Properties

Label 2475.2.a.bd.1.1
Level $2475$
Weight $2$
Character 2475.1
Self dual yes
Analytic conductor $19.763$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.7629745003\)
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: no (minimal twist has level 825)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.12489\) of defining polynomial
Character \(\chi\) \(=\) 2475.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.12489 q^{2} +2.51514 q^{4} +3.64002 q^{7} -1.09461 q^{8} +O(q^{10})\) \(q-2.12489 q^{2} +2.51514 q^{4} +3.64002 q^{7} -1.09461 q^{8} -1.00000 q^{11} +1.51514 q^{13} -7.73463 q^{14} -2.70436 q^{16} -1.15516 q^{17} +2.60975 q^{19} +2.12489 q^{22} +5.73463 q^{23} -3.21949 q^{26} +9.15516 q^{28} -6.24977 q^{29} +5.51514 q^{31} +7.93567 q^{32} +2.45459 q^{34} +0.454586 q^{37} -5.54541 q^{38} -4.12489 q^{41} +11.7044 q^{43} -2.51514 q^{44} -12.1854 q^{46} +3.48486 q^{47} +6.24977 q^{49} +3.81078 q^{52} +12.5601 q^{53} -3.98440 q^{56} +13.2800 q^{58} +7.73463 q^{59} -12.0147 q^{61} -11.7190 q^{62} -11.4537 q^{64} -14.2645 q^{67} -2.90539 q^{68} -8.51514 q^{71} -9.21949 q^{73} -0.965943 q^{74} +6.56387 q^{76} -3.64002 q^{77} +5.09461 q^{79} +8.76491 q^{82} +14.7493 q^{83} -24.8704 q^{86} +1.09461 q^{88} +10.4995 q^{89} +5.51514 q^{91} +14.4234 q^{92} -7.40493 q^{94} -6.77959 q^{97} -13.2800 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{2} + 8 q^{4} + 3 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{2} + 8 q^{4} + 3 q^{7} + 6 q^{8} - 3 q^{11} + 5 q^{13} - 6 q^{14} + 10 q^{16} + 4 q^{17} - q^{19} - 2 q^{22} + 8 q^{26} + 20 q^{28} - 2 q^{29} + 17 q^{31} + 34 q^{32} + 6 q^{34} - 18 q^{38} - 4 q^{41} + 17 q^{43} - 8 q^{44} - 30 q^{46} + 10 q^{47} + 2 q^{49} + 30 q^{52} + 6 q^{53} + 22 q^{56} + 24 q^{58} + 6 q^{59} - 3 q^{61} + 16 q^{62} + 34 q^{64} + 7 q^{67} - 18 q^{68} - 26 q^{71} - 10 q^{73} - 14 q^{74} - 24 q^{76} - 3 q^{77} + 6 q^{79} + 10 q^{82} - 6 q^{83} - 28 q^{86} - 6 q^{88} - 2 q^{89} + 17 q^{91} - 26 q^{92} + 2 q^{94} + 29 q^{97} - 24 q^{98}+O(q^{100}) \) 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 \(-0.770554\pi\)
−0.751260 + 0.660006i \(0.770554\pi\)
\(3\) 0 0
\(4\) 2.51514 1.25757
\(5\) 0 0
\(6\) 0 0
\(7\) 3.64002 1.37580 0.687900 0.725806i \(-0.258533\pi\)
0.687900 + 0.725806i \(0.258533\pi\)
\(8\) −1.09461 −0.387003
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 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 \(-0.544737\pi\)
−0.140084 + 0.990140i \(0.544737\pi\)
\(18\) 0 0
\(19\) 2.60975 0.598717 0.299359 0.954141i \(-0.403227\pi\)
0.299359 + 0.954141i \(0.403227\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 2.12489 0.453027
\(23\) 5.73463 1.19575 0.597877 0.801588i \(-0.296011\pi\)
0.597877 + 0.801588i \(0.296011\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −3.21949 −0.631395
\(27\) 0 0
\(28\) 9.15516 1.73016
\(29\) −6.24977 −1.16055 −0.580277 0.814419i \(-0.697056\pi\)
−0.580277 + 0.814419i \(0.697056\pi\)
\(30\) 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
\(33\) 0 0
\(34\) 2.45459 0.420958
\(35\) 0 0
\(36\) 0 0
\(37\) 0.454586 0.0747335 0.0373667 0.999302i \(-0.488103\pi\)
0.0373667 + 0.999302i \(0.488103\pi\)
\(38\) −5.54541 −0.899585
\(39\) 0 0
\(40\) 0 0
\(41\) −4.12489 −0.644199 −0.322099 0.946706i \(-0.604389\pi\)
−0.322099 + 0.946706i \(0.604389\pi\)
\(42\) 0 0
\(43\) 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 \(-0.418201\pi\)
0.254160 + 0.967162i \(0.418201\pi\)
\(48\) 0 0
\(49\) 6.24977 0.892824
\(50\) 0 0
\(51\) 0 0
\(52\) 3.81078 0.528460
\(53\) 12.5601 1.72526 0.862631 0.505834i \(-0.168815\pi\)
0.862631 + 0.505834i \(0.168815\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.98440 −0.532438
