Properties

Label 546.8.a.l.1.3
Level $546$
Weight $8$
Character 546.1
Self dual yes
Analytic conductor $170.562$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 546.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(170.562223914\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 2 x^{4} - 122890 x^{3} - 6160660 x^{2} + 3465881625 x + 278845474950\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{3}\cdot 5 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(315.005\) of defining polynomial
Character \(\chi\) \(=\) 546.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -17.2693 q^{5} -216.000 q^{6} -343.000 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} -27.0000 q^{3} +64.0000 q^{4} -17.2693 q^{5} -216.000 q^{6} -343.000 q^{7} +512.000 q^{8} +729.000 q^{9} -138.154 q^{10} -6366.67 q^{11} -1728.00 q^{12} -2197.00 q^{13} -2744.00 q^{14} +466.270 q^{15} +4096.00 q^{16} +31491.6 q^{17} +5832.00 q^{18} +46684.6 q^{19} -1105.23 q^{20} +9261.00 q^{21} -50933.4 q^{22} -36298.3 q^{23} -13824.0 q^{24} -77826.8 q^{25} -17576.0 q^{26} -19683.0 q^{27} -21952.0 q^{28} +42877.1 q^{29} +3730.16 q^{30} +96053.5 q^{31} +32768.0 q^{32} +171900. q^{33} +251933. q^{34} +5923.35 q^{35} +46656.0 q^{36} +246253. q^{37} +373477. q^{38} +59319.0 q^{39} -8841.86 q^{40} +29230.7 q^{41} +74088.0 q^{42} -290329. q^{43} -407467. q^{44} -12589.3 q^{45} -290386. q^{46} +985829. q^{47} -110592. q^{48} +117649. q^{49} -622614. q^{50} -850273. q^{51} -140608. q^{52} -1.38599e6 q^{53} -157464. q^{54} +109948. q^{55} -175616. q^{56} -1.26048e6 q^{57} +343017. q^{58} -1.70085e6 q^{59} +29841.3 q^{60} -2.60810e6 q^{61} +768428. q^{62} -250047. q^{63} +262144. q^{64} +37940.6 q^{65} +1.37520e6 q^{66} -3.89646e6 q^{67} +2.01546e6 q^{68} +980054. q^{69} +47386.8 q^{70} +1.70688e6 q^{71} +373248. q^{72} +2.08824e6 q^{73} +1.97003e6 q^{74} +2.10132e6 q^{75} +2.98782e6 q^{76} +2.18377e6 q^{77} +474552. q^{78} +4.85807e6 q^{79} -70734.9 q^{80} +531441. q^{81} +233845. q^{82} +1.90272e6 q^{83} +592704. q^{84} -543836. q^{85} -2.32263e6 q^{86} -1.15768e6 q^{87} -3.25973e6 q^{88} -1.67512e6 q^{89} -100714. q^{90} +753571. q^{91} -2.32309e6 q^{92} -2.59344e6 q^{93} +7.88663e6 q^{94} -806208. q^{95} -884736. q^{96} -4.89220e6 q^{97} +941192. q^{98} -4.64130e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 40 q^{2} - 135 q^{3} + 320 q^{4} - 250 q^{5} - 1080 q^{6} - 1715 q^{7} + 2560 q^{8} + 3645 q^{9} + O(q^{10}) \) \( 5 q + 40 q^{2} - 135 q^{3} + 320 q^{4} - 250 q^{5} - 1080 q^{6} - 1715 q^{7} + 2560 q^{8} + 3645 q^{9} - 2000 q^{10} + 659 q^{11} - 8640 q^{12} - 10985 q^{13} - 13720 q^{14} + 6750 q^{15} + 20480 q^{16} + 24575 q^{17} + 29160 q^{18} - 6446 q^{19} - 16000 q^{20} + 46305 q^{21} + 5272 q^{22} + 30268 q^{23} - 69120 q^{24} + 38965 q^{25} - 87880 q^{26} - 98415 q^{27} - 109760 q^{28} + 130950 q^{29} + 54000 q^{30} + 262979 q^{31} + 163840 q^{32} - 17793 q^{33} + 196600 q^{34} + 85750 q^{35} + 233280 q^{36} - 101549 q^{37} - 51568 q^{38} + 296595 q^{39} - 128000 q^{40} - 247328 q^{41} + 370440 q^{42} - 19092 q^{43} + 42176 q^{44} - 182250 q^{45} + 242144 q^{46} - 126419 q^{47} - 552960 q^{48} + 588245 q^{49} + 311720 q^{50} - 663525 q^{51} - 703040 q^{52} - 302793 q^{53} - 787320 q^{54} + 943985 q^{55} - 878080 q^{56} + 174042 q^{57} + 1047600 q^{58} - 2798636 q^{59} + 432000 q^{60} - 2493751 q^{61} + 2103832 q^{62} - 1250235 q^{63} + 1310720 q^{64} + 549250 q^{65} - 142344 q^{66} + 160188 q^{67} + 1572800 q^{68} - 817236 q^{69} + 686000 q^{70} + 3846088 q^{71} + 1866240 q^{72} + 5655872 q^{73} - 812392 q^{74} - 1052055 q^{75} - 412544 q^{76} - 226037 q^{77} + 2372760 q^{78} + 5647991 q^{79} - 1024000 q^{80} + 2657205 q^{81} - 1978624 q^{82} - 4607669 q^{83} + 2963520 q^{84} + 3873935 q^{85} - 152736 q^{86} - 3535650 q^{87} + 337408 q^{88} - 17424029 q^{89} - 1458000 q^{90} + 3767855 q^{91} + 1937152 q^{92} - 7100433 q^{93} - 1011352 q^{94} - 24593720 q^{95} - 4423680 q^{96} - 18380577 q^{97} + 4705960 q^{98} + 480411 q^{99} + O(q^{100}) \)

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
\(5\) −17.2693 −0.0617844 −0.0308922 0.999523i \(-0.509835\pi\)
−0.0308922 + 0.999523i \(0.509835\pi\)
\(6\) −216.000 −0.408248
\(7\) −343.000 −0.377964
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) −138.154 −0.0436881
\(11\) −6366.67 −1.44224 −0.721121 0.692810i \(-0.756373\pi\)
−0.721121 + 0.692810i \(0.756373\pi\)
\(12\) −1728.00 −0.288675
\(13\) −2197.00 −0.277350
\(14\) −2744.00 −0.267261
\(15\) 466.270 0.0356712
\(16\) 4096.00 0.250000
\(17\) 31491.6 1.55462 0.777308 0.629120i \(-0.216585\pi\)
0.777308 + 0.629120i \(0.216585\pi\)
\(18\) 5832.00 0.235702
\(19\) 46684.6 1.56148 0.780740 0.624857i \(-0.214843\pi\)
0.780740 + 0.624857i \(0.214843\pi\)
\(20\) −1105.23 −0.0308922
\(21\) 9261.00 0.218218
\(22\) −50933.4 −1.01982
\(23\) −36298.3 −0.622069 −0.311035 0.950399i \(-0.600676\pi\)
−0.311035 + 0.950399i \(0.600676\pi\)
\(24\) −13824.0 −0.204124
\(25\) −77826.8 −0.996183
\(26\) −17576.0 −0.196116
\(27\) −19683.0 −0.192450
\(28\) −21952.0 −0.188982
\(29\) 42877.1 0.326462 0.163231 0.986588i \(-0.447808\pi\)
0.163231 + 0.986588i \(0.447808\pi\)
\(30\) 3730.16 0.0252234
\(31\) 96053.5 0.579091 0.289546 0.957164i \(-0.406496\pi\)
0.289546 + 0.957164i \(0.406496\pi\)
\(32\) 32768.0 0.176777
\(33\) 171900. 0.832678
\(34\) 251933. 1.09928
\(35\) 5923.35 0.0233523
\(36\) 46656.0 0.166667
\(37\) 246253. 0.799238 0.399619 0.916681i \(-0.369143\pi\)
0.399619 + 0.916681i \(0.369143\pi\)
\(38\) 373477. 1.10413
\(39\) 59319.0 0.160128
\(40\) −8841.86 −0.0218441
