Properties

Label 3525.2.a.bd.1.8
Level $3525$
Weight $2$
Character 3525.1
Self dual yes
Analytic conductor $28.147$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.1472667125\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 3 x^{7} - 7 x^{6} + 24 x^{5} + 8 x^{4} - 47 x^{3} + 8 x^{2} + 13 x + 2\)
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 1.8
Root \(-2.25864\) of defining polynomial
Character \(\chi\) \(=\) 3525.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.25864 q^{2} +1.00000 q^{3} +3.10144 q^{4} +2.25864 q^{6} -3.65257 q^{7} +2.48774 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.25864 q^{2} +1.00000 q^{3} +3.10144 q^{4} +2.25864 q^{6} -3.65257 q^{7} +2.48774 q^{8} +1.00000 q^{9} -5.39045 q^{11} +3.10144 q^{12} -3.76717 q^{13} -8.24983 q^{14} -0.583969 q^{16} +6.40426 q^{17} +2.25864 q^{18} -8.07640 q^{19} -3.65257 q^{21} -12.1751 q^{22} +4.76354 q^{23} +2.48774 q^{24} -8.50866 q^{26} +1.00000 q^{27} -11.3282 q^{28} -3.78248 q^{29} -5.26069 q^{31} -6.29446 q^{32} -5.39045 q^{33} +14.4649 q^{34} +3.10144 q^{36} -3.48931 q^{37} -18.2416 q^{38} -3.76717 q^{39} +6.70344 q^{41} -8.24983 q^{42} -5.28901 q^{43} -16.7181 q^{44} +10.7591 q^{46} -1.00000 q^{47} -0.583969 q^{48} +6.34130 q^{49} +6.40426 q^{51} -11.6836 q^{52} +9.62076 q^{53} +2.25864 q^{54} -9.08666 q^{56} -8.07640 q^{57} -8.54325 q^{58} +10.0449 q^{59} -4.68111 q^{61} -11.8820 q^{62} -3.65257 q^{63} -13.0489 q^{64} -12.1751 q^{66} -14.1087 q^{67} +19.8624 q^{68} +4.76354 q^{69} +12.0784 q^{71} +2.48774 q^{72} +11.5541 q^{73} -7.88109 q^{74} -25.0484 q^{76} +19.6890 q^{77} -8.50866 q^{78} +1.64874 q^{79} +1.00000 q^{81} +15.1406 q^{82} -10.7120 q^{83} -11.3282 q^{84} -11.9459 q^{86} -3.78248 q^{87} -13.4100 q^{88} -7.50610 q^{89} +13.7599 q^{91} +14.7738 q^{92} -5.26069 q^{93} -2.25864 q^{94} -6.29446 q^{96} -11.7481 q^{97} +14.3227 q^{98} -5.39045 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 3q^{2} + 8q^{3} + 7q^{4} - 3q^{6} - 8q^{7} - 6q^{8} + 8q^{9} + O(q^{10}) \) \( 8q - 3q^{2} + 8q^{3} + 7q^{4} - 3q^{6} - 8q^{7} - 6q^{8} + 8q^{9} - 8q^{11} + 7q^{12} - 10q^{13} + q^{14} + 5q^{16} - 6q^{17} - 3q^{18} - 2q^{19} - 8q^{21} - 10q^{23} - 6q^{24} - 14q^{26} + 8q^{27} - 44q^{28} - 13q^{29} - 10q^{32} - 8q^{33} + 28q^{34} + 7q^{36} - 3q^{37} - 36q^{38} - 10q^{39} - 16q^{41} + q^{42} - 25q^{43} - 17q^{44} - 5q^{46} - 8q^{47} + 5q^{48} + 16q^{49} - 6q^{51} + 17q^{52} - 4q^{53} - 3q^{54} + 37q^{56} - 2q^{57} - 15q^{58} - 8q^{59} + 15q^{61} - 6q^{62} - 8q^{63} - 14q^{64} - 27q^{67} - 14q^{68} - 10q^{69} + 14q^{71} - 6q^{72} - 28q^{73} - 21q^{74} + 6q^{76} - 4q^{77} - 14q^{78} + 7q^{79} + 8q^{81} + 53q^{82} - 60q^{83} - 44q^{84} - 3q^{86} - 13q^{87} - 54q^{88} - 34q^{89} + 23q^{91} + 43q^{92} + 3q^{94} - 10q^{96} - 7q^{97} - 40q^{98} - 8q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.25864 1.59710 0.798548 0.601931i \(-0.205602\pi\)
0.798548 + 0.601931i \(0.205602\pi\)
\(3\) 1.00000 0.577350
\(4\) 3.10144 1.55072
\(5\) 0 0
\(6\) 2.25864 0.922084
\(7\) −3.65257 −1.38054 −0.690272 0.723550i \(-0.742509\pi\)
−0.690272 + 0.723550i \(0.742509\pi\)
\(8\) 2.48774 0.879549
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.39045 −1.62528 −0.812640 0.582766i \(-0.801971\pi\)
−0.812640 + 0.582766i \(0.801971\pi\)
\(12\) 3.10144 0.895307
\(13\) −3.76717 −1.04482 −0.522412 0.852693i \(-0.674968\pi\)
−0.522412 + 0.852693i \(0.674968\pi\)
\(14\) −8.24983 −2.20486
\(15\) 0 0
\(16\) −0.583969 −0.145992
\(17\) 6.40426 1.55326 0.776631 0.629956i \(-0.216927\pi\)
0.776631 + 0.629956i \(0.216927\pi\)
\(18\) 2.25864 0.532366
\(19\) −8.07640 −1.85285 −0.926427 0.376475i \(-0.877136\pi\)
−0.926427 + 0.376475i \(0.877136\pi\)
\(20\) 0 0
\(21\) −3.65257 −0.797057
\(22\) −12.1751 −2.59573
\(23\) 4.76354 0.993268 0.496634 0.867960i \(-0.334569\pi\)
0.496634 + 0.867960i \(0.334569\pi\)
\(24\) 2.48774 0.507808
\(25\) 0 0
\(26\) −8.50866 −1.66869
\(27\) 1.00000 0.192450
\(28\) −11.3282 −2.14083
\(29\) −3.78248 −0.702390 −0.351195 0.936302i \(-0.614225\pi\)
−0.351195 + 0.936302i \(0.614225\pi\)
\(30\) 0 0
\(31\) −5.26069 −0.944847 −0.472423 0.881372i \(-0.656621\pi\)
−0.472423 + 0.881372i \(0.656621\pi\)
\(32\) −6.29446 −1.11271
\(33\) −5.39045 −0.938356
\(34\) 14.4649 2.48071
\(35\) 0 0
\(36\) 3.10144 0.516906
\(37\) −3.48931 −0.573640 −0.286820 0.957985i \(-0.592598\pi\)
−0.286820 + 0.957985i \(0.592598\pi\)
\(38\) −18.2416 −2.95919
\(39\) −3.76717 −0.603230
\(40\) 0 0
\(41\) 6.70344 1.04690 0.523451 0.852056i \(-0.324644\pi\)
0.523451 + 0.852056i \(0.324644\pi\)
\(42\) −8.24983 −1.27298
\(43\) −5.28901 −0.806566 −0.403283 0.915075i \(-0.632131\pi\)
−0.403283 + 0.915075i \(0.632131\pi\)
\(44\) −16.7181 −2.52035
\(45\) 0 0
\(46\) 10.7591 1.58634
\(47\) −1.00000 −0.145865
\(48\) −0.583969 −0.0842887
