Properties

Label 6025.2.a.f.1.4
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 1.4
Root \(0.369356\) of defining polynomial
Character \(\chi\) \(=\) 6025.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.630644 q^{2} -2.33806 q^{3} -1.60229 q^{4} -1.47449 q^{6} +3.68231 q^{7} -2.27176 q^{8} +2.46654 q^{9} +O(q^{10})\) \(q+0.630644 q^{2} -2.33806 q^{3} -1.60229 q^{4} -1.47449 q^{6} +3.68231 q^{7} -2.27176 q^{8} +2.46654 q^{9} -4.96431 q^{11} +3.74625 q^{12} +1.69048 q^{13} +2.32223 q^{14} +1.77190 q^{16} -5.52260 q^{17} +1.55551 q^{18} +4.21489 q^{19} -8.60948 q^{21} -3.13071 q^{22} +2.77495 q^{23} +5.31152 q^{24} +1.06609 q^{26} +1.24727 q^{27} -5.90013 q^{28} -2.31253 q^{29} -0.199515 q^{31} +5.66096 q^{32} +11.6069 q^{33} -3.48280 q^{34} -3.95210 q^{36} -1.79089 q^{37} +2.65809 q^{38} -3.95245 q^{39} -12.1960 q^{41} -5.42952 q^{42} +5.34523 q^{43} +7.95425 q^{44} +1.75001 q^{46} +10.7345 q^{47} -4.14282 q^{48} +6.55944 q^{49} +12.9122 q^{51} -2.70864 q^{52} -1.32229 q^{53} +0.786587 q^{54} -8.36534 q^{56} -9.85467 q^{57} -1.45838 q^{58} +5.78578 q^{59} -0.0766435 q^{61} -0.125823 q^{62} +9.08256 q^{63} +0.0262532 q^{64} +7.31980 q^{66} -12.6166 q^{67} +8.84880 q^{68} -6.48800 q^{69} -5.20391 q^{71} -5.60338 q^{72} +14.0733 q^{73} -1.12941 q^{74} -6.75346 q^{76} -18.2801 q^{77} -2.49259 q^{78} +7.82844 q^{79} -10.3158 q^{81} -7.69137 q^{82} -2.55372 q^{83} +13.7949 q^{84} +3.37094 q^{86} +5.40683 q^{87} +11.2777 q^{88} -4.35327 q^{89} +6.22489 q^{91} -4.44627 q^{92} +0.466479 q^{93} +6.76964 q^{94} -13.2357 q^{96} -9.02693 q^{97} +4.13668 q^{98} -12.2446 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 4q^{2} + 3q^{3} + 2q^{4} - 5q^{6} + 7q^{7} + 6q^{8} - 2q^{9} + O(q^{10}) \) \( 7q + 4q^{2} + 3q^{3} + 2q^{4} - 5q^{6} + 7q^{7} + 6q^{8} - 2q^{9} - 18q^{11} - q^{12} + q^{13} - 6q^{14} + 4q^{16} + 2q^{17} - 8q^{18} - 6q^{19} - 2q^{21} - 10q^{22} + 22q^{23} - 3q^{24} + 8q^{26} - 3q^{27} - 9q^{28} - 16q^{29} - 18q^{31} + 6q^{32} - 4q^{33} + 11q^{34} - 7q^{36} - 8q^{37} - 16q^{38} - 9q^{39} - 15q^{41} - 19q^{42} - 14q^{43} - 4q^{44} + 11q^{46} + 10q^{47} - 31q^{48} + 6q^{49} + 13q^{51} - 27q^{52} - 15q^{53} + 16q^{54} + 13q^{56} - 14q^{57} - 17q^{58} - 18q^{59} + 4q^{61} - 13q^{62} + 16q^{63} + 2q^{64} + 16q^{66} - 18q^{67} + 15q^{68} + 26q^{69} - 50q^{71} - 30q^{72} + 10q^{74} - 20q^{76} - 17q^{77} + 32q^{78} - 15q^{79} - 9q^{81} - 45q^{82} + 24q^{83} + 6q^{84} - 23q^{86} - 12q^{87} - 8q^{88} - 13q^{89} - 12q^{91} + 10q^{92} - 14q^{93} - 32q^{94} - 15q^{96} - q^{97} - 9q^{98} - 20q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.630644 0.445933 0.222966 0.974826i \(-0.428426\pi\)
0.222966 + 0.974826i \(0.428426\pi\)
\(3\) −2.33806 −1.34988 −0.674940 0.737872i \(-0.735831\pi\)
−0.674940 + 0.737872i \(0.735831\pi\)
\(4\) −1.60229 −0.801144
\(5\) 0 0
\(6\) −1.47449 −0.601956
\(7\) 3.68231 1.39178 0.695892 0.718146i \(-0.255009\pi\)
0.695892 + 0.718146i \(0.255009\pi\)
\(8\) −2.27176 −0.803189
\(9\) 2.46654 0.822178
\(10\) 0 0
\(11\) −4.96431 −1.49679 −0.748397 0.663250i \(-0.769176\pi\)
−0.748397 + 0.663250i \(0.769176\pi\)
\(12\) 3.74625 1.08145
\(13\) 1.69048 0.468855 0.234428 0.972134i \(-0.424678\pi\)
0.234428 + 0.972134i \(0.424678\pi\)
\(14\) 2.32223 0.620642
\(15\) 0 0
\(16\) 1.77190 0.442975
\(17\) −5.52260 −1.33943 −0.669714 0.742619i \(-0.733583\pi\)
−0.669714 + 0.742619i \(0.733583\pi\)
\(18\) 1.55551 0.366636
\(19\) 4.21489 0.966961 0.483481 0.875355i \(-0.339372\pi\)
0.483481 + 0.875355i \(0.339372\pi\)
\(20\) 0 0
\(21\) −8.60948 −1.87874
\(22\) −3.13071 −0.667470
\(23\) 2.77495 0.578617 0.289308 0.957236i \(-0.406575\pi\)
0.289308 + 0.957236i \(0.406575\pi\)
\(24\) 5.31152 1.08421
\(25\) 0 0
\(26\) 1.06609 0.209078
\(27\) 1.24727 0.240038
\(28\) −5.90013 −1.11502
\(29\) −2.31253 −0.429425 −0.214713 0.976677i \(-0.568881\pi\)
−0.214713 + 0.976677i \(0.568881\pi\)
\(30\) 0 0
\(31\) −0.199515 −0.0358340 −0.0179170 0.999839i \(-0.505703\pi\)
−0.0179170 + 0.999839i \(0.505703\pi\)
\(32\) 5.66096 1.00073
\(33\) 11.6069 2.02049
\(34\) −3.48280 −0.597295
\(35\) 0 0
\(36\) −3.95210 −0.658683
\(37\) −1.79089 −0.294420 −0.147210 0.989105i \(-0.547029\pi\)
−0.147210 + 0.989105i \(0.547029\pi\)
\(38\) 2.65809 0.431200
\(39\) −3.95245 −0.632899
\(40\) 0 0
\(41\) −12.1960 −1.90470 −0.952351 0.305003i \(-0.901343\pi\)
−0.952351 + 0.305003i \(0.901343\pi\)
\(42\) −5.42952 −0.837793
\(43\) 5.34523 0.815139 0.407570 0.913174i \(-0.366376\pi\)
0.407570 + 0.913174i \(0.366376\pi\)
\(44\) 7.95425 1.19915
\(45\) 0 0
\(46\) 1.75001 0.258024
\(47\) 10.7345 1.56579 0.782893 0.622157i \(-0.213743\pi\)
0.782893 + 0.622157i \(0.213743\pi\)
\(48\) −4.14282 −0.597964
\(49\) 6.55944 0.937063
\(50\) 0 0
\(51\) 12.9122 1.80807
\(52\) −2.70864 −0.375621
\(53\) −1.32229 −0.181631 −0.0908154 0.995868i \(-0.528947\pi\)
