Properties

Label 91.8.a.d.1.5
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 1.5
Root \(3.66228\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.66228 q^{2} +42.3449 q^{3} -114.588 q^{4} -251.753 q^{5} -155.079 q^{6} -343.000 q^{7} +888.424 q^{8} -393.907 q^{9} +O(q^{10})\) \(q-3.66228 q^{2} +42.3449 q^{3} -114.588 q^{4} -251.753 q^{5} -155.079 q^{6} -343.000 q^{7} +888.424 q^{8} -393.907 q^{9} +921.992 q^{10} -2926.60 q^{11} -4852.21 q^{12} -2197.00 q^{13} +1256.16 q^{14} -10660.5 q^{15} +11413.6 q^{16} +27794.8 q^{17} +1442.60 q^{18} +34316.1 q^{19} +28847.8 q^{20} -14524.3 q^{21} +10718.0 q^{22} -21674.2 q^{23} +37620.3 q^{24} -14745.2 q^{25} +8046.03 q^{26} -109288. q^{27} +39303.6 q^{28} +115035. q^{29} +39041.7 q^{30} +13017.1 q^{31} -155518. q^{32} -123927. q^{33} -101793. q^{34} +86351.4 q^{35} +45136.9 q^{36} +481169. q^{37} -125675. q^{38} -93031.8 q^{39} -223664. q^{40} +316118. q^{41} +53192.1 q^{42} +75163.9 q^{43} +335353. q^{44} +99167.4 q^{45} +79377.0 q^{46} +500395. q^{47} +483307. q^{48} +117649. q^{49} +54001.2 q^{50} +1.17697e6 q^{51} +251749. q^{52} -466960. q^{53} +400245. q^{54} +736782. q^{55} -304730. q^{56} +1.45311e6 q^{57} -421289. q^{58} +1.34767e6 q^{59} +1.22156e6 q^{60} +2.22618e6 q^{61} -47672.2 q^{62} +135110. q^{63} -891386. q^{64} +553102. q^{65} +453855. q^{66} -2.36745e6 q^{67} -3.18495e6 q^{68} -917792. q^{69} -316243. q^{70} +619229. q^{71} -349957. q^{72} +4.50615e6 q^{73} -1.76217e6 q^{74} -624386. q^{75} -3.93221e6 q^{76} +1.00383e6 q^{77} +340709. q^{78} -188250. q^{79} -2.87340e6 q^{80} -3.76633e6 q^{81} -1.15771e6 q^{82} -5.10902e6 q^{83} +1.66431e6 q^{84} -6.99745e6 q^{85} -275271. q^{86} +4.87114e6 q^{87} -2.60007e6 q^{88} +1.60970e6 q^{89} -363179. q^{90} +753571. q^{91} +2.48360e6 q^{92} +551207. q^{93} -1.83259e6 q^{94} -8.63921e6 q^{95} -6.58540e6 q^{96} -811101. q^{97} -430864. q^{98} +1.15281e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{2} - 101 q^{3} + 361 q^{4} + 226 q^{5} + 1105 q^{6} - 3430 q^{7} + 291 q^{8} + 12247 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.66228 −0.323703 −0.161851 0.986815i \(-0.551747\pi\)
−0.161851 + 0.986815i \(0.551747\pi\)
\(3\) 42.3449 0.905476 0.452738 0.891644i \(-0.350447\pi\)
0.452738 + 0.891644i \(0.350447\pi\)
\(4\) −114.588 −0.895216
\(5\) −251.753 −0.900700 −0.450350 0.892852i \(-0.648701\pi\)
−0.450350 + 0.892852i \(0.648701\pi\)
\(6\) −155.079 −0.293105
\(7\) −343.000 −0.377964
\(8\) 888.424 0.613487
\(9\) −393.907 −0.180113
\(10\) 921.992 0.291559
\(11\) −2926.60 −0.662963 −0.331482 0.943462i \(-0.607549\pi\)
−0.331482 + 0.943462i \(0.607549\pi\)
\(12\) −4852.21 −0.810597
\(13\) −2197.00 −0.277350
\(14\) 1256.16 0.122348
\(15\) −10660.5 −0.815563
\(16\) 11413.6 0.696629
\(17\) 27794.8 1.37212 0.686061 0.727544i \(-0.259338\pi\)
0.686061 + 0.727544i \(0.259338\pi\)
\(18\) 1442.60 0.0583031
\(19\) 34316.1 1.14779 0.573893 0.818930i \(-0.305433\pi\)
0.573893 + 0.818930i \(0.305433\pi\)
\(20\) 28847.8 0.806322
\(21\) −14524.3 −0.342238
\(22\) 10718.0 0.214603
\(23\) −21674.2 −0.371446 −0.185723 0.982602i \(-0.559463\pi\)
−0.185723 + 0.982602i \(0.559463\pi\)
\(24\) 37620.3 0.555498
\(25\) −14745.2 −0.188739
\(26\) 8046.03 0.0897790
\(27\) −109288. −1.06856
\(28\) 39303.6 0.338360
\(29\) 115035. 0.875862 0.437931 0.899009i \(-0.355711\pi\)
0.437931 + 0.899009i \(0.355711\pi\)
\(30\) 39041.7 0.264000
\(31\) 13017.1 0.0784779 0.0392390 0.999230i \(-0.487507\pi\)
0.0392390 + 0.999230i \(0.487507\pi\)
\(32\) −155518. −0.838988
\(33\) −123927. −0.600298
\(34\) −101793. −0.444160
\(35\) 86351.4 0.340433
\(36\) 45136.9 0.161240
\(37\) 481169. 1.56168 0.780839 0.624733i \(-0.214792\pi\)
0.780839 + 0.624733i \(0.214792\pi\)
\(38\) −125675. −0.371542
\(39\) −93031.8 −0.251134
\(40\) −223664. −0.552568
\(41\) 316118. 0.716317 0.358158 0.933661i \(-0.383405\pi\)
0.358158 + 0.933661i \(0.383405\pi\)
\(42\) 53192.1 0.110783
\(43\) 75163.9 0.144168 0.0720841 0.997399i \(-0.477035\pi\)
0.0720841 + 0.997399i \(0.477035\pi\)
\(44\) 335353. 0.593496
\(45\) 99167.4 0.162228
\(46\) 79377.0 0.120238
\(47\) 500395. 0.703025 0.351512 0.936183i \(-0.385668\pi\)
0.351512 + 0.936183i \(0.385668\pi\)
\(48\) 483307. 0.630781
\(49\) 117649. 0.142857
\(50\) 54001.2 0.0610954
\(51\) 1.17697e6 1.24242
\(52\) 251749. 0.248288
\(53\) −466960. −0.430838 −0.215419 0.976522i \(-0.569112\pi\)
−0.215419 + 0.976522i \(0.569112\pi\)
\(54\) 400245. 0.345897
\(55\) 736782. 0.597131
\(56\) −304730. −0.231876
\(57\) 1.45311e6 1.03929
\(58\) −421289. −0.283519
\(59\) 1.34767e6 0.854280 0.427140 0.904186i \(-0.359521\pi\)
0.427140 + 0.904186i \(0.359521\pi\)
\(60\) 1.22156e6 0.730105
\(61\) 2.22618e6 1.25576 0.627879 0.778311i \(-0.283923\pi\)
0.627879 + 0.778311i \(0.283923\pi\)
\(62\) −47672.2 −0.0254035
\(63\) 135110. 0.0680763
\(64\) −891386. −0.425046
\(65\) 553102. 0.249809
\(66\) 453855. 0.194318
\(67\) −2.36745e6 −0.961654 −0.480827 0.876816i \(-0.659663\pi\)
−0.480827 + 0.876816i \(0.659663\pi\)
\(68\) −3.18495e6 −1.22835
\(69\) −917792. −0.336335
\(70\) −316243. −0.110199
\(71\) 619229. 0.205327 0.102664 0.994716i \(-0.467263\pi\)
0.102664 + 0.994716i \(0.467263\pi\)
\(72\) −349957. −0.110497
\(73\) 4.50615e6 1.35574 0.677868 0.735183i \(-0.262904\pi\)
0.677868 + 0.735183i \(0.262904\pi\)
\(74\) −1.76217e6 −0.505520