\(49\) 6.34130 0.905899
\(50\) 0 0
\(51\) 6.40426 0.896776
\(52\) −11.6836 −1.62023
\(53\) 9.62076 1.32151 0.660756 0.750601i \(-0.270236\pi\)
0.660756 + 0.750601i \(0.270236\pi\)
\(54\) 2.25864 0.307361
\(55\) 0 0
\(56\) −9.08666 −1.21426
\(57\) −8.07640 −1.06975
\(58\) −8.54325 −1.12178
\(59\) 10.0449 1.30774 0.653870 0.756607i \(-0.273144\pi\)
0.653870 + 0.756607i \(0.273144\pi\)
\(60\) 0 0
\(61\) −4.68111 −0.599355 −0.299677 0.954041i \(-0.596879\pi\)
−0.299677 + 0.954041i \(0.596879\pi\)
\(62\) −11.8820 −1.50901
\(63\) −3.65257 −0.460181
\(64\) −13.0489 −1.63112
\(65\) 0 0
\(66\) −12.1751 −1.49865
\(67\) −14.1087 −1.72365 −0.861824 0.507207i \(-0.830678\pi\)
−0.861824 + 0.507207i \(0.830678\pi\)
\(68\) 19.8624 2.40867
\(69\) 4.76354 0.573463
\(70\) 0 0
\(71\) 12.0784 1.43344 0.716721 0.697360i \(-0.245642\pi\)
0.716721 + 0.697360i \(0.245642\pi\)
\(72\) 2.48774 0.293183
\(73\) 11.5541 1.35231 0.676155 0.736759i \(-0.263645\pi\)
0.676155 + 0.736759i \(0.263645\pi\)
\(74\) −7.88109 −0.916158
\(75\) 0 0
\(76\) −25.0484 −2.87325
\(77\) 19.6890 2.24377
\(78\) −8.50866 −0.963417
\(79\) 1.64874 0.185498 0.0927491 0.995690i \(-0.470435\pi\)
0.0927491 + 0.995690i \(0.470435\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 15.1406 1.67200
\(83\) −10.7120 −1.17579 −0.587895 0.808937i \(-0.700043\pi\)
−0.587895 + 0.808937i \(0.700043\pi\)
\(84\) −11.3282 −1.23601
\(85\) 0 0
\(86\) −11.9459 −1.28816
\(87\) −3.78248 −0.405525
\(88\) −13.4100 −1.42951
\(89\) −7.50610 −0.795645 −0.397822 0.917462i \(-0.630234\pi\)
−0.397822 + 0.917462i \(0.630234\pi\)
\(90\) 0 0
\(91\) 13.7599 1.44243
\(92\) 14.7738 1.54028
\(93\) −5.26069 −0.545508
\(94\) −2.25864 −0.232960
\(95\) 0 0
\(96\) −6.29446 −0.642425
\(97\) −11.7481 −1.19284 −0.596420 0.802673i \(-0.703411\pi\)
−0.596420 + 0.802673i \(0.703411\pi\)
\(98\) 14.3227 1.44681
\(99\) −5.39045 −0.541760
\(100\) 0 0
\(101\) 9.79905 0.975042 0.487521 0.873111i \(-0.337901\pi\)
0.487521 + 0.873111i \(0.337901\pi\)
\(102\) 14.4649 1.43224
\(103\) 7.91135 0.779528 0.389764 0.920915i \(-0.372557\pi\)
0.389764 + 0.920915i \(0.372557\pi\)
\(104\) −9.37174 −0.918975
\(105\) 0 0
\(106\) 21.7298 2.11058
\(107\) −16.0171 −1.54843 −0.774217 0.632921i \(-0.781856\pi\)
−0.774217 + 0.632921i \(0.781856\pi\)
\(108\) 3.10144 0.298436
\(109\) −6.27573 −0.601106 −0.300553 0.953765i \(-0.597171\pi\)
−0.300553 + 0.953765i \(0.597171\pi\)
\(110\) 0 0
\(111\) −3.48931 −0.331191
\(112\) 2.13299 0.201549
\(113\) −15.5200 −1.45999 −0.729997 0.683450i \(-0.760479\pi\)
−0.729997 + 0.683450i \(0.760479\pi\)
\(114\) −18.2416 −1.70849
\(115\) 0 0
\(116\) −11.7311 −1.08921
\(117\) −3.76717 −0.348275
\(118\) 22.6879 2.08859
\(119\) −23.3920 −2.14434
\(120\) 0 0
\(121\) 18.0569 1.64154
\(122\) −10.5729 −0.957227
\(123\) 6.70344 0.604429
\(124\) −16.3157 −1.46519
\(125\) 0 0
\(126\) −8.24983 −0.734954
\(127\) −16.1501 −1.43309 −0.716543 0.697543i \(-0.754277\pi\)
−0.716543 + 0.697543i \(0.754277\pi\)
\(128\) −16.8839 −1.49234
\(129\) −5.28901 −0.465671
\(130\) 0 0
\(131\) 14.0586 1.22831 0.614155 0.789186i \(-0.289497\pi\)
0.614155 + 0.789186i \(0.289497\pi\)
\(132\) −16.7181 −1.45513
\(133\) 29.4997 2.55794
\(134\) −31.8664 −2.75283
\(135\) 0 0
\(136\) 15.9321 1.36617
\(137\) 20.6494 1.76419 0.882097 0.471069i \(-0.156132\pi\)
0.882097 + 0.471069i \(0.156132\pi\)
\(138\) 10.7591 0.915876
\(139\) −3.85467 −0.326949 −0.163474 0.986548i \(-0.552270\pi\)
−0.163474 + 0.986548i \(0.552270\pi\)
\(140\) 0 0
\(141\) −1.00000 −0.0842152
\(142\) 27.2807 2.28935
\(143\) 20.3067 1.69813
\(144\) −0.583969 −0.0486641
\(145\) 0 0
\(146\) 26.0966 2.15977
\(147\) 6.34130 0.523021
\(148\) −10.8219 −0.889553
\(149\) −9.31652 −0.763239 −0.381620 0.924319i \(-0.624634\pi\)
−0.381620 + 0.924319i \(0.624634\pi\)
\(150\) 0 0
\(151\) 10.9014 0.887146 0.443573 0.896238i \(-0.353711\pi\)
0.443573 + 0.896238i \(0.353711\pi\)
\(152\) −20.0920 −1.62968
\(153\) 6.40426 0.517754
\(154\) 44.4703 3.58352
\(155\) 0 0
\(156\) −11.6836 −0.935439
\(157\) −8.68544 −0.693174 −0.346587 0.938018i \(-0.612659\pi\)
−0.346587 + 0.938018i \(0.612659\pi\)
\(158\) 3.72391 0.296258
\(159\) 9.62076 0.762976
\(160\) 0 0
\(161\) −17.3992 −1.37125
\(162\) 2.25864 0.177455
\(163\) 14.4978 1.13556 0.567778 0.823182i \(-0.307803\pi\)
0.567778 + 0.823182i \(0.307803\pi\)
\(164\) 20.7903 1.62345
\(165\) 0 0
\(166\) −24.1944 −1.87785
\(167\) −14.1218 −1.09278 −0.546388 0.837532i \(-0.683998\pi\)
−0.546388 + 0.837532i \(0.683998\pi\)
\(168\) −9.08666 −0.701051
\(169\) 1.19157 0.0916591
\(170\) 0 0
\(171\) −8.07640 −0.617618
\(172\) −16.4035 −1.25076
\(173\) −6.32710 −0.481041 −0.240520 0.970644i \(-0.577318\pi\)
−0.240520 + 0.970644i \(0.577318\pi\)
\(174\) −8.54325 −0.647662
\(175\) 0 0
\(176\) 3.14785 0.237278
\(177\) 10.0449 0.755024
