Properties

Label 825.6.a.t.1.2
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $10$
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: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - x^{9} - 271 x^{8} + 309 x^{7} + 24456 x^{6} - 33410 x^{5} - 822204 x^{4} + 1367872 x^{3} + 7443872 x^{2} - 12856224 x - 7036608 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(9.15502\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-9.15502 q^{2} +9.00000 q^{3} +51.8143 q^{4} -82.3952 q^{6} +226.656 q^{7} -181.401 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-9.15502 q^{2} +9.00000 q^{3} +51.8143 q^{4} -82.3952 q^{6} +226.656 q^{7} -181.401 q^{8} +81.0000 q^{9} +121.000 q^{11} +466.329 q^{12} -60.5015 q^{13} -2075.04 q^{14} +2.66672 q^{16} +1984.50 q^{17} -741.556 q^{18} -1094.38 q^{19} +2039.90 q^{21} -1107.76 q^{22} -4636.35 q^{23} -1632.61 q^{24} +553.893 q^{26} +729.000 q^{27} +11744.0 q^{28} +8565.55 q^{29} -409.809 q^{31} +5780.41 q^{32} +1089.00 q^{33} -18168.1 q^{34} +4196.96 q^{36} -10085.7 q^{37} +10019.1 q^{38} -544.514 q^{39} +9021.70 q^{41} -18675.3 q^{42} +10712.1 q^{43} +6269.53 q^{44} +42445.8 q^{46} -17017.3 q^{47} +24.0005 q^{48} +34565.8 q^{49} +17860.5 q^{51} -3134.85 q^{52} +28394.2 q^{53} -6674.01 q^{54} -41115.5 q^{56} -9849.44 q^{57} -78417.8 q^{58} +30227.5 q^{59} -36839.5 q^{61} +3751.81 q^{62} +18359.1 q^{63} -53005.0 q^{64} -9969.81 q^{66} -11315.1 q^{67} +102825. q^{68} -41727.1 q^{69} +31917.4 q^{71} -14693.4 q^{72} -20881.1 q^{73} +92334.8 q^{74} -56704.7 q^{76} +27425.3 q^{77} +4985.03 q^{78} +77091.6 q^{79} +6561.00 q^{81} -82593.8 q^{82} +99048.9 q^{83} +105696. q^{84} -98069.0 q^{86} +77090.0 q^{87} -21949.5 q^{88} +130505. q^{89} -13713.0 q^{91} -240229. q^{92} -3688.28 q^{93} +155794. q^{94} +52023.6 q^{96} +74343.6 q^{97} -316450. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - q^{2} + 90 q^{3} + 223 q^{4} - 9 q^{6} + 188 q^{7} + 177 q^{8} + 810 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - q^{2} + 90 q^{3} + 223 q^{4} - 9 q^{6} + 188 q^{7} + 177 q^{8} + 810 q^{9} + 1210 q^{11} + 2007 q^{12} - 1102 q^{13} + 684 q^{14} + 7019 q^{16} + 1106 q^{17} - 81 q^{18} + 2586 q^{19} + 1692 q^{21} - 121 q^{22} + 2206 q^{23} + 1593 q^{24} + 12001 q^{26} + 7290 q^{27} + 14452 q^{28} + 5824 q^{29} - 4586 q^{31} - 10627 q^{32} + 10890 q^{33} + 7426 q^{34} + 18063 q^{36} - 18362 q^{37} + 44001 q^{38} - 9918 q^{39} - 5474 q^{41} + 6156 q^{42} + 20496 q^{43} + 26983 q^{44} + 19981 q^{46} - 14970 q^{47} + 63171 q^{48} + 68582 q^{49} + 9954 q^{51} - 58603 q^{52} + 61980 q^{53} - 729 q^{54} + 7132 q^{56} + 23274 q^{57} + 7161 q^{58} + 61190 q^{59} + 8230 q^{61} - 4509 q^{62} + 15228 q^{63} + 152223 q^{64} - 1089 q^{66} - 11930 q^{67} + 105598 q^{68} + 19854 q^{69} + 59822 q^{71} + 14337 q^{72} - 20680 q^{73} + 132564 q^{74} + 68165 q^{76} + 22748 q^{77} + 108009 q^{78} + 234494 q^{79} + 65610 q^{81} + 151948 q^{82} + 185478 q^{83} + 130068 q^{84} - 17825 q^{86} + 52416 q^{87} + 21417 q^{88} + 181834 q^{89} + 206274 q^{91} + 98373 q^{92} - 41274 q^{93} - 64998 q^{94} - 95643 q^{96} + 304358 q^{97} + 153453 q^{98} + 98010 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −9.15502 −1.61839 −0.809197 0.587538i \(-0.800097\pi\)
−0.809197 + 0.587538i \(0.800097\pi\)
\(3\) 9.00000 0.577350
\(4\) 51.8143 1.61920
\(5\) 0 0
\(6\) −82.3952 −0.934380
\(7\) 226.656 1.74832 0.874161 0.485636i \(-0.161412\pi\)
0.874161 + 0.485636i \(0.161412\pi\)
\(8\) −181.401 −1.00211
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) 466.329 0.934844
\(13\) −60.5015 −0.0992906 −0.0496453 0.998767i \(-0.515809\pi\)
−0.0496453 + 0.998767i \(0.515809\pi\)
\(14\) −2075.04 −2.82947
\(15\) 0 0
\(16\) 2.66672 0.00260422
\(17\) 1984.50 1.66544 0.832718 0.553697i \(-0.186783\pi\)
0.832718 + 0.553697i \(0.186783\pi\)
\(18\) −741.556 −0.539465
\(19\) −1094.38 −0.695481 −0.347740 0.937591i \(-0.613051\pi\)
−0.347740 + 0.937591i \(0.613051\pi\)
\(20\) 0 0
\(21\) 2039.90 1.00939
\(22\) −1107.76 −0.487964
\(23\) −4636.35 −1.82750 −0.913748 0.406282i \(-0.866825\pi\)
−0.913748 + 0.406282i \(0.866825\pi\)
\(24\) −1632.61 −0.578566
\(25\) 0 0
\(26\) 553.893 0.160691
\(27\) 729.000 0.192450
\(28\) 11744.0 2.83088
\(29\) 8565.55 1.89130 0.945650 0.325186i \(-0.105427\pi\)
0.945650 + 0.325186i \(0.105427\pi\)
\(30\) 0 0
\(31\) −409.809 −0.0765908 −0.0382954 0.999266i \(-0.512193\pi\)
−0.0382954 + 0.999266i \(0.512193\pi\)
\(32\) 5780.41 0.997892
\(33\) 1089.00 0.174078
\(34\) −18168.1 −2.69533
\(35\) 0 0
\(36\) 4196.96 0.539733
\(37\) −10085.7 −1.21116 −0.605581 0.795784i \(-0.707059\pi\)
−0.605581 + 0.795784i \(0.707059\pi\)
\(38\) 10019.1 1.12556
\(39\) −544.514 −0.0573254
\(40\) 0 0
\(41\) 9021.70 0.838163 0.419082 0.907949i \(-0.362352\pi\)
0.419082 + 0.907949i \(0.362352\pi\)
\(42\) −18675.3 −1.63360
\(43\) 10712.1 0.883490 0.441745 0.897141i \(-0.354360\pi\)
0.441745 + 0.897141i \(0.354360\pi\)
\(44\) 6269.53 0.488207
\(45\) 0 0
\(46\) 42445.8 2.95761
