Properties

Label 825.6.a.h.1.2
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $0$
Dimension $3$
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: \(3\)
Coefficient field: 3.3.788.1
Defining polynomial: \( x^{3} - x^{2} - 7x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.35386\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.08127 q^{2} +9.00000 q^{3} -30.8308 q^{4} -9.73147 q^{6} +139.508 q^{7} +67.9374 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-1.08127 q^{2} +9.00000 q^{3} -30.8308 q^{4} -9.73147 q^{6} +139.508 q^{7} +67.9374 q^{8} +81.0000 q^{9} -121.000 q^{11} -277.478 q^{12} +646.331 q^{13} -150.846 q^{14} +913.128 q^{16} +1378.32 q^{17} -87.5832 q^{18} +1908.90 q^{19} +1255.57 q^{21} +130.834 q^{22} -343.726 q^{23} +611.436 q^{24} -698.861 q^{26} +729.000 q^{27} -4301.15 q^{28} +53.5092 q^{29} +634.133 q^{31} -3161.34 q^{32} -1089.00 q^{33} -1490.34 q^{34} -2497.30 q^{36} -11674.2 q^{37} -2064.04 q^{38} +5816.98 q^{39} +18866.5 q^{41} -1357.62 q^{42} -13379.3 q^{43} +3730.53 q^{44} +371.662 q^{46} +2252.95 q^{47} +8218.15 q^{48} +2655.48 q^{49} +12404.9 q^{51} -19926.9 q^{52} +8900.71 q^{53} -788.249 q^{54} +9477.80 q^{56} +17180.1 q^{57} -57.8581 q^{58} -11276.7 q^{59} +11852.4 q^{61} -685.672 q^{62} +11300.1 q^{63} -25801.8 q^{64} +1177.51 q^{66} +57333.6 q^{67} -42494.7 q^{68} -3093.53 q^{69} +31075.8 q^{71} +5502.93 q^{72} -56010.8 q^{73} +12623.0 q^{74} -58853.0 q^{76} -16880.5 q^{77} -6289.75 q^{78} -883.036 q^{79} +6561.00 q^{81} -20399.9 q^{82} -93986.3 q^{83} -38710.3 q^{84} +14466.7 q^{86} +481.583 q^{87} -8220.42 q^{88} +21739.9 q^{89} +90168.4 q^{91} +10597.4 q^{92} +5707.20 q^{93} -2436.06 q^{94} -28452.0 q^{96} +158059. q^{97} -2871.30 q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 27 q^{3} - 20 q^{4} - 18 q^{6} - 152 q^{7} + 24 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} + 27 q^{3} - 20 q^{4} - 18 q^{6} - 152 q^{7} + 24 q^{8} + 243 q^{9} - 363 q^{11} - 180 q^{12} + 546 q^{13} - 8 q^{14} - 1360 q^{16} + 314 q^{17} - 162 q^{18} + 1808 q^{19} - 1368 q^{21} + 242 q^{22} - 4288 q^{23} + 216 q^{24} + 812 q^{26} + 2187 q^{27} - 5888 q^{28} + 5582 q^{29} + 6328 q^{31} + 736 q^{32} - 3267 q^{33} + 11596 q^{34} - 1620 q^{36} - 16866 q^{37} - 9584 q^{38} + 4914 q^{39} + 23282 q^{41} - 72 q^{42} - 20572 q^{43} + 2420 q^{44} + 16592 q^{46} - 3432 q^{47} - 12240 q^{48} + 11531 q^{49} + 2826 q^{51} - 21816 q^{52} - 16138 q^{53} - 1458 q^{54} + 15648 q^{56} + 16272 q^{57} - 17460 q^{58} + 21972 q^{59} + 8322 q^{61} - 5056 q^{62} - 12312 q^{63} + 22208 q^{64} + 2178 q^{66} + 84332 q^{67} - 59832 q^{68} - 38592 q^{69} + 50528 q^{71} + 1944 q^{72} + 53838 q^{73} + 79212 q^{74} - 52448 q^{76} + 18392 q^{77} + 7308 q^{78} + 6364 q^{79} + 19683 q^{81} + 68020 q^{82} - 96272 q^{83} - 52992 q^{84} + 143152 q^{86} + 50238 q^{87} - 2904 q^{88} - 38938 q^{89} + 104968 q^{91} - 24000 q^{92} + 56952 q^{93} + 49088 q^{94} + 6624 q^{96} + 103242 q^{97} - 9554 q^{98} - 29403 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.08127 −0.191144 −0.0955720 0.995423i \(-0.530468\pi\)
−0.0955720 + 0.995423i \(0.530468\pi\)
\(3\) 9.00000 0.577350
\(4\) −30.8308 −0.963464
\(5\) 0 0
\(6\) −9.73147 −0.110357
\(7\) 139.508 1.07610 0.538052 0.842912i \(-0.319161\pi\)
0.538052 + 0.842912i \(0.319161\pi\)
\(8\) 67.9374 0.375304
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) −277.478 −0.556256
\(13\) 646.331 1.06071 0.530355 0.847776i \(-0.322059\pi\)
0.530355 + 0.847776i \(0.322059\pi\)
\(14\) −150.846 −0.205691
\(15\) 0 0
\(16\) 913.128 0.891727
\(17\) 1378.32 1.15672 0.578358 0.815783i \(-0.303694\pi\)
0.578358 + 0.815783i \(0.303694\pi\)
\(18\) −87.5832 −0.0637147
\(19\) 1908.90 1.21311 0.606553 0.795043i \(-0.292552\pi\)
0.606553 + 0.795043i \(0.292552\pi\)
\(20\) 0 0
\(21\) 1255.57 0.621289
\(22\) 130.834 0.0576321
\(23\) −343.726 −0.135485 −0.0677427 0.997703i \(-0.521580\pi\)
−0.0677427 + 0.997703i \(0.521580\pi\)
\(24\) 611.436 0.216682
\(25\) 0 0
\(26\) −698.861 −0.202748
\(27\) 729.000 0.192450
\(28\) −4301.15 −1.03679
\(29\) 53.5092 0.0118150 0.00590750 0.999983i \(-0.498120\pi\)
0.00590750 + 0.999983i \(0.498120\pi\)
\(30\) 0 0
\(31\) 634.133 0.118516 0.0592579 0.998243i \(-0.481127\pi\)
0.0592579 + 0.998243i \(0.481127\pi\)
\(32\) −3161.34 −0.545753
\(33\) −1089.00 −0.174078
\(34\) −1490.34 −0.221099
\(35\) 0 0
\(36\) −2497.30 −0.321155
\(37\) −11674.2 −1.40192 −0.700960 0.713201i \(-0.747245\pi\)
−0.700960 + 0.713201i \(0.747245\pi\)
\(38\) −2064.04 −0.231878
\(39\) 5816.98 0.612401
\(40\) 0 0
\(41\) 18866.5 1.75280 0.876399 0.481585i \(-0.159939\pi\)
0.876399 + 0.481585i \(0.159939\pi\)
\(42\) −1357.62 −0.118756
\(43\) −13379.3 −1.10347 −0.551736 0.834018i \(-0.686035\pi\)
−0.551736 + 0.834018i \(0.686035\pi\)
\(44\) 3730.53 0.290495
\(45\) 0 0
\(46\) 371.662 0.0258972
\(47\) 2252.95 0.148767 0.0743835 0.997230i \(-0.476301\pi\)
0.0743835 + 0.997230i \(0.476301\pi\)
\(48\) 8218.15 0.514839
