Properties

Label 825.4.a.m.1.2
Level $825$
Weight $4$
Character 825.1
Self dual yes
Analytic conductor $48.677$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.56155 q^{2} -3.00000 q^{3} -1.43845 q^{4} -7.68466 q^{6} -6.24621 q^{7} -24.1771 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.56155 q^{2} -3.00000 q^{3} -1.43845 q^{4} -7.68466 q^{6} -6.24621 q^{7} -24.1771 q^{8} +9.00000 q^{9} -11.0000 q^{11} +4.31534 q^{12} +49.1231 q^{13} -16.0000 q^{14} -50.4233 q^{16} -82.7083 q^{17} +23.0540 q^{18} -130.354 q^{19} +18.7386 q^{21} -28.1771 q^{22} +185.693 q^{23} +72.5312 q^{24} +125.831 q^{26} -27.0000 q^{27} +8.98485 q^{28} -8.90720 q^{29} +5.26137 q^{31} +64.2547 q^{32} +33.0000 q^{33} -211.862 q^{34} -12.9460 q^{36} +416.894 q^{37} -333.909 q^{38} -147.369 q^{39} -298.479 q^{41} +48.0000 q^{42} +513.633 q^{43} +15.8229 q^{44} +475.663 q^{46} -557.295 q^{47} +151.270 q^{48} -303.985 q^{49} +248.125 q^{51} -70.6610 q^{52} +168.064 q^{53} -69.1619 q^{54} +151.015 q^{56} +391.062 q^{57} -22.8163 q^{58} +618.773 q^{59} +786.405 q^{61} +13.4773 q^{62} -56.2159 q^{63} +567.978 q^{64} +84.5312 q^{66} +339.015 q^{67} +118.972 q^{68} -557.080 q^{69} +1120.71 q^{71} -217.594 q^{72} +123.430 q^{73} +1067.90 q^{74} +187.508 q^{76} +68.7083 q^{77} -377.494 q^{78} -309.835 q^{79} +81.0000 q^{81} -764.570 q^{82} +1021.22 q^{83} -26.9545 q^{84} +1315.70 q^{86} +26.7216 q^{87} +265.948 q^{88} -141.879 q^{89} -306.833 q^{91} -267.110 q^{92} -15.7841 q^{93} -1427.54 q^{94} -192.764 q^{96} -798.345 q^{97} -778.673 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 6 q^{3} - 7 q^{4} - 3 q^{6} + 4 q^{7} - 3 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 6 q^{3} - 7 q^{4} - 3 q^{6} + 4 q^{7} - 3 q^{8} + 18 q^{9} - 22 q^{11} + 21 q^{12} + 90 q^{13} - 32 q^{14} - 39 q^{16} + 16 q^{17} + 9 q^{18} - 170 q^{19} - 12 q^{21} - 11 q^{22} + 124 q^{23} + 9 q^{24} + 62 q^{26} - 54 q^{27} - 48 q^{28} - 158 q^{29} + 60 q^{31} - 123 q^{32} + 66 q^{33} - 366 q^{34} - 63 q^{36} + 372 q^{37} - 272 q^{38} - 270 q^{39} + 38 q^{41} + 96 q^{42} + 516 q^{43} + 77 q^{44} + 572 q^{46} - 224 q^{47} + 117 q^{48} - 542 q^{49} - 48 q^{51} - 298 q^{52} - 472 q^{53} - 27 q^{54} + 368 q^{56} + 510 q^{57} + 210 q^{58} + 248 q^{59} + 72 q^{61} - 72 q^{62} + 36 q^{63} + 769 q^{64} + 33 q^{66} + 744 q^{67} - 430 q^{68} - 372 q^{69} + 2060 q^{71} - 27 q^{72} + 486 q^{73} + 1138 q^{74} + 408 q^{76} - 44 q^{77} - 186 q^{78} + 642 q^{79} + 162 q^{81} - 1290 q^{82} + 286 q^{83} + 144 q^{84} + 1312 q^{86} + 474 q^{87} + 33 q^{88} + 244 q^{89} + 112 q^{91} + 76 q^{92} - 180 q^{93} - 1948 q^{94} + 369 q^{96} + 168 q^{97} - 407 q^{98} - 198 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.56155 0.905646 0.452823 0.891601i \(-0.350417\pi\)
0.452823 + 0.891601i \(0.350417\pi\)
\(3\) −3.00000 −0.577350
\(4\) −1.43845 −0.179806
\(5\) 0 0
\(6\) −7.68466 −0.522875
\(7\) −6.24621 −0.337264 −0.168632 0.985679i \(-0.553935\pi\)
−0.168632 + 0.985679i \(0.553935\pi\)
\(8\) −24.1771 −1.06849
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 4.31534 0.103811
\(13\) 49.1231 1.04802 0.524011 0.851711i \(-0.324435\pi\)
0.524011 + 0.851711i \(0.324435\pi\)
\(14\) −16.0000 −0.305441
\(15\) 0 0
\(16\) −50.4233 −0.787864
\(17\) −82.7083 −1.17998 −0.589992 0.807409i \(-0.700869\pi\)
−0.589992 + 0.807409i \(0.700869\pi\)
\(18\) 23.0540 0.301882
\(19\) −130.354 −1.57396 −0.786981 0.616977i \(-0.788357\pi\)
−0.786981 + 0.616977i \(0.788357\pi\)
\(20\) 0 0
\(21\) 18.7386 0.194719
\(22\) −28.1771 −0.273062
\(23\) 185.693 1.68347 0.841733 0.539895i \(-0.181536\pi\)
0.841733 + 0.539895i \(0.181536\pi\)
\(24\) 72.5312 0.616891
\(25\) 0 0
\(26\) 125.831 0.949137
\(27\) −27.0000 −0.192450
\(28\) 8.98485 0.0606420
\(29\) −8.90720 −0.0570354 −0.0285177 0.999593i \(-0.509079\pi\)
−0.0285177 + 0.999593i \(0.509079\pi\)
\(30\) 0 0
\(31\) 5.26137 0.0304829 0.0152414 0.999884i \(-0.495148\pi\)
0.0152414 + 0.999884i \(0.495148\pi\)
\(32\) 64.2547 0.354961
\(33\) 33.0000 0.174078
\(34\) −211.862 −1.06865
\(35\) 0 0
\(36\) −12.9460 −0.0599353
\(37\) 416.894 1.85235 0.926175 0.377094i \(-0.123077\pi\)
0.926175 + 0.377094i \(0.123077\pi\)
\(38\) −333.909 −1.42545
\(39\) −147.369 −0.605076
\(40\) 0 0
\(41\) −298.479 −1.13694 −0.568471 0.822703i \(-0.692465\pi\)
−0.568471 + 0.822703i \(0.692465\pi\)
\(42\) 48.0000 0.176347
\(43\) 513.633 1.82159 0.910793 0.412863i \(-0.135471\pi\)
0.910793 + 0.412863i \(0.135471\pi\)
\(44\) 15.8229 0.0542135
\(45\) 0 0
\(46\) 475.663 1.52462
\(47\) −557.295 −1.72957 −0.864786 0.502140i \(-0.832546\pi\)
−0.864786 + 0.502140i \(0.832546\pi\)
\(48\) 151.270 0.454873
\(49\) −303.985 −0.886253
\(50\) 0 0
\(51\) 248.125 0.681264
\(52\) −70.6610 −0.188441
\(53\) 168.064 0.435574 0.217787 0.975996i \(-0.430116\pi\)
0.217787 + 0.975996i \(0.430116\pi\)
\(54\) −69.1619 −0.174292
