Properties

Label 4275.2.a.br.1.6
Level $4275$
Weight $2$
Character 4275.1
Self dual yes
Analytic conductor $34.136$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4275.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(34.1360468641\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.66064384.1
Defining polynomial: \( x^{6} - 9x^{4} + 13x^{2} - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(1.30397\) of defining polynomial
Character \(\chi\) \(=\) 4275.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.41987 q^{2} +3.85577 q^{4} -3.18676 q^{7} +4.49073 q^{8} +O(q^{10})\) \(q+2.41987 q^{2} +3.85577 q^{4} -3.18676 q^{7} +4.49073 q^{8} -4.15544 q^{11} +2.07086 q^{13} -7.71155 q^{14} +3.15544 q^{16} -5.79470 q^{17} -1.00000 q^{19} -10.0556 q^{22} +2.60794 q^{23} +5.01121 q^{26} -12.2874 q^{28} -6.00000 q^{29} +2.59933 q^{31} -1.34571 q^{32} -14.0224 q^{34} +4.30266 q^{37} -2.41987 q^{38} +0.599328 q^{41} +3.18676 q^{43} -16.0224 q^{44} +6.31087 q^{46} -11.7086 q^{47} +3.15544 q^{49} +7.98476 q^{52} -11.7503 q^{53} -14.3109 q^{56} -14.5192 q^{58} +1.71155 q^{59} -8.75476 q^{61} +6.29004 q^{62} -9.56732 q^{64} +4.76228 q^{67} -22.3430 q^{68} -13.7115 q^{71} -2.72714 q^{73} +10.4119 q^{74} -3.85577 q^{76} +13.2424 q^{77} -1.40067 q^{79} +1.45030 q^{82} +7.07154 q^{83} +7.71155 q^{86} -18.6609 q^{88} -16.5353 q^{89} -6.59933 q^{91} +10.0556 q^{92} -28.3333 q^{94} +2.07086 q^{97} +7.63575 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 8 q^{4} - 2 q^{11} - 16 q^{14} - 4 q^{16} - 6 q^{19} - 8 q^{26} - 36 q^{29} - 8 q^{34} - 12 q^{41} - 20 q^{44} - 8 q^{46} - 4 q^{49} - 40 q^{56} - 20 q^{59} - 14 q^{61} - 12 q^{64} - 52 q^{71} + 40 q^{74} - 8 q^{76} - 24 q^{79} + 16 q^{86} - 24 q^{89} - 24 q^{91} - 48 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.41987 1.71111 0.855553 0.517715i \(-0.173217\pi\)
0.855553 + 0.517715i \(0.173217\pi\)
\(3\) 0 0
\(4\) 3.85577 1.92789
\(5\) 0 0
\(6\) 0 0
\(7\) −3.18676 −1.20448 −0.602241 0.798314i \(-0.705725\pi\)
−0.602241 + 0.798314i \(0.705725\pi\)
\(8\) 4.49073 1.58771
\(9\) 0 0
\(10\) 0 0
\(11\) −4.15544 −1.25291 −0.626456 0.779457i \(-0.715495\pi\)
−0.626456 + 0.779457i \(0.715495\pi\)
\(12\) 0 0
\(13\) 2.07086 0.574353 0.287176 0.957878i \(-0.407283\pi\)
0.287176 + 0.957878i \(0.407283\pi\)
\(14\) −7.71155 −2.06100
\(15\) 0 0
\(16\) 3.15544 0.788859
\(17\) −5.79470 −1.40542 −0.702710 0.711476i \(-0.748027\pi\)
−0.702710 + 0.711476i \(0.748027\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) −10.0556 −2.14386
\(23\) 2.60794 0.543793 0.271896 0.962327i \(-0.412349\pi\)
0.271896 + 0.962327i \(0.412349\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 5.01121 0.982779
\(27\) 0 0
\(28\) −12.2874 −2.32210
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 2.59933 0.466853 0.233427 0.972374i \(-0.425006\pi\)
0.233427 + 0.972374i \(0.425006\pi\)
\(32\) −1.34571 −0.237890
\(33\) 0 0
\(34\) −14.0224 −2.40482
\(35\) 0 0
\(36\) 0 0
\(37\) 4.30266 0.707353 0.353677 0.935368i \(-0.384931\pi\)
0.353677 + 0.935368i \(0.384931\pi\)
\(38\) −2.41987 −0.392555
\(39\) 0 0
\(40\) 0 0
\(41\) 0.599328 0.0935993 0.0467997 0.998904i \(-0.485098\pi\)
0.0467997 + 0.998904i \(0.485098\pi\)
\(42\) 0 0
\(43\) 3.18676 0.485976 0.242988 0.970029i \(-0.421872\pi\)
0.242988 + 0.970029i \(0.421872\pi\)
\(44\) −16.0224 −2.41547
\(45\) 0 0
\(46\) 6.31087 0.930487
\(47\) −11.7086 −1.70787 −0.853937 0.520376i \(-0.825792\pi\)
−0.853937 + 0.520376i \(0.825792\pi\)
\(48\) 0 0
\(49\) 3.15544 0.450777
\(50\) 0 0
\(51\) 0 0
\(52\) 7.98476 1.10729
\(53\) −11.7503 −1.61403 −0.807017 0.590529i \(-0.798919\pi\)
−0.807017 + 0.590529i \(0.798919\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −14.3109 −1.91237
\(57\) 0 0
\(58\) −14.5192 −1.90647
\(59\) 1.71155 0.222824 0.111412 0.993774i \(-0.464463\pi\)
0.111412 + 0.993774i \(0.464463\pi\)
\(60\) 0 0
\(61\) −8.75476 −1.12093 −0.560466 0.828177i \(-0.689378\pi\)
−0.560466 + 0.828177i \(0.689378\pi\)
\(62\) 6.29004 0.798836
\(63\) 0 0
\(64\) −9.56732 −1.19591
\(65\) 0 0
\(66\) 0 0
\(67\) 4.76228 0.581805 0.290902 0.956753i \(-0.406044\pi\)
0.290902 + 0.956753i \(0.406044\pi\)
\(68\) −22.3430 −2.70949
\(69\) 0 0
\(70\) 0 0
\(71\) −13.7115 −1.62726 −0.813631 0.581382i \(-0.802512\pi\)
−0.813631 + 0.581382i \(0.802512\pi\)
\(72\) 0 0
\(73\) −2.72714 −0.319188 −0.159594 0.987183i \(-0.551018\pi\)
