Properties

Label 33.6.a.e.1.1
Level $33$
Weight $6$
Character 33.1
Self dual yes
Analytic conductor $5.293$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 33.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.29266605383\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 33.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.62772 q^{2} +9.00000 q^{3} -18.8397 q^{4} +57.7228 q^{5} +32.6495 q^{6} +251.081 q^{7} -184.432 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+3.62772 q^{2} +9.00000 q^{3} -18.8397 q^{4} +57.7228 q^{5} +32.6495 q^{6} +251.081 q^{7} -184.432 q^{8} +81.0000 q^{9} +209.402 q^{10} -121.000 q^{11} -169.557 q^{12} -277.549 q^{13} +910.853 q^{14} +519.505 q^{15} -66.1983 q^{16} +704.489 q^{17} +293.845 q^{18} -2861.18 q^{19} -1087.48 q^{20} +2259.73 q^{21} -438.954 q^{22} -1066.85 q^{23} -1659.89 q^{24} +206.923 q^{25} -1006.87 q^{26} +729.000 q^{27} -4730.29 q^{28} -3937.44 q^{29} +1884.62 q^{30} -644.350 q^{31} +5661.67 q^{32} -1089.00 q^{33} +2555.69 q^{34} +14493.1 q^{35} -1526.01 q^{36} -9042.34 q^{37} -10379.6 q^{38} -2497.94 q^{39} -10645.9 q^{40} +18219.0 q^{41} +8197.68 q^{42} -4054.54 q^{43} +2279.60 q^{44} +4675.55 q^{45} -3870.24 q^{46} +20750.8 q^{47} -595.785 q^{48} +46234.9 q^{49} +750.659 q^{50} +6340.40 q^{51} +5228.92 q^{52} -26485.9 q^{53} +2644.61 q^{54} -6984.46 q^{55} -46307.4 q^{56} -25750.7 q^{57} -14283.9 q^{58} +4293.12 q^{59} -9787.30 q^{60} -6831.76 q^{61} -2337.52 q^{62} +20337.6 q^{63} +22657.3 q^{64} -16020.9 q^{65} -3950.59 q^{66} -56749.5 q^{67} -13272.3 q^{68} -9601.68 q^{69} +52577.0 q^{70} +3187.09 q^{71} -14939.0 q^{72} -7397.14 q^{73} -32803.1 q^{74} +1862.31 q^{75} +53903.7 q^{76} -30380.9 q^{77} -9061.82 q^{78} +24393.7 q^{79} -3821.15 q^{80} +6561.00 q^{81} +66093.5 q^{82} +102795. q^{83} -42572.6 q^{84} +40665.1 q^{85} -14708.7 q^{86} -35437.0 q^{87} +22316.3 q^{88} +49599.4 q^{89} +16961.6 q^{90} -69687.4 q^{91} +20099.1 q^{92} -5799.15 q^{93} +75278.1 q^{94} -165156. q^{95} +50955.1 q^{96} -92279.5 q^{97} +167727. q^{98} -9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 13q^{2} + 18q^{3} + 37q^{4} + 58q^{5} + 117q^{6} + 146q^{7} + 39q^{8} + 162q^{9} + O(q^{10}) \) \( 2q + 13q^{2} + 18q^{3} + 37q^{4} + 58q^{5} + 117q^{6} + 146q^{7} + 39q^{8} + 162q^{9} + 212q^{10} - 242q^{11} + 333q^{12} - 130q^{13} - 74q^{14} + 522q^{15} + 241q^{16} - 728q^{17} + 1053q^{18} - 828q^{19} - 1072q^{20} + 1314q^{21} - 1573q^{22} - 238q^{23} + 351q^{24} - 2918q^{25} + 376q^{26} + 1458q^{27} - 10598q^{28} + 696q^{29} + 1908q^{30} - 10480q^{31} + 1391q^{32} - 2178q^{33} - 10870q^{34} + 14464q^{35} + 2997q^{36} - 1908q^{37} + 8676q^{38} - 1170q^{39} - 10584q^{40} + 36484q^{41} - 666q^{42} + 9768q^{43} - 4477q^{44} + 4698q^{45} + 3898q^{46} + 43742q^{47} + 2169q^{48} + 40470q^{49} - 28537q^{50} - 6552q^{51} + 13468q^{52} - 12174q^{53} + 9477q^{54} - 7018q^{55} - 69786q^{56} - 7452q^{57} + 29142q^{58} - 2788q^{59} - 9648q^{60} - 25302q^{61} - 94520q^{62} + 11826q^{63} - 27199q^{64} - 15980q^{65} - 14157q^{66} - 40520q^{67} - 93262q^{68} - 2142q^{69} + 52304q^{70} + 31386q^{71} + 3159q^{72} - 46780q^{73} + 34062q^{74} - 26262q^{75} + 167436q^{76} - 17666q^{77} + 3384q^{78} - 16850q^{79} - 3736q^{80} + 13122q^{81} + 237278q^{82} + 79440q^{83} - 95382q^{84} + 40268q^{85} + 114840q^{86} + 6264q^{87} - 4719q^{88} - 54204q^{89} + 17172q^{90} - 85192q^{91} + 66382q^{92} - 94320q^{93} + 290758q^{94} - 164592q^{95} + 12519q^{96} - 241568q^{97} + 113697q^{98} - 19602q^{99} + O(q^{100}) \)

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 \(-0.396099\pi\)
0.320648 + 0.947198i \(0.396099\pi\)
\(3\) 9.00000 0.577350
\(4\) −18.8397 −0.588739
\(5\) 57.7228 1.03258 0.516289 0.856415i \(-0.327313\pi\)
0.516289 + 0.856415i \(0.327313\pi\)
\(6\) 32.6495 0.370252
\(7\) 251.081 1.93673 0.968366 0.249534i \(-0.0802775\pi\)
0.968366 + 0.249534i \(0.0802775\pi\)
\(8\) −184.432 −1.01885
\(9\) 81.0000 0.333333
\(10\) 209.402 0.662188
\(11\) −121.000 −0.301511
\(12\) −169.557 −0.339909
\(13\) −277.549 −0.455492 −0.227746 0.973721i \(-0.573136\pi\)
−0.227746 + 0.973721i \(0.573136\pi\)
\(14\) 910.853 1.24202
\(15\) 519.505 0.596159
\(16\) −66.1983 −0.0646468
\(17\) 704.489 0.591224 0.295612 0.955308i \(-0.404477\pi\)
0.295612 + 0.955308i \(0.404477\pi\)
\(18\) 293.845 0.213765
\(19\) −2861.18 −1.81828 −0.909142 0.416486i \(-0.863261\pi\)
−0.909142 + 0.416486i \(0.863261\pi\)
\(20\) −1087.48 −0.607919
\(21\) 2259.73 1.11817
