Properties

Label 975.2.c.i.274.6
Level $975$
Weight $2$
Character 975.274
Analytic conductor $7.785$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 975.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.78541419707\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.399424.1
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 195)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.6
Root \(0.264658 - 1.38923i\) of defining polynomial
Character \(\chi\) \(=\) 975.274
Dual form 975.2.c.i.274.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.77846i q^{2} +1.00000i q^{3} -5.71982 q^{4} -2.77846 q^{6} -2.71982i q^{7} -10.3354i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+2.77846i q^{2} +1.00000i q^{3} -5.71982 q^{4} -2.77846 q^{6} -2.71982i q^{7} -10.3354i q^{8} -1.00000 q^{9} -2.71982 q^{11} -5.71982i q^{12} -1.00000i q^{13} +7.55691 q^{14} +17.2767 q^{16} -2.83709i q^{17} -2.77846i q^{18} +3.55691 q^{19} +2.71982 q^{21} -7.55691i q^{22} +4.83709i q^{23} +10.3354 q^{24} +2.77846 q^{26} -1.00000i q^{27} +15.5569i q^{28} -6.00000 q^{29} +7.55691 q^{31} +27.3319i q^{32} -2.71982i q^{33} +7.88273 q^{34} +5.71982 q^{36} -4.27674i q^{37} +9.88273i q^{38} +1.00000 q^{39} +2.83709 q^{41} +7.55691i q^{42} -11.1138i q^{43} +15.5569 q^{44} -13.4396 q^{46} -11.5569i q^{47} +17.2767i q^{48} -0.397442 q^{49} +2.83709 q^{51} +5.71982i q^{52} -1.16291i q^{53} +2.77846 q^{54} -28.1104 q^{56} +3.55691i q^{57} -16.6707i q^{58} +2.11727 q^{59} +6.60256 q^{61} +20.9966i q^{62} +2.71982i q^{63} -41.3871 q^{64} +7.55691 q^{66} +1.88273i q^{67} +16.2277i q^{68} -4.83709 q^{69} -6.71982 q^{71} +10.3354i q^{72} -9.11383i q^{73} +11.8827 q^{74} -20.3449 q^{76} +7.39744i q^{77} +2.77846i q^{78} -10.2767 q^{79} +1.00000 q^{81} +7.88273i q^{82} -2.11727i q^{83} -15.5569 q^{84} +30.8793 q^{86} -6.00000i q^{87} +28.1104i q^{88} -1.16291 q^{89} -2.71982 q^{91} -27.6673i q^{92} +7.55691i q^{93} +32.1104 q^{94} -27.3319 q^{96} -10.8371i q^{97} -1.10428i q^{98} +2.71982 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 16 q^{4} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 16 q^{4} - 6 q^{9} + 2 q^{11} + 12 q^{14} + 52 q^{16} - 12 q^{19} - 2 q^{21} + 12 q^{24} - 36 q^{29} + 12 q^{31} + 44 q^{34} + 16 q^{36} + 6 q^{39} + 2 q^{41} + 60 q^{44} - 44 q^{46} - 24 q^{49} + 2 q^{51} - 32 q^{56} + 16 q^{59} + 18 q^{61} - 60 q^{64} + 12 q^{66} - 14 q^{69} - 22 q^{71} + 68 q^{74} + 8 q^{76} - 10 q^{79} + 6 q^{81} - 60 q^{84} + 112 q^{86} - 22 q^{89} + 2 q^{91} + 56 q^{94} - 44 q^{96} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/975\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(326\) \(352\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.77846i 1.96467i 0.187142 + 0.982333i \(0.440078\pi\)
−0.187142 + 0.982333i \(0.559922\pi\)
\(3\) 1.00000i 0.577350i
\(4\) −5.71982 −2.85991
\(5\) 0 0
\(6\) −2.77846 −1.13430
\(7\) − 2.71982i − 1.02800i −0.857791 0.513998i \(-0.828164\pi\)
0.857791 0.513998i \(-0.171836\pi\)
\(8\) − 10.3354i − 3.65411i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −2.71982 −0.820058 −0.410029 0.912073i \(-0.634481\pi\)
−0.410029 + 0.912073i \(0.634481\pi\)
\(12\) − 5.71982i − 1.65117i
\(13\) − 1.00000i − 0.277350i
\(14\) 7.55691 2.01967
\(15\) 0 0
\(16\) 17.2767 4.31918
\(17\) − 2.83709i − 0.688095i −0.938952 0.344048i \(-0.888202\pi\)
0.938952 0.344048i \(-0.111798\pi\)
\(18\) − 2.77846i − 0.654889i
\(19\) 3.55691 0.816012 0.408006 0.912979i \(-0.366224\pi\)
0.408006 + 0.912979i \(0.366224\pi\)
\(20\) 0 0
\(21\) 2.71982 0.593514
\(22\) − 7.55691i − 1.61114i
\(23\) 4.83709i 1.00860i 0.863528 + 0.504302i \(0.168250\pi\)
−0.863528 + 0.504302i \(0.831750\pi\)
\(24\) 10.3354 2.10970
\(25\) 0 0
\(26\) 2.77846 0.544900
\(27\) − 1.00000i − 0.192450i
\(28\) 15.5569i 2.93998i
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 7.55691 1.35726 0.678631 0.734479i \(-0.262574\pi\)
0.678631 + 0.734479i \(0.262574\pi\)
\(32\) 27.3319i 4.83165i
\(33\) − 2.71982i − 0.473461i
\(34\) 7.88273 1.35188
\(35\) 0 0
\(36\) 5.71982 0.953304
\(37\) − 4.27674i − 0.703091i −0.936171 0.351546i \(-0.885656\pi\)
0.936171 0.351546i \(-0.114344\pi\)
\(38\) 9.88273i 1.60319i
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 2.83709 0.443079 0.221540 0.975151i \(-0.428892\pi\)
0.221540 + 0.975151i \(0.428892\pi\)
\(42\) 7.55691i 1.16606i
\(43\) − 11.1138i − 1.69484i −0.530921 0.847421i \(-0.678154\pi\)
0.530921 0.847421i \(-0.321846\pi\)
\(44\) 15.5569 2.34529
\(45\) 0 0
\(46\) −13.4396 −1.98157
\(47\) − 11.5569i − 1.68575i −0.538110 0.842875i \(-0.680862\pi\)
0.538110 0.842875i \(-0.319138\pi\)
\(48\) 17.2767i 2.49368i
\(49\) −0.397442 −0.0567775
\(50\) 0 0
\(51\) 2.83709 0.397272
\(52\) 5.71982i 0.793197i
\(53\) − 1.16291i − 0.159738i −0.996805 0.0798690i \(-0.974550\pi\)
0.996805 0.0798690i \(-0.0254502\pi\)
\(54\) 2.77846 0.378100
\(55\) 0 0
\(56\) −28.1104 −3.75641
\(57\) 3.55691i 0.471125i
\(58\) − 16.6707i − 2.18898i
\(59\) 2.11727 0.275645 0.137822 0.990457i \(-0.455990\pi\)
