Properties

Label 825.6.a.q.1.6
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(132.316651346\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - x^{7} - 176x^{6} + 272x^{5} + 9055x^{4} - 15851x^{3} - 118840x^{2} + 149572x - 33248 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(5.43413\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+6.43413 q^{2} -9.00000 q^{3} +9.39797 q^{4} -57.9071 q^{6} -9.96708 q^{7} -145.424 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+6.43413 q^{2} -9.00000 q^{3} +9.39797 q^{4} -57.9071 q^{6} -9.96708 q^{7} -145.424 q^{8} +81.0000 q^{9} -121.000 q^{11} -84.5818 q^{12} +813.596 q^{13} -64.1295 q^{14} -1236.41 q^{16} -616.038 q^{17} +521.164 q^{18} +1989.66 q^{19} +89.7037 q^{21} -778.529 q^{22} -545.944 q^{23} +1308.82 q^{24} +5234.78 q^{26} -729.000 q^{27} -93.6704 q^{28} -670.788 q^{29} -1229.55 q^{31} -3301.66 q^{32} +1089.00 q^{33} -3963.67 q^{34} +761.236 q^{36} +9820.34 q^{37} +12801.8 q^{38} -7322.36 q^{39} +5587.84 q^{41} +577.165 q^{42} -6305.91 q^{43} -1137.15 q^{44} -3512.67 q^{46} -10680.8 q^{47} +11127.7 q^{48} -16707.7 q^{49} +5544.35 q^{51} +7646.15 q^{52} +21821.5 q^{53} -4690.48 q^{54} +1449.46 q^{56} -17907.0 q^{57} -4315.93 q^{58} -2157.19 q^{59} -6743.79 q^{61} -7911.10 q^{62} -807.334 q^{63} +18321.9 q^{64} +7006.76 q^{66} -38984.7 q^{67} -5789.51 q^{68} +4913.49 q^{69} +25270.9 q^{71} -11779.4 q^{72} -23958.4 q^{73} +63185.3 q^{74} +18698.8 q^{76} +1206.02 q^{77} -47113.0 q^{78} -101544. q^{79} +6561.00 q^{81} +35952.9 q^{82} -14547.0 q^{83} +843.033 q^{84} -40573.0 q^{86} +6037.09 q^{87} +17596.3 q^{88} -80447.7 q^{89} -8109.18 q^{91} -5130.76 q^{92} +11066.0 q^{93} -68721.6 q^{94} +29715.0 q^{96} -78631.2 q^{97} -107499. q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 9 q^{2} - 72 q^{3} + 107 q^{4} - 81 q^{6} - 66 q^{7} + 207 q^{8} + 648 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 9 q^{2} - 72 q^{3} + 107 q^{4} - 81 q^{6} - 66 q^{7} + 207 q^{8} + 648 q^{9} - 968 q^{11} - 963 q^{12} + 382 q^{13} + 1048 q^{14} - 325 q^{16} + 288 q^{17} + 729 q^{18} - 988 q^{19} + 594 q^{21} - 1089 q^{22} + 5972 q^{23} - 1863 q^{24} - 14579 q^{26} - 5832 q^{27} + 5942 q^{28} + 1032 q^{29} - 4682 q^{31} + 3863 q^{32} + 8712 q^{33} + 2206 q^{34} + 8667 q^{36} - 17200 q^{37} + 11011 q^{38} - 3438 q^{39} - 13220 q^{41} - 9432 q^{42} + 22872 q^{43} - 12947 q^{44} + 9101 q^{46} + 6700 q^{47} + 2925 q^{48} - 43466 q^{49} - 2592 q^{51} + 5009 q^{52} + 6224 q^{53} - 6561 q^{54} + 20992 q^{56} + 8892 q^{57} + 33015 q^{58} - 77556 q^{59} + 11554 q^{61} + 12135 q^{62} - 5346 q^{63} - 149917 q^{64} + 9801 q^{66} + 20894 q^{67} + 91776 q^{68} - 53748 q^{69} - 21648 q^{71} + 16767 q^{72} + 64660 q^{73} - 179522 q^{74} + 24401 q^{76} + 7986 q^{77} + 131211 q^{78} - 22660 q^{79} + 52488 q^{81} - 56080 q^{82} + 100390 q^{83} - 53478 q^{84} + 47271 q^{86} - 9288 q^{87} - 25047 q^{88} - 25578 q^{89} + 73250 q^{91} + 95311 q^{92} + 42138 q^{93} - 170120 q^{94} - 34767 q^{96} + 142828 q^{97} + 303397 q^{98} - 78408 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 6.43413 1.13740 0.568702 0.822544i \(-0.307446\pi\)
0.568702 + 0.822544i \(0.307446\pi\)
\(3\) −9.00000 −0.577350
\(4\) 9.39797 0.293687
\(5\) 0 0
\(6\) −57.9071 −0.656680
\(7\) −9.96708 −0.0768817 −0.0384408 0.999261i \(-0.512239\pi\)
−0.0384408 + 0.999261i \(0.512239\pi\)
\(8\) −145.424 −0.803363
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) −84.5818 −0.169560
\(13\) 813.596 1.33521 0.667606 0.744515i \(-0.267319\pi\)
0.667606 + 0.744515i \(0.267319\pi\)
\(14\) −64.1295 −0.0874455
\(15\) 0 0
\(16\) −1236.41 −1.20743
\(17\) −616.038 −0.516994 −0.258497 0.966012i \(-0.583227\pi\)
−0.258497 + 0.966012i \(0.583227\pi\)
\(18\) 521.164 0.379134
\(19\) 1989.66 1.26443 0.632217 0.774792i \(-0.282145\pi\)
0.632217 + 0.774792i \(0.282145\pi\)
\(20\) 0 0
\(21\) 89.7037 0.0443877
\(22\) −778.529 −0.342940
\(23\) −545.944 −0.215193 −0.107597 0.994195i \(-0.534315\pi\)
−0.107597 + 0.994195i \(0.534315\pi\)
\(24\) 1308.82 0.463822
\(25\) 0 0
\(26\) 5234.78 1.51868
\(27\) −729.000 −0.192450
\(28\) −93.6704 −0.0225791
\(29\) −670.788 −0.148112 −0.0740560 0.997254i \(-0.523594\pi\)
−0.0740560 + 0.997254i \(0.523594\pi\)
\(30\) 0 0
\(31\) −1229.55 −0.229796 −0.114898 0.993377i \(-0.536654\pi\)
−0.114898 + 0.993377i \(0.536654\pi\)
\(32\) −3301.66 −0.569977
\(33\) 1089.00 0.174078
\(34\) −3963.67 −0.588031
\(35\) 0 0
\(36\) 761.236 0.0978956
\(37\) 9820.34 1.17929 0.589647 0.807661i \(-0.299267\pi\)
0.589647 + 0.807661i \(0.299267\pi\)
\(38\) 12801.8 1.43817
\(39\) −7322.36 −0.770885
\(40\) 0 0
\(41\) 5587.84 0.519140 0.259570 0.965724i \(-0.416419\pi\)
0.259570 + 0.965724i \(0.416419\pi\)
\(42\) 577.165 0.0504867
\(43\) −6305.91 −0.520088 −0.260044 0.965597i \(-0.583737\pi\)
−0.260044 + 0.965597i \(0.583737\pi\)
\(44\) −1137.15 −0.0885499
\(45\) 0 0
\(46\) −3512.67 −0.244761
\(47\) −10680.8 −0.705276 −0.352638 0.935760i \(-0.614715\pi\)