\(47\) −17017.3 −1.12369 −0.561844 0.827243i \(-0.689908\pi\)
−0.561844 + 0.827243i \(0.689908\pi\)
\(48\) 24.0005 0.00150355
\(49\) 34565.8 2.05663
\(50\) 0 0
\(51\) 17860.5 0.961540
\(52\) −3134.85 −0.160771
\(53\) 28394.2 1.38848 0.694241 0.719742i \(-0.255740\pi\)
0.694241 + 0.719742i \(0.255740\pi\)
\(54\) −6674.01 −0.311460
\(55\) 0 0
\(56\) −41115.5 −1.75200
\(57\) −9849.44 −0.401536
\(58\) −78417.8 −3.06087
\(59\) 30227.5 1.13050 0.565252 0.824918i \(-0.308779\pi\)
0.565252 + 0.824918i \(0.308779\pi\)
\(60\) 0 0
\(61\) −36839.5 −1.26762 −0.633810 0.773489i \(-0.718510\pi\)
−0.633810 + 0.773489i \(0.718510\pi\)
\(62\) 3751.81 0.123954
\(63\) 18359.1 0.582774
\(64\) −53005.0 −1.61759
\(65\) 0 0
\(66\) −9969.81 −0.281726
\(67\) −11315.1 −0.307945 −0.153972 0.988075i \(-0.549207\pi\)
−0.153972 + 0.988075i \(0.549207\pi\)
\(68\) 102825. 2.69667
\(69\) −41727.1 −1.05510
\(70\) 0 0
\(71\) 31917.4 0.751417 0.375709 0.926738i \(-0.377399\pi\)
0.375709 + 0.926738i \(0.377399\pi\)
\(72\) −14693.4 −0.334035
\(73\) −20881.1 −0.458612 −0.229306 0.973354i \(-0.573646\pi\)
−0.229306 + 0.973354i \(0.573646\pi\)
\(74\) 92334.8 1.96014
\(75\) 0 0
\(76\) −56704.7 −1.12612
\(77\) 27425.3 0.527139
\(78\) 4985.03 0.0927751
\(79\) 77091.6 1.38976 0.694879 0.719126i \(-0.255458\pi\)
0.694879 + 0.719126i \(0.255458\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −82593.8 −1.35648
\(83\) 99048.9 1.57817 0.789086 0.614283i \(-0.210555\pi\)
0.789086 + 0.614283i \(0.210555\pi\)
\(84\) 105696. 1.63441
\(85\) 0 0
\(86\) −98069.0 −1.42983
\(87\) 77090.0 1.09194
\(88\) −21949.5 −0.302146
\(89\) 130505. 1.74643 0.873217 0.487332i \(-0.162030\pi\)
0.873217 + 0.487332i \(0.162030\pi\)
\(90\) 0 0
\(91\) −13713.0 −0.173592
\(92\) −240229. −2.95908
\(93\) −3688.28 −0.0442197
\(94\) 155794. 1.81857
\(95\) 0 0
\(96\) 52023.6 0.576133
\(97\) 74343.6 0.802259 0.401129 0.916021i \(-0.368618\pi\)
0.401129 + 0.916021i \(0.368618\pi\)
\(98\) −316450. −3.32844
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) 7288.41 0.0710934 0.0355467 0.999368i \(-0.488683\pi\)
0.0355467 + 0.999368i \(0.488683\pi\)
\(102\) −163513. −1.55615
\(103\) 55373.1 0.514287 0.257144 0.966373i \(-0.417219\pi\)
0.257144 + 0.966373i \(0.417219\pi\)
\(104\) 10975.0 0.0994997
\(105\) 0 0
\(106\) −259950. −2.24711
\(107\) −227505. −1.92102 −0.960510 0.278245i \(-0.910247\pi\)
−0.960510 + 0.278245i \(0.910247\pi\)
\(108\) 37772.7 0.311615
\(109\) −116049. −0.935566 −0.467783 0.883843i \(-0.654947\pi\)
−0.467783 + 0.883843i \(0.654947\pi\)
\(110\) 0 0
\(111\) −90771.3 −0.699264
\(112\) 604.428 0.00455302
\(113\) −213268. −1.57119 −0.785597 0.618738i \(-0.787644\pi\)
−0.785597 + 0.618738i \(0.787644\pi\)
\(114\) 90171.8 0.649843
\(115\) 0 0
\(116\) 443819. 3.06239
\(117\) −4900.62 −0.0330969
\(118\) −276733. −1.82960
\(119\) 449797. 2.91172
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 337266. 2.05151
\(123\) 81195.3 0.483914
\(124\) −21234.0 −0.124016
\(125\) 0 0
\(126\) −168078. −0.943158
\(127\) −76087.7 −0.418606 −0.209303 0.977851i \(-0.567119\pi\)
−0.209303 + 0.977851i \(0.567119\pi\)
\(128\) 300289. 1.62000
\(129\) 96408.5 0.510083
\(130\) 0 0
\(131\) 147751. 0.752233 0.376116 0.926572i \(-0.377259\pi\)
0.376116 + 0.926572i \(0.377259\pi\)
\(132\) 56425.8 0.281866
\(133\) −248048. −1.21592
\(134\) 103590. 0.498376
\(135\) 0 0
\(136\) −359989. −1.66894
\(137\) −231518. −1.05386 −0.526931 0.849908i \(-0.676657\pi\)
−0.526931 + 0.849908i \(0.676657\pi\)
\(138\) 382012. 1.70758
\(139\) −121702. −0.534270 −0.267135 0.963659i \(-0.586077\pi\)
−0.267135 + 0.963659i \(0.586077\pi\)
\(140\) 0 0
\(141\) −153156. −0.648762
\(142\) −292204. −1.21609
\(143\) −7320.69 −0.0299372
\(144\) 216.005 0.000868074 0
\(145\) 0 0
\(146\) 191167. 0.742215
\(147\) 311092. 1.18740
\(148\) −522584. −1.96111
\(149\) −301130. −1.11119 −0.555594 0.831454i \(-0.687509\pi\)
−0.555594 + 0.831454i \(0.687509\pi\)
\(150\) 0 0
\(151\) 259623. 0.926619 0.463310 0.886196i \(-0.346662\pi\)
0.463310 + 0.886196i \(0.346662\pi\)
\(152\) 198522. 0.696946
\(153\) 160744. 0.555146
\(154\) −251079. −0.853118
\(155\) 0 0
\(156\) −28213.6 −0.0928212
\(157\) −14890.4 −0.0482122 −0.0241061 0.999709i \(-0.507674\pi\)
−0.0241061 + 0.999709i \(0.507674\pi\)
\(158\) −705775. −2.24918
\(159\) 255548. 0.801641
\(160\) 0 0
\(161\) −1.05085e6 −3.19505
\(162\) −60066.1 −0.179822
\(163\) −200766. −0.591862 −0.295931 0.955209i \(-0.595630\pi\)
−0.295931 + 0.955209i \(0.595630\pi\)
\(164\) 467453. 1.35715
\(165\) 0 0
\(166\) −906794. −2.55410
\(167\) 89683.6 0.248841 0.124420 0.992230i \(-0.460293\pi\)
0.124420 + 0.992230i \(0.460293\pi\)
\(168\) −370039. −1.01152
\(169\) −367633. −0.990141
\(170\) 0 0
\(171\) −88645.0 −0.231827
\(172\) 555038. 1.43054
\(173\) 413325. 1.04997 0.524984 0.851112i \(-0.324071\pi\)
0.524984 + 0.851112i \(0.324071\pi\)
\(174\) −705760. −1.76719
\(175\) 0 0
\(176\) 322.674 0.000785203 0
\(177\) 272047. 0.652697
\(178\) −1.19478e6 −2.82642