\(75\) −624386. −0.170899
\(76\) −3.93221e6 −1.02752
\(77\) 1.00383e6 0.250577
\(78\) 340709. 0.0812928
\(79\) −188250. −0.0429577 −0.0214789 0.999769i \(-0.506837\pi\)
−0.0214789 + 0.999769i \(0.506837\pi\)
\(80\) −2.87340e6 −0.627454
\(81\) −3.76633e6 −0.787446
\(82\) −1.15771e6 −0.231874
\(83\) −5.10902e6 −0.980763 −0.490381 0.871508i \(-0.663143\pi\)
−0.490381 + 0.871508i \(0.663143\pi\)
\(84\) 1.66431e6 0.306377
\(85\) −6.99745e6 −1.23587
\(86\) −275271. −0.0466677
\(87\) 4.87114e6 0.793072
\(88\) −2.60007e6 −0.406720
\(89\) 1.60970e6 0.242036 0.121018 0.992650i \(-0.461384\pi\)
0.121018 + 0.992650i \(0.461384\pi\)
\(90\) −363179. −0.0525136
\(91\) 753571. 0.104828
\(92\) 2.48360e6 0.332524
\(93\) 551207. 0.0710599
\(94\) −1.83259e6 −0.227571
\(95\) −8.63921e6 −1.03381
\(96\) −6.58540e6 −0.759684
\(97\) −811101. −0.0902348 −0.0451174 0.998982i \(-0.514366\pi\)
−0.0451174 + 0.998982i \(0.514366\pi\)
\(98\) −430864. −0.0462433
\(99\) 1.15281e6 0.119408
\(100\) 1.68962e6 0.168962
\(101\) −2.55347e6 −0.246608 −0.123304 0.992369i \(-0.539349\pi\)
−0.123304 + 0.992369i \(0.539349\pi\)
\(102\) −4.31040e6 −0.402176
\(103\) 1.20270e7 1.08449 0.542245 0.840220i \(-0.317574\pi\)
0.542245 + 0.840220i \(0.317574\pi\)
\(104\) −1.95187e6 −0.170151
\(105\) 3.65654e6 0.308254
\(106\) 1.71014e6 0.139464
\(107\) 4.99152e6 0.393904 0.196952 0.980413i \(-0.436896\pi\)
0.196952 + 0.980413i \(0.436896\pi\)
\(108\) 1.25231e7 0.956596
\(109\) −1.63927e7 −1.21243 −0.606217 0.795299i \(-0.707314\pi\)
−0.606217 + 0.795299i \(0.707314\pi\)
\(110\) −2.69830e6 −0.193293
\(111\) 2.03751e7 1.41406
\(112\) −3.91485e6 −0.263301
\(113\) −6.66918e6 −0.434808 −0.217404 0.976082i \(-0.569759\pi\)
−0.217404 + 0.976082i \(0.569759\pi\)
\(114\) −5.32171e6 −0.336422
\(115\) 5.45655e6 0.334561
\(116\) −1.31816e7 −0.784086
\(117\) 865414. 0.0499544
\(118\) −4.93553e6 −0.276533
\(119\) −9.53363e6 −0.518614
\(120\) −9.47103e6 −0.500337
\(121\) −1.09222e7 −0.560480
\(122\) −8.15290e6 −0.406493
\(123\) 1.33860e7 0.648608
\(124\) −1.49160e6 −0.0702547
\(125\) 2.33804e7 1.07070
\(126\) −494811. −0.0220365
\(127\) −4.30152e7 −1.86341 −0.931707 0.363212i \(-0.881680\pi\)
−0.931707 + 0.363212i \(0.881680\pi\)
\(128\) 2.31708e7 0.976577
\(129\) 3.18281e6 0.130541
\(130\) −2.02562e6 −0.0808640
\(131\) −647043. −0.0251469 −0.0125734 0.999921i \(-0.504002\pi\)
−0.0125734 + 0.999921i \(0.504002\pi\)
\(132\) 1.42005e7 0.537396
\(133\) −1.17704e7 −0.433822
\(134\) 8.67026e6 0.311290
\(135\) 2.75137e7 0.962456
\(136\) 2.46936e7 0.841780
\(137\) −1.45963e7 −0.484976 −0.242488 0.970154i \(-0.577963\pi\)
−0.242488 + 0.970154i \(0.577963\pi\)
\(138\) 3.36121e6 0.108873
\(139\) −1.58494e7 −0.500567 −0.250284 0.968173i \(-0.580524\pi\)
−0.250284 + 0.968173i \(0.580524\pi\)
\(140\) −9.89481e6 −0.304761
\(141\) 2.11892e7 0.636572
\(142\) −2.26779e6 −0.0664651
\(143\) 6.42975e6 0.183873
\(144\) −4.49589e6 −0.125472
\(145\) −2.89604e7 −0.788889
\(146\) −1.65028e7 −0.438856
\(147\) 4.98184e6 0.129354
\(148\) −5.51360e7 −1.39804
\(149\) −4.00703e7 −0.992363 −0.496182 0.868219i \(-0.665265\pi\)
−0.496182 + 0.868219i \(0.665265\pi\)
\(150\) 2.28668e6 0.0553204
\(151\) 7.97551e7 1.88512 0.942561 0.334035i \(-0.108410\pi\)
0.942561 + 0.334035i \(0.108410\pi\)
\(152\) 3.04873e7 0.704152
\(153\) −1.09486e7 −0.247137
\(154\) −3.67629e6 −0.0811124
\(155\) −3.27709e6 −0.0706851
\(156\) 1.06603e7 0.224819
\(157\) −1.82222e7 −0.375796 −0.187898 0.982189i \(-0.560167\pi\)
−0.187898 + 0.982189i \(0.560167\pi\)
\(158\) 689426. 0.0139055
\(159\) −1.97734e7 −0.390114
\(160\) 3.91522e7 0.755677
\(161\) 7.43425e6 0.140393
\(162\) 1.37934e7 0.254899
\(163\) 9.13519e7 1.65219 0.826097 0.563528i \(-0.190556\pi\)
0.826097 + 0.563528i \(0.190556\pi\)
\(164\) −3.62232e7 −0.641258
\(165\) 3.11990e7 0.540688
\(166\) 1.87107e7 0.317476
\(167\) 6.09136e7 1.01206 0.506030 0.862516i \(-0.331112\pi\)
0.506030 + 0.862516i \(0.331112\pi\)
\(168\) −1.29037e7 −0.209958
\(169\) 4.82681e6 0.0769231
\(170\) 2.56266e7 0.400055
\(171\) −1.35174e7 −0.206731
\(172\) −8.61286e6 −0.129062
\(173\) 1.26181e8 1.85282 0.926408 0.376520i \(-0.122880\pi\)
0.926408 + 0.376520i \(0.122880\pi\)
\(174\) −1.78395e7 −0.256720
\(175\) 5.05762e6 0.0713367
\(176\) −3.34030e7 −0.461839
\(177\) 5.70668e7 0.773530
\(178\) −5.89517e6 −0.0783478
\(179\) 2.72460e7 0.355073 0.177536 0.984114i \(-0.443187\pi\)
0.177536 + 0.984114i \(0.443187\pi\)
\(180\) −1.13634e7 −0.145229
\(181\) 8.01601e7 1.00481 0.502404 0.864633i \(-0.332449\pi\)
0.502404 + 0.864633i \(0.332449\pi\)
\(182\) −2.75979e6 −0.0339333
\(183\) 9.42674e7 1.13706
\(184\) −1.92559e7 −0.227877
\(185\) −1.21136e8 −1.40660
\(186\) −2.01868e6 −0.0230023
\(187\) −8.13445e7 −0.909667
\(188\) −5.73391e7 −0.629359
\(189\) 3.74859e7 0.403879
\(190\) 3.16392e7 0.334648
\(191\) 6.94392e7 0.721088 0.360544 0.932742i \(-0.382591\pi\)
0.360544 + 0.932742i \(0.382591\pi\)
\(192\) −3.77457e7 −0.384869
\(193\) 4.35489e7 0.436040 0.218020 0.975944i \(-0.430040\pi\)
0.218020 + 0.975944i \(0.430040\pi\)
\(194\) 2.97048e6 0.0292093
\(195\) 2.34211e7 0.226196
\(196\) −1.34811e7 −0.127888
\(197\) 8.12640e7 0.757298 0.378649 0.925540i \(-0.376389\pi\)
0.378649 + 0.925540i \(0.376389\pi\)
\(198\) −4.22191e6 −0.0386528
\(199\) −3.16714e6 −0.0284893 −0.0142446 0.999899i \(-0.504534\pi\)
