Properties

Label 49.6.a.g.1.2
Level $49$
Weight $6$
Character 49.1
Self dual yes
Analytic conductor $7.859$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.85880717084\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{113})\)
Defining polynomial: \( x^{4} - 2x^{3} - 59x^{2} + 60x + 674 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-6.22929\) of defining polynomial
Character \(\chi\) \(=\) 49.1

$q$-expansion

\(f(q)\) \(=\) \(q-7.81507 q^{2} +23.5186 q^{3} +29.0754 q^{4} -74.2753 q^{5} -183.799 q^{6} +22.8562 q^{8} +310.123 q^{9} +O(q^{10})\) \(q-7.81507 q^{2} +23.5186 q^{3} +29.0754 q^{4} -74.2753 q^{5} -183.799 q^{6} +22.8562 q^{8} +310.123 q^{9} +580.467 q^{10} -424.219 q^{11} +683.811 q^{12} -252.233 q^{13} -1746.85 q^{15} -1109.03 q^{16} -1104.35 q^{17} -2423.64 q^{18} -6.47100 q^{19} -2159.58 q^{20} +3315.30 q^{22} -3612.39 q^{23} +537.546 q^{24} +2391.82 q^{25} +1971.22 q^{26} +1578.65 q^{27} -5005.02 q^{29} +13651.8 q^{30} +2821.69 q^{31} +7935.79 q^{32} -9977.03 q^{33} +8630.55 q^{34} +9016.95 q^{36} -2046.88 q^{37} +50.5713 q^{38} -5932.17 q^{39} -1697.65 q^{40} +9393.81 q^{41} +10320.8 q^{43} -12334.3 q^{44} -23034.5 q^{45} +28231.1 q^{46} +17035.6 q^{47} -26082.9 q^{48} -18692.3 q^{50} -25972.7 q^{51} -7333.78 q^{52} -39506.7 q^{53} -12337.3 q^{54} +31509.0 q^{55} -152.189 q^{57} +39114.6 q^{58} +33949.8 q^{59} -50790.3 q^{60} +28295.2 q^{61} -22051.7 q^{62} -26529.7 q^{64} +18734.7 q^{65} +77971.2 q^{66} +56100.9 q^{67} -32109.3 q^{68} -84958.2 q^{69} -15537.4 q^{71} +7088.26 q^{72} -78219.5 q^{73} +15996.5 q^{74} +56252.3 q^{75} -188.147 q^{76} +46360.3 q^{78} -45335.5 q^{79} +82373.9 q^{80} -38232.4 q^{81} -73413.3 q^{82} +1381.82 q^{83} +82025.7 q^{85} -80657.6 q^{86} -117711. q^{87} -9696.05 q^{88} -68879.4 q^{89} +180016. q^{90} -105031. q^{92} +66362.1 q^{93} -133134. q^{94} +480.635 q^{95} +186638. q^{96} +108857. q^{97} -131560. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{2} + 10 q^{4} - 270 q^{8} + 220 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{2} + 10 q^{4} - 270 q^{8} + 220 q^{9} - 1952 q^{11} - 4096 q^{15} - 1566 q^{16} - 5974 q^{18} + 3524 q^{22} - 7136 q^{23} + 2764 q^{25} - 3352 q^{29} + 25608 q^{30} + 27810 q^{32} + 27670 q^{36} - 9208 q^{37} + 2464 q^{39} + 20448 q^{43} + 1900 q^{44} + 56712 q^{46} - 43070 q^{50} - 67408 q^{51} - 102920 q^{53} - 15576 q^{57} + 96972 q^{58} - 87080 q^{60} - 40318 q^{64} - 63168 q^{65} - 22896 q^{67} - 153824 q^{71} + 77358 q^{72} + 17596 q^{74} + 133056 q^{78} - 90688 q^{79} - 17204 q^{81} + 272656 q^{85} - 161860 q^{86} + 154812 q^{88} - 212200 q^{92} + 247760 q^{93} + 108224 q^{95} - 42272 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.81507 −1.38152 −0.690761 0.723083i \(-0.742724\pi\)
−0.690761 + 0.723083i \(0.742724\pi\)
\(3\) 23.5186 1.50872 0.754359 0.656462i \(-0.227948\pi\)
0.754359 + 0.656462i \(0.227948\pi\)
\(4\) 29.0754 0.908605
\(5\) −74.2753 −1.32868 −0.664339 0.747432i \(-0.731287\pi\)
−0.664339 + 0.747432i \(0.731287\pi\)
\(6\) −183.799 −2.08433
\(7\) 0 0
\(8\) 22.8562 0.126264
\(9\) 310.123 1.27623
\(10\) 580.467 1.83560
\(11\) −424.219 −1.05708 −0.528541 0.848908i \(-0.677261\pi\)
−0.528541 + 0.848908i \(0.677261\pi\)
\(12\) 683.811 1.37083
\(13\) −252.233 −0.413946 −0.206973 0.978347i \(-0.566361\pi\)
−0.206973 + 0.978347i \(0.566361\pi\)
\(14\) 0 0
\(15\) −1746.85 −2.00460
\(16\) −1109.03 −1.08304
\(17\) −1104.35 −0.926794 −0.463397 0.886151i \(-0.653370\pi\)
−0.463397 + 0.886151i \(0.653370\pi\)
\(18\) −2423.64 −1.76314
\(19\) −6.47100 −0.00411232 −0.00205616 0.999998i \(-0.500654\pi\)
−0.00205616 + 0.999998i \(0.500654\pi\)
\(20\) −2159.58 −1.20724
\(21\) 0 0
\(22\) 3315.30 1.46038
\(23\) −3612.39 −1.42388 −0.711942 0.702239i \(-0.752184\pi\)
−0.711942 + 0.702239i \(0.752184\pi\)
\(24\) 537.546 0.190497
\(25\) 2391.82 0.765383
\(26\) 1971.22 0.571876
\(27\) 1578.65 0.416751
\(28\) 0 0
\(29\) −5005.02 −1.10512 −0.552561 0.833472i \(-0.686350\pi\)
−0.552561 + 0.833472i \(0.686350\pi\)
\(30\) 13651.8 2.76940
\(31\) 2821.69 0.527357 0.263679 0.964611i \(-0.415064\pi\)
0.263679 + 0.964611i \(0.415064\pi\)
\(32\) 7935.79 1.36998
\(33\) −9977.03 −1.59484
\(34\) 8630.55 1.28039
\(35\) 0 0
\(36\) 9016.95 1.15959
\(37\) −2046.88 −0.245803 −0.122902 0.992419i \(-0.539220\pi\)
−0.122902 + 0.992419i \(0.539220\pi\)
\(38\) 50.5713 0.00568127
\(39\) −5932.17 −0.624528
\(40\) −1697.65 −0.167764
\(41\) 9393.81 0.872734 0.436367 0.899769i \(-0.356265\pi\)
0.436367 + 0.899769i \(0.356265\pi\)
\(42\) 0 0
\(43\) 10320.8 0.851218 0.425609 0.904907i \(-0.360060\pi\)
0.425609 + 0.904907i \(0.360060\pi\)
\(44\) −12334.3 −0.960470
\(45\) −23034.5 −1.69570
\(46\) 28231.1 1.96713
\(47\) 17035.6 1.12490 0.562448 0.826833i \(-0.309860\pi\)
0.562448 + 0.826833i \(0.309860\pi\)
\(48\) −26082.9 −1.63400
\(49\) 0 0
\(50\) −18692.3 −1.05739
\(51\) −25972.7 −1.39827
\(52\) −7333.78 −0.376114
\(53\) −39506.7 −1.93188 −0.965941 0.258761i \(-0.916686\pi\)
−0.965941 + 0.258761i \(0.916686\pi\)
\(54\) −12337.3 −0.575750
\(55\) 31509.0 1.40452
\(56\) 0 0
\(57\) −152.189 −0.00620433
