Properties

Label 825.6.a.c.1.2
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.62772 q^{2} -9.00000 q^{3} -18.8397 q^{4} +32.6495 q^{6} -251.081 q^{7} +184.432 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-3.62772 q^{2} -9.00000 q^{3} -18.8397 q^{4} +32.6495 q^{6} -251.081 q^{7} +184.432 q^{8} +81.0000 q^{9} -121.000 q^{11} +169.557 q^{12} +277.549 q^{13} +910.853 q^{14} -66.1983 q^{16} -704.489 q^{17} -293.845 q^{18} -2861.18 q^{19} +2259.73 q^{21} +438.954 q^{22} +1066.85 q^{23} -1659.89 q^{24} -1006.87 q^{26} -729.000 q^{27} +4730.29 q^{28} -3937.44 q^{29} -644.350 q^{31} -5661.67 q^{32} +1089.00 q^{33} +2555.69 q^{34} -1526.01 q^{36} +9042.34 q^{37} +10379.6 q^{38} -2497.94 q^{39} +18219.0 q^{41} -8197.68 q^{42} +4054.54 q^{43} +2279.60 q^{44} -3870.24 q^{46} -20750.8 q^{47} +595.785 q^{48} +46234.9 q^{49} +6340.40 q^{51} -5228.92 q^{52} +26485.9 q^{53} +2644.61 q^{54} -46307.4 q^{56} +25750.7 q^{57} +14283.9 q^{58} +4293.12 q^{59} -6831.76 q^{61} +2337.52 q^{62} -20337.6 q^{63} +22657.3 q^{64} -3950.59 q^{66} +56749.5 q^{67} +13272.3 q^{68} -9601.68 q^{69} +3187.09 q^{71} +14939.0 q^{72} +7397.14 q^{73} -32803.1 q^{74} +53903.7 q^{76} +30380.9 q^{77} +9061.82 q^{78} +24393.7 q^{79} +6561.00 q^{81} -66093.5 q^{82} -102795. q^{83} -42572.6 q^{84} -14708.7 q^{86} +35437.0 q^{87} -22316.3 q^{88} +49599.4 q^{89} -69687.4 q^{91} -20099.1 q^{92} +5799.15 q^{93} +75278.1 q^{94} +50955.1 q^{96} +92279.5 q^{97} -167727. q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 13 q^{2} - 18 q^{3} + 37 q^{4} + 117 q^{6} - 146 q^{7} - 39 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 13 q^{2} - 18 q^{3} + 37 q^{4} + 117 q^{6} - 146 q^{7} - 39 q^{8} + 162 q^{9} - 242 q^{11} - 333 q^{12} + 130 q^{13} - 74 q^{14} + 241 q^{16} + 728 q^{17} - 1053 q^{18} - 828 q^{19} + 1314 q^{21} + 1573 q^{22} + 238 q^{23} + 351 q^{24} + 376 q^{26} - 1458 q^{27} + 10598 q^{28} + 696 q^{29} - 10480 q^{31} - 1391 q^{32} + 2178 q^{33} - 10870 q^{34} + 2997 q^{36} + 1908 q^{37} - 8676 q^{38} - 1170 q^{39} + 36484 q^{41} + 666 q^{42} - 9768 q^{43} - 4477 q^{44} + 3898 q^{46} - 43742 q^{47} - 2169 q^{48} + 40470 q^{49} - 6552 q^{51} - 13468 q^{52} + 12174 q^{53} + 9477 q^{54} - 69786 q^{56} + 7452 q^{57} - 29142 q^{58} - 2788 q^{59} - 25302 q^{61} + 94520 q^{62} - 11826 q^{63} - 27199 q^{64} - 14157 q^{66} + 40520 q^{67} + 93262 q^{68} - 2142 q^{69} + 31386 q^{71} - 3159 q^{72} + 46780 q^{73} + 34062 q^{74} + 167436 q^{76} + 17666 q^{77} - 3384 q^{78} - 16850 q^{79} + 13122 q^{81} - 237278 q^{82} - 79440 q^{83} - 95382 q^{84} + 114840 q^{86} - 6264 q^{87} + 4719 q^{88} - 54204 q^{89} - 85192 q^{91} - 66382 q^{92} + 94320 q^{93} + 290758 q^{94} + 12519 q^{96} + 241568 q^{97} - 113697 q^{98} - 19602 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.62772 −0.641296 −0.320648 0.947198i \(-0.603901\pi\)
−0.320648 + 0.947198i \(0.603901\pi\)
\(3\) −9.00000 −0.577350
\(4\) −18.8397 −0.588739
\(5\) 0 0
\(6\) 32.6495 0.370252
\(7\) −251.081 −1.93673 −0.968366 0.249534i \(-0.919722\pi\)
−0.968366 + 0.249534i \(0.919722\pi\)
\(8\) 184.432 1.01885
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −121.000 −0.301511
\(12\) 169.557 0.339909
\(13\) 277.549 0.455492 0.227746 0.973721i \(-0.426864\pi\)
0.227746 + 0.973721i \(0.426864\pi\)
\(14\) 910.853 1.24202
\(15\) 0 0
\(16\) −66.1983 −0.0646468
\(17\) −704.489 −0.591224 −0.295612 0.955308i \(-0.595523\pi\)
−0.295612 + 0.955308i \(0.595523\pi\)
\(18\) −293.845 −0.213765
\(19\) −2861.18 −1.81828 −0.909142 0.416486i \(-0.863261\pi\)
−0.909142 + 0.416486i \(0.863261\pi\)
\(20\) 0 0
\(21\) 2259.73 1.11817
\(22\) 438.954 0.193358
\(23\) 1066.85 0.420518 0.210259 0.977646i \(-0.432569\pi\)
0.210259 + 0.977646i \(0.432569\pi\)
\(24\) −1659.89 −0.588235
\(25\) 0 0
\(26\) −1006.87 −0.292105
\(27\) −729.000 −0.192450
\(28\) 4730.29 1.14023
\(29\) −3937.44 −0.869399 −0.434700 0.900575i \(-0.643145\pi\)
−0.434700 + 0.900575i \(0.643145\pi\)
\(30\) 0 0
\(31\) −644.350 −0.120425 −0.0602126 0.998186i \(-0.519178\pi\)
−0.0602126 + 0.998186i \(0.519178\pi\)
\(32\) −5661.67 −0.977395
\(33\) 1089.00 0.174078
\(34\) 2555.69 0.379149
\(35\) 0 0
\(36\) −1526.01 −0.196246
\(37\) 9042.34 1.08587 0.542934 0.839776i \(-0.317314\pi\)
0.542934 + 0.839776i \(0.317314\pi\)
\(38\) 10379.6 1.16606
\(39\) −2497.94 −0.262979
\(40\) 0 0
\(41\) 18219.0 1.69264 0.846322 0.532672i \(-0.178812\pi\)
0.846322 + 0.532672i \(0.178812\pi\)
\(42\) −8197.68 −0.717080
\(43\) 4054.54 0.334403 0.167202 0.985923i \(-0.446527\pi\)
0.167202 + 0.985923i \(0.446527\pi\)
\(44\) 2279.60 0.177512
\(45\) 0 0
\(46\) −3870.24 −0.269677
\(47\) −20750.8 −1.37022 −0.685110 0.728439i \(-0.740246\pi\)
−0.685110 + 0.728439i \(0.740246\pi\)
\(48\) 595.785 0.0373238
\(49\) 46234.9 2.75093
\(50\) 0 0
\(51\) 6340.40 0.341343
\(52\) −5228.92 −0.268166
\(53\) 26485.9 1.29517 0.647583 0.761995i \(-0.275780\pi\)
0.647583 + 0.761995i \(0.275780\pi\)
\(54\) 2644.61 0.123417
