Properties

Label 825.2.a.i.1.3
Level $825$
Weight $2$
Character 825.1
Self dual yes
Analytic conductor $6.588$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \( x^{3} - x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.12489\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.12489 q^{2} -1.00000 q^{3} +2.51514 q^{4} -2.12489 q^{6} +3.64002 q^{7} +1.09461 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.12489 q^{2} -1.00000 q^{3} +2.51514 q^{4} -2.12489 q^{6} +3.64002 q^{7} +1.09461 q^{8} +1.00000 q^{9} +1.00000 q^{11} -2.51514 q^{12} +1.51514 q^{13} +7.73463 q^{14} -2.70436 q^{16} +1.15516 q^{17} +2.12489 q^{18} +2.60975 q^{19} -3.64002 q^{21} +2.12489 q^{22} -5.73463 q^{23} -1.09461 q^{24} +3.21949 q^{26} -1.00000 q^{27} +9.15516 q^{28} +6.24977 q^{29} +5.51514 q^{31} -7.93567 q^{32} -1.00000 q^{33} +2.45459 q^{34} +2.51514 q^{36} +0.454586 q^{37} +5.54541 q^{38} -1.51514 q^{39} +4.12489 q^{41} -7.73463 q^{42} +11.7044 q^{43} +2.51514 q^{44} -12.1854 q^{46} -3.48486 q^{47} +2.70436 q^{48} +6.24977 q^{49} -1.15516 q^{51} +3.81078 q^{52} -12.5601 q^{53} -2.12489 q^{54} +3.98440 q^{56} -2.60975 q^{57} +13.2800 q^{58} -7.73463 q^{59} -12.0147 q^{61} +11.7190 q^{62} +3.64002 q^{63} -11.4537 q^{64} -2.12489 q^{66} -14.2645 q^{67} +2.90539 q^{68} +5.73463 q^{69} +8.51514 q^{71} +1.09461 q^{72} -9.21949 q^{73} +0.965943 q^{74} +6.56387 q^{76} +3.64002 q^{77} -3.21949 q^{78} +5.09461 q^{79} +1.00000 q^{81} +8.76491 q^{82} -14.7493 q^{83} -9.15516 q^{84} +24.8704 q^{86} -6.24977 q^{87} +1.09461 q^{88} -10.4995 q^{89} +5.51514 q^{91} -14.4234 q^{92} -5.51514 q^{93} -7.40493 q^{94} +7.93567 q^{96} -6.77959 q^{97} +13.2800 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} - 3 q^{3} + 8 q^{4} + 2 q^{6} + 3 q^{7} - 6 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} - 3 q^{3} + 8 q^{4} + 2 q^{6} + 3 q^{7} - 6 q^{8} + 3 q^{9} + 3 q^{11} - 8 q^{12} + 5 q^{13} + 6 q^{14} + 10 q^{16} - 4 q^{17} - 2 q^{18} - q^{19} - 3 q^{21} - 2 q^{22} + 6 q^{24} - 8 q^{26} - 3 q^{27} + 20 q^{28} + 2 q^{29} + 17 q^{31} - 34 q^{32} - 3 q^{33} + 6 q^{34} + 8 q^{36} + 18 q^{38} - 5 q^{39} + 4 q^{41} - 6 q^{42} + 17 q^{43} + 8 q^{44} - 30 q^{46} - 10 q^{47} - 10 q^{48} + 2 q^{49} + 4 q^{51} + 30 q^{52} - 6 q^{53} + 2 q^{54} - 22 q^{56} + q^{57} + 24 q^{58} - 6 q^{59} - 3 q^{61} - 16 q^{62} + 3 q^{63} + 34 q^{64} + 2 q^{66} + 7 q^{67} + 18 q^{68} + 26 q^{71} - 6 q^{72} - 10 q^{73} + 14 q^{74} - 24 q^{76} + 3 q^{77} + 8 q^{78} + 6 q^{79} + 3 q^{81} + 10 q^{82} + 6 q^{83} - 20 q^{84} + 28 q^{86} - 2 q^{87} - 6 q^{88} + 2 q^{89} + 17 q^{91} + 26 q^{92} - 17 q^{93} + 2 q^{94} + 34 q^{96} + 29 q^{97} + 24 q^{98} + 3 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.12489 1.50252 0.751260 0.660006i \(-0.229446\pi\)
0.751260 + 0.660006i \(0.229446\pi\)
\(3\) −1.00000 −0.577350
\(4\) 2.51514 1.25757
\(5\) 0 0
\(6\) −2.12489 −0.867481
\(7\) 3.64002 1.37580 0.687900 0.725806i \(-0.258533\pi\)
0.687900 + 0.725806i \(0.258533\pi\)
\(8\) 1.09461 0.387003
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) −2.51514 −0.726058
\(13\) 1.51514 0.420224 0.210112 0.977677i \(-0.432617\pi\)
0.210112 + 0.977677i \(0.432617\pi\)
\(14\) 7.73463 2.06717
\(15\) 0 0
\(16\) −2.70436 −0.676089
\(17\) 1.15516 0.280168 0.140084 0.990140i \(-0.455263\pi\)
0.140084 + 0.990140i \(0.455263\pi\)
\(18\) 2.12489 0.500840
\(19\) 2.60975 0.598717 0.299359 0.954141i \(-0.403227\pi\)
0.299359 + 0.954141i \(0.403227\pi\)
\(20\) 0 0
\(21\) −3.64002 −0.794318
\(22\) 2.12489 0.453027
\(23\) −5.73463 −1.19575 −0.597877 0.801588i \(-0.703989\pi\)
−0.597877 + 0.801588i \(0.703989\pi\)
\(24\) −1.09461 −0.223436
\(25\) 0 0
\(26\) 3.21949 0.631395
\(27\) −1.00000 −0.192450
\(28\) 9.15516 1.73016
\(29\) 6.24977 1.16055 0.580277 0.814419i \(-0.302944\pi\)
0.580277 + 0.814419i \(0.302944\pi\)
\(30\) 0 0
\(31\) 5.51514 0.990548 0.495274 0.868737i \(-0.335068\pi\)
0.495274 + 0.868737i \(0.335068\pi\)
\(32\) −7.93567 −1.40284
\(33\) −1.00000 −0.174078
\(34\) 2.45459 0.420958
\(35\) 0 0
\(36\) 2.51514 0.419190
\(37\) 0.454586 0.0747335 0.0373667 0.999302i \(-0.488103\pi\)
0.0373667 + 0.999302i \(0.488103\pi\)
\(38\) 5.54541 0.899585
\(39\) −1.51514 −0.242616
\(40\) 0 0
\(41\) 4.12489 0.644199 0.322099 0.946706i \(-0.395611\pi\)
0.322099 + 0.946706i \(0.395611\pi\)
\(42\) −7.73463 −1.19348
\(43\) 11.7044 1.78490 0.892449 0.451149i \(-0.148986\pi\)
0.892449 + 0.451149i \(0.148986\pi\)
\(44\) 2.51514 0.379171
\(45\) 0 0
\(46\) −12.1854 −1.79664
\(47\) −3.48486 −0.508319 −0.254160 0.967162i \(-0.581799\pi\)
−0.254160 + 0.967162i \(0.581799\pi\)
\(48\) 2.70436 0.390340
\(49\) 6.24977 0.892824
\(50\) 0 0
\(51\) −1.15516 −0.161755
\(52\) 3.81078 0.528460
\(53\) −12.5601 −1.72526 −0.862631 0.505834i \(-0.831185\pi\)
−0.862631 + 0.505834i \(0.831185\pi\)
\(54\) −2.12489 −0.289160
