Properties

Label 6025.2.a.f.1.2
Level $6025$
Weight $2$
Character 6025.1
Self dual yes
Analytic conductor $48.110$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6025 = 5^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6025.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.1098672178\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: 7.7.31056073.1
Defining polynomial: \(x^{7} - 3 x^{6} - 3 x^{5} + 11 x^{4} + x^{3} - 9 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 241)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.48734\) of defining polynomial
Character \(\chi\) \(=\) 6025.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.487343 q^{2} +0.815004 q^{3} -1.76250 q^{4} -0.397187 q^{6} +4.61392 q^{7} +1.83363 q^{8} -2.33577 q^{9} +O(q^{10})\) \(q-0.487343 q^{2} +0.815004 q^{3} -1.76250 q^{4} -0.397187 q^{6} +4.61392 q^{7} +1.83363 q^{8} -2.33577 q^{9} +1.93974 q^{11} -1.43644 q^{12} +3.85571 q^{13} -2.24856 q^{14} +2.63139 q^{16} -5.40289 q^{17} +1.13832 q^{18} -4.17145 q^{19} +3.76036 q^{21} -0.945321 q^{22} +1.42545 q^{23} +1.49441 q^{24} -1.87905 q^{26} -4.34867 q^{27} -8.13202 q^{28} -4.85744 q^{29} -7.24699 q^{31} -4.94964 q^{32} +1.58090 q^{33} +2.63306 q^{34} +4.11678 q^{36} -7.12597 q^{37} +2.03293 q^{38} +3.14242 q^{39} +9.18955 q^{41} -1.83259 q^{42} -2.93624 q^{43} -3.41879 q^{44} -0.694685 q^{46} -2.48170 q^{47} +2.14459 q^{48} +14.2883 q^{49} -4.40338 q^{51} -6.79568 q^{52} -5.64997 q^{53} +2.11930 q^{54} +8.46022 q^{56} -3.39975 q^{57} +2.36724 q^{58} -11.9783 q^{59} -13.9214 q^{61} +3.53177 q^{62} -10.7771 q^{63} -2.85060 q^{64} -0.770440 q^{66} +7.30682 q^{67} +9.52258 q^{68} +1.16175 q^{69} -14.8844 q^{71} -4.28293 q^{72} -0.240264 q^{73} +3.47279 q^{74} +7.35217 q^{76} +8.94983 q^{77} -1.53144 q^{78} -3.15128 q^{79} +3.46313 q^{81} -4.47847 q^{82} +2.46821 q^{83} -6.62763 q^{84} +1.43096 q^{86} -3.95883 q^{87} +3.55677 q^{88} +5.55181 q^{89} +17.7900 q^{91} -2.51235 q^{92} -5.90632 q^{93} +1.20944 q^{94} -4.03398 q^{96} -5.27964 q^{97} -6.96330 q^{98} -4.53079 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 4q^{2} + 3q^{3} + 2q^{4} - 5q^{6} + 7q^{7} + 6q^{8} - 2q^{9} + O(q^{10}) \) \( 7q + 4q^{2} + 3q^{3} + 2q^{4} - 5q^{6} + 7q^{7} + 6q^{8} - 2q^{9} - 18q^{11} - q^{12} + q^{13} - 6q^{14} + 4q^{16} + 2q^{17} - 8q^{18} - 6q^{19} - 2q^{21} - 10q^{22} + 22q^{23} - 3q^{24} + 8q^{26} - 3q^{27} - 9q^{28} - 16q^{29} - 18q^{31} + 6q^{32} - 4q^{33} + 11q^{34} - 7q^{36} - 8q^{37} - 16q^{38} - 9q^{39} - 15q^{41} - 19q^{42} - 14q^{43} - 4q^{44} + 11q^{46} + 10q^{47} - 31q^{48} + 6q^{49} + 13q^{51} - 27q^{52} - 15q^{53} + 16q^{54} + 13q^{56} - 14q^{57} - 17q^{58} - 18q^{59} + 4q^{61} - 13q^{62} + 16q^{63} + 2q^{64} + 16q^{66} - 18q^{67} + 15q^{68} + 26q^{69} - 50q^{71} - 30q^{72} + 10q^{74} - 20q^{76} - 17q^{77} + 32q^{78} - 15q^{79} - 9q^{81} - 45q^{82} + 24q^{83} + 6q^{84} - 23q^{86} - 12q^{87} - 8q^{88} - 13q^{89} - 12q^{91} + 10q^{92} - 14q^{93} - 32q^{94} - 15q^{96} - q^{97} - 9q^{98} - 20q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.487343 −0.344604 −0.172302 0.985044i \(-0.555120\pi\)
−0.172302 + 0.985044i \(0.555120\pi\)
\(3\) 0.815004 0.470543 0.235271 0.971930i \(-0.424402\pi\)
0.235271 + 0.971930i \(0.424402\pi\)
\(4\) −1.76250 −0.881248
\(5\) 0 0
\(6\) −0.397187 −0.162151
\(7\) 4.61392 1.74390 0.871950 0.489596i \(-0.162856\pi\)
0.871950 + 0.489596i \(0.162856\pi\)
\(8\) 1.83363 0.648285
\(9\) −2.33577 −0.778590
\(10\) 0 0
\(11\) 1.93974 0.584855 0.292427 0.956288i \(-0.405537\pi\)
0.292427 + 0.956288i \(0.405537\pi\)
\(12\) −1.43644 −0.414665
\(13\) 3.85571 1.06938 0.534691 0.845048i \(-0.320428\pi\)
0.534691 + 0.845048i \(0.320428\pi\)
\(14\) −2.24856 −0.600954
\(15\) 0 0
\(16\) 2.63139 0.657847
\(17\) −5.40289 −1.31039 −0.655197 0.755458i \(-0.727414\pi\)
−0.655197 + 0.755458i \(0.727414\pi\)
\(18\) 1.13832 0.268305
\(19\) −4.17145 −0.956997 −0.478498 0.878088i \(-0.658819\pi\)
−0.478498 + 0.878088i \(0.658819\pi\)
\(20\) 0 0
\(21\) 3.76036 0.820579
\(22\) −0.945321 −0.201543
\(23\) 1.42545 0.297227 0.148614 0.988895i \(-0.452519\pi\)
0.148614 + 0.988895i \(0.452519\pi\)
\(24\) 1.49441 0.305046
\(25\) 0 0
\(26\) −1.87905 −0.368513
\(27\) −4.34867 −0.836902
\(28\) −8.13202 −1.53681
\(29\) −4.85744 −0.902003 −0.451002 0.892523i \(-0.648933\pi\)
−0.451002 + 0.892523i \(0.648933\pi\)
\(30\) 0 0
\(31\) −7.24699 −1.30160 −0.650799 0.759250i \(-0.725566\pi\)
−0.650799 + 0.759250i \(0.725566\pi\)
\(32\) −4.94964 −0.874982
\(33\) 1.58090 0.275199
\(34\) 2.63306 0.451567
\(35\) 0 0
\(36\) 4.11678 0.686131
\(37\) −7.12597 −1.17150 −0.585751 0.810491i \(-0.699200\pi\)
−0.585751 + 0.810491i \(0.699200\pi\)
\(38\) 2.03293 0.329785
\(39\) 3.14242 0.503190
\(40\) 0 0
\(41\) 9.18955 1.43517 0.717583 0.696473i \(-0.245248\pi\)
0.717583 + 0.696473i \(0.245248\pi\)
\(42\) −1.83259 −0.282775
