Properties

Label 275.2.b.c.199.3
Level $275$
Weight $2$
Character 275.199
Analytic conductor $2.196$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.19588605559\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{13})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.3
Root \(1.30278i\) of defining polynomial
Character \(\chi\) \(=\) 275.199
Dual form 275.2.b.c.199.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.30278i q^{2} +2.30278i q^{3} +0.302776 q^{4} -3.00000 q^{6} -0.697224i q^{7} +3.00000i q^{8} -2.30278 q^{9} +O(q^{10})\) \(q+1.30278i q^{2} +2.30278i q^{3} +0.302776 q^{4} -3.00000 q^{6} -0.697224i q^{7} +3.00000i q^{8} -2.30278 q^{9} -1.00000 q^{11} +0.697224i q^{12} +5.00000i q^{13} +0.908327 q^{14} -3.30278 q^{16} -6.90833i q^{17} -3.00000i q^{18} +1.00000 q^{19} +1.60555 q^{21} -1.30278i q^{22} -7.30278i q^{23} -6.90833 q^{24} -6.51388 q^{26} +1.60555i q^{27} -0.211103i q^{28} -0.908327 q^{29} +10.2111 q^{31} +1.69722i q^{32} -2.30278i q^{33} +9.00000 q^{34} -0.697224 q^{36} -2.39445i q^{37} +1.30278i q^{38} -11.5139 q^{39} -5.60555 q^{41} +2.09167i q^{42} +7.21110i q^{43} -0.302776 q^{44} +9.51388 q^{46} +3.00000i q^{47} -7.60555i q^{48} +6.51388 q^{49} +15.9083 q^{51} +1.51388i q^{52} -1.30278i q^{53} -2.09167 q^{54} +2.09167 q^{56} +2.30278i q^{57} -1.18335i q^{58} +14.2111 q^{59} -7.90833 q^{61} +13.3028i q^{62} +1.60555i q^{63} -8.81665 q^{64} +3.00000 q^{66} +4.00000i q^{67} -2.09167i q^{68} +16.8167 q^{69} -2.60555 q^{71} -6.90833i q^{72} -7.90833i q^{73} +3.11943 q^{74} +0.302776 q^{76} +0.697224i q^{77} -15.0000i q^{78} +10.9083 q^{79} -10.6056 q^{81} -7.30278i q^{82} -3.51388i q^{83} +0.486122 q^{84} -9.39445 q^{86} -2.09167i q^{87} -3.00000i q^{88} -1.69722 q^{89} +3.48612 q^{91} -2.21110i q^{92} +23.5139i q^{93} -3.90833 q^{94} -3.90833 q^{96} -15.3028i q^{97} +8.48612i q^{98} +2.30278 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 6 q^{4} - 12 q^{6} - 2 q^{9} + O(q^{10}) \) \( 4 q - 6 q^{4} - 12 q^{6} - 2 q^{9} - 4 q^{11} - 18 q^{14} - 6 q^{16} + 4 q^{19} - 8 q^{21} - 6 q^{24} + 10 q^{26} + 18 q^{29} + 12 q^{31} + 36 q^{34} - 10 q^{36} - 10 q^{39} - 8 q^{41} + 6 q^{44} + 2 q^{46} - 10 q^{49} + 42 q^{51} - 30 q^{54} + 30 q^{56} + 28 q^{59} - 10 q^{61} + 8 q^{64} + 12 q^{66} + 24 q^{69} + 4 q^{71} - 38 q^{74} - 6 q^{76} + 22 q^{79} - 28 q^{81} + 38 q^{84} - 52 q^{86} - 14 q^{89} + 50 q^{91} + 6 q^{94} + 6 q^{96} + 2 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.30278i 0.921201i 0.887607 + 0.460601i \(0.152366\pi\)
−0.887607 + 0.460601i \(0.847634\pi\)
\(3\) 2.30278i 1.32951i 0.747062 + 0.664754i \(0.231464\pi\)
−0.747062 + 0.664754i \(0.768536\pi\)
\(4\) 0.302776 0.151388
\(5\) 0 0
\(6\) −3.00000 −1.22474
\(7\) − 0.697224i − 0.263526i −0.991281 0.131763i \(-0.957936\pi\)
0.991281 0.131763i \(-0.0420638\pi\)
\(8\) 3.00000i 1.06066i
\(9\) −2.30278 −0.767592
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0.697224i 0.201271i
\(13\) 5.00000i 1.38675i 0.720577 + 0.693375i \(0.243877\pi\)
−0.720577 + 0.693375i \(0.756123\pi\)
\(14\) 0.908327 0.242761
\(15\) 0 0
\(16\) −3.30278 −0.825694
\(17\) − 6.90833i − 1.67552i −0.546042 0.837758i \(-0.683866\pi\)
0.546042 0.837758i \(-0.316134\pi\)
\(18\) − 3.00000i − 0.707107i
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 1.60555 0.350360
\(22\) − 1.30278i − 0.277753i
\(23\) − 7.30278i − 1.52273i −0.648321 0.761367i \(-0.724529\pi\)
0.648321 0.761367i \(-0.275471\pi\)
\(24\) −6.90833 −1.41016
\(25\) 0 0
\(26\) −6.51388 −1.27748
\(27\) 1.60555i 0.308988i
\(28\) − 0.211103i − 0.0398946i
\(29\) −0.908327 −0.168672 −0.0843360 0.996437i \(-0.526877\pi\)
−0.0843360 + 0.996437i \(0.526877\pi\)
\(30\) 0 0
\(31\) 10.2111 1.83397 0.916984 0.398924i \(-0.130616\pi\)
0.916984 + 0.398924i \(0.130616\pi\)
\(32\) 1.69722i 0.300030i
\(33\) − 2.30278i − 0.400862i
\(34\) 9.00000 1.54349
\(35\) 0 0
\(36\) −0.697224 −0.116204
\(37\) − 2.39445i − 0.393645i −0.980439 0.196822i \(-0.936938\pi\)
0.980439 0.196822i \(-0.0630623\pi\)
\(38\) 1.30278i 0.211338i
\(39\) −11.5139 −1.84370
\(40\) 0 0
\(41\) −5.60555 −0.875440 −0.437720 0.899111i \(-0.644214\pi\)
−0.437720 + 0.899111i \(0.644214\pi\)
\(42\) 2.09167i 0.322752i
\(43\) 7.21110i 1.09968i 0.835269 + 0.549841i \(0.185312\pi\)
−0.835269 + 0.549841i \(0.814688\pi\)
\(44\) −0.302776 −0.0456451
\(45\) 0 0
\(46\) 9.51388 1.40274
\(47\) 3.00000i 0.437595i 0.975770 + 0.218797i \(0.0702134\pi\)
−0.975770 + 0.218797i \(0.929787\pi\)
\(48\) − 7.60555i − 1.09777i
\(49\) 6.51388 0.930554
\(50\) 0 0
\(51\) 15.9083 2.22761
\(52\) 1.51388i 0.209937i
\(53\) − 1.30278i − 0.178950i −0.995989 0.0894750i \(-0.971481\pi\)
0.995989 0.0894750i \(-0.0285189\pi\)
\(54\) −2.09167 −0.284641
\(55\) 0 0
\(56\) 2.09167 0.279512
\(57\) 2.30278i 0.305010i
\(58\) − 1.18335i − 0.155381i
\(59\) 14.2111 1.85013 0.925064 0.379811i \(-0.124011\pi\)
0.925064 + 0.379811i \(0.124011\pi\)
\(60\) 0 0
\(61\) −7.90833 −1.01256 −0.506279 0.862370i \(-0.668979\pi\)
−0.506279 + 0.862370i \(0.668979\pi\)
\(62\) 13.3028i 1.68945i
