Properties

Label 405.4.a.n.1.2
Level $405$
Weight $4$
Character 405.1
Self dual yes
Analytic conductor $23.896$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.8957735523\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \( x^{7} - 2x^{6} - 44x^{5} + 74x^{4} + 479x^{3} - 460x^{2} - 1200x + 288 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 45)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-3.04174\) of defining polynomial
Character \(\chi\) \(=\) 405.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.04174 q^{2} +1.25219 q^{4} +5.00000 q^{5} -13.7122 q^{7} +20.5251 q^{8} +O(q^{10})\) \(q-3.04174 q^{2} +1.25219 q^{4} +5.00000 q^{5} -13.7122 q^{7} +20.5251 q^{8} -15.2087 q^{10} -31.8068 q^{11} +58.2432 q^{13} +41.7090 q^{14} -72.4495 q^{16} -109.055 q^{17} +129.695 q^{19} +6.26094 q^{20} +96.7481 q^{22} -79.6684 q^{23} +25.0000 q^{25} -177.161 q^{26} -17.1703 q^{28} -9.03537 q^{29} -33.3809 q^{31} +56.1719 q^{32} +331.717 q^{34} -68.5611 q^{35} -22.1645 q^{37} -394.500 q^{38} +102.625 q^{40} -121.740 q^{41} +10.1417 q^{43} -39.8281 q^{44} +242.331 q^{46} +441.492 q^{47} -154.975 q^{49} -76.0435 q^{50} +72.9314 q^{52} +593.610 q^{53} -159.034 q^{55} -281.445 q^{56} +27.4833 q^{58} -442.460 q^{59} +144.576 q^{61} +101.536 q^{62} +408.736 q^{64} +291.216 q^{65} +862.721 q^{67} -136.557 q^{68} +208.545 q^{70} +818.541 q^{71} +495.052 q^{73} +67.4187 q^{74} +162.403 q^{76} +436.142 q^{77} +1170.53 q^{79} -362.248 q^{80} +370.300 q^{82} +424.043 q^{83} -545.274 q^{85} -30.8485 q^{86} -652.838 q^{88} -1031.37 q^{89} -798.643 q^{91} -99.7598 q^{92} -1342.90 q^{94} +648.477 q^{95} +1598.71 q^{97} +471.394 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 2 q^{2} + 36 q^{4} + 35 q^{5} + 22 q^{7} + 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 2 q^{2} + 36 q^{4} + 35 q^{5} + 22 q^{7} + 18 q^{8} + 10 q^{10} + 23 q^{11} + 96 q^{13} - 21 q^{14} + 324 q^{16} + 161 q^{17} + 279 q^{19} + 180 q^{20} + 311 q^{22} + 96 q^{23} + 175 q^{25} - 358 q^{26} + 337 q^{28} - 296 q^{29} + 244 q^{31} - 314 q^{32} + 125 q^{34} + 110 q^{35} + 404 q^{37} + 305 q^{38} + 90 q^{40} - 47 q^{41} + 525 q^{43} + 55 q^{44} + 717 q^{46} + 164 q^{47} + 1225 q^{49} + 50 q^{50} + 1682 q^{52} + 506 q^{53} + 115 q^{55} - 981 q^{56} + 1183 q^{58} - 85 q^{59} + 828 q^{61} - 786 q^{62} + 2236 q^{64} + 480 q^{65} + 1093 q^{67} + 2473 q^{68} - 105 q^{70} + 328 q^{71} + 2085 q^{73} - 1316 q^{74} + 2789 q^{76} + 24 q^{77} + 2110 q^{79} + 1620 q^{80} - 62 q^{82} + 1290 q^{83} + 805 q^{85} - 2569 q^{86} + 2271 q^{88} - 3048 q^{89} + 3338 q^{91} + 2763 q^{92} - 517 q^{94} + 1395 q^{95} + 1787 q^{97} + 1279 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.04174 −1.07542 −0.537709 0.843131i \(-0.680710\pi\)
−0.537709 + 0.843131i \(0.680710\pi\)
\(3\) 0 0
\(4\) 1.25219 0.156523
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) −13.7122 −0.740390 −0.370195 0.928954i \(-0.620709\pi\)
−0.370195 + 0.928954i \(0.620709\pi\)
\(8\) 20.5251 0.907090
\(9\) 0 0
\(10\) −15.2087 −0.480941
\(11\) −31.8068 −0.871829 −0.435914 0.899988i \(-0.643575\pi\)
−0.435914 + 0.899988i \(0.643575\pi\)
\(12\) 0 0
\(13\) 58.2432 1.24260 0.621298 0.783574i \(-0.286606\pi\)
0.621298 + 0.783574i \(0.286606\pi\)
\(14\) 41.7090 0.796229
\(15\) 0 0
\(16\) −72.4495 −1.13202
\(17\) −109.055 −1.55586 −0.777932 0.628348i \(-0.783731\pi\)
−0.777932 + 0.628348i \(0.783731\pi\)
\(18\) 0 0
\(19\) 129.695 1.56601 0.783004 0.622017i \(-0.213687\pi\)
0.783004 + 0.622017i \(0.213687\pi\)
\(20\) 6.26094 0.0699994
\(21\) 0 0
\(22\) 96.7481 0.937580
\(23\) −79.6684 −0.722261 −0.361131 0.932515i \(-0.617609\pi\)
−0.361131 + 0.932515i \(0.617609\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −177.161 −1.33631
\(27\) 0 0
\(28\) −17.1703 −0.115888
\(29\) −9.03537 −0.0578561 −0.0289280 0.999581i \(-0.509209\pi\)
−0.0289280 + 0.999581i \(0.509209\pi\)
\(30\) 0 0
\(31\) −33.3809 −0.193400 −0.0966998 0.995314i \(-0.530829\pi\)
−0.0966998 + 0.995314i \(0.530829\pi\)
\(32\) 56.1719 0.310309
\(33\) 0 0
\(34\) 331.717 1.67320
\(35\) −68.5611 −0.331113
\(36\) 0 0
\(37\) −22.1645 −0.0984817 −0.0492408 0.998787i \(-0.515680\pi\)
−0.0492408 + 0.998787i \(0.515680\pi\)
\(38\) −394.500 −1.68411
\(39\) 0 0
\(40\) 102.625 0.405663
\(41\) −121.740 −0.463720 −0.231860 0.972749i \(-0.574481\pi\)
−0.231860 + 0.972749i \(0.574481\pi\)
\(42\) 0 0
\(43\) 10.1417 0.0359674 0.0179837 0.999838i \(-0.494275\pi\)
0.0179837 + 0.999838i \(0.494275\pi\)
\(44\) −39.8281 −0.136462
\(45\) 0 0
\(46\) 242.331 0.776733
\(47\) 441.492 1.37018 0.685088 0.728460i \(-0.259764\pi\)
0.685088 + 0.728460i \(0.259764\pi\)
\(48\) 0 0
\(49\) −154.975 −0.451822
