Properties

Label 3025.2.a.y.1.1
Level $3025$
Weight $2$
Character 3025.1
Self dual yes
Analytic conductor $24.155$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3025 = 5^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.1547466114\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 605)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.93185\) of defining polynomial
Character \(\chi\) \(=\) 3025.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.93185 q^{2} +0.517638 q^{3} +1.73205 q^{4} -1.00000 q^{6} -0.896575 q^{7} +0.517638 q^{8} -2.73205 q^{9} +O(q^{10})\) \(q-1.93185 q^{2} +0.517638 q^{3} +1.73205 q^{4} -1.00000 q^{6} -0.896575 q^{7} +0.517638 q^{8} -2.73205 q^{9} +0.896575 q^{12} +4.24264 q^{13} +1.73205 q^{14} -4.46410 q^{16} -1.03528 q^{17} +5.27792 q^{18} +6.19615 q^{19} -0.464102 q^{21} -6.31319 q^{23} +0.267949 q^{24} -8.19615 q^{26} -2.96713 q^{27} -1.55291 q^{28} -6.92820 q^{29} -5.26795 q^{31} +7.58871 q^{32} +2.00000 q^{34} -4.73205 q^{36} +6.69213 q^{37} -11.9700 q^{38} +2.19615 q^{39} -1.73205 q^{41} +0.896575 q^{42} +10.6945 q^{43} +12.1962 q^{46} +4.10394 q^{47} -2.31079 q^{48} -6.19615 q^{49} -0.535898 q^{51} +7.34847 q^{52} +11.9700 q^{53} +5.73205 q^{54} -0.464102 q^{56} +3.20736 q^{57} +13.3843 q^{58} -4.73205 q^{59} -8.26795 q^{61} +10.1769 q^{62} +2.44949 q^{63} -5.73205 q^{64} -14.9372 q^{67} -1.79315 q^{68} -3.26795 q^{69} +2.19615 q^{71} -1.41421 q^{72} +4.89898 q^{73} -12.9282 q^{74} +10.7321 q^{76} -4.24264 q^{78} -5.46410 q^{79} +6.66025 q^{81} +3.34607 q^{82} -9.89949 q^{83} -0.803848 q^{84} -20.6603 q^{86} -3.58630 q^{87} -6.46410 q^{89} -3.80385 q^{91} -10.9348 q^{92} -2.72689 q^{93} -7.92820 q^{94} +3.92820 q^{96} +0.656339 q^{97} +11.9700 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{6} - 4 q^{9} - 4 q^{16} + 4 q^{19} + 12 q^{21} + 8 q^{24} - 12 q^{26} - 28 q^{31} + 8 q^{34} - 12 q^{36} - 12 q^{39} + 28 q^{46} - 4 q^{49} - 16 q^{51} + 16 q^{54} + 12 q^{56} - 12 q^{59} - 40 q^{61} - 16 q^{64} - 20 q^{69} - 12 q^{71} - 24 q^{74} + 36 q^{76} - 8 q^{79} - 8 q^{81} - 24 q^{84} - 48 q^{86} - 12 q^{89} - 36 q^{91} - 4 q^{94} - 12 q^{96}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.93185 −1.36603 −0.683013 0.730406i \(-0.739331\pi\)
−0.683013 + 0.730406i \(0.739331\pi\)
\(3\) 0.517638 0.298858 0.149429 0.988772i \(-0.452256\pi\)
0.149429 + 0.988772i \(0.452256\pi\)
\(4\) 1.73205 0.866025
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) −0.896575 −0.338874 −0.169437 0.985541i \(-0.554195\pi\)
−0.169437 + 0.985541i \(0.554195\pi\)
\(8\) 0.517638 0.183013
\(9\) −2.73205 −0.910684
\(10\) 0 0
\(11\) 0 0
\(12\) 0.896575 0.258819
\(13\) 4.24264 1.17670 0.588348 0.808608i \(-0.299778\pi\)
0.588348 + 0.808608i \(0.299778\pi\)
\(14\) 1.73205 0.462910
\(15\) 0 0
\(16\) −4.46410 −1.11603
\(17\) −1.03528 −0.251091 −0.125546 0.992088i \(-0.540068\pi\)
−0.125546 + 0.992088i \(0.540068\pi\)
\(18\) 5.27792 1.24402
\(19\) 6.19615 1.42149 0.710747 0.703447i \(-0.248357\pi\)
0.710747 + 0.703447i \(0.248357\pi\)
\(20\) 0 0
\(21\) −0.464102 −0.101275
\(22\) 0 0
\(23\) −6.31319 −1.31639 −0.658196 0.752847i \(-0.728680\pi\)
−0.658196 + 0.752847i \(0.728680\pi\)
\(24\) 0.267949 0.0546949
\(25\) 0 0
\(26\) −8.19615 −1.60740
\(27\) −2.96713 −0.571024
\(28\) −1.55291 −0.293473
\(29\) −6.92820 −1.28654 −0.643268 0.765641i \(-0.722422\pi\)
−0.643268 + 0.765641i \(0.722422\pi\)
\(30\) 0 0
\(31\) −5.26795 −0.946152 −0.473076 0.881022i \(-0.656856\pi\)
−0.473076 + 0.881022i \(0.656856\pi\)
\(32\) 7.58871 1.34151
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) −4.73205 −0.788675
\(37\) 6.69213 1.10018 0.550090 0.835106i \(-0.314594\pi\)
0.550090 + 0.835106i \(0.314594\pi\)
\(38\) −11.9700 −1.94180
\(39\) 2.19615 0.351666
\(40\) 0 0
\(41\) −1.73205 −0.270501 −0.135250 0.990811i \(-0.543184\pi\)
−0.135250 + 0.990811i \(0.543184\pi\)
\(42\) 0.896575 0.138345
\(43\) 10.6945 1.63090 0.815451 0.578827i \(-0.196489\pi\)
0.815451 + 0.578827i \(0.196489\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 12.1962 1.79822
\(47\) 4.10394 0.598621 0.299311 0.954156i \(-0.403243\pi\)
0.299311 + 0.954156i \(0.403243\pi\)
\(48\) −2.31079 −0.333534
\(49\) −6.19615 −0.885165
\(50\) 0 0
\(51\) −0.535898 −0.0750408
\(52\) 7.34847 1.01905
\(53\) 11.9700 1.64421 0.822106 0.569334i \(-0.192799\pi\)
0.822106 + 0.569334i \(0.192799\pi\)
\(54\) 5.73205 0.780033
\(55\) 0 0
\(56\) −0.464102 −0.0620182
\(57\) 3.20736 0.424826
\(58\) 13.3843 1.75744
\(59\) −4.73205 −0.616061 −0.308030 0.951377i \(-0.599670\pi\)
−0.308030 + 0.951377i \(0.599670\pi\)
\(60\) 0 0
\(61\) −8.26795 −1.05860 −0.529301 0.848434i \(-0.677546\pi\)
−0.529301 + 0.848434i \(0.677546\pi\)
\(62\) 10.1769 1.29247
\(63\) 2.44949 0.308607
\(64\) −5.73205 −0.716506
\(65\) 0 0
\(66\) 0 0
