Properties

Label 150.6.c.g.49.1
Level $150$
Weight $6$
Character 150.49
Analytic conductor $24.058$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 150.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(24.0575729719\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 150.49
Dual form 150.6.c.g.49.2

$q$-expansion

\(f(q)\) \(=\) \(q-4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} -47.0000i q^{7} +64.0000i q^{8} -81.0000 q^{9} +O(q^{10})\) \(q-4.00000i q^{2} +9.00000i q^{3} -16.0000 q^{4} +36.0000 q^{6} -47.0000i q^{7} +64.0000i q^{8} -81.0000 q^{9} +222.000 q^{11} -144.000i q^{12} +101.000i q^{13} -188.000 q^{14} +256.000 q^{16} -162.000i q^{17} +324.000i q^{18} -1685.00 q^{19} +423.000 q^{21} -888.000i q^{22} +306.000i q^{23} -576.000 q^{24} +404.000 q^{26} -729.000i q^{27} +752.000i q^{28} -7890.00 q^{29} -8593.00 q^{31} -1024.00i q^{32} +1998.00i q^{33} -648.000 q^{34} +1296.00 q^{36} -8642.00i q^{37} +6740.00i q^{38} -909.000 q^{39} -18168.0 q^{41} -1692.00i q^{42} +14351.0i q^{43} -3552.00 q^{44} +1224.00 q^{46} +1098.00i q^{47} +2304.00i q^{48} +14598.0 q^{49} +1458.00 q^{51} -1616.00i q^{52} +17916.0i q^{53} -2916.00 q^{54} +3008.00 q^{56} -15165.0i q^{57} +31560.0i q^{58} -17610.0 q^{59} -21853.0 q^{61} +34372.0i q^{62} +3807.00i q^{63} -4096.00 q^{64} +7992.00 q^{66} -107.000i q^{67} +2592.00i q^{68} -2754.00 q^{69} -40728.0 q^{71} -5184.00i q^{72} +34706.0i q^{73} -34568.0 q^{74} +26960.0 q^{76} -10434.0i q^{77} +3636.00i q^{78} +69160.0 q^{79} +6561.00 q^{81} +72672.0i q^{82} -108534. i q^{83} -6768.00 q^{84} +57404.0 q^{86} -71010.0i q^{87} +14208.0i q^{88} -35040.0 q^{89} +4747.00 q^{91} -4896.00i q^{92} -77337.0i q^{93} +4392.00 q^{94} +9216.00 q^{96} +823.000i q^{97} -58392.0i q^{98} -17982.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 32q^{4} + 72q^{6} - 162q^{9} + O(q^{10}) \) \( 2q - 32q^{4} + 72q^{6} - 162q^{9} + 444q^{11} - 376q^{14} + 512q^{16} - 3370q^{19} + 846q^{21} - 1152q^{24} + 808q^{26} - 15780q^{29} - 17186q^{31} - 1296q^{34} + 2592q^{36} - 1818q^{39} - 36336q^{41} - 7104q^{44} + 2448q^{46} + 29196q^{49} + 2916q^{51} - 5832q^{54} + 6016q^{56} - 35220q^{59} - 43706q^{61} - 8192q^{64} + 15984q^{66} - 5508q^{69} - 81456q^{71} - 69136q^{74} + 53920q^{76} + 138320q^{79} + 13122q^{81} - 13536q^{84} + 114808q^{86} - 70080q^{89} + 9494q^{91} + 8784q^{94} + 18432q^{96} - 35964q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 4.00000i − 0.707107i
\(3\) 9.00000i 0.577350i
\(4\) −16.0000 −0.500000
\(5\) 0 0
\(6\) 36.0000 0.408248
\(7\) − 47.0000i − 0.362537i −0.983434 0.181269i \(-0.941980\pi\)
0.983434 0.181269i \(-0.0580204\pi\)
\(8\) 64.0000i 0.353553i
\(9\) −81.0000 −0.333333
\(10\) 0 0
\(11\) 222.000 0.553186 0.276593 0.960987i \(-0.410795\pi\)
0.276593 + 0.960987i \(0.410795\pi\)
\(12\) − 144.000i − 0.288675i
\(13\) 101.000i 0.165754i 0.996560 + 0.0828768i \(0.0264108\pi\)
−0.996560 + 0.0828768i \(0.973589\pi\)
\(14\) −188.000 −0.256353
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) − 162.000i − 0.135954i −0.997687 0.0679771i \(-0.978346\pi\)
0.997687 0.0679771i \(-0.0216545\pi\)
\(18\) 324.000i 0.235702i
\(19\) −1685.00 −1.07082 −0.535409 0.844593i \(-0.679843\pi\)
−0.535409 + 0.844593i \(0.679843\pi\)
\(20\) 0 0
\(21\) 423.000 0.209311
\(22\) − 888.000i − 0.391162i
\(23\) 306.000i 0.120615i 0.998180 + 0.0603076i \(0.0192082\pi\)
−0.998180 + 0.0603076i \(0.980792\pi\)
\(24\) −576.000 −0.204124
\(25\) 0 0
\(26\) 404.000 0.117206
\(27\) − 729.000i − 0.192450i
\(28\) 752.000i 0.181269i
\(29\) −7890.00 −1.74214 −0.871068 0.491163i \(-0.836572\pi\)
−0.871068 + 0.491163i \(0.836572\pi\)
\(30\) 0 0
\(31\) −8593.00 −1.60598 −0.802991 0.595991i \(-0.796759\pi\)
−0.802991 + 0.595991i \(0.796759\pi\)
\(32\) − 1024.00i − 0.176777i
\(33\) 1998.00i 0.319382i
\(34\) −648.000 −0.0961342
\(35\) 0 0
\(36\) 1296.00 0.166667
\(37\) − 8642.00i − 1.03779i −0.854838 0.518896i \(-0.826343\pi\)
0.854838 0.518896i \(-0.173657\pi\)
\(38\) 6740.00i 0.757183i
\(39\) −909.000 −0.0956979
\(40\) 0 0
\(41\) −18168.0 −1.68790 −0.843951 0.536420i \(-0.819777\pi\)
−0.843951 + 0.536420i \(0.819777\pi\)
\(42\) − 1692.00i − 0.148005i
\(43\) 14351.0i 1.18362i 0.806079 + 0.591808i \(0.201586\pi\)
−0.806079 + 0.591808i \(0.798414\pi\)
\(44\) −3552.00 −0.276593
\(45\) 0 0
\(46\) 1224.00 0.0852878
\(47\) 1098.00i 0.0725033i 0.999343 + 0.0362516i \(0.0115418\pi\)
−0.999343 + 0.0362516i \(0.988458\pi\)
\(48\) 2304.00i 0.144338i
\(49\) 14598.0 0.868567
\(50\) 0 0
\(51\) 1458.00 0.0784932
\(52\) − 1616.00i − 0.0828768i
\(53\) 17916.0i 0.876095i 0.898952 + 0.438048i \(0.144330\pi\)
−0.898952 + 0.438048i \(0.855670\pi\)
\(54\) −2916.00 −0.136083
\(55\) 0 0
\(56\) 3008.00 0.128176
\(57\) − 15165.0i − 0.618237i
\(58\) 31560.0i 1.23188i
\(59\) −17610.0 −0.658612 −0.329306 0.944223i \(-0.606815\pi\)
−0.329306 + 0.944223i \(0.606815\pi\)
\(60\) 0 0
\(61\) −21853.0 −0.751946 −0.375973 0.926631i \(-0.622691\pi\)
−0.375973 + 0.926631i \(0.622691\pi\)
\(62\) 34372.0i 1.13560i