\(55\) 0 0
\(56\) 3.98440 0.532438
\(57\) −2.60975 −0.345669
\(58\) 13.2800 1.74376
\(59\) −7.73463 −1.00696 −0.503482 0.864006i \(-0.667948\pi\)
−0.503482 + 0.864006i \(0.667948\pi\)
\(60\) 0 0
\(61\) −12.0147 −1.53832 −0.769161 0.639055i \(-0.779326\pi\)
−0.769161 + 0.639055i \(0.779326\pi\)
\(62\) 11.7190 1.48832
\(63\) 3.64002 0.458600
\(64\) −11.4537 −1.43171
\(65\) 0 0
\(66\) −2.12489 −0.261555
\(67\) −14.2645 −1.74268 −0.871340 0.490680i \(-0.836749\pi\)
−0.871340 + 0.490680i \(0.836749\pi\)
\(68\) 2.90539 0.352330
\(69\) 5.73463 0.690369
\(70\) 0 0
\(71\) 8.51514 1.01056 0.505280 0.862955i \(-0.331389\pi\)
0.505280 + 0.862955i \(0.331389\pi\)
\(72\) 1.09461 0.129001
\(73\) −9.21949 −1.07906 −0.539530 0.841966i \(-0.681398\pi\)
−0.539530 + 0.841966i \(0.681398\pi\)
\(74\) 0.965943 0.112289
\(75\) 0 0
\(76\) 6.56387 0.752928
\(77\) 3.64002 0.414819
\(78\) −3.21949 −0.364536
\(79\) 5.09461 0.573188 0.286594 0.958052i \(-0.407477\pi\)
0.286594 + 0.958052i \(0.407477\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 8.76491 0.967922
\(83\) −14.7493 −1.61895 −0.809474 0.587156i \(-0.800247\pi\)
−0.809474 + 0.587156i \(0.800247\pi\)
\(84\) −9.15516 −0.998910
\(85\) 0 0
\(86\) 24.8704 2.68185
\(87\) −6.24977 −0.670046
\(88\) 1.09461 0.116686
\(89\) −10.4995 −1.11295 −0.556475 0.830865i \(-0.687846\pi\)
−0.556475 + 0.830865i \(0.687846\pi\)
\(90\) 0 0
\(91\) 5.51514 0.578144
\(92\) −14.4234 −1.50374
\(93\) −5.51514 −0.571893
\(94\) −7.40493 −0.763760
\(95\) 0 0
\(96\) 7.93567 0.809931
\(97\) −6.77959 −0.688363 −0.344181 0.938903i \(-0.611844\pi\)
−0.344181 + 0.938903i \(0.611844\pi\)
\(98\) 13.2800 1.34149
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 7.40493 0.736818 0.368409 0.929664i \(-0.379903\pi\)
0.368409 + 0.929664i \(0.379903\pi\)
\(102\) −2.45459 −0.243040
\(103\) −16.4995 −1.62575 −0.812874 0.582439i \(-0.802098\pi\)
−0.812874 + 0.582439i \(0.802098\pi\)
\(104\) 1.65848 0.162628
\(105\) 0 0
\(106\) −26.6888 −2.59224
\(107\) 3.93945 0.380841 0.190420 0.981703i \(-0.439015\pi\)
0.190420 + 0.981703i \(0.439015\pi\)
\(108\) −2.51514 −0.242019
\(109\) 6.73463 0.645061 0.322530 0.946559i \(-0.395467\pi\)
0.322530 + 0.946559i \(0.395467\pi\)
\(110\) 0 0
\(111\) −0.454586 −0.0431474
\(112\) −9.84392 −0.930163
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −5.54541 −0.519376
\(115\) 0 0
\(116\) 15.7190 1.45948
\(117\) 1.51514 0.140075
\(118\) −16.4352 −1.51298
\(119\) 4.20482 0.385455
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −25.5298 −2.31136
\(123\) −4.12489 −0.371928
\(124\) 13.8713 1.24568
\(125\) 0 0
\(126\) 7.73463 0.689056
\(127\) 8.06433 0.715594 0.357797 0.933799i \(-0.383528\pi\)
0.357797 + 0.933799i \(0.383528\pi\)
\(128\) −8.46640 −0.748331
\(129\) −11.7044 −1.03051
\(130\) 0 0
\(131\) −12.8099 −1.11920 −0.559602 0.828762i \(-0.689046\pi\)
−0.559602 + 0.828762i \(0.689046\pi\)
\(132\) −2.51514 −0.218915
\(133\) 9.49954 0.823715
\(134\) −30.3103 −2.61841
\(135\) 0 0
\(136\) 1.26445 0.108426
\(137\) 22.8099 1.94878 0.974389 0.224868i \(-0.0721952\pi\)
0.974389 + 0.224868i \(0.0721952\pi\)
\(138\) 12.1854 1.03729
\(139\) 7.59037 0.643807 0.321903 0.946773i \(-0.395677\pi\)
0.321903 + 0.946773i \(0.395677\pi\)
\(140\) 0 0
\(141\) 3.48486 0.293478
\(142\) 18.0937 1.51839
\(143\) 1.51514 0.126702
\(144\) −2.70436 −0.225363
\(145\) 0 0
\(146\) −19.5904 −1.62131
\(147\) −6.24977 −0.515472
\(148\) 1.14335 0.0939825
\(149\) −1.81456 −0.148655 −0.0743274 0.997234i \(-0.523681\pi\)
−0.0743274 + 0.997234i \(0.523681\pi\)
\(150\) 0 0
\(151\) 24.3250 1.97954 0.989770 0.142670i \(-0.0455687\pi\)
0.989770 + 0.142670i \(0.0455687\pi\)
\(152\) 2.85665 0.231705
\(153\) 1.15516 0.0933893
\(154\) 7.73463 0.623274
\(155\) 0 0
\(156\) −3.81078 −0.305107
\(157\) 9.76491 0.779325 0.389662 0.920958i \(-0.372592\pi\)
0.389662 + 0.920958i \(0.372592\pi\)
\(158\) 10.8255 0.861227
\(159\) 12.5601 0.996080
\(160\) 0 0
\(161\) −20.8742 −1.64512
\(162\) 2.12489 0.166947
\(163\) −6.98440 −0.547061 −0.273530 0.961863i \(-0.588191\pi\)
−0.273530 + 0.961863i \(0.588191\pi\)
\(164\) 10.3747 0.810125
\(165\) 0 0
\(166\) −31.3406 −2.43250
\(167\) −6.31032 −0.488307 −0.244154 0.969737i \(-0.578510\pi\)
−0.244154 + 0.969737i \(0.578510\pi\)
\(168\) −3.98440 −0.307403
\(169\) −10.7044 −0.823412
\(170\) 0 0
\(171\) 2.60975 0.199572
\(172\) 29.4381 2.24463
\(173\) −12.8448 −0.976575 −0.488287 0.872683i \(-0.662378\pi\)
−0.488287 + 0.872683i \(0.662378\pi\)
\(174\) −13.2800 −1.00676
\(175\) 0 0
\(176\) −2.70436 −0.203849
\(177\) 7.73463 0.581371
\(178\) −22.3103 −1.67223
\(179\) 13.4849 1.00791 0.503953 0.863731i \(-0.331878\pi\)
0.503953 + 0.863731i \(0.331878\pi\)
\(180\) 0 0
\(181\) −23.0899 −1.71626 −0.858130 0.513433i \(-0.828374\pi\)
−0.858130 + 0.513433i \(0.828374\pi\)
\(182\) 11.7190 0.868673
\(183\) 12.0147 0.888151
