Properties

Label 177.8.a.c.1.12
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1669 x^{15} + 2385 x^{14} + 1108684 x^{13} - 848131 x^{12} - 377920980 x^{11} + 12724944 x^{10} + 71331230512 x^{9} + 50741131904 x^{8} - 7480805165760 x^{7} - 10751966150272 x^{6} + 413177144536320 x^{5} + 886760582981376 x^{4} - 10454479722123264 x^{3} - 29180140031461376 x^{2} + 79787300207378432 x + 248723246810300416\)
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(9.14085\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+9.14085 q^{2} -27.0000 q^{3} -44.4448 q^{4} -27.9722 q^{5} -246.803 q^{6} -1415.76 q^{7} -1576.29 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+9.14085 q^{2} -27.0000 q^{3} -44.4448 q^{4} -27.9722 q^{5} -246.803 q^{6} -1415.76 q^{7} -1576.29 q^{8} +729.000 q^{9} -255.690 q^{10} -4887.09 q^{11} +1200.01 q^{12} -5210.02 q^{13} -12941.3 q^{14} +755.249 q^{15} -8719.72 q^{16} +5796.28 q^{17} +6663.68 q^{18} +24940.5 q^{19} +1243.22 q^{20} +38225.5 q^{21} -44672.2 q^{22} -63003.2 q^{23} +42559.9 q^{24} -77342.6 q^{25} -47624.1 q^{26} -19683.0 q^{27} +62923.2 q^{28} +86556.7 q^{29} +6903.62 q^{30} +268352. q^{31} +122060. q^{32} +131951. q^{33} +52983.0 q^{34} +39601.9 q^{35} -32400.3 q^{36} -235277. q^{37} +227977. q^{38} +140671. q^{39} +44092.3 q^{40} -585710. q^{41} +349414. q^{42} -806765. q^{43} +217206. q^{44} -20391.7 q^{45} -575903. q^{46} +557072. q^{47} +235432. q^{48} +1.18083e6 q^{49} -706977. q^{50} -156500. q^{51} +231559. q^{52} +2.07376e6 q^{53} -179919. q^{54} +136703. q^{55} +2.23165e6 q^{56} -673393. q^{57} +791202. q^{58} -205379. q^{59} -33566.9 q^{60} -1.17120e6 q^{61} +2.45297e6 q^{62} -1.03209e6 q^{63} +2.23185e6 q^{64} +145736. q^{65} +1.20615e6 q^{66} +4.57510e6 q^{67} -257615. q^{68} +1.70109e6 q^{69} +361995. q^{70} +798821. q^{71} -1.14912e6 q^{72} -2.35826e6 q^{73} -2.15063e6 q^{74} +2.08825e6 q^{75} -1.10847e6 q^{76} +6.91895e6 q^{77} +1.28585e6 q^{78} +117864. q^{79} +243910. q^{80} +531441. q^{81} -5.35389e6 q^{82} +2.44537e6 q^{83} -1.69893e6 q^{84} -162135. q^{85} -7.37452e6 q^{86} -2.33703e6 q^{87} +7.70349e6 q^{88} -5.83686e6 q^{89} -186398. q^{90} +7.37614e6 q^{91} +2.80017e6 q^{92} -7.24551e6 q^{93} +5.09211e6 q^{94} -697640. q^{95} -3.29561e6 q^{96} -6.38002e6 q^{97} +1.07938e7 q^{98} -3.56269e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + 6521q^{10} - 1764q^{11} - 31482q^{12} + 18192q^{13} - 7827q^{14} + 8586q^{15} + 139226q^{16} - 15507q^{17} + 1458q^{18} + 52083q^{19} + 721q^{20} - 84915q^{21} - 234434q^{22} + 63823q^{23} - 63585q^{24} + 202153q^{25} - 367956q^{26} - 334611q^{27} + 182306q^{28} - 502955q^{29} - 176067q^{30} + 347531q^{31} - 243908q^{32} + 47628q^{33} - 330872q^{34} + 92641q^{35} + 850014q^{36} + 447615q^{37} + 775669q^{38} - 491184q^{39} + 2203270q^{40} + 940335q^{41} + 211329q^{42} + 478562q^{43} - 596924q^{44} - 231822q^{45} - 3078663q^{46} + 703121q^{47} - 3759102q^{48} + 1895082q^{49} - 876967q^{50} + 418689q^{51} + 6278296q^{52} - 1005974q^{53} - 39366q^{54} + 5212846q^{55} + 3425294q^{56} - 1406241q^{57} + 6710166q^{58} - 3491443q^{59} - 19467q^{60} + 11510749q^{61} + 5996234q^{62} + 2292705q^{63} + 29496941q^{64} + 11094180q^{65} + 6329718q^{66} + 14007144q^{67} + 19688159q^{68} - 1723221q^{69} + 30909708q^{70} + 5229074q^{71} + 1716795q^{72} + 5452211q^{73} + 12819662q^{74} - 5458131q^{75} + 41929340q^{76} + 9930777q^{77} + 9934812q^{78} + 15275654q^{79} + 36576105q^{80} + 9034497q^{81} + 32025935q^{82} + 7826609q^{83} - 4922262q^{84} + 11836945q^{85} + 51649136q^{86} + 13579785q^{87} + 30223741q^{88} - 6436185q^{89} + 4753809q^{90} + 11633535q^{91} + 43357972q^{92} - 9383337q^{93} - 4494252q^{94} + 23741055q^{95} + 6585516q^{96} + 26377540q^{97} + 26517816q^{98} - 1285956q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 9.14085 0.807945 0.403972 0.914771i \(-0.367629\pi\)
0.403972 + 0.914771i \(0.367629\pi\)
\(3\) −27.0000 −0.577350
\(4\) −44.4448 −0.347225
\(5\) −27.9722 −0.100076 −0.0500382 0.998747i \(-0.515934\pi\)
−0.0500382 + 0.998747i \(0.515934\pi\)
\(6\) −246.803 −0.466467
\(7\) −1415.76 −1.56008 −0.780040 0.625730i \(-0.784801\pi\)
−0.780040 + 0.625730i \(0.784801\pi\)
\(8\) −1576.29 −1.08848
\(9\) 729.000 0.333333
\(10\) −255.690 −0.0808562
\(11\) −4887.09 −1.10707 −0.553536 0.832825i \(-0.686722\pi\)
−0.553536 + 0.832825i \(0.686722\pi\)
\(12\) 1200.01 0.200471
\(13\) −5210.02 −0.657715 −0.328858 0.944380i \(-0.606664\pi\)
−0.328858 + 0.944380i \(0.606664\pi\)
\(14\) −12941.3 −1.26046
\(15\) 755.249 0.0577791
\(16\) −8719.72 −0.532210
\(17\) 5796.28 0.286140 0.143070 0.989713i \(-0.454303\pi\)
0.143070 + 0.989713i \(0.454303\pi\)
\(18\) 6663.68 0.269315
\(19\) 24940.5 0.834194 0.417097 0.908862i \(-0.363048\pi\)
0.417097 + 0.908862i \(0.363048\pi\)
\(20\) 1243.22 0.0347490
\(21\) 38225.5 0.900712
\(22\) −44672.2 −0.894454
\(23\) −63003.2 −1.07973 −0.539865 0.841751i \(-0.681525\pi\)
−0.539865 + 0.841751i \(0.681525\pi\)
\(24\) 42559.9 0.628436
\(25\) −77342.6 −0.989985
\(26\) −47624.1 −0.531398
\(27\) −19683.0 −0.192450
\(28\) 62923.2 0.541699
\(29\) 86556.7 0.659034 0.329517 0.944150i \(-0.393114\pi\)
0.329517 + 0.944150i \(0.393114\pi\)
\(30\) 6903.62 0.0466823
\(31\) 268352. 1.61785 0.808927 0.587909i \(-0.200049\pi\)
0.808927 + 0.587909i \(0.200049\pi\)
\(32\) 122060. 0.658488
\(33\) 131951. 0.639169
\(34\) 52983.0 0.231185
\(35\) 39601.9 0.156127
\(36\) −32400.3 −0.115742
\(37\) −235277. −0.763614 −0.381807 0.924242i \(-0.624698\pi\)
−0.381807 + 0.924242i \(0.624698\pi\)
\(38\) 227977. 0.673983
