Properties

Label 25.10.a.c.1.2
Level $25$
Weight $10$
Character 25.1
Self dual yes
Analytic conductor $12.876$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 25 = 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 25.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.8758959041\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \(x^{3} - x^{2} - 652 x + 4000\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(22.2334\) of defining polynomial
Character \(\chi\) \(=\) 25.1

$q$-expansion

\(f(q)\) \(=\) \(q-21.4187 q^{2} +210.171 q^{3} -53.2406 q^{4} -4501.59 q^{6} -9905.49 q^{7} +12106.7 q^{8} +24489.0 q^{9} +O(q^{10})\) \(q-21.4187 q^{2} +210.171 q^{3} -53.2406 q^{4} -4501.59 q^{6} -9905.49 q^{7} +12106.7 q^{8} +24489.0 q^{9} +36453.6 q^{11} -11189.6 q^{12} -164867. q^{13} +212162. q^{14} -232050. q^{16} -82357.1 q^{17} -524521. q^{18} -609617. q^{19} -2.08185e6 q^{21} -780787. q^{22} -1.88578e6 q^{23} +2.54448e6 q^{24} +3.53123e6 q^{26} +1.01008e6 q^{27} +527374. q^{28} +339235. q^{29} +547314. q^{31} -1.22842e6 q^{32} +7.66150e6 q^{33} +1.76398e6 q^{34} -1.30381e6 q^{36} -5.25687e6 q^{37} +1.30572e7 q^{38} -3.46503e7 q^{39} +2.05812e6 q^{41} +4.45904e7 q^{42} +6.76158e6 q^{43} -1.94081e6 q^{44} +4.03909e7 q^{46} -3.15241e7 q^{47} -4.87703e7 q^{48} +5.77651e7 q^{49} -1.73091e7 q^{51} +8.77761e6 q^{52} +4.89593e7 q^{53} -2.16345e7 q^{54} -1.19923e8 q^{56} -1.28124e8 q^{57} -7.26597e6 q^{58} +8.77960e7 q^{59} +3.84654e7 q^{61} -1.17227e7 q^{62} -2.42575e8 q^{63} +1.45121e8 q^{64} -1.64099e8 q^{66} +1.36116e8 q^{67} +4.38474e6 q^{68} -3.96337e8 q^{69} +3.49218e8 q^{71} +2.96481e8 q^{72} -1.61345e8 q^{73} +1.12595e8 q^{74} +3.24564e7 q^{76} -3.61091e8 q^{77} +7.42163e8 q^{78} -1.26975e8 q^{79} -2.69727e8 q^{81} -4.40822e7 q^{82} +2.87494e8 q^{83} +1.10839e8 q^{84} -1.44824e8 q^{86} +7.12976e7 q^{87} +4.41333e8 q^{88} -5.63133e8 q^{89} +1.63309e9 q^{91} +1.00400e8 q^{92} +1.15030e8 q^{93} +6.75205e8 q^{94} -2.58179e8 q^{96} +4.71704e8 q^{97} -1.23725e9 q^{98} +8.92711e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 33q^{2} - 89q^{3} + 341q^{4} + 3421q^{6} - 5258q^{7} + 105q^{8} + 58234q^{9} + O(q^{10}) \) \( 3q - 33q^{2} - 89q^{3} + 341q^{4} + 3421q^{6} - 5258q^{7} + 105q^{8} + 58234q^{9} - 54699q^{11} - 141853q^{12} - 215884q^{13} + 272922q^{14} - 699247q^{16} - 334983q^{17} - 2571854q^{18} + 818845q^{19} - 2375394q^{21} - 72761q^{22} - 3526854q^{23} + 2805735q^{24} - 1280004q^{26} - 6633395q^{27} + 428134q^{28} + 2175480q^{29} + 4274066q^{31} + 9464577q^{32} + 22122137q^{33} - 6838963q^{34} + 26795798q^{36} + 10305042q^{37} + 24180495q^{38} - 50414092q^{39} + 5926311q^{41} + 47560254q^{42} + 24429956q^{43} - 21995703q^{44} + 14223246q^{46} - 66858708q^{47} + 29557871q^{48} - 6453929q^{49} - 25634699q^{51} + 57862932q^{52} - 132620514q^{53} + 282250495q^{54} - 169538130q^{56} - 252946415q^{57} - 201908320q^{58} + 5670960q^{59} + 125306926q^{61} + 39831174q^{62} - 284323404q^{63} + 167542401q^{64} - 556915843q^{66} - 88829483q^{67} + 71162559q^{68} - 314274942q^{69} + 297550596q^{71} + 552918990q^{72} - 181321729q^{73} - 251507358q^{74} + 89414865q^{76} - 561214086q^{77} + 1439424692q^{78} - 310025170q^{79} + 1398847363q^{81} + 1368322979q^{82} + 731088801q^{83} + 38267082q^{84} + 33947196q^{86} - 1046385560q^{87} + 943671285q^{88} - 1103860035q^{89} + 1183187656q^{91} + 190024242q^{92} - 107386758q^{93} + 1727891132q^{94} - 2567180449q^{96} + 332236842q^{97} - 457927581q^{98} - 892234522q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −21.4187 −0.946580 −0.473290 0.880907i \(-0.656934\pi\)
−0.473290 + 0.880907i \(0.656934\pi\)
\(3\) 210.171 1.49805 0.749027 0.662539i \(-0.230521\pi\)
0.749027 + 0.662539i \(0.230521\pi\)
\(4\) −53.2406 −0.103986
\(5\) 0 0
\(6\) −4501.59 −1.41803
\(7\) −9905.49 −1.55932 −0.779659 0.626204i \(-0.784608\pi\)
−0.779659 + 0.626204i \(0.784608\pi\)
\(8\) 12106.7 1.04501
\(9\) 24489.0 1.24417
\(10\) 0 0
\(11\) 36453.6 0.750712 0.375356 0.926881i \(-0.377520\pi\)
0.375356 + 0.926881i \(0.377520\pi\)
\(12\) −11189.6 −0.155776
\(13\) −164867. −1.60099 −0.800494 0.599341i \(-0.795429\pi\)
−0.800494 + 0.599341i \(0.795429\pi\)
\(14\) 212162. 1.47602
\(15\) 0 0
\(16\) −232050. −0.885201
\(17\) −82357.1 −0.239156 −0.119578 0.992825i \(-0.538154\pi\)
−0.119578 + 0.992825i \(0.538154\pi\)
\(18\) −524521. −1.17771
\(19\) −609617. −1.07316 −0.536582 0.843848i \(-0.680285\pi\)
−0.536582 + 0.843848i \(0.680285\pi\)
\(20\) 0 0
\(21\) −2.08185e6 −2.33594
\(22\) −780787. −0.710609
\(23\) −1.88578e6 −1.40513 −0.702564 0.711620i \(-0.747962\pi\)
−0.702564 + 0.711620i \(0.747962\pi\)
\(24\) 2.54448e6 1.56548
\(25\) 0 0
\(26\) 3.53123e6 1.51546
\(27\) 1.01008e6 0.365778
\(28\) 527374. 0.162147
\(29\) 339235. 0.0890657 0.0445328 0.999008i \(-0.485820\pi\)
0.0445328 + 0.999008i \(0.485820\pi\)
\(30\) 0 0
\(31\) 547314. 0.106441 0.0532205 0.998583i \(-0.483051\pi\)
0.0532205 + 0.998583i \(0.483051\pi\)
\(32\) −1.22842e6 −0.207097
\(33\) 7.66150e6 1.12461
\(34\) 1.76398e6 0.226380
\(35\) 0 0
\(36\) −1.30381e6 −0.129376
\(37\) −5.25687e6 −0.461126 −0.230563 0.973057i \(-0.574057\pi\)
−0.230563 + 0.973057i \(0.574057\pi\)
\(38\) 1.30572e7 1.01584
\(39\) −3.46503e7 −2.39837
\(40\) 0 0
