Properties

Label 69.6.a.e.1.1
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: \(x^{5} - 113 x^{3} - 257 x^{2} + 1404 x + 2197\)
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 1.1
Root \(-1.42196\) of defining polynomial
Character \(\chi\) \(=\) 69.1

$q$-expansion

\(f(q)\) \(=\) \(q-9.32729 q^{2} +9.00000 q^{3} +54.9984 q^{4} +99.7124 q^{5} -83.9456 q^{6} +125.983 q^{7} -214.513 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-9.32729 q^{2} +9.00000 q^{3} +54.9984 q^{4} +99.7124 q^{5} -83.9456 q^{6} +125.983 q^{7} -214.513 q^{8} +81.0000 q^{9} -930.047 q^{10} +177.715 q^{11} +494.986 q^{12} -919.618 q^{13} -1175.08 q^{14} +897.412 q^{15} +240.875 q^{16} +1568.24 q^{17} -755.511 q^{18} +447.960 q^{19} +5484.02 q^{20} +1133.85 q^{21} -1657.60 q^{22} -529.000 q^{23} -1930.62 q^{24} +6817.57 q^{25} +8577.55 q^{26} +729.000 q^{27} +6928.86 q^{28} -1298.72 q^{29} -8370.42 q^{30} -6651.88 q^{31} +4617.70 q^{32} +1599.43 q^{33} -14627.4 q^{34} +12562.1 q^{35} +4454.87 q^{36} -8056.80 q^{37} -4178.25 q^{38} -8276.56 q^{39} -21389.6 q^{40} +17941.4 q^{41} -10575.7 q^{42} +10710.7 q^{43} +9774.04 q^{44} +8076.71 q^{45} +4934.14 q^{46} -17886.9 q^{47} +2167.87 q^{48} -935.293 q^{49} -63589.5 q^{50} +14114.1 q^{51} -50577.5 q^{52} -20343.9 q^{53} -6799.60 q^{54} +17720.4 q^{55} -27025.0 q^{56} +4031.64 q^{57} +12113.5 q^{58} +28488.5 q^{59} +49356.2 q^{60} +16222.3 q^{61} +62044.1 q^{62} +10204.6 q^{63} -50778.6 q^{64} -91697.4 q^{65} -14918.4 q^{66} +54128.0 q^{67} +86250.6 q^{68} -4761.00 q^{69} -117170. q^{70} +42505.6 q^{71} -17375.5 q^{72} +40936.2 q^{73} +75148.1 q^{74} +61358.1 q^{75} +24637.1 q^{76} +22389.1 q^{77} +77197.9 q^{78} -24760.6 q^{79} +24018.2 q^{80} +6561.00 q^{81} -167345. q^{82} -30825.2 q^{83} +62359.7 q^{84} +156373. q^{85} -99901.7 q^{86} -11688.4 q^{87} -38122.1 q^{88} -66531.6 q^{89} -75333.8 q^{90} -115856. q^{91} -29094.2 q^{92} -59866.9 q^{93} +166836. q^{94} +44667.2 q^{95} +41559.3 q^{96} -104506. q^{97} +8723.75 q^{98} +14394.9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 8 q^{2} + 45 q^{3} + 118 q^{4} + 94 q^{5} + 72 q^{6} + 272 q^{7} + 258 q^{8} + 405 q^{9} + O(q^{10}) \) \( 5 q + 8 q^{2} + 45 q^{3} + 118 q^{4} + 94 q^{5} + 72 q^{6} + 272 q^{7} + 258 q^{8} + 405 q^{9} - 172 q^{10} + 1100 q^{11} + 1062 q^{12} - 978 q^{13} - 344 q^{14} + 846 q^{15} + 1218 q^{16} + 2522 q^{17} + 648 q^{18} + 2060 q^{19} + 7720 q^{20} + 2448 q^{21} - 2572 q^{22} - 2645 q^{23} + 2322 q^{24} + 12035 q^{25} + 9280 q^{26} + 3645 q^{27} + 8072 q^{28} + 1526 q^{29} - 1548 q^{30} - 7392 q^{31} - 5086 q^{32} + 9900 q^{33} - 15608 q^{34} + 6056 q^{35} + 9558 q^{36} - 8210 q^{37} - 14276 q^{38} - 8802 q^{39} - 37472 q^{40} + 21250 q^{41} - 3096 q^{42} - 4548 q^{43} - 4260 q^{44} + 7614 q^{45} - 4232 q^{46} + 536 q^{47} + 10962 q^{48} - 27979 q^{49} - 81872 q^{50} + 22698 q^{51} - 76380 q^{52} - 11482 q^{53} + 5832 q^{54} - 77064 q^{55} - 28624 q^{56} + 18540 q^{57} - 79680 q^{58} + 74676 q^{59} + 69480 q^{60} - 44618 q^{61} + 64880 q^{62} + 22032 q^{63} - 137382 q^{64} - 24388 q^{65} - 23148 q^{66} - 1412 q^{67} + 80196 q^{68} - 23805 q^{69} - 222304 q^{70} + 37912 q^{71} + 20898 q^{72} + 46546 q^{73} + 111604 q^{74} + 108315 q^{75} - 79548 q^{76} + 157008 q^{77} + 83520 q^{78} + 50544 q^{79} + 69424 q^{80} + 32805 q^{81} - 233720 q^{82} + 89588 q^{83} + 72648 q^{84} + 147892 q^{85} + 77428 q^{86} + 13734 q^{87} + 54484 q^{88} + 280410 q^{89} - 13932 q^{90} - 27416 q^{91} - 62422 q^{92} - 66528 q^{93} + 113632 q^{94} + 203120 q^{95} - 45774 q^{96} + 90074 q^{97} + 32976 q^{98} + 89100 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −9.32729 −1.64885 −0.824424 0.565973i \(-0.808501\pi\)
−0.824424 + 0.565973i \(0.808501\pi\)
\(3\) 9.00000 0.577350
\(4\) 54.9984 1.71870
\(5\) 99.7124 1.78371 0.891855 0.452321i \(-0.149404\pi\)
0.891855 + 0.452321i \(0.149404\pi\)
\(6\) −83.9456 −0.951963
\(7\) 125.983 0.971777 0.485889 0.874021i \(-0.338496\pi\)
0.485889 + 0.874021i \(0.338496\pi\)
\(8\) −214.513 −1.18503
\(9\) 81.0000 0.333333
\(10\) −930.047 −2.94107
\(11\) 177.715 0.442835 0.221418 0.975179i \(-0.428932\pi\)
0.221418 + 0.975179i \(0.428932\pi\)
\(12\) 494.986 0.992292
\(13\) −919.618 −1.50921 −0.754604 0.656180i \(-0.772171\pi\)
−0.754604 + 0.656180i \(0.772171\pi\)
\(14\) −1175.08 −1.60231
\(15\) 897.412 1.02983
\(16\) 240.875 0.235229
\(17\) 1568.24 1.31610 0.658051 0.752973i \(-0.271381\pi\)
0.658051 + 0.752973i \(0.271381\pi\)
\(18\) −755.511 −0.549616
\(19\) 447.960 0.284679 0.142339 0.989818i \(-0.454538\pi\)
0.142339 + 0.989818i \(0.454538\pi\)
\(20\) 5484.02 3.06566
\(21\) 1133.85 0.561056
\(22\) −1657.60 −0.730168
\(23\) −529.000 −0.208514
\(24\) −1930.62 −0.684176
\(25\) 6817.57 2.18162
\(26\) 8577.55 2.48846
\(27\) 729.000 0.192450
\(28\) 6928.86 1.67019
\(29\) −1298.72 −0.286760 −0.143380 0.989668i \(-0.545797\pi\)
−0.143380 + 0.989668i \(0.545797\pi\)
\(30\) −8370.42 −1.69803
\(31\) −6651.88 −1.24320 −0.621599 0.783336i \(-0.713517\pi\)
−0.621599 + 0.783336i \(0.713517\pi\)
\(32\) 4617.70 0.797169
\(33\) 1599.43 0.255671
\(34\) −14627.4 −2.17005
\(35\) 12562.1 1.73337
\(36\) 4454.87 0.572900
\(37\) −8056.80 −0.967516 −0.483758 0.875202i \(-0.660728\pi\)
−0.483758 + 0.875202i \(0.660728\pi\)
\(38\) −4178.25 −0.469392
\(39\) −8276.56 −0.871342
\(40\) −21389.6 −2.11374
\(41\) 17941.4 1.66685 0.833427 0.552630i \(-0.186376\pi\)
