Properties

Label 546.8.a.l.1.2
Level $546$
Weight $8$
Character 546.1
Self dual yes
Analytic conductor $170.562$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(170.562223914\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \( x^{5} - 2x^{4} - 122890x^{3} - 6160660x^{2} + 3465881625x + 278845474950 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-231.824\) of defining polynomial
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -223.551 q^{5} -216.000 q^{6} -343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -223.551 q^{5} -216.000 q^{6} -343.000 q^{7} +512.000 q^{8} +729.000 q^{9} -1788.41 q^{10} +3707.45 q^{11} -1728.00 q^{12} -2197.00 q^{13} -2744.00 q^{14} +6035.88 q^{15} +4096.00 q^{16} -5322.98 q^{17} +5832.00 q^{18} -847.191 q^{19} -14307.3 q^{20} +9261.00 q^{21} +29659.6 q^{22} -6779.82 q^{23} -13824.0 q^{24} -28150.0 q^{25} -17576.0 q^{26} -19683.0 q^{27} -21952.0 q^{28} +114612. q^{29} +48287.0 q^{30} -107586. q^{31} +32768.0 q^{32} -100101. q^{33} -42583.9 q^{34} +76678.0 q^{35} +46656.0 q^{36} +500720. q^{37} -6777.53 q^{38} +59319.0 q^{39} -114458. q^{40} -193490. q^{41} +74088.0 q^{42} +528743. q^{43} +237277. q^{44} -162969. q^{45} -54238.6 q^{46} -270082. q^{47} -110592. q^{48} +117649. q^{49} -225200. q^{50} +143721. q^{51} -140608. q^{52} +166841. q^{53} -157464. q^{54} -828803. q^{55} -175616. q^{56} +22874.2 q^{57} +916897. q^{58} +2.73954e6 q^{59} +386296. q^{60} -6760.78 q^{61} -860691. q^{62} -250047. q^{63} +262144. q^{64} +491142. q^{65} -800808. q^{66} +1.33318e6 q^{67} -340671. q^{68} +183055. q^{69} +613424. q^{70} -1.97912e6 q^{71} +373248. q^{72} -1.51276e6 q^{73} +4.00576e6 q^{74} +760049. q^{75} -54220.2 q^{76} -1.27165e6 q^{77} +474552. q^{78} +1.03335e6 q^{79} -915665. q^{80} +531441. q^{81} -1.54792e6 q^{82} -6.46458e6 q^{83} +592704. q^{84} +1.18996e6 q^{85} +4.22994e6 q^{86} -3.09453e6 q^{87} +1.89821e6 q^{88} -1.08910e7 q^{89} -1.30375e6 q^{90} +753571. q^{91} -433909. q^{92} +2.90483e6 q^{93} -2.16065e6 q^{94} +189390. q^{95} -884736. q^{96} +7.47976e6 q^{97} +941192. q^{98} +2.70273e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 40 q^{2} - 135 q^{3} + 320 q^{4} - 250 q^{5} - 1080 q^{6} - 1715 q^{7} + 2560 q^{8} + 3645 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 40 q^{2} - 135 q^{3} + 320 q^{4} - 250 q^{5} - 1080 q^{6} - 1715 q^{7} + 2560 q^{8} + 3645 q^{9} - 2000 q^{10} + 659 q^{11} - 8640 q^{12} - 10985 q^{13} - 13720 q^{14} + 6750 q^{15} + 20480 q^{16} + 24575 q^{17} + 29160 q^{18} - 6446 q^{19} - 16000 q^{20} + 46305 q^{21} + 5272 q^{22} + 30268 q^{23} - 69120 q^{24} + 38965 q^{25} - 87880 q^{26} - 98415 q^{27} - 109760 q^{28} + 130950 q^{29} + 54000 q^{30} + 262979 q^{31} + 163840 q^{32} - 17793 q^{33} + 196600 q^{34} + 85750 q^{35} + 233280 q^{36} - 101549 q^{37} - 51568 q^{38} + 296595 q^{39} - 128000 q^{40} - 247328 q^{41} + 370440 q^{42} - 19092 q^{43} + 42176 q^{44} - 182250 q^{45} + 242144 q^{46} - 126419 q^{47} - 552960 q^{48} + 588245 q^{49} + 311720 q^{50} - 663525 q^{51} - 703040 q^{52} - 302793 q^{53} - 787320 q^{54} + 943985 q^{55} - 878080 q^{56} + 174042 q^{57} + 1047600 q^{58} - 2798636 q^{59} + 432000 q^{60} - 2493751 q^{61} + 2103832 q^{62} - 1250235 q^{63} + 1310720 q^{64} + 549250 q^{65} - 142344 q^{66} + 160188 q^{67} + 1572800 q^{68} - 817236 q^{69} + 686000 q^{70} + 3846088 q^{71} + 1866240 q^{72} + 5655872 q^{73} - 812392 q^{74} - 1052055 q^{75} - 412544 q^{76} - 226037 q^{77} + 2372760 q^{78} + 5647991 q^{79} - 1024000 q^{80} + 2657205 q^{81} - 1978624 q^{82} - 4607669 q^{83} + 2963520 q^{84} + 3873935 q^{85} - 152736 q^{86} - 3535650 q^{87} + 337408 q^{88} - 17424029 q^{89} - 1458000 q^{90} + 3767855 q^{91} + 1937152 q^{92} - 7100433 q^{93} - 1011352 q^{94} - 24593720 q^{95} - 4423680 q^{96} - 18380577 q^{97} + 4705960 q^{98} + 480411 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) −27.0000 −0.577350
\(4\) 64.0000 0.500000
\(5\) −223.551 −0.799800 −0.399900 0.916559i \(-0.630955\pi\)
−0.399900 + 0.916559i \(0.630955\pi\)
\(6\) −216.000 −0.408248
\(7\) −343.000 −0.377964
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) −1788.41 −0.565544
\(11\) 3707.45 0.839848 0.419924 0.907559i \(-0.362057\pi\)
0.419924 + 0.907559i \(0.362057\pi\)
\(12\) −1728.00 −0.288675
\(13\) −2197.00 −0.277350
\(14\) −2744.00 −0.267261
\(15\) 6035.88 0.461765
\(16\) 4096.00 0.250000
\(17\) −5322.98 −0.262775 −0.131387 0.991331i \(-0.541943\pi\)
−0.131387 + 0.991331i \(0.541943\pi\)
\(18\) 5832.00 0.235702
\(19\) −847.191 −0.0283363 −0.0141682 0.999900i \(-0.504510\pi\)
−0.0141682 + 0.999900i \(0.504510\pi\)
\(20\) −14307.3 −0.399900
\(21\) 9261.00 0.218218
\(22\) 29659.6 0.593862
\(23\) −6779.82 −0.116191 −0.0580953 0.998311i \(-0.518503\pi\)
−0.0580953 + 0.998311i \(0.518503\pi\)
\(24\) −13824.0 −0.204124
\(25\) −28150.0 −0.360319
\(26\) −17576.0 −0.196116
\(27\) −19683.0 −0.192450
\(28\) −21952.0 −0.188982
\(29\) 114612. 0.872645 0.436322 0.899790i \(-0.356281\pi\)
0.436322 + 0.899790i \(0.356281\pi\)
\(30\) 48287.0 0.326517
\(31\) −107586. −0.648621 −0.324311 0.945951i \(-0.605132\pi\)
−0.324311 + 0.945951i \(0.605132\pi\)
\(32\) 32768.0 0.176777
\(33\) −100101. −0.484886
\(34\) −42583.9 −0.185810
\(35\) 76678.0 0.302296
\(36\) 46656.0 0.166667
\(37\) 500720. 1.62513 0.812567 0.582868i \(-0.198070\pi\)
0.812567 + 0.582868i \(0.198070\pi\)
\(38\) −6777.53 −0.0200368
\(39\) 59319.0 0.160128
\(40\) −114458. −0.282772
\(41\) −193490. −0.438446 −0.219223 0.975675i \(-0.570352\pi\)
−0.219223 + 0.975675i \(0.570352\pi\)
\(42\) 74088.0 0.154303