\(49\) 2655.48 0.157998
\(50\) 0 0
\(51\) 12404.9 0.667830
\(52\) −19926.9 −1.02196
\(53\) 8900.71 0.435246 0.217623 0.976033i \(-0.430170\pi\)
0.217623 + 0.976033i \(0.430170\pi\)
\(54\) −788.249 −0.0367857
\(55\) 0 0
\(56\) 9477.80 0.403866
\(57\) 17180.1 0.700387
\(58\) −57.8581 −0.00225837
\(59\) −11276.7 −0.421746 −0.210873 0.977514i \(-0.567631\pi\)
−0.210873 + 0.977514i \(0.567631\pi\)
\(60\) 0 0
\(61\) 11852.4 0.407834 0.203917 0.978988i \(-0.434633\pi\)
0.203917 + 0.978988i \(0.434633\pi\)
\(62\) −685.672 −0.0226536
\(63\) 11300.1 0.358701
\(64\) −25801.8 −0.787409
\(65\) 0 0
\(66\) 1177.51 0.0332739
\(67\) 57333.6 1.56035 0.780175 0.625561i \(-0.215130\pi\)
0.780175 + 0.625561i \(0.215130\pi\)
\(68\) −42494.7 −1.11445
\(69\) −3093.53 −0.0782226
\(70\) 0 0
\(71\) 31075.8 0.731605 0.365803 0.930692i \(-0.380794\pi\)
0.365803 + 0.930692i \(0.380794\pi\)
\(72\) 5502.93 0.125101
\(73\) −56010.8 −1.23017 −0.615084 0.788462i \(-0.710878\pi\)
−0.615084 + 0.788462i \(0.710878\pi\)
\(74\) 12623.0 0.267969
\(75\) 0 0
\(76\) −58853.0 −1.16878
\(77\) −16880.5 −0.324457
\(78\) −6289.75 −0.117057
\(79\) −883.036 −0.0159188 −0.00795941 0.999968i \(-0.502534\pi\)
−0.00795941 + 0.999968i \(0.502534\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −20399.9 −0.335037
\(83\) −93986.3 −1.49751 −0.748754 0.662848i \(-0.769348\pi\)
−0.748754 + 0.662848i \(0.769348\pi\)
\(84\) −38710.3 −0.598589
\(85\) 0 0
\(86\) 14466.7 0.210922
\(87\) 481.583 0.00682139
\(88\) −8220.42 −0.113159
\(89\) 21739.9 0.290926 0.145463 0.989364i \(-0.453533\pi\)
0.145463 + 0.989364i \(0.453533\pi\)
\(90\) 0 0
\(91\) 90168.4 1.14143
\(92\) 10597.4 0.130535
\(93\) 5707.20 0.0684251
\(94\) −2436.06 −0.0284359
\(95\) 0 0
\(96\) −28452.0 −0.315090
\(97\) 158059. 1.70565 0.852825 0.522197i \(-0.174888\pi\)
0.852825 + 0.522197i \(0.174888\pi\)
\(98\) −2871.30 −0.0302004
\(99\) −9801.00 −0.100504
\(100\) 0 0
\(101\) −156968. −1.53111 −0.765556 0.643369i \(-0.777536\pi\)
−0.765556 + 0.643369i \(0.777536\pi\)
\(102\) −13413.0 −0.127652
\(103\) −169126. −1.57078 −0.785391 0.618999i \(-0.787538\pi\)
−0.785391 + 0.618999i \(0.787538\pi\)
\(104\) 43910.0 0.398089
\(105\) 0 0
\(106\) −9624.11 −0.0831948
\(107\) 13538.5 0.114317 0.0571587 0.998365i \(-0.481796\pi\)
0.0571587 + 0.998365i \(0.481796\pi\)
\(108\) −22475.7 −0.185419
\(109\) −74736.4 −0.602513 −0.301256 0.953543i \(-0.597406\pi\)
−0.301256 + 0.953543i \(0.597406\pi\)
\(110\) 0 0
\(111\) −105068. −0.809399
\(112\) 127389. 0.959590
\(113\) 205914. 1.51701 0.758507 0.651665i \(-0.225929\pi\)
0.758507 + 0.651665i \(0.225929\pi\)
\(114\) −18576.4 −0.133875
\(115\) 0 0
\(116\) −1649.73 −0.0113833
\(117\) 52352.8 0.353570
\(118\) 12193.2 0.0806142
\(119\) 192286. 1.24475
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −12815.7 −0.0779550
\(123\) 169799. 1.01198
\(124\) −19550.9 −0.114186
\(125\) 0 0
\(126\) −12218.6 −0.0685636
\(127\) 214682. 1.18110 0.590550 0.807001i \(-0.298911\pi\)
0.590550 + 0.807001i \(0.298911\pi\)
\(128\) 129062. 0.696261
\(129\) −120414. −0.637090
\(130\) 0 0
\(131\) 253067. 1.28842 0.644209 0.764849i \(-0.277187\pi\)
0.644209 + 0.764849i \(0.277187\pi\)
\(132\) 33574.8 0.167718
\(133\) 266307. 1.30543
\(134\) −61993.3 −0.298252
\(135\) 0 0
\(136\) 93639.2 0.434121
\(137\) −239398. −1.08973 −0.544865 0.838524i \(-0.683419\pi\)
−0.544865 + 0.838524i \(0.683419\pi\)
\(138\) 3344.96 0.0149518
\(139\) −166398. −0.730484 −0.365242 0.930913i \(-0.619014\pi\)
−0.365242 + 0.930913i \(0.619014\pi\)
\(140\) 0 0
\(141\) 20276.5 0.0858907
\(142\) −33601.5 −0.139842
\(143\) −78206.1 −0.319816
\(144\) 73963.4 0.297242
\(145\) 0 0
\(146\) 60563.0 0.235139
\(147\) 23899.3 0.0912203
\(148\) 359926. 1.35070
\(149\) 295512. 1.09046 0.545229 0.838287i \(-0.316443\pi\)
0.545229 + 0.838287i \(0.316443\pi\)
\(150\) 0 0
\(151\) 428373. 1.52890 0.764452 0.644681i \(-0.223010\pi\)
0.764452 + 0.644681i \(0.223010\pi\)
\(152\) 129686. 0.455284
\(153\) 111644. 0.385572
\(154\) 18252.4 0.0620181
\(155\) 0 0
\(156\) −179342. −0.590027
\(157\) 289003. 0.935735 0.467867 0.883799i \(-0.345022\pi\)
0.467867 + 0.883799i \(0.345022\pi\)
\(158\) 954.804 0.00304279
\(159\) 80106.4 0.251290
\(160\) 0 0
\(161\) −47952.5 −0.145796
\(162\) −7094.24 −0.0212382
\(163\) −288223. −0.849688 −0.424844 0.905267i \(-0.639671\pi\)
−0.424844 + 0.905267i \(0.639671\pi\)
\(164\) −581671. −1.68876
\(165\) 0 0
\(166\) 101625. 0.286240
\(167\) 180912. 0.501969 0.250985 0.967991i \(-0.419246\pi\)
0.250985 + 0.967991i \(0.419246\pi\)
\(168\) 85300.2 0.233172
\(169\) 46451.1 0.125106
\(170\) 0 0
\(171\) 154621. 0.404369
\(172\) 412495. 1.06316
\(173\) −218579. −0.555255 −0.277628 0.960689i \(-0.589548\pi\)
−0.277628 + 0.960689i \(0.589548\pi\)
\(174\) −520.723 −0.00130387
\(175\) 0 0
\(176\) −110489. −0.268866
\(177\) −101490. −0.243495
\(178\) −23506.8 −0.0556089
\(179\) 310916. 0.725287 0.362643 0.931928i \(-0.381874\pi\)
0.362643 + 0.931928i \(0.381874\pi\)
\(180\) 0 0
