Properties

Label 177.6.a.b.1.2
Level $177$
Weight $6$
Character 177.1
Self dual yes
Analytic conductor $28.388$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.3879361069\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 4 x^{11} - 283 x^{10} + 1045 x^{9} + 27968 x^{8} - 94393 x^{7} - 1130486 x^{6} + 3566264 x^{5} + 15496192 x^{4} - 53008480 x^{3} - 16576192 x^{2} + 120303168 x - 50564480\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(8.94324\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-8.94324 q^{2} -9.00000 q^{3} +47.9816 q^{4} +48.3356 q^{5} +80.4892 q^{6} -32.5863 q^{7} -142.927 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-8.94324 q^{2} -9.00000 q^{3} +47.9816 q^{4} +48.3356 q^{5} +80.4892 q^{6} -32.5863 q^{7} -142.927 q^{8} +81.0000 q^{9} -432.277 q^{10} -607.250 q^{11} -431.834 q^{12} -114.450 q^{13} +291.427 q^{14} -435.021 q^{15} -257.180 q^{16} +1477.59 q^{17} -724.402 q^{18} +1197.95 q^{19} +2319.22 q^{20} +293.277 q^{21} +5430.79 q^{22} -815.109 q^{23} +1286.34 q^{24} -788.665 q^{25} +1023.55 q^{26} -729.000 q^{27} -1563.54 q^{28} +7054.74 q^{29} +3890.50 q^{30} -2013.45 q^{31} +6873.69 q^{32} +5465.25 q^{33} -13214.4 q^{34} -1575.08 q^{35} +3886.51 q^{36} +2888.36 q^{37} -10713.6 q^{38} +1030.05 q^{39} -6908.46 q^{40} +7977.13 q^{41} -2622.85 q^{42} -11709.8 q^{43} -29136.8 q^{44} +3915.19 q^{45} +7289.72 q^{46} +2217.46 q^{47} +2314.62 q^{48} -15745.1 q^{49} +7053.22 q^{50} -13298.3 q^{51} -5491.47 q^{52} -16868.6 q^{53} +6519.62 q^{54} -29351.8 q^{55} +4657.46 q^{56} -10781.6 q^{57} -63092.2 q^{58} -3481.00 q^{59} -20873.0 q^{60} -11356.2 q^{61} +18006.8 q^{62} -2639.49 q^{63} -53243.2 q^{64} -5532.00 q^{65} -48877.1 q^{66} -29351.9 q^{67} +70896.9 q^{68} +7335.98 q^{69} +14086.3 q^{70} +61841.8 q^{71} -11577.1 q^{72} -6951.65 q^{73} -25831.3 q^{74} +7097.99 q^{75} +57479.7 q^{76} +19788.1 q^{77} -9211.96 q^{78} -74447.1 q^{79} -12431.0 q^{80} +6561.00 q^{81} -71341.4 q^{82} -64837.6 q^{83} +14071.9 q^{84} +71420.1 q^{85} +104724. q^{86} -63492.7 q^{87} +86792.4 q^{88} -6479.60 q^{89} -35014.5 q^{90} +3729.49 q^{91} -39110.2 q^{92} +18121.1 q^{93} -19831.3 q^{94} +57903.9 q^{95} -61863.2 q^{96} +34765.8 q^{97} +140812. q^{98} -49187.3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} + O(q^{10}) \) \( 12q - 4q^{2} - 108q^{3} + 198q^{4} + 36q^{5} + 36q^{6} - 411q^{7} - 69q^{8} + 972q^{9} - 863q^{10} + 492q^{11} - 1782q^{12} - 974q^{13} - 967q^{14} - 324q^{15} + 6370q^{16} - 1463q^{17} - 324q^{18} - 3189q^{19} - 835q^{20} + 3699q^{21} - 2726q^{22} - 2617q^{23} + 621q^{24} + 8642q^{25} + 2414q^{26} - 8748q^{27} - 20458q^{28} - 1963q^{29} + 7767q^{30} - 11929q^{31} - 14382q^{32} - 4428q^{33} - 20744q^{34} + 1829q^{35} + 16038q^{36} - 28105q^{37} - 23475q^{38} + 8766q^{39} - 100576q^{40} - 7585q^{41} + 8703q^{42} - 33146q^{43} + 26014q^{44} + 2916q^{45} - 142851q^{46} - 79215q^{47} - 57330q^{48} - 32569q^{49} - 136019q^{50} + 13167q^{51} - 248218q^{52} - 12220q^{53} + 2916q^{54} - 117770q^{55} - 186728q^{56} + 28701q^{57} - 188072q^{58} - 41772q^{59} + 7515q^{60} - 54195q^{61} + 36230q^{62} - 33291q^{63} + 45197q^{64} + 42368q^{65} + 24534q^{66} + 24224q^{67} - 209639q^{68} + 23553q^{69} - 35684q^{70} + 60254q^{71} - 5589q^{72} - 15385q^{73} + 214638q^{74} - 77778q^{75} - 167504q^{76} - 17169q^{77} - 21726q^{78} - 27054q^{79} + 216899q^{80} + 78732q^{81} + 37917q^{82} - 117595q^{83} + 184122q^{84} - 121585q^{85} + 306756q^{86} + 17667q^{87} - 105799q^{88} - 36033q^{89} - 69903q^{90} - 32217q^{91} - 30906q^{92} + 107361q^{93} + 128392q^{94} - 50721q^{95} + 129438q^{96} - 196914q^{97} + 574100q^{98} + 39852q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.94324 −1.58096 −0.790478 0.612490i \(-0.790168\pi\)
−0.790478 + 0.612490i \(0.790168\pi\)
\(3\) −9.00000 −0.577350
\(4\) 47.9816 1.49942
\(5\) 48.3356 0.864654 0.432327 0.901717i \(-0.357693\pi\)
0.432327 + 0.901717i \(0.357693\pi\)
\(6\) 80.4892 0.912766
\(7\) −32.5863 −0.251357 −0.125678 0.992071i \(-0.540111\pi\)
−0.125678 + 0.992071i \(0.540111\pi\)
\(8\) −142.927 −0.789567
\(9\) 81.0000 0.333333
\(10\) −432.277 −1.36698
\(11\) −607.250 −1.51316 −0.756582 0.653899i \(-0.773132\pi\)
−0.756582 + 0.653899i \(0.773132\pi\)
\(12\) −431.834 −0.865693
\(13\) −114.450 −0.187826 −0.0939131 0.995580i \(-0.529938\pi\)
−0.0939131 + 0.995580i \(0.529938\pi\)
\(14\) 291.427 0.397384
\(15\) −435.021 −0.499208
\(16\) −257.180 −0.251153
\(17\) 1477.59 1.24003 0.620013 0.784592i \(-0.287127\pi\)
0.620013 + 0.784592i \(0.287127\pi\)
\(18\) −724.402 −0.526985
\(19\) 1197.95 0.761300 0.380650 0.924719i \(-0.375700\pi\)
0.380650 + 0.924719i \(0.375700\pi\)
\(20\) 2319.22 1.29648
\(21\) 293.277 0.145121
\(22\) 5430.79 2.39225
\(23\) −815.109 −0.321289 −0.160645 0.987012i \(-0.551357\pi\)
−0.160645 + 0.987012i \(0.551357\pi\)
\(24\) 1286.34 0.455857
\(25\) −788.665 −0.252373
\(26\) 1023.55 0.296945
\(27\) −729.000 −0.192450
\(28\) −1563.54 −0.376890
\(29\) 7054.74 1.55771 0.778854 0.627205i \(-0.215801\pi\)
0.778854 + 0.627205i \(0.215801\pi\)
\(30\) 3890.50 0.789227
\(31\) −2013.45 −0.376302 −0.188151 0.982140i \(-0.560249\pi\)
−0.188151 + 0.982140i \(0.560249\pi\)
\(32\) 6873.69 1.18663
\(33\) 5465.25 0.873626
\(34\) −13214.4 −1.96043
\(35\) −1575.08 −0.217337
\(36\) 3886.51 0.499808
\(37\) 2888.36 0.346855 0.173427 0.984847i \(-0.444516\pi\)
0.173427 + 0.984847i \(0.444516\pi\)
\(38\) −10713.6 −1.20358
