Properties

Label 3450.2.a.bp.1.2
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.5483886973\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.3368.1
Defining polynomial: \(x^{3} - x^{2} - 15 x + 11\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.723686\) of defining polynomial
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +0.723686 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +0.723686 q^{7} -1.00000 q^{8} +1.00000 q^{9} +5.51445 q^{11} +1.00000 q^{12} +4.96183 q^{13} -0.723686 q^{14} +1.00000 q^{16} +4.23814 q^{17} -1.00000 q^{18} +4.96183 q^{19} +0.723686 q^{21} -5.51445 q^{22} -1.00000 q^{23} -1.00000 q^{24} -4.96183 q^{26} +1.00000 q^{27} +0.723686 q^{28} -1.00000 q^{29} +6.00000 q^{31} -1.00000 q^{32} +5.51445 q^{33} -4.23814 q^{34} +1.00000 q^{36} -4.23814 q^{37} -4.96183 q^{38} +4.96183 q^{39} +1.03817 q^{41} -0.723686 q^{42} -0.961825 q^{43} +5.51445 q^{44} +1.00000 q^{46} -7.96183 q^{47} +1.00000 q^{48} -6.47628 q^{49} +4.23814 q^{51} +4.96183 q^{52} -6.47628 q^{53} -1.00000 q^{54} -0.723686 q^{56} +4.96183 q^{57} +1.00000 q^{58} -9.02891 q^{59} +7.44737 q^{61} -6.00000 q^{62} +0.723686 q^{63} +1.00000 q^{64} -5.51445 q^{66} -11.9237 q^{67} +4.23814 q^{68} -1.00000 q^{69} -0.514453 q^{71} -1.00000 q^{72} -0.447372 q^{73} +4.23814 q^{74} +4.96183 q^{76} +3.99073 q^{77} -4.96183 q^{78} -5.51445 q^{79} +1.00000 q^{81} -1.03817 q^{82} +6.17106 q^{83} +0.723686 q^{84} +0.961825 q^{86} -1.00000 q^{87} -5.51445 q^{88} -2.23814 q^{89} +3.59080 q^{91} -1.00000 q^{92} +6.00000 q^{93} +7.96183 q^{94} -1.00000 q^{96} +2.55263 q^{97} +6.47628 q^{98} +5.51445 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{6} + q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{6} + q^{7} - 3q^{8} + 3q^{9} + 3q^{11} + 3q^{12} - q^{13} - q^{14} + 3q^{16} - 2q^{17} - 3q^{18} - q^{19} + q^{21} - 3q^{22} - 3q^{23} - 3q^{24} + q^{26} + 3q^{27} + q^{28} - 3q^{29} + 18q^{31} - 3q^{32} + 3q^{33} + 2q^{34} + 3q^{36} + 2q^{37} + q^{38} - q^{39} + 19q^{41} - q^{42} + 13q^{43} + 3q^{44} + 3q^{46} - 8q^{47} + 3q^{48} + 10q^{49} - 2q^{51} - q^{52} + 10q^{53} - 3q^{54} - q^{56} - q^{57} + 3q^{58} + 20q^{61} - 18q^{62} + q^{63} + 3q^{64} - 3q^{66} - 4q^{67} - 2q^{68} - 3q^{69} + 12q^{71} - 3q^{72} + q^{73} - 2q^{74} - q^{76} - 31q^{77} + q^{78} - 3q^{79} + 3q^{81} - 19q^{82} + 15q^{83} + q^{84} - 13q^{86} - 3q^{87} - 3q^{88} + 8q^{89} + 29q^{91} - 3q^{92} + 18q^{93} + 8q^{94} - 3q^{96} + 10q^{97} - 10q^{98} + 3q^{99} + O(q^{100}) \)

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
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 0.723686 0.273528 0.136764 0.990604i \(-0.456330\pi\)
0.136764 + 0.990604i \(0.456330\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.51445 1.66267 0.831335 0.555771i \(-0.187577\pi\)
0.831335 + 0.555771i \(0.187577\pi\)
\(12\) 1.00000 0.288675
\(13\) 4.96183 1.37616 0.688081 0.725634i \(-0.258453\pi\)
0.688081 + 0.725634i \(0.258453\pi\)
\(14\) −0.723686 −0.193413
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.23814 1.02790 0.513950 0.857820i \(-0.328182\pi\)
0.513950 + 0.857820i \(0.328182\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.96183 1.13832 0.569160 0.822227i \(-0.307268\pi\)
0.569160 + 0.822227i \(0.307268\pi\)
\(20\) 0 0
\(21\) 0.723686 0.157921
\(22\) −5.51445 −1.17569
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −4.96183 −0.973094
\(27\) 1.00000 0.192450
\(28\) 0.723686 0.136764
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) −1.00000 −0.176777
\(33\) 5.51445 0.959943
\(34\) −4.23814 −0.726835
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −4.23814 −0.696746 −0.348373 0.937356i \(-0.613266\pi\)
−0.348373 + 0.937356i \(0.613266\pi\)
\(38\) −4.96183 −0.804914
\(39\) 4.96183 0.794528
\(40\) 0 0
\(41\) 1.03817 0.162136 0.0810678 0.996709i \(-0.474167\pi\)
0.0810678 + 0.996709i \(0.474167\pi\)
\(42\) −0.723686 −0.111667
\(43\) −0.961825 −0.146677 −0.0733385 0.997307i \(-0.523365\pi\)
−0.0733385 + 0.997307i \(0.523365\pi\)
\(44\) 5.51445 0.831335
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) −7.96183 −1.16135 −0.580676 0.814135i \(-0.697212\pi\)
−0.580676 + 0.814135i \(0.697212\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.47628 −0.925183
\(50\) 0 0
\(51\) 4.23814 0.593458
\(52\) 4.96183 0.688081
\(53\) −6.47628 −0.889585 −0.444793 0.895634i \(-0.646723\pi\)
−0.444793 + 0.895634i \(0.646723\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −0.723686 −0.0967066
\(57\) 4.96183 0.657210
\(58\) 1.00000 0.131306
\(59\) −9.02891 −1.17546 −0.587732 0.809056i \(-0.699979\pi\)
−0.587732 + 0.809056i \(0.699979\pi\)
\(60\) 0 0
\(61\) 7.44737 0.953538 0.476769 0.879029i \(-0.341808\pi\)
0.476769 + 0.879029i \(0.341808\pi\)
\(62\) −6.00000 −0.762001
\(63\) 0.723686 0.0911759