\(41\) 29230.7 0.0662362 0.0331181 0.999451i \(-0.489456\pi\)
0.0331181 + 0.999451i \(0.489456\pi\)
\(42\) 74088.0 0.154303
\(43\) −290329. −0.556866 −0.278433 0.960456i \(-0.589815\pi\)
−0.278433 + 0.960456i \(0.589815\pi\)
\(44\) −407467. −0.721121
\(45\) −12589.3 −0.0205948
\(46\) −290386. −0.439869
\(47\) 985829. 1.38503 0.692515 0.721404i \(-0.256503\pi\)
0.692515 + 0.721404i \(0.256503\pi\)
\(48\) −110592. −0.144338
\(49\) 117649. 0.142857
\(50\) −622614. −0.704408
\(51\) −850273. −0.897558
\(52\) −140608. −0.138675
\(53\) −1.38599e6 −1.27877 −0.639387 0.768885i \(-0.720812\pi\)
−0.639387 + 0.768885i \(0.720812\pi\)
\(54\) −157464. −0.136083
\(55\) 109948. 0.0891080
\(56\) −175616. −0.133631
\(57\) −1.26048e6 −0.901520
\(58\) 343017. 0.230843
\(59\) −1.70085e6 −1.07816 −0.539080 0.842254i \(-0.681228\pi\)
−0.539080 + 0.842254i \(0.681228\pi\)
\(60\) 29841.3 0.0178356
\(61\) −2.60810e6 −1.47119 −0.735596 0.677421i \(-0.763098\pi\)
−0.735596 + 0.677421i \(0.763098\pi\)
\(62\) 768428. 0.409479
\(63\) −250047. −0.125988
\(64\) 262144. 0.125000
\(65\) 37940.6 0.0171359
\(66\) 1.37520e6 0.588792
\(67\) −3.89646e6 −1.58274 −0.791368 0.611340i \(-0.790631\pi\)
−0.791368 + 0.611340i \(0.790631\pi\)
\(68\) 2.01546e6 0.777308
\(69\) 980054. 0.359152
\(70\) 47386.8 0.0165126
\(71\) 1.70688e6 0.565978 0.282989 0.959123i \(-0.408674\pi\)
0.282989 + 0.959123i \(0.408674\pi\)
\(72\) 373248. 0.117851
\(73\) 2.08824e6 0.628275 0.314138 0.949377i \(-0.398285\pi\)
0.314138 + 0.949377i \(0.398285\pi\)
\(74\) 1.97003e6 0.565146
\(75\) 2.10132e6 0.575146
\(76\) 2.98782e6 0.780740
\(77\) 2.18377e6 0.545116
\(78\) 474552. 0.113228
\(79\) 4.85807e6 1.10859 0.554293 0.832322i \(-0.312989\pi\)
0.554293 + 0.832322i \(0.312989\pi\)
\(80\) −70734.9 −0.0154461
\(81\) 531441. 0.111111
\(82\) 233845. 0.0468360
\(83\) 1.90272e6 0.365259 0.182630 0.983182i \(-0.441539\pi\)
0.182630 + 0.983182i \(0.441539\pi\)
\(84\) 592704. 0.109109
\(85\) −543836. −0.0960510
\(86\) −2.32263e6 −0.393764
\(87\) −1.15768e6 −0.188483
\(88\) −3.25973e6 −0.509909
\(89\) −1.67512e6 −0.251873 −0.125936 0.992038i \(-0.540193\pi\)
−0.125936 + 0.992038i \(0.540193\pi\)
\(90\) −100714. −0.0145627
\(91\) 753571. 0.104828
\(92\) −2.32309e6 −0.311035
\(93\) −2.59344e6 −0.334339
\(94\) 7.88663e6 0.979364
\(95\) −806208. −0.0964750
\(96\) −884736. −0.102062
\(97\) −4.89220e6 −0.544255 −0.272128 0.962261i \(-0.587727\pi\)
−0.272128 + 0.962261i \(0.587727\pi\)
\(98\) 941192. 0.101015
\(99\) −4.64130e6 −0.480747
\(100\) −4.98091e6 −0.498091
\(101\) −2.64383e6 −0.255334 −0.127667 0.991817i \(-0.540749\pi\)
−0.127667 + 0.991817i \(0.540749\pi\)
\(102\) −6.80218e6 −0.634669
\(103\) −2.56466e6 −0.231260 −0.115630 0.993292i \(-0.536889\pi\)
−0.115630 + 0.993292i \(0.536889\pi\)
\(104\) −1.12486e6 −0.0980581
\(105\) −159931. −0.0134825
\(106\) −1.10879e7 −0.904229
\(107\) 7.33860e6 0.579122 0.289561 0.957160i \(-0.406491\pi\)
0.289561 + 0.957160i \(0.406491\pi\)
\(108\) −1.25971e6 −0.0962250
\(109\) −2.02890e7 −1.50061 −0.750306 0.661091i \(-0.770094\pi\)
−0.750306 + 0.661091i \(0.770094\pi\)
\(110\) 879581. 0.0630088
\(111\) −6.64884e6 −0.461440
\(112\) −1.40493e6 −0.0944911
\(113\) 1.91208e6 0.124662 0.0623308 0.998056i \(-0.480147\pi\)
0.0623308 + 0.998056i \(0.480147\pi\)
\(114\) −1.00839e7 −0.637471
\(115\) 626845. 0.0384342
\(116\) 2.74413e6 0.163231
\(117\) −1.60161e6 −0.0924500
\(118\) −1.36068e7 −0.762375
\(119\) −1.08016e7 −0.587590
\(120\) 238730. 0.0126117
\(121\) 2.10473e7 1.08006
\(122\) −2.08648e7 −1.04029
\(123\) −789228. −0.0382415
\(124\) 6.14742e6 0.289546
\(125\) 2.69317e6 0.123333
\(126\) −2.00038e6 −0.0890871
\(127\) 1.90374e7 0.824699 0.412349 0.911026i \(-0.364708\pi\)
0.412349 + 0.911026i \(0.364708\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) 7.83888e6 0.321507
\(130\) 303524. 0.0121169
\(131\) −2.24107e7 −0.870974 −0.435487 0.900195i \(-0.643424\pi\)
−0.435487 + 0.900195i \(0.643424\pi\)
\(132\) 1.10016e7 0.416339
\(133\) −1.60128e7 −0.590184
\(134\) −3.11717e7 −1.11916
\(135\) 339911. 0.0118904
\(136\) 1.61237e7 0.549640
\(137\) −2.33530e7 −0.775926 −0.387963 0.921675i \(-0.626821\pi\)
−0.387963 + 0.921675i \(0.626821\pi\)
\(138\) 7.84043e6 0.253959
\(139\) 3.49584e7 1.10408 0.552039 0.833819i \(-0.313850\pi\)
0.552039 + 0.833819i \(0.313850\pi\)
\(140\) 379095. 0.0116761
\(141\) −2.66174e7 −0.799647
\(142\) 1.36551e7 0.400207
\(143\) 1.39876e7 0.400006
\(144\) 2.98598e6 0.0833333
\(145\) −740455. −0.0201702
\(146\) 1.67059e7 0.444258
\(147\) −3.17652e6 −0.0824786
\(148\) 1.57602e7 0.399619
\(149\) −2.53533e6 −0.0627888 −0.0313944 0.999507i \(-0.509995\pi\)
−0.0313944 + 0.999507i \(0.509995\pi\)
\(150\) 1.68106e7 0.406690
\(151\) −7.46151e7 −1.76363 −0.881815 0.471595i \(-0.843678\pi\)
−0.881815 + 0.471595i \(0.843678\pi\)
\(152\) 2.39025e7 0.552066
\(153\) 2.29574e7 0.518205
\(154\) 1.74701e7 0.385455
\(155\) −1.65877e6 −0.0357788
\(156\) 3.79642e6 0.0800641
\(157\) −5.00343e7 −1.03186 −0.515928 0.856632i \(-0.672553\pi\)
−0.515928 + 0.856632i \(0.672553\pi\)
\(158\) 3.88646e7 0.783888
\(159\) 3.74216e7 0.738300
\(160\) −565879. −0.0109220
\(161\) 1.24503e7 0.235120
\(162\) 4.25153e6 0.0785674
\(163\) −4.72558e7 −0.854671 −0.427335 0.904093i \(-0.640548\pi\)
−0.427335 + 0.904093i \(0.640548\pi\)
\(164\) 1.87076e6 0.0331181
\(165\) −2.96859e6 −0.0514465
\(166\) 1.52217e7 0.258277
\(167\) −1.04290e8 −1.73275 −0.866375 0.499394i \(-0.833556\pi\)
−0.866375 + 0.499394i \(0.833556\pi\)
\(168\) 4.74163e6 0.0771517
\(169\) 4.82681e6 0.0769231
\(170\) −4.35069e6 −0.0679183