−0.0908154 + 0.995868i \(0.528947\pi\)
\(54\) 0.786587 0.107041
\(55\) 0 0
\(56\) −8.36534 −1.11787
\(57\) −9.85467 −1.30528
\(58\) −1.45838 −0.191495
\(59\) 5.78578 0.753244 0.376622 0.926367i \(-0.377086\pi\)
0.376622 + 0.926367i \(0.377086\pi\)
\(60\) 0 0
\(61\) −0.0766435 −0.00981320 −0.00490660 0.999988i \(-0.501562\pi\)
−0.00490660 + 0.999988i \(0.501562\pi\)
\(62\) −0.125823 −0.0159795
\(63\) 9.08256 1.14429
\(64\) 0.0262532 0.00328165
\(65\) 0 0
\(66\) 7.31980 0.901005
\(67\) −12.6166 −1.54136 −0.770681 0.637221i \(-0.780084\pi\)
−0.770681 + 0.637221i \(0.780084\pi\)
\(68\) 8.84880 1.07307
\(69\) −6.48800 −0.781064
\(70\) 0 0
\(71\) −5.20391 −0.617591 −0.308796 0.951128i \(-0.599926\pi\)
−0.308796 + 0.951128i \(0.599926\pi\)
\(72\) −5.60338 −0.660365
\(73\) 14.0733 1.64715 0.823577 0.567204i \(-0.191975\pi\)
0.823577 + 0.567204i \(0.191975\pi\)
\(74\) −1.12941 −0.131291
\(75\) 0 0
\(76\) −6.75346 −0.774675
\(77\) −18.2801 −2.08322
\(78\) −2.49259 −0.282230
\(79\) 7.82844 0.880769 0.440384 0.897809i \(-0.354842\pi\)
0.440384 + 0.897809i \(0.354842\pi\)
\(80\) 0 0
\(81\) −10.3158 −1.14620
\(82\) −7.69137 −0.849370
\(83\) −2.55372 −0.280307 −0.140153 0.990130i \(-0.544760\pi\)
−0.140153 + 0.990130i \(0.544760\pi\)
\(84\) 13.7949 1.50514
\(85\) 0 0
\(86\) 3.37094 0.363497
\(87\) 5.40683 0.579673
\(88\) 11.2777 1.20221
\(89\) −4.35327 −0.461446 −0.230723 0.973019i \(-0.574109\pi\)
−0.230723 + 0.973019i \(0.574109\pi\)
\(90\) 0 0
\(91\) 6.22489 0.652546
\(92\) −4.44627 −0.463555
\(93\) 0.466479 0.0483716
\(94\) 6.76964 0.698235
\(95\) 0 0
\(96\) −13.2357 −1.35086
\(97\) −9.02693 −0.916546 −0.458273 0.888811i \(-0.651532\pi\)
−0.458273 + 0.888811i \(0.651532\pi\)
\(98\) 4.13668 0.417867
\(99\) −12.2446 −1.23063
\(100\) 0 0
\(101\) 6.94323 0.690877 0.345439 0.938441i \(-0.387730\pi\)
0.345439 + 0.938441i \(0.387730\pi\)
\(102\) 8.14300 0.806277
\(103\) 0.105009 0.0103469 0.00517343 0.999987i \(-0.498353\pi\)
0.00517343 + 0.999987i \(0.498353\pi\)
\(104\) −3.84037 −0.376580
\(105\) 0 0
\(106\) −0.833897 −0.0809952
\(107\) 18.1842 1.75793 0.878966 0.476885i \(-0.158234\pi\)
0.878966 + 0.476885i \(0.158234\pi\)
\(108\) −1.99849 −0.192305
\(109\) 14.6576 1.40394 0.701972 0.712205i \(-0.252303\pi\)
0.701972 + 0.712205i \(0.252303\pi\)
\(110\) 0 0
\(111\) 4.18720 0.397432
\(112\) 6.52470 0.616526
\(113\) −1.08836 −0.102384 −0.0511922 0.998689i \(-0.516302\pi\)
−0.0511922 + 0.998689i \(0.516302\pi\)
\(114\) −6.21479 −0.582068
\(115\) 0 0
\(116\) 3.70533 0.344031
\(117\) 4.16963 0.385483
\(118\) 3.64877 0.335896
\(119\) −20.3360 −1.86420
\(120\) 0 0
\(121\) 13.6443 1.24040
\(122\) −0.0483348 −0.00437603
\(123\) 28.5151 2.57112
\(124\) 0.319681 0.0287082
\(125\) 0 0
\(126\) 5.72786 0.510279
\(127\) −0.364538 −0.0323475 −0.0161737 0.999869i \(-0.505148\pi\)
−0.0161737 + 0.999869i \(0.505148\pi\)
\(128\) −11.3054 −0.999263
\(129\) −12.4975 −1.10034
\(130\) 0 0
\(131\) 5.48865 0.479546 0.239773 0.970829i \(-0.422927\pi\)
0.239773 + 0.970829i \(0.422927\pi\)
\(132\) −18.5975 −1.61871
\(133\) 15.5205 1.34580
\(134\) −7.95658 −0.687344
\(135\) 0 0
\(136\) 12.5460 1.07581
\(137\) 17.1533 1.46551 0.732754 0.680493i \(-0.238234\pi\)
0.732754 + 0.680493i \(0.238234\pi\)
\(138\) −4.09162 −0.348302
\(139\) −19.6074 −1.66308 −0.831539 0.555467i \(-0.812540\pi\)
−0.831539 + 0.555467i \(0.812540\pi\)
\(140\) 0 0
\(141\) −25.0979 −2.11362
\(142\) −3.28182 −0.275404
\(143\) −8.39207 −0.701780
\(144\) 4.37046 0.364205
\(145\) 0 0
\(146\) 8.87525 0.734521
\(147\) −15.3364 −1.26492
\(148\) 2.86951 0.235873
\(149\) 19.0508 1.56070 0.780350 0.625343i \(-0.215041\pi\)
0.780350 + 0.625343i \(0.215041\pi\)
\(150\) 0 0
\(151\) −4.76802 −0.388016 −0.194008 0.981000i \(-0.562149\pi\)
−0.194008 + 0.981000i \(0.562149\pi\)
\(152\) −9.57522 −0.776653
\(153\) −13.6217 −1.10125
\(154\) −11.5283 −0.928974
\(155\) 0 0
\(156\) 6.33297 0.507043
\(157\) −6.12724 −0.489007 −0.244503 0.969648i \(-0.578625\pi\)
−0.244503 + 0.969648i \(0.578625\pi\)
\(158\) 4.93696 0.392764
\(159\) 3.09160 0.245180
\(160\) 0 0
\(161\) 10.2182 0.805310
\(162\) −6.50561 −0.511129
\(163\) 1.20414 0.0943159 0.0471579 0.998887i \(-0.484984\pi\)
0.0471579 + 0.998887i \(0.484984\pi\)
\(164\) 19.5416 1.52594
\(165\) 0 0
\(166\) −1.61049 −0.124998
\(167\) 0.900391 0.0696744 0.0348372 0.999393i \(-0.488909\pi\)
0.0348372 + 0.999393i \(0.488909\pi\)
\(168\) 19.5587 1.50899
\(169\) −10.1423 −0.780175
\(170\) 0 0
\(171\) 10.3962 0.795015
\(172\) −8.56459 −0.653044
\(173\) −12.3564 −0.939436 −0.469718 0.882817i \(-0.655644\pi\)
−0.469718 + 0.882817i \(0.655644\pi\)
\(174\) 3.40979 0.258495
\(175\) 0 0
\(176\) −8.79626 −0.663043
\(177\) −13.5275 −1.01679
\(178\) −2.74537 −0.205774
\(179\) −10.0258 −0.749367 −0.374683 0.927153i \(-0.622249\pi\)
−0.374683 + 0.927153i \(0.622249\pi\)