\(55\) 0 0
\(56\) −46307.4 −1.97324
\(57\) 25750.7 1.04979
\(58\) 14283.9 0.557542
\(59\) 4293.12 0.160562 0.0802810 0.996772i \(-0.474418\pi\)
0.0802810 + 0.996772i \(0.474418\pi\)
\(60\) 0 0
\(61\) −6831.76 −0.235076 −0.117538 0.993068i \(-0.537500\pi\)
−0.117538 + 0.993068i \(0.537500\pi\)
\(62\) 2337.52 0.0772282
\(63\) −20337.6 −0.645577
\(64\) 22657.3 0.691446
\(65\) 0 0
\(66\) −3950.59 −0.111635
\(67\) 56749.5 1.54445 0.772227 0.635347i \(-0.219143\pi\)
0.772227 + 0.635347i \(0.219143\pi\)
\(68\) 13272.3 0.348077
\(69\) −9601.68 −0.242786
\(70\) 0 0
\(71\) 3187.09 0.0750323 0.0375161 0.999296i \(-0.488055\pi\)
0.0375161 + 0.999296i \(0.488055\pi\)
\(72\) 14939.0 0.339617
\(73\) 7397.14 0.162464 0.0812319 0.996695i \(-0.474115\pi\)
0.0812319 + 0.996695i \(0.474115\pi\)
\(74\) −32803.1 −0.696362
\(75\) 0 0
\(76\) 53903.7 1.07050
\(77\) 30380.9 0.583947
\(78\) 9061.82 0.168647
\(79\) 24393.7 0.439754 0.219877 0.975528i \(-0.429434\pi\)
0.219877 + 0.975528i \(0.429434\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −66093.5 −1.08549
\(83\) −102795. −1.63786 −0.818932 0.573890i \(-0.805434\pi\)
−0.818932 + 0.573890i \(0.805434\pi\)
\(84\) −42572.6 −0.658312
\(85\) 0 0
\(86\) −14708.7 −0.214451
\(87\) 35437.0 0.501948
\(88\) −22316.3 −0.307196
\(89\) 49599.4 0.663745 0.331873 0.943324i \(-0.392320\pi\)
0.331873 + 0.943324i \(0.392320\pi\)
\(90\) 0 0
\(91\) −69687.4 −0.882166
\(92\) −20099.1 −0.247576
\(93\) 5799.15 0.0695275
\(94\) 75278.1 0.878717
\(95\) 0 0
\(96\) 50955.1 0.564299
\(97\) 92279.5 0.995808 0.497904 0.867232i \(-0.334103\pi\)
0.497904 + 0.867232i \(0.334103\pi\)
\(98\) −167727. −1.76416
\(99\) −9801.00 −0.100504
\(100\) 0 0
\(101\) −35546.1 −0.346728 −0.173364 0.984858i \(-0.555464\pi\)
−0.173364 + 0.984858i \(0.555464\pi\)
\(102\) −23001.2 −0.218902
\(103\) 59876.8 0.556116 0.278058 0.960564i \(-0.410309\pi\)
0.278058 + 0.960564i \(0.410309\pi\)
\(104\) 51188.9 0.464079
\(105\) 0 0
\(106\) −96083.5 −0.830586
\(107\) −89253.8 −0.753646 −0.376823 0.926285i \(-0.622983\pi\)
−0.376823 + 0.926285i \(0.622983\pi\)
\(108\) 13734.1 0.113303
\(109\) 22796.0 0.183777 0.0918887 0.995769i \(-0.470710\pi\)
0.0918887 + 0.995769i \(0.470710\pi\)
\(110\) 0 0
\(111\) −81381.1 −0.626926
\(112\) 16621.2 0.125203
\(113\) 166064. 1.22343 0.611717 0.791077i \(-0.290479\pi\)
0.611717 + 0.791077i \(0.290479\pi\)
\(114\) −93416.1 −0.673224
\(115\) 0 0
\(116\) 74180.1 0.511850
\(117\) 22481.5 0.151831
\(118\) −15574.2 −0.102968
\(119\) 176884. 1.14504
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) 24783.7 0.150753
\(123\) −163971. −0.977248
\(124\) 12139.3 0.0708991
\(125\) 0 0
\(126\) 73779.1 0.414006
\(127\) −159190. −0.875802 −0.437901 0.899023i \(-0.644278\pi\)
−0.437901 + 0.899023i \(0.644278\pi\)
\(128\) 98979.2 0.533973
\(129\) −36490.9 −0.193068
\(130\) 0 0
\(131\) 232174. 1.18205 0.591025 0.806654i \(-0.298724\pi\)
0.591025 + 0.806654i \(0.298724\pi\)
\(132\) −20516.4 −0.102486
\(133\) 718390. 3.52153
\(134\) −205871. −0.990452
\(135\) 0 0
\(136\) −129930. −0.602369
\(137\) −68205.4 −0.310468 −0.155234 0.987878i \(-0.549613\pi\)
−0.155234 + 0.987878i \(0.549613\pi\)
\(138\) 34832.2 0.155698
\(139\) 298162. 1.30893 0.654464 0.756093i \(-0.272894\pi\)
0.654464 + 0.756093i \(0.272894\pi\)
\(140\) 0 0
\(141\) 186757. 0.791097
\(142\) −11561.9 −0.0481179
\(143\) −33583.4 −0.137336
\(144\) −5362.06 −0.0215489
\(145\) 0 0
\(146\) −26834.7 −0.104187
\(147\) −416114. −1.58825
\(148\) −170355. −0.639293
\(149\) 83131.5 0.306761 0.153380 0.988167i \(-0.450984\pi\)
0.153380 + 0.988167i \(0.450984\pi\)
\(150\) 0 0
\(151\) 167598. 0.598171 0.299085 0.954226i \(-0.403318\pi\)
0.299085 + 0.954226i \(0.403318\pi\)
\(152\) −527694. −1.85256
\(153\) −57063.6 −0.197075
\(154\) −110213. −0.374483
\(155\) 0 0
\(156\) 47060.3 0.154826
\(157\) 63259.7 0.204823 0.102411 0.994742i \(-0.467344\pi\)
0.102411 + 0.994742i \(0.467344\pi\)
\(158\) −88493.4 −0.282012
\(159\) −238373. −0.747765
\(160\) 0 0
\(161\) −267867. −0.814431
\(162\) −23801.5 −0.0712551
\(163\) −237727. −0.700825 −0.350412 0.936596i \(-0.613959\pi\)
−0.350412 + 0.936596i \(0.613959\pi\)
\(164\) −343240. −0.996526
\(165\) 0 0
\(166\) 372912. 1.05036
\(167\) −487880. −1.35370 −0.676849 0.736122i \(-0.736655\pi\)
−0.676849 + 0.736122i \(0.736655\pi\)
\(168\) 416767. 1.13925
\(169\) −294260. −0.792527
\(170\) 0 0
\(171\) −231756. −0.606095
\(172\) −76386.1 −0.196876
\(173\) −325942. −0.827991 −0.413996 0.910279i \(-0.635867\pi\)
−0.413996 + 0.910279i \(0.635867\pi\)
\(174\) −128555. −0.321897
\(175\) 0 0
\(176\) 8009.99 0.0194917
\(177\) −38638.1 −0.0927005
\(178\) −179933. −0.425657
\(179\) 462906. 1.07984 0.539921 0.841716i \(-0.318454\pi\)
0.539921 + 0.841716i \(0.318454\pi\)
\(180\) 0 0
\(181\) 23901.3 0.0542281 0.0271141 0.999632i \(-0.491368\pi\)
0.0271141 + 0.999632i \(0.491368\pi\)
\(182\) 252806. 0.565730
\(183\) 61485.8 0.135721
\(184\) 196762. 0.428446
\(185\) 0 0
