Properties

Label 69.6.a.e.1.3
Level $69$
Weight $6$
Character 69.1
Self dual yes
Analytic conductor $11.066$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 69.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.0664835671\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \( x^{5} - 113x^{3} - 257x^{2} + 1404x + 2197 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-7.90234\) of defining polynomial
Character \(\chi\) \(=\) 69.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.24792 q^{2} +9.00000 q^{3} -26.9469 q^{4} +53.3906 q^{5} +20.2313 q^{6} +89.8688 q^{7} -132.508 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+2.24792 q^{2} +9.00000 q^{3} -26.9469 q^{4} +53.3906 q^{5} +20.2313 q^{6} +89.8688 q^{7} -132.508 q^{8} +81.0000 q^{9} +120.018 q^{10} +225.013 q^{11} -242.522 q^{12} +725.747 q^{13} +202.018 q^{14} +480.515 q^{15} +564.433 q^{16} +44.9540 q^{17} +182.081 q^{18} +1212.50 q^{19} -1438.71 q^{20} +808.819 q^{21} +505.810 q^{22} -529.000 q^{23} -1192.57 q^{24} -274.446 q^{25} +1631.42 q^{26} +729.000 q^{27} -2421.68 q^{28} +2592.49 q^{29} +1080.16 q^{30} +585.421 q^{31} +5509.05 q^{32} +2025.11 q^{33} +101.053 q^{34} +4798.15 q^{35} -2182.70 q^{36} -3849.04 q^{37} +2725.59 q^{38} +6531.73 q^{39} -7074.66 q^{40} -4299.92 q^{41} +1818.16 q^{42} -20567.4 q^{43} -6063.39 q^{44} +4324.64 q^{45} -1189.15 q^{46} -5221.50 q^{47} +5079.90 q^{48} -8730.61 q^{49} -616.931 q^{50} +404.586 q^{51} -19556.6 q^{52} -15753.7 q^{53} +1638.73 q^{54} +12013.6 q^{55} -11908.3 q^{56} +10912.5 q^{57} +5827.70 q^{58} -13780.6 q^{59} -12948.4 q^{60} -18988.5 q^{61} +1315.98 q^{62} +7279.37 q^{63} -5677.98 q^{64} +38748.1 q^{65} +4552.29 q^{66} +1951.43 q^{67} -1211.37 q^{68} -4761.00 q^{69} +10785.8 q^{70} +75281.4 q^{71} -10733.1 q^{72} -4177.11 q^{73} -8652.32 q^{74} -2470.01 q^{75} -32673.0 q^{76} +20221.6 q^{77} +14682.8 q^{78} -95896.7 q^{79} +30135.4 q^{80} +6561.00 q^{81} -9665.87 q^{82} +13366.7 q^{83} -21795.1 q^{84} +2400.12 q^{85} -46233.9 q^{86} +23332.4 q^{87} -29815.9 q^{88} +50932.1 q^{89} +9721.43 q^{90} +65222.0 q^{91} +14254.9 q^{92} +5268.79 q^{93} -11737.5 q^{94} +64735.9 q^{95} +49581.4 q^{96} +44280.7 q^{97} -19625.7 q^{98} +18226.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 8 q^{2} + 45 q^{3} + 118 q^{4} + 94 q^{5} + 72 q^{6} + 272 q^{7} + 258 q^{8} + 405 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 8 q^{2} + 45 q^{3} + 118 q^{4} + 94 q^{5} + 72 q^{6} + 272 q^{7} + 258 q^{8} + 405 q^{9} - 172 q^{10} + 1100 q^{11} + 1062 q^{12} - 978 q^{13} - 344 q^{14} + 846 q^{15} + 1218 q^{16} + 2522 q^{17} + 648 q^{18} + 2060 q^{19} + 7720 q^{20} + 2448 q^{21} - 2572 q^{22} - 2645 q^{23} + 2322 q^{24} + 12035 q^{25} + 9280 q^{26} + 3645 q^{27} + 8072 q^{28} + 1526 q^{29} - 1548 q^{30} - 7392 q^{31} - 5086 q^{32} + 9900 q^{33} - 15608 q^{34} + 6056 q^{35} + 9558 q^{36} - 8210 q^{37} - 14276 q^{38} - 8802 q^{39} - 37472 q^{40} + 21250 q^{41} - 3096 q^{42} - 4548 q^{43} - 4260 q^{44} + 7614 q^{45} - 4232 q^{46} + 536 q^{47} + 10962 q^{48} - 27979 q^{49} - 81872 q^{50} + 22698 q^{51} - 76380 q^{52} - 11482 q^{53} + 5832 q^{54} - 77064 q^{55} - 28624 q^{56} + 18540 q^{57} - 79680 q^{58} + 74676 q^{59} + 69480 q^{60} - 44618 q^{61} + 64880 q^{62} + 22032 q^{63} - 137382 q^{64} - 24388 q^{65} - 23148 q^{66} - 1412 q^{67} + 80196 q^{68} - 23805 q^{69} - 222304 q^{70} + 37912 q^{71} + 20898 q^{72} + 46546 q^{73} + 111604 q^{74} + 108315 q^{75} - 79548 q^{76} + 157008 q^{77} + 83520 q^{78} + 50544 q^{79} + 69424 q^{80} + 32805 q^{81} - 233720 q^{82} + 89588 q^{83} + 72648 q^{84} + 147892 q^{85} + 77428 q^{86} + 13734 q^{87} + 54484 q^{88} + 280410 q^{89} - 13932 q^{90} - 27416 q^{91} - 62422 q^{92} - 66528 q^{93} + 113632 q^{94} + 203120 q^{95} - 45774 q^{96} + 90074 q^{97} + 32976 q^{98} + 89100 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.24792 0.397380 0.198690 0.980062i \(-0.436331\pi\)
0.198690 + 0.980062i \(0.436331\pi\)
\(3\) 9.00000 0.577350
\(4\) −26.9469 −0.842090
\(5\) 53.3906 0.955080 0.477540 0.878610i \(-0.341529\pi\)
0.477540 + 0.878610i \(0.341529\pi\)
\(6\) 20.2313 0.229427
\(7\) 89.8688 0.693208 0.346604 0.938012i \(-0.387335\pi\)
0.346604 + 0.938012i \(0.387335\pi\)
\(8\) −132.508 −0.732009
\(9\) 81.0000 0.333333
\(10\) 120.018 0.379529
\(11\) 225.013 0.560693 0.280347 0.959899i \(-0.409551\pi\)
0.280347 + 0.959899i \(0.409551\pi\)
\(12\) −242.522 −0.486181
\(13\) 725.747 1.19104 0.595521 0.803340i \(-0.296946\pi\)
0.595521 + 0.803340i \(0.296946\pi\)
\(14\) 202.018 0.275467
\(15\) 480.515 0.551416
\(16\) 564.433 0.551204
\(17\) 44.9540 0.0377265 0.0188632 0.999822i \(-0.493995\pi\)
0.0188632 + 0.999822i \(0.493995\pi\)
\(18\) 182.081 0.132460
\(19\) 1212.50 0.770542 0.385271 0.922803i \(-0.374108\pi\)
0.385271 + 0.922803i \(0.374108\pi\)
\(20\) −1438.71 −0.804263
\(21\) 808.819 0.400224
\(22\) 505.810 0.222808
\(23\) −529.000 −0.208514
\(24\) −1192.57 −0.422625
\(25\) −274.446 −0.0878226
\(26\) 1631.42 0.473296
\(27\) 729.000 0.192450
\(28\) −2421.68 −0.583743
\(29\) 2592.49 0.572430 0.286215 0.958165i \(-0.407603\pi\)
0.286215 + 0.958165i \(0.407603\pi\)
\(30\) 1080.16 0.219121
\(31\) 585.421 0.109412 0.0547059 0.998503i \(-0.482578\pi\)
0.0547059 + 0.998503i \(0.482578\pi\)
\(32\) 5509.05 0.951046
\(33\) 2025.11 0.323716
\(34\) 101.053 0.0149917
\(35\) 4798.15 0.662069
\(36\) −2182.70 −0.280697
\(37\) −3849.04 −0.462219 −0.231109 0.972928i \(-0.574236\pi\)