\(43\) −2.93624 −0.447772 −0.223886 0.974615i \(-0.571874\pi\)
−0.223886 + 0.974615i \(0.571874\pi\)
\(44\) −3.41879 −0.515402
\(45\) 0 0
\(46\) −0.694685 −0.102426
\(47\) −2.48170 −0.361994 −0.180997 0.983484i \(-0.557932\pi\)
−0.180997 + 0.983484i \(0.557932\pi\)
\(48\) 2.14459 0.309545
\(49\) 14.2883 2.04118
\(50\) 0 0
\(51\) −4.40338 −0.616596
\(52\) −6.79568 −0.942391
\(53\) −5.64997 −0.776083 −0.388041 0.921642i \(-0.626848\pi\)
−0.388041 + 0.921642i \(0.626848\pi\)
\(54\) 2.11930 0.288400
\(55\) 0 0
\(56\) 8.46022 1.13054
\(57\) −3.39975 −0.450308
\(58\) 2.36724 0.310834
\(59\) −11.9783 −1.55944 −0.779718 0.626131i \(-0.784638\pi\)
−0.779718 + 0.626131i \(0.784638\pi\)
\(60\) 0 0
\(61\) −13.9214 −1.78245 −0.891223 0.453565i \(-0.850152\pi\)
−0.891223 + 0.453565i \(0.850152\pi\)
\(62\) 3.53177 0.448535
\(63\) −10.7771 −1.35778
\(64\) −2.85060 −0.356325
\(65\) 0 0
\(66\) −0.770440 −0.0948346
\(67\) 7.30682 0.892670 0.446335 0.894866i \(-0.352729\pi\)
0.446335 + 0.894866i \(0.352729\pi\)
\(68\) 9.52258 1.15478
\(69\) 1.16175 0.139858
\(70\) 0 0
\(71\) −14.8844 −1.76645 −0.883226 0.468948i \(-0.844633\pi\)
−0.883226 + 0.468948i \(0.844633\pi\)
\(72\) −4.28293 −0.504748
\(73\) −0.240264 −0.0281208 −0.0140604 0.999901i \(-0.504476\pi\)
−0.0140604 + 0.999901i \(0.504476\pi\)
\(74\) 3.47279 0.403704
\(75\) 0 0
\(76\) 7.35217 0.843352
\(77\) 8.94983 1.01993
\(78\) −1.53144 −0.173401
\(79\) −3.15128 −0.354546 −0.177273 0.984162i \(-0.556728\pi\)
−0.177273 + 0.984162i \(0.556728\pi\)
\(80\) 0 0
\(81\) 3.46313 0.384792
\(82\) −4.47847 −0.494564
\(83\) 2.46821 0.270922 0.135461 0.990783i \(-0.456749\pi\)
0.135461 + 0.990783i \(0.456749\pi\)
\(84\) −6.62763 −0.723134
\(85\) 0 0
\(86\) 1.43096 0.154304
\(87\) −3.95883 −0.424431
\(88\) 3.55677 0.379153
\(89\) 5.55181 0.588491 0.294246 0.955730i \(-0.404932\pi\)
0.294246 + 0.955730i \(0.404932\pi\)
\(90\) 0 0
\(91\) 17.7900 1.86489
\(92\) −2.51235 −0.261931
\(93\) −5.90632 −0.612457
\(94\) 1.20944 0.124744
\(95\) 0 0
\(96\) −4.03398 −0.411716
\(97\) −5.27964 −0.536067 −0.268033 0.963410i \(-0.586374\pi\)
−0.268033 + 0.963410i \(0.586374\pi\)
\(98\) −6.96330 −0.703400
\(99\) −4.53079 −0.455362
\(100\) 0 0
\(101\) −12.4239 −1.23623 −0.618113 0.786089i \(-0.712103\pi\)
−0.618113 + 0.786089i \(0.712103\pi\)
\(102\) 2.14596 0.212481
\(103\) 9.80533 0.966148 0.483074 0.875580i \(-0.339520\pi\)
0.483074 + 0.875580i \(0.339520\pi\)
\(104\) 7.06994 0.693264
\(105\) 0 0
\(106\) 2.75347 0.267441
\(107\) 9.15031 0.884593 0.442297 0.896869i \(-0.354164\pi\)
0.442297 + 0.896869i \(0.354164\pi\)
\(108\) 7.66452 0.737519
\(109\) 3.24534 0.310847 0.155424 0.987848i \(-0.450326\pi\)
0.155424 + 0.987848i \(0.450326\pi\)
\(110\) 0 0
\(111\) −5.80769 −0.551241
\(112\) 12.1410 1.14722
\(113\) 0.842874 0.0792909 0.0396455 0.999214i \(-0.487377\pi\)
0.0396455 + 0.999214i \(0.487377\pi\)
\(114\) 1.65684 0.155178
\(115\) 0 0
\(116\) 8.56121 0.794889
\(117\) −9.00605 −0.832610
\(118\) 5.83752 0.537387
\(119\) −24.9285 −2.28520
\(120\) 0 0
\(121\) −7.23740 −0.657945
\(122\) 6.78448 0.614238
\(123\) 7.48952 0.675307
\(124\) 12.7728 1.14703
\(125\) 0 0
\(126\) 5.25213 0.467897
\(127\) −11.3416 −1.00641 −0.503203 0.864168i \(-0.667845\pi\)
−0.503203 + 0.864168i \(0.667845\pi\)
\(128\) 11.2885 0.997773
\(129\) −2.39305 −0.210696
\(130\) 0 0
\(131\) 14.1261 1.23421 0.617103 0.786882i \(-0.288306\pi\)
0.617103 + 0.786882i \(0.288306\pi\)
\(132\) −2.78633 −0.242519
\(133\) −19.2468 −1.66891
\(134\) −3.56093 −0.307617
\(135\) 0 0
\(136\) −9.90689 −0.849509
\(137\) 12.9555 1.10686 0.553430 0.832896i \(-0.313319\pi\)
0.553430 + 0.832896i \(0.313319\pi\)
\(138\) −0.566170 −0.0481956
\(139\) 8.23606 0.698574 0.349287 0.937016i \(-0.386424\pi\)
0.349287 + 0.937016i \(0.386424\pi\)
\(140\) 0 0
\(141\) −2.02260 −0.170333
\(142\) 7.25380 0.608726
\(143\) 7.47909 0.625433
\(144\) −6.14631 −0.512193
\(145\) 0 0
\(146\) 0.117091 0.00969054
\(147\) 11.6450 0.960464
\(148\) 12.5595 1.03238
\(149\) 4.48606 0.367513 0.183756 0.982972i \(-0.441174\pi\)
0.183756 + 0.982972i \(0.441174\pi\)
\(150\) 0 0
\(151\) 0.0933624 0.00759773 0.00379886 0.999993i \(-0.498791\pi\)
0.00379886 + 0.999993i \(0.498791\pi\)
\(152\) −7.64889 −0.620407
\(153\) 12.6199 1.02026
\(154\) −4.36164 −0.351471
\(155\) 0 0
\(156\) −5.53850 −0.443435
\(157\) 10.1337 0.808760 0.404380 0.914591i \(-0.367487\pi\)
0.404380 + 0.914591i \(0.367487\pi\)
\(158\) 1.53575 0.122178
\(159\) −4.60474 −0.365180
\(160\) 0 0
\(161\) 6.57693 0.518334
\(162\) −1.68773 −0.132601
\(163\) −7.49804 −0.587292 −0.293646 0.955914i \(-0.594869\pi\)
−0.293646 + 0.955914i \(0.594869\pi\)
\(164\) −16.1965 −1.26474
\(165\) 0 0
\(166\) −1.20287 −0.0933606
\(167\) −17.7922 −1.37680 −0.688399 0.725332i \(-0.741686\pi\)
−0.688399 + 0.725332i \(0.741686\pi\)
\(168\) 6.89511 0.531969
\(169\) 1.86650 0.143577
\(170\) 0 0
\(171\) 9.74355 0.745108
