Properties

Label 570.8.a.c.1.2
Level $570$
Weight $8$
Character 570.1
Self dual yes
Analytic conductor $178.059$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 570.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(178.059464526\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 2087 x^{2} + 44517 x - 205110\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(28.1583\) of defining polynomial
Character \(\chi\) \(=\) 570.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +125.000 q^{5} +216.000 q^{6} -374.399 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} +125.000 q^{5} +216.000 q^{6} -374.399 q^{7} +512.000 q^{8} +729.000 q^{9} +1000.00 q^{10} +5921.73 q^{11} +1728.00 q^{12} -7458.14 q^{13} -2995.19 q^{14} +3375.00 q^{15} +4096.00 q^{16} -23081.3 q^{17} +5832.00 q^{18} -6859.00 q^{19} +8000.00 q^{20} -10108.8 q^{21} +47373.9 q^{22} -106830. q^{23} +13824.0 q^{24} +15625.0 q^{25} -59665.1 q^{26} +19683.0 q^{27} -23961.6 q^{28} -55953.5 q^{29} +27000.0 q^{30} -147150. q^{31} +32768.0 q^{32} +159887. q^{33} -184650. q^{34} -46799.9 q^{35} +46656.0 q^{36} -573493. q^{37} -54872.0 q^{38} -201370. q^{39} +64000.0 q^{40} +263485. q^{41} -80870.3 q^{42} -109369. q^{43} +378991. q^{44} +91125.0 q^{45} -854640. q^{46} +413556. q^{47} +110592. q^{48} -683368. q^{49} +125000. q^{50} -623194. q^{51} -477321. q^{52} -318813. q^{53} +157464. q^{54} +740217. q^{55} -191692. q^{56} -185193. q^{57} -447628. q^{58} +143102. q^{59} +216000. q^{60} +2.66024e6 q^{61} -1.17720e6 q^{62} -272937. q^{63} +262144. q^{64} -932267. q^{65} +1.27909e6 q^{66} +439857. q^{67} -1.47720e6 q^{68} -2.88441e6 q^{69} -374399. q^{70} +643837. q^{71} +373248. q^{72} +2.69771e6 q^{73} -4.58795e6 q^{74} +421875. q^{75} -438976. q^{76} -2.21709e6 q^{77} -1.61096e6 q^{78} +1.65048e6 q^{79} +512000. q^{80} +531441. q^{81} +2.10788e6 q^{82} -7.08061e6 q^{83} -646962. q^{84} -2.88516e6 q^{85} -874951. q^{86} -1.51075e6 q^{87} +3.03193e6 q^{88} -4.97662e6 q^{89} +729000. q^{90} +2.79232e6 q^{91} -6.83712e6 q^{92} -3.97305e6 q^{93} +3.30845e6 q^{94} -857375. q^{95} +884736. q^{96} -6.37924e6 q^{97} -5.46695e6 q^{98} +4.31694e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 32q^{2} + 108q^{3} + 256q^{4} + 500q^{5} + 864q^{6} - 1316q^{7} + 2048q^{8} + 2916q^{9} + O(q^{10}) \) \( 4q + 32q^{2} + 108q^{3} + 256q^{4} + 500q^{5} + 864q^{6} - 1316q^{7} + 2048q^{8} + 2916q^{9} + 4000q^{10} - 6570q^{11} + 6912q^{12} - 17578q^{13} - 10528q^{14} + 13500q^{15} + 16384q^{16} - 28466q^{17} + 23328q^{18} - 27436q^{19} + 32000q^{20} - 35532q^{21} - 52560q^{22} - 32136q^{23} + 55296q^{24} + 62500q^{25} - 140624q^{26} + 78732q^{27} - 84224q^{28} - 159122q^{29} + 108000q^{30} - 67974q^{31} + 131072q^{32} - 177390q^{33} - 227728q^{34} - 164500q^{35} + 186624q^{36} - 823702q^{37} - 219488q^{38} - 474606q^{39} + 256000q^{40} - 781924q^{41} - 284256q^{42} - 1115638q^{43} - 420480q^{44} + 364500q^{45} - 257088q^{46} - 209160q^{47} + 442368q^{48} - 1159308q^{49} + 500000q^{50} - 768582q^{51} - 1124992q^{52} - 848424q^{53} + 629856q^{54} - 821250q^{55} - 673792q^{56} - 740772q^{57} - 1272976q^{58} - 3677830q^{59} + 864000q^{60} - 1161072q^{61} - 543792q^{62} - 959364q^{63} + 1048576q^{64} - 2197250q^{65} - 1419120q^{66} - 6154740q^{67} - 1821824q^{68} - 867672q^{69} - 1316000q^{70} - 4456224q^{71} + 1492992q^{72} + 1057792q^{73} - 6589616q^{74} + 1687500q^{75} - 1755904q^{76} + 2402388q^{77} - 3796848q^{78} + 2910090q^{79} + 2048000q^{80} + 2125764q^{81} - 6255392q^{82} - 1767198q^{83} - 2274048q^{84} - 3558250q^{85} - 8925104q^{86} - 4296294q^{87} - 3363840q^{88} - 3677360q^{89} + 2916000q^{90} - 6727732q^{91} - 2056704q^{92} - 1835298q^{93} - 1673280q^{94} - 3429500q^{95} + 3538944q^{96} - 10419094q^{97} - 9274464q^{98} - 4789530q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) 27.0000 0.577350
\(4\) 64.0000 0.500000
\(5\) 125.000 0.447214
\(6\) 216.000 0.408248
\(7\) −374.399 −0.412565 −0.206282 0.978493i \(-0.566137\pi\)
−0.206282 + 0.978493i \(0.566137\pi\)
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) 1000.00 0.316228
\(11\) 5921.73 1.34145 0.670725 0.741706i \(-0.265983\pi\)
0.670725 + 0.741706i \(0.265983\pi\)
\(12\) 1728.00 0.288675
\(13\) −7458.14 −0.941518 −0.470759 0.882262i \(-0.656020\pi\)
−0.470759 + 0.882262i \(0.656020\pi\)
\(14\) −2995.19 −0.291727
\(15\) 3375.00 0.258199
\(16\) 4096.00 0.250000
\(17\) −23081.3 −1.13943 −0.569716 0.821842i \(-0.692947\pi\)
−0.569716 + 0.821842i \(0.692947\pi\)
\(18\) 5832.00 0.235702
\(19\) −6859.00 −0.229416
\(20\) 8000.00 0.223607
\(21\) −10108.8 −0.238194
\(22\) 47373.9 0.948548
\(23\) −106830. −1.83082 −0.915411 0.402521i \(-0.868134\pi\)
−0.915411 + 0.402521i \(0.868134\pi\)
\(24\) 13824.0 0.204124
\(25\) 15625.0 0.200000
\(26\) −59665.1 −0.665754
\(27\) 19683.0 0.192450
\(28\) −23961.6 −0.206282
\(29\) −55953.5 −0.426024 −0.213012 0.977050i \(-0.568327\pi\)
−0.213012 + 0.977050i \(0.568327\pi\)
\(30\) 27000.0 0.182574
\(31\) −147150. −0.887145 −0.443573 0.896238i \(-0.646289\pi\)
−0.443573 + 0.896238i \(0.646289\pi\)
\(32\) 32768.0 0.176777
\(33\) 159887. 0.774487
\(34\) −184650. −0.805700
\(35\) −46799.9 −0.184504
\(36\) 46656.0 0.166667
\(37\) −573493. −1.86133 −0.930663 0.365878i \(-0.880769\pi\)
−0.930663 + 0.365878i \(0.880769\pi\)
\(38\) −54872.0 −0.162221
\(39\) −201370. −0.543586
\(40\) 64000.0 0.158114
\(41\) 263485. 0.597052 0.298526 0.954401i \(-0.403505\pi\)
0.298526 + 0.954401i \(0.403505\pi\)
\(42\) −80870.3 −0.168429
\(43\) −109369. −0.209775 −0.104888 0.994484i \(-0.533448\pi\)
