Properties

Label 531.8.a.c.1.3
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(165.876448532\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1669 x^{15} + 2385 x^{14} + 1108684 x^{13} - 848131 x^{12} - 377920980 x^{11} + 12724944 x^{10} + 71331230512 x^{9} + 50741131904 x^{8} - 7480805165760 x^{7} - 10751966150272 x^{6} + 413177144536320 x^{5} + 886760582981376 x^{4} - 10454479722123264 x^{3} - 29180140031461376 x^{2} + 79787300207378432 x + 248723246810300416\)
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(16.5905\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q-16.5905 q^{2} +147.246 q^{4} -174.998 q^{5} +1111.07 q^{7} -319.307 q^{8} +O(q^{10})\) \(q-16.5905 q^{2} +147.246 q^{4} -174.998 q^{5} +1111.07 q^{7} -319.307 q^{8} +2903.31 q^{10} -3459.76 q^{11} -368.734 q^{13} -18433.2 q^{14} -13550.0 q^{16} -12018.6 q^{17} -20398.0 q^{19} -25767.8 q^{20} +57399.3 q^{22} -27686.4 q^{23} -47500.8 q^{25} +6117.50 q^{26} +163601. q^{28} -141983. q^{29} +224521. q^{31} +265674. q^{32} +199395. q^{34} -194434. q^{35} +433168. q^{37} +338413. q^{38} +55878.1 q^{40} -286012. q^{41} -152357. q^{43} -509436. q^{44} +459333. q^{46} -449388. q^{47} +410931. q^{49} +788065. q^{50} -54294.7 q^{52} -141405. q^{53} +605449. q^{55} -354773. q^{56} +2.35558e6 q^{58} +205379. q^{59} -522147. q^{61} -3.72492e6 q^{62} -2.67327e6 q^{64} +64527.6 q^{65} +1.89319e6 q^{67} -1.76969e6 q^{68} +3.22578e6 q^{70} -3.16299e6 q^{71} +3.37201e6 q^{73} -7.18650e6 q^{74} -3.00352e6 q^{76} -3.84403e6 q^{77} -1.64311e6 q^{79} +2.37123e6 q^{80} +4.74509e6 q^{82} +6.36325e6 q^{83} +2.10322e6 q^{85} +2.52769e6 q^{86} +1.10473e6 q^{88} -5.43034e6 q^{89} -409689. q^{91} -4.07672e6 q^{92} +7.45560e6 q^{94} +3.56959e6 q^{95} -2.91053e6 q^{97} -6.81757e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q - 2q^{2} + 1166q^{4} + 318q^{5} + 3145q^{7} - 2355q^{8} + O(q^{10}) \) \( 17q - 2q^{2} + 1166q^{4} + 318q^{5} + 3145q^{7} - 2355q^{8} + 6521q^{10} + 1764q^{11} + 18192q^{13} + 7827q^{14} + 139226q^{16} + 15507q^{17} + 52083q^{19} - 721q^{20} - 234434q^{22} - 63823q^{23} + 202153q^{25} + 367956q^{26} + 182306q^{28} + 502955q^{29} + 347531q^{31} + 243908q^{32} - 330872q^{34} - 92641q^{35} + 447615q^{37} - 775669q^{38} + 2203270q^{40} - 940335q^{41} + 478562q^{43} + 596924q^{44} - 3078663q^{46} - 703121q^{47} + 1895082q^{49} + 876967q^{50} + 6278296q^{52} + 1005974q^{53} + 5212846q^{55} - 3425294q^{56} + 6710166q^{58} + 3491443q^{59} + 11510749q^{61} - 5996234q^{62} + 29496941q^{64} - 11094180q^{65} + 14007144q^{67} - 19688159q^{68} + 30909708q^{70} - 5229074q^{71} + 5452211q^{73} - 12819662q^{74} + 41929340q^{76} - 9930777q^{77} + 15275654q^{79} - 36576105q^{80} + 32025935q^{82} - 7826609q^{83} + 11836945q^{85} - 51649136q^{86} + 30223741q^{88} + 6436185q^{89} + 11633535q^{91} - 43357972q^{92} - 4494252q^{94} - 23741055q^{95} + 26377540q^{97} - 26517816q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −16.5905 −1.46641 −0.733206 0.680007i \(-0.761977\pi\)
−0.733206 + 0.680007i \(0.761977\pi\)
\(3\) 0 0
\(4\) 147.246 1.15036
\(5\) −174.998 −0.626091 −0.313045 0.949738i \(-0.601349\pi\)
−0.313045 + 0.949738i \(0.601349\pi\)
\(6\) 0 0
\(7\) 1111.07 1.22433 0.612164 0.790731i \(-0.290299\pi\)
0.612164 + 0.790731i \(0.290299\pi\)
\(8\) −319.307 −0.220493
\(9\) 0 0
\(10\) 2903.31 0.918107
\(11\) −3459.76 −0.783738 −0.391869 0.920021i \(-0.628171\pi\)
−0.391869 + 0.920021i \(0.628171\pi\)
\(12\) 0 0
\(13\) −368.734 −0.0465491 −0.0232746 0.999729i \(-0.507409\pi\)
−0.0232746 + 0.999729i \(0.507409\pi\)
\(14\) −18433.2 −1.79537
\(15\) 0 0
\(16\) −13550.0 −0.827029
\(17\) −12018.6 −0.593310 −0.296655 0.954985i \(-0.595871\pi\)
−0.296655 + 0.954985i \(0.595871\pi\)
\(18\) 0 0
\(19\) −20398.0 −0.682259 −0.341129 0.940016i \(-0.610809\pi\)
−0.341129 + 0.940016i \(0.610809\pi\)
\(20\) −25767.8 −0.720231
\(21\) 0 0
\(22\) 57399.3 1.14928
\(23\) −27686.4 −0.474481 −0.237241 0.971451i \(-0.576243\pi\)
−0.237241 + 0.971451i \(0.576243\pi\)
\(24\) 0 0
\(25\) −47500.8 −0.608010
\(26\) 6117.50 0.0682602
\(27\) 0 0
\(28\) 163601. 1.40842
\(29\) −141983. −1.08105 −0.540523 0.841329i \(-0.681773\pi\)
−0.540523 + 0.841329i \(0.681773\pi\)
\(30\) 0 0
\(31\) 224521. 1.35360 0.676801 0.736166i \(-0.263366\pi\)
0.676801 + 0.736166i \(0.263366\pi\)
\(32\) 265674. 1.43326
\(33\) 0 0
\(34\) 199395. 0.870037
\(35\) −194434. −0.766541
\(36\) 0 0
\(37\) 433168. 1.40589 0.702944 0.711246i \(-0.251869\pi\)
0.702944 + 0.711246i \(0.251869\pi\)
\(38\) 338413. 1.00047
\(39\) 0 0
\(40\) 55878.1 0.138048
\(41\) −286012. −0.648097 −0.324049 0.946040i \(-0.605044\pi\)
−0.324049 + 0.946040i \(0.605044\pi\)
\(42\) 0 0
\(43\) −152357. −0.292229 −0.146115 0.989268i \(-0.546677\pi\)
−0.146115 + 0.989268i \(0.546677\pi\)