\(179\) 36462.3 0.0850572 0.0425286 0.999095i \(-0.486459\pi\)
0.0425286 + 0.999095i \(0.486459\pi\)
\(180\) 0 0
\(181\) 143714. 0.326064 0.163032 0.986621i \(-0.447873\pi\)
0.163032 + 0.986621i \(0.447873\pi\)
\(182\) 125543. 0.280940
\(183\) −331556. −0.731861
\(184\) 841036. 1.83134
\(185\) 0 0
\(186\) 33766.2 0.0715650
\(187\) 240124. 0.502148
\(188\) −881740. −1.81947
\(189\) 165232. 0.336465
\(190\) 0 0
\(191\) 144930. 0.287458 0.143729 0.989617i \(-0.454091\pi\)
0.143729 + 0.989617i \(0.454091\pi\)
\(192\) −477045. −0.933913
\(193\) 177102. 0.342239 0.171120 0.985250i \(-0.445262\pi\)
0.171120 + 0.985250i \(0.445262\pi\)
\(194\) −680617. −1.29837
\(195\) 0 0
\(196\) 1.79100e6 3.33009
\(197\) −1.01364e6 −1.86087 −0.930437 0.366453i \(-0.880572\pi\)
−0.930437 + 0.366453i \(0.880572\pi\)
\(198\) −89728.3 −0.162655
\(199\) 235430. 0.421434 0.210717 0.977547i \(-0.432420\pi\)
0.210717 + 0.977547i \(0.432420\pi\)
\(200\) 0 0
\(201\) −101836. −0.177792
\(202\) −66725.5 −0.115057
\(203\) 1.94143e6 3.30660
\(204\) 925429. 1.55692
\(205\) 0 0
\(206\) −506942. −0.832320
\(207\) −375544. −0.609165
\(208\) −161.341 −0.000258575 0
\(209\) −132420. −0.209695
\(210\) 0 0
\(211\) −919022. −1.42108 −0.710542 0.703655i \(-0.751550\pi\)
−0.710542 + 0.703655i \(0.751550\pi\)
\(212\) 1.47123e6 2.24823
\(213\) 287256. 0.433831
\(214\) 2.08281e6 3.10897
\(215\) 0 0
\(216\) −132241. −0.192855
\(217\) −92885.4 −0.133905
\(218\) 1.06243e6 1.51411
\(219\) −187930. −0.264780
\(220\) 0 0
\(221\) −120065. −0.165362
\(222\) 831013. 1.13168
\(223\) 1.28777e6 1.73411 0.867053 0.498216i \(-0.166011\pi\)
0.867053 + 0.498216i \(0.166011\pi\)
\(224\) 1.31016e6 1.74464
\(225\) 0 0
\(226\) 1.95247e6 2.54281
\(227\) −545866. −0.703107 −0.351554 0.936168i \(-0.614346\pi\)
−0.351554 + 0.936168i \(0.614346\pi\)
\(228\) −510342. −0.650166
\(229\) −542716. −0.683886 −0.341943 0.939721i \(-0.611085\pi\)
−0.341943 + 0.939721i \(0.611085\pi\)
\(230\) 0 0
\(231\) 246828. 0.304344
\(232\) −1.55380e6 −1.89528
\(233\) 126501. 0.152653 0.0763265 0.997083i \(-0.475681\pi\)
0.0763265 + 0.997083i \(0.475681\pi\)
\(234\) 44865.3 0.0535637
\(235\) 0 0
\(236\) 1.56622e6 1.83051
\(237\) 693825. 0.802378
\(238\) −4.11790e6 −4.71231
\(239\) 1.36942e6 1.55075 0.775377 0.631499i \(-0.217560\pi\)
0.775377 + 0.631499i \(0.217560\pi\)
\(240\) 0 0
\(241\) 1.28969e6 1.43036 0.715178 0.698942i \(-0.246346\pi\)
0.715178 + 0.698942i \(0.246346\pi\)
\(242\) −134039. −0.147127
\(243\) 59049.0 0.0641500
\(244\) −1.90881e6 −2.05253
\(245\) 0 0
\(246\) −743344. −0.783163
\(247\) 66211.8 0.0690547
\(248\) 74339.5 0.0767522
\(249\) 891440. 0.911158
\(250\) 0 0
\(251\) 316251. 0.316845 0.158423 0.987371i \(-0.449359\pi\)
0.158423 + 0.987371i \(0.449359\pi\)
\(252\) 951265. 0.943626
\(253\) −560998. −0.551011
\(254\) 696585. 0.677469
\(255\) 0 0
\(256\) −1.05299e6 −1.00421
\(257\) −100900. −0.0952922 −0.0476461 0.998864i \(-0.515172\pi\)
−0.0476461 + 0.998864i \(0.515172\pi\)
\(258\) −882621. −0.825515
\(259\) −2.28598e6 −2.11750
\(260\) 0 0
\(261\) 693810. 0.630433
\(262\) −1.35266e6 −1.21741
\(263\) −346997. −0.309340 −0.154670 0.987966i \(-0.549431\pi\)
−0.154670 + 0.987966i \(0.549431\pi\)
\(264\) −197545. −0.174444
\(265\) 0 0
\(266\) 2.27088e6 1.96784
\(267\) 1.17454e6 1.00830
\(268\) −586287. −0.498624
\(269\) 1.12964e6 0.951831 0.475915 0.879491i \(-0.342117\pi\)
0.475915 + 0.879491i \(0.342117\pi\)
\(270\) 0 0
\(271\) 756638. 0.625842 0.312921 0.949779i \(-0.398692\pi\)
0.312921 + 0.949779i \(0.398692\pi\)
\(272\) 5292.11 0.00433717
\(273\) −123417. −0.100223
\(274\) 2.11955e6 1.70556
\(275\) 0 0
\(276\) −2.16206e6 −1.70842
\(277\) 1.61681e6 1.26607 0.633036 0.774122i \(-0.281808\pi\)
0.633036 + 0.774122i \(0.281808\pi\)
\(278\) 1.11419e6 0.864660
\(279\) −33194.5 −0.0255303
\(280\) 0 0
\(281\) 1.28139e6 0.968091 0.484046 0.875043i \(-0.339167\pi\)
0.484046 + 0.875043i \(0.339167\pi\)
\(282\) 1.40214e6 1.04995
\(283\) −1.11420e6 −0.826986 −0.413493 0.910507i \(-0.635691\pi\)
−0.413493 + 0.910507i \(0.635691\pi\)
\(284\) 1.65378e6 1.21669
\(285\) 0 0
\(286\) 67021.0 0.0484502
\(287\) 2.04482e6 1.46538
\(288\) 468213. 0.332631
\(289\) 2.51837e6 1.77368
\(290\) 0 0
\(291\) 669093. 0.463184
\(292\) −1.08194e6 −0.742584
\(293\) 1.03577e6 0.704843 0.352421 0.935841i \(-0.385358\pi\)
0.352421 + 0.935841i \(0.385358\pi\)
\(294\) −2.84805e6 −1.92167
\(295\) 0 0
\(296\) 1.82955e6 1.21371
\(297\) 88209.0 0.0580259
\(298\) 2.75685e6 1.79834
\(299\) 280506. 0.181453
\(300\) 0 0
\(301\) 2.42795e6 1.54462
\(302\) −2.37686e6 −1.49963
\(303\) 65595.7 0.0410458
\(304\) −2918.42 −0.00181119
\(305\) 0 0
\(306\) −1.47162e6 −0.898444
\(307\) 2.52316e6 1.52791 0.763957 0.645267i \(-0.223254\pi\)
0.763957 + 0.645267i \(0.223254\pi\)
\(308\) 1.42103e6 0.853542
\(309\) 498358. 0.296924
\(310\) 0 0
\(311\) 1.75014e6 1.02606 0.513028 0.858372i \(-0.328524\pi\)
0.513028 + 0.858372i \(0.328524\pi\)
\(312\) 98775.1 0.0574462
