Properties

Label 1805.2.a.l.1.1
Level $1805$
Weight $2$
Character 1805.1
Self dual yes
Analytic conductor $14.413$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1805.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4129975648\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
Defining polynomial: \( x^{4} - 5x^{2} + 5 \) 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 1.1
Root \(-1.90211\) of defining polynomial
Character \(\chi\) \(=\) 1805.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.90211 q^{2} -1.90211 q^{3} +1.61803 q^{4} -1.00000 q^{5} +3.61803 q^{6} -4.23607 q^{7} +0.726543 q^{8} +0.618034 q^{9} +O(q^{10})\) \(q-1.90211 q^{2} -1.90211 q^{3} +1.61803 q^{4} -1.00000 q^{5} +3.61803 q^{6} -4.23607 q^{7} +0.726543 q^{8} +0.618034 q^{9} +1.90211 q^{10} -5.85410 q^{11} -3.07768 q^{12} +3.07768 q^{13} +8.05748 q^{14} +1.90211 q^{15} -4.61803 q^{16} +5.23607 q^{17} -1.17557 q^{18} -1.61803 q^{20} +8.05748 q^{21} +11.1352 q^{22} +4.09017 q^{23} -1.38197 q^{24} +1.00000 q^{25} -5.85410 q^{26} +4.53077 q^{27} -6.85410 q^{28} -2.80017 q^{29} -3.61803 q^{30} +1.90211 q^{31} +7.33094 q^{32} +11.1352 q^{33} -9.95959 q^{34} +4.23607 q^{35} +1.00000 q^{36} -2.80017 q^{37} -5.85410 q^{39} -0.726543 q^{40} -6.88191 q^{41} -15.3262 q^{42} +0.381966 q^{43} -9.47214 q^{44} -0.618034 q^{45} -7.77997 q^{46} -1.47214 q^{47} +8.78402 q^{48} +10.9443 q^{49} -1.90211 q^{50} -9.95959 q^{51} +4.97980 q^{52} +11.1352 q^{53} -8.61803 q^{54} +5.85410 q^{55} -3.07768 q^{56} +5.32624 q^{58} +14.0413 q^{59} +3.07768 q^{60} +3.94427 q^{61} -3.61803 q^{62} -2.61803 q^{63} -4.70820 q^{64} -3.07768 q^{65} -21.1803 q^{66} +5.98385 q^{67} +8.47214 q^{68} -7.77997 q^{69} -8.05748 q^{70} +0.171513 q^{71} +0.449028 q^{72} -1.00000 q^{73} +5.32624 q^{74} -1.90211 q^{75} +24.7984 q^{77} +11.1352 q^{78} -5.25731 q^{79} +4.61803 q^{80} -10.4721 q^{81} +13.0902 q^{82} -8.76393 q^{83} +13.0373 q^{84} -5.23607 q^{85} -0.726543 q^{86} +5.32624 q^{87} -4.25325 q^{88} +7.77997 q^{89} +1.17557 q^{90} -13.0373 q^{91} +6.61803 q^{92} -3.61803 q^{93} +2.80017 q^{94} -13.9443 q^{96} -2.62866 q^{97} -20.8172 q^{98} -3.61803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 4 q^{5} + 10 q^{6} - 8 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 4 q^{5} + 10 q^{6} - 8 q^{7} - 2 q^{9} - 10 q^{11} - 14 q^{16} + 12 q^{17} - 2 q^{20} - 6 q^{23} - 10 q^{24} + 4 q^{25} - 10 q^{26} - 14 q^{28} - 10 q^{30} + 8 q^{35} + 4 q^{36} - 10 q^{39} - 30 q^{42} + 6 q^{43} - 20 q^{44} + 2 q^{45} + 12 q^{47} + 8 q^{49} - 30 q^{54} + 10 q^{55} - 10 q^{58} - 20 q^{61} - 10 q^{62} - 6 q^{63} + 8 q^{64} - 40 q^{66} + 16 q^{68} - 4 q^{73} - 10 q^{74} + 50 q^{77} + 14 q^{80} - 24 q^{81} + 30 q^{82} - 44 q^{83} - 12 q^{85} - 10 q^{87} + 22 q^{92} - 10 q^{93} - 20 q^{96} - 10 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.90211 −1.34500 −0.672499 0.740098i \(-0.734779\pi\)
−0.672499 + 0.740098i \(0.734779\pi\)
\(3\) −1.90211 −1.09819 −0.549093 0.835761i \(-0.685027\pi\)
−0.549093 + 0.835761i \(0.685027\pi\)
\(4\) 1.61803 0.809017
\(5\) −1.00000 −0.447214
\(6\) 3.61803 1.47706
\(7\) −4.23607 −1.60108 −0.800542 0.599277i \(-0.795455\pi\)
−0.800542 + 0.599277i \(0.795455\pi\)
\(8\) 0.726543 0.256872
\(9\) 0.618034 0.206011
\(10\) 1.90211 0.601501
\(11\) −5.85410 −1.76508 −0.882539 0.470239i \(-0.844168\pi\)
−0.882539 + 0.470239i \(0.844168\pi\)
\(12\) −3.07768 −0.888451
\(13\) 3.07768 0.853596 0.426798 0.904347i \(-0.359642\pi\)
0.426798 + 0.904347i \(0.359642\pi\)
\(14\) 8.05748 2.15345
\(15\) 1.90211 0.491123
\(16\) −4.61803 −1.15451
\(17\) 5.23607 1.26993 0.634967 0.772540i \(-0.281014\pi\)
0.634967 + 0.772540i \(0.281014\pi\)
\(18\) −1.17557 −0.277085
\(19\) 0 0
\(20\) −1.61803 −0.361803
\(21\) 8.05748 1.75829
\(22\) 11.1352 2.37402
\(23\) 4.09017 0.852859 0.426430 0.904521i \(-0.359771\pi\)
0.426430 + 0.904521i \(0.359771\pi\)
\(24\) −1.38197 −0.282093
\(25\) 1.00000 0.200000
\(26\) −5.85410 −1.14808
\(27\) 4.53077 0.871947
\(28\) −6.85410 −1.29530
\(29\) −2.80017 −0.519978 −0.259989 0.965612i \(-0.583719\pi\)
−0.259989 + 0.965612i \(0.583719\pi\)
\(30\) −3.61803 −0.660560
\(31\) 1.90211 0.341630 0.170815 0.985303i \(-0.445360\pi\)
0.170815 + 0.985303i \(0.445360\pi\)
\(32\) 7.33094 1.29594
\(33\) 11.1352 1.93838
\(34\) −9.95959 −1.70806
\(35\) 4.23607 0.716026
\(36\) 1.00000 0.166667
\(37\) −2.80017 −0.460345 −0.230172 0.973150i \(-0.573929\pi\)
−0.230172 + 0.973150i \(0.573929\pi\)
\(38\) 0 0
\(39\) −5.85410 −0.937407
\(40\) −0.726543 −0.114876
\(41\) −6.88191 −1.07477 −0.537387 0.843336i \(-0.680588\pi\)
−0.537387 + 0.843336i \(0.680588\pi\)
\(42\) −15.3262 −2.36489
\(43\) 0.381966 0.0582493 0.0291246 0.999576i \(-0.490728\pi\)
0.0291246 + 0.999576i \(0.490728\pi\)
\(44\) −9.47214 −1.42798
\(45\) −0.618034 −0.0921311
\(46\) −7.77997 −1.14709
\(47\) −1.47214 −0.214733 −0.107367 0.994220i \(-0.534242\pi\)
−0.107367 + 0.994220i \(0.534242\pi\)
\(48\) 8.78402 1.26786
\(49\) 10.9443 1.56347
\(50\) −1.90211 −0.268999
\(51\) −9.95959 −1.39462
\(52\) 4.97980 0.690574
\(53\) 11.1352 1.52953 0.764766 0.644308i \(-0.222854\pi\)
