Properties

Label 29.6.a.a.1.4
Level $29$
Weight $6$
Character 29.1
Self dual yes
Analytic conductor $4.651$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 29 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 29.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.65113077458\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.3257317.1
Defining polynomial: \( x^{4} - 34x^{2} - 27x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.275208\) of defining polynomial
Character \(\chi\) \(=\) 29.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.91663 q^{2} -18.2828 q^{3} +3.00648 q^{4} -97.7313 q^{5} -108.173 q^{6} +139.558 q^{7} -171.544 q^{8} +91.2623 q^{9} +O(q^{10})\) \(q+5.91663 q^{2} -18.2828 q^{3} +3.00648 q^{4} -97.7313 q^{5} -108.173 q^{6} +139.558 q^{7} -171.544 q^{8} +91.2623 q^{9} -578.240 q^{10} +533.092 q^{11} -54.9671 q^{12} -675.965 q^{13} +825.715 q^{14} +1786.81 q^{15} -1111.17 q^{16} -268.994 q^{17} +539.965 q^{18} -2649.15 q^{19} -293.828 q^{20} -2551.52 q^{21} +3154.11 q^{22} +794.438 q^{23} +3136.31 q^{24} +6426.40 q^{25} -3999.43 q^{26} +2774.20 q^{27} +419.580 q^{28} -841.000 q^{29} +10571.9 q^{30} -4231.04 q^{31} -1084.97 q^{32} -9746.44 q^{33} -1591.54 q^{34} -13639.2 q^{35} +274.379 q^{36} -2689.54 q^{37} -15674.0 q^{38} +12358.6 q^{39} +16765.2 q^{40} +1395.36 q^{41} -15096.4 q^{42} -23810.5 q^{43} +1602.73 q^{44} -8919.18 q^{45} +4700.39 q^{46} +11267.5 q^{47} +20315.3 q^{48} +2669.53 q^{49} +38022.6 q^{50} +4917.98 q^{51} -2032.28 q^{52} -3396.67 q^{53} +16413.9 q^{54} -52099.8 q^{55} -23940.4 q^{56} +48433.9 q^{57} -4975.88 q^{58} -2785.38 q^{59} +5372.00 q^{60} +41551.7 q^{61} -25033.5 q^{62} +12736.4 q^{63} +29138.0 q^{64} +66062.9 q^{65} -57666.1 q^{66} +8574.14 q^{67} -808.728 q^{68} -14524.6 q^{69} -80698.2 q^{70} -6995.03 q^{71} -15655.5 q^{72} -4994.73 q^{73} -15913.0 q^{74} -117493. q^{75} -7964.62 q^{76} +74397.5 q^{77} +73121.0 q^{78} -23856.6 q^{79} +108596. q^{80} -72896.9 q^{81} +8255.84 q^{82} +43076.9 q^{83} -7671.11 q^{84} +26289.2 q^{85} -140878. q^{86} +15375.9 q^{87} -91448.7 q^{88} +13806.4 q^{89} -52771.5 q^{90} -94336.6 q^{91} +2388.47 q^{92} +77355.4 q^{93} +66665.5 q^{94} +258905. q^{95} +19836.3 q^{96} -176400. q^{97} +15794.6 q^{98} +48651.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 28 q^{3} + 10 q^{4} - 68 q^{5} - 194 q^{6} - 208 q^{7} - 504 q^{8} - 280 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 28 q^{3} + 10 q^{4} - 68 q^{5} - 194 q^{6} - 208 q^{7} - 504 q^{8} - 280 q^{9} - 788 q^{10} - 124 q^{11} + 20 q^{12} - 460 q^{13} + 768 q^{14} + 932 q^{15} - 414 q^{16} + 184 q^{17} + 3208 q^{18} - 2392 q^{19} + 2822 q^{20} + 992 q^{21} + 5538 q^{22} - 1192 q^{23} + 6786 q^{24} + 1824 q^{25} + 4724 q^{26} + 2468 q^{27} + 44 q^{28} - 3364 q^{29} + 8186 q^{30} - 19212 q^{31} + 6552 q^{32} - 10580 q^{33} - 7612 q^{34} - 22944 q^{35} - 7468 q^{36} - 10928 q^{37} - 456 q^{38} - 8732 q^{39} - 20 q^{40} - 1120 q^{41} + 1844 q^{42} - 21420 q^{43} - 1932 q^{44} - 8344 q^{45} - 7588 q^{46} + 23772 q^{47} + 33060 q^{48} + 10452 q^{49} + 43240 q^{50} + 12744 q^{51} - 29062 q^{52} + 8860 q^{53} + 35410 q^{54} - 52652 q^{55} + 34304 q^{56} + 48944 q^{57} - 10840 q^{59} + 25200 q^{60} + 49448 q^{61} + 18518 q^{62} + 27488 q^{63} - 20734 q^{64} + 97836 q^{65} - 47744 q^{66} - 7840 q^{67} + 20724 q^{68} + 58792 q^{69} - 77496 q^{70} - 48744 q^{71} + 8088 q^{72} - 74992 q^{73} - 35920 q^{74} - 90448 q^{75} - 140792 q^{76} + 128656 q^{77} + 2982 q^{78} - 106076 q^{79} + 58638 q^{80} - 59692 q^{81} - 234132 q^{82} + 62888 q^{83} - 59832 q^{84} + 23848 q^{85} - 216014 q^{86} + 23548 q^{87} - 39426 q^{88} + 107568 q^{89} + 41552 q^{90} - 268896 q^{91} - 26268 q^{92} + 221460 q^{93} + 30542 q^{94} + 147352 q^{95} - 78606 q^{96} - 49520 q^{97} + 242304 q^{98} + 166720 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.91663 1.04592 0.522961 0.852357i \(-0.324827\pi\)
0.522961 + 0.852357i \(0.324827\pi\)
\(3\) −18.2828 −1.17284 −0.586422 0.810005i \(-0.699464\pi\)
−0.586422 + 0.810005i \(0.699464\pi\)
\(4\) 3.00648 0.0939526
\(5\) −97.7313 −1.74827 −0.874135 0.485683i \(-0.838571\pi\)
−0.874135 + 0.485683i \(0.838571\pi\)
\(6\) −108.173 −1.22670
\(7\) 139.558 1.07649 0.538246 0.842788i \(-0.319087\pi\)
0.538246 + 0.842788i \(0.319087\pi\)
\(8\) −171.544 −0.947655
\(9\) 91.2623 0.375565
\(10\) −578.240 −1.82855
\(11\) 533.092 1.32838 0.664188 0.747566i \(-0.268778\pi\)
0.664188 + 0.747566i \(0.268778\pi\)
\(12\) −54.9671 −0.110192
\(13\) −675.965 −1.10934 −0.554672 0.832069i \(-0.687156\pi\)
−0.554672 + 0.832069i \(0.687156\pi\)
\(14\) 825.715 1.12593
\(15\) 1786.81 2.05045
\(16\) −1111.17 −1.08513
\(17\) −268.994 −0.225746 −0.112873 0.993609i \(-0.536005\pi\)
−0.112873 + 0.993609i \(0.536005\pi\)
\(18\) 539.965 0.392812
\(19\) −2649.15 −1.68354 −0.841768 0.539840i \(-0.818485\pi\)
−0.841768 + 0.539840i \(0.818485\pi\)
\(20\) −293.828 −0.164255
\(21\) −2551.52 −1.26256
\(22\) 3154.11 1.38938
\(23\) 794.438 0.313141 0.156571 0.987667i \(-0.449956\pi\)
0.156571 + 0.987667i \(0.449956\pi\)
\(24\) 3136.31 1.11145
\(25\) 6426.40 2.05645
\(26\) −3999.43 −1.16029
\(27\) 2774.20 0.732366
\(28\) 419.580 0.101139
\(29\) −841.000 −0.185695
\(30\) 10571.9 2.14461
\(31\) −4231.04 −0.790756 −0.395378 0.918518i \(-0.629386\pi\)
−0.395378 + 0.918518i \(0.629386\pi\)
\(32\) −1084.97 −0.187302
\(33\) −9746.44 −1.55798
\(34\) −1591.54 −0.236113
\(35\) −13639.2 −1.88200
\(36\) 274.379 0.0352853