\(39\) 140671. 0.379732
\(40\) 44092.3 0.108931
\(41\) −585710. −1.32721 −0.663605 0.748083i \(-0.730974\pi\)
−0.663605 + 0.748083i \(0.730974\pi\)
\(42\) 349414. 0.727726
\(43\) −806765. −1.54742 −0.773709 0.633541i \(-0.781601\pi\)
−0.773709 + 0.633541i \(0.781601\pi\)
\(44\) 217206. 0.384403
\(45\) −20391.7 −0.0333588
\(46\) −575903. −0.872363
\(47\) 557072. 0.782652 0.391326 0.920252i \(-0.372016\pi\)
0.391326 + 0.920252i \(0.372016\pi\)
\(48\) 235432. 0.307271
\(49\) 1.18083e6 1.43385
\(50\) −706977. −0.799853
\(51\) −156500. −0.165203
\(52\) 231559. 0.228375
\(53\) 2.07376e6 1.91334 0.956671 0.291170i \(-0.0940446\pi\)
0.956671 + 0.291170i \(0.0940446\pi\)
\(54\) −179919. −0.155489
\(55\) 136703. 0.110792
\(56\) 2.23165e6 1.69812
\(57\) −673393. −0.481622
\(58\) 791202. 0.532463
\(59\) −205379. −0.130189
\(60\) −33566.9 −0.0200624
\(61\) −1.17120e6 −0.660656 −0.330328 0.943866i \(-0.607159\pi\)
−0.330328 + 0.943866i \(0.607159\pi\)
\(62\) 2.45297e6 1.30714
\(63\) −1.03209e6 −0.520026
\(64\) 2.23185e6 1.06423
\(65\) 145736. 0.0658217
\(66\) 1.20615e6 0.516413
\(67\) 4.57510e6 1.85840 0.929199 0.369579i \(-0.120498\pi\)
0.929199 + 0.369579i \(0.120498\pi\)
\(68\) −257615. −0.0993549
\(69\) 1.70109e6 0.623383
\(70\) 361995. 0.126142
\(71\) 798821. 0.264878 0.132439 0.991191i \(-0.457719\pi\)
0.132439 + 0.991191i \(0.457719\pi\)
\(72\) −1.14912e6 −0.362828
\(73\) −2.35826e6 −0.709516 −0.354758 0.934958i \(-0.615437\pi\)
−0.354758 + 0.934958i \(0.615437\pi\)
\(74\) −2.15063e6 −0.616958
\(75\) 2.08825e6 0.571568
\(76\) −1.10847e6 −0.289653
\(77\) 6.91895e6 1.72712
\(78\) 1.28585e6 0.306803
\(79\) 117864. 0.0268958 0.0134479 0.999910i \(-0.495719\pi\)
0.0134479 + 0.999910i \(0.495719\pi\)
\(80\) 243910. 0.0532616
\(81\) 531441. 0.111111
\(82\) −5.35389e6 −1.07231
\(83\) 2.44537e6 0.469431 0.234715 0.972064i \(-0.424584\pi\)
0.234715 + 0.972064i \(0.424584\pi\)
\(84\) −1.69893e6 −0.312750
\(85\) −162135. −0.0286358
\(86\) −7.37452e6 −1.25023
\(87\) −2.33703e6 −0.380493
\(88\) 7.70349e6 1.20503
\(89\) −5.83686e6 −0.877636 −0.438818 0.898576i \(-0.644603\pi\)
−0.438818 + 0.898576i \(0.644603\pi\)
\(90\) −186398. −0.0269521
\(91\) 7.37614e6 1.02609
\(92\) 2.80017e6 0.374910
\(93\) −7.24551e6 −0.934069
\(94\) 5.09211e6 0.632340
\(95\) −697640. −0.0834831
\(96\) −3.29561e6 −0.380178
\(97\) −6.38002e6 −0.709776 −0.354888 0.934909i \(-0.615481\pi\)
−0.354888 + 0.934909i \(0.615481\pi\)
\(98\) 1.07938e7 1.15847
\(99\) −3.56269e6 −0.369024
\(100\) 3.43748e6 0.343748
\(101\) −1.63929e7 −1.58319 −0.791593 0.611048i \(-0.790748\pi\)
−0.791593 + 0.611048i \(0.790748\pi\)
\(102\) −1.43054e6 −0.133475
\(103\) −1.41494e7 −1.27587 −0.637935 0.770090i \(-0.720211\pi\)
−0.637935 + 0.770090i \(0.720211\pi\)
\(104\) 8.21252e6 0.715912
\(105\) −1.06925e6 −0.0901400
\(106\) 1.89559e7 1.54588
\(107\) 1.58657e7 1.25204 0.626019 0.779808i \(-0.284683\pi\)
0.626019 + 0.779808i \(0.284683\pi\)
\(108\) 874807. 0.0668235
\(109\) −8.25579e6 −0.610613 −0.305307 0.952254i \(-0.598759\pi\)
−0.305307 + 0.952254i \(0.598759\pi\)
\(110\) 1.24958e6 0.0895136
\(111\) 6.35249e6 0.440873
\(112\) 1.23450e7 0.830289
\(113\) 2.60775e7 1.70016 0.850082 0.526651i \(-0.176552\pi\)
0.850082 + 0.526651i \(0.176552\pi\)
\(114\) −6.15538e6 −0.389124
\(115\) 1.76234e6 0.108055
\(116\) −3.84700e6 −0.228833
\(117\) −3.79811e6 −0.219238
\(118\) −1.87734e6 −0.105185
\(119\) −8.20615e6 −0.446401
\(120\) −1.19049e6 −0.0628916
\(121\) 4.39649e6 0.225610
\(122\) −1.07057e7 −0.533774
\(123\) 1.58142e7 0.766265
\(124\) −1.19269e7 −0.561760
\(125\) 4.34877e6 0.199150
\(126\) −9.43418e6 −0.420153
\(127\) 1.28036e7 0.554649 0.277325 0.960776i \(-0.410552\pi\)
0.277325 + 0.960776i \(0.410552\pi\)
\(128\) 4.77741e6 0.201353
\(129\) 2.17827e7 0.893402
\(130\) 1.33215e6 0.0531803
\(131\) 2.45517e6 0.0954184 0.0477092 0.998861i \(-0.484808\pi\)
0.0477092 + 0.998861i \(0.484808\pi\)
\(132\) −5.86456e6 −0.221935
\(133\) −3.53097e7 −1.30141
\(134\) 4.18203e7 1.50148
\(135\) 550577. 0.0192597
\(136\) −9.13664e6 −0.311459
\(137\) −1.15632e7 −0.384198 −0.192099 0.981376i \(-0.561529\pi\)
−0.192099 + 0.981376i \(0.561529\pi\)
\(138\) 1.55494e7 0.503659
\(139\) −1.38898e7 −0.438678 −0.219339 0.975649i \(-0.570390\pi\)
−0.219339 + 0.975649i \(0.570390\pi\)
\(140\) −1.76010e6 −0.0542112
\(141\) −1.50409e7 −0.451865
\(142\) 7.30191e6 0.214007
\(143\) 2.54619e7 0.728139
\(144\) −6.35668e6 −0.177403
\(145\) −2.42118e6 −0.0659537
\(146\) −2.15565e7 −0.573250
\(147\) −3.18825e7 −0.827832
\(148\) 1.04569e7 0.265146
\(149\) 6.33739e6 0.156949 0.0784745 0.996916i \(-0.474995\pi\)
0.0784745 + 0.996916i \(0.474995\pi\)
\(150\) 1.90884e7 0.461795
\(151\) −6.02966e6 −0.142519 −0.0712596 0.997458i \(-0.522702\pi\)
−0.0712596 + 0.997458i \(0.522702\pi\)
\(152\) −3.93135e7 −0.908007
\(153\) 4.22549e6 0.0953799
\(154\) 6.32451e7 1.39542
\(155\) −7.50640e6 −0.161909
\(156\) −6.25208e6 −0.131853
\(157\) 3.30649e6 0.0681896 0.0340948 0.999419i \(-0.489145\pi\)
0.0340948 + 0.999419i \(0.489145\pi\)
\(158\) 1.07737e6 0.0217303
\(159\) −5.59915e7 −1.10467
\(160\) −3.41428e6 −0.0658990
\(161\) 8.91975e7 1.68447
\(162\) 4.85782e6 0.0897717
\(163\) 3.35536e7 0.606851 0.303425 0.952855i \(-0.401870\pi\)
0.303425 + 0.952855i \(0.401870\pi\)
\(164\) 2.60318e7 0.460840
\(165\) −3.69097e6 −0.0639657
\(166\) 2.23528e7 0.379274
\(167\) 4.17102e7 0.693002 0.346501 0.938050i \(-0.387370\pi\)
0.346501 + 0.938050i \(0.387370\pi\)
\(168\) −6.02546e7 −0.980410
\(169\) −3.56042e7 −0.567411
\(170\) −1.48205e6 −0.0231362
\(171\) 1.81816e7 0.278065