0.764766 + 0.644308i \(0.222854\pi\)
\(54\) −8.61803 −1.17277
\(55\) 5.85410 0.789367
\(56\) −3.07768 −0.411273
\(57\) 0 0
\(58\) 5.32624 0.699369
\(59\) 14.0413 1.82803 0.914013 0.405685i \(-0.132967\pi\)
0.914013 + 0.405685i \(0.132967\pi\)
\(60\) 3.07768 0.397327
\(61\) 3.94427 0.505012 0.252506 0.967595i \(-0.418745\pi\)
0.252506 + 0.967595i \(0.418745\pi\)
\(62\) −3.61803 −0.459491
\(63\) −2.61803 −0.329841
\(64\) −4.70820 −0.588525
\(65\) −3.07768 −0.381740
\(66\) −21.1803 −2.60712
\(67\) 5.98385 0.731044 0.365522 0.930803i \(-0.380890\pi\)
0.365522 + 0.930803i \(0.380890\pi\)
\(68\) 8.47214 1.02740
\(69\) −7.77997 −0.936598
\(70\) −8.05748 −0.963053
\(71\) 0.171513 0.0203549 0.0101774 0.999948i \(-0.496760\pi\)
0.0101774 + 0.999948i \(0.496760\pi\)
\(72\) 0.449028 0.0529185
\(73\) −1.00000 −0.117041 −0.0585206 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) 5.32624 0.619163
\(75\) −1.90211 −0.219637
\(76\) 0 0
\(77\) 24.7984 2.82604
\(78\) 11.1352 1.26081
\(79\) −5.25731 −0.591494 −0.295747 0.955266i \(-0.595568\pi\)
−0.295747 + 0.955266i \(0.595568\pi\)
\(80\) 4.61803 0.516312
\(81\) −10.4721 −1.16357
\(82\) 13.0902 1.44557
\(83\) −8.76393 −0.961967 −0.480983 0.876730i \(-0.659720\pi\)
−0.480983 + 0.876730i \(0.659720\pi\)
\(84\) 13.0373 1.42248
\(85\) −5.23607 −0.567931
\(86\) −0.726543 −0.0783451
\(87\) 5.32624 0.571033
\(88\) −4.25325 −0.453398
\(89\) 7.77997 0.824675 0.412337 0.911031i \(-0.364713\pi\)
0.412337 + 0.911031i \(0.364713\pi\)
\(90\) 1.17557 0.123916
\(91\) −13.0373 −1.36668
\(92\) 6.61803 0.689978
\(93\) −3.61803 −0.375173
\(94\) 2.80017 0.288815
\(95\) 0 0
\(96\) −13.9443 −1.42318
\(97\) −2.62866 −0.266900 −0.133450 0.991056i \(-0.542605\pi\)
−0.133450 + 0.991056i \(0.542605\pi\)
\(98\) −20.8172 −2.10286
\(99\) −3.61803 −0.363626
\(100\) 1.61803 0.161803
\(101\) 3.29180 0.327546 0.163773 0.986498i \(-0.447634\pi\)
0.163773 + 0.986498i \(0.447634\pi\)
\(102\) 18.9443 1.87576
\(103\) −14.9394 −1.47202 −0.736011 0.676970i \(-0.763293\pi\)
−0.736011 + 0.676970i \(0.763293\pi\)
\(104\) 2.23607 0.219265
\(105\) −8.05748 −0.786330
\(106\) −21.1803 −2.05722
\(107\) −8.33499 −0.805774 −0.402887 0.915250i \(-0.631993\pi\)
−0.402887 + 0.915250i \(0.631993\pi\)
\(108\) 7.33094 0.705420
\(109\) −2.17963 −0.208770 −0.104385 0.994537i \(-0.533287\pi\)
−0.104385 + 0.994537i \(0.533287\pi\)
\(110\) −11.1352 −1.06170
\(111\) 5.32624 0.505544
\(112\) 19.5623 1.84846
\(113\) 18.4661 1.73714 0.868572 0.495562i \(-0.165038\pi\)
0.868572 + 0.495562i \(0.165038\pi\)
\(114\) 0 0
\(115\) −4.09017 −0.381410
\(116\) −4.53077 −0.420671
\(117\) 1.90211 0.175850
\(118\) −26.7082 −2.45869
\(119\) −22.1803 −2.03327
\(120\) 1.38197 0.126156
\(121\) 23.2705 2.11550
\(122\) −7.50245 −0.679240
\(123\) 13.0902 1.18030
\(124\) 3.07768 0.276384
\(125\) −1.00000 −0.0894427
\(126\) 4.97980 0.443636
\(127\) 9.23305 0.819301 0.409650 0.912243i \(-0.365651\pi\)
0.409650 + 0.912243i \(0.365651\pi\)
\(128\) −5.70634 −0.504374
\(129\) −0.726543 −0.0639685
\(130\) 5.85410 0.513439
\(131\) −15.6180 −1.36455 −0.682277 0.731094i \(-0.739010\pi\)
−0.682277 + 0.731094i \(0.739010\pi\)
\(132\) 18.0171 1.56818
\(133\) 0 0
\(134\) −11.3820 −0.983252
\(135\) −4.53077 −0.389946
\(136\) 3.80423 0.326210
\(137\) 9.76393 0.834189 0.417095 0.908863i \(-0.363048\pi\)
0.417095 + 0.908863i \(0.363048\pi\)
\(138\) 14.7984 1.25972
\(139\) −22.0902 −1.87366 −0.936832 0.349780i \(-0.886256\pi\)
−0.936832 + 0.349780i \(0.886256\pi\)
\(140\) 6.85410 0.579277
\(141\) 2.80017 0.235817
\(142\) −0.326238 −0.0273773
\(143\) −18.0171 −1.50666
\(144\) −2.85410 −0.237842
\(145\) 2.80017 0.232541
\(146\) 1.90211 0.157420
\(147\) −20.8172 −1.71698
\(148\) −4.53077 −0.372427
\(149\) 7.09017 0.580849 0.290425 0.956898i \(-0.406203\pi\)
0.290425 + 0.956898i \(0.406203\pi\)
\(150\) 3.61803 0.295411
\(151\) 0.620541 0.0504989 0.0252495 0.999681i \(-0.491962\pi\)
0.0252495 + 0.999681i \(0.491962\pi\)
\(152\) 0 0
\(153\) 3.23607 0.261621
\(154\) −47.1693 −3.80101
\(155\) −1.90211 −0.152781
\(156\) −9.47214 −0.758378
\(157\) −16.2705 −1.29853 −0.649264 0.760563i \(-0.724923\pi\)
−0.649264 + 0.760563i \(0.724923\pi\)
\(158\) 10.0000 0.795557
\(159\) −21.1803 −1.67971
\(160\) −7.33094 −0.579562
\(161\) −17.3262 −1.36550
\(162\) 19.9192 1.56500
\(163\) −25.1803 −1.97228 −0.986138 0.165926i \(-0.946939\pi\)
−0.986138 + 0.165926i \(0.946939\pi\)
\(164\) −11.1352 −0.869510
\(165\) −11.1352 −0.866871
\(166\) 16.6700 1.29384
\(167\) 0.171513 0.0132721 0.00663605 0.999978i \(-0.497888\pi\)
0.00663605 + 0.999978i \(0.497888\pi\)
\(168\) 5.85410 0.451654
\(169\) −3.52786 −0.271374
\(170\) 9.95959 0.763866
\(171\) 0 0
\(172\) 0.618034 0.0471246
\(173\) −13.2088 −1.00425 −0.502123 0.864796i \(-0.667447\pi\)
−0.502123 + 0.864796i \(0.667447\pi\)
\(174\) −10.1311 −0.768037
\(175\) −4.23607 −0.320217
\(176\) 27.0344 2.03780
\(177\) −26.7082 −2.00751
\(178\) −14.7984 −1.10919
\(179\) 10.1311 0.757234 0.378617 0.925553i \(-0.376400\pi\)
0.378617 + 0.925553i \(0.376400\pi\)
\(180\) −1.00000 −0.0745356