\(37\) −2689.54 −0.322979 −0.161489 0.986874i \(-0.551630\pi\)
−0.161489 + 0.986874i \(0.551630\pi\)
\(38\) −15674.0 −1.76085
\(39\) 12358.6 1.30109
\(40\) 16765.2 1.65676
\(41\) 1395.36 0.129636 0.0648182 0.997897i \(-0.479353\pi\)
0.0648182 + 0.997897i \(0.479353\pi\)
\(42\) −15096.4 −1.32054
\(43\) −23810.5 −1.96380 −0.981901 0.189395i \(-0.939347\pi\)
−0.981901 + 0.189395i \(0.939347\pi\)
\(44\) 1602.73 0.124804
\(45\) −8919.18 −0.656589
\(46\) 4700.39 0.327521
\(47\) 11267.5 0.744016 0.372008 0.928229i \(-0.378669\pi\)
0.372008 + 0.928229i \(0.378669\pi\)
\(48\) 20315.3 1.27268
\(49\) 2669.53 0.158834
\(50\) 38022.6 2.15089
\(51\) 4917.98 0.264766
\(52\) −2032.28 −0.104226
\(53\) −3396.67 −0.166097 −0.0830487 0.996545i \(-0.526466\pi\)
−0.0830487 + 0.996545i \(0.526466\pi\)
\(54\) 16413.9 0.765997
\(55\) −52099.8 −2.32236
\(56\) −23940.4 −1.02014
\(57\) 48433.9 1.97453
\(58\) −4975.88 −0.194223
\(59\) −2785.38 −0.104173 −0.0520865 0.998643i \(-0.516587\pi\)
−0.0520865 + 0.998643i \(0.516587\pi\)
\(60\) 5372.00 0.192645
\(61\) 41551.7 1.42976 0.714881 0.699246i \(-0.246481\pi\)
0.714881 + 0.699246i \(0.246481\pi\)
\(62\) −25033.5 −0.827069
\(63\) 12736.4 0.404292
\(64\) 29138.0 0.889222
\(65\) 66062.9 1.93943
\(66\) −57666.1 −1.62952
\(67\) 8574.14 0.233348 0.116674 0.993170i \(-0.462777\pi\)
0.116674 + 0.993170i \(0.462777\pi\)
\(68\) −808.728 −0.0212095
\(69\) −14524.6 −0.367266
\(70\) −80698.2 −1.96842
\(71\) −6995.03 −0.164681 −0.0823405 0.996604i \(-0.526240\pi\)
−0.0823405 + 0.996604i \(0.526240\pi\)
\(72\) −15655.5 −0.355906
\(73\) −4994.73 −0.109699 −0.0548497 0.998495i \(-0.517468\pi\)
−0.0548497 + 0.998495i \(0.517468\pi\)
\(74\) −15913.0 −0.337811
\(75\) −117493. −2.41190
\(76\) −7964.62 −0.158173
\(77\) 74397.5 1.42998
\(78\) 73121.0 1.36084
\(79\) −23856.6 −0.430071 −0.215036 0.976606i \(-0.568987\pi\)
−0.215036 + 0.976606i \(0.568987\pi\)
\(80\) 108596. 1.89709
\(81\) −72896.9 −1.23452
\(82\) 8255.84 0.135590
\(83\) 43076.9 0.686356 0.343178 0.939270i \(-0.388497\pi\)
0.343178 + 0.939270i \(0.388497\pi\)
\(84\) −7671.11 −0.118621
\(85\) 26289.2 0.394666
\(86\) −140878. −2.05398
\(87\) 15375.9 0.217792
\(88\) −91448.7 −1.25884
\(89\) 13806.4 0.184759 0.0923793 0.995724i \(-0.470553\pi\)
0.0923793 + 0.995724i \(0.470553\pi\)
\(90\) −52771.5 −0.686741
\(91\) −94336.6 −1.19420
\(92\) 2388.47 0.0294205
\(93\) 77355.4 0.927435
\(94\) 66665.5 0.778183
\(95\) 258905. 2.94327
\(96\) 19836.3 0.219676
\(97\) −176400. −1.90357 −0.951786 0.306762i \(-0.900755\pi\)
−0.951786 + 0.306762i \(0.900755\pi\)
\(98\) 15794.6 0.166128
\(99\) 48651.2 0.498891
\(100\) 19320.9 0.193209
\(101\) −112043. −1.09290 −0.546452 0.837490i \(-0.684022\pi\)
−0.546452 + 0.837490i \(0.684022\pi\)
\(102\) 29097.9 0.276924
\(103\) 38255.6 0.355305 0.177653 0.984093i \(-0.443150\pi\)
0.177653 + 0.984093i \(0.443150\pi\)
\(104\) 115958. 1.05127
\(105\) 249364. 2.20729
\(106\) −20096.8 −0.173725
\(107\) −19410.6 −0.163900 −0.0819499 0.996636i \(-0.526115\pi\)
−0.0819499 + 0.996636i \(0.526115\pi\)
\(108\) 8340.58 0.0688077
\(109\) −51029.2 −0.411389 −0.205694 0.978616i \(-0.565945\pi\)
−0.205694 + 0.978616i \(0.565945\pi\)
\(110\) −308255. −2.42901
\(111\) 49172.5 0.378804
\(112\) −155073. −1.16813
\(113\) 45687.3 0.336588 0.168294 0.985737i \(-0.446174\pi\)
0.168294 + 0.985737i \(0.446174\pi\)
\(114\) 286566. 2.06520
\(115\) −77641.5 −0.547456
\(116\) −2528.45 −0.0174466
\(117\) −61690.1 −0.416630
\(118\) −16480.1 −0.108957
\(119\) −37540.4 −0.243014
\(120\) −306515. −1.94312
\(121\) 123136. 0.764581
\(122\) 245846. 1.49542
\(123\) −25511.2 −0.152043
\(124\) −12720.6 −0.0742937
\(125\) −322650. −1.84696
\(126\) 75356.6 0.422858
\(127\) 267210. 1.47009 0.735044 0.678019i \(-0.237161\pi\)
0.735044 + 0.678019i \(0.237161\pi\)
\(128\) 207118. 1.11736
\(129\) 435324. 2.30323
\(130\) 390870. 2.02849
\(131\) −165523. −0.842713 −0.421356 0.906895i \(-0.638446\pi\)
−0.421356 + 0.906895i \(0.638446\pi\)
\(132\) −29302.5 −0.146376
\(133\) −369711. −1.81231
\(134\) 50730.0 0.244063
\(135\) −271126. −1.28037
\(136\) 46144.3 0.213930
\(137\) 91000.4 0.414230 0.207115 0.978317i \(-0.433593\pi\)
0.207115 + 0.978317i \(0.433593\pi\)
\(138\) −85936.6 −0.384132
\(139\) −400431. −1.75788 −0.878942 0.476928i \(-0.841750\pi\)
−0.878942 + 0.476928i \(0.841750\pi\)
\(140\) −41006.1 −0.176819
\(141\) −206002. −0.872616
\(142\) −41387.0 −0.172244
\(143\) −360352. −1.47362
\(144\) −101408. −0.407535
\(145\) 82192.0 0.324646
\(146\) −29551.9 −0.114737
\(147\) −48806.6 −0.186288
\(148\) −8086.07 −0.0303447
\(149\) −18312.8 −0.0675756 −0.0337878 0.999429i \(-0.510757\pi\)
−0.0337878 + 0.999429i \(0.510757\pi\)
\(150\) −695162. −2.52265
\(151\) −17899.8 −0.0638859 −0.0319430 0.999490i \(-0.510169\pi\)
−0.0319430 + 0.999490i \(0.510169\pi\)
\(152\) 454445. 1.59541
\(153\) −24549.0 −0.0847824
\(154\) 440182. 1.49565
\(155\) 413505. 1.38246
\(156\) 37155.8 0.122241
\(157\) −413598. −1.33915 −0.669576 0.742744i \(-0.733524\pi\)
−0.669576 + 0.742744i \(0.733524\pi\)
\(158\) −141151. −0.449821
\(159\) 62100.7 0.194807
\(160\) 106035. 0.327454
\(161\) 110870. 0.337094
\(162\) −431304. −1.29121
\(163\) 178276. 0.525562 0.262781 0.964856i \(-0.415360\pi\)
0.262781 + 0.964856i \(0.415360\pi\)
\(164\) 4195.13 0.0121797
\(165\) 952532. 2.72377
\(166\) 254870. 0.717875
\(167\) −452143. −1.25454 −0.627269 0.778802i \(-0.715828\pi\)
−0.627269 + 0.778802i \(0.715828\pi\)
\(168\) 437698. 1.19647