\(58\) 39114.6 1.52675
\(59\) 33949.8 1.26972 0.634859 0.772628i \(-0.281058\pi\)
0.634859 + 0.772628i \(0.281058\pi\)
\(60\) −50790.3 −1.82139
\(61\) 28295.2 0.973618 0.486809 0.873508i \(-0.338161\pi\)
0.486809 + 0.873508i \(0.338161\pi\)
\(62\) −22051.7 −0.728556
\(63\) 0 0
\(64\) −26529.7 −0.809621
\(65\) 18734.7 0.550001
\(66\) 77971.2 2.20330
\(67\) 56100.9 1.52680 0.763402 0.645924i \(-0.223528\pi\)
0.763402 + 0.645924i \(0.223528\pi\)
\(68\) −32109.3 −0.842090
\(69\) −84958.2 −2.14824
\(70\) 0 0
\(71\) −15537.4 −0.365791 −0.182895 0.983132i \(-0.558547\pi\)
−0.182895 + 0.983132i \(0.558547\pi\)
\(72\) 7088.26 0.161142
\(73\) −78219.5 −1.71794 −0.858970 0.512025i \(-0.828895\pi\)
−0.858970 + 0.512025i \(0.828895\pi\)
\(74\) 15996.5 0.339583
\(75\) 56252.3 1.15475
\(76\) −188.147 −0.00373648
\(77\) 0 0
\(78\) 46360.3 0.862800
\(79\) −45335.5 −0.817279 −0.408640 0.912696i \(-0.633997\pi\)
−0.408640 + 0.912696i \(0.633997\pi\)
\(80\) 82373.9 1.43901
\(81\) −38232.4 −0.647469
\(82\) −73413.3 −1.20570
\(83\) 1381.82 0.0220169 0.0110085 0.999939i \(-0.496496\pi\)
0.0110085 + 0.999939i \(0.496496\pi\)
\(84\) 0 0
\(85\) 82025.7 1.23141
\(86\) −80657.6 −1.17598
\(87\) −117711. −1.66732
\(88\) −9696.05 −0.133471
\(89\) −68879.4 −0.921753 −0.460876 0.887464i \(-0.652465\pi\)
−0.460876 + 0.887464i \(0.652465\pi\)
\(90\) 180016. 2.34264
\(91\) 0 0
\(92\) −105031. −1.29375
\(93\) 66362.1 0.795633
\(94\) −133134. −1.55407
\(95\) 480.635 0.00546395
\(96\) 186638. 2.06692
\(97\) 108857. 1.17470 0.587351 0.809332i \(-0.300171\pi\)
0.587351 + 0.809332i \(0.300171\pi\)
\(98\) 0 0
\(99\) −131560. −1.34908
\(100\) 69543.1 0.695431
\(101\) 17972.3 0.175307 0.0876535 0.996151i \(-0.472063\pi\)
0.0876535 + 0.996151i \(0.472063\pi\)
\(102\) 202978. 1.93174
\(103\) 31773.7 0.295103 0.147552 0.989054i \(-0.452861\pi\)
0.147552 + 0.989054i \(0.452861\pi\)
\(104\) −5765.11 −0.0522666
\(105\) 0 0
\(106\) 308747. 2.66894
\(107\) 8229.36 0.0694875 0.0347438 0.999396i \(-0.488938\pi\)
0.0347438 + 0.999396i \(0.488938\pi\)
\(108\) 45899.8 0.378662
\(109\) −11068.7 −0.0892338 −0.0446169 0.999004i \(-0.514207\pi\)
−0.0446169 + 0.999004i \(0.514207\pi\)
\(110\) −246245. −1.94038
\(111\) −48139.6 −0.370847
\(112\) 0 0
\(113\) 65184.3 0.480228 0.240114 0.970745i \(-0.422815\pi\)
0.240114 + 0.970745i \(0.422815\pi\)
\(114\) 1189.37 0.00857143
\(115\) 268311. 1.89188
\(116\) −145523. −1.00412
\(117\) −78223.5 −0.528290
\(118\) −265320. −1.75414
\(119\) 0 0
\(120\) −39926.4 −0.253109
\(121\) 18910.9 0.117422
\(122\) −221129. −1.34508
\(123\) 220929. 1.31671
\(124\) 82041.7 0.479160
\(125\) 54456.9 0.311730
\(126\) 0 0
\(127\) −194777. −1.07159 −0.535796 0.844348i \(-0.679988\pi\)
−0.535796 + 0.844348i \(0.679988\pi\)
\(128\) −46614.1 −0.251473
\(129\) 242730. 1.28425
\(130\) −146413. −0.759839
\(131\) −236503. −1.20409 −0.602046 0.798462i \(-0.705647\pi\)
−0.602046 + 0.798462i \(0.705647\pi\)
\(132\) −290086. −1.44908
\(133\) 0 0
\(134\) −438433. −2.10931
\(135\) −117255. −0.553727
\(136\) −25241.2 −0.117021
\(137\) −200903. −0.914503 −0.457252 0.889337i \(-0.651166\pi\)
−0.457252 + 0.889337i \(0.651166\pi\)
\(138\) 663954. 2.96784
\(139\) 52985.2 0.232604 0.116302 0.993214i \(-0.462896\pi\)
0.116302 + 0.993214i \(0.462896\pi\)
\(140\) 0 0
\(141\) 400653. 1.69715
\(142\) 121426. 0.505348
\(143\) 107002. 0.437575
\(144\) −343938. −1.38221
\(145\) 371749. 1.46835
\(146\) 611291. 2.37337
\(147\) 0 0
\(148\) −59513.7 −0.223338
\(149\) −100770. −0.371849 −0.185925 0.982564i \(-0.559528\pi\)
−0.185925 + 0.982564i \(0.559528\pi\)
\(150\) −439616. −1.59531
\(151\) −457904. −1.63430 −0.817150 0.576425i \(-0.804447\pi\)
−0.817150 + 0.576425i \(0.804447\pi\)
\(152\) −147.903 −0.000519239 0
\(153\) −342484. −1.18280
\(154\) 0 0
\(155\) −209582. −0.700688
\(156\) −172480. −0.567450
\(157\) −179037. −0.579688 −0.289844 0.957074i \(-0.593603\pi\)
−0.289844 + 0.957074i \(0.593603\pi\)
\(158\) 354300. 1.12909
\(159\) −929141. −2.91467
\(160\) −589433. −1.82027
\(161\) 0 0
\(162\) 298789. 0.894494
\(163\) 243610. 0.718168 0.359084 0.933305i \(-0.383089\pi\)
0.359084 + 0.933305i \(0.383089\pi\)
\(164\) 273128. 0.792971
\(165\) 741047. 2.11902
\(166\) −10799.0 −0.0304169
\(167\) 117033. 0.324725 0.162362 0.986731i \(-0.448089\pi\)
0.162362 + 0.986731i \(0.448089\pi\)
\(168\) 0 0
\(169\) −307671. −0.828648
\(170\) −641037. −1.70122
\(171\) −2006.81 −0.00524826
\(172\) 300080. 0.773421
\(173\) 269733. 0.685203 0.342602 0.939481i \(-0.388692\pi\)
0.342602 + 0.939481i \(0.388692\pi\)
\(174\) 919919. 2.30344
\(175\) 0 0
\(176\) 470474. 1.14486
\(177\) 798451. 1.91564
\(178\) 538298. 1.27342
\(179\) 376525. 0.878336 0.439168 0.898405i \(-0.355273\pi\)
0.439168 + 0.898405i \(0.355273\pi\)
\(180\) −669737. −1.54072
\(181\) −434641. −0.986131 −0.493065 0.869992i \(-0.664124\pi\)
−0.493065 + 0.869992i \(0.664124\pi\)
\(182\) 0 0
\(183\) 665464. 1.46891
\(184\) −82565.5 −0.179785
\(185\) 152032. 0.326593
\(186\) −518625. −1.09919
\(187\) 468485. 0.979697
\(188\) 495316. 1.02209
\(189\) 0 0
\(190\) −3756.20 −0.00754857