\(63\) 1.60555i 0.202280i
\(64\) −8.81665 −1.10208
\(65\) 0 0
\(66\) 3.00000 0.369274
\(67\) 4.00000i 0.488678i 0.969690 + 0.244339i \(0.0785709\pi\)
−0.969690 + 0.244339i \(0.921429\pi\)
\(68\) − 2.09167i − 0.253653i
\(69\) 16.8167 2.02449
\(70\) 0 0
\(71\) −2.60555 −0.309222 −0.154611 0.987975i \(-0.549412\pi\)
−0.154611 + 0.987975i \(0.549412\pi\)
\(72\) − 6.90833i − 0.814154i
\(73\) − 7.90833i − 0.925600i −0.886463 0.462800i \(-0.846845\pi\)
0.886463 0.462800i \(-0.153155\pi\)
\(74\) 3.11943 0.362626
\(75\) 0 0
\(76\) 0.302776 0.0347307
\(77\) 0.697224i 0.0794561i
\(78\) − 15.0000i − 1.69842i
\(79\) 10.9083 1.22728 0.613641 0.789585i \(-0.289704\pi\)
0.613641 + 0.789585i \(0.289704\pi\)
\(80\) 0 0
\(81\) −10.6056 −1.17839
\(82\) − 7.30278i − 0.806457i
\(83\) − 3.51388i − 0.385698i −0.981228 0.192849i \(-0.938227\pi\)
0.981228 0.192849i \(-0.0617728\pi\)
\(84\) 0.486122 0.0530402
\(85\) 0 0
\(86\) −9.39445 −1.01303
\(87\) − 2.09167i − 0.224251i
\(88\) − 3.00000i − 0.319801i
\(89\) −1.69722 −0.179905 −0.0899527 0.995946i \(-0.528672\pi\)
−0.0899527 + 0.995946i \(0.528672\pi\)
\(90\) 0 0
\(91\) 3.48612 0.365445
\(92\) − 2.21110i − 0.230523i
\(93\) 23.5139i 2.43828i
\(94\) −3.90833 −0.403113
\(95\) 0 0
\(96\) −3.90833 −0.398892
\(97\) − 15.3028i − 1.55376i −0.629648 0.776881i \(-0.716801\pi\)
0.629648 0.776881i \(-0.283199\pi\)
\(98\) 8.48612i 0.857228i
\(99\) 2.30278 0.231438
\(100\) 0 0
\(101\) −0.513878 −0.0511328 −0.0255664 0.999673i \(-0.508139\pi\)
−0.0255664 + 0.999673i \(0.508139\pi\)
\(102\) 20.7250i 2.05208i
\(103\) 2.90833i 0.286566i 0.989682 + 0.143283i \(0.0457659\pi\)
−0.989682 + 0.143283i \(0.954234\pi\)
\(104\) −15.0000 −1.47087
\(105\) 0 0
\(106\) 1.69722 0.164849
\(107\) 3.00000i 0.290021i 0.989430 + 0.145010i \(0.0463216\pi\)
−0.989430 + 0.145010i \(0.953678\pi\)
\(108\) 0.486122i 0.0467771i
\(109\) −11.5139 −1.10283 −0.551415 0.834231i \(-0.685912\pi\)
−0.551415 + 0.834231i \(0.685912\pi\)
\(110\) 0 0
\(111\) 5.51388 0.523354
\(112\) 2.30278i 0.217592i
\(113\) − 10.8167i − 1.01755i −0.860901 0.508773i \(-0.830099\pi\)
0.860901 0.508773i \(-0.169901\pi\)
\(114\) −3.00000 −0.280976
\(115\) 0 0
\(116\) −0.275019 −0.0255349
\(117\) − 11.5139i − 1.06446i
\(118\) 18.5139i 1.70434i
\(119\) −4.81665 −0.441542
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 10.3028i − 0.932769i
\(123\) − 12.9083i − 1.16390i
\(124\) 3.09167 0.277640
\(125\) 0 0
\(126\) −2.09167 −0.186341
\(127\) − 8.11943i − 0.720483i −0.932859 0.360241i \(-0.882694\pi\)
0.932859 0.360241i \(-0.117306\pi\)
\(128\) − 8.09167i − 0.715210i
\(129\) −16.6056 −1.46204
\(130\) 0 0
\(131\) −9.90833 −0.865695 −0.432847 0.901467i \(-0.642491\pi\)
−0.432847 + 0.901467i \(0.642491\pi\)
\(132\) − 0.697224i − 0.0606856i
\(133\) − 0.697224i − 0.0604570i
\(134\) −5.21110 −0.450171
\(135\) 0 0
\(136\) 20.7250 1.77715
\(137\) 12.9083i 1.10283i 0.834230 + 0.551416i \(0.185912\pi\)
−0.834230 + 0.551416i \(0.814088\pi\)
\(138\) 21.9083i 1.86496i
\(139\) 6.21110 0.526819 0.263409 0.964684i \(-0.415153\pi\)
0.263409 + 0.964684i \(0.415153\pi\)
\(140\) 0 0
\(141\) −6.90833 −0.581786
\(142\) − 3.39445i − 0.284856i
\(143\) − 5.00000i − 0.418121i
\(144\) 7.60555 0.633796
\(145\) 0 0
\(146\) 10.3028 0.852664
\(147\) 15.0000i 1.23718i
\(148\) − 0.724981i − 0.0595930i
\(149\) −17.2111 −1.40999 −0.704994 0.709213i \(-0.749050\pi\)
−0.704994 + 0.709213i \(0.749050\pi\)
\(150\) 0 0
\(151\) 0.816654 0.0664583 0.0332292 0.999448i \(-0.489421\pi\)
0.0332292 + 0.999448i \(0.489421\pi\)
\(152\) 3.00000i 0.243332i
\(153\) 15.9083i 1.28611i
\(154\) −0.908327 −0.0731951
\(155\) 0 0
\(156\) −3.48612 −0.279113
\(157\) − 19.2111i − 1.53321i −0.642117 0.766606i \(-0.721944\pi\)
0.642117 0.766606i \(-0.278056\pi\)
\(158\) 14.2111i 1.13057i
\(159\) 3.00000 0.237915
\(160\) 0 0
\(161\) −5.09167 −0.401280
\(162\) − 13.8167i − 1.08554i
\(163\) 9.30278i 0.728650i 0.931272 + 0.364325i \(0.118700\pi\)
−0.931272 + 0.364325i \(0.881300\pi\)
\(164\) −1.69722 −0.132531
\(165\) 0 0
\(166\) 4.57779 0.355306
\(167\) − 13.4222i − 1.03864i −0.854579 0.519321i \(-0.826185\pi\)
0.854579 0.519321i \(-0.173815\pi\)
\(168\) 4.81665i 0.371613i
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) −2.30278 −0.176098
\(172\) 2.18335i 0.166479i
\(173\) − 4.81665i − 0.366203i −0.983094 0.183102i \(-0.941386\pi\)
0.983094 0.183102i \(-0.0586138\pi\)
\(174\) 2.72498 0.206580
\(175\) 0 0
\(176\) 3.30278 0.248956
\(177\) 32.7250i 2.45976i
\(178\) − 2.21110i − 0.165729i
\(179\) −12.5139 −0.935331 −0.467666 0.883905i \(-0.654905\pi\)
−0.467666 + 0.883905i \(0.654905\pi\)
\(180\) 0 0
\(181\) −19.9083 −1.47977 −0.739887 0.672731i \(-0.765121\pi\)
−0.739887 + 0.672731i \(0.765121\pi\)
\(182\) 4.54163i 0.336648i
\(183\) − 18.2111i − 1.34620i
\(184\) 21.9083 1.61510
\(185\) 0 0
\(186\) −30.6333 −2.24614
\(187\) 6.90833i 0.505187i
\(188\) 0.908327i 0.0662465i
\(189\) 1.11943 0.0814265
\(190\) 0 0
\(191\) 10.3028 0.745483 0.372741 0.927935i \(-0.378418\pi\)
0.372741 + 0.927935i \(0.378418\pi\)
