Properties

Label 177.6.a.d.1.13
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 1.13
Root \(10.6702\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+10.6702 q^{2} -9.00000 q^{3} +81.8525 q^{4} -61.2778 q^{5} -96.0315 q^{6} +186.685 q^{7} +531.935 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+10.6702 q^{2} -9.00000 q^{3} +81.8525 q^{4} -61.2778 q^{5} -96.0315 q^{6} +186.685 q^{7} +531.935 q^{8} +81.0000 q^{9} -653.844 q^{10} -424.892 q^{11} -736.673 q^{12} +983.052 q^{13} +1991.96 q^{14} +551.500 q^{15} +3056.56 q^{16} +1043.17 q^{17} +864.284 q^{18} -302.603 q^{19} -5015.74 q^{20} -1680.17 q^{21} -4533.67 q^{22} +4122.49 q^{23} -4787.42 q^{24} +629.966 q^{25} +10489.3 q^{26} -729.000 q^{27} +15280.7 q^{28} -8426.58 q^{29} +5884.60 q^{30} +10049.9 q^{31} +15592.1 q^{32} +3824.03 q^{33} +11130.8 q^{34} -11439.6 q^{35} +6630.06 q^{36} +10289.5 q^{37} -3228.83 q^{38} -8847.47 q^{39} -32595.8 q^{40} +1605.50 q^{41} -17927.7 q^{42} -4917.05 q^{43} -34778.5 q^{44} -4963.50 q^{45} +43987.7 q^{46} -24046.6 q^{47} -27509.0 q^{48} +18044.3 q^{49} +6721.85 q^{50} -9388.55 q^{51} +80465.3 q^{52} -6856.60 q^{53} -7778.55 q^{54} +26036.4 q^{55} +99304.4 q^{56} +2723.43 q^{57} -89913.0 q^{58} +3481.00 q^{59} +45141.7 q^{60} -31403.4 q^{61} +107234. q^{62} +15121.5 q^{63} +68560.1 q^{64} -60239.3 q^{65} +40803.0 q^{66} -61147.0 q^{67} +85386.3 q^{68} -37102.4 q^{69} -122063. q^{70} -7454.90 q^{71} +43086.7 q^{72} -4324.62 q^{73} +109790. q^{74} -5669.70 q^{75} -24768.9 q^{76} -79321.0 q^{77} -94404.0 q^{78} -30676.1 q^{79} -187299. q^{80} +6561.00 q^{81} +17130.9 q^{82} -29240.6 q^{83} -137526. q^{84} -63923.3 q^{85} -52465.8 q^{86} +75839.2 q^{87} -226015. q^{88} +23916.5 q^{89} -52961.4 q^{90} +183521. q^{91} +337436. q^{92} -90449.4 q^{93} -256582. q^{94} +18542.9 q^{95} -140329. q^{96} +31653.6 q^{97} +192536. q^{98} -34416.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 117q^{3} + 246q^{4} - 14q^{5} + 373q^{7} + 123q^{8} + 1053q^{9} + O(q^{10}) \) \( 13q - 117q^{3} + 246q^{4} - 14q^{5} + 373q^{7} + 123q^{8} + 1053q^{9} + 137q^{10} + 250q^{11} - 2214q^{12} + 1054q^{13} - 575q^{14} + 126q^{15} + 922q^{16} + 271q^{17} + 671q^{19} - 5491q^{20} - 3357q^{21} + 1094q^{22} + 3975q^{23} - 1107q^{24} + 15569q^{25} + 4622q^{26} - 9477q^{27} + 21214q^{28} - 10613q^{29} - 1233q^{30} + 25597q^{31} + 15966q^{32} - 2250q^{33} + 31796q^{34} + 6729q^{35} + 19926q^{36} + 17585q^{37} + 34903q^{38} - 9486q^{39} + 31382q^{40} + 12537q^{41} + 5175q^{42} + 26644q^{43} + 6654q^{44} - 1134q^{45} + 149005q^{46} + 52087q^{47} - 8298q^{48} + 95384q^{49} + 121821q^{50} - 2439q^{51} + 263630q^{52} + 20014q^{53} + 120932q^{55} + 126688q^{56} - 6039q^{57} + 86066q^{58} + 45253q^{59} + 49419q^{60} - 11667q^{61} + 164794q^{62} + 30213q^{63} + 151893q^{64} - 28674q^{65} - 9846q^{66} + 1106q^{67} - 4043q^{68} - 35775q^{69} + 56066q^{70} + 21230q^{71} + 9963q^{72} + 81131q^{73} + 102042q^{74} - 140121q^{75} + 73900q^{76} - 104655q^{77} - 41598q^{78} - 13470q^{79} - 191969q^{80} + 85293q^{81} + 79909q^{82} - 76149q^{83} - 190926q^{84} + 10035q^{85} - 321496q^{86} + 95517q^{87} - 276779q^{88} - 190205q^{89} + 11097q^{90} + 80601q^{91} + 45672q^{92} - 230373q^{93} + 36768q^{94} + 9875q^{95} - 143694q^{96} + 160850q^{97} - 116644q^{98} + 20250q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.6702 1.88624 0.943119 0.332456i \(-0.107877\pi\)
0.943119 + 0.332456i \(0.107877\pi\)
\(3\) −9.00000 −0.577350
\(4\) 81.8525 2.55789
\(5\) −61.2778 −1.09617 −0.548085 0.836423i \(-0.684643\pi\)
−0.548085 + 0.836423i \(0.684643\pi\)
\(6\) −96.0315 −1.08902
\(7\) 186.685 1.44001 0.720004 0.693970i \(-0.244140\pi\)
0.720004 + 0.693970i \(0.244140\pi\)
\(8\) 531.935 2.93855
\(9\) 81.0000 0.333333
\(10\) −653.844 −2.06764
\(11\) −424.892 −1.05876 −0.529379 0.848385i \(-0.677575\pi\)
−0.529379 + 0.848385i \(0.677575\pi\)
\(12\) −736.673 −1.47680
\(13\) 983.052 1.61331 0.806656 0.591021i \(-0.201275\pi\)
0.806656 + 0.591021i \(0.201275\pi\)
\(14\) 1991.96 2.71620
\(15\) 551.500 0.632874
\(16\) 3056.56 2.98492
\(17\) 1043.17 0.875455 0.437727 0.899108i \(-0.355783\pi\)
0.437727 + 0.899108i \(0.355783\pi\)
\(18\) 864.284 0.628746
\(19\) −302.603 −0.192305 −0.0961523 0.995367i \(-0.530654\pi\)
−0.0961523 + 0.995367i \(0.530654\pi\)
\(20\) −5015.74 −2.80389
\(21\) −1680.17 −0.831388
\(22\) −4533.67 −1.99707
\(23\) 4122.49 1.62495 0.812475 0.582996i \(-0.198120\pi\)
0.812475 + 0.582996i \(0.198120\pi\)
\(24\) −4787.42 −1.69658
\(25\) 629.966 0.201589
\(26\) 10489.3 3.04309
\(27\) −729.000 −0.192450
\(28\) 15280.7 3.68338
\(29\) −8426.58 −1.86061 −0.930307 0.366782i \(-0.880460\pi\)
−0.930307 + 0.366782i \(0.880460\pi\)
\(30\) 5884.60 1.19375
\(31\) 10049.9 1.87827 0.939137 0.343544i \(-0.111628\pi\)
0.939137 + 0.343544i \(0.111628\pi\)
\(32\) 15592.1 2.69171
\(33\) 3824.03 0.611274
\(34\) 11130.8 1.65132
\(35\) −11439.6 −1.57849
\(36\) 6630.06 0.852631
\(37\) 10289.5 1.23563 0.617815 0.786323i \(-0.288018\pi\)
0.617815 + 0.786323i \(0.288018\pi\)
\(38\) −3228.83 −0.362732
\(39\) −8847.47 −0.931446
\(40\) −32595.8 −3.22116
\(41\) 1605.50 0.149159 0.0745796 0.997215i \(-0.476239\pi\)
0.0745796 + 0.997215i \(0.476239\pi\)
\(42\) −17927.7 −1.56820
\(43\) −4917.05 −0.405540 −0.202770 0.979226i \(-0.564994\pi\)
−0.202770 + 0.979226i \(0.564994\pi\)
\(44\) −34778.5 −2.70819