\(171\) 3.40331e7 0.520493
\(172\) −1.85811e7 −0.278433
\(173\) −1.17020e8 −1.71830 −0.859148 0.511728i \(-0.829006\pi\)
−0.859148 + 0.511728i \(0.829006\pi\)
\(174\) −9.26145e6 −0.133277
\(175\) 2.66946e7 0.376522
\(176\) −2.60779e7 −0.360560
\(177\) 4.59229e7 0.622476
\(178\) −1.34010e7 −0.178101
\(179\) −2.74389e7 −0.357586 −0.178793 0.983887i \(-0.557219\pi\)
−0.178793 + 0.983887i \(0.557219\pi\)
\(180\) −805714. −0.0102974
\(181\) −1.24054e8 −1.55502 −0.777509 0.628872i \(-0.783517\pi\)
−0.777509 + 0.628872i \(0.783517\pi\)
\(182\) 6.02857e6 0.0741249
\(183\) 7.04186e7 0.849393
\(184\) −1.85847e7 −0.219935
\(185\) −4.25261e6 −0.0493804
\(186\) −2.07475e7 −0.236413
\(187\) −2.00496e8 −2.24213
\(188\) 6.30931e7 0.692515
\(189\) 6.75127e6 0.0727393
\(190\) −6.44967e6 −0.0682181
\(191\) 1.77095e8 1.83903 0.919517 0.393050i \(-0.128580\pi\)
0.919517 + 0.393050i \(0.128580\pi\)
\(192\) −7.07789e6 −0.0721688
\(193\) 2.29976e7 0.230267 0.115134 0.993350i \(-0.463270\pi\)
0.115134 + 0.993350i \(0.463270\pi\)
\(194\) −3.91376e7 −0.384847
\(195\) −1.02439e6 −0.00989342
\(196\) 7.52954e6 0.0714286
\(197\) −1.11687e8 −1.04081 −0.520405 0.853919i \(-0.674219\pi\)
−0.520405 + 0.853919i \(0.674219\pi\)
\(198\) −3.71304e7 −0.339940
\(199\) 4.87890e7 0.438870 0.219435 0.975627i \(-0.429579\pi\)
0.219435 + 0.975627i \(0.429579\pi\)
\(200\) −3.98473e7 −0.352204
\(201\) 1.05204e8 0.913793
\(202\) −2.11506e7 −0.180548
\(203\) −1.47068e7 −0.123391
\(204\) −5.44174e7 −0.448779
\(205\) −504792. −0.00409236
\(206\) −2.05173e7 −0.163525
\(207\) −2.64615e7 −0.207356
\(208\) −8.99891e6 −0.0693375
\(209\) −2.97225e8 −2.25203
\(210\) −1.27944e6 −0.00953353
\(211\) 1.78179e8 1.30577 0.652885 0.757457i \(-0.273558\pi\)
0.652885 + 0.757457i \(0.273558\pi\)
\(212\) −8.87031e7 −0.639387
\(213\) −4.60858e7 −0.326767
\(214\) 5.87088e7 0.409501
\(215\) 5.01376e6 0.0344056
\(216\) −1.00777e7 −0.0680414
\(217\) −3.29463e7 −0.218876
\(218\) −1.62312e8 −1.06109
\(219\) −5.63824e7 −0.362735
\(220\) 7.03665e6 0.0445540
\(221\) −6.91870e7 −0.431173
\(222\) −5.31907e7 −0.326287
\(223\) −1.42968e8 −0.863317 −0.431659 0.902037i \(-0.642071\pi\)
−0.431659 + 0.902037i \(0.642071\pi\)
\(224\) −1.12394e7 −0.0668153
\(225\) −5.67357e7 −0.332061
\(226\) 1.52967e7 0.0881490
\(227\) 3.04658e8 1.72871 0.864354 0.502884i \(-0.167728\pi\)
0.864354 + 0.502884i \(0.167728\pi\)
\(228\) −8.06710e7 −0.450760
\(229\) −2.13410e8 −1.17433 −0.587165 0.809467i \(-0.699756\pi\)
−0.587165 + 0.809467i \(0.699756\pi\)
\(230\) 5.01476e6 0.0271771
\(231\) −5.89617e7 −0.314723
\(232\) 2.19531e7 0.115422
\(233\) −6.43656e7 −0.333356 −0.166678 0.986011i \(-0.553304\pi\)
−0.166678 + 0.986011i \(0.553304\pi\)
\(234\) −1.28129e7 −0.0653720
\(235\) −1.70245e7 −0.0855732
\(236\) −1.08854e8 −0.539080
\(237\) −1.31168e8 −0.640042
\(238\) −8.64129e7 −0.415489
\(239\) −6.62558e7 −0.313929 −0.156964 0.987604i \(-0.550171\pi\)
−0.156964 + 0.987604i \(0.550171\pi\)
\(240\) 1.90984e6 0.00891781
\(241\) −2.98304e8 −1.37278 −0.686389 0.727235i \(-0.740805\pi\)
−0.686389 + 0.727235i \(0.740805\pi\)
\(242\) 1.68378e8 0.763717
\(243\) −1.43489e7 −0.0641500
\(244\) −1.66918e8 −0.735596
\(245\) −2.03171e6 −0.00882634
\(246\) −6.31382e6 −0.0270408
\(247\) −1.02566e8 −0.433076
\(248\) 4.91794e7 0.204740
\(249\) −5.13734e7 −0.210883
\(250\) 2.15454e7 0.0872095
\(251\) −3.10746e8 −1.24036 −0.620180 0.784460i \(-0.712940\pi\)
−0.620180 + 0.784460i \(0.712940\pi\)
\(252\) −1.60030e7 −0.0629941
\(253\) 2.31099e8 0.897174
\(254\) 1.52299e8 0.583150
\(255\) 1.46836e7 0.0554551
\(256\) 1.67772e7 0.0625000
\(257\) 5.93193e6 0.0217987 0.0108993 0.999941i \(-0.496531\pi\)
0.0108993 + 0.999941i \(0.496531\pi\)
\(258\) 6.27110e7 0.227340
\(259\) −8.44649e7 −0.302083
\(260\) 2.42820e6 0.00856795
\(261\) 3.12574e7 0.108821
\(262\) −1.79285e8 −0.615872
\(263\) 3.35505e8 1.13724 0.568622 0.822599i \(-0.307477\pi\)
0.568622 + 0.822599i \(0.307477\pi\)
\(264\) 8.80128e7 0.294396
\(265\) 2.39350e7 0.0790082
\(266\) −1.28103e8 −0.417323
\(267\) 4.52282e7 0.145419
\(268\) −2.49374e8 −0.791368
\(269\) −2.67432e7 −0.0837684 −0.0418842 0.999122i \(-0.513336\pi\)
−0.0418842 + 0.999122i \(0.513336\pi\)
\(270\) 2.71929e6 0.00840779
\(271\) −2.99466e8 −0.914018 −0.457009 0.889462i \(-0.651079\pi\)
−0.457009 + 0.889462i \(0.651079\pi\)
\(272\) 1.28989e8 0.388654
\(273\) −2.03464e7 −0.0605228
\(274\) −1.86824e8 −0.548663
\(275\) 4.95497e8 1.43674
\(276\) 6.27235e7 0.179576
\(277\) 2.92186e7 0.0826001 0.0413000 0.999147i \(-0.486850\pi\)
0.0413000 + 0.999147i \(0.486850\pi\)
\(278\) 2.79667e8 0.780701
\(279\) 7.00230e7 0.193030
\(280\) 3.03276e6 0.00825628
\(281\) −7.21369e8 −1.93948 −0.969740 0.244138i \(-0.921495\pi\)
−0.969740 + 0.244138i \(0.921495\pi\)
\(282\) −2.12939e8 −0.565436
\(283\) −6.21761e8 −1.63069 −0.815344 0.578977i \(-0.803452\pi\)
−0.815344 + 0.578977i \(0.803452\pi\)
\(284\) 1.09240e8 0.282989
\(285\) 2.17676e7 0.0556999
\(286\) 1.11901e8 0.282847
\(287\) −1.00261e7 −0.0250349
\(288\) 2.38879e7 0.0589256
\(289\) 5.81381e8 1.41683
\(290\) −5.92364e6 −0.0142625
\(291\) 1.32089e8 0.314226
\(292\) 1.33647e8 0.314138
\(293\) 4.75845e8 1.10517 0.552585 0.833456i \(-0.313641\pi\)
0.552585 + 0.833456i \(0.313641\pi\)
\(294\) −2.54122e7 −0.0583212
\(295\) 2.93724e7 0.0666135
\(296\) 1.26082e8 0.282573
\(297\) 1.25315e8 0.277559
\(298\) −2.02826e7 −0.0443984
\(299\) 7.97474e7 0.172531
\(300\) 1.34485e8 0.287573
\(301\) 9.95828e7 0.210476
\(302\) −5.96921e8 −1.24708
\(303\) 7.13833e7 0.147417
\(304\) 1.91220e8 0.390370
\(305\) 4.50399e7 0.0908966
\(306\) 1.83659e8 0.366427