\(184\) −6.27718 −0.462760
\(185\) 0 0
\(186\) −11.7190 −0.859281
\(187\) 1.15516 0.0844738
\(188\) −8.76491 −0.639247
\(189\) −3.64002 −0.264773
\(190\) 0 0
\(191\) −7.98440 −0.577731 −0.288866 0.957370i \(-0.593278\pi\)
−0.288866 + 0.957370i \(0.593278\pi\)
\(192\) 11.4537 0.826597
\(193\) 11.7649 0.846857 0.423428 0.905930i \(-0.360827\pi\)
0.423428 + 0.905930i \(0.360827\pi\)
\(194\) −14.4058 −1.03428
\(195\) 0 0
\(196\) 15.7190 1.12279
\(197\) −3.81456 −0.271776 −0.135888 0.990724i \(-0.543389\pi\)
−0.135888 + 0.990724i \(0.543389\pi\)
\(198\) 2.12489 0.151009
\(199\) −12.0752 −0.855990 −0.427995 0.903781i \(-0.640780\pi\)
−0.427995 + 0.903781i \(0.640780\pi\)
\(200\) 0 0
\(201\) 14.2645 1.00614
\(202\) 15.7346 1.10708
\(203\) 22.7493 1.59669
\(204\) −2.90539 −0.203418
\(205\) 0 0
\(206\) −35.0596 −2.44272
\(207\) −5.73463 −0.398585
\(208\) −4.09747 −0.284109
\(209\) 2.60975 0.180520
\(210\) 0 0
\(211\) 10.2645 0.706634 0.353317 0.935504i \(-0.385054\pi\)
0.353317 + 0.935504i \(0.385054\pi\)
\(212\) −31.5904 −2.16964
\(213\) −8.51514 −0.583448
\(214\) 8.37088 0.572221
\(215\) 0 0
\(216\) −1.09461 −0.0744787
\(217\) 20.0752 1.36280
\(218\) 14.3103 0.969217
\(219\) 9.21949 0.622996
\(220\) 0 0
\(221\) 1.75023 0.117733
\(222\) −0.965943 −0.0648298
\(223\) 12.9239 0.865445 0.432723 0.901527i \(-0.357553\pi\)
0.432723 + 0.901527i \(0.357553\pi\)
\(224\) −28.8860 −1.93003
\(225\) 0 0
\(226\) −12.7493 −0.848072
\(227\) 22.8099 1.51394 0.756972 0.653447i \(-0.226678\pi\)
0.756972 + 0.653447i \(0.226678\pi\)
\(228\) −6.56387 −0.434703
\(229\) 14.7796 0.976663 0.488331 0.872658i \(-0.337606\pi\)
0.488331 + 0.872658i \(0.337606\pi\)
\(230\) 0 0
\(231\) −3.64002 −0.239496
\(232\) 6.84106 0.449137
\(233\) 4.96594 0.325330 0.162665 0.986681i \(-0.447991\pi\)
0.162665 + 0.986681i \(0.447991\pi\)
\(234\) 3.21949 0.210465
\(235\) 0 0
\(236\) −19.4537 −1.26633
\(237\) −5.09461 −0.330930
\(238\) 8.93475 0.579154
\(239\) −14.9991 −0.970210 −0.485105 0.874456i \(-0.661219\pi\)
−0.485105 + 0.874456i \(0.661219\pi\)
\(240\) 0 0
\(241\) 5.04496 0.324974 0.162487 0.986711i \(-0.448048\pi\)
0.162487 + 0.986711i \(0.448048\pi\)
\(242\) 2.12489 0.136593
\(243\) −1.00000 −0.0641500
\(244\) −30.2186 −1.93455
\(245\) 0 0
\(246\) −8.76491 −0.558830
\(247\) 3.95413 0.251595
\(248\) 6.03692 0.383345
\(249\) 14.7493 0.934700
\(250\) 0 0
\(251\) −3.03028 −0.191269 −0.0956347 0.995417i \(-0.530488\pi\)
−0.0956347 + 0.995417i \(0.530488\pi\)
\(252\) 9.15516 0.576721
\(253\) −5.73463 −0.360533
\(254\) 17.1358 1.07519
\(255\) 0 0
\(256\) 4.91721 0.307325
\(257\) −13.6509 −0.851521 −0.425761 0.904836i \(-0.639993\pi\)
−0.425761 + 0.904836i \(0.639993\pi\)
\(258\) −24.8704 −1.54836
\(259\) 1.65470 0.102818
\(260\) 0 0
\(261\) 6.24977 0.386851
\(262\) −27.2195 −1.68163
\(263\) 12.5601 0.774489 0.387244 0.921977i \(-0.373427\pi\)
0.387244 + 0.921977i \(0.373427\pi\)
\(264\) −1.09461 −0.0673685
\(265\) 0 0
\(266\) 20.1854 1.23765
\(267\) 10.4995 0.642562
\(268\) −35.8771 −2.19154
\(269\) −24.6888 −1.50530 −0.752650 0.658421i \(-0.771225\pi\)
−0.752650 + 0.658421i \(0.771225\pi\)
\(270\) 0 0
\(271\) 7.56479 0.459528 0.229764 0.973246i \(-0.426205\pi\)
0.229764 + 0.973246i \(0.426205\pi\)
\(272\) −3.12397 −0.189418
\(273\) −5.51514 −0.333791
\(274\) 48.4683 2.92808
\(275\) 0 0
\(276\) 14.4234 0.868186
\(277\) 1.92477 0.115648 0.0578241 0.998327i \(-0.481584\pi\)
0.0578241 + 0.998327i \(0.481584\pi\)
\(278\) 16.1287 0.967333
\(279\) 5.51514 0.330183
\(280\) 0 0
\(281\) −1.87511 −0.111860 −0.0559300 0.998435i \(-0.517812\pi\)
−0.0559300 + 0.998435i \(0.517812\pi\)
\(282\) 7.40493 0.440957
\(283\) 30.1396 1.79161 0.895806 0.444446i \(-0.146599\pi\)
0.895806 + 0.444446i \(0.146599\pi\)
\(284\) 21.4167 1.27085
\(285\) 0 0
\(286\) 3.21949 0.190373
\(287\) 15.0147 0.886289
\(288\) −7.93567 −0.467614
\(289\) −15.6656 −0.921506
\(290\) 0 0
\(291\) 6.77959 0.397427
\(292\) −23.1883 −1.35699
\(293\) −29.1552 −1.70326 −0.851631 0.524141i \(-0.824386\pi\)
−0.851631 + 0.524141i \(0.824386\pi\)
\(294\) −13.2800 −0.774508
\(295\) 0 0
\(296\) 0.497594 0.0289221
\(297\) −1.00000 −0.0580259
\(298\) −3.85574 −0.223357
\(299\) −8.68876 −0.502484
\(300\) 0 0
\(301\) 42.6041 2.45566
\(302\) 51.6878 2.97430
\(303\) −7.40493 −0.425402
\(304\) −7.05769 −0.404786
\(305\) 0 0
\(306\) 2.45459 0.140319
\(307\) 27.8548 1.58976 0.794879 0.606768i \(-0.207534\pi\)
0.794879 + 0.606768i \(0.207534\pi\)
\(308\) 9.15516 0.521664
\(309\) 16.4995 0.938626
\(310\) 0 0
\(311\) 23.9083 1.35571 0.677856 0.735194i \(-0.262909\pi\)
0.677856 + 0.735194i \(0.262909\pi\)
\(312\) −1.65848 −0.0938932
\(313\) 28.3094 1.60014 0.800071 0.599905i \(-0.204795\pi\)
0.800071 + 0.599905i \(0.204795\pi\)
\(314\) 20.7493 1.17095
\(315\) 0 0
\(316\) 12.8136 0.720824
\(317\) −8.80986 −0.494811 −0.247406 0.968912i \(-0.579578\pi\)