−0.0142446 + 0.999899i \(0.504534\pi\)
\(200\) −1.31000e7 −0.115789
\(201\) −1.00249e8 −0.870755
\(202\) 9.35154e6 0.0798276
\(203\) −3.94569e7 −0.331045
\(204\) −1.34866e8 −1.11224
\(205\) −7.95837e7 −0.645187
\(206\) −4.40461e7 −0.351053
\(207\) 8.53762e6 0.0669022
\(208\) −2.50756e7 −0.193210
\(209\) −1.00430e8 −0.760940
\(210\) −1.33913e7 −0.0997826
\(211\) −1.68103e8 −1.23193 −0.615966 0.787772i \(-0.711234\pi\)
−0.615966 + 0.787772i \(0.711234\pi\)
\(212\) 5.35079e7 0.385694
\(213\) 2.62212e7 0.185919
\(214\) −1.82804e7 −0.127508
\(215\) −1.89228e7 −0.129852
\(216\) −9.70944e7 −0.655550
\(217\) −4.46486e6 −0.0296619
\(218\) 6.00347e7 0.392469
\(219\) 1.90812e8 1.22759
\(220\) −8.44262e7 −0.534562
\(221\) −6.10653e7 −0.380558
\(222\) −7.46192e7 −0.457736
\(223\) 7.63326e7 0.460938 0.230469 0.973080i \(-0.425974\pi\)
0.230469 + 0.973080i \(0.425974\pi\)
\(224\) 5.33427e7 0.317108
\(225\) 5.80826e6 0.0339944
\(226\) 2.44244e7 0.140749
\(227\) 4.57199e7 0.259427 0.129714 0.991552i \(-0.458594\pi\)
0.129714 + 0.991552i \(0.458594\pi\)
\(228\) −1.66509e8 −0.930392
\(229\) 1.99587e8 1.09827 0.549133 0.835735i \(-0.314958\pi\)
0.549133 + 0.835735i \(0.314958\pi\)
\(230\) −1.99834e7 −0.108298
\(231\) 4.25069e7 0.226891
\(232\) 1.02200e8 0.537330
\(233\) 6.01329e7 0.311434 0.155717 0.987802i \(-0.450231\pi\)
0.155717 + 0.987802i \(0.450231\pi\)
\(234\) −3.16939e6 −0.0161704
\(235\) −1.25976e8 −0.633215
\(236\) −1.54426e8 −0.764765
\(237\) −7.97145e6 −0.0388972
\(238\) 3.49148e7 0.167877
\(239\) 2.93560e8 1.39093 0.695464 0.718561i \(-0.255199\pi\)
0.695464 + 0.718561i \(0.255199\pi\)
\(240\) −1.21674e8 −0.568144
\(241\) −4.20654e8 −1.93582 −0.967910 0.251297i \(-0.919143\pi\)
−0.967910 + 0.251297i \(0.919143\pi\)
\(242\) 4.00000e7 0.181429
\(243\) 7.95285e7 0.355550
\(244\) −2.55093e8 −1.12418
\(245\) −2.96185e7 −0.128671
\(246\) −4.90232e7 −0.209956
\(247\) −7.53926e7 −0.318339
\(248\) 1.15647e7 0.0481452
\(249\) −2.16341e8 −0.888057
\(250\) −8.56256e7 −0.346588
\(251\) 4.62621e8 1.84658 0.923289 0.384106i \(-0.125490\pi\)
0.923289 + 0.384106i \(0.125490\pi\)
\(252\) −1.54820e7 −0.0609430
\(253\) 6.34318e7 0.246255
\(254\) 1.57534e8 0.603192
\(255\) −2.96306e8 −1.11905
\(256\) 2.92394e7 0.108925
\(257\) −2.21165e8 −0.812738 −0.406369 0.913709i \(-0.633205\pi\)
−0.406369 + 0.913709i \(0.633205\pi\)
\(258\) −1.16563e7 −0.0422565
\(259\) −1.65041e8 −0.590259
\(260\) −6.33787e7 −0.223633
\(261\) −4.53130e7 −0.157754
\(262\) 2.36965e6 0.00814011
\(263\) 5.16193e8 1.74971 0.874857 0.484381i \(-0.160955\pi\)
0.874857 + 0.484381i \(0.160955\pi\)
\(264\) −1.10100e8 −0.368275
\(265\) 1.17559e8 0.388056
\(266\) 4.31067e7 0.140430
\(267\) 6.81626e7 0.219158
\(268\) 2.71280e8 0.860888
\(269\) −5.39449e8 −1.68973 −0.844865 0.534979i \(-0.820320\pi\)
−0.844865 + 0.534979i \(0.820320\pi\)
\(270\) −1.00763e8 −0.311550
\(271\) −1.64435e8 −0.501882 −0.250941 0.968002i \(-0.580740\pi\)
−0.250941 + 0.968002i \(0.580740\pi\)
\(272\) 3.17238e8 0.955860
\(273\) 3.19099e7 0.0949197
\(274\) 5.34557e7 0.156988
\(275\) 4.31535e7 0.125127
\(276\) 1.05168e8 0.301093
\(277\) −4.88376e8 −1.38062 −0.690312 0.723512i \(-0.742527\pi\)
−0.690312 + 0.723512i \(0.742527\pi\)
\(278\) 5.80451e7 0.162035
\(279\) −5.12752e6 −0.0141349
\(280\) 7.67167e7 0.208851
\(281\) 5.32145e8 1.43073 0.715365 0.698751i \(-0.246261\pi\)
0.715365 + 0.698751i \(0.246261\pi\)
\(282\) −7.76008e7 −0.206060
\(283\) −6.50933e8 −1.70720 −0.853599 0.520930i \(-0.825585\pi\)
−0.853599 + 0.520930i \(0.825585\pi\)
\(284\) −7.09560e7 −0.183813
\(285\) −3.65827e8 −0.936091
\(286\) −2.35475e7 −0.0595202
\(287\) −1.08428e8 −0.270742
\(288\) 6.12596e7 0.151113
\(289\) 3.62215e8 0.882721
\(290\) 1.06061e8 0.255366
\(291\) −3.43460e7 −0.0817054
\(292\) −5.16349e8 −1.21368
\(293\) 6.98043e8 1.62123 0.810616 0.585578i \(-0.199132\pi\)
0.810616 + 0.585578i \(0.199132\pi\)
\(294\) −1.82449e7 −0.0418722
\(295\) −3.39279e8 −0.769450
\(296\) 4.27482e8 0.958069
\(297\) 3.19844e8 0.708419
\(298\) 1.46749e8 0.321231
\(299\) 4.76182e7 0.103021
\(300\) 7.15470e7 0.152991
\(301\) −2.57812e7 −0.0544905
\(302\) −2.92086e8 −0.610219
\(303\) −1.08127e8 −0.223297
\(304\) 3.91670e8 0.799581
\(305\) −5.60448e8 −1.13106
\(306\) 4.00968e7 0.0799990
\(307\) 2.24284e6 0.00442398 0.00221199 0.999998i \(-0.499296\pi\)
0.00221199 + 0.999998i \(0.499296\pi\)
\(308\) −1.15026e8 −0.224320
\(309\) 5.09281e8 0.981981
\(310\) 1.20016e7 0.0228810
\(311\) 1.35544e8 0.255517 0.127758 0.991805i \(-0.459222\pi\)
0.127758 + 0.991805i \(0.459222\pi\)
\(312\) −8.26517e7 −0.154067
\(313\) 4.00803e8 0.738798 0.369399 0.929271i \(-0.379564\pi\)
0.369399 + 0.929271i \(0.379564\pi\)
\(314\) 6.67347e7 0.121646
\(315\) −3.40144e7 −0.0613164
\(316\) 2.15712e7 0.0384564
\(317\) 9.32010e8 1.64329 0.821643 0.570002i \(-0.193058\pi\)
0.821643 + 0.570002i \(0.193058\pi\)
\(318\) 7.24157e7 0.126281
\(319\) −3.36661e8 −0.580665
\(320\) 2.24409e8 0.382839
\(321\) 2.11366e8 0.356670
\(322\) −2.72263e7 −0.0454457
\(323\) 9.53812e8 1.57490
\(324\) 4.31575e8 0.704935
\(325\) 3.23953e7 0.0523468
\(326\) −3.34556e8 −0.534820
\(327\) −6.94149e8 −1.09783
\(328\) 2.80846e8 0.439451
\(329\) −1.71636e8 −0.265718
\(330\) −1.14259e8 −0.175022
\(331\) 4.77925e8 0.724372 0.362186 0.932106i \(-0.382031\pi\)
0.362186 + 0.932106i \(0.382031\pi\)
\(332\) 5.85430e8 0.877995