\(180\) 0 0
\(181\) −17.0491 −1.26725 −0.633623 0.773642i \(-0.718433\pi\)
−0.633623 + 0.773642i \(0.718433\pi\)
\(182\) 3.92569 0.290992
\(183\) 0.179197 0.0132467
\(184\) −6.30402 −0.464739
\(185\) 0 0
\(186\) 0.294182 0.0215705
\(187\) 27.4159 2.00485
\(188\) −17.1997 −1.25442
\(189\) 4.59286 0.334081
\(190\) 0 0
\(191\) −23.7744 −1.72026 −0.860129 0.510077i \(-0.829617\pi\)
−0.860129 + 0.510077i \(0.829617\pi\)
\(192\) −0.0613816 −0.00442984
\(193\) −7.49373 −0.539411 −0.269705 0.962943i \(-0.586926\pi\)
−0.269705 + 0.962943i \(0.586926\pi\)
\(194\) −5.69278 −0.408718
\(195\) 0 0
\(196\) −10.5101 −0.750723
\(197\) −1.85647 −0.132268 −0.0661339 0.997811i \(-0.521066\pi\)
−0.0661339 + 0.997811i \(0.521066\pi\)
\(198\) −7.72201 −0.548779
\(199\) 3.46667 0.245746 0.122873 0.992422i \(-0.460789\pi\)
0.122873 + 0.992422i \(0.460789\pi\)
\(200\) 0 0
\(201\) 29.4984 2.08066
\(202\) 4.37871 0.308085
\(203\) −8.51545 −0.597667
\(204\) −20.6890 −1.44852
\(205\) 0 0
\(206\) 0.0662234 0.00461400
\(207\) 6.84451 0.475726
\(208\) 2.99537 0.207691
\(209\) −20.9240 −1.44734
\(210\) 0 0
\(211\) −2.33201 −0.160542 −0.0802711 0.996773i \(-0.525579\pi\)
−0.0802711 + 0.996773i \(0.525579\pi\)
\(212\) 2.11869 0.145512
\(213\) 12.1671 0.833674
\(214\) 11.4678 0.783919
\(215\) 0 0
\(216\) −2.83351 −0.192796
\(217\) −0.734678 −0.0498732
\(218\) 9.24373 0.626065
\(219\) −32.9042 −2.22346
\(220\) 0 0
\(221\) −9.33587 −0.627998
\(222\) 2.64063 0.177228
\(223\) 1.75535 0.117547 0.0587734 0.998271i \(-0.481281\pi\)
0.0587734 + 0.998271i \(0.481281\pi\)
\(224\) 20.8455 1.39280
\(225\) 0 0
\(226\) −0.686369 −0.0456566
\(227\) −4.35631 −0.289139 −0.144569 0.989495i \(-0.546180\pi\)
−0.144569 + 0.989495i \(0.546180\pi\)
\(228\) 15.7900 1.04572
\(229\) 6.24160 0.412457 0.206228 0.978504i \(-0.433881\pi\)
0.206228 + 0.978504i \(0.433881\pi\)
\(230\) 0 0
\(231\) 42.7401 2.81209
\(232\) 5.25351 0.344910
\(233\) −13.8315 −0.906135 −0.453067 0.891476i \(-0.649670\pi\)
−0.453067 + 0.891476i \(0.649670\pi\)
\(234\) 2.62956 0.171899
\(235\) 0 0
\(236\) −9.27048 −0.603457
\(237\) −18.3034 −1.18893
\(238\) −12.8248 −0.831306
\(239\) −19.3924 −1.25439 −0.627194 0.778863i \(-0.715797\pi\)
−0.627194 + 0.778863i \(0.715797\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157
\(242\) 8.60473 0.553133
\(243\) 20.3772 1.30720
\(244\) 0.122805 0.00786178
\(245\) 0 0
\(246\) 17.9829 1.14655
\(247\) 7.12519 0.453365
\(248\) 0.453251 0.0287815
\(249\) 5.97075 0.378381
\(250\) 0 0
\(251\) −14.4441 −0.911705 −0.455852 0.890055i \(-0.650666\pi\)
−0.455852 + 0.890055i \(0.650666\pi\)
\(252\) −14.5529 −0.916745
\(253\) −13.7757 −0.866071
\(254\) −0.229894 −0.0144248
\(255\) 0 0
\(256\) −7.18218 −0.448886
\(257\) −28.9004 −1.80276 −0.901379 0.433031i \(-0.857444\pi\)
−0.901379 + 0.433031i \(0.857444\pi\)
\(258\) −7.88146 −0.490678
\(259\) −6.59460 −0.409769
\(260\) 0 0
\(261\) −5.70393 −0.353064
\(262\) 3.46139 0.213845
\(263\) −26.3078 −1.62221 −0.811103 0.584903i \(-0.801133\pi\)
−0.811103 + 0.584903i \(0.801133\pi\)
\(264\) −26.3680 −1.62284
\(265\) 0 0
\(266\) 9.78794 0.600137
\(267\) 10.1782 0.622897
\(268\) 20.2154 1.23485
\(269\) −9.42450 −0.574622 −0.287311 0.957837i \(-0.592761\pi\)
−0.287311 + 0.957837i \(0.592761\pi\)
\(270\) 0 0
\(271\) −5.86519 −0.356285 −0.178142 0.984005i \(-0.557009\pi\)
−0.178142 + 0.984005i \(0.557009\pi\)
\(272\) −9.78551 −0.593334
\(273\) −14.5542 −0.880859
\(274\) 10.8177 0.653519
\(275\) 0 0
\(276\) 10.3956 0.625744
\(277\) −5.60395 −0.336708 −0.168354 0.985727i \(-0.553845\pi\)
−0.168354 + 0.985727i \(0.553845\pi\)
\(278\) −12.3653 −0.741621
\(279\) −0.492111 −0.0294619
\(280\) 0 0
\(281\) −4.71669 −0.281374 −0.140687 0.990054i \(-0.544931\pi\)
−0.140687 + 0.990054i \(0.544931\pi\)
\(282\) −15.8278 −0.942534
\(283\) −19.2068 −1.14173 −0.570864 0.821044i \(-0.693392\pi\)
−0.570864 + 0.821044i \(0.693392\pi\)
\(284\) 8.33817 0.494779
\(285\) 0 0
\(286\) −5.29241 −0.312947
\(287\) −44.9097 −2.65094
\(288\) 13.9630 0.822776
\(289\) 13.4992 0.794069
\(290\) 0 0
\(291\) 21.1055 1.23723
\(292\) −22.5495 −1.31961
\(293\) −13.0140 −0.760289 −0.380144 0.924927i \(-0.624126\pi\)
−0.380144 + 0.924927i \(0.624126\pi\)
\(294\) −9.67181 −0.564071
\(295\) 0 0
\(296\) 4.06847 0.236475
\(297\) −6.19185 −0.359288
\(298\) 12.0143 0.695968
\(299\) 4.69100 0.271288
\(300\) 0 0
\(301\) 19.6828 1.13450
\(302\) −3.00692 −0.173029
\(303\) −16.2337 −0.932602
\(304\) 7.46836 0.428340
\(305\) 0 0
\(306\) −8.59045 −0.491083
\(307\) 11.5372 0.658463 0.329232 0.944249i \(-0.393210\pi\)
0.329232 + 0.944249i \(0.393210\pi\)
\(308\) 29.2900 1.66896
\(309\) −0.245518 −0.0139670
\(310\) 0 0
\(311\) −23.1542 −1.31296 −0.656478 0.754345i \(-0.727955\pi\)
−0.656478 + 0.754345i \(0.727955\pi\)
\(312\) 8.97903 0.508338