−0.231109 + 0.972928i \(0.574236\pi\)
\(38\) 2725.59 0.306198
\(39\) 6531.73 0.687648
\(40\) −7074.66 −0.699127
\(41\) −4299.92 −0.399485 −0.199743 0.979848i \(-0.564011\pi\)
−0.199743 + 0.979848i \(0.564011\pi\)
\(42\) 1818.16 0.159041
\(43\) −20567.4 −1.69632 −0.848162 0.529737i \(-0.822291\pi\)
−0.848162 + 0.529737i \(0.822291\pi\)
\(44\) −6063.39 −0.472154
\(45\) 4324.64 0.318360
\(46\) −1189.15 −0.0828594
\(47\) −5221.50 −0.344787 −0.172393 0.985028i \(-0.555150\pi\)
−0.172393 + 0.985028i \(0.555150\pi\)
\(48\) 5079.90 0.318238
\(49\) −8730.61 −0.519462
\(50\) −616.931 −0.0348989
\(51\) 404.586 0.0217814
\(52\) −19556.6 −1.00296
\(53\) −15753.7 −0.770360 −0.385180 0.922841i \(-0.625861\pi\)
−0.385180 + 0.922841i \(0.625861\pi\)
\(54\) 1638.73 0.0764757
\(55\) 12013.6 0.535507
\(56\) −11908.3 −0.507434
\(57\) 10912.5 0.444873
\(58\) 5827.70 0.227472
\(59\) −13780.6 −0.515392 −0.257696 0.966226i \(-0.582963\pi\)
−0.257696 + 0.966226i \(0.582963\pi\)
\(60\) −12948.4 −0.464341
\(61\) −18988.5 −0.653380 −0.326690 0.945131i \(-0.605933\pi\)
−0.326690 + 0.945131i \(0.605933\pi\)
\(62\) 1315.98 0.0434780
\(63\) 7279.37 0.231069
\(64\) −5677.98 −0.173278
\(65\) 38748.1 1.13754
\(66\) 4552.29 0.128638
\(67\) 1951.43 0.0531088 0.0265544 0.999647i \(-0.491546\pi\)
0.0265544 + 0.999647i \(0.491546\pi\)
\(68\) −1211.37 −0.0317691
\(69\) −4761.00 −0.120386
\(70\) 10785.8 0.263093
\(71\) 75281.4 1.77232 0.886160 0.463380i \(-0.153363\pi\)
0.886160 + 0.463380i \(0.153363\pi\)
\(72\) −10733.1 −0.244003
\(73\) −4177.11 −0.0917422 −0.0458711 0.998947i \(-0.514606\pi\)
−0.0458711 + 0.998947i \(0.514606\pi\)
\(74\) −8652.32 −0.183676
\(75\) −2470.01 −0.0507044
\(76\) −32673.0 −0.648865
\(77\) 20221.6 0.388677
\(78\) 14682.8 0.273257
\(79\) −95896.7 −1.72877 −0.864383 0.502835i \(-0.832290\pi\)
−0.864383 + 0.502835i \(0.832290\pi\)
\(80\) 30135.4 0.526444
\(81\) 6561.00 0.111111
\(82\) −9665.87 −0.158747
\(83\) 13366.7 0.212976 0.106488 0.994314i \(-0.466039\pi\)
0.106488 + 0.994314i \(0.466039\pi\)
\(84\) −21795.1 −0.337024
\(85\) 2400.12 0.0360318
\(86\) −46233.9 −0.674084
\(87\) 23332.4 0.330492
\(88\) −29815.9 −0.410432
\(89\) 50932.1 0.681580 0.340790 0.940139i \(-0.389305\pi\)
0.340790 + 0.940139i \(0.389305\pi\)
\(90\) 9721.43 0.126510
\(91\) 65222.0 0.825640
\(92\) 14254.9 0.175588
\(93\) 5268.79 0.0631690
\(94\) −11737.5 −0.137011
\(95\) 64735.9 0.735929
\(96\) 49581.4 0.549087
\(97\) 44280.7 0.477843 0.238921 0.971039i \(-0.423206\pi\)
0.238921 + 0.971039i \(0.423206\pi\)
\(98\) −19625.7 −0.206424
\(99\) 18226.0 0.186898
\(100\) 7395.45 0.0739545
\(101\) 198402. 1.93527 0.967637 0.252346i \(-0.0812021\pi\)
0.967637 + 0.252346i \(0.0812021\pi\)
\(102\) 909.476 0.00865548
\(103\) 201915. 1.87532 0.937658 0.347559i \(-0.112989\pi\)
0.937658 + 0.347559i \(0.112989\pi\)
\(104\) −96167.1 −0.871853
\(105\) 43183.3 0.382246
\(106\) −35413.1 −0.306125
\(107\) 37688.4 0.318235 0.159117 0.987260i \(-0.449135\pi\)
0.159117 + 0.987260i \(0.449135\pi\)
\(108\) −19644.3 −0.162060
\(109\) −86493.8 −0.697298 −0.348649 0.937253i \(-0.613360\pi\)
−0.348649 + 0.937253i \(0.613360\pi\)
\(110\) 27005.5 0.212799
\(111\) −34641.3 −0.266862
\(112\) 50724.9 0.382099
\(113\) −150822. −1.11114 −0.555571 0.831469i \(-0.687500\pi\)
−0.555571 + 0.831469i \(0.687500\pi\)
\(114\) 24530.3 0.176783
\(115\) −28243.6 −0.199148
\(116\) −69859.5 −0.482037
\(117\) 58785.5 0.397014
\(118\) −30977.6 −0.204806
\(119\) 4039.96 0.0261523
\(120\) −63672.0 −0.403641
\(121\) −110420. −0.685623
\(122\) −42684.6 −0.259640
\(123\) −38699.3 −0.230643
\(124\) −15775.3 −0.0921346
\(125\) −181498. −1.03896
\(126\) 16363.4 0.0918222
\(127\) −47605.4 −0.261907 −0.130953 0.991389i \(-0.541804\pi\)
−0.130953 + 0.991389i \(0.541804\pi\)
\(128\) −189053. −1.01990
\(129\) −185107. −0.979373
\(130\) 87102.5 0.452035
\(131\) −260443. −1.32597 −0.662985 0.748632i \(-0.730711\pi\)
−0.662985 + 0.748632i \(0.730711\pi\)
\(132\) −54570.5 −0.272598
\(133\) 108966. 0.534146
\(134\) 4386.66 0.0211043
\(135\) 38921.7 0.183805
\(136\) −5956.75 −0.0276161
\(137\) 121613. 0.553580 0.276790 0.960930i \(-0.410729\pi\)
0.276790 + 0.960930i \(0.410729\pi\)
\(138\) −10702.3 −0.0478389
\(139\) 61341.2 0.269287 0.134643 0.990894i \(-0.457011\pi\)
0.134643 + 0.990894i \(0.457011\pi\)
\(140\) −129295. −0.557521
\(141\) −46993.5 −0.199063
\(142\) 169227. 0.704284
\(143\) 163302. 0.667809
\(144\) 45719.1 0.183735
\(145\) 138415. 0.546716
\(146\) −9389.81 −0.0364565
\(147\) −78575.4 −0.299912
\(148\) 103719. 0.389230
\(149\) −171257. −0.631950 −0.315975 0.948767i \(-0.602332\pi\)
−0.315975 + 0.948767i \(0.602332\pi\)
\(150\) −5552.38 −0.0201489
\(151\) −459607. −1.64038 −0.820189 0.572092i \(-0.806132\pi\)
−0.820189 + 0.572092i \(0.806132\pi\)
\(152\) −160665. −0.564043
\(153\) 3641.27 0.0125755
\(154\) 45456.5 0.154452
\(155\) 31256.0 0.104497
\(156\) −176010. −0.579061
\(157\) 387631. 1.25508 0.627538 0.778586i \(-0.284063\pi\)
0.627538 + 0.778586i \(0.284063\pi\)
\(158\) −215568. −0.686976
\(159\) −141784. −0.444767
\(160\) 294131. 0.908325
\(161\) −47540.6 −0.144544
\(162\) 14748.6 0.0441533
\(163\) −286971. −0.845998 −0.422999 0.906130i \(-0.639023\pi\)
−0.422999 + 0.906130i \(0.639023\pi\)
\(164\) 115869. 0.336402
\(165\) 108122. 0.309175
\(166\) 30047.3 0.0846322
\(167\) 580711. 1.61127 0.805636 0.592411i \(-0.201824\pi\)
0.805636 + 0.592411i \(0.201824\pi\)
\(168\) −107175. −0.292967
\(169\) 155416. 0.418581