\(45\) −4963.50 −0.365390
\(46\) 43987.7 3.06504
\(47\) −24046.6 −1.58785 −0.793926 0.608015i \(-0.791966\pi\)
−0.793926 + 0.608015i \(0.791966\pi\)
\(48\) −27509.0 −1.72334
\(49\) 18044.3 1.07362
\(50\) 6721.85 0.380245
\(51\) −9388.55 −0.505444
\(52\) 80465.3 4.12668
\(53\) −6856.60 −0.335289 −0.167644 0.985848i \(-0.553616\pi\)
−0.167644 + 0.985848i \(0.553616\pi\)
\(54\) −7778.55 −0.363007
\(55\) 26036.4 1.16058
\(56\) 99304.4 4.23154
\(57\) 2723.43 0.111027
\(58\) −89913.0 −3.50956
\(59\) 3481.00 0.130189
\(60\) 45141.7 1.61882
\(61\) −31403.4 −1.08057 −0.540283 0.841483i \(-0.681683\pi\)
−0.540283 + 0.841483i \(0.681683\pi\)
\(62\) 107234. 3.54287
\(63\) 15121.5 0.480002
\(64\) 68560.1 2.09229
\(65\) −60239.3 −1.76846
\(66\) 40803.0 1.15301
\(67\) −61147.0 −1.66413 −0.832066 0.554676i \(-0.812842\pi\)
−0.832066 + 0.554676i \(0.812842\pi\)
\(68\) 85386.3 2.23932
\(69\) −37102.4 −0.938166
\(70\) −122063. −2.97741
\(71\) −7454.90 −0.175508 −0.0877538 0.996142i \(-0.527969\pi\)
−0.0877538 + 0.996142i \(0.527969\pi\)
\(72\) 43086.7 0.979518
\(73\) −4324.62 −0.0949820 −0.0474910 0.998872i \(-0.515123\pi\)
−0.0474910 + 0.998872i \(0.515123\pi\)
\(74\) 109790. 2.33069
\(75\) −5669.70 −0.116388
\(76\) −24768.9 −0.491894
\(77\) −79321.0 −1.52462
\(78\) −94404.0 −1.75693
\(79\) −30676.1 −0.553010 −0.276505 0.961013i \(-0.589176\pi\)
−0.276505 + 0.961013i \(0.589176\pi\)
\(80\) −187299. −3.27198
\(81\) 6561.00 0.111111
\(82\) 17130.9 0.281350
\(83\) −29240.6 −0.465898 −0.232949 0.972489i \(-0.574837\pi\)
−0.232949 + 0.972489i \(0.574837\pi\)
\(84\) −137526. −2.12660
\(85\) −63923.3 −0.959648
\(86\) −52465.8 −0.764944
\(87\) 75839.2 1.07423
\(88\) −226015. −3.11122
\(89\) 23916.5 0.320053 0.160027 0.987113i \(-0.448842\pi\)
0.160027 + 0.987113i \(0.448842\pi\)
\(90\) −52961.4 −0.689212
\(91\) 183521. 2.32318
\(92\) 337436. 4.15645
\(93\) −90449.4 −1.08442
\(94\) −256582. −2.99506
\(95\) 18542.9 0.210799
\(96\) −140329. −1.55406
\(97\) 31653.6 0.341581 0.170790 0.985307i \(-0.445368\pi\)
0.170790 + 0.985307i \(0.445368\pi\)
\(98\) 192536. 2.02510
\(99\) −34416.2 −0.352919
\(100\) 51564.3 0.515643
\(101\) −90035.5 −0.878234 −0.439117 0.898430i \(-0.644709\pi\)
−0.439117 + 0.898430i \(0.644709\pi\)
\(102\) −100177. −0.953388
\(103\) 98739.3 0.917059 0.458529 0.888679i \(-0.348376\pi\)
0.458529 + 0.888679i \(0.348376\pi\)
\(104\) 522920. 4.74080
\(105\) 102957. 0.911343
\(106\) −73161.1 −0.632435
\(107\) −98628.3 −0.832803 −0.416401 0.909181i \(-0.636709\pi\)
−0.416401 + 0.909181i \(0.636709\pi\)
\(108\) −59670.5 −0.492267
\(109\) 57081.3 0.460180 0.230090 0.973169i \(-0.426098\pi\)
0.230090 + 0.973169i \(0.426098\pi\)
\(110\) 277813. 2.18913
\(111\) −92605.2 −0.713392
\(112\) 570614. 4.29830
\(113\) 70846.0 0.521938 0.260969 0.965347i \(-0.415958\pi\)
0.260969 + 0.965347i \(0.415958\pi\)
\(114\) 29059.5 0.209424
\(115\) −252617. −1.78122
\(116\) −689737. −4.75925
\(117\) 79627.2 0.537771
\(118\) 37142.9 0.245567
\(119\) 194745. 1.26066
\(120\) 293362. 1.85974
\(121\) 19482.1 0.120969
\(122\) −335079. −2.03821
\(123\) −14449.5 −0.0861172
\(124\) 822612. 4.80442
\(125\) 152890. 0.875194
\(126\) 161349. 0.905398
\(127\) −347695. −1.91289 −0.956444 0.291916i \(-0.905707\pi\)
−0.956444 + 0.291916i \(0.905707\pi\)
\(128\) 232602. 1.25484
\(129\) 44253.4 0.234138
\(130\) −642763. −3.33574
\(131\) 107422. 0.546909 0.273455 0.961885i \(-0.411834\pi\)
0.273455 + 0.961885i \(0.411834\pi\)
\(132\) 313006. 1.56357
\(133\) −56491.5 −0.276920
\(134\) −652449. −3.13895
\(135\) 44671.5 0.210958
\(136\) 554900. 2.57257
\(137\) 191277. 0.870686 0.435343 0.900265i \(-0.356627\pi\)
0.435343 + 0.900265i \(0.356627\pi\)
\(138\) −395889. −1.76960
\(139\) 23303.1 0.102300 0.0511502 0.998691i \(-0.483711\pi\)
0.0511502 + 0.998691i \(0.483711\pi\)
\(140\) −936364. −4.03761
\(141\) 216420. 0.916746
\(142\) −79545.1 −0.331049
\(143\) −417691. −1.70811
\(144\) 247581. 0.994973
\(145\) 516362. 2.03955
\(146\) −46144.5 −0.179159
\(147\) −162399. −0.619855
\(148\) 842219. 3.16061
\(149\) −16697.0 −0.0616132 −0.0308066 0.999525i \(-0.509808\pi\)
−0.0308066 + 0.999525i \(0.509808\pi\)
\(150\) −60496.6 −0.219535
\(151\) −141376. −0.504583 −0.252291 0.967651i \(-0.581184\pi\)
−0.252291 + 0.967651i \(0.581184\pi\)
\(152\) −160965. −0.565097
\(153\) 84497.0 0.291818
\(154\) −846369. −2.87579
\(155\) −615837. −2.05891
\(156\) −724188. −2.38254
\(157\) −6851.37 −0.0221834 −0.0110917 0.999938i \(-0.503531\pi\)
−0.0110917 + 0.999938i \(0.503531\pi\)
\(158\) −327319. −1.04311
\(159\) 61709.4 0.193579
\(160\) −955447. −2.95057
\(161\) 769608. 2.33994
\(162\) 70007.0 0.209582
\(163\) −275831. −0.813158 −0.406579 0.913616i \(-0.633278\pi\)
−0.406579 + 0.913616i \(0.633278\pi\)
\(164\) 131414. 0.381533
\(165\) −234328. −0.670061
\(166\) −312002. −0.878794
\(167\) 546461. 1.51624 0.758120 0.652115i \(-0.226118\pi\)
0.758120 + 0.652115i \(0.226118\pi\)
\(168\) −893739. −2.44308
\(169\) 595099. 1.60277
\(170\) −682072. −1.81012
\(171\) −24510.9 −0.0641015
\(172\) −402473. −1.03733
\(173\) −563301. −1.43095 −0.715476 0.698637i \(-0.753790\pi\)
−0.715476 + 0.698637i \(0.753790\pi\)
\(174\) 809217. 2.02625
\(175\) 117605. 0.290290
\(176\) −1.29871e6 −3.16031
\(177\) −31329.0 −0.0751646
\(178\) 255193. 0.603696
\(179\) 603180. 1.40707 0.703533 0.710663i \(-0.251605\pi\)
0.703533 + 0.710663i \(0.251605\pi\)