\(191\) 565940. 1.12250 0.561250 0.827646i \(-0.310320\pi\)
0.561250 + 0.827646i \(0.310320\pi\)
\(192\) −623940. −1.22149
\(193\) 514461. 0.994167 0.497084 0.867703i \(-0.334404\pi\)
0.497084 + 0.867703i \(0.334404\pi\)
\(194\) −850727. −1.62288
\(195\) 440614. 0.829796
\(196\) 0 0
\(197\) −298541. −0.548073 −0.274037 0.961719i \(-0.588359\pi\)
−0.274037 + 0.961719i \(0.588359\pi\)
\(198\) 1.02815e6 1.86378
\(199\) −591919. −1.05957 −0.529785 0.848132i \(-0.677727\pi\)
−0.529785 + 0.848132i \(0.677727\pi\)
\(200\) 54668.1 0.0966404
\(201\) 1.31941e6 2.30351
\(202\) −140455. −0.242191
\(203\) 0 0
\(204\) −755165. −1.27048
\(205\) −697728. −1.15958
\(206\) −248313. −0.407692
\(207\) −1.12029e6 −1.81720
\(208\) 279736. 0.448321
\(209\) 2745.12 0.00434706
\(210\) 0 0
\(211\) −140535. −0.217309 −0.108655 0.994080i \(-0.534654\pi\)
−0.108655 + 0.994080i \(0.534654\pi\)
\(212\) −1.14867e6 −1.75532
\(213\) −365418. −0.551875
\(214\) −64313.1 −0.0959986
\(215\) −766579. −1.13099
\(216\) 36082.0 0.0526206
\(217\) 0 0
\(218\) 86502.5 0.123278
\(219\) −1.83961e6 −2.59189
\(220\) 916136. 1.27615
\(221\) 278553. 0.383643
\(222\) 376215. 0.512334
\(223\) 490.526 0.000660541 0 0.000330271 1.00000i \(-0.499895\pi\)
0.000330271 1.00000i \(0.499895\pi\)
\(224\) 0 0
\(225\) 741761. 0.976804
\(226\) −509420. −0.663445
\(227\) 593898. 0.764975 0.382488 0.923961i \(-0.375067\pi\)
0.382488 + 0.923961i \(0.375067\pi\)
\(228\) −4424.94 −0.00563729
\(229\) −35880.3 −0.0452135 −0.0226067 0.999744i \(-0.507197\pi\)
−0.0226067 + 0.999744i \(0.507197\pi\)
\(230\) −2.09687e6 −2.61368
\(231\) 0 0
\(232\) −114396. −0.139537
\(233\) 1.24822e6 1.50626 0.753131 0.657871i \(-0.228543\pi\)
0.753131 + 0.657871i \(0.228543\pi\)
\(234\) 611322. 0.729845
\(235\) −1.26532e6 −1.49462
\(236\) 987102. 1.15367
\(237\) −1.06623e6 −1.23304
\(238\) 0 0
\(239\) −576943. −0.653339 −0.326669 0.945139i \(-0.605926\pi\)
−0.326669 + 0.945139i \(0.605926\pi\)
\(240\) 1.93732e6 2.17106
\(241\) −1.38241e6 −1.53318 −0.766592 0.642135i \(-0.778049\pi\)
−0.766592 + 0.642135i \(0.778049\pi\)
\(242\) −147790. −0.162221
\(243\) −1.28278e6 −1.39360
\(244\) 822694. 0.884634
\(245\) 0 0
\(246\) −1.72658e6 −1.81906
\(247\) 1632.20 0.00170228
\(248\) 64493.2 0.0665863
\(249\) 32498.5 0.0332173
\(250\) −425585. −0.430662
\(251\) 323217. 0.323824 0.161912 0.986805i \(-0.448234\pi\)
0.161912 + 0.986805i \(0.448234\pi\)
\(252\) 0 0
\(253\) 1.53244e6 1.50516
\(254\) 1.52220e6 1.48043
\(255\) 1.92913e6 1.85785
\(256\) 1.21324e6 1.15704
\(257\) 1.84601e6 1.74342 0.871711 0.490021i \(-0.163011\pi\)
0.871711 + 0.490021i \(0.163011\pi\)
\(258\) −1.89695e6 −1.77422
\(259\) 0 0
\(260\) 544719. 0.499734
\(261\) −1.55217e6 −1.41039
\(262\) 1.84829e6 1.66348
\(263\) −458222. −0.408495 −0.204248 0.978919i \(-0.565475\pi\)
−0.204248 + 0.978919i \(0.565475\pi\)
\(264\) −228037. −0.201371
\(265\) 2.93437e6 2.56685
\(266\) 0 0
\(267\) −1.61995e6 −1.39066
\(268\) 1.63115e6 1.38726
\(269\) 416958. 0.351327 0.175663 0.984450i \(-0.443793\pi\)
0.175663 + 0.984450i \(0.443793\pi\)
\(270\) 916354. 0.764987
\(271\) −900379. −0.744735 −0.372368 0.928085i \(-0.621454\pi\)
−0.372368 + 0.928085i \(0.621454\pi\)
\(272\) 1.22476e6 1.00376
\(273\) 0 0
\(274\) 1.57007e6 1.26341
\(275\) −1.01466e6 −0.809073
\(276\) −2.47019e6 −1.95190
\(277\) −447641. −0.350535 −0.175267 0.984521i \(-0.556079\pi\)
−0.175267 + 0.984521i \(0.556079\pi\)
\(278\) −414083. −0.321348
\(279\) 875072. 0.673029
\(280\) 0 0
\(281\) −768521. −0.580617 −0.290309 0.956933i \(-0.593758\pi\)
−0.290309 + 0.956933i \(0.593758\pi\)
\(282\) −3.13113e6 −2.34465
\(283\) −2.13220e6 −1.58256 −0.791282 0.611452i \(-0.790586\pi\)
−0.791282 + 0.611452i \(0.790586\pi\)
\(284\) −451756. −0.332359
\(285\) 11303.9 0.00824356
\(286\) −836230. −0.604520
\(287\) 0 0
\(288\) 2.46107e6 1.74841
\(289\) −200275. −0.141053
\(290\) −2.90525e6 −2.02856
\(291\) 2.56017e6 1.77229
\(292\) −2.27426e6 −1.56093
\(293\) −2.42669e6 −1.65138 −0.825688 0.564128i \(-0.809213\pi\)
−0.825688 + 0.564128i \(0.809213\pi\)
\(294\) 0 0
\(295\) −2.52163e6 −1.68704
\(296\) −46783.9 −0.0310361
\(297\) −669693. −0.440539
\(298\) 787528. 0.513719
\(299\) 911164. 0.589411
\(300\) 1.63556e6 1.04921
\(301\) 0 0
\(302\) 3.57855e6 2.25782
\(303\) 422682. 0.264489
\(304\) 7176.56 0.00445382
\(305\) −2.10164e6 −1.29362
\(306\) 2.67654e6 1.63407
\(307\) 2.44328e6 1.47954 0.739772 0.672857i \(-0.234933\pi\)
0.739772 + 0.672857i \(0.234933\pi\)
\(308\) 0 0
\(309\) 747271. 0.445228
\(310\) 1.63790e6 0.968016
\(311\) −1.15465e6 −0.676938 −0.338469 0.940978i \(-0.609909\pi\)
−0.338469 + 0.940978i \(0.609909\pi\)
\(312\) −135587. −0.0788555
\(313\) −1.65706e6 −0.956044 −0.478022 0.878348i \(-0.658646\pi\)
−0.478022 + 0.878348i \(0.658646\pi\)
\(314\) 1.39919e6 0.800852
\(315\) 0 0
\(316\) −1.31815e6 −0.742584
\(317\) 821361. 0.459077 0.229539 0.973300i \(-0.426278\pi\)
0.229539 + 0.973300i \(0.426278\pi\)
\(318\) 7.26130e6 4.02668
\(319\) 2.12322e6 1.16821
\(320\) 1.97050e6 1.07572
\(321\) 193543. 0.104837
\(322\) 0 0
\(323\) 7146.23 0.00381128
\(324\) −1.11162e6 −0.588294