−0.247406 + 0.968912i \(0.579578\pi\)
\(318\) 26.6888 1.49663
\(319\) 6.24977 0.349920
\(320\) 0 0
\(321\) −3.93945 −0.219879
\(322\) −44.3553 −2.47182
\(323\) 3.01468 0.167741
\(324\) 2.51514 0.139730
\(325\) 0 0
\(326\) −14.8411 −0.821970
\(327\) −6.73463 −0.372426
\(328\) 4.51514 0.249307
\(329\) −12.6850 −0.699346
\(330\) 0 0
\(331\) 32.2498 1.77261 0.886304 0.463104i \(-0.153264\pi\)
0.886304 + 0.463104i \(0.153264\pi\)
\(332\) −37.0966 −2.03594
\(333\) 0.454586 0.0249112
\(334\) −13.4087 −0.733692
\(335\) 0 0
\(336\) 9.84392 0.537030
\(337\) 28.9844 1.57888 0.789441 0.613827i \(-0.210371\pi\)
0.789441 + 0.613827i \(0.210371\pi\)
\(338\) −22.7455 −1.23719
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 5.51514 0.298661
\(342\) 5.54541 0.299862
\(343\) −2.73085 −0.147452
\(344\) 12.8117 0.690760
\(345\) 0 0
\(346\) −27.2938 −1.46732
\(347\) −35.7190 −1.91750 −0.958749 0.284253i \(-0.908254\pi\)
−0.958749 + 0.284253i \(0.908254\pi\)
\(348\) −15.7190 −0.842629
\(349\) −23.2800 −1.24615 −0.623076 0.782161i \(-0.714117\pi\)
−0.623076 + 0.782161i \(0.714117\pi\)
\(350\) 0 0
\(351\) −1.51514 −0.0808721
\(352\) −7.93567 −0.422972
\(353\) −9.75023 −0.518952 −0.259476 0.965750i \(-0.583550\pi\)
−0.259476 + 0.965750i \(0.583550\pi\)
\(354\) 16.4352 0.873521
\(355\) 0 0
\(356\) −26.4078 −1.39961
\(357\) −4.20482 −0.222542
\(358\) 28.6538 1.51440
\(359\) 33.9007 1.78921 0.894605 0.446858i \(-0.147457\pi\)
0.894605 + 0.446858i \(0.147457\pi\)
\(360\) 0 0
\(361\) −12.1892 −0.641538
\(362\) −49.0634 −2.57872
\(363\) −1.00000 −0.0524864
\(364\) 13.8713 0.727055
\(365\) 0 0
\(366\) 25.5298 1.33446
\(367\) 1.88601 0.0984491 0.0492245 0.998788i \(-0.484325\pi\)
0.0492245 + 0.998788i \(0.484325\pi\)
\(368\) 15.5085 0.808436
\(369\) 4.12489 0.214733
\(370\) 0 0
\(371\) −45.7190 −2.37361
\(372\) −13.8713 −0.719195
\(373\) −16.3250 −0.845277 −0.422638 0.906298i \(-0.638896\pi\)
−0.422638 + 0.906298i \(0.638896\pi\)
\(374\) 2.45459 0.126924
\(375\) 0 0
\(376\) −3.81456 −0.196721
\(377\) 9.46927 0.487692
\(378\) −7.73463 −0.397827
\(379\) −26.0440 −1.33779 −0.668896 0.743356i \(-0.733233\pi\)
−0.668896 + 0.743356i \(0.733233\pi\)
\(380\) 0 0
\(381\) −8.06433 −0.413148
\(382\) −16.9659 −0.868053
\(383\) 12.4702 0.637197 0.318598 0.947890i \(-0.396788\pi\)
0.318598 + 0.947890i \(0.396788\pi\)
\(384\) 8.46640 0.432049
\(385\) 0 0
\(386\) 24.9991 1.27242
\(387\) 11.7044 0.594966
\(388\) −17.0516 −0.865664
\(389\) −18.0899 −0.917195 −0.458597 0.888644i \(-0.651648\pi\)
−0.458597 + 0.888644i \(0.651648\pi\)
\(390\) 0 0
\(391\) −6.62443 −0.335012
\(392\) 6.84106 0.345526
\(393\) 12.8099 0.646172
\(394\) −8.10551 −0.408350
\(395\) 0 0
\(396\) 2.51514 0.126390
\(397\) −15.2342 −0.764581 −0.382291 0.924042i \(-0.624865\pi\)
−0.382291 + 0.924042i \(0.624865\pi\)
\(398\) −25.6585 −1.28614
\(399\) −9.49954 −0.475572
\(400\) 0 0
\(401\) 2.74931 0.137294 0.0686471 0.997641i \(-0.478132\pi\)
0.0686471 + 0.997641i \(0.478132\pi\)
\(402\) 30.3103 1.51174
\(403\) 8.35620 0.416252
\(404\) 18.6244 0.926600
\(405\) 0 0
\(406\) 48.3397 2.39906
\(407\) 0.454586 0.0225330
\(408\) −1.26445 −0.0625996
\(409\) −3.98532 −0.197061 −0.0985307 0.995134i \(-0.531414\pi\)
−0.0985307 + 0.995134i \(0.531414\pi\)
\(410\) 0 0
\(411\) −22.8099 −1.12513
\(412\) −41.4986 −2.04449
\(413\) −28.1542 −1.38538
\(414\) −12.1854 −0.598882
\(415\) 0 0
\(416\) −12.0236 −0.589507
\(417\) −7.59037 −0.371702
\(418\) 5.54541 0.271235
\(419\) −5.13578 −0.250899 −0.125450 0.992100i \(-0.540037\pi\)
−0.125450 + 0.992100i \(0.540037\pi\)
\(420\) 0 0
\(421\) −8.94657 −0.436029 −0.218014 0.975946i \(-0.569958\pi\)
−0.218014 + 0.975946i \(0.569958\pi\)
\(422\) 21.8108 1.06173
\(423\) −3.48486 −0.169440
\(424\) −13.7484 −0.667681
\(425\) 0 0
\(426\) −18.0937 −0.876642
\(427\) −43.7337 −2.11642
\(428\) 9.90826 0.478934
\(429\) −1.51514 −0.0731516
\(430\) 0 0
\(431\) 22.7493 1.09580 0.547898 0.836545i \(-0.315428\pi\)
0.547898 + 0.836545i \(0.315428\pi\)
\(432\) 2.70436 0.130113
\(433\) 7.58325 0.364428 0.182214 0.983259i \(-0.441674\pi\)
0.182214 + 0.983259i \(0.441674\pi\)
\(434\) 42.6576 2.04763
\(435\) 0 0
\(436\) 16.9385 0.811209
\(437\) −14.9659 −0.715918
\(438\) 19.5904 0.936064
\(439\) 17.3903 0.829991 0.414996 0.909823i \(-0.363783\pi\)
0.414996 + 0.909823i \(0.363783\pi\)
\(440\) 0 0
\(441\) 6.24977 0.297608
\(442\) 3.71904 0.176897
\(443\) 10.6438 0.505702 0.252851 0.967505i \(-0.418632\pi\)
0.252851 + 0.967505i \(0.418632\pi\)
\(444\) −1.14335 −0.0542608
\(445\) 0 0
\(446\) 27.4617 1.30035
\(447\) 1.81456 0.0858259
\(448\) −41.6916 −1.96974
\(449\) −26.9310 −1.27095 −0.635476 0.772121i \(-0.719196\pi\)
−0.635476 + 0.772121i \(0.719196\pi\)
\(450\) 0 0
\(451\) 4.12489 0.194233
\(452\) −15.0908 −0.709813
\(453\) −24.3250 −1.14289
\(454\) 48.4683 2.27473
\(455\) 0 0
