Properties

Label 387.6.a.c.1.7
Level $387$
Weight $6$
Character 387.1
Self dual yes
Analytic conductor $62.069$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 387.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(62.0685382676\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 4x^{7} - 173x^{6} + 462x^{5} + 9118x^{4} - 14192x^{3} - 167688x^{2} + 106368x + 681984 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-6.09504\) of defining polynomial
Character \(\chi\) \(=\) 387.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.09504 q^{2} +33.5297 q^{4} +63.3756 q^{5} +223.489 q^{7} +12.3830 q^{8} +O(q^{10})\) \(q+8.09504 q^{2} +33.5297 q^{4} +63.3756 q^{5} +223.489 q^{7} +12.3830 q^{8} +513.028 q^{10} +631.897 q^{11} +28.5724 q^{13} +1809.15 q^{14} -972.709 q^{16} +1743.07 q^{17} -2027.92 q^{19} +2124.97 q^{20} +5115.23 q^{22} -2980.86 q^{23} +891.469 q^{25} +231.295 q^{26} +7493.51 q^{28} -766.139 q^{29} -8355.33 q^{31} -8270.38 q^{32} +14110.3 q^{34} +14163.7 q^{35} +14892.6 q^{37} -16416.1 q^{38} +784.783 q^{40} +5342.20 q^{41} -1849.00 q^{43} +21187.3 q^{44} -24130.2 q^{46} +6282.09 q^{47} +33140.1 q^{49} +7216.48 q^{50} +958.024 q^{52} +915.172 q^{53} +40046.8 q^{55} +2767.47 q^{56} -6201.92 q^{58} +14644.5 q^{59} -21324.9 q^{61} -67636.7 q^{62} -35822.4 q^{64} +1810.79 q^{65} -12868.9 q^{67} +58444.8 q^{68} +114656. q^{70} -56454.6 q^{71} -25591.3 q^{73} +120556. q^{74} -67995.4 q^{76} +141222. q^{77} +5795.13 q^{79} -61646.1 q^{80} +43245.4 q^{82} +7857.24 q^{83} +110468. q^{85} -14967.7 q^{86} +7824.80 q^{88} +7560.11 q^{89} +6385.60 q^{91} -99947.5 q^{92} +50853.8 q^{94} -128520. q^{95} +111712. q^{97} +268271. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 12 q^{2} + 122 q^{4} + 212 q^{5} - 136 q^{7} + 666 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 12 q^{2} + 122 q^{4} + 212 q^{5} - 136 q^{7} + 666 q^{8} - 617 q^{10} + 532 q^{11} - 2492 q^{13} + 4240 q^{14} + 1882 q^{16} + 2534 q^{17} - 1678 q^{19} + 2607 q^{20} + 11502 q^{22} + 2488 q^{23} + 4378 q^{25} - 4586 q^{26} + 18640 q^{28} + 4360 q^{29} + 5704 q^{31} + 18294 q^{32} + 30007 q^{34} - 5640 q^{35} - 3772 q^{37} + 6559 q^{38} + 14869 q^{40} + 10698 q^{41} - 14792 q^{43} + 356 q^{44} - 19389 q^{46} + 77864 q^{47} + 7188 q^{49} - 26877 q^{50} - 60736 q^{52} + 62352 q^{53} - 49552 q^{55} + 144528 q^{56} + 52951 q^{58} + 26224 q^{59} - 82540 q^{61} + 9023 q^{62} + 153858 q^{64} + 5000 q^{65} + 27784 q^{67} - 40507 q^{68} + 185910 q^{70} + 9504 q^{71} + 14260 q^{73} + 15239 q^{74} + 1279 q^{76} + 218140 q^{77} + 160248 q^{79} + 1291 q^{80} - 47781 q^{82} + 77176 q^{83} + 141096 q^{85} - 22188 q^{86} + 129544 q^{88} + 265692 q^{89} + 401148 q^{91} - 190391 q^{92} + 248737 q^{94} - 135884 q^{95} + 144742 q^{97} + 292244 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.09504 1.43101 0.715507 0.698605i \(-0.246196\pi\)
0.715507 + 0.698605i \(0.246196\pi\)
\(3\) 0 0
\(4\) 33.5297 1.04780
\(5\) 63.3756 1.13370 0.566849 0.823822i \(-0.308162\pi\)
0.566849 + 0.823822i \(0.308162\pi\)
\(6\) 0 0
\(7\) 223.489 1.72389 0.861946 0.507000i \(-0.169245\pi\)
0.861946 + 0.507000i \(0.169245\pi\)
\(8\) 12.3830 0.0684073
\(9\) 0 0
\(10\) 513.028 1.62234
\(11\) 631.897 1.57458 0.787289 0.616584i \(-0.211484\pi\)
0.787289 + 0.616584i \(0.211484\pi\)
\(12\) 0 0
\(13\) 28.5724 0.0468909 0.0234454 0.999725i \(-0.492536\pi\)
0.0234454 + 0.999725i \(0.492536\pi\)
\(14\) 1809.15 2.46692
\(15\) 0 0
\(16\) −972.709 −0.949911
\(17\) 1743.07 1.46283 0.731414 0.681933i \(-0.238861\pi\)
0.731414 + 0.681933i \(0.238861\pi\)
\(18\) 0 0
\(19\) −2027.92 −1.28874 −0.644371 0.764713i \(-0.722881\pi\)
−0.644371 + 0.764713i \(0.722881\pi\)
\(20\) 2124.97 1.18789
\(21\) 0 0
\(22\) 5115.23 2.25324
\(23\) −2980.86 −1.17496 −0.587479 0.809239i \(-0.699880\pi\)
−0.587479 + 0.809239i \(0.699880\pi\)
\(24\) 0 0
\(25\) 891.469 0.285270
\(26\) 231.295 0.0671015
\(27\) 0 0
\(28\) 7493.51 1.80630
\(29\) −766.139 −0.169166 −0.0845829 0.996416i \(-0.526956\pi\)
−0.0845829 + 0.996416i \(0.526956\pi\)
\(30\) 0 0
\(31\) −8355.33 −1.56156 −0.780781 0.624805i \(-0.785178\pi\)
−0.780781 + 0.624805i \(0.785178\pi\)
\(32\) −8270.38 −1.42774
\(33\) 0 0
\(34\) 14110.3 2.09333
\(35\) 14163.7 1.95437
\(36\) 0 0
\(37\) 14892.6 1.78841 0.894204 0.447661i \(-0.147743\pi\)
0.894204 + 0.447661i \(0.147743\pi\)
\(38\) −16416.1 −1.84421
\(39\) 0 0
\(40\) 784.783 0.0775532
\(41\) 5342.20 0.496319 0.248159 0.968719i \(-0.420174\pi\)
0.248159 + 0.968719i \(0.420174\pi\)
\(42\) 0 0
\(43\) −1849.00 −0.152499
\(44\) 21187.3 1.64985
\(45\) 0 0
\(46\) −24130.2 −1.68138
\(47\) 6282.09 0.414820 0.207410 0.978254i \(-0.433497\pi\)
0.207410 + 0.978254i \(0.433497\pi\)