−0.352638 + 0.935760i \(0.614715\pi\)
\(48\) 11127.7 0.697113
\(49\) −16707.7 −0.994089
\(50\) 0 0
\(51\) 5544.35 0.298487
\(52\) 7646.15 0.392134
\(53\) 21821.5 1.06708 0.533538 0.845776i \(-0.320862\pi\)
0.533538 + 0.845776i \(0.320862\pi\)
\(54\) −4690.48 −0.218893
\(55\) 0 0
\(56\) 1449.46 0.0617639
\(57\) −17907.0 −0.730021
\(58\) −4315.93 −0.168463
\(59\) −2157.19 −0.0806788 −0.0403394 0.999186i \(-0.512844\pi\)
−0.0403394 + 0.999186i \(0.512844\pi\)
\(60\) 0 0
\(61\) −6743.79 −0.232049 −0.116024 0.993246i \(-0.537015\pi\)
−0.116024 + 0.993246i \(0.537015\pi\)
\(62\) −7911.10 −0.261371
\(63\) −807.334 −0.0256272
\(64\) 18321.9 0.559141
\(65\) 0 0
\(66\) 7006.76 0.197997
\(67\) −38984.7 −1.06098 −0.530490 0.847691i \(-0.677992\pi\)
−0.530490 + 0.847691i \(0.677992\pi\)
\(68\) −5789.51 −0.151834
\(69\) 4913.49 0.124242
\(70\) 0 0
\(71\) 25270.9 0.594941 0.297471 0.954731i \(-0.403857\pi\)
0.297471 + 0.954731i \(0.403857\pi\)
\(72\) −11779.4 −0.267788
\(73\) −23958.4 −0.526201 −0.263100 0.964768i \(-0.584745\pi\)
−0.263100 + 0.964768i \(0.584745\pi\)
\(74\) 63185.3 1.34133
\(75\) 0 0
\(76\) 18698.8 0.371347
\(77\) 1206.02 0.0231807
\(78\) −47113.0 −0.876808
\(79\) −101544. −1.83057 −0.915284 0.402810i \(-0.868034\pi\)
−0.915284 + 0.402810i \(0.868034\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 35952.9 0.590472
\(83\) −14547.0 −0.231781 −0.115891 0.993262i \(-0.536972\pi\)
−0.115891 + 0.993262i \(0.536972\pi\)
\(84\) 843.033 0.0130361
\(85\) 0 0
\(86\) −40573.0 −0.591550
\(87\) 6037.09 0.0855125
\(88\) 17596.3 0.242223
\(89\) −80447.7 −1.07656 −0.538281 0.842766i \(-0.680926\pi\)
−0.538281 + 0.842766i \(0.680926\pi\)
\(90\) 0 0
\(91\) −8109.18 −0.102653
\(92\) −5130.76 −0.0631993
\(93\) 11066.0 0.132673
\(94\) −68721.6 −0.802184
\(95\) 0 0
\(96\) 29715.0 0.329077
\(97\) −78631.2 −0.848527 −0.424264 0.905539i \(-0.639467\pi\)
−0.424264 + 0.905539i \(0.639467\pi\)
\(98\) −107499. −1.13068
\(99\) −9801.00 −0.100504
\(100\) 0 0
\(101\) 22024.1 0.214830 0.107415 0.994214i \(-0.465743\pi\)
0.107415 + 0.994214i \(0.465743\pi\)
\(102\) 35673.0 0.339500
\(103\) −85005.4 −0.789503 −0.394751 0.918788i \(-0.629169\pi\)
−0.394751 + 0.918788i \(0.629169\pi\)
\(104\) −118317. −1.07266
\(105\) 0 0
\(106\) 140403. 1.21370
\(107\) −129131. −1.09036 −0.545181 0.838319i \(-0.683539\pi\)
−0.545181 + 0.838319i \(0.683539\pi\)
\(108\) −6851.12 −0.0565200
\(109\) −153961. −1.24121 −0.620605 0.784124i \(-0.713113\pi\)
−0.620605 + 0.784124i \(0.713113\pi\)
\(110\) 0 0
\(111\) −88383.0 −0.680866
\(112\) 12323.4 0.0928296
\(113\) 139946. 1.03102 0.515508 0.856885i \(-0.327603\pi\)
0.515508 + 0.856885i \(0.327603\pi\)
\(114\) −115216. −0.830328
\(115\) 0 0
\(116\) −6304.04 −0.0434985
\(117\) 65901.3 0.445071
\(118\) −13879.7 −0.0917643
\(119\) 6140.10 0.0397474
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −43390.4 −0.263933
\(123\) −50290.6 −0.299726
\(124\) −11555.3 −0.0674881
\(125\) 0 0
\(126\) −5194.49 −0.0291485
\(127\) −119781. −0.658992 −0.329496 0.944157i \(-0.606879\pi\)
−0.329496 + 0.944157i \(0.606879\pi\)
\(128\) 223539. 1.20595
\(129\) 56753.2 0.300273
\(130\) 0 0
\(131\) 167797. 0.854293 0.427147 0.904182i \(-0.359519\pi\)
0.427147 + 0.904182i \(0.359519\pi\)
\(132\) 10234.4 0.0511243
\(133\) −19831.2 −0.0972118
\(134\) −250832. −1.20676
\(135\) 0 0
\(136\) 89586.9 0.415334
\(137\) 23349.3 0.106285 0.0531425 0.998587i \(-0.483076\pi\)
0.0531425 + 0.998587i \(0.483076\pi\)
\(138\) 31614.0 0.141313
\(139\) −108551. −0.476537 −0.238269 0.971199i \(-0.576580\pi\)
−0.238269 + 0.971199i \(0.576580\pi\)
\(140\) 0 0
\(141\) 96127.2 0.407191
\(142\) 162596. 0.676688
\(143\) −98445.1 −0.402582
\(144\) −100149. −0.402478
\(145\) 0 0
\(146\) −154152. −0.598502
\(147\) 150369. 0.573938
\(148\) 92291.3 0.346343
\(149\) −276956. −1.02199 −0.510993 0.859585i \(-0.670722\pi\)
−0.510993 + 0.859585i \(0.670722\pi\)
\(150\) 0 0
\(151\) 326462. 1.16517 0.582587 0.812769i \(-0.302041\pi\)
0.582587 + 0.812769i \(0.302041\pi\)
\(152\) −289346. −1.01580
\(153\) −49899.1 −0.172331
\(154\) 7759.66 0.0263658
\(155\) 0 0
\(156\) −68815.4 −0.226399
\(157\) −366289. −1.18597 −0.592986 0.805213i \(-0.702051\pi\)
−0.592986 + 0.805213i \(0.702051\pi\)
\(158\) −653346. −2.08209
\(159\) −196394. −0.616077
\(160\) 0 0
\(161\) 5441.46 0.0165444
\(162\) 42214.3 0.126378
\(163\) −642723. −1.89476 −0.947381 0.320107i \(-0.896281\pi\)
−0.947381 + 0.320107i \(0.896281\pi\)
\(164\) 52514.4 0.152464
\(165\) 0 0
\(166\) −93597.3 −0.263629
\(167\) 259189. 0.719159 0.359579 0.933115i \(-0.382920\pi\)
0.359579 + 0.933115i \(0.382920\pi\)
\(168\) −13045.1 −0.0356594
\(169\) 290645. 0.782792
\(170\) 0 0
\(171\) 161163. 0.421478
\(172\) −59262.8 −0.152743
\(173\) 242721. 0.616585 0.308292 0.951292i \(-0.400242\pi\)
0.308292 + 0.951292i \(0.400242\pi\)
\(174\) 38843.4 0.0972622
\(175\) 0 0
\(176\) 149606. 0.364055
\(177\) 19414.8 0.0465799
\(178\) −517611. −1.22448
