Properties

Label 387.6.a.e.1.2
Level $387$
Weight $6$
Character 387.1
Self dual yes
Analytic conductor $62.069$
Analytic rank $1$
Dimension $10$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [387,6,Mod(1,387)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(387, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("387.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 387.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(62.0685382676\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} - 256 x^{8} + 266 x^{7} + 21986 x^{6} - 10450 x^{5} - 719484 x^{4} + 384582 x^{3} + 8437093 x^{2} - 5752252 x - 22734604 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-8.57770\) of defining polynomial
Character \(\chi\) \(=\) 387.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-9.57770 q^{2} +59.7324 q^{4} -28.1028 q^{5} +195.604 q^{7} -265.613 q^{8} +O(q^{10})\) \(q-9.57770 q^{2} +59.7324 q^{4} -28.1028 q^{5} +195.604 q^{7} -265.613 q^{8} +269.160 q^{10} -72.8476 q^{11} +301.666 q^{13} -1873.44 q^{14} +632.525 q^{16} +1207.20 q^{17} -2350.21 q^{19} -1678.65 q^{20} +697.713 q^{22} -516.070 q^{23} -2335.23 q^{25} -2889.27 q^{26} +11683.9 q^{28} -1531.55 q^{29} +1126.13 q^{31} +2441.48 q^{32} -11562.2 q^{34} -5497.02 q^{35} -9339.18 q^{37} +22509.6 q^{38} +7464.47 q^{40} -19704.2 q^{41} +1849.00 q^{43} -4351.37 q^{44} +4942.77 q^{46} +13797.6 q^{47} +21453.9 q^{49} +22366.2 q^{50} +18019.2 q^{52} +3351.03 q^{53} +2047.22 q^{55} -51954.9 q^{56} +14668.7 q^{58} -2511.18 q^{59} +49249.3 q^{61} -10785.7 q^{62} -43624.6 q^{64} -8477.66 q^{65} +9116.54 q^{67} +72108.8 q^{68} +52648.8 q^{70} -43397.3 q^{71} +80067.4 q^{73} +89447.9 q^{74} -140384. q^{76} -14249.3 q^{77} -65991.7 q^{79} -17775.7 q^{80} +188721. q^{82} +76880.9 q^{83} -33925.6 q^{85} -17709.2 q^{86} +19349.3 q^{88} +75722.6 q^{89} +59007.1 q^{91} -30826.1 q^{92} -132150. q^{94} +66047.5 q^{95} -67921.7 q^{97} -205479. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 8 q^{2} + 202 q^{4} - 138 q^{5} + 60 q^{7} - 294 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 8 q^{2} + 202 q^{4} - 138 q^{5} + 60 q^{7} - 294 q^{8} - 17 q^{10} - 745 q^{11} + 1917 q^{13} - 1936 q^{14} + 5354 q^{16} - 4017 q^{17} - 2404 q^{19} - 1311 q^{20} - 5836 q^{22} - 1733 q^{23} + 7120 q^{25} + 1484 q^{26} - 15028 q^{28} - 6996 q^{29} - 4899 q^{31} + 7554 q^{32} - 27033 q^{34} - 7084 q^{35} + 1466 q^{37} - 13905 q^{38} - 93211 q^{40} - 10297 q^{41} + 18490 q^{43} + 36140 q^{44} + 17991 q^{46} - 48592 q^{47} + 29458 q^{49} - 983 q^{50} + 14232 q^{52} - 127165 q^{53} + 106672 q^{55} + 7780 q^{56} - 10305 q^{58} - 99372 q^{59} + 17408 q^{61} - 28265 q^{62} + 47202 q^{64} - 54484 q^{65} - 2021 q^{67} - 192151 q^{68} - 33194 q^{70} - 11286 q^{71} + 49892 q^{73} + 125431 q^{74} - 249803 q^{76} - 98144 q^{77} - 91524 q^{79} - 12251 q^{80} - 158909 q^{82} + 105203 q^{83} - 87212 q^{85} - 14792 q^{86} - 461824 q^{88} + 62682 q^{89} - 295304 q^{91} - 183783 q^{92} + 7259 q^{94} + 305340 q^{95} + 108383 q^{97} - 354656 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −9.57770 −1.69311 −0.846557 0.532297i \(-0.821329\pi\)
−0.846557 + 0.532297i \(0.821329\pi\)
\(3\) 0 0
\(4\) 59.7324 1.86664
\(5\) −28.1028 −0.502718 −0.251359 0.967894i \(-0.580878\pi\)
−0.251359 + 0.967894i \(0.580878\pi\)
\(6\) 0 0
\(7\) 195.604 1.50880 0.754401 0.656413i \(-0.227927\pi\)
0.754401 + 0.656413i \(0.227927\pi\)
\(8\) −265.613 −1.46732
\(9\) 0 0
\(10\) 269.160 0.851160
\(11\) −72.8476 −0.181524 −0.0907619 0.995873i \(-0.528930\pi\)
−0.0907619 + 0.995873i \(0.528930\pi\)
\(12\) 0 0
\(13\) 301.666 0.495072 0.247536 0.968879i \(-0.420379\pi\)
0.247536 + 0.968879i \(0.420379\pi\)
\(14\) −1873.44 −2.55458
\(15\) 0 0
\(16\) 632.525 0.617700
\(17\) 1207.20 1.01311 0.506554 0.862208i \(-0.330919\pi\)
0.506554 + 0.862208i \(0.330919\pi\)
\(18\) 0 0
\(19\) −2350.21 −1.49356 −0.746780 0.665071i \(-0.768401\pi\)
−0.746780 + 0.665071i \(0.768401\pi\)
\(20\) −1678.65 −0.938393
\(21\) 0 0
\(22\) 697.713 0.307341
\(23\) −516.070 −0.203418 −0.101709 0.994814i \(-0.532431\pi\)
−0.101709 + 0.994814i \(0.532431\pi\)
\(24\) 0 0
\(25\) −2335.23 −0.747274
\(26\) −2889.27 −0.838213
\(27\) 0 0
\(28\) 11683.9 2.81639
\(29\) −1531.55 −0.338171 −0.169085 0.985601i \(-0.554081\pi\)
−0.169085 + 0.985601i \(0.554081\pi\)
\(30\) 0 0
\(31\) 1126.13 0.210467 0.105234 0.994448i \(-0.466441\pi\)
0.105234 + 0.994448i \(0.466441\pi\)
\(32\) 2441.48 0.421481
\(33\) 0 0
\(34\) −11562.2 −1.71531
\(35\) −5497.02 −0.758503
\(36\) 0 0
\(37\) −9339.18 −1.12151 −0.560756 0.827981i \(-0.689490\pi\)
−0.560756 + 0.827981i \(0.689490\pi\)
\(38\) 22509.6 2.52877
\(39\) 0 0
\(40\) 7464.47 0.737647
\(41\) −19704.2 −1.83062 −0.915310 0.402750i \(-0.868054\pi\)