0.137822 + 0.990457i \(0.455990\pi\)
\(60\) 0 0
\(61\) 6.60256 0.845371 0.422685 0.906276i \(-0.361088\pi\)
0.422685 + 0.906276i \(0.361088\pi\)
\(62\) 20.9966i 2.66657i
\(63\) 2.71982i 0.342666i
\(64\) −41.3871 −5.17339
\(65\) 0 0
\(66\) 7.55691 0.930192
\(67\) 1.88273i 0.230013i 0.993365 + 0.115006i \(0.0366888\pi\)
−0.993365 + 0.115006i \(0.963311\pi\)
\(68\) 16.2277i 1.96789i
\(69\) −4.83709 −0.582317
\(70\) 0 0
\(71\) −6.71982 −0.797496 −0.398748 0.917060i \(-0.630555\pi\)
−0.398748 + 0.917060i \(0.630555\pi\)
\(72\) 10.3354i 1.21804i
\(73\) − 9.11383i − 1.06669i −0.845897 0.533346i \(-0.820934\pi\)
0.845897 0.533346i \(-0.179066\pi\)
\(74\) 11.8827 1.38134
\(75\) 0 0
\(76\) −20.3449 −2.33372
\(77\) 7.39744i 0.843017i
\(78\) 2.77846i 0.314598i
\(79\) −10.2767 −1.15622 −0.578112 0.815958i \(-0.696210\pi\)
−0.578112 + 0.815958i \(0.696210\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 7.88273i 0.870502i
\(83\) − 2.11727i − 0.232400i −0.993226 0.116200i \(-0.962929\pi\)
0.993226 0.116200i \(-0.0370714\pi\)
\(84\) −15.5569 −1.69740
\(85\) 0 0
\(86\) 30.8793 3.32980
\(87\) − 6.00000i − 0.643268i
\(88\) 28.1104i 2.99658i
\(89\) −1.16291 −0.123268 −0.0616341 0.998099i \(-0.519631\pi\)
−0.0616341 + 0.998099i \(0.519631\pi\)
\(90\) 0 0
\(91\) −2.71982 −0.285115
\(92\) − 27.6673i − 2.88452i
\(93\) 7.55691i 0.783616i
\(94\) 32.1104 3.31193
\(95\) 0 0
\(96\) −27.3319 −2.78955
\(97\) − 10.8371i − 1.10034i −0.835053 0.550170i \(-0.814563\pi\)
0.835053 0.550170i \(-0.185437\pi\)
\(98\) − 1.10428i − 0.111549i
\(99\) 2.71982 0.273353
\(100\) 0 0
\(101\) 7.67418 0.763610 0.381805 0.924243i \(-0.375303\pi\)
0.381805 + 0.924243i \(0.375303\pi\)
\(102\) 7.88273i 0.780507i
\(103\) − 3.76547i − 0.371023i −0.982642 0.185511i \(-0.940606\pi\)
0.982642 0.185511i \(-0.0593941\pi\)
\(104\) −10.3354 −1.01347
\(105\) 0 0
\(106\) 3.23109 0.313832
\(107\) − 12.6026i − 1.21834i −0.793041 0.609168i \(-0.791504\pi\)
0.793041 0.609168i \(-0.208496\pi\)
\(108\) 5.71982i 0.550390i
\(109\) −11.4396 −1.09572 −0.547860 0.836570i \(-0.684557\pi\)
−0.547860 + 0.836570i \(0.684557\pi\)
\(110\) 0 0
\(111\) 4.27674 0.405930
\(112\) − 46.9897i − 4.44011i
\(113\) 13.1138i 1.23365i 0.787102 + 0.616823i \(0.211580\pi\)
−0.787102 + 0.616823i \(0.788420\pi\)
\(114\) −9.88273 −0.925603
\(115\) 0 0
\(116\) 34.3189 3.18643
\(117\) 1.00000i 0.0924500i
\(118\) 5.88273i 0.541550i
\(119\) −7.71639 −0.707360
\(120\) 0 0
\(121\) −3.60256 −0.327505
\(122\) 18.3449i 1.66087i
\(123\) 2.83709i 0.255812i
\(124\) −43.2242 −3.88165
\(125\) 0 0
\(126\) −7.55691 −0.673223
\(127\) − 13.4396i − 1.19258i −0.802771 0.596288i \(-0.796642\pi\)
0.802771 0.596288i \(-0.203358\pi\)
\(128\) − 60.3285i − 5.33234i
\(129\) 11.1138 0.978518
\(130\) 0 0
\(131\) −9.43965 −0.824746 −0.412373 0.911015i \(-0.635300\pi\)
−0.412373 + 0.911015i \(0.635300\pi\)
\(132\) 15.5569i 1.35406i
\(133\) − 9.67418i − 0.838858i
\(134\) −5.23109 −0.451898
\(135\) 0 0
\(136\) −29.3224 −2.51437
\(137\) 1.76547i 0.150834i 0.997152 + 0.0754170i \(0.0240288\pi\)
−0.997152 + 0.0754170i \(0.975971\pi\)
\(138\) − 13.4396i − 1.14406i
\(139\) 6.27674 0.532386 0.266193 0.963920i \(-0.414234\pi\)
0.266193 + 0.963920i \(0.414234\pi\)
\(140\) 0 0
\(141\) 11.5569 0.973268
\(142\) − 18.6707i − 1.56681i
\(143\) 2.71982i 0.227443i
\(144\) −17.2767 −1.43973
\(145\) 0 0
\(146\) 25.3224 2.09570
\(147\) − 0.397442i − 0.0327805i
\(148\) 24.4622i 2.01078i
\(149\) 20.8302 1.70648 0.853239 0.521520i \(-0.174635\pi\)
0.853239 + 0.521520i \(0.174635\pi\)
\(150\) 0 0
\(151\) 4.99656 0.406614 0.203307 0.979115i \(-0.434831\pi\)
0.203307 + 0.979115i \(0.434831\pi\)
\(152\) − 36.7620i − 2.98179i
\(153\) 2.83709i 0.229365i
\(154\) −20.5535 −1.65625
\(155\) 0 0
\(156\) −5.71982 −0.457952
\(157\) 8.87930i 0.708645i 0.935123 + 0.354322i \(0.115288\pi\)
−0.935123 + 0.354322i \(0.884712\pi\)
\(158\) − 28.5535i − 2.27159i
\(159\) 1.16291 0.0922247
\(160\) 0 0
\(161\) 13.1560 1.03684
\(162\) 2.77846i 0.218296i
\(163\) 13.8337i 1.08354i 0.840528 + 0.541768i \(0.182245\pi\)
−0.840528 + 0.541768i \(0.817755\pi\)
\(164\) −16.2277 −1.26717
\(165\) 0 0
\(166\) 5.88273 0.456589
\(167\) − 9.88273i − 0.764749i −0.924007 0.382374i \(-0.875106\pi\)
0.924007 0.382374i \(-0.124894\pi\)
\(168\) − 28.1104i − 2.16876i
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) −3.55691 −0.272004
\(172\) 63.5691i 4.84710i
\(173\) − 13.1138i − 0.997026i −0.866882 0.498513i \(-0.833880\pi\)
0.866882 0.498513i \(-0.166120\pi\)
\(174\) 16.6707 1.26381
\(175\) 0 0
\(176\) −46.9897 −3.54198
\(177\) 2.11727i 0.159143i
\(178\) − 3.23109i − 0.242181i
\(179\) −8.55348 −0.639317 −0.319658 0.947533i \(-0.603568\pi\)
−0.319658 + 0.947533i \(0.603568\pi\)
\(180\) 0 0
\(181\) 3.72326 0.276748 0.138374 0.990380i \(-0.455812\pi\)
0.138374 + 0.990380i \(0.455812\pi\)
\(182\) − 7.55691i − 0.560156i
\(183\) 6.60256i 0.488075i
\(184\) 49.9931 3.68554
\(185\) 0 0
\(186\) −20.9966 −1.53954
\(187\) 7.71639i 0.564278i
\(188\) 66.1035i 4.82109i