\(22\) −438.954 −0.193358
\(23\) −1066.85 −0.420518 −0.210259 0.977646i \(-0.567431\pi\)
−0.210259 + 0.977646i \(0.567431\pi\)
\(24\) −1659.89 −0.588235
\(25\) 206.923 0.0662154
\(26\) −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 \(0.643145\pi\)
\(30\) 1884.62 0.382314
\(31\) −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
\(35\) 14493.1 1.99982
\(36\) −1526.01 −0.196246
\(37\) −9042.34 −1.08587 −0.542934 0.839776i \(-0.682686\pi\)
−0.542934 + 0.839776i \(0.682686\pi\)
\(38\) −10379.6 −1.16606
\(39\) −2497.94 −0.262979
\(40\) −10645.9 −1.05204
\(41\) 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 \(-0.553473\pi\)
−0.167202 + 0.985923i \(0.553473\pi\)
\(44\) 2279.60 0.177512
\(45\) 4675.55 0.344192
\(46\) −3870.24 −0.269677
\(47\) 20750.8 1.37022 0.685110 0.728439i \(-0.259754\pi\)
0.685110 + 0.728439i \(0.259754\pi\)
\(48\) −595.785 −0.0373238
\(49\) 46234.9 2.75093
\(50\) 750.659 0.0424637
\(51\) 6340.40 0.341343
\(52\) 5228.92 0.268166
\(53\) −26485.9 −1.29517 −0.647583 0.761995i \(-0.724220\pi\)
−0.647583 + 0.761995i \(0.724220\pi\)
\(54\) 2644.61 0.123417
\(55\) −6984.46 −0.311334
\(56\) −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 \(0.474418\pi\)
\(60\) −9787.30 −0.350982
\(61\) −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
\(65\) −16020.9 −0.470331
\(66\) −3950.59 −0.111635
\(67\) −56749.5 −1.54445 −0.772227 0.635347i \(-0.780857\pi\)
−0.772227 + 0.635347i \(0.780857\pi\)
\(68\) −13272.3 −0.348077
\(69\) −9601.68 −0.242786
\(70\) 52577.0 1.28248
\(71\) 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 \(-0.525885\pi\)
−0.0812319 + 0.996695i \(0.525885\pi\)
\(74\) −32803.1 −0.696362
\(75\) 1862.31 0.0382295
\(76\) 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 \(0.429434\pi\)
\(80\) −3821.15 −0.0667528
\(81\) 6561.00 0.111111
\(82\) 66093.5 1.08549
\(83\) 102795. 1.63786 0.818932 0.573890i \(-0.194566\pi\)
0.818932 + 0.573890i \(0.194566\pi\)
\(84\) −42572.6 −0.658312
\(85\) 40665.1 0.610484
\(86\) −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 \(0.392320\pi\)
\(90\) 16961.6 0.220729
\(91\) −69687.4 −0.882166
\(92\) 20099.1 0.247576
\(93\) −5799.15 −0.0695275
\(94\) 75278.1 0.878717
\(95\) −165156. −1.87752
\(96\) 50955.1 0.564299
\(97\) −92279.5 −0.995808 −0.497904 0.867232i \(-0.665897\pi\)
−0.497904 + 0.867232i \(0.665897\pi\)
\(98\) 167727. 1.76416
\(99\) −9801.00 −0.100504
\(100\) −3898.36 −0.0389836
\(101\) −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 \(-0.589691\pi\)
−0.278058 + 0.960564i \(0.589691\pi\)
\(104\) 51188.9 0.464079
\(105\) 130438. 1.15460
\(106\) −96083.5 −0.830586
\(107\) 89253.8 0.753646 0.376823 0.926285i \(-0.377017\pi\)
0.376823 + 0.926285i \(0.377017\pi\)
\(108\) −13734.1 −0.113303
\(109\) 22796.0 0.183777 0.0918887 0.995769i \(-0.470710\pi\)
0.0918887 + 0.995769i \(0.470710\pi\)
\(110\) −25337.7 −0.199657
\(111\) −81381.1 −0.626926
\(112\) −16621.2 −0.125203
\(113\) −166064. −1.22343 −0.611717 0.791077i \(-0.709521\pi\)
−0.611717 + 0.791077i \(0.709521\pi\)
\(114\) −93416.1 −0.673224
\(115\) −61581.7 −0.434218
\(116\) 74180.1 0.511850
\(117\) −22481.5 −0.151831
\(118\) 15574.2 0.102968
\(119\) 176884. 1.14504
\(120\) −95813.4 −0.607398
\(121\) 14641.0 0.0909091
\(122\) −24783.7 −0.150753
\(123\) 163971. 0.977248
\(124\) 12139.3 0.0708991
\(125\) −168440. −0.964205
\(126\) 73779.1 0.414006
\(127\) 159190. 0.875802 0.437901 0.899023i \(-0.355722\pi\)
0.437901 + 0.899023i \(0.355722\pi\)
\(128\) −98979.2 −0.533973
\(129\) −36490.9 −0.193068
\(130\) −58119.3 −0.301621
\(131\) 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
\(135\) 42079.9 0.198720
\(136\) −129930. −0.602369
\(137\) 68205.4 0.310468 0.155234 0.987878i \(-0.450387\pi\)
0.155234 + 0.987878i \(0.450387\pi\)
\(138\) −34832.2 −0.155698
\(139\) 298162. 1.30893 0.654464 0.756093i \(-0.272894\pi\)
0.654464 + 0.756093i \(0.272894\pi\)
\(140\) −273046. −1.17738
\(141\) 186757. 0.791097
\(142\) 11561.9 0.0481179
\(143\) 33583.4 0.137336
\(144\) −5362.06 −0.0215489
\(145\) −227280. −0.897722
\(146\) −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 \(0.450984\pi\)
\(150\) 6755.93 0.0245164
\(151\) 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
\(155\) −37193.7 −0.124348
\(156\) 47060.3 0.154826
\(157\) −63259.7 −0.204823 −0.102411 0.994742i \(-0.532656\pi\)
−0.102411 + 0.994742i \(0.532656\pi\)
\(158\) 88493.4 0.282012
\(159\) −238373. −0.747765
\(160\) 326808. 1.00924
\(161\) −267867. −0.814431