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −5.51445 −0.678782
\(67\) −11.9237 −1.45671 −0.728353 0.685202i \(-0.759714\pi\)
−0.728353 + 0.685202i \(0.759714\pi\)
\(68\) 4.23814 0.513950
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −0.514453 −0.0610544 −0.0305272 0.999534i \(-0.509719\pi\)
−0.0305272 + 0.999534i \(0.509719\pi\)
\(72\) −1.00000 −0.117851
\(73\) −0.447372 −0.0523609 −0.0261805 0.999657i \(-0.508334\pi\)
−0.0261805 + 0.999657i \(0.508334\pi\)
\(74\) 4.23814 0.492674
\(75\) 0 0
\(76\) 4.96183 0.569160
\(77\) 3.99073 0.454786
\(78\) −4.96183 −0.561816
\(79\) −5.51445 −0.620424 −0.310212 0.950667i \(-0.600400\pi\)
−0.310212 + 0.950667i \(0.600400\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −1.03817 −0.114647
\(83\) 6.17106 0.677362 0.338681 0.940901i \(-0.390019\pi\)
0.338681 + 0.940901i \(0.390019\pi\)
\(84\) 0.723686 0.0789606
\(85\) 0 0
\(86\) 0.961825 0.103716
\(87\) −1.00000 −0.107211
\(88\) −5.51445 −0.587843
\(89\) −2.23814 −0.237242 −0.118621 0.992940i \(-0.537847\pi\)
−0.118621 + 0.992940i \(0.537847\pi\)
\(90\) 0 0
\(91\) 3.59080 0.376418
\(92\) −1.00000 −0.104257
\(93\) 6.00000 0.622171
\(94\) 7.96183 0.821200
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 2.55263 0.259180 0.129590 0.991568i \(-0.458634\pi\)
0.129590 + 0.991568i \(0.458634\pi\)
\(98\) 6.47628 0.654203
\(99\) 5.51445 0.554223
\(100\) 0 0
\(101\) −1.06708 −0.106179 −0.0530893 0.998590i \(-0.516907\pi\)
−0.0530893 + 0.998590i \(0.516907\pi\)
\(102\) −4.23814 −0.419638
\(103\) −5.75259 −0.566820 −0.283410 0.958999i \(-0.591466\pi\)
−0.283410 + 0.958999i \(0.591466\pi\)
\(104\) −4.96183 −0.486547
\(105\) 0 0
\(106\) 6.47628 0.629032
\(107\) −15.9237 −1.53940 −0.769699 0.638407i \(-0.779594\pi\)
−0.769699 + 0.638407i \(0.779594\pi\)
\(108\) 1.00000 0.0962250
\(109\) 17.2670 1.65388 0.826942 0.562288i \(-0.190079\pi\)
0.826942 + 0.562288i \(0.190079\pi\)
\(110\) 0 0
\(111\) −4.23814 −0.402266
\(112\) 0.723686 0.0683819
\(113\) 6.23814 0.586835 0.293417 0.955984i \(-0.405207\pi\)
0.293417 + 0.955984i \(0.405207\pi\)
\(114\) −4.96183 −0.464718
\(115\) 0 0
\(116\) −1.00000 −0.0928477
\(117\) 4.96183 0.458721
\(118\) 9.02891 0.831178
\(119\) 3.06708 0.281159
\(120\) 0 0
\(121\) 19.4092 1.76447
\(122\) −7.44737 −0.674253
\(123\) 1.03817 0.0936091
\(124\) 6.00000 0.538816
\(125\) 0 0
\(126\) −0.723686 −0.0644711
\(127\) 12.9526 1.14935 0.574677 0.818380i \(-0.305128\pi\)
0.574677 + 0.818380i \(0.305128\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −0.961825 −0.0846840
\(130\) 0 0
\(131\) 11.5815 1.01188 0.505942 0.862568i \(-0.331145\pi\)
0.505942 + 0.862568i \(0.331145\pi\)
\(132\) 5.51445 0.479972
\(133\) 3.59080 0.311362
\(134\) 11.9237 1.03005
\(135\) 0 0
\(136\) −4.23814 −0.363417
\(137\) −22.2381 −1.89993 −0.949966 0.312353i \(-0.898883\pi\)
−0.949966 + 0.312353i \(0.898883\pi\)
\(138\) 1.00000 0.0851257
\(139\) 18.9907 1.61077 0.805386 0.592750i \(-0.201958\pi\)
0.805386 + 0.592750i \(0.201958\pi\)
\(140\) 0 0
\(141\) −7.96183 −0.670507
\(142\) 0.514453 0.0431720
\(143\) 27.3618 2.28810
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 0.447372 0.0370248
\(147\) −6.47628 −0.534154
\(148\) −4.23814 −0.348373
\(149\) −5.92365 −0.485284 −0.242642 0.970116i \(-0.578014\pi\)
−0.242642 + 0.970116i \(0.578014\pi\)
\(150\) 0 0
\(151\) 6.47628 0.527032 0.263516 0.964655i \(-0.415118\pi\)
0.263516 + 0.964655i \(0.415118\pi\)
\(152\) −4.96183 −0.402457
\(153\) 4.23814 0.342633
\(154\) −3.99073 −0.321582
\(155\) 0 0
\(156\) 4.96183 0.397264
\(157\) −9.37102 −0.747889 −0.373944 0.927451i \(-0.621995\pi\)
−0.373944 + 0.927451i \(0.621995\pi\)
\(158\) 5.51445 0.438706
\(159\) −6.47628 −0.513602
\(160\) 0 0
\(161\) −0.723686 −0.0570344
\(162\) −1.00000 −0.0785674
\(163\) 13.9237 1.09058 0.545292 0.838246i \(-0.316419\pi\)
0.545292 + 0.838246i \(0.316419\pi\)
\(164\) 1.03817 0.0810678
\(165\) 0 0
\(166\) −6.17106 −0.478967
\(167\) −9.61971 −0.744396 −0.372198 0.928153i \(-0.621396\pi\)
−0.372198 + 0.928153i \(0.621396\pi\)
\(168\) −0.723686 −0.0558336
\(169\) 11.6197 0.893824
\(170\) 0 0
\(171\) 4.96183 0.379440
\(172\) −0.961825 −0.0733385
\(173\) −1.51445 −0.115142 −0.0575709 0.998341i \(-0.518336\pi\)
−0.0575709 + 0.998341i \(0.518336\pi\)
\(174\) 1.00000 0.0758098
\(175\) 0 0
\(176\) 5.51445 0.415668
\(177\) −9.02891 −0.678654
\(178\) 2.23814 0.167756
\(179\) 10.4763 0.783034 0.391517 0.920171i \(-0.371950\pi\)
0.391517 + 0.920171i \(0.371950\pi\)
\(180\) 0 0
\(181\) −21.7433 −1.61617 −0.808084 0.589067i \(-0.799495\pi\)
−0.808084 + 0.589067i \(0.799495\pi\)
\(182\) −3.59080 −0.266168
\(183\) 7.44737 0.550526
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) −6.00000 −0.439941
\(187\) 23.3710 1.70906
\(188\) −7.96183 −0.580676
\(189\) 0.723686 0.0526404
\(190\) 0 0