−0.159594 + 0.987183i \(0.551018\pi\)
\(74\) 10.4119 1.21036
\(75\) 0 0
\(76\) −3.85577 −0.442287
\(77\) 13.2424 1.50911
\(78\) 0 0
\(79\) −1.40067 −0.157588 −0.0787939 0.996891i \(-0.525107\pi\)
−0.0787939 + 0.996891i \(0.525107\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1.45030 0.160158
\(83\) 7.07154 0.776203 0.388101 0.921617i \(-0.373131\pi\)
0.388101 + 0.921617i \(0.373131\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 7.71155 0.831557
\(87\) 0 0
\(88\) −18.6609 −1.98926
\(89\) −16.5353 −1.75274 −0.876370 0.481639i \(-0.840042\pi\)
−0.876370 + 0.481639i \(0.840042\pi\)
\(90\) 0 0
\(91\) −6.59933 −0.691798
\(92\) 10.0556 1.04837
\(93\) 0 0
\(94\) −28.3333 −2.92236
\(95\) 0 0
\(96\) 0 0
\(97\) 2.07086 0.210264 0.105132 0.994458i \(-0.466474\pi\)
0.105132 + 0.994458i \(0.466474\pi\)
\(98\) 7.63575 0.771327
\(99\) 0 0
\(100\) 0 0
\(101\) 1.71155 0.170305 0.0851525 0.996368i \(-0.472862\pi\)
0.0851525 + 0.996368i \(0.472862\pi\)
\(102\) 0 0
\(103\) 5.75296 0.566856 0.283428 0.958994i \(-0.408528\pi\)
0.283428 + 0.958994i \(0.408528\pi\)
\(104\) 9.29966 0.911907
\(105\) 0 0
\(106\) −28.4343 −2.76178
\(107\) 15.4324 1.49191 0.745955 0.665996i \(-0.231993\pi\)
0.745955 + 0.665996i \(0.231993\pi\)
\(108\) 0 0
\(109\) 11.7115 1.12176 0.560881 0.827896i \(-0.310462\pi\)
0.560881 + 0.827896i \(0.310462\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −10.0556 −0.950167
\(113\) 10.5927 0.996477 0.498239 0.867040i \(-0.333980\pi\)
0.498239 + 0.867040i \(0.333980\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −23.1346 −2.14800
\(117\) 0 0
\(118\) 4.14172 0.381276
\(119\) 18.4663 1.69280
\(120\) 0 0
\(121\) 6.26765 0.569787
\(122\) −21.1854 −1.91804
\(123\) 0 0
\(124\) 10.0224 0.900040
\(125\) 0 0
\(126\) 0 0
\(127\) −6.07484 −0.539055 −0.269528 0.962993i \(-0.586868\pi\)
−0.269528 + 0.962993i \(0.586868\pi\)
\(128\) −20.4602 −1.80845
\(129\) 0 0
\(130\) 0 0
\(131\) 13.5785 1.18636 0.593181 0.805069i \(-0.297872\pi\)
0.593181 + 0.805069i \(0.297872\pi\)
\(132\) 0 0
\(133\) 3.18676 0.276327
\(134\) 11.5241 0.995530
\(135\) 0 0
\(136\) −26.0224 −2.23140
\(137\) 7.94302 0.678618 0.339309 0.940675i \(-0.389807\pi\)
0.339309 + 0.940675i \(0.389807\pi\)
\(138\) 0 0
\(139\) 3.26765 0.277159 0.138579 0.990351i \(-0.455746\pi\)
0.138579 + 0.990351i \(0.455746\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −33.1802 −2.78442
\(143\) −8.60532 −0.719613
\(144\) 0 0
\(145\) 0 0
\(146\) −6.59933 −0.546164
\(147\) 0 0
\(148\) 16.5901 1.36370
\(149\) −8.44389 −0.691751 −0.345875 0.938280i \(-0.612418\pi\)
−0.345875 + 0.938280i \(0.612418\pi\)
\(150\) 0 0
\(151\) 0.887783 0.0722468 0.0361234 0.999347i \(-0.488499\pi\)
0.0361234 + 0.999347i \(0.488499\pi\)
\(152\) −4.49073 −0.364246
\(153\) 0 0
\(154\) 32.0448 2.58225
\(155\) 0 0
\(156\) 0 0
\(157\) 4.14172 0.330545 0.165273 0.986248i \(-0.447150\pi\)
0.165273 + 0.986248i \(0.447150\pi\)
\(158\) −3.38944 −0.269650
\(159\) 0 0
\(160\) 0 0
\(161\) −8.31087 −0.654989
\(162\) 0 0
\(163\) −24.7126 −1.93564 −0.967819 0.251647i \(-0.919028\pi\)
−0.967819 + 0.251647i \(0.919028\pi\)
\(164\) 2.31087 0.180449
\(165\) 0 0
\(166\) 17.1122 1.32817
\(167\) 3.60464 0.278935 0.139468 0.990227i \(-0.455461\pi\)
0.139468 + 0.990227i \(0.455461\pi\)
\(168\) 0 0
\(169\) −8.71155 −0.670119
\(170\) 0 0
\(171\) 0 0
\(172\) 12.2874 0.936907
\(173\) 22.4205 1.70460 0.852300 0.523054i \(-0.175207\pi\)
0.852300 + 0.523054i \(0.175207\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −13.1122 −0.988371
\(177\) 0 0
\(178\) −40.0133 −2.99912
\(179\) 5.13464 0.383781 0.191890 0.981416i \(-0.438538\pi\)
0.191890 + 0.981416i \(0.438538\pi\)
\(180\) 0 0
\(181\) 20.8462 1.54948 0.774742 0.632277i \(-0.217880\pi\)
0.774742 + 0.632277i \(0.217880\pi\)
\(182\) −15.9695 −1.18374
\(183\) 0 0
\(184\) 11.7115 0.863387
\(185\) 0 0
\(186\) 0 0
\(187\) 24.0795 1.76087
\(188\) −45.1457 −3.29259
\(189\) 0 0
\(190\) 0 0
\(191\) 5.26765 0.381154 0.190577 0.981672i \(-0.438964\pi\)
0.190577 + 0.981672i \(0.438964\pi\)
\(192\) 0 0
\(193\) 2.07086 0.149064 0.0745318 0.997219i \(-0.476254\pi\)
0.0745318 + 0.997219i \(0.476254\pi\)
\(194\) 5.01121 0.359784
\(195\) 0 0
\(196\) 12.1666 0.869046
\(197\) −10.4318 −0.743232 −0.371616 0.928387i \(-0.621196\pi\)
−0.371616 + 0.928387i \(0.621196\pi\)
\(198\) 0 0
\(199\) −2.73235 −0.193691 −0.0968455 0.995299i \(-0.530875\pi\)
