Properties

Label 825.6.a.v.1.1
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $13$
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: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \( x^{13} - 306 x^{11} - 206 x^{10} + 34574 x^{9} + 39928 x^{8} - 1788312 x^{7} - 2591628 x^{6} + 42852537 x^{5} + 63733360 x^{4} - 448113518 x^{3} + \cdots + 522579400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2}\cdot 5^{7} \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-9.75306\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-10.7531 q^{2} -9.00000 q^{3} +83.6283 q^{4} +96.7775 q^{6} -41.3813 q^{7} -555.162 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-10.7531 q^{2} -9.00000 q^{3} +83.6283 q^{4} +96.7775 q^{6} -41.3813 q^{7} -555.162 q^{8} +81.0000 q^{9} +121.000 q^{11} -752.654 q^{12} +13.6514 q^{13} +444.975 q^{14} +3293.58 q^{16} -1143.92 q^{17} -870.998 q^{18} -1425.16 q^{19} +372.432 q^{21} -1301.12 q^{22} +2533.88 q^{23} +4996.46 q^{24} -146.795 q^{26} -729.000 q^{27} -3460.65 q^{28} +8594.21 q^{29} +434.427 q^{31} -17650.9 q^{32} -1089.00 q^{33} +12300.6 q^{34} +6773.89 q^{36} -14403.2 q^{37} +15324.8 q^{38} -122.863 q^{39} +12386.5 q^{41} -4004.78 q^{42} +2699.39 q^{43} +10119.0 q^{44} -27246.9 q^{46} +1691.93 q^{47} -29642.3 q^{48} -15094.6 q^{49} +10295.3 q^{51} +1141.65 q^{52} +9465.19 q^{53} +7838.98 q^{54} +22973.3 q^{56} +12826.4 q^{57} -92414.0 q^{58} -41470.4 q^{59} -17867.3 q^{61} -4671.41 q^{62} -3351.88 q^{63} +84406.7 q^{64} +11710.1 q^{66} -51700.6 q^{67} -95664.1 q^{68} -22804.9 q^{69} -16645.8 q^{71} -44968.1 q^{72} +49707.4 q^{73} +154878. q^{74} -119184. q^{76} -5007.14 q^{77} +1321.15 q^{78} +71497.6 q^{79} +6561.00 q^{81} -133193. q^{82} +33539.3 q^{83} +31145.8 q^{84} -29026.7 q^{86} -77347.9 q^{87} -67174.6 q^{88} -76978.3 q^{89} -564.914 q^{91} +211904. q^{92} -3909.84 q^{93} -18193.4 q^{94} +158858. q^{96} -17713.7 q^{97} +162313. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 13 q^{2} - 117 q^{3} + 209 q^{4} + 117 q^{6} - 304 q^{7} - 399 q^{8} + 1053 q^{9} + 1573 q^{11} - 1881 q^{12} - 986 q^{13} - 610 q^{14} + 3501 q^{16} - 1476 q^{17} - 1053 q^{18} + 270 q^{19} + 2736 q^{21} - 1573 q^{22} - 9084 q^{23} + 3591 q^{24} + 2652 q^{26} - 9477 q^{27} - 10920 q^{28} + 11952 q^{29} + 19096 q^{31} - 11661 q^{32} - 14157 q^{33} - 1302 q^{34} + 16929 q^{36} - 39964 q^{37} - 1574 q^{38} + 8874 q^{39} + 35184 q^{41} + 5490 q^{42} + 96 q^{43} + 25289 q^{44} - 4120 q^{46} - 34984 q^{47} - 31509 q^{48} + 14557 q^{49} + 13284 q^{51} - 39002 q^{52} - 22984 q^{53} + 9477 q^{54} + 59802 q^{56} - 2430 q^{57} - 18896 q^{58} - 9192 q^{59} + 5438 q^{61} - 272 q^{62} - 24624 q^{63} + 106557 q^{64} + 14157 q^{66} - 71508 q^{67} - 127948 q^{68} + 81756 q^{69} + 101700 q^{71} - 32319 q^{72} - 77390 q^{73} + 13676 q^{74} + 139966 q^{76} - 36784 q^{77} - 23868 q^{78} + 93954 q^{79} + 85293 q^{81} - 53284 q^{82} - 185918 q^{83} + 98280 q^{84} + 370930 q^{86} - 107568 q^{87} - 48279 q^{88} - 18418 q^{89} + 174536 q^{91} - 274264 q^{92} - 171864 q^{93} + 64520 q^{94} + 104949 q^{96} - 94312 q^{97} - 145677 q^{98} + 127413 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −10.7531 −1.90089 −0.950445 0.310892i \(-0.899372\pi\)
−0.950445 + 0.310892i \(0.899372\pi\)
\(3\) −9.00000 −0.577350
\(4\) 83.6283 2.61338
\(5\) 0 0
\(6\) 96.7775 1.09748
\(7\) −41.3813 −0.319197 −0.159599 0.987182i \(-0.551020\pi\)
−0.159599 + 0.987182i \(0.551020\pi\)
\(8\) −555.162 −3.06687
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) −752.654 −1.50884
\(13\) 13.6514 0.0224037 0.0112019 0.999937i \(-0.496434\pi\)
0.0112019 + 0.999937i \(0.496434\pi\)
\(14\) 444.975 0.606759
\(15\) 0 0
\(16\) 3293.58 3.21639
\(17\) −1143.92 −0.960005 −0.480002 0.877267i \(-0.659364\pi\)
−0.480002 + 0.877267i \(0.659364\pi\)
\(18\) −870.998 −0.633630
\(19\) −1425.16 −0.905689 −0.452844 0.891590i \(-0.649591\pi\)
−0.452844 + 0.891590i \(0.649591\pi\)
\(20\) 0 0
\(21\) 372.432 0.184289
\(22\) −1301.12 −0.573140
\(23\) 2533.88 0.998770 0.499385 0.866380i \(-0.333559\pi\)
0.499385 + 0.866380i \(0.333559\pi\)
\(24\) 4996.46 1.77066
\(25\) 0 0
\(26\) −146.795 −0.0425870
\(27\) −729.000 −0.192450
\(28\) −3460.65 −0.834185
\(29\) 8594.21 1.89763 0.948814 0.315837i \(-0.102285\pi\)
0.948814 + 0.315837i \(0.102285\pi\)
\(30\) 0 0
\(31\) 434.427 0.0811918 0.0405959 0.999176i \(-0.487074\pi\)
0.0405959 + 0.999176i \(0.487074\pi\)
\(32\) −17650.9 −3.04714
\(33\) −1089.00 −0.174078
\(34\) 12300.6 1.82486
\(35\) 0 0
\(36\) 6773.89 0.871128
\(37\) −14403.2 −1.72963 −0.864815 0.502090i \(-0.832564\pi\)
−0.864815 + 0.502090i \(0.832564\pi\)
\(38\) 15324.8 1.72162
\(39\) −122.863 −0.0129348
\(40\) 0 0
\(41\) 12386.5 1.15077 0.575387 0.817881i \(-0.304851\pi\)
0.575387 + 0.817881i \(0.304851\pi\)
\(42\) −4004.78 −0.350312
\(43\) 2699.39 0.222635 0.111318 0.993785i \(-0.464493\pi\)
0.111318 + 0.993785i \(0.464493\pi\)
\(44\) 10119.0 0.787965
\(45\) 0 0
\(46\) −27246.9 −1.89855