\(179\) −214184. −0.499638 −0.249819 0.968293i \(-0.580371\pi\)
−0.249819 + 0.968293i \(0.580371\pi\)
\(180\) 0 0
\(181\) −247705. −0.562003 −0.281002 0.959707i \(-0.590667\pi\)
−0.281002 + 0.959707i \(0.590667\pi\)
\(182\) −52175.5 −0.116758
\(183\) 60694.1 0.133973
\(184\) 79393.4 0.172878
\(185\) 0 0
\(186\) 71199.9 0.150903
\(187\) 74540.6 0.155880
\(188\) −100378. −0.207130
\(189\) 7266.00 0.0147959
\(190\) 0 0
\(191\) −410762. −0.814718 −0.407359 0.913268i \(-0.633550\pi\)
−0.407359 + 0.913268i \(0.633550\pi\)
\(192\) −164897. −0.322820
\(193\) 150990. 0.291780 0.145890 0.989301i \(-0.453395\pi\)
0.145890 + 0.989301i \(0.453395\pi\)
\(194\) −505923. −0.965118
\(195\) 0 0
\(196\) −157018. −0.291951
\(197\) 151624. 0.278357 0.139178 0.990267i \(-0.455554\pi\)
0.139178 + 0.990267i \(0.455554\pi\)
\(198\) −63060.9 −0.114313
\(199\) 464676. 0.831797 0.415898 0.909411i \(-0.363467\pi\)
0.415898 + 0.909411i \(0.363467\pi\)
\(200\) 0 0
\(201\) 350862. 0.612557
\(202\) 141706. 0.244349
\(203\) 6685.80 0.0113871
\(204\) 52105.6 0.0876615
\(205\) 0 0
\(206\) −546936. −0.897983
\(207\) −44221.4 −0.0717310
\(208\) −1.00594e6 −1.61218
\(209\) −240749. −0.381241
\(210\) 0 0
\(211\) −1.17253e6 −1.81308 −0.906541 0.422117i \(-0.861287\pi\)
−0.906541 + 0.422117i \(0.861287\pi\)
\(212\) 205078. 0.313386
\(213\) −227438. −0.343490
\(214\) −830844. −1.24018
\(215\) 0 0
\(216\) 106014. 0.154607
\(217\) 12255.1 0.0176671
\(218\) −990606. −1.41176
\(219\) 215626. 0.303802
\(220\) 0 0
\(221\) −501206. −0.690297
\(222\) −568668. −0.774419
\(223\) 494387. 0.665740 0.332870 0.942973i \(-0.391983\pi\)
0.332870 + 0.942973i \(0.391983\pi\)
\(224\) 32907.9 0.0438208
\(225\) 0 0
\(226\) 900432. 1.17268
\(227\) 219419. 0.282624 0.141312 0.989965i \(-0.454868\pi\)
0.141312 + 0.989965i \(0.454868\pi\)
\(228\) −168289. −0.214397
\(229\) −1.23652e6 −1.55816 −0.779080 0.626925i \(-0.784313\pi\)
−0.779080 + 0.626925i \(0.784313\pi\)
\(230\) 0 0
\(231\) −10854.2 −0.0133834
\(232\) 97548.8 0.118988
\(233\) −84155.4 −0.101553 −0.0507764 0.998710i \(-0.516170\pi\)
−0.0507764 + 0.998710i \(0.516170\pi\)
\(234\) 424017. 0.506225
\(235\) 0 0
\(236\) −20273.3 −0.0236943
\(237\) 913894. 1.05688
\(238\) 39506.2 0.0452088
\(239\) −113266. −0.128264 −0.0641320 0.997941i \(-0.520428\pi\)
−0.0641320 + 0.997941i \(0.520428\pi\)
\(240\) 0 0
\(241\) −616377. −0.683603 −0.341801 0.939772i \(-0.611037\pi\)
−0.341801 + 0.939772i \(0.611037\pi\)
\(242\) 94202.0 0.103400
\(243\) −59049.0 −0.0641500
\(244\) −63378.0 −0.0681497
\(245\) 0 0
\(246\) −323576. −0.340909
\(247\) 1.61878e6 1.68829
\(248\) 178807. 0.184610
\(249\) 130923. 0.133819
\(250\) 0 0
\(251\) −1.34829e6 −1.35082 −0.675412 0.737441i \(-0.736034\pi\)
−0.675412 + 0.737441i \(0.736034\pi\)
\(252\) −7587.30 −0.00752638
\(253\) 66059.2 0.0648831
\(254\) −770689. −0.749540
\(255\) 0 0
\(256\) 851975. 0.812506
\(257\) −715746. −0.675968 −0.337984 0.941152i \(-0.609745\pi\)
−0.337984 + 0.941152i \(0.609745\pi\)
\(258\) 365157. 0.341531
\(259\) −97880.1 −0.0906661
\(260\) 0 0
\(261\) −54333.8 −0.0493707
\(262\) 1.07963e6 0.971676
\(263\) 2.10033e6 1.87239 0.936197 0.351475i \(-0.114320\pi\)
0.936197 + 0.351475i \(0.114320\pi\)
\(264\) −158367. −0.139848
\(265\) 0 0
\(266\) −127596. −0.110569
\(267\) 724029. 0.621553
\(268\) −366377. −0.311596
\(269\) −1.41437e6 −1.19174 −0.595871 0.803080i \(-0.703193\pi\)
−0.595871 + 0.803080i \(0.703193\pi\)
\(270\) 0 0
\(271\) 190480. 0.157553 0.0787763 0.996892i \(-0.474899\pi\)
0.0787763 + 0.996892i \(0.474899\pi\)
\(272\) 761678. 0.624236
\(273\) 72982.6 0.0592670
\(274\) 150232. 0.120889
\(275\) 0 0
\(276\) 46176.9 0.0364881
\(277\) 998049. 0.781542 0.390771 0.920488i \(-0.372208\pi\)
0.390771 + 0.920488i \(0.372208\pi\)
\(278\) −698430. −0.542015
\(279\) −99593.8 −0.0765988
\(280\) 0 0
\(281\) −1.43088e6 −1.08103 −0.540515 0.841334i \(-0.681771\pi\)
−0.540515 + 0.841334i \(0.681771\pi\)
\(282\) 618495. 0.463141
\(283\) −2.40496e6 −1.78501 −0.892506 0.451035i \(-0.851055\pi\)
−0.892506 + 0.451035i \(0.851055\pi\)
\(284\) 237495. 0.174726
\(285\) 0 0
\(286\) −633408. −0.457898
\(287\) −55694.5 −0.0399124
\(288\) −267435. −0.189992
\(289\) −1.04035e6 −0.732717
\(290\) 0 0
\(291\) 707681. 0.489897
\(292\) −225161. −0.154538
\(293\) 689179. 0.468990 0.234495 0.972117i \(-0.424656\pi\)
0.234495 + 0.972117i \(0.424656\pi\)
\(294\) 967492. 0.652799
\(295\) 0 0
\(296\) −1.42812e6 −0.947401
\(297\) 88209.0 0.0580259
\(298\) −1.78197e6 −1.16241
\(299\) −444177. −0.287328
\(300\) 0 0
\(301\) 62851.5 0.0399852
\(302\) 2.10050e6 1.32527
\(303\) −198217. −0.124032
\(304\) −2.46005e6 −1.52672
\(305\) 0 0
\(306\) −321057. −0.196010
\(307\) −986212. −0.597206 −0.298603 0.954377i \(-0.596521\pi\)
−0.298603 + 0.954377i \(0.596521\pi\)
\(308\) 11334.1 0.00680786
\(309\) 765049. 0.455819
\(310\) 0 0
\(311\) −1.70281e6 −0.998308 −0.499154 0.866513i \(-0.666356\pi\)
−0.499154 + 0.866513i \(0.666356\pi\)
\(312\) 1.06485e6 0.619301
