Properties

Label 475.2.a.j.1.1
Level $475$
Weight $2$
Character 475.1
Self dual yes
Analytic conductor $3.793$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.79289409601\)
Analytic rank: \(0\)
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 1.1
Root \(-1.30397\) of defining polynomial
Character \(\chi\) \(=\) 475.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.41987 q^{2} +0.537080 q^{3} +3.85577 q^{4} -1.29966 q^{6} -3.18676 q^{7} -4.49073 q^{8} -2.71155 q^{9} +O(q^{10})\) \(q-2.41987 q^{2} +0.537080 q^{3} +3.85577 q^{4} -1.29966 q^{6} -3.18676 q^{7} -4.49073 q^{8} -2.71155 q^{9} +4.15544 q^{11} +2.07086 q^{12} +2.07086 q^{13} +7.71155 q^{14} +3.15544 q^{16} +5.79470 q^{17} +6.56159 q^{18} -1.00000 q^{19} -1.71155 q^{21} -10.0556 q^{22} -2.60794 q^{23} -2.41188 q^{24} -5.01121 q^{26} -3.06756 q^{27} -12.2874 q^{28} +6.00000 q^{29} +2.59933 q^{31} +1.34571 q^{32} +2.23180 q^{33} -14.0224 q^{34} -10.4551 q^{36} +4.30266 q^{37} +2.41987 q^{38} +1.11222 q^{39} -0.599328 q^{41} +4.14172 q^{42} +3.18676 q^{43} +16.0224 q^{44} +6.31087 q^{46} +11.7086 q^{47} +1.69472 q^{48} +3.15544 q^{49} +3.11222 q^{51} +7.98476 q^{52} +11.7503 q^{53} +7.42309 q^{54} +14.3109 q^{56} -0.537080 q^{57} -14.5192 q^{58} -1.71155 q^{59} -8.75476 q^{61} -6.29004 q^{62} +8.64104 q^{63} -9.56732 q^{64} -5.40067 q^{66} +4.76228 q^{67} +22.3430 q^{68} -1.40067 q^{69} +13.7115 q^{71} +12.1768 q^{72} -2.72714 q^{73} -10.4119 q^{74} -3.85577 q^{76} -13.2424 q^{77} -2.69142 q^{78} -1.40067 q^{79} +6.48711 q^{81} +1.45030 q^{82} -7.07154 q^{83} -6.59933 q^{84} -7.71155 q^{86} +3.22248 q^{87} -18.6609 q^{88} +16.5353 q^{89} -6.59933 q^{91} -10.0556 q^{92} +1.39605 q^{93} -28.3333 q^{94} +0.722754 q^{96} +2.07086 q^{97} -7.63575 q^{98} -11.2677 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 8 q^{4} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 8 q^{4} + 14 q^{9} + 2 q^{11} + 16 q^{14} - 4 q^{16} - 6 q^{19} + 20 q^{21} + 8 q^{24} + 8 q^{26} + 36 q^{29} - 8 q^{34} - 32 q^{36} - 8 q^{39} + 12 q^{41} + 20 q^{44} - 8 q^{46} - 4 q^{49} + 4 q^{51} - 16 q^{54} + 40 q^{56} + 20 q^{59} - 14 q^{61} - 12 q^{64} - 48 q^{66} - 24 q^{69} + 52 q^{71} - 40 q^{74} - 8 q^{76} - 24 q^{79} + 38 q^{81} - 24 q^{84} - 16 q^{86} + 24 q^{89} - 24 q^{91} - 48 q^{94} - 64 q^{96} - 30 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.41987 −1.71111 −0.855553 0.517715i \(-0.826783\pi\)
−0.855553 + 0.517715i \(0.826783\pi\)
\(3\) 0.537080 0.310083 0.155042 0.987908i \(-0.450449\pi\)
0.155042 + 0.987908i \(0.450449\pi\)
\(4\) 3.85577 1.92789
\(5\) 0 0
\(6\) −1.29966 −0.530586
\(7\) −3.18676 −1.20448 −0.602241 0.798314i \(-0.705725\pi\)
−0.602241 + 0.798314i \(0.705725\pi\)
\(8\) −4.49073 −1.58771
\(9\) −2.71155 −0.903848
\(10\) 0 0
\(11\) 4.15544 1.25291 0.626456 0.779457i \(-0.284505\pi\)
0.626456 + 0.779457i \(0.284505\pi\)
\(12\) 2.07086 0.597805
\(13\) 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 \(-0.251973\pi\)
0.702710 + 0.711476i \(0.251973\pi\)
\(18\) 6.56159 1.54658
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −1.71155 −0.373490
\(22\) −10.0556 −2.14386
\(23\) −2.60794 −0.543793 −0.271896 0.962327i \(-0.587651\pi\)
−0.271896 + 0.962327i \(0.587651\pi\)
\(24\) −2.41188 −0.492323
\(25\) 0 0
\(26\) −5.01121 −0.982779
\(27\) −3.06756 −0.590352
\(28\) −12.2874 −2.32210
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 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
\(33\) 2.23180 0.388507
\(34\) −14.0224 −2.40482
\(35\) 0 0
\(36\) −10.4551 −1.74252
\(37\) 4.30266 0.707353 0.353677 0.935368i \(-0.384931\pi\)
0.353677 + 0.935368i \(0.384931\pi\)
\(38\) 2.41987 0.392555
\(39\) 1.11222 0.178097
\(40\) 0 0
\(41\) −0.599328 −0.0935993 −0.0467997 0.998904i \(-0.514902\pi\)
−0.0467997 + 0.998904i \(0.514902\pi\)
\(42\) 4.14172 0.639081
\(43\) 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 \(-0.174208\pi\)
0.853937 + 0.520376i \(0.174208\pi\)
\(48\) 1.69472 0.244612
\(49\) 3.15544 0.450777
\(50\) 0 0
\(51\) 3.11222 0.435798
\(52\) 7.98476 1.10729
\(53\) 11.7503 1.61403 0.807017 0.590529i \(-0.201081\pi\)
0.807017 + 0.590529i \(0.201081\pi\)
\(54\) 7.42309 1.01015
\(55\) 0 0
\(56\) 14.3109 1.91237
\(57\) −0.537080 −0.0711380