\(191\) 8.40920 0.608468 0.304234 0.952597i \(-0.401600\pi\)
0.304234 + 0.952597i \(0.401600\pi\)
\(192\) 1.00000 0.0721688
\(193\) 12.9907 0.935093 0.467547 0.883968i \(-0.345138\pi\)
0.467547 + 0.883968i \(0.345138\pi\)
\(194\) −2.55263 −0.183268
\(195\) 0 0
\(196\) −6.47628 −0.462591
\(197\) −20.5815 −1.46637 −0.733187 0.680027i \(-0.761968\pi\)
−0.733187 + 0.680027i \(0.761968\pi\)
\(198\) −5.51445 −0.391895
\(199\) −4.64734 −0.329441 −0.164720 0.986340i \(-0.552672\pi\)
−0.164720 + 0.986340i \(0.552672\pi\)
\(200\) 0 0
\(201\) −11.9237 −0.841029
\(202\) 1.06708 0.0750796
\(203\) −0.723686 −0.0507928
\(204\) 4.23814 0.296729
\(205\) 0 0
\(206\) 5.75259 0.400802
\(207\) −1.00000 −0.0695048
\(208\) 4.96183 0.344041
\(209\) 27.3618 1.89265
\(210\) 0 0
\(211\) −23.3328 −1.60630 −0.803150 0.595777i \(-0.796844\pi\)
−0.803150 + 0.595777i \(0.796844\pi\)
\(212\) −6.47628 −0.444793
\(213\) −0.514453 −0.0352498
\(214\) 15.9237 1.08852
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 4.34212 0.294762
\(218\) −17.2670 −1.16947
\(219\) −0.447372 −0.0302306
\(220\) 0 0
\(221\) 21.0289 1.41456
\(222\) 4.23814 0.284445
\(223\) −25.2947 −1.69386 −0.846928 0.531707i \(-0.821551\pi\)
−0.846928 + 0.531707i \(0.821551\pi\)
\(224\) −0.723686 −0.0483533
\(225\) 0 0
\(226\) −6.23814 −0.414955
\(227\) −7.13288 −0.473426 −0.236713 0.971580i \(-0.576070\pi\)
−0.236713 + 0.971580i \(0.576070\pi\)
\(228\) 4.96183 0.328605
\(229\) −7.58154 −0.501002 −0.250501 0.968116i \(-0.580595\pi\)
−0.250501 + 0.968116i \(0.580595\pi\)
\(230\) 0 0
\(231\) 3.99073 0.262571
\(232\) 1.00000 0.0656532
\(233\) −24.8855 −1.63030 −0.815151 0.579249i \(-0.803346\pi\)
−0.815151 + 0.579249i \(0.803346\pi\)
\(234\) −4.96183 −0.324365
\(235\) 0 0
\(236\) −9.02891 −0.587732
\(237\) −5.51445 −0.358202
\(238\) −3.06708 −0.198809
\(239\) −18.0960 −1.17053 −0.585266 0.810841i \(-0.699010\pi\)
−0.585266 + 0.810841i \(0.699010\pi\)
\(240\) 0 0
\(241\) 26.8762 1.73125 0.865624 0.500694i \(-0.166922\pi\)
0.865624 + 0.500694i \(0.166922\pi\)
\(242\) −19.4092 −1.24767
\(243\) 1.00000 0.0641500
\(244\) 7.44737 0.476769
\(245\) 0 0
\(246\) −1.03817 −0.0661916
\(247\) 24.6197 1.56651
\(248\) −6.00000 −0.381000
\(249\) 6.17106 0.391075
\(250\) 0 0
\(251\) −8.79077 −0.554868 −0.277434 0.960745i \(-0.589484\pi\)
−0.277434 + 0.960745i \(0.589484\pi\)
\(252\) 0.723686 0.0455879
\(253\) −5.51445 −0.346691
\(254\) −12.9526 −0.812716
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 10.9526 0.683202 0.341601 0.939845i \(-0.389031\pi\)
0.341601 + 0.939845i \(0.389031\pi\)
\(258\) 0.961825 0.0598806
\(259\) −3.06708 −0.190579
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) −11.5815 −0.715510
\(263\) −20.8762 −1.28728 −0.643641 0.765327i \(-0.722577\pi\)
−0.643641 + 0.765327i \(0.722577\pi\)
\(264\) −5.51445 −0.339391
\(265\) 0 0
\(266\) −3.59080 −0.220166
\(267\) −2.23814 −0.136972
\(268\) −11.9237 −0.728353
\(269\) 21.3618 1.30245 0.651225 0.758885i \(-0.274256\pi\)
0.651225 + 0.758885i \(0.274256\pi\)
\(270\) 0 0
\(271\) 11.0289 0.669958 0.334979 0.942226i \(-0.391271\pi\)
0.334979 + 0.942226i \(0.391271\pi\)
\(272\) 4.23814 0.256975
\(273\) 3.59080 0.217325
\(274\) 22.2381 1.34346
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) 15.4381 0.927586 0.463793 0.885944i \(-0.346488\pi\)
0.463793 + 0.885944i \(0.346488\pi\)
\(278\) −18.9907 −1.13899
\(279\) 6.00000 0.359211
\(280\) 0 0
\(281\) −2.16179 −0.128962 −0.0644808 0.997919i \(-0.520539\pi\)
−0.0644808 + 0.997919i \(0.520539\pi\)
\(282\) 7.96183 0.474120
\(283\) 16.9526 1.00772 0.503862 0.863784i \(-0.331912\pi\)
0.503862 + 0.863784i \(0.331912\pi\)
\(284\) −0.514453 −0.0305272
\(285\) 0 0
\(286\) −27.3618 −1.61793
\(287\) 0.751312 0.0443486
\(288\) −1.00000 −0.0589256
\(289\) 0.961825 0.0565780
\(290\) 0 0
\(291\) 2.55263 0.149638
\(292\) −0.447372 −0.0261805
\(293\) 13.0289 0.761157 0.380578 0.924749i \(-0.375725\pi\)
0.380578 + 0.924749i \(0.375725\pi\)
\(294\) 6.47628 0.377704
\(295\) 0 0
\(296\) 4.23814 0.246337
\(297\) 5.51445 0.319981
\(298\) 5.92365 0.343148
\(299\) −4.96183 −0.286950
\(300\) 0 0
\(301\) −0.696059 −0.0401202
\(302\) −6.47628 −0.372668
\(303\) −1.06708 −0.0613022
\(304\) 4.96183 0.284580
\(305\) 0 0
\(306\) −4.23814 −0.242278
\(307\) 27.0671 1.54480 0.772400 0.635136i \(-0.219056\pi\)
0.772400 + 0.635136i \(0.219056\pi\)
\(308\) 3.99073 0.227393
\(309\) −5.75259 −0.327254
\(310\) 0 0
\(311\) −31.4670 −1.78433 −0.892165 0.451709i \(-0.850814\pi\)
−0.892165 + 0.451709i \(0.850814\pi\)
\(312\) −4.96183 −0.280908
\(313\) 27.2947 1.54279 0.771393 0.636359i \(-0.219560\pi\)
0.771393 + 0.636359i \(0.219560\pi\)
\(314\) 9.37102 0.528837
\(315\) 0 0
\(316\) −5.51445 −0.310212
\(317\) −11.0000 −0.617822 −0.308911 0.951091i \(-0.599964\pi\)
−0.308911 + 0.951091i \(0.599964\pi\)
