Properties

Label 3822.2.a.bk.1.1
Level $3822$
Weight $2$
Character 3822.1
Self dual yes
Analytic conductor $30.519$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3822.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.5188236525\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.64575\) of defining polynomial
Character \(\chi\) \(=\) 3822.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.64575 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.64575 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +3.64575 q^{10} +0.645751 q^{11} +1.00000 q^{12} +1.00000 q^{13} -3.64575 q^{15} +1.00000 q^{16} -6.64575 q^{17} -1.00000 q^{18} +5.00000 q^{19} -3.64575 q^{20} -0.645751 q^{22} -2.35425 q^{23} -1.00000 q^{24} +8.29150 q^{25} -1.00000 q^{26} +1.00000 q^{27} +4.29150 q^{29} +3.64575 q^{30} +3.29150 q^{31} -1.00000 q^{32} +0.645751 q^{33} +6.64575 q^{34} +1.00000 q^{36} +5.64575 q^{37} -5.00000 q^{38} +1.00000 q^{39} +3.64575 q^{40} -2.35425 q^{41} -5.29150 q^{43} +0.645751 q^{44} -3.64575 q^{45} +2.35425 q^{46} +3.00000 q^{47} +1.00000 q^{48} -8.29150 q^{50} -6.64575 q^{51} +1.00000 q^{52} -3.00000 q^{53} -1.00000 q^{54} -2.35425 q^{55} +5.00000 q^{57} -4.29150 q^{58} -7.93725 q^{59} -3.64575 q^{60} -11.9373 q^{61} -3.29150 q^{62} +1.00000 q^{64} -3.64575 q^{65} -0.645751 q^{66} +7.58301 q^{67} -6.64575 q^{68} -2.35425 q^{69} +16.2915 q^{71} -1.00000 q^{72} -13.6458 q^{73} -5.64575 q^{74} +8.29150 q^{75} +5.00000 q^{76} -1.00000 q^{78} -10.0000 q^{79} -3.64575 q^{80} +1.00000 q^{81} +2.35425 q^{82} -13.2915 q^{83} +24.2288 q^{85} +5.29150 q^{86} +4.29150 q^{87} -0.645751 q^{88} +16.9373 q^{89} +3.64575 q^{90} -2.35425 q^{92} +3.29150 q^{93} -3.00000 q^{94} -18.2288 q^{95} -1.00000 q^{96} +0.937254 q^{97} +0.645751 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{10} - 4 q^{11} + 2 q^{12} + 2 q^{13} - 2 q^{15} + 2 q^{16} - 8 q^{17} - 2 q^{18} + 10 q^{19} - 2 q^{20} + 4 q^{22} - 10 q^{23} - 2 q^{24} + 6 q^{25} - 2 q^{26} + 2 q^{27} - 2 q^{29} + 2 q^{30} - 4 q^{31} - 2 q^{32} - 4 q^{33} + 8 q^{34} + 2 q^{36} + 6 q^{37} - 10 q^{38} + 2 q^{39} + 2 q^{40} - 10 q^{41} - 4 q^{44} - 2 q^{45} + 10 q^{46} + 6 q^{47} + 2 q^{48} - 6 q^{50} - 8 q^{51} + 2 q^{52} - 6 q^{53} - 2 q^{54} - 10 q^{55} + 10 q^{57} + 2 q^{58} - 2 q^{60} - 8 q^{61} + 4 q^{62} + 2 q^{64} - 2 q^{65} + 4 q^{66} - 6 q^{67} - 8 q^{68} - 10 q^{69} + 22 q^{71} - 2 q^{72} - 22 q^{73} - 6 q^{74} + 6 q^{75} + 10 q^{76} - 2 q^{78} - 20 q^{79} - 2 q^{80} + 2 q^{81} + 10 q^{82} - 16 q^{83} + 22 q^{85} - 2 q^{87} + 4 q^{88} + 18 q^{89} + 2 q^{90} - 10 q^{92} - 4 q^{93} - 6 q^{94} - 10 q^{95} - 2 q^{96} - 14 q^{97} - 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −3.64575 −1.63043 −0.815215 0.579159i \(-0.803381\pi\)
−0.815215 + 0.579159i \(0.803381\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 3.64575 1.15289
\(11\) 0.645751 0.194701 0.0973507 0.995250i \(-0.468963\pi\)
0.0973507 + 0.995250i \(0.468963\pi\)
\(12\) 1.00000 0.288675
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) −3.64575 −0.941329
\(16\) 1.00000 0.250000
\(17\) −6.64575 −1.61183 −0.805916 0.592030i \(-0.798327\pi\)
−0.805916 + 0.592030i \(0.798327\pi\)
\(18\) −1.00000 −0.235702
\(19\) 5.00000 1.14708 0.573539 0.819178i \(-0.305570\pi\)
0.573539 + 0.819178i \(0.305570\pi\)
\(20\) −3.64575 −0.815215
\(21\) 0 0
\(22\) −0.645751 −0.137675
\(23\) −2.35425 −0.490895 −0.245447 0.969410i \(-0.578935\pi\)
−0.245447 + 0.969410i \(0.578935\pi\)
\(24\) −1.00000 −0.204124
\(25\) 8.29150 1.65830
\(26\) −1.00000 −0.196116
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 4.29150 0.796912 0.398456 0.917187i \(-0.369546\pi\)
0.398456 + 0.917187i \(0.369546\pi\)
\(30\) 3.64575 0.665620
\(31\) 3.29150 0.591171 0.295586 0.955316i \(-0.404485\pi\)
0.295586 + 0.955316i \(0.404485\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.645751 0.112411
\(34\) 6.64575 1.13974
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 5.64575 0.928156 0.464078 0.885794i \(-0.346386\pi\)
0.464078 + 0.885794i \(0.346386\pi\)
\(38\) −5.00000 −0.811107
\(39\) 1.00000 0.160128
\(40\) 3.64575 0.576444
\(41\) −2.35425 −0.367672 −0.183836 0.982957i \(-0.558852\pi\)
−0.183836 + 0.982957i \(0.558852\pi\)
\(42\) 0 0
\(43\) −5.29150 −0.806947 −0.403473 0.914991i \(-0.632197\pi\)
−0.403473 + 0.914991i \(0.632197\pi\)
\(44\) 0.645751 0.0973507
\(45\) −3.64575 −0.543477
\(46\) 2.35425 0.347115
\(47\) 3.00000 0.437595 0.218797 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −8.29150 −1.17260
\(51\) −6.64575 −0.930591
\(52\) 1.00000 0.138675
\(53\) −3.00000 −0.412082 −0.206041 0.978543i \(-0.566058\pi\)
−0.206041 + 0.978543i \(0.566058\pi\)
\(54\) −1.00000 −0.136083
\(55\) −2.35425 −0.317447
\(56\) 0 0
\(57\) 5.00000 0.662266
\(58\) −4.29150 −0.563502
\(59\) −7.93725 −1.03334 −0.516671 0.856184i \(-0.672829\pi\)
