Properties

Label 8034.2.a.bb.1.14
Level $8034$
Weight $2$
Character 8034.1
Self dual yes
Analytic conductor $64.152$
Analytic rank $1$
Dimension $14$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8034.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Defining polynomial: \(x^{14} - 6 x^{13} - 29 x^{12} + 207 x^{11} + 269 x^{10} - 2601 x^{9} - 847 x^{8} + 14851 x^{7} + 678 x^{6} - 39390 x^{5} - 3280 x^{4} + 42456 x^{3} + 10816 x^{2} - 7296 x - 2048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.14
Root \(-3.69005\) of defining polynomial
Character \(\chi\) \(=\) 8034.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.69005 q^{5} -1.00000 q^{6} -0.801879 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.69005 q^{5} -1.00000 q^{6} -0.801879 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.69005 q^{10} -2.09517 q^{11} +1.00000 q^{12} -1.00000 q^{13} +0.801879 q^{14} +3.69005 q^{15} +1.00000 q^{16} -7.28473 q^{17} -1.00000 q^{18} +6.41973 q^{19} +3.69005 q^{20} -0.801879 q^{21} +2.09517 q^{22} -6.18478 q^{23} -1.00000 q^{24} +8.61644 q^{25} +1.00000 q^{26} +1.00000 q^{27} -0.801879 q^{28} -8.16717 q^{29} -3.69005 q^{30} -4.92193 q^{31} -1.00000 q^{32} -2.09517 q^{33} +7.28473 q^{34} -2.95897 q^{35} +1.00000 q^{36} +7.56623 q^{37} -6.41973 q^{38} -1.00000 q^{39} -3.69005 q^{40} -12.6172 q^{41} +0.801879 q^{42} -1.69130 q^{43} -2.09517 q^{44} +3.69005 q^{45} +6.18478 q^{46} -1.40161 q^{47} +1.00000 q^{48} -6.35699 q^{49} -8.61644 q^{50} -7.28473 q^{51} -1.00000 q^{52} +10.3169 q^{53} -1.00000 q^{54} -7.73128 q^{55} +0.801879 q^{56} +6.41973 q^{57} +8.16717 q^{58} +11.6675 q^{59} +3.69005 q^{60} +3.80363 q^{61} +4.92193 q^{62} -0.801879 q^{63} +1.00000 q^{64} -3.69005 q^{65} +2.09517 q^{66} -1.82701 q^{67} -7.28473 q^{68} -6.18478 q^{69} +2.95897 q^{70} -12.5083 q^{71} -1.00000 q^{72} -9.57442 q^{73} -7.56623 q^{74} +8.61644 q^{75} +6.41973 q^{76} +1.68007 q^{77} +1.00000 q^{78} -16.2346 q^{79} +3.69005 q^{80} +1.00000 q^{81} +12.6172 q^{82} -9.35341 q^{83} -0.801879 q^{84} -26.8810 q^{85} +1.69130 q^{86} -8.16717 q^{87} +2.09517 q^{88} +3.88578 q^{89} -3.69005 q^{90} +0.801879 q^{91} -6.18478 q^{92} -4.92193 q^{93} +1.40161 q^{94} +23.6891 q^{95} -1.00000 q^{96} +10.4617 q^{97} +6.35699 q^{98} -2.09517 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + O(q^{10}) \) \( 14q - 14q^{2} + 14q^{3} + 14q^{4} - 6q^{5} - 14q^{6} - 4q^{7} - 14q^{8} + 14q^{9} + 6q^{10} - 8q^{11} + 14q^{12} - 14q^{13} + 4q^{14} - 6q^{15} + 14q^{16} - 4q^{17} - 14q^{18} - q^{19} - 6q^{20} - 4q^{21} + 8q^{22} - 9q^{23} - 14q^{24} + 24q^{25} + 14q^{26} + 14q^{27} - 4q^{28} - 10q^{29} + 6q^{30} - 5q^{31} - 14q^{32} - 8q^{33} + 4q^{34} - 16q^{35} + 14q^{36} - 4q^{37} + q^{38} - 14q^{39} + 6q^{40} - 24q^{41} + 4q^{42} - 8q^{44} - 6q^{45} + 9q^{46} - 32q^{47} + 14q^{48} + 24q^{49} - 24q^{50} - 4q^{51} - 14q^{52} - 5q^{53} - 14q^{54} - 8q^{55} + 4q^{56} - q^{57} + 10q^{58} - 13q^{59} - 6q^{60} + 2q^{61} + 5q^{62} - 4q^{63} + 14q^{64} + 6q^{65} + 8q^{66} - 16q^{67} - 4q^{68} - 9q^{69} + 16q^{70} - 29q^{71} - 14q^{72} + 4q^{74} + 24q^{75} - q^{76} - 9q^{77} + 14q^{78} - 21q^{79} - 6q^{80} + 14q^{81} + 24q^{82} - 40q^{83} - 4q^{84} - 7q^{85} - 10q^{87} + 8q^{88} - 48q^{89} + 6q^{90} + 4q^{91} - 9q^{92} - 5q^{93} + 32q^{94} - 26q^{95} - 14q^{96} + 18q^{97} - 24q^{98} - 8q^{99} + O(q^{100}) \)

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
\(5\) 3.69005 1.65024 0.825119 0.564958i \(-0.191108\pi\)
0.825119 + 0.564958i \(0.191108\pi\)
\(6\) −1.00000 −0.408248
\(7\) −0.801879 −0.303082 −0.151541 0.988451i \(-0.548424\pi\)
−0.151541 + 0.988451i \(0.548424\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −3.69005 −1.16690
\(11\) −2.09517 −0.631718 −0.315859 0.948806i \(-0.602293\pi\)
−0.315859 + 0.948806i \(0.602293\pi\)
\(12\) 1.00000 0.288675
\(13\) −1.00000 −0.277350
\(14\) 0.801879 0.214311
\(15\) 3.69005 0.952766
\(16\) 1.00000 0.250000
\(17\) −7.28473 −1.76681 −0.883403 0.468613i \(-0.844754\pi\)
−0.883403 + 0.468613i \(0.844754\pi\)
\(18\) −1.00000 −0.235702
\(19\) 6.41973 1.47279 0.736393 0.676554i \(-0.236527\pi\)
0.736393 + 0.676554i \(0.236527\pi\)
\(20\) 3.69005 0.825119
\(21\) −0.801879 −0.174984
\(22\) 2.09517 0.446692
\(23\) −6.18478 −1.28962 −0.644808 0.764345i \(-0.723062\pi\)
−0.644808 + 0.764345i \(0.723062\pi\)
\(24\) −1.00000 −0.204124
\(25\) 8.61644 1.72329
\(26\) 1.00000 0.196116
\(27\) 1.00000 0.192450
\(28\) −0.801879 −0.151541
\(29\) −8.16717 −1.51660 −0.758302 0.651903i \(-0.773971\pi\)
−0.758302 + 0.651903i \(0.773971\pi\)
\(30\) −3.69005 −0.673707
\(31\) −4.92193 −0.884004 −0.442002 0.897014i \(-0.645732\pi\)
−0.442002 + 0.897014i \(0.645732\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.09517 −0.364723
\(34\) 7.28473 1.24932
\(35\) −2.95897 −0.500157
\(36\) 1.00000 0.166667
\(37\) 7.56623 1.24388 0.621940 0.783065i \(-0.286345\pi\)
0.621940 + 0.783065i \(0.286345\pi\)
\(38\) −6.41973 −1.04142
\(39\) −1.00000 −0.160128
\(40\) −3.69005 −0.583448
\(41\) −12.6172 −1.97047 −0.985234 0.171213i \(-0.945231\pi\)
−0.985234 + 0.171213i \(0.945231\pi\)
\(42\) 0.801879 0.123733
\(43\) −1.69130 −0.257921 −0.128960 0.991650i \(-0.541164\pi\)
−0.128960 + 0.991650i \(0.541164\pi\)