\(41\) 2.05812e6 0.113748 0.0568739 0.998381i \(-0.481887\pi\)
0.0568739 + 0.998381i \(0.481887\pi\)
\(42\) 4.45904e7 2.21116
\(43\) 6.76158e6 0.301606 0.150803 0.988564i \(-0.451814\pi\)
0.150803 + 0.988564i \(0.451814\pi\)
\(44\) −1.94081e6 −0.0780632
\(45\) 0 0
\(46\) 4.03909e7 1.33007
\(47\) −3.15241e7 −0.942330 −0.471165 0.882045i \(-0.656166\pi\)
−0.471165 + 0.882045i \(0.656166\pi\)
\(48\) −4.87703e7 −1.32608
\(49\) 5.77651e7 1.43147
\(50\) 0 0
\(51\) −1.73091e7 −0.358268
\(52\) 8.77761e6 0.166480
\(53\) 4.89593e7 0.852303 0.426151 0.904652i \(-0.359869\pi\)
0.426151 + 0.904652i \(0.359869\pi\)
\(54\) −2.16345e7 −0.346238
\(55\) 0 0
\(56\) −1.19923e8 −1.62950
\(57\) −1.28124e8 −1.60766
\(58\) −7.26597e6 −0.0843078
\(59\) 8.77960e7 0.943281 0.471640 0.881791i \(-0.343662\pi\)
0.471640 + 0.881791i \(0.343662\pi\)
\(60\) 0 0
\(61\) 3.84654e7 0.355701 0.177851 0.984057i \(-0.443086\pi\)
0.177851 + 0.984057i \(0.443086\pi\)
\(62\) −1.17227e7 −0.100755
\(63\) −2.42575e8 −1.94005
\(64\) 1.45121e8 1.08124
\(65\) 0 0
\(66\) −1.64099e8 −1.06453
\(67\) 1.36116e8 0.825227 0.412613 0.910906i \(-0.364616\pi\)
0.412613 + 0.910906i \(0.364616\pi\)
\(68\) 4.38474e6 0.0248687
\(69\) −3.96337e8 −2.10496
\(70\) 0 0
\(71\) 3.49218e8 1.63092 0.815462 0.578810i \(-0.196483\pi\)
0.815462 + 0.578810i \(0.196483\pi\)
\(72\) 2.96481e8 1.30017
\(73\) −1.61345e8 −0.664971 −0.332486 0.943108i \(-0.607887\pi\)
−0.332486 + 0.943108i \(0.607887\pi\)
\(74\) 1.12595e8 0.436492
\(75\) 0 0
\(76\) 3.24564e7 0.111594
\(77\) −3.61091e8 −1.17060
\(78\) 7.42163e8 2.27025
\(79\) −1.26975e8 −0.366772 −0.183386 0.983041i \(-0.558706\pi\)
−0.183386 + 0.983041i \(0.558706\pi\)
\(80\) 0 0
\(81\) −2.69727e8 −0.696213
\(82\) −4.40822e7 −0.107671
\(83\) 2.87494e8 0.664931 0.332466 0.943115i \(-0.392119\pi\)
0.332466 + 0.943115i \(0.392119\pi\)
\(84\) 1.10839e8 0.242904
\(85\) 0 0
\(86\) −1.44824e8 −0.285494
\(87\) 7.12976e7 0.133425
\(88\) 4.41333e8 0.784502
\(89\) −5.63133e8 −0.951385 −0.475692 0.879612i \(-0.657802\pi\)
−0.475692 + 0.879612i \(0.657802\pi\)
\(90\) 0 0
\(91\) 1.63309e9 2.49645
\(92\) 1.00400e8 0.146113
\(93\) 1.15030e8 0.159454
\(94\) 6.75205e8 0.891991
\(95\) 0 0
\(96\) −2.58179e8 −0.310242
\(97\) 4.71704e8 0.541000 0.270500 0.962720i \(-0.412811\pi\)
0.270500 + 0.962720i \(0.412811\pi\)
\(98\) −1.23725e9 −1.35500
\(99\) 8.92711e8 0.934012
\(100\) 0 0
\(101\) −1.25673e9 −1.20170 −0.600851 0.799361i \(-0.705172\pi\)
−0.600851 + 0.799361i \(0.705172\pi\)
\(102\) 3.70738e8 0.339130
\(103\) −1.95398e9 −1.71062 −0.855309 0.518119i \(-0.826633\pi\)
−0.855309 + 0.518119i \(0.826633\pi\)
\(104\) −1.99599e9 −1.67305
\(105\) 0 0
\(106\) −1.04864e9 −0.806773
\(107\) −2.12306e9 −1.56580 −0.782899 0.622148i \(-0.786260\pi\)
−0.782899 + 0.622148i \(0.786260\pi\)
\(108\) −5.37771e7 −0.0380356
\(109\) −1.18863e8 −0.0806544 −0.0403272 0.999187i \(-0.512840\pi\)
−0.0403272 + 0.999187i \(0.512840\pi\)
\(110\) 0 0
\(111\) −1.10484e9 −0.690791
\(112\) 2.29857e9 1.38031
\(113\) 1.85455e9 1.07001 0.535003 0.844850i \(-0.320311\pi\)
0.535003 + 0.844850i \(0.320311\pi\)
\(114\) 2.74425e9 1.52178
\(115\) 0 0
\(116\) −1.80611e7 −0.00926154
\(117\) −4.03742e9 −1.99190
\(118\) −1.88047e9 −0.892891
\(119\) 8.15787e8 0.372920
\(120\) 0 0
\(121\) −1.02908e9 −0.436432
\(122\) −8.23877e8 −0.336700
\(123\) 4.32557e8 0.170400
\(124\) −2.91393e7 −0.0110683
\(125\) 0 0
\(126\) 5.19564e9 1.83642
\(127\) 5.39208e9 1.83924 0.919622 0.392805i \(-0.128495\pi\)
0.919622 + 0.392805i \(0.128495\pi\)
\(128\) −2.47934e9 −0.816379
\(129\) 1.42109e9 0.451823
\(130\) 0 0
\(131\) 3.83690e9 1.13831 0.569154 0.822231i \(-0.307271\pi\)
0.569154 + 0.822231i \(0.307271\pi\)
\(132\) −4.07903e8 −0.116943
\(133\) 6.03856e9 1.67340
\(134\) −2.91543e9 −0.781143
\(135\) 0 0
\(136\) −9.97072e8 −0.249920
\(137\) 2.55408e9 0.619430 0.309715 0.950829i \(-0.399766\pi\)
0.309715 + 0.950829i \(0.399766\pi\)
\(138\) 8.48901e9 1.99251
\(139\) −4.09121e9 −0.929577 −0.464789 0.885422i \(-0.653870\pi\)
−0.464789 + 0.885422i \(0.653870\pi\)
\(140\) 0 0
\(141\) −6.62547e9 −1.41166
\(142\) −7.47978e9 −1.54380
\(143\) −6.00999e9 −1.20188
\(144\) −5.68267e9 −1.10134
\(145\) 0 0
\(146\) 3.45580e9 0.629449
\(147\) 1.21406e10 2.14443
\(148\) 2.79879e8 0.0479504
\(149\) −6.13643e9 −1.01995 −0.509974 0.860190i \(-0.670345\pi\)
−0.509974 + 0.860190i \(0.670345\pi\)
\(150\) 0 0
\(151\) −5.83754e9 −0.913763 −0.456882 0.889528i \(-0.651034\pi\)
−0.456882 + 0.889528i \(0.651034\pi\)
\(152\) −7.38046e9 −1.12147
\(153\) −2.01684e9 −0.297550
\(154\) 7.73408e9 1.10807
\(155\) 0 0
\(156\) 1.84480e9 0.249396
\(157\) 1.35031e9 0.177371 0.0886857 0.996060i \(-0.471733\pi\)
0.0886857 + 0.996060i \(0.471733\pi\)
\(158\) 2.71963e9 0.347179
\(159\) 1.02898e10 1.27680
\(160\) 0 0
\(161\) 1.86796e10 2.19104
\(162\) 5.77720e9 0.659022
\(163\) 4.50806e9 0.500202 0.250101 0.968220i \(-0.419536\pi\)
0.250101 + 0.968220i \(0.419536\pi\)
\(164\) −1.09575e8 −0.0118281
\(165\) 0 0
\(166\) −6.15773e9 −0.629411
\(167\) −1.81836e10 −1.80908 −0.904538 0.426393i \(-0.859784\pi\)
−0.904538 + 0.426393i \(0.859784\pi\)
\(168\) −2.52043e10 −2.44109
\(169\) 1.65766e10 1.56316
\(170\) 0 0
\(171\) −1.49289e10 −1.33520
\(172\) −3.59991e8 −0.0313627
\(173\) −9.77143e9 −0.829375 −0.414687 0.909964i \(-0.636109\pi\)