\(325\) −603298. −0.316828
\(326\) −1.90383e6 −0.992166
\(327\) −260319. −0.134629
\(328\) 214707. 0.110195
\(329\) 0 0
\(330\) −5.79134e6 −2.92748
\(331\) −95670.7 −0.0479964 −0.0239982 0.999712i \(-0.507640\pi\)
−0.0239982 + 0.999712i \(0.507640\pi\)
\(332\) 40177.0 0.0200047
\(333\) −634785. −0.313701
\(334\) −914618. −0.448615
\(335\) −4.16691e6 −2.02863
\(336\) 0 0
\(337\) 2.37020e6 1.13687 0.568435 0.822728i \(-0.307549\pi\)
0.568435 + 0.822728i \(0.307549\pi\)
\(338\) 2.40447e6 1.14480
\(339\) 1.53304e6 0.724528
\(340\) 2.38493e6 1.11887
\(341\) −1.19701e6 −0.557460
\(342\) 15683.4 0.00725060
\(343\) 0 0
\(344\) 235894. 0.107478
\(345\) 6.31029e6 2.85431
\(346\) −2.10799e6 −0.946624
\(347\) 490571. 0.218715 0.109358 0.994002i \(-0.465121\pi\)
0.109358 + 0.994002i \(0.465121\pi\)
\(348\) −3.42249e6 −1.51493
\(349\) −4.21208e6 −1.85111 −0.925557 0.378607i \(-0.876403\pi\)
−0.925557 + 0.378607i \(0.876403\pi\)
\(350\) 0 0
\(351\) −398188. −0.172512
\(352\) −3.36651e6 −1.44818
\(353\) 3.17378e6 1.35563 0.677814 0.735234i \(-0.262928\pi\)
0.677814 + 0.735234i \(0.262928\pi\)
\(354\) −6.23995e6 −2.64651
\(355\) 1.15405e6 0.486018
\(356\) −2.00269e6 −0.837509
\(357\) 0 0
\(358\) −2.94257e6 −1.21344
\(359\) −3.40098e6 −1.39273 −0.696366 0.717687i \(-0.745201\pi\)
−0.696366 + 0.717687i \(0.745201\pi\)
\(360\) −526483. −0.214105
\(361\) −2.47606e6 −0.999983
\(362\) 3.39675e6 1.36236
\(363\) 444757. 0.177156
\(364\) 0 0
\(365\) 5.80978e6 2.28259
\(366\) −5.20065e6 −2.02934
\(367\) 1.96872e6 0.762988 0.381494 0.924371i \(-0.375410\pi\)
0.381494 + 0.924371i \(0.375410\pi\)
\(368\) 4.00626e6 1.54213
\(369\) 2.91324e6 1.11381
\(370\) −1.18814e6 −0.451196
\(371\) 0 0
\(372\) 1.92950e6 0.722917
\(373\) −3.47889e6 −1.29470 −0.647349 0.762194i \(-0.724122\pi\)
−0.647349 + 0.762194i \(0.724122\pi\)
\(374\) −3.66124e6 −1.35347
\(375\) 1.28075e6 0.470312
\(376\) 389369. 0.142034
\(377\) 1.26243e6 0.457462
\(378\) 0 0
\(379\) −421294. −0.150656 −0.0753281 0.997159i \(-0.524000\pi\)
−0.0753281 + 0.997159i \(0.524000\pi\)
\(380\) 13974.7 0.00496457
\(381\) −4.58089e6 −1.61673
\(382\) −4.42286e6 −1.55076
\(383\) 2.66910e6 0.929754 0.464877 0.885375i \(-0.346099\pi\)
0.464877 + 0.885375i \(0.346099\pi\)
\(384\) −1.09630e6 −0.379402
\(385\) 0 0
\(386\) −4.02055e6 −1.37346
\(387\) 3.20071e6 1.08635
\(388\) 3.16506e6 1.06734
\(389\) 3.10178e6 1.03929 0.519645 0.854382i \(-0.326064\pi\)
0.519645 + 0.854382i \(0.326064\pi\)
\(390\) −3.44343e6 −1.14638
\(391\) 3.98933e6 1.31965
\(392\) 0 0
\(393\) −5.56223e6 −1.81663
\(394\) 2.33312e6 0.757176
\(395\) 3.36731e6 1.08590
\(396\) −3.82516e6 −1.22578
\(397\) 613257. 0.195284 0.0976419 0.995222i \(-0.468870\pi\)
0.0976419 + 0.995222i \(0.468870\pi\)
\(398\) 4.62589e6 1.46382
\(399\) 0 0
\(400\) −2.65262e6 −0.828942
\(401\) 2.82223e6 0.876459 0.438229 0.898863i \(-0.355606\pi\)
0.438229 + 0.898863i \(0.355606\pi\)
\(402\) −1.03113e7 −3.18236
\(403\) −711724. −0.218298
\(404\) 522550. 0.159285
\(405\) 2.83973e6 0.860278
\(406\) 0 0
\(407\) 868324. 0.259834
\(408\) −593637. −0.176551
\(409\) 2.28350e6 0.674983 0.337492 0.941329i \(-0.390422\pi\)
0.337492 + 0.941329i \(0.390422\pi\)
\(410\) 5.45279e6 1.60199
\(411\) −4.72496e6 −1.37973
\(412\) 923831. 0.268132
\(413\) 0 0
\(414\) 8.75511e6 2.51050
\(415\) −102635. −0.0292534
\(416\) −2.00167e6 −0.567100
\(417\) 1.24614e6 0.350934
\(418\) −21453.3 −0.00600556
\(419\) −2.65270e6 −0.738163 −0.369082 0.929397i \(-0.620328\pi\)
−0.369082 + 0.929397i \(0.620328\pi\)
\(420\) 0 0
\(421\) 2.93674e6 0.807532 0.403766 0.914862i \(-0.367701\pi\)
0.403766 + 0.914862i \(0.367701\pi\)
\(422\) 1.09829e6 0.300218
\(423\) 5.28314e6 1.43562
\(424\) −902974. −0.243927
\(425\) −2.64140e6 −0.709353
\(426\) 2.85577e6 0.762428
\(427\) 0 0
\(428\) 239272. 0.0631367
\(429\) 2.51654e6 0.660177
\(430\) 5.99087e6 1.56249
\(431\) −2.44565e6 −0.634164 −0.317082 0.948398i \(-0.602703\pi\)
−0.317082 + 0.948398i \(0.602703\pi\)
\(432\) −1.75078e6 −0.451358
\(433\) −2.11718e6 −0.542673 −0.271336 0.962485i \(-0.587466\pi\)
−0.271336 + 0.962485i \(0.587466\pi\)
\(434\) 0 0
\(435\) 8.74301e6 2.21533
\(436\) −321826. −0.0810783
\(437\) 23375.7 0.00585547
\(438\) 1.43767e7 3.58075
\(439\) 4.64764e6 1.15099 0.575495 0.817806i \(-0.304810\pi\)
0.575495 + 0.817806i \(0.304810\pi\)
\(440\) 720178. 0.177340
\(441\) 0 0
\(442\) −2.17691e6 −0.530012
\(443\) 4.42925e6 1.07231 0.536155 0.844119i \(-0.319876\pi\)
0.536155 + 0.844119i \(0.319876\pi\)
\(444\) −1.39968e6 −0.336954
\(445\) 5.11604e6 1.22471
\(446\) −3833.50 −0.000912553 0
\(447\) −2.36998e6 −0.561016
\(448\) 0 0
\(449\) −6.70171e6 −1.56881 −0.784404 0.620250i \(-0.787031\pi\)
−0.784404 + 0.620250i \(0.787031\pi\)
\(450\) −5.79691e6 −1.34948
\(451\) −3.98503e6 −0.922551
\(452\) 1.89526e6 0.436337
\(453\) −1.07692e7 −2.46570
\(454\) −4.64136e6 −1.05683
\(455\) 0 0
\(456\) −3478.46 −0.000783385 0
\(457\) −5.88344e6 −1.31777 −0.658887 0.752242i \(-0.728972\pi\)
−0.658887 + 0.752242i \(0.728972\pi\)
\(458\) 280407. 0.0624634
\(459\) −1.74338e6 −0.386242
\(460\) 7.80124e6 1.71897