\(44\) −2.09517 −0.315859
\(45\) 3.69005 0.550080
\(46\) 6.18478 0.911896
\(47\) −1.40161 −0.204446 −0.102223 0.994762i \(-0.532595\pi\)
−0.102223 + 0.994762i \(0.532595\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.35699 −0.908141
\(50\) −8.61644 −1.21855
\(51\) −7.28473 −1.02007
\(52\) −1.00000 −0.138675
\(53\) 10.3169 1.41714 0.708568 0.705642i \(-0.249341\pi\)
0.708568 + 0.705642i \(0.249341\pi\)
\(54\) −1.00000 −0.136083
\(55\) −7.73128 −1.04249
\(56\) 0.801879 0.107156
\(57\) 6.41973 0.850313
\(58\) 8.16717 1.07240
\(59\) 11.6675 1.51898 0.759491 0.650518i \(-0.225448\pi\)
0.759491 + 0.650518i \(0.225448\pi\)
\(60\) 3.69005 0.476383
\(61\) 3.80363 0.487005 0.243503 0.969900i \(-0.421704\pi\)
0.243503 + 0.969900i \(0.421704\pi\)
\(62\) 4.92193 0.625085
\(63\) −0.801879 −0.101027
\(64\) 1.00000 0.125000
\(65\) −3.69005 −0.457694
\(66\) 2.09517 0.257898
\(67\) −1.82701 −0.223205 −0.111603 0.993753i \(-0.535598\pi\)
−0.111603 + 0.993753i \(0.535598\pi\)
\(68\) −7.28473 −0.883403
\(69\) −6.18478 −0.744560
\(70\) 2.95897 0.353665
\(71\) −12.5083 −1.48446 −0.742230 0.670145i \(-0.766232\pi\)
−0.742230 + 0.670145i \(0.766232\pi\)
\(72\) −1.00000 −0.117851
\(73\) −9.57442 −1.12060 −0.560301 0.828289i \(-0.689314\pi\)
−0.560301 + 0.828289i \(0.689314\pi\)
\(74\) −7.56623 −0.879556
\(75\) 8.61644 0.994941
\(76\) 6.41973 0.736393
\(77\) 1.68007 0.191462
\(78\) 1.00000 0.113228
\(79\) −16.2346 −1.82654 −0.913268 0.407359i \(-0.866450\pi\)
−0.913268 + 0.407359i \(0.866450\pi\)
\(80\) 3.69005 0.412560
\(81\) 1.00000 0.111111
\(82\) 12.6172 1.39333
\(83\) −9.35341 −1.02667 −0.513335 0.858188i \(-0.671590\pi\)
−0.513335 + 0.858188i \(0.671590\pi\)
\(84\) −0.801879 −0.0874922
\(85\) −26.8810 −2.91565
\(86\) 1.69130 0.182378
\(87\) −8.16717 −0.875612
\(88\) 2.09517 0.223346
\(89\) 3.88578 0.411892 0.205946 0.978563i \(-0.433973\pi\)
0.205946 + 0.978563i \(0.433973\pi\)
\(90\) −3.69005 −0.388965
\(91\) 0.801879 0.0840598
\(92\) −6.18478 −0.644808
\(93\) −4.92193 −0.510380
\(94\) 1.40161 0.144565
\(95\) 23.6891 2.43045
\(96\) −1.00000 −0.102062
\(97\) 10.4617 1.06222 0.531111 0.847302i \(-0.321775\pi\)
0.531111 + 0.847302i \(0.321775\pi\)
\(98\) 6.35699 0.642153
\(99\) −2.09517 −0.210573
\(100\) 8.61644 0.861644
\(101\) 5.37036 0.534371 0.267185 0.963645i \(-0.413906\pi\)
0.267185 + 0.963645i \(0.413906\pi\)
\(102\) 7.28473 0.721296
\(103\) 1.00000 0.0985329
\(104\) 1.00000 0.0980581
\(105\) −2.95897 −0.288766
\(106\) −10.3169 −1.00207
\(107\) −5.84497 −0.565054 −0.282527 0.959259i \(-0.591173\pi\)
−0.282527 + 0.959259i \(0.591173\pi\)
\(108\) 1.00000 0.0962250
\(109\) 18.8090 1.80158 0.900789 0.434257i \(-0.142989\pi\)
0.900789 + 0.434257i \(0.142989\pi\)
\(110\) 7.73128 0.737149
\(111\) 7.56623 0.718155
\(112\) −0.801879 −0.0757704
\(113\) −8.04354 −0.756673 −0.378336 0.925668i \(-0.623504\pi\)
−0.378336 + 0.925668i \(0.623504\pi\)
\(114\) −6.41973 −0.601262
\(115\) −22.8221 −2.12817
\(116\) −8.16717 −0.758302
\(117\) −1.00000 −0.0924500
\(118\) −11.6675 −1.07408
\(119\) 5.84147 0.535487
\(120\) −3.69005 −0.336854
\(121\) −6.61026 −0.600932
\(122\) −3.80363 −0.344365
\(123\) −12.6172 −1.13765
\(124\) −4.92193 −0.442002
\(125\) 13.3448 1.19360
\(126\) 0.801879 0.0714371
\(127\) 6.32994 0.561692 0.280846 0.959753i \(-0.409385\pi\)
0.280846 + 0.959753i \(0.409385\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −1.69130 −0.148911
\(130\) 3.69005 0.323638
\(131\) −11.7846 −1.02963 −0.514813 0.857303i \(-0.672139\pi\)
−0.514813 + 0.857303i \(0.672139\pi\)
\(132\) −2.09517 −0.182361
\(133\) −5.14784 −0.446375
\(134\) 1.82701 0.157830
\(135\) 3.69005 0.317589
\(136\) 7.28473 0.624660
\(137\) 12.9694 1.10805 0.554027 0.832499i \(-0.313090\pi\)
0.554027 + 0.832499i \(0.313090\pi\)
\(138\) 6.18478 0.526483
\(139\) 17.3953 1.47545 0.737724 0.675103i \(-0.235901\pi\)
0.737724 + 0.675103i \(0.235901\pi\)
\(140\) −2.95897 −0.250079
\(141\) −1.40161 −0.118037
\(142\) 12.5083 1.04967
\(143\) 2.09517 0.175207
\(144\) 1.00000 0.0833333
\(145\) −30.1372 −2.50276
\(146\) 9.57442 0.792385
\(147\) −6.35699 −0.524316
\(148\) 7.56623 0.621940
\(149\) 13.0943 1.07273 0.536363 0.843988i \(-0.319798\pi\)
0.536363 + 0.843988i \(0.319798\pi\)
\(150\) −8.61644 −0.703530
\(151\) −20.0305 −1.63006 −0.815031 0.579417i \(-0.803280\pi\)
−0.815031 + 0.579417i \(0.803280\pi\)
\(152\) −6.41973 −0.520709
\(153\) −7.28473 −0.588936
\(154\) −1.68007 −0.135384
\(155\) −18.1621 −1.45882
\(156\) −1.00000 −0.0800641
\(157\) −18.5752 −1.48246 −0.741230 0.671251i \(-0.765757\pi\)
−0.741230 + 0.671251i \(0.765757\pi\)
\(158\) 16.2346 1.29156
\(159\) 10.3169 0.818184
\(160\) −3.69005 −0.291724
\(161\) 4.95944 0.390859
\(162\) −1.00000 −0.0785674
\(163\) 3.81533 0.298840 0.149420 0.988774i \(-0.452259\pi\)
0.149420 + 0.988774i \(0.452259\pi\)
\(164\) −12.6172 −0.985234
\(165\) −7.73128 −0.601879
\(166\) 9.35341 0.725965
\(167\) −11.8008 −0.913177 −0.456588 0.889678i \(-0.650929\pi\)
−0.456588 + 0.889678i \(0.650929\pi\)
\(168\) 0.801879 0.0618663
\(169\) 1.00000 0.0769231
\(170\) 26.8810 2.06168
\(171\) 6.41973 0.490929
\(172\) −1.69130 −0.128960
\(173\) −24.7683 −1.88310 −0.941549 0.336877i \(-0.890629\pi\)
−0.941549 + 0.336877i \(0.890629\pi\)