\(48\) 0 0
\(49\) 33140.1 1.97181
\(50\) 7216.48 0.408226
\(51\) 0 0
\(52\) 958.024 0.0491324
\(53\) 915.172 0.0447521 0.0223760 0.999750i \(-0.492877\pi\)
0.0223760 + 0.999750i \(0.492877\pi\)
\(54\) 0 0
\(55\) 40046.8 1.78510
\(56\) 2767.47 0.117927
\(57\) 0 0
\(58\) −6201.92 −0.242079
\(59\) 14644.5 0.547704 0.273852 0.961772i \(-0.411702\pi\)
0.273852 + 0.961772i \(0.411702\pi\)
\(60\) 0 0
\(61\) −21324.9 −0.733775 −0.366887 0.930265i \(-0.619577\pi\)
−0.366887 + 0.930265i \(0.619577\pi\)
\(62\) −67636.7 −2.23462
\(63\) 0 0
\(64\) −35822.4 −1.09321
\(65\) 1810.79 0.0531601
\(66\) 0 0
\(67\) −12868.9 −0.350232 −0.175116 0.984548i \(-0.556030\pi\)
−0.175116 + 0.984548i \(0.556030\pi\)
\(68\) 58444.8 1.53276
\(69\) 0 0
\(70\) 114656. 2.79674
\(71\) −56454.6 −1.32909 −0.664544 0.747249i \(-0.731374\pi\)
−0.664544 + 0.747249i \(0.731374\pi\)
\(72\) 0 0
\(73\) −25591.3 −0.562064 −0.281032 0.959698i \(-0.590677\pi\)
−0.281032 + 0.959698i \(0.590677\pi\)
\(74\) 120556. 2.55924
\(75\) 0 0
\(76\) −67995.4 −1.35035
\(77\) 141222. 2.71440
\(78\) 0 0
\(79\) 5795.13 0.104471 0.0522355 0.998635i \(-0.483365\pi\)
0.0522355 + 0.998635i \(0.483365\pi\)
\(80\) −61646.1 −1.07691
\(81\) 0 0
\(82\) 43245.4 0.710240
\(83\) 7857.24 0.125192 0.0625958 0.998039i \(-0.480062\pi\)
0.0625958 + 0.998039i \(0.480062\pi\)
\(84\) 0 0
\(85\) 110468. 1.65841
\(86\) −14967.7 −0.218228
\(87\) 0 0
\(88\) 7824.80 0.107713
\(89\) 7560.11 0.101170 0.0505852 0.998720i \(-0.483891\pi\)
0.0505852 + 0.998720i \(0.483891\pi\)
\(90\) 0 0
\(91\) 6385.60 0.0808348
\(92\) −99947.5 −1.23113
\(93\) 0 0
\(94\) 50853.8 0.593613
\(95\) −128520. −1.46104
\(96\) 0 0
\(97\) 111712. 1.20551 0.602754 0.797927i \(-0.294070\pi\)
0.602754 + 0.797927i \(0.294070\pi\)
\(98\) 268271. 2.82168
\(99\) 0 0
\(100\) 29890.7 0.298907
\(101\) −10492.3 −0.102345 −0.0511724 0.998690i \(-0.516296\pi\)
−0.0511724 + 0.998690i \(0.516296\pi\)
\(102\) 0 0
\(103\) −11286.6 −0.104826 −0.0524130 0.998625i \(-0.516691\pi\)
−0.0524130 + 0.998625i \(0.516691\pi\)
\(104\) 353.813 0.00320768
\(105\) 0 0
\(106\) 7408.36 0.0640409
\(107\) −39045.4 −0.329693 −0.164847 0.986319i \(-0.552713\pi\)
−0.164847 + 0.986319i \(0.552713\pi\)
\(108\) 0 0
\(109\) −14622.8 −0.117887 −0.0589433 0.998261i \(-0.518773\pi\)
−0.0589433 + 0.998261i \(0.518773\pi\)
\(110\) 324181. 2.55450
\(111\) 0 0
\(112\) −217389. −1.63755
\(113\) 859.121 0.00632934 0.00316467 0.999995i \(-0.498993\pi\)
0.00316467 + 0.999995i \(0.498993\pi\)
\(114\) 0 0
\(115\) −188914. −1.33205
\(116\) −25688.4 −0.177252
\(117\) 0 0
\(118\) 118548. 0.783772
\(119\) 389557. 2.52176
\(120\) 0 0
\(121\) 238242. 1.47930
\(122\) −172626. −1.05004
\(123\) 0 0
\(124\) −280152. −1.63621
\(125\) −141551. −0.810288
\(126\) 0 0
\(127\) 193396. 1.06399 0.531997 0.846746i \(-0.321442\pi\)
0.531997 + 0.846746i \(0.321442\pi\)
\(128\) −25331.5 −0.136658
\(129\) 0 0
\(130\) 14658.4 0.0760728
\(131\) 269599. 1.37259 0.686293 0.727325i \(-0.259237\pi\)
0.686293 + 0.727325i \(0.259237\pi\)
\(132\) 0 0
\(133\) −453216. −2.22165
\(134\) −104175. −0.501187
\(135\) 0 0
\(136\) 21584.6 0.100068
\(137\) −148987. −0.678183 −0.339091 0.940753i \(-0.610120\pi\)
−0.339091 + 0.940753i \(0.610120\pi\)
\(138\) 0 0
\(139\) −16355.6 −0.0718008 −0.0359004 0.999355i \(-0.511430\pi\)
−0.0359004 + 0.999355i \(0.511430\pi\)
\(140\) 474906. 2.04780
\(141\) 0 0
\(142\) −457003. −1.90194
\(143\) 18054.8 0.0738333
\(144\) 0 0
\(145\) −48554.5 −0.191783
\(146\) −207163. −0.804322
\(147\) 0 0
\(148\) 499345. 1.87390
\(149\) 150536. 0.555489 0.277744 0.960655i \(-0.410413\pi\)
0.277744 + 0.960655i \(0.410413\pi\)
\(150\) 0 0
\(151\) 428141. 1.52807 0.764037 0.645172i \(-0.223214\pi\)
0.764037 + 0.645172i \(0.223214\pi\)
\(152\) −25111.8 −0.0881594
\(153\) 0 0
\(154\) 1.14320e6 3.88435
\(155\) −529524. −1.77034
\(156\) 0 0
\(157\) −434190. −1.40582 −0.702912 0.711277i \(-0.748117\pi\)
−0.702912 + 0.711277i \(0.748117\pi\)
\(158\) 46911.8 0.149499
\(159\) 0 0
\(160\) −524140. −1.61863
\(161\) −666189. −2.02550
\(162\) 0 0
\(163\) −571830. −1.68577 −0.842884 0.538095i \(-0.819144\pi\)
−0.842884 + 0.538095i \(0.819144\pi\)
\(164\) 179122. 0.520044
\(165\) 0 0
\(166\) 63604.7 0.179151
\(167\) −605744. −1.68073 −0.840365 0.542020i \(-0.817660\pi\)
−0.840365 + 0.542020i \(0.817660\pi\)
\(168\) 0 0
\(169\) −370477. −0.997801
\(170\) 894246. 2.37320
\(171\) 0 0
\(172\) −61996.4 −0.159789
\(173\) 85238.8 0.216532 0.108266 0.994122i \(-0.465470\pi\)
0.108266 + 0.994122i \(0.465470\pi\)
\(174\) 0 0
\(175\) 199233. 0.491775
\(176\) −614652. −1.49571
\(177\) 0 0
\(178\) 61199.4 0.144776
\(179\) −77941.8 −0.181818 −0.0909092 0.995859i \(-0.528977\pi\)
