Properties

Label 546.6.a.i.1.1
Level $546$
Weight $6$
Character 546.1
Self dual yes
Analytic conductor $87.570$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(87.5695656179\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +24.0000 q^{5} +36.0000 q^{6} +49.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +24.0000 q^{5} +36.0000 q^{6} +49.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +96.0000 q^{10} +374.000 q^{11} +144.000 q^{12} -169.000 q^{13} +196.000 q^{14} +216.000 q^{15} +256.000 q^{16} +1786.00 q^{17} +324.000 q^{18} +584.000 q^{19} +384.000 q^{20} +441.000 q^{21} +1496.00 q^{22} -884.000 q^{23} +576.000 q^{24} -2549.00 q^{25} -676.000 q^{26} +729.000 q^{27} +784.000 q^{28} -5878.00 q^{29} +864.000 q^{30} +6196.00 q^{31} +1024.00 q^{32} +3366.00 q^{33} +7144.00 q^{34} +1176.00 q^{35} +1296.00 q^{36} -2690.00 q^{37} +2336.00 q^{38} -1521.00 q^{39} +1536.00 q^{40} +1848.00 q^{41} +1764.00 q^{42} +22164.0 q^{43} +5984.00 q^{44} +1944.00 q^{45} -3536.00 q^{46} -4190.00 q^{47} +2304.00 q^{48} +2401.00 q^{49} -10196.0 q^{50} +16074.0 q^{51} -2704.00 q^{52} -1082.00 q^{53} +2916.00 q^{54} +8976.00 q^{55} +3136.00 q^{56} +5256.00 q^{57} -23512.0 q^{58} -14286.0 q^{59} +3456.00 q^{60} -8390.00 q^{61} +24784.0 q^{62} +3969.00 q^{63} +4096.00 q^{64} -4056.00 q^{65} +13464.0 q^{66} -21508.0 q^{67} +28576.0 q^{68} -7956.00 q^{69} +4704.00 q^{70} +19266.0 q^{71} +5184.00 q^{72} +71926.0 q^{73} -10760.0 q^{74} -22941.0 q^{75} +9344.00 q^{76} +18326.0 q^{77} -6084.00 q^{78} -7184.00 q^{79} +6144.00 q^{80} +6561.00 q^{81} +7392.00 q^{82} +99134.0 q^{83} +7056.00 q^{84} +42864.0 q^{85} +88656.0 q^{86} -52902.0 q^{87} +23936.0 q^{88} -19940.0 q^{89} +7776.00 q^{90} -8281.00 q^{91} -14144.0 q^{92} +55764.0 q^{93} -16760.0 q^{94} +14016.0 q^{95} +9216.00 q^{96} -42322.0 q^{97} +9604.00 q^{98} +30294.0 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 0.707107
\(3\) 9.00000 0.577350
\(4\) 16.0000 0.500000
\(5\) 24.0000 0.429325 0.214663 0.976688i \(-0.431135\pi\)
0.214663 + 0.976688i \(0.431135\pi\)
\(6\) 36.0000 0.408248
\(7\) 49.0000 0.377964
\(8\) 64.0000 0.353553
\(9\) 81.0000 0.333333
\(10\) 96.0000 0.303579
\(11\) 374.000 0.931944 0.465972 0.884799i \(-0.345705\pi\)
0.465972 + 0.884799i \(0.345705\pi\)
\(12\) 144.000 0.288675
\(13\) −169.000 −0.277350
\(14\) 196.000 0.267261
\(15\) 216.000 0.247871
\(16\) 256.000 0.250000
\(17\) 1786.00 1.49885 0.749427 0.662087i \(-0.230329\pi\)
0.749427 + 0.662087i \(0.230329\pi\)
\(18\) 324.000 0.235702
\(19\) 584.000 0.371132 0.185566 0.982632i \(-0.440588\pi\)
0.185566 + 0.982632i \(0.440588\pi\)
\(20\) 384.000 0.214663
\(21\) 441.000 0.218218
\(22\) 1496.00 0.658984
\(23\) −884.000 −0.348444 −0.174222 0.984706i \(-0.555741\pi\)
−0.174222 + 0.984706i \(0.555741\pi\)
\(24\) 576.000 0.204124
\(25\) −2549.00 −0.815680
\(26\) −676.000 −0.196116
\(27\) 729.000 0.192450
\(28\) 784.000 0.188982
\(29\) −5878.00 −1.29788 −0.648940 0.760840i \(-0.724787\pi\)
−0.648940 + 0.760840i \(0.724787\pi\)
\(30\) 864.000 0.175271
\(31\) 6196.00 1.15800 0.578998 0.815329i \(-0.303444\pi\)
0.578998 + 0.815329i \(0.303444\pi\)
\(32\) 1024.00 0.176777
\(33\) 3366.00 0.538058
\(34\) 7144.00 1.05985
\(35\) 1176.00 0.162270
\(36\) 1296.00 0.166667
\(37\) −2690.00 −0.323034 −0.161517 0.986870i \(-0.551639\pi\)
−0.161517 + 0.986870i \(0.551639\pi\)
\(38\) 2336.00 0.262430
\(39\) −1521.00 −0.160128
\(40\) 1536.00 0.151789
\(41\) 1848.00 0.171689 0.0858445 0.996309i \(-0.472641\pi\)
0.0858445 + 0.996309i \(0.472641\pi\)
\(42\) 1764.00 0.154303
\(43\) 22164.0 1.82800 0.914002 0.405710i \(-0.132976\pi\)
0.914002 + 0.405710i \(0.132976\pi\)
\(44\) 5984.00 0.465972
\(45\) 1944.00 0.143108
\(46\) −3536.00 −0.246387
\(47\) −4190.00 −0.276675 −0.138337 0.990385i \(-0.544176\pi\)
−0.138337 + 0.990385i \(0.544176\pi\)
\(48\) 2304.00 0.144338
\(49\) 2401.00 0.142857
\(50\) −10196.0 −0.576773
\(51\) 16074.0 0.865363
\(52\) −2704.00 −0.138675
\(53\) −1082.00 −0.0529100 −0.0264550 0.999650i \(-0.508422\pi\)
−0.0264550 + 0.999650i \(0.508422\pi\)
\(54\) 2916.00 0.136083
\(55\) 8976.00 0.400107
\(56\) 3136.00 0.133631
\(57\) 5256.00 0.214273
\(58\) −23512.0 −0.917740
\(59\) −14286.0 −0.534294 −0.267147 0.963656i \(-0.586081\pi\)
−0.267147 + 0.963656i \(0.586081\pi\)
\(60\) 3456.00 0.123935
\(61\) −8390.00 −0.288694 −0.144347 0.989527i \(-0.546108\pi\)
−0.144347 + 0.989527i \(0.546108\pi\)
\(62\) 24784.0 0.818827
\(63\) 3969.00 0.125988
\(64\) 4096.00 0.125000
\(65\) −4056.00 −0.119073
\(66\) 13464.0 0.380465
\(67\) −21508.0 −0.585346 −0.292673 0.956213i \(-0.594545\pi\)
−0.292673 + 0.956213i \(0.594545\pi\)
\(68\) 28576.0 0.749427
\(69\) −7956.00 −0.201174
\(70\) 4704.00 0.114742
\(71\) 19266.0 0.453571 0.226786 0.973945i \(-0.427178\pi\)
0.226786 + 0.973945i \(0.427178\pi\)
\(72\) 5184.00 0.117851
\(73\) 71926.0 1.57972 0.789858 0.613290i \(-0.210154\pi\)
0.789858 + 0.613290i \(0.210154\pi\)
\(74\) −10760.0 −0.228419
\(75\) −22941.0 −0.470933
\(76\) 9344.00 0.185566
\(77\) 18326.0 0.352242
\(78\) −6084.00 −0.113228
\(79\) −7184.00 −0.129509 −0.0647543 0.997901i \(-0.520626\pi\)
−0.0647543 + 0.997901i \(0.520626\pi\)
\(80\) 6144.00 0.107331