−0.516671 + 0.856184i \(0.672829\pi\)
\(60\) −3.64575 −0.470664
\(61\) −11.9373 −1.52841 −0.764204 0.644974i \(-0.776868\pi\)
−0.764204 + 0.644974i \(0.776868\pi\)
\(62\) −3.29150 −0.418021
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.64575 −0.452200
\(66\) −0.645751 −0.0794865
\(67\) 7.58301 0.926412 0.463206 0.886251i \(-0.346699\pi\)
0.463206 + 0.886251i \(0.346699\pi\)
\(68\) −6.64575 −0.805916
\(69\) −2.35425 −0.283418
\(70\) 0 0
\(71\) 16.2915 1.93345 0.966723 0.255826i \(-0.0823474\pi\)
0.966723 + 0.255826i \(0.0823474\pi\)
\(72\) −1.00000 −0.117851
\(73\) −13.6458 −1.59711 −0.798557 0.601919i \(-0.794403\pi\)
−0.798557 + 0.601919i \(0.794403\pi\)
\(74\) −5.64575 −0.656305
\(75\) 8.29150 0.957420
\(76\) 5.00000 0.573539
\(77\) 0 0
\(78\) −1.00000 −0.113228
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) −3.64575 −0.407607
\(81\) 1.00000 0.111111
\(82\) 2.35425 0.259983
\(83\) −13.2915 −1.45893 −0.729466 0.684017i \(-0.760231\pi\)
−0.729466 + 0.684017i \(0.760231\pi\)
\(84\) 0 0
\(85\) 24.2288 2.62798
\(86\) 5.29150 0.570597
\(87\) 4.29150 0.460097
\(88\) −0.645751 −0.0688373
\(89\) 16.9373 1.79535 0.897673 0.440663i \(-0.145257\pi\)
0.897673 + 0.440663i \(0.145257\pi\)
\(90\) 3.64575 0.384296
\(91\) 0 0
\(92\) −2.35425 −0.245447
\(93\) 3.29150 0.341313
\(94\) −3.00000 −0.309426
\(95\) −18.2288 −1.87023
\(96\) −1.00000 −0.102062
\(97\) 0.937254 0.0951637 0.0475819 0.998867i \(-0.484849\pi\)
0.0475819 + 0.998867i \(0.484849\pi\)
\(98\) 0 0
\(99\) 0.645751 0.0649004
\(100\) 8.29150 0.829150
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 6.64575 0.658027
\(103\) −5.29150 −0.521387 −0.260694 0.965422i \(-0.583951\pi\)
−0.260694 + 0.965422i \(0.583951\pi\)
\(104\) −1.00000 −0.0980581
\(105\) 0 0
\(106\) 3.00000 0.291386
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 1.00000 0.0962250
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 2.35425 0.224469
\(111\) 5.64575 0.535871
\(112\) 0 0
\(113\) 15.2288 1.43260 0.716300 0.697792i \(-0.245834\pi\)
0.716300 + 0.697792i \(0.245834\pi\)
\(114\) −5.00000 −0.468293
\(115\) 8.58301 0.800369
\(116\) 4.29150 0.398456
\(117\) 1.00000 0.0924500
\(118\) 7.93725 0.730683
\(119\) 0 0
\(120\) 3.64575 0.332810
\(121\) −10.5830 −0.962091
\(122\) 11.9373 1.08075
\(123\) −2.35425 −0.212275
\(124\) 3.29150 0.295586
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −10.2288 −0.907655 −0.453828 0.891089i \(-0.649942\pi\)
−0.453828 + 0.891089i \(0.649942\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −5.29150 −0.465891
\(130\) 3.64575 0.319754
\(131\) 10.9373 0.955592 0.477796 0.878471i \(-0.341436\pi\)
0.477796 + 0.878471i \(0.341436\pi\)
\(132\) 0.645751 0.0562054
\(133\) 0 0
\(134\) −7.58301 −0.655072
\(135\) −3.64575 −0.313776
\(136\) 6.64575 0.569868
\(137\) −19.5203 −1.66773 −0.833864 0.551970i \(-0.813876\pi\)
−0.833864 + 0.551970i \(0.813876\pi\)
\(138\) 2.35425 0.200407
\(139\) −22.2288 −1.88542 −0.942709 0.333615i \(-0.891731\pi\)
−0.942709 + 0.333615i \(0.891731\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) −16.2915 −1.36715
\(143\) 0.645751 0.0540004
\(144\) 1.00000 0.0833333
\(145\) −15.6458 −1.29931
\(146\) 13.6458 1.12933
\(147\) 0 0
\(148\) 5.64575 0.464078
\(149\) −7.06275 −0.578603 −0.289301 0.957238i \(-0.593423\pi\)
−0.289301 + 0.957238i \(0.593423\pi\)
\(150\) −8.29150 −0.676998
\(151\) 6.06275 0.493379 0.246690 0.969095i \(-0.420657\pi\)
0.246690 + 0.969095i \(0.420657\pi\)
\(152\) −5.00000 −0.405554
\(153\) −6.64575 −0.537277
\(154\) 0 0
\(155\) −12.0000 −0.963863
\(156\) 1.00000 0.0800641
\(157\) 2.64575 0.211154 0.105577 0.994411i \(-0.466331\pi\)
0.105577 + 0.994411i \(0.466331\pi\)
\(158\) 10.0000 0.795557
\(159\) −3.00000 −0.237915
\(160\) 3.64575 0.288222
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 2.41699 0.189314 0.0946568 0.995510i \(-0.469825\pi\)
0.0946568 + 0.995510i \(0.469825\pi\)
\(164\) −2.35425 −0.183836
\(165\) −2.35425 −0.183278
\(166\) 13.2915 1.03162
\(167\) −6.87451 −0.531965 −0.265983 0.963978i \(-0.585696\pi\)
−0.265983 + 0.963978i \(0.585696\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) −24.2288 −1.85826
\(171\) 5.00000 0.382360
\(172\) −5.29150 −0.403473
\(173\) −22.2915 −1.69479 −0.847396 0.530962i \(-0.821831\pi\)
−0.847396 + 0.530962i \(0.821831\pi\)
\(174\) −4.29150 −0.325338
\(175\) 0 0
\(176\) 0.645751 0.0486753
\(177\) −7.93725 −0.596601
\(178\) −16.9373 −1.26950
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) −3.64575 −0.271738
\(181\) 14.6458 1.08861 0.544305 0.838887i \(-0.316793\pi\)
0.544305 + 0.838887i \(0.316793\pi\)
\(182\) 0 0
\(183\) −11.9373 −0.882427
\(184\) 2.35425 0.173558
\(185\) −20.5830 −1.51329
\(186\) −3.29150 −0.241345
\(187\) −4.29150 −0.313826
\(188\) 3.00000 0.218797
\(189\) 0 0
\(190\) 18.2288 1.32245
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 1.00000 0.0721688