\(180\) −406275. −0.934628
\(181\) 158219. 0.358973 0.179487 0.983760i \(-0.442556\pi\)
0.179487 + 0.983760i \(0.442556\pi\)
\(182\) 1.95820e6 4.38207
\(183\) 282630. 0.623865
\(184\) 2.19290e6 4.77501
\(185\) −630516. −1.35446
\(186\) −965110. −2.04548
\(187\) −443235. −0.926895
\(188\) −1.96828e6 −4.06155
\(189\) −136093. −0.277129
\(190\) 197855. 0.397616
\(191\) −512314. −1.01614 −0.508070 0.861316i \(-0.669641\pi\)
−0.508070 + 0.861316i \(0.669641\pi\)
\(192\) −617041. −1.20798
\(193\) −779315. −1.50598 −0.752991 0.658030i \(-0.771390\pi\)
−0.752991 + 0.658030i \(0.771390\pi\)
\(194\) 337749. 0.644302
\(195\) 542153. 1.02102
\(196\) 1.47697e6 2.74620
\(197\) −375338. −0.689059 −0.344530 0.938775i \(-0.611962\pi\)
−0.344530 + 0.938775i \(0.611962\pi\)
\(198\) −367227. −0.665690
\(199\) 373163. 0.667984 0.333992 0.942576i \(-0.391604\pi\)
0.333992 + 0.942576i \(0.391604\pi\)
\(200\) 335101. 0.592381
\(201\) 550323. 0.960788
\(202\) −960694. −1.65656
\(203\) −1.57312e6 −2.67930
\(204\) −768477. −1.29287
\(205\) −98381.4 −0.163504
\(206\) 1.05357e6 1.72979
\(207\) 333922. 0.541650
\(208\) 3.00476e6 4.81560
\(209\) 128574. 0.203604
\(210\) 1.09857e6 1.71901
\(211\) 968282. 1.49725 0.748627 0.662991i \(-0.230713\pi\)
0.748627 + 0.662991i \(0.230713\pi\)
\(212\) −561230. −0.857633
\(213\) 67094.1 0.101329
\(214\) −1.05238e6 −1.57086
\(215\) 301306. 0.444541
\(216\) −387781. −0.565525
\(217\) 1.87617e6 2.70473
\(218\) 609068. 0.868009
\(219\) 38921.6 0.0548379
\(220\) 2.13115e6 2.96864
\(221\) 1.02549e6 1.41238
\(222\) −988113. −1.34563
\(223\) −423194. −0.569872 −0.284936 0.958547i \(-0.591972\pi\)
−0.284936 + 0.958547i \(0.591972\pi\)
\(224\) 2.91081e6 3.87608
\(225\) 51027.3 0.0671964
\(226\) 755939. 0.984500
\(227\) 185430. 0.238845 0.119423 0.992844i \(-0.461896\pi\)
0.119423 + 0.992844i \(0.461896\pi\)
\(228\) 222920. 0.283995
\(229\) 418097. 0.526851 0.263426 0.964680i \(-0.415148\pi\)
0.263426 + 0.964680i \(0.415148\pi\)
\(230\) −2.69547e6 −3.35981
\(231\) 713889. 0.880239
\(232\) −4.48239e6 −5.46751
\(233\) 137239. 0.165611 0.0828055 0.996566i \(-0.473612\pi\)
0.0828055 + 0.996566i \(0.473612\pi\)
\(234\) 849636. 1.01436
\(235\) 1.47352e6 1.74056
\(236\) 284929. 0.333009
\(237\) 276085. 0.319280
\(238\) 2.07796e6 2.37791
\(239\) −207578. −0.235064 −0.117532 0.993069i \(-0.537498\pi\)
−0.117532 + 0.993069i \(0.537498\pi\)
\(240\) 1.68569e6 1.88908
\(241\) −507933. −0.563331 −0.281666 0.959513i \(-0.590887\pi\)
−0.281666 + 0.959513i \(0.590887\pi\)
\(242\) 207877. 0.228176
\(243\) −59049.0 −0.0641500
\(244\) −2.57044e6 −2.76397
\(245\) −1.10572e6 −1.17687
\(246\) −154178. −0.162437
\(247\) −297475. −0.310247
\(248\) 5.34591e6 5.51941
\(249\) 263165. 0.268986
\(250\) 1.63136e6 1.65082
\(251\) −58645.3 −0.0587556 −0.0293778 0.999568i \(-0.509353\pi\)
−0.0293778 + 0.999568i \(0.509353\pi\)
\(252\) 1.23773e6 1.22779
\(253\) −1.75161e6 −1.72043
\(254\) −3.70997e6 −3.60816
\(255\) 575310. 0.554053
\(256\) 287983. 0.274642
\(257\) 1586.01 0.00149787 0.000748935 1.00000i \(-0.499762\pi\)
0.000748935 1.00000i \(0.499762\pi\)
\(258\) 472192. 0.441641
\(259\) 1.92089e6 1.77932
\(260\) −4.93074e6 −4.52354
\(261\) −682553. −0.620205
\(262\) 1.14621e6 1.03160
\(263\) −1.39407e6 −1.24278 −0.621391 0.783501i \(-0.713432\pi\)
−0.621391 + 0.783501i \(0.713432\pi\)
\(264\) 2.03413e6 1.79626
\(265\) 420157. 0.367534
\(266\) −602774. −0.522337
\(267\) −215248. −0.184783
\(268\) −5.00504e6 −4.25667
\(269\) 749950. 0.631904 0.315952 0.948775i \(-0.397676\pi\)
0.315952 + 0.948775i \(0.397676\pi\)
\(270\) 476653. 0.397917
\(271\) 66861.0 0.0553031 0.0276516 0.999618i \(-0.491197\pi\)
0.0276516 + 0.999618i \(0.491197\pi\)
\(272\) 3.18852e6 2.61316
\(273\) −1.65169e6 −1.34129
\(274\) 2.04096e6 1.64232
\(275\) −267668. −0.213434
\(276\) −3.03693e6 −2.39973
\(277\) −398436. −0.312003 −0.156002 0.987757i \(-0.549860\pi\)
−0.156002 + 0.987757i \(0.549860\pi\)
\(278\) 248649. 0.192963
\(279\) 814044. 0.626091
\(280\) −6.08515e6 −4.63849
\(281\) 406894. 0.307408 0.153704 0.988117i \(-0.450880\pi\)
0.153704 + 0.988117i \(0.450880\pi\)
\(282\) 2.30924e6 1.72920
\(283\) 2.31675e6 1.71954 0.859772 0.510678i \(-0.170605\pi\)
0.859772 + 0.510678i \(0.170605\pi\)
\(284\) −610203. −0.448930
\(285\) −166886. −0.121705
\(286\) −4.45683e6 −3.22189
\(287\) 299723. 0.214790
\(288\) 1.26296e6 0.897237
\(289\) −331649. −0.233579
\(290\) 5.50967e6 3.84707
\(291\) −284882. −0.197212
\(292\) −353981. −0.242954
\(293\) 1.71243e6 1.16531 0.582657 0.812718i \(-0.302013\pi\)
0.582657 + 0.812718i \(0.302013\pi\)
\(294\) −1.73283e6 −1.16919
\(295\) −213308. −0.142709
\(296\) 5.47333e6 3.63097
\(297\) 309746. 0.203758
\(298\) −178160. −0.116217
\(299\) 4.05263e6 2.62155
\(300\) −464079. −0.297707
\(301\) −917940. −0.583980
\(302\) −1.50850e6 −0.951763
\(303\) 810319. 0.507048
\(304\) −924924. −0.574014
\(305\) 1.92433e6 1.18448
\(306\) 901597. 0.550439
\(307\) 518531. 0.313999 0.157000 0.987599i \(-0.449818\pi\)
0.157000 + 0.987599i \(0.449818\pi\)
\(308\) −6.49262e6 −3.89981
\(309\) −888654. −0.529464
\(310\) −6.57109e6 −3.88359
\(311\) −50432.7 −0.0295673 −0.0147836 0.999891i \(-0.504706\pi\)
−0.0147836 + 0.999891i \(0.504706\pi\)
\(312\) −4.70628e6 −2.73710
\(313\) 2.84888e6 1.64366 0.821832 0.569730i \(-0.192952\pi\)
0.821832 + 0.569730i \(0.192952\pi\)
\(314\) −73105.2 −0.0418432