−0.0909092 + 0.995859i \(0.528977\pi\)
\(180\) 0 0
\(181\) −347661. −0.788787 −0.394393 0.918942i \(-0.629045\pi\)
−0.394393 + 0.918942i \(0.629045\pi\)
\(182\) 51691.7 0.115676
\(183\) 0 0
\(184\) −36912.2 −0.0803757
\(185\) 943828. 2.02751
\(186\) 0 0
\(187\) 1.10144e6 2.30334
\(188\) 210637. 0.434650
\(189\) 0 0
\(190\) −1.04038e6 −2.09078
\(191\) 192163. 0.381142 0.190571 0.981673i \(-0.438966\pi\)
0.190571 + 0.981673i \(0.438966\pi\)
\(192\) 0 0
\(193\) −271724. −0.525091 −0.262546 0.964920i \(-0.584562\pi\)
−0.262546 + 0.964920i \(0.584562\pi\)
\(194\) 904313. 1.72510
\(195\) 0 0
\(196\) 1.11118e6 2.06606
\(197\) −76788.1 −0.140971 −0.0704853 0.997513i \(-0.522455\pi\)
−0.0704853 + 0.997513i \(0.522455\pi\)
\(198\) 0 0
\(199\) −694776. −1.24369 −0.621845 0.783140i \(-0.713617\pi\)
−0.621845 + 0.783140i \(0.713617\pi\)
\(200\) 11039.1 0.0195146
\(201\) 0 0
\(202\) −84935.3 −0.146457
\(203\) −171223. −0.291624
\(204\) 0 0
\(205\) 338565. 0.562675
\(206\) −91365.3 −0.150008
\(207\) 0 0
\(208\) −27792.6 −0.0445422
\(209\) −1.28143e6 −2.02923
\(210\) 0 0
\(211\) −433533. −0.670372 −0.335186 0.942152i \(-0.608799\pi\)
−0.335186 + 0.942152i \(0.608799\pi\)
\(212\) 30685.5 0.0468914
\(213\) 0 0
\(214\) −316074. −0.471796
\(215\) −117182. −0.172887
\(216\) 0 0
\(217\) −1.86732e6 −2.69196
\(218\) −118372. −0.168698
\(219\) 0 0
\(220\) 1.34276e6 1.87043
\(221\) 49803.8 0.0685933
\(222\) 0 0
\(223\) 1.04204e6 1.40320 0.701602 0.712569i \(-0.252469\pi\)
0.701602 + 0.712569i \(0.252469\pi\)
\(224\) −1.84834e6 −2.46128
\(225\) 0 0
\(226\) 6954.62 0.00905737
\(227\) −924970. −1.19141 −0.595707 0.803202i \(-0.703128\pi\)
−0.595707 + 0.803202i \(0.703128\pi\)
\(228\) 0 0
\(229\) −137140. −0.172812 −0.0864062 0.996260i \(-0.527538\pi\)
−0.0864062 + 0.996260i \(0.527538\pi\)
\(230\) −1.52927e6 −1.90618
\(231\) 0 0
\(232\) −9487.13 −0.0115722
\(233\) −428602. −0.517208 −0.258604 0.965983i \(-0.583262\pi\)
−0.258604 + 0.965983i \(0.583262\pi\)
\(234\) 0 0
\(235\) 398131. 0.470280
\(236\) 491027. 0.573886
\(237\) 0 0
\(238\) 3.15348e6 3.60868
\(239\) 1.17134e6 1.32645 0.663223 0.748422i \(-0.269188\pi\)
0.663223 + 0.748422i \(0.269188\pi\)
\(240\) 0 0
\(241\) 119252. 0.132258 0.0661291 0.997811i \(-0.478935\pi\)
0.0661291 + 0.997811i \(0.478935\pi\)
\(242\) 1.92858e6 2.11690
\(243\) 0 0
\(244\) −715018. −0.768852
\(245\) 2.10028e6 2.23543
\(246\) 0 0
\(247\) −57942.4 −0.0604302
\(248\) −103464. −0.106822
\(249\) 0 0
\(250\) −1.14586e6 −1.15953
\(251\) 141272. 0.141538 0.0707690 0.997493i \(-0.477455\pi\)
0.0707690 + 0.997493i \(0.477455\pi\)
\(252\) 0 0
\(253\) −1.88360e6 −1.85006
\(254\) 1.56555e6 1.52259
\(255\) 0 0
\(256\) 941257. 0.897652
\(257\) 196749. 0.185815 0.0929074 0.995675i \(-0.470384\pi\)
0.0929074 + 0.995675i \(0.470384\pi\)
\(258\) 0 0
\(259\) 3.32833e6 3.08302
\(260\) 60715.4 0.0557013
\(261\) 0 0
\(262\) 2.18241e6 1.96419
\(263\) 1.96879e6 1.75513 0.877565 0.479457i \(-0.159166\pi\)
0.877565 + 0.479457i \(0.159166\pi\)
\(264\) 0 0
\(265\) 57999.6 0.0507353
\(266\) −3.66880e6 −3.17922
\(267\) 0 0
\(268\) −431492. −0.366974
\(269\) 531007. 0.447424 0.223712 0.974655i \(-0.428182\pi\)
0.223712 + 0.974655i \(0.428182\pi\)
\(270\) 0 0
\(271\) −1.68769e6 −1.39595 −0.697975 0.716122i \(-0.745916\pi\)
−0.697975 + 0.716122i \(0.745916\pi\)
\(272\) −1.69550e6 −1.38956
\(273\) 0 0
\(274\) −1.20606e6 −0.970489
\(275\) 563316. 0.449180
\(276\) 0 0
\(277\) 13630.1 0.0106733 0.00533665 0.999986i \(-0.498301\pi\)
0.00533665 + 0.999986i \(0.498301\pi\)
\(278\) −132399. −0.102748
\(279\) 0 0
\(280\) 175390. 0.133693
\(281\) −1.12070e6 −0.846692 −0.423346 0.905968i \(-0.639145\pi\)
−0.423346 + 0.905968i \(0.639145\pi\)
\(282\) 0 0
\(283\) −1.52947e6 −1.13521 −0.567605 0.823301i \(-0.692130\pi\)
−0.567605 + 0.823301i \(0.692130\pi\)
\(284\) −1.89291e6 −1.39262
\(285\) 0 0
\(286\) 146154. 0.105657
\(287\) 1.19392e6 0.855600
\(288\) 0 0
\(289\) 1.61845e6 1.13987
\(290\) −393051. −0.274444
\(291\) 0 0
\(292\) −858070. −0.588932
\(293\) 1.58484e6 1.07849 0.539246 0.842148i \(-0.318709\pi\)
0.539246 + 0.842148i \(0.318709\pi\)
\(294\) 0 0
\(295\) 928107. 0.620930
\(296\) 184416. 0.122340
\(297\) 0 0
\(298\) 1.21860e6 0.794912
\(299\) −85170.4 −0.0550948
\(300\) 0 0
\(301\) −413230. −0.262891
\(302\) 3.46582e6 2.18670
\(303\) 0 0
\(304\) 1.97257e6 1.22419
\(305\) −1.35148e6 −0.831879
\(306\) 0 0
\(307\) 695751. 0.421316 0.210658 0.977560i \(-0.432439\pi\)
0.210658 + 0.977560i \(0.432439\pi\)
\(308\) 4.73512e6 2.84416
\(309\) 0 0
\(310\) −4.28652e6 −2.53338
\(311\) 988763. 0.579684 0.289842 0.957075i \(-0.406397\pi\)
0.289842 + 0.957075i \(0.406397\pi\)
\(312\) 0 0