\(313\) 2.92333e6 1.68662 0.843309 0.537428i \(-0.180604\pi\)
0.843309 + 0.537428i \(0.180604\pi\)
\(314\) −2.35675e6 −1.34893
\(315\) 0 0
\(316\) −954306. −0.537613
\(317\) 1.69795e6 0.949022 0.474511 0.880250i \(-0.342625\pi\)
0.474511 + 0.880250i \(0.342625\pi\)
\(318\) −1.26362e6 −0.700728
\(319\) 81165.3 0.0446574
\(320\) 0 0
\(321\) 1.16218e6 0.629520
\(322\) 35011.1 0.0188177
\(323\) −1.22571e6 −0.653704
\(324\) 61660.1 0.0326319
\(325\) 0 0
\(326\) −4.13536e6 −2.15511
\(327\) 1.38565e6 0.716612
\(328\) −812608. −0.417058
\(329\) 106456. 0.0542228
\(330\) 0 0
\(331\) 1.44074e6 0.722795 0.361397 0.932412i \(-0.382300\pi\)
0.361397 + 0.932412i \(0.382300\pi\)
\(332\) −136712. −0.0680711
\(333\) 795447. 0.393098
\(334\) 1.66765e6 0.817973
\(335\) 0 0
\(336\) −110911. −0.0535952
\(337\) 967187. 0.463912 0.231956 0.972726i \(-0.425487\pi\)
0.231956 + 0.972726i \(0.425487\pi\)
\(338\) 1.87005e6 0.890351
\(339\) −1.25952e6 −0.595257
\(340\) 0 0
\(341\) 148776. 0.0692862
\(342\) 1.03694e6 0.479390
\(343\) 334043. 0.153309
\(344\) 917032. 0.417819
\(345\) 0 0
\(346\) 1.56170e6 0.701306
\(347\) 2.81126e6 1.25336 0.626682 0.779275i \(-0.284413\pi\)
0.626682 + 0.779275i \(0.284413\pi\)
\(348\) 56736.4 0.0251139
\(349\) −1.64221e6 −0.721713 −0.360856 0.932621i \(-0.617516\pi\)
−0.360856 + 0.932621i \(0.617516\pi\)
\(350\) 0 0
\(351\) −593111. −0.256962
\(352\) 399501. 0.171855
\(353\) 2.89812e6 1.23788 0.618941 0.785438i \(-0.287562\pi\)
0.618941 + 0.785438i \(0.287562\pi\)
\(354\) 124917. 0.0529802
\(355\) 0 0
\(356\) −756045. −0.316172
\(357\) −55260.9 −0.0229482
\(358\) −1.37809e6 −0.568290
\(359\) 3.03267e6 1.24191 0.620954 0.783847i \(-0.286745\pi\)
0.620954 + 0.783847i \(0.286745\pi\)
\(360\) 0 0
\(361\) 1.48267e6 0.598792
\(362\) −1.59377e6 −0.639224
\(363\) −131769. −0.0524864
\(364\) −76209.8 −0.0301479
\(365\) 0 0
\(366\) 390514. 0.152382
\(367\) 1.16998e6 0.453431 0.226716 0.973961i \(-0.427201\pi\)
0.226716 + 0.973961i \(0.427201\pi\)
\(368\) 675012. 0.259832
\(369\) 452615. 0.173047
\(370\) 0 0
\(371\) −217497. −0.0820387
\(372\) 103998. 0.0389643
\(373\) 1.47636e6 0.549438 0.274719 0.961525i \(-0.411415\pi\)
0.274719 + 0.961525i \(0.411415\pi\)
\(374\) 479604. 0.177298
\(375\) 0 0
\(376\) 1.55325e6 0.566593
\(377\) −545750. −0.197761
\(378\) 46750.4 0.0168289
\(379\) 4.95103e6 1.77051 0.885253 0.465110i \(-0.153985\pi\)
0.885253 + 0.465110i \(0.153985\pi\)
\(380\) 0 0
\(381\) 1.07803e6 0.380469
\(382\) −2.64290e6 −0.926663
\(383\) 1.41125e6 0.491594 0.245797 0.969321i \(-0.420950\pi\)
0.245797 + 0.969321i \(0.420950\pi\)
\(384\) −2.01185e6 −0.696253
\(385\) 0 0
\(386\) 971492. 0.331872
\(387\) −510779. −0.173363
\(388\) −738974. −0.249201
\(389\) 1.35721e6 0.454750 0.227375 0.973807i \(-0.426986\pi\)
0.227375 + 0.973807i \(0.426986\pi\)
\(390\) 0 0
\(391\) 336322. 0.111253
\(392\) 2.42970e6 0.798615
\(393\) −1.51018e6 −0.493226
\(394\) 975566. 0.316604
\(395\) 0 0
\(396\) −92109.5 −0.0295166
\(397\) 1.77467e6 0.565122 0.282561 0.959249i \(-0.408816\pi\)
0.282561 + 0.959249i \(0.408816\pi\)
\(398\) 2.98978e6 0.946089
\(399\) 178480. 0.0561252
\(400\) 0 0
\(401\) 3.69680e6 1.14806 0.574030 0.818834i \(-0.305379\pi\)
0.574030 + 0.818834i \(0.305379\pi\)
\(402\) 2.25749e6 0.696725
\(403\) −1.00036e6 −0.306827
\(404\) 206982. 0.0630928
\(405\) 0 0
\(406\) 43017.3 0.0129517
\(407\) −1.18826e6 −0.355571
\(408\) −806282. −0.239793
\(409\) −824495. −0.243714 −0.121857 0.992548i \(-0.538885\pi\)
−0.121857 + 0.992548i \(0.538885\pi\)
\(410\) 0 0
\(411\) −210144. −0.0613637
\(412\) −798879. −0.231866
\(413\) 21500.9 0.00620272
\(414\) −284526. −0.0815871
\(415\) 0 0
\(416\) −2.68622e6 −0.761041
\(417\) 976959. 0.275129
\(418\) −1.54901e6 −0.433625
\(419\) −5.87905e6 −1.63596 −0.817979 0.575249i \(-0.804905\pi\)
−0.817979 + 0.575249i \(0.804905\pi\)
\(420\) 0 0
\(421\) 3.48720e6 0.958895 0.479448 0.877570i \(-0.340837\pi\)
0.479448 + 0.877570i \(0.340837\pi\)
\(422\) −7.54420e6 −2.06221
\(423\) −865145. −0.235092
\(424\) −3.17338e6 −0.857250
\(425\) 0 0
\(426\) −1.46336e6 −0.390686
\(427\) 67215.9 0.0178403
\(428\) −1.21357e6 −0.320225
\(429\) 886006. 0.232431
\(430\) 0 0
\(431\) −4.41226e6 −1.14411 −0.572055 0.820215i \(-0.693854\pi\)
−0.572055 + 0.820215i \(0.693854\pi\)
\(432\) 901345. 0.232371
\(433\) 989722. 0.253684 0.126842 0.991923i \(-0.459516\pi\)
0.126842 + 0.991923i \(0.459516\pi\)
\(434\) 78850.6 0.0200947
\(435\) 0 0
\(436\) −1.44692e6 −0.364527
\(437\) −1.08624e6 −0.272097
\(438\) 1.38736e6 0.345545
\(439\) −2.69582e6 −0.667621 −0.333811 0.942640i \(-0.608335\pi\)
−0.333811 + 0.942640i \(0.608335\pi\)
\(440\) 0 0
\(441\) −1.35332e6 −0.331363
\(442\) −3.22482e6 −0.785146
\(443\) 1.84727e6 0.447221 0.223610 0.974679i \(-0.428216\pi\)
0.223610 + 0.974679i \(0.428216\pi\)
\(444\) −830621. −0.199961
\(445\) 0 0
\(446\) 3.18095e6 0.757215
\(447\) 2.49260e6 0.590044
\(448\) −182616. −0.0429877
\(449\) −806296. −0.188746 −0.0943732 0.995537i \(-0.530085\pi\)