−0.0968455 + 0.995299i \(0.530875\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 4.14172 0.291410
\(203\) 19.1206 1.34200
\(204\) 0 0
\(205\) 0 0
\(206\) 13.9214 0.969951
\(207\) 0 0
\(208\) 6.53446 0.453083
\(209\) 4.15544 0.287438
\(210\) 0 0
\(211\) −15.7340 −1.08317 −0.541585 0.840646i \(-0.682176\pi\)
−0.541585 + 0.840646i \(0.682176\pi\)
\(212\) −45.3066 −3.11167
\(213\) 0 0
\(214\) 37.3445 2.55282
\(215\) 0 0
\(216\) 0 0
\(217\) −8.28343 −0.562316
\(218\) 28.3404 1.91946
\(219\) 0 0
\(220\) 0 0
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 18.8219 1.26041 0.630203 0.776430i \(-0.282972\pi\)
0.630203 + 0.776430i \(0.282972\pi\)
\(224\) 4.28845 0.286534
\(225\) 0 0
\(226\) 25.6330 1.70508
\(227\) 14.4418 0.958533 0.479267 0.877669i \(-0.340903\pi\)
0.479267 + 0.877669i \(0.340903\pi\)
\(228\) 0 0
\(229\) −4.17785 −0.276080 −0.138040 0.990427i \(-0.544080\pi\)
−0.138040 + 0.990427i \(0.544080\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −26.9444 −1.76898
\(233\) −12.0847 −0.791697 −0.395849 0.918316i \(-0.629550\pi\)
−0.395849 + 0.918316i \(0.629550\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 6.59933 0.429580
\(237\) 0 0
\(238\) 44.6861 2.89657
\(239\) 11.3541 0.734435 0.367218 0.930135i \(-0.380310\pi\)
0.367218 + 0.930135i \(0.380310\pi\)
\(240\) 0 0
\(241\) −3.40067 −0.219057 −0.109528 0.993984i \(-0.534934\pi\)
−0.109528 + 0.993984i \(0.534934\pi\)
\(242\) 15.1669 0.974966
\(243\) 0 0
\(244\) −33.7564 −2.16103
\(245\) 0 0
\(246\) 0 0
\(247\) −2.07086 −0.131766
\(248\) 11.6729 0.741229
\(249\) 0 0
\(250\) 0 0
\(251\) −3.04322 −0.192086 −0.0960432 0.995377i \(-0.530619\pi\)
−0.0960432 + 0.995377i \(0.530619\pi\)
\(252\) 0 0
\(253\) −10.8371 −0.681324
\(254\) −14.7003 −0.922381
\(255\) 0 0
\(256\) −30.3765 −1.89853
\(257\) −17.2881 −1.07840 −0.539201 0.842177i \(-0.681274\pi\)
−0.539201 + 0.842177i \(0.681274\pi\)
\(258\) 0 0
\(259\) −13.7115 −0.851994
\(260\) 0 0
\(261\) 0 0
\(262\) 32.8583 2.02999
\(263\) 1.19336 0.0735859 0.0367930 0.999323i \(-0.488286\pi\)
0.0367930 + 0.999323i \(0.488286\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 7.71155 0.472825
\(267\) 0 0
\(268\) 18.3623 1.12165
\(269\) −22.1089 −1.34800 −0.674000 0.738731i \(-0.735425\pi\)
−0.674000 + 0.738731i \(0.735425\pi\)
\(270\) 0 0
\(271\) −4.08644 −0.248234 −0.124117 0.992268i \(-0.539610\pi\)
−0.124117 + 0.992268i \(0.539610\pi\)
\(272\) −18.2848 −1.10868
\(273\) 0 0
\(274\) 19.2211 1.16119
\(275\) 0 0
\(276\) 0 0
\(277\) −10.0199 −0.602037 −0.301019 0.953618i \(-0.597327\pi\)
−0.301019 + 0.953618i \(0.597327\pi\)
\(278\) 7.90730 0.474248
\(279\) 0 0
\(280\) 0 0
\(281\) −0.599328 −0.0357529 −0.0178765 0.999840i \(-0.505691\pi\)
−0.0178765 + 0.999840i \(0.505691\pi\)
\(282\) 0 0
\(283\) 5.41856 0.322100 0.161050 0.986946i \(-0.448512\pi\)
0.161050 + 0.986946i \(0.448512\pi\)
\(284\) −52.8686 −3.13717
\(285\) 0 0
\(286\) −20.8238 −1.23133
\(287\) −1.90991 −0.112739
\(288\) 0 0
\(289\) 16.5785 0.975207
\(290\) 0 0
\(291\) 0 0
\(292\) −10.5152 −0.615358
\(293\) −3.46691 −0.202539 −0.101269 0.994859i \(-0.532290\pi\)
−0.101269 + 0.994859i \(0.532290\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 19.3221 1.12307
\(297\) 0 0
\(298\) −20.4331 −1.18366
\(299\) 5.40067 0.312329
\(300\) 0 0
\(301\) −10.1554 −0.585350
\(302\) 2.14832 0.123622
\(303\) 0 0
\(304\) −3.15544 −0.180977
\(305\) 0 0
\(306\) 0 0
\(307\) −16.5901 −0.946846 −0.473423 0.880835i \(-0.656982\pi\)
−0.473423 + 0.880835i \(0.656982\pi\)
\(308\) 51.0596 2.90939
\(309\) 0 0
\(310\) 0 0
\(311\) 4.15544 0.235633 0.117817 0.993035i \(-0.462411\pi\)
0.117817 + 0.993035i \(0.462411\pi\)
\(312\) 0 0
\(313\) −0.919237 −0.0519583 −0.0259792 0.999662i \(-0.508270\pi\)
−0.0259792 + 0.999662i \(0.508270\pi\)
\(314\) 10.0224 0.565598
\(315\) 0 0
\(316\) −5.40067 −0.303812
\(317\) 26.7292 1.50126 0.750630 0.660723i \(-0.229750\pi\)
0.750630 + 0.660723i \(0.229750\pi\)
\(318\) 0 0
\(319\) 24.9326 1.39596
\(320\) 0 0
\(321\) 0 0
\(322\) −20.1112 −1.12076
\(323\) 5.79470 0.322426
\(324\) 0 0
\(325\) 0 0
\(326\) −59.8012 −3.31208
\(327\) 0 0
\(328\) 2.69142 0.148609
\(329\) 37.3125 2.05710
\(330\) 0 0
\(331\) 8.00000 0.439720 0.219860 0.975531i \(-0.429440\pi\)
0.219860 + 0.975531i \(0.429440\pi\)
\(332\) 27.2663 1.49643
\(333\) 0 0
\(334\) 8.72275 0.477288
\(335\) 0 0
\(336\) 0 0