\(170\) 5395.27 0.0143183
\(171\) 98212.2 0.256847
\(172\) 554228. 1.42846
\(173\) −191878. −0.487427 −0.243714 0.969847i \(-0.578366\pi\)
−0.243714 + 0.969847i \(0.578366\pi\)
\(174\) 52449.3 0.131331
\(175\) −24664.1 −0.0608793
\(176\) 127005. 0.309056
\(177\) −124025. −0.297562
\(178\) 114491. 0.270846
\(179\) 18757.8 0.0437571 0.0218785 0.999761i \(-0.493035\pi\)
0.0218785 + 0.999761i \(0.493035\pi\)
\(180\) −116535. −0.268088
\(181\) 47967.6 0.108831 0.0544154 0.998518i \(-0.482670\pi\)
0.0544154 + 0.998518i \(0.482670\pi\)
\(182\) 146614. 0.328092
\(183\) −170897. −0.377229
\(184\) 70096.6 0.152634
\(185\) −205502. −0.441456
\(186\) 11843.8 0.0251021
\(187\) 10115.2 0.0211530
\(188\) 140703. 0.290341
\(189\) 65514.3 0.133408
\(190\) 145521. 0.292443
\(191\) −14294.3 −0.0283518 −0.0141759 0.999900i \(-0.504512\pi\)
−0.0141759 + 0.999900i \(0.504512\pi\)
\(192\) −51101.8 −0.100042
\(193\) −422764. −0.816968 −0.408484 0.912765i \(-0.633942\pi\)
−0.408484 + 0.912765i \(0.633942\pi\)
\(194\) 99539.3 0.189885
\(195\) 348733. 0.656759
\(196\) 235262. 0.437434
\(197\) −311005. −0.570955 −0.285477 0.958385i \(-0.592152\pi\)
−0.285477 + 0.958385i \(0.592152\pi\)
\(198\) 40970.6 0.0742693
\(199\) 60066.4 0.107522 0.0537612 0.998554i \(-0.482879\pi\)
0.0537612 + 0.998554i \(0.482879\pi\)
\(200\) 36366.2 0.0642869
\(201\) 17562.9 0.0306624
\(202\) 445991. 0.769038
\(203\) 232984. 0.396813
\(204\) −10902.3 −0.0183419
\(205\) −229575. −0.381540
\(206\) 453887. 0.745212
\(207\) −42849.0 −0.0695048
\(208\) 409636. 0.656507
\(209\) 272827. 0.432038
\(210\) 97072.5 0.151897
\(211\) −1.08896e6 −1.68385 −0.841926 0.539593i \(-0.818578\pi\)
−0.841926 + 0.539593i \(0.818578\pi\)
\(212\) 424514. 0.648712
\(213\) 677533. 1.02325
\(214\) 84720.3 0.126460
\(215\) −1.09811e6 −1.62012
\(216\) −96598.1 −0.140875
\(217\) 52611.1 0.0758452
\(218\) −194431. −0.277092
\(219\) −37594.0 −0.0529674
\(220\) −323728. −0.450945
\(221\) 32625.2 0.0449338
\(222\) −77870.9 −0.106046
\(223\) 365915. 0.492741 0.246370 0.969176i \(-0.420762\pi\)
0.246370 + 0.969176i \(0.420762\pi\)
\(224\) 495091. 0.659273
\(225\) −22230.1 −0.0292742
\(226\) −339036. −0.441545
\(227\) 1.08830e6 1.40179 0.700894 0.713265i \(-0.252784\pi\)
0.700894 + 0.713265i \(0.252784\pi\)
\(228\) −294057. −0.374623
\(229\) −1.45107e6 −1.82853 −0.914263 0.405121i \(-0.867229\pi\)
−0.914263 + 0.405121i \(0.867229\pi\)
\(230\) −63489.3 −0.0791373
\(231\) 181994. 0.224403
\(232\) −343525. −0.419023
\(233\) −1.27334e6 −1.53657 −0.768286 0.640106i \(-0.778890\pi\)
−0.768286 + 0.640106i \(0.778890\pi\)
\(234\) 132145. 0.157765
\(235\) −278779. −0.329299
\(236\) 371344. 0.434006
\(237\) −863070. −0.998103
\(238\) 9081.50 0.0103924
\(239\) 799348. 0.905193 0.452596 0.891715i \(-0.350498\pi\)
0.452596 + 0.891715i \(0.350498\pi\)
\(240\) 271219. 0.303943
\(241\) 727560. 0.806913 0.403456 0.914999i \(-0.367809\pi\)
0.403456 + 0.914999i \(0.367809\pi\)
\(242\) −248216. −0.272453
\(243\) 59049.0 0.0641500
\(244\) 511681. 0.550205
\(245\) −466132. −0.496128
\(246\) −86992.8 −0.0916528
\(247\) 879966. 0.917748
\(248\) −77572.9 −0.0800904
\(249\) 120301. 0.122962
\(250\) −407993. −0.412860
\(251\) 960924. 0.962730 0.481365 0.876520i \(-0.340141\pi\)
0.481365 + 0.876520i \(0.340141\pi\)
\(252\) −196156. −0.194581
\(253\) −119032. −0.116913
\(254\) −107013. −0.104076
\(255\) 21601.1 0.0208030
\(256\) −243281. −0.232010
\(257\) −1.57691e6 −1.48927 −0.744637 0.667470i \(-0.767377\pi\)
−0.744637 + 0.667470i \(0.767377\pi\)
\(258\) −416105. −0.389183
\(259\) −345908. −0.320414
\(260\) −1.04414e6 −0.957911
\(261\) 209992. 0.190810
\(262\) −585454. −0.526914
\(263\) 1.42682e6 1.27198 0.635991 0.771696i \(-0.280591\pi\)
0.635991 + 0.771696i \(0.280591\pi\)
\(264\) −268343. −0.236963
\(265\) −841101. −0.735755
\(266\) 244946. 0.212259
\(267\) 458389. 0.393510
\(268\) −52585.0 −0.0447224
\(269\) −341906. −0.288089 −0.144044 0.989571i \(-0.546011\pi\)
−0.144044 + 0.989571i \(0.546011\pi\)
\(270\) 87492.9 0.0730404
\(271\) 723709. 0.598605 0.299303 0.954158i \(-0.403246\pi\)
0.299303 + 0.954158i \(0.403246\pi\)
\(272\) 25373.5 0.0207950
\(273\) 586998. 0.476683
\(274\) 273377. 0.219981
\(275\) −61753.7 −0.0492415
\(276\) 128294. 0.101376
\(277\) −1.82928e6 −1.43245 −0.716226 0.697868i \(-0.754132\pi\)
−0.716226 + 0.697868i \(0.754132\pi\)
\(278\) 137890. 0.107009
\(279\) 47419.1 0.0364706
\(280\) −635791. −0.484640
\(281\) 1.14935e6 0.868337 0.434169 0.900832i \(-0.357042\pi\)
0.434169 + 0.900832i \(0.357042\pi\)
\(282\) −105638. −0.0791035
\(283\) −1.83262e6 −1.36021 −0.680105 0.733115i \(-0.738066\pi\)
−0.680105 + 0.733115i \(0.738066\pi\)
\(284\) −2.02860e6 −1.49245
\(285\) 582623. 0.424889
\(286\) 367090. 0.265374
\(287\) −386429. −0.276926
\(288\) 446233. 0.317015
\(289\) −1.41784e6 −0.998577
\(290\) 311145. 0.217254
\(291\) 398526. 0.275883
\(292\) 112560. 0.0772552
\(293\) 1.33169e6 0.906221 0.453111 0.891454i \(-0.350314\pi\)
0.453111 + 0.891454i \(0.350314\pi\)
\(294\) −176631. −0.119179
\(295\) −735754. −0.492241
\(296\) 510027. 0.338348
\(297\) 164034. 0.107905
\(298\) −384972. −0.251124
\(299\) −383920. −0.248349
\(300\) 66559.0 0.0426976
\(301\) −1.84837e6 −1.17591
\(302\) −1.03316e6 −0.651853
\(303\) 1.78562e6 1.11733
\(304\) 684373. 0.424726
\(305\) −1.01381e6 −0.624031
\(306\) 8185.29 0.00499724
\(307\) 3.00299e6 1.81848 0.909239 0.416275i \(-0.136665\pi\)
0.909239 + 0.416275i \(0.136665\pi\)
\(308\) −544909. −0.327301
