Properties

Label 825.6.a.m.1.6
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \( x^{7} - 2x^{6} - 152x^{5} + 358x^{4} + 5771x^{3} - 13444x^{2} - 51316x + 92576 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-6.33680\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.33680 q^{2} +9.00000 q^{3} -3.51852 q^{4} +48.0312 q^{6} +73.9998 q^{7} -189.555 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+5.33680 q^{2} +9.00000 q^{3} -3.51852 q^{4} +48.0312 q^{6} +73.9998 q^{7} -189.555 q^{8} +81.0000 q^{9} +121.000 q^{11} -31.6667 q^{12} -66.6080 q^{13} +394.923 q^{14} -899.027 q^{16} -808.876 q^{17} +432.281 q^{18} -900.233 q^{19} +665.998 q^{21} +645.753 q^{22} +3372.25 q^{23} -1706.00 q^{24} -355.474 q^{26} +729.000 q^{27} -260.370 q^{28} -5030.88 q^{29} +2599.04 q^{31} +1267.84 q^{32} +1089.00 q^{33} -4316.81 q^{34} -285.000 q^{36} -6950.65 q^{37} -4804.37 q^{38} -599.472 q^{39} +197.788 q^{41} +3554.30 q^{42} -13340.3 q^{43} -425.741 q^{44} +17997.0 q^{46} -3822.58 q^{47} -8091.25 q^{48} -11331.0 q^{49} -7279.88 q^{51} +234.362 q^{52} +4858.62 q^{53} +3890.53 q^{54} -14027.1 q^{56} -8102.10 q^{57} -26848.8 q^{58} -6936.31 q^{59} -10538.4 q^{61} +13870.5 q^{62} +5993.99 q^{63} +35535.1 q^{64} +5811.78 q^{66} -30768.1 q^{67} +2846.05 q^{68} +30350.3 q^{69} +13974.9 q^{71} -15354.0 q^{72} -13733.4 q^{73} -37094.3 q^{74} +3167.49 q^{76} +8953.98 q^{77} -3199.26 q^{78} +28954.9 q^{79} +6561.00 q^{81} +1055.56 q^{82} -103828. q^{83} -2343.33 q^{84} -71194.4 q^{86} -45277.9 q^{87} -22936.2 q^{88} -63069.9 q^{89} -4928.98 q^{91} -11865.3 q^{92} +23391.3 q^{93} -20400.4 q^{94} +11410.6 q^{96} -62964.9 q^{97} -60471.5 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 9 q^{2} + 63 q^{3} + 95 q^{4} - 81 q^{6} - 65 q^{7} - 207 q^{8} + 567 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 9 q^{2} + 63 q^{3} + 95 q^{4} - 81 q^{6} - 65 q^{7} - 207 q^{8} + 567 q^{9} + 847 q^{11} + 855 q^{12} + 113 q^{13} - 1212 q^{14} + 499 q^{16} - 1030 q^{17} - 729 q^{18} - 3803 q^{19} - 585 q^{21} - 1089 q^{22} - 514 q^{23} - 1863 q^{24} - 12111 q^{26} + 5103 q^{27} + 342 q^{28} - 2698 q^{29} - 17233 q^{31} - 9943 q^{32} + 7623 q^{33} + 4090 q^{34} + 7695 q^{36} - 23182 q^{37} + 11943 q^{38} + 1017 q^{39} - 16158 q^{41} - 10908 q^{42} + 4249 q^{43} + 11495 q^{44} - 28769 q^{46} - 7580 q^{47} + 4491 q^{48} + 37140 q^{49} - 9270 q^{51} + 23887 q^{52} - 20574 q^{53} - 6561 q^{54} - 73276 q^{56} - 34227 q^{57} - 8733 q^{58} - 364 q^{59} - 28127 q^{61} + 71917 q^{62} - 5265 q^{63} + 43379 q^{64} - 9801 q^{66} + 21493 q^{67} - 160660 q^{68} - 4626 q^{69} - 177084 q^{71} - 16767 q^{72} - 78670 q^{73} + 196750 q^{74} - 32701 q^{76} - 7865 q^{77} - 108999 q^{78} - 187432 q^{79} + 45927 q^{81} - 179552 q^{82} + 44592 q^{83} + 3078 q^{84} - 110433 q^{86} - 24282 q^{87} - 25047 q^{88} - 151168 q^{89} - 230153 q^{91} + 44767 q^{92} - 155097 q^{93} + 54040 q^{94} - 89487 q^{96} + 55589 q^{97} - 478341 q^{98} + 68607 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.33680 0.943423 0.471711 0.881753i \(-0.343636\pi\)
0.471711 + 0.881753i \(0.343636\pi\)
\(3\) 9.00000 0.577350
\(4\) −3.51852 −0.109954
\(5\) 0 0
\(6\) 48.0312 0.544685
\(7\) 73.9998 0.570802 0.285401 0.958408i \(-0.407873\pi\)
0.285401 + 0.958408i \(0.407873\pi\)
\(8\) −189.555 −1.04716
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) −31.6667 −0.0634819
\(13\) −66.6080 −0.109312 −0.0546560 0.998505i \(-0.517406\pi\)
−0.0546560 + 0.998505i \(0.517406\pi\)
\(14\) 394.923 0.538508
\(15\) 0 0
\(16\) −899.027 −0.877956
\(17\) −808.876 −0.678828 −0.339414 0.940637i \(-0.610229\pi\)
−0.339414 + 0.940637i \(0.610229\pi\)
\(18\) 432.281 0.314474
\(19\) −900.233 −0.572099 −0.286049 0.958215i \(-0.592342\pi\)
−0.286049 + 0.958215i \(0.592342\pi\)
\(20\) 0 0
\(21\) 665.998 0.329553
\(22\) 645.753 0.284453
\(23\) 3372.25 1.32923 0.664615 0.747186i \(-0.268595\pi\)
0.664615 + 0.747186i \(0.268595\pi\)
\(24\) −1706.00 −0.604576
\(25\) 0 0
\(26\) −355.474 −0.103127
\(27\) 729.000 0.192450
\(28\) −260.370 −0.0627619
\(29\) −5030.88 −1.11083 −0.555417 0.831572i \(-0.687441\pi\)
−0.555417 + 0.831572i \(0.687441\pi\)
\(30\) 0 0
\(31\) 2599.04 0.485745 0.242872 0.970058i \(-0.421910\pi\)
0.242872 + 0.970058i \(0.421910\pi\)
\(32\) 1267.84 0.218872
\(33\) 1089.00 0.174078
\(34\) −4316.81 −0.640422
\(35\) 0 0
\(36\) −285.000 −0.0366513
\(37\) −6950.65 −0.834682 −0.417341 0.908750i \(-0.637038\pi\)
−0.417341 + 0.908750i \(0.637038\pi\)
\(38\) −4804.37 −0.539731
\(39\) −599.472 −0.0631113
\(40\) 0 0
\(41\) 197.788 0.0183756 0.00918778 0.999958i \(-0.497075\pi\)
0.00918778 + 0.999958i \(0.497075\pi\)
\(42\) 3554.30 0.310908
\(43\) −13340.3 −1.10025 −0.550127 0.835081i \(-0.685421\pi\)
−0.550127 + 0.835081i \(0.685421\pi\)
\(44\) −425.741 −0.0331523
\(45\) 0 0
\(46\) 17997.0 1.25403
\(47\) −3822.58 −0.252413 −0.126207 0.992004i \(-0.540280\pi\)
−0.126207 + 0.992004i \(0.540280\pi\)
\(48\) −8091.25 −0.506888