\(174\) 8.16717 0.619151
\(175\) −6.90934 −0.522297
\(176\) −2.09517 −0.157930
\(177\) 11.6675 0.876985
\(178\) −3.88578 −0.291252
\(179\) −13.6899 −1.02323 −0.511616 0.859214i \(-0.670953\pi\)
−0.511616 + 0.859214i \(0.670953\pi\)
\(180\) 3.69005 0.275040
\(181\) −26.3059 −1.95530 −0.977652 0.210229i \(-0.932579\pi\)
−0.977652 + 0.210229i \(0.932579\pi\)
\(182\) −0.801879 −0.0594392
\(183\) 3.80363 0.281173
\(184\) 6.18478 0.455948
\(185\) 27.9197 2.05270
\(186\) 4.92193 0.360893
\(187\) 15.2628 1.11612
\(188\) −1.40161 −0.102223
\(189\) −0.801879 −0.0583281
\(190\) −23.6891 −1.71859
\(191\) 27.0533 1.95751 0.978753 0.205044i \(-0.0657337\pi\)
0.978753 + 0.205044i \(0.0657337\pi\)
\(192\) 1.00000 0.0721688
\(193\) −6.88418 −0.495534 −0.247767 0.968820i \(-0.579697\pi\)
−0.247767 + 0.968820i \(0.579697\pi\)
\(194\) −10.4617 −0.751105
\(195\) −3.69005 −0.264250
\(196\) −6.35699 −0.454071
\(197\) −12.1767 −0.867558 −0.433779 0.901019i \(-0.642820\pi\)
−0.433779 + 0.901019i \(0.642820\pi\)
\(198\) 2.09517 0.148897
\(199\) 6.86514 0.486657 0.243328 0.969944i \(-0.421761\pi\)
0.243328 + 0.969944i \(0.421761\pi\)
\(200\) −8.61644 −0.609274
\(201\) −1.82701 −0.128868
\(202\) −5.37036 −0.377857
\(203\) 6.54908 0.459655
\(204\) −7.28473 −0.510033
\(205\) −46.5579 −3.25174
\(206\) −1.00000 −0.0696733
\(207\) −6.18478 −0.429872
\(208\) −1.00000 −0.0693375
\(209\) −13.4504 −0.930386
\(210\) 2.95897 0.204188
\(211\) −25.0346 −1.72345 −0.861727 0.507371i \(-0.830617\pi\)
−0.861727 + 0.507371i \(0.830617\pi\)
\(212\) 10.3169 0.708568
\(213\) −12.5083 −0.857054
\(214\) 5.84497 0.399554
\(215\) −6.24097 −0.425631
\(216\) −1.00000 −0.0680414
\(217\) 3.94679 0.267926
\(218\) −18.8090 −1.27391
\(219\) −9.57442 −0.646979
\(220\) −7.73128 −0.521243
\(221\) 7.28473 0.490024
\(222\) −7.56623 −0.507812
\(223\) 11.4805 0.768788 0.384394 0.923169i \(-0.374410\pi\)
0.384394 + 0.923169i \(0.374410\pi\)
\(224\) 0.801879 0.0535778
\(225\) 8.61644 0.574429
\(226\) 8.04354 0.535048
\(227\) −6.32844 −0.420033 −0.210017 0.977698i \(-0.567352\pi\)
−0.210017 + 0.977698i \(0.567352\pi\)
\(228\) 6.41973 0.425157
\(229\) −1.09206 −0.0721656 −0.0360828 0.999349i \(-0.511488\pi\)
−0.0360828 + 0.999349i \(0.511488\pi\)
\(230\) 22.8221 1.50485
\(231\) 1.68007 0.110541
\(232\) 8.16717 0.536201
\(233\) −6.15204 −0.403033 −0.201517 0.979485i \(-0.564587\pi\)
−0.201517 + 0.979485i \(0.564587\pi\)
\(234\) 1.00000 0.0653720
\(235\) −5.17200 −0.337384
\(236\) 11.6675 0.759491
\(237\) −16.2346 −1.05455
\(238\) −5.84147 −0.378646
\(239\) −3.86903 −0.250267 −0.125133 0.992140i \(-0.539936\pi\)
−0.125133 + 0.992140i \(0.539936\pi\)
\(240\) 3.69005 0.238191
\(241\) 8.24640 0.531197 0.265599 0.964084i \(-0.414430\pi\)
0.265599 + 0.964084i \(0.414430\pi\)
\(242\) 6.61026 0.424923
\(243\) 1.00000 0.0641500
\(244\) 3.80363 0.243503
\(245\) −23.4576 −1.49865
\(246\) 12.6172 0.804440
\(247\) −6.41973 −0.408477
\(248\) 4.92193 0.312543
\(249\) −9.35341 −0.592748
\(250\) −13.3448 −0.844002
\(251\) −20.8393 −1.31537 −0.657683 0.753294i \(-0.728464\pi\)
−0.657683 + 0.753294i \(0.728464\pi\)
\(252\) −0.801879 −0.0505136
\(253\) 12.9582 0.814673
\(254\) −6.32994 −0.397176
\(255\) −26.8810 −1.68335
\(256\) 1.00000 0.0625000
\(257\) 16.9120 1.05494 0.527472 0.849573i \(-0.323140\pi\)
0.527472 + 0.849573i \(0.323140\pi\)
\(258\) 1.69130 0.105296
\(259\) −6.06720 −0.376997
\(260\) −3.69005 −0.228847
\(261\) −8.16717 −0.505535
\(262\) 11.7846 0.728055
\(263\) 18.1348 1.11824 0.559120 0.829086i \(-0.311139\pi\)
0.559120 + 0.829086i \(0.311139\pi\)
\(264\) 2.09517 0.128949
\(265\) 38.0699 2.33861
\(266\) 5.14784 0.315635
\(267\) 3.88578 0.237806
\(268\) −1.82701 −0.111603
\(269\) 0.636581 0.0388130 0.0194065 0.999812i \(-0.493822\pi\)
0.0194065 + 0.999812i \(0.493822\pi\)
\(270\) −3.69005 −0.224569
\(271\) −6.65381 −0.404190 −0.202095 0.979366i \(-0.564775\pi\)
−0.202095 + 0.979366i \(0.564775\pi\)
\(272\) −7.28473 −0.441702
\(273\) 0.801879 0.0485319
\(274\) −12.9694 −0.783513
\(275\) −18.0529 −1.08863
\(276\) −6.18478 −0.372280
\(277\) −18.6247 −1.11905 −0.559526 0.828813i \(-0.689017\pi\)
−0.559526 + 0.828813i \(0.689017\pi\)
\(278\) −17.3953 −1.04330
\(279\) −4.92193 −0.294668
\(280\) 2.95897 0.176832
\(281\) −15.8147 −0.943427 −0.471713 0.881752i \(-0.656364\pi\)
−0.471713 + 0.881752i \(0.656364\pi\)
\(282\) 1.40161 0.0834646
\(283\) 1.14296 0.0679421 0.0339710 0.999423i \(-0.489185\pi\)
0.0339710 + 0.999423i \(0.489185\pi\)
\(284\) −12.5083 −0.742230
\(285\) 23.6891 1.40322
\(286\) −2.09517 −0.123890
\(287\) 10.1174 0.597213
\(288\) −1.00000 −0.0589256
\(289\) 36.0673 2.12161
\(290\) 30.1372 1.76972
\(291\) 10.4617 0.613274
\(292\) −9.57442 −0.560301
\(293\) 19.8586 1.16015 0.580077 0.814561i \(-0.303022\pi\)
0.580077 + 0.814561i \(0.303022\pi\)
\(294\) 6.35699 0.370747
\(295\) 43.0537 2.50668
\(296\) −7.56623 −0.439778
\(297\) −2.09517 −0.121574
\(298\) −13.0943 −0.758532
\(299\) 6.18478 0.357675
\(300\) 8.61644 0.497470
\(301\) 1.35622 0.0781711
\(302\) 20.0305 1.15263
\(303\) 5.37036 0.308519
\(304\) 6.41973 0.368197
\(305\) 14.0356 0.803675
\(306\) 7.28473 0.416440
\(307\) 14.4202 0.823004 0.411502 0.911409i \(-0.365004\pi\)
0.411502 + 0.911409i \(0.365004\pi\)