\(172\) 5.17511 0.394599
\(173\) −18.4927 −1.40597 −0.702987 0.711203i \(-0.748151\pi\)
−0.702987 + 0.711203i \(0.748151\pi\)
\(174\) 1.92931 0.146260
\(175\) 0 0
\(176\) 5.10421 0.384745
\(177\) −9.76232 −0.733781
\(178\) −2.70564 −0.202796
\(179\) −17.3387 −1.29595 −0.647977 0.761660i \(-0.724385\pi\)
−0.647977 + 0.761660i \(0.724385\pi\)
\(180\) 0 0
\(181\) 17.7124 1.31655 0.658275 0.752778i \(-0.271287\pi\)
0.658275 + 0.752778i \(0.271287\pi\)
\(182\) −8.66981 −0.642649
\(183\) −11.3460 −0.838717
\(184\) 2.61375 0.192688
\(185\) 0 0
\(186\) 2.87841 0.211055
\(187\) −10.4802 −0.766390
\(188\) 4.37400 0.319006
\(189\) −20.0644 −1.45947
\(190\) 0 0
\(191\) 13.2440 0.958303 0.479151 0.877732i \(-0.340944\pi\)
0.479151 + 0.877732i \(0.340944\pi\)
\(192\) −2.32325 −0.167666
\(193\) 25.5030 1.83574 0.917872 0.396877i \(-0.129906\pi\)
0.917872 + 0.396877i \(0.129906\pi\)
\(194\) 2.57300 0.184731
\(195\) 0 0
\(196\) −25.1831 −1.79879
\(197\) 15.0303 1.07087 0.535433 0.844578i \(-0.320149\pi\)
0.535433 + 0.844578i \(0.320149\pi\)
\(198\) 2.20805 0.156919
\(199\) −19.3801 −1.37382 −0.686911 0.726742i \(-0.741034\pi\)
−0.686911 + 0.726742i \(0.741034\pi\)
\(200\) 0 0
\(201\) 5.95508 0.420039
\(202\) 6.05472 0.426008
\(203\) −22.4118 −1.57300
\(204\) 7.76094 0.543374
\(205\) 0 0
\(206\) −4.77856 −0.332938
\(207\) −3.32953 −0.231418
\(208\) 10.1459 0.703489
\(209\) −8.09154 −0.559704
\(210\) 0 0
\(211\) 27.9383 1.92335 0.961675 0.274192i \(-0.0884105\pi\)
0.961675 + 0.274192i \(0.0884105\pi\)
\(212\) 9.95805 0.683922
\(213\) −12.1308 −0.831191
\(214\) −4.45934 −0.304834
\(215\) 0 0
\(216\) −7.97384 −0.542551
\(217\) −33.4370 −2.26985
\(218\) −1.58160 −0.107119
\(219\) −0.195816 −0.0132320
\(220\) 0 0
\(221\) −20.8320 −1.40131
\(222\) 2.83034 0.189960
\(223\) 0.387500 0.0259489 0.0129745 0.999916i \(-0.495870\pi\)
0.0129745 + 0.999916i \(0.495870\pi\)
\(224\) −22.8373 −1.52588
\(225\) 0 0
\(226\) −0.410769 −0.0273240
\(227\) −10.2778 −0.682161 −0.341081 0.940034i \(-0.610793\pi\)
−0.341081 + 0.940034i \(0.610793\pi\)
\(228\) 5.99204 0.396833
\(229\) −10.5346 −0.696147 −0.348074 0.937467i \(-0.613164\pi\)
−0.348074 + 0.937467i \(0.613164\pi\)
\(230\) 0 0
\(231\) 7.29414 0.479919
\(232\) −8.90673 −0.584755
\(233\) −25.6725 −1.68186 −0.840932 0.541141i \(-0.817993\pi\)
−0.840932 + 0.541141i \(0.817993\pi\)
\(234\) 4.38904 0.286920
\(235\) 0 0
\(236\) 21.1116 1.37425
\(237\) −2.56830 −0.166829
\(238\) 12.1488 0.787487
\(239\) −25.1312 −1.62560 −0.812801 0.582542i \(-0.802058\pi\)
−0.812801 + 0.582542i \(0.802058\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157
\(242\) 3.52710 0.226730
\(243\) 15.8685 1.01796
\(244\) 24.5363 1.57078
\(245\) 0 0
\(246\) −3.64997 −0.232713
\(247\) −16.0839 −1.02339
\(248\) −13.2883 −0.843806
\(249\) 2.01160 0.127480
\(250\) 0 0
\(251\) 2.52238 0.159211 0.0796055 0.996826i \(-0.474634\pi\)
0.0796055 + 0.996826i \(0.474634\pi\)
\(252\) 18.9945 1.19654
\(253\) 2.76501 0.173835
\(254\) 5.52726 0.346811
\(255\) 0 0
\(256\) 0.199817 0.0124886
\(257\) −3.92348 −0.244740 −0.122370 0.992485i \(-0.539049\pi\)
−0.122370 + 0.992485i \(0.539049\pi\)
\(258\) 1.16623 0.0726066
\(259\) −32.8787 −2.04298
\(260\) 0 0
\(261\) 11.3458 0.702290
\(262\) −6.88428 −0.425312
\(263\) −6.24984 −0.385382 −0.192691 0.981260i \(-0.561721\pi\)
−0.192691 + 0.981260i \(0.561721\pi\)
\(264\) 2.89878 0.178407
\(265\) 0 0
\(266\) 9.37978 0.575111
\(267\) 4.52475 0.276910
\(268\) −12.8782 −0.786664
\(269\) 20.4943 1.24956 0.624780 0.780801i \(-0.285189\pi\)
0.624780 + 0.780801i \(0.285189\pi\)
\(270\) 0 0
\(271\) 15.3179 0.930499 0.465249 0.885180i \(-0.345965\pi\)
0.465249 + 0.885180i \(0.345965\pi\)
\(272\) −14.2171 −0.862039
\(273\) 14.4989 0.877512
\(274\) −6.31376 −0.381428
\(275\) 0 0
\(276\) −2.04758 −0.123250
\(277\) −3.26388 −0.196108 −0.0980539 0.995181i \(-0.531262\pi\)
−0.0980539 + 0.995181i \(0.531262\pi\)
\(278\) −4.01379 −0.240731
\(279\) 16.9273 1.01341
\(280\) 0 0
\(281\) −12.4791 −0.744442 −0.372221 0.928144i \(-0.621404\pi\)
−0.372221 + 0.928144i \(0.621404\pi\)
\(282\) 0.985700 0.0586976
\(283\) 15.8115 0.939899 0.469949 0.882693i \(-0.344272\pi\)
0.469949 + 0.882693i \(0.344272\pi\)
\(284\) 26.2337 1.55668
\(285\) 0 0
\(286\) −3.64488 −0.215526
\(287\) 42.3999 2.50279
\(288\) 11.5612 0.681252
\(289\) 12.1913 0.717133
\(290\) 0 0
\(291\) −4.30293 −0.252242
\(292\) 0.423465 0.0247814
\(293\) −18.7147 −1.09333 −0.546663 0.837353i \(-0.684102\pi\)
−0.546663 + 0.837353i \(0.684102\pi\)
\(294\) −5.67512 −0.330979
\(295\) 0 0
\(296\) −13.0664 −0.759467
\(297\) −8.43530 −0.489466
\(298\) −2.18625 −0.126646
\(299\) 5.49613 0.317849
\(300\) 0 0
\(301\) −13.5476 −0.780870
\(302\) −0.0454995 −0.00261820
\(303\) −10.1255 −0.581697
\(304\) −10.9767 −0.629557
\(305\) 0 0
\(306\) −6.15023 −0.351585
\(307\) −5.23477 −0.298764 −0.149382 0.988780i \(-0.547728\pi\)