−0.104888 + 0.994484i \(0.533448\pi\)
\(44\) 378991. 0.670725
\(45\) 91125.0 0.149071
\(46\) −854640. −1.29459
\(47\) 413556. 0.581021 0.290511 0.956872i \(-0.406175\pi\)
0.290511 + 0.956872i \(0.406175\pi\)
\(48\) 110592. 0.144338
\(49\) −683368. −0.829790
\(50\) 125000. 0.141421
\(51\) −623194. −0.657851
\(52\) −477321. −0.470759
\(53\) −318813. −0.294151 −0.147075 0.989125i \(-0.546986\pi\)
−0.147075 + 0.989125i \(0.546986\pi\)
\(54\) 157464. 0.136083
\(55\) 740217. 0.599915
\(56\) −191692. −0.145864
\(57\) −185193. −0.132453
\(58\) −447628. −0.301245
\(59\) 143102. 0.0907119 0.0453559 0.998971i \(-0.485558\pi\)
0.0453559 + 0.998971i \(0.485558\pi\)
\(60\) 216000. 0.129099
\(61\) 2.66024e6 1.50061 0.750304 0.661093i \(-0.229907\pi\)
0.750304 + 0.661093i \(0.229907\pi\)
\(62\) −1.17720e6 −0.627306
\(63\) −272937. −0.137522
\(64\) 262144. 0.125000
\(65\) −932267. −0.421060
\(66\) 1.27909e6 0.547645
\(67\) 439857. 0.178669 0.0893346 0.996002i \(-0.471526\pi\)
0.0893346 + 0.996002i \(0.471526\pi\)
\(68\) −1.47720e6 −0.569716
\(69\) −2.88441e6 −1.05703
\(70\) −374399. −0.130464
\(71\) 643837. 0.213487 0.106743 0.994287i \(-0.465958\pi\)
0.106743 + 0.994287i \(0.465958\pi\)
\(72\) 373248. 0.117851
\(73\) 2.69771e6 0.811644 0.405822 0.913952i \(-0.366985\pi\)
0.405822 + 0.913952i \(0.366985\pi\)
\(74\) −4.58795e6 −1.31616
\(75\) 421875. 0.115470
\(76\) −438976. −0.114708
\(77\) −2.21709e6 −0.553435
\(78\) −1.61096e6 −0.384373
\(79\) 1.65048e6 0.376631 0.188316 0.982109i \(-0.439697\pi\)
0.188316 + 0.982109i \(0.439697\pi\)
\(80\) 512000. 0.111803
\(81\) 531441. 0.111111
\(82\) 2.10788e6 0.422180
\(83\) −7.08061e6 −1.35924 −0.679621 0.733563i \(-0.737856\pi\)
−0.679621 + 0.733563i \(0.737856\pi\)
\(84\) −646962. −0.119097
\(85\) −2.88516e6 −0.509570
\(86\) −874951. −0.148333
\(87\) −1.51075e6 −0.245965
\(88\) 3.03193e6 0.474274
\(89\) −4.97662e6 −0.748289 −0.374145 0.927370i \(-0.622064\pi\)
−0.374145 + 0.927370i \(0.622064\pi\)
\(90\) 729000. 0.105409
\(91\) 2.79232e6 0.388437
\(92\) −6.83712e6 −0.915411
\(93\) −3.97305e6 −0.512193
\(94\) 3.30845e6 0.410844
\(95\) −857375. −0.102598
\(96\) 884736. 0.102062
\(97\) −6.37924e6 −0.709689 −0.354845 0.934925i \(-0.615466\pi\)
−0.354845 + 0.934925i \(0.615466\pi\)
\(98\) −5.46695e6 −0.586750
\(99\) 4.31694e6 0.447150
\(100\) 1.00000e6 0.100000
\(101\) 7.85742e6 0.758849 0.379424 0.925223i \(-0.376122\pi\)
0.379424 + 0.925223i \(0.376122\pi\)
\(102\) −4.98555e6 −0.465171
\(103\) −1.18113e7 −1.06504 −0.532521 0.846417i \(-0.678755\pi\)
−0.532521 + 0.846417i \(0.678755\pi\)
\(104\) −3.81857e6 −0.332877
\(105\) −1.26360e6 −0.106524
\(106\) −2.55050e6 −0.207996
\(107\) −1.24680e7 −0.983906 −0.491953 0.870622i \(-0.663717\pi\)
−0.491953 + 0.870622i \(0.663717\pi\)
\(108\) 1.25971e6 0.0962250
\(109\) 1.37661e6 0.101817 0.0509084 0.998703i \(-0.483788\pi\)
0.0509084 + 0.998703i \(0.483788\pi\)
\(110\) 5.92173e6 0.424204
\(111\) −1.54843e7 −1.07464
\(112\) −1.53354e6 −0.103141
\(113\) −2.63038e7 −1.71492 −0.857459 0.514552i \(-0.827958\pi\)
−0.857459 + 0.514552i \(0.827958\pi\)
\(114\) −1.48154e6 −0.0936586
\(115\) −1.33538e7 −0.818768
\(116\) −3.58103e6 −0.213012
\(117\) −5.43698e6 −0.313839
\(118\) 1.14482e6 0.0641430
\(119\) 8.64161e6 0.470089
\(120\) 1.72800e6 0.0912871
\(121\) 1.55798e7 0.799488
\(122\) 2.12820e7 1.06109
\(123\) 7.11410e6 0.344708
\(124\) −9.41761e6 −0.443573
\(125\) 1.95312e6 0.0894427
\(126\) −2.18350e6 −0.0972424
\(127\) 1.33085e7 0.576521 0.288260 0.957552i \(-0.406923\pi\)
0.288260 + 0.957552i \(0.406923\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) −2.95296e6 −0.121114
\(130\) −7.45814e6 −0.297734
\(131\) −2.57743e6 −0.100170 −0.0500849 0.998745i \(-0.515949\pi\)
−0.0500849 + 0.998745i \(0.515949\pi\)
\(132\) 1.02328e7 0.387243
\(133\) 2.56800e6 0.0946488
\(134\) 3.51886e6 0.126338
\(135\) 2.46038e6 0.0860663
\(136\) −1.18176e7 −0.402850
\(137\) 3.15345e7 1.04776 0.523882 0.851791i \(-0.324483\pi\)
0.523882 + 0.851791i \(0.324483\pi\)
\(138\) −2.30753e7 −0.747430
\(139\) 3.63938e7 1.14941 0.574706 0.818360i \(-0.305116\pi\)
0.574706 + 0.818360i \(0.305116\pi\)
\(140\) −2.99519e6 −0.0922522
\(141\) 1.11660e7 0.335453
\(142\) 5.15069e6 0.150958
\(143\) −4.41651e7 −1.26300
\(144\) 2.98598e6 0.0833333
\(145\) −6.99419e6 −0.190524
\(146\) 2.15817e7 0.573919
\(147\) −1.84509e7 −0.479080
\(148\) −3.67036e7 −0.930663
\(149\) 1.95241e7 0.483526 0.241763 0.970335i \(-0.422274\pi\)
0.241763 + 0.970335i \(0.422274\pi\)
\(150\) 3.37500e6 0.0816497
\(151\) −2.01132e7 −0.475403 −0.237702 0.971338i \(-0.576394\pi\)
−0.237702 + 0.971338i \(0.576394\pi\)
\(152\) −3.51181e6 −0.0811107
\(153\) −1.68262e7 −0.379811
\(154\) −1.77367e7 −0.391337
\(155\) −1.83938e7 −0.396743
\(156\) −1.28877e7 −0.271793
\(157\) 3.00256e7 0.619217 0.309608 0.950864i \(-0.399802\pi\)
0.309608 + 0.950864i \(0.399802\pi\)
\(158\) 1.32039e7 0.266319
\(159\) −8.60794e6 −0.169828
\(160\) 4.09600e6 0.0790569
\(161\) 3.99971e7 0.755332
\(162\) 4.25153e6 0.0785674
\(163\) −2.02207e7 −0.365713 −0.182856 0.983140i \(-0.558534\pi\)
−0.182856 + 0.983140i \(0.558534\pi\)
\(164\) 1.68630e7 0.298526
\(165\) 1.99859e7 0.346361
\(166\) −5.66448e7 −0.961130
\(167\) −3.95339e7 −0.656843 −0.328422 0.944531i \(-0.606517\pi\)
−0.328422 + 0.944531i \(0.606517\pi\)
\(168\) −5.17570e6 −0.0842144
\(169\) −7.12472e6 −0.113544
\(170\) −2.30813e7 −0.360320
\(171\) −5.00021e6 −0.0764719
\(172\) −6.99961e6 −0.104888
\(173\) −7.50134e7 −1.10148 −0.550741 0.834676i \(-0.685655\pi\)