\(44\) −509436. −0.901583
\(45\) 0 0
\(46\) 459333. 0.695785
\(47\) −449388. −0.631363 −0.315682 0.948865i \(-0.602233\pi\)
−0.315682 + 0.948865i \(0.602233\pi\)
\(48\) 0 0
\(49\) 410931. 0.498980
\(50\) 788065. 0.891593
\(51\) 0 0
\(52\) −54294.7 −0.0535483
\(53\) −141405. −0.130467 −0.0652335 0.997870i \(-0.520779\pi\)
−0.0652335 + 0.997870i \(0.520779\pi\)
\(54\) 0 0
\(55\) 605449. 0.490691
\(56\) −354773. −0.269955
\(57\) 0 0
\(58\) 2.35558e6 1.58526
\(59\) 205379. 0.130189
\(60\) 0 0
\(61\) −522147. −0.294536 −0.147268 0.989097i \(-0.547048\pi\)
−0.147268 + 0.989097i \(0.547048\pi\)
\(62\) −3.72492e6 −1.98494
\(63\) 0 0
\(64\) −2.67327e6 −1.27472
\(65\) 64527.6 0.0291440
\(66\) 0 0
\(67\) 1.89319e6 0.769010 0.384505 0.923123i \(-0.374372\pi\)
0.384505 + 0.923123i \(0.374372\pi\)
\(68\) −1.76969e6 −0.682522
\(69\) 0 0
\(70\) 3.22578e6 1.12406
\(71\) −3.16299e6 −1.04880 −0.524400 0.851472i \(-0.675711\pi\)
−0.524400 + 0.851472i \(0.675711\pi\)
\(72\) 0 0
\(73\) 3.37201e6 1.01452 0.507258 0.861794i \(-0.330659\pi\)
0.507258 + 0.861794i \(0.330659\pi\)
\(74\) −7.18650e6 −2.06161
\(75\) 0 0
\(76\) −3.00352e6 −0.784844
\(77\) −3.84403e6 −0.959553
\(78\) 0 0
\(79\) −1.64311e6 −0.374949 −0.187475 0.982269i \(-0.560030\pi\)
−0.187475 + 0.982269i \(0.560030\pi\)
\(80\) 2.37123e6 0.517795
\(81\) 0 0
\(82\) 4.74509e6 0.950377
\(83\) 6.36325e6 1.22153 0.610767 0.791810i \(-0.290861\pi\)
0.610767 + 0.791810i \(0.290861\pi\)
\(84\) 0 0
\(85\) 2.10322e6 0.371466
\(86\) 2.52769e6 0.428528
\(87\) 0 0
\(88\) 1.10473e6 0.172809
\(89\) −5.43034e6 −0.816511 −0.408256 0.912868i \(-0.633863\pi\)
−0.408256 + 0.912868i \(0.633863\pi\)
\(90\) 0 0
\(91\) −409689. −0.0569914
\(92\) −4.07672e6 −0.545825
\(93\) 0 0
\(94\) 7.45560e6 0.925838
\(95\) 3.56959e6 0.427156
\(96\) 0 0
\(97\) −2.91053e6 −0.323796 −0.161898 0.986807i \(-0.551762\pi\)
−0.161898 + 0.986807i \(0.551762\pi\)
\(98\) −6.81757e6 −0.731709
\(99\) 0 0
\(100\) −6.99432e6 −0.699432
\(101\) 1.20381e7 1.16261 0.581305 0.813686i \(-0.302542\pi\)
0.581305 + 0.813686i \(0.302542\pi\)
\(102\) 0 0
\(103\) −4.50334e6 −0.406074 −0.203037 0.979171i \(-0.565081\pi\)
−0.203037 + 0.979171i \(0.565081\pi\)
\(104\) 117739. 0.0102637
\(105\) 0 0
\(106\) 2.34599e6 0.191318
\(107\) 1.38674e6 0.109434 0.0547171 0.998502i \(-0.482574\pi\)
0.0547171 + 0.998502i \(0.482574\pi\)
\(108\) 0 0
\(109\) −1.51322e7 −1.11920 −0.559602 0.828761i \(-0.689046\pi\)
−0.559602 + 0.828761i \(0.689046\pi\)
\(110\) −1.00447e7 −0.719555
\(111\) 0 0
\(112\) −1.50550e7 −1.01256
\(113\) 1.74988e7 1.14086 0.570432 0.821345i \(-0.306776\pi\)
0.570432 + 0.821345i \(0.306776\pi\)
\(114\) 0 0
\(115\) 4.84506e6 0.297068
\(116\) −2.09065e7 −1.24359
\(117\) 0 0
\(118\) −3.40735e6 −0.190910
\(119\) −1.33535e7 −0.726407
\(120\) 0 0
\(121\) −7.51726e6 −0.385754
\(122\) 8.66270e6 0.431911
\(123\) 0 0
\(124\) 3.30599e7 1.55713
\(125\) 2.19842e7 1.00676
\(126\) 0 0
\(127\) −2.00254e7 −0.867495 −0.433748 0.901034i \(-0.642809\pi\)
−0.433748 + 0.901034i \(0.642809\pi\)
\(128\) 1.03448e7 0.436000
\(129\) 0 0
\(130\) −1.07055e6 −0.0427370
\(131\) −1.77758e7 −0.690845 −0.345422 0.938447i \(-0.612264\pi\)
−0.345422 + 0.938447i \(0.612264\pi\)
\(132\) 0 0
\(133\) −2.26635e7 −0.835308
\(134\) −3.14090e7 −1.12769
\(135\) 0 0
\(136\) 3.83762e6 0.130821
\(137\) −4.37881e7 −1.45490 −0.727451 0.686159i \(-0.759295\pi\)
−0.727451 + 0.686159i \(0.759295\pi\)
\(138\) 0 0
\(139\) −2.03284e7 −0.642024 −0.321012 0.947075i \(-0.604023\pi\)
−0.321012 + 0.947075i \(0.604023\pi\)
\(140\) −2.86298e7 −0.881799
\(141\) 0 0
\(142\) 5.24757e7 1.53797
\(143\) 1.27573e6 0.0364823
\(144\) 0 0
\(145\) 2.48467e7 0.676832
\(146\) −5.59435e7 −1.48770
\(147\) 0 0
\(148\) 6.37824e7 1.61728
\(149\) −6.14768e7 −1.52251 −0.761254 0.648454i \(-0.775416\pi\)
−0.761254 + 0.648454i \(0.775416\pi\)
\(150\) 0 0
\(151\) 6.11417e6 0.144517 0.0722583 0.997386i \(-0.476979\pi\)
0.0722583 + 0.997386i \(0.476979\pi\)
\(152\) 6.51322e6 0.150433
\(153\) 0 0
\(154\) 6.37745e7 1.40710
\(155\) −3.92906e7 −0.847477
\(156\) 0 0
\(157\) 3.59609e7 0.741621 0.370810 0.928709i \(-0.379080\pi\)
0.370810 + 0.928709i \(0.379080\pi\)
\(158\) 2.72601e7 0.549830
\(159\) 0 0
\(160\) −4.64923e7 −0.897349
\(161\) −3.07615e7 −0.580921
\(162\) 0 0
\(163\) −3.88451e7 −0.702553 −0.351277 0.936272i \(-0.614252\pi\)
−0.351277 + 0.936272i \(0.614252\pi\)
\(164\) −4.21142e7 −0.745547
\(165\) 0 0
\(166\) −1.05570e8 −1.79127
\(167\) 8.48288e7 1.40940 0.704702 0.709503i \(-0.251081\pi\)
0.704702 + 0.709503i \(0.251081\pi\)
\(168\) 0 0
\(169\) −6.26126e7 −0.997833
\(170\) −3.48936e7 −0.544722
\(171\) 0 0
\(172\) −2.24341e7 −0.336169
\(173\) 2.04205e7 0.299850 0.149925 0.988697i \(-0.452097\pi\)
0.149925 + 0.988697i \(0.452097\pi\)
\(174\) 0 0