0.833427 + 0.552630i \(0.186376\pi\)
\(42\) −10575.7 −0.925096
\(43\) 10710.7 0.883377 0.441688 0.897169i \(-0.354380\pi\)
0.441688 + 0.897169i \(0.354380\pi\)
\(44\) 9774.04 0.761101
\(45\) 8076.71 0.594570
\(46\) 4934.14 0.343809
\(47\) −17886.9 −1.18111 −0.590555 0.806997i \(-0.701091\pi\)
−0.590555 + 0.806997i \(0.701091\pi\)
\(48\) 2167.87 0.135810
\(49\) −935.293 −0.0556490
\(50\) −63589.5 −3.59716
\(51\) 14114.1 0.759852
\(52\) −50577.5 −2.59388
\(53\) −20343.9 −0.994820 −0.497410 0.867516i \(-0.665716\pi\)
−0.497410 + 0.867516i \(0.665716\pi\)
\(54\) −6799.60 −0.317321
\(55\) 17720.4 0.789890
\(56\) −27025.0 −1.15158
\(57\) 4031.64 0.164359
\(58\) 12113.5 0.472824
\(59\) 28488.5 1.06547 0.532734 0.846283i \(-0.321165\pi\)
0.532734 + 0.846283i \(0.321165\pi\)
\(60\) 49356.2 1.76996
\(61\) 16222.3 0.558196 0.279098 0.960263i \(-0.409965\pi\)
0.279098 + 0.960263i \(0.409965\pi\)
\(62\) 62044.1 2.04984
\(63\) 10204.6 0.323926
\(64\) −50778.6 −1.54964
\(65\) −91697.4 −2.69199
\(66\) −14918.4 −0.421563
\(67\) 54128.0 1.47311 0.736555 0.676378i \(-0.236451\pi\)
0.736555 + 0.676378i \(0.236451\pi\)
\(68\) 86250.6 2.26198
\(69\) −4761.00 −0.120386
\(70\) −117170. −2.85806
\(71\) 42505.6 1.00069 0.500346 0.865826i \(-0.333206\pi\)
0.500346 + 0.865826i \(0.333206\pi\)
\(72\) −17375.5 −0.395009
\(73\) 40936.2 0.899084 0.449542 0.893259i \(-0.351587\pi\)
0.449542 + 0.893259i \(0.351587\pi\)
\(74\) 75148.1 1.59529
\(75\) 61358.1 1.25956
\(76\) 24637.1 0.489277
\(77\) 22389.1 0.430337
\(78\) 77197.9 1.43671
\(79\) −24760.6 −0.446369 −0.223185 0.974776i \(-0.571645\pi\)
−0.223185 + 0.974776i \(0.571645\pi\)
\(80\) 24018.2 0.419581
\(81\) 6561.00 0.111111
\(82\) −167345. −2.74839
\(83\) −30825.2 −0.491147 −0.245573 0.969378i \(-0.578976\pi\)
−0.245573 + 0.969378i \(0.578976\pi\)
\(84\) 62359.7 0.964287
\(85\) 156373. 2.34755
\(86\) −99901.7 −1.45655
\(87\) −11688.4 −0.165561
\(88\) −38122.1 −0.524772
\(89\) −66531.6 −0.890334 −0.445167 0.895448i \(-0.646856\pi\)
−0.445167 + 0.895448i \(0.646856\pi\)
\(90\) −75333.8 −0.980356
\(91\) −115856. −1.46661
\(92\) −29094.2 −0.358374
\(93\) −59866.9 −0.717761
\(94\) 166836. 1.94747
\(95\) 44667.2 0.507784
\(96\) 41559.3 0.460246
\(97\) −104506. −1.12774 −0.563872 0.825862i \(-0.690689\pi\)
−0.563872 + 0.825862i \(0.690689\pi\)
\(98\) 8723.75 0.0917568
\(99\) 14394.9 0.147612
\(100\) 374956. 3.74956
\(101\) −203117. −1.98127 −0.990635 0.136540i \(-0.956402\pi\)
−0.990635 + 0.136540i \(0.956402\pi\)
\(102\) −131647. −1.25288
\(103\) −148132. −1.37580 −0.687901 0.725805i \(-0.741468\pi\)
−0.687901 + 0.725805i \(0.741468\pi\)
\(104\) 197270. 1.78845
\(105\) 113059. 1.00076
\(106\) 189754. 1.64031
\(107\) −43750.9 −0.369426 −0.184713 0.982793i \(-0.559136\pi\)
−0.184713 + 0.982793i \(0.559136\pi\)
\(108\) 40093.8 0.330764
\(109\) 189052. 1.52410 0.762052 0.647516i \(-0.224192\pi\)
0.762052 + 0.647516i \(0.224192\pi\)
\(110\) −165283. −1.30241
\(111\) −72511.2 −0.558596
\(112\) 30346.1 0.228591
\(113\) −82382.3 −0.606929 −0.303465 0.952843i \(-0.598143\pi\)
−0.303465 + 0.952843i \(0.598143\pi\)
\(114\) −37604.3 −0.271003
\(115\) −52747.9 −0.371929
\(116\) −71427.2 −0.492855
\(117\) −74489.1 −0.503069
\(118\) −265721. −1.75679
\(119\) 197571. 1.27896
\(120\) −192506. −1.22037
\(121\) −129468. −0.803897
\(122\) −151310. −0.920381
\(123\) 161473. 0.962358
\(124\) −365843. −2.13668
\(125\) 368195. 2.10767
\(126\) −95181.5 −0.534104
\(127\) −95322.4 −0.524428 −0.262214 0.965010i \(-0.584453\pi\)
−0.262214 + 0.965010i \(0.584453\pi\)
\(128\) 325861. 1.75795
\(129\) 96396.1 0.510018
\(130\) 855288. 4.43868
\(131\) 181975. 0.926474 0.463237 0.886234i \(-0.346688\pi\)
0.463237 + 0.886234i \(0.346688\pi\)
\(132\) 87966.4 0.439422
\(133\) 56435.3 0.276644
\(134\) −504868. −2.42893
\(135\) 72690.4 0.343275
\(136\) −336407. −1.55962
\(137\) 175024. 0.796705 0.398352 0.917232i \(-0.369582\pi\)
0.398352 + 0.917232i \(0.369582\pi\)
\(138\) 44407.2 0.198498
\(139\) 199672. 0.876556 0.438278 0.898839i \(-0.355588\pi\)
0.438278 + 0.898839i \(0.355588\pi\)
\(140\) 690894. 2.97914
\(141\) −160982. −0.681914
\(142\) −396462. −1.64999
\(143\) −163430. −0.668331
\(144\) 19510.9 0.0784098
\(145\) −129498. −0.511497
\(146\) −381824. −1.48245
\(147\) −8417.64 −0.0321290
\(148\) −443111. −1.66287
\(149\) 71174.8 0.262640 0.131320 0.991340i \(-0.458079\pi\)
0.131320 + 0.991340i \(0.458079\pi\)
\(150\) −572305. −2.07682
\(151\) 133491. 0.476440 0.238220 0.971211i \(-0.423436\pi\)
0.238220 + 0.971211i \(0.423436\pi\)
\(152\) −96093.1 −0.337352
\(153\) 127027. 0.438701
\(154\) −208829. −0.709561
\(155\) −663276. −2.21751
\(156\) −455198. −1.49757
\(157\) −361773. −1.17135 −0.585675 0.810546i \(-0.699171\pi\)
−0.585675 + 0.810546i \(0.699171\pi\)
\(158\) 230950. 0.735995
\(159\) −183095. −0.574360
\(160\) 460442. 1.42192
\(161\) −66645.0 −0.202630
\(162\) −61196.4 −0.183205
\(163\) −627275. −1.84922 −0.924611 0.380912i \(-0.875610\pi\)
−0.924611 + 0.380912i \(0.875610\pi\)
\(164\) 986750. 2.86482
\(165\) 159484. 0.456043
\(166\) 287516. 0.809826
\(167\) 9849.85 0.0273299 0.0136650 0.999907i \(-0.495650\pi\)
0.0136650 + 0.999907i \(0.495650\pi\)
\(168\) −243225. −0.664866
\(169\) 474405. 1.27771
\(170\) −1.45854e6 −3.87075
\(171\) 36284.7 0.0948929
\(172\) 589070. 1.51826
\(173\) −250617. −0.636641 −0.318320 0.947983i \(-0.603119\pi\)
−0.318320 + 0.947983i \(0.603119\pi\)
\(174\) 109021. 0.272985