\(39\) 1030.05 0.108441
\(40\) −6908.46 −0.682702
\(41\) 7977.13 0.741117 0.370559 0.928809i \(-0.379166\pi\)
0.370559 + 0.928809i \(0.379166\pi\)
\(42\) −2622.85 −0.229430
\(43\) −11709.8 −0.965784 −0.482892 0.875680i \(-0.660414\pi\)
−0.482892 + 0.875680i \(0.660414\pi\)
\(44\) −29136.8 −2.26887
\(45\) 3915.19 0.288218
\(46\) 7289.72 0.507944
\(47\) 2217.46 0.146424 0.0732119 0.997316i \(-0.476675\pi\)
0.0732119 + 0.997316i \(0.476675\pi\)
\(48\) 2314.62 0.145003
\(49\) −15745.1 −0.936820
\(50\) 7053.22 0.398990
\(51\) −13298.3 −0.715929
\(52\) −5491.47 −0.281631
\(53\) −16868.6 −0.824878 −0.412439 0.910985i \(-0.635323\pi\)
−0.412439 + 0.910985i \(0.635323\pi\)
\(54\) 6519.62 0.304255
\(55\) −29351.8 −1.30836
\(56\) 4657.46 0.198463
\(57\) −10781.6 −0.439537
\(58\) −63092.2 −2.46267
\(59\) −3481.00 −0.130189
\(60\) −20873.0 −0.748525
\(61\) −11356.2 −0.390758 −0.195379 0.980728i \(-0.562594\pi\)
−0.195379 + 0.980728i \(0.562594\pi\)
\(62\) 18006.8 0.594917
\(63\) −2639.49 −0.0837855
\(64\) −53243.2 −1.62486
\(65\) −5532.00 −0.162405
\(66\) −48877.1 −1.38116
\(67\) −29351.9 −0.798821 −0.399410 0.916772i \(-0.630785\pi\)
−0.399410 + 0.916772i \(0.630785\pi\)
\(68\) 70896.9 1.85932
\(69\) 7335.98 0.185496
\(70\) 14086.3 0.343600
\(71\) 61841.8 1.45592 0.727958 0.685622i \(-0.240470\pi\)
0.727958 + 0.685622i \(0.240470\pi\)
\(72\) −11577.1 −0.263189
\(73\) −6951.65 −0.152680 −0.0763398 0.997082i \(-0.524323\pi\)
−0.0763398 + 0.997082i \(0.524323\pi\)
\(74\) −25831.3 −0.548362
\(75\) 7097.99 0.145708
\(76\) 57479.7 1.14151
\(77\) 19788.1 0.380344
\(78\) −9211.96 −0.171441
\(79\) −74447.1 −1.34208 −0.671042 0.741419i \(-0.734153\pi\)
−0.671042 + 0.741419i \(0.734153\pi\)
\(80\) −12431.0 −0.217160
\(81\) 6561.00 0.111111
\(82\) −71341.4 −1.17167
\(83\) −64837.6 −1.03307 −0.516537 0.856265i \(-0.672779\pi\)
−0.516537 + 0.856265i \(0.672779\pi\)
\(84\) 14071.9 0.217598
\(85\) 71420.1 1.07219
\(86\) 104724. 1.52686
\(87\) −63492.7 −0.899343
\(88\) 86792.4 1.19474
\(89\) −6479.60 −0.0867108 −0.0433554 0.999060i \(-0.513805\pi\)
−0.0433554 + 0.999060i \(0.513805\pi\)
\(90\) −35014.5 −0.455660
\(91\) 3729.49 0.0472114
\(92\) −39110.2 −0.481749
\(93\) 18121.1 0.217258
\(94\) −19831.3 −0.231490
\(95\) 57903.9 0.658262
\(96\) −61863.2 −0.685100
\(97\) 34765.8 0.375166 0.187583 0.982249i \(-0.439935\pi\)
0.187583 + 0.982249i \(0.439935\pi\)
\(98\) 140812. 1.48107
\(99\) −49187.3 −0.504388
\(100\) −37841.4 −0.378414
\(101\) −10413.0 −0.101571 −0.0507857 0.998710i \(-0.516173\pi\)
−0.0507857 + 0.998710i \(0.516173\pi\)
\(102\) 118930. 1.13185
\(103\) −187352. −1.74007 −0.870033 0.492993i \(-0.835903\pi\)
−0.870033 + 0.492993i \(0.835903\pi\)
\(104\) 16357.9 0.148301
\(105\) 14175.7 0.125479
\(106\) 150860. 1.30410
\(107\) 18946.3 0.159979 0.0799897 0.996796i \(-0.474511\pi\)
0.0799897 + 0.996796i \(0.474511\pi\)
\(108\) −34978.6 −0.288564
\(109\) 12701.4 0.102396 0.0511981 0.998689i \(-0.483696\pi\)
0.0511981 + 0.998689i \(0.483696\pi\)
\(110\) 262501. 2.06847
\(111\) −25995.3 −0.200257
\(112\) 8380.57 0.0631289
\(113\) −90034.0 −0.663301 −0.331650 0.943402i \(-0.607605\pi\)
−0.331650 + 0.943402i \(0.607605\pi\)
\(114\) 96422.3 0.694889
\(115\) −39398.8 −0.277804
\(116\) 338497. 2.33566
\(117\) −9270.42 −0.0626087
\(118\) 31131.4 0.205823
\(119\) −48149.1 −0.311689
\(120\) 62176.2 0.394158
\(121\) 207702. 1.28967
\(122\) 101561. 0.617772
\(123\) −71794.1 −0.427884
\(124\) −96608.5 −0.564236
\(125\) −189170. −1.08287
\(126\) 23605.6 0.132461
\(127\) −75773.2 −0.416875 −0.208438 0.978036i \(-0.566838\pi\)
−0.208438 + 0.978036i \(0.566838\pi\)
\(128\) 256209. 1.38220
\(129\) 105389. 0.557595
\(130\) 49474.0 0.256755
\(131\) −279838. −1.42472 −0.712358 0.701816i \(-0.752373\pi\)
−0.712358 + 0.701816i \(0.752373\pi\)
\(132\) 262231. 1.30993
\(133\) −39036.9 −0.191358
\(134\) 262501. 1.26290
\(135\) −35236.7 −0.166403
\(136\) −211187. −0.979083
\(137\) −309329. −1.40805 −0.704026 0.710174i \(-0.748616\pi\)
−0.704026 + 0.710174i \(0.748616\pi\)
\(138\) −65607.4 −0.293262
\(139\) 440414. 1.93341 0.966706 0.255889i \(-0.0823682\pi\)
0.966706 + 0.255889i \(0.0823682\pi\)
\(140\) −75574.8 −0.325880
\(141\) −19957.2 −0.0845378
\(142\) −553066. −2.30174
\(143\) 69499.6 0.284212
\(144\) −20831.6 −0.0837176
\(145\) 340995. 1.34688
\(146\) 62170.3 0.241380
\(147\) 141706. 0.540873
\(148\) 138588. 0.520082
\(149\) −35329.4 −0.130368 −0.0651840 0.997873i \(-0.520763\pi\)
−0.0651840 + 0.997873i \(0.520763\pi\)
\(150\) −63479.0 −0.230357
\(151\) −277734. −0.991258 −0.495629 0.868534i \(-0.665062\pi\)
−0.495629 + 0.868534i \(0.665062\pi\)
\(152\) −171220. −0.601097
\(153\) 119684. 0.413342
\(154\) −176969. −0.601307
\(155\) −97321.4 −0.325371
\(156\) 49423.2 0.162600
\(157\) 123296. 0.399208 0.199604 0.979877i \(-0.436034\pi\)
0.199604 + 0.979877i \(0.436034\pi\)
\(158\) 665798. 2.12178
\(159\) 151818. 0.476244
\(160\) 332244. 1.02602
\(161\) 26561.4 0.0807582
\(162\) −58676.6 −0.175662
\(163\) −516622. −1.52301 −0.761507 0.648157i \(-0.775540\pi\)
−0.761507 + 0.648157i \(0.775540\pi\)
\(164\) 382755. 1.11125
\(165\) 264167. 0.755384
\(166\) 579858. 1.63325
\(167\) 265123. 0.735625 0.367812 0.929900i \(-0.380107\pi\)
0.367812 + 0.929900i \(0.380107\pi\)
\(168\) −41917.2 −0.114583
\(169\) −358194. −0.964721
\(170\) −638727. −1.69509
\(171\) 97034.2 0.253767
\(172\) −561856. −1.44812
\(173\) −571531. −1.45186 −0.725930 0.687769i \(-0.758590\pi\)
