Properties

Label 7623.2.a.cc.1.2
Level 7623
Weight 2
Character 7623.1
Self dual yes
Analytic conductor 60.870
Analytic rank 1
Dimension 3
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 7623.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(60.8699614608\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2541)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.470683\)
Character \(\chi\) = 7623.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.470683 q^{2} -1.77846 q^{4} -3.24914 q^{5} -1.00000 q^{7} -1.77846 q^{8} +O(q^{10})\) \(q+0.470683 q^{2} -1.77846 q^{4} -3.24914 q^{5} -1.00000 q^{7} -1.77846 q^{8} -1.52932 q^{10} -2.47068 q^{13} -0.470683 q^{14} +2.71982 q^{16} +3.24914 q^{17} -5.30777 q^{19} +5.77846 q^{20} +8.86469 q^{23} +5.55691 q^{25} -1.16291 q^{26} +1.77846 q^{28} +2.47068 q^{29} -6.49828 q^{31} +4.83709 q^{32} +1.52932 q^{34} +3.24914 q^{35} +1.77846 q^{37} -2.49828 q^{38} +5.77846 q^{40} +1.28018 q^{41} -4.89572 q^{43} +4.17246 q^{46} -3.96896 q^{47} +1.00000 q^{49} +2.61555 q^{50} +4.39400 q^{52} +9.77846 q^{53} +1.77846 q^{56} +1.16291 q^{58} -2.41205 q^{59} +4.74742 q^{61} -3.05863 q^{62} -3.16291 q^{64} +8.02760 q^{65} +14.8337 q^{67} -5.77846 q^{68} +1.52932 q^{70} +3.19051 q^{71} +8.98195 q^{73} +0.837090 q^{74} +9.43965 q^{76} -12.3810 q^{79} -8.83709 q^{80} +0.602558 q^{82} +13.0828 q^{83} -10.5569 q^{85} -2.30434 q^{86} -7.86469 q^{89} +2.47068 q^{91} -15.7655 q^{92} -1.86813 q^{94} +17.2457 q^{95} -14.8517 q^{97} +0.470683 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + q^{2} + 3q^{4} - q^{5} - 3q^{7} + 3q^{8} + O(q^{10}) \) \( 3q + q^{2} + 3q^{4} - q^{5} - 3q^{7} + 3q^{8} - 5q^{10} - 7q^{13} - q^{14} - q^{16} + q^{17} - 8q^{19} + 9q^{20} + 2q^{23} - 11q^{26} - 3q^{28} + 7q^{29} - 2q^{31} + 7q^{32} + 5q^{34} + q^{35} - 3q^{37} + 10q^{38} + 9q^{40} + 13q^{41} - 8q^{43} - 20q^{46} + 6q^{47} + 3q^{49} - 8q^{50} - 11q^{52} + 21q^{53} - 3q^{56} + 11q^{58} - 6q^{59} - 12q^{61} - 10q^{62} - 17q^{64} + 7q^{65} + 2q^{67} - 9q^{68} + 5q^{70} + 4q^{73} - 5q^{74} + 10q^{76} - 18q^{79} - 19q^{80} - 9q^{82} - 12q^{83} - 15q^{85} + 36q^{86} + q^{89} + 7q^{91} - 44q^{92} - 16q^{94} + 8q^{95} - 25q^{97} + q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.470683 0.332823 0.166412 0.986056i \(-0.446782\pi\)
0.166412 + 0.986056i \(0.446782\pi\)
\(3\) 0 0
\(4\) −1.77846 −0.889229
\(5\) −3.24914 −1.45306 −0.726530 0.687135i \(-0.758868\pi\)
−0.726530 + 0.687135i \(0.758868\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −1.77846 −0.628780
\(9\) 0 0
\(10\) −1.52932 −0.483612
\(11\) 0 0
\(12\) 0 0
\(13\) −2.47068 −0.685244 −0.342622 0.939473i \(-0.611315\pi\)
−0.342622 + 0.939473i \(0.611315\pi\)
\(14\) −0.470683 −0.125795
\(15\) 0 0
\(16\) 2.71982 0.679956
\(17\) 3.24914 0.788032 0.394016 0.919104i \(-0.371085\pi\)
0.394016 + 0.919104i \(0.371085\pi\)
\(18\) 0 0
\(19\) −5.30777 −1.21769 −0.608843 0.793290i \(-0.708366\pi\)
−0.608843 + 0.793290i \(0.708366\pi\)
\(20\) 5.77846 1.29210
\(21\) 0 0
\(22\) 0 0
\(23\) 8.86469 1.84842 0.924208 0.381890i \(-0.124727\pi\)
0.924208 + 0.381890i \(0.124727\pi\)
\(24\) 0 0
\(25\) 5.55691 1.11138
\(26\) −1.16291 −0.228065
\(27\) 0 0
\(28\) 1.77846 0.336097
\(29\) 2.47068 0.458794 0.229397 0.973333i \(-0.426325\pi\)
0.229397 + 0.973333i \(0.426325\pi\)
\(30\) 0 0
\(31\) −6.49828 −1.16713 −0.583563 0.812068i \(-0.698342\pi\)
−0.583563 + 0.812068i \(0.698342\pi\)
\(32\) 4.83709 0.855085
\(33\) 0 0
\(34\) 1.52932 0.262276
\(35\) 3.24914 0.549205
\(36\) 0 0
\(37\) 1.77846 0.292377 0.146188 0.989257i \(-0.453299\pi\)
0.146188 + 0.989257i \(0.453299\pi\)
\(38\) −2.49828 −0.405275
\(39\) 0 0
\(40\) 5.77846 0.913654
\(41\) 1.28018 0.199930 0.0999650 0.994991i \(-0.468127\pi\)
0.0999650 + 0.994991i \(0.468127\pi\)
\(42\) 0 0
\(43\) −4.89572 −0.746591 −0.373295 0.927713i \(-0.621772\pi\)
−0.373295 + 0.927713i \(0.621772\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 4.17246 0.615196
\(47\) −3.96896 −0.578933 −0.289466 0.957188i \(-0.593478\pi\)
−0.289466 + 0.957188i \(0.593478\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 2.61555 0.369894
\(51\) 0 0
\(52\) 4.39400 0.609339
\(53\) 9.77846 1.34317 0.671587 0.740926i \(-0.265613\pi\)
0.671587 + 0.740926i \(0.265613\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.77846 0.237656
\(57\) 0 0
\(58\) 1.16291 0.152698
\(59\) −2.41205 −0.314022 −0.157011 0.987597i \(-0.550186\pi\)
−0.157011 + 0.987597i \(0.550186\pi\)
\(60\) 0 0
