Properties

Label 7728.2.a.bn.1.1
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -0.618034 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -0.618034 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.23607 q^{11} -2.38197 q^{13} -0.618034 q^{15} +6.70820 q^{17} +3.47214 q^{19} +1.00000 q^{21} +1.00000 q^{23} -4.61803 q^{25} +1.00000 q^{27} -8.23607 q^{29} -6.70820 q^{31} -2.23607 q^{33} -0.618034 q^{35} -11.0000 q^{37} -2.38197 q^{39} -1.47214 q^{41} +1.61803 q^{43} -0.618034 q^{45} +7.23607 q^{47} +1.00000 q^{49} +6.70820 q^{51} -13.0902 q^{53} +1.38197 q^{55} +3.47214 q^{57} -9.38197 q^{59} -4.85410 q^{61} +1.00000 q^{63} +1.47214 q^{65} +5.09017 q^{67} +1.00000 q^{69} -4.38197 q^{71} -12.7082 q^{73} -4.61803 q^{75} -2.23607 q^{77} +9.47214 q^{79} +1.00000 q^{81} +9.18034 q^{83} -4.14590 q^{85} -8.23607 q^{87} +11.6180 q^{89} -2.38197 q^{91} -6.70820 q^{93} -2.14590 q^{95} +10.4164 q^{97} -2.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} + O(q^{10}) \) \( 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} - 7 q^{13} + q^{15} - 2 q^{19} + 2 q^{21} + 2 q^{23} - 7 q^{25} + 2 q^{27} - 12 q^{29} + q^{35} - 22 q^{37} - 7 q^{39} + 6 q^{41} + q^{43} + q^{45} + 10 q^{47} + 2 q^{49} - 15 q^{53} + 5 q^{55} - 2 q^{57} - 21 q^{59} - 3 q^{61} + 2 q^{63} - 6 q^{65} - q^{67} + 2 q^{69} - 11 q^{71} - 12 q^{73} - 7 q^{75} + 10 q^{79} + 2 q^{81} - 4 q^{83} - 15 q^{85} - 12 q^{87} + 21 q^{89} - 7 q^{91} - 11 q^{95} - 6 q^{97} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −0.618034 −0.276393 −0.138197 0.990405i \(-0.544131\pi\)
−0.138197 + 0.990405i \(0.544131\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.23607 −0.674200 −0.337100 0.941469i \(-0.609446\pi\)
−0.337100 + 0.941469i \(0.609446\pi\)
\(12\) 0 0
\(13\) −2.38197 −0.660639 −0.330319 0.943869i \(-0.607156\pi\)
−0.330319 + 0.943869i \(0.607156\pi\)
\(14\) 0 0
\(15\) −0.618034 −0.159576
\(16\) 0 0
\(17\) 6.70820 1.62698 0.813489 0.581580i \(-0.197565\pi\)
0.813489 + 0.581580i \(0.197565\pi\)
\(18\) 0 0
\(19\) 3.47214 0.796563 0.398281 0.917263i \(-0.369607\pi\)
0.398281 + 0.917263i \(0.369607\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −4.61803 −0.923607
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −8.23607 −1.52940 −0.764700 0.644387i \(-0.777113\pi\)
−0.764700 + 0.644387i \(0.777113\pi\)
\(30\) 0 0
\(31\) −6.70820 −1.20483 −0.602414 0.798183i \(-0.705795\pi\)
−0.602414 + 0.798183i \(0.705795\pi\)
\(32\) 0 0
\(33\) −2.23607 −0.389249
\(34\) 0 0
\(35\) −0.618034 −0.104467
\(36\) 0 0
\(37\) −11.0000 −1.80839 −0.904194 0.427121i \(-0.859528\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 0 0
\(39\) −2.38197 −0.381420
\(40\) 0 0
\(41\) −1.47214 −0.229909 −0.114955 0.993371i \(-0.536672\pi\)
−0.114955 + 0.993371i \(0.536672\pi\)
\(42\) 0 0
\(43\) 1.61803 0.246748 0.123374 0.992360i \(-0.460629\pi\)
0.123374 + 0.992360i \(0.460629\pi\)
\(44\) 0 0
\(45\) −0.618034 −0.0921311
\(46\) 0 0
\(47\) 7.23607 1.05549 0.527744 0.849403i \(-0.323038\pi\)
0.527744 + 0.849403i \(0.323038\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 6.70820 0.939336
\(52\) 0 0
\(53\) −13.0902 −1.79807 −0.899037 0.437874i \(-0.855732\pi\)
−0.899037 + 0.437874i \(0.855732\pi\)
\(54\) 0 0
\(55\) 1.38197 0.186344
\(56\) 0 0
\(57\) 3.47214 0.459896
\(58\) 0 0
\(59\) −9.38197 −1.22143 −0.610714 0.791851i \(-0.709117\pi\)
−0.610714 + 0.791851i \(0.709117\pi\)
\(60\) 0 0
\(61\) −4.85410 −0.621504 −0.310752 0.950491i \(-0.600581\pi\)
−0.310752 + 0.950491i \(0.600581\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 1.47214 0.182596
\(66\) 0 0
\(67\) 5.09017 0.621863 0.310932 0.950432i \(-0.399359\pi\)
0.310932 + 0.950432i \(0.399359\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −4.38197 −0.520044 −0.260022 0.965603i \(-0.583730\pi\)
−0.260022 + 0.965603i \(0.583730\pi\)
\(72\) 0 0
\(73\) −12.7082 −1.48738 −0.743691 0.668523i \(-0.766927\pi\)
−0.743691 + 0.668523i \(0.766927\pi\)
\(74\) 0 0
\(75\) −4.61803 −0.533245
\(76\) 0 0
\(77\) −2.23607 −0.254824
\(78\) 0 0
\(79\) 9.47214 1.06570 0.532849 0.846210i \(-0.321121\pi\)