−0.550741 + 0.834676i \(0.685655\pi\)
\(174\) −1.20860e7 −0.173924
\(175\) −5.84999e6 −0.0825129
\(176\) 2.42554e7 0.335363
\(177\) 3.86376e6 0.0523725
\(178\) −3.98130e7 −0.529120
\(179\) 8.16785e7 1.06444 0.532221 0.846605i \(-0.321358\pi\)
0.532221 + 0.846605i \(0.321358\pi\)
\(180\) 5.83200e6 0.0745356
\(181\) 1.09477e8 1.37230 0.686150 0.727461i \(-0.259300\pi\)
0.686150 + 0.727461i \(0.259300\pi\)
\(182\) 2.23386e7 0.274666
\(183\) 7.18266e7 0.866376
\(184\) −5.46970e7 −0.647293
\(185\) −7.16867e7 −0.832410
\(186\) −3.17844e7 −0.362175
\(187\) −1.36681e8 −1.52849
\(188\) 2.64676e7 0.290511
\(189\) −7.36930e6 −0.0793981
\(190\) −6.85900e6 −0.0725476
\(191\) −1.63486e8 −1.69771 −0.848856 0.528624i \(-0.822708\pi\)
−0.848856 + 0.528624i \(0.822708\pi\)
\(192\) 7.07789e6 0.0721688
\(193\) 1.38881e8 1.39057 0.695286 0.718733i \(-0.255278\pi\)
0.695286 + 0.718733i \(0.255278\pi\)
\(194\) −5.10340e7 −0.501826
\(195\) −2.51712e7 −0.243099
\(196\) −4.37356e7 −0.414895
\(197\) −1.04332e8 −0.972266 −0.486133 0.873885i \(-0.661593\pi\)
−0.486133 + 0.873885i \(0.661593\pi\)
\(198\) 3.45356e7 0.316183
\(199\) −1.56385e8 −1.40672 −0.703362 0.710832i \(-0.748319\pi\)
−0.703362 + 0.710832i \(0.748319\pi\)
\(200\) 8.00000e6 0.0707107
\(201\) 1.18761e7 0.103155
\(202\) 6.28594e7 0.536587
\(203\) 2.09490e7 0.175763
\(204\) −3.98844e7 −0.328926
\(205\) 3.29356e7 0.267010
\(206\) −9.44902e7 −0.753098
\(207\) −7.78791e7 −0.610274
\(208\) −3.05485e7 −0.235379
\(209\) −4.06172e7 −0.307750
\(210\) −1.01088e7 −0.0753236
\(211\) 5.40663e7 0.396221 0.198111 0.980180i \(-0.436519\pi\)
0.198111 + 0.980180i \(0.436519\pi\)
\(212\) −2.04040e7 −0.147075
\(213\) 1.73836e7 0.123257
\(214\) −9.97440e7 −0.695727
\(215\) −1.36711e7 −0.0938143
\(216\) 1.00777e7 0.0680414
\(217\) 5.50929e7 0.366005
\(218\) 1.10129e7 0.0719953
\(219\) 7.28382e7 0.468603
\(220\) 4.73739e7 0.299957
\(221\) 1.72143e8 1.07280
\(222\) −1.23875e8 −0.759883
\(223\) −6.40665e7 −0.386869 −0.193434 0.981113i \(-0.561963\pi\)
−0.193434 + 0.981113i \(0.561963\pi\)
\(224\) −1.22683e7 −0.0729318
\(225\) 1.13906e7 0.0666667
\(226\) −2.10430e8 −1.21263
\(227\) −2.23685e8 −1.26925 −0.634625 0.772820i \(-0.718845\pi\)
−0.634625 + 0.772820i \(0.718845\pi\)
\(228\) −1.18524e7 −0.0662266
\(229\) −2.32304e7 −0.127830 −0.0639151 0.997955i \(-0.520359\pi\)
−0.0639151 + 0.997955i \(0.520359\pi\)
\(230\) −1.06830e8 −0.578957
\(231\) −5.98615e7 −0.319526
\(232\) −2.86482e7 −0.150622
\(233\) 2.26495e8 1.17304 0.586519 0.809935i \(-0.300498\pi\)
0.586519 + 0.809935i \(0.300498\pi\)
\(234\) −4.34959e7 −0.221918
\(235\) 5.16945e7 0.259841
\(236\) 9.15854e6 0.0453559
\(237\) 4.45631e7 0.217448
\(238\) 6.91329e7 0.332403
\(239\) 2.61223e8 1.23771 0.618855 0.785506i \(-0.287597\pi\)
0.618855 + 0.785506i \(0.287597\pi\)
\(240\) 1.38240e7 0.0645497
\(241\) −2.44282e7 −0.112417 −0.0562084 0.998419i \(-0.517901\pi\)
−0.0562084 + 0.998419i \(0.517901\pi\)
\(242\) 1.24638e8 0.565324
\(243\) 1.43489e7 0.0641500
\(244\) 1.70256e8 0.750304
\(245\) −8.54210e7 −0.371094
\(246\) 5.69128e7 0.243746
\(247\) 5.11554e7 0.215999
\(248\) −7.53409e7 −0.313653
\(249\) −1.91176e8 −0.784759
\(250\) 1.56250e7 0.0632456
\(251\) 1.07657e8 0.429719 0.214860 0.976645i \(-0.431071\pi\)
0.214860 + 0.976645i \(0.431071\pi\)
\(252\) −1.74680e7 −0.0687608
\(253\) −6.32619e8 −2.45596
\(254\) 1.06468e8 0.407662
\(255\) −7.78993e7 −0.294200
\(256\) 1.67772e7 0.0625000
\(257\) 3.03168e7 0.111408 0.0557042 0.998447i \(-0.482260\pi\)
0.0557042 + 0.998447i \(0.482260\pi\)
\(258\) −2.36237e7 −0.0856403
\(259\) 2.14716e8 0.767917
\(260\) −5.96651e7 −0.210530
\(261\) −4.07901e7 −0.142008
\(262\) −2.06194e7 −0.0708307
\(263\) −3.25186e8 −1.10227 −0.551133 0.834418i \(-0.685804\pi\)
−0.551133 + 0.834418i \(0.685804\pi\)
\(264\) 8.18621e7 0.273822
\(265\) −3.98516e7 −0.131548
\(266\) 2.05440e7 0.0669268
\(267\) −1.34369e8 −0.432025
\(268\) 2.81509e7 0.0893346
\(269\) −4.74661e7 −0.148679 −0.0743396 0.997233i \(-0.523685\pi\)
−0.0743396 + 0.997233i \(0.523685\pi\)
\(270\) 1.96830e7 0.0608581
\(271\) 8.20294e7 0.250367 0.125183 0.992134i \(-0.460048\pi\)
0.125183 + 0.992134i \(0.460048\pi\)
\(272\) −9.45409e7 −0.284858
\(273\) 7.53927e7 0.224264
\(274\) 2.52276e8 0.740881
\(275\) 9.25271e7 0.268290
\(276\) −1.84602e8 −0.528513
\(277\) 4.03592e8 1.14094 0.570471 0.821318i \(-0.306761\pi\)
0.570471 + 0.821318i \(0.306761\pi\)
\(278\) 2.91151e8 0.812758
\(279\) −1.07272e8 −0.295715
\(280\) −2.39616e7 −0.0652322
\(281\) −2.80665e8 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(282\) 8.93282e7 0.237201
\(283\) −2.92789e8 −0.767895 −0.383948 0.923355i \(-0.625436\pi\)
−0.383948 + 0.923355i \(0.625436\pi\)
\(284\) 4.12055e7 0.106743
\(285\) −2.31491e7 −0.0592349
\(286\) −3.53321e8 −0.893075
\(287\) −9.86486e7 −0.246323
\(288\) 2.38879e7 0.0589256
\(289\) 1.22406e8 0.298305
\(290\) −5.59535e7 −0.134721
\(291\) −1.72240e8 −0.409739
\(292\) 1.72654e8 0.405822
\(293\) −3.66539e8 −0.851302 −0.425651 0.904887i \(-0.639955\pi\)
−0.425651 + 0.904887i \(0.639955\pi\)
\(294\) −1.47608e8 −0.338761
\(295\) 1.78878e7 0.0405676
\(296\) −2.93629e8 −0.658078
\(297\) 1.16557e8 0.258162
\(298\) 1.56193e8 0.341905
\(299\) 7.96753e8 1.72375
\(300\) 2.70000e7 0.0577350
\(301\) 4.09476e7 0.0865458
\(302\) −1.60906e8 −0.336161
\(303\) 2.12150e8 0.438122
\(304\) −2.80945e7 −0.0573539
\(305\) 3.32531e8 0.671092
\(306\) −1.34610e8 −0.268567
\(307\) 3.74875e7 0.0739439 0.0369719 0.999316i \(-0.488229\pi\)
0.0369719 + 0.999316i \(0.488229\pi\)
\(308\) −1.41894e8 −0.276717