\(181\) −423637. −0.961163 −0.480582 0.876950i \(-0.659574\pi\)
−0.480582 + 0.876950i \(0.659574\pi\)
\(182\) −97496.7 −0.218178
\(183\) 106672. 0.235463
\(184\) −23351.8 −0.0508483
\(185\) 0 0
\(186\) −6171.04 −0.0130791
\(187\) −166776. −0.348763
\(188\) −69460.3 −0.143332
\(189\) 101701. 0.207096
\(190\) 0 0
\(191\) 861089. 1.70791 0.853954 0.520348i \(-0.174198\pi\)
0.853954 + 0.520348i \(0.174198\pi\)
\(192\) −232216. −0.454611
\(193\) −430583. −0.832077 −0.416038 0.909347i \(-0.636582\pi\)
−0.416038 + 0.909347i \(0.636582\pi\)
\(194\) −170905. −0.326025
\(195\) 0 0
\(196\) −81870.6 −0.152226
\(197\) 184755. 0.339181 0.169590 0.985515i \(-0.445756\pi\)
0.169590 + 0.985515i \(0.445756\pi\)
\(198\) 10597.6 0.0192107
\(199\) 322993. 0.578177 0.289089 0.957302i \(-0.406648\pi\)
0.289089 + 0.957302i \(0.406648\pi\)
\(200\) 0 0
\(201\) 516002. 0.900869
\(202\) 169725. 0.292663
\(203\) 7464.96 0.0127142
\(204\) −382452. −0.643431
\(205\) 0 0
\(206\) 182871. 0.300246
\(207\) −27841.8 −0.0451618
\(208\) 590183. 0.945864
\(209\) −230977. −0.365765
\(210\) 0 0
\(211\) −394106. −0.609406 −0.304703 0.952447i \(-0.598557\pi\)
−0.304703 + 0.952447i \(0.598557\pi\)
\(212\) −274417. −0.419344
\(213\) 279682. 0.422393
\(214\) −14638.9 −0.0218511
\(215\) 0 0
\(216\) 49526.3 0.0722274
\(217\) 88466.6 0.127535
\(218\) 80810.6 0.115167
\(219\) −504097. −0.710238
\(220\) 0 0
\(221\) 890849. 1.22694
\(222\) 113607. 0.154712
\(223\) −445038. −0.599287 −0.299643 0.954051i \(-0.596868\pi\)
−0.299643 + 0.954051i \(0.596868\pi\)
\(224\) −441032. −0.587286
\(225\) 0 0
\(226\) −222649. −0.289968
\(227\) −1.15887e6 −1.49270 −0.746349 0.665555i \(-0.768195\pi\)
−0.746349 + 0.665555i \(0.768195\pi\)
\(228\) −529677. −0.674798
\(229\) 885853. 1.11628 0.558140 0.829747i \(-0.311515\pi\)
0.558140 + 0.829747i \(0.311515\pi\)
\(230\) 0 0
\(231\) −151924. −0.187326
\(232\) 3635.27 0.00443422
\(233\) −750392. −0.905520 −0.452760 0.891632i \(-0.649561\pi\)
−0.452760 + 0.891632i \(0.649561\pi\)
\(234\) −56607.8 −0.0675828
\(235\) 0 0
\(236\) 347669. 0.406337
\(237\) −7947.32 −0.00919073
\(238\) −207914. −0.237926
\(239\) 617415. 0.699169 0.349585 0.936905i \(-0.386323\pi\)
0.349585 + 0.936905i \(0.386323\pi\)
\(240\) 0 0
\(241\) 1.20275e6 1.33393 0.666966 0.745088i \(-0.267593\pi\)
0.666966 + 0.745088i \(0.267593\pi\)
\(242\) −15830.9 −0.0173767
\(243\) 59049.0 0.0641500
\(244\) −365421. −0.392933
\(245\) 0 0
\(246\) −183599. −0.193434
\(247\) 1.23378e6 1.28675
\(248\) 43081.3 0.0444795
\(249\) −845877. −0.864587
\(250\) 0 0
\(251\) 1.55775e6 1.56068 0.780338 0.625358i \(-0.215047\pi\)
0.780338 + 0.625358i \(0.215047\pi\)
\(252\) −348393. −0.345596
\(253\) 41590.8 0.0408504
\(254\) −232130. −0.225760
\(255\) 0 0
\(256\) 686107. 0.654323
\(257\) −92156.0 −0.0870344 −0.0435172 0.999053i \(-0.513856\pi\)
−0.0435172 + 0.999053i \(0.513856\pi\)
\(258\) 130200. 0.121776
\(259\) −1.62865e6 −1.50861
\(260\) 0 0
\(261\) 4334.24 0.00393833
\(262\) −273635. −0.246274
\(263\) 862122. 0.768563 0.384281 0.923216i \(-0.374449\pi\)
0.384281 + 0.923216i \(0.374449\pi\)
\(264\) −73983.8 −0.0653321
\(265\) 0 0
\(266\) −287950. −0.249525
\(267\) 195659. 0.167966
\(268\) −1.76764e6 −1.50334
\(269\) −1.25052e6 −1.05368 −0.526841 0.849964i \(-0.676624\pi\)
−0.526841 + 0.849964i \(0.676624\pi\)
\(270\) 0 0
\(271\) 844654. 0.698644 0.349322 0.937003i \(-0.386412\pi\)
0.349322 + 0.937003i \(0.386412\pi\)
\(272\) 1.25858e6 1.03147
\(273\) 811515. 0.659007
\(274\) 258855. 0.208295
\(275\) 0 0
\(276\) 95376.2 0.0753646
\(277\) 1.52990e6 1.19802 0.599008 0.800743i \(-0.295562\pi\)
0.599008 + 0.800743i \(0.295562\pi\)
\(278\) 179922. 0.139628
\(279\) 51364.8 0.0395053
\(280\) 0 0
\(281\) 1.13284e6 0.855858 0.427929 0.903812i \(-0.359243\pi\)
0.427929 + 0.903812i \(0.359243\pi\)
\(282\) −21924.5 −0.0164175
\(283\) 411138. 0.305156 0.152578 0.988291i \(-0.451242\pi\)
0.152578 + 0.988291i \(0.451242\pi\)
\(284\) −958094. −0.704875
\(285\) 0 0
\(286\) 84562.2 0.0611310
\(287\) 2.63203e6 1.88619
\(288\) −256068. −0.181918
\(289\) 479901. 0.337993
\(290\) 0 0
\(291\) 1.42253e6 0.984757
\(292\) 1.72686e6 1.18522
\(293\) −314013. −0.213687 −0.106843 0.994276i \(-0.534074\pi\)
−0.106843 + 0.994276i \(0.534074\pi\)
\(294\) −25841.7 −0.0174362
\(295\) 0 0
\(296\) −793115. −0.526147
\(297\) −88209.0 −0.0580259
\(298\) −319529. −0.208435
\(299\) −222161. −0.143711
\(300\) 0 0
\(301\) −1.86652e6 −1.18745
\(302\) −463189. −0.292241
\(303\) −1.41271e6 −0.883988
\(304\) 1.74307e6 1.08176
\(305\) 0 0
\(306\) −120717. −0.0736998
\(307\) −797600. −0.482991 −0.241496 0.970402i \(-0.577638\pi\)
−0.241496 + 0.970402i \(0.577638\pi\)
\(308\) 520439. 0.312603
\(309\) −1.52213e6 −0.906892
\(310\) 0 0
\(311\) −618368. −0.362532 −0.181266 0.983434i \(-0.558020\pi\)
−0.181266 + 0.983434i \(0.558020\pi\)
\(312\) 395190. 0.229837
\(313\) 2.60175e6 1.50108 0.750541 0.660824i \(-0.229793\pi\)
0.750541 + 0.660824i \(0.229793\pi\)
\(314\) −312491. −0.178860