\(178\) −16.9535 −1.27072
\(179\) 8.85078 0.661538 0.330769 0.943712i \(-0.392692\pi\)
0.330769 + 0.943712i \(0.392692\pi\)
\(180\) 0 0
\(181\) 13.6795 1.01679 0.508395 0.861124i \(-0.330239\pi\)
0.508395 + 0.861124i \(0.330239\pi\)
\(182\) 31.0785 2.30369
\(183\) −4.68111 −0.346038
\(184\) 11.8505 0.873628
\(185\) 0 0
\(186\) −11.8820 −0.871228
\(187\) −34.5218 −2.52449
\(188\) −3.10144 −0.226195
\(189\) −3.65257 −0.265686
\(190\) 0 0
\(191\) 15.7719 1.14121 0.570607 0.821223i \(-0.306708\pi\)
0.570607 + 0.821223i \(0.306708\pi\)
\(192\) −13.0489 −0.941727
\(193\) 7.82923 0.563561 0.281780 0.959479i \(-0.409075\pi\)
0.281780 + 0.959479i \(0.409075\pi\)
\(194\) −26.5347 −1.90508
\(195\) 0 0
\(196\) 19.6671 1.40479
\(197\) 10.6587 0.759401 0.379701 0.925109i \(-0.376027\pi\)
0.379701 + 0.925109i \(0.376027\pi\)
\(198\) −12.1751 −0.865243
\(199\) −6.40218 −0.453839 −0.226919 0.973914i \(-0.572865\pi\)
−0.226919 + 0.973914i \(0.572865\pi\)
\(200\) 0 0
\(201\) −14.1087 −0.995149
\(202\) 22.1325 1.55724
\(203\) 13.8158 0.969679
\(204\) 19.8624 1.39065
\(205\) 0 0
\(206\) 17.8689 1.24498
\(207\) 4.76354 0.331089
\(208\) 2.19991 0.152536
\(209\) 43.5354 3.01141
\(210\) 0 0
\(211\) 7.13642 0.491291 0.245646 0.969360i \(-0.421000\pi\)
0.245646 + 0.969360i \(0.421000\pi\)
\(212\) 29.8382 2.04929
\(213\) 12.0784 0.827599
\(214\) −36.1768 −2.47300
\(215\) 0 0
\(216\) 2.48774 0.169269
\(217\) 19.2150 1.30440
\(218\) −14.1746 −0.960024
\(219\) 11.5541 0.780756
\(220\) 0 0
\(221\) −24.1259 −1.62289
\(222\) −7.88109 −0.528944
\(223\) −5.41572 −0.362664 −0.181332 0.983422i \(-0.558041\pi\)
−0.181332 + 0.983422i \(0.558041\pi\)
\(224\) 22.9910 1.53615
\(225\) 0 0
\(226\) −35.0539 −2.33175
\(227\) −22.8812 −1.51868 −0.759339 0.650695i \(-0.774478\pi\)
−0.759339 + 0.650695i \(0.774478\pi\)
\(228\) −25.0484 −1.65887
\(229\) 25.9203 1.71286 0.856431 0.516261i \(-0.172677\pi\)
0.856431 + 0.516261i \(0.172677\pi\)
\(230\) 0 0
\(231\) 19.6890 1.29544
\(232\) −9.40984 −0.617786
\(233\) 1.38698 0.0908638 0.0454319 0.998967i \(-0.485534\pi\)
0.0454319 + 0.998967i \(0.485534\pi\)
\(234\) −8.50866 −0.556229
\(235\) 0 0
\(236\) 31.1538 2.02794
\(237\) 1.64874 0.107097
\(238\) −52.8341 −3.42472
\(239\) −20.8507 −1.34872 −0.674361 0.738402i \(-0.735581\pi\)
−0.674361 + 0.738402i \(0.735581\pi\)
\(240\) 0 0
\(241\) −19.2621 −1.24078 −0.620389 0.784294i \(-0.713025\pi\)
−0.620389 + 0.784294i \(0.713025\pi\)
\(242\) 40.7840 2.62169
\(243\) 1.00000 0.0641500
\(244\) −14.5182 −0.929430
\(245\) 0 0
\(246\) 15.1406 0.965332
\(247\) 30.4252 1.93591
\(248\) −13.0872 −0.831040
\(249\) −10.7120 −0.678843
\(250\) 0 0
\(251\) 10.0252 0.632784 0.316392 0.948629i \(-0.397529\pi\)
0.316392 + 0.948629i \(0.397529\pi\)
\(252\) −11.3282 −0.713611
\(253\) −25.6776 −1.61434
\(254\) −36.4771 −2.28878
\(255\) 0 0
\(256\) −12.0367 −0.752293
\(257\) 11.1976 0.698488 0.349244 0.937032i \(-0.386438\pi\)
0.349244 + 0.937032i \(0.386438\pi\)
\(258\) −11.9459 −0.743722
\(259\) 12.7450 0.791935
\(260\) 0 0
\(261\) −3.78248 −0.234130
\(262\) 31.7534 1.96173
\(263\) −0.764798 −0.0471595 −0.0235797 0.999722i \(-0.507506\pi\)
−0.0235797 + 0.999722i \(0.507506\pi\)
\(264\) −13.4100 −0.825331
\(265\) 0 0
\(266\) 66.6290 4.08528
\(267\) −7.50610 −0.459366
\(268\) −43.7571 −2.67289
\(269\) 2.34969 0.143263 0.0716316 0.997431i \(-0.477179\pi\)
0.0716316 + 0.997431i \(0.477179\pi\)
\(270\) 0 0
\(271\) −15.8548 −0.963111 −0.481556 0.876416i \(-0.659928\pi\)
−0.481556 + 0.876416i \(0.659928\pi\)
\(272\) −3.73989 −0.226764
\(273\) 13.7599 0.832785
\(274\) 46.6394 2.81759
\(275\) 0 0
\(276\) 14.7738 0.889280
\(277\) −17.8932 −1.07510 −0.537548 0.843233i \(-0.680649\pi\)
−0.537548 + 0.843233i \(0.680649\pi\)
\(278\) −8.70630 −0.522169
\(279\) −5.26069 −0.314949
\(280\) 0 0
\(281\) −21.7136 −1.29533 −0.647663 0.761927i \(-0.724253\pi\)
−0.647663 + 0.761927i \(0.724253\pi\)
\(282\) −2.25864 −0.134500
\(283\) −14.5961 −0.867648 −0.433824 0.900998i \(-0.642836\pi\)
−0.433824 + 0.900998i \(0.642836\pi\)
\(284\) 37.4604 2.22286
\(285\) 0 0
\(286\) 45.8655 2.71208
\(287\) −24.4848 −1.44529
\(288\) −6.29446 −0.370904
\(289\) 24.0145 1.41262
\(290\) 0 0
\(291\) −11.7481 −0.688686
\(292\) 35.8344 2.09705
\(293\) 6.51441 0.380576 0.190288 0.981728i \(-0.439058\pi\)
0.190288 + 0.981728i \(0.439058\pi\)
\(294\) 14.3227 0.835316
\(295\) 0 0
\(296\) −8.68051 −0.504545
\(297\) −5.39045 −0.312785
\(298\) −21.0426 −1.21897
\(299\) −17.9451 −1.03779
\(300\) 0 0
\(301\) 19.3185 1.11350
\(302\) 24.6224 1.41686
\(303\) 9.79905 0.562941
\(304\) 4.71637 0.270502
\(305\) 0 0
\(306\) 14.4649 0.826903
\(307\) −30.2614 −1.72711 −0.863555 0.504254i \(-0.831767\pi\)
−0.863555 + 0.504254i \(0.831767\pi\)
\(308\) 61.0642 3.47945
\(309\) 7.91135 0.450061
\(310\) 0 0