\(313\) −3.07551e6 −1.77442 −0.887208 0.461369i \(-0.847358\pi\)
−0.887208 + 0.461369i \(0.847358\pi\)
\(314\) −3.51479e6 −2.01176
\(315\) 0 0
\(316\) 194309. 0.109465
\(317\) −904527. −0.505561 −0.252780 0.967524i \(-0.581345\pi\)
−0.252780 + 0.967524i \(0.581345\pi\)
\(318\) 0 0
\(319\) −484120. −0.266365
\(320\) −2.27027e6 −1.23937
\(321\) 0 0
\(322\) −5.39283e6 −2.89852
\(323\) −3.53481e6 −1.88521
\(324\) 0 0
\(325\) 25471.4 0.0133766
\(326\) −4.62899e6 −2.41236
\(327\) 0 0
\(328\) 66152.7 0.0339518
\(329\) 1.40398e6 0.715105
\(330\) 0 0
\(331\) −2.09309e6 −1.05007 −0.525034 0.851081i \(-0.675947\pi\)
−0.525034 + 0.851081i \(0.675947\pi\)
\(332\) 263451. 0.131176
\(333\) 0 0
\(334\) −4.90353e6 −2.40515
\(335\) −815577. −0.397057
\(336\) 0 0
\(337\) −647484. −0.310566 −0.155283 0.987870i \(-0.549629\pi\)
−0.155283 + 0.987870i \(0.549629\pi\)
\(338\) −2.99902e6 −1.42787
\(339\) 0 0
\(340\) 3.70397e6 1.73768
\(341\) −5.27970e6 −2.45880
\(342\) 0 0
\(343\) 3.65027e6 1.67529
\(344\) −22896.2 −0.0104320
\(345\) 0 0
\(346\) 690012. 0.309860
\(347\) 462018. 0.205985 0.102992 0.994682i \(-0.467158\pi\)
0.102992 + 0.994682i \(0.467158\pi\)
\(348\) 0 0
\(349\) 2.85075e6 1.25284 0.626420 0.779486i \(-0.284520\pi\)
0.626420 + 0.779486i \(0.284520\pi\)
\(350\) 1.61280e6 0.703737
\(351\) 0 0
\(352\) −5.22602e6 −2.24810
\(353\) 809444. 0.345740 0.172870 0.984945i \(-0.444696\pi\)
0.172870 + 0.984945i \(0.444696\pi\)
\(354\) 0 0
\(355\) −3.57785e6 −1.50678
\(356\) 253488. 0.106007
\(357\) 0 0
\(358\) −630942. −0.260185
\(359\) −3.46118e6 −1.41739 −0.708694 0.705516i \(-0.750715\pi\)
−0.708694 + 0.705516i \(0.750715\pi\)
\(360\) 0 0
\(361\) 1.63635e6 0.660856
\(362\) −2.81433e6 −1.12877
\(363\) 0 0
\(364\) 214107. 0.0846990
\(365\) −1.62187e6 −0.637210
\(366\) 0 0
\(367\) 2.13602e6 0.827827 0.413914 0.910316i \(-0.364162\pi\)
0.413914 + 0.910316i \(0.364162\pi\)
\(368\) 2.89951e6 1.11611
\(369\) 0 0
\(370\) 7.64033e6 2.90140
\(371\) 204531. 0.0771478
\(372\) 0 0
\(373\) 2.37193e6 0.882734 0.441367 0.897327i \(-0.354494\pi\)
0.441367 + 0.897327i \(0.354494\pi\)
\(374\) 8.91622e6 3.29611
\(375\) 0 0
\(376\) 77791.4 0.0283767
\(377\) −21890.4 −0.00793233
\(378\) 0 0
\(379\) −5.47562e6 −1.95810 −0.979051 0.203616i \(-0.934731\pi\)
−0.979051 + 0.203616i \(0.934731\pi\)
\(380\) −4.30925e6 −1.53089
\(381\) 0 0
\(382\) 1.55557e6 0.545419
\(383\) −3.16966e6 −1.10412 −0.552059 0.833805i \(-0.686158\pi\)
−0.552059 + 0.833805i \(0.686158\pi\)
\(384\) 0 0
\(385\) 8.95001e6 3.07731
\(386\) −2.19962e6 −0.751413
\(387\) 0 0
\(388\) 3.74567e6 1.26314
\(389\) 311235. 0.104283 0.0521416 0.998640i \(-0.483395\pi\)
0.0521416 + 0.998640i \(0.483395\pi\)
\(390\) 0 0
\(391\) −5.19587e6 −1.71876
\(392\) 410376. 0.134886
\(393\) 0 0
\(394\) −621603. −0.201731
\(395\) 367270. 0.118438
\(396\) 0 0
\(397\) −496314. −0.158045 −0.0790224 0.996873i \(-0.525180\pi\)
−0.0790224 + 0.996873i \(0.525180\pi\)
\(398\) −5.62424e6 −1.77974
\(399\) 0 0
\(400\) −867140. −0.270981
\(401\) 5.63670e6 1.75051 0.875253 0.483665i \(-0.160695\pi\)
0.875253 + 0.483665i \(0.160695\pi\)
\(402\) 0 0
\(403\) −238732. −0.0732230
\(404\) −351802. −0.107237
\(405\) 0 0
\(406\) −1.38606e6 −0.417318
\(407\) 9.41059e6 2.81599
\(408\) 0 0
\(409\) −720515. −0.212978 −0.106489 0.994314i \(-0.533961\pi\)
−0.106489 + 0.994314i \(0.533961\pi\)
\(410\) 2.74070e6 0.805197
\(411\) 0 0
\(412\) −378435. −0.109837
\(413\) 3.27289e6 0.944182
\(414\) 0 0
\(415\) 497958. 0.141929
\(416\) −236305. −0.0669482
\(417\) 0 0
\(418\) −1.03733e7 −2.90385
\(419\) 3.94151e6 1.09680 0.548400 0.836216i \(-0.315237\pi\)
0.548400 + 0.836216i \(0.315237\pi\)
\(420\) 0 0
\(421\) 4.39506e6 1.20854 0.604268 0.796781i \(-0.293466\pi\)
0.604268 + 0.796781i \(0.293466\pi\)
\(422\) −3.50947e6 −0.959312
\(423\) 0 0
\(424\) 11332.6 0.00306137
\(425\) 1.55390e6 0.417301
\(426\) 0 0
\(427\) −4.76588e6 −1.26495
\(428\) −1.30918e6 −0.345454
\(429\) 0 0
\(430\) −948589. −0.247404
\(431\) −1.24913e6 −0.323903 −0.161951 0.986799i \(-0.551779\pi\)
−0.161951 + 0.986799i \(0.551779\pi\)
\(432\) 0 0
\(433\) 971902. 0.249117 0.124558 0.992212i \(-0.460249\pi\)
0.124558 + 0.992212i \(0.460249\pi\)
\(434\) −1.51160e7 −3.85224
\(435\) 0 0
\(436\) −490299. −0.123522
\(437\) 6.04494e6 1.51422
\(438\) 0 0
\(439\) −1.32453e6 −0.328021 −0.164010 0.986459i \(-0.552443\pi\)
−0.164010 + 0.986459i \(0.552443\pi\)
\(440\) 495902. 0.122114
\(441\) 0 0
\(442\) 403164. 0.0981581
\(443\) −4.76446e6 −1.15346 −0.576732 0.816933i \(-0.695672\pi\)
−0.576732 + 0.816933i \(0.695672\pi\)
\(444\) 0 0
\(445\) 479127. 0.114697
\(446\) 8.43533e6 2.00801
\(447\) 0 0
\(448\) −8.00589e6 −1.88458
\(449\) 5.40007e6 1.26411 0.632053 0.774926i \(-0.282213\pi\)