\(61\) 4.74742 0.607845 0.303923 0.952697i \(-0.401704\pi\)
0.303923 + 0.952697i \(0.401704\pi\)
\(62\) −3.05863 −0.388447
\(63\) 0 0
\(64\) −3.16291 −0.395364
\(65\) 8.02760 0.995701
\(66\) 0 0
\(67\) 14.8337 1.81222 0.906110 0.423043i \(-0.139038\pi\)
0.906110 + 0.423043i \(0.139038\pi\)
\(68\) −5.77846 −0.700741
\(69\) 0 0
\(70\) 1.52932 0.182788
\(71\) 3.19051 0.378644 0.189322 0.981915i \(-0.439371\pi\)
0.189322 + 0.981915i \(0.439371\pi\)
\(72\) 0 0
\(73\) 8.98195 1.05126 0.525629 0.850714i \(-0.323830\pi\)
0.525629 + 0.850714i \(0.323830\pi\)
\(74\) 0.837090 0.0973098
\(75\) 0 0
\(76\) 9.43965 1.08280
\(77\) 0 0
\(78\) 0 0
\(79\) −12.3810 −1.39297 −0.696486 0.717570i \(-0.745254\pi\)
−0.696486 + 0.717570i \(0.745254\pi\)
\(80\) −8.83709 −0.988017
\(81\) 0 0
\(82\) 0.602558 0.0665414
\(83\) 13.0828 1.43602 0.718012 0.696031i \(-0.245052\pi\)
0.718012 + 0.696031i \(0.245052\pi\)
\(84\) 0 0
\(85\) −10.5569 −1.14506
\(86\) −2.30434 −0.248483
\(87\) 0 0
\(88\) 0 0
\(89\) −7.86469 −0.833655 −0.416828 0.908986i \(-0.636858\pi\)
−0.416828 + 0.908986i \(0.636858\pi\)
\(90\) 0 0
\(91\) 2.47068 0.258998
\(92\) −15.7655 −1.64366
\(93\) 0 0
\(94\) −1.86813 −0.192682
\(95\) 17.2457 1.76937
\(96\) 0 0
\(97\) −14.8517 −1.50796 −0.753981 0.656897i \(-0.771869\pi\)
−0.753981 + 0.656897i \(0.771869\pi\)
\(98\) 0.470683 0.0475462
\(99\) 0 0
\(100\) −9.88273 −0.988273
\(101\) −6.97240 −0.693780 −0.346890 0.937906i \(-0.612762\pi\)
−0.346890 + 0.937906i \(0.612762\pi\)
\(102\) 0 0
\(103\) 11.3078 1.11419 0.557094 0.830449i \(-0.311916\pi\)
0.557094 + 0.830449i \(0.311916\pi\)
\(104\) 4.39400 0.430868
\(105\) 0 0
\(106\) 4.60256 0.447040
\(107\) 15.9233 1.53937 0.769683 0.638427i \(-0.220414\pi\)
0.769683 + 0.638427i \(0.220414\pi\)
\(108\) 0 0
\(109\) 14.4948 1.38835 0.694177 0.719804i \(-0.255768\pi\)
0.694177 + 0.719804i \(0.255768\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.71982 −0.256999
\(113\) 1.66119 0.156272 0.0781358 0.996943i \(-0.475103\pi\)
0.0781358 + 0.996943i \(0.475103\pi\)
\(114\) 0 0
\(115\) −28.8026 −2.68586
\(116\) −4.39400 −0.407973
\(117\) 0 0
\(118\) −1.13531 −0.104514
\(119\) −3.24914 −0.297848
\(120\) 0 0
\(121\) 0 0
\(122\) 2.23453 0.202305
\(123\) 0 0
\(124\) 11.5569 1.03784
\(125\) −1.80949 −0.161846
\(126\) 0 0
\(127\) 21.5405 1.91141 0.955705 0.294328i \(-0.0950957\pi\)
0.955705 + 0.294328i \(0.0950957\pi\)
\(128\) −11.1629 −0.986671
\(129\) 0 0
\(130\) 3.77846 0.331393
\(131\) 12.4121 1.08445 0.542223 0.840235i \(-0.317583\pi\)
0.542223 + 0.840235i \(0.317583\pi\)
\(132\) 0 0
\(133\) 5.30777 0.460242
\(134\) 6.98195 0.603149
\(135\) 0 0
\(136\) −5.77846 −0.495499
\(137\) −11.0992 −0.948270 −0.474135 0.880452i \(-0.657239\pi\)
−0.474135 + 0.880452i \(0.657239\pi\)
\(138\) 0 0
\(139\) −13.1905 −1.11880 −0.559402 0.828896i \(-0.688969\pi\)
−0.559402 + 0.828896i \(0.688969\pi\)
\(140\) −5.77846 −0.488369
\(141\) 0 0
\(142\) 1.50172 0.126021
\(143\) 0 0
\(144\) 0 0
\(145\) −8.02760 −0.666656
\(146\) 4.22766 0.349883
\(147\) 0 0
\(148\) −3.16291 −0.259990
\(149\) −12.6431 −1.03577 −0.517883 0.855451i \(-0.673280\pi\)
−0.517883 + 0.855451i \(0.673280\pi\)
\(150\) 0 0
\(151\) −23.2147 −1.88918 −0.944591 0.328248i \(-0.893542\pi\)
−0.944591 + 0.328248i \(0.893542\pi\)
\(152\) 9.43965 0.765657
\(153\) 0 0
\(154\) 0 0
\(155\) 21.1138 1.69590
\(156\) 0 0
\(157\) −21.0518 −1.68011 −0.840057 0.542499i \(-0.817478\pi\)
−0.840057 + 0.542499i \(0.817478\pi\)
\(158\) −5.82754 −0.463614
\(159\) 0 0
\(160\) −15.7164 −1.24249
\(161\) −8.86469 −0.698635
\(162\) 0 0
\(163\) 6.49828 0.508985 0.254492 0.967075i \(-0.418092\pi\)
0.254492 + 0.967075i \(0.418092\pi\)
\(164\) −2.27674 −0.177783
\(165\) 0 0
\(166\) 6.15785 0.477942
\(167\) −16.9103 −1.30856 −0.654280 0.756252i \(-0.727028\pi\)
−0.654280 + 0.756252i \(0.727028\pi\)
\(168\) 0 0
\(169\) −6.89572 −0.530440
\(170\) −4.96896 −0.381102
\(171\) 0 0
\(172\) 8.70683 0.663890
\(173\) −18.9103 −1.43773 −0.718863 0.695152i \(-0.755337\pi\)
−0.718863 + 0.695152i \(0.755337\pi\)
\(174\) 0 0
\(175\) −5.55691 −0.420063
\(176\) 0 0
\(177\) 0 0
\(178\) −3.70178 −0.277460
\(179\) −15.1690 −1.13379 −0.566893 0.823791i \(-0.691855\pi\)
−0.566893 + 0.823791i \(0.691855\pi\)
\(180\) 0 0
\(181\) −15.5078 −1.15269 −0.576344 0.817207i \(-0.695521\pi\)
−0.576344 + 0.817207i \(0.695521\pi\)
\(182\) 1.16291 0.0862006
\(183\) 0 0
\(184\) −15.7655 −1.16225
\(185\) −5.77846 −0.424841
\(186\) 0 0
\(187\) 0 0