−0.149382 + 0.988780i \(0.547728\pi\)
\(308\) −15.7740 −0.898809
\(309\) 7.99138 0.454614
\(310\) 0 0
\(311\) −26.6878 −1.51332 −0.756662 0.653806i \(-0.773171\pi\)
−0.756662 + 0.653806i \(0.773171\pi\)
\(312\) 5.76202 0.326210
\(313\) −6.24141 −0.352786 −0.176393 0.984320i \(-0.556443\pi\)
−0.176393 + 0.984320i \(0.556443\pi\)
\(314\) −4.93861 −0.278702
\(315\) 0 0
\(316\) 5.55412 0.312443
\(317\) −6.89526 −0.387276 −0.193638 0.981073i \(-0.562029\pi\)
−0.193638 + 0.981073i \(0.562029\pi\)
\(318\) 2.24409 0.125842
\(319\) −9.42218 −0.527541
\(320\) 0 0
\(321\) 7.45753 0.416239
\(322\) −3.20522 −0.178620
\(323\) 22.5379 1.25404
\(324\) −6.10375 −0.339097
\(325\) 0 0
\(326\) 3.65412 0.202383
\(327\) 2.64497 0.146267
\(328\) 16.8502 0.930397
\(329\) −11.4504 −0.631281
\(330\) 0 0
\(331\) −13.5697 −0.745860 −0.372930 0.927859i \(-0.621647\pi\)
−0.372930 + 0.927859i \(0.621647\pi\)
\(332\) −4.35022 −0.238749
\(333\) 16.6446 0.912119
\(334\) 8.67089 0.474450
\(335\) 0 0
\(336\) 9.89497 0.539815
\(337\) 20.2288 1.10193 0.550966 0.834528i \(-0.314260\pi\)
0.550966 + 0.834528i \(0.314260\pi\)
\(338\) −0.909627 −0.0494772
\(339\) 0.686946 0.0373098
\(340\) 0 0
\(341\) −14.0573 −0.761245
\(342\) −4.74845 −0.256767
\(343\) 33.6276 1.81572
\(344\) −5.38397 −0.290284
\(345\) 0 0
\(346\) 9.01229 0.484504
\(347\) −1.52458 −0.0818437 −0.0409219 0.999162i \(-0.513029\pi\)
−0.0409219 + 0.999162i \(0.513029\pi\)
\(348\) 6.97742 0.374029
\(349\) 16.5585 0.886358 0.443179 0.896433i \(-0.353851\pi\)
0.443179 + 0.896433i \(0.353851\pi\)
\(350\) 0 0
\(351\) −16.7672 −0.894968
\(352\) −9.60104 −0.511737
\(353\) −34.6802 −1.84584 −0.922920 0.384992i \(-0.874204\pi\)
−0.922920 + 0.384992i \(0.874204\pi\)
\(354\) 4.75760 0.252864
\(355\) 0 0
\(356\) −9.78505 −0.518607
\(357\) −20.3168 −1.07528
\(358\) 8.44989 0.446590
\(359\) 18.1341 0.957084 0.478542 0.878065i \(-0.341165\pi\)
0.478542 + 0.878065i \(0.341165\pi\)
\(360\) 0 0
\(361\) −1.59899 −0.0841573
\(362\) −8.63200 −0.453688
\(363\) −5.89850 −0.309591
\(364\) −31.3547 −1.64343
\(365\) 0 0
\(366\) 5.52937 0.289025
\(367\) 10.9879 0.573566 0.286783 0.957996i \(-0.407414\pi\)
0.286783 + 0.957996i \(0.407414\pi\)
\(368\) 3.75092 0.195530
\(369\) −21.4647 −1.11741
\(370\) 0 0
\(371\) −26.0685 −1.35341
\(372\) 10.4099 0.539727
\(373\) −5.23646 −0.271134 −0.135567 0.990768i \(-0.543286\pi\)
−0.135567 + 0.990768i \(0.543286\pi\)
\(374\) 5.10747 0.264101
\(375\) 0 0
\(376\) −4.55052 −0.234675
\(377\) −18.7289 −0.964586
\(378\) 9.77827 0.502940
\(379\) 8.07101 0.414580 0.207290 0.978280i \(-0.433536\pi\)
0.207290 + 0.978280i \(0.433536\pi\)
\(380\) 0 0
\(381\) −9.24346 −0.473557
\(382\) −6.45438 −0.330235
\(383\) −2.46503 −0.125957 −0.0629785 0.998015i \(-0.520060\pi\)
−0.0629785 + 0.998015i \(0.520060\pi\)
\(384\) 9.20017 0.469494
\(385\) 0 0
\(386\) −12.4287 −0.632604
\(387\) 6.85838 0.348631
\(388\) 9.30535 0.472408
\(389\) −28.7159 −1.45595 −0.727977 0.685601i \(-0.759539\pi\)
−0.727977 + 0.685601i \(0.759539\pi\)
\(390\) 0 0
\(391\) −7.70157 −0.389485
\(392\) 26.1994 1.32327
\(393\) 11.5128 0.580746
\(394\) −7.32492 −0.369024
\(395\) 0 0
\(396\) 7.98550 0.401287
\(397\) −12.0605 −0.605301 −0.302651 0.953102i \(-0.597871\pi\)
−0.302651 + 0.953102i \(0.597871\pi\)
\(398\) 9.44478 0.473424
\(399\) −15.6862 −0.785291
\(400\) 0 0
\(401\) −31.7082 −1.58343 −0.791716 0.610890i \(-0.790812\pi\)
−0.791716 + 0.610890i \(0.790812\pi\)
\(402\) −2.90217 −0.144747
\(403\) −27.9423 −1.39190
\(404\) 21.8971 1.08942
\(405\) 0 0
\(406\) 10.9223 0.542063
\(407\) −13.8225 −0.685158
\(408\) −8.07415 −0.399730
\(409\) −14.1893 −0.701617 −0.350809 0.936447i \(-0.614093\pi\)
−0.350809 + 0.936447i \(0.614093\pi\)
\(410\) 0 0
\(411\) 10.5587 0.520825
\(412\) −17.2819 −0.851416
\(413\) −55.2668 −2.71950
\(414\) 1.62262 0.0797476
\(415\) 0 0
\(416\) −19.0844 −0.935689
\(417\) 6.71242 0.328709
\(418\) 3.94336 0.192876
\(419\) 31.1014 1.51941 0.759703 0.650271i \(-0.225345\pi\)
0.759703 + 0.650271i \(0.225345\pi\)
\(420\) 0 0
\(421\) −27.4415 −1.33742 −0.668709 0.743524i \(-0.733153\pi\)
−0.668709 + 0.743524i \(0.733153\pi\)
\(422\) −13.6155 −0.662794
\(423\) 5.79669 0.281845
\(424\) −10.3599 −0.503123
\(425\) 0 0
\(426\) 5.91188 0.286431
\(427\) −64.2321 −3.10841
\(428\) −16.1274 −0.779546
\(429\) 6.09548 0.294293
\(430\) 0 0
\(431\) −14.1005 −0.679199 −0.339599 0.940570i \(-0.610291\pi\)
−0.339599 + 0.940570i \(0.610291\pi\)
\(432\) −11.4430 −0.550553
\(433\) 17.7403 0.852546 0.426273 0.904595i \(-0.359826\pi\)
0.426273 + 0.904595i \(0.359826\pi\)
\(434\) 16.2953 0.782200
\(435\) 0 0
\(436\) −5.71991 −0.273934
\(437\) −5.94621 −0.284446
\(438\) 0.0954297 0.00455981
\(439\) −2.52447 −0.120486 −0.0602431 0.998184i \(-0.519188\pi\)
−0.0602431 + 0.998184i \(0.519188\pi\)
\(440\) 0 0
\(441\) −33.3741 −1.58924
\(442\) 10.1523 0.482897
