Properties

Label 7728.2.a.bw.1.3
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.11491\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.47283 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.47283 q^{5} +1.00000 q^{7} +1.00000 q^{9} -4.58774 q^{11} +3.11491 q^{13} +1.47283 q^{15} -6.58774 q^{17} +6.81756 q^{19} +1.00000 q^{21} -1.00000 q^{23} -2.83076 q^{25} +1.00000 q^{27} -7.30359 q^{29} -1.64207 q^{31} -4.58774 q^{33} +1.47283 q^{35} -8.10170 q^{37} +3.11491 q^{39} -9.87189 q^{41} -1.11491 q^{43} +1.47283 q^{45} -10.2298 q^{47} +1.00000 q^{49} -6.58774 q^{51} -5.47283 q^{53} -6.75698 q^{55} +6.81756 q^{57} -12.9868 q^{59} -7.49228 q^{61} +1.00000 q^{63} +4.58774 q^{65} +8.64832 q^{67} -1.00000 q^{69} +6.96735 q^{71} +15.0474 q^{73} -2.83076 q^{75} -4.58774 q^{77} +2.69641 q^{79} +1.00000 q^{81} -10.1017 q^{83} -9.70265 q^{85} -7.30359 q^{87} +11.9519 q^{89} +3.11491 q^{91} -1.64207 q^{93} +10.0411 q^{95} +7.04737 q^{97} -4.58774 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} - q^{5} + 3 q^{7} + 3 q^{9} - 2 q^{11} + 3 q^{13} - q^{15} - 8 q^{17} - 4 q^{19} + 3 q^{21} - 3 q^{23} - 4 q^{25} + 3 q^{27} - 12 q^{29} - 4 q^{31} - 2 q^{33} - q^{35} + 2 q^{37} + 3 q^{39} - 16 q^{41} + 3 q^{43} - q^{45} - 18 q^{47} + 3 q^{49} - 8 q^{51} - 11 q^{53} - 13 q^{55} - 4 q^{57} - 19 q^{59} - 9 q^{61} + 3 q^{63} + 2 q^{65} - 3 q^{67} - 3 q^{69} + 9 q^{71} + 8 q^{73} - 4 q^{75} - 2 q^{77} + 18 q^{79} + 3 q^{81} - 4 q^{83} - 11 q^{85} - 12 q^{87} - 3 q^{89} + 3 q^{91} - 4 q^{93} + 21 q^{95} - 16 q^{97} - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.47283 0.658671 0.329336 0.944213i \(-0.393175\pi\)
0.329336 + 0.944213i \(0.393175\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.58774 −1.38326 −0.691628 0.722254i \(-0.743106\pi\)
−0.691628 + 0.722254i \(0.743106\pi\)
\(12\) 0 0
\(13\) 3.11491 0.863920 0.431960 0.901893i \(-0.357822\pi\)
0.431960 + 0.901893i \(0.357822\pi\)
\(14\) 0 0
\(15\) 1.47283 0.380284
\(16\) 0 0
\(17\) −6.58774 −1.59776 −0.798881 0.601489i \(-0.794574\pi\)
−0.798881 + 0.601489i \(0.794574\pi\)
\(18\) 0 0
\(19\) 6.81756 1.56405 0.782027 0.623244i \(-0.214186\pi\)
0.782027 + 0.623244i \(0.214186\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −2.83076 −0.566152
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −7.30359 −1.35624 −0.678122 0.734950i \(-0.737206\pi\)
−0.678122 + 0.734950i \(0.737206\pi\)
\(30\) 0 0
\(31\) −1.64207 −0.294925 −0.147463 0.989068i \(-0.547111\pi\)
−0.147463 + 0.989068i \(0.547111\pi\)
\(32\) 0 0
\(33\) −4.58774 −0.798623
\(34\) 0 0
\(35\) 1.47283 0.248954
\(36\) 0 0
\(37\) −8.10170 −1.33191 −0.665956 0.745991i \(-0.731976\pi\)
−0.665956 + 0.745991i \(0.731976\pi\)
\(38\) 0 0
\(39\) 3.11491 0.498784
\(40\) 0 0
\(41\) −9.87189 −1.54173 −0.770865 0.636999i \(-0.780176\pi\)
−0.770865 + 0.636999i \(0.780176\pi\)
\(42\) 0 0
\(43\) −1.11491 −0.170022 −0.0850109 0.996380i \(-0.527093\pi\)
−0.0850109 + 0.996380i \(0.527093\pi\)
\(44\) 0 0
\(45\) 1.47283 0.219557
\(46\) 0 0
\(47\) −10.2298 −1.49217 −0.746086 0.665850i \(-0.768069\pi\)
−0.746086 + 0.665850i \(0.768069\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −6.58774 −0.922468
\(52\) 0 0
\(53\) −5.47283 −0.751752 −0.375876 0.926670i \(-0.622658\pi\)
−0.375876 + 0.926670i \(0.622658\pi\)
\(54\) 0 0
\(55\) −6.75698 −0.911111
\(56\) 0 0
\(57\) 6.81756 0.903007
\(58\) 0 0
\(59\) −12.9868 −1.69074 −0.845368 0.534184i \(-0.820619\pi\)
−0.845368 + 0.534184i \(0.820619\pi\)
\(60\) 0 0
\(61\) −7.49228 −0.959288 −0.479644 0.877463i \(-0.659234\pi\)
−0.479644 + 0.877463i \(0.659234\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 4.58774 0.569039
\(66\) 0 0
\(67\) 8.64832 1.05656 0.528280 0.849070i \(-0.322837\pi\)
0.528280 + 0.849070i \(0.322837\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 6.96735 0.826872 0.413436 0.910533i \(-0.364328\pi\)
0.413436 + 0.910533i \(0.364328\pi\)
\(72\) 0 0
\(73\) 15.0474 1.76116 0.880581 0.473896i \(-0.157153\pi\)
0.880581 + 0.473896i \(0.157153\pi\)
\(74\) 0 0
\(75\) −2.83076 −0.326868
\(76\) 0 0
\(77\) −4.58774 −0.522822
\(78\) 0 0