0.532849 + 0.846210i \(0.321121\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 9.18034 1.00767 0.503837 0.863799i \(-0.331921\pi\)
0.503837 + 0.863799i \(0.331921\pi\)
\(84\) 0 0
\(85\) −4.14590 −0.449686
\(86\) 0 0
\(87\) −8.23607 −0.882999
\(88\) 0 0
\(89\) 11.6180 1.23151 0.615755 0.787938i \(-0.288851\pi\)
0.615755 + 0.787938i \(0.288851\pi\)
\(90\) 0 0
\(91\) −2.38197 −0.249698
\(92\) 0 0
\(93\) −6.70820 −0.695608
\(94\) 0 0
\(95\) −2.14590 −0.220164
\(96\) 0 0
\(97\) 10.4164 1.05763 0.528813 0.848738i \(-0.322637\pi\)
0.528813 + 0.848738i \(0.322637\pi\)
\(98\) 0 0
\(99\) −2.23607 −0.224733
\(100\) 0 0
\(101\) 4.14590 0.412532 0.206266 0.978496i \(-0.433869\pi\)
0.206266 + 0.978496i \(0.433869\pi\)
\(102\) 0 0
\(103\) 7.41641 0.730760 0.365380 0.930858i \(-0.380939\pi\)
0.365380 + 0.930858i \(0.380939\pi\)
\(104\) 0 0
\(105\) −0.618034 −0.0603139
\(106\) 0 0
\(107\) −3.32624 −0.321560 −0.160780 0.986990i \(-0.551401\pi\)
−0.160780 + 0.986990i \(0.551401\pi\)
\(108\) 0 0
\(109\) 19.2705 1.84578 0.922890 0.385064i \(-0.125820\pi\)
0.922890 + 0.385064i \(0.125820\pi\)
\(110\) 0 0
\(111\) −11.0000 −1.04407
\(112\) 0 0
\(113\) −0.909830 −0.0855896 −0.0427948 0.999084i \(-0.513626\pi\)
−0.0427948 + 0.999084i \(0.513626\pi\)
\(114\) 0 0
\(115\) −0.618034 −0.0576320
\(116\) 0 0
\(117\) −2.38197 −0.220213
\(118\) 0 0
\(119\) 6.70820 0.614940
\(120\) 0 0
\(121\) −6.00000 −0.545455
\(122\) 0 0
\(123\) −1.47214 −0.132738
\(124\) 0 0
\(125\) 5.94427 0.531672
\(126\) 0 0
\(127\) −22.2705 −1.97619 −0.988094 0.153851i \(-0.950832\pi\)
−0.988094 + 0.153851i \(0.950832\pi\)
\(128\) 0 0
\(129\) 1.61803 0.142460
\(130\) 0 0
\(131\) −15.1803 −1.32631 −0.663156 0.748481i \(-0.730784\pi\)
−0.663156 + 0.748481i \(0.730784\pi\)
\(132\) 0 0
\(133\) 3.47214 0.301072
\(134\) 0 0
\(135\) −0.618034 −0.0531919
\(136\) 0 0
\(137\) −16.4164 −1.40255 −0.701274 0.712892i \(-0.747385\pi\)
−0.701274 + 0.712892i \(0.747385\pi\)
\(138\) 0 0
\(139\) 11.3820 0.965406 0.482703 0.875784i \(-0.339655\pi\)
0.482703 + 0.875784i \(0.339655\pi\)
\(140\) 0 0
\(141\) 7.23607 0.609387
\(142\) 0 0
\(143\) 5.32624 0.445402
\(144\) 0 0
\(145\) 5.09017 0.422716
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −19.2361 −1.57588 −0.787940 0.615752i \(-0.788852\pi\)
−0.787940 + 0.615752i \(0.788852\pi\)
\(150\) 0 0
\(151\) 15.2361 1.23989 0.619947 0.784644i \(-0.287154\pi\)
0.619947 + 0.784644i \(0.287154\pi\)
\(152\) 0 0
\(153\) 6.70820 0.542326
\(154\) 0 0
\(155\) 4.14590 0.333007
\(156\) 0 0
\(157\) −0.291796 −0.0232879 −0.0116439 0.999932i \(-0.503706\pi\)
−0.0116439 + 0.999932i \(0.503706\pi\)
\(158\) 0 0
\(159\) −13.0902 −1.03812
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 0.618034 0.0484082 0.0242041 0.999707i \(-0.492295\pi\)
0.0242041 + 0.999707i \(0.492295\pi\)
\(164\) 0 0
\(165\) 1.38197 0.107586
\(166\) 0 0
\(167\) 7.18034 0.555631 0.277816 0.960634i \(-0.410390\pi\)
0.277816 + 0.960634i \(0.410390\pi\)
\(168\) 0 0
\(169\) −7.32624 −0.563557
\(170\) 0 0
\(171\) 3.47214 0.265521
\(172\) 0 0
\(173\) −3.47214 −0.263982 −0.131991 0.991251i \(-0.542137\pi\)
−0.131991 + 0.991251i \(0.542137\pi\)
\(174\) 0 0
\(175\) −4.61803 −0.349091
\(176\) 0 0
\(177\) −9.38197 −0.705192
\(178\) 0 0
\(179\) −1.85410 −0.138582 −0.0692910 0.997596i \(-0.522074\pi\)
−0.0692910 + 0.997596i \(0.522074\pi\)
\(180\) 0 0
\(181\) −1.94427 −0.144517 −0.0722583 0.997386i \(-0.523021\pi\)
−0.0722583 + 0.997386i \(0.523021\pi\)
\(182\) 0 0
\(183\) −4.85410 −0.358826
\(184\) 0 0
\(185\) 6.79837 0.499826
\(186\) 0 0
\(187\) −15.0000 −1.09691
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −16.1803 −1.17077 −0.585384 0.810756i \(-0.699056\pi\)
−0.585384 + 0.810756i \(0.699056\pi\)
\(192\) 0 0
\(193\) 8.29180 0.596857 0.298428 0.954432i \(-0.403538\pi\)
0.298428 + 0.954432i \(0.403538\pi\)
\(194\) 0 0
\(195\) 1.47214 0.105422
\(196\) 0 0
\(197\) −25.5066 −1.81727 −0.908634 0.417593i \(-0.862874\pi\)
−0.908634 + 0.417593i \(0.862874\pi\)
\(198\) 0 0
\(199\) 6.90983 0.489825 0.244912 0.969545i \(-0.421241\pi\)
0.244912 + 0.969545i \(0.421241\pi\)
\(200\) 0 0