−0.725930 + 0.687769i \(0.758590\pi\)
\(174\) 567830. 1.42182
\(175\) 25699.7 0.0634356
\(176\) 156173. 0.380035
\(177\) 31329.0 0.0751646
\(178\) 57948.6 0.137086
\(179\) 70392.7 0.164208 0.0821041 0.996624i \(-0.473836\pi\)
0.0821041 + 0.996624i \(0.473836\pi\)
\(180\) 187857. 0.432161
\(181\) −69490.3 −0.157662 −0.0788312 0.996888i \(-0.525119\pi\)
−0.0788312 + 0.996888i \(0.525119\pi\)
\(182\) −33353.8 −0.0746391
\(183\) 102206. 0.225604
\(184\) 116501. 0.253679
\(185\) 139611. 0.299909
\(186\) −162061. −0.343476
\(187\) −897265. −1.87636
\(188\) 106397. 0.219551
\(189\) 23755.4 0.0483736
\(190\) −517848. −1.04068
\(191\) −790909. −1.56871 −0.784356 0.620311i \(-0.787007\pi\)
−0.784356 + 0.620311i \(0.787007\pi\)
\(192\) 479189. 0.938110
\(193\) −970964. −1.87633 −0.938166 0.346185i \(-0.887477\pi\)
−0.938166 + 0.346185i \(0.887477\pi\)
\(194\) −310919. −0.593121
\(195\) 49788.0 0.0937644
\(196\) −755476. −1.40469
\(197\) −619957. −1.13814 −0.569070 0.822289i \(-0.692697\pi\)
−0.569070 + 0.822289i \(0.692697\pi\)
\(198\) 439894. 0.797416
\(199\) −175483. −0.314124 −0.157062 0.987589i \(-0.550202\pi\)
−0.157062 + 0.987589i \(0.550202\pi\)
\(200\) 112721. 0.199265
\(201\) 264167. 0.461199
\(202\) 93125.7 0.160580
\(203\) −229888. −0.391540
\(204\) −638072. −1.07348
\(205\) 385580. 0.640810
\(206\) 1.67554e6 2.75097
\(207\) −66023.8 −0.107096
\(208\) 29434.2 0.0471731
\(209\) −727458. −1.15197
\(210\) −126777. −0.198377
\(211\) 561184. 0.867759 0.433880 0.900971i \(-0.357144\pi\)
0.433880 + 0.900971i \(0.357144\pi\)
\(212\) −809383. −1.23684
\(213\) −556576. −0.840573
\(214\) −169441. −0.252921
\(215\) −566003. −0.835069
\(216\) 104194. 0.151952
\(217\) 65611.0 0.0945861
\(218\) −113591. −0.161884
\(219\) 62564.9 0.0881496
\(220\) −1.40835e6 −1.96179
\(221\) −169109. −0.232909
\(222\) 232482. 0.316597
\(223\) −346056. −0.465998 −0.232999 0.972477i \(-0.574854\pi\)
−0.232999 + 0.972477i \(0.574854\pi\)
\(224\) −223988. −0.298267
\(225\) −63881.9 −0.0841243
\(226\) 805196. 1.04865
\(227\) −69353.3 −0.0893310 −0.0446655 0.999002i \(-0.514222\pi\)
−0.0446655 + 0.999002i \(0.514222\pi\)
\(228\) −517317. −0.659052
\(229\) 1.03395e6 1.30289 0.651447 0.758694i \(-0.274162\pi\)
0.651447 + 0.758694i \(0.274162\pi\)
\(230\) 352353. 0.439196
\(231\) −178093. −0.219592
\(232\) −1.00831e6 −1.22991
\(233\) 1.08238e6 1.30614 0.653069 0.757298i \(-0.273481\pi\)
0.653069 + 0.757298i \(0.273481\pi\)
\(234\) 82907.6 0.0989817
\(235\) 107182. 0.126606
\(236\) −167024. −0.195208
\(237\) 670024. 0.774853
\(238\) 430609. 0.492766
\(239\) 526608. 0.596338 0.298169 0.954513i \(-0.403624\pi\)
0.298169 + 0.954513i \(0.403624\pi\)
\(240\) 111879. 0.125378
\(241\) 1.09725e6 1.21692 0.608461 0.793584i \(-0.291787\pi\)
0.608461 + 0.793584i \(0.291787\pi\)
\(242\) −1.85753e6 −2.03891
\(243\) −59049.0 −0.0641500
\(244\) −544888. −0.585912
\(245\) −761051. −0.810025
\(246\) 642072. 0.676466
\(247\) −137105. −0.142992
\(248\) 287776. 0.297116
\(249\) 583538. 0.596446
\(250\) 1.69179e6 1.71197
\(251\) 729854. 0.731226 0.365613 0.930767i \(-0.380859\pi\)
0.365613 + 0.930767i \(0.380859\pi\)
\(252\) −126647. −0.125630
\(253\) 494975. 0.486163
\(254\) 677658. 0.659062
\(255\) −642781. −0.619031
\(256\) −587557. −0.560338
\(257\) 722806. 0.682636 0.341318 0.939948i \(-0.389127\pi\)
0.341318 + 0.939948i \(0.389127\pi\)
\(258\) −942515. −0.881534
\(259\) −94121.1 −0.0871842
\(260\) −265434. −0.243513
\(261\) 571434. 0.519236
\(262\) 2.50266e6 2.25241
\(263\) −1.30671e6 −1.16490 −0.582452 0.812865i \(-0.697906\pi\)
−0.582452 + 0.812865i \(0.697906\pi\)
\(264\) −781131. −0.689786
\(265\) −815356. −0.713235
\(266\) 349116. 0.302528
\(267\) 58316.4 0.0500625
\(268\) −1.40835e6 −1.19777
\(269\) 2.09858e6 1.76825 0.884127 0.467246i \(-0.154754\pi\)
0.884127 + 0.467246i \(0.154754\pi\)
\(270\) 315130. 0.263076
\(271\) 1.04873e6 0.867440 0.433720 0.901048i \(-0.357201\pi\)
0.433720 + 0.901048i \(0.357201\pi\)
\(272\) −380006. −0.311436
\(273\) −33565.4 −0.0272575
\(274\) 2.76640e6 2.22607
\(275\) 478917. 0.381881
\(276\) 351992. 0.278138
\(277\) −191493. −0.149953 −0.0749763 0.997185i \(-0.523888\pi\)
−0.0749763 + 0.997185i \(0.523888\pi\)
\(278\) −3.93873e6 −3.05664
\(279\) −163090. −0.125434
\(280\) 225121. 0.171602
\(281\) −1.21812e6 −0.920292 −0.460146 0.887843i \(-0.652203\pi\)
−0.460146 + 0.887843i \(0.652203\pi\)
\(282\) 178482. 0.133651
\(283\) 798367. 0.592566 0.296283 0.955100i \(-0.404253\pi\)
0.296283 + 0.955100i \(0.404253\pi\)
\(284\) 2.96727e6 2.18303
\(285\) −521135. −0.380048
\(286\) −621552. −0.449327
\(287\) −259945. −0.186285
\(288\) 556769. 0.395543
\(289\) 763404. 0.537662
\(290\) −3.04960e6 −2.12936
\(291\) −312893. −0.216602
\(292\) −333551. −0.228931
\(293\) −1.05915e6 −0.720754 −0.360377 0.932807i \(-0.617352\pi\)
−0.360377 + 0.932807i \(0.617352\pi\)
\(294\) −1.26731e6 −0.855097
\(295\) −168256. −0.112568
\(296\) −412825. −0.273865
\(297\) 442685. 0.291209
\(298\) 315960. 0.206106
\(299\) 93288.9 0.0603465
\(300\) 340572. 0.218477
\(301\) 381581. 0.242756
\(302\) 2.48384e6 1.56714
\(303\) 93716.8 0.0586423
\(304\) −308090. −0.191203
\(305\) −548909. −0.337871
\(306\) −1.07037e6 −0.653475
\(307\) 80118.4 0.0485161 0.0242581 0.999706i \(-0.492278\pi\)
0.0242581 + 0.999706i \(0.492278\pi\)
\(308\) 949462. 0.570296
\(309\) 1.68617e6 1.00463
\(310\) 870369. 0.514398