\(43\) 528743. 1.01416 0.507078 0.861900i \(-0.330726\pi\)
0.507078 + 0.861900i \(0.330726\pi\)
\(44\) 237277. 0.419924
\(45\) −162969. −0.266600
\(46\) −54238.6 −0.0821592
\(47\) −270082. −0.379448 −0.189724 0.981837i \(-0.560759\pi\)
−0.189724 + 0.981837i \(0.560759\pi\)
\(48\) −110592. −0.144338
\(49\) 117649. 0.142857
\(50\) −225200. −0.254784
\(51\) 143721. 0.151713
\(52\) −140608. −0.138675
\(53\) 166841. 0.153935 0.0769674 0.997034i \(-0.475476\pi\)
0.0769674 + 0.997034i \(0.475476\pi\)
\(54\) −157464. −0.136083
\(55\) −828803. −0.671710
\(56\) −175616. −0.133631
\(57\) 22874.2 0.0163600
\(58\) 916897. 0.617053
\(59\) 2.73954e6 1.73659 0.868293 0.496052i \(-0.165218\pi\)
0.868293 + 0.496052i \(0.165218\pi\)
\(60\) 386296. 0.230882
\(61\) −6760.78 −0.00381366 −0.00190683 0.999998i \(-0.500607\pi\)
−0.00190683 + 0.999998i \(0.500607\pi\)
\(62\) −860691. −0.458645
\(63\) −250047. −0.125988
\(64\) 262144. 0.125000
\(65\) 491142. 0.221825
\(66\) −800808. −0.342866
\(67\) 1.33318e6 0.541535 0.270767 0.962645i \(-0.412723\pi\)
0.270767 + 0.962645i \(0.412723\pi\)
\(68\) −340671. −0.131387
\(69\) 183055. 0.0670827
\(70\) 613424. 0.213756
\(71\) −1.97912e6 −0.656249 −0.328125 0.944634i \(-0.606417\pi\)
−0.328125 + 0.944634i \(0.606417\pi\)
\(72\) 373248. 0.117851
\(73\) −1.51276e6 −0.455135 −0.227568 0.973762i \(-0.573077\pi\)
−0.227568 + 0.973762i \(0.573077\pi\)
\(74\) 4.00576e6 1.14914
\(75\) 760049. 0.208031
\(76\) −54220.2 −0.0141682
\(77\) −1.27165e6 −0.317433
\(78\) 474552. 0.113228
\(79\) 1.03335e6 0.235805 0.117902 0.993025i \(-0.462383\pi\)
0.117902 + 0.993025i \(0.462383\pi\)
\(80\) −915665. −0.199950
\(81\) 531441. 0.111111
\(82\) −1.54792e6 −0.310028
\(83\) −6.46458e6 −1.24099 −0.620493 0.784212i \(-0.713068\pi\)
−0.620493 + 0.784212i \(0.713068\pi\)
\(84\) 592704. 0.109109
\(85\) 1.18996e6 0.210167
\(86\) 4.22994e6 0.717117
\(87\) −3.09453e6 −0.503822
\(88\) 1.89821e6 0.296931
\(89\) −1.08910e7 −1.63758 −0.818791 0.574092i \(-0.805355\pi\)
−0.818791 + 0.574092i \(0.805355\pi\)
\(90\) −1.30375e6 −0.188515
\(91\) 753571. 0.104828
\(92\) −433909. −0.0580953
\(93\) 2.90483e6 0.374482
\(94\) −2.16065e6 −0.268310
\(95\) 189390. 0.0226634
\(96\) −884736. −0.102062
\(97\) 7.47976e6 0.832122 0.416061 0.909337i \(-0.363410\pi\)
0.416061 + 0.909337i \(0.363410\pi\)
\(98\) 941192. 0.101015
\(99\) 2.70273e6 0.279949
\(100\) −1.80160e6 −0.180160
\(101\) −6.63393e6 −0.640687 −0.320343 0.947301i \(-0.603798\pi\)
−0.320343 + 0.947301i \(0.603798\pi\)
\(102\) 1.14976e6 0.107277
\(103\) 9.42132e6 0.849535 0.424768 0.905302i \(-0.360356\pi\)
0.424768 + 0.905302i \(0.360356\pi\)
\(104\) −1.12486e6 −0.0980581
\(105\) −2.07031e6 −0.174531
\(106\) 1.33473e6 0.108848
\(107\) −2.00792e7 −1.58454 −0.792270 0.610170i \(-0.791101\pi\)
−0.792270 + 0.610170i \(0.791101\pi\)
\(108\) −1.25971e6 −0.0962250
\(109\) −1.99527e7 −1.47573 −0.737867 0.674947i \(-0.764167\pi\)
−0.737867 + 0.674947i \(0.764167\pi\)
\(110\) −6.63043e6 −0.474971
\(111\) −1.35194e7 −0.938271
\(112\) −1.40493e6 −0.0944911
\(113\) −1.26125e7 −0.822295 −0.411148 0.911569i \(-0.634872\pi\)
−0.411148 + 0.911569i \(0.634872\pi\)
\(114\) 182993. 0.0115683
\(115\) 1.51564e6 0.0929293
\(116\) 7.33517e6 0.436322
\(117\) −1.60161e6 −0.0924500
\(118\) 2.19163e7 1.22795
\(119\) 1.82578e6 0.0993196
\(120\) 3.09037e6 0.163259
\(121\) −5.74201e6 −0.294656
\(122\) −54086.3 −0.00269667
\(123\) 5.22424e6 0.253137
\(124\) −6.88553e6 −0.324311
\(125\) 2.37579e7 1.08798
\(126\) −2.00038e6 −0.0890871
\(127\) −3.28621e7 −1.42358 −0.711790 0.702392i \(-0.752115\pi\)
−0.711790 + 0.702392i \(0.752115\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) −1.42761e7 −0.585523
\(130\) 3.92913e6 0.156854
\(131\) −1.60922e7 −0.625413 −0.312707 0.949850i \(-0.601236\pi\)
−0.312707 + 0.949850i \(0.601236\pi\)
\(132\) −6.40647e6 −0.242443
\(133\) 290587. 0.0107101
\(134\) 1.06654e7 0.382923
\(135\) 4.40015e6 0.153922
\(136\) −2.72537e6 −0.0929050
\(137\) 4.48542e6 0.149032 0.0745162 0.997220i \(-0.476259\pi\)
0.0745162 + 0.997220i \(0.476259\pi\)
\(138\) 1.46444e6 0.0474346
\(139\) −3.78403e7 −1.19510 −0.597548 0.801833i \(-0.703858\pi\)
−0.597548 + 0.801833i \(0.703858\pi\)
\(140\) 4.90739e6 0.151148
\(141\) 7.29220e6 0.219074
\(142\) −1.58330e7 −0.464038
\(143\) −8.14526e6 −0.232932
\(144\) 2.98598e6 0.0833333
\(145\) −2.56216e7 −0.697942
\(146\) −1.21021e7 −0.321829
\(147\) −3.17652e6 −0.0824786
\(148\) 3.20461e7 0.812567
\(149\) 6.07663e7 1.50491 0.752456 0.658643i \(-0.228869\pi\)
0.752456 + 0.658643i \(0.228869\pi\)
\(150\) 6.08039e6 0.147100
\(151\) −3.29904e7 −0.779773 −0.389886 0.920863i \(-0.627486\pi\)
−0.389886 + 0.920863i \(0.627486\pi\)
\(152\) −433762. −0.0100184
\(153\) −3.88046e6 −0.0875916
\(154\) −1.01732e7 −0.224459
\(155\) 2.40510e7 0.518768
\(156\) 3.79642e6 0.0800641
\(157\) −2.58988e7 −0.534110 −0.267055 0.963681i \(-0.586051\pi\)
−0.267055 + 0.963681i \(0.586051\pi\)
\(158\) 8.26680e6 0.166739
\(159\) −4.50471e6 −0.0888744
\(160\) −7.32532e6 −0.141386
\(161\) 2.32548e6 0.0439159
\(162\) 4.25153e6 0.0785674
\(163\) −4.88552e6 −0.0883596 −0.0441798 0.999024i \(-0.514067\pi\)
−0.0441798 + 0.999024i \(0.514067\pi\)
\(164\) −1.23834e7 −0.219223
\(165\) 2.23777e7 0.387812
\(166\) −5.17167e7 −0.877510
\(167\) −8.22156e7 −1.36599 −0.682993 0.730425i \(-0.739322\pi\)
−0.682993 + 0.730425i \(0.739322\pi\)
\(168\) 4.74163e6 0.0771517
\(169\) 4.82681e6 0.0769231
\(170\) 9.51967e6 0.148611
\(171\) −617602. −0.00944545
\(172\) 3.38395e7 0.507078
\(173\) 6.15776e7 0.904194 0.452097 0.891969i \(-0.350676\pi\)