\(201\) 5.09017 0.359033
\(202\) 0 0
\(203\) −8.23607 −0.578059
\(204\) 0 0
\(205\) 0.909830 0.0635453
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −7.76393 −0.537042
\(210\) 0 0
\(211\) −10.4164 −0.717095 −0.358548 0.933511i \(-0.616728\pi\)
−0.358548 + 0.933511i \(0.616728\pi\)
\(212\) 0 0
\(213\) −4.38197 −0.300247
\(214\) 0 0
\(215\) −1.00000 −0.0681994
\(216\) 0 0
\(217\) −6.70820 −0.455383
\(218\) 0 0
\(219\) −12.7082 −0.858741
\(220\) 0 0
\(221\) −15.9787 −1.07484
\(222\) 0 0
\(223\) 17.8541 1.19560 0.597800 0.801646i \(-0.296042\pi\)
0.597800 + 0.801646i \(0.296042\pi\)
\(224\) 0 0
\(225\) −4.61803 −0.307869
\(226\) 0 0
\(227\) −24.3262 −1.61459 −0.807295 0.590149i \(-0.799069\pi\)
−0.807295 + 0.590149i \(0.799069\pi\)
\(228\) 0 0
\(229\) −16.3262 −1.07887 −0.539434 0.842028i \(-0.681362\pi\)
−0.539434 + 0.842028i \(0.681362\pi\)
\(230\) 0 0
\(231\) −2.23607 −0.147122
\(232\) 0 0
\(233\) −11.0902 −0.726541 −0.363271 0.931684i \(-0.618340\pi\)
−0.363271 + 0.931684i \(0.618340\pi\)
\(234\) 0 0
\(235\) −4.47214 −0.291730
\(236\) 0 0
\(237\) 9.47214 0.615281
\(238\) 0 0
\(239\) 4.79837 0.310381 0.155191 0.987885i \(-0.450401\pi\)
0.155191 + 0.987885i \(0.450401\pi\)
\(240\) 0 0
\(241\) −11.0000 −0.708572 −0.354286 0.935137i \(-0.615276\pi\)
−0.354286 + 0.935137i \(0.615276\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −0.618034 −0.0394847
\(246\) 0 0
\(247\) −8.27051 −0.526240
\(248\) 0 0
\(249\) 9.18034 0.581780
\(250\) 0 0
\(251\) −23.1246 −1.45961 −0.729806 0.683654i \(-0.760390\pi\)
−0.729806 + 0.683654i \(0.760390\pi\)
\(252\) 0 0
\(253\) −2.23607 −0.140580
\(254\) 0 0
\(255\) −4.14590 −0.259626
\(256\) 0 0
\(257\) −3.23607 −0.201860 −0.100930 0.994894i \(-0.532182\pi\)
−0.100930 + 0.994894i \(0.532182\pi\)
\(258\) 0 0
\(259\) −11.0000 −0.683507
\(260\) 0 0
\(261\) −8.23607 −0.509800
\(262\) 0 0
\(263\) 0.0557281 0.00343634 0.00171817 0.999999i \(-0.499453\pi\)
0.00171817 + 0.999999i \(0.499453\pi\)
\(264\) 0 0
\(265\) 8.09017 0.496975
\(266\) 0 0
\(267\) 11.6180 0.711012
\(268\) 0 0
\(269\) −32.0902 −1.95657 −0.978286 0.207259i \(-0.933546\pi\)
−0.978286 + 0.207259i \(0.933546\pi\)
\(270\) 0 0
\(271\) −21.9443 −1.33302 −0.666510 0.745496i \(-0.732213\pi\)
−0.666510 + 0.745496i \(0.732213\pi\)
\(272\) 0 0
\(273\) −2.38197 −0.144163
\(274\) 0 0
\(275\) 10.3262 0.622696
\(276\) 0 0
\(277\) 4.27051 0.256590 0.128295 0.991736i \(-0.459050\pi\)
0.128295 + 0.991736i \(0.459050\pi\)
\(278\) 0 0
\(279\) −6.70820 −0.401610
\(280\) 0 0
\(281\) −3.70820 −0.221213 −0.110606 0.993864i \(-0.535279\pi\)
−0.110606 + 0.993864i \(0.535279\pi\)
\(282\) 0 0
\(283\) 26.2148 1.55831 0.779154 0.626833i \(-0.215649\pi\)
0.779154 + 0.626833i \(0.215649\pi\)
\(284\) 0 0
\(285\) −2.14590 −0.127112
\(286\) 0 0
\(287\) −1.47214 −0.0868974
\(288\) 0 0
\(289\) 28.0000 1.64706
\(290\) 0 0
\(291\) 10.4164 0.610621
\(292\) 0 0
\(293\) 12.0000 0.701047 0.350524 0.936554i \(-0.386004\pi\)
0.350524 + 0.936554i \(0.386004\pi\)
\(294\) 0 0
\(295\) 5.79837 0.337594
\(296\) 0 0
\(297\) −2.23607 −0.129750
\(298\) 0 0
\(299\) −2.38197 −0.137753
\(300\) 0 0
\(301\) 1.61803 0.0932619
\(302\) 0 0
\(303\) 4.14590 0.238176
\(304\) 0 0
\(305\) 3.00000 0.171780
\(306\) 0 0
\(307\) −16.1246 −0.920280 −0.460140 0.887846i \(-0.652201\pi\)
−0.460140 + 0.887846i \(0.652201\pi\)
\(308\) 0 0
\(309\) 7.41641 0.421905
\(310\) 0 0
\(311\) −15.3262 −0.869071 −0.434536 0.900655i \(-0.643087\pi\)
−0.434536 + 0.900655i \(0.643087\pi\)
\(312\) 0 0
\(313\) −2.47214 −0.139733 −0.0698667 0.997556i \(-0.522257\pi\)
−0.0698667 + 0.997556i \(0.522257\pi\)
\(314\) 0 0
\(315\) −0.618034 −0.0348223
\(316\) 0 0
\(317\) 14.5066 0.814771 0.407385 0.913256i \(-0.366441\pi\)
0.407385 + 0.913256i \(0.366441\pi\)
\(318\) 0 0
\(319\) 18.4164 1.03112
\(320\) 0 0
\(321\) −3.32624 −0.185652
\(322\) 0 0
\(323\) 23.2918 1.29599
\(324\) 0 0
\(325\) 11.0000 0.610170
\(326\) 0 0
\(327\) 19.2705 1.06566
\(328\) 0 0
\(329\) 7.23607 0.398937
\(330\) 0 0
\(331\) 13.4164 0.737432 0.368716 0.929542i \(-0.379797\pi\)
0.368716 + 0.929542i \(0.379797\pi\)
\(332\) 0 0
\(333\) −11.0000 −0.602796