\(313\) 7.88902 0.445914 0.222957 0.974828i \(-0.428429\pi\)
0.222957 + 0.974828i \(0.428429\pi\)
\(314\) −3.86411 −0.218064
\(315\) 0 0
\(316\) −12.5434 −0.705623
\(317\) 2.38325 0.133857 0.0669283 0.997758i \(-0.478680\pi\)
0.0669283 + 0.997758i \(0.478680\pi\)
\(318\) 1.94970 0.109334
\(319\) 11.4801 0.642761
\(320\) 0 0
\(321\) −42.5157 −2.37300
\(322\) 6.44407 0.359114
\(323\) −23.2772 −1.29518
\(324\) 16.5289 0.918272
\(325\) 0 0
\(326\) 0.759387 0.0420585
\(327\) −34.2704 −1.89516
\(328\) 27.7065 1.52984
\(329\) 39.5278 2.17924
\(330\) 0 0
\(331\) −3.05598 −0.167972 −0.0839860 0.996467i \(-0.526765\pi\)
−0.0839860 + 0.996467i \(0.526765\pi\)
\(332\) 4.09179 0.224566
\(333\) −4.41728 −0.242066
\(334\) 0.567827 0.0310701
\(335\) 0 0
\(336\) −15.2551 −0.832237
\(337\) 16.1755 0.881134 0.440567 0.897720i \(-0.354777\pi\)
0.440567 + 0.897720i \(0.354777\pi\)
\(338\) −6.39617 −0.347906
\(339\) 2.54466 0.138207
\(340\) 0 0
\(341\) 0.990454 0.0536361
\(342\) 6.55628 0.354523
\(343\) −1.62227 −0.0875943
\(344\) −12.1431 −0.654711
\(345\) 0 0
\(346\) −7.79246 −0.418925
\(347\) −24.4716 −1.31370 −0.656851 0.754020i \(-0.728112\pi\)
−0.656851 + 0.754020i \(0.728112\pi\)
\(348\) −8.66329 −0.464401
\(349\) −13.0468 −0.698380 −0.349190 0.937052i \(-0.613543\pi\)
−0.349190 + 0.937052i \(0.613543\pi\)
\(350\) 0 0
\(351\) 2.10849 0.112543
\(352\) −28.1028 −1.49788
\(353\) −23.6788 −1.26029 −0.630147 0.776476i \(-0.717005\pi\)
−0.630147 + 0.776476i \(0.717005\pi\)
\(354\) −8.53104 −0.453420
\(355\) 0 0
\(356\) 6.97519 0.369685
\(357\) 47.5468 2.51644
\(358\) −6.32274 −0.334167
\(359\) −25.5499 −1.34847 −0.674236 0.738516i \(-0.735527\pi\)
−0.674236 + 0.738516i \(0.735527\pi\)
\(360\) 0 0
\(361\) −1.23473 −0.0649857
\(362\) −10.7519 −0.565107
\(363\) −31.9013 −1.67439
\(364\) −9.97406 −0.522783
\(365\) 0 0
\(366\) 0.113010 0.00590712
\(367\) −8.35766 −0.436266 −0.218133 0.975919i \(-0.569997\pi\)
−0.218133 + 0.975919i \(0.569997\pi\)
\(368\) 4.91693 0.256313
\(369\) −30.0820 −1.56601
\(370\) 0 0
\(371\) −4.86910 −0.252791
\(372\) −0.747433 −0.0387526
\(373\) 24.6934 1.27858 0.639288 0.768968i \(-0.279229\pi\)
0.639288 + 0.768968i \(0.279229\pi\)
\(374\) 17.2897 0.894028
\(375\) 0 0
\(376\) −24.3862 −1.25762
\(377\) −3.90928 −0.201338
\(378\) 2.89646 0.148978
\(379\) 18.6593 0.958465 0.479232 0.877688i \(-0.340915\pi\)
0.479232 + 0.877688i \(0.340915\pi\)
\(380\) 0 0
\(381\) 0.852312 0.0436652
\(382\) −14.9932 −0.767119
\(383\) −11.9780 −0.612049 −0.306025 0.952024i \(-0.598999\pi\)
−0.306025 + 0.952024i \(0.598999\pi\)
\(384\) 26.4327 1.34889
\(385\) 0 0
\(386\) −4.72588 −0.240541
\(387\) 13.1842 0.670190
\(388\) 14.4637 0.734285
\(389\) −7.62670 −0.386689 −0.193344 0.981131i \(-0.561933\pi\)
−0.193344 + 0.981131i \(0.561933\pi\)
\(390\) 0 0
\(391\) −15.3249 −0.775016
\(392\) −14.9015 −0.752639
\(393\) −12.8328 −0.647330
\(394\) −1.17077 −0.0589826
\(395\) 0 0
\(396\) 19.6194 0.985914
\(397\) 33.4013 1.67636 0.838181 0.545392i \(-0.183619\pi\)
0.838181 + 0.545392i \(0.183619\pi\)
\(398\) 2.18624 0.109586
\(399\) −36.2880 −1.81667
\(400\) 0 0
\(401\) −1.73530 −0.0866566 −0.0433283 0.999061i \(-0.513796\pi\)
−0.0433283 + 0.999061i \(0.513796\pi\)
\(402\) 18.6030 0.927833
\(403\) −0.337277 −0.0168010
\(404\) −11.1251 −0.553492
\(405\) 0 0
\(406\) −5.37022 −0.266519
\(407\) 8.89050 0.440686
\(408\) −29.3334 −1.45222
\(409\) −3.81908 −0.188841 −0.0944207 0.995532i \(-0.530100\pi\)
−0.0944207 + 0.995532i \(0.530100\pi\)
\(410\) 0 0
\(411\) −40.1056 −1.97826
\(412\) −0.168255 −0.00828932
\(413\) 21.3050 1.04835
\(414\) 4.31645 0.212142
\(415\) 0 0
\(416\) 9.56976 0.469196
\(417\) 45.8433 2.24496
\(418\) −13.1956 −0.645418
\(419\) 0.640088 0.0312703 0.0156352 0.999878i \(-0.495023\pi\)
0.0156352 + 0.999878i \(0.495023\pi\)
\(420\) 0 0
\(421\) 13.2765 0.647057 0.323528 0.946218i \(-0.395131\pi\)
0.323528 + 0.946218i \(0.395131\pi\)
\(422\) −1.47067 −0.0715910
\(423\) 26.4770 1.28735
\(424\) 3.00394 0.145884
\(425\) 0 0
\(426\) 7.67310 0.371763
\(427\) −0.282226 −0.0136579
\(428\) −29.1363 −1.40836
\(429\) 19.6212 0.947320
\(430\) 0 0
\(431\) −26.8180 −1.29178 −0.645888 0.763432i \(-0.723513\pi\)
−0.645888 + 0.763432i \(0.723513\pi\)
\(432\) 2.21005 0.106331
\(433\) 20.7392 0.996664 0.498332 0.866986i \(-0.333946\pi\)
0.498332 + 0.866986i \(0.333946\pi\)
\(434\) −0.463320 −0.0222401
\(435\) 0 0
\(436\) −23.4857 −1.12476
\(437\) 11.6961 0.559500
\(438\) −20.7509 −0.991515
\(439\) −12.7382 −0.607960 −0.303980 0.952678i \(-0.598316\pi\)
−0.303980 + 0.952678i \(0.598316\pi\)
\(440\) 0 0
\(441\) 16.1791 0.770433
\(442\) −5.88761 −0.280045
\(443\) −23.4058 −1.11204 −0.556022 0.831167i \(-0.687673\pi\)
−0.556022 + 0.831167i \(0.687673\pi\)
\(444\) −6.70910 −0.318400
\(445\) 0 0
\(446\) 1.10700 0.0524180
\(447\) −44.5419 −2.10676
\(448\) 0.0966726 0.00456735