\(47\) 1691.93 0.111722 0.0558608 0.998439i \(-0.482210\pi\)
0.0558608 + 0.998439i \(0.482210\pi\)
\(48\) −29642.3 −1.85698
\(49\) −15094.6 −0.898113
\(50\) 0 0
\(51\) 10295.3 0.554259
\(52\) 1141.65 0.0585495
\(53\) 9465.19 0.462850 0.231425 0.972853i \(-0.425661\pi\)
0.231425 + 0.972853i \(0.425661\pi\)
\(54\) 7838.98 0.365826
\(55\) 0 0
\(56\) 22973.3 0.978935
\(57\) 12826.4 0.522900
\(58\) −92414.0 −3.60718
\(59\) −41470.4 −1.55099 −0.775493 0.631356i \(-0.782499\pi\)
−0.775493 + 0.631356i \(0.782499\pi\)
\(60\) 0 0
\(61\) −17867.3 −0.614802 −0.307401 0.951580i \(-0.599459\pi\)
−0.307401 + 0.951580i \(0.599459\pi\)
\(62\) −4671.41 −0.154337
\(63\) −3351.88 −0.106399
\(64\) 84406.7 2.57589
\(65\) 0 0
\(66\) 11710.1 0.330903
\(67\) −51700.6 −1.40705 −0.703523 0.710672i \(-0.748391\pi\)
−0.703523 + 0.710672i \(0.748391\pi\)
\(68\) −95664.1 −2.50886
\(69\) −22804.9 −0.576640
\(70\) 0 0
\(71\) −16645.8 −0.391885 −0.195943 0.980615i \(-0.562777\pi\)
−0.195943 + 0.980615i \(0.562777\pi\)
\(72\) −44968.1 −1.02229
\(73\) 49707.4 1.09173 0.545863 0.837874i \(-0.316202\pi\)
0.545863 + 0.837874i \(0.316202\pi\)
\(74\) 154878. 3.28784
\(75\) 0 0
\(76\) −119184. −2.36691
\(77\) −5007.14 −0.0962416
\(78\) 1321.15 0.0245876
\(79\) 71497.6 1.28891 0.644456 0.764641i \(-0.277084\pi\)
0.644456 + 0.764641i \(0.277084\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −133193. −2.18750
\(83\) 33539.3 0.534390 0.267195 0.963642i \(-0.413903\pi\)
0.267195 + 0.963642i \(0.413903\pi\)
\(84\) 31145.8 0.481617
\(85\) 0 0
\(86\) −29026.7 −0.423205
\(87\) −77347.9 −1.09560
\(88\) −67174.6 −0.924695
\(89\) −76978.3 −1.03013 −0.515066 0.857150i \(-0.672233\pi\)
−0.515066 + 0.857150i \(0.672233\pi\)
\(90\) 0 0
\(91\) −564.914 −0.00715120
\(92\) 211904. 2.61017
\(93\) −3909.84 −0.0468761
\(94\) −18193.4 −0.212371
\(95\) 0 0
\(96\) 158858. 1.75927
\(97\) −17713.7 −0.191152 −0.0955761 0.995422i \(-0.530469\pi\)
−0.0955761 + 0.995422i \(0.530469\pi\)
\(98\) 162313. 1.70721
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) −2788.38 −0.0271987 −0.0135994 0.999908i \(-0.504329\pi\)
−0.0135994 + 0.999908i \(0.504329\pi\)
\(102\) −110706. −1.05359
\(103\) −3106.69 −0.0288539 −0.0144269 0.999896i \(-0.504592\pi\)
−0.0144269 + 0.999896i \(0.504592\pi\)
\(104\) −7578.76 −0.0687092
\(105\) 0 0
\(106\) −101780. −0.879826
\(107\) 21599.2 0.182381 0.0911904 0.995833i \(-0.470933\pi\)
0.0911904 + 0.995833i \(0.470933\pi\)
\(108\) −60965.0 −0.502946
\(109\) 190632. 1.53684 0.768421 0.639945i \(-0.221043\pi\)
0.768421 + 0.639945i \(0.221043\pi\)
\(110\) 0 0
\(111\) 129628. 0.998603
\(112\) −136293. −1.02666
\(113\) 239503. 1.76447 0.882236 0.470808i \(-0.156038\pi\)
0.882236 + 0.470808i \(0.156038\pi\)
\(114\) −137923. −0.993975
\(115\) 0 0
\(116\) 718719. 4.95923
\(117\) 1105.77 0.00746791
\(118\) 445934. 2.94826
\(119\) 47336.9 0.306431
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 192129. 1.16867
\(123\) −111479. −0.664400
\(124\) 36330.3 0.212185
\(125\) 0 0
\(126\) 36043.0 0.202253
\(127\) 112217. 0.617373 0.308687 0.951164i \(-0.400111\pi\)
0.308687 + 0.951164i \(0.400111\pi\)
\(128\) −342801. −1.84934
\(129\) −24294.5 −0.128539
\(130\) 0 0
\(131\) 334801. 1.70455 0.852273 0.523097i \(-0.175224\pi\)
0.852273 + 0.523097i \(0.175224\pi\)
\(132\) −91071.2 −0.454932
\(133\) 58974.9 0.289093
\(134\) 555939. 2.67464
\(135\) 0 0
\(136\) 635061. 2.94420
\(137\) −913.914 −0.00416010 −0.00208005 0.999998i \(-0.500662\pi\)
−0.00208005 + 0.999998i \(0.500662\pi\)
\(138\) 245222. 1.09613
\(139\) 270355. 1.18685 0.593427 0.804888i \(-0.297774\pi\)
0.593427 + 0.804888i \(0.297774\pi\)
\(140\) 0 0
\(141\) −15227.4 −0.0645025
\(142\) 178993. 0.744931
\(143\) 1651.82 0.00675498
\(144\) 266780. 1.07213
\(145\) 0 0
\(146\) −534507. −2.07525
\(147\) 135851. 0.518526
\(148\) −1.20451e6 −4.52019
\(149\) −533592. −1.96899 −0.984496 0.175407i \(-0.943876\pi\)
−0.984496 + 0.175407i \(0.943876\pi\)
\(150\) 0 0
\(151\) −109266. −0.389980 −0.194990 0.980805i \(-0.562467\pi\)
−0.194990 + 0.980805i \(0.562467\pi\)
\(152\) 791193. 2.77763
\(153\) −92657.5 −0.320002
\(154\) 53842.0 0.182945
\(155\) 0 0
\(156\) −10274.8 −0.0338036
\(157\) 362759. 1.17454 0.587272 0.809390i \(-0.300202\pi\)
0.587272 + 0.809390i \(0.300202\pi\)
\(158\) −768818. −2.45008
\(159\) −85186.7 −0.267226
\(160\) 0 0
\(161\) −104855. −0.318805
\(162\) −70550.8 −0.211210
\(163\) −339033. −0.999478 −0.499739 0.866176i \(-0.666571\pi\)
−0.499739 + 0.866176i \(0.666571\pi\)
\(164\) 1.03586e6 3.00742
\(165\) 0 0
\(166\) −360650. −1.01582
\(167\) −459806. −1.27580 −0.637901 0.770118i \(-0.720197\pi\)
−0.637901 + 0.770118i \(0.720197\pi\)
\(168\) −206760. −0.565188
\(169\) −371107. −0.999498
\(170\) 0 0
\(171\) −115438. −0.301896
\(172\) 225745. 0.581831
\(173\) 527699. 1.34051 0.670256 0.742130i \(-0.266184\pi\)
0.670256 + 0.742130i \(0.266184\pi\)
\(174\) 831726. 2.08261
\(175\) 0 0
\(176\) 398524. 0.969778
\(177\) 373234. 0.895463
\(178\) 827752. 1.95817