\(169\) 85635.9 0.230642
\(170\) 155543. 0.412790
\(171\) −241767. −0.632277
\(172\) −71586.0 −0.184504
\(173\) 754374. 1.91634 0.958168 0.286206i \(-0.0923943\pi\)
0.958168 + 0.286206i \(0.0923943\pi\)
\(174\) 90973.3 0.227793
\(175\) 896858. 2.21375
\(176\) −592356. −1.44145
\(177\) 50924.7 0.122179
\(178\) 81687.2 0.193243
\(179\) −352813. −0.823024 −0.411512 0.911404i \(-0.634999\pi\)
−0.411512 + 0.911404i \(0.634999\pi\)
\(180\) −26815.4 −0.0616883
\(181\) 227087. 0.515224 0.257612 0.966248i \(-0.417064\pi\)
0.257612 + 0.966248i \(0.417064\pi\)
\(182\) −558154. −1.24904
\(183\) −759683. −1.67689
\(184\) −136281. −0.296750
\(185\) 262852. 0.564654
\(186\) 457683. 0.970024
\(187\) −143399. −0.299876
\(188\) 33875.5 0.0699023
\(189\) 387162. 0.788385
\(190\) 1.53184e6 3.07844
\(191\) 183415. 0.363790 0.181895 0.983318i \(-0.441777\pi\)
0.181895 + 0.983318i \(0.441777\pi\)
\(192\) −532726. −1.04292
\(193\) 997005. 1.92665 0.963327 0.268329i \(-0.0864714\pi\)
0.963327 + 0.268329i \(0.0864714\pi\)
\(194\) −1.04369e6 −1.99099
\(195\) −1.20782e6 −2.27465
\(196\) 8025.90 0.0149229
\(197\) −861288. −1.58119 −0.790593 0.612342i \(-0.790227\pi\)
−0.790593 + 0.612342i \(0.790227\pi\)
\(198\) 287851. 0.521801
\(199\) −837743. −1.49961 −0.749805 0.661659i \(-0.769853\pi\)
−0.749805 + 0.661659i \(0.769853\pi\)
\(200\) −1.10241e6 −1.94880
\(201\) −156760. −0.273681
\(202\) −662918. −1.14309
\(203\) −117369. −0.199899
\(204\) 14785.8 0.0248754
\(205\) −136371. −0.226640
\(206\) 226344. 0.371621
\(207\) 72502.2 0.117605
\(208\) 751111. 1.20378
\(209\) −1.41224e6 −2.23637
\(210\) 1.47539e6 2.30866
\(211\) −637594. −0.985912 −0.492956 0.870054i \(-0.664084\pi\)
−0.492956 + 0.870054i \(0.664084\pi\)
\(212\) −10212.0 −0.0156053
\(213\) 127889. 0.193145
\(214\) −114845. −0.171426
\(215\) 2.32703e6 3.43326
\(216\) −475896. −0.694030
\(217\) −590477. −0.851243
\(218\) −301921. −0.430281
\(219\) 91317.8 0.128660
\(220\) −156637. −0.218192
\(221\) 181831. 0.250430
\(222\) 290935. 0.396199
\(223\) 45526.9 0.0613064 0.0306532 0.999530i \(-0.490241\pi\)
0.0306532 + 0.999530i \(0.490241\pi\)
\(224\) −151416. −0.201629
\(225\) 586488. 0.772330
\(226\) 270315. 0.352045
\(227\) 1.33313e6 1.71715 0.858575 0.512688i \(-0.171350\pi\)
0.858575 + 0.512688i \(0.171350\pi\)
\(228\) 145616. 0.185512
\(229\) 830883. 1.04701 0.523506 0.852022i \(-0.324624\pi\)
0.523506 + 0.852022i \(0.324624\pi\)
\(230\) −459376. −0.572596
\(231\) −1.36020e6 −1.67715
\(232\) 144268. 0.175975
\(233\) 767627. 0.926319 0.463159 0.886275i \(-0.346716\pi\)
0.463159 + 0.886275i \(0.346716\pi\)
\(234\) −364997. −0.435763
\(235\) −1.10119e6 −1.30074
\(236\) −8374.22 −0.00978733
\(237\) 436166. 0.504407
\(238\) −222113. −0.254174
\(239\) −1.22858e6 −1.39126 −0.695631 0.718399i \(-0.744875\pi\)
−0.695631 + 0.718399i \(0.744875\pi\)
\(240\) −1.98544e6 −2.22500
\(241\) 304030. 0.337190 0.168595 0.985685i \(-0.446077\pi\)
0.168595 + 0.985685i \(0.446077\pi\)
\(242\) 728553. 0.799692
\(243\) 658633. 0.715530
\(244\) 124924. 0.134330
\(245\) −260897. −0.277685
\(246\) −150940. −0.159026
\(247\) 1.79073e6 1.86762
\(248\) 725809. 0.749364
\(249\) −787569. −0.804989
\(250\) −1.90900e6 −1.93177
\(251\) 850953. 0.852553 0.426276 0.904593i \(-0.359825\pi\)
0.426276 + 0.904593i \(0.359825\pi\)
\(252\) 38291.8 0.0379843
\(253\) 423509. 0.415969
\(254\) 1.58098e6 1.53760
\(255\) −480641. −0.462882
\(256\) 293022. 0.279448
\(257\) 28252.8 0.0266826 0.0133413 0.999911i \(-0.495753\pi\)
0.0133413 + 0.999911i \(0.495753\pi\)
\(258\) 2.57565e6 2.40900
\(259\) −375348. −0.347684
\(260\) 198617. 0.182215
\(261\) −76751.6 −0.0697406
\(262\) −979337. −0.881412
\(263\) 393978. 0.351223 0.175611 0.984460i \(-0.443810\pi\)
0.175611 + 0.984460i \(0.443810\pi\)
\(264\) 1.67194e6 1.47643
\(265\) 331960. 0.290383
\(266\) −2.18744e6 −1.89554
\(267\) −252420. −0.216693
\(268\) 25778.0 0.0219236
\(269\) 453727. 0.382309 0.191154 0.981560i \(-0.438777\pi\)
0.191154 + 0.981560i \(0.438777\pi\)
\(270\) −1.60415e6 −1.33917
\(271\) −1.54312e6 −1.27637 −0.638186 0.769882i \(-0.720315\pi\)
−0.638186 + 0.769882i \(0.720315\pi\)
\(272\) 298898. 0.244963
\(273\) 1.72474e6 1.40061
\(274\) 538416. 0.433253
\(275\) 3.42587e6 2.73174
\(276\) −43667.9 −0.0345056
\(277\) −1.13023e6 −0.885050 −0.442525 0.896756i \(-0.645917\pi\)
−0.442525 + 0.896756i \(0.645917\pi\)
\(278\) −2.36920e6 −1.83861
\(279\) −386134. −0.296980
\(280\) 2.33972e6 1.78348
\(281\) −1.17984e6 −0.891371 −0.445685 0.895190i \(-0.647040\pi\)
−0.445685 + 0.895190i \(0.647040\pi\)
\(282\) −1.21884e6 −0.912688
\(283\) 345340. 0.256319 0.128160 0.991754i \(-0.459093\pi\)
0.128160 + 0.991754i \(0.459093\pi\)
\(284\) −21030.5 −0.0154722
\(285\) −4.73351e6 −3.45200
\(286\) −2.13207e6 −1.54130
\(287\) 194734. 0.139553
\(288\) −99016.6 −0.0703440
\(289\) −1.34750e6 −0.949039
\(290\) 486300. 0.339554
\(291\) 3.22509e6 2.23260
\(292\) −15016.6 −0.0103066
\(293\) 2.40825e6 1.63883 0.819414 0.573202i \(-0.194299\pi\)
0.819414 + 0.573202i \(0.194299\pi\)
\(294\) −288770. −0.194843
\(295\) 272219. 0.182123
\(296\) 461374. 0.306072
\(297\) 1.47890e6 0.972856
\(298\) −108350. −0.0706788
\(299\) −537013. −0.347381
\(300\) −353241. −0.226604
\(301\) −3.32296e6 −2.11402
\(302\) −105906. −0.0668197
\(303\) 2.04847e6 1.28181
\(304\) 2.94365e6 1.82685
\(305\) −4.06090e6 −2.49961
\(306\) −145248. −0.0886758
\(307\) 595416. 0.360558 0.180279 0.983616i \(-0.442300\pi\)
0.180279 + 0.983616i \(0.442300\pi\)