\(50\) −76.0435 −0.215084
\(51\) 0 0
\(52\) 72.9314 0.194495
\(53\) 593.610 1.53846 0.769232 0.638969i \(-0.220639\pi\)
0.769232 + 0.638969i \(0.220639\pi\)
\(54\) 0 0
\(55\) −159.034 −0.389894
\(56\) −281.445 −0.671601
\(57\) 0 0
\(58\) 27.4833 0.0622195
\(59\) −442.460 −0.976328 −0.488164 0.872752i \(-0.662333\pi\)
−0.488164 + 0.872752i \(0.662333\pi\)
\(60\) 0 0
\(61\) 144.576 0.303460 0.151730 0.988422i \(-0.451516\pi\)
0.151730 + 0.988422i \(0.451516\pi\)
\(62\) 101.536 0.207985
\(63\) 0 0
\(64\) 408.736 0.798312
\(65\) 291.216 0.555706
\(66\) 0 0
\(67\) 862.721 1.57311 0.786553 0.617523i \(-0.211864\pi\)
0.786553 + 0.617523i \(0.211864\pi\)
\(68\) −136.557 −0.243529
\(69\) 0 0
\(70\) 208.545 0.356084
\(71\) 818.541 1.36821 0.684105 0.729384i \(-0.260193\pi\)
0.684105 + 0.729384i \(0.260193\pi\)
\(72\) 0 0
\(73\) 495.052 0.793719 0.396859 0.917879i \(-0.370100\pi\)
0.396859 + 0.917879i \(0.370100\pi\)
\(74\) 67.4187 0.105909
\(75\) 0 0
\(76\) 162.403 0.245117
\(77\) 436.142 0.645494
\(78\) 0 0
\(79\) 1170.53 1.66702 0.833510 0.552505i \(-0.186328\pi\)
0.833510 + 0.552505i \(0.186328\pi\)
\(80\) −362.248 −0.506256
\(81\) 0 0
\(82\) 370.300 0.498693
\(83\) 424.043 0.560781 0.280390 0.959886i \(-0.409536\pi\)
0.280390 + 0.959886i \(0.409536\pi\)
\(84\) 0 0
\(85\) −545.274 −0.695804
\(86\) −30.8485 −0.0386799
\(87\) 0 0
\(88\) −652.838 −0.790827
\(89\) −1031.37 −1.22837 −0.614183 0.789163i \(-0.710514\pi\)
−0.614183 + 0.789163i \(0.710514\pi\)
\(90\) 0 0
\(91\) −798.643 −0.920006
\(92\) −99.7598 −0.113051
\(93\) 0 0
\(94\) −1342.90 −1.47351
\(95\) 648.477 0.700340
\(96\) 0 0
\(97\) 1598.71 1.67345 0.836725 0.547623i \(-0.184467\pi\)
0.836725 + 0.547623i \(0.184467\pi\)
\(98\) 471.394 0.485897
\(99\) 0 0
\(100\) 31.3047 0.0313047
\(101\) 214.000 0.210830 0.105415 0.994428i \(-0.466383\pi\)
0.105415 + 0.994428i \(0.466383\pi\)
\(102\) 0 0
\(103\) 1672.32 1.59979 0.799897 0.600138i \(-0.204888\pi\)
0.799897 + 0.600138i \(0.204888\pi\)
\(104\) 1195.45 1.12715
\(105\) 0 0
\(106\) −1805.61 −1.65449
\(107\) 600.699 0.542727 0.271363 0.962477i \(-0.412525\pi\)
0.271363 + 0.962477i \(0.412525\pi\)
\(108\) 0 0
\(109\) −771.570 −0.678009 −0.339005 0.940785i \(-0.610090\pi\)
−0.339005 + 0.940785i \(0.610090\pi\)
\(110\) 483.740 0.419299
\(111\) 0 0
\(112\) 993.444 0.838140
\(113\) 1166.68 0.971254 0.485627 0.874166i \(-0.338591\pi\)
0.485627 + 0.874166i \(0.338591\pi\)
\(114\) 0 0
\(115\) −398.342 −0.323005
\(116\) −11.3140 −0.00905583
\(117\) 0 0
\(118\) 1345.85 1.04996
\(119\) 1495.38 1.15195
\(120\) 0 0
\(121\) −319.327 −0.239915
\(122\) −439.763 −0.326347
\(123\) 0 0
\(124\) −41.7991 −0.0302716
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 1630.10 1.13896 0.569479 0.822006i \(-0.307145\pi\)
0.569479 + 0.822006i \(0.307145\pi\)
\(128\) −1692.64 −1.16883
\(129\) 0 0
\(130\) −885.803 −0.597616
\(131\) −258.825 −0.172623 −0.0863115 0.996268i \(-0.527508\pi\)
−0.0863115 + 0.996268i \(0.527508\pi\)
\(132\) 0 0
\(133\) −1778.41 −1.15946
\(134\) −2624.17 −1.69175
\(135\) 0 0
\(136\) −2238.36 −1.41131
\(137\) −1052.77 −0.656527 −0.328264 0.944586i \(-0.606463\pi\)
−0.328264 + 0.944586i \(0.606463\pi\)
\(138\) 0 0
\(139\) 2384.50 1.45504 0.727519 0.686088i \(-0.240673\pi\)
0.727519 + 0.686088i \(0.240673\pi\)
\(140\) −85.8514 −0.0518269
\(141\) 0 0
\(142\) −2489.79 −1.47140
\(143\) −1852.53 −1.08333
\(144\) 0 0
\(145\) −45.1769 −0.0258740
\(146\) −1505.82 −0.853579
\(147\) 0 0
\(148\) −27.7541 −0.0154147
\(149\) −1146.29 −0.630253 −0.315126 0.949050i \(-0.602047\pi\)
−0.315126 + 0.949050i \(0.602047\pi\)
\(150\) 0 0
\(151\) −1166.12 −0.628458 −0.314229 0.949347i \(-0.601746\pi\)
−0.314229 + 0.949347i \(0.601746\pi\)
\(152\) 2662.01 1.42051
\(153\) 0 0
\(154\) −1326.63 −0.694175
\(155\) −166.904 −0.0864909
\(156\) 0 0
\(157\) −3475.62 −1.76678 −0.883390 0.468638i \(-0.844745\pi\)
−0.883390 + 0.468638i \(0.844745\pi\)
\(158\) −3560.44 −1.79274
\(159\) 0 0
\(160\) 280.860 0.138774
\(161\) 1092.43 0.534755
\(162\) 0 0
\(163\) −2223.32 −1.06837 −0.534183 0.845369i \(-0.679381\pi\)
−0.534183 + 0.845369i \(0.679381\pi\)
\(164\) −152.441 −0.0725831
\(165\) 0 0
\(166\) −1289.83 −0.603074
\(167\) 1587.98 0.735819 0.367909 0.929862i \(-0.380074\pi\)
0.367909 + 0.929862i \(0.380074\pi\)
\(168\) 0 0
\(169\) 1195.26 0.544044
\(170\) 1658.58 0.748280
\(171\) 0 0
\(172\) 12.6993 0.00562974
\(173\) −1507.80 −0.662634 −0.331317 0.943520i \(-0.607493\pi\)
−0.331317 + 0.943520i \(0.607493\pi\)
\(174\) 0 0
\(175\) −342.806 −0.148078
\(176\) 2304.39 0.986931
\(177\) 0 0
\(178\) 3137.15 1.32101
\(179\) −1572.66 −0.656683 −0.328341 0.944559i \(-0.606490\pi\)
−0.328341 + 0.944559i \(0.606490\pi\)
