Properties

Label 177.6.a.d.1.2
Level $177$
Weight $6$
Character 177.1
Self dual yes
Analytic conductor $28.388$
Analytic rank $0$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.3879361069\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 331 x^{11} - 41 x^{10} + 41990 x^{9} + 7229 x^{8} - 2592364 x^{7} - 312100 x^{6} + 81977088 x^{5} - 3773728 x^{4} - 1245415104 x^{3} + 453320896 x^{2} + 6872784896 x - 6400833792\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-9.47242\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-9.47242 q^{2} -9.00000 q^{3} +57.7268 q^{4} -42.5216 q^{5} +85.2518 q^{6} +243.042 q^{7} -243.695 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-9.47242 q^{2} -9.00000 q^{3} +57.7268 q^{4} -42.5216 q^{5} +85.2518 q^{6} +243.042 q^{7} -243.695 q^{8} +81.0000 q^{9} +402.782 q^{10} +382.698 q^{11} -519.541 q^{12} +434.802 q^{13} -2302.20 q^{14} +382.694 q^{15} +461.122 q^{16} +67.7470 q^{17} -767.266 q^{18} +2843.46 q^{19} -2454.63 q^{20} -2187.38 q^{21} -3625.08 q^{22} -1477.96 q^{23} +2193.25 q^{24} -1316.92 q^{25} -4118.63 q^{26} -729.000 q^{27} +14030.0 q^{28} -413.606 q^{29} -3625.04 q^{30} -7561.45 q^{31} +3430.29 q^{32} -3444.29 q^{33} -641.728 q^{34} -10334.5 q^{35} +4675.87 q^{36} +2856.10 q^{37} -26934.4 q^{38} -3913.22 q^{39} +10362.3 q^{40} +4308.73 q^{41} +20719.8 q^{42} +991.892 q^{43} +22091.9 q^{44} -3444.25 q^{45} +13999.9 q^{46} -6818.58 q^{47} -4150.10 q^{48} +42262.4 q^{49} +12474.4 q^{50} -609.723 q^{51} +25099.7 q^{52} +27976.7 q^{53} +6905.39 q^{54} -16272.9 q^{55} -59228.0 q^{56} -25591.1 q^{57} +3917.85 q^{58} +3481.00 q^{59} +22091.7 q^{60} +32959.1 q^{61} +71625.3 q^{62} +19686.4 q^{63} -47249.0 q^{64} -18488.5 q^{65} +32625.7 q^{66} -14427.0 q^{67} +3910.81 q^{68} +13301.7 q^{69} +97893.0 q^{70} +31564.2 q^{71} -19739.3 q^{72} +21278.4 q^{73} -27054.2 q^{74} +11852.2 q^{75} +164144. q^{76} +93011.8 q^{77} +37067.7 q^{78} -51036.5 q^{79} -19607.6 q^{80} +6561.00 q^{81} -40814.1 q^{82} -17809.1 q^{83} -126270. q^{84} -2880.71 q^{85} -9395.62 q^{86} +3722.46 q^{87} -93261.6 q^{88} -38824.8 q^{89} +32625.4 q^{90} +105675. q^{91} -85318.0 q^{92} +68053.1 q^{93} +64588.5 q^{94} -120908. q^{95} -30872.6 q^{96} -83290.5 q^{97} -400327. q^{98} +30998.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 117q^{3} + 246q^{4} - 14q^{5} + 373q^{7} + 123q^{8} + 1053q^{9} + O(q^{10}) \) \( 13q - 117q^{3} + 246q^{4} - 14q^{5} + 373q^{7} + 123q^{8} + 1053q^{9} + 137q^{10} + 250q^{11} - 2214q^{12} + 1054q^{13} - 575q^{14} + 126q^{15} + 922q^{16} + 271q^{17} + 671q^{19} - 5491q^{20} - 3357q^{21} + 1094q^{22} + 3975q^{23} - 1107q^{24} + 15569q^{25} + 4622q^{26} - 9477q^{27} + 21214q^{28} - 10613q^{29} - 1233q^{30} + 25597q^{31} + 15966q^{32} - 2250q^{33} + 31796q^{34} + 6729q^{35} + 19926q^{36} + 17585q^{37} + 34903q^{38} - 9486q^{39} + 31382q^{40} + 12537q^{41} + 5175q^{42} + 26644q^{43} + 6654q^{44} - 1134q^{45} + 149005q^{46} + 52087q^{47} - 8298q^{48} + 95384q^{49} + 121821q^{50} - 2439q^{51} + 263630q^{52} + 20014q^{53} + 120932q^{55} + 126688q^{56} - 6039q^{57} + 86066q^{58} + 45253q^{59} + 49419q^{60} - 11667q^{61} + 164794q^{62} + 30213q^{63} + 151893q^{64} - 28674q^{65} - 9846q^{66} + 1106q^{67} - 4043q^{68} - 35775q^{69} + 56066q^{70} + 21230q^{71} + 9963q^{72} + 81131q^{73} + 102042q^{74} - 140121q^{75} + 73900q^{76} - 104655q^{77} - 41598q^{78} - 13470q^{79} - 191969q^{80} + 85293q^{81} + 79909q^{82} - 76149q^{83} - 190926q^{84} + 10035q^{85} - 321496q^{86} + 95517q^{87} - 276779q^{88} - 190205q^{89} + 11097q^{90} + 80601q^{91} + 45672q^{92} - 230373q^{93} + 36768q^{94} + 9875q^{95} - 143694q^{96} + 160850q^{97} - 116644q^{98} + 20250q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −9.47242 −1.67450 −0.837252 0.546818i \(-0.815839\pi\)
−0.837252 + 0.546818i \(0.815839\pi\)
\(3\) −9.00000 −0.577350
\(4\) 57.7268 1.80396
\(5\) −42.5216 −0.760649 −0.380324 0.924853i \(-0.624188\pi\)
−0.380324 + 0.924853i \(0.624188\pi\)
\(6\) 85.2518 0.966775
\(7\) 243.042 1.87472 0.937359 0.348364i \(-0.113263\pi\)
0.937359 + 0.348364i \(0.113263\pi\)
\(8\) −243.695 −1.34624
\(9\) 81.0000 0.333333
\(10\) 402.782 1.27371
\(11\) 382.698 0.953619 0.476810 0.879007i \(-0.341793\pi\)
0.476810 + 0.879007i \(0.341793\pi\)
\(12\) −519.541 −1.04152
\(13\) 434.802 0.713565 0.356782 0.934188i \(-0.383874\pi\)
0.356782 + 0.934188i \(0.383874\pi\)
\(14\) −2302.20 −3.13922
\(15\) 382.694 0.439161
\(16\) 461.122 0.450315
\(17\) 67.7470 0.0568549 0.0284274 0.999596i \(-0.490950\pi\)
0.0284274 + 0.999596i \(0.490950\pi\)
\(18\) −767.266 −0.558168
\(19\) 2843.46 1.80702 0.903509 0.428568i \(-0.140982\pi\)
0.903509 + 0.428568i \(0.140982\pi\)
\(20\) −2454.63 −1.37218
\(21\) −2187.38 −1.08237
\(22\) −3625.08 −1.59684
\(23\) −1477.96 −0.582564 −0.291282 0.956637i \(-0.594082\pi\)
−0.291282 + 0.956637i \(0.594082\pi\)
\(24\) 2193.25 0.777249
\(25\) −1316.92 −0.421413
\(26\) −4118.63 −1.19487
\(27\) −729.000 −0.192450
\(28\) 14030.0 3.38192
\(29\) −413.606 −0.0913255 −0.0456628 0.998957i \(-0.514540\pi\)
−0.0456628 + 0.998957i \(0.514540\pi\)
\(30\) −3625.04 −0.735376
\(31\) −7561.45 −1.41319 −0.706596 0.707617i \(-0.749770\pi\)
−0.706596 + 0.707617i \(0.749770\pi\)
\(32\) 3430.29 0.592182
\(33\) −3444.29 −0.550572
\(34\) −641.728 −0.0952037
\(35\) −10334.5 −1.42600
\(36\) 4675.87 0.601320
\(37\) 2856.10 0.342980 0.171490 0.985186i \(-0.445142\pi\)
0.171490 + 0.985186i \(0.445142\pi\)