0.452097 + 0.891969i \(0.350676\pi\)
\(174\) −2.47562e7 −0.356256
\(175\) 9.65543e6 0.136188
\(176\) 1.51857e7 0.209962
\(177\) −7.39677e7 −1.00262
\(178\) −8.71281e7 −1.15795
\(179\) 8.02038e7 1.04522 0.522612 0.852571i \(-0.324958\pi\)
0.522612 + 0.852571i \(0.324958\pi\)
\(180\) −1.04300e7 −0.133300
\(181\) 4.88590e7 0.612449 0.306224 0.951959i \(-0.400934\pi\)
0.306224 + 0.951959i \(0.400934\pi\)
\(182\) 6.02857e6 0.0741249
\(183\) 182541. 0.00220182
\(184\) −3.47127e6 −0.0410796
\(185\) −1.11936e8 −1.29978
\(186\) 2.32387e7 0.264799
\(187\) −1.97347e7 −0.220691
\(188\) −1.72852e7 −0.189724
\(189\) 6.75127e6 0.0727393
\(190\) 1.51512e6 0.0160255
\(191\) −8.29971e7 −0.861879 −0.430939 0.902381i \(-0.641818\pi\)
−0.430939 + 0.902381i \(0.641818\pi\)
\(192\) −7.07789e6 −0.0721688
\(193\) −6.75634e7 −0.676489 −0.338245 0.941058i \(-0.609833\pi\)
−0.338245 + 0.941058i \(0.609833\pi\)
\(194\) 5.98381e7 0.588399
\(195\) −1.32608e7 −0.128071
\(196\) 7.52954e6 0.0714286
\(197\) 2.08766e7 0.194548 0.0972742 0.995258i \(-0.468988\pi\)
0.0972742 + 0.995258i \(0.468988\pi\)
\(198\) 2.16218e7 0.197954
\(199\) −6.60055e7 −0.593737 −0.296869 0.954918i \(-0.595942\pi\)
−0.296869 + 0.954918i \(0.595942\pi\)
\(200\) −1.44128e7 −0.127392
\(201\) −3.59958e7 −0.312655
\(202\) −5.30714e7 −0.453034
\(203\) −3.93119e7 −0.329829
\(204\) 9.19812e6 0.0758566
\(205\) 4.32550e7 0.350669
\(206\) 7.53706e7 0.600712
\(207\) −4.94249e6 −0.0387302
\(208\) −8.99891e6 −0.0693375
\(209\) −3.14092e6 −0.0237982
\(210\) −1.65624e7 −0.123412
\(211\) 3.57733e7 0.262162 0.131081 0.991372i \(-0.458155\pi\)
0.131081 + 0.991372i \(0.458155\pi\)
\(212\) 1.06778e7 0.0769674
\(213\) 5.34364e7 0.378886
\(214\) −1.60634e8 −1.12044
\(215\) −1.18201e8 −0.811123
\(216\) −1.00777e7 −0.0680414
\(217\) 3.69021e7 0.245156
\(218\) −1.59621e8 −1.04350
\(219\) 4.08446e7 0.262772
\(220\) −5.30434e7 −0.335855
\(221\) 1.16946e7 0.0728807
\(222\) −1.08156e8 −0.663458
\(223\) −2.51998e8 −1.52170 −0.760852 0.648925i \(-0.775219\pi\)
−0.760852 + 0.648925i \(0.775219\pi\)
\(224\) −1.12394e7 −0.0668153
\(225\) −2.05213e7 −0.120106
\(226\) −1.00900e8 −0.581451
\(227\) −9.82486e7 −0.557488 −0.278744 0.960365i \(-0.589918\pi\)
−0.278744 + 0.960365i \(0.589918\pi\)
\(228\) 1.46395e6 0.00818000
\(229\) −2.73707e8 −1.50613 −0.753063 0.657949i \(-0.771424\pi\)
−0.753063 + 0.657949i \(0.771424\pi\)
\(230\) 1.21251e7 0.0657109
\(231\) 3.43347e7 0.183270
\(232\) 5.86814e7 0.308526
\(233\) −4.03236e7 −0.208840 −0.104420 0.994533i \(-0.533299\pi\)
−0.104420 + 0.994533i \(0.533299\pi\)
\(234\) −1.28129e7 −0.0653720
\(235\) 6.03770e7 0.303483
\(236\) 1.75331e8 0.868293
\(237\) −2.79005e7 −0.136142
\(238\) 1.46063e7 0.0702296
\(239\) 2.14738e7 0.101746 0.0508729 0.998705i \(-0.483800\pi\)
0.0508729 + 0.998705i \(0.483800\pi\)
\(240\) 2.47230e7 0.115441
\(241\) −9.25323e7 −0.425827 −0.212914 0.977071i \(-0.568295\pi\)
−0.212914 + 0.977071i \(0.568295\pi\)
\(242\) −4.59361e7 −0.208353
\(243\) −1.43489e7 −0.0641500
\(244\) −432690. −0.00190683
\(245\) −2.63006e7 −0.114257
\(246\) 4.17939e7 0.178995
\(247\) 1.86128e6 0.00785909
\(248\) −5.50842e7 −0.229322
\(249\) 1.74544e8 0.716484
\(250\) 1.90063e8 0.769321
\(251\) 8.47223e7 0.338174 0.169087 0.985601i \(-0.445918\pi\)
0.169087 + 0.985601i \(0.445918\pi\)
\(252\) −1.60030e7 −0.0629941
\(253\) −2.51358e7 −0.0975824
\(254\) −2.62897e8 −1.00662
\(255\) −3.21289e7 −0.121340
\(256\) 1.67772e7 0.0625000
\(257\) −2.14537e8 −0.788380 −0.394190 0.919029i \(-0.628975\pi\)
−0.394190 + 0.919029i \(0.628975\pi\)
\(258\) −1.14208e8 −0.414028
\(259\) −1.71747e8 −0.614243
\(260\) 3.14331e7 0.110912
\(261\) 8.35522e7 0.290882
\(262\) −1.28738e8 −0.442234
\(263\) −1.16062e6 −0.00393409 −0.00196705 0.999998i \(-0.500626\pi\)
−0.00196705 + 0.999998i \(0.500626\pi\)
\(264\) −5.12517e7 −0.171433
\(265\) −3.72975e7 −0.123117
\(266\) 2.32469e6 0.00757321
\(267\) 2.94057e8 0.945459
\(268\) 8.53234e7 0.270767
\(269\) 1.37359e8 0.430253 0.215126 0.976586i \(-0.430984\pi\)
0.215126 + 0.976586i \(0.430984\pi\)
\(270\) 3.52012e7 0.108839
\(271\) −1.44269e8 −0.440332 −0.220166 0.975462i \(-0.570660\pi\)
−0.220166 + 0.975462i \(0.570660\pi\)
\(272\) −2.18029e7 −0.0656937
\(273\) −2.03464e7 −0.0605228
\(274\) 3.58833e7 0.105382
\(275\) −1.04364e8 −0.302613
\(276\) 1.17155e7 0.0335413
\(277\) −1.86550e8 −0.527371 −0.263685 0.964609i \(-0.584938\pi\)
−0.263685 + 0.964609i \(0.584938\pi\)
\(278\) −3.02722e8 −0.845060
\(279\) −7.84304e7 −0.216207
\(280\) 3.92591e7 0.106878
\(281\) −1.43449e8 −0.385680 −0.192840 0.981230i \(-0.561770\pi\)
−0.192840 + 0.981230i \(0.561770\pi\)
\(282\) 5.83376e7 0.154909
\(283\) 1.55861e8 0.408775 0.204388 0.978890i \(-0.434480\pi\)
0.204388 + 0.978890i \(0.434480\pi\)
\(284\) −1.26664e8 −0.328125
\(285\) −5.11354e6 −0.0130847
\(286\) −6.51621e7 −0.164708
\(287\) 6.63672e7 0.165717
\(288\) 2.38879e7 0.0589256
\(289\) −3.82005e8 −0.930949
\(290\) −2.04973e8 −0.493519
\(291\) −2.01954e8 −0.480426
\(292\) −9.68167e7 −0.227568
\(293\) −2.48565e8 −0.577302 −0.288651 0.957434i \(-0.593207\pi\)
−0.288651 + 0.957434i \(0.593207\pi\)
\(294\) −2.54122e7 −0.0583212
\(295\) −6.12428e8 −1.38892
\(296\) 2.56369e8 0.574571
\(297\) −7.29737e7 −0.161629
\(298\) 4.86131e8 1.06413
\(299\) 1.48953e7 0.0322255
\(300\) 4.86431e7 0.104015
\(301\) −1.81359e8 −0.383315
\(302\) −2.63923e8 −0.551383
\(303\) 1.79116e8 0.369901
\(304\) −3.47009e6 −0.00708409
\(305\) 1.51138e6 0.00305017
\(306\) −3.10436e7 −0.0619366
\(307\) 3.67829e8 0.725540 0.362770 0.931879i \(-0.381831\pi\)