\(55\) 0 0
\(56\) 151.015 0.360362
\(57\) 391.062 0.908728
\(58\) −22.8163 −0.0516539
\(59\) 618.773 1.36538 0.682689 0.730709i \(-0.260810\pi\)
0.682689 + 0.730709i \(0.260810\pi\)
\(60\) 0 0
\(61\) 786.405 1.65064 0.825319 0.564667i \(-0.190996\pi\)
0.825319 + 0.564667i \(0.190996\pi\)
\(62\) 13.4773 0.0276067
\(63\) −56.2159 −0.112421
\(64\) 567.978 1.10933
\(65\) 0 0
\(66\) 84.5312 0.157653
\(67\) 339.015 0.618169 0.309084 0.951035i \(-0.399977\pi\)
0.309084 + 0.951035i \(0.399977\pi\)
\(68\) 118.972 0.212168
\(69\) −557.080 −0.971949
\(70\) 0 0
\(71\) 1120.71 1.87329 0.936645 0.350280i \(-0.113913\pi\)
0.936645 + 0.350280i \(0.113913\pi\)
\(72\) −217.594 −0.356162
\(73\) 123.430 0.197896 0.0989478 0.995093i \(-0.468452\pi\)
0.0989478 + 0.995093i \(0.468452\pi\)
\(74\) 1067.90 1.67757
\(75\) 0 0
\(76\) 187.508 0.283008
\(77\) 68.7083 0.101689
\(78\) −377.494 −0.547985
\(79\) −309.835 −0.441255 −0.220628 0.975358i \(-0.570811\pi\)
−0.220628 + 0.975358i \(0.570811\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −764.570 −1.02967
\(83\) 1021.22 1.35053 0.675263 0.737577i \(-0.264030\pi\)
0.675263 + 0.737577i \(0.264030\pi\)
\(84\) −26.9545 −0.0350117
\(85\) 0 0
\(86\) 1315.70 1.64971
\(87\) 26.7216 0.0329294
\(88\) 265.948 0.322161
\(89\) −141.879 −0.168979 −0.0844894 0.996424i \(-0.526926\pi\)
−0.0844894 + 0.996424i \(0.526926\pi\)
\(90\) 0 0
\(91\) −306.833 −0.353460
\(92\) −267.110 −0.302697
\(93\) −15.7841 −0.0175993
\(94\) −1427.54 −1.56638
\(95\) 0 0
\(96\) −192.764 −0.204937
\(97\) −798.345 −0.835666 −0.417833 0.908524i \(-0.637210\pi\)
−0.417833 + 0.908524i \(0.637210\pi\)
\(98\) −778.673 −0.802631
\(99\) −99.0000 −0.100504
\(100\) 0 0
\(101\) 241.400 0.237823 0.118912 0.992905i \(-0.462059\pi\)
0.118912 + 0.992905i \(0.462059\pi\)
\(102\) 635.585 0.616983
\(103\) −1168.38 −1.11771 −0.558853 0.829267i \(-0.688758\pi\)
−0.558853 + 0.829267i \(0.688758\pi\)
\(104\) −1187.65 −1.11980
\(105\) 0 0
\(106\) 430.506 0.394476
\(107\) 2106.82 1.90350 0.951748 0.306882i \(-0.0992857\pi\)
0.951748 + 0.306882i \(0.0992857\pi\)
\(108\) 38.8381 0.0346037
\(109\) 493.792 0.433914 0.216957 0.976181i \(-0.430387\pi\)
0.216957 + 0.976181i \(0.430387\pi\)
\(110\) 0 0
\(111\) −1250.68 −1.06945
\(112\) 314.955 0.265718
\(113\) −170.000 −0.141524 −0.0707622 0.997493i \(-0.522543\pi\)
−0.0707622 + 0.997493i \(0.522543\pi\)
\(114\) 1001.73 0.822986
\(115\) 0 0
\(116\) 12.8125 0.0102553
\(117\) 442.108 0.349341
\(118\) 1585.02 1.23655
\(119\) 516.614 0.397966
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 2014.42 1.49489
\(123\) 895.437 0.656414
\(124\) −7.56820 −0.00548100
\(125\) 0 0
\(126\) −144.000 −0.101814
\(127\) 948.182 0.662500 0.331250 0.943543i \(-0.392530\pi\)
0.331250 + 0.943543i \(0.392530\pi\)
\(128\) 940.868 0.649702
\(129\) −1540.90 −1.05169
\(130\) 0 0
\(131\) −1484.84 −0.990312 −0.495156 0.868804i \(-0.664889\pi\)
−0.495156 + 0.868804i \(0.664889\pi\)
\(132\) −47.4688 −0.0313002
\(133\) 814.220 0.530841
\(134\) 868.405 0.559842
\(135\) 0 0
\(136\) 1999.65 1.26080
\(137\) −684.928 −0.427134 −0.213567 0.976928i \(-0.568508\pi\)
−0.213567 + 0.976928i \(0.568508\pi\)
\(138\) −1426.99 −0.880242
\(139\) 830.483 0.506767 0.253384 0.967366i \(-0.418457\pi\)
0.253384 + 0.967366i \(0.418457\pi\)
\(140\) 0 0
\(141\) 1671.89 0.998569
\(142\) 2870.75 1.69654
\(143\) −540.354 −0.315991
\(144\) −453.810 −0.262621
\(145\) 0 0
\(146\) 316.172 0.179223
\(147\) 911.955 0.511679
\(148\) −599.680 −0.333063
\(149\) −1213.64 −0.667285 −0.333642 0.942700i \(-0.608278\pi\)
−0.333642 + 0.942700i \(0.608278\pi\)
\(150\) 0 0
\(151\) 30.8466 0.0166242 0.00831212 0.999965i \(-0.497354\pi\)
0.00831212 + 0.999965i \(0.497354\pi\)
\(152\) 3151.58 1.68176
\(153\) −744.375 −0.393328
\(154\) 176.000 0.0920941
\(155\) 0 0
\(156\) 211.983 0.108796
\(157\) −345.239 −0.175497 −0.0877485 0.996143i \(-0.527967\pi\)
−0.0877485 + 0.996143i \(0.527967\pi\)
\(158\) −793.659 −0.399621
\(159\) −504.193 −0.251479
\(160\) 0 0
\(161\) −1159.88 −0.567772
\(162\) 207.486 0.100627
\(163\) −1921.49 −0.923331 −0.461665 0.887054i \(-0.652748\pi\)
−0.461665 + 0.887054i \(0.652748\pi\)
\(164\) 429.346 0.204429
\(165\) 0 0
\(166\) 2615.91 1.22310
\(167\) −172.297 −0.0798369 −0.0399185 0.999203i \(-0.512710\pi\)
−0.0399185 + 0.999203i \(0.512710\pi\)
\(168\) −453.045 −0.208055
\(169\) 216.080 0.0983521
\(170\) 0 0
\(171\) −1173.19 −0.524654
\(172\) −738.833 −0.327532
\(173\) 1025.29 0.450587 0.225293 0.974291i \(-0.427666\pi\)
0.225293 + 0.974291i \(0.427666\pi\)
\(174\) 68.4488 0.0298224
\(175\) 0 0
\(176\) 554.656 0.237550
\(177\) −1856.32 −0.788302
\(178\) −363.430 −0.153035
\(179\) 1658.29 0.692437 0.346219 0.938154i \(-0.387466\pi\)
0.346219 + 0.938154i \(0.387466\pi\)
\(180\) 0 0
\(181\) −2021.81 −0.830275 −0.415137 0.909759i \(-0.636266\pi\)