\(311\) −14.8690 −0.843145 −0.421572 0.906795i \(-0.638522\pi\)
−0.421572 + 0.906795i \(0.638522\pi\)
\(312\) −9.37174 −0.530571
\(313\) −19.0799 −1.07846 −0.539231 0.842158i \(-0.681285\pi\)
−0.539231 + 0.842158i \(0.681285\pi\)
\(314\) −19.6172 −1.10707
\(315\) 0 0
\(316\) 5.11347 0.287655
\(317\) 11.9688 0.672234 0.336117 0.941820i \(-0.390886\pi\)
0.336117 + 0.941820i \(0.390886\pi\)
\(318\) 21.7298 1.21855
\(319\) 20.3893 1.14158
\(320\) 0 0
\(321\) −16.0171 −0.893989
\(322\) −39.2984 −2.19002
\(323\) −51.7234 −2.87797
\(324\) 3.10144 0.172302
\(325\) 0 0
\(326\) 32.7453 1.81359
\(327\) −6.27573 −0.347049
\(328\) 16.6764 0.920802
\(329\) 3.65257 0.201373
\(330\) 0 0
\(331\) −16.0901 −0.884390 −0.442195 0.896919i \(-0.645800\pi\)
−0.442195 + 0.896919i \(0.645800\pi\)
\(332\) −33.2224 −1.82332
\(333\) −3.48931 −0.191213
\(334\) −31.8960 −1.74527
\(335\) 0 0
\(336\) 2.13299 0.116364
\(337\) 9.68906 0.527797 0.263898 0.964550i \(-0.414992\pi\)
0.263898 + 0.964550i \(0.414992\pi\)
\(338\) 2.69132 0.146388
\(339\) −15.5200 −0.842928
\(340\) 0 0
\(341\) 28.3574 1.53564
\(342\) −18.2416 −0.986395
\(343\) 2.40596 0.129910
\(344\) −13.1577 −0.709415
\(345\) 0 0
\(346\) −14.2906 −0.768268
\(347\) −3.87840 −0.208203 −0.104102 0.994567i \(-0.533197\pi\)
−0.104102 + 0.994567i \(0.533197\pi\)
\(348\) −11.7311 −0.628855
\(349\) −26.0674 −1.39536 −0.697679 0.716411i \(-0.745784\pi\)
−0.697679 + 0.716411i \(0.745784\pi\)
\(350\) 0 0
\(351\) −3.76717 −0.201077
\(352\) 33.9299 1.80847
\(353\) −32.9876 −1.75575 −0.877877 0.478886i \(-0.841041\pi\)
−0.877877 + 0.478886i \(0.841041\pi\)
\(354\) 22.6879 1.20585
\(355\) 0 0
\(356\) −23.2797 −1.23382
\(357\) −23.3920 −1.23804
\(358\) 19.9907 1.05654
\(359\) −0.425406 −0.0224520 −0.0112260 0.999937i \(-0.503573\pi\)
−0.0112260 + 0.999937i \(0.503573\pi\)
\(360\) 0 0
\(361\) 46.2283 2.43307
\(362\) 30.8970 1.62391
\(363\) 18.0569 0.947742
\(364\) 42.6753 2.23680
\(365\) 0 0
\(366\) −10.5729 −0.552655
\(367\) 1.97076 0.102873 0.0514365 0.998676i \(-0.483620\pi\)
0.0514365 + 0.998676i \(0.483620\pi\)
\(368\) −2.78176 −0.145009
\(369\) 6.70344 0.348967
\(370\) 0 0
\(371\) −35.1405 −1.82441
\(372\) −16.3157 −0.845928
\(373\) −3.91613 −0.202770 −0.101385 0.994847i \(-0.532327\pi\)
−0.101385 + 0.994847i \(0.532327\pi\)
\(374\) −77.9722 −4.03185
\(375\) 0 0
\(376\) −2.48774 −0.128295
\(377\) 14.2493 0.733874
\(378\) −8.24983 −0.424326
\(379\) −6.09374 −0.313014 −0.156507 0.987677i \(-0.550023\pi\)
−0.156507 + 0.987677i \(0.550023\pi\)
\(380\) 0 0
\(381\) −16.1501 −0.827393
\(382\) 35.6230 1.82263
\(383\) 27.9056 1.42591 0.712956 0.701209i \(-0.247356\pi\)
0.712956 + 0.701209i \(0.247356\pi\)
\(384\) −16.8839 −0.861603
\(385\) 0 0
\(386\) 17.6834 0.900061
\(387\) −5.28901 −0.268855
\(388\) −36.4360 −1.84976
\(389\) 7.08100 0.359021 0.179511 0.983756i \(-0.442549\pi\)
0.179511 + 0.983756i \(0.442549\pi\)
\(390\) 0 0
\(391\) 30.5070 1.54280
\(392\) 15.7755 0.796783
\(393\) 14.0586 0.709165
\(394\) 24.0741 1.21284
\(395\) 0 0
\(396\) −16.7181 −0.840117
\(397\) −12.8045 −0.642637 −0.321319 0.946971i \(-0.604126\pi\)
−0.321319 + 0.946971i \(0.604126\pi\)
\(398\) −14.4602 −0.724824
\(399\) 29.4997 1.47683
\(400\) 0 0
\(401\) 17.9986 0.898808 0.449404 0.893329i \(-0.351636\pi\)
0.449404 + 0.893329i \(0.351636\pi\)
\(402\) −31.8664 −1.58935
\(403\) 19.8179 0.987200
\(404\) 30.3911 1.51202
\(405\) 0 0
\(406\) 31.2049 1.54867
\(407\) 18.8090 0.932326
\(408\) 15.9321 0.788759
\(409\) 28.9900 1.43346 0.716731 0.697350i \(-0.245637\pi\)
0.716731 + 0.697350i \(0.245637\pi\)
\(410\) 0 0
\(411\) 20.6494 1.01856
\(412\) 24.5365 1.20883
\(413\) −36.6899 −1.80539
\(414\) 10.7591 0.528781
\(415\) 0 0
\(416\) 23.7123 1.16259
\(417\) −3.85467 −0.188764
\(418\) 98.3306 4.80951
\(419\) −0.494504 −0.0241581 −0.0120791 0.999927i \(-0.503845\pi\)
−0.0120791 + 0.999927i \(0.503845\pi\)
\(420\) 0 0
\(421\) −24.4814 −1.19315 −0.596575 0.802557i \(-0.703472\pi\)
−0.596575 + 0.802557i \(0.703472\pi\)
\(422\) 16.1186 0.784640
\(423\) −1.00000 −0.0486217
\(424\) 23.9340 1.16234
\(425\) 0 0
\(426\) 27.2807 1.32175
\(427\) 17.0981 0.827435
\(428\) −49.6761 −2.40118
\(429\) 20.3067 0.980418
\(430\) 0 0
\(431\) 20.5802 0.991311 0.495656 0.868519i \(-0.334928\pi\)
0.495656 + 0.868519i \(0.334928\pi\)
\(432\) −0.583969 −0.0280962
\(433\) 10.8674 0.522254 0.261127 0.965304i \(-0.415906\pi\)
0.261127 + 0.965304i \(0.415906\pi\)
\(434\) 43.3998 2.08326
\(435\) 0 0
\(436\) −19.4638 −0.932145
\(437\) −38.4723 −1.84038
\(438\) 26.0966 1.24694
\(439\) −4.05108 −0.193347 −0.0966737 0.995316i \(-0.530820\pi\)
−0.0966737 + 0.995316i \(0.530820\pi\)
\(440\) 0 0
\(441\) 6.34130 0.301966
\(442\) −54.4917 −2.59191
\(443\) −19.8347 −0.942376 −0.471188 0.882033i \(-0.656175\pi\)
−0.471188 + 0.882033i \(0.656175\pi\)