\(309\) −3.18905e8 −0.614902
\(310\) −1.47150e8 −0.280540
\(311\) −7.55525e8 −1.42425 −0.712127 0.702050i \(-0.752268\pi\)
−0.712127 + 0.702050i \(0.752268\pi\)
\(312\) −1.03101e8 −0.192187
\(313\) −1.71267e8 −0.315695 −0.157848 0.987463i \(-0.550455\pi\)
−0.157848 + 0.987463i \(0.550455\pi\)
\(314\) 2.40205e8 0.437852
\(315\) −3.41171e7 −0.0615015
\(316\) 1.05631e8 0.188316
\(317\) −7.40590e8 −1.30578 −0.652891 0.757452i \(-0.726444\pi\)
−0.652891 + 0.757452i \(0.726444\pi\)
\(318\) −6.88636e7 −0.120087
\(319\) −3.31342e8 −0.571491
\(320\) 3.27680e7 0.0559017
\(321\) −3.36636e8 −0.568059
\(322\) 3.19977e8 0.534100
\(323\) 1.58314e8 0.261404
\(324\) 3.40122e7 0.0555556
\(325\) −1.16533e8 −0.188304
\(326\) −1.61766e8 −0.258598
\(327\) 3.71686e7 0.0587839
\(328\) 1.34904e8 0.211090
\(329\) −1.54835e8 −0.239709
\(330\) 1.59887e8 0.244914
\(331\) 1.73430e8 0.262861 0.131430 0.991325i \(-0.458043\pi\)
0.131430 + 0.991325i \(0.458043\pi\)
\(332\) −4.53159e8 −0.679621
\(333\) −4.18077e8 −0.620442
\(334\) −3.16271e8 −0.464458
\(335\) 5.49822e7 0.0799033
\(336\) −4.14056e7 −0.0595486
\(337\) 2.37247e8 0.337673 0.168837 0.985644i \(-0.445999\pi\)
0.168837 + 0.985644i \(0.445999\pi\)
\(338\) −5.69977e7 −0.0802877
\(339\) −7.10202e8 −0.990108
\(340\) −1.84650e8 −0.254785
\(341\) −8.71384e8 −1.19006
\(342\) −4.00017e7 −0.0540738
\(343\) 5.64187e8 0.754907
\(344\) −5.59969e7 −0.0741667
\(345\) −3.60551e8 −0.472716
\(346\) −6.00107e8 −0.778865
\(347\) 9.60686e8 1.23432 0.617161 0.786837i \(-0.288283\pi\)
0.617161 + 0.786837i \(0.288283\pi\)
\(348\) −9.66877e7 −0.122983
\(349\) −1.05309e9 −1.32610 −0.663050 0.748575i \(-0.730739\pi\)
−0.663050 + 0.748575i \(0.730739\pi\)
\(350\) −4.67999e7 −0.0583454
\(351\) −1.46798e8 −0.181195
\(352\) 1.94043e8 0.237137
\(353\) 8.40523e8 1.01704 0.508520 0.861050i \(-0.330193\pi\)
0.508520 + 0.861050i \(0.330193\pi\)
\(354\) 3.09101e7 0.0370330
\(355\) 8.04796e7 0.0954743
\(356\) −3.18504e8 −0.374145
\(357\) 2.33323e8 0.271406
\(358\) 6.53428e8 0.752674
\(359\) 1.04415e9 1.19106 0.595528 0.803335i \(-0.296943\pi\)
0.595528 + 0.803335i \(0.296943\pi\)
\(360\) 4.66560e7 0.0527046
\(361\) 4.70459e7 0.0526316
\(362\) 8.75818e8 0.970362
\(363\) 4.20654e8 0.461585
\(364\) 1.78709e8 0.194218
\(365\) 3.37214e8 0.362978
\(366\) 5.74613e8 0.612620
\(367\) −9.38878e8 −0.991467 −0.495733 0.868475i \(-0.665101\pi\)
−0.495733 + 0.868475i \(0.665101\pi\)
\(368\) −4.37576e8 −0.457705
\(369\) 1.92081e8 0.199017
\(370\) −5.73493e8 −0.588603
\(371\) 1.19363e8 0.121356
\(372\) −2.54275e8 −0.256097
\(373\) −6.14840e8 −0.613453 −0.306727 0.951798i \(-0.599234\pi\)
−0.306727 + 0.951798i \(0.599234\pi\)
\(374\) −1.09345e9 −1.08081
\(375\) 5.27344e7 0.0516398
\(376\) 2.11741e8 0.205422
\(377\) 4.17309e8 0.401110
\(378\) −5.89544e7 −0.0561429
\(379\) 1.42113e9 1.34090 0.670451 0.741954i \(-0.266101\pi\)
0.670451 + 0.741954i \(0.266101\pi\)
\(380\) −5.48720e7 −0.0512989
\(381\) 3.59328e8 0.332854
\(382\) −1.30789e9 −1.20046
\(383\) 8.66189e8 0.787802 0.393901 0.919153i \(-0.371125\pi\)
0.393901 + 0.919153i \(0.371125\pi\)
\(384\) 5.66231e7 0.0510310
\(385\) −2.77137e8 −0.247504
\(386\) 1.11105e9 0.983283
\(387\) −7.97299e7 −0.0699251
\(388\) −4.08272e8 −0.354845
\(389\) −4.18499e8 −0.360471 −0.180236 0.983623i \(-0.557686\pi\)
−0.180236 + 0.983623i \(0.557686\pi\)
\(390\) −2.01370e8 −0.171897
\(391\) 2.46577e9 2.08610
\(392\) −3.49884e8 −0.293375
\(393\) −6.95905e7 −0.0578331
\(394\) −8.34654e8 −0.687496
\(395\) 2.06310e8 0.168435
\(396\) 2.76284e8 0.223575
\(397\) −1.15119e9 −0.923377 −0.461688 0.887042i \(-0.652756\pi\)
−0.461688 + 0.887042i \(0.652756\pi\)
\(398\) −1.25108e9 −0.994704
\(399\) 6.93361e7 0.0546455
\(400\) 6.40000e7 0.0500000
\(401\) 1.09099e9 0.844923 0.422462 0.906381i \(-0.361166\pi\)
0.422462 + 0.906381i \(0.361166\pi\)
\(402\) 9.50092e7 0.0729414
\(403\) 1.09747e9 0.835263
\(404\) 5.02875e8 0.379424
\(405\) 6.64301e7 0.0496904
\(406\) 1.67592e8 0.124283
\(407\) −3.39608e9 −2.49688
\(408\) −3.19075e8 −0.232586
\(409\) −1.14121e9 −0.824769 −0.412384 0.911010i \(-0.635304\pi\)
−0.412384 + 0.911010i \(0.635304\pi\)
\(410\) 2.63485e8 0.188805
\(411\) 8.51431e8 0.604927
\(412\) −7.55922e8 −0.532521
\(413\) −5.35773e7 −0.0374245
\(414\) −6.23033e8 −0.431529
\(415\) −8.85076e8 −0.607872
\(416\) −2.44388e8 −0.166438
\(417\) 9.82634e8 0.663614
\(418\) −3.24937e8 −0.217612
\(419\) −1.35451e9 −0.899563 −0.449782 0.893139i \(-0.648498\pi\)
−0.449782 + 0.893139i \(0.648498\pi\)
\(420\) −8.08703e7 −0.0532619
\(421\) −1.87657e8 −0.122568 −0.0612841 0.998120i \(-0.519520\pi\)
−0.0612841 + 0.998120i \(0.519520\pi\)
\(422\) 4.32530e8 0.280171
\(423\) 3.01483e8 0.193674
\(424\) −1.63232e8 −0.103998
\(425\) −3.60645e8 −0.227886
\(426\) 1.39069e8 0.0871557
\(427\) −9.95994e8 −0.619097
\(428\) −7.97952e8 −0.491953
\(429\) −1.19246e9 −0.729193
\(430\) −1.09369e8 −0.0663367
\(431\) −2.15324e9 −1.29545 −0.647727 0.761873i \(-0.724280\pi\)
−0.647727 + 0.761873i \(0.724280\pi\)
\(432\) 8.06216e7 0.0481125
\(433\) −1.14521e9 −0.677916 −0.338958 0.940802i \(-0.610074\pi\)
−0.338958 + 0.940802i \(0.610074\pi\)
\(434\) 4.40743e8 0.258804
\(435\) −1.88843e8 −0.109999
\(436\) 8.81033e7 0.0509084
\(437\) 7.32747e8 0.420019
\(438\) 5.82706e8 0.331352
\(439\) 1.37097e9 0.773394 0.386697 0.922207i \(-0.373616\pi\)
0.386697 + 0.922207i \(0.373616\pi\)
\(440\) 3.78991e8 0.212102
\(441\) −4.98175e8 −0.276597
\(442\) 1.37715e9 0.758581
\(443\) 7.88121e7 0.0430705 0.0215353 0.999768i \(-0.493145\pi\)
