Properties

Label 729.2.a.c.1.1
Level $729$
Weight $2$
Character 729.1
Self dual yes
Analytic conductor $5.821$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 729 = 3^{6} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 729.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.82109430735\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{36})^+\)
Defining polynomial: \( x^{6} - 6x^{4} + 9x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.96962\) of defining polynomial
Character \(\chi\) \(=\) 729.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.96962 q^{2} +1.87939 q^{4} +3.70167 q^{5} -2.34730 q^{7} +0.237565 q^{8} +O(q^{10})\) \(q-1.96962 q^{2} +1.87939 q^{4} +3.70167 q^{5} -2.34730 q^{7} +0.237565 q^{8} -7.29086 q^{10} -2.17853 q^{11} -4.71688 q^{13} +4.62327 q^{14} -4.22668 q^{16} -2.93512 q^{17} -6.22668 q^{19} +6.95686 q^{20} +4.29086 q^{22} +0.519030 q^{23} +8.70233 q^{25} +9.29044 q^{26} -4.41147 q^{28} -3.49276 q^{29} -4.30541 q^{31} +7.84981 q^{32} +5.78106 q^{34} -8.68891 q^{35} +2.41147 q^{37} +12.2642 q^{38} +0.879385 q^{40} +2.49860 q^{41} +1.06418 q^{43} -4.09429 q^{44} -1.02229 q^{46} -0.237565 q^{47} -1.49020 q^{49} -17.1403 q^{50} -8.86484 q^{52} +4.66717 q^{53} -8.06418 q^{55} -0.557635 q^{56} +6.87939 q^{58} -13.3122 q^{59} -3.67499 q^{61} +8.48000 q^{62} -7.00774 q^{64} -17.4603 q^{65} +14.2986 q^{67} -5.51622 q^{68} +17.1138 q^{70} +1.20307 q^{71} -4.68004 q^{73} -4.74968 q^{74} -11.7023 q^{76} +5.11365 q^{77} -12.8007 q^{79} -15.6458 q^{80} -4.92127 q^{82} -11.3040 q^{83} -10.8648 q^{85} -2.09602 q^{86} -0.517541 q^{88} +0.699287 q^{89} +11.0719 q^{91} +0.975457 q^{92} +0.467911 q^{94} -23.0491 q^{95} -7.08647 q^{97} +2.93512 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 12 q^{7} - 12 q^{10} - 12 q^{13} - 12 q^{16} - 24 q^{19} - 6 q^{22} - 6 q^{28} - 30 q^{31} - 6 q^{37} - 6 q^{40} - 12 q^{43} + 6 q^{46} - 6 q^{49} - 6 q^{52} - 30 q^{55} + 30 q^{58} - 12 q^{61} + 6 q^{64} + 6 q^{67} + 30 q^{70} + 12 q^{73} - 18 q^{76} - 48 q^{79} - 12 q^{82} - 18 q^{85} + 42 q^{88} + 12 q^{94} - 12 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.96962 −1.39273 −0.696364 0.717689i \(-0.745200\pi\)
−0.696364 + 0.717689i \(0.745200\pi\)
\(3\) 0 0
\(4\) 1.87939 0.939693
\(5\) 3.70167 1.65544 0.827718 0.561145i \(-0.189639\pi\)
0.827718 + 0.561145i \(0.189639\pi\)
\(6\) 0 0
\(7\) −2.34730 −0.887195 −0.443597 0.896226i \(-0.646298\pi\)
−0.443597 + 0.896226i \(0.646298\pi\)
\(8\) 0.237565 0.0839918
\(9\) 0 0
\(10\) −7.29086 −2.30557
\(11\) −2.17853 −0.656850 −0.328425 0.944530i \(-0.606518\pi\)
−0.328425 + 0.944530i \(0.606518\pi\)
\(12\) 0 0
\(13\) −4.71688 −1.30823 −0.654114 0.756396i \(-0.726958\pi\)
−0.654114 + 0.756396i \(0.726958\pi\)
\(14\) 4.62327 1.23562
\(15\) 0 0
\(16\) −4.22668 −1.05667
\(17\) −2.93512 −0.711871 −0.355936 0.934510i \(-0.615838\pi\)
−0.355936 + 0.934510i \(0.615838\pi\)
\(18\) 0 0
\(19\) −6.22668 −1.42850 −0.714249 0.699891i \(-0.753232\pi\)
−0.714249 + 0.699891i \(0.753232\pi\)
\(20\) 6.95686 1.55560
\(21\) 0 0
\(22\) 4.29086 0.914814
\(23\) 0.519030 0.108225 0.0541126 0.998535i \(-0.482767\pi\)
0.0541126 + 0.998535i \(0.482767\pi\)
\(24\) 0 0
\(25\) 8.70233 1.74047
\(26\) 9.29044 1.82201
\(27\) 0 0
\(28\) −4.41147 −0.833690
\(29\) −3.49276 −0.648588 −0.324294 0.945956i \(-0.605127\pi\)
−0.324294 + 0.945956i \(0.605127\pi\)
\(30\) 0 0
\(31\) −4.30541 −0.773274 −0.386637 0.922232i \(-0.626363\pi\)
−0.386637 + 0.922232i \(0.626363\pi\)
\(32\) 7.84981 1.38766
\(33\) 0 0
\(34\) 5.78106 0.991443
\(35\) −8.68891 −1.46869
\(36\) 0 0
\(37\) 2.41147 0.396444 0.198222 0.980157i \(-0.436483\pi\)
0.198222 + 0.980157i \(0.436483\pi\)
\(38\) 12.2642 1.98951
\(39\) 0 0
\(40\) 0.879385 0.139043
\(41\) 2.49860 0.390215 0.195108 0.980782i \(-0.437494\pi\)
0.195108 + 0.980782i \(0.437494\pi\)
\(42\) 0 0
\(43\) 1.06418 0.162286 0.0811428 0.996702i \(-0.474143\pi\)
0.0811428 + 0.996702i \(0.474143\pi\)
\(44\) −4.09429 −0.617237
\(45\) 0 0
\(46\) −1.02229 −0.150728
\(47\) −0.237565 −0.0346524 −0.0173262 0.999850i \(-0.505515\pi\)
−0.0173262 + 0.999850i \(0.505515\pi\)
\(48\) 0 0
\(49\) −1.49020 −0.212886
\(50\) −17.1403 −2.42400
\(51\) 0 0
\(52\) −8.86484 −1.22933
\(53\) 4.66717 0.641085 0.320543 0.947234i \(-0.396135\pi\)
0.320543 + 0.947234i \(0.396135\pi\)
\(54\) 0 0
\(55\) −8.06418 −1.08737
\(56\) −0.557635 −0.0745171
\(57\) 0 0
\(58\) 6.87939 0.903308
\(59\) −13.3122 −1.73310 −0.866549 0.499092i \(-0.833667\pi\)
−0.866549 + 0.499092i \(0.833667\pi\)
\(60\) 0 0
\(61\) −3.67499 −0.470535 −0.235267 0.971931i \(-0.575597\pi\)
−0.235267 + 0.971931i \(0.575597\pi\)