\(449\) 35.8887 1.69369 0.846846 0.531838i \(-0.178498\pi\)
0.846846 + 0.531838i \(0.178498\pi\)
\(450\) 0 0
\(451\) 60.5449 2.85095
\(452\) 1.74387 0.0820247
\(453\) 11.1479 0.523775
\(454\) −2.74728 −0.128936
\(455\) 0 0
\(456\) 22.3875 1.04839
\(457\) 23.4014 1.09467 0.547336 0.836913i \(-0.315642\pi\)
0.547336 + 0.836913i \(0.315642\pi\)
\(458\) 3.93623 0.183928
\(459\) −6.88820 −0.321514
\(460\) 0 0
\(461\) 17.9146 0.834366 0.417183 0.908822i \(-0.363017\pi\)
0.417183 + 0.908822i \(0.363017\pi\)
\(462\) 26.9538 1.25400
\(463\) 32.0159 1.48790 0.743952 0.668233i \(-0.232949\pi\)
0.743952 + 0.668233i \(0.232949\pi\)
\(464\) −4.09757 −0.190225
\(465\) 0 0
\(466\) −8.72279 −0.404075
\(467\) 12.2949 0.568940 0.284470 0.958685i \(-0.408182\pi\)
0.284470 + 0.958685i \(0.408182\pi\)
\(468\) −6.68095 −0.308827
\(469\) −46.4583 −2.14524
\(470\) 0 0
\(471\) 14.3259 0.660101
\(472\) −13.1439 −0.604997
\(473\) −26.5353 −1.22010
\(474\) −11.5429 −0.530184
\(475\) 0 0
\(476\) 32.5841 1.49349
\(477\) −3.26148 −0.149333
\(478\) −12.2297 −0.559373
\(479\) 15.1555 0.692472 0.346236 0.938148i \(-0.387460\pi\)
0.346236 + 0.938148i \(0.387460\pi\)
\(480\) 0 0
\(481\) −3.02746 −0.138040
\(482\) −0.630644 −0.0287251
\(483\) −23.8909 −1.08707
\(484\) −21.8622 −0.993735
\(485\) 0 0
\(486\) 12.8508 0.582922
\(487\) 15.5008 0.702407 0.351204 0.936299i \(-0.385773\pi\)
0.351204 + 0.936299i \(0.385773\pi\)
\(488\) 0.174116 0.00788186
\(489\) −2.81536 −0.127315
\(490\) 0 0
\(491\) 12.5838 0.567901 0.283950 0.958839i \(-0.408355\pi\)
0.283950 + 0.958839i \(0.408355\pi\)
\(492\) −45.6894 −2.05984
\(493\) 12.7712 0.575184
\(494\) 4.49346 0.202170
\(495\) 0 0
\(496\) −0.353521 −0.0158736
\(497\) −19.1625 −0.859553
\(498\) 3.76542 0.168732
\(499\) −41.9638 −1.87856 −0.939280 0.343152i \(-0.888505\pi\)
−0.939280 + 0.343152i \(0.888505\pi\)
\(500\) 0 0
\(501\) −2.10517 −0.0940521
\(502\) −9.10910 −0.406559
\(503\) −7.23458 −0.322574 −0.161287 0.986908i \(-0.551564\pi\)
−0.161287 + 0.986908i \(0.551564\pi\)
\(504\) −20.6334 −0.919085
\(505\) 0 0
\(506\) −8.68756 −0.386209
\(507\) 23.7133 1.05314
\(508\) 0.584094 0.0259150
\(509\) 43.2945 1.91899 0.959497 0.281718i \(-0.0909041\pi\)
0.959497 + 0.281718i \(0.0909041\pi\)
\(510\) 0 0
\(511\) 51.8223 2.29248
\(512\) 18.0813 0.799090
\(513\) 5.25712 0.232108
\(514\) −18.2259 −0.803909
\(515\) 0 0
\(516\) 20.0245 0.881531
\(517\) −53.2893 −2.34366
\(518\) −4.15885 −0.182729
\(519\) 28.8899 1.26813
\(520\) 0 0
\(521\) −22.7560 −0.996961 −0.498480 0.866901i \(-0.666108\pi\)
−0.498480 + 0.866901i \(0.666108\pi\)
\(522\) −3.59715 −0.157443
\(523\) −36.7529 −1.60709 −0.803547 0.595241i \(-0.797056\pi\)
−0.803547 + 0.595241i \(0.797056\pi\)
\(524\) −8.79440 −0.384185
\(525\) 0 0
\(526\) −16.5908 −0.723395
\(527\) 1.10184 0.0479970
\(528\) 20.5662 0.895029
\(529\) −15.2997 −0.665203
\(530\) 0 0
\(531\) 14.2708 0.619301
\(532\) −24.8684 −1.07818
\(533\) −20.6172 −0.893030
\(534\) 6.41884 0.277770
\(535\) 0 0
\(536\) 28.6619 1.23801
\(537\) 23.4410 1.01156
\(538\) −5.94351 −0.256243
\(539\) −32.5631 −1.40259
\(540\) 0 0
\(541\) −19.7726 −0.850089 −0.425045 0.905172i \(-0.639742\pi\)
−0.425045 + 0.905172i \(0.639742\pi\)
\(542\) −3.69885 −0.158879
\(543\) 39.8617 1.71063
\(544\) −31.2633 −1.34040
\(545\) 0 0
\(546\) −9.17851 −0.392804
\(547\) 9.94319 0.425140 0.212570 0.977146i \(-0.431817\pi\)
0.212570 + 0.977146i \(0.431817\pi\)
\(548\) −27.4846 −1.17408
\(549\) −0.189044 −0.00806820
\(550\) 0 0
\(551\) −9.74703 −0.415238
\(552\) 14.7392 0.627342
\(553\) 28.8268 1.22584
\(554\) −3.53410 −0.150149
\(555\) 0 0
\(556\) 31.4167 1.33236
\(557\) 7.84163 0.332260 0.166130 0.986104i \(-0.446873\pi\)
0.166130 + 0.986104i \(0.446873\pi\)
\(558\) −0.310347 −0.0131380
\(559\) 9.03601 0.382182
\(560\) 0 0
\(561\) −64.1001 −2.70631
\(562\) −2.97455 −0.125474
\(563\) 19.2105 0.809625 0.404813 0.914400i \(-0.367337\pi\)
0.404813 + 0.914400i \(0.367337\pi\)
\(564\) 40.2140 1.69332
\(565\) 0 0
\(566\) −12.1127 −0.509134
\(567\) −37.9861 −1.59526
\(568\) 11.8221 0.496043
\(569\) 25.1338 1.05366 0.526831 0.849970i \(-0.323380\pi\)
0.526831 + 0.849970i \(0.323380\pi\)
\(570\) 0 0
\(571\) −18.1912 −0.761277 −0.380638 0.924724i \(-0.624296\pi\)
−0.380638 + 0.924724i \(0.624296\pi\)
\(572\) 13.4465 0.562227
\(573\) 55.5861 2.32214
\(574\) −28.3220 −1.18214
\(575\) 0 0
\(576\) 0.0647545 0.00269810
\(577\) −28.9417 −1.20486 −0.602429 0.798172i \(-0.705800\pi\)
−0.602429 + 0.798172i \(0.705800\pi\)
\(578\) 8.51317 0.354101
\(579\) 17.5208 0.728140
\(580\) 0 0
\(581\) −9.40358 −0.390126
\(582\) 13.3101 0.551721
\(583\) 6.56427 0.271864
\(584\) −31.9712 −1.32298
\(585\) 0 0
\(586\) −8.20724 −0.339038
\(587\) 21.9772 0.907098 0.453549 0.891231i \(-0.350158\pi\)
0.453549 + 0.891231i \(0.350158\pi\)
\(588\) 24.5733 1.01339