\(79\) 2.69641 0.303369 0.151685 0.988429i \(-0.451530\pi\)
0.151685 + 0.988429i \(0.451530\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −10.1017 −1.10881 −0.554403 0.832248i \(-0.687053\pi\)
−0.554403 + 0.832248i \(0.687053\pi\)
\(84\) 0 0
\(85\) −9.70265 −1.05240
\(86\) 0 0
\(87\) −7.30359 −0.783027
\(88\) 0 0
\(89\) 11.9519 1.26690 0.633450 0.773784i \(-0.281638\pi\)
0.633450 + 0.773784i \(0.281638\pi\)
\(90\) 0 0
\(91\) 3.11491 0.326531
\(92\) 0 0
\(93\) −1.64207 −0.170275
\(94\) 0 0
\(95\) 10.0411 1.03020
\(96\) 0 0
\(97\) 7.04737 0.715552 0.357776 0.933807i \(-0.383535\pi\)
0.357776 + 0.933807i \(0.383535\pi\)
\(98\) 0 0
\(99\) −4.58774 −0.461085
\(100\) 0 0
\(101\) 0.0411284 0.00409243 0.00204622 0.999998i \(-0.499349\pi\)
0.00204622 + 0.999998i \(0.499349\pi\)
\(102\) 0 0
\(103\) 16.9193 1.66710 0.833552 0.552441i \(-0.186303\pi\)
0.833552 + 0.552441i \(0.186303\pi\)
\(104\) 0 0
\(105\) 1.47283 0.143734
\(106\) 0 0
\(107\) 4.39905 0.425273 0.212636 0.977131i \(-0.431795\pi\)
0.212636 + 0.977131i \(0.431795\pi\)
\(108\) 0 0
\(109\) −14.6483 −1.40305 −0.701527 0.712643i \(-0.747498\pi\)
−0.701527 + 0.712643i \(0.747498\pi\)
\(110\) 0 0
\(111\) −8.10170 −0.768980
\(112\) 0 0
\(113\) −18.2709 −1.71879 −0.859393 0.511316i \(-0.829158\pi\)
−0.859393 + 0.511316i \(0.829158\pi\)
\(114\) 0 0
\(115\) −1.47283 −0.137342
\(116\) 0 0
\(117\) 3.11491 0.287973
\(118\) 0 0
\(119\) −6.58774 −0.603897
\(120\) 0 0
\(121\) 10.0474 0.913397
\(122\) 0 0
\(123\) −9.87189 −0.890118
\(124\) 0 0
\(125\) −11.5334 −1.03158
\(126\) 0 0
\(127\) −3.68320 −0.326831 −0.163416 0.986557i \(-0.552251\pi\)
−0.163416 + 0.986557i \(0.552251\pi\)
\(128\) 0 0
\(129\) −1.11491 −0.0981621
\(130\) 0 0
\(131\) 15.1949 1.32759 0.663794 0.747916i \(-0.268945\pi\)
0.663794 + 0.747916i \(0.268945\pi\)
\(132\) 0 0
\(133\) 6.81756 0.591157
\(134\) 0 0
\(135\) 1.47283 0.126761
\(136\) 0 0
\(137\) −8.01945 −0.685148 −0.342574 0.939491i \(-0.611299\pi\)
−0.342574 + 0.939491i \(0.611299\pi\)
\(138\) 0 0
\(139\) 11.5551 0.980090 0.490045 0.871697i \(-0.336980\pi\)
0.490045 + 0.871697i \(0.336980\pi\)
\(140\) 0 0
\(141\) −10.2298 −0.861506
\(142\) 0 0
\(143\) −14.2904 −1.19502
\(144\) 0 0
\(145\) −10.7570 −0.893319
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 1.40530 0.115126 0.0575632 0.998342i \(-0.481667\pi\)
0.0575632 + 0.998342i \(0.481667\pi\)
\(150\) 0 0
\(151\) −14.6894 −1.19541 −0.597705 0.801716i \(-0.703921\pi\)
−0.597705 + 0.801716i \(0.703921\pi\)
\(152\) 0 0
\(153\) −6.58774 −0.532587
\(154\) 0 0
\(155\) −2.41850 −0.194259
\(156\) 0 0
\(157\) −0.689445 −0.0550237 −0.0275119 0.999621i \(-0.508758\pi\)
−0.0275119 + 0.999621i \(0.508758\pi\)
\(158\) 0 0
\(159\) −5.47283 −0.434024
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −19.6762 −1.54116 −0.770581 0.637342i \(-0.780034\pi\)
−0.770581 + 0.637342i \(0.780034\pi\)
\(164\) 0 0
\(165\) −6.75698 −0.526030
\(166\) 0 0
\(167\) 15.9108 1.23121 0.615607 0.788054i \(-0.288911\pi\)
0.615607 + 0.788054i \(0.288911\pi\)
\(168\) 0 0
\(169\) −3.29735 −0.253642
\(170\) 0 0
\(171\) 6.81756 0.521352
\(172\) 0 0
\(173\) 10.6266 0.807928 0.403964 0.914775i \(-0.367632\pi\)
0.403964 + 0.914775i \(0.367632\pi\)
\(174\) 0 0
\(175\) −2.83076 −0.213985
\(176\) 0 0
\(177\) −12.9868 −0.976147
\(178\) 0 0
\(179\) 1.95887 0.146413 0.0732065 0.997317i \(-0.476677\pi\)
0.0732065 + 0.997317i \(0.476677\pi\)
\(180\) 0 0
\(181\) 6.58774 0.489663 0.244831 0.969566i \(-0.421267\pi\)
0.244831 + 0.969566i \(0.421267\pi\)
\(182\) 0 0
\(183\) −7.49228 −0.553845
\(184\) 0 0
\(185\) −11.9325 −0.877292
\(186\) 0 0
\(187\) 30.2229 2.21011
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 5.77018 0.417516 0.208758 0.977967i \(-0.433058\pi\)
0.208758 + 0.977967i \(0.433058\pi\)
\(192\) 0 0
\(193\) −17.1491 −1.23442 −0.617209 0.786799i \(-0.711737\pi\)
−0.617209 + 0.786799i \(0.711737\pi\)
\(194\) 0 0
\(195\) 4.58774 0.328535
\(196\) 0 0
\(197\) −6.73753 −0.480029 −0.240015 0.970769i \(-0.577152\pi\)
−0.240015 + 0.970769i \(0.577152\pi\)
\(198\) 0 0
\(199\) −3.85021 −0.272934 −0.136467 0.990645i \(-0.543575\pi\)