\(193\) 18.9373 1.36313 0.681567 0.731756i \(-0.261299\pi\)
0.681567 + 0.731756i \(0.261299\pi\)
\(194\) −0.937254 −0.0672909
\(195\) −3.64575 −0.261078
\(196\) 0 0
\(197\) −26.8118 −1.91026 −0.955129 0.296189i \(-0.904284\pi\)
−0.955129 + 0.296189i \(0.904284\pi\)
\(198\) −0.645751 −0.0458915
\(199\) −22.2288 −1.57575 −0.787877 0.615832i \(-0.788820\pi\)
−0.787877 + 0.615832i \(0.788820\pi\)
\(200\) −8.29150 −0.586298
\(201\) 7.58301 0.534864
\(202\) 0 0
\(203\) 0 0
\(204\) −6.64575 −0.465296
\(205\) 8.58301 0.599463
\(206\) 5.29150 0.368676
\(207\) −2.35425 −0.163632
\(208\) 1.00000 0.0693375
\(209\) 3.22876 0.223338
\(210\) 0 0
\(211\) −6.35425 −0.437445 −0.218722 0.975787i \(-0.570189\pi\)
−0.218722 + 0.975787i \(0.570189\pi\)
\(212\) −3.00000 −0.206041
\(213\) 16.2915 1.11628
\(214\) 12.0000 0.820303
\(215\) 19.2915 1.31567
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 4.00000 0.270914
\(219\) −13.6458 −0.922094
\(220\) −2.35425 −0.158723
\(221\) −6.64575 −0.447042
\(222\) −5.64575 −0.378918
\(223\) −20.5203 −1.37414 −0.687069 0.726592i \(-0.741103\pi\)
−0.687069 + 0.726592i \(0.741103\pi\)
\(224\) 0 0
\(225\) 8.29150 0.552767
\(226\) −15.2288 −1.01300
\(227\) −4.70850 −0.312514 −0.156257 0.987716i \(-0.549943\pi\)
−0.156257 + 0.987716i \(0.549943\pi\)
\(228\) 5.00000 0.331133
\(229\) 26.4575 1.74836 0.874181 0.485601i \(-0.161399\pi\)
0.874181 + 0.485601i \(0.161399\pi\)
\(230\) −8.58301 −0.565947
\(231\) 0 0
\(232\) −4.29150 −0.281751
\(233\) 0.645751 0.0423046 0.0211523 0.999776i \(-0.493267\pi\)
0.0211523 + 0.999776i \(0.493267\pi\)
\(234\) −1.00000 −0.0653720
\(235\) −10.9373 −0.713468
\(236\) −7.93725 −0.516671
\(237\) −10.0000 −0.649570
\(238\) 0 0
\(239\) 3.00000 0.194054 0.0970269 0.995282i \(-0.469067\pi\)
0.0970269 + 0.995282i \(0.469067\pi\)
\(240\) −3.64575 −0.235332
\(241\) −13.8745 −0.893736 −0.446868 0.894600i \(-0.647461\pi\)
−0.446868 + 0.894600i \(0.647461\pi\)
\(242\) 10.5830 0.680301
\(243\) 1.00000 0.0641500
\(244\) −11.9373 −0.764204
\(245\) 0 0
\(246\) 2.35425 0.150101
\(247\) 5.00000 0.318142
\(248\) −3.29150 −0.209011
\(249\) −13.2915 −0.842315
\(250\) 12.0000 0.758947
\(251\) −13.2915 −0.838952 −0.419476 0.907766i \(-0.637786\pi\)
−0.419476 + 0.907766i \(0.637786\pi\)
\(252\) 0 0
\(253\) −1.52026 −0.0955779
\(254\) 10.2288 0.641809
\(255\) 24.2288 1.51726
\(256\) 1.00000 0.0625000
\(257\) −15.8745 −0.990225 −0.495112 0.868829i \(-0.664873\pi\)
−0.495112 + 0.868829i \(0.664873\pi\)
\(258\) 5.29150 0.329435
\(259\) 0 0
\(260\) −3.64575 −0.226100
\(261\) 4.29150 0.265637
\(262\) −10.9373 −0.675706
\(263\) −3.64575 −0.224807 −0.112403 0.993663i \(-0.535855\pi\)
−0.112403 + 0.993663i \(0.535855\pi\)
\(264\) −0.645751 −0.0397432
\(265\) 10.9373 0.671870
\(266\) 0 0
\(267\) 16.9373 1.03654
\(268\) 7.58301 0.463206
\(269\) 6.87451 0.419146 0.209573 0.977793i \(-0.432793\pi\)
0.209573 + 0.977793i \(0.432793\pi\)
\(270\) 3.64575 0.221873
\(271\) −22.6458 −1.37563 −0.687816 0.725885i \(-0.741430\pi\)
−0.687816 + 0.725885i \(0.741430\pi\)
\(272\) −6.64575 −0.402958
\(273\) 0 0
\(274\) 19.5203 1.17926
\(275\) 5.35425 0.322873
\(276\) −2.35425 −0.141709
\(277\) 18.5203 1.11277 0.556387 0.830923i \(-0.312187\pi\)
0.556387 + 0.830923i \(0.312187\pi\)
\(278\) 22.2288 1.33319
\(279\) 3.29150 0.197057
\(280\) 0 0
\(281\) −2.58301 −0.154089 −0.0770446 0.997028i \(-0.524548\pi\)
−0.0770446 + 0.997028i \(0.524548\pi\)
\(282\) −3.00000 −0.178647
\(283\) −11.0627 −0.657612 −0.328806 0.944397i \(-0.606646\pi\)
−0.328806 + 0.944397i \(0.606646\pi\)
\(284\) 16.2915 0.966723
\(285\) −18.2288 −1.07978
\(286\) −0.645751 −0.0381841
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) 27.1660 1.59800
\(290\) 15.6458 0.918750
\(291\) 0.937254 0.0549428
\(292\) −13.6458 −0.798557
\(293\) 7.52026 0.439338 0.219669 0.975574i \(-0.429502\pi\)
0.219669 + 0.975574i \(0.429502\pi\)
\(294\) 0 0
\(295\) 28.9373 1.68479
\(296\) −5.64575 −0.328153
\(297\) 0.645751 0.0374703
\(298\) 7.06275 0.409134
\(299\) −2.35425 −0.136150
\(300\) 8.29150 0.478710
\(301\) 0 0
\(302\) −6.06275 −0.348872
\(303\) 0 0
\(304\) 5.00000 0.286770
\(305\) 43.5203 2.49196
\(306\) 6.64575 0.379912
\(307\) −9.58301 −0.546931 −0.273465 0.961882i \(-0.588170\pi\)
−0.273465 + 0.961882i \(0.588170\pi\)
\(308\) 0 0
\(309\) −5.29150 −0.301023
\(310\) 12.0000 0.681554
\(311\) −1.06275 −0.0602628 −0.0301314 0.999546i \(-0.509593\pi\)
−0.0301314 + 0.999546i \(0.509593\pi\)
\(312\) −1.00000 −0.0566139
\(313\) 13.1660 0.744187 0.372093 0.928195i \(-0.378640\pi\)
0.372093 + 0.928195i \(0.378640\pi\)
\(314\) −2.64575 −0.149308
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) −13.2915 −0.746525 −0.373263 0.927726i \(-0.621761\pi\)
−0.373263 + 0.927726i \(0.621761\pi\)
\(318\) 3.00000 0.168232
\(319\) 2.77124 0.155160
\(320\) −3.64575 −0.203804