\(181\) −6.71040 −0.498780 −0.249390 0.968403i \(-0.580230\pi\)
−0.249390 + 0.968403i \(0.580230\pi\)
\(182\) 24.7984 1.83818
\(183\) −7.50245 −0.554597
\(184\) 2.97168 0.219075
\(185\) 2.80017 0.205873
\(186\) 6.88191 0.504606
\(187\) −30.6525 −2.24153
\(188\) −2.38197 −0.173723
\(189\) −19.1926 −1.39606
\(190\) 0 0
\(191\) 13.0000 0.940647 0.470323 0.882494i \(-0.344137\pi\)
0.470323 + 0.882494i \(0.344137\pi\)
\(192\) 8.95554 0.646310
\(193\) 1.45309 0.104595 0.0522977 0.998632i \(-0.483346\pi\)
0.0522977 + 0.998632i \(0.483346\pi\)
\(194\) 5.00000 0.358979
\(195\) 5.85410 0.419221
\(196\) 17.7082 1.26487
\(197\) 14.8885 1.06076 0.530382 0.847759i \(-0.322048\pi\)
0.530382 + 0.847759i \(0.322048\pi\)
\(198\) 6.88191 0.489076
\(199\) 13.4164 0.951064 0.475532 0.879698i \(-0.342256\pi\)
0.475532 + 0.879698i \(0.342256\pi\)
\(200\) 0.726543 0.0513743
\(201\) −11.3820 −0.802822
\(202\) −6.26137 −0.440548
\(203\) 11.8617 0.832529
\(204\) −16.1150 −1.12827
\(205\) 6.88191 0.480653
\(206\) 28.4164 1.97986
\(207\) 2.52786 0.175699
\(208\) −14.2128 −0.985484
\(209\) 0 0
\(210\) 15.3262 1.05761
\(211\) −1.73060 −0.119139 −0.0595697 0.998224i \(-0.518973\pi\)
−0.0595697 + 0.998224i \(0.518973\pi\)
\(212\) 18.0171 1.23742
\(213\) −0.326238 −0.0223535
\(214\) 15.8541 1.08376
\(215\) −0.381966 −0.0260499
\(216\) 3.29180 0.223978
\(217\) −8.05748 −0.546977
\(218\) 4.14590 0.280796
\(219\) 1.90211 0.128533
\(220\) 9.47214 0.638611
\(221\) 16.1150 1.08401
\(222\) −10.1311 −0.679955
\(223\) −14.8334 −0.993317 −0.496659 0.867946i \(-0.665440\pi\)
−0.496659 + 0.867946i \(0.665440\pi\)
\(224\) −31.0543 −2.07491
\(225\) 0.618034 0.0412023
\(226\) −35.1246 −2.33645
\(227\) 17.3965 1.15465 0.577324 0.816515i \(-0.304097\pi\)
0.577324 + 0.816515i \(0.304097\pi\)
\(228\) 0 0
\(229\) −18.1459 −1.19911 −0.599557 0.800332i \(-0.704657\pi\)
−0.599557 + 0.800332i \(0.704657\pi\)
\(230\) 7.77997 0.512996
\(231\) −47.1693 −3.10351
\(232\) −2.03444 −0.133568
\(233\) 1.76393 0.115559 0.0577795 0.998329i \(-0.481598\pi\)
0.0577795 + 0.998329i \(0.481598\pi\)
\(234\) −3.61803 −0.236518
\(235\) 1.47214 0.0960316
\(236\) 22.7194 1.47890
\(237\) 10.0000 0.649570
\(238\) 42.1895 2.73474
\(239\) −23.2148 −1.50164 −0.750820 0.660507i \(-0.770341\pi\)
−0.750820 + 0.660507i \(0.770341\pi\)
\(240\) −8.78402 −0.567006
\(241\) −6.32688 −0.407550 −0.203775 0.979018i \(-0.565321\pi\)
−0.203775 + 0.979018i \(0.565321\pi\)
\(242\) −44.2631 −2.84534
\(243\) 6.32688 0.405870
\(244\) 6.38197 0.408564
\(245\) −10.9443 −0.699204
\(246\) −24.8990 −1.58750
\(247\) 0 0
\(248\) 1.38197 0.0877549
\(249\) 16.6700 1.05642
\(250\) 1.90211 0.120300
\(251\) −3.00000 −0.189358 −0.0946792 0.995508i \(-0.530183\pi\)
−0.0946792 + 0.995508i \(0.530183\pi\)
\(252\) −4.23607 −0.266847
\(253\) −23.9443 −1.50536
\(254\) −17.5623 −1.10196
\(255\) 9.95959 0.623694
\(256\) 20.2705 1.26691
\(257\) −3.24920 −0.202679 −0.101340 0.994852i \(-0.532313\pi\)
−0.101340 + 0.994852i \(0.532313\pi\)
\(258\) 1.38197 0.0860374
\(259\) 11.8617 0.737051
\(260\) −4.97980 −0.308834
\(261\) −1.73060 −0.107121
\(262\) 29.7073 1.83532
\(263\) −6.00000 −0.369976 −0.184988 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) 8.09017 0.497916
\(265\) −11.1352 −0.684028
\(266\) 0 0
\(267\) −14.7984 −0.905646
\(268\) 9.68208 0.591427
\(269\) 19.6417 1.19757 0.598787 0.800908i \(-0.295650\pi\)
0.598787 + 0.800908i \(0.295650\pi\)
\(270\) 8.61803 0.524477
\(271\) 8.61803 0.523508 0.261754 0.965135i \(-0.415699\pi\)
0.261754 + 0.965135i \(0.415699\pi\)
\(272\) −24.1803 −1.46615
\(273\) 24.7984 1.50087
\(274\) −18.5721 −1.12198
\(275\) −5.85410 −0.353016
\(276\) −12.5882 −0.757724
\(277\) 24.8328 1.49206 0.746030 0.665913i \(-0.231958\pi\)
0.746030 + 0.665913i \(0.231958\pi\)
\(278\) 42.0180 2.52007
\(279\) 1.17557 0.0703796
\(280\) 3.07768 0.183927
\(281\) −25.0705 −1.49558 −0.747790 0.663935i \(-0.768885\pi\)
−0.747790 + 0.663935i \(0.768885\pi\)
\(282\) −5.32624 −0.317173
\(283\) −28.2705 −1.68051 −0.840254 0.542193i \(-0.817594\pi\)
−0.840254 + 0.542193i \(0.817594\pi\)
\(284\) 0.277515 0.0164675
\(285\) 0 0
\(286\) 34.2705 2.02646
\(287\) 29.1522 1.72080
\(288\) 4.53077 0.266978
\(289\) 10.4164 0.612730
\(290\) −5.32624 −0.312767
\(291\) 5.00000 0.293105
\(292\) −1.61803 −0.0946883
\(293\) 13.8293 0.807918 0.403959 0.914777i \(-0.367634\pi\)
0.403959 + 0.914777i \(0.367634\pi\)
\(294\) 39.5967 2.30933
\(295\) −14.0413 −0.817518
\(296\) −2.03444 −0.118250
\(297\) −26.5236 −1.53905
\(298\) −13.4863 −0.781241
\(299\) 12.5882 0.727997
\(300\) −3.07768 −0.177690
\(301\) −1.61803 −0.0932619
\(302\) −1.18034 −0.0679209
\(303\) −6.26137 −0.359706
\(304\) 0 0
\(305\) −3.94427 −0.225848
\(306\) −6.15537 −0.351879
\(307\) 18.3601 1.04787 0.523933 0.851759i \(-0.324464\pi\)
0.523933 + 0.851759i \(0.324464\pi\)
\(308\) 40.1246 2.28631
\(309\) 28.4164 1.61655
\(310\) 3.61803 0.205491
\(311\) −2.76393 −0.156728 −0.0783641 0.996925i \(-0.524970\pi\)
−0.0783641 + 0.996925i \(0.524970\pi\)
\(312\) −4.25325 −0.240793
\(313\) 3.27051 0.184860 0.0924301 0.995719i \(-0.470537\pi\)