−0.415137 + 0.909759i \(0.636266\pi\)
\(182\) −785.970 −0.320110
\(183\) −2359.22 −0.952996
\(184\) −4489.52 −1.79876
\(185\) 0 0
\(186\) −40.4318 −0.0159387
\(187\) 909.792 0.355778
\(188\) 801.640 0.310987
\(189\) 168.648 0.0649064
\(190\) 0 0
\(191\) 1440.22 0.545607 0.272803 0.962070i \(-0.412049\pi\)
0.272803 + 0.962070i \(0.412049\pi\)
\(192\) −1703.93 −0.640473
\(193\) 2798.05 1.04356 0.521782 0.853079i \(-0.325267\pi\)
0.521782 + 0.853079i \(0.325267\pi\)
\(194\) −2045.00 −0.756817
\(195\) 0 0
\(196\) 437.266 0.159354
\(197\) −458.943 −0.165982 −0.0829908 0.996550i \(-0.526447\pi\)
−0.0829908 + 0.996550i \(0.526447\pi\)
\(198\) −253.594 −0.0910208
\(199\) −2371.04 −0.844615 −0.422308 0.906453i \(-0.638780\pi\)
−0.422308 + 0.906453i \(0.638780\pi\)
\(200\) 0 0
\(201\) −1017.05 −0.356900
\(202\) 618.358 0.215384
\(203\) 55.6363 0.0192360
\(204\) −356.915 −0.122495
\(205\) 0 0
\(206\) −2992.86 −1.01225
\(207\) 1671.24 0.561155
\(208\) −2476.95 −0.825699
\(209\) 1433.90 0.474568
\(210\) 0 0
\(211\) 4319.87 1.40944 0.704721 0.709484i \(-0.251072\pi\)
0.704721 + 0.709484i \(0.251072\pi\)
\(212\) −241.752 −0.0783187
\(213\) −3362.12 −1.08154
\(214\) 5396.73 1.72389
\(215\) 0 0
\(216\) 652.781 0.205630
\(217\) −32.8636 −0.0102808
\(218\) 1264.87 0.392973
\(219\) −370.290 −0.114255
\(220\) 0 0
\(221\) −4062.89 −1.23665
\(222\) −3203.69 −0.968547
\(223\) 3837.73 1.15244 0.576219 0.817295i \(-0.304527\pi\)
0.576219 + 0.817295i \(0.304527\pi\)
\(224\) −401.349 −0.119715
\(225\) 0 0
\(226\) −435.464 −0.128171
\(227\) 5003.71 1.46303 0.731515 0.681825i \(-0.238813\pi\)
0.731515 + 0.681825i \(0.238813\pi\)
\(228\) −562.523 −0.163395
\(229\) −277.375 −0.0800412 −0.0400206 0.999199i \(-0.512742\pi\)
−0.0400206 + 0.999199i \(0.512742\pi\)
\(230\) 0 0
\(231\) −206.125 −0.0587101
\(232\) 215.350 0.0609415
\(233\) −2269.91 −0.638225 −0.319113 0.947717i \(-0.603385\pi\)
−0.319113 + 0.947717i \(0.603385\pi\)
\(234\) 1132.48 0.316379
\(235\) 0 0
\(236\) −890.072 −0.245503
\(237\) 929.505 0.254759
\(238\) 1323.33 0.360416
\(239\) −1617.11 −0.437665 −0.218832 0.975762i \(-0.570225\pi\)
−0.218832 + 0.975762i \(0.570225\pi\)
\(240\) 0 0
\(241\) 5646.63 1.50926 0.754629 0.656151i \(-0.227817\pi\)
0.754629 + 0.656151i \(0.227817\pi\)
\(242\) 309.948 0.0823314
\(243\) −243.000 −0.0641500
\(244\) −1131.20 −0.296794
\(245\) 0 0
\(246\) 2293.71 0.594478
\(247\) −6403.40 −1.64955
\(248\) −127.204 −0.0325705
\(249\) −3063.66 −0.779726
\(250\) 0 0
\(251\) −6217.61 −1.56355 −0.781777 0.623558i \(-0.785687\pi\)
−0.781777 + 0.623558i \(0.785687\pi\)
\(252\) 80.8636 0.0202140
\(253\) −2042.62 −0.507584
\(254\) 2428.82 0.599991
\(255\) 0 0
\(256\) −2133.74 −0.520933
\(257\) −7712.75 −1.87202 −0.936008 0.351980i \(-0.885509\pi\)
−0.936008 + 0.351980i \(0.885509\pi\)
\(258\) −3947.09 −0.952462
\(259\) −2604.01 −0.624730
\(260\) 0 0
\(261\) −80.1648 −0.0190118
\(262\) −3803.49 −0.896871
\(263\) −206.347 −0.0483798 −0.0241899 0.999707i \(-0.507701\pi\)
−0.0241899 + 0.999707i \(0.507701\pi\)
\(264\) −797.844 −0.186000
\(265\) 0 0
\(266\) 2085.67 0.480753
\(267\) 425.636 0.0975600
\(268\) −487.655 −0.111150
\(269\) 1712.47 0.388146 0.194073 0.980987i \(-0.437830\pi\)
0.194073 + 0.980987i \(0.437830\pi\)
\(270\) 0 0
\(271\) −477.081 −0.106940 −0.0534698 0.998569i \(-0.517028\pi\)
−0.0534698 + 0.998569i \(0.517028\pi\)
\(272\) 4170.43 0.929666
\(273\) 920.500 0.204070
\(274\) −1754.48 −0.386832
\(275\) 0 0
\(276\) 801.329 0.174762
\(277\) −4283.48 −0.929130 −0.464565 0.885539i \(-0.653789\pi\)
−0.464565 + 0.885539i \(0.653789\pi\)
\(278\) 2127.33 0.458952
\(279\) 47.3523 0.0101610
\(280\) 0 0
\(281\) 3477.79 0.738319 0.369160 0.929366i \(-0.379646\pi\)
0.369160 + 0.929366i \(0.379646\pi\)
\(282\) 4282.62 0.904350
\(283\) 6568.27 1.37966 0.689829 0.723973i \(-0.257686\pi\)
0.689829 + 0.723973i \(0.257686\pi\)
\(284\) −1612.08 −0.336829
\(285\) 0 0
\(286\) −1384.15 −0.286176
\(287\) 1864.36 0.383449
\(288\) 578.292 0.118320
\(289\) 1927.67 0.392360
\(290\) 0 0
\(291\) 2395.03 0.482472
\(292\) −177.547 −0.0355828
\(293\) −8352.29 −1.66534 −0.832672 0.553766i \(-0.813190\pi\)
−0.832672 + 0.553766i \(0.813190\pi\)
\(294\) 2336.02 0.463399
\(295\) 0 0
\(296\) −10079.3 −1.97921
\(297\) 297.000 0.0580259
\(298\) −3108.81 −0.604324
\(299\) 9121.83 1.76431
\(300\) 0 0
\(301\) −3208.26 −0.614355
\(302\) 79.0152 0.0150557
\(303\) −724.199 −0.137307
\(304\) 6572.89 1.24007
\(305\) 0 0
\(306\) −1906.76 −0.356216
\(307\) −5383.89 −1.00090 −0.500448 0.865767i \(-0.666831\pi\)
−0.500448 + 0.865767i \(0.666831\pi\)
\(308\) −98.8333 −0.0182843
\(309\) 3505.14 0.645308
\(310\) 0 0
\(311\) −1790.41 −0.326447 −0.163223 0.986589i \(-0.552189\pi\)
−0.163223 + 0.986589i \(0.552189\pi\)
\(312\) 3562.96 0.646516
\(313\) 809.076 0.146108 0.0730538 0.997328i \(-0.476726\pi\)
0.0730538 + 0.997328i \(0.476726\pi\)