−0.915310 + 0.402750i \(0.868054\pi\)
\(42\) 0 0
\(43\) 1849.00 0.152499
\(44\) −4351.37 −0.338839
\(45\) 0 0
\(46\) 4942.77 0.344410
\(47\) 13797.6 0.911088 0.455544 0.890213i \(-0.349445\pi\)
0.455544 + 0.890213i \(0.349445\pi\)
\(48\) 0 0
\(49\) 21453.9 1.27649
\(50\) 22366.2 1.26522
\(51\) 0 0
\(52\) 18019.2 0.924120
\(53\) 3351.03 0.163866 0.0819330 0.996638i \(-0.473891\pi\)
0.0819330 + 0.996638i \(0.473891\pi\)
\(54\) 0 0
\(55\) 2047.22 0.0912554
\(56\) −51954.9 −2.21389
\(57\) 0 0
\(58\) 14668.7 0.572562
\(59\) −2511.18 −0.0939177 −0.0469588 0.998897i \(-0.514953\pi\)
−0.0469588 + 0.998897i \(0.514953\pi\)
\(60\) 0 0
\(61\) 49249.3 1.69463 0.847316 0.531088i \(-0.178217\pi\)
0.847316 + 0.531088i \(0.178217\pi\)
\(62\) −10785.7 −0.356345
\(63\) 0 0
\(64\) −43624.6 −1.33132
\(65\) −8477.66 −0.248882
\(66\) 0 0
\(67\) 9116.54 0.248109 0.124055 0.992275i \(-0.460410\pi\)
0.124055 + 0.992275i \(0.460410\pi\)
\(68\) 72108.8 1.89111
\(69\) 0 0
\(70\) 52648.8 1.28423
\(71\) −43397.3 −1.02168 −0.510842 0.859675i \(-0.670666\pi\)
−0.510842 + 0.859675i \(0.670666\pi\)
\(72\) 0 0
\(73\) 80067.4 1.75852 0.879262 0.476338i \(-0.158036\pi\)
0.879262 + 0.476338i \(0.158036\pi\)
\(74\) 89447.9 1.89885
\(75\) 0 0
\(76\) −140384. −2.78794
\(77\) −14249.3 −0.273884
\(78\) 0 0
\(79\) −65991.7 −1.18966 −0.594828 0.803853i \(-0.702780\pi\)
−0.594828 + 0.803853i \(0.702780\pi\)
\(80\) −17775.7 −0.310529
\(81\) 0 0
\(82\) 188721. 3.09945
\(83\) 76880.9 1.22496 0.612482 0.790484i \(-0.290171\pi\)
0.612482 + 0.790484i \(0.290171\pi\)
\(84\) 0 0
\(85\) −33925.6 −0.509308
\(86\) −17709.2 −0.258198
\(87\) 0 0
\(88\) 19349.3 0.266353
\(89\) 75722.6 1.01333 0.506665 0.862143i \(-0.330878\pi\)
0.506665 + 0.862143i \(0.330878\pi\)
\(90\) 0 0
\(91\) 59007.1 0.746965
\(92\) −30826.1 −0.379708
\(93\) 0 0
\(94\) −132150. −1.54258
\(95\) 66047.5 0.750840
\(96\) 0 0
\(97\) −67921.7 −0.732958 −0.366479 0.930426i \(-0.619437\pi\)
−0.366479 + 0.930426i \(0.619437\pi\)
\(98\) −205479. −2.16124
\(99\) 0 0
\(100\) −139489. −1.39489
\(101\) −130824. −1.27610 −0.638051 0.769994i \(-0.720259\pi\)
−0.638051 + 0.769994i \(0.720259\pi\)
\(102\) 0 0
\(103\) 41945.9 0.389580 0.194790 0.980845i \(-0.437597\pi\)
0.194790 + 0.980845i \(0.437597\pi\)
\(104\) −80126.4 −0.726428
\(105\) 0 0
\(106\) −32095.2 −0.277444
\(107\) 5277.45 0.0445620 0.0222810 0.999752i \(-0.492907\pi\)
0.0222810 + 0.999752i \(0.492907\pi\)
\(108\) 0 0
\(109\) −138680. −1.11802 −0.559008 0.829163i \(-0.688818\pi\)
−0.559008 + 0.829163i \(0.688818\pi\)
\(110\) −19607.7 −0.154506
\(111\) 0 0
\(112\) 123724. 0.931988
\(113\) −251325. −1.85157 −0.925783 0.378055i \(-0.876593\pi\)
−0.925783 + 0.378055i \(0.876593\pi\)
\(114\) 0 0
\(115\) 14503.0 0.102262
\(116\) −91483.1 −0.631242
\(117\) 0 0
\(118\) 24051.3 0.159013
\(119\) 236132. 1.52858
\(120\) 0 0
\(121\) −155744. −0.967049
\(122\) −471695. −2.86921
\(123\) 0 0
\(124\) 67266.5 0.392866
\(125\) 153448. 0.878387
\(126\) 0 0
\(127\) −175695. −0.966606 −0.483303 0.875453i \(-0.660563\pi\)
−0.483303 + 0.875453i \(0.660563\pi\)
\(128\) 339696. 1.83259
\(129\) 0 0
\(130\) 81196.5 0.421385
\(131\) −213758. −1.08829 −0.544144 0.838992i \(-0.683146\pi\)
−0.544144 + 0.838992i \(0.683146\pi\)
\(132\) 0 0
\(133\) −459711. −2.25349
\(134\) −87315.5 −0.420077
\(135\) 0 0
\(136\) −320647. −1.48655
\(137\) −126026. −0.573664 −0.286832 0.957981i \(-0.592602\pi\)
−0.286832 + 0.957981i \(0.592602\pi\)
\(138\) 0 0
\(139\) 181105. 0.795049 0.397524 0.917592i \(-0.369869\pi\)
0.397524 + 0.917592i \(0.369869\pi\)
\(140\) −328350. −1.41585
\(141\) 0 0
\(142\) 415646. 1.72983
\(143\) −21975.7 −0.0898673
\(144\) 0 0
\(145\) 43040.8 0.170004
\(146\) −766862. −2.97738
\(147\) 0 0
\(148\) −557852. −2.09346
\(149\) −187084. −0.690354 −0.345177 0.938538i \(-0.612181\pi\)
−0.345177 + 0.938538i \(0.612181\pi\)
\(150\) 0 0
\(151\) −396158. −1.41392 −0.706962 0.707252i \(-0.749935\pi\)
−0.706962 + 0.707252i \(0.749935\pi\)
\(152\) 624247. 2.19153
\(153\) 0 0
\(154\) 136475. 0.463717
\(155\) −31647.4 −0.105806
\(156\) 0 0
\(157\) 504286. 1.63278 0.816390 0.577501i \(-0.195972\pi\)
0.816390 + 0.577501i \(0.195972\pi\)
\(158\) 632049. 2.01423
\(159\) 0 0
\(160\) −68612.4 −0.211886
\(161\) −100945. −0.306917
\(162\) 0 0
\(163\) −466039. −1.37389 −0.686946 0.726708i \(-0.741049\pi\)
−0.686946 + 0.726708i \(0.741049\pi\)
\(164\) −1.17698e6 −3.41711
\(165\) 0 0
\(166\) −736343. −2.07401
\(167\) 142267. 0.394742 0.197371 0.980329i \(-0.436760\pi\)
0.197371 + 0.980329i \(0.436760\pi\)
\(168\) 0 0
\(169\) −280291. −0.754904
\(170\) 324930. 0.862317
\(171\) 0 0
\(172\) 110445. 0.284660
\(173\) −526184. −1.33666 −0.668332 0.743863i \(-0.732991\pi\)