−0.414687 + 0.909964i \(0.636109\pi\)
\(174\) −1.52710e9 −0.126298
\(175\) 0 0
\(176\) −8.45906e9 −0.664531
\(177\) 1.84522e10 1.41309
\(178\) 1.20616e10 0.900562
\(179\) 6.65866e9 0.484784 0.242392 0.970178i \(-0.422068\pi\)
0.242392 + 0.970178i \(0.422068\pi\)
\(180\) 0 0
\(181\) −7.32873e9 −0.507546 −0.253773 0.967264i \(-0.581672\pi\)
−0.253773 + 0.967264i \(0.581672\pi\)
\(182\) −3.49785e10 −2.36309
\(183\) 8.08432e9 0.532860
\(184\) −2.28306e10 −1.46837
\(185\) 0 0
\(186\) −2.46378e9 −0.150936
\(187\) −3.00221e9 −0.179537
\(188\) 1.67836e9 0.0979887
\(189\) −1.00053e10 −0.570364
\(190\) 0 0
\(191\) 8.06956e9 0.438732 0.219366 0.975643i \(-0.429601\pi\)
0.219366 + 0.975643i \(0.429601\pi\)
\(192\) 3.05003e10 1.61975
\(193\) −3.12603e9 −0.162176 −0.0810878 0.996707i \(-0.525839\pi\)
−0.0810878 + 0.996707i \(0.525839\pi\)
\(194\) −1.01033e10 −0.512100
\(195\) 0 0
\(196\) −3.07545e9 −0.148853
\(197\) 2.40133e10 1.13594 0.567968 0.823051i \(-0.307730\pi\)
0.567968 + 0.823051i \(0.307730\pi\)
\(198\) −1.91207e10 −0.884117
\(199\) 9.73605e9 0.440093 0.220046 0.975489i \(-0.429379\pi\)
0.220046 + 0.975489i \(0.429379\pi\)
\(200\) 0 0
\(201\) 2.86077e10 1.23623
\(202\) 2.69175e10 1.13751
\(203\) −3.36029e9 −0.138882
\(204\) 9.21546e8 0.0372547
\(205\) 0 0
\(206\) 4.18517e10 1.61924
\(207\) −4.61808e10 −1.74822
\(208\) 3.82574e10 1.41720
\(209\) −2.22227e10 −0.805637
\(210\) 0 0
\(211\) −2.39305e10 −0.831151 −0.415576 0.909559i \(-0.636420\pi\)
−0.415576 + 0.909559i \(0.636420\pi\)
\(212\) −2.60662e9 −0.0886272
\(213\) 7.33956e10 2.44321
\(214\) 4.54732e10 1.48215
\(215\) 0 0
\(216\) 1.22287e10 0.382242
\(217\) −5.42141e9 −0.165975
\(218\) 2.54589e9 0.0763459
\(219\) −3.39101e10 −0.996163
\(220\) 0 0
\(221\) 1.35779e10 0.382885
\(222\) 2.36643e10 0.653890
\(223\) −2.02289e10 −0.547773 −0.273886 0.961762i \(-0.588309\pi\)
−0.273886 + 0.961762i \(0.588309\pi\)
\(224\) 1.21681e10 0.322930
\(225\) 0 0
\(226\) −3.97220e10 −1.01285
\(227\) −4.42152e10 −1.10524 −0.552618 0.833434i \(-0.686371\pi\)
−0.552618 + 0.833434i \(0.686371\pi\)
\(228\) 6.82140e9 0.167173
\(229\) −4.89483e10 −1.17619 −0.588096 0.808791i \(-0.700122\pi\)
−0.588096 + 0.808791i \(0.700122\pi\)
\(230\) 0 0
\(231\) −7.58909e10 −1.75362
\(232\) 4.10702e9 0.0930746
\(233\) −6.68132e10 −1.48512 −0.742559 0.669781i \(-0.766388\pi\)
−0.742559 + 0.669781i \(0.766388\pi\)
\(234\) 8.64761e10 1.88549
\(235\) 0 0
\(236\) −4.67431e9 −0.0980876
\(237\) −2.66865e10 −0.549444
\(238\) −1.74731e10 −0.352999
\(239\) −2.21153e10 −0.438431 −0.219216 0.975676i \(-0.570350\pi\)
−0.219216 + 0.975676i \(0.570350\pi\)
\(240\) 0 0
\(241\) 9.97637e10 1.90500 0.952502 0.304532i \(-0.0985002\pi\)
0.952502 + 0.304532i \(0.0985002\pi\)
\(242\) 2.20416e10 0.413118
\(243\) −7.65703e10 −1.40874
\(244\) −2.04792e9 −0.0369878
\(245\) 0 0
\(246\) −9.26480e9 −0.161298
\(247\) 1.00506e11 1.71812
\(248\) 6.62616e9 0.111232
\(249\) 6.04229e10 0.996104
\(250\) 0 0
\(251\) −4.97276e10 −0.790799 −0.395399 0.918509i \(-0.629394\pi\)
−0.395399 + 0.918509i \(0.629394\pi\)
\(252\) 1.29149e10 0.201738
\(253\) −6.87435e10 −1.05485
\(254\) −1.15491e11 −1.74099
\(255\) 0 0
\(256\) −2.11977e10 −0.308467
\(257\) −5.81308e10 −0.831204 −0.415602 0.909547i \(-0.636429\pi\)
−0.415602 + 0.909547i \(0.636429\pi\)
\(258\) −3.04379e10 −0.427686
\(259\) 5.20718e10 0.719041
\(260\) 0 0
\(261\) 8.30753e9 0.110813
\(262\) −8.21814e10 −1.07750
\(263\) 3.23476e10 0.416909 0.208454 0.978032i \(-0.433157\pi\)
0.208454 + 0.978032i \(0.433157\pi\)
\(264\) 9.27555e10 1.17523
\(265\) 0 0
\(266\) −1.29338e11 −1.58401
\(267\) −1.18354e11 −1.42523
\(268\) −7.24691e9 −0.0858116
\(269\) 6.24220e9 0.0726863 0.0363431 0.999339i \(-0.488429\pi\)
0.0363431 + 0.999339i \(0.488429\pi\)
\(270\) 0 0
\(271\) 8.05816e10 0.907557 0.453779 0.891114i \(-0.350076\pi\)
0.453779 + 0.891114i \(0.350076\pi\)
\(272\) 1.91110e10 0.211701
\(273\) 3.43228e11 3.73982
\(274\) −5.47050e10 −0.586340
\(275\) 0 0
\(276\) 2.11012e10 0.218885
\(277\) 7.07976e10 0.722536 0.361268 0.932462i \(-0.382344\pi\)
0.361268 + 0.932462i \(0.382344\pi\)
\(278\) 8.76284e10 0.879920
\(279\) 1.34031e10 0.132430
\(280\) 0 0
\(281\) 8.15619e10 0.780385 0.390193 0.920733i \(-0.372408\pi\)
0.390193 + 0.920733i \(0.372408\pi\)
\(282\) 1.41909e11 1.33625
\(283\) −1.89301e11 −1.75434 −0.877168 0.480184i \(-0.840570\pi\)
−0.877168 + 0.480184i \(0.840570\pi\)
\(284\) −1.85926e10 −0.169593
\(285\) 0 0
\(286\) 1.28726e11 1.13768
\(287\) −2.03867e10 −0.177369
\(288\) −3.00828e10 −0.257663
\(289\) −1.11805e11 −0.942805
\(290\) 0 0
\(291\) 9.91387e10 0.810447
\(292\) 8.59011e9 0.0691474
\(293\) 5.39489e10 0.427640 0.213820 0.976873i \(-0.431409\pi\)
0.213820 + 0.976873i \(0.431409\pi\)
\(294\) −2.60035e11 −2.02987
\(295\) 0 0
\(296\) −6.36433e10 −0.481881
\(297\) 3.68209e10 0.274594
\(298\) 1.31434e11 0.965462
\(299\) 3.10903e11 2.24959
\(300\) 0 0
\(301\) −6.69768e10 −0.470300
\(302\) 1.25032e11 0.864950
\(303\) −2.64129e11 −1.80022
\(304\) 1.41462e11 0.949966
\(305\) 0 0
\(306\) 4.31980e10 0.281655
\(307\) 1.30037e11 0.835496 0.417748 0.908563i \(-0.362819\pi\)
0.417748 + 0.908563i \(0.362819\pi\)
\(308\) 1.92247e10 0.121725
\(309\) −4.10671e11 −2.56260
\(310\) 0 0