0.362770 + 0.931879i \(0.381831\pi\)
\(308\) −8.13859e7 −0.158716
\(309\) −2.54376e8 −0.490479
\(310\) 1.92408e8 0.366824
\(311\) 4.48616e7 0.0845694 0.0422847 0.999106i \(-0.486536\pi\)
0.0422847 + 0.999106i \(0.486536\pi\)
\(312\) 3.03713e7 0.0566139
\(313\) 3.44280e8 0.634610 0.317305 0.948324i \(-0.397222\pi\)
0.317305 + 0.948324i \(0.397222\pi\)
\(314\) −2.07190e8 −0.377673
\(315\) 5.58983e7 0.100765
\(316\) 6.61344e7 0.117902
\(317\) −2.45491e8 −0.432840 −0.216420 0.976300i \(-0.569438\pi\)
−0.216420 + 0.976300i \(0.569438\pi\)
\(318\) −3.60376e7 −0.0628437
\(319\) 4.24918e8 0.732888
\(320\) −5.86026e7 −0.0999750
\(321\) 5.42139e8 0.914835
\(322\) 1.86038e7 0.0310532
\(323\) 4.50959e6 0.00744608
\(324\) 3.40122e7 0.0555556
\(325\) 6.18455e7 0.0999346
\(326\) −3.90841e7 −0.0624797
\(327\) 5.38722e8 0.852015
\(328\) −9.90671e7 −0.155014
\(329\) 9.26380e7 0.143418
\(330\) 1.79022e8 0.274225
\(331\) 3.78784e8 0.574108 0.287054 0.957914i \(-0.407324\pi\)
0.287054 + 0.957914i \(0.407324\pi\)
\(332\) −4.13733e8 −0.620493
\(333\) 3.65025e8 0.541711
\(334\) −6.57725e8 −0.965898
\(335\) −2.98033e8 −0.433120
\(336\) 3.79331e7 0.0545545
\(337\) 1.21640e9 1.73130 0.865648 0.500654i \(-0.166907\pi\)
0.865648 + 0.500654i \(0.166907\pi\)
\(338\) 3.86145e7 0.0543928
\(339\) 3.40538e8 0.474752
\(340\) 7.61573e7 0.105084
\(341\) −3.98871e8 −0.544743
\(342\) −4.94082e6 −0.00667894
\(343\) −4.03536e7 −0.0539949
\(344\) 2.70716e8 0.358558
\(345\) −4.09222e7 −0.0536527
\(346\) 4.92621e8 0.639362
\(347\) 5.22321e8 0.671096 0.335548 0.942023i \(-0.391079\pi\)
0.335548 + 0.942023i \(0.391079\pi\)
\(348\) −1.98050e8 −0.251911
\(349\) 3.39559e8 0.427588 0.213794 0.976879i \(-0.431418\pi\)
0.213794 + 0.976879i \(0.431418\pi\)
\(350\) 7.72435e7 0.0962994
\(351\) 4.32436e7 0.0533761
\(352\) 1.21486e8 0.148465
\(353\) 5.05955e8 0.612209 0.306105 0.951998i \(-0.400974\pi\)
0.306105 + 0.951998i \(0.400974\pi\)
\(354\) −5.91741e8 −0.708958
\(355\) 4.42435e8 0.524869
\(356\) −6.97025e8 −0.818791
\(357\) −4.92962e7 −0.0573422
\(358\) 6.41630e8 0.739085
\(359\) −1.16756e8 −0.133183 −0.0665913 0.997780i \(-0.521212\pi\)
−0.0665913 + 0.997780i \(0.521212\pi\)
\(360\) −8.34400e7 −0.0942574
\(361\) −8.93154e8 −0.999197
\(362\) 3.90872e8 0.433066
\(363\) 1.55034e8 0.170120
\(364\) 4.82285e7 0.0524142
\(365\) 3.38179e8 0.364017
\(366\) 1.46033e6 0.00155692
\(367\) −1.43967e9 −1.52031 −0.760157 0.649740i \(-0.774878\pi\)
−0.760157 + 0.649740i \(0.774878\pi\)
\(368\) −2.77702e7 −0.0290476
\(369\) −1.41054e8 −0.146149
\(370\) −8.95492e8 −0.919085
\(371\) −5.72264e7 −0.0581819
\(372\) 1.85909e8 0.187241
\(373\) −1.15250e8 −0.114990 −0.0574950 0.998346i \(-0.518311\pi\)
−0.0574950 + 0.998346i \(0.518311\pi\)
\(374\) −1.57877e8 −0.156052
\(375\) −6.41463e8 −0.628148
\(376\) −1.38282e8 −0.134155
\(377\) −2.51803e8 −0.242028
\(378\) 5.40102e7 0.0514344
\(379\) 1.15820e8 0.109282 0.0546408 0.998506i \(-0.482599\pi\)
0.0546408 + 0.998506i \(0.482599\pi\)
\(380\) 1.21210e7 0.0113317
\(381\) 8.87276e8 0.821905
\(382\) −6.63977e8 −0.609440
\(383\) −6.45665e8 −0.587234 −0.293617 0.955923i \(-0.594859\pi\)
−0.293617 + 0.955923i \(0.594859\pi\)
\(384\) −5.66231e7 −0.0510310
\(385\) 2.84280e8 0.253883
\(386\) −5.40507e8 −0.478350
\(387\) 3.85454e8 0.338052
\(388\) 4.78705e8 0.416061
\(389\) −3.15611e8 −0.271850 −0.135925 0.990719i \(-0.543401\pi\)
−0.135925 + 0.990719i \(0.543401\pi\)
\(390\) −1.06087e8 −0.0905596
\(391\) 3.60889e7 0.0305320
\(392\) 6.02363e7 0.0505076
\(393\) 4.34491e8 0.361083
\(394\) 1.67013e8 0.137566
\(395\) −2.31006e8 −0.188597
\(396\) 1.72975e8 0.139975
\(397\) −5.55156e8 −0.445295 −0.222648 0.974899i \(-0.571470\pi\)
−0.222648 + 0.974899i \(0.571470\pi\)
\(398\) −5.28044e8 −0.419836
\(399\) −7.84584e6 −0.00618350
\(400\) −1.15302e8 −0.0900799
\(401\) −2.99200e8 −0.231716 −0.115858 0.993266i \(-0.536962\pi\)
−0.115858 + 0.993266i \(0.536962\pi\)
\(402\) −2.87967e8 −0.221081
\(403\) 2.36367e8 0.179895
\(404\) −4.24571e8 −0.320343
\(405\) −1.18804e8 −0.0888667
\(406\) −3.14496e8 −0.233224
\(407\) 1.85639e9 1.36486
\(408\) 7.35849e7 0.0536387
\(409\) −3.78639e8 −0.273649 −0.136824 0.990595i \(-0.543690\pi\)
−0.136824 + 0.990595i \(0.543690\pi\)
\(410\) 3.46040e8 0.247960
\(411\) −1.21106e8 −0.0860439
\(412\) 6.02964e8 0.424768
\(413\) −9.39663e8 −0.656368
\(414\) −3.95399e7 −0.0273864
\(415\) 1.44516e9 0.992542
\(416\) −7.19913e7 −0.0490290
\(417\) 1.02169e9 0.689989
\(418\) −2.51273e7 −0.0168279
\(419\) 6.14549e8 0.408139 0.204069 0.978956i \(-0.434583\pi\)
0.204069 + 0.978956i \(0.434583\pi\)
\(420\) −1.32500e8 −0.0872654
\(421\) −1.34731e9 −0.879996 −0.439998 0.897999i \(-0.645021\pi\)
−0.439998 + 0.897999i \(0.645021\pi\)
\(422\) 2.86186e8 0.185377
\(423\) −1.96889e8 −0.126483
\(424\) 8.54226e7 0.0544242
\(425\) 1.49842e8 0.0946829
\(426\) 4.27491e8 0.267913
\(427\) 2.31895e6 0.00144143
\(428\) −1.28507e9 −0.792270
\(429\) 2.19922e8 0.134483
\(430\) −9.45608e8 −0.573550
\(431\) −1.75053e9 −1.05317 −0.526585 0.850122i \(-0.676528\pi\)
−0.526585 + 0.850122i \(0.676528\pi\)
\(432\) −8.06216e7 −0.0481125
\(433\) 2.79533e8 0.165472 0.0827361 0.996571i \(-0.473634\pi\)
0.0827361 + 0.996571i \(0.473634\pi\)
\(434\) 2.95217e8 0.173351
\(435\) 6.91784e8 0.402957
\(436\) −1.27697e9 −0.737867
\(437\) 5.74381e6 0.00329242
\(438\) 3.26756e8 0.185808
\(439\) 9.31660e8 0.525571 0.262786 0.964854i \(-0.415359\pi\)
0.262786 + 0.964854i \(0.415359\pi\)
\(440\) −4.24347e8 −0.237485
\(441\) 8.57661e7 0.0476190
\(442\) 9.35568e7 0.0515344
\(443\) −2.72631e9 −1.48992 −0.744958 0.667112i \(-0.767530\pi\)