\(162\) 23801.5 0.0712551
\(163\) 237727. 0.700825 0.350412 0.936596i \(-0.386041\pi\)
0.350412 + 0.936596i \(0.386041\pi\)
\(164\) −343240. −0.996526
\(165\) −62860.1 −0.179749
\(166\) 372912. 1.05036
\(167\) 487880. 1.35370 0.676849 0.736122i \(-0.263345\pi\)
0.676849 + 0.736122i \(0.263345\pi\)
\(168\) −416767. −1.13925
\(169\) −294260. −0.792527
\(170\) 147521. 0.391501
\(171\) −231756. −0.606095
\(172\) 76386.1 0.196876
\(173\) 325942. 0.827991 0.413996 0.910279i \(-0.364133\pi\)
0.413996 + 0.910279i \(0.364133\pi\)
\(174\) −128555. −0.321897
\(175\) 51954.6 0.128242
\(176\) 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 \(0.318454\pi\)
\(180\) −88085.7 −0.202640
\(181\) 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
\(185\) −521950. −1.12124
\(186\) −21037.7 −0.0445877
\(187\) −85243.1 −0.178261
\(188\) −390938. −0.806703
\(189\) 183038. 0.372724
\(190\) −599138. −1.20405
\(191\) −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 \(-0.471741\pi\)
0.0886627 + 0.996062i \(0.471741\pi\)
\(194\) −334764. −0.638608
\(195\) −144188. −0.271546
\(196\) −871049. −1.61958
\(197\) 485247. 0.890836 0.445418 0.895323i \(-0.353055\pi\)
0.445418 + 0.895323i \(0.353055\pi\)
\(198\) −35555.3 −0.0644527
\(199\) 692632. 1.23985 0.619926 0.784660i \(-0.287162\pi\)
0.619926 + 0.784660i \(0.287162\pi\)
\(200\) −38163.2 −0.0674637
\(201\) −510745. −0.891690
\(202\) −128951. −0.222355
\(203\) −988619. −1.68379
\(204\) −119451. −0.200962
\(205\) 1.05165e6 1.74778
\(206\) −217216. −0.356635
\(207\) −86415.1 −0.140173
\(208\) 18373.3 0.0294461
\(209\) 346203. 0.548233
\(210\) 473193. 0.740440
\(211\) −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
\(215\) −234039. −0.345297
\(216\) −134451. −0.196078
\(217\) −161784. −0.233231
\(218\) 82697.4 0.117856
\(219\) −66574.2 −0.0937985
\(220\) 131585. 0.183294
\(221\) −195530. −0.269298
\(222\) −295228. −0.402045
\(223\) 1.13133e6 1.52345 0.761726 0.647899i \(-0.224352\pi\)
0.761726 + 0.647899i \(0.224352\pi\)
\(224\) 1.42154e6 1.89295
\(225\) 16760.8 0.0220718
\(226\) −602435. −0.784583
\(227\) −820354. −1.05666 −0.528332 0.849038i \(-0.677182\pi\)
−0.528332 + 0.849038i \(0.677182\pi\)
\(228\) 485133. 0.618051
\(229\) −1.00301e6 −1.26392 −0.631958 0.775003i \(-0.717748\pi\)
−0.631958 + 0.775003i \(0.717748\pi\)
\(230\) −223401. −0.278462
\(231\) −273428. −0.337142
\(232\) 726191. 0.885790
\(233\) 734.569 0.000886427 0 0.000443214 1.00000i \(-0.499859\pi\)
0.000443214 1.00000i \(0.499859\pi\)
\(234\) −81556.4 −0.0973685
\(235\) 1.19780e6 1.41486
\(236\) −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 \(0.705813\pi\)
\(240\) −34390.4 −0.0385397
\(241\) −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
\(245\) 2.66881e6 2.84055
\(246\) 594841. 0.626705
\(247\) 794118. 0.828214
\(248\) 118839. 0.122696
\(249\) 925158. 0.945622
\(250\) −611051. −0.618341
\(251\) 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
\(255\) 365986. 0.352463
\(256\) −1.08410e6 −1.03388
\(257\) −1.65015e6 −1.55844 −0.779220 0.626751i \(-0.784384\pi\)
−0.779220 + 0.626751i \(0.784384\pi\)
\(258\) −132379. −0.123814
\(259\) −2.27036e6 −2.10303
\(260\) 301828. 0.276902
\(261\) −318933. −0.289800
\(262\) 842262. 0.758044
\(263\) −868163. −0.773948 −0.386974 0.922091i \(-0.626480\pi\)
−0.386974 + 0.922091i \(0.626480\pi\)
\(264\) 200846. 0.177359
\(265\) −1.52884e6 −1.33736
\(266\) −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 \(0.463505\pi\)
\(270\) 152654. 0.127438
\(271\) −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
\(275\) −25037.7 −0.0199647
\(276\) 180892. 0.142938
\(277\) 811445. 0.635418 0.317709 0.948188i \(-0.397086\pi\)
0.317709 + 0.948188i \(0.397086\pi\)
\(278\) 1.08165e6 0.839411
\(279\) −52192.3 −0.0401417
\(280\) −2.67300e6 −2.03753
\(281\) −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 \(-0.467698\pi\)
0.101305 + 0.994855i \(0.467698\pi\)
\(284\) −60043.6 −0.0441744
\(285\) −1.48640e6 −1.08399
\(286\) 121831. 0.0880731
\(287\) 4.57446e6 3.27820
\(288\) 458596. 0.325798
\(289\) −923553. −0.650455
\(290\) −824509. −0.575706
\(291\) −830515. −0.574930
\(292\) 139360. 0.0956488
\(293\) −713441. −0.485500 −0.242750 0.970089i \(-0.578049\pi\)
−0.242750 + 0.970089i \(0.578049\pi\)
\(294\) 1.50954e6 1.01854
\(295\) 247811. 0.165793
\(296\) 1.66770e6 1.10634
\(297\) −88209.0 −0.0580259
\(298\) 301578. 0.196725
\(299\) 296104. 0.191543
\(300\) −35085.3 −0.0225072
\(301\) −1.01802e6 −0.647649