\(311\) 2.09580e9 0.0127036 0.00635182 0.999980i \(-0.497978\pi\)
0.00635182 + 0.999980i \(0.497978\pi\)
\(312\) −4.19500e11 −2.50632
\(313\) 2.43788e11 1.43570 0.717848 0.696200i \(-0.245127\pi\)
0.717848 + 0.696200i \(0.245127\pi\)
\(314\) −2.89217e10 −0.167896
\(315\) 0 0
\(316\) 6.76022e9 0.0381390
\(317\) 3.46483e11 1.92715 0.963573 0.267445i \(-0.0861794\pi\)
0.963573 + 0.267445i \(0.0861794\pi\)
\(318\) −2.20395e11 −1.20859
\(319\) 1.23664e10 0.0668626
\(320\) 0 0
\(321\) −4.46207e11 −2.34565
\(322\) −4.00092e11 −2.07400
\(323\) 5.02063e10 0.256653
\(324\) 1.43604e10 0.0723961
\(325\) 0 0
\(326\) −9.65566e10 −0.473481
\(327\) −2.49816e10 −0.120825
\(328\) 2.49170e10 0.118868
\(329\) 3.12262e11 1.46939
\(330\) 0 0
\(331\) −1.43303e11 −0.656189 −0.328094 0.944645i \(-0.606406\pi\)
−0.328094 + 0.944645i \(0.606406\pi\)
\(332\) −1.53063e10 −0.0691433
\(333\) −1.28735e11 −0.573718
\(334\) 3.89470e11 1.71244
\(335\) 0 0
\(336\) 4.83094e11 2.06778
\(337\) −3.34820e11 −1.41409 −0.707044 0.707170i \(-0.749972\pi\)
−0.707044 + 0.707170i \(0.749972\pi\)
\(338\) −3.55048e11 −1.47966
\(339\) 3.89773e11 1.60293
\(340\) 0 0
\(341\) 1.99515e10 0.0799064
\(342\) 3.19757e11 1.26387
\(343\) −1.72469e11 −0.672804
\(344\) 8.18604e10 0.315182
\(345\) 0 0
\(346\) 2.09291e11 0.785070
\(347\) 9.22101e10 0.341425 0.170713 0.985321i \(-0.445393\pi\)
0.170713 + 0.985321i \(0.445393\pi\)
\(348\) −3.79592e9 −0.0138743
\(349\) −4.71738e11 −1.70211 −0.851053 0.525080i \(-0.824035\pi\)
−0.851053 + 0.525080i \(0.824035\pi\)
\(350\) 0 0
\(351\) −1.66528e11 −0.585606
\(352\) −4.47805e10 −0.155470
\(353\) 4.01029e11 1.37464 0.687322 0.726353i \(-0.258786\pi\)
0.687322 + 0.726353i \(0.258786\pi\)
\(354\) −3.95222e11 −1.33760
\(355\) 0 0
\(356\) 2.99816e10 0.0989303
\(357\) 1.71455e11 0.558654
\(358\) −1.42620e11 −0.458887
\(359\) −2.36110e10 −0.0750220 −0.0375110 0.999296i \(-0.511943\pi\)
−0.0375110 + 0.999296i \(0.511943\pi\)
\(360\) 0 0
\(361\) 4.89457e10 0.151681
\(362\) 1.56972e11 0.480433
\(363\) −2.16284e11 −0.653799
\(364\) −8.69465e10 −0.259595
\(365\) 0 0
\(366\) −1.73155e11 −0.504395
\(367\) −1.23035e11 −0.354022 −0.177011 0.984209i \(-0.556643\pi\)
−0.177011 + 0.984209i \(0.556643\pi\)
\(368\) 4.37596e11 1.24382
\(369\) 5.04012e10 0.141521
\(370\) 0 0
\(371\) −4.84966e11 −1.32901
\(372\) −6.12424e9 −0.0165809
\(373\) 2.39248e11 0.639968 0.319984 0.947423i \(-0.396322\pi\)
0.319984 + 0.947423i \(0.396322\pi\)
\(374\) 6.43034e10 0.169946
\(375\) 0 0
\(376\) −3.81653e11 −0.984745
\(377\) −5.59287e10 −0.142593
\(378\) 2.14300e11 0.539896
\(379\) −5.90629e11 −1.47041 −0.735205 0.677845i \(-0.762914\pi\)
−0.735205 + 0.677845i \(0.762914\pi\)
\(380\) 0 0
\(381\) 1.13326e12 2.75529
\(382\) −1.72839e11 −0.415295
\(383\) 5.30525e10 0.125983 0.0629914 0.998014i \(-0.479936\pi\)
0.0629914 + 0.998014i \(0.479936\pi\)
\(384\) −5.21087e11 −1.22298
\(385\) 0 0
\(386\) 6.69554e10 0.153512
\(387\) 1.65584e11 0.375249
\(388\) −2.51138e10 −0.0562562
\(389\) 3.10395e11 0.687293 0.343646 0.939099i \(-0.388338\pi\)
0.343646 + 0.939099i \(0.388338\pi\)
\(390\) 0 0
\(391\) 1.55307e11 0.336045
\(392\) 6.99345e11 1.49591
\(393\) 8.06407e11 1.70525
\(394\) −5.14333e11 −1.07525
\(395\) 0 0
\(396\) −4.75285e10 −0.0971238
\(397\) 2.09351e11 0.422979 0.211489 0.977380i \(-0.432169\pi\)
0.211489 + 0.977380i \(0.432169\pi\)
\(398\) −2.08533e11 −0.416583
\(399\) 1.26913e12 2.50685
\(400\) 0 0
\(401\) 4.43868e11 0.857244 0.428622 0.903484i \(-0.358999\pi\)
0.428622 + 0.903484i \(0.358999\pi\)
\(402\) −6.12739e11 −1.17020
\(403\) −9.02338e10 −0.170411
\(404\) 6.69092e10 0.124960
\(405\) 0 0
\(406\) 7.19730e10 0.131463
\(407\) −1.91632e11 −0.346172
\(408\) −2.09556e11 −0.374394
\(409\) 4.56031e11 0.805823 0.402912 0.915239i \(-0.367998\pi\)
0.402912 + 0.915239i \(0.367998\pi\)
\(410\) 0 0
\(411\) 5.36794e11 0.927940
\(412\) 1.04031e11 0.177880
\(413\) −8.69663e11 −1.47087
\(414\) 9.89132e11 1.65483
\(415\) 0 0
\(416\) 2.02526e11 0.331559
\(417\) −8.59856e11 −1.39256
\(418\) 4.75982e11 0.762600
\(419\) −1.12392e12 −1.78144 −0.890720 0.454552i \(-0.849799\pi\)
−0.890720 + 0.454552i \(0.849799\pi\)
\(420\) 0 0
\(421\) −1.02822e12 −1.59521 −0.797605 0.603180i \(-0.793900\pi\)
−0.797605 + 0.603180i \(0.793900\pi\)
\(422\) 5.12559e11 0.786751
\(423\) −7.71994e11 −1.17242
\(424\) 5.92736e11 0.890666
\(425\) 0 0
\(426\) −1.57204e12 −2.31270
\(427\) −3.81018e11 −0.554652
\(428\) 1.13033e11 0.162820
\(429\) −1.26313e12 −1.80048
\(430\) 0 0
\(431\) −1.37714e12 −1.92234 −0.961171 0.275953i \(-0.911007\pi\)
−0.961171 + 0.275953i \(0.911007\pi\)
\(432\) −2.34389e11 −0.323787
\(433\) −1.94790e11 −0.266299 −0.133150 0.991096i \(-0.542509\pi\)
−0.133150 + 0.991096i \(0.542509\pi\)
\(434\) 1.16119e11 0.157109
\(435\) 0 0
\(436\) 6.32835e9 0.00838689
\(437\) 1.14961e12 1.50793
\(438\) 7.26309e11 0.942948
\(439\) −5.32533e11 −0.684315 −0.342157 0.939643i \(-0.611158\pi\)
−0.342157 + 0.939643i \(0.611158\pi\)
\(440\) 0 0
\(441\) 1.41461e12 1.78099
\(442\) −2.90821e11 −0.362432
\(443\) −7.26633e11 −0.896393 −0.448196 0.893935i \(-0.647933\pi\)
−0.448196 + 0.893935i \(0.647933\pi\)
\(444\) 5.88225e10 0.0718323
\(445\) 0 0
\(446\) 4.33276e11 0.518511
\(447\) −1.28970e12 −1.52794
\(448\) −1.43749e12 −1.68599