\(175\) −5.27767e7 −0.744404
\(176\) 4.68799e7 0.648175
\(177\) 0 0
\(178\) 9.00924e7 1.19734
\(179\) 2.15476e7 0.280811 0.140405 0.990094i \(-0.455159\pi\)
0.140405 + 0.990094i \(0.455159\pi\)
\(180\) 0 0
\(181\) −6.19200e7 −0.776168 −0.388084 0.921624i \(-0.626863\pi\)
−0.388084 + 0.921624i \(0.626863\pi\)
\(182\) 6.79696e6 0.0835728
\(183\) 0 0
\(184\) 8.84047e6 0.104620
\(185\) −7.58034e7 −0.880213
\(186\) 0 0
\(187\) 4.15814e7 0.465000
\(188\) −6.61708e7 −0.726296
\(189\) 0 0
\(190\) −5.92215e7 −0.626386
\(191\) 1.83254e8 1.90300 0.951498 0.307656i \(-0.0995445\pi\)
0.951498 + 0.307656i \(0.0995445\pi\)
\(192\) 0 0
\(193\) −2.51674e7 −0.251993 −0.125996 0.992031i \(-0.540213\pi\)
−0.125996 + 0.992031i \(0.540213\pi\)
\(194\) 4.82873e7 0.474818
\(195\) 0 0
\(196\) 6.05081e7 0.574007
\(197\) −4.11501e7 −0.383477 −0.191738 0.981446i \(-0.561412\pi\)
−0.191738 + 0.981446i \(0.561412\pi\)
\(198\) 0 0
\(199\) 1.20872e8 1.08728 0.543638 0.839320i \(-0.317047\pi\)
0.543638 + 0.839320i \(0.317047\pi\)
\(200\) 1.51674e7 0.134062
\(201\) 0 0
\(202\) −1.99719e8 −1.70486
\(203\) −1.57753e8 −1.32355
\(204\) 0 0
\(205\) 5.00514e7 0.405768
\(206\) 7.47129e7 0.595471
\(207\) 0 0
\(208\) 4.99636e6 0.0384975
\(209\) 7.05720e7 0.534712
\(210\) 0 0
\(211\) −1.24225e8 −0.910379 −0.455189 0.890395i \(-0.650428\pi\)
−0.455189 + 0.890395i \(0.650428\pi\)
\(212\) −2.08214e7 −0.150084
\(213\) 0 0
\(214\) −2.30068e7 −0.160475
\(215\) 2.66622e7 0.182962
\(216\) 0 0
\(217\) 2.49458e8 1.65725
\(218\) 2.51052e8 1.64121
\(219\) 0 0
\(220\) 8.91502e7 0.564473
\(221\) 4.43166e6 0.0276181
\(222\) 0 0
\(223\) −4.32331e6 −0.0261065 −0.0130533 0.999915i \(-0.504155\pi\)
−0.0130533 + 0.999915i \(0.504155\pi\)
\(224\) 2.95182e8 1.75478
\(225\) 0 0
\(226\) −2.90315e8 −1.67298
\(227\) 1.29928e8 0.737244 0.368622 0.929579i \(-0.379830\pi\)
0.368622 + 0.929579i \(0.379830\pi\)
\(228\) 0 0
\(229\) −6.03820e7 −0.332264 −0.166132 0.986103i \(-0.553128\pi\)
−0.166132 + 0.986103i \(0.553128\pi\)
\(230\) −8.03821e7 −0.435624
\(231\) 0 0
\(232\) 4.53363e7 0.238362
\(233\) 2.38228e8 1.23381 0.616904 0.787038i \(-0.288387\pi\)
0.616904 + 0.787038i \(0.288387\pi\)
\(234\) 0 0
\(235\) 7.86419e7 0.395291
\(236\) 3.02413e7 0.149764
\(237\) 0 0
\(238\) 2.21541e8 1.06521
\(239\) 3.08535e7 0.146188 0.0730940 0.997325i \(-0.476713\pi\)
0.0730940 + 0.997325i \(0.476713\pi\)
\(240\) 0 0
\(241\) 9.76207e6 0.0449244 0.0224622 0.999748i \(-0.492849\pi\)
0.0224622 + 0.999748i \(0.492849\pi\)
\(242\) 1.24715e8 0.565674
\(243\) 0 0
\(244\) −7.68842e7 −0.338823
\(245\) −7.19120e7 −0.312407
\(246\) 0 0
\(247\) 7.52142e6 0.0317585
\(248\) −7.16912e7 −0.298459
\(249\) 0 0
\(250\) −3.64730e8 −1.47632
\(251\) −2.04882e8 −0.817798 −0.408899 0.912580i \(-0.634087\pi\)
−0.408899 + 0.912580i \(0.634087\pi\)
\(252\) 0 0
\(253\) 9.57882e7 0.371869
\(254\) 3.32232e8 1.27211
\(255\) 0 0
\(256\) 1.70553e8 0.635360
\(257\) 7.58629e7 0.278781 0.139391 0.990237i \(-0.455486\pi\)
0.139391 + 0.990237i \(0.455486\pi\)
\(258\) 0 0
\(259\) 4.81280e8 1.72127
\(260\) 9.50145e6 0.0335261
\(261\) 0 0
\(262\) 2.94911e8 1.01306
\(263\) 3.76186e8 1.27514 0.637569 0.770393i \(-0.279940\pi\)
0.637569 + 0.770393i \(0.279940\pi\)
\(264\) 0 0
\(265\) 2.47456e7 0.0816842
\(266\) 3.76000e8 1.22491
\(267\) 0 0
\(268\) 2.78765e8 0.884640
\(269\) 5.43241e8 1.70161 0.850804 0.525483i \(-0.176115\pi\)
0.850804 + 0.525483i \(0.176115\pi\)
\(270\) 0 0
\(271\) 5.59119e8 1.70652 0.853261 0.521484i \(-0.174621\pi\)
0.853261 + 0.521484i \(0.174621\pi\)
\(272\) 1.62852e8 0.490685
\(273\) 0 0
\(274\) 7.26468e8 2.13349
\(275\) 1.64341e8 0.476521
\(276\) 0 0
\(277\) 4.08022e8 1.15346 0.576732 0.816933i \(-0.304328\pi\)
0.576732 + 0.816933i \(0.304328\pi\)
\(278\) 3.37259e8 0.941472
\(279\) 0 0
\(280\) 6.20844e7 0.169017
\(281\) −3.46395e8 −0.931321 −0.465660 0.884964i \(-0.654183\pi\)
−0.465660 + 0.884964i \(0.654183\pi\)
\(282\) 0 0
\(283\) 1.32435e8 0.347337 0.173669 0.984804i \(-0.444438\pi\)
0.173669 + 0.984804i \(0.444438\pi\)
\(284\) −4.65738e8 −1.20650
\(285\) 0 0
\(286\) −2.11651e7 −0.0534981
\(287\) −3.17779e8 −0.793484
\(288\) 0 0
\(289\) −2.65892e8 −0.647983
\(290\) −4.12221e8 −0.992515
\(291\) 0 0
\(292\) 4.96516e8 1.16706
\(293\) 6.11894e8 1.42115 0.710574 0.703622i \(-0.248435\pi\)
0.710574 + 0.703622i \(0.248435\pi\)
\(294\) 0 0
\(295\) −3.59408e7 −0.0815101
\(296\) −1.38314e8 −0.309988
\(297\) 0 0
\(298\) 1.01993e9 2.23262
\(299\) 1.02089e7 0.0220867
\(300\) 0 0
\(301\) −1.69279e8 −0.357785
\(302\) −1.01437e8 −0.211921
\(303\) 0 0
\(304\) 2.76393e8 0.564248
\(305\) 9.13745e7 0.184406
\(306\) 0 0
\(307\) 1.61294e8 0.318151 0.159075 0.987266i \(-0.449149\pi\)
0.159075 + 0.987266i \(0.449149\pi\)
\(308\) −5.66019e8 −1.10383
\(309\) 0 0