−0.0943732 + 0.995537i \(0.530085\pi\)
\(450\) 0 0
\(451\) −676129. −0.156527
\(452\) 1.31521e6 0.302796
\(453\) −2.93816e6 −0.672713
\(454\) 1.41177e6 0.321457
\(455\) 0 0
\(456\) 2.60411e6 0.586472
\(457\) 4.78463e6 1.07166 0.535830 0.844326i \(-0.319999\pi\)
0.535830 + 0.844326i \(0.319999\pi\)
\(458\) −7.95592e6 −1.77226
\(459\) 449092. 0.0994955
\(460\) 0 0
\(461\) −8.05029e6 −1.76425 −0.882123 0.471019i \(-0.843886\pi\)
−0.882123 + 0.471019i \(0.843886\pi\)
\(462\) −69837.0 −0.0152223
\(463\) −646347. −0.140124 −0.0700622 0.997543i \(-0.522320\pi\)
−0.0700622 + 0.997543i \(0.522320\pi\)
\(464\) 829371. 0.178836
\(465\) 0 0
\(466\) −541466. −0.115507
\(467\) −5.97613e6 −1.26803 −0.634013 0.773323i \(-0.718593\pi\)
−0.634013 + 0.773323i \(0.718593\pi\)
\(468\) 619338. 0.130711
\(469\) 388564. 0.0815699
\(470\) 0 0
\(471\) 3.29660e6 0.684721
\(472\) 313709. 0.0648144
\(473\) 763015. 0.156812
\(474\) 5.88011e6 1.20210
\(475\) 0 0
\(476\) 57704.5 0.0116733
\(477\) 1.76754e6 0.355692
\(478\) −728768. −0.145888
\(479\) −7.53887e6 −1.50130 −0.750650 0.660700i \(-0.770260\pi\)
−0.750650 + 0.660700i \(0.770260\pi\)
\(480\) 0 0
\(481\) 7.98979e6 1.57461
\(482\) −3.96585e6 −0.777532
\(483\) −48973.2 −0.00955191
\(484\) 137596. 0.0266988
\(485\) 0 0
\(486\) −379929. −0.0729645
\(487\) −6.72146e6 −1.28423 −0.642113 0.766610i \(-0.721942\pi\)
−0.642113 + 0.766610i \(0.721942\pi\)
\(488\) 980711. 0.186420
\(489\) 5.78451e6 1.09394
\(490\) 0 0
\(491\) −2.38900e6 −0.447212 −0.223606 0.974680i \(-0.571783\pi\)
−0.223606 + 0.974680i \(0.571783\pi\)
\(492\) −472629. −0.0880254
\(493\) 413231. 0.0765730
\(494\) 1.04155e7 1.92026
\(495\) 0 0
\(496\) 1.52024e6 0.277464
\(497\) −251877. −0.0457401
\(498\) 842376. 0.152206
\(499\) −4.78068e6 −0.859485 −0.429742 0.902952i \(-0.641396\pi\)
−0.429742 + 0.902952i \(0.641396\pi\)
\(500\) 0 0
\(501\) −2.33270e6 −0.415206
\(502\) −8.67506e6 −1.53643
\(503\) 7.98014e6 1.40634 0.703170 0.711021i \(-0.251767\pi\)
0.703170 + 0.711021i \(0.251767\pi\)
\(504\) 117406. 0.0205880
\(505\) 0 0
\(506\) 425033. 0.0737983
\(507\) −2.61581e6 −0.451945
\(508\) −1.12570e6 −0.193537
\(509\) 888027. 0.151926 0.0759630 0.997111i \(-0.475797\pi\)
0.0759630 + 0.997111i \(0.475797\pi\)
\(510\) 0 0
\(511\) 238796. 0.0404552
\(512\) −1.67153e6 −0.281798
\(513\) −1.45047e6 −0.243340
\(514\) −4.60520e6 −0.768848
\(515\) 0 0
\(516\) 533365. 0.0881861
\(517\) 1.29238e6 0.212649
\(518\) −629773. −0.103124
\(519\) −2.18449e6 −0.355985
\(520\) 0 0
\(521\) 4.40559e6 0.711066 0.355533 0.934664i \(-0.384299\pi\)
0.355533 + 0.934664i \(0.384299\pi\)
\(522\) −349591. −0.0561543
\(523\) 3.95891e6 0.632879 0.316440 0.948613i \(-0.397513\pi\)
0.316440 + 0.948613i \(0.397513\pi\)
\(524\) 1.57696e6 0.250895
\(525\) 0 0
\(526\) 1.35138e7 2.12967
\(527\) 757452. 0.118803
\(528\) −1.34645e6 −0.210187
\(529\) −6.13829e6 −0.953692
\(530\) 0 0
\(531\) −174733. −0.0268929
\(532\) −186373. −0.0285498
\(533\) 4.54625e6 0.693162
\(534\) 4.65850e6 0.706956
\(535\) 0 0
\(536\) 5.66932e6 0.852352
\(537\) 1.92766e6 0.288466
\(538\) −9.10023e6 −1.35549
\(539\) 2.02163e6 0.299729
\(540\) 0 0
\(541\) 8.12527e6 1.19356 0.596780 0.802405i \(-0.296446\pi\)
0.596780 + 0.802405i \(0.296446\pi\)
\(542\) 1.22557e6 0.179201
\(543\) 2.22935e6 0.324473
\(544\) 2.03395e6 0.294675
\(545\) 0 0
\(546\) 469579. 0.0674104
\(547\) −1.49316e6 −0.213373 −0.106686 0.994293i \(-0.534024\pi\)
−0.106686 + 0.994293i \(0.534024\pi\)
\(548\) 219436. 0.0312145
\(549\) −546247. −0.0773496
\(550\) 0 0
\(551\) −1.33464e6 −0.187278
\(552\) −714541. −0.0998112
\(553\) 1.01210e6 0.140737
\(554\) 6.42157e6 0.888929
\(555\) 0 0
\(556\) −1.02016e6 −0.139953
\(557\) 3.54818e6 0.484582 0.242291 0.970204i \(-0.422101\pi\)
0.242291 + 0.970204i \(0.422101\pi\)
\(558\) −640799. −0.0871237
\(559\) −5.13046e6 −0.694428
\(560\) 0 0
\(561\) −670866. −0.0899971
\(562\) −9.20647e6 −1.22957
\(563\) 6.70997e6 0.892174 0.446087 0.894990i \(-0.352817\pi\)
0.446087 + 0.894990i \(0.352817\pi\)
\(564\) 903401. 0.119587
\(565\) 0 0
\(566\) −1.54738e7 −2.03028
\(567\) −65394.0 −0.00854241
\(568\) −3.67500e6 −0.477954
\(569\) 6.17393e6 0.799431 0.399716 0.916639i \(-0.369109\pi\)
0.399716 + 0.916639i \(0.369109\pi\)
\(570\) 0 0
\(571\) 1.82693e6 0.234494 0.117247 0.993103i \(-0.462593\pi\)
0.117247 + 0.993103i \(0.462593\pi\)
\(572\) −925184. −0.118233
\(573\) 3.69686e6 0.470377
\(574\) −358345. −0.0453965
\(575\) 0 0
\(576\) 1.48408e6 0.186380
\(577\) −7.62349e6 −0.953267 −0.476633 0.879102i \(-0.658143\pi\)
−0.476633 + 0.879102i \(0.658143\pi\)
\(578\) −6.69377e6 −0.833395
\(579\) −1.35891e6 −0.168460
\(580\) 0 0
\(581\) 144991. 0.0178197
\(582\) 4.55331e6 0.557211
\(583\) −2.64041e6 −0.321736
\(584\) 3.48414e6 0.422730
\(585\) 0 0
\(586\) 4.43427e6 0.533430
\(587\) −1.53545e7 −1.83925 −0.919623 0.392803i \(-0.871505\pi\)
−0.919623 + 0.392803i \(0.871505\pi\)
\(588\) 1.41316e6 0.168558
\(589\) −2.44640e6 −0.290562
\(590\) 0 0