−0.668332 + 0.743863i \(0.732991\pi\)
\(174\) 0 0
\(175\) −456781. −1.12749
\(176\) −46077.9 −0.112127
\(177\) 0 0
\(178\) −725249. −1.71568
\(179\) −692470. −1.61536 −0.807678 0.589624i \(-0.799276\pi\)
−0.807678 + 0.589624i \(0.799276\pi\)
\(180\) 0 0
\(181\) 286664. 0.650393 0.325197 0.945646i \(-0.394569\pi\)
0.325197 + 0.945646i \(0.394569\pi\)
\(182\) −565152. −1.26470
\(183\) 0 0
\(184\) 137075. 0.298479
\(185\) 262457. 0.563805
\(186\) 0 0
\(187\) −87941.4 −0.183903
\(188\) 824167. 1.70067
\(189\) 0 0
\(190\) −632584. −1.27126
\(191\) 692472. 1.37347 0.686734 0.726909i \(-0.259044\pi\)
0.686734 + 0.726909i \(0.259044\pi\)
\(192\) 0 0
\(193\) 855425. 1.65306 0.826530 0.562893i \(-0.190312\pi\)
0.826530 + 0.562893i \(0.190312\pi\)
\(194\) 650534. 1.24098
\(195\) 0 0
\(196\) 1.28149e6 2.38274
\(197\) 749649. 1.37623 0.688117 0.725600i \(-0.258438\pi\)
0.688117 + 0.725600i \(0.258438\pi\)
\(198\) 0 0
\(199\) 990082. 1.77230 0.886152 0.463394i \(-0.153369\pi\)
0.886152 + 0.463394i \(0.153369\pi\)
\(200\) 620268. 1.09649
\(201\) 0 0
\(202\) 1.25300e6 2.16059
\(203\) −299577. −0.510233
\(204\) 0 0
\(205\) 553742. 0.920286
\(206\) −401746. −0.659604
\(207\) 0 0
\(208\) 190811. 0.305806
\(209\) 171207. 0.271117
\(210\) 0 0
\(211\) −274555. −0.424545 −0.212272 0.977211i \(-0.568086\pi\)
−0.212272 + 0.977211i \(0.568086\pi\)
\(212\) 200165. 0.305879
\(213\) 0 0
\(214\) −50545.8 −0.0754486
\(215\) −51962.1 −0.0766638
\(216\) 0 0
\(217\) 220276. 0.317554
\(218\) 1.32824e6 1.89293
\(219\) 0 0
\(220\) 122286. 0.170341
\(221\) 364170. 0.501561
\(222\) 0 0
\(223\) −568751. −0.765878 −0.382939 0.923774i \(-0.625088\pi\)
−0.382939 + 0.923774i \(0.625088\pi\)
\(224\) 477563. 0.635932
\(225\) 0 0
\(226\) 2.40712e6 3.13492
\(227\) −607473. −0.782461 −0.391230 0.920293i \(-0.627950\pi\)
−0.391230 + 0.920293i \(0.627950\pi\)
\(228\) 0 0
\(229\) 104213. 0.131320 0.0656600 0.997842i \(-0.479085\pi\)
0.0656600 + 0.997842i \(0.479085\pi\)
\(230\) −138906. −0.173141
\(231\) 0 0
\(232\) 406799. 0.496204
\(233\) −1.28050e6 −1.54521 −0.772606 0.634885i \(-0.781047\pi\)
−0.772606 + 0.634885i \(0.781047\pi\)
\(234\) 0 0
\(235\) −387752. −0.458020
\(236\) −149999. −0.175310
\(237\) 0 0
\(238\) −2.26161e6 −2.58806
\(239\) −221000. −0.250264 −0.125132 0.992140i \(-0.539935\pi\)
−0.125132 + 0.992140i \(0.539935\pi\)
\(240\) 0 0
\(241\) −480522. −0.532931 −0.266465 0.963845i \(-0.585856\pi\)
−0.266465 + 0.963845i \(0.585856\pi\)
\(242\) 1.49167e6 1.63733
\(243\) 0 0
\(244\) 2.94178e6 3.16327
\(245\) −602915. −0.641713
\(246\) 0 0
\(247\) −708979. −0.739420
\(248\) −299115. −0.308822
\(249\) 0 0
\(250\) −1.46968e6 −1.48721
\(251\) −722732. −0.724091 −0.362045 0.932161i \(-0.617921\pi\)
−0.362045 + 0.932161i \(0.617921\pi\)
\(252\) 0 0
\(253\) 37594.5 0.0369252
\(254\) 1.68275e6 1.63657
\(255\) 0 0
\(256\) −1.85752e6 −1.77147
\(257\) 1.34492e6 1.27018 0.635089 0.772439i \(-0.280963\pi\)
0.635089 + 0.772439i \(0.280963\pi\)
\(258\) 0 0
\(259\) −1.82678e6 −1.69214
\(260\) −506391. −0.464572
\(261\) 0 0
\(262\) 2.04731e6 1.84260
\(263\) 18066.1 0.0161055 0.00805275 0.999968i \(-0.497437\pi\)
0.00805275 + 0.999968i \(0.497437\pi\)
\(264\) 0 0
\(265\) −94173.4 −0.0823784
\(266\) 4.40297e6 3.81542
\(267\) 0 0
\(268\) 544553. 0.463130
\(269\) 549502. 0.463008 0.231504 0.972834i \(-0.425635\pi\)
0.231504 + 0.972834i \(0.425635\pi\)
\(270\) 0 0
\(271\) 13732.4 0.0113586 0.00567929 0.999984i \(-0.498192\pi\)
0.00567929 + 0.999984i \(0.498192\pi\)
\(272\) 763582. 0.625797
\(273\) 0 0
\(274\) 1.20704e6 0.971278
\(275\) 170116. 0.135648
\(276\) 0 0
\(277\) −501657. −0.392833 −0.196416 0.980521i \(-0.562930\pi\)
−0.196416 + 0.980521i \(0.562930\pi\)
\(278\) −1.73457e6 −1.34611
\(279\) 0 0
\(280\) 1.46008e6 1.11296
\(281\) −605657. −0.457574 −0.228787 0.973477i \(-0.573476\pi\)
−0.228787 + 0.973477i \(0.573476\pi\)
\(282\) 0 0
\(283\) −1.85484e6 −1.37671 −0.688353 0.725376i \(-0.741666\pi\)
−0.688353 + 0.725376i \(0.741666\pi\)
\(284\) −2.59223e6 −1.90711
\(285\) 0 0
\(286\) 210476. 0.152156
\(287\) −3.85421e6 −2.76204
\(288\) 0 0
\(289\) 37467.4 0.0263882
\(290\) −412232. −0.287837
\(291\) 0 0
\(292\) 4.78262e6 3.28253
\(293\) −950968. −0.647138 −0.323569 0.946205i \(-0.604883\pi\)
−0.323569 + 0.946205i \(0.604883\pi\)
\(294\) 0 0
\(295\) 70571.1 0.0472141
\(296\) 2.48061e6 1.64562
\(297\) 0 0
\(298\) 1.79184e6 1.16885
\(299\) −155681. −0.100706
\(300\) 0 0
\(301\) 361672. 0.230090
\(302\) 3.79428e6 2.39393
\(303\) 0 0
\(304\) −1.48657e6 −0.922572
\(305\) −1.38404e6 −0.851923
\(306\) 0 0
\(307\) −736219. −0.445822 −0.222911 0.974839i \(-0.571556\pi\)
−0.222911 + 0.974839i \(0.571556\pi\)
\(308\) −851144. −0.511242
\(309\) 0 0
\(310\) 303110. 0.179141