\(67\) −14.9372 −1.82487 −0.912433 0.409226i \(-0.865799\pi\)
−0.912433 + 0.409226i \(0.865799\pi\)
\(68\) −1.79315 −0.217451
\(69\) −3.26795 −0.393415
\(70\) 0 0
\(71\) 2.19615 0.260635 0.130318 0.991472i \(-0.458400\pi\)
0.130318 + 0.991472i \(0.458400\pi\)
\(72\) −1.41421 −0.166667
\(73\) 4.89898 0.573382 0.286691 0.958023i \(-0.407445\pi\)
0.286691 + 0.958023i \(0.407445\pi\)
\(74\) −12.9282 −1.50287
\(75\) 0 0
\(76\) 10.7321 1.23105
\(77\) 0 0
\(78\) −4.24264 −0.480384
\(79\) −5.46410 −0.614759 −0.307380 0.951587i \(-0.599452\pi\)
−0.307380 + 0.951587i \(0.599452\pi\)
\(80\) 0 0
\(81\) 6.66025 0.740028
\(82\) 3.34607 0.369511
\(83\) −9.89949 −1.08661 −0.543305 0.839535i \(-0.682827\pi\)
−0.543305 + 0.839535i \(0.682827\pi\)
\(84\) −0.803848 −0.0877070
\(85\) 0 0
\(86\) −20.6603 −2.22785
\(87\) −3.58630 −0.384492
\(88\) 0 0
\(89\) −6.46410 −0.685193 −0.342597 0.939483i \(-0.611306\pi\)
−0.342597 + 0.939483i \(0.611306\pi\)
\(90\) 0 0
\(91\) −3.80385 −0.398752
\(92\) −10.9348 −1.14003
\(93\) −2.72689 −0.282765
\(94\) −7.92820 −0.817732
\(95\) 0 0
\(96\) 3.92820 0.400921
\(97\) 0.656339 0.0666411 0.0333206 0.999445i \(-0.489392\pi\)
0.0333206 + 0.999445i \(0.489392\pi\)
\(98\) 11.9700 1.20916
\(99\) 0 0
\(100\) 0 0
\(101\) −1.39230 −0.138540 −0.0692698 0.997598i \(-0.522067\pi\)
−0.0692698 + 0.997598i \(0.522067\pi\)
\(102\) 1.03528 0.102508
\(103\) 4.24264 0.418040 0.209020 0.977911i \(-0.432973\pi\)
0.209020 + 0.977911i \(0.432973\pi\)
\(104\) 2.19615 0.215350
\(105\) 0 0
\(106\) −23.1244 −2.24604
\(107\) −4.38134 −0.423560 −0.211780 0.977317i \(-0.567926\pi\)
−0.211780 + 0.977317i \(0.567926\pi\)
\(108\) −5.13922 −0.494521
\(109\) −1.19615 −0.114571 −0.0572853 0.998358i \(-0.518244\pi\)
−0.0572853 + 0.998358i \(0.518244\pi\)
\(110\) 0 0
\(111\) 3.46410 0.328798
\(112\) 4.00240 0.378192
\(113\) 2.82843 0.266076 0.133038 0.991111i \(-0.457527\pi\)
0.133038 + 0.991111i \(0.457527\pi\)
\(114\) −6.19615 −0.580323
\(115\) 0 0
\(116\) −12.0000 −1.11417
\(117\) −11.5911 −1.07160
\(118\) 9.14162 0.841554
\(119\) 0.928203 0.0850883
\(120\) 0 0
\(121\) 0 0
\(122\) 15.9725 1.44608
\(123\) −0.896575 −0.0808415
\(124\) −9.12436 −0.819391
\(125\) 0 0
\(126\) −4.73205 −0.421565
\(127\) −3.34607 −0.296915 −0.148458 0.988919i \(-0.547431\pi\)
−0.148458 + 0.988919i \(0.547431\pi\)
\(128\) −4.10394 −0.362740
\(129\) 5.53590 0.487409
\(130\) 0 0
\(131\) 21.1244 1.84564 0.922822 0.385227i \(-0.125877\pi\)
0.922822 + 0.385227i \(0.125877\pi\)
\(132\) 0 0
\(133\) −5.55532 −0.481707
\(134\) 28.8564 2.49281
\(135\) 0 0
\(136\) −0.535898 −0.0459529
\(137\) −13.7632 −1.17587 −0.587935 0.808908i \(-0.700059\pi\)
−0.587935 + 0.808908i \(0.700059\pi\)
\(138\) 6.31319 0.537415
\(139\) 16.5885 1.40701 0.703507 0.710688i \(-0.251616\pi\)
0.703507 + 0.710688i \(0.251616\pi\)
\(140\) 0 0
\(141\) 2.12436 0.178903
\(142\) −4.24264 −0.356034
\(143\) 0 0
\(144\) 12.1962 1.01635
\(145\) 0 0
\(146\) −9.46410 −0.783255
\(147\) −3.20736 −0.264539
\(148\) 11.5911 0.952783
\(149\) −13.7321 −1.12497 −0.562487 0.826806i \(-0.690155\pi\)
−0.562487 + 0.826806i \(0.690155\pi\)
\(150\) 0 0
\(151\) −6.19615 −0.504236 −0.252118 0.967697i \(-0.581127\pi\)
−0.252118 + 0.967697i \(0.581127\pi\)
\(152\) 3.20736 0.260152
\(153\) 2.82843 0.228665
\(154\) 0 0
\(155\) 0 0
\(156\) 3.80385 0.304552
\(157\) −12.0716 −0.963417 −0.481709 0.876331i \(-0.659984\pi\)
−0.481709 + 0.876331i \(0.659984\pi\)
\(158\) 10.5558 0.839777
\(159\) 6.19615 0.491387
\(160\) 0 0
\(161\) 5.66025 0.446091
\(162\) −12.8666 −1.01090
\(163\) −7.58871 −0.594393 −0.297197 0.954816i \(-0.596052\pi\)
−0.297197 + 0.954816i \(0.596052\pi\)
\(164\) −3.00000 −0.234261
\(165\) 0 0
\(166\) 19.1244 1.48434
\(167\) −1.93185 −0.149491 −0.0747456 0.997203i \(-0.523814\pi\)
−0.0747456 + 0.997203i \(0.523814\pi\)
\(168\) −0.240237 −0.0185347
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) −16.9282 −1.29453
\(172\) 18.5235 1.41240
\(173\) −7.62587 −0.579784 −0.289892 0.957059i \(-0.593619\pi\)
−0.289892 + 0.957059i \(0.593619\pi\)
\(174\) 6.92820 0.525226
\(175\) 0 0
\(176\) 0 0
\(177\) −2.44949 −0.184115
\(178\) 12.4877 0.935992
\(179\) −8.53590 −0.638003 −0.319002 0.947754i \(-0.603348\pi\)
−0.319002 + 0.947754i \(0.603348\pi\)
\(180\) 0 0
\(181\) 3.39230 0.252148 0.126074 0.992021i \(-0.459762\pi\)
0.126074 + 0.992021i \(0.459762\pi\)
\(182\) 7.34847 0.544705
\(183\) −4.27981 −0.316372
\(184\) −3.26795 −0.240916
\(185\) 0 0
\(186\) 5.26795 0.386265
\(187\) 0 0
\(188\) 7.10823 0.518421
\(189\) 2.66025 0.193505
\(190\) 0 0
\(191\) −10.7321 −0.776544 −0.388272 0.921545i \(-0.626928\pi\)
−0.388272 + 0.921545i \(0.626928\pi\)
\(192\) −2.96713 −0.214134
\(193\) 10.2784 0.739858 0.369929 0.929060i \(-0.379382\pi\)
0.369929 + 0.929060i \(0.379382\pi\)
\(194\) −1.26795 −0.0910334
\(195\) 0 0
\(196\) −10.7321 −0.766575