\(38\) −26934.4 −3.02586
\(39\) −3913.22 −0.411977
\(40\) 10362.3 1.02401
\(41\) 4308.73 0.400304 0.200152 0.979765i \(-0.435856\pi\)
0.200152 + 0.979765i \(0.435856\pi\)
\(42\) 20719.8 1.81243
\(43\) 991.892 0.0818075 0.0409038 0.999163i \(-0.486976\pi\)
0.0409038 + 0.999163i \(0.486976\pi\)
\(44\) 22091.9 1.72029
\(45\) −3444.25 −0.253550
\(46\) 13999.9 0.975506
\(47\) −6818.58 −0.450246 −0.225123 0.974330i \(-0.572278\pi\)
−0.225123 + 0.974330i \(0.572278\pi\)
\(48\) −4150.10 −0.259989
\(49\) 42262.4 2.51457
\(50\) 12474.4 0.705658
\(51\) −609.723 −0.0328252
\(52\) 25099.7 1.28724
\(53\) 27976.7 1.36806 0.684032 0.729452i \(-0.260225\pi\)
0.684032 + 0.729452i \(0.260225\pi\)
\(54\) 6905.39 0.322258
\(55\) −16272.9 −0.725370
\(56\) −59228.0 −2.52381
\(57\) −25591.1 −1.04328
\(58\) 3917.85 0.152925
\(59\) 3481.00 0.130189
\(60\) 22091.7 0.792229
\(61\) 32959.1 1.13410 0.567049 0.823684i \(-0.308085\pi\)
0.567049 + 0.823684i \(0.308085\pi\)
\(62\) 71625.3 2.36639
\(63\) 19686.4 0.624906
\(64\) −47249.0 −1.44193
\(65\) −18488.5 −0.542772
\(66\) 32625.7 0.921935
\(67\) −14427.0 −0.392636 −0.196318 0.980540i \(-0.562899\pi\)
−0.196318 + 0.980540i \(0.562899\pi\)
\(68\) 3910.81 0.102564
\(69\) 13301.7 0.336344
\(70\) 97893.0 2.38785
\(71\) 31564.2 0.743102 0.371551 0.928412i \(-0.378826\pi\)
0.371551 + 0.928412i \(0.378826\pi\)
\(72\) −19739.3 −0.448745
\(73\) 21278.4 0.467339 0.233670 0.972316i \(-0.424927\pi\)
0.233670 + 0.972316i \(0.424927\pi\)
\(74\) −27054.2 −0.574321
\(75\) 11852.2 0.243303
\(76\) 164144. 3.25979
\(77\) 93011.8 1.78777
\(78\) 37067.7 0.689856
\(79\) −51036.5 −0.920054 −0.460027 0.887905i \(-0.652160\pi\)
−0.460027 + 0.887905i \(0.652160\pi\)
\(80\) −19607.6 −0.342531
\(81\) 6561.00 0.111111
\(82\) −40814.1 −0.670310
\(83\) −17809.1 −0.283757 −0.141879 0.989884i \(-0.545314\pi\)
−0.141879 + 0.989884i \(0.545314\pi\)
\(84\) −126270. −1.95255
\(85\) −2880.71 −0.0432466
\(86\) −9395.62 −0.136987
\(87\) 3722.46 0.0527268
\(88\) −93261.6 −1.28380
\(89\) −38824.8 −0.519559 −0.259779 0.965668i \(-0.583650\pi\)
−0.259779 + 0.965668i \(0.583650\pi\)
\(90\) 32625.4 0.424570
\(91\) 105675. 1.33773
\(92\) −85318.0 −1.05092
\(93\) 68053.1 0.815907
\(94\) 64588.5 0.753938
\(95\) −120908. −1.37451
\(96\) −30872.6 −0.341897
\(97\) −83290.5 −0.898806 −0.449403 0.893329i \(-0.648363\pi\)
−0.449403 + 0.893329i \(0.648363\pi\)
\(98\) −400327. −4.21066
\(99\) 30998.6 0.317873
\(100\) −76021.3 −0.760213
\(101\) −151049. −1.47338 −0.736689 0.676232i \(-0.763612\pi\)
−0.736689 + 0.676232i \(0.763612\pi\)
\(102\) 5775.55 0.0549659
\(103\) 183578. 1.70501 0.852506 0.522718i \(-0.175082\pi\)
0.852506 + 0.522718i \(0.175082\pi\)
\(104\) −105959. −0.960626
\(105\) 93010.7 0.823303
\(106\) −265007. −2.29083
\(107\) −158219. −1.33597 −0.667987 0.744173i \(-0.732844\pi\)
−0.667987 + 0.744173i \(0.732844\pi\)
\(108\) −42082.8 −0.347172
\(109\) −133410. −1.07553 −0.537766 0.843094i \(-0.680732\pi\)
−0.537766 + 0.843094i \(0.680732\pi\)
\(110\) 154144. 1.21463
\(111\) −25704.9 −0.198020
\(112\) 112072. 0.844213
\(113\) 247000. 1.81971 0.909854 0.414930i \(-0.136194\pi\)
0.909854 + 0.414930i \(0.136194\pi\)
\(114\) 242410. 1.74698
\(115\) 62845.3 0.443127
\(116\) −23876.1 −0.164748
\(117\) 35219.0 0.237855
\(118\) −32973.5 −0.218002
\(119\) 16465.4 0.106587
\(120\) −93260.5 −0.591214
\(121\) −14592.9 −0.0906103
\(122\) −312202. −1.89905
\(123\) −38778.6 −0.231115
\(124\) −436498. −2.54934
\(125\) 188877. 1.08120
\(126\) −186478. −1.04641
\(127\) 295605. 1.62631 0.813155 0.582048i \(-0.197748\pi\)
0.813155 + 0.582048i \(0.197748\pi\)
\(128\) 337793. 1.82233
\(129\) −8927.03 −0.0472316
\(130\) 175131. 0.908874
\(131\) 27171.8 0.138338 0.0691689 0.997605i \(-0.477965\pi\)
0.0691689 + 0.997605i \(0.477965\pi\)
\(132\) −198827. −0.993211
\(133\) 691079. 3.38765
\(134\) 136659. 0.657470
\(135\) 30998.2 0.146387
\(136\) −16509.6 −0.0765401
\(137\) 153128. 0.697035 0.348517 0.937302i \(-0.386685\pi\)
0.348517 + 0.937302i \(0.386685\pi\)
\(138\) −125999. −0.563209
\(139\) −152753. −0.670585 −0.335292 0.942114i \(-0.608835\pi\)
−0.335292 + 0.942114i \(0.608835\pi\)
\(140\) −596579. −2.57245
\(141\) 61367.2 0.259949
\(142\) −298989. −1.24433
\(143\) 166398. 0.680469
\(144\) 37350.9 0.150105
\(145\) 17587.2 0.0694667
\(146\) −201558. −0.782561
\(147\) −380361. −1.45179
\(148\) 164873. 0.618722
\(149\) −209034. −0.771349 −0.385675 0.922635i \(-0.626031\pi\)
−0.385675 + 0.922635i \(0.626031\pi\)
\(150\) −112269. −0.407412
\(151\) 468814. 1.67324 0.836620 0.547783i \(-0.184528\pi\)
0.836620 + 0.547783i \(0.184528\pi\)
\(152\) −692935. −2.43267
\(153\) 5487.51 0.0189516
\(154\) −881047. −2.99362
\(155\) 321525. 1.07494
\(156\) −225897. −0.743190
\(157\) 145781. 0.472012 0.236006 0.971752i \(-0.424162\pi\)
0.236006 + 0.971752i \(0.424162\pi\)
\(158\) 483439. 1.54063
\(159\) −251790. −0.789852
\(160\) −145861. −0.450443
\(161\) −359207. −1.09214
\(162\) −62148.6 −0.186056
\(163\) −361526. −1.06579 −0.532894 0.846182i \(-0.678895\pi\)
−0.532894 + 0.846182i \(0.678895\pi\)
\(164\) 248729. 0.722132
\(165\) 146456. 0.418792
\(166\) 168695. 0.475153
\(167\) −405904. −1.12624 −0.563122 0.826374i \(-0.690400\pi\)
−0.563122 + 0.826374i \(0.690400\pi\)
\(168\) 533052. 1.45712
\(169\) −182240. −0.490826
\(170\) 27287.3 0.0724166
\(171\) 230320. 0.602340
\(172\) 57258.7 0.147578
\(173\) 184103. 0.467677 0.233839 0.972275i \(-0.424871\pi\)