\(456\) −2.85665 −0.133775
\(457\) −15.7796 −0.738138 −0.369069 0.929402i \(-0.620323\pi\)
−0.369069 + 0.929402i \(0.620323\pi\)
\(458\) 31.4049 1.46746
\(459\) −1.15516 −0.0539183
\(460\) 0 0
\(461\) 8.18922 0.381410 0.190705 0.981647i \(-0.438923\pi\)
0.190705 + 0.981647i \(0.438923\pi\)
\(462\) −7.73463 −0.359848
\(463\) 16.0899 0.747762 0.373881 0.927477i \(-0.378027\pi\)
0.373881 + 0.927477i \(0.378027\pi\)
\(464\) −16.9016 −0.784638
\(465\) 0 0
\(466\) 10.5521 0.488815
\(467\) −29.4693 −1.36367 −0.681837 0.731504i \(-0.738819\pi\)
−0.681837 + 0.731504i \(0.738819\pi\)
\(468\) 3.81078 0.176153
\(469\) −51.9229 −2.39758
\(470\) 0 0
\(471\) −9.76491 −0.449943
\(472\) −8.46640 −0.389698
\(473\) 11.7044 0.538167
\(474\) −10.8255 −0.497230
\(475\) 0 0
\(476\) 10.5757 0.484736
\(477\) −12.5601 −0.575087
\(478\) −31.8713 −1.45776
\(479\) 32.2186 1.47210 0.736052 0.676925i \(-0.236688\pi\)
0.736052 + 0.676925i \(0.236688\pi\)
\(480\) 0 0
\(481\) 0.688760 0.0314048
\(482\) 10.7200 0.488280
\(483\) 20.8742 0.949809
\(484\) 2.51514 0.114324
\(485\) 0 0
\(486\) −2.12489 −0.0963868
\(487\) −35.8936 −1.62649 −0.813247 0.581919i \(-0.802302\pi\)
−0.813247 + 0.581919i \(0.802302\pi\)
\(488\) −13.1514 −0.595335
\(489\) 6.98440 0.315846
\(490\) 0 0
\(491\) −7.15894 −0.323079 −0.161539 0.986866i \(-0.551646\pi\)
−0.161539 + 0.986866i \(0.551646\pi\)
\(492\) −10.3747 −0.467726
\(493\) 7.21949 0.325150
\(494\) 8.40207 0.378027
\(495\) 0 0
\(496\) −14.9149 −0.669699
\(497\) 30.9953 1.39033
\(498\) 31.3406 1.40441
\(499\) 27.0743 1.21201 0.606006 0.795460i \(-0.292771\pi\)
0.606006 + 0.795460i \(0.292771\pi\)
\(500\) 0 0
\(501\) 6.31032 0.281924
\(502\) −6.43899 −0.287386
\(503\) 26.9991 1.20383 0.601915 0.798560i \(-0.294405\pi\)
0.601915 + 0.798560i \(0.294405\pi\)
\(504\) 3.98440 0.177479
\(505\) 0 0
\(506\) −12.1854 −0.541709
\(507\) 10.7044 0.475397
\(508\) 20.2829 0.899909
\(509\) 15.5904 0.691031 0.345515 0.938413i \(-0.387704\pi\)
0.345515 + 0.938413i \(0.387704\pi\)
\(510\) 0 0
\(511\) −33.5592 −1.48457
\(512\) 27.3813 1.21009
\(513\) −2.60975 −0.115223
\(514\) −29.0066 −1.27943
\(515\) 0 0
\(516\) −29.4381 −1.29594
\(517\) −3.48486 −0.153264
\(518\) 3.51605 0.154487
\(519\) 12.8448 0.563826
\(520\) 0 0
\(521\) 11.1589 0.488882 0.244441 0.969664i \(-0.421396\pi\)
0.244441 + 0.969664i \(0.421396\pi\)
\(522\) 13.2800 0.581252
\(523\) −10.5786 −0.462568 −0.231284 0.972886i \(-0.574293\pi\)
−0.231284 + 0.972886i \(0.574293\pi\)
\(524\) −32.2186 −1.40748
\(525\) 0 0
\(526\) 26.6888 1.16369
\(527\) 6.37088 0.277520
\(528\) 2.70436 0.117692
\(529\) 9.88601 0.429827
\(530\) 0 0
\(531\) −7.73463 −0.335654
\(532\) 23.8927 1.03588
\(533\) 6.24977 0.270708
\(534\) 22.3103 0.965462
\(535\) 0 0
\(536\) −15.6140 −0.674422
\(537\) −13.4849 −0.581915
\(538\) −52.4608 −2.26175
\(539\) 6.24977 0.269197
\(540\) 0 0
\(541\) −11.2947 −0.485598 −0.242799 0.970077i \(-0.578066\pi\)
−0.242799 + 0.970077i \(0.578066\pi\)
\(542\) 16.0743 0.690451
\(543\) 23.0899 0.990883
\(544\) −9.16698 −0.393031
\(545\) 0 0
\(546\) −11.7190 −0.501528
\(547\) 6.09369 0.260547 0.130274 0.991478i \(-0.458414\pi\)
0.130274 + 0.991478i \(0.458414\pi\)
\(548\) 57.3700 2.45072
\(549\) −12.0147 −0.512774
\(550\) 0 0
\(551\) 16.3103 0.694843
\(552\) 6.27718 0.267175
\(553\) 18.5445 0.788592
\(554\) 4.08991 0.173764
\(555\) 0 0
\(556\) 19.0908 0.809631
\(557\) −5.90069 −0.250020 −0.125010 0.992155i \(-0.539896\pi\)
−0.125010 + 0.992155i \(0.539896\pi\)
\(558\) 11.7190 0.496106
\(559\) 17.7337 0.750056
\(560\) 0 0
\(561\) −1.15516 −0.0487710
\(562\) −3.98440 −0.168072
\(563\) 3.03028 0.127711 0.0638555 0.997959i \(-0.479660\pi\)
0.0638555 + 0.997959i \(0.479660\pi\)
\(564\) 8.76491 0.369069
\(565\) 0 0
\(566\) 64.0431 2.69193
\(567\) 3.64002 0.152867
\(568\) 9.32075 0.391090
\(569\) 13.4049 0.561964 0.280982 0.959713i \(-0.409340\pi\)
0.280982 + 0.959713i \(0.409340\pi\)
\(570\) 0 0
\(571\) 26.8851 1.12511 0.562553 0.826761i \(-0.309819\pi\)
0.562553 + 0.826761i \(0.309819\pi\)
\(572\) 3.81078 0.159337
\(573\) 7.98440 0.333553
\(574\) 31.9045 1.33167
\(575\) 0 0
\(576\) −11.4537 −0.477236
\(577\) 2.03028 0.0845215 0.0422607 0.999107i \(-0.486544\pi\)
0.0422607 + 0.999107i \(0.486544\pi\)
\(578\) −33.2876 −1.38458
\(579\) −11.7649 −0.488933
\(580\) 0 0
\(581\) −53.6878 −2.22735
\(582\) 14.4058 0.597142
\(583\) −12.5601 −0.520186
\(584\) −10.0917 −0.417599
\(585\) 0 0
\(586\) −61.9514 −2.55919
\(587\) 21.8245 0.900795 0.450398 0.892828i \(-0.351282\pi\)
0.450398 + 0.892828i \(0.351282\pi\)
\(588\) −15.7190 −0.648242
\(589\) 14.3931 0.593058
\(590\) 0 0
\(591\) 3.81456 0.156910
\(592\) −1.22936 −0.0505265
\(593\) 8.06811 0.331318 0.165659 0.986183i \(-0.447025\pi\)
0.165659 + 0.986183i \(0.447025\pi\)
\(594\) −2.12489 −0.0871851
\(595\) 0 0
\(596\) −4.56387 −0.186944