\(309\) 1.81723e6 1.08271
\(310\) 70260.9 0.0415250
\(311\) 2.75093e6 1.61279 0.806397 0.591374i \(-0.201414\pi\)
0.806397 + 0.591374i \(0.201414\pi\)
\(312\) −865504. −0.503364
\(313\) −119894. −0.0691727 −0.0345864 0.999402i \(-0.511011\pi\)
−0.0345864 + 0.999402i \(0.511011\pi\)
\(314\) 871364. 0.498741
\(315\) 388650. 0.220690
\(316\) 2.58412e6 1.45578
\(317\) 514254. 0.287428 0.143714 0.989619i \(-0.454095\pi\)
0.143714 + 0.989619i \(0.454095\pi\)
\(318\) −318718. −0.176741
\(319\) 583343. 0.320957
\(320\) −303151. −0.165494
\(321\) 339195. 0.183733
\(322\) −106867. −0.0574388
\(323\) 54506.6 0.0290698
\(324\) −176798. −0.0935655
\(325\) −199178. −0.104600
\(326\) −645088. −0.336182
\(327\) −778444. −0.402585
\(328\) 569773. 0.292427
\(329\) −469250. −0.239009
\(330\) 243049. 0.122860
\(331\) 2.46021e6 1.23425 0.617124 0.786866i \(-0.288298\pi\)
0.617124 + 0.786866i \(0.288298\pi\)
\(332\) −360192. −0.179345
\(333\) −311772. −0.154073
\(334\) 1.30539e6 0.640286
\(335\) 104188. 0.0507231
\(336\) 456524. 0.220605
\(337\) −1.42694e6 −0.684432 −0.342216 0.939621i \(-0.611177\pi\)
−0.342216 + 0.939621i \(0.611177\pi\)
\(338\) 349363. 0.166335
\(339\) −1.35740e6 −0.641518
\(340\) −64675.7 −0.0303420
\(341\) 131727. 0.0613465
\(342\) 220773. 0.102066
\(343\) −2.29503e6 −1.05330
\(344\) 2.72534e6 1.24172
\(345\) −254193. −0.114978
\(346\) −431326. −0.193694
\(347\) 1.63440e6 0.728676 0.364338 0.931267i \(-0.381295\pi\)
0.364338 + 0.931267i \(0.381295\pi\)
\(348\) −628735. −0.278304
\(349\) −2.65498e6 −1.16680 −0.583402 0.812184i \(-0.698279\pi\)
−0.583402 + 0.812184i \(0.698279\pi\)
\(350\) −55442.8 −0.0241922
\(351\) 529070. 0.229216
\(352\) 1.23961e6 0.533245
\(353\) 934019. 0.398950 0.199475 0.979903i \(-0.436076\pi\)
0.199475 + 0.979903i \(0.436076\pi\)
\(354\) −278799. −0.118245
\(355\) 4.01932e6 1.69271
\(356\) −1.37246e6 −0.573952
\(357\) 36359.7 0.0150990
\(358\) 42165.9 0.0173882
\(359\) 2.98227e6 1.22127 0.610634 0.791913i \(-0.290915\pi\)
0.610634 + 0.791913i \(0.290915\pi\)
\(360\) −573048. −0.233042
\(361\) −1.00595e6 −0.406265
\(362\) 107827. 0.0432471
\(363\) −993783. −0.395845
\(364\) −1.75753e6 −0.695263
\(365\) −223019. −0.0876211
\(366\) −384161. −0.149903
\(367\) 4.17728e6 1.61893 0.809466 0.587167i \(-0.199757\pi\)
0.809466 + 0.587167i \(0.199757\pi\)
\(368\) −298585. −0.114934
\(369\) −348294. −0.133162
\(370\) −461952. −0.175426
\(371\) −1.41577e6 −0.534020
\(372\) −141977. −0.0531939
\(373\) −1.12351e6 −0.418124 −0.209062 0.977902i \(-0.567041\pi\)
−0.209062 + 0.977902i \(0.567041\pi\)
\(374\) 22738.2 0.00840576
\(375\) −1.63349e6 −0.599842
\(376\) 691889. 0.252387
\(377\) 1.88149e6 0.681788
\(378\) 147271. 0.0530136
\(379\) 3.45021e6 1.23381 0.616904 0.787039i \(-0.288387\pi\)
0.616904 + 0.787039i \(0.288387\pi\)
\(380\) −1.74443e6 −0.619718
\(381\) −428448. −0.151212
\(382\) −32132.5 −0.0112664
\(383\) −573250. −0.199686 −0.0998430 0.995003i \(-0.531834\pi\)
−0.0998430 + 0.995003i \(0.531834\pi\)
\(384\) −1.70148e6 −0.588841
\(385\) 1.07964e6 0.371218
\(386\) −950339. −0.324646
\(387\) −1.66596e6 −0.565441
\(388\) −1.19323e6 −0.402386
\(389\) −5.10842e6 −1.71164 −0.855820 0.517274i \(-0.826947\pi\)
−0.855820 + 0.517274i \(0.826947\pi\)
\(390\) 783922. 0.260983
\(391\) −23780.7 −0.00786651
\(392\) 1.15687e6 0.380251
\(393\) −2.34398e6 −0.765550
\(394\) −699114. −0.226886
\(395\) −5.11998e6 −1.65111
\(396\) −491134. −0.157385
\(397\) −417826. −0.133051 −0.0665256 0.997785i \(-0.521191\pi\)
−0.0665256 + 0.997785i \(0.521191\pi\)
\(398\) 135024. 0.0427272
\(399\) 980690. 0.308389
\(400\) −154906. −0.0484082
\(401\) 3.79619e6 1.17893 0.589463 0.807795i \(-0.299339\pi\)
0.589463 + 0.807795i \(0.299339\pi\)
\(402\) 39479.9 0.0121846
\(403\) 424868. 0.130314
\(404\) −5.34631e6 −1.62967
\(405\) 350296. 0.106120
\(406\) 523729. 0.157685
\(407\) −866082. −0.259163
\(408\) −53610.8 −0.0159442
\(409\) 4.95389e6 1.46433 0.732164 0.681129i \(-0.238511\pi\)
0.732164 + 0.681129i \(0.238511\pi\)
\(410\) −516066. −0.151616
\(411\) 1.09452e6 0.319609
\(412\) −5.44096e6 −1.57918
\(413\) −1.23844e6 −0.357274
\(414\) −96321.0 −0.0276198
\(415\) 713658. 0.203409
\(416\) 3.99818e6 1.13274
\(417\) 552071. 0.155473
\(418\) 613293. 0.171683
\(419\) −2.38845e6 −0.664631 −0.332316 0.943168i \(-0.607830\pi\)
−0.332316 + 0.943168i \(0.607830\pi\)
\(420\) −1.16365e6 −0.321885
\(421\) −582907. −0.160285 −0.0801427 0.996783i \(-0.525538\pi\)
−0.0801427 + 0.996783i \(0.525538\pi\)
\(422\) −2.44788e6 −0.669128
\(423\) −422942. −0.114929
\(424\) 2.08749e6 0.563910
\(425\) −12337.4 −0.00331324
\(426\) 1.52304e6 0.406618
\(427\) −1.70647e6 −0.452929
\(428\) −1.01558e6 −0.267982
\(429\) 1.46972e6 0.385560
\(430\) −2.46845e6 −0.643804
\(431\) −1.17212e6 −0.303935 −0.151967 0.988386i \(-0.548561\pi\)
−0.151967 + 0.988386i \(0.548561\pi\)
\(432\) 411472. 0.106079
\(433\) −5.34133e6 −1.36908 −0.684542 0.728974i \(-0.739998\pi\)
−0.684542 + 0.728974i \(0.739998\pi\)
\(434\) 118265. 0.0301393
\(435\) 1.24573e6 0.315647
\(436\) 2.33074e6 0.587188
\(437\) −641410. −0.160669
\(438\) −84508.3 −0.0210482
\(439\) 3.68652e6 0.912967 0.456483 0.889732i \(-0.349109\pi\)
0.456483 + 0.889732i \(0.349109\pi\)
\(440\) −1.59189e6 −0.391996
\(441\) −707179. −0.173154
\(442\) 73338.9 0.0178558
\(443\) −705619. −0.170829 −0.0854143 0.996346i \(-0.527221\pi\)
−0.0854143 + 0.996346i \(0.527221\pi\)
\(444\) 933475. 0.224722
\(445\) 2.71930e6 0.650963