\(57\) 0 0
\(58\) 13.2800 1.74376
\(59\) 7.73463 1.00696 0.503482 0.864006i \(-0.332052\pi\)
0.503482 + 0.864006i \(0.332052\pi\)
\(60\) 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
\(63\) 0 0
\(64\) −11.4537 −1.43171
\(65\) 0 0
\(66\) 0 0
\(67\) −14.2645 −1.74268 −0.871340 0.490680i \(-0.836749\pi\)
−0.871340 + 0.490680i \(0.836749\pi\)
\(68\) −2.90539 −0.352330
\(69\) 0 0
\(70\) 0 0
\(71\) −8.51514 −1.01056 −0.505280 0.862955i \(-0.668611\pi\)
−0.505280 + 0.862955i \(0.668611\pi\)
\(72\) 0 0
\(73\) −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
\(78\) 0 0
\(79\) 5.09461 0.573188 0.286594 0.958052i \(-0.407477\pi\)
0.286594 + 0.958052i \(0.407477\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 8.76491 0.967922
\(83\) 14.7493 1.61895 0.809474 0.587156i \(-0.199753\pi\)
0.809474 + 0.587156i \(0.199753\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −24.8704 −2.68185
\(87\) 0 0
\(88\) 1.09461 0.116686
\(89\) 10.4995 1.11295 0.556475 0.830865i \(-0.312154\pi\)
0.556475 + 0.830865i \(0.312154\pi\)
\(90\) 0 0
\(91\) 5.51514 0.578144
\(92\) 14.4234 1.50374
\(93\) 0 0
\(94\) −7.40493 −0.763760
\(95\) 0 0
\(96\) 0 0
\(97\) −6.77959 −0.688363 −0.344181 0.938903i \(-0.611844\pi\)
−0.344181 + 0.938903i \(0.611844\pi\)
\(98\) −13.2800 −1.34149
\(99\) 0 0
\(100\) 0 0
\(101\) −7.40493 −0.736818 −0.368409 0.929664i \(-0.620097\pi\)
−0.368409 + 0.929664i \(0.620097\pi\)
\(102\) 0 0
\(103\) −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 \(-0.560985\pi\)
−0.190420 + 0.981703i \(0.560985\pi\)
\(108\) 0 0
\(109\) 6.73463 0.645061 0.322530 0.946559i \(-0.395467\pi\)
0.322530 + 0.946559i \(0.395467\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −9.84392 −0.930163
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −15.7190 −1.45948
\(117\) 0 0
\(118\) −16.4352 −1.51298
\(119\) −4.20482 −0.385455
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 25.5298 2.31136
\(123\) 0 0
\(124\) 13.8713 1.24568
\(125\) 0 0
\(126\) 0 0
\(127\) 8.06433 0.715594 0.357797 0.933799i \(-0.383528\pi\)
0.357797 + 0.933799i \(0.383528\pi\)
\(128\) 8.46640 0.748331
\(129\) 0 0
\(130\) 0 0
\(131\) 12.8099 1.11920 0.559602 0.828762i \(-0.310954\pi\)
0.559602 + 0.828762i \(0.310954\pi\)
\(132\) 0 0
\(133\) 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 \(-0.927805\pi\)
−0.974389 + 0.224868i \(0.927805\pi\)
\(138\) 0 0
\(139\) 7.59037 0.643807 0.321903 0.946773i \(-0.395677\pi\)
0.321903 + 0.946773i \(0.395677\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 18.0937 1.51839
\(143\) −1.51514 −0.126702
\(144\) 0 0
\(145\) 0 0
\(146\) 19.5904 1.62131
\(147\) 0 0
\(148\) 1.14335 0.0939825
\(149\) 1.81456 0.148655 0.0743274 0.997234i \(-0.476319\pi\)
0.0743274 + 0.997234i \(0.476319\pi\)
\(150\) 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
\(153\) 0 0
\(154\) 7.73463 0.623274
\(155\) 0 0
\(156\) 0 0
\(157\) 9.76491 0.779325 0.389662 0.920958i \(-0.372592\pi\)
0.389662 + 0.920958i \(0.372592\pi\)
\(158\) −10.8255 −0.861227
\(159\) 0 0
\(160\) 0 0
\(161\) 20.8742 1.64512
\(162\) 0 0
\(163\) −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 \(-0.421490\pi\)
0.244154 + 0.969737i \(0.421490\pi\)
\(168\) 0 0
\(169\) −10.7044 −0.823412
\(170\) 0 0
\(171\) 0 0
\(172\) 29.4381 2.24463
\(173\) 12.8448 0.976575 0.488287 0.872683i \(-0.337622\pi\)
0.488287 + 0.872683i \(0.337622\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.70436 0.203849
\(177\) 0 0
\(178\) −22.3103 −1.67223
\(179\) −13.4849 −1.00791 −0.503953 0.863731i \(-0.668122\pi\)
−0.503953 + 0.863731i \(0.668122\pi\)
\(180\) 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
\(183\) 0 0
\(184\) −6.27718 −0.462760
\(185\) 0 0