\(192\) − 20.3028i − 1.46523i
\(193\) 13.2111i 0.950956i 0.879728 + 0.475478i \(0.157725\pi\)
−0.879728 + 0.475478i \(0.842275\pi\)
\(194\) 19.9361 1.43133
\(195\) 0 0
\(196\) 1.97224 0.140875
\(197\) − 13.3028i − 0.947784i −0.880583 0.473892i \(-0.842849\pi\)
0.880583 0.473892i \(-0.157151\pi\)
\(198\) 3.00000i 0.213201i
\(199\) 6.48612 0.459789 0.229894 0.973216i \(-0.426162\pi\)
0.229894 + 0.973216i \(0.426162\pi\)
\(200\) 0 0
\(201\) −9.21110 −0.649701
\(202\) − 0.669468i − 0.0471036i
\(203\) 0.633308i 0.0444495i
\(204\) 4.81665 0.337233
\(205\) 0 0
\(206\) −3.78890 −0.263985
\(207\) 16.8167i 1.16884i
\(208\) − 16.5139i − 1.14503i
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) −25.2389 −1.73751 −0.868757 0.495238i \(-0.835081\pi\)
−0.868757 + 0.495238i \(0.835081\pi\)
\(212\) − 0.394449i − 0.0270908i
\(213\) − 6.00000i − 0.411113i
\(214\) −3.90833 −0.267168
\(215\) 0 0
\(216\) −4.81665 −0.327732
\(217\) − 7.11943i − 0.483298i
\(218\) − 15.0000i − 1.01593i
\(219\) 18.2111 1.23059
\(220\) 0 0
\(221\) 34.5416 2.32352
\(222\) 7.18335i 0.482115i
\(223\) − 22.6333i − 1.51564i −0.652465 0.757819i \(-0.726265\pi\)
0.652465 0.757819i \(-0.273735\pi\)
\(224\) 1.18335 0.0790656
\(225\) 0 0
\(226\) 14.0917 0.937364
\(227\) − 1.69722i − 0.112649i −0.998413 0.0563244i \(-0.982062\pi\)
0.998413 0.0563244i \(-0.0179381\pi\)
\(228\) 0.697224i 0.0461748i
\(229\) 18.7250 1.23738 0.618691 0.785635i \(-0.287663\pi\)
0.618691 + 0.785635i \(0.287663\pi\)
\(230\) 0 0
\(231\) −1.60555 −0.105638
\(232\) − 2.72498i − 0.178904i
\(233\) 15.9083i 1.04219i 0.853499 + 0.521095i \(0.174476\pi\)
−0.853499 + 0.521095i \(0.825524\pi\)
\(234\) 15.0000 0.980581
\(235\) 0 0
\(236\) 4.30278 0.280087
\(237\) 25.1194i 1.63168i
\(238\) − 6.27502i − 0.406749i
\(239\) −21.1194 −1.36610 −0.683051 0.730371i \(-0.739347\pi\)
−0.683051 + 0.730371i \(0.739347\pi\)
\(240\) 0 0
\(241\) 21.9361 1.41303 0.706514 0.707699i \(-0.250267\pi\)
0.706514 + 0.707699i \(0.250267\pi\)
\(242\) 1.30278i 0.0837456i
\(243\) − 19.6056i − 1.25770i
\(244\) −2.39445 −0.153289
\(245\) 0 0
\(246\) 16.8167 1.07219
\(247\) 5.00000i 0.318142i
\(248\) 30.6333i 1.94522i
\(249\) 8.09167 0.512789
\(250\) 0 0
\(251\) 6.90833 0.436050 0.218025 0.975943i \(-0.430039\pi\)
0.218025 + 0.975943i \(0.430039\pi\)
\(252\) 0.486122i 0.0306228i
\(253\) 7.30278i 0.459122i
\(254\) 10.5778 0.663710
\(255\) 0 0
\(256\) −7.09167 −0.443230
\(257\) 18.0000i 1.12281i 0.827541 + 0.561405i \(0.189739\pi\)
−0.827541 + 0.561405i \(0.810261\pi\)
\(258\) − 21.6333i − 1.34683i
\(259\) −1.66947 −0.103736
\(260\) 0 0
\(261\) 2.09167 0.129471
\(262\) − 12.9083i − 0.797479i
\(263\) − 22.8167i − 1.40694i −0.710727 0.703468i \(-0.751634\pi\)
0.710727 0.703468i \(-0.248366\pi\)
\(264\) 6.90833 0.425178
\(265\) 0 0
\(266\) 0.908327 0.0556931
\(267\) − 3.90833i − 0.239186i
\(268\) 1.21110i 0.0739799i
\(269\) −8.72498 −0.531971 −0.265986 0.963977i \(-0.585697\pi\)
−0.265986 + 0.963977i \(0.585697\pi\)
\(270\) 0 0
\(271\) −0.211103 −0.0128236 −0.00641178 0.999979i \(-0.502041\pi\)
−0.00641178 + 0.999979i \(0.502041\pi\)
\(272\) 22.8167i 1.38346i
\(273\) 8.02776i 0.485862i
\(274\) −16.8167 −1.01593
\(275\) 0 0
\(276\) 5.09167 0.306483
\(277\) − 14.3944i − 0.864879i −0.901663 0.432439i \(-0.857653\pi\)
0.901663 0.432439i \(-0.142347\pi\)
\(278\) 8.09167i 0.485306i
\(279\) −23.5139 −1.40774
\(280\) 0 0
\(281\) −1.18335 −0.0705925 −0.0352963 0.999377i \(-0.511237\pi\)
−0.0352963 + 0.999377i \(0.511237\pi\)
\(282\) − 9.00000i − 0.535942i
\(283\) 6.30278i 0.374661i 0.982297 + 0.187331i \(0.0599836\pi\)
−0.982297 + 0.187331i \(0.940016\pi\)
\(284\) −0.788897 −0.0468125
\(285\) 0 0
\(286\) 6.51388 0.385174
\(287\) 3.90833i 0.230701i
\(288\) − 3.90833i − 0.230300i
\(289\) −30.7250 −1.80735
\(290\) 0 0
\(291\) 35.2389 2.06574
\(292\) − 2.39445i − 0.140125i
\(293\) 0.788897i 0.0460879i 0.999734 + 0.0230439i \(0.00733576\pi\)
−0.999734 + 0.0230439i \(0.992664\pi\)
\(294\) −19.5416 −1.13969
\(295\) 0 0
\(296\) 7.18335 0.417524
\(297\) − 1.60555i − 0.0931635i
\(298\) − 22.4222i − 1.29888i
\(299\) 36.5139 2.11165
\(300\) 0 0
\(301\) 5.02776 0.289795
\(302\) 1.06392i 0.0612215i
\(303\) − 1.18335i − 0.0679815i
\(304\) −3.30278 −0.189427
\(305\) 0 0
\(306\) −20.7250 −1.18477
\(307\) 16.9083i 0.965009i 0.875893 + 0.482505i \(0.160273\pi\)
−0.875893 + 0.482505i \(0.839727\pi\)
\(308\) 0.211103i 0.0120287i
\(309\) −6.69722 −0.380992
\(310\) 0 0
\(311\) 4.81665 0.273127 0.136564 0.990631i \(-0.456394\pi\)
0.136564 + 0.990631i \(0.456394\pi\)
\(312\) − 34.5416i − 1.95553i
\(313\) 0.183346i 0.0103633i 0.999987 + 0.00518167i \(0.00164938\pi\)
−0.999987 + 0.00518167i \(0.998351\pi\)
\(314\) 25.0278 1.41240
\(315\) 0 0
\(316\) 3.30278 0.185796
\(317\) 0.908327i 0.0510167i 0.999675 + 0.0255084i \(0.00812044\pi\)
−0.999675 + 0.0255084i \(0.991880\pi\)
\(318\) 3.90833i 0.219168i
\(319\) 0.908327 0.0508565
\(320\) 0 0
\(321\) −6.90833 −0.385585
\(322\) − 6.63331i − 0.369660i
\(323\) − 6.90833i − 0.384390i
\(324\) −3.21110 −0.178395