0.233839 + 0.972275i \(0.424871\pi\)
\(174\) −35260.7 −0.0882912
\(175\) −320066. −0.790031
\(176\) 176471. 0.429429
\(177\) −31329.0 −0.0751646
\(178\) 367765. 0.870003
\(179\) 636770. 1.48542 0.742711 0.669612i \(-0.233540\pi\)
0.742711 + 0.669612i \(0.233540\pi\)
\(180\) −198825. −0.457394
\(181\) 543803. 1.23380 0.616901 0.787041i \(-0.288388\pi\)
0.616901 + 0.787041i \(0.288388\pi\)
\(182\) −1.00100e6 −2.24004
\(183\) −296632. −0.654771
\(184\) 360172. 0.784269
\(185\) −121446. −0.260887
\(186\) −644627. −1.36624
\(187\) 25926.7 0.0542179
\(188\) −393615. −0.812226
\(189\) −177178. −0.360790
\(190\) 1.14529e6 2.30162
\(191\) −379624. −0.752958 −0.376479 0.926425i \(-0.622865\pi\)
−0.376479 + 0.926425i \(0.622865\pi\)
\(192\) 425241. 0.832496
\(193\) 388085. 0.749951 0.374976 0.927035i \(-0.377651\pi\)
0.374976 + 0.927035i \(0.377651\pi\)
\(194\) 788962. 1.50505
\(195\) 166396. 0.313370
\(196\) 2.43967e6 4.53619
\(197\) 985373. 1.80899 0.904493 0.426489i \(-0.140250\pi\)
0.904493 + 0.426489i \(0.140250\pi\)
\(198\) −293632. −0.532280
\(199\) 345186. 0.617903 0.308952 0.951078i \(-0.400022\pi\)
0.308952 + 0.951078i \(0.400022\pi\)
\(200\) 320925. 0.567321
\(201\) 129843. 0.226689
\(202\) 1.43080e6 2.46717
\(203\) −100524. −0.171210
\(204\) −35197.3 −0.0592153
\(205\) −183214. −0.304491
\(206\) −1.73893e6 −2.85505
\(207\) −119715. −0.194188
\(208\) 200497. 0.321328
\(209\) 1.08819e6 1.72321
\(210\) −881037. −1.37862
\(211\) −873858. −1.35125 −0.675624 0.737247i \(-0.736125\pi\)
−0.675624 + 0.737247i \(0.736125\pi\)
\(212\) 1.61500e6 2.46793
\(213\) −284078. −0.429030
\(214\) 1.49871e6 2.23709
\(215\) −42176.8 −0.0622268
\(216\) 177653. 0.259083
\(217\) −1.83775e6 −2.64934
\(218\) 1.26372e6 1.80098
\(219\) −191506. −0.269818
\(220\) −939384. −1.30854
\(221\) 29456.5 0.0405696
\(222\) 243487. 0.331584
\(223\) −1.00517e6 −1.35356 −0.676778 0.736187i \(-0.736624\pi\)
−0.676778 + 0.736187i \(0.736624\pi\)
\(224\) 833703. 1.11018
\(225\) −106670. −0.140471
\(226\) −2.33969e6 −3.04711
\(227\) 1.20719e6 1.55493 0.777467 0.628924i \(-0.216504\pi\)
0.777467 + 0.628924i \(0.216504\pi\)
\(228\) −1.47729e6 −1.88204
\(229\) −924824. −1.16539 −0.582694 0.812692i \(-0.698001\pi\)
−0.582694 + 0.812692i \(0.698001\pi\)
\(230\) −595297. −0.742018
\(231\) −837106. −1.03217
\(232\) 100794. 0.122946
\(233\) 750812. 0.906027 0.453014 0.891504i \(-0.350349\pi\)
0.453014 + 0.891504i \(0.350349\pi\)
\(234\) −333609. −0.398289
\(235\) 289937. 0.342479
\(236\) 200947. 0.234856
\(237\) 459329. 0.531194
\(238\) −155967. −0.178480
\(239\) 1.55346e6 1.75916 0.879580 0.475752i \(-0.157824\pi\)
0.879580 + 0.475752i \(0.157824\pi\)
\(240\) 176469. 0.197761
\(241\) 571502. 0.633833 0.316917 0.948453i \(-0.397352\pi\)
0.316917 + 0.948453i \(0.397352\pi\)
\(242\) 138230. 0.151727
\(243\) −59049.0 −0.0641500
\(244\) 1.90262e6 2.04587
\(245\) −1.79706e6 −1.91271
\(246\) 367327. 0.387004
\(247\) 1.23634e6 1.28942
\(248\) 1.84269e6 1.90249
\(249\) 160282. 0.163827
\(250\) −1.78912e6 −1.81047
\(251\) 710664. 0.712000 0.356000 0.934486i \(-0.384140\pi\)
0.356000 + 0.934486i \(0.384140\pi\)
\(252\) 1.13643e6 1.12731
\(253\) −565614. −0.555545
\(254\) −2.80010e6 −2.72326
\(255\) 25926.4 0.0249684
\(256\) −1.68775e6 −1.60957
\(257\) 1.30557e6 1.23301 0.616504 0.787352i \(-0.288548\pi\)
0.616504 + 0.787352i \(0.288548\pi\)
\(258\) 84560.6 0.0790895
\(259\) 694152. 0.642991
\(260\) −1.06728e6 −0.979140
\(261\) −33502.1 −0.0304418
\(262\) −257383. −0.231647
\(263\) 1.90024e6 1.69402 0.847012 0.531574i \(-0.178399\pi\)
0.847012 + 0.531574i \(0.178399\pi\)
\(264\) 839354. 0.741200
\(265\) −1.18961e6 −1.04062
\(266\) −6.54619e6 −5.67263
\(267\) 349423. 0.299967
\(268\) −832826. −0.708300
\(269\) −1.46393e6 −1.23350 −0.616749 0.787160i \(-0.711551\pi\)
−0.616749 + 0.787160i \(0.711551\pi\)
\(270\) −293628. −0.245125
\(271\) 107612. 0.0890095 0.0445048 0.999009i \(-0.485829\pi\)
0.0445048 + 0.999009i \(0.485829\pi\)
\(272\) 31239.6 0.0256026
\(273\) −951076. −0.772340
\(274\) −1.45050e6 −1.16719
\(275\) −503982. −0.401868
\(276\) 767862. 0.606751
\(277\) −471552. −0.369258 −0.184629 0.982808i \(-0.559108\pi\)
−0.184629 + 0.982808i \(0.559108\pi\)
\(278\) 1.44694e6 1.12290
\(279\) −612478. −0.471064
\(280\) 2.51847e6 1.91974
\(281\) −1.46229e6 −1.10476 −0.552380 0.833592i \(-0.686280\pi\)
−0.552380 + 0.833592i \(0.686280\pi\)
\(282\) −581296. −0.435286
\(283\) −270851. −0.201031 −0.100516 0.994935i \(-0.532049\pi\)
−0.100516 + 0.994935i \(0.532049\pi\)
\(284\) 1.82210e6 1.34053
\(285\) 1.08817e6 0.793572
\(286\) −1.57619e6 −1.13945
\(287\) 1.04720e6 0.750457
\(288\) 277853. 0.197394
\(289\) −1.41527e6 −0.996768
\(290\) −166593. −0.116322
\(291\) 749614. 0.518926
\(292\) 1.22833e6 0.843061
\(293\) −127463. −0.0867391 −0.0433695 0.999059i \(-0.513809\pi\)
−0.0433695 + 0.999059i \(0.513809\pi\)
\(294\) 3.60294e6 2.43102
\(295\) −148018. −0.0990281
\(296\) −696016. −0.461732
\(297\) −278987. −0.183524
\(298\) 1.98006e6 1.29163
\(299\) −642621. −0.415697
\(300\) 684192. 0.438909
\(301\) 241071. 0.153366
\(302\) −4.44080e6 −2.80185
\(303\) 1.35944e6 0.850655
\(304\) 1.31118e6 0.813727
\(305\) −1.40147e6 −0.862650
\(306\) −51980.0 −0.0317346
\(307\) 1.10268e6 0.667736 0.333868 0.942620i \(-0.391646\pi\)
0.333868 + 0.942620i \(0.391646\pi\)
\(308\) 5.36927e6 3.22506
\(309\) −1.65220e6 −0.984389
\(310\) −3.04562e6 −1.80000