\(49\) −11331.0 −0.674185
\(50\) 0 0
\(51\) −7279.88 −0.391921
\(52\) 234.362 0.0120193
\(53\) 4858.62 0.237588 0.118794 0.992919i \(-0.462097\pi\)
0.118794 + 0.992919i \(0.462097\pi\)
\(54\) 3890.53 0.181562
\(55\) 0 0
\(56\) −14027.1 −0.597719
\(57\) −8102.10 −0.330301
\(58\) −26848.8 −1.04799
\(59\) −6936.31 −0.259417 −0.129708 0.991552i \(-0.541404\pi\)
−0.129708 + 0.991552i \(0.541404\pi\)
\(60\) 0 0
\(61\) −10538.4 −0.362619 −0.181309 0.983426i \(-0.558033\pi\)
−0.181309 + 0.983426i \(0.558033\pi\)
\(62\) 13870.5 0.458263
\(63\) 5993.99 0.190267
\(64\) 35535.1 1.08444
\(65\) 0 0
\(66\) 5811.78 0.164229
\(67\) −30768.1 −0.837362 −0.418681 0.908133i \(-0.637507\pi\)
−0.418681 + 0.908133i \(0.637507\pi\)
\(68\) 2846.05 0.0746397
\(69\) 30350.3 0.767432
\(70\) 0 0
\(71\) 13974.9 0.329006 0.164503 0.986377i \(-0.447398\pi\)
0.164503 + 0.986377i \(0.447398\pi\)
\(72\) −15354.0 −0.349052
\(73\) −13733.4 −0.301627 −0.150813 0.988562i \(-0.548189\pi\)
−0.150813 + 0.988562i \(0.548189\pi\)
\(74\) −37094.3 −0.787458
\(75\) 0 0
\(76\) 3167.49 0.0629045
\(77\) 8953.98 0.172103
\(78\) −3199.26 −0.0595407
\(79\) 28954.9 0.521980 0.260990 0.965342i \(-0.415951\pi\)
0.260990 + 0.965342i \(0.415951\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 1055.56 0.0173359
\(83\) −103828. −1.65432 −0.827161 0.561965i \(-0.810046\pi\)
−0.827161 + 0.561965i \(0.810046\pi\)
\(84\) −2343.33 −0.0362356
\(85\) 0 0
\(86\) −71194.4 −1.03801
\(87\) −45277.9 −0.641340
\(88\) −22936.2 −0.315729
\(89\) −63069.9 −0.844010 −0.422005 0.906594i \(-0.638673\pi\)
−0.422005 + 0.906594i \(0.638673\pi\)
\(90\) 0 0
\(91\) −4928.98 −0.0623955
\(92\) −11865.3 −0.146154
\(93\) 23391.3 0.280445
\(94\) −20400.4 −0.238132
\(95\) 0 0
\(96\) 11410.6 0.126366
\(97\) −62964.9 −0.679468 −0.339734 0.940522i \(-0.610337\pi\)
−0.339734 + 0.940522i \(0.610337\pi\)
\(98\) −60471.5 −0.636041
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) 32233.2 0.314413 0.157206 0.987566i \(-0.449751\pi\)
0.157206 + 0.987566i \(0.449751\pi\)
\(102\) −38851.3 −0.369748
\(103\) 45990.8 0.427148 0.213574 0.976927i \(-0.431490\pi\)
0.213574 + 0.976927i \(0.431490\pi\)
\(104\) 12625.9 0.114467
\(105\) 0 0
\(106\) 25929.5 0.224145
\(107\) 40102.2 0.338617 0.169309 0.985563i \(-0.445847\pi\)
0.169309 + 0.985563i \(0.445847\pi\)
\(108\) −2565.00 −0.0211606
\(109\) −44635.8 −0.359847 −0.179923 0.983681i \(-0.557585\pi\)
−0.179923 + 0.983681i \(0.557585\pi\)
\(110\) 0 0
\(111\) −62555.8 −0.481904
\(112\) −66527.9 −0.501139
\(113\) −251257. −1.85106 −0.925532 0.378669i \(-0.876382\pi\)
−0.925532 + 0.378669i \(0.876382\pi\)
\(114\) −43239.3 −0.311614
\(115\) 0 0
\(116\) 17701.3 0.122140
\(117\) −5395.25 −0.0364373
\(118\) −37017.7 −0.244740
\(119\) −59856.7 −0.387476
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −56241.4 −0.342103
\(123\) 1780.09 0.0106091
\(124\) −9144.77 −0.0534095
\(125\) 0 0
\(126\) 31988.7 0.179503
\(127\) 214460. 1.17988 0.589939 0.807448i \(-0.299152\pi\)
0.589939 + 0.807448i \(0.299152\pi\)
\(128\) 149073. 0.804218
\(129\) −120062. −0.635232
\(130\) 0 0
\(131\) 157394. 0.801325 0.400662 0.916226i \(-0.368780\pi\)
0.400662 + 0.916226i \(0.368780\pi\)
\(132\) −3831.67 −0.0191405
\(133\) −66617.1 −0.326555
\(134\) −164203. −0.789986
\(135\) 0 0
\(136\) 153327. 0.710838
\(137\) −385559. −1.75505 −0.877525 0.479532i \(-0.840807\pi\)
−0.877525 + 0.479532i \(0.840807\pi\)
\(138\) 161973. 0.724012
\(139\) −181540. −0.796959 −0.398480 0.917177i \(-0.630462\pi\)
−0.398480 + 0.917177i \(0.630462\pi\)
\(140\) 0 0
\(141\) −34403.2 −0.145731
\(142\) 74581.5 0.310392
\(143\) −8059.57 −0.0329588
\(144\) −72821.2 −0.292652
\(145\) 0 0
\(146\) −73292.3 −0.284561
\(147\) −101979. −0.389241
\(148\) 24456.0 0.0917765
\(149\) −189099. −0.697790 −0.348895 0.937162i \(-0.613443\pi\)
−0.348895 + 0.937162i \(0.613443\pi\)
\(150\) 0 0
\(151\) 182551. 0.651541 0.325770 0.945449i \(-0.394376\pi\)
0.325770 + 0.945449i \(0.394376\pi\)
\(152\) 170644. 0.599076
\(153\) −65519.0 −0.226276
\(154\) 47785.6 0.162366
\(155\) 0 0
\(156\) 2109.26 0.00693933
\(157\) −299449. −0.969559 −0.484779 0.874636i \(-0.661100\pi\)
−0.484779 + 0.874636i \(0.661100\pi\)
\(158\) 154526. 0.492447
\(159\) 43727.6 0.137171
\(160\) 0 0
\(161\) 249546. 0.758728
\(162\) 35014.8 0.104825
\(163\) −105810. −0.311930 −0.155965 0.987763i \(-0.549849\pi\)
−0.155965 + 0.987763i \(0.549849\pi\)
\(164\) −695.922 −0.00202046
\(165\) 0 0
\(166\) −554111. −1.56072
\(167\) 44979.5 0.124802 0.0624012 0.998051i \(-0.480124\pi\)
0.0624012 + 0.998051i \(0.480124\pi\)
\(168\) −126244. −0.345093
\(169\) −366856. −0.988051
\(170\) 0 0
\(171\) −72918.9 −0.190700
\(172\) 46938.0 0.120977
\(173\) −445425. −1.13151 −0.565756 0.824573i \(-0.691415\pi\)
−0.565756 + 0.824573i \(0.691415\pi\)
\(174\) −241639. −0.605055
\(175\) 0 0
\(176\) −108782. −0.264714
\(177\) −62426.8 −0.149774
\(178\) −336592. −0.796258
\(179\) −552254. −1.28827 −0.644134 0.764913i \(-0.722782\pi\)
