Properties

Label 150.6.a.m.1.1
Level $150$
Weight $6$
Character 150.1
Self dual yes
Analytic conductor $24.058$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 150.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.0575729719\)
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 \(\chi\) \(=\) 150.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +36.0000 q^{6} +47.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} +36.0000 q^{6} +47.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +222.000 q^{11} +144.000 q^{12} +101.000 q^{13} +188.000 q^{14} +256.000 q^{16} +162.000 q^{17} +324.000 q^{18} +1685.00 q^{19} +423.000 q^{21} +888.000 q^{22} +306.000 q^{23} +576.000 q^{24} +404.000 q^{26} +729.000 q^{27} +752.000 q^{28} +7890.00 q^{29} -8593.00 q^{31} +1024.00 q^{32} +1998.00 q^{33} +648.000 q^{34} +1296.00 q^{36} +8642.00 q^{37} +6740.00 q^{38} +909.000 q^{39} -18168.0 q^{41} +1692.00 q^{42} +14351.0 q^{43} +3552.00 q^{44} +1224.00 q^{46} -1098.00 q^{47} +2304.00 q^{48} -14598.0 q^{49} +1458.00 q^{51} +1616.00 q^{52} +17916.0 q^{53} +2916.00 q^{54} +3008.00 q^{56} +15165.0 q^{57} +31560.0 q^{58} +17610.0 q^{59} -21853.0 q^{61} -34372.0 q^{62} +3807.00 q^{63} +4096.00 q^{64} +7992.00 q^{66} +107.000 q^{67} +2592.00 q^{68} +2754.00 q^{69} -40728.0 q^{71} +5184.00 q^{72} +34706.0 q^{73} +34568.0 q^{74} +26960.0 q^{76} +10434.0 q^{77} +3636.00 q^{78} -69160.0 q^{79} +6561.00 q^{81} -72672.0 q^{82} -108534. q^{83} +6768.00 q^{84} +57404.0 q^{86} +71010.0 q^{87} +14208.0 q^{88} +35040.0 q^{89} +4747.00 q^{91} +4896.00 q^{92} -77337.0 q^{93} -4392.00 q^{94} +9216.00 q^{96} -823.000 q^{97} -58392.0 q^{98} +17982.0 q^{99} +O(q^{100})\)

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
\(5\) 0 0
\(6\) 36.0000 0.408248
\(7\) 47.0000 0.362537 0.181269 0.983434i \(-0.441980\pi\)
0.181269 + 0.983434i \(0.441980\pi\)
\(8\) 64.0000 0.353553
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 222.000 0.553186 0.276593 0.960987i \(-0.410795\pi\)
0.276593 + 0.960987i \(0.410795\pi\)
\(12\) 144.000 0.288675
\(13\) 101.000 0.165754 0.0828768 0.996560i \(-0.473589\pi\)
0.0828768 + 0.996560i \(0.473589\pi\)
\(14\) 188.000 0.256353
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) 162.000 0.135954 0.0679771 0.997687i \(-0.478346\pi\)
0.0679771 + 0.997687i \(0.478346\pi\)
\(18\) 324.000 0.235702
\(19\) 1685.00 1.07082 0.535409 0.844593i \(-0.320157\pi\)
0.535409 + 0.844593i \(0.320157\pi\)
\(20\) 0 0
\(21\) 423.000 0.209311
\(22\) 888.000 0.391162
\(23\) 306.000 0.120615 0.0603076 0.998180i \(-0.480792\pi\)
0.0603076 + 0.998180i \(0.480792\pi\)
\(24\) 576.000 0.204124
\(25\) 0 0
\(26\) 404.000 0.117206
\(27\) 729.000 0.192450
\(28\) 752.000 0.181269
\(29\) 7890.00 1.74214 0.871068 0.491163i \(-0.163428\pi\)
0.871068 + 0.491163i \(0.163428\pi\)
\(30\) 0 0
\(31\) −8593.00 −1.60598 −0.802991 0.595991i \(-0.796759\pi\)
−0.802991 + 0.595991i \(0.796759\pi\)
\(32\) 1024.00 0.176777
\(33\) 1998.00 0.319382
\(34\) 648.000 0.0961342
\(35\) 0 0
\(36\) 1296.00 0.166667
\(37\) 8642.00 1.03779 0.518896 0.854838i \(-0.326343\pi\)
0.518896 + 0.854838i \(0.326343\pi\)
\(38\) 6740.00 0.757183
\(39\) 909.000 0.0956979
\(40\) 0 0
\(41\) −18168.0 −1.68790 −0.843951 0.536420i \(-0.819777\pi\)
−0.843951 + 0.536420i \(0.819777\pi\)
\(42\) 1692.00 0.148005
\(43\) 14351.0 1.18362 0.591808 0.806079i \(-0.298414\pi\)
0.591808 + 0.806079i \(0.298414\pi\)
\(44\) 3552.00 0.276593
\(45\) 0 0
\(46\) 1224.00 0.0852878
\(47\) −1098.00 −0.0725033 −0.0362516 0.999343i \(-0.511542\pi\)
−0.0362516 + 0.999343i \(0.511542\pi\)
\(48\) 2304.00 0.144338
\(49\) −14598.0 −0.868567
\(50\) 0 0
\(51\) 1458.00 0.0784932
\(52\) 1616.00 0.0828768
\(53\) 17916.0 0.876095 0.438048 0.898952i \(-0.355670\pi\)
0.438048 + 0.898952i \(0.355670\pi\)
\(54\) 2916.00 0.136083
\(55\) 0 0
\(56\) 3008.00 0.128176
\(57\) 15165.0 0.618237
\(58\) 31560.0 1.23188
\(59\) 17610.0 0.658612 0.329306 0.944223i \(-0.393185\pi\)
0.329306 + 0.944223i \(0.393185\pi\)
\(60\) 0 0
\(61\) −21853.0 −0.751946 −0.375973 0.926631i \(-0.622691\pi\)
−0.375973 + 0.926631i \(0.622691\pi\)
\(62\) −34372.0 −1.13560
\(63\) 3807.00 0.120846
\(64\) 4096.00 0.125000
\(65\) 0 0
\(66\) 7992.00 0.225837
\(67\) 107.000 0.00291204 0.00145602 0.999999i \(-0.499537\pi\)
0.00145602 + 0.999999i \(0.499537\pi\)
\(68\) 2592.00 0.0679771
\(69\) 2754.00 0.0696372
\(70\) 0 0
\(71\) −40728.0 −0.958842 −0.479421 0.877585i \(-0.659153\pi\)
−0.479421 + 0.877585i \(0.659153\pi\)
\(72\) 5184.00 0.117851
\(73\) 34706.0 0.762250 0.381125 0.924524i \(-0.375537\pi\)
0.381125 + 0.924524i \(0.375537\pi\)
\(74\) 34568.0 0.733829
\(75\) 0 0
\(76\) 26960.0 0.535409
\(77\) 10434.0 0.200551
\(78\) 3636.00 0.0676686
\(79\) −69160.0 −1.24677 −0.623386 0.781914i \(-0.714244\pi\)