\(180\) 0 0
\(181\) 1984.41 0.814918 0.407459 0.913223i \(-0.366415\pi\)
0.407459 + 0.913223i \(0.366415\pi\)
\(182\) 2429.27 0.989391
\(183\) 0 0
\(184\) −1635.20 −0.655156
\(185\) −110.823 −0.0440423
\(186\) 0 0
\(187\) 3468.69 1.35645
\(188\) 552.831 0.214465
\(189\) 0 0
\(190\) −1972.50 −0.753158
\(191\) 2348.35 0.889639 0.444819 0.895620i \(-0.353268\pi\)
0.444819 + 0.895620i \(0.353268\pi\)
\(192\) 0 0
\(193\) −234.169 −0.0873362 −0.0436681 0.999046i \(-0.513904\pi\)
−0.0436681 + 0.999046i \(0.513904\pi\)
\(194\) −4862.87 −1.79966
\(195\) 0 0
\(196\) −194.058 −0.0707207
\(197\) 717.846 0.259616 0.129808 0.991539i \(-0.458564\pi\)
0.129808 + 0.991539i \(0.458564\pi\)
\(198\) 0 0
\(199\) 1701.45 0.606094 0.303047 0.952976i \(-0.401996\pi\)
0.303047 + 0.952976i \(0.401996\pi\)
\(200\) 513.127 0.181418
\(201\) 0 0
\(202\) −650.933 −0.226730
\(203\) 123.895 0.0428361
\(204\) 0 0
\(205\) −608.698 −0.207382
\(206\) −5086.77 −1.72045
\(207\) 0 0
\(208\) −4219.69 −1.40665
\(209\) −4125.19 −1.36529
\(210\) 0 0
\(211\) −4938.66 −1.61133 −0.805667 0.592368i \(-0.798193\pi\)
−0.805667 + 0.592368i \(0.798193\pi\)
\(212\) 743.311 0.240806
\(213\) 0 0
\(214\) −1827.17 −0.583658
\(215\) 50.7086 0.0160851
\(216\) 0 0
\(217\) 457.726 0.143191
\(218\) 2346.92 0.729143
\(219\) 0 0
\(220\) −199.141 −0.0610275
\(221\) −6351.70 −1.93331
\(222\) 0 0
\(223\) −3053.15 −0.916833 −0.458416 0.888737i \(-0.651583\pi\)
−0.458416 + 0.888737i \(0.651583\pi\)
\(224\) −770.242 −0.229750
\(225\) 0 0
\(226\) −3548.73 −1.04450
\(227\) 839.953 0.245593 0.122797 0.992432i \(-0.460814\pi\)
0.122797 + 0.992432i \(0.460814\pi\)
\(228\) 0 0
\(229\) 1401.51 0.404429 0.202214 0.979341i \(-0.435186\pi\)
0.202214 + 0.979341i \(0.435186\pi\)
\(230\) 1211.65 0.347365
\(231\) 0 0
\(232\) −185.452 −0.0524807
\(233\) 2856.99 0.803295 0.401647 0.915794i \(-0.368438\pi\)
0.401647 + 0.915794i \(0.368438\pi\)
\(234\) 0 0
\(235\) 2207.46 0.612761
\(236\) −554.043 −0.152818
\(237\) 0 0
\(238\) −4548.57 −1.23882
\(239\) 1709.83 0.462761 0.231381 0.972863i \(-0.425676\pi\)
0.231381 + 0.972863i \(0.425676\pi\)
\(240\) 0 0
\(241\) −4332.45 −1.15800 −0.579000 0.815328i \(-0.696557\pi\)
−0.579000 + 0.815328i \(0.696557\pi\)
\(242\) 971.309 0.258009
\(243\) 0 0
\(244\) 181.036 0.0474987
\(245\) −774.875 −0.202061
\(246\) 0 0
\(247\) 7553.86 1.94591
\(248\) −685.146 −0.175431
\(249\) 0 0
\(250\) −380.218 −0.0961883
\(251\) 1724.73 0.433722 0.216861 0.976202i \(-0.430418\pi\)
0.216861 + 0.976202i \(0.430418\pi\)
\(252\) 0 0
\(253\) 2534.00 0.629688
\(254\) −4958.33 −1.22486
\(255\) 0 0
\(256\) 1878.70 0.458666
\(257\) 5853.25 1.42068 0.710342 0.703856i \(-0.248540\pi\)
0.710342 + 0.703856i \(0.248540\pi\)
\(258\) 0 0
\(259\) 303.925 0.0729149
\(260\) 364.657 0.0869810
\(261\) 0 0
\(262\) 787.277 0.185642
\(263\) −392.360 −0.0919923 −0.0459962 0.998942i \(-0.514646\pi\)
−0.0459962 + 0.998942i \(0.514646\pi\)
\(264\) 0 0
\(265\) 2968.05 0.688022
\(266\) 5409.47 1.24690
\(267\) 0 0
\(268\) 1080.29 0.246228
\(269\) 4610.33 1.04497 0.522485 0.852648i \(-0.325005\pi\)
0.522485 + 0.852648i \(0.325005\pi\)
\(270\) 0 0
\(271\) −1155.72 −0.259058 −0.129529 0.991576i \(-0.541347\pi\)
−0.129529 + 0.991576i \(0.541347\pi\)
\(272\) 7900.97 1.76128
\(273\) 0 0
\(274\) 3202.25 0.706041
\(275\) −795.170 −0.174366
\(276\) 0 0
\(277\) −608.410 −0.131970 −0.0659852 0.997821i \(-0.521019\pi\)
−0.0659852 + 0.997821i \(0.521019\pi\)
\(278\) −7253.02 −1.56477
\(279\) 0 0
\(280\) −1407.22 −0.300349
\(281\) −2749.24 −0.583650 −0.291825 0.956472i \(-0.594263\pi\)
−0.291825 + 0.956472i \(0.594263\pi\)
\(282\) 0 0
\(283\) 1021.56 0.214578 0.107289 0.994228i \(-0.465783\pi\)
0.107289 + 0.994228i \(0.465783\pi\)
\(284\) 1024.97 0.214157
\(285\) 0 0
\(286\) 5634.91 1.16503
\(287\) 1669.32 0.343334
\(288\) 0 0
\(289\) 6979.96 1.42071
\(290\) 137.416 0.0278254
\(291\) 0 0
\(292\) 619.898 0.124236
\(293\) 5232.50 1.04330 0.521648 0.853161i \(-0.325317\pi\)
0.521648 + 0.853161i \(0.325317\pi\)
\(294\) 0 0
\(295\) −2212.30 −0.436627
\(296\) −454.929 −0.0893317
\(297\) 0 0
\(298\) 3486.71 0.677785
\(299\) −4640.14 −0.897479
\(300\) 0 0
\(301\) −139.065 −0.0266299
\(302\) 3547.02 0.675855
\(303\) 0 0
\(304\) −9396.36 −1.77276
\(305\) 722.881 0.135712
\(306\) 0 0
\(307\) −4912.68 −0.913296 −0.456648 0.889648i \(-0.650950\pi\)
−0.456648 + 0.889648i \(0.650950\pi\)
\(308\) 546.132 0.101035
\(309\) 0 0
\(310\) 507.680 0.0930138
\(311\) 828.860 0.151126 0.0755632 0.997141i \(-0.475925\pi\)
0.0755632 + 0.997141i \(0.475925\pi\)
\(312\) 0 0
\(313\) 9237.21 1.66811 0.834054 0.551683i \(-0.186014\pi\)
0.834054 + 0.551683i \(0.186014\pi\)
\(314\) 10571.9 1.90003