\(315\) −926612. −0.526164
\(316\) −2.51092e6 −1.41454
\(317\) 1.74502e6 0.975331 0.487666 0.873031i \(-0.337849\pi\)
0.487666 + 0.873031i \(0.337849\pi\)
\(318\) 658450. 0.365136
\(319\) 3.58038e6 1.96994
\(320\) −4.20121e6 −2.29351
\(321\) 887655. 0.480819
\(322\) 8.21185e6 4.41368
\(323\) −315667. −0.168354
\(324\) 537035. 0.284210
\(325\) 619290. 0.325226
\(326\) −2.94317e6 −1.53381
\(327\) −513732. −0.265685
\(328\) 854021. 0.438313
\(329\) −4.48915e6 −2.28652
\(330\) −2.50032e6 −1.26389
\(331\) −515970. −0.258854 −0.129427 0.991589i \(-0.541314\pi\)
−0.129427 + 0.991589i \(0.541314\pi\)
\(332\) −2.39341e6 −1.19172
\(333\) 833447. 0.411877
\(334\) 5.83083e6 2.85999
\(335\) 3.74695e6 1.82417
\(336\) −5.13552e6 −2.48163
\(337\) −2.01691e6 −0.967414 −0.483707 0.875230i \(-0.660710\pi\)
−0.483707 + 0.875230i \(0.660710\pi\)
\(338\) 6.34981e6 3.02321
\(339\) −637614. −0.301341
\(340\) −5.23228e6 −2.45467
\(341\) −4.27013e6 −1.98864
\(342\) −261535. −0.120911
\(343\) 230992. 0.106014
\(344\) −2.61555e6 −1.19170
\(345\) 2.27355e6 1.02839
\(346\) −6.01052e6 −2.69912
\(347\) 1.32354e6 0.590084 0.295042 0.955484i \(-0.404666\pi\)
0.295042 + 0.955484i \(0.404666\pi\)
\(348\) 6.20763e6 2.74775
\(349\) −1.20078e6 −0.527714 −0.263857 0.964562i \(-0.584995\pi\)
−0.263857 + 0.964562i \(0.584995\pi\)
\(350\) 1.25487e6 0.547556
\(351\) −716645. −0.310482
\(352\) −6.62494e6 −2.84987
\(353\) −2.56612e6 −1.09607 −0.548037 0.836454i \(-0.684625\pi\)
−0.548037 + 0.836454i \(0.684625\pi\)
\(354\) −334286. −0.141778
\(355\) 456820. 0.192386
\(356\) 1.95762e6 0.818661
\(357\) −1.75270e6 −0.727843
\(358\) 6.43603e6 2.65406
\(359\) 1.98664e6 0.813547 0.406773 0.913529i \(-0.366654\pi\)
0.406773 + 0.913529i \(0.366654\pi\)
\(360\) −2.64026e6 −1.07372
\(361\) −2.38453e6 −0.963019
\(362\) 1.68822e6 0.677109
\(363\) −175339. −0.0698413
\(364\) 1.50217e7 5.94244
\(365\) 265003. 0.104116
\(366\) 3.01571e6 1.17676
\(367\) −1.10023e6 −0.426400 −0.213200 0.977009i \(-0.568389\pi\)
−0.213200 + 0.977009i \(0.568389\pi\)
\(368\) 1.26006e7 4.85035
\(369\) 130045. 0.0497198
\(370\) −6.72771e6 −2.55484
\(371\) −1.28003e6 −0.482818
\(372\) −7.40351e6 −2.77383
\(373\) −2.20944e6 −0.822263 −0.411131 0.911576i \(-0.634866\pi\)
−0.411131 + 0.911576i \(0.634866\pi\)
\(374\) −4.72940e6 −1.74834
\(375\) −1.37601e6 −0.505294
\(376\) −1.27913e7 −4.66599
\(377\) −8.28377e6 −3.00175
\(378\) −1.45214e6 −0.522732
\(379\) −560573. −0.200463 −0.100231 0.994964i \(-0.531958\pi\)
−0.100231 + 0.994964i \(0.531958\pi\)
\(380\) 1.51778e6 0.539200
\(381\) 3.12926e6 1.10441
\(382\) −5.46648e6 −1.91668
\(383\) −3.40354e6 −1.18559 −0.592795 0.805354i \(-0.701975\pi\)
−0.592795 + 0.805354i \(0.701975\pi\)
\(384\) −2.09342e6 −0.724484
\(385\) 4.86061e6 1.67124
\(386\) −8.31543e6 −2.84064
\(387\) −398281. −0.135180
\(388\) 2.59092e6 0.873726
\(389\) −3.10419e6 −1.04010 −0.520048 0.854137i \(-0.674086\pi\)
−0.520048 + 0.854137i \(0.674086\pi\)
\(390\) 5.78487e6 1.92589
\(391\) 4.30047e6 1.42257
\(392\) 9.59842e6 3.15489
\(393\) −966798. −0.315758
\(394\) −4.00492e6 −1.29973
\(395\) 1.87976e6 0.606193
\(396\) −2.81706e6 −0.902730
\(397\) −3.18351e6 −1.01375 −0.506875 0.862020i \(-0.669199\pi\)
−0.506875 + 0.862020i \(0.669199\pi\)
\(398\) 3.98171e6 1.25998
\(399\) 508424. 0.159880
\(400\) 1.92553e6 0.601728
\(401\) 311471. 0.0967289 0.0483644 0.998830i \(-0.484599\pi\)
0.0483644 + 0.998830i \(0.484599\pi\)
\(402\) 5.87204e6 1.81227
\(403\) 9.87961e6 3.03024
\(404\) −7.36963e6 −2.24643
\(405\) −402044. −0.121797
\(406\) −1.67854e7 −5.05379
\(407\) −4.37191e6 −1.30823
\(408\) −4.99410e6 −1.48527
\(409\) −559438. −0.165365 −0.0826825 0.996576i \(-0.526349\pi\)
−0.0826825 + 0.996576i \(0.526349\pi\)
\(410\) −1.04975e6 −0.308407
\(411\) −1.72149e6 −0.502691
\(412\) 8.08206e6 2.34574
\(413\) 649851. 0.187473
\(414\) 3.56300e6 1.02168
\(415\) 1.79180e6 0.510703
\(416\) 1.53278e7 4.34257
\(417\) −209728. −0.0590632
\(418\) 1.37190e6 0.384046
\(419\) 142773. 0.0397294 0.0198647 0.999803i \(-0.493676\pi\)
0.0198647 + 0.999803i \(0.493676\pi\)
\(420\) 8.42728e6 2.33112
\(421\) −3.82502e6 −1.05179 −0.525895 0.850550i \(-0.676270\pi\)
−0.525895 + 0.850550i \(0.676270\pi\)
\(422\) 1.03317e7 2.82418
\(423\) −1.94778e6 −0.529284
\(424\) −3.64727e6 −0.985265
\(425\) 657163. 0.176482
\(426\) 715906. 0.191131
\(427\) −5.86254e6 −1.55602
\(428\) −8.07298e6 −2.13022
\(429\) 3.75922e6 0.986176
\(430\) 3.21499e6 0.838509
\(431\) −4.27196e6 −1.10773 −0.553865 0.832606i \(-0.686848\pi\)
−0.553865 + 0.832606i \(0.686848\pi\)
\(432\) −2.22823e6 −0.574448
\(433\) −3.40429e6 −0.872583 −0.436292 0.899805i \(-0.643708\pi\)
−0.436292 + 0.899805i \(0.643708\pi\)
\(434\) 2.00191e7 5.10176
\(435\) −4.64726e6 −1.17753
\(436\) 4.67225e6 1.17709
\(437\) −1.24748e6 −0.312486
\(438\) 415300. 0.103437
\(439\) 6.97376e6 1.72705 0.863527 0.504303i \(-0.168250\pi\)
0.863527 + 0.504303i \(0.168250\pi\)
\(440\) 1.38497e7 3.41042
\(441\) 1.46159e6 0.357873
\(442\) 1.09422e7 2.66409
\(443\) 4.62500e6 1.11970 0.559851 0.828593i \(-0.310858\pi\)
0.559851 + 0.828593i \(0.310858\pi\)
\(444\) −7.57997e6 −1.82478
\(445\) −1.46555e6 −0.350833
\(446\) −4.51555e6 −1.07491
\(447\) 150273. 0.0355724
\(448\) 1.27992e7 3.01291
\(449\) −4.22998e6 −0.990199 −0.495100 0.868836i \(-0.664868\pi\)
−0.495100 + 0.868836i \(0.664868\pi\)