\(337\) 22.5040 1.22587 0.612935 0.790133i \(-0.289989\pi\)
0.612935 + 0.790133i \(0.289989\pi\)
\(338\) −21.0808 −1.14664
\(339\) 0 0
\(340\) 0 0
\(341\) −10.8013 −0.584926
\(342\) 0 0
\(343\) 12.2517 0.661530
\(344\) 14.3109 0.771591
\(345\) 0 0
\(346\) 54.2547 2.91675
\(347\) 14.4543 0.775946 0.387973 0.921671i \(-0.373175\pi\)
0.387973 + 0.921671i \(0.373175\pi\)
\(348\) 0 0
\(349\) −13.3541 −0.714828 −0.357414 0.933946i \(-0.616342\pi\)
−0.357414 + 0.933946i \(0.616342\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.59201 0.298055
\(353\) −17.6410 −0.938937 −0.469469 0.882949i \(-0.655554\pi\)
−0.469469 + 0.882949i \(0.655554\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −63.7564 −3.37908
\(357\) 0 0
\(358\) 12.4252 0.656690
\(359\) −12.4663 −0.657947 −0.328973 0.944339i \(-0.606703\pi\)
−0.328973 + 0.944339i \(0.606703\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 50.4451 2.65133
\(363\) 0 0
\(364\) −25.4455 −1.33371
\(365\) 0 0
\(366\) 0 0
\(367\) −16.4291 −0.857594 −0.428797 0.903401i \(-0.641062\pi\)
−0.428797 + 0.903401i \(0.641062\pi\)
\(368\) 8.22918 0.428976
\(369\) 0 0
\(370\) 0 0
\(371\) 37.4455 1.94407
\(372\) 0 0
\(373\) −5.29334 −0.274079 −0.137039 0.990566i \(-0.543759\pi\)
−0.137039 + 0.990566i \(0.543759\pi\)
\(374\) 58.2693 3.01303
\(375\) 0 0
\(376\) −52.5801 −2.71161
\(377\) −12.4252 −0.639928
\(378\) 0 0
\(379\) 14.5353 0.746629 0.373314 0.927705i \(-0.378221\pi\)
0.373314 + 0.927705i \(0.378221\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 12.7470 0.652195
\(383\) −0.453598 −0.0231778 −0.0115889 0.999933i \(-0.503689\pi\)
−0.0115889 + 0.999933i \(0.503689\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 5.01121 0.255064
\(387\) 0 0
\(388\) 7.98476 0.405365
\(389\) −16.1554 −0.819113 −0.409557 0.912285i \(-0.634317\pi\)
−0.409557 + 0.912285i \(0.634317\pi\)
\(390\) 0 0
\(391\) −15.1122 −0.764258
\(392\) 14.1702 0.715704
\(393\) 0 0
\(394\) −25.2435 −1.27175
\(395\) 0 0
\(396\) 0 0
\(397\) 32.7563 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(398\) −6.61192 −0.331426
\(399\) 0 0
\(400\) 0 0
\(401\) −12.0864 −0.603568 −0.301784 0.953376i \(-0.597582\pi\)
−0.301784 + 0.953376i \(0.597582\pi\)
\(402\) 0 0
\(403\) 5.38284 0.268138
\(404\) 6.59933 0.328329
\(405\) 0 0
\(406\) 46.2693 2.29631
\(407\) −17.8794 −0.886251
\(408\) 0 0
\(409\) −19.1346 −0.946147 −0.473073 0.881023i \(-0.656855\pi\)
−0.473073 + 0.881023i \(0.656855\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 22.1821 1.09283
\(413\) −5.45428 −0.268388
\(414\) 0 0
\(415\) 0 0
\(416\) −2.78678 −0.136633
\(417\) 0 0
\(418\) 10.0556 0.491836
\(419\) −8.04484 −0.393016 −0.196508 0.980502i \(-0.562960\pi\)
−0.196508 + 0.980502i \(0.562960\pi\)
\(420\) 0 0
\(421\) −29.3591 −1.43087 −0.715437 0.698678i \(-0.753772\pi\)
−0.715437 + 0.698678i \(0.753772\pi\)
\(422\) −38.0742 −1.85342
\(423\) 0 0
\(424\) −52.7676 −2.56262
\(425\) 0 0
\(426\) 0 0
\(427\) 27.8993 1.35014
\(428\) 59.5040 2.87623
\(429\) 0 0
\(430\) 0 0
\(431\) 32.0448 1.54355 0.771773 0.635898i \(-0.219370\pi\)
0.771773 + 0.635898i \(0.219370\pi\)
\(432\) 0 0
\(433\) 0.482831 0.0232034 0.0116017 0.999933i \(-0.496307\pi\)
0.0116017 + 0.999933i \(0.496307\pi\)
\(434\) −20.0448 −0.962183
\(435\) 0 0
\(436\) 45.1571 2.16263
\(437\) −2.60794 −0.124755
\(438\) 0 0
\(439\) −27.3591 −1.30578 −0.652889 0.757454i \(-0.726443\pi\)
−0.652889 + 0.757454i \(0.726443\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −29.0384 −1.38122
\(443\) −23.3815 −1.11089 −0.555444 0.831554i \(-0.687452\pi\)
−0.555444 + 0.831554i \(0.687452\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 45.5465 2.15669
\(447\) 0 0
\(448\) 30.4887 1.44046
\(449\) −23.1346 −1.09179 −0.545895 0.837853i \(-0.683810\pi\)
−0.545895 + 0.837853i \(0.683810\pi\)
\(450\) 0 0
\(451\) −2.49047 −0.117272
\(452\) 40.8430 1.92109
\(453\) 0 0
\(454\) 34.9472 1.64015
\(455\) 0 0
\(456\) 0 0
\(457\) 21.2503 0.994049 0.497025 0.867736i \(-0.334426\pi\)
0.497025 + 0.867736i \(0.334426\pi\)
\(458\) −10.1099 −0.472403
\(459\) 0 0
\(460\) 0 0
\(461\) −31.5785 −1.47076 −0.735379 0.677656i \(-0.762996\pi\)
−0.735379 + 0.677656i \(0.762996\pi\)
\(462\) 0 0
\(463\) −15.6119 −0.725547 −0.362774 0.931877i \(-0.618170\pi\)
−0.362774 + 0.931877i \(0.618170\pi\)
\(464\) −18.9326 −0.878925
\(465\) 0 0
\(466\) −29.2435 −1.35468
\(467\) 29.8264 1.38020 0.690101 0.723713i \(-0.257566\pi\)