\(315\) 0 0
\(316\) 27224.7 0.0153372
\(317\) 729701. 0.407846 0.203923 0.978987i \(-0.434631\pi\)
0.203923 + 0.978987i \(0.434631\pi\)
\(318\) −86617.0 −0.0480325
\(319\) −6474.61 −0.00356235
\(320\) 0 0
\(321\) 121847. 0.0660011
\(322\) 51849.8 0.0278681
\(323\) 2.63107e6 1.40322
\(324\) −202281. −0.107052
\(325\) 0 0
\(326\) 311648. 0.162413
\(327\) −672628. −0.347861
\(328\) 1.28174e6 0.657833
\(329\) 314304. 0.160089
\(330\) 0 0
\(331\) 2.87129e6 1.44048 0.720239 0.693726i \(-0.244032\pi\)
0.720239 + 0.693726i \(0.244032\pi\)
\(332\) 2.89768e6 1.44280
\(333\) −945611. −0.467307
\(334\) −195616. −0.0959484
\(335\) 0 0
\(336\) 1.14650e6 0.554020
\(337\) −2.16558e6 −1.03872 −0.519361 0.854555i \(-0.673830\pi\)
−0.519361 + 0.854555i \(0.673830\pi\)
\(338\) −50226.3 −0.0239133
\(339\) 1.85323e6 0.875849
\(340\) 0 0
\(341\) −76730.1 −0.0357338
\(342\) −167187. −0.0772927
\(343\) −1.97425e6 −0.906081
\(344\) −908953. −0.414138
\(345\) 0 0
\(346\) 236343. 0.106134
\(347\) −392930. −0.175183 −0.0875913 0.996156i \(-0.527917\pi\)
−0.0875913 + 0.996156i \(0.527917\pi\)
\(348\) −14847.6 −0.00657216
\(349\) 247840. 0.108920 0.0544601 0.998516i \(-0.482656\pi\)
0.0544601 + 0.998516i \(0.482656\pi\)
\(350\) 0 0
\(351\) 471175. 0.204134
\(352\) 382522. 0.164551
\(353\) 2.90847e6 1.24230 0.621151 0.783691i \(-0.286665\pi\)
0.621151 + 0.783691i \(0.286665\pi\)
\(354\) 109738. 0.0465426
\(355\) 0 0
\(356\) −670261. −0.280297
\(357\) 1.73058e6 0.718655
\(358\) −336185. −0.138634
\(359\) 758403. 0.310573 0.155287 0.987869i \(-0.450370\pi\)
0.155287 + 0.987869i \(0.450370\pi\)
\(360\) 0 0
\(361\) 1.16779e6 0.471627
\(362\) 458067. 0.183721
\(363\) 131769. 0.0524864
\(364\) −2.77997e6 −1.09973
\(365\) 0 0
\(366\) −115342. −0.0450073
\(367\) 2.29924e6 0.891085 0.445542 0.895261i \(-0.353011\pi\)
0.445542 + 0.895261i \(0.353011\pi\)
\(368\) −313866. −0.120816
\(369\) 1.52819e6 0.584266
\(370\) 0 0
\(371\) 1.24172e6 0.468370
\(372\) −175958. −0.0659251
\(373\) 19163.9 0.00713203 0.00356601 0.999994i \(-0.498865\pi\)
0.00356601 + 0.999994i \(0.498865\pi\)
\(374\) 180331. 0.0666640
\(375\) 0 0
\(376\) 153059. 0.0558329
\(377\) 34584.7 0.0125323
\(378\) −109967. −0.0395852
\(379\) 346535. 0.123922 0.0619610 0.998079i \(-0.480265\pi\)
0.0619610 + 0.998079i \(0.480265\pi\)
\(380\) 0 0
\(381\) 1.93214e6 0.681909
\(382\) −931073. −0.326457
\(383\) 5.25066e6 1.82901 0.914507 0.404571i \(-0.132579\pi\)
0.914507 + 0.404571i \(0.132579\pi\)
\(384\) 1.16155e6 0.401987
\(385\) 0 0
\(386\) 465578. 0.159047
\(387\) −1.08372e6 −0.367824
\(388\) −4.87309e6 −1.64333
\(389\) −4.34474e6 −1.45576 −0.727880 0.685704i \(-0.759494\pi\)
−0.727880 + 0.685704i \(0.759494\pi\)
\(390\) 0 0
\(391\) −473763. −0.156718
\(392\) 180406. 0.0592974
\(393\) 2.27760e6 0.743869
\(394\) −199771. −0.0648324
\(395\) 0 0
\(396\) 302173. 0.0968318
\(397\) −3.90748e6 −1.24429 −0.622144 0.782903i \(-0.713738\pi\)
−0.622144 + 0.782903i \(0.713738\pi\)
\(398\) −349244. −0.110515
\(399\) 2.39676e6 0.753689
\(400\) 0 0
\(401\) −3.05315e6 −0.948171 −0.474085 0.880479i \(-0.657221\pi\)
−0.474085 + 0.880479i \(0.657221\pi\)
\(402\) −557940. −0.172196
\(403\) 409860. 0.125711
\(404\) 4.83945e6 1.47517
\(405\) 0 0
\(406\) −8071.67 −0.00243023
\(407\) 1.41258e6 0.422695
\(408\) 842753. 0.250640
\(409\) −5.46050e6 −1.61408 −0.807038 0.590499i \(-0.798931\pi\)
−0.807038 + 0.590499i \(0.798931\pi\)
\(410\) 0 0
\(411\) −2.15458e6 −0.629156
\(412\) 5.21428e6 1.51339
\(413\) −1.57318e6 −0.453842
\(414\) 30104.6 0.00863241
\(415\) 0 0
\(416\) −2.04327e6 −0.578886
\(417\) −1.49758e6 −0.421745
\(418\) 249749. 0.0699139
\(419\) 3.02804e6 0.842609 0.421305 0.906919i \(-0.361572\pi\)
0.421305 + 0.906919i \(0.361572\pi\)
\(420\) 0 0
\(421\) −3.31510e6 −0.911572 −0.455786 0.890089i \(-0.650642\pi\)
−0.455786 + 0.890089i \(0.650642\pi\)
\(422\) 426136. 0.116484
\(423\) 182489. 0.0495890
\(424\) 604691. 0.163350
\(425\) 0 0
\(426\) −302413. −0.0807378
\(427\) 1.65351e6 0.438871
\(428\) −417404. −0.110141
\(429\) −703855. −0.184646
\(430\) 0 0
\(431\) −3.02139e6 −0.783455 −0.391728 0.920081i \(-0.628122\pi\)
−0.391728 + 0.920081i \(0.628122\pi\)
\(432\) 665670. 0.171613
\(433\) 739663. 0.189589 0.0947947 0.995497i \(-0.469781\pi\)
0.0947947 + 0.995497i \(0.469781\pi\)
\(434\) −95656.7 −0.0243776
\(435\) 0 0
\(436\) 2.30419e6 0.580499
\(437\) −656138. −0.164358
\(438\) 545067. 0.135758
\(439\) 89371.4 0.0221329 0.0110664 0.999939i \(-0.496477\pi\)
0.0110664 + 0.999939i \(0.496477\pi\)
\(440\) 0 0
\(441\) 215094. 0.0526661
\(442\) −963252. −0.234522
\(443\) −5.06164e6 −1.22541 −0.612706 0.790311i \(-0.709919\pi\)
−0.612706 + 0.790311i \(0.709919\pi\)
\(444\) 3.23933e6 0.779826
\(445\) 0 0
\(446\) 481208. 0.114550
\(447\) 2.65961e6 0.629577
\(448\) −3.59956e6 −0.847334
\(449\) 6.64598e6 1.55576 0.777881 0.628411i \(-0.216294\pi\)
0.777881 + 0.628411i \(0.216294\pi\)
\(450\) 0 0
\(451\) −2.28285e6 −0.528489
\(452\) −6.34850e6 −1.46159