\(310\) 6.51853e8 1.24275
\(311\) −5.52921e8 −1.04232 −0.521160 0.853459i \(-0.674501\pi\)
−0.521160 + 0.853459i \(0.674501\pi\)
\(312\) 0 0
\(313\) 3.45307e8 0.636503 0.318252 0.948006i \(-0.396904\pi\)
0.318252 + 0.948006i \(0.396904\pi\)
\(314\) −5.96611e8 −1.08752
\(315\) 0 0
\(316\) −2.41942e8 −0.431327
\(317\) 5.61715e8 0.990395 0.495198 0.868780i \(-0.335096\pi\)
0.495198 + 0.868780i \(0.335096\pi\)
\(318\) 0 0
\(319\) 4.91227e8 0.847257
\(320\) 4.67817e8 0.798088
\(321\) 0 0
\(322\) 5.10350e8 0.851869
\(323\) 2.45154e8 0.404791
\(324\) 0 0
\(325\) 1.75152e7 0.0283023
\(326\) 6.44461e8 1.03023
\(327\) 0 0
\(328\) 9.13256e7 0.142901
\(329\) −4.99302e8 −0.772996
\(330\) 0 0
\(331\) −7.48834e8 −1.13498 −0.567489 0.823381i \(-0.692085\pi\)
−0.567489 + 0.823381i \(0.692085\pi\)
\(332\) 9.36965e8 1.40521
\(333\) 0 0
\(334\) −1.40736e9 −2.06677
\(335\) −3.31304e8 −0.481470
\(336\) 0 0
\(337\) 3.77520e8 0.537322 0.268661 0.963235i \(-0.413419\pi\)
0.268661 + 0.963235i \(0.413419\pi\)
\(338\) 1.03878e9 1.46323
\(339\) 0 0
\(340\) 3.09692e8 0.427321
\(341\) −7.76787e8 −1.06087
\(342\) 0 0
\(343\) −4.58440e8 −0.613413
\(344\) 4.86488e7 0.0644344
\(345\) 0 0
\(346\) −3.38787e8 −0.439703
\(347\) −2.25075e8 −0.289185 −0.144592 0.989491i \(-0.546187\pi\)
−0.144592 + 0.989491i \(0.546187\pi\)
\(348\) 0 0
\(349\) 2.00272e8 0.252192 0.126096 0.992018i \(-0.459755\pi\)
0.126096 + 0.992018i \(0.459755\pi\)
\(350\) 8.75594e8 1.09160
\(351\) 0 0
\(352\) −9.19168e8 −1.12330
\(353\) 9.69527e6 0.0117314 0.00586568 0.999983i \(-0.498133\pi\)
0.00586568 + 0.999983i \(0.498133\pi\)
\(354\) 0 0
\(355\) 5.53515e8 0.656645
\(356\) −7.99598e8 −0.939284
\(357\) 0 0
\(358\) −3.57487e8 −0.411784
\(359\) −1.59435e9 −1.81867 −0.909334 0.416067i \(-0.863408\pi\)
−0.909334 + 0.416067i \(0.863408\pi\)
\(360\) 0 0
\(361\) −4.77795e8 −0.534523
\(362\) 1.02729e9 1.13818
\(363\) 0 0
\(364\) −6.03252e7 −0.0655607
\(365\) −5.90094e8 −0.635179
\(366\) 0 0
\(367\) 4.36226e8 0.460660 0.230330 0.973113i \(-0.426019\pi\)
0.230330 + 0.973113i \(0.426019\pi\)
\(368\) 3.75152e8 0.392410
\(369\) 0 0
\(370\) 1.25762e9 1.29075
\(371\) −1.57111e8 −0.159734
\(372\) 0 0
\(373\) 8.71065e8 0.869100 0.434550 0.900648i \(-0.356907\pi\)
0.434550 + 0.900648i \(0.356907\pi\)
\(374\) −6.89858e8 −0.681882
\(375\) 0 0
\(376\) 1.43493e8 0.139211
\(377\) 5.23540e7 0.0503217
\(378\) 0 0
\(379\) 1.10371e9 1.04140 0.520702 0.853739i \(-0.325670\pi\)
0.520702 + 0.853739i \(0.325670\pi\)
\(380\) 5.25610e8 0.491384
\(381\) 0 0
\(382\) −3.04029e9 −2.79057
\(383\) −3.23348e8 −0.294086 −0.147043 0.989130i \(-0.546976\pi\)
−0.147043 + 0.989130i \(0.546976\pi\)
\(384\) 0 0
\(385\) 6.72696e8 0.600767
\(386\) 4.17541e8 0.369525
\(387\) 0 0
\(388\) −4.28565e8 −0.372483
\(389\) −1.09065e9 −0.939423 −0.469711 0.882820i \(-0.655642\pi\)
−0.469711 + 0.882820i \(0.655642\pi\)
\(390\) 0 0
\(391\) 3.32751e8 0.281515
\(392\) −1.31213e8 −0.110021
\(393\) 0 0
\(394\) 6.82702e8 0.562334
\(395\) 2.87541e8 0.234752
\(396\) 0 0
\(397\) −2.19755e9 −1.76268 −0.881338 0.472487i \(-0.843356\pi\)
−0.881338 + 0.472487i \(0.843356\pi\)
\(398\) −2.00533e9 −1.59439
\(399\) 0 0
\(400\) 6.43638e8 0.502842
\(401\) 2.55403e9 1.97797 0.988987 0.148004i \(-0.0472848\pi\)
0.988987 + 0.148004i \(0.0472848\pi\)
\(402\) 0 0
\(403\) −8.27885e7 −0.0630089
\(404\) 1.77257e9 1.33742
\(405\) 0 0
\(406\) 2.61721e9 1.94087
\(407\) −1.49866e9 −1.10185
\(408\) 0 0
\(409\) −2.88636e8 −0.208602 −0.104301 0.994546i \(-0.533261\pi\)
−0.104301 + 0.994546i \(0.533261\pi\)
\(410\) −8.30380e8 −0.595022
\(411\) 0 0
\(412\) −6.63101e8 −0.467132
\(413\) 2.28190e8 0.159394
\(414\) 0 0
\(415\) −1.11355e9 −0.764791
\(416\) −9.79631e7 −0.0667169
\(417\) 0 0
\(418\) −1.17083e9 −0.784108
\(419\) 5.30993e8 0.352647 0.176323 0.984332i \(-0.443580\pi\)
0.176323 + 0.984332i \(0.443580\pi\)
\(420\) 0 0
\(421\) −2.15095e9 −1.40489 −0.702445 0.711738i \(-0.747908\pi\)
−0.702445 + 0.711738i \(0.747908\pi\)
\(422\) 2.06097e9 1.33499
\(423\) 0 0
\(424\) 4.51518e7 0.0287670
\(425\) 5.70892e8 0.360739
\(426\) 0 0
\(427\) −5.80141e8 −0.360609
\(428\) 2.04193e8 0.125889
\(429\) 0 0
\(430\) −4.42340e8 −0.268298
\(431\) 3.30976e8 0.199125 0.0995624 0.995031i \(-0.468256\pi\)
0.0995624 + 0.995031i \(0.468256\pi\)
\(432\) 0 0
\(433\) 2.05100e9 1.21411 0.607056 0.794659i \(-0.292350\pi\)
0.607056 + 0.794659i \(0.292350\pi\)
\(434\) −4.13865e9 −2.43021
\(435\) 0 0
\(436\) −2.22816e9 −1.28749
\(437\) 5.64746e8 0.323719
\(438\) 0 0
\(439\) 1.57643e9 0.889299 0.444649 0.895705i \(-0.353328\pi\)
0.444649 + 0.895705i \(0.353328\pi\)
\(440\) −1.93324e8 −0.108194
\(441\) 0 0
\(442\) −7.35237e7 −0.0404995
\(443\) −3.27614e8 −0.179040 −0.0895199 0.995985i \(-0.528533\pi\)
−0.0895199 + 0.995985i \(0.528533\pi\)
\(444\) 0 0