\(589\) −0.840934 −0.0346501
\(590\) 0 0
\(591\) 4.34054 0.178546
\(592\) −3.17327 −0.130421
\(593\) −15.0138 −0.616544 −0.308272 0.951298i \(-0.599751\pi\)
−0.308272 + 0.951298i \(0.599751\pi\)
\(594\) −3.90486 −0.160218
\(595\) 0 0
\(596\) −30.5248 −1.25035
\(597\) −8.10529 −0.331728
\(598\) 2.95835 0.120976
\(599\) −31.7086 −1.29558 −0.647789 0.761820i \(-0.724306\pi\)
−0.647789 + 0.761820i \(0.724306\pi\)
\(600\) 0 0
\(601\) 5.63682 0.229931 0.114965 0.993369i \(-0.463324\pi\)
0.114965 + 0.993369i \(0.463324\pi\)
\(602\) 12.4129 0.505910
\(603\) −31.1193 −1.26727
\(604\) 7.63973 0.310856
\(605\) 0 0
\(606\) −10.2377 −0.415878
\(607\) −39.7534 −1.61354 −0.806770 0.590866i \(-0.798786\pi\)
−0.806770 + 0.590866i \(0.798786\pi\)
\(608\) 23.8603 0.967664
\(609\) 19.9096 0.806780
\(610\) 0 0
\(611\) 18.1465 0.734127
\(612\) 21.8259 0.882259
\(613\) −30.3066 −1.22407 −0.612035 0.790831i \(-0.709649\pi\)
−0.612035 + 0.790831i \(0.709649\pi\)
\(614\) 7.27588 0.293631
\(615\) 0 0
\(616\) 41.5281 1.67322
\(617\) −18.2240 −0.733670 −0.366835 0.930286i \(-0.619559\pi\)
−0.366835 + 0.930286i \(0.619559\pi\)
\(618\) −0.154834 −0.00622836
\(619\) −26.3431 −1.05882 −0.529409 0.848366i \(-0.677586\pi\)
−0.529409 + 0.848366i \(0.677586\pi\)
\(620\) 0 0
\(621\) 3.46112 0.138890
\(622\) −14.6021 −0.585491
\(623\) −16.0301 −0.642233
\(624\) −7.00335 −0.280359
\(625\) 0 0
\(626\) 4.97516 0.198848
\(627\) 48.9216 1.95374
\(628\) 9.81759 0.391765
\(629\) 9.89035 0.394354
\(630\) 0 0
\(631\) 18.4187 0.733236 0.366618 0.930372i \(-0.380516\pi\)
0.366618 + 0.930372i \(0.380516\pi\)
\(632\) −17.7844 −0.707424
\(633\) 5.45238 0.216713
\(634\) 1.50298 0.0596910
\(635\) 0 0
\(636\) −4.95364 −0.196425
\(637\) 11.0886 0.439347
\(638\) 7.23985 0.286629
\(639\) −12.8356 −0.507770
\(640\) 0 0
\(641\) −12.6956 −0.501446 −0.250723 0.968059i \(-0.580668\pi\)
−0.250723 + 0.968059i \(0.580668\pi\)
\(642\) −26.8123 −1.05820
\(643\) 35.1591 1.38654 0.693270 0.720678i \(-0.256169\pi\)
0.693270 + 0.720678i \(0.256169\pi\)
\(644\) −16.3725 −0.645169
\(645\) 0 0
\(646\) −14.6796 −0.577561
\(647\) −19.7535 −0.776588 −0.388294 0.921535i \(-0.626936\pi\)
−0.388294 + 0.921535i \(0.626936\pi\)
\(648\) 23.4351 0.920616
\(649\) −28.7224 −1.12745
\(650\) 0 0
\(651\) 1.71772 0.0673228
\(652\) −1.92939 −0.0755606
\(653\) 44.0129 1.72236 0.861178 0.508303i \(-0.169727\pi\)
0.861178 + 0.508303i \(0.169727\pi\)
\(654\) −21.6124 −0.845113
\(655\) 0 0
\(656\) −21.6102 −0.843736
\(657\) 34.7123 1.35426
\(658\) 24.9280 0.971793
\(659\) −32.2032 −1.25446 −0.627229 0.778835i \(-0.715811\pi\)
−0.627229 + 0.778835i \(0.715811\pi\)
\(660\) 0 0
\(661\) 29.5485 1.14930 0.574651 0.818398i \(-0.305138\pi\)
0.574651 + 0.818398i \(0.305138\pi\)
\(662\) −1.92724 −0.0749043
\(663\) 21.8278 0.847723
\(664\) 5.80143 0.225139
\(665\) 0 0
\(666\) −2.78573 −0.107945
\(667\) −6.41714 −0.248473
\(668\) −1.44269 −0.0558192
\(669\) −4.10412 −0.158674
\(670\) 0 0
\(671\) 0.380482 0.0146883
\(672\) −48.7380 −1.88011
\(673\) −28.4424 −1.09637 −0.548186 0.836356i \(-0.684681\pi\)
−0.548186 + 0.836356i \(0.684681\pi\)
\(674\) 10.2010 0.392927
\(675\) 0 0
\(676\) 16.2508 0.625032
\(677\) −16.9259 −0.650514 −0.325257 0.945626i \(-0.605451\pi\)
−0.325257 + 0.945626i \(0.605451\pi\)
\(678\) 1.60477 0.0616310
\(679\) −33.2400 −1.27563
\(680\) 0 0
\(681\) 10.1853 0.390303
\(682\) 0.624625 0.0239181
\(683\) 29.0324 1.11090 0.555448 0.831551i \(-0.312547\pi\)
0.555448 + 0.831551i \(0.312547\pi\)
\(684\) −16.6576 −0.636921
\(685\) 0 0
\(686\) −1.02307 −0.0390612
\(687\) −14.5933 −0.556767
\(688\) 9.47121 0.361087
\(689\) −2.23531 −0.0851586
\(690\) 0 0
\(691\) −21.5798 −0.820935 −0.410467 0.911875i \(-0.634634\pi\)
−0.410467 + 0.911875i \(0.634634\pi\)
\(692\) 19.7984 0.752623
\(693\) −45.0886 −1.71277
\(694\) −15.4329 −0.585823
\(695\) 0 0
\(696\) −12.2830 −0.465587
\(697\) 67.3540 2.55121
\(698\) −8.22790 −0.311431
\(699\) 32.3390 1.22317
\(700\) 0 0
\(701\) −13.2247 −0.499489 −0.249744 0.968312i \(-0.580347\pi\)
−0.249744 + 0.968312i \(0.580347\pi\)
\(702\) 1.32971 0.0501867
\(703\) −7.54838 −0.284692
\(704\) −0.130329 −0.00491196
\(705\) 0 0
\(706\) −14.9329 −0.562006
\(707\) 25.5672 0.961552
\(708\) 21.6750 0.814595
\(709\) 18.8314 0.707228 0.353614 0.935391i \(-0.384953\pi\)
0.353614 + 0.935391i \(0.384953\pi\)
\(710\) 0 0
\(711\) 19.3091 0.724149
\(712\) 9.88960 0.370628
\(713\) −0.553644 −0.0207341
\(714\) 29.9851 1.12216
\(715\) 0 0
\(716\) 16.0643 0.600351
\(717\) 45.3406 1.69328
\(718\) −16.1129 −0.601328
\(719\) −37.7936 −1.40946 −0.704731 0.709475i \(-0.748932\pi\)
−0.704731 + 0.709475i \(0.748932\pi\)
\(720\) 0 0
\(721\) 0.386677 0.0144006
\(722\) −0.778675 −0.0289793
\(723\) 2.33806 0.0869535
\(724\) 27.3175 1.01525
\(725\) 0 0
\(726\) −20.1184 −0.746664
\(727\) 9.96692 0.369653 0.184826 0.982771i \(-0.440828\pi\)