\(318\) 6.47628 0.363172
\(319\) −5.51445 −0.308750
\(320\) 0 0
\(321\) −15.9237 −0.888772
\(322\) 0.723686 0.0403294
\(323\) 21.0289 1.17008
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −13.9237 −0.771160
\(327\) 17.2670 0.954870
\(328\) −1.03817 −0.0573236
\(329\) −5.76186 −0.317662
\(330\) 0 0
\(331\) −8.51445 −0.467997 −0.233998 0.972237i \(-0.575181\pi\)
−0.233998 + 0.972237i \(0.575181\pi\)
\(332\) 6.17106 0.338681
\(333\) −4.23814 −0.232249
\(334\) 9.61971 0.526367
\(335\) 0 0
\(336\) 0.723686 0.0394803
\(337\) −1.52372 −0.0830024 −0.0415012 0.999138i \(-0.513214\pi\)
−0.0415012 + 0.999138i \(0.513214\pi\)
\(338\) −11.6197 −0.632029
\(339\) 6.23814 0.338809
\(340\) 0 0
\(341\) 33.0867 1.79175
\(342\) −4.96183 −0.268305
\(343\) −9.75259 −0.526591
\(344\) 0.961825 0.0518581
\(345\) 0 0
\(346\) 1.51445 0.0814175
\(347\) 25.5815 1.37329 0.686644 0.726993i \(-0.259083\pi\)
0.686644 + 0.726993i \(0.259083\pi\)
\(348\) −1.00000 −0.0536056
\(349\) −26.8855 −1.43915 −0.719573 0.694417i \(-0.755663\pi\)
−0.719573 + 0.694417i \(0.755663\pi\)
\(350\) 0 0
\(351\) 4.96183 0.264843
\(352\) −5.51445 −0.293921
\(353\) 10.6197 0.565230 0.282615 0.959233i \(-0.408798\pi\)
0.282615 + 0.959233i \(0.408798\pi\)
\(354\) 9.02891 0.479881
\(355\) 0 0
\(356\) −2.23814 −0.118621
\(357\) 3.06708 0.162327
\(358\) −10.4763 −0.553689
\(359\) −5.43810 −0.287012 −0.143506 0.989649i \(-0.545838\pi\)
−0.143506 + 0.989649i \(0.545838\pi\)
\(360\) 0 0
\(361\) 5.61971 0.295774
\(362\) 21.7433 1.14280
\(363\) 19.4092 1.01872
\(364\) 3.59080 0.188209
\(365\) 0 0
\(366\) −7.44737 −0.389280
\(367\) 8.56190 0.446927 0.223464 0.974712i \(-0.428264\pi\)
0.223464 + 0.974712i \(0.428264\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 1.03817 0.0540452
\(370\) 0 0
\(371\) −4.68679 −0.243326
\(372\) 6.00000 0.311086
\(373\) −26.2960 −1.36155 −0.680776 0.732491i \(-0.738357\pi\)
−0.680776 + 0.732491i \(0.738357\pi\)
\(374\) −23.3710 −1.20849
\(375\) 0 0
\(376\) 7.96183 0.410600
\(377\) −4.96183 −0.255547
\(378\) −0.723686 −0.0372224
\(379\) −6.34212 −0.325773 −0.162886 0.986645i \(-0.552080\pi\)
−0.162886 + 0.986645i \(0.552080\pi\)
\(380\) 0 0
\(381\) 12.9526 0.663580
\(382\) −8.40920 −0.430252
\(383\) 35.4959 1.81376 0.906878 0.421393i \(-0.138459\pi\)
0.906878 + 0.421393i \(0.138459\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −12.9907 −0.661211
\(387\) −0.961825 −0.0488923
\(388\) 2.55263 0.129590
\(389\) −9.84730 −0.499278 −0.249639 0.968339i \(-0.580312\pi\)
−0.249639 + 0.968339i \(0.580312\pi\)
\(390\) 0 0
\(391\) −4.23814 −0.214332
\(392\) 6.47628 0.327101
\(393\) 11.5815 0.584211
\(394\) 20.5815 1.03688
\(395\) 0 0
\(396\) 5.51445 0.277112
\(397\) −18.8762 −0.947370 −0.473685 0.880694i \(-0.657076\pi\)
−0.473685 + 0.880694i \(0.657076\pi\)
\(398\) 4.64734 0.232950
\(399\) 3.59080 0.179765
\(400\) 0 0
\(401\) 33.5052 1.67317 0.836585 0.547838i \(-0.184549\pi\)
0.836585 + 0.547838i \(0.184549\pi\)
\(402\) 11.9237 0.594698
\(403\) 29.7710 1.48300
\(404\) −1.06708 −0.0530893
\(405\) 0 0
\(406\) 0.723686 0.0359159
\(407\) −23.3710 −1.15846
\(408\) −4.23814 −0.209819
\(409\) −25.5341 −1.26258 −0.631290 0.775547i \(-0.717474\pi\)
−0.631290 + 0.775547i \(0.717474\pi\)
\(410\) 0 0
\(411\) −22.2381 −1.09693
\(412\) −5.75259 −0.283410
\(413\) −6.53409 −0.321522
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) −4.96183 −0.243273
\(417\) 18.9907 0.929980
\(418\) −27.3618 −1.33831
\(419\) −37.2000 −1.81734 −0.908669 0.417518i \(-0.862900\pi\)
−0.908669 + 0.417518i \(0.862900\pi\)
\(420\) 0 0
\(421\) −11.5815 −0.564449 −0.282225 0.959348i \(-0.591072\pi\)
−0.282225 + 0.959348i \(0.591072\pi\)
\(422\) 23.3328 1.13583
\(423\) −7.96183 −0.387117
\(424\) 6.47628 0.314516
\(425\) 0 0
\(426\) 0.514453 0.0249254
\(427\) 5.38956 0.260819
\(428\) −15.9237 −0.769699
\(429\) 27.3618 1.32104
\(430\) 0 0
\(431\) 7.37102 0.355050 0.177525 0.984116i \(-0.443191\pi\)
0.177525 + 0.984116i \(0.443191\pi\)
\(432\) 1.00000 0.0481125
\(433\) 19.3710 0.930912 0.465456 0.885071i \(-0.345890\pi\)
0.465456 + 0.885071i \(0.345890\pi\)
\(434\) −4.34212 −0.208428
\(435\) 0 0
\(436\) 17.2670 0.826942
\(437\) −4.96183 −0.237356
\(438\) 0.447372 0.0213763
\(439\) 17.5052 0.835477 0.417738 0.908567i \(-0.362823\pi\)
0.417738 + 0.908567i \(0.362823\pi\)
\(440\) 0 0
\(441\) −6.47628 −0.308394
\(442\) −21.0289 −1.00024
\(443\) −17.3710 −0.825322 −0.412661 0.910885i \(-0.635401\pi\)
−0.412661 + 0.910885i \(0.635401\pi\)
\(444\) −4.23814 −0.201133
\(445\) 0 0
\(446\) 25.2947 1.19774
\(447\) −5.92365 −0.280179
\(448\) 0.723686 0.0341909
\(449\) −16.8947 −0.797312 −0.398656 0.917100i \(-0.630523\pi\)
−0.398656 + 0.917100i \(0.630523\pi\)
\(450\) 0 0
\(451\) 5.72497 0.269578
\(452\) 6.23814 0.293417
\(453\) 6.47628 0.304282
\(454\) 7.13288 0.334763