\(308\) 1.68007 0.0957311
\(309\) 1.00000 0.0568880
\(310\) 18.1621 1.03154
\(311\) −16.9322 −0.960138 −0.480069 0.877231i \(-0.659388\pi\)
−0.480069 + 0.877231i \(0.659388\pi\)
\(312\) 1.00000 0.0566139
\(313\) 8.02812 0.453776 0.226888 0.973921i \(-0.427145\pi\)
0.226888 + 0.973921i \(0.427145\pi\)
\(314\) 18.5752 1.04826
\(315\) −2.95897 −0.166719
\(316\) −16.2346 −0.913268
\(317\) 26.3844 1.48189 0.740947 0.671563i \(-0.234377\pi\)
0.740947 + 0.671563i \(0.234377\pi\)
\(318\) −10.3169 −0.578543
\(319\) 17.1116 0.958066
\(320\) 3.69005 0.206280
\(321\) −5.84497 −0.326234
\(322\) −4.95944 −0.276379
\(323\) −46.7660 −2.60213
\(324\) 1.00000 0.0555556
\(325\) −8.61644 −0.477954
\(326\) −3.81533 −0.211312
\(327\) 18.8090 1.04014
\(328\) 12.6172 0.696666
\(329\) 1.12392 0.0619638
\(330\) 7.73128 0.425593
\(331\) 19.3054 1.06112 0.530560 0.847648i \(-0.321982\pi\)
0.530560 + 0.847648i \(0.321982\pi\)
\(332\) −9.35341 −0.513335
\(333\) 7.56623 0.414627
\(334\) 11.8008 0.645714
\(335\) −6.74176 −0.368342
\(336\) −0.801879 −0.0437461
\(337\) 7.12603 0.388179 0.194090 0.980984i \(-0.437825\pi\)
0.194090 + 0.980984i \(0.437825\pi\)
\(338\) −1.00000 −0.0543928
\(339\) −8.04354 −0.436865
\(340\) −26.8810 −1.45783
\(341\) 10.3123 0.558441
\(342\) −6.41973 −0.347139
\(343\) 10.7107 0.578323
\(344\) 1.69130 0.0911888
\(345\) −22.8221 −1.22870
\(346\) 24.7683 1.33155
\(347\) 33.6190 1.80476 0.902381 0.430939i \(-0.141818\pi\)
0.902381 + 0.430939i \(0.141818\pi\)
\(348\) −8.16717 −0.437806
\(349\) 0.865602 0.0463346 0.0231673 0.999732i \(-0.492625\pi\)
0.0231673 + 0.999732i \(0.492625\pi\)
\(350\) 6.90934 0.369320
\(351\) −1.00000 −0.0533761
\(352\) 2.09517 0.111673
\(353\) 32.6448 1.73751 0.868754 0.495243i \(-0.164921\pi\)
0.868754 + 0.495243i \(0.164921\pi\)
\(354\) −11.6675 −0.620122
\(355\) −46.1562 −2.44972
\(356\) 3.88578 0.205946
\(357\) 5.84147 0.309163
\(358\) 13.6899 0.723534
\(359\) −13.5526 −0.715277 −0.357639 0.933860i \(-0.616418\pi\)
−0.357639 + 0.933860i \(0.616418\pi\)
\(360\) −3.69005 −0.194483
\(361\) 22.2129 1.16910
\(362\) 26.3059 1.38261
\(363\) −6.61026 −0.346948
\(364\) 0.801879 0.0420299
\(365\) −35.3301 −1.84926
\(366\) −3.80363 −0.198819
\(367\) −3.18662 −0.166340 −0.0831702 0.996535i \(-0.526505\pi\)
−0.0831702 + 0.996535i \(0.526505\pi\)
\(368\) −6.18478 −0.322404
\(369\) −12.6172 −0.656823
\(370\) −27.9197 −1.45148
\(371\) −8.27291 −0.429508
\(372\) −4.92193 −0.255190
\(373\) 12.2027 0.631834 0.315917 0.948787i \(-0.397688\pi\)
0.315917 + 0.948787i \(0.397688\pi\)
\(374\) −15.2628 −0.789219
\(375\) 13.3448 0.689124
\(376\) 1.40161 0.0722825
\(377\) 8.16717 0.420630
\(378\) 0.801879 0.0412442
\(379\) −4.04966 −0.208017 −0.104008 0.994576i \(-0.533167\pi\)
−0.104008 + 0.994576i \(0.533167\pi\)
\(380\) 23.6891 1.21522
\(381\) 6.32994 0.324293
\(382\) −27.0533 −1.38417
\(383\) −8.52524 −0.435620 −0.217810 0.975991i \(-0.569891\pi\)
−0.217810 + 0.975991i \(0.569891\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 6.19955 0.315958
\(386\) 6.88418 0.350396
\(387\) −1.69130 −0.0859736
\(388\) 10.4617 0.531111
\(389\) −2.55183 −0.129383 −0.0646914 0.997905i \(-0.520606\pi\)
−0.0646914 + 0.997905i \(0.520606\pi\)
\(390\) 3.69005 0.186853
\(391\) 45.0544 2.27850
\(392\) 6.35699 0.321076
\(393\) −11.7846 −0.594455
\(394\) 12.1767 0.613456
\(395\) −59.9065 −3.01422
\(396\) −2.09517 −0.105286
\(397\) −15.6618 −0.786042 −0.393021 0.919530i \(-0.628570\pi\)
−0.393021 + 0.919530i \(0.628570\pi\)
\(398\) −6.86514 −0.344118
\(399\) −5.14784 −0.257715
\(400\) 8.61644 0.430822
\(401\) −10.6910 −0.533883 −0.266942 0.963713i \(-0.586013\pi\)
−0.266942 + 0.963713i \(0.586013\pi\)
\(402\) 1.82701 0.0911231
\(403\) 4.92193 0.245179
\(404\) 5.37036 0.267185
\(405\) 3.69005 0.183360
\(406\) −6.54908 −0.325025
\(407\) −15.8525 −0.785782
\(408\) 7.28473 0.360648
\(409\) 14.4409 0.714055 0.357028 0.934094i \(-0.383790\pi\)
0.357028 + 0.934094i \(0.383790\pi\)
\(410\) 46.5579 2.29933
\(411\) 12.9694 0.639736
\(412\) 1.00000 0.0492665
\(413\) −9.35594 −0.460376
\(414\) 6.18478 0.303965
\(415\) −34.5145 −1.69425
\(416\) 1.00000 0.0490290
\(417\) 17.3953 0.851850
\(418\) 13.4504 0.657882
\(419\) −14.0266 −0.685244 −0.342622 0.939473i \(-0.611315\pi\)
−0.342622 + 0.939473i \(0.611315\pi\)
\(420\) −2.95897 −0.144383
\(421\) 21.9429 1.06943 0.534715 0.845032i \(-0.320419\pi\)
0.534715 + 0.845032i \(0.320419\pi\)
\(422\) 25.0346 1.21867
\(423\) −1.40161 −0.0681486
\(424\) −10.3169 −0.501033
\(425\) −62.7685 −3.04472
\(426\) 12.5083 0.606029
\(427\) −3.05005 −0.147602
\(428\) −5.84497 −0.282527
\(429\) 2.09517 0.101156
\(430\) 6.24097 0.300966
\(431\) 1.88113 0.0906108 0.0453054 0.998973i \(-0.485574\pi\)
0.0453054 + 0.998973i \(0.485574\pi\)
\(432\) 1.00000 0.0481125
\(433\) 15.2492 0.732831 0.366415 0.930451i \(-0.380585\pi\)
0.366415 + 0.930451i \(0.380585\pi\)
\(434\) −3.94679 −0.189452
\(435\) −30.1372 −1.44497
\(436\) 18.8090 0.900789
\(437\) −39.7046 −1.89933
\(438\) 9.57442 0.457484
\(439\) 38.1006 1.81844 0.909220 0.416315i \(-0.136679\pi\)
0.909220 + 0.416315i \(0.136679\pi\)
\(440\) 7.73128 0.368574
\(441\) −6.35699 −0.302714
\(442\) −7.28473 −0.346499
\(443\) −2.12579 −0.100999 −0.0504996 0.998724i \(-0.516081\pi\)