\(63\) 3807.00i 0.120846i
\(64\) −4096.00 −0.125000
\(65\) 0 0
\(66\) 7992.00 0.225837
\(67\) − 107.000i − 0.00291204i −0.999999 0.00145602i \(-0.999537\pi\)
0.999999 0.00145602i \(-0.000463465\pi\)
\(68\) 2592.00i 0.0679771i
\(69\) −2754.00 −0.0696372
\(70\) 0 0
\(71\) −40728.0 −0.958842 −0.479421 0.877585i \(-0.659153\pi\)
−0.479421 + 0.877585i \(0.659153\pi\)
\(72\) − 5184.00i − 0.117851i
\(73\) 34706.0i 0.762250i 0.924524 + 0.381125i \(0.124463\pi\)
−0.924524 + 0.381125i \(0.875537\pi\)
\(74\) −34568.0 −0.733829
\(75\) 0 0
\(76\) 26960.0 0.535409
\(77\) − 10434.0i − 0.200551i
\(78\) 3636.00i 0.0676686i
\(79\) 69160.0 1.24677 0.623386 0.781914i \(-0.285756\pi\)
0.623386 + 0.781914i \(0.285756\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 72672.0i 1.19353i
\(83\) − 108534.i − 1.72930i −0.502374 0.864650i \(-0.667540\pi\)
0.502374 0.864650i \(-0.332460\pi\)
\(84\) −6768.00 −0.104656
\(85\) 0 0
\(86\) 57404.0 0.836943
\(87\) − 71010.0i − 1.00582i
\(88\) 14208.0i 0.195581i
\(89\) −35040.0 −0.468910 −0.234455 0.972127i \(-0.575330\pi\)
−0.234455 + 0.972127i \(0.575330\pi\)
\(90\) 0 0
\(91\) 4747.00 0.0600919
\(92\) − 4896.00i − 0.0603076i
\(93\) − 77337.0i − 0.927214i
\(94\) 4392.00 0.0512676
\(95\) 0 0
\(96\) 9216.00 0.102062
\(97\) 823.000i 0.00888118i 0.999990 + 0.00444059i \(0.00141349\pi\)
−0.999990 + 0.00444059i \(0.998587\pi\)
\(98\) − 58392.0i − 0.614169i
\(99\) −17982.0 −0.184395
\(100\) 0 0
\(101\) −33828.0 −0.329969 −0.164984 0.986296i \(-0.552757\pi\)
−0.164984 + 0.986296i \(0.552757\pi\)
\(102\) − 5832.00i − 0.0555031i
\(103\) − 133444.i − 1.23938i −0.784845 0.619692i \(-0.787257\pi\)
0.784845 0.619692i \(-0.212743\pi\)
\(104\) −6464.00 −0.0586028
\(105\) 0 0
\(106\) 71664.0 0.619493
\(107\) − 81252.0i − 0.686080i −0.939321 0.343040i \(-0.888543\pi\)
0.939321 0.343040i \(-0.111457\pi\)
\(108\) 11664.0i 0.0962250i
\(109\) 217015. 1.74954 0.874769 0.484540i \(-0.161013\pi\)
0.874769 + 0.484540i \(0.161013\pi\)
\(110\) 0 0
\(111\) 77778.0 0.599169
\(112\) − 12032.0i − 0.0906343i
\(113\) − 138324.i − 1.01906i −0.860452 0.509532i \(-0.829819\pi\)
0.860452 0.509532i \(-0.170181\pi\)
\(114\) −60660.0 −0.437160
\(115\) 0 0
\(116\) 126240. 0.871068
\(117\) − 8181.00i − 0.0552512i
\(118\) 70440.0i 0.465709i
\(119\) −7614.00 −0.0492885
\(120\) 0 0
\(121\) −111767. −0.693985
\(122\) 87412.0i 0.531706i
\(123\) − 163512.i − 0.974511i
\(124\) 137488. 0.802991
\(125\) 0 0
\(126\) 15228.0 0.0854509
\(127\) 256048.i 1.40868i 0.709863 + 0.704340i \(0.248757\pi\)
−0.709863 + 0.704340i \(0.751243\pi\)
\(128\) 16384.0i 0.0883883i
\(129\) −129159. −0.683361
\(130\) 0 0
\(131\) 118452. 0.603065 0.301533 0.953456i \(-0.402502\pi\)
0.301533 + 0.953456i \(0.402502\pi\)
\(132\) − 31968.0i − 0.159691i
\(133\) 79195.0i 0.388212i
\(134\) −428.000 −0.00205912
\(135\) 0 0
\(136\) 10368.0 0.0480671
\(137\) 13218.0i 0.0601678i 0.999547 + 0.0300839i \(0.00957745\pi\)
−0.999547 + 0.0300839i \(0.990423\pi\)
\(138\) 11016.0i 0.0492409i
\(139\) 350740. 1.53974 0.769872 0.638199i \(-0.220320\pi\)
0.769872 + 0.638199i \(0.220320\pi\)
\(140\) 0 0
\(141\) −9882.00 −0.0418598
\(142\) 162912.i 0.678004i
\(143\) 22422.0i 0.0916926i
\(144\) −20736.0 −0.0833333
\(145\) 0 0
\(146\) 138824. 0.538992
\(147\) 131382.i 0.501467i
\(148\) 138272.i 0.518896i
\(149\) −109890. −0.405502 −0.202751 0.979230i \(-0.564988\pi\)
−0.202751 + 0.979230i \(0.564988\pi\)
\(150\) 0 0
\(151\) −172603. −0.616036 −0.308018 0.951381i \(-0.599666\pi\)
−0.308018 + 0.951381i \(0.599666\pi\)
\(152\) − 107840.i − 0.378592i
\(153\) 13122.0i 0.0453181i
\(154\) −41736.0 −0.141811
\(155\) 0 0
\(156\) 14544.0 0.0478489
\(157\) 349993.i 1.13321i 0.823990 + 0.566605i \(0.191743\pi\)
−0.823990 + 0.566605i \(0.808257\pi\)
\(158\) − 276640.i − 0.881601i
\(159\) −161244. −0.505814
\(160\) 0 0
\(161\) 14382.0 0.0437275
\(162\) − 26244.0i − 0.0785674i
\(163\) 192581.i 0.567733i 0.958864 + 0.283867i \(0.0916173\pi\)
−0.958864 + 0.283867i \(0.908383\pi\)
\(164\) 290688. 0.843951
\(165\) 0 0
\(166\) −434136. −1.22280
\(167\) − 580692.i − 1.61122i −0.592447 0.805610i \(-0.701838\pi\)
0.592447 0.805610i \(-0.298162\pi\)
\(168\) 27072.0i 0.0740026i
\(169\) 361092. 0.972526
\(170\) 0 0
\(171\) 136485. 0.356940
\(172\) − 229616.i − 0.591808i
\(173\) 738126.i 1.87506i 0.347904 + 0.937530i \(0.386894\pi\)
−0.347904 + 0.937530i \(0.613106\pi\)
\(174\) −284040. −0.711224
\(175\) 0 0
\(176\) 56832.0 0.138297
\(177\) − 158490.i − 0.380250i
\(178\) 140160.i 0.331569i
\(179\) −497370. −1.16024 −0.580119 0.814532i \(-0.696994\pi\)
−0.580119 + 0.814532i \(0.696994\pi\)
\(180\) 0 0
\(181\) −333163. −0.755893 −0.377947 0.925827i \(-0.623370\pi\)
−0.377947 + 0.925827i \(0.623370\pi\)
\(182\) − 18988.0i − 0.0424914i
\(183\) − 196677.i − 0.434136i
\(184\) −19584.0 −0.0426439
\(185\) 0 0
\(186\) −309348. −0.655639
\(187\) − 35964.0i − 0.0752080i
\(188\) − 17568.0i − 0.0362516i
\(189\) −34263.0 −0.0697703
\(190\) 0 0
\(191\) −40638.0 −0.0806026 −0.0403013 0.999188i \(-0.512832\pi\)