\(315\) 0 0
\(316\) 1465.72 0.260928
\(317\) 9098.41 1.61204 0.806021 0.591887i \(-0.201617\pi\)
0.806021 + 0.591887i \(0.201617\pi\)
\(318\) 0 0
\(319\) 287.386 0.0504406
\(320\) 2043.68 0.357016
\(321\) 0 0
\(322\) −3322.89 −0.575086
\(323\) −14143.9 −2.43649
\(324\) 0 0
\(325\) 1456.08 0.248519
\(326\) 6762.76 1.14894
\(327\) 0 0
\(328\) −2498.72 −0.420636
\(329\) −6053.84 −1.01447
\(330\) 0 0
\(331\) 5963.45 0.990275 0.495137 0.868815i \(-0.335118\pi\)
0.495137 + 0.868815i \(0.335118\pi\)
\(332\) 530.982 0.0877754
\(333\) 0 0
\(334\) −4830.23 −0.791313
\(335\) 4313.60 0.703515
\(336\) 0 0
\(337\) 596.854 0.0964768 0.0482384 0.998836i \(-0.484639\pi\)
0.0482384 + 0.998836i \(0.484639\pi\)
\(338\) −3635.69 −0.585075
\(339\) 0 0
\(340\) −682.786 −0.108910
\(341\) 1061.74 0.168611
\(342\) 0 0
\(343\) 6828.34 1.07492
\(344\) 208.160 0.0326256
\(345\) 0 0
\(346\) 4586.32 0.712608
\(347\) 4562.10 0.705782 0.352891 0.935664i \(-0.385199\pi\)
0.352891 + 0.935664i \(0.385199\pi\)
\(348\) 0 0
\(349\) 675.267 0.103571 0.0517854 0.998658i \(-0.483509\pi\)
0.0517854 + 0.998658i \(0.483509\pi\)
\(350\) 1042.73 0.159246
\(351\) 0 0
\(352\) −1786.65 −0.270536
\(353\) 3361.83 0.506890 0.253445 0.967350i \(-0.418436\pi\)
0.253445 + 0.967350i \(0.418436\pi\)
\(354\) 0 0
\(355\) 4092.70 0.611882
\(356\) −1291.46 −0.192268
\(357\) 0 0
\(358\) 4783.63 0.706208
\(359\) −7701.61 −1.13224 −0.566122 0.824322i \(-0.691557\pi\)
−0.566122 + 0.824322i \(0.691557\pi\)
\(360\) 0 0
\(361\) 9961.87 1.45238
\(362\) −6036.07 −0.876378
\(363\) 0 0
\(364\) −1000.05 −0.144003
\(365\) 2475.26 0.354962
\(366\) 0 0
\(367\) 5909.50 0.840526 0.420263 0.907402i \(-0.361938\pi\)
0.420263 + 0.907402i \(0.361938\pi\)
\(368\) 5771.94 0.817617
\(369\) 0 0
\(370\) 337.093 0.0473639
\(371\) −8139.72 −1.13906
\(372\) 0 0
\(373\) −8424.38 −1.16943 −0.584716 0.811238i \(-0.698794\pi\)
−0.584716 + 0.811238i \(0.698794\pi\)
\(374\) −10550.8 −1.45875
\(375\) 0 0
\(376\) 9061.67 1.24287
\(377\) −526.249 −0.0718917
\(378\) 0 0
\(379\) −3510.24 −0.475749 −0.237875 0.971296i \(-0.576451\pi\)
−0.237875 + 0.971296i \(0.576451\pi\)
\(380\) 812.014 0.109620
\(381\) 0 0
\(382\) −7143.09 −0.956733
\(383\) −5732.03 −0.764733 −0.382367 0.924011i \(-0.624891\pi\)
−0.382367 + 0.924011i \(0.624891\pi\)
\(384\) 0 0
\(385\) 2180.71 0.288674
\(386\) 712.283 0.0939229
\(387\) 0 0
\(388\) 2001.89 0.261934
\(389\) −14367.2 −1.87261 −0.936306 0.351184i \(-0.885779\pi\)
−0.936306 + 0.351184i \(0.885779\pi\)
\(390\) 0 0
\(391\) 8688.23 1.12374
\(392\) −3180.88 −0.409843
\(393\) 0 0
\(394\) −2183.50 −0.279196
\(395\) 5852.63 0.745514
\(396\) 0 0
\(397\) −9014.42 −1.13960 −0.569800 0.821784i \(-0.692979\pi\)
−0.569800 + 0.821784i \(0.692979\pi\)
\(398\) −5175.37 −0.651804
\(399\) 0 0
\(400\) −1811.24 −0.226405
\(401\) −10511.1 −1.30897 −0.654487 0.756073i \(-0.727115\pi\)
−0.654487 + 0.756073i \(0.727115\pi\)
\(402\) 0 0
\(403\) −1944.21 −0.240317
\(404\) 267.969 0.0329998
\(405\) 0 0
\(406\) −376.857 −0.0460667
\(407\) 704.982 0.0858591
\(408\) 0 0
\(409\) 3318.20 0.401160 0.200580 0.979677i \(-0.435717\pi\)
0.200580 + 0.979677i \(0.435717\pi\)
\(410\) 1851.50 0.223022
\(411\) 0 0
\(412\) 2094.06 0.250405
\(413\) 6067.11 0.722864
\(414\) 0 0
\(415\) 2120.22 0.250789
\(416\) 3271.63 0.385589
\(417\) 0 0
\(418\) 12547.8 1.46826
\(419\) 486.138 0.0566812 0.0283406 0.999598i \(-0.490978\pi\)
0.0283406 + 0.999598i \(0.490978\pi\)
\(420\) 0 0
\(421\) 5051.93 0.584836 0.292418 0.956291i \(-0.405540\pi\)
0.292418 + 0.956291i \(0.405540\pi\)
\(422\) 15022.1 1.73286
\(423\) 0 0
\(424\) 12183.9 1.39553
\(425\) −2726.37 −0.311173
\(426\) 0 0
\(427\) −1982.46 −0.224679
\(428\) 752.188 0.0849495
\(429\) 0 0
\(430\) −154.242 −0.0172982
\(431\) −6944.24 −0.776084 −0.388042 0.921642i \(-0.626848\pi\)
−0.388042 + 0.921642i \(0.626848\pi\)
\(432\) 0 0
\(433\) −13738.3 −1.52476 −0.762378 0.647131i \(-0.775968\pi\)
−0.762378 + 0.647131i \(0.775968\pi\)
\(434\) −1392.28 −0.153990
\(435\) 0 0
\(436\) −966.150 −0.106124
\(437\) −10332.6 −1.13107
\(438\) 0 0
\(439\) 8481.83 0.922132 0.461066 0.887366i \(-0.347467\pi\)
0.461066 + 0.887366i \(0.347467\pi\)
\(440\) −3264.19 −0.353668
\(441\) 0 0
\(442\) 19320.2 2.07912
\(443\) 558.195 0.0598660 0.0299330 0.999552i \(-0.490471\pi\)
0.0299330 + 0.999552i \(0.490471\pi\)
\(444\) 0 0
\(445\) −5156.83 −0.549342
\(446\) 9286.88 0.985979
\(447\) 0 0
\(448\) −5604.68 −0.591063
\(449\) −14775.0 −1.55295 −0.776476 0.630147i \(-0.782995\pi\)
−0.776476 + 0.630147i \(0.782995\pi\)
\(450\) 0 0
\(451\) 3872.15 0.404285
\(452\) 1460.90 0.152024
\(453\) 0 0
\(454\) −2554.92 −0.264115
\(455\) −3993.22 −0.411439
\(456\) 0 0