−0.623386 + 0.781914i \(0.714244\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −72672.0 −1.19353
\(83\) −108534. −1.72930 −0.864650 0.502374i \(-0.832460\pi\)
−0.864650 + 0.502374i \(0.832460\pi\)
\(84\) 6768.00 0.104656
\(85\) 0 0
\(86\) 57404.0 0.836943
\(87\) 71010.0 1.00582
\(88\) 14208.0 0.195581
\(89\) 35040.0 0.468910 0.234455 0.972127i \(-0.424670\pi\)
0.234455 + 0.972127i \(0.424670\pi\)
\(90\) 0 0
\(91\) 4747.00 0.0600919
\(92\) 4896.00 0.0603076
\(93\) −77337.0 −0.927214
\(94\) −4392.00 −0.0512676
\(95\) 0 0
\(96\) 9216.00 0.102062
\(97\) −823.000 −0.00888118 −0.00444059 0.999990i \(-0.501413\pi\)
−0.00444059 + 0.999990i \(0.501413\pi\)
\(98\) −58392.0 −0.614169
\(99\) 17982.0 0.184395
\(100\) 0 0
\(101\) −33828.0 −0.329969 −0.164984 0.986296i \(-0.552757\pi\)
−0.164984 + 0.986296i \(0.552757\pi\)
\(102\) 5832.00 0.0555031
\(103\) −133444. −1.23938 −0.619692 0.784845i \(-0.712743\pi\)
−0.619692 + 0.784845i \(0.712743\pi\)
\(104\) 6464.00 0.0586028
\(105\) 0 0
\(106\) 71664.0 0.619493
\(107\) 81252.0 0.686080 0.343040 0.939321i \(-0.388543\pi\)
0.343040 + 0.939321i \(0.388543\pi\)
\(108\) 11664.0 0.0962250
\(109\) −217015. −1.74954 −0.874769 0.484540i \(-0.838987\pi\)
−0.874769 + 0.484540i \(0.838987\pi\)
\(110\) 0 0
\(111\) 77778.0 0.599169
\(112\) 12032.0 0.0906343
\(113\) −138324. −1.01906 −0.509532 0.860452i \(-0.670181\pi\)
−0.509532 + 0.860452i \(0.670181\pi\)
\(114\) 60660.0 0.437160
\(115\) 0 0
\(116\) 126240. 0.871068
\(117\) 8181.00 0.0552512
\(118\) 70440.0 0.465709
\(119\) 7614.00 0.0492885
\(120\) 0 0
\(121\) −111767. −0.693985
\(122\) −87412.0 −0.531706
\(123\) −163512. −0.974511
\(124\) −137488. −0.802991
\(125\) 0 0
\(126\) 15228.0 0.0854509
\(127\) −256048. −1.40868 −0.704340 0.709863i \(-0.748757\pi\)
−0.704340 + 0.709863i \(0.748757\pi\)
\(128\) 16384.0 0.0883883
\(129\) 129159. 0.683361
\(130\) 0 0
\(131\) 118452. 0.603065 0.301533 0.953456i \(-0.402502\pi\)
0.301533 + 0.953456i \(0.402502\pi\)
\(132\) 31968.0 0.159691
\(133\) 79195.0 0.388212
\(134\) 428.000 0.00205912
\(135\) 0 0
\(136\) 10368.0 0.0480671
\(137\) −13218.0 −0.0601678 −0.0300839 0.999547i \(-0.509577\pi\)
−0.0300839 + 0.999547i \(0.509577\pi\)
\(138\) 11016.0 0.0492409
\(139\) −350740. −1.53974 −0.769872 0.638199i \(-0.779680\pi\)
−0.769872 + 0.638199i \(0.779680\pi\)
\(140\) 0 0
\(141\) −9882.00 −0.0418598
\(142\) −162912. −0.678004
\(143\) 22422.0 0.0916926
\(144\) 20736.0 0.0833333
\(145\) 0 0
\(146\) 138824. 0.538992
\(147\) −131382. −0.501467
\(148\) 138272. 0.518896
\(149\) 109890. 0.405502 0.202751 0.979230i \(-0.435012\pi\)
0.202751 + 0.979230i \(0.435012\pi\)
\(150\) 0 0
\(151\) −172603. −0.616036 −0.308018 0.951381i \(-0.599666\pi\)
−0.308018 + 0.951381i \(0.599666\pi\)
\(152\) 107840. 0.378592
\(153\) 13122.0 0.0453181
\(154\) 41736.0 0.141811
\(155\) 0 0
\(156\) 14544.0 0.0478489
\(157\) −349993. −1.13321 −0.566605 0.823990i \(-0.691743\pi\)
−0.566605 + 0.823990i \(0.691743\pi\)
\(158\) −276640. −0.881601
\(159\) 161244. 0.505814
\(160\) 0 0
\(161\) 14382.0 0.0437275
\(162\) 26244.0 0.0785674
\(163\) 192581. 0.567733 0.283867 0.958864i \(-0.408383\pi\)
0.283867 + 0.958864i \(0.408383\pi\)
\(164\) −290688. −0.843951
\(165\) 0 0
\(166\) −434136. −1.22280
\(167\) 580692. 1.61122 0.805610 0.592447i \(-0.201838\pi\)
0.805610 + 0.592447i \(0.201838\pi\)
\(168\) 27072.0 0.0740026
\(169\) −361092. −0.972526
\(170\) 0 0
\(171\) 136485. 0.356940
\(172\) 229616. 0.591808
\(173\) 738126. 1.87506 0.937530 0.347904i \(-0.113106\pi\)
0.937530 + 0.347904i \(0.113106\pi\)
\(174\) 284040. 0.711224
\(175\) 0 0
\(176\) 56832.0 0.138297
\(177\) 158490. 0.380250
\(178\) 140160. 0.331569
\(179\) 497370. 1.16024 0.580119 0.814532i \(-0.303006\pi\)
0.580119 + 0.814532i \(0.303006\pi\)
\(180\) 0 0
\(181\) −333163. −0.755893 −0.377947 0.925827i \(-0.623370\pi\)
−0.377947 + 0.925827i \(0.623370\pi\)
\(182\) 18988.0 0.0424914
\(183\) −196677. −0.434136
\(184\) 19584.0 0.0426439
\(185\) 0 0
\(186\) −309348. −0.655639
\(187\) 35964.0 0.0752080
\(188\) −17568.0 −0.0362516
\(189\) 34263.0 0.0697703
\(190\) 0 0
\(191\) −40638.0 −0.0806026 −0.0403013 0.999188i \(-0.512832\pi\)
−0.0403013 + 0.999188i \(0.512832\pi\)
\(192\) 36864.0 0.0721688
\(193\) 494651. 0.955885 0.477942 0.878391i \(-0.341383\pi\)
0.477942 + 0.878391i \(0.341383\pi\)
\(194\) −3292.00 −0.00627994
\(195\) 0 0
\(196\) −233568. −0.434283
\(197\) 552342. 1.01401 0.507005 0.861943i \(-0.330752\pi\)
0.507005 + 0.861943i \(0.330752\pi\)
\(198\) 71928.0 0.130387
\(199\) 685625. 1.22731 0.613655 0.789575i \(-0.289699\pi\)
0.613655 + 0.789575i \(0.289699\pi\)
\(200\) 0 0
\(201\) 963.000 0.00168126
\(202\) −135312. −0.233323
\(203\) 370830. 0.631589
\(204\) 23328.0 0.0392466
\(205\) 0 0
\(206\) −533776. −0.876377
\(207\) 24786.0 0.0402050
\(208\) 25856.0 0.0414384
\(209\) 374070. 0.592362
\(210\) 0 0