\(186\) −21037.7 −0.0445877
\(187\) 85243.1 0.178261
\(188\) 390938. 0.806703
\(189\) 183038. 0.372724
\(190\) 0 0
\(191\) −565986. −1.12259 −0.561297 0.827615i \(-0.689697\pi\)
−0.561297 + 0.827615i \(0.689697\pi\)
\(192\) −203916. −0.399207
\(193\) −91762.3 −0.177325 −0.0886627 0.996062i \(-0.528259\pi\)
−0.0886627 + 0.996062i \(0.528259\pi\)
\(194\) −334764. −0.638608
\(195\) 0 0
\(196\) −871049. −1.61958
\(197\) −485247. −0.890836 −0.445418 0.895323i \(-0.646945\pi\)
−0.445418 + 0.895323i \(0.646945\pi\)
\(198\) 35555.3 0.0644527
\(199\) 692632. 1.23985 0.619926 0.784660i \(-0.287162\pi\)
0.619926 + 0.784660i \(0.287162\pi\)
\(200\) 0 0
\(201\) −510745. −0.891690
\(202\) 128951. 0.222355
\(203\) 988619. 1.68379
\(204\) −119451. −0.200962
\(205\) 0 0
\(206\) −217216. −0.356635
\(207\) 86415.1 0.140173
\(208\) −18373.3 −0.0294461
\(209\) 346203. 0.548233
\(210\) 0 0
\(211\) −441020. −0.681949 −0.340975 0.940073i \(-0.610757\pi\)
−0.340975 + 0.940073i \(0.610757\pi\)
\(212\) −498986. −0.762516
\(213\) −28683.8 −0.0433199
\(214\) 323788. 0.483310
\(215\) 0 0
\(216\) −134451. −0.196078
\(217\) 161784. 0.233231
\(218\) −82697.4 −0.117856
\(219\) −66574.2 −0.0937985
\(220\) 0 0
\(221\) −195530. −0.269298
\(222\) 295228. 0.402045
\(223\) −1.13133e6 −1.52345 −0.761726 0.647899i \(-0.775648\pi\)
−0.761726 + 0.647899i \(0.775648\pi\)
\(224\) 1.42154e6 1.89295
\(225\) 0 0
\(226\) −602435. −0.784583
\(227\) 820354. 1.05666 0.528332 0.849038i \(-0.322818\pi\)
0.528332 + 0.849038i \(0.322818\pi\)
\(228\) −485133. −0.618051
\(229\) −1.00301e6 −1.26392 −0.631958 0.775003i \(-0.717748\pi\)
−0.631958 + 0.775003i \(0.717748\pi\)
\(230\) 0 0
\(231\) −273428. −0.337142
\(232\) −726191. −0.885790
\(233\) −734.569 −0.000886427 0 −0.000443214 1.00000i \(-0.500141\pi\)
−0.000443214 1.00000i \(0.500141\pi\)
\(234\) −81556.4 −0.0973685
\(235\) 0 0
\(236\) −80880.9 −0.0945291
\(237\) −219543. −0.253892
\(238\) −641685. −0.734311
\(239\) −1.06403e6 −1.20492 −0.602460 0.798149i \(-0.705813\pi\)
−0.602460 + 0.798149i \(0.705813\pi\)
\(240\) 0 0
\(241\) −661636. −0.733798 −0.366899 0.930261i \(-0.619581\pi\)
−0.366899 + 0.930261i \(0.619581\pi\)
\(242\) −53113.4 −0.0582996
\(243\) −59049.0 −0.0641500
\(244\) 128708. 0.138398
\(245\) 0 0
\(246\) 594841. 0.626705
\(247\) −794118. −0.828214
\(248\) −118839. −0.122696
\(249\) 925158. 0.945622
\(250\) 0 0
\(251\) 1.34864e6 1.35118 0.675590 0.737277i \(-0.263889\pi\)
0.675590 + 0.737277i \(0.263889\pi\)
\(252\) 383153. 0.380077
\(253\) −129089. −0.126791
\(254\) 577496. 0.561648
\(255\) 0 0
\(256\) −1.08410e6 −1.03388
\(257\) 1.65015e6 1.55844 0.779220 0.626751i \(-0.215616\pi\)
0.779220 + 0.626751i \(0.215616\pi\)
\(258\) 132379. 0.123814
\(259\) −2.27036e6 −2.10303
\(260\) 0 0
\(261\) −318933. −0.289800
\(262\) −842262. −0.758044
\(263\) 868163. 0.773948 0.386974 0.922091i \(-0.373520\pi\)
0.386974 + 0.922091i \(0.373520\pi\)
\(264\) 200846. 0.177359
\(265\) 0 0
\(266\) −2.60612e6 −2.25834
\(267\) −446395. −0.383213
\(268\) −1.06914e6 −0.909280
\(269\) 271547. 0.228804 0.114402 0.993435i \(-0.463505\pi\)
0.114402 + 0.993435i \(0.463505\pi\)
\(270\) 0 0
\(271\) −752770. −0.622643 −0.311322 0.950305i \(-0.600772\pi\)
−0.311322 + 0.950305i \(0.600772\pi\)
\(272\) 46635.9 0.0382207
\(273\) 627186. 0.509319
\(274\) 247430. 0.199102
\(275\) 0 0
\(276\) 180892. 0.142938
\(277\) −811445. −0.635418 −0.317709 0.948188i \(-0.602914\pi\)
−0.317709 + 0.948188i \(0.602914\pi\)
\(278\) −1.08165e6 −0.839411
\(279\) −52192.3 −0.0401417
\(280\) 0 0
\(281\) −1.72395e6 −1.30244 −0.651221 0.758888i \(-0.725743\pi\)
−0.651221 + 0.758888i \(0.725743\pi\)
\(282\) −677503. −0.507327
\(283\) −272979. −0.202611 −0.101305 0.994855i \(-0.532302\pi\)
−0.101305 + 0.994855i \(0.532302\pi\)
\(284\) −60043.6 −0.0441744
\(285\) 0 0
\(286\) 121831. 0.0880731
\(287\) −4.57446e6 −3.27820
\(288\) −458596. −0.325798
\(289\) −923553. −0.650455
\(290\) 0 0
\(291\) −830515. −0.574930
\(292\) −139360. −0.0956488
\(293\) 713441. 0.485500 0.242750 0.970089i \(-0.421951\pi\)
0.242750 + 0.970089i \(0.421951\pi\)
\(294\) 1.50954e6 1.01854
\(295\) 0 0
\(296\) 1.66770e6 1.10634
\(297\) 88209.0 0.0580259
\(298\) −301578. −0.196725
\(299\) 296104. 0.191543
\(300\) 0 0
\(301\) −1.01802e6 −0.647649
\(302\) −607997. −0.383605
\(303\) 319915. 0.200184
\(304\) 189405. 0.117546
\(305\) 0 0
\(306\) 207011. 0.126383
\(307\) 2.67734e6 1.62128 0.810640 0.585545i \(-0.199119\pi\)
0.810640 + 0.585545i \(0.199119\pi\)
\(308\) −572365. −0.343792
\(309\) −538891. −0.321074
\(310\) 0 0
\(311\) −1.28078e6 −0.750886 −0.375443 0.926846i \(-0.622509\pi\)
−0.375443 + 0.926846i \(0.622509\pi\)
\(312\) −460700. −0.267936
\(313\) −1.36808e6 −0.789313 −0.394656 0.918829i \(-0.629136\pi\)
−0.394656 + 0.918829i \(0.629136\pi\)
\(314\) −229488. −0.131352
\(315\) 0 0
\(316\) −459569. −0.258900
\(317\) 7686.28 0.00429604 0.00214802 0.999998i \(-0.499316\pi\)
0.00214802 + 0.999998i \(0.499316\pi\)
\(318\) 864752. 0.479539