\(307\) 2.16226e8 0.426505 0.213253 0.976997i \(-0.431594\pi\)
0.213253 + 0.976997i \(0.431594\pi\)
\(308\) 1.39761e8 0.272558
\(309\) 6.92459e7 0.133518
\(310\) −1.32702e7 −0.0252994
\(311\) −3.92463e8 −0.739839 −0.369919 0.929064i \(-0.620615\pi\)
−0.369919 + 0.929064i \(0.620615\pi\)
\(312\) 3.03713e7 0.0566139
\(313\) 4.30979e8 0.794422 0.397211 0.917727i \(-0.369978\pi\)
0.397211 + 0.917727i \(0.369978\pi\)
\(314\) −4.00275e8 −0.729633
\(315\) 4.31813e6 0.00778410
\(316\) 3.10917e8 0.554293
\(317\) −2.52760e8 −0.445657 −0.222829 0.974858i \(-0.571529\pi\)
−0.222829 + 0.974858i \(0.571529\pi\)
\(318\) 2.99373e8 0.522057
\(319\) −2.72984e8 −0.470837
\(320\) −4.52703e6 −0.00772305
\(321\) −1.98142e8 −0.334356
\(322\) 9.96025e7 0.166255
\(323\) 1.47017e9 2.42750
\(324\) 3.40122e7 0.0555556
\(325\) 1.70985e8 0.276291
\(326\) −3.78047e8 −0.604344
\(327\) 5.47804e8 0.866379
\(328\) 1.49661e7 0.0234180
\(329\) −3.38139e8 −0.523492
\(330\) −2.37487e7 −0.0363782
\(331\) −1.03391e9 −1.56705 −0.783526 0.621359i \(-0.786581\pi\)
−0.783526 + 0.621359i \(0.786581\pi\)
\(332\) 1.21774e8 0.182630
\(333\) 1.79519e8 0.266413
\(334\) −8.34322e8 −1.22524
\(335\) 6.72890e7 0.0977884
\(336\) 3.79331e7 0.0545545
\(337\) −3.98599e8 −0.567325 −0.283662 0.958924i \(-0.591549\pi\)
−0.283662 + 0.958924i \(0.591549\pi\)
\(338\) 3.86145e7 0.0543928
\(339\) −5.16263e7 −0.0719734
\(340\) −3.48055e7 −0.0480255
\(341\) −6.11541e8 −0.835189
\(342\) 2.72265e8 0.368044
\(343\) −4.03536e7 −0.0539949
\(344\) −1.48648e8 −0.196882
\(345\) −1.69248e7 −0.0221900
\(346\) −9.36158e8 −1.21502
\(347\) −9.63106e8 −1.23743 −0.618715 0.785615i \(-0.712347\pi\)
−0.618715 + 0.785615i \(0.712347\pi\)
\(348\) −7.40916e7 −0.0942414
\(349\) 1.32415e9 1.66743 0.833717 0.552191i \(-0.186208\pi\)
0.833717 + 0.552191i \(0.186208\pi\)
\(350\) 2.13557e8 0.266241
\(351\) 4.32436e7 0.0533761
\(352\) −2.08623e8 −0.254955
\(353\) −1.32391e9 −1.60194 −0.800969 0.598706i \(-0.795682\pi\)
−0.800969 + 0.598706i \(0.795682\pi\)
\(354\) 3.67383e8 0.440157
\(355\) −2.94766e7 −0.0349686
\(356\) −1.07208e8 −0.125936
\(357\) 2.91643e8 0.339245
\(358\) −2.19511e8 −0.252852
\(359\) 4.14851e8 0.473218 0.236609 0.971605i \(-0.423964\pi\)
0.236609 + 0.971605i \(0.423964\pi\)
\(360\) −6.44571e6 −0.00728136
\(361\) 1.28558e9 1.43822
\(362\) −9.92431e8 −1.09956
\(363\) −5.68277e8 −0.623573
\(364\) 4.82285e7 0.0524142
\(365\) −3.60623e7 −0.0388176
\(366\) 5.63349e8 0.600611
\(367\) 2.03316e8 0.214704 0.107352 0.994221i \(-0.465763\pi\)
0.107352 + 0.994221i \(0.465763\pi\)
\(368\) −1.48678e8 −0.155517
\(369\) 2.13092e7 0.0220787
\(370\) −3.40209e7 −0.0349172
\(371\) 4.75393e8 0.483331
\(372\) −1.65980e8 −0.167169
\(373\) −2.01932e8 −0.201477 −0.100738 0.994913i \(-0.532120\pi\)
−0.100738 + 0.994913i \(0.532120\pi\)
\(374\) −1.60397e9 −1.58543
\(375\) −7.27156e7 −0.0712063
\(376\) 5.04745e8 0.489682
\(377\) −9.42010e7 −0.0905442
\(378\) 5.40102e7 0.0514344
\(379\) 1.81680e9 1.71423 0.857115 0.515125i \(-0.172254\pi\)
0.857115 + 0.515125i \(0.172254\pi\)
\(380\) −5.15973e7 −0.0482375
\(381\) −5.14011e8 −0.476140
\(382\) 1.41676e9 1.30039
\(383\) 4.97022e7 0.0452043 0.0226021 0.999745i \(-0.492805\pi\)
0.0226021 + 0.999745i \(0.492805\pi\)
\(384\) −5.66231e7 −0.0510310
\(385\) −3.77120e7 −0.0336796
\(386\) 1.83981e8 0.162824
\(387\) −2.11650e8 −0.185622
\(388\) −3.13101e8 −0.272128
\(389\) −7.80289e8 −0.672097 −0.336048 0.941845i \(-0.609091\pi\)
−0.336048 + 0.941845i \(0.609091\pi\)
\(390\) −8.19516e6 −0.00699570
\(391\) −1.14309e9 −0.967079
\(392\) 6.02363e7 0.0505076
\(393\) 6.05088e8 0.502857
\(394\) −8.93498e8 −0.735964
\(395\) −8.38953e7 −0.0684932
\(396\) −2.97043e8 −0.240374
\(397\) 2.22245e9 1.78265 0.891324 0.453367i \(-0.149777\pi\)
0.891324 + 0.453367i \(0.149777\pi\)
\(398\) 3.90312e8 0.310328
\(399\) 4.32346e8 0.340743
\(400\) −3.18778e8 −0.249046
\(401\) −8.88297e8 −0.687944 −0.343972 0.938980i \(-0.611773\pi\)
−0.343972 + 0.938980i \(0.611773\pi\)
\(402\) 8.41636e8 0.646149
\(403\) −2.11029e8 −0.160611
\(404\) −1.69205e8 −0.127667
\(405\) −9.17759e6 −0.00686493
\(406\) −1.17655e8 −0.0872506
\(407\) −1.56781e9 −1.15269
\(408\) −4.35340e8 −0.317335
\(409\) −1.53205e9 −1.10724 −0.553620 0.832769i \(-0.686754\pi\)
−0.553620 + 0.832769i \(0.686754\pi\)
\(410\) −4.03833e6 −0.00289374
\(411\) 6.30531e8 0.447981
\(412\) −1.64138e8 −0.115630
\(413\) 5.83391e8 0.407506
\(414\) −2.11692e8 −0.146623
\(415\) −3.28585e7 −0.0225673
\(416\) −7.19913e7 −0.0490290
\(417\) −9.43876e8 −0.637439
\(418\) −2.37780e9 −1.59243
\(419\) −5.56107e8 −0.369325 −0.184663 0.982802i \(-0.559119\pi\)
−0.184663 + 0.982802i \(0.559119\pi\)
\(420\) −1.02356e7 −0.00674123
\(421\) 1.74619e9 1.14053 0.570263 0.821462i \(-0.306841\pi\)
0.570263 + 0.821462i \(0.306841\pi\)
\(422\) 1.42543e9 0.923319
\(423\) 7.18669e8 0.461677
\(424\) −7.09625e8 −0.452115
\(425\) −2.45089e9 −1.54868
\(426\) −3.68687e8 −0.231059
\(427\) 8.94577e8 0.556058
\(428\) 4.69671e8 0.289561
\(429\) −3.77664e8 −0.230943
\(430\) 4.01101e7 0.0243284
\(431\) 1.55597e9 0.936118 0.468059 0.883697i \(-0.344953\pi\)
0.468059 + 0.883697i \(0.344953\pi\)
\(432\) −8.06216e7 −0.0481125
\(433\) 2.68061e9 1.58681 0.793407 0.608692i \(-0.208305\pi\)
0.793407 + 0.608692i \(0.208305\pi\)
\(434\) −2.63571e8 −0.154769
\(435\) 1.99923e7 0.0116453
\(436\) −1.29850e9 −0.750306
\(437\) −1.69457e9 −0.971348
\(438\) −4.51059e8 −0.256492
\(439\) 2.20605e9 1.24448 0.622242 0.782825i \(-0.286222\pi\)
0.622242 + 0.782825i \(0.286222\pi\)
\(440\) 5.62932e7 0.0315044
\(441\) 8.57661e7 0.0476190