\(186\) 0 0
\(187\) 1.15516 0.0844738
\(188\) 8.76491 0.639247
\(189\) 0 0
\(190\) 0 0
\(191\) 7.98440 0.577731 0.288866 0.957370i \(-0.406722\pi\)
0.288866 + 0.957370i \(0.406722\pi\)
\(192\) 0 0
\(193\) 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 \(-0.456611\pi\)
0.135888 + 0.990724i \(0.456611\pi\)
\(198\) 0 0
\(199\) −12.0752 −0.855990 −0.427995 0.903781i \(-0.640780\pi\)
−0.427995 + 0.903781i \(0.640780\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 15.7346 1.10708
\(203\) −22.7493 −1.59669
\(204\) 0 0
\(205\) 0 0
\(206\) 35.0596 2.44272
\(207\) 0 0
\(208\) −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
\(213\) 0 0
\(214\) 8.37088 0.572221
\(215\) 0 0
\(216\) 0 0
\(217\) 20.0752 1.36280
\(218\) −14.3103 −0.969217
\(219\) 0 0
\(220\) 0 0
\(221\) −1.75023 −0.117733
\(222\) 0 0
\(223\) 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 \(-0.773322\pi\)
−0.756972 + 0.653447i \(0.773322\pi\)
\(228\) 0 0
\(229\) 14.7796 0.976663 0.488331 0.872658i \(-0.337606\pi\)
0.488331 + 0.872658i \(0.337606\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.84106 0.449137
\(233\) −4.96594 −0.325330 −0.162665 0.986681i \(-0.552009\pi\)
−0.162665 + 0.986681i \(0.552009\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 19.4537 1.26633
\(237\) 0 0
\(238\) 8.93475 0.579154
\(239\) 14.9991 0.970210 0.485105 0.874456i \(-0.338781\pi\)
0.485105 + 0.874456i \(0.338781\pi\)
\(240\) 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
\(243\) 0 0
\(244\) −30.2186 −1.93455
\(245\) 0 0
\(246\) 0 0
\(247\) 3.95413 0.251595
\(248\) −6.03692 −0.383345
\(249\) 0 0
\(250\) 0 0
\(251\) 3.03028 0.191269 0.0956347 0.995417i \(-0.469512\pi\)
0.0956347 + 0.995417i \(0.469512\pi\)
\(252\) 0 0
\(253\) −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 \(-0.360007\pi\)
0.425761 + 0.904836i \(0.360007\pi\)
\(258\) 0 0
\(259\) 1.65470 0.102818
\(260\) 0 0
\(261\) 0 0
\(262\) −27.2195 −1.68163
\(263\) −12.5601 −0.774489 −0.387244 0.921977i \(-0.626573\pi\)
−0.387244 + 0.921977i \(0.626573\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −20.1854 −1.23765
\(267\) 0 0
\(268\) −35.8771 −2.19154
\(269\) 24.6888 1.50530 0.752650 0.658421i \(-0.228775\pi\)
0.752650 + 0.658421i \(0.228775\pi\)
\(270\) 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
\(273\) 0 0
\(274\) 48.4683 2.92808
\(275\) 0 0
\(276\) 0 0
\(277\) 1.92477 0.115648 0.0578241 0.998327i \(-0.481584\pi\)
0.0578241 + 0.998327i \(0.481584\pi\)
\(278\) −16.1287 −0.967333
\(279\) 0 0
\(280\) 0 0
\(281\) 1.87511 0.111860 0.0559300 0.998435i \(-0.482188\pi\)
0.0559300 + 0.998435i \(0.482188\pi\)
\(282\) 0 0
\(283\) 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
\(288\) 0 0
\(289\) −15.6656 −0.921506
\(290\) 0 0
\(291\) 0 0
\(292\) −23.1883 −1.35699
\(293\) 29.1552 1.70326 0.851631 0.524141i \(-0.175614\pi\)
0.851631 + 0.524141i \(0.175614\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −0.497594 −0.0289221
\(297\) 0 0
\(298\) −3.85574 −0.223357
\(299\) 8.68876 0.502484
\(300\) 0 0
\(301\) 42.6041 2.45566
\(302\) −51.6878 −2.97430
\(303\) 0 0
\(304\) −7.05769 −0.404786
\(305\) 0 0
\(306\) 0 0
\(307\) 27.8548 1.58976 0.794879 0.606768i \(-0.207534\pi\)
0.794879 + 0.606768i \(0.207534\pi\)
\(308\) −9.15516 −0.521664
\(309\) 0 0
\(310\) 0 0
\(311\) −23.9083 −1.35571 −0.677856 0.735194i \(-0.737091\pi\)
−0.677856 + 0.735194i \(0.737091\pi\)
\(312\) 0 0
\(313\) 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 \(-0.420422\pi\)
0.247406 + 0.968912i \(0.420422\pi\)
\(318\) 0 0
\(319\) 6.24977 0.349920
\(320\) 0 0
\(321\) 0 0
\(322\) −44.3553 −2.47182
\(323\) −3.01468 −0.167741
\(324\) 0 0
\(325\) 0 0
\(326\) 14.8411 0.821970
\(327\) 0 0