\(319\) 476431. 0.262134
\(320\) 0 0
\(321\) 803284. 0.435117
\(322\) 971746. 0.522292
\(323\) 2.01567e6 1.07501
\(324\) −123607. −0.0654155
\(325\) 0 0
\(326\) 862406. 0.449436
\(327\) −205164. −0.106104
\(328\) 3.36017e6 1.72455
\(329\) 5.21014e6 2.65375
\(330\) 0 0
\(331\) 958347. 0.480787 0.240393 0.970676i \(-0.422724\pi\)
0.240393 + 0.970676i \(0.422724\pi\)
\(332\) 1.93663e6 0.964275
\(333\) 732430. 0.361956
\(334\) 1.76989e6 0.868121
\(335\) 0 0
\(336\) −149590. −0.0722862
\(337\) −4.08768e6 −1.96066 −0.980330 0.197365i \(-0.936762\pi\)
−0.980330 + 0.197365i \(0.936762\pi\)
\(338\) 1.06749e6 0.508244
\(339\) −1.49458e6 −0.706350
\(340\) 0 0
\(341\) 77966.3 0.0363096
\(342\) 840745. 0.388686
\(343\) −7.38880e6 −3.39108
\(344\) 747787. 0.340707
\(345\) 0 0
\(346\) 1.18243e6 0.530988
\(347\) −84250.2 −0.0375619 −0.0187809 0.999824i \(-0.505979\pi\)
−0.0187809 + 0.999824i \(0.505979\pi\)
\(348\) −667621. −0.295517
\(349\) 1.11859e6 0.491597 0.245798 0.969321i \(-0.420950\pi\)
0.245798 + 0.969321i \(0.420950\pi\)
\(350\) 0 0
\(351\) −202333. −0.0876595
\(352\) 685063. 0.294696
\(353\) 3.73570e6 1.59564 0.797820 0.602896i \(-0.205987\pi\)
0.797820 + 0.602896i \(0.205987\pi\)
\(354\) 140168. 0.0594485
\(355\) 0 0
\(356\) −934436. −0.390773
\(357\) −1.59196e6 −0.661090
\(358\) −1.67929e6 −0.692499
\(359\) 1.45342e6 0.595189 0.297595 0.954692i \(-0.403816\pi\)
0.297595 + 0.954692i \(0.403816\pi\)
\(360\) 0 0
\(361\) 5.71027e6 2.30616
\(362\) −86707.1 −0.0347763
\(363\) −131769. −0.0524864
\(364\) 1.31289e6 0.519366
\(365\) 0 0
\(366\) −223053. −0.0870374
\(367\) −25363.4 −0.00982975 −0.00491487 0.999988i \(-0.501564\pi\)
−0.00491487 + 0.999988i \(0.501564\pi\)
\(368\) −70623.8 −0.0271851
\(369\) 1.47574e6 0.564214
\(370\) 0 0
\(371\) −6.65013e6 −2.50839
\(372\) −109254. −0.0409336
\(373\) 1.72695e6 0.642699 0.321350 0.946961i \(-0.395864\pi\)
0.321350 + 0.946961i \(0.395864\pi\)
\(374\) −309238. −0.114318
\(375\) 0 0
\(376\) −3.82711e6 −1.39605
\(377\) −1.09283e6 −0.396005
\(378\) −664012. −0.239027
\(379\) −4.29401e6 −1.53555 −0.767777 0.640718i \(-0.778637\pi\)
−0.767777 + 0.640718i \(0.778637\pi\)
\(380\) 0 0
\(381\) 1.43271e6 0.505644
\(382\) 2.05324e6 0.719915
\(383\) −1.29297e6 −0.450392 −0.225196 0.974313i \(-0.572302\pi\)
−0.225196 + 0.974313i \(0.572302\pi\)
\(384\) −890813. −0.308289
\(385\) 0 0
\(386\) 332888. 0.113718
\(387\) 328418. 0.111468
\(388\) −1.73851e6 −0.586272
\(389\) 1.55257e6 0.520209 0.260104 0.965581i \(-0.416243\pi\)
0.260104 + 0.965581i \(0.416243\pi\)
\(390\) 0 0
\(391\) −751586. −0.248620
\(392\) 8.52719e6 2.80279
\(393\) −2.08957e6 −0.682456
\(394\) 1.76034e6 0.571290
\(395\) 0 0
\(396\) 184647. 0.0591705
\(397\) −819569. −0.260981 −0.130491 0.991450i \(-0.541655\pi\)
−0.130491 + 0.991450i \(0.541655\pi\)
\(398\) −2.51268e6 −0.795113
\(399\) −6.46551e6 −2.03316
\(400\) 0 0
\(401\) 1.38956e6 0.431536 0.215768 0.976445i \(-0.430774\pi\)
0.215768 + 0.976445i \(0.430774\pi\)
\(402\) 1.85284e6 0.571838
\(403\) −178839. −0.0548528
\(404\) 669677. 0.204132
\(405\) 0 0
\(406\) −3.58643e6 −1.07981
\(407\) −1.09412e6 −0.327401
\(408\) 1.16937e6 0.347778
\(409\) −3.53091e6 −1.04371 −0.521854 0.853035i \(-0.674759\pi\)
−0.521854 + 0.853035i \(0.674759\pi\)
\(410\) 0 0
\(411\) 613849. 0.179249
\(412\) −1.12806e6 −0.327407
\(413\) −1.07792e6 −0.310966
\(414\) −313490. −0.0898923
\(415\) 0 0
\(416\) −1.57139e6 −0.445196
\(417\) −2.68346e6 −0.755710
\(418\) −1.25593e6 −0.351580
\(419\) −6.69977e6 −1.86434 −0.932170 0.362021i \(-0.882087\pi\)
−0.932170 + 0.362021i \(0.882087\pi\)
\(420\) 0 0
\(421\) −5.01124e6 −1.37797 −0.688986 0.724775i \(-0.741944\pi\)
−0.688986 + 0.724775i \(0.741944\pi\)
\(422\) 1.59990e6 0.437331
\(423\) −1.68082e6 −0.456740
\(424\) 4.88485e6 1.31958
\(425\) 0 0
\(426\) 104057. 0.0277809
\(427\) 1.71533e6 0.455279
\(428\) 1.68151e6 0.443701
\(429\) 302251. 0.0792910
\(430\) 0 0
\(431\) −1.14741e6 −0.297526 −0.148763 0.988873i \(-0.547529\pi\)
−0.148763 + 0.988873i \(0.547529\pi\)
\(432\) 48258.6 0.0124413
\(433\) 1.13393e6 0.290649 0.145324 0.989384i \(-0.453577\pi\)
0.145324 + 0.989384i \(0.453577\pi\)
\(434\) −586908. −0.149570
\(435\) 0 0
\(436\) −429469. −0.108197
\(437\) −3.05246e6 −0.764622
\(438\) 241513. 0.0601526
\(439\) 7.44692e6 1.84423 0.922115 0.386915i \(-0.126459\pi\)
0.922115 + 0.386915i \(0.126459\pi\)
\(440\) 0 0
\(441\) 3.74503e6 0.916977
\(442\) 709328. 0.172700
\(443\) −6.72521e6 −1.62816 −0.814079 0.580754i \(-0.802758\pi\)
−0.814079 + 0.580754i \(0.802758\pi\)
\(444\) 1.53319e6 0.369096
\(445\) 0 0
\(446\) 4.10416e6 0.976984
\(447\) −748183. −0.177108
\(448\) −5.68883e6 −1.33915
\(449\) 2.17793e6 0.509834 0.254917 0.966963i \(-0.417952\pi\)
0.254917 + 0.966963i \(0.417952\pi\)
\(450\) 0 0
\(451\) −2.20450e6 −0.510351
\(452\) −3.12860e6 −0.720284
\(453\) −1.50838e6 −0.345354
\(454\) −2.97601e6 −0.677634
\(455\) 0 0
\(456\) 4.74924e6 1.06958