\(443\) 16.8382 0.800008 0.400004 0.916513i \(-0.369009\pi\)
0.400004 + 0.916513i \(0.369009\pi\)
\(444\) 10.2360 0.485780
\(445\) 0 0
\(446\) −0.188846 −0.00894211
\(447\) 3.65616 0.172930
\(448\) −13.1524 −0.621394
\(449\) −30.5688 −1.44263 −0.721314 0.692608i \(-0.756462\pi\)
−0.721314 + 0.692608i \(0.756462\pi\)
\(450\) 0 0
\(451\) 17.8254 0.839364
\(452\) −1.48556 −0.0698750
\(453\) 0.0760907 0.00357505
\(454\) 5.00882 0.235075
\(455\) 0 0
\(456\) −6.23387 −0.291928
\(457\) 32.7043 1.52984 0.764922 0.644123i \(-0.222777\pi\)
0.764922 + 0.644123i \(0.222777\pi\)
\(458\) 5.13398 0.239895
\(459\) 23.4954 1.09667
\(460\) 0 0
\(461\) 12.3932 0.577210 0.288605 0.957448i \(-0.406809\pi\)
0.288605 + 0.957448i \(0.406809\pi\)
\(462\) −3.55475 −0.165382
\(463\) −12.1379 −0.564094 −0.282047 0.959401i \(-0.591013\pi\)
−0.282047 + 0.959401i \(0.591013\pi\)
\(464\) −12.7818 −0.593380
\(465\) 0 0
\(466\) 12.5113 0.579577
\(467\) −5.38203 −0.249051 −0.124525 0.992216i \(-0.539741\pi\)
−0.124525 + 0.992216i \(0.539741\pi\)
\(468\) 15.8731 0.733736
\(469\) 33.7131 1.55673
\(470\) 0 0
\(471\) 8.25903 0.380556
\(472\) −21.9637 −1.01096
\(473\) −5.69555 −0.261882
\(474\) 1.25165 0.0574900
\(475\) 0 0
\(476\) 43.9365 2.01382
\(477\) 13.1970 0.604250
\(478\) 12.2475 0.560188
\(479\) −25.3752 −1.15942 −0.579711 0.814822i \(-0.696834\pi\)
−0.579711 + 0.814822i \(0.696834\pi\)
\(480\) 0 0
\(481\) −27.4757 −1.25278
\(482\) 0.487343 0.0221979
\(483\) 5.36022 0.243898
\(484\) 12.7559 0.579813
\(485\) 0 0
\(486\) −7.73339 −0.350794
\(487\) 10.4243 0.472369 0.236185 0.971708i \(-0.424103\pi\)
0.236185 + 0.971708i \(0.424103\pi\)
\(488\) −25.5266 −1.15553
\(489\) −6.11093 −0.276346
\(490\) 0 0
\(491\) 34.2858 1.54730 0.773649 0.633614i \(-0.218429\pi\)
0.773649 + 0.633614i \(0.218429\pi\)
\(492\) −13.2002 −0.595113
\(493\) 26.2442 1.18198
\(494\) 7.83839 0.352666
\(495\) 0 0
\(496\) −19.0696 −0.856252
\(497\) −68.6754 −3.08051
\(498\) −0.980341 −0.0439301
\(499\) 15.9672 0.714792 0.357396 0.933953i \(-0.383665\pi\)
0.357396 + 0.933953i \(0.383665\pi\)
\(500\) 0 0
\(501\) −14.5007 −0.647842
\(502\) −1.22926 −0.0548647
\(503\) 4.53174 0.202060 0.101030 0.994883i \(-0.467786\pi\)
0.101030 + 0.994883i \(0.467786\pi\)
\(504\) −19.7611 −0.880230
\(505\) 0 0
\(506\) −1.34751 −0.0599041
\(507\) 1.52121 0.0675591
\(508\) 19.9896 0.886893
\(509\) −4.08798 −0.181197 −0.0905983 0.995888i \(-0.528878\pi\)
−0.0905983 + 0.995888i \(0.528878\pi\)
\(510\) 0 0
\(511\) −1.10856 −0.0490398
\(512\) −22.6744 −1.00208
\(513\) 18.1403 0.800913
\(514\) 1.91208 0.0843384
\(515\) 0 0
\(516\) 4.21774 0.185675
\(517\) −4.81387 −0.211714
\(518\) 16.0232 0.704019
\(519\) −15.0716 −0.661571
\(520\) 0 0
\(521\) 34.2884 1.50220 0.751102 0.660186i \(-0.229523\pi\)
0.751102 + 0.660186i \(0.229523\pi\)
\(522\) −5.52932 −0.242012
\(523\) −33.7724 −1.47676 −0.738382 0.674382i \(-0.764410\pi\)
−0.738382 + 0.674382i \(0.764410\pi\)
\(524\) −24.8973 −1.08764
\(525\) 0 0
\(526\) 3.04582 0.132804
\(527\) 39.1547 1.70561
\(528\) 4.15995 0.181039
\(529\) −20.9681 −0.911656
\(530\) 0 0
\(531\) 27.9784 1.21416
\(532\) 33.9223 1.47072
\(533\) 35.4322 1.53474
\(534\) −2.20511 −0.0954243
\(535\) 0 0
\(536\) 13.3980 0.578705
\(537\) −14.1311 −0.609801
\(538\) −9.98777 −0.430603
\(539\) 27.7156 1.19380
\(540\) 0 0
\(541\) −31.0330 −1.33421 −0.667106 0.744963i \(-0.732467\pi\)
−0.667106 + 0.744963i \(0.732467\pi\)
\(542\) −7.46510 −0.320653
\(543\) 14.4356 0.619492
\(544\) 26.7424 1.14657
\(545\) 0 0
\(546\) −7.06593 −0.302394
\(547\) 33.0104 1.41142 0.705712 0.708499i \(-0.250627\pi\)
0.705712 + 0.708499i \(0.250627\pi\)
\(548\) −22.8340 −0.975418
\(549\) 32.5171 1.38779
\(550\) 0 0
\(551\) 20.2626 0.863214
\(552\) 2.13021 0.0906679
\(553\) −14.5398 −0.618293
\(554\) 1.59063 0.0675795
\(555\) 0 0
\(556\) −14.5160 −0.615617
\(557\) −15.4826 −0.656019 −0.328010 0.944674i \(-0.606378\pi\)
−0.328010 + 0.944674i \(0.606378\pi\)
\(558\) −8.24940 −0.349225
\(559\) −11.3213 −0.478840
\(560\) 0 0
\(561\) −8.54142 −0.360619
\(562\) 6.08161 0.256537
\(563\) −6.21072 −0.261751 −0.130875 0.991399i \(-0.541779\pi\)
−0.130875 + 0.991399i \(0.541779\pi\)
\(564\) 3.56482 0.150106
\(565\) 0 0
\(566\) −7.70565 −0.323893
\(567\) 15.9786 0.671038
\(568\) −27.2924 −1.14516
\(569\) −20.3014 −0.851078 −0.425539 0.904940i \(-0.639916\pi\)
−0.425539 + 0.904940i \(0.639916\pi\)
\(570\) 0 0
\(571\) −38.5209 −1.61205 −0.806024 0.591882i \(-0.798385\pi\)
−0.806024 + 0.591882i \(0.798385\pi\)
\(572\) −13.1819 −0.551161
\(573\) 10.7939 0.450922
\(574\) −20.6633 −0.862469
\(575\) 0 0
\(576\) 6.65834 0.277431
\(577\) −19.3481 −0.805472 −0.402736 0.915316i \(-0.631941\pi\)
−0.402736 + 0.915316i \(0.631941\pi\)
\(578\) −5.94133 −0.247127
\(579\) 20.7850 0.863795
\(580\) 0 0
\(581\) 11.3881 0.472460
\(582\) 2.09700 0.0869236
\(583\) −10.9595 −0.453896
\(584\) −0.440555 −0.0182303