0.632053 + 0.774926i \(0.282213\pi\)
\(450\) 0 0
\(451\) 3.37572e6 0.781493
\(452\) 28806.1 0.00663190
\(453\) 0 0
\(454\) −7.48767e6 −1.70493
\(455\) 404692. 0.0916422
\(456\) 0 0
\(457\) 2.12752e6 0.476523 0.238261 0.971201i \(-0.423423\pi\)
0.238261 + 0.971201i \(0.423423\pi\)
\(458\) −1.11015e6 −0.247297
\(459\) 0 0
\(460\) −6.33423e6 −1.39572
\(461\) −2.80905e6 −0.615612 −0.307806 0.951449i \(-0.599595\pi\)
−0.307806 + 0.951449i \(0.599595\pi\)
\(462\) 0 0
\(463\) −2.85342e6 −0.618605 −0.309302 0.950964i \(-0.600096\pi\)
−0.309302 + 0.950964i \(0.600096\pi\)
\(464\) 745230. 0.160692
\(465\) 0 0
\(466\) −3.46955e6 −0.740132
\(467\) 559463. 0.118708 0.0593539 0.998237i \(-0.481096\pi\)
0.0593539 + 0.998237i \(0.481096\pi\)
\(468\) 0 0
\(469\) −2.87606e6 −0.603762
\(470\) 3.22289e6 0.672978
\(471\) 0 0
\(472\) 181344. 0.0374669
\(473\) −1.16838e6 −0.240121
\(474\) 0 0
\(475\) −1.80782e6 −0.367640
\(476\) 1.30617e7 2.64231
\(477\) 0 0
\(478\) 9.48208e6 1.89816
\(479\) −5.46747e6 −1.08880 −0.544399 0.838826i \(-0.683242\pi\)
−0.544399 + 0.838826i \(0.683242\pi\)
\(480\) 0 0
\(481\) 425517. 0.0838600
\(482\) 965349. 0.189263
\(483\) 0 0
\(484\) 7.98819e6 1.55001
\(485\) 7.07981e6 1.36668
\(486\) 0 0
\(487\) 9.34759e6 1.78598 0.892991 0.450074i \(-0.148603\pi\)
0.892991 + 0.450074i \(0.148603\pi\)
\(488\) −264067. −0.0501955
\(489\) 0 0
\(490\) 1.70018e7 3.19893
\(491\) 1.04964e7 1.96488 0.982438 0.186589i \(-0.0597433\pi\)
0.982438 + 0.186589i \(0.0597433\pi\)
\(492\) 0 0
\(493\) −1.33544e6 −0.247460
\(494\) −469046. −0.0864766
\(495\) 0 0
\(496\) 8.12730e6 1.48335
\(497\) −1.26170e7 −2.29120
\(498\) 0 0
\(499\) 5.74842e6 1.03347 0.516734 0.856146i \(-0.327148\pi\)
0.516734 + 0.856146i \(0.327148\pi\)
\(500\) −4.74618e6 −0.849022
\(501\) 0 0
\(502\) 1.14361e6 0.202543
\(503\) −3.10341e6 −0.546915 −0.273457 0.961884i \(-0.588167\pi\)
−0.273457 + 0.961884i \(0.588167\pi\)
\(504\) 0 0
\(505\) −664953. −0.116028
\(506\) −1.52478e7 −2.64747
\(507\) 0 0
\(508\) 6.48453e6 1.11486
\(509\) 7.92714e6 1.35620 0.678098 0.734972i \(-0.262805\pi\)
0.678098 + 0.734972i \(0.262805\pi\)
\(510\) 0 0
\(511\) −5.71937e6 −0.968938
\(512\) 8.43012e6 1.42121
\(513\) 0 0
\(514\) 1.59269e6 0.265904
\(515\) −715293. −0.118841
\(516\) 0 0
\(517\) 3.96963e6 0.653166
\(518\) 2.69429e7 4.41185
\(519\) 0 0
\(520\) 22423.1 0.00363654
\(521\) 4.37136e6 0.705541 0.352771 0.935710i \(-0.385240\pi\)
0.352771 + 0.935710i \(0.385240\pi\)
\(522\) 0 0
\(523\) 1.11680e6 0.178534 0.0892671 0.996008i \(-0.471548\pi\)
0.0892671 + 0.996008i \(0.471548\pi\)
\(524\) 9.03957e6 1.43820
\(525\) 0 0
\(526\) 1.59374e7 2.51162
\(527\) −1.45640e7 −2.28430
\(528\) 0 0
\(529\) 2.44920e6 0.380527
\(530\) 469509. 0.0726030
\(531\) 0 0
\(532\) −1.51962e7 −2.32786
\(533\) 152640. 0.0232728
\(534\) 0 0
\(535\) −2.47453e6 −0.373773
\(536\) −159357. −0.0239584
\(537\) 0 0
\(538\) 4.29852e6 0.640271
\(539\) 2.09411e7 3.10476
\(540\) 0 0
\(541\) −6212.12 −0.000912528 0 −0.000456264 1.00000i \(-0.500145\pi\)
−0.000456264 1.00000i \(0.500145\pi\)
\(542\) −1.36619e7 −1.99763
\(543\) 0 0
\(544\) −1.44159e7 −2.08855
\(545\) −926730. −0.133648
\(546\) 0 0
\(547\) −9.82839e6 −1.40448 −0.702238 0.711942i \(-0.747816\pi\)
−0.702238 + 0.711942i \(0.747816\pi\)
\(548\) −4.99549e6 −0.710602
\(549\) 0 0
\(550\) 4.56007e6 0.642783
\(551\) 1.55367e6 0.218011
\(552\) 0 0
\(553\) 1.29515e6 0.180097
\(554\) 110336. 0.0152737
\(555\) 0 0
\(556\) −548398. −0.0752332
\(557\) 6.27045e6 0.856369 0.428184 0.903691i \(-0.359153\pi\)
0.428184 + 0.903691i \(0.359153\pi\)
\(558\) 0 0
\(559\) −52830.4 −0.00715079
\(560\) −1.37772e7 −1.85648
\(561\) 0 0
\(562\) −9.07215e6 −1.21163
\(563\) 6.71129e6 0.892350 0.446175 0.894946i \(-0.352786\pi\)
0.446175 + 0.894946i \(0.352786\pi\)
\(564\) 0 0
\(565\) 54447.3 0.00717555
\(566\) −1.23812e7 −1.62450
\(567\) 0 0
\(568\) −699080. −0.0909193
\(569\) 7.01697e6 0.908592 0.454296 0.890851i \(-0.349891\pi\)
0.454296 + 0.890851i \(0.349891\pi\)
\(570\) 0 0
\(571\) 1.61627e6 0.207455 0.103727 0.994606i \(-0.466923\pi\)
0.103727 + 0.994606i \(0.466923\pi\)
\(572\) 605372. 0.0773628
\(573\) 0 0
\(574\) 9.66484e6 1.22438
\(575\) −2.65735e6 −0.335181
\(576\) 0 0
\(577\) 1.08712e7 1.35937 0.679686 0.733503i \(-0.262116\pi\)
0.679686 + 0.733503i \(0.262116\pi\)
\(578\) 1.31014e7 1.63117
\(579\) 0 0
\(580\) −1.62802e6 −0.200951
\(581\) 1.75600e6 0.215817
\(582\) 0 0
\(583\) 578294. 0.0704656
\(584\) −316898. −0.0384493
\(585\) 0 0
\(586\) 1.28294e7 1.54334
\(587\) −8.65320e6 −1.03653 −0.518264 0.855221i \(-0.673422\pi\)
−0.518264 + 0.855221i \(0.673422\pi\)
\(588\) 0 0
\(589\) 1.69439e7 2.01245
\(590\) 7.51306e6 0.888560
\(591\) 0 0