\(591\) −1.36461e6 −0.160709
\(592\) −1.21420e7 −1.42392
\(593\) 8.62212e6 1.00688 0.503439 0.864031i \(-0.332068\pi\)
0.503439 + 0.864031i \(0.332068\pi\)
\(594\) 567548. 0.0659988
\(595\) 0 0
\(596\) −2.60282e6 −0.300144
\(597\) −4.18208e6 −0.480238
\(598\) −2.85789e6 −0.326808
\(599\) −2.18726e6 −0.249077 −0.124539 0.992215i \(-0.539745\pi\)
−0.124539 + 0.992215i \(0.539745\pi\)
\(600\) 0 0
\(601\) −3.77734e6 −0.426579 −0.213290 0.976989i \(-0.568418\pi\)
−0.213290 + 0.976989i \(0.568418\pi\)
\(602\) 404395. 0.0454793
\(603\) −3.15776e6 −0.353660
\(604\) 3.06808e6 0.342196
\(605\) 0 0
\(606\) −1.27535e6 −0.141075
\(607\) −1.39714e7 −1.53910 −0.769550 0.638587i \(-0.779519\pi\)
−0.769550 + 0.638587i \(0.779519\pi\)
\(608\) −6.56920e6 −0.720698
\(609\) −60172.2 −0.00657434
\(610\) 0 0
\(611\) −8.68986e6 −0.941694
\(612\) −468950. −0.0506114
\(613\) −8.72631e6 −0.937949 −0.468974 0.883212i \(-0.655376\pi\)
−0.468974 + 0.883212i \(0.655376\pi\)
\(614\) −6.34541e6 −0.679264
\(615\) 0 0
\(616\) −175384. −0.0186225
\(617\) −8.27444e6 −0.875035 −0.437518 0.899210i \(-0.644142\pi\)
−0.437518 + 0.899210i \(0.644142\pi\)
\(618\) 4.92242e6 0.518451
\(619\) 1.25043e7 1.31169 0.655846 0.754894i \(-0.272312\pi\)
0.655846 + 0.754894i \(0.272312\pi\)
\(620\) 0 0
\(621\) 397993. 0.0414139
\(622\) −1.09561e7 −1.13548
\(623\) 801829. 0.0827678
\(624\) 9.05347e6 0.930794
\(625\) 0 0
\(626\) 1.88091e7 1.91837
\(627\) 2.16675e6 0.220110
\(628\) −3.44237e6 −0.348304
\(629\) −6.04970e6 −0.609688
\(630\) 0 0
\(631\) 1.59310e7 1.59283 0.796416 0.604750i \(-0.206727\pi\)
0.796416 + 0.604750i \(0.206727\pi\)
\(632\) 1.47669e7 1.47061
\(633\) 1.05528e7 1.04678
\(634\) 1.09248e7 1.07942
\(635\) 0 0
\(636\) −1.84570e6 −0.180934
\(637\) −1.35933e7 −1.32732
\(638\) 522228. 0.0507935
\(639\) 2.04694e6 0.198314
\(640\) 0 0
\(641\) −5.06724e6 −0.487109 −0.243555 0.969887i \(-0.578313\pi\)
−0.243555 + 0.969887i \(0.578313\pi\)
\(642\) 7.47759e6 0.716019
\(643\) 5.24812e6 0.500583 0.250292 0.968170i \(-0.419474\pi\)
0.250292 + 0.968170i \(0.419474\pi\)
\(644\) 51138.7 0.00485887
\(645\) 0 0
\(646\) −7.88637e6 −0.743526
\(647\) 1.34119e7 1.25959 0.629797 0.776760i \(-0.283138\pi\)
0.629797 + 0.776760i \(0.283138\pi\)
\(648\) −954129. −0.0892626
\(649\) 261021. 0.0243256
\(650\) 0 0
\(651\) −110296. −0.0102001
\(652\) −6.04029e6 −0.556467
\(653\) 4.94854e6 0.454144 0.227072 0.973878i \(-0.427085\pi\)
0.227072 + 0.973878i \(0.427085\pi\)
\(654\) 8.91545e6 0.815077
\(655\) 0 0
\(656\) −6.90888e6 −0.626828
\(657\) −1.94063e6 −0.175400
\(658\) 684954. 0.0616732
\(659\) 1.09443e7 0.981687 0.490844 0.871248i \(-0.336689\pi\)
0.490844 + 0.871248i \(0.336689\pi\)
\(660\) 0 0
\(661\) −9.35940e6 −0.833191 −0.416595 0.909092i \(-0.636777\pi\)
−0.416595 + 0.909092i \(0.636777\pi\)
\(662\) 9.26989e6 0.822109
\(663\) 4.51086e6 0.398543
\(664\) 2.11549e6 0.186205
\(665\) 0 0
\(666\) 5.11801e6 0.447111
\(667\) 366212. 0.0318727
\(668\) 2.43585e6 0.211207
\(669\) −4.44948e6 −0.384365
\(670\) 0 0
\(671\) 815999. 0.0699654
\(672\) −296171. −0.0253000
\(673\) −2.66957e6 −0.227197 −0.113599 0.993527i \(-0.536238\pi\)
−0.113599 + 0.993527i \(0.536238\pi\)
\(674\) 6.22300e6 0.527655
\(675\) 0 0
\(676\) 2.73148e6 0.229896
\(677\) −1.36620e7 −1.14563 −0.572814 0.819685i \(-0.694148\pi\)
−0.572814 + 0.819685i \(0.694148\pi\)
\(678\) −8.10389e6 −0.677048
\(679\) 783724. 0.0652362
\(680\) 0 0
\(681\) −1.97477e6 −0.163173
\(682\) 957243. 0.0788064
\(683\) −5.01005e6 −0.410951 −0.205475 0.978662i \(-0.565874\pi\)
−0.205475 + 0.978662i \(0.565874\pi\)
\(684\) 1.51460e6 0.123782
\(685\) 0 0
\(686\) 2.14928e6 0.174374
\(687\) 1.11287e7 0.899604
\(688\) 7.79671e6 0.627972
\(689\) 1.77539e7 1.42477
\(690\) 0 0
\(691\) 1.37818e6 0.109802 0.0549009 0.998492i \(-0.482516\pi\)
0.0549009 + 0.998492i \(0.482516\pi\)
\(692\) 2.28109e6 0.181083
\(693\) 97687.4 0.00772690
\(694\) 1.80880e7 1.42558
\(695\) 0 0
\(696\) −877939. −0.0686976
\(697\) −3.44232e6 −0.268392
\(698\) −1.05662e7 −0.820879
\(699\) 757399. 0.0586316
\(700\) 0 0
\(701\) 1.98776e7 1.52781 0.763906 0.645328i \(-0.223279\pi\)
0.763906 + 0.645328i \(0.223279\pi\)
\(702\) −3.81615e6 −0.292269
\(703\) 1.95392e7 1.49114
\(704\) −2.21695e6 −0.168587
\(705\) 0 0
\(706\) 1.86468e7 1.40797
\(707\) −219516. −0.0165165
\(708\) 182459. 0.0136799
\(709\) −5.68213e6 −0.424518 −0.212259 0.977213i \(-0.568082\pi\)
−0.212259 + 0.977213i \(0.568082\pi\)
\(710\) 0 0
\(711\) −8.22505e6 −0.610189
\(712\) 1.16991e7 0.864870
\(713\) 671267. 0.0494506
\(714\) −355556. −0.0261013
\(715\) 0 0
\(716\) −2.01290e6 −0.146737
\(717\) 1.01939e6 0.0740533
\(718\) 1.95126e7 1.41255
\(719\) −4.90403e6 −0.353778 −0.176889 0.984231i \(-0.556603\pi\)
−0.176889 + 0.984231i \(0.556603\pi\)
\(720\) 0 0
\(721\) 847256. 0.0606983
\(722\) 9.53967e6 0.681068
\(723\) 5.54739e6 0.394678
\(724\) −2.32793e6 −0.165053
\(725\) 0 0
\(726\) −847818. −0.0596982
\(727\) −2.17098e7 −1.52342 −0.761711 0.647917i \(-0.775641\pi\)