\(313\) 840872. 0.485142 0.242571 0.970134i \(-0.422009\pi\)
0.242571 + 0.970134i \(0.422009\pi\)
\(314\) 136322. 0.0780263
\(315\) 0 0
\(316\) 3.99445e6 2.25029
\(317\) 1.34573e6 0.752157 0.376079 0.926588i \(-0.377272\pi\)
0.376079 + 0.926588i \(0.377272\pi\)
\(318\) −2.33955e6 −1.29737
\(319\) 1.03643e6 0.570248
\(320\) 0 0
\(321\) −2.04755e6 −1.10910
\(322\) 9.62059e6 5.17085
\(323\) −2.17180e6 −1.15828
\(324\) 339954. 0.179911
\(325\) 0 0
\(326\) 1.83801e6 0.957865
\(327\) −1.04444e6 −0.540149
\(328\) −1.63654e6 −0.839928
\(329\) −3.85707e6 −1.96457
\(330\) 0 0
\(331\) −1.71783e6 −0.861808 −0.430904 0.902398i \(-0.641805\pi\)
−0.430904 + 0.902398i \(0.641805\pi\)
\(332\) 5.13215e6 2.55537
\(333\) −816942. −0.403720
\(334\) −821055. −0.402723
\(335\) 0 0
\(336\) 5439.85 0.00262869
\(337\) −1.88053e6 −0.901996 −0.450998 0.892525i \(-0.648932\pi\)
−0.450998 + 0.892525i \(0.648932\pi\)
\(338\) 3.36568e6 1.60244
\(339\) −1.91941e6 −0.907130
\(340\) 0 0
\(341\) −49586.8 −0.0230930
\(342\) 811546. 0.375187
\(343\) 4.02512e6 1.84733
\(344\) −1.94317e6 −0.885350
\(345\) 0 0
\(346\) −3.78399e6 −1.69926
\(347\) 777998. 0.346860 0.173430 0.984846i \(-0.444515\pi\)
0.173430 + 0.984846i \(0.444515\pi\)
\(348\) 3.99437e6 1.76807
\(349\) −36672.3 −0.0161166 −0.00805831 0.999968i \(-0.502565\pi\)
−0.00805831 + 0.999968i \(0.502565\pi\)
\(350\) 0 0
\(351\) −44105.6 −0.0191085
\(352\) 699429. 0.300876
\(353\) −2.53998e6 −1.08491 −0.542455 0.840085i \(-0.682505\pi\)
−0.542455 + 0.840085i \(0.682505\pi\)
\(354\) −2.49060e6 −1.05632
\(355\) 0 0
\(356\) 6.76203e6 2.82782
\(357\) 4.04818e6 1.68108
\(358\) −333813. −0.137656
\(359\) 460599. 0.188620 0.0943098 0.995543i \(-0.469936\pi\)
0.0943098 + 0.995543i \(0.469936\pi\)
\(360\) 0 0
\(361\) −1.27843e6 −0.516306
\(362\) −1.31570e6 −0.527699
\(363\) 131769. 0.0524864
\(364\) −710531. −0.281080
\(365\) 0 0
\(366\) 3.03540e6 1.18444
\(367\) −918849. −0.356106 −0.178053 0.984021i \(-0.556980\pi\)
−0.178053 + 0.984021i \(0.556980\pi\)
\(368\) −12363.9 −0.00475921
\(369\) 730757. 0.279388
\(370\) 0 0
\(371\) 6.43571e6 2.42751
\(372\) −191106. −0.0716005
\(373\) 476713. 0.177413 0.0887064 0.996058i \(-0.471727\pi\)
0.0887064 + 0.996058i \(0.471727\pi\)
\(374\) −2.19834e6 −0.812673
\(375\) 0 0
\(376\) 3.08695e6 1.12606
\(377\) −518229. −0.187788
\(378\) −1.51270e6 −0.544532
\(379\) 340866. 0.121895 0.0609474 0.998141i \(-0.480588\pi\)
0.0609474 + 0.998141i \(0.480588\pi\)
\(380\) 0 0
\(381\) −684790. −0.241682
\(382\) −1.32684e6 −0.465221
\(383\) 322302. 0.112271 0.0561353 0.998423i \(-0.482122\pi\)
0.0561353 + 0.998423i \(0.482122\pi\)
\(384\) 2.70260e6 0.935307
\(385\) 0 0
\(386\) −1.62137e6 −0.553878
\(387\) 867676. 0.294497
\(388\) 3.85207e6 1.29902
\(389\) 2.91721e6 0.977447 0.488724 0.872439i \(-0.337463\pi\)
0.488724 + 0.872439i \(0.337463\pi\)
\(390\) 0 0
\(391\) −9.20082e6 −3.04358
\(392\) −6.27025e6 −2.06096
\(393\) 1.32976e6 0.434302
\(394\) 9.27986e6 3.01163
\(395\) 0 0
\(396\) 507832. 0.162736
\(397\) 2.07740e6 0.661520 0.330760 0.943715i \(-0.392695\pi\)
0.330760 + 0.943715i \(0.392695\pi\)
\(398\) −2.15537e6 −0.682047
\(399\) −2.23243e6 −0.702014
\(400\) 0 0
\(401\) −2.82127e6 −0.876161 −0.438080 0.898936i \(-0.644341\pi\)
−0.438080 + 0.898936i \(0.644341\pi\)
\(402\) 932313. 0.287738
\(403\) 24794.1 0.00760475
\(404\) 377644. 0.115114
\(405\) 0 0
\(406\) −1.77738e7 −5.35138
\(407\) −1.22037e6 −0.365179
\(408\) −3.23990e6 −0.963566
\(409\) −2.31242e6 −0.683530 −0.341765 0.939785i \(-0.611025\pi\)
−0.341765 + 0.939785i \(0.611025\pi\)
\(410\) 0 0
\(411\) −2.08366e6 −0.608447
\(412\) 2.86912e6 0.832733
\(413\) 6.85123e6 1.97649
\(414\) 3.43811e6 0.985869
\(415\) 0 0
\(416\) −349723. −0.0990812
\(417\) −1.09532e6 −0.308461
\(418\) 1.21231e6 0.339370
\(419\) 5.50334e6 1.53141 0.765705 0.643192i \(-0.222390\pi\)
0.765705 + 0.643192i \(0.222390\pi\)
\(420\) 0 0
\(421\) −4.12010e6 −1.13293 −0.566464 0.824086i \(-0.691689\pi\)
−0.566464 + 0.824086i \(0.691689\pi\)
\(422\) 8.41366e6 2.29987
\(423\) −1.37840e6 −0.374563
\(424\) −5.15073e6 −1.39141
\(425\) 0 0
\(426\) −2.62984e6 −0.702109
\(427\) −8.34988e6 −2.21621
\(428\) −1.17880e7 −3.11051
\(429\) −65886.2 −0.0172843
\(430\) 0 0
\(431\) 721675. 0.187132 0.0935661 0.995613i \(-0.470173\pi\)
0.0935661 + 0.995613i \(0.470173\pi\)
\(432\) 1944.04 0.000501183 0
\(433\) −1.55374e6 −0.398253 −0.199126 0.979974i \(-0.563810\pi\)
−0.199126 + 0.979974i \(0.563810\pi\)
\(434\) 850368. 0.216712
\(435\) 0 0
\(436\) −6.01299e6 −1.51487
\(437\) 5.07394e6 1.27099
\(438\) 1.72050e6 0.428518
\(439\) −6.59983e6 −1.63445 −0.817224 0.576320i \(-0.804488\pi\)
−0.817224 + 0.576320i \(0.804488\pi\)
\(440\) 0 0
\(441\) 2.79983e6 0.685543
\(442\) 1.09920e6 0.267621
\(443\) −1.41708e6 −0.343071 −0.171536 0.985178i \(-0.554873\pi\)
−0.171536 + 0.985178i \(0.554873\pi\)
\(444\) −4.70326e6 −1.13225
\(445\) 0 0
\(446\) −1.17895e7 −2.80647
\(447\) −2.71017e6 −0.641545
\(448\) −1.20139e7 −2.82806