0.0924301 + 0.995719i \(0.470537\pi\)
\(314\) 30.9483 1.74652
\(315\) 2.61803 0.147510
\(316\) −8.50651 −0.478528
\(317\) −16.4580 −0.924373 −0.462186 0.886783i \(-0.652935\pi\)
−0.462186 + 0.886783i \(0.652935\pi\)
\(318\) 40.2874 2.25921
\(319\) 16.3925 0.917802
\(320\) 4.70820 0.263197
\(321\) 15.8541 0.884890
\(322\) 32.9565 1.83659
\(323\) 0 0
\(324\) −16.9443 −0.941348
\(325\) 3.07768 0.170719
\(326\) 47.8959 2.65271
\(327\) 4.14590 0.229269
\(328\) −5.00000 −0.276079
\(329\) 6.23607 0.343806
\(330\) 21.1803 1.16594
\(331\) −19.9192 −1.09486 −0.547429 0.836852i \(-0.684393\pi\)
−0.547429 + 0.836852i \(0.684393\pi\)
\(332\) −14.1803 −0.778247
\(333\) −1.73060 −0.0948363
\(334\) −0.326238 −0.0178509
\(335\) −5.98385 −0.326933
\(336\) −37.2097 −2.02996
\(337\) 28.4257 1.54845 0.774223 0.632913i \(-0.218141\pi\)
0.774223 + 0.632913i \(0.218141\pi\)
\(338\) 6.71040 0.364997
\(339\) −35.1246 −1.90771
\(340\) −8.47214 −0.459466
\(341\) −11.1352 −0.603003
\(342\) 0 0
\(343\) −16.7082 −0.902158
\(344\) 0.277515 0.0149626
\(345\) 7.77997 0.418859
\(346\) 25.1246 1.35071
\(347\) 16.4721 0.884271 0.442135 0.896948i \(-0.354221\pi\)
0.442135 + 0.896948i \(0.354221\pi\)
\(348\) 8.61803 0.461975
\(349\) −2.56231 −0.137157 −0.0685785 0.997646i \(-0.521846\pi\)
−0.0685785 + 0.997646i \(0.521846\pi\)
\(350\) 8.05748 0.430690
\(351\) 13.9443 0.744290
\(352\) −42.9161 −2.28743
\(353\) 3.67376 0.195535 0.0977673 0.995209i \(-0.468830\pi\)
0.0977673 + 0.995209i \(0.468830\pi\)
\(354\) 50.8020 2.70010
\(355\) −0.171513 −0.00910299
\(356\) 12.5882 0.667176
\(357\) 42.1895 2.23291
\(358\) −19.2705 −1.01848
\(359\) 7.38197 0.389605 0.194803 0.980842i \(-0.437593\pi\)
0.194803 + 0.980842i \(0.437593\pi\)
\(360\) −0.449028 −0.0236659
\(361\) 0 0
\(362\) 12.7639 0.670857
\(363\) −44.2631 −2.32321
\(364\) −21.0948 −1.10567
\(365\) 1.00000 0.0523424
\(366\) 14.2705 0.745931
\(367\) 31.0689 1.62178 0.810891 0.585197i \(-0.198983\pi\)
0.810891 + 0.585197i \(0.198983\pi\)
\(368\) −18.8885 −0.984633
\(369\) −4.25325 −0.221416
\(370\) −5.32624 −0.276898
\(371\) −47.1693 −2.44891
\(372\) −5.85410 −0.303521
\(373\) 5.42882 0.281094 0.140547 0.990074i \(-0.455114\pi\)
0.140547 + 0.990074i \(0.455114\pi\)
\(374\) 58.3045 3.01485
\(375\) 1.90211 0.0982247
\(376\) −1.06957 −0.0551588
\(377\) −8.61803 −0.443851
\(378\) 36.5066 1.87770
\(379\) −12.8658 −0.660870 −0.330435 0.943829i \(-0.607195\pi\)
−0.330435 + 0.943829i \(0.607195\pi\)
\(380\) 0 0
\(381\) −17.5623 −0.899744
\(382\) −24.7275 −1.26517
\(383\) −4.25325 −0.217331 −0.108666 0.994078i \(-0.534658\pi\)
−0.108666 + 0.994078i \(0.534658\pi\)
\(384\) 10.8541 0.553896
\(385\) −24.7984 −1.26384
\(386\) −2.76393 −0.140680
\(387\) 0.236068 0.0120000
\(388\) −4.25325 −0.215926
\(389\) −20.8541 −1.05734 −0.528672 0.848826i \(-0.677310\pi\)
−0.528672 + 0.848826i \(0.677310\pi\)
\(390\) −11.1352 −0.563851
\(391\) 21.4164 1.08307
\(392\) 7.95148 0.401610
\(393\) 29.7073 1.49853
\(394\) −28.3197 −1.42673
\(395\) 5.25731 0.264524
\(396\) −5.85410 −0.294180
\(397\) −23.8328 −1.19613 −0.598067 0.801446i \(-0.704064\pi\)
−0.598067 + 0.801446i \(0.704064\pi\)
\(398\) −25.5195 −1.27918
\(399\) 0 0
\(400\) −4.61803 −0.230902
\(401\) 9.06154 0.452512 0.226256 0.974068i \(-0.427351\pi\)
0.226256 + 0.974068i \(0.427351\pi\)
\(402\) 21.6498 1.07979
\(403\) 5.85410 0.291614
\(404\) 5.32624 0.264990
\(405\) 10.4721 0.520365
\(406\) −22.5623 −1.11975
\(407\) 16.3925 0.812545
\(408\) −7.23607 −0.358239
\(409\) 21.7153 1.07375 0.536876 0.843661i \(-0.319604\pi\)
0.536876 + 0.843661i \(0.319604\pi\)
\(410\) −13.0902 −0.646477
\(411\) −18.5721 −0.916094
\(412\) −24.1724 −1.19089
\(413\) −59.4800 −2.92682
\(414\) −4.80828 −0.236314
\(415\) 8.76393 0.430205
\(416\) 22.5623 1.10621
\(417\) 42.0180 2.05763
\(418\) 0 0
\(419\) 26.1803 1.27899 0.639497 0.768794i \(-0.279143\pi\)
0.639497 + 0.768794i \(0.279143\pi\)
\(420\) −13.0373 −0.636154
\(421\) −11.5842 −0.564579 −0.282289 0.959329i \(-0.591094\pi\)
−0.282289 + 0.959329i \(0.591094\pi\)
\(422\) 3.29180 0.160242
\(423\) −0.909830 −0.0442375
\(424\) 8.09017 0.392893
\(425\) 5.23607 0.253987
\(426\) 0.620541 0.0300653
\(427\) −16.7082 −0.808567
\(428\) −13.4863 −0.651885
\(429\) 34.2705 1.65460
\(430\) 0.726543 0.0350370
\(431\) 6.88191 0.331490 0.165745 0.986169i \(-0.446997\pi\)
0.165745 + 0.986169i \(0.446997\pi\)
\(432\) −20.9232 −1.00667
\(433\) −6.04937 −0.290714 −0.145357 0.989379i \(-0.546433\pi\)
−0.145357 + 0.989379i \(0.546433\pi\)
\(434\) 15.3262 0.735683
\(435\) −5.32624 −0.255374
\(436\) −3.52671 −0.168899
\(437\) 0 0
\(438\) −3.61803 −0.172876
\(439\) 5.64083 0.269222 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(440\) 4.25325 0.202766
\(441\) 6.76393 0.322092
\(442\) −30.6525 −1.45799
\(443\) 34.7771 1.65231 0.826155 0.563443i \(-0.190524\pi\)
0.826155 + 0.563443i \(0.190524\pi\)
\(444\) 8.61803 0.408994
\(445\) −7.77997 −0.368806
\(446\) 28.2148 1.33601
\(447\) −13.4863 −0.637880
\(448\) 19.9443 0.942278
\(449\) 5.64083 0.266207 0.133104 0.991102i \(-0.457506\pi\)
0.133104 + 0.991102i \(0.457506\pi\)