\(457\) 3.72380e6 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(458\) 3.63865e6 0.810544
\(459\) 513572. 0.113781
\(460\) 0 0
\(461\) 4.74202e6 1.03923 0.519615 0.854401i \(-0.326076\pi\)
0.519615 + 0.854401i \(0.326076\pi\)
\(462\) 991919. 0.216208
\(463\) 4.82803e6 1.04669 0.523344 0.852121i \(-0.324684\pi\)
0.523344 + 0.852121i \(0.324684\pi\)
\(464\) 260652. 0.0562039
\(465\) 0 0
\(466\) 2664.81 0.000568462 0
\(467\) −7.56254e6 −1.60463 −0.802316 0.596900i \(-0.796399\pi\)
−0.802316 + 0.596900i \(0.796399\pi\)
\(468\) −423543. −0.0893887
\(469\) −1.42487e7 −2.99119
\(470\) 0 0
\(471\) −569337. −0.118254
\(472\) 791788. 0.163589
\(473\) −490599. −0.100826
\(474\) 796441. 0.162820
\(475\) 0 0
\(476\) −3.33243e6 −0.674131
\(477\) 2.14536e6 0.431722
\(478\) 3.85999e6 0.772710
\(479\) −1.71897e6 −0.342319 −0.171159 0.985243i \(-0.554751\pi\)
−0.171159 + 0.985243i \(0.554751\pi\)
\(480\) 0 0
\(481\) 2.50969e6 0.494604
\(482\) 2.40023e6 0.470582
\(483\) 2.41080e6 0.470212
\(484\) −275831. −0.0535218
\(485\) 0 0
\(486\) 214213. 0.0411392
\(487\) 3.08125e6 0.588715 0.294357 0.955695i \(-0.404894\pi\)
0.294357 + 0.955695i \(0.404894\pi\)
\(488\) −1.25999e6 −0.239508
\(489\) 2.13954e6 0.404621
\(490\) 0 0
\(491\) −1.05909e6 −0.198258 −0.0991290 0.995075i \(-0.531606\pi\)
−0.0991290 + 0.995075i \(0.531606\pi\)
\(492\) 3.08916e6 0.575344
\(493\) 2.77388e6 0.514009
\(494\) 2.88084e6 0.531131
\(495\) 0 0
\(496\) 42654.9 0.00778510
\(497\) −800218. −0.145317
\(498\) −3.35621e6 −0.606423
\(499\) 2.26415e6 0.407056 0.203528 0.979069i \(-0.434759\pi\)
0.203528 + 0.979069i \(0.434759\pi\)
\(500\) 0 0
\(501\) 4.39092e6 0.781558
\(502\) −4.89250e6 −0.866507
\(503\) 7.65222e6 1.34855 0.674276 0.738480i \(-0.264456\pi\)
0.674276 + 0.738480i \(0.264456\pi\)
\(504\) −3.75090e6 −0.657748
\(505\) 0 0
\(506\) 468299. 0.0813106
\(507\) 2.64834e6 0.457566
\(508\) 2.99908e6 0.515619
\(509\) 489107. 0.0836777 0.0418388 0.999124i \(-0.486678\pi\)
0.0418388 + 0.999124i \(0.486678\pi\)
\(510\) 0 0
\(511\) −1.85728e6 −0.314649
\(512\) 765484. 0.129051
\(513\) 2.08580e6 0.349929
\(514\) −5.98627e6 −0.999421
\(515\) 0 0
\(516\) 687475. 0.113667
\(517\) 2.51085e6 0.413137
\(518\) 8.23624e6 1.34867
\(519\) 2.93348e6 0.478041
\(520\) 0 0
\(521\) 1.36556e6 0.220402 0.110201 0.993909i \(-0.464851\pi\)
0.110201 + 0.993909i \(0.464851\pi\)
\(522\) 1.15700e6 0.185847
\(523\) 5.63994e6 0.901613 0.450807 0.892622i \(-0.351136\pi\)
0.450807 + 0.892622i \(0.351136\pi\)
\(524\) −4.37408e6 −0.695919
\(525\) 0 0
\(526\) −3.14945e6 −0.496330
\(527\) 453937. 0.0711982
\(528\) −72089.9 −0.0112536
\(529\) −5.29817e6 −0.823164
\(530\) 0 0
\(531\) 347742. 0.0535207
\(532\) −1.35342e7 −2.07326
\(533\) 5.05667e6 0.770986
\(534\) 1.61939e6 0.245753
\(535\) 0 0
\(536\) 1.04664e7 1.57357
\(537\) −4.16615e6 −0.623447
\(538\) −985095. −0.146731
\(539\) −5.59442e6 −0.829437
\(540\) 0 0
\(541\) 8.24934e6 1.21179 0.605893 0.795546i \(-0.292816\pi\)
0.605893 + 0.795546i \(0.292816\pi\)
\(542\) 2.73084e6 0.399299
\(543\) −215111. −0.0313086
\(544\) 3.98859e6 0.577859
\(545\) 0 0
\(546\) −2.27526e6 −0.326624
\(547\) 4.74537e6 0.678112 0.339056 0.940766i \(-0.389892\pi\)
0.339056 + 0.940766i \(0.389892\pi\)
\(548\) 1.28497e6 0.182785
\(549\) −553372. −0.0783586
\(550\) 0 0
\(551\) 1.12657e7 1.58082
\(552\) −1.77086e6 −0.247363
\(553\) −6.12480e6 −0.851685
\(554\) 2.94369e6 0.407491
\(555\) 0 0
\(556\) −5.61728e6 −0.770617
\(557\) −2.99690e6 −0.409293 −0.204647 0.978836i \(-0.565605\pi\)
−0.204647 + 0.978836i \(0.565605\pi\)
\(558\) 189339. 0.0257427
\(559\) 1.12533e6 0.152318
\(560\) 0 0
\(561\) −767188. −0.102919
\(562\) 6.25400e6 0.835251
\(563\) 3.89491e6 0.517876 0.258938 0.965894i \(-0.416627\pi\)
0.258938 + 0.965894i \(0.416627\pi\)
\(564\) −3.51844e6 −0.465750
\(565\) 0 0
\(566\) 990290. 0.129934
\(567\) −1.64735e6 −0.215192
\(568\) 587801. 0.0764468
\(569\) 2.08127e6 0.269493 0.134747 0.990880i \(-0.456978\pi\)
0.134747 + 0.990880i \(0.456978\pi\)
\(570\) 0 0
\(571\) 1.28958e7 1.65523 0.827615 0.561297i \(-0.189697\pi\)
0.827615 + 0.561297i \(0.189697\pi\)
\(572\) 632700. 0.0808551
\(573\) 5.09387e6 0.648129
\(574\) 1.65948e7 2.10229
\(575\) 0 0
\(576\) 1.83524e6 0.230482
\(577\) −1.26163e6 −0.157759 −0.0788795 0.996884i \(-0.525134\pi\)
−0.0788795 + 0.996884i \(0.525134\pi\)
\(578\) 3.35039e6 0.417134
\(579\) 825861. 0.102379
\(580\) 0 0
\(581\) 2.58100e7 3.17210
\(582\) 3.01288e6 0.368701
\(583\) −3.20480e6 −0.390508
\(584\) 1.36427e6 0.165527
\(585\) 0 0
\(586\) −2.58816e6 −0.311349
\(587\) 1.25673e7 1.50538 0.752689 0.658376i \(-0.228756\pi\)
0.752689 + 0.658376i \(0.228756\pi\)
\(588\) 7.83945e6 0.935065
\(589\) 1.84360e6 0.218967
\(590\) 0 0
\(591\) 4.36723e6 0.514324
\(592\) −598588. −0.0701978
\(593\) 4.32620e6 0.505207 0.252604 0.967570i \(-0.418713\pi\)
0.252604 + 0.967570i \(0.418713\pi\)
\(594\) −319997. −0.0372118
\(595\) 0 0
\(596\) −1.56617e6 −0.180602
\(597\) −6.23369e6 −0.715829