\(449\) −3.90859e6 −0.914964 −0.457482 0.889219i \(-0.651249\pi\)
−0.457482 + 0.889219i \(0.651249\pi\)
\(450\) 0 0
\(451\) 1.09163e6 0.252716
\(452\) −1.10504e7 −2.54408
\(453\) 2.33661e6 0.534984
\(454\) 4.99741e6 1.13790
\(455\) 0 0
\(456\) 1.78669e6 0.402382
\(457\) 4.85837e6 1.08818 0.544090 0.839027i \(-0.316875\pi\)
0.544090 + 0.839027i \(0.316875\pi\)
\(458\) 4.96857e6 1.10680
\(459\) 1.44670e6 0.320513
\(460\) 0 0
\(461\) 905464. 0.198435 0.0992176 0.995066i \(-0.468366\pi\)
0.0992176 + 0.995066i \(0.468366\pi\)
\(462\) −2.25971e6 −0.492548
\(463\) −3.54245e6 −0.767981 −0.383991 0.923337i \(-0.625451\pi\)
−0.383991 + 0.923337i \(0.625451\pi\)
\(464\) 22842.0 0.00492537
\(465\) 0 0
\(466\) −1.15812e6 −0.247052
\(467\) −7.14306e6 −1.51563 −0.757813 0.652472i \(-0.773732\pi\)
−0.757813 + 0.652472i \(0.773732\pi\)
\(468\) −253923. −0.0535904
\(469\) −2.56464e6 −0.538387
\(470\) 0 0
\(471\) −134013. −0.0278353
\(472\) −5.48329e6 −1.13289
\(473\) 1.29616e6 0.266382
\(474\) −6.35198e6 −1.29856
\(475\) 0 0
\(476\) 2.33060e7 4.71465
\(477\) 2.29993e6 0.462828
\(478\) −1.25371e7 −2.50973
\(479\) 4.49284e6 0.894711 0.447355 0.894356i \(-0.352366\pi\)
0.447355 + 0.894356i \(0.352366\pi\)
\(480\) 0 0
\(481\) 610201. 0.120257
\(482\) −1.18072e7 −2.31488
\(483\) −9.45768e6 −1.84466
\(484\) 758614. 0.147200
\(485\) 0 0
\(486\) −540595. −0.103820
\(487\) −2.83972e6 −0.542567 −0.271283 0.962500i \(-0.587448\pi\)
−0.271283 + 0.962500i \(0.587448\pi\)
\(488\) 6.68271e6 1.27029
\(489\) −1.80689e6 −0.341712
\(490\) 0 0
\(491\) −604654. −0.113189 −0.0565944 0.998397i \(-0.518024\pi\)
−0.0565944 + 0.998397i \(0.518024\pi\)
\(492\) 4.20708e6 0.783552
\(493\) 1.69983e7 3.14984
\(494\) −606170. −0.111758
\(495\) 0 0
\(496\) −1092.85 −0.000199460 0
\(497\) 7.23425e6 1.31372
\(498\) −8.16115e6 −1.47461
\(499\) 979877. 0.176165 0.0880826 0.996113i \(-0.471926\pi\)
0.0880826 + 0.996113i \(0.471926\pi\)
\(500\) 0 0
\(501\) 807153. 0.143668
\(502\) −2.89528e6 −0.512781
\(503\) 6.83985e6 1.20539 0.602693 0.797973i \(-0.294094\pi\)
0.602693 + 0.797973i \(0.294094\pi\)
\(504\) −3.33035e6 −0.584001
\(505\) 0 0
\(506\) 5.13595e6 0.891752
\(507\) −3.30869e6 −0.571658
\(508\) −3.94244e6 −0.677806
\(509\) 3.42217e6 0.585473 0.292737 0.956193i \(-0.405434\pi\)
0.292737 + 0.956193i \(0.405434\pi\)
\(510\) 0 0
\(511\) −4.73281e6 −0.801802
\(512\) 30894.6 0.00520844
\(513\) −797805. −0.133845
\(514\) 923739. 0.154220
\(515\) 0 0
\(516\) 4.99534e6 0.825925
\(517\) −2.05909e6 −0.338805
\(518\) 2.09282e7 3.42695
\(519\) 3.71992e6 0.606199
\(520\) 0 0
\(521\) 3.25613e6 0.525542 0.262771 0.964858i \(-0.415364\pi\)
0.262771 + 0.964858i \(0.415364\pi\)
\(522\) −6.35184e6 −1.02029
\(523\) −2.77887e6 −0.444237 −0.222118 0.975020i \(-0.571297\pi\)
−0.222118 + 0.975020i \(0.571297\pi\)
\(524\) 7.65562e6 1.21801
\(525\) 0 0
\(526\) 3.17676e6 0.500634
\(527\) −813264. −0.127557
\(528\) 2904.06 0.000453337 0
\(529\) 1.50594e7 2.33974
\(530\) 0 0
\(531\) 2.44843e6 0.376835
\(532\) −1.28524e7 −1.96882
\(533\) −545826. −0.0832217
\(534\) −1.07530e7 −1.63183
\(535\) 0 0
\(536\) 2.05257e6 0.308594
\(537\) 328161. 0.0491078
\(538\) −1.03419e7 −1.54044
\(539\) 4.18246e6 0.620097
\(540\) 0 0
\(541\) −882770. −0.129675 −0.0648373 0.997896i \(-0.520653\pi\)
−0.0648373 + 0.997896i \(0.520653\pi\)
\(542\) −6.92703e6 −1.01286
\(543\) 1.29342e6 0.188253
\(544\) 1.14712e7 1.66193
\(545\) 0 0
\(546\) 1.12989e6 0.162201
\(547\) 6.88538e6 0.983920 0.491960 0.870618i \(-0.336281\pi\)
0.491960 + 0.870618i \(0.336281\pi\)
\(548\) −1.19960e7 −1.70641
\(549\) −2.98400e6 −0.422540
\(550\) 0 0
\(551\) −9.37399e6 −1.31536
\(552\) 7.56932e6 1.05733
\(553\) 1.74732e7 2.42975
\(554\) −1.48019e7 −2.04900
\(555\) 0 0
\(556\) −6.30591e6 −0.865090
\(557\) 1.75983e6 0.240344 0.120172 0.992753i \(-0.461655\pi\)
0.120172 + 0.992753i \(0.461655\pi\)
\(558\) 303896. 0.0413180
\(559\) −648096. −0.0877222
\(560\) 0 0
\(561\) 2.16112e6 0.289915
\(562\) −1.17312e7 −1.56675
\(563\) 1.06993e7 1.42260 0.711301 0.702888i \(-0.248107\pi\)
0.711301 + 0.702888i \(0.248107\pi\)
\(564\) −7.93566e6 −1.05047
\(565\) 0 0
\(566\) 1.02006e7 1.33839
\(567\) 1.48709e6 0.194258
\(568\) −5.78983e6 −0.753000
\(569\) −1.02640e6 −0.132904 −0.0664518 0.997790i \(-0.521168\pi\)
−0.0664518 + 0.997790i \(0.521168\pi\)
\(570\) 0 0
\(571\) −1.40704e7 −1.80599 −0.902995 0.429650i \(-0.858637\pi\)
−0.902995 + 0.429650i \(0.858637\pi\)
\(572\) −379316. −0.0484743
\(573\) 1.30437e6 0.165964
\(574\) −1.87203e7 −2.37156
\(575\) 0 0
\(576\) −4.29341e6 −0.539195
\(577\) 1.54276e6 0.192912 0.0964562 0.995337i \(-0.469249\pi\)
0.0964562 + 0.995337i \(0.469249\pi\)
\(578\) −2.30557e7 −2.87051
\(579\) 1.59392e6 0.197592
\(580\) 0 0
\(581\) 2.24500e7 2.75915
\(582\) −6.12555e6 −0.749614
\(583\) 3.43570e6 0.418643
\(584\) 3.78784e6 0.459578
\(585\) 0 0
\(586\) −9.48245e6 −1.14071
\(587\) −3.85126e6 −0.461325 −0.230663 0.973034i \(-0.574089\pi\)
−0.230663 + 0.973034i \(0.574089\pi\)
\(588\) 1.61190e7 1.92263