\(585\) 0 0
\(586\) 9.12050 0.376764
\(587\) 20.0183 0.826243 0.413122 0.910676i \(-0.364438\pi\)
0.413122 + 0.910676i \(0.364438\pi\)
\(588\) −20.5243 −0.846407
\(589\) 30.2305 1.24562
\(590\) 0 0
\(591\) 12.2498 0.503888
\(592\) −18.7512 −0.770668
\(593\) −12.5904 −0.517027 −0.258513 0.966008i \(-0.583233\pi\)
−0.258513 + 0.966008i \(0.583233\pi\)
\(594\) 4.11089 0.168672
\(595\) 0 0
\(596\) −7.90667 −0.323870
\(597\) −15.7949 −0.646441
\(598\) −2.67850 −0.109532
\(599\) 24.4282 0.998107 0.499054 0.866571i \(-0.333681\pi\)
0.499054 + 0.866571i \(0.333681\pi\)
\(600\) 0 0
\(601\) 9.02256 0.368038 0.184019 0.982923i \(-0.441089\pi\)
0.184019 + 0.982923i \(0.441089\pi\)
\(602\) 6.60233 0.269091
\(603\) −17.0670 −0.695024
\(604\) −0.164551 −0.00669548
\(605\) 0 0
\(606\) 4.93462 0.200455
\(607\) 19.0044 0.771364 0.385682 0.922632i \(-0.373966\pi\)
0.385682 + 0.922632i \(0.373966\pi\)
\(608\) 20.6472 0.837355
\(609\) −18.2657 −0.740165
\(610\) 0 0
\(611\) −9.56873 −0.387110
\(612\) −22.2426 −0.899102
\(613\) −3.38617 −0.136766 −0.0683831 0.997659i \(-0.521784\pi\)
−0.0683831 + 0.997659i \(0.521784\pi\)
\(614\) 2.55113 0.102955
\(615\) 0 0
\(616\) 16.4106 0.661204
\(617\) 33.9920 1.36847 0.684234 0.729263i \(-0.260137\pi\)
0.684234 + 0.729263i \(0.260137\pi\)
\(618\) −3.89455 −0.156662
\(619\) 29.6477 1.19164 0.595822 0.803117i \(-0.296826\pi\)
0.595822 + 0.803117i \(0.296826\pi\)
\(620\) 0 0
\(621\) −6.19882 −0.248750
\(622\) 13.0061 0.521497
\(623\) 25.6156 1.02627
\(624\) 8.26892 0.331022
\(625\) 0 0
\(626\) 3.04171 0.121571
\(627\) −6.59464 −0.263364
\(628\) −17.8607 −0.712718
\(629\) 38.5008 1.53513
\(630\) 0 0
\(631\) 14.6241 0.582176 0.291088 0.956696i \(-0.405983\pi\)
0.291088 + 0.956696i \(0.405983\pi\)
\(632\) −5.77827 −0.229847
\(633\) 22.7698 0.905018
\(634\) 3.36036 0.133457
\(635\) 0 0
\(636\) 8.11585 0.321814
\(637\) 55.0915 2.18280
\(638\) 4.59184 0.181793
\(639\) 34.7665 1.37534
\(640\) 0 0
\(641\) 4.17547 0.164921 0.0824606 0.996594i \(-0.473722\pi\)
0.0824606 + 0.996594i \(0.473722\pi\)
\(642\) −3.63438 −0.143437
\(643\) −23.1928 −0.914635 −0.457317 0.889304i \(-0.651190\pi\)
−0.457317 + 0.889304i \(0.651190\pi\)
\(644\) −11.5918 −0.456781
\(645\) 0 0
\(646\) −10.9837 −0.432148
\(647\) 31.0975 1.22257 0.611284 0.791411i \(-0.290653\pi\)
0.611284 + 0.791411i \(0.290653\pi\)
\(648\) 6.35008 0.249455
\(649\) −23.2347 −0.912043
\(650\) 0 0
\(651\) −27.2513 −1.06806
\(652\) 13.2153 0.517550
\(653\) 31.5439 1.23441 0.617204 0.786803i \(-0.288265\pi\)
0.617204 + 0.786803i \(0.288265\pi\)
\(654\) −1.28901 −0.0504041
\(655\) 0 0
\(656\) 24.1813 0.944120
\(657\) 0.561202 0.0218946
\(658\) 5.58027 0.217542
\(659\) 25.2754 0.984590 0.492295 0.870428i \(-0.336158\pi\)
0.492295 + 0.870428i \(0.336158\pi\)
\(660\) 0 0
\(661\) −21.7628 −0.846476 −0.423238 0.906018i \(-0.639107\pi\)
−0.423238 + 0.906018i \(0.639107\pi\)
\(662\) 6.61312 0.257026
\(663\) −16.9781 −0.659377
\(664\) 4.52578 0.175634
\(665\) 0 0
\(666\) −8.11164 −0.314320
\(667\) −6.92404 −0.268100
\(668\) 31.3586 1.21330
\(669\) 0.315814 0.0122101
\(670\) 0 0
\(671\) −27.0038 −1.04247
\(672\) −18.6125 −0.717991
\(673\) 37.2710 1.43669 0.718346 0.695686i \(-0.244899\pi\)
0.718346 + 0.695686i \(0.244899\pi\)
\(674\) −9.85835 −0.379730
\(675\) 0 0
\(676\) −3.28970 −0.126527
\(677\) 33.5515 1.28949 0.644745 0.764398i \(-0.276964\pi\)
0.644745 + 0.764398i \(0.276964\pi\)
\(678\) −0.334778 −0.0128571
\(679\) −24.3599 −0.934846
\(680\) 0 0
\(681\) −8.37644 −0.320986
\(682\) 6.85073 0.262328
\(683\) −8.94737 −0.342362 −0.171181 0.985240i \(-0.554758\pi\)
−0.171181 + 0.985240i \(0.554758\pi\)
\(684\) −17.1730 −0.656625
\(685\) 0 0
\(686\) −16.3882 −0.625704
\(687\) −8.58575 −0.327567
\(688\) −7.72638 −0.294566
\(689\) −21.7846 −0.829929
\(690\) 0 0
\(691\) 7.04969 0.268183 0.134092 0.990969i \(-0.457188\pi\)
0.134092 + 0.990969i \(0.457188\pi\)
\(692\) 32.5933 1.23901
\(693\) −20.9047 −0.794105
\(694\) 0.742994 0.0282036
\(695\) 0 0
\(696\) −7.25902 −0.275152
\(697\) −49.6502 −1.88063
\(698\) −8.06969 −0.305442
\(699\) −20.9232 −0.791389
\(700\) 0 0
\(701\) −5.19179 −0.196091 −0.0980456 0.995182i \(-0.531259\pi\)
−0.0980456 + 0.995182i \(0.531259\pi\)
\(702\) 8.17139 0.308409
\(703\) 29.7256 1.12112
\(704\) −5.52943 −0.208398
\(705\) 0 0
\(706\) 16.9012 0.636083
\(707\) −57.3230 −2.15586
\(708\) 17.2061 0.646643
\(709\) −16.3067 −0.612410 −0.306205 0.951966i \(-0.599059\pi\)
−0.306205 + 0.951966i \(0.599059\pi\)
\(710\) 0 0
\(711\) 7.36066 0.276046
\(712\) 10.1800 0.381510
\(713\) −10.3302 −0.386870
\(714\) 9.90128 0.370546
\(715\) 0 0
\(716\) 30.5594 1.14206
\(717\) −20.4820 −0.764915
\(718\) −8.83756 −0.329815
\(719\) −27.4968 −1.02546 −0.512729 0.858551i \(-0.671365\pi\)
−0.512729 + 0.858551i \(0.671365\pi\)
\(720\) 0 0
\(721\) 45.2410 1.68486
\(722\) 0.779256 0.0290009
\(723\) −0.815004 −0.0303103