\(302\) 607997. 0.383605
\(303\) −319915. −0.200184
\(304\) 189405. 0.117546
\(305\) −394348. −0.242734
\(306\) 207011. 0.126383
\(307\) −2.67734e6 −1.62128 −0.810640 0.585545i \(-0.800881\pi\)
−0.810640 + 0.585545i \(0.800881\pi\)
\(308\) 572365. 0.343792
\(309\) −538891. −0.321074
\(310\) −134928. −0.0797441
\(311\) −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 \(-0.370864\pi\)
0.394656 + 0.918829i \(0.370864\pi\)
\(314\) −229488. −0.131352
\(315\) 1.17394e6 0.666608
\(316\) −459569. −0.258900
\(317\) −7686.28 −0.00429604 −0.00214802 0.999998i \(-0.500684\pi\)
−0.00214802 + 0.999998i \(0.500684\pi\)
\(318\) −864752. −0.479539
\(319\) 476431. 0.262134
\(320\) 1.30784e6 0.713971
\(321\) 803284. 0.435117
\(322\) −971746. −0.522292
\(323\) −2.01567e6 −1.07501
\(324\) −123607. −0.0654155
\(325\) −57431.3 −0.0301606
\(326\) 862406. 0.449436
\(327\) 205164. 0.106104
\(328\) −3.36017e6 −1.72455
\(329\) 5.21014e6 2.65375
\(330\) −228039. −0.115272
\(331\) 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
\(335\) −3.27574e6 −1.59477
\(336\) −149590. −0.0722862
\(337\) 4.08768e6 1.96066 0.980330 0.197365i \(-0.0632384\pi\)
0.980330 + 0.197365i \(0.0632384\pi\)
\(338\) −1.06749e6 −0.508244
\(339\) −1.49458e6 −0.706350
\(340\) −766116. −0.359416
\(341\) 77966.3 0.0363096
\(342\) −840745. −0.388686
\(343\) 7.38880e6 3.39108
\(344\) 747787. 0.340707
\(345\) −554236. −0.250696
\(346\) 1.18243e6 0.530988
\(347\) 84250.2 0.0375619 0.0187809 0.999824i \(-0.494021\pi\)
0.0187809 + 0.999824i \(0.494021\pi\)
\(348\) 667621. 0.295517
\(349\) 1.11859e6 0.491597 0.245798 0.969321i \(-0.420950\pi\)
0.245798 + 0.969321i \(0.420950\pi\)
\(350\) 188477. 0.0822408
\(351\) −202333. −0.0876595
\(352\) −685063. −0.294696
\(353\) −3.73570e6 −1.59564 −0.797820 0.602896i \(-0.794013\pi\)
−0.797820 + 0.602896i \(0.794013\pi\)
\(354\) 140168. 0.0594485
\(355\) 183968. 0.0774766
\(356\) −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 \(0.403816\pi\)
\(360\) −862321. −0.350681
\(361\) 5.71027e6 2.30616
\(362\) 86707.1 0.0347763
\(363\) 131769. 0.0524864
\(364\) 1.31289e6 0.519366
\(365\) −426984. −0.167756
\(366\) −223053. −0.0870374
\(367\) 25363.4 0.00982975 0.00491487 0.999988i \(-0.498436\pi\)
0.00491487 + 0.999988i \(0.498436\pi\)
\(368\) 70623.8 0.0271851
\(369\) 1.47574e6 0.564214
\(370\) −1.89349e6 −0.719048
\(371\) −6.65013e6 −2.50839
\(372\) 109254. 0.0409336
\(373\) −1.72695e6 −0.642699 −0.321350 0.946961i \(-0.604136\pi\)
−0.321350 + 0.946961i \(0.604136\pi\)
\(374\) −309238. −0.114318
\(375\) −1.51596e6 −0.556684
\(376\) −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 \(0.778637\pi\)
\(380\) 3.11147e6 1.10537
\(381\) 1.43271e6 0.505644
\(382\) −2.05324e6 −0.719915
\(383\) 1.29297e6 0.450392 0.225196 0.974313i \(-0.427698\pi\)
0.225196 + 0.974313i \(0.427698\pi\)
\(384\) −890813. −0.308289
\(385\) −1.75367e6 −0.602970
\(386\) 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 \(0.416243\pi\)
\(390\) −523074. −0.174141
\(391\) −751586. −0.248620
\(392\) −8.52719e6 −2.80279
\(393\) 2.08957e6 0.682456
\(394\) 1.76034e6 0.571290
\(395\) 1.40807e6 0.454080
\(396\) 184647. 0.0591705
\(397\) 819569. 0.260981 0.130491 0.991450i \(-0.458345\pi\)
0.130491 + 0.991450i \(0.458345\pi\)
\(398\) 2.51268e6 0.795113
\(399\) −6.46551e6 −2.03316
\(400\) −13698.0 −0.00428061
\(401\) 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
\(405\) 378719. 0.114731
\(406\) −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 \(0.674759\pi\)
\(410\) 3.81510e6 1.12085
\(411\) 613849. 0.179249
\(412\) 1.12806e6 0.327407
\(413\) 1.07792e6 0.310966
\(414\) −313490. −0.0898923
\(415\) 5.93363e6 1.69122
\(416\) −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 \(0.882087\pi\)
\(420\) −2.45741e6 −0.679758
\(421\) −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
\(425\) 145775. 0.0391481
\(426\) 104057. 0.0277809
\(427\) −1.71533e6 −0.455279
\(428\) −1.68151e6 −0.443701
\(429\) 302251. 0.0792910
\(430\) −849029. −0.221438
\(431\) −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 \(-0.546423\pi\)
−0.145324 + 0.989384i \(0.546423\pi\)
\(434\) −586908. −0.149570
\(435\) −2.04552e6 −0.518300
\(436\) −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 \(0.126459\pi\)
\(440\) 1.28816e6 0.317203
\(441\) 3.74503e6 0.916977
\(442\) −709328. −0.172700
\(443\) 6.72521e6 1.62816 0.814079 0.580754i \(-0.197242\pi\)
0.814079 + 0.580754i \(0.197242\pi\)
\(444\) 1.53319e6 0.369096