\(81\) 6561.00 0.111111
\(82\) 7392.00 0.121402
\(83\) 99134.0 1.57953 0.789764 0.613411i \(-0.210203\pi\)
0.789764 + 0.613411i \(0.210203\pi\)
\(84\) 7056.00 0.109109
\(85\) 42864.0 0.643495
\(86\) 88656.0 1.29259
\(87\) −52902.0 −0.749331
\(88\) 23936.0 0.329492
\(89\) −19940.0 −0.266840 −0.133420 0.991060i \(-0.542596\pi\)
−0.133420 + 0.991060i \(0.542596\pi\)
\(90\) 7776.00 0.101193
\(91\) −8281.00 −0.104828
\(92\) −14144.0 −0.174222
\(93\) 55764.0 0.668569
\(94\) −16760.0 −0.195639
\(95\) 14016.0 0.159336
\(96\) 9216.00 0.102062
\(97\) −42322.0 −0.456706 −0.228353 0.973578i \(-0.573334\pi\)
−0.228353 + 0.973578i \(0.573334\pi\)
\(98\) 9604.00 0.101015
\(99\) 30294.0 0.310648
\(100\) −40784.0 −0.407840
\(101\) −162462. −1.58470 −0.792352 0.610064i \(-0.791144\pi\)
−0.792352 + 0.610064i \(0.791144\pi\)
\(102\) 64296.0 0.611904
\(103\) −120600. −1.12009 −0.560047 0.828461i \(-0.689217\pi\)
−0.560047 + 0.828461i \(0.689217\pi\)
\(104\) −10816.0 −0.0980581
\(105\) 10584.0 0.0936864
\(106\) −4328.00 −0.0374130
\(107\) 42992.0 0.363018 0.181509 0.983389i \(-0.441902\pi\)
0.181509 + 0.983389i \(0.441902\pi\)
\(108\) 11664.0 0.0962250
\(109\) 204522. 1.64882 0.824411 0.565992i \(-0.191507\pi\)
0.824411 + 0.565992i \(0.191507\pi\)
\(110\) 35904.0 0.282918
\(111\) −24210.0 −0.186504
\(112\) 12544.0 0.0944911
\(113\) 79046.0 0.582350 0.291175 0.956670i \(-0.405954\pi\)
0.291175 + 0.956670i \(0.405954\pi\)
\(114\) 21024.0 0.151514
\(115\) −21216.0 −0.149596
\(116\) −94048.0 −0.648940
\(117\) −13689.0 −0.0924500
\(118\) −57144.0 −0.377803
\(119\) 87514.0 0.566513
\(120\) 13824.0 0.0876356
\(121\) −21175.0 −0.131480
\(122\) −33560.0 −0.204137
\(123\) 16632.0 0.0991247
\(124\) 99136.0 0.578998
\(125\) −136176. −0.779517
\(126\) 15876.0 0.0890871
\(127\) 125904. 0.692676 0.346338 0.938110i \(-0.387425\pi\)
0.346338 + 0.938110i \(0.387425\pi\)
\(128\) 16384.0 0.0883883
\(129\) 199476. 1.05540
\(130\) −16224.0 −0.0841976
\(131\) 159568. 0.812396 0.406198 0.913785i \(-0.366854\pi\)
0.406198 + 0.913785i \(0.366854\pi\)
\(132\) 53856.0 0.269029
\(133\) 28616.0 0.140275
\(134\) −86032.0 −0.413902
\(135\) 17496.0 0.0826236
\(136\) 114304. 0.529925
\(137\) −1824.00 −0.00830278 −0.00415139 0.999991i \(-0.501321\pi\)
−0.00415139 + 0.999991i \(0.501321\pi\)
\(138\) −31824.0 −0.142252
\(139\) 135268. 0.593824 0.296912 0.954905i \(-0.404043\pi\)
0.296912 + 0.954905i \(0.404043\pi\)
\(140\) 18816.0 0.0811348
\(141\) −37710.0 −0.159738
\(142\) 77064.0 0.320723
\(143\) −63206.0 −0.258475
\(144\) 20736.0 0.0833333
\(145\) −141072. −0.557212
\(146\) 287704. 1.11703
\(147\) 21609.0 0.0824786
\(148\) −43040.0 −0.161517
\(149\) −138700. −0.511812 −0.255906 0.966702i \(-0.582374\pi\)
−0.255906 + 0.966702i \(0.582374\pi\)
\(150\) −91764.0 −0.333000
\(151\) −206196. −0.735932 −0.367966 0.929839i \(-0.619946\pi\)
−0.367966 + 0.929839i \(0.619946\pi\)
\(152\) 37376.0 0.131215
\(153\) 144666. 0.499618
\(154\) 73304.0 0.249073
\(155\) 148704. 0.497157
\(156\) −24336.0 −0.0800641
\(157\) −40802.0 −0.132109 −0.0660545 0.997816i \(-0.521041\pi\)
−0.0660545 + 0.997816i \(0.521041\pi\)
\(158\) −28736.0 −0.0915764
\(159\) −9738.00 −0.0305476
\(160\) 24576.0 0.0758947
\(161\) −43316.0 −0.131699
\(162\) 26244.0 0.0785674
\(163\) −76904.0 −0.226715 −0.113357 0.993554i \(-0.536161\pi\)
−0.113357 + 0.993554i \(0.536161\pi\)
\(164\) 29568.0 0.0858445
\(165\) 80784.0 0.231002
\(166\) 396536. 1.11690
\(167\) 287434. 0.797530 0.398765 0.917053i \(-0.369439\pi\)
0.398765 + 0.917053i \(0.369439\pi\)
\(168\) 28224.0 0.0771517
\(169\) 28561.0 0.0769231
\(170\) 171456. 0.455020
\(171\) 47304.0 0.123711
\(172\) 354624. 0.914002
\(173\) 113466. 0.288238 0.144119 0.989560i \(-0.453965\pi\)
0.144119 + 0.989560i \(0.453965\pi\)
\(174\) −211608. −0.529857
\(175\) −124901. −0.308298
\(176\) 95744.0 0.232986
\(177\) −128574. −0.308475
\(178\) −79760.0 −0.188684
\(179\) 87524.0 0.204171 0.102086 0.994776i \(-0.467448\pi\)
0.102086 + 0.994776i \(0.467448\pi\)
\(180\) 31104.0 0.0715542
\(181\) −320810. −0.727866 −0.363933 0.931425i \(-0.618566\pi\)
−0.363933 + 0.931425i \(0.618566\pi\)
\(182\) −33124.0 −0.0741249
\(183\) −75510.0 −0.166677
\(184\) −56576.0 −0.123193
\(185\) −64560.0 −0.138687
\(186\) 223056. 0.472750
\(187\) 667964. 1.39685
\(188\) −67040.0 −0.138337
\(189\) 35721.0 0.0727393
\(190\) 56064.0 0.112668
\(191\) −609128. −1.20816 −0.604081 0.796923i \(-0.706460\pi\)
−0.604081 + 0.796923i \(0.706460\pi\)
\(192\) 36864.0 0.0721688
\(193\) −16118.0 −0.0311471 −0.0155736 0.999879i \(-0.504957\pi\)
−0.0155736 + 0.999879i \(0.504957\pi\)
\(194\) −169288. −0.322940
\(195\) −36504.0 −0.0687470
\(196\) 38416.0 0.0714286
\(197\) −267048. −0.490257 −0.245128 0.969491i \(-0.578830\pi\)
−0.245128 + 0.969491i \(0.578830\pi\)
\(198\) 121176. 0.219661
\(199\) 227872. 0.407904 0.203952 0.978981i \(-0.434621\pi\)
0.203952 + 0.978981i \(0.434621\pi\)
\(200\) −163136. −0.288386
\(201\) −193572. −0.337950
\(202\) −649848. −1.12056
\(203\) −288022. −0.490553
\(204\) 257184. 0.432682
\(205\) 44352.0 0.0737104
\(206\) −482400. −0.792026
\(207\) −71604.0 −0.116148
\(208\) −43264.0 −0.0693375
\(209\) 218416. 0.345875
\(210\) 42336.0 0.0662463