\(58\) −14.5192 −1.90647
\(59\) −1.71155 −0.222824 −0.111412 0.993774i \(-0.535537\pi\)
−0.111412 + 0.993774i \(0.535537\pi\)
\(60\) 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
\(63\) 8.64104 1.08867
\(64\) −9.56732 −1.19591
\(65\) 0 0
\(66\) −5.40067 −0.664777
\(67\) 4.76228 0.581805 0.290902 0.956753i \(-0.406044\pi\)
0.290902 + 0.956753i \(0.406044\pi\)
\(68\) 22.3430 2.70949
\(69\) −1.40067 −0.168621
\(70\) 0 0
\(71\) 13.7115 1.62726 0.813631 0.581382i \(-0.197488\pi\)
0.813631 + 0.581382i \(0.197488\pi\)
\(72\) 12.1768 1.43505
\(73\) −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
\(78\) −2.69142 −0.304743
\(79\) −1.40067 −0.157588 −0.0787939 0.996891i \(-0.525107\pi\)
−0.0787939 + 0.996891i \(0.525107\pi\)
\(80\) 0 0
\(81\) 6.48711 0.720790
\(82\) 1.45030 0.160158
\(83\) −7.07154 −0.776203 −0.388101 0.921617i \(-0.626869\pi\)
−0.388101 + 0.921617i \(0.626869\pi\)
\(84\) −6.59933 −0.720046
\(85\) 0 0
\(86\) −7.71155 −0.831557
\(87\) 3.22248 0.345486
\(88\) −18.6609 −1.98926
\(89\) 16.5353 1.75274 0.876370 0.481639i \(-0.159958\pi\)
0.876370 + 0.481639i \(0.159958\pi\)
\(90\) 0 0
\(91\) −6.59933 −0.691798
\(92\) −10.0556 −1.04837
\(93\) 1.39605 0.144763
\(94\) −28.3333 −2.92236
\(95\) 0 0
\(96\) 0.722754 0.0737658
\(97\) 2.07086 0.210264 0.105132 0.994458i \(-0.466474\pi\)
0.105132 + 0.994458i \(0.466474\pi\)
\(98\) −7.63575 −0.771327
\(99\) −11.2677 −1.13244
\(100\) 0 0
\(101\) −1.71155 −0.170305 −0.0851525 0.996368i \(-0.527138\pi\)
−0.0851525 + 0.996368i \(0.527138\pi\)
\(102\) −7.53116 −0.745696
\(103\) 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 \(-0.768007\pi\)
−0.745955 + 0.665996i \(0.768007\pi\)
\(108\) −11.8278 −1.13813
\(109\) 11.7115 1.12176 0.560881 0.827896i \(-0.310462\pi\)
0.560881 + 0.827896i \(0.310462\pi\)
\(110\) 0 0
\(111\) 2.31087 0.219338
\(112\) −10.0556 −0.950167
\(113\) −10.5927 −0.996477 −0.498239 0.867040i \(-0.666020\pi\)
−0.498239 + 0.867040i \(0.666020\pi\)
\(114\) 1.29966 0.121725
\(115\) 0 0
\(116\) 23.1346 2.14800
\(117\) −5.61523 −0.519128
\(118\) 4.14172 0.381276
\(119\) −18.4663 −1.69280
\(120\) 0 0
\(121\) 6.26765 0.569787
\(122\) 21.1854 1.91804
\(123\) −0.321887 −0.0290236
\(124\) 10.0224 0.900040
\(125\) 0 0
\(126\) −20.9102 −1.86283
\(127\) −6.07484 −0.539055 −0.269528 0.962993i \(-0.586868\pi\)
−0.269528 + 0.962993i \(0.586868\pi\)
\(128\) 20.4602 1.80845
\(129\) 1.71155 0.150693
\(130\) 0 0
\(131\) −13.5785 −1.18636 −0.593181 0.805069i \(-0.702128\pi\)
−0.593181 + 0.805069i \(0.702128\pi\)
\(132\) 8.60532 0.748997
\(133\) 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 \(-0.610193\pi\)
−0.339309 + 0.940675i \(0.610193\pi\)
\(138\) 3.38944 0.288529
\(139\) 3.26765 0.277159 0.138579 0.990351i \(-0.455746\pi\)
0.138579 + 0.990351i \(0.455746\pi\)
\(140\) 0 0
\(141\) 6.28845 0.529583
\(142\) −33.1802 −2.78442
\(143\) 8.60532 0.719613
\(144\) −8.55611 −0.713009
\(145\) 0 0
\(146\) 6.59933 0.546164
\(147\) 1.69472 0.139778
\(148\) 16.5901 1.36370
\(149\) 8.44389 0.691751 0.345875 0.938280i \(-0.387582\pi\)
0.345875 + 0.938280i \(0.387582\pi\)
\(150\) 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
\(153\) −15.7126 −1.27029
\(154\) 32.0448 2.58225
\(155\) 0 0
\(156\) 4.28845 0.343351
\(157\) 4.14172 0.330545 0.165273 0.986248i \(-0.447150\pi\)
0.165273 + 0.986248i \(0.447150\pi\)
\(158\) 3.38944 0.269650
\(159\) 6.31087 0.500485
\(160\) 0 0
\(161\) 8.31087 0.654989
\(162\) −15.6980 −1.23335
\(163\) −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 \(-0.544539\pi\)
−0.139468 + 0.990227i \(0.544539\pi\)
\(168\) 7.68608 0.592994
\(169\) −8.71155 −0.670119
\(170\) 0 0
\(171\) 2.71155 0.207357
\(172\) 12.2874 0.936907
\(173\) −22.4205 −1.70460 −0.852300 0.523054i \(-0.824793\pi\)
−0.852300 + 0.523054i \(0.824793\pi\)
\(174\) −7.79798 −0.591164
\(175\) 0 0
\(176\) 13.1122 0.988371
\(177\) −0.919237 −0.0690941
\(178\) −40.0133 −2.99912
\(179\) −5.13464 −0.383781 −0.191890 0.981416i \(-0.561462\pi\)
−0.191890 + 0.981416i \(0.561462\pi\)
\(180\) 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
\(183\) −4.70201 −0.347583
\(184\) 11.7115 0.863387
\(185\) 0 0
\(186\) −3.37825 −0.247706
\(187\) 24.0795 1.76087