\(445\) 2.86302e6 0.685368
\(446\) 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 \(0.417952\pi\)
\(450\) 60803.4 0.0141546
\(451\) −2.20450e6 −0.510351
\(452\) 3.12860e6 0.720284
\(453\) 1.50838e6 0.345354
\(454\) −2.97601e6 −0.677634
\(455\) −4.02255e6 −0.910905
\(456\) 4.74924e6 1.06958
\(457\) −3.72380e6 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(458\) −3.63865e6 −0.810544
\(459\) 513572. 0.113781
\(460\) 1.16018e6 0.255641
\(461\) 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 \(-0.675316\pi\)
−0.523344 + 0.852121i \(0.675316\pi\)
\(464\) 260652. 0.0562039
\(465\) −334743. −0.0717925
\(466\) 2664.81 0.000568462 0
\(467\) 7.56254e6 1.60463 0.802316 0.596900i \(-0.203601\pi\)
0.802316 + 0.596900i \(0.203601\pi\)
\(468\) 423543. 0.0893887
\(469\) −1.42487e7 −2.99119
\(470\) 4.34526e6 0.907343
\(471\) −569337. −0.118254
\(472\) −791788. −0.163589
\(473\) 490599. 0.100826
\(474\) 796441. 0.162820
\(475\) −592045. −0.120398
\(476\) −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 \(0.554751\pi\)
\(480\) 2.94127e6 0.582682
\(481\) 2.50969e6 0.494604
\(482\) −2.40023e6 −0.470582
\(483\) −2.41080e6 −0.470212
\(484\) −275831. −0.0535218
\(485\) −5.32663e6 −1.02825
\(486\) 214213. 0.0411392
\(487\) −3.08125e6 −0.588715 −0.294357 0.955695i \(-0.595106\pi\)
−0.294357 + 0.955695i \(0.595106\pi\)
\(488\) 1.25999e6 0.239508
\(489\) 2.13954e6 0.404621
\(490\) 9.68168e6 1.82163
\(491\) −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
\(495\) −565741. −0.103778
\(496\) 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 \(0.434759\pi\)
\(500\) 3.17334e6 0.567665
\(501\) 4.39092e6 0.781558
\(502\) 4.89250e6 0.866507
\(503\) −7.65222e6 −1.34855 −0.674276 0.738480i \(-0.735544\pi\)
−0.674276 + 0.738480i \(0.735544\pi\)
\(504\) −3.75090e6 −0.657748
\(505\) −2.05182e6 −0.358023
\(506\) 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 \(0.486678\pi\)
\(510\) 1.32769e6 0.226033
\(511\) −1.85728e6 −0.314649
\(512\) −765484. −0.129051
\(513\) −2.08580e6 −0.349929
\(514\) −5.98627e6 −0.999421
\(515\) −3.45626e6 −0.574233
\(516\) 687475. 0.113667
\(517\) −2.51085e6 −0.413137
\(518\) −8.23624e6 −1.34867
\(519\) 2.93348e6 0.478041
\(520\) 2.95477e6 0.479198
\(521\) 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 \(-0.648864\pi\)
−0.450807 + 0.892622i \(0.648864\pi\)
\(524\) −4.37408e6 −0.695919
\(525\) 467591. 0.0740403
\(526\) −3.14945e6 −0.496330
\(527\) −453937. −0.0711982
\(528\) 72089.9 0.0112536
\(529\) −5.29817e6 −0.823164
\(530\) −5.54621e6 −0.857644
\(531\) 347742. 0.0535207
\(532\) 1.35342e7 2.07326
\(533\) −5.05667e6 −0.770986
\(534\) 1.61939e6 0.245753
\(535\) 5.15198e6 0.778197
\(536\) 1.04664e7 1.57357
\(537\) 4.16615e6 0.623447
\(538\) 985095. 0.146731
\(539\) −5.59442e6 −0.829437
\(540\) −792771. −0.116994
\(541\) 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
\(545\) 1.31585e6 0.189764
\(546\) −2.27526e6 −0.326624
\(547\) −4.74537e6 −0.678112 −0.339056 0.940766i \(-0.610108\pi\)
−0.339056 + 0.940766i \(0.610108\pi\)
\(548\) −1.28497e6 −0.182785
\(549\) −553372. −0.0783586
\(550\) −90829.7 −0.0128033
\(551\) 1.12657e7 1.58082
\(552\) 1.77086e6 0.247363
\(553\) 6.12480e6 0.851685
\(554\) 2.94369e6 0.407491
\(555\) −4.69755e6 −0.647349
\(556\) −5.61728e6 −0.770617
\(557\) 2.99690e6 0.409293 0.204647 0.978836i \(-0.434395\pi\)
0.204647 + 0.978836i \(0.434395\pi\)
\(558\) −189339. −0.0257427
\(559\) 1.12533e6 0.152318
\(560\) −959420. −0.129282
\(561\) −767188. −0.102919
\(562\) −6.25400e6 −0.835251
\(563\) −3.89491e6 −0.517876 −0.258938 0.965894i \(-0.583373\pi\)
−0.258938 + 0.965894i \(0.583373\pi\)
\(564\) −3.51844e6 −0.465750
\(565\) −9.58571e6 −1.26329
\(566\) 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 \(0.456978\pi\)
\(570\) −5.39224e6 −0.695156
\(571\) 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
\(575\) −220757. −0.0278448
\(576\) 1.83524e6 0.230482
\(577\) 1.26163e6 0.157759 0.0788795 0.996884i \(-0.474866\pi\)
0.0788795 + 0.996884i \(0.474866\pi\)
\(578\) −3.35039e6 −0.417134
\(579\) 825861. 0.102379
\(580\) 4.28188e6 0.528524
\(581\) 2.58100e7 3.17210
\(582\) −3.01288e6 −0.368701
\(583\) 3.20480e6 0.390508
\(584\) 1.36427e6 0.165527
\(585\) −1.29769e6 −0.156777
\(586\) −2.58816e6 −0.311349
\(587\) −1.25673e7 −1.50538 −0.752689 0.658376i \(-0.771244\pi\)
−0.752689 + 0.658376i \(0.771244\pi\)
\(588\) −7.83945e6 −0.935065
\(589\) 1.84360e6 0.218967
\(590\) 898988. 0.106322
\(591\) 4.36723e6 0.514324