\(333\) −1.89536e8 −0.281278
\(334\) −2.23083e8 −0.327607
\(335\) 5.96013e8 0.866162
\(336\) −1.65774e8 −0.238413
\(337\) 3.00880e8 0.428242 0.214121 0.976807i \(-0.431311\pi\)
0.214121 + 0.976807i \(0.431311\pi\)
\(338\) −1.76771e7 −0.0249002
\(339\) −2.82406e8 −0.393709
\(340\) 8.01821e8 1.10637
\(341\) −3.80958e7 −0.0520280
\(342\) 4.95044e7 0.0669195
\(343\) −4.03536e7 −0.0539949
\(344\) 6.67774e7 0.0884454
\(345\) 2.31057e8 0.302937
\(346\) −4.62110e8 −0.599762
\(347\) 1.04913e8 0.134796 0.0673980 0.997726i \(-0.478530\pi\)
0.0673980 + 0.997726i \(0.478530\pi\)
\(348\) −5.58172e8 −0.709971
\(349\) −1.54627e8 −0.194713 −0.0973567 0.995250i \(-0.531039\pi\)
−0.0973567 + 0.995250i \(0.531039\pi\)
\(350\) −1.85224e7 −0.0230919
\(351\) 2.40106e8 0.296366
\(352\) 4.55140e8 0.556218
\(353\) 1.29968e9 1.57262 0.786312 0.617830i \(-0.211988\pi\)
0.786312 + 0.617830i \(0.211988\pi\)
\(354\) −2.08995e8 −0.250394
\(355\) −1.55893e8 −0.184939
\(356\) −1.84452e8 −0.216675
\(357\) −4.03701e8 −0.469592
\(358\) −9.97825e7 −0.114938
\(359\) −1.35313e9 −1.54351 −0.771755 0.635920i \(-0.780621\pi\)
−0.771755 + 0.635920i \(0.780621\pi\)
\(360\) 8.81028e7 0.0995247
\(361\) 2.83726e8 0.317413
\(362\) −2.93569e8 −0.325259
\(363\) −4.62498e8 −0.507501
\(364\) −8.63500e7 −0.0938442
\(365\) −1.13444e9 −1.22111
\(366\) −3.45234e8 −0.368069
\(367\) −1.45759e8 −0.153923 −0.0769617 0.997034i \(-0.524522\pi\)
−0.0769617 + 0.997034i \(0.524522\pi\)
\(368\) −2.47380e8 −0.258760
\(369\) −1.24521e8 −0.129018
\(370\) 4.43633e8 0.455322
\(371\) 1.60167e8 0.162842
\(372\) −6.31616e7 −0.0636140
\(373\) 8.21893e8 0.820039 0.410019 0.912077i \(-0.365522\pi\)
0.410019 + 0.912077i \(0.365522\pi\)
\(374\) 2.97906e8 0.294462
\(375\) 9.90041e8 0.969491
\(376\) 4.44563e8 0.431297
\(377\) −2.52731e8 −0.242920
\(378\) −1.37284e8 −0.130737
\(379\) −3.42342e8 −0.323015 −0.161508 0.986871i \(-0.551636\pi\)
−0.161508 + 0.986871i \(0.551636\pi\)
\(380\) 9.89947e8 0.925485
\(381\) −1.82148e9 −1.68728
\(382\) −2.54306e8 −0.233418
\(383\) −3.24840e8 −0.295443 −0.147722 0.989029i \(-0.547194\pi\)
−0.147722 + 0.989029i \(0.547194\pi\)
\(384\) 9.81166e8 0.884267
\(385\) −2.52716e8 −0.225694
\(386\) −1.59488e8 −0.141147
\(387\) −2.96076e7 −0.0259666
\(388\) 9.29422e7 0.0807796
\(389\) −7.62926e8 −0.657141 −0.328571 0.944479i \(-0.606567\pi\)
−0.328571 + 0.944479i \(0.606567\pi\)
\(390\) −8.57745e7 −0.0732204
\(391\) −6.02431e8 −0.509669
\(392\) 1.04522e8 0.0876410
\(393\) −2.73990e7 −0.0227699
\(394\) −2.97612e8 −0.245139
\(395\) 4.73927e7 0.0386920
\(396\) −1.32098e8 −0.106896
\(397\) 3.17876e8 0.254971 0.127486 0.991840i \(-0.459309\pi\)
0.127486 + 0.991840i \(0.459309\pi\)
\(398\) 1.15990e7 0.00922207
\(399\) −4.98418e8 −0.392816
\(400\) −1.68296e8 −0.131481
\(401\) −1.32306e8 −0.102464 −0.0512322 0.998687i \(-0.516315\pi\)
−0.0512322 + 0.998687i \(0.516315\pi\)
\(402\) 3.67142e8 0.281866
\(403\) −2.85985e7 −0.0217659
\(404\) 2.92597e8 0.220767
\(405\) 9.48187e8 0.709253
\(406\) 1.44502e8 0.107160
\(407\) −1.40819e9 −1.03533
\(408\) 1.04565e9 0.762211
\(409\) −1.26662e9 −0.915405 −0.457702 0.889105i \(-0.651328\pi\)
−0.457702 + 0.889105i \(0.651328\pi\)
\(410\) 2.91458e8 0.208849
\(411\) −6.18078e8 −0.439134
\(412\) −1.37814e9 −0.970854
\(413\) −4.62249e8 −0.322887
\(414\) −3.12672e7 −0.0216564
\(415\) 1.28621e9 0.883373
\(416\) 3.41673e8 0.232693
\(417\) −6.71144e8 −0.453252
\(418\) 3.67802e8 0.246319
\(419\) −8.44907e8 −0.561125 −0.280563 0.959836i \(-0.590521\pi\)
−0.280563 + 0.959836i \(0.590521\pi\)
\(420\) −4.18995e8 −0.275954
\(421\) 9.22607e8 0.602601 0.301300 0.953529i \(-0.402579\pi\)
0.301300 + 0.953529i \(0.402579\pi\)
\(422\) 6.15641e8 0.398780
\(423\) −1.97109e8 −0.126624
\(424\) −4.14859e8 −0.264314
\(425\) −4.09842e8 −0.258973
\(426\) −9.60294e7 −0.0601826
\(427\) −7.63580e8 −0.474632
\(428\) −5.71967e8 −0.352629
\(429\) 2.72267e8 0.166493
\(430\) 6.93005e7 0.0420336
\(431\) 6.54823e8 0.393961 0.196981 0.980407i \(-0.436886\pi\)
0.196981 + 0.980407i \(0.436886\pi\)
\(432\) −1.24737e9 −0.744393
\(433\) −3.21139e9 −1.90101 −0.950507 0.310702i \(-0.899436\pi\)
−0.950507 + 0.310702i \(0.899436\pi\)
\(434\) 1.63516e7 0.00960163
\(435\) −1.22632e9 −0.714320
\(436\) 1.87840e9 1.08539
\(437\) −7.43775e8 −0.426340
\(438\) −6.98809e8 −0.397374
\(439\) −2.88389e9 −1.62687 −0.813434 0.581658i \(-0.802404\pi\)
−0.813434 + 0.581658i \(0.802404\pi\)
\(440\) 6.54575e8 0.366332
\(441\) −4.63428e7 −0.0257304
\(442\) 2.23638e8 0.123188
\(443\) 1.64951e9 0.901452 0.450726 0.892662i \(-0.351165\pi\)
0.450726 + 0.892662i \(0.351165\pi\)
\(444\) −2.33473e9 −1.26589
\(445\) −4.05247e8 −0.218002
\(446\) −2.79551e8 −0.149207
\(447\) −1.69677e9 −0.898561
\(448\) 3.05745e8 0.160652
\(449\) 8.94046e8 0.466120 0.233060 0.972462i \(-0.425126\pi\)
0.233060 + 0.972462i \(0.425126\pi\)
\(450\) −2.12715e7 −0.0110041
\(451\) −9.25151e8 −0.474892
\(452\) 7.64206e8 0.389248
\(453\) 3.37722e9 1.70693
\(454\) −1.67439e8 −0.0839773
\(455\) −1.89714e8 −0.0944190
\(456\) 1.29098e9 0.637593
\(457\) 2.84806e9 1.39586 0.697930 0.716166i \(-0.254105\pi\)
0.697930 + 0.716166i \(0.254105\pi\)
\(458\) −7.30943e8 −0.355512
\(459\) −3.03765e9 −1.46620
\(460\) −6.25254e8 −0.299505
\(461\) −8.96558e8 −0.426212 −0.213106 0.977029i \(-0.568358\pi\)
−0.213106 + 0.977029i \(0.568358\pi\)