0.0215353 + 0.999768i \(0.493145\pi\)
\(444\) −9.90997e8 −0.537319
\(445\) −6.22077e8 −0.334645
\(446\) −5.12532e8 −0.273558
\(447\) 5.27152e8 0.279164
\(448\) −9.81465e7 −0.0515706
\(449\) 1.36078e9 0.709456 0.354728 0.934970i \(-0.384574\pi\)
0.354728 + 0.934970i \(0.384574\pi\)
\(450\) 9.11250e7 0.0471405
\(451\) 1.56029e9 0.800916
\(452\) −1.68344e9 −0.857459
\(453\) −5.43057e8 −0.274474
\(454\) −1.78948e9 −0.897495
\(455\) 3.49040e8 0.173714
\(456\) −9.48188e7 −0.0468293
\(457\) 1.65255e8 0.0809932 0.0404966 0.999180i \(-0.487106\pi\)
0.0404966 + 0.999180i \(0.487106\pi\)
\(458\) −1.85844e8 −0.0903896
\(459\) −4.54309e8 −0.219284
\(460\) −8.54640e8 −0.409384
\(461\) −5.86002e8 −0.278578 −0.139289 0.990252i \(-0.544482\pi\)
−0.139289 + 0.990252i \(0.544482\pi\)
\(462\) −4.78892e8 −0.225939
\(463\) 2.78831e7 0.0130559 0.00652795 0.999979i \(-0.497922\pi\)
0.00652795 + 0.999979i \(0.497922\pi\)
\(464\) −2.29186e8 −0.106506
\(465\) −4.96632e8 −0.229060
\(466\) 1.81196e9 0.829463
\(467\) 3.18096e9 1.44527 0.722636 0.691229i \(-0.242930\pi\)
0.722636 + 0.691229i \(0.242930\pi\)
\(468\) −3.47967e8 −0.156920
\(469\) −1.64682e8 −0.0737126
\(470\) 4.13556e8 0.183735
\(471\) 8.10691e8 0.357505
\(472\) 7.32683e7 0.0320715
\(473\) −6.47653e8 −0.281403
\(474\) 3.56505e8 0.153759
\(475\) −1.07172e8 −0.0458831
\(476\) 5.53063e8 0.235045
\(477\) −2.32414e8 −0.0980503
\(478\) 2.08978e9 0.875192
\(479\) −2.55588e9 −1.06259 −0.531296 0.847186i \(-0.678295\pi\)
−0.531296 + 0.847186i \(0.678295\pi\)
\(480\) 1.10592e8 0.0456435
\(481\) 4.27719e9 1.75247
\(482\) −1.95425e8 −0.0794906
\(483\) 1.07992e9 0.436091
\(484\) 9.97105e8 0.399744
\(485\) −7.97406e8 −0.317383
\(486\) 1.14791e8 0.0453609
\(487\) 1.83949e9 0.721681 0.360840 0.932628i \(-0.382490\pi\)
0.360840 + 0.932628i \(0.382490\pi\)
\(488\) 1.36205e9 0.530545
\(489\) −5.45959e8 −0.211144
\(490\) −6.83368e8 −0.262403
\(491\) −1.52967e9 −0.583195 −0.291597 0.956541i \(-0.594187\pi\)
−0.291597 + 0.956541i \(0.594187\pi\)
\(492\) 4.55302e8 0.172354
\(493\) 1.29148e9 0.485426
\(494\) 4.09243e8 0.152734
\(495\) 5.39618e8 0.199972
\(496\) −6.02727e8 −0.221786
\(497\) −2.41052e8 −0.0880772
\(498\) −1.52941e9 −0.554909
\(499\) −8.44562e7 −0.0304284 −0.0152142 0.999884i \(-0.504843\pi\)
−0.0152142 + 0.999884i \(0.504843\pi\)
\(500\) 1.25000e8 0.0447214
\(501\) −1.06741e9 −0.379229
\(502\) 8.61257e8 0.303857
\(503\) 4.73320e9 1.65832 0.829158 0.559015i \(-0.188821\pi\)
0.829158 + 0.559015i \(0.188821\pi\)
\(504\) −1.39744e8 −0.0486212
\(505\) 9.82178e8 0.339367
\(506\) −5.06095e9 −1.73662
\(507\) −1.92367e8 −0.0655547
\(508\) 8.51742e8 0.288260
\(509\) −4.53559e8 −0.152448 −0.0762240 0.997091i \(-0.524286\pi\)
−0.0762240 + 0.997091i \(0.524286\pi\)
\(510\) −6.23194e8 −0.208031
\(511\) −1.01002e9 −0.334855
\(512\) 1.34218e8 0.0441942
\(513\) −1.35006e8 −0.0441511
\(514\) 2.42535e8 0.0787776
\(515\) −1.47641e9 −0.476301
\(516\) −1.88989e8 −0.0605569
\(517\) 2.44897e9 0.779411
\(518\) 1.71772e9 0.542999
\(519\) −2.02536e9 −0.635941
\(520\) −4.77321e8 −0.148867
\(521\) 9.49189e8 0.294050 0.147025 0.989133i \(-0.453030\pi\)
0.147025 + 0.989133i \(0.453030\pi\)
\(522\) −3.26321e8 −0.100415
\(523\) 3.07593e9 0.940199 0.470100 0.882613i \(-0.344218\pi\)
0.470100 + 0.882613i \(0.344218\pi\)
\(524\) −1.64955e8 −0.0500849
\(525\) −1.57950e8 −0.0476389
\(526\) −2.60148e9 −0.779419
\(527\) 3.39641e9 1.01084
\(528\) 6.54896e8 0.193622
\(529\) 8.00783e9 2.35191
\(530\) −3.18813e8 −0.0930187
\(531\) 1.04321e8 0.0302373
\(532\) 1.64352e8 0.0473244
\(533\) −1.96511e9 −0.562136
\(534\) −1.07495e9 −0.305488
\(535\) −1.55850e9 −0.440016
\(536\) 2.25207e8 0.0631691
\(537\) 2.20532e9 0.614556
\(538\) −3.79729e8 −0.105132
\(539\) −4.04672e9 −1.11312
\(540\) 1.57464e8 0.0430331
\(541\) 3.56121e9 0.966956 0.483478 0.875356i \(-0.339373\pi\)
0.483478 + 0.875356i \(0.339373\pi\)
\(542\) 6.56235e8 0.177036
\(543\) 2.95589e9 0.792297
\(544\) −7.56327e8 −0.201425
\(545\) 1.72077e8 0.0455338
\(546\) 6.03141e8 0.158579
\(547\) −4.71882e9 −1.23276 −0.616379 0.787450i \(-0.711401\pi\)
−0.616379 + 0.787450i \(0.711401\pi\)
\(548\) 2.01821e9 0.523882
\(549\) 1.93932e9 0.500202
\(550\) 7.40217e8 0.189710
\(551\) 3.83785e8 0.0977367
\(552\) −1.47682e9 −0.373715
\(553\) −6.17940e8 −0.155385
\(554\) 3.22873e9 0.806767
\(555\) −1.93554e9 −0.480592
\(556\) 2.32921e9 0.574706
\(557\) 6.30254e9 1.54534 0.772668 0.634811i \(-0.218922\pi\)
0.772668 + 0.634811i \(0.218922\pi\)
\(558\) −8.58179e8 −0.209102
\(559\) 8.15688e8 0.197507
\(560\) −1.91692e8 −0.0461261
\(561\) −3.69039e9 −0.882475
\(562\) −2.24532e9 −0.533582
\(563\) −3.51725e9 −0.830660 −0.415330 0.909671i \(-0.636334\pi\)
−0.415330 + 0.909671i \(0.636334\pi\)
\(564\) 7.14625e8 0.167726
\(565\) −3.28797e9 −0.766935
\(566\) −2.34231e9 −0.542984
\(567\) −1.98971e8 −0.0458405
\(568\) 3.29644e8 0.0754790
\(569\) −4.02350e9 −0.915610 −0.457805 0.889053i \(-0.651364\pi\)
−0.457805 + 0.889053i \(0.651364\pi\)
\(570\) −1.85193e8 −0.0418854
\(571\) −4.83785e9 −1.08749 −0.543746 0.839250i \(-0.682994\pi\)
−0.543746 + 0.839250i \(0.682994\pi\)
\(572\) −2.82657e9 −0.631500
\(573\) −4.41413e9 −0.980175
\(574\) −7.89189e8 −0.174176
\(575\) −1.66922e9 −0.366164
\(576\) 1.91103e8 0.0416667
\(577\) 4.99425e9 1.08232 0.541159 0.840920i \(-0.317986\pi\)
0.541159 + 0.840920i \(0.317986\pi\)
\(578\) 9.79250e8 0.210934
\(579\) 3.74980e9 0.802847
\(580\) −4.47628e8 −0.0952620
\(581\) 2.65097e9 0.560775
\(582\) −1.37792e9 −0.289729
\(583\) −1.88792e9 −0.394589
\(584\) 1.38123e9 0.286959