\(179\) 789810. 1.84243 0.921213 0.389059i \(-0.127200\pi\)
0.921213 + 0.389059i \(0.127200\pi\)
\(180\) 0 0
\(181\) −375428. −0.851786 −0.425893 0.904773i \(-0.640040\pi\)
−0.425893 + 0.904773i \(0.640040\pi\)
\(182\) 6074.56 0.0135937
\(183\) 160806. 0.354956
\(184\) −1.40671e6 −3.06309
\(185\) 0 0
\(186\) 42042.7 0.0891063
\(187\) −138414. −0.289452
\(188\) 141493. 0.291972
\(189\) 30167.0 0.0614295
\(190\) 0 0
\(191\) 339151. 0.672681 0.336341 0.941740i \(-0.390811\pi\)
0.336341 + 0.941740i \(0.390811\pi\)
\(192\) −759660. −1.48719
\(193\) −359337. −0.694398 −0.347199 0.937792i \(-0.612867\pi\)
−0.347199 + 0.937792i \(0.612867\pi\)
\(194\) 190476. 0.363359
\(195\) 0 0
\(196\) −1.26233e6 −2.34711
\(197\) 604048. 1.10893 0.554467 0.832206i \(-0.312922\pi\)
0.554467 + 0.832206i \(0.312922\pi\)
\(198\) −105391. −0.191047
\(199\) −1.03446e6 −1.85174 −0.925868 0.377846i \(-0.876665\pi\)
−0.925868 + 0.377846i \(0.876665\pi\)
\(200\) 0 0
\(201\) 465305. 0.812358
\(202\) 29983.6 0.0517018
\(203\) −355639. −0.605717
\(204\) 860976. 1.44849
\(205\) 0 0
\(206\) 33406.4 0.0548481
\(207\) 205244. 0.332923
\(208\) 44962.2 0.0720591
\(209\) −172444. −0.273075
\(210\) 0 0
\(211\) −758371. −1.17267 −0.586335 0.810069i \(-0.699430\pi\)
−0.586335 + 0.810069i \(0.699430\pi\)
\(212\) 791558. 1.20960
\(213\) 149812. 0.226255
\(214\) −232258. −0.346686
\(215\) 0 0
\(216\) 404713. 0.590218
\(217\) −17977.1 −0.0259162
\(218\) −2.04988e6 −2.92137
\(219\) −447367. −0.630309
\(220\) 0 0
\(221\) −15616.2 −0.0215077
\(222\) −1.39390e6 −1.89823
\(223\) 1.08373e6 1.45935 0.729677 0.683792i \(-0.239671\pi\)
0.729677 + 0.683792i \(0.239671\pi\)
\(224\) 730418. 0.972638
\(225\) 0 0
\(226\) −2.57539e6 −3.35407
\(227\) 635159. 0.818122 0.409061 0.912507i \(-0.365856\pi\)
0.409061 + 0.912507i \(0.365856\pi\)
\(228\) 1.07265e6 1.36654
\(229\) 622829. 0.784838 0.392419 0.919787i \(-0.371638\pi\)
0.392419 + 0.919787i \(0.371638\pi\)
\(230\) 0 0
\(231\) 45064.2 0.0555651
\(232\) −4.77118e6 −5.81977
\(233\) −1.31801e6 −1.59048 −0.795239 0.606296i \(-0.792655\pi\)
−0.795239 + 0.606296i \(0.792655\pi\)
\(234\) −11890.4 −0.0141957
\(235\) 0 0
\(236\) −3.46810e6 −4.05332
\(237\) −643478. −0.744154
\(238\) −509016. −0.582491
\(239\) −240029. −0.271812 −0.135906 0.990722i \(-0.543395\pi\)
−0.135906 + 0.990722i \(0.543395\pi\)
\(240\) 0 0
\(241\) 659027. 0.730904 0.365452 0.930830i \(-0.380914\pi\)
0.365452 + 0.930830i \(0.380914\pi\)
\(242\) −157436. −0.172808
\(243\) −59049.0 −0.0641500
\(244\) −1.49422e6 −1.60671
\(245\) 0 0
\(246\) 1.19874e6 1.26295
\(247\) −19455.5 −0.0202908
\(248\) −241177. −0.249004
\(249\) −301854. −0.308530
\(250\) 0 0
\(251\) 1.72716e6 1.73040 0.865202 0.501424i \(-0.167190\pi\)
0.865202 + 0.501424i \(0.167190\pi\)
\(252\) −280312. −0.278062
\(253\) 306599. 0.301141
\(254\) −1.20667e6 −1.17356
\(255\) 0 0
\(256\) 985142. 0.939505
\(257\) 889184. 0.839767 0.419883 0.907578i \(-0.362071\pi\)
0.419883 + 0.907578i \(0.362071\pi\)
\(258\) 261240. 0.244338
\(259\) 596021. 0.552093
\(260\) 0 0
\(261\) 696131. 0.632542
\(262\) −3.60014e6 −3.24015
\(263\) 352666. 0.314394 0.157197 0.987567i \(-0.449754\pi\)
0.157197 + 0.987567i \(0.449754\pi\)
\(264\) 604571. 0.533873
\(265\) 0 0
\(266\) −634160. −0.549535
\(267\) 692804. 0.594747
\(268\) −4.32363e6 −3.67715
\(269\) −506736. −0.426973 −0.213487 0.976946i \(-0.568482\pi\)
−0.213487 + 0.976946i \(0.568482\pi\)
\(270\) 0 0
\(271\) −1.33528e6 −1.10446 −0.552230 0.833691i \(-0.686223\pi\)
−0.552230 + 0.833691i \(0.686223\pi\)
\(272\) −3.76760e6 −3.08775
\(273\) 5084.23 0.00412875
\(274\) 9827.37 0.00790790
\(275\) 0 0
\(276\) −1.90713e6 −1.50698
\(277\) 1.72199e6 1.34844 0.674218 0.738532i \(-0.264481\pi\)
0.674218 + 0.738532i \(0.264481\pi\)
\(278\) −2.90714e6 −2.25608
\(279\) 35188.5 0.0270639
\(280\) 0 0
\(281\) −1.20590e6 −0.911061 −0.455530 0.890220i \(-0.650550\pi\)
−0.455530 + 0.890220i \(0.650550\pi\)
\(282\) 163741. 0.122612
\(283\) −920183. −0.682980 −0.341490 0.939885i \(-0.610932\pi\)
−0.341490 + 0.939885i \(0.610932\pi\)
\(284\) −1.39206e6 −1.02415
\(285\) 0 0
\(286\) −17762.2 −0.0128405
\(287\) −512571. −0.367324
\(288\) −1.42972e6 −1.01571
\(289\) −111304. −0.0783911
\(290\) 0 0
\(291\) 159423. 0.110362
\(292\) 4.15694e6 2.85310
\(293\) 125447. 0.0853674 0.0426837 0.999089i \(-0.486409\pi\)
0.0426837 + 0.999089i \(0.486409\pi\)
\(294\) −1.46082e6 −0.985661
\(295\) 0 0
\(296\) 7.99608e6 5.30454
\(297\) −88209.0 −0.0580259
\(298\) 5.73775e6 3.74284
\(299\) 34591.1 0.0223762
\(300\) 0 0
\(301\) −111704. −0.0710646
\(302\) 1.17494e6 0.741310
\(303\) 25095.4 0.0157032
\(304\) −4.69388e6 −2.91305
\(305\) 0 0
\(306\) 996352. 0.608288
\(307\) −922678. −0.558733 −0.279366 0.960185i \(-0.590124\pi\)
−0.279366 + 0.960185i \(0.590124\pi\)
\(308\) −418738. −0.251516
\(309\) 27960.2 0.0166588
\(310\) 0 0
\(311\) −1.60136e6 −0.938831 −0.469416 0.882977i \(-0.655535\pi\)
−0.469416 + 0.882977i \(0.655535\pi\)
\(312\) 68208.9 0.0396693
\(313\) −177127. −0.102193 −0.0510967 0.998694i \(-0.516272\pi\)