\(444\) −10.8219 −0.513584
\(445\) 0 0
\(446\) −12.2321 −0.579209
\(447\) −9.31652 −0.440656
\(448\) 47.6622 2.25183
\(449\) −12.3888 −0.584663 −0.292331 0.956317i \(-0.594431\pi\)
−0.292331 + 0.956317i \(0.594431\pi\)
\(450\) 0 0
\(451\) −36.1346 −1.70151
\(452\) −48.1341 −2.26404
\(453\) 10.9014 0.512194
\(454\) −51.6803 −2.42548
\(455\) 0 0
\(456\) −20.0920 −0.940894
\(457\) 0.726902 0.0340030 0.0170015 0.999855i \(-0.494588\pi\)
0.0170015 + 0.999855i \(0.494588\pi\)
\(458\) 58.5445 2.73561
\(459\) 6.40426 0.298925
\(460\) 0 0
\(461\) −5.78513 −0.269440 −0.134720 0.990884i \(-0.543014\pi\)
−0.134720 + 0.990884i \(0.543014\pi\)
\(462\) 44.4703 2.06894
\(463\) −23.2529 −1.08066 −0.540328 0.841455i \(-0.681700\pi\)
−0.540328 + 0.841455i \(0.681700\pi\)
\(464\) 2.20885 0.102544
\(465\) 0 0
\(466\) 3.13267 0.145118
\(467\) 19.5028 0.902481 0.451241 0.892402i \(-0.350982\pi\)
0.451241 + 0.892402i \(0.350982\pi\)
\(468\) −11.6836 −0.540076
\(469\) 51.5330 2.37957
\(470\) 0 0
\(471\) −8.68544 −0.400204
\(472\) 24.9892 1.15022
\(473\) 28.5101 1.31090
\(474\) 3.72391 0.171045
\(475\) 0 0
\(476\) −72.5489 −3.32527
\(477\) 9.62076 0.440504
\(478\) −47.0942 −2.15404
\(479\) 15.9371 0.728185 0.364093 0.931363i \(-0.381379\pi\)
0.364093 + 0.931363i \(0.381379\pi\)
\(480\) 0 0
\(481\) 13.1448 0.599353
\(482\) −43.5060 −1.98164
\(483\) −17.3992 −0.791691
\(484\) 56.0023 2.54556
\(485\) 0 0
\(486\) 2.25864 0.102454
\(487\) −21.5979 −0.978695 −0.489347 0.872089i \(-0.662765\pi\)
−0.489347 + 0.872089i \(0.662765\pi\)
\(488\) −11.6454 −0.527162
\(489\) 14.4978 0.655613
\(490\) 0 0
\(491\) −19.5726 −0.883298 −0.441649 0.897188i \(-0.645606\pi\)
−0.441649 + 0.897188i \(0.645606\pi\)
\(492\) 20.7903 0.937299
\(493\) −24.2240 −1.09099
\(494\) 68.7194 3.09183
\(495\) 0 0
\(496\) 3.07208 0.137940
\(497\) −44.1173 −1.97893
\(498\) −24.1944 −1.08418
\(499\) −30.3032 −1.35656 −0.678278 0.734805i \(-0.737274\pi\)
−0.678278 + 0.734805i \(0.737274\pi\)
\(500\) 0 0
\(501\) −14.1218 −0.630915
\(502\) 22.6432 1.01062
\(503\) 23.6646 1.05515 0.527577 0.849507i \(-0.323101\pi\)
0.527577 + 0.849507i \(0.323101\pi\)
\(504\) −9.08666 −0.404752
\(505\) 0 0
\(506\) −57.9964 −2.57825
\(507\) 1.19157 0.0529194
\(508\) −50.0884 −2.22231
\(509\) 12.2650 0.543638 0.271819 0.962348i \(-0.412375\pi\)
0.271819 + 0.962348i \(0.412375\pi\)
\(510\) 0 0
\(511\) −42.2024 −1.86692
\(512\) 6.58130 0.290855
\(513\) −8.07640 −0.356582
\(514\) 25.2913 1.11555
\(515\) 0 0
\(516\) −16.4035 −0.722125
\(517\) 5.39045 0.237072
\(518\) 28.7863 1.26480
\(519\) −6.32710 −0.277729
\(520\) 0 0
\(521\) −29.7739 −1.30442 −0.652210 0.758038i \(-0.726158\pi\)
−0.652210 + 0.758038i \(0.726158\pi\)
\(522\) −8.54325 −0.373928
\(523\) −36.9743 −1.61677 −0.808386 0.588653i \(-0.799658\pi\)
−0.808386 + 0.588653i \(0.799658\pi\)
\(524\) 43.6020 1.90476
\(525\) 0 0
\(526\) −1.72740 −0.0753182
\(527\) −33.6908 −1.46759
\(528\) 3.14785 0.136993
\(529\) −0.308648 −0.0134195
\(530\) 0 0
\(531\) 10.0449 0.435914
\(532\) 91.4913 3.96665
\(533\) −25.2530 −1.09383
\(534\) −16.9535 −0.733652
\(535\) 0 0
\(536\) −35.0987 −1.51603
\(537\) 8.85078 0.381939
\(538\) 5.30710 0.228805
\(539\) −34.1824 −1.47234
\(540\) 0 0
\(541\) 17.3797 0.747212 0.373606 0.927587i \(-0.378121\pi\)
0.373606 + 0.927587i \(0.378121\pi\)
\(542\) −35.8102 −1.53818
\(543\) 13.6795 0.587044
\(544\) −40.3113 −1.72833
\(545\) 0 0
\(546\) 31.0785 1.33004
\(547\) −3.14180 −0.134334 −0.0671668 0.997742i \(-0.521396\pi\)
−0.0671668 + 0.997742i \(0.521396\pi\)
\(548\) 64.0426 2.73577
\(549\) −4.68111 −0.199785
\(550\) 0 0
\(551\) 30.5489 1.30143
\(552\) 11.8505 0.504389
\(553\) −6.02216 −0.256088
\(554\) −40.4141 −1.71703
\(555\) 0 0
\(556\) −11.9550 −0.507005
\(557\) 4.70436 0.199330 0.0996650 0.995021i \(-0.468223\pi\)
0.0996650 + 0.995021i \(0.468223\pi\)
\(558\) −11.8820 −0.503004
\(559\) 19.9246 0.842721
\(560\) 0 0
\(561\) −34.5218 −1.45751
\(562\) −49.0432 −2.06876
\(563\) −9.19564 −0.387550 −0.193775 0.981046i \(-0.562073\pi\)
−0.193775 + 0.981046i \(0.562073\pi\)
\(564\) −3.10144 −0.130594
\(565\) 0 0
\(566\) −32.9673 −1.38572
\(567\) −3.65257 −0.153394
\(568\) 30.0479 1.26078
\(569\) 4.54858 0.190686 0.0953432 0.995444i \(-0.469605\pi\)
0.0953432 + 0.995444i \(0.469605\pi\)
\(570\) 0 0
\(571\) −8.85447 −0.370548 −0.185274 0.982687i \(-0.559317\pi\)
−0.185274 + 0.982687i \(0.559317\pi\)
\(572\) 62.9800 2.63333
\(573\) 15.7719 0.658881
\(574\) −55.3023 −2.30827
\(575\) 0 0
\(576\) −13.0489 −0.543706
\(577\) 8.34516 0.347414 0.173707 0.984797i \(-0.444425\pi\)
0.173707 + 0.984797i \(0.444425\pi\)
\(578\) 54.2401 2.25609
\(579\) 7.82923 0.325372
\(580\) 0 0
\(581\) 39.1262 1.62323
\(582\) −26.5347 −1.09990
\(583\) −51.8602 −2.14783
\(584\) 28.7437 1.18942
\(585\) 0 0
\(586\) 14.7137 0.607817