\(62\) 8.48000 1.07696
\(63\) 0 0
\(64\) −7.00774 −0.875968
\(65\) −17.4603 −2.16569
\(66\) 0 0
\(67\) 14.2986 1.74685 0.873426 0.486957i \(-0.161893\pi\)
0.873426 + 0.486957i \(0.161893\pi\)
\(68\) −5.51622 −0.668940
\(69\) 0 0
\(70\) 17.1138 2.04549
\(71\) 1.20307 0.142778 0.0713891 0.997449i \(-0.477257\pi\)
0.0713891 + 0.997449i \(0.477257\pi\)
\(72\) 0 0
\(73\) −4.68004 −0.547758 −0.273879 0.961764i \(-0.588307\pi\)
−0.273879 + 0.961764i \(0.588307\pi\)
\(74\) −4.74968 −0.552139
\(75\) 0 0
\(76\) −11.7023 −1.34235
\(77\) 5.11365 0.582754
\(78\) 0 0
\(79\) −12.8007 −1.44019 −0.720093 0.693877i \(-0.755901\pi\)
−0.720093 + 0.693877i \(0.755901\pi\)
\(80\) −15.6458 −1.74925
\(81\) 0 0
\(82\) −4.92127 −0.543464
\(83\) −11.3040 −1.24077 −0.620385 0.784297i \(-0.713024\pi\)
−0.620385 + 0.784297i \(0.713024\pi\)
\(84\) 0 0
\(85\) −10.8648 −1.17846
\(86\) −2.09602 −0.226020
\(87\) 0 0
\(88\) −0.517541 −0.0551701
\(89\) 0.699287 0.0741242 0.0370621 0.999313i \(-0.488200\pi\)
0.0370621 + 0.999313i \(0.488200\pi\)
\(90\) 0 0
\(91\) 11.0719 1.16065
\(92\) 0.975457 0.101698
\(93\) 0 0
\(94\) 0.467911 0.0482613
\(95\) −23.0491 −2.36479
\(96\) 0 0
\(97\) −7.08647 −0.719522 −0.359761 0.933045i \(-0.617142\pi\)
−0.359761 + 0.933045i \(0.617142\pi\)
\(98\) 2.93512 0.296492
\(99\) 0 0
\(100\) 16.3550 1.63550
\(101\) 4.67712 0.465391 0.232696 0.972550i \(-0.425245\pi\)
0.232696 + 0.972550i \(0.425245\pi\)
\(102\) 0 0
\(103\) −13.6236 −1.34237 −0.671187 0.741288i \(-0.734215\pi\)
−0.671187 + 0.741288i \(0.734215\pi\)
\(104\) −1.12056 −0.109880
\(105\) 0 0
\(106\) −9.19253 −0.892858
\(107\) 11.6340 1.12470 0.562350 0.826900i \(-0.309898\pi\)
0.562350 + 0.826900i \(0.309898\pi\)
\(108\) 0 0
\(109\) 14.6040 1.39881 0.699405 0.714725i \(-0.253448\pi\)
0.699405 + 0.714725i \(0.253448\pi\)
\(110\) 15.8833 1.51442
\(111\) 0 0
\(112\) 9.92127 0.937472
\(113\) 4.69583 0.441746 0.220873 0.975303i \(-0.429109\pi\)
0.220873 + 0.975303i \(0.429109\pi\)
\(114\) 0 0
\(115\) 1.92127 0.179160
\(116\) −6.56423 −0.609474
\(117\) 0 0
\(118\) 26.2199 2.41374
\(119\) 6.88960 0.631568
\(120\) 0 0
\(121\) −6.25402 −0.568548
\(122\) 7.23832 0.655327
\(123\) 0 0
\(124\) −8.09152 −0.726640
\(125\) 13.7048 1.22579
\(126\) 0 0
\(127\) 6.09152 0.540535 0.270267 0.962785i \(-0.412888\pi\)
0.270267 + 0.962785i \(0.412888\pi\)
\(128\) −1.89706 −0.167678
\(129\) 0 0
\(130\) 34.3901 3.01621
\(131\) 10.3484 0.904144 0.452072 0.891981i \(-0.350685\pi\)
0.452072 + 0.891981i \(0.350685\pi\)
\(132\) 0 0
\(133\) 14.6159 1.26736
\(134\) −28.1627 −2.43289
\(135\) 0 0
\(136\) −0.697281 −0.0597914
\(137\) 19.0847 1.63051 0.815257 0.579100i \(-0.196596\pi\)
0.815257 + 0.579100i \(0.196596\pi\)
\(138\) 0 0
\(139\) −23.3037 −1.97659 −0.988295 0.152555i \(-0.951250\pi\)
−0.988295 + 0.152555i \(0.951250\pi\)
\(140\) −16.3298 −1.38012
\(141\) 0 0
\(142\) −2.36959 −0.198851
\(143\) 10.2759 0.859310
\(144\) 0 0
\(145\) −12.9290 −1.07370
\(146\) 9.21789 0.762878
\(147\) 0 0
\(148\) 4.53209 0.372535
\(149\) 15.5060 1.27030 0.635149 0.772390i \(-0.280939\pi\)
0.635149 + 0.772390i \(0.280939\pi\)
\(150\) 0 0
\(151\) −5.90167 −0.480271 −0.240136 0.970739i \(-0.577192\pi\)
−0.240136 + 0.970739i \(0.577192\pi\)
\(152\) −1.47924 −0.119982
\(153\) 0 0
\(154\) −10.0719 −0.811618
\(155\) −15.9372 −1.28011
\(156\) 0 0
\(157\) −1.21213 −0.0967388 −0.0483694 0.998830i \(-0.515402\pi\)
−0.0483694 + 0.998830i \(0.515402\pi\)
\(158\) 25.2124 2.00579
\(159\) 0 0
\(160\) 29.0574 2.29719
\(161\) −1.21832 −0.0960168
\(162\) 0 0
\(163\) 2.77332 0.217223 0.108612 0.994084i \(-0.465360\pi\)
0.108612 + 0.994084i \(0.465360\pi\)
\(164\) 4.69583 0.366682
\(165\) 0 0
\(166\) 22.2645 1.72806
\(167\) −3.82807 −0.296225 −0.148113 0.988971i \(-0.547320\pi\)
−0.148113 + 0.988971i \(0.547320\pi\)
\(168\) 0 0
\(169\) 9.24897 0.711459
\(170\) 21.3996 1.64127
\(171\) 0 0
\(172\) 2.00000 0.152499
\(173\) −7.02941 −0.534436 −0.267218 0.963636i \(-0.586104\pi\)
−0.267218 + 0.963636i \(0.586104\pi\)
\(174\) 0 0
\(175\) −20.4270 −1.54413
\(176\) 9.20794 0.694074
\(177\) 0 0
\(178\) −1.37733 −0.103235
\(179\) −14.3854 −1.07521 −0.537607 0.843195i \(-0.680672\pi\)
−0.537607 + 0.843195i \(0.680672\pi\)
\(180\) 0 0
\(181\) 13.2003 0.981169 0.490584 0.871394i \(-0.336783\pi\)
0.490584 + 0.871394i \(0.336783\pi\)
\(182\) −21.8074 −1.61647
\(183\) 0 0
\(184\) 0.123303 0.00909003
\(185\) 8.92647 0.656287
\(186\) 0 0
\(187\) 6.39424 0.467593
\(188\) −0.446476 −0.0325626
\(189\) 0 0
\(190\) 45.3979 3.29351
\(191\) 13.4233 0.971279 0.485639 0.874159i \(-0.338587\pi\)
0.485639 + 0.874159i \(0.338587\pi\)
\(192\) 0 0