\(175\) 858898. 2.12005
\(176\) 42807.1 0.104168
\(177\) 256397. 0.615148
\(178\) 620560. 1.46802
\(179\) 742331. 1.73167 0.865835 0.500330i \(-0.166788\pi\)
0.865835 + 0.500330i \(0.166788\pi\)
\(180\) 444206. 1.02189
\(181\) −292957. −0.664673 −0.332336 0.943161i \(-0.607837\pi\)
−0.332336 + 0.943161i \(0.607837\pi\)
\(182\) 1.08062e6 2.41822
\(183\) 146000. 0.322275
\(184\) 113477. 0.247095
\(185\) −803363. −1.72577
\(186\) 558397. 1.18348
\(187\) 278699. 0.582817
\(188\) −983751. −2.02997
\(189\) 91841.6 0.187019
\(190\) −416624. −0.837259
\(191\) 57674.2 0.114393 0.0571963 0.998363i \(-0.481784\pi\)
0.0571963 + 0.998363i \(0.481784\pi\)
\(192\) −457008. −0.894685
\(193\) −178690. −0.345308 −0.172654 0.984983i \(-0.555234\pi\)
−0.172654 + 0.984983i \(0.555234\pi\)
\(194\) 974754. 1.85948
\(195\) −825276. −1.55422
\(196\) −51439.6 −0.0956439
\(197\) −525992. −0.965636 −0.482818 0.875721i \(-0.660387\pi\)
−0.482818 + 0.875721i \(0.660387\pi\)
\(198\) −134266. −0.243389
\(199\) −184283. −0.329877 −0.164939 0.986304i \(-0.552743\pi\)
−0.164939 + 0.986304i \(0.552743\pi\)
\(200\) −1.46246e6 −2.58528
\(201\) 487152. 0.850500
\(202\) 1.89453e6 3.26681
\(203\) −163616. −0.278667
\(204\) 776255. 1.30596
\(205\) 1.78898e6 2.97318
\(206\) 1.38167e6 2.26849
\(207\) −42849.0 −0.0695048
\(208\) −221513. −0.355010
\(209\) 79609.1 0.126066
\(210\) −1.05453e6 −1.65010
\(211\) −313460. −0.484703 −0.242352 0.970188i \(-0.577919\pi\)
−0.242352 + 0.970188i \(0.577919\pi\)
\(212\) −1.11888e6 −1.70980
\(213\) 382550. 0.577749
\(214\) 408077. 0.609127
\(215\) 1.06799e6 1.57569
\(216\) −156380. −0.228059
\(217\) −838024. −1.20811
\(218\) −1.76334e6 −2.51302
\(219\) 368426. 0.519086
\(220\) 974593. 1.35758
\(221\) −1.44218e6 −1.98627
\(222\) 676333. 0.921039
\(223\) −1.00274e6 −1.35028 −0.675141 0.737689i \(-0.735917\pi\)
−0.675141 + 0.737689i \(0.735917\pi\)
\(224\) 581751. 0.774671
\(225\) 552223. 0.727208
\(226\) 768404. 1.00073
\(227\) −1.02818e6 −1.32435 −0.662176 0.749348i \(-0.730367\pi\)
−0.662176 + 0.749348i \(0.730367\pi\)
\(228\) 221734. 0.282484
\(229\) −83843.9 −0.105653 −0.0528266 0.998604i \(-0.516823\pi\)
−0.0528266 + 0.998604i \(0.516823\pi\)
\(230\) 491995. 0.613255
\(231\) 201502. 0.248455
\(232\) 278591. 0.339819
\(233\) −241626. −0.291577 −0.145789 0.989316i \(-0.546572\pi\)
−0.145789 + 0.989316i \(0.546572\pi\)
\(234\) 694781. 0.829485
\(235\) −1.78355e6 −2.10676
\(236\) 1.56682e6 1.83122
\(237\) −222846. −0.257711
\(238\) −1.84281e6 −2.10881
\(239\) 1.32779e6 1.50361 0.751806 0.659384i \(-0.229183\pi\)
0.751806 + 0.659384i \(0.229183\pi\)
\(240\) 216164. 0.242245
\(241\) −1.21493e6 −1.34744 −0.673721 0.738986i \(-0.735305\pi\)
−0.673721 + 0.738986i \(0.735305\pi\)
\(242\) 1.20759e6 1.32550
\(243\) 59049.0 0.0641500
\(244\) 892199. 0.959372
\(245\) −93260.3 −0.0992617
\(246\) −1.50610e6 −1.58678
\(247\) −411952. −0.429639
\(248\) 1.42691e6 1.47322
\(249\) −277427. −0.283564
\(250\) −3.43427e6 −3.47523
\(251\) 1.21766e6 1.21995 0.609977 0.792419i \(-0.291179\pi\)
0.609977 + 0.792419i \(0.291179\pi\)
\(252\) 561238. 0.556731
\(253\) −94011.2 −0.0923376
\(254\) 889100. 0.864702
\(255\) 1.40736e6 1.35536
\(256\) −1.41448e6 −1.34896
\(257\) 1.21439e6 1.14690 0.573450 0.819241i \(-0.305605\pi\)
0.573450 + 0.819241i \(0.305605\pi\)
\(258\) −899115. −0.840942
\(259\) −1.01502e6 −0.940210
\(260\) −5.04321e6 −4.62672
\(261\) −105196. −0.0955867
\(262\) −1.69733e6 −1.52761
\(263\) 93687.8 0.0835206 0.0417603 0.999128i \(-0.486703\pi\)
0.0417603 + 0.999128i \(0.486703\pi\)
\(264\) −343099. −0.302977
\(265\) −2.02854e6 −1.77447
\(266\) −526388. −0.456144
\(267\) −598784. −0.514034
\(268\) 2.97695e6 2.53183
\(269\) −323647. −0.272703 −0.136352 0.990660i \(-0.543538\pi\)
−0.136352 + 0.990660i \(0.543538\pi\)
\(270\) −678004. −0.566009
\(271\) 1.76235e6 1.45771 0.728853 0.684670i \(-0.240054\pi\)
0.728853 + 0.684670i \(0.240054\pi\)
\(272\) 377749. 0.309586
\(273\) −1.04271e6 −0.846750
\(274\) −1.63250e6 −1.31364
\(275\) 1.21158e6 0.966100
\(276\) −261847. −0.206907
\(277\) −1.83478e6 −1.43676 −0.718381 0.695650i \(-0.755117\pi\)
−0.718381 + 0.695650i \(0.755117\pi\)
\(278\) −1.86240e6 −1.44531
\(279\) −538803. −0.414399
\(280\) −2.69472e6 −2.05409
\(281\) −1.71252e6 −1.29381 −0.646903 0.762572i \(-0.723936\pi\)
−0.646903 + 0.762572i \(0.723936\pi\)
\(282\) 1.50153e6 1.12437
\(283\) 1.53094e6 1.13629 0.568147 0.822927i \(-0.307660\pi\)
0.568147 + 0.822927i \(0.307660\pi\)
\(284\) 2.33774e6 1.71989
\(285\) 402004. 0.293169
\(286\) 1.52436e6 1.10198
\(287\) 2.26031e6 1.61981
\(288\) 374034. 0.265723
\(289\) 1.03951e6 0.732125
\(290\) 1.20787e6 0.843381
\(291\) −940550. −0.651103
\(292\) 2.25142e6 1.54526
\(293\) −2.02997e6 −1.38140 −0.690701 0.723141i \(-0.742698\pi\)
−0.690701 + 0.723141i \(0.742698\pi\)
\(294\) 78513.8 0.0529758
\(295\) 2.84066e6 1.90049
\(296\) 1.72829e6 1.14653
\(297\) 129554. 0.0852237
\(298\) −663868. −0.433053
\(299\) 486478. 0.314692
\(300\) 3.37460e6 2.16481
\(301\) 1.34936e6 0.858445
\(302\) −1.24511e6 −0.785578
\(303\) −1.82806e6 −1.14389
\(304\) 107902. 0.0669648
\(305\) 1.61756e6 0.995661
\(306\) −1.18482e6 −0.723351
\(307\) 396907. 0.240349 0.120175 0.992753i \(-0.461655\pi\)
0.120175 + 0.992753i \(0.461655\pi\)
\(308\) 1.23136e6 0.739621
\(309\) −1.33319e6 −0.794319
\(310\) 6.18657e6 3.65633
\(311\) 1.27993e6 0.750385 0.375192 0.926947i \(-0.377577\pi\)