\(461\) 1.54764e6 0.339171 0.169585 0.985515i \(-0.445757\pi\)
0.169585 + 0.985515i \(0.445757\pi\)
\(462\) 0 0
\(463\) −3.93764e6 −0.853656 −0.426828 0.904333i \(-0.640369\pi\)
−0.426828 + 0.904333i \(0.640369\pi\)
\(464\) 5.55074e6 1.19689
\(465\) −4.92907e6 −1.05714
\(466\) −9.75491e6 −2.08093
\(467\) −6.81586e6 −1.44620 −0.723100 0.690743i \(-0.757284\pi\)
−0.723100 + 0.690743i \(0.757284\pi\)
\(468\) −2.27438e6 −0.480007
\(469\) 0 0
\(470\) 9.88860e6 2.06486
\(471\) −4.21070e6 −0.874585
\(472\) 775964. 0.160320
\(473\) −4.37827e6 −0.899807
\(474\) 8.33263e6 1.70348
\(475\) −15477.5 −0.00314750
\(476\) 0 0
\(477\) −1.22519e7 −2.46552
\(478\) 4.50885e6 0.902602
\(479\) −5.20406e6 −1.03634 −0.518171 0.855277i \(-0.673387\pi\)
−0.518171 + 0.855277i \(0.673387\pi\)
\(480\) −1.38626e7 −2.74627
\(481\) 516291. 0.101749
\(482\) 1.08036e7 2.11813
\(483\) 0 0
\(484\) 549840. 0.106690
\(485\) −8.08540e6 −1.56080
\(486\) 1.00251e7 1.92529
\(487\) −154998. −0.0296145 −0.0148073 0.999890i \(-0.504713\pi\)
−0.0148073 + 0.999890i \(0.504713\pi\)
\(488\) 646723. 0.122933
\(489\) 5.72936e6 1.08351
\(490\) 0 0
\(491\) −1.61951e6 −0.303165 −0.151583 0.988445i \(-0.548437\pi\)
−0.151583 + 0.988445i \(0.548437\pi\)
\(492\) 6.42359e6 1.19637
\(493\) 5.52727e6 1.02422
\(494\) −12755.8 −0.00235174
\(495\) 9.77168e6 1.79249
\(496\) −3.12935e6 −0.571150
\(497\) 0 0
\(498\) −253978. −0.0458905
\(499\) 4.10674e6 0.738322 0.369161 0.929366i \(-0.379645\pi\)
0.369161 + 0.929366i \(0.379645\pi\)
\(500\) 1.58336e6 0.283239
\(501\) 2.75244e6 0.489918
\(502\) −2.52596e6 −0.447371
\(503\) −3.40748e6 −0.600501 −0.300250 0.953860i \(-0.597070\pi\)
−0.300250 + 0.953860i \(0.597070\pi\)
\(504\) 0 0
\(505\) −1.33490e6 −0.232927
\(506\) −1.19762e7 −2.07941
\(507\) −7.23599e6 −1.25020
\(508\) −5.66322e6 −0.973654
\(509\) 1.07091e7 1.83213 0.916067 0.401025i \(-0.131346\pi\)
0.916067 + 0.401025i \(0.131346\pi\)
\(510\) −1.50763e7 −2.56666
\(511\) 0 0
\(512\) −7.98992e6 −1.34700
\(513\) −10215.4 −0.00171381
\(514\) −1.44267e7 −2.40858
\(515\) −2.36000e6 −0.392097
\(516\) 7.05746e6 1.16687
\(517\) −7.22682e6 −1.18911
\(518\) 0 0
\(519\) 6.34374e6 1.03378
\(520\) 428205. 0.0694454
\(521\) 7.92001e6 1.27830 0.639148 0.769084i \(-0.279287\pi\)
0.639148 + 0.769084i \(0.279287\pi\)
\(522\) 1.21303e7 1.94848
\(523\) −8.32746e6 −1.33125 −0.665623 0.746288i \(-0.731834\pi\)
−0.665623 + 0.746288i \(0.731834\pi\)
\(524\) −6.87643e6 −1.09404
\(525\) 0 0
\(526\) 3.58104e6 0.564345
\(527\) −3.11612e6 −0.488752
\(528\) 1.10649e7 1.72728
\(529\) 6.61298e6 1.02744
\(530\) −2.29323e7 −3.54616
\(531\) 1.05286e7 1.62045
\(532\) 0 0
\(533\) −2.36943e6 −0.361265
\(534\) 1.26600e7 1.92124
\(535\) −611239. −0.0923265
\(536\) 1.28226e6 0.192780
\(537\) 8.85532e6 1.32516
\(538\) −3.25856e6 −0.485366
\(539\) 0 0
\(540\) −3.40922e6 −0.503119
\(541\) 623261. 0.0915539 0.0457770 0.998952i \(-0.485424\pi\)
0.0457770 + 0.998952i \(0.485424\pi\)
\(542\) 7.03653e6 1.02887
\(543\) −1.02221e7 −1.48779
\(544\) −8.76386e6 −1.26969
\(545\) 822129. 0.118563
\(546\) 0 0
\(547\) 1.05691e7 1.51032 0.755159 0.655541i \(-0.227559\pi\)
0.755159 + 0.655541i \(0.227559\pi\)
\(548\) −5.84133e6 −0.830922
\(549\) 8.77502e6 1.24256
\(550\) 7.92962e6 1.11775
\(551\) 32387.5 0.00454462
\(552\) −1.94182e6 −0.271245
\(553\) 0 0
\(554\) 3.49835e6 0.484272
\(555\) 3.57559e6 0.492737
\(556\) 1.54056e6 0.211345
\(557\) −1.35398e7 −1.84916 −0.924579 0.380991i \(-0.875583\pi\)
−0.924579 + 0.380991i \(0.875583\pi\)
\(558\) −6.83875e6 −0.929804
\(559\) −2.60324e6 −0.352359
\(560\) 0 0
\(561\) 1.10181e7 1.47809
\(562\) 6.00605e6 0.802136
\(563\) −1.39757e7 −1.85824 −0.929122 0.369774i \(-0.879435\pi\)
−0.929122 + 0.369774i \(0.879435\pi\)
\(564\) 1.16491e7 1.54204
\(565\) −4.84159e6 −0.638068
\(566\) 1.66633e7 2.18635
\(567\) 0 0
\(568\) −355127. −0.0461862
\(569\) −5.61993e6 −0.727697 −0.363848 0.931458i \(-0.618537\pi\)
−0.363848 + 0.931458i \(0.618537\pi\)
\(570\) −88340.5 −0.0113887
\(571\) 8.70790e6 1.11769 0.558847 0.829271i \(-0.311244\pi\)
0.558847 + 0.829271i \(0.311244\pi\)
\(572\) 3.11113e6 0.397583
\(573\) 1.33101e7 1.69354
\(574\) 0 0
\(575\) −8.64019e6 −1.08982
\(576\) −8.22747e6 −1.03326
\(577\) −6.63992e6 −0.830278 −0.415139 0.909758i \(-0.636267\pi\)
−0.415139 + 0.909758i \(0.636267\pi\)
\(578\) 1.56517e6 0.194868
\(579\) 1.20994e7 1.49992
\(580\) 1.08087e7 1.33415
\(581\) 0 0
\(582\) −2.00079e7 −2.44846
\(583\) 1.67595e7 2.04216
\(584\) −1.78780e6 −0.216914
\(585\) 5.81007e6 0.701927
\(586\) 1.89648e7 2.28141
\(587\) −1.36652e7 −1.63689 −0.818446 0.574583i \(-0.805164\pi\)
−0.818446 + 0.574583i \(0.805164\pi\)
\(588\) 0 0
\(589\) −18259.2 −0.00216866
\(590\) 1.97067e7 2.33069
\(591\) −7.02126e6 −0.826888
\(592\) 2.27006e6 0.266215
\(593\) −7.02589e6 −0.820474 −0.410237 0.911979i \(-0.634554\pi\)
−0.410237 + 0.911979i \(0.634554\pi\)
\(594\) 5.23370e6 0.608615
\(595\) 0 0
\(596\) −2.92994e6 −0.337864
\(597\) −1.39211e7 −1.59859
\(598\) −7.12081e6 −0.814285
\(599\) −3.58663e6 −0.408432 −0.204216 0.978926i \(-0.565464\pi\)
−0.204216 + 0.978926i \(0.565464\pi\)