\(592\) −1.44862e7 −1.69883
\(593\) −5.13574e6 −0.599745 −0.299872 0.953979i \(-0.596944\pi\)
−0.299872 + 0.953979i \(0.596944\pi\)
\(594\) 0 0
\(595\) 2.46884e7 2.85891
\(596\) 5.04743e6 0.582043
\(597\) 0 0
\(598\) −689458. −0.0788415
\(599\) −7.56853e6 −0.861875 −0.430937 0.902382i \(-0.641817\pi\)
−0.430937 + 0.902382i \(0.641817\pi\)
\(600\) 0 0
\(601\) 2.56891e6 0.290110 0.145055 0.989424i \(-0.453664\pi\)
0.145055 + 0.989424i \(0.453664\pi\)
\(602\) −3.34512e6 −0.376201
\(603\) 0 0
\(604\) 1.43554e7 1.60112
\(605\) 1.50987e7 1.67707
\(606\) 0 0
\(607\) 2.62172e6 0.288812 0.144406 0.989519i \(-0.453873\pi\)
0.144406 + 0.989519i \(0.453873\pi\)
\(608\) 1.67716e7 1.83999
\(609\) 0 0
\(610\) −1.09403e7 −1.19043
\(611\) 179494. 0.0194513
\(612\) 0 0
\(613\) 537167. 0.0577376 0.0288688 0.999583i \(-0.490810\pi\)
0.0288688 + 0.999583i \(0.490810\pi\)
\(614\) 5.63213e6 0.602909
\(615\) 0 0
\(616\) 1.74875e6 0.185685
\(617\) −1.15522e7 −1.22166 −0.610830 0.791762i \(-0.709164\pi\)
−0.610830 + 0.791762i \(0.709164\pi\)
\(618\) 0 0
\(619\) 2.20752e6 0.231568 0.115784 0.993274i \(-0.463062\pi\)
0.115784 + 0.993274i \(0.463062\pi\)
\(620\) −1.77548e7 −1.85497
\(621\) 0 0
\(622\) 8.00408e6 0.829536
\(623\) 1.68960e6 0.174407
\(624\) 0 0
\(625\) −1.17567e7 −1.20389
\(626\) −2.48963e7 −2.53922
\(627\) 0 0
\(628\) −1.45583e7 −1.47303
\(629\) 2.59589e7 2.61613
\(630\) 0 0
\(631\) −1.43943e7 −1.43919 −0.719594 0.694395i \(-0.755672\pi\)
−0.719594 + 0.694395i \(0.755672\pi\)
\(632\) 71761.3 0.00714657
\(633\) 0 0
\(634\) −7.32219e6 −0.723465
\(635\) 1.22566e7 1.20625
\(636\) 0 0
\(637\) 946893. 0.0924597
\(638\) −3.91897e6 −0.381172
\(639\) 0 0
\(640\) −1.60540e6 −0.154929
\(641\) −4.90224e6 −0.471248 −0.235624 0.971844i \(-0.575713\pi\)
−0.235624 + 0.971844i \(0.575713\pi\)
\(642\) 0 0
\(643\) −1.80253e7 −1.71931 −0.859656 0.510874i \(-0.829322\pi\)
−0.859656 + 0.510874i \(0.829322\pi\)
\(644\) −2.23371e7 −2.12233
\(645\) 0 0
\(646\) −2.86144e7 −2.69776
\(647\) 1.24242e7 1.16683 0.583415 0.812174i \(-0.301716\pi\)
0.583415 + 0.812174i \(0.301716\pi\)
\(648\) 0 0
\(649\) 9.25383e6 0.862402
\(650\) 206192. 0.0191421
\(651\) 0 0
\(652\) −1.91733e7 −1.76635
\(653\) 1.28553e7 1.17977 0.589887 0.807486i \(-0.299172\pi\)
0.589887 + 0.807486i \(0.299172\pi\)
\(654\) 0 0
\(655\) 1.70860e7 1.55610
\(656\) −5.19641e6 −0.471459
\(657\) 0 0
\(658\) 1.13652e7 1.02333
\(659\) 5.18218e6 0.464835 0.232417 0.972616i \(-0.425336\pi\)
0.232417 + 0.972616i \(0.425336\pi\)
\(660\) 0 0
\(661\) 6.47130e6 0.576087 0.288043 0.957617i \(-0.406995\pi\)
0.288043 + 0.957617i \(0.406995\pi\)
\(662\) −1.69436e7 −1.50266
\(663\) 0 0
\(664\) 97296.6 0.00856402
\(665\) −2.87229e7 −2.51868
\(666\) 0 0
\(667\) 2.28375e6 0.198763
\(668\) −2.03104e7 −1.76108
\(669\) 0 0
\(670\) −6.60213e6 −0.568194
\(671\) −1.34751e7 −1.15539
\(672\) 0 0
\(673\) 6.21171e6 0.528657 0.264328 0.964433i \(-0.414850\pi\)
0.264328 + 0.964433i \(0.414850\pi\)
\(674\) −5.24141e6 −0.444425
\(675\) 0 0
\(676\) −1.24220e7 −1.04550
\(677\) −1.85078e7 −1.55197 −0.775984 0.630752i \(-0.782747\pi\)
−0.775984 + 0.630752i \(0.782747\pi\)
\(678\) 0 0
\(679\) 2.49663e7 2.07817
\(680\) 1.36793e6 0.113447
\(681\) 0 0
\(682\) −4.27394e7 −3.51858
\(683\) 4.91590e6 0.403228 0.201614 0.979465i \(-0.435381\pi\)
0.201614 + 0.979465i \(0.435381\pi\)
\(684\) 0 0
\(685\) −9.44214e6 −0.768854
\(686\) 2.95491e7 2.39736
\(687\) 0 0
\(688\) 1.79854e6 0.144860
\(689\) 26148.7 0.00209846
\(690\) 0 0
\(691\) −1.07645e7 −0.857630 −0.428815 0.903392i \(-0.641069\pi\)
−0.428815 + 0.903392i \(0.641069\pi\)
\(692\) 2.85803e6 0.226883
\(693\) 0 0
\(694\) 3.74005e6 0.294767
\(695\) −1.03655e6 −0.0814004
\(696\) 0 0
\(697\) 9.31185e6 0.726029
\(698\) 2.30769e7 1.79283
\(699\) 0 0
\(700\) 6.68023e6 0.515284
\(701\) −9.02819e6 −0.693914 −0.346957 0.937881i \(-0.612785\pi\)
−0.346957 + 0.937881i \(0.612785\pi\)
\(702\) 0 0
\(703\) −3.02010e7 −2.30480
\(704\) −2.26360e7 −1.72135
\(705\) 0 0
\(706\) 6.55248e6 0.494760
\(707\) −2.34490e6 −0.176431
\(708\) 0 0
\(709\) 4.91059e6 0.366875 0.183438 0.983031i \(-0.441277\pi\)
0.183438 + 0.983031i \(0.441277\pi\)
\(710\) −2.89628e7 −2.15623
\(711\) 0 0
\(712\) 93617.2 0.00692079
\(713\) 2.49061e7 1.83477
\(714\) 0 0
\(715\) 1.14423e6 0.0837047
\(716\) −2.61336e6 −0.190510
\(717\) 0 0
\(718\) −2.80184e7 −2.02830
\(719\) −1.44751e7 −1.04424 −0.522119 0.852873i \(-0.674858\pi\)
−0.522119 + 0.852873i \(0.674858\pi\)
\(720\) 0 0
\(721\) −2.52242e6 −0.180709
\(722\) 1.32463e7 0.945695
\(723\) 0 0
\(724\) −1.16570e7 −0.826493
\(725\) −682989. −0.0482579
\(726\) 0 0
\(727\) 1.55189e7 1.08899 0.544495 0.838764i \(-0.316721\pi\)
0.544495 + 0.838764i \(0.316721\pi\)