−0.0403013 + 0.999188i \(0.512832\pi\)
\(192\) − 36864.0i − 0.0721688i
\(193\) 494651.i 0.955885i 0.878391 + 0.477942i \(0.158617\pi\)
−0.878391 + 0.477942i \(0.841383\pi\)
\(194\) 3292.00 0.00627994
\(195\) 0 0
\(196\) −233568. −0.434283
\(197\) − 552342.i − 1.01401i −0.861943 0.507005i \(-0.830752\pi\)
0.861943 0.507005i \(-0.169248\pi\)
\(198\) 71928.0i 0.130387i
\(199\) −685625. −1.22731 −0.613655 0.789575i \(-0.710301\pi\)
−0.613655 + 0.789575i \(0.710301\pi\)
\(200\) 0 0
\(201\) 963.000 0.00168126
\(202\) 135312.i 0.233323i
\(203\) 370830.i 0.631589i
\(204\) −23328.0 −0.0392466
\(205\) 0 0
\(206\) −533776. −0.876377
\(207\) − 24786.0i − 0.0402050i
\(208\) 25856.0i 0.0414384i
\(209\) −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.i − 0.438048i
\(213\) − 366552.i − 0.553588i
\(214\) −325008. −0.485132
\(215\) 0 0
\(216\) 46656.0 0.0680414
\(217\) 403871.i 0.582228i
\(218\) − 868060.i − 1.23711i
\(219\) −312354. −0.440085
\(220\) 0 0
\(221\) 16362.0 0.0225349
\(222\) − 311112.i − 0.423676i
\(223\) 169271.i 0.227940i 0.993484 + 0.113970i \(0.0363568\pi\)
−0.993484 + 0.113970i \(0.963643\pi\)
\(224\) −48128.0 −0.0640882
\(225\) 0 0
\(226\) −553296. −0.720587
\(227\) 46488.0i 0.0598792i 0.999552 + 0.0299396i \(0.00953150\pi\)
−0.999552 + 0.0299396i \(0.990468\pi\)
\(228\) 242640.i 0.309119i
\(229\) 90115.0 0.113556 0.0567778 0.998387i \(-0.481917\pi\)
0.0567778 + 0.998387i \(0.481917\pi\)
\(230\) 0 0
\(231\) 93906.0 0.115788
\(232\) − 504960.i − 0.615938i
\(233\) 1.06414e6i 1.28413i 0.766652 + 0.642063i \(0.221921\pi\)
−0.766652 + 0.642063i \(0.778079\pi\)
\(234\) −32724.0 −0.0390685
\(235\) 0 0
\(236\) 281760. 0.329306
\(237\) 622440.i 0.719825i
\(238\) 30456.0i 0.0348522i
\(239\) −1.15158e6 −1.30407 −0.652033 0.758191i \(-0.726084\pi\)
−0.652033 + 0.758191i \(0.726084\pi\)
\(240\) 0 0
\(241\) 856217. 0.949601 0.474801 0.880093i \(-0.342520\pi\)
0.474801 + 0.880093i \(0.342520\pi\)
\(242\) 447068.i 0.490722i
\(243\) 59049.0i 0.0641500i
\(244\) 349648. 0.375973
\(245\) 0 0
\(246\) −654048. −0.689084
\(247\) − 170185.i − 0.177492i
\(248\) − 549952.i − 0.567800i
\(249\) 976806. 0.998412
\(250\) 0 0
\(251\) −207708. −0.208098 −0.104049 0.994572i \(-0.533180\pi\)
−0.104049 + 0.994572i \(0.533180\pi\)
\(252\) − 60912.0i − 0.0604229i
\(253\) 67932.0i 0.0667226i
\(254\) 1.02419e6 0.996087
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) 1.45319e6i 1.37243i 0.727401 + 0.686213i \(0.240728\pi\)
−0.727401 + 0.686213i \(0.759272\pi\)
\(258\) 516636.i 0.483209i
\(259\) −406174. −0.376238
\(260\) 0 0
\(261\) 639090. 0.580712
\(262\) − 473808.i − 0.426431i
\(263\) 169296.i 0.150924i 0.997149 + 0.0754618i \(0.0240431\pi\)
−0.997149 + 0.0754618i \(0.975957\pi\)
\(264\) −127872. −0.112919
\(265\) 0 0
\(266\) 316780. 0.274507
\(267\) − 315360.i − 0.270725i
\(268\) 1712.00i 0.00145602i
\(269\) 1.58109e6 1.33222 0.666110 0.745854i \(-0.267958\pi\)
0.666110 + 0.745854i \(0.267958\pi\)
\(270\) 0 0
\(271\) 822512. 0.680329 0.340165 0.940366i \(-0.389517\pi\)
0.340165 + 0.940366i \(0.389517\pi\)
\(272\) − 41472.0i − 0.0339886i
\(273\) 42723.0i 0.0346941i
\(274\) 52872.0 0.0425451
\(275\) 0 0
\(276\) 44064.0 0.0348186
\(277\) 546823.i 0.428201i 0.976812 + 0.214100i \(0.0686820\pi\)
−0.976812 + 0.214100i \(0.931318\pi\)
\(278\) − 1.40296e6i − 1.08876i
\(279\) 696033. 0.535327
\(280\) 0 0
\(281\) −1.09250e6 −0.825382 −0.412691 0.910871i \(-0.635411\pi\)
−0.412691 + 0.910871i \(0.635411\pi\)
\(282\) 39528.0i 0.0295993i
\(283\) − 2.48480e6i − 1.84427i −0.386865 0.922136i \(-0.626442\pi\)
0.386865 0.922136i \(-0.373558\pi\)
\(284\) 651648. 0.479421
\(285\) 0 0
\(286\) 89688.0 0.0648365
\(287\) 853896.i 0.611928i
\(288\) 82944.0i 0.0589256i
\(289\) 1.39361e6 0.981516
\(290\) 0 0
\(291\) −7407.00 −0.00512755
\(292\) − 555296.i − 0.381125i
\(293\) − 341394.i − 0.232320i −0.993231 0.116160i \(-0.962941\pi\)
0.993231 0.116160i \(-0.0370586\pi\)
\(294\) 525528. 0.354591
\(295\) 0 0
\(296\) 553088. 0.366915
\(297\) − 161838.i − 0.106461i
\(298\) 439560.i 0.286733i
\(299\) −30906.0 −0.0199924
\(300\) 0 0
\(301\) 674497. 0.429105
\(302\) 690412.i 0.435603i
\(303\) − 304452.i − 0.190508i
\(304\) −431360. −0.267705
\(305\) 0 0
\(306\) 52488.0 0.0320447
\(307\) − 2.02898e6i − 1.22866i −0.789050 0.614329i \(-0.789427\pi\)
0.789050 0.614329i \(-0.210573\pi\)
\(308\) 166944.i 0.100275i
\(309\) 1.20100e6 0.715559
\(310\) 0 0
\(311\) −206598. −0.121123 −0.0605613 0.998164i \(-0.519289\pi\)
−0.0605613 + 0.998164i \(0.519289\pi\)
\(312\) − 58176.0i − 0.0338343i
\(313\) − 3.34223e6i − 1.92830i −0.265352 0.964152i \(-0.585488\pi\)
0.265352 0.964152i \(-0.414512\pi\)
\(314\) 1.39997e6 0.801300
\(315\) 0 0
\(316\) −1.10656e6 −0.623386
\(317\) 2.53289e6i 1.41569i 0.706368 + 0.707844i \(0.250332\pi\)
−0.706368 + 0.707844i \(0.749668\pi\)
\(318\) 644976.i 0.357664i
\(319\) −1.75158e6 −0.963725
\(320\) 0 0
\(321\) 731268. 0.396108
\(322\) − 57528.0i − 0.0309200i
\(323\) 272970.i 0.145582i
\(324\) −104976. −0.0555556
\(325\) 0 0