\(455\) 0 0
\(456\) −4.96183 −0.232359
\(457\) 21.9237 1.02555 0.512773 0.858524i \(-0.328618\pi\)
0.512773 + 0.858524i \(0.328618\pi\)
\(458\) 7.58154 0.354262
\(459\) 4.23814 0.197819
\(460\) 0 0
\(461\) −2.65788 −0.123790 −0.0618950 0.998083i \(-0.519714\pi\)
−0.0618950 + 0.998083i \(0.519714\pi\)
\(462\) −3.99073 −0.185666
\(463\) −12.0000 −0.557687 −0.278844 0.960337i \(-0.589951\pi\)
−0.278844 + 0.960337i \(0.589951\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 0 0
\(466\) 24.8855 1.15280
\(467\) −15.8289 −0.732476 −0.366238 0.930521i \(-0.619354\pi\)
−0.366238 + 0.930521i \(0.619354\pi\)
\(468\) 4.96183 0.229360
\(469\) −8.62898 −0.398449
\(470\) 0 0
\(471\) −9.37102 −0.431794
\(472\) 9.02891 0.415589
\(473\) −5.30394 −0.243875
\(474\) 5.51445 0.253287
\(475\) 0 0
\(476\) 3.06708 0.140579
\(477\) −6.47628 −0.296528
\(478\) 18.0960 0.827691
\(479\) 27.5723 1.25981 0.629905 0.776673i \(-0.283094\pi\)
0.629905 + 0.776673i \(0.283094\pi\)
\(480\) 0 0
\(481\) −21.0289 −0.958836
\(482\) −26.8762 −1.22418
\(483\) −0.723686 −0.0329288
\(484\) 19.4092 0.882236
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 16.6868 0.756151 0.378075 0.925775i \(-0.376586\pi\)
0.378075 + 0.925775i \(0.376586\pi\)
\(488\) −7.44737 −0.337127
\(489\) 13.9237 0.629649
\(490\) 0 0
\(491\) −13.4288 −0.606035 −0.303017 0.952985i \(-0.597994\pi\)
−0.303017 + 0.952985i \(0.597994\pi\)
\(492\) 1.03817 0.0468045
\(493\) −4.23814 −0.190876
\(494\) −24.6197 −1.10769
\(495\) 0 0
\(496\) 6.00000 0.269408
\(497\) −0.372303 −0.0167001
\(498\) −6.17106 −0.276532
\(499\) −8.99073 −0.402480 −0.201240 0.979542i \(-0.564497\pi\)
−0.201240 + 0.979542i \(0.564497\pi\)
\(500\) 0 0
\(501\) −9.61971 −0.429777
\(502\) 8.79077 0.392351
\(503\) 18.9618 0.845466 0.422733 0.906254i \(-0.361071\pi\)
0.422733 + 0.906254i \(0.361071\pi\)
\(504\) −0.723686 −0.0322355
\(505\) 0 0
\(506\) 5.51445 0.245147
\(507\) 11.6197 0.516049
\(508\) 12.9526 0.574677
\(509\) 26.2276 1.16252 0.581259 0.813719i \(-0.302560\pi\)
0.581259 + 0.813719i \(0.302560\pi\)
\(510\) 0 0
\(511\) −0.323757 −0.0143222
\(512\) −1.00000 −0.0441942
\(513\) 4.96183 0.219070
\(514\) −10.9526 −0.483097
\(515\) 0 0
\(516\) −0.961825 −0.0423420
\(517\) −43.9051 −1.93094
\(518\) 3.06708 0.134760
\(519\) −1.51445 −0.0664771
\(520\) 0 0
\(521\) 32.2381 1.41238 0.706189 0.708023i \(-0.250413\pi\)
0.706189 + 0.708023i \(0.250413\pi\)
\(522\) 1.00000 0.0437688
\(523\) 5.59080 0.244469 0.122234 0.992501i \(-0.460994\pi\)
0.122234 + 0.992501i \(0.460994\pi\)
\(524\) 11.5815 0.505942
\(525\) 0 0
\(526\) 20.8762 0.910246
\(527\) 25.4288 1.10770
\(528\) 5.51445 0.239986
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −9.02891 −0.391821
\(532\) 3.59080 0.155681
\(533\) 5.15124 0.223125
\(534\) 2.23814 0.0968538
\(535\) 0 0
\(536\) 11.9237 0.515023
\(537\) 10.4763 0.452085
\(538\) −21.3618 −0.920971
\(539\) −35.7131 −1.53827
\(540\) 0 0
\(541\) 23.6486 1.01673 0.508367 0.861141i \(-0.330249\pi\)
0.508367 + 0.861141i \(0.330249\pi\)
\(542\) −11.0289 −0.473732
\(543\) −21.7433 −0.933095
\(544\) −4.23814 −0.181709
\(545\) 0 0
\(546\) −3.59080 −0.153672
\(547\) 39.3328 1.68175 0.840876 0.541229i \(-0.182041\pi\)
0.840876 + 0.541229i \(0.182041\pi\)
\(548\) −22.2381 −0.949966
\(549\) 7.44737 0.317846
\(550\) 0 0
\(551\) −4.96183 −0.211381
\(552\) 1.00000 0.0425628
\(553\) −3.99073 −0.169703
\(554\) −15.4381 −0.655902
\(555\) 0 0
\(556\) 18.9907 0.805386
\(557\) −34.2658 −1.45189 −0.725944 0.687754i \(-0.758597\pi\)
−0.725944 + 0.687754i \(0.758597\pi\)
\(558\) −6.00000 −0.254000
\(559\) −4.77241 −0.201851
\(560\) 0 0
\(561\) 23.3710 0.986725
\(562\) 2.16179 0.0911896
\(563\) 20.5434 0.865799 0.432900 0.901442i \(-0.357490\pi\)
0.432900 + 0.901442i \(0.357490\pi\)
\(564\) −7.96183 −0.335253
\(565\) 0 0
\(566\) −16.9526 −0.712569
\(567\) 0.723686 0.0303920
\(568\) 0.514453 0.0215860
\(569\) −27.4474 −1.15065 −0.575327 0.817924i \(-0.695125\pi\)
−0.575327 + 0.817924i \(0.695125\pi\)
\(570\) 0 0
\(571\) −5.58154 −0.233580 −0.116790 0.993157i \(-0.537260\pi\)
−0.116790 + 0.993157i \(0.537260\pi\)
\(572\) 27.3618 1.14405
\(573\) 8.40920 0.351299
\(574\) −0.751312 −0.0313592
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 4.56190 0.189914 0.0949571 0.995481i \(-0.469729\pi\)
0.0949571 + 0.995481i \(0.469729\pi\)
\(578\) −0.961825 −0.0400067
\(579\) 12.9907 0.539876
\(580\) 0 0
\(581\) 4.46591 0.185277
\(582\) −2.55263 −0.105810
\(583\) −35.7131 −1.47909
\(584\) 0.447372 0.0185124
\(585\) 0 0
\(586\) −13.0289 −0.538219
\(587\) −23.4474 −0.967777 −0.483888 0.875130i \(-0.660776\pi\)
−0.483888 + 0.875130i \(0.660776\pi\)
\(588\) −6.47628 −0.267077
\(589\) 29.7710 1.22669
\(590\) 0 0
\(591\) −20.5815 −0.846611
\(592\) −4.23814 −0.174186
\(593\) 14.4092 0.591715 0.295857 0.955232i \(-0.404395\pi\)