−0.0504996 + 0.998724i \(0.516081\pi\)
\(444\) 7.56623 0.359077
\(445\) 14.3387 0.679720
\(446\) −11.4805 −0.543615
\(447\) 13.0943 0.619338
\(448\) −0.801879 −0.0378852
\(449\) −22.5617 −1.06475 −0.532377 0.846507i \(-0.678701\pi\)
−0.532377 + 0.846507i \(0.678701\pi\)
\(450\) −8.61644 −0.406183
\(451\) 26.4351 1.24478
\(452\) −8.04354 −0.378336
\(453\) −20.0305 −0.941117
\(454\) 6.32844 0.297008
\(455\) 2.95897 0.138719
\(456\) −6.41973 −0.300631
\(457\) −28.9729 −1.35529 −0.677647 0.735387i \(-0.737000\pi\)
−0.677647 + 0.735387i \(0.737000\pi\)
\(458\) 1.09206 0.0510288
\(459\) −7.28473 −0.340022
\(460\) −22.8221 −1.06409
\(461\) −26.4125 −1.23015 −0.615077 0.788467i \(-0.710875\pi\)
−0.615077 + 0.788467i \(0.710875\pi\)
\(462\) −1.68007 −0.0781641
\(463\) −32.9808 −1.53275 −0.766374 0.642394i \(-0.777941\pi\)
−0.766374 + 0.642394i \(0.777941\pi\)
\(464\) −8.16717 −0.379151
\(465\) −18.1621 −0.842249
\(466\) 6.15204 0.284988
\(467\) 22.2234 1.02838 0.514188 0.857678i \(-0.328093\pi\)
0.514188 + 0.857678i \(0.328093\pi\)
\(468\) −1.00000 −0.0462250
\(469\) 1.46504 0.0676494
\(470\) 5.17200 0.238567
\(471\) −18.5752 −0.855898
\(472\) −11.6675 −0.537041
\(473\) 3.54356 0.162933
\(474\) 16.2346 0.745680
\(475\) 55.3152 2.53804
\(476\) 5.84147 0.267743
\(477\) 10.3169 0.472379
\(478\) 3.86903 0.176965
\(479\) 12.2928 0.561673 0.280837 0.959756i \(-0.409388\pi\)
0.280837 + 0.959756i \(0.409388\pi\)
\(480\) −3.69005 −0.168427
\(481\) −7.56623 −0.344990
\(482\) −8.24640 −0.375613
\(483\) 4.95944 0.225662
\(484\) −6.61026 −0.300466
\(485\) 38.6041 1.75292
\(486\) −1.00000 −0.0453609
\(487\) 11.6049 0.525867 0.262934 0.964814i \(-0.415310\pi\)
0.262934 + 0.964814i \(0.415310\pi\)
\(488\) −3.80363 −0.172182
\(489\) 3.81533 0.172535
\(490\) 23.4576 1.05971
\(491\) 11.2148 0.506115 0.253057 0.967451i \(-0.418564\pi\)
0.253057 + 0.967451i \(0.418564\pi\)
\(492\) −12.6172 −0.568825
\(493\) 59.4956 2.67955
\(494\) 6.41973 0.288837
\(495\) −7.73128 −0.347495
\(496\) −4.92193 −0.221001
\(497\) 10.0301 0.449913
\(498\) 9.35341 0.419136
\(499\) −8.47929 −0.379585 −0.189793 0.981824i \(-0.560782\pi\)
−0.189793 + 0.981824i \(0.560782\pi\)
\(500\) 13.3448 0.596799
\(501\) −11.8008 −0.527223
\(502\) 20.8393 0.930105
\(503\) 18.4500 0.822644 0.411322 0.911490i \(-0.365067\pi\)
0.411322 + 0.911490i \(0.365067\pi\)
\(504\) 0.801879 0.0357185
\(505\) 19.8169 0.881839
\(506\) −12.9582 −0.576061
\(507\) 1.00000 0.0444116
\(508\) 6.32994 0.280846
\(509\) 2.89416 0.128281 0.0641406 0.997941i \(-0.479569\pi\)
0.0641406 + 0.997941i \(0.479569\pi\)
\(510\) 26.8810 1.19031
\(511\) 7.67753 0.339634
\(512\) −1.00000 −0.0441942
\(513\) 6.41973 0.283438
\(514\) −16.9120 −0.745958
\(515\) 3.69005 0.162603
\(516\) −1.69130 −0.0744553
\(517\) 2.93661 0.129152
\(518\) 6.06720 0.266577
\(519\) −24.7683 −1.08721
\(520\) 3.69005 0.161819
\(521\) −17.4473 −0.764379 −0.382190 0.924084i \(-0.624830\pi\)
−0.382190 + 0.924084i \(0.624830\pi\)
\(522\) 8.16717 0.357467
\(523\) 26.5858 1.16252 0.581259 0.813719i \(-0.302560\pi\)
0.581259 + 0.813719i \(0.302560\pi\)
\(524\) −11.7846 −0.514813
\(525\) −6.90934 −0.301548
\(526\) −18.1348 −0.790716
\(527\) 35.8549 1.56186
\(528\) −2.09517 −0.0911806
\(529\) 15.2515 0.663107
\(530\) −38.0699 −1.65365
\(531\) 11.6675 0.506327
\(532\) −5.14784 −0.223187
\(533\) 12.6172 0.546510
\(534\) −3.88578 −0.168154
\(535\) −21.5682 −0.932475
\(536\) 1.82701 0.0789150
\(537\) −13.6899 −0.590763
\(538\) −0.636581 −0.0274450
\(539\) 13.3190 0.573689
\(540\) 3.69005 0.158794
\(541\) 23.5105 1.01079 0.505397 0.862887i \(-0.331346\pi\)
0.505397 + 0.862887i \(0.331346\pi\)
\(542\) 6.65381 0.285806
\(543\) −26.3059 −1.12890
\(544\) 7.28473 0.312330
\(545\) 69.4062 2.97303
\(546\) −0.801879 −0.0343173
\(547\) 28.0184 1.19798 0.598990 0.800756i \(-0.295569\pi\)
0.598990 + 0.800756i \(0.295569\pi\)
\(548\) 12.9694 0.554027
\(549\) 3.80363 0.162335
\(550\) 18.0529 0.769779
\(551\) −52.4310 −2.23363
\(552\) 6.18478 0.263242
\(553\) 13.0182 0.553590
\(554\) 18.6247 0.791289
\(555\) 27.9197 1.18513
\(556\) 17.3953 0.737724
\(557\) −9.47001 −0.401257 −0.200629 0.979667i \(-0.564298\pi\)
−0.200629 + 0.979667i \(0.564298\pi\)
\(558\) 4.92193 0.208362
\(559\) 1.69130 0.0715344
\(560\) −2.95897 −0.125039
\(561\) 15.2628 0.644394
\(562\) 15.8147 0.667104
\(563\) −38.8473 −1.63722 −0.818608 0.574352i \(-0.805254\pi\)
−0.818608 + 0.574352i \(0.805254\pi\)
\(564\) −1.40161 −0.0590184
\(565\) −29.6810 −1.24869
\(566\) −1.14296 −0.0480423
\(567\) −0.801879 −0.0336758
\(568\) 12.5083 0.524836
\(569\) 2.91023 0.122003 0.0610017 0.998138i \(-0.480570\pi\)
0.0610017 + 0.998138i \(0.480570\pi\)
\(570\) −23.6891 −0.992227
\(571\) −6.33484 −0.265105 −0.132552 0.991176i \(-0.542317\pi\)
−0.132552 + 0.991176i \(0.542317\pi\)
\(572\) 2.09517 0.0876035
\(573\) 27.0533 1.13017
\(574\) −10.1174 −0.422293
\(575\) −53.2908 −2.22238
\(576\) 1.00000 0.0416667
\(577\) 34.2196 1.42458 0.712290 0.701886i \(-0.247658\pi\)
0.712290 + 0.701886i \(0.247658\pi\)
\(578\) −36.0673 −1.50020
\(579\) −6.88418 −0.286097
\(580\) −30.1372 −1.25138
\(581\) 7.50030 0.311165
\(582\) −10.4617 −0.433651
\(583\) −21.6157 −0.895230
\(584\) 9.57442 0.396192