\(189\) −2.71982 −0.197838
\(190\) 0 0
\(191\) −4.23453 −0.306400 −0.153200 0.988195i \(-0.548958\pi\)
−0.153200 + 0.988195i \(0.548958\pi\)
\(192\) − 41.3871i − 2.98686i
\(193\) 23.3906i 1.68369i 0.539719 + 0.841845i \(0.318530\pi\)
−0.539719 + 0.841845i \(0.681470\pi\)
\(194\) 30.1104 2.16180
\(195\) 0 0
\(196\) 2.27330 0.162379
\(197\) 14.5535i 1.03689i 0.855110 + 0.518446i \(0.173489\pi\)
−0.855110 + 0.518446i \(0.826511\pi\)
\(198\) 7.55691i 0.537047i
\(199\) 15.1138 1.07139 0.535695 0.844411i \(-0.320050\pi\)
0.535695 + 0.844411i \(0.320050\pi\)
\(200\) 0 0
\(201\) −1.88273 −0.132798
\(202\) 21.3224i 1.50024i
\(203\) 16.3189i 1.14537i
\(204\) −16.2277 −1.13616
\(205\) 0 0
\(206\) 10.4622 0.728935
\(207\) − 4.83709i − 0.336201i
\(208\) − 17.2767i − 1.19793i
\(209\) −9.67418 −0.669177
\(210\) 0 0
\(211\) −18.2277 −1.25484 −0.627422 0.778680i \(-0.715890\pi\)
−0.627422 + 0.778680i \(0.715890\pi\)
\(212\) 6.65164i 0.456836i
\(213\) − 6.71982i − 0.460435i
\(214\) 35.0157 2.39362
\(215\) 0 0
\(216\) −10.3354 −0.703233
\(217\) − 20.5535i − 1.39526i
\(218\) − 31.7846i − 2.15272i
\(219\) 9.11383 0.615855
\(220\) 0 0
\(221\) −2.83709 −0.190843
\(222\) 11.8827i 0.797517i
\(223\) − 10.1173i − 0.677502i −0.940876 0.338751i \(-0.889996\pi\)
0.940876 0.338751i \(-0.110004\pi\)
\(224\) 74.3380 4.96692
\(225\) 0 0
\(226\) −36.4362 −2.42370
\(227\) 11.3224i 0.751493i 0.926723 + 0.375746i \(0.122614\pi\)
−0.926723 + 0.375746i \(0.877386\pi\)
\(228\) − 20.3449i − 1.34738i
\(229\) 6.23453 0.411990 0.205995 0.978553i \(-0.433957\pi\)
0.205995 + 0.978553i \(0.433957\pi\)
\(230\) 0 0
\(231\) −7.39744 −0.486716
\(232\) 62.0122i 4.07130i
\(233\) − 6.83709i − 0.447913i −0.974599 0.223956i \(-0.928103\pi\)
0.974599 0.223956i \(-0.0718973\pi\)
\(234\) −2.77846 −0.181633
\(235\) 0 0
\(236\) −12.1104 −0.788319
\(237\) − 10.2767i − 0.667546i
\(238\) − 21.4396i − 1.38973i
\(239\) −1.28018 −0.0828077 −0.0414039 0.999142i \(-0.513183\pi\)
−0.0414039 + 0.999142i \(0.513183\pi\)
\(240\) 0 0
\(241\) −6.00000 −0.386494 −0.193247 0.981150i \(-0.561902\pi\)
−0.193247 + 0.981150i \(0.561902\pi\)
\(242\) − 10.0096i − 0.643438i
\(243\) 1.00000i 0.0641500i
\(244\) −37.7655 −2.41769
\(245\) 0 0
\(246\) −7.88273 −0.502585
\(247\) − 3.55691i − 0.226321i
\(248\) − 78.1035i − 4.95958i
\(249\) 2.11727 0.134176
\(250\) 0 0
\(251\) 18.2277 1.15052 0.575260 0.817971i \(-0.304901\pi\)
0.575260 + 0.817971i \(0.304901\pi\)
\(252\) − 15.5569i − 0.979993i
\(253\) − 13.1560i − 0.827113i
\(254\) 37.3415 2.34301
\(255\) 0 0
\(256\) 84.8459 5.30287
\(257\) 1.11383i 0.0694787i 0.999396 + 0.0347394i \(0.0110601\pi\)
−0.999396 + 0.0347394i \(0.988940\pi\)
\(258\) 30.8793i 1.92246i
\(259\) −11.6320 −0.722776
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) − 26.2277i − 1.62035i
\(263\) − 8.00000i − 0.493301i −0.969104 0.246651i \(-0.920670\pi\)
0.969104 0.246651i \(-0.0793300\pi\)
\(264\) −28.1104 −1.73007
\(265\) 0 0
\(266\) 26.8793 1.64808
\(267\) − 1.16291i − 0.0711689i
\(268\) − 10.7689i − 0.657816i
\(269\) −15.6742 −0.955672 −0.477836 0.878449i \(-0.658579\pi\)
−0.477836 + 0.878449i \(0.658579\pi\)
\(270\) 0 0
\(271\) 0.443086 0.0269155 0.0134578 0.999909i \(-0.495716\pi\)
0.0134578 + 0.999909i \(0.495716\pi\)
\(272\) − 49.0157i − 2.97201i
\(273\) − 2.71982i − 0.164611i
\(274\) −4.90528 −0.296339
\(275\) 0 0
\(276\) 27.6673 1.66538
\(277\) − 4.87930i − 0.293168i −0.989198 0.146584i \(-0.953172\pi\)
0.989198 0.146584i \(-0.0468279\pi\)
\(278\) 17.4396i 1.04596i
\(279\) −7.55691 −0.452421
\(280\) 0 0
\(281\) −9.11383 −0.543685 −0.271843 0.962342i \(-0.587633\pi\)
−0.271843 + 0.962342i \(0.587633\pi\)
\(282\) 32.1104i 1.91215i
\(283\) − 33.3415i − 1.98195i −0.134063 0.990973i \(-0.542802\pi\)
0.134063 0.990973i \(-0.457198\pi\)
\(284\) 38.4362 2.28077
\(285\) 0 0
\(286\) −7.55691 −0.446850
\(287\) − 7.71639i − 0.455484i
\(288\) − 27.3319i − 1.61055i
\(289\) 8.95092 0.526525
\(290\) 0 0
\(291\) 10.8371 0.635281
\(292\) 52.1295i 3.05065i
\(293\) 29.4328i 1.71948i 0.510731 + 0.859740i \(0.329375\pi\)
−0.510731 + 0.859740i \(0.670625\pi\)
\(294\) 1.10428 0.0644027
\(295\) 0 0
\(296\) −44.2017 −2.56917
\(297\) 2.71982i 0.157820i
\(298\) 57.8759i 3.35266i
\(299\) 4.83709 0.279736
\(300\) 0 0
\(301\) −30.2277 −1.74229
\(302\) 13.8827i 0.798862i
\(303\) 7.67418i 0.440870i
\(304\) 61.4519 3.52451
\(305\) 0 0
\(306\) −7.88273 −0.450626
\(307\) − 21.8337i − 1.24611i −0.782177 0.623056i \(-0.785891\pi\)
0.782177 0.623056i \(-0.214109\pi\)
\(308\) − 42.3121i − 2.41095i
\(309\) 3.76547 0.214210
\(310\) 0 0
\(311\) 25.1070 1.42368 0.711842 0.702339i \(-0.247861\pi\)
0.711842 + 0.702339i \(0.247861\pi\)
\(312\) − 10.3354i − 0.585125i
\(313\) − 8.22766i − 0.465055i −0.972590 0.232527i \(-0.925300\pi\)
0.972590 0.232527i \(-0.0746995\pi\)
\(314\) −24.6707 −1.39225
\(315\) 0 0
\(316\) 58.7811 3.30670
\(317\) 27.6742i 1.55434i 0.629293 + 0.777168i \(0.283345\pi\)
−0.629293 + 0.777168i \(0.716655\pi\)
\(318\) 3.23109i 0.181191i
\(319\) 16.3189 0.913685