\(311\) −256033. −0.150105 −0.0750525 0.997180i \(-0.523912\pi\)
−0.0750525 + 0.997180i \(0.523912\pi\)
\(312\) 953631. 0.554618
\(313\) 1.51228e6 0.872510 0.436255 0.899823i \(-0.356305\pi\)
0.436255 + 0.899823i \(0.356305\pi\)
\(314\) −1.38090e6 −0.790385
\(315\) −837097. −0.475334
\(316\) −2.94617e6 −1.65974
\(317\) 2.98841e6 1.67029 0.835146 0.550028i \(-0.185383\pi\)
0.835146 + 0.550028i \(0.185383\pi\)
\(318\) 2.38506e6 1.32261
\(319\) −158286. −0.0870898
\(320\) 2.00910e6 1.09680
\(321\) 1.42397e6 0.771325
\(322\) 3.40256e6 1.82880
\(323\) 192636. 0.102738
\(324\) 378745. 0.200440
\(325\) −572598. −0.300705
\(326\) 3.42453e6 1.78466
\(327\) 1.20069e6 0.620959
\(328\) −1.05001e6 −0.538903
\(329\) −1.65720e6 −0.844084
\(330\) −1.38730e6 −0.701269
\(331\) −2.69754e6 −1.35331 −0.676655 0.736300i \(-0.736571\pi\)
−0.676655 + 0.736300i \(0.736571\pi\)
\(332\) −1.02806e6 −0.511887
\(333\) 231344. 0.114327
\(334\) 3.84489e6 1.88590
\(335\) 613460. 0.298658
\(336\) −1.00865e6 −0.487407
\(337\) 1.34712e6 0.646149 0.323075 0.946373i \(-0.395283\pi\)
0.323075 + 0.946373i \(0.395283\pi\)
\(338\) 1.72626e6 0.821889
\(339\) −2.22300e6 −1.05061
\(340\) −166294. −0.0780152
\(341\) −2.89376e6 −1.34765
\(342\) −2.18169e6 −1.00862
\(343\) 6.18673e6 2.83939
\(344\) −241719. −0.110132
\(345\) −565608. −0.255840
\(346\) −1.74390e6 −0.783127
\(347\) −858085. −0.382566 −0.191283 0.981535i \(-0.561265\pi\)
−0.191283 + 0.981535i \(0.561265\pi\)
\(348\) 214885. 0.0951171
\(349\) −2.15268e6 −0.946054 −0.473027 0.881048i \(-0.656839\pi\)
−0.473027 + 0.881048i \(0.656839\pi\)
\(350\) 3.03180e6 1.32291
\(351\) −316971. −0.137326
\(352\) 1.31277e6 0.564716
\(353\) −3.43309e6 −1.46639 −0.733194 0.680020i \(-0.761971\pi\)
−0.733194 + 0.680020i \(0.761971\pi\)
\(354\) 296761. 0.125863
\(355\) −1.34216e6 −0.565240
\(356\) −2.24123e6 −0.937264
\(357\) −148188. −0.0615380
\(358\) −6.03175e6 −2.48734
\(359\) −2.78940e6 −1.14229 −0.571144 0.820850i \(-0.693500\pi\)
−0.571144 + 0.820850i \(0.693500\pi\)
\(360\) 839345. 0.341338
\(361\) 5.60915e6 2.26532
\(362\) −5.15113e6 −2.06600
\(363\) 131336. 0.0523139
\(364\) 6.10028e6 2.41322
\(365\) −904791. −0.355481
\(366\) 2.80982e6 1.09642
\(367\) 843940. 0.327074 0.163537 0.986537i \(-0.447710\pi\)
0.163537 + 0.986537i \(0.447710\pi\)
\(368\) −681521. −0.262337
\(369\) 349007. 0.133435
\(370\) 1.15039e6 0.436857
\(371\) 6.79951e6 2.56474
\(372\) 3.92848e6 1.47186
\(373\) −4.21025e6 −1.56688 −0.783440 0.621467i \(-0.786537\pi\)
−0.783440 + 0.621467i \(0.786537\pi\)
\(374\) −245588. −0.0907881
\(375\) −1.69990e6 −0.624229
\(376\) 1.66165e6 0.606137
\(377\) −179837. −0.0651666
\(378\) 1.67830e6 0.604144
\(379\) 3.19942e6 1.14412 0.572061 0.820211i \(-0.306144\pi\)
0.572061 + 0.820211i \(0.306144\pi\)
\(380\) −6.97964e6 −2.47956
\(381\) −2.66045e6 −0.938950
\(382\) 3.59596e6 1.26083
\(383\) −391078. −0.136228 −0.0681140 0.997678i \(-0.521698\pi\)
−0.0681140 + 0.997678i \(0.521698\pi\)
\(384\) −3.04014e6 −1.05212
\(385\) −3.95501e6 −1.35986
\(386\) −3.67610e6 −1.25580
\(387\) 80343.3 0.0272692
\(388\) −4.80809e6 −1.62141
\(389\) 3.96468e6 1.32841 0.664207 0.747548i \(-0.268769\pi\)
0.664207 + 0.747548i \(0.268769\pi\)
\(390\) −1.57617e6 −0.524738
\(391\) −100128. −0.0331216
\(392\) −1.02991e7 −3.38520
\(393\) −244546. −0.0798693
\(394\) −9.33387e6 −3.02915
\(395\) 2.17015e6 0.699838
\(396\) 1.78945e6 0.573431
\(397\) −4.83817e6 −1.54065 −0.770327 0.637649i \(-0.779907\pi\)
−0.770327 + 0.637649i \(0.779907\pi\)
\(398\) −3.26975e6 −1.03468
\(399\) −6.21971e6 −1.95586
\(400\) −607259. −0.189768
\(401\) 939464. 0.291755 0.145878 0.989303i \(-0.453399\pi\)
0.145878 + 0.989303i \(0.453399\pi\)
\(402\) −1.22993e6 −0.379591
\(403\) −3.28774e6 −1.00840
\(404\) −8.71956e6 −2.65791
\(405\) −278984. −0.0845166
\(406\) 952202. 0.286691
\(407\) 1.09302e6 0.327072
\(408\) 148586. 0.0441904
\(409\) 1.92053e6 0.567691 0.283846 0.958870i \(-0.408390\pi\)
0.283846 + 0.958870i \(0.408390\pi\)
\(410\) 1.73548e6 0.509871
\(411\) −1.37816e6 −0.402433
\(412\) 1.05974e7 3.07577
\(413\) 846029. 0.244068
\(414\) 1.13399e6 0.325169
\(415\) 757272. 0.215840
\(416\) 1.49150e6 0.422560
\(417\) 1.37478e6 0.387162
\(418\) −1.03078e7 −2.88552
\(419\) −683307. −0.190143 −0.0950716 0.995470i \(-0.530308\pi\)
−0.0950716 + 0.995470i \(0.530308\pi\)
\(420\) 5.36921e6 1.48521
\(421\) −645392. −0.177467 −0.0887336 0.996055i \(-0.528282\pi\)
−0.0887336 + 0.996055i \(0.528282\pi\)
\(422\) 8.27755e6 2.26267
\(423\) −552305. −0.150082
\(424\) −6.81777e6 −1.84174
\(425\) −89217.1 −0.0239594
\(426\) 2.69090e6 0.718413
\(427\) 8.01044e6 2.12611
\(428\) −9.13345e6 −2.41005
\(429\) −1.49758e6 −0.392869
\(430\) 399516. 0.104199
\(431\) 4.43313e6 1.14952 0.574761 0.818322i \(-0.305095\pi\)
0.574761 + 0.818322i \(0.305095\pi\)
\(432\) −336158. −0.0866631
\(433\) 7.41745e6 1.90123 0.950616 0.310371i \(-0.100453\pi\)
0.950616 + 0.310371i \(0.100453\pi\)
\(434\) 1.74079e7 4.43632
\(435\) −158285. −0.0401066
\(436\) −7.70135e6 −1.94022
\(437\) −4.20252e6 −1.05271
\(438\) 1.81402e6 0.451812
\(439\) −1.71298e6 −0.424219 −0.212110 0.977246i \(-0.568033\pi\)
−0.212110 + 0.977246i \(0.568033\pi\)
\(440\) 3.96563e6 0.976518
\(441\) 3.42325e6 0.838190
\(442\) −279025. −0.0679340
\(443\) −5.52257e6 −1.33700 −0.668501 0.743712i \(-0.733064\pi\)
−0.668501 + 0.743712i \(0.733064\pi\)
\(444\) −1.48386e6 −0.357220
\(445\) 1.65089e6 0.395202
\(446\) 9.52137e6 2.26653
