Properties

Label 91.8.a.d.1.6
Level $91$
Weight $8$
Character 91.1
Self dual yes
Analytic conductor $28.427$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.4270373191\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \( x^{10} - 3 x^{9} - 816 x^{8} + 2298 x^{7} + 213848 x^{6} - 507132 x^{5} - 19919976 x^{4} + 24331248 x^{3} + 727257184 x^{2} - 56397312 x - 7335224320 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-4.83483\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.83483 q^{2} +62.4820 q^{3} -104.624 q^{4} +314.530 q^{5} +302.090 q^{6} -343.000 q^{7} -1124.70 q^{8} +1717.00 q^{9} +O(q^{10})\) \(q+4.83483 q^{2} +62.4820 q^{3} -104.624 q^{4} +314.530 q^{5} +302.090 q^{6} -343.000 q^{7} -1124.70 q^{8} +1717.00 q^{9} +1520.70 q^{10} +8467.65 q^{11} -6537.15 q^{12} -2197.00 q^{13} -1658.35 q^{14} +19652.4 q^{15} +7954.20 q^{16} +11951.4 q^{17} +8301.41 q^{18} +6886.88 q^{19} -32907.5 q^{20} -21431.3 q^{21} +40939.6 q^{22} +103273. q^{23} -70273.5 q^{24} +20803.8 q^{25} -10622.1 q^{26} -29366.4 q^{27} +35886.2 q^{28} -101393. q^{29} +95016.1 q^{30} +240887. q^{31} +182419. q^{32} +529076. q^{33} +57783.1 q^{34} -107884. q^{35} -179640. q^{36} -494398. q^{37} +33296.9 q^{38} -137273. q^{39} -353751. q^{40} -517724. q^{41} -103617. q^{42} +670384. q^{43} -885923. q^{44} +540048. q^{45} +499309. q^{46} +390703. q^{47} +496995. q^{48} +117649. q^{49} +100583. q^{50} +746750. q^{51} +229860. q^{52} +316966. q^{53} -141982. q^{54} +2.66333e6 q^{55} +385772. q^{56} +430306. q^{57} -490216. q^{58} +2.76489e6 q^{59} -2.05613e6 q^{60} +50060.8 q^{61} +1.16465e6 q^{62} -588932. q^{63} -136175. q^{64} -691021. q^{65} +2.55799e6 q^{66} -710978. q^{67} -1.25041e6 q^{68} +6.45272e6 q^{69} -521599. q^{70} -4.81274e6 q^{71} -1.93111e6 q^{72} +319628. q^{73} -2.39033e6 q^{74} +1.29987e6 q^{75} -720536. q^{76} -2.90440e6 q^{77} -663691. q^{78} +395221. q^{79} +2.50183e6 q^{80} -5.58996e6 q^{81} -2.50311e6 q^{82} -5.55444e6 q^{83} +2.24224e6 q^{84} +3.75908e6 q^{85} +3.24119e6 q^{86} -6.33521e6 q^{87} -9.52355e6 q^{88} -1.22040e7 q^{89} +2.61104e6 q^{90} +753571. q^{91} -1.08049e7 q^{92} +1.50511e7 q^{93} +1.88898e6 q^{94} +2.16613e6 q^{95} +1.13979e7 q^{96} -5.15456e6 q^{97} +568813. q^{98} +1.45390e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{2} - 101 q^{3} + 361 q^{4} + 226 q^{5} + 1105 q^{6} - 3430 q^{7} + 291 q^{8} + 12247 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 3 q^{2} - 101 q^{3} + 361 q^{4} + 226 q^{5} + 1105 q^{6} - 3430 q^{7} + 291 q^{8} + 12247 q^{9} + 2548 q^{10} + 451 q^{11} - 16241 q^{12} - 21970 q^{13} + 1029 q^{14} + 27184 q^{15} + 11897 q^{16} - 8654 q^{17} + 159348 q^{18} + 10130 q^{19} - 82012 q^{20} + 34643 q^{21} - 57863 q^{22} - 52155 q^{23} - 49227 q^{24} + 47190 q^{25} + 6591 q^{26} - 155171 q^{27} - 123823 q^{28} + 520154 q^{29} + 1070236 q^{30} + 692605 q^{31} + 149835 q^{32} + 436053 q^{33} + 1059060 q^{34} - 77518 q^{35} + 2843742 q^{36} - 20511 q^{37} + 1905286 q^{38} + 221897 q^{39} + 636320 q^{40} + 355049 q^{41} - 379015 q^{42} + 1256772 q^{43} - 687913 q^{44} + 1259926 q^{45} + 4043075 q^{46} + 1260721 q^{47} + 1128551 q^{48} + 1176490 q^{49} + 609035 q^{50} + 1411976 q^{51} - 793117 q^{52} + 928854 q^{53} + 6642607 q^{54} + 3423196 q^{55} - 99813 q^{56} + 3014966 q^{57} + 1612588 q^{58} + 3144446 q^{59} + 7738848 q^{60} + 6322923 q^{61} + 6545331 q^{62} - 4200721 q^{63} - 6629943 q^{64} - 496522 q^{65} - 14343317 q^{66} + 3944507 q^{67} - 1787356 q^{68} - 148281 q^{69} - 873964 q^{70} + 6032248 q^{71} + 9760866 q^{72} + 1248533 q^{73} - 8263279 q^{74} + 1573413 q^{75} + 1788254 q^{76} - 154693 q^{77} - 2427685 q^{78} - 14947605 q^{79} - 9147616 q^{80} + 25716334 q^{81} - 6987095 q^{82} - 14177784 q^{83} + 5570663 q^{84} - 11788444 q^{85} + 8748840 q^{86} - 29484448 q^{87} - 15390723 q^{88} + 6734836 q^{89} + 5994972 q^{90} + 7535710 q^{91} - 24493215 q^{92} + 17307847 q^{93} - 22760149 q^{94} - 9329708 q^{95} - 36488483 q^{96} - 12365397 q^{97} - 352947 q^{98} - 43198042 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.83483 0.427342 0.213671 0.976906i \(-0.431458\pi\)
0.213671 + 0.976906i \(0.431458\pi\)
\(3\) 62.4820 1.33607 0.668037 0.744128i \(-0.267135\pi\)
0.668037 + 0.744128i \(0.267135\pi\)
\(4\) −104.624 −0.817378
\(5\) 314.530 1.12530 0.562648 0.826697i \(-0.309783\pi\)
0.562648 + 0.826697i \(0.309783\pi\)
\(6\) 302.090 0.570961
\(7\) −343.000 −0.377964
\(8\) −1124.70 −0.776643
\(9\) 1717.00 0.785094
\(10\) 1520.70 0.480886
\(11\) 8467.65 1.91818 0.959088 0.283109i \(-0.0913658\pi\)
0.959088 + 0.283109i \(0.0913658\pi\)
\(12\) −6537.15 −1.09208
\(13\) −2197.00 −0.277350
\(14\) −1658.35 −0.161520
\(15\) 19652.4 1.50348
\(16\) 7954.20 0.485486
\(17\) 11951.4 0.589996 0.294998 0.955498i \(-0.404681\pi\)
0.294998 + 0.955498i \(0.404681\pi\)
\(18\) 8301.41 0.335504
\(19\) 6886.88 0.230348 0.115174 0.993345i \(-0.463257\pi\)
0.115174 + 0.993345i \(0.463257\pi\)
\(20\) −32907.5 −0.919792
\(21\) −21431.3 −0.504989
\(22\) 40939.6 0.819718
\(23\) 103273. 1.76987 0.884934 0.465717i \(-0.154204\pi\)
0.884934 + 0.465717i \(0.154204\pi\)
\(24\) −70273.5 −1.03765
\(25\) 20803.8 0.266289
\(26\) −10622.1 −0.118523
\(27\) −29366.4 −0.287130
\(28\) 35886.2 0.308940
\(29\) −101393. −0.771993 −0.385996 0.922500i \(-0.626142\pi\)
−0.385996 + 0.922500i \(0.626142\pi\)
\(30\) 95016.1 0.642500
\(31\) 240887. 1.45227 0.726137 0.687551i \(-0.241314\pi\)
0.726137 + 0.687551i \(0.241314\pi\)
\(32\) 182419. 0.984112
\(33\) 529076. 2.56282
\(34\) 57783.1 0.252130
\(35\) −107884. −0.425322