−0.0510967 + 0.998694i \(0.516272\pi\)
\(314\) −3.90077e6 −2.23268
\(315\) 0 0
\(316\) 5.97922e6 3.36842
\(317\) −578254. −0.323199 −0.161600 0.986856i \(-0.551665\pi\)
−0.161600 + 0.986856i \(0.551665\pi\)
\(318\) 916018. 0.507968
\(319\) 1.03990e6 0.572156
\(320\) 0 0
\(321\) −194393. −0.105298
\(322\) 1.12751e6 0.606013
\(323\) 1.63027e6 0.869466
\(324\) 548685. 0.290376
\(325\) 0 0
\(326\) 3.64565e6 1.89990
\(327\) −1.71569e6 −0.887296
\(328\) −6.87653e6 −3.52927
\(329\) −70014.2 −0.0356612
\(330\) 0 0
\(331\) −3.23768e6 −1.62429 −0.812145 0.583456i \(-0.801700\pi\)
−0.812145 + 0.583456i \(0.801700\pi\)
\(332\) 2.80483e6 1.39657
\(333\) −1.16666e6 −0.576543
\(334\) 4.94432e6 2.42516
\(335\) 0 0
\(336\) 1.22663e6 0.592744
\(337\) −1.82330e6 −0.874546 −0.437273 0.899329i \(-0.644056\pi\)
−0.437273 + 0.899329i \(0.644056\pi\)
\(338\) 3.99053e6 1.89994
\(339\) −2.15553e6 −1.01872
\(340\) 0 0
\(341\) 52565.6 0.0244802
\(342\) 1.24131e6 0.573872
\(343\) 1.32013e6 0.605872
\(344\) −1.49860e6 −0.682792
\(345\) 0 0
\(346\) −5.67438e6 −2.54817
\(347\) −564150. −0.251519 −0.125759 0.992061i \(-0.540137\pi\)
−0.125759 + 0.992061i \(0.540137\pi\)
\(348\) −6.46847e6 −2.86321
\(349\) 2.72653e6 1.19825 0.599124 0.800657i \(-0.295516\pi\)
0.599124 + 0.800657i \(0.295516\pi\)
\(350\) 0 0
\(351\) −9951.90 −0.00431160
\(352\) −2.13576e6 −0.918747
\(353\) −694975. −0.296847 −0.148423 0.988924i \(-0.547420\pi\)
−0.148423 + 0.988924i \(0.547420\pi\)
\(354\) −4.01340e6 −1.70218
\(355\) 0 0
\(356\) −6.43756e6 −2.69213
\(357\) −426032. −0.176918
\(358\) −8.49287e6 −3.50225
\(359\) −306112. −0.125356 −0.0626779 0.998034i \(-0.519964\pi\)
−0.0626779 + 0.998034i \(0.519964\pi\)
\(360\) 0 0
\(361\) −445023. −0.179728
\(362\) 4.03700e6 1.61915
\(363\) −131769. −0.0524864
\(364\) −47242.8 −0.0186888
\(365\) 0 0
\(366\) −1.72916e6 −0.674733
\(367\) −1.51571e6 −0.587423 −0.293712 0.955894i \(-0.594891\pi\)
−0.293712 + 0.955894i \(0.594891\pi\)
\(368\) 8.34553e6 3.21244
\(369\) 1.00331e6 0.383592
\(370\) 0 0
\(371\) −391682. −0.147740
\(372\) −326973. −0.122505
\(373\) −1.05811e6 −0.393783 −0.196892 0.980425i \(-0.563085\pi\)
−0.196892 + 0.980425i \(0.563085\pi\)
\(374\) 1.48838e6 0.550217
\(375\) 0 0
\(376\) −939294. −0.342635
\(377\) 117323. 0.0425139
\(378\) −324387. −0.116771
\(379\) 812383. 0.290511 0.145256 0.989394i \(-0.453600\pi\)
0.145256 + 0.989394i \(0.453600\pi\)
\(380\) 0 0
\(381\) −1.00995e6 −0.356441
\(382\) −3.64691e6 −1.27869
\(383\) 1.22178e6 0.425593 0.212797 0.977096i \(-0.431743\pi\)
0.212797 + 0.977096i \(0.431743\pi\)
\(384\) 3.08521e6 1.06772
\(385\) 0 0
\(386\) 3.86397e6 1.31997
\(387\) 218650. 0.0742118
\(388\) −1.48136e6 −0.499554
\(389\) −4.08811e6 −1.36977 −0.684887 0.728649i \(-0.740148\pi\)
−0.684887 + 0.728649i \(0.740148\pi\)
\(390\) 0 0
\(391\) −2.89855e6 −0.958824
\(392\) 8.37994e6 2.75439
\(393\) −3.01321e6 −0.984120
\(394\) −6.49536e6 −2.10796
\(395\) 0 0
\(396\) 819641. 0.262655
\(397\) −15256.4 −0.00485822 −0.00242911 0.999997i \(-0.500773\pi\)
−0.00242911 + 0.999997i \(0.500773\pi\)
\(398\) 1.11236e7 3.51995
\(399\) −530774. −0.166908
\(400\) 0 0
\(401\) −4.75411e6 −1.47642 −0.738208 0.674574i \(-0.764328\pi\)
−0.738208 + 0.674574i \(0.764328\pi\)
\(402\) −5.00345e6 −1.54420
\(403\) 5930.55 0.00181900
\(404\) −233188. −0.0710807
\(405\) 0 0
\(406\) 3.82421e6 1.15140
\(407\) −1.74278e6 −0.521503
\(408\) −5.71555e6 −1.69984
\(409\) −6.12814e6 −1.81143 −0.905713 0.423891i \(-0.860664\pi\)
−0.905713 + 0.423891i \(0.860664\pi\)
\(410\) 0 0
\(411\) 8225.23 0.00240184
\(412\) −259807. −0.0754063
\(413\) 1.71610e6 0.495071
\(414\) −2.20700e6 −0.632851
\(415\) 0 0
\(416\) −240961. −0.0682673
\(417\) −2.43320e6 −0.685231
\(418\) 1.85430e6 0.519086
\(419\) −3.61815e6 −1.00682 −0.503410 0.864048i \(-0.667921\pi\)
−0.503410 + 0.864048i \(0.667921\pi\)
\(420\) 0 0
\(421\) −1.49527e6 −0.411164 −0.205582 0.978640i \(-0.565909\pi\)
−0.205582 + 0.978640i \(0.565909\pi\)
\(422\) 8.15481e6 2.22912
\(423\) 137046. 0.0372406
\(424\) −5.25471e6 −1.41950
\(425\) 0 0
\(426\) −1.61094e6 −0.430086
\(427\) 739374. 0.196243
\(428\) 1.80631e6 0.476631
\(429\) −14866.4 −0.00389999
\(430\) 0 0
\(431\) 538580. 0.139655 0.0698276 0.997559i \(-0.477755\pi\)
0.0698276 + 0.997559i \(0.477755\pi\)
\(432\) −2.40102e6 −0.618995
\(433\) 5.50572e6 1.41122 0.705610 0.708600i \(-0.250673\pi\)
0.705610 + 0.708600i \(0.250673\pi\)
\(434\) 193309. 0.0492638
\(435\) 0 0
\(436\) 1.59422e7 4.01636
\(437\) −3.61117e6 −0.904575
\(438\) 4.81056e6 1.19815
\(439\) 4.55179e6 1.12725 0.563626 0.826030i \(-0.309406\pi\)
0.563626 + 0.826030i \(0.309406\pi\)
\(440\) 0 0
\(441\) −1.22266e6 −0.299371
\(442\) 167921. 0.0408837
\(443\) 1.92800e6 0.466764 0.233382 0.972385i \(-0.425021\pi\)
0.233382 + 0.972385i \(0.425021\pi\)
\(444\) 1.08406e7 2.60973
\(445\) 0 0
\(446\) −1.16535e7 −2.77407
\(447\) 4.80233e6 1.13680
\(448\) −3.49286e6 −0.822216
\(449\) 2.25763e6 0.528491 0.264246 0.964455i \(-0.414877\pi\)