\(446\) 822548. 0.195805
\(447\) −1.54131e6 −0.364857
\(448\) −510273. −0.120118
\(449\) 442688. 0.103629 0.0518146 0.998657i \(-0.483500\pi\)
0.0518146 + 0.998657i \(0.483500\pi\)
\(450\) −49971.4 −0.0116330
\(451\) −967537. −0.223989
\(452\) 4.06419e6 0.935682
\(453\) −4.13646e6 −0.947073
\(454\) 2.44640e6 0.557042
\(455\) 3.48224e6 0.788552
\(456\) −1.44599e6 −0.325651
\(457\) 4.85276e6 1.08692 0.543461 0.839434i \(-0.317113\pi\)
0.543461 + 0.839434i \(0.317113\pi\)
\(458\) −3.26190e6 −0.726619
\(459\) 32771.5 0.00726046
\(460\) 761077. 0.167700
\(461\) 8.87833e6 1.94571 0.972857 0.231408i \(-0.0743331\pi\)
0.972857 + 0.231408i \(0.0743331\pi\)
\(462\) 409109. 0.0891731
\(463\) −3.58364e6 −0.776911 −0.388456 0.921467i \(-0.626991\pi\)
−0.388456 + 0.921467i \(0.626991\pi\)
\(464\) 1.46329e6 0.315526
\(465\) 281304. 0.0603314
\(466\) −2.86235e6 −0.610603
\(467\) −6.13857e6 −1.30249 −0.651246 0.758867i \(-0.725753\pi\)
−0.651246 + 0.758867i \(0.725753\pi\)
\(468\) −1.58409e6 −0.334321
\(469\) 175373. 0.0368155
\(470\) −626672. −0.130857
\(471\) 3.48868e6 0.724618
\(472\) 1.82603e6 0.377271
\(473\) −4.62793e6 −0.951117
\(474\) −1.94011e6 −0.396626
\(475\) −332764. −0.0676710
\(476\) −108864. −0.0220226
\(477\) −1.27605e6 −0.256787
\(478\) 1.79687e6 0.359705
\(479\) 505577. 0.100681 0.0503406 0.998732i \(-0.483969\pi\)
0.0503406 + 0.998732i \(0.483969\pi\)
\(480\) 2.64718e6 0.524422
\(481\) −2.79343e6 −0.550522
\(482\) 1.63550e6 0.320651
\(483\) −427865. −0.0834525
\(484\) 2.97548e6 0.577356
\(485\) 2.36417e6 0.456378
\(486\) 132737. 0.0254919
\(487\) −751626. −0.143608 −0.0718041 0.997419i \(-0.522876\pi\)
−0.0718041 + 0.997419i \(0.522876\pi\)
\(488\) 2.51612e6 0.478280
\(489\) −2.58274e6 −0.488437
\(490\) −1.04783e6 −0.197151
\(491\) −2.19083e6 −0.410115 −0.205057 0.978750i \(-0.565738\pi\)
−0.205057 + 0.978750i \(0.565738\pi\)
\(492\) 1.04282e6 0.194222
\(493\) 116543. 0.0215957
\(494\) 1.97809e6 0.364694
\(495\) 973098. 0.178502
\(496\) 330431. 0.0603083
\(497\) 6.76545e6 1.22859
\(498\) 270426. 0.0488624
\(499\) 2.21183e6 0.397649 0.198824 0.980035i \(-0.436288\pi\)
0.198824 + 0.980035i \(0.436288\pi\)
\(500\) 4.89081e6 0.874895
\(501\) 5.22640e6 0.930268
\(502\) 2.16008e6 0.382569
\(503\) −1.05756e7 −1.86373 −0.931866 0.362803i \(-0.881820\pi\)
−0.931866 + 0.362803i \(0.881820\pi\)
\(504\) −964573. −0.169145
\(505\) 1.05928e7 1.84834
\(506\) −267573. −0.0464587
\(507\) 1.39874e6 0.241668
\(508\) 1.28282e6 0.220549
\(509\) −1.00676e7 −1.72239 −0.861195 0.508275i \(-0.830283\pi\)
−0.861195 + 0.508275i \(0.830283\pi\)
\(510\) 48557.5 0.00826667
\(511\) −375392. −0.0635964
\(512\) 5.50282e6 0.927707
\(513\) 883910. 0.148291
\(514\) −3.54477e6 −0.591807
\(515\) 1.07803e7 1.79108
\(516\) 4.98805e6 0.824720
\(517\) −1.17490e6 −0.193320
\(518\) −777573. −0.127326
\(519\) −1.72690e6 −0.281416
\(520\) −5.13442e6 −0.832689
\(521\) 7.71005e6 1.24441 0.622204 0.782855i \(-0.286237\pi\)
0.622204 + 0.782855i \(0.286237\pi\)
\(522\) 472044. 0.0758239
\(523\) 153540. 0.0245453 0.0122726 0.999925i \(-0.496093\pi\)
0.0122726 + 0.999925i \(0.496093\pi\)
\(524\) 7.01811e6 1.11659
\(525\) −221977. −0.0351487
\(526\) 3.20738e6 0.505460
\(527\) 26317.0 0.00412772
\(528\) 1.14304e6 0.178434
\(529\) 279841. 0.0434783
\(530\) −1.89073e6 −0.292374
\(531\) −1.11623e6 −0.171797
\(532\) −2.93628e6 −0.449799
\(533\) −3.12066e6 −0.475804
\(534\) 1.03042e6 0.156373
\(535\) 2.01220e6 0.303940
\(536\) −258580. −0.0388761
\(537\) 168820. 0.0252632
\(538\) −768577. −0.114481
\(539\) −1.96450e6 −0.291259
\(540\) −1.04882e6 −0.154780
\(541\) 267983. 0.0393653 0.0196827 0.999806i \(-0.493734\pi\)
0.0196827 + 0.999806i \(0.493734\pi\)
\(542\) 1.62684e6 0.237873
\(543\) 431708. 0.0628334
\(544\) 247654. 0.0358796
\(545\) −4.61795e6 −0.665976
\(546\) 1.31952e6 0.189424
\(547\) 6.44942e6 0.921621 0.460810 0.887499i \(-0.347559\pi\)
0.460810 + 0.887499i \(0.347559\pi\)
\(548\) −3.27710e6 −0.466164
\(549\) −1.53807e6 −0.217793
\(550\) −138817. −0.0195676
\(551\) 3.14338e6 0.441081
\(552\) 630869. 0.0881235
\(553\) −8.61812e6 −1.19839
\(554\) −4.11207e6 −0.569227
\(555\) −1.84952e6 −0.254875
\(556\) −1.65295e6 −0.226764
\(557\) −1.36650e7 −1.86626 −0.933132 0.359534i \(-0.882936\pi\)
−0.933132 + 0.359534i \(0.882936\pi\)
\(558\) 106594. 0.0144927
\(559\) −1.49268e7 −2.02039
\(560\) 2.70823e6 0.364935
\(561\) 91037.0 0.0122127
\(562\) 2.58366e6 0.345059
\(563\) 564712. 0.0750855 0.0375427 0.999295i \(-0.488047\pi\)
0.0375427 + 0.999295i \(0.488047\pi\)
\(564\) 1.26633e6 0.167629
\(565\) −8.05250e6 −1.06123
\(566\) −4.11957e6 −0.540519
\(567\) 589629. 0.0770231
\(568\) −9.97537e6 −1.29735
\(569\) 5.68519e6 0.736147 0.368074 0.929797i \(-0.380017\pi\)
0.368074 + 0.929797i \(0.380017\pi\)
\(570\) 1.30969e6 0.168842
\(571\) 5.94914e6 0.763597 0.381798 0.924246i \(-0.375305\pi\)
0.381798 + 0.924246i \(0.375305\pi\)
\(572\) −4.40049e6 −0.562355
\(573\) −128649. −0.0163689
\(574\) −868660. −0.110045
\(575\) 145182. 0.0183123
\(576\) −459916. −0.0577594
\(577\) −7.28162e6 −0.910518 −0.455259 0.890359i \(-0.650453\pi\)
−0.455259 + 0.890359i \(0.650453\pi\)
\(578\) −3.18718e6 −0.396814
\(579\) −3.80488e6 −0.471677
\(580\) −3.72984e6 −0.460384
\(581\) 1.20125e6 0.147637
\(582\) 895854. 0.109630
\(583\) −3.54479e6 −0.431935
\(584\) 553500. 0.0671561
\(585\) 3.13859e6 0.379180
\(586\) 2.99353e6 0.360114
\(587\) 9.14151e6 1.09502 0.547511 0.836799i \(-0.315575\pi\)