\(592\) 598588. 0.0701978
\(593\) −4.32620e6 −0.505207 −0.252604 0.967570i \(-0.581287\pi\)
−0.252604 + 0.967570i \(0.581287\pi\)
\(594\) −319997. −0.0372118
\(595\) 1.02102e7 1.18234
\(596\) −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 \(0.189991\pi\)
\(600\) −343469. −0.0389502
\(601\) −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
\(605\) 845120. 0.0938706
\(606\) −1.16056e6 −0.128377
\(607\) −4.99233e6 −0.549961 −0.274980 0.961450i \(-0.588671\pi\)
−0.274980 + 0.961450i \(0.588671\pi\)
\(608\) −1.61991e7 −1.77718
\(609\) −8.89757e6 −0.972139
\(610\) −1.43058e6 −0.155664
\(611\) −5.75936e6 −0.624125
\(612\) −1.07506e6 −0.116026
\(613\) 2.99353e6 0.321760 0.160880 0.986974i \(-0.448567\pi\)
0.160880 + 0.986974i \(0.448567\pi\)
\(614\) −9.71265e6 −1.03972
\(615\) 9.46488e6 1.00908
\(616\) 5.60320e6 0.594955
\(617\) 6.27781e6 0.663889 0.331945 0.943299i \(-0.392295\pi\)
0.331945 + 0.943299i \(0.392295\pi\)
\(618\) −1.95494e6 −0.205903
\(619\) 3.96477e6 0.415902 0.207951 0.978139i \(-0.433320\pi\)
0.207951 + 0.978139i \(0.433320\pi\)
\(620\) 700716. 0.0732088
\(621\) −777736. −0.0809288
\(622\) −4.64631e6 −0.481540
\(623\) 1.24535e7 1.28550
\(624\) 165359. 0.0170007
\(625\) −1.03694e7 −1.06183
\(626\) 4.96299e6 0.506183
\(627\) 3.11583e6 0.316523
\(628\) 1.19179e6 0.120587
\(629\) −6.37023e6 −0.641990
\(630\) 4.25874e6 0.427493
\(631\) −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
\(635\) 9.18888e6 0.904333
\(636\) 4.49087e6 0.440239
\(637\) −1.28324e7 −1.25303
\(638\) 1.72836e6 0.168105
\(639\) 258154. 0.0250108
\(640\) −5.71336e6 −0.551368
\(641\) −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 \(-0.0942512\pi\)
0.956482 + 0.291791i \(0.0942512\pi\)
\(644\) 5.04652e6 0.479488
\(645\) −2.10635e6 −0.199357
\(646\) −7.31229e6 −0.689401
\(647\) 128900. 0.0121057 0.00605287 0.999982i \(-0.498073\pi\)
0.00605287 + 0.999982i \(0.498073\pi\)
\(648\) −1.21006e6 −0.113206
\(649\) −519467. −0.0484113
\(650\) −208345. −0.0193419
\(651\) −1.45606e6 −0.134656
\(652\) −4.47869e6 −0.412603
\(653\) 3.03250e6 0.278303 0.139151 0.990271i \(-0.455563\pi\)
0.139151 + 0.990271i \(0.455563\pi\)
\(654\) 744277. 0.0680441
\(655\) 1.34017e7 1.22056
\(656\) −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 \(0.752981\pi\)
\(660\) 1.18426e6 0.105825
\(661\) 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
\(665\) −4.14675e7 −3.63625
\(666\) −2.65705e6 −0.232121
\(667\) 4.20067e6 0.365598
\(668\) −9.19149e6 −0.796975
\(669\) 1.01820e7 0.879565
\(670\) −1.18835e7 −1.02272
\(671\) 826643. 0.0708780
\(672\) 1.27939e7 1.09290
\(673\) −5.66035e6 −0.481732 −0.240866 0.970558i \(-0.577431\pi\)
−0.240866 + 0.970558i \(0.577431\pi\)
\(674\) 1.48290e7 1.25736
\(675\) 150847. 0.0127432
\(676\) 5.54375e6 0.466592
\(677\) −1.53601e7 −1.28802 −0.644012 0.765016i \(-0.722731\pi\)
−0.644012 + 0.765016i \(0.722731\pi\)
\(678\) −5.42192e6 −0.452979
\(679\) −2.31697e7 −1.92861
\(680\) −7.49994e6 −0.621993
\(681\) −7.38319e6 −0.610065
\(682\) 282840. 0.0232852
\(683\) 1.47069e6 0.120634 0.0603170 0.998179i \(-0.480789\pi\)
0.0603170 + 0.998179i \(0.480789\pi\)
\(684\) 4.36620e6 0.356832
\(685\) 3.93701e6 0.320583
\(686\) 2.68045e7 2.17469
\(687\) −9.02712e6 −0.729722
\(688\) 268404. 0.0216181
\(689\) 7.35114e6 0.589939
\(690\) −2.01061e6 −0.160770
\(691\) −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
\(695\) 1.72108e7 1.35157
\(696\) 6.53571e6 0.511411
\(697\) 1.28351e7 1.00073
\(698\) 4.05794e6 0.315259
\(699\) 6611.12 0.000511779 0
\(700\) −978806. −0.0755008
\(701\) −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
\(705\) 1.07802e7 0.816869
\(706\) −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 \(0.687018\pi\)
\(710\) 667383. 0.0496854
\(711\) 1.97589e6 0.146585
\(712\) −9.14772e6 −0.676258
\(713\) 687427. 0.0506410
\(714\) 5.77517e6 0.423954
\(715\) 1.93853e6 0.141810
\(716\) −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 \(0.334819\pi\)
\(720\) −309513. −0.0222509
\(721\) −1.50339e7 −1.07705
\(722\) 2.07153e7 1.47893
\(723\) −5.95473e6 −0.423659
\(724\) −450292. −0.0319262
\(725\) −814748. −0.0575676
\(726\) 478021. 0.0336593
\(727\) −8.20549e6 −0.575796 −0.287898 0.957661i \(-0.592956\pi\)
−0.287898 + 0.957661i \(0.592956\pi\)
\(728\) 1.28526e7 0.898797
\(729\) 531441. 0.0370370
\(730\) −1.54898e6 −0.107582
\(731\) −2.85638e6 −0.197707
\(732\) 1.15837e6 0.0799043
\(733\) 1.20636e7 0.829312 0.414656 0.909978i \(-0.363902\pi\)