0.264246 + 0.964455i \(0.414877\pi\)
\(450\) 0 0
\(451\) 1.49877e6 0.346972
\(452\) 2.00292e7 4.61124
\(453\) 983394. 0.225155
\(454\) −6.82991e6 −1.55516
\(455\) 0 0
\(456\) −7.12074e6 −1.60366
\(457\) 1.40364e6 0.314388 0.157194 0.987568i \(-0.449755\pi\)
0.157194 + 0.987568i \(0.449755\pi\)
\(458\) −6.69732e6 −1.49189
\(459\) 833918. 0.184753
\(460\) 0 0
\(461\) 1.28230e6 0.281021 0.140510 0.990079i \(-0.455126\pi\)
0.140510 + 0.990079i \(0.455126\pi\)
\(462\) −484578. −0.105623
\(463\) 5.16733e6 1.12025 0.560124 0.828409i \(-0.310754\pi\)
0.560124 + 0.828409i \(0.310754\pi\)
\(464\) 2.83057e7 6.10351
\(465\) 0 0
\(466\) 1.41726e7 3.02332
\(467\) −1.85528e6 −0.393657 −0.196828 0.980438i \(-0.563064\pi\)
−0.196828 + 0.980438i \(0.563064\pi\)
\(468\) 92473.4 0.0195165
\(469\) 2.13944e6 0.449125
\(470\) 0 0
\(471\) −3.26483e6 −0.678123
\(472\) 2.30228e7 4.75667
\(473\) 326626. 0.0671271
\(474\) 6.91936e6 1.41456
\(475\) 0 0
\(476\) 3.95870e6 0.800821
\(477\) 766681. 0.154283
\(478\) 2.58104e6 0.516685
\(479\) 2.77689e6 0.552993 0.276497 0.961015i \(-0.410827\pi\)
0.276497 + 0.961015i \(0.410827\pi\)
\(480\) 0 0
\(481\) −196624. −0.0387502
\(482\) −7.08655e6 −1.38937
\(483\) 943695. 0.184062
\(484\) 1.22440e6 0.237580
\(485\) 0 0
\(486\) 634957. 0.121942
\(487\) −8.55421e6 −1.63440 −0.817198 0.576356i \(-0.804474\pi\)
−0.817198 + 0.576356i \(0.804474\pi\)
\(488\) 9.91927e6 1.88552
\(489\) 3.05130e6 0.577049
\(490\) 0 0
\(491\) −9.34368e6 −1.74910 −0.874550 0.484936i \(-0.838843\pi\)
−0.874550 + 0.484936i \(0.838843\pi\)
\(492\) −9.32278e6 −1.73633
\(493\) −9.83109e6 −1.82173
\(494\) 209206. 0.0385706
\(495\) 0 0
\(496\) 1.43082e6 0.261144
\(497\) 688825. 0.125089
\(498\) 3.24585e6 0.586483
\(499\) 1.03633e7 1.86314 0.931569 0.363564i \(-0.118440\pi\)
0.931569 + 0.363564i \(0.118440\pi\)
\(500\) 0 0
\(501\) 4.13825e6 0.736585
\(502\) −1.85722e7 −3.28931
\(503\) −5.19173e6 −0.914938 −0.457469 0.889225i \(-0.651244\pi\)
−0.457469 + 0.889225i \(0.651244\pi\)
\(504\) 1.86084e6 0.326312
\(505\) 0 0
\(506\) −3.29688e6 −0.572435
\(507\) 3.33996e6 0.577060
\(508\) 9.38448e6 1.61343
\(509\) 591024. 0.101114 0.0505569 0.998721i \(-0.483900\pi\)
0.0505569 + 0.998721i \(0.483900\pi\)
\(510\) 0 0
\(511\) −2.05696e6 −0.348476
\(512\) 376329. 0.0634444
\(513\) 1.03894e6 0.174300
\(514\) −9.56145e6 −1.59630
\(515\) 0 0
\(516\) −2.03171e6 −0.335921
\(517\) 204723. 0.0336854
\(518\) −6.40905e6 −1.04947
\(519\) −4.74929e6 −0.773945
\(520\) 0 0
\(521\) 4.41573e6 0.712702 0.356351 0.934352i \(-0.384021\pi\)
0.356351 + 0.934352i \(0.384021\pi\)
\(522\) −7.48554e6 −1.20239
\(523\) −7.91473e6 −1.26527 −0.632633 0.774452i \(-0.718026\pi\)
−0.632633 + 0.774452i \(0.718026\pi\)
\(524\) 2.79988e7 4.45463
\(525\) 0 0
\(526\) −3.79224e6 −0.597629
\(527\) −496949. −0.0779445
\(528\) −3.58671e6 −0.559902
\(529\) −15819.3 −0.00245780
\(530\) 0 0
\(531\) −3.35910e6 −0.516996
\(532\) 4.93197e6 0.755512
\(533\) 169094. 0.0257816
\(534\) −7.44977e6 −1.13055
\(535\) 0 0
\(536\) 2.87022e7 4.31522
\(537\) −7.10829e6 −1.06372
\(538\) 5.44896e6 0.811629
\(539\) −1.82645e6 −0.270791
\(540\) 0 0
\(541\) 1.37134e6 0.201443 0.100722 0.994915i \(-0.467885\pi\)
0.100722 + 0.994915i \(0.467885\pi\)
\(542\) 1.43584e7 2.09946
\(543\) 3.37886e6 0.491779
\(544\) 2.01912e7 2.92527
\(545\) 0 0
\(546\) −54671.0 −0.00784830
\(547\) −7.82390e6 −1.11803 −0.559017 0.829156i \(-0.688821\pi\)
−0.559017 + 0.829156i \(0.688821\pi\)
\(548\) −76429.1 −0.0108719
\(549\) −1.44726e6 −0.204934
\(550\) 0 0
\(551\) −1.22481e7 −1.71866
\(552\) 1.26604e7 1.76848
\(553\) −2.95866e6 −0.411417
\(554\) −1.85166e7 −2.56323
\(555\) 0 0
\(556\) 2.26093e7 3.10171
\(557\) −8.25549e6 −1.12747 −0.563735 0.825956i \(-0.690636\pi\)
−0.563735 + 0.825956i \(0.690636\pi\)
\(558\) −378385. −0.0514456
\(559\) 36850.5 0.00498786
\(560\) 0 0
\(561\) 1.24573e6 0.167115
\(562\) 1.29672e7 1.73183
\(563\) −1.20969e7 −1.60843 −0.804214 0.594340i \(-0.797413\pi\)
−0.804214 + 0.594340i \(0.797413\pi\)
\(564\) −1.27344e6 −0.168570
\(565\) 0 0
\(566\) 9.89478e6 1.29827
\(567\) −271503. −0.0354663
\(568\) 9.24112e6 1.20186
\(569\) −7.97678e6 −1.03287 −0.516437 0.856325i \(-0.672742\pi\)
−0.516437 + 0.856325i \(0.672742\pi\)
\(570\) 0 0
\(571\) −8.94027e6 −1.14752 −0.573760 0.819023i \(-0.694516\pi\)
−0.573760 + 0.819023i \(0.694516\pi\)
\(572\) 138139. 0.0176533
\(573\) −3.05236e6 −0.388373
\(574\) 5.51171e6 0.698242
\(575\) 0 0
\(576\) 6.83694e6 0.858629
\(577\) 4.92430e6 0.615751 0.307875 0.951427i \(-0.400382\pi\)
0.307875 + 0.951427i \(0.400382\pi\)
\(578\) 1.19686e6 0.149013
\(579\) 3.23403e6 0.400911
\(580\) 0 0
\(581\) −1.38790e6 −0.170576
\(582\) −1.71428e6 −0.209786
\(583\) 1.14529e6 0.139554
\(584\) −2.75957e7 −3.34818
\(585\) 0 0
\(586\) −1.34894e6 −0.162274
\(587\) −1.22885e7 −1.47199 −0.735993 0.676989i \(-0.763285\pi\)
−0.735993 + 0.676989i \(0.763285\pi\)
\(588\) 1.13610e7 1.35511
\(589\) −619126. −0.0735345
\(590\) 0 0
\(591\) −5.43643e6 −0.640244