\(450\) 544470. 0.126748
\(451\) −682163. −0.157924
\(452\) 5.79893e6 1.33506
\(453\) 1.27238e6 0.291321
\(454\) 1.97857e6 0.450519
\(455\) −1.12458e7 −2.54660
\(456\) 1.44869e6 0.326259
\(457\) 8.46533e6 1.89607 0.948033 0.318173i \(-0.103069\pi\)
0.948033 + 0.318173i \(0.103069\pi\)
\(458\) 4.46116e6 0.993767
\(459\) −760473. −0.168481
\(460\) −2.06774e7 −4.55617
\(461\) −5.61651e6 −1.23088 −0.615438 0.788185i \(-0.711021\pi\)
−0.615438 + 0.788185i \(0.711021\pi\)
\(462\) 7.61732e6 1.66034
\(463\) 692123. 0.150048 0.0750241 0.997182i \(-0.476097\pi\)
0.0750241 + 0.997182i \(0.476097\pi\)
\(464\) −2.57563e7 −5.55378
\(465\) 5.54254e6 1.18871
\(466\) 1.46437e6 0.312382
\(467\) 6.20510e6 1.31661 0.658304 0.752752i \(-0.271274\pi\)
0.658304 + 0.752752i \(0.271274\pi\)
\(468\) 6.51769e6 1.37556
\(469\) −1.14152e7 −2.39636
\(470\) 1.57228e7 3.28310
\(471\) 61662.3 0.0128076
\(472\) 1.85167e6 0.382567
\(473\) 2.08921e6 0.429368
\(474\) 2.94588e6 0.602238
\(475\) −190630. −0.0387665
\(476\) 1.59404e7 3.22463
\(477\) −555385. −0.111763
\(478\) −2.21489e6 −0.443387
\(479\) −4.54829e6 −0.905752 −0.452876 0.891573i \(-0.649602\pi\)
−0.452876 + 0.891573i \(0.649602\pi\)
\(480\) 8.59902e6 1.70352
\(481\) 1.01151e7 1.99346
\(482\) −5.41973e6 −1.06258
\(483\) −6.92647e6 −1.35097
\(484\) 1.59466e6 0.309425
\(485\) −1.93966e6 −0.374430
\(486\) −630063. −0.121002
\(487\) −3.94237e6 −0.753242 −0.376621 0.926367i \(-0.622914\pi\)
−0.376621 + 0.926367i \(0.622914\pi\)
\(488\) −1.67045e7 −3.17530
\(489\) 2.48248e6 0.469477
\(490\) −1.17982e7 −2.21986
\(491\) 557676. 0.104395 0.0521973 0.998637i \(-0.483378\pi\)
0.0521973 + 0.998637i \(0.483378\pi\)
\(492\) −1.18273e6 −0.220278
\(493\) −8.79037e6 −1.62888
\(494\) −3.17411e6 −0.585200
\(495\) 2.10895e6 0.386860
\(496\) 3.07182e7 5.60649
\(497\) −1.39172e6 −0.252732
\(498\) 2.80802e6 0.507372
\(499\) −4.02783e6 −0.724136 −0.362068 0.932152i \(-0.617929\pi\)
−0.362068 + 0.932152i \(0.617929\pi\)
\(500\) 1.25144e7 2.23865
\(501\) −4.91815e6 −0.875402
\(502\) −625755. −0.110827
\(503\) 9.17895e6 1.61761 0.808803 0.588079i \(-0.200116\pi\)
0.808803 + 0.588079i \(0.200116\pi\)
\(504\) 8.04365e6 1.41051
\(505\) 5.51717e6 0.962694
\(506\) −1.86900e7 −3.24514
\(507\) −5.35589e6 −0.925362
\(508\) −2.84597e7 −4.89296
\(509\) −8.23834e6 −1.40944 −0.704718 0.709487i \(-0.748927\pi\)
−0.704718 + 0.709487i \(0.748927\pi\)
\(510\) 6.13865e6 1.04508
\(511\) −807343. −0.136775
\(512\) −4.37045e6 −0.736803
\(513\) 220598. 0.0370090
\(514\) 16923.0 0.00282534
\(515\) −6.05053e6 −1.00525
\(516\) 3.62226e6 0.598901
\(517\) 1.02172e7 1.68115
\(518\) 2.04962e7 3.35621
\(519\) 5.06971e6 0.826161
\(520\) −3.20434e7 −5.19673
\(521\) −4.78219e6 −0.771850 −0.385925 0.922530i \(-0.626118\pi\)
−0.385925 + 0.922530i \(0.626118\pi\)
\(522\) −7.28296e6 −1.16985
\(523\) −1.22841e7 −1.96376 −0.981881 0.189501i \(-0.939313\pi\)
−0.981881 + 0.189501i \(0.939313\pi\)
\(524\) 8.79277e6 1.39893
\(525\) −1.05845e6 −0.167599
\(526\) −1.48749e7 −2.34418
\(527\) 1.04838e7 1.64434
\(528\) 1.16884e7 1.82460
\(529\) 1.05586e7 1.64046
\(530\) 4.48315e6 0.693256
\(531\) 281961. 0.0433963
\(532\) −4.62398e6 −0.708331
\(533\) 1.57829e6 0.240640
\(534\) −2.29674e6 −0.348544
\(535\) 6.04373e6 0.912894
\(536\) −3.25262e7 −4.89014
\(537\) −5.42862e6 −0.812370
\(538\) 8.00209e6 1.19192
\(539\) −7.66689e6 −1.13670
\(540\) 3.65648e6 0.539608
\(541\) −7.08864e6 −1.04128 −0.520642 0.853775i \(-0.674307\pi\)
−0.520642 + 0.853775i \(0.674307\pi\)
\(542\) 713418. 0.104315
\(543\) −1.42397e6 −0.207253
\(544\) 1.62652e7 2.35647
\(545\) −3.49782e6 −0.504436
\(546\) −1.76238e7 −2.52999
\(547\) 1.55919e6 0.222807 0.111404 0.993775i \(-0.464465\pi\)
0.111404 + 0.993775i \(0.464465\pi\)
\(548\) 1.56565e7 2.22712
\(549\) −2.54367e6 −0.360189
\(550\) −2.85606e6 −0.402588
\(551\) 2.54991e6 0.357805
\(552\) −1.97361e7 −2.75685
\(553\) −5.72678e6 −0.796338
\(554\) −4.25138e6 −0.588512
\(555\) 5.67464e6 0.781999
\(556\) 1.90742e6 0.261674
\(557\) 2.40850e6 0.328934 0.164467 0.986383i \(-0.447410\pi\)
0.164467 + 0.986383i \(0.447410\pi\)
\(558\) 8.68599e6 1.18096
\(559\) −4.83372e6 −0.654262
\(560\) −3.49659e7 −4.71167
\(561\) 3.98912e6 0.535143
\(562\) 4.34163e6 0.579845
\(563\) −1.96787e6 −0.261652 −0.130826 0.991405i \(-0.541763\pi\)
−0.130826 + 0.991405i \(0.541763\pi\)
\(564\) 1.77145e7 2.34494
\(565\) −4.34129e6 −0.572133
\(566\) 2.47201e7 3.24347
\(567\) 1.22484e6 0.160001
\(568\) −3.96552e6 −0.515739
\(569\) 5.44328e6 0.704823 0.352411 0.935845i \(-0.385362\pi\)
0.352411 + 0.935845i \(0.385362\pi\)
\(570\) −1.78070e6 −0.229564
\(571\) 1.26908e7 1.62892 0.814461 0.580218i \(-0.197033\pi\)
0.814461 + 0.580218i \(0.197033\pi\)
\(572\) −3.41891e7 −4.36915
\(573\) 4.61083e6 0.586668
\(574\) 3.19809e6 0.405146
\(575\) 2.59703e6 0.327573
\(576\) 5.55337e6 0.697430
\(577\) 4.17811e6 0.522445 0.261222 0.965279i \(-0.415874\pi\)
0.261222 + 0.965279i \(0.415874\pi\)
\(578\) −3.53875e6 −0.440585
\(579\) 7.01384e6 0.869480
\(580\) 4.22655e7 5.21695
\(581\) −5.45878e6 −0.670896
\(582\) −3.03974e6 −0.371988
\(583\) 2.91331e6 0.354990
\(584\) −2.30042e6 −0.279110
\(585\) −4.87938e6 −0.589488
\(586\) 1.82719e7 2.19806
\(587\) 1.07114e6 0.128307 0.0641537 0.997940i \(-0.479565\pi\)
0.0641537 + 0.997940i \(0.479565\pi\)
\(588\) −1.32928e7 −1.58552
\(589\) −3.04114e6 −0.361201
\(590\) −2.27603e6 −0.269183