\(328\) 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 0
\(334\) −13.4087 −0.733692
\(335\) 0 0
\(336\) 0 0
\(337\) 28.9844 1.57888 0.789441 0.613827i \(-0.210371\pi\)
0.789441 + 0.613827i \(0.210371\pi\)
\(338\) 22.7455 1.23719
\(339\) 0 0
\(340\) 0 0
\(341\) −5.51514 −0.298661
\(342\) 0 0
\(343\) −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 \(-0.0917457\pi\)
0.958749 + 0.284253i \(0.0917457\pi\)
\(348\) 0 0
\(349\) −23.2800 −1.24615 −0.623076 0.782161i \(-0.714117\pi\)
−0.623076 + 0.782161i \(0.714117\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −7.93567 −0.422972
\(353\) 9.75023 0.518952 0.259476 0.965750i \(-0.416450\pi\)
0.259476 + 0.965750i \(0.416450\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 26.4078 1.39961
\(357\) 0 0
\(358\) 28.6538 1.51440
\(359\) −33.9007 −1.78921 −0.894605 0.446858i \(-0.852543\pi\)
−0.894605 + 0.446858i \(0.852543\pi\)
\(360\) 0 0
\(361\) −12.1892 −0.641538
\(362\) 49.0634 2.57872
\(363\) 0 0
\(364\) 13.8713 0.727055
\(365\) 0 0
\(366\) 0 0
\(367\) 1.88601 0.0984491 0.0492245 0.998788i \(-0.484325\pi\)
0.0492245 + 0.998788i \(0.484325\pi\)
\(368\) −15.5085 −0.808436
\(369\) 0 0
\(370\) 0 0
\(371\) 45.7190 2.37361
\(372\) 0 0
\(373\) −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
\(378\) 0 0
\(379\) −26.0440 −1.33779 −0.668896 0.743356i \(-0.733233\pi\)
−0.668896 + 0.743356i \(0.733233\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −16.9659 −0.868053
\(383\) −12.4702 −0.637197 −0.318598 0.947890i \(-0.603212\pi\)
−0.318598 + 0.947890i \(0.603212\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −24.9991 −1.27242
\(387\) 0 0
\(388\) −17.0516 −0.865664
\(389\) 18.0899 0.917195 0.458597 0.888644i \(-0.348352\pi\)
0.458597 + 0.888644i \(0.348352\pi\)
\(390\) 0 0
\(391\) −6.62443 −0.335012
\(392\) −6.84106 −0.345526
\(393\) 0 0
\(394\) −8.10551 −0.408350
\(395\) 0 0
\(396\) 0 0
\(397\) −15.2342 −0.764581 −0.382291 0.924042i \(-0.624865\pi\)
−0.382291 + 0.924042i \(0.624865\pi\)
\(398\) 25.6585 1.28614
\(399\) 0 0
\(400\) 0 0
\(401\) −2.74931 −0.137294 −0.0686471 0.997641i \(-0.521868\pi\)
−0.0686471 + 0.997641i \(0.521868\pi\)
\(402\) 0 0
\(403\) 8.35620 0.416252
\(404\) −18.6244 −0.926600
\(405\) 0 0
\(406\) 48.3397 2.39906
\(407\) −0.454586 −0.0225330
\(408\) 0 0
\(409\) −3.98532 −0.197061 −0.0985307 0.995134i \(-0.531414\pi\)
−0.0985307 + 0.995134i \(0.531414\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −41.4986 −2.04449
\(413\) 28.1542 1.38538
\(414\) 0 0
\(415\) 0 0
\(416\) 12.0236 0.589507
\(417\) 0 0
\(418\) 5.54541 0.271235
\(419\) 5.13578 0.250899 0.125450 0.992100i \(-0.459963\pi\)
0.125450 + 0.992100i \(0.459963\pi\)
\(420\) 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
\(423\) 0 0
\(424\) −13.7484 −0.667681
\(425\) 0 0
\(426\) 0 0
\(427\) −43.7337 −2.11642
\(428\) −9.90826 −0.478934
\(429\) 0 0
\(430\) 0 0
\(431\) −22.7493 −1.09580 −0.547898 0.836545i \(-0.684572\pi\)
−0.547898 + 0.836545i \(0.684572\pi\)
\(432\) 0 0
\(433\) 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
\(438\) 0 0
\(439\) 17.3903 0.829991 0.414996 0.909823i \(-0.363783\pi\)
0.414996 + 0.909823i \(0.363783\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 3.71904 0.176897
\(443\) −10.6438 −0.505702 −0.252851 0.967505i \(-0.581368\pi\)
−0.252851 + 0.967505i \(0.581368\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −27.4617 −1.30035
\(447\) 0 0
\(448\) −41.6916 −1.96974
\(449\) 26.9310 1.27095 0.635476 0.772121i \(-0.280804\pi\)
0.635476 + 0.772121i \(0.280804\pi\)
\(450\) 0 0
\(451\) 4.12489 0.194233
\(452\) 15.0908 0.709813
\(453\) 0 0
\(454\) 48.4683 2.27473
\(455\) 0 0
\(456\) 0 0
\(457\) −15.7796 −0.738138 −0.369069 0.929402i \(-0.620323\pi\)
