 + 0.980218i \(0.436581\pi\)
\(920\) 0 0
\(921\) 12.0000 0.395413
\(922\) − 7.00000i − 0.230533i
\(923\) 50.0000i 1.64577i
\(924\) 15.0000 0.493464
\(925\) 0 0
\(926\) 4.00000 0.131448
\(927\) − 19.0000i − 0.624042i
\(928\) − 5.00000i − 0.164133i
\(929\) 43.0000 1.41078 0.705392 0.708817i \(-0.250771\pi\)
0.705392 + 0.708817i \(0.250771\pi\)
\(930\) 0 0
\(931\) −18.0000 −0.589926
\(932\) − 7.00000i − 0.229293i
\(933\) 28.0000i 0.916679i
\(934\) −21.0000 −0.687141
\(935\) 0 0
\(936\) 5.00000 0.163430
\(937\) 20.0000i 0.653372i 0.945133 + 0.326686i \(0.105932\pi\)
−0.945133 + 0.326686i \(0.894068\pi\)
\(938\) − 60.0000i − 1.95907i
\(939\) 0 0
\(940\) 0 0
\(941\) −48.0000 −1.56476 −0.782378 0.622804i \(-0.785993\pi\)
−0.782378 + 0.622804i \(0.785993\pi\)
\(942\) − 4.00000i − 0.130327i
\(943\) − 7.00000i − 0.227951i
\(944\) 10.0000 0.325472
\(945\) 0 0
\(946\) 21.0000 0.682769
\(947\) − 32.0000i − 1.03986i −0.854209 0.519930i \(-0.825958\pi\)
0.854209 0.519930i \(-0.174042\pi\)
\(948\) − 5.00000i − 0.162392i
\(949\) −75.0000 −2.43460
\(950\) 0 0
\(951\) 9.00000 0.291845
\(952\) − 30.0000i − 0.972306i
\(953\) − 12.0000i − 0.388718i −0.980930 0.194359i \(-0.937737\pi\)
0.980930 0.194359i \(-0.0622627\pi\)
\(954\) −8.00000 −0.259010
\(955\) 0 0
\(956\) 24.0000 0.776215
\(957\) − 15.0000i − 0.484881i
\(958\) − 25.0000i − 0.807713i
\(959\) 40.0000 1.29167
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) − 20.0000i − 0.644826i
\(963\) − 12.0000i − 0.386695i
\(964\) 18.0000 0.579741
\(965\) 0 0
\(966\) 5.00000 0.160872
\(967\) 8.00000i 0.257263i 0.991692 + 0.128631i \(0.0410584\pi\)
−0.991692 + 0.128631i \(0.958942\pi\)
\(968\) − 2.00000i − 0.0642824i
\(969\) 6.00000 0.192748
\(970\) 0 0
\(971\) −49.0000 −1.57248 −0.786242 0.617918i \(-0.787976\pi\)
−0.786242 + 0.617918i \(0.787976\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) 10.0000i 0.320585i
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) −12.0000 −0.384111
\(977\) − 8.00000i − 0.255943i −0.991778 0.127971i \(-0.959153\pi\)
0.991778 0.127971i \(-0.0408466\pi\)
\(978\) 12.0000i 0.383718i
\(979\) 42.0000 1.34233
\(980\) 0 0
\(981\) 12.0000 0.383131
\(982\) 32.0000i 1.02116i
\(983\) − 31.0000i − 0.988746i −0.869250 0.494373i \(-0.835398\pi\)
0.869250 0.494373i \(-0.164602\pi\)
\(984\) 7.00000 0.223152
\(985\) 0 0
\(986\) −30.0000 −0.955395
\(987\) 30.0000i 0.954911i
\(988\) − 5.00000i − 0.159071i
\(989\) 7.00000 0.222587
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 8.00000i 0.254000i
\(993\) 4.00000i 0.126936i
\(994\) −50.0000 −1.58590
\(995\) 0 0
\(996\) −9.00000 −0.285176
\(997\) 1.00000i 0.0316703i 0.999875 + 0.0158352i \(0.00504070\pi\)
−0.999875 + 0.0158352i \(0.994959\pi\)
\(998\) − 6.00000i − 0.189927i
\(999\) −4.00000 −0.126554
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 3450.2.d.l.2899.1 2
5.2 odd 4 3450.2.a.bb.1.1 yes 1
5.3 odd 4 3450.2.a.a.1.1 1
5.4 even 2 inner 3450.2.d.l.2899.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3450.2.a.a.1.1 1 5.3 odd 4
3450.2.a.bb.1.1 yes 1 5.2 odd 4
3450.2.d.l.2899.1 2 1.1 even 1 trivial
3450.2.d.l.2899.2 2 5.4 even 2 inner