0.414656 + 0.909978i \(0.363902\pi\)
\(734\) 92011.3 0.00630378
\(735\) 2.40193e7 1.63999
\(736\) −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 \(0.611986\pi\)
\(740\) 9.83335e6 0.660119
\(741\) 7.14706e6 0.478170
\(742\) −2.41248e7 −1.60862
\(743\) 2.41980e7 1.60808 0.804039 0.594577i \(-0.202680\pi\)
0.804039 + 0.594577i \(0.202680\pi\)
\(744\) 1.06955e6 0.0708383
\(745\) 4.79858e6 0.316754
\(746\) −6.26489e6 −0.412161
\(747\) 8.32642e6 0.545955
\(748\) 1.60595e6 0.104949
\(749\) 2.24100e7 1.45961
\(750\) −5.49946e6 −0.356999
\(751\) 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
\(755\) 9.67420e6 0.617657
\(756\) −3.44838e6 −0.219437
\(757\) −2.08747e7 −1.32398 −0.661989 0.749513i \(-0.730288\pi\)
−0.661989 + 0.749513i \(0.730288\pi\)
\(758\) −1.55775e7 −0.984744
\(759\) 1.16180e6 0.0732028
\(760\) 3.04600e7 1.91291
\(761\) 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
\(765\) 3.29387e6 0.203495
\(766\) 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 \(0.529863\pi\)
\(770\) −6.36182e6 −0.386682
\(771\) −1.48513e7 −0.899765
\(772\) −1.72877e6 −0.104398
\(773\) −3.02076e7 −1.81831 −0.909153 0.416461i \(-0.863270\pi\)
−0.909153 + 0.416461i \(0.863270\pi\)
\(774\) −1.19141e6 −0.0714838
\(775\) −133331. −0.00797401
\(776\) 1.70193e7 1.01458
\(777\) −2.04333e7 −1.21419
\(778\) 5.63229e6 0.333608
\(779\) −5.21280e7 −3.07771
\(780\) 2.71645e6 0.159870
\(781\) −385638. −0.0226231
\(782\) −2.72654e6 −0.159439
\(783\) −2.87040e6 −0.167316
\(784\) −3.06067e6 −0.177839
\(785\) −3.65153e6 −0.211495
\(786\) 7.58036e6 0.437657
\(787\) −2.07854e7 −1.19625 −0.598126 0.801402i \(-0.704088\pi\)
−0.598126 + 0.801402i \(0.704088\pi\)
\(788\) −9.14190e6 −0.524470
\(789\) −7.81346e6 −0.446839
\(790\) 5.10809e6 0.291200
\(791\) −4.16957e7 −2.36946
\(792\) 1.80762e6 0.102399
\(793\) 1.89615e6 0.107075
\(794\) 2.97317e6 0.167366
\(795\) −1.37596e7 −0.772125
\(796\) −1.30490e7 −0.729950
\(797\) −7.95535e6 −0.443622 −0.221811 0.975090i \(-0.571197\pi\)
−0.221811 + 0.975090i \(0.571197\pi\)
\(798\) −2.34551e7 −1.30385
\(799\) 1.46187e7 0.810106
\(800\) 1.17153e6 0.0647186
\(801\) 4.01755e6 0.221248
\(802\) 5.04094e6 0.276742
\(803\) 895054. 0.0489847
\(804\) 9.62226e6 0.524973
\(805\) −1.54620e7 −0.840963
\(806\) 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 \(0.805085\pi\)
\(810\) 1.37389e6 0.0735764
\(811\) 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
\(815\) 1.37223e7 0.723655
\(816\) −419723. −0.0220667
\(817\) 1.16008e7 0.608040
\(818\) −1.28092e7 −0.669325
\(819\) −5.64468e6 −0.294055
\(820\) −1.98128e7 −1.02899
\(821\) 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 \(-0.432014\pi\)
0.211964 + 0.977278i \(0.432014\pi\)
\(824\) 1.10432e7 0.566600
\(825\) −225339. −0.0115266
\(826\) 3.91040e6 0.199421
\(827\) 2.21035e7 1.12382 0.561909 0.827199i \(-0.310067\pi\)
0.561909 + 0.827199i \(0.310067\pi\)
\(828\) 1.62803e6 0.0825252
\(829\) −5.57875e6 −0.281936 −0.140968 0.990014i \(-0.545021\pi\)
−0.140968 + 0.990014i \(0.545021\pi\)
\(830\) 2.15256e7 1.08457
\(831\) 7.30300e6 0.366859
\(832\) −6.28851e6 −0.314948
\(833\) 3.25720e7 1.62641
\(834\) 9.73484e6 0.484634
\(835\) 2.81618e7 1.39780
\(836\) −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 \(0.694138\pi\)
\(840\) −2.40570e7 −1.17637
\(841\) −5.00769e6 −0.244145
\(842\) −1.81794e7 −0.883688
\(843\) −1.55155e7 −0.751965
\(844\) 8.30866e6 0.401490
\(845\) −1.69855e7 −0.818345
\(846\) 6.09753e6 0.292906
\(847\) 3.67608e6 0.176067
\(848\) 1.75332e6 0.0837284
\(849\) 2.45681e6 0.116977
\(850\) 528831. 0.0251055
\(851\) 9.64685e6 0.456627
\(852\) −540393. −0.0255041
\(853\) 8.94855e6 0.421095 0.210548 0.977584i \(-0.432475\pi\)
0.210548 + 0.977584i \(0.432475\pi\)
\(854\) −6.22273e6 −0.291969
\(855\) −1.33776e7 −0.625839
\(856\) −1.64612e7 −0.767853
\(857\) −2.03190e6 −0.0945039 −0.0472520 0.998883i \(-0.515046\pi\)
−0.0472520 + 0.998883i \(0.515046\pi\)
\(858\) 1.09648e6 0.0508490
\(859\) −1.92359e7 −0.889467 −0.444734 0.895663i \(-0.646702\pi\)
−0.444734 + 0.895663i \(0.646702\pi\)
\(860\) 4.40922e6 0.203290
\(861\) 4.11701e7 1.89267
\(862\) −4.16247e6 −0.190802
\(863\) 2.00380e7 0.915856 0.457928 0.888989i \(-0.348592\pi\)
0.457928 + 0.888989i \(0.348592\pi\)
\(864\) 4.12736e6 0.188100
\(865\) 1.88143e7 0.854965
\(866\) −4.11360e6 −0.186392
\(867\) −8.31197e6 −0.375540
\(868\) 3.04796e6 0.137312
\(869\) −2.95164e6 −0.132591
\(870\) −7.42058e6 −0.332384
\(871\) 1.57507e7 0.703486