\(591\) 3.37804e6 0.397828
\(592\) 3.14503e7 3.68826
\(593\) −7.74188e6 −0.904085 −0.452043 0.891996i \(-0.649305\pi\)
−0.452043 + 0.891996i \(0.649305\pi\)
\(594\) 3.30504e6 0.384336
\(595\) −1.19335e7 −1.38190
\(596\) −1.36670e6 −0.157600
\(597\) −3.35847e6 −0.385661
\(598\) 4.32422e7 4.94487
\(599\) 3.00377e6 0.342058 0.171029 0.985266i \(-0.445291\pi\)
0.171029 + 0.985266i \(0.445291\pi\)
\(600\) −3.01591e6 −0.342011
\(601\) 5.32845e6 0.601748 0.300874 0.953664i \(-0.402722\pi\)
0.300874 + 0.953664i \(0.402722\pi\)
\(602\) −9.79458e6 −1.10153
\(603\) −4.95291e6 −0.554711
\(604\) −1.15720e7 −1.29067
\(605\) −1.19382e6 −0.132602
\(606\) 8.64624e6 0.956414
\(607\) 1.59774e7 1.76009 0.880046 0.474888i \(-0.157512\pi\)
0.880046 + 0.474888i \(0.157512\pi\)
\(608\) −4.71821e6 −0.517629
\(609\) 1.41581e7 1.54689
\(610\) 2.05329e7 2.23422
\(611\) −2.36391e7 −2.56170
\(612\) 6.91629e6 0.746440
\(613\) 8.04897e6 0.865145 0.432573 0.901599i \(-0.357606\pi\)
0.432573 + 0.901599i \(0.357606\pi\)
\(614\) 5.53281e6 0.592277
\(615\) 885433. 0.0943991
\(616\) −4.21936e7 −4.48018
\(617\) −1.00376e7 −1.06150 −0.530748 0.847529i \(-0.678089\pi\)
−0.530748 + 0.847529i \(0.678089\pi\)
\(618\) −9.48209e6 −0.998695
\(619\) −3.31323e6 −0.347556 −0.173778 0.984785i \(-0.555598\pi\)
−0.173778 + 0.984785i \(0.555598\pi\)
\(620\) −5.04078e7 −5.26646
\(621\) −3.00530e6 −0.312722
\(622\) −538126. −0.0557709
\(623\) 4.46485e6 0.460879
\(624\) −2.70428e7 −2.78029
\(625\) −1.13374e7 −1.16095
\(626\) 3.03980e7 3.10034
\(627\) −1.15716e6 −0.117551
\(628\) −560802. −0.0567427
\(629\) 1.07337e7 1.08174
\(630\) −9.88710e6 −0.992471
\(631\) 1.30632e7 1.30610 0.653048 0.757316i \(-0.273490\pi\)
0.653048 + 0.757316i \(0.273490\pi\)
\(632\) −1.63177e7 −1.62505
\(633\) −8.71454e6 −0.864441
\(634\) 1.86197e7 1.83971
\(635\) 2.13060e7 2.09685
\(636\) 5.05107e6 0.495155
\(637\) 1.77385e7 1.73208
\(638\) 3.82033e7 3.71577
\(639\) −603847. −0.0585025
\(640\) −1.42534e7 −1.37552
\(641\) −3.24010e6 −0.311468 −0.155734 0.987799i \(-0.549774\pi\)
−0.155734 + 0.987799i \(0.549774\pi\)
\(642\) 9.47143e6 0.906939
\(643\) −1.43038e7 −1.36435 −0.682174 0.731189i \(-0.738966\pi\)
−0.682174 + 0.731189i \(0.738966\pi\)
\(644\) 6.29944e7 5.98531
\(645\) −2.71175e6 −0.256656
\(646\) −3.36823e6 −0.317556
\(647\) −1.69491e7 −1.59179 −0.795896 0.605434i \(-0.793000\pi\)
−0.795896 + 0.605434i \(0.793000\pi\)
\(648\) 3.49003e6 0.326506
\(649\) −1.47905e6 −0.137839
\(650\) 6.60793e6 0.613454
\(651\) −1.68855e7 −1.56157
\(652\) −2.25775e7 −2.07997
\(653\) 4.66123e6 0.427777 0.213889 0.976858i \(-0.431387\pi\)
0.213889 + 0.976858i \(0.431387\pi\)
\(654\) −5.48161e6 −0.501145
\(655\) −6.58258e6 −0.599506
\(656\) 4.90730e6 0.445228
\(657\) −350295. −0.0316607
\(658\) −4.79000e7 −4.31291
\(659\) 1.84395e7 1.65400 0.827001 0.562201i \(-0.190045\pi\)
0.827001 + 0.562201i \(0.190045\pi\)
\(660\) −1.91803e7 −1.71394
\(661\) −1.59041e6 −0.141581 −0.0707905 0.997491i \(-0.522552\pi\)
−0.0707905 + 0.997491i \(0.522552\pi\)
\(662\) −5.50549e6 −0.488260
\(663\) −9.22944e6 −0.815439
\(664\) −1.55541e7 −1.36907
\(665\) 3.46168e6 0.303551
\(666\) 8.89302e6 0.776897
\(667\) −3.47385e7 −3.02341
\(668\) 4.47292e7 3.87838
\(669\) 3.80875e6 0.329016
\(670\) 3.99806e7 3.44082
\(671\) 1.33430e7 1.14406
\(672\) −2.61973e7 −2.23786
\(673\) −1.37250e7 −1.16809 −0.584044 0.811722i \(-0.698530\pi\)
−0.584044 + 0.811722i \(0.698530\pi\)
\(674\) −2.15208e7 −1.82477
\(675\) −459245. −0.0387959
\(676\) 4.87104e7 4.09972
\(677\) 1.55559e7 1.30444 0.652219 0.758031i \(-0.273838\pi\)
0.652219 + 0.758031i \(0.273838\pi\)
\(678\) −6.80345e6 −0.568401
\(679\) 5.90925e6 0.491878
\(680\) −3.40030e7 −2.81998
\(681\) −1.66887e6 −0.137897
\(682\) −4.55630e7 −3.75104
\(683\) −1.03021e7 −0.845033 −0.422517 0.906355i \(-0.638853\pi\)
−0.422517 + 0.906355i \(0.638853\pi\)
\(684\) −2.00628e6 −0.163965
\(685\) −1.17210e7 −0.954420
\(686\) 2.46472e6 0.199967
\(687\) −3.76287e6 −0.304178
\(688\) −1.50292e7 −1.21050
\(689\) −6.74040e6 −0.540926
\(690\) 2.42592e7 1.93979
\(691\) 5.02324e6 0.400211 0.200106 0.979774i \(-0.435871\pi\)
0.200106 + 0.979774i \(0.435871\pi\)
\(692\) −4.61076e7 −3.66022
\(693\) −6.42500e6 −0.508206
\(694\) 1.41224e7 1.11304
\(695\) −1.42797e6 −0.112139
\(696\) 4.03415e7 3.15667
\(697\) 1.67481e6 0.130582
\(698\) −1.28125e7 −0.995393
\(699\) −1.23515e6 −0.0956155
\(700\) 9.62630e6 0.742530
\(701\) −1.22395e7 −0.940739 −0.470370 0.882469i \(-0.655879\pi\)
−0.470370 + 0.882469i \(0.655879\pi\)
\(702\) −7.64673e6 −0.585643
\(703\) −3.11363e6 −0.237617
\(704\) −2.91306e7 −2.21523
\(705\) −1.32617e7 −1.00491
\(706\) −2.73809e7 −2.06745
\(707\) −1.68083e7 −1.26466
\(708\) −2.56436e6 −0.192263
\(709\) 3.66774e6 0.274021 0.137010 0.990570i \(-0.456251\pi\)
0.137010 + 0.990570i \(0.456251\pi\)
\(710\) 4.87435e6 0.362886
\(711\) −2.48477e6 −0.184337
\(712\) 1.27220e7 0.940493
\(713\) 4.14307e7 3.05210
\(714\) −1.87016e7 −1.37288
\(715\) 2.55952e7 1.87238
\(716\) 4.93718e7 3.59912
\(717\) 1.86820e6 0.135714
\(718\) 2.11978e7 1.53454
\(719\) 1.96481e7 1.41742 0.708709 0.705501i \(-0.249278\pi\)
0.708709 + 0.705501i \(0.249278\pi\)
\(720\) −1.51712e7 −1.09066
\(721\) 1.84332e7 1.32057
\(722\) −2.54433e7 −1.81648
\(723\) 4.57140e6 0.325240
\(724\) 1.29506e7 0.918215
\(725\) −5.30846e6 −0.375080
\(726\) −1.87090e6 −0.131737