\(445\) 9.50297e8 0.511210
\(446\) 7.17261e7 0.0382829
\(447\) 0 0
\(448\) −2.97019e9 −1.56067
\(449\) 3.76746e7 0.0196420 0.00982101 0.999952i \(-0.496874\pi\)
0.00982101 + 0.999952i \(0.496874\pi\)
\(450\) 0 0
\(451\) 9.89531e8 0.507939
\(452\) 2.57664e9 1.31241
\(453\) 0 0
\(454\) −2.15557e9 −1.08110
\(455\) 7.16946e7 0.0356818
\(456\) 0 0
\(457\) 1.95746e8 0.0959372 0.0479686 0.998849i \(-0.484725\pi\)
0.0479686 + 0.998849i \(0.484725\pi\)
\(458\) 1.00177e9 0.487236
\(459\) 0 0
\(460\) 7.13417e8 0.341736
\(461\) 1.06215e9 0.504931 0.252465 0.967606i \(-0.418759\pi\)
0.252465 + 0.967606i \(0.418759\pi\)
\(462\) 0 0
\(463\) 1.12424e9 0.526412 0.263206 0.964740i \(-0.415220\pi\)
0.263206 + 0.964740i \(0.415220\pi\)
\(464\) 1.92388e9 0.894056
\(465\) 0 0
\(466\) −3.95234e9 −1.80927
\(467\) 1.93669e9 0.879936 0.439968 0.898013i \(-0.354990\pi\)
0.439968 + 0.898013i \(0.354990\pi\)
\(468\) 0 0
\(469\) 2.10346e9 0.941521
\(470\) −1.30471e9 −0.579659
\(471\) 0 0
\(472\) −6.55790e7 −0.0287057
\(473\) 5.27119e8 0.229031
\(474\) 0 0
\(475\) 9.68919e8 0.414820
\(476\) −1.96625e9 −0.835631
\(477\) 0 0
\(478\) −5.11877e8 −0.214372
\(479\) −2.67237e9 −1.11102 −0.555510 0.831510i \(-0.687477\pi\)
−0.555510 + 0.831510i \(0.687477\pi\)
\(480\) 0 0
\(481\) −1.59724e8 −0.0654428
\(482\) −1.61958e8 −0.0658777
\(483\) 0 0
\(484\) −1.10689e9 −0.443757
\(485\) 5.09337e8 0.202726
\(486\) 0 0
\(487\) 9.86225e8 0.386923 0.193462 0.981108i \(-0.438029\pi\)
0.193462 + 0.981108i \(0.438029\pi\)
\(488\) 1.66725e8 0.0649430
\(489\) 0 0
\(490\) 1.19306e9 0.458116
\(491\) 3.33799e9 1.27262 0.636312 0.771431i \(-0.280459\pi\)
0.636312 + 0.771431i \(0.280459\pi\)
\(492\) 0 0
\(493\) 1.70644e9 0.641395
\(494\) −1.24784e8 −0.0465711
\(495\) 0 0
\(496\) −3.04227e9 −1.11947
\(497\) −3.51430e9 −1.28408
\(498\) 0 0
\(499\) 4.31622e9 1.55508 0.777539 0.628835i \(-0.216468\pi\)
0.777539 + 0.628835i \(0.216468\pi\)
\(500\) 3.23710e9 1.15814
\(501\) 0 0
\(502\) 3.39911e9 1.19923
\(503\) 3.79261e9 1.32877 0.664386 0.747389i \(-0.268693\pi\)
0.664386 + 0.747389i \(0.268693\pi\)
\(504\) 0 0
\(505\) −2.10664e9 −0.727899
\(506\) −1.58918e9 −0.545313
\(507\) 0 0
\(508\) −2.94866e9 −0.997934
\(509\) −1.85516e9 −0.623548 −0.311774 0.950156i \(-0.600923\pi\)
−0.311774 + 0.950156i \(0.600923\pi\)
\(510\) 0 0
\(511\) 3.74653e9 1.24210
\(512\) −4.15370e9 −1.36770
\(513\) 0 0
\(514\) −1.25861e9 −0.408808
\(515\) 7.88075e8 0.254239
\(516\) 0 0
\(517\) 1.55477e9 0.494824
\(518\) −7.98470e9 −2.52409
\(519\) 0 0
\(520\) −2.06041e7 −0.00642603
\(521\) 2.97325e8 0.0921083 0.0460542 0.998939i \(-0.485335\pi\)
0.0460542 + 0.998939i \(0.485335\pi\)
\(522\) 0 0
\(523\) 2.48461e9 0.759456 0.379728 0.925098i \(-0.376018\pi\)
0.379728 + 0.925098i \(0.376018\pi\)
\(524\) −2.61743e9 −0.794722
\(525\) 0 0
\(526\) −6.24113e9 −1.86988
\(527\) −2.69842e9 −0.803106
\(528\) 0 0
\(529\) −2.63829e9 −0.774867
\(530\) −4.10544e8 −0.119783
\(531\) 0 0
\(532\) −3.33712e9 −0.960907
\(533\) 1.05462e8 0.0301684
\(534\) 0 0
\(535\) −2.42677e8 −0.0685157
\(536\) −6.04509e8 −0.169561
\(537\) 0 0
\(538\) −9.01267e9 −2.49526
\(539\) −1.42172e9 −0.391070
\(540\) 0 0
\(541\) −4.16187e9 −1.13005 −0.565026 0.825073i \(-0.691134\pi\)
−0.565026 + 0.825073i \(0.691134\pi\)
\(542\) −9.27610e9 −2.50246
\(543\) 0 0
\(544\) −3.19303e9 −0.850367
\(545\) 2.64810e9 0.700724
\(546\) 0 0
\(547\) 6.75557e9 1.76484 0.882422 0.470459i \(-0.155912\pi\)
0.882422 + 0.470459i \(0.155912\pi\)
\(548\) −6.44764e9 −1.67367
\(549\) 0 0
\(550\) −2.72651e9 −0.698776
\(551\) 2.89617e9 0.737552
\(552\) 0 0
\(553\) −1.82561e9 −0.459061
\(554\) −6.76931e9 −1.69145
\(555\) 0 0
\(556\) −2.99328e9 −0.738560
\(557\) −3.44498e9 −0.844681 −0.422341 0.906437i \(-0.638791\pi\)
−0.422341 + 0.906437i \(0.638791\pi\)
\(558\) 0 0
\(559\) 5.61793e7 0.0136030
\(560\) 2.63460e9 0.633951
\(561\) 0 0
\(562\) 5.74688e9 1.36570
\(563\) 2.05492e9 0.485306 0.242653 0.970113i \(-0.421982\pi\)
0.242653 + 0.970113i \(0.421982\pi\)
\(564\) 0 0
\(565\) −3.06225e9 −0.714284
\(566\) −2.19717e9 −0.509339
\(567\) 0 0
\(568\) 1.00996e9 0.231253
\(569\) −7.09854e8 −0.161538 −0.0807692 0.996733i \(-0.525738\pi\)
−0.0807692 + 0.996733i \(0.525738\pi\)
\(570\) 0 0
\(571\) −3.56358e7 −0.00801051 −0.00400525 0.999992i \(-0.501275\pi\)
−0.00400525 + 0.999992i \(0.501275\pi\)
\(572\) 1.87847e8 0.0419679
\(573\) 0 0
\(574\) 5.27212e9 1.16357
\(575\) 1.31513e9 0.288490
\(576\) 0 0
\(577\) −1.37806e9 −0.298644 −0.149322 0.988789i \(-0.547709\pi\)
−0.149322 + 0.988789i \(0.547709\pi\)
\(578\) 4.41130e9 0.950209
\(579\) 0 0
\(580\) 3.65859e9 0.778602
\(581\) 7.07001e9 1.49556
\(582\) 0 0
\(583\) 4.89228e8 0.102252
\(584\) −1.07671e9 −0.223693
\(585\) 0 0
\(586\) −1.01517e10 −2.08399