\(308\) 223675. 0.134351
\(309\) −699420. −0.416718
\(310\) 2.44655e6 1.44594
\(311\) 999900. 0.586213 0.293107 0.956080i \(-0.405311\pi\)
0.293107 + 0.956080i \(0.405311\pi\)
\(312\) −2.12004e6 −1.23298
\(313\) −365270. −0.210743 −0.105371 0.994433i \(-0.533603\pi\)
−0.105371 + 0.994433i \(0.533603\pi\)
\(314\) −2.44711e6 −1.40065
\(315\) −1.24475e6 −0.706813
\(316\) −71724.5 −0.0404064
\(317\) 1.60635e6 0.897823 0.448912 0.893576i \(-0.351812\pi\)
0.448912 + 0.893576i \(0.351812\pi\)
\(318\) 367427. 0.203752
\(319\) −448331. −0.246673
\(320\) −2.84770e6 −1.55460
\(321\) 354880. 0.192229
\(322\) 655979. 0.352574
\(323\) 712606. 0.380052
\(324\) −219164. −0.115986
\(325\) −4.34403e6 −2.28131
\(326\) 1.05479e6 0.549697
\(327\) 932959. 0.482495
\(328\) −239366. −0.122851
\(329\) 1.57247e6 0.800928
\(330\) 5.63578e6 2.84885
\(331\) −535124. −0.268463 −0.134232 0.990950i \(-0.542857\pi\)
−0.134232 + 0.990950i \(0.542857\pi\)
\(332\) 129510. 0.0644850
\(333\) −245454. −0.121299
\(334\) −2.67516e6 −1.31215
\(335\) −837961. −0.407955
\(336\) 2.83517e6 1.37003
\(337\) 1.93303e6 0.927178 0.463589 0.886050i \(-0.346561\pi\)
0.463589 + 0.886050i \(0.346561\pi\)
\(338\) 506676. 0.241234
\(339\) −835293. −0.394766
\(340\) 79038.0 0.0370799
\(341\) −2.25553e6 −1.05042
\(342\) −1.43045e6 −0.661312
\(343\) −1.97300e6 −0.905508
\(344\) 4.08455e6 1.86101
\(345\) 1.41951e6 0.642081
\(346\) 4.46335e6 2.00434
\(347\) −2.83701e6 −1.26485 −0.632423 0.774623i \(-0.717940\pi\)
−0.632423 + 0.774623i \(0.717940\pi\)
\(348\) 46227.3 0.0204621
\(349\) 1.11395e6 0.489555 0.244778 0.969579i \(-0.421285\pi\)
0.244778 + 0.969579i \(0.421285\pi\)
\(350\) 5.30638e6 2.31541
\(351\) −1.87526e6 −0.812445
\(352\) −578388. −0.248807
\(353\) −3.49053e6 −1.49092 −0.745461 0.666549i \(-0.767771\pi\)
−0.745461 + 0.666549i \(0.767771\pi\)
\(354\) 301303. 0.127789
\(355\) 683633. 0.287907
\(356\) 41508.7 0.0173586
\(357\) 686345. 0.285018
\(358\) −2.08746e6 −0.860819
\(359\) 469676. 0.192337 0.0961684 0.995365i \(-0.469341\pi\)
0.0961684 + 0.995365i \(0.469341\pi\)
\(360\) 1.53003e6 0.622220
\(361\) 4.54189e6 1.83429
\(362\) 1.34359e6 0.538884
\(363\) −2.25128e6 −0.896735
\(364\) −283621. −0.112198
\(365\) 488141. 0.191784
\(366\) −4.49476e6 −1.75390
\(367\) 2.70229e6 1.04729 0.523645 0.851936i \(-0.324572\pi\)
0.523645 + 0.851936i \(0.324572\pi\)
\(368\) −882755. −0.339798
\(369\) 127344. 0.0486869
\(370\) 1.55520e6 0.590584
\(371\) −474033. −0.178803
\(372\) 232568. 0.0871349
\(373\) 2.77666e6 1.03336 0.516678 0.856180i \(-0.327168\pi\)
0.516678 + 0.856180i \(0.327168\pi\)
\(374\) −848438. −0.313647
\(375\) 5.89897e6 2.16620
\(376\) −1.93287e6 −0.705071
\(377\) 568487. 0.206000
\(378\) 2.29070e6 0.824590
\(379\) −4.41837e6 −1.58002 −0.790012 0.613091i \(-0.789926\pi\)
−0.790012 + 0.613091i \(0.789926\pi\)
\(380\) 778393. 0.276528
\(381\) −4.88536e6 −1.72419
\(382\) 1.08520e6 0.380496
\(383\) 1.37796e6 0.479999 0.239999 0.970773i \(-0.422853\pi\)
0.239999 + 0.970773i \(0.422853\pi\)
\(384\) −3.78670e6 −1.31049
\(385\) −7.27096e6 −2.50000
\(386\) 5.89891e6 2.01513
\(387\) −2.17300e6 −0.737535
\(388\) −530344. −0.178846
\(389\) 649334. 0.217568 0.108784 0.994065i \(-0.465304\pi\)
0.108784 + 0.994065i \(0.465304\pi\)
\(390\) −7.14621e6 −2.37911
\(391\) −213699. −0.0706906
\(392\) −457941. −0.150520
\(393\) 3.02623e6 0.988371
\(394\) −5.09592e6 −1.65380
\(395\) 2.33154e6 0.751881
\(396\) 146269. 0.0468721
\(397\) −2.52909e6 −0.805356 −0.402678 0.915342i \(-0.631921\pi\)
−0.402678 + 0.915342i \(0.631921\pi\)
\(398\) −4.95661e6 −1.56847
\(399\) 6.75936e6 2.12556
\(400\) −7.14082e6 −2.23151
\(401\) −3.64231e6 −1.13114 −0.565569 0.824701i \(-0.691344\pi\)
−0.565569 + 0.824701i \(0.691344\pi\)
\(402\) −927488. −0.286248
\(403\) 2.86003e6 0.877220
\(404\) −336856. −0.102681
\(405\) 7.12431e6 2.15827
\(406\) −694426. −0.209079
\(407\) −1.43377e6 −0.429037
\(408\) −843650. −0.250906
\(409\) 2.68522e6 0.793729 0.396865 0.917877i \(-0.370098\pi\)
0.396865 + 0.917877i \(0.370098\pi\)
\(410\) −806854. −0.237047
\(411\) −1.66375e6 −0.485828
\(412\) 115015. 0.0333819
\(413\) −388724. −0.112141
\(414\) 428969. 0.123006
\(415\) −4.20996e6 −1.19994
\(416\) 733401. 0.207782
\(417\) 7.32101e6 2.06173
\(418\) −8.35570e6 −2.33906
\(419\) −4.33500e6 −1.20630 −0.603148 0.797629i \(-0.706087\pi\)
−0.603148 + 0.797629i \(0.706087\pi\)
\(420\) 749708. 0.207381
\(421\) 3.38962e6 0.932063 0.466032 0.884768i \(-0.345683\pi\)
0.466032 + 0.884768i \(0.345683\pi\)
\(422\) −3.77241e6 −1.03119
\(423\) 1.02830e6 0.279426
\(424\) 582677. 0.157403
\(425\) −1.72867e6 −0.464236
\(426\) 756672. 0.202015
\(427\) 5.79888e6 1.53913
\(428\) −58357.6 −0.0153988
\(429\) 6.58826e6 1.72833
\(430\) 1.37682e7 3.59092
\(431\) 290831. 0.0754132 0.0377066 0.999289i \(-0.487995\pi\)
0.0377066 + 0.999289i \(0.487995\pi\)
\(432\) −3.08260e6 −0.794709
\(433\) −2.64162e6 −0.677097 −0.338549 0.940949i \(-0.609936\pi\)
−0.338549 + 0.940949i \(0.609936\pi\)
\(434\) −3.49363e6 −0.890333
\(435\) −1.50270e6 −0.380759
\(436\) −153418. −0.0386511
\(437\) −2.10458e6 −0.527185
\(438\) 540293. 0.134569
\(439\) 5.65486e6 1.40043 0.700214 0.713933i \(-0.253088\pi\)
0.700214 + 0.713933i \(0.253088\pi\)
\(440\) 8.93740e6 2.20079
\(441\) 243627. 0.0596526
\(442\) 1.07583e6 0.261931
\(443\) 1.31778e6 0.319031 0.159515 0.987195i \(-0.449007\pi\)
0.159515 + 0.987195i \(0.449007\pi\)
\(444\) 147836. 0.0355896
\(445\) −1.34932e6 −0.323008
\(446\) 269366. 0.0641217