\(311\) 2.98406e6 1.74947 0.874736 0.484600i \(-0.161035\pi\)
0.874736 + 0.484600i \(0.161035\pi\)
\(312\) −147221. −0.0856218
\(313\) 678550. 0.391491 0.195745 0.980655i \(-0.437287\pi\)
0.195745 + 0.980655i \(0.437287\pi\)
\(314\) −1.10266e6 −0.631130
\(315\) −127582. −0.0724455
\(316\) −3.57209e6 −2.01235
\(317\) 1.81380e6 1.01377 0.506887 0.862013i \(-0.330796\pi\)
0.506887 + 0.862013i \(0.330796\pi\)
\(318\) −1.35774e6 −0.752921
\(319\) −4.28399e6 −2.35707
\(320\) −2.57355e6 −1.40494
\(321\) −170516. −0.0923642
\(322\) −237545. −0.127675
\(323\) 1.77008e6 0.944031
\(324\) 314807. 0.166603
\(325\) 90262.4 0.0474022
\(326\) 4.62027e6 2.40782
\(327\) −114312. −0.0591185
\(328\) −1.14015e6 −0.585161
\(329\) −72258.9 −0.0368046
\(330\) −2.36250e6 −1.19423
\(331\) −2.91425e6 −1.46203 −0.731016 0.682360i \(-0.760954\pi\)
−0.731016 + 0.682360i \(0.760954\pi\)
\(332\) −3.11101e6 −1.54902
\(333\) 233957. 0.115618
\(334\) −2.37106e6 −1.16299
\(335\) −1.41874e6 −0.690704
\(336\) −75425.1 −0.0364475
\(337\) −1.92879e6 −0.925145 −0.462572 0.886582i \(-0.653073\pi\)
−0.462572 + 0.886582i \(0.653073\pi\)
\(338\) 3.20342e6 1.52518
\(339\) 810306. 0.382957
\(340\) 3.42685e6 1.60767
\(341\) 1.22267e6 0.569407
\(342\) −867801. −0.401194
\(343\) 1.06075e6 0.486833
\(344\) 1.67365e6 0.762551
\(345\) 354589. 0.160390
\(346\) 5.11134e6 2.29533
\(347\) −2.63458e6 −1.17459 −0.587297 0.809372i \(-0.699808\pi\)
−0.587297 + 0.809372i \(0.699808\pi\)
\(348\) −3.04648e6 −1.34850
\(349\) −338492. −0.148759 −0.0743797 0.997230i \(-0.523698\pi\)
−0.0743797 + 0.997230i \(0.523698\pi\)
\(350\) −229839. −0.100289
\(351\) 83433.8 0.0361472
\(352\) −4.17405e6 −1.79556
\(353\) −1.24130e6 −0.530200 −0.265100 0.964221i \(-0.585405\pi\)
−0.265100 + 0.964221i \(0.585405\pi\)
\(354\) −280183. −0.118832
\(355\) 2.98916e6 1.25886
\(356\) −310901. −0.130016
\(357\) 433342. 0.179953
\(358\) −629539. −0.259606
\(359\) −1.85276e6 −0.758723 −0.379361 0.925249i \(-0.623856\pi\)
−0.379361 + 0.925249i \(0.623856\pi\)
\(360\) −559585. −0.227567
\(361\) −1.04101e6 −0.420422
\(362\) 621469. 0.249257
\(363\) −1.86932e6 −0.744589
\(364\) 178947. 0.0707898
\(365\) −336013. −0.132015
\(366\) −914050. −0.356671
\(367\) 2.21751e6 0.859410 0.429705 0.902969i \(-0.358618\pi\)
0.429705 + 0.902969i \(0.358618\pi\)
\(368\) 209630. 0.0806927
\(369\) 646147. 0.247039
\(370\) −1.24857e6 −0.474144
\(371\) 549687. 0.207339
\(372\) 869476. 0.325762
\(373\) −1.99276e6 −0.741623 −0.370811 0.928708i \(-0.620920\pi\)
−0.370811 + 0.928708i \(0.620920\pi\)
\(374\) 8.02445e6 2.96645
\(375\) 1.70253e6 0.625195
\(376\) −316935. −0.115611
\(377\) −807413. −0.292578
\(378\) −212451. −0.0764766
\(379\) 1.04393e6 0.373312 0.186656 0.982425i \(-0.440235\pi\)
0.186656 + 0.982425i \(0.440235\pi\)
\(380\) 2.77832e6 0.987013
\(381\) 681959. 0.240683
\(382\) 7.07329e6 2.48007
\(383\) 1.07017e6 0.372783 0.186391 0.982476i \(-0.440321\pi\)
0.186391 + 0.982476i \(0.440321\pi\)
\(384\) −2.30588e6 −0.798012
\(385\) 956469. 0.328866
\(386\) 8.68356e6 2.96640
\(387\) −948497. −0.321928
\(388\) 1.66812e6 0.562533
\(389\) 1.68363e6 0.564122 0.282061 0.959396i \(-0.408982\pi\)
0.282061 + 0.959396i \(0.408982\pi\)
\(390\) −445266. −0.148237
\(391\) −1.20439e6 −0.398407
\(392\) 2.25040e6 0.739682
\(393\) 2.51854e6 0.822560
\(394\) 5.54442e6 1.79935
\(395\) −3.59845e6 −1.16044
\(396\) −2.36008e6 −0.756291
\(397\) −812204. −0.258636 −0.129318 0.991603i \(-0.541279\pi\)
−0.129318 + 0.991603i \(0.541279\pi\)
\(398\) 1.56938e6 0.496617
\(399\) 351332. 0.110481
\(400\) 202829. 0.0633841
\(401\) −2.25492e6 −0.700279 −0.350139 0.936698i \(-0.613866\pi\)
−0.350139 + 0.936698i \(0.613866\pi\)
\(402\) −2.36251e6 −0.729136
\(403\) 230439. 0.0706794
\(404\) −499631. −0.152299
\(405\) 317130. 0.0960727
\(406\) 2.05594e6 0.619008
\(407\) −1.75396e6 −0.524848
\(408\) 1.90068e6 0.565274
\(409\) 949748. 0.280737 0.140369 0.990099i \(-0.455171\pi\)
0.140369 + 0.990099i \(0.455171\pi\)
\(410\) −3.44833e6 −1.01309
\(411\) 2.78396e6 0.812939
\(412\) −8.98945e6 −2.60910
\(413\) 113433. 0.0327238
\(414\) 590467. 0.169315
\(415\) −3.13397e6 −0.893252
\(416\) −786691. −0.222880
\(417\) −3.96373e6 −1.11626
\(418\) 6.50583e6 1.82122
\(419\) −328196. −0.0913267 −0.0456634 0.998957i \(-0.514540\pi\)
−0.0456634 + 0.998957i \(0.514540\pi\)
\(420\) 680174. 0.188147
\(421\) 6.14766e6 1.69046 0.845230 0.534403i \(-0.179463\pi\)
0.845230 + 0.534403i \(0.179463\pi\)
\(422\) −5.01880e6 −1.37189
\(423\) 179614. 0.0488079
\(424\) 2.41098e6 0.651297
\(425\) −1.16532e6 −0.312949
\(426\) 4.97759e6 1.32891
\(427\) 370057. 0.0982197
\(428\) 909071. 0.239877
\(429\) −625496. −0.164090
\(430\) 5.06190e6 1.32021
\(431\) 3.28062e6 0.850674 0.425337 0.905035i \(-0.360156\pi\)
0.425337 + 0.905035i \(0.360156\pi\)
\(432\) 187485. 0.0483344
\(433\) −1.79870e6 −0.461040 −0.230520 0.973068i \(-0.574043\pi\)
−0.230520 + 0.973068i \(0.574043\pi\)
\(434\) −586775. −0.149536
\(435\) −3.06896e6 −0.777621
\(436\) 609431. 0.153535
\(437\) −976463. −0.244598
\(438\) −559533. −0.139361
\(439\) 1.62646e6 0.402794 0.201397 0.979510i \(-0.435452\pi\)
0.201397 + 0.979510i \(0.435452\pi\)
\(440\) 4.19517e6 1.03304
\(441\) −1.27536e6 −0.312273
\(442\) 1.51238e6 0.368219
\(443\) 3.82966e6 0.927151 0.463576 0.886057i \(-0.346566\pi\)
0.463576 + 0.886057i \(0.346566\pi\)
\(444\) −1.24729e6 −0.300269
\(445\) −313196. −0.0749748
\(446\) 3.09486e6 0.736723
\(447\) 317965. 0.0752680