\(598\) −1.07418e6 −0.122836
\(599\) 1.45262e7 1.65419 0.827097 0.562060i \(-0.189991\pi\)
0.827097 + 0.562060i \(0.189991\pi\)
\(600\) 0 0
\(601\) −380688. −0.0429915 −0.0214958 0.999769i \(-0.506843\pi\)
−0.0214958 + 0.999769i \(0.506843\pi\)
\(602\) 3.69309e6 0.415335
\(603\) 4.59671e6 0.514818
\(604\) −3.15748e6 −0.352167
\(605\) 0 0
\(606\) −1.16056e6 −0.128377
\(607\) 4.99233e6 0.549961 0.274980 0.961450i \(-0.411329\pi\)
0.274980 + 0.961450i \(0.411329\pi\)
\(608\) 1.61991e7 1.77718
\(609\) −8.89757e6 −0.972139
\(610\) 0 0
\(611\) −5.75936e6 −0.624125
\(612\) 1.07506e6 0.116026
\(613\) −2.99353e6 −0.321760 −0.160880 0.986974i \(-0.551433\pi\)
−0.160880 + 0.986974i \(0.551433\pi\)
\(614\) −9.71265e6 −1.03972
\(615\) 0 0
\(616\) 5.60320e6 0.594955
\(617\) −6.27781e6 −0.663889 −0.331945 0.943299i \(-0.607705\pi\)
−0.331945 + 0.943299i \(0.607705\pi\)
\(618\) 1.95494e6 0.205903
\(619\) 3.96477e6 0.415902 0.207951 0.978139i \(-0.433320\pi\)
0.207951 + 0.978139i \(0.433320\pi\)
\(620\) 0 0
\(621\) −777736. −0.0809288
\(622\) 4.64631e6 0.481540
\(623\) −1.24535e7 −1.28550
\(624\) 165359. 0.0170007
\(625\) 0 0
\(626\) 4.96299e6 0.506183
\(627\) −3.11583e6 −0.316523
\(628\) −1.19179e6 −0.120587
\(629\) −6.37023e6 −0.641990
\(630\) 0 0
\(631\) −9.62587e6 −0.962425 −0.481212 0.876604i \(-0.659803\pi\)
−0.481212 + 0.876604i \(0.659803\pi\)
\(632\) 4.49898e6 0.448044
\(633\) 3.96918e6 0.393724
\(634\) −27883.7 −0.00275503
\(635\) 0 0
\(636\) 4.49087e6 0.440239
\(637\) 1.28324e7 1.25303
\(638\) −1.72836e6 −0.168105
\(639\) 258154. 0.0250108
\(640\) 0 0
\(641\) −2.20090e6 −0.211571 −0.105785 0.994389i \(-0.533736\pi\)
−0.105785 + 0.994389i \(0.533736\pi\)
\(642\) −2.91409e6 −0.279039
\(643\) −2.00555e7 −1.91296 −0.956482 0.291791i \(-0.905749\pi\)
−0.956482 + 0.291791i \(0.905749\pi\)
\(644\) 5.04652e6 0.479488
\(645\) 0 0
\(646\) −7.31229e6 −0.689401
\(647\) −128900. −0.0121057 −0.00605287 0.999982i \(-0.501927\pi\)
−0.00605287 + 0.999982i \(0.501927\pi\)
\(648\) 1.21006e6 0.113206
\(649\) −519467. −0.0484113
\(650\) 0 0
\(651\) −1.45606e6 −0.134656
\(652\) 4.47869e6 0.412603
\(653\) −3.03250e6 −0.278303 −0.139151 0.990271i \(-0.544437\pi\)
−0.139151 + 0.990271i \(0.544437\pi\)
\(654\) 744277. 0.0680441
\(655\) 0 0
\(656\) −1.20607e6 −0.109424
\(657\) 599168. 0.0541546
\(658\) −1.89009e7 −1.70184
\(659\) −1.59132e7 −1.42740 −0.713698 0.700454i \(-0.752981\pi\)
−0.713698 + 0.700454i \(0.752981\pi\)
\(660\) 0 0
\(661\) 6.50892e6 0.579436 0.289718 0.957112i \(-0.406439\pi\)
0.289718 + 0.957112i \(0.406439\pi\)
\(662\) −3.47661e6 −0.308327
\(663\) 1.75977e6 0.155479
\(664\) −1.89587e7 −1.66874
\(665\) 0 0
\(666\) −2.65705e6 −0.232121
\(667\) −4.20067e6 −0.365598
\(668\) 9.19149e6 0.796975
\(669\) 1.01820e7 0.879565
\(670\) 0 0
\(671\) 826643. 0.0708780
\(672\) −1.27939e7 −1.09290
\(673\) 5.66035e6 0.481732 0.240866 0.970558i \(-0.422569\pi\)
0.240866 + 0.970558i \(0.422569\pi\)
\(674\) 1.48290e7 1.25736
\(675\) 0 0
\(676\) 5.54375e6 0.466592
\(677\) 1.53601e7 1.28802 0.644012 0.765016i \(-0.277269\pi\)
0.644012 + 0.765016i \(0.277269\pi\)
\(678\) 5.42192e6 0.452979
\(679\) −2.31697e7 −1.92861
\(680\) 0 0
\(681\) −7.38319e6 −0.610065
\(682\) −282840. −0.0232852
\(683\) −1.47069e6 −0.120634 −0.0603170 0.998179i \(-0.519211\pi\)
−0.0603170 + 0.998179i \(0.519211\pi\)
\(684\) 4.36620e6 0.356832
\(685\) 0 0
\(686\) 2.68045e7 2.17469
\(687\) 9.02712e6 0.729722
\(688\) −268404. −0.0216181
\(689\) 7.35114e6 0.589939
\(690\) 0 0
\(691\) −3.65617e6 −0.291294 −0.145647 0.989337i \(-0.546526\pi\)
−0.145647 + 0.989337i \(0.546526\pi\)
\(692\) 6.14064e6 0.487471
\(693\) 2.46085e6 0.194649
\(694\) 305636. 0.0240883
\(695\) 0 0
\(696\) 6.53571e6 0.511411
\(697\) −1.28351e7 −1.00073
\(698\) −4.05794e6 −0.315259
\(699\) 6611.12 0.000511779 0
\(700\) 0 0
\(701\) −5.21232e6 −0.400623 −0.200312 0.979732i \(-0.564195\pi\)
−0.200312 + 0.979732i \(0.564195\pi\)
\(702\) 734008. 0.0562157
\(703\) −2.58718e7 −1.97442
\(704\) −2.74153e6 −0.208479
\(705\) 0 0
\(706\) −1.35521e7 −1.02328
\(707\) 8.92498e6 0.671519
\(708\) 727928. 0.0545764
\(709\) −1.48388e7 −1.10862 −0.554311 0.832310i \(-0.687018\pi\)
−0.554311 + 0.832310i \(0.687018\pi\)
\(710\) 0 0
\(711\) 1.97589e6 0.146585
\(712\) 9.14772e6 0.676258
\(713\) −687427. −0.0506410
\(714\) 5.77517e6 0.423954
\(715\) 0 0
\(716\) −8.72099e6 −0.635745
\(717\) 9.57625e6 0.695661
\(718\) −5.27260e6 −0.381693
\(719\) 1.37497e7 0.991906 0.495953 0.868349i \(-0.334819\pi\)
0.495953 + 0.868349i \(0.334819\pi\)
\(720\) 0 0
\(721\) −1.50339e7 −1.07705
\(722\) −2.07153e7 −1.47893
\(723\) 5.95473e6 0.423659
\(724\) −450292. −0.0319262
\(725\) 0 0
\(726\) 478021. 0.0336593
\(727\) 8.20549e6 0.575796 0.287898 0.957661i \(-0.407044\pi\)
0.287898 + 0.957661i \(0.407044\pi\)
\(728\) −1.28526e7 −0.898797
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −2.85638e6 −0.197707
\(732\) −1.15837e6 −0.0799043
\(733\) −1.20636e7 −0.829312 −0.414656 0.909978i \(-0.636098\pi\)