\(585\) −3.69005 −0.152565
\(586\) −19.8586 −0.820353
\(587\) −3.59839 −0.148522 −0.0742608 0.997239i \(-0.523660\pi\)
−0.0742608 + 0.997239i \(0.523660\pi\)
\(588\) −6.35699 −0.262158
\(589\) −31.5974 −1.30195
\(590\) −43.0537 −1.77249
\(591\) −12.1767 −0.500885
\(592\) 7.56623 0.310970
\(593\) −11.5291 −0.473443 −0.236722 0.971578i \(-0.576073\pi\)
−0.236722 + 0.971578i \(0.576073\pi\)
\(594\) 2.09517 0.0859659
\(595\) 21.5553 0.883681
\(596\) 13.0943 0.536363
\(597\) 6.86514 0.280971
\(598\) −6.18478 −0.252914
\(599\) −10.3969 −0.424807 −0.212403 0.977182i \(-0.568129\pi\)
−0.212403 + 0.977182i \(0.568129\pi\)
\(600\) −8.61644 −0.351765
\(601\) 13.9017 0.567060 0.283530 0.958963i \(-0.408494\pi\)
0.283530 + 0.958963i \(0.408494\pi\)
\(602\) −1.35622 −0.0552753
\(603\) −1.82701 −0.0744017
\(604\) −20.0305 −0.815031
\(605\) −24.3921 −0.991682
\(606\) −5.37036 −0.218156
\(607\) −23.6585 −0.960267 −0.480134 0.877195i \(-0.659412\pi\)
−0.480134 + 0.877195i \(0.659412\pi\)
\(608\) −6.41973 −0.260354
\(609\) 6.54908 0.265382
\(610\) −14.0356 −0.568284
\(611\) 1.40161 0.0567031
\(612\) −7.28473 −0.294468
\(613\) 11.6937 0.472303 0.236152 0.971716i \(-0.424114\pi\)
0.236152 + 0.971716i \(0.424114\pi\)
\(614\) −14.4202 −0.581952
\(615\) −46.5579 −1.87739
\(616\) −1.68007 −0.0676921
\(617\) −26.1720 −1.05365 −0.526823 0.849975i \(-0.676617\pi\)
−0.526823 + 0.849975i \(0.676617\pi\)
\(618\) −1.00000 −0.0402259
\(619\) −23.6493 −0.950545 −0.475273 0.879839i \(-0.657651\pi\)
−0.475273 + 0.879839i \(0.657651\pi\)
\(620\) −18.1621 −0.729409
\(621\) −6.18478 −0.248187
\(622\) 16.9322 0.678920
\(623\) −3.11593 −0.124837
\(624\) −1.00000 −0.0400320
\(625\) 6.16086 0.246434
\(626\) −8.02812 −0.320868
\(627\) −13.4504 −0.537158
\(628\) −18.5752 −0.741230
\(629\) −55.1179 −2.19770
\(630\) 2.95897 0.117888
\(631\) −11.2193 −0.446632 −0.223316 0.974746i \(-0.571688\pi\)
−0.223316 + 0.974746i \(0.571688\pi\)
\(632\) 16.2346 0.645778
\(633\) −25.0346 −0.995037
\(634\) −26.3844 −1.04786
\(635\) 23.3578 0.926925
\(636\) 10.3169 0.409092
\(637\) 6.35699 0.251873
\(638\) −17.1116 −0.677455
\(639\) −12.5083 −0.494820
\(640\) −3.69005 −0.145862
\(641\) 39.6666 1.56674 0.783368 0.621558i \(-0.213500\pi\)
0.783368 + 0.621558i \(0.213500\pi\)
\(642\) 5.84497 0.230683
\(643\) −43.2448 −1.70541 −0.852705 0.522393i \(-0.825039\pi\)
−0.852705 + 0.522393i \(0.825039\pi\)
\(644\) 4.95944 0.195429
\(645\) −6.24097 −0.245738
\(646\) 46.7660 1.83998
\(647\) 30.1059 1.18359 0.591793 0.806090i \(-0.298420\pi\)
0.591793 + 0.806090i \(0.298420\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −24.4455 −0.959568
\(650\) 8.61644 0.337965
\(651\) 3.94679 0.154687
\(652\) 3.81533 0.149420
\(653\) −16.3241 −0.638813 −0.319406 0.947618i \(-0.603483\pi\)
−0.319406 + 0.947618i \(0.603483\pi\)
\(654\) −18.8090 −0.735491
\(655\) −43.4857 −1.69913
\(656\) −12.6172 −0.492617
\(657\) −9.57442 −0.373534
\(658\) −1.12392 −0.0438150
\(659\) 7.27050 0.283219 0.141609 0.989923i \(-0.454772\pi\)
0.141609 + 0.989923i \(0.454772\pi\)
\(660\) −7.73128 −0.300940
\(661\) −14.1712 −0.551195 −0.275597 0.961273i \(-0.588876\pi\)
−0.275597 + 0.961273i \(0.588876\pi\)
\(662\) −19.3054 −0.750325
\(663\) 7.28473 0.282915
\(664\) 9.35341 0.362983
\(665\) −18.9958 −0.736625
\(666\) −7.56623 −0.293185
\(667\) 50.5121 1.95584
\(668\) −11.8008 −0.456588
\(669\) 11.4805 0.443860
\(670\) 6.74176 0.260457
\(671\) −7.96926 −0.307650
\(672\) 0.801879 0.0309332
\(673\) 40.3104 1.55385 0.776925 0.629593i \(-0.216778\pi\)
0.776925 + 0.629593i \(0.216778\pi\)
\(674\) −7.12603 −0.274484
\(675\) 8.61644 0.331647
\(676\) 1.00000 0.0384615
\(677\) 28.8923 1.11042 0.555210 0.831710i \(-0.312638\pi\)
0.555210 + 0.831710i \(0.312638\pi\)
\(678\) 8.04354 0.308910
\(679\) −8.38900 −0.321940
\(680\) 26.8810 1.03084
\(681\) −6.32844 −0.242506
\(682\) −10.3123 −0.394878
\(683\) −11.9995 −0.459148 −0.229574 0.973291i \(-0.573733\pi\)
−0.229574 + 0.973291i \(0.573733\pi\)
\(684\) 6.41973 0.245464
\(685\) 47.8579 1.82855
\(686\) −10.7107 −0.408936
\(687\) −1.09206 −0.0416648
\(688\) −1.69130 −0.0644802
\(689\) −10.3169 −0.393043
\(690\) 22.8221 0.868823
\(691\) −16.3492 −0.621954 −0.310977 0.950417i \(-0.600656\pi\)
−0.310977 + 0.950417i \(0.600656\pi\)
\(692\) −24.7683 −0.941549
\(693\) 1.68007 0.0638207
\(694\) −33.6190 −1.27616
\(695\) 64.1894 2.43484
\(696\) 8.16717 0.309576
\(697\) 91.9126 3.48144
\(698\) −0.865602 −0.0327635
\(699\) −6.15204 −0.232691
\(700\) −6.90934 −0.261149
\(701\) 33.1110 1.25059 0.625293 0.780390i \(-0.284980\pi\)
0.625293 + 0.780390i \(0.284980\pi\)
\(702\) 1.00000 0.0377426
\(703\) 48.5731 1.83197
\(704\) −2.09517 −0.0789648
\(705\) −5.17200 −0.194789
\(706\) −32.6448 −1.22860
\(707\) −4.30638 −0.161958
\(708\) 11.6675 0.438492
\(709\) −44.6576 −1.67715 −0.838576 0.544784i \(-0.816612\pi\)
−0.838576 + 0.544784i \(0.816612\pi\)
\(710\) 46.1562 1.73221
\(711\) −16.2346 −0.608845
\(712\) −3.88578 −0.145626
\(713\) 30.4410 1.14003
\(714\) −5.84147 −0.218612
\(715\) 7.73128 0.289133
\(716\) −13.6899 −0.511616
\(717\) −3.86903 −0.144491
\(718\) 13.5526 0.505777
\(719\) −33.5215 −1.25014 −0.625070 0.780568i \(-0.714930\pi\)
−0.625070 + 0.780568i \(0.714930\pi\)