\(587\) 18.7043 0.772010 0.386005 0.922497i \(-0.373855\pi\)
0.386005 + 0.922497i \(0.373855\pi\)
\(588\) 19.6671 0.811058
\(589\) 42.4874 1.75066
\(590\) 0 0
\(591\) 10.6587 0.438440
\(592\) 2.03765 0.0837470
\(593\) 19.5188 0.801542 0.400771 0.916178i \(-0.368742\pi\)
0.400771 + 0.916178i \(0.368742\pi\)
\(594\) −12.1751 −0.499548
\(595\) 0 0
\(596\) −28.8946 −1.18357
\(597\) −6.40218 −0.262024
\(598\) −40.5314 −1.65745
\(599\) −10.2208 −0.417610 −0.208805 0.977957i \(-0.566957\pi\)
−0.208805 + 0.977957i \(0.566957\pi\)
\(600\) 0 0
\(601\) −23.3384 −0.951992 −0.475996 0.879448i \(-0.657912\pi\)
−0.475996 + 0.879448i \(0.657912\pi\)
\(602\) 43.6335 1.77837
\(603\) −14.1087 −0.574550
\(604\) 33.8101 1.37571
\(605\) 0 0
\(606\) 22.1325 0.899071
\(607\) 31.6937 1.28641 0.643204 0.765695i \(-0.277605\pi\)
0.643204 + 0.765695i \(0.277605\pi\)
\(608\) 50.8366 2.06169
\(609\) 13.8158 0.559845
\(610\) 0 0
\(611\) 3.76717 0.152403
\(612\) 19.8624 0.802890
\(613\) 34.6718 1.40038 0.700191 0.713955i \(-0.253098\pi\)
0.700191 + 0.713955i \(0.253098\pi\)
\(614\) −68.3495 −2.75836
\(615\) 0 0
\(616\) 48.9811 1.97351
\(617\) −45.0825 −1.81495 −0.907476 0.420104i \(-0.861994\pi\)
−0.907476 + 0.420104i \(0.861994\pi\)
\(618\) 17.8689 0.718791
\(619\) 15.5285 0.624141 0.312071 0.950059i \(-0.398977\pi\)
0.312071 + 0.950059i \(0.398977\pi\)
\(620\) 0 0
\(621\) 4.76354 0.191154
\(622\) −33.5837 −1.34658
\(623\) 27.4166 1.09842
\(624\) 2.19991 0.0880669
\(625\) 0 0
\(626\) −43.0947 −1.72241
\(627\) 43.5354 1.73864
\(628\) −26.9373 −1.07492
\(629\) −22.3465 −0.891012
\(630\) 0 0
\(631\) 11.8534 0.471878 0.235939 0.971768i \(-0.424183\pi\)
0.235939 + 0.971768i \(0.424183\pi\)
\(632\) 4.10165 0.163155
\(633\) 7.13642 0.283647
\(634\) 27.0331 1.07362
\(635\) 0 0
\(636\) 29.8382 1.18316
\(637\) −23.8887 −0.946506
\(638\) 46.0519 1.82321
\(639\) 12.0784 0.477814
\(640\) 0 0
\(641\) 11.3123 0.446807 0.223404 0.974726i \(-0.428283\pi\)
0.223404 + 0.974726i \(0.428283\pi\)
\(642\) −36.1768 −1.42779
\(643\) 3.71679 0.146576 0.0732879 0.997311i \(-0.476651\pi\)
0.0732879 + 0.997311i \(0.476651\pi\)
\(644\) −53.9625 −2.12642
\(645\) 0 0
\(646\) −116.824 −4.59639
\(647\) −20.9097 −0.822047 −0.411023 0.911625i \(-0.634829\pi\)
−0.411023 + 0.911625i \(0.634829\pi\)
\(648\) 2.48774 0.0977277
\(649\) −54.1467 −2.12545
\(650\) 0 0
\(651\) 19.2150 0.753097
\(652\) 44.9640 1.76093
\(653\) 31.2980 1.22479 0.612393 0.790553i \(-0.290207\pi\)
0.612393 + 0.790553i \(0.290207\pi\)
\(654\) −14.1746 −0.554270
\(655\) 0 0
\(656\) −3.91461 −0.152840
\(657\) 11.5541 0.450770
\(658\) 8.24983 0.321612
\(659\) 11.6158 0.452487 0.226243 0.974071i \(-0.427356\pi\)
0.226243 + 0.974071i \(0.427356\pi\)
\(660\) 0 0
\(661\) −9.93633 −0.386478 −0.193239 0.981152i \(-0.561899\pi\)
−0.193239 + 0.981152i \(0.561899\pi\)
\(662\) −36.3416 −1.41246
\(663\) −24.1259 −0.936974
\(664\) −26.6486 −1.03417
\(665\) 0 0
\(666\) −7.88109 −0.305386
\(667\) −18.0180 −0.697661
\(668\) −43.7978 −1.69459
\(669\) −5.41572 −0.209384
\(670\) 0 0
\(671\) 25.2333 0.974119
\(672\) 22.9910 0.886896
\(673\) 22.5946 0.870957 0.435479 0.900199i \(-0.356579\pi\)
0.435479 + 0.900199i \(0.356579\pi\)
\(674\) 21.8841 0.842943
\(675\) 0 0
\(676\) 3.69557 0.142137
\(677\) −30.0102 −1.15338 −0.576692 0.816962i \(-0.695657\pi\)
−0.576692 + 0.816962i \(0.695657\pi\)
\(678\) −35.0539 −1.34624
\(679\) 42.9108 1.64677
\(680\) 0 0
\(681\) −22.8812 −0.876809
\(682\) 64.0491 2.45257
\(683\) 2.71242 0.103788 0.0518939 0.998653i \(-0.483474\pi\)
0.0518939 + 0.998653i \(0.483474\pi\)
\(684\) −25.0484 −0.957751
\(685\) 0 0
\(686\) 5.43419 0.207479
\(687\) 25.9203 0.988921
\(688\) 3.08862 0.117753
\(689\) −36.2430 −1.38075
\(690\) 0 0
\(691\) 12.4120 0.472174 0.236087 0.971732i \(-0.424135\pi\)
0.236087 + 0.971732i \(0.424135\pi\)
\(692\) −19.6231 −0.745958
\(693\) 19.6890 0.747923
\(694\) −8.75989 −0.332521
\(695\) 0 0
\(696\) −9.40984 −0.356679
\(697\) 42.9306 1.62611
\(698\) −58.8768 −2.22852
\(699\) 1.38698 0.0524603
\(700\) 0 0
\(701\) 38.1760 1.44189 0.720943 0.692994i \(-0.243709\pi\)
0.720943 + 0.692994i \(0.243709\pi\)
\(702\) −8.50866 −0.321139
\(703\) 28.1811 1.06287
\(704\) 70.3396 2.65102
\(705\) 0 0
\(706\) −74.5070 −2.80411
\(707\) −35.7918 −1.34609
\(708\) 31.1538 1.17083
\(709\) −21.2535 −0.798194 −0.399097 0.916909i \(-0.630676\pi\)
−0.399097 + 0.916909i \(0.630676\pi\)
\(710\) 0 0
\(711\) 1.64874 0.0618327
\(712\) −18.6732 −0.699809
\(713\) −25.0595 −0.938486
\(714\) −52.8341 −1.97727
\(715\) 0 0
\(716\) 27.4501 1.02586
\(717\) −20.8507 −0.778685
\(718\) −0.960836 −0.0358581
\(719\) 25.3271 0.944543 0.472272 0.881453i \(-0.343434\pi\)
0.472272 + 0.881453i \(0.343434\pi\)
\(720\) 0 0
\(721\) −28.8968 −1.07617
\(722\) 104.413 3.88584
\(723\) −19.2621 −0.716364