\(314\) −884.347 −0.158938
\(315\) 0 0
\(316\) 445.682 0.0793403
\(317\) −10744.5 −1.90370 −0.951849 0.306567i \(-0.900820\pi\)
−0.951849 + 0.306567i \(0.900820\pi\)
\(318\) −1291.52 −0.227751
\(319\) 97.9792 0.0171968
\(320\) 0 0
\(321\) −6320.46 −1.09898
\(322\) −2971.09 −0.514200
\(323\) 10781.4 1.85725
\(324\) −116.514 −0.0199784
\(325\) 0 0
\(326\) −4922.00 −0.836210
\(327\) −1481.37 −0.250521
\(328\) 7216.35 1.21481
\(329\) 3480.98 0.583322
\(330\) 0 0
\(331\) 3399.12 0.564449 0.282224 0.959348i \(-0.408928\pi\)
0.282224 + 0.959348i \(0.408928\pi\)
\(332\) −1468.97 −0.242832
\(333\) 3752.05 0.617450
\(334\) −441.349 −0.0723039
\(335\) 0 0
\(336\) −944.864 −0.153412
\(337\) 11840.0 1.91384 0.956919 0.290356i \(-0.0937736\pi\)
0.956919 + 0.290356i \(0.0937736\pi\)
\(338\) 553.499 0.0890721
\(339\) 510.000 0.0817091
\(340\) 0 0
\(341\) −57.8750 −0.00919093
\(342\) −3005.18 −0.475151
\(343\) 4041.20 0.636165
\(344\) −12418.1 −1.94634
\(345\) 0 0
\(346\) 2626.34 0.408072
\(347\) 2076.67 0.321272 0.160636 0.987014i \(-0.448645\pi\)
0.160636 + 0.987014i \(0.448645\pi\)
\(348\) −38.4376 −0.00592090
\(349\) 5837.37 0.895322 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(350\) 0 0
\(351\) −1326.32 −0.201692
\(352\) −706.802 −0.107025
\(353\) 2423.64 0.365431 0.182715 0.983166i \(-0.441511\pi\)
0.182715 + 0.983166i \(0.441511\pi\)
\(354\) −4755.06 −0.713922
\(355\) 0 0
\(356\) 204.085 0.0303834
\(357\) −1549.84 −0.229765
\(358\) 4247.79 0.627103
\(359\) −3882.22 −0.570740 −0.285370 0.958417i \(-0.592116\pi\)
−0.285370 + 0.958417i \(0.592116\pi\)
\(360\) 0 0
\(361\) 10133.2 1.47736
\(362\) −5178.96 −0.751935
\(363\) −363.000 −0.0524864
\(364\) 441.363 0.0635542
\(365\) 0 0
\(366\) −6043.26 −0.863077
\(367\) −5666.65 −0.805986 −0.402993 0.915203i \(-0.632030\pi\)
−0.402993 + 0.915203i \(0.632030\pi\)
\(368\) −9363.26 −1.32634
\(369\) −2686.31 −0.378981
\(370\) 0 0
\(371\) −1049.77 −0.146903
\(372\) 22.7046 0.00316446
\(373\) −174.771 −0.0242608 −0.0121304 0.999926i \(-0.503861\pi\)
−0.0121304 + 0.999926i \(0.503861\pi\)
\(374\) 2330.48 0.322209
\(375\) 0 0
\(376\) 13473.8 1.84802
\(377\) −437.550 −0.0597744
\(378\) 432.000 0.0587822
\(379\) 252.686 0.0342470 0.0171235 0.999853i \(-0.494549\pi\)
0.0171235 + 0.999853i \(0.494549\pi\)
\(380\) 0 0
\(381\) −2844.55 −0.382495
\(382\) 3689.21 0.494126
\(383\) 11014.5 1.46950 0.734748 0.678340i \(-0.237300\pi\)
0.734748 + 0.678340i \(0.237300\pi\)
\(384\) −2822.61 −0.375105
\(385\) 0 0
\(386\) 7167.35 0.945099
\(387\) 4622.69 0.607196
\(388\) 1148.38 0.150258
\(389\) 8099.40 1.05567 0.527835 0.849347i \(-0.323004\pi\)
0.527835 + 0.849347i \(0.323004\pi\)
\(390\) 0 0
\(391\) −15358.4 −1.98646
\(392\) 7349.47 0.946949
\(393\) 4454.51 0.571757
\(394\) −1175.61 −0.150320
\(395\) 0 0
\(396\) 142.406 0.0180712
\(397\) 424.353 0.0536465 0.0268232 0.999640i \(-0.491461\pi\)
0.0268232 + 0.999640i \(0.491461\pi\)
\(398\) −6073.54 −0.764922
\(399\) −2442.66 −0.306481
\(400\) 0 0
\(401\) −5904.18 −0.735263 −0.367632 0.929972i \(-0.619831\pi\)
−0.367632 + 0.929972i \(0.619831\pi\)
\(402\) −2605.22 −0.323225
\(403\) 258.455 0.0319468
\(404\) −347.241 −0.0427620
\(405\) 0 0
\(406\) 142.515 0.0174210
\(407\) −4585.83 −0.558504
\(408\) −5998.94 −0.727921
\(409\) 1370.47 0.165686 0.0828430 0.996563i \(-0.473600\pi\)
0.0828430 + 0.996563i \(0.473600\pi\)
\(410\) 0 0
\(411\) 2054.78 0.246606
\(412\) 1680.65 0.200970
\(413\) −3864.98 −0.460493
\(414\) 4280.97 0.508208
\(415\) 0 0
\(416\) 3156.39 0.372007
\(417\) −2491.45 −0.292582
\(418\) 3673.00 0.429790
\(419\) 1268.73 0.147927 0.0739635 0.997261i \(-0.476435\pi\)
0.0739635 + 0.997261i \(0.476435\pi\)
\(420\) 0 0
\(421\) −12241.9 −1.41719 −0.708594 0.705617i \(-0.750670\pi\)
−0.708594 + 0.705617i \(0.750670\pi\)
\(422\) 11065.6 1.27646
\(423\) −5015.66 −0.576524
\(424\) −4063.31 −0.465405
\(425\) 0 0
\(426\) −8612.26 −0.979496
\(427\) −4912.05 −0.556700
\(428\) −3030.55 −0.342260
\(429\) 1621.06 0.182437
\(430\) 0 0
\(431\) 8050.11 0.899675 0.449838 0.893110i \(-0.351482\pi\)
0.449838 + 0.893110i \(0.351482\pi\)
\(432\) 1361.43 0.151624
\(433\) 16565.7 1.83856 0.919282 0.393600i \(-0.128770\pi\)
0.919282 + 0.393600i \(0.128770\pi\)
\(434\) −84.1819 −0.00931073
\(435\) 0 0
\(436\) −710.293 −0.0780203
\(437\) −24205.9 −2.64971
\(438\) −948.517 −0.103475
\(439\) 4705.80 0.511607 0.255804 0.966729i \(-0.417660\pi\)
0.255804 + 0.966729i \(0.417660\pi\)
\(440\) 0 0
\(441\) −2735.86 −0.295418
\(442\) −10407.3 −1.11997
\(443\) −15094.0 −1.61882 −0.809408 0.587246i \(-0.800212\pi\)
−0.809408 + 0.587246i \(0.800212\pi\)
\(444\) 1799.04 0.192294
\(445\) 0 0
\(446\) 9830.56 1.04370
\(447\) 3640.93 0.385257
\(448\) −3547.71 −0.374138
\(449\) 973.478 0.102319 0.0511595 0.998690i \(-0.483708\pi\)
0.0511595 + 0.998690i \(0.483708\pi\)
\(450\) 0 0