\(211\) −341948. −0.528754 −0.264377 0.964419i \(-0.585166\pi\)
−0.264377 + 0.964419i \(0.585166\pi\)
\(212\) −17312.0 −0.0264550
\(213\) 173394. 0.261870
\(214\) 171968. 0.256693
\(215\) 531936. 0.784808
\(216\) 46656.0 0.0680414
\(217\) 303604. 0.437681
\(218\) 818088. 1.16589
\(219\) 647334. 0.912049
\(220\) 143616. 0.200053
\(221\) −301834. −0.415707
\(222\) −96840.0 −0.131878
\(223\) 316596. 0.426327 0.213164 0.977016i \(-0.431623\pi\)
0.213164 + 0.977016i \(0.431623\pi\)
\(224\) 50176.0 0.0668153
\(225\) −206469. −0.271893
\(226\) 316184. 0.411783
\(227\) 984474. 1.26806 0.634030 0.773309i \(-0.281400\pi\)
0.634030 + 0.773309i \(0.281400\pi\)
\(228\) 84096.0 0.107137
\(229\) −51734.0 −0.0651910 −0.0325955 0.999469i \(-0.510377\pi\)
−0.0325955 + 0.999469i \(0.510377\pi\)
\(230\) −84864.0 −0.105780
\(231\) 164934. 0.203367
\(232\) −376192. −0.458870
\(233\) 739926. 0.892891 0.446446 0.894811i \(-0.352690\pi\)
0.446446 + 0.894811i \(0.352690\pi\)
\(234\) −54756.0 −0.0653720
\(235\) −100560. −0.118783
\(236\) −228576. −0.267147
\(237\) −64656.0 −0.0747718
\(238\) 350056. 0.400585
\(239\) −142014. −0.160819 −0.0804093 0.996762i \(-0.525623\pi\)
−0.0804093 + 0.996762i \(0.525623\pi\)
\(240\) 55296.0 0.0619677
\(241\) −174810. −0.193876 −0.0969379 0.995290i \(-0.530905\pi\)
−0.0969379 + 0.995290i \(0.530905\pi\)
\(242\) −84700.0 −0.0929705
\(243\) 59049.0 0.0641500
\(244\) −134240. −0.144347
\(245\) 57624.0 0.0613322
\(246\) 66528.0 0.0700917
\(247\) −98696.0 −0.102934
\(248\) 396544. 0.409414
\(249\) 892206. 0.911941
\(250\) −544704. −0.551202
\(251\) −428960. −0.429766 −0.214883 0.976640i \(-0.568937\pi\)
−0.214883 + 0.976640i \(0.568937\pi\)
\(252\) 63504.0 0.0629941
\(253\) −330616. −0.324730
\(254\) 503616. 0.489796
\(255\) 385776. 0.371522
\(256\) 65536.0 0.0625000
\(257\) −1.57731e6 −1.48965 −0.744827 0.667258i \(-0.767468\pi\)
−0.744827 + 0.667258i \(0.767468\pi\)
\(258\) 797904. 0.746279
\(259\) −131810. −0.122095
\(260\) −64896.0 −0.0595367
\(261\) −476118. −0.432627
\(262\) 638272. 0.574450
\(263\) −1.02049e6 −0.909746 −0.454873 0.890556i \(-0.650315\pi\)
−0.454873 + 0.890556i \(0.650315\pi\)
\(264\) 215424. 0.190232
\(265\) −25968.0 −0.0227156
\(266\) 114464. 0.0991893
\(267\) −179460. −0.154060
\(268\) −344128. −0.292673
\(269\) 575954. 0.485296 0.242648 0.970114i \(-0.421984\pi\)
0.242648 + 0.970114i \(0.421984\pi\)
\(270\) 69984.0 0.0584237
\(271\) −1.11443e6 −0.921787 −0.460893 0.887456i \(-0.652471\pi\)
−0.460893 + 0.887456i \(0.652471\pi\)
\(272\) 457216. 0.374713
\(273\) −74529.0 −0.0605228
\(274\) −7296.00 −0.00587095
\(275\) −953326. −0.760168
\(276\) −127296. −0.100587
\(277\) 218942. 0.171447 0.0857235 0.996319i \(-0.472680\pi\)
0.0857235 + 0.996319i \(0.472680\pi\)
\(278\) 541072. 0.419897
\(279\) 501876. 0.385999
\(280\) 75264.0 0.0573710
\(281\) 480496. 0.363015 0.181507 0.983390i \(-0.441902\pi\)
0.181507 + 0.983390i \(0.441902\pi\)
\(282\) −150840. −0.112952
\(283\) −1.31389e6 −0.975200 −0.487600 0.873067i \(-0.662127\pi\)
−0.487600 + 0.873067i \(0.662127\pi\)
\(284\) 308256. 0.226786
\(285\) 126144. 0.0919929
\(286\) −252824. −0.182769
\(287\) 90552.0 0.0648923
\(288\) 82944.0 0.0589256
\(289\) 1.76994e6 1.24656
\(290\) −564288. −0.394009
\(291\) −380898. −0.263679
\(292\) 1.15082e6 0.789858
\(293\) 502236. 0.341774 0.170887 0.985291i \(-0.445337\pi\)
0.170887 + 0.985291i \(0.445337\pi\)
\(294\) 86436.0 0.0583212
\(295\) −342864. −0.229386
\(296\) −172160. −0.114210
\(297\) 272646. 0.179353
\(298\) −554800. −0.361906
\(299\) 149396. 0.0966409
\(300\) −367056. −0.235467
\(301\) 1.08604e6 0.690920
\(302\) −824784. −0.520383
\(303\) −1.46216e6 −0.914930
\(304\) 149504. 0.0927831
\(305\) −201360. −0.123943
\(306\) 578664. 0.353283
\(307\) 355796. 0.215454 0.107727 0.994180i \(-0.465643\pi\)
0.107727 + 0.994180i \(0.465643\pi\)
\(308\) 293216. 0.176121
\(309\) −1.08540e6 −0.646686
\(310\) 594816. 0.351543
\(311\) −1.93790e6 −1.13614 −0.568068 0.822982i \(-0.692309\pi\)
−0.568068 + 0.822982i \(0.692309\pi\)
\(312\) −97344.0 −0.0566139
\(313\) −2.71943e6 −1.56898 −0.784490 0.620141i \(-0.787075\pi\)
−0.784490 + 0.620141i \(0.787075\pi\)
\(314\) −163208. −0.0934152
\(315\) 95256.0 0.0540899
\(316\) −114944. −0.0647543
\(317\) −1.32195e6 −0.738869 −0.369435 0.929257i \(-0.620449\pi\)
−0.369435 + 0.929257i \(0.620449\pi\)
\(318\) −38952.0 −0.0216004
\(319\) −2.19837e6 −1.20955
\(320\) 98304.0 0.0536656
\(321\) 386928. 0.209589
\(322\) −173264. −0.0931255
\(323\) 1.04302e6 0.556273
\(324\) 104976. 0.0555556
\(325\) 430781. 0.226229
\(326\) −307616. −0.160312
\(327\) 1.84070e6 0.951948
\(328\) 118272. 0.0607012
\(329\) −205310. −0.104573
\(330\) 323136. 0.163343
\(331\) −1.68150e6 −0.843581 −0.421791 0.906693i \(-0.638598\pi\)
−0.421791 + 0.906693i \(0.638598\pi\)
\(332\) 1.58614e6 0.789764
\(333\) −217890. −0.107678
\(334\) 1.14974e6 0.563939
\(335\) −516192. −0.251304
\(336\) 112896. 0.0545545
\(337\) 2.19209e6 1.05144 0.525720 0.850658i \(-0.323796\pi\)
0.525720 + 0.850658i \(0.323796\pi\)
\(338\) 114244. 0.0543928
\(339\) 711414. 0.336220
\(340\) 685824. 0.321748
\(341\) 2.31730e6 1.07919
\(342\) 189216. 0.0874767