0.375192 + 0.926947i \(0.377577\pi\)
\(312\) 1.77543e6 1.03256
\(313\) −2.41300e6 −1.39218 −0.696091 0.717953i \(-0.745079\pi\)
−0.696091 + 0.717953i \(0.745079\pi\)
\(314\) 3.37436e6 1.93138
\(315\) 1.01753e6 0.577790
\(316\) −1.36180e6 −0.767175
\(317\) 1.03063e6 0.576043 0.288022 0.957624i \(-0.407002\pi\)
0.288022 + 0.957624i \(0.407002\pi\)
\(318\) 1.70778e6 0.947032
\(319\) −230801. −0.126988
\(320\) −5.06326e6 −2.76411
\(321\) −393758. −0.213288
\(322\) 621617. 0.334105
\(323\) 702507. 0.374666
\(324\) 360844. 0.190967
\(325\) −6.26956e6 −3.29252
\(326\) 5.85078e6 3.04909
\(327\) 1.70147e6 0.879942
\(328\) −3.84867e6 −1.97527
\(329\) −2.25345e6 −1.14778
\(330\) −1.48755e6 −0.751946
\(331\) 231656. 0.116218 0.0581089 0.998310i \(-0.481493\pi\)
0.0581089 + 0.998310i \(0.481493\pi\)
\(332\) −1.69534e6 −0.844134
\(333\) −652601. −0.322505
\(334\) −91872.4 −0.0450629
\(335\) 5.39723e6 2.62760
\(336\) 273115. 0.131977
\(337\) −351073. −0.168393 −0.0841963 0.996449i \(-0.526832\pi\)
−0.0841963 + 0.996449i \(0.526832\pi\)
\(338\) −4.42491e6 −2.10675
\(339\) −741441. −0.350411
\(340\) 8.60026e6 4.03473
\(341\) −1.18214e6 −0.550532
\(342\) −338438. −0.156464
\(343\) −2.23523e6 −1.02586
\(344\) −2.29758e6 −1.04683
\(345\) −474731. −0.214734
\(346\) 2.33757e6 1.04972
\(347\) −550941. −0.245630 −0.122815 0.992430i \(-0.539192\pi\)
−0.122815 + 0.992430i \(0.539192\pi\)
\(348\) −642845. −0.284550
\(349\) −1.30593e6 −0.573928 −0.286964 0.957941i \(-0.592646\pi\)
−0.286964 + 0.957941i \(0.592646\pi\)
\(350\) −8.01119e6 −3.49564
\(351\) −670402. −0.290447
\(352\) 820634. 0.353015
\(353\) 2.13442e6 0.911682 0.455841 0.890061i \(-0.349339\pi\)
0.455841 + 0.890061i \(0.349339\pi\)
\(354\) −2.39149e6 −1.01429
\(355\) 4.23834e6 1.78494
\(356\) −3.65913e6 −1.53022
\(357\) 1.77814e6 0.738407
\(358\) −6.92394e6 −2.85526
\(359\) 1.06632e6 0.436670 0.218335 0.975874i \(-0.429937\pi\)
0.218335 + 0.975874i \(0.429937\pi\)
\(360\) −1.73256e6 −0.704582
\(361\) −2.27543e6 −0.918958
\(362\) 2.73250e6 1.09594
\(363\) −1.16522e6 −0.464130
\(364\) −6.37191e6 −2.52067
\(365\) 4.08185e6 1.60371
\(366\) −1.36179e6 −0.531382
\(367\) 1.34294e6 0.520464 0.260232 0.965546i \(-0.416201\pi\)
0.260232 + 0.965546i \(0.416201\pi\)
\(368\) −127423. −0.0490487
\(369\) 1.45326e6 0.555618
\(370\) 7.49320e6 2.84553
\(371\) −2.56299e6 −0.966744
\(372\) −3.29259e6 −1.23362
\(373\) 3.22658e6 1.20080 0.600400 0.799700i \(-0.295008\pi\)
0.600400 + 0.799700i \(0.295008\pi\)
\(374\) −2.59951e6 −0.960976
\(375\) 3.31376e6 1.21687
\(376\) 3.83697e6 1.39965
\(377\) 1.19432e6 0.432781
\(378\) −856633. −0.308365
\(379\) −2.41939e6 −0.865184 −0.432592 0.901590i \(-0.642401\pi\)
−0.432592 + 0.901590i \(0.642401\pi\)
\(380\) 2.45662e6 0.872729
\(381\) −857902. −0.302779
\(382\) −537944. −0.188616
\(383\) 1.03280e6 0.359764 0.179882 0.983688i \(-0.442428\pi\)
0.179882 + 0.983688i \(0.442428\pi\)
\(384\) 2.93275e6 1.01495
\(385\) 2.23247e6 0.767597
\(386\) 1.66669e6 0.569360
\(387\) 867565. 0.294459
\(388\) −5.74764e6 −1.93825
\(389\) 2.56135e6 0.858212 0.429106 0.903254i \(-0.358829\pi\)
0.429106 + 0.903254i \(0.358829\pi\)
\(390\) 7.69759e6 2.56267
\(391\) −829598. −0.274426
\(392\) 200632. 0.0659456
\(393\) 1.63777e6 0.534900
\(394\) 4.90608e6 1.59219
\(395\) −2.46894e6 −0.796193
\(396\) 791697. 0.253700
\(397\) −2.07970e6 −0.662255 −0.331128 0.943586i \(-0.607429\pi\)
−0.331128 + 0.943586i \(0.607429\pi\)
\(398\) 1.71886e6 0.543917
\(399\) 507918. 0.159721
\(400\) 1.64218e6 0.513182
\(401\) −1.97329e6 −0.612817 −0.306408 0.951900i \(-0.599127\pi\)
−0.306408 + 0.951900i \(0.599127\pi\)
\(402\) −4.54381e6 −1.40235
\(403\) 6.11719e6 1.87624
\(404\) −1.11711e7 −3.40521
\(405\) 654213. 0.198190
\(406\) 1.52609e6 0.459480
\(407\) −1.43181e6 −0.428450
\(408\) −3.02766e6 −0.900445
\(409\) 1.68806e6 0.498975 0.249488 0.968378i \(-0.419738\pi\)
0.249488 + 0.968378i \(0.419738\pi\)
\(410\) −1.66864e7 −4.90233
\(411\) 1.57522e6 0.459978
\(412\) −8.14702e6 −2.36459
\(413\) 3.58907e6 1.03540
\(414\) 399665. 0.114603
\(415\) −3.07366e6 −0.876064
\(416\) −4.24652e6 −1.20309
\(417\) 1.79705e6 0.506080
\(418\) −742538. −0.207863
\(419\) 1.72868e6 0.481039 0.240520 0.970644i \(-0.422682\pi\)
0.240520 + 0.970644i \(0.422682\pi\)
\(420\) 6.21804e6 1.72001
\(421\) 932870. 0.256517 0.128258 0.991741i \(-0.459061\pi\)
0.128258 + 0.991741i \(0.459061\pi\)
\(422\) 2.92373e6 0.799202
\(423\) −1.44884e6 −0.393703
\(424\) 4.36403e6 1.17889
\(425\) 1.06916e7 2.87124
\(426\) −3.56816e6 −0.952621
\(427\) 2.04373e6 0.542443
\(428\) −2.40623e6 −0.634932
\(429\) −1.47087e6 −0.385861
\(430\) −9.96144e6 −2.59807
\(431\) −7.62939e6 −1.97832 −0.989161 0.146838i \(-0.953091\pi\)
−0.989161 + 0.146838i \(0.953091\pi\)
\(432\) 175598. 0.0452699
\(433\) 4.68013e6 1.19960 0.599802 0.800148i \(-0.295246\pi\)
0.599802 + 0.800148i \(0.295246\pi\)
\(434\) 7.81650e6 1.99199
\(435\) −1.16548e6 −0.295313
\(436\) 1.03976e7 2.61948
\(437\) −236971. −0.0593596
\(438\) −3.43641e6 −0.855895
\(439\) 1.21981e6 0.302087 0.151043 0.988527i \(-0.451737\pi\)
0.151043 + 0.988527i \(0.451737\pi\)
\(440\) −3.80125e6 −0.936041
\(441\) −75758.7 −0.0185497
\(442\) 1.34516e7 3.27506
\(443\) −1.82127e6 −0.440926 −0.220463 0.975395i \(-0.570757\pi\)
−0.220463 + 0.975395i \(0.570757\pi\)
\(444\) −3.98800e6 −0.960058
\(445\) −6.63403e6 −1.58810
\(446\) 9.35281e6 2.22641
\(447\) 640573. 0.151635
\(448\) −6.39724e6 −1.50591