\(462\) −1.55672e8 −0.0734453
\(463\) 7.62965e8 0.357249 0.178625 0.983917i \(-0.442835\pi\)
0.178625 + 0.983917i \(0.442835\pi\)
\(464\) 1.31296e9 0.610151
\(465\) −1.38768e8 −0.0640037
\(466\) −2.20223e8 −0.100812
\(467\) 2.52211e9 1.14592 0.572960 0.819583i \(-0.305795\pi\)
0.572960 + 0.819583i \(0.305795\pi\)
\(468\) −9.91658e7 −0.0447200
\(469\) 8.12035e8 0.363471
\(470\) 4.61360e8 0.204973
\(471\) −7.71617e8 −0.340274
\(472\) 1.19730e9 0.524090
\(473\) −2.19975e8 −0.0955783
\(474\) 2.91937e7 0.0125911
\(475\) −5.06000e8 −0.216632
\(476\) 1.09244e9 0.464271
\(477\) 1.83939e8 0.0775996
\(478\) −1.07510e9 −0.450247
\(479\) 7.30274e7 0.0303607 0.0151803 0.999885i \(-0.495168\pi\)
0.0151803 + 0.999885i \(0.495168\pi\)
\(480\) 1.65790e9 0.684247
\(481\) −1.05713e9 −0.433131
\(482\) 1.54055e9 0.626631
\(483\) 3.14803e8 0.127123
\(484\) 1.25155e9 0.501750
\(485\) 2.04197e8 0.0812745
\(486\) −2.91256e8 −0.115093
\(487\) −1.94137e9 −0.761653 −0.380826 0.924647i \(-0.624360\pi\)
−0.380826 + 0.924647i \(0.624360\pi\)
\(488\) 1.97779e9 0.770391
\(489\) 3.86829e9 1.49602
\(490\) 1.08471e8 0.0416513
\(491\) 4.02561e9 1.53478 0.767391 0.641180i \(-0.221555\pi\)
0.767391 + 0.641180i \(0.221555\pi\)
\(492\) −1.53387e9 −0.580644
\(493\) 3.19737e9 1.20179
\(494\) 2.76109e8 0.103047
\(495\) −2.90224e8 −0.107551
\(496\) 1.48571e8 0.0546700
\(497\) −2.12396e8 −0.0776065
\(498\) 7.92301e8 0.287467
\(499\) −2.79089e9 −1.00552 −0.502761 0.864426i \(-0.667682\pi\)
−0.502761 + 0.864426i \(0.667682\pi\)
\(500\) −2.67911e9 −0.958506
\(501\) 2.57938e9 0.916397
\(502\) −1.69425e9 −0.597743
\(503\) 4.23835e9 1.48494 0.742470 0.669879i \(-0.233654\pi\)
0.742470 + 0.669879i \(0.233654\pi\)
\(504\) 1.20035e8 0.0417639
\(505\) 6.42846e8 0.222120
\(506\) −2.32305e8 −0.0797135
\(507\) 2.04391e8 0.0696520
\(508\) 4.92902e9 1.66816
\(509\) 3.80443e9 1.27873 0.639363 0.768905i \(-0.279198\pi\)
0.639363 + 0.768905i \(0.279198\pi\)
\(510\) 1.08516e9 0.362240
\(511\) −1.54561e9 −0.512420
\(512\) −3.07295e9 −1.01184
\(513\) −3.75035e9 −1.22648
\(514\) 8.09968e8 0.263086
\(515\) −3.02783e9 −0.976801
\(516\) −3.64711e8 −0.116862
\(517\) −1.46446e9 −0.466080
\(518\) 6.04426e8 0.191068
\(519\) 5.34312e9 1.67768
\(520\) 4.91389e8 0.153255
\(521\) 7.33979e8 0.227380 0.113690 0.993516i \(-0.463733\pi\)
0.113690 + 0.993516i \(0.463733\pi\)
\(522\) 1.65949e8 0.0510655
\(523\) −3.33450e9 −1.01924 −0.509619 0.860400i \(-0.670213\pi\)
−0.509619 + 0.860400i \(0.670213\pi\)
\(524\) 7.41432e7 0.0225119
\(525\) 2.14164e8 0.0645936
\(526\) −1.89044e9 −0.566388
\(527\) 3.61808e8 0.107681
\(528\) −1.41445e9 −0.418185
\(529\) −2.93505e9 −0.862028
\(530\) −4.30533e8 −0.125615
\(531\) −5.30855e8 −0.153867
\(532\) 1.34875e9 0.388365
\(533\) −6.94510e8 −0.198671
\(534\) −2.49631e8 −0.0709420
\(535\) −1.25663e9 −0.354789
\(536\) −2.10330e9 −0.589962
\(537\) 1.15373e9 0.321510
\(538\) 1.97561e9 0.546971
\(539\) −3.44312e8 −0.0947091
\(540\) −3.15273e9 −0.861606
\(541\) −7.29442e8 −0.198062 −0.0990308 0.995084i \(-0.531574\pi\)
−0.0990308 + 0.995084i \(0.531574\pi\)
\(542\) 6.02207e8 0.162461
\(543\) 3.39437e9 0.909830
\(544\) −4.32260e9 −1.15119
\(545\) 4.12692e9 1.09204
\(546\) −1.16863e8 −0.0307258
\(547\) −7.28347e9 −1.90275 −0.951377 0.308028i \(-0.900331\pi\)
−0.951377 + 0.308028i \(0.900331\pi\)
\(548\) 1.67255e9 0.434158
\(549\) −8.76908e8 −0.226178
\(550\) −1.58040e8 −0.0405040
\(551\) 3.94755e9 1.00530
\(552\) −8.15389e8 −0.206337
\(553\) 6.45699e7 0.0162365
\(554\) 1.78857e9 0.446912
\(555\) −5.12949e9 −1.27365
\(556\) 1.81615e9 0.448116
\(557\) 1.25258e9 0.307124 0.153562 0.988139i \(-0.450926\pi\)
0.153562 + 0.988139i \(0.450926\pi\)
\(558\) 1.87784e7 0.00457551
\(559\) −1.65135e8 −0.0399851
\(560\) 9.85578e8 0.237155
\(561\) −3.44453e9 −0.823682
\(562\) −1.94886e9 −0.463131
\(563\) −6.94124e9 −1.63930 −0.819649 0.572866i \(-0.805832\pi\)
−0.819649 + 0.572866i \(0.805832\pi\)
\(564\) −2.42802e9 −0.569870
\(565\) 1.67899e9 0.391632
\(566\) 2.38390e9 0.552625
\(567\) 1.29185e9 0.297627
\(568\) 5.50138e8 0.125966
\(569\) 2.86322e9 0.651570 0.325785 0.945444i \(-0.394371\pi\)
0.325785 + 0.945444i \(0.394371\pi\)
\(570\) 1.33976e9 0.303016
\(571\) 5.45875e9 1.22706 0.613532 0.789670i \(-0.289748\pi\)
0.613532 + 0.789670i \(0.289748\pi\)
\(572\) −7.36770e8 −0.164606
\(573\) 2.94040e9 0.652928
\(574\) 3.97095e8 0.0876401
\(575\) 3.19591e8 0.0701063
\(576\) 3.51123e8 0.0765563
\(577\) 4.92177e9 1.06661 0.533306 0.845923i \(-0.320950\pi\)
0.533306 + 0.845923i \(0.320950\pi\)
\(578\) −1.32653e9 −0.285739
\(579\) 1.84407e9 0.394824
\(580\) 3.31850e9 0.706227
\(581\) 1.75239e9 0.370693
\(582\) 1.25785e8 0.0264483
\(583\) 1.36661e9 0.285630
\(584\) 4.00337e9 0.831727
\(585\) −2.17871e8 −0.0449939
\(586\) −2.55643e9 −0.524798
\(587\) −6.14056e9 −1.25307 −0.626534 0.779394i \(-0.715527\pi\)
−0.626534 + 0.779394i \(0.715527\pi\)
\(588\) −5.70857e8 −0.115800
\(589\) 4.46696e8 0.0900759
\(590\) 1.24254e9 0.249073
\(591\) 3.44112e9 0.685715
\(592\) 5.49185e9 1.08791
\(593\) 9.20031e7 0.0181180 0.00905901 0.999959i \(-0.497116\pi\)
0.00905901 + 0.999959i \(0.497116\pi\)
\(594\) −1.17136e9 −0.229317
\(595\) 2.40012e9 0.467115
\(596\) 4.59156e9 0.888380
\(597\) −1.34112e8 −0.0257964
\(598\) −1.74391e8 −0.0333481
\(599\) −1.30152e9 −0.247432 −0.123716 0.992318i \(-0.539481\pi\)