−0.761711 + 0.647917i \(0.775641\pi\)
\(728\) 1.17927e6 0.0824680
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 3.88468e6 0.268882
\(732\) 570402. 0.0393462
\(733\) 1.22771e7 0.843985 0.421992 0.906599i \(-0.361331\pi\)
0.421992 + 0.906599i \(0.361331\pi\)
\(734\) 7.52777e6 0.515734
\(735\) 0 0
\(736\) 1.80252e6 0.122655
\(737\) 4.71715e6 0.319898
\(738\) 2.91218e6 0.196824
\(739\) −1.86424e7 −1.25571 −0.627857 0.778329i \(-0.716068\pi\)
−0.627857 + 0.778329i \(0.716068\pi\)
\(740\) 0 0
\(741\) −1.45690e7 −0.974733
\(742\) −1.39940e6 −0.0933111
\(743\) 9.40769e6 0.625188 0.312594 0.949887i \(-0.398802\pi\)
0.312594 + 0.949887i \(0.398802\pi\)
\(744\) −1.60926e6 −0.106585
\(745\) 0 0
\(746\) 9.49906e6 0.624933
\(747\) −1.17831e6 −0.0772605
\(748\) 700531. 0.0457797
\(749\) 1.28706e6 0.0838288
\(750\) 0 0
\(751\) −3.32683e6 −0.215244 −0.107622 0.994192i \(-0.534324\pi\)
−0.107622 + 0.994192i \(0.534324\pi\)
\(752\) 1.32059e7 0.851575
\(753\) 1.21346e7 0.779899
\(754\) −3.51142e6 −0.224934
\(755\) 0 0
\(756\) 68285.7 0.00434535
\(757\) −3.52278e6 −0.223432 −0.111716 0.993740i \(-0.535635\pi\)
−0.111716 + 0.993740i \(0.535635\pi\)
\(758\) 3.18555e7 2.01378
\(759\) −594533. −0.0374603
\(760\) 0 0
\(761\) 3.43213e6 0.214833 0.107417 0.994214i \(-0.465742\pi\)
0.107417 + 0.994214i \(0.465742\pi\)
\(762\) 6.93620e6 0.432747
\(763\) 1.53454e6 0.0954263
\(764\) −3.86033e6 −0.239272
\(765\) 0 0
\(766\) 9.08014e6 0.559140
\(767\) −1.75508e6 −0.107723
\(768\) −7.66777e6 −0.469101
\(769\) −1.40196e7 −0.854908 −0.427454 0.904037i \(-0.640589\pi\)
−0.427454 + 0.904037i \(0.640589\pi\)
\(770\) 0 0
\(771\) 6.44171e6 0.390270
\(772\) 1.41900e6 0.0856920
\(773\) −3.40453e6 −0.204931 −0.102466 0.994737i \(-0.532673\pi\)
−0.102466 + 0.994737i \(0.532673\pi\)
\(774\) −3.28641e6 −0.197183
\(775\) 0 0
\(776\) 1.14349e7 0.681675
\(777\) 880921. 0.0523461
\(778\) 8.73246e6 0.517235
\(779\) 1.11179e7 0.656418
\(780\) 0 0
\(781\) −3.05777e6 −0.179382
\(782\) 2.16394e6 0.126540
\(783\) 489004. 0.0285042
\(784\) 2.06576e7 1.20030
\(785\) 0 0
\(786\) −9.71667e6 −0.560998
\(787\) 5.77229e6 0.332209 0.166105 0.986108i \(-0.446881\pi\)
0.166105 + 0.986108i \(0.446881\pi\)
\(788\) 1.42496e6 0.0817497
\(789\) −1.89029e7 −1.08103
\(790\) 0 0
\(791\) −1.39486e6 −0.0792662
\(792\) 1.42530e6 0.0807410
\(793\) −5.48672e6 −0.309835
\(794\) 1.14185e7 0.642771
\(795\) 0 0
\(796\) 4.36701e6 0.244288
\(797\) 3.55488e7 1.98234 0.991171 0.132589i \(-0.0423291\pi\)
0.991171 + 0.132589i \(0.0423291\pi\)
\(798\) 1.14837e6 0.0638370
\(799\) 6.57978e6 0.364624
\(800\) 0 0
\(801\) −6.51626e6 −0.358854
\(802\) 2.37857e7 1.30581
\(803\) 2.89897e6 0.158655
\(804\) 3.29739e6 0.179900
\(805\) 0 0
\(806\) −6.43644e6 −0.348986
\(807\) 1.27293e7 0.688052
\(808\) −3.20284e6 −0.172587
\(809\) 8.35527e6 0.448838 0.224419 0.974493i \(-0.427952\pi\)
0.224419 + 0.974493i \(0.427952\pi\)
\(810\) 0 0
\(811\) 3.83660e6 0.204831 0.102415 0.994742i \(-0.467343\pi\)
0.102415 + 0.994742i \(0.467343\pi\)
\(812\) 62832.9 0.00334424
\(813\) −1.71432e6 −0.0909630
\(814\) −7.64542e6 −0.404427
\(815\) 0 0
\(816\) −6.85510e6 −0.360403
\(817\) −1.25466e7 −0.657616
\(818\) −5.30491e6 −0.277201
\(819\) −656843. −0.0342178
\(820\) 0 0
\(821\) −9.99705e6 −0.517624 −0.258812 0.965928i \(-0.583331\pi\)
−0.258812 + 0.965928i \(0.583331\pi\)
\(822\) −1.35209e6 −0.0697953
\(823\) −2.67331e7 −1.37578 −0.687890 0.725815i \(-0.741463\pi\)
−0.687890 + 0.725815i \(0.741463\pi\)
\(824\) 1.23619e7 0.634257
\(825\) 0 0
\(826\) 138340. 0.00705500
\(827\) 951161. 0.0483605 0.0241802 0.999708i \(-0.492302\pi\)
0.0241802 + 0.999708i \(0.492302\pi\)
\(828\) −415592. −0.0210664
\(829\) −1.44000e7 −0.727741 −0.363870 0.931450i \(-0.618545\pi\)
−0.363870 + 0.931450i \(0.618545\pi\)
\(830\) 0 0
\(831\) −8.98244e6 −0.451223
\(832\) 1.49066e7 0.746572
\(833\) 1.02926e7 0.513938
\(834\) 6.28587e6 0.312932
\(835\) 0 0
\(836\) −2.26256e6 −0.111965
\(837\) 896344. 0.0442243
\(838\) −3.78265e7 −1.86074
\(839\) 3.42903e6 0.168177 0.0840884 0.996458i \(-0.473202\pi\)
0.0840884 + 0.996458i \(0.473202\pi\)
\(840\) 0 0
\(841\) −2.00612e7 −0.978063
\(842\) 2.24371e7 1.09065
\(843\) 1.28779e7 0.624133
\(844\) −1.10194e7 −0.532478
\(845\) 0 0
\(846\) −5.56645e6 −0.267395
\(847\) −145928. −0.00698924
\(848\) −2.69804e7 −1.28843
\(849\) 2.16446e7 1.03058
\(850\) 0 0
\(851\) −5.36135e6 −0.253776
\(852\) −2.13745e6 −0.100878
\(853\) 1.57990e7 0.743460 0.371730 0.928341i \(-0.378765\pi\)
0.371730 + 0.928341i \(0.378765\pi\)
\(854\) 432476. 0.0202916
\(855\) 0 0
\(856\) 1.87788e7 0.875956
\(857\) 5.06147e6 0.235410 0.117705 0.993049i \(-0.462446\pi\)
0.117705 + 0.993049i \(0.462446\pi\)
\(858\) 5.70067e6 0.264367
\(859\) −2.48302e7 −1.14815 −0.574074 0.818803i \(-0.694638\pi\)
−0.574074 + 0.818803i \(0.694638\pi\)
\(860\) 0 0
\(861\) 501250. 0.0230434
\(862\) −2.83890e7 −1.30131
\(863\) 1.03123e7 0.471336 0.235668 0.971834i \(-0.424272\pi\)