\(188\) 7.05863 0.514804
\(189\) 0 0
\(190\) 8.11727 0.588888
\(191\) −0.483673 −0.0349974 −0.0174987 0.999847i \(-0.505570\pi\)
−0.0174987 + 0.999847i \(0.505570\pi\)
\(192\) 0 0
\(193\) −18.2311 −1.31230 −0.656151 0.754629i \(-0.727817\pi\)
−0.656151 + 0.754629i \(0.727817\pi\)
\(194\) −6.99045 −0.501885
\(195\) 0 0
\(196\) −1.77846 −0.127033
\(197\) 2.10428 0.149923 0.0749617 0.997186i \(-0.476117\pi\)
0.0749617 + 0.997186i \(0.476117\pi\)
\(198\) 0 0
\(199\) −2.63016 −0.186447 −0.0932234 0.995645i \(-0.529717\pi\)
−0.0932234 + 0.995645i \(0.529717\pi\)
\(200\) −9.88273 −0.698815
\(201\) 0 0
\(202\) −3.28179 −0.230906
\(203\) −2.47068 −0.173408
\(204\) 0 0
\(205\) −4.15947 −0.290510
\(206\) 5.32238 0.370828
\(207\) 0 0
\(208\) −6.71982 −0.465936
\(209\) 0 0
\(210\) 0 0
\(211\) −16.9509 −1.16695 −0.583475 0.812131i \(-0.698307\pi\)
−0.583475 + 0.812131i \(0.698307\pi\)
\(212\) −17.3906 −1.19439
\(213\) 0 0
\(214\) 7.49484 0.512337
\(215\) 15.9069 1.08484
\(216\) 0 0
\(217\) 6.49828 0.441132
\(218\) 6.82248 0.462077
\(219\) 0 0
\(220\) 0 0
\(221\) −8.02760 −0.539995
\(222\) 0 0
\(223\) −2.11727 −0.141783 −0.0708913 0.997484i \(-0.522584\pi\)
−0.0708913 + 0.997484i \(0.522584\pi\)
\(224\) −4.83709 −0.323192
\(225\) 0 0
\(226\) 0.781895 0.0520109
\(227\) −28.4052 −1.88532 −0.942659 0.333758i \(-0.891683\pi\)
−0.942659 + 0.333758i \(0.891683\pi\)
\(228\) 0 0
\(229\) 13.8337 0.914153 0.457077 0.889427i \(-0.348896\pi\)
0.457077 + 0.889427i \(0.348896\pi\)
\(230\) −13.5569 −0.893916
\(231\) 0 0
\(232\) −4.39400 −0.288481
\(233\) 10.0276 0.656930 0.328465 0.944516i \(-0.393469\pi\)
0.328465 + 0.944516i \(0.393469\pi\)
\(234\) 0 0
\(235\) 12.8957 0.841224
\(236\) 4.28973 0.279238
\(237\) 0 0
\(238\) −1.52932 −0.0991309
\(239\) −5.55691 −0.359447 −0.179723 0.983717i \(-0.557520\pi\)
−0.179723 + 0.983717i \(0.557520\pi\)
\(240\) 0 0
\(241\) −21.3078 −1.37255 −0.686277 0.727340i \(-0.740756\pi\)
−0.686277 + 0.727340i \(0.740756\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −8.44309 −0.540513
\(245\) −3.24914 −0.207580
\(246\) 0 0
\(247\) 13.1138 0.834413
\(248\) 11.5569 0.733865
\(249\) 0 0
\(250\) −0.851698 −0.0538661
\(251\) 29.9931 1.89315 0.946575 0.322485i \(-0.104518\pi\)
0.946575 + 0.322485i \(0.104518\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 10.1387 0.636162
\(255\) 0 0
\(256\) 1.07162 0.0669764
\(257\) 14.4543 0.901632 0.450816 0.892617i \(-0.351133\pi\)
0.450816 + 0.892617i \(0.351133\pi\)
\(258\) 0 0
\(259\) −1.77846 −0.110508
\(260\) −14.2767 −0.885406
\(261\) 0 0
\(262\) 5.84215 0.360929
\(263\) 13.1284 0.809534 0.404767 0.914420i \(-0.367353\pi\)
0.404767 + 0.914420i \(0.367353\pi\)
\(264\) 0 0
\(265\) −31.7716 −1.95171
\(266\) 2.49828 0.153179
\(267\) 0 0
\(268\) −26.3810 −1.61148
\(269\) −8.95436 −0.545957 −0.272978 0.962020i \(-0.588009\pi\)
−0.272978 + 0.962020i \(0.588009\pi\)
\(270\) 0 0
\(271\) −9.38445 −0.570065 −0.285032 0.958518i \(-0.592004\pi\)
−0.285032 + 0.958518i \(0.592004\pi\)
\(272\) 8.83709 0.535827
\(273\) 0 0
\(274\) −5.22422 −0.315607
\(275\) 0 0
\(276\) 0 0
\(277\) 20.7846 1.24882 0.624412 0.781095i \(-0.285339\pi\)
0.624412 + 0.781095i \(0.285339\pi\)
\(278\) −6.20855 −0.372364
\(279\) 0 0
\(280\) −5.77846 −0.345329
\(281\) −11.0878 −0.661446 −0.330723 0.943728i \(-0.607293\pi\)
−0.330723 + 0.943728i \(0.607293\pi\)
\(282\) 0 0
\(283\) 2.17246 0.129139 0.0645697 0.997913i \(-0.479433\pi\)
0.0645697 + 0.997913i \(0.479433\pi\)
\(284\) −5.67418 −0.336701
\(285\) 0 0
\(286\) 0 0
\(287\) −1.28018 −0.0755664
\(288\) 0 0
\(289\) −6.44309 −0.379005
\(290\) −3.77846 −0.221879
\(291\) 0 0
\(292\) −15.9740 −0.934809
\(293\) 15.8647 0.926825 0.463412 0.886143i \(-0.346625\pi\)
0.463412 + 0.886143i \(0.346625\pi\)
\(294\) 0 0
\(295\) 7.83709 0.456293
\(296\) −3.16291 −0.183840
\(297\) 0 0
\(298\) −5.95092 −0.344727
\(299\) −21.9018 −1.26662
\(300\) 0 0
\(301\) 4.89572 0.282185
\(302\) −10.9268 −0.628764
\(303\) 0 0
\(304\) −14.4362 −0.827973
\(305\) −15.4250 −0.883235
\(306\) 0 0
\(307\) 9.59750 0.547758 0.273879 0.961764i \(-0.411693\pi\)
0.273879 + 0.961764i \(0.411693\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 9.93793 0.564436
\(311\) 31.0828 1.76254 0.881272 0.472610i \(-0.156688\pi\)
0.881272 + 0.472610i \(0.156688\pi\)
\(312\) 0 0
\(313\) 17.8888 1.01114 0.505569 0.862786i \(-0.331283\pi\)
0.505569 + 0.862786i \(0.331283\pi\)
\(314\) −9.90871 −0.559181
\(315\) 0 0
\(316\) 22.0191 1.23867