\(453\) 3.85536e6 0.882713
\(454\) 1.25306e6 0.285320
\(455\) 0 0
\(456\) 1.16717e6 0.262858
\(457\) 176313. 0.0394905 0.0197453 0.999805i \(-0.493714\pi\)
0.0197453 + 0.999805i \(0.493714\pi\)
\(458\) −957850. −0.213370
\(459\) 1.00479e6 0.222610
\(460\) 0 0
\(461\) 8.43740e6 1.84908 0.924542 0.381081i \(-0.124448\pi\)
0.924542 + 0.381081i \(0.124448\pi\)
\(462\) 164272. 0.0358062
\(463\) −5.65739e6 −1.22649 −0.613244 0.789893i \(-0.710136\pi\)
−0.613244 + 0.789893i \(0.710136\pi\)
\(464\) 48860.8 0.0105357
\(465\) 0 0
\(466\) 811379. 0.173085
\(467\) 2.54278e6 0.539532 0.269766 0.962926i \(-0.413054\pi\)
0.269766 + 0.962926i \(0.413054\pi\)
\(468\) −1.61408e6 −0.340652
\(469\) 7.99850e6 1.67910
\(470\) 0 0
\(471\) 2.60102e6 0.540247
\(472\) −766107. −0.158283
\(473\) 1.61889e6 0.332710
\(474\) 8593.23 0.00175675
\(475\) 0 0
\(476\) −5.92835e6 −1.19927
\(477\) 720958. 0.145082
\(478\) −667594. −0.133642
\(479\) 6.89136e6 1.37235 0.686177 0.727434i \(-0.259287\pi\)
0.686177 + 0.727434i \(0.259287\pi\)
\(480\) 0 0
\(481\) −7.54540e6 −1.48703
\(482\) −1.30051e6 −0.254973
\(483\) −431573. −0.0841756
\(484\) −451394. −0.0875876
\(485\) 0 0
\(486\) −63848.2 −0.0122619
\(487\) −7.89949e6 −1.50930 −0.754652 0.656125i \(-0.772194\pi\)
−0.754652 + 0.656125i \(0.772194\pi\)
\(488\) 805223. 0.153062
\(489\) −2.59401e6 −0.490568
\(490\) 0 0
\(491\) −2.29000e6 −0.428679 −0.214340 0.976759i \(-0.568760\pi\)
−0.214340 + 0.976759i \(0.568760\pi\)
\(492\) −5.23504e6 −0.975005
\(493\) 73752.6 0.0136666
\(494\) −1.33406e6 −0.245955
\(495\) 0 0
\(496\) 579045. 0.105684
\(497\) 4.33533e6 0.787283
\(498\) 914625. 0.165261
\(499\) −5.96310e6 −1.07206 −0.536032 0.844198i \(-0.680077\pi\)
−0.536032 + 0.844198i \(0.680077\pi\)
\(500\) 0 0
\(501\) 1.62821e6 0.289812
\(502\) −1.68435e6 −0.298314
\(503\) −2.17453e6 −0.383218 −0.191609 0.981471i \(-0.561371\pi\)
−0.191609 + 0.981471i \(0.561371\pi\)
\(504\) 767702. 0.134622
\(505\) 0 0
\(506\) −44971.1 −0.00780831
\(507\) 418060. 0.0722301
\(508\) −6.61883e6 −1.13795
\(509\) −3.33531e6 −0.570613 −0.285307 0.958436i \(-0.592095\pi\)
−0.285307 + 0.958436i \(0.592095\pi\)
\(510\) 0 0
\(511\) −7.81395e6 −1.32379
\(512\) −4.87184e6 −0.821331
\(513\) 1.39159e6 0.233462
\(514\) 99645.9 0.0166361
\(515\) 0 0
\(516\) 3.71245e6 0.613814
\(517\) −272607. −0.0448549
\(518\) 1.76101e6 0.288362
\(519\) −1.96721e6 −0.320577
\(520\) 0 0
\(521\) 9.81327e6 1.58387 0.791935 0.610605i \(-0.209074\pi\)
0.791935 + 0.610605i \(0.209074\pi\)
\(522\) −4686.51 −0.000752789 0
\(523\) −7.96466e6 −1.27325 −0.636624 0.771175i \(-0.719670\pi\)
−0.636624 + 0.771175i \(0.719670\pi\)
\(524\) −7.80227e6 −1.24134
\(525\) 0 0
\(526\) −932190. −0.146906
\(527\) 874036. 0.137089
\(528\) −994397. −0.155230
\(529\) −6.31820e6 −0.981644
\(530\) 0 0
\(531\) −913410. −0.140582
\(532\) −8.21046e6 −1.25773
\(533\) 1.21940e7 1.85921
\(534\) −211561. −0.0321058
\(535\) 0 0
\(536\) 3.89509e6 0.585607
\(537\) 2.79824e6 0.418745
\(538\) 1.35215e6 0.201405
\(539\) −321313. −0.0476383
\(540\) 0 0
\(541\) −5.50616e6 −0.808827 −0.404413 0.914576i \(-0.632524\pi\)
−0.404413 + 0.914576i \(0.632524\pi\)
\(542\) −913303. −0.133542
\(543\) −3.81273e6 −0.554928
\(544\) −4.35733e6 −0.631281
\(545\) 0 0
\(546\) −877470. −0.125965
\(547\) −2.92918e6 −0.418580 −0.209290 0.977854i \(-0.567115\pi\)
−0.209290 + 0.977854i \(0.567115\pi\)
\(548\) 7.38084e6 1.04992
\(549\) 960047. 0.135945
\(550\) 0 0
\(551\) 102144. 0.0143328
\(552\) −210167. −0.0293573
\(553\) −123191. −0.0171303
\(554\) −1.65424e6 −0.228994
\(555\) 0 0
\(556\) 5.13018e6 0.703795
\(557\) −4.05453e6 −0.553736 −0.276868 0.960908i \(-0.589296\pi\)
−0.276868 + 0.960908i \(0.589296\pi\)
\(558\) −55539.4 −0.00755120
\(559\) −8.64745e6 −1.17047
\(560\) 0 0
\(561\) −1.50099e6 −0.201358
\(562\) −1.22491e6 −0.163592
\(563\) 1.65845e6 0.220512 0.110256 0.993903i \(-0.464833\pi\)
0.110256 + 0.993903i \(0.464833\pi\)
\(564\) −625143. −0.0827526
\(565\) 0 0
\(566\) −444553. −0.0583288
\(567\) 915312. 0.119567
\(568\) 2.11121e6 0.274575
\(569\) 1.05475e7 1.36575 0.682873 0.730537i \(-0.260730\pi\)
0.682873 + 0.730537i \(0.260730\pi\)
\(570\) 0 0
\(571\) −1.46769e7 −1.88384 −0.941918 0.335843i \(-0.890979\pi\)
−0.941918 + 0.335843i \(0.890979\pi\)
\(572\) 2.41116e6 0.308131
\(573\) 7.74980e6 0.986062
\(574\) −2.84595e6 −0.360534
\(575\) 0 0
\(576\) −2.08995e6 −0.262470
\(577\) 8.25271e6 1.03195 0.515973 0.856605i \(-0.327430\pi\)
0.515973 + 0.856605i \(0.327430\pi\)
\(578\) −518905. −0.0646053
\(579\) −3.87525e6 −0.480400
\(580\) 0 0
\(581\) −1.31118e7 −1.61147
\(582\) −1.53815e6 −0.188230
\(583\) −1.07699e6 −0.131232
\(584\) −3.80522e6 −0.461687
\(585\) 0 0
\(586\) 339534. 0.0408450
\(587\) −1.01063e7 −1.21059 −0.605296 0.796000i \(-0.706945\pi\)
−0.605296 + 0.796000i \(0.706945\pi\)
\(588\) −736835. −0.0878875
\(589\) 1.21050e6 0.143772
\(590\) 0 0
\(591\) 1.66280e6 0.195826
\(592\) −1.06600e7 −1.25013