\(325\) 0 0
\(326\) −12.1194 −0.671233
\(327\) − 26.5139i − 1.46622i
\(328\) − 16.8167i − 0.928544i
\(329\) 2.09167 0.115318
\(330\) 0 0
\(331\) −21.6056 −1.18755 −0.593774 0.804632i \(-0.702363\pi\)
−0.593774 + 0.804632i \(0.702363\pi\)
\(332\) − 1.06392i − 0.0583900i
\(333\) 5.51388i 0.302159i
\(334\) 17.4861 0.956798
\(335\) 0 0
\(336\) −5.30278 −0.289290
\(337\) 30.8444i 1.68020i 0.542430 + 0.840101i \(0.317504\pi\)
−0.542430 + 0.840101i \(0.682496\pi\)
\(338\) − 15.6333i − 0.850340i
\(339\) 24.9083 1.35283
\(340\) 0 0
\(341\) −10.2111 −0.552962
\(342\) − 3.00000i − 0.162221i
\(343\) − 9.42221i − 0.508751i
\(344\) −21.6333 −1.16639
\(345\) 0 0
\(346\) 6.27502 0.337347
\(347\) 12.5139i 0.671780i 0.941901 + 0.335890i \(0.109037\pi\)
−0.941901 + 0.335890i \(0.890963\pi\)
\(348\) − 0.633308i − 0.0339489i
\(349\) 5.18335 0.277458 0.138729 0.990330i \(-0.455698\pi\)
0.138729 + 0.990330i \(0.455698\pi\)
\(350\) 0 0
\(351\) −8.02776 −0.428490
\(352\) − 1.69722i − 0.0904624i
\(353\) 18.6333i 0.991751i 0.868394 + 0.495875i \(0.165153\pi\)
−0.868394 + 0.495875i \(0.834847\pi\)
\(354\) −42.6333 −2.26593
\(355\) 0 0
\(356\) −0.513878 −0.0272355
\(357\) − 11.0917i − 0.587034i
\(358\) − 16.3028i − 0.861628i
\(359\) −0.788897 −0.0416364 −0.0208182 0.999783i \(-0.506627\pi\)
−0.0208182 + 0.999783i \(0.506627\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) − 25.9361i − 1.36317i
\(363\) 2.30278i 0.120864i
\(364\) 1.05551 0.0553239
\(365\) 0 0
\(366\) 23.7250 1.24012
\(367\) 20.6972i 1.08039i 0.841541 + 0.540193i \(0.181649\pi\)
−0.841541 + 0.540193i \(0.818351\pi\)
\(368\) 24.1194i 1.25731i
\(369\) 12.9083 0.671981
\(370\) 0 0
\(371\) −0.908327 −0.0471580
\(372\) 7.11943i 0.369125i
\(373\) 27.4222i 1.41987i 0.704268 + 0.709934i \(0.251275\pi\)
−0.704268 + 0.709934i \(0.748725\pi\)
\(374\) −9.00000 −0.465379
\(375\) 0 0
\(376\) −9.00000 −0.464140
\(377\) − 4.54163i − 0.233906i
\(378\) 1.45837i 0.0750102i
\(379\) −3.18335 −0.163518 −0.0817588 0.996652i \(-0.526054\pi\)
−0.0817588 + 0.996652i \(0.526054\pi\)
\(380\) 0 0
\(381\) 18.6972 0.957888
\(382\) 13.4222i 0.686740i
\(383\) 21.6333i 1.10541i 0.833377 + 0.552705i \(0.186404\pi\)
−0.833377 + 0.552705i \(0.813596\pi\)
\(384\) 18.6333 0.950877
\(385\) 0 0
\(386\) −17.2111 −0.876022
\(387\) − 16.6056i − 0.844108i
\(388\) − 4.63331i − 0.235221i
\(389\) −12.0000 −0.608424 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) −50.4500 −2.55136
\(392\) 19.5416i 0.987002i
\(393\) − 22.8167i − 1.15095i
\(394\) 17.3305 0.873100
\(395\) 0 0
\(396\) 0.697224 0.0350368
\(397\) − 21.6972i − 1.08895i −0.838776 0.544476i \(-0.816728\pi\)
0.838776 0.544476i \(-0.183272\pi\)
\(398\) 8.44996i 0.423558i
\(399\) 1.60555 0.0803781
\(400\) 0 0
\(401\) −12.7889 −0.638647 −0.319324 0.947646i \(-0.603456\pi\)
−0.319324 + 0.947646i \(0.603456\pi\)
\(402\) − 12.0000i − 0.598506i
\(403\) 51.0555i 2.54326i
\(404\) −0.155590 −0.00774088
\(405\) 0 0
\(406\) −0.825058 −0.0409469
\(407\) 2.39445i 0.118688i
\(408\) 47.7250i 2.36274i
\(409\) 6.21110 0.307119 0.153560 0.988139i \(-0.450926\pi\)
0.153560 + 0.988139i \(0.450926\pi\)
\(410\) 0 0
\(411\) −29.7250 −1.46623
\(412\) 0.880571i 0.0433826i
\(413\) − 9.90833i − 0.487557i
\(414\) −21.9083 −1.07674
\(415\) 0 0
\(416\) −8.48612 −0.416066
\(417\) 14.3028i 0.700410i
\(418\) − 1.30278i − 0.0637208i
\(419\) −6.39445 −0.312389 −0.156195 0.987726i \(-0.549923\pi\)
−0.156195 + 0.987726i \(0.549923\pi\)
\(420\) 0 0
\(421\) 0.697224 0.0339806 0.0169903 0.999856i \(-0.494592\pi\)
0.0169903 + 0.999856i \(0.494592\pi\)
\(422\) − 32.8806i − 1.60060i
\(423\) − 6.90833i − 0.335894i
\(424\) 3.90833 0.189805
\(425\) 0 0
\(426\) 7.81665 0.378718
\(427\) 5.51388i 0.266835i
\(428\) 0.908327i 0.0439056i
\(429\) 11.5139 0.555895
\(430\) 0 0
\(431\) 33.0000 1.58955 0.794777 0.606902i \(-0.207588\pi\)
0.794777 + 0.606902i \(0.207588\pi\)
\(432\) − 5.30278i − 0.255130i
\(433\) 5.00000i 0.240285i 0.992757 + 0.120142i \(0.0383351\pi\)
−0.992757 + 0.120142i \(0.961665\pi\)
\(434\) 9.27502 0.445215
\(435\) 0 0
\(436\) −3.48612 −0.166955
\(437\) − 7.30278i − 0.349339i
\(438\) 23.7250i 1.13362i
\(439\) −24.3028 −1.15991 −0.579954 0.814649i \(-0.696930\pi\)
−0.579954 + 0.814649i \(0.696930\pi\)
\(440\) 0 0
\(441\) −15.0000 −0.714286
\(442\) 45.0000i 2.14043i
\(443\) − 8.60555i − 0.408862i −0.978881 0.204431i \(-0.934466\pi\)
0.978881 0.204431i \(-0.0655344\pi\)
\(444\) 1.66947 0.0792294
\(445\) 0 0
\(446\) 29.4861 1.39621
\(447\) − 39.6333i − 1.87459i
\(448\) 6.14719i 0.290427i
\(449\) −23.4861 −1.10838 −0.554189 0.832391i \(-0.686972\pi\)
−0.554189 + 0.832391i \(0.686972\pi\)
\(450\) 0 0
\(451\) 5.60555 0.263955
\(452\) − 3.27502i − 0.154044i
\(453\) 1.88057i 0.0883569i
\(454\) 2.21110 0.103772
\(455\) 0 0
\(456\) −6.90833 −0.323512
\(457\) 20.6972i 0.968175i 0.875020 + 0.484088i \(0.160848\pi\)
−0.875020 + 0.484088i \(0.839152\pi\)
\(458\) 24.3944i 1.13988i
\(459\) 11.0917 0.517715
\(460\) 0 0
\(461\) 32.2111 1.50022 0.750110 0.661313i \(-0.230000\pi\)