0.295857 + 0.955232i \(0.404395\pi\)
\(594\) −5.51445 −0.226261
\(595\) 0 0
\(596\) −5.92365 −0.242642
\(597\) −4.64734 −0.190203
\(598\) 4.96183 0.202904
\(599\) 27.9815 1.14329 0.571646 0.820500i \(-0.306305\pi\)
0.571646 + 0.820500i \(0.306305\pi\)
\(600\) 0 0
\(601\) −6.38029 −0.260257 −0.130129 0.991497i \(-0.541539\pi\)
−0.130129 + 0.991497i \(0.541539\pi\)
\(602\) 0.696059 0.0283693
\(603\) −11.9237 −0.485569
\(604\) 6.47628 0.263516
\(605\) 0 0
\(606\) 1.06708 0.0433472
\(607\) 41.6946 1.69233 0.846166 0.532920i \(-0.178905\pi\)
0.846166 + 0.532920i \(0.178905\pi\)
\(608\) −4.96183 −0.201229
\(609\) −0.723686 −0.0293252
\(610\) 0 0
\(611\) −39.5052 −1.59821
\(612\) 4.23814 0.171317
\(613\) −10.8486 −0.438170 −0.219085 0.975706i \(-0.570307\pi\)
−0.219085 + 0.975706i \(0.570307\pi\)
\(614\) −27.0671 −1.09234
\(615\) 0 0
\(616\) −3.99073 −0.160791
\(617\) 17.5052 0.704732 0.352366 0.935862i \(-0.385377\pi\)
0.352366 + 0.935862i \(0.385377\pi\)
\(618\) 5.75259 0.231403
\(619\) 22.3421 0.898005 0.449003 0.893530i \(-0.351779\pi\)
0.449003 + 0.893530i \(0.351779\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 31.4670 1.26171
\(623\) −1.61971 −0.0648923
\(624\) 4.96183 0.198632
\(625\) 0 0
\(626\) −27.2947 −1.09091
\(627\) 27.3618 1.09272
\(628\) −9.37102 −0.373944
\(629\) −17.9618 −0.716185
\(630\) 0 0
\(631\) −13.4105 −0.533863 −0.266931 0.963716i \(-0.586010\pi\)
−0.266931 + 0.963716i \(0.586010\pi\)
\(632\) 5.51445 0.219353
\(633\) −23.3328 −0.927397
\(634\) 11.0000 0.436866
\(635\) 0 0
\(636\) −6.47628 −0.256801
\(637\) −32.1342 −1.27320
\(638\) 5.51445 0.218319
\(639\) −0.514453 −0.0203515
\(640\) 0 0
\(641\) 35.9538 1.42009 0.710046 0.704156i \(-0.248674\pi\)
0.710046 + 0.704156i \(0.248674\pi\)
\(642\) 15.9237 0.628456
\(643\) 2.48555 0.0980204 0.0490102 0.998798i \(-0.484393\pi\)
0.0490102 + 0.998798i \(0.484393\pi\)
\(644\) −0.723686 −0.0285172
\(645\) 0 0
\(646\) −21.0289 −0.827371
\(647\) 44.4381 1.74704 0.873521 0.486786i \(-0.161831\pi\)
0.873521 + 0.486786i \(0.161831\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −49.7895 −1.95441
\(650\) 0 0
\(651\) 4.34212 0.170181
\(652\) 13.9237 0.545292
\(653\) −21.5526 −0.843420 −0.421710 0.906731i \(-0.638570\pi\)
−0.421710 + 0.906731i \(0.638570\pi\)
\(654\) −17.2670 −0.675195
\(655\) 0 0
\(656\) 1.03817 0.0405339
\(657\) −0.447372 −0.0174536
\(658\) 5.76186 0.224621
\(659\) −7.33413 −0.285697 −0.142849 0.989745i \(-0.545626\pi\)
−0.142849 + 0.989745i \(0.545626\pi\)
\(660\) 0 0
\(661\) −46.1618 −1.79549 −0.897743 0.440520i \(-0.854794\pi\)
−0.897743 + 0.440520i \(0.854794\pi\)
\(662\) 8.51445 0.330924
\(663\) 21.0289 0.816695
\(664\) −6.17106 −0.239483
\(665\) 0 0
\(666\) 4.23814 0.164225
\(667\) 1.00000 0.0387202
\(668\) −9.61971 −0.372198
\(669\) −25.2947 −0.977949
\(670\) 0 0
\(671\) 41.0682 1.58542
\(672\) −0.723686 −0.0279168
\(673\) −37.8184 −1.45779 −0.728896 0.684624i \(-0.759966\pi\)
−0.728896 + 0.684624i \(0.759966\pi\)
\(674\) 1.52372 0.0586916
\(675\) 0 0
\(676\) 11.6197 0.446912
\(677\) 2.21051 0.0849569 0.0424785 0.999097i \(-0.486475\pi\)
0.0424785 + 0.999097i \(0.486475\pi\)
\(678\) −6.23814 −0.239574
\(679\) 1.84730 0.0708929
\(680\) 0 0
\(681\) −7.13288 −0.273333
\(682\) −33.0867 −1.26696
\(683\) −29.6393 −1.13412 −0.567059 0.823677i \(-0.691919\pi\)
−0.567059 + 0.823677i \(0.691919\pi\)
\(684\) 4.96183 0.189720
\(685\) 0 0
\(686\) 9.75259 0.372356
\(687\) −7.58154 −0.289254
\(688\) −0.961825 −0.0366692
\(689\) −32.1342 −1.22421
\(690\) 0 0
\(691\) −40.1512 −1.52743 −0.763713 0.645556i \(-0.776626\pi\)
−0.763713 + 0.645556i \(0.776626\pi\)
\(692\) −1.51445 −0.0575709
\(693\) 3.99073 0.151595
\(694\) −25.5815 −0.971062
\(695\) 0 0
\(696\) 1.00000 0.0379049
\(697\) 4.39993 0.166659
\(698\) 26.8855 1.01763
\(699\) −24.8855 −0.941255
\(700\) 0 0
\(701\) −40.8762 −1.54387 −0.771937 0.635700i \(-0.780712\pi\)
−0.771937 + 0.635700i \(0.780712\pi\)
\(702\) −4.96183 −0.187272
\(703\) −21.0289 −0.793120
\(704\) 5.51445 0.207834
\(705\) 0 0
\(706\) −10.6197 −0.399678
\(707\) −0.772232 −0.0290428
\(708\) −9.02891 −0.339327
\(709\) −25.0565 −0.941018 −0.470509 0.882395i \(-0.655930\pi\)
−0.470509 + 0.882395i \(0.655930\pi\)
\(710\) 0 0
\(711\) −5.51445 −0.206808
\(712\) 2.23814 0.0838778
\(713\) −6.00000 −0.224702
\(714\) −3.06708 −0.114783
\(715\) 0 0
\(716\) 10.4763 0.391517
\(717\) −18.0960 −0.675807
\(718\) 5.43810 0.202948
\(719\) −34.5145 −1.28717 −0.643586 0.765374i \(-0.722554\pi\)
−0.643586 + 0.765374i \(0.722554\pi\)
\(720\) 0 0
\(721\) −4.16307 −0.155041
\(722\) −5.61971 −0.209144
\(723\) 26.8762 0.999537
\(724\) −21.7433 −0.808084
\(725\) 0 0
\(726\) −19.4092 −0.720343
\(727\) 37.0289 1.37333 0.686663 0.726976i \(-0.259075\pi\)
0.686663 + 0.726976i \(0.259075\pi\)
\(728\) −3.59080 −0.133084