−0.369069 + 0.929402i \(0.620323\pi\)
\(458\) −31.4049 −1.46746
\(459\) 0 0
\(460\) 0 0
\(461\) −8.18922 −0.381410 −0.190705 0.981647i \(-0.561077\pi\)
−0.190705 + 0.981647i \(0.561077\pi\)
\(462\) 0 0
\(463\) 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 \(-0.261181\pi\)
0.681837 + 0.731504i \(0.261181\pi\)
\(468\) 0 0
\(469\) −51.9229 −2.39758
\(470\) 0 0
\(471\) 0 0
\(472\) −8.46640 −0.389698
\(473\) −11.7044 −0.538167
\(474\) 0 0
\(475\) 0 0
\(476\) −10.5757 −0.484736
\(477\) 0 0
\(478\) −31.8713 −1.45776
\(479\) −32.2186 −1.47210 −0.736052 0.676925i \(-0.763312\pi\)
−0.736052 + 0.676925i \(0.763312\pi\)
\(480\) 0 0
\(481\) 0.688760 0.0314048
\(482\) −10.7200 −0.488280
\(483\) 0 0
\(484\) 2.51514 0.114324
\(485\) 0 0
\(486\) 0 0
\(487\) −35.8936 −1.62649 −0.813247 0.581919i \(-0.802302\pi\)
−0.813247 + 0.581919i \(0.802302\pi\)
\(488\) 13.1514 0.595335
\(489\) 0 0
\(490\) 0 0
\(491\) 7.15894 0.323079 0.161539 0.986866i \(-0.448354\pi\)
0.161539 + 0.986866i \(0.448354\pi\)
\(492\) 0 0
\(493\) 7.21949 0.325150
\(494\) −8.40207 −0.378027
\(495\) 0 0
\(496\) −14.9149 −0.669699
\(497\) −30.9953 −1.39033
\(498\) 0 0
\(499\) 27.0743 1.21201 0.606006 0.795460i \(-0.292771\pi\)
0.606006 + 0.795460i \(0.292771\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −6.43899 −0.287386
\(503\) −26.9991 −1.20383 −0.601915 0.798560i \(-0.705595\pi\)
−0.601915 + 0.798560i \(0.705595\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 12.1854 0.541709
\(507\) 0 0
\(508\) 20.2829 0.899909
\(509\) −15.5904 −0.691031 −0.345515 0.938413i \(-0.612296\pi\)
−0.345515 + 0.938413i \(0.612296\pi\)
\(510\) 0 0
\(511\) −33.5592 −1.48457
\(512\) −27.3813 −1.21009
\(513\) 0 0
\(514\) −29.0066 −1.27943
\(515\) 0 0
\(516\) 0 0
\(517\) −3.48486 −0.153264
\(518\) −3.51605 −0.154487
\(519\) 0 0
\(520\) 0 0
\(521\) −11.1589 −0.488882 −0.244441 0.969664i \(-0.578604\pi\)
−0.244441 + 0.969664i \(0.578604\pi\)
\(522\) 0 0
\(523\) −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
\(528\) 0 0
\(529\) 9.88601 0.429827
\(530\) 0 0
\(531\) 0 0
\(532\) 23.8927 1.03588
\(533\) −6.24977 −0.270708
\(534\) 0 0
\(535\) 0 0
\(536\) 15.6140 0.674422
\(537\) 0 0
\(538\) −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
\(543\) 0 0
\(544\) −9.16698 −0.393031
\(545\) 0 0
\(546\) 0 0
\(547\) 6.09369 0.260547 0.130274 0.991478i \(-0.458414\pi\)
0.130274 + 0.991478i \(0.458414\pi\)
\(548\) −57.3700 −2.45072
\(549\) 0 0
\(550\) 0 0
\(551\) −16.3103 −0.694843
\(552\) 0 0
\(553\) 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 \(-0.460104\pi\)
0.125010 + 0.992155i \(0.460104\pi\)
\(558\) 0 0
\(559\) 17.7337 0.750056
\(560\) 0 0
\(561\) 0 0
\(562\) −3.98440 −0.168072
\(563\) −3.03028 −0.127711 −0.0638555 0.997959i \(-0.520340\pi\)
−0.0638555 + 0.997959i \(0.520340\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −64.0431 −2.69193
\(567\) 0 0
\(568\) 9.32075 0.391090
\(569\) −13.4049 −0.561964 −0.280982 0.959713i \(-0.590660\pi\)
−0.280982 + 0.959713i \(0.590660\pi\)
\(570\) 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
\(573\) 0 0
\(574\) 31.9045 1.33167
\(575\) 0 0
\(576\) 0 0
\(577\) 2.03028 0.0845215 0.0422607 0.999107i \(-0.486544\pi\)
0.0422607 + 0.999107i \(0.486544\pi\)
\(578\) 33.2876 1.38458
\(579\) 0 0
\(580\) 0 0
\(581\) 53.6878 2.22735
\(582\) 0 0
\(583\) −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 \(-0.648718\pi\)
−0.450398 + 0.892828i \(0.648718\pi\)
\(588\) 0 0
\(589\) 14.3931 0.593058
\(590\) 0 0
\(591\) 0 0
\(592\) −1.22936 −0.0505265
\(593\) −8.06811 −0.331318 −0.165659 0.986183i \(-0.552975\pi\)
−0.165659 + 0.986183i \(0.552975\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 4.56387 0.186944