\(447\) 1.88131e6 0.445339
\(448\) −1.14835e7 −2.70321
\(449\) 3.46045e6 0.810059 0.405030 0.914304i \(-0.367261\pi\)
0.405030 + 0.914304i \(0.367261\pi\)
\(450\) 1.01043e6 0.235219
\(451\) 1.64894e6 0.381737
\(452\) 1.42585e7 3.28268
\(453\) −4.21933e6 −0.966046
\(454\) −1.14350e7 −2.60374
\(455\) −4.49347e6 −1.01755
\(456\) 6.23642e6 1.40450
\(457\) −892817. −0.199973 −0.0999866 0.994989i \(-0.531880\pi\)
−0.0999866 + 0.994989i \(0.531880\pi\)
\(458\) 8.76032e6 1.95145
\(459\) −49387.6 −0.0109417
\(460\) 3.62786e6 0.799384
\(461\) −2.39311e6 −0.524458 −0.262229 0.965006i \(-0.584458\pi\)
−0.262229 + 0.965006i \(0.584458\pi\)
\(462\) 7.92942e6 1.72837
\(463\) 8.67234e6 1.88011 0.940057 0.341018i \(-0.110772\pi\)
0.940057 + 0.341018i \(0.110772\pi\)
\(464\) −190723. −0.0411252
\(465\) −2.89372e6 −0.620618
\(466\) −7.11201e6 −1.51715
\(467\) 7.47827e6 1.58675 0.793376 0.608732i \(-0.208321\pi\)
0.793376 + 0.608732i \(0.208321\pi\)
\(468\) 2.03308e6 0.429081
\(469\) −3.50638e6 −0.736082
\(470\) −2.74640e6 −0.573482
\(471\) −1.31203e6 −0.272516
\(472\) −848301. −0.175265
\(473\) 379596. 0.0780132
\(474\) −4.35096e6 −0.889485
\(475\) −3.74459e6 −0.761502
\(476\) 950492. 0.192279
\(477\) 2.26611e6 0.456021
\(478\) −1.47150e7 −2.94572
\(479\) 1.51970e6 0.302636 0.151318 0.988485i \(-0.451648\pi\)
0.151318 + 0.988485i \(0.451648\pi\)
\(480\) 1.31275e6 0.260063
\(481\) 1.24184e6 0.244738
\(482\) −5.41351e6 −1.06136
\(483\) 3.23286e6 0.630550
\(484\) −842400. −0.163457
\(485\) 3.54164e6 0.683676
\(486\) 559337. 0.107419
\(487\) −2.51702e6 −0.480910 −0.240455 0.970660i \(-0.577297\pi\)
−0.240455 + 0.970660i \(0.577297\pi\)
\(488\) −8.03195e6 −1.52676
\(489\) 3.25374e6 0.615333
\(490\) 1.70225e7 3.20283
\(491\) −5.68351e6 −1.06393 −0.531965 0.846767i \(-0.678546\pi\)
−0.531965 + 0.846767i \(0.678546\pi\)
\(492\) −2.23856e6 −0.416923
\(493\) −28020.6 −0.00519230
\(494\) −1.17111e7 −2.15915
\(495\) −1.31811e6 −0.241790
\(496\) −3.48675e6 −0.636381
\(497\) 7.67142e6 1.39311
\(498\) −1.51826e6 −0.274330
\(499\) −2.41309e6 −0.433833 −0.216916 0.976190i \(-0.569600\pi\)
−0.216916 + 0.976190i \(0.569600\pi\)
\(500\) 1.09033e7 1.95044
\(501\) 3.65314e6 0.650237
\(502\) −6.73171e6 −1.19225
\(503\) 5.40338e6 0.952237 0.476119 0.879381i \(-0.342043\pi\)
0.476119 + 0.879381i \(0.342043\pi\)
\(504\) −4.79747e6 −0.841271
\(505\) 6.42283e6 1.12072
\(506\) 5.35774e6 0.930261
\(507\) 1.64016e6 0.283378
\(508\) 1.70643e7 2.93380
\(509\) −4.13477e6 −0.707386 −0.353693 0.935362i \(-0.615074\pi\)
−0.353693 + 0.935362i \(0.615074\pi\)
\(510\) −245586. −0.0418097
\(511\) 5.17155e6 0.876129
\(512\) 5.17772e6 0.872898
\(513\) −2.07288e6 −0.347761
\(514\) −1.23669e7 −2.06468
\(515\) −7.80602e6 −1.29692
\(516\) −515328. −0.0852040
\(517\) −2.60946e6 −0.429363
\(518\) −6.57530e6 −1.07669
\(519\) −1.65693e6 −0.270014
\(520\) 4.50554e6 0.730699
\(521\) 3.96698e6 0.640273 0.320137 0.947371i \(-0.396271\pi\)
0.320137 + 0.947371i \(0.396271\pi\)
\(522\) 317346. 0.0509750
\(523\) 8.11326e6 1.29700 0.648502 0.761213i \(-0.275396\pi\)
0.648502 + 0.761213i \(0.275396\pi\)
\(524\) 1.56854e6 0.249556
\(525\) 2.88059e6 0.456125
\(526\) −1.79999e7 −2.83665
\(527\) −512266. −0.0803468
\(528\) −1.58824e6 −0.247931
\(529\) −4.25197e6 −0.660619
\(530\) 1.12685e7 1.74252
\(531\) 281961. 0.0433963
\(532\) 3.98938e7 6.11119
\(533\) 1.87345e6 0.285643
\(534\) −3.30989e6 −0.502296
\(535\) 6.72771e6 1.01621
\(536\) 3.51579e6 0.528581
\(537\) −5.73093e6 −0.857609
\(538\) 1.38669e7 2.06550
\(539\) 1.61738e7 2.39794
\(540\) 1.78943e6 0.264076
\(541\) −7.71908e6 −1.13389 −0.566947 0.823754i \(-0.691876\pi\)
−0.566947 + 0.823754i \(0.691876\pi\)
\(542\) −1.01934e6 −0.149047
\(543\) −4.89423e6 −0.712336
\(544\) 232392. 0.0336685
\(545\) 5.67282e6 0.818102
\(546\) 9.00899e6 1.29329
\(547\) −147862. −0.0211295 −0.0105647 0.999944i \(-0.503363\pi\)
−0.0105647 + 0.999944i \(0.503363\pi\)
\(548\) 8.83961e6 1.25742
\(549\) 2.66968e6 0.378032
\(550\) 4.77393e6 0.672929
\(551\) −1.17607e6 −0.165027
\(552\) −3.24155e6 −0.452798
\(553\) −1.24040e7 −1.72484
\(554\) 4.46674e6 0.618324
\(555\) 1.09301e6 0.150623
\(556\) −8.81796e6 −1.20971
\(557\) −8.43178e6 −1.15155 −0.575773 0.817610i \(-0.695299\pi\)
−0.575773 + 0.817610i \(0.695299\pi\)
\(558\) 5.80165e6 0.788798
\(559\) 431277. 0.0583749
\(560\) −4.76548e6 −0.642150
\(561\) −233340. −0.0313027
\(562\) 1.38514e7 1.84992
\(563\) −1.36993e7 −1.82149 −0.910746 0.412966i \(-0.864493\pi\)
−0.910746 + 0.412966i \(0.864493\pi\)
\(564\) 3.54253e6 0.468939
\(565\) −1.05028e7 −1.38416
\(566\) 2.56561e6 0.336628
\(567\) 1.59460e6 0.208302
\(568\) −7.69202e6 −1.00039
\(569\) 3.13483e6 0.405913 0.202957 0.979188i \(-0.434945\pi\)
0.202957 + 0.979188i \(0.434945\pi\)
\(570\) −1.03076e7 −1.32884
\(571\) −4.13353e6 −0.530556 −0.265278 0.964172i \(-0.585464\pi\)
−0.265278 + 0.964172i \(0.585464\pi\)
\(572\) 9.60562e6 1.22754
\(573\) 3.41662e6 0.434720
\(574\) −9.91954e6 −1.25664
\(575\) 1.94635e6 0.245500
\(576\) −3.82717e6 −0.480642
\(577\) 1.21527e7 1.51961 0.759807 0.650149i \(-0.225293\pi\)
0.759807 + 0.650149i \(0.225293\pi\)
\(578\) 1.34060e7 1.66909
\(579\) −3.49276e6 −0.432985
\(580\) 1.01525e6 0.125315
\(581\) −4.32836e6 −0.531965
\(582\) −7.10066e6 −0.868943
\(583\) 1.07066e7 1.30461
\(584\) −5.18544e6 −0.629148
\(585\) −1.49757e6 −0.180924
\(586\) 1.20738e6 0.145245
\(587\) −2.01425e6 −0.241278 −0.120639 0.992696i \(-0.538494\pi\)