\(211\) 749477. 1.15892 0.579458 0.815002i \(-0.303264\pi\)
0.579458 + 0.815002i \(0.303264\pi\)
\(212\) 286656. 0.438048
\(213\) −366552. −0.553588
\(214\) 325008. 0.485132
\(215\) 0 0
\(216\) 46656.0 0.0680414
\(217\) −403871. −0.582228
\(218\) −868060. −1.23711
\(219\) 312354. 0.440085
\(220\) 0 0
\(221\) 16362.0 0.0225349
\(222\) 311112. 0.423676
\(223\) 169271. 0.227940 0.113970 0.993484i \(-0.463643\pi\)
0.113970 + 0.993484i \(0.463643\pi\)
\(224\) 48128.0 0.0640882
\(225\) 0 0
\(226\) −553296. −0.720587
\(227\) −46488.0 −0.0598792 −0.0299396 0.999552i \(-0.509532\pi\)
−0.0299396 + 0.999552i \(0.509532\pi\)
\(228\) 242640. 0.309119
\(229\) −90115.0 −0.113556 −0.0567778 0.998387i \(-0.518083\pi\)
−0.0567778 + 0.998387i \(0.518083\pi\)
\(230\) 0 0
\(231\) 93906.0 0.115788
\(232\) 504960. 0.615938
\(233\) 1.06414e6 1.28413 0.642063 0.766652i \(-0.278079\pi\)
0.642063 + 0.766652i \(0.278079\pi\)
\(234\) 32724.0 0.0390685
\(235\) 0 0
\(236\) 281760. 0.329306
\(237\) −622440. −0.719825
\(238\) 30456.0 0.0348522
\(239\) 1.15158e6 1.30407 0.652033 0.758191i \(-0.273916\pi\)
0.652033 + 0.758191i \(0.273916\pi\)
\(240\) 0 0
\(241\) 856217. 0.949601 0.474801 0.880093i \(-0.342520\pi\)
0.474801 + 0.880093i \(0.342520\pi\)
\(242\) −447068. −0.490722
\(243\) 59049.0 0.0641500
\(244\) −349648. −0.375973
\(245\) 0 0
\(246\) −654048. −0.689084
\(247\) 170185. 0.177492
\(248\) −549952. −0.567800
\(249\) −976806. −0.998412
\(250\) 0 0
\(251\) −207708. −0.208098 −0.104049 0.994572i \(-0.533180\pi\)
−0.104049 + 0.994572i \(0.533180\pi\)
\(252\) 60912.0 0.0604229
\(253\) 67932.0 0.0667226
\(254\) −1.02419e6 −0.996087
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) −1.45319e6 −1.37243 −0.686213 0.727401i \(-0.740728\pi\)
−0.686213 + 0.727401i \(0.740728\pi\)
\(258\) 516636. 0.483209
\(259\) 406174. 0.376238
\(260\) 0 0
\(261\) 639090. 0.580712
\(262\) 473808. 0.426431
\(263\) 169296. 0.150924 0.0754618 0.997149i \(-0.475957\pi\)
0.0754618 + 0.997149i \(0.475957\pi\)
\(264\) 127872. 0.112919
\(265\) 0 0
\(266\) 316780. 0.274507
\(267\) 315360. 0.270725
\(268\) 1712.00 0.00145602
\(269\) −1.58109e6 −1.33222 −0.666110 0.745854i \(-0.732042\pi\)
−0.666110 + 0.745854i \(0.732042\pi\)
\(270\) 0 0
\(271\) 822512. 0.680329 0.340165 0.940366i \(-0.389517\pi\)
0.340165 + 0.940366i \(0.389517\pi\)
\(272\) 41472.0 0.0339886
\(273\) 42723.0 0.0346941
\(274\) −52872.0 −0.0425451
\(275\) 0 0
\(276\) 44064.0 0.0348186
\(277\) −546823. −0.428201 −0.214100 0.976812i \(-0.568682\pi\)
−0.214100 + 0.976812i \(0.568682\pi\)
\(278\) −1.40296e6 −1.08876
\(279\) −696033. −0.535327
\(280\) 0 0
\(281\) −1.09250e6 −0.825382 −0.412691 0.910871i \(-0.635411\pi\)
−0.412691 + 0.910871i \(0.635411\pi\)
\(282\) −39528.0 −0.0295993
\(283\) −2.48480e6 −1.84427 −0.922136 0.386865i \(-0.873558\pi\)
−0.922136 + 0.386865i \(0.873558\pi\)
\(284\) −651648. −0.479421
\(285\) 0 0
\(286\) 89688.0 0.0648365
\(287\) −853896. −0.611928
\(288\) 82944.0 0.0589256
\(289\) −1.39361e6 −0.981516
\(290\) 0 0
\(291\) −7407.00 −0.00512755
\(292\) 555296. 0.381125
\(293\) −341394. −0.232320 −0.116160 0.993231i \(-0.537059\pi\)
−0.116160 + 0.993231i \(0.537059\pi\)
\(294\) −525528. −0.354591
\(295\) 0 0
\(296\) 553088. 0.366915
\(297\) 161838. 0.106461
\(298\) 439560. 0.286733
\(299\) 30906.0 0.0199924
\(300\) 0 0
\(301\) 674497. 0.429105
\(302\) −690412. −0.435603
\(303\) −304452. −0.190508
\(304\) 431360. 0.267705
\(305\) 0 0
\(306\) 52488.0 0.0320447
\(307\) 2.02898e6 1.22866 0.614329 0.789050i \(-0.289427\pi\)
0.614329 + 0.789050i \(0.289427\pi\)
\(308\) 166944. 0.100275
\(309\) −1.20100e6 −0.715559
\(310\) 0 0
\(311\) −206598. −0.121123 −0.0605613 0.998164i \(-0.519289\pi\)
−0.0605613 + 0.998164i \(0.519289\pi\)
\(312\) 58176.0 0.0338343
\(313\) −3.34223e6 −1.92830 −0.964152 0.265352i \(-0.914512\pi\)
−0.964152 + 0.265352i \(0.914512\pi\)
\(314\) −1.39997e6 −0.801300
\(315\) 0 0
\(316\) −1.10656e6 −0.623386
\(317\) −2.53289e6 −1.41569 −0.707844 0.706368i \(-0.750332\pi\)
−0.707844 + 0.706368i \(0.750332\pi\)
\(318\) 644976. 0.357664
\(319\) 1.75158e6 0.963725
\(320\) 0 0
\(321\) 731268. 0.396108
\(322\) 57528.0 0.0309200
\(323\) 272970. 0.145582
\(324\) 104976. 0.0555556
\(325\) 0 0
\(326\) 770324. 0.401448
\(327\) −1.95314e6 −1.01010
\(328\) −1.16275e6 −0.596764
\(329\) −51606.0 −0.0262851
\(330\) 0 0
\(331\) 602132. 0.302080 0.151040 0.988528i \(-0.451738\pi\)
0.151040 + 0.988528i \(0.451738\pi\)
\(332\) −1.73654e6 −0.864650
\(333\) 700002. 0.345930
\(334\) 2.32277e6 1.13930
\(335\) 0 0
\(336\) 108288. 0.0523278
\(337\) 209777. 0.100620 0.0503099 0.998734i \(-0.483979\pi\)
0.0503099 + 0.998734i \(0.483979\pi\)
\(338\) −1.44437e6 −0.687680
\(339\) −1.24492e6 −0.588357
\(340\) 0 0
\(341\) −1.90765e6 −0.888407
\(342\) 545940. 0.252394
\(343\) −1.47603e6 −0.677425