\(197\) 2.55103 0.181753 0.0908765 0.995862i \(-0.471033\pi\)
0.0908765 + 0.995862i \(0.471033\pi\)
\(198\) 0 0
\(199\) −2.00000 −0.141776 −0.0708881 0.997484i \(-0.522583\pi\)
−0.0708881 + 0.997484i \(0.522583\pi\)
\(200\) 0 0
\(201\) −7.73205 −0.545377
\(202\) 2.68973 0.189248
\(203\) 6.21166 0.435973
\(204\) −0.928203 −0.0649872
\(205\) 0 0
\(206\) −8.19615 −0.571053
\(207\) 17.2480 1.19882
\(208\) −18.9396 −1.31322
\(209\) 0 0
\(210\) 0 0
\(211\) 11.1244 0.765832 0.382916 0.923783i \(-0.374920\pi\)
0.382916 + 0.923783i \(0.374920\pi\)
\(212\) 20.7327 1.42393
\(213\) 1.13681 0.0778931
\(214\) 8.46410 0.578594
\(215\) 0 0
\(216\) −1.53590 −0.104505
\(217\) 4.72311 0.320626
\(218\) 2.31079 0.156506
\(219\) 2.53590 0.171360
\(220\) 0 0
\(221\) −4.39230 −0.295458
\(222\) −6.69213 −0.449146
\(223\) −5.79555 −0.388099 −0.194050 0.980992i \(-0.562162\pi\)
−0.194050 + 0.980992i \(0.562162\pi\)
\(224\) −6.80385 −0.454601
\(225\) 0 0
\(226\) −5.46410 −0.363467
\(227\) −21.5278 −1.42885 −0.714424 0.699713i \(-0.753311\pi\)
−0.714424 + 0.699713i \(0.753311\pi\)
\(228\) 5.55532 0.367910
\(229\) −17.9282 −1.18473 −0.592365 0.805670i \(-0.701805\pi\)
−0.592365 + 0.805670i \(0.701805\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3.58630 −0.235452
\(233\) −13.0053 −0.852007 −0.426004 0.904721i \(-0.640079\pi\)
−0.426004 + 0.904721i \(0.640079\pi\)
\(234\) 22.3923 1.46383
\(235\) 0 0
\(236\) −8.19615 −0.533524
\(237\) −2.82843 −0.183726
\(238\) −1.79315 −0.116233
\(239\) −11.6603 −0.754239 −0.377120 0.926165i \(-0.623085\pi\)
−0.377120 + 0.926165i \(0.623085\pi\)
\(240\) 0 0
\(241\) −10.1244 −0.652167 −0.326084 0.945341i \(-0.605729\pi\)
−0.326084 + 0.945341i \(0.605729\pi\)
\(242\) 0 0
\(243\) 12.3490 0.792188
\(244\) −14.3205 −0.916777
\(245\) 0 0
\(246\) 1.73205 0.110432
\(247\) 26.2880 1.67267
\(248\) −2.72689 −0.173158
\(249\) −5.12436 −0.324743
\(250\) 0 0
\(251\) −14.1962 −0.896053 −0.448027 0.894020i \(-0.647873\pi\)
−0.448027 + 0.894020i \(0.647873\pi\)
\(252\) 4.24264 0.267261
\(253\) 0 0
\(254\) 6.46410 0.405594
\(255\) 0 0
\(256\) 19.3923 1.21202
\(257\) 5.00052 0.311924 0.155962 0.987763i \(-0.450152\pi\)
0.155962 + 0.987763i \(0.450152\pi\)
\(258\) −10.6945 −0.665813
\(259\) −6.00000 −0.372822
\(260\) 0 0
\(261\) 18.9282 1.17163
\(262\) −40.8091 −2.52120
\(263\) −21.4906 −1.32517 −0.662584 0.748988i \(-0.730540\pi\)
−0.662584 + 0.748988i \(0.730540\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 10.7321 0.658024
\(267\) −3.34607 −0.204776
\(268\) −25.8719 −1.58038
\(269\) −19.7321 −1.20308 −0.601542 0.798841i \(-0.705447\pi\)
−0.601542 + 0.798841i \(0.705447\pi\)
\(270\) 0 0
\(271\) 4.53590 0.275536 0.137768 0.990465i \(-0.456007\pi\)
0.137768 + 0.990465i \(0.456007\pi\)
\(272\) 4.62158 0.280224
\(273\) −1.96902 −0.119170
\(274\) 26.5885 1.60627
\(275\) 0 0
\(276\) −5.66025 −0.340707
\(277\) 22.7017 1.36402 0.682008 0.731345i \(-0.261107\pi\)
0.682008 + 0.731345i \(0.261107\pi\)
\(278\) −32.0464 −1.92202
\(279\) 14.3923 0.861645
\(280\) 0 0
\(281\) 17.3205 1.03325 0.516627 0.856210i \(-0.327187\pi\)
0.516627 + 0.856210i \(0.327187\pi\)
\(282\) −4.10394 −0.244386
\(283\) −30.1146 −1.79013 −0.895063 0.445939i \(-0.852870\pi\)
−0.895063 + 0.445939i \(0.852870\pi\)
\(284\) 3.80385 0.225717
\(285\) 0 0
\(286\) 0 0
\(287\) 1.55291 0.0916656
\(288\) −20.7327 −1.22169
\(289\) −15.9282 −0.936953
\(290\) 0 0
\(291\) 0.339746 0.0199163
\(292\) 8.48528 0.496564
\(293\) −9.41902 −0.550265 −0.275133 0.961406i \(-0.588722\pi\)
−0.275133 + 0.961406i \(0.588722\pi\)
\(294\) 6.19615 0.361367
\(295\) 0 0
\(296\) 3.46410 0.201347
\(297\) 0 0
\(298\) 26.5283 1.53674
\(299\) −26.7846 −1.54899
\(300\) 0 0
\(301\) −9.58846 −0.552669
\(302\) 11.9700 0.688799
\(303\) −0.720710 −0.0414037
\(304\) −27.6603 −1.58642
\(305\) 0 0
\(306\) −5.46410 −0.312362
\(307\) −17.6269 −1.00602 −0.503010 0.864280i \(-0.667774\pi\)
−0.503010 + 0.864280i \(0.667774\pi\)
\(308\) 0 0
\(309\) 2.19615 0.124935
\(310\) 0 0
\(311\) −28.0526 −1.59071 −0.795357 0.606141i \(-0.792717\pi\)
−0.795357 + 0.606141i \(0.792717\pi\)
\(312\) 1.13681 0.0643593
\(313\) 17.1464 0.969173 0.484587 0.874743i \(-0.338970\pi\)
0.484587 + 0.874743i \(0.338970\pi\)
\(314\) 23.3205 1.31605
\(315\) 0 0
\(316\) −9.46410 −0.532397
\(317\) 6.96953 0.391448 0.195724 0.980659i \(-0.437294\pi\)
0.195724 + 0.980659i \(0.437294\pi\)
\(318\) −11.9700 −0.671247
\(319\) 0 0
\(320\) 0 0
\(321\) −2.26795 −0.126585
\(322\) −10.9348 −0.609371
\(323\) −6.41473 −0.356925
\(324\) 11.5359 0.640883
\(325\) 0 0
\(326\) 14.6603 0.811956
\(327\) −0.619174 −0.0342404
\(328\) −0.896575 −0.0495051
\(329\) −3.67949 −0.202857
\(330\) 0 0
\(331\) −7.80385 −0.428938 −0.214469 0.976731i \(-0.568802\pi\)
−0.214469 + 0.976731i \(0.568802\pi\)
\(332\) −17.1464 −0.941033