\(36\) −179640. −0.641719
\(37\) −494398. −1.60461 −0.802307 0.596912i \(-0.796394\pi\)
−0.802307 + 0.596912i \(0.796394\pi\)
\(38\) 33296.9 0.0984376
\(39\) −137273. −0.370560
\(40\) −353751. −0.873952
\(41\) −517724. −1.17315 −0.586577 0.809893i \(-0.699525\pi\)
−0.586577 + 0.809893i \(0.699525\pi\)
\(42\) −103617. −0.215803
\(43\) 670384. 1.28583 0.642916 0.765937i \(-0.277724\pi\)
0.642916 + 0.765937i \(0.277724\pi\)
\(44\) −885923. −1.56788
\(45\) 540048. 0.883463
\(46\) 499309. 0.756339
\(47\) 390703. 0.548914 0.274457 0.961599i \(-0.411502\pi\)
0.274457 + 0.961599i \(0.411502\pi\)
\(48\) 496995. 0.648645
\(49\) 117649. 0.142857
\(50\) 100583. 0.113797
\(51\) 746750. 0.788278
\(52\) 229860. 0.226700
\(53\) 316966. 0.292447 0.146224 0.989252i \(-0.453288\pi\)
0.146224 + 0.989252i \(0.453288\pi\)
\(54\) −141982. −0.122703
\(55\) 2.66333e6 2.15851
\(56\) 385772. 0.293543
\(57\) 430306. 0.307762
\(58\) −490216. −0.329905
\(59\) 2.76489e6 1.75265 0.876326 0.481719i \(-0.159987\pi\)
0.876326 + 0.481719i \(0.159987\pi\)
\(60\) −2.05613e6 −1.22891
\(61\) 50060.8 0.0282386 0.0141193 0.999900i \(-0.495506\pi\)
0.0141193 + 0.999900i \(0.495506\pi\)
\(62\) 1.16465e6 0.620618
\(63\) −588932. −0.296738
\(64\) −136175. −0.0649334
\(65\) −691021. −0.312101
\(66\) 2.55799e6 1.09520
\(67\) −710978. −0.288798 −0.144399 0.989520i \(-0.546125\pi\)
−0.144399 + 0.989520i \(0.546125\pi\)
\(68\) −1.25041e6 −0.482250
\(69\) 6.45272e6 2.36467
\(70\) −521599. −0.181758
\(71\) −4.81274e6 −1.59584 −0.797918 0.602766i \(-0.794065\pi\)
−0.797918 + 0.602766i \(0.794065\pi\)
\(72\) −1.93111e6 −0.609738
\(73\) 319628. 0.0961645 0.0480822 0.998843i \(-0.484689\pi\)
0.0480822 + 0.998843i \(0.484689\pi\)
\(74\) −2.39033e6 −0.685720
\(75\) 1.29987e6 0.355782
\(76\) −720536. −0.188282
\(77\) −2.90440e6 −0.725002
\(78\) −663691. −0.158356
\(79\) 395221. 0.0901872 0.0450936 0.998983i \(-0.485641\pi\)
0.0450936 + 0.998983i \(0.485641\pi\)
\(80\) 2.50183e6 0.546315
\(81\) −5.58996e6 −1.16872
\(82\) −2.50311e6 −0.501339
\(83\) −5.55444e6 −1.06627 −0.533135 0.846030i \(-0.678986\pi\)
−0.533135 + 0.846030i \(0.678986\pi\)
\(84\) 2.24224e6 0.412767
\(85\) 3.75908e6 0.663919
\(86\) 3.24119e6 0.549490
\(87\) −6.33521e6 −1.03144
\(88\) −9.52355e6 −1.48974
\(89\) −1.22040e7 −1.83500 −0.917500 0.397735i \(-0.869796\pi\)
−0.917500 + 0.397735i \(0.869796\pi\)
\(90\) 2.61104e6 0.377541
\(91\) 753571. 0.104828
\(92\) −1.08049e7 −1.44665
\(93\) 1.50511e7 1.94034
\(94\) 1.88898e6 0.234574
\(95\) 2.16613e6 0.259210
\(96\) 1.13979e7 1.31485
\(97\) −5.15456e6 −0.573443 −0.286721 0.958014i \(-0.592565\pi\)
−0.286721 + 0.958014i \(0.592565\pi\)
\(98\) 568813. 0.0610489
\(99\) 1.45390e7 1.50595
\(100\) −2.17659e6 −0.217659
\(101\) 5.66908e6 0.547504 0.273752 0.961800i \(-0.411735\pi\)
0.273752 + 0.961800i \(0.411735\pi\)
\(102\) 3.61041e6 0.336865
\(103\) −8.15115e6 −0.735003 −0.367501 0.930023i \(-0.619787\pi\)
−0.367501 + 0.930023i \(0.619787\pi\)
\(104\) 2.47096e6 0.215402
\(105\) −6.74079e6 −0.568261
\(106\) 1.53248e6 0.124975
\(107\) −1.74326e7 −1.37568 −0.687841 0.725861i \(-0.741441\pi\)
−0.687841 + 0.725861i \(0.741441\pi\)
\(108\) 3.07245e6 0.234694
\(109\) 3.93590e6 0.291106 0.145553 0.989350i \(-0.453504\pi\)
0.145553 + 0.989350i \(0.453504\pi\)
\(110\) 1.28767e7 0.922424
\(111\) −3.08910e7 −2.14388
\(112\) −2.72829e6 −0.183496
\(113\) −2.61919e7 −1.70762 −0.853812 0.520581i \(-0.825715\pi\)
−0.853812 + 0.520581i \(0.825715\pi\)
\(114\) 2.08046e6 0.131520
\(115\) 3.24825e7 1.99162
\(116\) 1.06081e7 0.631010
\(117\) −3.77225e6 −0.217746
\(118\) 1.33678e7 0.748982
\(119\) −4.09934e6 −0.222997
\(120\) −2.21031e7 −1.16767
\(121\) 5.22139e7 2.67940
\(122\) 242036. 0.0120676
\(123\) −3.23485e7 −1.56742
\(124\) −2.52027e7 −1.18706
\(125\) −1.80292e7 −0.825641
\(126\) −2.84738e6 −0.126809
\(127\) 5.87479e6 0.254495 0.127247 0.991871i \(-0.459386\pi\)
0.127247 + 0.991871i \(0.459386\pi\)
\(128\) −2.40080e7 −1.01186
\(129\) 4.18870e7 1.71797
\(130\) −3.34097e6 −0.133374
\(131\) −1.07603e6 −0.0418190 −0.0209095 0.999781i \(-0.506656\pi\)
−0.0209095 + 0.999781i \(0.506656\pi\)
\(132\) −5.53542e7 −2.09480
\(133\) −2.36220e6 −0.0870634
\(134\) −3.43746e6 −0.123416
\(135\) −9.23662e6 −0.323106
\(136\) −1.34418e7 −0.458216
\(137\) −3.97143e7 −1.31955 −0.659773 0.751465i \(-0.729348\pi\)
−0.659773 + 0.751465i \(0.729348\pi\)
\(138\) 3.11978e7 1.01053
\(139\) 2.73886e6 0.0865003 0.0432501 0.999064i \(-0.486229\pi\)
0.0432501 + 0.999064i \(0.486229\pi\)
\(140\) 1.12873e7 0.347649
\(141\) 2.44119e7 0.733389
\(142\) −2.32688e7 −0.681968
\(143\) −1.86034e7 −0.532006
\(144\) 1.36574e7 0.381152
\(145\) −3.18910e7 −0.868720
\(146\) 1.54535e6 0.0410952
\(147\) 7.35095e6 0.190868
\(148\) 5.17261e7 1.31158
\(149\) 5.33444e7 1.32110 0.660552 0.750781i \(-0.270322\pi\)
0.660552 + 0.750781i \(0.270322\pi\)
\(150\) 6.28462e6 0.152041
\(151\) −3.23075e7 −0.763632 −0.381816 0.924238i \(-0.624701\pi\)
−0.381816 + 0.924238i \(0.624701\pi\)
\(152\) −7.74567e6 −0.178898
\(153\) 2.05206e7 0.463202
\(154\) −1.40423e7 −0.309824
\(155\) 7.57662e7 1.63424
\(156\) 1.43621e7 0.302888
\(157\) 9.76430e6 0.201369 0.100684 0.994918i \(-0.467897\pi\)
0.100684 + 0.994918i \(0.467897\pi\)
\(158\) 1.91082e6 0.0385408
\(159\) 1.98047e7 0.390731
\(160\) 5.73761e7 1.10742
\(161\) −3.54227e7 −0.668947
\(162\) −2.70265e7 −0.499444
\(163\) 1.91293e7 0.345972 0.172986 0.984924i \(-0.444658\pi\)
0.172986 + 0.984924i \(0.444658\pi\)
\(164\) 5.41666e7 0.958911
\(165\) 1.66410e8 2.88393
\(166\) −2.68548e7 −0.455662