\(334\) 0 0
\(335\) −3.14590 −0.171879
\(336\) 0 0
\(337\) −7.50658 −0.408909 −0.204455 0.978876i \(-0.565542\pi\)
−0.204455 + 0.978876i \(0.565542\pi\)
\(338\) 0 0
\(339\) −0.909830 −0.0494152
\(340\) 0 0
\(341\) 15.0000 0.812296
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −0.618034 −0.0332738
\(346\) 0 0
\(347\) −5.18034 −0.278095 −0.139048 0.990286i \(-0.544404\pi\)
−0.139048 + 0.990286i \(0.544404\pi\)
\(348\) 0 0
\(349\) −11.8541 −0.634536 −0.317268 0.948336i \(-0.602765\pi\)
−0.317268 + 0.948336i \(0.602765\pi\)
\(350\) 0 0
\(351\) −2.38197 −0.127140
\(352\) 0 0
\(353\) 30.3050 1.61297 0.806485 0.591255i \(-0.201367\pi\)
0.806485 + 0.591255i \(0.201367\pi\)
\(354\) 0 0
\(355\) 2.70820 0.143737
\(356\) 0 0
\(357\) 6.70820 0.355036
\(358\) 0 0
\(359\) −23.0902 −1.21865 −0.609326 0.792920i \(-0.708560\pi\)
−0.609326 + 0.792920i \(0.708560\pi\)
\(360\) 0 0
\(361\) −6.94427 −0.365488
\(362\) 0 0
\(363\) −6.00000 −0.314918
\(364\) 0 0
\(365\) 7.85410 0.411102
\(366\) 0 0
\(367\) −10.8541 −0.566580 −0.283290 0.959034i \(-0.591426\pi\)
−0.283290 + 0.959034i \(0.591426\pi\)
\(368\) 0 0
\(369\) −1.47214 −0.0766363
\(370\) 0 0
\(371\) −13.0902 −0.679608
\(372\) 0 0
\(373\) 24.4164 1.26423 0.632117 0.774873i \(-0.282186\pi\)
0.632117 + 0.774873i \(0.282186\pi\)
\(374\) 0 0
\(375\) 5.94427 0.306961
\(376\) 0 0
\(377\) 19.6180 1.01038
\(378\) 0 0
\(379\) −7.41641 −0.380955 −0.190478 0.981692i \(-0.561004\pi\)
−0.190478 + 0.981692i \(0.561004\pi\)
\(380\) 0 0
\(381\) −22.2705 −1.14095
\(382\) 0 0
\(383\) 0.708204 0.0361875 0.0180938 0.999836i \(-0.494240\pi\)
0.0180938 + 0.999836i \(0.494240\pi\)
\(384\) 0 0
\(385\) 1.38197 0.0704315
\(386\) 0 0
\(387\) 1.61803 0.0822493
\(388\) 0 0
\(389\) 5.29180 0.268305 0.134152 0.990961i \(-0.457169\pi\)
0.134152 + 0.990961i \(0.457169\pi\)
\(390\) 0 0
\(391\) 6.70820 0.339248
\(392\) 0 0
\(393\) −15.1803 −0.765747
\(394\) 0 0
\(395\) −5.85410 −0.294552
\(396\) 0 0
\(397\) 22.0689 1.10761 0.553803 0.832648i \(-0.313176\pi\)
0.553803 + 0.832648i \(0.313176\pi\)
\(398\) 0 0
\(399\) 3.47214 0.173824
\(400\) 0 0
\(401\) 33.1803 1.65695 0.828474 0.560028i \(-0.189210\pi\)
0.828474 + 0.560028i \(0.189210\pi\)
\(402\) 0 0
\(403\) 15.9787 0.795956
\(404\) 0 0
\(405\) −0.618034 −0.0307104
\(406\) 0 0
\(407\) 24.5967 1.21922
\(408\) 0 0
\(409\) 6.41641 0.317271 0.158635 0.987337i \(-0.449291\pi\)
0.158635 + 0.987337i \(0.449291\pi\)
\(410\) 0 0
\(411\) −16.4164 −0.809762
\(412\) 0 0
\(413\) −9.38197 −0.461656
\(414\) 0 0
\(415\) −5.67376 −0.278514
\(416\) 0 0
\(417\) 11.3820 0.557377
\(418\) 0 0
\(419\) −15.6738 −0.765713 −0.382857 0.923808i \(-0.625060\pi\)
−0.382857 + 0.923808i \(0.625060\pi\)
\(420\) 0 0
\(421\) −10.2705 −0.500554 −0.250277 0.968174i \(-0.580522\pi\)
−0.250277 + 0.968174i \(0.580522\pi\)
\(422\) 0 0
\(423\) 7.23607 0.351830
\(424\) 0 0
\(425\) −30.9787 −1.50269
\(426\) 0 0
\(427\) −4.85410 −0.234906
\(428\) 0 0
\(429\) 5.32624 0.257153
\(430\) 0 0
\(431\) −0.673762 −0.0324540 −0.0162270 0.999868i \(-0.505165\pi\)
−0.0162270 + 0.999868i \(0.505165\pi\)
\(432\) 0 0
\(433\) 19.1246 0.919070 0.459535 0.888160i \(-0.348016\pi\)
0.459535 + 0.888160i \(0.348016\pi\)
\(434\) 0 0
\(435\) 5.09017 0.244055
\(436\) 0 0
\(437\) 3.47214 0.166095
\(438\) 0 0
\(439\) −23.6525 −1.12887 −0.564436 0.825477i \(-0.690906\pi\)
−0.564436 + 0.825477i \(0.690906\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 9.52786 0.452682 0.226341 0.974048i \(-0.427324\pi\)
0.226341 + 0.974048i \(0.427324\pi\)
\(444\) 0 0
\(445\) −7.18034 −0.340381
\(446\) 0 0
\(447\) −19.2361 −0.909835
\(448\) 0 0
\(449\) −18.4377 −0.870129 −0.435064 0.900399i \(-0.643274\pi\)
−0.435064 + 0.900399i \(0.643274\pi\)
\(450\) 0 0
\(451\) 3.29180 0.155005
\(452\) 0 0
\(453\) 15.2361 0.715853
\(454\) 0 0
\(455\) 1.47214 0.0690148
\(456\) 0 0
\(457\) −14.3262 −0.670153 −0.335077 0.942191i \(-0.608762\pi\)
−0.335077 + 0.942191i \(0.608762\pi\)
\(458\) 0 0
\(459\) 6.70820 0.313112
\(460\) 0 0
\(461\) 25.4508 1.18536 0.592682 0.805436i \(-0.298069\pi\)
0.592682 + 0.805436i \(0.298069\pi\)
\(462\) 0 0