\(311\) 1.02220e6 0.599286 0.299643 0.954051i \(-0.403133\pi\)
0.299643 + 0.954051i \(0.403133\pi\)
\(312\) 0 0
\(313\) −60474.7 −0.0348909 −0.0174455 0.999848i \(-0.505553\pi\)
−0.0174455 + 0.999848i \(0.505553\pi\)
\(314\) −4.82990e6 −2.76448
\(315\) 0 0
\(316\) −3.94184e6 −2.22066
\(317\) −842265. −0.470761 −0.235381 0.971903i \(-0.575634\pi\)
−0.235381 + 0.971903i \(0.575634\pi\)
\(318\) 0 0
\(319\) 111570. 0.0613860
\(320\) 1.22597e6 0.669277
\(321\) 0 0
\(322\) 966825. 0.519646
\(323\) −2.83717e6 −1.51314
\(324\) 0 0
\(325\) −704460. −0.369954
\(326\) 4.46358e6 2.32616
\(327\) 0 0
\(328\) 5.23368e6 2.68610
\(329\) 2.69887e6 1.37465
\(330\) 0 0
\(331\) −3.50942e6 −1.76062 −0.880308 0.474402i \(-0.842664\pi\)
−0.880308 + 0.474402i \(0.842664\pi\)
\(332\) 4.59228e6 2.28657
\(333\) 0 0
\(334\) −1.36259e6 −0.668344
\(335\) −256200. −0.124729
\(336\) 0 0
\(337\) −1.49987e6 −0.719414 −0.359707 0.933065i \(-0.617123\pi\)
−0.359707 + 0.933065i \(0.617123\pi\)
\(338\) 2.68454e6 1.27814
\(339\) 0 0
\(340\) −2.02646e6 −0.950694
\(341\) −82036.0 −0.0382048
\(342\) 0 0
\(343\) 908952. 0.417163
\(344\) −491118. −0.223764
\(345\) 0 0
\(346\) 5.03963e6 2.26313
\(347\) −2.16958e6 −0.967280 −0.483640 0.875267i \(-0.660686\pi\)
−0.483640 + 0.875267i \(0.660686\pi\)
\(348\) 0 0
\(349\) 393576. 0.172968 0.0864838 0.996253i \(-0.472437\pi\)
0.0864838 + 0.996253i \(0.472437\pi\)
\(350\) 4.37491e6 1.90897
\(351\) 0 0
\(352\) −177856. −0.0765089
\(353\) −272744. −0.116498 −0.0582491 0.998302i \(-0.518552\pi\)
−0.0582491 + 0.998302i \(0.518552\pi\)
\(354\) 0 0
\(355\) 1.21959e6 0.513619
\(356\) 4.52309e6 1.89152
\(357\) 0 0
\(358\) 6.63227e6 2.73498
\(359\) 3.96827e6 1.62504 0.812521 0.582932i \(-0.198094\pi\)
0.812521 + 0.582932i \(0.198094\pi\)
\(360\) 0 0
\(361\) 3.04739e6 1.23072
\(362\) −2.74558e6 −1.10119
\(363\) 0 0
\(364\) 3.52464e6 1.39431
\(365\) −2.25012e6 −0.884042
\(366\) 0 0
\(367\) −4.90395e6 −1.90056 −0.950278 0.311404i \(-0.899201\pi\)
−0.950278 + 0.311404i \(0.899201\pi\)
\(368\) −326427. −0.125651
\(369\) 0 0
\(370\) −2.51374e6 −0.954587
\(371\) 655475. 0.247242
\(372\) 0 0
\(373\) 2.52633e6 0.940196 0.470098 0.882614i \(-0.344219\pi\)
0.470098 + 0.882614i \(0.344219\pi\)
\(374\) 842277. 0.311369
\(375\) 0 0
\(376\) −3.66483e6 −1.33686
\(377\) −462016. −0.167419
\(378\) 0 0
\(379\) −607631. −0.217291 −0.108646 0.994081i \(-0.534651\pi\)
−0.108646 + 0.994081i \(0.534651\pi\)
\(380\) 3.94518e6 1.40155
\(381\) 0 0
\(382\) −6.63229e6 −2.32544
\(383\) 3.94370e6 1.37375 0.686874 0.726777i \(-0.258982\pi\)
0.686874 + 0.726777i \(0.258982\pi\)
\(384\) 0 0
\(385\) 400445. 0.137686
\(386\) −8.19301e6 −2.79882
\(387\) 0 0
\(388\) −4.05713e6 −1.36817
\(389\) 2.42464e6 0.812405 0.406203 0.913783i \(-0.366853\pi\)
0.406203 + 0.913783i \(0.366853\pi\)
\(390\) 0 0
\(391\) −622998. −0.206084
\(392\) −5.69843e6 −1.87301
\(393\) 0 0
\(394\) −7.17991e6 −2.33012
\(395\) 1.85455e6 0.598062
\(396\) 0 0
\(397\) −2.94322e6 −0.937232 −0.468616 0.883402i \(-0.655247\pi\)
−0.468616 + 0.883402i \(0.655247\pi\)
\(398\) −9.48271e6 −3.00072
\(399\) 0 0
\(400\) −1.47709e6 −0.461591
\(401\) 5.82714e6 1.80965 0.904824 0.425785i \(-0.140002\pi\)
0.904824 + 0.425785i \(0.140002\pi\)
\(402\) 0 0
\(403\) 339715. 0.104196
\(404\) −7.81446e6 −2.38202
\(405\) 0 0
\(406\) 2.86926e6 0.863883
\(407\) 680337. 0.203581
\(408\) 0 0
\(409\) 1.22650e6 0.362542 0.181271 0.983433i \(-0.441979\pi\)
0.181271 + 0.983433i \(0.441979\pi\)
\(410\) −5.30358e6 −1.55815
\(411\) 0 0
\(412\) 2.50553e6 0.727205
\(413\) −491196. −0.141703
\(414\) 0 0
\(415\) −2.16057e6 −0.615812
\(416\) 736511. 0.208663
\(417\) 0 0
\(418\) −1.63977e6 −0.459032
\(419\) −1.13490e6 −0.315809 −0.157904 0.987454i \(-0.550474\pi\)
−0.157904 + 0.987454i \(0.550474\pi\)
\(420\) 0 0
\(421\) −5.21555e6 −1.43415 −0.717075 0.696996i \(-0.754520\pi\)
−0.717075 + 0.696996i \(0.754520\pi\)
\(422\) 2.62961e6 0.718803
\(423\) 0 0
\(424\) −890078. −0.240444
\(425\) −2.81909e6 −0.757070
\(426\) 0 0
\(427\) 9.63336e6 2.55687
\(428\) 315235. 0.0831811
\(429\) 0 0
\(430\) 497677. 0.129801
\(431\) −4.94938e6 −1.28339 −0.641693 0.766961i \(-0.721768\pi\)
−0.641693 + 0.766961i \(0.721768\pi\)
\(432\) 0 0
\(433\) −2.02994e6 −0.520311 −0.260156 0.965567i \(-0.583774\pi\)
−0.260156 + 0.965567i \(0.583774\pi\)
\(434\) −2.10973e6 −0.537655
\(435\) 0 0
\(436\) −8.28369e6 −2.08693
\(437\) 1.21287e6 0.303817
\(438\) 0 0
\(439\) −4.05767e6 −1.00488 −0.502442 0.864611i \(-0.667565\pi\)
−0.502442 + 0.864611i \(0.667565\pi\)
\(440\) −543769. −0.133901
\(441\) 0 0
\(442\) −3.48792e6 −0.849201
\(443\) −3.73717e6 −0.904761 −0.452380 0.891825i \(-0.649425\pi\)
−0.452380 + 0.891825i \(0.649425\pi\)
\(444\) 0 0
\(445\) −2.12802e6 −0.509419