\(188\) 45.1457 3.29259
\(189\) 9.77557 0.711068
\(190\) 0 0
\(191\) −5.26765 −0.381154 −0.190577 0.981672i \(-0.561036\pi\)
−0.190577 + 0.981672i \(0.561036\pi\)
\(192\) −5.13842 −0.370833
\(193\) 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 \(-0.378804\pi\)
0.371616 + 0.928387i \(0.378804\pi\)
\(198\) 27.2663 1.93773
\(199\) −2.73235 −0.193691 −0.0968455 0.995299i \(-0.530875\pi\)
−0.0968455 + 0.995299i \(0.530875\pi\)
\(200\) 0 0
\(201\) 2.55773 0.180408
\(202\) 4.14172 0.291410
\(203\) −19.1206 −1.34200
\(204\) 12.0000 0.840168
\(205\) 0 0
\(206\) −13.9214 −0.969951
\(207\) 7.07154 0.491506
\(208\) 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
\(213\) 7.36420 0.504586
\(214\) 37.3445 2.55282
\(215\) 0 0
\(216\) 13.7756 0.937309
\(217\) −8.28343 −0.562316
\(218\) −28.3404 −1.91946
\(219\) −1.46469 −0.0989748
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) −5.59201 −0.375311
\(223\) 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 \(-0.659097\pi\)
−0.479267 + 0.877669i \(0.659097\pi\)
\(228\) −2.07086 −0.137146
\(229\) −4.17785 −0.276080 −0.138040 0.990427i \(-0.544080\pi\)
−0.138040 + 0.990427i \(0.544080\pi\)
\(230\) 0 0
\(231\) −7.11222 −0.467950
\(232\) −26.9444 −1.76898
\(233\) 12.0847 0.791697 0.395849 0.918316i \(-0.370450\pi\)
0.395849 + 0.918316i \(0.370450\pi\)
\(234\) 13.5881 0.888283
\(235\) 0 0
\(236\) −6.59933 −0.429580
\(237\) −0.752273 −0.0488654
\(238\) 44.6861 2.89657
\(239\) −11.3541 −0.734435 −0.367218 0.930135i \(-0.619690\pi\)
−0.367218 + 0.930135i \(0.619690\pi\)
\(240\) 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
\(243\) 12.6868 0.813857
\(244\) −33.7564 −2.16103
\(245\) 0 0
\(246\) 0.778925 0.0496625
\(247\) −2.07086 −0.131766
\(248\) −11.6729 −0.741229
\(249\) −3.79798 −0.240687
\(250\) 0 0
\(251\) 3.04322 0.192086 0.0960432 0.995377i \(-0.469381\pi\)
0.0960432 + 0.995377i \(0.469381\pi\)
\(252\) 33.3179 2.09883
\(253\) −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 \(-0.318726\pi\)
0.539201 + 0.842177i \(0.318726\pi\)
\(258\) −4.14172 −0.257852
\(259\) −13.7115 −0.851994
\(260\) 0 0
\(261\) −16.2693 −1.00704
\(262\) 32.8583 2.02999
\(263\) −1.19336 −0.0735859 −0.0367930 0.999323i \(-0.511714\pi\)
−0.0367930 + 0.999323i \(0.511714\pi\)
\(264\) −10.0224 −0.616837
\(265\) 0 0
\(266\) −7.71155 −0.472825
\(267\) 8.88078 0.543495
\(268\) 18.3623 1.12165
\(269\) 22.1089 1.34800 0.674000 0.738731i \(-0.264575\pi\)
0.674000 + 0.738731i \(0.264575\pi\)
\(270\) 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
\(273\) −3.54437 −0.214515
\(274\) 19.2211 1.16119
\(275\) 0 0
\(276\) −5.40067 −0.325082
\(277\) −10.0199 −0.602037 −0.301019 0.953618i \(-0.597327\pi\)
−0.301019 + 0.953618i \(0.597327\pi\)
\(278\) −7.90730 −0.474248
\(279\) −7.04820 −0.421964
\(280\) 0 0
\(281\) 0.599328 0.0357529 0.0178765 0.999840i \(-0.494309\pi\)
0.0178765 + 0.999840i \(0.494309\pi\)
\(282\) −15.2172 −0.906174
\(283\) 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
\(288\) −3.64895 −0.215017
\(289\) 16.5785 0.975207
\(290\) 0 0
\(291\) 1.11222 0.0651993
\(292\) −10.5152 −0.615358
\(293\) 3.46691 0.202539 0.101269 0.994859i \(-0.467710\pi\)
0.101269 + 0.994859i \(0.467710\pi\)
\(294\) −4.10101 −0.239176
\(295\) 0 0
\(296\) −19.3221 −1.12307
\(297\) −12.7470 −0.739658
\(298\) −20.4331 −1.18366
\(299\) −5.40067 −0.312329
\(300\) 0 0
\(301\) −10.1554 −0.585350
\(302\) −2.14832 −0.123622
\(303\) −0.919237 −0.0528088
\(304\) −3.15544 −0.180977
\(305\) 0 0
\(306\) 38.0224 2.17360
\(307\) −16.5901 −0.946846 −0.473423 0.880835i \(-0.656982\pi\)
−0.473423 + 0.880835i \(0.656982\pi\)
\(308\) −51.0596 −2.90939
\(309\) 3.08980 0.175772
\(310\) 0 0
\(311\) −4.15544 −0.235633 −0.117817 0.993035i \(-0.537589\pi\)
−0.117817 + 0.993035i \(0.537589\pi\)
\(312\) −4.99466 −0.282767
\(313\) −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 \(-0.770250\pi\)
−0.750630 + 0.660723i \(0.770250\pi\)
\(318\) −15.2715 −0.856383
\(319\) 24.9326 1.39596
\(320\) 0 0
\(321\) −8.28845 −0.462616
\(322\) −20.1112 −1.12076
\(323\) −5.79470 −0.322426
\(324\) 25.0128 1.38960
\(325\) 0 0
\(326\) 59.8012 3.31208
\(327\) 6.29004 0.347840
\(328\) 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