0.184826 + 0.982771i \(0.440828\pi\)
\(728\) −14.1415 −0.524118
\(729\) −16.6957 −0.618359
\(730\) 0 0
\(731\) −29.5196 −1.09182
\(732\) −0.287126 −0.0106125
\(733\) −46.8282 −1.72964 −0.864819 0.502083i \(-0.832567\pi\)
−0.864819 + 0.502083i \(0.832567\pi\)
\(734\) −5.27071 −0.194546
\(735\) 0 0
\(736\) 15.7089 0.579037
\(737\) 62.6326 2.30710
\(738\) −18.9710 −0.698333
\(739\) 31.1213 1.14482 0.572408 0.819969i \(-0.306009\pi\)
0.572408 + 0.819969i \(0.306009\pi\)
\(740\) 0 0
\(741\) −16.6591 −0.611989
\(742\) −3.07067 −0.112728
\(743\) 22.3465 0.819815 0.409907 0.912127i \(-0.365561\pi\)
0.409907 + 0.912127i \(0.365561\pi\)
\(744\) −1.05973 −0.0388516
\(745\) 0 0
\(746\) 15.5728 0.570159
\(747\) −6.29883 −0.230462
\(748\) −43.9282 −1.60617
\(749\) 66.9599 2.44666
\(750\) 0 0
\(751\) −41.3430 −1.50863 −0.754313 0.656515i \(-0.772030\pi\)
−0.754313 + 0.656515i \(0.772030\pi\)
\(752\) 19.0204 0.693604
\(753\) 33.7713 1.23069
\(754\) −2.46537 −0.0897834
\(755\) 0 0
\(756\) −7.35908 −0.267647
\(757\) 5.70532 0.207364 0.103682 0.994611i \(-0.466938\pi\)
0.103682 + 0.994611i \(0.466938\pi\)
\(758\) 11.7674 0.427411
\(759\) 32.2084 1.16909
\(760\) 0 0
\(761\) 15.1656 0.549754 0.274877 0.961479i \(-0.411363\pi\)
0.274877 + 0.961479i \(0.411363\pi\)
\(762\) 0.537505 0.0194718
\(763\) 53.9739 1.95399
\(764\) 38.0935 1.37817
\(765\) 0 0
\(766\) −7.55388 −0.272933
\(767\) 9.78075 0.353162
\(768\) 16.7924 0.605943
\(769\) 5.23538 0.188793 0.0943964 0.995535i \(-0.469908\pi\)
0.0943964 + 0.995535i \(0.469908\pi\)
\(770\) 0 0
\(771\) 67.5710 2.43351
\(772\) 12.0071 0.432145
\(773\) 27.0422 0.972641 0.486321 0.873780i \(-0.338339\pi\)
0.486321 + 0.873780i \(0.338339\pi\)
\(774\) 8.31453 0.298860
\(775\) 0 0
\(776\) 20.5070 0.736160
\(777\) 15.4186 0.553139
\(778\) −4.80973 −0.172437
\(779\) −51.4050 −1.84177
\(780\) 0 0
\(781\) 25.8338 0.924407
\(782\) −9.66459 −0.345605
\(783\) −2.88435 −0.103078
\(784\) 11.6227 0.415096
\(785\) 0 0
\(786\) −8.09294 −0.288666
\(787\) 25.6221 0.913329 0.456665 0.889639i \(-0.349044\pi\)
0.456665 + 0.889639i \(0.349044\pi\)
\(788\) 2.97459 0.105966
\(789\) 61.5092 2.18979
\(790\) 0 0
\(791\) −4.00769 −0.142497
\(792\) 27.8169 0.988431
\(793\) −0.129565 −0.00460097
\(794\) 21.0643 0.747545
\(795\) 0 0
\(796\) −5.55460 −0.196878
\(797\) −28.8151 −1.02068 −0.510341 0.859972i \(-0.670481\pi\)
−0.510341 + 0.859972i \(0.670481\pi\)
\(798\) −22.8848 −0.810114
\(799\) −59.2823 −2.09726
\(800\) 0 0
\(801\) −10.7375 −0.379391
\(802\) −1.09436 −0.0386430
\(803\) −69.8642 −2.46545
\(804\) −47.2649 −1.66690
\(805\) 0 0
\(806\) −0.212702 −0.00749210
\(807\) 22.0351 0.775671
\(808\) −15.7734 −0.554905
\(809\) 20.6265 0.725188 0.362594 0.931947i \(-0.381891\pi\)
0.362594 + 0.931947i \(0.381891\pi\)
\(810\) 0 0
\(811\) 45.2088 1.58749 0.793747 0.608248i \(-0.208127\pi\)
0.793747 + 0.608248i \(0.208127\pi\)
\(812\) 13.6442 0.478817
\(813\) 13.7132 0.480942
\(814\) 5.60675 0.196516
\(815\) 0 0
\(816\) 22.8791 0.800930
\(817\) 22.5295 0.788208
\(818\) −2.40848 −0.0842106
\(819\) 15.3539 0.536509
\(820\) 0 0
\(821\) 23.2674 0.812039 0.406019 0.913864i \(-0.366917\pi\)
0.406019 + 0.913864i \(0.366917\pi\)
\(822\) −25.2924 −0.882172
\(823\) −2.12031 −0.0739095 −0.0369547 0.999317i \(-0.511766\pi\)
−0.0369547 + 0.999317i \(0.511766\pi\)
\(824\) −0.238556 −0.00831049
\(825\) 0 0
\(826\) 13.4359 0.467495
\(827\) 16.5417 0.575213 0.287606 0.957749i \(-0.407141\pi\)
0.287606 + 0.957749i \(0.407141\pi\)
\(828\) −10.9669 −0.381125
\(829\) 2.61675 0.0908835 0.0454418 0.998967i \(-0.485530\pi\)
0.0454418 + 0.998967i \(0.485530\pi\)
\(830\) 0 0
\(831\) 13.1024 0.454516
\(832\) 0.0443806 0.00153862
\(833\) −36.2252 −1.25513
\(834\) 28.9108 1.00110
\(835\) 0 0
\(836\) 33.5263 1.15953
\(837\) −0.248850 −0.00860152
\(838\) 0.403668 0.0139445
\(839\) 40.9480 1.41368 0.706841 0.707373i \(-0.250120\pi\)
0.706841 + 0.707373i \(0.250120\pi\)
\(840\) 0 0
\(841\) −23.6522 −0.815594
\(842\) 8.37275 0.288544
\(843\) 11.0279 0.379821
\(844\) 3.73655 0.128617
\(845\) 0 0
\(846\) 16.6976 0.574074
\(847\) 50.2428 1.72636
\(848\) −2.34297 −0.0804580
\(849\) 44.9068 1.54120
\(850\) 0 0
\(851\) −4.96961 −0.170356
\(852\) −19.4952 −0.667893
\(853\) −17.3933 −0.595537 −0.297768 0.954638i \(-0.596242\pi\)
−0.297768 + 0.954638i \(0.596242\pi\)
\(854\) −0.177984 −0.00609049
\(855\) 0 0
\(856\) −41.3101 −1.41195
\(857\) −16.7101 −0.570805 −0.285402 0.958408i \(-0.592127\pi\)
−0.285402 + 0.958408i \(0.592127\pi\)
\(858\) 12.3740 0.422441
\(859\) 13.9877 0.477255 0.238628 0.971111i \(-0.423302\pi\)
0.238628 + 0.971111i \(0.423302\pi\)
\(860\) 0 0
\(861\) 105.002 3.57845
\(862\) −16.9126 −0.576045
\(863\) 23.1575 0.788290 0.394145 0.919048i \(-0.371041\pi\)
0.394145 + 0.919048i \(0.371041\pi\)
\(864\) 7.06078 0.240212