\(448\) 1.73500e6 0.408418
\(449\) −4.45166e6 −1.04209 −0.521046 0.853529i \(-0.674458\pi\)
−0.521046 + 0.853529i \(0.674458\pi\)
\(450\) 571311. 0.132997
\(451\) −4.84411e6 −1.12143
\(452\) −4.31997e6 −0.994569
\(453\) 2.49961e6 0.572303
\(454\) 620243. 0.141228
\(455\) 180268. 0.0408215
\(456\) 1.54098e6 0.347044
\(457\) −1.17886e6 −0.264042 −0.132021 0.991247i \(-0.542147\pi\)
−0.132021 + 0.991247i \(0.542147\pi\)
\(458\) −9.24683e6 −2.05982
\(459\) −1.07716e6 −0.238643
\(460\) −1.89042e6 −0.416546
\(461\) 1.73221e6 0.379620 0.189810 0.981821i \(-0.439213\pi\)
0.189810 + 0.981821i \(0.439213\pi\)
\(462\) 1.59272e6 0.347165
\(463\) 1.35619e6 0.294015 0.147007 0.989135i \(-0.453036\pi\)
0.147007 + 0.989135i \(0.453036\pi\)
\(464\) −1.81434e6 −0.391223
\(465\) 875893. 0.187853
\(466\) −9.67997e6 −2.06495
\(467\) −3.59227e6 −0.762214 −0.381107 0.924531i \(-0.624457\pi\)
−0.381107 + 0.924531i \(0.624457\pi\)
\(468\) −444809. −0.0938770
\(469\) 956471. 0.200789
\(470\) −958558. −0.200158
\(471\) −1.10966e6 −0.230483
\(472\) 497528. 0.102793
\(473\) 7.11080e6 1.46139
\(474\) −5.99218e6 −1.22501
\(475\) −944784. −0.192131
\(476\) −2.31027e6 −0.467353
\(477\) −1.36636e6 −0.274959
\(478\) −4.70958e6 −0.942784
\(479\) −5.70119e6 −1.13534 −0.567671 0.823255i \(-0.692156\pi\)
−0.567671 + 0.823255i \(0.692156\pi\)
\(480\) −2.99020e6 −0.592375
\(481\) −330572. −0.0651484
\(482\) −9.81296e6 −1.92390
\(483\) −239053. −0.0466257
\(484\) 9.96586e6 1.93375
\(485\) 1.68043e6 0.324389
\(486\) 528089. 0.101418
\(487\) −2.32159e6 −0.443571 −0.221786 0.975095i \(-0.571189\pi\)
−0.221786 + 0.975095i \(0.571189\pi\)
\(488\) 1.62310e6 0.308530
\(489\) 4.64960e6 0.879312
\(490\) 6.80626e6 1.28061
\(491\) −7.27492e6 −1.36183 −0.680917 0.732360i \(-0.738419\pi\)
−0.680917 + 0.732360i \(0.738419\pi\)
\(492\) −3.44479e6 −0.641580
\(493\) 1.04240e7 1.93160
\(494\) 1.22617e6 0.226064
\(495\) −2.37750e6 −0.436121
\(496\) 517820. 0.0945093
\(497\) −2.01520e6 −0.365954
\(498\) −5.21872e6 −0.942955
\(499\) 2.13472e6 0.383786 0.191893 0.981416i \(-0.438537\pi\)
0.191893 + 0.981416i \(0.438537\pi\)
\(500\) −9.07665e6 −1.62368
\(501\) −2.38611e6 −0.424713
\(502\) −6.52726e6 −1.15604
\(503\) 6.73329e6 1.18661 0.593304 0.804979i \(-0.297823\pi\)
0.593304 + 0.804979i \(0.297823\pi\)
\(504\) 377254. 0.0661543
\(505\) −503318. −0.0878242
\(506\) −4.42668e6 −0.768603
\(507\) 3.22375e6 0.556982
\(508\) −3.63571e6 −0.625073
\(509\) −1.65266e6 −0.282742 −0.141371 0.989957i \(-0.545151\pi\)
−0.141371 + 0.989957i \(0.545151\pi\)
\(510\) 5.74854e6 0.978661
\(511\) 226529. 0.0383770
\(512\) −2.94403e6 −0.496327
\(513\) −873308. −0.146512
\(514\) −6.46423e6 −1.07922
\(515\) −9.05579e6 −1.50456
\(516\) 5.05671e6 0.836072
\(517\) −1.34655e6 −0.221563
\(518\) 841748. 0.137834
\(519\) 5.14378e6 0.838231
\(520\) 790671. 0.128229
\(521\) −4.60489e6 −0.743233 −0.371617 0.928386i \(-0.621196\pi\)
−0.371617 + 0.928386i \(0.621196\pi\)
\(522\) −5.11047e6 −0.820890
\(523\) −797422. −0.127478 −0.0637388 0.997967i \(-0.520302\pi\)
−0.0637388 + 0.997967i \(0.520302\pi\)
\(524\) −1.34271e7 −2.13625
\(525\) −231297. −0.0366245
\(526\) 1.16862e7 1.84166
\(527\) −2.97505e6 −0.466624
\(528\) −1.40556e6 −0.219414
\(529\) −5.77194e6 −0.896773
\(530\) 7.29192e6 1.12759
\(531\) −281961. −0.0433963
\(532\) −1.87305e6 −0.286926
\(533\) −912979. −0.139201
\(534\) −521537. −0.0791466
\(535\) 915780. 0.138327
\(536\) 4.19518e6 0.630722
\(537\) −633534. −0.0948057
\(538\) −1.87681e7 −2.79553
\(539\) 9.56124e6 1.41756
\(540\) −1.69071e6 −0.249508
\(541\) −7.75717e6 −1.13949 −0.569745 0.821822i \(-0.692958\pi\)
−0.569745 + 0.821822i \(0.692958\pi\)
\(542\) −9.37902e6 −1.37138
\(543\) 625413. 0.0910264
\(544\) 1.01565e7 1.47145
\(545\) 613929. 0.0885374
\(546\) 300184. 0.0430929
\(547\) 6.87407e6 0.982304 0.491152 0.871074i \(-0.336576\pi\)
0.491152 + 0.871074i \(0.336576\pi\)
\(548\) −1.48421e7 −2.11127
\(549\) −919851. −0.130253
\(550\) −4.28307e6 −0.603738
\(551\) 8.45125e6 1.18588
\(552\) −1.04851e6 −0.146462
\(553\) 2.42596e6 0.337342
\(554\) 1.71257e6 0.237068
\(555\) −1.25650e6 −0.173153
\(556\) 2.11318e7 2.89900
\(557\) −1.03785e7 −1.41741 −0.708707 0.705503i \(-0.750721\pi\)
−0.708707 + 0.705503i \(0.750721\pi\)
\(558\) 1.45855e6 0.198306
\(559\) 1.34019e6 0.181399
\(560\) 405080. 0.0545847
\(561\) 8.07538e6 1.08332
\(562\) 1.08940e7 1.45494
\(563\) −1.17737e7 −1.56546 −0.782730 0.622361i \(-0.786173\pi\)
−0.782730 + 0.622361i \(0.786173\pi\)
\(564\) −957575. −0.126758
\(565\) −4.35185e6 −0.573526
\(566\) −7.13999e6 −0.936821
\(567\) −213799. −0.0279285
\(568\) −8.83885e6 −1.14954
\(569\) −1.71606e6 −0.222204 −0.111102 0.993809i \(-0.535438\pi\)
−0.111102 + 0.993809i \(0.535438\pi\)
\(570\) 4.66063e6 0.600839
\(571\) −1.38827e7 −1.78190 −0.890950 0.454102i \(-0.849960\pi\)
−0.890950 + 0.454102i \(0.849960\pi\)
\(572\) 3.33470e6 0.426154
\(573\) 7.11819e6 0.905697
\(574\) 2.32475e6 0.294508
\(575\) 642848. 0.0810846
\(576\) −4.31270e6 −0.541618
\(577\) 4.75324e6 0.594360 0.297180 0.954821i \(-0.403954\pi\)
0.297180 + 0.954821i \(0.403954\pi\)
\(578\) −6.82730e6 −0.850021
\(579\) 8.73868e6 1.08330
\(580\) 1.63615e7 2.01954
\(581\) 2.11282e6 0.259670
\(582\) 2.79827e6 0.342439
\(583\) 1.02435e7 1.24818
\(584\) 993578. 0.120551
\(585\) −448092. −0.0541349
\(586\) 9.47221e6 1.13948
\(587\) −1.92855e6 −0.231013 −0.115506 0.993307i \(-0.536849\pi\)