\(343\) 117649. 0.0539949
\(344\) 1.41850e6 0.646297
\(345\) −190944. −0.0863691
\(346\) 453864. 0.203815
\(347\) −15628.0 −0.00696754 −0.00348377 0.999994i \(-0.501109\pi\)
−0.00348377 + 0.999994i \(0.501109\pi\)
\(348\) −846432. −0.374666
\(349\) 525182. 0.230806 0.115403 0.993319i \(-0.463184\pi\)
0.115403 + 0.993319i \(0.463184\pi\)
\(350\) −499604. −0.218000
\(351\) −123201. −0.0533761
\(352\) 382976. 0.164746
\(353\) −4.58393e6 −1.95795 −0.978974 0.203986i \(-0.934610\pi\)
−0.978974 + 0.203986i \(0.934610\pi\)
\(354\) −514296. −0.218125
\(355\) 462384. 0.194730
\(356\) −319040. −0.133420
\(357\) 787626. 0.327077
\(358\) 350096. 0.144371
\(359\) 1.22984e6 0.503632 0.251816 0.967775i \(-0.418972\pi\)
0.251816 + 0.967775i \(0.418972\pi\)
\(360\) 124416. 0.0505964
\(361\) −2.13504e6 −0.862261
\(362\) −1.28324e6 −0.514679
\(363\) −190575. −0.0759101
\(364\) −132496. −0.0524142
\(365\) 1.72622e6 0.678211
\(366\) −302040. −0.117859
\(367\) −3.37209e6 −1.30687 −0.653437 0.756981i \(-0.726674\pi\)
−0.653437 + 0.756981i \(0.726674\pi\)
\(368\) −226304. −0.0871109
\(369\) 149688. 0.0572296
\(370\) −258240. −0.0980662
\(371\) −53018.0 −0.0199981
\(372\) 892224. 0.334285
\(373\) −2.92194e6 −1.08743 −0.543713 0.839271i \(-0.682982\pi\)
−0.543713 + 0.839271i \(0.682982\pi\)
\(374\) 2.67186e6 0.987720
\(375\) −1.22558e6 −0.450054
\(376\) −268160. −0.0978193
\(377\) 993382. 0.359967
\(378\) 142884. 0.0514344
\(379\) 862640. 0.308483 0.154242 0.988033i \(-0.450707\pi\)
0.154242 + 0.988033i \(0.450707\pi\)
\(380\) 224256. 0.0796682
\(381\) 1.13314e6 0.399917
\(382\) −2.43651e6 −0.854299
\(383\) −208506. −0.0726309 −0.0363155 0.999340i \(-0.511562\pi\)
−0.0363155 + 0.999340i \(0.511562\pi\)
\(384\) 147456. 0.0510310
\(385\) 439824. 0.151226
\(386\) −64472.0 −0.0220243
\(387\) 1.79528e6 0.609334
\(388\) −677152. −0.228353
\(389\) −2.89455e6 −0.969854 −0.484927 0.874555i \(-0.661154\pi\)
−0.484927 + 0.874555i \(0.661154\pi\)
\(390\) −146016. −0.0486115
\(391\) −1.57882e6 −0.522266
\(392\) 153664. 0.0505076
\(393\) 1.43611e6 0.469037
\(394\) −1.06819e6 −0.346664
\(395\) −172416. −0.0556013
\(396\) 484704. 0.155324
\(397\) −2.58856e6 −0.824294 −0.412147 0.911117i \(-0.635221\pi\)
−0.412147 + 0.911117i \(0.635221\pi\)
\(398\) 911488. 0.288432
\(399\) 257544. 0.0809877
\(400\) −652544. −0.203920
\(401\) 4.23658e6 1.31569 0.657846 0.753152i \(-0.271468\pi\)
0.657846 + 0.753152i \(0.271468\pi\)
\(402\) −774288. −0.238967
\(403\) −1.04712e6 −0.321170
\(404\) −2.59939e6 −0.792352
\(405\) 157464. 0.0477028
\(406\) −1.15209e6 −0.346873
\(407\) −1.00606e6 −0.301049
\(408\) 1.02874e6 0.305952
\(409\) −4.39615e6 −1.29946 −0.649731 0.760164i \(-0.725119\pi\)
−0.649731 + 0.760164i \(0.725119\pi\)
\(410\) 177408. 0.0521211
\(411\) −16416.0 −0.00479361
\(412\) −1.92960e6 −0.560047
\(413\) −700014. −0.201944
\(414\) −286416. −0.0821290
\(415\) 2.37922e6 0.678131
\(416\) −173056. −0.0490290
\(417\) 1.21741e6 0.342845
\(418\) 873664. 0.244570
\(419\) −4.06963e6 −1.13245 −0.566226 0.824250i \(-0.691597\pi\)
−0.566226 + 0.824250i \(0.691597\pi\)
\(420\) 169344. 0.0468432
\(421\) −1.05229e6 −0.289356 −0.144678 0.989479i \(-0.546215\pi\)
−0.144678 + 0.989479i \(0.546215\pi\)
\(422\) −1.36779e6 −0.373886
\(423\) −339390. −0.0922249
\(424\) −69248.0 −0.0187065
\(425\) −4.55251e6 −1.22258
\(426\) 693576. 0.185170
\(427\) −411110. −0.109116
\(428\) 687872. 0.181509
\(429\) −568854. −0.149230
\(430\) 2.12774e6 0.554943
\(431\) −7746.00 −0.00200856 −0.00100428 0.999999i \(-0.500320\pi\)
−0.00100428 + 0.999999i \(0.500320\pi\)
\(432\) 186624. 0.0481125
\(433\) −739858. −0.189639 −0.0948197 0.995494i \(-0.530227\pi\)
−0.0948197 + 0.995494i \(0.530227\pi\)
\(434\) 1.21442e6 0.309488
\(435\) −1.26965e6 −0.321707
\(436\) 3.27235e6 0.824411
\(437\) −516256. −0.129319
\(438\) 2.58934e6 0.644916
\(439\) −599792. −0.148539 −0.0742693 0.997238i \(-0.523662\pi\)
−0.0742693 + 0.997238i \(0.523662\pi\)
\(440\) 574464. 0.141459
\(441\) 194481. 0.0476190
\(442\) −1.20734e6 −0.293949
\(443\) 6.68431e6 1.61826 0.809128 0.587632i \(-0.199940\pi\)
0.809128 + 0.587632i \(0.199940\pi\)
\(444\) −387360. −0.0932518
\(445\) −478560. −0.114561
\(446\) 1.26638e6 0.301459
\(447\) −1.24830e6 −0.295495
\(448\) 200704. 0.0472456
\(449\) −4.51972e6 −1.05802 −0.529012 0.848615i \(-0.677437\pi\)
−0.529012 + 0.848615i \(0.677437\pi\)
\(450\) −825876. −0.192258
\(451\) 691152. 0.160005
\(452\) 1.26474e6 0.291175
\(453\) −1.85576e6 −0.424891
\(454\) 3.93790e6 0.896653
\(455\) −198744. −0.0450055
\(456\) 336384. 0.0757571
\(457\) −6.03042e6 −1.35069 −0.675347 0.737500i \(-0.736006\pi\)
−0.675347 + 0.737500i \(0.736006\pi\)
\(458\) −206936. −0.0460970
\(459\) 1.30199e6 0.288454
\(460\) −339456. −0.0747978
\(461\) −8.98766e6 −1.96967 −0.984837 0.173480i \(-0.944499\pi\)
−0.984837 + 0.173480i \(0.944499\pi\)
\(462\) 659736. 0.143802
\(463\) 1.40153e6 0.303843 0.151922 0.988393i \(-0.451454\pi\)
0.151922 + 0.988393i \(0.451454\pi\)
\(464\) −1.50477e6 −0.324470
\(465\) 1.33834e6 0.287034
\(466\) 2.95970e6 0.631369
\(467\) −5.62831e6 −1.19422 −0.597112 0.802158i \(-0.703685\pi\)
−0.597112 + 0.802158i \(0.703685\pi\)
\(468\) −219024. −0.0462250