\(457\) −14207.9 −1.45431 −0.727154 0.686474i \(-0.759158\pi\)
−0.727154 + 0.686474i \(0.759158\pi\)
\(458\) −4263.02 −0.434930
\(459\) 0 0
\(460\) −498.799 −0.0505579
\(461\) 7702.51 0.778182 0.389091 0.921199i \(-0.372789\pi\)
0.389091 + 0.921199i \(0.372789\pi\)
\(462\) 0 0
\(463\) 7909.07 0.793878 0.396939 0.917845i \(-0.370072\pi\)
0.396939 + 0.917845i \(0.370072\pi\)
\(464\) 654.608 0.0654945
\(465\) 0 0
\(466\) −8690.22 −0.863877
\(467\) −11639.8 −1.15337 −0.576685 0.816967i \(-0.695654\pi\)
−0.576685 + 0.816967i \(0.695654\pi\)
\(468\) 0 0
\(469\) −11829.8 −1.16471
\(470\) −6714.52 −0.658974
\(471\) 0 0
\(472\) −9081.53 −0.885618
\(473\) −322.576 −0.0313574
\(474\) 0 0
\(475\) 3242.38 0.313202
\(476\) 1872.50 0.180307
\(477\) 0 0
\(478\) −5200.87 −0.497662
\(479\) 18695.9 1.78337 0.891687 0.452653i \(-0.149522\pi\)
0.891687 + 0.452653i \(0.149522\pi\)
\(480\) 0 0
\(481\) −1290.93 −0.122373
\(482\) 13178.2 1.24533
\(483\) 0 0
\(484\) −399.857 −0.0375523
\(485\) 7993.56 0.748390
\(486\) 0 0
\(487\) −2249.54 −0.209315 −0.104658 0.994508i \(-0.533375\pi\)
−0.104658 + 0.994508i \(0.533375\pi\)
\(488\) 2967.44 0.275266
\(489\) 0 0
\(490\) 2356.97 0.217300
\(491\) −1988.35 −0.182756 −0.0913778 0.995816i \(-0.529127\pi\)
−0.0913778 + 0.995816i \(0.529127\pi\)
\(492\) 0 0
\(493\) 985.351 0.0900162
\(494\) −22976.9 −2.09267
\(495\) 0 0
\(496\) 2418.43 0.218933
\(497\) −11224.0 −1.01301
\(498\) 0 0
\(499\) 2413.90 0.216556 0.108278 0.994121i \(-0.465466\pi\)
0.108278 + 0.994121i \(0.465466\pi\)
\(500\) 156.523 0.0139999
\(501\) 0 0
\(502\) −5246.19 −0.466432
\(503\) 8758.56 0.776391 0.388196 0.921577i \(-0.373098\pi\)
0.388196 + 0.921577i \(0.373098\pi\)
\(504\) 0 0
\(505\) 1070.00 0.0942860
\(506\) −7707.77 −0.677178
\(507\) 0 0
\(508\) 2041.19 0.178274
\(509\) 11848.0 1.03174 0.515869 0.856667i \(-0.327469\pi\)
0.515869 + 0.856667i \(0.327469\pi\)
\(510\) 0 0
\(511\) −6788.27 −0.587662
\(512\) 7826.64 0.675570
\(513\) 0 0
\(514\) −17804.1 −1.52783
\(515\) 8361.60 0.715449
\(516\) 0 0
\(517\) −14042.5 −1.19456
\(518\) −924.460 −0.0784140
\(519\) 0 0
\(520\) 5977.23 0.504075
\(521\) −3816.55 −0.320933 −0.160466 0.987041i \(-0.551300\pi\)
−0.160466 + 0.987041i \(0.551300\pi\)
\(522\) 0 0
\(523\) −12158.9 −1.01658 −0.508288 0.861187i \(-0.669722\pi\)
−0.508288 + 0.861187i \(0.669722\pi\)
\(524\) −324.097 −0.0270195
\(525\) 0 0
\(526\) 1193.46 0.0989302
\(527\) 3640.35 0.300903
\(528\) 0 0
\(529\) −5819.94 −0.478338
\(530\) −9028.04 −0.739911
\(531\) 0 0
\(532\) −2226.90 −0.181482
\(533\) −7090.50 −0.576217
\(534\) 0 0
\(535\) 3003.49 0.242715
\(536\) 17707.4 1.42695
\(537\) 0 0
\(538\) −14023.4 −1.12378
\(539\) 4929.26 0.393911
\(540\) 0 0
\(541\) 14919.8 1.18568 0.592840 0.805321i \(-0.298007\pi\)
0.592840 + 0.805321i \(0.298007\pi\)
\(542\) 3515.39 0.278596
\(543\) 0 0
\(544\) −6125.82 −0.482799
\(545\) −3857.85 −0.303215
\(546\) 0 0
\(547\) −1027.16 −0.0802889 −0.0401444 0.999194i \(-0.512782\pi\)
−0.0401444 + 0.999194i \(0.512782\pi\)
\(548\) −1318.27 −0.102762
\(549\) 0 0
\(550\) 2418.70 0.187516
\(551\) −1171.85 −0.0906031
\(552\) 0 0
\(553\) −16050.5 −1.23425
\(554\) 1850.63 0.141923
\(555\) 0 0
\(556\) 2985.84 0.227748
\(557\) 10590.8 0.805648 0.402824 0.915277i \(-0.368029\pi\)
0.402824 + 0.915277i \(0.368029\pi\)
\(558\) 0 0
\(559\) 590.685 0.0446929
\(560\) 4967.22 0.374827
\(561\) 0 0
\(562\) 8362.46 0.627668
\(563\) 2192.07 0.164093 0.0820467 0.996628i \(-0.473854\pi\)
0.0820467 + 0.996628i \(0.473854\pi\)
\(564\) 0 0
\(565\) 5833.38 0.434358
\(566\) −3107.32 −0.230761
\(567\) 0 0
\(568\) 16800.6 1.24109
\(569\) 16569.8 1.22081 0.610405 0.792089i \(-0.291007\pi\)
0.610405 + 0.792089i \(0.291007\pi\)
\(570\) 0 0
\(571\) 7279.33 0.533503 0.266752 0.963765i \(-0.414050\pi\)
0.266752 + 0.963765i \(0.414050\pi\)
\(572\) −2319.71 −0.169567
\(573\) 0 0
\(574\) −5077.64 −0.369227
\(575\) −1991.71 −0.144452
\(576\) 0 0
\(577\) −17938.0 −1.29422 −0.647112 0.762395i \(-0.724024\pi\)
−0.647112 + 0.762395i \(0.724024\pi\)
\(578\) −21231.2 −1.52786
\(579\) 0 0
\(580\) −56.5699 −0.00404989
\(581\) −5814.58 −0.415197
\(582\) 0 0
\(583\) −18880.8 −1.34128
\(584\) 10161.0 0.719974
\(585\) 0 0
\(586\) −15915.9 −1.12198
\(587\) −11509.7 −0.809293 −0.404646 0.914473i \(-0.632605\pi\)
−0.404646 + 0.914473i \(0.632605\pi\)
\(588\) 0 0
\(589\) −4329.35 −0.302865
\(590\) 6729.24 0.469557
\(591\) 0 0
\(592\) 1605.81 0.111484
\(593\) 9612.80 0.665684 0.332842 0.942983i \(-0.391992\pi\)
0.332842 + 0.942983i \(0.391992\pi\)
\(594\) 0 0
\(595\) 7476.92 0.515166
\(596\) −1435.37 −0.0986493
\(597\) 0 0
\(598\) 14114.1 0.965165
\(599\) −7371.99 −0.502857 −0.251429 0.967876i \(-0.580900\pi\)