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) −33.2288 −1.84890
\(324\) 1.00000 0.0555556
\(325\) 8.29150 0.459930
\(326\) −2.41699 −0.133865
\(327\) −4.00000 −0.221201
\(328\) 2.35425 0.129992
\(329\) 0 0
\(330\) 2.35425 0.129597
\(331\) 11.4170 0.627535 0.313767 0.949500i \(-0.398409\pi\)
0.313767 + 0.949500i \(0.398409\pi\)
\(332\) −13.2915 −0.729466
\(333\) 5.64575 0.309385
\(334\) 6.87451 0.376156
\(335\) −27.6458 −1.51045
\(336\) 0 0
\(337\) −15.5830 −0.848860 −0.424430 0.905461i \(-0.639526\pi\)
−0.424430 + 0.905461i \(0.639526\pi\)
\(338\) −1.00000 −0.0543928
\(339\) 15.2288 0.827113
\(340\) 24.2288 1.31399
\(341\) 2.12549 0.115102
\(342\) −5.00000 −0.270369
\(343\) 0 0
\(344\) 5.29150 0.285299
\(345\) 8.58301 0.462093
\(346\) 22.2915 1.19840
\(347\) 0.228757 0.0122803 0.00614015 0.999981i \(-0.498046\pi\)
0.00614015 + 0.999981i \(0.498046\pi\)
\(348\) 4.29150 0.230049
\(349\) 18.9373 1.01369 0.506844 0.862038i \(-0.330812\pi\)
0.506844 + 0.862038i \(0.330812\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) −0.645751 −0.0344187
\(353\) −20.5830 −1.09552 −0.547761 0.836635i \(-0.684520\pi\)
−0.547761 + 0.836635i \(0.684520\pi\)
\(354\) 7.93725 0.421860
\(355\) −59.3948 −3.15235
\(356\) 16.9373 0.897673
\(357\) 0 0
\(358\) 6.00000 0.317110
\(359\) −6.00000 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 3.64575 0.192148
\(361\) 6.00000 0.315789
\(362\) −14.6458 −0.769764
\(363\) −10.5830 −0.555464
\(364\) 0 0
\(365\) 49.7490 2.60398
\(366\) 11.9373 0.623970
\(367\) −0.583005 −0.0304326 −0.0152163 0.999884i \(-0.504844\pi\)
−0.0152163 + 0.999884i \(0.504844\pi\)
\(368\) −2.35425 −0.122724
\(369\) −2.35425 −0.122557
\(370\) 20.5830 1.07006
\(371\) 0 0
\(372\) 3.29150 0.170656
\(373\) 14.6458 0.758328 0.379164 0.925329i \(-0.376212\pi\)
0.379164 + 0.925329i \(0.376212\pi\)
\(374\) 4.29150 0.221908
\(375\) −12.0000 −0.619677
\(376\) −3.00000 −0.154713
\(377\) 4.29150 0.221024
\(378\) 0 0
\(379\) −9.16601 −0.470826 −0.235413 0.971895i \(-0.575644\pi\)
−0.235413 + 0.971895i \(0.575644\pi\)
\(380\) −18.2288 −0.935115
\(381\) −10.2288 −0.524035
\(382\) 0 0
\(383\) 25.7490 1.31571 0.657857 0.753143i \(-0.271463\pi\)
0.657857 + 0.753143i \(0.271463\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −18.9373 −0.963881
\(387\) −5.29150 −0.268982
\(388\) 0.937254 0.0475819
\(389\) 7.70850 0.390836 0.195418 0.980720i \(-0.437394\pi\)
0.195418 + 0.980720i \(0.437394\pi\)
\(390\) 3.64575 0.184610
\(391\) 15.6458 0.791240
\(392\) 0 0
\(393\) 10.9373 0.551711
\(394\) 26.8118 1.35076
\(395\) 36.4575 1.83438
\(396\) 0.645751 0.0324502
\(397\) −14.9373 −0.749679 −0.374840 0.927090i \(-0.622302\pi\)
−0.374840 + 0.927090i \(0.622302\pi\)
\(398\) 22.2288 1.11423
\(399\) 0 0
\(400\) 8.29150 0.414575
\(401\) −18.2288 −0.910301 −0.455150 0.890415i \(-0.650414\pi\)
−0.455150 + 0.890415i \(0.650414\pi\)
\(402\) −7.58301 −0.378206
\(403\) 3.29150 0.163961
\(404\) 0 0
\(405\) −3.64575 −0.181159
\(406\) 0 0
\(407\) 3.64575 0.180713
\(408\) 6.64575 0.329014
\(409\) −26.9373 −1.33196 −0.665981 0.745969i \(-0.731987\pi\)
−0.665981 + 0.745969i \(0.731987\pi\)
\(410\) −8.58301 −0.423884
\(411\) −19.5203 −0.962863
\(412\) −5.29150 −0.260694
\(413\) 0 0
\(414\) 2.35425 0.115705
\(415\) 48.4575 2.37869
\(416\) −1.00000 −0.0490290
\(417\) −22.2288 −1.08855
\(418\) −3.22876 −0.157924
\(419\) 30.4575 1.48795 0.743973 0.668209i \(-0.232939\pi\)
0.743973 + 0.668209i \(0.232939\pi\)
\(420\) 0 0
\(421\) −24.3542 −1.18695 −0.593477 0.804851i \(-0.702245\pi\)
−0.593477 + 0.804851i \(0.702245\pi\)
\(422\) 6.35425 0.309320
\(423\) 3.00000 0.145865
\(424\) 3.00000 0.145693
\(425\) −55.1033 −2.67290
\(426\) −16.2915 −0.789326
\(427\) 0 0
\(428\) −12.0000 −0.580042
\(429\) 0.645751 0.0311772
\(430\) −19.2915 −0.930319
\(431\) 20.5830 0.991448 0.495724 0.868480i \(-0.334903\pi\)
0.495724 + 0.868480i \(0.334903\pi\)
\(432\) 1.00000 0.0481125
\(433\) −33.5830 −1.61390 −0.806948 0.590622i \(-0.798882\pi\)
−0.806948 + 0.590622i \(0.798882\pi\)
\(434\) 0 0
\(435\) −15.6458 −0.750156
\(436\) −4.00000 −0.191565
\(437\) −11.7712 −0.563095
\(438\) 13.6458 0.652019
\(439\) 23.6458 1.12855 0.564275 0.825587i \(-0.309156\pi\)
0.564275 + 0.825587i \(0.309156\pi\)
\(440\) 2.35425 0.112234
\(441\) 0 0
\(442\) 6.64575 0.316106
\(443\) 19.0627 0.905698 0.452849 0.891587i \(-0.350408\pi\)
0.452849 + 0.891587i \(0.350408\pi\)
\(444\) 5.64575 0.267935
\(445\) −61.7490 −2.92718
\(446\) 20.5203 0.971662
\(447\) −7.06275 −0.334056
\(448\) 0 0
\(449\) 12.0000 0.566315 0.283158 0.959073i \(-0.408618\pi\)
0.283158 + 0.959073i \(0.408618\pi\)
\(450\) −8.29150 −0.390865
\(451\) −1.52026 −0.0715862
\(452\) 15.2288 0.716300
\(453\) 6.06275 0.284853
\(454\) 4.70850 0.220981
\(455\) 0 0
\(456\) −5.00000 −0.234146
\(457\) 32.2288 1.50760 0.753799 0.657105i \(-0.228219\pi\)
0.753799 + 0.657105i \(0.228219\pi\)