\(344\) 918464. 0.418472
\(345\) 0 0
\(346\) 2.95250e6 1.32587
\(347\) 4.02166e6 1.79301 0.896503 0.443037i \(-0.146099\pi\)
0.896503 + 0.443037i \(0.146099\pi\)
\(348\) 1.13616e6 0.502911
\(349\) 8330.00 0.00366085 0.00183042 0.999998i \(-0.499417\pi\)
0.00183042 + 0.999998i \(0.499417\pi\)
\(350\) 0 0
\(351\) 73629.0 0.0318993
\(352\) 227328. 0.0977904
\(353\) 1.95001e6 0.832912 0.416456 0.909156i \(-0.363272\pi\)
0.416456 + 0.909156i \(0.363272\pi\)
\(354\) 633960. 0.268877
\(355\) 0 0
\(356\) 560640. 0.234455
\(357\) 68526.0 0.0284567
\(358\) 1.98948e6 0.820412
\(359\) 2.27088e6 0.929947 0.464973 0.885325i \(-0.346064\pi\)
0.464973 + 0.885325i \(0.346064\pi\)
\(360\) 0 0
\(361\) 363126. 0.146652
\(362\) −1.33265e6 −0.534497
\(363\) −1.00590e6 −0.400673
\(364\) 75952.0 0.0300459
\(365\) 0 0
\(366\) −786708. −0.306981
\(367\) 2.86154e6 1.10901 0.554503 0.832181i \(-0.312908\pi\)
0.554503 + 0.832181i \(0.312908\pi\)
\(368\) 78336.0 0.0301538
\(369\) −1.47161e6 −0.562634
\(370\) 0 0
\(371\) 842052. 0.317617
\(372\) −1.23739e6 −0.463607
\(373\) 615311. 0.228993 0.114497 0.993424i \(-0.463474\pi\)
0.114497 + 0.993424i \(0.463474\pi\)
\(374\) 143856. 0.0531801
\(375\) 0 0
\(376\) −70272.0 −0.0256338
\(377\) 796890. 0.288765
\(378\) 137052. 0.0493351
\(379\) 5.39878e6 1.93062 0.965311 0.261103i \(-0.0840863\pi\)
0.965311 + 0.261103i \(0.0840863\pi\)
\(380\) 0 0
\(381\) −2.30443e6 −0.813301
\(382\) −162552. −0.0569946
\(383\) 1.08688e6 0.378602 0.189301 0.981919i \(-0.439378\pi\)
0.189301 + 0.981919i \(0.439378\pi\)
\(384\) 147456. 0.0510310
\(385\) 0 0
\(386\) 1.97860e6 0.675913
\(387\) 1.16243e6 0.394539
\(388\) −13168.0 −0.00444059
\(389\) −3.48432e6 −1.16747 −0.583733 0.811946i \(-0.698408\pi\)
−0.583733 + 0.811946i \(0.698408\pi\)
\(390\) 0 0
\(391\) 49572.0 0.0163981
\(392\) −934272. −0.307085
\(393\) 1.06607e6 0.348180
\(394\) 2.20937e6 0.717014
\(395\) 0 0
\(396\) 287712. 0.0921977
\(397\) 3.26591e6 1.03999 0.519993 0.854170i \(-0.325935\pi\)
0.519993 + 0.854170i \(0.325935\pi\)
\(398\) 2.74250e6 0.867839
\(399\) 712755. 0.224134
\(400\) 0 0
\(401\) −4.27319e6 −1.32706 −0.663531 0.748149i \(-0.730943\pi\)
−0.663531 + 0.748149i \(0.730943\pi\)
\(402\) 3852.00 0.00118883
\(403\) −867893. −0.266197
\(404\) −541248. −0.164984
\(405\) 0 0
\(406\) 1.48332e6 0.446601
\(407\) 1.91852e6 0.574092
\(408\) 93312.0 0.0277515
\(409\) −1.45188e6 −0.429162 −0.214581 0.976706i \(-0.568839\pi\)
−0.214581 + 0.976706i \(0.568839\pi\)
\(410\) 0 0
\(411\) −118962. −0.0347379
\(412\) −2.13510e6 −0.619692
\(413\) 827670. 0.238771
\(414\) 99144.0 0.0284293
\(415\) 0 0
\(416\) 103424. 0.0293014
\(417\) −3.15666e6 −0.888971
\(418\) 1.49628e6 0.418863
\(419\) 559380. 0.155658 0.0778291 0.996967i \(-0.475201\pi\)
0.0778291 + 0.996967i \(0.475201\pi\)
\(420\) 0 0
\(421\) −3.91470e6 −1.07645 −0.538224 0.842802i \(-0.680905\pi\)
−0.538224 + 0.842802i \(0.680905\pi\)
\(422\) 2.99791e6 0.819478
\(423\) −88938.0 −0.0241678
\(424\) 1.14662e6 0.309746
\(425\) 0 0
\(426\) −1.46621e6 −0.391446
\(427\) −1.02709e6 −0.272608
\(428\) 1.30003e6 0.343040
\(429\) 201798. 0.0529387
\(430\) 0 0
\(431\) −3.57500e6 −0.927006 −0.463503 0.886095i \(-0.653408\pi\)
−0.463503 + 0.886095i \(0.653408\pi\)
\(432\) 186624. 0.0481125
\(433\) 7.15969e6 1.83516 0.917581 0.397548i \(-0.130139\pi\)
0.917581 + 0.397548i \(0.130139\pi\)
\(434\) −1.61548e6 −0.411698
\(435\) 0 0
\(436\) −3.47224e6 −0.874769
\(437\) 515610. 0.129157
\(438\) 1.24942e6 0.311187
\(439\) 1.71790e6 0.425437 0.212719 0.977114i \(-0.431768\pi\)
0.212719 + 0.977114i \(0.431768\pi\)
\(440\) 0 0
\(441\) −1.18244e6 −0.289522
\(442\) 65448.0 0.0159346
\(443\) 3.39670e6 0.822332 0.411166 0.911560i \(-0.365122\pi\)
0.411166 + 0.911560i \(0.365122\pi\)
\(444\) 1.24445e6 0.299584
\(445\) 0 0
\(446\) 677084. 0.161178
\(447\) 989010. 0.234116
\(448\) 192512. 0.0453172
\(449\) −3.39606e6 −0.794986 −0.397493 0.917605i \(-0.630120\pi\)
−0.397493 + 0.917605i \(0.630120\pi\)
\(450\) 0 0
\(451\) −4.03330e6 −0.933724
\(452\) −2.21318e6 −0.509532
\(453\) −1.55343e6 −0.355668
\(454\) −185952. −0.0423410
\(455\) 0 0
\(456\) 970560. 0.218580
\(457\) −4.52814e6 −1.01421 −0.507106 0.861883i \(-0.669285\pi\)
−0.507106 + 0.861883i \(0.669285\pi\)
\(458\) −360460. −0.0802959
\(459\) 118098. 0.0261644
\(460\) 0 0
\(461\) −1.27895e6 −0.280285 −0.140143 0.990131i \(-0.544756\pi\)
−0.140143 + 0.990131i \(0.544756\pi\)
\(462\) 375624. 0.0818744
\(463\) −7.19862e6 −1.56062 −0.780310 0.625393i \(-0.784939\pi\)
−0.780310 + 0.625393i \(0.784939\pi\)
\(464\) 2.01984e6 0.435534
\(465\) 0 0
\(466\) 4.25654e6 0.908014
\(467\) 4.83034e6 1.02491 0.512455 0.858714i \(-0.328736\pi\)
0.512455 + 0.858714i \(0.328736\pi\)
\(468\) 130896. 0.0276256
\(469\) 5029.00 0.00105572
\(470\) 0 0
\(471\) −3.14994e6 −0.654259
\(472\) 1.12704e6 0.232854