\(724\) 42.4261 1.57675
\(725\) 0 0
\(726\) 40.7840 1.51363
\(727\) 38.4277 1.42521 0.712603 0.701567i \(-0.247516\pi\)
0.712603 + 0.701567i \(0.247516\pi\)
\(728\) 34.2310 1.26868
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −33.8722 −1.25281
\(732\) −14.5182 −0.536607
\(733\) −39.9589 −1.47592 −0.737959 0.674846i \(-0.764210\pi\)
−0.737959 + 0.674846i \(0.764210\pi\)
\(734\) 4.45123 0.164298
\(735\) 0 0
\(736\) −29.9839 −1.10522
\(737\) 76.0520 2.80141
\(738\) 15.1406 0.557335
\(739\) 6.27640 0.230881 0.115441 0.993314i \(-0.463172\pi\)
0.115441 + 0.993314i \(0.463172\pi\)
\(740\) 0 0
\(741\) 30.4252 1.11770
\(742\) −79.3696 −2.91375
\(743\) −18.5595 −0.680882 −0.340441 0.940266i \(-0.610576\pi\)
−0.340441 + 0.940266i \(0.610576\pi\)
\(744\) −13.0872 −0.479801
\(745\) 0 0
\(746\) −8.84511 −0.323843
\(747\) −10.7120 −0.391930
\(748\) −107.067 −3.91476
\(749\) 58.5037 2.13768
\(750\) 0 0
\(751\) 46.3145 1.69004 0.845019 0.534736i \(-0.179589\pi\)
0.845019 + 0.534736i \(0.179589\pi\)
\(752\) 0.583969 0.0212952
\(753\) 10.0252 0.365338
\(754\) 32.1839 1.17207
\(755\) 0 0
\(756\) −11.3282 −0.412003
\(757\) −28.8540 −1.04872 −0.524358 0.851498i \(-0.675695\pi\)
−0.524358 + 0.851498i \(0.675695\pi\)
\(758\) −13.7635 −0.499914
\(759\) −25.6776 −0.932039
\(760\) 0 0
\(761\) 2.16914 0.0786313 0.0393156 0.999227i \(-0.487482\pi\)
0.0393156 + 0.999227i \(0.487482\pi\)
\(762\) −36.4771 −1.32143
\(763\) 22.9226 0.829852
\(764\) 48.9155 1.76970
\(765\) 0 0
\(766\) 63.0287 2.27732
\(767\) −37.8410 −1.36636
\(768\) −12.0367 −0.434337
\(769\) −30.4385 −1.09764 −0.548821 0.835940i \(-0.684923\pi\)
−0.548821 + 0.835940i \(0.684923\pi\)
\(770\) 0 0
\(771\) 11.1976 0.403272
\(772\) 24.2819 0.873923
\(773\) 0.878615 0.0316016 0.0158008 0.999875i \(-0.494970\pi\)
0.0158008 + 0.999875i \(0.494970\pi\)
\(774\) −11.9459 −0.429388
\(775\) 0 0
\(776\) −29.2262 −1.04916
\(777\) 12.7450 0.457224
\(778\) 15.9934 0.573391
\(779\) −54.1397 −1.93976
\(780\) 0 0
\(781\) −65.1080 −2.32975
\(782\) 68.9041 2.46401
\(783\) −3.78248 −0.135175
\(784\) −3.70312 −0.132254
\(785\) 0 0
\(786\) 31.7534 1.13260
\(787\) −17.9038 −0.638202 −0.319101 0.947721i \(-0.603381\pi\)
−0.319101 + 0.947721i \(0.603381\pi\)
\(788\) 33.0573 1.17762
\(789\) −0.764798 −0.0272275
\(790\) 0 0
\(791\) 56.6878 2.01559
\(792\) −13.4100 −0.476505
\(793\) 17.6345 0.626221
\(794\) −28.9206 −1.02635
\(795\) 0 0
\(796\) −19.8560 −0.703776
\(797\) −2.03099 −0.0719413 −0.0359707 0.999353i \(-0.511452\pi\)
−0.0359707 + 0.999353i \(0.511452\pi\)
\(798\) 66.6290 2.35864
\(799\) −6.40426 −0.226566
\(800\) 0 0
\(801\) −7.50610 −0.265215
\(802\) 40.6523 1.43548
\(803\) −62.2820 −2.19788
\(804\) −43.7571 −1.54320
\(805\) 0 0
\(806\) 44.7614 1.57665
\(807\) 2.34969 0.0827130
\(808\) 24.3775 0.857598
\(809\) −3.21240 −0.112942 −0.0564710 0.998404i \(-0.517985\pi\)
−0.0564710 + 0.998404i \(0.517985\pi\)
\(810\) 0 0
\(811\) 2.93628 0.103107 0.0515533 0.998670i \(-0.483583\pi\)
0.0515533 + 0.998670i \(0.483583\pi\)
\(812\) 42.8488 1.50370
\(813\) −15.8548 −0.556053
\(814\) 42.4826 1.48901
\(815\) 0 0
\(816\) −3.73989 −0.130922
\(817\) 42.7162 1.49445
\(818\) 65.4778 2.28938
\(819\) 13.7599 0.480809
\(820\) 0 0
\(821\) −13.8856 −0.484611 −0.242305 0.970200i \(-0.577904\pi\)
−0.242305 + 0.970200i \(0.577904\pi\)
\(822\) 46.6394 1.62673
\(823\) 10.2213 0.356294 0.178147 0.984004i \(-0.442990\pi\)
0.178147 + 0.984004i \(0.442990\pi\)
\(824\) 19.6814 0.685634
\(825\) 0 0
\(826\) −82.8691 −2.88339
\(827\) −27.1692 −0.944765 −0.472383 0.881394i \(-0.656606\pi\)
−0.472383 + 0.881394i \(0.656606\pi\)
\(828\) 14.7738 0.513426
\(829\) −7.41476 −0.257525 −0.128763 0.991675i \(-0.541101\pi\)
−0.128763 + 0.991675i \(0.541101\pi\)
\(830\) 0 0
\(831\) −17.8932 −0.620707
\(832\) 49.1576 1.70423
\(833\) 40.6113 1.40710
\(834\) −8.70630 −0.301474
\(835\) 0 0
\(836\) 135.022 4.66984
\(837\) −5.26069 −0.181836
\(838\) −1.11691 −0.0385829
\(839\) 6.99598 0.241528 0.120764 0.992681i \(-0.461466\pi\)
0.120764 + 0.992681i \(0.461466\pi\)
\(840\) 0 0
\(841\) −14.6928 −0.506649
\(842\) −55.2946 −1.90558
\(843\) −21.7136 −0.747857
\(844\) 22.1331 0.761854
\(845\) 0 0
\(846\) −2.25864 −0.0776535
\(847\) −65.9542 −2.26621
\(848\) −5.61823 −0.192931
\(849\) −14.5961 −0.500937
\(850\) 0 0
\(851\) −16.6215 −0.569778
\(852\) 37.4604 1.28337
\(853\) 25.3316 0.867338 0.433669 0.901072i \(-0.357219\pi\)
0.433669 + 0.901072i \(0.357219\pi\)
\(854\) 38.6184 1.32149
\(855\) 0 0
\(856\) −39.8465 −1.36192
\(857\) −11.2384 −0.383898 −0.191949 0.981405i \(-0.561481\pi\)
−0.191949 + 0.981405i \(0.561481\pi\)
\(858\) 45.8655 1.56582
\(859\) −41.7711 −1.42521 −0.712605 0.701566i \(-0.752485\pi\)
−0.712605 + 0.701566i \(0.752485\pi\)
\(860\) 0 0
\(861\) −24.4848 −0.834441
\(862\) 46.4831 1.58322