\(589\) 448487. 0.0532675
\(590\) 0 0
\(591\) −9.12273e6 −1.07438
\(592\) −26895.8 −0.00315413
\(593\) −304507. −0.0355599 −0.0177799 0.999842i \(-0.505660\pi\)
−0.0177799 + 0.999842i \(0.505660\pi\)
\(594\) −807555. −0.0939087
\(595\) 0 0
\(596\) −1.56028e7 −1.79923
\(597\) 2.11887e6 0.243315
\(598\) −2.56804e6 −0.293662
\(599\) 5.46626e6 0.622476 0.311238 0.950332i \(-0.399256\pi\)
0.311238 + 0.950332i \(0.399256\pi\)
\(600\) 0 0
\(601\) −4.68185e6 −0.528726 −0.264363 0.964423i \(-0.585162\pi\)
−0.264363 + 0.964423i \(0.585162\pi\)
\(602\) −2.22279e7 −2.49981
\(603\) −916527. −0.102648
\(604\) 1.34522e7 1.50038
\(605\) 0 0
\(606\) −600530. −0.0664283
\(607\) −6.36108e6 −0.700744 −0.350372 0.936611i \(-0.613945\pi\)
−0.350372 + 0.936611i \(0.613945\pi\)
\(608\) −6.32597e6 −0.694014
\(609\) 1.74729e7 1.90907
\(610\) 0 0
\(611\) 1.02957e6 0.111572
\(612\) 8.32886e6 0.898891
\(613\) 8.61598e6 0.926091 0.463045 0.886335i \(-0.346757\pi\)
0.463045 + 0.886335i \(0.346757\pi\)
\(614\) −2.30996e7 −2.47277
\(615\) 0 0
\(616\) −4.97497e6 −0.528249
\(617\) 1.46873e6 0.155321 0.0776605 0.996980i \(-0.475255\pi\)
0.0776605 + 0.996980i \(0.475255\pi\)
\(618\) −4.56248e6 −0.480540
\(619\) 1.01690e7 1.06672 0.533362 0.845887i \(-0.320928\pi\)
0.533362 + 0.845887i \(0.320928\pi\)
\(620\) 0 0
\(621\) −3.37990e6 −0.351702
\(622\) −1.60225e7 −1.66056
\(623\) 2.95797e7 3.05333
\(624\) −1452.07 −0.000149288 0
\(625\) 0 0
\(626\) −7.69819e6 −0.785151
\(627\) −1.19178e6 −0.121068
\(628\) −771535. −0.0780650
\(629\) −2.00150e7 −2.01711
\(630\) 0 0
\(631\) 5.17219e6 0.517131 0.258566 0.965994i \(-0.416750\pi\)
0.258566 + 0.965994i \(0.416750\pi\)
\(632\) −1.39845e7 −1.39269
\(633\) −8.27120e6 −0.820463
\(634\) −1.23201e7 −1.21729
\(635\) 0 0
\(636\) 1.32411e7 1.29802
\(637\) −2.09128e6 −0.204204
\(638\) −9.48855e6 −0.922887
\(639\) 2.58531e6 0.250472
\(640\) 0 0
\(641\) −1.09775e7 −1.05526 −0.527629 0.849475i \(-0.676919\pi\)
−0.527629 + 0.849475i \(0.676919\pi\)
\(642\) 1.87453e7 1.79496
\(643\) −348856. −0.0332750 −0.0166375 0.999862i \(-0.505296\pi\)
−0.0166375 + 0.999862i \(0.505296\pi\)
\(644\) −5.44493e7 −5.17342
\(645\) 0 0
\(646\) 1.98829e7 1.87455
\(647\) −1.90506e7 −1.78915 −0.894577 0.446915i \(-0.852523\pi\)
−0.894577 + 0.446915i \(0.852523\pi\)
\(648\) −1.19017e6 −0.111345
\(649\) 3.65753e6 0.340860
\(650\) 0 0
\(651\) −835969. −0.0773103
\(652\) −1.04025e7 −0.958342
\(653\) 3.51919e6 0.322968 0.161484 0.986875i \(-0.448372\pi\)
0.161484 + 0.986875i \(0.448372\pi\)
\(654\) 9.56186e6 0.874174
\(655\) 0 0
\(656\) 24058.4 0.00218276
\(657\) −1.69137e6 −0.152871
\(658\) 3.53115e7 3.17945
\(659\) 1.68730e7 1.51349 0.756743 0.653712i \(-0.226789\pi\)
0.756743 + 0.653712i \(0.226789\pi\)
\(660\) 0 0
\(661\) 1.61331e6 0.143620 0.0718098 0.997418i \(-0.477123\pi\)
0.0718098 + 0.997418i \(0.477123\pi\)
\(662\) 1.57268e7 1.39474
\(663\) −1.08059e6 −0.0954719
\(664\) −1.79675e7 −1.58150
\(665\) 0 0
\(666\) 7.47912e6 0.653378
\(667\) −3.97129e7 −3.45634
\(668\) 4.64690e6 0.402923
\(669\) 1.15899e7 1.00119
\(670\) 0 0
\(671\) −4.45758e6 −0.382202
\(672\) 1.17915e7 1.00727
\(673\) −1.03328e7 −0.879387 −0.439693 0.898148i \(-0.644913\pi\)
−0.439693 + 0.898148i \(0.644913\pi\)
\(674\) 1.72162e7 1.45978
\(675\) 0 0
\(676\) −1.90486e7 −1.60323
\(677\) 1.19467e7 1.00179 0.500895 0.865508i \(-0.333004\pi\)
0.500895 + 0.865508i \(0.333004\pi\)
\(678\) 1.75723e7 1.46809
\(679\) 1.68504e7 1.40261
\(680\) 0 0
\(681\) −4.91279e6 −0.405939
\(682\) 453968. 0.0373736
\(683\) −1.23358e7 −1.01185 −0.505926 0.862577i \(-0.668849\pi\)
−0.505926 + 0.862577i \(0.668849\pi\)
\(684\) −4.59308e6 −0.375374
\(685\) 0 0
\(686\) −3.68501e7 −2.98970
\(687\) −4.88444e6 −0.394842
\(688\) 28566.1 0.00230080
\(689\) −1.71789e6 −0.137863
\(690\) 0 0
\(691\) 2.00464e7 1.59714 0.798568 0.601904i \(-0.205591\pi\)
0.798568 + 0.601904i \(0.205591\pi\)
\(692\) 2.14161e7 1.70011
\(693\) 2.22145e6 0.175713
\(694\) −7.12259e6 −0.561357
\(695\) 0 0
\(696\) −1.39842e7 −1.09424
\(697\) 1.79035e7 1.39591
\(698\) 335735. 0.0260830
\(699\) 1.13851e6 0.0881342
\(700\) 0 0
\(701\) 1.38095e7 1.06141 0.530706 0.847556i \(-0.321927\pi\)
0.530706 + 0.847556i \(0.321927\pi\)
\(702\) 403788. 0.0309250
\(703\) 1.10376e7 0.842339
\(704\) −6.41361e6 −0.487720
\(705\) 0 0
\(706\) 2.32536e7 1.75581
\(707\) 1.65196e6 0.124294
\(708\) 1.40960e7 1.05685
\(709\) −9.12564e6 −0.681785 −0.340893 0.940102i \(-0.610729\pi\)
−0.340893 + 0.940102i \(0.610729\pi\)
\(710\) 0 0
\(711\) 6.24442e6 0.463253
\(712\) −2.36737e7 −1.75011
\(713\) 1.90001e6 0.139969
\(714\) −3.70611e7 −2.72065
\(715\) 0 0
\(716\) 1.88927e6 0.137724
\(717\) 1.23248e7 0.895328
\(718\) −4.21679e6 −0.305261
\(719\) 1.20353e7 0.868232 0.434116 0.900857i \(-0.357061\pi\)
0.434116 + 0.900857i \(0.357061\pi\)
\(720\) 0 0
\(721\) 1.25506e7 0.899140
\(722\) 1.17040e7 0.835587
\(723\) 1.16073e7 0.825817
\(724\) 7.44644e6 0.527961
\(725\) 0 0
\(726\) −1.20635e6 −0.0849436