\(597\) 0 0
\(598\) −18.4626 −0.754993
\(599\) −7.61353 −0.311080 −0.155540 0.987830i \(-0.549712\pi\)
−0.155540 + 0.987830i \(0.549712\pi\)
\(600\) 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
\(603\) 0 0
\(604\) 61.1807 2.48941
\(605\) 0 0
\(606\) 0 0
\(607\) 17.5298 0.711513 0.355757 0.934579i \(-0.384223\pi\)
0.355757 + 0.934579i \(0.384223\pi\)
\(608\) 20.7101 0.839905
\(609\) 0 0
\(610\) 0 0
\(611\) 5.28005 0.213608
\(612\) 0 0
\(613\) 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 \(-0.604116\pi\)
−0.321288 + 0.946982i \(0.604116\pi\)
\(618\) 0 0
\(619\) −9.23417 −0.371153 −0.185576 0.982630i \(-0.559415\pi\)
−0.185576 + 0.982630i \(0.559415\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 50.8023 2.03699
\(623\) 38.2186 1.53119
\(624\) 0 0
\(625\) 0 0
\(626\) −60.1542 −2.40425
\(627\) 0 0
\(628\) 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
\(633\) 0 0
\(634\) −18.7200 −0.743464
\(635\) 0 0
\(636\) 0 0
\(637\) 9.46927 0.375186
\(638\) −13.2800 −0.525762
\(639\) 0 0
\(640\) 0 0
\(641\) −13.9612 −0.551436 −0.275718 0.961239i \(-0.588916\pi\)
−0.275718 + 0.961239i \(0.588916\pi\)
\(642\) 0 0
\(643\) 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 \(-0.698000\pi\)
−0.582691 + 0.812694i \(0.698000\pi\)
\(648\) 0 0
\(649\) −7.73463 −0.303611
\(650\) 0 0
\(651\) 0 0
\(652\) −17.5667 −0.687967
\(653\) −9.90069 −0.387444 −0.193722 0.981056i \(-0.562056\pi\)
−0.193722 + 0.981056i \(0.562056\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 11.1552 0.435536
\(657\) 0 0
\(658\) −26.9541 −1.05078
\(659\) 5.28005 0.205681 0.102841 0.994698i \(-0.467207\pi\)
0.102841 + 0.994698i \(0.467207\pi\)
\(660\) 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
\(663\) 0 0
\(664\) −16.1447 −0.626537
\(665\) 0 0
\(666\) 0 0
\(667\) −35.8401 −1.38774
\(668\) 15.8713 0.614080
\(669\) 0 0
\(670\) 0 0
\(671\) 12.0147 0.463822
\(672\) 0 0
\(673\) −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 \(-0.402512\pi\)
0.301502 + 0.953466i \(0.402512\pi\)
\(678\) 0 0
\(679\) −24.6779 −0.947049
\(680\) 0 0
\(681\) 0 0
\(682\) 11.7190 0.448745
\(683\) 15.6353 0.598269 0.299135 0.954211i \(-0.403302\pi\)
0.299135 + 0.954211i \(0.403302\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 5.80275 0.221550
\(687\) 0 0
\(688\) −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
\(693\) 0 0
\(694\) −75.8989 −2.88108
\(695\) 0 0
\(696\) 0 0
\(697\) 4.76491 0.180484
\(698\) 49.4674 1.87237
\(699\) 0 0
\(700\) 0 0
\(701\) −3.24507 −0.122565 −0.0612824 0.998120i \(-0.519519\pi\)
−0.0612824 + 0.998120i \(0.519519\pi\)
\(702\) 0 0
\(703\) 1.18635 0.0447442
\(704\) 11.4537 0.431676
\(705\) 0 0
\(706\) −20.7181 −0.779737
\(707\) −26.9541 −1.01371
\(708\) 0 0
\(709\) −26.7190 −1.00345 −0.501727 0.865026i \(-0.667302\pi\)
−0.501727 + 0.865026i \(0.667302\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −11.4929 −0.430714
\(713\) 31.6273 1.18445
\(714\) 0 0
\(715\) 0 0
\(716\) −33.9163 −1.26751
\(717\) 0 0
\(718\) 72.0351 2.68833
\(719\) 6.78807 0.253152 0.126576 0.991957i \(-0.459601\pi\)
0.126576 + 0.991957i \(0.459601\pi\)
\(720\) 0 0
\(721\) −60.0587 −2.23670
\(722\) 25.9007 0.963924
\(723\) 0 0
\(724\) −58.0743 −2.15831
\(725\) 0 0
\(726\) 0 0
\(727\) −19.9154 −0.738620 −0.369310 0.929306i \(-0.620406\pi\)
−0.369310 + 0.929306i \(0.620406\pi\)
\(728\) −6.03692 −0.223743
\(729\) 0 0
\(730\) 0 0
\(731\) −13.5204 −0.500071
\(732\) 0 0
\(733\) −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
\(738\) 0 0
\(739\) −2.25355 −0.0828982 −0.0414491 0.999141i \(-0.513197\pi\)
−0.0414491 + 0.999141i \(0.513197\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −97.1477 −3.56640