\(863\) 30.0343 1.02238 0.511190 0.859468i \(-0.329205\pi\)
0.511190 + 0.859468i \(0.329205\pi\)
\(864\) −6.29446 −0.214142
\(865\) 0 0
\(866\) 24.5455 0.834090
\(867\) 24.0145 0.815577
\(868\) 59.5942 2.02276
\(869\) −8.88746 −0.301487
\(870\) 0 0
\(871\) 53.1498 1.80091
\(872\) −15.6124 −0.528702
\(873\) −11.7481 −0.397613
\(874\) −86.8949 −2.93926
\(875\) 0 0
\(876\) 35.8344 1.21073
\(877\) 8.46607 0.285879 0.142939 0.989731i \(-0.454345\pi\)
0.142939 + 0.989731i \(0.454345\pi\)
\(878\) −9.14991 −0.308794
\(879\) 6.51441 0.219726
\(880\) 0 0
\(881\) −12.4949 −0.420965 −0.210482 0.977598i \(-0.567503\pi\)
−0.210482 + 0.977598i \(0.567503\pi\)
\(882\) 14.3227 0.482270
\(883\) 8.80299 0.296244 0.148122 0.988969i \(-0.452677\pi\)
0.148122 + 0.988969i \(0.452677\pi\)
\(884\) −74.8250 −2.51664
\(885\) 0 0
\(886\) −44.7994 −1.50507
\(887\) 56.7788 1.90645 0.953223 0.302269i \(-0.0977441\pi\)
0.953223 + 0.302269i \(0.0977441\pi\)
\(888\) −8.68051 −0.291299
\(889\) 58.9893 1.97844
\(890\) 0 0
\(891\) −5.39045 −0.180587
\(892\) −16.7965 −0.562389
\(893\) 8.07640 0.270266
\(894\) −21.0426 −0.703771
\(895\) 0 0
\(896\) 61.6697 2.06024
\(897\) −17.9451 −0.599169
\(898\) −27.9817 −0.933763
\(899\) 19.8985 0.663651
\(900\) 0 0
\(901\) 61.6138 2.05265
\(902\) −81.6148 −2.71748
\(903\) 19.3185 0.642879
\(904\) −38.6096 −1.28414
\(905\) 0 0
\(906\) 24.6224 0.818024
\(907\) 39.1164 1.29884 0.649419 0.760430i \(-0.275012\pi\)
0.649419 + 0.760430i \(0.275012\pi\)
\(908\) −70.9645 −2.35504
\(909\) 9.79905 0.325014
\(910\) 0 0
\(911\) 20.2215 0.669969 0.334984 0.942224i \(-0.391269\pi\)
0.334984 + 0.942224i \(0.391269\pi\)
\(912\) 4.71637 0.156175
\(913\) 57.7422 1.91099
\(914\) 1.64181 0.0543061
\(915\) 0 0
\(916\) 80.3902 2.65617
\(917\) −51.3502 −1.69573
\(918\) 14.4649 0.477413
\(919\) −30.6670 −1.01161 −0.505806 0.862647i \(-0.668805\pi\)
−0.505806 + 0.862647i \(0.668805\pi\)
\(920\) 0 0
\(921\) −30.2614 −0.997148
\(922\) −13.0665 −0.430322
\(923\) −45.5014 −1.49770
\(924\) 61.0642 2.00886
\(925\) 0 0
\(926\) −52.5199 −1.72591
\(927\) 7.91135 0.259843
\(928\) 23.8087 0.781558
\(929\) 17.3908 0.570573 0.285286 0.958442i \(-0.407911\pi\)
0.285286 + 0.958442i \(0.407911\pi\)
\(930\) 0 0
\(931\) −51.2149 −1.67850
\(932\) 4.30162 0.140904
\(933\) −14.8690 −0.486790
\(934\) 44.0497 1.44135
\(935\) 0 0
\(936\) −9.37174 −0.306325
\(937\) 25.7353 0.840736 0.420368 0.907354i \(-0.361901\pi\)
0.420368 + 0.907354i \(0.361901\pi\)
\(938\) 116.394 3.80041
\(939\) −19.0799 −0.622651
\(940\) 0 0
\(941\) 5.20220 0.169587 0.0847934 0.996399i \(-0.472977\pi\)
0.0847934 + 0.996399i \(0.472977\pi\)
\(942\) −19.6172 −0.639164
\(943\) 31.9322 1.03985
\(944\) −5.86594 −0.190920
\(945\) 0 0
\(946\) 64.3940 2.09363
\(947\) −15.3966 −0.500323 −0.250162 0.968204i \(-0.580484\pi\)
−0.250162 + 0.968204i \(0.580484\pi\)
\(948\) 5.11347 0.166078
\(949\) −43.5264 −1.41293
\(950\) 0 0
\(951\) 11.9688 0.388115
\(952\) −58.1933 −1.88606
\(953\) 25.8939 0.838785 0.419393 0.907805i \(-0.362243\pi\)
0.419393 + 0.907805i \(0.362243\pi\)
\(954\) 21.7298 0.703528
\(955\) 0 0
\(956\) −64.6672 −2.09149
\(957\) 20.3893 0.659092
\(958\) 35.9961 1.16298
\(959\) −75.4233 −2.43554
\(960\) 0 0
\(961\) −3.32519 −0.107264
\(962\) 29.6894 0.957225
\(963\) −16.0171 −0.516145
\(964\) −59.7400 −1.92410
\(965\) 0 0
\(966\) −39.2984 −1.26441
\(967\) 59.1183 1.90112 0.950559 0.310545i \(-0.100512\pi\)
0.950559 + 0.310545i \(0.100512\pi\)
\(968\) 44.9209 1.44381
\(969\) −51.7234 −1.66159
\(970\) 0 0
\(971\) −6.55439 −0.210340 −0.105170 0.994454i \(-0.533539\pi\)
−0.105170 + 0.994454i \(0.533539\pi\)
\(972\) 3.10144 0.0994786
\(973\) 14.0795 0.451367
\(974\) −48.7818 −1.56307
\(975\) 0 0
\(976\) 2.73362 0.0875012
\(977\) −12.4405 −0.398007 −0.199004 0.979999i \(-0.563771\pi\)
−0.199004 + 0.979999i \(0.563771\pi\)
\(978\) 32.7453 1.04708
\(979\) 40.4612 1.29315
\(980\) 0 0
\(981\) −6.27573 −0.200369
\(982\) −44.2073 −1.41071
\(983\) −11.5633 −0.368812 −0.184406 0.982850i \(-0.559036\pi\)
−0.184406 + 0.982850i \(0.559036\pi\)
\(984\) 16.6764 0.531625
\(985\) 0 0
\(986\) −54.7132 −1.74242
\(987\) 3.65257 0.116263
\(988\) 94.3617 3.00205
\(989\) −25.1944 −0.801136
\(990\) 0 0
\(991\) −10.5006 −0.333562 −0.166781 0.985994i \(-0.553337\pi\)
−0.166781 + 0.985994i \(0.553337\pi\)
\(992\) 33.1132 1.05134
\(993\) −16.0901 −0.510603
\(994\) −99.6448 −3.16054
\(995\) 0 0
\(996\) −33.2224 −1.05269
\(997\) −31.0404 −0.983059 −0.491529 0.870861i \(-0.663562\pi\)
−0.491529 + 0.870861i \(0.663562\pi\)
\(998\) −68.4438 −2.16655
\(999\) −3.48931 −0.110397
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 3525.2.a.bd.1.8 8
5.4 even 2 3525.2.a.be.1.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3525.2.a.bd.1.8 8 1.1 even 1 trivial
3525.2.a.be.1.1 yes 8 5.4 even 2