\(172\) 3.58565e7 0.537302
\(173\) −9.03552e7 −1.32676 −0.663379 0.748284i \(-0.730878\pi\)
−0.663379 + 0.748284i \(0.730878\pi\)
\(174\) −2.13625e7 −0.307418
\(175\) 1.09499e8 1.54445
\(176\) 4.26141e7 0.589195
\(177\) 5.54523e6 0.0751646
\(178\) −5.33539e7 −0.709081
\(179\) −2.78323e6 −0.0362713 −0.0181357 0.999836i \(-0.505773\pi\)
−0.0181357 + 0.999836i \(0.505773\pi\)
\(180\) 906306. 0.0115830
\(181\) −1.28005e8 −1.60455 −0.802274 0.596955i \(-0.796377\pi\)
−0.802274 + 0.596955i \(0.796377\pi\)
\(182\) 6.74243e7 0.829022
\(183\) 3.16223e7 0.381430
\(184\) 9.93115e7 1.17527
\(185\) 6.58122e6 0.0764197
\(186\) −6.62302e7 −0.754676
\(187\) −2.83270e7 −0.316778
\(188\) −2.47590e7 −0.271757
\(189\) 2.78664e7 0.300237
\(190\) −6.37702e6 −0.0674497
\(191\) −1.75929e8 −1.82693 −0.913464 0.406920i \(-0.866603\pi\)
−0.913464 + 0.406920i \(0.866603\pi\)
\(192\) −6.02601e7 −0.614434
\(193\) −9.04189e7 −0.905333 −0.452667 0.891680i \(-0.649527\pi\)
−0.452667 + 0.891680i \(0.649527\pi\)
\(194\) −5.83188e7 −0.573460
\(195\) −3.93486e6 −0.0380022
\(196\) −5.24820e7 −0.497868
\(197\) −7.90918e7 −0.737055 −0.368527 0.929617i \(-0.620138\pi\)
−0.368527 + 0.929617i \(0.620138\pi\)
\(198\) −3.25660e7 −0.298151
\(199\) 1.80077e8 1.61985 0.809923 0.586537i \(-0.199509\pi\)
0.809923 + 0.586537i \(0.199509\pi\)
\(200\) 1.21915e8 1.07758
\(201\) −1.23528e8 −1.07295
\(202\) −1.49845e8 −1.27913
\(203\) −1.22544e8 −1.02814
\(204\) 6.95560e6 0.0573626
\(205\) 1.63836e7 0.132822
\(206\) −1.29337e8 −1.03083
\(207\) −4.59294e7 −0.359910
\(208\) 4.54300e7 0.350042
\(209\) −1.21886e8 −0.923513
\(210\) −9.77387e6 −0.0728281
\(211\) 2.21867e8 1.62594 0.812968 0.582308i \(-0.197850\pi\)
0.812968 + 0.582308i \(0.197850\pi\)
\(212\) −9.21678e7 −0.664361
\(213\) −2.15682e7 −0.152927
\(214\) 1.45026e8 1.01158
\(215\) 2.25670e7 0.154860
\(216\) 3.10262e7 0.209479
\(217\) −3.79923e8 −2.52398
\(218\) −7.54650e7 −0.493342
\(219\) 6.36731e7 0.409639
\(220\) −6.07572e6 −0.0384697
\(221\) −3.01988e7 −0.188199
\(222\) 5.80671e7 0.356201
\(223\) 2.20216e8 1.32978 0.664892 0.746940i \(-0.268478\pi\)
0.664892 + 0.746940i \(0.268478\pi\)
\(224\) −1.72807e8 −1.02729
\(225\) −5.63827e7 −0.329995
\(226\) 2.38370e8 1.37364
\(227\) 1.55341e8 0.881444 0.440722 0.897644i \(-0.354723\pi\)
0.440722 + 0.897644i \(0.354723\pi\)
\(228\) 2.99288e7 0.167231
\(229\) −2.23695e8 −1.23093 −0.615464 0.788165i \(-0.711031\pi\)
−0.615464 + 0.788165i \(0.711031\pi\)
\(230\) 1.61093e7 0.0873029
\(231\) −1.86812e8 −0.997154
\(232\) −1.36439e8 −0.717347
\(233\) 1.96538e8 1.01789 0.508946 0.860798i \(-0.330035\pi\)
0.508946 + 0.860798i \(0.330035\pi\)
\(234\) −3.47179e7 −0.177133
\(235\) −1.55825e7 −0.0783250
\(236\) 9.12803e6 0.0452049
\(237\) −3.18231e6 −0.0155283
\(238\) −7.50112e7 −0.360667
\(239\) −3.22046e8 −1.52589 −0.762947 0.646461i \(-0.776248\pi\)
−0.762947 + 0.646461i \(0.776248\pi\)
\(240\) −6.58556e6 −0.0307506
\(241\) 1.17931e8 0.542711 0.271355 0.962479i \(-0.412528\pi\)
0.271355 + 0.962479i \(0.412528\pi\)
\(242\) 4.01877e7 0.182280
\(243\) −1.43489e7 −0.0641500
\(244\) 5.20536e7 0.229396
\(245\) −3.30305e7 −0.143494
\(246\) 1.44555e8 0.619100
\(247\) −1.29940e8 −0.548662
\(248\) −4.23002e8 −1.76101
\(249\) −6.60250e7 −0.271026
\(250\) 3.97514e7 0.160903
\(251\) −4.16401e8 −1.66209 −0.831043 0.556208i \(-0.812256\pi\)
−0.831043 + 0.556208i \(0.812256\pi\)
\(252\) 4.58710e7 0.180566
\(253\) 3.07903e8 1.19534
\(254\) 1.17036e8 0.448126
\(255\) 4.37764e6 0.0165329
\(256\) −2.42008e8 −0.901549
\(257\) 3.87165e8 1.42276 0.711378 0.702810i \(-0.248072\pi\)
0.711378 + 0.702810i \(0.248072\pi\)
\(258\) 1.99112e8 0.721820
\(259\) 3.33096e8 1.19130
\(260\) −6.47720e6 −0.0228550
\(261\) 6.30998e7 0.219678
\(262\) 2.24424e7 0.0770928
\(263\) −1.55540e8 −0.527227 −0.263613 0.964628i \(-0.584914\pi\)
−0.263613 + 0.964628i \(0.584914\pi\)
\(264\) −2.07994e8 −0.695725
\(265\) −5.80076e7 −0.191480
\(266\) −3.22761e8 −1.05147
\(267\) 1.57595e8 0.506703
\(268\) −2.03339e8 −0.645283
\(269\) 2.06155e8 0.645743 0.322872 0.946443i \(-0.395352\pi\)
0.322872 + 0.946443i \(0.395352\pi\)
\(270\) 5.03274e6 0.0155608
\(271\) 2.53618e8 0.774082 0.387041 0.922062i \(-0.373497\pi\)
0.387041 + 0.922062i \(0.373497\pi\)
\(272\) −5.05420e7 −0.152286
\(273\) −1.99156e8 −0.592412
\(274\) −1.05697e8 −0.310410
\(275\) 3.77980e8 1.09598
\(276\) −7.56045e7 −0.216454
\(277\) 5.09014e8 1.43896 0.719482 0.694511i \(-0.244379\pi\)
0.719482 + 0.694511i \(0.244379\pi\)
\(278\) −1.26965e8 −0.354427
\(279\) 1.95629e8 0.539285
\(280\) −6.24242e7 −0.169942
\(281\) −1.26418e8 −0.339888 −0.169944 0.985454i \(-0.554359\pi\)
−0.169944 + 0.985454i \(0.554359\pi\)
\(282\) −1.37487e8 −0.365082
\(283\) 2.56537e8 0.672819 0.336409 0.941716i \(-0.390787\pi\)
0.336409 + 0.941716i \(0.390787\pi\)
\(284\) −3.55035e7 −0.0919722
\(285\) 1.88363e7 0.0481990
\(286\) 2.32743e8 0.588296
\(287\) 8.29226e8 2.07055
\(288\) 8.89816e7 0.219496
\(289\) −3.76742e8 −0.918124
\(290\) −2.21316e7 −0.0532869
\(291\) 1.72261e8 0.409789
\(292\) 1.04813e8 0.246362
\(293\) 9.81288e7 0.227908 0.113954 0.993486i \(-0.463648\pi\)
0.113954 + 0.993486i \(0.463648\pi\)
\(294\) −2.91434e8 −0.668843
\(295\) 5.74490e6 0.0130288
\(296\) 3.70866e8 0.831181
\(297\) 9.61926e7 0.213056
\(298\) 5.79292e7 0.126806
\(299\) 3.28248e8 0.710155
\(300\) −9.28118e7 −0.198463
\(301\) 1.14219e9 2.41409
\(302\) −5.51162e7 −0.115148
\(303\) 4.42609e8 0.914053
\(304\) −2.17474e8 −0.443966
\(305\) 3.27609e7 0.0661160
\(306\) 3.86246e7 0.0770617