\(585\) −6.79623e8 −0.140353
\(586\) −2.93231e9 −0.601962
\(587\) 1.44381e8 0.0294631 0.0147315 0.999891i \(-0.495311\pi\)
0.0147315 + 0.999891i \(0.495311\pi\)
\(588\) −1.18086e9 −0.239540
\(589\) 1.00930e9 0.203525
\(590\) 1.43102e8 0.0286856
\(591\) −2.81696e9 −0.561338
\(592\) −2.34903e9 −0.465331
\(593\) −1.34802e9 −0.265463 −0.132732 0.991152i \(-0.542375\pi\)
−0.132732 + 0.991152i \(0.542375\pi\)
\(594\) 9.32460e8 0.182548
\(595\) 1.08020e9 0.210230
\(596\) 1.24954e9 0.241763
\(597\) −4.22239e9 −0.812173
\(598\) 6.37402e9 1.21888
\(599\) 6.25437e9 1.18902 0.594511 0.804088i \(-0.297346\pi\)
0.594511 + 0.804088i \(0.297346\pi\)
\(600\) 2.16000e8 0.0408248
\(601\) −3.97378e9 −0.746695 −0.373348 0.927692i \(-0.621790\pi\)
−0.373348 + 0.927692i \(0.621790\pi\)
\(602\) 3.27581e8 0.0611971
\(603\) 3.20656e8 0.0595564
\(604\) −1.28725e9 −0.237702
\(605\) 1.94747e9 0.357542
\(606\) 1.69720e9 0.309799
\(607\) 9.26887e9 1.68216 0.841078 0.540913i \(-0.181921\pi\)
0.841078 + 0.540913i \(0.181921\pi\)
\(608\) −2.24756e8 −0.0405554
\(609\) 5.65622e8 0.101477
\(610\) 2.66024e9 0.474534
\(611\) −3.08436e9 −0.547042
\(612\) −1.07688e9 −0.189905
\(613\) −3.19069e9 −0.559465 −0.279733 0.960078i \(-0.590246\pi\)
−0.279733 + 0.960078i \(0.590246\pi\)
\(614\) 2.99900e8 0.0522862
\(615\) 8.89262e8 0.154158
\(616\) −1.13515e9 −0.195669
\(617\) 7.46815e9 1.28001 0.640007 0.768369i \(-0.278931\pi\)
0.640007 + 0.768369i \(0.278931\pi\)
\(618\) −2.55124e9 −0.434801
\(619\) −3.35558e9 −0.568657 −0.284329 0.958727i \(-0.591771\pi\)
−0.284329 + 0.958727i \(0.591771\pi\)
\(620\) −1.17720e9 −0.198372
\(621\) −2.10274e9 −0.352342
\(622\) −6.04420e9 −1.00710
\(623\) 1.86324e9 0.308718
\(624\) −8.24810e8 −0.135896
\(625\) 2.44141e8 0.0400000
\(626\) −1.37013e9 −0.223230
\(627\) −1.09666e9 −0.177679
\(628\) 1.92164e9 0.309608
\(629\) 1.32370e10 2.12085
\(630\) −2.72937e8 −0.0434881
\(631\) −5.25921e9 −0.833331 −0.416666 0.909060i \(-0.636801\pi\)
−0.416666 + 0.909060i \(0.636801\pi\)
\(632\) 8.45048e8 0.133159
\(633\) 1.45979e9 0.228759
\(634\) −5.92472e9 −0.923327
\(635\) 1.66356e9 0.257828
\(636\) −5.50908e8 −0.0849140
\(637\) 5.09665e9 0.781263
\(638\) −2.65074e9 −0.404105
\(639\) 4.69357e8 0.0711623
\(640\) 2.62144e8 0.0395285
\(641\) 7.25521e9 1.08804 0.544022 0.839071i \(-0.316901\pi\)
0.544022 + 0.839071i \(0.316901\pi\)
\(642\) −2.69309e9 −0.401678
\(643\) −1.87494e8 −0.0278131 −0.0139066 0.999903i \(-0.504427\pi\)
−0.0139066 + 0.999903i \(0.504427\pi\)
\(644\) 2.55981e9 0.377666
\(645\) −3.69120e8 −0.0541637
\(646\) 1.26652e9 0.184840
\(647\) 1.32861e10 1.92856 0.964281 0.264883i \(-0.0853332\pi\)
0.964281 + 0.264883i \(0.0853332\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) 8.47413e8 0.121685
\(650\) −9.32267e8 −0.133151
\(651\) 1.48751e9 0.211313
\(652\) −1.29413e9 −0.182856
\(653\) 9.78357e9 1.37499 0.687497 0.726187i \(-0.258709\pi\)
0.687497 + 0.726187i \(0.258709\pi\)
\(654\) 2.97349e8 0.0415665
\(655\) −3.22178e8 −0.0447973
\(656\) 1.07923e9 0.149263
\(657\) 1.96663e9 0.270548
\(658\) −1.23868e9 −0.169500
\(659\) −5.72695e9 −0.779515 −0.389758 0.920918i \(-0.627441\pi\)
−0.389758 + 0.920918i \(0.627441\pi\)
\(660\) 1.27909e9 0.173180
\(661\) −2.10644e9 −0.283690 −0.141845 0.989889i \(-0.545303\pi\)
−0.141845 + 0.989889i \(0.545303\pi\)
\(662\) 1.38744e9 0.185871
\(663\) 4.64787e9 0.619379
\(664\) −3.62527e9 −0.480565
\(665\) 3.21001e8 0.0423282
\(666\) −3.34461e9 −0.438719
\(667\) 5.97752e9 0.779975
\(668\) −2.53017e9 −0.328422
\(669\) −1.72979e9 −0.223359
\(670\) 4.39857e8 0.0565002
\(671\) 1.57533e10 2.01299
\(672\) −3.31245e8 −0.0421072
\(673\) 7.08743e9 0.896264 0.448132 0.893967i \(-0.352089\pi\)
0.448132 + 0.893967i \(0.352089\pi\)
\(674\) 1.89798e9 0.238771
\(675\) 3.07547e8 0.0384900
\(676\) −4.55982e8 −0.0567720
\(677\) 9.58227e9 1.18688 0.593442 0.804877i \(-0.297769\pi\)
0.593442 + 0.804877i \(0.297769\pi\)
\(678\) −5.68161e9 −0.700112
\(679\) 2.38838e9 0.292793
\(680\) −1.47720e9 −0.180160
\(681\) −6.03951e9 −0.732802
\(682\) −6.97107e9 −0.841500
\(683\) 3.41360e9 0.409959 0.204980 0.978766i \(-0.434287\pi\)
0.204980 + 0.978766i \(0.434287\pi\)
\(684\) −3.20014e8 −0.0382360
\(685\) 3.94181e9 0.468575
\(686\) 4.51349e9 0.533800
\(687\) −6.27222e8 −0.0738028
\(688\) −4.47975e8 −0.0524438
\(689\) 2.37775e9 0.276948
\(690\) −2.88441e9 −0.334261
\(691\) 1.40284e10 1.61747 0.808734 0.588175i \(-0.200153\pi\)
0.808734 + 0.588175i \(0.200153\pi\)
\(692\) −4.80085e9 −0.550741
\(693\) −1.61626e9 −0.184478
\(694\) 7.68549e9 0.872797
\(695\) 4.54923e9 0.514033
\(696\) −7.73502e8 −0.0869619
\(697\) −6.08157e9 −0.680301
\(698\) −8.42472e9 −0.937695
\(699\) 6.11535e9 0.677254
\(700\) −3.74399e8 −0.0412565
\(701\) 1.41883e9 0.155567 0.0777833 0.996970i \(-0.475216\pi\)
0.0777833 + 0.996970i \(0.475216\pi\)
\(702\) −1.17439e9 −0.128124
\(703\) 3.93359e9 0.427017
\(704\) 1.55235e9 0.167681
\(705\) 1.39575e9 0.150019
\(706\) 6.72418e9 0.719156
\(707\) −2.94181e9 −0.313074
\(708\) 2.47280e8 0.0261863
\(709\) −1.77639e10 −1.87187 −0.935935 0.352173i \(-0.885443\pi\)
−0.935935 + 0.352173i \(0.885443\pi\)
\(710\) 6.43837e8 0.0675105
\(711\) 1.20320e9 0.125544
\(712\) −2.54803e9 −0.264560
\(713\) 1.57201e10 1.62420
\(714\) 1.86659e9 0.191913
\(715\) −5.52064e9 −0.564830
\(716\) 5.22742e9 0.532221
\(717\) 7.05302e9 0.714592
\(718\) 8.35319e9 0.842203
\(719\) −8.42004e9 −0.844818 −0.422409 0.906405i \(-0.638815\pi\)
−0.422409 + 0.906405i \(0.638815\pi\)
\(720\) 3.73248e8 0.0372678
\(721\) 4.42213e9 0.439398
\(722\) 3.76367e8 0.0372161