\(333\) −18.2832 −1.00192
\(334\) 3.73205 0.204209
\(335\) 0 0
\(336\) 2.07180 0.113026
\(337\) −7.82894 −0.426470 −0.213235 0.977001i \(-0.568400\pi\)
−0.213235 + 0.977001i \(0.568400\pi\)
\(338\) −9.65926 −0.525394
\(339\) 1.46410 0.0795191
\(340\) 0 0
\(341\) 0 0
\(342\) 32.7028 1.76836
\(343\) 11.8313 0.638833
\(344\) 5.53590 0.298476
\(345\) 0 0
\(346\) 14.7321 0.792000
\(347\) 2.96713 0.159284 0.0796419 0.996824i \(-0.474622\pi\)
0.0796419 + 0.996824i \(0.474622\pi\)
\(348\) −6.21166 −0.332980
\(349\) 27.3205 1.46243 0.731217 0.682145i \(-0.238953\pi\)
0.731217 + 0.682145i \(0.238953\pi\)
\(350\) 0 0
\(351\) −12.5885 −0.671922
\(352\) 0 0
\(353\) 1.69161 0.0900356 0.0450178 0.998986i \(-0.485666\pi\)
0.0450178 + 0.998986i \(0.485666\pi\)
\(354\) 4.73205 0.251506
\(355\) 0 0
\(356\) −11.1962 −0.593395
\(357\) 0.480473 0.0254293
\(358\) 16.4901 0.871528
\(359\) 18.3397 0.967935 0.483967 0.875086i \(-0.339195\pi\)
0.483967 + 0.875086i \(0.339195\pi\)
\(360\) 0 0
\(361\) 19.3923 1.02065
\(362\) −6.55343 −0.344441
\(363\) 0 0
\(364\) −6.58846 −0.345329
\(365\) 0 0
\(366\) 8.26795 0.432173
\(367\) 29.6341 1.54689 0.773444 0.633865i \(-0.218532\pi\)
0.773444 + 0.633865i \(0.218532\pi\)
\(368\) 28.1827 1.46913
\(369\) 4.73205 0.246341
\(370\) 0 0
\(371\) −10.7321 −0.557180
\(372\) −4.72311 −0.244882
\(373\) −34.1170 −1.76651 −0.883255 0.468892i \(-0.844653\pi\)
−0.883255 + 0.468892i \(0.844653\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 2.12436 0.109555
\(377\) −29.3939 −1.51386
\(378\) −5.13922 −0.264333
\(379\) −5.46410 −0.280672 −0.140336 0.990104i \(-0.544818\pi\)
−0.140336 + 0.990104i \(0.544818\pi\)
\(380\) 0 0
\(381\) −1.73205 −0.0887357
\(382\) 20.7327 1.06078
\(383\) −23.7642 −1.21430 −0.607148 0.794589i \(-0.707686\pi\)
−0.607148 + 0.794589i \(0.707686\pi\)
\(384\) −2.12436 −0.108408
\(385\) 0 0
\(386\) −19.8564 −1.01066
\(387\) −29.2180 −1.48523
\(388\) 1.13681 0.0577129
\(389\) −18.1244 −0.918941 −0.459471 0.888193i \(-0.651961\pi\)
−0.459471 + 0.888193i \(0.651961\pi\)
\(390\) 0 0
\(391\) 6.53590 0.330535
\(392\) −3.20736 −0.161996
\(393\) 10.9348 0.551586
\(394\) −4.92820 −0.248279
\(395\) 0 0
\(396\) 0 0
\(397\) −16.0096 −0.803500 −0.401750 0.915749i \(-0.631598\pi\)
−0.401750 + 0.915749i \(0.631598\pi\)
\(398\) 3.86370 0.193670
\(399\) −2.87564 −0.143962
\(400\) 0 0
\(401\) −32.3205 −1.61401 −0.807005 0.590545i \(-0.798913\pi\)
−0.807005 + 0.590545i \(0.798913\pi\)
\(402\) 14.9372 0.744999
\(403\) −22.3500 −1.11333
\(404\) −2.41154 −0.119979
\(405\) 0 0
\(406\) −12.0000 −0.595550
\(407\) 0 0
\(408\) −0.277401 −0.0137334
\(409\) −1.87564 −0.0927446 −0.0463723 0.998924i \(-0.514766\pi\)
−0.0463723 + 0.998924i \(0.514766\pi\)
\(410\) 0 0
\(411\) −7.12436 −0.351419
\(412\) 7.34847 0.362033
\(413\) 4.24264 0.208767
\(414\) −33.3205 −1.63761
\(415\) 0 0
\(416\) 32.1962 1.57855
\(417\) 8.58682 0.420498
\(418\) 0 0
\(419\) −29.6603 −1.44900 −0.724499 0.689276i \(-0.757929\pi\)
−0.724499 + 0.689276i \(0.757929\pi\)
\(420\) 0 0
\(421\) 21.0526 1.02604 0.513019 0.858377i \(-0.328527\pi\)
0.513019 + 0.858377i \(0.328527\pi\)
\(422\) −21.4906 −1.04615
\(423\) −11.2122 −0.545154
\(424\) 6.19615 0.300912
\(425\) 0 0
\(426\) −2.19615 −0.106404
\(427\) 7.41284 0.358732
\(428\) −7.58871 −0.366814
\(429\) 0 0
\(430\) 0 0
\(431\) −18.9282 −0.911739 −0.455870 0.890047i \(-0.650672\pi\)
−0.455870 + 0.890047i \(0.650672\pi\)
\(432\) 13.2456 0.637277
\(433\) 20.2523 0.973261 0.486631 0.873608i \(-0.338226\pi\)
0.486631 + 0.873608i \(0.338226\pi\)
\(434\) −9.12436 −0.437983
\(435\) 0 0
\(436\) −2.07180 −0.0992211
\(437\) −39.1175 −1.87124
\(438\) −4.89898 −0.234082
\(439\) −20.2487 −0.966418 −0.483209 0.875505i \(-0.660529\pi\)
−0.483209 + 0.875505i \(0.660529\pi\)
\(440\) 0 0
\(441\) 16.9282 0.806105
\(442\) 8.48528 0.403604
\(443\) −9.17878 −0.436097 −0.218049 0.975938i \(-0.569969\pi\)
−0.218049 + 0.975938i \(0.569969\pi\)
\(444\) 6.00000 0.284747
\(445\) 0 0
\(446\) 11.1962 0.530153
\(447\) −7.10823 −0.336208
\(448\) 5.13922 0.242805
\(449\) −10.5167 −0.496312 −0.248156 0.968720i \(-0.579825\pi\)
−0.248156 + 0.968720i \(0.579825\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 4.89898 0.230429
\(453\) −3.20736 −0.150695
\(454\) 41.5885 1.95184
\(455\) 0 0
\(456\) 1.66025 0.0777485
\(457\) −21.2132 −0.992312 −0.496156 0.868233i \(-0.665256\pi\)
−0.496156 + 0.868233i \(0.665256\pi\)
\(458\) 34.6346 1.61837
\(459\) 3.07180 0.143379
\(460\) 0 0
\(461\) −33.0000 −1.53696 −0.768482 0.639872i \(-0.778987\pi\)
−0.768482 + 0.639872i \(0.778987\pi\)
\(462\) 0 0
\(463\) −17.8671 −0.830356 −0.415178 0.909740i \(-0.636281\pi\)
−0.415178 + 0.909740i \(0.636281\pi\)
\(464\) 30.9282 1.43581
\(465\) 0 0
\(466\) 25.1244 1.16386
\(467\) −17.7656 −0.822094 −0.411047 0.911614i \(-0.634837\pi\)