\(307\) 4.01381e8 0.791722 0.395861 0.918311i \(-0.370446\pi\)
0.395861 + 0.918311i \(0.370446\pi\)
\(308\) −3.07512e8 −0.599700
\(309\) 3.82032e8 0.736624
\(310\) −6.86149e7 −0.130813
\(311\) −9.27378e8 −1.74822 −0.874109 0.485730i \(-0.838554\pi\)
−0.874109 + 0.485730i \(0.838554\pi\)
\(312\) −2.21738e8 −0.413332
\(313\) 8.01269e8 1.47697 0.738487 0.674268i \(-0.235541\pi\)
0.738487 + 0.674268i \(0.235541\pi\)
\(314\) 3.02241e7 0.0550934
\(315\) 2.88698e7 0.0520423
\(316\) −5.23842e6 −0.00933890
\(317\) −2.96984e8 −0.523632 −0.261816 0.965118i \(-0.584321\pi\)
−0.261816 + 0.965118i \(0.584321\pi\)
\(318\) −5.11810e8 −0.892512
\(319\) −4.23011e8 −0.729598
\(320\) −6.24299e7 −0.106504
\(321\) −4.28375e8 −0.722864
\(322\) 8.15341e8 1.36095
\(323\) 1.44562e8 0.238696
\(324\) −2.36198e7 −0.0385806
\(325\) 4.02957e8 0.651128
\(326\) 3.06708e8 0.490302
\(327\) 2.22906e8 0.352538
\(328\) 9.23251e8 1.44465
\(329\) −7.88681e8 −1.22100
\(330\) −3.37386e7 −0.0516807
\(331\) 4.65841e8 0.706057 0.353029 0.935612i \(-0.385152\pi\)
0.353029 + 0.935612i \(0.385152\pi\)
\(332\) −1.08684e8 −0.162998
\(333\) −1.71517e8 −0.254538
\(334\) 3.81267e8 0.559908
\(335\) −1.27976e8 −0.185982
\(336\) −3.33316e8 −0.479368
\(337\) 9.05809e8 1.28924 0.644618 0.764505i \(-0.277017\pi\)
0.644618 + 0.764505i \(0.277017\pi\)
\(338\) −3.25453e8 −0.458436
\(339\) −7.04091e8 −0.981590
\(340\) 7.20605e6 0.00994308
\(341\) −1.31146e9 −1.79108
\(342\) 1.66195e8 0.224661
\(343\) −5.05839e8 −0.676836
\(344\) 1.27170e9 1.68434
\(345\) −4.75831e7 −0.0623859
\(346\) −8.25923e8 −1.07195
\(347\) 8.14436e8 1.04642 0.523208 0.852205i \(-0.324735\pi\)
0.523208 + 0.852205i \(0.324735\pi\)
\(348\) 1.03869e8 0.132117
\(349\) 1.34997e9 1.69994 0.849972 0.526827i \(-0.176619\pi\)
0.849972 + 0.526827i \(0.176619\pi\)
\(350\) 1.00091e9 1.24783
\(351\) 1.02549e8 0.126577
\(352\) −5.96517e8 −0.728994
\(353\) 6.77178e8 0.819391 0.409695 0.912222i \(-0.365635\pi\)
0.409695 + 0.912222i \(0.365635\pi\)
\(354\) 5.06882e7 0.0607289
\(355\) −2.23448e7 −0.0265080
\(356\) 2.59418e8 0.304737
\(357\) 2.21566e8 0.257730
\(358\) −2.54411e7 −0.0293052
\(359\) 7.72908e8 0.881652 0.440826 0.897592i \(-0.354685\pi\)
0.440826 + 0.897592i \(0.354685\pi\)
\(360\) 3.21433e7 0.0363105
\(361\) −2.71844e8 −0.304120
\(362\) −1.17008e9 −1.29639
\(363\) −1.18705e8 −0.130256
\(364\) −3.27831e8 −0.356283
\(365\) 6.59658e7 0.0710058
\(366\) 2.89055e8 0.308174
\(367\) 1.86496e9 1.96942 0.984712 0.174192i \(-0.0557312\pi\)
0.984712 + 0.174192i \(0.0557312\pi\)
\(368\) 5.49371e8 0.574643
\(369\) −4.26983e8 −0.442403
\(370\) 6.01580e7 0.0617429
\(371\) −2.93595e9 −2.98497
\(372\) 3.22025e8 0.324332
\(373\) 1.15094e9 1.14834 0.574171 0.818735i \(-0.305324\pi\)
0.574171 + 0.818735i \(0.305324\pi\)
\(374\) −2.58933e8 −0.255939
\(375\) −1.17417e8 −0.114980
\(376\) −8.78109e8 −0.851904
\(377\) −4.50962e8 −0.433457
\(378\) 2.54723e8 0.242575
\(379\) 8.25715e7 0.0779100 0.0389550 0.999241i \(-0.487597\pi\)
0.0389550 + 0.999241i \(0.487597\pi\)
\(380\) 3.10065e7 0.0289874
\(381\) −3.45697e8 −0.320227
\(382\) −1.60814e9 −1.47606
\(383\) −6.72581e8 −0.611714 −0.305857 0.952077i \(-0.598943\pi\)
−0.305857 + 0.952077i \(0.598943\pi\)
\(384\) −1.28990e8 −0.116251
\(385\) −1.93538e8 −0.172844
\(386\) −8.26505e8 −0.731459
\(387\) −5.88132e8 −0.515806
\(388\) 2.83559e8 0.246452
\(389\) −9.07409e8 −0.781591 −0.390795 0.920478i \(-0.627800\pi\)
−0.390795 + 0.920478i \(0.627800\pi\)
\(390\) −3.59680e7 −0.0307037
\(391\) −3.65185e8 −0.308954
\(392\) −1.86134e9 −1.56072
\(393\) −6.62896e7 −0.0550899
\(394\) −7.22967e8 −0.595500
\(395\) −3.29690e6 −0.00269163
\(396\) 1.58343e8 0.128134
\(397\) 2.34468e9 1.88069 0.940345 0.340223i \(-0.110503\pi\)
0.940345 + 0.340223i \(0.110503\pi\)
\(398\) 1.64606e9 1.30875
\(399\) 9.53363e8 0.751369
\(400\) 6.74406e8 0.526879
\(401\) 1.03275e9 0.799819 0.399910 0.916555i \(-0.369042\pi\)
0.399910 + 0.916555i \(0.369042\pi\)
\(402\) −1.12915e9 −0.866882
\(403\) −1.39812e9 −1.06409
\(404\) 7.28581e8 0.549722
\(405\) −1.48656e7 −0.0111196
\(406\) −1.12015e9 −0.830684
\(407\) 1.14982e9 0.845376
\(408\) 2.46689e8 0.179821
\(409\) −4.25915e8 −0.307816 −0.153908 0.988085i \(-0.549186\pi\)
−0.153908 + 0.988085i \(0.549186\pi\)
\(410\) 1.49760e8 0.107313
\(411\) 3.12205e8 0.221817
\(412\) 6.28865e8 0.443014
\(413\) 2.90767e8 0.203105
\(414\) −4.19833e8 −0.290788
\(415\) −6.84024e7 −0.0469789
\(416\) −6.35934e8 −0.433097
\(417\) 3.75026e8 0.253271
\(418\) −1.11415e9 −0.746148
\(419\) −9.70564e8 −0.644577 −0.322289 0.946641i \(-0.604452\pi\)
−0.322289 + 0.946641i \(0.604452\pi\)
\(420\) 4.75227e7 0.0312989
\(421\) 3.26959e6 0.00213553 0.00106777 0.999999i \(-0.499660\pi\)
0.00106777 + 0.999999i \(0.499660\pi\)
\(422\) 2.02805e9 1.31367
\(423\) 4.06106e8 0.260884
\(424\) −3.26885e9 −2.08264
\(425\) −4.48299e8 −0.283274
\(426\) −1.97152e8 −0.123557
\(427\) 1.65813e9 1.03068
\(428\) −7.05150e8 −0.434739
\(429\) −6.87470e8 −0.420391
\(430\) 2.06281e8 0.125118
\(431\) 9.95591e8 0.598978 0.299489 0.954100i \(-0.403184\pi\)
0.299489 + 0.954100i \(0.403184\pi\)
\(432\) 1.71630e8 0.102424
\(433\) −1.28699e9 −0.761847 −0.380924 0.924606i \(-0.624394\pi\)
−0.380924 + 0.924606i \(0.624394\pi\)
\(434\) −3.47282e9 −2.03924
\(435\) 6.53719e7 0.0380784
\(436\) 3.66927e8 0.212020
\(437\) −1.57133e9 −0.900705
\(438\) 5.82027e8 0.330966
\(439\) 5.89758e8 0.332697 0.166348 0.986067i \(-0.446802\pi\)
0.166348 + 0.986067i \(0.446802\pi\)
\(440\) −2.15483e8 −0.120595
\(441\) 8.60828e8 0.477949
\(442\) −2.76042e8 −0.152054
\(443\) 3.59302e8 0.196357 0.0981786 0.995169i \(-0.468698\pi\)