−0.251429 + 0.967876i \(0.580900\pi\)
\(600\) 0 0
\(601\) −24190.6 −1.64185 −0.820927 0.571033i \(-0.806543\pi\)
−0.820927 + 0.571033i \(0.806543\pi\)
\(602\) 423.001 0.0286383
\(603\) 0 0
\(604\) −1460.20 −0.0983684
\(605\) −1596.63 −0.107293
\(606\) 0 0
\(607\) −1317.73 −0.0881138 −0.0440569 0.999029i \(-0.514028\pi\)
−0.0440569 + 0.999029i \(0.514028\pi\)
\(608\) 7285.23 0.485946
\(609\) 0 0
\(610\) −2198.82 −0.145947
\(611\) 25713.9 1.70257
\(612\) 0 0
\(613\) −4137.52 −0.272615 −0.136307 0.990667i \(-0.543523\pi\)
−0.136307 + 0.990667i \(0.543523\pi\)
\(614\) 14943.1 0.982174
\(615\) 0 0
\(616\) 8951.86 0.585521
\(617\) 7791.61 0.508393 0.254197 0.967153i \(-0.418189\pi\)
0.254197 + 0.967153i \(0.418189\pi\)
\(618\) 0 0
\(619\) 21956.4 1.42569 0.712844 0.701322i \(-0.247407\pi\)
0.712844 + 0.701322i \(0.247407\pi\)
\(620\) −208.996 −0.0135379
\(621\) 0 0
\(622\) −2521.18 −0.162524
\(623\) 14142.3 0.909471
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) −28097.2 −1.79391
\(627\) 0 0
\(628\) −4352.13 −0.276543
\(629\) 2417.15 0.153224
\(630\) 0 0
\(631\) −11152.7 −0.703618 −0.351809 0.936072i \(-0.614433\pi\)
−0.351809 + 0.936072i \(0.614433\pi\)
\(632\) 24025.2 1.51214
\(633\) 0 0
\(634\) −27675.0 −1.73362
\(635\) 8150.48 0.509357
\(636\) 0 0
\(637\) −9026.23 −0.561432
\(638\) −874.155 −0.0542447
\(639\) 0 0
\(640\) −8463.22 −0.522716
\(641\) −4809.44 −0.296352 −0.148176 0.988961i \(-0.547340\pi\)
−0.148176 + 0.988961i \(0.547340\pi\)
\(642\) 0 0
\(643\) −2553.03 −0.156581 −0.0782904 0.996931i \(-0.524946\pi\)
−0.0782904 + 0.996931i \(0.524946\pi\)
\(644\) 1367.93 0.0837018
\(645\) 0 0
\(646\) 43022.1 2.62025
\(647\) −8446.00 −0.513210 −0.256605 0.966516i \(-0.582604\pi\)
−0.256605 + 0.966516i \(0.582604\pi\)
\(648\) 0 0
\(649\) 14073.2 0.851191
\(650\) −4429.01 −0.267262
\(651\) 0 0
\(652\) −2784.01 −0.167224
\(653\) −8094.87 −0.485110 −0.242555 0.970138i \(-0.577985\pi\)
−0.242555 + 0.970138i \(0.577985\pi\)
\(654\) 0 0
\(655\) −1294.12 −0.0771994
\(656\) 8819.97 0.524942
\(657\) 0 0
\(658\) 18414.2 1.09097
\(659\) −2175.33 −0.128587 −0.0642936 0.997931i \(-0.520479\pi\)
−0.0642936 + 0.997931i \(0.520479\pi\)
\(660\) 0 0
\(661\) −8876.95 −0.522350 −0.261175 0.965291i \(-0.584110\pi\)
−0.261175 + 0.965291i \(0.584110\pi\)
\(662\) −18139.3 −1.06496
\(663\) 0 0
\(664\) 8703.53 0.508679
\(665\) −8892.06 −0.518525
\(666\) 0 0
\(667\) 719.834 0.0417872
\(668\) 1988.45 0.115173
\(669\) 0 0
\(670\) −13120.9 −0.756572
\(671\) −4598.51 −0.264565
\(672\) 0 0
\(673\) −9167.92 −0.525108 −0.262554 0.964917i \(-0.584565\pi\)
−0.262554 + 0.964917i \(0.584565\pi\)
\(674\) −1815.47 −0.103753
\(675\) 0 0
\(676\) 1496.70 0.0851557
\(677\) 18877.9 1.07169 0.535846 0.844316i \(-0.319993\pi\)
0.535846 + 0.844316i \(0.319993\pi\)
\(678\) 0 0
\(679\) −21921.9 −1.23901
\(680\) −11191.8 −0.631156
\(681\) 0 0
\(682\) −3229.54 −0.181328
\(683\) 27207.5 1.52426 0.762128 0.647426i \(-0.224155\pi\)
0.762128 + 0.647426i \(0.224155\pi\)
\(684\) 0 0
\(685\) −5263.85 −0.293608
\(686\) −20770.1 −1.15598
\(687\) 0 0
\(688\) −734.762 −0.0407159
\(689\) 34573.7 1.91169
\(690\) 0 0
\(691\) −17407.2 −0.958325 −0.479162 0.877726i \(-0.659060\pi\)
−0.479162 + 0.877726i \(0.659060\pi\)
\(692\) −1888.04 −0.103718
\(693\) 0 0
\(694\) −13876.7 −0.759011
\(695\) 11922.5 0.650713
\(696\) 0 0
\(697\) 13276.3 0.721486
\(698\) −2053.99 −0.111382
\(699\) 0 0
\(700\) −429.257 −0.0231777
\(701\) −18543.0 −0.999086 −0.499543 0.866289i \(-0.666499\pi\)
−0.499543 + 0.866289i \(0.666499\pi\)
\(702\) 0 0
\(703\) −2874.63 −0.154223
\(704\) −13000.6 −0.695991
\(705\) 0 0
\(706\) −10225.8 −0.545118
\(707\) −2934.42 −0.156096
\(708\) 0 0
\(709\) 26233.8 1.38961 0.694803 0.719200i \(-0.255492\pi\)
0.694803 + 0.719200i \(0.255492\pi\)
\(710\) −12448.9 −0.658029
\(711\) 0 0
\(712\) −21168.9 −1.11424
\(713\) 2659.40 0.139685
\(714\) 0 0
\(715\) −9262.65 −0.484480
\(716\) −1969.27 −0.102786
\(717\) 0 0
\(718\) 23426.3 1.21763
\(719\) 3043.06 0.157840 0.0789199 0.996881i \(-0.474853\pi\)
0.0789199 + 0.996881i \(0.474853\pi\)
\(720\) 0 0
\(721\) −22931.2 −1.18447
\(722\) −30301.4 −1.56192
\(723\) 0 0
\(724\) 2484.86 0.127554
\(725\) −225.884 −0.0115712
\(726\) 0 0
\(727\) 21942.5 1.11940 0.559698 0.828696i \(-0.310917\pi\)
0.559698 + 0.828696i \(0.310917\pi\)
\(728\) −16392.2 −0.834528
\(729\) 0 0
\(730\) −7529.10 −0.381732
\(731\) −1106.00 −0.0559603
\(732\) 0 0
\(733\) −26576.9 −1.33921 −0.669604 0.742718i \(-0.733536\pi\)
−0.669604 + 0.742718i \(0.733536\pi\)
\(734\) −17975.2 −0.903917
\(735\) 0 0
\(736\) −4475.13 −0.224124
\(737\) −27440.4 −1.37148
\(738\) 0 0
\(739\) 28787.7 1.43298 0.716489 0.697598i \(-0.245748\pi\)