\(723\) −6.59560e8 −0.0649038
\(724\) 7.00654e9 0.686150
\(725\) −8.74274e8 −0.0852049
\(726\) 3.36523e9 0.326390
\(727\) 5.47118e9 0.528093 0.264047 0.964510i \(-0.414943\pi\)
0.264047 + 0.964510i \(0.414943\pi\)
\(728\) 1.42967e9 0.137333
\(729\) 3.87420e8 0.0370370
\(730\) 2.69771e9 0.256664
\(731\) 2.52437e9 0.239025
\(732\) 4.59690e9 0.433188
\(733\) 1.87447e10 1.75798 0.878991 0.476839i \(-0.158217\pi\)
0.878991 + 0.476839i \(0.158217\pi\)
\(734\) −7.51103e9 −0.701073
\(735\) −2.30637e9 −0.214251
\(736\) −3.50061e9 −0.323647
\(737\) 2.60472e9 0.239676
\(738\) 1.53664e9 0.140727
\(739\) −4.17756e9 −0.380774 −0.190387 0.981709i \(-0.560974\pi\)
−0.190387 + 0.981709i \(0.560974\pi\)
\(740\) −4.58795e9 −0.416205
\(741\) 1.38119e9 0.124707
\(742\) 9.54906e8 0.0858118
\(743\) −1.13969e10 −1.01936 −0.509678 0.860365i \(-0.670235\pi\)
−0.509678 + 0.860365i \(0.670235\pi\)
\(744\) −2.03420e9 −0.181088
\(745\) 2.44052e9 0.216239
\(746\) −4.91872e9 −0.433777
\(747\) −5.16176e9 −0.453081
\(748\) −8.74759e9 −0.764246
\(749\) 4.66801e9 0.405925
\(750\) 4.21875e8 0.0365148
\(751\) −1.16463e10 −1.00334 −0.501671 0.865059i \(-0.667281\pi\)
−0.501671 + 0.865059i \(0.667281\pi\)
\(752\) 1.69393e9 0.145255
\(753\) 2.90674e9 0.248099
\(754\) 3.33847e9 0.283627
\(755\) −2.51415e9 −0.212607
\(756\) −4.71635e8 −0.0396990
\(757\) −8.42845e9 −0.706175 −0.353087 0.935590i \(-0.614868\pi\)
−0.353087 + 0.935590i \(0.614868\pi\)
\(758\) 1.13691e10 0.948161
\(759\) −1.70807e10 −1.41795
\(760\) −4.38976e8 −0.0362738
\(761\) 1.19543e10 0.983284 0.491642 0.870797i \(-0.336397\pi\)
0.491642 + 0.870797i \(0.336397\pi\)
\(762\) 2.87463e9 0.235364
\(763\) −5.15403e8 −0.0420060
\(764\) −1.04631e10 −0.848856
\(765\) −2.10328e9 −0.169857
\(766\) 6.92952e9 0.557060
\(767\) −1.06728e9 −0.0854068
\(768\) 4.52985e8 0.0360844
\(769\) −3.72184e9 −0.295132 −0.147566 0.989052i \(-0.547144\pi\)
−0.147566 + 0.989052i \(0.547144\pi\)
\(770\) −2.21709e9 −0.175011
\(771\) 8.18554e8 0.0643217
\(772\) 8.88841e9 0.695286
\(773\) 1.32038e10 1.02818 0.514091 0.857736i \(-0.328129\pi\)
0.514091 + 0.857736i \(0.328129\pi\)
\(774\) −6.37839e8 −0.0494445
\(775\) −2.29922e9 −0.177429
\(776\) −3.26617e9 −0.250913
\(777\) 5.79732e9 0.443357
\(778\) −3.34799e9 −0.254892
\(779\) −1.80724e9 −0.136973
\(780\) −1.61096e9 −0.121549
\(781\) 3.81263e9 0.286382
\(782\) 1.97262e10 1.47509
\(783\) −1.10133e9 −0.0819885
\(784\) −2.79908e9 −0.207448
\(785\) 3.75320e9 0.276922
\(786\) −5.56724e8 −0.0408941
\(787\) −1.29308e10 −0.945613 −0.472806 0.881166i \(-0.656759\pi\)
−0.472806 + 0.881166i \(0.656759\pi\)
\(788\) −6.67723e9 −0.486133
\(789\) −8.78001e9 −0.636393
\(790\) 1.65048e9 0.119101
\(791\) 9.84811e9 0.707514
\(792\) 2.21028e9 0.158091
\(793\) −1.98405e10 −1.41285
\(794\) −9.20949e9 −0.652926
\(795\) −1.07599e9 −0.0759494
\(796\) −1.00086e10 −0.703362
\(797\) −1.46124e10 −1.02239 −0.511196 0.859464i \(-0.670798\pi\)
−0.511196 + 0.859464i \(0.670798\pi\)
\(798\) 5.54689e8 0.0386402
\(799\) −9.54540e9 −0.662035
\(800\) 5.12000e8 0.0353553
\(801\) −3.62796e9 −0.249430
\(802\) 8.72796e9 0.597451
\(803\) 1.59751e10 1.08878
\(804\) 7.60073e8 0.0515774
\(805\) 4.99964e9 0.337795
\(806\) 8.77972e9 0.590620
\(807\) −1.28158e9 −0.0858400
\(808\) 4.02300e9 0.268294
\(809\) 1.37916e9 0.0915787 0.0457894 0.998951i \(-0.485420\pi\)
0.0457894 + 0.998951i \(0.485420\pi\)
\(810\) 5.31441e8 0.0351364
\(811\) 2.16079e10 1.42246 0.711229 0.702961i \(-0.248139\pi\)
0.711229 + 0.702961i \(0.248139\pi\)
\(812\) 1.34073e9 0.0878813
\(813\) 2.21479e9 0.144549
\(814\) −2.71686e10 −1.76556
\(815\) −2.52759e9 −0.163552
\(816\) −2.55260e9 −0.164463
\(817\) 7.50161e8 0.0481257
\(818\) −9.12964e9 −0.583200
\(819\) 2.03560e9 0.129479
\(820\) 2.10788e9 0.133505
\(821\) 9.12023e9 0.575181 0.287591 0.957753i \(-0.407146\pi\)
0.287591 + 0.957753i \(0.407146\pi\)
\(822\) 6.81145e9 0.427748
\(823\) −9.66665e9 −0.604473 −0.302236 0.953233i \(-0.597733\pi\)
−0.302236 + 0.953233i \(0.597733\pi\)
\(824\) −6.04737e9 −0.376549
\(825\) 2.49823e9 0.154897
\(826\) −4.28619e8 −0.0264631
\(827\) 2.43768e10 1.49867 0.749337 0.662189i \(-0.230372\pi\)
0.749337 + 0.662189i \(0.230372\pi\)
\(828\) −4.98426e9 −0.305137
\(829\) 2.03825e10 1.24256 0.621280 0.783588i \(-0.286613\pi\)
0.621280 + 0.783588i \(0.286613\pi\)
\(830\) −7.08061e9 −0.429830
\(831\) 1.08970e10 0.658723
\(832\) −1.95511e9 −0.117690
\(833\) 1.57730e10 0.945490
\(834\) 7.86107e9 0.469246
\(835\) −4.94173e9 −0.293749
\(836\) −2.59950e9 −0.153875
\(837\) −2.89636e9 −0.170731
\(838\) −1.08360e10 −0.636087
\(839\) −1.69741e10 −0.992249 −0.496124 0.868251i \(-0.665244\pi\)
−0.496124 + 0.868251i \(0.665244\pi\)
\(840\) −6.46962e8 −0.0376618
\(841\) −1.41191e10 −0.818503
\(842\) −1.50126e9 −0.0866688
\(843\) −7.57796e9 −0.435668
\(844\) 3.46024e9 0.198111
\(845\) −8.90590e8 −0.0507784
\(846\) 2.41186e9 0.136948
\(847\) −5.83305e9 −0.329841
\(848\) −1.30586e9 −0.0735377
\(849\) −7.90530e9 −0.443344
\(850\) −2.88516e9 −0.161140
\(851\) 6.12663e10 3.40776
\(852\) 1.11255e9 0.0616284
\(853\) −1.83567e10 −1.01268 −0.506341 0.862334i \(-0.669002\pi\)
−0.506341 + 0.862334i \(0.669002\pi\)
\(854\) −7.96795e9 −0.437768
\(855\) −6.25026e8 −0.0341993
\(856\) −6.38362e9 −0.347863
\(857\) −5.02108e9 −0.272499 −0.136249 0.990675i \(-0.543505\pi\)
−0.136249 + 0.990675i \(0.543505\pi\)
\(858\) −9.53966e9 −0.515617
\(859\) 1.98706e10 1.06963 0.534817 0.844968i \(-0.320381\pi\)
0.534817 + 0.844968i \(0.320381\pi\)
\(860\) −8.74951e8 −0.0469072
\(861\) −2.66351e9 −0.142214