−0.136467 + 0.990645i \(0.543575\pi\)
\(200\) 0 0
\(201\) 8.64832 0.610005
\(202\) 0 0
\(203\) −7.30359 −0.512612
\(204\) 0 0
\(205\) −14.5397 −1.01549
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) −31.2772 −2.16349
\(210\) 0 0
\(211\) −3.53341 −0.243250 −0.121625 0.992576i \(-0.538811\pi\)
−0.121625 + 0.992576i \(0.538811\pi\)
\(212\) 0 0
\(213\) 6.96735 0.477395
\(214\) 0 0
\(215\) −1.64207 −0.111988
\(216\) 0 0
\(217\) −1.64207 −0.111471
\(218\) 0 0
\(219\) 15.0474 1.01681
\(220\) 0 0
\(221\) −20.5202 −1.38034
\(222\) 0 0
\(223\) 10.4185 0.697674 0.348837 0.937183i \(-0.386577\pi\)
0.348837 + 0.937183i \(0.386577\pi\)
\(224\) 0 0
\(225\) −2.83076 −0.188717
\(226\) 0 0
\(227\) 24.0147 1.59391 0.796957 0.604037i \(-0.206442\pi\)
0.796957 + 0.604037i \(0.206442\pi\)
\(228\) 0 0
\(229\) 5.13435 0.339288 0.169644 0.985505i \(-0.445738\pi\)
0.169644 + 0.985505i \(0.445738\pi\)
\(230\) 0 0
\(231\) −4.58774 −0.301851
\(232\) 0 0
\(233\) 0.500759 0.0328058 0.0164029 0.999865i \(-0.494779\pi\)
0.0164029 + 0.999865i \(0.494779\pi\)
\(234\) 0 0
\(235\) −15.0668 −0.982851
\(236\) 0 0
\(237\) 2.69641 0.175150
\(238\) 0 0
\(239\) 19.9714 1.29184 0.645920 0.763405i \(-0.276474\pi\)
0.645920 + 0.763405i \(0.276474\pi\)
\(240\) 0 0
\(241\) −19.0474 −1.22695 −0.613475 0.789715i \(-0.710229\pi\)
−0.613475 + 0.789715i \(0.710229\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 1.47283 0.0940959
\(246\) 0 0
\(247\) 21.2361 1.35122
\(248\) 0 0
\(249\) −10.1017 −0.640169
\(250\) 0 0
\(251\) 13.9736 0.882005 0.441003 0.897506i \(-0.354623\pi\)
0.441003 + 0.897506i \(0.354623\pi\)
\(252\) 0 0
\(253\) 4.58774 0.288429
\(254\) 0 0
\(255\) −9.70265 −0.607603
\(256\) 0 0
\(257\) 22.3246 1.39257 0.696284 0.717767i \(-0.254835\pi\)
0.696284 + 0.717767i \(0.254835\pi\)
\(258\) 0 0
\(259\) −8.10170 −0.503415
\(260\) 0 0
\(261\) −7.30359 −0.452081
\(262\) 0 0
\(263\) −23.2772 −1.43533 −0.717666 0.696387i \(-0.754790\pi\)
−0.717666 + 0.696387i \(0.754790\pi\)
\(264\) 0 0
\(265\) −8.06058 −0.495157
\(266\) 0 0
\(267\) 11.9519 0.731445
\(268\) 0 0
\(269\) −1.60095 −0.0976114 −0.0488057 0.998808i \(-0.515542\pi\)
−0.0488057 + 0.998808i \(0.515542\pi\)
\(270\) 0 0
\(271\) −27.3594 −1.66197 −0.830984 0.556296i \(-0.812222\pi\)
−0.830984 + 0.556296i \(0.812222\pi\)
\(272\) 0 0
\(273\) 3.11491 0.188523
\(274\) 0 0
\(275\) 12.9868 0.783133
\(276\) 0 0
\(277\) 10.7375 0.645156 0.322578 0.946543i \(-0.395451\pi\)
0.322578 + 0.946543i \(0.395451\pi\)
\(278\) 0 0
\(279\) −1.64207 −0.0983084
\(280\) 0 0
\(281\) 0.689445 0.0411289 0.0205644 0.999789i \(-0.493454\pi\)
0.0205644 + 0.999789i \(0.493454\pi\)
\(282\) 0 0
\(283\) −33.1902 −1.97295 −0.986476 0.163903i \(-0.947592\pi\)
−0.986476 + 0.163903i \(0.947592\pi\)
\(284\) 0 0
\(285\) 10.0411 0.594785
\(286\) 0 0
\(287\) −9.87189 −0.582719
\(288\) 0 0
\(289\) 26.3983 1.55284
\(290\) 0 0
\(291\) 7.04737 0.413124
\(292\) 0 0
\(293\) 16.0947 0.940265 0.470132 0.882596i \(-0.344206\pi\)
0.470132 + 0.882596i \(0.344206\pi\)
\(294\) 0 0
\(295\) −19.1274 −1.11364
\(296\) 0 0
\(297\) −4.58774 −0.266208
\(298\) 0 0
\(299\) −3.11491 −0.180140
\(300\) 0 0
\(301\) −1.11491 −0.0642622
\(302\) 0 0
\(303\) 0.0411284 0.00236277
\(304\) 0 0
\(305\) −11.0349 −0.631856
\(306\) 0 0
\(307\) 6.56133 0.374475 0.187238 0.982315i \(-0.440047\pi\)
0.187238 + 0.982315i \(0.440047\pi\)
\(308\) 0 0
\(309\) 16.9193 0.962503
\(310\) 0 0
\(311\) 23.2097 1.31610 0.658049 0.752975i \(-0.271382\pi\)
0.658049 + 0.752975i \(0.271382\pi\)
\(312\) 0 0
\(313\) 11.1755 0.631676 0.315838 0.948813i \(-0.397714\pi\)
0.315838 + 0.948813i \(0.397714\pi\)
\(314\) 0 0
\(315\) 1.47283 0.0829848
\(316\) 0 0
\(317\) −12.0411 −0.676297 −0.338149 0.941093i \(-0.609801\pi\)
−0.338149 + 0.941093i \(0.609801\pi\)
\(318\) 0 0
\(319\) 33.5070 1.87603
\(320\) 0 0
\(321\) 4.39905 0.245531
\(322\) 0 0
\(323\) −44.9123 −2.49899
\(324\) 0 0
\(325\) −8.81756 −0.489110
\(326\) 0 0
\(327\) −14.6483 −0.810054
\(328\) 0 0
\(329\) −10.2298 −0.563988
\(330\) 0 0
\(331\) −19.0668 −1.04801 −0.524004 0.851716i \(-0.675562\pi\)