\(587\) −2.82850e9 −0.577195 −0.288597 0.957451i \(-0.593189\pi\)
−0.288597 + 0.957451i \(0.593189\pi\)
\(588\) 0 0
\(589\) −4.57977e9 −0.923506
\(590\) 5.96278e8 0.119527
\(591\) 0 0
\(592\) −5.86945e9 −1.16271
\(593\) 4.23981e9 0.834940 0.417470 0.908691i \(-0.362917\pi\)
0.417470 + 0.908691i \(0.362917\pi\)
\(594\) 0 0
\(595\) 2.33683e9 0.454797
\(596\) −9.05224e9 −1.75143
\(597\) 0 0
\(598\) −1.69372e8 −0.0323882
\(599\) 7.47951e9 1.42193 0.710967 0.703225i \(-0.248258\pi\)
0.710967 + 0.703225i \(0.248258\pi\)
\(600\) 0 0
\(601\) 2.42694e9 0.456034 0.228017 0.973657i \(-0.426776\pi\)
0.228017 + 0.973657i \(0.426776\pi\)
\(602\) 2.80844e9 0.524659
\(603\) 0 0
\(604\) 9.00288e8 0.166246
\(605\) 1.31550e9 0.241517
\(606\) 0 0
\(607\) −6.09411e9 −1.10599 −0.552993 0.833186i \(-0.686514\pi\)
−0.552993 + 0.833186i \(0.686514\pi\)
\(608\) −5.41921e9 −0.977852
\(609\) 0 0
\(610\) −1.51595e9 −0.270415
\(611\) 1.65705e8 0.0293894
\(612\) 0 0
\(613\) −6.89396e9 −1.20881 −0.604403 0.796678i \(-0.706588\pi\)
−0.604403 + 0.796678i \(0.706588\pi\)
\(614\) −2.67595e9 −0.466540
\(615\) 0 0
\(616\) 1.22743e9 0.211574
\(617\) 1.05989e10 1.81661 0.908304 0.418311i \(-0.137378\pi\)
0.908304 + 0.418311i \(0.137378\pi\)
\(618\) 0 0
\(619\) 7.11095e9 1.20507 0.602533 0.798094i \(-0.294158\pi\)
0.602533 + 0.798094i \(0.294158\pi\)
\(620\) −5.78540e9 −0.974905
\(621\) 0 0
\(622\) 9.17326e9 1.52847
\(623\) −6.03349e9 −0.999678
\(624\) 0 0
\(625\) −1.36187e8 −0.0223130
\(626\) −5.72884e9 −0.933375
\(627\) 0 0
\(628\) 5.29511e9 0.853133
\(629\) −5.20607e9 −0.834128
\(630\) 0 0
\(631\) −3.97705e9 −0.630170 −0.315085 0.949063i \(-0.602033\pi\)
−0.315085 + 0.949063i \(0.602033\pi\)
\(632\) 5.24658e8 0.0826735
\(633\) 0 0
\(634\) −9.31916e9 −1.45233
\(635\) 3.50439e9 0.543131
\(636\) 0 0
\(637\) −1.51524e8 −0.0232271
\(638\) −8.14973e9 −1.24243
\(639\) 0 0
\(640\) −1.81031e9 −0.272976
\(641\) −5.39534e8 −0.0809125 −0.0404563 0.999181i \(-0.512881\pi\)
−0.0404563 + 0.999181i \(0.512881\pi\)
\(642\) 0 0
\(643\) 6.57420e8 0.0975225 0.0487612 0.998810i \(-0.484473\pi\)
0.0487612 + 0.998810i \(0.484473\pi\)
\(644\) −4.52952e9 −0.668269
\(645\) 0 0
\(646\) −4.06725e9 −0.593590
\(647\) 2.49688e9 0.362437 0.181219 0.983443i \(-0.441996\pi\)
0.181219 + 0.983443i \(0.441996\pi\)
\(648\) 0 0
\(649\) −7.10561e8 −0.102034
\(650\) −2.90586e8 −0.0415029
\(651\) 0 0
\(652\) −5.71979e9 −0.808191
\(653\) 9.29237e9 1.30596 0.652981 0.757374i \(-0.273518\pi\)
0.652981 + 0.757374i \(0.273518\pi\)
\(654\) 0 0
\(655\) 3.11073e9 0.432532
\(656\) 3.87547e9 0.535995
\(657\) 0 0
\(658\) 8.28369e9 1.13353
\(659\) 6.58505e9 0.896315 0.448157 0.893955i \(-0.352080\pi\)
0.448157 + 0.893955i \(0.352080\pi\)
\(660\) 0 0
\(661\) −3.64989e9 −0.491558 −0.245779 0.969326i \(-0.579044\pi\)
−0.245779 + 0.969326i \(0.579044\pi\)
\(662\) 1.24236e10 1.66434
\(663\) 0 0
\(664\) −2.03183e9 −0.269339
\(665\) 3.96607e9 0.522979
\(666\) 0 0
\(667\) 3.93100e9 0.512936
\(668\) 1.24907e10 1.62132
\(669\) 0 0
\(670\) 5.49651e9 0.706033
\(671\) 1.80650e9 0.230839
\(672\) 0 0
\(673\) 7.02119e9 0.887887 0.443944 0.896055i \(-0.353579\pi\)
0.443944 + 0.896055i \(0.353579\pi\)
\(674\) −6.26326e9 −0.787936
\(675\) 0 0
\(676\) −9.21947e9 −1.14787
\(677\) −8.84175e9 −1.09516 −0.547580 0.836753i \(-0.684451\pi\)
−0.547580 + 0.836753i \(0.684451\pi\)
\(678\) 0 0
\(679\) −3.23380e9 −0.396433
\(680\) −6.71575e8 −0.0819055
\(681\) 0 0
\(682\) 1.28873e10 1.55567
\(683\) −3.28325e9 −0.394304 −0.197152 0.980373i \(-0.563169\pi\)
−0.197152 + 0.980373i \(0.563169\pi\)
\(684\) 0 0
\(685\) 7.66281e9 0.910901
\(686\) 7.60577e9 0.899516
\(687\) 0 0
\(688\) 2.06445e9 0.241682
\(689\) 5.21410e7 0.00607312
\(690\) 0 0
\(691\) 3.72206e9 0.429151 0.214576 0.976707i \(-0.431163\pi\)
0.214576 + 0.976707i \(0.431163\pi\)
\(692\) 3.00684e9 0.344936
\(693\) 0 0
\(694\) 3.73413e9 0.424063
\(695\) 3.55742e9 0.401965
\(696\) 0 0
\(697\) 3.43745e9 0.384523
\(698\) −3.32263e9 −0.369818
\(699\) 0 0
\(700\) −7.77117e9 −0.856334
\(701\) −2.70122e9 −0.296174 −0.148087 0.988974i \(-0.547312\pi\)
−0.148087 + 0.988974i \(0.547312\pi\)
\(702\) 0 0
\(703\) −8.83575e9 −0.959179
\(704\) 9.24887e9 0.999044
\(705\) 0 0
\(706\) −1.60850e8 −0.0172030
\(707\) 1.33752e10 1.42342
\(708\) 0 0
\(709\) 4.56678e8 0.0481225 0.0240613 0.999710i \(-0.492340\pi\)
0.0240613 + 0.999710i \(0.492340\pi\)
\(710\) −9.18312e9 −0.962911
\(711\) 0 0
\(712\) 1.73395e9 0.180035
\(713\) −6.21617e9 −0.642258
\(714\) 0 0
\(715\) −2.23250e8 −0.0228412
\(716\) 3.17281e9 0.323034
\(717\) 0 0
\(718\) 2.64512e10 2.66692
\(719\) −4.28395e9 −0.429827 −0.214913 0.976633i \(-0.568947\pi\)
−0.214913 + 0.976633i \(0.568947\pi\)
\(720\) 0 0
\(721\) −5.00352e9 −0.497167
\(722\) 7.92689e9 0.783831
\(723\) 0 0
\(724\) −9.11749e9 −0.892874