\(320\) 0 0
\(321\) 12.6026 0.703406
\(322\) 36.5535i 2.03705i
\(323\) − 10.0913i − 0.561494i
\(324\) −5.71982 −0.317768
\(325\) 0 0
\(326\) −38.4362 −2.12878
\(327\) − 11.4396i − 0.632614i
\(328\) − 29.3224i − 1.61906i
\(329\) −31.4328 −1.73294
\(330\) 0 0
\(331\) −13.2311 −0.727247 −0.363623 0.931546i \(-0.618460\pi\)
−0.363623 + 0.931546i \(0.618460\pi\)
\(332\) 12.1104i 0.664644i
\(333\) 4.27674i 0.234364i
\(334\) 27.4588 1.50248
\(335\) 0 0
\(336\) 46.9897 2.56350
\(337\) − 4.32582i − 0.235642i −0.993035 0.117821i \(-0.962409\pi\)
0.993035 0.117821i \(-0.0375910\pi\)
\(338\) − 2.77846i − 0.151128i
\(339\) −13.1138 −0.712245
\(340\) 0 0
\(341\) −20.5535 −1.11303
\(342\) − 9.88273i − 0.534397i
\(343\) − 17.9578i − 0.969630i
\(344\) −114.866 −6.19314
\(345\) 0 0
\(346\) 36.4362 1.95882
\(347\) 6.27674i 0.336953i 0.985706 + 0.168476i \(0.0538847\pi\)
−0.985706 + 0.168476i \(0.946115\pi\)
\(348\) 34.3189i 1.83969i
\(349\) −17.6673 −0.945709 −0.472855 0.881140i \(-0.656776\pi\)
−0.472855 + 0.881140i \(0.656776\pi\)
\(350\) 0 0
\(351\) −1.00000 −0.0533761
\(352\) − 74.3380i − 3.96223i
\(353\) 13.7655i 0.732662i 0.930485 + 0.366331i \(0.119386\pi\)
−0.930485 + 0.366331i \(0.880614\pi\)
\(354\) −5.88273 −0.312664
\(355\) 0 0
\(356\) 6.65164 0.352536
\(357\) − 7.71639i − 0.408394i
\(358\) − 23.7655i − 1.25604i
\(359\) −0.996562 −0.0525965 −0.0262983 0.999654i \(-0.508372\pi\)
−0.0262983 + 0.999654i \(0.508372\pi\)
\(360\) 0 0
\(361\) −6.34836 −0.334124
\(362\) 10.3449i 0.543717i
\(363\) − 3.60256i − 0.189085i
\(364\) 15.5569 0.815404
\(365\) 0 0
\(366\) −18.3449 −0.958905
\(367\) 14.2277i 0.742678i 0.928497 + 0.371339i \(0.121101\pi\)
−0.928497 + 0.371339i \(0.878899\pi\)
\(368\) 83.5691i 4.35634i
\(369\) −2.83709 −0.147693
\(370\) 0 0
\(371\) −3.16291 −0.164210
\(372\) − 43.2242i − 2.24107i
\(373\) − 15.6742i − 0.811578i −0.913967 0.405789i \(-0.866997\pi\)
0.913967 0.405789i \(-0.133003\pi\)
\(374\) −21.4396 −1.10862
\(375\) 0 0
\(376\) −119.445 −6.15991
\(377\) 6.00000i 0.309016i
\(378\) − 7.55691i − 0.388686i
\(379\) −26.2017 −1.34589 −0.672945 0.739693i \(-0.734971\pi\)
−0.672945 + 0.739693i \(0.734971\pi\)
\(380\) 0 0
\(381\) 13.4396 0.688534
\(382\) − 11.7655i − 0.601974i
\(383\) − 22.4362i − 1.14644i −0.819403 0.573218i \(-0.805695\pi\)
0.819403 0.573218i \(-0.194305\pi\)
\(384\) 60.3285 3.07863
\(385\) 0 0
\(386\) −64.9897 −3.30789
\(387\) 11.1138i 0.564948i
\(388\) 61.9862i 3.14687i
\(389\) −31.6742 −1.60594 −0.802972 0.596016i \(-0.796749\pi\)
−0.802972 + 0.596016i \(0.796749\pi\)
\(390\) 0 0
\(391\) 13.7233 0.694015
\(392\) 4.10771i 0.207471i
\(393\) − 9.43965i − 0.476167i
\(394\) −40.4362 −2.03715
\(395\) 0 0
\(396\) −15.5569 −0.781764
\(397\) 17.9509i 0.900931i 0.892794 + 0.450465i \(0.148742\pi\)
−0.892794 + 0.450465i \(0.851258\pi\)
\(398\) 41.9931i 2.10493i
\(399\) 9.67418 0.484315
\(400\) 0 0
\(401\) 13.5829 0.678297 0.339149 0.940733i \(-0.389861\pi\)
0.339149 + 0.940733i \(0.389861\pi\)
\(402\) − 5.23109i − 0.260903i
\(403\) − 7.55691i − 0.376437i
\(404\) −43.8950 −2.18386
\(405\) 0 0
\(406\) −45.3415 −2.25026
\(407\) 11.6320i 0.576576i
\(408\) − 29.3224i − 1.45167i
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) −1.76547 −0.0870841
\(412\) 21.5378i 1.06109i
\(413\) − 5.75859i − 0.283362i
\(414\) 13.4396 0.660523
\(415\) 0 0
\(416\) 27.3319 1.34006
\(417\) 6.27674i 0.307373i
\(418\) − 26.8793i − 1.31471i
\(419\) −12.3189 −0.601820 −0.300910 0.953653i \(-0.597290\pi\)
−0.300910 + 0.953653i \(0.597290\pi\)
\(420\) 0 0
\(421\) 22.7880 1.11062 0.555310 0.831644i \(-0.312600\pi\)
0.555310 + 0.831644i \(0.312600\pi\)
\(422\) − 50.6448i − 2.46535i
\(423\) 11.5569i 0.561916i
\(424\) −12.0191 −0.583699
\(425\) 0 0
\(426\) 18.6707 0.904600
\(427\) − 17.9578i − 0.869039i
\(428\) 72.0844i 3.48433i
\(429\) −2.71982 −0.131314
\(430\) 0 0
\(431\) −8.99656 −0.433349 −0.216675 0.976244i \(-0.569521\pi\)
−0.216675 + 0.976244i \(0.569521\pi\)
\(432\) − 17.2767i − 0.831227i
\(433\) 20.3258i 0.976797i 0.872621 + 0.488398i \(0.162419\pi\)
−0.872621 + 0.488398i \(0.837581\pi\)
\(434\) 57.1070 2.74122
\(435\) 0 0
\(436\) 65.4328 3.13366
\(437\) 17.2051i 0.823032i
\(438\) 25.3224i 1.20995i
\(439\) 25.3906 1.21183 0.605913 0.795531i \(-0.292808\pi\)
0.605913 + 0.795531i \(0.292808\pi\)
\(440\) 0 0
\(441\) 0.397442 0.0189258
\(442\) − 7.88273i − 0.374943i
\(443\) − 10.9284i − 0.519223i −0.965713 0.259611i \(-0.916406\pi\)
0.965713 0.259611i \(-0.0835945\pi\)
\(444\) −24.4622 −1.16092
\(445\) 0 0
\(446\) 28.1104 1.33107
\(447\) 20.8302i 0.985235i
\(448\) 112.566i 5.31823i
\(449\) −2.83709 −0.133891 −0.0669453 0.997757i \(-0.521325\pi\)
−0.0669453 + 0.997757i \(0.521325\pi\)
\(450\) 0 0
\(451\) −7.71639 −0.363350
\(452\) − 75.0088i − 3.52812i
\(453\) 4.99656i 0.234759i
\(454\) −31.4588 −1.47643
\(455\) 0 0
\(456\) 36.7620 1.72154
\(457\) − 13.7164i − 0.641625i −0.947143 0.320813i \(-0.896044\pi\)
0.947143 0.320813i \(-0.103956\pi\)
\(458\) 17.3224i 0.809422i
\(459\) −2.83709 −0.132424
