Properties

Label 6762.2.a.cf.1.2
Level $6762$
Weight $2$
Character 6762.1
Self dual yes
Analytic conductor $53.995$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6762.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(53.9948418468\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.3132.1
Defining polynomial: \(x^{3} - 15 x - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 966)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.404409\) of defining polynomial
Character \(\chi\) \(=\) 6762.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -0.404409 q^{11} +1.00000 q^{12} -4.12043 q^{13} -1.00000 q^{15} +1.00000 q^{16} -5.31161 q^{17} -1.00000 q^{18} +2.71602 q^{19} -1.00000 q^{20} +0.404409 q^{22} -1.00000 q^{23} -1.00000 q^{24} -4.00000 q^{25} +4.12043 q^{26} +1.00000 q^{27} +9.12043 q^{29} +1.00000 q^{30} +9.83645 q^{31} -1.00000 q^{32} -0.404409 q^{33} +5.31161 q^{34} +1.00000 q^{36} -7.90720 q^{37} -2.71602 q^{38} -4.12043 q^{39} +1.00000 q^{40} +10.7160 q^{41} +0.808819 q^{43} -0.404409 q^{44} -1.00000 q^{45} +1.00000 q^{46} +10.8365 q^{47} +1.00000 q^{48} +4.00000 q^{50} -5.31161 q^{51} -4.12043 q^{52} -0.191181 q^{53} -1.00000 q^{54} +0.404409 q^{55} +2.71602 q^{57} -9.12043 q^{58} +8.31161 q^{59} -1.00000 q^{60} -10.0000 q^{61} -9.83645 q^{62} +1.00000 q^{64} +4.12043 q^{65} +0.404409 q^{66} +2.83645 q^{67} -5.31161 q^{68} -1.00000 q^{69} -6.92925 q^{71} -1.00000 q^{72} -4.59559 q^{73} +7.90720 q^{74} -4.00000 q^{75} +2.71602 q^{76} +4.12043 q^{78} -9.92925 q^{79} -1.00000 q^{80} +1.00000 q^{81} -10.7160 q^{82} +4.40441 q^{83} +5.31161 q^{85} -0.808819 q^{86} +9.12043 q^{87} +0.404409 q^{88} -7.90720 q^{89} +1.00000 q^{90} -1.00000 q^{92} +9.83645 q^{93} -10.8365 q^{94} -2.71602 q^{95} -1.00000 q^{96} -15.3613 q^{97} -0.404409 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} - 3q^{6} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} - 3q^{6} - 3q^{8} + 3q^{9} + 3q^{10} + 3q^{12} + 3q^{13} - 3q^{15} + 3q^{16} - 3q^{17} - 3q^{18} - 6q^{19} - 3q^{20} - 3q^{23} - 3q^{24} - 12q^{25} - 3q^{26} + 3q^{27} + 12q^{29} + 3q^{30} - 3q^{32} + 3q^{34} + 3q^{36} - 12q^{37} + 6q^{38} + 3q^{39} + 3q^{40} + 18q^{41} - 3q^{45} + 3q^{46} + 3q^{47} + 3q^{48} + 12q^{50} - 3q^{51} + 3q^{52} - 3q^{53} - 3q^{54} - 6q^{57} - 12q^{58} + 12q^{59} - 3q^{60} - 30q^{61} + 3q^{64} - 3q^{65} - 21q^{67} - 3q^{68} - 3q^{69} - 3q^{71} - 3q^{72} - 15q^{73} + 12q^{74} - 12q^{75} - 6q^{76} - 3q^{78} - 12q^{79} - 3q^{80} + 3q^{81} - 18q^{82} + 12q^{83} + 3q^{85} + 12q^{87} - 12q^{89} + 3q^{90} - 3q^{92} - 3q^{94} + 6q^{95} - 3q^{96} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214 −0.223607 0.974679i \(-0.571783\pi\)
−0.223607 + 0.974679i \(0.571783\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −0.404409 −0.121934 −0.0609670 0.998140i \(-0.519418\pi\)
−0.0609670 + 0.998140i \(0.519418\pi\)
\(12\) 1.00000 0.288675
\(13\) −4.12043 −1.14280 −0.571401 0.820671i \(-0.693600\pi\)
−0.571401 + 0.820671i \(0.693600\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) −5.31161 −1.28826 −0.644128 0.764918i \(-0.722779\pi\)
−0.644128 + 0.764918i \(0.722779\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.71602 0.623098 0.311549 0.950230i \(-0.399152\pi\)
0.311549 + 0.950230i \(0.399152\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 0.404409 0.0862204
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) −4.00000 −0.800000
\(26\) 4.12043 0.808083
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 9.12043 1.69362 0.846811 0.531894i \(-0.178520\pi\)
0.846811 + 0.531894i \(0.178520\pi\)
\(30\) 1.00000 0.182574
\(31\) 9.83645 1.76668 0.883340 0.468734i \(-0.155290\pi\)
0.883340 + 0.468734i \(0.155290\pi\)
\(32\) −1.00000 −0.176777
\(33\) −0.404409 −0.0703986
\(34\) 5.31161 0.910934
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −7.90720 −1.29994 −0.649968 0.759961i \(-0.725218\pi\)
−0.649968 + 0.759961i \(0.725218\pi\)
\(38\) −2.71602 −0.440597
\(39\) −4.12043 −0.659797
\(40\) 1.00000 0.158114
\(41\) 10.7160 1.67356 0.836781 0.547538i \(-0.184435\pi\)
0.836781 + 0.547538i \(0.184435\pi\)
\(42\) 0 0
\(43\) 0.808819 0.123344 0.0616718 0.998096i \(-0.480357\pi\)
0.0616718 + 0.998096i \(0.480357\pi\)
\(44\) −0.404409 −0.0609670
\(45\) −1.00000 −0.149071
\(46\) 1.00000 0.147442
\(47\) 10.8365 1.58066 0.790330 0.612682i \(-0.209909\pi\)
0.790330 + 0.612682i \(0.209909\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) 4.00000 0.565685
\(51\) −5.31161 −0.743775
\(52\) −4.12043 −0.571401
\(53\) −0.191181 −0.0262608 −0.0131304 0.999914i \(-0.504180\pi\)
−0.0131304 + 0.999914i \(0.504180\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0.404409 0.0545305
\(56\) 0 0
\(57\) 2.71602 0.359746
\(58\) −9.12043 −1.19757
\(59\) 8.31161 1.08208 0.541040 0.840997i \(-0.318031\pi\)
0.541040 + 0.840997i \(0.318031\pi\)
\(60\) −1.00000 −0.129099
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) −9.83645 −1.24923