\(326\) 770324. 0.401448
\(327\) 1.95314e6i 1.01010i
\(328\) − 1.16275e6i − 0.596764i
\(329\) 51606.0 0.0262851
\(330\) 0 0
\(331\) 602132. 0.302080 0.151040 0.988528i \(-0.451738\pi\)
0.151040 + 0.988528i \(0.451738\pi\)
\(332\) 1.73654e6i 0.864650i
\(333\) 700002.i 0.345930i
\(334\) −2.32277e6 −1.13930
\(335\) 0 0
\(336\) 108288. 0.0523278
\(337\) − 209777.i − 0.100620i −0.998734 0.0503099i \(-0.983979\pi\)
0.998734 0.0503099i \(-0.0160209\pi\)
\(338\) − 1.44437e6i − 0.687680i
\(339\) 1.24492e6 0.588357
\(340\) 0 0
\(341\) −1.90765e6 −0.888407
\(342\) − 545940.i − 0.252394i
\(343\) − 1.47603e6i − 0.677425i
\(344\) −918464. −0.418472
\(345\) 0 0
\(346\) 2.95250e6 1.32587
\(347\) − 4.02166e6i − 1.79301i −0.443037 0.896503i \(-0.646099\pi\)
0.443037 0.896503i \(-0.353901\pi\)
\(348\) 1.13616e6i 0.502911i
\(349\) −8330.00 −0.00366085 −0.00183042 0.999998i \(-0.500583\pi\)
−0.00183042 + 0.999998i \(0.500583\pi\)
\(350\) 0 0
\(351\) 73629.0 0.0318993
\(352\) − 227328.i − 0.0977904i
\(353\) 1.95001e6i 0.832912i 0.909156 + 0.416456i \(0.136728\pi\)
−0.909156 + 0.416456i \(0.863272\pi\)
\(354\) −633960. −0.268877
\(355\) 0 0
\(356\) 560640. 0.234455
\(357\) − 68526.0i − 0.0284567i
\(358\) 1.98948e6i 0.820412i
\(359\) −2.27088e6 −0.929947 −0.464973 0.885325i \(-0.653936\pi\)
−0.464973 + 0.885325i \(0.653936\pi\)
\(360\) 0 0
\(361\) 363126. 0.146652
\(362\) 1.33265e6i 0.534497i
\(363\) − 1.00590e6i − 0.400673i
\(364\) −75952.0 −0.0300459
\(365\) 0 0
\(366\) −786708. −0.306981
\(367\) − 2.86154e6i − 1.10901i −0.832181 0.554503i \(-0.812908\pi\)
0.832181 0.554503i \(-0.187092\pi\)
\(368\) 78336.0i 0.0301538i
\(369\) 1.47161e6 0.562634
\(370\) 0 0
\(371\) 842052. 0.317617
\(372\) 1.23739e6i 0.463607i
\(373\) 615311.i 0.228993i 0.993424 + 0.114497i \(0.0365255\pi\)
−0.993424 + 0.114497i \(0.963474\pi\)
\(374\) −143856. −0.0531801
\(375\) 0 0
\(376\) −70272.0 −0.0256338
\(377\) − 796890.i − 0.288765i
\(378\) 137052.i 0.0493351i
\(379\) −5.39878e6 −1.93062 −0.965311 0.261103i \(-0.915914\pi\)
−0.965311 + 0.261103i \(0.915914\pi\)
\(380\) 0 0
\(381\) −2.30443e6 −0.813301
\(382\) 162552.i 0.0569946i
\(383\) 1.08688e6i 0.378602i 0.981919 + 0.189301i \(0.0606222\pi\)
−0.981919 + 0.189301i \(0.939378\pi\)
\(384\) −147456. −0.0510310
\(385\) 0 0
\(386\) 1.97860e6 0.675913
\(387\) − 1.16243e6i − 0.394539i
\(388\) − 13168.0i − 0.00444059i
\(389\) 3.48432e6 1.16747 0.583733 0.811946i \(-0.301592\pi\)
0.583733 + 0.811946i \(0.301592\pi\)
\(390\) 0 0
\(391\) 49572.0 0.0163981
\(392\) 934272.i 0.307085i
\(393\) 1.06607e6i 0.348180i
\(394\) −2.20937e6 −0.717014
\(395\) 0 0
\(396\) 287712. 0.0921977
\(397\) − 3.26591e6i − 1.03999i −0.854170 0.519993i \(-0.825935\pi\)
0.854170 0.519993i \(-0.174065\pi\)
\(398\) 2.74250e6i 0.867839i
\(399\) −712755. −0.224134
\(400\) 0 0
\(401\) −4.27319e6 −1.32706 −0.663531 0.748149i \(-0.730943\pi\)
−0.663531 + 0.748149i \(0.730943\pi\)
\(402\) − 3852.00i − 0.00118883i
\(403\) − 867893.i − 0.266197i
\(404\) 541248. 0.164984
\(405\) 0 0
\(406\) 1.48332e6 0.446601
\(407\) − 1.91852e6i − 0.574092i
\(408\) 93312.0i 0.0277515i
\(409\) 1.45188e6 0.429162 0.214581 0.976706i \(-0.431161\pi\)
0.214581 + 0.976706i \(0.431161\pi\)
\(410\) 0 0
\(411\) −118962. −0.0347379
\(412\) 2.13510e6i 0.619692i
\(413\) 827670.i 0.238771i
\(414\) −99144.0 −0.0284293
\(415\) 0 0
\(416\) 103424. 0.0293014
\(417\) 3.15666e6i 0.888971i
\(418\) 1.49628e6i 0.418863i
\(419\) −559380. −0.155658 −0.0778291 0.996967i \(-0.524799\pi\)
−0.0778291 + 0.996967i \(0.524799\pi\)
\(420\) 0 0
\(421\) −3.91470e6 −1.07645 −0.538224 0.842802i \(-0.680905\pi\)
−0.538224 + 0.842802i \(0.680905\pi\)
\(422\) − 2.99791e6i − 0.819478i
\(423\) − 88938.0i − 0.0241678i
\(424\) −1.14662e6 −0.309746
\(425\) 0 0
\(426\) −1.46621e6 −0.391446
\(427\) 1.02709e6i 0.272608i
\(428\) 1.30003e6i 0.343040i
\(429\) −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.i − 0.0481125i
\(433\) 7.15969e6i 1.83516i 0.397548 + 0.917581i \(0.369861\pi\)
−0.397548 + 0.917581i \(0.630139\pi\)
\(434\) 1.61548e6 0.411698
\(435\) 0 0
\(436\) −3.47224e6 −0.874769
\(437\) − 515610.i − 0.129157i
\(438\) 1.24942e6i 0.311187i
\(439\) −1.71790e6 −0.425437 −0.212719 0.977114i \(-0.568232\pi\)
−0.212719 + 0.977114i \(0.568232\pi\)
\(440\) 0 0
\(441\) −1.18244e6 −0.289522
\(442\) − 65448.0i − 0.0159346i
\(443\) 3.39670e6i 0.822332i 0.911560 + 0.411166i \(0.134878\pi\)
−0.911560 + 0.411166i \(0.865122\pi\)
\(444\) −1.24445e6 −0.299584
\(445\) 0 0
\(446\) 677084. 0.161178
\(447\) − 989010.i − 0.234116i
\(448\) 192512.i 0.0453172i
\(449\) 3.39606e6 0.794986 0.397493 0.917605i \(-0.369880\pi\)
0.397493 + 0.917605i \(0.369880\pi\)
\(450\) 0 0
\(451\) −4.03330e6 −0.933724
\(452\) 2.21318e6i 0.509532i
\(453\) − 1.55343e6i − 0.355668i
\(454\) 185952. 0.0423410
\(455\) 0 0
\(456\) 970560. 0.218580
\(457\) 4.52814e6i 1.01421i 0.861883 + 0.507106i \(0.169285\pi\)
−0.861883 + 0.507106i \(0.830715\pi\)
\(458\) − 360460.i − 0.0802959i
\(459\) −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.i − 0.0818744i