\(460\) 0 0
\(461\) −19.6251 −0.914032 −0.457016 0.889458i \(-0.651082\pi\)
−0.457016 + 0.889458i \(0.651082\pi\)
\(462\) − 20.5535i − 0.956234i
\(463\) − 27.0388i − 1.25660i −0.777972 0.628299i \(-0.783751\pi\)
0.777972 0.628299i \(-0.216249\pi\)
\(464\) −103.660 −4.81231
\(465\) 0 0
\(466\) 18.9966 0.879999
\(467\) 28.9215i 1.33833i 0.743115 + 0.669164i \(0.233348\pi\)
−0.743115 + 0.669164i \(0.766652\pi\)
\(468\) − 5.71982i − 0.264399i
\(469\) 5.12070 0.236452
\(470\) 0 0
\(471\) −8.87930 −0.409136
\(472\) − 21.8827i − 1.00723i
\(473\) 30.2277i 1.38987i
\(474\) 28.5535 1.31150
\(475\) 0 0
\(476\) 44.1364 2.02299
\(477\) 1.16291i 0.0532460i
\(478\) − 3.55691i − 0.162689i
\(479\) 12.1595 0.555580 0.277790 0.960642i \(-0.410398\pi\)
0.277790 + 0.960642i \(0.410398\pi\)
\(480\) 0 0
\(481\) −4.27674 −0.195002
\(482\) − 16.6707i − 0.759332i
\(483\) 13.1560i 0.598620i
\(484\) 20.6060 0.936636
\(485\) 0 0
\(486\) −2.77846 −0.126033
\(487\) − 0.159472i − 0.00722636i −0.999993 0.00361318i \(-0.998850\pi\)
0.999993 0.00361318i \(-0.00115011\pi\)
\(488\) − 68.2399i − 3.08907i
\(489\) −13.8337 −0.625579
\(490\) 0 0
\(491\) −42.2277 −1.90571 −0.952854 0.303430i \(-0.901868\pi\)
−0.952854 + 0.303430i \(0.901868\pi\)
\(492\) − 16.2277i − 0.731599i
\(493\) 17.0225i 0.766657i
\(494\) 9.88273 0.444645
\(495\) 0 0
\(496\) 130.559 5.86226
\(497\) 18.2767i 0.819824i
\(498\) 5.88273i 0.263612i
\(499\) −7.79145 −0.348793 −0.174397 0.984676i \(-0.555797\pi\)
−0.174397 + 0.984676i \(0.555797\pi\)
\(500\) 0 0
\(501\) 9.88273 0.441528
\(502\) 50.6448i 2.26039i
\(503\) − 27.3484i − 1.21940i −0.792631 0.609702i \(-0.791289\pi\)
0.792631 0.609702i \(-0.208711\pi\)
\(504\) 28.1104 1.25214
\(505\) 0 0
\(506\) 36.5535 1.62500
\(507\) − 1.00000i − 0.0444116i
\(508\) 76.8724i 3.41066i
\(509\) 33.4819 1.48406 0.742029 0.670368i \(-0.233864\pi\)
0.742029 + 0.670368i \(0.233864\pi\)
\(510\) 0 0
\(511\) −24.7880 −1.09656
\(512\) 115.084i 5.08603i
\(513\) − 3.55691i − 0.157042i
\(514\) −3.09472 −0.136502
\(515\) 0 0
\(516\) −63.5691 −2.79848
\(517\) 31.4328i 1.38241i
\(518\) − 32.3189i − 1.42001i
\(519\) 13.1138 0.575633
\(520\) 0 0
\(521\) −17.3484 −0.760045 −0.380023 0.924977i \(-0.624084\pi\)
−0.380023 + 0.924977i \(0.624084\pi\)
\(522\) 16.6707i 0.729659i
\(523\) 4.00000i 0.174908i 0.996169 + 0.0874539i \(0.0278730\pi\)
−0.996169 + 0.0874539i \(0.972127\pi\)
\(524\) 53.9931 2.35870
\(525\) 0 0
\(526\) 22.2277 0.969172
\(527\) − 21.4396i − 0.933926i
\(528\) − 46.9897i − 2.04496i
\(529\) −0.397442 −0.0172801
\(530\) 0 0
\(531\) −2.11727 −0.0918815
\(532\) 55.3346i 2.39906i
\(533\) − 2.83709i − 0.122888i
\(534\) 3.23109 0.139823
\(535\) 0 0
\(536\) 19.4588 0.840490
\(537\) − 8.55348i − 0.369110i
\(538\) − 43.5500i − 1.87758i
\(539\) 1.08097 0.0465608
\(540\) 0 0
\(541\) −32.6448 −1.40351 −0.701754 0.712419i \(-0.747599\pi\)
−0.701754 + 0.712419i \(0.747599\pi\)
\(542\) 1.23109i 0.0528800i
\(543\) 3.72326i 0.159780i
\(544\) 77.5432 3.32464
\(545\) 0 0
\(546\) 7.55691 0.323406
\(547\) 34.2277i 1.46347i 0.681590 + 0.731734i \(0.261289\pi\)
−0.681590 + 0.731734i \(0.738711\pi\)
\(548\) − 10.0982i − 0.431372i
\(549\) −6.60256 −0.281790
\(550\) 0 0
\(551\) −21.3415 −0.909178
\(552\) 49.9931i 2.12785i
\(553\) 27.9509i 1.18859i
\(554\) 13.5569 0.575978
\(555\) 0 0
\(556\) −35.9018 −1.52258
\(557\) 6.65164i 0.281839i 0.990021 + 0.140919i \(0.0450059\pi\)
−0.990021 + 0.140919i \(0.954994\pi\)
\(558\) − 20.9966i − 0.888855i
\(559\) −11.1138 −0.470065
\(560\) 0 0
\(561\) −7.71639 −0.325786
\(562\) − 25.3224i − 1.06816i
\(563\) − 40.2699i − 1.69717i −0.529057 0.848586i \(-0.677454\pi\)
0.529057 0.848586i \(-0.322546\pi\)
\(564\) −66.1035 −2.78346
\(565\) 0 0
\(566\) 92.6379 3.89386
\(567\) − 2.71982i − 0.114222i
\(568\) 69.4519i 2.91414i
\(569\) 13.4328 0.563131 0.281566 0.959542i \(-0.409146\pi\)
0.281566 + 0.959542i \(0.409146\pi\)
\(570\) 0 0
\(571\) −35.7164 −1.49468 −0.747342 0.664439i \(-0.768670\pi\)
−0.747342 + 0.664439i \(0.768670\pi\)
\(572\) − 15.5569i − 0.650467i
\(573\) − 4.23453i − 0.176900i
\(574\) 21.4396 0.894874
\(575\) 0 0
\(576\) 41.3871 1.72446
\(577\) − 13.7164i − 0.571021i −0.958376 0.285510i \(-0.907837\pi\)
0.958376 0.285510i \(-0.0921631\pi\)
\(578\) 24.8697i 1.03444i
\(579\) −23.3906 −0.972079
\(580\) 0 0
\(581\) −5.75859 −0.238907
\(582\) 30.1104i 1.24812i
\(583\) 3.16291i 0.130994i
\(584\) −94.1948 −3.89781
\(585\) 0 0
\(586\) −81.7777 −3.37821
\(587\) 30.6707i 1.26592i 0.774186 + 0.632959i \(0.218160\pi\)
−0.774186 + 0.632959i \(0.781840\pi\)
\(588\) 2.27330i 0.0937493i
\(589\) 26.8793 1.10754
\(590\) 0 0
\(591\) −14.5535 −0.598650
\(592\) − 73.8881i − 3.03678i
\(593\) 45.6673i 1.87533i 0.347538 + 0.937666i \(0.387018\pi\)
−0.347538 + 0.937666i \(0.612982\pi\)
\(594\) −7.55691 −0.310064
\(595\) 0 0
\(596\) −119.145 −4.88038
\(597\) 15.1138i 0.618568i
\(598\) 13.4396i 0.549588i
\(599\) 40.2208 1.64338 0.821688 0.569937i \(-0.193032\pi\)
0.821688 + 0.569937i \(0.193032\pi\)
\(600\) 0 0