\(167\) 6.64443e7 1.10395 0.551976 0.833860i \(-0.313874\pi\)
0.551976 + 0.833860i \(0.313874\pi\)
\(168\) 2.41038e7 0.392196
\(169\) 4.82681e6 0.0769231
\(170\) 1.81745e7 0.283721
\(171\) 1.18248e7 0.180845
\(172\) −7.01386e7 −1.05101
\(173\) 8.10105e7 1.18954 0.594771 0.803895i \(-0.297243\pi\)
0.594771 + 0.803895i \(0.297243\pi\)
\(174\) −3.06297e7 −0.440778
\(175\) −7.13571e6 −0.100648
\(176\) 6.73534e7 0.931247
\(177\) 1.72756e8 2.34167
\(178\) −5.90041e7 −0.784174
\(179\) −6.95288e7 −0.906106 −0.453053 0.891484i \(-0.649665\pi\)
−0.453053 + 0.891484i \(0.649665\pi\)
\(180\) −5.65022e7 −0.722124
\(181\) −1.08438e8 −1.35927 −0.679637 0.733548i \(-0.737863\pi\)
−0.679637 + 0.733548i \(0.737863\pi\)
\(182\) 3.64339e6 0.0447977
\(183\) 3.12790e6 0.0377289
\(184\) −1.16151e8 −1.37455
\(185\) −1.55503e8 −1.80566
\(186\) 7.27696e7 0.829192
\(187\) 1.01201e8 1.13172
\(188\) −4.08771e7 −0.448670
\(189\) 1.00727e7 0.108525
\(190\) 1.04728e7 0.110771
\(191\) 6.71166e7 0.696968 0.348484 0.937315i \(-0.386697\pi\)
0.348484 + 0.937315i \(0.386697\pi\)
\(192\) −8.50850e6 −0.0867558
\(193\) 8.36893e7 0.837953 0.418976 0.907997i \(-0.362389\pi\)
0.418976 + 0.907997i \(0.362389\pi\)
\(194\) −2.49214e7 −0.245056
\(195\) −4.31764e7 −0.416990
\(196\) −1.23090e7 −0.116768
\(197\) −9.70987e7 −0.904860 −0.452430 0.891800i \(-0.649443\pi\)
−0.452430 + 0.891800i \(0.649443\pi\)
\(198\) 7.02934e7 0.643556
\(199\) 2.29633e7 0.206561 0.103280 0.994652i \(-0.467066\pi\)
0.103280 + 0.994652i \(0.467066\pi\)
\(200\) −2.33980e7 −0.206811
\(201\) −4.44233e7 −0.385856
\(202\) 2.74090e7 0.233972
\(203\) 3.47777e7 0.291786
\(204\) −7.81283e7 −0.644322
\(205\) −1.62840e8 −1.32014
\(206\) −3.94094e7 −0.314098
\(207\) 1.77320e8 1.38951
\(208\) −1.74754e7 −0.134650
\(209\) 5.83157e7 0.441848
\(210\) −3.25905e7 −0.242842
\(211\) 7.86028e6 0.0576036 0.0288018 0.999585i \(-0.490831\pi\)
0.0288018 + 0.999585i \(0.490831\pi\)
\(212\) −3.31624e7 −0.239040
\(213\) −3.00710e8 −2.13215
\(214\) −8.42835e7 −0.587887
\(215\) 2.10856e8 1.44694
\(216\) 3.30284e7 0.222997
\(217\) −8.26244e7 −0.548908
\(218\) 1.90294e7 0.124402
\(219\) 1.99710e7 0.128483
\(220\) −2.78649e8 −1.76432
\(221\) −2.62573e7 −0.163635
\(222\) −1.49353e8 −0.916172
\(223\) 1.68887e8 1.01983 0.509916 0.860224i \(-0.329676\pi\)
0.509916 + 0.860224i \(0.329676\pi\)
\(224\) −6.25696e7 −0.371959
\(225\) 3.57202e7 0.209062
\(226\) −1.26633e8 −0.729740
\(227\) −4.61537e6 −0.0261888 −0.0130944 0.999914i \(-0.504168\pi\)
−0.0130944 + 0.999914i \(0.504168\pi\)
\(228\) −4.50205e7 −0.251558
\(229\) −3.10653e8 −1.70943 −0.854716 0.519096i \(-0.826269\pi\)
−0.854716 + 0.519096i \(0.826269\pi\)
\(230\) 1.57047e8 0.851105
\(231\) −1.81473e8 −0.968657
\(232\) 1.14036e8 0.599563
\(233\) 9.25305e7 0.479225 0.239612 0.970869i \(-0.422980\pi\)
0.239612 + 0.970869i \(0.422980\pi\)
\(234\) −1.82382e7 −0.0930521
\(235\) 1.22888e8 0.617690
\(236\) −2.89275e8 −1.43258
\(237\) 2.46942e7 0.120497
\(238\) −1.98196e7 −0.0952963
\(239\) 2.39451e8 1.13455 0.567275 0.823528i \(-0.307998\pi\)
0.567275 + 0.823528i \(0.307998\pi\)
\(240\) 1.56319e8 0.729917
\(241\) 1.76231e8 0.811005 0.405502 0.914094i \(-0.367097\pi\)
0.405502 + 0.914094i \(0.367097\pi\)
\(242\) 2.52445e8 1.14502
\(243\) −2.85047e8 −1.27437
\(244\) −5.23759e6 −0.0230817
\(245\) 3.70041e7 0.160756
\(246\) −1.56399e8 −0.669826
\(247\) −1.51305e7 −0.0638871
\(248\) −2.70926e8 −1.12790
\(249\) −3.47053e8 −1.42462
\(250\) −8.71681e7 −0.352832
\(251\) 2.42360e8 0.967393 0.483696 0.875236i \(-0.339294\pi\)
0.483696 + 0.875236i \(0.339294\pi\)
\(252\) 6.16166e7 0.242547
\(253\) 8.74482e8 3.39492
\(254\) 2.84036e7 0.108756
\(255\) 2.34875e8 0.887046
\(256\) −9.86440e7 −0.367477
\(257\) −1.66761e8 −0.612816 −0.306408 0.951900i \(-0.599127\pi\)
−0.306408 + 0.951900i \(0.599127\pi\)
\(258\) 2.02516e8 0.734160
\(259\) 1.69578e8 0.606487
\(260\) 7.22977e7 0.255104
\(261\) −1.74091e8 −0.606087
\(262\) −5.20240e6 −0.0178710
\(263\) −3.88876e8 −1.31815 −0.659077 0.752076i \(-0.729053\pi\)
−0.659077 + 0.752076i \(0.729053\pi\)
\(264\) −5.95051e8 −1.99040
\(265\) 9.96952e7 0.329089
\(266\) −1.14208e7 −0.0372059
\(267\) −7.62529e8 −2.45170
\(268\) 7.43857e7 0.236057
\(269\) −7.59623e7 −0.237939 −0.118969 0.992898i \(-0.537959\pi\)
−0.118969 + 0.992898i \(0.537959\pi\)
\(270\) −4.46574e7 −0.138077
\(271\) 4.51789e8 1.37893 0.689466 0.724318i \(-0.257845\pi\)
0.689466 + 0.724318i \(0.257845\pi\)
\(272\) 9.50642e7 0.286435
\(273\) 4.70846e7 0.140059
\(274\) −1.92012e8 −0.563898
\(275\) 1.76160e8 0.510789
\(276\) −6.75113e8 −1.93283
\(277\) −1.85269e8 −0.523750 −0.261875 0.965102i \(-0.584341\pi\)
−0.261875 + 0.965102i \(0.584341\pi\)
\(278\) 1.32419e7 0.0369652
\(279\) 4.13604e8 1.14017
\(280\) 1.21337e8 0.330323
\(281\) −1.75327e8 −0.471385 −0.235693 0.971828i \(-0.575736\pi\)
−0.235693 + 0.971828i \(0.575736\pi\)
\(282\) 1.18027e8 0.313408
\(283\) 5.10613e8 1.33918 0.669591 0.742730i \(-0.266469\pi\)
0.669591 + 0.742730i \(0.266469\pi\)
\(284\) 5.03530e8 1.30440
\(285\) 1.35344e8 0.346323
\(286\) −8.99443e7 −0.227349
\(287\) 1.77579e8 0.443411
\(288\) 3.13213e8 0.772621
\(289\) −2.67502e8 −0.651905
\(290\) −1.54187e8 −0.371241
\(291\) −3.22067e8 −0.766162
\(292\) −3.34409e7 −0.0786028
\(293\) 6.87785e8 1.59741 0.798704 0.601724i \(-0.205519\pi\)
0.798704 + 0.601724i \(0.205519\pi\)
\(294\) 3.55406e7 0.0815659
\(295\) 8.69639e8 1.97225
\(296\) 5.56049e8 1.24621
\(297\) −2.48665e8 −0.550765
\(298\) 2.57911e8 0.564563
\(299\) −2.26891e8 −0.490873
\(300\) −1.35998e8 −0.290809
\(301\) −2.29942e8 −0.485999