\(597\) 12.0752 0.494206
\(598\) −18.4626 −0.754993
\(599\) 7.61353 0.311080 0.155540 0.987830i \(-0.450288\pi\)
0.155540 + 0.987830i \(0.450288\pi\)
\(600\) 0 0
\(601\) 3.57569 0.145855 0.0729277 0.997337i \(-0.476766\pi\)
0.0729277 + 0.997337i \(0.476766\pi\)
\(602\) 90.5289 3.68968
\(603\) −14.2645 −0.580893
\(604\) 61.1807 2.48941
\(605\) 0 0
\(606\) −15.7346 −0.639176
\(607\) 17.5298 0.711513 0.355757 0.934579i \(-0.384223\pi\)
0.355757 + 0.934579i \(0.384223\pi\)
\(608\) −20.7101 −0.839905
\(609\) −22.7493 −0.921849
\(610\) 0 0
\(611\) −5.28005 −0.213608
\(612\) 2.90539 0.117443
\(613\) 12.5601 0.507297 0.253649 0.967296i \(-0.418369\pi\)
0.253649 + 0.967296i \(0.418369\pi\)
\(614\) 59.1883 2.38865
\(615\) 0 0
\(616\) 3.98440 0.160536
\(617\) 15.9612 0.642576 0.321288 0.946982i \(-0.395884\pi\)
0.321288 + 0.946982i \(0.395884\pi\)
\(618\) 35.0596 1.41031
\(619\) −9.23417 −0.371153 −0.185576 0.982630i \(-0.559415\pi\)
−0.185576 + 0.982630i \(0.559415\pi\)
\(620\) 0 0
\(621\) 5.73463 0.230123
\(622\) 50.8023 2.03699
\(623\) −38.2186 −1.53119
\(624\) 4.09747 0.164030
\(625\) 0 0
\(626\) 60.1542 2.40425
\(627\) −2.60975 −0.104223
\(628\) 24.5601 0.980054
\(629\) 0.525120 0.0209379
\(630\) 0 0
\(631\) 29.2342 1.16379 0.581897 0.813262i \(-0.302311\pi\)
0.581897 + 0.813262i \(0.302311\pi\)
\(632\) 5.57661 0.221826
\(633\) −10.2645 −0.407975
\(634\) −18.7200 −0.743464
\(635\) 0 0
\(636\) 31.5904 1.25264
\(637\) 9.46927 0.375186
\(638\) 13.2800 0.525762
\(639\) 8.51514 0.336854
\(640\) 0 0
\(641\) 13.9612 0.551436 0.275718 0.961239i \(-0.411084\pi\)
0.275718 + 0.961239i \(0.411084\pi\)
\(642\) −8.37088 −0.330372
\(643\) 12.6206 0.497710 0.248855 0.968541i \(-0.419946\pi\)
0.248855 + 0.968541i \(0.419946\pi\)
\(644\) −52.5015 −2.06885
\(645\) 0 0
\(646\) 6.40585 0.252035
\(647\) 29.6429 1.16538 0.582691 0.812694i \(-0.302000\pi\)
0.582691 + 0.812694i \(0.302000\pi\)
\(648\) 1.09461 0.0430003
\(649\) −7.73463 −0.303611
\(650\) 0 0
\(651\) −20.0752 −0.786810
\(652\) −17.5667 −0.687967
\(653\) 9.90069 0.387444 0.193722 0.981056i \(-0.437944\pi\)
0.193722 + 0.981056i \(0.437944\pi\)
\(654\) −14.3103 −0.559578
\(655\) 0 0
\(656\) −11.1552 −0.435536
\(657\) −9.21949 −0.359687
\(658\) −26.9541 −1.05078
\(659\) −5.28005 −0.205681 −0.102841 0.994698i \(-0.532793\pi\)
−0.102841 + 0.994698i \(0.532793\pi\)
\(660\) 0 0
\(661\) −26.8548 −1.04453 −0.522266 0.852783i \(-0.674913\pi\)
−0.522266 + 0.852783i \(0.674913\pi\)
\(662\) 68.5271 2.66338
\(663\) −1.75023 −0.0679733
\(664\) −16.1447 −0.626537
\(665\) 0 0
\(666\) 0.965943 0.0374295
\(667\) −35.8401 −1.38774
\(668\) −15.8713 −0.614080
\(669\) −12.9239 −0.499665
\(670\) 0 0
\(671\) −12.0147 −0.463822
\(672\) 28.8860 1.11430
\(673\) −3.81834 −0.147186 −0.0735932 0.997288i \(-0.523447\pi\)
−0.0735932 + 0.997288i \(0.523447\pi\)
\(674\) 61.5885 2.37230
\(675\) 0 0
\(676\) −26.9229 −1.03550
\(677\) −15.6897 −0.603003 −0.301502 0.953466i \(-0.597488\pi\)
−0.301502 + 0.953466i \(0.597488\pi\)
\(678\) 12.7493 0.489634
\(679\) −24.6779 −0.947049
\(680\) 0 0
\(681\) −22.8099 −0.874076
\(682\) 11.7190 0.448745
\(683\) −15.6353 −0.598269 −0.299135 0.954211i \(-0.596698\pi\)
−0.299135 + 0.954211i \(0.596698\pi\)
\(684\) 6.56387 0.250976
\(685\) 0 0
\(686\) −5.80275 −0.221550
\(687\) −14.7796 −0.563876
\(688\) −31.6528 −1.20675
\(689\) −19.0303 −0.724996
\(690\) 0 0
\(691\) −31.4305 −1.19567 −0.597836 0.801618i \(-0.703973\pi\)
−0.597836 + 0.801618i \(0.703973\pi\)
\(692\) −32.3065 −1.22811
\(693\) 3.64002 0.138273
\(694\) −75.8989 −2.88108
\(695\) 0 0
\(696\) −6.84106 −0.259310
\(697\) 4.76491 0.180484
\(698\) −49.4674 −1.87237
\(699\) −4.96594 −0.187829
\(700\) 0 0
\(701\) 3.24507 0.122565 0.0612824 0.998120i \(-0.480481\pi\)
0.0612824 + 0.998120i \(0.480481\pi\)
\(702\) −3.21949 −0.121512
\(703\) 1.18635 0.0447442
\(704\) −11.4537 −0.431676
\(705\) 0 0
\(706\) −20.7181 −0.779737
\(707\) 26.9541 1.01371
\(708\) 19.4537 0.731114
\(709\) −26.7190 −1.00345 −0.501727 0.865026i \(-0.667302\pi\)
−0.501727 + 0.865026i \(0.667302\pi\)
\(710\) 0 0
\(711\) 5.09461 0.191063
\(712\) −11.4929 −0.430714
\(713\) −31.6273 −1.18445
\(714\) −8.93475 −0.334375
\(715\) 0 0
\(716\) 33.9163 1.26751
\(717\) 14.9991 0.560151
\(718\) 72.0351 2.68833
\(719\) −6.78807 −0.253152 −0.126576 0.991957i \(-0.540399\pi\)
−0.126576 + 0.991957i \(0.540399\pi\)
\(720\) 0 0
\(721\) −60.0587 −2.23670
\(722\) −25.9007 −0.963924
\(723\) −5.04496 −0.187624
\(724\) −58.0743 −2.15831
\(725\) 0 0
\(726\) −2.12489 −0.0788619
\(727\) −19.9154 −0.738620 −0.369310 0.929306i \(-0.620406\pi\)
−0.369310 + 0.929306i \(0.620406\pi\)
\(728\) 6.03692 0.223743
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 13.5204 0.500071
\(732\) 30.2186 1.11691
\(733\) −31.3388 −1.15752 −0.578762 0.815497i \(-0.696464\pi\)
−0.578762 + 0.815497i \(0.696464\pi\)