\(728\) 79073.2 0.00552969
\(729\) 0 0
\(730\) −1.31291e7 −0.911857
\(731\) −3.22294e6 −0.223079
\(732\) 0 0
\(733\) 1.03870e6 0.0714050 0.0357025 0.999362i \(-0.488633\pi\)
0.0357025 + 0.999362i \(0.488633\pi\)
\(734\) 1.72912e7 1.18463
\(735\) 0 0
\(736\) 2.46529e7 1.67754
\(737\) −8.13184e6 −0.551467
\(738\) 0 0
\(739\) 1.33918e6 0.0902046 0.0451023 0.998982i \(-0.485639\pi\)
0.0451023 + 0.998982i \(0.485639\pi\)
\(740\) 3.16463e7 2.12443
\(741\) 0 0
\(742\) 1.65568e6 0.110400
\(743\) −1.56518e7 −1.04014 −0.520070 0.854124i \(-0.674094\pi\)
−0.520070 + 0.854124i \(0.674094\pi\)
\(744\) 0 0
\(745\) 9.54032e6 0.629756
\(746\) 1.92009e7 1.26321
\(747\) 0 0
\(748\) 3.69310e7 2.41345
\(749\) −8.72620e6 −0.568356
\(750\) 0 0
\(751\) 1.92274e7 1.24400 0.622002 0.783016i \(-0.286320\pi\)
0.622002 + 0.783016i \(0.286320\pi\)
\(752\) −6.11065e6 −0.394042
\(753\) 0 0
\(754\) −177204. −0.0113513
\(755\) 2.71337e7 1.73237
\(756\) 0 0
\(757\) −2.73627e7 −1.73548 −0.867739 0.497021i \(-0.834427\pi\)
−0.867739 + 0.497021i \(0.834427\pi\)
\(758\) −4.43254e7 −2.80207
\(759\) 0 0
\(760\) −1.59147e6 −0.0999461
\(761\) −5.86301e6 −0.366994 −0.183497 0.983020i \(-0.558742\pi\)
−0.183497 + 0.983020i \(0.558742\pi\)
\(762\) 0 0
\(763\) −3.26803e6 −0.203224
\(764\) 6.44317e6 0.399361
\(765\) 0 0
\(766\) −2.56585e7 −1.58001
\(767\) 418430. 0.0256823
\(768\) 0 0
\(769\) −1.36162e7 −0.830308 −0.415154 0.909751i \(-0.636272\pi\)
−0.415154 + 0.909751i \(0.636272\pi\)
\(770\) 7.24507e7 4.40368
\(771\) 0 0
\(772\) −9.11083e6 −0.550192
\(773\) 9.44787e6 0.568703 0.284351 0.958720i \(-0.408222\pi\)
0.284351 + 0.958720i \(0.408222\pi\)
\(774\) 0 0
\(775\) −7.44851e6 −0.445467
\(776\) 1.38333e6 0.0824655
\(777\) 0 0
\(778\) 2.51946e6 0.149231
\(779\) −1.08335e7 −0.639627
\(780\) 0 0
\(781\) −3.56735e7 −2.09275
\(782\) −4.20608e7 −2.45958
\(783\) 0 0
\(784\) −3.22357e7 −1.87304
\(785\) −2.75171e7 −1.59378
\(786\) 0 0
\(787\) 2.97678e7 1.71320 0.856602 0.515977i \(-0.172571\pi\)
0.856602 + 0.515977i \(0.172571\pi\)
\(788\) −2.57468e6 −0.147709
\(789\) 0 0
\(790\) 2.97307e6 0.169487
\(791\) 192004. 0.0109111
\(792\) 0 0
\(793\) −609304. −0.0344073
\(794\) −4.01768e6 −0.226164
\(795\) 0 0
\(796\) −2.32956e7 −1.30314
\(797\) 1.80924e7 1.00891 0.504453 0.863439i \(-0.331694\pi\)
0.504453 + 0.863439i \(0.331694\pi\)
\(798\) 0 0
\(799\) 1.09501e7 0.606810
\(800\) −7.37279e6 −0.407293
\(801\) 0 0
\(802\) 4.56293e7 2.50500
\(803\) −1.61711e7 −0.885013
\(804\) 0 0
\(805\) −4.22201e7 −2.29631
\(806\) −1.93254e6 −0.104783
\(807\) 0 0
\(808\) −129926. −0.00700113
\(809\) 1.57182e7 0.844366 0.422183 0.906511i \(-0.361264\pi\)
0.422183 + 0.906511i \(0.361264\pi\)
\(810\) 0 0
\(811\) −1.57562e6 −0.0841202 −0.0420601 0.999115i \(-0.513392\pi\)
−0.0420601 + 0.999115i \(0.513392\pi\)
\(812\) −5.74106e6 −0.305564
\(813\) 0 0
\(814\) 7.61791e7 4.02972
\(815\) −3.62401e7 −1.91115
\(816\) 0 0
\(817\) 3.74962e6 0.196531
\(818\) −5.83260e6 −0.304775
\(819\) 0 0
\(820\) 1.13520e7 0.589573
\(821\) −2.61911e7 −1.35611 −0.678056 0.735010i \(-0.737177\pi\)
−0.678056 + 0.735010i \(0.737177\pi\)
\(822\) 0 0
\(823\) −3.66910e7 −1.88825 −0.944126 0.329585i \(-0.893091\pi\)
−0.944126 + 0.329585i \(0.893091\pi\)
\(824\) −139762. −0.00717086
\(825\) 0 0
\(826\) 2.64942e7 1.35114
\(827\) −1.38402e7 −0.703685 −0.351843 0.936059i \(-0.614445\pi\)
−0.351843 + 0.936059i \(0.614445\pi\)
\(828\) 0 0
\(829\) −2.83219e7 −1.43132 −0.715660 0.698449i \(-0.753874\pi\)
−0.715660 + 0.698449i \(0.753874\pi\)
\(830\) 4.03099e6 0.203103
\(831\) 0 0
\(832\) −1.02353e6 −0.0512617
\(833\) 5.77657e7 2.88441
\(834\) 0 0
\(835\) −3.83894e7 −1.90544
\(836\) −4.29661e7 −2.12623
\(837\) 0 0
\(838\) 3.19067e7 1.56954
\(839\) −2.80231e7 −1.37439 −0.687197 0.726471i \(-0.741159\pi\)
−0.687197 + 0.726471i \(0.741159\pi\)
\(840\) 0 0
\(841\) −1.99242e7 −0.971383
\(842\) 3.55782e7 1.72943
\(843\) 0 0
\(844\) −1.45362e7 −0.702418
\(845\) −2.34792e7 −1.13120
\(846\) 0 0
\(847\) 5.32444e7 2.55015
\(848\) −890197. −0.0425105
\(849\) 0 0
\(850\) 1.25789e7 0.597164
\(851\) −4.43928e7 −2.10130
\(852\) 0 0
\(853\) −2.38300e7 −1.12138 −0.560689 0.828027i \(-0.689463\pi\)
−0.560689 + 0.828027i \(0.689463\pi\)
\(854\) −3.85800e7 −1.81016
\(855\) 0 0
\(856\) −483501. −0.0225534
\(857\) 3.83129e7 1.78194 0.890971 0.454061i \(-0.150025\pi\)
0.890971 + 0.454061i \(0.150025\pi\)
\(858\) 0 0
\(859\) 2.68650e7 1.24224 0.621118 0.783717i \(-0.286679\pi\)
0.621118 + 0.783717i \(0.286679\pi\)
\(860\) −3.92906e6 −0.181152
\(861\) 0 0
\(862\) −1.01118e7 −0.463510
\(863\) 1.63558e7 0.747558 0.373779 0.927518i \(-0.378062\pi\)
0.373779 + 0.927518i \(0.378062\pi\)
\(864\) 0 0