\(469\) −1.05389e6 −0.221240
\(470\) −402240. −0.0839925
\(471\) −367218. −0.0762732
\(472\) −914304. −0.188902
\(473\) 8.28934e6 1.70360
\(474\) −258624. −0.0528717
\(475\) −1.48862e6 −0.302725
\(476\) 1.40022e6 0.283257
\(477\) −87642.0 −0.0176367
\(478\) −568056. −0.113716
\(479\) 3.70610e6 0.738038 0.369019 0.929422i \(-0.379694\pi\)
0.369019 + 0.929422i \(0.379694\pi\)
\(480\) 221184. 0.0438178
\(481\) 454610. 0.0895935
\(482\) −699240. −0.137091
\(483\) −389844. −0.0760367
\(484\) −338800. −0.0657400
\(485\) −1.01573e6 −0.196075
\(486\) 236196. 0.0453609
\(487\) −6.22468e6 −1.18931 −0.594655 0.803981i \(-0.702711\pi\)
−0.594655 + 0.803981i \(0.702711\pi\)
\(488\) −536960. −0.102069
\(489\) −692136. −0.130894
\(490\) 230496. 0.0433684
\(491\) 4.55595e6 0.852855 0.426428 0.904522i \(-0.359772\pi\)
0.426428 + 0.904522i \(0.359772\pi\)
\(492\) 266112. 0.0495623
\(493\) −1.04981e7 −1.94533
\(494\) −394784. −0.0727850
\(495\) 727056. 0.133369
\(496\) 1.58618e6 0.289499
\(497\) 944034. 0.171434
\(498\) 3.56882e6 0.644840
\(499\) −2.44377e6 −0.439349 −0.219674 0.975573i \(-0.570499\pi\)
−0.219674 + 0.975573i \(0.570499\pi\)
\(500\) −2.17882e6 −0.389758
\(501\) 2.58691e6 0.460454
\(502\) −1.71584e6 −0.303891
\(503\) −2.06452e6 −0.363831 −0.181916 0.983314i \(-0.558230\pi\)
−0.181916 + 0.983314i \(0.558230\pi\)
\(504\) 254016. 0.0445435
\(505\) −3.89909e6 −0.680353
\(506\) −1.32246e6 −0.229619
\(507\) 257049. 0.0444116
\(508\) 2.01446e6 0.346338
\(509\) −2.26445e6 −0.387407 −0.193704 0.981060i \(-0.562050\pi\)
−0.193704 + 0.981060i \(0.562050\pi\)
\(510\) 1.54310e6 0.262706
\(511\) 3.52437e6 0.597076
\(512\) 262144. 0.0441942
\(513\) 425736. 0.0714245
\(514\) −6.30926e6 −1.05334
\(515\) −2.89440e6 −0.480884
\(516\) 3.19162e6 0.527699
\(517\) −1.56706e6 −0.257845
\(518\) −527240. −0.0863344
\(519\) 1.02119e6 0.166414
\(520\) −259584. −0.0420988
\(521\) 1.13926e7 1.83877 0.919383 0.393363i \(-0.128688\pi\)
0.919383 + 0.393363i \(0.128688\pi\)
\(522\) −1.90447e6 −0.305913
\(523\) −8.15304e6 −1.30336 −0.651682 0.758493i \(-0.725936\pi\)
−0.651682 + 0.758493i \(0.725936\pi\)
\(524\) 2.55309e6 0.406198
\(525\) −1.12411e6 −0.177996
\(526\) −4.08197e6 −0.643288
\(527\) 1.10661e7 1.73567
\(528\) 861696. 0.134515
\(529\) −5.65489e6 −0.878587
\(530\) −103872. −0.0160623
\(531\) −1.15717e6 −0.178098
\(532\) 457856. 0.0701374
\(533\) −312312. −0.0476179
\(534\) −717840. −0.108937
\(535\) 1.03181e6 0.155853
\(536\) −1.37651e6 −0.206951
\(537\) 787716. 0.117878
\(538\) 2.30382e6 0.343156
\(539\) 897974. 0.133135
\(540\) 279936. 0.0413118
\(541\) −5.17796e6 −0.760616 −0.380308 0.924860i \(-0.624182\pi\)
−0.380308 + 0.924860i \(0.624182\pi\)
\(542\) −4.45773e6 −0.651801
\(543\) −2.88729e6 −0.420234
\(544\) 1.82886e6 0.264962
\(545\) 4.90853e6 0.707881
\(546\) −298116. −0.0427960
\(547\) 3.97011e6 0.567328 0.283664 0.958924i \(-0.408450\pi\)
0.283664 + 0.958924i \(0.408450\pi\)
\(548\) −29184.0 −0.00415139
\(549\) −679590. −0.0962312
\(550\) −3.81330e6 −0.537520
\(551\) −3.43275e6 −0.481685
\(552\) −509184. −0.0711258
\(553\) −352016. −0.0489496
\(554\) 875768. 0.121231
\(555\) −581040. −0.0800707
\(556\) 2.16429e6 0.296912
\(557\) 204396. 0.0279148 0.0139574 0.999903i \(-0.495557\pi\)
0.0139574 + 0.999903i \(0.495557\pi\)
\(558\) 2.00750e6 0.272942
\(559\) −3.74572e6 −0.506997
\(560\) 301056. 0.0405674
\(561\) 6.01168e6 0.806470
\(562\) 1.92198e6 0.256690
\(563\) 1.07479e7 1.42907 0.714537 0.699598i \(-0.246638\pi\)
0.714537 + 0.699598i \(0.246638\pi\)
\(564\) −603360. −0.0798691
\(565\) 1.89710e6 0.250017
\(566\) −5.25557e6 −0.689570
\(567\) 321489. 0.0419961
\(568\) 1.23302e6 0.160362
\(569\) 1.39894e7 1.81142 0.905709 0.423899i \(-0.139339\pi\)
0.905709 + 0.423899i \(0.139339\pi\)
\(570\) 504576. 0.0650488
\(571\) −1.23318e7 −1.58284 −0.791420 0.611273i \(-0.790658\pi\)
−0.791420 + 0.611273i \(0.790658\pi\)
\(572\) −1.01130e6 −0.129237
\(573\) −5.48215e6 −0.697533
\(574\) 362208. 0.0458858
\(575\) 2.25332e6 0.284219
\(576\) 331776. 0.0416667
\(577\) −1.00343e7 −1.25472 −0.627358 0.778731i \(-0.715864\pi\)
−0.627358 + 0.778731i \(0.715864\pi\)
\(578\) 7.07976e6 0.881452
\(579\) −145062. −0.0179828
\(580\) −2.25715e6 −0.278606
\(581\) 4.85757e6 0.597006
\(582\) −1.52359e6 −0.186449
\(583\) −404668. −0.0493091
\(584\) 4.60326e6 0.558514
\(585\) −328536. −0.0396911
\(586\) 2.00894e6 0.241671
\(587\) 1.13866e6 0.136395 0.0681977 0.997672i \(-0.478275\pi\)
0.0681977 + 0.997672i \(0.478275\pi\)
\(588\) 345744. 0.0412393
\(589\) 3.61846e6 0.429770
\(590\) −1.37146e6 −0.162200
\(591\) −2.40343e6 −0.283050
\(592\) −688640. −0.0807585
\(593\) 4.92581e6 0.575229 0.287615 0.957746i \(-0.407138\pi\)
0.287615 + 0.957746i \(0.407138\pi\)
\(594\) 1.09058e6 0.126822
\(595\) 2.10034e6 0.243218
\(596\) −2.21920e6 −0.255906
\(597\) 2.05085e6 0.235504
\(598\) 597584. 0.0683354
\(599\) −1.08550e7 −1.23613 −0.618066 0.786127i \(-0.712083\pi\)
−0.618066 + 0.786127i \(0.712083\pi\)
\(600\) −1.46822e6 −0.166500
\(601\) −1.20410e7 −1.35981 −0.679905 0.733300i \(-0.737979\pi\)
−0.679905 + 0.733300i \(0.737979\pi\)
\(602\) 4.34414e6 0.488554
\(603\) −1.74215e6 −0.195115
\(604\) −3.29914e6 −0.367966