\(593\) −1.48922e7 −1.73909 −0.869543 0.493857i \(-0.835587\pi\)
−0.869543 + 0.493857i \(0.835587\pi\)
\(594\) 95378.1 0.0110913
\(595\) 0 0
\(596\) −9.11088e6 −1.05062
\(597\) 2.90694e6 0.333811
\(598\) 240217. 0.0274695
\(599\) 1.59013e7 1.81078 0.905390 0.424580i \(-0.139578\pi\)
0.905390 + 0.424580i \(0.139578\pi\)
\(600\) 0 0
\(601\) −2.84828e6 −0.321660 −0.160830 0.986982i \(-0.551417\pi\)
−0.160830 + 0.986982i \(0.551417\pi\)
\(602\) 2.01822e6 0.226974
\(603\) 4.64402e6 0.520117
\(604\) −1.32071e7 −1.47304
\(605\) 0 0
\(606\) 1.52753e6 0.168969
\(607\) −1.32016e6 −0.145430 −0.0727149 0.997353i \(-0.523166\pi\)
−0.0727149 + 0.997353i \(0.523166\pi\)
\(608\) −6.03467e6 −0.662056
\(609\) 67184.6 0.00734052
\(610\) 0 0
\(611\) 1.45615e6 0.157799
\(612\) −3.44207e6 −0.371485
\(613\) 978527. 0.105177 0.0525886 0.998616i \(-0.483253\pi\)
0.0525886 + 0.998616i \(0.483253\pi\)
\(614\) 862424. 0.0923209
\(615\) 0 0
\(616\) −1.14681e6 −0.121770
\(617\) −1.29643e7 −1.37100 −0.685500 0.728073i \(-0.740416\pi\)
−0.685500 + 0.728073i \(0.740416\pi\)
\(618\) 1.64584e6 0.173347
\(619\) 1.27406e7 1.33648 0.668240 0.743946i \(-0.267048\pi\)
0.668240 + 0.743946i \(0.267048\pi\)
\(620\) 0 0
\(621\) −250576. −0.0260742
\(622\) 668626. 0.0692958
\(623\) 3.03289e6 0.313067
\(624\) 5.31165e6 0.546095
\(625\) 0 0
\(626\) −2.81320e6 −0.286923
\(627\) −2.07879e6 −0.211175
\(628\) −8.91020e6 −0.901547
\(629\) −1.60908e7 −1.62162
\(630\) 0 0
\(631\) −1.93988e7 −1.93955 −0.969775 0.243999i \(-0.921541\pi\)
−0.969775 + 0.243999i \(0.921541\pi\)
\(632\) −59991.1 −0.00597440
\(633\) −3.54695e6 −0.351840
\(634\) −789006. −0.0779574
\(635\) 0 0
\(636\) −2.46975e6 −0.242108
\(637\) 1.71632e6 0.167590
\(638\) 7000.83 0.000680923 0
\(639\) 2.51714e6 0.243868
\(640\) 0 0
\(641\) 6.77844e6 0.651605 0.325803 0.945438i \(-0.394366\pi\)
0.325803 + 0.945438i \(0.394366\pi\)
\(642\) −131750. −0.0126157
\(643\) 1.74257e7 1.66212 0.831060 0.556182i \(-0.187734\pi\)
0.831060 + 0.556182i \(0.187734\pi\)
\(644\) 1.47842e6 0.140470
\(645\) 0 0
\(646\) −2.84491e6 −0.268217
\(647\) −792517. −0.0744300 −0.0372150 0.999307i \(-0.511849\pi\)
−0.0372150 + 0.999307i \(0.511849\pi\)
\(648\) 445737. 0.0417005
\(649\) 1.36448e6 0.127161
\(650\) 0 0
\(651\) 796200. 0.0736325
\(652\) 8.88616e6 0.818644
\(653\) 9.09663e6 0.834829 0.417414 0.908716i \(-0.362936\pi\)
0.417414 + 0.908716i \(0.362936\pi\)
\(654\) 727295. 0.0664915
\(655\) 0 0
\(656\) 1.72275e7 1.56302
\(657\) −4.53687e6 −0.410056
\(658\) −339849. −0.0306000
\(659\) −8.15992e6 −0.731935 −0.365968 0.930628i \(-0.619262\pi\)
−0.365968 + 0.930628i \(0.619262\pi\)
\(660\) 0 0
\(661\) −1.85172e6 −0.164843 −0.0824216 0.996598i \(-0.526265\pi\)
−0.0824216 + 0.996598i \(0.526265\pi\)
\(662\) −3.10465e6 −0.275339
\(663\) 8.01765e6 0.708375
\(664\) −6.38518e6 −0.562022
\(665\) 0 0
\(666\) 1.02246e6 0.0893229
\(667\) −18392.5 −0.00160076
\(668\) −5.57768e6 −0.483629
\(669\) −4.00534e6 −0.345998
\(670\) 0 0
\(671\) −1.43414e6 −0.122966
\(672\) −3.96929e6 −0.339070
\(673\) −1.42740e6 −0.121481 −0.0607405 0.998154i \(-0.519346\pi\)
−0.0607405 + 0.998154i \(0.519346\pi\)
\(674\) 2.34158e6 0.198545
\(675\) 0 0
\(676\) −1.43213e6 −0.120535
\(677\) −941868. −0.0789802 −0.0394901 0.999220i \(-0.512573\pi\)
−0.0394901 + 0.999220i \(0.512573\pi\)
\(678\) −2.00385e6 −0.167413
\(679\) 2.20505e7 1.83545
\(680\) 0 0
\(681\) −1.04299e7 −0.861809
\(682\) 82966.3 0.00683031
\(683\) −9.56089e6 −0.784236 −0.392118 0.919915i \(-0.628257\pi\)
−0.392118 + 0.919915i \(0.628257\pi\)
\(684\) −4.76709e6 −0.389595
\(685\) 0 0
\(686\) 2.13471e6 0.173192
\(687\) 7.97268e6 0.644484
\(688\) −1.22170e7 −0.983996
\(689\) 5.75281e6 0.461670
\(690\) 0 0
\(691\) −1.53765e7 −1.22507 −0.612536 0.790443i \(-0.709851\pi\)
−0.612536 + 0.790443i \(0.709851\pi\)
\(692\) 6.73897e6 0.534968
\(693\) −1.36732e6 −0.108152
\(694\) 424865. 0.0334851
\(695\) 0 0
\(696\) 32717.5 0.00256010
\(697\) 2.60040e7 2.02749
\(698\) −267983. −0.0208194
\(699\) −6.75352e6 −0.522802
\(700\) 0 0
\(701\) 1.72261e7 1.32401 0.662006 0.749499i \(-0.269705\pi\)
0.662006 + 0.749499i \(0.269705\pi\)
\(702\) −509470. −0.0390190
\(703\) −2.22849e7 −1.70068
\(704\) 3.12202e6 0.237413
\(705\) 0 0
\(706\) −3.14485e6 −0.237459
\(707\) −2.18983e7 −1.64764
\(708\) 3.12902e6 0.234599
\(709\) 2.47878e7 1.85192 0.925959 0.377625i \(-0.123259\pi\)
0.925959 + 0.377625i \(0.123259\pi\)
\(710\) 0 0
\(711\) −71525.9 −0.00530627
\(712\) 1.47695e6 0.109186
\(713\) −217968. −0.0160572
\(714\) −1.87123e6 −0.137367
\(715\) 0 0
\(716\) −9.58579e6 −0.698788
\(717\) 5.55673e6 0.403665
\(718\) −820042. −0.0593642
\(719\) 7.44502e6 0.537086 0.268543 0.963268i \(-0.413458\pi\)
0.268543 + 0.963268i \(0.413458\pi\)
\(720\) 0 0
\(721\) −2.35944e7 −1.69032
\(722\) −1.26271e6 −0.0901486
\(723\) 1.08248e7 0.770146
\(724\) 1.30611e7 0.926046
\(725\) 0 0
\(726\) −142478. −0.0100325
\(727\) −3.56658e6 −0.250274 −0.125137 0.992139i \(-0.539937\pi\)
−0.125137 + 0.992139i \(0.539937\pi\)