\(317\) 14.9268 0.838370 0.419185 0.907901i \(-0.362316\pi\)
0.419185 + 0.907901i \(0.362316\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 10.2767 0.574487
\(321\) 0 0
\(322\) −4.17246 −0.232522
\(323\) −17.2457 −0.959577
\(324\) 0 0
\(325\) −13.7294 −0.761569
\(326\) 3.05863 0.169402
\(327\) 0 0
\(328\) −2.27674 −0.125712
\(329\) 3.96896 0.218816
\(330\) 0 0
\(331\) −6.86974 −0.377595 −0.188798 0.982016i \(-0.560459\pi\)
−0.188798 + 0.982016i \(0.560459\pi\)
\(332\) −23.2672 −1.27695
\(333\) 0 0
\(334\) −7.95941 −0.435520
\(335\) −48.1966 −2.63326
\(336\) 0 0
\(337\) −14.5113 −0.790479 −0.395240 0.918578i \(-0.629338\pi\)
−0.395240 + 0.918578i \(0.629338\pi\)
\(338\) −3.24570 −0.176543
\(339\) 0 0
\(340\) 18.7750 1.01822
\(341\) 0 0
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 8.70683 0.469441
\(345\) 0 0
\(346\) −8.90078 −0.478509
\(347\) −31.4734 −1.68958 −0.844789 0.535099i \(-0.820274\pi\)
−0.844789 + 0.535099i \(0.820274\pi\)
\(348\) 0 0
\(349\) 22.6578 1.21284 0.606421 0.795144i \(-0.292605\pi\)
0.606421 + 0.795144i \(0.292605\pi\)
\(350\) −2.61555 −0.139807
\(351\) 0 0
\(352\) 0 0
\(353\) −15.2802 −0.813282 −0.406641 0.913588i \(-0.633300\pi\)
−0.406641 + 0.913588i \(0.633300\pi\)
\(354\) 0 0
\(355\) −10.3664 −0.550192
\(356\) 13.9870 0.741310
\(357\) 0 0
\(358\) −7.13981 −0.377351
\(359\) −16.4768 −0.869612 −0.434806 0.900524i \(-0.643183\pi\)
−0.434806 + 0.900524i \(0.643183\pi\)
\(360\) 0 0
\(361\) 9.17246 0.482761
\(362\) −7.29928 −0.383642
\(363\) 0 0
\(364\) −4.39400 −0.230308
\(365\) −29.1836 −1.52754
\(366\) 0 0
\(367\) 15.9525 0.832716 0.416358 0.909201i \(-0.363306\pi\)
0.416358 + 0.909201i \(0.363306\pi\)
\(368\) 24.1104 1.25684
\(369\) 0 0
\(370\) −2.71982 −0.141397
\(371\) −9.77846 −0.507672
\(372\) 0 0
\(373\) 0.723262 0.0374491 0.0187245 0.999825i \(-0.494039\pi\)
0.0187245 + 0.999825i \(0.494039\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 7.05863 0.364021
\(377\) −6.10428 −0.314386
\(378\) 0 0
\(379\) −2.42666 −0.124649 −0.0623245 0.998056i \(-0.519851\pi\)
−0.0623245 + 0.998056i \(0.519851\pi\)
\(380\) −30.6707 −1.57338
\(381\) 0 0
\(382\) −0.227657 −0.0116479
\(383\) 21.2553 1.08609 0.543046 0.839703i \(-0.317271\pi\)
0.543046 + 0.839703i \(0.317271\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −8.58107 −0.436765
\(387\) 0 0
\(388\) 26.4131 1.34092
\(389\) 4.78189 0.242452 0.121226 0.992625i \(-0.461317\pi\)
0.121226 + 0.992625i \(0.461317\pi\)
\(390\) 0 0
\(391\) 28.8026 1.45661
\(392\) −1.77846 −0.0898256
\(393\) 0 0
\(394\) 0.990448 0.0498981
\(395\) 40.2277 2.02407
\(396\) 0 0
\(397\) 8.66463 0.434865 0.217433 0.976075i \(-0.430232\pi\)
0.217433 + 0.976075i \(0.430232\pi\)
\(398\) −1.23797 −0.0620539
\(399\) 0 0
\(400\) 15.1138 0.755691
\(401\) −15.1629 −0.757200 −0.378600 0.925560i \(-0.623594\pi\)
−0.378600 + 0.925560i \(0.623594\pi\)
\(402\) 0 0
\(403\) 16.0552 0.799766
\(404\) 12.4001 0.616929
\(405\) 0 0
\(406\) −1.16291 −0.0577142
\(407\) 0 0
\(408\) 0 0
\(409\) 17.0242 0.841791 0.420895 0.907109i \(-0.361716\pi\)
0.420895 + 0.907109i \(0.361716\pi\)
\(410\) −1.95779 −0.0966886
\(411\) 0 0
\(412\) −20.1104 −0.990768
\(413\) 2.41205 0.118689
\(414\) 0 0
\(415\) −42.5078 −2.08663
\(416\) −11.9509 −0.585942
\(417\) 0 0
\(418\) 0 0
\(419\) −7.76041 −0.379121 −0.189560 0.981869i \(-0.560706\pi\)
−0.189560 + 0.981869i \(0.560706\pi\)
\(420\) 0 0
\(421\) −15.9966 −0.779625 −0.389812 0.920894i \(-0.627460\pi\)
−0.389812 + 0.920894i \(0.627460\pi\)
\(422\) −7.97852 −0.388388
\(423\) 0 0
\(424\) −17.3906 −0.844561
\(425\) 18.0552 0.875806
\(426\) 0 0
\(427\) −4.74742 −0.229744
\(428\) −28.3189 −1.36885
\(429\) 0 0
\(430\) 7.48711 0.361061
\(431\) −26.1725 −1.26068 −0.630342 0.776318i \(-0.717085\pi\)
−0.630342 + 0.776318i \(0.717085\pi\)
\(432\) 0 0
\(433\) −20.3173 −0.976388 −0.488194 0.872735i \(-0.662344\pi\)
−0.488194 + 0.872735i \(0.662344\pi\)
\(434\) 3.05863 0.146819
\(435\) 0 0
\(436\) −25.7785 −1.23456
\(437\) −47.0518 −2.25079
\(438\) 0 0
\(439\) −18.7880 −0.896703 −0.448351 0.893857i \(-0.647989\pi\)
−0.448351 + 0.893857i \(0.647989\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −3.77846 −0.179723
\(443\) 34.9820 1.66204 0.831021 0.556240i \(-0.187757\pi\)
0.831021 + 0.556240i \(0.187757\pi\)
\(444\) 0 0
\(445\) 25.5535 1.21135
\(446\) −0.996562 −0.0471886
\(447\) 0 0
\(448\) 3.16291 0.149433
\(449\) 8.77502 0.414119 0.207059 0.978328i \(-0.433611\pi\)