\(601\) 17.3974 0.709656 0.354828 0.934932i \(-0.384539\pi\)
0.354828 + 0.934932i \(0.384539\pi\)
\(602\) − 83.9862i − 3.42302i
\(603\) − 1.88273i − 0.0766708i
\(604\) −28.5795 −1.16288
\(605\) 0 0
\(606\) −21.3224 −0.866163
\(607\) − 14.2277i − 0.577483i −0.957407 0.288741i \(-0.906763\pi\)
0.957407 0.288741i \(-0.0932368\pi\)
\(608\) 97.2173i 3.94268i
\(609\) −16.3189 −0.661277
\(610\) 0 0
\(611\) −11.5569 −0.467543
\(612\) − 16.2277i − 0.655964i
\(613\) 40.8302i 1.64912i 0.565777 + 0.824558i \(0.308576\pi\)
−0.565777 + 0.824558i \(0.691424\pi\)
\(614\) 60.6639 2.44819
\(615\) 0 0
\(616\) 76.4553 3.08047
\(617\) 5.11383i 0.205875i 0.994688 + 0.102937i \(0.0328242\pi\)
−0.994688 + 0.102937i \(0.967176\pi\)
\(618\) 10.4622i 0.420851i
\(619\) −11.5569 −0.464512 −0.232256 0.972655i \(-0.574611\pi\)
−0.232256 + 0.972655i \(0.574611\pi\)
\(620\) 0 0
\(621\) 4.83709 0.194106
\(622\) 69.7586i 2.79706i
\(623\) 3.16291i 0.126719i
\(624\) 17.2767 0.691623
\(625\) 0 0
\(626\) 22.8602 0.913677
\(627\) − 9.67418i − 0.386350i
\(628\) − 50.7880i − 2.02666i
\(629\) −12.1335 −0.483794
\(630\) 0 0
\(631\) 35.2242 1.40225 0.701127 0.713036i \(-0.252681\pi\)
0.701127 + 0.713036i \(0.252681\pi\)
\(632\) 106.214i 4.22496i
\(633\) − 18.2277i − 0.724484i
\(634\) −76.8915 −3.05375
\(635\) 0 0
\(636\) −6.65164 −0.263755
\(637\) 0.397442i 0.0157472i
\(638\) 45.3415i 1.79509i
\(639\) 6.71982 0.265832
\(640\) 0 0
\(641\) −21.9018 −0.865071 −0.432535 0.901617i \(-0.642381\pi\)
−0.432535 + 0.901617i \(0.642381\pi\)
\(642\) 35.0157i 1.38196i
\(643\) − 7.50783i − 0.296080i −0.988981 0.148040i \(-0.952704\pi\)
0.988981 0.148040i \(-0.0472964\pi\)
\(644\) −75.2502 −2.96527
\(645\) 0 0
\(646\) 28.0382 1.10315
\(647\) − 14.0422i − 0.552056i −0.961150 0.276028i \(-0.910982\pi\)
0.961150 0.276028i \(-0.0890183\pi\)
\(648\) − 10.3354i − 0.406012i
\(649\) −5.75859 −0.226044
\(650\) 0 0
\(651\) 20.5535 0.805554
\(652\) − 79.1261i − 3.09882i
\(653\) − 7.99312i − 0.312795i −0.987694 0.156398i \(-0.950012\pi\)
0.987694 0.156398i \(-0.0499881\pi\)
\(654\) 31.7846 1.24288
\(655\) 0 0
\(656\) 49.0157 1.91374
\(657\) 9.11383i 0.355564i
\(658\) − 87.3346i − 3.40466i
\(659\) 25.3415 0.987164 0.493582 0.869699i \(-0.335687\pi\)
0.493582 + 0.869699i \(0.335687\pi\)
\(660\) 0 0
\(661\) 27.4396 1.06728 0.533639 0.845712i \(-0.320824\pi\)
0.533639 + 0.845712i \(0.320824\pi\)
\(662\) − 36.7620i − 1.42880i
\(663\) − 2.83709i − 0.110183i
\(664\) −21.8827 −0.849215
\(665\) 0 0
\(666\) −11.8827 −0.460447
\(667\) − 29.0225i − 1.12376i
\(668\) 56.5275i 2.18711i
\(669\) 10.1173 0.391156
\(670\) 0 0
\(671\) −17.9578 −0.693253
\(672\) 74.3380i 2.86765i
\(673\) − 27.1070i − 1.04490i −0.852671 0.522448i \(-0.825019\pi\)
0.852671 0.522448i \(-0.174981\pi\)
\(674\) 12.0191 0.462959
\(675\) 0 0
\(676\) 5.71982 0.219993
\(677\) − 36.5957i − 1.40649i −0.710949 0.703243i \(-0.751735\pi\)
0.710949 0.703243i \(-0.248265\pi\)
\(678\) − 36.4362i − 1.39932i
\(679\) −29.4750 −1.13115
\(680\) 0 0
\(681\) −11.3224 −0.433875
\(682\) − 57.1070i − 2.18674i
\(683\) 13.4656i 0.515248i 0.966245 + 0.257624i \(0.0829396\pi\)
−0.966245 + 0.257624i \(0.917060\pi\)
\(684\) 20.3449 0.777908
\(685\) 0 0
\(686\) 49.8950 1.90500
\(687\) 6.23453i 0.237862i
\(688\) − 192.011i − 7.32034i
\(689\) −1.16291 −0.0443033
\(690\) 0 0
\(691\) 29.5500 1.12414 0.562068 0.827091i \(-0.310006\pi\)
0.562068 + 0.827091i \(0.310006\pi\)
\(692\) 75.0088i 2.85141i
\(693\) − 7.39744i − 0.281006i
\(694\) −17.4396 −0.662000
\(695\) 0 0
\(696\) −62.0122 −2.35057
\(697\) − 8.04908i − 0.304881i
\(698\) − 49.0878i − 1.85800i
\(699\) 6.83709 0.258603
\(700\) 0 0
\(701\) 43.6604 1.64903 0.824516 0.565839i \(-0.191448\pi\)
0.824516 + 0.565839i \(0.191448\pi\)
\(702\) − 2.77846i − 0.104866i
\(703\) − 15.2120i − 0.573731i
\(704\) 112.566 4.24248
\(705\) 0 0
\(706\) −38.2468 −1.43944
\(707\) − 20.8724i − 0.784988i
\(708\) − 12.1104i − 0.455136i
\(709\) 26.7880 1.00604 0.503022 0.864273i \(-0.332221\pi\)
0.503022 + 0.864273i \(0.332221\pi\)
\(710\) 0 0
\(711\) 10.2767 0.385408
\(712\) 12.0191i 0.450435i
\(713\) 36.5535i 1.36894i
\(714\) 21.4396 0.802359
\(715\) 0 0
\(716\) 48.9244 1.82839
\(717\) − 1.28018i − 0.0478091i
\(718\) − 2.76891i − 0.103335i
\(719\) 34.8793 1.30078 0.650389 0.759601i \(-0.274606\pi\)
0.650389 + 0.759601i \(0.274606\pi\)
\(720\) 0 0
\(721\) −10.2414 −0.381410
\(722\) − 17.6386i − 0.656443i
\(723\) − 6.00000i − 0.223142i
\(724\) −21.2964 −0.791475
\(725\) 0 0
\(726\) 10.0096 0.371489
\(727\) 37.4396i 1.38856i 0.719705 + 0.694280i \(0.244277\pi\)
−0.719705 + 0.694280i \(0.755723\pi\)
\(728\) 28.1104i 1.04184i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −31.5309 −1.16621
\(732\) − 37.7655i − 1.39585i
\(733\) 47.1560i 1.74175i 0.491506 + 0.870874i \(0.336446\pi\)
−0.491506 + 0.870874i \(0.663554\pi\)
\(734\) −39.5309 −1.45911
\(735\) 0 0
\(736\) −132.207 −4.87322
\(737\) − 5.12070i − 0.188624i
\(738\) − 7.88273i − 0.290167i
\(739\) 31.7914 1.16947 0.584734 0.811225i \(-0.301199\pi\)