\(872\) −4.20431e6 −0.187242
\(873\) −7.47464e6 −0.331936
\(874\) 1.10735e7 0.490349
\(875\) −4.22921e7 −1.86741
\(876\) 1.25424e6 0.0552229
\(877\) 1.98487e7 0.871430 0.435715 0.900085i \(-0.356496\pi\)
0.435715 + 0.900085i \(0.356496\pi\)
\(878\) 2.70153e7 1.18270
\(879\) −6.42097e6 −0.280304
\(880\) 462359. 0.0201267
\(881\) −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 \(-0.187602\pi\)
0.831291 + 0.555838i \(0.187602\pi\)
\(884\) 3.68372e6 0.158546
\(885\) 2.23030e6 0.0957204
\(886\) 2.43972e7 1.04413
\(887\) −2.68797e6 −0.114714 −0.0573569 0.998354i \(-0.518267\pi\)
−0.0573569 + 0.998354i \(0.518267\pi\)
\(888\) 1.50093e7 0.638745
\(889\) 3.99696e7 1.69619
\(890\) 1.03862e7 0.439524
\(891\) −793881. −0.0335013
\(892\) −2.13139e7 −0.896916
\(893\) −5.93719e7 −2.49145
\(894\) 2.71420e6 0.113579
\(895\) 2.67202e7 1.11502
\(896\) −2.48519e7 −1.03416
\(897\) 2.66493e6 0.110587
\(898\) 7.90093e6 0.326955
\(899\) 2.53709e6 0.104698
\(900\) −315767. −0.0129945
\(901\) −1.86590e7 −0.765733
\(902\) −7.99731e6 −0.327286
\(903\) −9.16218e6 −0.373921
\(904\) 3.06276e7 1.24650
\(905\) 1.37965e6 0.0559947
\(906\) 5.47197e6 0.221474
\(907\) −7.78340e6 −0.314160 −0.157080 0.987586i \(-0.550208\pi\)
−0.157080 + 0.987586i \(0.550208\pi\)
\(908\) 1.54552e7 0.622099
\(909\) −2.87924e6 −0.115576
\(910\) −1.45927e7 −0.584160
\(911\) 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
\(915\) −3.54913e6 −0.140142
\(916\) 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 \(0.192560\pi\)
\(920\) 1.13576e7 0.442404
\(921\) −2.40961e7 −0.936047
\(922\) 1.72027e7 0.666454
\(923\) −884572. −0.0341766
\(924\) 5.15128e6 0.198489
\(925\) −1.87107e6 −0.0719011
\(926\) −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 \(0.442599\pi\)
\(930\) −1.21435e6 −0.0460403
\(931\) −1.32287e8 −5.00197
\(932\) −13839.0 −0.000521874 0
\(933\) −1.15270e7 −0.433524
\(934\) 2.74348e7 1.02904
\(935\) −4.92047e6 −0.184068
\(936\) 4.14630e6 0.154693
\(937\) 2.17869e7 0.810674 0.405337 0.914167i \(-0.367154\pi\)
0.405337 + 0.914167i \(0.367154\pi\)
\(938\) −5.16904e7 −1.91824
\(939\) 1.23127e7 0.455710
\(940\) −2.25660e7 −0.832983
\(941\) 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
\(945\) 1.05655e7 0.384867
\(946\) 1.77976e6 0.0646595
\(947\) 2.42335e7 0.878094 0.439047 0.898464i \(-0.355316\pi\)
0.439047 + 0.898464i \(0.355316\pi\)
\(948\) −4.13612e6 −0.149476
\(949\) 2.05307e6 0.0740010
\(950\) −2.14777e6 −0.0772110
\(951\) −69176.5 −0.00248032
\(952\) −3.26231e7 −1.16663
\(953\) 5.56092e7 1.98342 0.991710 0.128499i \(-0.0410159\pi\)
0.991710 + 0.128499i \(0.0410159\pi\)
\(954\) −7.78277e6 −0.276862
\(955\) −3.26703e7 −1.15916
\(956\) 2.00459e7 0.709383
\(957\) 4.28788e6 0.151343
\(958\) −6.23595e6 −0.219528
\(959\) 1.71251e7 0.601294
\(960\) 1.17706e7 0.412212
\(961\) −2.82140e7 −0.985498
\(962\) 9.10446e6 0.317188
\(963\) 7.22956e6 0.251215
\(964\) 1.24650e7 0.432016
\(965\) 5.29678e6 0.183102
\(966\) −8.74571e6 −0.301545
\(967\) −2.09122e7 −0.719173 −0.359586 0.933112i \(-0.617082\pi\)
−0.359586 + 0.933112i \(0.617082\pi\)
\(968\) −2.70027e6 −0.0926229
\(969\) −1.81410e7 −0.620659
\(970\) −1.93235e7 −0.659412
\(971\) 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
\(975\) −516882. −0.0174132
\(976\) 452251. 0.0151969
\(977\) −3.45472e7 −1.15792 −0.578958 0.815358i \(-0.696540\pi\)
−0.578958 + 0.815358i \(0.696540\pi\)
\(978\) 7.76166e6 0.259482
\(979\) −6.00153e6 −0.200127
\(980\) −5.02794e7 −1.67234
\(981\) 1.84648e6 0.0612592
\(982\) −3.84210e6 −0.127142
\(983\) −3.40232e7 −1.12303 −0.561516 0.827466i \(-0.689782\pi\)
−0.561516 + 0.827466i \(0.689782\pi\)
\(984\) −3.02415e7 −0.995671
\(985\) 2.80099e7 0.919857
\(986\) −1.00629e7 −0.329632
\(987\) 4.68913e7 1.53214
\(988\) −1.49609e7 −0.487602
\(989\) 4.32560e6 0.140623
\(990\) −2.05235e6 −0.0665524
\(991\) −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
\(995\) 3.99807e7 1.28024
\(996\) −1.74297e7 −0.556725
\(997\) −5.86494e6 −0.186864 −0.0934320 0.995626i \(-0.529784\pi\)
−0.0934320 + 0.995626i \(0.529784\pi\)
\(998\) 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 33.6.a.e.1.1 2
3.2 odd 2 99.6.a.d.1.2 2
4.3 odd 2 528.6.a.o.1.2 2
5.4 even 2 825.6.a.c.1.2 2
11.10 odd 2 363.6.a.f.1.2 2
33.32 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 1.1 even 1 trivial
99.6.a.d.1.2 2 3.2 odd 2
363.6.a.f.1.2 2 11.10 odd 2
528.6.a.o.1.2 2 4.3 odd 2
825.6.a.c.1.2 2 5.4 even 2
1089.6.a.p.1.1 2 33.32 even 2