\(463\) −17.2918 −0.803618 −0.401809 0.915724i \(-0.631618\pi\)
−0.401809 + 0.915724i \(0.631618\pi\)
\(464\) 0 0
\(465\) 4.14590 0.192261
\(466\) 0 0
\(467\) 15.1803 0.702462 0.351231 0.936289i \(-0.385763\pi\)
0.351231 + 0.936289i \(0.385763\pi\)
\(468\) 0 0
\(469\) 5.09017 0.235042
\(470\) 0 0
\(471\) −0.291796 −0.0134453
\(472\) 0 0
\(473\) −3.61803 −0.166357
\(474\) 0 0
\(475\) −16.0344 −0.735711
\(476\) 0 0
\(477\) −13.0902 −0.599358
\(478\) 0 0
\(479\) −21.0000 −0.959514 −0.479757 0.877401i \(-0.659275\pi\)
−0.479757 + 0.877401i \(0.659275\pi\)
\(480\) 0 0
\(481\) 26.2016 1.19469
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) −6.43769 −0.292321
\(486\) 0 0
\(487\) 29.2918 1.32734 0.663669 0.748026i \(-0.268998\pi\)
0.663669 + 0.748026i \(0.268998\pi\)
\(488\) 0 0
\(489\) 0.618034 0.0279485
\(490\) 0 0
\(491\) −9.79837 −0.442194 −0.221097 0.975252i \(-0.570964\pi\)
−0.221097 + 0.975252i \(0.570964\pi\)
\(492\) 0 0
\(493\) −55.2492 −2.48830
\(494\) 0 0
\(495\) 1.38197 0.0621148
\(496\) 0 0
\(497\) −4.38197 −0.196558
\(498\) 0 0
\(499\) 38.3951 1.71880 0.859401 0.511302i \(-0.170837\pi\)
0.859401 + 0.511302i \(0.170837\pi\)
\(500\) 0 0
\(501\) 7.18034 0.320794
\(502\) 0 0
\(503\) −36.3262 −1.61971 −0.809853 0.586632i \(-0.800453\pi\)
−0.809853 + 0.586632i \(0.800453\pi\)
\(504\) 0 0
\(505\) −2.56231 −0.114021
\(506\) 0 0
\(507\) −7.32624 −0.325370
\(508\) 0 0
\(509\) −15.5967 −0.691314 −0.345657 0.938361i \(-0.612344\pi\)
−0.345657 + 0.938361i \(0.612344\pi\)
\(510\) 0 0
\(511\) −12.7082 −0.562178
\(512\) 0 0
\(513\) 3.47214 0.153299
\(514\) 0 0
\(515\) −4.58359 −0.201977
\(516\) 0 0
\(517\) −16.1803 −0.711611
\(518\) 0 0
\(519\) −3.47214 −0.152410
\(520\) 0 0
\(521\) 3.52786 0.154559 0.0772793 0.997009i \(-0.475377\pi\)
0.0772793 + 0.997009i \(0.475377\pi\)
\(522\) 0 0
\(523\) −21.4164 −0.936474 −0.468237 0.883603i \(-0.655111\pi\)
−0.468237 + 0.883603i \(0.655111\pi\)
\(524\) 0 0
\(525\) −4.61803 −0.201548
\(526\) 0 0
\(527\) −45.0000 −1.96023
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −9.38197 −0.407143
\(532\) 0 0
\(533\) 3.50658 0.151887
\(534\) 0 0
\(535\) 2.05573 0.0888769
\(536\) 0 0
\(537\) −1.85410 −0.0800104
\(538\) 0 0
\(539\) −2.23607 −0.0963143
\(540\) 0 0
\(541\) −5.12461 −0.220324 −0.110162 0.993914i \(-0.535137\pi\)
−0.110162 + 0.993914i \(0.535137\pi\)
\(542\) 0 0
\(543\) −1.94427 −0.0834367
\(544\) 0 0
\(545\) −11.9098 −0.510161
\(546\) 0 0
\(547\) −12.7984 −0.547219 −0.273609 0.961841i \(-0.588218\pi\)
−0.273609 + 0.961841i \(0.588218\pi\)
\(548\) 0 0
\(549\) −4.85410 −0.207168
\(550\) 0 0
\(551\) −28.5967 −1.21826
\(552\) 0 0
\(553\) 9.47214 0.402796
\(554\) 0 0
\(555\) 6.79837 0.288575
\(556\) 0 0
\(557\) −5.81966 −0.246587 −0.123293 0.992370i \(-0.539346\pi\)
−0.123293 + 0.992370i \(0.539346\pi\)
\(558\) 0 0
\(559\) −3.85410 −0.163011
\(560\) 0 0
\(561\) −15.0000 −0.633300
\(562\) 0 0
\(563\) 23.5066 0.990684 0.495342 0.868698i \(-0.335043\pi\)
0.495342 + 0.868698i \(0.335043\pi\)
\(564\) 0 0
\(565\) 0.562306 0.0236564
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −16.3607 −0.685875 −0.342938 0.939358i \(-0.611422\pi\)
−0.342938 + 0.939358i \(0.611422\pi\)
\(570\) 0 0
\(571\) 44.4164 1.85877 0.929384 0.369113i \(-0.120339\pi\)
0.929384 + 0.369113i \(0.120339\pi\)
\(572\) 0 0
\(573\) −16.1803 −0.675943
\(574\) 0 0
\(575\) −4.61803 −0.192585
\(576\) 0 0
\(577\) −36.8328 −1.53337 −0.766685 0.642023i \(-0.778095\pi\)
−0.766685 + 0.642023i \(0.778095\pi\)
\(578\) 0 0
\(579\) 8.29180 0.344595
\(580\) 0 0
\(581\) 9.18034 0.380865
\(582\) 0 0
\(583\) 29.2705 1.21226
\(584\) 0 0
\(585\) 1.47214 0.0608653
\(586\) 0 0
\(587\) 31.0344 1.28093 0.640464 0.767988i \(-0.278742\pi\)
0.640464 + 0.767988i \(0.278742\pi\)
\(588\) 0 0
\(589\) −23.2918 −0.959722
\(590\) 0 0
\(591\) −25.5066 −1.04920
\(592\) 0 0
\(593\) −32.0689 −1.31691 −0.658456 0.752620i \(-0.728790\pi\)
−0.658456 + 0.752620i \(0.728790\pi\)
\(594\) 0 0
\(595\) −4.14590 −0.169965
\(596\) 0 0
\(597\) 6.90983 0.282801
\(598\) 0 0
\(599\) 15.5066 0.633582 0.316791 0.948495i \(-0.397395\pi\)
0.316791 + 0.948495i \(0.397395\pi\)