\(447\) 334810. 0.0792556
\(448\) 4.06646e6 0.957241
\(449\) −6.50617e6 −1.52303 −0.761517 0.648145i \(-0.775545\pi\)
−0.761517 + 0.648145i \(0.775545\pi\)
\(450\) 3.47003e6 0.807797
\(451\) 743857. 0.172206
\(452\) 137358. 0.0316234
\(453\) 327259. 0.0749283
\(454\) 7.88764e6 1.79601
\(455\) 9.21963e6 2.08778
\(456\) −8.30855e6 −1.87117
\(457\) −3.50691e6 −0.785477 −0.392739 0.919650i \(-0.628472\pi\)
−0.392739 + 0.919650i \(0.628472\pi\)
\(458\) 4.91603e6 1.09509
\(459\) −746244. −0.165329
\(460\) −233428. −0.0514349
\(461\) −1.22247e6 −0.267908 −0.133954 0.990988i \(-0.542767\pi\)
−0.133954 + 0.990988i \(0.542767\pi\)
\(462\) −8.04778e6 −1.75417
\(463\) 382178. 0.0828540 0.0414270 0.999142i \(-0.486810\pi\)
0.0414270 + 0.999142i \(0.486810\pi\)
\(464\) 934493. 0.201503
\(465\) −7.56004e6 −1.62141
\(466\) 4.54176e6 0.968857
\(467\) −1.81409e6 −0.384916 −0.192458 0.981305i \(-0.561646\pi\)
−0.192458 + 0.981305i \(0.561646\pi\)
\(468\) −185470. −0.0391435
\(469\) 1.19659e6 0.251197
\(470\) −6.51531e6 −1.36047
\(471\) 7.56175e6 1.57062
\(472\) 477816. 0.0987200
\(473\) −1.26932e7 −2.60867
\(474\) 2.58063e6 0.527570
\(475\) −1.70245e7 −3.46210
\(476\) −112865. −0.0228318
\(477\) −309987. −0.0623804
\(478\) −7.26906e6 −1.45515
\(479\) 4.87397e6 0.970609 0.485304 0.874345i \(-0.338709\pi\)
0.485304 + 0.874345i \(0.338709\pi\)
\(480\) −1.93863e6 −0.384053
\(481\) 1.81804e6 0.358294
\(482\) 1.79883e6 0.352674
\(483\) −2.02703e6 −0.395359
\(484\) 370208. 0.0718344
\(485\) 1.72398e7 3.32796
\(486\) 3.89689e6 0.748389
\(487\) −1.84954e6 −0.353379 −0.176690 0.984267i \(-0.556539\pi\)
−0.176690 + 0.984267i \(0.556539\pi\)
\(488\) −7.12793e6 −1.35492
\(489\) −3.25939e6 −0.616403
\(490\) −1.54363e6 −0.290437
\(491\) −5.47088e6 −1.02413 −0.512063 0.858948i \(-0.671119\pi\)
−0.512063 + 0.858948i \(0.671119\pi\)
\(492\) −76699.0 −0.0142849
\(493\) 226224. 0.0419201
\(494\) 1.05951e7 1.95338
\(495\) −4.75475e6 −0.872196
\(496\) 4.70140e6 0.858070
\(497\) −976215. −0.177278
\(498\) −4.65975e6 −0.841956
\(499\) 5.27760e6 0.948823 0.474411 0.880303i \(-0.342661\pi\)
0.474411 + 0.880303i \(0.342661\pi\)
\(500\) −970044. −0.173527
\(501\) 8.26645e6 1.47138
\(502\) 5.03477e6 0.891703
\(503\) 6.07721e6 1.07099 0.535494 0.844539i \(-0.320125\pi\)
0.535494 + 0.844539i \(0.320125\pi\)
\(504\) −2.18485e6 −0.383130
\(505\) 1.09501e7 1.91069
\(506\) 2.50574e6 0.435071
\(507\) −1.56567e6 −0.270508
\(508\) 803363. 0.138119
\(509\) 4.28972e6 0.733896 0.366948 0.930242i \(-0.380403\pi\)
0.366948 + 0.930242i \(0.380403\pi\)
\(510\) −2.84377e6 −0.484138
\(511\) −697056. −0.118091
\(512\) −4.89407e6 −0.825078
\(513\) −7.34926e6 −1.23296
\(514\) 167161. 0.0279079
\(515\) −3.73876e6 −0.621169
\(516\) 1.30879e6 0.216395
\(517\) 6.00661e6 0.988333
\(518\) −2.22079e6 −0.363650
\(519\) −1.37921e7 −2.24756
\(520\) −1.13327e7 −1.83791
\(521\) 3.12541e6 0.504444 0.252222 0.967669i \(-0.418839\pi\)
0.252222 + 0.967669i \(0.418839\pi\)
\(522\) −454110. −0.0729433
\(523\) −6.57035e6 −1.05035 −0.525176 0.850994i \(-0.676000\pi\)
−0.525176 + 0.850994i \(0.676000\pi\)
\(524\) −497642. −0.0791751
\(525\) −1.63971e7 −2.59639
\(526\) 2.33102e6 0.367352
\(527\) 1.13813e6 0.178510
\(528\) 1.08299e7 1.69060
\(529\) −5.80521e6 −0.901942
\(530\) 1.96409e6 0.303718
\(531\) −254201. −0.0391237
\(532\) −1.11153e6 −0.170271
\(533\) −943216. −0.143811
\(534\) −1.49347e6 −0.226644
\(535\) 1.89702e6 0.286541
\(536\) −1.47084e6 −0.221133
\(537\) 6.45043e6 0.965279
\(538\) 2.68453e6 0.399865
\(539\) 1.42311e6 0.210992
\(540\) −815136. −0.120294
\(541\) −6.72381e6 −0.987694 −0.493847 0.869549i \(-0.664410\pi\)
−0.493847 + 0.869549i \(0.664410\pi\)
\(542\) −9.13008e6 −1.33499
\(543\) −4.15180e6 −0.604278
\(544\) 291850. 0.0422827
\(545\) 4.98715e6 0.719219
\(546\) 1.02046e7 1.46493
\(547\) 1.19150e6 0.170264 0.0851322 0.996370i \(-0.472869\pi\)
0.0851322 + 0.996370i \(0.472869\pi\)
\(548\) 273591. 0.0389180
\(549\) 3.79210e6 0.536969
\(550\) 2.02696e7 2.85718
\(551\) 2.22793e6 0.312625
\(552\) 2.49160e6 0.348042
\(553\) −3.32939e6 −0.462968
\(554\) −6.68716e6 −0.925693
\(555\) −4.80569e6 −0.662252
\(556\) −1.20389e6 −0.165158
\(557\) 2.43064e6 0.331958 0.165979 0.986129i \(-0.446922\pi\)
0.165979 + 0.986129i \(0.446922\pi\)
\(558\) −2.28461e6 −0.310618
\(559\) 1.60951e7 2.17853
\(560\) 1.51555e7 2.04220
\(561\) 2.62174e6 0.351708
\(562\) −6.98069e6 −0.932304
\(563\) −1.05702e7 −1.40544 −0.702721 0.711466i \(-0.748032\pi\)
−0.702721 + 0.711466i \(0.748032\pi\)
\(564\) −619341. −0.0819846
\(565\) −4.46508e6 −0.588447
\(566\) 2.04325e6 0.268090
\(567\) −1.01734e7 −1.32895
\(568\) 1.19995e6 0.156061
\(569\) 5.17877e6 0.670573 0.335286 0.942116i \(-0.391167\pi\)
0.335286 + 0.942116i \(0.391167\pi\)
\(570\) −2.80064e7 −3.61053
\(571\) −3.92258e6 −0.503479 −0.251739 0.967795i \(-0.581003\pi\)
−0.251739 + 0.967795i \(0.581003\pi\)
\(572\) −1.08339e6 −0.138451
\(573\) −3.35335e6 −0.426670
\(574\) 1.15217e6 0.145961
\(575\) 5.10538e6 0.643959
\(576\) 2.65920e6 0.333961
\(577\) 1.12133e7 1.40215 0.701074 0.713089i \(-0.252704\pi\)
0.701074 + 0.713089i \(0.252704\pi\)
\(578\) −7.97265e6 −0.992620
\(579\) −1.82281e7 −2.25967
\(580\) 247109. 0.0305013
\(581\) 6.01174e6 0.738857
\(582\) 1.90817e7 2.33512
\(583\) −1.81074e6 −0.220640
\(584\) 856814. 0.103957
\(585\) 6.02905e6 0.728382
\(586\) 1.42487e7 1.71409
\(587\) 1.03337e7 1.23783 0.618915 0.785458i \(-0.287573\pi\)
0.618915 + 0.785458i \(0.287573\pi\)