−0.115506 + 0.993307i \(0.536849\pi\)
\(588\) 6.79928e6 0.810998
\(589\) −2.41202e6 −0.286479
\(590\) 1.50476e6 0.177966
\(591\) 5.57961e6 0.657105
\(592\) −742830. −0.0871135
\(593\) 1.17565e7 1.37291 0.686453 0.727174i \(-0.259167\pi\)
0.686453 + 0.727174i \(0.259167\pi\)
\(594\) −3.95904e6 −0.460388
\(595\) −2.32732e6 −0.269503
\(596\) −1.69516e6 −0.195477
\(597\) 1.57934e6 0.181360
\(598\) −834305. −0.0954052
\(599\) 1.36231e7 1.55135 0.775673 0.631136i \(-0.217411\pi\)
0.775673 + 0.631136i \(0.217411\pi\)
\(600\) −1.01449e6 −0.115046
\(601\) 5.50257e6 0.621411 0.310706 0.950506i \(-0.399435\pi\)
0.310706 + 0.950506i \(0.399435\pi\)
\(602\) −3.41257e6 −0.383787
\(603\) −2.37750e6 −0.266274
\(604\) −1.33261e7 −1.48632
\(605\) 1.00394e7 1.11512
\(606\) −838132. −0.0927109
\(607\) 529691. 0.0583513 0.0291757 0.999574i \(-0.490712\pi\)
0.0291757 + 0.999574i \(0.490712\pi\)
\(608\) 8.23436e6 0.903380
\(609\) 2.06899e6 0.226056
\(610\) 4.90902e6 0.534159
\(611\) −253788. −0.0275022
\(612\) 5.74265e6 0.619774
\(613\) −6.94496e6 −0.746480 −0.373240 0.927735i \(-0.621753\pi\)
−0.373240 + 0.927735i \(0.621753\pi\)
\(614\) −716518. −0.0767019
\(615\) −3.47022e6 −0.369972
\(616\) −2.82824e6 −0.300307
\(617\) 1.72574e7 1.82500 0.912501 0.409075i \(-0.134148\pi\)
0.912501 + 0.409075i \(0.134148\pi\)
\(618\) −1.50798e7 −1.58827
\(619\) −1.09446e7 −1.14809 −0.574043 0.818825i \(-0.694626\pi\)
−0.574043 + 0.818825i \(0.694626\pi\)
\(620\) −4.66963e6 −0.487869
\(621\) 594214. 0.0618321
\(622\) −2.66872e7 −2.76584
\(623\) 211146. 0.0217953
\(624\) −264908. −0.0272354
\(625\) −6.67905e6 −0.683935
\(626\) −6.06844e6 −0.618930
\(627\) 6.54712e6 0.665091
\(628\) 5.91592e6 0.598581
\(629\) 4.26780e6 0.430108
\(630\) 1.14099e6 0.114533
\(631\) −1.74968e7 −1.74939 −0.874694 0.484676i \(-0.838937\pi\)
−0.874694 + 0.484676i \(0.838937\pi\)
\(632\) 1.06405e7 1.05967
\(633\) −5.05066e6 −0.501001
\(634\) −1.62212e7 −1.60273
\(635\) −3.66255e6 −0.360453
\(636\) 7.28444e6 0.714091
\(637\) 1.80202e6 0.175959
\(638\) 3.83128e7 3.72642
\(639\) 5.00919e6 0.485305
\(640\) 1.23840e7 1.19512
\(641\) −1.02308e6 −0.0983478 −0.0491739 0.998790i \(-0.515659\pi\)
−0.0491739 + 0.998790i \(0.515659\pi\)
\(642\) 1.52497e6 0.146024
\(643\) 1.59762e7 1.52387 0.761933 0.647655i \(-0.224250\pi\)
0.761933 + 0.647655i \(0.224250\pi\)
\(644\) 1.27446e6 0.121091
\(645\) 5.09402e6 0.482127
\(646\) −1.58302e7 −1.49247
\(647\) 1.48605e7 1.39564 0.697820 0.716273i \(-0.254153\pi\)
0.697820 + 0.716273i \(0.254153\pi\)
\(648\) −937743. −0.0877296
\(649\) 2.11384e6 0.196997
\(650\) −807239. −0.0749408
\(651\) −590499. −0.0546093
\(652\) −2.47883e7 −2.28364
\(653\) 1.12385e7 1.03140 0.515699 0.856770i \(-0.327532\pi\)
0.515699 + 0.856770i \(0.327532\pi\)
\(654\) 1.02232e6 0.0934638
\(655\) −1.35261e7 −1.23189
\(656\) −2.05156e6 −0.186134
\(657\) −563084. −0.0508932
\(658\) 646229. 0.0581864
\(659\) 1.66209e7 1.49088 0.745439 0.666574i \(-0.232240\pi\)
0.745439 + 0.666574i \(0.232240\pi\)
\(660\) 1.26751e7 1.13264
\(661\) 5.12621e6 0.456344 0.228172 0.973621i \(-0.426725\pi\)
0.228172 + 0.973621i \(0.426725\pi\)
\(662\) 2.60628e7 2.31141
\(663\) 1.52198e6 0.134470
\(664\) 9.26703e6 0.815681
\(665\) −1.88687e6 −0.165458
\(666\) −2.09234e6 −0.182787
\(667\) −5.75038e6 −0.500475
\(668\) 1.27210e7 1.10301
\(669\) 3.11450e6 0.269044
\(670\) 1.26882e7 1.09197
\(671\) 6.89605e6 0.591281
\(672\) 2.01589e6 0.172204
\(673\) 2.11025e6 0.179596 0.0897978 0.995960i \(-0.471378\pi\)
0.0897978 + 0.995960i \(0.471378\pi\)
\(674\) 1.72496e7 1.46261
\(675\) 574937. 0.0485692
\(676\) −1.71867e7 −1.44653
\(677\) 4.85784e6 0.407354 0.203677 0.979038i \(-0.434711\pi\)
0.203677 + 0.979038i \(0.434711\pi\)
\(678\) −7.24676e6 −0.605438
\(679\) −1.13289e6 −0.0943005
\(680\) −1.02078e7 −0.846568
\(681\) 624179. 0.0515753
\(682\) −1.09346e7 −0.900208
\(683\) −1.60898e7 −1.31977 −0.659887 0.751365i \(-0.729396\pi\)
−0.659887 + 0.751365i \(0.729396\pi\)
\(684\) 4.65585e6 0.380504
\(685\) −1.49516e7 −1.21748
\(686\) −9.48658e6 −0.769661
\(687\) −9.30552e6 −0.752227
\(688\) 3.01154e6 0.242559
\(689\) 1.93061e6 0.154934
\(690\) −3.17118e6 −0.253570
\(691\) 5.42126e6 0.431922 0.215961 0.976402i \(-0.430712\pi\)
0.215961 + 0.976402i \(0.430712\pi\)
\(692\) −2.74229e7 −2.17695
\(693\) 1.60283e6 0.126781
\(694\) 2.35617e7 1.85698
\(695\) 2.12877e7 1.67173
\(696\) 9.07481e6 0.710091
\(697\) 1.17869e7 0.919004
\(698\) 3.02721e6 0.235182
\(699\) −9.74140e6 −0.754099
\(700\) 1.23311e6 0.0951168
\(701\) 1.47704e7 1.13527 0.567633 0.823282i \(-0.307859\pi\)
0.567633 + 0.823282i \(0.307859\pi\)
\(702\) −746169. −0.0571471
\(703\) 3.46012e6 0.264060
\(704\) 3.23320e7 2.45867
\(705\) −964642. −0.0730960
\(706\) 1.11012e7 0.838223
\(707\) 339321. 0.0255306
\(708\) 1.50321e6 0.112704
\(709\) 2.41386e7 1.80341 0.901707 0.432347i \(-0.142314\pi\)
0.901707 + 0.432347i \(0.142314\pi\)
\(710\) −2.67328e7 −1.99021
\(711\) −6.03021e6 −0.447362
\(712\) 926108. 0.0684639
\(713\) 1.64118e6 0.120902
\(714\) −3.87548e6 −0.284499
\(715\) 3.35931e6 0.245745
\(716\) 3.37755e6 0.246218
\(717\) −4.73947e6 −0.344296
\(718\) 1.65697e7 1.19951
\(719\) 1.09977e7 0.793374 0.396687 0.917954i \(-0.370160\pi\)
0.396687 + 0.917954i \(0.370160\pi\)
\(720\) −1.00691e6 −0.0723868
\(721\) 6.10512e6 0.437377
\(722\) 9.30997e6 0.664669
\(723\) −9.87524e6 −0.702590
\(724\) −3.33425e6 −0.236403
\(725\) −5.56383e6 −0.393123
\(726\) 1.67178e7 1.17716