\(450\) −1.17557 −0.0554169
\(451\) 40.2874 1.89706
\(452\) 29.8788 1.40538
\(453\) −1.18034 −0.0554572
\(454\) −33.0902 −1.55300
\(455\) 13.0373 0.611197
\(456\) 0 0
\(457\) 5.23607 0.244933 0.122466 0.992473i \(-0.460920\pi\)
0.122466 + 0.992473i \(0.460920\pi\)
\(458\) 34.5155 1.61281
\(459\) 23.7234 1.10731
\(460\) −6.61803 −0.308567
\(461\) −8.00000 −0.372597 −0.186299 0.982493i \(-0.559649\pi\)
−0.186299 + 0.982493i \(0.559649\pi\)
\(462\) 89.7214 4.17422
\(463\) 0.145898 0.00678046 0.00339023 0.999994i \(-0.498921\pi\)
0.00339023 + 0.999994i \(0.498921\pi\)
\(464\) 12.9313 0.600319
\(465\) 3.61803 0.167782
\(466\) −3.35520 −0.155427
\(467\) −9.18034 −0.424815 −0.212408 0.977181i \(-0.568130\pi\)
−0.212408 + 0.977181i \(0.568130\pi\)
\(468\) 3.07768 0.142266
\(469\) −25.3480 −1.17046
\(470\) −2.80017 −0.129162
\(471\) 30.9483 1.42602
\(472\) 10.2016 0.469568
\(473\) −2.23607 −0.102815
\(474\) −19.0211 −0.873669
\(475\) 0 0
\(476\) −35.8885 −1.64495
\(477\) 6.88191 0.315101
\(478\) 44.1571 2.01970
\(479\) −11.4377 −0.522602 −0.261301 0.965257i \(-0.584151\pi\)
−0.261301 + 0.965257i \(0.584151\pi\)
\(480\) 13.9443 0.636466
\(481\) −8.61803 −0.392949
\(482\) 12.0344 0.548154
\(483\) 32.9565 1.49957
\(484\) 37.6525 1.71148
\(485\) 2.62866 0.119361
\(486\) −12.0344 −0.545893
\(487\) −21.0292 −0.952926 −0.476463 0.879195i \(-0.658081\pi\)
−0.476463 + 0.879195i \(0.658081\pi\)
\(488\) 2.86568 0.129723
\(489\) 47.8959 2.16593
\(490\) 20.8172 0.940427
\(491\) 37.2705 1.68199 0.840997 0.541039i \(-0.181969\pi\)
0.840997 + 0.541039i \(0.181969\pi\)
\(492\) 21.1803 0.954883
\(493\) −14.6619 −0.660338
\(494\) 0 0
\(495\) 3.61803 0.162619
\(496\) −8.78402 −0.394414
\(497\) −0.726543 −0.0325899
\(498\) −31.7082 −1.42088
\(499\) −29.5279 −1.32185 −0.660924 0.750453i \(-0.729836\pi\)
−0.660924 + 0.750453i \(0.729836\pi\)
\(500\) −1.61803 −0.0723607
\(501\) −0.326238 −0.0145752
\(502\) 5.70634 0.254686
\(503\) 31.6525 1.41131 0.705657 0.708554i \(-0.250652\pi\)
0.705657 + 0.708554i \(0.250652\pi\)
\(504\) −1.90211 −0.0847268
\(505\) −3.29180 −0.146483
\(506\) 45.5447 2.02471
\(507\) 6.71040 0.298019
\(508\) 14.9394 0.662828
\(509\) 30.1563 1.33665 0.668327 0.743868i \(-0.267011\pi\)
0.668327 + 0.743868i \(0.267011\pi\)
\(510\) −18.9443 −0.838866
\(511\) 4.23607 0.187393
\(512\) −27.1441 −1.19961
\(513\) 0 0
\(514\) 6.18034 0.272603
\(515\) 14.9394 0.658308
\(516\) −1.17557 −0.0517516
\(517\) 8.61803 0.379021
\(518\) −22.5623 −0.991331
\(519\) 25.1246 1.10285
\(520\) −2.23607 −0.0980581
\(521\) −32.6789 −1.43169 −0.715845 0.698259i \(-0.753958\pi\)
−0.715845 + 0.698259i \(0.753958\pi\)
\(522\) 3.29180 0.144078
\(523\) −6.53888 −0.285925 −0.142963 0.989728i \(-0.545663\pi\)
−0.142963 + 0.989728i \(0.545663\pi\)
\(524\) −25.2705 −1.10395
\(525\) 8.05748 0.351657
\(526\) 11.4127 0.497616
\(527\) 9.95959 0.433847
\(528\) −51.4226 −2.23788
\(529\) −6.27051 −0.272631
\(530\) 21.1803 0.920015
\(531\) 8.67802 0.376594
\(532\) 0 0
\(533\) −21.1803 −0.917422
\(534\) 28.1482 1.21809
\(535\) 8.33499 0.360353
\(536\) 4.34752 0.187784
\(537\) −19.2705 −0.831584
\(538\) −37.3607 −1.61073
\(539\) −64.0689 −2.75964
\(540\) −7.33094 −0.315473
\(541\) −24.5967 −1.05750 −0.528748 0.848779i \(-0.677338\pi\)
−0.528748 + 0.848779i \(0.677338\pi\)
\(542\) −16.3925 −0.704117
\(543\) 12.7639 0.547753
\(544\) 38.3853 1.64576
\(545\) 2.17963 0.0933650
\(546\) −47.1693 −2.01866
\(547\) −21.2663 −0.909280 −0.454640 0.890675i \(-0.650232\pi\)
−0.454640 + 0.890675i \(0.650232\pi\)
\(548\) 15.7984 0.674873
\(549\) 2.43769 0.104038
\(550\) 11.1352 0.474805
\(551\) 0 0
\(552\) −5.65248 −0.240585
\(553\) 22.2703 0.947031
\(554\) −47.2348 −2.00682
\(555\) −5.32624 −0.226086
\(556\) −35.7426 −1.51583
\(557\) −31.0689 −1.31643 −0.658215 0.752830i \(-0.728688\pi\)
−0.658215 + 0.752830i \(0.728688\pi\)
\(558\) −2.23607 −0.0946603
\(559\) 1.17557 0.0497213
\(560\) −19.5623 −0.826658
\(561\) 58.3045 2.46162
\(562\) 47.6869 2.01155
\(563\) −4.35926 −0.183721 −0.0918604 0.995772i \(-0.529281\pi\)
−0.0918604 + 0.995772i \(0.529281\pi\)
\(564\) 4.53077 0.190780
\(565\) −18.4661 −0.776875
\(566\) 53.7737 2.26028
\(567\) 44.3607 1.86297
\(568\) 0.124612 0.00522859
\(569\) −25.7970 −1.08147 −0.540734 0.841194i \(-0.681853\pi\)
−0.540734 + 0.841194i \(0.681853\pi\)
\(570\) 0 0
\(571\) −25.8541 −1.08196 −0.540980 0.841035i \(-0.681947\pi\)
−0.540980 + 0.841035i \(0.681947\pi\)
\(572\) −29.1522 −1.21892
\(573\) −24.7275 −1.03300
\(574\) −55.4508 −2.31447
\(575\) 4.09017 0.170572
\(576\) −2.90983 −0.121243
\(577\) 8.52786 0.355020 0.177510 0.984119i \(-0.443196\pi\)
0.177510 + 0.984119i \(0.443196\pi\)
\(578\) −19.8132 −0.824120
\(579\) −2.76393 −0.114865
\(580\) 4.53077 0.188130
\(581\) 37.1246 1.54019
\(582\) −9.51057 −0.394226
\(583\) −65.1864 −2.69974
\(584\) −0.726543 −0.0300645
\(585\) −1.90211 −0.0786427
\(586\) −26.3050 −1.08665
\(587\) 36.4164 1.50307 0.751533 0.659695i \(-0.229315\pi\)
0.751533 + 0.659695i \(0.229315\pi\)
\(588\) −33.6830 −1.38906
\(589\) 0 0
\(590\) 26.7082 1.09956
\(591\) −28.3197 −1.16492