\(588\) −146736. −0.0175023
\(589\) 1.12086e7 1.33127
\(590\) 1.61062e6 0.190486
\(591\) 1.57468e7 1.85448
\(592\) 2.98853e6 0.350472
\(593\) −1.58653e7 −1.85272 −0.926362 0.376634i \(-0.877081\pi\)
−0.926362 + 0.376634i \(0.877081\pi\)
\(594\) 8.75012e6 1.01753
\(595\) 3.66887e6 0.424855
\(596\) −55057.2 −0.00634890
\(597\) 1.53163e7 1.75881
\(598\) −3.17730e6 −0.363334
\(599\) −9.66314e6 −1.10040 −0.550201 0.835032i \(-0.685449\pi\)
−0.550201 + 0.835032i \(0.685449\pi\)
\(600\) 2.01552e7 2.28564
\(601\) −1.07881e7 −1.21831 −0.609155 0.793051i \(-0.708491\pi\)
−0.609155 + 0.793051i \(0.708491\pi\)
\(602\) −1.96607e7 −2.21110
\(603\) 782495. 0.0876372
\(604\) −53815.4 −0.00600225
\(605\) −1.20343e7 −1.33669
\(606\) 1.21200e7 1.34067
\(607\) 6.76786e6 0.745555 0.372778 0.927921i \(-0.378405\pi\)
0.372778 + 0.927921i \(0.378405\pi\)
\(608\) 2.87424e6 0.315329
\(609\) 2.14583e6 0.234451
\(610\) −2.40268e7 −2.61440
\(611\) −7.61643e6 −0.825370
\(612\) −73806.3 −0.00796553
\(613\) 2.91439e6 0.313253 0.156627 0.987658i \(-0.449938\pi\)
0.156627 + 0.987658i \(0.449938\pi\)
\(614\) 3.52286e6 0.377115
\(615\) 2.49324e6 0.265813
\(616\) −1.27624e7 −1.35513
\(617\) −1.01418e7 −1.07251 −0.536254 0.844056i \(-0.680161\pi\)
−0.536254 + 0.844056i \(0.680161\pi\)
\(618\) −4.13821e6 −0.435854
\(619\) −6.89280e6 −0.723051 −0.361526 0.932362i \(-0.617744\pi\)
−0.361526 + 0.932362i \(0.617744\pi\)
\(620\) 1.24320e6 0.129885
\(621\) 2.20393e6 0.229334
\(622\) 5.91603e6 0.613133
\(623\) 1.92680e6 0.198891
\(624\) −1.37324e7 −1.41184
\(625\) 1.14505e7 1.17253
\(626\) −2.16116e6 −0.220420
\(627\) 2.58198e7 2.62291
\(628\) −1.24348e6 −0.125817
\(629\) 723472. 0.0729113
\(630\) −7.36470e6 −0.739271
\(631\) −7.52515e6 −0.752388 −0.376194 0.926541i \(-0.622767\pi\)
−0.376194 + 0.926541i \(0.622767\pi\)
\(632\) 4.09245e6 0.407559
\(633\) 1.16570e7 1.15632
\(634\) 9.50415e6 0.939053
\(635\) −2.61148e7 −2.57011
\(636\) 186705. 0.0183026
\(637\) −1.80451e6 −0.176202
\(638\) −2.65261e6 −0.258001
\(639\) −638382. −0.0618484
\(640\) −2.02419e7 −1.95345
\(641\) −278549. −0.0267767 −0.0133883 0.999910i \(-0.504262\pi\)
−0.0133883 + 0.999910i \(0.504262\pi\)
\(642\) 2.09969e6 0.201057
\(643\) −3.63623e6 −0.346836 −0.173418 0.984848i \(-0.555481\pi\)
−0.173418 + 0.984848i \(0.555481\pi\)
\(644\) 333330. 0.0316709
\(645\) −4.25448e7 −4.02668
\(646\) 4.21622e6 0.397505
\(647\) −1.08234e7 −1.01649 −0.508243 0.861214i \(-0.669705\pi\)
−0.508243 + 0.861214i \(0.669705\pi\)
\(648\) 1.25050e7 1.16989
\(649\) −1.48487e6 −0.138381
\(650\) −2.57020e7 −2.38607
\(651\) 1.07956e7 0.998376
\(652\) 535984. 0.0493779
\(653\) 8.86232e6 0.813326 0.406663 0.913578i \(-0.366692\pi\)
0.406663 + 0.913578i \(0.366692\pi\)
\(654\) 5.51997e6 0.504652
\(655\) 1.61768e7 1.47329
\(656\) −1.55048e6 −0.140672
\(657\) −455830. −0.0411993
\(658\) 9.30373e6 0.837708
\(659\) −4.94619e6 −0.443667 −0.221833 0.975085i \(-0.571204\pi\)
−0.221833 + 0.975085i \(0.571204\pi\)
\(660\) 2.86377e6 0.255905
\(661\) −6.34440e6 −0.564790 −0.282395 0.959298i \(-0.591129\pi\)
−0.282395 + 0.959298i \(0.591129\pi\)
\(662\) −3.16613e6 −0.280791
\(663\) −3.32438e6 −0.293716
\(664\) −7.38958e6 −0.650429
\(665\) 3.61323e7 3.16841
\(666\) −1.45226e6 −0.126870
\(667\) −668122. −0.0581489
\(668\) −1.35936e6 −0.117867
\(669\) −832361. −0.0719029
\(670\) −4.95790e6 −0.426689
\(671\) 2.21509e7 1.89926
\(672\) 2.76832e6 0.236479
\(673\) 1.00239e7 0.853095 0.426548 0.904465i \(-0.359730\pi\)
0.426548 + 0.904465i \(0.359730\pi\)
\(674\) 1.14370e7 0.969756
\(675\) 1.78281e7 1.50607
\(676\) 257463. 0.0216695
\(677\) 3.44231e6 0.288654 0.144327 0.989530i \(-0.453898\pi\)
0.144327 + 0.989530i \(0.453898\pi\)
\(678\) −4.94212e6 −0.412894
\(679\) −2.46181e7 −2.04918
\(680\) −4.50975e6 −0.374007
\(681\) −2.43734e7 −2.01395
\(682\) −1.33452e7 −1.09866
\(683\) 2.51352e6 0.206172 0.103086 0.994672i \(-0.467128\pi\)
0.103086 + 0.994672i \(0.467128\pi\)
\(684\) −726869. −0.0594041
\(685\) −8.89359e6 −0.724187
\(686\) −1.16735e7 −0.947090
\(687\) −1.51909e7 −1.22798
\(688\) 2.64575e7 2.13097
\(689\) 2.29603e6 0.184259
\(690\) 8.39869e6 0.671566
\(691\) −2.01696e7 −1.60695 −0.803474 0.595340i \(-0.797017\pi\)
−0.803474 + 0.595340i \(0.797017\pi\)
\(692\) 2.26802e6 0.180045
\(693\) 6.78968e6 0.537052
\(694\) −1.67856e7 −1.32293
\(695\) 3.91346e7 3.07326
\(696\) −2.63764e6 −0.206391
\(697\) −375345. −0.0292650
\(698\) 6.59082e6 0.512037
\(699\) −1.40344e7 −1.08643
\(700\) 2.69639e6 0.207988
\(701\) 1.67322e6 0.128605 0.0643025 0.997930i \(-0.479518\pi\)
0.0643025 + 0.997930i \(0.479518\pi\)
\(702\) −1.10952e7 −0.849754
\(703\) 7.12499e6 0.543746
\(704\) 1.55333e7 1.18122
\(705\) 2.01328e7 1.52557
\(706\) −2.06522e7 −1.55939
\(707\) −1.56366e7 −1.17650
\(708\) 153104. 0.0114790
\(709\) 6.28913e6 0.469867 0.234933 0.972011i \(-0.424513\pi\)
0.234933 + 0.972011i \(0.424513\pi\)
\(710\) 4.04480e6 0.301128
\(711\) −2.17721e6 −0.161520
\(712\) −2.36840e6 −0.175087
\(713\) −3.36130e6 −0.247619
\(714\) 4.06085e6 0.298107
\(715\) 3.52177e7 2.57629
\(716\) −1.06073e6 −0.0773253
\(717\) 2.24620e7 1.63173
\(718\) 2.77890e6 0.201169
\(719\) 2.87519e6 0.207417 0.103709 0.994608i \(-0.466929\pi\)
0.103709 + 0.994608i \(0.466929\pi\)
\(720\) 9.91071e6 0.712481
\(721\) 5.33888e6 0.382483
\(722\) 2.68726e7 1.91852
\(723\) −5.55854e6 −0.395471
\(724\) 682734. 0.0484067
\(725\) −5.40461e6 −0.381873
\(726\) −1.33200e7 −0.937914
\(727\) 2.76368e6 0.193933 0.0969665 0.995288i \(-0.469086\pi\)