\(605\) −508200. −0.0564477
\(606\) −5.84863e6 −0.646953
\(607\) −1.39099e7 −1.53233 −0.766166 0.642642i \(-0.777838\pi\)
−0.766166 + 0.642642i \(0.777838\pi\)
\(608\) 598016. 0.0656076
\(609\) −2.59220e6 −0.283221
\(610\) −805440. −0.0876413
\(611\) 708110. 0.0767357
\(612\) 2.31466e6 0.249809
\(613\) 4.47612e6 0.481117 0.240558 0.970635i \(-0.422669\pi\)
0.240558 + 0.970635i \(0.422669\pi\)
\(614\) 1.42318e6 0.152349
\(615\) 399168. 0.0425567
\(616\) 1.17286e6 0.124536
\(617\) 7.73177e6 0.817647 0.408824 0.912613i \(-0.365939\pi\)
0.408824 + 0.912613i \(0.365939\pi\)
\(618\) −4.34160e6 −0.457276
\(619\) 1.29248e7 1.35580 0.677901 0.735153i \(-0.262890\pi\)
0.677901 + 0.735153i \(0.262890\pi\)
\(620\) 2.37926e6 0.248578
\(621\) −644436. −0.0670580
\(622\) −7.75160e6 −0.803370
\(623\) −977060. −0.100856
\(624\) −389376. −0.0400320
\(625\) 4.69740e6 0.481014
\(626\) −1.08777e7 −1.10944
\(627\) 1.96574e6 0.199691
\(628\) −652832. −0.0660545
\(629\) −4.80434e6 −0.484180
\(630\) 381024. 0.0382473
\(631\) 1.33124e7 1.33102 0.665509 0.746390i \(-0.268215\pi\)
0.665509 + 0.746390i \(0.268215\pi\)
\(632\) −459776. −0.0457882
\(633\) −3.07753e6 −0.305276
\(634\) −5.28781e6 −0.522459
\(635\) 3.02170e6 0.297383
\(636\) −155808. −0.0152738
\(637\) −405769. −0.0396214
\(638\) −8.79349e6 −0.855282
\(639\) 1.56055e6 0.151190
\(640\) 393216. 0.0379473
\(641\) 1.03522e7 0.995150 0.497575 0.867421i \(-0.334224\pi\)
0.497575 + 0.867421i \(0.334224\pi\)
\(642\) 1.54771e6 0.148201
\(643\) −6.11730e6 −0.583489 −0.291744 0.956496i \(-0.594236\pi\)
−0.291744 + 0.956496i \(0.594236\pi\)
\(644\) −693056. −0.0658497
\(645\) 4.78742e6 0.453109
\(646\) 4.17210e6 0.393344
\(647\) 1.14021e7 1.07084 0.535420 0.844586i \(-0.320153\pi\)
0.535420 + 0.844586i \(0.320153\pi\)
\(648\) 419904. 0.0392837
\(649\) −5.34296e6 −0.497933
\(650\) 1.72312e6 0.159968
\(651\) 2.73244e6 0.252696
\(652\) −1.23046e6 −0.113357
\(653\) −2.63309e6 −0.241648 −0.120824 0.992674i \(-0.538554\pi\)
−0.120824 + 0.992674i \(0.538554\pi\)
\(654\) 7.36279e6 0.673129
\(655\) 3.82963e6 0.348782
\(656\) 473088. 0.0429222
\(657\) 5.82601e6 0.526572
\(658\) −821240. −0.0739444
\(659\) 2.71632e6 0.243651 0.121825 0.992552i \(-0.461125\pi\)
0.121825 + 0.992552i \(0.461125\pi\)
\(660\) 1.29254e6 0.115501
\(661\) 2.14041e7 1.90543 0.952714 0.303868i \(-0.0982783\pi\)
0.952714 + 0.303868i \(0.0982783\pi\)
\(662\) −6.72600e6 −0.596502
\(663\) −2.71651e6 −0.240009
\(664\) 6.34458e6 0.558448
\(665\) 686784. 0.0602235
\(666\) −871560. −0.0761398
\(667\) 5.19615e6 0.452238
\(668\) 4.59894e6 0.398765
\(669\) 2.84936e6 0.246140
\(670\) −2.06477e6 −0.177699
\(671\) −3.13786e6 −0.269046
\(672\) 451584. 0.0385758
\(673\) −8.01013e6 −0.681713 −0.340857 0.940115i \(-0.610717\pi\)
−0.340857 + 0.940115i \(0.610717\pi\)
\(674\) 8.76838e6 0.743480
\(675\) −1.85822e6 −0.156978
\(676\) 456976. 0.0384615
\(677\) 7.94655e6 0.666357 0.333178 0.942864i \(-0.391879\pi\)
0.333178 + 0.942864i \(0.391879\pi\)
\(678\) 2.84566e6 0.237743
\(679\) −2.07378e6 −0.172619
\(680\) 2.74330e6 0.227510
\(681\) 8.86027e6 0.732114
\(682\) 9.26922e6 0.763101
\(683\) 1.11470e7 0.914333 0.457166 0.889381i \(-0.348864\pi\)
0.457166 + 0.889381i \(0.348864\pi\)
\(684\) 756864. 0.0618554
\(685\) −43776.0 −0.00356459
\(686\) 470596. 0.0381802
\(687\) −465606. −0.0376380
\(688\) 5.67398e6 0.457001
\(689\) 182858. 0.0146746
\(690\) −763776. −0.0610722
\(691\) −2.45717e7 −1.95768 −0.978838 0.204637i \(-0.934399\pi\)
−0.978838 + 0.204637i \(0.934399\pi\)
\(692\) 1.81546e6 0.144119
\(693\) 1.48441e6 0.117414
\(694\) −62512.0 −0.00492680
\(695\) 3.24643e6 0.254944
\(696\) −3.38573e6 −0.264929
\(697\) 3.30053e6 0.257337
\(698\) 2.10073e6 0.163204
\(699\) 6.65933e6 0.515511
\(700\) −1.99842e6 −0.154149
\(701\) 359214. 0.0276095 0.0138047 0.999905i \(-0.495606\pi\)
0.0138047 + 0.999905i \(0.495606\pi\)
\(702\) −492804. −0.0377426
\(703\) −1.57096e6 −0.119888
\(704\) 1.53190e6 0.116493
\(705\) −905040. −0.0685796
\(706\) −1.83357e7 −1.38448
\(707\) −7.96064e6 −0.598962
\(708\) −2.05718e6 −0.154238
\(709\) 2.20095e7 1.64435 0.822177 0.569232i \(-0.192759\pi\)
0.822177 + 0.569232i \(0.192759\pi\)
\(710\) 1.84954e6 0.137695
\(711\) −581904. −0.0431695
\(712\) −1.27616e6 −0.0943420
\(713\) −5.47726e6 −0.403497
\(714\) 3.15050e6 0.231278
\(715\) −1.51694e6 −0.110970
\(716\) 1.40038e6 0.102086
\(717\) −1.27813e6 −0.0928487
\(718\) 4.91937e6 0.356122
\(719\) 7.60177e6 0.548394 0.274197 0.961674i \(-0.411588\pi\)
0.274197 + 0.961674i \(0.411588\pi\)
\(720\) 497664. 0.0357771
\(721\) −5.90940e6 −0.423356
\(722\) −8.54017e6 −0.609710
\(723\) −1.57329e6 −0.111934
\(724\) −5.13296e6 −0.363933
\(725\) 1.49830e7 1.05865
\(726\) −762300. −0.0536765
\(727\) −9.43667e6 −0.662190 −0.331095 0.943597i \(-0.607418\pi\)
−0.331095 + 0.943597i \(0.607418\pi\)
\(728\) −529984. −0.0370625
\(729\) 531441. 0.0370370
\(730\) 6.90490e6 0.479568
\(731\) 3.95849e7 2.73991
\(732\) −1.20816e6 −0.0833387
\(733\) 5.03323e6 0.346009 0.173004 0.984921i \(-0.444653\pi\)
0.173004 + 0.984921i \(0.444653\pi\)
\(734\) −1.34884e7 −0.924100
\(735\) 518616. 0.0354101
\(736\) −905216. −0.0615967
\(737\) −8.04399e6 −0.545510