\(724\) −31.2180 −1.16021
\(725\) 0 0
\(726\) 2.87460 0.106686
\(727\) 15.2319 0.564919 0.282460 0.959279i \(-0.408850\pi\)
0.282460 + 0.959279i \(0.408850\pi\)
\(728\) 32.6201 1.20898
\(729\) 2.54349 0.0942032
\(730\) 0 0
\(731\) 15.8642 0.586758
\(732\) 19.9972 0.739118
\(733\) −6.76475 −0.249862 −0.124931 0.992165i \(-0.539871\pi\)
−0.124931 + 0.992165i \(0.539871\pi\)
\(734\) −5.35490 −0.197653
\(735\) 0 0
\(736\) −7.05548 −0.260068
\(737\) 14.1734 0.522082
\(738\) 10.4607 0.385062
\(739\) 17.9519 0.660372 0.330186 0.943916i \(-0.392889\pi\)
0.330186 + 0.943916i \(0.392889\pi\)
\(740\) 0 0
\(741\) −13.1084 −0.481551
\(742\) 12.7043 0.466390
\(743\) 17.2096 0.631357 0.315679 0.948866i \(-0.397768\pi\)
0.315679 + 0.948866i \(0.397768\pi\)
\(744\) −10.8300 −0.397047
\(745\) 0 0
\(746\) 2.55195 0.0934336
\(747\) −5.76518 −0.210937
\(748\) 18.4714 0.675380
\(749\) 42.2188 1.54264
\(750\) 0 0
\(751\) 4.98853 0.182034 0.0910170 0.995849i \(-0.470988\pi\)
0.0910170 + 0.995849i \(0.470988\pi\)
\(752\) −6.53032 −0.238136
\(753\) 2.05575 0.0749156
\(754\) 9.12739 0.332400
\(755\) 0 0
\(756\) 35.3635 1.28616
\(757\) 19.5585 0.710864 0.355432 0.934702i \(-0.384334\pi\)
0.355432 + 0.934702i \(0.384334\pi\)
\(758\) −3.93335 −0.142866
\(759\) 2.25349 0.0817966
\(760\) 0 0
\(761\) −41.0235 −1.48710 −0.743550 0.668681i \(-0.766859\pi\)
−0.743550 + 0.668681i \(0.766859\pi\)
\(762\) 4.50474 0.163189
\(763\) 14.9738 0.542087
\(764\) −23.3425 −0.844503
\(765\) 0 0
\(766\) 1.20131 0.0434053
\(767\) −46.1847 −1.66763
\(768\) 0.162852 0.00587641
\(769\) 11.7068 0.422159 0.211079 0.977469i \(-0.432302\pi\)
0.211079 + 0.977469i \(0.432302\pi\)
\(770\) 0 0
\(771\) −3.19765 −0.115161
\(772\) −44.9489 −1.61775
\(773\) 6.66278 0.239643 0.119822 0.992795i \(-0.461768\pi\)
0.119822 + 0.992795i \(0.461768\pi\)
\(774\) −3.34238 −0.120140
\(775\) 0 0
\(776\) −9.68090 −0.347524
\(777\) −26.7962 −0.961309
\(778\) 13.9945 0.501727
\(779\) −38.3338 −1.37345
\(780\) 0 0
\(781\) −28.8719 −1.03312
\(782\) 3.75331 0.134218
\(783\) 21.1234 0.754888
\(784\) 37.5980 1.34279
\(785\) 0 0
\(786\) −5.61071 −0.200127
\(787\) −2.88890 −0.102978 −0.0514891 0.998674i \(-0.516397\pi\)
−0.0514891 + 0.998674i \(0.516397\pi\)
\(788\) −26.4909 −0.943698
\(789\) −5.09364 −0.181338
\(790\) 0 0
\(791\) 3.88896 0.138275
\(792\) −8.30778 −0.295204
\(793\) −53.6767 −1.90612
\(794\) 5.87762 0.208589
\(795\) 0 0
\(796\) 34.1574 1.21068
\(797\) −18.0912 −0.640824 −0.320412 0.947278i \(-0.603821\pi\)
−0.320412 + 0.947278i \(0.603821\pi\)
\(798\) 7.64455 0.270614
\(799\) 13.4084 0.474355
\(800\) 0 0
\(801\) −12.9678 −0.458193
\(802\) 15.4528 0.545656
\(803\) −0.466051 −0.0164466
\(804\) −10.4958 −0.370159
\(805\) 0 0
\(806\) 13.6175 0.479655
\(807\) 16.7029 0.587971
\(808\) −22.7809 −0.801428
\(809\) 8.04121 0.282714 0.141357 0.989959i \(-0.454853\pi\)
0.141357 + 0.989959i \(0.454853\pi\)
\(810\) 0 0
\(811\) 24.5111 0.860701 0.430350 0.902662i \(-0.358390\pi\)
0.430350 + 0.902662i \(0.358390\pi\)
\(812\) 39.5008 1.38621
\(813\) 12.4842 0.437839
\(814\) 6.73632 0.236108
\(815\) 0 0
\(816\) −11.5870 −0.405626
\(817\) 12.2484 0.428517
\(818\) 6.91507 0.241780
\(819\) −41.5532 −1.45199
\(820\) 0 0
\(821\) −10.6539 −0.371823 −0.185912 0.982566i \(-0.559524\pi\)
−0.185912 + 0.982566i \(0.559524\pi\)
\(822\) −5.14574 −0.179478
\(823\) −25.6473 −0.894009 −0.447004 0.894532i \(-0.647509\pi\)
−0.447004 + 0.894532i \(0.647509\pi\)
\(824\) 17.9793 0.626339
\(825\) 0 0
\(826\) 26.9339 0.937150
\(827\) −18.0150 −0.626442 −0.313221 0.949680i \(-0.601408\pi\)
−0.313221 + 0.949680i \(0.601408\pi\)
\(828\) 5.86828 0.203937
\(829\) 10.1189 0.351444 0.175722 0.984440i \(-0.443774\pi\)
0.175722 + 0.984440i \(0.443774\pi\)
\(830\) 0 0
\(831\) −2.66008 −0.0922771
\(832\) −10.9911 −0.381047
\(833\) −77.1981 −2.67476
\(834\) −3.27125 −0.113274
\(835\) 0 0
\(836\) 14.2613 0.493238
\(837\) 31.5148 1.08931
\(838\) −15.1571 −0.523593
\(839\) −46.0365 −1.58936 −0.794679 0.607030i \(-0.792361\pi\)
−0.794679 + 0.607030i \(0.792361\pi\)
\(840\) 0 0
\(841\) −5.40531 −0.186390
\(842\) 13.3734 0.460879
\(843\) −10.1705 −0.350292
\(844\) −49.2411 −1.69495
\(845\) 0 0
\(846\) −2.82498 −0.0971247
\(847\) −33.3928 −1.14739
\(848\) −14.8673 −0.510544
\(849\) 12.8865 0.442262
\(850\) 0 0
\(851\) −10.1577 −0.348202
\(852\) 21.3805 0.732485
\(853\) 55.0001 1.88317 0.941584 0.336780i \(-0.109338\pi\)
0.941584 + 0.336780i \(0.109338\pi\)
\(854\) 31.3031 1.07117
\(855\) 0 0
\(856\) 16.7783 0.573469
\(857\) 31.5202 1.07671 0.538354 0.842719i \(-0.319046\pi\)
0.538354 + 0.842719i \(0.319046\pi\)
\(858\) −2.97059 −0.101414
\(859\) −37.2208 −1.26996 −0.634979 0.772529i \(-0.718991\pi\)
−0.634979 + 0.772529i \(0.718991\pi\)
\(860\) 0 0
\(861\) 34.5561 1.17767
\(862\) 6.87180 0.234054
\(863\) −30.3224 −1.03218 −0.516092 0.856533i \(-0.672614\pi\)
−0.516092 + 0.856533i \(0.672614\pi\)