\(728\) 6.12580e6 0.428385
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −1.84409e7 −1.27641
\(732\) −3.28879e6 −0.226860
\(733\) −1.38091e7 −0.949305 −0.474652 0.880173i \(-0.657426\pi\)
−0.474652 + 0.880173i \(0.657426\pi\)
\(734\) −2.48611e6 −0.170326
\(735\) 0 0
\(736\) 1.08663e6 0.0739416
\(737\) −6.93737e6 −0.470463
\(738\) −1.65239e6 −0.111679
\(739\) −7.18607e6 −0.484039 −0.242020 0.970271i \(-0.577810\pi\)
−0.242020 + 0.970271i \(0.577810\pi\)
\(740\) 0 0
\(741\) 1.11040e7 0.742908
\(742\) −1.34264e6 −0.0895261
\(743\) −2.18843e6 −0.145432 −0.0727161 0.997353i \(-0.523167\pi\)
−0.0727161 + 0.997353i \(0.523167\pi\)
\(744\) 387732. 0.0256803
\(745\) 0 0
\(746\) −20721.5 −0.00136324
\(747\) −7.61289e6 −0.499170
\(748\) 5.14186e6 0.336021
\(749\) 1.88873e6 0.123017
\(750\) 0 0
\(751\) −1.41771e6 −0.0917250 −0.0458625 0.998948i \(-0.514604\pi\)
−0.0458625 + 0.998948i \(0.514604\pi\)
\(752\) 2.05723e6 0.132660
\(753\) 1.40197e7 0.901056
\(754\) −37395.5 −0.00239547
\(755\) 0 0
\(756\) −3.13554e6 −0.199530
\(757\) 2.05513e7 1.30347 0.651734 0.758448i \(-0.274042\pi\)
0.651734 + 0.758448i \(0.274042\pi\)
\(758\) −374699. −0.0236870
\(759\) 374318. 0.0235850
\(760\) 0 0
\(761\) −1.30587e7 −0.817409 −0.408705 0.912667i \(-0.634019\pi\)
−0.408705 + 0.912667i \(0.634019\pi\)
\(762\) −2.08917e6 −0.130343
\(763\) −1.04263e7 −0.648366
\(764\) −2.65481e7 −1.64551
\(765\) 0 0
\(766\) −5.67740e6 −0.349605
\(767\) −7.28846e6 −0.447350
\(768\) 6.17497e6 0.377774
\(769\) 1.82667e7 1.11389 0.556946 0.830549i \(-0.311973\pi\)
0.556946 + 0.830549i \(0.311973\pi\)
\(770\) 0 0
\(771\) −829404. −0.0502493
\(772\) 1.32752e7 0.801676
\(773\) 1.49182e7 0.897981 0.448990 0.893537i \(-0.351784\pi\)
0.448990 + 0.893537i \(0.351784\pi\)
\(774\) 1.17180e6 0.0703074
\(775\) 0 0
\(776\) 1.07381e7 0.640138
\(777\) −1.46578e7 −0.870996
\(778\) 4.69786e6 0.278260
\(779\) 3.60143e7 2.12633
\(780\) 0 0
\(781\) −3.76018e6 −0.220587
\(782\) 512268. 0.0299558
\(783\) 39008.2 0.00227380
\(784\) 2.42479e6 0.140891
\(785\) 0 0
\(786\) −2.46271e6 −0.142186
\(787\) 1.60565e7 0.924087 0.462043 0.886857i \(-0.347116\pi\)
0.462043 + 0.886857i \(0.347116\pi\)
\(788\) −5.69616e6 −0.326788
\(789\) 7.75910e6 0.443730
\(790\) 0 0
\(791\) 2.87266e7 1.63246
\(792\) −665854. −0.0377195
\(793\) 7.66060e6 0.432593
\(794\) 4.22506e6 0.237838
\(795\) 0 0
\(796\) −9.95816e6 −0.557053
\(797\) −2.10741e7 −1.17518 −0.587588 0.809160i \(-0.699922\pi\)
−0.587588 + 0.809160i \(0.699922\pi\)
\(798\) −2.59155e6 −0.144063
\(799\) 3.10528e6 0.172081
\(800\) 0 0
\(801\) 1.76093e6 0.0969755
\(802\) 3.30129e6 0.181237
\(803\) 6.77730e6 0.370909
\(804\) −1.59088e7 −0.867955
\(805\) 0 0
\(806\) −443171. −0.0240289
\(807\) −1.12547e7 −0.608344
\(808\) −1.06640e7 −0.574633
\(809\) 2.38181e7 1.27949 0.639744 0.768588i \(-0.279040\pi\)
0.639744 + 0.768588i \(0.279040\pi\)
\(810\) 0 0
\(811\) 2.93046e7 1.56453 0.782264 0.622947i \(-0.214065\pi\)
0.782264 + 0.622947i \(0.214065\pi\)
\(812\) −230151. −0.0122496
\(813\) 7.60189e6 0.403362
\(814\) −1.52739e6 −0.0807956
\(815\) 0 0
\(816\) 1.13272e7 0.595522
\(817\) −2.55397e7 −1.33863
\(818\) 5.90430e6 0.308521
\(819\) 7.30364e6 0.380478
\(820\) 0 0
\(821\) −3.14594e6 −0.162889 −0.0814446 0.996678i \(-0.525953\pi\)
−0.0814446 + 0.996678i \(0.525953\pi\)
\(822\) 2.32969e6 0.120259
\(823\) −2.44990e7 −1.26081 −0.630404 0.776267i \(-0.717111\pi\)
−0.630404 + 0.776267i \(0.717111\pi\)
\(824\) −1.14899e7 −0.589522
\(825\) 0 0
\(826\) 1.70104e6 0.0867492
\(827\) −1.34163e7 −0.682135 −0.341067 0.940039i \(-0.610788\pi\)
−0.341067 + 0.940039i \(0.610788\pi\)
\(828\) 858386. 0.0435118
\(829\) 1.42349e7 0.719394 0.359697 0.933069i \(-0.382880\pi\)
0.359697 + 0.933069i \(0.382880\pi\)
\(830\) 0 0
\(831\) 1.37691e7 0.691675
\(832\) −1.66765e7 −0.835213
\(833\) 3.66009e6 0.182759
\(834\) 1.61929e6 0.0806140
\(835\) 0 0
\(836\) 7.12121e6 0.352402
\(837\) 462283. 0.0228084
\(838\) −3.27414e6 −0.161060
\(839\) 1.15794e7 0.567913 0.283957 0.958837i \(-0.408353\pi\)
0.283957 + 0.958837i \(0.408353\pi\)
\(840\) 0 0
\(841\) −2.05083e7 −0.999860
\(842\) 3.58453e6 0.174242
\(843\) 1.01955e7 0.494130
\(844\) 1.21506e7 0.587140
\(845\) 0 0
\(846\) −197320. −0.00947865
\(847\) 2.04254e6 0.0978276
\(848\) 8.12749e6 0.388121
\(849\) 3.70025e6 0.176182
\(850\) 0 0
\(851\) 4.01273e6 0.189940
\(852\) −8.62285e6 −0.406960
\(853\) −6.61595e6 −0.311329 −0.155664 0.987810i \(-0.549752\pi\)
−0.155664 + 0.987810i \(0.549752\pi\)
\(854\) −1.78790e6 −0.0838876
\(855\) 0 0
\(856\) 919772. 0.0429038
\(857\) 9.97147e6 0.463775 0.231887 0.972743i \(-0.425510\pi\)
0.231887 + 0.972743i \(0.425510\pi\)
\(858\) 761060. 0.0352940
\(859\) −3.79757e7 −1.75599 −0.877996 0.478667i \(-0.841120\pi\)
−0.877996 + 0.478667i \(0.841120\pi\)
\(860\) 0 0
\(861\) 2.36883e7 1.08899
\(862\) 3.26695e6 0.149753
\(863\) −8.18531e6 −0.374118 −0.187059 0.982349i \(-0.559896\pi\)
−0.187059 + 0.982349i \(0.559896\pi\)