\(446\) 5.44733e6 1.29672
\(447\) 0 0
\(448\) −8.53313e6 −2.00869
\(449\) 2.08643e6 0.488414 0.244207 0.969723i \(-0.421472\pi\)
0.244207 + 0.969723i \(0.421472\pi\)
\(450\) 0 0
\(451\) 1.43540e6 0.332301
\(452\) −1.50122e7 −3.45620
\(453\) 0 0
\(454\) 5.81820e6 1.32480
\(455\) −1.65826e6 −0.375513
\(456\) 0 0
\(457\) 383012. 0.0857871 0.0428936 0.999080i \(-0.486342\pi\)
0.0428936 + 0.999080i \(0.486342\pi\)
\(458\) −998117. −0.222340
\(459\) 0 0
\(460\) 866300. 0.190886
\(461\) 767599. 0.168222 0.0841109 0.996456i \(-0.473195\pi\)
0.0841109 + 0.996456i \(0.473195\pi\)
\(462\) 0 0
\(463\) 2.87956e6 0.624272 0.312136 0.950037i \(-0.398955\pi\)
0.312136 + 0.950037i \(0.398955\pi\)
\(464\) −968743. −0.208888
\(465\) 0 0
\(466\) 1.22642e7 2.61622
\(467\) −4.73971e6 −1.00568 −0.502840 0.864379i \(-0.667712\pi\)
−0.502840 + 0.864379i \(0.667712\pi\)
\(468\) 0 0
\(469\) 1.78323e6 0.374348
\(470\) 3.71378e6 0.775481
\(471\) 0 0
\(472\) 667001. 0.137807
\(473\) −134695. −0.0276821
\(474\) 0 0
\(475\) 5.48829e6 1.11610
\(476\) 1.41048e7 2.85331
\(477\) 0 0
\(478\) 2.11668e6 0.423726
\(479\) −5.41230e6 −1.07781 −0.538906 0.842366i \(-0.681162\pi\)
−0.538906 + 0.842366i \(0.681162\pi\)
\(480\) 0 0
\(481\) −2.81731e6 −0.555229
\(482\) 4.60230e6 0.902313
\(483\) 0 0
\(484\) −9.30298e6 −1.80513
\(485\) 1.90879e6 0.368471
\(486\) 0 0
\(487\) 5.16701e6 0.987227 0.493614 0.869681i \(-0.335676\pi\)
0.493614 + 0.869681i \(0.335676\pi\)
\(488\) −1.30813e7 −2.48657
\(489\) 0 0
\(490\) 5.77454e6 1.08649
\(491\) −1.68112e6 −0.314698 −0.157349 0.987543i \(-0.550295\pi\)
−0.157349 + 0.987543i \(0.550295\pi\)
\(492\) 0 0
\(493\) −1.84888e6 −0.342603
\(494\) 6.79039e6 1.25192
\(495\) 0 0
\(496\) 712306. 0.130006
\(497\) −8.48868e6 −1.54152
\(498\) 0 0
\(499\) 3.57956e6 0.643545 0.321772 0.946817i \(-0.395721\pi\)
0.321772 + 0.946817i \(0.395721\pi\)
\(500\) 9.16581e6 1.63963
\(501\) 0 0
\(502\) 6.92211e6 1.22597
\(503\) 1.07972e6 0.190279 0.0951397 0.995464i \(-0.469670\pi\)
0.0951397 + 0.995464i \(0.469670\pi\)
\(504\) 0 0
\(505\) 3.67653e6 0.641520
\(506\) −360069. −0.0625186
\(507\) 0 0
\(508\) −1.04947e7 −1.80430
\(509\) −1.11760e6 −0.191202 −0.0956012 0.995420i \(-0.530477\pi\)
−0.0956012 + 0.995420i \(0.530477\pi\)
\(510\) 0 0
\(511\) 1.56615e7 2.65327
\(512\) 6.92051e6 1.16671
\(513\) 0 0
\(514\) −1.28813e7 −2.15056
\(515\) −1.17880e6 −0.195849
\(516\) 0 0
\(517\) −1.00513e6 −0.165384
\(518\) 1.74964e7 2.86499
\(519\) 0 0
\(520\) 2.25178e6 0.365188
\(521\) 6.50399e6 1.04975 0.524874 0.851180i \(-0.324112\pi\)
0.524874 + 0.851180i \(0.324112\pi\)
\(522\) 0 0
\(523\) 437675. 0.0699677 0.0349838 0.999388i \(-0.488862\pi\)
0.0349838 + 0.999388i \(0.488862\pi\)
\(524\) −1.27683e7 −2.03144
\(525\) 0 0
\(526\) −173031. −0.0272685
\(527\) 1.35946e6 0.213226
\(528\) 0 0
\(529\) −6.17001e6 −0.958621
\(530\) 901965. 0.139476
\(531\) 0 0
\(532\) −2.74596e7 −4.20645
\(533\) −5.94407e6 −0.906288
\(534\) 0 0
\(535\) −148311. −0.0224021
\(536\) −2.42147e6 −0.364055
\(537\) 0 0
\(538\) −5.26297e6 −0.783926
\(539\) −1.56287e6 −0.231713
\(540\) 0 0
\(541\) 6.56956e6 0.965035 0.482518 0.875886i \(-0.339722\pi\)
0.482518 + 0.875886i \(0.339722\pi\)
\(542\) −131525. −0.0192314
\(543\) 0 0
\(544\) 2.94735e6 0.427006
\(545\) 3.89730e6 0.562047
\(546\) 0 0
\(547\) −2.84947e6 −0.407189 −0.203595 0.979055i \(-0.565262\pi\)
−0.203595 + 0.979055i \(0.565262\pi\)
\(548\) −7.52781e6 −1.07082
\(549\) 0 0
\(550\) −1.62932e6 −0.229668
\(551\) 3.59946e6 0.505078
\(552\) 0 0
\(553\) −1.29082e7 −1.79496
\(554\) 4.80472e6 0.665111
\(555\) 0 0
\(556\) 1.08178e7 1.48407
\(557\) −1.14478e7 −1.56344 −0.781722 0.623627i \(-0.785658\pi\)
−0.781722 + 0.623627i \(0.785658\pi\)
\(558\) 0 0
\(559\) 557781. 0.0754977
\(560\) −3.47700e6 −0.468527
\(561\) 0 0
\(562\) 5.80080e6 0.774725
\(563\) −3.98082e6 −0.529300 −0.264650 0.964345i \(-0.585256\pi\)
−0.264650 + 0.964345i \(0.585256\pi\)
\(564\) 0 0
\(565\) 7.06293e6 0.930816
\(566\) 1.77652e7 2.33092
\(567\) 0 0
\(568\) 1.15269e7 1.49914
\(569\) −8.49999e6 −1.10062 −0.550311 0.834960i \(-0.685491\pi\)
−0.550311 + 0.834960i \(0.685491\pi\)
\(570\) 0 0
\(571\) 5.24847e6 0.673663 0.336831 0.941565i \(-0.390645\pi\)
0.336831 + 0.941565i \(0.390645\pi\)
\(572\) −1.31266e6 −0.167750
\(573\) 0 0
\(574\) 3.69145e7 4.67646
\(575\) 1.20514e6 0.152009
\(576\) 0 0
\(577\) 1.94309e6 0.242970 0.121485 0.992593i \(-0.461234\pi\)
0.121485 + 0.992593i \(0.461234\pi\)
\(578\) −358852. −0.0446782
\(579\) 0 0
\(580\) 2.57093e6 0.317337
\(581\) 1.50382e7 1.84823
\(582\) 0 0
\(583\) −244115. −0.0297456
\(584\) −2.12669e7 −2.58031
\(585\) 0 0
\(586\) 9.10809e6 1.09568
\(587\) −924441. −0.110735 −0.0553674 0.998466i \(-0.517633\pi\)