\(473\) 3.18592e6 0.654760
\(474\) −2.48976e6 −0.508993
\(475\) 0 0
\(476\) 121824. 0.0246442
\(477\) 1.45120e6 0.292032
\(478\) 4.60632e6 0.922113
\(479\) 748650. 0.149087 0.0745435 0.997218i \(-0.476250\pi\)
0.0745435 + 0.997218i \(0.476250\pi\)
\(480\) 0 0
\(481\) 872842. 0.172018
\(482\) 3.42487e6 0.671469
\(483\) 129438. 0.0252461
\(484\) −1.78827e6 −0.346993
\(485\) 0 0
\(486\) 236196. 0.0453609
\(487\) −5.16394e6 −0.986641 −0.493320 0.869848i \(-0.664217\pi\)
−0.493320 + 0.869848i \(0.664217\pi\)
\(488\) −1.39859e6 −0.265853
\(489\) 1.73323e6 0.327781
\(490\) 0 0
\(491\) 8.54287e6 1.59919 0.799595 0.600539i \(-0.205047\pi\)
0.799595 + 0.600539i \(0.205047\pi\)
\(492\) −2.61619e6 −0.487256
\(493\) 1.27818e6 0.236851
\(494\) 680740. 0.125506
\(495\) 0 0
\(496\) −2.19981e6 −0.401495
\(497\) −1.91422e6 −0.347616
\(498\) −3.90722e6 −0.705984
\(499\) −4.20588e6 −0.756145 −0.378072 0.925776i \(-0.623413\pi\)
−0.378072 + 0.925776i \(0.623413\pi\)
\(500\) 0 0
\(501\) 5.22623e6 0.930238
\(502\) −830832. −0.147148
\(503\) −8.18342e6 −1.44217 −0.721083 0.692849i \(-0.756355\pi\)
−0.721083 + 0.692849i \(0.756355\pi\)
\(504\) 243648. 0.0427254
\(505\) 0 0
\(506\) 271728. 0.0471800
\(507\) −3.24983e6 −0.561488
\(508\) −4.09677e6 −0.704340
\(509\) 3.85923e6 0.660247 0.330123 0.943938i \(-0.392910\pi\)
0.330123 + 0.943938i \(0.392910\pi\)
\(510\) 0 0
\(511\) 1.63118e6 0.276344
\(512\) 262144. 0.0441942
\(513\) 1.22837e6 0.206079
\(514\) −5.81275e6 −0.970452
\(515\) 0 0
\(516\) 2.06654e6 0.341681
\(517\) −243756. −0.0401078
\(518\) 1.62470e6 0.266040
\(519\) 6.64313e6 1.08257
\(520\) 0 0
\(521\) 4.55410e6 0.735036 0.367518 0.930016i \(-0.380208\pi\)
0.367518 + 0.930016i \(0.380208\pi\)
\(522\) 2.55636e6 0.410625
\(523\) 4.82224e6 0.770894 0.385447 0.922730i \(-0.374047\pi\)
0.385447 + 0.922730i \(0.374047\pi\)
\(524\) 1.89523e6 0.301533
\(525\) 0 0
\(526\) 677184. 0.106719
\(527\) −1.39207e6 −0.218340
\(528\) 511488. 0.0798455
\(529\) −6.34271e6 −0.985452
\(530\) 0 0
\(531\) 1.42641e6 0.219537
\(532\) 1.26712e6 0.194106
\(533\) −1.83497e6 −0.279776
\(534\) 1.26144e6 0.191432
\(535\) 0 0
\(536\) 6848.00 0.00102956
\(537\) 4.47633e6 0.669864
\(538\) −6.32436e6 −0.942022
\(539\) −3.24076e6 −0.480479
\(540\) 0 0
\(541\) 362537. 0.0532549 0.0266274 0.999645i \(-0.491523\pi\)
0.0266274 + 0.999645i \(0.491523\pi\)
\(542\) 3.29005e6 0.481065
\(543\) −2.99847e6 −0.436415
\(544\) 165888. 0.0240335
\(545\) 0 0
\(546\) 170892. 0.0245324
\(547\) 3.11439e6 0.445046 0.222523 0.974927i \(-0.428571\pi\)
0.222523 + 0.974927i \(0.428571\pi\)
\(548\) −211488. −0.0300839
\(549\) −1.77009e6 −0.250649
\(550\) 0 0
\(551\) 1.32947e7 1.86551
\(552\) 176256. 0.0246205
\(553\) −3.25052e6 −0.452002
\(554\) −2.18729e6 −0.302784
\(555\) 0 0
\(556\) −5.61184e6 −0.769872
\(557\) −7.99304e6 −1.09163 −0.545813 0.837907i \(-0.683779\pi\)
−0.545813 + 0.837907i \(0.683779\pi\)
\(558\) −2.78413e6 −0.378534
\(559\) 1.44945e6 0.196189
\(560\) 0 0
\(561\) 323676. 0.0434214
\(562\) −4.36999e6 −0.583633
\(563\) −1.23236e7 −1.63857 −0.819286 0.573385i \(-0.805630\pi\)
−0.819286 + 0.573385i \(0.805630\pi\)
\(564\) −158112. −0.0209299
\(565\) 0 0
\(566\) −9.93920e6 −1.30410
\(567\) 308367. 0.0402819
\(568\) −2.60659e6 −0.339002
\(569\) 1.01364e7 1.31252 0.656258 0.754537i \(-0.272138\pi\)
0.656258 + 0.754537i \(0.272138\pi\)
\(570\) 0 0
\(571\) 6.53084e6 0.838260 0.419130 0.907926i \(-0.362335\pi\)
0.419130 + 0.907926i \(0.362335\pi\)
\(572\) 358752. 0.0458463
\(573\) −365742. −0.0465359
\(574\) −3.41558e6 −0.432698
\(575\) 0 0
\(576\) 331776. 0.0416667
\(577\) −1.24453e6 −0.155621 −0.0778103 0.996968i \(-0.524793\pi\)
−0.0778103 + 0.996968i \(0.524793\pi\)
\(578\) −5.57445e6 −0.694037
\(579\) 4.45186e6 0.551880
\(580\) 0 0
\(581\) −5.10110e6 −0.626936
\(582\) −29628.0 −0.00362573
\(583\) 3.97735e6 0.484644
\(584\) 2.22118e6 0.269496
\(585\) 0 0
\(586\) −1.36558e6 −0.164275
\(587\) −1.33403e6 −0.159797 −0.0798987 0.996803i \(-0.525460\pi\)
−0.0798987 + 0.996803i \(0.525460\pi\)
\(588\) −2.10211e6 −0.250734
\(589\) −1.44792e7 −1.71972
\(590\) 0 0
\(591\) 4.97108e6 0.585439
\(592\) 2.21235e6 0.259448
\(593\) −1.19401e7 −1.39435 −0.697177 0.716899i \(-0.745561\pi\)
−0.697177 + 0.716899i \(0.745561\pi\)
\(594\) 647352. 0.0752791
\(595\) 0 0
\(596\) 1.75824e6 0.202751
\(597\) 6.17062e6 0.708587
\(598\) 123624. 0.0141368
\(599\) −7.16430e6 −0.815843 −0.407922 0.913017i \(-0.633746\pi\)
−0.407922 + 0.913017i \(0.633746\pi\)
\(600\) 0 0
\(601\) 1.15163e6 0.130055 0.0650273 0.997883i \(-0.479287\pi\)
0.0650273 + 0.997883i \(0.479287\pi\)
\(602\) 2.69799e6 0.303423
\(603\) 8667.00 0.000970679 0
\(604\) −2.76165e6 −0.308018
\(605\) 0 0
\(606\) −1.21781e6 −0.134709
\(607\) −1.34268e7 −1.47911 −0.739554 0.673097i \(-0.764963\pi\)
−0.739554 + 0.673097i \(0.764963\pi\)
\(608\) 1.72544e6 0.189296