\(193\) 15.0051 1.08009 0.540044 0.841637i \(-0.318408\pi\)
0.540044 + 0.841637i \(0.318408\pi\)
\(194\) 13.9576 1.00210
\(195\) 0 0
\(196\) −2.80066 −0.200047
\(197\) −22.3212 −1.59032 −0.795158 0.606402i \(-0.792612\pi\)
−0.795158 + 0.606402i \(0.792612\pi\)
\(198\) 0 0
\(199\) −9.10101 −0.645154 −0.322577 0.946543i \(-0.604549\pi\)
−0.322577 + 0.946543i \(0.604549\pi\)
\(200\) 2.06737 0.146185
\(201\) 0 0
\(202\) −9.21213 −0.648163
\(203\) 8.19853 0.575424
\(204\) 0 0
\(205\) 9.24897 0.645976
\(206\) 26.8333 1.86956
\(207\) 0 0
\(208\) 19.9368 1.38237
\(209\) 13.5650 0.938310
\(210\) 0 0
\(211\) −5.97771 −0.411523 −0.205761 0.978602i \(-0.565967\pi\)
−0.205761 + 0.978602i \(0.565967\pi\)
\(212\) 8.77141 0.602423
\(213\) 0 0
\(214\) −22.9145 −1.56640
\(215\) 3.93923 0.268653
\(216\) 0 0
\(217\) 10.1061 0.686045
\(218\) −28.7643 −1.94816
\(219\) 0 0
\(220\) −15.1557 −1.02180
\(221\) 13.8446 0.931290
\(222\) 0 0
\(223\) −8.90167 −0.596100 −0.298050 0.954550i \(-0.596336\pi\)
−0.298050 + 0.954550i \(0.596336\pi\)
\(224\) −18.4258 −1.23113
\(225\) 0 0
\(226\) −9.24897 −0.615232
\(227\) −10.6685 −0.708092 −0.354046 0.935228i \(-0.615194\pi\)
−0.354046 + 0.935228i \(0.615194\pi\)
\(228\) 0 0
\(229\) −8.27126 −0.546580 −0.273290 0.961932i \(-0.588112\pi\)
−0.273290 + 0.961932i \(0.588112\pi\)
\(230\) −3.78417 −0.249521
\(231\) 0 0
\(232\) −0.829755 −0.0544761
\(233\) 12.7393 0.834579 0.417290 0.908774i \(-0.362980\pi\)
0.417290 + 0.908774i \(0.362980\pi\)
\(234\) 0 0
\(235\) −0.879385 −0.0573648
\(236\) −25.0187 −1.62858
\(237\) 0 0
\(238\) −13.5699 −0.879603
\(239\) 15.0156 0.971277 0.485638 0.874160i \(-0.338587\pi\)
0.485638 + 0.874160i \(0.338587\pi\)
\(240\) 0 0
\(241\) 0.795607 0.0512496 0.0256248 0.999672i \(-0.491842\pi\)
0.0256248 + 0.999672i \(0.491842\pi\)
\(242\) 12.3180 0.791832
\(243\) 0 0
\(244\) −6.90673 −0.442158
\(245\) −5.51622 −0.352419
\(246\) 0 0
\(247\) 29.3705 1.86880
\(248\) −1.02281 −0.0649487
\(249\) 0 0
\(250\) −26.9932 −1.70720
\(251\) 8.31499 0.524837 0.262419 0.964954i \(-0.415480\pi\)
0.262419 + 0.964954i \(0.415480\pi\)
\(252\) 0 0
\(253\) −1.13072 −0.0710877
\(254\) −11.9980 −0.752818
\(255\) 0 0
\(256\) 17.7520 1.10950
\(257\) −25.6202 −1.59815 −0.799074 0.601233i \(-0.794676\pi\)
−0.799074 + 0.601233i \(0.794676\pi\)
\(258\) 0 0
\(259\) −5.66044 −0.351723
\(260\) −32.8147 −2.03508
\(261\) 0 0
\(262\) −20.3824 −1.25923
\(263\) 28.0650 1.73056 0.865281 0.501287i \(-0.167140\pi\)
0.865281 + 0.501287i \(0.167140\pi\)
\(264\) 0 0
\(265\) 17.2763 1.06128
\(266\) −28.7876 −1.76508
\(267\) 0 0
\(268\) 26.8726 1.64150
\(269\) −30.1710 −1.83956 −0.919778 0.392439i \(-0.871631\pi\)
−0.919778 + 0.392439i \(0.871631\pi\)
\(270\) 0 0
\(271\) −19.0000 −1.15417 −0.577084 0.816685i \(-0.695809\pi\)
−0.577084 + 0.816685i \(0.695809\pi\)
\(272\) 12.4058 0.752213
\(273\) 0 0
\(274\) −37.5895 −2.27086
\(275\) −18.9583 −1.14323
\(276\) 0 0
\(277\) 21.1215 1.26907 0.634535 0.772894i \(-0.281191\pi\)
0.634535 + 0.772894i \(0.281191\pi\)
\(278\) 45.8992 2.75285
\(279\) 0 0
\(280\) −2.06418 −0.123358
\(281\) −1.74730 −0.104235 −0.0521175 0.998641i \(-0.516597\pi\)
−0.0521175 + 0.998641i \(0.516597\pi\)
\(282\) 0 0
\(283\) −7.28817 −0.433237 −0.216618 0.976256i \(-0.569503\pi\)
−0.216618 + 0.976256i \(0.569503\pi\)
\(284\) 2.26103 0.134168
\(285\) 0 0
\(286\) −20.2395 −1.19679
\(287\) −5.86495 −0.346197
\(288\) 0 0
\(289\) −8.38507 −0.493239
\(290\) 25.4652 1.49537
\(291\) 0 0
\(292\) −8.79561 −0.514724
\(293\) −15.3509 −0.896809 −0.448404 0.893831i \(-0.648008\pi\)
−0.448404 + 0.893831i \(0.648008\pi\)
\(294\) 0 0
\(295\) −49.2772 −2.86903
\(296\) 0.572881 0.0332980
\(297\) 0 0
\(298\) −30.5408 −1.76918
\(299\) −2.44820 −0.141583
\(300\) 0 0
\(301\) −2.49794 −0.143979
\(302\) 11.6240 0.668888
\(303\) 0 0
\(304\) 26.3182 1.50945
\(305\) −13.6036 −0.778940
\(306\) 0 0
\(307\) 16.7638 0.956762 0.478381 0.878152i \(-0.341224\pi\)
0.478381 + 0.878152i \(0.341224\pi\)
\(308\) 9.61051 0.547610
\(309\) 0 0
\(310\) 31.3901 1.78284
\(311\) 16.0231 0.908589 0.454295 0.890852i \(-0.349891\pi\)
0.454295 + 0.890852i \(0.349891\pi\)
\(312\) 0 0
\(313\) 34.4056 1.94472 0.972360 0.233488i \(-0.0750138\pi\)
0.972360 + 0.233488i \(0.0750138\pi\)
\(314\) 2.38744 0.134731
\(315\) 0 0
\(316\) −24.0574 −1.35333
\(317\) 16.3111 0.916123 0.458061 0.888921i \(-0.348544\pi\)
0.458061 + 0.888921i \(0.348544\pi\)
\(318\) 0 0
\(319\) 7.60906 0.426026
\(320\) −25.9403 −1.45011
\(321\) 0 0
\(322\) 2.39961 0.133725
\(323\) 18.2761 1.01691
\(324\) 0 0
\(325\) −41.0479 −2.27693
\(326\) −5.46237 −0.302533
\(327\) 0 0
\(328\) 0.593578 0.0327749
\(329\) 0.557635 0.0307434