0.584734 + 0.811225i \(0.301199\pi\)
\(740\) 0 0
\(741\) 3.55691 0.130667
\(742\) − 8.78801i − 0.322618i
\(743\) − 16.6776i − 0.611842i −0.952057 0.305921i \(-0.901036\pi\)
0.952057 0.305921i \(-0.0989644\pi\)
\(744\) 78.1035 2.86341
\(745\) 0 0
\(746\) 43.5500 1.59448
\(747\) 2.11727i 0.0774667i
\(748\) − 44.1364i − 1.61379i
\(749\) −34.2767 −1.25244
\(750\) 0 0
\(751\) 16.1855 0.590616 0.295308 0.955402i \(-0.404578\pi\)
0.295308 + 0.955402i \(0.404578\pi\)
\(752\) − 199.666i − 7.28106i
\(753\) 18.2277i 0.664253i
\(754\) −16.6707 −0.607113
\(755\) 0 0
\(756\) 15.5569 0.565800
\(757\) 12.3258i 0.447990i 0.974590 + 0.223995i \(0.0719099\pi\)
−0.974590 + 0.223995i \(0.928090\pi\)
\(758\) − 72.8002i − 2.64422i
\(759\) 13.1560 0.477534
\(760\) 0 0
\(761\) −0.00687569 −0.000249244 0 −0.000124622 1.00000i \(-0.500040\pi\)
−0.000124622 1.00000i \(0.500040\pi\)
\(762\) 37.3415i 1.35274i
\(763\) 31.1138i 1.12640i
\(764\) 24.2208 0.876277
\(765\) 0 0
\(766\) 62.3380 2.25237
\(767\) − 2.11727i − 0.0764501i
\(768\) 84.8459i 3.06161i
\(769\) 20.3258 0.732968 0.366484 0.930424i \(-0.380561\pi\)
0.366484 + 0.930424i \(0.380561\pi\)
\(770\) 0 0
\(771\) −1.11383 −0.0401136
\(772\) − 133.790i − 4.81520i
\(773\) − 9.90184i − 0.356144i −0.984017 0.178072i \(-0.943014\pi\)
0.984017 0.178072i \(-0.0569861\pi\)
\(774\) −30.8793 −1.10993
\(775\) 0 0
\(776\) −112.005 −4.02076
\(777\) − 11.6320i − 0.417295i
\(778\) − 88.0054i − 3.15514i
\(779\) 10.0913 0.361558
\(780\) 0 0
\(781\) 18.2767 0.653993
\(782\) 38.1295i 1.36351i
\(783\) 6.00000i 0.214423i
\(784\) −6.86651 −0.245232
\(785\) 0 0
\(786\) 26.2277 0.935510
\(787\) 36.3449i 1.29556i 0.761829 + 0.647778i \(0.224302\pi\)
−0.761829 + 0.647778i \(0.775698\pi\)
\(788\) − 83.2433i − 2.96542i
\(789\) 8.00000 0.284808
\(790\) 0 0
\(791\) 35.6673 1.26818
\(792\) − 28.1104i − 0.998859i
\(793\) − 6.60256i − 0.234464i
\(794\) −49.8759 −1.77003
\(795\) 0 0
\(796\) −86.4484 −3.06408
\(797\) 18.8371i 0.667244i 0.942707 + 0.333622i \(0.108271\pi\)
−0.942707 + 0.333622i \(0.891729\pi\)
\(798\) 26.8793i 0.951517i
\(799\) −32.7880 −1.15996
\(800\) 0 0
\(801\) 1.16291 0.0410894
\(802\) 37.7395i 1.33263i
\(803\) 24.7880i 0.874750i
\(804\) 10.7689 0.379790
\(805\) 0 0
\(806\) 20.9966 0.739572
\(807\) − 15.6742i − 0.551757i
\(808\) − 79.3155i − 2.79031i
\(809\) −32.2277 −1.13306 −0.566532 0.824040i \(-0.691715\pi\)
−0.566532 + 0.824040i \(0.691715\pi\)
\(810\) 0 0
\(811\) −23.0034 −0.807760 −0.403880 0.914812i \(-0.632339\pi\)
−0.403880 + 0.914812i \(0.632339\pi\)
\(812\) − 93.3415i − 3.27564i
\(813\) 0.443086i 0.0155397i
\(814\) −32.3189 −1.13278
\(815\) 0 0
\(816\) 49.0157 1.71589
\(817\) − 39.5309i − 1.38301i
\(818\) 38.8984i 1.36005i
\(819\) 2.71982 0.0950383
\(820\) 0 0
\(821\) 49.9372 1.74282 0.871410 0.490556i \(-0.163206\pi\)
0.871410 + 0.490556i \(0.163206\pi\)
\(822\) − 4.90528i − 0.171091i
\(823\) − 28.2345i − 0.984194i −0.870540 0.492097i \(-0.836231\pi\)
0.870540 0.492097i \(-0.163769\pi\)
\(824\) −38.9175 −1.35576
\(825\) 0 0
\(826\) 16.0000 0.556711
\(827\) − 9.55004i − 0.332087i −0.986118 0.166044i \(-0.946901\pi\)
0.986118 0.166044i \(-0.0530993\pi\)
\(828\) 27.6673i 0.961505i
\(829\) 37.9862 1.31932 0.659658 0.751565i \(-0.270701\pi\)
0.659658 + 0.751565i \(0.270701\pi\)
\(830\) 0 0
\(831\) 4.87930 0.169261
\(832\) 41.3871i 1.43484i
\(833\) 1.12758i 0.0390683i
\(834\) −17.4396 −0.603886
\(835\) 0 0
\(836\) 55.3346 1.91379
\(837\) − 7.55691i − 0.261205i
\(838\) − 34.2277i − 1.18237i
\(839\) 4.72670 0.163184 0.0815919 0.996666i \(-0.474000\pi\)
0.0815919 + 0.996666i \(0.474000\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 63.3155i 2.18200i
\(843\) − 9.11383i − 0.313897i
\(844\) 104.259 3.58874
\(845\) 0 0
\(846\) −32.1104 −1.10398
\(847\) 9.79832i 0.336674i
\(848\) − 20.0913i − 0.689938i
\(849\) 33.3415 1.14428
\(850\) 0 0
\(851\) 20.6870 0.709140
\(852\) 38.4362i 1.31680i
\(853\) 48.3611i 1.65585i 0.560836 + 0.827927i \(0.310480\pi\)
−0.560836 + 0.827927i \(0.689520\pi\)
\(854\) 49.8950 1.70737
\(855\) 0 0
\(856\) −130.252 −4.45193
\(857\) 6.83709i 0.233551i 0.993158 + 0.116775i \(0.0372557\pi\)
−0.993158 + 0.116775i \(0.962744\pi\)
\(858\) − 7.55691i − 0.257989i
\(859\) −12.6026 −0.429994 −0.214997 0.976615i \(-0.568974\pi\)
−0.214997 + 0.976615i \(0.568974\pi\)
\(860\) 0 0
\(861\) 7.71639 0.262974
\(862\) − 24.9966i − 0.851386i
\(863\) − 8.20855i − 0.279422i −0.990192 0.139711i \(-0.955383\pi\)
0.990192 0.139711i \(-0.0446174\pi\)
\(864\) 27.3319 0.929851
\(865\) 0 0
\(866\) −56.4744 −1.91908
\(867\) 8.95092i 0.303989i
\(868\) 117.562i 3.99032i
\(869\) 27.9509 0.948170
\(870\) 0 0
\(871\) 1.88273 0.0637940
\(872\) 118.233i 4.00387i
\(873\) 10.8371i 0.366780i
\(874\) −47.8037 −1.61698
\(875\) 0 0
\(876\) −52.1295 −1.76129
\(877\) 13.5309i 0.456907i 0.973555 + 0.228454i \(0.0733669\pi\)
−0.973555 + 0.228454i \(0.926633\pi\)
\(878\) 70.5466i 2.38083i
\(879\) −29.4328 −0.992743
\(880\) 0 0
\(881\) −9.34836 −0.314954 −0.157477 0.987523i \(-0.550336\pi\)