\(458\) −26.4575 −1.23628
\(459\) −6.64575 −0.310197
\(460\) 8.58301 0.400185
\(461\) −36.4575 −1.69800 −0.848998 0.528396i \(-0.822794\pi\)
−0.848998 + 0.528396i \(0.822794\pi\)
\(462\) 0 0
\(463\) 25.1660 1.16956 0.584782 0.811191i \(-0.301180\pi\)
0.584782 + 0.811191i \(0.301180\pi\)
\(464\) 4.29150 0.199228
\(465\) −12.0000 −0.556487
\(466\) −0.645751 −0.0299139
\(467\) −37.5203 −1.73623 −0.868115 0.496363i \(-0.834669\pi\)
−0.868115 + 0.496363i \(0.834669\pi\)
\(468\) 1.00000 0.0462250
\(469\) 0 0
\(470\) 10.9373 0.504498
\(471\) 2.64575 0.121910
\(472\) 7.93725 0.365342
\(473\) −3.41699 −0.157114
\(474\) 10.0000 0.459315
\(475\) 41.4575 1.90220
\(476\) 0 0
\(477\) −3.00000 −0.137361
\(478\) −3.00000 −0.137217
\(479\) −34.7490 −1.58772 −0.793862 0.608099i \(-0.791933\pi\)
−0.793862 + 0.608099i \(0.791933\pi\)
\(480\) 3.64575 0.166405
\(481\) 5.64575 0.257424
\(482\) 13.8745 0.631967
\(483\) 0 0
\(484\) −10.5830 −0.481046
\(485\) −3.41699 −0.155158
\(486\) −1.00000 −0.0453609
\(487\) −23.9373 −1.08470 −0.542350 0.840152i \(-0.682465\pi\)
−0.542350 + 0.840152i \(0.682465\pi\)
\(488\) 11.9373 0.540374
\(489\) 2.41699 0.109300
\(490\) 0 0
\(491\) 11.1660 0.503915 0.251957 0.967738i \(-0.418926\pi\)
0.251957 + 0.967738i \(0.418926\pi\)
\(492\) −2.35425 −0.106138
\(493\) −28.5203 −1.28449
\(494\) −5.00000 −0.224961
\(495\) −2.35425 −0.105816
\(496\) 3.29150 0.147793
\(497\) 0 0
\(498\) 13.2915 0.595606
\(499\) −29.2915 −1.31127 −0.655634 0.755079i \(-0.727599\pi\)
−0.655634 + 0.755079i \(0.727599\pi\)
\(500\) −12.0000 −0.536656
\(501\) −6.87451 −0.307130
\(502\) 13.2915 0.593229
\(503\) 36.4575 1.62556 0.812780 0.582571i \(-0.197953\pi\)
0.812780 + 0.582571i \(0.197953\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.52026 0.0675838
\(507\) 1.00000 0.0444116
\(508\) −10.2288 −0.453828
\(509\) 24.0000 1.06378 0.531891 0.846813i \(-0.321482\pi\)
0.531891 + 0.846813i \(0.321482\pi\)
\(510\) −24.2288 −1.07287
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 5.00000 0.220755
\(514\) 15.8745 0.700195
\(515\) 19.2915 0.850085
\(516\) −5.29150 −0.232945
\(517\) 1.93725 0.0852003
\(518\) 0 0
\(519\) −22.2915 −0.978488
\(520\) 3.64575 0.159877
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) −4.29150 −0.187834
\(523\) −26.7085 −1.16788 −0.583941 0.811796i \(-0.698490\pi\)
−0.583941 + 0.811796i \(0.698490\pi\)
\(524\) 10.9373 0.477796
\(525\) 0 0
\(526\) 3.64575 0.158962
\(527\) −21.8745 −0.952869
\(528\) 0.645751 0.0281027
\(529\) −17.4575 −0.759022
\(530\) −10.9373 −0.475084
\(531\) −7.93725 −0.344447
\(532\) 0 0
\(533\) −2.35425 −0.101974
\(534\) −16.9373 −0.732947
\(535\) 43.7490 1.89143
\(536\) −7.58301 −0.327536
\(537\) −6.00000 −0.258919
\(538\) −6.87451 −0.296381
\(539\) 0 0
\(540\) −3.64575 −0.156888
\(541\) 41.6458 1.79049 0.895245 0.445574i \(-0.147000\pi\)
0.895245 + 0.445574i \(0.147000\pi\)
\(542\) 22.6458 0.972718
\(543\) 14.6458 0.628509
\(544\) 6.64575 0.284934
\(545\) 14.5830 0.624667
\(546\) 0 0
\(547\) −20.9373 −0.895212 −0.447606 0.894231i \(-0.647723\pi\)
−0.447606 + 0.894231i \(0.647723\pi\)
\(548\) −19.5203 −0.833864
\(549\) −11.9373 −0.509470
\(550\) −5.35425 −0.228306
\(551\) 21.4575 0.914121
\(552\) 2.35425 0.100203
\(553\) 0 0
\(554\) −18.5203 −0.786850
\(555\) −20.5830 −0.873700
\(556\) −22.2288 −0.942709
\(557\) 11.1660 0.473119 0.236560 0.971617i \(-0.423980\pi\)
0.236560 + 0.971617i \(0.423980\pi\)
\(558\) −3.29150 −0.139340
\(559\) −5.29150 −0.223807
\(560\) 0 0
\(561\) −4.29150 −0.181187
\(562\) 2.58301 0.108958
\(563\) −27.0405 −1.13962 −0.569811 0.821776i \(-0.692984\pi\)
−0.569811 + 0.821776i \(0.692984\pi\)
\(564\) 3.00000 0.126323
\(565\) −55.5203 −2.33575
\(566\) 11.0627 0.465002
\(567\) 0 0
\(568\) −16.2915 −0.683576
\(569\) −33.2288 −1.39302 −0.696511 0.717546i \(-0.745265\pi\)
−0.696511 + 0.717546i \(0.745265\pi\)
\(570\) 18.2288 0.763519
\(571\) −16.0000 −0.669579 −0.334790 0.942293i \(-0.608665\pi\)
−0.334790 + 0.942293i \(0.608665\pi\)
\(572\) 0.645751 0.0270002
\(573\) 0 0
\(574\) 0 0
\(575\) −19.5203 −0.814051
\(576\) 1.00000 0.0416667
\(577\) −12.5830 −0.523837 −0.261919 0.965090i \(-0.584355\pi\)
−0.261919 + 0.965090i \(0.584355\pi\)
\(578\) −27.1660 −1.12996
\(579\) 18.9373 0.787005
\(580\) −15.6458 −0.649654
\(581\) 0 0
\(582\) −0.937254 −0.0388504
\(583\) −1.93725 −0.0802329
\(584\) 13.6458 0.564665
\(585\) −3.64575 −0.150733
\(586\) −7.52026 −0.310659
\(587\) 31.9373 1.31819 0.659096 0.752059i \(-0.270939\pi\)
0.659096 + 0.752059i \(0.270939\pi\)
\(588\) 0 0
\(589\) 16.4575 0.678120
\(590\) −28.9373 −1.19133
\(591\) −26.8118 −1.10289
\(592\) 5.64575 0.232039
\(593\) 43.5203 1.78716 0.893581 0.448901i \(-0.148184\pi\)
0.893581 + 0.448901i \(0.148184\pi\)
\(594\) −0.645751 −0.0264955
\(595\) 0 0
\(596\) −7.06275 −0.289301
\(597\) −22.2288 −0.909762
\(598\) 2.35425 0.0962724