\(609\) 3.33747e6 0.364648
\(610\) 0 0
\(611\) −110898. −0.0120177
\(612\) 209952. 0.0226590
\(613\) 1.20184e7 1.29180 0.645900 0.763422i \(-0.276482\pi\)
0.645900 + 0.763422i \(0.276482\pi\)
\(614\) 8.11591e6 0.868793
\(615\) 0 0
\(616\) 667776. 0.0709054
\(617\) −6.98519e6 −0.738695 −0.369348 0.929291i \(-0.620419\pi\)
−0.369348 + 0.929291i \(0.620419\pi\)
\(618\) −4.80398e6 −0.505977
\(619\) −8.20625e6 −0.860832 −0.430416 0.902631i \(-0.641633\pi\)
−0.430416 + 0.902631i \(0.641633\pi\)
\(620\) 0 0
\(621\) 223074. 0.0232124
\(622\) −826392. −0.0856466
\(623\) 1.64688e6 0.169997
\(624\) 232704. 0.0239245
\(625\) 0 0
\(626\) −1.33689e7 −1.36352
\(627\) 3.36663e6 0.342000
\(628\) −5.59989e6 −0.566605
\(629\) 1.40000e6 0.141092
\(630\) 0 0
\(631\) −1.07686e7 −1.07668 −0.538338 0.842729i \(-0.680947\pi\)
−0.538338 + 0.842729i \(0.680947\pi\)
\(632\) −4.42624e6 −0.440801
\(633\) 6.74529e6 0.669101
\(634\) −1.01316e7 −1.00104
\(635\) 0 0
\(636\) 2.57990e6 0.252907
\(637\) −1.47440e6 −0.143968
\(638\) 7.00632e6 0.681457
\(639\) −3.29897e6 −0.319614
\(640\) 0 0
\(641\) 1.92571e7 1.85117 0.925585 0.378539i \(-0.123573\pi\)
0.925585 + 0.378539i \(0.123573\pi\)
\(642\) 2.92507e6 0.280091
\(643\) 1.00999e7 0.963364 0.481682 0.876346i \(-0.340026\pi\)
0.481682 + 0.876346i \(0.340026\pi\)
\(644\) 230112. 0.0218637
\(645\) 0 0
\(646\) 1.09188e6 0.102942
\(647\) 7.52113e6 0.706354 0.353177 0.935556i \(-0.385101\pi\)
0.353177 + 0.935556i \(0.385101\pi\)
\(648\) 419904. 0.0392837
\(649\) 3.90942e6 0.364335
\(650\) 0 0
\(651\) −3.63484e6 −0.336150
\(652\) 3.08130e6 0.283867
\(653\) −2.67197e6 −0.245216 −0.122608 0.992455i \(-0.539126\pi\)
−0.122608 + 0.992455i \(0.539126\pi\)
\(654\) −7.81254e6 −0.714246
\(655\) 0 0
\(656\) −4.65101e6 −0.421976
\(657\) 2.81119e6 0.254083
\(658\) −206424. −0.0185864
\(659\) 6.99948e6 0.627845 0.313922 0.949449i \(-0.398357\pi\)
0.313922 + 0.949449i \(0.398357\pi\)
\(660\) 0 0
\(661\) 408122. 0.0363318 0.0181659 0.999835i \(-0.494217\pi\)
0.0181659 + 0.999835i \(0.494217\pi\)
\(662\) 2.40853e6 0.213603
\(663\) 147258. 0.0130105
\(664\) −6.94618e6 −0.611400
\(665\) 0 0
\(666\) 2.80001e6 0.244610
\(667\) 2.41434e6 0.210128
\(668\) 9.29107e6 0.805610
\(669\) 1.52344e6 0.131601
\(670\) 0 0
\(671\) −4.85137e6 −0.415966
\(672\) 433152. 0.0370013
\(673\) 1.74939e7 1.48885 0.744423 0.667709i \(-0.232725\pi\)
0.744423 + 0.667709i \(0.232725\pi\)
\(674\) 839108. 0.0711489
\(675\) 0 0
\(676\) −5.77747e6 −0.486263
\(677\) −8.67440e6 −0.727391 −0.363695 0.931518i \(-0.618485\pi\)
−0.363695 + 0.931518i \(0.618485\pi\)
\(678\) −4.97966e6 −0.416031
\(679\) −38681.0 −0.00321976
\(680\) 0 0
\(681\) −418392. −0.0345713
\(682\) −7.63058e6 −0.628198
\(683\) 1.18478e7 0.971822 0.485911 0.874008i \(-0.338488\pi\)
0.485911 + 0.874008i \(0.338488\pi\)
\(684\) 2.18376e6 0.178470
\(685\) 0 0
\(686\) −5.90414e6 −0.479012
\(687\) −811035. −0.0655613
\(688\) 3.67386e6 0.295904
\(689\) 1.80952e6 0.145216
\(690\) 0 0
\(691\) 9.47775e6 0.755110 0.377555 0.925987i \(-0.376765\pi\)
0.377555 + 0.925987i \(0.376765\pi\)
\(692\) 1.18100e7 0.937530
\(693\) 845154. 0.0668502
\(694\) 1.60866e7 1.26785
\(695\) 0 0
\(696\) 4.54464e6 0.355612
\(697\) −2.94322e6 −0.229478
\(698\) 33320.0 0.00258861
\(699\) 9.57722e6 0.741390
\(700\) 0 0
\(701\) −2.28147e7 −1.75355 −0.876777 0.480898i \(-0.840311\pi\)
−0.876777 + 0.480898i \(0.840311\pi\)
\(702\) 294516. 0.0225562
\(703\) 1.45618e7 1.11129
\(704\) 909312. 0.0691483
\(705\) 0 0
\(706\) 7.80002e6 0.588958
\(707\) −1.58992e6 −0.119626
\(708\) 2.53584e6 0.190125
\(709\) 1.27436e7 0.952090 0.476045 0.879421i \(-0.342070\pi\)
0.476045 + 0.879421i \(0.342070\pi\)
\(710\) 0 0
\(711\) −5.60196e6 −0.415591
\(712\) 2.24256e6 0.165785
\(713\) −2.62946e6 −0.193706
\(714\) 274104. 0.0201219
\(715\) 0 0
\(716\) 7.95792e6 0.580119
\(717\) 1.03642e7 0.752902
\(718\) 9.08352e6 0.657572
\(719\) 2.44929e6 0.176692 0.0883462 0.996090i \(-0.471842\pi\)
0.0883462 + 0.996090i \(0.471842\pi\)
\(720\) 0 0
\(721\) −6.27187e6 −0.449323
\(722\) 1.45250e6 0.103699
\(723\) 7.70595e6 0.548252
\(724\) −5.33061e6 −0.377947
\(725\) 0 0
\(726\) −4.02361e6 −0.283318
\(727\) −415033. −0.0291237 −0.0145619 0.999894i \(-0.504635\pi\)
−0.0145619 + 0.999894i \(0.504635\pi\)
\(728\) 303808. 0.0212457
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 2.32486e6 0.160918
\(732\) −3.14683e6 −0.217068
\(733\) −1.72877e7 −1.18844 −0.594221 0.804302i \(-0.702539\pi\)
−0.594221 + 0.804302i \(0.702539\pi\)
\(734\) 1.14461e7 0.784186
\(735\) 0 0
\(736\) 313344. 0.0213219
\(737\) 23754.0 0.00161090
\(738\) −5.88643e6 −0.397843
\(739\) 5.18834e6 0.349476 0.174738 0.984615i \(-0.444092\pi\)
0.174738 + 0.984615i \(0.444092\pi\)
\(740\) 0 0
\(741\) 1.53166e6 0.102475
\(742\) 3.36821e6 0.224589