\(449\) 9.34830e11 1.08549 0.542743 0.839899i \(-0.317386\pi\)
0.542743 + 0.839899i \(0.317386\pi\)
\(450\) 0 0
\(451\) 7.50258e10 0.0853918
\(452\) −9.87374e10 −0.111265
\(453\) −1.22688e12 −1.36887
\(454\) 9.47031e11 1.04620
\(455\) 0 0
\(456\) −1.55116e12 −1.68002
\(457\) 1.09190e12 1.17101 0.585503 0.810670i \(-0.300897\pi\)
0.585503 + 0.810670i \(0.300897\pi\)
\(458\) 1.04841e12 1.11336
\(459\) −8.31870e10 −0.0874779
\(460\) 0 0
\(461\) 9.93683e11 1.02469 0.512347 0.858779i \(-0.328776\pi\)
0.512347 + 0.858779i \(0.328776\pi\)
\(462\) 1.62548e12 1.65994
\(463\) 6.95734e11 0.703605 0.351802 0.936074i \(-0.385569\pi\)
0.351802 + 0.936074i \(0.385569\pi\)
\(464\) −7.87197e10 −0.0788411
\(465\) 0 0
\(466\) 1.43105e12 1.40578
\(467\) −1.87957e12 −1.82866 −0.914328 0.404975i \(-0.867280\pi\)
−0.914328 + 0.404975i \(0.867280\pi\)
\(468\) 2.14955e11 0.207129
\(469\) −1.34830e12 −1.28679
\(470\) 0 0
\(471\) 2.83795e11 0.265712
\(472\) 1.06292e12 0.985739
\(473\) 2.46484e11 0.226419
\(474\) 5.71589e11 0.520093
\(475\) 0 0
\(476\) −4.34330e10 −0.0387783
\(477\) 1.19896e12 1.06041
\(478\) 4.73679e11 0.415010
\(479\) −2.29011e11 −0.198768 −0.0993838 0.995049i \(-0.531687\pi\)
−0.0993838 + 0.995049i \(0.531687\pi\)
\(480\) 0 0
\(481\) 8.66683e11 0.738256
\(482\) −2.13681e12 −1.80324
\(483\) 3.92591e12 3.28230
\(484\) 5.47890e10 0.0453826
\(485\) 0 0
\(486\) 1.64003e12 1.33349
\(487\) 4.81410e11 0.387824 0.193912 0.981019i \(-0.437882\pi\)
0.193912 + 0.981019i \(0.437882\pi\)
\(488\) 4.65689e11 0.371712
\(489\) 9.47465e11 0.749330
\(490\) 0 0
\(491\) −1.70320e12 −1.32251 −0.661257 0.750160i \(-0.729977\pi\)
−0.661257 + 0.750160i \(0.729977\pi\)
\(492\) −2.30296e10 −0.0177192
\(493\) −2.79384e10 −0.0213006
\(494\) −2.15270e12 −1.62634
\(495\) 0 0
\(496\) −1.27004e11 −0.0942217
\(497\) −3.45917e12 −2.54313
\(498\) −1.29418e12 −0.942892
\(499\) 2.06075e12 1.48789 0.743947 0.668238i \(-0.232951\pi\)
0.743947 + 0.668238i \(0.232951\pi\)
\(500\) 0 0
\(501\) −3.82168e12 −2.71010
\(502\) 1.06510e12 0.748555
\(503\) 5.97493e11 0.416176 0.208088 0.978110i \(-0.433276\pi\)
0.208088 + 0.978110i \(0.433276\pi\)
\(504\) −2.93679e12 −2.02738
\(505\) 0 0
\(506\) 1.47239e12 0.998497
\(507\) 3.48392e12 2.34170
\(508\) −2.87077e11 −0.191255
\(509\) −8.46938e11 −0.559270 −0.279635 0.960106i \(-0.590213\pi\)
−0.279635 + 0.960106i \(0.590213\pi\)
\(510\) 0 0
\(511\) 1.59820e12 1.03690
\(512\) 1.72345e12 1.10837
\(513\) −6.15760e11 −0.392540
\(514\) 1.24508e12 0.786801
\(515\) 0 0
\(516\) −7.56597e10 −0.0469830
\(517\) −1.14917e12 −0.707418
\(518\) −1.11531e12 −0.680630
\(519\) −2.05367e12 −1.24245
\(520\) 0 0
\(521\) 7.31013e11 0.434666 0.217333 0.976098i \(-0.430264\pi\)
0.217333 + 0.976098i \(0.430264\pi\)
\(522\) −1.77936e11 −0.104893
\(523\) −1.04453e12 −0.610467 −0.305233 0.952278i \(-0.598734\pi\)
−0.305233 + 0.952278i \(0.598734\pi\)
\(524\) −2.04279e11 −0.118368
\(525\) 0 0
\(526\) −6.92842e11 −0.394638
\(527\) −4.50751e10 −0.0254559
\(528\) −1.77785e12 −0.995504
\(529\) 1.75502e12 0.974386
\(530\) 0 0
\(531\) 2.15003e12 1.17360
\(532\) −3.21496e11 −0.174010
\(533\) −3.39315e11 −0.182109
\(534\) 2.53500e12 1.34909
\(535\) 0 0
\(536\) 1.64792e12 0.862371
\(537\) 1.39946e12 0.726233
\(538\) −1.33700e11 −0.0688034
\(539\) 2.10575e12 1.07462
\(540\) 0 0
\(541\) −1.88034e12 −0.943733 −0.471866 0.881670i \(-0.656420\pi\)
−0.471866 + 0.881670i \(0.656420\pi\)
\(542\) −1.72595e12 −0.859076
\(543\) −1.54029e12 −0.760331
\(544\) 1.01169e11 0.0495284
\(545\) 0 0
\(546\) −7.35148e12 −3.54004
\(547\) 1.08624e12 0.518780 0.259390 0.965773i \(-0.416479\pi\)
0.259390 + 0.965773i \(0.416479\pi\)
\(548\) −1.35981e11 −0.0644117
\(549\) 9.41977e11 0.442553
\(550\) 0 0
\(551\) −2.06804e11 −0.0955821
\(552\) −4.79834e12 −2.19971
\(553\) 1.25775e12 0.571914
\(554\) −1.51639e12 −0.683938
\(555\) 0 0
\(556\) 2.17819e11 0.0966626
\(557\) −2.11965e12 −0.933072 −0.466536 0.884502i \(-0.654498\pi\)
−0.466536 + 0.884502i \(0.654498\pi\)
\(558\) −2.87078e11 −0.125356
\(559\) −1.11476e12 −0.482868
\(560\) 0 0
\(561\) −6.30978e11 −0.268956
\(562\) −1.74695e12 −0.738697
\(563\) −3.04489e12 −1.27727 −0.638636 0.769509i \(-0.720501\pi\)
−0.638636 + 0.769509i \(0.720501\pi\)
\(564\) 3.52744e11 0.146792
\(565\) 0 0
\(566\) 4.05456e12 1.66062
\(567\) 2.67178e12 1.08562
\(568\) 4.22788e12 1.70433
\(569\) 3.25275e12 1.30090 0.650452 0.759548i \(-0.274580\pi\)
0.650452 + 0.759548i \(0.274580\pi\)
\(570\) 0 0
\(571\) −1.35916e12 −0.535066 −0.267533 0.963549i \(-0.586208\pi\)
−0.267533 + 0.963549i \(0.586208\pi\)
\(572\) 3.19975e11 0.124978
\(573\) 1.69599e12 0.657245
\(574\) 4.36655e11 0.167894
\(575\) 0 0
\(576\) 3.55386e12 1.34524
\(577\) 2.35782e12 0.885564 0.442782 0.896629i \(-0.353992\pi\)
0.442782 + 0.896629i \(0.353992\pi\)
\(578\) 2.39472e12 0.892440
\(579\) −6.57002e11 −0.242948
\(580\) 0 0
\(581\) −2.84777e12 −1.03684
\(582\) −2.12342e12 −0.767154
\(583\) 1.78474e12 0.639834
\(584\) −1.95336e12 −0.694902
\(585\) 0 0
\(586\) −1.15551e12 −0.404796
\(587\) 2.37065e12 0.824130 0.412065 0.911154i \(-0.364808\pi\)
0.412065 + 0.911154i \(0.364808\pi\)
\(588\) −6.46371e11 −0.222989
\(589\) −3.33652e11 −0.114229
\(590\) 0 0
\(591\) 5.04690e12 1.70169
\(592\) 1.21986e12 0.408189