0.690101 + 0.723713i \(0.257566\pi\)
\(468\) 0 0
\(469\) −15.1762 −0.700774
\(470\) 0 0
\(471\) 0 0
\(472\) 7.68608 0.353781
\(473\) −13.2424 −0.608885
\(474\) 0 0
\(475\) 0 0
\(476\) 71.2019 3.26353
\(477\) 0 0
\(478\) 27.4754 1.25670
\(479\) −4.53531 −0.207223 −0.103612 0.994618i \(-0.533040\pi\)
−0.103612 + 0.994618i \(0.533040\pi\)
\(480\) 0 0
\(481\) 8.91020 0.406270
\(482\) −8.22918 −0.374829
\(483\) 0 0
\(484\) 24.1666 1.09848
\(485\) 0 0
\(486\) 0 0
\(487\) 39.2550 1.77881 0.889407 0.457116i \(-0.151118\pi\)
0.889407 + 0.457116i \(0.151118\pi\)
\(488\) −39.3153 −1.77972
\(489\) 0 0
\(490\) 0 0
\(491\) −38.8910 −1.75513 −0.877564 0.479460i \(-0.840832\pi\)
−0.877564 + 0.479460i \(0.840832\pi\)
\(492\) 0 0
\(493\) 34.7682 1.56588
\(494\) −5.01121 −0.225465
\(495\) 0 0
\(496\) 8.20202 0.368281
\(497\) 43.6954 1.96001
\(498\) 0 0
\(499\) 4.73235 0.211849 0.105924 0.994374i \(-0.466220\pi\)
0.105924 + 0.994374i \(0.466220\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −7.36420 −0.328680
\(503\) −1.85567 −0.0827400 −0.0413700 0.999144i \(-0.513172\pi\)
−0.0413700 + 0.999144i \(0.513172\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −26.2244 −1.16582
\(507\) 0 0
\(508\) −23.4232 −1.03924
\(509\) −22.8878 −1.01448 −0.507242 0.861804i \(-0.669335\pi\)
−0.507242 + 0.861804i \(0.669335\pi\)
\(510\) 0 0
\(511\) 8.69074 0.384456
\(512\) −32.5867 −1.44014
\(513\) 0 0
\(514\) −41.8350 −1.84526
\(515\) 0 0
\(516\) 0 0
\(517\) 48.6543 2.13982
\(518\) −33.1802 −1.45785
\(519\) 0 0
\(520\) 0 0
\(521\) 29.7340 1.30267 0.651334 0.758791i \(-0.274210\pi\)
0.651334 + 0.758791i \(0.274210\pi\)
\(522\) 0 0
\(523\) 30.6497 1.34022 0.670109 0.742263i \(-0.266248\pi\)
0.670109 + 0.742263i \(0.266248\pi\)
\(524\) 52.3557 2.28717
\(525\) 0 0
\(526\) 2.88778 0.125913
\(527\) −15.0623 −0.656125
\(528\) 0 0
\(529\) −16.1987 −0.704289
\(530\) 0 0
\(531\) 0 0
\(532\) 12.2874 0.532727
\(533\) 1.24112 0.0537590
\(534\) 0 0
\(535\) 0 0
\(536\) 21.3861 0.923739
\(537\) 0 0
\(538\) −53.5006 −2.30657
\(539\) −13.1122 −0.564783
\(540\) 0 0
\(541\) 2.21946 0.0954220 0.0477110 0.998861i \(-0.484807\pi\)
0.0477110 + 0.998861i \(0.484807\pi\)
\(542\) −9.88865 −0.424754
\(543\) 0 0
\(544\) 7.79798 0.334336
\(545\) 0 0
\(546\) 0 0
\(547\) −14.4297 −0.616970 −0.308485 0.951229i \(-0.599822\pi\)
−0.308485 + 0.951229i \(0.599822\pi\)
\(548\) 30.6265 1.30830
\(549\) 0 0
\(550\) 0 0
\(551\) 6.00000 0.255609
\(552\) 0 0
\(553\) 4.46360 0.189812
\(554\) −24.2469 −1.03015
\(555\) 0 0
\(556\) 12.5993 0.534331
\(557\) −19.4610 −0.824588 −0.412294 0.911051i \(-0.635272\pi\)
−0.412294 + 0.911051i \(0.635272\pi\)
\(558\) 0 0
\(559\) 6.59933 0.279122
\(560\) 0 0
\(561\) 0 0
\(562\) −1.45030 −0.0611771
\(563\) 12.3649 0.521118 0.260559 0.965458i \(-0.416093\pi\)
0.260559 + 0.965458i \(0.416093\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 13.1122 0.551148
\(567\) 0 0
\(568\) −61.5748 −2.58362
\(569\) −15.0898 −0.632597 −0.316299 0.948660i \(-0.602440\pi\)
−0.316299 + 0.948660i \(0.602440\pi\)
\(570\) 0 0
\(571\) 17.1571 0.718000 0.359000 0.933337i \(-0.383118\pi\)
0.359000 + 0.933337i \(0.383118\pi\)
\(572\) −33.1802 −1.38733
\(573\) 0 0
\(574\) −4.62175 −0.192908
\(575\) 0 0
\(576\) 0 0
\(577\) −22.5165 −0.937374 −0.468687 0.883364i \(-0.655273\pi\)
−0.468687 + 0.883364i \(0.655273\pi\)
\(578\) 40.1179 1.66868
\(579\) 0 0
\(580\) 0 0
\(581\) −22.5353 −0.934922
\(582\) 0 0
\(583\) 48.8278 2.02224
\(584\) −12.2469 −0.506778
\(585\) 0 0
\(586\) −8.38946 −0.346566
\(587\) −7.49544 −0.309370 −0.154685 0.987964i \(-0.549436\pi\)
−0.154685 + 0.987964i \(0.549436\pi\)
\(588\) 0 0
\(589\) −2.59933 −0.107103
\(590\) 0 0
\(591\) 0 0
\(592\) 13.5768 0.558002
\(593\) −27.8094 −1.14199 −0.570997 0.820952i \(-0.693443\pi\)
−0.570997 + 0.820952i \(0.693443\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −32.5577 −1.33362
\(597\) 0 0
\(598\) 13.0689 0.534428
\(599\) 45.4903 1.85869 0.929343 0.369219i \(-0.120375\pi\)
0.929343 + 0.369219i \(0.120375\pi\)
\(600\) 0 0
\(601\) −16.5993 −0.677101 −0.338550 0.940948i \(-0.609937\pi\)
−0.338550 + 0.940948i \(0.609937\pi\)
\(602\) −24.5748 −1.00160
\(603\) 0 0
\(604\) 3.42309 0.139284
\(605\) 0 0
\(606\) 0 0
\(607\) −5.08417 −0.206360 −0.103180 0.994663i \(-0.532902\pi\)
−0.103180 + 0.994663i \(0.532902\pi\)
\(608\) 1.34571 0.0545758
\(609\) 0 0
\(610\) 0 0