−0.644134 + 0.764913i \(0.722782\pi\)
\(180\) 0 0
\(181\) 52408.5 0.118906 0.0594532 0.998231i \(-0.481064\pi\)
0.0594532 + 0.998231i \(0.481064\pi\)
\(182\) −26305.0 −0.0588654
\(183\) −94845.6 −0.209358
\(184\) −639228. −1.39191
\(185\) 0 0
\(186\) 124835. 0.264578
\(187\) −97874.0 −0.204674
\(188\) 13449.8 0.0277538
\(189\) 53945.9 0.109851
\(190\) 0 0
\(191\) 135687. 0.269125 0.134562 0.990905i \(-0.457037\pi\)
0.134562 + 0.990905i \(0.457037\pi\)
\(192\) 319816. 0.626105
\(193\) −561304. −1.08469 −0.542344 0.840157i \(-0.682463\pi\)
−0.542344 + 0.840157i \(0.682463\pi\)
\(194\) −336031. −0.641026
\(195\) 0 0
\(196\) 39868.5 0.0741292
\(197\) 774322. 1.42153 0.710765 0.703429i \(-0.248349\pi\)
0.710765 + 0.703429i \(0.248349\pi\)
\(198\) 52306.0 0.0948175
\(199\) −494275. −0.884780 −0.442390 0.896823i \(-0.645869\pi\)
−0.442390 + 0.896823i \(0.645869\pi\)
\(200\) 0 0
\(201\) −276913. −0.483451
\(202\) 172022. 0.296624
\(203\) −372284. −0.634066
\(204\) 25614.4 0.0430933
\(205\) 0 0
\(206\) 245444. 0.402981
\(207\) 273152. 0.443077
\(208\) 59882.4 0.0959712
\(209\) −108928. −0.172494
\(210\) 0 0
\(211\) −30102.4 −0.0465474 −0.0232737 0.999729i \(-0.507409\pi\)
−0.0232737 + 0.999729i \(0.507409\pi\)
\(212\) −17095.2 −0.0261237
\(213\) 125774. 0.189952
\(214\) 214018. 0.319459
\(215\) 0 0
\(216\) −138186. −0.201525
\(217\) 192328. 0.277264
\(218\) −238213. −0.339487
\(219\) −123600. −0.174144
\(220\) 0 0
\(221\) 53877.6 0.0742041
\(222\) −333848. −0.454639
\(223\) 539464. 0.726441 0.363221 0.931703i \(-0.381677\pi\)
0.363221 + 0.931703i \(0.381677\pi\)
\(224\) 93820.0 0.124932
\(225\) 0 0
\(226\) −1.34091e6 −1.74634
\(227\) 972036. 1.25204 0.626019 0.779808i \(-0.284683\pi\)
0.626019 + 0.779808i \(0.284683\pi\)
\(228\) 28507.4 0.0363179
\(229\) 172701. 0.217624 0.108812 0.994062i \(-0.465295\pi\)
0.108812 + 0.994062i \(0.465295\pi\)
\(230\) 0 0
\(231\) 80585.8 0.0993639
\(232\) 953631. 1.16322
\(233\) 18356.9 0.0221519 0.0110759 0.999939i \(-0.496474\pi\)
0.0110759 + 0.999939i \(0.496474\pi\)
\(234\) −28793.4 −0.0343758
\(235\) 0 0
\(236\) 24405.6 0.0285239
\(237\) 260594. 0.301365
\(238\) −319443. −0.365554
\(239\) −50802.6 −0.0575296 −0.0287648 0.999586i \(-0.509157\pi\)
−0.0287648 + 0.999586i \(0.509157\pi\)
\(240\) 0 0
\(241\) 57284.6 0.0635324 0.0317662 0.999495i \(-0.489887\pi\)
0.0317662 + 0.999495i \(0.489887\pi\)
\(242\) 78136.1 0.0857657
\(243\) 59049.0 0.0641500
\(244\) 37079.6 0.0398713
\(245\) 0 0
\(246\) 9500.01 0.0100089
\(247\) 59962.7 0.0625373
\(248\) −492661. −0.508650
\(249\) −934454. −0.955123
\(250\) 0 0
\(251\) 275746. 0.276265 0.138132 0.990414i \(-0.455890\pi\)
0.138132 + 0.990414i \(0.455890\pi\)
\(252\) −21090.0 −0.0209206
\(253\) 408042. 0.400778
\(254\) 1.14453e6 1.11312
\(255\) 0 0
\(256\) −341550. −0.325727
\(257\) −55323.5 −0.0522489 −0.0261244 0.999659i \(-0.508317\pi\)
−0.0261244 + 0.999659i \(0.508317\pi\)
\(258\) −640749. −0.599293
\(259\) −514347. −0.476438
\(260\) 0 0
\(261\) −407501. −0.370278
\(262\) 839978. 0.755988
\(263\) 1.50652e6 1.34303 0.671515 0.740991i \(-0.265644\pi\)
0.671515 + 0.740991i \(0.265644\pi\)
\(264\) −206426. −0.182286
\(265\) 0 0
\(266\) −355522. −0.308080
\(267\) −567630. −0.487289
\(268\) 108258. 0.0920711
\(269\) 300252. 0.252991 0.126496 0.991967i \(-0.459627\pi\)
0.126496 + 0.991967i \(0.459627\pi\)
\(270\) 0 0
\(271\) −1.88992e6 −1.56322 −0.781608 0.623770i \(-0.785600\pi\)
−0.781608 + 0.623770i \(0.785600\pi\)
\(272\) 727202. 0.595981
\(273\) −44360.8 −0.0360241
\(274\) −2.05765e6 −1.65575
\(275\) 0 0
\(276\) −106788. −0.0843821
\(277\) 493383. 0.386354 0.193177 0.981164i \(-0.438121\pi\)
0.193177 + 0.981164i \(0.438121\pi\)
\(278\) −968845. −0.751869
\(279\) 210522. 0.161915
\(280\) 0 0
\(281\) 1.39118e6 1.05104 0.525518 0.850783i \(-0.323872\pi\)
0.525518 + 0.850783i \(0.323872\pi\)
\(282\) −183603. −0.137486
\(283\) 1.92592e6 1.42946 0.714730 0.699400i \(-0.246549\pi\)
0.714730 + 0.699400i \(0.246549\pi\)
\(284\) −49171.1 −0.0361755
\(285\) 0 0
\(286\) −43012.3 −0.0310941
\(287\) 14636.3 0.0104888
\(288\) 102695. 0.0729572
\(289\) −765577. −0.539193
\(290\) 0 0
\(291\) −566684. −0.392291
\(292\) 48321.1 0.0331650
\(293\) −1.53666e6 −1.04571 −0.522853 0.852423i \(-0.675132\pi\)
−0.522853 + 0.852423i \(0.675132\pi\)
\(294\) −544243. −0.367219
\(295\) 0 0
\(296\) 1.31753e6 0.874042
\(297\) 88209.0 0.0580259
\(298\) −1.00919e6 −0.658311
\(299\) −224619. −0.145301
\(300\) 0 0
\(301\) −987177. −0.628028
\(302\) 974238. 0.614678
\(303\) 290099. 0.181526
\(304\) 809334. 0.502278
\(305\) 0 0
\(306\) −349662. −0.213474
\(307\) 2.17758e6 1.31865 0.659324 0.751859i \(-0.270842\pi\)
0.659324 + 0.751859i \(0.270842\pi\)
\(308\) −31504.8 −0.0189234
\(309\) 413917. 0.246614
\(310\) 0 0
\(311\) 624920. 0.366373 0.183186 0.983078i \(-0.441359\pi\)
0.183186 + 0.983078i \(0.441359\pi\)
\(312\) 113633. 0.0660874
\(313\) 757413. 0.436990 0.218495 0.975838i \(-0.429885\pi\)