0.716489 + 0.697598i \(0.245748\pi\)
\(740\) −138.771 −0.00689366
\(741\) 0 0
\(742\) 24758.9 1.22497
\(743\) 4503.02 0.222341 0.111171 0.993801i \(-0.464540\pi\)
0.111171 + 0.993801i \(0.464540\pi\)
\(744\) 0 0
\(745\) −5731.44 −0.281858
\(746\) 25624.8 1.25763
\(747\) 0 0
\(748\) 4343.45 0.212316
\(749\) −8236.92 −0.401830
\(750\) 0 0
\(751\) 5705.65 0.277233 0.138617 0.990346i \(-0.455734\pi\)
0.138617 + 0.990346i \(0.455734\pi\)
\(752\) −31985.9 −1.55107
\(753\) 0 0
\(754\) 1600.71 0.0773136
\(755\) −5830.58 −0.281055
\(756\) 0 0
\(757\) 17397.6 0.835305 0.417652 0.908607i \(-0.362853\pi\)
0.417652 + 0.908607i \(0.362853\pi\)
\(758\) 10677.2 0.511629
\(759\) 0 0
\(760\) 13310.0 0.635271
\(761\) 35514.4 1.69171 0.845857 0.533410i \(-0.179090\pi\)
0.845857 + 0.533410i \(0.179090\pi\)
\(762\) 0 0
\(763\) 10579.9 0.501992
\(764\) 2940.58 0.139249
\(765\) 0 0
\(766\) 17435.3 0.822408
\(767\) −25770.3 −1.21318
\(768\) 0 0
\(769\) 31263.7 1.46605 0.733027 0.680199i \(-0.238107\pi\)
0.733027 + 0.680199i \(0.238107\pi\)
\(770\) −6633.16 −0.310445
\(771\) 0 0
\(772\) −293.224 −0.0136702
\(773\) 6676.47 0.310654 0.155327 0.987863i \(-0.450357\pi\)
0.155327 + 0.987863i \(0.450357\pi\)
\(774\) 0 0
\(775\) −834.522 −0.0386799
\(776\) 32813.7 1.51797
\(777\) 0 0
\(778\) 43701.3 2.01384
\(779\) −15789.1 −0.726189
\(780\) 0 0
\(781\) −26035.2 −1.19284
\(782\) −26427.3 −1.20849
\(783\) 0 0
\(784\) 11227.9 0.511473
\(785\) −17378.1 −0.790128
\(786\) 0 0
\(787\) 9433.83 0.427293 0.213646 0.976911i \(-0.431466\pi\)
0.213646 + 0.976911i \(0.431466\pi\)
\(788\) 898.878 0.0406361
\(789\) 0 0
\(790\) −17802.2 −0.801739
\(791\) −15997.7 −0.719107
\(792\) 0 0
\(793\) 8420.57 0.377078
\(794\) 27419.5 1.22555
\(795\) 0 0
\(796\) 2130.54 0.0948679
\(797\) 23476.3 1.04338 0.521689 0.853136i \(-0.325302\pi\)
0.521689 + 0.853136i \(0.325302\pi\)
\(798\) 0 0
\(799\) −48146.9 −2.13181
\(800\) 1404.30 0.0620618
\(801\) 0 0
\(802\) 31972.0 1.40769
\(803\) −15746.0 −0.691987
\(804\) 0 0
\(805\) 5462.16 0.239150
\(806\) 5913.78 0.258442
\(807\) 0 0
\(808\) 4392.38 0.191242
\(809\) 33269.8 1.44586 0.722932 0.690919i \(-0.242794\pi\)
0.722932 + 0.690919i \(0.242794\pi\)
\(810\) 0 0
\(811\) 27892.8 1.20771 0.603853 0.797096i \(-0.293631\pi\)
0.603853 + 0.797096i \(0.293631\pi\)
\(812\) 155.140 0.00670485
\(813\) 0 0
\(814\) −2144.37 −0.0923345
\(815\) −11116.6 −0.477788
\(816\) 0 0
\(817\) 1315.33 0.0563252
\(818\) −10093.1 −0.431415
\(819\) 0 0
\(820\) −762.204 −0.0324601
\(821\) 3697.32 0.157171 0.0785856 0.996907i \(-0.474960\pi\)
0.0785856 + 0.996907i \(0.474960\pi\)
\(822\) 0 0
\(823\) 7144.51 0.302602 0.151301 0.988488i \(-0.451654\pi\)
0.151301 + 0.988488i \(0.451654\pi\)
\(824\) 34324.6 1.45116
\(825\) 0 0
\(826\) −18454.6 −0.777381
\(827\) −19866.9 −0.835357 −0.417679 0.908595i \(-0.637156\pi\)
−0.417679 + 0.908595i \(0.637156\pi\)
\(828\) 0 0
\(829\) −36736.5 −1.53910 −0.769548 0.638589i \(-0.779518\pi\)
−0.769548 + 0.638589i \(0.779518\pi\)
\(830\) −6449.15 −0.269703
\(831\) 0 0
\(832\) 23806.1 0.991979
\(833\) 16900.8 0.702974
\(834\) 0 0
\(835\) 7939.91 0.329068
\(836\) −5165.52 −0.213700
\(837\) 0 0
\(838\) −1478.71 −0.0609560
\(839\) −30012.3 −1.23497 −0.617485 0.786583i \(-0.711848\pi\)
−0.617485 + 0.786583i \(0.711848\pi\)
\(840\) 0 0
\(841\) −24307.4 −0.996653
\(842\) −15366.6 −0.628943
\(843\) 0 0
\(844\) −6184.13 −0.252212
\(845\) 5976.32 0.243304
\(846\) 0 0
\(847\) 4378.68 0.177631
\(848\) −43006.8 −1.74158
\(849\) 0 0
\(850\) 8292.92 0.334641
\(851\) 1765.81 0.0711295
\(852\) 0 0
\(853\) −5381.02 −0.215994 −0.107997 0.994151i \(-0.534444\pi\)
−0.107997 + 0.994151i \(0.534444\pi\)
\(854\) 6030.13 0.241624
\(855\) 0 0
\(856\) 12329.4 0.492302
\(857\) −14639.8 −0.583530 −0.291765 0.956490i \(-0.594242\pi\)
−0.291765 + 0.956490i \(0.594242\pi\)
\(858\) 0 0
\(859\) 29540.0 1.17333 0.586667 0.809828i \(-0.300440\pi\)
0.586667 + 0.809828i \(0.300440\pi\)
\(860\) 63.4966 0.00251769
\(861\) 0 0
\(862\) 21122.6 0.834615
\(863\) −3387.63 −0.133622 −0.0668112 0.997766i \(-0.521283\pi\)
−0.0668112 + 0.997766i \(0.521283\pi\)
\(864\) 0 0
\(865\) −7538.98 −0.296339
\(866\) 41788.3 1.63975
\(867\) 0 0
\(868\) 573.159 0.0224128
\(869\) −37230.7 −1.45336
\(870\) 0 0
\(871\) 50247.6 1.95474
\(872\) −15836.5 −0.615015
\(873\) 0 0
\(874\) 31429.2 1.21637
\(875\) −1714.03 −0.0662225
\(876\) 0 0
\(877\) −244.104 −0.00939888 −0.00469944 0.999989i \(-0.501496\pi\)
−0.00469944 + 0.999989i \(0.501496\pi\)
\(878\) −25799.5 −0.991677
\(879\) 0 0
\(880\) 11521.9 0.441369
\(881\) −13910.2 −0.531948 −0.265974 0.963980i \(-0.585693\pi\)
−0.265974 + 0.963980i \(0.585693\pi\)