\(592\) −4.74380e7 −5.56317
\(593\) 8.20846e6 0.958572 0.479286 0.877659i \(-0.340896\pi\)
0.479286 + 0.877659i \(0.340896\pi\)
\(594\) 948517. 0.110301
\(595\) 0 0
\(596\) −4.46234e7 −5.14573
\(597\) 9.31010e6 1.06910
\(598\) −371960. −0.0425347
\(599\) −2.00442e6 −0.228256 −0.114128 0.993466i \(-0.536407\pi\)
−0.114128 + 0.993466i \(0.536407\pi\)
\(600\) 0 0
\(601\) −871399. −0.0984081 −0.0492041 0.998789i \(-0.515668\pi\)
−0.0492041 + 0.998789i \(0.515668\pi\)
\(602\) 1.20116e6 0.135086
\(603\) −4.18775e6 −0.469015
\(604\) −9.13773e6 −1.01917
\(605\) 0 0
\(606\) −269853. −0.0298501
\(607\) 1.04538e7 1.15160 0.575802 0.817589i \(-0.304690\pi\)
0.575802 + 0.817589i \(0.304690\pi\)
\(608\) 2.51554e7 2.75976
\(609\) 3.20075e6 0.349711
\(610\) 0 0
\(611\) 23097.3 0.00250298
\(612\) −7.74879e6 −0.836287
\(613\) 5.79441e6 0.622814 0.311407 0.950277i \(-0.399200\pi\)
0.311407 + 0.950277i \(0.399200\pi\)
\(614\) 9.92161e6 1.06209
\(615\) 0 0
\(616\) 2.77977e6 0.295160
\(617\) −7.10900e6 −0.751788 −0.375894 0.926663i \(-0.622664\pi\)
−0.375894 + 0.926663i \(0.622664\pi\)
\(618\) −300657. −0.0316665
\(619\) −1.13332e7 −1.18885 −0.594424 0.804152i \(-0.702620\pi\)
−0.594424 + 0.804152i \(0.702620\pi\)
\(620\) 0 0
\(621\) −1.84720e6 −0.192213
\(622\) 1.72195e7 1.78462
\(623\) 3.18546e6 0.328815
\(624\) −404660. −0.0416034
\(625\) 0 0
\(626\) 1.90465e6 0.194259
\(627\) 1.55200e6 0.157660
\(628\) 3.03369e7 3.06953
\(629\) 1.64761e7 1.66045
\(630\) 0 0
\(631\) 3.77282e6 0.377218 0.188609 0.982052i \(-0.439602\pi\)
0.188609 + 0.982052i \(0.439602\pi\)
\(632\) −3.96927e7 −3.95292
\(633\) 6.82534e6 0.677041
\(634\) 6.21800e6 0.614366
\(635\) 0 0
\(636\) −7.12402e6 −0.698365
\(637\) −206063. −0.0201211
\(638\) −1.11821e7 −1.08761
\(639\) −1.34831e6 −0.130628
\(640\) 0 0
\(641\) 1.76552e6 0.169718 0.0848591 0.996393i \(-0.472956\pi\)
0.0848591 + 0.996393i \(0.472956\pi\)
\(642\) 2.09032e6 0.200159
\(643\) −3.29197e6 −0.313999 −0.157000 0.987599i \(-0.550182\pi\)
−0.157000 + 0.987599i \(0.550182\pi\)
\(644\) −8.76885e6 −0.833159
\(645\) 0 0
\(646\) −1.75304e7 −1.65276
\(647\) −1.62220e7 −1.52350 −0.761752 0.647868i \(-0.775661\pi\)
−0.761752 + 0.647868i \(0.775661\pi\)
\(648\) −3.64242e6 −0.340763
\(649\) −5.01792e6 −0.467640
\(650\) 0 0
\(651\) 161794. 0.0149627
\(652\) −2.83528e7 −2.61202
\(653\) −3.42354e6 −0.314190 −0.157095 0.987583i \(-0.550213\pi\)
−0.157095 + 0.987583i \(0.550213\pi\)
\(654\) 1.84489e7 1.68665
\(655\) 0 0
\(656\) 4.07961e7 3.70134
\(657\) 4.02630e6 0.363909
\(658\) 752867. 0.0677881
\(659\) 5.40108e6 0.484470 0.242235 0.970218i \(-0.422119\pi\)
0.242235 + 0.970218i \(0.422119\pi\)
\(660\) 0 0
\(661\) 4.42083e6 0.393550 0.196775 0.980449i \(-0.436953\pi\)
0.196775 + 0.980449i \(0.436953\pi\)
\(662\) 3.48149e7 3.08760
\(663\) 140545. 0.0124175
\(664\) −1.86197e7 −1.63890
\(665\) 0 0
\(666\) 1.25451e7 1.09595
\(667\) 2.17767e7 1.89529
\(668\) −3.84528e7 −3.33416
\(669\) −9.75360e6 −0.842558
\(670\) 0 0
\(671\) −2.16195e6 −0.185370
\(672\) −6.57376e6 −0.561553
\(673\) −1.15029e7 −0.978969 −0.489484 0.872012i \(-0.662815\pi\)
−0.489484 + 0.872012i \(0.662815\pi\)
\(674\) 1.96060e7 1.66242
\(675\) 0 0
\(676\) −3.10350e7 −2.61207
\(677\) 2.08105e7 1.74506 0.872530 0.488560i \(-0.162478\pi\)
0.872530 + 0.488560i \(0.162478\pi\)
\(678\) 2.31785e7 1.93647
\(679\) 733014. 0.0610152
\(680\) 0 0
\(681\) −5.71643e6 −0.472343
\(682\) −565241. −0.0465343
\(683\) −2.29333e7 −1.88111 −0.940556 0.339638i \(-0.889695\pi\)
−0.940556 + 0.339638i \(0.889695\pi\)
\(684\) −9.65386e6 −0.788971
\(685\) 0 0
\(686\) −1.41954e7 −1.15170
\(687\) −5.60546e6 −0.453126
\(688\) 8.89066e6 0.716082
\(689\) 129214. 0.0103696
\(690\) 0 0
\(691\) −1.87802e7 −1.49626 −0.748128 0.663554i \(-0.769047\pi\)
−0.748128 + 0.663554i \(0.769047\pi\)
\(692\) 4.41306e7 3.50327
\(693\) −405578. −0.0320805
\(694\) 6.06634e6 0.478110
\(695\) 0 0
\(696\) 4.29406e7 3.36004
\(697\) −1.41692e7 −1.10475
\(698\) −2.93185e7 −2.27774
\(699\) 1.18621e7 0.918263
\(700\) 0 0
\(701\) −4.01030e6 −0.308235 −0.154118 0.988053i \(-0.549253\pi\)
−0.154118 + 0.988053i \(0.549253\pi\)
\(702\) 107013. 0.00819588
\(703\) 2.05268e7 1.56651
\(704\) 1.02132e7 0.776659
\(705\) 0 0
\(706\) 7.47310e6 0.564273
\(707\) 115387. 0.00868176
\(708\) 3.12129e7 2.34019
\(709\) −326851. −0.0244193 −0.0122097 0.999925i \(-0.503887\pi\)
−0.0122097 + 0.999925i \(0.503887\pi\)
\(710\) 0 0
\(711\) 5.79130e6 0.429638
\(712\) 4.27354e7 3.15928
\(713\) 1.10078e6 0.0810919
\(714\) 4.58115e6 0.336301
\(715\) 0 0
\(716\) 6.60504e7 4.81496
\(717\) 2.16026e6 0.156931
\(718\) 3.29164e6 0.238287
\(719\) −1.34067e7 −0.967161 −0.483581 0.875300i \(-0.660664\pi\)
−0.483581 + 0.875300i \(0.660664\pi\)
\(720\) 0 0
\(721\) 128559. 0.00921008
\(722\) 4.78536e6 0.341643
\(723\) −5.93124e6 −0.421988
\(724\) −3.13964e7 −2.22604
\(725\) 0 0
\(726\) 1.41692e6 0.0997709
\(727\) 2.14838e7 1.50756 0.753780 0.657127i \(-0.228229\pi\)
0.753780 + 0.657127i \(0.228229\pi\)
\(728\) 313619. 0.0219318