\(600\) 0 0
\(601\) 0.909830 0.0371127 0.0185564 0.999828i \(-0.494093\pi\)
0.0185564 + 0.999828i \(0.494093\pi\)
\(602\) 0 0
\(603\) 5.09017 0.207288
\(604\) 0 0
\(605\) 3.70820 0.150760
\(606\) 0 0
\(607\) 33.7984 1.37183 0.685917 0.727680i \(-0.259401\pi\)
0.685917 + 0.727680i \(0.259401\pi\)
\(608\) 0 0
\(609\) −8.23607 −0.333742
\(610\) 0 0
\(611\) −17.2361 −0.697297
\(612\) 0 0
\(613\) −11.6525 −0.470639 −0.235320 0.971918i \(-0.575614\pi\)
−0.235320 + 0.971918i \(0.575614\pi\)
\(614\) 0 0
\(615\) 0.909830 0.0366879
\(616\) 0 0
\(617\) −2.67376 −0.107642 −0.0538208 0.998551i \(-0.517140\pi\)
−0.0538208 + 0.998551i \(0.517140\pi\)
\(618\) 0 0
\(619\) 19.6180 0.788515 0.394258 0.919000i \(-0.371002\pi\)
0.394258 + 0.919000i \(0.371002\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 11.6180 0.465467
\(624\) 0 0
\(625\) 19.4164 0.776656
\(626\) 0 0
\(627\) −7.76393 −0.310062
\(628\) 0 0
\(629\) −73.7902 −2.94221
\(630\) 0 0
\(631\) −16.1246 −0.641911 −0.320955 0.947094i \(-0.604004\pi\)
−0.320955 + 0.947094i \(0.604004\pi\)
\(632\) 0 0
\(633\) −10.4164 −0.414015
\(634\) 0 0
\(635\) 13.7639 0.546205
\(636\) 0 0
\(637\) −2.38197 −0.0943769
\(638\) 0 0
\(639\) −4.38197 −0.173348
\(640\) 0 0
\(641\) 17.7984 0.702994 0.351497 0.936189i \(-0.385673\pi\)
0.351497 + 0.936189i \(0.385673\pi\)
\(642\) 0 0
\(643\) 11.9787 0.472394 0.236197 0.971705i \(-0.424099\pi\)
0.236197 + 0.971705i \(0.424099\pi\)
\(644\) 0 0
\(645\) −1.00000 −0.0393750
\(646\) 0 0
\(647\) 24.0344 0.944891 0.472446 0.881360i \(-0.343371\pi\)
0.472446 + 0.881360i \(0.343371\pi\)
\(648\) 0 0
\(649\) 20.9787 0.823487
\(650\) 0 0
\(651\) −6.70820 −0.262915
\(652\) 0 0
\(653\) 27.7426 1.08565 0.542827 0.839845i \(-0.317354\pi\)
0.542827 + 0.839845i \(0.317354\pi\)
\(654\) 0 0
\(655\) 9.38197 0.366584
\(656\) 0 0
\(657\) −12.7082 −0.495794
\(658\) 0 0
\(659\) −9.65248 −0.376007 −0.188004 0.982168i \(-0.560202\pi\)
−0.188004 + 0.982168i \(0.560202\pi\)
\(660\) 0 0
\(661\) 14.4164 0.560733 0.280367 0.959893i \(-0.409544\pi\)
0.280367 + 0.959893i \(0.409544\pi\)
\(662\) 0 0
\(663\) −15.9787 −0.620562
\(664\) 0 0
\(665\) −2.14590 −0.0832144
\(666\) 0 0
\(667\) −8.23607 −0.318902
\(668\) 0 0
\(669\) 17.8541 0.690279
\(670\) 0 0
\(671\) 10.8541 0.419018
\(672\) 0 0
\(673\) 12.4164 0.478617 0.239309 0.970944i \(-0.423079\pi\)
0.239309 + 0.970944i \(0.423079\pi\)
\(674\) 0 0
\(675\) −4.61803 −0.177748
\(676\) 0 0
\(677\) −1.25735 −0.0483240 −0.0241620 0.999708i \(-0.507692\pi\)
−0.0241620 + 0.999708i \(0.507692\pi\)
\(678\) 0 0
\(679\) 10.4164 0.399745
\(680\) 0 0
\(681\) −24.3262 −0.932183
\(682\) 0 0
\(683\) −26.0132 −0.995366 −0.497683 0.867359i \(-0.665816\pi\)
−0.497683 + 0.867359i \(0.665816\pi\)
\(684\) 0 0
\(685\) 10.1459 0.387655
\(686\) 0 0
\(687\) −16.3262 −0.622885
\(688\) 0 0
\(689\) 31.1803 1.18788
\(690\) 0 0
\(691\) 7.72949 0.294044 0.147022 0.989133i \(-0.453031\pi\)
0.147022 + 0.989133i \(0.453031\pi\)
\(692\) 0 0
\(693\) −2.23607 −0.0849412
\(694\) 0 0
\(695\) −7.03444 −0.266832
\(696\) 0 0
\(697\) −9.87539 −0.374057
\(698\) 0 0
\(699\) −11.0902 −0.419469
\(700\) 0 0
\(701\) 37.4508 1.41450 0.707250 0.706964i \(-0.249936\pi\)
0.707250 + 0.706964i \(0.249936\pi\)
\(702\) 0 0
\(703\) −38.1935 −1.44049
\(704\) 0 0
\(705\) −4.47214 −0.168430
\(706\) 0 0
\(707\) 4.14590 0.155923
\(708\) 0 0
\(709\) −2.56231 −0.0962294 −0.0481147 0.998842i \(-0.515321\pi\)
−0.0481147 + 0.998842i \(0.515321\pi\)
\(710\) 0 0
\(711\) 9.47214 0.355233
\(712\) 0 0
\(713\) −6.70820 −0.251224
\(714\) 0 0
\(715\) −3.29180 −0.123106
\(716\) 0 0
\(717\) 4.79837 0.179199
\(718\) 0 0
\(719\) 3.00000 0.111881 0.0559406 0.998434i \(-0.482184\pi\)
0.0559406 + 0.998434i \(0.482184\pi\)
\(720\) 0 0
\(721\) 7.41641 0.276201
\(722\) 0 0
\(723\) −11.0000 −0.409094
\(724\) 0 0
\(725\) 38.0344 1.41256
\(726\) 0 0
\(727\) −18.8885 −0.700537 −0.350269 0.936649i \(-0.613910\pi\)
−0.350269 + 0.936649i \(0.613910\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 10.8541 0.401453
\(732\) 0 0
\(733\) 25.5967 0.945437 0.472719 0.881213i \(-0.343273\pi\)