−0.411047 + 0.911614i \(0.634837\pi\)
\(468\) −20.0764 −0.928032
\(469\) 13.3923 0.618399
\(470\) 0 0
\(471\) −6.24871 −0.287925
\(472\) −2.44949 −0.112747
\(473\) 0 0
\(474\) 5.46410 0.250974
\(475\) 0 0
\(476\) 1.60770 0.0736886
\(477\) −32.7028 −1.49736
\(478\) 22.5259 1.03031
\(479\) 9.46410 0.432426 0.216213 0.976346i \(-0.430629\pi\)
0.216213 + 0.976346i \(0.430629\pi\)
\(480\) 0 0
\(481\) 28.3923 1.29458
\(482\) 19.5588 0.890877
\(483\) 2.92996 0.133318
\(484\) 0 0
\(485\) 0 0
\(486\) −23.8564 −1.08215
\(487\) 12.7279 0.576757 0.288379 0.957516i \(-0.406884\pi\)
0.288379 + 0.957516i \(0.406884\pi\)
\(488\) −4.27981 −0.193738
\(489\) −3.92820 −0.177639
\(490\) 0 0
\(491\) −15.7128 −0.709109 −0.354555 0.935035i \(-0.615368\pi\)
−0.354555 + 0.935035i \(0.615368\pi\)
\(492\) −1.55291 −0.0700108
\(493\) 7.17260 0.323038
\(494\) −50.7846 −2.28491
\(495\) 0 0
\(496\) 23.5167 1.05593
\(497\) −1.96902 −0.0883225
\(498\) 9.89949 0.443607
\(499\) 38.3923 1.71868 0.859338 0.511408i \(-0.170876\pi\)
0.859338 + 0.511408i \(0.170876\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) 27.4249 1.22403
\(503\) 34.2557 1.52739 0.763693 0.645580i \(-0.223384\pi\)
0.763693 + 0.645580i \(0.223384\pi\)
\(504\) 1.26795 0.0564789
\(505\) 0 0
\(506\) 0 0
\(507\) 2.58819 0.114946
\(508\) −5.79555 −0.257136
\(509\) −21.9282 −0.971951 −0.485975 0.873973i \(-0.661535\pi\)
−0.485975 + 0.873973i \(0.661535\pi\)
\(510\) 0 0
\(511\) −4.39230 −0.194304
\(512\) −29.2552 −1.29291
\(513\) −18.3848 −0.811708
\(514\) −9.66025 −0.426096
\(515\) 0 0
\(516\) 9.58846 0.422108
\(517\) 0 0
\(518\) 11.5911 0.509284
\(519\) −3.94744 −0.173273
\(520\) 0 0
\(521\) 39.2487 1.71952 0.859759 0.510701i \(-0.170614\pi\)
0.859759 + 0.510701i \(0.170614\pi\)
\(522\) −36.5665 −1.60047
\(523\) −8.66115 −0.378726 −0.189363 0.981907i \(-0.560642\pi\)
−0.189363 + 0.981907i \(0.560642\pi\)
\(524\) 36.5885 1.59837
\(525\) 0 0
\(526\) 41.5167 1.81021
\(527\) 5.45378 0.237570
\(528\) 0 0
\(529\) 16.8564 0.732887
\(530\) 0 0
\(531\) 12.9282 0.561036
\(532\) −9.62209 −0.417171
\(533\) −7.34847 −0.318298
\(534\) 6.46410 0.279729
\(535\) 0 0
\(536\) −7.73205 −0.333974
\(537\) −4.41851 −0.190673
\(538\) 38.1194 1.64344
\(539\) 0 0
\(540\) 0 0
\(541\) −23.3923 −1.00571 −0.502857 0.864370i \(-0.667718\pi\)
−0.502857 + 0.864370i \(0.667718\pi\)
\(542\) −8.76268 −0.376389
\(543\) 1.75599 0.0753566
\(544\) −7.85641 −0.336841
\(545\) 0 0
\(546\) 3.80385 0.162790
\(547\) 20.2523 0.865924 0.432962 0.901412i \(-0.357468\pi\)
0.432962 + 0.901412i \(0.357468\pi\)
\(548\) −23.8386 −1.01833
\(549\) 22.5885 0.964052
\(550\) 0 0
\(551\) −42.9282 −1.82880
\(552\) −1.69161 −0.0719999
\(553\) 4.89898 0.208326
\(554\) −43.8564 −1.86328
\(555\) 0 0
\(556\) 28.7321 1.21851
\(557\) 6.48906 0.274950 0.137475 0.990505i \(-0.456101\pi\)
0.137475 + 0.990505i \(0.456101\pi\)
\(558\) −27.8038 −1.17703
\(559\) 45.3731 1.91908
\(560\) 0 0
\(561\) 0 0
\(562\) −33.4607 −1.41145
\(563\) −6.07296 −0.255945 −0.127972 0.991778i \(-0.540847\pi\)
−0.127972 + 0.991778i \(0.540847\pi\)
\(564\) 3.67949 0.154935
\(565\) 0 0
\(566\) 58.1769 2.44536
\(567\) −5.97142 −0.250776
\(568\) 1.13681 0.0476996
\(569\) 5.87564 0.246320 0.123160 0.992387i \(-0.460697\pi\)
0.123160 + 0.992387i \(0.460697\pi\)
\(570\) 0 0
\(571\) 35.7128 1.49453 0.747267 0.664524i \(-0.231365\pi\)
0.747267 + 0.664524i \(0.231365\pi\)
\(572\) 0 0
\(573\) −5.55532 −0.232077
\(574\) −3.00000 −0.125218
\(575\) 0 0
\(576\) 15.6603 0.652511
\(577\) 20.7327 0.863115 0.431557 0.902085i \(-0.357964\pi\)
0.431557 + 0.902085i \(0.357964\pi\)
\(578\) 30.7709 1.27990
\(579\) 5.32051 0.221113
\(580\) 0 0
\(581\) 8.87564 0.368224
\(582\) −0.656339 −0.0272061
\(583\) 0 0
\(584\) 2.53590 0.104936
\(585\) 0 0
\(586\) 18.1962 0.751676
\(587\) 35.9473 1.48370 0.741852 0.670564i \(-0.233948\pi\)
0.741852 + 0.670564i \(0.233948\pi\)
\(588\) −5.55532 −0.229097
\(589\) −32.6410 −1.34495
\(590\) 0 0
\(591\) 1.32051 0.0543184
\(592\) −29.8744 −1.22783
\(593\) 30.2533 1.24235 0.621177 0.783670i \(-0.286655\pi\)
0.621177 + 0.783670i \(0.286655\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −23.7846 −0.974256
\(597\) −1.03528 −0.0423710
\(598\) 51.7439 2.11597
\(599\) 20.4449 0.835354 0.417677 0.908595i \(-0.362844\pi\)
0.417677 + 0.908595i \(0.362844\pi\)
\(600\) 0 0
\(601\) 20.0000 0.815817 0.407909 0.913023i \(-0.366258\pi\)
0.407909 + 0.913023i \(0.366258\pi\)
\(602\) 18.5235 0.754961
\(603\) 40.8091 1.66188
\(604\) −10.7321 −0.436681
\(605\) 0 0
\(606\) 1.39230 0.0565585
\(607\) 31.4916 1.27821 0.639103 0.769121i \(-0.279306\pi\)
0.639103 + 0.769121i \(0.279306\pi\)
\(608\) 47.0208 1.90694
\(609\) 3.21539 0.130294
\(610\) 0 0
\(611\) 17.4115 0.704396
\(612\) 4.89898 0.198030
\(613\) 6.69213 0.270293 0.135146 0.990826i \(-0.456850\pi\)