\(882\) 0 0
\(883\) −8805.87 −0.335607 −0.167803 0.985820i \(-0.553667\pi\)
−0.167803 + 0.985820i \(0.553667\pi\)
\(884\) −7953.52 −0.302608
\(885\) 0 0
\(886\) −1697.89 −0.0643810
\(887\) −13280.8 −0.502735 −0.251368 0.967892i \(-0.580880\pi\)
−0.251368 + 0.967892i \(0.580880\pi\)
\(888\) 0 0
\(889\) −22352.2 −0.843273
\(890\) 15685.8 0.590773
\(891\) 0 0
\(892\) −3823.11 −0.143506
\(893\) 57259.5 2.14571
\(894\) 0 0
\(895\) −7863.30 −0.293677
\(896\) 23209.9 0.865389
\(897\) 0 0
\(898\) 44941.7 1.67007
\(899\) 301.609 0.0111893
\(900\) 0 0
\(901\) −64736.1 −2.39364
\(902\) −11778.1 −0.434775
\(903\) 0 0
\(904\) 23946.2 0.881015
\(905\) 9922.06 0.364442
\(906\) 0 0
\(907\) −1084.24 −0.0396932 −0.0198466 0.999803i \(-0.506318\pi\)
−0.0198466 + 0.999803i \(0.506318\pi\)
\(908\) 1051.78 0.0384411
\(909\) 0 0
\(910\) 12146.3 0.442469
\(911\) −50188.7 −1.82528 −0.912638 0.408769i \(-0.865958\pi\)
−0.912638 + 0.408769i \(0.865958\pi\)
\(912\) 0 0
\(913\) −13487.5 −0.488905
\(914\) 43216.8 1.56399
\(915\) 0 0
\(916\) 1754.95 0.0633026
\(917\) 3549.06 0.127808
\(918\) 0 0
\(919\) 30376.1 1.09033 0.545166 0.838328i \(-0.316467\pi\)
0.545166 + 0.838328i \(0.316467\pi\)
\(920\) −8176.01 −0.292995
\(921\) 0 0
\(922\) −23429.1 −0.836871
\(923\) 47674.4 1.70013
\(924\) 0 0
\(925\) −554.113 −0.0196963
\(926\) −24057.3 −0.853750
\(927\) 0 0
\(928\) −507.534 −0.0179533
\(929\) 46197.6 1.63153 0.815766 0.578382i \(-0.196316\pi\)
0.815766 + 0.578382i \(0.196316\pi\)
\(930\) 0 0
\(931\) −20099.5 −0.707557
\(932\) 3577.49 0.125734
\(933\) 0 0
\(934\) 35405.1 1.24035
\(935\) 17343.4 0.606621
\(936\) 0 0
\(937\) 37004.7 1.29017 0.645085 0.764111i \(-0.276822\pi\)
0.645085 + 0.764111i \(0.276822\pi\)
\(938\) 35983.2 1.25255
\(939\) 0 0
\(940\) 2764.16 0.0959115
\(941\) −43814.4 −1.51786 −0.758932 0.651170i \(-0.774278\pi\)
−0.758932 + 0.651170i \(0.774278\pi\)
\(942\) 0 0
\(943\) 9698.80 0.334927
\(944\) 32056.0 1.10523
\(945\) 0 0
\(946\) 981.191 0.0337223
\(947\) −4609.73 −0.158180 −0.0790898 0.996867i \(-0.525201\pi\)
−0.0790898 + 0.996867i \(0.525201\pi\)
\(948\) 0 0
\(949\) 28833.4 0.986272
\(950\) −9862.49 −0.336822
\(951\) 0 0
\(952\) 30692.9 1.04492
\(953\) −4281.80 −0.145542 −0.0727708 0.997349i \(-0.523184\pi\)
−0.0727708 + 0.997349i \(0.523184\pi\)
\(954\) 0 0
\(955\) 11741.8 0.397859
\(956\) 2141.03 0.0724330
\(957\) 0 0
\(958\) −56868.0 −1.91787
\(959\) 14435.8 0.486086
\(960\) 0 0
\(961\) −28676.7 −0.962597
\(962\) 3926.68 0.131602
\(963\) 0 0
\(964\) −5425.05 −0.181254
\(965\) −1170.85 −0.0390579
\(966\) 0 0
\(967\) −36229.8 −1.20483 −0.602416 0.798182i \(-0.705795\pi\)
−0.602416 + 0.798182i \(0.705795\pi\)
\(968\) −6554.21 −0.217624
\(969\) 0 0
\(970\) −24314.3 −0.804832
\(971\) −844.928 −0.0279249 −0.0139624 0.999903i \(-0.504445\pi\)
−0.0139624 + 0.999903i \(0.504445\pi\)
\(972\) 0 0
\(973\) −32696.7 −1.07730
\(974\) 6842.53 0.225101
\(975\) 0 0
\(976\) −10474.5 −0.343524
\(977\) −45844.9 −1.50124 −0.750619 0.660735i \(-0.770245\pi\)
−0.750619 + 0.660735i \(0.770245\pi\)
\(978\) 0 0
\(979\) 32804.5 1.07093
\(980\) −970.288 −0.0316273
\(981\) 0 0
\(982\) 6048.05 0.196539
\(983\) 40202.0 1.30442 0.652210 0.758039i \(-0.273842\pi\)
0.652210 + 0.758039i \(0.273842\pi\)
\(984\) 0 0
\(985\) 3589.23 0.116104
\(986\) −2997.18 −0.0968050
\(987\) 0 0
\(988\) 9458.86 0.304581
\(989\) −807.974 −0.0259778
\(990\) 0 0
\(991\) 1797.91 0.0576313 0.0288157 0.999585i \(-0.490826\pi\)
0.0288157 + 0.999585i \(0.490826\pi\)
\(992\) −1875.07 −0.0600136
\(993\) 0 0
\(994\) 34140.5 1.08941
\(995\) 8507.26 0.271053
\(996\) 0 0
\(997\) 9132.07 0.290086 0.145043 0.989425i \(-0.453668\pi\)
0.145043 + 0.989425i \(0.453668\pi\)
\(998\) −7342.47 −0.232888
\(999\) 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 405.4.a.n.1.2 7
3.2 odd 2 405.4.a.m.1.6 7
5.4 even 2 2025.4.a.ba.1.6 7
9.2 odd 6 45.4.e.c.31.2 yes 14
9.4 even 3 135.4.e.c.46.6 14
9.5 odd 6 45.4.e.c.16.2 14
9.7 even 3 135.4.e.c.91.6 14
15.14 odd 2 2025.4.a.bb.1.2 7
45.2 even 12 225.4.k.d.49.11 28
45.14 odd 6 225.4.e.d.151.6 14
45.23 even 12 225.4.k.d.124.11 28
45.29 odd 6 225.4.e.d.76.6 14
45.32 even 12 225.4.k.d.124.4 28
45.38 even 12 225.4.k.d.49.4 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
45.4.e.c.16.2 14 9.5 odd 6
45.4.e.c.31.2 yes 14 9.2 odd 6
135.4.e.c.46.6 14 9.4 even 3
135.4.e.c.91.6 14 9.7 even 3
225.4.e.d.76.6 14 45.29 odd 6
225.4.e.d.151.6 14 45.14 odd 6
225.4.k.d.49.4 28 45.38 even 12
225.4.k.d.49.11 28 45.2 even 12
225.4.k.d.124.4 28 45.32 even 12
225.4.k.d.124.11 28 45.23 even 12
405.4.a.m.1.6 7 3.2 odd 2
405.4.a.n.1.2 7 1.1 even 1 trivial
2025.4.a.ba.1.6 7 5.4 even 2
2025.4.a.bb.1.2 7 15.14 odd 2