\(442\) −5.53496e8 −0.304885
\(443\) 3.03233e9 1.65716 0.828578 0.559874i \(-0.189151\pi\)
0.828578 + 0.559874i \(0.189151\pi\)
\(444\) −4.25526e8 −0.230720
\(445\) 2.89281e7 0.0155618
\(446\) −1.14374e9 −0.610457
\(447\) 6.84538e7 0.0362511
\(448\) −8.99154e7 −0.0472456
\(449\) 1.13559e9 0.592051 0.296025 0.955180i \(-0.404339\pi\)
0.296025 + 0.955180i \(0.404339\pi\)
\(450\) −4.53886e8 −0.234803
\(451\) −1.86102e8 −0.0955285
\(452\) 1.22373e8 0.0623308
\(453\) 2.01461e9 1.01823
\(454\) 2.43726e9 1.22238
\(455\) −1.30136e7 −0.00647676
\(456\) −6.45368e8 −0.318736
\(457\) −1.90034e9 −0.931377 −0.465688 0.884949i \(-0.654193\pi\)
−0.465688 + 0.884949i \(0.654193\pi\)
\(458\) −1.70728e9 −0.830377
\(459\) −6.19849e8 −0.299186
\(460\) 4.01181e7 0.0192171
\(461\) 1.65878e9 0.788562 0.394281 0.918990i \(-0.370994\pi\)
0.394281 + 0.918990i \(0.370994\pi\)
\(462\) −4.71694e8 −0.222543
\(463\) −3.84728e9 −1.80144 −0.900721 0.434397i \(-0.856961\pi\)
−0.900721 + 0.434397i \(0.856961\pi\)
\(464\) 1.75625e8 0.0816154
\(465\) 4.47868e7 0.0206569
\(466\) −5.14924e8 −0.235718
\(467\) −2.44108e9 −1.10911 −0.554553 0.832149i \(-0.687111\pi\)
−0.554553 + 0.832149i \(0.687111\pi\)
\(468\) −1.02503e8 −0.0462250
\(469\) 1.33649e9 0.598218
\(470\) −1.36196e8 −0.0605094
\(471\) 1.35093e9 0.595743
\(472\) −8.70834e8 −0.381187
\(473\) 1.84843e9 0.803135
\(474\) −1.04934e9 −0.452578
\(475\) −3.63331e9 −1.55552
\(476\) −6.91303e8 −0.293795
\(477\) −1.01038e9 −0.426258
\(478\) −5.30046e8 −0.221981
\(479\) −6.79391e8 −0.282452 −0.141226 0.989977i \(-0.545104\pi\)
−0.141226 + 0.989977i \(0.545104\pi\)
\(480\) 1.52787e7 0.00630584
\(481\) −5.41018e8 −0.221669
\(482\) −2.38644e9 −0.970700
\(483\) −3.36159e8 −0.135747
\(484\) 1.34703e9 0.540030
\(485\) 8.44846e7 0.0336265
\(486\) −1.14791e8 −0.0453609
\(487\) −1.52178e8 −0.0597036 −0.0298518 0.999554i \(-0.509504\pi\)
−0.0298518 + 0.999554i \(0.509504\pi\)
\(488\) −1.33535e9 −0.520145
\(489\) 1.27591e9 0.493444
\(490\) −1.62537e7 −0.00624116
\(491\) 1.26029e9 0.480489 0.240245 0.970712i \(-0.422772\pi\)
0.240245 + 0.970712i \(0.422772\pi\)
\(492\) −5.05106e7 −0.0191207
\(493\) 1.35027e9 0.507523
\(494\) −8.20529e8 −0.306231
\(495\) 8.01518e7 0.0297027
\(496\) 3.93435e8 0.144773
\(497\) −5.85461e8 −0.213919
\(498\) −4.10987e8 −0.149116
\(499\) −1.27347e9 −0.458814 −0.229407 0.973331i \(-0.573679\pi\)
−0.229407 + 0.973331i \(0.573679\pi\)
\(500\) 1.72363e8 0.0616664
\(501\) 2.81584e9 1.00040
\(502\) −2.48597e9 −0.877067
\(503\) 9.56647e8 0.335169 0.167584 0.985858i \(-0.446403\pi\)
0.167584 + 0.985858i \(0.446403\pi\)
\(504\) −1.28024e8 −0.0445435
\(505\) 4.56569e7 0.0157756
\(506\) 1.84879e9 0.634398
\(507\) −1.30324e8 −0.0444116
\(508\) 1.21840e9 0.412349
\(509\) −2.17508e9 −0.731075 −0.365538 0.930797i \(-0.619115\pi\)
−0.365538 + 0.930797i \(0.619115\pi\)
\(510\) 1.17469e8 0.0392126
\(511\) −7.16265e8 −0.237466
\(512\) 1.34218e8 0.0441942
\(513\) −9.18893e8 −0.300507
\(514\) 4.74554e7 0.0154140
\(515\) 4.42898e7 0.0142882
\(516\) 5.01688e8 0.160753
\(517\) −6.27645e9 −1.99755
\(518\) −6.75719e8 −0.213605
\(519\) 3.15953e9 0.992058
\(520\) 1.94256e7 0.00605846
\(521\) 2.31046e9 0.715758 0.357879 0.933768i \(-0.383500\pi\)
0.357879 + 0.933768i \(0.383500\pi\)
\(522\) 2.50059e8 0.0769478
\(523\) 1.91977e9 0.586804 0.293402 0.955989i \(-0.405212\pi\)
0.293402 + 0.955989i \(0.405212\pi\)
\(524\) −1.43428e9 −0.435487
\(525\) −7.20754e8 −0.217385
\(526\) 2.68404e9 0.804153
\(527\) 3.02488e9 0.900265
\(528\) 7.04103e8 0.208170
\(529\) −2.08726e9 −0.613030
\(530\) 1.91480e8 0.0558672
\(531\) −1.23992e9 −0.359387
\(532\) −1.02482e9 −0.295092
\(533\) −6.42198e7 −0.0183706
\(534\) 3.61826e8 0.102827
\(535\) −1.26732e8 −0.0357807
\(536\) −1.99499e9 −0.559582
\(537\) 7.40850e8 0.206453
\(538\) −2.13945e8 −0.0592332
\(539\) −7.49032e8 −0.206034
\(540\) 2.17543e7 0.00594520
\(541\) −2.09153e8 −0.0567904 −0.0283952 0.999597i \(-0.509040\pi\)
−0.0283952 + 0.999597i \(0.509040\pi\)
\(542\) −2.39573e9 −0.646308
\(543\) 3.34946e9 0.897790
\(544\) 1.03192e9 0.274820
\(545\) 3.50376e8 0.0927144
\(546\) −1.62771e8 −0.0427960
\(547\) −1.08327e9 −0.282996 −0.141498 0.989939i \(-0.545192\pi\)
−0.141498 + 0.989939i \(0.545192\pi\)
\(548\) −1.49459e9 −0.387963
\(549\) −1.90130e9 −0.490397
\(550\) 3.96398e9 1.01593
\(551\) 2.00170e9 0.509763
\(552\) 5.01788e8 0.126979
\(553\) −1.66632e9 −0.419006
\(554\) 2.33749e8 0.0584071
\(555\) 1.14820e8 0.0285098
\(556\) 2.23734e9 0.552039
\(557\) 4.70813e9 1.15440 0.577199 0.816604i \(-0.304146\pi\)
0.577199 + 0.816604i \(0.304146\pi\)
\(558\) 5.60184e8 0.136493
\(559\) 6.37853e8 0.154447
\(560\) 2.42621e7 0.00583807
\(561\) 5.41340e9 1.29450
\(562\) −5.77095e9 −1.37142
\(563\) −7.11679e7 −0.0168076 −0.00840378 0.999965i \(-0.502675\pi\)
−0.00840378 + 0.999965i \(0.502675\pi\)
\(564\) −1.70351e9 −0.399824
\(565\) −3.30203e7 −0.00770213
\(566\) −4.97409e9 −1.15307
\(567\) −1.82284e8 −0.0419961
\(568\) 8.73924e8 0.200103
\(569\) 3.37971e9 0.769107 0.384553 0.923103i \(-0.374355\pi\)
0.384553 + 0.923103i \(0.374355\pi\)
\(570\) 1.74141e8 0.0393858
\(571\) −2.83549e9 −0.637385 −0.318693 0.947858i \(-0.603244\pi\)
−0.318693 + 0.947858i \(0.603244\pi\)
\(572\) 8.95205e8 0.200003
\(573\) −4.78157e9 −1.06177
\(574\) −8.02089e7 −0.0177024
\(575\) 2.82498e9 0.619695
\(576\) 1.91103e8 0.0416667
\(577\) −1.62806e9 −0.352822 −0.176411 0.984317i \(-0.556449\pi\)
−0.176411 + 0.984317i \(0.556449\pi\)
\(578\) 4.65105e9 1.00185
\(579\) −6.20936e8 −0.132945
\(580\) −4.73891e7 −0.0100851
\(581\) −6.52632e8 −0.138055
\(582\) 1.05671e9 0.222191