−0.0553674 + 0.998466i \(0.517633\pi\)
\(588\) 0 0
\(589\) −2.64665e6 −0.314346
\(590\) −675909. −0.0799389
\(591\) 0 0
\(592\) −5.90726e6 −0.692759
\(593\) 1.09140e7 1.27453 0.637263 0.770646i \(-0.280066\pi\)
0.637263 + 0.770646i \(0.280066\pi\)
\(594\) 0 0
\(595\) −6.63598e6 −0.768445
\(596\) −1.11750e7 −1.28864
\(597\) 0 0
\(598\) 1.49106e6 0.170508
\(599\) 6.47200e6 0.737006 0.368503 0.929626i \(-0.379870\pi\)
0.368503 + 0.929626i \(0.379870\pi\)
\(600\) 0 0
\(601\) −9.66608e6 −1.09160 −0.545801 0.837915i \(-0.683775\pi\)
−0.545801 + 0.837915i \(0.683775\pi\)
\(602\) −3.46398e6 −0.389569
\(603\) 0 0
\(604\) −2.36635e7 −2.63928
\(605\) 4.37685e6 0.486153
\(606\) 0 0
\(607\) 1.50825e7 1.66151 0.830753 0.556641i \(-0.187910\pi\)
0.830753 + 0.556641i \(0.187910\pi\)
\(608\) −5.73799e6 −0.629507
\(609\) 0 0
\(610\) 1.32560e7 1.44240
\(611\) 4.16228e6 0.451054
\(612\) 0 0
\(613\) −2.07592e6 −0.223131 −0.111565 0.993757i \(-0.535586\pi\)
−0.111565 + 0.993757i \(0.535586\pi\)
\(614\) 7.05129e6 0.754827
\(615\) 0 0
\(616\) 3.78479e6 0.401875
\(617\) −1.05100e7 −1.11145 −0.555727 0.831365i \(-0.687560\pi\)
−0.555727 + 0.831365i \(0.687560\pi\)
\(618\) 0 0
\(619\) −6.33902e6 −0.664960 −0.332480 0.943110i \(-0.607885\pi\)
−0.332480 + 0.943110i \(0.607885\pi\)
\(620\) −1.89038e6 −0.197501
\(621\) 0 0
\(622\) −9.79030e6 −1.01466
\(623\) 1.48116e7 1.52891
\(624\) 0 0
\(625\) 2.98529e6 0.305694
\(626\) 579208. 0.0590744
\(627\) 0 0
\(628\) 3.01222e7 3.04781
\(629\) −1.12742e7 −1.13621
\(630\) 0 0
\(631\) −1.15541e7 −1.15521 −0.577606 0.816316i \(-0.696013\pi\)
−0.577606 + 0.816316i \(0.696013\pi\)
\(632\) 1.75283e7 1.74560
\(633\) 0 0
\(634\) 8.06697e6 0.797053
\(635\) 4.93751e6 0.485930
\(636\) 0 0
\(637\) 6.47191e6 0.631952
\(638\) −1.06858e6 −0.103934
\(639\) 0 0
\(640\) −9.54640e6 −0.921276
\(641\) 3.92627e6 0.377429 0.188714 0.982032i \(-0.439568\pi\)
0.188714 + 0.982032i \(0.439568\pi\)
\(642\) 0 0
\(643\) −1.87570e7 −1.78911 −0.894553 0.446961i \(-0.852506\pi\)
−0.894553 + 0.446961i \(0.852506\pi\)
\(644\) −6.02971e6 −0.572904
\(645\) 0 0
\(646\) 2.71736e7 2.56192
\(647\) 9.03463e6 0.848496 0.424248 0.905546i \(-0.360538\pi\)
0.424248 + 0.905546i \(0.360538\pi\)
\(648\) 0 0
\(649\) 182933. 0.0170483
\(650\) 6.74711e6 0.626375
\(651\) 0 0
\(652\) −2.78376e7 −2.56456
\(653\) −1.74423e7 −1.60074 −0.800371 0.599505i \(-0.795364\pi\)
−0.800371 + 0.599505i \(0.795364\pi\)
\(654\) 0 0
\(655\) 6.00720e6 0.547103
\(656\) −1.24634e7 −1.13077
\(657\) 0 0
\(658\) −2.58490e7 −2.32744
\(659\) −1.74636e7 −1.56646 −0.783230 0.621732i \(-0.786429\pi\)
−0.783230 + 0.621732i \(0.786429\pi\)
\(660\) 0 0
\(661\) 5.12626e6 0.456348 0.228174 0.973620i \(-0.426724\pi\)
0.228174 + 0.973620i \(0.426724\pi\)
\(662\) 3.36121e7 2.98093
\(663\) 0 0
\(664\) −2.04206e7 −1.79741
\(665\) 1.29192e7 1.13287
\(666\) 0 0
\(667\) 790386. 0.0687899
\(668\) 8.49796e6 0.736841
\(669\) 0 0
\(670\) 2.45381e6 0.211181
\(671\) −3.58770e6 −0.307616
\(672\) 0 0
\(673\) 4.98529e6 0.424280 0.212140 0.977239i \(-0.431957\pi\)
0.212140 + 0.977239i \(0.431957\pi\)
\(674\) 1.43653e7 1.21805
\(675\) 0 0
\(676\) −1.67424e7 −1.40913
\(677\) −8.15170e6 −0.683560 −0.341780 0.939780i \(-0.611030\pi\)
−0.341780 + 0.939780i \(0.611030\pi\)
\(678\) 0 0
\(679\) −1.32857e7 −1.10589
\(680\) 9.01108e6 0.747317
\(681\) 0 0
\(682\) 785716. 0.0646852
\(683\) 3.28051e6 0.269085 0.134543 0.990908i \(-0.457043\pi\)
0.134543 + 0.990908i \(0.457043\pi\)
\(684\) 0 0
\(685\) 3.54167e6 0.288391
\(686\) −8.70567e6 −0.706305
\(687\) 0 0
\(688\) 1.16954e6 0.0941984
\(689\) 1.01089e6 0.0811254
\(690\) 0 0
\(691\) 1.54825e7 1.23352 0.616758 0.787153i \(-0.288446\pi\)
0.616758 + 0.787153i \(0.288446\pi\)
\(692\) −3.14302e7 −2.49507
\(693\) 0 0
\(694\) 2.07796e7 1.63772
\(695\) −5.08956e6 −0.399685
\(696\) 0 0
\(697\) −2.37868e7 −1.85462
\(698\) −3.76955e6 −0.292854
\(699\) 0 0
\(700\) −2.72846e7 −2.10462
\(701\) 4.91809e6 0.378008 0.189004 0.981976i \(-0.439474\pi\)
0.189004 + 0.981976i \(0.439474\pi\)
\(702\) 0 0
\(703\) 2.19490e7 1.67505
\(704\) 3.17795e6 0.241666
\(705\) 0 0
\(706\) 2.61227e6 0.197245
\(707\) −2.55898e7 −1.92539
\(708\) 0 0
\(709\) −8.69925e6 −0.649930 −0.324965 0.945726i \(-0.605352\pi\)
−0.324965 + 0.945726i \(0.605352\pi\)
\(710\) −1.16808e7 −0.869616
\(711\) 0 0
\(712\) −2.01129e7 −1.48688
\(713\) −581162. −0.0428128
\(714\) 0 0
\(715\) 617578. 0.0451779
\(716\) −4.13629e7 −3.01529
\(717\) 0 0
\(718\) −3.80069e7 −2.75138
\(719\) 8.92512e6 0.643861 0.321930 0.946763i \(-0.395668\pi\)
0.321930 + 0.946763i \(0.395668\pi\)
\(720\) 0 0
\(721\) 8.20479e6 0.587799
\(722\) −2.91870e7 −2.08376
\(723\) 0 0
\(724\) 1.71231e7 1.21405
\(725\) 3.57652e6 0.252706
\(726\) 0 0