\(593\) −4.56707e12 −1.51667 −0.758336 0.651864i \(-0.773987\pi\)
−0.758336 + 0.651864i \(0.773987\pi\)
\(594\) −7.88655e11 −0.259925
\(595\) 0 0
\(596\) 3.26707e11 0.106060
\(597\) 2.04624e12 0.659283
\(598\) −6.65912e12 −2.12942
\(599\) −5.30493e12 −1.68368 −0.841839 0.539729i \(-0.818527\pi\)
−0.841839 + 0.539729i \(0.818527\pi\)
\(600\) 0 0
\(601\) −2.71307e12 −0.848255 −0.424128 0.905602i \(-0.639419\pi\)
−0.424128 + 0.905602i \(0.639419\pi\)
\(602\) 1.43455e12 0.445177
\(603\) 3.33335e12 1.02672
\(604\) 3.10794e11 0.0950182
\(605\) 0 0
\(606\) 5.65730e12 1.70405
\(607\) 4.00734e12 1.19814 0.599069 0.800698i \(-0.295538\pi\)
0.599069 + 0.800698i \(0.295538\pi\)
\(608\) 7.48868e11 0.222249
\(609\) −7.06237e11 −0.208052
\(610\) 0 0
\(611\) 5.19728e12 1.50866
\(612\) 1.07378e11 0.0309409
\(613\) 2.24954e12 0.643459 0.321730 0.946832i \(-0.395736\pi\)
0.321730 + 0.946832i \(0.395736\pi\)
\(614\) −2.78522e12 −0.790864
\(615\) 0 0
\(616\) −4.37162e12 −1.22329
\(617\) −1.19664e12 −0.332414 −0.166207 0.986091i \(-0.553152\pi\)
−0.166207 + 0.986091i \(0.553152\pi\)
\(618\) 8.79602e12 2.42571
\(619\) −4.36723e12 −1.19563 −0.597817 0.801633i \(-0.703965\pi\)
−0.597817 + 0.801633i \(0.703965\pi\)
\(620\) 0 0
\(621\) −1.90478e12 −0.513965
\(622\) −4.48893e10 −0.0120250
\(623\) 5.57811e12 1.48351
\(624\) 8.04060e12 2.12304
\(625\) 0 0
\(626\) −5.22161e12 −1.35900
\(627\) −4.67058e12 −1.20689
\(628\) −7.18911e10 −0.0184441
\(629\) 4.32940e11 0.110281
\(630\) 0 0
\(631\) 2.78790e12 0.700077 0.350039 0.936735i \(-0.386168\pi\)
0.350039 + 0.936735i \(0.386168\pi\)
\(632\) −1.53725e12 −0.383281
\(633\) −5.02950e12 −1.24511
\(634\) −7.42120e12 −1.82420
\(635\) 0 0
\(636\) −5.47837e11 −0.132768
\(637\) −9.52355e12 −2.29177
\(638\) −2.64871e11 −0.0632909
\(639\) 8.55199e12 2.02915
\(640\) 0 0
\(641\) −2.28295e12 −0.534115 −0.267058 0.963681i \(-0.586051\pi\)
−0.267058 + 0.963681i \(0.586051\pi\)
\(642\) 9.55716e12 2.22035
\(643\) 2.29995e12 0.530603 0.265301 0.964166i \(-0.414529\pi\)
0.265301 + 0.964166i \(0.414529\pi\)
\(644\) −9.94512e11 −0.227837
\(645\) 0 0
\(646\) −1.07535e12 −0.242943
\(647\) −1.85647e12 −0.416503 −0.208251 0.978075i \(-0.566777\pi\)
−0.208251 + 0.978075i \(0.566777\pi\)
\(648\) −3.26551e12 −0.727550
\(649\) 3.20048e12 0.708132
\(650\) 0 0
\(651\) −1.13942e12 −0.248640
\(652\) −2.40012e11 −0.0520138
\(653\) −7.75397e12 −1.66884 −0.834420 0.551129i \(-0.814197\pi\)
−0.834420 + 0.551129i \(0.814197\pi\)
\(654\) 5.35073e11 0.114370
\(655\) 0 0
\(656\) −4.77587e11 −0.100690
\(657\) −3.95117e12 −0.827336
\(658\) −6.68824e12 −1.39090
\(659\) 7.42166e12 1.53291 0.766455 0.642298i \(-0.222019\pi\)
0.766455 + 0.642298i \(0.222019\pi\)
\(660\) 0 0
\(661\) 5.79783e12 1.18130 0.590648 0.806929i \(-0.298872\pi\)
0.590648 + 0.806929i \(0.298872\pi\)
\(662\) 3.06936e12 0.621135
\(663\) 2.85369e12 0.573583
\(664\) 3.48060e12 0.694861
\(665\) 0 0
\(666\) 2.75734e12 0.543070
\(667\) −6.39724e11 −0.125149
\(668\) 9.68108e11 0.188118
\(669\) −4.25153e12 −0.820593
\(670\) 0 0
\(671\) 1.40220e12 0.267029
\(672\) 2.55739e12 0.483766
\(673\) 6.32456e12 1.18840 0.594200 0.804318i \(-0.297469\pi\)
0.594200 + 0.804318i \(0.297469\pi\)
\(674\) 7.17139e12 1.33855
\(675\) 0 0
\(676\) −8.82546e11 −0.162546
\(677\) 4.95066e12 0.905762 0.452881 0.891571i \(-0.350396\pi\)
0.452881 + 0.891571i \(0.350396\pi\)
\(678\) −8.34843e12 −1.51730
\(679\) −4.67246e12 −0.843591
\(680\) 0 0
\(681\) −9.29277e12 −1.65571
\(682\) −4.27336e11 −0.0756379
\(683\) −7.32611e12 −1.28819 −0.644095 0.764945i \(-0.722766\pi\)
−0.644095 + 0.764945i \(0.722766\pi\)
\(684\) 7.94824e11 0.138841
\(685\) 0 0
\(686\) 3.69406e12 0.636863
\(687\) −1.02875e13 −1.76200
\(688\) −1.56903e12 −0.266982
\(689\) −8.07176e12 −1.36453
\(690\) 0 0
\(691\) −5.66806e12 −0.945765 −0.472882 0.881126i \(-0.656786\pi\)
−0.472882 + 0.881126i \(0.656786\pi\)
\(692\) 5.20237e11 0.0862430
\(693\) −8.84274e12 −1.45642
\(694\) −1.97502e12 −0.323187
\(695\) 0 0
\(696\) 8.63178e11 0.139431
\(697\) −1.69501e11 −0.0272034
\(698\) 1.01040e13 1.61118
\(699\) −1.40422e13 −2.22479
\(700\) 0 0
\(701\) −4.20985e12 −0.658470 −0.329235 0.944248i \(-0.606791\pi\)
−0.329235 + 0.944248i \(0.606791\pi\)
\(702\) 3.56681e12 0.554323
\(703\) 3.20468e12 0.494863
\(704\) 5.29018e12 0.811696
\(705\) 0 0
\(706\) −8.58952e12 −1.30121
\(707\) 1.24486e13 1.87384
\(708\) −9.82407e11 −0.146941
\(709\) 1.19654e13 1.77835 0.889176 0.457566i \(-0.151279\pi\)
0.889176 + 0.457566i \(0.151279\pi\)
\(710\) 0 0
\(711\) −3.10949e12 −0.456326
\(712\) −6.81769e12 −0.994208
\(713\) −1.03211e12 −0.149563
\(714\) −3.67234e12 −0.528811
\(715\) 0 0
\(716\) −3.54511e11 −0.0504105
\(717\) −4.64799e12 −0.656794
\(718\) 5.05715e11 0.0710143
\(719\) −4.07118e11 −0.0568120 −0.0284060 0.999596i \(-0.509043\pi\)
−0.0284060 + 0.999596i \(0.509043\pi\)
\(720\) 0 0
\(721\) 1.93551e13 2.66740
\(722\) −1.04835e12 −0.143579
\(723\) 2.09675e13 2.85380
\(724\) 3.90186e11 0.0527774
\(725\) 0 0
\(726\) 4.63251e12 0.618873
\(727\) −6.44424e12 −0.855592 −0.427796 0.903875i \(-0.640710\pi\)
−0.427796 + 0.903875i \(0.640710\pi\)
\(728\) 1.97713e13 2.60882
\(729\) −1.07838e13 −1.41416
\(730\) 0 0
\(731\) −5.56864e11 −0.0721308
\(732\) −4.30414e11 −0.0554098
\(733\) −5.46122e12 −0.698749 −0.349375 0.936983i \(-0.613606\pi\)