\(583\) 8.82412e9 1.84430
\(584\) 1.06918e9 0.222129
\(585\) 2.76587e7 0.00571197
\(586\) 3.80676e9 0.781473
\(587\) 7.21774e8 0.147288 0.0736441 0.997285i \(-0.476537\pi\)
0.0736441 + 0.997285i \(0.476537\pi\)
\(588\) −2.03297e8 −0.0412393
\(589\) 4.48422e9 0.904239
\(590\) 2.34979e8 0.0471028
\(591\) 3.01556e9 0.600912
\(592\) 1.00865e9 0.199809
\(593\) 9.08076e9 1.78826 0.894130 0.447807i \(-0.147795\pi\)
0.894130 + 0.447807i \(0.147795\pi\)
\(594\) 1.00252e9 0.196264
\(595\) 1.86536e8 0.0363039
\(596\) −1.62261e8 −0.0313944
\(597\) −1.31730e9 −0.253382
\(598\) 6.37979e8 0.121998
\(599\) 2.15705e9 0.410078 0.205039 0.978754i \(-0.434268\pi\)
0.205039 + 0.978754i \(0.434268\pi\)
\(600\) 1.07588e9 0.203345
\(601\) −3.45569e9 −0.649343 −0.324671 0.945827i \(-0.605254\pi\)
−0.324671 + 0.945827i \(0.605254\pi\)
\(602\) 7.96663e8 0.148829
\(603\) −2.84052e9 −0.527579
\(604\) −4.77537e9 −0.881815
\(605\) −3.63471e8 −0.0667308
\(606\) 5.71066e8 0.104240
\(607\) 3.11495e9 0.565316 0.282658 0.959221i \(-0.408784\pi\)
0.282658 + 0.959221i \(0.408784\pi\)
\(608\) 1.52976e9 0.276033
\(609\) 3.97085e8 0.0712398
\(610\) 3.60319e8 0.0642736
\(611\) −2.16587e9 −0.384138
\(612\) 1.46927e9 0.259103
\(613\) 3.69814e9 0.648443 0.324221 0.945981i \(-0.394898\pi\)
0.324221 + 0.945981i \(0.394898\pi\)
\(614\) 1.72981e9 0.301585
\(615\) 1.36294e7 0.00236272
\(616\) 1.11809e9 0.192728
\(617\) −6.49779e8 −0.111370 −0.0556849 0.998448i \(-0.517734\pi\)
−0.0556849 + 0.998448i \(0.517734\pi\)
\(618\) 5.53967e8 0.0944114
\(619\) 3.98935e9 0.676060 0.338030 0.941135i \(-0.390240\pi\)
0.338030 + 0.941135i \(0.390240\pi\)
\(620\) −1.06161e8 −0.0178894
\(621\) 7.14459e8 0.119717
\(622\) −3.13970e9 −0.523145
\(623\) 5.74566e8 0.0951989
\(624\) 2.42971e8 0.0400320
\(625\) 6.03371e9 0.988563
\(626\) 3.44784e9 0.561741
\(627\) 8.02509e9 1.30021
\(628\) −3.20220e9 −0.515928
\(629\) 7.75490e9 1.24251
\(630\) 3.45450e7 0.00550419
\(631\) −8.73103e9 −1.38345 −0.691724 0.722162i \(-0.743148\pi\)
−0.691724 + 0.722162i \(0.743148\pi\)
\(632\) 2.48733e9 0.391944
\(633\) −4.81082e9 −0.753887
\(634\) −2.02208e9 −0.315127
\(635\) −3.28762e8 −0.0509535
\(636\) 2.39499e9 0.369150
\(637\) −2.58475e8 −0.0396214
\(638\) −2.18387e9 −0.332932
\(639\) 1.24432e9 0.188659
\(640\) −3.62163e7 −0.00546102
\(641\) −3.80807e9 −0.571087 −0.285543 0.958366i \(-0.592174\pi\)
−0.285543 + 0.958366i \(0.592174\pi\)
\(642\) −1.58514e9 −0.236426
\(643\) −6.90209e9 −1.02386 −0.511932 0.859026i \(-0.671070\pi\)
−0.511932 + 0.859026i \(0.671070\pi\)
\(644\) 7.96820e8 0.117560
\(645\) −1.35372e8 −0.0198641
\(646\) 1.17614e10 1.71650
\(647\) −1.48243e9 −0.215184 −0.107592 0.994195i \(-0.534314\pi\)
−0.107592 + 0.994195i \(0.534314\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) 1.08287e10 1.55497
\(650\) 1.36788e9 0.195367
\(651\) 8.89551e8 0.126368
\(652\) −3.02437e9 −0.427335
\(653\) 5.44881e9 0.765783 0.382892 0.923793i \(-0.374928\pi\)
0.382892 + 0.923793i \(0.374928\pi\)
\(654\) 4.38243e9 0.612622
\(655\) 3.87015e8 0.0538126
\(656\) 1.19729e8 0.0165590
\(657\) 1.52233e9 0.209425
\(658\) −2.70512e9 −0.370165
\(659\) 5.31741e9 0.723771 0.361885 0.932223i \(-0.382133\pi\)
0.361885 + 0.932223i \(0.382133\pi\)
\(660\) −1.89990e8 −0.0257233
\(661\) 6.77864e9 0.912930 0.456465 0.889741i \(-0.349115\pi\)
0.456465 + 0.889741i \(0.349115\pi\)
\(662\) −8.27125e9 −1.10807
\(663\) 1.86805e9 0.248938
\(664\) 9.74192e8 0.129139
\(665\) 2.76530e8 0.0364641
\(666\) 1.43615e9 0.188382
\(667\) −1.55637e9 −0.203082
\(668\) −6.67457e9 −0.866375
\(669\) 3.86012e9 0.498436
\(670\) 5.38312e8 0.0691468
\(671\) 1.66049e10 2.12181
\(672\) 3.03464e8 0.0385758
\(673\) −1.46642e10 −1.85441 −0.927204 0.374556i \(-0.877795\pi\)
−0.927204 + 0.374556i \(0.877795\pi\)
\(674\) −3.18879e9 −0.401159
\(675\) 1.53186e9 0.191715
\(676\) 3.08916e8 0.0384615
\(677\) 6.31710e9 0.782452 0.391226 0.920295i \(-0.372051\pi\)
0.391226 + 0.920295i \(0.372051\pi\)
\(678\) −4.13010e8 −0.0508928
\(679\) 1.67802e9 0.205709
\(680\) −2.78444e8 −0.0339591
\(681\) −8.22576e9 −0.998070
\(682\) −4.89233e9 −0.590568
\(683\) 2.83532e9 0.340510 0.170255 0.985400i \(-0.445541\pi\)
0.170255 + 0.985400i \(0.445541\pi\)
\(684\) 2.17812e9 0.260247
\(685\) 4.03289e8 0.0479401
\(686\) −3.22829e8 −0.0381802
\(687\) 5.76206e9 0.678000
\(688\) −1.18919e9 −0.139217
\(689\) 3.04501e9 0.354668
\(690\) −1.35398e8 −0.0156907
\(691\) 6.93920e9 0.800085 0.400042 0.916497i \(-0.368995\pi\)
0.400042 + 0.916497i \(0.368995\pi\)
\(692\) −7.48926e9 −0.859148
\(693\) 1.59197e9 0.181705
\(694\) −7.70485e9 −0.874996
\(695\) −6.03705e8 −0.0682147
\(696\) −5.92733e8 −0.0666387
\(697\) 9.20520e8 0.102972
\(698\) 1.05932e10 1.17905
\(699\) 1.73787e9 0.192463
\(700\) 1.70845e9 0.188261
\(701\) −4.76326e9 −0.522265 −0.261133 0.965303i \(-0.584096\pi\)
−0.261133 + 0.965303i \(0.584096\pi\)
\(702\) 3.45948e8 0.0377426
\(703\) 1.14962e10 1.24799
\(704\) −1.66898e9 −0.180280
\(705\) 4.59662e8 0.0494057
\(706\) −1.05912e10 −1.13274
\(707\) 9.06832e8 0.0965071
\(708\) 2.93907e9 0.311238
\(709\) 1.20409e9 0.126881 0.0634406 0.997986i \(-0.479793\pi\)
0.0634406 + 0.997986i \(0.479793\pi\)
\(710\) −2.35813e8 −0.0247265
\(711\) 3.54153e9 0.369528
\(712\) −8.57662e8 −0.0890504
\(713\) −3.48658e9 −0.360235
\(714\) 2.33315e9 0.239882
\(715\) −2.41555e8 −0.0247141
\(716\) −1.75609e9 −0.178793
\(717\) 1.78891e9 0.181247
\(718\) 3.31881e9 0.334616
\(719\) 3.42723e9 0.343868 0.171934 0.985108i \(-0.444998\pi\)
0.171934 + 0.985108i \(0.444998\pi\)
\(720\) −5.15657e7 −0.00514870
\(721\) 8.79679e8 0.0874080