0.135146 + 0.990826i \(0.456850\pi\)
\(614\) 34.0526 1.37425
\(615\) 0 0
\(616\) 0 0
\(617\) −1.69161 −0.0681019 −0.0340509 0.999420i \(-0.510841\pi\)
−0.0340509 + 0.999420i \(0.510841\pi\)
\(618\) −4.24264 −0.170664
\(619\) −28.7846 −1.15695 −0.578476 0.815700i \(-0.696352\pi\)
−0.578476 + 0.815700i \(0.696352\pi\)
\(620\) 0 0
\(621\) 18.7321 0.751691
\(622\) 54.1934 2.17296
\(623\) 5.79555 0.232194
\(624\) −9.80385 −0.392468
\(625\) 0 0
\(626\) −33.1244 −1.32392
\(627\) 0 0
\(628\) −20.9086 −0.834344
\(629\) −6.92820 −0.276246
\(630\) 0 0
\(631\) 19.3205 0.769137 0.384569 0.923096i \(-0.374350\pi\)
0.384569 + 0.923096i \(0.374350\pi\)
\(632\) −2.82843 −0.112509
\(633\) 5.75839 0.228875
\(634\) −13.4641 −0.534728
\(635\) 0 0
\(636\) 10.7321 0.425553
\(637\) −26.2880 −1.04157
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 19.8564 0.784281 0.392140 0.919905i \(-0.371735\pi\)
0.392140 + 0.919905i \(0.371735\pi\)
\(642\) 4.38134 0.172918
\(643\) −27.6651 −1.09100 −0.545502 0.838109i \(-0.683661\pi\)
−0.545502 + 0.838109i \(0.683661\pi\)
\(644\) 9.80385 0.386326
\(645\) 0 0
\(646\) 12.3923 0.487569
\(647\) 18.9767 0.746053 0.373026 0.927821i \(-0.378320\pi\)
0.373026 + 0.927821i \(0.378320\pi\)
\(648\) 3.44760 0.135435
\(649\) 0 0
\(650\) 0 0
\(651\) 2.44486 0.0958218
\(652\) −13.1440 −0.514760
\(653\) −31.9449 −1.25010 −0.625050 0.780584i \(-0.714922\pi\)
−0.625050 + 0.780584i \(0.714922\pi\)
\(654\) 1.19615 0.0467733
\(655\) 0 0
\(656\) 7.73205 0.301886
\(657\) −13.3843 −0.522170
\(658\) 7.10823 0.277108
\(659\) 11.3205 0.440984 0.220492 0.975389i \(-0.429234\pi\)
0.220492 + 0.975389i \(0.429234\pi\)
\(660\) 0 0
\(661\) −1.58846 −0.0617838 −0.0308919 0.999523i \(-0.509835\pi\)
−0.0308919 + 0.999523i \(0.509835\pi\)
\(662\) 15.0759 0.585941
\(663\) −2.27362 −0.0883003
\(664\) −5.12436 −0.198864
\(665\) 0 0
\(666\) 35.3205 1.36864
\(667\) 43.7391 1.69358
\(668\) −3.34607 −0.129463
\(669\) −3.00000 −0.115987
\(670\) 0 0
\(671\) 0 0
\(672\) −3.52193 −0.135861
\(673\) −1.61729 −0.0623418 −0.0311709 0.999514i \(-0.509924\pi\)
−0.0311709 + 0.999514i \(0.509924\pi\)
\(674\) 15.1244 0.582568
\(675\) 0 0
\(676\) 8.66025 0.333087
\(677\) −9.04008 −0.347439 −0.173719 0.984795i \(-0.555579\pi\)
−0.173719 + 0.984795i \(0.555579\pi\)
\(678\) −2.82843 −0.108625
\(679\) −0.588457 −0.0225829
\(680\) 0 0
\(681\) −11.1436 −0.427023
\(682\) 0 0
\(683\) 22.4887 0.860507 0.430253 0.902708i \(-0.358424\pi\)
0.430253 + 0.902708i \(0.358424\pi\)
\(684\) −29.3205 −1.12110
\(685\) 0 0
\(686\) −22.8564 −0.872662
\(687\) −9.28032 −0.354066
\(688\) −47.7415 −1.82013
\(689\) 50.7846 1.93474
\(690\) 0 0
\(691\) −27.9090 −1.06171 −0.530854 0.847464i \(-0.678129\pi\)
−0.530854 + 0.847464i \(0.678129\pi\)
\(692\) −13.2084 −0.502108
\(693\) 0 0
\(694\) −5.73205 −0.217586
\(695\) 0 0
\(696\) −1.85641 −0.0703669
\(697\) 1.79315 0.0679204
\(698\) −52.7792 −1.99772
\(699\) −6.73205 −0.254630
\(700\) 0 0
\(701\) 10.1436 0.383118 0.191559 0.981481i \(-0.438646\pi\)
0.191559 + 0.981481i \(0.438646\pi\)
\(702\) 24.3190 0.917863
\(703\) 41.4655 1.56390
\(704\) 0 0
\(705\) 0 0
\(706\) −3.26795 −0.122991
\(707\) 1.24831 0.0469474
\(708\) −4.24264 −0.159448
\(709\) 23.3923 0.878516 0.439258 0.898361i \(-0.355241\pi\)
0.439258 + 0.898361i \(0.355241\pi\)
\(710\) 0 0
\(711\) 14.9282 0.559851
\(712\) −3.34607 −0.125399
\(713\) 33.2576 1.24551
\(714\) −0.928203 −0.0347371
\(715\) 0 0
\(716\) −14.7846 −0.552527
\(717\) −6.03579 −0.225411
\(718\) −35.4297 −1.32222
\(719\) −39.8038 −1.48443 −0.742217 0.670160i \(-0.766225\pi\)
−0.742217 + 0.670160i \(0.766225\pi\)
\(720\) 0 0
\(721\) −3.80385 −0.141663
\(722\) −37.4631 −1.39423
\(723\) −5.24075 −0.194906
\(724\) 5.87564 0.218367
\(725\) 0 0
\(726\) 0 0
\(727\) 4.00240 0.148441 0.0742205 0.997242i \(-0.476353\pi\)
0.0742205 + 0.997242i \(0.476353\pi\)
\(728\) −1.96902 −0.0729766
\(729\) −13.5885 −0.503276
\(730\) 0 0
\(731\) −11.0718 −0.409505
\(732\) −7.41284 −0.273986
\(733\) 34.7733 1.28438 0.642191 0.766545i \(-0.278026\pi\)
0.642191 + 0.766545i \(0.278026\pi\)
\(734\) −57.2487 −2.11309
\(735\) 0 0
\(736\) −47.9090 −1.76595
\(737\) 0 0
\(738\) −9.14162 −0.336508
\(739\) −25.0718 −0.922281 −0.461140 0.887327i \(-0.652560\pi\)
−0.461140 + 0.887327i \(0.652560\pi\)
\(740\) 0 0
\(741\) 13.6077 0.499891
\(742\) 20.7327 0.761122
\(743\) −8.04197 −0.295031 −0.147516 0.989060i \(-0.547128\pi\)
−0.147516 + 0.989060i \(0.547128\pi\)
\(744\) −1.41154 −0.0517497
\(745\) 0 0
\(746\) 65.9090 2.41310
\(747\) 27.0459 0.989559
\(748\) 0 0
\(749\) 3.92820 0.143533
\(750\) 0 0
\(751\) −20.6410 −0.753201 −0.376601 0.926376i \(-0.622907\pi\)
−0.376601 + 0.926376i \(0.622907\pi\)
\(752\) −18.3204 −0.668076
\(753\) −7.34847 −0.267793
\(754\) 56.7846 2.06797
\(755\) 0 0
\(756\) 4.60770 0.167580