\(865\) 0 0
\(866\) 13.0791 0.444445
\(867\) −31.5619 −1.07190
\(868\) 1.17716 0.0399556
\(869\) −38.8628 −1.31833
\(870\) 0 0
\(871\) −21.3281 −0.722676
\(872\) −33.2986 −1.12763
\(873\) −22.2652 −0.753564
\(874\) 7.37608 0.249499
\(875\) 0 0
\(876\) 52.7221 1.78131
\(877\) −40.3104 −1.36119 −0.680594 0.732661i \(-0.738278\pi\)
−0.680594 + 0.732661i \(0.738278\pi\)
\(878\) −8.03326 −0.271110
\(879\) 30.4276 1.02630
\(880\) 0 0
\(881\) −23.2156 −0.782152 −0.391076 0.920358i \(-0.627897\pi\)
−0.391076 + 0.920358i \(0.627897\pi\)
\(882\) 10.2033 0.343562
\(883\) 0.920055 0.0309623 0.0154812 0.999880i \(-0.495072\pi\)
0.0154812 + 0.999880i \(0.495072\pi\)
\(884\) 14.9587 0.503117
\(885\) 0 0
\(886\) −14.7608 −0.495897
\(887\) 28.9082 0.970643 0.485321 0.874336i \(-0.338703\pi\)
0.485321 + 0.874336i \(0.338703\pi\)
\(888\) −9.51233 −0.319213
\(889\) −1.34234 −0.0450207
\(890\) 0 0
\(891\) 51.2109 1.71563
\(892\) −2.81257 −0.0941720
\(893\) 45.2446 1.51405
\(894\) −28.0901 −0.939473
\(895\) 0 0
\(896\) −41.6299 −1.39076
\(897\) −10.9679 −0.366206
\(898\) 22.6330 0.755273
\(899\) 0.461384 0.0153880
\(900\) 0 0
\(901\) 7.30250 0.243282
\(902\) 38.1823 1.27133
\(903\) −46.0196 −1.53144
\(904\) 2.47250 0.0822341
\(905\) 0 0
\(906\) 7.03037 0.233568
\(907\) −36.3884 −1.20826 −0.604128 0.796887i \(-0.706478\pi\)
−0.604128 + 0.796887i \(0.706478\pi\)
\(908\) 6.98007 0.231642
\(909\) 17.1257 0.568024
\(910\) 0 0
\(911\) −10.8289 −0.358776 −0.179388 0.983778i \(-0.557412\pi\)
−0.179388 + 0.983778i \(0.557412\pi\)
\(912\) −17.4615 −0.578208
\(913\) 12.6774 0.419562
\(914\) 14.7580 0.488150
\(915\) 0 0
\(916\) −10.0008 −0.330437
\(917\) 20.2110 0.667424
\(918\) −4.34401 −0.143374
\(919\) 28.3998 0.936823 0.468412 0.883510i \(-0.344826\pi\)
0.468412 + 0.883510i \(0.344826\pi\)
\(920\) 0 0
\(921\) −26.9747 −0.888847
\(922\) 11.2977 0.372071
\(923\) −8.79712 −0.289561
\(924\) −68.4820 −2.25289
\(925\) 0 0
\(926\) 20.1906 0.663505
\(927\) 0.259009 0.00850696
\(928\) −13.0911 −0.429737
\(929\) −32.8431 −1.07755 −0.538774 0.842451i \(-0.681112\pi\)
−0.538774 + 0.842451i \(0.681112\pi\)
\(930\) 0 0
\(931\) 27.6473 0.906104
\(932\) 22.1621 0.725944
\(933\) 54.1361 1.77233
\(934\) 7.75371 0.253709
\(935\) 0 0
\(936\) −9.47242 −0.309616
\(937\) −4.11911 −0.134566 −0.0672828 0.997734i \(-0.521433\pi\)
−0.0672828 + 0.997734i \(0.521433\pi\)
\(938\) −29.2986 −0.956635
\(939\) −18.4450 −0.601930
\(940\) 0 0
\(941\) −8.26409 −0.269402 −0.134701 0.990886i \(-0.543007\pi\)
−0.134701 + 0.990886i \(0.543007\pi\)
\(942\) 9.03452 0.294361
\(943\) −33.8434 −1.10209
\(944\) 10.2518 0.333668
\(945\) 0 0
\(946\) −16.7344 −0.544081
\(947\) −43.3860 −1.40986 −0.704928 0.709279i \(-0.749021\pi\)
−0.704928 + 0.709279i \(0.749021\pi\)
\(948\) 29.3273 0.952506
\(949\) 23.7907 0.772277
\(950\) 0 0
\(951\) −5.57218 −0.180690
\(952\) 46.1985 1.49730
\(953\) 26.7790 0.867457 0.433728 0.901044i \(-0.357198\pi\)
0.433728 + 0.901044i \(0.357198\pi\)
\(954\) −2.05684 −0.0665925
\(955\) 0 0
\(956\) 31.0722 1.00495
\(957\) −26.8412 −0.867651
\(958\) 9.55772 0.308796
\(959\) 63.1640 2.03967
\(960\) 0 0
\(961\) −30.9602 −0.998716
\(962\) −1.90925 −0.0615567
\(963\) 44.8519 1.44533
\(964\) 1.60229 0.0516062
\(965\) 0 0
\(966\) −15.0666 −0.484761
\(967\) −7.46107 −0.239932 −0.119966 0.992778i \(-0.538279\pi\)
−0.119966 + 0.992778i \(0.538279\pi\)
\(968\) −30.9967 −0.996272
\(969\) 54.4234 1.74833
\(970\) 0 0
\(971\) −21.4365 −0.687931 −0.343966 0.938982i \(-0.611770\pi\)
−0.343966 + 0.938982i \(0.611770\pi\)
\(972\) −32.6501 −1.04725
\(973\) −72.2006 −2.31465
\(974\) 9.77548 0.313226
\(975\) 0 0
\(976\) −0.135805 −0.00434700
\(977\) 30.9584 0.990446 0.495223 0.868766i \(-0.335086\pi\)
0.495223 + 0.868766i \(0.335086\pi\)
\(978\) −1.77549 −0.0567740
\(979\) 21.6110 0.690690
\(980\) 0 0
\(981\) 36.1535 1.15429
\(982\) 7.93592 0.253246
\(983\) −13.6397 −0.435040 −0.217520 0.976056i \(-0.569797\pi\)
−0.217520 + 0.976056i \(0.569797\pi\)
\(984\) −64.7796 −2.06510
\(985\) 0 0
\(986\) 8.05406 0.256494
\(987\) −92.4183 −2.94171
\(988\) −11.4166 −0.363211
\(989\) 14.8327 0.471653
\(990\) 0 0
\(991\) −46.0458 −1.46269 −0.731347 0.682006i \(-0.761108\pi\)
−0.731347 + 0.682006i \(0.761108\pi\)
\(992\) −1.12945 −0.0358600
\(993\) 7.14508 0.226742
\(994\) −12.0847 −0.383303
\(995\) 0 0
\(996\) −9.56685 −0.303137
\(997\) −57.1775 −1.81083 −0.905415 0.424527i \(-0.860440\pi\)
−0.905415 + 0.424527i \(0.860440\pi\)
\(998\) −26.4643 −0.837711
\(999\) −2.23373 −0.0706719
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.1.4 7
5.4 even 2 241.2.a.a.1.4 7
15.14 odd 2 2169.2.a.e.1.4 7
20.19 odd 2 3856.2.a.j.1.1 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
241.2.a.a.1.4 7 5.4 even 2
2169.2.a.e.1.4 7 15.14 odd 2
3856.2.a.j.1.1 7 20.19 odd 2
6025.2.a.f.1.4 7 1.1 even 1 trivial