\(463\) − 7.19862e6i − 1.56062i −0.625393 0.780310i \(-0.715061\pi\)
0.625393 0.780310i \(-0.284939\pi\)
\(464\) −2.01984e6 −0.435534
\(465\) 0 0
\(466\) 4.25654e6 0.908014
\(467\) − 4.83034e6i − 1.02491i −0.858714 0.512455i \(-0.828736\pi\)
0.858714 0.512455i \(-0.171264\pi\)
\(468\) 130896.i 0.0276256i
\(469\) −5029.00 −0.00105572
\(470\) 0 0
\(471\) −3.14994e6 −0.654259
\(472\) − 1.12704e6i − 0.232854i
\(473\) 3.18592e6i 0.654760i
\(474\) 2.48976e6 0.508993
\(475\) 0 0
\(476\) 121824. 0.0246442
\(477\) − 1.45120e6i − 0.292032i
\(478\) 4.60632e6i 0.922113i
\(479\) −748650. −0.149087 −0.0745435 0.997218i \(-0.523750\pi\)
−0.0745435 + 0.997218i \(0.523750\pi\)
\(480\) 0 0
\(481\) 872842. 0.172018
\(482\) − 3.42487e6i − 0.671469i
\(483\) 129438.i 0.0252461i
\(484\) 1.78827e6 0.346993
\(485\) 0 0
\(486\) 236196. 0.0453609
\(487\) 5.16394e6i 0.986641i 0.869848 + 0.493320i \(0.164217\pi\)
−0.869848 + 0.493320i \(0.835783\pi\)
\(488\) − 1.39859e6i − 0.265853i
\(489\) −1.73323e6 −0.327781
\(490\) 0 0
\(491\) 8.54287e6 1.59919 0.799595 0.600539i \(-0.205047\pi\)
0.799595 + 0.600539i \(0.205047\pi\)
\(492\) 2.61619e6i 0.487256i
\(493\) 1.27818e6i 0.236851i
\(494\) −680740. −0.125506
\(495\) 0 0
\(496\) −2.19981e6 −0.401495
\(497\) 1.91422e6i 0.347616i
\(498\) − 3.90722e6i − 0.705984i
\(499\) 4.20588e6 0.756145 0.378072 0.925776i \(-0.376587\pi\)
0.378072 + 0.925776i \(0.376587\pi\)
\(500\) 0 0
\(501\) 5.22623e6 0.930238
\(502\) 830832.i 0.147148i
\(503\) − 8.18342e6i − 1.44217i −0.692849 0.721083i \(-0.743645\pi\)
0.692849 0.721083i \(-0.256355\pi\)
\(504\) −243648. −0.0427254
\(505\) 0 0
\(506\) 271728. 0.0471800
\(507\) 3.24983e6i 0.561488i
\(508\) − 4.09677e6i − 0.704340i
\(509\) −3.85923e6 −0.660247 −0.330123 0.943938i \(-0.607090\pi\)
−0.330123 + 0.943938i \(0.607090\pi\)
\(510\) 0 0
\(511\) 1.63118e6 0.276344
\(512\) − 262144.i − 0.0441942i
\(513\) 1.22837e6i 0.206079i
\(514\) 5.81275e6 0.970452
\(515\) 0 0
\(516\) 2.06654e6 0.341681
\(517\) 243756.i 0.0401078i
\(518\) 1.62470e6i 0.266040i
\(519\) −6.64313e6 −1.08257
\(520\) 0 0
\(521\) 4.55410e6 0.735036 0.367518 0.930016i \(-0.380208\pi\)
0.367518 + 0.930016i \(0.380208\pi\)
\(522\) − 2.55636e6i − 0.410625i
\(523\) 4.82224e6i 0.770894i 0.922730 + 0.385447i \(0.125953\pi\)
−0.922730 + 0.385447i \(0.874047\pi\)
\(524\) −1.89523e6 −0.301533
\(525\) 0 0
\(526\) 677184. 0.106719
\(527\) 1.39207e6i 0.218340i
\(528\) 511488.i 0.0798455i
\(529\) 6.34271e6 0.985452
\(530\) 0 0
\(531\) 1.42641e6 0.219537
\(532\) − 1.26712e6i − 0.194106i
\(533\) − 1.83497e6i − 0.279776i
\(534\) −1.26144e6 −0.191432
\(535\) 0 0
\(536\) 6848.00 0.00102956
\(537\) − 4.47633e6i − 0.669864i
\(538\) − 6.32436e6i − 0.942022i
\(539\) 3.24076e6 0.480479
\(540\) 0 0
\(541\) 362537. 0.0532549 0.0266274 0.999645i \(-0.491523\pi\)
0.0266274 + 0.999645i \(0.491523\pi\)
\(542\) − 3.29005e6i − 0.481065i
\(543\) − 2.99847e6i − 0.436415i
\(544\) −165888. −0.0240335
\(545\) 0 0
\(546\) 170892. 0.0245324
\(547\) − 3.11439e6i − 0.445046i −0.974927 0.222523i \(-0.928571\pi\)
0.974927 0.222523i \(-0.0714293\pi\)
\(548\) − 211488.i − 0.0300839i
\(549\) 1.77009e6 0.250649
\(550\) 0 0
\(551\) 1.32947e7 1.86551
\(552\) − 176256.i − 0.0246205i
\(553\) − 3.25052e6i − 0.452002i
\(554\) 2.18729e6 0.302784
\(555\) 0 0
\(556\) −5.61184e6 −0.769872
\(557\) 7.99304e6i 1.09163i 0.837907 + 0.545813i \(0.183779\pi\)
−0.837907 + 0.545813i \(0.816221\pi\)
\(558\) − 2.78413e6i − 0.378534i
\(559\) −1.44945e6 −0.196189
\(560\) 0 0
\(561\) 323676. 0.0434214
\(562\) 4.36999e6i 0.583633i
\(563\) − 1.23236e7i − 1.63857i −0.573385 0.819286i \(-0.694370\pi\)
0.573385 0.819286i \(-0.305630\pi\)
\(564\) 158112. 0.0209299
\(565\) 0 0
\(566\) −9.93920e6 −1.30410
\(567\) − 308367.i − 0.0402819i
\(568\) − 2.60659e6i − 0.339002i
\(569\) −1.01364e7 −1.31252 −0.656258 0.754537i \(-0.727862\pi\)
−0.656258 + 0.754537i \(0.727862\pi\)
\(570\) 0 0
\(571\) 6.53084e6 0.838260 0.419130 0.907926i \(-0.362335\pi\)
0.419130 + 0.907926i \(0.362335\pi\)
\(572\) − 358752.i − 0.0458463i
\(573\) − 365742.i − 0.0465359i
\(574\) 3.41558e6 0.432698
\(575\) 0 0
\(576\) 331776. 0.0416667
\(577\) 1.24453e6i 0.155621i 0.996968 + 0.0778103i \(0.0247928\pi\)
−0.996968 + 0.0778103i \(0.975207\pi\)
\(578\) − 5.57445e6i − 0.694037i
\(579\) −4.45186e6 −0.551880
\(580\) 0 0
\(581\) −5.10110e6 −0.626936
\(582\) 29628.0i 0.00362573i
\(583\) 3.97735e6i 0.484644i
\(584\) −2.22118e6 −0.269496
\(585\) 0 0
\(586\) −1.36558e6 −0.164275
\(587\) 1.33403e6i 0.159797i 0.996803 + 0.0798987i \(0.0254597\pi\)
−0.996803 + 0.0798987i \(0.974540\pi\)
\(588\) − 2.10211e6i − 0.250734i
\(589\) 1.44792e7 1.71972
\(590\) 0 0
\(591\) 4.97108e6 0.585439
\(592\) − 2.21235e6i − 0.259448i
\(593\) − 1.19401e7i − 1.39435i −0.716899 0.697177i \(-0.754439\pi\)
0.716899 0.697177i \(-0.245561\pi\)
\(594\) −647352. −0.0752791
\(595\) 0 0
\(596\) 1.75824e6 0.202751
\(597\) − 6.17062e6i − 0.708587i
\(598\) 123624.i 0.0141368i
\(599\) 7.16430e6 0.815843 0.407922 0.913017i \(-0.366254\pi\)