0.472719 + 0.881213i \(0.343273\pi\)
\(734\) 0 0
\(735\) −0.618034 −0.0227965
\(736\) 0 0
\(737\) −11.3820 −0.419260
\(738\) 0 0
\(739\) −45.2492 −1.66452 −0.832260 0.554386i \(-0.812953\pi\)
−0.832260 + 0.554386i \(0.812953\pi\)
\(740\) 0 0
\(741\) −8.27051 −0.303825
\(742\) 0 0
\(743\) −11.6738 −0.428269 −0.214134 0.976804i \(-0.568693\pi\)
−0.214134 + 0.976804i \(0.568693\pi\)
\(744\) 0 0
\(745\) 11.8885 0.435563
\(746\) 0 0
\(747\) 9.18034 0.335891
\(748\) 0 0
\(749\) −3.32624 −0.121538
\(750\) 0 0
\(751\) −32.9787 −1.20341 −0.601705 0.798718i \(-0.705512\pi\)
−0.601705 + 0.798718i \(0.705512\pi\)
\(752\) 0 0
\(753\) −23.1246 −0.842708
\(754\) 0 0
\(755\) −9.41641 −0.342698
\(756\) 0 0
\(757\) −17.0000 −0.617876 −0.308938 0.951082i \(-0.599973\pi\)
−0.308938 + 0.951082i \(0.599973\pi\)
\(758\) 0 0
\(759\) −2.23607 −0.0811641
\(760\) 0 0
\(761\) −14.4721 −0.524615 −0.262307 0.964984i \(-0.584483\pi\)
−0.262307 + 0.964984i \(0.584483\pi\)
\(762\) 0 0
\(763\) 19.2705 0.697639
\(764\) 0 0
\(765\) −4.14590 −0.149895
\(766\) 0 0
\(767\) 22.3475 0.806922
\(768\) 0 0
\(769\) 33.2361 1.19852 0.599262 0.800553i \(-0.295461\pi\)
0.599262 + 0.800553i \(0.295461\pi\)
\(770\) 0 0
\(771\) −3.23607 −0.116544
\(772\) 0 0
\(773\) −19.9443 −0.717346 −0.358673 0.933463i \(-0.616771\pi\)
−0.358673 + 0.933463i \(0.616771\pi\)
\(774\) 0 0
\(775\) 30.9787 1.11279
\(776\) 0 0
\(777\) −11.0000 −0.394623
\(778\) 0 0
\(779\) −5.11146 −0.183137
\(780\) 0 0
\(781\) 9.79837 0.350613
\(782\) 0 0
\(783\) −8.23607 −0.294333
\(784\) 0 0
\(785\) 0.180340 0.00643661
\(786\) 0 0
\(787\) 48.5755 1.73153 0.865764 0.500452i \(-0.166833\pi\)
0.865764 + 0.500452i \(0.166833\pi\)
\(788\) 0 0
\(789\) 0.0557281 0.00198397
\(790\) 0 0
\(791\) −0.909830 −0.0323498
\(792\) 0 0
\(793\) 11.5623 0.410590
\(794\) 0 0
\(795\) 8.09017 0.286929
\(796\) 0 0
\(797\) 44.1803 1.56495 0.782474 0.622683i \(-0.213957\pi\)
0.782474 + 0.622683i \(0.213957\pi\)
\(798\) 0 0
\(799\) 48.5410 1.71726
\(800\) 0 0
\(801\) 11.6180 0.410503
\(802\) 0 0
\(803\) 28.4164 1.00279
\(804\) 0 0
\(805\) −0.618034 −0.0217828
\(806\) 0 0
\(807\) −32.0902 −1.12963
\(808\) 0 0
\(809\) −9.43769 −0.331812 −0.165906 0.986142i \(-0.553055\pi\)
−0.165906 + 0.986142i \(0.553055\pi\)
\(810\) 0 0
\(811\) −8.52786 −0.299454 −0.149727 0.988727i \(-0.547839\pi\)
−0.149727 + 0.988727i \(0.547839\pi\)
\(812\) 0 0
\(813\) −21.9443 −0.769619
\(814\) 0 0
\(815\) −0.381966 −0.0133797
\(816\) 0 0
\(817\) 5.61803 0.196550
\(818\) 0 0
\(819\) −2.38197 −0.0832326
\(820\) 0 0
\(821\) −29.1246 −1.01646 −0.508228 0.861223i \(-0.669699\pi\)
−0.508228 + 0.861223i \(0.669699\pi\)
\(822\) 0 0
\(823\) 39.8541 1.38923 0.694613 0.719383i \(-0.255575\pi\)
0.694613 + 0.719383i \(0.255575\pi\)
\(824\) 0 0
\(825\) 10.3262 0.359513
\(826\) 0 0
\(827\) 28.2148 0.981124 0.490562 0.871406i \(-0.336792\pi\)
0.490562 + 0.871406i \(0.336792\pi\)
\(828\) 0 0
\(829\) −30.4164 −1.05641 −0.528203 0.849118i \(-0.677134\pi\)
−0.528203 + 0.849118i \(0.677134\pi\)
\(830\) 0 0
\(831\) 4.27051 0.148142
\(832\) 0 0
\(833\) 6.70820 0.232425
\(834\) 0 0
\(835\) −4.43769 −0.153573
\(836\) 0 0
\(837\) −6.70820 −0.231869
\(838\) 0 0
\(839\) −16.7426 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(840\) 0 0
\(841\) 38.8328 1.33906
\(842\) 0 0
\(843\) −3.70820 −0.127717
\(844\) 0 0
\(845\) 4.52786 0.155763
\(846\) 0 0
\(847\) −6.00000 −0.206162
\(848\) 0 0
\(849\) 26.2148 0.899689
\(850\) 0 0
\(851\) −11.0000 −0.377075
\(852\) 0 0
\(853\) −47.1246 −1.61352 −0.806758 0.590882i \(-0.798780\pi\)
−0.806758 + 0.590882i \(0.798780\pi\)
\(854\) 0 0
\(855\) −2.14590 −0.0733882
\(856\) 0 0
\(857\) 39.7082 1.35641 0.678203 0.734874i \(-0.262759\pi\)
0.678203 + 0.734874i \(0.262759\pi\)
\(858\) 0 0
\(859\) 14.0000 0.477674 0.238837 0.971060i \(-0.423234\pi\)
0.238837 + 0.971060i \(0.423234\pi\)
\(860\) 0 0
\(861\) −1.47214 −0.0501703
\(862\) 0 0
\(863\) 8.06888 0.274668 0.137334 0.990525i \(-0.456147\pi\)
0.137334 + 0.990525i \(0.456147\pi\)
\(864\) 0 0
\(865\) 2.14590 0.0729627
\(866\) 0 0
\(867\) 28.0000 0.950930
\(868\) 0 0
\(869\) −21.1803 −0.718494
\(870\) 0 0