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −3.08788e6 −0.213731
\(732\) 1.34479e7 0.927637
\(733\) −8.60145e6 −0.591305 −0.295653 0.955296i \(-0.595537\pi\)
−0.295653 + 0.955296i \(0.595537\pi\)
\(734\) 1.62985e7 1.11663
\(735\) 0 0
\(736\) −4.47252e7 −3.04339
\(737\) −6.25577e6 −0.424240
\(738\) −1.07886e7 −0.729165
\(739\) −4.69199e6 −0.316043 −0.158022 0.987436i \(-0.550512\pi\)
−0.158022 + 0.987436i \(0.550512\pi\)
\(740\) 0 0
\(741\) 175099. 0.0117149
\(742\) 4.21178e6 0.280838
\(743\) 1.84011e7 1.22284 0.611422 0.791305i \(-0.290598\pi\)
0.611422 + 0.791305i \(0.290598\pi\)
\(744\) 2.17059e6 0.143763
\(745\) 0 0
\(746\) 1.13779e7 0.748539
\(747\) 2.71668e6 0.178130
\(748\) −1.15754e7 −0.756450
\(749\) −893804. −0.0582154
\(750\) 0 0
\(751\) 4.90697e6 0.317478 0.158739 0.987321i \(-0.449257\pi\)
0.158739 + 0.987321i \(0.449257\pi\)
\(752\) 5.57251e6 0.359341
\(753\) −1.55444e7 −0.999049
\(754\) −1.26158e6 −0.0808143
\(755\) 0 0
\(756\) 2.52281e6 0.160539
\(757\) −1.77340e7 −1.12478 −0.562390 0.826872i \(-0.690118\pi\)
−0.562390 + 0.826872i \(0.690118\pi\)
\(758\) −8.73560e6 −0.552230
\(759\) −2.75939e6 −0.173864
\(760\) 0 0
\(761\) 7.16267e6 0.448346 0.224173 0.974549i \(-0.428032\pi\)
0.224173 + 0.974549i \(0.428032\pi\)
\(762\) 1.08600e7 0.677554
\(763\) −7.88859e6 −0.490556
\(764\) 2.83626e7 1.75797
\(765\) 0 0
\(766\) −1.31378e7 −0.809006
\(767\) −566131. −0.0347479
\(768\) −8.86628e6 −0.542423
\(769\) −1.90012e7 −1.15869 −0.579343 0.815084i \(-0.696691\pi\)
−0.579343 + 0.815084i \(0.696691\pi\)
\(770\) 0 0
\(771\) −8.00265e6 −0.484840
\(772\) −3.00507e7 −1.81473
\(773\) 6.66234e6 0.401031 0.200516 0.979690i \(-0.435738\pi\)
0.200516 + 0.979690i \(0.435738\pi\)
\(774\) −2.35116e6 −0.141068
\(775\) 0 0
\(776\) 9.83395e6 0.586238
\(777\) −5.36419e6 −0.318751
\(778\) 4.39597e7 2.60379
\(779\) −1.76528e7 −1.04224
\(780\) 0 0
\(781\) −2.01414e6 −0.118158
\(782\) 3.11683e7 1.82262
\(783\) −6.26518e6 −0.365198
\(784\) −4.97153e7 −2.88868
\(785\) 0 0
\(786\) 3.24012e7 1.87070
\(787\) 9.20508e6 0.529774 0.264887 0.964279i \(-0.414665\pi\)
0.264887 + 0.964279i \(0.414665\pi\)
\(788\) 5.05155e7 2.89807
\(789\) −3.17400e6 −0.181516
\(790\) 0 0
\(791\) −9.91094e6 −0.563214
\(792\) −5.44114e6 −0.308232
\(793\) −243915. −0.0137739
\(794\) 164053. 0.00923494
\(795\) 0 0
\(796\) −8.65098e7 −4.83930
\(797\) 2.71663e7 1.51490 0.757450 0.652893i \(-0.226445\pi\)
0.757450 + 0.652893i \(0.226445\pi\)
\(798\) 5.70744e6 0.317274
\(799\) −1.93543e6 −0.107253
\(800\) 0 0
\(801\) −6.23524e6 −0.343378
\(802\) 5.11213e7 2.80650
\(803\) 6.01460e6 0.329168
\(804\) 3.89127e7 2.12300
\(805\) 0 0
\(806\) −63771.6 −0.00345772
\(807\) 4.56062e6 0.246513
\(808\) 1.54800e6 0.0834149
\(809\) −2.31559e7 −1.24392 −0.621958 0.783051i \(-0.713662\pi\)
−0.621958 + 0.783051i \(0.713662\pi\)
\(810\) 0 0
\(811\) 1.33212e6 0.0711200 0.0355600 0.999368i \(-0.488679\pi\)
0.0355600 + 0.999368i \(0.488679\pi\)
\(812\) −2.97415e7 −1.58297
\(813\) 1.20176e7 0.637661
\(814\) 1.87402e7 0.991320
\(815\) 0 0
\(816\) 3.39084e7 1.78271
\(817\) −3.84705e6 −0.201638
\(818\) 6.58963e7 3.44332
\(819\) −45758.1 −0.00238373
\(820\) 0 0
\(821\) 1.06084e7 0.549277 0.274638 0.961548i \(-0.411442\pi\)
0.274638 + 0.961548i \(0.411442\pi\)
\(822\) −88446.4 −0.00456563
\(823\) −1.83916e7 −0.946496 −0.473248 0.880929i \(-0.656919\pi\)
−0.473248 + 0.880929i \(0.656919\pi\)
\(824\) 1.72471e6 0.0884910
\(825\) 0 0
\(826\) −1.84533e7 −0.941075
\(827\) −1.95386e7 −0.993414 −0.496707 0.867918i \(-0.665458\pi\)
−0.496707 + 0.867918i \(0.665458\pi\)
\(828\) 1.71642e7 0.870057
\(829\) −3.37798e7 −1.70714 −0.853572 0.520974i \(-0.825569\pi\)
−0.853572 + 0.520974i \(0.825569\pi\)
\(830\) 0 0
\(831\) −1.54979e7 −0.778520
\(832\) 1.15227e6 0.0577095
\(833\) 1.72670e7 0.862193
\(834\) 2.61643e7 1.30255
\(835\) 0 0
\(836\) −1.44212e7 −0.713651
\(837\) −316697. −0.0156254
\(838\) 3.89062e7 1.91385
\(839\) 2.77050e7 1.35879 0.679396 0.733772i \(-0.262242\pi\)
0.679396 + 0.733772i \(0.262242\pi\)
\(840\) 0 0
\(841\) 5.33493e7 2.60099
\(842\) 1.60788e7 0.781578
\(843\) 1.08531e7 0.526001
\(844\) −6.34213e7 −3.06463
\(845\) 0 0
\(846\) −1.47367e6 −0.0707902
\(847\) −605864. −0.0290179
\(848\) 3.11744e7 1.48870
\(849\) 8.28165e6 0.394319
\(850\) 0 0
\(851\) −3.64958e7 −1.72750
\(852\) 1.25285e7 0.591291
\(853\) 1.04555e7 0.492007 0.246003 0.969269i \(-0.420883\pi\)
0.246003 + 0.969269i \(0.420883\pi\)
\(854\) −7.95053e6 −0.373037
\(855\) 0 0
\(856\) −1.19911e7 −0.559337
\(857\) −2.53866e7 −1.18074 −0.590369 0.807134i \(-0.701018\pi\)
−0.590369 + 0.807134i \(0.701018\pi\)
\(858\) 159860. 0.00741345
\(859\) 1.84775e6 0.0854396 0.0427198 0.999087i \(-0.486398\pi\)
0.0427198 + 0.999087i \(0.486398\pi\)
\(860\) 0 0
\(861\) 4.61314e6 0.212075
\(862\) −5.79138e6 −0.265469
\(863\) −2.89194e7 −1.32179 −0.660894 0.750479i \(-0.729823\pi\)
−0.660894 + 0.750479i \(0.729823\pi\)
\(864\) 1.28675e7 0.586422
\(865\) 0 0
\(866\) −5.92034e7 −2.68257