\(727\) 1.33205e7 0.934729 0.467365 0.884065i \(-0.345204\pi\)
0.467365 + 0.884065i \(0.345204\pi\)
\(728\) −1.56730e7 −1.09604
\(729\) 0 0
\(730\) 2.15510e7 1.49678
\(731\) 2.23211e6 0.154498
\(732\) 0 0
\(733\) 1.41244e7 0.970981 0.485490 0.874242i \(-0.338641\pi\)
0.485490 + 0.874242i \(0.338641\pi\)
\(734\) 4.69685e7 3.21786
\(735\) 0 0
\(736\) −1.25997e6 −0.0857368
\(737\) −664118. −0.0450377
\(738\) 0 0
\(739\) −1.65506e7 −1.11482 −0.557408 0.830239i \(-0.688204\pi\)
−0.557408 + 0.830239i \(0.688204\pi\)
\(740\) 1.56772e7 1.05242
\(741\) 0 0
\(742\) −6.27795e6 −0.418608
\(743\) −1.32085e7 −0.877771 −0.438885 0.898543i \(-0.644627\pi\)
−0.438885 + 0.898543i \(0.644627\pi\)
\(744\) 0 0
\(745\) 5.25759e6 0.347054
\(746\) −2.41965e7 −1.59186
\(747\) 0 0
\(748\) −5.25296e6 −0.343281
\(749\) 1.03229e6 0.0672353
\(750\) 0 0
\(751\) 1.62740e6 0.105291 0.0526457 0.998613i \(-0.483235\pi\)
0.0526457 + 0.998613i \(0.483235\pi\)
\(752\) 8.72735e6 0.562779
\(753\) 0 0
\(754\) 4.42506e6 0.283459
\(755\) 1.11331e7 0.710805
\(756\) 0 0
\(757\) −1.41817e7 −0.899472 −0.449736 0.893162i \(-0.648482\pi\)
−0.449736 + 0.893162i \(0.648482\pi\)
\(758\) 5.81971e6 0.367899
\(759\) 0 0
\(760\) −1.75431e7 −1.10172
\(761\) 631434. 0.0395245 0.0197622 0.999805i \(-0.493709\pi\)
0.0197622 + 0.999805i \(0.493709\pi\)
\(762\) 0 0
\(763\) −2.71264e7 −1.68686
\(764\) 4.13630e7 2.56377
\(765\) 0 0
\(766\) −3.77716e7 −2.32591
\(767\) −757537. −0.0464960
\(768\) 0 0
\(769\) 2920.58 0.000178096 0 8.90480e−5 1.00000i \(-0.499972\pi\)
8.90480e−5 1.00000i \(0.499972\pi\)
\(770\) −3.83534e6 −0.233119
\(771\) 0 0
\(772\) 5.10966e7 3.08566
\(773\) 8.64224e6 0.520209 0.260104 0.965580i \(-0.416243\pi\)
0.260104 + 0.965580i \(0.416243\pi\)
\(774\) 0 0
\(775\) −2.62978e6 −0.157277
\(776\) 1.80409e7 1.07548
\(777\) 0 0
\(778\) −2.32225e7 −1.37550
\(779\) 4.63089e7 2.73414
\(780\) 0 0
\(781\) 3.16139e6 0.185460
\(782\) 5.96689e6 0.348924
\(783\) 0 0
\(784\) 1.35701e7 0.788485
\(785\) −1.41718e7 −0.820828
\(786\) 0 0
\(787\) 6.88838e6 0.396443 0.198221 0.980157i \(-0.436484\pi\)
0.198221 + 0.980157i \(0.436484\pi\)
\(788\) 4.47783e7 2.56893
\(789\) 0 0
\(790\) −1.77623e7 −1.01259
\(791\) −4.91601e7 −2.79365
\(792\) 0 0
\(793\) 1.48568e7 0.838965
\(794\) 2.81893e7 1.58684
\(795\) 0 0
\(796\) 5.91400e7 3.30825
\(797\) −2.93617e7 −1.63733 −0.818664 0.574273i \(-0.805285\pi\)
−0.818664 + 0.574273i \(0.805285\pi\)
\(798\) 0 0
\(799\) 1.66565e7 0.923030
\(800\) −5.70142e6 −0.314962
\(801\) 0 0
\(802\) −5.58106e7 −3.06394
\(803\) −5.83272e6 −0.319214
\(804\) 0 0
\(805\) 2.83685e6 0.154293
\(806\) −3.25369e6 −0.176416
\(807\) 0 0
\(808\) 3.47487e7 1.87245
\(809\) 2.84195e6 0.152667 0.0763334 0.997082i \(-0.475679\pi\)
0.0763334 + 0.997082i \(0.475679\pi\)
\(810\) 0 0
\(811\) 1.67390e6 0.0893670 0.0446835 0.999001i \(-0.485772\pi\)
0.0446835 + 0.999001i \(0.485772\pi\)
\(812\) −1.78945e7 −0.952420
\(813\) 0 0
\(814\) −6.51607e6 −0.344687
\(815\) 1.30970e7 0.690681
\(816\) 0 0
\(817\) −4.34554e6 −0.227766
\(818\) −1.17470e7 −0.613825
\(819\) 0 0
\(820\) 3.30763e7 1.71784
\(821\) −3.75970e7 −1.94668 −0.973340 0.229365i \(-0.926335\pi\)
−0.973340 + 0.229365i \(0.926335\pi\)
\(822\) 0 0
\(823\) 1.67263e7 0.860797 0.430398 0.902639i \(-0.358373\pi\)
0.430398 + 0.902639i \(0.358373\pi\)
\(824\) −1.11414e7 −0.571638
\(825\) 0 0
\(826\) 4.70453e6 0.239920
\(827\) 2.69145e7 1.36843 0.684216 0.729279i \(-0.260144\pi\)
0.684216 + 0.729279i \(0.260144\pi\)
\(828\) 0 0
\(829\) −7.34223e6 −0.371058 −0.185529 0.982639i \(-0.559400\pi\)
−0.185529 + 0.982639i \(0.559400\pi\)
\(830\) 2.06933e7 1.04264
\(831\) 0 0
\(832\) −1.31600e7 −0.659097
\(833\) 2.58991e7 1.29322
\(834\) 0 0
\(835\) −3.99811e6 −0.198444
\(836\) 1.02266e7 0.506077
\(837\) 0 0
\(838\) 1.08698e7 0.534701
\(839\) 1.44619e7 0.709284 0.354642 0.935002i \(-0.384603\pi\)
0.354642 + 0.935002i \(0.384603\pi\)
\(840\) 0 0
\(841\) −1.81655e7 −0.885641
\(842\) 4.99530e7 2.42818
\(843\) 0 0
\(844\) −1.63998e7 −0.792471
\(845\) 7.87695e6 0.379504
\(846\) 0 0
\(847\) −3.04642e7 −1.45909
\(848\) 2.11961e6 0.101220
\(849\) 0 0
\(850\) 2.70004e7 1.28181
\(851\) 4.81967e6 0.228136
\(852\) 0 0
\(853\) −3.00835e6 −0.141565 −0.0707825 0.997492i \(-0.522550\pi\)
−0.0707825 + 0.997492i \(0.522550\pi\)
\(854\) −9.22655e7 −4.32907
\(855\) 0 0
\(856\) −1.40176e6 −0.0653866
\(857\) −5.35883e6 −0.249240 −0.124620 0.992205i \(-0.539771\pi\)
−0.124620 + 0.992205i \(0.539771\pi\)
\(858\) 0 0
\(859\) 2.59898e7 1.20176 0.600882 0.799338i \(-0.294816\pi\)
0.600882 + 0.799338i \(0.294816\pi\)
\(860\) −3.10382e6 −0.143104
\(861\) 0 0
\(862\) 4.74037e7 2.17292
\(863\) 4.32743e7 1.97789 0.988947 0.148266i \(-0.0473693\pi\)
0.988947 + 0.148266i \(0.0473693\pi\)