\(451\) 3283.27 0.342801
\(452\) 244.536 0.0254469
\(453\) −92.5398 −0.00959801
\(454\) 12817.3 1.32499
\(455\) 0 0
\(456\) −9454.75 −0.970963
\(457\) −62.6577 −0.00641358 −0.00320679 0.999995i \(-0.501021\pi\)
−0.00320679 + 0.999995i \(0.501021\pi\)
\(458\) −710.510 −0.0724890
\(459\) 2233.12 0.227088
\(460\) 0 0
\(461\) −11866.2 −1.19884 −0.599419 0.800436i \(-0.704602\pi\)
−0.599419 + 0.800436i \(0.704602\pi\)
\(462\) −528.000 −0.0531705
\(463\) 13144.8 1.31942 0.659711 0.751519i \(-0.270679\pi\)
0.659711 + 0.751519i \(0.270679\pi\)
\(464\) 449.131 0.0449361
\(465\) 0 0
\(466\) −5814.48 −0.578006
\(467\) 10176.8 1.00840 0.504201 0.863586i \(-0.331787\pi\)
0.504201 + 0.863586i \(0.331787\pi\)
\(468\) −635.949 −0.0628136
\(469\) −2117.56 −0.208486
\(470\) 0 0
\(471\) 1035.72 0.101323
\(472\) −14960.1 −1.45889
\(473\) −5649.96 −0.549229
\(474\) 2380.98 0.230721
\(475\) 0 0
\(476\) −743.121 −0.0715565
\(477\) 1512.58 0.145191
\(478\) −4142.30 −0.396369
\(479\) 3431.25 0.327302 0.163651 0.986518i \(-0.447673\pi\)
0.163651 + 0.986518i \(0.447673\pi\)
\(480\) 0 0
\(481\) 20479.1 1.94130
\(482\) 14464.1 1.36685
\(483\) 3479.64 0.327803
\(484\) −174.052 −0.0163460
\(485\) 0 0
\(486\) −622.457 −0.0580972
\(487\) 2833.20 0.263624 0.131812 0.991275i \(-0.457921\pi\)
0.131812 + 0.991275i \(0.457921\pi\)
\(488\) −19013.0 −1.76368
\(489\) 5764.48 0.533085
\(490\) 0 0
\(491\) −2667.29 −0.245159 −0.122580 0.992459i \(-0.539117\pi\)
−0.122580 + 0.992459i \(0.539117\pi\)
\(492\) −1288.04 −0.118027
\(493\) 736.700 0.0673008
\(494\) −16402.7 −1.49391
\(495\) 0 0
\(496\) −265.295 −0.0240164
\(497\) −7000.18 −0.631793
\(498\) −7847.74 −0.706156
\(499\) −11137.0 −0.999120 −0.499560 0.866279i \(-0.666505\pi\)
−0.499560 + 0.866279i \(0.666505\pi\)
\(500\) 0 0
\(501\) 516.892 0.0460939
\(502\) −15926.7 −1.41603
\(503\) −8780.30 −0.778319 −0.389159 0.921170i \(-0.627234\pi\)
−0.389159 + 0.921170i \(0.627234\pi\)
\(504\) 1359.14 0.120121
\(505\) 0 0
\(506\) −5232.29 −0.459691
\(507\) −648.239 −0.0567836
\(508\) −1363.91 −0.119121
\(509\) 13597.4 1.18408 0.592039 0.805910i \(-0.298323\pi\)
0.592039 + 0.805910i \(0.298323\pi\)
\(510\) 0 0
\(511\) −770.969 −0.0667430
\(512\) −12992.6 −1.12148
\(513\) 3519.56 0.302909
\(514\) −19756.6 −1.69538
\(515\) 0 0
\(516\) 2216.50 0.189101
\(517\) 6130.25 0.521486
\(518\) −6670.30 −0.565784
\(519\) −3075.88 −0.260146
\(520\) 0 0
\(521\) 14001.3 1.17736 0.588682 0.808364i \(-0.299647\pi\)
0.588682 + 0.808364i \(0.299647\pi\)
\(522\) −205.346 −0.0172180
\(523\) 14749.8 1.23320 0.616602 0.787275i \(-0.288509\pi\)
0.616602 + 0.787275i \(0.288509\pi\)
\(524\) 2135.86 0.178064
\(525\) 0 0
\(526\) −528.568 −0.0438149
\(527\) −435.159 −0.0359693
\(528\) −1663.97 −0.137150
\(529\) 22315.0 1.83406
\(530\) 0 0
\(531\) 5568.95 0.455126
\(532\) −1171.21 −0.0954483
\(533\) −14662.2 −1.19154
\(534\) 1090.29 0.0883548
\(535\) 0 0
\(536\) −8196.40 −0.660505
\(537\) −4974.86 −0.399779
\(538\) 4386.59 0.351523
\(539\) 3343.83 0.267215
\(540\) 0 0
\(541\) 1484.06 0.117939 0.0589694 0.998260i \(-0.481219\pi\)
0.0589694 + 0.998260i \(0.481219\pi\)
\(542\) −1222.07 −0.0968493
\(543\) 6065.42 0.479359
\(544\) −5314.40 −0.418847
\(545\) 0 0
\(546\) 2357.91 0.184815
\(547\) 16562.2 1.29460 0.647302 0.762234i \(-0.275897\pi\)
0.647302 + 0.762234i \(0.275897\pi\)
\(548\) 985.233 0.0768012
\(549\) 7077.65 0.550212
\(550\) 0 0
\(551\) 1161.09 0.0897716
\(552\) 13468.6 1.03851
\(553\) 1935.30 0.148819
\(554\) −10972.3 −0.841463
\(555\) 0 0
\(556\) −1194.61 −0.0911197
\(557\) 8821.52 0.671059 0.335529 0.942030i \(-0.391085\pi\)
0.335529 + 0.942030i \(0.391085\pi\)
\(558\) 121.295 0.00920223
\(559\) 25231.2 1.90906
\(560\) 0 0
\(561\) −2729.37 −0.205409
\(562\) 8908.55 0.668656
\(563\) 5985.53 0.448064 0.224032 0.974582i \(-0.428078\pi\)
0.224032 + 0.974582i \(0.428078\pi\)
\(564\) −2404.92 −0.179549
\(565\) 0 0
\(566\) 16825.0 1.24948
\(567\) −505.943 −0.0374737
\(568\) −27095.5 −2.00158
\(569\) −3453.08 −0.254413 −0.127206 0.991876i \(-0.540601\pi\)
−0.127206 + 0.991876i \(0.540601\pi\)
\(570\) 0 0
\(571\) −21484.5 −1.57460 −0.787302 0.616568i \(-0.788523\pi\)
−0.787302 + 0.616568i \(0.788523\pi\)
\(572\) 777.271 0.0568170
\(573\) −4320.67 −0.315006
\(574\) 4775.67 0.347269
\(575\) 0 0
\(576\) 5111.80 0.369777
\(577\) 13294.4 0.959189 0.479594 0.877490i \(-0.340784\pi\)
0.479594 + 0.877490i \(0.340784\pi\)
\(578\) 4937.82 0.355340
\(579\) −8394.14 −0.602502
\(580\) 0 0
\(581\) −6378.77 −0.455483
\(582\) 6135.01 0.436949
\(583\) −1848.71 −0.131330
\(584\) −2984.18 −0.211449
\(585\) 0 0
\(586\) −21394.8 −1.50821
\(587\) −6695.73 −0.470805 −0.235402 0.971898i \(-0.575641\pi\)
−0.235402 + 0.971898i \(0.575641\pi\)
\(588\) −1311.80 −0.0920028
\(589\) −685.841 −0.0479789
\(590\) 0 0
\(591\) 1376.83 0.0958295
\(592\) −21021.2 −1.45940