−0.157477 + 0.987523i \(0.550336\pi\)
\(882\) 1.10428i 0.0371829i
\(883\) 55.1001i 1.85427i 0.374733 + 0.927133i \(0.377734\pi\)
−0.374733 + 0.927133i \(0.622266\pi\)
\(884\) 16.2277 0.545795
\(885\) 0 0
\(886\) 30.3640 1.02010
\(887\) 0.133492i 0.00448223i 0.999997 + 0.00224112i \(0.000713370\pi\)
−0.999997 + 0.00224112i \(0.999287\pi\)
\(888\) − 44.2017i − 1.48331i
\(889\) −36.5535 −1.22596
\(890\) 0 0
\(891\) −2.71982 −0.0911175
\(892\) 57.8690i 1.93760i
\(893\) − 41.1070i − 1.37559i
\(894\) −57.8759 −1.93566
\(895\) 0 0
\(896\) −164.083 −5.48162
\(897\) 4.83709i 0.161506i
\(898\) − 7.88273i − 0.263050i
\(899\) −45.3415 −1.51222
\(900\) 0 0
\(901\) −3.29928 −0.109915
\(902\) − 21.4396i − 0.713862i
\(903\) − 30.2277i − 1.00591i
\(904\) 135.536 4.50787
\(905\) 0 0
\(906\) −13.8827 −0.461223
\(907\) − 58.5466i − 1.94401i −0.234964 0.972004i \(-0.575497\pi\)
0.234964 0.972004i \(-0.424503\pi\)
\(908\) − 64.7620i − 2.14920i
\(909\) −7.67418 −0.254537
\(910\) 0 0
\(911\) 50.4622 1.67189 0.835943 0.548816i \(-0.184921\pi\)
0.835943 + 0.548816i \(0.184921\pi\)
\(912\) 61.4519i 2.03487i
\(913\) 5.75859i 0.190582i
\(914\) 38.1104 1.26058
\(915\) 0 0
\(916\) −35.6604 −1.17825
\(917\) 25.6742i 0.847836i
\(918\) − 7.88273i − 0.260169i
\(919\) −56.9735 −1.87938 −0.939691 0.342026i \(-0.888887\pi\)
−0.939691 + 0.342026i \(0.888887\pi\)
\(920\) 0 0
\(921\) 21.8337 0.719443
\(922\) − 54.5275i − 1.79577i
\(923\) 6.71982i 0.221186i
\(924\) 42.3121 1.39196
\(925\) 0 0
\(926\) 75.1261 2.46880
\(927\) 3.76547i 0.123674i
\(928\) − 163.992i − 5.38329i
\(929\) 36.5957 1.20067 0.600333 0.799750i \(-0.295035\pi\)
0.600333 + 0.799750i \(0.295035\pi\)
\(930\) 0 0
\(931\) −1.41367 −0.0463311
\(932\) 39.1070i 1.28099i
\(933\) 25.1070i 0.821965i
\(934\) −80.3572 −2.62937
\(935\) 0 0
\(936\) 10.3354 0.337822
\(937\) − 47.1070i − 1.53892i −0.638697 0.769459i \(-0.720526\pi\)
0.638697 0.769459i \(-0.279474\pi\)
\(938\) 14.2277i 0.464549i
\(939\) 8.22766 0.268499
\(940\) 0 0
\(941\) −36.3611 −1.18534 −0.592670 0.805446i \(-0.701926\pi\)
−0.592670 + 0.805446i \(0.701926\pi\)
\(942\) − 24.6707i − 0.803816i
\(943\) 13.7233i 0.446891i
\(944\) 36.5795 1.19056
\(945\) 0 0
\(946\) −83.9862 −2.73063
\(947\) 44.5795i 1.44864i 0.689465 + 0.724319i \(0.257846\pi\)
−0.689465 + 0.724319i \(0.742154\pi\)
\(948\) 58.7811i 1.90912i
\(949\) −9.11383 −0.295847
\(950\) 0 0
\(951\) −27.6742 −0.897397
\(952\) 79.7517i 2.58477i
\(953\) − 19.8596i − 0.643317i −0.946856 0.321658i \(-0.895760\pi\)
0.946856 0.321658i \(-0.104240\pi\)
\(954\) −3.23109 −0.104611
\(955\) 0 0
\(956\) 7.32238 0.236823
\(957\) 16.3189i 0.527517i
\(958\) 33.7846i 1.09153i
\(959\) 4.80176 0.155057
\(960\) 0 0
\(961\) 26.1070 0.842160
\(962\) − 11.8827i − 0.383115i
\(963\) 12.6026i 0.406112i
\(964\) 34.3189 1.10534
\(965\) 0 0
\(966\) −36.5535 −1.17609
\(967\) 47.4068i 1.52450i 0.647283 + 0.762250i \(0.275905\pi\)
−0.647283 + 0.762250i \(0.724095\pi\)
\(968\) 37.2338i 1.19674i
\(969\) 10.0913 0.324179
\(970\) 0 0
\(971\) −10.6448 −0.341607 −0.170803 0.985305i \(-0.554636\pi\)
−0.170803 + 0.985305i \(0.554636\pi\)
\(972\) − 5.71982i − 0.183463i
\(973\) − 17.0716i − 0.547291i
\(974\) 0.443086 0.0141974
\(975\) 0 0
\(976\) 114.071 3.65131
\(977\) − 1.21199i − 0.0387750i −0.999812 0.0193875i \(-0.993828\pi\)
0.999812 0.0193875i \(-0.00617162\pi\)
\(978\) − 38.4362i − 1.22905i
\(979\) 3.16291 0.101087
\(980\) 0 0
\(981\) 11.4396 0.365240
\(982\) − 117.328i − 3.74408i
\(983\) − 51.8759i − 1.65458i −0.561773 0.827291i \(-0.689881\pi\)
0.561773 0.827291i \(-0.310119\pi\)
\(984\) 29.3224 0.934763
\(985\) 0 0
\(986\) −47.2964 −1.50622
\(987\) − 31.4328i − 1.00052i
\(988\) 20.3449i 0.647258i
\(989\) 53.7586 1.70942
\(990\) 0 0
\(991\) −21.6251 −0.686944 −0.343472 0.939163i \(-0.611603\pi\)
−0.343472 + 0.939163i \(0.611603\pi\)
\(992\) 206.545i 6.55781i
\(993\) − 13.2311i − 0.419876i
\(994\) −50.7811 −1.61068
\(995\) 0 0
\(996\) −12.1104 −0.383732
\(997\) 23.2051i 0.734913i 0.930041 + 0.367457i \(0.119771\pi\)
−0.930041 + 0.367457i \(0.880229\pi\)
\(998\) − 21.6482i − 0.685262i
\(999\) −4.27674 −0.135310
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 975.2.c.i.274.6 6
3.2 odd 2 2925.2.c.w.2224.1 6
5.2 odd 4 975.2.a.o.1.1 3
5.3 odd 4 195.2.a.e.1.3 3
5.4 even 2 inner 975.2.c.i.274.1 6
15.2 even 4 2925.2.a.bh.1.3 3
15.8 even 4 585.2.a.n.1.1 3
15.14 odd 2 2925.2.c.w.2224.6 6
20.3 even 4 3120.2.a.bj.1.3 3
35.13 even 4 9555.2.a.bq.1.3 3
60.23 odd 4 9360.2.a.dd.1.3 3
65.38 odd 4 2535.2.a.bc.1.1 3
195.38 even 4 7605.2.a.bx.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
195.2.a.e.1.3 3 5.3 odd 4
585.2.a.n.1.1 3 15.8 even 4
975.2.a.o.1.1 3 5.2 odd 4
975.2.c.i.274.1 6 5.4 even 2 inner
975.2.c.i.274.6 6 1.1 even 1 trivial
2535.2.a.bc.1.1 3 65.38 odd 4
2925.2.a.bh.1.3 3 15.2 even 4
2925.2.c.w.2224.1 6 3.2 odd 2
2925.2.c.w.2224.6 6 15.14 odd 2
3120.2.a.bj.1.3 3 20.3 even 4
7605.2.a.bx.1.3 3 195.38 even 4
9360.2.a.dd.1.3 3 60.23 odd 4
9555.2.a.bq.1.3 3 35.13 even 4