\(864\) 0 0
\(865\) 1.47872e7 0.671965
\(866\) 1.94421e7 0.880947
\(867\) 0 0
\(868\) 1.31576e7 0.592758
\(869\) 4.80734e6 0.215951
\(870\) 0 0
\(871\) 2.75015e6 0.122832
\(872\) 3.68352e7 1.64048
\(873\) 0 0
\(874\) −1.16165e7 −0.514397
\(875\) 3.00150e7 1.32531
\(876\) 0 0
\(877\) 1.03640e7 0.455017 0.227508 0.973776i \(-0.426942\pi\)
0.227508 + 0.973776i \(0.426942\pi\)
\(878\) 3.88632e7 1.70138
\(879\) 0 0
\(880\) 1.29492e6 0.0563684
\(881\) 2.43175e7 1.05555 0.527776 0.849384i \(-0.323026\pi\)
0.527776 + 0.849384i \(0.323026\pi\)
\(882\) 0 0
\(883\) 1.78896e7 0.772147 0.386073 0.922468i \(-0.373831\pi\)
0.386073 + 0.922468i \(0.373831\pi\)
\(884\) 2.17528e7 0.936233
\(885\) 0 0
\(886\) 3.57935e7 1.53186
\(887\) 2.17492e7 0.928185 0.464092 0.885787i \(-0.346381\pi\)
0.464092 + 0.885787i \(0.346381\pi\)
\(888\) 0 0
\(889\) −3.43666e7 −1.45842
\(890\) 2.03815e7 0.862505
\(891\) 0 0
\(892\) −3.39729e7 −1.42962
\(893\) −3.24274e7 −1.36076
\(894\) 0 0
\(895\) 1.94603e7 0.812069
\(896\) 6.64458e7 2.76502
\(897\) 0 0
\(898\) −1.99832e7 −0.826941
\(899\) −1.72472e6 −0.0711738
\(900\) 0 0
\(901\) 4.04536e6 0.166014
\(902\) −1.37478e7 −0.562624
\(903\) 0 0
\(904\) 6.67551e7 2.71684
\(905\) −8.05605e6 −0.326965
\(906\) 0 0
\(907\) 2.98499e7 1.20483 0.602414 0.798184i \(-0.294206\pi\)
0.602414 + 0.798184i \(0.294206\pi\)
\(908\) −3.62859e7 −1.46057
\(909\) 0 0
\(910\) 1.58824e7 0.635787
\(911\) 1.09193e7 0.435912 0.217956 0.975959i \(-0.430061\pi\)
0.217956 + 0.975959i \(0.430061\pi\)
\(912\) 0 0
\(913\) −5.60060e6 −0.222360
\(914\) −3.66838e6 −0.145247
\(915\) 0 0
\(916\) 6.22487e6 0.245127
\(917\) −4.18119e7 −1.64201
\(918\) 0 0
\(919\) 9.16828e6 0.358096 0.179048 0.983840i \(-0.442698\pi\)
0.179048 + 0.983840i \(0.442698\pi\)
\(920\) −3.85219e6 −0.150051
\(921\) 0 0
\(922\) −7.35184e6 −0.284819
\(923\) −1.30915e7 −0.505807
\(924\) 0 0
\(925\) 2.18091e7 0.838078
\(926\) −2.75796e7 −1.05697
\(927\) 0 0
\(928\) −3.73924e6 −0.142532
\(929\) 1.38913e6 0.0528083 0.0264041 0.999651i \(-0.491594\pi\)
0.0264041 + 0.999651i \(0.491594\pi\)
\(930\) 0 0
\(931\) −5.04212e7 −1.90651
\(932\) −7.64871e7 −2.88435
\(933\) 0 0
\(934\) 4.53956e7 1.70273
\(935\) 2.47140e6 0.0924515
\(936\) 0 0
\(937\) −2.52918e7 −0.941090 −0.470545 0.882376i \(-0.655943\pi\)
−0.470545 + 0.882376i \(0.655943\pi\)
\(938\) −1.70793e7 −0.633814
\(939\) 0 0
\(940\) −2.31614e7 −0.854958
\(941\) −2.69608e7 −0.992563 −0.496282 0.868162i \(-0.665302\pi\)
−0.496282 + 0.868162i \(0.665302\pi\)
\(942\) 0 0
\(943\) 1.01687e7 0.372381
\(944\) −1.58838e6 −0.0580129
\(945\) 0 0
\(946\) 1.29007e6 0.0468690
\(947\) 3.19275e6 0.115689 0.0578443 0.998326i \(-0.481577\pi\)
0.0578443 + 0.998326i \(0.481577\pi\)
\(948\) 0 0
\(949\) 2.41536e7 0.870596
\(950\) −5.25652e7 −1.88969
\(951\) 0 0
\(952\) −6.27198e7 −2.24291
\(953\) 3.59618e7 1.28265 0.641327 0.767268i \(-0.278384\pi\)
0.641327 + 0.767268i \(0.278384\pi\)
\(954\) 0 0
\(955\) −1.94604e7 −0.690468
\(956\) −1.32009e7 −0.467152
\(957\) 0 0
\(958\) 5.18374e7 1.82486
\(959\) −2.46511e7 −0.865545
\(960\) 0 0
\(961\) −2.73610e7 −0.955704
\(962\) 2.69834e7 0.940067
\(963\) 0 0
\(964\) −2.87027e7 −0.994789
\(965\) −2.40398e7 −0.831023
\(966\) 0 0
\(967\) −1.05454e7 −0.362659 −0.181330 0.983422i \(-0.558040\pi\)
−0.181330 + 0.983422i \(0.558040\pi\)
\(968\) 4.13677e7 1.41897
\(969\) 0 0
\(970\) −1.82818e7 −0.623864
\(971\) 1.47340e7 0.501500 0.250750 0.968052i \(-0.419323\pi\)
0.250750 + 0.968052i \(0.419323\pi\)
\(972\) 0 0
\(973\) 3.54249e7 1.19957
\(974\) −4.94881e7 −1.67149
\(975\) 0 0
\(976\) 3.11514e7 1.04677
\(977\) −4.83492e7 −1.62051 −0.810257 0.586075i \(-0.800673\pi\)
−0.810257 + 0.586075i \(0.800673\pi\)
\(978\) 0 0
\(979\) −5.51621e6 −0.183943
\(980\) −3.60136e7 −1.19785
\(981\) 0 0
\(982\) 1.61012e7 0.532820
\(983\) 1.25654e7 0.414754 0.207377 0.978261i \(-0.433507\pi\)
0.207377 + 0.978261i \(0.433507\pi\)
\(984\) 0 0
\(985\) −2.10672e7 −0.691858
\(986\) 1.77080e7 0.580067
\(987\) 0 0
\(988\) −4.23490e7 −1.38023
\(989\) −954214. −0.0310209
\(990\) 0 0
\(991\) −4.25363e7 −1.37586 −0.687932 0.725776i \(-0.741481\pi\)
−0.687932 + 0.725776i \(0.741481\pi\)
\(992\) 2.74942e6 0.0887080
\(993\) 0 0
\(994\) 8.13021e7 2.60997
\(995\) −2.78241e7 −0.890970
\(996\) 0 0
\(997\) −3.70772e7 −1.18132 −0.590662 0.806919i \(-0.701133\pi\)
−0.590662 + 0.806919i \(0.701133\pi\)
\(998\) −3.42840e7 −1.08960
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 387.6.a.e.1.2 10
3.2 odd 2 43.6.a.b.1.9 10
12.11 even 2 688.6.a.h.1.7 10
15.14 odd 2 1075.6.a.b.1.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.6.a.b.1.9 10 3.2 odd 2
387.6.a.e.1.2 10 1.1 even 1 trivial
688.6.a.h.1.7 10 12.11 even 2
1075.6.a.b.1.2 10 15.14 odd 2