0.0981786 + 0.995169i \(0.468698\pi\)
\(444\) −2.82335e8 −0.153082
\(445\) 1.63270e8 0.0878306
\(446\) 2.01296e9 1.07439
\(447\) −1.71110e8 −0.0906146
\(448\) −3.15977e9 −1.66028
\(449\) 2.56647e9 1.33806 0.669028 0.743237i \(-0.266710\pi\)
0.669028 + 0.743237i \(0.266710\pi\)
\(450\) −5.15386e8 −0.266618
\(451\) 2.86242e9 1.46932
\(452\) −1.15901e9 −0.590340
\(453\) 1.62801e8 0.0822835
\(454\) 1.41995e9 0.712158
\(455\) −2.06327e8 −0.102687
\(456\) 1.06146e9 0.524238
\(457\) −1.46001e9 −0.715565 −0.357782 0.933805i \(-0.616467\pi\)
−0.357782 + 0.933805i \(0.616467\pi\)
\(458\) −2.04477e9 −0.994522
\(459\) −1.14088e8 −0.0550676
\(460\) −7.83268e7 −0.0375196
\(461\) −4.94351e7 −0.0235008 −0.0117504 0.999931i \(-0.503740\pi\)
−0.0117504 + 0.999931i \(0.503740\pi\)
\(462\) −1.70762e9 −0.805645
\(463\) −2.80354e9 −1.31272 −0.656361 0.754447i \(-0.727905\pi\)
−0.656361 + 0.754447i \(0.727905\pi\)
\(464\) −7.54750e8 −0.350744
\(465\) 2.02673e8 0.0934782
\(466\) 1.79653e9 0.822401
\(467\) 2.95995e9 1.34485 0.672426 0.740164i \(-0.265252\pi\)
0.672426 + 0.740164i \(0.265252\pi\)
\(468\) 1.68806e8 0.0761251
\(469\) −6.47725e9 −2.89925
\(470\) −1.42438e8 −0.0632823
\(471\) −8.92752e7 −0.0393693
\(472\) 3.23737e8 0.141708
\(473\) 3.94274e9 1.71310
\(474\) −2.90891e7 −0.0125460
\(475\) −1.92896e9 −0.825839
\(476\) 3.64721e8 0.155002
\(477\) 1.51177e9 0.637781
\(478\) −2.94377e9 −1.23284
\(479\) 2.16259e9 0.899084 0.449542 0.893259i \(-0.351587\pi\)
0.449542 + 0.893259i \(0.351587\pi\)
\(480\) 9.21855e7 0.0380468
\(481\) 1.22580e9 0.502241
\(482\) 1.07799e9 0.438480
\(483\) −2.40833e9 −0.972526
\(484\) −1.95401e8 −0.0783374
\(485\) 1.78463e8 0.0710317
\(486\) −1.31161e8 −0.0518297
\(487\) −2.92186e9 −1.14633 −0.573163 0.819441i \(-0.694284\pi\)
−0.573163 + 0.819441i \(0.694284\pi\)
\(488\) 1.84615e9 0.719113
\(489\) −9.05946e8 −0.350366
\(490\) −3.01927e8 −0.115935
\(491\) 1.26732e9 0.483171 0.241585 0.970380i \(-0.422333\pi\)
0.241585 + 0.970380i \(0.422333\pi\)
\(492\) −7.02858e8 −0.266066
\(493\) 5.01707e8 0.188576
\(494\) −1.18777e9 −0.443289
\(495\) 9.96562e7 0.0369306
\(496\) −2.33996e9 −0.861038
\(497\) −1.13094e9 −0.413230
\(498\) −6.03525e8 −0.218974
\(499\) −2.05947e9 −0.742000 −0.371000 0.928633i \(-0.620985\pi\)
−0.371000 + 0.928633i \(0.620985\pi\)
\(500\) −1.93280e8 −0.0691500
\(501\) −1.12618e9 −0.400105
\(502\) −3.80626e9 −1.34287
\(503\) 1.28852e9 0.451443 0.225721 0.974192i \(-0.427526\pi\)
0.225721 + 0.974192i \(0.427526\pi\)
\(504\) 1.62687e9 0.566040
\(505\) 4.58546e8 0.158440
\(506\) 2.81449e9 0.965769
\(507\) 9.61313e8 0.327595
\(508\) −5.69053e8 −0.192588
\(509\) 7.14570e7 0.0240177 0.0120089 0.999928i \(-0.496177\pi\)
0.0120089 + 0.999928i \(0.496177\pi\)
\(510\) 4.00153e7 0.0133577
\(511\) 3.33874e9 1.10690
\(512\) −2.82367e9 −0.929755
\(513\) −4.90903e8 −0.160541
\(514\) 3.53902e9 1.14951
\(515\) 3.95788e8 0.127684
\(516\) −9.68126e8 −0.310212
\(517\) −2.72246e9 −0.866453
\(518\) 3.04478e9 0.962503
\(519\) 2.43959e9 0.766004
\(520\) −2.29722e8 −0.0716459
\(521\) −1.07670e8 −0.0333551 −0.0166776 0.999861i \(-0.505309\pi\)
−0.0166776 + 0.999861i \(0.505309\pi\)
\(522\) 5.76786e8 0.177488
\(523\) 1.67231e9 0.511165 0.255582 0.966787i \(-0.417733\pi\)
0.255582 + 0.966787i \(0.417733\pi\)
\(524\) −1.09120e8 −0.0331317
\(525\) −2.95646e9 −0.891691
\(526\) −1.42177e9 −0.425970
\(527\) 1.55545e9 0.462933
\(528\) −1.15058e9 −0.340172
\(529\) 5.64582e8 0.165818
\(530\) −5.30239e8 −0.154706
\(531\) −1.49721e8 −0.0433963
\(532\) 1.56933e9 0.451882
\(533\) 3.05156e9 0.872926
\(534\) 1.44055e9 0.409388
\(535\) −4.43800e8 −0.125299
\(536\) −7.21170e9 −2.02284
\(537\) 7.51472e7 0.0209412
\(538\) 1.88443e9 0.521725
\(539\) −5.77085e9 −1.58737
\(540\) −2.44703e7 −0.00668745
\(541\) 1.37337e9 0.372903 0.186452 0.982464i \(-0.440301\pi\)
0.186452 + 0.982464i \(0.440301\pi\)
\(542\) 2.31828e9 0.625416
\(543\) 3.45614e9 0.926387
\(544\) 7.07493e8 0.188420
\(545\) 2.30933e8 0.0611079
\(546\) −1.82045e9 −0.478636
\(547\) −3.99898e9 −1.04470 −0.522352 0.852730i \(-0.674945\pi\)
−0.522352 + 0.852730i \(0.674945\pi\)
\(548\) 5.13923e8 0.133403
\(549\) −8.53803e8 −0.220219
\(550\) 3.45506e9 0.885495
\(551\) 2.15877e9 0.549762
\(552\) −2.68141e9 −0.678542
\(553\) −1.66867e8 −0.0419596
\(554\) 4.65282e9 1.16260
\(555\) −1.77693e8 −0.0441209
\(556\) 6.17331e8 0.152320
\(557\) −2.15460e9 −0.528291 −0.264146 0.964483i \(-0.585090\pi\)
−0.264146 + 0.964483i \(0.585090\pi\)
\(558\) 1.78821e9 0.435712
\(559\) 4.20327e9 1.01776
\(560\) −3.45318e8 −0.0830923
\(561\) 7.64828e8 0.182892
\(562\) −1.15557e9 −0.274611
\(563\) −5.78892e9 −1.36716 −0.683578 0.729878i \(-0.739577\pi\)
−0.683578 + 0.729878i \(0.739577\pi\)
\(564\) 6.68492e8 0.156899
\(565\) −7.29444e8 −0.170146
\(566\) 2.34497e9 0.543600
\(567\) −7.52393e8 −0.173342
\(568\) −1.25918e9 −0.288315
\(569\) −2.74050e9 −0.623643 −0.311822 0.950141i \(-0.600939\pi\)
−0.311822 + 0.950141i \(0.600939\pi\)
\(570\) 1.72180e8 0.0389421
\(571\) 8.65695e9 1.94598 0.972991 0.230844i \(-0.0741488\pi\)
0.972991 + 0.230844i \(0.0741488\pi\)
\(572\) −1.13165e9 −0.252828
\(573\) 4.75009e9 1.05478
\(574\) 7.57983e9 1.67289
\(575\) 4.87283e9 1.06892
\(576\) 1.62702e9 0.354744
\(577\) −5.33758e9 −1.15672 −0.578361 0.815781i \(-0.696308\pi\)
−0.578361 + 0.815781i \(0.696308\pi\)
\(578\) −3.44374e9 −0.741794
\(579\) 2.44131e9 0.522694
\(580\) 1.07609e8 0.0229008
\(581\) −3.46206e9 −0.732349
\(582\) 1.57461e9 0.331087
\(583\) −1.01347e10 −2.11821
\(584\) 3.71731e9 0.772297