\(871\) −12.1246 −0.410827
\(872\) 0 0
\(873\) 10.4164 0.352542
\(874\) 0 0
\(875\) 5.94427 0.200953
\(876\) 0 0
\(877\) 16.5836 0.559988 0.279994 0.960002i \(-0.409667\pi\)
0.279994 + 0.960002i \(0.409667\pi\)
\(878\) 0 0
\(879\) 12.0000 0.404750
\(880\) 0 0
\(881\) 1.81966 0.0613059 0.0306530 0.999530i \(-0.490241\pi\)
0.0306530 + 0.999530i \(0.490241\pi\)
\(882\) 0 0
\(883\) 3.90983 0.131576 0.0657881 0.997834i \(-0.479044\pi\)
0.0657881 + 0.997834i \(0.479044\pi\)
\(884\) 0 0
\(885\) 5.79837 0.194910
\(886\) 0 0
\(887\) −9.79837 −0.328997 −0.164499 0.986377i \(-0.552601\pi\)
−0.164499 + 0.986377i \(0.552601\pi\)
\(888\) 0 0
\(889\) −22.2705 −0.746929
\(890\) 0 0
\(891\) −2.23607 −0.0749111
\(892\) 0 0
\(893\) 25.1246 0.840763
\(894\) 0 0
\(895\) 1.14590 0.0383031
\(896\) 0 0
\(897\) −2.38197 −0.0795315
\(898\) 0 0
\(899\) 55.2492 1.84266
\(900\) 0 0
\(901\) −87.8115 −2.92543
\(902\) 0 0
\(903\) 1.61803 0.0538448
\(904\) 0 0
\(905\) 1.20163 0.0399434
\(906\) 0 0
\(907\) −48.9787 −1.62631 −0.813156 0.582046i \(-0.802252\pi\)
−0.813156 + 0.582046i \(0.802252\pi\)
\(908\) 0 0
\(909\) 4.14590 0.137511
\(910\) 0 0
\(911\) 7.59675 0.251691 0.125846 0.992050i \(-0.459836\pi\)
0.125846 + 0.992050i \(0.459836\pi\)
\(912\) 0 0
\(913\) −20.5279 −0.679373
\(914\) 0 0
\(915\) 3.00000 0.0991769
\(916\) 0 0
\(917\) −15.1803 −0.501299
\(918\) 0 0
\(919\) 50.1246 1.65346 0.826729 0.562600i \(-0.190199\pi\)
0.826729 + 0.562600i \(0.190199\pi\)
\(920\) 0 0
\(921\) −16.1246 −0.531324
\(922\) 0 0
\(923\) 10.4377 0.343561
\(924\) 0 0
\(925\) 50.7984 1.67024
\(926\) 0 0
\(927\) 7.41641 0.243587
\(928\) 0 0
\(929\) 28.9098 0.948501 0.474250 0.880390i \(-0.342719\pi\)
0.474250 + 0.880390i \(0.342719\pi\)
\(930\) 0 0
\(931\) 3.47214 0.113795
\(932\) 0 0
\(933\) −15.3262 −0.501759
\(934\) 0 0
\(935\) 9.27051 0.303178
\(936\) 0 0
\(937\) 2.70820 0.0884732 0.0442366 0.999021i \(-0.485914\pi\)
0.0442366 + 0.999021i \(0.485914\pi\)
\(938\) 0 0
\(939\) −2.47214 −0.0806751
\(940\) 0 0
\(941\) −15.0557 −0.490803 −0.245401 0.969422i \(-0.578920\pi\)
−0.245401 + 0.969422i \(0.578920\pi\)
\(942\) 0 0
\(943\) −1.47214 −0.0479393
\(944\) 0 0
\(945\) −0.618034 −0.0201046
\(946\) 0 0
\(947\) 18.9443 0.615606 0.307803 0.951450i \(-0.400406\pi\)
0.307803 + 0.951450i \(0.400406\pi\)
\(948\) 0 0
\(949\) 30.2705 0.982622
\(950\) 0 0
\(951\) 14.5066 0.470408
\(952\) 0 0
\(953\) 12.1591 0.393870 0.196935 0.980417i \(-0.436901\pi\)
0.196935 + 0.980417i \(0.436901\pi\)
\(954\) 0 0
\(955\) 10.0000 0.323592
\(956\) 0 0
\(957\) 18.4164 0.595318
\(958\) 0 0
\(959\) −16.4164 −0.530113
\(960\) 0 0
\(961\) 14.0000 0.451613
\(962\) 0 0
\(963\) −3.32624 −0.107187
\(964\) 0 0
\(965\) −5.12461 −0.164967
\(966\) 0 0
\(967\) −50.3607 −1.61949 −0.809745 0.586782i \(-0.800395\pi\)
−0.809745 + 0.586782i \(0.800395\pi\)
\(968\) 0 0
\(969\) 23.2918 0.748240
\(970\) 0 0
\(971\) 19.6869 0.631783 0.315892 0.948795i \(-0.397696\pi\)
0.315892 + 0.948795i \(0.397696\pi\)
\(972\) 0 0
\(973\) 11.3820 0.364889
\(974\) 0 0
\(975\) 11.0000 0.352282
\(976\) 0 0
\(977\) 49.7984 1.59319 0.796596 0.604513i \(-0.206632\pi\)
0.796596 + 0.604513i \(0.206632\pi\)
\(978\) 0 0
\(979\) −25.9787 −0.830283
\(980\) 0 0
\(981\) 19.2705 0.615260
\(982\) 0 0
\(983\) 7.40325 0.236127 0.118064 0.993006i \(-0.462331\pi\)
0.118064 + 0.993006i \(0.462331\pi\)
\(984\) 0 0
\(985\) 15.7639 0.502281
\(986\) 0 0
\(987\) 7.23607 0.230327
\(988\) 0 0
\(989\) 1.61803 0.0514505
\(990\) 0 0
\(991\) 35.5755 1.13009 0.565046 0.825059i \(-0.308858\pi\)
0.565046 + 0.825059i \(0.308858\pi\)
\(992\) 0 0
\(993\) 13.4164 0.425757
\(994\) 0 0
\(995\) −4.27051 −0.135384
\(996\) 0 0
\(997\) 57.7214 1.82805 0.914027 0.405654i \(-0.132956\pi\)
0.914027 + 0.405654i \(0.132956\pi\)
\(998\) 0 0
\(999\) −11.0000 −0.348025
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.bn.1.1 2
4.3 odd 2 483.2.a.d.1.1 2
12.11 even 2 1449.2.a.h.1.2 2
28.27 even 2 3381.2.a.r.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.d.1.1 2 4.3 odd 2
1449.2.a.h.1.2 2 12.11 even 2
3381.2.a.r.1.1 2 28.27 even 2
7728.2.a.bn.1.1 2 1.1 even 1 trivial