−0.744958 + 0.667112i \(0.767530\pi\)
\(444\) −8.65244e8 −0.469136
\(445\) 2.43470e9 1.30974
\(446\) −2.01598e9 −1.07601
\(447\) −1.64069e9 −0.868861
\(448\) −8.99154e7 −0.0472456
\(449\) −8.07255e8 −0.420871 −0.210435 0.977608i \(-0.567488\pi\)
−0.210435 + 0.977608i \(0.567488\pi\)
\(450\) −1.64171e8 −0.0849281
\(451\) −7.17355e8 −0.368228
\(452\) −8.07202e8 −0.411148
\(453\) 8.90740e8 0.450202
\(454\) −7.85989e8 −0.394204
\(455\) −1.68462e8 −0.0838419
\(456\) 1.17116e7 0.00578413
\(457\) −1.56428e9 −0.766669 −0.383334 0.923610i \(-0.625224\pi\)
−0.383334 + 0.923610i \(0.625224\pi\)
\(458\) −2.18965e9 −1.06499
\(459\) 1.04772e8 0.0505711
\(460\) 9.70007e7 0.0464646
\(461\) −3.03297e8 −0.144183 −0.0720916 0.997398i \(-0.522967\pi\)
−0.0720916 + 0.997398i \(0.522967\pi\)
\(462\) 2.74677e8 0.129591
\(463\) 7.12250e7 0.0333502 0.0166751 0.999861i \(-0.494692\pi\)
0.0166751 + 0.999861i \(0.494692\pi\)
\(464\) 4.69451e8 0.218161
\(465\) −6.49378e8 −0.299511
\(466\) −3.22589e8 −0.147672
\(467\) 1.56816e9 0.712493 0.356246 0.934392i \(-0.384056\pi\)
0.356246 + 0.934392i \(0.384056\pi\)
\(468\) −1.02503e8 −0.0462250
\(469\) −4.57280e8 −0.204681
\(470\) 4.83016e8 0.214595
\(471\) 6.99268e8 0.308369
\(472\) 1.40265e9 0.613976
\(473\) 1.96029e9 0.851737
\(474\) −2.23204e8 −0.0962669
\(475\) 2.38484e7 0.0102101
\(476\) 1.16850e8 0.0496598
\(477\) 1.21627e8 0.0513116
\(478\) 1.71791e8 0.0719451
\(479\) 3.95349e9 1.64364 0.821819 0.569749i \(-0.192960\pi\)
0.821819 + 0.569749i \(0.192960\pi\)
\(480\) 1.97784e8 0.0816293
\(481\) −1.10008e9 −0.450731
\(482\) −7.40258e8 −0.301105
\(483\) −6.27879e7 −0.0253549
\(484\) −3.67489e8 −0.147328
\(485\) −1.67211e9 −0.665531
\(486\) −1.14791e8 −0.0453609
\(487\) 3.30775e9 1.29772 0.648861 0.760907i \(-0.275246\pi\)
0.648861 + 0.760907i \(0.275246\pi\)
\(488\) −3.46152e6 −0.00134833
\(489\) 1.31909e8 0.0510145
\(490\) −2.10404e8 −0.0807920
\(491\) 9.54688e8 0.363979 0.181989 0.983300i \(-0.441746\pi\)
0.181989 + 0.983300i \(0.441746\pi\)
\(492\) 3.34351e8 0.126568
\(493\) −6.10078e8 −0.229309
\(494\) 1.48902e7 0.00555721
\(495\) −6.04198e8 −0.223903
\(496\) −4.40674e8 −0.162155
\(497\) 6.78840e8 0.248039
\(498\) 1.39635e9 0.506631
\(499\) −5.78352e8 −0.208372 −0.104186 0.994558i \(-0.533224\pi\)
−0.104186 + 0.994558i \(0.533224\pi\)
\(500\) 1.52050e9 0.543992
\(501\) 2.21982e9 0.788653
\(502\) 6.77779e8 0.239125
\(503\) −3.78503e9 −1.32611 −0.663057 0.748568i \(-0.730742\pi\)
−0.663057 + 0.748568i \(0.730742\pi\)
\(504\) −1.28024e8 −0.0445435
\(505\) 1.48302e9 0.512422
\(506\) −2.01087e8 −0.0690012
\(507\) −1.30324e8 −0.0444116
\(508\) −2.10317e9 −0.711790
\(509\) 4.97723e7 0.0167292 0.00836460 0.999965i \(-0.497337\pi\)
0.00836460 + 0.999965i \(0.497337\pi\)
\(510\) −2.57031e8 −0.0858005
\(511\) 5.18877e8 0.172025
\(512\) 1.34218e8 0.0441942
\(513\) 1.66753e7 0.00545333
\(514\) −1.71629e9 −0.557469
\(515\) −2.10615e9 −0.679459
\(516\) −9.13668e8 −0.292762
\(517\) −1.00131e9 −0.318679
\(518\) −1.37398e9 −0.434335
\(519\) −1.66260e9 −0.522037
\(520\) 2.51464e8 0.0784269
\(521\) −3.23429e9 −1.00195 −0.500976 0.865461i \(-0.667025\pi\)
−0.500976 + 0.865461i \(0.667025\pi\)
\(522\) 6.68418e8 0.205684
\(523\) −5.04124e9 −1.54093 −0.770463 0.637485i \(-0.779975\pi\)
−0.770463 + 0.637485i \(0.779975\pi\)
\(524\) −1.02990e9 −0.312707
\(525\) −2.60697e8 −0.0786281
\(526\) −9.28495e6 −0.00278182
\(527\) 5.72680e8 0.170441
\(528\) −4.10014e8 −0.121222
\(529\) −3.35886e9 −0.986500
\(530\) −2.98380e8 −0.0870570
\(531\) 1.99713e9 0.578862
\(532\) 1.85975e7 0.00535507
\(533\) 4.25098e8 0.121603
\(534\) 2.35246e9 0.668540
\(535\) 4.48873e9 1.26732
\(536\) 6.82587e8 0.191462
\(537\) −2.16550e9 −0.603460
\(538\) 1.09887e9 0.304235
\(539\) 4.36177e8 0.119978
\(540\) 2.81610e8 0.0769608
\(541\) 1.15462e9 0.313508 0.156754 0.987638i \(-0.449897\pi\)
0.156754 + 0.987638i \(0.449897\pi\)
\(542\) −1.15415e9 −0.311362
\(543\) −1.31919e9 −0.353597
\(544\) −1.74424e8 −0.0464525
\(545\) 4.46044e9 1.18029
\(546\) −1.62771e8 −0.0427960
\(547\) 4.59022e9 1.19916 0.599581 0.800314i \(-0.295334\pi\)
0.599581 + 0.800314i \(0.295334\pi\)
\(548\) 2.87067e8 0.0745162
\(549\) −4.92861e6 −0.00127122
\(550\) −8.34916e8 −0.213980
\(551\) −9.70984e7 −0.0247276
\(552\) 9.37243e7 0.0237173
\(553\) −3.54439e8 −0.0891259
\(554\) −1.49240e9 −0.372907
\(555\) 3.02228e9 0.750430
\(556\) −2.42178e9 −0.597548
\(557\) 5.53417e9 1.35694 0.678468 0.734630i \(-0.262644\pi\)
0.678468 + 0.734630i \(0.262644\pi\)
\(558\) −6.27444e8 −0.152882
\(559\) −1.16165e9 −0.281276
\(560\) 3.14073e8 0.0755740
\(561\) 5.32836e8 0.127416
\(562\) −1.14760e9 −0.272717
\(563\) −1.51012e9 −0.356641 −0.178321 0.983972i \(-0.557066\pi\)
−0.178321 + 0.983972i \(0.557066\pi\)
\(564\) 4.66701e8 0.109537
\(565\) 2.81954e9 0.657672
\(566\) 1.24689e9 0.289048
\(567\) −1.82284e8 −0.0419961
\(568\) −1.01331e9 −0.232019
\(569\) 2.00672e9 0.456660 0.228330 0.973584i \(-0.426673\pi\)
0.228330 + 0.973584i \(0.426673\pi\)
\(570\) −4.09083e7 −0.00925230
\(571\) −4.95810e9 −1.11452 −0.557261 0.830337i \(-0.688148\pi\)
−0.557261 + 0.830337i \(0.688148\pi\)
\(572\) −5.21297e8 −0.116466
\(573\) 2.24092e9 0.497606
\(574\) 5.30938e8 0.117180
\(575\) 1.90852e8 0.0418657
\(576\) 1.91103e8 0.0416667
\(577\) 5.46115e8 0.118350 0.0591750 0.998248i \(-0.481153\pi\)
0.0591750 + 0.998248i \(0.481153\pi\)
\(578\) −3.05604e9 −0.658281
\(579\) 1.82421e9 0.390571
\(580\) −1.63979e9 −0.348971
\(581\) 2.21735e9 0.469049
\(582\) −1.61563e9 −0.339712
\(583\) 6.18554e8 0.129282