\(727\) 2.37157e7 1.66418 0.832089 0.554642i \(-0.187145\pi\)
0.832089 + 0.554642i \(0.187145\pi\)
\(728\) 2.48755e6 0.173957
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 2.12580e7 1.47140
\(732\) −1.71793e7 −1.18503
\(733\) −2.43268e7 −1.67234 −0.836172 0.548467i \(-0.815212\pi\)
−0.836172 + 0.548467i \(0.815212\pi\)
\(734\) 8.41208e6 0.576320
\(735\) 0 0
\(736\) −2.68000e7 −1.82364
\(737\) −1.36913e6 −0.0928489
\(738\) −6.69010e6 −0.452159
\(739\) 2.33006e7 1.56948 0.784739 0.619826i \(-0.212797\pi\)
0.784739 + 0.619826i \(0.212797\pi\)
\(740\) 0 0
\(741\) 595906. 0.0398687
\(742\) −5.89191e7 −3.92867
\(743\) 5.15145e6 0.342340 0.171170 0.985242i \(-0.445245\pi\)
0.171170 + 0.985242i \(0.445245\pi\)
\(744\) 669056. 0.0443129
\(745\) 0 0
\(746\) −4.36431e6 −0.287124
\(747\) 8.02296e6 0.526057
\(748\) 1.24419e7 0.813077
\(749\) −5.15653e7 −3.35856
\(750\) 0 0
\(751\) −1.59164e7 −1.02978 −0.514891 0.857256i \(-0.672168\pi\)
−0.514891 + 0.857256i \(0.672168\pi\)
\(752\) −45380.5 −0.00292634
\(753\) 2.84626e6 0.182931
\(754\) 4.74440e6 0.303915
\(755\) 0 0
\(756\) 8.56138e6 0.544803
\(757\) −4.92277e6 −0.312227 −0.156113 0.987739i \(-0.549897\pi\)
−0.156113 + 0.987739i \(0.549897\pi\)
\(758\) −3.12063e6 −0.197274
\(759\) −5.04898e6 −0.318126
\(760\) 0 0
\(761\) −2.61608e7 −1.63753 −0.818765 0.574129i \(-0.805341\pi\)
−0.818765 + 0.574129i \(0.805341\pi\)
\(762\) 6.26926e6 0.391137
\(763\) −2.63031e7 −1.63567
\(764\) 7.50945e6 0.465452
\(765\) 0 0
\(766\) −2.95068e6 −0.181698
\(767\) −1.82881e6 −0.112248
\(768\) −9.47691e6 −0.579781
\(769\) 2.56323e6 0.156305 0.0781524 0.996941i \(-0.475098\pi\)
0.0781524 + 0.996941i \(0.475098\pi\)
\(770\) 0 0
\(771\) −908098. −0.0550170
\(772\) 9.17642e6 0.554153
\(773\) 5.47140e6 0.329344 0.164672 0.986348i \(-0.447343\pi\)
0.164672 + 0.986348i \(0.447343\pi\)
\(774\) −7.94359e6 −0.476611
\(775\) 0 0
\(776\) −1.34860e7 −0.803948
\(777\) −2.05738e7 −1.22254
\(778\) −2.67071e7 −1.58189
\(779\) −9.87319e6 −0.582926
\(780\) 0 0
\(781\) 3.86200e6 0.226561
\(782\) 8.42336e7 4.92571
\(783\) 6.24429e6 0.363981
\(784\) 92177.4 0.00535592
\(785\) 0 0
\(786\) −1.21740e7 −0.702871
\(787\) 1.29522e7 0.745429 0.372714 0.927946i \(-0.378427\pi\)
0.372714 + 0.927946i \(0.378427\pi\)
\(788\) −5.25209e7 −3.01312
\(789\) −3.12297e6 −0.178597
\(790\) 0 0
\(791\) −4.83384e7 −2.74695
\(792\) −1.77791e6 −0.100715
\(793\) 2.22885e6 0.125863
\(794\) −1.90186e7 −1.07060
\(795\) 0 0
\(796\) 1.21987e7 0.682386
\(797\) 8.84349e6 0.493149 0.246574 0.969124i \(-0.420695\pi\)
0.246574 + 0.969124i \(0.420695\pi\)
\(798\) 2.04379e7 1.13614
\(799\) −3.37708e7 −1.87143
\(800\) 0 0
\(801\) 1.05709e7 0.582145
\(802\) 2.58288e7 1.41797
\(803\) −2.52661e6 −0.138277
\(804\) −5.27658e6 −0.287881
\(805\) 0 0
\(806\) −226990. −0.0123075
\(807\) 1.01668e7 0.549540
\(808\) −1.32212e6 −0.0712431
\(809\) 2.89181e7 1.55345 0.776727 0.629838i \(-0.216879\pi\)
0.776727 + 0.629838i \(0.216879\pi\)
\(810\) 0 0
\(811\) 3.10413e7 1.65725 0.828624 0.559805i \(-0.189124\pi\)
0.828624 + 0.559805i \(0.189124\pi\)
\(812\) 1.00594e8 5.35404
\(813\) 6.80974e6 0.361330
\(814\) 1.11725e7 0.591003
\(815\) 0 0
\(816\) 47629.0 0.00250407
\(817\) −1.17231e7 −0.614450
\(818\) 2.11702e7 1.10622
\(819\) −1.11075e6 −0.0578640
\(820\) 0 0
\(821\) −2.95235e7 −1.52866 −0.764329 0.644827i \(-0.776929\pi\)
−0.764329 + 0.644827i \(0.776929\pi\)
\(822\) 1.90760e7 0.984707
\(823\) 2.12030e7 1.09118 0.545592 0.838051i \(-0.316305\pi\)
0.545592 + 0.838051i \(0.316305\pi\)
\(824\) −1.00447e7 −0.515371
\(825\) 0 0
\(826\) −6.27231e7 −3.19873
\(827\) −1.92124e7 −0.976829 −0.488414 0.872612i \(-0.662424\pi\)
−0.488414 + 0.872612i \(0.662424\pi\)
\(828\) −1.94586e7 −0.986359
\(829\) −4.31098e6 −0.217866 −0.108933 0.994049i \(-0.534743\pi\)
−0.108933 + 0.994049i \(0.534743\pi\)
\(830\) 0 0
\(831\) 1.45513e7 0.730968
\(832\) 3.20689e6 0.160611
\(833\) 6.85957e7 3.42519
\(834\) 1.00277e7 0.499212
\(835\) 0 0
\(836\) −6.86127e6 −0.339538
\(837\) −298751. −0.0147399
\(838\) −5.03832e7 −2.47842
\(839\) −1.03289e7 −0.506583 −0.253291 0.967390i \(-0.581513\pi\)
−0.253291 + 0.967390i \(0.581513\pi\)
\(840\) 0 0
\(841\) 5.28576e7 2.57702
\(842\) 3.77196e7 1.83352
\(843\) 1.15325e7 0.558928
\(844\) −4.76185e7 −2.30102
\(845\) 0 0
\(846\) 1.26193e7 0.606190
\(847\) 3.31846e6 0.158938
\(848\) 75719.6 0.00361592
\(849\) −1.00278e7 −0.477461
\(850\) 0 0
\(851\) 4.67608e7 2.21339
\(852\) 1.48840e7 0.702458
\(853\) −3.79900e6 −0.178771 −0.0893855 0.995997i \(-0.528490\pi\)
−0.0893855 + 0.995997i \(0.528490\pi\)
\(854\) 7.64433e7 3.58670
\(855\) 0 0
\(856\) 4.12696e7 1.92507
\(857\) −3.34577e7 −1.55612 −0.778062 0.628188i \(-0.783797\pi\)
−0.778062 + 0.628188i \(0.783797\pi\)
\(858\) 603189. 0.0279728
\(859\) −7.44861e6 −0.344423 −0.172212 0.985060i \(-0.555091\pi\)
−0.172212 + 0.985060i \(0.555091\pi\)
\(860\) 0 0
\(861\) 1.84034e7 0.846037
\(862\) −6.60695e6 −0.302854
\(863\) −3.47712e7 −1.58925 −0.794626 0.607099i \(-0.792333\pi\)