\(725\) 6.74432e9 0.657287
\(726\) 0 0
\(727\) 9.46884e9 0.913958 0.456979 0.889477i \(-0.348931\pi\)
0.456979 + 0.889477i \(0.348931\pi\)
\(728\) 1.30817e8 0.0125662
\(729\) 0 0
\(730\) 9.78998e9 0.931433
\(731\) 1.83112e9 0.173383
\(732\) 0 0
\(733\) −7.31501e9 −0.686043 −0.343021 0.939328i \(-0.611450\pi\)
−0.343021 + 0.939328i \(0.611450\pi\)
\(734\) −7.23723e9 −0.675517
\(735\) 0 0
\(736\) −7.35556e9 −0.680054
\(737\) −6.54997e9 −0.602703
\(738\) 0 0
\(739\) −9.28262e9 −0.846087 −0.423043 0.906109i \(-0.639038\pi\)
−0.423043 + 0.906109i \(0.639038\pi\)
\(740\) −1.11618e10 −1.01256
\(741\) 0 0
\(742\) 2.60656e9 0.234236
\(743\) 8.15117e9 0.729053 0.364526 0.931193i \(-0.381231\pi\)
0.364526 + 0.931193i \(0.381231\pi\)
\(744\) 0 0
\(745\) 1.07583e10 0.953228
\(746\) −1.44515e10 −1.27446
\(747\) 0 0
\(748\) 6.12270e9 0.534919
\(749\) 1.54077e9 0.133983
\(750\) 0 0
\(751\) −1.21007e10 −1.04248 −0.521242 0.853409i \(-0.674531\pi\)
−0.521242 + 0.853409i \(0.674531\pi\)
\(752\) 6.08923e9 0.522156
\(753\) 0 0
\(754\) −8.68582e8 −0.0737923
\(755\) −1.06996e9 −0.0904805
\(756\) 0 0
\(757\) −2.15810e10 −1.80816 −0.904078 0.427367i \(-0.859441\pi\)
−0.904078 + 0.427367i \(0.859441\pi\)
\(758\) −1.83112e10 −1.52713
\(759\) 0 0
\(760\) −1.13980e9 −0.0941847
\(761\) −2.06853e9 −0.170144 −0.0850719 0.996375i \(-0.527112\pi\)
−0.0850719 + 0.996375i \(0.527112\pi\)
\(762\) 0 0
\(763\) −1.68129e10 −1.37027
\(764\) 2.69835e10 2.18913
\(765\) 0 0
\(766\) 5.36452e9 0.431251
\(767\) −7.57302e7 −0.00606018
\(768\) 0 0
\(769\) 1.90437e10 1.51011 0.755055 0.655662i \(-0.227610\pi\)
0.755055 + 0.655662i \(0.227610\pi\)
\(770\) −1.11604e10 −0.880972
\(771\) 0 0
\(772\) −3.70581e9 −0.289883
\(773\) −7.43503e9 −0.578968 −0.289484 0.957183i \(-0.593484\pi\)
−0.289484 + 0.957183i \(0.593484\pi\)
\(774\) 0 0
\(775\) −1.06649e10 −0.823003
\(776\) 9.29355e8 0.0713946
\(777\) 0 0
\(778\) 1.80945e10 1.37758
\(779\) 5.83405e9 0.442170
\(780\) 0 0
\(781\) 1.09432e10 0.821986
\(782\) −5.52053e9 −0.412816
\(783\) 0 0
\(784\) −5.56814e9 −0.412671
\(785\) −6.29308e9 −0.464322
\(786\) 0 0
\(787\) 9.17424e9 0.670901 0.335451 0.942058i \(-0.391111\pi\)
0.335451 + 0.942058i \(0.391111\pi\)
\(788\) −6.05920e9 −0.441137
\(789\) 0 0
\(790\) −4.77046e9 −0.344243
\(791\) 1.94424e10 1.39679
\(792\) 0 0
\(793\) 1.92533e8 0.0137104
\(794\) 3.64586e10 2.58481
\(795\) 0 0
\(796\) 1.77979e10 1.25076
\(797\) 2.11317e10 1.47853 0.739265 0.673415i \(-0.235173\pi\)
0.739265 + 0.673415i \(0.235173\pi\)
\(798\) 0 0
\(799\) 5.40101e9 0.374594
\(800\) −1.26197e10 −0.871436
\(801\) 0 0
\(802\) −4.23727e10 −2.90052
\(803\) −1.16663e10 −0.795115
\(804\) 0 0
\(805\) 5.38319e9 0.363709
\(806\) 1.37351e9 0.0923970
\(807\) 0 0
\(808\) −3.84386e9 −0.256347
\(809\) 2.83173e9 0.188032 0.0940161 0.995571i \(-0.470029\pi\)
0.0940161 + 0.995571i \(0.470029\pi\)
\(810\) 0 0
\(811\) 2.43178e10 1.60085 0.800427 0.599430i \(-0.204606\pi\)
0.800427 + 0.599430i \(0.204606\pi\)
\(812\) −2.32286e10 −1.52257
\(813\) 0 0
\(814\) 2.48635e10 1.61576
\(815\) 6.79780e9 0.439862
\(816\) 0 0
\(817\) 3.10778e9 0.199376
\(818\) 4.78863e9 0.305897
\(819\) 0 0
\(820\) 7.36988e9 0.466780
\(821\) 2.00983e10 1.26753 0.633765 0.773525i \(-0.281509\pi\)
0.633765 + 0.773525i \(0.281509\pi\)
\(822\) 0 0
\(823\) 1.63406e10 1.02180 0.510902 0.859639i \(-0.329312\pi\)
0.510902 + 0.859639i \(0.329312\pi\)
\(824\) 1.43795e9 0.0895362
\(825\) 0 0
\(826\) −3.78580e9 −0.233737
\(827\) 9.17251e9 0.563922 0.281961 0.959426i \(-0.409015\pi\)
0.281961 + 0.959426i \(0.409015\pi\)
\(828\) 0 0
\(829\) −2.43239e10 −1.48283 −0.741417 0.671045i \(-0.765846\pi\)
−0.741417 + 0.671045i \(0.765846\pi\)
\(830\) 1.84745e10 1.12150
\(831\) 0 0
\(832\) 9.85727e8 0.0593369
\(833\) −4.93881e9 −0.296050
\(834\) 0 0
\(835\) −1.48448e10 −0.882415
\(836\) 1.03915e10 0.615113
\(837\) 0 0
\(838\) −8.80946e9 −0.517125
\(839\) −1.35860e9 −0.0794193 −0.0397096 0.999211i \(-0.512643\pi\)
−0.0397096 + 0.999211i \(0.512643\pi\)
\(840\) 0 0
\(841\) 2.90934e9 0.168659
\(842\) 3.56854e10 2.06015
\(843\) 0 0
\(844\) −1.82917e10 −1.04726
\(845\) 1.09571e10 0.624734
\(846\) 0 0
\(847\) −8.35219e9 −0.472290
\(848\) 1.91605e9 0.107900
\(849\) 0 0
\(850\) −9.47142e9 −0.528992
\(851\) −1.19929e10 −0.667067
\(852\) 0 0
\(853\) −1.40851e10 −0.777031 −0.388516 0.921442i \(-0.627012\pi\)
−0.388516 + 0.921442i \(0.627012\pi\)
\(854\) 9.62486e9 0.528800
\(855\) 0 0
\(856\) −4.42797e8 −0.0241294
\(857\) −2.25854e10 −1.22573 −0.612865 0.790188i \(-0.709983\pi\)
−0.612865 + 0.790188i \(0.709983\pi\)
\(858\) 0 0
\(859\) 3.28621e10 1.76897 0.884483 0.466572i \(-0.154511\pi\)
0.884483 + 0.466572i \(0.154511\pi\)
\(860\) 3.92591e9 0.210473
\(861\) 0 0
\(862\) −5.49107e9 −0.291999
\(863\) 7.68268e9 0.406888 0.203444 0.979087i \(-0.434787\pi\)