−0.414656 + 0.909978i \(0.636098\pi\)
\(734\) 92011.3 0.00630378
\(735\) 0 0
\(736\) −6.04017e6 −0.411012
\(737\) −6.86668e6 −0.465670
\(738\) −5.35357e6 −0.361828
\(739\) −1.02319e7 −0.689203 −0.344602 0.938749i \(-0.611986\pi\)
−0.344602 + 0.938749i \(0.611986\pi\)
\(740\) 0 0
\(741\) 7.14706e6 0.478170
\(742\) 2.41248e7 1.60862
\(743\) −2.41980e7 −1.60808 −0.804039 0.594577i \(-0.797320\pi\)
−0.804039 + 0.594577i \(0.797320\pi\)
\(744\) 1.06955e6 0.0708383
\(745\) 0 0
\(746\) −6.26489e6 −0.412161
\(747\) −8.32642e6 −0.545955
\(748\) −1.60595e6 −0.104949
\(749\) 2.24100e7 1.45961
\(750\) 0 0
\(751\) 1.10121e7 0.712476 0.356238 0.934395i \(-0.384059\pi\)
0.356238 + 0.934395i \(0.384059\pi\)
\(752\) 1.37367e6 0.0885803
\(753\) −1.21378e7 −0.780104
\(754\) 3.96449e6 0.253956
\(755\) 0 0
\(756\) −3.44838e6 −0.219437
\(757\) 2.08747e7 1.32398 0.661989 0.749513i \(-0.269712\pi\)
0.661989 + 0.749513i \(0.269712\pi\)
\(758\) 1.55775e7 0.984744
\(759\) 1.16180e6 0.0732028
\(760\) 0 0
\(761\) 2.85723e7 1.78848 0.894240 0.447587i \(-0.147716\pi\)
0.894240 + 0.447587i \(0.147716\pi\)
\(762\) −5.19746e6 −0.324268
\(763\) −5.72365e6 −0.355928
\(764\) 1.06630e7 0.660915
\(765\) 0 0
\(766\) 4.69052e6 0.288835
\(767\) 1.19155e6 0.0731347
\(768\) 9.75692e6 0.596911
\(769\) −3.07246e6 −0.187357 −0.0936787 0.995602i \(-0.529863\pi\)
−0.0936787 + 0.995602i \(0.529863\pi\)
\(770\) 0 0
\(771\) −1.48513e7 −0.899765
\(772\) 1.72877e6 0.104398
\(773\) 3.02076e7 1.81831 0.909153 0.416461i \(-0.136730\pi\)
0.909153 + 0.416461i \(0.136730\pi\)
\(774\) −1.19141e6 −0.0714838
\(775\) 0 0
\(776\) 1.70193e7 1.01458
\(777\) 2.04333e7 1.21419
\(778\) −5.63229e6 −0.333608
\(779\) −5.21280e7 −3.07771
\(780\) 0 0
\(781\) −385638. −0.0226231
\(782\) 2.72654e6 0.159439
\(783\) 2.87040e6 0.167316
\(784\) −3.06067e6 −0.177839
\(785\) 0 0
\(786\) 7.58036e6 0.437657
\(787\) 2.07854e7 1.19625 0.598126 0.801402i \(-0.295912\pi\)
0.598126 + 0.801402i \(0.295912\pi\)
\(788\) 9.14190e6 0.524470
\(789\) −7.81346e6 −0.446839
\(790\) 0 0
\(791\) −4.16957e7 −2.36946
\(792\) −1.80762e6 −0.102399
\(793\) −1.89615e6 −0.107075
\(794\) 2.97317e6 0.167366
\(795\) 0 0
\(796\) −1.30490e7 −0.729950
\(797\) 7.95535e6 0.443622 0.221811 0.975090i \(-0.428803\pi\)
0.221811 + 0.975090i \(0.428803\pi\)
\(798\) 2.34551e7 1.30385
\(799\) 1.46187e7 0.810106
\(800\) 0 0
\(801\) 4.01755e6 0.221248
\(802\) −5.04094e6 −0.276742
\(803\) −895054. −0.0489847
\(804\) 9.62226e6 0.524973
\(805\) 0 0
\(806\) 648776. 0.0351769
\(807\) −2.44392e6 −0.132100
\(808\) −6.55584e6 −0.353265
\(809\) −3.04660e7 −1.63661 −0.818303 0.574787i \(-0.805085\pi\)
−0.818303 + 0.574787i \(0.805085\pi\)
\(810\) 0 0
\(811\) 1.28041e7 0.683594 0.341797 0.939774i \(-0.388964\pi\)
0.341797 + 0.939774i \(0.388964\pi\)
\(812\) −1.86252e7 −0.991316
\(813\) 6.77493e6 0.359483
\(814\) 3.96917e6 0.209961
\(815\) 0 0
\(816\) −419723. −0.0220667
\(817\) −1.16008e7 −0.608040
\(818\) 1.28092e7 0.669325
\(819\) −5.64468e6 −0.294055
\(820\) 0 0
\(821\) 1.13635e7 0.588373 0.294186 0.955748i \(-0.404951\pi\)
0.294186 + 0.955748i \(0.404951\pi\)
\(822\) −2.22687e6 −0.114952
\(823\) −8.23741e6 −0.423927 −0.211964 0.977278i \(-0.567986\pi\)
−0.211964 + 0.977278i \(0.567986\pi\)
\(824\) 1.10432e7 0.566600
\(825\) 0 0
\(826\) 3.91040e6 0.199421
\(827\) −2.21035e7 −1.12382 −0.561909 0.827199i \(-0.689933\pi\)
−0.561909 + 0.827199i \(0.689933\pi\)
\(828\) −1.62803e6 −0.0825252
\(829\) −5.57875e6 −0.281936 −0.140968 0.990014i \(-0.545021\pi\)
−0.140968 + 0.990014i \(0.545021\pi\)
\(830\) 0 0
\(831\) 7.30300e6 0.366859
\(832\) 6.28851e6 0.314948
\(833\) −3.25720e7 −1.62641
\(834\) 9.73484e6 0.484634
\(835\) 0 0
\(836\) −6.52235e6 −0.322766
\(837\) 469731. 0.0231758
\(838\) 2.43049e7 1.19559
\(839\) −2.33576e7 −1.14558 −0.572788 0.819704i \(-0.694138\pi\)
−0.572788 + 0.819704i \(0.694138\pi\)
\(840\) 0 0
\(841\) −5.00769e6 −0.244145
\(842\) 1.81794e7 0.883688
\(843\) 1.55155e7 0.751965
\(844\) 8.30866e6 0.401490
\(845\) 0 0
\(846\) 6.09753e6 0.292906
\(847\) −3.67608e6 −0.176067
\(848\) −1.75332e6 −0.0837284
\(849\) 2.45681e6 0.116977
\(850\) 0 0
\(851\) 9.64685e6 0.456627
\(852\) 540393. 0.0255041
\(853\) −8.94855e6 −0.421095 −0.210548 0.977584i \(-0.567525\pi\)
−0.210548 + 0.977584i \(0.567525\pi\)
\(854\) −6.22273e6 −0.291969
\(855\) 0 0
\(856\) −1.64612e7 −0.767853
\(857\) 2.03190e6 0.0945039 0.0472520 0.998883i \(-0.484954\pi\)
0.0472520 + 0.998883i \(0.484954\pi\)
\(858\) −1.09648e6 −0.0508490
\(859\) −1.92359e7 −0.889467 −0.444734 0.895663i \(-0.646702\pi\)
−0.444734 + 0.895663i \(0.646702\pi\)
\(860\) 0 0
\(861\) 4.11701e7 1.89267
\(862\) 4.16247e6 0.190802
\(863\) −2.00380e7 −0.915856 −0.457928 0.888989i \(-0.651408\pi\)
−0.457928 + 0.888989i \(0.651408\pi\)
\(864\) 4.12736e6 0.188100
\(865\) 0 0
\(866\) −4.11360e6 −0.186392
\(867\) 8.31197e6 0.375540
\(868\) −3.04796e6 −0.137312
\(869\) −2.95164e6 −0.132591
\(870\) 0 0
\(871\) 1.57507e7 0.703486