−0.524004 + 0.851716i \(0.675562\pi\)
\(332\) 0 0
\(333\) −8.10170 −0.443971
\(334\) 0 0
\(335\) 12.7375 0.695926
\(336\) 0 0
\(337\) −19.1149 −1.04126 −0.520628 0.853784i \(-0.674302\pi\)
−0.520628 + 0.853784i \(0.674302\pi\)
\(338\) 0 0
\(339\) −18.2709 −0.992341
\(340\) 0 0
\(341\) 7.53341 0.407957
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −1.47283 −0.0792947
\(346\) 0 0
\(347\) 8.50548 0.456598 0.228299 0.973591i \(-0.426684\pi\)
0.228299 + 0.973591i \(0.426684\pi\)
\(348\) 0 0
\(349\) −3.24302 −0.173595 −0.0867974 0.996226i \(-0.527663\pi\)
−0.0867974 + 0.996226i \(0.527663\pi\)
\(350\) 0 0
\(351\) 3.11491 0.166261
\(352\) 0 0
\(353\) 2.58774 0.137732 0.0688658 0.997626i \(-0.478062\pi\)
0.0688658 + 0.997626i \(0.478062\pi\)
\(354\) 0 0
\(355\) 10.2617 0.544637
\(356\) 0 0
\(357\) −6.58774 −0.348660
\(358\) 0 0
\(359\) 27.5940 1.45635 0.728177 0.685389i \(-0.240368\pi\)
0.728177 + 0.685389i \(0.240368\pi\)
\(360\) 0 0
\(361\) 27.4791 1.44627
\(362\) 0 0
\(363\) 10.0474 0.527350
\(364\) 0 0
\(365\) 22.1623 1.16003
\(366\) 0 0
\(367\) −23.4659 −1.22491 −0.612454 0.790506i \(-0.709818\pi\)
−0.612454 + 0.790506i \(0.709818\pi\)
\(368\) 0 0
\(369\) −9.87189 −0.513910
\(370\) 0 0
\(371\) −5.47283 −0.284135
\(372\) 0 0
\(373\) 1.33000 0.0688649 0.0344324 0.999407i \(-0.489038\pi\)
0.0344324 + 0.999407i \(0.489038\pi\)
\(374\) 0 0
\(375\) −11.5334 −0.595583
\(376\) 0 0
\(377\) −22.7500 −1.17169
\(378\) 0 0
\(379\) 4.00000 0.205466 0.102733 0.994709i \(-0.467241\pi\)
0.102733 + 0.994709i \(0.467241\pi\)
\(380\) 0 0
\(381\) −3.68320 −0.188696
\(382\) 0 0
\(383\) 5.42474 0.277192 0.138596 0.990349i \(-0.455741\pi\)
0.138596 + 0.990349i \(0.455741\pi\)
\(384\) 0 0
\(385\) −6.75698 −0.344368
\(386\) 0 0
\(387\) −1.11491 −0.0566739
\(388\) 0 0
\(389\) 33.4806 1.69753 0.848767 0.528767i \(-0.177346\pi\)
0.848767 + 0.528767i \(0.177346\pi\)
\(390\) 0 0
\(391\) 6.58774 0.333156
\(392\) 0 0
\(393\) 15.1949 0.766483
\(394\) 0 0
\(395\) 3.97136 0.199821
\(396\) 0 0
\(397\) 20.9457 1.05123 0.525616 0.850722i \(-0.323835\pi\)
0.525616 + 0.850722i \(0.323835\pi\)
\(398\) 0 0
\(399\) 6.81756 0.341305
\(400\) 0 0
\(401\) −20.1017 −1.00383 −0.501916 0.864917i \(-0.667371\pi\)
−0.501916 + 0.864917i \(0.667371\pi\)
\(402\) 0 0
\(403\) −5.11491 −0.254792
\(404\) 0 0
\(405\) 1.47283 0.0731857
\(406\) 0 0
\(407\) 37.1685 1.84238
\(408\) 0 0
\(409\) 1.53341 0.0758222 0.0379111 0.999281i \(-0.487930\pi\)
0.0379111 + 0.999281i \(0.487930\pi\)
\(410\) 0 0
\(411\) −8.01945 −0.395570
\(412\) 0 0
\(413\) −12.9868 −0.639038
\(414\) 0 0
\(415\) −14.8781 −0.730339
\(416\) 0 0
\(417\) 11.5551 0.565855
\(418\) 0 0
\(419\) 25.1468 1.22850 0.614252 0.789110i \(-0.289458\pi\)
0.614252 + 0.789110i \(0.289458\pi\)
\(420\) 0 0
\(421\) −28.1164 −1.37031 −0.685155 0.728397i \(-0.740266\pi\)
−0.685155 + 0.728397i \(0.740266\pi\)
\(422\) 0 0
\(423\) −10.2298 −0.497391
\(424\) 0 0
\(425\) 18.6483 0.904576
\(426\) 0 0
\(427\) −7.49228 −0.362577
\(428\) 0 0
\(429\) −14.2904 −0.689947
\(430\) 0 0
\(431\) 32.5202 1.56644 0.783222 0.621743i \(-0.213575\pi\)
0.783222 + 0.621743i \(0.213575\pi\)
\(432\) 0 0
\(433\) −24.4721 −1.17605 −0.588027 0.808841i \(-0.700095\pi\)
−0.588027 + 0.808841i \(0.700095\pi\)
\(434\) 0 0
\(435\) −10.7570 −0.515758
\(436\) 0 0
\(437\) −6.81756 −0.326128
\(438\) 0 0
\(439\) −16.9651 −0.809701 −0.404850 0.914383i \(-0.632676\pi\)
−0.404850 + 0.914383i \(0.632676\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 7.02792 0.333907 0.166953 0.985965i \(-0.446607\pi\)
0.166953 + 0.985965i \(0.446607\pi\)
\(444\) 0 0
\(445\) 17.6032 0.834471
\(446\) 0 0
\(447\) 1.40530 0.0664683
\(448\) 0 0
\(449\) −9.47283 −0.447051 −0.223525 0.974698i \(-0.571757\pi\)
−0.223525 + 0.974698i \(0.571757\pi\)
\(450\) 0 0
\(451\) 45.2897 2.13261
\(452\) 0 0
\(453\) −14.6894 −0.690170
\(454\) 0 0
\(455\) 4.58774 0.215077
\(456\) 0 0
\(457\) 4.53965 0.212356 0.106178 0.994347i \(-0.466139\pi\)
0.106178 + 0.994347i \(0.466139\pi\)
\(458\) 0 0
\(459\) −6.58774 −0.307489
\(460\) 0 0
\(461\) −26.3098 −1.22537 −0.612686 0.790327i \(-0.709911\pi\)