\(330\) 0 0
\(331\) −31.5800 −1.73579 −0.867896 0.496746i \(-0.834528\pi\)
−0.867896 + 0.496746i \(0.834528\pi\)
\(332\) −21.2445 −1.16594
\(333\) 0 0
\(334\) 7.53983 0.412561
\(335\) 52.9286 2.89180
\(336\) 0 0
\(337\) 6.00505 0.327116 0.163558 0.986534i \(-0.447703\pi\)
0.163558 + 0.986534i \(0.447703\pi\)
\(338\) −18.2169 −0.990870
\(339\) 0 0
\(340\) −20.4192 −1.10739
\(341\) 9.37944 0.507925
\(342\) 0 0
\(343\) 19.9290 1.07607
\(344\) 0.252811 0.0136307
\(345\) 0 0
\(346\) 13.8452 0.744325
\(347\) −23.0304 −1.23634 −0.618168 0.786046i \(-0.712125\pi\)
−0.618168 + 0.786046i \(0.712125\pi\)
\(348\) 0 0
\(349\) −11.5662 −0.619126 −0.309563 0.950879i \(-0.600183\pi\)
−0.309563 + 0.950879i \(0.600183\pi\)
\(350\) 40.2332 2.15056
\(351\) 0 0
\(352\) −17.1010 −0.911487
\(353\) 2.13463 0.113615 0.0568073 0.998385i \(-0.481908\pi\)
0.0568073 + 0.998385i \(0.481908\pi\)
\(354\) 0 0
\(355\) 4.45336 0.236360
\(356\) 1.31423 0.0696540
\(357\) 0 0
\(358\) 28.3337 1.49748
\(359\) −24.2235 −1.27847 −0.639234 0.769012i \(-0.720749\pi\)
−0.639234 + 0.769012i \(0.720749\pi\)
\(360\) 0 0
\(361\) 19.7716 1.04061
\(362\) −25.9995 −1.36650
\(363\) 0 0
\(364\) 20.8084 1.09066
\(365\) −17.3240 −0.906778
\(366\) 0 0
\(367\) −2.83481 −0.147976 −0.0739879 0.997259i \(-0.523573\pi\)
−0.0739879 + 0.997259i \(0.523573\pi\)
\(368\) −2.19377 −0.114358
\(369\) 0 0
\(370\) −17.5817 −0.914030
\(371\) −10.9552 −0.568767
\(372\) 0 0
\(373\) −28.8334 −1.49294 −0.746468 0.665421i \(-0.768252\pi\)
−0.746468 + 0.665421i \(0.768252\pi\)
\(374\) −12.5942 −0.651230
\(375\) 0 0
\(376\) −0.0564370 −0.00291052
\(377\) 16.4749 0.848501
\(378\) 0 0
\(379\) 32.1985 1.65393 0.826963 0.562256i \(-0.190066\pi\)
0.826963 + 0.562256i \(0.190066\pi\)
\(380\) −43.3181 −2.22217
\(381\) 0 0
\(382\) −26.4388 −1.35273
\(383\) −0.672250 −0.0343504 −0.0171752 0.999852i \(-0.505467\pi\)
−0.0171752 + 0.999852i \(0.505467\pi\)
\(384\) 0 0
\(385\) 18.9290 0.964712
\(386\) −29.5542 −1.50427
\(387\) 0 0
\(388\) −13.3182 −0.676129
\(389\) 4.25930 0.215955 0.107978 0.994153i \(-0.465563\pi\)
0.107978 + 0.994153i \(0.465563\pi\)
\(390\) 0 0
\(391\) −1.52341 −0.0770424
\(392\) −0.354019 −0.0178807
\(393\) 0 0
\(394\) 43.9641 2.21488
\(395\) −47.3838 −2.38414
\(396\) 0 0
\(397\) −8.86484 −0.444913 −0.222457 0.974943i \(-0.571408\pi\)
−0.222457 + 0.974943i \(0.571408\pi\)
\(398\) 17.9255 0.898524
\(399\) 0 0
\(400\) −36.7820 −1.83910
\(401\) −37.0839 −1.85188 −0.925941 0.377667i \(-0.876726\pi\)
−0.925941 + 0.377667i \(0.876726\pi\)
\(402\) 0 0
\(403\) 20.3081 1.01162
\(404\) 8.79012 0.437325
\(405\) 0 0
\(406\) −16.1480 −0.801410
\(407\) −5.25346 −0.260404
\(408\) 0 0
\(409\) −3.38919 −0.167584 −0.0837922 0.996483i \(-0.526703\pi\)
−0.0837922 + 0.996483i \(0.526703\pi\)
\(410\) −18.2169 −0.899669
\(411\) 0 0
\(412\) −25.6040 −1.26142
\(413\) 31.2476 1.53760
\(414\) 0 0
\(415\) −41.8435 −2.05402
\(416\) −37.0266 −1.81538
\(417\) 0 0
\(418\) −26.7178 −1.30681
\(419\) 20.6564 1.00913 0.504565 0.863374i \(-0.331653\pi\)
0.504565 + 0.863374i \(0.331653\pi\)
\(420\) 0 0
\(421\) 27.5330 1.34188 0.670939 0.741513i \(-0.265891\pi\)
0.670939 + 0.741513i \(0.265891\pi\)
\(422\) 11.7738 0.573139
\(423\) 0 0
\(424\) 1.10876 0.0538459
\(425\) −25.5424 −1.23899
\(426\) 0 0
\(427\) 8.62630 0.417456
\(428\) 21.8647 1.05687
\(429\) 0 0
\(430\) −7.75877 −0.374161
\(431\) 2.58110 0.124327 0.0621636 0.998066i \(-0.480200\pi\)
0.0621636 + 0.998066i \(0.480200\pi\)
\(432\) 0 0
\(433\) −27.0137 −1.29820 −0.649098 0.760704i \(-0.724854\pi\)
−0.649098 + 0.760704i \(0.724854\pi\)
\(434\) −19.9051 −0.955474
\(435\) 0 0
\(436\) 27.4466 1.31445
\(437\) −3.23183 −0.154599
\(438\) 0 0
\(439\) 11.7101 0.558891 0.279446 0.960162i \(-0.409849\pi\)
0.279446 + 0.960162i \(0.409849\pi\)
\(440\) −1.91576 −0.0913305
\(441\) 0 0
\(442\) −27.2686 −1.29703
\(443\) −2.08077 −0.0988606 −0.0494303 0.998778i \(-0.515741\pi\)
−0.0494303 + 0.998778i \(0.515741\pi\)
\(444\) 0 0
\(445\) 2.58853 0.122708
\(446\) 17.5329 0.830206
\(447\) 0 0
\(448\) 16.4492 0.777154
\(449\) −10.5508 −0.497924 −0.248962 0.968513i \(-0.580089\pi\)
−0.248962 + 0.968513i \(0.580089\pi\)
\(450\) 0 0
\(451\) −5.44326 −0.256313
\(452\) 8.82526 0.415106
\(453\) 0 0
\(454\) 21.0128 0.986179
\(455\) 40.9845 1.92139
\(456\) 0 0
\(457\) −7.90941 −0.369987 −0.184993 0.982740i \(-0.559226\pi\)
−0.184993 + 0.982740i \(0.559226\pi\)
\(458\) 16.2912 0.761238
\(459\) 0 0
\(460\) 3.61081 0.168355
\(461\) 39.0928 1.82073 0.910366 0.413804i \(-0.135800\pi\)
0.910366 + 0.413804i \(0.135800\pi\)
\(462\) 0 0
\(463\) 23.6928 1.10110 0.550550 0.834802i \(-0.314418\pi\)