−0.794626 + 0.607099i \(0.792333\pi\)
\(864\) 4.21392e6 0.192044
\(865\) 0 0
\(866\) 1.42245e7 0.644529
\(867\) 2.26653e7 1.02403
\(868\) −4.81280e6 −0.216819
\(869\) 9.32809e6 0.419028
\(870\) 0 0
\(871\) 684584. 0.0305760
\(872\) 2.10513e7 0.937536
\(873\) 6.02183e6 0.267420
\(874\) −4.64520e7 −2.05696
\(875\) 0 0
\(876\) −9.73745e6 −0.428731
\(877\) −3.96598e7 −1.74121 −0.870605 0.491982i \(-0.836273\pi\)
−0.870605 + 0.491982i \(0.836273\pi\)
\(878\) 6.04215e7 2.64518
\(879\) 9.32189e6 0.406941
\(880\) 0 0
\(881\) −4.15251e7 −1.80248 −0.901240 0.433320i \(-0.857342\pi\)
−0.901240 + 0.433320i \(0.857342\pi\)
\(882\) −2.56325e7 −1.10948
\(883\) −1.44451e7 −0.623473 −0.311737 0.950169i \(-0.600911\pi\)
−0.311737 + 0.950169i \(0.600911\pi\)
\(884\) −6.22109e6 −0.267754
\(885\) 0 0
\(886\) 1.29734e7 0.555224
\(887\) 1.93342e7 0.825122 0.412561 0.910930i \(-0.364634\pi\)
0.412561 + 0.910930i \(0.364634\pi\)
\(888\) 1.64660e7 0.700737
\(889\) −1.72457e7 −0.731858
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) 6.67249e7 2.80786
\(893\) 1.86234e7 0.781504
\(894\) 2.48116e7 1.03827
\(895\) 0 0
\(896\) 6.80622e7 2.83228
\(897\) 2.52455e6 0.104762
\(898\) 3.57832e7 1.48077
\(899\) −3.51024e6 −0.144856
\(900\) 0 0
\(901\) 5.63483e7 2.31243
\(902\) −9.99385e6 −0.408993
\(903\) 2.18515e7 0.891789
\(904\) 3.86870e7 1.57450
\(905\) 0 0
\(906\) −2.13917e7 −0.865815
\(907\) −1.72690e7 −0.697025 −0.348513 0.937304i \(-0.613313\pi\)
−0.348513 + 0.937304i \(0.613313\pi\)
\(908\) −2.82837e7 −1.13847
\(909\) 590361. 0.0236978
\(910\) 0 0
\(911\) 4.42672e6 0.176720 0.0883601 0.996089i \(-0.471837\pi\)
0.0883601 + 0.996089i \(0.471837\pi\)
\(912\) −26265.7 −0.00104569
\(913\) 1.19849e7 0.475837
\(914\) −4.44785e7 −1.76110
\(915\) 0 0
\(916\) −2.81205e7 −1.10735
\(917\) 3.34886e7 1.31514
\(918\) −1.32445e7 −0.518717
\(919\) 1.62676e7 0.635382 0.317691 0.948194i \(-0.397093\pi\)
0.317691 + 0.948194i \(0.397093\pi\)
\(920\) 0 0
\(921\) 2.27085e7 0.882142
\(922\) −8.28954e6 −0.321146
\(923\) −1.93105e6 −0.0746087
\(924\) 1.27892e7 0.492793
\(925\) 0 0
\(926\) 3.24312e7 1.24290
\(927\) 4.48522e6 0.171429
\(928\) 4.95124e7 1.88731
\(929\) 2.99303e7 1.13782 0.568908 0.822401i \(-0.307366\pi\)
0.568908 + 0.822401i \(0.307366\pi\)
\(930\) 0 0
\(931\) −3.78282e7 −1.43035
\(932\) 6.55458e6 0.247175
\(933\) 1.57512e7 0.592394
\(934\) 6.53948e7 2.45288
\(935\) 0 0
\(936\) 888976. 0.0331666
\(937\) −1.50548e7 −0.560179 −0.280089 0.959974i \(-0.590364\pi\)
−0.280089 + 0.959974i \(0.590364\pi\)
\(938\) 2.34793e7 0.871322
\(939\) 7.56784e6 0.280097
\(940\) 0 0
\(941\) −1.96722e7 −0.724233 −0.362117 0.932133i \(-0.617946\pi\)
−0.362117 + 0.932133i \(0.617946\pi\)
\(942\) 1.22689e6 0.0450485
\(943\) −4.18277e7 −1.53174
\(944\) 80608.4 0.00294409
\(945\) 0 0
\(946\) −1.18664e7 −0.431111
\(947\) −1.92584e7 −0.697824 −0.348912 0.937155i \(-0.613449\pi\)
−0.348912 + 0.937155i \(0.613449\pi\)
\(948\) 3.59501e7 1.29921
\(949\) 1.26334e6 0.0455359
\(950\) 0 0
\(951\) 1.21115e7 0.434258
\(952\) −8.15935e7 −2.91785
\(953\) −1.49922e7 −0.534729 −0.267364 0.963596i \(-0.586153\pi\)
−0.267364 + 0.963596i \(0.586153\pi\)
\(954\) −2.10559e7 −0.749037
\(955\) 0 0
\(956\) 7.09557e7 2.51098
\(957\) 9.32789e6 0.329233
\(958\) −4.11321e7 −1.44799
\(959\) −5.24749e7 −1.84249
\(960\) 0 0
\(961\) −2.84612e7 −0.994134
\(962\) −5.58640e6 −0.194623
\(963\) −1.84279e7 −0.640340
\(964\) 6.68247e7 2.31603
\(965\) 0 0
\(966\) 8.65853e7 2.98539
\(967\) 2.53960e7 0.873370 0.436685 0.899614i \(-0.356152\pi\)
0.436685 + 0.899614i \(0.356152\pi\)
\(968\) −2.65589e6 −0.0911006
\(969\) −1.95462e7 −0.668733
\(970\) 0 0
\(971\) 3.80328e7 1.29453 0.647263 0.762267i \(-0.275914\pi\)
0.647263 + 0.762267i \(0.275914\pi\)
\(972\) 3.05958e6 0.103872
\(973\) −2.75845e7 −0.934077
\(974\) 2.59977e7 0.878086
\(975\) 0 0
\(976\) −98240.8 −0.00330117
\(977\) −7.60558e6 −0.254915 −0.127458 0.991844i \(-0.540682\pi\)
−0.127458 + 0.991844i \(0.540682\pi\)
\(978\) 1.65421e7 0.553024
\(979\) 1.57911e7 0.526570
\(980\) 0 0
\(981\) −9.39995e6 −0.311855
\(982\) 5.53562e6 0.183184
\(983\) −3.96257e7 −1.30796 −0.653979 0.756513i \(-0.726901\pi\)
−0.653979 + 0.756513i \(0.726901\pi\)
\(984\) −1.47289e7 −0.484933
\(985\) 0 0
\(986\) −1.55620e8 −5.09768
\(987\) −3.47136e7 −1.13424
\(988\) 3.43072e6 0.111813
\(989\) −4.96648e7 −1.61457
\(990\) 0 0
\(991\) 5.43979e7 1.75954 0.879768 0.475403i \(-0.157698\pi\)
0.879768 + 0.475403i \(0.157698\pi\)
\(992\) −2.36886e6 −0.0764294
\(993\) −1.54605e7 −0.497565
\(994\) −6.62297e7 −2.12612
\(995\) 0 0
\(996\) 4.61894e7 1.47535
\(997\) −4.80608e7 −1.53127 −0.765637 0.643273i \(-0.777576\pi\)
−0.765637 + 0.643273i \(0.777576\pi\)
\(998\) −8.97079e6 −0.285105
\(999\) −7.35248e6 −0.233088
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.t.1.2 10
5.4 even 2 825.6.a.u.1.9 yes 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.6.a.t.1.2 10 1.1 even 1 trivial
825.6.a.u.1.9 yes 10 5.4 even 2