\(738\) 598752. 0.0404675
\(739\) −5.53779e6 −0.373014 −0.186507 0.982454i \(-0.559717\pi\)
−0.186507 + 0.982454i \(0.559717\pi\)
\(740\) −1.03296e6 −0.0693433
\(741\) −888264. −0.0594287
\(742\) −212072. −0.0141408
\(743\) 1.59329e7 1.05882 0.529411 0.848366i \(-0.322413\pi\)
0.529411 + 0.848366i \(0.322413\pi\)
\(744\) 3.56890e6 0.236375
\(745\) −3.32880e6 −0.219734
\(746\) −1.16878e7 −0.768926
\(747\) 8.02985e6 0.526509
\(748\) 1.06874e7 0.698424
\(749\) 2.10661e6 0.137208
\(750\) −4.90234e6 −0.318236
\(751\) 1.76926e7 1.14470 0.572351 0.820009i \(-0.306032\pi\)
0.572351 + 0.820009i \(0.306032\pi\)
\(752\) −1.07264e6 −0.0691687
\(753\) −3.86064e6 −0.248126
\(754\) 3.97353e6 0.254535
\(755\) −4.94870e6 −0.315954
\(756\) 571536. 0.0363696
\(757\) 8.76182e6 0.555718 0.277859 0.960622i \(-0.410375\pi\)
0.277859 + 0.960622i \(0.410375\pi\)
\(758\) 3.45056e6 0.218131
\(759\) −2.97554e6 −0.187483
\(760\) 897024. 0.0563339
\(761\) 7.26804e6 0.454942 0.227471 0.973785i \(-0.426954\pi\)
0.227471 + 0.973785i \(0.426954\pi\)
\(762\) 4.53254e6 0.282784
\(763\) 1.00216e7 0.623196
\(764\) −9.74605e6 −0.604081
\(765\) 3.47198e6 0.214498
\(766\) −834024. −0.0513578
\(767\) 2.41433e6 0.148187
\(768\) 589824. 0.0360844
\(769\) −4.01863e6 −0.245054 −0.122527 0.992465i \(-0.539100\pi\)
−0.122527 + 0.992465i \(0.539100\pi\)
\(770\) 1.75930e6 0.106933
\(771\) −1.41958e7 −0.860052
\(772\) −257888. −0.0155736
\(773\) 1.77194e7 1.06660 0.533298 0.845927i \(-0.320952\pi\)
0.533298 + 0.845927i \(0.320952\pi\)
\(774\) 7.18114e6 0.430865
\(775\) −1.57936e7 −0.944554
\(776\) −2.70861e6 −0.161470
\(777\) −1.18629e6 −0.0704918
\(778\) −1.15782e7 −0.685790
\(779\) 1.07923e6 0.0637193
\(780\) −584064. −0.0343735
\(781\) 7.20548e6 0.422703
\(782\) −6.31530e6 −0.369298
\(783\) −4.28506e6 −0.249777
\(784\) 614656. 0.0357143
\(785\) −979248. −0.0567177
\(786\) 5.74445e6 0.331659
\(787\) 6.58989e6 0.379264 0.189632 0.981855i \(-0.439271\pi\)
0.189632 + 0.981855i \(0.439271\pi\)
\(788\) −4.27277e6 −0.245128
\(789\) −9.18443e6 −0.525242
\(790\) −689664. −0.0393160
\(791\) 3.87325e6 0.220107
\(792\) 1.93882e6 0.109831
\(793\) 1.41791e6 0.0800692
\(794\) −1.03542e7 −0.582864
\(795\) −233712. −0.0131148
\(796\) 3.64595e6 0.203952
\(797\) 3.97198e6 0.221494 0.110747 0.993849i \(-0.464676\pi\)
0.110747 + 0.993849i \(0.464676\pi\)
\(798\) 1.03018e6 0.0572670
\(799\) −7.48334e6 −0.414695
\(800\) −2.61018e6 −0.144193
\(801\) −1.61514e6 −0.0889465
\(802\) 1.69463e7 0.930335
\(803\) 2.69003e7 1.47221
\(804\) −3.09715e6 −0.168975
\(805\) −1.03958e6 −0.0565418
\(806\) −4.18850e6 −0.227102
\(807\) 5.18359e6 0.280186
\(808\) −1.03976e7 −0.560278
\(809\) 3.61744e7 1.94326 0.971628 0.236513i \(-0.0760045\pi\)
0.971628 + 0.236513i \(0.0760045\pi\)
\(810\) 629856. 0.0337310
\(811\) −1.61022e7 −0.859672 −0.429836 0.902907i \(-0.641429\pi\)
−0.429836 + 0.902907i \(0.641429\pi\)
\(812\) −4.60835e6 −0.245276
\(813\) −1.00299e7 −0.532194
\(814\) −4.02424e6 −0.212874
\(815\) −1.84570e6 −0.0973344
\(816\) 4.11494e6 0.216341
\(817\) 1.29438e7 0.678431
\(818\) −1.75846e7 −0.918859
\(819\) −670761. −0.0349428
\(820\) 709632. 0.0368552
\(821\) 2.33984e7 1.21151 0.605756 0.795650i \(-0.292871\pi\)
0.605756 + 0.795650i \(0.292871\pi\)
\(822\) −65664.0 −0.00338959
\(823\) 3.45541e7 1.77828 0.889140 0.457636i \(-0.151304\pi\)
0.889140 + 0.457636i \(0.151304\pi\)
\(824\) −7.71840e6 −0.396013
\(825\) −8.57993e6 −0.438883
\(826\) −2.80006e6 −0.142796
\(827\) −2.16044e7 −1.09845 −0.549224 0.835675i \(-0.685076\pi\)
−0.549224 + 0.835675i \(0.685076\pi\)
\(828\) −1.14566e6 −0.0580740
\(829\) 1.26865e7 0.641145 0.320572 0.947224i \(-0.396125\pi\)
0.320572 + 0.947224i \(0.396125\pi\)
\(830\) 9.51686e6 0.479511
\(831\) 1.97048e6 0.0989849
\(832\) −692224. −0.0346688
\(833\) 4.28819e6 0.214122
\(834\) 4.86965e6 0.242428
\(835\) 6.89842e6 0.342400
\(836\) 3.49466e6 0.172937
\(837\) 4.51688e6 0.222856
\(838\) −1.62785e7 −0.800765
\(839\) −322154. −0.0158001 −0.00790003 0.999969i \(-0.502515\pi\)
−0.00790003 + 0.999969i \(0.502515\pi\)
\(840\) 677376. 0.0331231
\(841\) 1.40397e7 0.684493
\(842\) −4.20918e6 −0.204605
\(843\) 4.32446e6 0.209587
\(844\) −5.47117e6 −0.264377
\(845\) 685464. 0.0330250
\(846\) −1.35756e6 −0.0652128
\(847\) −1.03758e6 −0.0496948
\(848\) −276992. −0.0132275
\(849\) −1.18250e7 −0.563032
\(850\) −1.82101e7 −0.864498
\(851\) 2.37796e6 0.112559
\(852\) 2.77430e6 0.130935
\(853\) 2.80762e7 1.32119 0.660595 0.750742i \(-0.270304\pi\)
0.660595 + 0.750742i \(0.270304\pi\)
\(854\) −1.64444e6 −0.0771566
\(855\) 1.13530e6 0.0531121
\(856\) 2.75149e6 0.128346
\(857\) 2.46095e7 1.14459 0.572296 0.820047i \(-0.306053\pi\)
0.572296 + 0.820047i \(0.306053\pi\)
\(858\) −2.27542e6 −0.105522
\(859\) −1.82440e7 −0.843599 −0.421800 0.906689i \(-0.638601\pi\)
−0.421800 + 0.906689i \(0.638601\pi\)
\(860\) 8.51098e6 0.392404
\(861\) 814968. 0.0374656
\(862\) −30984.0 −0.00142026
\(863\) −1.15851e7 −0.529506 −0.264753 0.964316i \(-0.585290\pi\)
−0.264753 + 0.964316i \(0.585290\pi\)
\(864\) 746496. 0.0340207
\(865\) 2.72318e6 0.123748
\(866\) −2.95943e6 −0.134095
\(867\) 1.59295e7 0.719703
\(868\) 4.85766e6 0.218841
\(869\) −2.68682e6 −0.120695