0.547511 + 0.836799i \(0.315575\pi\)
\(588\) 2.11736e6 0.252553
\(589\) 709821. 0.0843064
\(590\) −1.65391e6 −0.195606
\(591\) −2.79904e6 −0.329641
\(592\) −2.17252e6 −0.254777
\(593\) 6.14133e6 0.717176 0.358588 0.933496i \(-0.383258\pi\)
0.358588 + 0.933496i \(0.383258\pi\)
\(594\) 368735. 0.0428794
\(595\) 215696. 0.0249775
\(596\) 4.61484e6 0.532159
\(597\) 540598. 0.0620781
\(598\) −863021. −0.0986890
\(599\) −6.53791e6 −0.744512 −0.372256 0.928130i \(-0.621416\pi\)
−0.372256 + 0.928130i \(0.621416\pi\)
\(600\) 327295. 0.0371161
\(601\) −9.92849e6 −1.12124 −0.560618 0.828075i \(-0.689436\pi\)
−0.560618 + 0.828075i \(0.689436\pi\)
\(602\) −4.15498e6 −0.467281
\(603\) 158066. 0.0177029
\(604\) 1.23850e7 1.38135
\(605\) −5.89540e6 −0.654825
\(606\) 4.01392e6 0.444004
\(607\) −2.59619e6 −0.285999 −0.143000 0.989723i \(-0.545675\pi\)
−0.143000 + 0.989723i \(0.545675\pi\)
\(608\) 6.67970e6 0.732821
\(609\) 2.09685e6 0.229100
\(610\) −2.27896e6 −0.247977
\(611\) −3.78949e6 −0.410656
\(612\) −98120.9 −0.0105897
\(613\) −4.12434e6 −0.443306 −0.221653 0.975126i \(-0.571145\pi\)
−0.221653 + 0.975126i \(0.571145\pi\)
\(614\) 6.75047e6 0.722626
\(615\) −2.06618e6 −0.220282
\(616\) −2.67952e6 −0.284515
\(617\) 1.17375e7 1.24126 0.620631 0.784103i \(-0.286876\pi\)
0.620631 + 0.784103i \(0.286876\pi\)
\(618\) 4.08499e6 0.430249
\(619\) 6.90815e6 0.724662 0.362331 0.932050i \(-0.381981\pi\)
0.362331 + 0.932050i \(0.381981\pi\)
\(620\) −842251. −0.0879959
\(621\) −385641. −0.0401286
\(622\) 6.18387e6 0.640892
\(623\) 4.57721e6 0.472477
\(624\) 3.68672e6 0.379035
\(625\) −8.83266e6 −0.904465
\(626\) −269511. −0.0274878
\(627\) 2.45544e6 0.249437
\(628\) −1.04455e7 −1.05689
\(629\) −173030. −0.0174379
\(630\) 873653. 0.0876976
\(631\) 793455. 0.0793321 0.0396660 0.999213i \(-0.487371\pi\)
0.0396660 + 0.999213i \(0.487371\pi\)
\(632\) 1.27071e7 1.26547
\(633\) −9.80060e6 −0.972173
\(634\) 1.15600e6 0.114218
\(635\) −2.54168e6 −0.250142
\(636\) 3.82062e6 0.374534
\(637\) −6.33621e6 −0.618702
\(638\) 1.31131e6 0.127542
\(639\) 6.09780e6 0.590773
\(640\) −1.00937e7 −0.974089
\(641\) 3.78009e6 0.363377 0.181689 0.983356i \(-0.441844\pi\)
0.181689 + 0.983356i \(0.441844\pi\)
\(642\) 762483. 0.0730117
\(643\) 1.21266e7 1.15667 0.578336 0.815799i \(-0.303702\pi\)
0.578336 + 0.815799i \(0.303702\pi\)
\(644\) 1.28107e6 0.121719
\(645\) −9.88296e6 −0.935380
\(646\) 122526. 0.0115518
\(647\) −6.56812e6 −0.616851 −0.308425 0.951249i \(-0.599802\pi\)
−0.308425 + 0.951249i \(0.599802\pi\)
\(648\) −869383. −0.0813343
\(649\) −3.10081e6 −0.288977
\(650\) −447736. −0.0415660
\(651\) 473500. 0.0437892
\(652\) 7.73297e6 0.712406
\(653\) −9.97904e6 −0.915811 −0.457905 0.889001i \(-0.651400\pi\)
−0.457905 + 0.889001i \(0.651400\pi\)
\(654\) −1.74988e6 −0.159979
\(655\) −1.39052e7 −1.26641
\(656\) −2.42702e6 −0.220198
\(657\) −338346. −0.0305807
\(658\) −1.05484e6 −0.0949773
\(659\) 1.22411e7 1.09801 0.549006 0.835818i \(-0.315006\pi\)
0.549006 + 0.835818i \(0.315006\pi\)
\(660\) −2.91355e6 −0.260353
\(661\) −7.26433e6 −0.646684 −0.323342 0.946282i \(-0.604806\pi\)
−0.323342 + 0.946282i \(0.604806\pi\)
\(662\) 5.53035e6 0.490465
\(663\) 293627. 0.0259425
\(664\) −1.77120e6 −0.155900
\(665\) 5.81773e6 0.510152
\(666\) −700838. −0.0612254
\(667\) −1.37143e6 −0.119360
\(668\) −1.56483e7 −1.35683
\(669\) 3.29324e6 0.284484
\(670\) 234206. 0.0201563
\(671\) −4.27265e6 −0.366346
\(672\) 4.45582e6 0.380631
\(673\) 1.76906e7 1.50558 0.752791 0.658260i \(-0.228707\pi\)
0.752791 + 0.658260i \(0.228707\pi\)
\(674\) −3.20764e6 −0.271979
\(675\) −200071. −0.0169015
\(676\) −4.18798e6 −0.352482
\(677\) 8.88669e6 0.745193 0.372596 0.927994i \(-0.378468\pi\)
0.372596 + 0.927994i \(0.378468\pi\)
\(678\) −3.05133e6 −0.254926
\(679\) 3.97945e6 0.331244
\(680\) −318035. −0.0263756
\(681\) 9.79467e6 0.809323
\(682\) 296112. 0.0243778
\(683\) −3.93961e6 −0.323148 −0.161574 0.986861i \(-0.551657\pi\)
−0.161574 + 0.986861i \(0.551657\pi\)
\(684\) −2.64651e6 −0.216288
\(685\) 6.49301e6 0.528713
\(686\) −5.15905e6 −0.418561
\(687\) −1.30597e7 −1.05570
\(688\) −1.16089e7 −0.935021
\(689\) −1.14332e7 −0.917531
\(690\) −571404. −0.0456899
\(691\) 2.19400e7 1.74800 0.873999 0.485927i \(-0.161518\pi\)
0.873999 + 0.485927i \(0.161518\pi\)
\(692\) 5.17051e6 0.410457
\(693\) 1.63795e6 0.129559
\(694\) 3.67399e6 0.289561
\(695\) 3.27504e6 0.257190
\(696\) −3.09172e6 −0.241923
\(697\) −193299. −0.0150712
\(698\) −5.96818e6 −0.463664
\(699\) −1.14600e7 −0.887141
\(700\) 664620. 0.0512658
\(701\) 1.20994e7 0.929969 0.464984 0.885319i \(-0.346060\pi\)
0.464984 + 0.885319i \(0.346060\pi\)
\(702\) 1.18931e6 0.0910858
\(703\) −4.66694e6 −0.356159
\(704\) −1.27762e6 −0.0971559
\(705\) −2.50901e6 −0.190121
\(706\) 2.09960e6 0.158535
\(707\) 1.78301e7 1.34155
\(708\) 3.34209e6 0.250574
\(709\) 1.95715e7 1.46221 0.731103 0.682267i \(-0.239006\pi\)
0.731103 + 0.682267i \(0.239006\pi\)
\(710\) 9.03510e6 0.672647
\(711\) −7.76763e6 −0.576255
\(712\) −6.74890e6 −0.498923
\(713\) −309688. −0.0228139
\(714\) 81733.5 0.00600005
\(715\) 8.71881e6 0.637811
\(716\) −505463. −0.0368474
\(717\) 7.19413e6 0.522613
\(718\) 6.70390e6 0.485307
\(719\) −1.98434e7 −1.43151 −0.715753 0.698354i \(-0.753916\pi\)
−0.715753 + 0.698354i \(0.753916\pi\)
\(720\) 2.44097e6 0.175481
\(721\) 1.81458e7 1.29998
\(722\) −2.26130e6 −0.161441
\(723\) 6.54804e6 0.465871
\(724\) −1.29258e6 −0.0916452
\(725\) −711497. −0.0502722
\(726\) −2.23394e6 −0.157301