0.218495 + 0.975838i \(0.429885\pi\)
\(314\) −1.59810e6 −0.914704
\(315\) 0 0
\(316\) −101878. −0.0573937
\(317\) 3.10220e6 1.73389 0.866944 0.498406i \(-0.166081\pi\)
0.866944 + 0.498406i \(0.166081\pi\)
\(318\) 233366. 0.129410
\(319\) −608737. −0.334929
\(320\) 0 0
\(321\) 360920. 0.195501
\(322\) 1.33178e6 0.715801
\(323\) 728177. 0.388357
\(324\) −23085.0 −0.0122171
\(325\) 0 0
\(326\) −564687. −0.294282
\(327\) −401723. −0.207758
\(328\) −37491.8 −0.0192421
\(329\) −282870. −0.144078
\(330\) 0 0
\(331\) −1.59132e6 −0.798339 −0.399169 0.916877i \(-0.630701\pi\)
−0.399169 + 0.916877i \(0.630701\pi\)
\(332\) 365322. 0.181899
\(333\) −563003. −0.278227
\(334\) 240047. 0.117741
\(335\) 0 0
\(336\) −598751. −0.289333
\(337\) −2.25270e6 −1.08051 −0.540254 0.841502i \(-0.681672\pi\)
−0.540254 + 0.841502i \(0.681672\pi\)
\(338\) −1.95784e6 −0.932150
\(339\) −2.26131e6 −1.06871
\(340\) 0 0
\(341\) 314483. 0.146458
\(342\) −389154. −0.179910
\(343\) −2.08221e6 −0.955628
\(344\) 2.52872e6 1.15214
\(345\) 0 0
\(346\) −2.37714e6 −1.06749
\(347\) 171210. 0.0763318 0.0381659 0.999271i \(-0.487848\pi\)
0.0381659 + 0.999271i \(0.487848\pi\)
\(348\) 159311. 0.0705178
\(349\) 3.09886e6 1.36188 0.680940 0.732339i \(-0.261571\pi\)
0.680940 + 0.732339i \(0.261571\pi\)
\(350\) 0 0
\(351\) −48557.2 −0.0210371
\(352\) 153409. 0.0659923
\(353\) −2.33605e6 −0.997805 −0.498903 0.866658i \(-0.666263\pi\)
−0.498903 + 0.866658i \(0.666263\pi\)
\(354\) −333159. −0.141301
\(355\) 0 0
\(356\) 221913. 0.0928021
\(357\) −538710. −0.223710
\(358\) −2.94727e6 −1.21538
\(359\) 161176. 0.0660029 0.0330015 0.999455i \(-0.489493\pi\)
0.0330015 + 0.999455i \(0.489493\pi\)
\(360\) 0 0
\(361\) −1.66568e6 −0.672703
\(362\) 279694. 0.112179
\(363\) 131769. 0.0524864
\(364\) 17342.7 0.00686063
\(365\) 0 0
\(366\) −506172. −0.197513
\(367\) 2.18840e6 0.848129 0.424064 0.905632i \(-0.360603\pi\)
0.424064 + 0.905632i \(0.360603\pi\)
\(368\) −3.03175e6 −1.16701
\(369\) 16020.8 0.00612519
\(370\) 0 0
\(371\) 359537. 0.135615
\(372\) −82302.9 −0.0308360
\(373\) 3.63629e6 1.35328 0.676639 0.736315i \(-0.263436\pi\)
0.676639 + 0.736315i \(0.263436\pi\)
\(374\) −522334. −0.193094
\(375\) 0 0
\(376\) 724591. 0.264316
\(377\) 335097. 0.121428
\(378\) 287899. 0.103636
\(379\) 1.67725e6 0.599791 0.299895 0.953972i \(-0.403048\pi\)
0.299895 + 0.953972i \(0.403048\pi\)
\(380\) 0 0
\(381\) 1.93014e6 0.681203
\(382\) 724134. 0.253899
\(383\) −2.08391e6 −0.725908 −0.362954 0.931807i \(-0.618232\pi\)
−0.362954 + 0.931807i \(0.618232\pi\)
\(384\) 1.34166e6 0.464315
\(385\) 0 0
\(386\) −2.99557e6 −1.02332
\(387\) −1.08056e6 −0.366752
\(388\) 221543. 0.0747101
\(389\) −5.80280e6 −1.94430 −0.972151 0.234356i \(-0.924702\pi\)
−0.972151 + 0.234356i \(0.924702\pi\)
\(390\) 0 0
\(391\) −2.72773e6 −0.902319
\(392\) 2.14786e6 0.705976
\(393\) 1.41654e6 0.462645
\(394\) 4.13241e6 1.34110
\(395\) 0 0
\(396\) −34485.0 −0.0110508
\(397\) 1.25839e6 0.400718 0.200359 0.979723i \(-0.435789\pi\)
0.200359 + 0.979723i \(0.435789\pi\)
\(398\) −2.63785e6 −0.834722
\(399\) −599554. −0.188537
\(400\) 0 0
\(401\) −1.29837e6 −0.403216 −0.201608 0.979466i \(-0.564617\pi\)
−0.201608 + 0.979466i \(0.564617\pi\)
\(402\) −1.47783e6 −0.456099
\(403\) −173117. −0.0530977
\(404\) −113413. −0.0345709
\(405\) 0 0
\(406\) −1.98681e6 −0.598193
\(407\) −841029. −0.251666
\(408\) 1.37994e6 0.410403
\(409\) −4.45805e6 −1.31776 −0.658880 0.752248i \(-0.728970\pi\)
−0.658880 + 0.752248i \(0.728970\pi\)
\(410\) 0 0
\(411\) −3.47003e6 −1.01328
\(412\) −161820. −0.0469665
\(413\) −513286. −0.148076
\(414\) 1.45776e6 0.418009
\(415\) 0 0
\(416\) −84448.3 −0.0239253
\(417\) −1.63386e6 −0.460125
\(418\) −581329. −0.162735
\(419\) 2.33339e6 0.649312 0.324656 0.945832i \(-0.394752\pi\)
0.324656 + 0.945832i \(0.394752\pi\)
\(420\) 0 0
\(421\) 4.04741e6 1.11294 0.556470 0.830868i \(-0.312156\pi\)
0.556470 + 0.830868i \(0.312156\pi\)
\(422\) −160651. −0.0439139
\(423\) −309629. −0.0841377
\(424\) −920978. −0.248791
\(425\) 0 0
\(426\) 671234. 0.179205
\(427\) −779840. −0.206983
\(428\) −141101. −0.0372322
\(429\) −72536.1 −0.0190288
\(430\) 0 0
\(431\) 2.38060e6 0.617295 0.308647 0.951177i \(-0.400124\pi\)
0.308647 + 0.951177i \(0.400124\pi\)
\(432\) −655391. −0.168963
\(433\) 5.02364e6 1.28765 0.643826 0.765172i \(-0.277346\pi\)
0.643826 + 0.765172i \(0.277346\pi\)
\(434\) 1.02642e6 0.261577
\(435\) 0 0
\(436\) 157052. 0.0395665
\(437\) −3.03581e6 −0.760451
\(438\) −659630. −0.164292
\(439\) −2.27414e6 −0.563192 −0.281596 0.959533i \(-0.590864\pi\)
−0.281596 + 0.959533i \(0.590864\pi\)
\(440\) 0 0
\(441\) −917813. −0.224728
\(442\) 287534. 0.0700058
\(443\) −93691.6 −0.0226825 −0.0113413 0.999936i \(-0.503610\pi\)
−0.0113413 + 0.999936i \(0.503610\pi\)
\(444\) 220104. 0.0529872
\(445\) 0 0
\(446\) 2.87901e6 0.685341
\(447\) −1.70190e6 −0.402869
\(448\) 2.62959e6 0.619003
\(449\) −5.96118e6 −1.39546 −0.697728 0.716363i \(-0.745806\pi\)
−0.697728 + 0.716363i \(0.745806\pi\)