−0.120639 + 0.992696i \(0.538494\pi\)
\(588\) −2.19570e7 −2.61897
\(589\) −2.15007e7 −2.55366
\(590\) 1.40208e6 0.165823
\(591\) −8.86836e6 −1.04442
\(592\) 1.31701e6 0.154449
\(593\) −9.67469e6 −1.12980 −0.564899 0.825160i \(-0.691085\pi\)
−0.564899 + 0.825160i \(0.691085\pi\)
\(594\) 2.64268e6 0.307312
\(595\) −700133. −0.0810752
\(596\) −1.20669e7 −1.39148
\(597\) −3.10667e6 −0.356747
\(598\) 6.08718e6 0.696087
\(599\) −3.89300e6 −0.443320 −0.221660 0.975124i \(-0.571148\pi\)
−0.221660 + 0.975124i \(0.571148\pi\)
\(600\) −2.88833e6 −0.327543
\(601\) −1.38351e6 −0.156241 −0.0781204 0.996944i \(-0.524892\pi\)
−0.0781204 + 0.996944i \(0.524892\pi\)
\(602\) −2.28353e6 −0.256812
\(603\) −1.16859e6 −0.130879
\(604\) 2.70631e7 3.01846
\(605\) 620512. 0.0689226
\(606\) −1.28772e7 −1.42442
\(607\) −2.91020e6 −0.320591 −0.160295 0.987069i \(-0.551245\pi\)
−0.160295 + 0.987069i \(0.551245\pi\)
\(608\) 9.75387e6 1.07008
\(609\) 904713. 0.0988479
\(610\) 1.32753e7 1.44451
\(611\) −2.96473e6 −0.321279
\(612\) 316776. 0.0341880
\(613\) 7.41582e6 0.797091 0.398546 0.917149i \(-0.369515\pi\)
0.398546 + 0.917149i \(0.369515\pi\)
\(614\) −1.04451e7 −1.11813
\(615\) 1.64893e6 0.175798
\(616\) −2.26665e7 −2.40676
\(617\) −532614. −0.0563248 −0.0281624 0.999603i \(-0.508966\pi\)
−0.0281624 + 0.999603i \(0.508966\pi\)
\(618\) 1.56503e7 1.64836
\(619\) 1.74793e6 0.183357 0.0916787 0.995789i \(-0.470777\pi\)
0.0916787 + 0.995789i \(0.470777\pi\)
\(620\) 1.85606e7 1.93915
\(621\) 1.07744e6 0.112115
\(622\) 2.42525e6 0.251351
\(623\) −9.43606e6 −0.974026
\(624\) −1.80447e6 −0.185519
\(625\) −3.91599e6 −0.400998
\(626\) −1.43249e7 −1.46102
\(627\) −9.79368e6 −0.994895
\(628\) 8.41548e6 0.851491
\(629\) 193492. 0.0195001
\(630\) 7.92933e6 0.795949
\(631\) −2.72819e6 −0.272773 −0.136386 0.990656i \(-0.543549\pi\)
−0.136386 + 0.990656i \(0.543549\pi\)
\(632\) 1.24373e7 1.23861
\(633\) 7.86472e6 0.780143
\(634\) −2.83075e7 −2.79691
\(635\) −1.25696e7 −1.23705
\(636\) −1.45350e7 −1.42486
\(637\) 1.83758e7 1.79431
\(638\) 1.49936e6 0.145832
\(639\) 2.55670e6 0.247701
\(640\) −1.43635e7 −1.38615
\(641\) −1.24990e7 −1.20152 −0.600759 0.799430i \(-0.705135\pi\)
−0.600759 + 0.799430i \(0.705135\pi\)
\(642\) −1.34884e7 −1.29159
\(643\) −1.21125e7 −1.15533 −0.577663 0.816275i \(-0.696035\pi\)
−0.577663 + 0.816275i \(0.696035\pi\)
\(644\) −2.07359e7 −1.97019
\(645\) 379591. 0.0359267
\(646\) −1.82473e6 −0.172035
\(647\) 4.87189e6 0.457548 0.228774 0.973480i \(-0.426528\pi\)
0.228774 + 0.973480i \(0.426528\pi\)
\(648\) −1.59888e6 −0.149582
\(649\) 1.33217e6 0.124151
\(650\) 5.42389e6 0.503532
\(651\) 1.65398e7 1.52960
\(652\) −2.08697e7 −1.92264
\(653\) −1.94196e7 −1.78220 −0.891100 0.453806i \(-0.850066\pi\)
−0.891100 + 0.453806i \(0.850066\pi\)
\(654\) −1.13735e7 −1.03980
\(655\) −1.15539e6 −0.105226
\(656\) 1.98685e6 0.180263
\(657\) 1.72355e6 0.155780
\(658\) 1.56977e7 1.41342
\(659\) −4.75216e6 −0.426263 −0.213132 0.977024i \(-0.568366\pi\)
−0.213132 + 0.977024i \(0.568366\pi\)
\(660\) 8.45446e6 0.755485
\(661\) −548417. −0.0488211 −0.0244105 0.999702i \(-0.507771\pi\)
−0.0244105 + 0.999702i \(0.507771\pi\)
\(662\) 2.55522e7 2.26612
\(663\) −265109. −0.0234229
\(664\) 4.33999e6 0.382004
\(665\) −2.93858e7 −2.57681
\(666\) −2.19139e6 −0.191440
\(667\) 611295. 0.0532030
\(668\) −2.34315e7 −2.03170
\(669\) 9.04651e6 0.781476
\(670\) −5.81095e6 −0.500104
\(671\) 1.26134e7 1.08150
\(672\) −7.50333e6 −0.640960
\(673\) −8.23444e6 −0.700803 −0.350402 0.936600i \(-0.613955\pi\)
−0.350402 + 0.936600i \(0.613955\pi\)
\(674\) −1.27605e7 −1.08198
\(675\) 960032. 0.0811010
\(676\) −1.05201e7 −0.885430
\(677\) 2.26732e7 1.90126 0.950630 0.310326i \(-0.100438\pi\)
0.950630 + 0.310326i \(0.100438\pi\)
\(678\) 2.10572e7 1.75925
\(679\) −2.02431e7 −1.68501
\(680\) 702013. 0.0582201
\(681\) −1.08647e7 −0.897741
\(682\) 2.74109e7 2.25664
\(683\) −8.80273e6 −0.722047 −0.361024 0.932557i \(-0.617573\pi\)
−0.361024 + 0.932557i \(0.617573\pi\)
\(684\) 1.32956e7 1.08660
\(685\) −6.51126e6 −0.530199
\(686\) −5.86033e7 −4.75457
\(687\) 8.32342e6 0.672837
\(688\) 457383. 0.0368391
\(689\) 1.21643e7 0.976202
\(690\) 5.35768e6 0.428404
\(691\) 5.96759e6 0.475449 0.237724 0.971333i \(-0.423599\pi\)
0.237724 + 0.971333i \(0.423599\pi\)
\(692\) 1.06277e7 0.843672
\(693\) 7.53395e6 0.595923
\(694\) 8.12814e6 0.640608
\(695\) 6.49531e6 0.510080
\(696\) −907143. −0.0709827
\(697\) 291904. 0.0227592
\(698\) 2.03911e7 1.58417
\(699\) −6.75731e6 −0.523095
\(700\) −1.84764e7 −1.42519
\(701\) −6.81387e6 −0.523720 −0.261860 0.965106i \(-0.584336\pi\)
−0.261860 + 0.965106i \(0.584336\pi\)
\(702\) 3.00248e6 0.229952
\(703\) 8.12119e6 0.619771
\(704\) −1.80821e7 −1.37505
\(705\) −2.60943e6 −0.197730
\(706\) 3.25197e7 2.45547
\(707\) −3.67112e7 −2.76217
\(708\) −1.80852e6 −0.135594
\(709\) 2.19624e7 1.64083 0.820414 0.571769i \(-0.193743\pi\)
0.820414 + 0.571769i \(0.193743\pi\)
\(710\) 1.27135e7 0.946496
\(711\) −4.13396e6 −0.306685
\(712\) 9.46140e6 0.699448
\(713\) 1.11755e7 0.823275
\(714\) 1.40370e6 0.103046
\(715\) −7.07551e6 −0.517598
\(716\) 3.67586e7 2.67964
\(717\) −1.39811e7 −1.01565
\(718\) 2.64224e7 1.91276
\(719\) 4.29024e6 0.309499 0.154750 0.987954i \(-0.450543\pi\)
0.154750 + 0.987954i \(0.450543\pi\)
\(720\) −1.58822e6 −0.114177
\(721\) 4.46171e7 3.19642
\(722\) −5.31322e7 −3.79328
\(723\) −5.14352e6 −0.365944
\(724\) 3.13920e7 2.22573