\(449\) −3.31712e6 −0.776506 −0.388253 0.921553i \(-0.626921\pi\)
−0.388253 + 0.921553i \(0.626921\pi\)
\(450\) −5.15075e6 −1.19905
\(451\) 3.18846e6 0.738142
\(452\) −4.53090e6 −1.04313
\(453\) 1.20142e6 0.275073
\(454\) 9.59012e6 2.18366
\(455\) −1.15523e7 −2.61602
\(456\) −864838. −0.194770
\(457\) −2.14897e6 −0.481326 −0.240663 0.970609i \(-0.577365\pi\)
−0.240663 + 0.970609i \(0.577365\pi\)
\(458\) 782036. 0.174206
\(459\) 1.14325e6 0.253284
\(460\) −2.90105e6 −0.639235
\(461\) −6.83686e6 −1.49832 −0.749160 0.662389i \(-0.769543\pi\)
−0.749160 + 0.662389i \(0.769543\pi\)
\(462\) −1.87946e6 −0.409665
\(463\) 3.26293e6 0.707385 0.353692 0.935362i \(-0.384926\pi\)
0.353692 + 0.935362i \(0.384926\pi\)
\(464\) −312828. −0.0674544
\(465\) −5.96948e6 −1.28028
\(466\) 2.25372e6 0.480767
\(467\) −4.30238e6 −0.912886 −0.456443 0.889753i \(-0.650877\pi\)
−0.456443 + 0.889753i \(0.650877\pi\)
\(468\) −4.09678e6 −0.864625
\(469\) 6.81920e6 1.43153
\(470\) 1.66357e7 3.47373
\(471\) −3.25595e6 −0.676279
\(472\) −6.11116e6 −1.26261
\(473\) 1.90345e6 0.391191
\(474\) 2.07855e6 0.424927
\(475\) 3.05400e6 0.621061
\(476\) 1.08661e7 2.19815
\(477\) −1.64786e6 −0.331607
\(478\) −1.23847e7 −2.47923
\(479\) 3.15612e6 0.628514 0.314257 0.949338i \(-0.398245\pi\)
0.314257 + 0.949338i \(0.398245\pi\)
\(480\) 4.14398e6 0.820946
\(481\) 7.40918e6 1.46018
\(482\) 1.13320e7 2.22173
\(483\) −599805. −0.116988
\(484\) −7.12055e6 −1.38166
\(485\) −1.04205e7 −2.01157
\(486\) −550767. −0.105774
\(487\) −1.04981e6 −0.200581 −0.100290 0.994958i \(-0.531977\pi\)
−0.100290 + 0.994958i \(0.531977\pi\)
\(488\) −3.47988e6 −0.661478
\(489\) −5.64548e6 −1.06765
\(490\) 869867. 0.163667
\(491\) 1.67278e6 0.313138 0.156569 0.987667i \(-0.449957\pi\)
0.156569 + 0.987667i \(0.449957\pi\)
\(492\) 8.88075e6 1.65401
\(493\) −2.03669e6 −0.377406
\(494\) 3.84240e6 0.708410
\(495\) 1.43535e6 0.263297
\(496\) −1.60227e6 −0.292437
\(497\) 5.35498e6 0.972449
\(498\) 2.58765e6 0.467554
\(499\) −3.07979e6 −0.553695 −0.276847 0.960914i \(-0.589290\pi\)
−0.276847 + 0.960914i \(0.589290\pi\)
\(500\) 2.02502e7 3.62246
\(501\) 88648.7 0.0157789
\(502\) −1.13575e7 −2.01152
\(503\) 4.06934e6 0.717140 0.358570 0.933503i \(-0.383264\pi\)
0.358570 + 0.933503i \(0.383264\pi\)
\(504\) −2.18902e6 −0.383861
\(505\) −2.02533e7 −3.53401
\(506\) 876870. 0.152251
\(507\) 4.26964e6 0.737686
\(508\) −5.24258e6 −0.901334
\(509\) 5.65269e6 0.967077 0.483538 0.875323i \(-0.339351\pi\)
0.483538 + 0.875323i \(0.339351\pi\)
\(510\) −1.31268e7 −2.23478
\(511\) 5.15726e6 0.873709
\(512\) 2.76575e6 0.466271
\(513\) 326563. 0.0547864
\(514\) −1.13270e7 −1.89106
\(515\) −1.47706e7 −2.45403
\(516\) 5.30163e6 0.876568
\(517\) −3.17877e6 −0.523038
\(518\) 9.46738e6 1.55026
\(519\) −2.25555e6 −0.367565
\(520\) 1.96703e7 3.19008
\(521\) −729056. −0.117670 −0.0588351 0.998268i \(-0.518739\pi\)
−0.0588351 + 0.998268i \(0.518739\pi\)
\(522\) 981193. 0.157608
\(523\) 5.74086e6 0.917747 0.458873 0.888502i \(-0.348253\pi\)
0.458873 + 0.888502i \(0.348253\pi\)
\(524\) 1.00083e7 1.59233
\(525\) 7.73008e6 1.22401
\(526\) −873853. −0.137713
\(527\) −1.04317e7 −1.63618
\(528\) 385264. 0.0601414
\(529\) 279841. 0.0434783
\(530\) 1.89208e7 2.92583
\(531\) 2.30757e6 0.355156
\(532\) 3.10385e6 0.475468
\(533\) −1.64993e7 −2.51563
\(534\) 5.58504e6 0.847565
\(535\) −4.36250e6 −0.658948
\(536\) −1.16111e7 −1.74567
\(537\) 6.68098e6 0.999780
\(538\) 3.01875e6 0.449647
\(539\) −166216. −0.0246434
\(540\) 3.99785e6 0.589987
\(541\) 1.12666e7 1.65500 0.827500 0.561466i \(-0.189762\pi\)
0.827500 + 0.561466i \(0.189762\pi\)
\(542\) −1.64380e7 −2.40354
\(543\) −2.63661e6 −0.383749
\(544\) 7.24165e6 1.04916
\(545\) 1.88508e7 2.71856
\(546\) 9.72562e6 1.39616
\(547\) 1.92802e6 0.275513 0.137757 0.990466i \(-0.456011\pi\)
0.137757 + 0.990466i \(0.456011\pi\)
\(548\) 9.62606e6 1.36930
\(549\) 1.31400e6 0.186065
\(550\) −1.13008e7 −1.59295
\(551\) −581772. −0.0816345
\(552\) 1.02130e6 0.142660
\(553\) −3.11942e6 −0.433771
\(554\) 1.71135e7 2.36900
\(555\) −7.23027e6 −0.996373
\(556\) 1.09816e7 1.50654
\(557\) −1.27385e7 −1.73973 −0.869865 0.493290i \(-0.835794\pi\)
−0.869865 + 0.493290i \(0.835794\pi\)
\(558\) 5.02557e6 0.683282
\(559\) −9.84974e6 −1.33320
\(560\) 3.02589e6 0.407739
\(561\) 2.50829e6 0.336489
\(562\) 1.59732e7 2.13329
\(563\) 1.30970e7 1.74140 0.870702 0.491810i \(-0.163665\pi\)
0.870702 + 0.491810i \(0.163665\pi\)
\(564\) −8.85376e6 −1.17201
\(565\) −8.21454e6 −1.08259
\(566\) −1.42795e7 −1.87358
\(567\) 826574. 0.107975
\(568\) −9.11799e6 −1.18585
\(569\) 1.32454e7 1.71509 0.857543 0.514413i \(-0.171990\pi\)
0.857543 + 0.514413i \(0.171990\pi\)
\(570\) −3.74961e6 −0.483392
\(571\) −4.22643e6 −0.542480 −0.271240 0.962512i \(-0.587434\pi\)
−0.271240 + 0.962512i \(0.587434\pi\)
\(572\) −8.98838e6 −1.14866
\(573\) 519068. 0.0660446
\(574\) −2.10826e7 −2.67082
\(575\) −3.60650e6 −0.454900
\(576\) −4.11307e6 −0.516547
\(577\) −224943. −0.0281276 −0.0140638 0.999901i \(-0.504477\pi\)
−0.0140638 + 0.999901i \(0.504477\pi\)
\(578\) −9.69584e6 −1.20716
\(579\) −1.60821e6 −0.199364
\(580\) −7.12219e6 −0.879110
\(581\) −3.88346e6 −0.477285
\(582\) 8.77279e6 1.07357
\(583\) −3.61542e6 −0.440542
\(584\) −8.78134e6 −1.06544
\(585\) −7.42749e6 −0.897330
\(586\) 1.89341e7 2.27772
\(587\) 4.80198e6 0.575209 0.287604 0.957749i \(-0.407141\pi\)
0.287604 + 0.957749i \(0.407141\pi\)