\(722\) 1.02847e10 1.01697
\(723\) 8.05422e9 0.792573
\(724\) −7.93945e9 −0.777509
\(725\) −3.33699e9 −0.325216
\(726\) −4.54622e9 −0.440932
\(727\) −3.19742e9 −0.308624 −0.154312 0.988022i \(-0.549316\pi\)
−0.154312 + 0.988022i \(0.549316\pi\)
\(728\) 3.85828e8 0.0370625
\(729\) 3.87420e8 0.0370370
\(730\) −2.88498e8 −0.0274482
\(731\) −9.14292e9 −0.865713
\(732\) 4.50679e9 0.424696
\(733\) −1.19932e10 −1.12479 −0.562395 0.826869i \(-0.690120\pi\)
−0.562395 + 0.826869i \(0.690120\pi\)
\(734\) 1.62653e9 0.151819
\(735\) 5.48562e7 0.00509589
\(736\) −1.18942e9 −0.109967
\(737\) 2.48075e10 2.28269
\(738\) 1.70473e8 0.0156120
\(739\) 1.84002e10 1.67713 0.838567 0.544798i \(-0.183394\pi\)
0.838567 + 0.544798i \(0.183394\pi\)
\(740\) −2.72167e8 −0.0246902
\(741\) 2.76928e9 0.250037
\(742\) 3.80315e9 0.341767
\(743\) 1.37821e10 1.23269 0.616346 0.787476i \(-0.288612\pi\)
0.616346 + 0.787476i \(0.288612\pi\)
\(744\) −1.32784e9 −0.118207
\(745\) 4.37832e7 0.00387936
\(746\) −1.61546e9 −0.142466
\(747\) 1.38708e9 0.121753
\(748\) −1.28318e10 −1.12107
\(749\) −2.51714e9 −0.218888
\(750\) −5.81725e8 −0.0503504
\(751\) −1.49296e10 −1.28620 −0.643100 0.765782i \(-0.722352\pi\)
−0.643100 + 0.765782i \(0.722352\pi\)
\(752\) 4.03796e9 0.346257
\(753\) 8.39014e9 0.716122
\(754\) −7.53608e8 −0.0640244
\(755\) 1.28855e9 0.108965
\(756\) 4.32081e8 0.0363696
\(757\) −1.44421e10 −1.21002 −0.605012 0.796216i \(-0.706832\pi\)
−0.605012 + 0.796216i \(0.706832\pi\)
\(758\) 1.45344e10 1.21214
\(759\) −6.23968e9 −0.517984
\(760\) −4.12779e8 −0.0341091
\(761\) 1.79186e10 1.47387 0.736933 0.675966i \(-0.236273\pi\)
0.736933 + 0.675966i \(0.236273\pi\)
\(762\) −4.11209e9 −0.336682
\(763\) 6.95914e9 0.567178
\(764\) 1.13341e10 0.919517
\(765\) −3.96456e8 −0.0320170
\(766\) 3.97617e8 0.0319643
\(767\) 3.73676e9 0.299028
\(768\) −4.52985e8 −0.0360844
\(769\) 5.34345e9 0.423721 0.211860 0.977300i \(-0.432048\pi\)
0.211860 + 0.977300i \(0.432048\pi\)
\(770\) −3.01696e8 −0.0238151
\(771\) −1.60162e8 −0.0125855
\(772\) 1.47185e9 0.115134
\(773\) −1.81113e10 −1.41033 −0.705166 0.709042i \(-0.749128\pi\)
−0.705166 + 0.709042i \(0.749128\pi\)
\(774\) −1.69320e9 −0.131255
\(775\) −7.47553e9 −0.576881
\(776\) −2.50480e9 −0.192423
\(777\) 2.28055e9 0.174408
\(778\) −6.24231e9 −0.475244
\(779\) 1.36462e9 0.103426
\(780\) −6.55613e7 −0.00494671
\(781\) −1.08672e10 −0.816276
\(782\) −9.14472e9 −0.683828
\(783\) −8.43950e8 −0.0628276
\(784\) 4.81890e8 0.0357143
\(785\) 8.64056e8 0.0637526
\(786\) 4.84070e9 0.355574
\(787\) −1.34001e10 −0.979930 −0.489965 0.871742i \(-0.662990\pi\)
−0.489965 + 0.871742i \(0.662990\pi\)
\(788\) −7.14798e9 −0.520405
\(789\) −9.05863e9 −0.656588
\(790\) −6.71162e8 −0.0484320
\(791\) −6.55845e8 −0.0471176
\(792\) −2.37635e9 −0.169970
\(793\) 5.72999e9 0.408035
\(794\) 1.77796e10 1.26052
\(795\) −6.46244e8 −0.0456154
\(796\) 3.12250e9 0.219435
\(797\) −7.54941e9 −0.528213 −0.264106 0.964494i \(-0.585077\pi\)
−0.264106 + 0.964494i \(0.585077\pi\)
\(798\) 3.45877e9 0.240941
\(799\) 3.10453e10 2.15319
\(800\) −2.55023e9 −0.176102
\(801\) −1.22116e9 −0.0839575
\(802\) −7.10638e9 −0.486450
\(803\) −1.32951e10 −0.906124
\(804\) 6.73308e9 0.456897
\(805\) −2.15008e8 −0.0145267
\(806\) −1.68824e9 −0.113569
\(807\) 7.22066e8 0.0483637
\(808\) −1.35364e9 −0.0902741
\(809\) 8.18752e9 0.543667 0.271833 0.962344i \(-0.412370\pi\)
0.271833 + 0.962344i \(0.412370\pi\)
\(810\) −7.34207e7 −0.00485424
\(811\) −1.11910e10 −0.736706 −0.368353 0.929686i \(-0.620078\pi\)
−0.368353 + 0.929686i \(0.620078\pi\)
\(812\) −9.41238e8 −0.0616955
\(813\) 8.08558e9 0.527709
\(814\) −1.25425e10 −0.815077
\(815\) 8.16073e8 0.0528053
\(816\) −3.48272e9 −0.224390
\(817\) −1.35539e10 −0.869535
\(818\) −1.22564e10 −0.782937
\(819\) 5.49353e8 0.0349428
\(820\) −3.23067e7 −0.00204618
\(821\) −2.92344e10 −1.84371 −0.921855 0.387535i \(-0.873327\pi\)
−0.921855 + 0.387535i \(0.873327\pi\)
\(822\) 5.04425e9 0.316771
\(823\) −2.95262e10 −1.84633 −0.923163 0.384410i \(-0.874405\pi\)
−0.923163 + 0.384410i \(0.874405\pi\)
\(824\) −1.31311e9 −0.0817627
\(825\) −1.33784e10 −0.829500
\(826\) 4.66713e9 0.288151
\(827\) 2.37879e9 0.146247 0.0731235 0.997323i \(-0.476703\pi\)
0.0731235 + 0.997323i \(0.476703\pi\)
\(828\) −1.69353e9 −0.103678
\(829\) 1.59710e10 0.973626 0.486813 0.873506i \(-0.338159\pi\)
0.486813 + 0.873506i \(0.338159\pi\)
\(830\) −2.62868e8 −0.0159575
\(831\) −7.88903e8 −0.0476892
\(832\) −5.75930e8 −0.0346688
\(833\) 3.70495e9 0.222088
\(834\) −7.55101e9 −0.450738
\(835\) 1.80101e9 0.107057
\(836\) −1.90224e10 −1.12601
\(837\) −1.89062e9 −0.111446
\(838\) −4.44885e9 −0.261152
\(839\) 2.27440e10 1.32953 0.664767 0.747051i \(-0.268531\pi\)
0.664767 + 0.747051i \(0.268531\pi\)
\(840\) −8.18845e7 −0.00476677
\(841\) −1.54114e10 −0.893423
\(842\) 1.39695e10 0.806473
\(843\) 1.94770e10 1.11976
\(844\) 1.14034e10 0.652885
\(845\) −8.33554e7 −0.00475264
\(846\) 5.74936e9 0.326455
\(847\) −7.21923e9 −0.408224
\(848\) −5.67700e9 −0.319693
\(849\) 1.67875e10 0.941478
\(850\) −1.96071e10 −1.09508
\(851\) −8.93857e9 −0.497181
\(852\) −2.94949e9 −0.163384
\(853\) −2.26033e10 −1.24696 −0.623478 0.781841i \(-0.714281\pi\)
−0.623478 + 0.781841i \(0.714281\pi\)
\(854\) 7.15662e9 0.393192
\(855\) −5.87726e8 −0.0321583
\(856\) 3.75737e9 0.204751
\(857\) −7.53555e8 −0.0408961 −0.0204481 0.999791i \(-0.506509\pi\)
−0.0204481 + 0.999791i \(0.506509\pi\)
\(858\) −3.02132e9 −0.163302
\(859\) −2.24138e10 −1.20653 −0.603266 0.797540i \(-0.706134\pi\)
−0.603266 + 0.797540i \(0.706134\pi\)
\(860\) 3.20881e8 0.0172028