\(865\) 5.40206e6 0.245482
\(866\) 7.86758e6 0.356489
\(867\) 0 0
\(868\) −6.26107e7 −2.82065
\(869\) 3.66192e6 0.164498
\(870\) 0 0
\(871\) −367696. −0.0164227
\(872\) −181075. −0.00806431
\(873\) 0 0
\(874\) 4.89341e7 2.16687
\(875\) −3.16351e7 −1.39685
\(876\) 0 0
\(877\) −2.30604e7 −1.01243 −0.506217 0.862406i \(-0.668957\pi\)
−0.506217 + 0.862406i \(0.668957\pi\)
\(878\) −1.07221e7 −0.469402
\(879\) 0 0
\(880\) −3.89539e7 −1.69568
\(881\) 3.15095e7 1.36774 0.683868 0.729606i \(-0.260296\pi\)
0.683868 + 0.729606i \(0.260296\pi\)
\(882\) 0 0
\(883\) 2.86348e7 1.23593 0.617963 0.786208i \(-0.287958\pi\)
0.617963 + 0.786208i \(0.287958\pi\)
\(884\) 1.66991e6 0.0718723
\(885\) 0 0
\(886\) −3.85685e7 −1.65063
\(887\) 2.45670e7 1.04844 0.524220 0.851583i \(-0.324357\pi\)
0.524220 + 0.851583i \(0.324357\pi\)
\(888\) 0 0
\(889\) 4.32219e7 1.83421
\(890\) 3.87855e6 0.164132
\(891\) 0 0
\(892\) 3.49392e7 1.47028
\(893\) −1.27396e7 −0.534596
\(894\) 0 0
\(895\) −4.93961e6 −0.206127
\(896\) −5.66130e6 −0.235584
\(897\) 0 0
\(898\) 4.37138e7 1.80895
\(899\) 6.40134e6 0.264163
\(900\) 0 0
\(901\) 1.59521e6 0.0654646
\(902\) 2.73266e7 1.11833
\(903\) 0 0
\(904\) 10638.5 0.000432973 0
\(905\) −2.20332e7 −0.894245
\(906\) 0 0
\(907\) 1.90832e7 0.770254 0.385127 0.922864i \(-0.374158\pi\)
0.385127 + 0.922864i \(0.374158\pi\)
\(908\) −3.10140e7 −1.24837
\(909\) 0 0
\(910\) 3.27600e6 0.131141
\(911\) 3.04616e7 1.21606 0.608032 0.793912i \(-0.291959\pi\)
0.608032 + 0.793912i \(0.291959\pi\)
\(912\) 0 0
\(913\) 4.96497e6 0.197124
\(914\) 1.72224e7 0.681911
\(915\) 0 0
\(916\) −4.59826e6 −0.181073
\(917\) 6.02522e7 2.36619
\(918\) 0 0
\(919\) 2.73866e7 1.06967 0.534834 0.844957i \(-0.320374\pi\)
0.534834 + 0.844957i \(0.320374\pi\)
\(920\) −2.33933e6 −0.0911218
\(921\) 0 0
\(922\) −2.27394e7 −0.880950
\(923\) −1.61304e6 −0.0623221
\(924\) 0 0
\(925\) 1.32763e7 0.510179
\(926\) −2.30986e7 −0.885232
\(927\) 0 0
\(928\) 6.33626e6 0.241525
\(929\) 2.65451e7 1.00913 0.504563 0.863375i \(-0.331654\pi\)
0.504563 + 0.863375i \(0.331654\pi\)
\(930\) 0 0
\(931\) −6.72054e7 −2.54115
\(932\) −1.43709e7 −0.541932
\(933\) 0 0
\(934\) 4.52888e6 0.169873
\(935\) 6.98046e7 2.61129
\(936\) 0 0
\(937\) −2.34660e7 −0.873151 −0.436576 0.899668i \(-0.643809\pi\)
−0.436576 + 0.899668i \(0.643809\pi\)
\(938\) −2.32818e7 −0.863993
\(939\) 0 0
\(940\) 1.33492e7 0.492761
\(941\) 2.18303e7 0.803686 0.401843 0.915709i \(-0.368370\pi\)
0.401843 + 0.915709i \(0.368370\pi\)
\(942\) 0 0
\(943\) −1.59244e7 −0.583154
\(944\) −1.42449e7 −0.520270
\(945\) 0 0
\(946\) −9.45806e6 −0.343617
\(947\) −1.87197e7 −0.678305 −0.339152 0.940731i \(-0.610140\pi\)
−0.339152 + 0.940731i \(0.610140\pi\)
\(948\) 0 0
\(949\) −731206. −0.0263557
\(950\) −1.46344e7 −0.526098
\(951\) 0 0
\(952\) 4.82390e6 0.172507
\(953\) −5.37413e7 −1.91680 −0.958399 0.285432i \(-0.907863\pi\)
−0.958399 + 0.285432i \(0.907863\pi\)
\(954\) 0 0
\(955\) 1.21785e7 0.432099
\(956\) 3.92748e7 1.38986
\(957\) 0 0
\(958\) −4.42594e7 −1.55809
\(959\) −3.32969e7 −1.16911
\(960\) 0 0
\(961\) 4.11823e7 1.43847
\(962\) 3.44458e6 0.120005
\(963\) 0 0
\(964\) 3.99848e6 0.138581
\(965\) −1.72207e7 −0.595295
\(966\) 0 0
\(967\) 1.78136e7 0.612611 0.306306 0.951933i \(-0.400907\pi\)
0.306306 + 0.951933i \(0.400907\pi\)
\(968\) 2.95016e6 0.101195
\(969\) 0 0
\(970\) 5.73114e7 1.95574
\(971\) −3.06231e7 −1.04232 −0.521160 0.853459i \(-0.674500\pi\)
−0.521160 + 0.853459i \(0.674500\pi\)
\(972\) 0 0
\(973\) −3.65529e6 −0.123777
\(974\) 7.56691e7 2.55577
\(975\) 0 0
\(976\) 2.07429e7 0.697021
\(977\) 1.93935e7 0.650011 0.325005 0.945712i \(-0.394634\pi\)
0.325005 + 0.945712i \(0.394634\pi\)
\(978\) 0 0
\(979\) 4.77721e6 0.159301
\(980\) 7.04217e7 2.34229
\(981\) 0 0
\(982\) 8.49685e7 2.81177
\(983\) −3.41089e7 −1.12586 −0.562929 0.826505i \(-0.690325\pi\)
−0.562929 + 0.826505i \(0.690325\pi\)
\(984\) 0 0
\(985\) −4.86650e6 −0.159818
\(986\) −1.08104e7 −0.354120
\(987\) 0 0
\(988\) −1.94279e6 −0.0633190
\(989\) 5.51162e6 0.179179
\(990\) 0 0
\(991\) −1.15321e7 −0.373013 −0.186507 0.982454i \(-0.559717\pi\)
−0.186507 + 0.982454i \(0.559717\pi\)
\(992\) 6.91017e7 2.22951
\(993\) 0 0
\(994\) −1.02135e8 −3.27875
\(995\) −4.40319e7 −1.40997
\(996\) 0 0
\(997\) 3.45745e7 1.10159 0.550793 0.834642i \(-0.314325\pi\)
0.550793 + 0.834642i \(0.314325\pi\)
\(998\) 4.65337e7 1.47891
\(999\) 0 0
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 387.6.a.c.1.7 8
3.2 odd 2 43.6.a.a.1.2 8
12.11 even 2 688.6.a.e.1.2 8
15.14 odd 2 1075.6.a.a.1.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.6.a.a.1.2 8 3.2 odd 2
387.6.a.c.1.7 8 1.1 even 1 trivial
688.6.a.e.1.2 8 12.11 even 2
1075.6.a.a.1.7 8 15.14 odd 2