−0.123716 + 0.992318i \(0.539481\pi\)
\(600\) −5.54720e8 −0.104844
\(601\) −9.25912e8 −0.173984 −0.0869920 0.996209i \(-0.527725\pi\)
−0.0869920 + 0.996209i \(0.527725\pi\)
\(602\) 9.44181e7 0.0176387
\(603\) 9.32555e8 0.173206
\(604\) −9.13896e9 −1.68759
\(605\) 2.74969e9 0.504824
\(606\) 3.95990e8 0.0722820
\(607\) −4.23987e9 −0.769471 −0.384736 0.923027i \(-0.625707\pi\)
−0.384736 + 0.923027i \(0.625707\pi\)
\(608\) −5.33678e9 −0.962979
\(609\) −1.67080e9 −0.299753
\(610\) 2.05252e9 0.366128
\(611\) −1.09937e9 −0.194984
\(612\) 1.25457e9 0.221241
\(613\) 7.05872e8 0.123770 0.0618848 0.998083i \(-0.480289\pi\)
0.0618848 + 0.998083i \(0.480289\pi\)
\(614\) −8.21389e6 −0.00143206
\(615\) −3.36996e9 −0.584201
\(616\) 8.91823e8 0.153726
\(617\) −2.81933e9 −0.483223 −0.241611 0.970373i \(-0.577676\pi\)
−0.241611 + 0.970373i \(0.577676\pi\)
\(618\) −1.86513e9 −0.317870
\(619\) 7.25478e8 0.122944 0.0614719 0.998109i \(-0.480421\pi\)
0.0614719 + 0.998109i \(0.480421\pi\)
\(620\) 3.75515e8 0.0632785
\(621\) 2.36874e9 0.396914
\(622\) −4.96401e8 −0.0827116
\(623\) −5.52127e8 −0.0914810
\(624\) −1.06182e9 −0.174947
\(625\) −4.73412e9 −0.775638
\(626\) −1.46785e9 −0.239151
\(627\) −4.25269e9 −0.689013
\(628\) 2.08804e9 0.336418
\(629\) 1.33740e10 2.14281
\(630\) 1.24570e8 0.0198483
\(631\) 6.24005e9 0.988748 0.494374 0.869249i \(-0.335397\pi\)
0.494374 + 0.869249i \(0.335397\pi\)
\(632\) −1.67246e8 −0.0263540
\(633\) −7.11831e9 −1.11549
\(634\) −3.41328e9 −0.531937
\(635\) 1.08292e10 1.67838
\(636\) 2.26579e9 0.349236
\(637\) −2.58475e8 −0.0396214
\(638\) 1.23295e9 0.187963
\(639\) −2.43919e8 −0.0369822
\(640\) −5.83333e9 −0.879603
\(641\) 6.88175e9 1.03204 0.516019 0.856577i \(-0.327413\pi\)
0.516019 + 0.856577i \(0.327413\pi\)
\(642\) −7.74081e8 −0.115455
\(643\) −6.87511e9 −1.01986 −0.509931 0.860215i \(-0.670329\pi\)
−0.509931 + 0.860215i \(0.670329\pi\)
\(644\) −8.51873e8 −0.125682
\(645\) −8.01283e8 −0.117578
\(646\) −3.49313e9 −0.509801
\(647\) −1.33004e9 −0.193064 −0.0965318 0.995330i \(-0.530775\pi\)
−0.0965318 + 0.995330i \(0.530775\pi\)
\(648\) −3.34610e9 −0.483088
\(649\) −3.94408e9 −0.566356
\(650\) −1.18641e8 −0.0169448
\(651\) −1.89064e8 −0.0268581
\(652\) −1.04678e10 −1.47907
\(653\) −1.24447e10 −1.74899 −0.874497 0.485031i \(-0.838808\pi\)
−0.874497 + 0.485031i \(0.838808\pi\)
\(654\) 2.54217e9 0.355371
\(655\) 1.62895e8 0.0226498
\(656\) 3.60803e9 0.499007
\(657\) −1.77500e9 −0.244186
\(658\) 6.28578e8 0.0860138
\(659\) −1.28083e10 −1.74338 −0.871691 0.490057i \(-0.836976\pi\)
−0.871691 + 0.490057i \(0.836976\pi\)
\(660\) −3.57502e9 −0.484033
\(661\) 3.13568e9 0.422306 0.211153 0.977453i \(-0.432278\pi\)
0.211153 + 0.977453i \(0.432278\pi\)
\(662\) −1.75029e9 −0.234481
\(663\) −2.58580e9 −0.344587
\(664\) −4.53897e9 −0.601685
\(665\) 2.96325e9 0.390744
\(666\) 6.94133e8 0.0910506
\(667\) −2.49328e9 −0.325335
\(668\) −6.97995e9 −0.906013
\(669\) 3.23230e9 0.417369
\(670\) −2.18277e9 −0.280379
\(671\) −6.51515e9 −0.832522
\(672\) 2.25879e9 0.287133
\(673\) −6.36507e9 −0.804916 −0.402458 0.915438i \(-0.631844\pi\)
−0.402458 + 0.915438i \(0.631844\pi\)
\(674\) −1.10191e9 −0.138623
\(675\) 1.61148e9 0.201680
\(676\) −5.53093e8 −0.0688628
\(677\) −3.81955e9 −0.473098 −0.236549 0.971620i \(-0.576016\pi\)
−0.236549 + 0.971620i \(0.576016\pi\)
\(678\) 1.03425e9 0.127445
\(679\) 2.78208e8 0.0341055
\(680\) −6.21670e9 −0.758191
\(681\) 1.93601e9 0.234905
\(682\) 1.39518e8 0.0168416
\(683\) −6.09849e9 −0.732403 −0.366202 0.930536i \(-0.619342\pi\)
−0.366202 + 0.930536i \(0.619342\pi\)
\(684\) 1.54892e9 0.185069
\(685\) 3.67466e9 0.436818
\(686\) 1.47786e8 0.0174783
\(687\) 8.45149e9 0.994454
\(688\) 8.57888e8 0.100432
\(689\) 1.02591e9 0.119493
\(690\) −8.46196e8 −0.0980617
\(691\) 3.92762e9 0.452852 0.226426 0.974028i \(-0.427296\pi\)
0.226426 + 0.974028i \(0.427296\pi\)
\(692\) −1.44588e10 −1.65867
\(693\) −3.95414e8 −0.0451321
\(694\) −3.84221e8 −0.0436338
\(695\) 3.99015e9 0.450861
\(696\) 4.32763e9 0.486540
\(697\) 8.78644e9 0.982874
\(698\) 5.66287e8 0.0630293
\(699\) 2.54632e9 0.281996
\(700\) −5.79541e8 −0.0638618
\(701\) 8.10924e9 0.889134 0.444567 0.895746i \(-0.353358\pi\)
0.444567 + 0.895746i \(0.353358\pi\)
\(702\) −8.79337e8 −0.0959347
\(703\) 1.65119e10 1.79247
\(704\) 2.60873e9 0.281790
\(705\) −5.33445e9 −0.573361
\(706\) −4.75979e9 −0.509063
\(707\) 8.75842e8 0.0932089
\(708\) −6.53915e9 −0.692477
\(709\) 1.39644e10 1.47150 0.735751 0.677253i \(-0.236830\pi\)
0.735751 + 0.677253i \(0.236830\pi\)
\(710\) 5.70924e8 0.0598651
\(711\) 7.41532e7 0.00773724
\(712\) 1.43010e9 0.148486
\(713\) −2.82135e8 −0.0291503
\(714\) 1.47847e9 0.152008
\(715\) −1.61871e9 −0.165614
\(716\) −3.12206e9 −0.317867
\(717\) 1.24308e10 1.25945
\(718\) 4.95555e9 0.499639
\(719\) 4.82788e9 0.484401 0.242201 0.970226i \(-0.422131\pi\)
0.242201 + 0.970226i \(0.422131\pi\)
\(720\) 1.13185e9 0.113013
\(721\) −4.12525e9 −0.409899
\(722\) −1.03909e9 −0.102747
\(723\) −1.78125e10 −1.75284
\(724\) −9.18536e9 −0.899521
\(725\) −1.69621e9 −0.165309
\(726\) 1.69380e9 0.164280
\(727\) −1.24688e10 −1.20352 −0.601761 0.798677i \(-0.705534\pi\)
−0.601761 + 0.798677i \(0.705534\pi\)
\(728\) 6.69491e8 0.0643109
\(729\) 1.16046e10 1.10939
\(730\) 4.15463e9 0.395278
\(731\) 2.08917e9 0.197817
\(732\) −1.08019e10 −1.01791