0.550550 + 0.834802i \(0.314418\pi\)
\(464\) 14.7628 0.685344
\(465\) 0 0
\(466\) −25.0915 −1.16234
\(467\) 34.7152 1.60643 0.803214 0.595691i \(-0.203122\pi\)
0.803214 + 0.595691i \(0.203122\pi\)
\(468\) 0 0
\(469\) −33.5631 −1.54980
\(470\) 1.73205 0.0798935
\(471\) 0 0
\(472\) −3.16250 −0.145566
\(473\) −2.31834 −0.106597
\(474\) 0 0
\(475\) −54.1867 −2.48625
\(476\) 12.9482 0.593480
\(477\) 0 0
\(478\) −29.5749 −1.35272
\(479\) 6.48173 0.296158 0.148079 0.988976i \(-0.452691\pi\)
0.148079 + 0.988976i \(0.452691\pi\)
\(480\) 0 0
\(481\) −11.3746 −0.518639
\(482\) −1.56704 −0.0713767
\(483\) 0 0
\(484\) −11.7537 −0.534260
\(485\) −26.2317 −1.19112
\(486\) 0 0
\(487\) 19.9828 0.905505 0.452753 0.891636i \(-0.350442\pi\)
0.452753 + 0.891636i \(0.350442\pi\)
\(488\) −0.873048 −0.0395210
\(489\) 0 0
\(490\) 10.8648 0.490823
\(491\) 26.2271 1.18361 0.591806 0.806081i \(-0.298415\pi\)
0.591806 + 0.806081i \(0.298415\pi\)
\(492\) 0 0
\(493\) 10.2517 0.461711
\(494\) −57.8486 −2.60273
\(495\) 0 0
\(496\) 18.1976 0.817096
\(497\) −2.82396 −0.126672
\(498\) 0 0
\(499\) 6.31727 0.282800 0.141400 0.989953i \(-0.454840\pi\)
0.141400 + 0.989953i \(0.454840\pi\)
\(500\) 25.7566 1.15187
\(501\) 0 0
\(502\) −16.3773 −0.730956
\(503\) 21.8261 0.973179 0.486589 0.873631i \(-0.338241\pi\)
0.486589 + 0.873631i \(0.338241\pi\)
\(504\) 0 0
\(505\) 17.3131 0.770425
\(506\) 2.22708 0.0990059
\(507\) 0 0
\(508\) 11.4483 0.507937
\(509\) −29.0931 −1.28953 −0.644765 0.764381i \(-0.723045\pi\)
−0.644765 + 0.764381i \(0.723045\pi\)
\(510\) 0 0
\(511\) 10.9855 0.485968
\(512\) −31.1704 −1.37755
\(513\) 0 0
\(514\) 50.4620 2.22579
\(515\) −50.4301 −2.22221
\(516\) 0 0
\(517\) 0.517541 0.0227614
\(518\) 11.1489 0.489855
\(519\) 0 0
\(520\) −4.14796 −0.181900
\(521\) 13.6949 0.599982 0.299991 0.953942i \(-0.403016\pi\)
0.299991 + 0.953942i \(0.403016\pi\)
\(522\) 0 0
\(523\) 13.1506 0.575038 0.287519 0.957775i \(-0.407170\pi\)
0.287519 + 0.957775i \(0.407170\pi\)
\(524\) 19.4486 0.849618
\(525\) 0 0
\(526\) −55.2772 −2.41020
\(527\) 12.6369 0.550472
\(528\) 0 0
\(529\) −22.7306 −0.988287
\(530\) −34.0277 −1.47807
\(531\) 0 0
\(532\) 27.4688 1.19093
\(533\) −11.7856 −0.510490
\(534\) 0 0
\(535\) 43.0651 1.86187
\(536\) 3.39684 0.146721
\(537\) 0 0
\(538\) 59.4252 2.56200
\(539\) 3.24644 0.139834
\(540\) 0 0
\(541\) 6.26083 0.269174 0.134587 0.990902i \(-0.457029\pi\)
0.134587 + 0.990902i \(0.457029\pi\)
\(542\) 37.4227 1.60744
\(543\) 0 0
\(544\) −23.0401 −0.987838
\(545\) 54.0592 2.31564
\(546\) 0 0
\(547\) −31.3783 −1.34164 −0.670819 0.741621i \(-0.734057\pi\)
−0.670819 + 0.741621i \(0.734057\pi\)
\(548\) 35.8674 1.53218
\(549\) 0 0
\(550\) 37.3405 1.59220
\(551\) 21.7483 0.926508
\(552\) 0 0
\(553\) 30.0469 1.27773
\(554\) −41.6013 −1.76747
\(555\) 0 0
\(556\) −43.7965 −1.85739
\(557\) 43.4392 1.84058 0.920290 0.391237i \(-0.127953\pi\)
0.920290 + 0.391237i \(0.127953\pi\)
\(558\) 0 0
\(559\) −5.01960 −0.212306
\(560\) 36.7252 1.55192
\(561\) 0 0
\(562\) 3.44150 0.145171
\(563\) 32.0176 1.34938 0.674691 0.738100i \(-0.264277\pi\)
0.674691 + 0.738100i \(0.264277\pi\)
\(564\) 0 0
\(565\) 17.3824 0.731282
\(566\) 14.3549 0.603381
\(567\) 0 0
\(568\) 0.285807 0.0119922
\(569\) 6.77194 0.283895 0.141947 0.989874i \(-0.454664\pi\)
0.141947 + 0.989874i \(0.454664\pi\)
\(570\) 0 0
\(571\) −21.1530 −0.885226 −0.442613 0.896713i \(-0.645948\pi\)
−0.442613 + 0.896713i \(0.645948\pi\)
\(572\) 19.3123 0.807487
\(573\) 0 0
\(574\) 11.5517 0.482158
\(575\) 4.51677 0.188362
\(576\) 0 0
\(577\) 11.9162 0.496079 0.248039 0.968750i \(-0.420214\pi\)
0.248039 + 0.968750i \(0.420214\pi\)
\(578\) 16.5154 0.686948
\(579\) 0 0
\(580\) −24.2986 −1.00894
\(581\) 26.5337 1.10081
\(582\) 0 0
\(583\) −10.1676 −0.421097
\(584\) −1.11181 −0.0460072
\(585\) 0 0
\(586\) 30.2354 1.24901
\(587\) 0.129862 0.00535996 0.00267998 0.999996i \(-0.499147\pi\)
0.00267998 + 0.999996i \(0.499147\pi\)
\(588\) 0 0
\(589\) 26.8084 1.10462
\(590\) 97.0572 3.99578
\(591\) 0 0
\(592\) −10.1925 −0.418911
\(593\) −26.2622 −1.07846 −0.539230 0.842158i \(-0.681285\pi\)
−0.539230 + 0.842158i \(0.681285\pi\)
\(594\) 0 0
\(595\) 25.5030 1.04552
\(596\) 29.1417 1.19369
\(597\) 0 0
\(598\) 4.82201 0.197187
\(599\) −27.5952 −1.12751 −0.563754 0.825943i \(-0.690643\pi\)
−0.563754 + 0.825943i \(0.690643\pi\)
\(600\) 0 0
\(601\) 1.33275 0.0543639 0.0271820 0.999631i \(-0.491347\pi\)
0.0271820 + 0.999631i \(0.491347\pi\)
\(602\) 4.91998 0.200524
\(603\) 0 0
\(604\) −11.0915 −0.451308
\(605\) −23.1503 −0.941194
\(606\) 0 0
\(607\) −12.5645 −0.509977 −0.254988 0.966944i \(-0.582072\pi\)