\(584\) −7.74534e8 −0.160915
\(585\) 3.58042e8 0.0739416
\(586\) −1.98852e9 −0.408214
\(587\) −3.28998e9 −0.671367 −0.335683 0.941975i \(-0.608967\pi\)
−0.335683 + 0.941975i \(0.608967\pi\)
\(588\) −2.03297e8 −0.0412393
\(589\) 9.11462e7 0.0183796
\(590\) −4.89942e9 −0.982116
\(591\) −5.63668e8 −0.112323
\(592\) 2.05095e9 0.406283
\(593\) 7.13873e9 1.40582 0.702910 0.711279i \(-0.251884\pi\)
0.702910 + 0.711279i \(0.251884\pi\)
\(594\) −5.83789e8 −0.114289
\(595\) −4.08156e8 −0.0794358
\(596\) 3.88905e9 0.752456
\(597\) 1.78215e9 0.342794
\(598\) 1.19162e8 0.0227869
\(599\) 8.67322e9 1.64887 0.824435 0.565956i \(-0.191493\pi\)
0.824435 + 0.565956i \(0.191493\pi\)
\(600\) 3.89145e8 0.0735499
\(601\) −6.48192e7 −0.0121799 −0.00608994 0.999981i \(-0.501939\pi\)
−0.00608994 + 0.999981i \(0.501939\pi\)
\(602\) −1.45087e9 −0.271045
\(603\) 9.71887e8 0.180512
\(604\) −2.11138e9 −0.389886
\(605\) 1.28363e9 0.235666
\(606\) 1.43293e9 0.261559
\(607\) 7.01589e9 1.27328 0.636638 0.771163i \(-0.280325\pi\)
0.636638 + 0.771163i \(0.280325\pi\)
\(608\) −2.77608e7 −0.00500921
\(609\) 1.06142e9 0.190427
\(610\) 1.20910e7 0.00215680
\(611\) 5.93369e8 0.105240
\(612\) −2.48349e8 −0.0437958
\(613\) 6.35211e9 1.11380 0.556899 0.830580i \(-0.311991\pi\)
0.556899 + 0.830580i \(0.311991\pi\)
\(614\) 2.94263e9 0.513034
\(615\) −1.16788e9 −0.202459
\(616\) −6.51087e8 −0.112229
\(617\) 4.08336e9 0.699874 0.349937 0.936773i \(-0.386203\pi\)
0.349937 + 0.936773i \(0.386203\pi\)
\(618\) −2.03501e9 −0.346821
\(619\) −8.14644e9 −1.38055 −0.690273 0.723549i \(-0.742509\pi\)
−0.690273 + 0.723549i \(0.742509\pi\)
\(620\) 1.53927e9 0.259384
\(621\) 1.33447e8 0.0223609
\(622\) 3.58893e8 0.0597996
\(623\) 3.73562e9 0.618948
\(624\) 2.42971e8 0.0400320
\(625\) −3.11188e9 −0.509850
\(626\) 2.75424e9 0.448737
\(627\) 8.48047e7 0.0137399
\(628\) −1.65752e9 −0.267055
\(629\) −2.66533e9 −0.427044
\(630\) 4.47186e8 0.0712519
\(631\) 1.12053e10 1.77550 0.887750 0.460325i \(-0.152267\pi\)
0.887750 + 0.460325i \(0.152267\pi\)
\(632\) 5.29075e8 0.0833696
\(633\) −9.65879e8 −0.151360
\(634\) −1.96393e9 −0.306064
\(635\) 7.34635e9 1.13858
\(636\) −2.88301e8 −0.0444372
\(637\) −2.58475e8 −0.0396214
\(638\) 3.39935e9 0.518230
\(639\) −1.44278e9 −0.218750
\(640\) −4.68820e8 −0.0706930
\(641\) 8.72371e9 1.30827 0.654136 0.756377i \(-0.273032\pi\)
0.654136 + 0.756377i \(0.273032\pi\)
\(642\) 4.33711e9 0.646886
\(643\) 6.26395e9 0.929202 0.464601 0.885520i \(-0.346198\pi\)
0.464601 + 0.885520i \(0.346198\pi\)
\(644\) 1.48831e8 0.0219580
\(645\) 3.19143e9 0.468302
\(646\) 3.60767e7 0.00526517
\(647\) 7.54789e9 1.09562 0.547811 0.836602i \(-0.315461\pi\)
0.547811 + 0.836602i \(0.315461\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) 1.01567e10 1.45847
\(650\) 4.94764e8 0.0706645
\(651\) −9.96357e8 −0.141541
\(652\) −3.12673e8 −0.0441798
\(653\) 1.04105e10 1.46310 0.731550 0.681788i \(-0.238797\pi\)
0.731550 + 0.681788i \(0.238797\pi\)
\(654\) 4.30977e9 0.602466
\(655\) 3.59744e9 0.500206
\(656\) −7.92537e8 −0.109611
\(657\) −1.10280e9 −0.151712
\(658\) 7.41104e8 0.101412
\(659\) −5.19413e9 −0.706990 −0.353495 0.935436i \(-0.615007\pi\)
−0.353495 + 0.935436i \(0.615007\pi\)
\(660\) 1.43217e9 0.193906
\(661\) −4.90935e9 −0.661178 −0.330589 0.943775i \(-0.607247\pi\)
−0.330589 + 0.943775i \(0.607247\pi\)
\(662\) 3.03027e9 0.405955
\(663\) −3.15754e8 −0.0420777
\(664\) −3.30987e9 −0.438755
\(665\) −6.49609e7 −0.00856597
\(666\) 2.92020e9 0.383048
\(667\) −7.77050e8 −0.101393
\(668\) −5.26180e9 −0.682993
\(669\) 6.80395e9 0.878556
\(670\) −2.38427e9 −0.306262
\(671\) −2.50652e7 −0.00320290
\(672\) 3.03464e8 0.0385758
\(673\) 4.04854e8 0.0511972 0.0255986 0.999672i \(-0.491851\pi\)
0.0255986 + 0.999672i \(0.491851\pi\)
\(674\) 9.73119e9 1.22421
\(675\) 5.54076e8 0.0693435
\(676\) 3.08916e8 0.0384615
\(677\) 3.66555e9 0.454025 0.227012 0.973892i \(-0.427104\pi\)
0.227012 + 0.973892i \(0.427104\pi\)
\(678\) 2.72431e9 0.335701
\(679\) −2.56556e9 −0.314512
\(680\) 6.09259e8 0.0743054
\(681\) 2.65271e9 0.321866
\(682\) −3.19096e9 −0.385191
\(683\) 6.67383e9 0.801499 0.400749 0.916188i \(-0.368750\pi\)
0.400749 + 0.916188i \(0.368750\pi\)
\(684\) −3.95266e7 −0.00472272
\(685\) −1.00272e9 −0.119196
\(686\) −3.22829e8 −0.0381802
\(687\) 7.39008e9 0.869562
\(688\) 2.16573e9 0.253539
\(689\) −3.66550e8 −0.0426939
\(690\) −3.27377e8 −0.0379382
\(691\) −9.07458e9 −1.04629 −0.523147 0.852243i \(-0.675242\pi\)
−0.523147 + 0.852243i \(0.675242\pi\)
\(692\) 3.94097e9 0.452097
\(693\) −9.27036e8 −0.105811
\(694\) 4.17857e9 0.474537
\(695\) 8.45924e9 0.955838
\(696\) −1.58440e9 −0.178128
\(697\) 1.02995e9 0.115213
\(698\) 2.71647e9 0.302351
\(699\) 1.08874e9 0.120574
\(700\) 6.17948e8 0.0680940
\(701\) −2.65100e9 −0.290668 −0.145334 0.989383i \(-0.546426\pi\)
−0.145334 + 0.989383i \(0.546426\pi\)
\(702\) 3.45948e8 0.0377426
\(703\) −4.24206e8 −0.0460503
\(704\) 9.71885e8 0.104981
\(705\) −1.63018e9 −0.175216
\(706\) 4.04764e9 0.432897
\(707\) 2.27544e9 0.242157
\(708\) −4.73393e9 −0.501309
\(709\) −1.59598e10 −1.68176 −0.840882 0.541219i \(-0.817963\pi\)
−0.840882 + 0.541219i \(0.817963\pi\)
\(710\) 3.53948e9 0.371138
\(711\) 7.53312e8 0.0786016
\(712\) −5.57620e9 −0.578973
\(713\) 7.29416e8 0.0753637
\(714\) −3.94369e8 −0.0405471
\(715\) 1.82088e9 0.186299
\(716\) 5.13304e9 0.522612
\(717\) −5.79793e8 −0.0587430
\(718\) −9.34045e8 −0.0941743
\(719\) 1.08770e10 1.09133 0.545666 0.838003i \(-0.316277\pi\)
0.545666 + 0.838003i \(0.316277\pi\)
\(720\) −6.67520e8 −0.0666500
\(721\) −3.23151e9 −0.321094