\(333\) −11.6669 −0.639340
\(334\) 8.72275 0.477288
\(335\) 0 0
\(336\) −5.40067 −0.294631
\(337\) 22.5040 1.22587 0.612935 0.790133i \(-0.289989\pi\)
0.612935 + 0.790133i \(0.289989\pi\)
\(338\) 21.0808 1.14664
\(339\) −5.68913 −0.308991
\(340\) 0 0
\(341\) 10.8013 0.584926
\(342\) −6.56159 −0.354810
\(343\) 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 \(-0.626825\pi\)
−0.387973 + 0.921671i \(0.626825\pi\)
\(348\) 12.4252 0.666058
\(349\) −13.3541 −0.714828 −0.357414 0.933946i \(-0.616342\pi\)
−0.357414 + 0.933946i \(0.616342\pi\)
\(350\) 0 0
\(351\) −6.35248 −0.339070
\(352\) 5.59201 0.298055
\(353\) 17.6410 0.938937 0.469469 0.882949i \(-0.344446\pi\)
0.469469 + 0.882949i \(0.344446\pi\)
\(354\) 2.22443 0.118227
\(355\) 0 0
\(356\) 63.7564 3.37908
\(357\) −9.91789 −0.524910
\(358\) 12.4252 0.656690
\(359\) 12.4663 0.657947 0.328973 0.944339i \(-0.393297\pi\)
0.328973 + 0.944339i \(0.393297\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −50.4451 −2.65133
\(363\) 3.36623 0.176681
\(364\) −25.4455 −1.33371
\(365\) 0 0
\(366\) 11.3783 0.594751
\(367\) −16.4291 −0.857594 −0.428797 0.903401i \(-0.641062\pi\)
−0.428797 + 0.903401i \(0.641062\pi\)
\(368\) −8.22918 −0.428976
\(369\) 1.62511 0.0845996
\(370\) 0 0
\(371\) −37.4455 −1.94407
\(372\) 5.38284 0.279087
\(373\) −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
\(378\) −23.6556 −1.21671
\(379\) 14.5353 0.746629 0.373314 0.927705i \(-0.378221\pi\)
0.373314 + 0.927705i \(0.378221\pi\)
\(380\) 0 0
\(381\) −3.26268 −0.167152
\(382\) 12.7470 0.652195
\(383\) 0.453598 0.0231778 0.0115889 0.999933i \(-0.496311\pi\)
0.0115889 + 0.999933i \(0.496311\pi\)
\(384\) 10.9888 0.560769
\(385\) 0 0
\(386\) −5.01121 −0.255064
\(387\) −8.64104 −0.439249
\(388\) 7.98476 0.405365
\(389\) 16.1554 0.819113 0.409557 0.912285i \(-0.365683\pi\)
0.409557 + 0.912285i \(0.365683\pi\)
\(390\) 0 0
\(391\) −15.1122 −0.764258
\(392\) −14.1702 −0.715704
\(393\) −7.29276 −0.367871
\(394\) −25.2435 −1.27175
\(395\) 0 0
\(396\) −43.4455 −2.18322
\(397\) 32.7563 1.64399 0.821995 0.569495i \(-0.192861\pi\)
0.821995 + 0.569495i \(0.192861\pi\)
\(398\) 6.61192 0.331426
\(399\) 1.71155 0.0856844
\(400\) 0 0
\(401\) 12.0864 0.603568 0.301784 0.953376i \(-0.402418\pi\)
0.301784 + 0.953376i \(0.402418\pi\)
\(402\) −6.18936 −0.308697
\(403\) 5.38284 0.268138
\(404\) −6.59933 −0.328329
\(405\) 0 0
\(406\) 46.2693 2.29631
\(407\) 17.8794 0.886251
\(408\) −13.9761 −0.691921
\(409\) −19.1346 −0.946147 −0.473073 0.881023i \(-0.656855\pi\)
−0.473073 + 0.881023i \(0.656855\pi\)
\(410\) 0 0
\(411\) −4.26604 −0.210428
\(412\) 22.1821 1.09283
\(413\) 5.45428 0.268388
\(414\) −17.1122 −0.841020
\(415\) 0 0
\(416\) 2.78678 0.136633
\(417\) 1.75499 0.0859423
\(418\) 10.0556 0.491836
\(419\) 8.04484 0.393016 0.196508 0.980502i \(-0.437040\pi\)
0.196508 + 0.980502i \(0.437040\pi\)
\(420\) 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
\(423\) −31.7484 −1.54366
\(424\) −52.7676 −2.56262
\(425\) 0 0
\(426\) −17.8204 −0.863401
\(427\) 27.8993 1.35014
\(428\) −59.5040 −2.87623
\(429\) 4.62175 0.223140
\(430\) 0 0
\(431\) −32.0448 −1.54355 −0.771773 0.635898i \(-0.780630\pi\)
−0.771773 + 0.635898i \(0.780630\pi\)
\(432\) −9.67948 −0.465704
\(433\) 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
\(438\) 3.54437 0.169356
\(439\) −27.3591 −1.30578 −0.652889 0.757454i \(-0.726443\pi\)
−0.652889 + 0.757454i \(0.726443\pi\)
\(440\) 0 0
\(441\) −8.55611 −0.407434
\(442\) −29.0384 −1.38122
\(443\) 23.3815 1.11089 0.555444 0.831554i \(-0.312548\pi\)
0.555444 + 0.831554i \(0.312548\pi\)
\(444\) 8.91020 0.422859
\(445\) 0 0
\(446\) −45.5465 −2.15669
\(447\) 4.53505 0.214500
\(448\) 30.4887 1.44046
\(449\) 23.1346 1.09179 0.545895 0.837853i \(-0.316190\pi\)
0.545895 + 0.837853i \(0.316190\pi\)
\(450\) 0 0
\(451\) −2.49047 −0.117272
\(452\) −40.8430 −1.92109
\(453\) 0.476811 0.0224025
\(454\) 34.9472 1.64015
\(455\) 0 0
\(456\) 2.41188 0.112947
\(457\) 21.2503 0.994049 0.497025 0.867736i \(-0.334426\pi\)
0.497025 + 0.867736i \(0.334426\pi\)
\(458\) 10.1099 0.472403
\(459\) −17.7756 −0.829692
\(460\) 0 0
\(461\) 31.5785 1.47076 0.735379 0.677656i \(-0.237004\pi\)
0.735379 + 0.677656i \(0.237004\pi\)