\(727\) 1.64307e6 0.115297 0.0576486 0.998337i \(-0.481640\pi\)
0.0576486 + 0.998337i \(0.481640\pi\)
\(728\) 9.76214e7 6.82679
\(729\) 531441. 0.0370370
\(730\) 2.82763e6 0.196388
\(731\) −5.12933e6 −0.355032
\(732\) 2.31340e7 1.59578
\(733\) −4.63590e6 −0.318694 −0.159347 0.987223i \(-0.550939\pi\)
−0.159347 + 0.987223i \(0.550939\pi\)
\(734\) −1.17396e7 −0.804291
\(735\) 9.95145e6 0.679467
\(736\) 6.42781e7 4.37390
\(737\) 2.59809e7 1.76191
\(738\) 1.38761e6 0.0937833
\(739\) 2.52043e7 1.69771 0.848855 0.528625i \(-0.177292\pi\)
0.848855 + 0.528625i \(0.177292\pi\)
\(740\) −5.16093e7 −3.46457
\(741\) 2.67727e6 0.179121
\(742\) −1.36581e7 −0.910710
\(743\) 2.59323e7 1.72333 0.861666 0.507475i \(-0.169421\pi\)
0.861666 + 0.507475i \(0.169421\pi\)
\(744\) −4.81132e7 −3.18663
\(745\) 1.02316e6 0.0675386
\(746\) −2.35751e7 −1.55098
\(747\) −2.36849e6 −0.155299
\(748\) −3.62799e7 −2.37090
\(749\) −1.84124e7 −1.19924
\(750\) −1.46823e7 −0.953104
\(751\) 2.11649e7 1.36936 0.684678 0.728846i \(-0.259943\pi\)
0.684678 + 0.728846i \(0.259943\pi\)
\(752\) −7.34999e7 −4.73961
\(753\) 527808. 0.0339225
\(754\) −8.83892e7 −5.66201
\(755\) 8.66319e6 0.553109
\(756\) −1.11396e7 −0.708867
\(757\) 1.49940e7 0.950992 0.475496 0.879718i \(-0.342269\pi\)
0.475496 + 0.879718i \(0.342269\pi\)
\(758\) −5.98140e6 −0.378120
\(759\) 1.57645e7 0.993291
\(760\) 9.86360e6 0.619443
\(761\) −2.32932e7 −1.45803 −0.729015 0.684497i \(-0.760022\pi\)
−0.729015 + 0.684497i \(0.760022\pi\)
\(762\) 3.33897e7 2.08317
\(763\) 1.06562e7 0.662663
\(764\) −4.19342e7 −2.59917
\(765\) −5.17779e6 −0.319883
\(766\) −3.63164e7 −2.23630
\(767\) 3.42201e6 0.210035
\(768\) −2.59185e6 −0.158565
\(769\) −5.03552e6 −0.307064 −0.153532 0.988144i \(-0.549065\pi\)
−0.153532 + 0.988144i \(0.549065\pi\)
\(770\) 5.18636e7 3.15236
\(771\) −14274.1 −0.000864795 0
\(772\) −6.37890e7 −3.85214
\(773\) −1.34665e7 −0.810599 −0.405300 0.914184i \(-0.632833\pi\)
−0.405300 + 0.914184i \(0.632833\pi\)
\(774\) −4.24973e6 −0.254981
\(775\) 6.33112e6 0.378640
\(776\) 1.68376e7 1.00375
\(777\) −1.72880e7 −1.02729
\(778\) −3.31222e7 −1.96187
\(779\) −485829. −0.0286840
\(780\) 4.43766e7 2.61167
\(781\) 3.16753e6 0.185820
\(782\) 4.58867e7 2.68331
\(783\) 6.14298e6 0.358075
\(784\) 5.51535e7 3.20467
\(785\) 419837. 0.0243168
\(786\) −1.03159e7 −0.595595
\(787\) 6.43085e6 0.370111 0.185055 0.982728i \(-0.440754\pi\)
0.185055 + 0.982728i \(0.440754\pi\)
\(788\) −3.07223e7 −1.76254
\(789\) 1.25466e7 0.717520
\(790\) 2.00574e7 1.14342
\(791\) 1.32259e7 0.751595
\(792\) −1.83072e7 −1.03707
\(793\) −3.08711e7 −1.74329
\(794\) −3.39686e7 −1.91217
\(795\) −3.78142e6 −0.212196
\(796\) 3.05444e7 1.70863
\(797\) 2.19054e7 1.22153 0.610765 0.791812i \(-0.290862\pi\)
0.610765 + 0.791812i \(0.290862\pi\)
\(798\) 5.42497e6 0.301571
\(799\) −2.50848e7 −1.39009
\(800\) 9.82247e6 0.542620
\(801\) 1.93723e6 0.106684
\(802\) 3.32344e6 0.182454
\(803\) 1.83750e6 0.100563
\(804\) 4.50453e7 2.45759
\(805\) −4.71599e7 −2.56497
\(806\) 1.05417e8 5.71575
\(807\) −6.74955e6 −0.364830
\(808\) −4.78930e7 −2.58074
\(809\) −3.19147e7 −1.71443 −0.857214 0.514961i \(-0.827806\pi\)
−0.857214 + 0.514961i \(0.827806\pi\)
\(810\) −4.28987e6 −0.229737
\(811\) 7.21928e6 0.385426 0.192713 0.981255i \(-0.438271\pi\)
0.192713 + 0.981255i \(0.438271\pi\)
\(812\) −1.28764e8 −6.85335
\(813\) −601749. −0.0319293
\(814\) −4.66490e7 −2.46764
\(815\) 1.69023e7 0.891359
\(816\) −2.86966e7 −1.50871
\(817\) 1.48792e6 0.0779872
\(818\) −5.96930e6 −0.311918
\(819\) 1.48652e7 0.774393
\(820\) −8.05277e6 −0.418225
\(821\) 1.80572e7 0.934958 0.467479 0.884004i \(-0.345162\pi\)
0.467479 + 0.884004i \(0.345162\pi\)
\(822\) −1.83686e7 −0.948194
\(823\) −4.54084e6 −0.233688 −0.116844 0.993150i \(-0.537278\pi\)
−0.116844 + 0.993150i \(0.537278\pi\)
\(824\) 5.25229e7 2.69483
\(825\) 2.40901e6 0.123226
\(826\) 6.93402e6 0.353619
\(827\) 2.72523e7 1.38561 0.692803 0.721127i \(-0.256376\pi\)
0.692803 + 0.721127i \(0.256376\pi\)
\(828\) 2.73323e7 1.38548
\(829\) 1.54051e7 0.778537 0.389269 0.921124i \(-0.372728\pi\)
0.389269 + 0.921124i \(0.372728\pi\)
\(830\) 1.91188e7 0.963308
\(831\) 3.58592e6 0.180135
\(832\) 6.73982e7 3.37552
\(833\) 1.88234e7 0.939906
\(834\) −2.23784e6 −0.111407
\(835\) −3.34859e7 −1.66206
\(836\) 1.05241e7 0.520797
\(837\) −7.32640e6 −0.361474
\(838\) 1.52341e6 0.0749390
\(839\) 1.50266e7 0.736979 0.368489 0.929632i \(-0.379875\pi\)
0.368489 + 0.929632i \(0.379875\pi\)
\(840\) 5.47664e7 2.67803
\(841\) 5.04961e7 2.46188
\(842\) −4.08137e7 −1.98393
\(843\) −3.66204e6 −0.177482
\(844\) 7.92564e7 3.82982
\(845\) −3.64663e7 −1.75691
\(846\) −2.07831e7 −0.998355
\(847\) 3.63702e6 0.174196
\(848\) −2.09576e7 −1.00081
\(849\) −2.08508e7 −0.992780
\(850\) 7.01205e6 0.332887
\(851\) 4.24182e7 2.00784
\(852\) 5.49182e6 0.259190
\(853\) 3.68547e7 1.73428 0.867142 0.498060i \(-0.165954\pi\)
0.867142 + 0.498060i \(0.165954\pi\)
\(854\) −6.25543e7 −2.93503
\(855\) 1.50197e6 0.0702662
\(856\) −5.24639e7 −2.44724
\(857\) −3.27506e7 −1.52324 −0.761618 0.648026i \(-0.775595\pi\)
−0.761618 + 0.648026i \(0.775595\pi\)
\(858\) 4.01115e7 1.86016
\(859\) 3.34950e7 1.54881 0.774403 0.632693i \(-0.218050\pi\)
0.774403 + 0.632693i \(0.218050\pi\)
\(860\) 2.46627e7 1.13709
\(861\) −2.69750e6 −0.124009
\(862\) −4.55826e7 −2.08944
\(863\) 1.12755e7 0.515358 0.257679 0.966231i \(-0.417042\pi\)