\(872\) 4.20431e6 0.187242
\(873\) 7.47464e6 0.331936
\(874\) 1.10735e7 0.490349
\(875\) 0 0
\(876\) 1.25424e6 0.0552229
\(877\) −1.98487e7 −0.871430 −0.435715 0.900085i \(-0.643504\pi\)
−0.435715 + 0.900085i \(0.643504\pi\)
\(878\) −2.70153e7 −1.18270
\(879\) −6.42097e6 −0.280304
\(880\) 0 0
\(881\) −2.93546e7 −1.27420 −0.637099 0.770782i \(-0.719866\pi\)
−0.637099 + 0.770782i \(0.719866\pi\)
\(882\) −1.35859e7 −0.588054
\(883\) −3.85199e7 −1.66258 −0.831291 0.555838i \(-0.812398\pi\)
−0.831291 + 0.555838i \(0.812398\pi\)
\(884\) 3.68372e6 0.158546
\(885\) 0 0
\(886\) 2.43972e7 1.04413
\(887\) 2.68797e6 0.114714 0.0573569 0.998354i \(-0.481733\pi\)
0.0573569 + 0.998354i \(0.481733\pi\)
\(888\) −1.50093e7 −0.638745
\(889\) 3.99696e7 1.69619
\(890\) 0 0
\(891\) −793881. −0.0335013
\(892\) 2.13139e7 0.896916
\(893\) 5.93719e7 2.49145
\(894\) 2.71420e6 0.113579
\(895\) 0 0
\(896\) −2.48519e7 −1.03416
\(897\) −2.66493e6 −0.110587
\(898\) −7.90093e6 −0.326955
\(899\) 2.53709e6 0.104698
\(900\) 0 0
\(901\) −1.86590e7 −0.765733
\(902\) 7.99731e6 0.327286
\(903\) 9.16218e6 0.373921
\(904\) 3.06276e7 1.24650
\(905\) 0 0
\(906\) 5.47197e6 0.221474
\(907\) 7.78340e6 0.314160 0.157080 0.987586i \(-0.449792\pi\)
0.157080 + 0.987586i \(0.449792\pi\)
\(908\) −1.54552e7 −0.622099
\(909\) −2.87924e6 −0.115576
\(910\) 0 0
\(911\) 3.15361e7 1.25896 0.629481 0.777016i \(-0.283268\pi\)
0.629481 + 0.777016i \(0.283268\pi\)
\(912\) −1.70465e6 −0.0678653
\(913\) 1.24382e7 0.493835
\(914\) −1.35089e7 −0.534878
\(915\) 0 0
\(916\) 1.88964e7 0.744117
\(917\) −5.82946e7 −2.28931
\(918\) −1.86310e6 −0.0729673
\(919\) 4.21184e7 1.64507 0.822533 0.568717i \(-0.192560\pi\)
0.822533 + 0.568717i \(0.192560\pi\)
\(920\) 0 0
\(921\) −2.40961e7 −0.936047
\(922\) −1.72027e7 −0.666454
\(923\) 884572. 0.0341766
\(924\) 5.15128e6 0.198489
\(925\) 0 0
\(926\) −1.75147e7 −0.671237
\(927\) 4.85002e6 0.185372
\(928\) 2.22925e7 0.849746
\(929\) 9.43595e6 0.358712 0.179356 0.983784i \(-0.442599\pi\)
0.179356 + 0.983784i \(0.442599\pi\)
\(930\) 0 0
\(931\) −1.32287e8 −5.00197
\(932\) 13839.0 0.000521874 0
\(933\) 1.15270e7 0.433524
\(934\) 2.74348e7 1.02904
\(935\) 0 0
\(936\) 4.14630e6 0.154693
\(937\) −2.17869e7 −0.810674 −0.405337 0.914167i \(-0.632846\pi\)
−0.405337 + 0.914167i \(0.632846\pi\)
\(938\) 5.16904e7 1.91824
\(939\) 1.23127e7 0.455710
\(940\) 0 0
\(941\) 2.18678e7 0.805065 0.402532 0.915406i \(-0.368130\pi\)
0.402532 + 0.915406i \(0.368130\pi\)
\(942\) 2.06540e6 0.0758361
\(943\) 1.94370e7 0.711787
\(944\) −284197. −0.0103798
\(945\) 0 0
\(946\) 1.77976e6 0.0646595
\(947\) −2.42335e7 −0.878094 −0.439047 0.898464i \(-0.644684\pi\)
−0.439047 + 0.898464i \(0.644684\pi\)
\(948\) 4.13612e6 0.149476
\(949\) 2.05307e6 0.0740010
\(950\) 0 0
\(951\) −69176.5 −0.00248032
\(952\) 3.26231e7 1.16663
\(953\) −5.56092e7 −1.98342 −0.991710 0.128499i \(-0.958984\pi\)
−0.991710 + 0.128499i \(0.958984\pi\)
\(954\) −7.78277e6 −0.276862
\(955\) 0 0
\(956\) 2.00459e7 0.709383
\(957\) −4.28788e6 −0.151343
\(958\) 6.23595e6 0.219528
\(959\) 1.71251e7 0.601294
\(960\) 0 0
\(961\) −2.82140e7 −0.985498
\(962\) −9.10446e6 −0.317188
\(963\) −7.22956e6 −0.251215
\(964\) 1.24650e7 0.432016
\(965\) 0 0
\(966\) −8.74571e6 −0.301545
\(967\) 2.09122e7 0.719173 0.359586 0.933112i \(-0.382918\pi\)
0.359586 + 0.933112i \(0.382918\pi\)
\(968\) 2.70027e6 0.0926229
\(969\) −1.81410e7 −0.620659
\(970\) 0 0
\(971\) 1.68184e7 0.572448 0.286224 0.958163i \(-0.407600\pi\)
0.286224 + 0.958163i \(0.407600\pi\)
\(972\) 1.11246e6 0.0377676
\(973\) −7.48630e7 −2.53504
\(974\) −1.11779e7 −0.377541
\(975\) 0 0
\(976\) 452251. 0.0151969
\(977\) 3.45472e7 1.15792 0.578958 0.815358i \(-0.303460\pi\)
0.578958 + 0.815358i \(0.303460\pi\)
\(978\) −7.76166e6 −0.259482
\(979\) −6.00153e6 −0.200127
\(980\) 0 0
\(981\) 1.84648e6 0.0612592
\(982\) 3.84210e6 0.127142
\(983\) 3.40232e7 1.12303 0.561516 0.827466i \(-0.310218\pi\)
0.561516 + 0.827466i \(0.310218\pi\)
\(984\) −3.02415e7 −0.995671
\(985\) 0 0
\(986\) −1.00629e7 −0.329632
\(987\) −4.68913e7 −1.53214
\(988\) 1.49609e7 0.487602
\(989\) 4.32560e6 0.140623
\(990\) 0 0
\(991\) −4.47458e7 −1.44733 −0.723665 0.690151i \(-0.757544\pi\)
−0.723665 + 0.690151i \(0.757544\pi\)
\(992\) 3.64810e6 0.117703
\(993\) −8.62512e6 −0.277583
\(994\) 2.90297e6 0.0931915
\(995\) 0 0
\(996\) −1.74297e7 −0.556725
\(997\) 5.86494e6 0.186864 0.0934320 0.995626i \(-0.470216\pi\)
0.0934320 + 0.995626i \(0.470216\pi\)
\(998\) −8.21371e6 −0.261044
\(999\) −6.59187e6 −0.208975
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.c.1.2 2
5.4 even 2 33.6.a.e.1.1 2
15.14 odd 2 99.6.a.d.1.2 2
20.19 odd 2 528.6.a.o.1.2 2
55.54 odd 2 363.6.a.f.1.2 2
165.164 even 2 1089.6.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.6.a.e.1.1 2 5.4 even 2
99.6.a.d.1.2 2 15.14 odd 2
363.6.a.f.1.2 2 55.54 odd 2
528.6.a.o.1.2 2 20.19 odd 2
825.6.a.c.1.2 2 1.1 even 1 trivial
1089.6.a.p.1.1 2 165.164 even 2