\(593\) 10239.6 0.709088 0.354544 0.935039i \(-0.384636\pi\)
0.354544 + 0.935039i \(0.384636\pi\)
\(594\) 760.781 0.0525509
\(595\) 0 0
\(596\) 1745.76 0.119982
\(597\) 7113.11 0.487639
\(598\) 23366.0 1.59784
\(599\) −23890.8 −1.62963 −0.814817 0.579719i \(-0.803162\pi\)
−0.814817 + 0.579719i \(0.803162\pi\)
\(600\) 0 0
\(601\) −11343.8 −0.769920 −0.384960 0.922933i \(-0.625785\pi\)
−0.384960 + 0.922933i \(0.625785\pi\)
\(602\) −8218.12 −0.556388
\(603\) 3051.14 0.206056
\(604\) −44.3712 −0.00298914
\(605\) 0 0
\(606\) −1855.07 −0.124352
\(607\) 26032.5 1.74074 0.870369 0.492399i \(-0.163880\pi\)
0.870369 + 0.492399i \(0.163880\pi\)
\(608\) −8375.87 −0.558695
\(609\) −166.909 −0.0111059
\(610\) 0 0
\(611\) −27376.1 −1.81263
\(612\) 1070.74 0.0707226
\(613\) 4568.13 0.300987 0.150493 0.988611i \(-0.451914\pi\)
0.150493 + 0.988611i \(0.451914\pi\)
\(614\) −13791.1 −0.906457
\(615\) 0 0
\(616\) −1661.17 −0.108653
\(617\) 12755.9 0.832308 0.416154 0.909294i \(-0.363378\pi\)
0.416154 + 0.909294i \(0.363378\pi\)
\(618\) 8978.59 0.584421
\(619\) −1138.94 −0.0739545 −0.0369772 0.999316i \(-0.511773\pi\)
−0.0369772 + 0.999316i \(0.511773\pi\)
\(620\) 0 0
\(621\) −5013.72 −0.323983
\(622\) −4586.24 −0.295645
\(623\) 886.205 0.0569904
\(624\) 7430.85 0.476718
\(625\) 0 0
\(626\) 2072.49 0.132322
\(627\) −4301.69 −0.273992
\(628\) 496.607 0.0315554
\(629\) −34480.6 −2.18574
\(630\) 0 0
\(631\) 7997.36 0.504548 0.252274 0.967656i \(-0.418822\pi\)
0.252274 + 0.967656i \(0.418822\pi\)
\(632\) 7490.91 0.471475
\(633\) −12959.6 −0.813742
\(634\) −27522.7 −1.72408
\(635\) 0 0
\(636\) 725.255 0.0452173
\(637\) −14932.7 −0.928814
\(638\) 250.979 0.0155742
\(639\) 10086.4 0.624430
\(640\) 0 0
\(641\) −573.115 −0.0353146 −0.0176573 0.999844i \(-0.505621\pi\)
−0.0176573 + 0.999844i \(0.505621\pi\)
\(642\) −16190.2 −0.995290
\(643\) 16027.8 0.983009 0.491504 0.870875i \(-0.336447\pi\)
0.491504 + 0.870875i \(0.336447\pi\)
\(644\) 1668.42 0.102089
\(645\) 0 0
\(646\) 27617.1 1.68201
\(647\) 2622.74 0.159367 0.0796837 0.996820i \(-0.474609\pi\)
0.0796837 + 0.996820i \(0.474609\pi\)
\(648\) −1958.34 −0.118721
\(649\) −6806.50 −0.411677
\(650\) 0 0
\(651\) 98.5908 0.00593560
\(652\) 2763.97 0.166020
\(653\) −3102.00 −0.185897 −0.0929484 0.995671i \(-0.529629\pi\)
−0.0929484 + 0.995671i \(0.529629\pi\)
\(654\) −3794.62 −0.226883
\(655\) 0 0
\(656\) 15050.3 0.895755
\(657\) 1110.87 0.0659652
\(658\) 8916.73 0.528283
\(659\) −20840.2 −1.23190 −0.615948 0.787787i \(-0.711227\pi\)
−0.615948 + 0.787787i \(0.711227\pi\)
\(660\) 0 0
\(661\) 18242.9 1.07348 0.536738 0.843749i \(-0.319657\pi\)
0.536738 + 0.843749i \(0.319657\pi\)
\(662\) 8707.03 0.511191
\(663\) 12188.7 0.713980
\(664\) −24690.2 −1.44302
\(665\) 0 0
\(666\) 9611.06 0.559191
\(667\) −1654.01 −0.0960171
\(668\) 247.841 0.0143551
\(669\) −11513.2 −0.665361
\(670\) 0 0
\(671\) −8650.46 −0.497686
\(672\) 1204.05 0.0691177
\(673\) 12746.6 0.730084 0.365042 0.930991i \(-0.381055\pi\)
0.365042 + 0.930991i \(0.381055\pi\)
\(674\) 30328.7 1.73326
\(675\) 0 0
\(676\) −310.819 −0.0176843
\(677\) 7683.11 0.436168 0.218084 0.975930i \(-0.430019\pi\)
0.218084 + 0.975930i \(0.430019\pi\)
\(678\) 1306.39 0.0739995
\(679\) 4986.63 0.281840
\(680\) 0 0
\(681\) −15011.1 −0.844681
\(682\) −148.250 −0.00832373
\(683\) 21397.1 1.19874 0.599368 0.800473i \(-0.295418\pi\)
0.599368 + 0.800473i \(0.295418\pi\)
\(684\) 1687.57 0.0943359
\(685\) 0 0
\(686\) 10351.8 0.576140
\(687\) 832.124 0.0462118
\(688\) −25899.0 −1.43516
\(689\) 8255.84 0.456491
\(690\) 0 0
\(691\) 26137.5 1.43895 0.719477 0.694516i \(-0.244381\pi\)
0.719477 + 0.694516i \(0.244381\pi\)
\(692\) −1474.83 −0.0810181
\(693\) 618.375 0.0338963
\(694\) 5319.50 0.290959
\(695\) 0 0
\(696\) −646.051 −0.0351846
\(697\) 24686.7 1.34157
\(698\) 14952.7 0.810845
\(699\) 6809.72 0.368479
\(700\) 0 0
\(701\) 13382.4 0.721036 0.360518 0.932752i \(-0.382600\pi\)
0.360518 + 0.932752i \(0.382600\pi\)
\(702\) −3397.45 −0.182662
\(703\) −54343.9 −2.91553
\(704\) −6247.76 −0.334476
\(705\) 0 0
\(706\) 6208.27 0.330951
\(707\) −1507.83 −0.0802092
\(708\) 2670.22 0.141741
\(709\) 18164.6 0.962179 0.481090 0.876671i \(-0.340241\pi\)
0.481090 + 0.876671i \(0.340241\pi\)
\(710\) 0 0
\(711\) −2788.52 −0.147085
\(712\) 3430.21 0.180552
\(713\) 977.000 0.0513169
\(714\) −3970.00 −0.208086
\(715\) 0 0
\(716\) −2385.36 −0.124504
\(717\) 4851.32 0.252686
\(718\) −9944.50 −0.516888
\(719\) 9665.62 0.501344 0.250672 0.968072i \(-0.419348\pi\)
0.250672 + 0.968072i \(0.419348\pi\)
\(720\) 0 0
\(721\) 7297.94 0.376962
\(722\) 25956.7 1.33796
\(723\) −16939.9 −0.871371
\(724\) 2908.26 0.149288
\(725\) 0 0
\(726\) −929.844 −0.0475341
\(727\) 29779.6 1.51921 0.759605 0.650385i \(-0.225392\pi\)
0.759605 + 0.650385i \(0.225392\pi\)
\(728\) 7418.33 0.377667
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −42481.7 −2.14944