\(722\) −7.14523e9 −0.706539
\(723\) 2.49837e9 0.245852
\(724\) 3.12698e9 0.306224
\(725\) −3.22633e9 −0.314431
\(726\) 1.24028e9 0.120293
\(727\) −1.85850e10 −1.79388 −0.896940 0.442153i \(-0.854215\pi\)
−0.896940 + 0.442153i \(0.854215\pi\)
\(728\) 3.85828e8 0.0370625
\(729\) 3.87420e8 0.0370370
\(730\) 2.70543e9 0.257399
\(731\) −2.81449e9 −0.266495
\(732\) 1.16826e7 0.00110091
\(733\) 1.47641e10 1.38466 0.692330 0.721581i \(-0.256584\pi\)
0.692330 + 0.721581i \(0.256584\pi\)
\(734\) −1.15174e10 −1.07502
\(735\) 7.10115e8 0.0659664
\(736\) −2.22161e8 −0.0205398
\(737\) 4.94269e9 0.454807
\(738\) −1.12844e9 −0.103343
\(739\) 1.82788e10 1.66607 0.833035 0.553220i \(-0.186601\pi\)
0.833035 + 0.553220i \(0.186601\pi\)
\(740\) −7.16393e9 −0.649891
\(741\) −5.02545e7 −0.00453745
\(742\) −4.57812e8 −0.0411408
\(743\) −1.17979e10 −1.05522 −0.527612 0.849486i \(-0.676912\pi\)
−0.527612 + 0.849486i \(0.676912\pi\)
\(744\) 1.48727e9 0.132399
\(745\) −1.35844e10 −1.20363
\(746\) −9.22000e8 −0.0813102
\(747\) −4.71268e9 −0.413662
\(748\) −1.26302e9 −0.110345
\(749\) 6.88717e9 0.598900
\(750\) −5.13170e9 −0.444168
\(751\) −6.79474e9 −0.585373 −0.292687 0.956208i \(-0.594549\pi\)
−0.292687 + 0.956208i \(0.594549\pi\)
\(752\) −1.10625e9 −0.0948620
\(753\) −2.28750e9 −0.195245
\(754\) −2.01442e9 −0.171140
\(755\) 7.37503e9 0.623663
\(756\) 4.32081e8 0.0363696
\(757\) −1.81738e10 −1.52268 −0.761341 0.648352i \(-0.775459\pi\)
−0.761341 + 0.648352i \(0.775459\pi\)
\(758\) 9.26562e8 0.0772738
\(759\) 6.78667e8 0.0563392
\(760\) 9.69679e7 0.00801273
\(761\) 2.81062e9 0.231183 0.115592 0.993297i \(-0.463124\pi\)
0.115592 + 0.993297i \(0.463124\pi\)
\(762\) 7.09821e9 0.581174
\(763\) 6.84376e9 0.557775
\(764\) −5.31182e9 −0.430939
\(765\) 8.67480e8 0.0700558
\(766\) −5.16532e9 −0.415237
\(767\) −6.01878e9 −0.481642
\(768\) −4.52985e8 −0.0360844
\(769\) −1.77729e10 −1.40934 −0.704671 0.709534i \(-0.748906\pi\)
−0.704671 + 0.709534i \(0.748906\pi\)
\(770\) 2.27424e9 0.179522
\(771\) 5.79249e9 0.455171
\(772\) −4.32406e9 −0.338245
\(773\) 8.49608e9 0.661592 0.330796 0.943702i \(-0.392683\pi\)
0.330796 + 0.943702i \(0.392683\pi\)
\(774\) 3.08363e9 0.239039
\(775\) 3.02855e9 0.233711
\(776\) 3.82964e9 0.294199
\(777\) 4.63717e9 0.354633
\(778\) −2.52489e9 −0.192227
\(779\) 1.63923e8 0.0124239
\(780\) −8.48693e8 −0.0640353
\(781\) −7.33750e9 −0.551150
\(782\) 2.88711e8 0.0215894
\(783\) −2.25591e9 −0.167941
\(784\) 4.81890e8 0.0357143
\(785\) 5.78970e9 0.427182
\(786\) 3.47593e9 0.255324
\(787\) 1.09652e10 0.801869 0.400935 0.916107i \(-0.368685\pi\)
0.400935 + 0.916107i \(0.368685\pi\)
\(788\) 1.33610e9 0.0972742
\(789\) 3.13367e7 0.00227135
\(790\) −1.84805e9 −0.133358
\(791\) 4.32610e9 0.310798
\(792\) 1.38380e9 0.0989770
\(793\) 1.48534e7 0.00105772
\(794\) −4.44124e9 −0.314871
\(795\) 1.00703e9 0.0710817
\(796\) −4.22435e9 −0.296869
\(797\) −1.08261e10 −0.757471 −0.378736 0.925505i \(-0.623641\pi\)
−0.378736 + 0.925505i \(0.623641\pi\)
\(798\) −6.27667e7 −0.00437239
\(799\) 1.43764e9 0.0997094
\(800\) −9.22418e8 −0.0636961
\(801\) −7.93955e9 −0.545861
\(802\) −2.39360e9 −0.163848
\(803\) −5.60848e9 −0.382244
\(804\) −2.30373e9 −0.156328
\(805\) −5.19863e8 −0.0351240
\(806\) 1.89094e9 0.127205
\(807\) −3.70869e9 −0.248407
\(808\) −3.39657e9 −0.226517
\(809\) −1.70536e10 −1.13239 −0.566196 0.824270i \(-0.691586\pi\)
−0.566196 + 0.824270i \(0.691586\pi\)
\(810\) −9.50433e8 −0.0628382
\(811\) −1.39453e10 −0.918023 −0.459011 0.888430i \(-0.651796\pi\)
−0.459011 + 0.888430i \(0.651796\pi\)
\(812\) −2.51596e9 −0.164914
\(813\) 3.89526e9 0.254226
\(814\) 1.48511e10 0.965105
\(815\) 1.09216e9 0.0706701
\(816\) 5.88679e8 0.0379283
\(817\) −4.47946e8 −0.0287375
\(818\) −3.02911e9 −0.193499
\(819\) 5.49353e8 0.0349428
\(820\) 2.76832e9 0.175335
\(821\) 1.11715e10 0.704545 0.352273 0.935897i \(-0.385409\pi\)
0.352273 + 0.935897i \(0.385409\pi\)
\(822\) −9.68850e8 −0.0608422
\(823\) −1.07780e9 −0.0673965 −0.0336983 0.999432i \(-0.510729\pi\)
−0.0336983 + 0.999432i \(0.510729\pi\)
\(824\) 4.82372e9 0.300356
\(825\) 2.81784e9 0.174714
\(826\) −7.51731e9 −0.464122
\(827\) −2.40146e10 −1.47641 −0.738204 0.674577i \(-0.764326\pi\)
−0.738204 + 0.674577i \(0.764326\pi\)
\(828\) −3.16319e8 −0.0193651
\(829\) −3.06915e10 −1.87102 −0.935508 0.353305i \(-0.885058\pi\)
−0.935508 + 0.353305i \(0.885058\pi\)
\(830\) 1.15613e10 0.701833
\(831\) 5.03685e9 0.304478
\(832\) −5.75930e8 −0.0346688
\(833\) −6.26244e8 −0.0375393
\(834\) 8.17350e9 0.487896
\(835\) 1.83794e10 1.09252
\(836\) −2.01019e8 −0.0118991
\(837\) 2.11762e9 0.124827
\(838\) 4.91640e9 0.288598
\(839\) −3.65874e9 −0.213877 −0.106939 0.994266i \(-0.534105\pi\)
−0.106939 + 0.994266i \(0.534105\pi\)
\(840\) −1.06000e9 −0.0617059
\(841\) −4.11394e9 −0.238491
\(842\) −1.07785e10 −0.622251
\(843\) 3.87313e9 0.222672
\(844\) 2.28949e9 0.131081
\(845\) −1.07904e9 −0.0615231
\(846\) −1.57512e9 −0.0894368
\(847\) 1.96951e9 0.111370
\(848\) 6.83380e8 0.0384837
\(849\) −4.20824e9 −0.236006
\(850\) 1.19873e9 0.0669509
\(851\) −3.39479e9 −0.188825
\(852\) 3.41993e9 0.189443
\(853\) −9.74468e9 −0.537583 −0.268792 0.963198i \(-0.586624\pi\)
−0.268792 + 0.963198i \(0.586624\pi\)
\(854\) 1.85516e7 0.00101924
\(855\) 1.38066e8 0.00755447
\(856\) −1.02806e10 −0.560220
\(857\) 3.19200e10 1.73233 0.866165 0.499759i \(-0.166578\pi\)
0.866165 + 0.499759i \(0.166578\pi\)
\(858\) 1.75938e9 0.0950940
\(859\) 2.02111e9 0.108796 0.0543980 0.998519i \(-0.482676\pi\)
0.0543980 + 0.998519i \(0.482676\pi\)
\(860\) −7.56486e9 −0.405561
\(861\) −1.79191e9 −0.0956767