\(600\) 1.28572e6 0.145803
\(601\) 1.58600e7 1.79108 0.895542 0.444977i \(-0.146788\pi\)
0.895542 + 0.444977i \(0.146788\pi\)
\(602\) 0 0
\(603\) 1.73982e7 1.94855
\(604\) −1.33137e7 −1.48493
\(605\) −1.40461e6 −0.156015
\(606\) −3.30329e6 −0.365397
\(607\) 6.44170e6 0.709625 0.354812 0.934938i \(-0.384545\pi\)
0.354812 + 0.934938i \(0.384545\pi\)
\(608\) −51352.5 −0.00563381
\(609\) 0 0
\(610\) 1.64244e7 1.78717
\(611\) −4.29694e6 −0.465647
\(612\) −9.95784e6 −1.07470
\(613\) −4.58865e6 −0.493212 −0.246606 0.969116i \(-0.579315\pi\)
−0.246606 + 0.969116i \(0.579315\pi\)
\(614\) −1.90944e7 −2.04402
\(615\) −1.64096e7 −1.74948
\(616\) 0 0
\(617\) 1.47104e6 0.155565 0.0777825 0.996970i \(-0.475216\pi\)
0.0777825 + 0.996970i \(0.475216\pi\)
\(618\) −5.83998e6 −0.615092
\(619\) −3.13569e6 −0.328932 −0.164466 0.986383i \(-0.552590\pi\)
−0.164466 + 0.986383i \(0.552590\pi\)
\(620\) −6.09367e6 −0.636649
\(621\) −5.70269e6 −0.593404
\(622\) 9.02366e6 0.935205
\(623\) 0 0
\(624\) 6.57898e6 0.676390
\(625\) −1.15193e7 −1.17957
\(626\) 1.29501e7 1.32080
\(627\) 64561.3 0.00655849
\(628\) −5.20557e6 −0.526707
\(629\) 2.26046e6 0.227809
\(630\) 0 0
\(631\) 484547. 0.0484465 0.0242233 0.999707i \(-0.492289\pi\)
0.0242233 + 0.999707i \(0.492289\pi\)
\(632\) −1.03620e6 −0.103193
\(633\) −3.30518e6 −0.327858
\(634\) −6.41899e6 −0.634226
\(635\) 1.44672e7 1.42380
\(636\) −2.70151e7 −2.64828
\(637\) 0 0
\(638\) −1.65932e7 −1.61390
\(639\) −4.81851e6 −0.466832
\(640\) 3.46228e6 0.334127
\(641\) 3.04085e6 0.292314 0.146157 0.989261i \(-0.453310\pi\)
0.146157 + 0.989261i \(0.453310\pi\)
\(642\) −1.51255e6 −0.144835
\(643\) 5.25888e6 0.501609 0.250805 0.968038i \(-0.419305\pi\)
0.250805 + 0.968038i \(0.419305\pi\)
\(644\) 0 0
\(645\) −1.80288e7 −1.70635
\(646\) −55848.3 −0.00526536
\(647\) 2.11970e7 1.99074 0.995368 0.0961386i \(-0.0306492\pi\)
0.995368 + 0.0961386i \(0.0306492\pi\)
\(648\) −873849. −0.0817521
\(649\) −1.44021e7 −1.34219
\(650\) 4.71481e6 0.437705
\(651\) 0 0
\(652\) 7.08305e6 0.652531
\(653\) 1.30106e7 1.19403 0.597013 0.802232i \(-0.296354\pi\)
0.597013 + 0.802232i \(0.296354\pi\)
\(654\) 2.03442e6 0.185992
\(655\) 1.75664e7 1.59985
\(656\) −1.04181e7 −0.945208
\(657\) −2.42577e7 −2.19248
\(658\) 0 0
\(659\) −1.59874e7 −1.43405 −0.717024 0.697049i \(-0.754496\pi\)
−0.717024 + 0.697049i \(0.754496\pi\)
\(660\) 2.15462e7 1.92536
\(661\) 4.03142e6 0.358884 0.179442 0.983769i \(-0.442571\pi\)
0.179442 + 0.983769i \(0.442571\pi\)
\(662\) 747673. 0.0663082
\(663\) 6.55117e6 0.578809
\(664\) 31583.3 0.00277995
\(665\) 0 0
\(666\) 4.96089e6 0.433385
\(667\) 1.80800e7 1.57357
\(668\) 3.40276e6 0.295047
\(669\) 11536.5 0.000996570 0
\(670\) 3.25647e7 2.80260
\(671\) −1.20034e7 −1.02919
\(672\) 0 0
\(673\) 2.98234e6 0.253816 0.126908 0.991914i \(-0.459495\pi\)
0.126908 + 0.991914i \(0.459495\pi\)
\(674\) −1.85233e7 −1.57061
\(675\) 3.77585e6 0.318974
\(676\) −8.94566e6 −0.752914
\(677\) 1.94696e6 0.163262 0.0816309 0.996663i \(-0.473987\pi\)
0.0816309 + 0.996663i \(0.473987\pi\)
\(678\) −1.19808e7 −1.00095
\(679\) 0 0
\(680\) 1.87480e6 0.155483
\(681\) 1.39676e7 1.15413
\(682\) 9.35476e6 0.770144
\(683\) 1.19940e7 0.983812 0.491906 0.870648i \(-0.336300\pi\)
0.491906 + 0.870648i \(0.336300\pi\)
\(684\) −58348.7 −0.00476860
\(685\) 1.49221e7 1.21508
\(686\) 0 0
\(687\) −843854. −0.0682143
\(688\) −1.14461e7 −0.921905
\(689\) 9.96490e6 0.799696
\(690\) −4.93154e7 −3.94330
\(691\) −8.66304e6 −0.690201 −0.345100 0.938566i \(-0.612155\pi\)
−0.345100 + 0.938566i \(0.612155\pi\)
\(692\) 7.84260e6 0.622579
\(693\) 0 0
\(694\) −3.83385e6 −0.302160
\(695\) −3.93549e6 −0.309056
\(696\) −2.69043e6 −0.210522
\(697\) −1.03740e7 −0.808845
\(698\) 3.29177e7 2.55736
\(699\) 2.93563e7 2.27252
\(700\) 0 0
\(701\) −8.13382e6 −0.625172 −0.312586 0.949889i \(-0.601195\pi\)
−0.312586 + 0.949889i \(0.601195\pi\)
\(702\) 3.11187e6 0.238330
\(703\) 13245.3 0.00101082
\(704\) 1.12544e7 0.855835
\(705\) −2.97586e7 −2.25497
\(706\) −2.48033e7 −1.87283
\(707\) 0 0
\(708\) 2.32152e7 1.74056
\(709\) 2.21326e7 1.65355 0.826773 0.562535i \(-0.190174\pi\)
0.826773 + 0.562535i \(0.190174\pi\)
\(710\) −9.01895e6 −0.671445
\(711\) −1.40596e7 −1.04303
\(712\) −1.57432e6 −0.116384
\(713\) −1.01930e7 −0.750896
\(714\) 0 0
\(715\) −7.94762e6 −0.581396
\(716\) 1.09476e7 0.798061
\(717\) −1.35689e7 −0.985704
\(718\) 2.65789e7 1.92409
\(719\) −7.23196e6 −0.521716 −0.260858 0.965377i \(-0.584005\pi\)
−0.260858 + 0.965377i \(0.584005\pi\)
\(720\) 2.55461e7 1.83651
\(721\) 0 0
\(722\) 1.93506e7 1.38150
\(723\) −3.25123e7 −2.31314
\(724\) −1.26374e7 −0.896004
\(725\) −1.19711e7 −0.845843
\(726\) −3.47581e6 −0.244745
\(727\) −1.70200e7 −1.19433 −0.597163 0.802120i \(-0.703705\pi\)
−0.597163 + 0.802120i \(0.703705\pi\)
\(728\) 0 0
\(729\) −2.08788e7 −1.45508
\(730\) −4.54039e7 −3.15345
\(731\) −1.13977e7 −0.788904
\(732\) 1.93486e7 1.33466
\(733\) 1.00011e6 0.0687525 0.0343763 0.999409i \(-0.489056\pi\)
0.0343763 + 0.999409i \(0.489056\pi\)
\(734\) −1.53857e7 −1.05409
\(735\) 0 0
\(736\) −2.86671e7 −1.95070