0.750110 + 0.661313i \(0.230000\pi\)
\(462\) − 2.09167i − 0.0973134i
\(463\) 11.7889i 0.547877i 0.961747 + 0.273938i \(0.0883264\pi\)
−0.961747 + 0.273938i \(0.911674\pi\)
\(464\) 3.00000 0.139272
\(465\) 0 0
\(466\) −20.7250 −0.960066
\(467\) 18.6333i 0.862247i 0.902293 + 0.431123i \(0.141883\pi\)
−0.902293 + 0.431123i \(0.858117\pi\)
\(468\) − 3.48612i − 0.161146i
\(469\) 2.78890 0.128779
\(470\) 0 0
\(471\) 44.2389 2.03842
\(472\) 42.6333i 1.96236i
\(473\) − 7.21110i − 0.331567i
\(474\) −32.7250 −1.50311
\(475\) 0 0
\(476\) −1.45837 −0.0668441
\(477\) 3.00000i 0.137361i
\(478\) − 27.5139i − 1.25846i
\(479\) 34.8167 1.59081 0.795407 0.606076i \(-0.207257\pi\)
0.795407 + 0.606076i \(0.207257\pi\)
\(480\) 0 0
\(481\) 11.9722 0.545887
\(482\) 28.5778i 1.30168i
\(483\) − 11.7250i − 0.533505i
\(484\) 0.302776 0.0137625
\(485\) 0 0
\(486\) 25.5416 1.15859
\(487\) − 4.21110i − 0.190823i −0.995438 0.0954116i \(-0.969583\pi\)
0.995438 0.0954116i \(-0.0304167\pi\)
\(488\) − 23.7250i − 1.07398i
\(489\) −21.4222 −0.968746
\(490\) 0 0
\(491\) −9.78890 −0.441767 −0.220883 0.975300i \(-0.570894\pi\)
−0.220883 + 0.975300i \(0.570894\pi\)
\(492\) − 3.90833i − 0.176201i
\(493\) 6.27502i 0.282613i
\(494\) −6.51388 −0.293073
\(495\) 0 0
\(496\) −33.7250 −1.51430
\(497\) 1.81665i 0.0814881i
\(498\) 10.5416i 0.472382i
\(499\) 3.48612 0.156060 0.0780301 0.996951i \(-0.475137\pi\)
0.0780301 + 0.996951i \(0.475137\pi\)
\(500\) 0 0
\(501\) 30.9083 1.38088
\(502\) 9.00000i 0.401690i
\(503\) 9.39445i 0.418878i 0.977822 + 0.209439i \(0.0671637\pi\)
−0.977822 + 0.209439i \(0.932836\pi\)
\(504\) −4.81665 −0.214551
\(505\) 0 0
\(506\) −9.51388 −0.422943
\(507\) − 27.6333i − 1.22724i
\(508\) − 2.45837i − 0.109072i
\(509\) 22.6972 1.00604 0.503018 0.864276i \(-0.332223\pi\)
0.503018 + 0.864276i \(0.332223\pi\)
\(510\) 0 0
\(511\) −5.51388 −0.243920
\(512\) − 25.4222i − 1.12351i
\(513\) 1.60555i 0.0708868i
\(514\) −23.4500 −1.03433
\(515\) 0 0
\(516\) −5.02776 −0.221335
\(517\) − 3.00000i − 0.131940i
\(518\) − 2.17494i − 0.0955615i
\(519\) 11.0917 0.486870
\(520\) 0 0
\(521\) −41.4500 −1.81596 −0.907978 0.419018i \(-0.862374\pi\)
−0.907978 + 0.419018i \(0.862374\pi\)
\(522\) 2.72498i 0.119269i
\(523\) − 32.4222i − 1.41772i −0.705347 0.708862i \(-0.749209\pi\)
0.705347 0.708862i \(-0.250791\pi\)
\(524\) −3.00000 −0.131056
\(525\) 0 0
\(526\) 29.7250 1.29607
\(527\) − 70.5416i − 3.07284i
\(528\) 7.60555i 0.330989i
\(529\) −30.3305 −1.31872
\(530\) 0 0
\(531\) −32.7250 −1.42014
\(532\) − 0.211103i − 0.00915246i
\(533\) − 28.0278i − 1.21402i
\(534\) 5.09167 0.220338
\(535\) 0 0
\(536\) −12.0000 −0.518321
\(537\) − 28.8167i − 1.24353i
\(538\) − 11.3667i − 0.490053i
\(539\) −6.51388 −0.280573
\(540\) 0 0
\(541\) 25.7250 1.10600 0.553002 0.833180i \(-0.313482\pi\)
0.553002 + 0.833180i \(0.313482\pi\)
\(542\) − 0.275019i − 0.0118131i
\(543\) − 45.8444i − 1.96737i
\(544\) 11.7250 0.502704
\(545\) 0 0
\(546\) −10.4584 −0.447577
\(547\) 7.11943i 0.304405i 0.988349 + 0.152202i \(0.0486366\pi\)
−0.988349 + 0.152202i \(0.951363\pi\)
\(548\) 3.90833i 0.166955i
\(549\) 18.2111 0.777231
\(550\) 0 0
\(551\) −0.908327 −0.0386960
\(552\) 50.4500i 2.14729i
\(553\) − 7.60555i − 0.323421i
\(554\) 18.7527 0.796727
\(555\) 0 0
\(556\) 1.88057 0.0797540
\(557\) − 19.4222i − 0.822945i −0.911422 0.411473i \(-0.865015\pi\)
0.911422 0.411473i \(-0.134985\pi\)
\(558\) − 30.6333i − 1.29681i
\(559\) −36.0555 −1.52499
\(560\) 0 0
\(561\) −15.9083 −0.671650
\(562\) − 1.54163i − 0.0650299i
\(563\) 8.09167i 0.341023i 0.985356 + 0.170512i \(0.0545421\pi\)
−0.985356 + 0.170512i \(0.945458\pi\)
\(564\) −2.09167 −0.0880753
\(565\) 0 0
\(566\) −8.21110 −0.345138
\(567\) 7.39445i 0.310538i
\(568\) − 7.81665i − 0.327980i
\(569\) 46.1472 1.93459 0.967295 0.253653i \(-0.0816321\pi\)
0.967295 + 0.253653i \(0.0816321\pi\)
\(570\) 0 0
\(571\) 22.3305 0.934504 0.467252 0.884124i \(-0.345244\pi\)
0.467252 + 0.884124i \(0.345244\pi\)
\(572\) − 1.51388i − 0.0632984i
\(573\) 23.7250i 0.991125i
\(574\) −5.09167 −0.212522
\(575\) 0 0
\(576\) 20.3028 0.845949
\(577\) − 44.3583i − 1.84666i −0.384008 0.923330i \(-0.625456\pi\)
0.384008 0.923330i \(-0.374544\pi\)
\(578\) − 40.0278i − 1.66494i
\(579\) −30.4222 −1.26430
\(580\) 0 0
\(581\) −2.44996 −0.101642
\(582\) 45.9083i 1.90296i
\(583\) 1.30278i 0.0539555i
\(584\) 23.7250 0.981747
\(585\) 0 0
\(586\) −1.02776 −0.0424562
\(587\) − 16.5416i − 0.682746i −0.939928 0.341373i \(-0.889108\pi\)
0.939928 0.341373i \(-0.110892\pi\)
\(588\) 4.54163i 0.187294i
\(589\) 10.2111 0.420741
\(590\) 0 0
\(591\) 30.6333 1.26009
\(592\) 7.90833i 0.325030i
\(593\) − 6.39445i − 0.262589i −0.991343 0.131294i \(-0.958087\pi\)
0.991343 0.131294i \(-0.0419133\pi\)
\(594\) 2.09167 0.0858224
\(595\) 0 0
\(596\) −5.21110 −0.213455
\(597\) 14.9361i 0.611293i
\(598\) 47.5694i 1.94526i
\(599\) 24.9083 1.01773 0.508863 0.860847i \(-0.330066\pi\)
0.508863 + 0.860847i \(0.330066\pi\)
\(600\) 0 0
\(601\) −1.90833 −0.0778423 −0.0389211 0.999242i \(-0.512392\pi\)
−0.0389211 + 0.999242i \(0.512392\pi\)