\(462\) 17.2106 0.800712
\(463\) −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 \(-0.742434\pi\)
−0.690101 + 0.723713i \(0.742434\pi\)
\(468\) −21.6510 −1.00082
\(469\) −15.1762 −0.700774
\(470\) 0 0
\(471\) 2.22443 0.102496
\(472\) 7.68608 0.353781
\(473\) 13.2424 0.608885
\(474\) 1.82040 0.0836139
\(475\) 0 0
\(476\) −71.2019 −3.26353
\(477\) −31.8616 −1.45884
\(478\) 27.4754 1.25670
\(479\) 4.53531 0.207223 0.103612 0.994618i \(-0.466960\pi\)
0.103612 + 0.994618i \(0.466960\pi\)
\(480\) 0 0
\(481\) 8.91020 0.406270
\(482\) 8.22918 0.374829
\(483\) 4.46360 0.203101
\(484\) 24.1666 1.09848
\(485\) 0 0
\(486\) −30.7003 −1.39260
\(487\) 39.2550 1.77881 0.889407 0.457116i \(-0.151118\pi\)
0.889407 + 0.457116i \(0.151118\pi\)
\(488\) 39.3153 1.77972
\(489\) −13.2726 −0.600209
\(490\) 0 0
\(491\) 38.8910 1.75513 0.877564 0.479460i \(-0.159168\pi\)
0.877564 + 0.479460i \(0.159168\pi\)
\(492\) −1.24112 −0.0559542
\(493\) 34.7682 1.56588
\(494\) 5.01121 0.225465
\(495\) 0 0
\(496\) 8.20202 0.368281
\(497\) −43.6954 −1.96001
\(498\) 9.19063 0.411842
\(499\) 4.73235 0.211849 0.105924 0.994374i \(-0.466220\pi\)
0.105924 + 0.994374i \(0.466220\pi\)
\(500\) 0 0
\(501\) −1.93598 −0.0864931
\(502\) −7.36420 −0.328680
\(503\) 1.85567 0.0827400 0.0413700 0.999144i \(-0.486828\pi\)
0.0413700 + 0.999144i \(0.486828\pi\)
\(504\) −38.8046 −1.72849
\(505\) 0 0
\(506\) 26.2244 1.16582
\(507\) −4.67880 −0.207793
\(508\) −23.4232 −1.03924
\(509\) 22.8878 1.01448 0.507242 0.861804i \(-0.330665\pi\)
0.507242 + 0.861804i \(0.330665\pi\)
\(510\) 0 0
\(511\) 8.69074 0.384456
\(512\) 32.5867 1.44014
\(513\) 3.06756 0.135436
\(514\) −41.8350 −1.84526
\(515\) 0 0
\(516\) 6.59933 0.290519
\(517\) 48.6543 2.13982
\(518\) 33.1802 1.45785
\(519\) −12.0416 −0.528568
\(520\) 0 0
\(521\) −29.7340 −1.30267 −0.651334 0.758791i \(-0.725790\pi\)
−0.651334 + 0.758791i \(0.725790\pi\)
\(522\) 39.3695 1.72316
\(523\) 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
\(528\) 7.04231 0.306477
\(529\) −16.1987 −0.704289
\(530\) 0 0
\(531\) 4.64093 0.201399
\(532\) 12.2874 0.532727
\(533\) −1.24112 −0.0537590
\(534\) −21.4903 −0.929978
\(535\) 0 0
\(536\) −21.3861 −0.923739
\(537\) −2.75771 −0.119004
\(538\) −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
\(543\) 11.1961 0.480469
\(544\) 7.79798 0.334336
\(545\) 0 0
\(546\) 8.57691 0.367058
\(547\) −14.4297 −0.616970 −0.308485 0.951229i \(-0.599822\pi\)
−0.308485 + 0.951229i \(0.599822\pi\)
\(548\) −30.6265 −1.30830
\(549\) 23.7389 1.01315
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 6.29004 0.267722
\(553\) 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 \(-0.364728\pi\)
0.412294 + 0.911051i \(0.364728\pi\)
\(558\) 17.0557 0.722026
\(559\) 6.59933 0.279122
\(560\) 0 0
\(561\) 12.9326 0.546016
\(562\) −1.45030 −0.0611771
\(563\) −12.3649 −0.521118 −0.260559 0.965458i \(-0.583907\pi\)
−0.260559 + 0.965458i \(0.583907\pi\)
\(564\) 24.2469 1.02098
\(565\) 0 0
\(566\) −13.1122 −0.551148
\(567\) −20.6729 −0.868179
\(568\) −61.5748 −2.58362
\(569\) 15.0898 0.632597 0.316299 0.948660i \(-0.397560\pi\)
0.316299 + 0.948660i \(0.397560\pi\)
\(570\) 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
\(573\) −2.82915 −0.118190
\(574\) −4.62175 −0.192908
\(575\) 0 0
\(576\) 25.9422 1.08093
\(577\) −22.5165 −0.937374 −0.468687 0.883364i \(-0.655273\pi\)
−0.468687 + 0.883364i \(0.655273\pi\)
\(578\) −40.1179 −1.66868
\(579\) 1.11222 0.0462222
\(580\) 0 0
\(581\) 22.5353 0.934922
\(582\) −2.69142 −0.111563
\(583\) 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 \(-0.450564\pi\)
0.154685 + 0.987964i \(0.450564\pi\)
\(588\) 6.53446 0.269477
\(589\) −2.59933 −0.107103
\(590\) 0 0
\(591\) 5.60269 0.230464
\(592\) 13.5768 0.558002
\(593\) 27.8094 1.14199 0.570997 0.820952i \(-0.306557\pi\)
0.570997 + 0.820952i \(0.306557\pi\)
\(594\) 30.8462 1.26563
\(595\) 0 0
\(596\) 32.5577 1.33362
\(597\) −1.46749 −0.0600603
\(598\) 13.0689 0.534428
\(599\) −45.4903 −1.85869 −0.929343 0.369219i \(-0.879625\pi\)
−0.929343 + 0.369219i \(0.879625\pi\)
\(600\) 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
\(603\) −12.9131 −0.525863
\(604\) 3.42309 0.139284
\(605\) 0 0
\(606\) 2.22443 0.0903614