0.0969665 + 0.995288i \(0.469086\pi\)
\(728\) 1.61829e7 1.13169
\(729\) 5.67227e6 0.395310
\(730\) 2.88815e6 0.200591
\(731\) 6.40490e6 0.443321
\(732\) −2.28397e6 −0.157548
\(733\) −1.06643e7 −0.733115 −0.366557 0.930395i \(-0.619464\pi\)
−0.366557 + 0.930395i \(0.619464\pi\)
\(734\) 1.59885e7 1.09538
\(735\) 4.76993e6 0.325682
\(736\) −861940. −0.0586520
\(737\) 4.57081e6 0.309973
\(738\) 753446. 0.0509227
\(739\) −3.98603e6 −0.268491 −0.134245 0.990948i \(-0.542861\pi\)
−0.134245 + 0.990948i \(0.542861\pi\)
\(740\) 790262. 0.0530508
\(741\) −3.27397e7 −2.19043
\(742\) −2.80468e6 −0.187014
\(743\) −2.19748e7 −1.46034 −0.730170 0.683266i \(-0.760559\pi\)
−0.730170 + 0.683266i \(0.760559\pi\)
\(744\) −1.32698e7 −0.878888
\(745\) 1.78974e6 0.118140
\(746\) 1.64285e7 1.08081
\(747\) 3.93130e6 0.257771
\(748\) −431127. −0.0281741
\(749\) −2.70891e6 −0.176437
\(750\) 3.49020e7 2.26567
\(751\) −1.84068e7 −1.19091 −0.595456 0.803388i \(-0.703028\pi\)
−0.595456 + 0.803388i \(0.703028\pi\)
\(752\) −1.25201e7 −0.807351
\(753\) −1.55578e7 −0.999912
\(754\) 3.36352e6 0.215460
\(755\) 1.74937e6 0.111690
\(756\) 1.16400e6 0.0740709
\(757\) 2.78199e7 1.76447 0.882237 0.470805i \(-0.156037\pi\)
0.882237 + 0.470805i \(0.156037\pi\)
\(758\) −2.61418e7 −1.65258
\(759\) −7.74295e6 −0.487867
\(760\) −4.44135e7 −2.78921
\(761\) 1.96481e6 0.122987 0.0614935 0.998107i \(-0.480414\pi\)
0.0614935 + 0.998107i \(0.480414\pi\)
\(762\) −2.89048e7 −1.80336
\(763\) −7.12155e6 −0.442857
\(764\) 551434. 0.0341791
\(765\) 2.39921e6 0.148223
\(766\) 8.15288e6 0.502041
\(767\) 1.88282e6 0.115564
\(768\) −5.35728e6 −0.327749
\(769\) −1.16064e7 −0.707754 −0.353877 0.935292i \(-0.615137\pi\)
−0.353877 + 0.935292i \(0.615137\pi\)
\(770\) −4.30196e7 −2.61481
\(771\) −516541. −0.0312946
\(772\) 2.99748e6 0.181014
\(773\) 2.10057e7 1.26441 0.632207 0.774800i \(-0.282149\pi\)
0.632207 + 0.774800i \(0.282149\pi\)
\(774\) −1.28568e7 −0.771404
\(775\) −2.71904e7 −1.62615
\(776\) 3.02604e7 1.80393
\(777\) 6.86243e6 0.407779
\(778\) 3.84187e6 0.227559
\(779\) −3.69652e6 −0.218248
\(780\) −3.63129e6 −0.213710
\(781\) −3.72900e6 −0.218758
\(782\) −1.26438e6 −0.0739368
\(783\) −2.33310e6 −0.135997
\(784\) −2.96630e6 −0.172355
\(785\) 4.04215e7 2.34120
\(786\) 1.79051e7 1.03376
\(787\) −8.11927e6 −0.467283 −0.233642 0.972323i \(-0.575064\pi\)
−0.233642 + 0.972323i \(0.575064\pi\)
\(788\) −2.58945e6 −0.148557
\(789\) −7.20304e6 −0.411930
\(790\) 1.37948e7 0.786409
\(791\) 6.37604e6 0.362335
\(792\) −8.34582e6 −0.472776
\(793\) −2.80875e7 −1.58610
\(794\) −1.49637e7 −0.842340
\(795\) −6.06918e6 −0.340575
\(796\) −2.51866e6 −0.140892
\(797\) −1.42204e7 −0.792985 −0.396493 0.918038i \(-0.629773\pi\)
−0.396493 + 0.918038i \(0.629773\pi\)
\(798\) 3.99926e7 2.22317
\(799\) −3.03089e6 −0.167959
\(800\) −6.97244e6 −0.385177
\(801\) 1.26000e6 0.0693889
\(802\) −2.15502e7 −1.18308
\(803\) −2.66265e6 −0.145722
\(804\) −471295. −0.0257130
\(805\) −1.08355e7 −0.589332
\(806\) 1.69218e7 0.917504
\(807\) −8.29542e6 −0.448389
\(808\) 1.92203e7 1.03570
\(809\) 2.40784e7 1.29347 0.646736 0.762714i \(-0.276134\pi\)
0.646736 + 0.762714i \(0.276134\pi\)
\(810\) 4.21519e7 2.25738
\(811\) 2.71158e7 1.44767 0.723836 0.689972i \(-0.242377\pi\)
0.723836 + 0.689972i \(0.242377\pi\)
\(812\) −352867. −0.0187811
\(813\) 2.82127e7 1.49699
\(814\) −8.48311e6 −0.448739
\(815\) −1.74231e7 −0.918824
\(816\) −5.46471e6 −0.287304
\(817\) 6.30776e7 3.30613
\(818\) 1.58875e7 0.830179
\(819\) −8.60937e6 −0.448499
\(820\) −409996. −0.0212934
\(821\) −1.57023e7 −0.813025 −0.406513 0.913645i \(-0.633255\pi\)
−0.406513 + 0.913645i \(0.633255\pi\)
\(822\) −9.84377e6 −0.508138
\(823\) −1.52411e7 −0.784364 −0.392182 0.919888i \(-0.628280\pi\)
−0.392182 + 0.919888i \(0.628280\pi\)
\(824\) −6.56250e6 −0.336707
\(825\) −6.26346e7 −3.20390
\(826\) −2.29993e6 −0.117291
\(827\) −2.27417e7 −1.15627 −0.578136 0.815940i \(-0.696220\pi\)
−0.578136 + 0.815940i \(0.696220\pi\)
\(828\) 217977. 0.0110493
\(829\) 1.10029e7 0.556060 0.278030 0.960572i \(-0.410319\pi\)
0.278030 + 0.960572i \(0.410319\pi\)
\(830\) −2.49088e7 −1.25504
\(831\) 2.06638e7 1.03803
\(832\) −1.96963e7 −0.986453
\(833\) −718089. −0.0358563
\(834\) 4.33157e7 2.15640
\(835\) 4.41885e7 2.19327
\(836\) −4.24588e6 −0.210113
\(837\) −1.17377e7 −0.579123
\(838\) −2.56486e7 −1.26169
\(839\) −2.33372e7 −1.14458 −0.572288 0.820053i \(-0.693944\pi\)
−0.572288 + 0.820053i \(0.693944\pi\)
\(840\) −4.27768e7 −2.09175
\(841\) 707281. 0.0344828
\(842\) 2.00551e7 0.974866
\(843\) 2.15709e7 1.04544
\(844\) −1.91692e6 −0.0926291
\(845\) −8.36931e6 −0.403225
\(846\) 6.08405e6 0.292258
\(847\) 1.71847e7 0.823065
\(848\) 3.77427e6 0.180237
\(849\) −6.31380e6 −0.300622
\(850\) −1.02279e7 −0.485555
\(851\) −2.13667e6 −0.101138
\(852\) 384496. 0.0181465
\(853\) 1.63119e7 0.767595 0.383797 0.923417i \(-0.374616\pi\)
0.383797 + 0.923417i \(0.374616\pi\)
\(854\) 3.43098e7 1.60981
\(855\) 2.36282e7 1.10539
\(856\) 3.32976e6 0.155321
\(857\) −1.11103e7 −0.516743 −0.258372 0.966046i \(-0.583186\pi\)
−0.258372 + 0.966046i \(0.583186\pi\)
\(858\) 3.89803e7 1.80770
\(859\) 2.94971e7 1.36394 0.681971 0.731379i \(-0.261123\pi\)
0.681971 + 0.731379i \(0.261123\pi\)
\(860\) 6.99619e6 0.322564
\(861\) −3.56030e6 −0.163674
\(862\) 1.72074e6 0.0788764
\(863\) 2.33489e7 1.06718 0.533592 0.845742i \(-0.320842\pi\)
0.533592 + 0.845742i \(0.320842\pi\)
\(864\) −3.00991e6 −0.137173
\(865\) −7.37260e7 −3.35027