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.12043 0.511077
\(66\) 0.404409 0.0497793
\(67\) 2.83645 0.346528 0.173264 0.984875i \(-0.444569\pi\)
0.173264 + 0.984875i \(0.444569\pi\)
\(68\) −5.31161 −0.644128
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −6.92925 −0.822351 −0.411175 0.911556i \(-0.634882\pi\)
−0.411175 + 0.911556i \(0.634882\pi\)
\(72\) −1.00000 −0.117851
\(73\) −4.59559 −0.537873 −0.268937 0.963158i \(-0.586672\pi\)
−0.268937 + 0.963158i \(0.586672\pi\)
\(74\) 7.90720 0.919194
\(75\) −4.00000 −0.461880
\(76\) 2.71602 0.311549
\(77\) 0 0
\(78\) 4.12043 0.466547
\(79\) −9.92925 −1.11713 −0.558564 0.829461i \(-0.688647\pi\)
−0.558564 + 0.829461i \(0.688647\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −10.7160 −1.18339
\(83\) 4.40441 0.483447 0.241723 0.970345i \(-0.422287\pi\)
0.241723 + 0.970345i \(0.422287\pi\)
\(84\) 0 0
\(85\) 5.31161 0.576125
\(86\) −0.808819 −0.0872172
\(87\) 9.12043 0.977813
\(88\) 0.404409 0.0431102
\(89\) −7.90720 −0.838162 −0.419081 0.907949i \(-0.637648\pi\)
−0.419081 + 0.907949i \(0.637648\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −1.00000 −0.104257
\(93\) 9.83645 1.01999
\(94\) −10.8365 −1.11769
\(95\) −2.71602 −0.278658
\(96\) −1.00000 −0.102062
\(97\) −15.3613 −1.55970 −0.779852 0.625965i \(-0.784705\pi\)
−0.779852 + 0.625965i \(0.784705\pi\)
\(98\) 0 0
\(99\) −0.404409 −0.0406447
\(100\) −4.00000 −0.400000
\(101\) 5.19118 0.516542 0.258271 0.966073i \(-0.416847\pi\)
0.258271 + 0.966073i \(0.416847\pi\)
\(102\) 5.31161 0.525928
\(103\) −16.8365 −1.65895 −0.829473 0.558548i \(-0.811359\pi\)
−0.829473 + 0.558548i \(0.811359\pi\)
\(104\) 4.12043 0.404042
\(105\) 0 0
\(106\) 0.191181 0.0185692
\(107\) −9.59559 −0.927641 −0.463820 0.885929i \(-0.653522\pi\)
−0.463820 + 0.885929i \(0.653522\pi\)
\(108\) 1.00000 0.0962250
\(109\) −3.19118 −0.305660 −0.152830 0.988253i \(-0.548839\pi\)
−0.152830 + 0.988253i \(0.548839\pi\)
\(110\) −0.404409 −0.0385589
\(111\) −7.90720 −0.750519
\(112\) 0 0
\(113\) −18.7437 −1.76325 −0.881627 0.471946i \(-0.843552\pi\)
−0.881627 + 0.471946i \(0.843552\pi\)
\(114\) −2.71602 −0.254379
\(115\) 1.00000 0.0932505
\(116\) 9.12043 0.846811
\(117\) −4.12043 −0.380934
\(118\) −8.31161 −0.765146
\(119\) 0 0
\(120\) 1.00000 0.0912871
\(121\) −10.8365 −0.985132
\(122\) 10.0000 0.905357
\(123\) 10.7160 0.966231
\(124\) 9.83645 0.883340
\(125\) 9.00000 0.804984
\(126\) 0 0
\(127\) −11.5956 −1.02894 −0.514471 0.857508i \(-0.672012\pi\)
−0.514471 + 0.857508i \(0.672012\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0.808819 0.0712125
\(130\) −4.12043 −0.361386
\(131\) −16.7657 −1.46483 −0.732413 0.680860i \(-0.761606\pi\)
−0.732413 + 0.680860i \(0.761606\pi\)
\(132\) −0.404409 −0.0351993
\(133\) 0 0
\(134\) −2.83645 −0.245032
\(135\) −1.00000 −0.0860663
\(136\) 5.31161 0.455467
\(137\) −9.64527 −0.824051 −0.412026 0.911172i \(-0.635179\pi\)
−0.412026 + 0.911172i \(0.635179\pi\)
\(138\) 1.00000 0.0851257
\(139\) 16.8641 1.43039 0.715197 0.698923i \(-0.246337\pi\)
0.715197 + 0.698923i \(0.246337\pi\)
\(140\) 0 0
\(141\) 10.8365 0.912594
\(142\) 6.92925 0.581490
\(143\) 1.66634 0.139346
\(144\) 1.00000 0.0833333
\(145\) −9.12043 −0.757411
\(146\) 4.59559 0.380334
\(147\) 0 0
\(148\) −7.90720 −0.649968
\(149\) −10.2132 −0.836700 −0.418350 0.908286i \(-0.637391\pi\)
−0.418350 + 0.908286i \(0.637391\pi\)
\(150\) 4.00000 0.326599
\(151\) 5.59559 0.455363 0.227681 0.973736i \(-0.426886\pi\)
0.227681 + 0.973736i \(0.426886\pi\)
\(152\) −2.71602 −0.220298
\(153\) −5.31161 −0.429418
\(154\) 0 0
\(155\) −9.83645 −0.790083
\(156\) −4.12043 −0.329899
\(157\) −22.7713 −1.81735 −0.908673 0.417508i \(-0.862904\pi\)
−0.908673 + 0.417508i \(0.862904\pi\)
\(158\) 9.92925 0.789929
\(159\) −0.191181 −0.0151617
\(160\) 1.00000 0.0790569
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 24.7713 1.94024 0.970119 0.242631i \(-0.0780105\pi\)
0.970119 + 0.242631i \(0.0780105\pi\)
\(164\) 10.7160 0.836781
\(165\) 0.404409 0.0314832
\(166\) −4.40441 −0.341848
\(167\) 19.0773 1.47625 0.738123 0.674666i \(-0.235712\pi\)
0.738123 + 0.674666i \(0.235712\pi\)
\(168\) 0 0
\(169\) 3.97795 0.305996
\(170\) −5.31161 −0.407382
\(171\) 2.71602 0.207699
\(172\) 0.808819 0.0616718
\(173\) −6.24086 −0.474484 −0.237242 0.971451i \(-0.576243\pi\)
−0.237242 + 0.971451i \(0.576243\pi\)
\(174\) −9.12043 −0.691418
\(175\) 0 0
\(176\) −0.404409 −0.0304835
\(177\) 8.31161 0.624739
\(178\) 7.90720 0.592670
\(179\) −11.2188 −0.838534 −0.419267 0.907863i \(-0.637713\pi\)
−0.419267 + 0.907863i \(0.637713\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −8.24086 −0.612538 −0.306269 0.951945i \(-0.599081\pi\)
−0.306269 + 0.951945i \(0.599081\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 1.00000 0.0737210
\(185\) 7.90720 0.581349
\(186\) −9.83645 −0.721244
\(187\) 2.14807 0.157082
\(188\) 10.8365 0.790330
\(189\) 0 0