\(725\) 544685. 0.0384858
\(726\) −1.24407e6 −0.0875998
\(727\) 2.05224e7 1.44010 0.720049 0.693923i \(-0.244119\pi\)
0.720049 + 0.693923i \(0.244119\pi\)
\(728\) −2.57525e7 −1.80090
\(729\) 531441. 0.0370370
\(730\) 8.57057e6 0.595254
\(731\) 67197.7 0.00465116
\(732\) −1.71236e7 −1.18118
\(733\) −2.27907e7 −1.56674 −0.783372 0.621553i \(-0.786502\pi\)
−0.783372 + 0.621553i \(0.786502\pi\)
\(734\) −7.99415e6 −0.547687
\(735\) 1.61736e7 1.10430
\(736\) −5.06984e6 −0.344984
\(737\) −5.52121e6 −0.374425
\(738\) −3.30594e6 −0.223437
\(739\) 1.16303e7 0.783390 0.391695 0.920095i \(-0.371889\pi\)
0.391695 + 0.920095i \(0.371889\pi\)
\(740\) −7.01067e6 −0.470631
\(741\) −1.11271e7 −0.744450
\(742\) −6.44078e7 −4.29466
\(743\) −1.36248e6 −0.0905437 −0.0452718 0.998975i \(-0.514415\pi\)
−0.0452718 + 0.998975i \(0.514415\pi\)
\(744\) −1.65842e7 −1.09840
\(745\) 8.88845e6 0.586726
\(746\) 3.98813e7 2.62375
\(747\) −1.44254e6 −0.0945858
\(748\) 1.49666e6 0.0978070
\(749\) −3.84538e7 −2.50458
\(750\) 1.61021e7 1.04527
\(751\) −3.68228e6 −0.238241 −0.119121 0.992880i \(-0.538008\pi\)
−0.119121 + 0.992880i \(0.538008\pi\)
\(752\) −3.14420e6 −0.202752
\(753\) −6.39597e6 −0.411073
\(754\) 1.70349e6 0.109122
\(755\) −1.99347e7 −1.27275
\(756\) −1.02279e7 −0.650851
\(757\) −1.07391e7 −0.681127 −0.340563 0.940222i \(-0.610618\pi\)
−0.340563 + 0.940222i \(0.610618\pi\)
\(758\) −3.03062e7 −1.91584
\(759\) 5.09053e6 0.320744
\(760\) 2.94647e7 1.85041
\(761\) 1.41023e7 0.882731 0.441366 0.897327i \(-0.354494\pi\)
0.441366 + 0.897327i \(0.354494\pi\)
\(762\) 2.52009e7 1.57228
\(763\) −3.24243e7 −2.01632
\(764\) −2.19145e7 −1.35831
\(765\) −233337. −0.0144155
\(766\) 3.70446e6 0.228114
\(767\) 1.51355e6 0.0928982
\(768\) 1.51898e7 0.929284
\(769\) 1.64536e7 1.00333 0.501665 0.865062i \(-0.332721\pi\)
0.501665 + 0.865062i \(0.332721\pi\)
\(770\) 3.74635e7 2.27710
\(771\) −1.17501e7 −0.711878
\(772\) 2.24029e7 1.35288
\(773\) 1.13373e7 0.682433 0.341216 0.939985i \(-0.389161\pi\)
0.341216 + 0.939985i \(0.389161\pi\)
\(774\) −761045. −0.0456623
\(775\) 9.95780e6 0.595537
\(776\) 2.02974e7 1.21000
\(777\) −6.24736e6 −0.371231
\(778\) −3.75551e7 −2.22443
\(779\) 1.22517e7 0.723356
\(780\) 9.60551e6 0.565307
\(781\) 1.20796e7 0.708637
\(782\) 948450. 0.0554623
\(783\) 301519. 0.0175756
\(784\) 1.94881e7 1.13235
\(785\) −6.19885e6 −0.359035
\(786\) 2.31645e6 0.133741
\(787\) 2.87599e7 1.65520 0.827601 0.561317i \(-0.189705\pi\)
0.827601 + 0.561317i \(0.189705\pi\)
\(788\) 5.68824e7 3.26334
\(789\) −1.71022e7 −0.978045
\(790\) −2.05566e7 −1.17188
\(791\) 6.00314e7 3.41144
\(792\) −7.55419e6 −0.427932
\(793\) 1.43307e7 0.809252
\(794\) 4.58292e7 2.57983
\(795\) 1.07065e7 0.600800
\(796\) 1.99265e7 1.11467
\(797\) −3.21674e7 −1.79378 −0.896892 0.442250i \(-0.854181\pi\)
−0.896892 + 0.442250i \(0.854181\pi\)
\(798\) 5.89158e7 3.27510
\(799\) −461939. −0.0255987
\(800\) −4.51740e6 −0.249553
\(801\) −3.14481e6 −0.173186
\(802\) −8.89899e6 −0.488546
\(803\) 8.14322e6 0.445663
\(804\) 7.49544e6 0.408937
\(805\) 1.52740e7 0.830739
\(806\) 3.11428e7 1.68857
\(807\) 1.31753e7 0.712161
\(808\) 3.68098e7 1.98351
\(809\) −1.90148e7 −1.02146 −0.510728 0.859742i \(-0.670624\pi\)
−0.510728 + 0.859742i \(0.670624\pi\)
\(810\) 2.64265e6 0.141523
\(811\) −4.76970e6 −0.254647 −0.127324 0.991861i \(-0.540639\pi\)
−0.127324 + 0.991861i \(0.540639\pi\)
\(812\) −5.80290e6 −0.308856
\(813\) −968506. −0.0513897
\(814\) −1.03536e7 −0.547684
\(815\) 1.53727e7 0.810690
\(816\) −281157. −0.0147817
\(817\) 2.82040e6 0.147828
\(818\) −1.81920e7 −0.950601
\(819\) 8.55969e6 0.445911
\(820\) −1.05763e7 −0.549289
\(821\) −2.63580e7 −1.36475 −0.682376 0.731001i \(-0.739053\pi\)
−0.682376 + 0.731001i \(0.739053\pi\)
\(822\) 1.30545e7 0.673875
\(823\) −3.92377e6 −0.201931 −0.100966 0.994890i \(-0.532193\pi\)
−0.100966 + 0.994890i \(0.532193\pi\)
\(824\) −4.47370e7 −2.29535
\(825\) 4.53584e6 0.232018
\(826\) −8.01394e6 −0.408692
\(827\) 143623. 0.00730230 0.00365115 0.999993i \(-0.498838\pi\)
0.00365115 + 0.999993i \(0.498838\pi\)
\(828\) −6.91076e6 −0.350308
\(829\) −2.78716e6 −0.140856 −0.0704281 0.997517i \(-0.522437\pi\)
−0.0704281 + 0.997517i \(0.522437\pi\)
\(830\) −7.17320e6 −0.361424
\(831\) 4.24397e6 0.213191
\(832\) −2.05440e7 −1.02891
\(833\) 2.86315e6 0.142966
\(834\) −1.30225e7 −0.648305
\(835\) 1.72597e7 0.856676
\(836\) 6.28175e7 3.10860
\(837\) 5.51230e6 0.271969
\(838\) 6.47257e6 0.318396
\(839\) 1.42188e7 0.697362 0.348681 0.937242i \(-0.386630\pi\)
0.348681 + 0.937242i \(0.386630\pi\)
\(840\) −2.26662e7 −1.10836
\(841\) −2.03401e7 −0.991660
\(842\) 6.11342e6 0.297169
\(843\) 1.31606e7 0.637833
\(844\) −5.04450e7 −2.43760
\(845\) 7.74914e6 0.373346
\(846\) 5.23167e6 0.251313
\(847\) −3.54668e6 −0.169869
\(848\) 1.29007e7 0.616059
\(849\) 2.43766e6 0.116065
\(850\) 845102. 0.0401201
\(851\) −4.22121e6 −0.199808
\(852\) −1.63989e7 −0.773954
\(853\) 3.54743e7 1.66933 0.834663 0.550761i \(-0.185662\pi\)
0.834663 + 0.550761i \(0.185662\pi\)
\(854\) −7.58782e7 −3.56018
\(855\) −9.79357e6 −0.458169
\(856\) 3.85570e7 1.79854
\(857\) −6.66554e6 −0.310016 −0.155008 0.987913i \(-0.549540\pi\)
−0.155008 + 0.987913i \(0.549540\pi\)
\(858\) 1.41857e7 0.657860
\(859\) −2.09784e7 −0.970038 −0.485019 0.874503i \(-0.661187\pi\)
−0.485019 + 0.874503i \(0.661187\pi\)
\(860\) −2.43473e6 −0.112255
\(861\) −9.42482e6 −0.433277
\(862\) −4.19925e7 −1.92488