\(611\) −24.2469 −0.980923
\(612\) 0 0
\(613\) 4.63706 0.187289 0.0936445 0.995606i \(-0.470148\pi\)
0.0936445 + 0.995606i \(0.470148\pi\)
\(614\) −40.1458 −1.62015
\(615\) 0 0
\(616\) 59.4679 2.39603
\(617\) −40.2874 −1.62191 −0.810955 0.585108i \(-0.801052\pi\)
−0.810955 + 0.585108i \(0.801052\pi\)
\(618\) 0 0
\(619\) −43.3815 −1.74365 −0.871825 0.489818i \(-0.837063\pi\)
−0.871825 + 0.489818i \(0.837063\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 10.0556 0.403194
\(623\) 52.6940 2.11114
\(624\) 0 0
\(625\) 0 0
\(626\) −2.22443 −0.0889062
\(627\) 0 0
\(628\) 15.9695 0.637253
\(629\) −24.9326 −0.994129
\(630\) 0 0
\(631\) 7.53369 0.299911 0.149956 0.988693i \(-0.452087\pi\)
0.149956 + 0.988693i \(0.452087\pi\)
\(632\) −6.29004 −0.250204
\(633\) 0 0
\(634\) 64.6812 2.56882
\(635\) 0 0
\(636\) 0 0
\(637\) 6.53446 0.258905
\(638\) 60.3337 2.38863
\(639\) 0 0
\(640\) 0 0
\(641\) 23.3075 0.920591 0.460296 0.887766i \(-0.347743\pi\)
0.460296 + 0.887766i \(0.347743\pi\)
\(642\) 0 0
\(643\) 1.87419 0.0739110 0.0369555 0.999317i \(-0.488234\pi\)
0.0369555 + 0.999317i \(0.488234\pi\)
\(644\) −32.0448 −1.26274
\(645\) 0 0
\(646\) 14.0224 0.551705
\(647\) 47.0371 1.84922 0.924609 0.380917i \(-0.124392\pi\)
0.924609 + 0.380917i \(0.124392\pi\)
\(648\) 0 0
\(649\) −7.11222 −0.279179
\(650\) 0 0
\(651\) 0 0
\(652\) −95.2861 −3.73169
\(653\) −24.1630 −0.945571 −0.472785 0.881178i \(-0.656751\pi\)
−0.472785 + 0.881178i \(0.656751\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.89114 0.0738367
\(657\) 0 0
\(658\) 90.2914 3.51992
\(659\) −10.2885 −0.400781 −0.200391 0.979716i \(-0.564221\pi\)
−0.200391 + 0.979716i \(0.564221\pi\)
\(660\) 0 0
\(661\) 5.93598 0.230883 0.115441 0.993314i \(-0.463172\pi\)
0.115441 + 0.993314i \(0.463172\pi\)
\(662\) 19.3590 0.752407
\(663\) 0 0
\(664\) 31.7564 1.23239
\(665\) 0 0
\(666\) 0 0
\(667\) −15.6476 −0.605879
\(668\) 13.8987 0.537755
\(669\) 0 0
\(670\) 0 0
\(671\) 36.3799 1.40443
\(672\) 0 0
\(673\) −40.7053 −1.56907 −0.784537 0.620082i \(-0.787099\pi\)
−0.784537 + 0.620082i \(0.787099\pi\)
\(674\) 54.4567 2.09759
\(675\) 0 0
\(676\) −33.5897 −1.29191
\(677\) −18.8761 −0.725469 −0.362734 0.931893i \(-0.618157\pi\)
−0.362734 + 0.931893i \(0.618157\pi\)
\(678\) 0 0
\(679\) −6.59933 −0.253259
\(680\) 0 0
\(681\) 0 0
\(682\) −26.1379 −1.00087
\(683\) 19.6576 0.752179 0.376089 0.926583i \(-0.377269\pi\)
0.376089 + 0.926583i \(0.377269\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 29.6475 1.13195
\(687\) 0 0
\(688\) 10.0556 0.383367
\(689\) −24.3333 −0.927025
\(690\) 0 0
\(691\) −48.2451 −1.83533 −0.917665 0.397354i \(-0.869928\pi\)
−0.917665 + 0.397354i \(0.869928\pi\)
\(692\) 86.4483 3.28627
\(693\) 0 0
\(694\) 34.9775 1.32773
\(695\) 0 0
\(696\) 0 0
\(697\) −3.47293 −0.131546
\(698\) −32.3152 −1.22315
\(699\) 0 0
\(700\) 0 0
\(701\) −0.512889 −0.0193715 −0.00968577 0.999953i \(-0.503083\pi\)
−0.00968577 + 0.999953i \(0.503083\pi\)
\(702\) 0 0
\(703\) −4.30266 −0.162278
\(704\) 39.7564 1.49838
\(705\) 0 0
\(706\) −42.6890 −1.60662
\(707\) −5.45428 −0.205129
\(708\) 0 0
\(709\) −8.84618 −0.332225 −0.166113 0.986107i \(-0.553122\pi\)
−0.166113 + 0.986107i \(0.553122\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −74.2556 −2.78285
\(713\) 6.77889 0.253871
\(714\) 0 0
\(715\) 0 0
\(716\) 19.7980 0.739885
\(717\) 0 0
\(718\) −30.1669 −1.12582
\(719\) −7.84456 −0.292553 −0.146276 0.989244i \(-0.546729\pi\)
−0.146276 + 0.989244i \(0.546729\pi\)
\(720\) 0 0
\(721\) −18.3333 −0.682767
\(722\) 2.41987 0.0900582
\(723\) 0 0
\(724\) 80.3781 2.98723
\(725\) 0 0
\(726\) 0 0
\(727\) −2.91130 −0.107974 −0.0539870 0.998542i \(-0.517193\pi\)
−0.0539870 + 0.998542i \(0.517193\pi\)
\(728\) −29.6358 −1.09838
\(729\) 0 0
\(730\) 0 0
\(731\) −18.4663 −0.683001
\(732\) 0 0
\(733\) −33.7775 −1.24760 −0.623800 0.781584i \(-0.714412\pi\)
−0.623800 + 0.781584i \(0.714412\pi\)
\(734\) −39.7564 −1.46743
\(735\) 0 0
\(736\) −3.50953 −0.129363
\(737\) −19.7893 −0.728950
\(738\) 0 0
\(739\) −35.8030 −1.31703 −0.658517 0.752566i \(-0.728816\pi\)
−0.658517 + 0.752566i \(0.728816\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 90.6133 3.32652
\(743\) 5.66948 0.207993 0.103996 0.994578i \(-0.466837\pi\)
0.103996 + 0.994578i \(0.466837\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −12.8092 −0.468978
\(747\) 0 0
\(748\) 92.8451 3.39475
\(749\) −49.1795 −1.79698