\(190\) 2.71602 0.197041
\(191\) 24.7713 1.79239 0.896194 0.443663i \(-0.146321\pi\)
0.896194 + 0.443663i \(0.146321\pi\)
\(192\) 1.00000 0.0721688
\(193\) −14.8585 −1.06954 −0.534769 0.844998i \(-0.679601\pi\)
−0.534769 + 0.844998i \(0.679601\pi\)
\(194\) 15.3613 1.10288
\(195\) 4.12043 0.295070
\(196\) 0 0
\(197\) −11.4320 −0.814499 −0.407250 0.913317i \(-0.633512\pi\)
−0.407250 + 0.913317i \(0.633512\pi\)
\(198\) 0.404409 0.0287401
\(199\) 8.62323 0.611284 0.305642 0.952146i \(-0.401129\pi\)
0.305642 + 0.952146i \(0.401129\pi\)
\(200\) 4.00000 0.282843
\(201\) 2.83645 0.200068
\(202\) −5.19118 −0.365250
\(203\) 0 0
\(204\) −5.31161 −0.371887
\(205\) −10.7160 −0.748439
\(206\) 16.8365 1.17305
\(207\) −1.00000 −0.0695048
\(208\) −4.12043 −0.285701
\(209\) −1.09838 −0.0759769
\(210\) 0 0
\(211\) 7.43204 0.511643 0.255821 0.966724i \(-0.417654\pi\)
0.255821 + 0.966724i \(0.417654\pi\)
\(212\) −0.191181 −0.0131304
\(213\) −6.92925 −0.474784
\(214\) 9.59559 0.655941
\(215\) −0.808819 −0.0551610
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 3.19118 0.216134
\(219\) −4.59559 −0.310541
\(220\) 0.404409 0.0272653
\(221\) 21.8861 1.47222
\(222\) 7.90720 0.530697
\(223\) −10.2188 −0.684303 −0.342151 0.939645i \(-0.611156\pi\)
−0.342151 + 0.939645i \(0.611156\pi\)
\(224\) 0 0
\(225\) −4.00000 −0.266667
\(226\) 18.7437 1.24681
\(227\) −12.2188 −0.810991 −0.405496 0.914097i \(-0.632901\pi\)
−0.405496 + 0.914097i \(0.632901\pi\)
\(228\) 2.71602 0.179873
\(229\) −3.85193 −0.254543 −0.127271 0.991868i \(-0.540622\pi\)
−0.127271 + 0.991868i \(0.540622\pi\)
\(230\) −1.00000 −0.0659380
\(231\) 0 0
\(232\) −9.12043 −0.598786
\(233\) −26.5304 −1.73807 −0.869033 0.494754i \(-0.835258\pi\)
−0.869033 + 0.494754i \(0.835258\pi\)
\(234\) 4.12043 0.269361
\(235\) −10.8365 −0.706892
\(236\) 8.31161 0.541040
\(237\) −9.92925 −0.644974
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) −1.00000 −0.0645497
\(241\) 14.7933 0.952923 0.476461 0.879195i \(-0.341919\pi\)
0.476461 + 0.879195i \(0.341919\pi\)
\(242\) 10.8365 0.696594
\(243\) 1.00000 0.0641500
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) −10.7160 −0.683229
\(247\) −11.1912 −0.712078
\(248\) −9.83645 −0.624615
\(249\) 4.40441 0.279118
\(250\) −9.00000 −0.569210
\(251\) −14.0220 −0.885064 −0.442532 0.896753i \(-0.645920\pi\)
−0.442532 + 0.896753i \(0.645920\pi\)
\(252\) 0 0
\(253\) 0.404409 0.0254250
\(254\) 11.5956 0.727572
\(255\) 5.31161 0.332626
\(256\) 1.00000 0.0625000
\(257\) −7.09838 −0.442785 −0.221393 0.975185i \(-0.571060\pi\)
−0.221393 + 0.975185i \(0.571060\pi\)
\(258\) −0.808819 −0.0503548
\(259\) 0 0
\(260\) 4.12043 0.255538
\(261\) 9.12043 0.564541
\(262\) 16.7657 1.03579
\(263\) −8.80882 −0.543175 −0.271588 0.962414i \(-0.587549\pi\)
−0.271588 + 0.962414i \(0.587549\pi\)
\(264\) 0.404409 0.0248897
\(265\) 0.191181 0.0117442
\(266\) 0 0
\(267\) −7.90720 −0.483913
\(268\) 2.83645 0.173264
\(269\) 4.49721 0.274199 0.137100 0.990557i \(-0.456222\pi\)
0.137100 + 0.990557i \(0.456222\pi\)
\(270\) 1.00000 0.0608581
\(271\) −15.2685 −0.927495 −0.463748 0.885967i \(-0.653496\pi\)
−0.463748 + 0.885967i \(0.653496\pi\)
\(272\) −5.31161 −0.322064
\(273\) 0 0
\(274\) 9.64527 0.582692
\(275\) 1.61764 0.0975472
\(276\) −1.00000 −0.0601929
\(277\) −2.21323 −0.132980 −0.0664900 0.997787i \(-0.521180\pi\)
−0.0664900 + 0.997787i \(0.521180\pi\)
\(278\) −16.8641 −1.01144
\(279\) 9.83645 0.588893
\(280\) 0 0
\(281\) 29.4110 1.75451 0.877256 0.480023i \(-0.159372\pi\)
0.877256 + 0.480023i \(0.159372\pi\)
\(282\) −10.8365 −0.645301
\(283\) −27.0773 −1.60958 −0.804790 0.593560i \(-0.797722\pi\)
−0.804790 + 0.593560i \(0.797722\pi\)
\(284\) −6.92925 −0.411175
\(285\) −2.71602 −0.160883
\(286\) −1.66634 −0.0985328
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 11.2132 0.659602
\(290\) 9.12043 0.535570
\(291\) −15.3613 −0.900495
\(292\) −4.59559 −0.268937
\(293\) 19.8641 1.16047 0.580236 0.814448i \(-0.302960\pi\)
0.580236 + 0.814448i \(0.302960\pi\)
\(294\) 0 0
\(295\) −8.31161 −0.483921
\(296\) 7.90720 0.459597
\(297\) −0.404409 −0.0234662
\(298\) 10.2132 0.591636
\(299\) 4.12043 0.238291
\(300\) −4.00000 −0.230940
\(301\) 0 0
\(302\) −5.59559 −0.321990
\(303\) 5.19118 0.298226
\(304\) 2.71602 0.155775
\(305\) 10.0000 0.572598
\(306\) 5.31161 0.303645
\(307\) 30.5745 1.74498 0.872490 0.488632i \(-0.162504\pi\)
0.872490 + 0.488632i \(0.162504\pi\)
\(308\) 0 0
\(309\) −16.8365 −0.957792
\(310\) 9.83645 0.558673
\(311\) 7.07075 0.400945 0.200473 0.979699i \(-0.435752\pi\)
0.200473 + 0.979699i \(0.435752\pi\)
\(312\) 4.12043 0.233273
\(313\) −7.68839 −0.434573 −0.217287 0.976108i \(-0.569721\pi\)
−0.217287 + 0.976108i \(0.569721\pi\)
\(314\) 22.7713 1.28506
\(315\) 0 0
\(316\) −9.92925 −0.558564
\(317\) −22.7933 −1.28020 −0.640101 0.768291i \(-0.721107\pi\)
−0.640101 + 0.768291i \(0.721107\pi\)
\(318\) 0.191181 0.0107209
\(319\) −3.68839 −0.206510