\(733\) 1.65314e10 1.55041 0.775205 0.631710i \(-0.217647\pi\)
0.775205 + 0.631710i \(0.217647\pi\)
\(734\) 5.33811e8 0.0498255
\(735\) −1.25419e9 −0.116509
\(736\) 3.37073e9 0.311639
\(737\) 6.92858e9 0.637541
\(738\) 4.56031e8 0.0417635
\(739\) 1.33868e10 1.22017 0.610084 0.792337i \(-0.291136\pi\)
0.610084 + 0.792337i \(0.291136\pi\)
\(740\) 1.38807e10 1.25921
\(741\) −3.19249e9 −0.288248
\(742\) −5.86578e8 −0.0527123
\(743\) −2.04467e10 −1.82878 −0.914392 0.404830i \(-0.867331\pi\)
−0.914392 + 0.404830i \(0.867331\pi\)
\(744\) 4.89706e8 0.0435943
\(745\) 1.00878e10 0.893822
\(746\) −3.01000e9 −0.265449
\(747\) 2.01248e9 0.176648
\(748\) 9.32108e9 0.814349
\(749\) −1.71209e9 −0.148882
\(750\) −3.62581e9 −0.313827
\(751\) 4.82638e8 0.0415797 0.0207899 0.999784i \(-0.493382\pi\)
0.0207899 + 0.999784i \(0.493382\pi\)
\(752\) 5.71129e9 0.489747
\(753\) 1.95897e10 1.67203
\(754\) 9.25573e8 0.0786341
\(755\) −2.00786e10 −1.69793
\(756\) −4.29542e9 −0.361559
\(757\) 2.12332e10 1.77902 0.889509 0.456917i \(-0.151046\pi\)
0.889509 + 0.456917i \(0.151046\pi\)
\(758\) 1.25375e9 0.104561
\(759\) 2.68601e9 0.222978
\(760\) −7.67528e9 −0.634230
\(761\) 1.25867e10 1.03530 0.517651 0.855592i \(-0.326807\pi\)
0.517651 + 0.855592i \(0.326807\pi\)
\(762\) 6.67076e9 0.546176
\(763\) 5.62270e9 0.458257
\(764\) −7.95688e9 −0.645529
\(765\) 2.75634e9 0.222596
\(766\) 1.18966e9 0.0956359
\(767\) −2.96082e9 −0.236935
\(768\) 1.23814e9 0.0986292
\(769\) −1.20255e10 −0.953592 −0.476796 0.879014i \(-0.658202\pi\)
−0.476796 + 0.879014i \(0.658202\pi\)
\(770\) 9.25518e8 0.0730579
\(771\) −9.36522e9 −0.735915
\(772\) −4.99016e9 −0.390350
\(773\) 1.38597e10 1.07926 0.539628 0.841904i \(-0.318565\pi\)
0.539628 + 0.841904i \(0.318565\pi\)
\(774\) 1.08431e8 0.00840546
\(775\) −1.91940e8 −0.0148119
\(776\) −7.20602e8 −0.0553579
\(777\) −6.98864e9 −0.534465
\(778\) 2.79405e9 0.212718
\(779\) 1.08479e10 0.822178
\(780\) −2.68377e9 −0.202495
\(781\) −1.81224e9 −0.136125
\(782\) 2.20627e9 0.164981
\(783\) −1.25719e10 −0.935915
\(784\) 1.34279e9 0.0995184
\(785\) 4.58750e9 0.338479
\(786\) 1.00343e8 0.00737068
\(787\) 2.07864e10 1.52008 0.760042 0.649874i \(-0.225178\pi\)
0.760042 + 0.649874i \(0.225178\pi\)
\(788\) −9.31186e9 −0.677945
\(789\) 2.18582e10 1.58432
\(790\) −1.73565e8 −0.0125247
\(791\) 2.28753e9 0.164342
\(792\) 1.02418e9 0.0732555
\(793\) −4.89092e9 −0.348285
\(794\) −1.16415e9 −0.0825349
\(795\) 4.97802e9 0.351376
\(796\) 3.62915e8 0.0255041
\(797\) −1.53592e10 −1.07465 −0.537323 0.843377i \(-0.680564\pi\)
−0.537323 + 0.843377i \(0.680564\pi\)
\(798\) 1.82535e9 0.127156
\(799\) 1.39084e10 0.964636
\(800\) 2.29315e9 0.158350
\(801\) −6.34072e8 −0.0435938
\(802\) 4.84541e8 0.0331681
\(803\) −1.31877e10 −0.898804
\(804\) 1.14874e10 0.779514
\(805\) −1.87160e9 −0.126452
\(806\) 1.04736e8 0.00704567
\(807\) −2.28429e10 −1.53001
\(808\) −2.26857e9 −0.151291
\(809\) 2.72236e10 1.80770 0.903850 0.427850i \(-0.140729\pi\)
0.903850 + 0.427850i \(0.140729\pi\)
\(810\) −3.47253e9 −0.229587
\(811\) −8.41383e9 −0.553886 −0.276943 0.960886i \(-0.589321\pi\)
−0.276943 + 0.960886i \(0.589321\pi\)
\(812\) 4.52128e9 0.296357
\(813\) −6.96299e9 −0.454442
\(814\) 5.15719e9 0.335141
\(815\) −2.29982e10 −1.48813
\(816\) 1.34334e10 0.865509
\(817\) 2.57934e9 0.165474
\(818\) 4.63870e9 0.296319
\(819\) −2.96837e8 −0.0188810
\(820\) 9.11931e9 0.577582
\(821\) −1.89305e10 −1.19388 −0.596939 0.802287i \(-0.703617\pi\)
−0.596939 + 0.802287i \(0.703617\pi\)
\(822\) 2.26358e9 0.142149
\(823\) −4.76908e9 −0.298219 −0.149109 0.988821i \(-0.547641\pi\)
−0.149109 + 0.988821i \(0.547641\pi\)
\(824\) 1.06850e10 0.665321
\(825\) 1.82733e9 0.113300
\(826\) 1.69289e9 0.104520
\(827\) −1.27860e10 −0.786079 −0.393039 0.919522i \(-0.628576\pi\)
−0.393039 + 0.919522i \(0.628576\pi\)
\(828\) −9.78306e8 −0.0598920
\(829\) 4.84354e9 0.295272 0.147636 0.989042i \(-0.452834\pi\)
0.147636 + 0.989042i \(0.452834\pi\)
\(830\) −4.71047e9 −0.285951
\(831\) −2.06803e10 −1.25012
\(832\) 1.95838e9 0.117887
\(833\) 3.27004e9 0.196018
\(834\) 2.45792e9 0.146719
\(835\) −1.53352e10 −0.911563
\(836\) 1.15080e10 0.681206
\(837\) −1.42261e9 −0.0838587
\(838\) 3.09429e9 0.181638
\(839\) 9.01419e9 0.526938 0.263469 0.964668i \(-0.415133\pi\)
0.263469 + 0.964668i \(0.415133\pi\)
\(840\) 3.24856e9 0.189110
\(841\) −4.01690e9 −0.232865
\(842\) −3.37885e9 −0.195064
\(843\) 2.25336e10 1.29549
\(844\) 1.92625e10 1.10285
\(845\) −1.21517e9 −0.0692846
\(846\) 7.21869e8 0.0409885
\(847\) 3.74630e9 0.211841
\(848\) −5.32968e9 −0.300134
\(849\) −2.75637e10 −1.54583
\(850\) 1.50096e9 0.0838304
\(851\) −1.04289e10 −0.580079
\(852\) −3.00463e9 −0.166438
\(853\) −1.27370e10 −0.702663 −0.351331 0.936251i \(-0.614271\pi\)
−0.351331 + 0.936251i \(0.614271\pi\)
\(854\) 2.79644e9 0.153640
\(855\) 3.40304e9 0.186203
\(856\) 4.43459e9 0.241655
\(857\) −3.24724e10 −1.76231 −0.881154 0.472830i \(-0.843232\pi\)
−0.881154 + 0.472830i \(0.843232\pi\)
\(858\) −9.97119e8 −0.0538941
\(859\) −1.83427e10 −0.987385 −0.493692 0.869637i \(-0.664353\pi\)
−0.493692 + 0.869637i \(0.664353\pi\)
\(860\) 2.16832e9 0.116246
\(861\) −4.59139e9 −0.245151
\(862\) −2.39815e9 −0.127526
\(863\) 1.03550e10 0.548418 0.274209 0.961670i \(-0.411584\pi\)
0.274209 + 0.961670i \(0.411584\pi\)
\(864\) 1.69963e10 0.896512
\(865\) −3.17665e10 −1.66883