\(592\) 12.9313 0.531472
\(593\) 20.7082 0.850384 0.425192 0.905103i \(-0.360207\pi\)
0.425192 + 0.905103i \(0.360207\pi\)
\(594\) 50.4508 2.07002
\(595\) 22.1803 0.909305
\(596\) 11.4721 0.469917
\(597\) −25.5195 −1.04444
\(598\) −23.9443 −0.979154
\(599\) −30.0503 −1.22782 −0.613911 0.789375i \(-0.710405\pi\)
−0.613911 + 0.789375i \(0.710405\pi\)
\(600\) −1.38197 −0.0564185
\(601\) 10.5146 0.428900 0.214450 0.976735i \(-0.431204\pi\)
0.214450 + 0.976735i \(0.431204\pi\)
\(602\) 3.07768 0.125437
\(603\) 3.69822 0.150603
\(604\) 1.00406 0.0408545
\(605\) −23.2705 −0.946081
\(606\) 11.9098 0.483804
\(607\) −16.2210 −0.658389 −0.329194 0.944262i \(-0.606777\pi\)
−0.329194 + 0.944262i \(0.606777\pi\)
\(608\) 0 0
\(609\) −22.5623 −0.914271
\(610\) 7.50245 0.303765
\(611\) −4.53077 −0.183295
\(612\) 5.23607 0.211656
\(613\) 26.5279 1.07145 0.535725 0.844392i \(-0.320038\pi\)
0.535725 + 0.844392i \(0.320038\pi\)
\(614\) −34.9230 −1.40938
\(615\) −13.0902 −0.527847
\(616\) 18.0171 0.725929
\(617\) 21.6180 0.870309 0.435155 0.900356i \(-0.356694\pi\)
0.435155 + 0.900356i \(0.356694\pi\)
\(618\) −54.0512 −2.17426
\(619\) 39.5410 1.58929 0.794644 0.607076i \(-0.207658\pi\)
0.794644 + 0.607076i \(0.207658\pi\)
\(620\) −3.07768 −0.123603
\(621\) 18.5316 0.743648
\(622\) 5.25731 0.210799
\(623\) −32.9565 −1.32037
\(624\) 27.0344 1.08224
\(625\) 1.00000 0.0400000
\(626\) −6.22088 −0.248636
\(627\) 0 0
\(628\) −26.3262 −1.05053
\(629\) −14.6619 −0.584607
\(630\) −4.97980 −0.198400
\(631\) −41.1803 −1.63936 −0.819682 0.572819i \(-0.805850\pi\)
−0.819682 + 0.572819i \(0.805850\pi\)
\(632\) −3.81966 −0.151938
\(633\) 3.29180 0.130837
\(634\) 31.3050 1.24328
\(635\) −9.23305 −0.366402
\(636\) −34.2705 −1.35891
\(637\) 33.6830 1.33457
\(638\) −31.1803 −1.23444
\(639\) 0.106001 0.00419334
\(640\) 5.70634 0.225563
\(641\) −36.6952 −1.44937 −0.724686 0.689079i \(-0.758015\pi\)
−0.724686 + 0.689079i \(0.758015\pi\)
\(642\) −30.1563 −1.19017
\(643\) −38.4721 −1.51719 −0.758596 0.651561i \(-0.774115\pi\)
−0.758596 + 0.651561i \(0.774115\pi\)
\(644\) −28.0344 −1.10471
\(645\) 0.726543 0.0286076
\(646\) 0 0
\(647\) 8.52786 0.335265 0.167632 0.985850i \(-0.446388\pi\)
0.167632 + 0.985850i \(0.446388\pi\)
\(648\) −7.60845 −0.298888
\(649\) −82.1994 −3.22661
\(650\) −5.85410 −0.229617
\(651\) 15.3262 0.600683
\(652\) −40.7426 −1.59561
\(653\) −13.7984 −0.539972 −0.269986 0.962864i \(-0.587019\pi\)
−0.269986 + 0.962864i \(0.587019\pi\)
\(654\) −7.88597 −0.308366
\(655\) 15.6180 0.610247
\(656\) 31.7809 1.24084
\(657\) −0.618034 −0.0241118
\(658\) −11.8617 −0.462417
\(659\) 7.26543 0.283021 0.141510 0.989937i \(-0.454804\pi\)
0.141510 + 0.989937i \(0.454804\pi\)
\(660\) −18.0171 −0.701314
\(661\) −29.3238 −1.14056 −0.570281 0.821450i \(-0.693166\pi\)
−0.570281 + 0.821450i \(0.693166\pi\)
\(662\) 37.8885 1.47258
\(663\) −30.6525 −1.19044
\(664\) −6.36737 −0.247102
\(665\) 0 0
\(666\) 3.29180 0.127555
\(667\) −11.4532 −0.443468
\(668\) 0.277515 0.0107374
\(669\) 28.2148 1.09085
\(670\) 11.3820 0.439724
\(671\) −23.0902 −0.891386
\(672\) 59.0689 2.27863
\(673\) 24.2380 0.934304 0.467152 0.884177i \(-0.345280\pi\)
0.467152 + 0.884177i \(0.345280\pi\)
\(674\) −54.0689 −2.08266
\(675\) 4.53077 0.174389
\(676\) −5.70820 −0.219546
\(677\) 33.5770 1.29047 0.645235 0.763985i \(-0.276760\pi\)
0.645235 + 0.763985i \(0.276760\pi\)
\(678\) 66.8110 2.56586
\(679\) 11.1352 0.427328
\(680\) −3.80423 −0.145885
\(681\) −33.0902 −1.26802
\(682\) 21.1803 0.811037
\(683\) 28.0422 1.07300 0.536502 0.843899i \(-0.319745\pi\)
0.536502 + 0.843899i \(0.319745\pi\)
\(684\) 0 0
\(685\) −9.76393 −0.373061
\(686\) 31.7809 1.21340
\(687\) 34.5155 1.31685
\(688\) −1.76393 −0.0672493
\(689\) 34.2705 1.30560
\(690\) −14.7984 −0.563364
\(691\) −12.2361 −0.465482 −0.232741 0.972539i \(-0.574769\pi\)
−0.232741 + 0.972539i \(0.574769\pi\)
\(692\) −21.3723 −0.812452
\(693\) 15.3262 0.582196
\(694\) −31.3319 −1.18934
\(695\) 22.0902 0.837928
\(696\) 3.86974 0.146682
\(697\) −36.0341 −1.36489
\(698\) 4.87380 0.184476
\(699\) −3.35520 −0.126905
\(700\) −6.85410 −0.259061
\(701\) −31.9098 −1.20522 −0.602609 0.798037i \(-0.705872\pi\)
−0.602609 + 0.798037i \(0.705872\pi\)
\(702\) −26.5236 −1.00107
\(703\) 0 0
\(704\) 27.5623 1.03879
\(705\) −2.80017 −0.105460
\(706\) −6.98791 −0.262993
\(707\) −13.9443 −0.524428
\(708\) −43.2148 −1.62411
\(709\) 36.1803 1.35878 0.679391 0.733777i \(-0.262244\pi\)
0.679391 + 0.733777i \(0.262244\pi\)
\(710\) 0.326238 0.0122435
\(711\) −3.24920 −0.121854
\(712\) 5.65248 0.211835
\(713\) 7.77997 0.291362
\(714\) −80.2492 −3.00325
\(715\) 18.0171 0.673800
\(716\) 16.3925 0.612616
\(717\) 44.1571 1.64908
\(718\) −14.0413 −0.524018
\(719\) −30.9098 −1.15274 −0.576371 0.817188i \(-0.695532\pi\)
−0.576371 + 0.817188i \(0.695532\pi\)
\(720\) 2.85410 0.106366
\(721\) 63.2843 2.35683
\(722\) 0 0
\(723\) 12.0344 0.447566
\(724\) −10.8576 −0.403521
\(725\) −2.80017 −0.103996
\(726\) 84.1935 3.12471
\(727\) 34.3607 1.27437 0.637184 0.770712i \(-0.280099\pi\)
0.637184 + 0.770712i \(0.280099\pi\)
\(728\) −9.47214 −0.351061