\(599\) −32.8118 −1.34065 −0.670326 0.742067i \(-0.733846\pi\)
−0.670326 + 0.742067i \(0.733846\pi\)
\(600\) −8.29150 −0.338499
\(601\) 14.8745 0.606744 0.303372 0.952872i \(-0.401888\pi\)
0.303372 + 0.952872i \(0.401888\pi\)
\(602\) 0 0
\(603\) 7.58301 0.308804
\(604\) 6.06275 0.246690
\(605\) 38.5830 1.56862
\(606\) 0 0
\(607\) 16.8118 0.682368 0.341184 0.939996i \(-0.389172\pi\)
0.341184 + 0.939996i \(0.389172\pi\)
\(608\) −5.00000 −0.202777
\(609\) 0 0
\(610\) −43.5203 −1.76208
\(611\) 3.00000 0.121367
\(612\) −6.64575 −0.268639
\(613\) 41.8745 1.69130 0.845648 0.533741i \(-0.179214\pi\)
0.845648 + 0.533741i \(0.179214\pi\)
\(614\) 9.58301 0.386739
\(615\) 8.58301 0.346100
\(616\) 0 0
\(617\) −10.7085 −0.431108 −0.215554 0.976492i \(-0.569156\pi\)
−0.215554 + 0.976492i \(0.569156\pi\)
\(618\) 5.29150 0.212855
\(619\) −41.7490 −1.67803 −0.839017 0.544105i \(-0.816869\pi\)
−0.839017 + 0.544105i \(0.816869\pi\)
\(620\) −12.0000 −0.481932
\(621\) −2.35425 −0.0944727
\(622\) 1.06275 0.0426122
\(623\) 0 0
\(624\) 1.00000 0.0400320
\(625\) 2.29150 0.0916601
\(626\) −13.1660 −0.526220
\(627\) 3.22876 0.128944
\(628\) 2.64575 0.105577
\(629\) −37.5203 −1.49603
\(630\) 0 0
\(631\) 14.4575 0.575545 0.287772 0.957699i \(-0.407085\pi\)
0.287772 + 0.957699i \(0.407085\pi\)
\(632\) 10.0000 0.397779
\(633\) −6.35425 −0.252559
\(634\) 13.2915 0.527873
\(635\) 37.2915 1.47987
\(636\) −3.00000 −0.118958
\(637\) 0 0
\(638\) −2.77124 −0.109715
\(639\) 16.2915 0.644482
\(640\) 3.64575 0.144111
\(641\) −21.4170 −0.845920 −0.422960 0.906148i \(-0.639009\pi\)
−0.422960 + 0.906148i \(0.639009\pi\)
\(642\) 12.0000 0.473602
\(643\) −10.8745 −0.428849 −0.214424 0.976741i \(-0.568788\pi\)
−0.214424 + 0.976741i \(0.568788\pi\)
\(644\) 0 0
\(645\) 19.2915 0.759602
\(646\) 33.2288 1.30737
\(647\) 25.7490 1.01230 0.506149 0.862446i \(-0.331069\pi\)
0.506149 + 0.862446i \(0.331069\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −5.12549 −0.201193
\(650\) −8.29150 −0.325219
\(651\) 0 0
\(652\) 2.41699 0.0946568
\(653\) 12.0000 0.469596 0.234798 0.972044i \(-0.424557\pi\)
0.234798 + 0.972044i \(0.424557\pi\)
\(654\) 4.00000 0.156412
\(655\) −39.8745 −1.55803
\(656\) −2.35425 −0.0919180
\(657\) −13.6458 −0.532371
\(658\) 0 0
\(659\) −16.9373 −0.659782 −0.329891 0.944019i \(-0.607012\pi\)
−0.329891 + 0.944019i \(0.607012\pi\)
\(660\) −2.35425 −0.0916390
\(661\) 15.5203 0.603668 0.301834 0.953360i \(-0.402401\pi\)
0.301834 + 0.953360i \(0.402401\pi\)
\(662\) −11.4170 −0.443734
\(663\) −6.64575 −0.258100
\(664\) 13.2915 0.515810
\(665\) 0 0
\(666\) −5.64575 −0.218768
\(667\) −10.1033 −0.391200
\(668\) −6.87451 −0.265983
\(669\) −20.5203 −0.793359
\(670\) 27.6458 1.06805
\(671\) −7.70850 −0.297583
\(672\) 0 0
\(673\) −6.58301 −0.253756 −0.126878 0.991918i \(-0.540496\pi\)
−0.126878 + 0.991918i \(0.540496\pi\)
\(674\) 15.5830 0.600235
\(675\) 8.29150 0.319140
\(676\) 1.00000 0.0384615
\(677\) 20.1660 0.775043 0.387521 0.921861i \(-0.373331\pi\)
0.387521 + 0.921861i \(0.373331\pi\)
\(678\) −15.2288 −0.584857
\(679\) 0 0
\(680\) −24.2288 −0.929130
\(681\) −4.70850 −0.180430
\(682\) −2.12549 −0.0813893
\(683\) 19.2915 0.738169 0.369084 0.929396i \(-0.379671\pi\)
0.369084 + 0.929396i \(0.379671\pi\)
\(684\) 5.00000 0.191180
\(685\) 71.1660 2.71911
\(686\) 0 0
\(687\) 26.4575 1.00942
\(688\) −5.29150 −0.201737
\(689\) −3.00000 −0.114291
\(690\) −8.58301 −0.326749
\(691\) −40.0405 −1.52321 −0.761607 0.648040i \(-0.775589\pi\)
−0.761607 + 0.648040i \(0.775589\pi\)
\(692\) −22.2915 −0.847396
\(693\) 0 0
\(694\) −0.228757 −0.00868348
\(695\) 81.0405 3.07404
\(696\) −4.29150 −0.162669
\(697\) 15.6458 0.592625
\(698\) −18.9373 −0.716786
\(699\) 0.645751 0.0244246
\(700\) 0 0
\(701\) 12.0000 0.453234 0.226617 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(702\) −1.00000 −0.0377426
\(703\) 28.2288 1.06467
\(704\) 0.645751 0.0243377
\(705\) −10.9373 −0.411921
\(706\) 20.5830 0.774652
\(707\) 0 0
\(708\) −7.93725 −0.298300
\(709\) 6.47974 0.243352 0.121676 0.992570i \(-0.461173\pi\)
0.121676 + 0.992570i \(0.461173\pi\)
\(710\) 59.3948 2.22905
\(711\) −10.0000 −0.375029
\(712\) −16.9373 −0.634750
\(713\) −7.74902 −0.290203
\(714\) 0 0
\(715\) −2.35425 −0.0880439
\(716\) −6.00000 −0.224231
\(717\) 3.00000 0.112037
\(718\) 6.00000 0.223918
\(719\) 44.5830 1.66267 0.831333 0.555775i \(-0.187578\pi\)
0.831333 + 0.555775i \(0.187578\pi\)
\(720\) −3.64575 −0.135869
\(721\) 0 0
\(722\) −6.00000 −0.223297
\(723\) −13.8745 −0.515998
\(724\) 14.6458 0.544305
\(725\) 35.5830 1.32152
\(726\) 10.5830 0.392772
\(727\) −16.4575 −0.610375 −0.305188 0.952292i \(-0.598719\pi\)
−0.305188 + 0.952292i \(0.598719\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −49.7490 −1.84129
\(731\) 35.1660 1.30066
\(732\) −11.9373 −0.441214
\(733\) 14.2288 0.525551 0.262776 0.964857i \(-0.415362\pi\)