\(588\) −462956. −0.0552201
\(589\) −2.97978e6 −0.353912
\(590\) −2.64957e7 −3.13361
\(591\) −4.73392e6 −0.557510
\(592\) −1.94068e6 −0.227588
\(593\) −9.07891e6 −1.06022 −0.530111 0.847928i \(-0.677850\pi\)
−0.530111 + 0.847928i \(0.677850\pi\)
\(594\) −1.20839e6 −0.140521
\(595\) 1.97003e7 2.28129
\(596\) 3.91450e6 0.451399
\(597\) −1.65855e6 −0.190455
\(598\) −4.53752e6 −0.518879
\(599\) −487665. −0.0555334 −0.0277667 0.999614i \(-0.508840\pi\)
−0.0277667 + 0.999614i \(0.508840\pi\)
\(600\) −1.31621e7 −1.49261
\(601\) −9.72337e6 −1.09807 −0.549036 0.835799i \(-0.685005\pi\)
−0.549036 + 0.835799i \(0.685005\pi\)
\(602\) −1.25859e7 −1.41545
\(603\) 4.38437e6 0.491036
\(604\) 7.34177e6 0.818858
\(605\) −1.29096e7 −1.43392
\(606\) 1.70508e7 1.88609
\(607\) −3.30773e6 −0.364384 −0.182192 0.983263i \(-0.558319\pi\)
−0.182192 + 0.983263i \(0.558319\pi\)
\(608\) 2.06854e6 0.226937
\(609\) −1.47254e6 −0.160888
\(610\) −1.50875e7 −1.64169
\(611\) 1.64491e7 1.78254
\(612\) 6.98630e6 0.753995
\(613\) −234065. −0.0251585 −0.0125793 0.999921i \(-0.504004\pi\)
−0.0125793 + 0.999921i \(0.504004\pi\)
\(614\) −3.70207e6 −0.396300
\(615\) 1.61009e7 1.71657
\(616\) −4.80274e6 −0.509961
\(617\) −9.19095e6 −0.971958 −0.485979 0.873970i \(-0.661537\pi\)
−0.485979 + 0.873970i \(0.661537\pi\)
\(618\) 1.24350e7 1.30971
\(619\) 7.15853e6 0.750926 0.375463 0.926837i \(-0.377484\pi\)
0.375463 + 0.926837i \(0.377484\pi\)
\(620\) −3.64791e7 −3.81123
\(621\) −385641. −0.0401286
\(622\) −1.19382e7 −1.23727
\(623\) −8.38185e6 −0.865206
\(624\) −1.99362e6 −0.204965
\(625\) 1.54088e7 1.57786
\(626\) 2.25067e7 2.29550
\(627\) 716482. 0.0727841
\(628\) −1.98969e7 −2.01320
\(629\) −1.26350e7 −1.27335
\(630\) −9.49078e6 −0.952687
\(631\) 1.00474e7 1.00457 0.502283 0.864703i \(-0.332493\pi\)
0.502283 + 0.864703i \(0.332493\pi\)
\(632\) 5.31147e6 0.528960
\(633\) −2.82114e6 −0.279844
\(634\) −9.61300e6 −0.949808
\(635\) −9.50483e6 −0.935427
\(636\) −1.00699e7 −0.987152
\(637\) 860112. 0.0839859
\(638\) 2.15275e6 0.209383
\(639\) 3.44295e6 0.333564
\(640\) 3.24924e7 3.13568
\(641\) 4.21517e6 0.405201 0.202600 0.979262i \(-0.435061\pi\)
0.202600 + 0.979262i \(0.435061\pi\)
\(642\) 3.67269e6 0.351680
\(643\) 9.48561e6 0.904770 0.452385 0.891823i \(-0.350573\pi\)
0.452385 + 0.891823i \(0.350573\pi\)
\(644\) −3.66537e6 −0.348259
\(645\) 9.61189e6 0.909724
\(646\) −6.55249e6 −0.617768
\(647\) 2.54033e6 0.238578 0.119289 0.992860i \(-0.461939\pi\)
0.119289 + 0.992860i \(0.461939\pi\)
\(648\) −1.40742e6 −0.131670
\(649\) 5.06284e6 0.471827
\(650\) 5.84781e7 5.42887
\(651\) −7.54222e6 −0.697504
\(652\) −3.44991e7 −3.17826
\(653\) 9.06671e6 0.832083 0.416042 0.909346i \(-0.363417\pi\)
0.416042 + 0.909346i \(0.363417\pi\)
\(654\) −1.58701e7 −1.45089
\(655\) 1.81452e7 1.65256
\(656\) 4.32164e6 0.392093
\(657\) 3.31583e6 0.299695
\(658\) 2.10185e7 1.89251
\(659\) −1.63938e7 −1.47050 −0.735251 0.677795i \(-0.762936\pi\)
−0.735251 + 0.677795i \(0.762936\pi\)
\(660\) 8.77134e6 0.783802
\(661\) −1.06203e7 −0.945435 −0.472717 0.881214i \(-0.656727\pi\)
−0.472717 + 0.881214i \(0.656727\pi\)
\(662\) −2.16072e6 −0.191626
\(663\) −1.29796e7 −1.14677
\(664\) 6.61241e6 0.582022
\(665\) 5.62730e6 0.493453
\(666\) 6.08700e6 0.531762
\(667\) 687020. 0.0597936
\(668\) 541726. 0.0469719
\(669\) −9.02462e6 −0.779586
\(670\) −5.03416e7 −4.33251
\(671\) 2.88294e6 0.247189
\(672\) 5.23576e6 0.447257
\(673\) −1.19056e7 −1.01324 −0.506620 0.862169i \(-0.669105\pi\)
−0.506620 + 0.862169i \(0.669105\pi\)
\(674\) 3.27456e6 0.277654
\(675\) 4.97001e6 0.419854
\(676\) 2.60915e7 2.19600
\(677\) 2.17878e7 1.82701 0.913507 0.406822i \(-0.133363\pi\)
0.913507 + 0.406822i \(0.133363\pi\)
\(678\) 6.91564e6 0.577774
\(679\) −1.31659e7 −1.09592
\(680\) −3.35440e7 −2.78190
\(681\) −9.25360e6 −0.764615
\(682\) 1.10262e7 0.907744
\(683\) 8.27503e6 0.678762 0.339381 0.940649i \(-0.389782\pi\)
0.339381 + 0.940649i \(0.389782\pi\)
\(684\) 1.99560e6 0.163092
\(685\) 1.74521e7 1.42109
\(686\) 2.08486e7 1.69148
\(687\) −754595. −0.0609989
\(688\) 2.57993e6 0.207796
\(689\) 1.87086e7 1.50139
\(690\) 4.42795e6 0.354063
\(691\) 8.37140e6 0.666965 0.333483 0.942756i \(-0.391776\pi\)
0.333483 + 0.942756i \(0.391776\pi\)
\(692\) −1.37835e7 −1.09419
\(693\) 1.81351e6 0.143446
\(694\) 5.13879e6 0.405007
\(695\) 1.99098e7 1.56352
\(696\) 2.50732e6 0.196194
\(697\) 2.81364e7 2.19375
\(698\) 1.21808e7 0.946320
\(699\) −2.17463e6 −0.168342
\(700\) 4.72380e7 3.64373
\(701\) 3.01421e6 0.231675 0.115837 0.993268i \(-0.463045\pi\)
0.115837 + 0.993268i \(0.463045\pi\)
\(702\) 6.25303e6 0.478903
\(703\) −3.60912e6 −0.275431
\(704\) −9.02412e6 −0.686236
\(705\) −1.60519e7 −1.21634
\(706\) −1.99084e7 −1.50323
\(707\) −2.55893e7 −1.92535
\(708\) 1.41014e7 1.05725
\(709\) −5.74628e6 −0.429310 −0.214655 0.976690i \(-0.568863\pi\)
−0.214655 + 0.976690i \(0.568863\pi\)
\(710\) −3.95322e7 −2.94310
\(711\) −2.00561e6 −0.148790
\(712\) 1.42719e7 1.05507
\(713\) 3.51885e6 0.259225
\(714\) −1.65852e7 −1.21752
\(715\) −1.62960e7 −1.19211
\(716\) 4.08270e7 2.97622
\(717\) 1.19501e7 0.868111
\(718\) −9.94592e6 −0.720002
\(719\) −1.64352e7 −1.18564 −0.592821 0.805334i \(-0.701986\pi\)
−0.592821 + 0.805334i \(0.701986\pi\)
\(720\) 1.94548e6 0.139860
\(721\) −1.86621e7 −1.33697
\(722\) 2.12236e7 1.51522
\(723\) −1.09344e7 −0.777945
\(724\) −1.61122e7 −1.14237
\(725\) −8.85408e6 −0.625603
\(726\) 1.08683e7 0.765280