\(866\) −1.56295e7 −0.708191
\(867\) 2.46361e7 1.11307
\(868\) −1.77526e6 −0.0799765
\(869\) −1.27178e7 −0.571296
\(870\) −8.89094e6 −0.398244
\(871\) −5.79582e6 −0.258863
\(872\) 8.75374e6 0.389855
\(873\) −1.60987e7 −0.714915
\(874\) −1.24520e7 −0.551394
\(875\) −4.50286e7 −1.98824
\(876\) 274545. 0.0120880
\(877\) −3.29425e7 −1.44630 −0.723149 0.690692i \(-0.757306\pi\)
−0.723149 + 0.690692i \(0.757306\pi\)
\(878\) 3.34577e7 1.46474
\(879\) −4.40297e7 −1.92209
\(880\) 5.78917e7 2.52005
\(881\) 3.22782e7 1.40110 0.700550 0.713603i \(-0.252938\pi\)
0.700550 + 0.713603i \(0.252938\pi\)
\(882\) 1.44145e6 0.0623920
\(883\) −1.31542e7 −0.567758 −0.283879 0.958860i \(-0.591621\pi\)
−0.283879 + 0.958860i \(0.591621\pi\)
\(884\) 546672. 0.0235286
\(885\) −4.97694e6 −0.213602
\(886\) 7.79679e6 0.333681
\(887\) −5.84285e6 −0.249354 −0.124677 0.992197i \(-0.539789\pi\)
−0.124677 + 0.992197i \(0.539789\pi\)
\(888\) −8.43523e6 −0.358975
\(889\) 3.72914e7 1.58254
\(890\) −7.98340e6 −0.337841
\(891\) −3.88608e7 −1.63990
\(892\) 136876. 0.00575990
\(893\) −2.98492e7 −1.25258
\(894\) 1.98095e6 0.0828952
\(895\) 3.44809e7 1.43887
\(896\) 2.89050e7 1.20283
\(897\) 9.81811e6 0.407424
\(898\) −3.84946e7 −1.59297
\(899\) 3.55830e6 0.146840
\(900\) 1.76327e6 0.0725625
\(901\) 913684. 0.0374959
\(902\) 4.40112e6 0.180114
\(903\) 6.07531e7 2.47941
\(904\) −7.83737e6 −0.318970
\(905\) −2.21935e7 −0.900751
\(906\) 1.93627e6 0.0783691
\(907\) −2.27296e7 −0.917431 −0.458716 0.888583i \(-0.651690\pi\)
−0.458716 + 0.888583i \(0.651690\pi\)
\(908\) 4.00804e6 0.161331
\(909\) −1.02253e7 −0.410457
\(910\) 5.45491e7 2.18366
\(911\) 2.85916e7 1.14141 0.570706 0.821154i \(-0.306670\pi\)
0.570706 + 0.821154i \(0.306670\pi\)
\(912\) −5.38183e7 −2.14261
\(913\) 2.29640e7 0.911738
\(914\) −2.07491e7 −0.821548
\(915\) 7.42448e7 2.93166
\(916\) 2.49804e6 0.0983695
\(917\) −2.31001e7 −0.907173
\(918\) −4.41525e6 −0.172921
\(919\) 4.07727e7 1.59250 0.796252 0.604966i \(-0.206813\pi\)
0.796252 + 0.604966i \(0.206813\pi\)
\(920\) 1.33189e7 0.518799
\(921\) −1.08859e7 −0.422878
\(922\) −7.23289e6 −0.280210
\(923\) 4.72840e6 0.182688
\(924\) −4.08941e6 −0.157573
\(925\) −1.72841e7 −0.664189
\(926\) 2.26121e6 0.0866588
\(927\) 3.49129e6 0.133440
\(928\) 912458. 0.0347811
\(929\) −4.12838e7 −1.56943 −0.784713 0.619860i \(-0.787190\pi\)
−0.784713 + 0.619860i \(0.787190\pi\)
\(930\) −4.47300e7 −1.69586
\(931\) −7.07198e6 −0.267403
\(932\) 2.30786e6 0.0870301
\(933\) −1.82810e7 −0.687537
\(934\) −1.07333e7 −0.402592
\(935\) 1.40146e7 0.524264
\(936\) 1.05826e7 0.394822
\(937\) 4.54057e7 1.68951 0.844755 0.535153i \(-0.179746\pi\)
0.844755 + 0.535153i \(0.179746\pi\)
\(938\) 7.07979e6 0.262732
\(939\) 6.67817e6 0.247169
\(940\) −3.31070e6 −0.122208
\(941\) 4.70900e7 1.73362 0.866812 0.498635i \(-0.166165\pi\)
0.866812 + 0.498635i \(0.166165\pi\)
\(942\) 4.47401e7 1.64274
\(943\) 1.10853e6 0.0405946
\(944\) 3.09503e6 0.113041
\(945\) −3.78379e7 −1.37831
\(946\) −7.51010e7 −2.72846
\(947\) 2.49040e6 0.0902390 0.0451195 0.998982i \(-0.485633\pi\)
0.0451195 + 0.998982i \(0.485633\pi\)
\(948\) 1.31133e6 0.0473904
\(949\) 3.37626e6 0.121694
\(950\) −1.00728e8 −3.62109
\(951\) −2.93686e7 −1.05301
\(952\) 6.43983e6 0.230294
\(953\) −1.43265e7 −0.510985 −0.255493 0.966811i \(-0.582238\pi\)
−0.255493 + 0.966811i \(0.582238\pi\)
\(954\) −1.83408e6 −0.0652450
\(955\) −1.79254e7 −0.636004
\(956\) −3.69371e6 −0.130713
\(957\) 8.19676e6 0.289309
\(958\) 2.88375e7 1.01518
\(959\) 1.26999e7 0.445916
\(960\) 5.20640e7 1.82331
\(961\) −1.07275e7 −0.374704
\(962\) 1.07566e7 0.374748
\(963\) −1.77145e6 −0.0615550
\(964\) 914063. 0.0316799
\(965\) −9.74385e7 −3.36831
\(966\) −1.19932e7 −0.413515
\(967\) −1.92616e7 −0.662407 −0.331204 0.943559i \(-0.607455\pi\)
−0.331204 + 0.943559i \(0.607455\pi\)
\(968\) −2.11233e7 −0.724559
\(969\) −1.30285e7 −0.445742
\(970\) 1.02002e8 3.48079
\(971\) −3.31255e7 −1.12749 −0.563747 0.825948i \(-0.690641\pi\)
−0.563747 + 0.825948i \(0.690641\pi\)
\(972\) 1.98017e6 0.0672259
\(973\) −5.58834e7 −1.89235
\(974\) −1.09430e7 −0.369607
\(975\) 7.94211e7 2.67562
\(976\) −4.61709e7 −1.55147
\(977\) −5.00959e7 −1.67906 −0.839529 0.543314i \(-0.817169\pi\)
−0.839529 + 0.543314i \(0.817169\pi\)
\(978\) −1.92846e7 −0.644709
\(979\) 7.36008e6 0.245429
\(980\) −784382. −0.0260893
\(981\) −4.65704e6 −0.154503
\(982\) −3.23692e7 −1.07116
\(983\) −6.69976e6 −0.221144 −0.110572 0.993868i \(-0.535268\pi\)
−0.110572 + 0.993868i \(0.535268\pi\)
\(984\) 4.37629e6 0.144085
\(985\) 8.41748e7 2.76434
\(986\) 1.33849e6 0.0438451
\(987\) −2.87493e7 −0.939364
\(988\) 5.38381e6 0.175468
\(989\) −1.89160e7 −0.614948
\(990\) −2.81321e7 −0.912249
\(991\) −3.54819e7 −1.14768 −0.573842 0.818966i \(-0.694548\pi\)
−0.573842 + 0.818966i \(0.694548\pi\)
\(992\) 4.59054e6 0.148110
\(993\) 9.78359e6 0.314866
\(994\) −5.77590e6 −0.185419
\(995\) 8.18737e7 2.62172
\(996\) −2.36781e6 −0.0756309
\(997\) −2.99768e7 −0.955097 −0.477549 0.878605i \(-0.658475\pi\)
−0.477549 + 0.878605i \(0.658475\pi\)
\(998\) 3.12256e7 0.992395
\(999\) −7.46132e6 −0.236538
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 29.6.a.a.1.4 4
3.2 odd 2 261.6.a.a.1.1 4
4.3 odd 2 464.6.a.i.1.4 4
5.4 even 2 725.6.a.a.1.1 4
29.28 even 2 841.6.a.a.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
29.6.a.a.1.4 4 1.1 even 1 trivial
261.6.a.a.1.1 4 3.2 odd 2
464.6.a.i.1.4 4 4.3 odd 2
725.6.a.a.1.1 4 5.4 even 2
841.6.a.a.1.1 4 29.28 even 2