\(585\) 1.06241e8 0.0219406
\(586\) 8.96981e8 0.184137
\(587\) −7.46525e9 −1.52339 −0.761695 0.647936i \(-0.775633\pi\)
−0.761695 + 0.647936i \(0.775633\pi\)
\(588\) 1.41701e9 0.287444
\(589\) 6.69283e9 1.34960
\(590\) 5.25133e7 0.0105266
\(591\) 2.13548e9 0.425539
\(592\) 2.05155e9 0.406403
\(593\) 3.09092e9 0.608690 0.304345 0.952562i \(-0.401562\pi\)
0.304345 + 0.952562i \(0.401562\pi\)
\(594\) 8.79283e8 0.172138
\(595\) 2.29544e8 0.0446742
\(596\) −2.81664e8 −0.0544966
\(597\) −4.86209e9 −0.935218
\(598\) 3.00047e9 0.573766
\(599\) 2.66060e9 0.505809 0.252904 0.967491i \(-0.418614\pi\)
0.252904 + 0.967491i \(0.418614\pi\)
\(600\) −3.29169e9 −0.622142
\(601\) 4.24214e8 0.0797122 0.0398561 0.999205i \(-0.487310\pi\)
0.0398561 + 0.999205i \(0.487310\pi\)
\(602\) 1.04406e10 1.95045
\(603\) 3.33525e9 0.619466
\(604\) 2.67987e8 0.0494862
\(605\) −1.22980e8 −0.0225782
\(606\) 4.04583e9 0.738505
\(607\) 1.13859e9 0.206637 0.103318 0.994648i \(-0.467054\pi\)
0.103318 + 0.994648i \(0.467054\pi\)
\(608\) 3.04423e9 0.549306
\(609\) 3.30868e9 0.593600
\(610\) 2.99463e8 0.0534181
\(611\) −2.90236e9 −0.514762
\(612\) −1.87801e8 −0.0331183
\(613\) −1.59219e9 −0.279179 −0.139590 0.990209i \(-0.544578\pi\)
−0.139590 + 0.990209i \(0.544578\pi\)
\(614\) 3.66896e9 0.639667
\(615\) −4.42357e8 −0.0766849
\(616\) −1.09063e10 −1.87994
\(617\) −5.01588e9 −0.859705 −0.429853 0.902899i \(-0.641434\pi\)
−0.429853 + 0.902899i \(0.641434\pi\)
\(618\) 3.49210e9 0.595151
\(619\) −1.05539e10 −1.78854 −0.894268 0.447532i \(-0.852303\pi\)
−0.894268 + 0.447532i \(0.852303\pi\)
\(620\) 3.33621e8 0.0562188
\(621\) 1.24009e9 0.207794
\(622\) −8.47703e9 −1.41246
\(623\) 8.26360e9 1.36918
\(624\) −1.22661e9 −0.202097
\(625\) 5.92074e9 0.970054
\(626\) 7.32428e9 1.19331
\(627\) 3.29093e9 0.533191
\(628\) −1.46956e8 −0.0236771
\(629\) −1.36373e9 −0.218500
\(630\) 2.63895e8 0.0420473
\(631\) −5.24953e9 −0.831797 −0.415899 0.909411i \(-0.636533\pi\)
−0.415899 + 0.909411i \(0.636533\pi\)
\(632\) −1.85787e8 −0.0292756
\(633\) −5.99040e9 −0.938735
\(634\) −2.71469e9 −0.423066
\(635\) −3.58144e8 −0.0555073
\(636\) 2.48853e9 0.383569
\(637\) −6.15218e9 −0.943063
\(638\) −3.86668e9 −0.589475
\(639\) 5.82341e8 0.0882926
\(640\) −1.33634e8 −0.0201506
\(641\) −4.54601e9 −0.681753 −0.340876 0.940108i \(-0.610724\pi\)
−0.340876 + 0.940108i \(0.610724\pi\)
\(642\) −3.91571e9 −0.584034
\(643\) 3.12857e9 0.464095 0.232048 0.972704i \(-0.425458\pi\)
0.232048 + 0.972704i \(0.425458\pi\)
\(644\) −3.96437e9 −0.584889
\(645\) −6.09309e8 −0.0894084
\(646\) 1.32142e9 0.192853
\(647\) −5.99209e9 −0.869788 −0.434894 0.900482i \(-0.643214\pi\)
−0.434894 + 0.900482i \(0.643214\pi\)
\(648\) −8.37707e8 −0.120943
\(649\) 1.00371e9 0.144129
\(650\) 3.68337e9 0.526076
\(651\) 1.02579e10 1.45722
\(652\) −1.49128e9 −0.210714
\(653\) 1.00926e10 1.41842 0.709210 0.704997i \(-0.249052\pi\)
0.709210 + 0.704997i \(0.249052\pi\)
\(654\) 2.03755e9 0.284831
\(655\) −6.86765e7 −0.00954913
\(656\) 5.10723e9 0.706353
\(657\) −1.71917e9 −0.236505
\(658\) −7.20921e9 −0.986500
\(659\) 1.22466e10 1.66693 0.833465 0.552573i \(-0.186354\pi\)
0.833465 + 0.552573i \(0.186354\pi\)
\(660\) 1.64045e8 0.0222105
\(661\) −6.31287e8 −0.0850201 −0.0425101 0.999096i \(-0.513535\pi\)
−0.0425101 + 0.999096i \(0.513535\pi\)
\(662\) 4.25819e9 0.570455
\(663\) 8.15367e8 0.108656
\(664\) −3.85462e9 −0.510968
\(665\) 9.87690e8 0.130240
\(666\) −1.56781e9 −0.205653
\(667\) −5.45335e9 −0.711579
\(668\) −1.85380e9 −0.240628
\(669\) −5.94582e9 −0.767751
\(670\) −1.16981e9 −0.150263
\(671\) 5.72375e9 0.731394
\(672\) 4.66580e9 0.593108
\(673\) 1.33894e10 1.69320 0.846599 0.532232i \(-0.178647\pi\)
0.846599 + 0.532232i \(0.178647\pi\)
\(674\) 8.27987e9 1.04163
\(675\) 1.52233e9 0.190523
\(676\) 1.58242e9 0.197019
\(677\) −1.86123e9 −0.230536 −0.115268 0.993334i \(-0.536773\pi\)
−0.115268 + 0.993334i \(0.536773\pi\)
\(678\) −6.43600e9 −0.793071
\(679\) 9.03258e9 1.10731
\(680\) 2.55572e8 0.0311696
\(681\) −4.19420e9 −0.508902
\(682\) −1.19879e10 −1.44710
\(683\) −1.29941e10 −1.56054 −0.780268 0.625445i \(-0.784918\pi\)
−0.780268 + 0.625445i \(0.784918\pi\)
\(684\) −8.08078e8 −0.0965510
\(685\) 3.23447e8 0.0384491
\(686\) −4.62380e9 −0.546846
\(687\) 6.03977e9 0.710677
\(688\) 7.03477e9 0.823550
\(689\) −1.08043e10 −1.25843
\(690\) −4.34950e8 −0.0504043
\(691\) −9.85622e9 −1.13642 −0.568208 0.822885i \(-0.692363\pi\)
−0.568208 + 0.822885i \(0.692363\pi\)
\(692\) 4.01582e9 0.460684
\(693\) 5.04392e9 0.575707
\(694\) 7.44464e9 0.845446
\(695\) 3.88529e8 0.0439012
\(696\) 3.68384e9 0.414161
\(697\) −3.39494e9 −0.379767
\(698\) 1.23399e10 1.37346
\(699\) −5.30654e9 −0.587680
\(700\) −4.86664e9 −0.536273
\(701\) −9.35979e9 −1.02625 −0.513125 0.858314i \(-0.671512\pi\)
−0.513125 + 0.858314i \(0.671512\pi\)
\(702\) 9.37384e8 0.102268
\(703\) −5.86793e9 −0.637002
\(704\) −1.09073e10 −1.17818
\(705\) 4.20728e8 0.0452209
\(706\) 6.18998e9 0.662023
\(707\) 2.32085e10 2.46990
\(708\) −2.46457e8 −0.0260990
\(709\) −9.52210e9 −1.00339 −0.501696 0.865044i \(-0.667290\pi\)
−0.501696 + 0.865044i \(0.667290\pi\)
\(710\) −2.04250e8 −0.0214170
\(711\) 8.59225e7 0.00896527
\(712\) 9.20060e9 0.955292
\(713\) −1.69071e10 −1.74685
\(714\) 2.02530e9 0.208231
\(715\) −7.12224e8 −0.0728694
\(716\) 1.23700e8 0.0125943
\(717\) 8.69523e9 0.880976
\(718\) 7.06504e9 0.712327
\(719\) −6.22106e9 −0.624185 −0.312092 0.950052i \(-0.601030\pi\)
−0.312092 + 0.950052i \(0.601030\pi\)
\(720\) 1.77810e8 0.0177539
\(721\) 2.00321e10 1.99046
\(722\) −2.48489e9 −0.245712