\(734\) 4.00756 0.147922
\(735\) 0 0
\(736\) 45.5081 1.67745
\(737\) −14.2645 −0.525438
\(738\) 8.76491 0.322641
\(739\) −2.25355 −0.0828982 −0.0414491 0.999141i \(-0.513197\pi\)
−0.0414491 + 0.999141i \(0.513197\pi\)
\(740\) 0 0
\(741\) −3.95413 −0.145259
\(742\) −97.1477 −3.56640
\(743\) −6.74931 −0.247608 −0.123804 0.992307i \(-0.539509\pi\)
−0.123804 + 0.992307i \(0.539509\pi\)
\(744\) −6.03692 −0.221324
\(745\) 0 0
\(746\) −34.6888 −1.27005
\(747\) −14.7493 −0.539649
\(748\) 2.90539 0.106232
\(749\) 14.3397 0.523961
\(750\) 0 0
\(751\) 22.4390 0.818810 0.409405 0.912353i \(-0.365736\pi\)
0.409405 + 0.912353i \(0.365736\pi\)
\(752\) 9.42431 0.343669
\(753\) 3.03028 0.110429
\(754\) 20.1211 0.732767
\(755\) 0 0
\(756\) −9.15516 −0.332970
\(757\) 25.4158 0.923754 0.461877 0.886944i \(-0.347176\pi\)
0.461877 + 0.886944i \(0.347176\pi\)
\(758\) −55.3406 −2.01006
\(759\) 5.73463 0.208154
\(760\) 0 0
\(761\) −30.7493 −1.11466 −0.557331 0.830291i \(-0.688174\pi\)
−0.557331 + 0.830291i \(0.688174\pi\)
\(762\) −17.1358 −0.620764
\(763\) 24.5142 0.887474
\(764\) −20.0819 −0.726537
\(765\) 0 0
\(766\) 26.4977 0.957401
\(767\) −11.7190 −0.423150
\(768\) −4.91721 −0.177434
\(769\) −16.2956 −0.587636 −0.293818 0.955861i \(-0.594926\pi\)
−0.293818 + 0.955861i \(0.594926\pi\)
\(770\) 0 0
\(771\) 13.6509 0.491626
\(772\) 29.5904 1.06498
\(773\) 48.7787 1.75445 0.877223 0.480082i \(-0.159393\pi\)
0.877223 + 0.480082i \(0.159393\pi\)
\(774\) 24.8704 0.893949
\(775\) 0 0
\(776\) −7.42100 −0.266398
\(777\) −1.65470 −0.0593621
\(778\) −38.4390 −1.37810
\(779\) 10.7649 0.385693
\(780\) 0 0
\(781\) 8.51514 0.304696
\(782\) −14.0761 −0.503362
\(783\) −6.24977 −0.223349
\(784\) −16.9016 −0.603629
\(785\) 0 0
\(786\) 27.2195 0.970887
\(787\) −46.2001 −1.64686 −0.823428 0.567421i \(-0.807941\pi\)
−0.823428 + 0.567421i \(0.807941\pi\)
\(788\) −9.59415 −0.341777
\(789\) −12.5601 −0.447151
\(790\) 0 0
\(791\) −21.8401 −0.776546
\(792\) 1.09461 0.0388952
\(793\) −18.2039 −0.646439
\(794\) −32.3709 −1.14880
\(795\) 0 0
\(796\) −30.3709 −1.07647
\(797\) 36.3784 1.28859 0.644295 0.764777i \(-0.277151\pi\)
0.644295 + 0.764777i \(0.277151\pi\)
\(798\) −20.1854 −0.714557
\(799\) −4.02558 −0.142415
\(800\) 0 0
\(801\) −10.4995 −0.370983
\(802\) 5.84197 0.206287
\(803\) −9.21949 −0.325349
\(804\) 35.8771 1.26529
\(805\) 0 0
\(806\) 17.7560 0.625427
\(807\) 24.6888 0.869086
\(808\) 8.10551 0.285151
\(809\) 11.3737 0.399879 0.199940 0.979808i \(-0.435925\pi\)
0.199940 + 0.979808i \(0.435925\pi\)
\(810\) 0 0
\(811\) −13.3903 −0.470195 −0.235098 0.971972i \(-0.575541\pi\)
−0.235098 + 0.971972i \(0.575541\pi\)
\(812\) 57.2177 2.00795
\(813\) −7.56479 −0.265309
\(814\) 0.965943 0.0338563
\(815\) 0 0
\(816\) 3.12397 0.109361
\(817\) 30.5454 1.06865
\(818\) −8.46835 −0.296089
\(819\) 5.51514 0.192715
\(820\) 0 0
\(821\) 32.0975 1.12021 0.560105 0.828422i \(-0.310761\pi\)
0.560105 + 0.828422i \(0.310761\pi\)
\(822\) −48.4683 −1.69053
\(823\) 16.7952 0.585443 0.292722 0.956198i \(-0.405439\pi\)
0.292722 + 0.956198i \(0.405439\pi\)
\(824\) −18.0606 −0.629169
\(825\) 0 0
\(826\) −59.8245 −2.08156
\(827\) −45.5904 −1.58533 −0.792666 0.609656i \(-0.791308\pi\)
−0.792666 + 0.609656i \(0.791308\pi\)
\(828\) −14.4234 −0.501248
\(829\) 12.9385 0.449374 0.224687 0.974431i \(-0.427864\pi\)
0.224687 + 0.974431i \(0.427864\pi\)
\(830\) 0 0
\(831\) −1.92477 −0.0667695
\(832\) −17.3539 −0.601638
\(833\) 7.21949 0.250141
\(834\) −16.1287 −0.558490
\(835\) 0 0
\(836\) 6.56387 0.227016
\(837\) −5.51514 −0.190631
\(838\) −10.9130 −0.376982
\(839\) −1.59037 −0.0549057 −0.0274528 0.999623i \(-0.508740\pi\)
−0.0274528 + 0.999623i \(0.508740\pi\)
\(840\) 0 0
\(841\) 10.0596 0.346884
\(842\) −19.0104 −0.655143
\(843\) 1.87511 0.0645824
\(844\) 25.8165 0.888641
\(845\) 0 0
\(846\) −7.40493 −0.254587
\(847\) 3.64002 0.125073
\(848\) 33.9670 1.16643
\(849\) −30.1396 −1.03439
\(850\) 0 0
\(851\) −2.60688 −0.0893628
\(852\) −21.4167 −0.733726
\(853\) 10.5161 0.360063 0.180031 0.983661i \(-0.442380\pi\)
0.180031 + 0.983661i \(0.442380\pi\)
\(854\) −92.9291 −3.17997
\(855\) 0 0
\(856\) 4.31216 0.147386
\(857\) −57.4637 −1.96292 −0.981460 0.191665i \(-0.938611\pi\)
−0.981460 + 0.191665i \(0.938611\pi\)
\(858\) −3.21949 −0.109912
\(859\) 32.7181 1.11633 0.558164 0.829731i \(-0.311506\pi\)
0.558164 + 0.829731i \(0.311506\pi\)
\(860\) 0 0
\(861\) −15.0147 −0.511699
\(862\) 48.3397 1.64646
\(863\) −43.1807 −1.46989 −0.734945 0.678127i \(-0.762792\pi\)
−0.734945 + 0.678127i \(0.762792\pi\)
\(864\) 7.93567 0.269977
\(865\) 0 0
\(866\) 16.1135 0.547560
\(867\) 15.6656 0.532032
\(868\) 50.4920 1.71381
\(869\) 5.09461 0.172823
\(870\) 0 0
\(871\) −21.6126 −0.732315
\(872\) 7.37179 0.249640
\(873\) −6.77959 −0.229454
\(874\) −31.8009 −1.07568
\(875\) 0 0
\(876\) 23.1883 0.783460