−0.612686 + 0.790327i \(0.709911\pi\)
\(462\) 0 0
\(463\) −25.0085 −1.16224 −0.581121 0.813817i \(-0.697386\pi\)
−0.581121 + 0.813817i \(0.697386\pi\)
\(464\) 0 0
\(465\) −2.41850 −0.112155
\(466\) 0 0
\(467\) 21.8066 1.00909 0.504544 0.863386i \(-0.331661\pi\)
0.504544 + 0.863386i \(0.331661\pi\)
\(468\) 0 0
\(469\) 8.64832 0.399342
\(470\) 0 0
\(471\) −0.689445 −0.0317680
\(472\) 0 0
\(473\) 5.11491 0.235184
\(474\) 0 0
\(475\) −19.2989 −0.885493
\(476\) 0 0
\(477\) −5.47283 −0.250584
\(478\) 0 0
\(479\) −4.80507 −0.219549 −0.109775 0.993957i \(-0.535013\pi\)
−0.109775 + 0.993957i \(0.535013\pi\)
\(480\) 0 0
\(481\) −25.2361 −1.15067
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) 10.3796 0.471314
\(486\) 0 0
\(487\) 26.6825 1.20910 0.604549 0.796568i \(-0.293353\pi\)
0.604549 + 0.796568i \(0.293353\pi\)
\(488\) 0 0
\(489\) −19.6762 −0.889790
\(490\) 0 0
\(491\) −28.7764 −1.29866 −0.649331 0.760506i \(-0.724951\pi\)
−0.649331 + 0.760506i \(0.724951\pi\)
\(492\) 0 0
\(493\) 48.1142 2.16695
\(494\) 0 0
\(495\) −6.75698 −0.303704
\(496\) 0 0
\(497\) 6.96735 0.312528
\(498\) 0 0
\(499\) 3.55509 0.159148 0.0795739 0.996829i \(-0.474644\pi\)
0.0795739 + 0.996829i \(0.474644\pi\)
\(500\) 0 0
\(501\) 15.9108 0.710841
\(502\) 0 0
\(503\) 3.60791 0.160869 0.0804343 0.996760i \(-0.474369\pi\)
0.0804343 + 0.996760i \(0.474369\pi\)
\(504\) 0 0
\(505\) 0.0605754 0.00269557
\(506\) 0 0
\(507\) −3.29735 −0.146440
\(508\) 0 0
\(509\) −10.3774 −0.459969 −0.229984 0.973194i \(-0.573868\pi\)
−0.229984 + 0.973194i \(0.573868\pi\)
\(510\) 0 0
\(511\) 15.0474 0.665657
\(512\) 0 0
\(513\) 6.81756 0.301002
\(514\) 0 0
\(515\) 24.9193 1.09807
\(516\) 0 0
\(517\) 46.9317 2.06406
\(518\) 0 0
\(519\) 10.6266 0.466458
\(520\) 0 0
\(521\) −43.3789 −1.90046 −0.950232 0.311544i \(-0.899154\pi\)
−0.950232 + 0.311544i \(0.899154\pi\)
\(522\) 0 0
\(523\) 0.715853 0.0313021 0.0156510 0.999878i \(-0.495018\pi\)
0.0156510 + 0.999878i \(0.495018\pi\)
\(524\) 0 0
\(525\) −2.83076 −0.123545
\(526\) 0 0
\(527\) 10.8176 0.471220
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −12.9868 −0.563579
\(532\) 0 0
\(533\) −30.7500 −1.33193
\(534\) 0 0
\(535\) 6.47908 0.280115
\(536\) 0 0
\(537\) 1.95887 0.0845315
\(538\) 0 0
\(539\) −4.58774 −0.197608
\(540\) 0 0
\(541\) −13.9177 −0.598371 −0.299185 0.954195i \(-0.596715\pi\)
−0.299185 + 0.954195i \(0.596715\pi\)
\(542\) 0 0
\(543\) 6.58774 0.282707
\(544\) 0 0
\(545\) −21.5745 −0.924152
\(546\) 0 0
\(547\) −12.6872 −0.542466 −0.271233 0.962514i \(-0.587431\pi\)
−0.271233 + 0.962514i \(0.587431\pi\)
\(548\) 0 0
\(549\) −7.49228 −0.319763
\(550\) 0 0
\(551\) −49.7927 −2.12124
\(552\) 0 0
\(553\) 2.69641 0.114663
\(554\) 0 0
\(555\) −11.9325 −0.506505
\(556\) 0 0
\(557\) 0.0653013 0.00276691 0.00138345 0.999999i \(-0.499560\pi\)
0.00138345 + 0.999999i \(0.499560\pi\)
\(558\) 0 0
\(559\) −3.47283 −0.146885
\(560\) 0 0
\(561\) 30.2229 1.27601
\(562\) 0 0
\(563\) −18.6678 −0.786752 −0.393376 0.919378i \(-0.628693\pi\)
−0.393376 + 0.919378i \(0.628693\pi\)
\(564\) 0 0
\(565\) −26.9101 −1.13211
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −37.7438 −1.58230 −0.791151 0.611621i \(-0.790518\pi\)
−0.791151 + 0.611621i \(0.790518\pi\)
\(570\) 0 0
\(571\) −14.1840 −0.593580 −0.296790 0.954943i \(-0.595916\pi\)
−0.296790 + 0.954943i \(0.595916\pi\)
\(572\) 0 0
\(573\) 5.77018 0.241053
\(574\) 0 0
\(575\) 2.83076 0.118051
\(576\) 0 0
\(577\) −14.2423 −0.592915 −0.296457 0.955046i \(-0.595805\pi\)
−0.296457 + 0.955046i \(0.595805\pi\)
\(578\) 0 0
\(579\) −17.1491 −0.712691
\(580\) 0 0
\(581\) −10.1017 −0.419089
\(582\) 0 0
\(583\) 25.1079 1.03986
\(584\) 0 0
\(585\) 4.58774 0.189680
\(586\) 0 0
\(587\) −34.8323 −1.43768 −0.718841 0.695175i \(-0.755327\pi\)
−0.718841 + 0.695175i \(0.755327\pi\)
\(588\) 0 0
\(589\) −11.1949 −0.461279
\(590\) 0 0
\(591\) −6.73753 −0.277145
\(592\) 0 0
\(593\) −12.6894 −0.521093 −0.260547 0.965461i \(-0.583903\pi\)
−0.260547 + 0.965461i \(0.583903\pi\)
\(594\) 0 0
\(595\) −9.70265 −0.397770
\(596\) 0 0
\(597\) −3.85021 −0.157578
\(598\) 0 0