\(607\) −5.08417 −0.206360 −0.103180 0.994663i \(-0.532902\pi\)
−0.103180 + 0.994663i \(0.532902\pi\)
\(608\) −1.34571 −0.0545758
\(609\) −10.2693 −0.416132
\(610\) 0 0
\(611\) 24.2469 0.980923
\(612\) −60.5842 −2.44897
\(613\) 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 \(-0.198948\pi\)
0.810955 + 0.585108i \(0.198948\pi\)
\(618\) −7.47691 −0.300766
\(619\) −43.3815 −1.74365 −0.871825 0.489818i \(-0.837063\pi\)
−0.871825 + 0.489818i \(0.837063\pi\)
\(620\) 0 0
\(621\) 8.00000 0.321029
\(622\) 10.0556 0.403194
\(623\) −52.6940 −2.11114
\(624\) 3.50953 0.140494
\(625\) 0 0
\(626\) 2.22443 0.0889062
\(627\) −2.23180 −0.0891296
\(628\) 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
\(633\) −8.45040 −0.335873
\(634\) 64.6812 2.56882
\(635\) 0 0
\(636\) 24.3333 0.964878
\(637\) 6.53446 0.258905
\(638\) −60.3337 −2.38863
\(639\) −37.1795 −1.47080
\(640\) 0 0
\(641\) −23.3075 −0.920591 −0.460296 0.887766i \(-0.652257\pi\)
−0.460296 + 0.887766i \(0.652257\pi\)
\(642\) 20.0570 0.791586
\(643\) 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 \(-0.875608\pi\)
−0.924609 + 0.380917i \(0.875608\pi\)
\(648\) −29.1319 −1.14441
\(649\) −7.11222 −0.279179
\(650\) 0 0
\(651\) −4.44887 −0.174365
\(652\) −95.2861 −3.73169
\(653\) 24.1630 0.945571 0.472785 0.881178i \(-0.343249\pi\)
0.472785 + 0.881178i \(0.343249\pi\)
\(654\) −15.2211 −0.595191
\(655\) 0 0
\(656\) −1.89114 −0.0738367
\(657\) 7.39477 0.288497
\(658\) 90.2914 3.51992
\(659\) 10.2885 0.400781 0.200391 0.979716i \(-0.435779\pi\)
0.200391 + 0.979716i \(0.435779\pi\)
\(660\) 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
\(663\) 6.44496 0.250302
\(664\) 31.7564 1.23239
\(665\) 0 0
\(666\) 28.2323 1.09398
\(667\) −15.6476 −0.605879
\(668\) −13.8987 −0.537755
\(669\) 10.1089 0.390831
\(670\) 0 0
\(671\) −36.3799 −1.40443
\(672\) −2.30324 −0.0888496
\(673\) −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 \(-0.381843\pi\)
0.362734 + 0.931893i \(0.381843\pi\)
\(678\) 13.7669 0.528716
\(679\) −6.59933 −0.253259
\(680\) 0 0
\(681\) −7.75638 −0.297225
\(682\) −26.1379 −1.00087
\(683\) −19.6576 −0.752179 −0.376089 0.926583i \(-0.622731\pi\)
−0.376089 + 0.926583i \(0.622731\pi\)
\(684\) 10.4551 0.399761
\(685\) 0 0
\(686\) −29.6475 −1.13195
\(687\) −2.24384 −0.0856079
\(688\) 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
\(693\) 35.9073 1.36401
\(694\) 34.9775 1.32773
\(695\) 0 0
\(696\) −14.4713 −0.548533
\(697\) −3.47293 −0.131546
\(698\) 32.3152 1.22315
\(699\) 6.49047 0.245492
\(700\) 0 0
\(701\) 0.512889 0.0193715 0.00968577 0.999953i \(-0.496917\pi\)
0.00968577 + 0.999953i \(0.496917\pi\)
\(702\) 15.3722 0.580185
\(703\) −4.30266 −0.162278
\(704\) −39.7564 −1.49838
\(705\) 0 0
\(706\) −42.6890 −1.60662
\(707\) 5.45428 0.205129
\(708\) −3.54437 −0.133205
\(709\) −8.84618 −0.332225 −0.166113 0.986107i \(-0.553122\pi\)
−0.166113 + 0.986107i \(0.553122\pi\)
\(710\) 0 0
\(711\) 3.79798 0.142436
\(712\) −74.2556 −2.78285
\(713\) −6.77889 −0.253871
\(714\) 24.0000 0.898177
\(715\) 0 0
\(716\) −19.7980 −0.739885
\(717\) −6.09806 −0.227736
\(718\) −30.1669 −1.12582
\(719\) 7.84456 0.292553 0.146276 0.989244i \(-0.453271\pi\)
0.146276 + 0.989244i \(0.453271\pi\)
\(720\) 0 0
\(721\) −18.3333 −0.682767
\(722\) −2.41987 −0.0900582
\(723\) −1.82643 −0.0679258
\(724\) 80.3781 2.98723
\(725\) 0 0
\(726\) −8.14584 −0.302321
\(727\) −2.91130 −0.107974 −0.0539870 0.998542i \(-0.517193\pi\)
−0.0539870 + 0.998542i \(0.517193\pi\)
\(728\) 29.6358 1.09838
\(729\) −12.6475 −0.468427
\(730\) 0 0
\(731\) 18.4663 0.683001
\(732\) −18.1299 −0.670100
\(733\) −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
\(738\) −3.93254 −0.144759
\(739\) −35.8030 −1.31703 −0.658517 0.752566i \(-0.728816\pi\)
−0.658517 + 0.752566i \(0.728816\pi\)
\(740\) 0 0
\(741\) −1.11222 −0.0408583
\(742\) 90.6133 3.32652
\(743\) −5.66948 −0.207993 −0.103996 0.994578i \(-0.533163\pi\)
−0.103996 + 0.994578i \(0.533163\pi\)
\(744\) −6.26927 −0.229843
\(745\) 0 0
\(746\) 12.8092 0.468978
\(747\) 19.1748 0.701569
\(748\) 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
\(753\) 1.63445 0.0595628
\(754\) −30.0673 −1.09498
\(755\) 0 0
\(756\) 37.6924 1.37086