\(743\) 6.74931 0.247608 0.123804 0.992307i \(-0.460491\pi\)
0.123804 + 0.992307i \(0.460491\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 34.6888 1.27005
\(747\) 0 0
\(748\) 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
\(753\) 0 0
\(754\) 20.1211 0.732767
\(755\) 0 0
\(756\) 0 0
\(757\) 25.4158 0.923754 0.461877 0.886944i \(-0.347176\pi\)
0.461877 + 0.886944i \(0.347176\pi\)
\(758\) 55.3406 2.01006
\(759\) 0 0
\(760\) 0 0
\(761\) 30.7493 1.11466 0.557331 0.830291i \(-0.311826\pi\)
0.557331 + 0.830291i \(0.311826\pi\)
\(762\) 0 0
\(763\) 24.5142 0.887474
\(764\) 20.0819 0.726537
\(765\) 0 0
\(766\) 26.4977 0.957401
\(767\) 11.7190 0.423150
\(768\) 0 0
\(769\) −16.2956 −0.587636 −0.293818 0.955861i \(-0.594926\pi\)
−0.293818 + 0.955861i \(0.594926\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 29.5904 1.06498
\(773\) −48.7787 −1.75445 −0.877223 0.480082i \(-0.840607\pi\)
−0.877223 + 0.480082i \(0.840607\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 7.42100 0.266398
\(777\) 0 0
\(778\) −38.4390 −1.37810
\(779\) −10.7649 −0.385693
\(780\) 0 0
\(781\) 8.51514 0.304696
\(782\) 14.0761 0.503362
\(783\) 0 0
\(784\) −16.9016 −0.603629
\(785\) 0 0
\(786\) 0 0
\(787\) −46.2001 −1.64686 −0.823428 0.567421i \(-0.807941\pi\)
−0.823428 + 0.567421i \(0.807941\pi\)
\(788\) 9.59415 0.341777
\(789\) 0 0
\(790\) 0 0
\(791\) 21.8401 0.776546
\(792\) 0 0
\(793\) −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 \(-0.722849\pi\)
−0.644295 + 0.764777i \(0.722849\pi\)
\(798\) 0 0
\(799\) −4.02558 −0.142415
\(800\) 0 0
\(801\) 0 0
\(802\) 5.84197 0.206287
\(803\) 9.21949 0.325349
\(804\) 0 0
\(805\) 0 0
\(806\) −17.7560 −0.625427
\(807\) 0 0
\(808\) 8.10551 0.285151
\(809\) −11.3737 −0.399879 −0.199940 0.979808i \(-0.564075\pi\)
−0.199940 + 0.979808i \(0.564075\pi\)
\(810\) 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
\(813\) 0 0
\(814\) 0.965943 0.0338563
\(815\) 0 0
\(816\) 0 0
\(817\) 30.5454 1.06865
\(818\) 8.46835 0.296089
\(819\) 0 0
\(820\) 0 0
\(821\) −32.0975 −1.12021 −0.560105 0.828422i \(-0.689239\pi\)
−0.560105 + 0.828422i \(0.689239\pi\)
\(822\) 0 0
\(823\) 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 \(-0.208692\pi\)
0.792666 + 0.609656i \(0.208692\pi\)
\(828\) 0 0
\(829\) 12.9385 0.449374 0.224687 0.974431i \(-0.427864\pi\)
0.224687 + 0.974431i \(0.427864\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −17.3539 −0.601638
\(833\) −7.21949 −0.250141
\(834\) 0 0
\(835\) 0 0
\(836\) −6.56387 −0.227016
\(837\) 0 0
\(838\) −10.9130 −0.376982
\(839\) 1.59037 0.0549057 0.0274528 0.999623i \(-0.491260\pi\)
0.0274528 + 0.999623i \(0.491260\pi\)
\(840\) 0 0
\(841\) 10.0596 0.346884
\(842\) 19.0104 0.655143
\(843\) 0 0
\(844\) 25.8165 0.888641
\(845\) 0 0
\(846\) 0 0
\(847\) 3.64002 0.125073
\(848\) −33.9670 −1.16643
\(849\) 0 0
\(850\) 0 0
\(851\) 2.60688 0.0893628
\(852\) 0 0
\(853\) 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 \(-0.0613886\pi\)
0.981460 + 0.191665i \(0.0613886\pi\)
\(858\) 0 0
\(859\) 32.7181 1.11633 0.558164 0.829731i \(-0.311506\pi\)
0.558164 + 0.829731i \(0.311506\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 48.3397 1.64646
\(863\) 43.1807 1.46989 0.734945 0.678127i \(-0.237208\pi\)
0.734945 + 0.678127i \(0.237208\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −16.1135 −0.547560
\(867\) 0 0
\(868\) 50.4920 1.71381
\(869\) −5.09461 −0.172823
\(870\) 0 0
\(871\) −21.6126 −0.732315
\(872\) −7.37179 −0.249640
\(873\) 0 0
\(874\) −31.8009 −1.07568
\(875\) 0 0
\(876\) 0 0
\(877\) −16.0752 −0.542822 −0.271411 0.962464i \(-0.587490\pi\)
−0.271411 + 0.962464i \(0.587490\pi\)
\(878\) −36.9523 −1.24708
\(879\) 0 0
\(880\) 0 0
\(881\) −31.2876 −1.05411 −0.527053 0.849832i \(-0.676703\pi\)