\(302\) −1.56201e8 −0.326332
\(303\) 3.54215e8 0.731506
\(304\) 5.47796e7 0.111831
\(305\) 1.57456e7 0.0317768
\(306\) 9.92137e7 0.197946
\(307\) −8.22599e8 −1.62257 −0.811286 0.584650i \(-0.801232\pi\)
−0.811286 + 0.584650i \(0.801232\pi\)
\(308\) 3.03872e8 0.592601
\(309\) −5.09301e8 −0.982018
\(310\) 3.66317e8 0.698378
\(311\) −3.26737e8 −0.615939 −0.307969 0.951396i \(-0.599649\pi\)
−0.307969 + 0.951396i \(0.599649\pi\)
\(312\) 1.54391e8 0.287793
\(313\) −3.79034e8 −0.698672 −0.349336 0.936998i \(-0.613593\pi\)
−0.349336 + 0.936998i \(0.613593\pi\)
\(314\) 4.72087e7 0.0860534
\(315\) −1.85236e8 −0.333918
\(316\) −4.13498e7 −0.0737171
\(317\) −1.32838e8 −0.234214 −0.117107 0.993119i \(-0.537362\pi\)
−0.117107 + 0.993119i \(0.537362\pi\)
\(318\) 9.57522e7 0.166976
\(319\) −8.58557e8 −1.48082
\(320\) −4.28311e7 −0.0730692
\(321\) −1.08922e9 −1.83801
\(322\) −1.71263e8 −0.285869
\(323\) 8.23081e7 0.135904
\(324\) 5.84846e8 0.955288
\(325\) −4.57060e7 −0.0738553
\(326\) 9.24866e7 0.147849
\(327\) 2.45923e8 0.388939
\(328\) 5.82284e8 0.911122
\(329\) −1.34011e8 −0.207470
\(330\) 8.04563e8 1.23243
\(331\) 2.57230e8 0.389873 0.194937 0.980816i \(-0.437550\pi\)
0.194937 + 0.980816i \(0.437550\pi\)
\(332\) 5.81131e8 0.871546
\(333\) −8.48882e8 −1.25977
\(334\) 3.21247e8 0.471765
\(335\) −2.23624e8 −0.324983
\(336\) −1.70469e8 −0.245165
\(337\) 3.24693e8 0.462135 0.231067 0.972938i \(-0.425778\pi\)
0.231067 + 0.972938i \(0.425778\pi\)
\(338\) 2.33368e7 0.0328725
\(339\) −1.63652e9 −2.28151
\(340\) −3.93292e8 −0.542673
\(341\) 2.03975e9 2.78571
\(342\) 5.71708e7 0.0772828
\(343\) −4.03536e7 −0.0539949
\(344\) −7.53980e8 −0.998632
\(345\) 2.02957e9 2.66096
\(346\) 3.91672e8 0.508342
\(347\) −3.43601e8 −0.441470 −0.220735 0.975334i \(-0.570846\pi\)
−0.220735 + 0.975334i \(0.570846\pi\)
\(348\) 6.62818e8 0.843076
\(349\) −1.00954e9 −1.27127 −0.635633 0.771991i \(-0.719261\pi\)
−0.635633 + 0.771991i \(0.719261\pi\)
\(350\) −3.44999e7 −0.0430111
\(351\) 6.45181e7 0.0796355
\(352\) 1.54466e9 1.88770
\(353\) 9.78169e7 0.118359 0.0591796 0.998247i \(-0.481152\pi\)
0.0591796 + 0.998247i \(0.481152\pi\)
\(354\) 8.35244e8 1.00070
\(355\) −1.51375e9 −1.79579
\(356\) 1.27683e9 1.49989
\(357\) −2.56135e8 −0.297941
\(358\) −3.36160e8 −0.387218
\(359\) 1.47626e9 1.68396 0.841978 0.539511i \(-0.181391\pi\)
0.841978 + 0.539511i \(0.181391\pi\)
\(360\) −6.07391e8 −0.686135
\(361\) −8.46443e8 −0.946940
\(362\) −5.24280e8 −0.580876
\(363\) 3.26243e9 3.57987
\(364\) −7.88419e7 −0.0856845
\(365\) 1.00532e8 0.108213
\(366\) 1.51229e7 0.0161232
\(367\) −1.45180e8 −0.153312 −0.0766559 0.997058i \(-0.524424\pi\)
−0.0766559 + 0.997058i \(0.524424\pi\)
\(368\) 8.21457e8 0.859246
\(369\) −8.88934e8 −0.921037
\(370\) −7.51829e8 −0.771637
\(371\) −1.08719e8 −0.110535
\(372\) −1.57472e9 −1.58600
\(373\) −1.71209e7 −0.0170823 −0.00854114 0.999964i \(-0.502719\pi\)
−0.00854114 + 0.999964i \(0.502719\pi\)
\(374\) 4.89287e8 0.483630
\(375\) −1.12650e9 −1.10312
\(376\) −4.39423e8 −0.426310
\(377\) 2.22759e8 0.214112
\(378\) 4.86997e7 0.0463773
\(379\) 3.62265e8 0.341814 0.170907 0.985287i \(-0.445330\pi\)
0.170907 + 0.985287i \(0.445330\pi\)
\(380\) −2.26630e8 −0.211872
\(381\) 3.67069e8 0.340024
\(382\) 3.24497e8 0.297844
\(383\) 3.36403e8 0.305960 0.152980 0.988229i \(-0.451113\pi\)
0.152980 + 0.988229i \(0.451113\pi\)
\(384\) −1.50007e9 −1.35192
\(385\) −9.13521e8 −0.815841
\(386\) 4.04623e8 0.358093
\(387\) 1.15105e9 1.00950
\(388\) 5.39293e8 0.468720
\(389\) −7.45679e7 −0.0642286 −0.0321143 0.999484i \(-0.510224\pi\)
−0.0321143 + 0.999484i \(0.510224\pi\)
\(390\) −2.08750e8 −0.178197
\(391\) 1.23426e9 1.04421
\(392\) −1.32320e8 −0.110949
\(393\) −6.72323e7 −0.0558733
\(394\) −4.69455e8 −0.386685
\(395\) 1.24309e8 0.101487
\(396\) −1.52113e9 −1.23093
\(397\) 1.66485e9 1.33539 0.667696 0.744434i \(-0.267281\pi\)
0.667696 + 0.744434i \(0.267281\pi\)
\(398\) 1.11023e8 0.0882722
\(399\) −1.47595e8 −0.116323
\(400\) 1.65478e8 0.129280
\(401\) 9.52020e8 0.737294 0.368647 0.929569i \(-0.379821\pi\)
0.368647 + 0.929569i \(0.379821\pi\)
\(402\) −2.14779e8 −0.164893
\(403\) −5.29230e8 −0.402788
\(404\) −5.93124e8 −0.447518
\(405\) −1.75821e9 −1.31516
\(406\) 1.68144e8 0.124692
\(407\) −4.18639e9 −3.07793
\(408\) −8.39869e8 −0.612211
\(409\) −1.66648e9 −1.20439 −0.602197 0.798348i \(-0.705708\pi\)
−0.602197 + 0.798348i \(0.705708\pi\)
\(410\) −7.87301e8 −0.564154
\(411\) −2.48143e9 −1.76301
\(412\) 8.52810e8 0.600775
\(413\) −9.48357e8 −0.662440
\(414\) 8.57314e8 0.593798
\(415\) −1.74704e9 −1.19987
\(416\) −4.00774e8 −0.272943
\(417\) 1.71129e8 0.115571
\(418\) 2.81946e8 0.188821
\(419\) 7.54096e8 0.500815 0.250408 0.968141i \(-0.419435\pi\)
0.250408 + 0.968141i \(0.419435\pi\)
\(420\) 7.05251e8 0.464484
\(421\) 9.66522e8 0.631284 0.315642 0.948878i \(-0.397780\pi\)
0.315642 + 0.948878i \(0.397780\pi\)
\(422\) 3.80031e7 0.0246165
\(423\) 6.70837e8 0.430949
\(424\) −3.56492e8 −0.227127
\(425\) 2.48636e8 0.157109
\(426\) −1.45388e9 −0.911160
\(427\) −1.71709e7 −0.0106732
\(428\) 1.82387e9 1.12445
\(429\) −1.16238e9 −0.710800
\(430\) 1.01945e9 0.618339
\(431\) 1.15524e9 0.695025 0.347512 0.937675i \(-0.387026\pi\)
0.347512 + 0.937675i \(0.387026\pi\)
\(432\) −2.33587e8 −0.139398
\(433\) −1.01905e9 −0.603237 −0.301619 0.953429i \(-0.597527\pi\)
−0.301619 + 0.953429i \(0.597527\pi\)
\(434\) −3.99475e8 −0.234572
\(435\) −1.99261e9 −1.16067
\(436\) −4.11791e8 −0.237944
\(437\) 7.11231e8 0.407686
\(438\) 9.65563e7 0.0549062
\(439\) 2.05507e9 1.15931 0.579655 0.814862i \(-0.303187\pi\)