−0.254988 + 0.966944i \(0.582072\pi\)
\(608\) −48.8783 −1.98228
\(609\) 0 0
\(610\) 26.7939 1.08485
\(611\) 1.12056 0.0453332
\(612\) 0 0
\(613\) −13.9982 −0.565384 −0.282692 0.959211i \(-0.591227\pi\)
−0.282692 + 0.959211i \(0.591227\pi\)
\(614\) −33.0183 −1.33251
\(615\) 0 0
\(616\) 1.21482 0.0489466
\(617\) −24.0467 −0.968084 −0.484042 0.875045i \(-0.660832\pi\)
−0.484042 + 0.875045i \(0.660832\pi\)
\(618\) 0 0
\(619\) 6.87164 0.276195 0.138097 0.990419i \(-0.455901\pi\)
0.138097 + 0.990419i \(0.455901\pi\)
\(620\) −29.9521 −1.20291
\(621\) 0 0
\(622\) −31.5594 −1.26542
\(623\) −1.64143 −0.0657626
\(624\) 0 0
\(625\) 7.21894 0.288758
\(626\) −67.7658 −2.70847
\(627\) 0 0
\(628\) −2.27807 −0.0909047
\(629\) −7.07797 −0.282217
\(630\) 0 0
\(631\) −35.3773 −1.40835 −0.704175 0.710027i \(-0.748683\pi\)
−0.704175 + 0.710027i \(0.748683\pi\)
\(632\) −3.04098 −0.120964
\(633\) 0 0
\(634\) −32.1266 −1.27591
\(635\) 22.5488 0.894821
\(636\) 0 0
\(637\) 7.02910 0.278503
\(638\) −14.9869 −0.593338
\(639\) 0 0
\(640\) −7.02229 −0.277580
\(641\) 19.1737 0.757314 0.378657 0.925537i \(-0.376386\pi\)
0.378657 + 0.925537i \(0.376386\pi\)
\(642\) 0 0
\(643\) 19.3764 0.764130 0.382065 0.924135i \(-0.375213\pi\)
0.382065 + 0.924135i \(0.375213\pi\)
\(644\) −2.28969 −0.0902263
\(645\) 0 0
\(646\) −35.9968 −1.41628
\(647\) −8.77141 −0.344840 −0.172420 0.985024i \(-0.555159\pi\)
−0.172420 + 0.985024i \(0.555159\pi\)
\(648\) 0 0
\(649\) 29.0009 1.13839
\(650\) 80.8485 3.17114
\(651\) 0 0
\(652\) 5.21213 0.204123
\(653\) −32.8094 −1.28393 −0.641965 0.766734i \(-0.721881\pi\)
−0.641965 + 0.766734i \(0.721881\pi\)
\(654\) 0 0
\(655\) 38.3063 1.49675
\(656\) −10.5608 −0.412329
\(657\) 0 0
\(658\) −1.09833 −0.0428172
\(659\) −18.6516 −0.726563 −0.363282 0.931679i \(-0.618344\pi\)
−0.363282 + 0.931679i \(0.618344\pi\)
\(660\) 0 0
\(661\) 36.4074 1.41608 0.708041 0.706171i \(-0.249579\pi\)
0.708041 + 0.706171i \(0.249579\pi\)
\(662\) 62.2004 2.41749
\(663\) 0 0
\(664\) −2.68542 −0.104215
\(665\) 54.1031 2.09803
\(666\) 0 0
\(667\) −1.81284 −0.0701936
\(668\) −7.19442 −0.278361
\(669\) 0 0
\(670\) −104.249 −4.02749
\(671\) 8.00607 0.309071
\(672\) 0 0
\(673\) 39.3901 1.51838 0.759189 0.650871i \(-0.225596\pi\)
0.759189 + 0.650871i \(0.225596\pi\)
\(674\) −11.8276 −0.455584
\(675\) 0 0
\(676\) 17.3824 0.668553
\(677\) −31.6754 −1.21738 −0.608692 0.793406i \(-0.708306\pi\)
−0.608692 + 0.793406i \(0.708306\pi\)
\(678\) 0 0
\(679\) 16.6340 0.638356
\(680\) −2.58110 −0.0989807
\(681\) 0 0
\(682\) −18.4739 −0.707402
\(683\) 29.0656 1.11217 0.556083 0.831127i \(-0.312304\pi\)
0.556083 + 0.831127i \(0.312304\pi\)
\(684\) 0 0
\(685\) 70.6451 2.69921
\(686\) −39.2525 −1.49867
\(687\) 0 0
\(688\) −4.49794 −0.171482
\(689\) −22.0145 −0.838685
\(690\) 0 0
\(691\) 5.35740 0.203805 0.101903 0.994794i \(-0.467507\pi\)
0.101903 + 0.994794i \(0.467507\pi\)
\(692\) −13.2110 −0.502206
\(693\) 0 0
\(694\) 45.3610 1.72188
\(695\) −86.2623 −3.27212
\(696\) 0 0
\(697\) −7.33368 −0.277783
\(698\) 22.7810 0.862275
\(699\) 0 0
\(700\) −38.3901 −1.45101
\(701\) 25.6536 0.968922 0.484461 0.874813i \(-0.339016\pi\)
0.484461 + 0.874813i \(0.339016\pi\)
\(702\) 0 0
\(703\) −15.0155 −0.566320
\(704\) 15.2665 0.575380
\(705\) 0 0
\(706\) −4.20439 −0.158234
\(707\) −10.9786 −0.412893
\(708\) 0 0
\(709\) −4.69047 −0.176154 −0.0880772 0.996114i \(-0.528072\pi\)
−0.0880772 + 0.996114i \(0.528072\pi\)
\(710\) −8.77141 −0.329185
\(711\) 0 0
\(712\) 0.166126 0.00622583
\(713\) −2.23463 −0.0836877
\(714\) 0 0
\(715\) 38.0378 1.42253
\(716\) −27.0357 −1.01037
\(717\) 0 0
\(718\) 47.7110 1.78056
\(719\) −39.0669 −1.45695 −0.728476 0.685072i \(-0.759771\pi\)
−0.728476 + 0.685072i \(0.759771\pi\)
\(720\) 0 0
\(721\) 31.9786 1.19095
\(722\) −38.9424 −1.44929
\(723\) 0 0
\(724\) 24.8084 0.921997
\(725\) −30.3951 −1.12885
\(726\) 0 0
\(727\) −10.8253 −0.401489 −0.200744 0.979644i \(-0.564336\pi\)
−0.200744 + 0.979644i \(0.564336\pi\)
\(728\) 2.63030 0.0974853
\(729\) 0 0
\(730\) 34.1215 1.26290
\(731\) −3.12349 −0.115526
\(732\) 0 0
\(733\) −3.41241 −0.126040 −0.0630201 0.998012i \(-0.520073\pi\)
−0.0630201 + 0.998012i \(0.520073\pi\)
\(734\) 5.58348 0.206090
\(735\) 0 0
\(736\) 4.07428 0.150180
\(737\) −31.1499 −1.14742
\(738\) 0 0
\(739\) 26.3010 0.967497 0.483748 0.875207i \(-0.339275\pi\)
0.483748 + 0.875207i \(0.339275\pi\)
\(740\) 16.7763 0.616708
\(741\) 0 0
\(742\) 21.5776 0.792139
\(743\) −28.9439 −1.06185 −0.530924 0.847419i \(-0.678155\pi\)
−0.530924 + 0.847419i \(0.678155\pi\)
\(744\) 0 0
\(745\) 57.3979 2.10289
\(746\) 56.7907 2.07925
\(747\) 0 0
\(748\) 12.0172 0.439394
\(749\) −27.3084 −0.997827