\(602\) 6.55004i 0.266960i
\(603\) − 9.21110i − 0.375105i
\(604\) 0.247263 0.0100610
\(605\) 0 0
\(606\) 1.54163 0.0626246
\(607\) − 7.21110i − 0.292690i −0.989234 0.146345i \(-0.953249\pi\)
0.989234 0.146345i \(-0.0467509\pi\)
\(608\) 1.69722i 0.0688315i
\(609\) −1.45837 −0.0590959
\(610\) 0 0
\(611\) −15.0000 −0.606835
\(612\) 4.81665i 0.194702i
\(613\) − 15.8806i − 0.641410i −0.947179 0.320705i \(-0.896080\pi\)
0.947179 0.320705i \(-0.103920\pi\)
\(614\) −22.0278 −0.888968
\(615\) 0 0
\(616\) −2.09167 −0.0842759
\(617\) − 3.39445i − 0.136655i −0.997663 0.0683277i \(-0.978234\pi\)
0.997663 0.0683277i \(-0.0217663\pi\)
\(618\) − 8.72498i − 0.350970i
\(619\) 11.4222 0.459097 0.229549 0.973297i \(-0.426275\pi\)
0.229549 + 0.973297i \(0.426275\pi\)
\(620\) 0 0
\(621\) 11.7250 0.470507
\(622\) 6.27502i 0.251605i
\(623\) 1.18335i 0.0474098i
\(624\) 38.0278 1.52233
\(625\) 0 0
\(626\) −0.238859 −0.00954672
\(627\) − 2.30278i − 0.0919640i
\(628\) − 5.81665i − 0.232110i
\(629\) −16.5416 −0.659558
\(630\) 0 0
\(631\) 6.93608 0.276121 0.138061 0.990424i \(-0.455913\pi\)
0.138061 + 0.990424i \(0.455913\pi\)
\(632\) 32.7250i 1.30173i
\(633\) − 58.1194i − 2.31004i
\(634\) −1.18335 −0.0469967
\(635\) 0 0
\(636\) 0.908327 0.0360175
\(637\) 32.5694i 1.29045i
\(638\) 1.18335i 0.0468491i
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) 27.7889 1.09760 0.548798 0.835955i \(-0.315086\pi\)
0.548798 + 0.835955i \(0.315086\pi\)
\(642\) − 9.00000i − 0.355202i
\(643\) − 22.0000i − 0.867595i −0.901010 0.433798i \(-0.857173\pi\)
0.901010 0.433798i \(-0.142827\pi\)
\(644\) −1.54163 −0.0607489
\(645\) 0 0
\(646\) 9.00000 0.354100
\(647\) 33.2389i 1.30675i 0.757033 + 0.653377i \(0.226648\pi\)
−0.757033 + 0.653377i \(0.773352\pi\)
\(648\) − 31.8167i − 1.24988i
\(649\) −14.2111 −0.557835
\(650\) 0 0
\(651\) 16.3944 0.642549
\(652\) 2.81665i 0.110309i
\(653\) 6.11943i 0.239472i 0.992806 + 0.119736i \(0.0382048\pi\)
−0.992806 + 0.119736i \(0.961795\pi\)
\(654\) 34.5416 1.35068
\(655\) 0 0
\(656\) 18.5139 0.722846
\(657\) 18.2111i 0.710483i
\(658\) 2.72498i 0.106231i
\(659\) 30.9083 1.20402 0.602009 0.798489i \(-0.294367\pi\)
0.602009 + 0.798489i \(0.294367\pi\)
\(660\) 0 0
\(661\) −8.81665 −0.342928 −0.171464 0.985190i \(-0.554850\pi\)
−0.171464 + 0.985190i \(0.554850\pi\)
\(662\) − 28.1472i − 1.09397i
\(663\) 79.5416i 3.08914i
\(664\) 10.5416 0.409095
\(665\) 0 0
\(666\) −7.18335 −0.278349
\(667\) 6.63331i 0.256843i
\(668\) − 4.06392i − 0.157238i
\(669\) 52.1194 2.01505
\(670\) 0 0
\(671\) 7.90833 0.305298
\(672\) 2.72498i 0.105118i
\(673\) 30.0278i 1.15748i 0.815510 + 0.578742i \(0.196456\pi\)
−0.815510 + 0.578742i \(0.803544\pi\)
\(674\) −40.1833 −1.54780
\(675\) 0 0
\(676\) −3.63331 −0.139743
\(677\) − 24.2389i − 0.931575i −0.884897 0.465788i \(-0.845771\pi\)
0.884897 0.465788i \(-0.154229\pi\)
\(678\) 32.4500i 1.24623i
\(679\) −10.6695 −0.409457
\(680\) 0 0
\(681\) 3.90833 0.149767
\(682\) − 13.3028i − 0.509390i
\(683\) 47.8444i 1.83072i 0.402642 + 0.915358i \(0.368092\pi\)
−0.402642 + 0.915358i \(0.631908\pi\)
\(684\) −0.697224 −0.0266590
\(685\) 0 0
\(686\) 12.2750 0.468662
\(687\) 43.1194i 1.64511i
\(688\) − 23.8167i − 0.908001i
\(689\) 6.51388 0.248159
\(690\) 0 0
\(691\) 27.5416 1.04773 0.523867 0.851800i \(-0.324489\pi\)
0.523867 + 0.851800i \(0.324489\pi\)
\(692\) − 1.45837i − 0.0554387i
\(693\) − 1.60555i − 0.0609898i
\(694\) −16.3028 −0.618845
\(695\) 0 0
\(696\) 6.27502 0.237854
\(697\) 38.7250i 1.46681i
\(698\) 6.75274i 0.255595i
\(699\) −36.6333 −1.38560
\(700\) 0 0
\(701\) −41.2111 −1.55652 −0.778261 0.627941i \(-0.783898\pi\)
−0.778261 + 0.627941i \(0.783898\pi\)
\(702\) − 10.4584i − 0.394726i
\(703\) − 2.39445i − 0.0903083i
\(704\) 8.81665 0.332290
\(705\) 0 0
\(706\) −24.2750 −0.913602
\(707\) 0.358288i 0.0134748i
\(708\) 9.90833i 0.372378i
\(709\) 31.6333 1.18801 0.594007 0.804460i \(-0.297545\pi\)
0.594007 + 0.804460i \(0.297545\pi\)
\(710\) 0 0
\(711\) −25.1194 −0.942052
\(712\) − 5.09167i − 0.190819i
\(713\) − 74.5694i − 2.79265i
\(714\) 14.4500 0.540776
\(715\) 0 0
\(716\) −3.78890 −0.141598
\(717\) − 48.6333i − 1.81624i
\(718\) − 1.02776i − 0.0383555i
\(719\) −7.18335 −0.267894 −0.133947 0.990989i \(-0.542765\pi\)
−0.133947 + 0.990989i \(0.542765\pi\)
\(720\) 0 0
\(721\) 2.02776 0.0755176
\(722\) − 23.4500i − 0.872717i
\(723\) 50.5139i 1.87863i
\(724\) −6.02776 −0.224020
\(725\) 0 0
\(726\) −3.00000 −0.111340
\(727\) 39.3305i 1.45869i 0.684147 + 0.729344i \(0.260175\pi\)
−0.684147 + 0.729344i \(0.739825\pi\)
\(728\) 10.4584i 0.387613i
\(729\) 13.3305 0.493723
\(730\) 0 0
\(731\) 49.8167 1.84254
\(732\) − 5.51388i − 0.203799i
\(733\) 19.6056i 0.724148i 0.932149 + 0.362074i \(0.117931\pi\)
−0.932149 + 0.362074i \(0.882069\pi\)
\(734\) −26.9638 −0.995253
\(735\) 0 0
\(736\) 12.3944 0.456865
\(737\) − 4.00000i − 0.147342i
\(738\) 16.8167i 0.619030i
\(739\) −35.1194 −1.29189 −0.645945 0.763384i \(-0.723536\pi\)
−0.645945 + 0.763384i \(0.723536\pi\)
\(740\) 0 0
\(741\) −11.5139 −0.422973