\(737\) −2.37991e7 −1.61396
\(738\) −2.27672e7 −1.53875
\(739\) 3.25979e6 0.219572 0.109786 0.993955i \(-0.464983\pi\)
0.109786 + 0.993955i \(0.464983\pi\)
\(740\) 4.42040e6 0.296744
\(741\) 38387.1 0.00256826
\(742\) 0 0
\(743\) 1.36125e7 0.904617 0.452309 0.891861i \(-0.350601\pi\)
0.452309 + 0.891861i \(0.350601\pi\)
\(744\) 1.51679e6 0.100460
\(745\) 7.48475e6 0.494068
\(746\) 2.71878e7 1.78865
\(747\) 428536. 0.0280986
\(748\) 1.36214e7 0.890158
\(749\) 0 0
\(750\) −1.00092e7 −0.649747
\(751\) 6.56544e6 0.424780 0.212390 0.977185i \(-0.431875\pi\)
0.212390 + 0.977185i \(0.431875\pi\)
\(752\) −1.88931e7 −1.21831
\(753\) 7.60160e6 0.488560
\(754\) −9.86600e6 −0.631994
\(755\) 3.40110e7 2.17146
\(756\) 0 0
\(757\) 2.62531e7 1.66510 0.832551 0.553948i \(-0.186879\pi\)
0.832551 + 0.553948i \(0.186879\pi\)
\(758\) 3.29244e6 0.208135
\(759\) 3.60409e7 2.27086
\(760\) 10985.5 0.000689901 0
\(761\) −5.25111e6 −0.328692 −0.164346 0.986403i \(-0.552551\pi\)
−0.164346 + 0.986403i \(0.552551\pi\)
\(762\) 3.58000e7 2.23355
\(763\) 0 0
\(764\) 1.64549e7 1.01991
\(765\) 2.54381e7 1.57156
\(766\) −2.08592e7 −1.28448
\(767\) −8.56327e6 −0.525595
\(768\) 2.85337e7 1.74564
\(769\) −1.77307e7 −1.08121 −0.540605 0.841277i \(-0.681805\pi\)
−0.540605 + 0.841277i \(0.681805\pi\)
\(770\) 0 0
\(771\) 4.34156e7 2.63033
\(772\) 1.49581e7 0.903305
\(773\) 3.82592e6 0.230296 0.115148 0.993348i \(-0.463266\pi\)
0.115148 + 0.993348i \(0.463266\pi\)
\(774\) −2.50138e7 −1.50082
\(775\) 6.74898e6 0.403631
\(776\) 2.48807e6 0.148323
\(777\) 0 0
\(778\) −2.42406e7 −1.43580
\(779\) −60787.3 −0.00358896
\(780\) 1.28110e7 0.753957
\(781\) 6.59126e6 0.386671
\(782\) −3.11769e7 −1.82312
\(783\) −7.90117e6 −0.460561
\(784\) 0 0
\(785\) 1.32980e7 0.770218
\(786\) 4.34692e7 2.50972
\(787\) 2.94263e7 1.69355 0.846777 0.531948i \(-0.178540\pi\)
0.846777 + 0.531948i \(0.178540\pi\)
\(788\) −8.68019e6 −0.497982
\(789\) −1.07767e7 −0.616304
\(790\) −2.63157e7 −1.50020
\(791\) 0 0
\(792\) −3.00697e6 −0.170340
\(793\) −7.13700e6 −0.403026
\(794\) −4.79265e6 −0.269789
\(795\) 6.90122e7 3.87265
\(796\) −1.72102e7 −0.962730
\(797\) 1.35805e7 0.757305 0.378653 0.925539i \(-0.376387\pi\)
0.378653 + 0.925539i \(0.376387\pi\)
\(798\) 0 0
\(799\) −1.88132e7 −1.04255
\(800\) 1.89810e7 1.04856
\(801\) −2.13611e7 −1.17637
\(802\) −2.20559e7 −1.21085
\(803\) 3.31822e7 1.81600
\(804\) 3.83624e7 2.09299
\(805\) 0 0
\(806\) 5.56218e6 0.301583
\(807\) 9.80625e6 0.530053
\(808\) 410778. 0.0221350
\(809\) −1.15714e7 −0.621604 −0.310802 0.950475i \(-0.600598\pi\)
−0.310802 + 0.950475i \(0.600598\pi\)
\(810\) −2.21927e7 −1.18849
\(811\) −3.52530e6 −0.188210 −0.0941052 0.995562i \(-0.529999\pi\)
−0.0941052 + 0.995562i \(0.529999\pi\)
\(812\) 0 0
\(813\) −2.11756e7 −1.12360
\(814\) −6.78602e6 −0.358966
\(815\) −1.80942e7 −0.954214
\(816\) 2.88046e7 1.51439
\(817\) −66785.7 −0.00350049
\(818\) −1.78457e7 −0.932505
\(819\) 0 0
\(820\) −2.02867e7 −1.05360
\(821\) 1.78951e7 0.926567 0.463283 0.886210i \(-0.346671\pi\)
0.463283 + 0.886210i \(0.346671\pi\)
\(822\) 3.69259e7 1.90612
\(823\) −3.61421e7 −1.86000 −0.930001 0.367557i \(-0.880194\pi\)
−0.930001 + 0.367557i \(0.880194\pi\)
\(824\) 726226. 0.0372610
\(825\) −2.38633e7 −1.22066
\(826\) 0 0
\(827\) −1.00605e7 −0.511512 −0.255756 0.966741i \(-0.582324\pi\)
−0.255756 + 0.966741i \(0.582324\pi\)
\(828\) −3.25727e7 −1.65112
\(829\) 2.03654e7 1.02921 0.514607 0.857426i \(-0.327938\pi\)
0.514607 + 0.857426i \(0.327938\pi\)
\(830\) 802102. 0.0404142
\(831\) −1.05279e7 −0.528858
\(832\) 6.69166e6 0.335140
\(833\) 0 0
\(834\) −9.73865e6 −0.484823
\(835\) −8.69263e6 −0.431454
\(836\) 79815.4 0.00394976
\(837\) 4.45446e6 0.219777
\(838\) 2.07310e7 1.01979
\(839\) −5.95014e6 −0.291825 −0.145912 0.989298i \(-0.546612\pi\)
−0.145912 + 0.989298i \(0.546612\pi\)
\(840\) 0 0
\(841\) 4.53905e6 0.221297
\(842\) −2.29508e7 −1.11562
\(843\) −1.80745e7 −0.875987
\(844\) −4.08611e6 −0.197448
\(845\) 2.28524e7 1.10101
\(846\) −4.12881e7 −1.98335
\(847\) 0 0
\(848\) 4.38143e7 2.09231
\(849\) −5.01462e7 −2.38764
\(850\) 2.06428e7 0.979987
\(851\) 7.39411e6 0.349995
\(852\) −1.06247e7 −0.501436
\(853\) 1.59836e7 0.752146 0.376073 0.926590i \(-0.377274\pi\)
0.376073 + 0.926590i \(0.377274\pi\)
\(854\) 0 0
\(855\) 149056. 0.00697325
\(856\) 188092. 0.00877378
\(857\) −2.34591e6 −0.109109 −0.0545544 0.998511i \(-0.517374\pi\)
−0.0545544 + 0.998511i \(0.517374\pi\)
\(858\) −1.96669e7 −0.912050
\(859\) −1.30223e7 −0.602152 −0.301076 0.953600i \(-0.597346\pi\)
−0.301076 + 0.953600i \(0.597346\pi\)
\(860\) −2.22886e7 −1.02763
\(861\) 0 0
\(862\) 1.91130e7 0.876112
\(863\) 1.96688e7 0.898983 0.449492 0.893285i \(-0.351605\pi\)
0.449492 + 0.893285i \(0.351605\pi\)
\(864\) 1.25278e7 0.570941
\(865\) −2.00345e7 −0.910414
\(866\) 1.65459e7 0.749715
\(867\) −4.71019e6 −0.212809
\(868\) 0 0
\(869\) 1.92322e7 0.863931
\(870\) −6.83273e7 −3.06053
\(871\) −1.41505e7 −0.632015
\(872\) −252988. −0.0112670
\(873\) 3.37592e7 1.49919
\(874\) −182683. −0.00808946
\(875\) 0 0
\(876\) −5.34874e7 −2.35500
\(877\) 2.34581e7 1.02990 0.514950 0.857221i \(-0.327811\pi\)