−0.349375 + 0.936983i \(0.613606\pi\)
\(734\) 2.63524e12 0.335110
\(735\) 0 0
\(736\) 2.31654e12 0.290998
\(737\) 4.96192e12 0.619507
\(738\) −1.07953e12 −0.133961
\(739\) −1.88201e12 −0.232125 −0.116062 0.993242i \(-0.537027\pi\)
−0.116062 + 0.993242i \(0.537027\pi\)
\(740\) 0 0
\(741\) 2.11234e13 2.57384
\(742\) 1.03873e13 1.25802
\(743\) −1.36704e13 −1.64563 −0.822813 0.568312i \(-0.807597\pi\)
−0.822813 + 0.568312i \(0.807597\pi\)
\(744\) 1.39263e12 0.166632
\(745\) 0 0
\(746\) −5.12437e12 −0.605781
\(747\) 7.04042e12 0.827287
\(748\) 1.59840e11 0.0186693
\(749\) 2.10300e13 2.44158
\(750\) 0 0
\(751\) 8.52585e12 0.978043 0.489021 0.872272i \(-0.337354\pi\)
0.489021 + 0.872272i \(0.337354\pi\)
\(752\) 7.31518e12 0.834152
\(753\) −1.04513e13 −1.18466
\(754\) 1.19792e12 0.134976
\(755\) 0 0
\(756\) 5.32688e11 0.0593096
\(757\) 7.50792e12 0.830975 0.415487 0.909599i \(-0.363611\pi\)
0.415487 + 0.909599i \(0.363611\pi\)
\(758\) 1.26505e13 1.39186
\(759\) −1.44479e13 −1.58022
\(760\) 0 0
\(761\) −1.61560e13 −1.74624 −0.873118 0.487509i \(-0.837906\pi\)
−0.873118 + 0.487509i \(0.837906\pi\)
\(762\) −2.42729e13 −2.60810
\(763\) 1.17740e12 0.125766
\(764\) −4.29628e11 −0.0456218
\(765\) 0 0
\(766\) −1.13631e12 −0.119253
\(767\) −1.44747e13 −1.51018
\(768\) −4.45514e12 −0.462100
\(769\) 1.11830e12 0.115317 0.0576583 0.998336i \(-0.481637\pi\)
0.0576583 + 0.998336i \(0.481637\pi\)
\(770\) 0 0
\(771\) −1.22174e13 −1.24519
\(772\) 1.66432e11 0.0168639
\(773\) 1.73165e13 1.74442 0.872212 0.489129i \(-0.162685\pi\)
0.872212 + 0.489129i \(0.162685\pi\)
\(774\) −3.54659e12 −0.355203
\(775\) 0 0
\(776\) 5.71078e12 0.565351
\(777\) 1.09440e13 1.07716
\(778\) −6.64825e12 −0.650578
\(779\) −1.25466e12 −0.122070
\(780\) 0 0
\(781\) 1.27302e13 1.22435
\(782\) −3.32648e12 −0.318093
\(783\) 3.42654e11 0.0325783
\(784\) −1.34044e13 −1.26714
\(785\) 0 0
\(786\) −1.72722e13 −1.61416
\(787\) 3.88515e11 0.0361012 0.0180506 0.999837i \(-0.494254\pi\)
0.0180506 + 0.999837i \(0.494254\pi\)
\(788\) −1.27848e12 −0.118121
\(789\) 6.79853e12 0.624552
\(790\) 0 0
\(791\) −1.83702e13 −1.66848
\(792\) 1.08078e13 0.976053
\(793\) −6.34166e12 −0.569474
\(794\) −4.48403e12 −0.400383
\(795\) 0 0
\(796\) −5.18353e11 −0.0457633
\(797\) 1.08432e12 0.0951910 0.0475955 0.998867i \(-0.484844\pi\)
0.0475955 + 0.998867i \(0.484844\pi\)
\(798\) −2.71831e13 −2.37294
\(799\) 2.59624e12 0.225363
\(800\) 0 0
\(801\) −1.37906e13 −1.18368
\(802\) −9.50706e12 −0.811450
\(803\) −5.88161e12 −0.499202
\(804\) −1.52309e12 −0.128551
\(805\) 0 0
\(806\) 1.93269e12 0.161307
\(807\) 1.31193e12 0.108888
\(808\) −1.52149e13 −1.25579
\(809\) 4.06803e12 0.333900 0.166950 0.985965i \(-0.446608\pi\)
0.166950 + 0.985965i \(0.446608\pi\)
\(810\) 0 0
\(811\) 4.87468e12 0.395688 0.197844 0.980234i \(-0.436606\pi\)
0.197844 + 0.980234i \(0.436606\pi\)
\(812\) 1.78904e11 0.0144417
\(813\) 1.69359e13 1.35957
\(814\) 4.10449e12 0.327680
\(815\) 0 0
\(816\) 4.01658e12 0.317140
\(817\) −4.12198e12 −0.323673
\(818\) −9.76758e12 −0.762777
\(819\) 3.99926e13 3.10600
\(820\) 0 0
\(821\) −1.03829e12 −0.0797582 −0.0398791 0.999205i \(-0.512697\pi\)
−0.0398791 + 0.999205i \(0.512697\pi\)
\(822\) −1.14974e13 −0.878370
\(823\) −1.10923e13 −0.842794 −0.421397 0.906876i \(-0.638460\pi\)
−0.421397 + 0.906876i \(0.638460\pi\)
\(824\) −2.36563e13 −1.78761
\(825\) 0 0
\(826\) 1.86270e13 1.39230
\(827\) 1.45526e13 1.08184 0.540922 0.841073i \(-0.318075\pi\)
0.540922 + 0.841073i \(0.318075\pi\)
\(828\) 2.45870e12 0.181789
\(829\) 1.37985e13 1.01470 0.507348 0.861741i \(-0.330626\pi\)
0.507348 + 0.861741i \(0.330626\pi\)
\(830\) 0 0
\(831\) 1.48796e13 1.08240
\(832\) −2.39256e13 −1.73104
\(833\) −4.75736e12 −0.342345
\(834\) 1.84170e13 1.31817
\(835\) 0 0
\(836\) 1.18315e12 0.0837746
\(837\) 5.52829e11 0.0389337
\(838\) 2.40728e13 1.68628
\(839\) 9.20234e11 0.0641165 0.0320582 0.999486i \(-0.489794\pi\)
0.0320582 + 0.999486i \(0.489794\pi\)
\(840\) 0 0
\(841\) −1.43921e13 −0.992067
\(842\) 2.20232e13 1.50999
\(843\) 1.71420e13 1.16906
\(844\) 1.27407e12 0.0864277
\(845\) 0 0
\(846\) 1.65351e13 1.10979
\(847\) 1.01936e13 0.680536
\(848\) −1.13610e13 −0.754460
\(849\) −3.97855e13 −2.62809
\(850\) 0 0
\(851\) 9.91330e12 0.647941
\(852\) −3.90762e12 −0.254059
\(853\) 7.46964e12 0.483091 0.241545 0.970390i \(-0.422346\pi\)
0.241545 + 0.970390i \(0.422346\pi\)
\(854\) 8.16090e12 0.525022
\(855\) 0 0
\(856\) −2.57033e13 −1.63628
\(857\) −2.05568e13 −1.30179 −0.650897 0.759166i \(-0.725607\pi\)
−0.650897 + 0.759166i \(0.725607\pi\)
\(858\) 2.70545e13 1.70430
\(859\) −1.05923e12 −0.0663775 −0.0331888 0.999449i \(-0.510566\pi\)
−0.0331888 + 0.999449i \(0.510566\pi\)
\(860\) 0 0
\(861\) −4.28469e12 −0.265709
\(862\) 2.94965e13 1.81965
\(863\) −1.53023e13 −0.939092 −0.469546 0.882908i \(-0.655582\pi\)
−0.469546 + 0.882908i \(0.655582\pi\)
\(864\) −1.24080e12 −0.0757514
\(865\) 0 0
\(866\) 4.17213e12 0.252074
\(867\) −2.34982e13 −1.41237
\(868\) 2.88639e11 0.0172590
\(869\) −4.62869e12 −0.275340
\(870\) 0 0
\(871\) −2.24410e13 −1.32118
\(872\) −1.43904e12 −0.0842847
\(873\) 1.15516e13 0.673095
\(874\) −2.46230e13 −1.42738
\(875\) 0 0
\(876\) 1.80539e12 0.103587
\(877\) 1.74558e11 0.00996416 0.00498208 0.999988i \(-0.498414\pi\)