\(861\) 2.70705e8 0.0144539
\(862\) 1.24478e10 0.661935
\(863\) 9.57096e9 0.506895 0.253447 0.967349i \(-0.418436\pi\)
0.253447 + 0.967349i \(0.418436\pi\)
\(864\) −6.44973e8 −0.0340207
\(865\) 2.02084e9 0.106164
\(866\) 2.14449e10 1.12205
\(867\) −1.56973e10 −0.818008
\(868\) −2.10857e9 −0.109438
\(869\) −3.09297e10 −1.59885
\(870\) 1.59938e8 0.00823446
\(871\) 8.56053e9 0.438972
\(872\) −1.03880e10 −0.530547
\(873\) −3.56641e9 −0.181418
\(874\) −1.35566e10 −0.686847
\(875\) −9.23758e8 −0.0466154
\(876\) −3.60847e9 −0.181367
\(877\) 1.32581e10 0.663715 0.331857 0.943330i \(-0.392325\pi\)
0.331857 + 0.943330i \(0.392325\pi\)
\(878\) 1.76484e10 0.879983
\(879\) −1.28478e10 −0.638070
\(880\) 4.50346e8 0.0222770
\(881\) −2.53389e10 −1.24845 −0.624226 0.781244i \(-0.714586\pi\)
−0.624226 + 0.781244i \(0.714586\pi\)
\(882\) 6.86129e8 0.0336718
\(883\) 5.43794e9 0.265810 0.132905 0.991129i \(-0.457569\pi\)
0.132905 + 0.991129i \(0.457569\pi\)
\(884\) −4.42797e9 −0.215586
\(885\) −7.93054e8 −0.0384593
\(886\) 2.42586e10 1.17179
\(887\) −8.74030e9 −0.420527 −0.210263 0.977645i \(-0.567432\pi\)
−0.210263 + 0.977645i \(0.567432\pi\)
\(888\) −3.40420e9 −0.163144
\(889\) −6.52984e9 −0.311707
\(890\) 2.31425e8 0.0110038
\(891\) −3.38351e9 −0.160249
\(892\) −9.14992e9 −0.431659
\(893\) 4.60231e10 2.16270
\(894\) 5.47631e8 0.0256334
\(895\) 4.73849e8 0.0220932
\(896\) −7.19323e8 −0.0334077
\(897\) −2.15318e9 −0.0996108
\(898\) 9.08471e9 0.418643
\(899\) 4.11849e9 0.189051
\(900\) −3.63109e9 −0.166030
\(901\) −4.36469e10 −1.98800
\(902\) −1.48882e9 −0.0675489
\(903\) −2.68874e9 −0.121518
\(904\) 9.78987e8 0.0440745
\(905\) 2.14232e9 0.0960758
\(906\) 1.61169e10 0.719999
\(907\) −3.97846e10 −1.77047 −0.885236 0.465141i \(-0.846004\pi\)
−0.885236 + 0.465141i \(0.846004\pi\)
\(908\) 1.94981e10 0.864354
\(909\) −1.92735e9 −0.0851112
\(910\) −1.04109e8 −0.00457976
\(911\) 1.92105e10 0.841829 0.420914 0.907100i \(-0.361709\pi\)
0.420914 + 0.907100i \(0.361709\pi\)
\(912\) −5.16294e9 −0.225380
\(913\) −1.21140e10 −0.526792
\(914\) −1.52027e10 −0.658583
\(915\) −1.21608e9 −0.0524792
\(916\) −1.36582e10 −0.587165
\(917\) 7.68686e9 0.329197
\(918\) −4.95879e9 −0.211556
\(919\) −1.42698e10 −0.606476 −0.303238 0.952915i \(-0.598068\pi\)
−0.303238 + 0.952915i \(0.598068\pi\)
\(920\) 3.20944e8 0.0135885
\(921\) −5.83811e9 −0.246243
\(922\) 1.32703e10 0.557598
\(923\) −3.75002e9 −0.156974
\(924\) −3.77355e9 −0.157361
\(925\) −1.91651e10 −0.796187
\(926\) −3.07783e10 −1.27381
\(927\) −1.86964e9 −0.0770866
\(928\) 1.40500e9 0.0577108
\(929\) 1.79166e10 0.733162 0.366581 0.930386i \(-0.380528\pi\)
0.366581 + 0.930386i \(0.380528\pi\)
\(930\) 3.58295e8 0.0146066
\(931\) 5.49240e9 0.223068
\(932\) −4.11940e9 −0.166678
\(933\) 1.05965e10 0.427146
\(934\) −1.95286e10 −0.784256
\(935\) 3.46242e9 0.138529
\(936\) −8.20026e8 −0.0326860
\(937\) −3.89744e10 −1.54771 −0.773857 0.633361i \(-0.781675\pi\)
−0.773857 + 0.633361i \(0.781675\pi\)
\(938\) 1.06919e10 0.423004
\(939\) −1.16364e10 −0.458660
\(940\) −1.08957e9 −0.0427866
\(941\) −7.73440e8 −0.0302596 −0.0151298 0.999886i \(-0.504816\pi\)
−0.0151298 + 0.999886i \(0.504816\pi\)
\(942\) 1.08074e10 0.421254
\(943\) −1.06102e9 −0.0412035
\(944\) −6.96667e9 −0.269540
\(945\) −1.16589e8 −0.00449415
\(946\) 1.47874e10 0.567902
\(947\) 3.31534e10 1.26854 0.634269 0.773113i \(-0.281301\pi\)
0.634269 + 0.773113i \(0.281301\pi\)
\(948\) −8.39475e9 −0.320021
\(949\) −4.58786e9 −0.174252
\(950\) −2.90665e10 −1.09992
\(951\) 6.82452e9 0.257300
\(952\) −5.53042e9 −0.207744
\(953\) 2.09417e10 0.783766 0.391883 0.920015i \(-0.371824\pi\)
0.391883 + 0.920015i \(0.371824\pi\)
\(954\) −8.08307e9 −0.301410
\(955\) −3.05830e9 −0.113624
\(956\) −4.24037e9 −0.156964
\(957\) 7.37057e9 0.271838
\(958\) −5.43512e9 −0.199724
\(959\) 8.01007e9 0.293273
\(960\) 1.22230e8 0.00445890
\(961\) −1.82863e10 −0.664653
\(962\) −4.32815e9 −0.156743
\(963\) 5.34984e9 0.193041
\(964\) −1.90915e10 −0.686389
\(965\) −3.97152e8 −0.0142269
\(966\) −2.68927e9 −0.0959874
\(967\) 1.60900e10 0.572220 0.286110 0.958197i \(-0.407638\pi\)
0.286110 + 0.958197i \(0.407638\pi\)
\(968\) 1.07762e10 0.381859
\(969\) −3.96946e10 −1.40152
\(970\) 6.75877e8 0.0237775
\(971\) 2.11517e10 0.741443 0.370722 0.928744i \(-0.379110\pi\)
0.370722 + 0.928744i \(0.379110\pi\)
\(972\) −9.18330e8 −0.0320750
\(973\) −1.19907e10 −0.417302
\(974\) −1.21742e9 −0.0422168
\(975\) −4.61661e9 −0.159517
\(976\) −1.06828e10 −0.367798
\(977\) 5.08868e10 1.74572 0.872859 0.487973i \(-0.162264\pi\)
0.872859 + 0.487973i \(0.162264\pi\)
\(978\) 1.02073e10 0.348918
\(979\) 1.06649e10 0.363261
\(980\) −1.30029e8 −0.00441317
\(981\) −1.47907e10 −0.500204
\(982\) 1.00823e10 0.339757
\(983\) 4.91736e9 0.165118 0.0825590 0.996586i \(-0.473691\pi\)
0.0825590 + 0.996586i \(0.473691\pi\)
\(984\) −4.04085e8 −0.0135204
\(985\) 1.92876e9 0.0643058
\(986\) 1.08021e10 0.358873
\(987\) 9.12976e9 0.302238
\(988\) −6.56423e9 −0.216538
\(989\) 1.05384e10 0.346409
\(990\) 6.41215e8 0.0210029
\(991\) 2.30540e10 0.752469 0.376234 0.926525i \(-0.377219\pi\)
0.376234 + 0.926525i \(0.377219\pi\)
\(992\) 3.14748e9 0.102370
\(993\) 2.79155e10 0.904738
\(994\) −4.68368e9 −0.151264
\(995\) −8.42550e8 −0.0271153
\(996\) −3.28790e9 −0.105441
\(997\) −2.53208e10 −0.809178 −0.404589 0.914499i \(-0.632585\pi\)
−0.404589 + 0.914499i \(0.632585\pi\)
\(998\) −1.01878e10 −0.324431
\(999\) −4.84700e9 −0.153813
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 546.8.a.l.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.8.a.l.1.3 5 1.1 even 1 trivial