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −4.07635 −0.150769
\(732\) 7.44737 0.275263
\(733\) 8.02763 0.296507 0.148254 0.988949i \(-0.452635\pi\)
0.148254 + 0.988949i \(0.452635\pi\)
\(734\) −8.56190 −0.316025
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) −65.7524 −2.42202
\(738\) −1.03817 −0.0382157
\(739\) −34.3432 −1.26334 −0.631668 0.775239i \(-0.717629\pi\)
−0.631668 + 0.775239i \(0.717629\pi\)
\(740\) 0 0
\(741\) 24.6197 0.904428
\(742\) 4.68679 0.172058
\(743\) −30.8855 −1.13308 −0.566539 0.824035i \(-0.691718\pi\)
−0.566539 + 0.824035i \(0.691718\pi\)
\(744\) −6.00000 −0.219971
\(745\) 0 0
\(746\) 26.2960 0.962763
\(747\) 6.17106 0.225787
\(748\) 23.3710 0.854529
\(749\) −11.5237 −0.421068
\(750\) 0 0
\(751\) 19.0658 0.695721 0.347860 0.937546i \(-0.386908\pi\)
0.347860 + 0.937546i \(0.386908\pi\)
\(752\) −7.96183 −0.290338
\(753\) −8.79077 −0.320353
\(754\) 4.96183 0.180699
\(755\) 0 0
\(756\) 0.723686 0.0263202
\(757\) −26.1040 −0.948765 −0.474383 0.880319i \(-0.657329\pi\)
−0.474383 + 0.880319i \(0.657329\pi\)
\(758\) 6.34212 0.230356
\(759\) −5.51445 −0.200162
\(760\) 0 0
\(761\) 16.9433 0.614194 0.307097 0.951678i \(-0.400642\pi\)
0.307097 + 0.951678i \(0.400642\pi\)
\(762\) −12.9526 −0.469222
\(763\) 12.4959 0.452383
\(764\) 8.40920 0.304234
\(765\) 0 0
\(766\) −35.4959 −1.28252
\(767\) −44.7999 −1.61763
\(768\) 1.00000 0.0360844
\(769\) −41.5815 −1.49947 −0.749734 0.661739i \(-0.769819\pi\)
−0.749734 + 0.661739i \(0.769819\pi\)
\(770\) 0 0
\(771\) 10.9526 0.394447
\(772\) 12.9907 0.467547
\(773\) 39.2183 1.41059 0.705293 0.708916i \(-0.250816\pi\)
0.705293 + 0.708916i \(0.250816\pi\)
\(774\) 0.961825 0.0345721
\(775\) 0 0
\(776\) −2.55263 −0.0916340
\(777\) −3.06708 −0.110031
\(778\) 9.84730 0.353043
\(779\) 5.15124 0.184562
\(780\) 0 0
\(781\) −2.83693 −0.101513
\(782\) 4.23814 0.151556
\(783\) −1.00000 −0.0357371
\(784\) −6.47628 −0.231296
\(785\) 0 0
\(786\) −11.5815 −0.413100
\(787\) −17.3039 −0.616819 −0.308409 0.951254i \(-0.599797\pi\)
−0.308409 + 0.951254i \(0.599797\pi\)
\(788\) −20.5815 −0.733187
\(789\) −20.8762 −0.743213
\(790\) 0 0
\(791\) 4.51445 0.160515
\(792\) −5.51445 −0.195948
\(793\) 36.9526 1.31222
\(794\) 18.8762 0.669892
\(795\) 0 0
\(796\) −4.64734 −0.164720
\(797\) 12.3236 0.436524 0.218262 0.975890i \(-0.429961\pi\)
0.218262 + 0.975890i \(0.429961\pi\)
\(798\) −3.59080 −0.127113
\(799\) −33.7433 −1.19375
\(800\) 0 0
\(801\) −2.23814 −0.0790808
\(802\) −33.5052 −1.18311
\(803\) −2.46701 −0.0870589
\(804\) −11.9237 −0.420515
\(805\) 0 0
\(806\) −29.7710 −1.04864
\(807\) 21.3618 0.751969
\(808\) 1.06708 0.0375398
\(809\) −21.3618 −0.751039 −0.375520 0.926814i \(-0.622536\pi\)
−0.375520 + 0.926814i \(0.622536\pi\)
\(810\) 0 0
\(811\) 47.0104 1.65076 0.825379 0.564579i \(-0.190962\pi\)
0.825379 + 0.564579i \(0.190962\pi\)
\(812\) −0.723686 −0.0253964
\(813\) 11.0289 0.386801
\(814\) 23.3710 0.819154
\(815\) 0 0
\(816\) 4.23814 0.148365
\(817\) −4.77241 −0.166965
\(818\) 25.5341 0.892779
\(819\) 3.59080 0.125473
\(820\) 0 0
\(821\) 30.4855 1.06395 0.531976 0.846759i \(-0.321449\pi\)
0.531976 + 0.846759i \(0.321449\pi\)
\(822\) 22.2381 0.775644
\(823\) −6.34212 −0.221072 −0.110536 0.993872i \(-0.535257\pi\)
−0.110536 + 0.993872i \(0.535257\pi\)
\(824\) 5.75259 0.200401
\(825\) 0 0
\(826\) 6.53409 0.227350
\(827\) 56.1340 1.95197 0.975985 0.217838i \(-0.0699004\pi\)
0.975985 + 0.217838i \(0.0699004\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −9.93292 −0.344985 −0.172492 0.985011i \(-0.555182\pi\)
−0.172492 + 0.985011i \(0.555182\pi\)
\(830\) 0 0
\(831\) 15.4381 0.535542
\(832\) 4.96183 0.172020
\(833\) −27.4474 −0.950995
\(834\) −18.9907 −0.657595
\(835\) 0 0
\(836\) 27.3618 0.946326
\(837\) 6.00000 0.207390
\(838\) 37.2000 1.28505
\(839\) 31.7802 1.09718 0.548588 0.836093i \(-0.315166\pi\)
0.548588 + 0.836093i \(0.315166\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 11.5815 0.399126
\(843\) −2.16179 −0.0744560
\(844\) −23.3328 −0.803150
\(845\) 0 0
\(846\) 7.96183 0.273733
\(847\) 14.0462 0.482632
\(848\) −6.47628 −0.222396
\(849\) 16.9526 0.581810
\(850\) 0 0
\(851\) 4.23814 0.145282
\(852\) −0.514453 −0.0176249
\(853\) −5.30394 −0.181603 −0.0908017 0.995869i \(-0.528943\pi\)
−0.0908017 + 0.995869i \(0.528943\pi\)
\(854\) −5.38956 −0.184427
\(855\) 0 0
\(856\) 15.9237 0.544259
\(857\) −19.6579 −0.671501 −0.335750 0.941951i \(-0.608990\pi\)
−0.335750 + 0.941951i \(0.608990\pi\)
\(858\) −27.3618 −0.934115
\(859\) 57.7524 1.97049 0.985244 0.171159i \(-0.0547510\pi\)
0.985244 + 0.171159i \(0.0547510\pi\)
\(860\) 0 0
\(861\) 0.751312 0.0256047
\(862\) −7.37102 −0.251058
\(863\) −27.3328 −0.930421 −0.465210 0.885200i \(-0.654021\pi\)
−0.465210 + 0.885200i \(0.654021\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −19.3710 −0.658254