\(867\) 1.00174e6 0.0452591
\(868\) −1.50340e6 −0.0677289
\(869\) 8.65121e6 0.388622
\(870\) 0 0
\(871\) −705788. −0.0315231
\(872\) −1.05832e8 −4.71329
\(873\) −1.43481e6 −0.0637174
\(874\) 3.88312e7 1.71950
\(875\) 0 0
\(876\) −3.74125e7 −1.64724
\(877\) −2.63977e7 −1.15896 −0.579478 0.814988i \(-0.696743\pi\)
−0.579478 + 0.814988i \(0.696743\pi\)
\(878\) −4.89457e7 −2.14278
\(879\) −1.12903e6 −0.0492869
\(880\) 0 0
\(881\) 1.08347e7 0.470304 0.235152 0.971959i \(-0.424441\pi\)
0.235152 + 0.971959i \(0.424441\pi\)
\(882\) 1.31474e7 0.569072
\(883\) −2.45316e7 −1.05883 −0.529413 0.848364i \(-0.677588\pi\)
−0.529413 + 0.848364i \(0.677588\pi\)
\(884\) −1.30595e6 −0.0562078
\(885\) 0 0
\(886\) −2.07319e7 −0.887268
\(887\) −2.84619e7 −1.21466 −0.607330 0.794450i \(-0.707759\pi\)
−0.607330 + 0.794450i \(0.707759\pi\)
\(888\) −7.19647e7 −3.06258
\(889\) −4.64367e6 −0.197064
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) 9.06308e7 3.81385
\(893\) −2.41127e6 −0.101185
\(894\) −5.16398e7 −2.16093
\(895\) 0 0
\(896\) 1.41855e7 0.590304
\(897\) −311320. −0.0129189
\(898\) −2.42765e7 −1.00460
\(899\) 3.73355e6 0.154072
\(900\) 0 0
\(901\) −1.08274e7 −0.444338
\(902\) −1.61164e7 −0.659555
\(903\) 1.00534e6 0.0410291
\(904\) −1.32963e8 −5.41139
\(905\) 0 0
\(906\) −1.05745e7 −0.427995
\(907\) −1.85629e7 −0.749253 −0.374626 0.927176i \(-0.622229\pi\)
−0.374626 + 0.927176i \(0.622229\pi\)
\(908\) 5.31173e7 2.13807
\(909\) −225859. −0.00906625
\(910\) 0 0
\(911\) 3.93363e7 1.57035 0.785177 0.619271i \(-0.212572\pi\)
0.785177 + 0.619271i \(0.212572\pi\)
\(912\) 4.22449e7 1.68185
\(913\) 4.05825e6 0.161125
\(914\) −1.50935e7 −0.597617
\(915\) 0 0
\(916\) 5.20861e7 2.05108
\(917\) −1.38545e7 −0.544086
\(918\) −8.96717e6 −0.351195
\(919\) 2.72458e7 1.06417 0.532084 0.846691i \(-0.321409\pi\)
0.532084 + 0.846691i \(0.321409\pi\)
\(920\) 0 0
\(921\) 8.30410e6 0.322584
\(922\) −1.37887e7 −0.534190
\(923\) −227239. −0.00877969
\(924\) 3.76864e6 0.145213
\(925\) 0 0
\(926\) −5.55646e7 −2.12947
\(927\) −251642. −0.00961796
\(928\) −1.51696e8 −5.78233
\(929\) −2.98863e7 −1.13614 −0.568071 0.822980i \(-0.692310\pi\)
−0.568071 + 0.822980i \(0.692310\pi\)
\(930\) 0 0
\(931\) 2.15122e7 0.813411
\(932\) −1.10223e8 −4.15653
\(933\) 1.44122e7 0.542035
\(934\) 1.99500e7 0.748298
\(935\) 0 0
\(936\) −613880. −0.0229031
\(937\) 3.91979e7 1.45852 0.729262 0.684234i \(-0.239863\pi\)
0.729262 + 0.684234i \(0.239863\pi\)
\(938\) −2.30055e7 −0.853737
\(939\) 1.59414e6 0.0590014
\(940\) 0 0
\(941\) 2.33041e7 0.857941 0.428971 0.903318i \(-0.358876\pi\)
0.428971 + 0.903318i \(0.358876\pi\)
\(942\) 3.51069e7 1.28904
\(943\) 3.13859e7 1.14936
\(944\) −1.36586e8 −4.98858
\(945\) 0 0
\(946\) −3.51223e6 −0.127601
\(947\) −2.41038e7 −0.873394 −0.436697 0.899609i \(-0.643852\pi\)
−0.436697 + 0.899609i \(0.643852\pi\)
\(948\) −5.38130e7 −1.94476
\(949\) 678578. 0.0244587
\(950\) 0 0
\(951\) 5.20428e6 0.186599
\(952\) −2.62796e7 −0.939782
\(953\) −3.46893e7 −1.23727 −0.618633 0.785680i \(-0.712313\pi\)
−0.618633 + 0.785680i \(0.712313\pi\)
\(954\) −8.24416e6 −0.293275
\(955\) 0 0
\(956\) −2.00732e7 −0.710349
\(957\) −9.35909e6 −0.330334
\(958\) −2.98601e7 −1.05118
\(959\) 37819.0 0.00132789
\(960\) 0 0
\(961\) −2.84404e7 −0.993408
\(962\) 2.11431e6 0.0736598
\(963\) 1.74954e6 0.0607936
\(964\) 5.51133e7 1.91013
\(965\) 0 0
\(966\) −1.01476e7 −0.349882
\(967\) −3.99380e7 −1.37347 −0.686737 0.726906i \(-0.740958\pi\)
−0.686737 + 0.726906i \(0.740958\pi\)
\(968\) −8.12813e6 −0.278806
\(969\) −1.46724e7 −0.501986
\(970\) 0 0
\(971\) −3.25227e6 −0.110698 −0.0553488 0.998467i \(-0.517627\pi\)
−0.0553488 + 0.998467i \(0.517627\pi\)
\(972\) −4.93817e6 −0.167649
\(973\) −1.11876e7 −0.378841
\(974\) 9.19839e7 3.10681
\(975\) 0 0
\(976\) −5.88476e7 −1.97744
\(977\) −2.38653e7 −0.799892 −0.399946 0.916539i \(-0.630971\pi\)
−0.399946 + 0.916539i \(0.630971\pi\)
\(978\) −3.28108e7 −1.09691
\(979\) −9.31437e6 −0.310597
\(980\) 0 0
\(981\) 1.54412e7 0.512281
\(982\) 1.00473e8 3.32485
\(983\) 2.22870e7 0.735645 0.367823 0.929896i \(-0.380103\pi\)
0.367823 + 0.929896i \(0.380103\pi\)
\(984\) 6.18888e7 2.03763
\(985\) 0 0
\(986\) 1.05714e8 3.46291
\(987\) 630128. 0.0205890
\(988\) −1.62703e6 −0.0530277
\(989\) 6.83991e6 0.222362
\(990\) 0 0
\(991\) −1.74834e7 −0.565513 −0.282756 0.959192i \(-0.591249\pi\)
−0.282756 + 0.959192i \(0.591249\pi\)
\(992\) −7.66803e6 −0.247403
\(993\) 2.91391e7 0.937784
\(994\) −7.40697e6 −0.237780
\(995\) 0 0
\(996\) −2.52435e7 −0.806308
\(997\) −4.50302e7 −1.43472 −0.717358 0.696705i \(-0.754649\pi\)
−0.717358 + 0.696705i \(0.754649\pi\)
\(998\) −1.11437e8 −3.54162
\(999\) 1.04999e7 0.332868
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.v.1.1 13
5.2 odd 4 165.6.c.b.34.1 26
5.3 odd 4 165.6.c.b.34.26 yes 26
5.4 even 2 825.6.a.y.1.13 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.c.b.34.1 26 5.2 odd 4
165.6.c.b.34.26 yes 26 5.3 odd 4
825.6.a.v.1.1 13 1.1 even 1 trivial
825.6.a.y.1.13 13 5.4 even 2