\(742\) − 1.18335i − 0.0434420i
\(743\) 40.6972i 1.49304i 0.665365 + 0.746518i \(0.268276\pi\)
−0.665365 + 0.746518i \(0.731724\pi\)
\(744\) −70.5416 −2.58618
\(745\) 0 0
\(746\) −35.7250 −1.30798
\(747\) 8.09167i 0.296059i
\(748\) 2.09167i 0.0764791i
\(749\) 2.09167 0.0764281
\(750\) 0 0
\(751\) −45.3305 −1.65413 −0.827067 0.562103i \(-0.809992\pi\)
−0.827067 + 0.562103i \(0.809992\pi\)
\(752\) − 9.90833i − 0.361320i
\(753\) 15.9083i 0.579732i
\(754\) 5.91673 0.215475
\(755\) 0 0
\(756\) 0.338936 0.0123270
\(757\) − 49.0555i − 1.78295i −0.453067 0.891476i \(-0.649670\pi\)
0.453067 0.891476i \(-0.350330\pi\)
\(758\) − 4.14719i − 0.150633i
\(759\) −16.8167 −0.610406
\(760\) 0 0
\(761\) −13.5778 −0.492195 −0.246097 0.969245i \(-0.579148\pi\)
−0.246097 + 0.969245i \(0.579148\pi\)
\(762\) 24.3583i 0.882408i
\(763\) 8.02776i 0.290624i
\(764\) 3.11943 0.112857
\(765\) 0 0
\(766\) −28.1833 −1.01831
\(767\) 71.0555i 2.56567i
\(768\) − 16.3305i − 0.589277i
\(769\) 5.18335 0.186916 0.0934581 0.995623i \(-0.470208\pi\)
0.0934581 + 0.995623i \(0.470208\pi\)
\(770\) 0 0
\(771\) −41.4500 −1.49278
\(772\) 4.00000i 0.143963i
\(773\) 3.11943i 0.112198i 0.998425 + 0.0560990i \(0.0178663\pi\)
−0.998425 + 0.0560990i \(0.982134\pi\)
\(774\) 21.6333 0.777593
\(775\) 0 0
\(776\) 45.9083 1.64801
\(777\) − 3.84441i − 0.137917i
\(778\) − 15.6333i − 0.560481i
\(779\) −5.60555 −0.200840
\(780\) 0 0
\(781\) 2.60555 0.0932340
\(782\) − 65.7250i − 2.35032i
\(783\) − 1.45837i − 0.0521177i
\(784\) −21.5139 −0.768353
\(785\) 0 0
\(786\) 29.7250 1.06025
\(787\) − 10.2111i − 0.363986i −0.983300 0.181993i \(-0.941745\pi\)
0.983300 0.181993i \(-0.0582549\pi\)
\(788\) − 4.02776i − 0.143483i
\(789\) 52.5416 1.87053
\(790\) 0 0
\(791\) −7.54163 −0.268150
\(792\) 6.90833i 0.245477i
\(793\) − 39.5416i − 1.40416i
\(794\) 28.2666 1.00314
\(795\) 0 0
\(796\) 1.96384 0.0696065
\(797\) − 3.51388i − 0.124468i −0.998062 0.0622340i \(-0.980178\pi\)
0.998062 0.0622340i \(-0.0198225\pi\)
\(798\) 2.09167i 0.0740444i
\(799\) 20.7250 0.733197
\(800\) 0 0
\(801\) 3.90833 0.138094
\(802\) − 16.6611i − 0.588323i
\(803\) 7.90833i 0.279079i
\(804\) −2.78890 −0.0983568
\(805\) 0 0
\(806\) −66.5139 −2.34285
\(807\) − 20.0917i − 0.707260i
\(808\) − 1.54163i − 0.0542345i
\(809\) −39.6333 −1.39343 −0.696716 0.717347i \(-0.745356\pi\)
−0.696716 + 0.717347i \(0.745356\pi\)
\(810\) 0 0
\(811\) 38.8722 1.36499 0.682493 0.730892i \(-0.260896\pi\)
0.682493 + 0.730892i \(0.260896\pi\)
\(812\) 0.191750i 0.00672911i
\(813\) − 0.486122i − 0.0170490i
\(814\) −3.11943 −0.109336
\(815\) 0 0
\(816\) −52.5416 −1.83933
\(817\) 7.21110i 0.252285i
\(818\) 8.09167i 0.282919i
\(819\) −8.02776 −0.280513
\(820\) 0 0
\(821\) 12.0000 0.418803 0.209401 0.977830i \(-0.432848\pi\)
0.209401 + 0.977830i \(0.432848\pi\)
\(822\) − 38.7250i − 1.35069i
\(823\) 18.4222i 0.642158i 0.947052 + 0.321079i \(0.104045\pi\)
−0.947052 + 0.321079i \(0.895955\pi\)
\(824\) −8.72498 −0.303949
\(825\) 0 0
\(826\) 12.9083 0.449138
\(827\) − 13.8167i − 0.480452i −0.970717 0.240226i \(-0.922778\pi\)
0.970717 0.240226i \(-0.0772216\pi\)
\(828\) 5.09167i 0.176948i
\(829\) −29.7527 −1.03336 −0.516678 0.856180i \(-0.672831\pi\)
−0.516678 + 0.856180i \(0.672831\pi\)
\(830\) 0 0
\(831\) 33.1472 1.14986
\(832\) − 44.0833i − 1.52831i
\(833\) − 45.0000i − 1.55916i
\(834\) −18.6333 −0.645219
\(835\) 0 0
\(836\) −0.302776 −0.0104717
\(837\) 16.3944i 0.566675i
\(838\) − 8.33053i − 0.287773i
\(839\) 9.11943 0.314838 0.157419 0.987532i \(-0.449683\pi\)
0.157419 + 0.987532i \(0.449683\pi\)
\(840\) 0 0
\(841\) −28.1749 −0.971550
\(842\) 0.908327i 0.0313030i
\(843\) − 2.72498i − 0.0938533i
\(844\) −7.64171 −0.263039
\(845\) 0 0
\(846\) 9.00000 0.309426
\(847\) − 0.697224i − 0.0239569i
\(848\) 4.30278i 0.147758i
\(849\) −14.5139 −0.498115
\(850\) 0 0
\(851\) −17.4861 −0.599417
\(852\) − 1.81665i − 0.0622375i
\(853\) − 12.7250i − 0.435695i −0.975983 0.217848i \(-0.930096\pi\)
0.975983 0.217848i \(-0.0699035\pi\)
\(854\) −7.18335 −0.245809
\(855\) 0 0
\(856\) −9.00000 −0.307614
\(857\) 3.00000i 0.102478i 0.998686 + 0.0512390i \(0.0163170\pi\)
−0.998686 + 0.0512390i \(0.983683\pi\)
\(858\) 15.0000i 0.512092i
\(859\) −41.3944 −1.41236 −0.706180 0.708032i \(-0.749583\pi\)
−0.706180 + 0.708032i \(0.749583\pi\)
\(860\) 0 0
\(861\) −9.00000 −0.306719
\(862\) 42.9916i 1.46430i
\(863\) − 12.3944i − 0.421912i −0.977496 0.210956i \(-0.932342\pi\)
0.977496 0.210956i \(-0.0676577\pi\)
\(864\) −2.72498 −0.0927057
\(865\) 0 0
\(866\) −6.51388 −0.221351
\(867\) − 70.7527i − 2.40289i
\(868\) − 2.15559i − 0.0731655i
\(869\) −10.9083 −0.370040
\(870\) 0 0
\(871\) −20.0000 −0.677674
\(872\) − 34.5416i − 1.16973i
\(873\) 35.2389i 1.19265i
\(874\) 9.51388 0.321812
\(875\) 0 0
\(876\) 5.51388 0.186297
\(877\) − 20.0000i − 0.675352i −0.941262 0.337676i \(-0.890359\pi\)
0.941262 0.337676i \(-0.109641\pi\)
\(878\) − 31.6611i − 1.06851i
\(879\) −1.81665 −0.0612742
\(880\) 0 0
\(881\) 19.5416 0.658374 0.329187 0.944265i \(-0.393225\pi\)
0.329187 + 0.944265i \(0.393225\pi\)