\(750\) 0 0
\(751\) −27.4679 −1.00232 −0.501159 0.865355i \(-0.667093\pi\)
−0.501159 + 0.865355i \(0.667093\pi\)
\(752\) −36.9457 −1.34727
\(753\) 0 0
\(754\) −30.0673 −1.09498
\(755\) 0 0
\(756\) 0 0
\(757\) −41.6370 −1.51332 −0.756662 0.653806i \(-0.773171\pi\)
−0.756662 + 0.653806i \(0.773171\pi\)
\(758\) 35.1736 1.27756
\(759\) 0 0
\(760\) 0 0
\(761\) 9.46967 0.343275 0.171638 0.985160i \(-0.445094\pi\)
0.171638 + 0.985160i \(0.445094\pi\)
\(762\) 0 0
\(763\) −37.3219 −1.35114
\(764\) 20.3109 0.734822
\(765\) 0 0
\(766\) −1.09765 −0.0396597
\(767\) 3.54437 0.127980
\(768\) 0 0
\(769\) 1.90858 0.0688253 0.0344127 0.999408i \(-0.489044\pi\)
0.0344127 + 0.999408i \(0.489044\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 7.98476 0.287378
\(773\) 28.4007 1.02150 0.510751 0.859729i \(-0.329367\pi\)
0.510751 + 0.859729i \(0.329367\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 9.29966 0.333838
\(777\) 0 0
\(778\) −39.0941 −1.40159
\(779\) −0.599328 −0.0214732
\(780\) 0 0
\(781\) 56.9775 2.03881
\(782\) −36.5696 −1.30773
\(783\) 0 0
\(784\) 9.95678 0.355599
\(785\) 0 0
\(786\) 0 0
\(787\) 15.6708 0.558605 0.279303 0.960203i \(-0.409897\pi\)
0.279303 + 0.960203i \(0.409897\pi\)
\(788\) −40.2225 −1.43287
\(789\) 0 0
\(790\) 0 0
\(791\) −33.7564 −1.20024
\(792\) 0 0
\(793\) −18.1299 −0.643811
\(794\) 79.2659 2.81304
\(795\) 0 0
\(796\) −10.5353 −0.373414
\(797\) 36.9225 1.30786 0.653932 0.756554i \(-0.273118\pi\)
0.653932 + 0.756554i \(0.273118\pi\)
\(798\) 0 0
\(799\) 67.8478 2.40028
\(800\) 0 0
\(801\) 0 0
\(802\) −29.2476 −1.03277
\(803\) 11.3325 0.399914
\(804\) 0 0
\(805\) 0 0
\(806\) 13.0258 0.458813
\(807\) 0 0
\(808\) 7.68608 0.270396
\(809\) −43.0465 −1.51343 −0.756716 0.653743i \(-0.773198\pi\)
−0.756716 + 0.653743i \(0.773198\pi\)
\(810\) 0 0
\(811\) 32.2469 1.13234 0.566170 0.824288i \(-0.308425\pi\)
0.566170 + 0.824288i \(0.308425\pi\)
\(812\) 73.7245 2.58722
\(813\) 0 0
\(814\) −43.2659 −1.51647
\(815\) 0 0
\(816\) 0 0
\(817\) −3.18676 −0.111491
\(818\) −46.3033 −1.61896
\(819\) 0 0
\(820\) 0 0
\(821\) −5.18121 −0.180826 −0.0904128 0.995904i \(-0.528819\pi\)
−0.0904128 + 0.995904i \(0.528819\pi\)
\(822\) 0 0
\(823\) −25.8517 −0.901133 −0.450567 0.892743i \(-0.648778\pi\)
−0.450567 + 0.892743i \(0.648778\pi\)
\(824\) 25.8350 0.900004
\(825\) 0 0
\(826\) −13.1987 −0.459240
\(827\) 9.97816 0.346974 0.173487 0.984836i \(-0.444496\pi\)
0.173487 + 0.984836i \(0.444496\pi\)
\(828\) 0 0
\(829\) 21.9808 0.763425 0.381713 0.924281i \(-0.375334\pi\)
0.381713 + 0.924281i \(0.375334\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −19.8126 −0.686877
\(833\) −18.2848 −0.633531
\(834\) 0 0
\(835\) 0 0
\(836\) 16.0224 0.554147
\(837\) 0 0
\(838\) −19.4675 −0.672492
\(839\) 16.1089 0.556140 0.278070 0.960561i \(-0.410305\pi\)
0.278070 + 0.960561i \(0.410305\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −71.0451 −2.44838
\(843\) 0 0
\(844\) −60.6666 −2.08823
\(845\) 0 0
\(846\) 0 0
\(847\) −19.9735 −0.686298
\(848\) −37.0775 −1.27324
\(849\) 0 0
\(850\) 0 0
\(851\) 11.2211 0.384653
\(852\) 0 0
\(853\) 44.9615 1.53945 0.769727 0.638373i \(-0.220392\pi\)
0.769727 + 0.638373i \(0.220392\pi\)
\(854\) 67.5128 2.31024
\(855\) 0 0
\(856\) 69.3029 2.36872
\(857\) 46.2431 1.57963 0.789817 0.613343i \(-0.210176\pi\)
0.789817 + 0.613343i \(0.210176\pi\)
\(858\) 0 0
\(859\) −10.7772 −0.367713 −0.183856 0.982953i \(-0.558858\pi\)
−0.183856 + 0.982953i \(0.558858\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 77.5443 2.64117
\(863\) −22.7966 −0.776007 −0.388003 0.921658i \(-0.626835\pi\)
−0.388003 + 0.921658i \(0.626835\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 1.16839 0.0397034
\(867\) 0 0
\(868\) −31.9390 −1.08408
\(869\) 5.82040 0.197444
\(870\) 0 0
\(871\) 9.86201 0.334161
\(872\) 52.5934 1.78104
\(873\) 0 0
\(874\) −6.31087 −0.213468
\(875\) 0 0
\(876\) 0 0
\(877\) −4.94644 −0.167029 −0.0835146 0.996507i \(-0.526615\pi\)
−0.0835146 + 0.996507i \(0.526615\pi\)
\(878\) −66.2054 −2.23432
\(879\) 0 0
\(880\) 0 0
\(881\) −2.53033 −0.0852490 −0.0426245 0.999091i \(-0.513572\pi\)
−0.0426245 + 0.999091i \(0.513572\pi\)
\(882\) 0 0
\(883\) 29.7430 1.00093 0.500465 0.865757i \(-0.333162\pi\)
0.500465 + 0.865757i \(0.333162\pi\)
\(884\) −46.2693 −1.55620
\(885\) 0 0
\(886\) −56.5801 −1.90085
\(887\) −45.5450 −1.52925 −0.764626 0.644474i \(-0.777076\pi\)