\(867\) 0.961825 0.0326653
\(868\) 4.34212 0.147381
\(869\) −30.4092 −1.03156
\(870\) 0 0
\(871\) −59.1631 −2.00466
\(872\) −17.2670 −0.584736
\(873\) 2.55263 0.0863934
\(874\) 4.96183 0.167836
\(875\) 0 0
\(876\) −0.447372 −0.0151153
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) −17.5052 −0.590771
\(879\) 13.0289 0.439454
\(880\) 0 0
\(881\) 19.4474 0.655199 0.327599 0.944817i \(-0.393760\pi\)
0.327599 + 0.944817i \(0.393760\pi\)
\(882\) 6.47628 0.218068
\(883\) −34.2854 −1.15380 −0.576898 0.816816i \(-0.695737\pi\)
−0.576898 + 0.816816i \(0.695737\pi\)
\(884\) 21.0289 0.707279
\(885\) 0 0
\(886\) 17.3710 0.583591
\(887\) −35.9433 −1.20686 −0.603429 0.797417i \(-0.706199\pi\)
−0.603429 + 0.797417i \(0.706199\pi\)
\(888\) 4.23814 0.142223
\(889\) 9.37358 0.314380
\(890\) 0 0
\(891\) 5.51445 0.184741
\(892\) −25.2947 −0.846928
\(893\) −39.5052 −1.32199
\(894\) 5.92365 0.198117
\(895\) 0 0
\(896\) −0.723686 −0.0241766
\(897\) −4.96183 −0.165671
\(898\) 16.8947 0.563785
\(899\) −6.00000 −0.200111
\(900\) 0 0
\(901\) −27.4474 −0.914405
\(902\) −5.72497 −0.190621
\(903\) −0.696059 −0.0231634
\(904\) −6.23814 −0.207477
\(905\) 0 0
\(906\) −6.47628 −0.215160
\(907\) 34.0671 1.13118 0.565589 0.824687i \(-0.308649\pi\)
0.565589 + 0.824687i \(0.308649\pi\)
\(908\) −7.13288 −0.236713
\(909\) −1.06708 −0.0353929
\(910\) 0 0
\(911\) 16.0671 0.532326 0.266163 0.963928i \(-0.414244\pi\)
0.266163 + 0.963928i \(0.414244\pi\)
\(912\) 4.96183 0.164302
\(913\) 34.0300 1.12623
\(914\) −21.9237 −0.725170
\(915\) 0 0
\(916\) −7.58154 −0.250501
\(917\) 8.38139 0.276778
\(918\) −4.23814 −0.139879
\(919\) −19.7619 −0.651884 −0.325942 0.945390i \(-0.605681\pi\)
−0.325942 + 0.945390i \(0.605681\pi\)
\(920\) 0 0
\(921\) 27.0671 0.891891
\(922\) 2.65788 0.0875328
\(923\) −2.55263 −0.0840208
\(924\) 3.99073 0.131285
\(925\) 0 0
\(926\) 12.0000 0.394344
\(927\) −5.75259 −0.188940
\(928\) 1.00000 0.0328266
\(929\) −6.33285 −0.207774 −0.103887 0.994589i \(-0.533128\pi\)
−0.103887 + 0.994589i \(0.533128\pi\)
\(930\) 0 0
\(931\) −32.1342 −1.05315
\(932\) −24.8855 −0.815151
\(933\) −31.4670 −1.03018
\(934\) 15.8289 0.517939
\(935\) 0 0
\(936\) −4.96183 −0.162182
\(937\) 25.8473 0.844395 0.422197 0.906504i \(-0.361259\pi\)
0.422197 + 0.906504i \(0.361259\pi\)
\(938\) 8.62898 0.281746
\(939\) 27.2947 0.890728
\(940\) 0 0
\(941\) −9.04744 −0.294938 −0.147469 0.989067i \(-0.547113\pi\)
−0.147469 + 0.989067i \(0.547113\pi\)
\(942\) 9.37102 0.305324
\(943\) −1.03817 −0.0338076
\(944\) −9.02891 −0.293866
\(945\) 0 0
\(946\) 5.30394 0.172446
\(947\) 0.971093 0.0315563 0.0157781 0.999876i \(-0.494977\pi\)
0.0157781 + 0.999876i \(0.494977\pi\)
\(948\) −5.51445 −0.179101
\(949\) −2.21978 −0.0720571
\(950\) 0 0
\(951\) −11.0000 −0.356699
\(952\) −3.06708 −0.0994047
\(953\) 5.19070 0.168143 0.0840716 0.996460i \(-0.473208\pi\)
0.0840716 + 0.996460i \(0.473208\pi\)
\(954\) 6.47628 0.209677
\(955\) 0 0
\(956\) −18.0960 −0.585266
\(957\) −5.51445 −0.178257
\(958\) −27.5723 −0.890820
\(959\) −16.0934 −0.519684
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 21.0289 0.677999
\(963\) −15.9237 −0.513132
\(964\) 26.8762 0.865624
\(965\) 0 0
\(966\) 0.723686 0.0232842
\(967\) 17.0289 0.547613 0.273806 0.961785i \(-0.411717\pi\)
0.273806 + 0.961785i \(0.411717\pi\)
\(968\) −19.4092 −0.623835
\(969\) 21.0289 0.675546
\(970\) 0 0
\(971\) −25.4683 −0.817316 −0.408658 0.912688i \(-0.634003\pi\)
−0.408658 + 0.912688i \(0.634003\pi\)
\(972\) 1.00000 0.0320750
\(973\) 13.7433 0.440591
\(974\) −16.6868 −0.534679
\(975\) 0 0
\(976\) 7.44737 0.238385
\(977\) −23.9513 −0.766269 −0.383135 0.923693i \(-0.625155\pi\)
−0.383135 + 0.923693i \(0.625155\pi\)
\(978\) −13.9237 −0.445229
\(979\) −12.3421 −0.394456
\(980\) 0 0
\(981\) 17.2670 0.551294
\(982\) 13.4288 0.428531
\(983\) 13.5723 0.432888 0.216444 0.976295i \(-0.430554\pi\)
0.216444 + 0.976295i \(0.430554\pi\)
\(984\) −1.03817 −0.0330958
\(985\) 0 0
\(986\) 4.23814 0.134970
\(987\) −5.76186 −0.183402
\(988\) 24.6197 0.783257
\(989\) 0.961825 0.0305843
\(990\) 0 0
\(991\) 25.9237 0.823492 0.411746 0.911299i \(-0.364919\pi\)
0.411746 + 0.911299i \(0.364919\pi\)
\(992\) −6.00000 −0.190500
\(993\) −8.51445 −0.270198
\(994\) 0.372303 0.0118087
\(995\) 0 0
\(996\) 6.17106 0.195537
\(997\) −2.06708 −0.0654651 −0.0327326 0.999464i \(-0.510421\pi\)
−0.0327326 + 0.999464i \(0.510421\pi\)
\(998\) 8.99073 0.284597
\(999\) −4.23814 −0.134089
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.a.bp.1.2 3
5.2 odd 4 3450.2.d.ba.2899.2 6
5.3 odd 4 3450.2.d.ba.2899.5 6
5.4 even 2 3450.2.a.bs.1.2 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3450.2.a.bp.1.2 3 1.1 even 1 trivial
3450.2.a.bs.1.2 yes 3 5.4 even 2
3450.2.d.ba.2899.2 6 5.2 odd 4
3450.2.d.ba.2899.5 6 5.3 odd 4