\(729\) 19.3820 0.717851
\(730\) −1.90211 −0.0704004
\(731\) 2.00000 0.0739727
\(732\) −12.1392 −0.448679
\(733\) −11.0000 −0.406294 −0.203147 0.979148i \(-0.565117\pi\)
−0.203147 + 0.979148i \(0.565117\pi\)
\(734\) −59.0965 −2.18129
\(735\) 20.8172 0.767856
\(736\) 29.9848 1.10525
\(737\) −35.0301 −1.29035
\(738\) 8.09017 0.297803
\(739\) −27.7639 −1.02131 −0.510656 0.859785i \(-0.670598\pi\)
−0.510656 + 0.859785i \(0.670598\pi\)
\(740\) 4.53077 0.166554
\(741\) 0 0
\(742\) 89.7214 3.29377
\(743\) −5.98385 −0.219526 −0.109763 0.993958i \(-0.535009\pi\)
−0.109763 + 0.993958i \(0.535009\pi\)
\(744\) −2.62866 −0.0963712
\(745\) −7.09017 −0.259764
\(746\) −10.3262 −0.378070
\(747\) −5.41641 −0.198176
\(748\) −49.5967 −1.81344
\(749\) 35.3076 1.29011
\(750\) −3.61803 −0.132112
\(751\) 50.4590 1.84127 0.920637 0.390419i \(-0.127670\pi\)
0.920637 + 0.390419i \(0.127670\pi\)
\(752\) 6.79837 0.247911
\(753\) 5.70634 0.207951
\(754\) 16.3925 0.596979
\(755\) −0.620541 −0.0225838
\(756\) −31.0543 −1.12944
\(757\) 2.97871 0.108263 0.0541316 0.998534i \(-0.482761\pi\)
0.0541316 + 0.998534i \(0.482761\pi\)
\(758\) 24.4721 0.888868
\(759\) 45.5447 1.65317
\(760\) 0 0
\(761\) −18.8197 −0.682212 −0.341106 0.940025i \(-0.610802\pi\)
−0.341106 + 0.940025i \(0.610802\pi\)
\(762\) 33.4055 1.21015
\(763\) 9.23305 0.334259
\(764\) 21.0344 0.760999
\(765\) −3.23607 −0.117000
\(766\) 8.09017 0.292310
\(767\) 43.2148 1.56040
\(768\) −38.5568 −1.39130
\(769\) −13.2705 −0.478547 −0.239273 0.970952i \(-0.576909\pi\)
−0.239273 + 0.970952i \(0.576909\pi\)
\(770\) 47.1693 1.69986
\(771\) 6.18034 0.222580
\(772\) 2.35114 0.0846194
\(773\) −32.5074 −1.16921 −0.584606 0.811318i \(-0.698751\pi\)
−0.584606 + 0.811318i \(0.698751\pi\)
\(774\) −0.449028 −0.0161400
\(775\) 1.90211 0.0683259
\(776\) −1.90983 −0.0685589
\(777\) −22.5623 −0.809418
\(778\) 39.6669 1.42213
\(779\) 0 0
\(780\) 9.47214 0.339157
\(781\) −1.00406 −0.0359280
\(782\) −40.7364 −1.45673
\(783\) −12.6869 −0.453393
\(784\) −50.5410 −1.80504
\(785\) 16.2705 0.580719
\(786\) −56.5066 −2.01552
\(787\) 20.1312 0.717599 0.358800 0.933415i \(-0.383186\pi\)
0.358800 + 0.933415i \(0.383186\pi\)
\(788\) 24.0902 0.858177
\(789\) 11.4127 0.406302
\(790\) −10.0000 −0.355784
\(791\) −78.2237 −2.78131
\(792\) −2.62866 −0.0934052
\(793\) 12.1392 0.431076
\(794\) 45.3327 1.60880
\(795\) 21.1803 0.751189
\(796\) 21.7082 0.769427
\(797\) −23.7234 −0.840326 −0.420163 0.907449i \(-0.638027\pi\)
−0.420163 + 0.907449i \(0.638027\pi\)
\(798\) 0 0
\(799\) −7.70820 −0.272697
\(800\) 7.33094 0.259188
\(801\) 4.80828 0.169892
\(802\) −17.2361 −0.608627
\(803\) 5.85410 0.206587
\(804\) −18.4164 −0.649497
\(805\) 17.3262 0.610670
\(806\) −11.1352 −0.392219
\(807\) −37.3607 −1.31516
\(808\) 2.39163 0.0841372
\(809\) −53.2148 −1.87093 −0.935466 0.353417i \(-0.885020\pi\)
−0.935466 + 0.353417i \(0.885020\pi\)
\(810\) −19.9192 −0.699889
\(811\) −13.1433 −0.461523 −0.230761 0.973010i \(-0.574122\pi\)
−0.230761 + 0.973010i \(0.574122\pi\)
\(812\) 19.1926 0.673530
\(813\) −16.3925 −0.574909
\(814\) −31.1803 −1.09287
\(815\) 25.1803 0.882029
\(816\) 45.9937 1.61010
\(817\) 0 0
\(818\) −41.3050 −1.44419
\(819\) −8.05748 −0.281551
\(820\) 11.1352 0.388857
\(821\) −27.4508 −0.958041 −0.479021 0.877804i \(-0.659008\pi\)
−0.479021 + 0.877804i \(0.659008\pi\)
\(822\) 35.3262 1.23214
\(823\) −16.7639 −0.584354 −0.292177 0.956364i \(-0.594380\pi\)
−0.292177 + 0.956364i \(0.594380\pi\)
\(824\) −10.8541 −0.378121
\(825\) 11.1352 0.387677
\(826\) 113.138 3.93657
\(827\) −9.23305 −0.321065 −0.160532 0.987031i \(-0.551321\pi\)
−0.160532 + 0.987031i \(0.551321\pi\)
\(828\) 4.09017 0.142143
\(829\) 25.9686 0.901925 0.450963 0.892543i \(-0.351081\pi\)
0.450963 + 0.892543i \(0.351081\pi\)
\(830\) −16.6700 −0.578624
\(831\) −47.2348 −1.63856
\(832\) −14.4904 −0.502363
\(833\) 57.3050 1.98550
\(834\) −79.9230 −2.76751
\(835\) −0.171513 −0.00593547
\(836\) 0 0
\(837\) 8.61803 0.297883
\(838\) −49.7980 −1.72024
\(839\) −4.08174 −0.140917 −0.0704587 0.997515i \(-0.522446\pi\)
−0.0704587 + 0.997515i \(0.522446\pi\)
\(840\) −5.85410 −0.201986
\(841\) −21.1591 −0.729623
\(842\) 22.0344 0.759357
\(843\) 47.6869 1.64242
\(844\) −2.80017 −0.0963858
\(845\) 3.52786 0.121362
\(846\) 1.73060 0.0594992
\(847\) −98.5755 −3.38709
\(848\) −51.4226 −1.76586
\(849\) 53.7737 1.84551
\(850\) −9.95959 −0.341611
\(851\) −11.4532 −0.392610
\(852\) −0.527864 −0.0180843
\(853\) −15.4721 −0.529756 −0.264878 0.964282i \(-0.585332\pi\)
−0.264878 + 0.964282i \(0.585332\pi\)
\(854\) 31.7809 1.08752
\(855\) 0 0
\(856\) −6.05573 −0.206981
\(857\) 6.22088 0.212501 0.106251 0.994339i \(-0.466115\pi\)
0.106251 + 0.994339i \(0.466115\pi\)
\(858\) −65.1864 −2.22543
\(859\) −22.7639 −0.776695 −0.388348 0.921513i \(-0.626954\pi\)
−0.388348 + 0.921513i \(0.626954\pi\)
\(860\) −0.618034 −0.0210748
\(861\) −55.4508 −1.88976
\(862\) −13.0902 −0.445853
\(863\) 36.3772 1.23829 0.619147 0.785275i \(-0.287479\pi\)
0.619147 + 0.785275i \(0.287479\pi\)
\(864\) 33.2148 1.12999
\(865\) 13.2088 0.449112
\(866\) 11.5066 0.391009