\(750\) 0 0
\(751\) −19.4483 −0.709679 −0.354839 0.934927i \(-0.615464\pi\)
−0.354839 + 0.934927i \(0.615464\pi\)
\(752\) 1.00411 0.0366161
\(753\) 0 0
\(754\) −32.4492 −1.18173
\(755\) −21.8460 −0.795058
\(756\) 0 0
\(757\) 6.59627 0.239745 0.119873 0.992789i \(-0.461751\pi\)
0.119873 + 0.992789i \(0.461751\pi\)
\(758\) −63.4187 −2.30347
\(759\) 0 0
\(760\) −5.47565 −0.198623
\(761\) −10.3297 −0.374451 −0.187226 0.982317i \(-0.559950\pi\)
−0.187226 + 0.982317i \(0.559950\pi\)
\(762\) 0 0
\(763\) −34.2799 −1.24102
\(764\) 25.2276 0.912703
\(765\) 0 0
\(766\) 1.32407 0.0478407
\(767\) 62.7920 2.26729
\(768\) 0 0
\(769\) 44.8590 1.61766 0.808828 0.588046i \(-0.200102\pi\)
0.808828 + 0.588046i \(0.200102\pi\)
\(770\) −37.2829 −1.34358
\(771\) 0 0
\(772\) 28.2003 1.01495
\(773\) −42.9355 −1.54428 −0.772141 0.635452i \(-0.780814\pi\)
−0.772141 + 0.635452i \(0.780814\pi\)
\(774\) 0 0
\(775\) −37.4671 −1.34586
\(776\) −1.68349 −0.0604339
\(777\) 0 0
\(778\) −8.38919 −0.300767
\(779\) −15.5580 −0.557422
\(780\) 0 0
\(781\) −2.62092 −0.0937839
\(782\) 3.00054 0.107299
\(783\) 0 0
\(784\) 6.29860 0.224950
\(785\) −4.48691 −0.160145
\(786\) 0 0
\(787\) 0.477407 0.0170177 0.00850885 0.999964i \(-0.497292\pi\)
0.00850885 + 0.999964i \(0.497292\pi\)
\(788\) −41.9501 −1.49441
\(789\) 0 0
\(790\) 93.3278 3.32045
\(791\) −11.0225 −0.391915
\(792\) 0 0
\(793\) 17.3345 0.615566
\(794\) 17.4603 0.619644
\(795\) 0 0
\(796\) −17.1043 −0.606246
\(797\) −11.4690 −0.406252 −0.203126 0.979153i \(-0.565110\pi\)
−0.203126 + 0.979153i \(0.565110\pi\)
\(798\) 0 0
\(799\) 0.697281 0.0246680
\(800\) 68.3116 2.41518
\(801\) 0 0
\(802\) 73.0411 2.57917
\(803\) 10.1956 0.359795
\(804\) 0 0
\(805\) −4.50980 −0.158950
\(806\) −39.9991 −1.40891
\(807\) 0 0
\(808\) 1.11112 0.0390890
\(809\) −42.7873 −1.50432 −0.752161 0.658979i \(-0.770988\pi\)
−0.752161 + 0.658979i \(0.770988\pi\)
\(810\) 0 0
\(811\) −31.2098 −1.09592 −0.547962 0.836503i \(-0.684596\pi\)
−0.547962 + 0.836503i \(0.684596\pi\)
\(812\) 15.4082 0.540722
\(813\) 0 0
\(814\) 10.3473 0.362673
\(815\) 10.2659 0.359599
\(816\) 0 0
\(817\) −6.62630 −0.231825
\(818\) 6.67539 0.233400
\(819\) 0 0
\(820\) 17.3824 0.607019
\(821\) −27.2511 −0.951070 −0.475535 0.879697i \(-0.657745\pi\)
−0.475535 + 0.879697i \(0.657745\pi\)
\(822\) 0 0
\(823\) −14.4406 −0.503367 −0.251683 0.967810i \(-0.580984\pi\)
−0.251683 + 0.967810i \(0.580984\pi\)
\(824\) −3.23649 −0.112748
\(825\) 0 0
\(826\) −61.5458 −2.14145
\(827\) −7.02757 −0.244373 −0.122186 0.992507i \(-0.538991\pi\)
−0.122186 + 0.992507i \(0.538991\pi\)
\(828\) 0 0
\(829\) −42.0806 −1.46152 −0.730760 0.682635i \(-0.760834\pi\)
−0.730760 + 0.682635i \(0.760834\pi\)
\(830\) 82.4156 2.86069
\(831\) 0 0
\(832\) 33.0547 1.14596
\(833\) 4.37392 0.151547
\(834\) 0 0
\(835\) −14.1702 −0.490382
\(836\) 25.4938 0.881723
\(837\) 0 0
\(838\) −40.6851 −1.40544
\(839\) 39.9213 1.37824 0.689118 0.724649i \(-0.257998\pi\)
0.689118 + 0.724649i \(0.257998\pi\)
\(840\) 0 0
\(841\) −16.8007 −0.579333
\(842\) −54.2295 −1.86887
\(843\) 0 0
\(844\) −11.2344 −0.386705
\(845\) 34.2366 1.17777
\(846\) 0 0
\(847\) 14.6800 0.504412
\(848\) −19.7266 −0.677416
\(849\) 0 0
\(850\) 50.3087 1.72557
\(851\) 1.25163 0.0429052
\(852\) 0 0
\(853\) −23.4810 −0.803975 −0.401988 0.915645i \(-0.631680\pi\)
−0.401988 + 0.915645i \(0.631680\pi\)
\(854\) −16.9905 −0.581402
\(855\) 0 0
\(856\) 2.76382 0.0944655
\(857\) 8.03472 0.274461 0.137230 0.990539i \(-0.456180\pi\)
0.137230 + 0.990539i \(0.456180\pi\)
\(858\) 0 0
\(859\) −24.2959 −0.828966 −0.414483 0.910057i \(-0.636038\pi\)
−0.414483 + 0.910057i \(0.636038\pi\)
\(860\) 7.40333 0.252452
\(861\) 0 0
\(862\) −5.08378 −0.173154
\(863\) −10.2828 −0.350029 −0.175015 0.984566i \(-0.555997\pi\)
−0.175015 + 0.984566i \(0.555997\pi\)
\(864\) 0 0
\(865\) −26.0205 −0.884725
\(866\) 53.2067 1.80804
\(867\) 0 0
\(868\) 18.9932 0.644671
\(869\) 27.8866 0.945987
\(870\) 0 0
\(871\) −67.4448 −2.28528
\(872\) 3.46940 0.117489
\(873\) 0 0
\(874\) 6.36547 0.215315
\(875\) −32.1692 −1.08752
\(876\) 0 0
\(877\) −26.2276 −0.885644 −0.442822 0.896610i \(-0.646023\pi\)
−0.442822 + 0.896610i \(0.646023\pi\)
\(878\) −23.0643 −0.778384
\(879\) 0 0
\(880\) 34.0847 1.14900
\(881\) −39.2326 −1.32178 −0.660890 0.750483i \(-0.729821\pi\)
−0.660890 + 0.750483i \(0.729821\pi\)
\(882\) 0 0
\(883\) 7.79830 0.262434 0.131217 0.991354i \(-0.458112\pi\)
0.131217 + 0.991354i \(0.458112\pi\)
\(884\) 26.0194 0.875126
\(885\) 0 0
\(886\) 4.09833 0.137686
\(887\) −22.8571 −0.767465 −0.383732 0.923444i \(-0.625361\pi\)
−0.383732 + 0.923444i \(0.625361\pi\)
\(888\) 0 0