0.579655 + 0.814862i \(0.303187\pi\)
\(440\) −2.99544e9 −1.67639
\(441\) 2.02004e8 0.112156
\(442\) −1.26950e8 −0.0699283
\(443\) 1.38350e9 0.756079 0.378040 0.925789i \(-0.376598\pi\)
0.378040 + 0.925789i \(0.376598\pi\)
\(444\) 3.23195e9 1.75236
\(445\) −3.83851e9 −2.06492
\(446\) 8.16539e8 0.435818
\(447\) 3.33306e9 1.76509
\(448\) 4.67081e7 0.0245425
\(449\) 1.79991e9 0.938404 0.469202 0.883091i \(-0.344542\pi\)
0.469202 + 0.883091i \(0.344542\pi\)
\(450\) 1.72701e8 0.0893411
\(451\) −4.38391e9 −2.25032
\(452\) 2.74031e9 1.39578
\(453\) −2.01864e9 −1.02027
\(454\) −2.23145e7 −0.0111916
\(455\) 2.37020e8 0.117963
\(456\) −4.83965e8 −0.239021
\(457\) 1.35945e9 0.666279 0.333139 0.942878i \(-0.391892\pi\)
0.333139 + 0.942878i \(0.391892\pi\)
\(458\) −1.50196e9 −0.730513
\(459\) −3.50971e8 −0.169405
\(460\) −3.39846e9 −1.62791
\(461\) 9.44068e8 0.448797 0.224399 0.974497i \(-0.427958\pi\)
0.224399 + 0.974497i \(0.427958\pi\)
\(462\) −8.77390e8 −0.413948
\(463\) −2.11591e8 −0.0990751 −0.0495375 0.998772i \(-0.515775\pi\)
−0.0495375 + 0.998772i \(0.515775\pi\)
\(464\) −8.06497e8 −0.374792
\(465\) 4.73403e9 2.18346
\(466\) 4.47369e8 0.204793
\(467\) −1.11706e9 −0.507535 −0.253768 0.967265i \(-0.581670\pi\)
−0.253768 + 0.967265i \(0.581670\pi\)
\(468\) 3.94670e8 0.177981
\(469\) 2.43866e8 0.109155
\(470\) 5.94140e8 0.263965
\(471\) 6.10093e8 0.269044
\(472\) −3.10967e9 −1.36118
\(473\) 5.67658e9 2.46645
\(474\) 1.19392e8 0.0514934
\(475\) 1.43273e8 0.0613392
\(476\) 4.28891e8 0.182273
\(477\) 5.44231e8 0.229599
\(478\) 1.15770e9 0.484842
\(479\) −2.45192e9 −1.01937 −0.509684 0.860361i \(-0.670238\pi\)
−0.509684 + 0.860361i \(0.670238\pi\)
\(480\) 3.58497e9 1.47959
\(481\) 1.08619e9 0.445040
\(482\) 8.52048e8 0.346577
\(483\) −2.21328e9 −0.893763
\(484\) −5.46285e9 −2.19008
\(485\) −1.62126e9 −0.645292
\(486\) −1.37815e9 −0.544592
\(487\) −3.55244e9 −1.39372 −0.696860 0.717207i \(-0.745420\pi\)
−0.696860 + 0.717207i \(0.745420\pi\)
\(488\) −5.63034e7 −0.0219313
\(489\) 1.19523e9 0.462245
\(490\) 1.78908e8 0.0686980
\(491\) −8.10999e8 −0.309197 −0.154598 0.987977i \(-0.549408\pi\)
−0.154598 + 0.987977i \(0.549408\pi\)
\(492\) 3.38444e9 1.28118
\(493\) −1.21179e9 −0.455472
\(494\) −7.31532e7 −0.0273017
\(495\) 4.57293e9 1.69464
\(496\) 1.91607e9 0.705058
\(497\) 1.65077e9 0.603169
\(498\) −1.67794e9 −0.608799
\(499\) −3.77182e9 −1.35894 −0.679468 0.733705i \(-0.737789\pi\)
−0.679468 + 0.733705i \(0.737789\pi\)
\(500\) 1.88630e9 0.674861
\(501\) 4.15157e9 1.47496
\(502\) 1.17177e9 0.413408
\(503\) −4.69213e9 −1.64393 −0.821963 0.569541i \(-0.807121\pi\)
−0.821963 + 0.569541i \(0.807121\pi\)
\(504\) 6.62371e8 0.230459
\(505\) 1.78309e9 0.616104
\(506\) 4.22797e9 1.45079
\(507\) 3.01589e8 0.102775
\(508\) −6.14646e8 −0.208019
\(509\) 2.19417e9 0.737494 0.368747 0.929530i \(-0.379787\pi\)
0.368747 + 0.929530i \(0.379787\pi\)
\(510\) 1.13558e9 0.379072
\(511\) −1.09632e8 −0.0363468
\(512\) 2.59609e9 0.854822
\(513\) −2.02243e8 −0.0661398
\(514\) −8.06263e8 −0.261882
\(515\) −2.56378e9 −0.827095
\(516\) −4.38240e9 −1.40423
\(517\) 3.30833e9 1.05291
\(518\) 8.19883e8 0.259178
\(519\) 5.06170e9 1.58932
\(520\) 7.77191e8 0.242391
\(521\) −2.61146e9 −0.809005 −0.404503 0.914537i \(-0.632555\pi\)
−0.404503 + 0.914537i \(0.632555\pi\)
\(522\) −8.41701e8 −0.259007
\(523\) −2.69127e9 −0.822625 −0.411312 0.911495i \(-0.634929\pi\)
−0.411312 + 0.911495i \(0.634929\pi\)
\(524\) 1.12579e8 0.0341819
\(525\) −4.45854e8 −0.134473
\(526\) −1.88015e9 −0.563303
\(527\) 2.87895e9 0.856835
\(528\) 4.20837e9 1.24422
\(529\) 7.26055e9 2.13243
\(530\) 4.82009e8 0.140634
\(531\) 4.74732e9 1.37600
\(532\) 2.47144e8 0.0711638
\(533\) 1.13744e9 0.325374
\(534\) −3.68670e9 −1.04771
\(535\) −5.48306e9 −1.54805
\(536\) 7.99637e8 0.224293
\(537\) −4.34430e9 −1.21063
\(538\) −3.67265e8 −0.101681
\(539\) 9.96210e8 0.274025
\(540\) 9.66376e8 0.264100
\(541\) −4.26647e9 −1.15845 −0.579227 0.815166i \(-0.696645\pi\)
−0.579227 + 0.815166i \(0.696645\pi\)
\(542\) 2.18432e9 0.589276
\(543\) −6.77544e9 −1.81609
\(544\) 2.18017e9 0.580622
\(545\) 1.23796e9 0.327580
\(546\) 2.27646e8 0.0598530
\(547\) 5.32779e9 1.39185 0.695923 0.718116i \(-0.254995\pi\)
0.695923 + 0.718116i \(0.254995\pi\)
\(548\) 4.15508e9 1.07857
\(549\) 8.59545e7 0.0221700
\(550\) 8.51701e8 0.218282
\(551\) −6.98278e8 −0.177827
\(552\) −7.25737e9 −1.83651
\(553\) −1.35561e8 −0.0340876
\(554\) −8.95744e8 −0.223820
\(555\) −9.71612e9 −2.41250
\(556\) −2.86551e8 −0.0707035
\(557\) 3.19673e9 0.783812 0.391906 0.920005i \(-0.371816\pi\)
0.391906 + 0.920005i \(0.371816\pi\)
\(558\) 1.99970e9 0.487244
\(559\) −1.47283e9 −0.356626
\(560\) −8.58128e8 −0.206488
\(561\) 6.32321e9 1.51206
\(562\) −8.47674e8 −0.201443
\(563\) 2.46611e9 0.582417 0.291208 0.956660i \(-0.405943\pi\)
0.291208 + 0.956660i \(0.405943\pi\)
\(564\) −2.55408e9 −0.599457
\(565\) −8.23813e9 −1.92158
\(566\) 2.46873e9 0.572289
\(567\) 1.91736e9 0.441735
\(568\) 5.41288e9 1.23939
\(569\) −7.64209e9 −1.73908 −0.869539 0.493864i \(-0.835584\pi\)
−0.869539 + 0.493864i \(0.835584\pi\)
\(570\) 6.54365e8 0.147999
\(571\) −2.39383e9 −0.538104 −0.269052 0.963126i \(-0.586710\pi\)
−0.269052 + 0.963126i \(0.586710\pi\)
\(572\) 1.94637e9 0.434850
\(573\) 4.19358e9 0.931201
\(574\) 8.58566e8 0.189488
\(575\) 2.14848e9 0.471296
\(576\) −2.33813e8 −0.0509788
\(577\) 4.29479e9 0.930736 0.465368 0.885117i \(-0.345922\pi\)
0.465368 + 0.885117i \(0.345922\pi\)
\(578\) −1.29333e9 −0.278587
\(579\) 5.22908e9 1.11957
\(580\) 3.33657e9 0.710073
\(581\) 1.90517e9 0.403012