\(320\) −1.00000 −0.0559017
\(321\) −9.59559 −0.535574
\(322\) 0 0
\(323\) −14.4265 −0.802709
\(324\) 1.00000 0.0555556
\(325\) 16.4817 0.914242
\(326\) −24.7713 −1.37195
\(327\) −3.19118 −0.176473
\(328\) −10.7160 −0.591693
\(329\) 0 0
\(330\) −0.404409 −0.0222620
\(331\) −20.5304 −1.12845 −0.564227 0.825620i \(-0.690826\pi\)
−0.564227 + 0.825620i \(0.690826\pi\)
\(332\) 4.40441 0.241723
\(333\) −7.90720 −0.433312
\(334\) −19.0773 −1.04386
\(335\) −2.83645 −0.154972
\(336\) 0 0
\(337\) 16.7933 0.914791 0.457396 0.889263i \(-0.348782\pi\)
0.457396 + 0.889263i \(0.348782\pi\)
\(338\) −3.97795 −0.216372
\(339\) −18.7437 −1.01802
\(340\) 5.31161 0.288063
\(341\) −3.97795 −0.215418
\(342\) −2.71602 −0.146866
\(343\) 0 0
\(344\) −0.808819 −0.0436086
\(345\) 1.00000 0.0538382
\(346\) 6.24086 0.335511
\(347\) 0.354728 0.0190428 0.00952141 0.999955i \(-0.496969\pi\)
0.00952141 + 0.999955i \(0.496969\pi\)
\(348\) 9.12043 0.488906
\(349\) −20.9292 −1.12032 −0.560159 0.828385i \(-0.689260\pi\)
−0.560159 + 0.828385i \(0.689260\pi\)
\(350\) 0 0
\(351\) −4.12043 −0.219932
\(352\) 0.404409 0.0215551
\(353\) 26.5304 1.41207 0.706036 0.708176i \(-0.250482\pi\)
0.706036 + 0.708176i \(0.250482\pi\)
\(354\) −8.31161 −0.441757
\(355\) 6.92925 0.367766
\(356\) −7.90720 −0.419081
\(357\) 0 0
\(358\) 11.2188 0.592933
\(359\) 4.86409 0.256717 0.128358 0.991728i \(-0.459029\pi\)
0.128358 + 0.991728i \(0.459029\pi\)
\(360\) 1.00000 0.0527046
\(361\) −11.6232 −0.611749
\(362\) 8.24086 0.433130
\(363\) −10.8365 −0.568766
\(364\) 0 0
\(365\) 4.59559 0.240544
\(366\) 10.0000 0.522708
\(367\) 25.9569 1.35494 0.677469 0.735551i \(-0.263077\pi\)
0.677469 + 0.735551i \(0.263077\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 10.7160 0.557854
\(370\) −7.90720 −0.411076
\(371\) 0 0
\(372\) 9.83645 0.509996
\(373\) −13.6729 −0.707956 −0.353978 0.935254i \(-0.615171\pi\)
−0.353978 + 0.935254i \(0.615171\pi\)
\(374\) −2.14807 −0.111074
\(375\) 9.00000 0.464758
\(376\) −10.8365 −0.558847
\(377\) −37.5801 −1.93547
\(378\) 0 0
\(379\) −22.4541 −1.15339 −0.576695 0.816960i \(-0.695658\pi\)
−0.576695 + 0.816960i \(0.695658\pi\)
\(380\) −2.71602 −0.139329
\(381\) −11.5956 −0.594060
\(382\) −24.7713 −1.26741
\(383\) −12.3889 −0.633045 −0.316522 0.948585i \(-0.602515\pi\)
−0.316522 + 0.948585i \(0.602515\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 14.8585 0.756278
\(387\) 0.808819 0.0411146
\(388\) −15.3613 −0.779852
\(389\) 28.5370 1.44688 0.723442 0.690386i \(-0.242559\pi\)
0.723442 + 0.690386i \(0.242559\pi\)
\(390\) −4.12043 −0.208646
\(391\) 5.31161 0.268620
\(392\) 0 0
\(393\) −16.7657 −0.845718
\(394\) 11.4320 0.575938
\(395\) 9.92925 0.499595
\(396\) −0.404409 −0.0203223
\(397\) −5.31161 −0.266582 −0.133291 0.991077i \(-0.542555\pi\)
−0.133291 + 0.991077i \(0.542555\pi\)
\(398\) −8.62323 −0.432243
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −6.21323 −0.310274 −0.155137 0.987893i \(-0.549582\pi\)
−0.155137 + 0.987893i \(0.549582\pi\)
\(402\) −2.83645 −0.141469
\(403\) −40.5304 −2.01896
\(404\) 5.19118 0.258271
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 3.19775 0.158506
\(408\) 5.31161 0.262964
\(409\) −28.1049 −1.38970 −0.694850 0.719155i \(-0.744529\pi\)
−0.694850 + 0.719155i \(0.744529\pi\)
\(410\) 10.7160 0.529227
\(411\) −9.64527 −0.475766
\(412\) −16.8365 −0.829473
\(413\) 0 0
\(414\) 1.00000 0.0491473
\(415\) −4.40441 −0.216204
\(416\) 4.12043 0.202021
\(417\) 16.8641 0.825838
\(418\) 1.09838 0.0537237
\(419\) 11.0056 0.537658 0.268829 0.963188i \(-0.413363\pi\)
0.268829 + 0.963188i \(0.413363\pi\)
\(420\) 0 0
\(421\) −23.4873 −1.14470 −0.572351 0.820009i \(-0.693968\pi\)
−0.572351 + 0.820009i \(0.693968\pi\)
\(422\) −7.43204 −0.361786
\(423\) 10.8365 0.526886
\(424\) 0.191181 0.00928459
\(425\) 21.2465 1.03060
\(426\) 6.92925 0.335723
\(427\) 0 0
\(428\) −9.59559 −0.463820
\(429\) 1.66634 0.0804517
\(430\) 0.808819 0.0390047
\(431\) 24.4817 1.17924 0.589622 0.807680i \(-0.299277\pi\)
0.589622 + 0.807680i \(0.299277\pi\)
\(432\) 1.00000 0.0481125
\(433\) −1.23527 −0.0593635 −0.0296818 0.999559i \(-0.509449\pi\)
−0.0296818 + 0.999559i \(0.509449\pi\)
\(434\) 0 0
\(435\) −9.12043 −0.437291
\(436\) −3.19118 −0.152830
\(437\) −2.71602 −0.129925
\(438\) 4.59559 0.219586
\(439\) 22.6453 1.08080 0.540400 0.841408i \(-0.318273\pi\)
0.540400 + 0.841408i \(0.318273\pi\)
\(440\) −0.404409 −0.0192795
\(441\) 0 0
\(442\) −21.8861 −1.04102
\(443\) −7.90162 −0.375417 −0.187709 0.982225i \(-0.560106\pi\)
−0.187709 + 0.982225i \(0.560106\pi\)
\(444\) −7.90720 −0.375259
\(445\) 7.90720 0.374837
\(446\) 10.2188 0.483875
\(447\) −10.2132 −0.483069
\(448\) 0 0
\(449\) 16.3889 0.773441 0.386721 0.922197i \(-0.373608\pi\)
0.386721 + 0.922197i \(0.373608\pi\)
\(450\) 4.00000 0.188562
\(451\) −4.33366 −0.204064
\(452\) −18.7437 −0.881627
\(453\) 5.59559 0.262904
\(454\) 12.2188 0.573457
\(455\) 0 0
\(456\) −2.71602 −0.127189