\(757\) −35.0779 −1.27493 −0.637465 0.770480i \(-0.720017\pi\)
−0.637465 + 0.770480i \(0.720017\pi\)
\(758\) 10.5558 0.383405
\(759\) 0 0
\(760\) 0 0
\(761\) 7.85641 0.284795 0.142397 0.989810i \(-0.454519\pi\)
0.142397 + 0.989810i \(0.454519\pi\)
\(762\) 3.34607 0.121215
\(763\) 1.07244 0.0388250
\(764\) −18.5885 −0.672507
\(765\) 0 0
\(766\) 45.9090 1.65876
\(767\) −20.0764 −0.724916
\(768\) 10.0382 0.362222
\(769\) 17.8564 0.643918 0.321959 0.946754i \(-0.395659\pi\)
0.321959 + 0.946754i \(0.395659\pi\)
\(770\) 0 0
\(771\) 2.58846 0.0932210
\(772\) 17.8028 0.640736
\(773\) 51.6424 1.85745 0.928723 0.370774i \(-0.120907\pi\)
0.928723 + 0.370774i \(0.120907\pi\)
\(774\) 56.4449 2.02887
\(775\) 0 0
\(776\) 0.339746 0.0121962
\(777\) −3.10583 −0.111421
\(778\) 35.0136 1.25530
\(779\) −10.7321 −0.384516
\(780\) 0 0
\(781\) 0 0
\(782\) −12.6264 −0.451519
\(783\) 20.5569 0.734642
\(784\) 27.6603 0.987866
\(785\) 0 0
\(786\) −21.1244 −0.753481
\(787\) 4.65874 0.166066 0.0830331 0.996547i \(-0.473539\pi\)
0.0830331 + 0.996547i \(0.473539\pi\)
\(788\) 4.41851 0.157403
\(789\) −11.1244 −0.396038
\(790\) 0 0
\(791\) −2.53590 −0.0901662
\(792\) 0 0
\(793\) −35.0779 −1.24565
\(794\) 30.9282 1.09760
\(795\) 0 0
\(796\) −3.46410 −0.122782
\(797\) −7.90327 −0.279948 −0.139974 0.990155i \(-0.544702\pi\)
−0.139974 + 0.990155i \(0.544702\pi\)
\(798\) 5.55532 0.196656
\(799\) −4.24871 −0.150309
\(800\) 0 0
\(801\) 17.6603 0.623994
\(802\) 62.4384 2.20478
\(803\) 0 0
\(804\) −13.3923 −0.472310
\(805\) 0 0
\(806\) 43.1769 1.52084
\(807\) −10.2141 −0.359552
\(808\) −0.720710 −0.0253545
\(809\) −21.4641 −0.754638 −0.377319 0.926083i \(-0.623154\pi\)
−0.377319 + 0.926083i \(0.623154\pi\)
\(810\) 0 0
\(811\) −32.9808 −1.15811 −0.579056 0.815288i \(-0.696579\pi\)
−0.579056 + 0.815288i \(0.696579\pi\)
\(812\) 10.7589 0.377564
\(813\) 2.34795 0.0823463
\(814\) 0 0
\(815\) 0 0
\(816\) 2.39230 0.0837474
\(817\) 66.2650 2.31832
\(818\) 3.62347 0.126692
\(819\) 10.3923 0.363137
\(820\) 0 0
\(821\) 18.8038 0.656259 0.328129 0.944633i \(-0.393582\pi\)
0.328129 + 0.944633i \(0.393582\pi\)
\(822\) 13.7632 0.480047
\(823\) −48.3978 −1.68704 −0.843521 0.537096i \(-0.819521\pi\)
−0.843521 + 0.537096i \(0.819521\pi\)
\(824\) 2.19615 0.0765066
\(825\) 0 0
\(826\) −8.19615 −0.285181
\(827\) −26.7314 −0.929540 −0.464770 0.885431i \(-0.653863\pi\)
−0.464770 + 0.885431i \(0.653863\pi\)
\(828\) 29.8744 1.03821
\(829\) 41.9808 1.45805 0.729026 0.684486i \(-0.239973\pi\)
0.729026 + 0.684486i \(0.239973\pi\)
\(830\) 0 0
\(831\) 11.7513 0.407648
\(832\) −24.3190 −0.843111
\(833\) 6.41473 0.222257
\(834\) −16.5885 −0.574411
\(835\) 0 0
\(836\) 0 0
\(837\) 15.6307 0.540275
\(838\) 57.2992 1.97937
\(839\) 2.78461 0.0961354 0.0480677 0.998844i \(-0.484694\pi\)
0.0480677 + 0.998844i \(0.484694\pi\)
\(840\) 0 0
\(841\) 19.0000 0.655172
\(842\) −40.6704 −1.40160
\(843\) 8.96575 0.308797
\(844\) 19.2679 0.663230
\(845\) 0 0
\(846\) 21.6603 0.744695
\(847\) 0 0
\(848\) −53.4355 −1.83498
\(849\) −15.5885 −0.534994
\(850\) 0 0
\(851\) −42.2487 −1.44827
\(852\) 1.96902 0.0674574
\(853\) 12.4233 0.425366 0.212683 0.977121i \(-0.431780\pi\)
0.212683 + 0.977121i \(0.431780\pi\)
\(854\) −14.3205 −0.490038
\(855\) 0 0
\(856\) −2.26795 −0.0775169
\(857\) 24.4206 0.834191 0.417095 0.908863i \(-0.363048\pi\)
0.417095 + 0.908863i \(0.363048\pi\)
\(858\) 0 0
\(859\) −11.8038 −0.402742 −0.201371 0.979515i \(-0.564540\pi\)
−0.201371 + 0.979515i \(0.564540\pi\)
\(860\) 0 0
\(861\) 0.803848 0.0273951
\(862\) 36.5665 1.24546
\(863\) 9.76079 0.332261 0.166131 0.986104i \(-0.446873\pi\)
0.166131 + 0.986104i \(0.446873\pi\)
\(864\) −22.5167 −0.766032
\(865\) 0 0
\(866\) −39.1244 −1.32950
\(867\) −8.24504 −0.280016
\(868\) 8.18067 0.277670
\(869\) 0 0
\(870\) 0 0
\(871\) −63.3731 −2.14731
\(872\) −0.619174 −0.0209679
\(873\) −1.79315 −0.0606890
\(874\) 75.5692 2.55617
\(875\) 0 0
\(876\) 4.39230 0.148402
\(877\) 55.9865 1.89053 0.945265 0.326302i \(-0.105803\pi\)
0.945265 + 0.326302i \(0.105803\pi\)
\(878\) 39.1175 1.32015
\(879\) −4.87564 −0.164451
\(880\) 0 0
\(881\) 38.9090 1.31088 0.655438 0.755249i \(-0.272484\pi\)
0.655438 + 0.755249i \(0.272484\pi\)
\(882\) −32.7028 −1.10116
\(883\) 47.0208 1.58238 0.791188 0.611574i \(-0.209463\pi\)
0.791188 + 0.611574i \(0.209463\pi\)
\(884\) −7.60770 −0.255874
\(885\) 0 0
\(886\) 17.7321 0.595720
\(887\) 8.34658 0.280251 0.140125 0.990134i \(-0.455249\pi\)
0.140125 + 0.990134i \(0.455249\pi\)
\(888\) 1.79315 0.0601742
\(889\) 3.00000 0.100617
\(890\) 0 0
\(891\) 0 0
\(892\) −10.0382 −0.336104
\(893\) 25.4286 0.850937
\(894\) 13.7321 0.459268
\(895\) 0 0
\(896\) 3.67949 0.122923
\(897\) −13.8647 −0.462930
\(898\) 20.3166 0.677975
\(899\) 36.4974 1.21726
\(900\) 0 0
\(901\) −12.3923 −0.412848
\(902\) 0 0
\(903\) −4.96335 −0.165170
\(904\) 1.46410 0.0486953