\(727\) −1.70462e7 −1.19616 −0.598082 0.801435i \(-0.704070\pi\)
−0.598082 + 0.801435i \(0.704070\pi\)
\(728\) −533045. −0.0372765
\(729\) 531441. 0.0370370
\(730\) 3.00504e6 0.208710
\(731\) −1.73023e7 −1.19760
\(732\) 4.90399e6 0.338276
\(733\) −337753. −0.0232188 −0.0116094 0.999933i \(-0.503695\pi\)
−0.0116094 + 0.999933i \(0.503695\pi\)
\(734\) −1.98317e7 −1.35869
\(735\) 6.84946e6 0.467668
\(736\) −5.60280e6 −0.381251
\(737\) 1.78240e7 1.20875
\(738\) −5.77865e6 −0.390558
\(739\) −805741. −0.0542731 −0.0271365 0.999632i \(-0.508639\pi\)
−0.0271365 + 0.999632i \(0.508639\pi\)
\(740\) 6.69875e6 0.449691
\(741\) 1.23395e6 0.0825565
\(742\) −4.91598e6 −0.327793
\(743\) −2.67091e7 −1.77495 −0.887477 0.460851i \(-0.847544\pi\)
−0.887477 + 0.460851i \(0.847544\pi\)
\(744\) −2.58999e6 −0.171540
\(745\) −1.70767e6 −0.112723
\(746\) 1.78217e7 1.17247
\(747\) −5.25185e6 −0.344358
\(748\) −4.30521e7 −2.81346
\(749\) −617389. −0.0402119
\(750\) −1.52261e7 −0.988406
\(751\) −3.82078e6 −0.247202 −0.123601 0.992332i \(-0.539444\pi\)
−0.123601 + 0.992332i \(0.539444\pi\)
\(752\) −570288. −0.0367747
\(753\) −6.56868e6 −0.422174
\(754\) 7.22089e6 0.462554
\(755\) −1.34245e7 −0.857096
\(756\) 1.13982e6 0.0725325
\(757\) 8.80899e6 0.558710 0.279355 0.960188i \(-0.409879\pi\)
0.279355 + 0.960188i \(0.409879\pi\)
\(758\) −9.33609e6 −0.590190
\(759\) −4.45478e6 −0.280686
\(760\) −8.27602e6 −0.519741
\(761\) 2.10564e6 0.131802 0.0659012 0.997826i \(-0.479008\pi\)
0.0659012 + 0.997826i \(0.479008\pi\)
\(762\) −6.09892e6 −0.380509
\(763\) −413891. −0.0257380
\(764\) −3.79491e7 −2.35216
\(765\) 5.78503e6 0.357398
\(766\) −9.57079e6 −0.589353
\(767\) 398399. 0.0244529
\(768\) 5.28801e6 0.323511
\(769\) −6.98302e6 −0.425821 −0.212911 0.977072i \(-0.568294\pi\)
−0.212911 + 0.977072i \(0.568294\pi\)
\(770\) −8.55393e6 −0.519923
\(771\) −6.50525e6 −0.394120
\(772\) −4.65884e7 −2.81342
\(773\) −1.43123e7 −0.861511 −0.430756 0.902469i \(-0.641753\pi\)
−0.430756 + 0.902469i \(0.641753\pi\)
\(774\) 8.48264e6 0.508954
\(775\) 1.58794e6 0.0949684
\(776\) −4.96897e6 −0.296219
\(777\) 847090. 0.0503358
\(778\) −1.50571e7 −0.891852
\(779\) 9.55623e6 0.564213
\(780\) 2.38890e6 0.140593
\(781\) −3.75534e7 −2.20304
\(782\) 1.07712e7 0.629864
\(783\) −5.14291e6 −0.299781
\(784\) 4.04934e6 0.235285
\(785\) 5.95958e6 0.345177
\(786\) −2.25239e7 −1.30043
\(787\) −1.83449e7 −1.05579 −0.527897 0.849308i \(-0.677019\pi\)
−0.527897 + 0.849308i \(0.677019\pi\)
\(788\) −2.97465e7 −1.70655
\(789\) 1.17604e7 0.672557
\(790\) 3.21818e7 1.83460
\(791\) 2.93388e6 0.166725
\(792\) 7.03018e6 0.398248
\(793\) 1.29971e6 0.0733946
\(794\) 7.26373e6 0.408892
\(795\) 7.33820e6 0.411786
\(796\) −8.41993e6 −0.471005
\(797\) −1.80668e7 −1.00748 −0.503740 0.863855i \(-0.668043\pi\)
−0.503740 + 0.863855i \(0.668043\pi\)
\(798\) −3.14205e6 −0.174665
\(799\) 3.27649e6 0.181569
\(800\) −5.42104e6 −0.299473
\(801\) −524847. −0.0289036
\(802\) 2.01663e7 1.10711
\(803\) 4.22139e6 0.231029
\(804\) 1.26752e7 0.691533
\(805\) 1.28386e6 0.0698279
\(806\) −2.06087e6 −0.111741
\(807\) −1.88872e7 −1.02090
\(808\) 1.48829e6 0.0801974
\(809\) 1.71475e7 0.921150 0.460575 0.887621i \(-0.347643\pi\)
0.460575 + 0.887621i \(0.347643\pi\)
\(810\) −2.83617e6 −0.151887
\(811\) −2.71687e7 −1.45050 −0.725249 0.688487i \(-0.758275\pi\)
−0.725249 + 0.688487i \(0.758275\pi\)
\(812\) −1.10304e7 −0.587085
\(813\) −9.43854e6 −0.500817
\(814\) 1.56861e7 0.829762
\(815\) −2.49712e7 −1.31688
\(816\) 3.42006e6 0.179808
\(817\) −1.40278e7 −0.735251
\(818\) −8.49383e6 −0.443834
\(819\) 302089. 0.0157371
\(820\) 1.85007e7 0.960846
\(821\) 2.36514e7 1.22462 0.612308 0.790619i \(-0.290241\pi\)
0.612308 + 0.790619i \(0.290241\pi\)
\(822\) −2.48976e7 −1.28522
\(823\) −3.65582e7 −1.88142 −0.940708 0.339216i \(-0.889838\pi\)
−0.940708 + 0.339216i \(0.889838\pi\)
\(824\) 2.67777e7 1.37390
\(825\) −4.31025e6 −0.220479
\(826\) −1.01446e6 −0.0517350
\(827\) 3.43682e7 1.74740 0.873702 0.486461i \(-0.161712\pi\)
0.873702 + 0.486461i \(0.161712\pi\)
\(828\) −3.16793e6 −0.160583
\(829\) −6.66101e6 −0.336631 −0.168315 0.985733i \(-0.553833\pi\)
−0.168315 + 0.985733i \(0.553833\pi\)
\(830\) 2.80278e7 1.41219
\(831\) 1.72344e6 0.0865751
\(832\) 6.09367e6 0.305190
\(833\) −2.32648e7 −1.16168
\(834\) 3.54486e7 1.76475
\(835\) 1.28149e7 0.636061
\(836\) −3.49045e7 −1.72729
\(837\) 1.46781e6 0.0724194
\(838\) 2.93513e6 0.144384
\(839\) 2.74300e7 1.34531 0.672654 0.739957i \(-0.265154\pi\)
0.672654 + 0.739957i \(0.265154\pi\)
\(840\) −2.02609e6 −0.0990743
\(841\) 2.92582e7 1.42645
\(842\) −5.49800e7 −2.67254
\(843\) 1.09631e7 0.531331
\(844\) 2.69265e7 1.30114
\(845\) −1.73136e7 −0.834151
\(846\) −1.60633e6 −0.0771632
\(847\) −6.76824e6 −0.324166
\(848\) 4.33828e6 0.207171
\(849\) −7.18530e6 −0.342118
\(850\) 1.04217e7 0.494758
\(851\) −2.35433e6 −0.111441
\(852\) −2.67054e7 −1.26038
\(853\) 3.05075e7 1.43560 0.717800 0.696249i \(-0.245149\pi\)
0.717800 + 0.696249i \(0.245149\pi\)
\(854\) −3.30950e6 −0.155281
\(855\) 4.69021e6 0.219421
\(856\) −2.70793e6 −0.126314
\(857\) 3.59631e7 1.67265 0.836324 0.548235i \(-0.184700\pi\)
0.836324 + 0.548235i \(0.184700\pi\)
\(858\) 5.59396e6 0.259419
\(859\) 2.39847e7 1.10905 0.554525 0.832167i \(-0.312900\pi\)
0.554525 + 0.832167i \(0.312900\pi\)
\(860\) −2.71577e7 −1.25212
\(861\) 2.33951e6 0.107552
\(862\) −2.93394e7 −1.34488
\(863\) −7.98871e6 −0.365132 −0.182566 0.983194i \(-0.558440\pi\)