\(457\) −31.0342 −1.45172 −0.725859 0.687843i \(-0.758558\pi\)
−0.725859 + 0.687843i \(0.758558\pi\)
\(458\) 3.85193 0.179989
\(459\) −5.31161 −0.247925
\(460\) 1.00000 0.0466252
\(461\) −7.61764 −0.354789 −0.177394 0.984140i \(-0.556767\pi\)
−0.177394 + 0.984140i \(0.556767\pi\)
\(462\) 0 0
\(463\) 8.80882 0.409381 0.204690 0.978827i \(-0.434381\pi\)
0.204690 + 0.978827i \(0.434381\pi\)
\(464\) 9.12043 0.423405
\(465\) −9.83645 −0.456155
\(466\) 26.5304 1.22900
\(467\) −0.808819 −0.0374277 −0.0187138 0.999825i \(-0.505957\pi\)
−0.0187138 + 0.999825i \(0.505957\pi\)
\(468\) −4.12043 −0.190467
\(469\) 0 0
\(470\) 10.8365 0.499848
\(471\) −22.7713 −1.04925
\(472\) −8.31161 −0.382573
\(473\) −0.327094 −0.0150398
\(474\) 9.92925 0.456066
\(475\) −10.8641 −0.498479
\(476\) 0 0
\(477\) −0.191181 −0.00875359
\(478\) 16.0000 0.731823
\(479\) 28.3337 1.29460 0.647299 0.762236i \(-0.275899\pi\)
0.647299 + 0.762236i \(0.275899\pi\)
\(480\) 1.00000 0.0456435
\(481\) 32.5811 1.48557
\(482\) −14.7933 −0.673818
\(483\) 0 0
\(484\) −10.8365 −0.492566
\(485\) 15.3613 0.697520
\(486\) −1.00000 −0.0453609
\(487\) 18.7005 0.847402 0.423701 0.905802i \(-0.360731\pi\)
0.423701 + 0.905802i \(0.360731\pi\)
\(488\) 10.0000 0.452679
\(489\) 24.7713 1.12020
\(490\) 0 0
\(491\) 5.09280 0.229835 0.114917 0.993375i \(-0.463340\pi\)
0.114917 + 0.993375i \(0.463340\pi\)
\(492\) 10.7160 0.483116
\(493\) −48.4442 −2.18182
\(494\) 11.1912 0.503515
\(495\) 0.404409 0.0181768
\(496\) 9.83645 0.441670
\(497\) 0 0
\(498\) −4.40441 −0.197366
\(499\) 19.4873 0.872372 0.436186 0.899857i \(-0.356329\pi\)
0.436186 + 0.899857i \(0.356329\pi\)
\(500\) 9.00000 0.402492
\(501\) 19.0773 0.852312
\(502\) 14.0220 0.625835
\(503\) 2.99441 0.133514 0.0667571 0.997769i \(-0.478735\pi\)
0.0667571 + 0.997769i \(0.478735\pi\)
\(504\) 0 0
\(505\) −5.19118 −0.231005
\(506\) −0.404409 −0.0179782
\(507\) 3.97795 0.176667
\(508\) −11.5956 −0.514471
\(509\) 11.7437 0.520528 0.260264 0.965537i \(-0.416190\pi\)
0.260264 + 0.965537i \(0.416190\pi\)
\(510\) −5.31161 −0.235202
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 2.71602 0.119915
\(514\) 7.09838 0.313096
\(515\) 16.8365 0.741903
\(516\) 0.808819 0.0356063
\(517\) −4.38236 −0.192736
\(518\) 0 0
\(519\) −6.24086 −0.273943
\(520\) −4.12043 −0.180693
\(521\) −5.21882 −0.228640 −0.114320 0.993444i \(-0.536469\pi\)
−0.114320 + 0.993444i \(0.536469\pi\)
\(522\) −9.12043 −0.399190
\(523\) 26.4541 1.15676 0.578378 0.815769i \(-0.303686\pi\)
0.578378 + 0.815769i \(0.303686\pi\)
\(524\) −16.7657 −0.732413
\(525\) 0 0
\(526\) 8.80882 0.384083
\(527\) −52.2474 −2.27593
\(528\) −0.404409 −0.0175997
\(529\) 1.00000 0.0434783
\(530\) −0.191181 −0.00830439
\(531\) 8.31161 0.360693
\(532\) 0 0
\(533\) −44.1546 −1.91255
\(534\) 7.90720 0.342178
\(535\) 9.59559 0.414854
\(536\) −2.83645 −0.122516
\(537\) −11.2188 −0.484128
\(538\) −4.49721 −0.193888
\(539\) 0 0
\(540\) −1.00000 −0.0430331
\(541\) −3.31161 −0.142377 −0.0711887 0.997463i \(-0.522679\pi\)
−0.0711887 + 0.997463i \(0.522679\pi\)
\(542\) 15.2685 0.655838
\(543\) −8.24086 −0.353649
\(544\) 5.31161 0.227734
\(545\) 3.19118 0.136695
\(546\) 0 0
\(547\) 3.95130 0.168945 0.0844726 0.996426i \(-0.473079\pi\)
0.0844726 + 0.996426i \(0.473079\pi\)
\(548\) −9.64527 −0.412026
\(549\) −10.0000 −0.426790
\(550\) −1.61764 −0.0689763
\(551\) 24.7713 1.05529
\(552\) 1.00000 0.0425628
\(553\) 0 0
\(554\) 2.21323 0.0940310
\(555\) 7.90720 0.335642
\(556\) 16.8641 0.715197
\(557\) 2.60118 0.110215 0.0551077 0.998480i \(-0.482450\pi\)
0.0551077 + 0.998480i \(0.482450\pi\)
\(558\) −9.83645 −0.416410
\(559\) −3.33268 −0.140957
\(560\) 0 0
\(561\) 2.14807 0.0906914
\(562\) −29.4110 −1.24063
\(563\) 25.4100 1.07090 0.535452 0.844566i \(-0.320141\pi\)
0.535452 + 0.844566i \(0.320141\pi\)
\(564\) 10.8365 0.456297
\(565\) 18.7437 0.788552
\(566\) 27.0773 1.13814
\(567\) 0 0
\(568\) 6.92925 0.290745
\(569\) −16.5956 −0.695723 −0.347862 0.937546i \(-0.613092\pi\)
−0.347862 + 0.937546i \(0.613092\pi\)
\(570\) 2.71602 0.113762
\(571\) 32.6574 1.36667 0.683335 0.730105i \(-0.260529\pi\)
0.683335 + 0.730105i \(0.260529\pi\)
\(572\) 1.66634 0.0696732
\(573\) 24.7713 1.03484
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 1.00000 0.0416667
\(577\) −0.545910 −0.0227265 −0.0113633 0.999935i \(-0.503617\pi\)
−0.0113633 + 0.999935i \(0.503617\pi\)
\(578\) −11.2132 −0.466409
\(579\) −14.8585 −0.617498
\(580\) −9.12043 −0.378705
\(581\) 0 0
\(582\) 15.3613 0.636746
\(583\) 0.0773155 0.00320208
\(584\) 4.59559 0.190167
\(585\) 4.12043 0.170359
\(586\) −19.8641 −0.820578
\(587\) −8.82097 −0.364080 −0.182040 0.983291i \(-0.558270\pi\)
−0.182040 + 0.983291i \(0.558270\pi\)
\(588\) 0 0
\(589\) 26.7160 1.10081
\(590\) 8.31161 0.342184
\(591\) −11.4320 −0.470251
\(592\) −7.90720 −0.324984
\(593\) 40.0553 1.64487 0.822436 0.568857i \(-0.192614\pi\)
0.822436 + 0.568857i \(0.192614\pi\)