\(870\) −5.07859e6 −0.227481
\(871\) 3.63485e6 0.162346
\(872\) 1.30894e7 0.582947
\(873\) −3.42808e6 −0.152235
\(874\) −2.06502e6 −0.0914422
\(875\) −6.67262e6 −0.294630
\(876\) 1.03573e7 0.456024
\(877\) 3.16646e7 1.39019 0.695097 0.718916i \(-0.255361\pi\)
0.695097 + 0.718916i \(0.255361\pi\)
\(878\) −2.39917e6 −0.105033
\(879\) 4.52012e6 0.197323
\(880\) 2.29786e6 0.100027
\(881\) −2.68256e7 −1.16442 −0.582210 0.813038i \(-0.697812\pi\)
−0.582210 + 0.813038i \(0.697812\pi\)
\(882\) 777924. 0.0336718
\(883\) −4.51779e7 −1.94995 −0.974977 0.222307i \(-0.928641\pi\)
−0.974977 + 0.222307i \(0.928641\pi\)
\(884\) −4.82934e6 −0.207854
\(885\) −3.08578e6 −0.132436
\(886\) 2.67372e7 1.14428
\(887\) 7.09987e6 0.302999 0.151500 0.988457i \(-0.451590\pi\)
0.151500 + 0.988457i \(0.451590\pi\)
\(888\) −1.54944e6 −0.0659390
\(889\) 6.16930e6 0.261807
\(890\) −1.91424e6 −0.0810068
\(891\) 2.45381e6 0.103549
\(892\) 5.06554e6 0.213164
\(893\) −2.44696e6 −0.102683
\(894\) −4.99320e6 −0.208947
\(895\) 2.10058e6 0.0876558
\(896\) 802816. 0.0334077
\(897\) 1.34456e6 0.0557957
\(898\) −1.80789e7 −0.748136
\(899\) −3.64201e7 −1.50294
\(900\) −3.30350e6 −0.135947
\(901\) −1.93245e6 −0.0793043
\(902\) 2.76461e6 0.113140
\(903\) 9.77432e6 0.398903
\(904\) 5.05894e6 0.205892
\(905\) −7.69944e6 −0.312491
\(906\) −7.42306e6 −0.300443
\(907\) 1.22356e7 0.493864 0.246932 0.969033i \(-0.420578\pi\)
0.246932 + 0.969033i \(0.420578\pi\)
\(908\) 1.57516e7 0.634030
\(909\) −1.31594e7 −0.528235
\(910\) −794976. −0.0318237
\(911\) −6.85780e6 −0.273772 −0.136886 0.990587i \(-0.543709\pi\)
−0.136886 + 0.990587i \(0.543709\pi\)
\(912\) 1.34554e6 0.0535683
\(913\) 3.70761e7 1.47203
\(914\) −2.41217e7 −0.955085
\(915\) −1.81224e6 −0.0715588
\(916\) −827744. −0.0325955
\(917\) 7.81883e6 0.307057
\(918\) 5.20798e6 0.203968
\(919\) 4.74950e7 1.85506 0.927532 0.373743i \(-0.121926\pi\)
0.927532 + 0.373743i \(0.121926\pi\)
\(920\) −1.35782e6 −0.0528900
\(921\) 3.20216e6 0.124393
\(922\) −3.59507e7 −1.39277
\(923\) −3.25595e6 −0.125798
\(924\) 2.63894e6 0.101683
\(925\) 6.85681e6 0.263492
\(926\) 5.60611e6 0.214849
\(927\) −9.76860e6 −0.373364
\(928\) −6.01907e6 −0.229435
\(929\) −9.27974e6 −0.352774 −0.176387 0.984321i \(-0.556441\pi\)
−0.176387 + 0.984321i \(0.556441\pi\)
\(930\) 5.35334e6 0.202963
\(931\) 1.40218e6 0.0530189
\(932\) 1.18388e7 0.446446
\(933\) −1.74411e7 −0.655949
\(934\) −2.25132e7 −0.844444
\(935\) 1.60311e7 0.599702
\(936\) −876096. −0.0326860
\(937\) 4.88738e7 1.81856 0.909279 0.416188i \(-0.136634\pi\)
0.909279 + 0.416188i \(0.136634\pi\)
\(938\) −4.21557e6 −0.156440
\(939\) −2.44749e7 −0.905852
\(940\) −1.60896e6 −0.0593917
\(941\) −3.95399e7 −1.45567 −0.727833 0.685754i \(-0.759473\pi\)
−0.727833 + 0.685754i \(0.759473\pi\)
\(942\) −1.46887e6 −0.0539333
\(943\) −1.63363e6 −0.0598239
\(944\) −3.65722e6 −0.133574
\(945\) 857304. 0.0312288
\(946\) 3.31573e7 1.20463
\(947\) −2.22841e7 −0.807459 −0.403729 0.914878i \(-0.632286\pi\)
−0.403729 + 0.914878i \(0.632286\pi\)
\(948\) −1.03450e6 −0.0373859
\(949\) −1.21555e7 −0.438134
\(950\) −5.95446e6 −0.214059
\(951\) −1.18976e7 −0.426586
\(952\) 5.60090e6 0.200293
\(953\) −4.94948e7 −1.76534 −0.882668 0.469998i \(-0.844255\pi\)
−0.882668 + 0.469998i \(0.844255\pi\)
\(954\) −350568. −0.0124710
\(955\) −1.46191e7 −0.518694
\(956\) −2.27222e6 −0.0804093
\(957\) −1.97853e7 −0.698335
\(958\) 1.48244e7 0.521872
\(959\) −89376.0 −0.00313816
\(960\) 884736. 0.0309839
\(961\) 9.76126e6 0.340955
\(962\) 1.81844e6 0.0633521
\(963\) 3.48235e6 0.121006
\(964\) −2.79696e6 −0.0969379
\(965\) −386832. −0.0133722
\(966\) −1.55938e6 −0.0537660
\(967\) −1.82132e7 −0.626353 −0.313177 0.949695i \(-0.601393\pi\)
−0.313177 + 0.949695i \(0.601393\pi\)
\(968\) −1.35520e6 −0.0464852
\(969\) 9.38722e6 0.321164
\(970\) −4.06291e6 −0.138646
\(971\) 7.90048e6 0.268909 0.134454 0.990920i \(-0.457072\pi\)
0.134454 + 0.990920i \(0.457072\pi\)
\(972\) 944784. 0.0320750
\(973\) 6.62813e6 0.224445
\(974\) −2.48987e7 −0.840969
\(975\) 3.87703e6 0.130613
\(976\) −2.14784e6 −0.0721734
\(977\) 1.11208e7 0.372733 0.186367 0.982480i \(-0.440329\pi\)
0.186367 + 0.982480i \(0.440329\pi\)
\(978\) −2.76854e6 −0.0925559
\(979\) −7.45756e6 −0.248680
\(980\) 921984. 0.0306661
\(981\) 1.65663e7 0.549607
\(982\) 1.82238e7 0.603060
\(983\) −2.35770e7 −0.778224 −0.389112 0.921191i \(-0.627218\pi\)
−0.389112 + 0.921191i \(0.627218\pi\)
\(984\) 1.06445e6 0.0350459
\(985\) −6.40915e6 −0.210480
\(986\) −4.19924e7 −1.37556
\(987\) −1.84779e6 −0.0603754
\(988\) −1.57914e6 −0.0514668
\(989\) −1.95930e7 −0.636956
\(990\) 2.90822e6 0.0943061
\(991\) −5.11700e7 −1.65513 −0.827564 0.561371i \(-0.810274\pi\)
−0.827564 + 0.561371i \(0.810274\pi\)
\(992\) 6.34470e6 0.204707
\(993\) −1.51335e7 −0.487042
\(994\) 3.77614e6 0.121222
\(995\) 5.46893e6 0.175124
\(996\) 1.42753e7 0.455971
\(997\) −4.57099e7 −1.45637 −0.728186 0.685380i \(-0.759636\pi\)
−0.728186 + 0.685380i \(0.759636\pi\)
\(998\) −9.77509e6 −0.310666
\(999\) −1.96101e6 −0.0621679
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 546.6.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.6.a.i.1.1 1 1.1 even 1 trivial