\(599\) −26.4504 −1.08074 −0.540368 0.841429i \(-0.681715\pi\)
−0.540368 + 0.841429i \(0.681715\pi\)
\(600\) 0 0
\(601\) −8.04113 −0.328004 −0.164002 0.986460i \(-0.552440\pi\)
−0.164002 + 0.986460i \(0.552440\pi\)
\(602\) 0 0
\(603\) 8.64832 0.352187
\(604\) 0 0
\(605\) 14.7981 0.601629
\(606\) 0 0
\(607\) −19.6398 −0.797156 −0.398578 0.917134i \(-0.630496\pi\)
−0.398578 + 0.917134i \(0.630496\pi\)
\(608\) 0 0
\(609\) −7.30359 −0.295957
\(610\) 0 0
\(611\) −31.8649 −1.28912
\(612\) 0 0
\(613\) 30.3315 1.22508 0.612539 0.790440i \(-0.290148\pi\)
0.612539 + 0.790440i \(0.290148\pi\)
\(614\) 0 0
\(615\) −14.5397 −0.586295
\(616\) 0 0
\(617\) −41.8821 −1.68611 −0.843056 0.537826i \(-0.819246\pi\)
−0.843056 + 0.537826i \(0.819246\pi\)
\(618\) 0 0
\(619\) −11.3014 −0.454240 −0.227120 0.973867i \(-0.572931\pi\)
−0.227120 + 0.973867i \(0.572931\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 11.9519 0.478843
\(624\) 0 0
\(625\) −2.83299 −0.113320
\(626\) 0 0
\(627\) −31.2772 −1.24909
\(628\) 0 0
\(629\) 53.3719 2.12808
\(630\) 0 0
\(631\) 17.1296 0.681920 0.340960 0.940078i \(-0.389248\pi\)
0.340960 + 0.940078i \(0.389248\pi\)
\(632\) 0 0
\(633\) −3.53341 −0.140440
\(634\) 0 0
\(635\) −5.42474 −0.215274
\(636\) 0 0
\(637\) 3.11491 0.123417
\(638\) 0 0
\(639\) 6.96735 0.275624
\(640\) 0 0
\(641\) 39.6134 1.56464 0.782318 0.622879i \(-0.214037\pi\)
0.782318 + 0.622879i \(0.214037\pi\)
\(642\) 0 0
\(643\) 13.4534 0.530550 0.265275 0.964173i \(-0.414537\pi\)
0.265275 + 0.964173i \(0.414537\pi\)
\(644\) 0 0
\(645\) −1.64207 −0.0646566
\(646\) 0 0
\(647\) −41.4853 −1.63096 −0.815478 0.578788i \(-0.803526\pi\)
−0.815478 + 0.578788i \(0.803526\pi\)
\(648\) 0 0
\(649\) 59.5801 2.33872
\(650\) 0 0
\(651\) −1.64207 −0.0643579
\(652\) 0 0
\(653\) 19.5815 0.766283 0.383142 0.923690i \(-0.374842\pi\)
0.383142 + 0.923690i \(0.374842\pi\)
\(654\) 0 0
\(655\) 22.3796 0.874444
\(656\) 0 0
\(657\) 15.0474 0.587054
\(658\) 0 0
\(659\) −15.4512 −0.601891 −0.300946 0.953641i \(-0.597302\pi\)
−0.300946 + 0.953641i \(0.597302\pi\)
\(660\) 0 0
\(661\) 40.9387 1.59233 0.796166 0.605079i \(-0.206858\pi\)
0.796166 + 0.605079i \(0.206858\pi\)
\(662\) 0 0
\(663\) −20.5202 −0.796939
\(664\) 0 0
\(665\) 10.0411 0.389378
\(666\) 0 0
\(667\) 7.30359 0.282796
\(668\) 0 0
\(669\) 10.4185 0.402803
\(670\) 0 0
\(671\) 34.3726 1.32694
\(672\) 0 0
\(673\) −30.0753 −1.15932 −0.579659 0.814859i \(-0.696814\pi\)
−0.579659 + 0.814859i \(0.696814\pi\)
\(674\) 0 0
\(675\) −2.83076 −0.108956
\(676\) 0 0
\(677\) 27.8697 1.07112 0.535559 0.844498i \(-0.320101\pi\)
0.535559 + 0.844498i \(0.320101\pi\)
\(678\) 0 0
\(679\) 7.04737 0.270453
\(680\) 0 0
\(681\) 24.0147 0.920246
\(682\) 0 0
\(683\) −12.2882 −0.470193 −0.235097 0.971972i \(-0.575541\pi\)
−0.235097 + 0.971972i \(0.575541\pi\)
\(684\) 0 0
\(685\) −11.8113 −0.451287
\(686\) 0 0
\(687\) 5.13435 0.195888
\(688\) 0 0
\(689\) −17.0474 −0.649453
\(690\) 0 0
\(691\) −7.68320 −0.292283 −0.146141 0.989264i \(-0.546685\pi\)
−0.146141 + 0.989264i \(0.546685\pi\)
\(692\) 0 0
\(693\) −4.58774 −0.174274
\(694\) 0 0
\(695\) 17.0187 0.645557
\(696\) 0 0
\(697\) 65.0335 2.46332
\(698\) 0 0
\(699\) 0.500759 0.0189404
\(700\) 0 0
\(701\) −11.7415 −0.443472 −0.221736 0.975107i \(-0.571172\pi\)
−0.221736 + 0.975107i \(0.571172\pi\)
\(702\) 0 0
\(703\) −55.2338 −2.08318
\(704\) 0 0
\(705\) −15.0668 −0.567449
\(706\) 0 0
\(707\) 0.0411284 0.00154679
\(708\) 0 0
\(709\) −4.32928 −0.162590 −0.0812948 0.996690i \(-0.525906\pi\)
−0.0812948 + 0.996690i \(0.525906\pi\)
\(710\) 0 0
\(711\) 2.69641 0.101123
\(712\) 0 0
\(713\) 1.64207 0.0614961
\(714\) 0 0
\(715\) −21.0474 −0.787127
\(716\) 0 0
\(717\) 19.9714 0.745844
\(718\) 0 0
\(719\) 24.9512 0.930523 0.465261 0.885173i \(-0.345960\pi\)
0.465261 + 0.885173i \(0.345960\pi\)
\(720\) 0 0
\(721\) 16.9193 0.630106
\(722\) 0 0
\(723\) −19.0474 −0.708379
\(724\) 0 0
\(725\) 20.6747 0.767840
\(726\) 0 0
\(727\) −16.7089 −0.619699 −0.309849 0.950786i \(-0.600279\pi\)
−0.309849 + 0.950786i \(0.600279\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 7.34472 0.271654
\(732\) 0 0