\(450\) 0 0
\(451\) 23932.4 0.00554044
\(452\) 884052. 0.203532
\(453\) 1.64296e6 0.376167
\(454\) 5.18756e6 1.18120
\(455\) 0 0
\(456\) 1.53580e6 0.345877
\(457\) 878215. 0.196703 0.0983514 0.995152i \(-0.468643\pi\)
0.0983514 + 0.995152i \(0.468643\pi\)
\(458\) 921673. 0.205311
\(459\) −589671. −0.130640
\(460\) 0 0
\(461\) −2960.95 −0.000648901 0 −0.000324450 1.00000i \(-0.500103\pi\)
−0.000324450 1.00000i \(0.500103\pi\)
\(462\) 430071. 0.0937421
\(463\) −1.73957e6 −0.377129 −0.188564 0.982061i \(-0.560383\pi\)
−0.188564 + 0.982061i \(0.560383\pi\)
\(464\) 4.52290e6 0.975264
\(465\) 0 0
\(466\) 97967.3 0.0208986
\(467\) 236495. 0.0501800 0.0250900 0.999685i \(-0.492013\pi\)
0.0250900 + 0.999685i \(0.492013\pi\)
\(468\) 18983.3 0.00400643
\(469\) −2.27683e6 −0.477968
\(470\) 0 0
\(471\) −2.69504e6 −0.559775
\(472\) 1.31481e6 0.271650
\(473\) −1.61417e6 −0.331739
\(474\) 1.39074e6 0.284315
\(475\) 0 0
\(476\) 210607. 0.0426045
\(477\) 393549. 0.0791958
\(478\) −271124. −0.0542747
\(479\) −1.23533e6 −0.246006 −0.123003 0.992406i \(-0.539252\pi\)
−0.123003 + 0.992406i \(0.539252\pi\)
\(480\) 0 0
\(481\) 462969. 0.0912408
\(482\) 305717. 0.0599379
\(483\) 2.24591e6 0.438052
\(484\) −51514.7 −0.00999580
\(485\) 0 0
\(486\) 315133. 0.0605206
\(487\) −4.77244e6 −0.911839 −0.455920 0.890021i \(-0.650690\pi\)
−0.455920 + 0.890021i \(0.650690\pi\)
\(488\) 1.99761e6 0.379718
\(489\) −952289. −0.180093
\(490\) 0 0
\(491\) 9.12682e6 1.70850 0.854251 0.519860i \(-0.174016\pi\)
0.854251 + 0.519860i \(0.174016\pi\)
\(492\) −6263.30 −0.00116652
\(493\) 4.06936e6 0.754065
\(494\) 320009. 0.0589991
\(495\) 0 0
\(496\) −2.33660e6 −0.426463
\(497\) 1.03414e6 0.187797
\(498\) −4.98700e6 −0.901085
\(499\) 8.42908e6 1.51541 0.757703 0.652600i \(-0.226322\pi\)
0.757703 + 0.652600i \(0.226322\pi\)
\(500\) 0 0
\(501\) 404815. 0.0720547
\(502\) 1.47160e6 0.260634
\(503\) 9.13714e6 1.61024 0.805120 0.593113i \(-0.202101\pi\)
0.805120 + 0.593113i \(0.202101\pi\)
\(504\) −1.13619e6 −0.199240
\(505\) 0 0
\(506\) 2.17764e6 0.378103
\(507\) −3.30171e6 −0.570451
\(508\) −754583. −0.129732
\(509\) 3.02787e6 0.518015 0.259008 0.965875i \(-0.416605\pi\)
0.259008 + 0.965875i \(0.416605\pi\)
\(510\) 0 0
\(511\) −1.01627e6 −0.172169
\(512\) −6.59312e6 −1.11152
\(513\) −656270. −0.110100
\(514\) −295251. −0.0492928
\(515\) 0 0
\(516\) 422442. 0.0698462
\(517\) −462532. −0.0761054
\(518\) −2.74497e6 −0.449483
\(519\) −4.00882e6 −0.653278
\(520\) 0 0
\(521\) 1.19443e7 1.92782 0.963912 0.266221i \(-0.0857751\pi\)
0.963912 + 0.266221i \(0.0857751\pi\)
\(522\) −2.17476e6 −0.349329
\(523\) 8.53138e6 1.36384 0.681922 0.731425i \(-0.261144\pi\)
0.681922 + 0.731425i \(0.261144\pi\)
\(524\) −553793. −0.0881087
\(525\) 0 0
\(526\) 8.04000e6 1.26704
\(527\) −2.10230e6 −0.329737
\(528\) −979041. −0.152833
\(529\) 4.93574e6 0.766854
\(530\) 0 0
\(531\) −561841. −0.0864723
\(532\) 234394. 0.0359060
\(533\) −13174.3 −0.00200867
\(534\) −3.02933e6 −0.459720
\(535\) 0 0
\(536\) 5.83225e6 0.876848
\(537\) −4.97029e6 −0.743782
\(538\) 1.60239e6 0.238678
\(539\) −1.37105e6 −0.203274
\(540\) 0 0
\(541\) −2.84250e6 −0.417548 −0.208774 0.977964i \(-0.566947\pi\)
−0.208774 + 0.977964i \(0.566947\pi\)
\(542\) −1.00861e7 −1.47477
\(543\) 471677. 0.0686507
\(544\) −1.02553e6 −0.148576
\(545\) 0 0
\(546\) −236745. −0.0339859
\(547\) 1.40460e6 0.200717 0.100358 0.994951i \(-0.468001\pi\)
0.100358 + 0.994951i \(0.468001\pi\)
\(548\) 1.35660e6 0.192974
\(549\) −853610. −0.120873
\(550\) 0 0
\(551\) 4.52897e6 0.635507
\(552\) −5.75306e6 −0.803620
\(553\) 2.14265e6 0.297947
\(554\) 2.63309e6 0.364495
\(555\) 0 0
\(556\) 638754. 0.0876287
\(557\) −4.17495e6 −0.570182 −0.285091 0.958500i \(-0.592024\pi\)
−0.285091 + 0.958500i \(0.592024\pi\)
\(558\) 1.12351e6 0.152754
\(559\) 888568. 0.120271
\(560\) 0 0
\(561\) −880866. −0.118169
\(562\) 7.42445e6 0.991570
\(563\) 7.54584e6 1.00331 0.501657 0.865067i \(-0.332724\pi\)
0.501657 + 0.865067i \(0.332724\pi\)
\(564\) 121049. 0.0160237
\(565\) 0 0
\(566\) 1.02783e7 1.34859
\(567\) 485513. 0.0634225
\(568\) −2.64903e6 −0.344521
\(569\) 1.62724e6 0.210703 0.105351 0.994435i \(-0.466403\pi\)
0.105351 + 0.994435i \(0.466403\pi\)
\(570\) 0 0
\(571\) 2.56277e6 0.328942 0.164471 0.986382i \(-0.447408\pi\)
0.164471 + 0.986382i \(0.447408\pi\)
\(572\) 28357.8 0.00362395
\(573\) 1.22118e6 0.155379
\(574\) 78111.0 0.00989538
\(575\) 0 0
\(576\) 2.87834e6 0.361482
\(577\) −6.16261e6 −0.770594 −0.385297 0.922793i \(-0.625901\pi\)
−0.385297 + 0.922793i \(0.625901\pi\)
\(578\) −4.08573e6 −0.508687
\(579\) −5.05173e6 −0.626245
\(580\) 0 0
\(581\) −7.68327e6 −0.944290
\(582\) −3.02428e6 −0.370096
\(583\) 587893. 0.0716353
\(584\) 2.60323e6 0.315850
\(585\) 0 0
\(586\) −8.20086e6 −0.986542
\(587\) −9.16105e6 −1.09736 −0.548681 0.836032i \(-0.684870\pi\)
−0.548681 + 0.836032i \(0.684870\pi\)
\(588\) 358816. 0.0427985
\(589\) −2.33974e6 −0.277894
\(590\) 0 0