\(727\) −4.65331e6 −0.326532 −0.163266 0.986582i \(-0.552203\pi\)
−0.163266 + 0.986582i \(0.552203\pi\)
\(728\) 2.48526e7 1.73798
\(729\) 531441. 0.0370370
\(730\) −3.80726e7 −2.64427
\(731\) 1.67969e7 1.16261
\(732\) 8.02979e6 0.553894
\(733\) 1.76748e7 1.21505 0.607527 0.794299i \(-0.292162\pi\)
0.607527 + 0.794299i \(0.292162\pi\)
\(734\) −1.25260e7 −0.858166
\(735\) −839343. −0.0573088
\(736\) −2.44276e6 −0.166221
\(737\) 9.61936e6 0.652345
\(738\) −1.35549e7 −0.916129
\(739\) 2.32944e7 1.56907 0.784533 0.620087i \(-0.212903\pi\)
0.784533 + 0.620087i \(0.212903\pi\)
\(740\) −4.41837e7 −2.96608
\(741\) −3.70757e6 −0.248052
\(742\) 2.39057e7 1.59401
\(743\) 1.77107e7 1.17696 0.588482 0.808511i \(-0.299726\pi\)
0.588482 + 0.808511i \(0.299726\pi\)
\(744\) 1.28422e7 0.850566
\(745\) 7.09701e6 0.468473
\(746\) −3.00953e7 −1.97994
\(747\) −2.49685e6 −0.163716
\(748\) 1.53280e7 1.00169
\(749\) −5.51186e6 −0.358999
\(750\) −3.09084e7 −2.00643
\(751\) −9.59704e6 −0.620922 −0.310461 0.950586i \(-0.600483\pi\)
−0.310461 + 0.950586i \(0.600483\pi\)
\(752\) −4.30851e6 −0.277832
\(753\) 1.09590e7 0.704340
\(754\) −1.11398e7 −0.713590
\(755\) 1.33107e7 0.849832
\(756\) 5.05114e6 0.321429
\(757\) 1.81520e7 1.15129 0.575645 0.817700i \(-0.304751\pi\)
0.575645 + 0.817700i \(0.304751\pi\)
\(758\) 2.25664e7 1.42656
\(759\) −846101. −0.0533111
\(760\) −9.58167e6 −0.601738
\(761\) 2.18570e7 1.36813 0.684067 0.729419i \(-0.260210\pi\)
0.684067 + 0.729419i \(0.260210\pi\)
\(762\) 8.00190e6 0.499236
\(763\) 2.38173e7 1.48109
\(764\) 3.17199e6 0.196607
\(765\) 1.26662e7 0.782515
\(766\) −9.63319e6 −0.593196
\(767\) −2.61986e7 −1.60801
\(768\) −1.27303e7 −0.778820
\(769\) −3.01818e6 −0.184047 −0.0920237 0.995757i \(-0.529334\pi\)
−0.0920237 + 0.995757i \(0.529334\pi\)
\(770\) −2.08229e7 −1.26565
\(771\) 1.09295e7 0.662163
\(772\) −9.82765e6 −0.593480
\(773\) 7.09462e6 0.427052 0.213526 0.976937i \(-0.431505\pi\)
0.213526 + 0.976937i \(0.431505\pi\)
\(774\) −8.09204e6 −0.485518
\(775\) −4.53497e7 −2.71219
\(776\) 2.24178e7 1.33641
\(777\) −9.13517e6 −0.542830
\(778\) −2.38904e7 −1.41506
\(779\) 8.03704e6 0.474518
\(780\) −4.53889e7 −2.67124
\(781\) 7.55388e6 0.443142
\(782\) 7.73790e6 0.452487
\(783\) −946763. −0.0551870
\(784\) −225289. −0.0130903
\(785\) −3.60732e7 −2.08935
\(786\) −1.52760e7 −0.881969
\(787\) 2.85125e7 1.64096 0.820482 0.571672i \(-0.193705\pi\)
0.820482 + 0.571672i \(0.193705\pi\)
\(788\) −2.89287e7 −1.65964
\(789\) 843190. 0.0482206
\(790\) 2.30286e7 1.31280
\(791\) −1.03788e7 −0.589800
\(792\) −3.08789e6 −0.174924
\(793\) −1.49183e7 −0.842435
\(794\) 1.93980e7 1.09196
\(795\) −1.82569e7 −1.02449
\(796\) −1.01353e7 −0.566960
\(797\) 7.59475e6 0.423514 0.211757 0.977322i \(-0.432081\pi\)
0.211757 + 0.977322i \(0.432081\pi\)
\(798\) −4.73750e6 −0.263355
\(799\) −2.80509e7 −1.55446
\(800\) 3.14815e7 1.73912
\(801\) −5.38906e6 −0.296778
\(802\) 1.84055e7 1.01044
\(803\) 7.27497e6 0.398146
\(804\) 2.67926e7 1.46175
\(805\) −6.64533e6 −0.361432
\(806\) −5.70568e7 −3.09364
\(807\) −2.91282e6 −0.157445
\(808\) 4.35713e7 2.34786
\(809\) 135394. 0.00727324 0.00363662 0.999993i \(-0.498842\pi\)
0.00363662 + 0.999993i \(0.498842\pi\)
\(810\) −6.10204e6 −0.326785
\(811\) −2.84394e7 −1.51834 −0.759168 0.650895i \(-0.774394\pi\)
−0.759168 + 0.650895i \(0.774394\pi\)
\(812\) −8.99862e6 −0.478945
\(813\) 1.58612e7 0.841607
\(814\) 1.33549e7 0.706449
\(815\) −6.25471e7 −3.29848
\(816\) 3.39974e6 0.178740
\(817\) 4.79795e6 0.251479
\(818\) −1.57450e7 −0.822735
\(819\) −9.38435e6 −0.488871
\(820\) 9.83913e7 5.11001
\(821\) −9.25492e6 −0.479198 −0.239599 0.970872i \(-0.577016\pi\)
−0.239599 + 0.970872i \(0.577016\pi\)
\(822\) −1.46925e7 −0.758433
\(823\) 8.81289e6 0.453543 0.226772 0.973948i \(-0.427183\pi\)
0.226772 + 0.973948i \(0.427183\pi\)
\(824\) 3.17762e7 1.63036
\(825\) 1.09043e7 0.557778
\(826\) −3.34763e7 −1.70721
\(827\) 2.40557e7 1.22308 0.611539 0.791214i \(-0.290551\pi\)
0.611539 + 0.791214i \(0.290551\pi\)
\(828\) −2.35663e6 −0.119458
\(829\) 3.34899e7 1.69249 0.846247 0.532791i \(-0.178857\pi\)
0.846247 + 0.532791i \(0.178857\pi\)
\(830\) 2.86689e7 1.44450
\(831\) −1.65130e7 −0.829515
\(832\) 4.66969e7 2.33873
\(833\) −1.46676e6 −0.0732398
\(834\) −1.67616e7 −0.834449
\(835\) 982153. 0.0487487
\(836\) 4.37838e6 0.216669
\(837\) −4.84922e6 −0.239254
\(838\) −1.61239e7 −0.793160
\(839\) 6.66922e6 0.327092 0.163546 0.986536i \(-0.447707\pi\)
0.163546 + 0.986536i \(0.447707\pi\)
\(840\) −2.42525e7 −1.18593
\(841\) −1.88245e7 −0.917769
\(842\) −8.70115e6 −0.422957
\(843\) −1.54127e7 −0.746979
\(844\) −1.72398e7 −0.833060
\(845\) 4.73040e7 2.27906
\(846\) 1.35137e7 0.649157
\(847\) −1.63108e7 −0.781209
\(848\) −4.90034e6 −0.234011
\(849\) 1.37784e7 0.656040
\(850\) −9.97235e7 −4.73424
\(851\) 4.26205e6 0.201741
\(852\) 2.10397e7 0.992978
\(853\) 2.91274e6 0.137066 0.0685330 0.997649i \(-0.478168\pi\)
0.0685330 + 0.997649i \(0.478168\pi\)
\(854\) −1.90625e7 −0.894405
\(855\) 3.61804e6 0.169261
\(856\) 9.38512e6 0.437779
\(857\) 1.52102e7 0.707429 0.353715 0.935353i \(-0.384918\pi\)
0.353715 + 0.935353i \(0.384918\pi\)
\(858\) 1.37192e7 0.636226
\(859\) 1.06621e7 0.493016 0.246508 0.969141i \(-0.420717\pi\)
0.246508 + 0.969141i \(0.420717\pi\)
\(860\) 5.87376e7 2.70814
\(861\) 2.03428e7 0.935198
\(862\) 7.11616e7 3.26195
\(863\) −3.79585e7 −1.73493 −0.867465 0.497498i \(-0.834252\pi\)