\(862\) −1.40042e10 −0.744704
\(863\) 1.61680e10 0.856284 0.428142 0.903711i \(-0.359168\pi\)
0.428142 + 0.903711i \(0.359168\pi\)
\(864\) −6.44973e8 −0.0340207
\(865\) −1.37657e10 −0.723175
\(866\) 2.23626e9 0.117006
\(867\) 1.03141e10 0.537484
\(868\) 2.36174e9 0.122578
\(869\) 3.83109e9 0.198040
\(870\) 5.53428e9 0.284933
\(871\) −2.92899e9 −0.150195
\(872\) −1.02158e10 −0.521751
\(873\) 5.45275e9 0.277374
\(874\) 4.59505e7 0.00232809
\(875\) −8.14895e9 −0.411219
\(876\) 2.61405e9 0.131386
\(877\) 1.14782e10 0.574612 0.287306 0.957839i \(-0.407240\pi\)
0.287306 + 0.957839i \(0.407240\pi\)
\(878\) 7.45328e9 0.371635
\(879\) 6.71125e9 0.333305
\(880\) −3.39478e9 −0.167928
\(881\) −2.78517e10 −1.37226 −0.686129 0.727480i \(-0.740691\pi\)
−0.686129 + 0.727480i \(0.740691\pi\)
\(882\) 6.86129e8 0.0336718
\(883\) −3.43236e10 −1.67776 −0.838882 0.544313i \(-0.816790\pi\)
−0.838882 + 0.544313i \(0.816790\pi\)
\(884\) 7.48454e8 0.0364403
\(885\) 1.65355e10 0.801894
\(886\) −2.18104e10 −1.05353
\(887\) 3.67327e10 1.76734 0.883670 0.468111i \(-0.155065\pi\)
0.883670 + 0.468111i \(0.155065\pi\)
\(888\) −6.92196e9 −0.331729
\(889\) 1.12717e10 0.538063
\(890\) 1.94776e10 0.926125
\(891\) 1.97029e9 0.0933164
\(892\) −1.61279e10 −0.760852
\(893\) 2.28811e8 0.0107522
\(894\) −1.31255e10 −0.614378
\(895\) −1.79296e10 −0.835970
\(896\) −7.19323e8 −0.0334077
\(897\) −4.02172e8 −0.0186054
\(898\) −6.45804e9 −0.297600
\(899\) −1.23307e10 −0.566016
\(900\) −1.31336e9 −0.0600532
\(901\) −8.88092e8 −0.0404502
\(902\) −5.73884e9 −0.260376
\(903\) 4.89669e9 0.221307
\(904\) −6.45762e9 −0.290725
\(905\) −1.09225e10 −0.489837
\(906\) 7.12592e9 0.318341
\(907\) 6.68095e9 0.297312 0.148656 0.988889i \(-0.452505\pi\)
0.148656 + 0.988889i \(0.452505\pi\)
\(908\) −6.28791e9 −0.278744
\(909\) −4.83613e9 −0.213562
\(910\) −1.34769e9 −0.0592851
\(911\) −3.25671e10 −1.42713 −0.713567 0.700587i \(-0.752922\pi\)
−0.713567 + 0.700587i \(0.752922\pi\)
\(912\) 9.36926e7 0.00409000
\(913\) −2.39671e10 −1.04224
\(914\) −1.25142e10 −0.542117
\(915\) −4.08072e7 −0.00176102
\(916\) −1.75172e10 −0.753063
\(917\) 5.51964e9 0.236384
\(918\) 8.38178e8 0.0357591
\(919\) 2.33408e10 0.991998 0.495999 0.868323i \(-0.334802\pi\)
0.495999 + 0.868323i \(0.334802\pi\)
\(920\) 7.76006e8 0.0328555
\(921\) −9.93138e9 −0.418891
\(922\) −2.42638e9 −0.101953
\(923\) 4.34814e9 0.182011
\(924\) 2.19742e9 0.0916349
\(925\) −1.40952e10 −0.585567
\(926\) 5.69800e8 0.0235822
\(927\) 6.86814e9 0.283178
\(928\) 3.75561e9 0.154263
\(929\) −2.81419e10 −1.15159 −0.575795 0.817594i \(-0.695307\pi\)
−0.575795 + 0.817594i \(0.695307\pi\)
\(930\) −5.19502e9 −0.211786
\(931\) −9.96712e7 −0.00404805
\(932\) −2.58071e9 −0.104420
\(933\) −1.21126e9 −0.0488262
\(934\) 1.25453e10 0.503809
\(935\) 4.41171e9 0.176509
\(936\) −8.20026e8 −0.0326860
\(937\) −1.69089e10 −0.671472 −0.335736 0.941956i \(-0.608985\pi\)
−0.335736 + 0.941956i \(0.608985\pi\)
\(938\) −3.65824e9 −0.144731
\(939\) −9.29556e9 −0.366392
\(940\) 3.86413e9 0.151741
\(941\) 1.28772e9 0.0503798 0.0251899 0.999683i \(-0.491981\pi\)
0.0251899 + 0.999683i \(0.491981\pi\)
\(942\) 5.59414e9 0.218050
\(943\) 1.31183e9 0.0509433
\(944\) 1.12212e10 0.434146
\(945\) −1.50925e9 −0.0581769
\(946\) 1.56823e10 0.602269
\(947\) 3.25118e10 1.24399 0.621993 0.783023i \(-0.286323\pi\)
0.621993 + 0.783023i \(0.286323\pi\)
\(948\) −1.78563e9 −0.0680710
\(949\) 3.32354e9 0.126232
\(950\) 1.90787e8 0.00721966
\(951\) 6.62825e9 0.249900
\(952\) 9.34801e8 0.0351148
\(953\) −5.91066e9 −0.221213 −0.110606 0.993864i \(-0.535279\pi\)
−0.110606 + 0.993864i \(0.535279\pi\)
\(954\) 9.73016e8 0.0362828
\(955\) 1.85541e10 0.689331
\(956\) 1.37432e9 0.0508729
\(957\) −1.14728e10 −0.423133
\(958\) 3.16279e10 1.16223
\(959\) −1.53850e9 −0.0563290
\(960\) 1.58227e9 0.0577206
\(961\) −1.59378e10 −0.579290
\(962\) −8.80066e9 −0.318715
\(963\) −1.46377e10 −0.528180
\(964\) −5.92207e9 −0.212914
\(965\) 1.51039e10 0.541056
\(966\) −5.02304e8 −0.0179286
\(967\) −3.91881e10 −1.39368 −0.696838 0.717229i \(-0.745410\pi\)
−0.696838 + 0.717229i \(0.745410\pi\)
\(968\) −2.93991e9 −0.104177
\(969\) −1.21759e8 −0.00429900
\(970\) −1.33769e10 −0.470602
\(971\) 3.00821e10 1.05449 0.527243 0.849715i \(-0.323226\pi\)
0.527243 + 0.849715i \(0.323226\pi\)
\(972\) −9.18330e8 −0.0320750
\(973\) 1.29792e10 0.451704
\(974\) 2.64620e10 0.917627
\(975\) −1.66983e9 −0.0576973
\(976\) −2.76922e7 −0.000953416 0
\(977\) −3.20746e10 −1.10035 −0.550175 0.835050i \(-0.685439\pi\)
−0.550175 + 0.835050i \(0.685439\pi\)
\(978\) 1.05527e9 0.0360727
\(979\) −4.03778e10 −1.37532
\(980\) −1.68324e9 −0.0571286
\(981\) −1.45455e10 −0.491911
\(982\) 7.63750e9 0.257372
\(983\) 5.21892e10 1.75244 0.876221 0.481910i \(-0.160057\pi\)
0.876221 + 0.481910i \(0.160057\pi\)
\(984\) 2.67481e9 0.0894974
\(985\) −4.66698e9 −0.155600
\(986\) −4.88063e9 −0.162146
\(987\) −2.50122e9 −0.0828024
\(988\) 1.19122e8 0.00392954
\(989\) −3.58478e9 −0.117835
\(990\) −4.83358e9 −0.158324
\(991\) −1.74474e10 −0.569474 −0.284737 0.958606i \(-0.591906\pi\)
−0.284737 + 0.958606i \(0.591906\pi\)
\(992\) −3.52539e9 −0.114661
\(993\) −1.02272e10 −0.331461
\(994\) 5.43072e9 0.175390
\(995\) 1.47556e10 0.474871
\(996\) 1.11708e10 0.358242
\(997\) −3.30188e9 −0.105519 −0.0527593 0.998607i \(-0.516802\pi\)
−0.0527593 + 0.998607i \(0.516802\pi\)
\(998\) −4.62681e9 −0.147341
\(999\) −9.85567e9 −0.312757
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 546.8.a.l.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.8.a.l.1.2 5 1.1 even 1 trivial