\(732\) 3393.61 0.171354
\(733\) −35029.5 −1.76513 −0.882567 0.470187i \(-0.844186\pi\)
−0.882567 + 0.470187i \(0.844186\pi\)
\(734\) −14515.4 −0.729937
\(735\) 0 0
\(736\) 11931.7 0.597564
\(737\) −3729.17 −0.186385
\(738\) −6881.13 −0.343222
\(739\) 23297.3 1.15968 0.579842 0.814729i \(-0.303114\pi\)
0.579842 + 0.814729i \(0.303114\pi\)
\(740\) 0 0
\(741\) 19210.2 0.952368
\(742\) −2689.03 −0.133042
\(743\) −21570.4 −1.06506 −0.532530 0.846411i \(-0.678759\pi\)
−0.532530 + 0.846411i \(0.678759\pi\)
\(744\) 381.613 0.0188046
\(745\) 0 0
\(746\) −447.684 −0.0219717
\(747\) 9190.99 0.450175
\(748\) −1308.69 −0.0639710
\(749\) −13159.6 −0.641980
\(750\) 0 0
\(751\) 28554.8 1.38746 0.693729 0.720236i \(-0.255966\pi\)
0.693729 + 0.720236i \(0.255966\pi\)
\(752\) 28100.7 1.36267
\(753\) 18652.8 0.902719
\(754\) −1120.81 −0.0541344
\(755\) 0 0
\(756\) −242.591 −0.0116706
\(757\) −7812.81 −0.375114 −0.187557 0.982254i \(-0.560057\pi\)
−0.187557 + 0.982254i \(0.560057\pi\)
\(758\) 647.268 0.0310156
\(759\) 6127.87 0.293054
\(760\) 0 0
\(761\) 2875.13 0.136956 0.0684778 0.997653i \(-0.478186\pi\)
0.0684778 + 0.997653i \(0.478186\pi\)
\(762\) −7286.45 −0.346405
\(763\) −3084.33 −0.146344
\(764\) −2071.69 −0.0981033
\(765\) 0 0
\(766\) 28214.3 1.33084
\(767\) 30396.0 1.43095
\(768\) 6401.22 0.300761
\(769\) −27657.7 −1.29696 −0.648479 0.761233i \(-0.724594\pi\)
−0.648479 + 0.761233i \(0.724594\pi\)
\(770\) 0 0
\(771\) 23138.2 1.08081
\(772\) −4024.84 −0.187639
\(773\) 3929.35 0.182832 0.0914160 0.995813i \(-0.470861\pi\)
0.0914160 + 0.995813i \(0.470861\pi\)
\(774\) 11841.3 0.549904
\(775\) 0 0
\(776\) 19301.6 0.892898
\(777\) 7812.02 0.360688
\(778\) 20747.0 0.956064
\(779\) 38908.0 1.78950
\(780\) 0 0
\(781\) −12327.8 −0.564818
\(782\) −39341.3 −1.79903
\(783\) 240.495 0.0109765
\(784\) 15327.9 0.698247
\(785\) 0 0
\(786\) 11410.5 0.517809
\(787\) 21125.7 0.956860 0.478430 0.878126i \(-0.341206\pi\)
0.478430 + 0.878126i \(0.341206\pi\)
\(788\) 660.166 0.0298445
\(789\) 619.040 0.0279321
\(790\) 0 0
\(791\) 1061.86 0.0477310
\(792\) 2393.53 0.107387
\(793\) 38630.7 1.72991
\(794\) 1087.00 0.0485847
\(795\) 0 0
\(796\) 3410.61 0.151867
\(797\) −11696.3 −0.519828 −0.259914 0.965632i \(-0.583694\pi\)
−0.259914 + 0.965632i \(0.583694\pi\)
\(798\) −6257.00 −0.277563
\(799\) 46093.0 2.04087
\(800\) 0 0
\(801\) −1276.91 −0.0563263
\(802\) −15123.9 −0.665888
\(803\) −1357.73 −0.0596678
\(804\) 1462.97 0.0641727
\(805\) 0 0
\(806\) 662.045 0.0289324
\(807\) −5137.42 −0.224096
\(808\) −5836.34 −0.254111
\(809\) 14310.2 0.621902 0.310951 0.950426i \(-0.399352\pi\)
0.310951 + 0.950426i \(0.399352\pi\)
\(810\) 0 0
\(811\) 21697.9 0.939477 0.469739 0.882806i \(-0.344348\pi\)
0.469739 + 0.882806i \(0.344348\pi\)
\(812\) −80.0299 −0.00345874
\(813\) 1431.24 0.0617416
\(814\) −11746.9 −0.505807
\(815\) 0 0
\(816\) −12511.3 −0.536743
\(817\) −66954.1 −2.86711
\(818\) 3510.54 0.150053
\(819\) −2761.50 −0.117820
\(820\) 0 0
\(821\) 3613.00 0.153587 0.0767934 0.997047i \(-0.475532\pi\)
0.0767934 + 0.997047i \(0.475532\pi\)
\(822\) 5263.44 0.223338
\(823\) 4763.98 0.201776 0.100888 0.994898i \(-0.467832\pi\)
0.100888 + 0.994898i \(0.467832\pi\)
\(824\) 28248.0 1.19425
\(825\) 0 0
\(826\) −9900.36 −0.417043
\(827\) 33571.7 1.41161 0.705806 0.708405i \(-0.250585\pi\)
0.705806 + 0.708405i \(0.250585\pi\)
\(828\) −2403.99 −0.100899
\(829\) 17980.5 0.753303 0.376652 0.926355i \(-0.377075\pi\)
0.376652 + 0.926355i \(0.377075\pi\)
\(830\) 0 0
\(831\) 12850.4 0.536434
\(832\) 27900.9 1.16261
\(833\) 25142.1 1.04576
\(834\) −6381.98 −0.264976
\(835\) 0 0
\(836\) −2062.58 −0.0853301
\(837\) −142.057 −0.00586643
\(838\) 3249.91 0.133969
\(839\) −40139.5 −1.65169 −0.825847 0.563895i \(-0.809302\pi\)
−0.825847 + 0.563895i \(0.809302\pi\)
\(840\) 0 0
\(841\) −24309.7 −0.996747
\(842\) −31358.4 −1.28347
\(843\) −10433.4 −0.426269
\(844\) −6213.91 −0.253426
\(845\) 0 0
\(846\) −12847.9 −0.522127
\(847\) −755.792 −0.0306603
\(848\) −8474.36 −0.343173
\(849\) −19704.8 −0.796546
\(850\) 0 0
\(851\) 77414.4 3.11837
\(852\) 4836.24 0.194468
\(853\) 15369.2 0.616919 0.308459 0.951237i \(-0.400187\pi\)
0.308459 + 0.951237i \(0.400187\pi\)
\(854\) −12582.5 −0.504173
\(855\) 0 0
\(856\) −50936.8 −2.03386
\(857\) −10324.9 −0.411541 −0.205770 0.978600i \(-0.565970\pi\)
−0.205770 + 0.978600i \(0.565970\pi\)
\(858\) 4152.44 0.165224
\(859\) −27112.5 −1.07691 −0.538455 0.842655i \(-0.680992\pi\)
−0.538455 + 0.842655i \(0.680992\pi\)
\(860\) 0 0
\(861\) −5593.09 −0.221384
\(862\) 20620.8 0.814787
\(863\) −30463.6 −1.20161 −0.600807 0.799394i \(-0.705154\pi\)
−0.600807 + 0.799394i \(0.705154\pi\)
\(864\) −1734.88 −0.0683122
\(865\) 0 0
\(866\) 42434.0 1.66509
\(867\) −5783.00 −0.226529
\(868\) 47.2726 0.00184854
\(869\) 3408.19 0.133044
\(870\) 0 0