\(727\) 2.50056e7 1.75469 0.877345 0.479860i \(-0.159312\pi\)
0.877345 + 0.479860i \(0.159312\pi\)
\(728\) −8.64242e6 −0.604376
\(729\) 531441. 0.0370370
\(730\) −501327. −0.0348188
\(731\) −924588. −0.0639963
\(732\) 4.60513e6 0.317661
\(733\) 2.53798e7 1.74473 0.872365 0.488855i \(-0.162585\pi\)
0.872365 + 0.488855i \(0.162585\pi\)
\(734\) 9.39018e6 0.643330
\(735\) −4.19519e6 −0.286440
\(736\) −2.91429e6 −0.198307
\(737\) 439097. 0.0297777
\(738\) −782935. −0.0529158
\(739\) 1.50131e7 1.01125 0.505625 0.862753i \(-0.331262\pi\)
0.505625 + 0.862753i \(0.331262\pi\)
\(740\) 5.53764e6 0.371745
\(741\) 7.91969e6 0.529862
\(742\) −3.18253e6 −0.212209
\(743\) −2.53859e7 −1.68702 −0.843512 0.537111i \(-0.819516\pi\)
−0.843512 + 0.537111i \(0.819516\pi\)
\(744\) −698156. −0.0462402
\(745\) −9.14351e6 −0.603563
\(746\) −2.52556e6 −0.166154
\(747\) 1.08271e6 0.0709919
\(748\) −272573. −0.0178127
\(749\) 3.38701e6 0.220603
\(750\) −3.67194e6 −0.238365
\(751\) 1.25539e7 0.812230 0.406115 0.913822i \(-0.366883\pi\)
0.406115 + 0.913822i \(0.366883\pi\)
\(752\) −2.94719e6 −0.190048
\(753\) 8.64831e6 0.555833
\(754\) 4.22944e6 0.270928
\(755\) −2.45387e7 −1.56669
\(756\) −1.76541e6 −0.112341
\(757\) 6.15033e6 0.390085 0.195042 0.980795i \(-0.437516\pi\)
0.195042 + 0.980795i \(0.437516\pi\)
\(758\) 7.75579e6 0.490290
\(759\) −1.07129e6 −0.0674995
\(760\) −8.57800e6 −0.538706
\(761\) −6.22830e6 −0.389859 −0.194930 0.980817i \(-0.562448\pi\)
−0.194930 + 0.980817i \(0.562448\pi\)
\(762\) −963117. −0.0600885
\(763\) −7.77309e6 −0.483373
\(764\) 385187. 0.0238747
\(765\) 194410. 0.0120106
\(766\) −1.28862e6 −0.0793511
\(767\) −1.00012e7 −0.613853
\(768\) −2.18953e6 −0.133951
\(769\) 8.72415e6 0.531995 0.265997 0.963974i \(-0.414299\pi\)
0.265997 + 0.963974i \(0.414299\pi\)
\(770\) 2.42695e6 0.147514
\(771\) −1.41922e7 −0.859832
\(772\) 1.13922e7 0.687960
\(773\) −7.23571e6 −0.435545 −0.217772 0.976000i \(-0.569879\pi\)
−0.217772 + 0.976000i \(0.569879\pi\)
\(774\) −3.74494e6 −0.224695
\(775\) −160666. −0.00960883
\(776\) −5.86753e6 −0.349785
\(777\) −3.11317e6 −0.184991
\(778\) −1.14833e7 −0.680171
\(779\) −5.21364e6 −0.307820
\(780\) −9.39725e6 −0.553050
\(781\) 1.69393e7 0.993728
\(782\) −53457.0 −0.00312599
\(783\) 1.88992e6 0.110164
\(784\) −4.92784e6 −0.286330
\(785\) 2.06959e7 1.19870
\(786\) −5.26908e6 −0.304214
\(787\) −1.35431e7 −0.779440 −0.389720 0.920933i \(-0.627428\pi\)
−0.389720 + 0.920933i \(0.627428\pi\)
\(788\) 8.38061e6 0.480795
\(789\) 1.28414e7 0.734379
\(790\) −1.15093e7 −0.656117
\(791\) −1.35542e7 −0.770253
\(792\) −2.41509e6 −0.136811
\(793\) −1.37809e7 −0.778203
\(794\) −939238. −0.0528718
\(795\) −7.56991e6 −0.424788
\(796\) −1.61860e6 −0.0905435
\(797\) 1.01044e7 0.563461 0.281731 0.959494i \(-0.409092\pi\)
0.281731 + 0.959494i \(0.409092\pi\)
\(798\) 2.20451e6 0.122548
\(799\) −234727. −0.0130076
\(800\) −1.51193e6 −0.0835233
\(801\) 4.12550e6 0.227193
\(802\) 8.53352e6 0.468481
\(803\) −939904. −0.0514392
\(804\) −473265. −0.0258205
\(805\) −2.53822e6 −0.138051
\(806\) 955068. 0.0517841
\(807\) −3.07716e6 −0.166328
\(808\) −2.62898e7 −1.41664
\(809\) −3.23912e6 −0.174003 −0.0870013 0.996208i \(-0.527728\pi\)
−0.0870013 + 0.996208i \(0.527728\pi\)
\(810\) 787436. 0.0421699
\(811\) 6.20038e6 0.331029 0.165515 0.986207i \(-0.447072\pi\)
0.165515 + 0.986207i \(0.447072\pi\)
\(812\) −6.27818e6 −0.334152
\(813\) 6.51338e6 0.345605
\(814\) −1.94688e6 −0.102986
\(815\) −1.53216e7 −0.807995
\(816\) 228362. 0.0120060
\(817\) −2.49379e7 −1.30709
\(818\) 1.11359e7 0.581894
\(819\) 5.28298e6 0.275213
\(820\) 6.18633e6 0.321291
\(821\) 1.75836e7 0.910439 0.455219 0.890379i \(-0.349561\pi\)
0.455219 + 0.890379i \(0.349561\pi\)
\(822\) 2.46039e6 0.127006
\(823\) −3.80649e7 −1.95896 −0.979478 0.201549i \(-0.935402\pi\)
−0.979478 + 0.201549i \(0.935402\pi\)
\(824\) −2.67552e7 −1.37275
\(825\) −555784. −0.0284296
\(826\) −2.78392e6 −0.141973
\(827\) 8.39081e6 0.426619 0.213310 0.976985i \(-0.431576\pi\)
0.213310 + 0.976985i \(0.431576\pi\)
\(828\) 1.15465e6 0.0585293
\(829\) 394283. 0.0199261 0.00996305 0.999950i \(-0.496829\pi\)
0.00996305 + 0.999950i \(0.496829\pi\)
\(830\) 1.60424e6 0.0808305
\(831\) −1.64635e7 −0.827027
\(832\) −4.12078e6 −0.206382
\(833\) −392476. −0.0195975
\(834\) 1.24101e6 0.0617817
\(835\) 3.10045e7 1.53889
\(836\) −7.35183e6 −0.363814
\(837\) 426772. 0.0210563
\(838\) −5.36904e6 −0.264111
\(839\) 1.63376e7 0.801276 0.400638 0.916236i \(-0.368788\pi\)
0.400638 + 0.916236i \(0.368788\pi\)
\(840\) −5.72212e6 −0.279807
\(841\) −1.37901e7 −0.672324
\(842\) −1.31033e6 −0.0636941
\(843\) 1.03442e7 0.501335
\(844\) 2.93439e7 1.41795
\(845\) 8.29775e6 0.399778
\(846\) −950738. −0.0456704
\(847\) −9.92334e6 −0.475280
\(848\) −8.89193e6 −0.424626
\(849\) −1.64936e7 −0.785317
\(850\) −27733.5 −0.00131661
\(851\) 2.03614e6 0.0963793
\(852\) −1.82574e7 −0.861668
\(853\) 2.38816e6 0.112380 0.0561901 0.998420i \(-0.482105\pi\)
0.0561901 + 0.998420i \(0.482105\pi\)
\(854\) −3.83601e6 −0.179985
\(855\) 5.24361e6 0.245310
\(856\) −4.99400e6 −0.232951
\(857\) −1.41990e7 −0.660398 −0.330199 0.943911i \(-0.607116\pi\)
−0.330199 + 0.943911i \(0.607116\pi\)
\(858\) 3.30381e6 0.153214
\(859\) 1.13234e7 0.523595 0.261797 0.965123i \(-0.415685\pi\)
0.261797 + 0.965123i \(0.415685\pi\)
\(860\) 2.95905e7 1.36429
\(861\) −3.47786e6 −0.159884
\(862\) −2.63484e6 −0.120777
\(863\) 2.55589e7 1.16819 0.584097 0.811684i \(-0.301449\pi\)