0.00498208 + 0.999988i \(0.498414\pi\)
\(878\) 1.14061e13 0.647759
\(879\) 1.13385e13 0.640629
\(880\) 0 0
\(881\) 8.59039e12 0.480420 0.240210 0.970721i \(-0.422784\pi\)
0.240210 + 0.970721i \(0.422784\pi\)
\(882\) −3.02990e13 −1.68585
\(883\) 1.70227e13 0.942336 0.471168 0.882043i \(-0.343833\pi\)
0.471168 + 0.882043i \(0.343833\pi\)
\(884\) −7.22898e11 −0.0398145
\(885\) 0 0
\(886\) 1.55635e13 0.848508
\(887\) 2.01246e13 1.09162 0.545810 0.837909i \(-0.316222\pi\)
0.545810 + 0.837909i \(0.316222\pi\)
\(888\) −1.33760e13 −0.721885
\(889\) −5.34112e13 −2.86797
\(890\) 0 0
\(891\) −9.83253e12 −0.522655
\(892\) 1.07700e12 0.0569604
\(893\) 1.92177e13 1.01127
\(894\) 2.76237e13 1.44632
\(895\) 0 0
\(896\) 2.45591e13 1.27299
\(897\) 6.53428e13 3.37002
\(898\) −2.00228e13 −1.02750
\(899\) 1.85668e11 0.00948023
\(900\) 0 0
\(901\) −4.03214e12 −0.203833
\(902\) −1.60695e12 −0.0808302
\(903\) −1.40766e13 −0.704535
\(904\) 2.24525e13 1.11817
\(905\) 0 0
\(906\) 2.62782e13 1.29574
\(907\) 2.10130e13 1.03099 0.515495 0.856893i \(-0.327608\pi\)
0.515495 + 0.856893i \(0.327608\pi\)
\(908\) 2.35404e12 0.114929
\(909\) −3.07761e13 −1.49512
\(910\) 0 0
\(911\) 1.95345e12 0.0939656 0.0469828 0.998896i \(-0.485039\pi\)
0.0469828 + 0.998896i \(0.485039\pi\)
\(912\) 2.97312e13 1.42310
\(913\) 1.04802e13 0.499172
\(914\) −2.33870e13 −1.10845
\(915\) 0 0
\(916\) 2.60604e12 0.122307
\(917\) −3.80064e13 −1.77499
\(918\) 1.78175e12 0.0828048
\(919\) −2.47656e12 −0.114533 −0.0572663 0.998359i \(-0.518238\pi\)
−0.0572663 + 0.998359i \(0.518238\pi\)
\(920\) 0 0
\(921\) 2.73301e13 1.25162
\(922\) −2.12834e13 −0.969954
\(923\) −5.75744e13 −2.61109
\(924\) 4.04048e12 0.182351
\(925\) 0 0
\(926\) −1.49017e13 −0.666018
\(927\) −4.78510e13 −2.12830
\(928\) −4.16725e11 −0.0184452
\(929\) 4.40844e13 1.94184 0.970922 0.239395i \(-0.0769490\pi\)
0.970922 + 0.239395i \(0.0769490\pi\)
\(930\) 0 0
\(931\) −3.52146e13 −1.53621
\(932\) 3.55718e12 0.154431
\(933\) 4.40477e11 0.0190308
\(934\) 4.02578e13 1.73097
\(935\) 0 0
\(936\) −4.88798e13 −2.08156
\(937\) −4.92250e12 −0.208621 −0.104310 0.994545i \(-0.533264\pi\)
−0.104310 + 0.994545i \(0.533264\pi\)
\(938\) 2.88787e13 1.21805
\(939\) 5.12372e13 2.15075
\(940\) 0 0
\(941\) −1.54680e13 −0.643103 −0.321552 0.946892i \(-0.604204\pi\)
−0.321552 + 0.946892i \(0.604204\pi\)
\(942\) −6.07852e12 −0.251518
\(943\) −3.88116e12 −0.159830
\(944\) −2.03731e13 −0.834993
\(945\) 0 0
\(946\) −5.27936e12 −0.214324
\(947\) −5.49107e12 −0.221862 −0.110931 0.993828i \(-0.535383\pi\)
−0.110931 + 0.993828i \(0.535383\pi\)
\(948\) 1.42080e12 0.0571343
\(949\) 2.66004e13 1.06461
\(950\) 0 0
\(951\) 7.28207e13 2.88697
\(952\) 9.87649e12 0.389705
\(953\) 1.76932e13 0.694845 0.347422 0.937709i \(-0.387057\pi\)
0.347422 + 0.937709i \(0.387057\pi\)
\(954\) −2.56802e13 −1.00376
\(955\) 0 0
\(956\) 1.17743e12 0.0455905
\(957\) 2.59905e12 0.100164
\(958\) 4.90510e12 0.188150
\(959\) −2.52994e13 −0.965888
\(960\) 0 0
\(961\) −2.61401e13 −0.988670
\(962\) −1.85632e13 −0.698819
\(963\) −5.19916e13 −1.94812
\(964\) −5.31148e12 −0.198093
\(965\) 0 0
\(966\) −8.40878e13 −3.10696
\(967\) −2.89231e13 −1.06372 −0.531858 0.846834i \(-0.678506\pi\)
−0.531858 + 0.846834i \(0.678506\pi\)
\(968\) −1.24588e13 −0.456076
\(969\) 1.05519e13 0.384481
\(970\) 0 0
\(971\) 2.69482e13 0.972843 0.486422 0.873724i \(-0.338302\pi\)
0.486422 + 0.873724i \(0.338302\pi\)
\(972\) 4.07665e12 0.146489
\(973\) 4.05255e13 1.44951
\(974\) −1.03112e13 −0.367107
\(975\) 0 0
\(976\) −8.92590e12 −0.314867
\(977\) −3.90598e13 −1.37153 −0.685764 0.727824i \(-0.740532\pi\)
−0.685764 + 0.727824i \(0.740532\pi\)
\(978\) −2.02934e13 −0.709301
\(979\) −2.05282e13 −0.714216
\(980\) 0 0
\(981\) −2.91084e12 −0.100348
\(982\) 3.64804e13 1.25187
\(983\) −9.03830e12 −0.308742 −0.154371 0.988013i \(-0.549335\pi\)
−0.154371 + 0.988013i \(0.549335\pi\)
\(984\) 5.23684e12 0.178070
\(985\) 0 0
\(986\) 5.98404e11 0.0201627
\(987\) 6.56285e13 2.20123
\(988\) −5.35098e12 −0.178660
\(989\) −1.27509e13 −0.423795
\(990\) 0 0
\(991\) 4.89096e13 1.61088 0.805439 0.592678i \(-0.201929\pi\)
0.805439 + 0.592678i \(0.201929\pi\)
\(992\) −6.72333e11 −0.0220436
\(993\) −3.01181e13 −0.983007
\(994\) 7.40909e13 2.40728
\(995\) 0 0
\(996\) −3.21695e12 −0.103580
\(997\) −4.01445e13 −1.28676 −0.643380 0.765547i \(-0.722468\pi\)
−0.643380 + 0.765547i \(0.722468\pi\)
\(998\) −4.41384e13 −1.40841
\(999\) −5.30984e12 −0.168670
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 25.10.a.c.1.2 3
3.2 odd 2 225.10.a.p.1.2 3
4.3 odd 2 400.10.a.y.1.1 3
5.2 odd 4 25.10.b.c.24.2 6
5.3 odd 4 25.10.b.c.24.5 6
5.4 even 2 25.10.a.d.1.2 yes 3
15.2 even 4 225.10.b.m.199.5 6
15.8 even 4 225.10.b.m.199.2 6
15.14 odd 2 225.10.a.m.1.2 3
20.3 even 4 400.10.c.q.49.2 6
20.7 even 4 400.10.c.q.49.5 6
20.19 odd 2 400.10.a.u.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.10.a.c.1.2 3 1.1 even 1 trivial
25.10.a.d.1.2 yes 3 5.4 even 2
25.10.b.c.24.2 6 5.2 odd 4
25.10.b.c.24.5 6 5.3 odd 4
225.10.a.m.1.2 3 15.14 odd 2
225.10.a.p.1.2 3 3.2 odd 2
225.10.b.m.199.2 6 15.8 even 4
225.10.b.m.199.5 6 15.2 even 4
400.10.a.u.1.3 3 20.19 odd 2
400.10.a.y.1.1 3 4.3 odd 2
400.10.c.q.49.2 6 20.3 even 4
400.10.c.q.49.5 6 20.7 even 4