−0.182566 + 0.983194i \(0.558440\pi\)
\(864\) −5.01092e6 −0.228367
\(865\) −2.76253e7 −1.25536
\(866\) 1.60862e7 0.728885
\(867\) −6.87063e6 −0.310420
\(868\) 3.14812e6 0.141825
\(869\) 4.52080e7 2.03079
\(870\) 2.74464e7 1.22939
\(871\) 3.35932e6 0.150039
\(872\) −1.81537e6 −0.0808487
\(873\) 2.81603e6 0.125055
\(874\) 8.73274e6 0.386698
\(875\) 6.16434e6 0.272186
\(876\) 3.00196e6 0.132174
\(877\) 1.96183e7 0.861316 0.430658 0.902515i \(-0.358281\pi\)
0.430658 + 0.902515i \(0.358281\pi\)
\(878\) −1.45459e7 −0.636800
\(879\) 9.53232e6 0.416128
\(880\) 7.54872e6 0.328599
\(881\) 2.55652e7 1.10971 0.554854 0.831947i \(-0.312774\pi\)
0.554854 + 0.831947i \(0.312774\pi\)
\(882\) 1.14058e7 0.493690
\(883\) 1.76039e7 0.759812 0.379906 0.925025i \(-0.375956\pi\)
0.379906 + 0.925025i \(0.375956\pi\)
\(884\) −8.11412e6 −0.349229
\(885\) 1.51431e6 0.0649914
\(886\) −3.42495e7 −1.46579
\(887\) 3.35376e7 1.43127 0.715637 0.698473i \(-0.246137\pi\)
0.715637 + 0.698473i \(0.246137\pi\)
\(888\) 3.71542e6 0.158116
\(889\) 2.46917e6 0.104784
\(890\) 2.80098e6 0.118532
\(891\) −3.98417e6 −0.168129
\(892\) −1.66043e7 −0.698729
\(893\) 2.65642e6 0.111472
\(894\) −2.84364e6 −0.118995
\(895\) 3.40248e6 0.141983
\(896\) −8.34892e6 −0.347424
\(897\) −839600. −0.0348411
\(898\) 3.98122e7 1.64750
\(899\) −1.42044e7 −0.586169
\(900\) −3.06515e6 −0.126138
\(901\) −2.49248e7 −1.02287
\(902\) 4.33221e7 1.77293
\(903\) −3.43423e6 −0.140155
\(904\) 1.28683e7 0.523720
\(905\) −3.35886e6 −0.136323
\(906\) −2.23546e7 −0.904787
\(907\) −4.58984e7 −1.85259 −0.926296 0.376798i \(-0.877025\pi\)
−0.926296 + 0.376798i \(0.877025\pi\)
\(908\) −3.32768e6 −0.133945
\(909\) −843451. −0.0338571
\(910\) −1.61218e6 −0.0645370
\(911\) −3.40938e7 −1.36107 −0.680534 0.732716i \(-0.738252\pi\)
−0.680534 + 0.732716i \(0.738252\pi\)
\(912\) 2.77281e6 0.110391
\(913\) 3.93727e7 1.56321
\(914\) 1.05429e7 0.417439
\(915\) 4.94018e6 0.195070
\(916\) 4.96103e7 1.95359
\(917\) 9.11889e6 0.358112
\(918\) 9.63330e6 0.377284
\(919\) −4.71301e7 −1.84081 −0.920406 0.390965i \(-0.872141\pi\)
−0.920406 + 0.390965i \(0.872141\pi\)
\(920\) 5.63115e6 0.219345
\(921\) −721065. −0.0280108
\(922\) −1.54916e7 −0.600162
\(923\) −7.07777e6 −0.273459
\(924\) −8.54516e6 −0.329261
\(925\) −2.27795e6 −0.0875367
\(926\) −1.21288e7 −0.464825
\(927\) −1.51755e7 −0.580022
\(928\) 4.84921e7 1.84842
\(929\) 3.41936e7 1.29989 0.649944 0.759982i \(-0.274792\pi\)
0.649944 + 0.759982i \(0.274792\pi\)
\(930\) −7.83332e6 −0.296988
\(931\) −1.88619e7 −0.713201
\(932\) 5.19342e7 1.95845
\(933\) −2.68566e7 −1.01006
\(934\) 3.21265e7 1.20503
\(935\) −4.33699e7 −1.62240
\(936\) 1.32499e6 0.0494338
\(937\) −4.54494e7 −1.69114 −0.845569 0.533866i \(-0.820739\pi\)
−0.845569 + 0.533866i \(0.820739\pi\)
\(938\) −8.55395e6 −0.317438
\(939\) −6.10695e6 −0.226027
\(940\) 5.14278e6 0.189836
\(941\) −5.33355e7 −1.96355 −0.981776 0.190043i \(-0.939137\pi\)
−0.981776 + 0.190043i \(0.939137\pi\)
\(942\) 9.92397e6 0.364383
\(943\) −6.50223e6 −0.238113
\(944\) 895245. 0.0326973
\(945\) 1.14823e6 0.0418265
\(946\) −6.35936e7 −2.31039
\(947\) 2.21988e7 0.804367 0.402183 0.915559i \(-0.368251\pi\)
0.402183 + 0.915559i \(0.368251\pi\)
\(948\) 3.21488e7 1.16183
\(949\) 795614. 0.0286772
\(950\) 8.44943e6 0.303752
\(951\) −1.63242e7 −0.585303
\(952\) 6.88180e6 0.246099
\(953\) −43754.2 −0.00156059 −0.000780293 1.00000i \(-0.500248\pi\)
−0.000780293 1.00000i \(0.500248\pi\)
\(954\) 1.22197e7 0.434699
\(955\) −3.82291e7 −1.35639
\(956\) 2.52675e7 0.894163
\(957\) 3.85559e7 1.36085
\(958\) 5.09871e7 1.79493
\(959\) 1.00799e7 0.353923
\(960\) 2.31619e7 0.811141
\(961\) −2.45752e7 −0.858397
\(962\) 2.95639e6 0.102997
\(963\) 1.53465e6 0.0533265
\(964\) 5.26477e7 1.82468
\(965\) −4.69322e7 −1.62238
\(966\) 2.13791e6 0.0737133
\(967\) 4.71228e7 1.62056 0.810280 0.586043i \(-0.199315\pi\)
0.810280 + 0.586043i \(0.199315\pi\)
\(968\) −2.96862e7 −1.01828
\(969\) −1.59307e7 −0.545037
\(970\) −1.50285e7 −0.512845
\(971\) −3.47681e7 −1.18340 −0.591701 0.806157i \(-0.701543\pi\)
−0.591701 + 0.806157i \(0.701543\pi\)
\(972\) −2.83326e6 −0.0961881
\(973\) −1.43515e7 −0.485976
\(974\) 2.07626e7 0.701267
\(975\) −812362. −0.0273677
\(976\) 2.92059e6 0.0981400
\(977\) −4.21133e7 −1.41151 −0.705753 0.708458i \(-0.749391\pi\)
−0.705753 + 0.708458i \(0.749391\pi\)
\(978\) −4.15825e7 −1.39015
\(979\) 3.93474e6 0.131208
\(980\) −3.65164e7 −1.21457
\(981\) 1.02881e6 0.0341321
\(982\) 6.50613e7 2.15300
\(983\) −1.65545e7 −0.546427 −0.273214 0.961953i \(-0.588087\pi\)
−0.273214 + 0.961953i \(0.588087\pi\)
\(984\) 1.02613e7 0.337843
\(985\) −2.99660e7 −0.984098
\(986\) −9.32242e7 −3.05377
\(987\) 650330. 0.0212491
\(988\) −6.57853e6 −0.214406
\(989\) 9.54480e6 0.310296
\(990\) 2.12625e7 0.689489
\(991\) 5.06855e6 0.163946 0.0819728 0.996635i \(-0.473878\pi\)
0.0819728 + 0.996635i \(0.473878\pi\)
\(992\) −1.38398e7 −0.446531
\(993\) 2.62283e7 0.844105
\(994\) 1.80224e7 0.578557
\(995\) −8.48207e6 −0.271609
\(996\) 2.79991e7 0.894325
\(997\) 3.70847e6 0.118156 0.0590782 0.998253i \(-0.481184\pi\)
0.0590782 + 0.998253i \(0.481184\pi\)
\(998\) −1.90913e7 −0.606749
\(999\) −2.10562e6 −0.0667522
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 177.6.a.b.1.2 12
3.2 odd 2 531.6.a.d.1.11 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.6.a.b.1.2 12 1.1 even 1 trivial
531.6.a.d.1.11 12 3.2 odd 2