\(723\) −3.18414e9 −0.313334
\(724\) 5.68917e9 0.557140
\(725\) −6.69452e9 −0.652433
\(726\) −1.08507e9 −0.105240
\(727\) 4.54280e9 0.438483 0.219242 0.975671i \(-0.429642\pi\)
0.219242 + 0.975671i \(0.429642\pi\)
\(728\) −1.16270e10 −1.11688
\(729\) 3.87420e8 0.0370370
\(730\) 6.02984e8 0.0573688
\(731\) −4.67624e9 −0.442778
\(732\) −1.40545e9 −0.132442
\(733\) 9.84479e9 0.923299 0.461649 0.887062i \(-0.347258\pi\)
0.461649 + 0.887062i \(0.347258\pi\)
\(734\) 1.70474e10 1.59119
\(735\) 8.91824e8 0.0828464
\(736\) −7.69016e9 −0.710989
\(737\) −2.23589e10 −2.05738
\(738\) −3.90299e9 −0.357437
\(739\) 8.69557e9 0.792579 0.396289 0.918126i \(-0.370298\pi\)
0.396289 + 0.918126i \(0.370298\pi\)
\(740\) −2.92501e8 −0.0265348
\(741\) 3.50839e9 0.316770
\(742\) −2.68370e10 −2.41169
\(743\) 1.41962e10 1.26973 0.634864 0.772624i \(-0.281056\pi\)
0.634864 + 0.772624i \(0.281056\pi\)
\(744\) 1.14210e10 1.01672
\(745\) −1.77271e8 −0.0157069
\(746\) 1.05206e10 0.927797
\(747\) 1.78268e9 0.156477
\(748\) 1.25899e9 0.109993
\(749\) −2.24621e10 −1.95328
\(750\) −1.07329e9 −0.0928971
\(751\) 1.57390e10 1.35593 0.677967 0.735092i \(-0.262861\pi\)
0.677967 + 0.735092i \(0.262861\pi\)
\(752\) −4.85751e9 −0.416535
\(753\) 1.12428e10 0.959606
\(754\) −4.12218e9 −0.350209
\(755\) 1.68663e8 0.0142628
\(756\) −1.23852e9 −0.104250
\(757\) 2.58251e9 0.216374 0.108187 0.994131i \(-0.465495\pi\)
0.108187 + 0.994131i \(0.465495\pi\)
\(758\) 7.54774e8 0.0629470
\(759\) −8.31337e9 −0.690130
\(760\) 1.09968e9 0.0908700
\(761\) −1.11863e9 −0.0920114 −0.0460057 0.998941i \(-0.514649\pi\)
−0.0460057 + 0.998941i \(0.514649\pi\)
\(762\) −3.15996e9 −0.258726
\(763\) 1.16882e10 0.952605
\(764\) 7.81914e9 0.634355
\(765\) −1.18196e8 −0.00954528
\(766\) −6.14796e9 −0.494231
\(767\) 1.07003e9 0.0856272
\(768\) 6.53421e9 0.520510
\(769\) 4.81105e9 0.381503 0.190751 0.981638i \(-0.438908\pi\)
0.190751 + 0.981638i \(0.438908\pi\)
\(770\) −1.76910e9 −0.139648
\(771\) −1.04535e10 −0.821428
\(772\) 4.01865e9 0.314354
\(773\) −9.07363e9 −0.706566 −0.353283 0.935516i \(-0.614935\pi\)
−0.353283 + 0.935516i \(0.614935\pi\)
\(774\) −5.37603e9 −0.416743
\(775\) −2.07551e10 −1.60165
\(776\) 1.00568e10 0.772579
\(777\) −8.99360e9 −0.687796
\(778\) −8.29449e9 −0.631482
\(779\) −1.46079e10 −1.10715
\(780\) 1.74884e8 0.0131953
\(781\) −3.90391e9 −0.293239
\(782\) −3.33810e9 −0.249618
\(783\) −1.70370e9 −0.126831
\(784\) −1.02965e10 −0.763107
\(785\) −9.24897e7 −0.00682417
\(786\) −6.05944e8 −0.0445096
\(787\) −9.67486e9 −0.707511 −0.353755 0.935338i \(-0.615095\pi\)
−0.353755 + 0.935338i \(0.615095\pi\)
\(788\) 3.51522e9 0.255924
\(789\) 4.19958e9 0.304395
\(790\) −3.01365e7 −0.00217469
\(791\) −3.69194e10 −2.65239
\(792\) 5.61584e9 0.401677
\(793\) 6.10196e9 0.434524
\(794\) 2.14324e10 1.51949
\(795\) 1.56620e9 0.110551
\(796\) −8.00351e9 −0.562451
\(797\) 5.79578e9 0.405516 0.202758 0.979229i \(-0.435010\pi\)
0.202758 + 0.979229i \(0.435010\pi\)
\(798\) 8.71455e9 0.607064
\(799\) 3.22895e9 0.223948
\(800\) −9.44041e9 −0.651893
\(801\) −4.25507e9 −0.292545
\(802\) 9.44026e9 0.646210
\(803\) 1.15251e10 0.785486
\(804\) 5.49017e9 0.372554
\(805\) −2.49505e9 −0.168575
\(806\) −1.27800e10 −0.859724
\(807\) −5.56617e9 −0.372820
\(808\) 2.58401e10 1.72327
\(809\) −5.99978e9 −0.398396 −0.199198 0.979959i \(-0.563834\pi\)
−0.199198 + 0.979959i \(0.563834\pi\)
\(810\) −1.35884e8 −0.00898402
\(811\) 2.58658e10 1.70275 0.851377 0.524553i \(-0.175768\pi\)
0.851377 + 0.524553i \(0.175768\pi\)
\(812\) 5.44643e9 0.356998
\(813\) −6.84768e9 −0.446917
\(814\) 1.05103e10 0.683017
\(815\) −9.38566e8 −0.0607314
\(816\) 1.36463e9 0.0879226
\(817\) −2.01211e10 −1.29085
\(818\) −3.89323e9 −0.248699
\(819\) 5.37721e9 0.342029
\(820\) −7.28166e8 −0.0461192
\(821\) 6.65337e9 0.419605 0.209802 0.977744i \(-0.432718\pi\)
0.209802 + 0.977744i \(0.432718\pi\)
\(822\) 2.85382e9 0.179216
\(823\) 2.32319e10 1.45273 0.726366 0.687308i \(-0.241208\pi\)
0.726366 + 0.687308i \(0.241208\pi\)
\(824\) 2.23035e10 1.38876
\(825\) −1.02055e10 −0.632767
\(826\) 2.65786e9 0.164098
\(827\) 4.17843e9 0.256888 0.128444 0.991717i \(-0.459002\pi\)
0.128444 + 0.991717i \(0.459002\pi\)
\(828\) 2.04132e9 0.124970
\(829\) 1.91468e10 1.16722 0.583612 0.812033i \(-0.301639\pi\)
0.583612 + 0.812033i \(0.301639\pi\)
\(830\) −6.25256e8 −0.0379564
\(831\) −1.37434e10 −0.830787
\(832\) −1.16280e10 −0.699961
\(833\) 6.84445e9 0.410281
\(834\) 3.42805e9 0.204629
\(835\) −1.16673e9 −0.0693531
\(836\) 5.41722e9 0.320667
\(837\) −5.28198e9 −0.311356
\(838\) −8.87178e9 −0.520783
\(839\) 8.57931e9 0.501517 0.250758 0.968050i \(-0.419320\pi\)
0.250758 + 0.968050i \(0.419320\pi\)
\(840\) 1.68545e9 0.0981159
\(841\) −9.75781e9 −0.565674
\(842\) 2.98869e7 0.00172539
\(843\) 3.41328e9 0.196234
\(844\) −9.86082e9 −0.564566
\(845\) 9.95927e8 0.0567844
\(846\) 3.71215e9 0.210780
\(847\) −6.22438e9 −0.351969
\(848\) −1.80826e10 −1.01830
\(849\) −6.92651e9 −0.388452
\(850\) −4.09784e9 −0.228870
\(851\) 1.48232e10 0.824497
\(852\) 9.58594e8 0.0531002
\(853\) 1.55006e10 0.855118 0.427559 0.903987i \(-0.359374\pi\)
0.427559 + 0.903987i \(0.359374\pi\)
\(854\) 1.51568e10 0.832729
\(855\) −5.08579e8 −0.0278277
\(856\) −2.50091e10 −1.36282
\(857\) −2.35700e10 −1.27916 −0.639582 0.768723i \(-0.720893\pi\)
−0.639582 + 0.768723i \(0.720893\pi\)
\(858\) −6.28406e9 −0.339653
\(859\) −1.55175e10 −0.835307 −0.417654 0.908606i \(-0.637147\pi\)
−0.417654 + 0.908606i \(0.637147\pi\)
\(860\) −1.00299e9 −0.0537712
\(861\) −2.23891e10 −1.19543
\(862\) 9.10055e9 0.483941