\(882\) − 19.5416i − 0.658001i
\(883\) 52.4500i 1.76508i 0.470236 + 0.882541i \(0.344169\pi\)
−0.470236 + 0.882541i \(0.655831\pi\)
\(884\) 10.4584 0.351753
\(885\) 0 0
\(886\) 11.2111 0.376644
\(887\) 3.23886i 0.108750i 0.998521 + 0.0543751i \(0.0173167\pi\)
−0.998521 + 0.0543751i \(0.982683\pi\)
\(888\) 16.5416i 0.555101i
\(889\) −5.66106 −0.189866
\(890\) 0 0
\(891\) 10.6056 0.355299
\(892\) − 6.85281i − 0.229449i
\(893\) 3.00000i 0.100391i
\(894\) 51.6333 1.72688
\(895\) 0 0
\(896\) −5.64171 −0.188476
\(897\) 84.0833i 2.80746i
\(898\) − 30.5971i − 1.02104i
\(899\) −9.27502 −0.309339
\(900\) 0 0
\(901\) −9.00000 −0.299833
\(902\) 7.30278i 0.243156i
\(903\) 11.5778i 0.385285i
\(904\) 32.4500 1.07927
\(905\) 0 0
\(906\) −2.44996 −0.0813945
\(907\) 4.00000i 0.132818i 0.997792 + 0.0664089i \(0.0211542\pi\)
−0.997792 + 0.0664089i \(0.978846\pi\)
\(908\) − 0.513878i − 0.0170536i
\(909\) 1.18335 0.0392491
\(910\) 0 0
\(911\) −24.7889 −0.821293 −0.410646 0.911795i \(-0.634697\pi\)
−0.410646 + 0.911795i \(0.634697\pi\)
\(912\) − 7.60555i − 0.251845i
\(913\) 3.51388i 0.116292i
\(914\) −26.9638 −0.891885
\(915\) 0 0
\(916\) 5.66947 0.187324
\(917\) 6.90833i 0.228133i
\(918\) 14.4500i 0.476920i
\(919\) −26.7889 −0.883684 −0.441842 0.897093i \(-0.645675\pi\)
−0.441842 + 0.897093i \(0.645675\pi\)
\(920\) 0 0
\(921\) −38.9361 −1.28299
\(922\) 41.9638i 1.38201i
\(923\) − 13.0278i − 0.428814i
\(924\) −0.486122 −0.0159922
\(925\) 0 0
\(926\) −15.3583 −0.504705
\(927\) − 6.69722i − 0.219966i
\(928\) − 1.54163i − 0.0506066i
\(929\) −53.6056 −1.75874 −0.879371 0.476138i \(-0.842036\pi\)
−0.879371 + 0.476138i \(0.842036\pi\)
\(930\) 0 0
\(931\) 6.51388 0.213484
\(932\) 4.81665i 0.157775i
\(933\) 11.0917i 0.363125i
\(934\) −24.2750 −0.794303
\(935\) 0 0
\(936\) 34.5416 1.12903
\(937\) 9.21110i 0.300914i 0.988617 + 0.150457i \(0.0480744\pi\)
−0.988617 + 0.150457i \(0.951926\pi\)
\(938\) 3.63331i 0.118632i
\(939\) −0.422205 −0.0137781
\(940\) 0 0
\(941\) 59.6056 1.94309 0.971543 0.236864i \(-0.0761197\pi\)
0.971543 + 0.236864i \(0.0761197\pi\)
\(942\) 57.6333i 1.87779i
\(943\) 40.9361i 1.33306i
\(944\) −46.9361 −1.52764
\(945\) 0 0
\(946\) 9.39445 0.305440
\(947\) − 6.63331i − 0.215554i −0.994175 0.107777i \(-0.965627\pi\)
0.994175 0.107777i \(-0.0343732\pi\)
\(948\) 7.60555i 0.247017i
\(949\) 39.5416 1.28358
\(950\) 0 0
\(951\) −2.09167 −0.0678271
\(952\) − 14.4500i − 0.468326i
\(953\) 37.2666i 1.20718i 0.797293 + 0.603592i \(0.206264\pi\)
−0.797293 + 0.603592i \(0.793736\pi\)
\(954\) −3.90833 −0.126537
\(955\) 0 0
\(956\) −6.39445 −0.206811
\(957\) 2.09167i 0.0676142i
\(958\) 45.3583i 1.46546i
\(959\) 9.00000 0.290625
\(960\) 0 0
\(961\) 73.2666 2.36344
\(962\) 15.5971i 0.502872i
\(963\) − 6.90833i − 0.222618i
\(964\) 6.64171 0.213915
\(965\) 0 0
\(966\) 15.2750 0.491466
\(967\) − 14.9083i − 0.479419i −0.970845 0.239710i \(-0.922948\pi\)
0.970845 0.239710i \(-0.0770523\pi\)
\(968\) 3.00000i 0.0964237i
\(969\) 15.9083 0.511049
\(970\) 0 0
\(971\) 45.3583 1.45562 0.727808 0.685781i \(-0.240539\pi\)
0.727808 + 0.685781i \(0.240539\pi\)
\(972\) − 5.93608i − 0.190400i
\(973\) − 4.33053i − 0.138830i
\(974\) 5.48612 0.175787
\(975\) 0 0
\(976\) 26.1194 0.836063
\(977\) 52.0278i 1.66452i 0.554389 + 0.832258i \(0.312952\pi\)
−0.554389 + 0.832258i \(0.687048\pi\)
\(978\) − 27.9083i − 0.892410i
\(979\) 1.69722 0.0542435
\(980\) 0 0
\(981\) 26.5139 0.846523
\(982\) − 12.7527i − 0.406956i
\(983\) 8.84441i 0.282093i 0.990003 + 0.141046i \(0.0450467\pi\)
−0.990003 + 0.141046i \(0.954953\pi\)
\(984\) 38.7250 1.23451
\(985\) 0 0
\(986\) −8.17494 −0.260343
\(987\) 4.81665i 0.153316i
\(988\) 1.51388i 0.0481629i
\(989\) 52.6611 1.67452
\(990\) 0 0
\(991\) −16.9083 −0.537111 −0.268555 0.963264i \(-0.586546\pi\)
−0.268555 + 0.963264i \(0.586546\pi\)
\(992\) 17.3305i 0.550245i
\(993\) − 49.7527i − 1.57886i
\(994\) −2.36669 −0.0750669
\(995\) 0 0
\(996\) 2.44996 0.0776300
\(997\) − 46.7250i − 1.47979i −0.672720 0.739897i \(-0.734874\pi\)
0.672720 0.739897i \(-0.265126\pi\)
\(998\) 4.54163i 0.143763i
\(999\) 3.84441 0.121632
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 275.2.b.c.199.3 4
3.2 odd 2 2475.2.c.k.199.2 4
4.3 odd 2 4400.2.b.y.4049.1 4
5.2 odd 4 275.2.a.f.1.1 yes 2
5.3 odd 4 275.2.a.e.1.2 2
5.4 even 2 inner 275.2.b.c.199.2 4
15.2 even 4 2475.2.a.o.1.2 2
15.8 even 4 2475.2.a.t.1.1 2
15.14 odd 2 2475.2.c.k.199.3 4
20.3 even 4 4400.2.a.bs.1.2 2
20.7 even 4 4400.2.a.bh.1.1 2
20.19 odd 2 4400.2.b.y.4049.4 4
55.32 even 4 3025.2.a.h.1.2 2
55.43 even 4 3025.2.a.n.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.a.e.1.2 2 5.3 odd 4
275.2.a.f.1.1 yes 2 5.2 odd 4
275.2.b.c.199.2 4 5.4 even 2 inner
275.2.b.c.199.3 4 1.1 even 1 trivial
2475.2.a.o.1.2 2 15.2 even 4
2475.2.a.t.1.1 2 15.8 even 4
2475.2.c.k.199.2 4 3.2 odd 2
2475.2.c.k.199.3 4 15.14 odd 2
3025.2.a.h.1.2 2 55.32 even 4
3025.2.a.n.1.1 2 55.43 even 4
4400.2.a.bh.1.1 2 20.7 even 4
4400.2.a.bs.1.2 2 20.3 even 4
4400.2.b.y.4049.1 4 4.3 odd 2
4400.2.b.y.4049.4 4 20.19 odd 2