\(591\) 6.96890e6 0.820721
\(592\) 6.24882e6 0.732814
\(593\) −6.23563e6 −0.728188 −0.364094 0.931362i \(-0.618621\pi\)
−0.364094 + 0.931362i \(0.618621\pi\)
\(594\) 470754. 0.0547429
\(595\) 0 0
\(596\) 665351. 0.0767247
\(597\) −4.44847e6 −0.510828
\(598\) −1.19875e6 −0.137080
\(599\) −1.48855e6 −0.169510 −0.0847552 0.996402i \(-0.527011\pi\)
−0.0847552 + 0.996402i \(0.527011\pi\)
\(600\) 0 0
\(601\) −3.40965e6 −0.385055 −0.192528 0.981292i \(-0.561668\pi\)
−0.192528 + 0.981292i \(0.561668\pi\)
\(602\) −5.26837e6 −0.592496
\(603\) −2.49221e6 −0.279121
\(604\) −642309. −0.0716394
\(605\) 0 0
\(606\) 1.54820e6 0.171256
\(607\) 4.05477e6 0.446678 0.223339 0.974741i \(-0.428304\pi\)
0.223339 + 0.974741i \(0.428304\pi\)
\(608\) −1.14135e6 −0.125216
\(609\) −3.35056e6 −0.366078
\(610\) 0 0
\(611\) 254614. 0.0275918
\(612\) 230530. 0.0248799
\(613\) −1.60665e6 −0.172691 −0.0863456 0.996265i \(-0.527519\pi\)
−0.0863456 + 0.996265i \(0.527519\pi\)
\(614\) 1.16213e7 1.24404
\(615\) 0 0
\(616\) −1.69728e6 −0.180219
\(617\) −1.01598e7 −1.07441 −0.537207 0.843451i \(-0.680520\pi\)
−0.537207 + 0.843451i \(0.680520\pi\)
\(618\) 2.20900e6 0.232661
\(619\) 3.04114e6 0.319014 0.159507 0.987197i \(-0.449009\pi\)
0.159507 + 0.987197i \(0.449009\pi\)
\(620\) 0 0
\(621\) 2.45837e6 0.255811
\(622\) 3.33508e6 0.345645
\(623\) −4.66717e6 −0.481763
\(624\) 538942. 0.0554090
\(625\) 0 0
\(626\) 4.04216e6 0.412267
\(627\) −980354. −0.0995896
\(628\) 1.05362e6 0.106607
\(629\) 5.62221e6 0.566605
\(630\) 0 0
\(631\) −1.07109e6 −0.107091 −0.0535455 0.998565i \(-0.517052\pi\)
−0.0535455 + 0.998565i \(0.517052\pi\)
\(632\) −5.48855e6 −0.546594
\(633\) −270922. −0.0268742
\(634\) 1.65558e7 1.63579
\(635\) 0 0
\(636\) −153857. −0.0150825
\(637\) 754737. 0.0736965
\(638\) −3.24871e6 −0.315980
\(639\) 1.13197e6 0.109669
\(640\) 0 0
\(641\) −902752. −0.0867807 −0.0433904 0.999058i \(-0.513816\pi\)
−0.0433904 + 0.999058i \(0.513816\pi\)
\(642\) 1.92616e6 0.184440
\(643\) 1.24048e7 1.18321 0.591607 0.806227i \(-0.298494\pi\)
0.591607 + 0.806227i \(0.298494\pi\)
\(644\) −878033. −0.0834250
\(645\) 0 0
\(646\) 3.88614e6 0.366384
\(647\) 1.32160e7 1.24119 0.620594 0.784132i \(-0.286891\pi\)
0.620594 + 0.784132i \(0.286891\pi\)
\(648\) −1.24367e6 −0.116351
\(649\) −839293. −0.0782172
\(650\) 0 0
\(651\) 1.73095e6 0.160079
\(652\) 372294. 0.0342979
\(653\) 1.61119e6 0.147864 0.0739321 0.997263i \(-0.476445\pi\)
0.0739321 + 0.997263i \(0.476445\pi\)
\(654\) −2.14391e6 −0.196003
\(655\) 0 0
\(656\) −177817. −0.0161329
\(657\) −1.11240e6 −0.100542
\(658\) −1.50962e6 −0.135926
\(659\) 2.72261e6 0.244215 0.122108 0.992517i \(-0.461035\pi\)
0.122108 + 0.992517i \(0.461035\pi\)
\(660\) 0 0
\(661\) −6.62010e6 −0.589333 −0.294666 0.955600i \(-0.595209\pi\)
−0.294666 + 0.955600i \(0.595209\pi\)
\(662\) −8.49256e6 −0.753171
\(663\) 484898. 0.0428417
\(664\) 1.96812e7 1.73233
\(665\) 0 0
\(666\) −3.00463e6 −0.262486
\(667\) −1.69654e7 −1.47655
\(668\) −158261. −0.0137225
\(669\) 4.85518e6 0.419411
\(670\) 0 0
\(671\) −1.27515e6 −0.109334
\(672\) 844380. 0.0721298
\(673\) 4.49731e6 0.382750 0.191375 0.981517i \(-0.438705\pi\)
0.191375 + 0.981517i \(0.438705\pi\)
\(674\) −1.20222e7 −1.01938
\(675\) 0 0
\(676\) 1.29079e6 0.108640
\(677\) 1.05097e6 0.0881288 0.0440644 0.999029i \(-0.485969\pi\)
0.0440644 + 0.999029i \(0.485969\pi\)
\(678\) −1.20682e7 −1.00825
\(679\) −4.65939e6 −0.387842
\(680\) 0 0
\(681\) 8.74832e6 0.722865
\(682\) 1.67834e6 0.138171
\(683\) 158619. 0.0130108 0.00650540 0.999979i \(-0.497929\pi\)
0.00650540 + 0.999979i \(0.497929\pi\)
\(684\) 256567. 0.0209682
\(685\) 0 0
\(686\) −1.11123e7 −0.901561
\(687\) 1.55431e6 0.125645
\(688\) 1.19933e7 0.965976
\(689\) −323623. −0.0259712
\(690\) 0 0
\(691\) −1.14939e7 −0.915741 −0.457870 0.889019i \(-0.651388\pi\)
−0.457870 + 0.889019i \(0.651388\pi\)
\(692\) 1.56724e6 0.124414
\(693\) 725272. 0.0573678
\(694\) 913714. 0.0720131
\(695\) 0 0
\(696\) 8.58268e6 0.671583
\(697\) −159986. −0.0124738
\(698\) 1.65380e7 1.28483
\(699\) 165212. 0.0127894
\(700\) 0 0
\(701\) −1.15908e7 −0.890879 −0.445440 0.895312i \(-0.646953\pi\)
−0.445440 + 0.895312i \(0.646953\pi\)
\(702\) −259140. −0.0198469
\(703\) 6.25721e6 0.477521
\(704\) 4.29975e6 0.326972
\(705\) 0 0
\(706\) −1.24671e7 −0.941352
\(707\) 2.38525e6 0.179467
\(708\) 219650. 0.0164683
\(709\) 8.43220e6 0.629978 0.314989 0.949095i \(-0.397999\pi\)
0.314989 + 0.949095i \(0.397999\pi\)
\(710\) 0 0
\(711\) 2.34534e6 0.173993
\(712\) 1.19552e7 0.883809
\(713\) 8.76461e6 0.645667
\(714\) −2.87499e6 −0.211053
\(715\) 0 0
\(716\) 1.94312e6 0.141650
\(717\) −457223. −0.0332147
\(718\) 860163. 0.0622687
\(719\) 2.06959e6 0.149301 0.0746505 0.997210i \(-0.476216\pi\)
0.0746505 + 0.997210i \(0.476216\pi\)
\(720\) 0 0
\(721\) 3.40331e6 0.243817
\(722\) −8.88940e6 −0.634643
\(723\) 515562. 0.0366805
\(724\) −184401. −0.0130742
\(725\) 0 0
\(726\) 703225. 0.0495168
\(727\) 1.69013e7 1.18600 0.593001 0.805202i \(-0.297943\pi\)