−0.867465 + 0.497498i \(0.834252\pi\)
\(864\) 3.36630e6 0.153415
\(865\) −2.49896e7 −1.13558
\(866\) −4.36529e7 −1.97796
\(867\) 9.35562e6 0.422693
\(868\) −4.60900e7 −2.07638
\(869\) −4.40034e6 −0.197668
\(870\) 1.08708e7 0.486926
\(871\) −4.97771e7 −2.22323
\(872\) −4.05540e7 −1.80610
\(873\) −8.46495e6 −0.375914
\(874\) 2.21029e6 0.0978750
\(875\) 4.63863e7 2.04819
\(876\) 2.02628e7 0.892154
\(877\) 2.72329e7 1.19562 0.597812 0.801636i \(-0.296037\pi\)
0.597812 + 0.801636i \(0.296037\pi\)
\(878\) −1.13775e7 −0.498095
\(879\) −1.82697e7 −0.797553
\(880\) 4.26840e6 0.185805
\(881\) 8.49018e6 0.368534 0.184267 0.982876i \(-0.441009\pi\)
0.184267 + 0.982876i \(0.441009\pi\)
\(882\) 706624. 0.0305856
\(883\) −2.58940e7 −1.11763 −0.558815 0.829292i \(-0.688744\pi\)
−0.558815 + 0.829292i \(0.688744\pi\)
\(884\) −7.93176e7 −3.41381
\(885\) 2.55660e7 1.09725
\(886\) 1.69875e7 0.727020
\(887\) −1.57892e7 −0.673832 −0.336916 0.941535i \(-0.609384\pi\)
−0.336916 + 0.941535i \(0.609384\pi\)
\(888\) 1.55546e7 0.661951
\(889\) −1.20090e7 −0.509627
\(890\) 6.18775e7 2.61853
\(891\) 1.16599e6 0.0492039
\(892\) −5.51489e7 −2.32073
\(893\) −8.01261e6 −0.336237
\(894\) −5.97481e6 −0.250023
\(895\) 7.40196e7 3.08880
\(896\) 4.10529e7 1.70834
\(897\) 4.37830e6 0.181687
\(898\) 3.09397e7 1.28034
\(899\) 8.63890e6 0.356500
\(900\) 3.03714e7 1.24985
\(901\) −3.19041e7 −1.30929
\(902\) −2.97397e7 −1.21708
\(903\) 1.21443e7 0.495624
\(904\) 1.76721e7 0.719227
\(905\) −2.92115e7 −1.18558
\(906\) −1.12060e7 −0.453554
\(907\) −1.12308e7 −0.453305 −0.226653 0.973976i \(-0.572778\pi\)
−0.226653 + 0.973976i \(0.572778\pi\)
\(908\) −5.65481e7 −2.27616
\(909\) −1.64525e7 −0.660423
\(910\) 1.07752e8 4.31341
\(911\) 2.61794e7 1.04511 0.522557 0.852604i \(-0.324978\pi\)
0.522557 + 0.852604i \(0.324978\pi\)
\(912\) 971120. 0.0386621
\(913\) −5.47811e6 −0.217497
\(914\) 2.00441e7 0.793634
\(915\) 1.45581e7 0.574845
\(916\) −4.61128e6 −0.181586
\(917\) 2.29257e7 0.900326
\(918\) −1.06634e7 −0.417627
\(919\) 4.07119e7 1.59013 0.795065 0.606524i \(-0.207437\pi\)
0.795065 + 0.606524i \(0.207437\pi\)
\(920\) 1.13151e7 0.440746
\(921\) 3.57217e6 0.138766
\(922\) 6.37694e7 2.47050
\(923\) −3.90889e7 −1.51025
\(924\) 1.10823e7 0.427020
\(925\) −5.49278e7 −2.11075
\(926\) −3.04343e7 −1.16637
\(927\) −1.19987e7 −0.458600
\(928\) −5.99707e6 −0.228596
\(929\) 8.89152e6 0.338015 0.169008 0.985615i \(-0.445944\pi\)
0.169008 + 0.985615i \(0.445944\pi\)
\(930\) 5.56791e7 2.11098
\(931\) −418973. −0.0158421
\(932\) −1.32890e7 −0.501134
\(933\) 1.15193e7 0.433235
\(934\) 4.01296e7 1.50521
\(935\) 2.77898e7 1.03958
\(936\) 1.59789e7 0.596151
\(937\) 4.07318e6 0.151560 0.0757799 0.997125i \(-0.475855\pi\)
0.0757799 + 0.997125i \(0.475855\pi\)
\(938\) −6.36047e7 −2.36038
\(939\) −2.17170e7 −0.803777
\(940\) −9.80922e7 −3.62089
\(941\) −1.08008e7 −0.397634 −0.198817 0.980037i \(-0.563710\pi\)
−0.198817 + 0.980037i \(0.563710\pi\)
\(942\) 3.03692e7 1.11508
\(943\) −9.49102e6 −0.347563
\(944\) 6.86218e6 0.250629
\(945\) 9.15775e6 0.333587
\(946\) −1.77540e7 −0.645014
\(947\) 6.93307e6 0.251218 0.125609 0.992080i \(-0.459912\pi\)
0.125609 + 0.992080i \(0.459912\pi\)
\(948\) −1.22562e7 −0.442929
\(949\) −3.76457e7 −1.35690
\(950\) −2.84855e7 −1.02404
\(951\) 9.27568e6 0.332579
\(952\) −4.23816e7 −1.51560
\(953\) 3.81395e7 1.36033 0.680163 0.733060i \(-0.261909\pi\)
0.680163 + 0.733060i \(0.261909\pi\)
\(954\) 1.53700e7 0.546769
\(955\) 5.75084e6 0.204043
\(956\) 7.30265e7 2.58426
\(957\) −2.07721e6 −0.0733163
\(958\) −2.94381e7 −1.03632
\(959\) 2.20501e7 0.774219
\(960\) −4.55693e7 −1.59586
\(961\) 1.56184e7 0.545542
\(962\) −6.91076e7 −2.40762
\(963\) −3.54382e6 −0.123142
\(964\) −6.68194e7 −2.31585
\(965\) −1.78176e7 −0.615929
\(966\) 5.59456e6 0.192896
\(967\) 1.52545e6 0.0524603 0.0262302 0.999656i \(-0.491650\pi\)
0.0262302 + 0.999656i \(0.491650\pi\)
\(968\) 2.77726e7 0.952639
\(969\) 6.32257e6 0.216314
\(970\) 9.71951e7 3.31677
\(971\) −4.37167e7 −1.48799 −0.743993 0.668187i \(-0.767071\pi\)
−0.743993 + 0.668187i \(0.767071\pi\)
\(972\) 3.24760e6 0.110255
\(973\) 2.51553e7 0.851817
\(974\) 9.79190e6 0.330727
\(975\) −5.64261e7 −1.90094
\(976\) 3.90754e6 0.131304
\(977\) 1.59223e6 0.0533666 0.0266833 0.999644i \(-0.491505\pi\)
0.0266833 + 0.999644i \(0.491505\pi\)
\(978\) 5.26570e7 1.76039
\(979\) −1.18237e7 −0.394271
\(980\) −5.12917e6 −0.170601
\(981\) 1.53132e7 0.508035
\(982\) −1.56025e7 −0.516316
\(983\) 1.98366e7 0.654762 0.327381 0.944892i \(-0.393834\pi\)
0.327381 + 0.944892i \(0.393834\pi\)
\(984\) −3.46380e7 −1.14042
\(985\) −5.24479e7 −1.72241
\(986\) 1.89968e7 0.622285
\(987\) −2.02810e7 −0.662669
\(988\) −2.26567e7 −0.738421
\(989\) −5.66595e6 −0.184197
\(990\) −1.33879e7 −0.434136
\(991\) 5.77793e7 1.86891 0.934454 0.356083i \(-0.115888\pi\)
0.934454 + 0.356083i \(0.115888\pi\)
\(992\) −3.07164e7 −0.991040
\(993\) 2.08490e6 0.0670984
\(994\) −4.99475e7 −1.60342
\(995\) −1.83753e7 −0.588405
\(996\) −1.52581e7 −0.487361
\(997\) −2.48258e7 −0.790980 −0.395490 0.918470i \(-0.629425\pi\)
−0.395490 + 0.918470i \(0.629425\pi\)
\(998\) 2.87261e7 0.912958
\(999\) −5.87341e6 −0.186199
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.1.1 5
3.2 odd 2 207.6.a.f.1.5 5
4.3 odd 2 1104.6.a.r.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.6.a.e.1.1 5 1.1 even 1 trivial
207.6.a.f.1.5 5 3.2 odd 2
1104.6.a.r.1.5 5 4.3 odd 2