−0.527053 + 0.849832i \(0.676703\pi\)
\(882\) 0 0
\(883\) 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 \(-0.354384\pi\)
0.441676 + 0.897174i \(0.354384\pi\)
\(888\) 0 0
\(889\) 29.3544 0.984514
\(890\) 0 0
\(891\) 0 0
\(892\) 32.5053 1.08836
\(893\) 9.09461 0.304339
\(894\) 0 0
\(895\) 0 0
\(896\) 30.8179 1.02955
\(897\) 0 0
\(898\) −57.2252 −1.90963
\(899\) −34.4683 −1.14958
\(900\) 0 0
\(901\) −14.5089 −0.483363
\(902\) −8.76491 −0.291840
\(903\) 0 0
\(904\) −6.56766 −0.218437
\(905\) 0 0
\(906\) 0 0
\(907\) −55.9301 −1.85713 −0.928563 0.371174i \(-0.878955\pi\)
−0.928563 + 0.371174i \(0.878955\pi\)
\(908\) −57.3700 −1.90389
\(909\) 0 0
\(910\) 0 0
\(911\) 15.3931 0.509997 0.254998 0.966941i \(-0.417925\pi\)
0.254998 + 0.966941i \(0.417925\pi\)
\(912\) 0 0
\(913\) −14.7493 −0.488131
\(914\) 33.5298 1.10907
\(915\) 0 0
\(916\) 37.1727 1.22822
\(917\) 46.6282 1.53980
\(918\) 0 0
\(919\) −51.2598 −1.69090 −0.845452 0.534052i \(-0.820669\pi\)
−0.845452 + 0.534052i \(0.820669\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 17.4012 0.573076
\(923\) −12.9016 −0.424662
\(924\) 0 0
\(925\) 0 0
\(926\) −34.1892 −1.12353
\(927\) 0 0
\(928\) −49.5961 −1.62807
\(929\) −14.8099 −0.485896 −0.242948 0.970039i \(-0.578114\pi\)
−0.242948 + 0.970039i \(0.578114\pi\)
\(930\) 0 0
\(931\) 16.3103 0.534549
\(932\) −12.4900 −0.409125
\(933\) 0 0
\(934\) −62.6188 −2.04895
\(935\) 0 0
\(936\) 0 0
\(937\) 27.2654 0.890721 0.445360 0.895351i \(-0.353076\pi\)
0.445360 + 0.895351i \(0.353076\pi\)
\(938\) 110.330 3.60241
\(939\) 0 0
\(940\) 0 0
\(941\) 49.5630 1.61571 0.807853 0.589384i \(-0.200629\pi\)
0.807853 + 0.589384i \(0.200629\pi\)
\(942\) 0 0
\(943\) −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 \(-0.427038\pi\)
0.227216 + 0.973844i \(0.427038\pi\)
\(948\) 0 0
\(949\) −13.9688 −0.453447
\(950\) 0 0
\(951\) 0 0
\(952\) 4.60263 0.149172
\(953\) −17.1240 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 37.7248 1.22011
\(957\) 0 0
\(958\) 68.4608 2.21187
\(959\) −83.0284 −2.68113
\(960\) 0 0
\(961\) −0.583252 −0.0188146
\(962\) −1.46354 −0.0471863
\(963\) 0 0
\(964\) 12.6888 0.408677
\(965\) 0 0
\(966\) 0 0
\(967\) 1.90826 0.0613653 0.0306827 0.999529i \(-0.490232\pi\)
0.0306827 + 0.999529i \(0.490232\pi\)
\(968\) −1.09461 −0.0351821
\(969\) 0 0
\(970\) 0 0
\(971\) −31.3856 −1.00721 −0.503605 0.863934i \(-0.667993\pi\)
−0.503605 + 0.863934i \(0.667993\pi\)
\(972\) 0 0
\(973\) 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 \(-0.730870\pi\)
−0.663360 + 0.748301i \(0.730870\pi\)
\(978\) 0 0
\(979\) −10.4995 −0.335567
\(980\) 0 0
\(981\) 0 0
\(982\) −15.2119 −0.485432
\(983\) 6.23601 0.198898 0.0994489 0.995043i \(-0.468292\pi\)
0.0994489 + 0.995043i \(0.468292\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −15.3406 −0.488544
\(987\) 0 0
\(988\) 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
\(993\) 0 0
\(994\) 65.8615 2.08900
\(995\) 0 0
\(996\) 0 0
\(997\) 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 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 2475.2.a.bd.1.1 3
3.2 odd 2 825.2.a.i.1.3 3
5.2 odd 4 2475.2.c.q.199.2 6
5.3 odd 4 2475.2.c.q.199.5 6
5.4 even 2 2475.2.a.z.1.3 3
15.2 even 4 825.2.c.f.199.5 6
15.8 even 4 825.2.c.f.199.2 6
15.14 odd 2 825.2.a.m.1.1 yes 3
33.32 even 2 9075.2.a.cj.1.1 3
165.164 even 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 3.2 odd 2
825.2.a.m.1.1 yes 3 15.14 odd 2
825.2.c.f.199.2 6 15.8 even 4
825.2.c.f.199.5 6 15.2 even 4
2475.2.a.z.1.3 3 5.4 even 2
2475.2.a.bd.1.1 3 1.1 even 1 trivial
2475.2.c.q.199.2 6 5.2 odd 4
2475.2.c.q.199.5 6 5.3 odd 4
9075.2.a.cd.1.3 3 165.164 even 2
9075.2.a.cj.1.1 3 33.32 even 2