\(864\) −2.30462e6 −0.105030
\(865\) 0 0
\(866\) −799779. −0.0362389
\(867\) 4.31911e6 0.195140
\(868\) −2.72750e6 −0.122876
\(869\) 106847. 0.00479970
\(870\) 0 0
\(871\) 3.70565e7 1.65508
\(872\) −5.07740e6 −0.226126
\(873\) 1.28028e7 0.568550
\(874\) 709465. 0.0314161
\(875\) 0 0
\(876\) 1.55417e7 0.684288
\(877\) 3.02517e7 1.32816 0.664081 0.747660i \(-0.268823\pi\)
0.664081 + 0.747660i \(0.268823\pi\)
\(878\) −96635.0 −0.00423056
\(879\) −2.82611e6 −0.123372
\(880\) 0 0
\(881\) 6.11414e6 0.265397 0.132698 0.991156i \(-0.457636\pi\)
0.132698 + 0.991156i \(0.457636\pi\)
\(882\) −232575. −0.0100668
\(883\) 7.72672e6 0.333498 0.166749 0.985999i \(-0.446673\pi\)
0.166749 + 0.985999i \(0.446673\pi\)
\(884\) −2.74656e7 −1.18211
\(885\) 0 0
\(886\) 5.47302e6 0.234230
\(887\) 2.48274e7 1.05955 0.529775 0.848138i \(-0.322276\pi\)
0.529775 + 0.848138i \(0.322276\pi\)
\(888\) −7.13803e6 −0.303771
\(889\) 2.99499e7 1.27099
\(890\) 0 0
\(891\) −793881. −0.0335013
\(892\) 1.37209e7 0.577391
\(893\) 4.30065e6 0.180470
\(894\) −2.87576e6 −0.120340
\(895\) 0 0
\(896\) 1.80051e7 0.749249
\(897\) −1.99945e6 −0.0829715
\(898\) −7.18613e6 −0.297375
\(899\) 33931.9 0.00140026
\(900\) 0 0
\(901\) 1.22680e7 0.503457
\(902\) 2.46838e6 0.101017
\(903\) −1.67987e7 −0.685575
\(904\) 1.39893e7 0.569342
\(905\) 0 0
\(906\) −4.16870e6 −0.168725
\(907\) 3.64608e6 0.147166 0.0735831 0.997289i \(-0.476557\pi\)
0.0735831 + 0.997289i \(0.476557\pi\)
\(908\) 3.57291e7 1.43816
\(909\) −1.27144e7 −0.510371
\(910\) 0 0
\(911\) −1.15898e7 −0.462678 −0.231339 0.972873i \(-0.574311\pi\)
−0.231339 + 0.972873i \(0.574311\pi\)
\(912\) 1.56876e7 0.624554
\(913\) 1.13723e7 0.451516
\(914\) −190642. −0.00754838
\(915\) 0 0
\(916\) −2.73116e7 −1.07550
\(917\) 3.53048e7 1.38647
\(918\) −1.08646e6 −0.0425506
\(919\) −3.48544e7 −1.36135 −0.680674 0.732586i \(-0.738313\pi\)
−0.680674 + 0.732586i \(0.738313\pi\)
\(920\) 0 0
\(921\) −7.17840e6 −0.278855
\(922\) −9.12314e6 −0.353441
\(923\) 2.00853e7 0.776021
\(924\) 4.68395e6 0.180481
\(925\) 0 0
\(926\) 6.11719e6 0.234436
\(927\) −1.36992e7 −0.523594
\(928\) −169161. −0.00644806
\(929\) −4.11213e7 −1.56325 −0.781624 0.623750i \(-0.785608\pi\)
−0.781624 + 0.623750i \(0.785608\pi\)
\(930\) 0 0
\(931\) 5.06903e6 0.191669
\(932\) 2.31352e7 0.872436
\(933\) −5.56532e6 −0.209308
\(934\) −2.74945e6 −0.103128
\(935\) 0 0
\(936\) 3.55671e6 0.132696
\(937\) 1.53222e6 0.0570126 0.0285063 0.999594i \(-0.490925\pi\)
0.0285063 + 0.999594i \(0.490925\pi\)
\(938\) −8.64857e6 −0.320950
\(939\) 2.34157e7 0.866650
\(940\) 0 0
\(941\) −1.91551e7 −0.705196 −0.352598 0.935775i \(-0.614702\pi\)
−0.352598 + 0.935775i \(0.614702\pi\)
\(942\) −2.81242e6 −0.103265
\(943\) −6.48491e6 −0.237479
\(944\) −1.02970e7 −0.376082
\(945\) 0 0
\(946\) −1.75047e6 −0.0635955
\(947\) −2.83184e7 −1.02611 −0.513054 0.858356i \(-0.671486\pi\)
−0.513054 + 0.858356i \(0.671486\pi\)
\(948\) 245023. 0.00885494
\(949\) −3.62015e7 −1.30485
\(950\) 0 0
\(951\) 6.56731e6 0.235470
\(952\) 1.30634e7 0.467159
\(953\) 5.47682e7 1.95342 0.976711 0.214559i \(-0.0688313\pi\)
0.976711 + 0.214559i \(0.0688313\pi\)
\(954\) −779553. −0.0277316
\(955\) 0 0
\(956\) −1.90354e7 −0.673624
\(957\) −58271.5 −0.00205673
\(958\) −7.45145e6 −0.262318
\(959\) −3.33979e7 −1.17266
\(960\) 0 0
\(961\) −2.82270e7 −0.985954
\(962\) 8.15865e6 0.284237
\(963\) 1.09662e6 0.0381058
\(964\) −3.70819e7 −1.28520
\(965\) 0 0
\(966\) 466648. 0.0160897
\(967\) −2.29671e7 −0.789843 −0.394922 0.918715i \(-0.629228\pi\)
−0.394922 + 0.918715i \(0.629228\pi\)
\(968\) 994671. 0.0341186
\(969\) 2.36796e7 0.810149
\(970\) 0 0
\(971\) 1.46281e7 0.497898 0.248949 0.968517i \(-0.419915\pi\)
0.248949 + 0.968517i \(0.419915\pi\)
\(972\) −1.82053e6 −0.0618062
\(973\) −2.32138e7 −0.786076
\(974\) 8.54152e6 0.288495
\(975\) 0 0
\(976\) 1.08228e7 0.363676
\(977\) 1.54714e7 0.518552 0.259276 0.965803i \(-0.416516\pi\)
0.259276 + 0.965803i \(0.416516\pi\)
\(978\) 2.80483e6 0.0937691
\(979\) −2.63053e6 −0.0877176
\(980\) 0 0
\(981\) −6.05365e6 −0.200838
\(982\) 2.47612e6 0.0819395
\(983\) −2.02694e7 −0.669047 −0.334524 0.942387i \(-0.608575\pi\)
−0.334524 + 0.942387i \(0.608575\pi\)
\(984\) 1.15357e7 0.379800
\(985\) 0 0
\(986\) −79746.8 −0.00261229
\(987\) 2.82874e6 0.0924272
\(988\) −3.80385e7 −1.23974
\(989\) 4.59881e6 0.149505
\(990\) 0 0
\(991\) −4.43452e7 −1.43437 −0.717187 0.696881i \(-0.754570\pi\)
−0.717187 + 0.696881i \(0.754570\pi\)
\(992\) −2.00471e6 −0.0646803
\(993\) 2.58416e7 0.831661
\(994\) −4.68768e6 −0.150484
\(995\) 0 0
\(996\) 2.60791e7 0.832998
\(997\) −2.32293e7 −0.740112 −0.370056 0.929009i \(-0.620662\pi\)
−0.370056 + 0.929009i \(0.620662\pi\)
\(998\) 6.44775e6 0.204919
\(999\) −8.51050e6 −0.269800
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.h.1.2 3
5.4 even 2 165.6.a.d.1.2 3
15.14 odd 2 495.6.a.c.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.d.1.2 3 5.4 even 2
495.6.a.c.1.2 3 15.14 odd 2
825.6.a.h.1.2 3 1.1 even 1 trivial