0.593001 + 0.805202i \(0.297943\pi\)
\(728\) 934315. 0.0653378
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 1.07906e7 0.746884
\(732\) 333716. 0.0230197
\(733\) 9.45646e6 0.650083 0.325041 0.945700i \(-0.394622\pi\)
0.325041 + 0.945700i \(0.394622\pi\)
\(734\) 1.16791e7 0.800144
\(735\) 0 0
\(736\) 4.27548e6 0.290931
\(737\) −3.72294e6 −0.252474
\(738\) 85500.1 0.00577864
\(739\) 1.41110e7 0.950486 0.475243 0.879855i \(-0.342360\pi\)
0.475243 + 0.879855i \(0.342360\pi\)
\(740\) 0 0
\(741\) 539665. 0.0361059
\(742\) 1.91878e6 0.127943
\(743\) 3.86361e6 0.256757 0.128378 0.991725i \(-0.459023\pi\)
0.128378 + 0.991725i \(0.459023\pi\)
\(744\) −4.43395e6 −0.293669
\(745\) 0 0
\(746\) 1.94062e7 1.27671
\(747\) −8.41008e6 −0.551441
\(748\) 344372. 0.0225047
\(749\) 2.96756e6 0.193283
\(750\) 0 0
\(751\) 5.68914e6 0.368084 0.184042 0.982918i \(-0.441082\pi\)
0.184042 + 0.982918i \(0.441082\pi\)
\(752\) 3.43660e6 0.221608
\(753\) 2.48172e6 0.159502
\(754\) 1.78835e6 0.114557
\(755\) 0 0
\(756\) −189810. −0.0120785
\(757\) 2.31729e7 1.46974 0.734870 0.678208i \(-0.237243\pi\)
0.734870 + 0.678208i \(0.237243\pi\)
\(758\) 8.95116e6 0.565856
\(759\) 3.67238e6 0.231389
\(760\) 0 0
\(761\) −1.16856e7 −0.731455 −0.365728 0.930722i \(-0.619180\pi\)
−0.365728 + 0.930722i \(0.619180\pi\)
\(762\) 1.03008e7 0.642662
\(763\) −3.30304e6 −0.205401
\(764\) −477417. −0.0295913
\(765\) 0 0
\(766\) −1.11214e7 −0.684838
\(767\) 462014. 0.0283574
\(768\) −3.07395e6 −0.188059
\(769\) 7.02112e6 0.428145 0.214072 0.976818i \(-0.431327\pi\)
0.214072 + 0.976818i \(0.431327\pi\)
\(770\) 0 0
\(771\) −497912. −0.0301659
\(772\) 1.97496e6 0.119266
\(773\) 4.45003e6 0.267864 0.133932 0.990991i \(-0.457240\pi\)
0.133932 + 0.990991i \(0.457240\pi\)
\(774\) −5.76674e6 −0.346002
\(775\) 0 0
\(776\) 1.19353e7 0.711509
\(777\) −4.62912e6 −0.275072
\(778\) −3.09684e7 −1.83430
\(779\) −178056. −0.0105126
\(780\) 0 0
\(781\) 1.69097e6 0.0991991
\(782\) −1.45574e7 −0.851268
\(783\) −3.66751e6 −0.213780
\(784\) 1.01869e7 0.591905
\(785\) 0 0
\(786\) 7.55980e6 0.436470
\(787\) 1.96811e7 1.13269 0.566347 0.824167i \(-0.308356\pi\)
0.566347 + 0.824167i \(0.308356\pi\)
\(788\) −2.72447e6 −0.156303
\(789\) 1.35587e7 0.775398
\(790\) 0 0
\(791\) −1.85929e7 −1.05659
\(792\) −1.85783e6 −0.105243
\(793\) 701941. 0.0396386
\(794\) 6.71578e6 0.378047
\(795\) 0 0
\(796\) 1.73912e6 0.0972850
\(797\) −3.00192e7 −1.67399 −0.836996 0.547208i \(-0.815690\pi\)
−0.836996 + 0.547208i \(0.815690\pi\)
\(798\) −3.19970e6 −0.177870
\(799\) 3.09199e6 0.171345
\(800\) 0 0
\(801\) −5.10867e6 −0.281337
\(802\) −6.92915e6 −0.380403
\(803\) −1.66174e6 −0.0909439
\(804\) 974323. 0.0531573
\(805\) 0 0
\(806\) −923889. −0.0500936
\(807\) 2.70227e6 0.146065
\(808\) −6.10998e6 −0.329239
\(809\) −8.99554e6 −0.483233 −0.241616 0.970372i \(-0.577678\pi\)
−0.241616 + 0.970372i \(0.577678\pi\)
\(810\) 0 0
\(811\) −2.96119e7 −1.58093 −0.790467 0.612505i \(-0.790162\pi\)
−0.790467 + 0.612505i \(0.790162\pi\)
\(812\) 1.30989e6 0.0697180
\(813\) −1.70092e7 −0.902523
\(814\) −4.48840e6 −0.237427
\(815\) 0 0
\(816\) 6.54481e6 0.344090
\(817\) 1.20093e7 0.629455
\(818\) −2.37917e7 −1.24320
\(819\) −399247. −0.0207985
\(820\) 0 0
\(821\) −2.05145e7 −1.06219 −0.531097 0.847311i \(-0.678220\pi\)
−0.531097 + 0.847311i \(0.678220\pi\)
\(822\) −1.85189e7 −0.955949
\(823\) 2.38536e7 1.22759 0.613796 0.789465i \(-0.289642\pi\)
0.613796 + 0.789465i \(0.289642\pi\)
\(824\) −8.71781e6 −0.447290
\(825\) 0 0
\(826\) −2.73930e6 −0.139698
\(827\) 1.89103e7 0.961468 0.480734 0.876866i \(-0.340370\pi\)
0.480734 + 0.876866i \(0.340370\pi\)
\(828\) −961093. −0.0487180
\(829\) 3.48075e7 1.75908 0.879541 0.475823i \(-0.157850\pi\)
0.879541 + 0.475823i \(0.157850\pi\)
\(830\) 0 0
\(831\) 4.44045e6 0.223061
\(832\) −2.36692e6 −0.118543
\(833\) 9.16540e6 0.457656
\(834\) −8.71961e6 −0.434092
\(835\) 0 0
\(836\) 383266. 0.0189664
\(837\) 1.89470e6 0.0934816
\(838\) 1.24529e7 0.612575
\(839\) −2.32033e7 −1.13800 −0.569002 0.822336i \(-0.692670\pi\)
−0.569002 + 0.822336i \(0.692670\pi\)
\(840\) 0 0
\(841\) 4.79863e6 0.233952
\(842\) 2.16002e7 1.04997
\(843\) 1.25206e7 0.606815
\(844\) 105916. 0.00511807
\(845\) 0 0
\(846\) −1.65243e6 −0.0793774
\(847\) 1.08343e6 0.0518911
\(848\) −4.36804e6 −0.208591
\(849\) 1.73333e7 0.825299
\(850\) 0 0
\(851\) −2.34393e7 −1.10949
\(852\) −442540. −0.0208859
\(853\) 2.30500e7 1.08467 0.542336 0.840162i \(-0.317540\pi\)
0.542336 + 0.840162i \(0.317540\pi\)
\(854\) −4.16185e6 −0.195273
\(855\) 0 0
\(856\) −7.60159e6 −0.354585
\(857\) −1.45809e7 −0.678158 −0.339079 0.940758i \(-0.610115\pi\)
−0.339079 + 0.940758i \(0.610115\pi\)
\(858\) −387111. −0.0179522
\(859\) 4.26393e6 0.197164 0.0985818 0.995129i \(-0.468569\pi\)
0.0985818 + 0.995129i \(0.468569\pi\)
\(860\) 0 0
\(861\) 131727. 0.00605572
\(862\) 1.27048e7 0.582370
\(863\) 9.80948e6 0.448352 0.224176 0.974549i \(-0.428031\pi\)
0.224176 + 0.974549i \(0.428031\pi\)