\(733\) 20.5249 0.758106 0.379053 0.925375i \(-0.376250\pi\)
0.379053 + 0.925375i \(0.376250\pi\)
\(734\) 0 0
\(735\) 1.47283 0.0543263
\(736\) 0 0
\(737\) −39.6762 −1.46149
\(738\) 0 0
\(739\) 25.0040 0.919787 0.459894 0.887974i \(-0.347888\pi\)
0.459894 + 0.887974i \(0.347888\pi\)
\(740\) 0 0
\(741\) 21.2361 0.780126
\(742\) 0 0
\(743\) 30.4574 1.11737 0.558687 0.829379i \(-0.311305\pi\)
0.558687 + 0.829379i \(0.311305\pi\)
\(744\) 0 0
\(745\) 2.06977 0.0758305
\(746\) 0 0
\(747\) −10.1017 −0.369602
\(748\) 0 0
\(749\) 4.39905 0.160738
\(750\) 0 0
\(751\) −15.4200 −0.562684 −0.281342 0.959607i \(-0.590780\pi\)
−0.281342 + 0.959607i \(0.590780\pi\)
\(752\) 0 0
\(753\) 13.9736 0.509226
\(754\) 0 0
\(755\) −21.6351 −0.787382
\(756\) 0 0
\(757\) 43.4247 1.57830 0.789150 0.614201i \(-0.210522\pi\)
0.789150 + 0.614201i \(0.210522\pi\)
\(758\) 0 0
\(759\) 4.58774 0.166524
\(760\) 0 0
\(761\) −2.20341 −0.0798735 −0.0399367 0.999202i \(-0.512716\pi\)
−0.0399367 + 0.999202i \(0.512716\pi\)
\(762\) 0 0
\(763\) −14.6483 −0.530305
\(764\) 0 0
\(765\) −9.70265 −0.350800
\(766\) 0 0
\(767\) −40.4527 −1.46066
\(768\) 0 0
\(769\) 23.2577 0.838696 0.419348 0.907826i \(-0.362259\pi\)
0.419348 + 0.907826i \(0.362259\pi\)
\(770\) 0 0
\(771\) 22.3246 0.803999
\(772\) 0 0
\(773\) 38.4527 1.38305 0.691523 0.722354i \(-0.256940\pi\)
0.691523 + 0.722354i \(0.256940\pi\)
\(774\) 0 0
\(775\) 4.64832 0.166972
\(776\) 0 0
\(777\) −8.10170 −0.290647
\(778\) 0 0
\(779\) −67.3022 −2.41135
\(780\) 0 0
\(781\) −31.9644 −1.14378
\(782\) 0 0
\(783\) −7.30359 −0.261009
\(784\) 0 0
\(785\) −1.01544 −0.0362425
\(786\) 0 0
\(787\) −41.6109 −1.48327 −0.741635 0.670804i \(-0.765949\pi\)
−0.741635 + 0.670804i \(0.765949\pi\)
\(788\) 0 0
\(789\) −23.2772 −0.828690
\(790\) 0 0
\(791\) −18.2709 −0.649640
\(792\) 0 0
\(793\) −23.3378 −0.828748
\(794\) 0 0
\(795\) −8.06058 −0.285879
\(796\) 0 0
\(797\) −4.24374 −0.150321 −0.0751604 0.997171i \(-0.523947\pi\)
−0.0751604 + 0.997171i \(0.523947\pi\)
\(798\) 0 0
\(799\) 67.3914 2.38414
\(800\) 0 0
\(801\) 11.9519 0.422300
\(802\) 0 0
\(803\) −69.0335 −2.43614
\(804\) 0 0
\(805\) −1.47283 −0.0519106
\(806\) 0 0
\(807\) −1.60095 −0.0563559
\(808\) 0 0
\(809\) −34.6803 −1.21929 −0.609646 0.792674i \(-0.708689\pi\)
−0.609646 + 0.792674i \(0.708689\pi\)
\(810\) 0 0
\(811\) 55.1949 1.93816 0.969078 0.246754i \(-0.0793641\pi\)
0.969078 + 0.246754i \(0.0793641\pi\)
\(812\) 0 0
\(813\) −27.3594 −0.959538
\(814\) 0 0
\(815\) −28.9798 −1.01512
\(816\) 0 0
\(817\) −7.60095 −0.265923
\(818\) 0 0
\(819\) 3.11491 0.108844
\(820\) 0 0
\(821\) −12.6506 −0.441507 −0.220754 0.975330i \(-0.570852\pi\)
−0.220754 + 0.975330i \(0.570852\pi\)
\(822\) 0 0
\(823\) −48.5521 −1.69242 −0.846211 0.532849i \(-0.821122\pi\)
−0.846211 + 0.532849i \(0.821122\pi\)
\(824\) 0 0
\(825\) 12.9868 0.452142
\(826\) 0 0
\(827\) −29.6065 −1.02952 −0.514759 0.857335i \(-0.672119\pi\)
−0.514759 + 0.857335i \(0.672119\pi\)
\(828\) 0 0
\(829\) 53.3550 1.85309 0.926547 0.376178i \(-0.122762\pi\)
0.926547 + 0.376178i \(0.122762\pi\)
\(830\) 0 0
\(831\) 10.7375 0.372481
\(832\) 0 0
\(833\) −6.58774 −0.228252
\(834\) 0 0
\(835\) 23.4339 0.810965
\(836\) 0 0
\(837\) −1.64207 −0.0567584
\(838\) 0 0
\(839\) 43.4270 1.49927 0.749633 0.661854i \(-0.230230\pi\)
0.749633 + 0.661854i \(0.230230\pi\)
\(840\) 0 0
\(841\) 24.3425 0.839396
\(842\) 0 0
\(843\) 0.689445 0.0237458
\(844\) 0 0
\(845\) −4.85645 −0.167067
\(846\) 0 0
\(847\) 10.0474 0.345232
\(848\) 0 0
\(849\) −33.1902 −1.13908
\(850\) 0 0
\(851\) 8.10170 0.277723
\(852\) 0 0
\(853\) −49.0963 −1.68102 −0.840512 0.541793i \(-0.817746\pi\)
−0.840512 + 0.541793i \(0.817746\pi\)
\(854\) 0 0
\(855\) 10.0411 0.343399
\(856\) 0 0
\(857\) 24.1770 0.825871 0.412935 0.910760i \(-0.364504\pi\)
0.412935 + 0.910760i \(0.364504\pi\)
\(858\) 0 0
\(859\) −48.2034 −1.64468 −0.822340 0.568997i \(-0.807332\pi\)
−0.822340 + 0.568997i \(0.807332\pi\)
\(860\) 0 0
\(861\) −9.87189 −0.336433
\(862\) 0 0
\(863\) −42.8929 −1.46009 −0.730045 0.683399i \(-0.760501\pi\)
−0.730045 + 0.683399i \(0.760501\pi\)
\(864\) 0 0
\(865\) 15.6513 0.532159