−0.764626 + 0.644474i \(0.777076\pi\)
\(888\) 0 0
\(889\) 19.3591 0.649282
\(890\) 0 0
\(891\) 0 0
\(892\) 72.5729 2.42992
\(893\) 11.7086 0.391813
\(894\) 0 0
\(895\) 0 0
\(896\) 65.2019 2.17824
\(897\) 0 0
\(898\) −55.9828 −1.86817
\(899\) −15.5960 −0.520155
\(900\) 0 0
\(901\) 68.0897 2.26840
\(902\) −6.02662 −0.200664
\(903\) 0 0
\(904\) 47.5689 1.58212
\(905\) 0 0
\(906\) 0 0
\(907\) 33.2034 1.10250 0.551250 0.834340i \(-0.314151\pi\)
0.551250 + 0.834340i \(0.314151\pi\)
\(908\) 55.6841 1.84794
\(909\) 0 0
\(910\) 0 0
\(911\) −55.7788 −1.84803 −0.924017 0.382351i \(-0.875114\pi\)
−0.924017 + 0.382351i \(0.875114\pi\)
\(912\) 0 0
\(913\) −29.3853 −0.972513
\(914\) 51.4231 1.70092
\(915\) 0 0
\(916\) −16.1089 −0.532252
\(917\) −43.2715 −1.42895
\(918\) 0 0
\(919\) −28.5769 −0.942665 −0.471333 0.881956i \(-0.656227\pi\)
−0.471333 + 0.881956i \(0.656227\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −76.4159 −2.51662
\(923\) −28.3947 −0.934622
\(924\) 0 0
\(925\) 0 0
\(926\) −37.7788 −1.24149
\(927\) 0 0
\(928\) 8.07426 0.265051
\(929\) 54.4937 1.78788 0.893940 0.448186i \(-0.147930\pi\)
0.893940 + 0.448186i \(0.147930\pi\)
\(930\) 0 0
\(931\) −3.15544 −0.103415
\(932\) −46.5960 −1.52630
\(933\) 0 0
\(934\) 72.1761 2.36167
\(935\) 0 0
\(936\) 0 0
\(937\) −37.1484 −1.21359 −0.606793 0.794860i \(-0.707544\pi\)
−0.606793 + 0.794860i \(0.707544\pi\)
\(938\) −36.7245 −1.19910
\(939\) 0 0
\(940\) 0 0
\(941\) −32.3973 −1.05612 −0.528061 0.849206i \(-0.677081\pi\)
−0.528061 + 0.849206i \(0.677081\pi\)
\(942\) 0 0
\(943\) 1.56301 0.0508986
\(944\) 5.40067 0.175777
\(945\) 0 0
\(946\) −32.0448 −1.04187
\(947\) −35.4662 −1.15250 −0.576249 0.817275i \(-0.695484\pi\)
−0.576249 + 0.817275i \(0.695484\pi\)
\(948\) 0 0
\(949\) −5.64752 −0.183326
\(950\) 0 0
\(951\) 0 0
\(952\) 82.9272 2.68769
\(953\) 14.3411 0.464553 0.232277 0.972650i \(-0.425383\pi\)
0.232277 + 0.972650i \(0.425383\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 43.7788 1.41591
\(957\) 0 0
\(958\) −10.9749 −0.354581
\(959\) −25.3125 −0.817383
\(960\) 0 0
\(961\) −24.2435 −0.782048
\(962\) 21.5615 0.695172
\(963\) 0 0
\(964\) −13.1122 −0.422316
\(965\) 0 0
\(966\) 0 0
\(967\) −27.9351 −0.898331 −0.449165 0.893449i \(-0.648279\pi\)
−0.449165 + 0.893449i \(0.648279\pi\)
\(968\) 28.1463 0.904657
\(969\) 0 0
\(970\) 0 0
\(971\) 42.5993 1.36708 0.683539 0.729914i \(-0.260440\pi\)
0.683539 + 0.729914i \(0.260440\pi\)
\(972\) 0 0
\(973\) −10.4132 −0.333833
\(974\) 94.9920 3.04374
\(975\) 0 0
\(976\) −27.6251 −0.884258
\(977\) −39.5597 −1.26563 −0.632813 0.774304i \(-0.718100\pi\)
−0.632813 + 0.774304i \(0.718100\pi\)
\(978\) 0 0
\(979\) 68.7114 2.19603
\(980\) 0 0
\(981\) 0 0
\(982\) −94.1112 −3.00321
\(983\) 39.7689 1.26843 0.634215 0.773157i \(-0.281323\pi\)
0.634215 + 0.773157i \(0.281323\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 84.1345 2.67939
\(987\) 0 0
\(988\) −7.98476 −0.254029
\(989\) 8.31087 0.264270
\(990\) 0 0
\(991\) 55.2019 1.75355 0.876773 0.480905i \(-0.159692\pi\)
0.876773 + 0.480905i \(0.159692\pi\)
\(992\) −3.49794 −0.111060
\(993\) 0 0
\(994\) 105.737 3.35378
\(995\) 0 0
\(996\) 0 0
\(997\) −11.6543 −0.369097 −0.184548 0.982823i \(-0.559082\pi\)
−0.184548 + 0.982823i \(0.559082\pi\)
\(998\) 11.4517 0.362496
\(999\) 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 4275.2.a.br.1.6 6
3.2 odd 2 475.2.a.j.1.1 6
5.2 odd 4 855.2.c.d.514.6 6
5.3 odd 4 855.2.c.d.514.1 6
5.4 even 2 inner 4275.2.a.br.1.1 6
12.11 even 2 7600.2.a.ck.1.3 6
15.2 even 4 95.2.b.b.39.1 6
15.8 even 4 95.2.b.b.39.6 yes 6
15.14 odd 2 475.2.a.j.1.6 6
57.56 even 2 9025.2.a.bx.1.6 6
60.23 odd 4 1520.2.d.h.609.3 6
60.47 odd 4 1520.2.d.h.609.4 6
60.59 even 2 7600.2.a.ck.1.4 6
285.113 odd 4 1805.2.b.e.1084.1 6
285.227 odd 4 1805.2.b.e.1084.6 6
285.284 even 2 9025.2.a.bx.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.b.b.39.1 6 15.2 even 4
95.2.b.b.39.6 yes 6 15.8 even 4
475.2.a.j.1.1 6 3.2 odd 2
475.2.a.j.1.6 6 15.14 odd 2
855.2.c.d.514.1 6 5.3 odd 4
855.2.c.d.514.6 6 5.2 odd 4
1520.2.d.h.609.3 6 60.23 odd 4
1520.2.d.h.609.4 6 60.47 odd 4
1805.2.b.e.1084.1 6 285.113 odd 4
1805.2.b.e.1084.6 6 285.227 odd 4
4275.2.a.br.1.1 6 5.4 even 2 inner
4275.2.a.br.1.6 6 1.1 even 1 trivial
7600.2.a.ck.1.3 6 12.11 even 2
7600.2.a.ck.1.4 6 60.59 even 2
9025.2.a.bx.1.1 6 285.284 even 2
9025.2.a.bx.1.6 6 57.56 even 2