\(866\) 1.17610e10 0.615364
\(867\) 1.53379e10 0.799283
\(868\) 5.11618e8 0.0265538
\(869\) 5.50934e8 0.0284794
\(870\) 4.49115e9 0.231228
\(871\) 5.20128e9 0.266715
\(872\) −1.45637e10 −0.743813
\(873\) 3.19499e8 0.0162525
\(874\) 2.72391e9 0.138008
\(875\) −8.01948e9 −0.404686
\(876\) −2.18648e10 −1.09896
\(877\) −1.19121e10 −0.596336 −0.298168 0.954513i \(-0.596376\pi\)
−0.298168 + 0.954513i \(0.596376\pi\)
\(878\) 1.05616e10 0.526622
\(879\) 2.95586e10 1.46799
\(880\) 8.40932e9 0.415979
\(881\) 1.16583e10 0.574406 0.287203 0.957870i \(-0.407275\pi\)
0.287203 + 0.957870i \(0.407275\pi\)
\(882\) 1.69720e8 0.00832902
\(883\) 1.60409e10 0.784092 0.392046 0.919946i \(-0.371767\pi\)
0.392046 + 0.919946i \(0.371767\pi\)
\(884\) 6.99733e9 0.340682
\(885\) −1.43668e10 −0.696719
\(886\) −6.04098e9 −0.291803
\(887\) −1.93804e10 −0.932461 −0.466231 0.884663i \(-0.654388\pi\)
−0.466231 + 0.884663i \(0.654388\pi\)
\(888\) 1.81017e10 0.867509
\(889\) 1.47542e10 0.704304
\(890\) 1.48413e9 0.0705678
\(891\) 1.10226e10 0.522048
\(892\) −8.74677e9 −0.412640
\(893\) 1.71716e10 0.806922
\(894\) 6.21406e9 0.290867
\(895\) −6.85927e9 −0.319814
\(896\) −7.94759e9 −0.369111
\(897\) 2.01639e9 0.0932826
\(898\) −3.27425e9 −0.150884
\(899\) 1.49742e9 0.0687359
\(900\) −6.65555e8 −0.0304323
\(901\) −1.29791e10 −0.591163
\(902\) 3.38816e9 0.153724
\(903\) −1.09170e9 −0.0493398
\(904\) −5.92506e9 −0.266749
\(905\) −2.01806e10 −0.905031
\(906\) −1.23683e10 −0.552539
\(907\) −4.28945e10 −1.90887 −0.954436 0.298417i \(-0.903541\pi\)
−0.954436 + 0.298417i \(0.903541\pi\)
\(908\) −5.23894e9 −0.232243
\(909\) 1.00583e9 0.0444173
\(910\) 6.94786e8 0.0305637
\(911\) 1.77908e10 0.779619 0.389809 0.920896i \(-0.372541\pi\)
0.389809 + 0.920896i \(0.372541\pi\)
\(912\) 1.65852e10 0.724001
\(913\) 1.49521e10 0.650210
\(914\) −1.04304e10 −0.451844
\(915\) −2.37321e10 −1.02415
\(916\) −2.28702e10 −0.983186
\(917\) 2.21936e8 0.00950462
\(918\) 1.11247e10 0.474614
\(919\) 2.58002e9 0.109652 0.0548262 0.998496i \(-0.482540\pi\)
0.0548262 + 0.998496i \(0.482540\pi\)
\(920\) 4.84773e9 0.205249
\(921\) 9.49727e7 0.00400581
\(922\) 3.28345e9 0.137966
\(923\) −1.36045e9 −0.0569476
\(924\) −4.87077e9 −0.203117
\(925\) −7.09495e9 −0.294750
\(926\) −2.79419e9 −0.115643
\(927\) −4.73751e9 −0.195331
\(928\) −1.78900e10 −0.734838
\(929\) −4.51111e10 −1.84598 −0.922992 0.384819i \(-0.874264\pi\)
−0.922992 + 0.384819i \(0.874264\pi\)
\(930\) 5.08208e8 0.0207182
\(931\) 4.03726e9 0.163969
\(932\) −6.89049e9 −0.278801
\(933\) 5.73961e9 0.231364
\(934\) −9.23666e9 −0.370938
\(935\) 2.04788e10 0.819337
\(936\) 7.68855e8 0.0306464
\(937\) 3.60079e10 1.42991 0.714957 0.699169i \(-0.246446\pi\)
0.714957 + 0.699169i \(0.246446\pi\)
\(938\) −2.97390e9 −0.117657
\(939\) 1.69720e10 0.668964
\(940\) 1.44353e10 0.566864
\(941\) −2.33906e10 −0.915117 −0.457559 0.889179i \(-0.651276\pi\)
−0.457559 + 0.889179i \(0.651276\pi\)
\(942\) 2.82588e9 0.110148
\(943\) −6.85159e9 −0.266073
\(944\) 1.53817e10 0.595116
\(945\) −9.43720e9 −0.363774
\(946\) 8.05610e8 0.0309390
\(947\) −1.85649e10 −0.710342 −0.355171 0.934801i \(-0.615577\pi\)
−0.355171 + 0.934801i \(0.615577\pi\)
\(948\) 9.13430e8 0.0348214
\(949\) −9.90000e9 −0.376014
\(950\) 1.85311e9 0.0701244
\(951\) 3.94659e10 1.48796
\(952\) −8.46991e9 −0.318163
\(953\) −1.84417e10 −0.690200 −0.345100 0.938566i \(-0.612155\pi\)
−0.345100 + 0.938566i \(0.612155\pi\)
\(954\) −6.73636e8 −0.0251192
\(955\) −1.74816e10 −0.649484
\(956\) −3.36384e10 −1.24518
\(957\) −1.42559e10 −0.525778
\(958\) −2.67447e8 −0.00982784
\(959\) 5.00652e9 0.183304
\(960\) 9.50260e9 0.346652
\(961\) −2.73432e10 −0.993841
\(962\) 3.87150e9 0.140206
\(963\) −1.96620e9 −0.0709472
\(964\) 4.82017e10 1.73298
\(965\) −1.09636e10 −0.392741
\(966\) −1.15290e9 −0.0411500
\(967\) −8.38242e9 −0.298110 −0.149055 0.988829i \(-0.547623\pi\)
−0.149055 + 0.988829i \(0.547623\pi\)
\(968\) −9.70351e9 −0.343847
\(969\) 4.03891e10 1.42604
\(970\) −7.47828e8 −0.0263088
\(971\) −2.38373e10 −0.835583 −0.417791 0.908543i \(-0.637196\pi\)
−0.417791 + 0.908543i \(0.637196\pi\)
\(972\) −9.11299e9 −0.318294
\(973\) 5.43636e9 0.189197
\(974\) 7.10984e9 0.246549
\(975\) 1.37178e9 0.0473988
\(976\) 2.54087e10 0.874797
\(977\) 8.95023e8 0.0307046 0.0153523 0.999882i \(-0.495113\pi\)
0.0153523 + 0.999882i \(0.495113\pi\)
\(978\) −1.41668e10 −0.484267
\(979\) −4.71095e9 −0.160461
\(980\) 3.39392e9 0.115189
\(981\) 6.45721e9 0.218375
\(982\) −1.47429e10 −0.496813
\(983\) −5.06403e10 −1.70043 −0.850215 0.526435i \(-0.823528\pi\)
−0.850215 + 0.526435i \(0.823528\pi\)
\(984\) 1.18924e10 0.397912
\(985\) −2.04585e10 −0.682098
\(986\) −1.17097e10 −0.389023
\(987\) −7.26789e9 −0.240602
\(988\) 8.63906e9 0.284982
\(989\) −1.62912e9 −0.0535507
\(990\) 1.06288e9 0.0348146
\(991\) 4.44477e10 1.45075 0.725374 0.688355i \(-0.241667\pi\)
0.725374 + 0.688355i \(0.241667\pi\)
\(992\) −2.02439e9 −0.0658420
\(993\) 2.02377e10 0.655901
\(994\) 7.77852e8 0.0251215
\(995\) 7.97338e8 0.0256603
\(996\) 2.47900e10 0.795003
\(997\) −3.96617e10 −1.26747 −0.633736 0.773550i \(-0.718479\pi\)
−0.633736 + 0.773550i \(0.718479\pi\)
\(998\) 1.02210e10 0.325490
\(999\) −5.25861e10 −1.66875
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.1.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.8.a.d.1.5 10 1.1 even 1 trivial