\(582\) −1.55714e9 −0.327414
\(583\) 2.68396e9 0.560965
\(584\) −3.59485e8 −0.0746854
\(585\) −1.18648e9 −0.245029
\(586\) 3.32532e9 0.682640
\(587\) −4.21222e9 −0.859563 −0.429782 0.902933i \(-0.641409\pi\)
−0.429782 + 0.902933i \(0.641409\pi\)
\(588\) −7.69089e8 −0.156011
\(589\) 1.65896e9 0.334529
\(590\) 4.20455e9 0.842826
\(591\) −6.06692e9 −1.20896
\(592\) −3.93254e9 −0.779018
\(593\) −1.53014e9 −0.301328 −0.150664 0.988585i \(-0.548141\pi\)
−0.150664 + 0.988585i \(0.548141\pi\)
\(594\) −1.20225e9 −0.235365
\(595\) −1.28936e9 −0.250938
\(596\) −5.58113e9 −1.07984
\(597\) 1.43479e9 0.275981
\(598\) −1.09698e9 −0.209771
\(599\) −5.16267e9 −0.981478 −0.490739 0.871307i \(-0.663273\pi\)
−0.490739 + 0.871307i \(0.663273\pi\)
\(600\) −1.46196e9 −0.276316
\(601\) −1.95738e9 −0.367803 −0.183901 0.982945i \(-0.558873\pi\)
−0.183901 + 0.982945i \(0.558873\pi\)
\(602\) −1.11173e9 −0.207688
\(603\) −1.22075e9 −0.226734
\(604\) 3.38016e9 0.624176
\(605\) 1.64228e10 3.01511
\(606\) 1.71257e9 0.312604
\(607\) 4.00519e9 0.726880 0.363440 0.931618i \(-0.381602\pi\)
0.363440 + 0.931618i \(0.381602\pi\)
\(608\) 1.25630e9 0.226688
\(609\) 2.17298e9 0.389847
\(610\) 7.61273e7 0.0135796
\(611\) −8.58374e8 −0.152241
\(612\) −2.14696e9 −0.378612
\(613\) 2.44078e9 0.427974 0.213987 0.976837i \(-0.431355\pi\)
0.213987 + 0.976837i \(0.431355\pi\)
\(614\) −3.97712e9 −0.693394
\(615\) −1.01745e10 −1.76381
\(616\) 3.26658e9 0.563068
\(617\) 7.35176e9 1.26007 0.630033 0.776569i \(-0.283042\pi\)
0.630033 + 0.776569i \(0.283042\pi\)
\(618\) −2.46238e9 −0.419658
\(619\) −4.08633e9 −0.692495 −0.346248 0.938143i \(-0.612544\pi\)
−0.346248 + 0.938143i \(0.612544\pi\)
\(620\) −7.92700e9 −1.33579
\(621\) −3.03277e9 −0.508182
\(622\) −1.57972e9 −0.263217
\(623\) 4.18596e9 0.693565
\(624\) −1.09190e9 −0.179902
\(625\) −7.29602e9 −1.19538
\(626\) −1.83257e9 −0.298572
\(627\) 3.64368e9 0.590342
\(628\) −1.02158e9 −0.164595
\(629\) −5.90876e9 −0.946715
\(630\) −8.95586e8 −0.142697
\(631\) 2.99656e9 0.474811 0.237406 0.971411i \(-0.423703\pi\)
0.237406 + 0.971411i \(0.423703\pi\)
\(632\) −4.44504e8 −0.0700432
\(633\) 4.91126e8 0.0769627
\(634\) −6.42247e8 −0.100090
\(635\) 1.84779e9 0.286382
\(636\) −2.07205e9 −0.319375
\(637\) −2.58475e8 −0.0396214
\(638\) −4.15097e9 −0.632816
\(639\) −8.26348e9 −1.25288
\(640\) −7.55122e9 −1.13864
\(641\) 3.86937e9 0.580279 0.290140 0.956984i \(-0.406298\pi\)
0.290140 + 0.956984i \(0.406298\pi\)
\(642\) −5.26620e9 −0.785461
\(643\) −6.05858e9 −0.898736 −0.449368 0.893347i \(-0.648351\pi\)
−0.449368 + 0.893347i \(0.648351\pi\)
\(644\) 3.70609e9 0.546783
\(645\) 1.31747e10 1.93322
\(646\) 3.97945e8 0.0580777
\(647\) −1.17641e10 −1.70763 −0.853817 0.520573i \(-0.825718\pi\)
−0.853817 + 0.520573i \(0.825718\pi\)
\(648\) 6.28702e9 0.907679
\(649\) 2.34121e10 3.36189
\(650\) −2.20981e8 −0.0315615
\(651\) −5.16254e9 −0.733381
\(652\) −2.00139e9 −0.282790
\(653\) −1.25932e9 −0.176987 −0.0884934 0.996077i \(-0.528205\pi\)
−0.0884934 + 0.996077i \(0.528205\pi\)
\(654\) 1.18899e9 0.166210
\(655\) −3.38442e8 −0.0470587
\(656\) −4.11808e9 −0.569550
\(657\) 5.48802e8 0.0754982
\(658\) −6.47920e8 −0.0886607
\(659\) 9.16362e9 1.24729 0.623646 0.781707i \(-0.285651\pi\)
0.623646 + 0.781707i \(0.285651\pi\)
\(660\) −1.74105e10 −2.35727
\(661\) 1.30199e10 1.75348 0.876742 0.480961i \(-0.159712\pi\)
0.876742 + 0.480961i \(0.159712\pi\)
\(662\) 1.24366e9 0.166609
\(663\) −1.64061e9 −0.218629
\(664\) 6.24708e9 0.828111
\(665\) −7.42982e8 −0.0979721
\(666\) −4.10420e9 −0.538355
\(667\) −1.04711e10 −1.36632
\(668\) −6.95170e9 −0.902346
\(669\) 1.05524e10 1.36257
\(670\) −1.08118e9 −0.138879
\(671\) 4.23898e8 0.0541667
\(672\) −3.90947e9 −0.496965
\(673\) −1.04391e10 −1.32011 −0.660053 0.751219i \(-0.729466\pi\)
−0.660053 + 0.751219i \(0.729466\pi\)
\(674\) 1.56984e9 0.197490
\(675\) −6.10935e8 −0.0764595
\(676\) −5.05002e8 −0.0628753
\(677\) −8.55686e9 −1.05987 −0.529937 0.848037i \(-0.677784\pi\)
−0.529937 + 0.848037i \(0.677784\pi\)
\(678\) −7.91230e9 −0.974987
\(679\) 1.76801e9 0.216741
\(680\) −4.22783e9 −0.515628
\(681\) −2.88378e8 −0.0349902
\(682\) 9.86184e9 1.19045
\(683\) 1.80307e9 0.216540 0.108270 0.994122i \(-0.465469\pi\)
0.108270 + 0.994122i \(0.465469\pi\)
\(684\) −1.23716e9 −0.147819
\(685\) −1.24913e10 −1.48488
\(686\) −1.95103e8 −0.0230743
\(687\) −1.94102e10 −2.28393
\(688\) 5.33237e9 0.624253
\(689\) −6.96375e8 −0.0811102
\(690\) 9.81263e9 1.13714
\(691\) 1.68072e9 0.193786 0.0968928 0.995295i \(-0.469110\pi\)
0.0968928 + 0.995295i \(0.469110\pi\)
\(692\) −8.47568e9 −0.972306
\(693\) −4.98686e9 −0.569195
\(694\) −1.66125e9 −0.188659
\(695\) 8.61451e8 0.0973384
\(696\) 7.12521e9 0.801060
\(697\) −6.18755e9 −0.692156
\(698\) −4.88097e9 −0.543266
\(699\) 5.78149e9 0.640280
\(700\) 7.46570e8 0.0822673
\(701\) −1.60557e10 −1.76041 −0.880207 0.474590i \(-0.842596\pi\)
−0.880207 + 0.474590i \(0.842596\pi\)
\(702\) 3.11934e8 0.0340316
\(703\) −3.40486e9 −0.369620
\(704\) −1.15308e9 −0.124554
\(705\) 7.67826e9 0.825279
\(706\) 4.72928e8 0.0505799
\(707\) −1.94449e9 −0.206937
\(708\) −1.80745e10 −1.91403
\(709\) −7.44466e9 −0.784482 −0.392241 0.919862i \(-0.628300\pi\)
−0.392241 + 0.919862i \(0.628300\pi\)
\(710\) −7.31871e9 −0.767416
\(711\) 6.78595e8 0.0708055
\(712\) 1.37258e10 1.42514
\(713\) 2.48772e10 2.57033
\(714\) −1.23837e9 −0.127323
\(715\) −5.85133e9 −0.598664
\(716\) 7.27441e9 0.740632
\(717\) 1.49614e10 1.51584
\(718\) 7.13744e9 0.719626
\(719\) 1.47292e10 1.47784 0.738922 0.673791i \(-0.235335\pi\)
0.738922 + 0.673791i \(0.235335\pi\)