0.407922 + 0.913017i \(0.366254\pi\)
\(600\) 0 0
\(601\) 1.15163e6 0.130055 0.0650273 0.997883i \(-0.479287\pi\)
0.0650273 + 0.997883i \(0.479287\pi\)
\(602\) − 2.69799e6i − 0.303423i
\(603\) 8667.00i 0 0.000970679i
\(604\) 2.76165e6 0.308018
\(605\) 0 0
\(606\) −1.21781e6 −0.134709
\(607\) 1.34268e7i 1.47911i 0.673097 + 0.739554i \(0.264963\pi\)
−0.673097 + 0.739554i \(0.735037\pi\)
\(608\) 1.72544e6i 0.189296i
\(609\) −3.33747e6 −0.364648
\(610\) 0 0
\(611\) −110898. −0.0120177
\(612\) − 209952.i − 0.0226590i
\(613\) 1.20184e7i 1.29180i 0.763422 + 0.645900i \(0.223518\pi\)
−0.763422 + 0.645900i \(0.776482\pi\)
\(614\) −8.11591e6 −0.868793
\(615\) 0 0
\(616\) 667776. 0.0709054
\(617\) 6.98519e6i 0.738695i 0.929291 + 0.369348i \(0.120419\pi\)
−0.929291 + 0.369348i \(0.879581\pi\)
\(618\) − 4.80398e6i − 0.505977i
\(619\) 8.20625e6 0.860832 0.430416 0.902631i \(-0.358367\pi\)
0.430416 + 0.902631i \(0.358367\pi\)
\(620\) 0 0
\(621\) 223074. 0.0232124
\(622\) 826392.i 0.0856466i
\(623\) 1.64688e6i 0.169997i
\(624\) −232704. −0.0239245
\(625\) 0 0
\(626\) −1.33689e7 −1.36352
\(627\) − 3.36663e6i − 0.342000i
\(628\) − 5.59989e6i − 0.566605i
\(629\) −1.40000e6 −0.141092
\(630\) 0 0
\(631\) −1.07686e7 −1.07668 −0.538338 0.842729i \(-0.680947\pi\)
−0.538338 + 0.842729i \(0.680947\pi\)
\(632\) 4.42624e6i 0.440801i
\(633\) 6.74529e6i 0.669101i
\(634\) 1.01316e7 1.00104
\(635\) 0 0
\(636\) 2.57990e6 0.252907
\(637\) 1.47440e6i 0.143968i
\(638\) 7.00632e6i 0.681457i
\(639\) 3.29897e6 0.319614
\(640\) 0 0
\(641\) 1.92571e7 1.85117 0.925585 0.378539i \(-0.123573\pi\)
0.925585 + 0.378539i \(0.123573\pi\)
\(642\) − 2.92507e6i − 0.280091i
\(643\) 1.00999e7i 0.963364i 0.876346 + 0.481682i \(0.159974\pi\)
−0.876346 + 0.481682i \(0.840026\pi\)
\(644\) −230112. −0.0218637
\(645\) 0 0
\(646\) 1.09188e6 0.102942
\(647\) − 7.52113e6i − 0.706354i −0.935556 0.353177i \(-0.885101\pi\)
0.935556 0.353177i \(-0.114899\pi\)
\(648\) 419904.i 0.0392837i
\(649\) −3.90942e6 −0.364335
\(650\) 0 0
\(651\) −3.63484e6 −0.336150
\(652\) − 3.08130e6i − 0.283867i
\(653\) − 2.67197e6i − 0.245216i −0.992455 0.122608i \(-0.960874\pi\)
0.992455 0.122608i \(-0.0391258\pi\)
\(654\) 7.81254e6 0.714246
\(655\) 0 0
\(656\) −4.65101e6 −0.421976
\(657\) − 2.81119e6i − 0.254083i
\(658\) − 206424.i − 0.0185864i
\(659\) −6.99948e6 −0.627845 −0.313922 0.949449i \(-0.601643\pi\)
−0.313922 + 0.949449i \(0.601643\pi\)
\(660\) 0 0
\(661\) 408122. 0.0363318 0.0181659 0.999835i \(-0.494217\pi\)
0.0181659 + 0.999835i \(0.494217\pi\)
\(662\) − 2.40853e6i − 0.213603i
\(663\) 147258.i 0.0130105i
\(664\) 6.94618e6 0.611400
\(665\) 0 0
\(666\) 2.80001e6 0.244610
\(667\) − 2.41434e6i − 0.210128i
\(668\) 9.29107e6i 0.805610i
\(669\) −1.52344e6 −0.131601
\(670\) 0 0
\(671\) −4.85137e6 −0.415966
\(672\) − 433152.i − 0.0370013i
\(673\) 1.74939e7i 1.48885i 0.667709 + 0.744423i \(0.267275\pi\)
−0.667709 + 0.744423i \(0.732725\pi\)
\(674\) −839108. −0.0711489
\(675\) 0 0
\(676\) −5.77747e6 −0.486263
\(677\) 8.67440e6i 0.727391i 0.931518 + 0.363695i \(0.118485\pi\)
−0.931518 + 0.363695i \(0.881515\pi\)
\(678\) − 4.97966e6i − 0.416031i
\(679\) 38681.0 0.00321976
\(680\) 0 0
\(681\) −418392. −0.0345713
\(682\) 7.63058e6i 0.628198i
\(683\) 1.18478e7i 0.971822i 0.874008 + 0.485911i \(0.161512\pi\)
−0.874008 + 0.485911i \(0.838488\pi\)
\(684\) −2.18376e6 −0.178470
\(685\) 0 0
\(686\) −5.90414e6 −0.479012
\(687\) 811035.i 0.0655613i
\(688\) 3.67386e6i 0.295904i
\(689\) −1.80952e6 −0.145216
\(690\) 0 0
\(691\) 9.47775e6 0.755110 0.377555 0.925987i \(-0.376765\pi\)
0.377555 + 0.925987i \(0.376765\pi\)
\(692\) − 1.18100e7i − 0.937530i
\(693\) 845154.i 0.0668502i
\(694\) −1.60866e7 −1.26785
\(695\) 0 0
\(696\) 4.54464e6 0.355612
\(697\) 2.94322e6i 0.229478i
\(698\) 33320.0i 0.00258861i
\(699\) −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.i − 0.0225562i
\(703\) 1.45618e7i 1.11129i
\(704\) −909312. −0.0691483
\(705\) 0 0
\(706\) 7.80002e6 0.588958
\(707\) 1.58992e6i 0.119626i
\(708\) 2.53584e6i 0.190125i
\(709\) −1.27436e7 −0.952090 −0.476045 0.879421i \(-0.657930\pi\)
−0.476045 + 0.879421i \(0.657930\pi\)
\(710\) 0 0
\(711\) −5.60196e6 −0.415591
\(712\) − 2.24256e6i − 0.165785i
\(713\) − 2.62946e6i − 0.193706i
\(714\) −274104. −0.0201219
\(715\) 0 0
\(716\) 7.95792e6 0.580119
\(717\) − 1.03642e7i − 0.752902i
\(718\) 9.08352e6i 0.657572i
\(719\) −2.44929e6 −0.176692 −0.0883462 0.996090i \(-0.528158\pi\)
−0.0883462 + 0.996090i \(0.528158\pi\)
\(720\) 0 0
\(721\) −6.27187e6 −0.449323
\(722\) − 1.45250e6i − 0.103699i
\(723\) 7.70595e6i 0.548252i
\(724\) 5.33061e6 0.377947
\(725\) 0 0
\(726\) −4.02361e6 −0.283318
\(727\) 415033.i 0.0291237i 0.999894 + 0.0145619i \(0.00463535\pi\)
−0.999894 + 0.0145619i \(0.995365\pi\)
\(728\) 303808.i 0.0212457i
\(729\) −531441. −0.0370370
\(730\) 0 0
\(731\) 2.32486e6 0.160918
\(732\) 3.14683e6i 0.217068i
\(733\) − 1.72877e7i − 1.18844i −0.804302 0.594221i \(-0.797461\pi\)
0.804302 0.594221i \(-0.202539\pi\)