0.584097 + 0.811684i \(0.301449\pi\)
\(864\) 4.01609e6 0.183029
\(865\) −1.02445e7 −0.465532
\(866\) −1.20069e7 −0.544046
\(867\) −1.27605e7 −0.576529
\(868\) −1.41770e6 −0.0638684
\(869\) −2.15780e7 −0.969307
\(870\) 2.80030e6 0.125431
\(871\) 1.41625e6 0.0632548
\(872\) 1.14611e7 0.510428
\(873\) 3.58673e6 0.159281
\(874\) −1.44184e6 −0.0638466
\(875\) −1.63110e7 −0.720214
\(876\) 1.01304e6 0.0446033
\(877\) 1.94757e7 0.855056 0.427528 0.904002i \(-0.359385\pi\)
0.427528 + 0.904002i \(0.359385\pi\)
\(878\) 8.28699e6 0.362794
\(879\) 1.19852e7 0.523207
\(880\) 6.78085e6 0.295174
\(881\) 9.76547e6 0.423890 0.211945 0.977282i \(-0.432020\pi\)
0.211945 + 0.977282i \(0.432020\pi\)
\(882\) −1.58968e6 −0.0688079
\(883\) 4.86226e6 0.209863 0.104932 0.994479i \(-0.466538\pi\)
0.104932 + 0.994479i \(0.466538\pi\)
\(884\) −879148. −0.0378383
\(885\) −6.62178e6 −0.284195
\(886\) −1.58617e6 −0.0678838
\(887\) −3.00831e7 −1.28385 −0.641925 0.766767i \(-0.721864\pi\)
−0.641925 + 0.766767i \(0.721864\pi\)
\(888\) 4.59024e6 0.195345
\(889\) −4.27824e6 −0.181556
\(890\) 6.11276e6 0.258680
\(891\) 1.47631e6 0.0622992
\(892\) −9.86027e6 −0.414932
\(893\) −6.33105e6 −0.265673
\(894\) −3.46475e6 −0.144987
\(895\) 1.00149e6 0.0417915
\(896\) −1.69900e7 −0.707005
\(897\) −3.45528e6 −0.143385
\(898\) 995127. 0.0411801
\(899\) 1.51770e6 0.0626306
\(900\) 599031. 0.0246515
\(901\) −708193. −0.0290630
\(902\) −2.17494e6 −0.0890085
\(903\) −1.66353e7 −0.678909
\(904\) 1.99851e7 0.813366
\(905\) 2.56102e6 0.103942
\(906\) −9.29842e6 −0.376347
\(907\) −3.27277e7 −1.32098 −0.660491 0.750834i \(-0.729652\pi\)
−0.660491 + 0.750834i \(0.729652\pi\)
\(908\) −2.93262e7 −1.18043
\(909\) 1.60706e7 0.645091
\(910\) 7.82779e6 0.313354
\(911\) −1.44550e6 −0.0577062 −0.0288531 0.999584i \(-0.509186\pi\)
−0.0288531 + 0.999584i \(0.509186\pi\)
\(912\) 6.15936e6 0.245216
\(913\) 3.00769e6 0.119414
\(914\) 1.09086e7 0.431921
\(915\) −9.12427e6 −0.360284
\(916\) 3.91019e7 1.53978
\(917\) −2.34057e7 −0.919174
\(918\) 73667.6 0.00288516
\(919\) 1.02478e7 0.400258 0.200129 0.979770i \(-0.435864\pi\)
0.200129 + 0.979770i \(0.435864\pi\)
\(920\) 3.74250e6 0.145778
\(921\) 2.70269e7 1.04990
\(922\) 1.99578e7 0.773187
\(923\) 5.46353e7 2.11091
\(924\) −4.90418e6 −0.188967
\(925\) 1.05635e6 0.0405933
\(926\) −8.05572e6 −0.308729
\(927\) 1.63551e7 0.625105
\(928\) 1.42821e7 0.544407
\(929\) 4.79655e7 1.82343 0.911716 0.410821i \(-0.134758\pi\)
0.911716 + 0.410821i \(0.134758\pi\)
\(930\) 632348. 0.0239745
\(931\) −1.05858e7 −0.400268
\(932\) 3.43124e7 1.29393
\(933\) 2.47584e7 0.931147
\(934\) −1.37990e7 −0.517583
\(935\) 540058. 0.0202028
\(936\) −7.78954e6 −0.290618
\(937\) −1.18919e7 −0.442489 −0.221244 0.975218i \(-0.571012\pi\)
−0.221244 + 0.975218i \(0.571012\pi\)
\(938\) 394224. 0.0146297
\(939\) −1.07904e6 −0.0399369
\(940\) 7.51222e6 0.277299
\(941\) 3.72086e7 1.36984 0.684919 0.728619i \(-0.259838\pi\)
0.684919 + 0.728619i \(0.259838\pi\)
\(942\) 7.84227e6 0.287948
\(943\) 2.27466e6 0.0832985
\(944\) −7.77822e6 −0.284086
\(945\) 3.49785e6 0.127415
\(946\) −1.04032e7 −0.377954
\(947\) 2.24159e7 0.812235 0.406118 0.913821i \(-0.366882\pi\)
0.406118 + 0.913821i \(0.366882\pi\)
\(948\) 2.32570e7 0.840492
\(949\) −3.03153e6 −0.109269
\(950\) −748027. −0.0268911
\(951\) 4.62829e6 0.165947
\(952\) −535326. −0.0191437
\(953\) 1.19242e7 0.425302 0.212651 0.977128i \(-0.431790\pi\)
0.212651 + 0.977128i \(0.431790\pi\)
\(954\) −2.86846e6 −0.102042
\(955\) −763182. −0.0270782
\(956\) −2.15399e7 −0.762253
\(957\) 5.25009e6 0.185305
\(958\) 1.13649e6 0.0400086
\(959\) 1.09292e7 0.383746
\(960\) −2.72836e6 −0.0955483
\(961\) −2.82864e7 −0.988029
\(962\) −6.27940e6 −0.218766
\(963\) 3.05276e6 0.106078
\(964\) −1.96055e7 −0.679493
\(965\) −2.25716e7 −0.780270
\(966\) −961806. −0.0331623
\(967\) 1.40918e7 0.484619 0.242309 0.970199i \(-0.422095\pi\)
0.242309 + 0.970199i \(0.422095\pi\)
\(968\) 1.46315e7 0.501882
\(969\) 490559. 0.0167835
\(970\) 5.31446e6 0.181355
\(971\) −2.06293e7 −0.702160 −0.351080 0.936346i \(-0.614185\pi\)
−0.351080 + 0.936346i \(0.614185\pi\)
\(972\) −1.59119e6 −0.0540201
\(973\) 5.51265e6 0.186672
\(974\) −1.68959e6 −0.0570669
\(975\) −1.79260e6 −0.0603911
\(976\) −1.07177e7 −0.360146
\(977\) 2.77805e7 0.931116 0.465558 0.885017i \(-0.345854\pi\)
0.465558 + 0.885017i \(0.345854\pi\)
\(978\) −5.80579e6 −0.194095
\(979\) 1.14604e7 0.382157
\(980\) 1.25608e7 0.417784
\(981\) −7.00600e6 −0.232433
\(982\) −4.92481e6 −0.162971
\(983\) −8.68777e6 −0.286764 −0.143382 0.989667i \(-0.545798\pi\)
−0.143382 + 0.989667i \(0.545798\pi\)
\(984\) 5.12795e6 0.168833
\(985\) −1.66047e7 −0.545307
\(986\) 261979. 0.00858171
\(987\) −4.22325e6 −0.137992
\(988\) −2.37123e7 −0.772826
\(989\) 1.08802e7 0.353708
\(990\) 2.18744e6 0.0709331
\(991\) −5.74667e7 −1.85880 −0.929399 0.369077i \(-0.879674\pi\)
−0.929399 + 0.369077i \(0.879674\pi\)
\(992\) 3.22511e6 0.104056
\(993\) 2.21419e7 0.712594
\(994\) 1.52082e7 0.488215
\(995\) 3.20698e6 0.102692
\(996\) −3.24173e6 −0.103545
\(997\) −3.52375e7 −1.12271 −0.561354 0.827576i \(-0.689719\pi\)
−0.561354 + 0.827576i \(0.689719\pi\)
\(998\) 4.97200e6 0.158017
\(999\) −2.80595e6 −0.0889541
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 69.6.a.e.1.3 5
3.2 odd 2 207.6.a.f.1.3 5
4.3 odd 2 1104.6.a.r.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.6.a.e.1.3 5 1.1 even 1 trivial
207.6.a.f.1.3 5 3.2 odd 2
1104.6.a.r.1.3 5 4.3 odd 2