\(743\) 4.79572e6 0.318700 0.159350 0.987222i \(-0.449060\pi\)
0.159350 + 0.987222i \(0.449060\pi\)
\(744\) −4.94957e6 −0.327820
\(745\) 0 0
\(746\) 2.46124e6 0.161923
\(747\) −8.79125e6 −0.576434
\(748\) 575424. 0.0376040
\(749\) 3.81884e6 0.248730
\(750\) 0 0
\(751\) −1.85654e7 −1.20117 −0.600585 0.799561i \(-0.705066\pi\)
−0.600585 + 0.799561i \(0.705066\pi\)
\(752\) −281088. −0.0181258
\(753\) −1.86937e6 −0.120146
\(754\) 3.18756e6 0.204188
\(755\) 0 0
\(756\) 548208. 0.0348852
\(757\) 2.82068e7 1.78902 0.894508 0.447053i \(-0.147526\pi\)
0.894508 + 0.447053i \(0.147526\pi\)
\(758\) 2.15951e7 1.36516
\(759\) 611388. 0.0385223
\(760\) 0 0
\(761\) 6.56161e6 0.410723 0.205361 0.978686i \(-0.434163\pi\)
0.205361 + 0.978686i \(0.434163\pi\)
\(762\) −9.21773e6 −0.575091
\(763\) −1.01997e7 −0.634273
\(764\) −650208. −0.0403013
\(765\) 0 0
\(766\) 4.34750e6 0.267712
\(767\) 1.77861e6 0.109167
\(768\) 589824. 0.0360844
\(769\) 2.20930e7 1.34722 0.673610 0.739087i \(-0.264743\pi\)
0.673610 + 0.739087i \(0.264743\pi\)
\(770\) 0 0
\(771\) −1.30787e7 −0.792371
\(772\) 7.91442e6 0.477942
\(773\) −3.00787e7 −1.81055 −0.905276 0.424824i \(-0.860336\pi\)
−0.905276 + 0.424824i \(0.860336\pi\)
\(774\) 4.64972e6 0.278981
\(775\) 0 0
\(776\) −52672.0 −0.00313997
\(777\) 3.65557e6 0.217221
\(778\) −1.39373e7 −0.825523
\(779\) −3.06131e7 −1.80744
\(780\) 0 0
\(781\) −9.04162e6 −0.530418
\(782\) 198288. 0.0115952
\(783\) 5.75181e6 0.335274
\(784\) −3.73709e6 −0.217142
\(785\) 0 0
\(786\) 4.26427e6 0.246200
\(787\) −3.28954e6 −0.189321 −0.0946605 0.995510i \(-0.530177\pi\)
−0.0946605 + 0.995510i \(0.530177\pi\)
\(788\) 8.83747e6 0.507005
\(789\) 1.52366e6 0.0871358
\(790\) 0 0
\(791\) −6.50123e6 −0.369449
\(792\) 1.15085e6 0.0651936
\(793\) −2.20715e6 −0.124638
\(794\) 1.30636e7 0.735381
\(795\) 0 0
\(796\) 1.09700e7 0.613655
\(797\) 6.71053e6 0.374206 0.187103 0.982340i \(-0.440090\pi\)
0.187103 + 0.982340i \(0.440090\pi\)
\(798\) 2.85102e6 0.158487
\(799\) −177876. −0.00985713
\(800\) 0 0
\(801\) 2.83824e6 0.156303
\(802\) −1.70928e7 −0.938374
\(803\) 7.70473e6 0.421666
\(804\) 15408.0 0.000840632 0
\(805\) 0 0
\(806\) −3.47157e6 −0.188230
\(807\) −1.42298e7 −0.769157
\(808\) −2.16499e6 −0.116662
\(809\) 8.74254e6 0.469641 0.234821 0.972039i \(-0.424550\pi\)
0.234821 + 0.972039i \(0.424550\pi\)
\(810\) 0 0
\(811\) −2.48410e7 −1.32622 −0.663112 0.748520i \(-0.730765\pi\)
−0.663112 + 0.748520i \(0.730765\pi\)
\(812\) 5.93328e6 0.315795
\(813\) 7.40261e6 0.392788
\(814\) 7.67410e6 0.405944
\(815\) 0 0
\(816\) 373248. 0.0196233
\(817\) 2.41814e7 1.26744
\(818\) −5.80750e6 −0.303463
\(819\) 384507. 0.0200306
\(820\) 0 0
\(821\) 2.12219e7 1.09882 0.549409 0.835554i \(-0.314853\pi\)
0.549409 + 0.835554i \(0.314853\pi\)
\(822\) −475848. −0.0245634
\(823\) −8.70659e6 −0.448073 −0.224036 0.974581i \(-0.571923\pi\)
−0.224036 + 0.974581i \(0.571923\pi\)
\(824\) −8.54042e6 −0.438189
\(825\) 0 0
\(826\) 3.31068e6 0.168837
\(827\) 3.71184e7 1.88723 0.943617 0.331040i \(-0.107400\pi\)
0.943617 + 0.331040i \(0.107400\pi\)
\(828\) 396576. 0.0201025
\(829\) 1.01765e6 0.0514295 0.0257147 0.999669i \(-0.491814\pi\)
0.0257147 + 0.999669i \(0.491814\pi\)
\(830\) 0 0
\(831\) −4.92141e6 −0.247222
\(832\) 413696. 0.0207192
\(833\) −2.36488e6 −0.118085
\(834\) −1.26266e7 −0.628598
\(835\) 0 0
\(836\) 5.98512e6 0.296181
\(837\) −6.26430e6 −0.309071
\(838\) 2.23752e6 0.110067
\(839\) −3.36194e7 −1.64887 −0.824433 0.565960i \(-0.808506\pi\)
−0.824433 + 0.565960i \(0.808506\pi\)
\(840\) 0 0
\(841\) 4.17410e7 2.03504
\(842\) −1.56588e7 −0.761164
\(843\) −9.83248e6 −0.476534
\(844\) 1.19916e7 0.579458
\(845\) 0 0
\(846\) −355752. −0.0170892
\(847\) −5.25305e6 −0.251596
\(848\) 4.58650e6 0.219024
\(849\) −2.23632e7 −1.06479
\(850\) 0 0
\(851\) 2.64445e6 0.125173
\(852\) −5.86483e6 −0.276794
\(853\) −3.52574e7 −1.65912 −0.829559 0.558419i \(-0.811408\pi\)
−0.829559 + 0.558419i \(0.811408\pi\)
\(854\) −4.10836e6 −0.192763
\(855\) 0 0
\(856\) 5.20013e6 0.242566
\(857\) −3.14941e7 −1.46480 −0.732398 0.680877i \(-0.761599\pi\)
−0.732398 + 0.680877i \(0.761599\pi\)
\(858\) 807192. 0.0374333
\(859\) 1.19344e7 0.551848 0.275924 0.961180i \(-0.411016\pi\)
0.275924 + 0.961180i \(0.411016\pi\)
\(860\) 0 0
\(861\) −7.68506e6 −0.353297
\(862\) −1.43000e7 −0.655492
\(863\) 8.70442e6 0.397844 0.198922 0.980015i \(-0.436256\pi\)
0.198922 + 0.980015i \(0.436256\pi\)
\(864\) 746496. 0.0340207
\(865\) 0 0
\(866\) 2.86388e7 1.29766
\(867\) −1.25425e7 −0.566679
\(868\) −6.46194e6 −0.291114
\(869\) −1.53535e7 −0.689697
\(870\) 0 0
\(871\) 10807.0 0.000482681 0
\(872\) −1.38890e7 −0.618555
\(873\) −66663.0 −0.00296039
\(874\) 2.06244e6 0.0913277
\(875\) 0 0
\(876\) 4.99766e6 0.220043
\(877\) 1.17999e7 0.518059 0.259029 0.965869i \(-0.416597\pi\)
0.259029 + 0.965869i \(0.416597\pi\)
\(878\) 6.87158e6 0.300829