\(720\) 3.69005 0.137520
\(721\) −0.801879 −0.0298635
\(722\) −22.2129 −0.826678
\(723\) 8.24640 0.306687
\(724\) −26.3059 −0.977652
\(725\) −70.3719 −2.61355
\(726\) 6.61026 0.245330
\(727\) 2.63467 0.0977143 0.0488572 0.998806i \(-0.484442\pi\)
0.0488572 + 0.998806i \(0.484442\pi\)
\(728\) −0.801879 −0.0297196
\(729\) 1.00000 0.0370370
\(730\) 35.3301 1.30762
\(731\) 12.3207 0.455696
\(732\) 3.80363 0.140586
\(733\) 3.95177 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(734\) 3.18662 0.117620
\(735\) −23.4576 −0.865246
\(736\) 6.18478 0.227974
\(737\) 3.82791 0.141003
\(738\) 12.6172 0.464444
\(739\) −45.8517 −1.68668 −0.843342 0.537377i \(-0.819415\pi\)
−0.843342 + 0.537377i \(0.819415\pi\)
\(740\) 27.9197 1.02635
\(741\) −6.41973 −0.235835
\(742\) 8.27291 0.303708
\(743\) 25.6074 0.939444 0.469722 0.882814i \(-0.344354\pi\)
0.469722 + 0.882814i \(0.344354\pi\)
\(744\) 4.92193 0.180447
\(745\) 48.3185 1.77025
\(746\) −12.2027 −0.446774
\(747\) −9.35341 −0.342223
\(748\) 15.2628 0.558062
\(749\) 4.68696 0.171258
\(750\) −13.3448 −0.487285
\(751\) −38.7288 −1.41323 −0.706617 0.707597i \(-0.749779\pi\)
−0.706617 + 0.707597i \(0.749779\pi\)
\(752\) −1.40161 −0.0511114
\(753\) −20.8393 −0.759427
\(754\) −8.16717 −0.297431
\(755\) −73.9137 −2.68999
\(756\) −0.801879 −0.0291641
\(757\) −10.4638 −0.380313 −0.190156 0.981754i \(-0.560899\pi\)
−0.190156 + 0.981754i \(0.560899\pi\)
\(758\) 4.04966 0.147090
\(759\) 12.9582 0.470352
\(760\) −23.6891 −0.859293
\(761\) −21.4493 −0.777536 −0.388768 0.921336i \(-0.627099\pi\)
−0.388768 + 0.921336i \(0.627099\pi\)
\(762\) −6.32994 −0.229310
\(763\) −15.0826 −0.546025
\(764\) 27.0533 0.978753
\(765\) −26.8810 −0.971884
\(766\) 8.52524 0.308030
\(767\) −11.6675 −0.421290
\(768\) 1.00000 0.0360844
\(769\) 26.1613 0.943401 0.471701 0.881759i \(-0.343640\pi\)
0.471701 + 0.881759i \(0.343640\pi\)
\(770\) −6.19955 −0.223416
\(771\) 16.9120 0.609072
\(772\) −6.88418 −0.247767
\(773\) 27.5564 0.991134 0.495567 0.868570i \(-0.334960\pi\)
0.495567 + 0.868570i \(0.334960\pi\)
\(774\) 1.69130 0.0607925
\(775\) −42.4095 −1.52339
\(776\) −10.4617 −0.375552
\(777\) −6.06720 −0.217660
\(778\) 2.55183 0.0914875
\(779\) −80.9987 −2.90208
\(780\) −3.69005 −0.132125
\(781\) 26.2070 0.937761
\(782\) −45.0544 −1.61114
\(783\) −8.16717 −0.291871
\(784\) −6.35699 −0.227035
\(785\) −68.5432 −2.44641
\(786\) 11.7846 0.420343
\(787\) 23.2778 0.829763 0.414882 0.909875i \(-0.363823\pi\)
0.414882 + 0.909875i \(0.363823\pi\)
\(788\) −12.1767 −0.433779
\(789\) 18.1348 0.645617
\(790\) 59.9065 2.13138
\(791\) 6.44995 0.229334
\(792\) 2.09517 0.0744487
\(793\) −3.80363 −0.135071
\(794\) 15.6618 0.555815
\(795\) 38.0699 1.35020
\(796\) 6.86514 0.243328
\(797\) 11.9373 0.422839 0.211420 0.977395i \(-0.432191\pi\)
0.211420 + 0.977395i \(0.432191\pi\)
\(798\) 5.14784 0.182232
\(799\) 10.2103 0.361216
\(800\) −8.61644 −0.304637
\(801\) 3.88578 0.137297
\(802\) 10.6910 0.377512
\(803\) 20.0601 0.707904
\(804\) −1.82701 −0.0644338
\(805\) 18.3006 0.645010
\(806\) −4.92193 −0.173368
\(807\) 0.636581 0.0224087
\(808\) −5.37036 −0.188929
\(809\) 46.2084 1.62460 0.812300 0.583239i \(-0.198215\pi\)
0.812300 + 0.583239i \(0.198215\pi\)
\(810\) −3.69005 −0.129655
\(811\) −23.2023 −0.814741 −0.407371 0.913263i \(-0.633554\pi\)
−0.407371 + 0.913263i \(0.633554\pi\)
\(812\) 6.54908 0.229828
\(813\) −6.65381 −0.233359
\(814\) 15.8525 0.555631
\(815\) 14.0787 0.493157
\(816\) −7.28473 −0.255017
\(817\) −10.8577 −0.379862
\(818\) −14.4409 −0.504913
\(819\) 0.801879 0.0280199
\(820\) −46.5579 −1.62587
\(821\) 14.9328 0.521157 0.260579 0.965453i \(-0.416087\pi\)
0.260579 + 0.965453i \(0.416087\pi\)
\(822\) −12.9694 −0.452361
\(823\) −17.3180 −0.603667 −0.301833 0.953361i \(-0.597599\pi\)
−0.301833 + 0.953361i \(0.597599\pi\)
\(824\) −1.00000 −0.0348367
\(825\) −18.0529 −0.628522
\(826\) 9.35594 0.325535
\(827\) −19.6705 −0.684010 −0.342005 0.939698i \(-0.611106\pi\)
−0.342005 + 0.939698i \(0.611106\pi\)
\(828\) −6.18478 −0.214936
\(829\) −21.9011 −0.760658 −0.380329 0.924851i \(-0.624189\pi\)
−0.380329 + 0.924851i \(0.624189\pi\)
\(830\) 34.5145 1.19802
\(831\) −18.6247 −0.646085
\(832\) −1.00000 −0.0346688
\(833\) 46.3090 1.60451
\(834\) −17.3953 −0.602349
\(835\) −43.5457 −1.50696
\(836\) −13.4504 −0.465193
\(837\) −4.92193 −0.170127
\(838\) 14.0266 0.484540
\(839\) −57.0194 −1.96853 −0.984264 0.176705i \(-0.943456\pi\)
−0.984264 + 0.176705i \(0.943456\pi\)
\(840\) 2.95897 0.102094
\(841\) 37.7026 1.30009
\(842\) −21.9429 −0.756201
\(843\) −15.8147 −0.544688
\(844\) −25.0346 −0.861727
\(845\) 3.69005 0.126941
\(846\) 1.40161 0.0481883
\(847\) 5.30063 0.182132
\(848\) 10.3169 0.354284
\(849\) 1.14296 0.0392264
\(850\) 62.7685 2.15294
\(851\) −46.7954 −1.60413
\(852\) −12.5083 −0.428527
\(853\) 29.7693 1.01928 0.509641 0.860387i \(-0.329778\pi\)
0.509641 + 0.860387i \(0.329778\pi\)
\(854\) 3.05005 0.104371
\(855\) 23.6891 0.810150
\(856\) 5.84497 0.199777
\(857\) 14.0815 0.481014 0.240507 0.970647i \(-0.422686\pi\)
0.240507 + 0.970647i \(0.422686\pi\)
\(858\) −2.09517 −0.0715280
\(859\) −36.9848 −1.26190 −0.630952 0.775822i \(-0.717335\pi\)
−0.630952 + 0.775822i \(0.717335\pi\)
\(860\) −6.24097 −0.212815