\(889\) −14.2986 −0.479560
\(890\) −5.09840 −0.170899
\(891\) 0 0
\(892\) −16.7297 −0.560151
\(893\) 1.47924 0.0495009
\(894\) 0 0
\(895\) −53.2499 −1.77995
\(896\) 4.45297 0.148763
\(897\) 0 0
\(898\) 20.7811 0.693473
\(899\) 15.0377 0.501537
\(900\) 0 0
\(901\) −13.6987 −0.456370
\(902\) 10.7211 0.356974
\(903\) 0 0
\(904\) 1.11556 0.0371031
\(905\) 48.8630 1.62426
\(906\) 0 0
\(907\) −37.8485 −1.25674 −0.628370 0.777915i \(-0.716278\pi\)
−0.628370 + 0.777915i \(0.716278\pi\)
\(908\) −20.0502 −0.665388
\(909\) 0 0
\(910\) −80.7238 −2.67597
\(911\) −52.7045 −1.74618 −0.873089 0.487561i \(-0.837887\pi\)
−0.873089 + 0.487561i \(0.837887\pi\)
\(912\) 0 0
\(913\) 24.6260 0.815001
\(914\) 15.5785 0.515291
\(915\) 0 0
\(916\) −15.5449 −0.513617
\(917\) −24.2908 −0.802152
\(918\) 0 0
\(919\) −49.1052 −1.61983 −0.809916 0.586545i \(-0.800488\pi\)
−0.809916 + 0.586545i \(0.800488\pi\)
\(920\) 0.456427 0.0150480
\(921\) 0 0
\(922\) −76.9977 −2.53579
\(923\) −5.67474 −0.186786
\(924\) 0 0
\(925\) 20.9855 0.689997
\(926\) −46.6658 −1.53353
\(927\) 0 0
\(928\) −27.4175 −0.900022
\(929\) −17.9507 −0.588943 −0.294472 0.955660i \(-0.595144\pi\)
−0.294472 + 0.955660i \(0.595144\pi\)
\(930\) 0 0
\(931\) 9.27900 0.304107
\(932\) 23.9420 0.784248
\(933\) 0 0
\(934\) −68.3756 −2.23732
\(935\) 23.6693 0.774070
\(936\) 0 0
\(937\) −29.7980 −0.973457 −0.486729 0.873553i \(-0.661810\pi\)
−0.486729 + 0.873553i \(0.661810\pi\)
\(938\) 66.1063 2.15845
\(939\) 0 0
\(940\) −1.65270 −0.0539052
\(941\) 30.0112 0.978339 0.489169 0.872189i \(-0.337300\pi\)
0.489169 + 0.872189i \(0.337300\pi\)
\(942\) 0 0
\(943\) 1.29685 0.0422311
\(944\) 56.2663 1.83131
\(945\) 0 0
\(946\) 4.56624 0.148461
\(947\) −45.6013 −1.48184 −0.740922 0.671591i \(-0.765611\pi\)
−0.740922 + 0.671591i \(0.765611\pi\)
\(948\) 0 0
\(949\) 22.0752 0.716592
\(950\) 106.727 3.46268
\(951\) 0 0
\(952\) 1.63673 0.0530466
\(953\) 13.1957 0.427451 0.213726 0.976894i \(-0.431440\pi\)
0.213726 + 0.976894i \(0.431440\pi\)
\(954\) 0 0
\(955\) 49.6887 1.60789
\(956\) 28.2201 0.912702
\(957\) 0 0
\(958\) −12.7665 −0.412467
\(959\) −44.7974 −1.44658
\(960\) 0 0
\(961\) −12.4635 −0.402047
\(962\) 22.4037 0.722323
\(963\) 0 0
\(964\) 1.49525 0.0481588
\(965\) 55.5437 1.78801
\(966\) 0 0
\(967\) −20.1548 −0.648133 −0.324067 0.946034i \(-0.605050\pi\)
−0.324067 + 0.946034i \(0.605050\pi\)
\(968\) −1.48574 −0.0477533
\(969\) 0 0
\(970\) 51.6664 1.65891
\(971\) 26.5839 0.853118 0.426559 0.904460i \(-0.359726\pi\)
0.426559 + 0.904460i \(0.359726\pi\)
\(972\) 0 0
\(973\) 54.7006 1.75362
\(974\) −39.3584 −1.26112
\(975\) 0 0
\(976\) 15.5330 0.497200
\(977\) −14.4041 −0.460828 −0.230414 0.973093i \(-0.574008\pi\)
−0.230414 + 0.973093i \(0.574008\pi\)
\(978\) 0 0
\(979\) −1.52341 −0.0486885
\(980\) −10.3671 −0.331165
\(981\) 0 0
\(982\) −51.6573 −1.64845
\(983\) 42.9881 1.37111 0.685554 0.728022i \(-0.259560\pi\)
0.685554 + 0.728022i \(0.259560\pi\)
\(984\) 0 0
\(985\) −82.6255 −2.63267
\(986\) −20.1918 −0.643039
\(987\) 0 0
\(988\) 55.1985 1.75610
\(989\) 0.552340 0.0175634
\(990\) 0 0
\(991\) −15.5794 −0.494895 −0.247447 0.968901i \(-0.579592\pi\)
−0.247447 + 0.968901i \(0.579592\pi\)
\(992\) −33.7966 −1.07304
\(993\) 0 0
\(994\) 5.56212 0.176420
\(995\) −33.6889 −1.06801
\(996\) 0 0
\(997\) 40.6186 1.28640 0.643201 0.765697i \(-0.277606\pi\)
0.643201 + 0.765697i \(0.277606\pi\)
\(998\) −12.4426 −0.393863
\(999\) 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 729.2.a.c.1.1 6
3.2 odd 2 inner 729.2.a.c.1.6 yes 6
9.2 odd 6 729.2.c.c.244.1 12
9.4 even 3 729.2.c.c.487.6 12
9.5 odd 6 729.2.c.c.487.1 12
9.7 even 3 729.2.c.c.244.6 12
27.2 odd 18 729.2.e.r.568.1 12
27.4 even 9 729.2.e.q.406.1 12
27.5 odd 18 729.2.e.m.649.2 12
27.7 even 9 729.2.e.q.325.1 12
27.11 odd 18 729.2.e.m.82.2 12
27.13 even 9 729.2.e.r.163.2 12
27.14 odd 18 729.2.e.r.163.1 12
27.16 even 9 729.2.e.m.82.1 12
27.20 odd 18 729.2.e.q.325.2 12
27.22 even 9 729.2.e.m.649.1 12
27.23 odd 18 729.2.e.q.406.2 12
27.25 even 9 729.2.e.r.568.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
729.2.a.c.1.1 6 1.1 even 1 trivial
729.2.a.c.1.6 yes 6 3.2 odd 2 inner
729.2.c.c.244.1 12 9.2 odd 6
729.2.c.c.244.6 12 9.7 even 3
729.2.c.c.487.1 12 9.5 odd 6
729.2.c.c.487.6 12 9.4 even 3
729.2.e.m.82.1 12 27.16 even 9
729.2.e.m.82.2 12 27.11 odd 18
729.2.e.m.649.1 12 27.22 even 9
729.2.e.m.649.2 12 27.5 odd 18
729.2.e.q.325.1 12 27.7 even 9
729.2.e.q.325.2 12 27.20 odd 18
729.2.e.q.406.1 12 27.4 even 9
729.2.e.q.406.2 12 27.23 odd 18
729.2.e.r.163.1 12 27.14 odd 18
729.2.e.r.163.2 12 27.13 even 9
729.2.e.r.568.1 12 27.2 odd 18
729.2.e.r.568.2 12 27.25 even 9