0.235668 + 0.971834i \(0.424272\pi\)
\(864\) 2.40691e6 0.109692
\(865\) 0 0
\(866\) 6.36800e6 0.288541
\(867\) 9.36318e6 0.423035
\(868\) 115173. 0.00518860
\(869\) 1.22868e7 0.551937
\(870\) 0 0
\(871\) −3.17178e7 −1.41663
\(872\) 2.23897e7 0.997142
\(873\) −6.36913e6 −0.282842
\(874\) −6.98904e6 −0.309484
\(875\) 0 0
\(876\) 2.02645e6 0.0892226
\(877\) 1.04098e7 0.457027 0.228514 0.973541i \(-0.426613\pi\)
0.228514 + 0.973541i \(0.426613\pi\)
\(878\) −1.73453e7 −0.759355
\(879\) −6.20261e6 −0.270771
\(880\) 0 0
\(881\) −1.99961e7 −0.867970 −0.433985 0.900920i \(-0.642893\pi\)
−0.433985 + 0.900920i \(0.642893\pi\)
\(882\) −8.70743e6 −0.376894
\(883\) −2.42565e7 −1.04695 −0.523475 0.852041i \(-0.675364\pi\)
−0.523475 + 0.852041i \(0.675364\pi\)
\(884\) −4.71032e6 −0.202731
\(885\) 0 0
\(886\) 1.18856e7 0.508671
\(887\) 3.28464e7 1.40178 0.700888 0.713271i \(-0.252787\pi\)
0.700888 + 0.713271i \(0.252787\pi\)
\(888\) 1.28530e7 0.546983
\(889\) 1.19387e6 0.0506644
\(890\) 0 0
\(891\) −793881. −0.0335013
\(892\) 4.64624e6 0.195519
\(893\) −2.12512e7 −0.891775
\(894\) 1.60377e7 0.671118
\(895\) 0 0
\(896\) −2.22803e6 −0.0927151
\(897\) 3.99760e6 0.165889
\(898\) −5.18781e6 −0.214681
\(899\) 824769. 0.0340356
\(900\) 0 0
\(901\) −1.34429e7 −0.551672
\(902\) −4.35030e6 −0.178034
\(903\) −565664. −0.0230855
\(904\) −2.03516e7 −0.828280
\(905\) 0 0
\(906\) −1.89045e7 −0.765146
\(907\) −1.54289e7 −0.622753 −0.311377 0.950287i \(-0.600790\pi\)
−0.311377 + 0.950287i \(0.600790\pi\)
\(908\) 2.06209e6 0.0830029
\(909\) 1.78395e6 0.0716101
\(910\) 0 0
\(911\) 3.82548e7 1.52718 0.763590 0.645701i \(-0.223435\pi\)
0.763590 + 0.645701i \(0.223435\pi\)
\(912\) 2.21404e7 0.881453
\(913\) 1.76019e6 0.0698847
\(914\) 3.07849e7 1.21891
\(915\) 0 0
\(916\) −1.16208e7 −0.457611
\(917\) −1.67245e6 −0.0656795
\(918\) 2.88951e6 0.113167
\(919\) 3.21575e7 1.25601 0.628005 0.778209i \(-0.283872\pi\)
0.628005 + 0.778209i \(0.283872\pi\)
\(920\) 0 0
\(921\) 8.87591e6 0.344797
\(922\) −5.17966e7 −2.00666
\(923\) 2.05603e7 0.794373
\(924\) −102007. −0.00393052
\(925\) 0 0
\(926\) −4.15868e6 −0.159378
\(927\) −6.88544e6 −0.263168
\(928\) 2.21471e6 0.0844204
\(929\) 2.16438e7 0.822802 0.411401 0.911455i \(-0.365040\pi\)
0.411401 + 0.911455i \(0.365040\pi\)
\(930\) 0 0
\(931\) −3.32426e7 −1.25696
\(932\) −790890. −0.0298247
\(933\) 1.53253e7 0.576374
\(934\) −3.84512e7 −1.44226
\(935\) 0 0
\(936\) −9.58364e6 −0.357554
\(937\) −2.80993e7 −1.04555 −0.522776 0.852470i \(-0.675104\pi\)
−0.522776 + 0.852470i \(0.675104\pi\)
\(938\) 2.50007e6 0.0927779
\(939\) −2.63100e7 −0.973770
\(940\) 0 0
\(941\) −1.13300e7 −0.417116 −0.208558 0.978010i \(-0.566877\pi\)
−0.208558 + 0.978010i \(0.566877\pi\)
\(942\) 2.12107e7 0.778804
\(943\) −3.05065e6 −0.111715
\(944\) 2.66718e6 0.0974144
\(945\) 0 0
\(946\) 4.90933e6 0.178359
\(947\) −3.04381e7 −1.10292 −0.551458 0.834203i \(-0.685928\pi\)
−0.551458 + 0.834203i \(0.685928\pi\)
\(948\) 8.58875e6 0.310391
\(949\) −1.94925e7 −0.702589
\(950\) 0 0
\(951\) −1.52815e7 −0.547918
\(952\) −892920. −0.0319316
\(953\) 2.97688e7 1.06177 0.530883 0.847445i \(-0.321860\pi\)
0.530883 + 0.847445i \(0.321860\pi\)
\(954\) 1.13726e7 0.404566
\(955\) 0 0
\(956\) −1.06447e6 −0.0376694
\(957\) −730488. −0.0257830
\(958\) −4.85060e7 −1.70758
\(959\) −232724. −0.00817137
\(960\) 0 0
\(961\) −2.71174e7 −0.947194
\(962\) 5.14073e7 1.79096
\(963\) −1.04596e7 −0.363454
\(964\) −5.79270e6 −0.200765
\(965\) 0 0
\(966\) −315100. −0.0108644
\(967\) 1.65127e7 0.567875 0.283937 0.958843i \(-0.408359\pi\)
0.283937 + 0.958843i \(0.408359\pi\)
\(968\) −2.12916e6 −0.0730330
\(969\) 1.10314e7 0.377416
\(970\) 0 0
\(971\) 3.43022e7 1.16755 0.583773 0.811917i \(-0.301576\pi\)
0.583773 + 0.811917i \(0.301576\pi\)
\(972\) −554941. −0.0188400
\(973\) 1.08194e6 0.0366370
\(974\) −4.32467e7 −1.46068
\(975\) 0 0
\(976\) 8.33811e6 0.280184
\(977\) −4.76980e7 −1.59869 −0.799344 0.600873i \(-0.794820\pi\)
−0.799344 + 0.600873i \(0.794820\pi\)
\(978\) 3.72182e7 1.24425
\(979\) 9.73417e6 0.324595
\(980\) 0 0
\(981\) −1.24709e7 −0.413736
\(982\) −1.53711e7 −0.508660
\(983\) −4.43331e6 −0.146334 −0.0731669 0.997320i \(-0.523311\pi\)
−0.0731669 + 0.997320i \(0.523311\pi\)
\(984\) 7.31347e6 0.240789
\(985\) 0 0
\(986\) 2.65878e6 0.0870944
\(987\) −958108. −0.0313056
\(988\) 1.52133e7 0.495827
\(989\) 3.44267e6 0.111919
\(990\) 0 0
\(991\) −9.98431e6 −0.322949 −0.161474 0.986877i \(-0.551625\pi\)
−0.161474 + 0.986877i \(0.551625\pi\)
\(992\) 4.05957e6 0.130979
\(993\) −1.29666e7 −0.417306
\(994\) −1.62061e6 −0.0520249
\(995\) 0 0
\(996\) 1.23041e6 0.0393009
\(997\) 3.98385e7 1.26930 0.634652 0.772798i \(-0.281144\pi\)
0.634652 + 0.772798i \(0.281144\pi\)
\(998\) −3.07595e7 −0.977581
\(999\) −7.15903e6 −0.226955
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 825.6.a.q.1.6 yes 8
5.4 even 2 825.6.a.p.1.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.p.1.3 8 5.4 even 2
825.6.a.q.1.6 yes 8 1.1 even 1 trivial