\(866\) 0 0
\(867\) 26.3983 0.896535
\(868\) 0 0
\(869\) −12.3704 −0.419638
\(870\) 0 0
\(871\) 26.9387 0.912783
\(872\) 0 0
\(873\) 7.04737 0.238517
\(874\) 0 0
\(875\) −11.5334 −0.389900
\(876\) 0 0
\(877\) −49.3091 −1.66505 −0.832525 0.553987i \(-0.813106\pi\)
−0.832525 + 0.553987i \(0.813106\pi\)
\(878\) 0 0
\(879\) 16.0947 0.542862
\(880\) 0 0
\(881\) 11.0404 0.371961 0.185980 0.982553i \(-0.440454\pi\)
0.185980 + 0.982553i \(0.440454\pi\)
\(882\) 0 0
\(883\) 55.9116 1.88157 0.940787 0.338997i \(-0.110088\pi\)
0.940787 + 0.338997i \(0.110088\pi\)
\(884\) 0 0
\(885\) −19.1274 −0.642960
\(886\) 0 0
\(887\) 46.0202 1.54521 0.772604 0.634888i \(-0.218954\pi\)
0.772604 + 0.634888i \(0.218954\pi\)
\(888\) 0 0
\(889\) −3.68320 −0.123531
\(890\) 0 0
\(891\) −4.58774 −0.153695
\(892\) 0 0
\(893\) −69.7423 −2.33384
\(894\) 0 0
\(895\) 2.88509 0.0964380
\(896\) 0 0
\(897\) −3.11491 −0.104004
\(898\) 0 0
\(899\) 11.9930 0.399990
\(900\) 0 0
\(901\) 36.0536 1.20112
\(902\) 0 0
\(903\) −1.11491 −0.0371018
\(904\) 0 0
\(905\) 9.70265 0.322527
\(906\) 0 0
\(907\) 17.4464 0.579299 0.289650 0.957133i \(-0.406461\pi\)
0.289650 + 0.957133i \(0.406461\pi\)
\(908\) 0 0
\(909\) 0.0411284 0.00136414
\(910\) 0 0
\(911\) −58.7144 −1.94530 −0.972648 0.232285i \(-0.925380\pi\)
−0.972648 + 0.232285i \(0.925380\pi\)
\(912\) 0 0
\(913\) 46.3440 1.53376
\(914\) 0 0
\(915\) −11.0349 −0.364802
\(916\) 0 0
\(917\) 15.1949 0.501781
\(918\) 0 0
\(919\) 1.64207 0.0541670 0.0270835 0.999633i \(-0.491378\pi\)
0.0270835 + 0.999633i \(0.491378\pi\)
\(920\) 0 0
\(921\) 6.56133 0.216203
\(922\) 0 0
\(923\) 21.7026 0.714351
\(924\) 0 0
\(925\) 22.9340 0.754065
\(926\) 0 0
\(927\) 16.9193 0.555701
\(928\) 0 0
\(929\) −1.45891 −0.0478654 −0.0239327 0.999714i \(-0.507619\pi\)
−0.0239327 + 0.999714i \(0.507619\pi\)
\(930\) 0 0
\(931\) 6.81756 0.223436
\(932\) 0 0
\(933\) 23.2097 0.759850
\(934\) 0 0
\(935\) 44.5132 1.45574
\(936\) 0 0
\(937\) −8.98207 −0.293431 −0.146716 0.989179i \(-0.546870\pi\)
−0.146716 + 0.989179i \(0.546870\pi\)
\(938\) 0 0
\(939\) 11.1755 0.364698
\(940\) 0 0
\(941\) 37.9861 1.23831 0.619155 0.785268i \(-0.287475\pi\)
0.619155 + 0.785268i \(0.287475\pi\)
\(942\) 0 0
\(943\) 9.87189 0.321473
\(944\) 0 0
\(945\) 1.47283 0.0479113
\(946\) 0 0
\(947\) 12.4985 0.406147 0.203074 0.979163i \(-0.434907\pi\)
0.203074 + 0.979163i \(0.434907\pi\)
\(948\) 0 0
\(949\) 46.8712 1.52150
\(950\) 0 0
\(951\) −12.0411 −0.390460
\(952\) 0 0
\(953\) 9.22357 0.298781 0.149390 0.988778i \(-0.452269\pi\)
0.149390 + 0.988778i \(0.452269\pi\)
\(954\) 0 0
\(955\) 8.49852 0.275006
\(956\) 0 0
\(957\) 33.5070 1.08313
\(958\) 0 0
\(959\) −8.01945 −0.258961
\(960\) 0 0
\(961\) −28.3036 −0.913019
\(962\) 0 0
\(963\) 4.39905 0.141758
\(964\) 0 0
\(965\) −25.2577 −0.813075
\(966\) 0 0
\(967\) −7.74378 −0.249023 −0.124512 0.992218i \(-0.539736\pi\)
−0.124512 + 0.992218i \(0.539736\pi\)
\(968\) 0 0
\(969\) −44.9123 −1.44279
\(970\) 0 0
\(971\) 31.8044 1.02065 0.510325 0.859982i \(-0.329525\pi\)
0.510325 + 0.859982i \(0.329525\pi\)
\(972\) 0 0
\(973\) 11.5551 0.370439
\(974\) 0 0
\(975\) −8.81756 −0.282388
\(976\) 0 0
\(977\) 47.7779 1.52855 0.764276 0.644889i \(-0.223097\pi\)
0.764276 + 0.644889i \(0.223097\pi\)
\(978\) 0 0
\(979\) −54.8323 −1.75245
\(980\) 0 0
\(981\) −14.6483 −0.467685
\(982\) 0 0
\(983\) −16.7089 −0.532931 −0.266465 0.963844i \(-0.585856\pi\)
−0.266465 + 0.963844i \(0.585856\pi\)
\(984\) 0 0
\(985\) −9.92327 −0.316182
\(986\) 0 0
\(987\) −10.2298 −0.325619
\(988\) 0 0
\(989\) 1.11491 0.0354520
\(990\) 0 0
\(991\) 21.7919 0.692241 0.346121 0.938190i \(-0.387499\pi\)
0.346121 + 0.938190i \(0.387499\pi\)
\(992\) 0 0
\(993\) −19.0668 −0.605067
\(994\) 0 0
\(995\) −5.67072 −0.179774
\(996\) 0 0
\(997\) 26.0070 0.823649 0.411824 0.911263i \(-0.364892\pi\)
0.411824 + 0.911263i \(0.364892\pi\)
\(998\) 0 0
\(999\) −8.10170 −0.256327
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 7728.2.a.bw.1.3 3
4.3 odd 2 3864.2.a.l.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.l.1.3 3 4.3 odd 2
7728.2.a.bw.1.3 3 1.1 even 1 trivial