\(734\) −1.14461e7 −0.784186
\(735\) 0 0
\(736\) 313344. 0.0213219
\(737\) − 23754.0i − 0.00161090i
\(738\) − 5.88643e6i − 0.397843i
\(739\) −5.18834e6 −0.349476 −0.174738 0.984615i \(-0.555908\pi\)
−0.174738 + 0.984615i \(0.555908\pi\)
\(740\) 0 0
\(741\) 1.53166e6 0.102475
\(742\) − 3.36821e6i − 0.224589i
\(743\) 4.79572e6i 0.318700i 0.987222 + 0.159350i \(0.0509398\pi\)
−0.987222 + 0.159350i \(0.949060\pi\)
\(744\) 4.94957e6 0.327820
\(745\) 0 0
\(746\) 2.46124e6 0.161923
\(747\) 8.79125e6i 0.576434i
\(748\) 575424.i 0.0376040i
\(749\) −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.i 0.0181258i
\(753\) − 1.86937e6i − 0.120146i
\(754\) −3.18756e6 −0.204188
\(755\) 0 0
\(756\) 548208. 0.0348852
\(757\) − 2.82068e7i − 1.78902i −0.447053 0.894508i \(-0.647526\pi\)
0.447053 0.894508i \(-0.352474\pi\)
\(758\) 2.15951e7i 1.36516i
\(759\) −611388. −0.0385223
\(760\) 0 0
\(761\) 6.56161e6 0.410723 0.205361 0.978686i \(-0.434163\pi\)
0.205361 + 0.978686i \(0.434163\pi\)
\(762\) 9.21773e6i 0.575091i
\(763\) − 1.01997e7i − 0.634273i
\(764\) 650208. 0.0403013
\(765\) 0 0
\(766\) 4.34750e6 0.267712
\(767\) − 1.77861e6i − 0.109167i
\(768\) 589824.i 0.0360844i
\(769\) −2.20930e7 −1.34722 −0.673610 0.739087i \(-0.735257\pi\)
−0.673610 + 0.739087i \(0.735257\pi\)
\(770\) 0 0
\(771\) −1.30787e7 −0.792371
\(772\) − 7.91442e6i − 0.477942i
\(773\) − 3.00787e7i − 1.81055i −0.424824 0.905276i \(-0.639664\pi\)
0.424824 0.905276i \(-0.360336\pi\)
\(774\) −4.64972e6 −0.278981
\(775\) 0 0
\(776\) −52672.0 −0.00313997
\(777\) − 3.65557e6i − 0.217221i
\(778\) − 1.39373e7i − 0.825523i
\(779\) 3.06131e7 1.80744
\(780\) 0 0
\(781\) −9.04162e6 −0.530418
\(782\) − 198288.i − 0.0115952i
\(783\) 5.75181e6i 0.335274i
\(784\) 3.73709e6 0.217142
\(785\) 0 0
\(786\) 4.26427e6 0.246200
\(787\) 3.28954e6i 0.189321i 0.995510 + 0.0946605i \(0.0301766\pi\)
−0.995510 + 0.0946605i \(0.969823\pi\)
\(788\) 8.83747e6i 0.507005i
\(789\) −1.52366e6 −0.0871358
\(790\) 0 0
\(791\) −6.50123e6 −0.369449
\(792\) − 1.15085e6i − 0.0651936i
\(793\) − 2.20715e6i − 0.124638i
\(794\) −1.30636e7 −0.735381
\(795\) 0 0
\(796\) 1.09700e7 0.613655
\(797\) − 6.71053e6i − 0.374206i −0.982340 0.187103i \(-0.940090\pi\)
0.982340 0.187103i \(-0.0599099\pi\)
\(798\) 2.85102e6i 0.158487i
\(799\) 177876. 0.00985713
\(800\) 0 0
\(801\) 2.83824e6 0.156303
\(802\) 1.70928e7i 0.938374i
\(803\) 7.70473e6i 0.421666i
\(804\) −15408.0 −0.000840632 0
\(805\) 0 0
\(806\) −3.47157e6 −0.188230
\(807\) 1.42298e7i 0.769157i
\(808\) − 2.16499e6i − 0.116662i
\(809\) −8.74254e6 −0.469641 −0.234821 0.972039i \(-0.575450\pi\)
−0.234821 + 0.972039i \(0.575450\pi\)
\(810\) 0 0
\(811\) −2.48410e7 −1.32622 −0.663112 0.748520i \(-0.730765\pi\)
−0.663112 + 0.748520i \(0.730765\pi\)
\(812\) − 5.93328e6i − 0.315795i
\(813\) 7.40261e6i 0.392788i
\(814\) −7.67410e6 −0.405944
\(815\) 0 0
\(816\) 373248. 0.0196233
\(817\) − 2.41814e7i − 1.26744i
\(818\) − 5.80750e6i − 0.303463i
\(819\) −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.i 0.0245634i
\(823\) − 8.70659e6i − 0.448073i −0.974581 0.224036i \(-0.928077\pi\)
0.974581 0.224036i \(-0.0719234\pi\)
\(824\) 8.54042e6 0.438189
\(825\) 0 0
\(826\) 3.31068e6 0.168837
\(827\) − 3.71184e7i − 1.88723i −0.331040 0.943617i \(-0.607400\pi\)
0.331040 0.943617i \(-0.392600\pi\)
\(828\) 396576.i 0.0201025i
\(829\) −1.01765e6 −0.0514295 −0.0257147 0.999669i \(-0.508186\pi\)
−0.0257147 + 0.999669i \(0.508186\pi\)
\(830\) 0 0
\(831\) −4.92141e6 −0.247222
\(832\) − 413696.i − 0.0207192i
\(833\) − 2.36488e6i − 0.118085i
\(834\) 1.26266e7 0.628598
\(835\) 0 0
\(836\) 5.98512e6 0.296181
\(837\) 6.26430e6i 0.309071i
\(838\) 2.23752e6i 0.110067i
\(839\) 3.36194e7 1.64887 0.824433 0.565960i \(-0.191494\pi\)
0.824433 + 0.565960i \(0.191494\pi\)
\(840\) 0 0
\(841\) 4.17410e7 2.03504
\(842\) 1.56588e7i 0.761164i
\(843\) − 9.83248e6i − 0.476534i
\(844\) −1.19916e7 −0.579458
\(845\) 0 0
\(846\) −355752. −0.0170892
\(847\) 5.25305e6i 0.251596i
\(848\) 4.58650e6i 0.219024i
\(849\) 2.23632e7 1.06479
\(850\) 0 0
\(851\) 2.64445e6 0.125173
\(852\) 5.86483e6i 0.276794i
\(853\) − 3.52574e7i − 1.65912i −0.558419 0.829559i \(-0.688592\pi\)
0.558419 0.829559i \(-0.311408\pi\)
\(854\) 4.10836e6 0.192763
\(855\) 0 0
\(856\) 5.20013e6 0.242566
\(857\) 3.14941e7i 1.46480i 0.680877 + 0.732398i \(0.261599\pi\)
−0.680877 + 0.732398i \(0.738401\pi\)
\(858\) 807192.i 0.0374333i
\(859\) −1.19344e7 −0.551848 −0.275924 0.961180i \(-0.588984\pi\)
−0.275924 + 0.961180i \(0.588984\pi\)
\(860\) 0 0
\(861\) −7.68506e6 −0.353297
\(862\) 1.43000e7i 0.655492i
\(863\) 8.70442e6i 0.397844i 0.980015 + 0.198922i \(0.0637440\pi\)
−0.980015 + 0.198922i \(0.936256\pi\)
\(864\) −746496. −0.0340207
\(865\) 0 0
\(866\) 2.86388e7 1.29766
\(867\) 1.25425e7i 0.566679i
\(868\) − 6.46194e6i − 0.291114i
\(869\) 1.53535e7 0.689697
\(870\) 0 0
\(871\) 10807.0 0.000482681 0
\(872\) 1.38890e7i 0.618555i
\(873\) − 66663.0i − 0.00296039i
\(874\) −2.06244e6 −0.0913277
\(875\) 0 0
\(876\) 4.99766e6 0.220043