0.207059 + 0.978328i \(0.433611\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −2.95436 −0.138961
\(453\) 0 0
\(454\) −13.3698 −0.627478
\(455\) −8.02760 −0.376340
\(456\) 0 0
\(457\) 31.9053 1.49247 0.746233 0.665685i \(-0.231861\pi\)
0.746233 + 0.665685i \(0.231861\pi\)
\(458\) 6.51127 0.304252
\(459\) 0 0
\(460\) 51.2242 2.38834
\(461\) −19.0958 −0.889379 −0.444690 0.895685i \(-0.646686\pi\)
−0.444690 + 0.895685i \(0.646686\pi\)
\(462\) 0 0
\(463\) 21.1430 0.982601 0.491300 0.870990i \(-0.336522\pi\)
0.491300 + 0.870990i \(0.336522\pi\)
\(464\) 6.71982 0.311960
\(465\) 0 0
\(466\) 4.71982 0.218642
\(467\) 7.05176 0.326316 0.163158 0.986600i \(-0.447832\pi\)
0.163158 + 0.986600i \(0.447832\pi\)
\(468\) 0 0
\(469\) −14.8337 −0.684954
\(470\) 6.06980 0.279979
\(471\) 0 0
\(472\) 4.28973 0.197451
\(473\) 0 0
\(474\) 0 0
\(475\) −29.4948 −1.35332
\(476\) 5.77846 0.264855
\(477\) 0 0
\(478\) −2.61555 −0.119632
\(479\) −20.7310 −0.947223 −0.473612 0.880734i \(-0.657050\pi\)
−0.473612 + 0.880734i \(0.657050\pi\)
\(480\) 0 0
\(481\) −4.39400 −0.200349
\(482\) −10.0292 −0.456818
\(483\) 0 0
\(484\) 0 0
\(485\) 48.2553 2.19116
\(486\) 0 0
\(487\) −26.4526 −1.19868 −0.599342 0.800493i \(-0.704571\pi\)
−0.599342 + 0.800493i \(0.704571\pi\)
\(488\) −8.44309 −0.382201
\(489\) 0 0
\(490\) −1.52932 −0.0690875
\(491\) 0.131874 0.00595140 0.00297570 0.999996i \(-0.499053\pi\)
0.00297570 + 0.999996i \(0.499053\pi\)
\(492\) 0 0
\(493\) 8.02760 0.361545
\(494\) 6.17246 0.277712
\(495\) 0 0
\(496\) −17.6742 −0.793594
\(497\) −3.19051 −0.143114
\(498\) 0 0
\(499\) −18.2441 −0.816717 −0.408359 0.912822i \(-0.633899\pi\)
−0.408359 + 0.912822i \(0.633899\pi\)
\(500\) 3.21811 0.143918
\(501\) 0 0
\(502\) 14.1173 0.630084
\(503\) 28.5224 1.27175 0.635876 0.771791i \(-0.280639\pi\)
0.635876 + 0.771791i \(0.280639\pi\)
\(504\) 0 0
\(505\) 22.6543 1.00810
\(506\) 0 0
\(507\) 0 0
\(508\) −38.3088 −1.69968
\(509\) 33.3725 1.47921 0.739605 0.673041i \(-0.235012\pi\)
0.739605 + 0.673041i \(0.235012\pi\)
\(510\) 0 0
\(511\) −8.98195 −0.397338
\(512\) 22.8302 1.00896
\(513\) 0 0
\(514\) 6.80338 0.300084
\(515\) −36.7405 −1.61898
\(516\) 0 0
\(517\) 0 0
\(518\) −0.837090 −0.0367796
\(519\) 0 0
\(520\) −14.2767 −0.626076
\(521\) −4.63971 −0.203269 −0.101635 0.994822i \(-0.532407\pi\)
−0.101635 + 0.994822i \(0.532407\pi\)
\(522\) 0 0
\(523\) −21.6267 −0.945670 −0.472835 0.881151i \(-0.656769\pi\)
−0.472835 + 0.881151i \(0.656769\pi\)
\(524\) −22.0743 −0.964320
\(525\) 0 0
\(526\) 6.17934 0.269432
\(527\) −21.1138 −0.919733
\(528\) 0 0
\(529\) 55.5827 2.41664
\(530\) −14.9544 −0.649576
\(531\) 0 0
\(532\) −9.43965 −0.409261
\(533\) −3.16291 −0.137001
\(534\) 0 0
\(535\) −51.7371 −2.23679
\(536\) −26.3810 −1.13949
\(537\) 0 0
\(538\) −4.21467 −0.181707
\(539\) 0 0
\(540\) 0 0
\(541\) −15.1043 −0.649384 −0.324692 0.945820i \(-0.605261\pi\)
−0.324692 + 0.945820i \(0.605261\pi\)
\(542\) −4.41711 −0.189731
\(543\) 0 0
\(544\) 15.7164 0.673834
\(545\) −47.0958 −2.01736
\(546\) 0 0
\(547\) −5.34836 −0.228679 −0.114340 0.993442i \(-0.536475\pi\)
−0.114340 + 0.993442i \(0.536475\pi\)
\(548\) 19.7395 0.843229
\(549\) 0 0
\(550\) 0 0
\(551\) −13.1138 −0.558668
\(552\) 0 0
\(553\) 12.3810 0.526494
\(554\) 9.78295 0.415638
\(555\) 0 0
\(556\) 23.4588 0.994873
\(557\) 1.76547 0.0748053 0.0374026 0.999300i \(-0.488092\pi\)
0.0374026 + 0.999300i \(0.488092\pi\)
\(558\) 0 0
\(559\) 12.0958 0.511597
\(560\) 8.83709 0.373435
\(561\) 0 0
\(562\) −5.21887 −0.220145
\(563\) −7.79650 −0.328583 −0.164292 0.986412i \(-0.552534\pi\)
−0.164292 + 0.986412i \(0.552534\pi\)
\(564\) 0 0
\(565\) −5.39744 −0.227072
\(566\) 1.02254 0.0429806
\(567\) 0 0
\(568\) −5.67418 −0.238083
\(569\) −42.2423 −1.77089 −0.885444 0.464746i \(-0.846146\pi\)
−0.885444 + 0.464746i \(0.846146\pi\)
\(570\) 0 0
\(571\) 1.64820 0.0689751 0.0344875 0.999405i \(-0.489020\pi\)
0.0344875 + 0.999405i \(0.489020\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −0.602558 −0.0251503
\(575\) 49.2603 2.05430
\(576\) 0 0
\(577\) 0.243026 0.0101173 0.00505866 0.999987i \(-0.498390\pi\)
0.00505866 + 0.999987i \(0.498390\pi\)
\(578\) −3.03265 −0.126142
\(579\) 0 0
\(580\) 14.2767 0.592809
\(581\) −13.0828 −0.542766
\(582\) 0 0
\(583\) 0 0
\(584\) −15.9740 −0.661010
\(585\) 0 0
\(586\) 7.46725 0.308469
\(587\) −5.85170 −0.241525 −0.120763 0.992681i \(-0.538534\pi\)
−0.120763 + 0.992681i \(0.538534\pi\)
\(588\) 0 0
\(589\) 34.4914 1.42119
\(590\) 3.68879 0.151865
\(591\) 0 0