\(871\) 16653.5 0.647855
\(872\) −11938.4 −0.463631
\(873\) −7185.10 −0.278555
\(874\) −62004.6 −2.39970
\(875\) 0 0
\(876\) 532.642 0.0205437
\(877\) 5086.12 0.195833 0.0979167 0.995195i \(-0.468782\pi\)
0.0979167 + 0.995195i \(0.468782\pi\)
\(878\) 12054.2 0.463335
\(879\) 25056.9 0.961487
\(880\) 0 0
\(881\) −10625.5 −0.406338 −0.203169 0.979144i \(-0.565124\pi\)
−0.203169 + 0.979144i \(0.565124\pi\)
\(882\) −7008.06 −0.267544
\(883\) −13112.2 −0.499728 −0.249864 0.968281i \(-0.580386\pi\)
−0.249864 + 0.968281i \(0.580386\pi\)
\(884\) 5844.25 0.222357
\(885\) 0 0
\(886\) −38664.0 −1.46607
\(887\) 14442.8 0.546719 0.273360 0.961912i \(-0.411865\pi\)
0.273360 + 0.961912i \(0.411865\pi\)
\(888\) 30237.8 1.14270
\(889\) −5922.54 −0.223437
\(890\) 0 0
\(891\) −891.000 −0.0335013
\(892\) −5520.38 −0.207215
\(893\) 72645.8 2.72228
\(894\) 9326.43 0.348906
\(895\) 0 0
\(896\) −5876.86 −0.219121
\(897\) −27365.5 −1.01863
\(898\) 2493.61 0.0926648
\(899\) −46.8641 −0.00173860
\(900\) 0 0
\(901\) −13900.3 −0.513970
\(902\) 8410.27 0.310456
\(903\) 9624.77 0.354698
\(904\) 4110.10 0.151217
\(905\) 0 0
\(906\) −237.045 −0.00869239
\(907\) 44981.9 1.64675 0.823374 0.567499i \(-0.192089\pi\)
0.823374 + 0.567499i \(0.192089\pi\)
\(908\) −7197.57 −0.263062
\(909\) 2172.60 0.0792745
\(910\) 0 0
\(911\) 6841.96 0.248830 0.124415 0.992230i \(-0.460295\pi\)
0.124415 + 0.992230i \(0.460295\pi\)
\(912\) −19718.7 −0.715954
\(913\) −11233.4 −0.407199
\(914\) −160.501 −0.00580843
\(915\) 0 0
\(916\) 398.989 0.0143919
\(917\) 9274.61 0.333996
\(918\) 5720.27 0.205661
\(919\) −4753.54 −0.170625 −0.0853127 0.996354i \(-0.527189\pi\)
−0.0853127 + 0.996354i \(0.527189\pi\)
\(920\) 0 0
\(921\) 16151.7 0.577868
\(922\) −30395.9 −1.08572
\(923\) 55052.7 1.96325
\(924\) 296.500 0.0105564
\(925\) 0 0
\(926\) 33671.2 1.19493
\(927\) −10515.4 −0.372569
\(928\) −572.330 −0.0202453
\(929\) −7507.93 −0.265153 −0.132576 0.991173i \(-0.542325\pi\)
−0.132576 + 0.991173i \(0.542325\pi\)
\(930\) 0 0
\(931\) 39625.7 1.39493
\(932\) 3265.14 0.114757
\(933\) 5371.24 0.188474
\(934\) 26068.3 0.913256
\(935\) 0 0
\(936\) −10688.9 −0.373266
\(937\) −8540.47 −0.297764 −0.148882 0.988855i \(-0.547567\pi\)
−0.148882 + 0.988855i \(0.547567\pi\)
\(938\) −5424.24 −0.188814
\(939\) −2427.23 −0.0843552
\(940\) 0 0
\(941\) 9101.13 0.315290 0.157645 0.987496i \(-0.449610\pi\)
0.157645 + 0.987496i \(0.449610\pi\)
\(942\) 2653.04 0.0917630
\(943\) −55425.5 −1.91400
\(944\) −31200.6 −1.07573
\(945\) 0 0
\(946\) −14472.7 −0.497407
\(947\) −47540.0 −1.63130 −0.815650 0.578546i \(-0.803620\pi\)
−0.815650 + 0.578546i \(0.803620\pi\)
\(948\) −1337.04 −0.0458072
\(949\) 6063.26 0.207399
\(950\) 0 0
\(951\) 32233.6 1.09910
\(952\) −12490.2 −0.425221
\(953\) −47370.7 −1.61016 −0.805082 0.593164i \(-0.797879\pi\)
−0.805082 + 0.593164i \(0.797879\pi\)
\(954\) 3874.55 0.131492
\(955\) 0 0
\(956\) 2326.12 0.0786947
\(957\) −293.938 −0.00992859
\(958\) 8789.33 0.296420
\(959\) 4278.20 0.144057
\(960\) 0 0
\(961\) −29763.3 −0.999071
\(962\) 52458.4 1.75813
\(963\) 18961.4 0.634498
\(964\) −8122.38 −0.271374
\(965\) 0 0
\(966\) 8913.27 0.296874
\(967\) 36171.6 1.20290 0.601448 0.798912i \(-0.294591\pi\)
0.601448 + 0.798912i \(0.294591\pi\)
\(968\) −2925.43 −0.0971351
\(969\) −32344.1 −1.07228
\(970\) 0 0
\(971\) −31713.2 −1.04812 −0.524060 0.851681i \(-0.675583\pi\)
−0.524060 + 0.851681i \(0.675583\pi\)
\(972\) 349.543 0.0115346
\(973\) −5187.37 −0.170914
\(974\) 7257.40 0.238750
\(975\) 0 0
\(976\) −39653.1 −1.30048
\(977\) −22800.5 −0.746626 −0.373313 0.927706i \(-0.621778\pi\)
−0.373313 + 0.927706i \(0.621778\pi\)
\(978\) 14766.0 0.482786
\(979\) 1560.67 0.0509490
\(980\) 0 0
\(981\) 4444.12 0.144638
\(982\) −6832.41 −0.222027
\(983\) 44597.4 1.44704 0.723518 0.690305i \(-0.242524\pi\)
0.723518 + 0.690305i \(0.242524\pi\)
\(984\) −21649.1 −0.701369
\(985\) 0 0
\(986\) 1887.10 0.0609507
\(987\) −10443.0 −0.336781
\(988\) 9210.95 0.296599
\(989\) 95378.1 3.06658
\(990\) 0 0
\(991\) −34788.1 −1.11512 −0.557558 0.830138i \(-0.688261\pi\)
−0.557558 + 0.830138i \(0.688261\pi\)
\(992\) 338.068 0.0108202
\(993\) −10197.4 −0.325885
\(994\) −17931.3 −0.572180
\(995\) 0 0
\(996\) 4406.92 0.140199
\(997\) 20360.5 0.646765 0.323383 0.946268i \(-0.395180\pi\)
0.323383 + 0.946268i \(0.395180\pi\)
\(998\) −28528.0 −0.904849
\(999\) −11256.1 −0.356485
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 825.4.a.m.1.2 2
3.2 odd 2 2475.4.a.n.1.1 2
5.2 odd 4 825.4.c.j.199.4 4
5.3 odd 4 825.4.c.j.199.1 4
5.4 even 2 165.4.a.c.1.1 2
15.14 odd 2 495.4.a.d.1.2 2
55.54 odd 2 1815.4.a.n.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.c.1.1 2 5.4 even 2
495.4.a.d.1.2 2 15.14 odd 2
825.4.a.m.1.2 2 1.1 even 1 trivial
825.4.c.j.199.1 4 5.3 odd 4
825.4.c.j.199.4 4 5.2 odd 4
1815.4.a.n.1.2 2 55.54 odd 2
2475.4.a.n.1.1 2 3.2 odd 2