\(757\) −41.6370 −1.51332 −0.756662 0.653806i \(-0.773171\pi\)
−0.756662 + 0.653806i \(0.773171\pi\)
\(758\) −35.1736 −1.27756
\(759\) −5.82040 −0.211267
\(760\) 0 0
\(761\) −9.46967 −0.343275 −0.171638 0.985160i \(-0.554906\pi\)
−0.171638 + 0.985160i \(0.554906\pi\)
\(762\) 7.89526 0.286015
\(763\) −37.3219 −1.35114
\(764\) −20.3109 −0.734822
\(765\) 0 0
\(766\) −1.09765 −0.0396597
\(767\) −3.54437 −0.127980
\(768\) −16.3146 −0.588703
\(769\) 1.90858 0.0688253 0.0344127 0.999408i \(-0.489044\pi\)
0.0344127 + 0.999408i \(0.489044\pi\)
\(770\) 0 0
\(771\) 9.28510 0.334395
\(772\) 7.98476 0.287378
\(773\) −28.4007 −1.02150 −0.510751 0.859729i \(-0.670633\pi\)
−0.510751 + 0.859729i \(0.670633\pi\)
\(774\) 20.9102 0.751602
\(775\) 0 0
\(776\) −9.29966 −0.333838
\(777\) −7.36420 −0.264189
\(778\) −39.0941 −1.40159
\(779\) 0.599328 0.0214732
\(780\) 0 0
\(781\) 56.9775 2.03881
\(782\) 36.5696 1.30773
\(783\) −18.4053 −0.657753
\(784\) 9.95678 0.355599
\(785\) 0 0
\(786\) 17.6475 0.629466
\(787\) 15.6708 0.558605 0.279303 0.960203i \(-0.409897\pi\)
0.279303 + 0.960203i \(0.409897\pi\)
\(788\) 40.2225 1.43287
\(789\) −0.640931 −0.0228178
\(790\) 0 0
\(791\) 33.7564 1.20024
\(792\) 50.6000 1.79799
\(793\) −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 \(-0.726882\pi\)
−0.653932 + 0.756554i \(0.726882\pi\)
\(798\) −4.14172 −0.146615
\(799\) 67.8478 2.40028
\(800\) 0 0
\(801\) −44.8362 −1.58421
\(802\) −29.2476 −1.03277
\(803\) −11.3325 −0.399914
\(804\) 9.86201 0.347806
\(805\) 0 0
\(806\) −13.0258 −0.458813
\(807\) 11.8742 0.417993
\(808\) 7.68608 0.270396
\(809\) 43.0465 1.51343 0.756716 0.653743i \(-0.226802\pi\)
0.756716 + 0.653743i \(0.226802\pi\)
\(810\) 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
\(813\) −2.19475 −0.0769731
\(814\) −43.2659 −1.51647
\(815\) 0 0
\(816\) 9.82040 0.343783
\(817\) −3.18676 −0.111491
\(818\) 46.3033 1.61896
\(819\) 17.8944 0.625280
\(820\) 0 0
\(821\) 5.18121 0.180826 0.0904128 0.995904i \(-0.471181\pi\)
0.0904128 + 0.995904i \(0.471181\pi\)
\(822\) 10.3233 0.360065
\(823\) −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 \(-0.555504\pi\)
−0.173487 + 0.984836i \(0.555504\pi\)
\(828\) 27.2663 0.947568
\(829\) 21.9808 0.763425 0.381713 0.924281i \(-0.375334\pi\)
0.381713 + 0.924281i \(0.375334\pi\)
\(830\) 0 0
\(831\) −5.38149 −0.186682
\(832\) −19.8126 −0.686877
\(833\) 18.2848 0.633531
\(834\) −4.24685 −0.147056
\(835\) 0 0
\(836\) −16.0224 −0.554147
\(837\) −7.97359 −0.275607
\(838\) −19.4675 −0.672492
\(839\) −16.1089 −0.556140 −0.278070 0.960561i \(-0.589695\pi\)
−0.278070 + 0.960561i \(0.589695\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 71.0451 2.44838
\(843\) 0.321887 0.0110864
\(844\) −60.6666 −2.08823
\(845\) 0 0
\(846\) 76.8270 2.64137
\(847\) −19.9735 −0.686298
\(848\) 37.0775 1.27324
\(849\) 2.91020 0.0998779
\(850\) 0 0
\(851\) −11.2211 −0.384653
\(852\) 28.3947 0.972785
\(853\) 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 \(-0.789824\pi\)
−0.789817 + 0.613343i \(0.789824\pi\)
\(858\) −11.1840 −0.381816
\(859\) −10.7772 −0.367713 −0.183856 0.982953i \(-0.558858\pi\)
−0.183856 + 0.982953i \(0.558858\pi\)
\(860\) 0 0
\(861\) 1.02578 0.0349584
\(862\) 77.5443 2.64117
\(863\) 22.7966 0.776007 0.388003 0.921658i \(-0.373165\pi\)
0.388003 + 0.921658i \(0.373165\pi\)
\(864\) −4.12804 −0.140439
\(865\) 0 0
\(866\) −1.16839 −0.0397034
\(867\) 8.90400 0.302396
\(868\) −31.9390 −1.08408
\(869\) −5.82040 −0.197444
\(870\) 0 0
\(871\) 9.86201 0.334161
\(872\) −52.5934 −1.78104
\(873\) −5.61523 −0.190047
\(874\) −6.31087 −0.213468
\(875\) 0 0
\(876\) −5.64752 −0.190812
\(877\) −4.94644 −0.167029 −0.0835146 0.996507i \(-0.526615\pi\)
−0.0835146 + 0.996507i \(0.526615\pi\)
\(878\) 66.2054 2.23432
\(879\) 1.86201 0.0628039
\(880\) 0 0
\(881\) 2.53033 0.0852490 0.0426245 0.999091i \(-0.486428\pi\)
0.0426245 + 0.999091i \(0.486428\pi\)
\(882\) 20.7047 0.697163
\(883\) 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 \(-0.222924\pi\)
0.764626 + 0.644474i \(0.222924\pi\)
\(888\) −10.3775 −0.348246
\(889\) 19.3591 0.649282
\(890\) 0 0
\(891\) 26.9568 0.903086
\(892\) 72.5729 2.42992
\(893\) −11.7086 −0.391813
\(894\) −10.9742 −0.367033
\(895\) 0 0
\(896\) −65.2019 −2.17824