\(594\) 0.404409 0.0165931
\(595\) 0 0
\(596\) −10.2132 −0.418350
\(597\) 8.62323 0.352925
\(598\) −4.12043 −0.168497
\(599\) −24.4166 −0.997634 −0.498817 0.866707i \(-0.666232\pi\)
−0.498817 + 0.866707i \(0.666232\pi\)
\(600\) 4.00000 0.163299
\(601\) 9.21323 0.375815 0.187908 0.982187i \(-0.439829\pi\)
0.187908 + 0.982187i \(0.439829\pi\)
\(602\) 0 0
\(603\) 2.83645 0.115509
\(604\) 5.59559 0.227681
\(605\) 10.8365 0.440564
\(606\) −5.19118 −0.210877
\(607\) −33.6950 −1.36764 −0.683818 0.729653i \(-0.739682\pi\)
−0.683818 + 0.729653i \(0.739682\pi\)
\(608\) −2.71602 −0.110149
\(609\) 0 0
\(610\) −10.0000 −0.404888
\(611\) −44.6509 −1.80638
\(612\) −5.31161 −0.214709
\(613\) −24.4265 −0.986575 −0.493288 0.869866i \(-0.664205\pi\)
−0.493288 + 0.869866i \(0.664205\pi\)
\(614\) −30.5745 −1.23389
\(615\) −10.7160 −0.432112
\(616\) 0 0
\(617\) −2.41000 −0.0970228 −0.0485114 0.998823i \(-0.515448\pi\)
−0.0485114 + 0.998823i \(0.515448\pi\)
\(618\) 16.8365 0.677261
\(619\) −45.8486 −1.84281 −0.921406 0.388602i \(-0.872958\pi\)
−0.921406 + 0.388602i \(0.872958\pi\)
\(620\) −9.83645 −0.395041
\(621\) −1.00000 −0.0401286
\(622\) −7.07075 −0.283511
\(623\) 0 0
\(624\) −4.12043 −0.164949
\(625\) 11.0000 0.440000
\(626\) 7.68839 0.307290
\(627\) −1.09838 −0.0438653
\(628\) −22.7713 −0.908673
\(629\) 42.0000 1.67465
\(630\) 0 0
\(631\) −33.5745 −1.33658 −0.668290 0.743901i \(-0.732974\pi\)
−0.668290 + 0.743901i \(0.732974\pi\)
\(632\) 9.92925 0.394964
\(633\) 7.43204 0.295397
\(634\) 22.7933 0.905239
\(635\) 11.5956 0.460157
\(636\) −0.191181 −0.00758083
\(637\) 0 0
\(638\) 3.68839 0.146025
\(639\) −6.92925 −0.274117
\(640\) 1.00000 0.0395285
\(641\) −33.4110 −1.31965 −0.659827 0.751417i \(-0.729370\pi\)
−0.659827 + 0.751417i \(0.729370\pi\)
\(642\) 9.59559 0.378708
\(643\) −11.5248 −0.454495 −0.227248 0.973837i \(-0.572973\pi\)
−0.227248 + 0.973837i \(0.572973\pi\)
\(644\) 0 0
\(645\) −0.808819 −0.0318472
\(646\) 14.4265 0.567601
\(647\) 48.3402 1.90045 0.950225 0.311564i \(-0.100853\pi\)
0.950225 + 0.311564i \(0.100853\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −3.36129 −0.131942
\(650\) −16.4817 −0.646466
\(651\) 0 0
\(652\) 24.7713 0.970119
\(653\) 25.7989 1.00959 0.504795 0.863239i \(-0.331568\pi\)
0.504795 + 0.863239i \(0.331568\pi\)
\(654\) 3.19118 0.124785
\(655\) 16.7657 0.655090
\(656\) 10.7160 0.418390
\(657\) −4.59559 −0.179291
\(658\) 0 0
\(659\) −14.1856 −0.552592 −0.276296 0.961073i \(-0.589107\pi\)
−0.276296 + 0.961073i \(0.589107\pi\)
\(660\) 0.404409 0.0157416
\(661\) 47.3017 1.83982 0.919912 0.392125i \(-0.128260\pi\)
0.919912 + 0.392125i \(0.128260\pi\)
\(662\) 20.5304 0.797938
\(663\) 21.8861 0.849987
\(664\) −4.40441 −0.170924
\(665\) 0 0
\(666\) 7.90720 0.306398
\(667\) −9.12043 −0.353145
\(668\) 19.0773 0.738123
\(669\) −10.2188 −0.395082
\(670\) 2.83645 0.109582
\(671\) 4.04409 0.156120
\(672\) 0 0
\(673\) 0.448503 0.0172885 0.00864425 0.999963i \(-0.497248\pi\)
0.00864425 + 0.999963i \(0.497248\pi\)
\(674\) −16.7933 −0.646855
\(675\) −4.00000 −0.153960
\(676\) 3.97795 0.152998
\(677\) 16.6176 0.638668 0.319334 0.947642i \(-0.396541\pi\)
0.319334 + 0.947642i \(0.396541\pi\)
\(678\) 18.7437 0.719846
\(679\) 0 0
\(680\) −5.31161 −0.203691
\(681\) −12.2188 −0.468226
\(682\) 3.97795 0.152324
\(683\) −27.2033 −1.04091 −0.520453 0.853890i \(-0.674237\pi\)
−0.520453 + 0.853890i \(0.674237\pi\)
\(684\) 2.71602 0.103850
\(685\) 9.64527 0.368527
\(686\) 0 0
\(687\) −3.85193 −0.146960
\(688\) 0.808819 0.0308359
\(689\) 0.787750 0.0300109
\(690\) −1.00000 −0.0380693
\(691\) −45.0122 −1.71234 −0.856172 0.516692i \(-0.827163\pi\)
−0.856172 + 0.516692i \(0.827163\pi\)
\(692\) −6.24086 −0.237242
\(693\) 0 0
\(694\) −0.354728 −0.0134653
\(695\) −16.8641 −0.639691
\(696\) −9.12043 −0.345709
\(697\) −56.9194 −2.15597
\(698\) 20.9292 0.792184
\(699\) −26.5304 −1.00347
\(700\) 0 0
\(701\) −35.6232 −1.34547 −0.672735 0.739883i \(-0.734881\pi\)
−0.672735 + 0.739883i \(0.734881\pi\)
\(702\) 4.12043 0.155516
\(703\) −21.4761 −0.809988
\(704\) −0.404409 −0.0152417
\(705\) −10.8365 −0.408124
\(706\) −26.5304 −0.998486
\(707\) 0 0
\(708\) 8.31161 0.312370
\(709\) −20.1922 −0.758332 −0.379166 0.925329i \(-0.623789\pi\)
−0.379166 + 0.925329i \(0.623789\pi\)
\(710\) −6.92925 −0.260050
\(711\) −9.92925 −0.372376
\(712\) 7.90720 0.296335
\(713\) −9.83645 −0.368378
\(714\) 0 0
\(715\) −1.66634 −0.0623176
\(716\) −11.2188 −0.419267
\(717\) −16.0000 −0.597531
\(718\) −4.86409 −0.181526
\(719\) −0.977953 −0.0364715 −0.0182358 0.999834i \(-0.505805\pi\)
−0.0182358 + 0.999834i \(0.505805\pi\)
\(720\) −1.00000 −0.0372678
\(721\) 0 0
\(722\) 11.6232 0.432572
\(723\) 14.7933 0.550170
\(724\) −8.24086 −0.306269
\(725\) −36.4817 −1.35490
\(726\) 10.8365 0.402178
\(727\) 36.9348 1.36984 0.684919 0.728620i \(-0.259838\pi\)
0.684919 + 0.728620i \(0.259838\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −4.59559 −0.170090