\(720\) 4.29565e9 0.428909
\(721\) 2.79585e9 0.277805
\(722\) −4.09240e9 −0.404667
\(723\) 1.10113e10 1.08356
\(724\) 1.13453e10 1.11104
\(725\) −2.10935e9 −0.205573
\(726\) 1.57733e10 1.52983
\(727\) −2.84326e9 −0.274439 −0.137219 0.990541i \(-0.543817\pi\)
−0.137219 + 0.990541i \(0.543817\pi\)
\(728\) −8.47541e8 −0.0814143
\(729\) −5.58509e9 −0.533930
\(730\) 4.86057e8 0.0462442
\(731\) 8.01205e9 0.758635
\(732\) −3.27255e8 −0.0308388
\(733\) 5.86487e9 0.550041 0.275020 0.961438i \(-0.411315\pi\)
0.275020 + 0.961438i \(0.411315\pi\)
\(734\) −7.01920e8 −0.0655166
\(735\) 2.31209e9 0.214783
\(736\) 1.88390e10 1.74175
\(737\) −6.02031e9 −0.553966
\(738\) −4.29784e9 −0.393598
\(739\) 1.71343e10 1.56175 0.780873 0.624690i \(-0.214775\pi\)
0.780873 + 0.624690i \(0.214775\pi\)
\(740\) 1.62694e10 1.47591
\(741\) −9.45382e8 −0.0853579
\(742\) −5.25640e8 −0.0472361
\(743\) 8.14736e9 0.728712 0.364356 0.931260i \(-0.381289\pi\)
0.364356 + 0.931260i \(0.381289\pi\)
\(744\) −1.69280e10 −1.50695
\(745\) 1.67784e10 1.48663
\(746\) −8.27766e7 −0.00729998
\(747\) −9.53699e9 −0.837123
\(748\) −1.05881e10 −0.925040
\(749\) 5.97937e9 0.519959
\(750\) −5.44644e9 −0.471409
\(751\) 1.07000e10 0.921815 0.460907 0.887448i \(-0.347524\pi\)
0.460907 + 0.887448i \(0.347524\pi\)
\(752\) 3.10773e9 0.266490
\(753\) 1.51431e10 1.29251
\(754\) 1.07700e9 0.0914992
\(755\) −1.01617e10 −0.859311
\(756\) −1.05385e9 −0.0887059
\(757\) −6.06443e9 −0.508106 −0.254053 0.967190i \(-0.581764\pi\)
−0.254053 + 0.967190i \(0.581764\pi\)
\(758\) 1.75149e9 0.146071
\(759\) 5.46394e10 4.53586
\(760\) −2.43624e9 −0.201313
\(761\) −1.38928e10 −1.14273 −0.571364 0.820697i \(-0.693586\pi\)
−0.571364 + 0.820697i \(0.693586\pi\)
\(762\) 1.77471e9 0.145307
\(763\) −1.35001e9 −0.110028
\(764\) −7.02203e9 −0.569687
\(765\) 6.45435e9 0.521239
\(766\) 1.62645e9 0.130749
\(767\) −6.07446e9 −0.486098
\(768\) −6.16347e9 −0.490977
\(769\) 5.83542e9 0.462733 0.231366 0.972867i \(-0.425680\pi\)
0.231366 + 0.972867i \(0.425680\pi\)
\(770\) −4.41671e9 −0.348644
\(771\) −1.04196e10 −0.818767
\(772\) −8.75595e9 −0.684924
\(773\) −1.21327e10 −0.944778 −0.472389 0.881390i \(-0.656608\pi\)
−0.472389 + 0.881390i \(0.656608\pi\)
\(774\) 5.56513e9 0.431402
\(775\) 5.01138e9 0.386724
\(776\) 5.79732e9 0.445360
\(777\) 1.05956e10 0.810312
\(778\) −3.60523e8 −0.0274476
\(779\) −3.56551e9 −0.270234
\(780\) 4.51731e9 0.340838
\(781\) −4.07526e10 −3.06109
\(782\) 5.96746e9 0.446237
\(783\) 2.97754e9 0.221662
\(784\) 9.35804e8 0.0693551
\(785\) 3.07116e9 0.226599
\(786\) −3.25057e8 −0.0238770
\(787\) 6.77656e9 0.495562 0.247781 0.968816i \(-0.420299\pi\)
0.247781 + 0.968816i \(0.420299\pi\)
\(788\) 1.01589e10 0.739613
\(789\) −2.42978e10 −1.76115
\(790\) 6.01011e8 0.0433698
\(791\) 8.98382e9 0.645421
\(792\) −1.63520e10 −1.16958
\(793\) −1.09984e8 −0.00783199
\(794\) 8.04927e9 0.570670
\(795\) 6.22916e9 0.439688
\(796\) −2.40252e9 −0.168838
\(797\) −1.31908e10 −0.922925 −0.461462 0.887160i \(-0.652675\pi\)
−0.461462 + 0.887160i \(0.652675\pi\)
\(798\) −7.13596e8 −0.0497098
\(799\) 4.66946e9 0.323857
\(800\) 3.79501e9 0.262058
\(801\) −2.09542e10 −1.44065
\(802\) 4.60285e9 0.315077
\(803\) 2.70650e9 0.184460
\(804\) 4.64777e9 0.315390
\(805\) −1.11415e10 −0.752763
\(806\) −2.55873e9 −0.172128
\(807\) −4.74628e9 −0.317904
\(808\) −6.37601e9 −0.425215
\(809\) 3.68694e8 0.0244820 0.0122410 0.999925i \(-0.496103\pi\)
0.0122410 + 0.999925i \(0.496103\pi\)
\(810\) −8.50063e9 −0.562022
\(811\) 9.58211e9 0.630795 0.315397 0.948960i \(-0.397862\pi\)
0.315397 + 0.948960i \(0.397862\pi\)
\(812\) −3.63859e9 −0.238499
\(813\) 2.82287e10 1.84236
\(814\) −2.02405e10 −1.31533
\(815\) 6.01672e9 0.389321
\(816\) 5.93980e9 0.382698
\(817\) 4.61685e9 0.296189
\(818\) −8.05714e9 −0.514688
\(819\) 1.29388e9 0.0823003
\(820\) 1.70370e10 1.07906
\(821\) −2.19021e9 −0.138129 −0.0690646 0.997612i \(-0.522001\pi\)
−0.0690646 + 0.997612i \(0.522001\pi\)
\(822\) −1.19973e10 −0.753409
\(823\) −1.01062e10 −0.631957 −0.315978 0.948766i \(-0.602333\pi\)
−0.315978 + 0.948766i \(0.602333\pi\)
\(824\) 9.16760e9 0.570835
\(825\) 1.10068e10 0.682452
\(826\) −4.58514e9 −0.283089
\(827\) −1.63226e10 −1.00351 −0.501753 0.865011i \(-0.667312\pi\)
−0.501753 + 0.865011i \(0.667312\pi\)
\(828\) −1.85521e10 −1.13576
\(829\) −2.02220e10 −1.23277 −0.616387 0.787444i \(-0.711404\pi\)
−0.616387 + 0.787444i \(0.711404\pi\)
\(830\) −8.44662e9 −0.512755
\(831\) −1.15760e10 −0.699768
\(832\) 2.99177e8 0.0180093
\(833\) 1.40607e9 0.0842851
\(834\) 8.27381e8 0.0493883
\(835\) 2.08987e10 1.24227
\(836\) −6.10124e9 −0.361157
\(837\) −7.07401e9 −0.416991
\(838\) 3.64592e9 0.214020
\(839\) −9.25923e9 −0.541263 −0.270631 0.962683i \(-0.587232\pi\)
−0.270631 + 0.962683i \(0.587232\pi\)
\(840\) 7.58136e9 0.441336
\(841\) −6.96942e9 −0.404027
\(842\) 4.67297e9 0.269774
\(843\) −1.09548e10 −0.629806
\(844\) −8.22377e8 −0.0470839
\(845\) 1.51817e9 0.0865612
\(846\) 3.24338e9 0.184163
\(847\) −1.79094e10 −1.01272
\(848\) 2.52121e9 0.141979
\(849\) 3.19041e10 1.78925
\(850\) 1.20211e9 0.0671395
\(851\) −5.10581e10 −2.83995
\(852\) 3.14616e10 1.74278
\(853\) −1.03242e10 −0.569552 −0.284776 0.958594i \(-0.591919\pi\)
−0.284776 + 0.958594i \(0.591919\pi\)
\(854\) −8.30182e7 −0.00456111
\(855\) 3.71924e9 0.203504
\(856\) 1.96064e10 1.06841
\(857\) 1.34846e10 0.731824 0.365912 0.930649i \(-0.380757\pi\)
0.365912 + 0.930649i \(0.380757\pi\)
\(858\) −5.61990e9 −0.303755
\(859\) −7.04555e9 −0.379262 −0.189631 0.981855i \(-0.560729\pi\)
−0.189631 + 0.981855i \(0.560729\pi\)