\(864\) 924256. 0.0421219
\(865\) 0 0
\(866\) 2.68102e7 1.21480
\(867\) −6.89019e6 −0.311303
\(868\) −676711. −0.0304863
\(869\) 3.50354e6 0.157383
\(870\) 0 0
\(871\) 2.04940e6 0.0915337
\(872\) 8.46096e6 0.376815
\(873\) −5.10016e6 −0.226489
\(874\) −1.62015e7 −0.717427
\(875\) 0 0
\(876\) 434890. 0.0191478
\(877\) 1.86437e7 0.818525 0.409263 0.912417i \(-0.365786\pi\)
0.409263 + 0.912417i \(0.365786\pi\)
\(878\) −1.21367e7 −0.531328
\(879\) −1.38300e7 −0.603738
\(880\) 0 0
\(881\) −3.42571e7 −1.48700 −0.743500 0.668736i \(-0.766836\pi\)
−0.743500 + 0.668736i \(0.766836\pi\)
\(882\) −4.89819e6 −0.212014
\(883\) 1.57969e7 0.681818 0.340909 0.940096i \(-0.389265\pi\)
0.340909 + 0.940096i \(0.389265\pi\)
\(884\) −189570. −0.00815902
\(885\) 0 0
\(886\) −500014. −0.0213992
\(887\) −2.96846e7 −1.26684 −0.633420 0.773808i \(-0.718349\pi\)
−0.633420 + 0.773808i \(0.718349\pi\)
\(888\) 1.18578e7 0.504628
\(889\) 1.58700e7 0.673477
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) −1.89812e6 −0.0798750
\(893\) 3.44121e6 0.144405
\(894\) −9.08268e6 −0.380076
\(895\) 0 0
\(896\) 1.10314e7 0.459049
\(897\) −2.02157e6 −0.0838895
\(898\) −3.18136e7 −1.31650
\(899\) −1.30754e7 −0.539582
\(900\) 0 0
\(901\) −3.93002e6 −0.161281
\(902\) 127722. 0.00522698
\(903\) −8.88460e6 −0.362592
\(904\) 4.76271e7 1.93835
\(905\) 0 0
\(906\) 8.76814e6 0.354885
\(907\) 2.52650e7 1.01977 0.509884 0.860243i \(-0.329688\pi\)
0.509884 + 0.860243i \(0.329688\pi\)
\(908\) −3.42013e6 −0.137666
\(909\) 2.61089e6 0.104804
\(910\) 0 0
\(911\) −2.64889e7 −1.05747 −0.528734 0.848787i \(-0.677333\pi\)
−0.528734 + 0.848787i \(0.677333\pi\)
\(912\) 7.28401e6 0.289990
\(913\) −1.25632e7 −0.498797
\(914\) 4.68686e6 0.185574
\(915\) 0 0
\(916\) −607653. −0.0239286
\(917\) 1.16471e7 0.457398
\(918\) −3.14696e6 −0.123249
\(919\) 4.82111e6 0.188303 0.0941517 0.995558i \(-0.469986\pi\)
0.0941517 + 0.995558i \(0.469986\pi\)
\(920\) 0 0
\(921\) 1.95983e7 0.761322
\(922\) −15802.0 −0.000612188 0
\(923\) −930843. −0.0359643
\(924\) −283543. −0.0109254
\(925\) 0 0
\(926\) −9.28375e6 −0.355792
\(927\) 3.72526e6 0.142383
\(928\) −6.37836e6 −0.243130
\(929\) −5.00384e7 −1.90224 −0.951118 0.308828i \(-0.900063\pi\)
−0.951118 + 0.308828i \(0.900063\pi\)
\(930\) 0 0
\(931\) 1.02006e7 0.385700
\(932\) −64589.3 −0.00243568
\(933\) 5.62428e6 0.211526
\(934\) 1.26213e6 0.0473409
\(935\) 0 0
\(936\) 1.02270e6 0.0381556
\(937\) 3.70330e7 1.37797 0.688985 0.724775i \(-0.258056\pi\)
0.688985 + 0.724775i \(0.258056\pi\)
\(938\) −1.21510e7 −0.450926
\(939\) 6.81672e6 0.252296
\(940\) 0 0
\(941\) 3.30398e7 1.21636 0.608182 0.793798i \(-0.291899\pi\)
0.608182 + 0.793798i \(0.291899\pi\)
\(942\) −1.43829e7 −0.528104
\(943\) 666992. 0.0244254
\(944\) 6.23593e6 0.227757
\(945\) 0 0
\(946\) −8.61452e6 −0.312970
\(947\) 410586. 0.0148775 0.00743873 0.999972i \(-0.497632\pi\)
0.00743873 + 0.999972i \(0.497632\pi\)
\(948\) −916905. −0.0331363
\(949\) 914752. 0.0329714
\(950\) 0 0
\(951\) 2.79198e7 1.00106
\(952\) 1.13462e7 0.405748
\(953\) −4.08733e7 −1.45783 −0.728915 0.684604i \(-0.759975\pi\)
−0.728915 + 0.684604i \(0.759975\pi\)
\(954\) 2.10029e6 0.0747151
\(955\) 0 0
\(956\) 178750. 0.00632560
\(957\) −5.47863e6 −0.193371
\(958\) −6.59272e6 −0.232087
\(959\) −2.85313e7 −1.00179
\(960\) 0 0
\(961\) −2.18742e7 −0.764052
\(962\) 2.47077e6 0.0860786
\(963\) 3.24828e6 0.112872
\(964\) −201557. −0.00698564
\(965\) 0 0
\(966\) 1.19860e7 0.413268
\(967\) 1.34815e7 0.463629 0.231815 0.972760i \(-0.425534\pi\)
0.231815 + 0.972760i \(0.425534\pi\)
\(968\) −2.77528e6 −0.0951960
\(969\) 6.55359e6 0.224218
\(970\) 0 0
\(971\) −4.78750e7 −1.62952 −0.814761 0.579797i \(-0.803132\pi\)
−0.814761 + 0.579797i \(0.803132\pi\)
\(972\) −207765. −0.00705354
\(973\) −1.34340e7 −0.454906
\(974\) −2.54696e7 −0.860250
\(975\) 0 0
\(976\) 9.47431e6 0.318363
\(977\) −1.03468e7 −0.346793 −0.173396 0.984852i \(-0.555474\pi\)
−0.173396 + 0.984852i \(0.555474\pi\)
\(978\) −5.08218e6 −0.169904
\(979\) −7.63146e6 −0.254479
\(980\) 0 0
\(981\) −3.61550e6 −0.119949
\(982\) 4.87080e7 1.61184
\(983\) 1.81075e7 0.597689 0.298844 0.954302i \(-0.403399\pi\)
0.298844 + 0.954302i \(0.403399\pi\)
\(984\) −337426. −0.0111094
\(985\) 0 0
\(986\) 2.17174e7 0.711402
\(987\) −2.54583e6 −0.0831835
\(988\) −210980. −0.00687621
\(989\) −4.49867e7 −1.46249
\(990\) 0 0
\(991\) −4.58378e7 −1.48265 −0.741327 0.671144i \(-0.765803\pi\)
−0.741327 + 0.671144i \(0.765803\pi\)
\(992\) 3.29516e6 0.106316
\(993\) −1.43219e7 −0.460921
\(994\) 5.51902e6 0.177172
\(995\) 0 0
\(996\) 3.28790e6 0.105019
\(997\) −3.02752e7 −0.964605 −0.482302 0.876005i \(-0.660199\pi\)
−0.482302 + 0.876005i \(0.660199\pi\)
\(998\) 4.49844e7 1.42967
\(999\) −5.06702e6 −0.160635
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.m.1.6 7
5.4 even 2 825.6.a.o.1.2 yes 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.m.1.6 7 1.1 even 1 trivial
825.6.a.o.1.2 yes 7 5.4 even 2