0.262776 + 0.964857i \(0.415362\pi\)
\(734\) 0.583005 0.0215191
\(735\) 0 0
\(736\) 2.35425 0.0867788
\(737\) 4.89674 0.180374
\(738\) 2.35425 0.0866611
\(739\) −17.2915 −0.636078 −0.318039 0.948078i \(-0.603024\pi\)
−0.318039 + 0.948078i \(0.603024\pi\)
\(740\) −20.5830 −0.756646
\(741\) 5.00000 0.183680
\(742\) 0 0
\(743\) −51.4575 −1.88779 −0.943897 0.330241i \(-0.892870\pi\)
−0.943897 + 0.330241i \(0.892870\pi\)
\(744\) −3.29150 −0.120672
\(745\) 25.7490 0.943371
\(746\) −14.6458 −0.536219
\(747\) −13.2915 −0.486311
\(748\) −4.29150 −0.156913
\(749\) 0 0
\(750\) 12.0000 0.438178
\(751\) −33.1660 −1.21024 −0.605122 0.796132i \(-0.706876\pi\)
−0.605122 + 0.796132i \(0.706876\pi\)
\(752\) 3.00000 0.109399
\(753\) −13.2915 −0.484369
\(754\) −4.29150 −0.156287
\(755\) −22.1033 −0.804420
\(756\) 0 0
\(757\) 26.6458 0.968456 0.484228 0.874942i \(-0.339100\pi\)
0.484228 + 0.874942i \(0.339100\pi\)
\(758\) 9.16601 0.332924
\(759\) −1.52026 −0.0551819
\(760\) 18.2288 0.661226
\(761\) 7.29150 0.264317 0.132158 0.991229i \(-0.457809\pi\)
0.132158 + 0.991229i \(0.457809\pi\)
\(762\) 10.2288 0.370549
\(763\) 0 0
\(764\) 0 0
\(765\) 24.2288 0.875993
\(766\) −25.7490 −0.930350
\(767\) −7.93725 −0.286598
\(768\) 1.00000 0.0360844
\(769\) 5.64575 0.203591 0.101795 0.994805i \(-0.467541\pi\)
0.101795 + 0.994805i \(0.467541\pi\)
\(770\) 0 0
\(771\) −15.8745 −0.571706
\(772\) 18.9373 0.681567
\(773\) 40.3320 1.45064 0.725321 0.688411i \(-0.241691\pi\)
0.725321 + 0.688411i \(0.241691\pi\)
\(774\) 5.29150 0.190199
\(775\) 27.2915 0.980340
\(776\) −0.937254 −0.0336455
\(777\) 0 0
\(778\) −7.70850 −0.276363
\(779\) −11.7712 −0.421749
\(780\) −3.64575 −0.130539
\(781\) 10.5203 0.376444
\(782\) −15.6458 −0.559491
\(783\) 4.29150 0.153366
\(784\) 0 0
\(785\) −9.64575 −0.344272
\(786\) −10.9373 −0.390119
\(787\) 52.6235 1.87583 0.937913 0.346871i \(-0.112756\pi\)
0.937913 + 0.346871i \(0.112756\pi\)
\(788\) −26.8118 −0.955129
\(789\) −3.64575 −0.129792
\(790\) −36.4575 −1.29710
\(791\) 0 0
\(792\) −0.645751 −0.0229458
\(793\) −11.9373 −0.423904
\(794\) 14.9373 0.530103
\(795\) 10.9373 0.387904
\(796\) −22.2288 −0.787877
\(797\) −45.8745 −1.62496 −0.812479 0.582990i \(-0.801883\pi\)
−0.812479 + 0.582990i \(0.801883\pi\)
\(798\) 0 0
\(799\) −19.9373 −0.705329
\(800\) −8.29150 −0.293149
\(801\) 16.9373 0.598448
\(802\) 18.2288 0.643680
\(803\) −8.81176 −0.310960
\(804\) 7.58301 0.267432
\(805\) 0 0
\(806\) −3.29150 −0.115938
\(807\) 6.87451 0.241994
\(808\) 0 0
\(809\) 23.8118 0.837177 0.418588 0.908176i \(-0.362525\pi\)
0.418588 + 0.908176i \(0.362525\pi\)
\(810\) 3.64575 0.128099
\(811\) 38.0000 1.33436 0.667180 0.744896i \(-0.267501\pi\)
0.667180 + 0.744896i \(0.267501\pi\)
\(812\) 0 0
\(813\) −22.6458 −0.794221
\(814\) −3.64575 −0.127784
\(815\) −8.81176 −0.308663
\(816\) −6.64575 −0.232648
\(817\) −26.4575 −0.925631
\(818\) 26.9373 0.941839
\(819\) 0 0
\(820\) 8.58301 0.299732
\(821\) −5.77124 −0.201418 −0.100709 0.994916i \(-0.532111\pi\)
−0.100709 + 0.994916i \(0.532111\pi\)
\(822\) 19.5203 0.680847
\(823\) 15.5203 0.541002 0.270501 0.962720i \(-0.412811\pi\)
0.270501 + 0.962720i \(0.412811\pi\)
\(824\) 5.29150 0.184338
\(825\) 5.35425 0.186411
\(826\) 0 0
\(827\) 35.3542 1.22939 0.614694 0.788766i \(-0.289280\pi\)
0.614694 + 0.788766i \(0.289280\pi\)
\(828\) −2.35425 −0.0818158
\(829\) 47.2288 1.64032 0.820161 0.572132i \(-0.193884\pi\)
0.820161 + 0.572132i \(0.193884\pi\)
\(830\) −48.4575 −1.68198
\(831\) 18.5203 0.642461
\(832\) 1.00000 0.0346688
\(833\) 0 0
\(834\) 22.2288 0.769719
\(835\) 25.0627 0.867332
\(836\) 3.22876 0.111669
\(837\) 3.29150 0.113771
\(838\) −30.4575 −1.05214
\(839\) 21.0000 0.725001 0.362500 0.931984i \(-0.381923\pi\)
0.362500 + 0.931984i \(0.381923\pi\)
\(840\) 0 0
\(841\) −10.5830 −0.364931
\(842\) 24.3542 0.839303
\(843\) −2.58301 −0.0889634
\(844\) −6.35425 −0.218722
\(845\) −3.64575 −0.125418
\(846\) −3.00000 −0.103142
\(847\) 0 0
\(848\) −3.00000 −0.103020
\(849\) −11.0627 −0.379672
\(850\) 55.1033 1.89003
\(851\) −13.2915 −0.455627
\(852\) 16.2915 0.558138
\(853\) −20.1033 −0.688323 −0.344161 0.938911i \(-0.611837\pi\)
−0.344161 + 0.938911i \(0.611837\pi\)
\(854\) 0 0
\(855\) −18.2288 −0.623410
\(856\) 12.0000 0.410152
\(857\) 13.9373 0.476088 0.238044 0.971254i \(-0.423494\pi\)
0.238044 + 0.971254i \(0.423494\pi\)
\(858\) −0.645751 −0.0220456
\(859\) 44.4575 1.51687 0.758435 0.651748i \(-0.225964\pi\)
0.758435 + 0.651748i \(0.225964\pi\)
\(860\) 19.2915 0.657835
\(861\) 0 0
\(862\) −20.5830 −0.701060
\(863\) 31.7490 1.08075 0.540375 0.841425i \(-0.318283\pi\)
0.540375 + 0.841425i \(0.318283\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 81.2693 2.76324
\(866\) 33.5830 1.14120
\(867\) 27.1660 0.922606
\(868\) 0 0
\(869\) −6.45751 −0.219056
\(870\) 15.6458 0.530441
\(871\) 7.58301 0.256940
\(872\) 4.00000 0.135457