\(877\) − 1.17999e7i − 0.518059i −0.965869 0.259029i \(-0.916597\pi\)
0.965869 0.259029i \(-0.0834026\pi\)
\(878\) 6.87158e6i 0.300829i
\(879\) 3.07255e6 0.134130
\(880\) 0 0
\(881\) −2.73840e7 −1.18866 −0.594330 0.804221i \(-0.702583\pi\)
−0.594330 + 0.804221i \(0.702583\pi\)
\(882\) 4.72975e6i 0.204723i
\(883\) − 8.80577e6i − 0.380072i −0.981777 0.190036i \(-0.939140\pi\)
0.981777 0.190036i \(-0.0608604\pi\)
\(884\) −261792. −0.0112675
\(885\) 0 0
\(886\) 1.35868e7 0.581477
\(887\) − 250122.i − 0.0106744i −0.999986 0.00533719i \(-0.998301\pi\)
0.999986 0.00533719i \(-0.00169889\pi\)
\(888\) 4.97779e6i 0.211838i
\(889\) 1.20343e7 0.510699
\(890\) 0 0
\(891\) 1.45654e6 0.0614651
\(892\) − 2.70834e6i − 0.113970i
\(893\) − 1.85013e6i − 0.0776379i
\(894\) −3.95604e6 −0.165545
\(895\) 0 0
\(896\) 770048. 0.0320441
\(897\) − 278154.i − 0.0115426i
\(898\) − 1.35842e7i − 0.562140i
\(899\) 6.77988e7 2.79784
\(900\) 0 0
\(901\) 2.90239e6 0.119109
\(902\) 1.61332e7i 0.660243i
\(903\) 6.07047e6i 0.247744i
\(904\) 8.85274e6 0.360294
\(905\) 0 0
\(906\) −6.21371e6 −0.251496
\(907\) 3.24955e7i 1.31161i 0.754929 + 0.655806i \(0.227671\pi\)
−0.754929 + 0.655806i \(0.772329\pi\)
\(908\) − 743808.i − 0.0299396i
\(909\) 2.74007e6 0.109990
\(910\) 0 0
\(911\) 4.24595e7 1.69504 0.847518 0.530766i \(-0.178096\pi\)
0.847518 + 0.530766i \(0.178096\pi\)
\(912\) − 3.88224e6i − 0.154559i
\(913\) − 2.40945e7i − 0.956625i
\(914\) 1.81126e7 0.717157
\(915\) 0 0
\(916\) −1.44184e6 −0.0567778
\(917\) − 5.56724e6i − 0.218634i
\(918\) 472392.i 0.0185010i
\(919\) −1.41629e7 −0.553176 −0.276588 0.960989i \(-0.589204\pi\)
−0.276588 + 0.960989i \(0.589204\pi\)
\(920\) 0 0
\(921\) 1.82608e7 0.709366
\(922\) 5.11579e6i 0.198192i
\(923\) − 4.11353e6i − 0.158932i
\(924\) −1.50250e6 −0.0578940
\(925\) 0 0
\(926\) −2.87945e7 −1.10352
\(927\) 1.08090e7i 0.413128i
\(928\) 8.07936e6i 0.307969i
\(929\) −4.37292e7 −1.66239 −0.831194 0.555982i \(-0.812342\pi\)
−0.831194 + 0.555982i \(0.812342\pi\)
\(930\) 0 0
\(931\) −2.45976e7 −0.930077
\(932\) − 1.70262e7i − 0.642063i
\(933\) − 1.85938e6i − 0.0699302i
\(934\) −1.93214e7 −0.724721
\(935\) 0 0
\(936\) 523584. 0.0195343
\(937\) − 5.73509e6i − 0.213398i −0.994291 0.106699i \(-0.965972\pi\)
0.994291 0.106699i \(-0.0340282\pi\)
\(938\) 20116.0i 0 0.000746508i
\(939\) 3.00801e7 1.11331
\(940\) 0 0
\(941\) −3.37395e7 −1.24212 −0.621061 0.783762i \(-0.713298\pi\)
−0.621061 + 0.783762i \(0.713298\pi\)
\(942\) 1.25997e7i 0.462631i
\(943\) − 5.55941e6i − 0.203587i
\(944\) −4.50816e6 −0.164653
\(945\) 0 0
\(946\) 1.27437e7 0.462985
\(947\) 3.07342e7i 1.11365i 0.830631 + 0.556823i \(0.187980\pi\)
−0.830631 + 0.556823i \(0.812020\pi\)
\(948\) − 9.95904e6i − 0.359912i
\(949\) −3.50531e6 −0.126346
\(950\) 0 0
\(951\) −2.27960e7 −0.817348
\(952\) − 487296.i − 0.0174261i
\(953\) − 2.51847e7i − 0.898264i −0.893465 0.449132i \(-0.851733\pi\)
0.893465 0.449132i \(-0.148267\pi\)
\(954\) −5.80478e6 −0.206498
\(955\) 0 0
\(956\) 1.84253e7 0.652033
\(957\) − 1.57642e7i − 0.556407i
\(958\) 2.99460e6i 0.105420i
\(959\) 621246. 0.0218131
\(960\) 0 0
\(961\) 4.52105e7 1.57918
\(962\) − 3.49137e6i − 0.121635i
\(963\) 6.58141e6i 0.228693i
\(964\) −1.36995e7 −0.474801
\(965\) 0 0
\(966\) 517752. 0.0178517
\(967\) − 1.44556e7i − 0.497130i −0.968615 0.248565i \(-0.920041\pi\)
0.968615 0.248565i \(-0.0799589\pi\)
\(968\) − 7.15309e6i − 0.245361i
\(969\) −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.i − 0.0320750i
\(973\) − 1.64848e7i − 0.558214i
\(974\) 2.06558e7 0.697660
\(975\) 0 0
\(976\) −5.59437e6 −0.187986
\(977\) 8.41568e6i 0.282067i 0.990005 + 0.141034i \(0.0450426\pi\)
−0.990005 + 0.141034i \(0.954957\pi\)
\(978\) 6.93292e6i 0.231776i
\(979\) −7.77888e6 −0.259394
\(980\) 0 0
\(981\) −1.75782e7 −0.583180
\(982\) − 3.41715e7i − 1.13080i
\(983\) 3.89409e7i 1.28535i 0.766138 + 0.642676i \(0.222176\pi\)
−0.766138 + 0.642676i \(0.777824\pi\)
\(984\) 1.04648e7 0.344542
\(985\) 0 0
\(986\) 5.11272e6 0.167479
\(987\) 464454.i 0.0151757i
\(988\) 2.72296e6i 0.0887460i
\(989\) −4.39141e6 −0.142762
\(990\) 0 0
\(991\) 4.84592e7 1.56745 0.783723 0.621111i \(-0.213318\pi\)
0.783723 + 0.621111i \(0.213318\pi\)
\(992\) 8.79923e6i 0.283900i
\(993\) 5.41919e6i 0.174406i
\(994\) 7.65686e6 0.245802
\(995\) 0 0
\(996\) −1.56289e7 −0.499206
\(997\) − 3.84733e7i − 1.22581i −0.790158 0.612903i \(-0.790002\pi\)
0.790158 0.612903i \(-0.209998\pi\)
\(998\) − 1.68235e7i − 0.534675i
\(999\) −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.c.g.49.1 2
3.2 odd 2 450.6.c.e.199.2 2
5.2 odd 4 150.6.a.m.1.1 yes 1
5.3 odd 4 150.6.a.a.1.1 1
5.4 even 2 inner 150.6.c.g.49.2 2
15.2 even 4 450.6.a.i.1.1 1
15.8 even 4 450.6.a.p.1.1 1
15.14 odd 2 450.6.c.e.199.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
150.6.a.a.1.1 1 5.3 odd 4
150.6.a.m.1.1 yes 1 5.2 odd 4
150.6.c.g.49.1 2 1.1 even 1 trivial
150.6.c.g.49.2 2 5.4 even 2 inner
450.6.a.i.1.1 1 15.2 even 4
450.6.a.p.1.1 1 15.8 even 4
450.6.c.e.199.1 2 15.14 odd 2
450.6.c.e.199.2 2 3.2 odd 2