0.514950 + 0.857221i \(0.327811\pi\)
\(878\) −3.63217e7 −1.59012
\(879\) −5.70724e7 −2.49146
\(880\) −3.49446e7 −1.52115
\(881\) 4.59257e6 0.199350 0.0996750 0.995020i \(-0.468220\pi\)
0.0996750 + 0.995020i \(0.468220\pi\)
\(882\) 0 0
\(883\) −1.23402e7 −0.532622 −0.266311 0.963887i \(-0.585805\pi\)
−0.266311 + 0.963887i \(0.585805\pi\)
\(884\) 8.09903e6 0.348580
\(885\) −5.93052e7 −2.54527
\(886\) −3.46149e7 −1.48142
\(887\) 1.36554e7 0.582769 0.291384 0.956606i \(-0.405884\pi\)
0.291384 + 0.956606i \(0.405884\pi\)
\(888\) −1.10029e6 −0.0468247
\(889\) 0 0
\(890\) −3.99822e7 −1.69197
\(891\) 1.62189e7 0.684428
\(892\) 14262.2 0.000600171 0
\(893\) −110237. −0.00462594
\(894\) 1.85215e7 0.775056
\(895\) −2.79665e7 −1.16703
\(896\) 0 0
\(897\) 2.14293e7 0.889255
\(898\) 5.23744e7 2.16735
\(899\) −1.41226e7 −0.582795
\(900\) 2.15670e7 0.887529
\(901\) 4.36291e7 1.79046
\(902\) 3.11433e7 1.27453
\(903\) 0 0
\(904\) 1.48987e6 0.0606355
\(905\) 3.22831e7 1.31025
\(906\) 8.41624e7 3.40642
\(907\) 7.39599e6 0.298523 0.149262 0.988798i \(-0.452310\pi\)
0.149262 + 0.988798i \(0.452310\pi\)
\(908\) 1.72678e7 0.695060
\(909\) 5.57362e6 0.223732
\(910\) 0 0
\(911\) −3.51041e7 −1.40140 −0.700699 0.713457i \(-0.747129\pi\)
−0.700699 + 0.713457i \(0.747129\pi\)
\(912\) 168783. 0.00671955
\(913\) −586195. −0.0232737
\(914\) 4.59795e7 1.82053
\(915\) −4.94275e7 −1.95171
\(916\) −1.04323e6 −0.0410812
\(917\) 0 0
\(918\) 1.36246e7 0.533602
\(919\) −4.81337e7 −1.88001 −0.940005 0.341159i \(-0.889180\pi\)
−0.940005 + 0.341159i \(0.889180\pi\)
\(920\) 6.13258e6 0.238877
\(921\) 5.74626e7 2.23221
\(922\) −1.20949e7 −0.468572
\(923\) 3.91905e6 0.151418
\(924\) 0 0
\(925\) −4.89577e6 −0.188134
\(926\) 3.07729e7 1.17935
\(927\) 9.85376e6 0.376619
\(928\) −3.97188e7 −1.51400
\(929\) −1.83602e7 −0.697971 −0.348986 0.937128i \(-0.613474\pi\)
−0.348986 + 0.937128i \(0.613474\pi\)
\(930\) 3.85210e7 1.46046
\(931\) 0 0
\(932\) 3.62924e7 1.36860
\(933\) −2.71557e7 −1.02131
\(934\) 5.32665e7 1.99796
\(935\) −3.47969e7 −1.30170
\(936\) −1.78789e6 −0.0667041
\(937\) 2.27081e7 0.844951 0.422475 0.906374i \(-0.361161\pi\)
0.422475 + 0.906374i \(0.361161\pi\)
\(938\) 0 0
\(939\) −3.89718e7 −1.44240
\(940\) −3.67898e7 −1.35802
\(941\) 4.15801e7 1.53078 0.765389 0.643568i \(-0.222547\pi\)
0.765389 + 0.643568i \(0.222547\pi\)
\(942\) 3.29069e7 1.20826
\(943\) −3.39340e7 −1.24267
\(944\) −3.76515e7 −1.37516
\(945\) 0 0
\(946\) 3.42165e7 1.24310
\(947\) 1.55341e7 0.562874 0.281437 0.959580i \(-0.409189\pi\)
0.281437 + 0.959580i \(0.409189\pi\)
\(948\) −3.10009e7 −1.12035
\(949\) 1.97296e7 0.711135
\(950\) 120958. 0.00434835
\(951\) 1.93172e7 0.692618
\(952\) 0 0
\(953\) 3.94908e7 1.40852 0.704262 0.709940i \(-0.251278\pi\)
0.704262 + 0.709940i \(0.251278\pi\)
\(954\) 9.57499e7 3.40618
\(955\) −4.20354e7 −1.49144
\(956\) −1.67748e7 −0.593627
\(957\) 4.99352e7 1.76249
\(958\) 4.06701e7 1.43173
\(959\) 0 0
\(960\) 4.63433e7 1.62296
\(961\) −2.06672e7 −0.721894
\(962\) −4.03485e6 −0.140569
\(963\) 2.55212e6 0.0886819
\(964\) −4.01941e7 −1.39306
\(965\) −3.82118e7 −1.32093
\(966\) 0 0
\(967\) −1.87472e7 −0.644718 −0.322359 0.946617i \(-0.604476\pi\)
−0.322359 + 0.946617i \(0.604476\pi\)
\(968\) 432231. 0.0148261
\(969\) 168069. 0.00575014
\(970\) 6.31880e7 2.15628
\(971\) −5.35423e7 −1.82242 −0.911211 0.411940i \(-0.864851\pi\)
−0.911211 + 0.411940i \(0.864851\pi\)
\(972\) −3.72974e7 −1.26623
\(973\) 0 0
\(974\) 1.21132e6 0.0409131
\(975\) −1.41887e7 −0.478004
\(976\) −3.13804e7 −1.05447
\(977\) 4.46104e7 1.49520 0.747600 0.664149i \(-0.231206\pi\)
0.747600 + 0.664149i \(0.231206\pi\)
\(978\) −4.47754e7 −1.49690
\(979\) 2.92200e7 0.974368
\(980\) 0 0
\(981\) −3.43265e6 −0.113883
\(982\) 1.26566e7 0.418829
\(983\) −2.00068e7 −0.660380 −0.330190 0.943914i \(-0.607113\pi\)
−0.330190 + 0.943914i \(0.607113\pi\)
\(984\) 5.04960e6 0.166253
\(985\) 2.21742e7 0.728213
\(986\) −4.31961e7 −1.41498
\(987\) 0 0
\(988\) 47456.9 0.00154670
\(989\) −3.72826e7 −1.21204
\(990\) −7.63664e7 −2.47636
\(991\) 3.24635e7 1.05005 0.525026 0.851086i \(-0.324056\pi\)
0.525026 + 0.851086i \(0.324056\pi\)
\(992\) 2.23923e7 0.722471
\(993\) −2.25004e6 −0.0724130
\(994\) 0 0
\(995\) 4.39649e7 1.40783
\(996\) 944905. 0.0301814
\(997\) 2.78940e7 0.888736 0.444368 0.895844i \(-0.353428\pi\)
0.444368 + 0.895844i \(0.353428\pi\)
\(998\) −3.20945e7 −1.02001
\(999\) −3.23130e6 −0.102439
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 49.6.a.g.1.2 yes 4
3.2 odd 2 441.6.a.z.1.4 4
4.3 odd 2 784.6.a.bf.1.1 4
7.2 even 3 49.6.c.h.18.3 8
7.3 odd 6 49.6.c.h.30.4 8
7.4 even 3 49.6.c.h.30.3 8
7.5 odd 6 49.6.c.h.18.4 8
7.6 odd 2 inner 49.6.a.g.1.1 4
21.20 even 2 441.6.a.z.1.3 4
28.27 even 2 784.6.a.bf.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.6.a.g.1.1 4 7.6 odd 2 inner
49.6.a.g.1.2 yes 4 1.1 even 1 trivial
49.6.c.h.18.3 8 7.2 even 3
49.6.c.h.18.4 8 7.5 odd 6
49.6.c.h.30.3 8 7.4 even 3
49.6.c.h.30.4 8 7.3 odd 6
441.6.a.z.1.3 4 21.20 even 2
441.6.a.z.1.4 4 3.2 odd 2
784.6.a.bf.1.1 4 4.3 odd 2
784.6.a.bf.1.4 4 28.27 even 2