\(897\) −2.90059 −0.0968480
\(898\) −55.9828 −1.86817
\(899\) 15.5960 0.520155
\(900\) 0 0
\(901\) 68.0897 2.26840
\(902\) 6.02662 0.200664
\(903\) −5.45428 −0.181507
\(904\) 47.5689 1.58212
\(905\) 0 0
\(906\) −1.15382 −0.0383331
\(907\) 33.2034 1.10250 0.551250 0.834340i \(-0.314151\pi\)
0.551250 + 0.834340i \(0.314151\pi\)
\(908\) −55.6841 −1.84794
\(909\) 4.64093 0.153930
\(910\) 0 0
\(911\) 55.7788 1.84803 0.924017 0.382351i \(-0.124886\pi\)
0.924017 + 0.382351i \(0.124886\pi\)
\(912\) −1.69472 −0.0561179
\(913\) −29.3853 −0.972513
\(914\) −51.4231 −1.70092
\(915\) 0 0
\(916\) −16.1089 −0.532252
\(917\) 43.2715 1.42895
\(918\) 43.0146 1.41969
\(919\) −28.5769 −0.942665 −0.471333 0.881956i \(-0.656227\pi\)
−0.471333 + 0.881956i \(0.656227\pi\)
\(920\) 0 0
\(921\) −8.91020 −0.293601
\(922\) −76.4159 −2.51662
\(923\) 28.3947 0.934622
\(924\) −27.4231 −0.902153
\(925\) 0 0
\(926\) 37.7788 1.24149
\(927\) −15.5994 −0.512352
\(928\) 8.07426 0.265051
\(929\) −54.4937 −1.78788 −0.893940 0.448186i \(-0.852070\pi\)
−0.893940 + 0.448186i \(0.852070\pi\)
\(930\) 0 0
\(931\) −3.15544 −0.103415
\(932\) 46.5960 1.52630
\(933\) −2.23180 −0.0730659
\(934\) 72.1761 2.36167
\(935\) 0 0
\(936\) 25.2165 0.824226
\(937\) −37.1484 −1.21359 −0.606793 0.794860i \(-0.707544\pi\)
−0.606793 + 0.794860i \(0.707544\pi\)
\(938\) 36.7245 1.19910
\(939\) −0.493704 −0.0161114
\(940\) 0 0
\(941\) 32.3973 1.05612 0.528061 0.849206i \(-0.322919\pi\)
0.528061 + 0.849206i \(0.322919\pi\)
\(942\) −5.38284 −0.175382
\(943\) 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 \(-0.304516\pi\)
0.576249 + 0.817275i \(0.304516\pi\)
\(948\) −2.90059 −0.0942069
\(949\) −5.64752 −0.183326
\(950\) 0 0
\(951\) −14.3557 −0.465516
\(952\) 82.9272 2.68769
\(953\) −14.3411 −0.464553 −0.232277 0.972650i \(-0.574617\pi\)
−0.232277 + 0.972650i \(0.574617\pi\)
\(954\) 77.1009 2.49623
\(955\) 0 0
\(956\) −43.7788 −1.41591
\(957\) 13.3908 0.432864
\(958\) −10.9749 −0.354581
\(959\) 25.3125 0.817383
\(960\) 0 0
\(961\) −24.2435 −0.782048
\(962\) −21.5615 −0.695172
\(963\) 41.8458 1.34846
\(964\) −13.1122 −0.422316
\(965\) 0 0
\(966\) −10.8013 −0.347528
\(967\) −27.9351 −0.898331 −0.449165 0.893449i \(-0.648279\pi\)
−0.449165 + 0.893449i \(0.648279\pi\)
\(968\) −28.1463 −0.904657
\(969\) −3.11222 −0.0999788
\(970\) 0 0
\(971\) −42.5993 −1.36708 −0.683539 0.729914i \(-0.739560\pi\)
−0.683539 + 0.729914i \(0.739560\pi\)
\(972\) 48.9173 1.56902
\(973\) −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 \(-0.281900\pi\)
0.632813 + 0.774304i \(0.281900\pi\)
\(978\) 32.1180 1.02702
\(979\) 68.7114 2.19603
\(980\) 0 0
\(981\) −31.7564 −1.01390
\(982\) −94.1112 −3.00321
\(983\) −39.7689 −1.26843 −0.634215 0.773157i \(-0.718677\pi\)
−0.634215 + 0.773157i \(0.718677\pi\)
\(984\) 1.44551 0.0460811
\(985\) 0 0
\(986\) −84.1345 −2.67939
\(987\) −20.0398 −0.637874
\(988\) −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
\(993\) 4.29664 0.136350
\(994\) 105.737 3.35378
\(995\) 0 0
\(996\) −14.6442 −0.464018
\(997\) −11.6543 −0.369097 −0.184548 0.982823i \(-0.559082\pi\)
−0.184548 + 0.982823i \(0.559082\pi\)
\(998\) −11.4517 −0.362496
\(999\) −13.1987 −0.417587
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 475.2.a.j.1.1 6
3.2 odd 2 4275.2.a.br.1.6 6
4.3 odd 2 7600.2.a.ck.1.3 6
5.2 odd 4 95.2.b.b.39.1 6
5.3 odd 4 95.2.b.b.39.6 yes 6
5.4 even 2 inner 475.2.a.j.1.6 6
15.2 even 4 855.2.c.d.514.6 6
15.8 even 4 855.2.c.d.514.1 6
15.14 odd 2 4275.2.a.br.1.1 6
19.18 odd 2 9025.2.a.bx.1.6 6
20.3 even 4 1520.2.d.h.609.3 6
20.7 even 4 1520.2.d.h.609.4 6
20.19 odd 2 7600.2.a.ck.1.4 6
95.18 even 4 1805.2.b.e.1084.1 6
95.37 even 4 1805.2.b.e.1084.6 6
95.94 odd 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 5.2 odd 4
95.2.b.b.39.6 yes 6 5.3 odd 4
475.2.a.j.1.1 6 1.1 even 1 trivial
475.2.a.j.1.6 6 5.4 even 2 inner
855.2.c.d.514.1 6 15.8 even 4
855.2.c.d.514.6 6 15.2 even 4
1520.2.d.h.609.3 6 20.3 even 4
1520.2.d.h.609.4 6 20.7 even 4
1805.2.b.e.1084.1 6 95.18 even 4
1805.2.b.e.1084.6 6 95.37 even 4
4275.2.a.br.1.1 6 15.14 odd 2
4275.2.a.br.1.6 6 3.2 odd 2
7600.2.a.ck.1.3 6 4.3 odd 2
7600.2.a.ck.1.4 6 20.19 odd 2
9025.2.a.bx.1.1 6 95.94 odd 2
9025.2.a.bx.1.6 6 19.18 odd 2