\(873\) 0.937254 0.0317212
\(874\) 11.7712 0.398168
\(875\) 0 0
\(876\) −13.6458 −0.461047
\(877\) −54.8118 −1.85086 −0.925431 0.378917i \(-0.876297\pi\)
−0.925431 + 0.378917i \(0.876297\pi\)
\(878\) −23.6458 −0.798005
\(879\) 7.52026 0.253652
\(880\) −2.35425 −0.0793617
\(881\) −10.7085 −0.360778 −0.180389 0.983595i \(-0.557736\pi\)
−0.180389 + 0.983595i \(0.557736\pi\)
\(882\) 0 0
\(883\) −29.0627 −0.978039 −0.489020 0.872273i \(-0.662645\pi\)
−0.489020 + 0.872273i \(0.662645\pi\)
\(884\) −6.64575 −0.223521
\(885\) 28.9373 0.972715
\(886\) −19.0627 −0.640425
\(887\) 20.5830 0.691110 0.345555 0.938399i \(-0.387691\pi\)
0.345555 + 0.938399i \(0.387691\pi\)
\(888\) −5.64575 −0.189459
\(889\) 0 0
\(890\) 61.7490 2.06983
\(891\) 0.645751 0.0216335
\(892\) −20.5203 −0.687069
\(893\) 15.0000 0.501956
\(894\) 7.06275 0.236214
\(895\) 21.8745 0.731184
\(896\) 0 0
\(897\) −2.35425 −0.0786061
\(898\) −12.0000 −0.400445
\(899\) 14.1255 0.471112
\(900\) 8.29150 0.276383
\(901\) 19.9373 0.664206
\(902\) 1.52026 0.0506191
\(903\) 0 0
\(904\) −15.2288 −0.506501
\(905\) −53.3948 −1.77490
\(906\) −6.06275 −0.201421
\(907\) −3.77124 −0.125222 −0.0626110 0.998038i \(-0.519943\pi\)
−0.0626110 + 0.998038i \(0.519943\pi\)
\(908\) −4.70850 −0.156257
\(909\) 0 0
\(910\) 0 0
\(911\) −37.7490 −1.25068 −0.625340 0.780352i \(-0.715040\pi\)
−0.625340 + 0.780352i \(0.715040\pi\)
\(912\) 5.00000 0.165567
\(913\) −8.58301 −0.284056
\(914\) −32.2288 −1.06603
\(915\) 43.5203 1.43874
\(916\) 26.4575 0.874181
\(917\) 0 0
\(918\) 6.64575 0.219342
\(919\) 47.8745 1.57923 0.789617 0.613600i \(-0.210279\pi\)
0.789617 + 0.613600i \(0.210279\pi\)
\(920\) −8.58301 −0.282973
\(921\) −9.58301 −0.315771
\(922\) 36.4575 1.20066
\(923\) 16.2915 0.536241
\(924\) 0 0
\(925\) 46.8118 1.53916
\(926\) −25.1660 −0.827006
\(927\) −5.29150 −0.173796
\(928\) −4.29150 −0.140875
\(929\) −13.7490 −0.451091 −0.225545 0.974233i \(-0.572416\pi\)
−0.225545 + 0.974233i \(0.572416\pi\)
\(930\) 12.0000 0.393496
\(931\) 0 0
\(932\) 0.645751 0.0211523
\(933\) −1.06275 −0.0347927
\(934\) 37.5203 1.22770
\(935\) 15.6458 0.511671
\(936\) −1.00000 −0.0326860
\(937\) −31.4575 −1.02767 −0.513836 0.857888i \(-0.671776\pi\)
−0.513836 + 0.857888i \(0.671776\pi\)
\(938\) 0 0
\(939\) 13.1660 0.429657
\(940\) −10.9373 −0.356734
\(941\) −17.1660 −0.559596 −0.279798 0.960059i \(-0.590267\pi\)
−0.279798 + 0.960059i \(0.590267\pi\)
\(942\) −2.64575 −0.0862032
\(943\) 5.54249 0.180488
\(944\) −7.93725 −0.258336
\(945\) 0 0
\(946\) 3.41699 0.111096
\(947\) −10.5203 −0.341862 −0.170931 0.985283i \(-0.554678\pi\)
−0.170931 + 0.985283i \(0.554678\pi\)
\(948\) −10.0000 −0.324785
\(949\) −13.6458 −0.442960
\(950\) −41.4575 −1.34506
\(951\) −13.2915 −0.431007
\(952\) 0 0
\(953\) −25.1033 −0.813174 −0.406587 0.913612i \(-0.633281\pi\)
−0.406587 + 0.913612i \(0.633281\pi\)
\(954\) 3.00000 0.0971286
\(955\) 0 0
\(956\) 3.00000 0.0970269
\(957\) 2.77124 0.0895816
\(958\) 34.7490 1.12269
\(959\) 0 0
\(960\) −3.64575 −0.117666
\(961\) −20.1660 −0.650516
\(962\) −5.64575 −0.182026
\(963\) −12.0000 −0.386695
\(964\) −13.8745 −0.446868
\(965\) −69.0405 −2.22249
\(966\) 0 0
\(967\) 9.10326 0.292741 0.146371 0.989230i \(-0.453241\pi\)
0.146371 + 0.989230i \(0.453241\pi\)
\(968\) 10.5830 0.340151
\(969\) −33.2288 −1.06746
\(970\) 3.41699 0.109713
\(971\) 16.9373 0.543542 0.271771 0.962362i \(-0.412391\pi\)
0.271771 + 0.962362i \(0.412391\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) 23.9373 0.766999
\(975\) 8.29150 0.265541
\(976\) −11.9373 −0.382102
\(977\) 20.1255 0.643872 0.321936 0.946762i \(-0.395666\pi\)
0.321936 + 0.946762i \(0.395666\pi\)
\(978\) −2.41699 −0.0772870
\(979\) 10.9373 0.349556
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) −11.1660 −0.356322
\(983\) −30.0405 −0.958144 −0.479072 0.877776i \(-0.659027\pi\)
−0.479072 + 0.877776i \(0.659027\pi\)
\(984\) 2.35425 0.0750507
\(985\) 97.7490 3.11454
\(986\) 28.5203 0.908270
\(987\) 0 0
\(988\) 5.00000 0.159071
\(989\) 12.4575 0.396126
\(990\) 2.35425 0.0748229
\(991\) 5.41699 0.172077 0.0860383 0.996292i \(-0.472579\pi\)
0.0860383 + 0.996292i \(0.472579\pi\)
\(992\) −3.29150 −0.104505
\(993\) 11.4170 0.362307
\(994\) 0 0
\(995\) 81.0405 2.56916
\(996\) −13.2915 −0.421157
\(997\) −1.22876 −0.0389151 −0.0194576 0.999811i \(-0.506194\pi\)
−0.0194576 + 0.999811i \(0.506194\pi\)
\(998\) 29.2915 0.927206
\(999\) 5.64575 0.178624
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 3822.2.a.bk.1.1 2
7.2 even 3 546.2.i.j.235.2 yes 4
7.4 even 3 546.2.i.j.79.2 4
7.6 odd 2 3822.2.a.bi.1.2 2
21.2 odd 6 1638.2.j.k.235.1 4
21.11 odd 6 1638.2.j.k.1171.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.i.j.79.2 4 7.4 even 3
546.2.i.j.235.2 yes 4 7.2 even 3
1638.2.j.k.235.1 4 21.2 odd 6
1638.2.j.k.1171.1 4 21.11 odd 6
3822.2.a.bi.1.2 2 7.6 odd 2
3822.2.a.bk.1.1 2 1.1 even 1 trivial