\(864\) 21.5244 0.732274
\(865\) 0 0
\(866\) −8.64563 −0.293791
\(867\) 9.93592 0.337442
\(868\) 58.9327 2.00031
\(869\) −6.11267 −0.207358
\(870\) 0 0
\(871\) 28.1730 0.954605
\(872\) 5.95075 0.201518
\(873\) 12.3320 0.417376
\(874\) 2.89784 0.0980210
\(875\) 0 0
\(876\) 0.345125 0.0116607
\(877\) −27.0037 −0.911849 −0.455925 0.890018i \(-0.650691\pi\)
−0.455925 + 0.890018i \(0.650691\pi\)
\(878\) 1.23028 0.0415200
\(879\) −15.2526 −0.514457
\(880\) 0 0
\(881\) −0.565506 −0.0190524 −0.00952619 0.999955i \(-0.503032\pi\)
−0.00952619 + 0.999955i \(0.503032\pi\)
\(882\) 16.2647 0.547660
\(883\) −23.8838 −0.803752 −0.401876 0.915694i \(-0.631642\pi\)
−0.401876 + 0.915694i \(0.631642\pi\)
\(884\) 36.7163 1.23490
\(885\) 0 0
\(886\) −8.20600 −0.275686
\(887\) 10.1717 0.341533 0.170767 0.985311i \(-0.445376\pi\)
0.170767 + 0.985311i \(0.445376\pi\)
\(888\) −10.6491 −0.357362
\(889\) −52.3294 −1.75507
\(890\) 0 0
\(891\) 6.71757 0.225047
\(892\) −0.682968 −0.0228675
\(893\) 10.3523 0.346427
\(894\) −1.78180 −0.0595924
\(895\) 0 0
\(896\) 52.0843 1.74001
\(897\) 4.47937 0.149562
\(898\) 14.8975 0.497135
\(899\) 35.2018 1.17405
\(900\) 0 0
\(901\) 30.5262 1.01697
\(902\) −8.68707 −0.289248
\(903\) −11.0413 −0.367432
\(904\) 1.54552 0.0514031
\(905\) 0 0
\(906\) −0.0370823 −0.00123198
\(907\) 44.1661 1.46651 0.733255 0.679954i \(-0.238000\pi\)
0.733255 + 0.679954i \(0.238000\pi\)
\(908\) 18.1146 0.601154
\(909\) 29.0194 0.962514
\(910\) 0 0
\(911\) −7.07316 −0.234344 −0.117172 0.993112i \(-0.537383\pi\)
−0.117172 + 0.993112i \(0.537383\pi\)
\(912\) −8.94605 −0.296233
\(913\) 4.78770 0.158450
\(914\) −15.9382 −0.527190
\(915\) 0 0
\(916\) 18.5672 0.613479
\(917\) 65.1769 2.15233
\(918\) −11.4503 −0.377917
\(919\) 23.6016 0.778544 0.389272 0.921123i \(-0.372727\pi\)
0.389272 + 0.921123i \(0.372727\pi\)
\(920\) 0 0
\(921\) −4.26636 −0.140581
\(922\) −6.03976 −0.198909
\(923\) −57.3899 −1.88901
\(924\) −12.8559 −0.422928
\(925\) 0 0
\(926\) 5.91530 0.194389
\(927\) −22.9030 −0.752233
\(928\) 24.0426 0.789236
\(929\) 31.8496 1.04495 0.522475 0.852654i \(-0.325009\pi\)
0.522475 + 0.852654i \(0.325009\pi\)
\(930\) 0 0
\(931\) −59.6029 −1.95341
\(932\) 45.2478 1.48214
\(933\) −21.7506 −0.712083
\(934\) 2.62290 0.0858238
\(935\) 0 0
\(936\) −16.5137 −0.539768
\(937\) −44.1729 −1.44307 −0.721533 0.692380i \(-0.756562\pi\)
−0.721533 + 0.692380i \(0.756562\pi\)
\(938\) −16.4299 −0.536454
\(939\) −5.08677 −0.166001
\(940\) 0 0
\(941\) −34.5303 −1.12566 −0.562828 0.826574i \(-0.690287\pi\)
−0.562828 + 0.826574i \(0.690287\pi\)
\(942\) −4.02498 −0.131141
\(943\) 13.0993 0.426571
\(944\) −31.5194 −1.02587
\(945\) 0 0
\(946\) 2.77569 0.0902454
\(947\) 3.46191 0.112497 0.0562485 0.998417i \(-0.482086\pi\)
0.0562485 + 0.998417i \(0.482086\pi\)
\(948\) 4.52662 0.147018
\(949\) −0.926389 −0.0300719
\(950\) 0 0
\(951\) −5.61966 −0.182230
\(952\) −45.7097 −1.48146
\(953\) −60.2596 −1.95200 −0.976000 0.217771i \(-0.930122\pi\)
−0.976000 + 0.217771i \(0.930122\pi\)
\(954\) −6.43148 −0.208227
\(955\) 0 0
\(956\) 44.2936 1.43256
\(957\) −7.67911 −0.248230
\(958\) 12.3664 0.399541
\(959\) 59.7755 1.93025
\(960\) 0 0
\(961\) 21.5188 0.694156
\(962\) 13.3901 0.431713
\(963\) −21.3730 −0.688735
\(964\) 1.76250 0.0567662
\(965\) 0 0
\(966\) −2.61227 −0.0840483
\(967\) 39.9168 1.28364 0.641819 0.766856i \(-0.278180\pi\)
0.641819 + 0.766856i \(0.278180\pi\)
\(968\) −13.2707 −0.426536
\(969\) 18.3685 0.590081
\(970\) 0 0
\(971\) 43.7104 1.40273 0.701366 0.712801i \(-0.252574\pi\)
0.701366 + 0.712801i \(0.252574\pi\)
\(972\) −27.9681 −0.897078
\(973\) 38.0006 1.21824
\(974\) −5.08020 −0.162780
\(975\) 0 0
\(976\) −36.6325 −1.17258
\(977\) 0.955388 0.0305656 0.0152828 0.999883i \(-0.495135\pi\)
0.0152828 + 0.999883i \(0.495135\pi\)
\(978\) 2.97812 0.0952298
\(979\) 10.7691 0.344182
\(980\) 0 0
\(981\) −7.58037 −0.242023
\(982\) −16.7090 −0.533205
\(983\) 48.7488 1.55485 0.777423 0.628978i \(-0.216526\pi\)
0.777423 + 0.628978i \(0.216526\pi\)
\(984\) 13.7330 0.437792
\(985\) 0 0
\(986\) −12.7899 −0.407315
\(987\) −9.33211 −0.297044
\(988\) 28.3478 0.901865
\(989\) −4.18547 −0.133090
\(990\) 0 0
\(991\) 0.488986 0.0155332 0.00776658 0.999970i \(-0.497528\pi\)
0.00776658 + 0.999970i \(0.497528\pi\)
\(992\) 35.8700 1.13887
\(993\) −11.0594 −0.350959
\(994\) 33.4685 1.06156
\(995\) 0 0
\(996\) −3.54544 −0.112342
\(997\) −39.5339 −1.25205 −0.626026 0.779802i \(-0.715320\pi\)
−0.626026 + 0.779802i \(0.715320\pi\)
\(998\) −7.78153 −0.246320
\(999\) 30.9885 0.980432
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 6025.2.a.f.1.2 7
5.4 even 2 241.2.a.a.1.6 7
15.14 odd 2 2169.2.a.e.1.2 7
20.19 odd 2 3856.2.a.j.1.5 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
241.2.a.a.1.6 7 5.4 even 2
2169.2.a.e.1.2 7 15.14 odd 2
3856.2.a.j.1.5 7 20.19 odd 2
6025.2.a.f.1.2 7 1.1 even 1 trivial