\(860\) −2.20607e10 −1.18270
\(861\) 1.10955e10 0.592430
\(862\) 5.58537e9 0.297014
\(863\) 5.76366e9 0.305254 0.152627 0.988284i \(-0.451227\pi\)
0.152627 + 0.988284i \(0.451227\pi\)
\(864\) −5.35699e9 −0.282568
\(865\) 2.54802e10 1.33859
\(866\) −4.92693e9 −0.257789
\(867\) −1.67141e10 −0.870993
\(868\) 8.64453e9 0.448665
\(869\) 3.34659e9 0.172995
\(870\) −9.63393e9 −0.496005
\(871\) 1.56202e9 0.0800982
\(872\) −4.42670e9 −0.226085
\(873\) −8.85038e9 −0.450207
\(874\) 3.43868e9 0.174221
\(875\) 6.18402e9 0.312063
\(876\) −2.08945e9 −0.105019
\(877\) 2.42411e10 1.21354 0.606771 0.794877i \(-0.292465\pi\)
0.606771 + 0.794877i \(0.292465\pi\)
\(878\) 9.93588e9 0.495423
\(879\) 4.29742e10 2.13426
\(880\) 2.11846e10 1.04793
\(881\) −1.64052e10 −0.808285 −0.404143 0.914696i \(-0.632430\pi\)
−0.404143 + 0.914696i \(0.632430\pi\)
\(882\) 9.76652e8 0.0479292
\(883\) 2.40299e10 1.17460 0.587299 0.809370i \(-0.300191\pi\)
0.587299 + 0.809370i \(0.300191\pi\)
\(884\) 2.74716e9 0.133752
\(885\) 5.43368e10 2.63507
\(886\) 6.68900e9 0.323105
\(887\) 2.04477e10 0.983813 0.491906 0.870648i \(-0.336300\pi\)
0.491906 + 0.870648i \(0.336300\pi\)
\(888\) 3.47430e10 1.66503
\(889\) −2.01505e9 −0.0961901
\(890\) −1.85585e10 −0.882427
\(891\) −4.73338e10 −2.24181
\(892\) −1.76697e10 −0.833589
\(893\) 2.69072e9 0.126441
\(894\) 1.61148e10 0.754299
\(895\) −2.18689e10 −1.01964
\(896\) 8.23474e9 0.382447
\(897\) −1.41766e10 −0.655843
\(898\) 8.70228e9 0.401020
\(899\) −2.44242e10 −1.12114
\(900\) −3.73721e9 −0.170883
\(901\) 3.78820e9 0.172543
\(902\) −2.11954e10 −0.961655
\(903\) −1.43672e10 −0.649330
\(904\) 2.94580e10 1.32621
\(905\) −3.41070e10 −1.52959
\(906\) −9.75977e9 −0.436004
\(907\) 1.70630e10 0.759329 0.379664 0.925124i \(-0.376040\pi\)
0.379664 + 0.925124i \(0.376040\pi\)
\(908\) 4.82880e8 0.0214062
\(909\) 9.73381e9 0.429843
\(910\) 1.14595e9 0.0504106
\(911\) −5.54686e9 −0.243071 −0.121536 0.992587i \(-0.538782\pi\)
−0.121536 + 0.992587i \(0.538782\pi\)
\(912\) 3.42274e9 0.149414
\(913\) −4.70331e10 −2.04529
\(914\) 6.57270e9 0.284729
\(915\) 9.83818e8 0.0424562
\(916\) 3.25019e10 1.39725
\(917\) 3.69077e8 0.0158061
\(918\) −1.69689e9 −0.0723941
\(919\) −1.28731e10 −0.547115 −0.273557 0.961856i \(-0.588200\pi\)
−0.273557 + 0.961856i \(0.588200\pi\)
\(920\) −3.65330e10 −1.54678
\(921\) −5.13976e10 −2.16788
\(922\) 4.56441e9 0.191790
\(923\) 1.05736e10 0.442605
\(924\) 1.89865e10 0.791759
\(925\) −1.02854e10 −0.427291
\(926\) −1.02301e9 −0.0423390
\(927\) −1.39955e10 −0.577046
\(928\) −1.84959e10 −0.759727
\(929\) 3.77350e10 1.54415 0.772075 0.635531i \(-0.219219\pi\)
0.772075 + 0.635531i \(0.219219\pi\)
\(930\) 2.28882e10 0.933085
\(931\) 8.10234e8 0.0329069
\(932\) −9.68096e9 −0.391708
\(933\) −2.04152e10 −0.822940
\(934\) −5.40078e9 −0.216891
\(935\) 3.18306e10 1.27351
\(936\) 4.24265e9 0.169111
\(937\) −1.35680e10 −0.538801 −0.269401 0.963028i \(-0.586826\pi\)
−0.269401 + 0.963028i \(0.586826\pi\)
\(938\) 1.17905e9 0.0466468
\(939\) −2.36828e10 −0.933478
\(940\) −1.28570e10 −0.504886
\(941\) 1.66279e10 0.650541 0.325270 0.945621i \(-0.394545\pi\)
0.325270 + 0.945621i \(0.394545\pi\)
\(942\) 2.94969e9 0.114974
\(943\) −5.34671e10 −2.07633
\(944\) 2.19925e10 0.850888
\(945\) 3.16816e9 0.122122
\(946\) 2.74453e10 1.05402
\(947\) −3.94921e10 −1.51107 −0.755536 0.655107i \(-0.772624\pi\)
−0.755536 + 0.655107i \(0.772624\pi\)
\(948\) −2.58362e9 −0.0984915
\(949\) −7.02222e8 −0.0266712
\(950\) 6.92703e8 0.0262128
\(951\) −8.29996e9 −0.312928
\(952\) 4.61053e9 0.173189
\(953\) −5.06925e10 −1.89723 −0.948613 0.316439i \(-0.897513\pi\)
−0.948613 + 0.316439i \(0.897513\pi\)
\(954\) 2.63126e9 0.0981172
\(955\) 2.11101e10 0.784295
\(956\) −2.50524e10 −0.927357
\(957\) −5.36443e10 −1.97848
\(958\) −1.18546e10 −0.435619
\(959\) 1.36220e10 0.498741
\(960\) −2.67617e9 −0.0976259
\(961\) 3.05142e10 1.10910
\(962\) 5.25155e9 0.190184
\(963\) −2.99317e10 −1.08004
\(964\) −1.84381e10 −0.662898
\(965\) 2.63228e10 0.942944
\(966\) −1.07008e10 −0.381943
\(967\) 4.34587e10 1.54555 0.772776 0.634679i \(-0.218868\pi\)
0.772776 + 0.634679i \(0.218868\pi\)
\(968\) −5.87249e10 −2.08093
\(969\) 5.14278e9 0.181578
\(970\) −7.83851e9 −0.275761
\(971\) −2.42378e10 −0.849623 −0.424812 0.905282i \(-0.639660\pi\)
−0.424812 + 0.905282i \(0.639660\pi\)
\(972\) 2.98229e10 1.04164
\(973\) −9.39428e8 −0.0326940
\(974\) −1.71754e10 −0.595596
\(975\) −2.85580e9 −0.0986762
\(976\) 3.98194e8 0.0137095
\(977\) 5.82477e10 1.99824 0.999120 0.0419334i \(-0.0133517\pi\)
0.999120 + 0.0419334i \(0.0133517\pi\)
\(978\) 5.77875e9 0.197537
\(979\) −1.03339e11 −3.51985
\(980\) −3.87153e9 −0.131399
\(981\) 6.75794e9 0.228546
\(982\) −3.92104e9 −0.132133
\(983\) −3.52029e10 −1.18207 −0.591033 0.806648i \(-0.701280\pi\)
−0.591033 + 0.806648i \(0.701280\pi\)
\(984\) 3.63823e10 1.21733
\(985\) −3.05404e10 −1.01823
\(986\) −5.85878e9 −0.194643
\(987\) −8.37328e9 −0.277195
\(988\) 1.58302e9 0.0522199
\(989\) 6.92328e10 2.27575
\(990\) 2.21093e10 0.724190
\(991\) 2.65405e10 0.866266 0.433133 0.901330i \(-0.357408\pi\)
0.433133 + 0.901330i \(0.357408\pi\)
\(992\) 4.39424e10 1.42920
\(993\) 1.60722e10 0.520900
\(994\) 7.98119e9 0.257760
\(995\) 7.22263e9 0.232442
\(996\) 3.63102e10 1.16445
\(997\) −3.04787e9 −0.0974009 −0.0487004 0.998813i \(-0.515508\pi\)
−0.0487004 + 0.998813i \(0.515508\pi\)
\(998\) −1.82361e10 −0.580731
\(999\) 1.45187e10 0.460732
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 91.8.a.d.1.6 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.8.a.d.1.6 10 1.1 even 1 trivial