\(905\) 0 0
\(906\) 6.19615 0.205853
\(907\) −8.06918 −0.267933 −0.133966 0.990986i \(-0.542771\pi\)
−0.133966 + 0.990986i \(0.542771\pi\)
\(908\) −37.2872 −1.23742
\(909\) 3.80385 0.126166
\(910\) 0 0
\(911\) 18.2487 0.604607 0.302303 0.953212i \(-0.402244\pi\)
0.302303 + 0.953212i \(0.402244\pi\)
\(912\) −14.3180 −0.474116
\(913\) 0 0
\(914\) 40.9808 1.35552
\(915\) 0 0
\(916\) −31.0526 −1.02601
\(917\) −18.9396 −0.625440
\(918\) −5.93426 −0.195860
\(919\) 32.9808 1.08793 0.543967 0.839106i \(-0.316921\pi\)
0.543967 + 0.839106i \(0.316921\pi\)
\(920\) 0 0
\(921\) −9.12436 −0.300658
\(922\) 63.7511 2.09953
\(923\) 9.31749 0.306689
\(924\) 0 0
\(925\) 0 0
\(926\) 34.5167 1.13429
\(927\) −11.5911 −0.380702
\(928\) −52.5761 −1.72589
\(929\) −16.1436 −0.529654 −0.264827 0.964296i \(-0.585315\pi\)
−0.264827 + 0.964296i \(0.585315\pi\)
\(930\) 0 0
\(931\) −38.3923 −1.25826
\(932\) −22.5259 −0.737860
\(933\) −14.5211 −0.475399
\(934\) 34.3205 1.12300
\(935\) 0 0
\(936\) −6.00000 −0.196116
\(937\) 22.7017 0.741634 0.370817 0.928706i \(-0.379078\pi\)
0.370817 + 0.928706i \(0.379078\pi\)
\(938\) −25.8719 −0.844749
\(939\) 8.87564 0.289646
\(940\) 0 0
\(941\) −27.0000 −0.880175 −0.440087 0.897955i \(-0.645053\pi\)
−0.440087 + 0.897955i \(0.645053\pi\)
\(942\) 12.0716 0.393313
\(943\) 10.9348 0.356085
\(944\) 21.1244 0.687539
\(945\) 0 0
\(946\) 0 0
\(947\) 14.3180 0.465273 0.232636 0.972564i \(-0.425265\pi\)
0.232636 + 0.972564i \(0.425265\pi\)
\(948\) −4.89898 −0.159111
\(949\) 20.7846 0.674697
\(950\) 0 0
\(951\) 3.60770 0.116988
\(952\) 0.480473 0.0155722
\(953\) 31.3901 1.01683 0.508413 0.861114i \(-0.330233\pi\)
0.508413 + 0.861114i \(0.330233\pi\)
\(954\) 63.1769 2.04543
\(955\) 0 0
\(956\) −20.1962 −0.653190
\(957\) 0 0
\(958\) −18.2832 −0.590705
\(959\) 12.3397 0.398471
\(960\) 0 0
\(961\) −3.24871 −0.104797
\(962\) −54.8497 −1.76843
\(963\) 11.9700 0.385729
\(964\) −17.5359 −0.564793
\(965\) 0 0
\(966\) −5.66025 −0.182116
\(967\) 4.24264 0.136434 0.0682171 0.997671i \(-0.478269\pi\)
0.0682171 + 0.997671i \(0.478269\pi\)
\(968\) 0 0
\(969\) −3.32051 −0.106670
\(970\) 0 0
\(971\) −11.0718 −0.355311 −0.177655 0.984093i \(-0.556851\pi\)
−0.177655 + 0.984093i \(0.556851\pi\)
\(972\) 21.3891 0.686055
\(973\) −14.8728 −0.476800
\(974\) −24.5885 −0.787865
\(975\) 0 0
\(976\) 36.9090 1.18143
\(977\) 14.9743 0.479072 0.239536 0.970888i \(-0.423005\pi\)
0.239536 + 0.970888i \(0.423005\pi\)
\(978\) 7.58871 0.242660
\(979\) 0 0
\(980\) 0 0
\(981\) 3.26795 0.104338
\(982\) 30.3548 0.968661
\(983\) 15.0115 0.478793 0.239396 0.970922i \(-0.423050\pi\)
0.239396 + 0.970922i \(0.423050\pi\)
\(984\) −0.464102 −0.0147950
\(985\) 0 0
\(986\) −13.8564 −0.441278
\(987\) −1.90465 −0.0606255
\(988\) 45.5322 1.44857
\(989\) −67.5167 −2.14690
\(990\) 0 0
\(991\) −45.9090 −1.45835 −0.729173 0.684329i \(-0.760095\pi\)
−0.729173 + 0.684329i \(0.760095\pi\)
\(992\) −39.9769 −1.26927
\(993\) −4.03957 −0.128192
\(994\) 3.80385 0.120651
\(995\) 0 0
\(996\) −8.87564 −0.281236
\(997\) 0.175865 0.00556971 0.00278486 0.999996i \(-0.499114\pi\)
0.00278486 + 0.999996i \(0.499114\pi\)
\(998\) −74.1682 −2.34775
\(999\) −19.8564 −0.628229
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 3025.2.a.y.1.1 4
5.2 odd 4 605.2.b.d.364.1 4
5.3 odd 4 605.2.b.d.364.4 yes 4
5.4 even 2 inner 3025.2.a.y.1.4 4
11.10 odd 2 3025.2.a.z.1.4 4
55.2 even 20 605.2.j.e.444.4 16
55.3 odd 20 605.2.j.f.9.4 16
55.7 even 20 605.2.j.e.269.1 16
55.8 even 20 605.2.j.e.9.1 16
55.13 even 20 605.2.j.e.444.1 16
55.17 even 20 605.2.j.e.124.1 16
55.18 even 20 605.2.j.e.269.4 16
55.27 odd 20 605.2.j.f.124.4 16
55.28 even 20 605.2.j.e.124.4 16
55.32 even 4 605.2.b.e.364.4 yes 4
55.37 odd 20 605.2.j.f.269.4 16
55.38 odd 20 605.2.j.f.124.1 16
55.42 odd 20 605.2.j.f.444.1 16
55.43 even 4 605.2.b.e.364.1 yes 4
55.47 odd 20 605.2.j.f.9.1 16
55.48 odd 20 605.2.j.f.269.1 16
55.52 even 20 605.2.j.e.9.4 16
55.53 odd 20 605.2.j.f.444.4 16
55.54 odd 2 3025.2.a.z.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
605.2.b.d.364.1 4 5.2 odd 4
605.2.b.d.364.4 yes 4 5.3 odd 4
605.2.b.e.364.1 yes 4 55.43 even 4
605.2.b.e.364.4 yes 4 55.32 even 4
605.2.j.e.9.1 16 55.8 even 20
605.2.j.e.9.4 16 55.52 even 20
605.2.j.e.124.1 16 55.17 even 20
605.2.j.e.124.4 16 55.28 even 20
605.2.j.e.269.1 16 55.7 even 20
605.2.j.e.269.4 16 55.18 even 20
605.2.j.e.444.1 16 55.13 even 20
605.2.j.e.444.4 16 55.2 even 20
605.2.j.f.9.1 16 55.47 odd 20
605.2.j.f.9.4 16 55.3 odd 20
605.2.j.f.124.1 16 55.38 odd 20
605.2.j.f.124.4 16 55.27 odd 20
605.2.j.f.269.1 16 55.48 odd 20
605.2.j.f.269.4 16 55.37 odd 20
605.2.j.f.444.1 16 55.42 odd 20
605.2.j.f.444.4 16 55.53 odd 20
3025.2.a.y.1.1 4 1.1 even 1 trivial
3025.2.a.y.1.4 4 5.4 even 2 inner
3025.2.a.z.1.1 4 55.54 odd 2
3025.2.a.z.1.4 4 11.10 odd 2