\(731\) −4.29613 −0.158898
\(732\) −10.0000 −0.369611
\(733\) 35.6354 1.31622 0.658111 0.752921i \(-0.271356\pi\)
0.658111 + 0.752921i \(0.271356\pi\)
\(734\) −25.9569 −0.958086
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −1.14709 −0.0422535
\(738\) −10.7160 −0.394462
\(739\) −33.8697 −1.24592 −0.622958 0.782255i \(-0.714069\pi\)
−0.622958 + 0.782255i \(0.714069\pi\)
\(740\) 7.90720 0.290675
\(741\) −11.1912 −0.411118
\(742\) 0 0
\(743\) 21.1537 0.776052 0.388026 0.921648i \(-0.373157\pi\)
0.388026 + 0.921648i \(0.373157\pi\)
\(744\) −9.83645 −0.360622
\(745\) 10.2132 0.374184
\(746\) 13.6729 0.500601
\(747\) 4.40441 0.161149
\(748\) 2.14807 0.0785411
\(749\) 0 0
\(750\) −9.00000 −0.328634
\(751\) 20.9789 0.765532 0.382766 0.923845i \(-0.374972\pi\)
0.382766 + 0.923845i \(0.374972\pi\)
\(752\) 10.8365 0.395165
\(753\) −14.0220 −0.510992
\(754\) 37.5801 1.36859
\(755\) −5.59559 −0.203644
\(756\) 0 0
\(757\) −4.95688 −0.180161 −0.0900805 0.995934i \(-0.528712\pi\)
−0.0900805 + 0.995934i \(0.528712\pi\)
\(758\) 22.4541 0.815569
\(759\) 0.404409 0.0146791
\(760\) 2.71602 0.0985205
\(761\) −19.5689 −0.709373 −0.354687 0.934985i \(-0.615412\pi\)
−0.354687 + 0.934985i \(0.615412\pi\)
\(762\) 11.5956 0.420064
\(763\) 0 0
\(764\) 24.7713 0.896194
\(765\) 5.31161 0.192042
\(766\) 12.3889 0.447630
\(767\) −34.2474 −1.23660
\(768\) 1.00000 0.0360844
\(769\) −4.92366 −0.177552 −0.0887759 0.996052i \(-0.528295\pi\)
−0.0887759 + 0.996052i \(0.528295\pi\)
\(770\) 0 0
\(771\) −7.09838 −0.255642
\(772\) −14.8585 −0.534769
\(773\) 2.02763 0.0729289 0.0364645 0.999335i \(-0.488390\pi\)
0.0364645 + 0.999335i \(0.488390\pi\)
\(774\) −0.808819 −0.0290724
\(775\) −39.3458 −1.41334
\(776\) 15.3613 0.551438
\(777\) 0 0
\(778\) −28.5370 −1.02310
\(779\) 29.1049 1.04279
\(780\) 4.12043 0.147535
\(781\) 2.80225 0.100273
\(782\) −5.31161 −0.189943
\(783\) 9.12043 0.325938
\(784\) 0 0
\(785\) 22.7713 0.812742
\(786\) 16.7657 0.598013
\(787\) 0.268497 0.00957088 0.00478544 0.999989i \(-0.498477\pi\)
0.00478544 + 0.999989i \(0.498477\pi\)
\(788\) −11.4320 −0.407250
\(789\) −8.80882 −0.313602
\(790\) −9.92925 −0.353267
\(791\) 0 0
\(792\) 0.404409 0.0143701
\(793\) 41.2043 1.46321
\(794\) 5.31161 0.188502
\(795\) 0.191181 0.00678050
\(796\) 8.62323 0.305642
\(797\) 16.0497 0.568509 0.284254 0.958749i \(-0.408254\pi\)
0.284254 + 0.958749i \(0.408254\pi\)
\(798\) 0 0
\(799\) −57.5590 −2.03629
\(800\) 4.00000 0.141421
\(801\) −7.90720 −0.279387
\(802\) 6.21323 0.219397
\(803\) 1.85850 0.0655850
\(804\) 2.83645 0.100034
\(805\) 0 0
\(806\) 40.5304 1.42762
\(807\) 4.49721 0.158309
\(808\) −5.19118 −0.182625
\(809\) −7.72161 −0.271477 −0.135739 0.990745i \(-0.543341\pi\)
−0.135739 + 0.990745i \(0.543341\pi\)
\(810\) 1.00000 0.0351364
\(811\) 23.2465 0.816293 0.408147 0.912916i \(-0.366175\pi\)
0.408147 + 0.912916i \(0.366175\pi\)
\(812\) 0 0
\(813\) −15.2685 −0.535490
\(814\) −3.19775 −0.112081
\(815\) −24.7713 −0.867700
\(816\) −5.31161 −0.185944
\(817\) 2.19677 0.0768552
\(818\) 28.1049 0.982667
\(819\) 0 0
\(820\) −10.7160 −0.374220
\(821\) −39.1204 −1.36531 −0.682656 0.730740i \(-0.739175\pi\)
−0.682656 + 0.730740i \(0.739175\pi\)
\(822\) 9.64527 0.336417
\(823\) 42.1546 1.46942 0.734709 0.678382i \(-0.237319\pi\)
0.734709 + 0.678382i \(0.237319\pi\)
\(824\) 16.8365 0.586526
\(825\) 1.61764 0.0563189
\(826\) 0 0
\(827\) 54.3294 1.88922 0.944608 0.328200i \(-0.106442\pi\)
0.944608 + 0.328200i \(0.106442\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −18.6397 −0.647383 −0.323691 0.946163i \(-0.604924\pi\)
−0.323691 + 0.946163i \(0.604924\pi\)
\(830\) 4.40441 0.152879
\(831\) −2.21323 −0.0767760
\(832\) −4.12043 −0.142850
\(833\) 0 0
\(834\) −16.8641 −0.583956
\(835\) −19.0773 −0.660198
\(836\) −1.09838 −0.0379884
\(837\) 9.83645 0.339998
\(838\) −11.0056 −0.380182
\(839\) −32.9569 −1.13780 −0.568899 0.822407i \(-0.692630\pi\)
−0.568899 + 0.822407i \(0.692630\pi\)
\(840\) 0 0
\(841\) 54.1823 1.86835
\(842\) 23.4873 0.809426
\(843\) 29.4110 1.01297
\(844\) 7.43204 0.255821
\(845\) −3.97795 −0.136846
\(846\) −10.8365 −0.372565
\(847\) 0 0
\(848\) −0.191181 −0.00656519
\(849\) −27.0773 −0.929291
\(850\) −21.2465 −0.728747
\(851\) 7.90720 0.271055
\(852\) −6.92925 −0.237392
\(853\) 25.0497 0.857685 0.428842 0.903379i \(-0.358922\pi\)
0.428842 + 0.903379i \(0.358922\pi\)
\(854\) 0 0
\(855\) −2.71602 −0.0928860
\(856\) 9.59559 0.327971
\(857\) 4.04870 0.138301 0.0691505 0.997606i \(-0.477971\pi\)
0.0691505 + 0.997606i \(0.477971\pi\)
\(858\) −1.66634 −0.0568879
\(859\) −47.7769 −1.63013 −0.815063 0.579372i \(-0.803298\pi\)
−0.815063 + 0.579372i \(0.803298\pi\)
\(860\) −0.808819 −0.0275805
\(861\) 0 0
\(862\) −24.4817 −0.833851
\(863\) −24.8852 −0.847101 −0.423550 0.905873i \(-0.639216\pi\)
−0.423550 + 0.905873i \(0.639216\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 6.24086 0.212196
\(866\) 1.23527 0.0419763