\(592\) 4.83709 0.198803
\(593\) −21.1250 −0.867500 −0.433750 0.901033i \(-0.642810\pi\)
−0.433750 + 0.901033i \(0.642810\pi\)
\(594\) 0 0
\(595\) 10.5569 0.432791
\(596\) 22.4853 0.921033
\(597\) 0 0
\(598\) −10.3088 −0.421559
\(599\) −9.26719 −0.378647 −0.189323 0.981915i \(-0.560629\pi\)
−0.189323 + 0.981915i \(0.560629\pi\)
\(600\) 0 0
\(601\) 38.5811 1.57375 0.786877 0.617109i \(-0.211696\pi\)
0.786877 + 0.617109i \(0.211696\pi\)
\(602\) 2.30434 0.0939177
\(603\) 0 0
\(604\) 41.2863 1.67992
\(605\) 0 0
\(606\) 0 0
\(607\) 19.5423 0.793198 0.396599 0.917992i \(-0.370190\pi\)
0.396599 + 0.917992i \(0.370190\pi\)
\(608\) −25.6742 −1.04123
\(609\) 0 0
\(610\) −7.26031 −0.293961
\(611\) 9.80605 0.396711
\(612\) 0 0
\(613\) 25.6087 1.03432 0.517162 0.855887i \(-0.326988\pi\)
0.517162 + 0.855887i \(0.326988\pi\)
\(614\) 4.51738 0.182307
\(615\) 0 0
\(616\) 0 0
\(617\) 0.899161 0.0361989 0.0180994 0.999836i \(-0.494238\pi\)
0.0180994 + 0.999836i \(0.494238\pi\)
\(618\) 0 0
\(619\) 11.2311 0.451416 0.225708 0.974195i \(-0.427531\pi\)
0.225708 + 0.974195i \(0.427531\pi\)
\(620\) −37.5500 −1.50805
\(621\) 0 0
\(622\) 14.6302 0.586616
\(623\) 7.86469 0.315092
\(624\) 0 0
\(625\) −21.9053 −0.876211
\(626\) 8.41998 0.336530
\(627\) 0 0
\(628\) 37.4396 1.49400
\(629\) 5.77846 0.230402
\(630\) 0 0
\(631\) −9.95436 −0.396277 −0.198138 0.980174i \(-0.563490\pi\)
−0.198138 + 0.980174i \(0.563490\pi\)
\(632\) 22.0191 0.875873
\(633\) 0 0
\(634\) 7.02578 0.279029
\(635\) −69.9881 −2.77739
\(636\) 0 0
\(637\) −2.47068 −0.0978920
\(638\) 0 0
\(639\) 0 0
\(640\) 36.2699 1.43369
\(641\) −42.0828 −1.66217 −0.831085 0.556145i \(-0.812280\pi\)
−0.831085 + 0.556145i \(0.812280\pi\)
\(642\) 0 0
\(643\) 2.87930 0.113548 0.0567742 0.998387i \(-0.481918\pi\)
0.0567742 + 0.998387i \(0.481918\pi\)
\(644\) 15.7655 0.621246
\(645\) 0 0
\(646\) −8.11727 −0.319370
\(647\) −4.55530 −0.179087 −0.0895436 0.995983i \(-0.528541\pi\)
−0.0895436 + 0.995983i \(0.528541\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −6.46219 −0.253468
\(651\) 0 0
\(652\) −11.5569 −0.452604
\(653\) 37.7440 1.47704 0.738518 0.674234i \(-0.235526\pi\)
0.738518 + 0.674234i \(0.235526\pi\)
\(654\) 0 0
\(655\) −40.3285 −1.57576
\(656\) 3.48185 0.135944
\(657\) 0 0
\(658\) 1.86813 0.0728271
\(659\) −22.2423 −0.866436 −0.433218 0.901289i \(-0.642622\pi\)
−0.433218 + 0.901289i \(0.642622\pi\)
\(660\) 0 0
\(661\) 14.3027 0.556311 0.278156 0.960536i \(-0.410277\pi\)
0.278156 + 0.960536i \(0.410277\pi\)
\(662\) −3.23347 −0.125673
\(663\) 0 0
\(664\) −23.2672 −0.902942
\(665\) −17.2457 −0.668760
\(666\) 0 0
\(667\) 21.9018 0.848043
\(668\) 30.0743 1.16361
\(669\) 0 0
\(670\) −22.6854 −0.876412
\(671\) 0 0
\(672\) 0 0
\(673\) −4.89572 −0.188716 −0.0943581 0.995538i \(-0.530080\pi\)
−0.0943581 + 0.995538i \(0.530080\pi\)
\(674\) −6.83021 −0.263090
\(675\) 0 0
\(676\) 12.2637 0.471683
\(677\) 23.4147 0.899901 0.449951 0.893053i \(-0.351442\pi\)
0.449951 + 0.893053i \(0.351442\pi\)
\(678\) 0 0
\(679\) 14.8517 0.569956
\(680\) 18.7750 0.719989
\(681\) 0 0
\(682\) 0 0
\(683\) 24.2829 0.929158 0.464579 0.885532i \(-0.346206\pi\)
0.464579 + 0.885532i \(0.346206\pi\)
\(684\) 0 0
\(685\) 36.0629 1.37789
\(686\) −0.470683 −0.0179708
\(687\) 0 0
\(688\) −13.3155 −0.507649
\(689\) −24.1595 −0.920403
\(690\) 0 0
\(691\) −9.17590 −0.349068 −0.174534 0.984651i \(-0.555842\pi\)
−0.174534 + 0.984651i \(0.555842\pi\)
\(692\) 33.6312 1.27847
\(693\) 0 0
\(694\) −14.8140 −0.562331
\(695\) 42.8578 1.62569
\(696\) 0 0
\(697\) 4.15947 0.157551
\(698\) 10.6646 0.403662
\(699\) 0 0
\(700\) 9.88273 0.373532
\(701\) 20.9621 0.791727 0.395864 0.918309i \(-0.370445\pi\)
0.395864 + 0.918309i \(0.370445\pi\)
\(702\) 0 0
\(703\) −9.43965 −0.356023
\(704\) 0 0
\(705\) 0 0
\(706\) −7.19213 −0.270679
\(707\) 6.97240 0.262224
\(708\) 0 0
\(709\) 41.0974 1.54345 0.771723 0.635959i \(-0.219395\pi\)
0.771723 + 0.635959i \(0.219395\pi\)
\(710\) −4.87930 −0.183117
\(711\) 0 0
\(712\) 13.9870 0.524185
\(713\) −57.6052 −2.15733
\(714\) 0 0
\(715\) 0 0
\(716\) 26.9775 1.00819
\(717\) 0 0
\(718\) −7.75536 −0.289427
\(719\) −10.0621 −0.375252 −0.187626 0.982241i \(-0.560079\pi\)
−0.187626 + 0.982241i \(0.560079\pi\)
\(720\) 0 0
\(721\) −11.3078 −0.421123
\(722\) 4.31733 0.160674
\(723\) 0 0
\(724\) 27.5800 1.02500
\(725\) 13.7294 0.509896
\(726\) 0 0
\(727\) 1.33699 0.0495862 0.0247931 0.999693i \(-0.492107\pi\)
0.0247931 + 0.999693i \(0.492107\pi\)