\(877\) −16.0752 −0.542822 −0.271411 0.962464i \(-0.587490\pi\)
−0.271411 + 0.962464i \(0.587490\pi\)
\(878\) 36.9523 1.24708
\(879\) 29.1552 0.983379
\(880\) 0 0
\(881\) 31.2876 1.05411 0.527053 0.849832i \(-0.323297\pi\)
0.527053 + 0.849832i \(0.323297\pi\)
\(882\) 13.2800 0.447162
\(883\) 24.7640 0.833375 0.416687 0.909050i \(-0.363191\pi\)
0.416687 + 0.909050i \(0.363191\pi\)
\(884\) 4.40207 0.148058
\(885\) 0 0
\(886\) 22.6169 0.759828
\(887\) −26.3085 −0.883353 −0.441676 0.897174i \(-0.645616\pi\)
−0.441676 + 0.897174i \(0.645616\pi\)
\(888\) −0.497594 −0.0166982
\(889\) 29.3544 0.984514
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 32.5053 1.08836
\(893\) −9.09461 −0.304339
\(894\) 3.85574 0.128955
\(895\) 0 0
\(896\) −30.8179 −1.02955
\(897\) 8.68876 0.290109
\(898\) −57.2252 −1.90963
\(899\) 34.4683 1.14958
\(900\) 0 0
\(901\) −14.5089 −0.483363
\(902\) 8.76491 0.291840
\(903\) −42.6041 −1.41778
\(904\) −6.56766 −0.218437
\(905\) 0 0
\(906\) −51.6878 −1.71721
\(907\) −55.9301 −1.85713 −0.928563 0.371174i \(-0.878955\pi\)
−0.928563 + 0.371174i \(0.878955\pi\)
\(908\) 57.3700 1.90389
\(909\) 7.40493 0.245606
\(910\) 0 0
\(911\) −15.3931 −0.509997 −0.254998 0.966941i \(-0.582075\pi\)
−0.254998 + 0.966941i \(0.582075\pi\)
\(912\) 7.05769 0.233703
\(913\) −14.7493 −0.488131
\(914\) −33.5298 −1.10907
\(915\) 0 0
\(916\) 37.1727 1.22822
\(917\) −46.6282 −1.53980
\(918\) −2.45459 −0.0810134
\(919\) −51.2598 −1.69090 −0.845452 0.534052i \(-0.820669\pi\)
−0.845452 + 0.534052i \(0.820669\pi\)
\(920\) 0 0
\(921\) −27.8548 −0.917848
\(922\) 17.4012 0.573076
\(923\) 12.9016 0.424662
\(924\) −9.15516 −0.301183
\(925\) 0 0
\(926\) 34.1892 1.12353
\(927\) −16.4995 −0.541916
\(928\) −49.5961 −1.62807
\(929\) 14.8099 0.485896 0.242948 0.970039i \(-0.421886\pi\)
0.242948 + 0.970039i \(0.421886\pi\)
\(930\) 0 0
\(931\) 16.3103 0.534549
\(932\) 12.4900 0.409125
\(933\) −23.9083 −0.782721
\(934\) −62.6188 −2.04895
\(935\) 0 0
\(936\) 1.65848 0.0542093
\(937\) 27.2654 0.890721 0.445360 0.895351i \(-0.353076\pi\)
0.445360 + 0.895351i \(0.353076\pi\)
\(938\) −110.330 −3.60241
\(939\) −28.3094 −0.923843
\(940\) 0 0
\(941\) −49.5630 −1.61571 −0.807853 0.589384i \(-0.799371\pi\)
−0.807853 + 0.589384i \(0.799371\pi\)
\(942\) −20.7493 −0.676049
\(943\) −23.6547 −0.770303
\(944\) 20.9172 0.680797
\(945\) 0 0
\(946\) 24.8704 0.808607
\(947\) −13.9844 −0.454432 −0.227216 0.973844i \(-0.572962\pi\)
−0.227216 + 0.973844i \(0.572962\pi\)
\(948\) −12.8136 −0.416168
\(949\) −13.9688 −0.453447
\(950\) 0 0
\(951\) 8.80986 0.285679
\(952\) 4.60263 0.149172
\(953\) 17.1240 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(954\) −26.6888 −0.864081
\(955\) 0 0
\(956\) −37.7248 −1.22011
\(957\) −6.24977 −0.202026
\(958\) 68.4608 2.21187
\(959\) 83.0284 2.68113
\(960\) 0 0
\(961\) −0.583252 −0.0188146
\(962\) 1.46354 0.0471863
\(963\) 3.93945 0.126947
\(964\) 12.6888 0.408677
\(965\) 0 0
\(966\) 44.3553 1.42711
\(967\) 1.90826 0.0613653 0.0306827 0.999529i \(-0.490232\pi\)
0.0306827 + 0.999529i \(0.490232\pi\)
\(968\) 1.09461 0.0351821
\(969\) −3.01468 −0.0968455
\(970\) 0 0
\(971\) 31.3856 1.00721 0.503605 0.863934i \(-0.332007\pi\)
0.503605 + 0.863934i \(0.332007\pi\)
\(972\) −2.51514 −0.0806731
\(973\) 27.6291 0.885749
\(974\) −76.2697 −2.44384
\(975\) 0 0
\(976\) 32.4920 1.04004
\(977\) 41.4693 1.32672 0.663360 0.748301i \(-0.269130\pi\)
0.663360 + 0.748301i \(0.269130\pi\)
\(978\) 14.8411 0.474565
\(979\) −10.4995 −0.335567
\(980\) 0 0
\(981\) 6.73463 0.215020
\(982\) −15.2119 −0.485432
\(983\) −6.23601 −0.198898 −0.0994489 0.995043i \(-0.531708\pi\)
−0.0994489 + 0.995043i \(0.531708\pi\)
\(984\) −4.51514 −0.143937
\(985\) 0 0
\(986\) 15.3406 0.488544
\(987\) 12.6850 0.403767
\(988\) 9.94518 0.316398
\(989\) −67.1202 −2.13430
\(990\) 0 0
\(991\) 27.2048 0.864189 0.432095 0.901828i \(-0.357775\pi\)
0.432095 + 0.901828i \(0.357775\pi\)
\(992\) −43.7663 −1.38958
\(993\) −32.2498 −1.02342
\(994\) 65.8615 2.08900
\(995\) 0 0
\(996\) 37.0966 1.17545
\(997\) 28.0606 0.888687 0.444343 0.895857i \(-0.353437\pi\)
0.444343 + 0.895857i \(0.353437\pi\)
\(998\) 57.5298 1.82107
\(999\) −0.454586 −0.0143825
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 825.2.a.i.1.3 3
3.2 odd 2 2475.2.a.bd.1.1 3
5.2 odd 4 825.2.c.f.199.5 6
5.3 odd 4 825.2.c.f.199.2 6
5.4 even 2 825.2.a.m.1.1 yes 3
11.10 odd 2 9075.2.a.cj.1.1 3
15.2 even 4 2475.2.c.q.199.2 6
15.8 even 4 2475.2.c.q.199.5 6
15.14 odd 2 2475.2.a.z.1.3 3
55.54 odd 2 9075.2.a.cd.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.a.i.1.3 3 1.1 even 1 trivial
825.2.a.m.1.1 yes 3 5.4 even 2
825.2.c.f.199.2 6 5.3 odd 4
825.2.c.f.199.5 6 5.2 odd 4
2475.2.a.z.1.3 3 15.14 odd 2
2475.2.a.bd.1.1 3 3.2 odd 2
2475.2.c.q.199.2 6 15.2 even 4
2475.2.c.q.199.5 6 15.8 even 4
9075.2.a.cd.1.3 3 55.54 odd 2
9075.2.a.cj.1.1 3 11.10 odd 2