\(867\) 11.2132 0.380821
\(868\) 0 0
\(869\) 4.01548 0.136216
\(870\) 9.12043 0.309212
\(871\) −11.6874 −0.396013
\(872\) 3.19118 0.108067
\(873\) −15.3613 −0.519901
\(874\) 2.71602 0.0918708
\(875\) 0 0
\(876\) −4.59559 −0.155271
\(877\) −4.06516 −0.137271 −0.0686354 0.997642i \(-0.521865\pi\)
−0.0686354 + 0.997642i \(0.521865\pi\)
\(878\) −22.6453 −0.764241
\(879\) 19.8641 0.669999
\(880\) 0.404409 0.0136326
\(881\) 12.2132 0.411474 0.205737 0.978607i \(-0.434041\pi\)
0.205737 + 0.978607i \(0.434041\pi\)
\(882\) 0 0
\(883\) −5.14248 −0.173058 −0.0865291 0.996249i \(-0.527578\pi\)
−0.0865291 + 0.996249i \(0.527578\pi\)
\(884\) 21.8861 0.736110
\(885\) −8.31161 −0.279392
\(886\) 7.90162 0.265460
\(887\) −30.3824 −1.02014 −0.510070 0.860133i \(-0.670380\pi\)
−0.510070 + 0.860133i \(0.670380\pi\)
\(888\) 7.90720 0.265348
\(889\) 0 0
\(890\) −7.90720 −0.265050
\(891\) −0.404409 −0.0135482
\(892\) −10.2188 −0.342151
\(893\) 29.4320 0.984906
\(894\) 10.2132 0.341581
\(895\) 11.2188 0.375004
\(896\) 0 0
\(897\) 4.12043 0.137577
\(898\) −16.3889 −0.546906
\(899\) 89.7127 2.99209
\(900\) −4.00000 −0.133333
\(901\) 1.01548 0.0338306
\(902\) 4.33366 0.144295
\(903\) 0 0
\(904\) 18.7437 0.623405
\(905\) 8.24086 0.273936
\(906\) −5.59559 −0.185901
\(907\) −13.8796 −0.460864 −0.230432 0.973088i \(-0.574014\pi\)
−0.230432 + 0.973088i \(0.574014\pi\)
\(908\) −12.2188 −0.405496
\(909\) 5.19118 0.172181
\(910\) 0 0
\(911\) −27.3017 −0.904546 −0.452273 0.891879i \(-0.649387\pi\)
−0.452273 + 0.891879i \(0.649387\pi\)
\(912\) 2.71602 0.0899365
\(913\) −1.78118 −0.0589486
\(914\) 31.0342 1.02652
\(915\) 10.0000 0.330590
\(916\) −3.85193 −0.127271
\(917\) 0 0
\(918\) 5.31161 0.175309
\(919\) 38.4541 1.26848 0.634242 0.773135i \(-0.281312\pi\)
0.634242 + 0.773135i \(0.281312\pi\)
\(920\) −1.00000 −0.0329690
\(921\) 30.5745 1.00746
\(922\) 7.61764 0.250874
\(923\) 28.5515 0.939784
\(924\) 0 0
\(925\) 31.6288 1.03995
\(926\) −8.80882 −0.289476
\(927\) −16.8365 −0.552982
\(928\) −9.12043 −0.299393
\(929\) −53.0674 −1.74109 −0.870543 0.492093i \(-0.836232\pi\)
−0.870543 + 0.492093i \(0.836232\pi\)
\(930\) 9.83645 0.322550
\(931\) 0 0
\(932\) −26.5304 −0.869033
\(933\) 7.07075 0.231486
\(934\) 0.808819 0.0264654
\(935\) −2.14807 −0.0702493
\(936\) 4.12043 0.134681
\(937\) 29.1645 0.952763 0.476382 0.879239i \(-0.341948\pi\)
0.476382 + 0.879239i \(0.341948\pi\)
\(938\) 0 0
\(939\) −7.68839 −0.250901
\(940\) −10.8365 −0.353446
\(941\) −26.0220 −0.848294 −0.424147 0.905593i \(-0.639426\pi\)
−0.424147 + 0.905593i \(0.639426\pi\)
\(942\) 22.7713 0.741929
\(943\) −10.7160 −0.348962
\(944\) 8.31161 0.270520
\(945\) 0 0
\(946\) 0.327094 0.0106347
\(947\) −12.9227 −0.419931 −0.209965 0.977709i \(-0.567335\pi\)
−0.209965 + 0.977709i \(0.567335\pi\)
\(948\) −9.92925 −0.322487
\(949\) 18.9358 0.614683
\(950\) 10.8641 0.352478
\(951\) −22.7933 −0.739125
\(952\) 0 0
\(953\) 12.5680 0.407116 0.203558 0.979063i \(-0.434749\pi\)
0.203558 + 0.979063i \(0.434749\pi\)
\(954\) 0.191181 0.00618972
\(955\) −24.7713 −0.801580
\(956\) −16.0000 −0.517477
\(957\) −3.68839 −0.119229
\(958\) −28.3337 −0.915419
\(959\) 0 0
\(960\) −1.00000 −0.0322749
\(961\) 65.7558 2.12116
\(962\) −32.5811 −1.05046
\(963\) −9.59559 −0.309214
\(964\) 14.7933 0.476461
\(965\) 14.8585 0.478312
\(966\) 0 0
\(967\) 12.8309 0.412613 0.206306 0.978487i \(-0.433856\pi\)
0.206306 + 0.978487i \(0.433856\pi\)
\(968\) 10.8365 0.348297
\(969\) −14.4265 −0.463445
\(970\) −15.3613 −0.493221
\(971\) 22.4044 0.718992 0.359496 0.933147i \(-0.382949\pi\)
0.359496 + 0.933147i \(0.382949\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −18.7005 −0.599204
\(975\) 16.4817 0.527838
\(976\) −10.0000 −0.320092
\(977\) −20.5094 −0.656153 −0.328076 0.944651i \(-0.606400\pi\)
−0.328076 + 0.944651i \(0.606400\pi\)
\(978\) −24.7713 −0.792099
\(979\) 3.19775 0.102200
\(980\) 0 0
\(981\) −3.19118 −0.101887
\(982\) −5.09280 −0.162518
\(983\) −4.53043 −0.144498 −0.0722491 0.997387i \(-0.523018\pi\)
−0.0722491 + 0.997387i \(0.523018\pi\)
\(984\) −10.7160 −0.341614
\(985\) 11.4320 0.364255
\(986\) 48.4442 1.54278
\(987\) 0 0
\(988\) −11.1912 −0.356039
\(989\) −0.808819 −0.0257189
\(990\) −0.404409 −0.0128530
\(991\) −31.4100 −0.997771 −0.498886 0.866668i \(-0.666257\pi\)
−0.498886 + 0.866668i \(0.666257\pi\)
\(992\) −9.83645 −0.312308
\(993\) −20.5304 −0.651513
\(994\) 0 0
\(995\) −8.62323 −0.273375
\(996\) 4.40441 0.139559
\(997\) −18.4752 −0.585114 −0.292557 0.956248i \(-0.594506\pi\)
−0.292557 + 0.956248i \(0.594506\pi\)
\(998\) −19.4873 −0.616860
\(999\) −7.90720 −0.250173
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 6762.2.a.cf.1.2 3
7.3 odd 6 966.2.i.k.415.3 yes 6
7.5 odd 6 966.2.i.k.277.3 6
7.6 odd 2 6762.2.a.ce.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
966.2.i.k.277.3 6 7.5 odd 6
966.2.i.k.415.3 yes 6 7.3 odd 6
6762.2.a.ce.1.2 3 7.6 odd 2
6762.2.a.cf.1.2 3 1.1 even 1 trivial