\(728\) −4.39400 −0.162853
\(729\) 0 0
\(730\) −13.7363 −0.508401
\(731\) −15.9069 −0.588338
\(732\) 0 0
\(733\) −9.73787 −0.359676 −0.179838 0.983696i \(-0.557557\pi\)
−0.179838 + 0.983696i \(0.557557\pi\)
\(734\) 7.50859 0.277147
\(735\) 0 0
\(736\) 42.8793 1.58055
\(737\) 0 0
\(738\) 0 0
\(739\) −13.1430 −0.483475 −0.241737 0.970342i \(-0.577717\pi\)
−0.241737 + 0.970342i \(0.577717\pi\)
\(740\) 10.2767 0.377780
\(741\) 0 0
\(742\) −4.60256 −0.168965
\(743\) −3.31465 −0.121603 −0.0608013 0.998150i \(-0.519366\pi\)
−0.0608013 + 0.998150i \(0.519366\pi\)
\(744\) 0 0
\(745\) 41.0794 1.50503
\(746\) 0.340427 0.0124639
\(747\) 0 0
\(748\) 0 0
\(749\) −15.9233 −0.581825
\(750\) 0 0
\(751\) −23.8854 −0.871591 −0.435795 0.900046i \(-0.643533\pi\)
−0.435795 + 0.900046i \(0.643533\pi\)
\(752\) −10.7949 −0.393649
\(753\) 0 0
\(754\) −2.87318 −0.104635
\(755\) 75.4277 2.74510
\(756\) 0 0
\(757\) −49.5535 −1.80105 −0.900526 0.434802i \(-0.856818\pi\)
−0.900526 + 0.434802i \(0.856818\pi\)
\(758\) −1.14219 −0.0414861
\(759\) 0 0
\(760\) −30.6707 −1.11254
\(761\) 15.3404 0.556090 0.278045 0.960568i \(-0.410314\pi\)
0.278045 + 0.960568i \(0.410314\pi\)
\(762\) 0 0
\(763\) −14.4948 −0.524749
\(764\) 0.860192 0.0311207
\(765\) 0 0
\(766\) 10.0045 0.361477
\(767\) 5.95941 0.215182
\(768\) 0 0
\(769\) −32.0613 −1.15616 −0.578080 0.815980i \(-0.696198\pi\)
−0.578080 + 0.815980i \(0.696198\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 32.4232 1.16694
\(773\) −44.7209 −1.60850 −0.804249 0.594292i \(-0.797432\pi\)
−0.804249 + 0.594292i \(0.797432\pi\)
\(774\) 0 0
\(775\) −36.1104 −1.29712
\(776\) 26.4131 0.948175
\(777\) 0 0
\(778\) 2.25076 0.0806936
\(779\) −6.79488 −0.243452
\(780\) 0 0
\(781\) 0 0
\(782\) 13.5569 0.484794
\(783\) 0 0
\(784\) 2.71982 0.0971366
\(785\) 68.4001 2.44130
\(786\) 0 0
\(787\) −48.7000 −1.73597 −0.867983 0.496594i \(-0.834584\pi\)
−0.867983 + 0.496594i \(0.834584\pi\)
\(788\) −3.74237 −0.133316
\(789\) 0 0
\(790\) 18.9345 0.673659
\(791\) −1.66119 −0.0590651
\(792\) 0 0
\(793\) −11.7294 −0.416522
\(794\) 4.07830 0.144733
\(795\) 0 0
\(796\) 4.67762 0.165794
\(797\) −46.5224 −1.64791 −0.823955 0.566656i \(-0.808237\pi\)
−0.823955 + 0.566656i \(0.808237\pi\)
\(798\) 0 0
\(799\) −12.8957 −0.456218
\(800\) 26.8793 0.950327
\(801\) 0 0
\(802\) −7.13693 −0.252014
\(803\) 0 0
\(804\) 0 0
\(805\) 28.8026 1.01516
\(806\) 7.55691 0.266181
\(807\) 0 0
\(808\) 12.4001 0.436235
\(809\) −3.65957 −0.128664 −0.0643319 0.997929i \(-0.520492\pi\)
−0.0643319 + 0.997929i \(0.520492\pi\)
\(810\) 0 0
\(811\) 2.95597 0.103798 0.0518992 0.998652i \(-0.483473\pi\)
0.0518992 + 0.998652i \(0.483473\pi\)
\(812\) 4.39400 0.154199
\(813\) 0 0
\(814\) 0 0
\(815\) −21.1138 −0.739585
\(816\) 0 0
\(817\) 25.9854 0.909114
\(818\) 8.01299 0.280168
\(819\) 0 0
\(820\) 7.39744 0.258330
\(821\) 3.23109 0.112766 0.0563830 0.998409i \(-0.482043\pi\)
0.0563830 + 0.998409i \(0.482043\pi\)
\(822\) 0 0
\(823\) 21.9448 0.764948 0.382474 0.923966i \(-0.375072\pi\)
0.382474 + 0.923966i \(0.375072\pi\)
\(824\) −20.1104 −0.700579
\(825\) 0 0
\(826\) 1.13531 0.0395026
\(827\) −32.4508 −1.12843 −0.564213 0.825629i \(-0.690820\pi\)
−0.564213 + 0.825629i \(0.690820\pi\)
\(828\) 0 0
\(829\) 2.94298 0.102214 0.0511070 0.998693i \(-0.483725\pi\)
0.0511070 + 0.998693i \(0.483725\pi\)
\(830\) −20.0077 −0.694479
\(831\) 0 0
\(832\) 7.81455 0.270921
\(833\) 3.24914 0.112576
\(834\) 0 0
\(835\) 54.9440 1.90142
\(836\) 0 0
\(837\) 0 0
\(838\) −3.65270 −0.126180
\(839\) −17.4017 −0.600775 −0.300387 0.953817i \(-0.597116\pi\)
−0.300387 + 0.953817i \(0.597116\pi\)
\(840\) 0 0
\(841\) −22.8957 −0.789508
\(842\) −7.52932 −0.259477
\(843\) 0 0
\(844\) 30.1465 1.03768
\(845\) 22.4052 0.770761
\(846\) 0 0
\(847\) 0 0
\(848\) 26.5957 0.913299
\(849\) 0 0
\(850\) 8.49828 0.291489
\(851\) 15.7655 0.540433
\(852\) 0 0
\(853\) 41.9440 1.43614 0.718068 0.695973i \(-0.245027\pi\)
0.718068 + 0.695973i \(0.245027\pi\)
\(854\) −2.23453 −0.0764641
\(855\) 0 0
\(856\) −28.3189 −0.967922
\(857\) −9.82248 −0.335530 −0.167765 0.985827i \(-0.553655\pi\)
−0.167765 + 0.985827i \(0.553655\pi\)
\(858\) 0 0
\(859\) 12.8533 0.438549 0.219275 0.975663i \(-0.429631\pi\)
0.219275 + 0.975663i \(0.429631\pi\)
\(860\) −28.2897 −0.964672
\(861\) 0 0
\(862\) −12.3189 −0.419585
\(863\) −30.5795 −1.04094 −0.520468 0.853881i \(-0.674243\pi\)
−0.520468 + 0.853881i \(0.674243\pi\)
\(864\) 0 0
\(865\) 61.4423 2.08910
\(866\) −9.56303 −0.324965