\(861\) 10.1174 0.344801
\(862\) −1.88113 −0.0640715
\(863\) −39.8626 −1.35694 −0.678470 0.734628i \(-0.737357\pi\)
−0.678470 + 0.734628i \(0.737357\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −91.3961 −3.10756
\(866\) −15.2492 −0.518190
\(867\) 36.0673 1.22491
\(868\) 3.94679 0.133963
\(869\) 34.0143 1.15386
\(870\) 30.1372 1.02175
\(871\) 1.82701 0.0619060
\(872\) −18.8090 −0.636954
\(873\) 10.4617 0.354074
\(874\) 39.7046 1.34303
\(875\) −10.7009 −0.361758
\(876\) −9.57442 −0.323490
\(877\) 56.4669 1.90675 0.953375 0.301787i \(-0.0975831\pi\)
0.953375 + 0.301787i \(0.0975831\pi\)
\(878\) −38.1006 −1.28583
\(879\) 19.8586 0.669815
\(880\) −7.73128 −0.260621
\(881\) 13.1832 0.444153 0.222077 0.975029i \(-0.428716\pi\)
0.222077 + 0.975029i \(0.428716\pi\)
\(882\) 6.35699 0.214051
\(883\) −36.4169 −1.22553 −0.612764 0.790266i \(-0.709942\pi\)
−0.612764 + 0.790266i \(0.709942\pi\)
\(884\) 7.28473 0.245012
\(885\) 43.0537 1.44723
\(886\) 2.12579 0.0714173
\(887\) 1.43736 0.0482619 0.0241310 0.999709i \(-0.492318\pi\)
0.0241310 + 0.999709i \(0.492318\pi\)
\(888\) −7.56623 −0.253906
\(889\) −5.07585 −0.170238
\(890\) −14.3387 −0.480635
\(891\) −2.09517 −0.0701909
\(892\) 11.4805 0.384394
\(893\) −8.99795 −0.301105
\(894\) −13.0943 −0.437938
\(895\) −50.5164 −1.68858
\(896\) 0.801879 0.0267889
\(897\) 6.18478 0.206504
\(898\) 22.5617 0.752895
\(899\) 40.1982 1.34069
\(900\) 8.61644 0.287215
\(901\) −75.1559 −2.50381
\(902\) −26.4351 −0.880193
\(903\) 1.35622 0.0451321
\(904\) 8.04354 0.267524
\(905\) −97.0701 −3.22672
\(906\) 20.0305 0.665470
\(907\) −32.2421 −1.07058 −0.535290 0.844668i \(-0.679798\pi\)
−0.535290 + 0.844668i \(0.679798\pi\)
\(908\) −6.32844 −0.210017
\(909\) 5.37036 0.178124
\(910\) −2.95897 −0.0980889
\(911\) 9.72458 0.322190 0.161095 0.986939i \(-0.448497\pi\)
0.161095 + 0.986939i \(0.448497\pi\)
\(912\) 6.41973 0.212578
\(913\) 19.5970 0.648566
\(914\) 28.9729 0.958338
\(915\) 14.0356 0.464002
\(916\) −1.09206 −0.0360828
\(917\) 9.44983 0.312061
\(918\) 7.28473 0.240432
\(919\) 12.4522 0.410760 0.205380 0.978682i \(-0.434157\pi\)
0.205380 + 0.978682i \(0.434157\pi\)
\(920\) 22.8221 0.752423
\(921\) 14.4202 0.475161
\(922\) 26.4125 0.869851
\(923\) 12.5083 0.411715
\(924\) 1.68007 0.0552704
\(925\) 65.1940 2.14356
\(926\) 32.9808 1.08382
\(927\) 1.00000 0.0328443
\(928\) 8.16717 0.268100
\(929\) 4.97961 0.163376 0.0816878 0.996658i \(-0.473969\pi\)
0.0816878 + 0.996658i \(0.473969\pi\)
\(930\) 18.1621 0.595560
\(931\) −40.8101 −1.33750
\(932\) −6.15204 −0.201517
\(933\) −16.9322 −0.554336
\(934\) −22.2234 −0.727172
\(935\) 56.3203 1.84187
\(936\) 1.00000 0.0326860
\(937\) −3.07753 −0.100539 −0.0502693 0.998736i \(-0.516008\pi\)
−0.0502693 + 0.998736i \(0.516008\pi\)
\(938\) −1.46504 −0.0478354
\(939\) 8.02812 0.261988
\(940\) −5.17200 −0.168692
\(941\) −40.1334 −1.30831 −0.654156 0.756360i \(-0.726976\pi\)
−0.654156 + 0.756360i \(0.726976\pi\)
\(942\) 18.5752 0.605212
\(943\) 78.0343 2.54115
\(944\) 11.6675 0.379746
\(945\) −2.95897 −0.0962553
\(946\) −3.54356 −0.115211
\(947\) 0.788239 0.0256143 0.0128072 0.999918i \(-0.495923\pi\)
0.0128072 + 0.999918i \(0.495923\pi\)
\(948\) −16.2346 −0.527276
\(949\) 9.57442 0.310799
\(950\) −55.3152 −1.79466
\(951\) 26.3844 0.855572
\(952\) −5.84147 −0.189323
\(953\) −50.7289 −1.64327 −0.821636 0.570013i \(-0.806938\pi\)
−0.821636 + 0.570013i \(0.806938\pi\)
\(954\) −10.3169 −0.334022
\(955\) 99.8278 3.23035
\(956\) −3.86903 −0.125133
\(957\) 17.1116 0.553140
\(958\) −12.2928 −0.397163
\(959\) −10.3999 −0.335831
\(960\) 3.69005 0.119096
\(961\) −6.77463 −0.218536
\(962\) 7.56623 0.243945
\(963\) −5.84497 −0.188351
\(964\) 8.24640 0.265599
\(965\) −25.4030 −0.817750
\(966\) −4.95944 −0.159567
\(967\) −30.7771 −0.989724 −0.494862 0.868972i \(-0.664781\pi\)
−0.494862 + 0.868972i \(0.664781\pi\)
\(968\) 6.61026 0.212462
\(969\) −46.7660 −1.50234
\(970\) −38.6041 −1.23950
\(971\) −34.1303 −1.09529 −0.547647 0.836709i \(-0.684476\pi\)
−0.547647 + 0.836709i \(0.684476\pi\)
\(972\) 1.00000 0.0320750
\(973\) −13.9489 −0.447181
\(974\) −11.6049 −0.371844
\(975\) −8.61644 −0.275947
\(976\) 3.80363 0.121751
\(977\) −7.28451 −0.233052 −0.116526 0.993188i \(-0.537176\pi\)
−0.116526 + 0.993188i \(0.537176\pi\)
\(978\) −3.81533 −0.122001
\(979\) −8.14138 −0.260200
\(980\) −23.4576 −0.749325
\(981\) 18.8090 0.600526
\(982\) −11.2148 −0.357877
\(983\) −45.7382 −1.45882 −0.729411 0.684076i \(-0.760206\pi\)
−0.729411 + 0.684076i \(0.760206\pi\)
\(984\) 12.6172 0.402220
\(985\) −44.9328 −1.43168
\(986\) −59.4956 −1.89473
\(987\) 1.12392 0.0357748
\(988\) −6.41973 −0.204239
\(989\) 10.4603 0.332619
\(990\) 7.73128 0.245716
\(991\) 19.9232 0.632882 0.316441 0.948612i \(-0.397512\pi\)
0.316441 + 0.948612i \(0.397512\pi\)
\(992\) 4.92193 0.156271
\(993\) 19.3054 0.612637
\(994\) −10.0301 −0.318137
\(995\) 25.3327 0.803100
\(996\) −9.35341 −0.296374
\(997\) 47.5548 1.50608 0.753038 0.657977i \(-0.228588\pi\)
0.753038 + 0.657977i \(0.228588\pi\)
\(998\) 8.47929 0.268407
\(999\) 7.56623 0.239385
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 8034.2.a.bb.1.14 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8034.2.a.bb.1.14 14 1.1 even 1 trivial