0.203444 + 0.979087i \(0.434787\pi\)
\(864\) 0 0
\(865\) −3.57353e9 −0.187733
\(866\) −3.40273e10 −1.78039
\(867\) 0 0
\(868\) 3.67318e10 1.90644
\(869\) 5.68477e9 0.293862
\(870\) 0 0
\(871\) −6.98083e8 −0.0357967
\(872\) 4.83183e9 0.246776
\(873\) 0 0
\(874\) −9.36945e9 −0.474705
\(875\) 2.44260e10 1.23261
\(876\) 0 0
\(877\) 3.95364e10 1.97924 0.989621 0.143702i \(-0.0459008\pi\)
0.989621 + 0.143702i \(0.0459008\pi\)
\(878\) −2.61538e10 −1.30408
\(879\) 0 0
\(880\) −8.20387e9 −0.405816
\(881\) −1.90800e10 −0.940074 −0.470037 0.882647i \(-0.655759\pi\)
−0.470037 + 0.882647i \(0.655759\pi\)
\(882\) 0 0
\(883\) −4.60772e9 −0.225228 −0.112614 0.993639i \(-0.535922\pi\)
−0.112614 + 0.993639i \(0.535922\pi\)
\(884\) 6.52546e8 0.0317708
\(885\) 0 0
\(886\) 5.43530e9 0.262546
\(887\) 1.44107e10 0.693349 0.346674 0.937986i \(-0.387311\pi\)
0.346674 + 0.937986i \(0.387311\pi\)
\(888\) 0 0
\(889\) −2.22495e10 −1.06210
\(890\) −1.57660e10 −0.749644
\(891\) 0 0
\(892\) −6.36591e8 −0.0300320
\(893\) 9.16660e9 0.430753
\(894\) 0 0
\(895\) −3.77078e9 −0.175813
\(896\) 1.14938e10 0.533807
\(897\) 0 0
\(898\) −6.25042e8 −0.0288033
\(899\) −3.18782e10 −1.46330
\(900\) 0 0
\(901\) 1.69949e9 0.0774074
\(902\) −1.64169e10 −0.744847
\(903\) 0 0
\(904\) −5.58750e9 −0.251552
\(905\) 1.08358e10 0.485951
\(906\) 0 0
\(907\) 3.03408e10 1.35021 0.675105 0.737721i \(-0.264098\pi\)
0.675105 + 0.737721i \(0.264098\pi\)
\(908\) 1.91314e10 0.848098
\(909\) 0 0
\(910\) −1.18945e9 −0.0523242
\(911\) 3.82951e9 0.167814 0.0839072 0.996474i \(-0.473260\pi\)
0.0839072 + 0.996474i \(0.473260\pi\)
\(912\) 0 0
\(913\) −2.20153e10 −0.957363
\(914\) −3.24754e9 −0.140683
\(915\) 0 0
\(916\) −8.89104e9 −0.382224
\(917\) −1.97502e10 −0.845821
\(918\) 0 0
\(919\) 2.78817e10 1.18499 0.592495 0.805574i \(-0.298143\pi\)
0.592495 + 0.805574i \(0.298143\pi\)
\(920\) −1.54706e9 −0.0655014
\(921\) 0 0
\(922\) −1.76216e10 −0.740436
\(923\) 1.16630e9 0.0488208
\(924\) 0 0
\(925\) −2.05758e10 −0.854794
\(926\) −1.86518e10 −0.771936
\(927\) 0 0
\(928\) −3.77212e10 −1.54942
\(929\) −1.44369e10 −0.590770 −0.295385 0.955378i \(-0.595448\pi\)
−0.295385 + 0.955378i \(0.595448\pi\)
\(930\) 0 0
\(931\) −8.38216e9 −0.340433
\(932\) 3.50782e10 1.41933
\(933\) 0 0
\(934\) −3.21308e10 −1.29035
\(935\) −7.27664e9 −0.291132
\(936\) 0 0
\(937\) −3.25299e10 −1.29180 −0.645898 0.763424i \(-0.723517\pi\)
−0.645898 + 0.763424i \(0.723517\pi\)
\(938\) −3.48976e10 −1.38066
\(939\) 0 0
\(940\) 1.15797e10 0.454727
\(941\) 1.74946e9 0.0684448 0.0342224 0.999414i \(-0.489105\pi\)
0.0342224 + 0.999414i \(0.489105\pi\)
\(942\) 0 0
\(943\) 7.91863e9 0.307510
\(944\) −2.78290e9 −0.107670
\(945\) 0 0
\(946\) −8.74520e9 −0.335854
\(947\) 2.65135e10 1.01448 0.507239 0.861805i \(-0.330666\pi\)
0.507239 + 0.861805i \(0.330666\pi\)
\(948\) 0 0
\(949\) −1.24337e9 −0.0472248
\(950\) −1.60749e10 −0.608297
\(951\) 0 0
\(952\) 4.26386e9 0.160167
\(953\) −3.31597e10 −1.24104 −0.620519 0.784191i \(-0.713078\pi\)
−0.620519 + 0.784191i \(0.713078\pi\)
\(954\) 0 0
\(955\) −3.20691e10 −1.19145
\(956\) 4.54307e9 0.168169
\(957\) 0 0
\(958\) 4.43361e10 1.62921
\(959\) −4.86516e10 −1.78128
\(960\) 0 0
\(961\) 2.28970e10 0.832236
\(962\) 2.64991e9 0.0959661
\(963\) 0 0
\(964\) 1.43743e9 0.0516793
\(965\) 4.40424e9 0.157770
\(966\) 0 0
\(967\) 1.58334e9 0.0563095 0.0281547 0.999604i \(-0.491037\pi\)
0.0281547 + 0.999604i \(0.491037\pi\)
\(968\) 2.40032e9 0.0850559
\(969\) 0 0
\(970\) −8.45017e9 −0.297279
\(971\) 7.55884e9 0.264965 0.132482 0.991185i \(-0.457705\pi\)
0.132482 + 0.991185i \(0.457705\pi\)
\(972\) 0 0
\(973\) −2.25863e10 −0.786048
\(974\) −1.63620e10 −0.567388
\(975\) 0 0
\(976\) 7.07511e9 0.243590
\(977\) −1.24479e9 −0.0427036 −0.0213518 0.999772i \(-0.506797\pi\)
−0.0213518 + 0.999772i \(0.506797\pi\)
\(978\) 0 0
\(979\) 1.87877e10 0.639931
\(980\) −1.05888e10 −0.359381
\(981\) 0 0
\(982\) −5.53792e10 −1.86619
\(983\) −1.03091e10 −0.346166 −0.173083 0.984907i \(-0.555373\pi\)
−0.173083 + 0.984907i \(0.555373\pi\)
\(984\) 0 0
\(985\) 7.20117e9 0.240091
\(986\) −2.83107e10 −0.940550
\(987\) 0 0
\(988\) 1.10750e9 0.0365338
\(989\) 4.21823e9 0.138657
\(990\) 0 0
\(991\) −1.42062e10 −0.463683 −0.231842 0.972754i \(-0.574475\pi\)
−0.231842 + 0.972754i \(0.574475\pi\)
\(992\) 5.96494e10 1.94006
\(993\) 0 0
\(994\) 5.83041e10 1.88298
\(995\) −2.11523e10 −0.680733
\(996\) 0 0
\(997\) 8.79297e9 0.280998 0.140499 0.990081i \(-0.455129\pi\)
0.140499 + 0.990081i \(0.455129\pi\)
\(998\) −7.16085e10 −2.28038
\(999\) 0 0
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 531.8.a.c.1.3 17
3.2 odd 2 177.8.a.c.1.15 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.c.1.15 17 3.2 odd 2
531.8.a.c.1.3 17 1.1 even 1 trivial