\(862\) −1.72259e10 −0.916024
\(863\) −5.01834e9 −0.265780 −0.132890 0.991131i \(-0.542426\pi\)
−0.132890 + 0.991131i \(0.542426\pi\)
\(864\) 6.44973e8 0.0340207
\(865\) −9.37667e9 −0.492598
\(866\) −9.16164e9 −0.479359
\(867\) 3.30497e9 0.172227
\(868\) 3.52595e9 0.183002
\(869\) 9.77373e9 0.505232
\(870\) −1.51075e9 −0.0777811
\(871\) −3.28052e9 −0.168220
\(872\) 7.04826e8 0.0359977
\(873\) −4.65047e9 −0.236563
\(874\) 5.86198e9 0.296998
\(875\) −7.31249e8 −0.0369009
\(876\) 4.66165e9 0.234301
\(877\) 1.63182e10 0.816908 0.408454 0.912779i \(-0.366068\pi\)
0.408454 + 0.912779i \(0.366068\pi\)
\(878\) 1.09677e10 0.546872
\(879\) −9.89656e9 −0.491500
\(880\) 3.03193e9 0.149979
\(881\) 3.03510e10 1.49540 0.747700 0.664037i \(-0.231158\pi\)
0.747700 + 0.664037i \(0.231158\pi\)
\(882\) −3.98540e9 −0.195583
\(883\) −1.64109e10 −0.802175 −0.401088 0.916040i \(-0.631368\pi\)
−0.401088 + 0.916040i \(0.631368\pi\)
\(884\) 1.10172e10 0.536398
\(885\) 4.82970e8 0.0234217
\(886\) 6.30497e8 0.0304555
\(887\) 1.98008e10 0.952684 0.476342 0.879260i \(-0.341962\pi\)
0.476342 + 0.879260i \(0.341962\pi\)
\(888\) −7.92797e9 −0.379942
\(889\) −4.98268e9 −0.237852
\(890\) −4.97662e9 −0.236630
\(891\) 3.14705e9 0.149050
\(892\) −4.10025e9 −0.193434
\(893\) −2.83658e9 −0.133295
\(894\) 4.21721e9 0.197399
\(895\) 1.02098e10 0.476033
\(896\) −7.85172e8 −0.0364659
\(897\) 2.15123e10 0.995208
\(898\) 1.08862e10 0.501661
\(899\) 8.23357e9 0.377946
\(900\) 7.29000e8 0.0333333
\(901\) 7.35860e9 0.335165
\(902\) 1.24823e10 0.566333
\(903\) 1.10559e9 0.0499672
\(904\) −1.34675e10 −0.606315
\(905\) 1.36847e10 0.613711
\(906\) −4.34445e9 −0.194083
\(907\) −1.36134e10 −0.605817 −0.302909 0.953020i \(-0.597958\pi\)
−0.302909 + 0.953020i \(0.597958\pi\)
\(908\) −1.43159e10 −0.634625
\(909\) 5.72806e9 0.252950
\(910\) 2.79232e9 0.122835
\(911\) −2.91953e10 −1.27938 −0.639690 0.768633i \(-0.720937\pi\)
−0.639690 + 0.768633i \(0.720937\pi\)
\(912\) −7.58551e8 −0.0331133
\(913\) −4.19295e10 −1.82336
\(914\) 1.32204e9 0.0572708
\(915\) 8.97832e9 0.387455
\(916\) −1.48675e9 −0.0639151
\(917\) 9.64987e8 0.0413265
\(918\) −3.63447e9 −0.155057
\(919\) −2.16107e10 −0.918468 −0.459234 0.888315i \(-0.651876\pi\)
−0.459234 + 0.888315i \(0.651876\pi\)
\(920\) −6.83712e9 −0.289478
\(921\) 1.01216e9 0.0426915
\(922\) −4.68802e9 −0.196984
\(923\) −4.80182e9 −0.201002
\(924\) −3.83114e9 −0.159763
\(925\) −8.96084e9 −0.372265
\(926\) 2.23065e8 0.00923192
\(927\) −8.61042e9 −0.355014
\(928\) −1.83349e9 −0.0753112
\(929\) 1.51410e10 0.619584 0.309792 0.950804i \(-0.399741\pi\)
0.309792 + 0.950804i \(0.399741\pi\)
\(930\) −3.97305e9 −0.161970
\(931\) 4.68722e9 0.190367
\(932\) 1.44957e10 0.586519
\(933\) −2.03992e10 −0.822294
\(934\) 2.54477e10 1.02196
\(935\) −1.70851e10 −0.683562
\(936\) −2.78373e9 −0.110959
\(937\) 3.32582e9 0.132072 0.0660360 0.997817i \(-0.478965\pi\)
0.0660360 + 0.997817i \(0.478965\pi\)
\(938\) −1.31746e9 −0.0521227
\(939\) −4.62421e9 −0.182267
\(940\) 3.30845e9 0.129920
\(941\) −3.79162e10 −1.48341 −0.741704 0.670727i \(-0.765982\pi\)
−0.741704 + 0.670727i \(0.765982\pi\)
\(942\) 6.48552e9 0.252794
\(943\) −2.81481e10 −1.09310
\(944\) 5.86146e8 0.0226780
\(945\) −9.21163e8 −0.0355079
\(946\) −5.18123e9 −0.198982
\(947\) −6.77198e9 −0.259114 −0.129557 0.991572i \(-0.541355\pi\)
−0.129557 + 0.991572i \(0.541355\pi\)
\(948\) 2.85204e9 0.108724
\(949\) −2.01199e10 −0.764177
\(950\) −8.57375e8 −0.0324443
\(951\) −1.99959e10 −0.753893
\(952\) 4.42450e9 0.166202
\(953\) −1.60541e10 −0.600844 −0.300422 0.953806i \(-0.597128\pi\)
−0.300422 + 0.953806i \(0.597128\pi\)
\(954\) −1.85932e9 −0.0693320
\(955\) −2.04358e10 −0.759240
\(956\) 1.67183e10 0.618855
\(957\) −8.94623e9 −0.329950
\(958\) −2.04471e10 −0.751366
\(959\) −1.18065e10 −0.432270
\(960\) 8.84736e8 0.0322749
\(961\) −5.85946e9 −0.212974
\(962\) 3.42175e10 1.23918
\(963\) −9.08918e9 −0.327969
\(964\) −1.56340e9 −0.0562084
\(965\) 1.73602e10 0.621883
\(966\) 8.63937e9 0.308363
\(967\) −1.57689e10 −0.560802 −0.280401 0.959883i \(-0.590467\pi\)
−0.280401 + 0.959883i \(0.590467\pi\)
\(968\) 7.97684e9 0.282662
\(969\) 4.27449e9 0.150921
\(970\) −6.37924e9 −0.224423
\(971\) 6.26780e9 0.219709 0.109854 0.993948i \(-0.464962\pi\)
0.109854 + 0.993948i \(0.464962\pi\)
\(972\) 9.18330e8 0.0320750
\(973\) −1.36258e10 −0.474207
\(974\) 1.47159e10 0.510305
\(975\) −3.14640e9 −0.108717
\(976\) 1.08964e10 0.375152
\(977\) 1.62372e10 0.557031 0.278515 0.960432i \(-0.410158\pi\)
0.278515 + 0.960432i \(0.410158\pi\)
\(978\) −4.36767e9 −0.149302
\(979\) −2.94702e10 −1.00379
\(980\) −5.46695e9 −0.185547
\(981\) 1.00355e9 0.0339389
\(982\) −1.22374e10 −0.412381
\(983\) 4.51810e10 1.51711 0.758557 0.651607i \(-0.225905\pi\)
0.758557 + 0.651607i \(0.225905\pi\)
\(984\) 3.64242e9 0.121873
\(985\) −1.30415e10 −0.434810
\(986\) 1.03318e10 0.343248
\(987\) −4.18055e9 −0.138396
\(988\) 3.27394e9 0.108000
\(989\) 1.16839e10 0.384061
\(990\) 4.31694e9 0.141401
\(991\) 3.68239e10 1.20191 0.600954 0.799283i \(-0.294787\pi\)
0.600954 + 0.799283i \(0.294787\pi\)
\(992\) −4.82181e9 −0.156827
\(993\) 4.68260e9 0.151763
\(994\) −1.92842e9 −0.0622800
\(995\) −1.95481e10 −0.629106
\(996\) −1.22353e10 −0.392380
\(997\) 6.39507e9 0.204368 0.102184 0.994766i \(-0.467417\pi\)
0.102184 + 0.994766i \(0.467417\pi\)
\(998\) −6.75650e8 −0.0215162
\(999\) −1.12881e10 −0.358212
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 570.8.a.c.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.8.a.c.1.2 4 1.1 even 1 trivial