\(863\) −1.31058e10 −0.694104 −0.347052 0.937846i \(-0.612817\pi\)
−0.347052 + 0.937846i \(0.612817\pi\)
\(864\) −2.40250e9 −0.126726
\(865\) 2.52743e9 0.132777
\(866\) −1.17642e10 −0.615531
\(867\) 1.01720e10 0.530079
\(868\) 1.68856e10 0.876390
\(869\) −5.76010e8 −0.0297756
\(870\) 5.97555e8 0.0307652
\(871\) −2.38364e10 −1.22230
\(872\) 1.30135e10 0.664642
\(873\) −4.65104e9 −0.236592
\(874\) −1.43633e10 −0.727720
\(875\) −6.15681e9 −0.310690
\(876\) −2.82994e9 −0.142237
\(877\) −3.93591e10 −1.97037 −0.985183 0.171509i \(-0.945136\pi\)
−0.985183 + 0.171509i \(0.945136\pi\)
\(878\) 5.39089e9 0.268800
\(879\) −2.64948e9 −0.131583
\(880\) −1.19201e9 −0.0589644
\(881\) −2.83704e10 −1.39781 −0.698907 0.715213i \(-0.746330\pi\)
−0.698907 + 0.715213i \(0.746330\pi\)
\(882\) 7.86871e9 0.386156
\(883\) −2.71601e10 −1.32760 −0.663802 0.747908i \(-0.731058\pi\)
−0.663802 + 0.747908i \(0.731058\pi\)
\(884\) 1.34218e9 0.0653473
\(885\) −1.55112e8 −0.00752220
\(886\) 3.28433e9 0.158646
\(887\) −2.72456e10 −1.31088 −0.655441 0.755246i \(-0.727517\pi\)
−0.655441 + 0.755246i \(0.727517\pi\)
\(888\) −1.00134e10 −0.479883
\(889\) −1.81268e10 −0.865297
\(890\) 1.49242e9 0.0709622
\(891\) −2.59720e9 −0.123008
\(892\) −9.78744e9 −0.461734
\(893\) 1.38936e10 0.652884
\(894\) −1.56409e9 −0.0732116
\(895\) 7.78530e7 0.00362990
\(896\) −6.76366e9 −0.314126
\(897\) −8.86270e9 −0.410008
\(898\) 2.34597e10 1.08108
\(899\) 2.32277e10 1.06622
\(900\) 2.50592e9 0.114583
\(901\) 1.20201e10 0.547484
\(902\) 2.61650e10 1.18713
\(903\) −3.08390e10 −1.39378
\(904\) −4.11057e10 −1.85060
\(905\) 3.58059e9 0.160577
\(906\) 1.48814e9 0.0664805
\(907\) −2.89542e10 −1.28850 −0.644251 0.764814i \(-0.722831\pi\)
−0.644251 + 0.764814i \(0.722831\pi\)
\(908\) −6.90409e9 −0.306059
\(909\) −1.19505e10 −0.527729
\(910\) −1.88600e9 −0.0829655
\(911\) 4.90888e9 0.215114 0.107557 0.994199i \(-0.465697\pi\)
0.107557 + 0.994199i \(0.465697\pi\)
\(912\) 5.87180e9 0.256324
\(913\) −1.19508e10 −0.519694
\(914\) −1.33457e10 −0.578137
\(915\) −8.84545e8 −0.0381721
\(916\) 9.94210e9 0.427409
\(917\) −3.47593e9 −0.148860
\(918\) −1.04286e9 −0.0444916
\(919\) −8.65519e8 −0.0367851 −0.0183926 0.999831i \(-0.505855\pi\)
−0.0183926 + 0.999831i \(0.505855\pi\)
\(920\) −2.77796e9 −0.117617
\(921\) −1.08373e10 −0.457101
\(922\) −4.51879e8 −0.0189873
\(923\) −4.16188e9 −0.174214
\(924\) 8.30281e9 0.346237
\(925\) 1.81969e10 0.755966
\(926\) −2.56267e10 −1.06061
\(927\) −1.03149e10 −0.425290
\(928\) 1.05651e10 0.433966
\(929\) 1.75729e10 0.719100 0.359550 0.933126i \(-0.382930\pi\)
0.359550 + 0.933126i \(0.382930\pi\)
\(930\) 1.85260e9 0.0755252
\(931\) 2.94506e10 1.19611
\(932\) −8.73512e9 −0.353438
\(933\) 2.50392e10 1.00933
\(934\) 2.70564e10 1.08657
\(935\) 7.92367e8 0.0317019
\(936\) 5.98693e9 0.238637
\(937\) −1.98729e9 −0.0789174 −0.0394587 0.999221i \(-0.512563\pi\)
−0.0394587 + 0.999221i \(0.512563\pi\)
\(938\) −5.92076e10 −2.34243
\(939\) −2.16343e10 −0.852731
\(940\) 6.92562e8 0.0271964
\(941\) 9.16263e9 0.358473 0.179237 0.983806i \(-0.442637\pi\)
0.179237 + 0.983806i \(0.442637\pi\)
\(942\) −8.16051e8 −0.0318082
\(943\) 3.69016e10 1.43303
\(944\) 1.79085e9 0.0692878
\(945\) −7.79484e8 −0.0300467
\(946\) 3.60400e10 1.38409
\(947\) −8.66497e9 −0.331545 −0.165772 0.986164i \(-0.553012\pi\)
−0.165772 + 0.986164i \(0.553012\pi\)
\(948\) 1.41437e8 0.00539182
\(949\) 1.22866e10 0.466660
\(950\) −1.76323e10 −0.667233
\(951\) 8.01858e9 0.302319
\(952\) 1.29353e10 0.485900
\(953\) 4.09329e10 1.53196 0.765979 0.642866i \(-0.222255\pi\)
0.765979 + 0.642866i \(0.222255\pi\)
\(954\) 1.38189e10 0.515292
\(955\) 4.92113e9 0.182832
\(956\) 1.43133e10 0.529829
\(957\) 1.14213e10 0.421234
\(958\) 1.97679e10 0.726411
\(959\) 1.63707e10 0.599379
\(960\) 1.68561e9 0.0614903
\(961\) 4.45004e10 1.61745
\(962\) 1.12049e10 0.405783
\(963\) 1.15661e10 0.417346
\(964\) −5.24142e9 −0.188443
\(965\) 2.52921e9 0.0906024
\(966\) −2.20142e10 −0.785748
\(967\) 4.75373e10 1.69060 0.845302 0.534289i \(-0.179421\pi\)
0.845302 + 0.534289i \(0.179421\pi\)
\(968\) −6.93016e9 −0.245572
\(969\) −3.90318e9 −0.137811
\(970\) 1.63131e9 0.0573897
\(971\) 4.18209e10 1.46597 0.732986 0.680244i \(-0.238126\pi\)
0.732986 + 0.680244i \(0.238126\pi\)
\(972\) 6.37735e8 0.0222745
\(973\) 1.96647e10 0.684372
\(974\) −2.67083e10 −0.926168
\(975\) −1.08798e10 −0.375929
\(976\) 1.02125e10 0.351608
\(977\) −3.63532e10 −1.24713 −0.623565 0.781771i \(-0.714316\pi\)
−0.623565 + 0.781771i \(0.714316\pi\)
\(978\) −8.28112e9 −0.283076
\(979\) 2.85253e10 0.971606
\(980\) 1.46804e9 0.0498248
\(981\) −6.01847e9 −0.203538
\(982\) 1.15844e10 0.390375
\(983\) 1.22719e10 0.412072 0.206036 0.978544i \(-0.433944\pi\)
0.206036 + 0.978544i \(0.433944\pi\)
\(984\) −2.49278e10 −0.834066
\(985\) 2.21237e9 0.0737617
\(986\) 4.58603e9 0.152359
\(987\) 2.12944e10 0.704944
\(988\) 5.77518e9 0.190509
\(989\) 5.08288e10 1.67079
\(990\) 9.10943e8 0.0298379
\(991\) −3.42031e10 −1.11637 −0.558185 0.829717i \(-0.688502\pi\)
−0.558185 + 0.829717i \(0.688502\pi\)
\(992\) 3.27550e10 1.06534
\(993\) −1.25777e10 −0.407642
\(994\) −1.03378e10 −0.333867
\(995\) −5.03716e9 −0.162108
\(996\) 2.93447e9 0.0941070
\(997\) 6.42072e9 0.205187 0.102594 0.994723i \(-0.467286\pi\)
0.102594 + 0.994723i \(0.467286\pi\)
\(998\) −1.88253e10 −0.599495
\(999\) 4.63096e9 0.146958
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 177.8.a.c.1.12 17
3.2 odd 2 531.8.a.c.1.6 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.c.1.12 17 1.1 even 1 trivial
531.8.a.c.1.6 17 3.2 odd 2