\(867\) −19.8132 −0.672891
\(868\) −13.0373 −0.442514
\(869\) 30.7768 1.04403
\(870\) 10.1311 0.343477
\(871\) 18.4164 0.624016
\(872\) −1.58359 −0.0536272
\(873\) −1.62460 −0.0549843
\(874\) 0 0
\(875\) 4.23607 0.143205
\(876\) 3.07768 0.103985
\(877\) −11.9677 −0.404121 −0.202060 0.979373i \(-0.564764\pi\)
−0.202060 + 0.979373i \(0.564764\pi\)
\(878\) −10.7295 −0.362103
\(879\) −26.3050 −0.887244
\(880\) −27.0344 −0.911331
\(881\) 10.3262 0.347900 0.173950 0.984755i \(-0.444347\pi\)
0.173950 + 0.984755i \(0.444347\pi\)
\(882\) −12.8658 −0.433213
\(883\) −5.78522 −0.194688 −0.0973440 0.995251i \(-0.531035\pi\)
−0.0973440 + 0.995251i \(0.531035\pi\)
\(884\) 26.0746 0.876982
\(885\) 26.7082 0.897786
\(886\) −66.1500 −2.22235
\(887\) −14.3188 −0.480780 −0.240390 0.970676i \(-0.577275\pi\)
−0.240390 + 0.970676i \(0.577275\pi\)
\(888\) 3.86974 0.129860
\(889\) −39.1118 −1.31177
\(890\) 14.7984 0.496043
\(891\) 61.3050 2.05379
\(892\) −24.0009 −0.803610
\(893\) 0 0
\(894\) 25.6525 0.857947
\(895\) −10.1311 −0.338646
\(896\) 24.1724 0.807545
\(897\) −23.9443 −0.799476
\(898\) −10.7295 −0.358048
\(899\) −5.32624 −0.177640
\(900\) 1.00000 0.0333333
\(901\) 58.3045 1.94240
\(902\) −76.6312 −2.55154
\(903\) 3.07768 0.102419
\(904\) 13.4164 0.446223
\(905\) 6.71040 0.223061
\(906\) 2.24514 0.0745898
\(907\) −51.9371 −1.72454 −0.862272 0.506446i \(-0.830959\pi\)
−0.862272 + 0.506446i \(0.830959\pi\)
\(908\) 28.1482 0.934130
\(909\) 2.03444 0.0674782
\(910\) −24.7984 −0.822058
\(911\) 34.3035 1.13653 0.568264 0.822847i \(-0.307615\pi\)
0.568264 + 0.822847i \(0.307615\pi\)
\(912\) 0 0
\(913\) 51.3050 1.69795
\(914\) −9.95959 −0.329434
\(915\) 7.50245 0.248023
\(916\) −29.3607 −0.970104
\(917\) 66.1591 2.18476
\(918\) −45.1246 −1.48933
\(919\) 7.36068 0.242806 0.121403 0.992603i \(-0.461261\pi\)
0.121403 + 0.992603i \(0.461261\pi\)
\(920\) −2.97168 −0.0979735
\(921\) −34.9230 −1.15075
\(922\) 15.2169 0.501142
\(923\) 0.527864 0.0173749
\(924\) −76.3215 −2.51079
\(925\) −2.80017 −0.0920690
\(926\) −0.277515 −0.00911969
\(927\) −9.23305 −0.303253
\(928\) −20.5279 −0.673860
\(929\) 37.5623 1.23238 0.616190 0.787598i \(-0.288675\pi\)
0.616190 + 0.787598i \(0.288675\pi\)
\(930\) −6.88191 −0.225667
\(931\) 0 0
\(932\) 2.85410 0.0934892
\(933\) 5.25731 0.172117
\(934\) 17.4620 0.571376
\(935\) 30.6525 1.00244
\(936\) 1.38197 0.0451710
\(937\) 23.2016 0.757964 0.378982 0.925404i \(-0.376274\pi\)
0.378982 + 0.925404i \(0.376274\pi\)
\(938\) 48.2148 1.57427
\(939\) −6.22088 −0.203011
\(940\) 2.38197 0.0776912
\(941\) −50.7615 −1.65478 −0.827389 0.561629i \(-0.810175\pi\)
−0.827389 + 0.561629i \(0.810175\pi\)
\(942\) −58.8673 −1.91800
\(943\) −28.1482 −0.916631
\(944\) −64.8434 −2.11047
\(945\) 19.1926 0.624337
\(946\) 4.25325 0.138285
\(947\) 1.41641 0.0460271 0.0230135 0.999735i \(-0.492674\pi\)
0.0230135 + 0.999735i \(0.492674\pi\)
\(948\) 16.1803 0.525513
\(949\) −3.07768 −0.0999058
\(950\) 0 0
\(951\) 31.3050 1.01513
\(952\) −16.1150 −0.522289
\(953\) 42.2300 1.36796 0.683982 0.729499i \(-0.260247\pi\)
0.683982 + 0.729499i \(0.260247\pi\)
\(954\) −13.0902 −0.423810
\(955\) −13.0000 −0.420670
\(956\) −37.5623 −1.21485
\(957\) −31.1803 −1.00792
\(958\) 21.7558 0.702898
\(959\) −41.3607 −1.33561
\(960\) −8.95554 −0.289039
\(961\) −27.3820 −0.883289
\(962\) 16.3925 0.528515
\(963\) −5.15131 −0.165999
\(964\) −10.2371 −0.329715
\(965\) −1.45309 −0.0467765
\(966\) −62.6869 −2.01692
\(967\) −29.0557 −0.934369 −0.467185 0.884160i \(-0.654732\pi\)
−0.467185 + 0.884160i \(0.654732\pi\)
\(968\) 16.9070 0.543412
\(969\) 0 0
\(970\) −5.00000 −0.160540
\(971\) −27.4216 −0.880002 −0.440001 0.897997i \(-0.645022\pi\)
−0.440001 + 0.897997i \(0.645022\pi\)
\(972\) 10.2371 0.328355
\(973\) 93.5755 2.99989
\(974\) 40.0000 1.28168
\(975\) −5.85410 −0.187481
\(976\) −18.2148 −0.583041
\(977\) −36.7607 −1.17608 −0.588039 0.808832i \(-0.700100\pi\)
−0.588039 + 0.808832i \(0.700100\pi\)
\(978\) −91.1033 −2.91316
\(979\) −45.5447 −1.45562
\(980\) −17.7082 −0.565668
\(981\) −1.34708 −0.0430091
\(982\) −70.8927 −2.26228
\(983\) −19.6417 −0.626472 −0.313236 0.949675i \(-0.601413\pi\)
−0.313236 + 0.949675i \(0.601413\pi\)
\(984\) 9.51057 0.303186
\(985\) −14.8885 −0.474388
\(986\) 27.8885 0.888152
\(987\) −11.8617 −0.377562
\(988\) 0 0
\(989\) 1.56231 0.0496784
\(990\) −6.88191 −0.218721
\(991\) 57.6839 1.83239 0.916195 0.400732i \(-0.131244\pi\)
0.916195 + 0.400732i \(0.131244\pi\)
\(992\) 13.9443 0.442731
\(993\) 37.8885 1.20236
\(994\) 1.38197 0.0438333
\(995\) −13.4164 −0.425329
\(996\) 26.9726 0.854660
\(997\) −36.9443 −1.17004 −0.585018 0.811020i \(-0.698913\pi\)
−0.585018 + 0.811020i \(0.698913\pi\)
\(998\) 56.1653 1.77788
\(999\) −12.6869 −0.401396
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 1805.2.a.l.1.1 4
5.4 even 2 9025.2.a.bl.1.4 4
19.18 odd 2 inner 1805.2.a.l.1.4 yes 4
95.94 odd 2 9025.2.a.bl.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1805.2.a.l.1.1 4 1.1 even 1 trivial
1805.2.a.l.1.4 yes 4 19.18 odd 2 inner
9025.2.a.bl.1.1 4 95.94 odd 2
9025.2.a.bl.1.4 4 5.4 even 2