\(879\) −3.07255e6 −0.134130
\(880\) 0 0
\(881\) −2.73840e7 −1.18866 −0.594330 0.804221i \(-0.702583\pi\)
−0.594330 + 0.804221i \(0.702583\pi\)
\(882\) −4.72975e6 −0.204723
\(883\) −8.80577e6 −0.380072 −0.190036 0.981777i \(-0.560860\pi\)
−0.190036 + 0.981777i \(0.560860\pi\)
\(884\) 261792. 0.0112675
\(885\) 0 0
\(886\) 1.35868e7 0.581477
\(887\) 250122. 0.0106744 0.00533719 0.999986i \(-0.498301\pi\)
0.00533719 + 0.999986i \(0.498301\pi\)
\(888\) 4.97779e6 0.211838
\(889\) −1.20343e7 −0.510699
\(890\) 0 0
\(891\) 1.45654e6 0.0614651
\(892\) 2.70834e6 0.113970
\(893\) −1.85013e6 −0.0776379
\(894\) 3.95604e6 0.165545
\(895\) 0 0
\(896\) 770048. 0.0320441
\(897\) 278154. 0.0115426
\(898\) −1.35842e7 −0.562140
\(899\) −6.77988e7 −2.79784
\(900\) 0 0
\(901\) 2.90239e6 0.119109
\(902\) −1.61332e7 −0.660243
\(903\) 6.07047e6 0.247744
\(904\) −8.85274e6 −0.360294
\(905\) 0 0
\(906\) −6.21371e6 −0.251496
\(907\) −3.24955e7 −1.31161 −0.655806 0.754929i \(-0.727671\pi\)
−0.655806 + 0.754929i \(0.727671\pi\)
\(908\) −743808. −0.0299396
\(909\) −2.74007e6 −0.109990
\(910\) 0 0
\(911\) 4.24595e7 1.69504 0.847518 0.530766i \(-0.178096\pi\)
0.847518 + 0.530766i \(0.178096\pi\)
\(912\) 3.88224e6 0.154559
\(913\) −2.40945e7 −0.956625
\(914\) −1.81126e7 −0.717157
\(915\) 0 0
\(916\) −1.44184e6 −0.0567778
\(917\) 5.56724e6 0.218634
\(918\) 472392. 0.0185010
\(919\) 1.41629e7 0.553176 0.276588 0.960989i \(-0.410796\pi\)
0.276588 + 0.960989i \(0.410796\pi\)
\(920\) 0 0
\(921\) 1.82608e7 0.709366
\(922\) −5.11579e6 −0.198192
\(923\) −4.11353e6 −0.158932
\(924\) 1.50250e6 0.0578940
\(925\) 0 0
\(926\) −2.87945e7 −1.10352
\(927\) −1.08090e7 −0.413128
\(928\) 8.07936e6 0.307969
\(929\) 4.37292e7 1.66239 0.831194 0.555982i \(-0.187658\pi\)
0.831194 + 0.555982i \(0.187658\pi\)
\(930\) 0 0
\(931\) −2.45976e7 −0.930077
\(932\) 1.70262e7 0.642063
\(933\) −1.85938e6 −0.0699302
\(934\) 1.93214e7 0.724721
\(935\) 0 0
\(936\) 523584. 0.0195343
\(937\) 5.73509e6 0.213398 0.106699 0.994291i \(-0.465972\pi\)
0.106699 + 0.994291i \(0.465972\pi\)
\(938\) 20116.0 0.000746508 0
\(939\) −3.00801e7 −1.11331
\(940\) 0 0
\(941\) −3.37395e7 −1.24212 −0.621061 0.783762i \(-0.713298\pi\)
−0.621061 + 0.783762i \(0.713298\pi\)
\(942\) −1.25997e7 −0.462631
\(943\) −5.55941e6 −0.203587
\(944\) 4.50816e6 0.164653
\(945\) 0 0
\(946\) 1.27437e7 0.462985
\(947\) −3.07342e7 −1.11365 −0.556823 0.830631i \(-0.687980\pi\)
−0.556823 + 0.830631i \(0.687980\pi\)
\(948\) −9.95904e6 −0.359912
\(949\) 3.50531e6 0.126346
\(950\) 0 0
\(951\) −2.27960e7 −0.817348
\(952\) 487296. 0.0174261
\(953\) −2.51847e7 −0.898264 −0.449132 0.893465i \(-0.648267\pi\)
−0.449132 + 0.893465i \(0.648267\pi\)
\(954\) 5.80478e6 0.206498
\(955\) 0 0
\(956\) 1.84253e7 0.652033
\(957\) 1.57642e7 0.556407
\(958\) 2.99460e6 0.105420
\(959\) −621246. −0.0218131
\(960\) 0 0
\(961\) 4.52105e7 1.57918
\(962\) 3.49137e6 0.121635
\(963\) 6.58141e6 0.228693
\(964\) 1.36995e7 0.474801
\(965\) 0 0
\(966\) 517752. 0.0178517
\(967\) 1.44556e7 0.497130 0.248565 0.968615i \(-0.420041\pi\)
0.248565 + 0.968615i \(0.420041\pi\)
\(968\) −7.15309e6 −0.245361
\(969\) 2.45673e6 0.0840520
\(970\) 0 0
\(971\) −1.06974e7 −0.364109 −0.182054 0.983288i \(-0.558275\pi\)
−0.182054 + 0.983288i \(0.558275\pi\)
\(972\) 944784. 0.0320750
\(973\) −1.64848e7 −0.558214
\(974\) −2.06558e7 −0.697660
\(975\) 0 0
\(976\) −5.59437e6 −0.187986
\(977\) −8.41568e6 −0.282067 −0.141034 0.990005i \(-0.545043\pi\)
−0.141034 + 0.990005i \(0.545043\pi\)
\(978\) 6.93292e6 0.231776
\(979\) 7.77888e6 0.259394
\(980\) 0 0
\(981\) −1.75782e7 −0.583180
\(982\) 3.41715e7 1.13080
\(983\) 3.89409e7 1.28535 0.642676 0.766138i \(-0.277824\pi\)
0.642676 + 0.766138i \(0.277824\pi\)
\(984\) −1.04648e7 −0.344542
\(985\) 0 0
\(986\) 5.11272e6 0.167479
\(987\) −464454. −0.0151757
\(988\) 2.72296e6 0.0887460
\(989\) 4.39141e6 0.142762
\(990\) 0 0
\(991\) 4.84592e7 1.56745 0.783723 0.621111i \(-0.213318\pi\)
0.783723 + 0.621111i \(0.213318\pi\)
\(992\) −8.79923e6 −0.283900
\(993\) 5.41919e6 0.174406
\(994\) −7.65686e6 −0.245802
\(995\) 0 0
\(996\) −1.56289e7 −0.499206
\(997\) 3.84733e7 1.22581 0.612903 0.790158i \(-0.290002\pi\)
0.612903 + 0.790158i \(0.290002\pi\)
\(998\) −1.68235e7 −0.534675
\(999\) 6.30002e6 0.199723
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 150.6.a.m.1.1 yes 1
3.2 odd 2 450.6.a.i.1.1 1
5.2 odd 4 150.6.c.g.49.2 2
5.3 odd 4 150.6.c.g.49.1 2
5.4 even 2 150.6.a.a.1.1 1
15.2 even 4 450.6.c.e.199.1 2
15.8 even 4 450.6.c.e.199.2 2
15.14 odd 2 450.6.a.p.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
150.6.a.a.1.1 1 5.4 even 2
150.6.a.m.1.1 yes 1 1.1 even 1 trivial
150.6.c.g.49.1 2 5.3 odd 4
150.6.c.g.49.2 2 5.2 odd 4
450.6.a.i.1.1 1 3.2 odd 2
450.6.a.p.1.1 1 15.14 odd 2
450.6.c.e.199.1 2 15.2 even 4
450.6.c.e.199.2 2 15.8 even 4