0.257679 + 0.966231i \(0.417042\pi\)
\(864\) −1.13666e7 −0.518020
\(865\) 3.45178e7 1.56857
\(866\) −3.63244e7 −1.64590
\(867\) 2.98484e6 0.134857
\(868\) 1.53569e8 6.91840
\(869\) 1.30340e7 0.585503
\(870\) −4.95870e7 −2.22111
\(871\) −6.01107e7 −2.68476
\(872\) 3.03636e7 1.35226
\(873\) 2.56394e6 0.113860
\(874\) −1.33108e7 −0.589422
\(875\) 2.85423e7 1.26029
\(876\) 3.18583e6 0.140269
\(877\) −3.74225e6 −0.164299 −0.0821493 0.996620i \(-0.526178\pi\)
−0.0821493 + 0.996620i \(0.526178\pi\)
\(878\) 7.44112e7 3.25763
\(879\) −1.54118e7 −0.672794
\(880\) 7.95818e7 3.46423
\(881\) 1.20043e7 0.521073 0.260537 0.965464i \(-0.416101\pi\)
0.260537 + 0.965464i \(0.416101\pi\)
\(882\) 1.55954e7 0.675034
\(883\) −1.38235e7 −0.596647 −0.298324 0.954465i \(-0.596427\pi\)
−0.298324 + 0.954465i \(0.596427\pi\)
\(884\) 8.39392e7 3.61272
\(885\) 1.91977e6 0.0823932
\(886\) 4.93495e7 2.11202
\(887\) 1.58414e7 0.676059 0.338030 0.941135i \(-0.390240\pi\)
0.338030 + 0.941135i \(0.390240\pi\)
\(888\) −4.92600e7 −2.09634
\(889\) −6.49095e7 −2.75457
\(890\) −1.56376e7 −0.661754
\(891\) −2.78772e6 −0.117640
\(892\) −3.46395e7 −1.45767
\(893\) 7.27659e6 0.305351
\(894\) 1.60344e6 0.0670980
\(895\) −3.69615e7 −1.54238
\(896\) 4.34234e7 1.80698
\(897\) −3.64736e7 −1.51355
\(898\) −4.51346e7 −1.86775
\(899\) −8.46865e7 −3.49474
\(900\) 4.17671e6 0.171881
\(901\) −7.15262e6 −0.293530
\(902\) −7.27880e6 −0.297881
\(903\) 8.26146e6 0.337161
\(904\) 3.76855e7 1.53374
\(905\) −9.69531e6 −0.393496
\(906\) 1.35765e7 0.549501
\(907\) −3.64791e7 −1.47240 −0.736200 0.676764i \(-0.763382\pi\)
−0.736200 + 0.676764i \(0.763382\pi\)
\(908\) 1.51780e7 0.610940
\(909\) −7.29287e6 −0.292745
\(910\) −1.19994e8 −4.80349
\(911\) −3.55848e7 −1.42059 −0.710294 0.703905i \(-0.751438\pi\)
−0.710294 + 0.703905i \(0.751438\pi\)
\(912\) 8.32432e6 0.331407
\(913\) 1.24241e7 0.493273
\(914\) 9.03265e7 3.57643
\(915\) −1.73190e7 −0.683863
\(916\) 3.42223e7 1.34763
\(917\) 2.00541e7 0.787553
\(918\) −8.11437e6 −0.317796
\(919\) 3.19136e7 1.24649 0.623243 0.782028i \(-0.285815\pi\)
0.623243 + 0.782028i \(0.285815\pi\)
\(920\) −1.34376e8 −5.23422
\(921\) −4.66678e6 −0.181288
\(922\) −5.99291e7 −2.32172
\(923\) −7.32856e6 −0.283149
\(924\) 5.84336e7 2.25156
\(925\) 6.48202e6 0.249090
\(926\) 7.38507e6 0.283026
\(927\) 7.99788e6 0.305686
\(928\) −1.31388e8 −5.00824
\(929\) −3.37958e6 −0.128477 −0.0642383 0.997935i \(-0.520462\pi\)
−0.0642383 + 0.997935i \(0.520462\pi\)
\(930\) 5.91398e7 2.24219
\(931\) −5.46028e6 −0.206462
\(932\) 1.12334e7 0.423615
\(933\) 453894. 0.0170707
\(934\) 6.62095e7 2.48344
\(935\) 2.71605e7 1.01603
\(936\) 4.23565e7 1.58027
\(937\) 3.48856e7 1.29807 0.649034 0.760760i \(-0.275173\pi\)
0.649034 + 0.760760i \(0.275173\pi\)
\(938\) −1.21802e8 −4.52011
\(939\) −2.56399e7 −0.948970
\(940\) 1.20612e8 4.45215
\(941\) −4.74186e7 −1.74572 −0.872859 0.487972i \(-0.837737\pi\)
−0.872859 + 0.487972i \(0.837737\pi\)
\(942\) 657947. 0.0241582
\(943\) 6.61865e6 0.242376
\(944\) 1.06399e7 0.388603
\(945\) 8.33950e6 0.303781
\(946\) 2.22923e7 0.809891
\(947\) 3.31556e7 1.20139 0.600693 0.799480i \(-0.294892\pi\)
0.600693 + 0.799480i \(0.294892\pi\)
\(948\) 2.25983e7 0.816684
\(949\) −4.25133e6 −0.153236
\(950\) −2.03405e6 −0.0731229
\(951\) −1.57052e7 −0.563108
\(952\) 1.03592e8 3.70452
\(953\) 4.68158e7 1.66978 0.834892 0.550413i \(-0.185530\pi\)
0.834892 + 0.550413i \(0.185530\pi\)
\(954\) −5.92605e6 −0.210812
\(955\) 3.13935e7 1.11386
\(956\) −1.69908e7 −0.601269
\(957\) −3.22235e7 −1.13735
\(958\) −4.85310e7 −1.70846
\(959\) 3.57086e7 1.25379
\(960\) 3.78109e7 1.32416
\(961\) 7.23719e7 2.52791
\(962\) 1.07930e8 3.76013
\(963\) −7.98890e6 −0.277601
\(964\) −4.15756e7 −1.44094
\(965\) 4.77547e7 1.65081
\(966\) −7.39066e7 −2.54824
\(967\) −3.49546e7 −1.20209 −0.601047 0.799214i \(-0.705250\pi\)
−0.601047 + 0.799214i \(0.705250\pi\)
\(968\) 1.03632e7 0.355473
\(969\) 2.84101e6 0.0971992
\(970\) −2.06965e7 −0.706265
\(971\) 5.44571e6 0.185356 0.0926780 0.995696i \(-0.470457\pi\)
0.0926780 + 0.995696i \(0.470457\pi\)
\(972\) −4.83331e6 −0.164089
\(973\) 4.35035e6 0.147313
\(974\) −4.20657e7 −1.42079
\(975\) −5.57361e6 −0.187769
\(976\) −9.59862e7 −3.22540
\(977\) 2.76151e7 0.925573 0.462786 0.886470i \(-0.346850\pi\)
0.462786 + 0.886470i \(0.346850\pi\)
\(978\) 2.64885e7 0.885545
\(979\) −1.01619e7 −0.338859
\(980\) −9.05057e7 −3.01031
\(981\) 4.62359e6 0.153393
\(982\) 5.95050e6 0.196913
\(983\) −8.85517e6 −0.292289 −0.146145 0.989263i \(-0.546687\pi\)
−0.146145 + 0.989263i \(0.546687\pi\)
\(984\) −7.68619e6 −0.253060
\(985\) 2.29999e7 0.755326
\(986\) −9.37948e7 −3.07246
\(987\) 4.04024e7 1.32012
\(988\) −2.43491e7 −0.793579
\(989\) −2.02705e7 −0.658982
\(990\) 2.25029e7 0.729709
\(991\) 3.63535e7 1.17588 0.587938 0.808906i \(-0.299940\pi\)
0.587938 + 0.808906i \(0.299940\pi\)
\(992\) 1.56699e8 5.05577
\(993\) 4.64373e6 0.149449
\(994\) −1.48499e7 −0.476713
\(995\) −2.28666e7 −0.732224
\(996\) 2.15407e7 0.688038
\(997\) 3.59054e6 0.114399 0.0571994 0.998363i \(-0.481783\pi\)
0.0571994 + 0.998363i \(0.481783\pi\)
\(998\) −4.29777e7 −1.36589
\(999\) −7.50102e6 −0.237797
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.1.13 13
3.2 odd 2 531.6.a.e.1.1 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.d.1.13 13 1.1 even 1 trivial
531.6.a.e.1.1 13 3.2 odd 2