\(863\) −3.29411e7 −1.50560 −0.752801 0.658248i \(-0.771298\pi\)
−0.752801 + 0.658248i \(0.771298\pi\)
\(864\) −2.50068e6 −0.113966
\(865\) −7.82836e6 −0.355738
\(866\) −7.02612e7 −3.18362
\(867\) 1.27374e7 0.575484
\(868\) −1.06087e8 −4.77930
\(869\) −1.95316e7 −0.877381
\(870\) 1.49934e6 0.0671586
\(871\) −6.27291e6 −0.280171
\(872\) 3.25114e7 1.44792
\(873\) −6.74653e6 −0.299602
\(874\) 3.98081e7 1.76276
\(875\) 4.59051e7 2.02694
\(876\) −1.10550e7 −0.486742
\(877\) 1.74947e7 0.768081 0.384041 0.923316i \(-0.374532\pi\)
0.384041 + 0.923316i \(0.374532\pi\)
\(878\) 1.62260e7 0.710356
\(879\) 1.14717e6 0.0500788
\(880\) −7.50381e6 −0.326644
\(881\) −2.47900e7 −1.07606 −0.538031 0.842925i \(-0.680832\pi\)
−0.538031 + 0.842925i \(0.680832\pi\)
\(882\) −3.24265e7 −1.40355
\(883\) 1.85125e7 0.799028 0.399514 0.916727i \(-0.369179\pi\)
0.399514 + 0.916727i \(0.369179\pi\)
\(884\) 1.70043e6 0.0731860
\(885\) 1.33216e6 0.0571739
\(886\) 5.23121e7 2.23881
\(887\) −3.23247e7 −1.37951 −0.689756 0.724042i \(-0.742282\pi\)
−0.689756 + 0.724042i \(0.742282\pi\)
\(888\) 6.26414e6 0.266581
\(889\) 7.18445e7 3.04887
\(890\) −1.56380e7 −0.661767
\(891\) 2.51088e6 0.105958
\(892\) −5.80251e7 −2.44176
\(893\) −1.93883e7 −0.813602
\(894\) −1.78205e7 −0.745721
\(895\) −2.70764e7 −1.12988
\(896\) 8.20980e7 3.41635
\(897\) 5.78359e6 0.240003
\(898\) −3.27789e7 −1.35645
\(899\) 3.12746e6 0.129060
\(900\) −6.15772e6 −0.253404
\(901\) 1.89534e6 0.0777811
\(902\) −1.56195e7 −0.639220
\(903\) −2.16964e6 −0.0885460
\(904\) −6.01927e7 −2.44975
\(905\) −2.31234e7 −0.938490
\(906\) 3.99672e7 1.61765
\(907\) 5.10131e6 0.205903 0.102952 0.994686i \(-0.467171\pi\)
0.102952 + 0.994686i \(0.467171\pi\)
\(908\) 6.96873e7 2.80504
\(909\) −1.22350e7 −0.491126
\(910\) 4.25641e7 1.70388
\(911\) 3.65998e7 1.46111 0.730555 0.682854i \(-0.239262\pi\)
0.730555 + 0.682854i \(0.239262\pi\)
\(912\) −1.18006e7 −0.469805
\(913\) −6.81552e6 −0.270596
\(914\) 8.45714e6 0.334856
\(915\) 1.26132e7 0.498051
\(916\) −5.33871e7 −2.10231
\(917\) 6.60389e6 0.259344
\(918\) 467820. 0.0183220
\(919\) −4.17959e7 −1.63247 −0.816233 0.577722i \(-0.803942\pi\)
−0.816233 + 0.577722i \(0.803942\pi\)
\(920\) −1.53151e7 −0.596553
\(921\) −9.92415e6 −0.385518
\(922\) 2.26686e7 0.878207
\(923\) 1.37242e7 0.530251
\(924\) −4.83234e7 −1.86199
\(925\) −3.76124e6 −0.144536
\(926\) −8.21481e7 −3.14826
\(927\) 1.48698e7 0.568337
\(928\) −1.41879e6 −0.0540813
\(929\) −3.62772e6 −0.137910 −0.0689549 0.997620i \(-0.521966\pi\)
−0.0689549 + 0.997620i \(0.521966\pi\)
\(930\) 2.74106e7 1.03923
\(931\) 1.20171e8 4.54388
\(932\) 4.33419e7 1.63444
\(933\) 2.30430e6 0.0866631
\(934\) −7.08374e7 −2.65702
\(935\) −1.10244e6 −0.0412408
\(936\) −8.58267e6 −0.320209
\(937\) −2.61792e7 −0.974110 −0.487055 0.873371i \(-0.661929\pi\)
−0.487055 + 0.873371i \(0.661929\pi\)
\(938\) 3.32139e7 1.23257
\(939\) −1.36105e7 −0.503744
\(940\) 1.67371e7 0.617819
\(941\) 2.74091e7 1.00907 0.504535 0.863391i \(-0.331664\pi\)
0.504535 + 0.863391i \(0.331664\pi\)
\(942\) 1.24281e7 0.456329
\(943\) −6.36814e6 −0.233203
\(944\) 1.60517e6 0.0586260
\(945\) 7.53387e6 0.274434
\(946\) −3.59569e6 −0.130633
\(947\) −2.02712e7 −0.734523 −0.367262 0.930118i \(-0.619705\pi\)
−0.367262 + 0.930118i \(0.619705\pi\)
\(948\) 2.65156e7 0.958252
\(949\) 9.25190e6 0.333477
\(950\) 3.54704e7 1.27514
\(951\) −2.68957e7 −0.964344
\(952\) −4.01252e6 −0.143491
\(953\) −5.47512e7 −1.95282 −0.976408 0.215936i \(-0.930720\pi\)
−0.976408 + 0.215936i \(0.930720\pi\)
\(954\) −2.14656e7 −0.763609
\(955\) 1.61422e7 0.572737
\(956\) 8.96762e7 3.17346
\(957\) 1.42458e6 0.0502813
\(958\) −1.43953e7 −0.506765
\(959\) 3.72166e7 1.30674
\(960\) −1.80819e7 −0.633237
\(961\) 2.85464e7 0.997110
\(962\) −1.17632e7 −0.409815
\(963\) −1.28157e7 −0.445325
\(964\) 3.29909e7 1.14341
\(965\) −1.65020e7 −0.570450
\(966\) −3.06230e7 −1.05586
\(967\) 3.51750e6 0.120967 0.0604836 0.998169i \(-0.480736\pi\)
0.0604836 + 0.998169i \(0.480736\pi\)
\(968\) 3.55621e6 0.121983
\(969\) −1.73372e6 −0.0593157
\(970\) −3.35479e7 −1.14482
\(971\) 5.13829e7 1.74892 0.874462 0.485094i \(-0.161215\pi\)
0.874462 + 0.485094i \(0.161215\pi\)
\(972\) −3.40871e6 −0.115724
\(973\) −3.71255e7 −1.25716
\(974\) 2.38423e7 0.805286
\(975\) 5.15338e6 0.173612
\(976\) 1.51982e7 0.510701
\(977\) 1.29861e7 0.435253 0.217627 0.976032i \(-0.430169\pi\)
0.217627 + 0.976032i \(0.430169\pi\)
\(978\) −3.08207e7 −1.03038
\(979\) −1.48582e7 −0.495461
\(980\) −1.03739e8 −3.45045
\(981\) −1.08062e7 −0.358511
\(982\) 5.38366e7 1.78155
\(983\) 2.68171e7 0.885171 0.442586 0.896726i \(-0.354061\pi\)
0.442586 + 0.896726i \(0.354061\pi\)
\(984\) 9.45013e6 0.311136
\(985\) −4.18996e7 −1.37600
\(986\) 265423. 0.00869452
\(987\) 1.49148e7 0.487332
\(988\) 7.13700e7 2.32607
\(989\) −1.46598e6 −0.0476582
\(990\) 1.24857e7 0.404878
\(991\) 1.52984e7 0.494838 0.247419 0.968909i \(-0.420418\pi\)
0.247419 + 0.968909i \(0.420418\pi\)
\(992\) −2.59379e7 −0.836867
\(993\) 2.42778e7 0.781334
\(994\) −7.26669e7 −2.33276
\(995\) −1.46778e7 −0.470007
\(996\) 9.25256e6 0.295538
\(997\) 5.96740e7 1.90129 0.950643 0.310286i \(-0.100425\pi\)
0.950643 + 0.310286i \(0.100425\pi\)
\(998\) 2.28578e7 0.726454
\(999\) −2.08210e6 −0.0660065
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 177.6.a.d.1.2 13
3.2 odd 2 531.6.a.e.1.12 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.d.1.2 13 1.1 even 1 trivial
531.6.a.e.1.12 13 3.2 odd 2