\(867\) 0 0
\(868\) −11.5569 −0.392267
\(869\) 0 0
\(870\) 0 0
\(871\) −36.6493 −1.24181
\(872\) −25.7785 −0.872969
\(873\) 0 0
\(874\) −22.1465 −0.749116
\(875\) 1.80949 0.0611720
\(876\) 0 0
\(877\) −6.45608 −0.218006 −0.109003 0.994041i \(-0.534766\pi\)
−0.109003 + 0.994041i \(0.534766\pi\)
\(878\) −8.84320 −0.298444
\(879\) 0 0
\(880\) 0 0
\(881\) 13.4819 0.454215 0.227108 0.973870i \(-0.427073\pi\)
0.227108 + 0.973870i \(0.427073\pi\)
\(882\) 0 0
\(883\) 58.9372 1.98339 0.991697 0.128598i \(-0.0410477\pi\)
0.991697 + 0.128598i \(0.0410477\pi\)
\(884\) 14.2767 0.480179
\(885\) 0 0
\(886\) 16.4654 0.553167
\(887\) 1.79650 0.0603207 0.0301603 0.999545i \(-0.490398\pi\)
0.0301603 + 0.999545i \(0.490398\pi\)
\(888\) 0 0
\(889\) −21.5405 −0.722445
\(890\) 12.0276 0.403166
\(891\) 0 0
\(892\) 3.76547 0.126077
\(893\) 21.0664 0.704959
\(894\) 0 0
\(895\) 49.2863 1.64746
\(896\) 11.1629 0.372927
\(897\) 0 0
\(898\) 4.13026 0.137828
\(899\) −16.0552 −0.535471
\(900\) 0 0
\(901\) 31.7716 1.05846
\(902\) 0 0
\(903\) 0 0
\(904\) −2.95436 −0.0982604
\(905\) 50.3871 1.67492
\(906\) 0 0
\(907\) −4.87930 −0.162014 −0.0810072 0.996714i \(-0.525814\pi\)
−0.0810072 + 0.996714i \(0.525814\pi\)
\(908\) 50.5174 1.67648
\(909\) 0 0
\(910\) −3.77846 −0.125255
\(911\) 18.1364 0.600885 0.300442 0.953800i \(-0.402866\pi\)
0.300442 + 0.953800i \(0.402866\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 15.0173 0.496728
\(915\) 0 0
\(916\) −24.6026 −0.812891
\(917\) −12.4121 −0.409882
\(918\) 0 0
\(919\) 1.73625 0.0572737 0.0286368 0.999590i \(-0.490883\pi\)
0.0286368 + 0.999590i \(0.490883\pi\)
\(920\) 51.2242 1.68881
\(921\) 0 0
\(922\) −8.98807 −0.296006
\(923\) −7.88273 −0.259463
\(924\) 0 0
\(925\) 9.88273 0.324942
\(926\) 9.95168 0.327033
\(927\) 0 0
\(928\) 11.9509 0.392308
\(929\) 46.5354 1.52678 0.763389 0.645939i \(-0.223534\pi\)
0.763389 + 0.645939i \(0.223534\pi\)
\(930\) 0 0
\(931\) −5.30777 −0.173955
\(932\) −17.8337 −0.584161
\(933\) 0 0
\(934\) 3.31915 0.108606
\(935\) 0 0
\(936\) 0 0
\(937\) −32.2069 −1.05215 −0.526077 0.850437i \(-0.676338\pi\)
−0.526077 + 0.850437i \(0.676338\pi\)
\(938\) −6.98195 −0.227969
\(939\) 0 0
\(940\) −22.9345 −0.748041
\(941\) 5.13369 0.167354 0.0836768 0.996493i \(-0.473334\pi\)
0.0836768 + 0.996493i \(0.473334\pi\)
\(942\) 0 0
\(943\) 11.3484 0.369553
\(944\) −6.56035 −0.213521
\(945\) 0 0
\(946\) 0 0
\(947\) 52.5726 1.70838 0.854190 0.519962i \(-0.174054\pi\)
0.854190 + 0.519962i \(0.174054\pi\)
\(948\) 0 0
\(949\) −22.1916 −0.720369
\(950\) −13.8827 −0.450415
\(951\) 0 0
\(952\) 5.77846 0.187281
\(953\) −29.1077 −0.942891 −0.471446 0.881895i \(-0.656268\pi\)
−0.471446 + 0.881895i \(0.656268\pi\)
\(954\) 0 0
\(955\) 1.57152 0.0508533
\(956\) 9.88273 0.319630
\(957\) 0 0
\(958\) −9.75774 −0.315258
\(959\) 11.0992 0.358413
\(960\) 0 0
\(961\) 11.2277 0.362182
\(962\) −2.06819 −0.0666810
\(963\) 0 0
\(964\) 37.8950 1.22051
\(965\) 59.2354 1.90685
\(966\) 0 0
\(967\) −53.3319 −1.71504 −0.857520 0.514451i \(-0.827996\pi\)
−0.857520 + 0.514451i \(0.827996\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 22.7129 0.729269
\(971\) 3.85170 0.123607 0.0618034 0.998088i \(-0.480315\pi\)
0.0618034 + 0.998088i \(0.480315\pi\)
\(972\) 0 0
\(973\) 13.1905 0.422868
\(974\) −12.4508 −0.398950
\(975\) 0 0
\(976\) 12.9122 0.413308
\(977\) −24.0568 −0.769646 −0.384823 0.922990i \(-0.625737\pi\)
−0.384823 + 0.922990i \(0.625737\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 5.77846 0.184586
\(981\) 0 0
\(982\) 0.0620710 0.00198077
\(983\) −18.5604 −0.591983 −0.295992 0.955191i \(-0.595650\pi\)
−0.295992 + 0.955191i \(0.595650\pi\)
\(984\) 0 0
\(985\) −6.83709 −0.217848
\(986\) 3.77846 0.120331
\(987\) 0 0
\(988\) −23.3224 −0.741984
\(989\) −43.3991 −1.38001
\(990\) 0 0
\(991\) 26.2637 0.834295 0.417148 0.908839i \(-0.363030\pi\)
0.417148 + 0.908839i \(0.363030\pi\)
\(992\) −31.4328 −0.997992
\(993\) 0 0
\(994\) −1.50172 −0.0476316
\(995\) 8.54574 0.270918
\(996\) 0 0
\(997\) 28.0130 0.887180 0.443590 0.896230i \(-0.353705\pi\)
0.443590 + 0.896230i \(0.353705\pi\)
\(998\) −8.58719 −0.271823
\(999\) 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 7623.2.a.cc.1.2 3
3.2 odd 2 2541.2.a.bh.1.2 3
11.10 odd 2 7623.2.a.ca.1.2 3
33.32 even 2 2541.2.a.bj.1.2 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2541.2.a.bh.1.2 3 3.2 odd 2
2541.2.a.bj.1.2 yes 3 33.32 even 2
7623.2.a.ca.1.2 3 11.10 odd 2
7623.2.a.cc.1.2 3 1.1 even 1 trivial