Properties

Label 825.2.c.e.199.2
Level $825$
Weight $2$
Character 825.199
Analytic conductor $6.588$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.2
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.2.c.e.199.3

$q$-expansion

\(f(q)\) \(=\) \(q-0.414214i q^{2} -1.00000i q^{3} +1.82843 q^{4} -0.414214 q^{6} +4.82843i q^{7} -1.58579i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-0.414214i q^{2} -1.00000i q^{3} +1.82843 q^{4} -0.414214 q^{6} +4.82843i q^{7} -1.58579i q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.82843i q^{12} +5.65685i q^{13} +2.00000 q^{14} +3.00000 q^{16} +6.82843i q^{17} +0.414214i q^{18} +1.17157 q^{19} +4.82843 q^{21} +0.414214i q^{22} -4.00000i q^{23} -1.58579 q^{24} +2.34315 q^{26} +1.00000i q^{27} +8.82843i q^{28} -0.828427 q^{29} -4.41421i q^{32} +1.00000i q^{33} +2.82843 q^{34} -1.82843 q^{36} -0.343146i q^{37} -0.485281i q^{38} +5.65685 q^{39} -0.828427 q^{41} -2.00000i q^{42} -3.17157i q^{43} -1.82843 q^{44} -1.65685 q^{46} +4.00000i q^{47} -3.00000i q^{48} -16.3137 q^{49} +6.82843 q^{51} +10.3431i q^{52} -13.3137i q^{53} +0.414214 q^{54} +7.65685 q^{56} -1.17157i q^{57} +0.343146i q^{58} +4.00000 q^{59} -0.343146 q^{61} -4.82843i q^{63} +4.17157 q^{64} +0.414214 q^{66} -5.65685i q^{67} +12.4853i q^{68} -4.00000 q^{69} +13.6569 q^{71} +1.58579i q^{72} -11.3137i q^{73} -0.142136 q^{74} +2.14214 q^{76} -4.82843i q^{77} -2.34315i q^{78} +8.48528 q^{79} +1.00000 q^{81} +0.343146i q^{82} -10.0000i q^{83} +8.82843 q^{84} -1.31371 q^{86} +0.828427i q^{87} +1.58579i q^{88} +7.65685 q^{89} -27.3137 q^{91} -7.31371i q^{92} +1.65685 q^{94} -4.41421 q^{96} -0.343146i q^{97} +6.75736i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 4q^{6} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{4} + 4q^{6} - 4q^{9} - 4q^{11} + 8q^{14} + 12q^{16} + 16q^{19} + 8q^{21} - 12q^{24} + 32q^{26} + 8q^{29} + 4q^{36} + 8q^{41} + 4q^{44} + 16q^{46} - 20q^{49} + 16q^{51} - 4q^{54} + 8q^{56} + 16q^{59} - 24q^{61} + 28q^{64} - 4q^{66} - 16q^{69} + 32q^{71} + 56q^{74} - 48q^{76} + 4q^{81} + 24q^{84} + 40q^{86} + 8q^{89} - 64q^{91} - 16q^{94} - 12q^{96} + 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.414214i − 0.292893i −0.989219 0.146447i \(-0.953216\pi\)
0.989219 0.146447i \(-0.0467837\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) 1.82843 0.914214
\(5\) 0 0
\(6\) −0.414214 −0.169102
\(7\) 4.82843i 1.82497i 0.409106 + 0.912487i \(0.365841\pi\)
−0.409106 + 0.912487i \(0.634159\pi\)
\(8\) − 1.58579i − 0.560660i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) − 1.82843i − 0.527821i
\(13\) 5.65685i 1.56893i 0.620174 + 0.784465i \(0.287062\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 6.82843i 1.65614i 0.560627 + 0.828068i \(0.310560\pi\)
−0.560627 + 0.828068i \(0.689440\pi\)
\(18\) 0.414214i 0.0976311i
\(19\) 1.17157 0.268777 0.134389 0.990929i \(-0.457093\pi\)
0.134389 + 0.990929i \(0.457093\pi\)
\(20\) 0 0
\(21\) 4.82843 1.05365
\(22\) 0.414214i 0.0883106i
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) −1.58579 −0.323697
\(25\) 0 0
\(26\) 2.34315 0.459529
\(27\) 1.00000i 0.192450i
\(28\) 8.82843i 1.66842i
\(29\) −0.828427 −0.153835 −0.0769175 0.997037i \(-0.524508\pi\)
−0.0769175 + 0.997037i \(0.524508\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) − 4.41421i − 0.780330i
\(33\) 1.00000i 0.174078i
\(34\) 2.82843 0.485071
\(35\) 0 0
\(36\) −1.82843 −0.304738
\(37\) − 0.343146i − 0.0564128i −0.999602 0.0282064i \(-0.991020\pi\)
0.999602 0.0282064i \(-0.00897957\pi\)
\(38\) − 0.485281i − 0.0787230i
\(39\) 5.65685 0.905822
\(40\) 0 0
\(41\) −0.828427 −0.129379 −0.0646893 0.997905i \(-0.520606\pi\)
−0.0646893 + 0.997905i \(0.520606\pi\)
\(42\) − 2.00000i − 0.308607i
\(43\) − 3.17157i − 0.483660i −0.970319 0.241830i \(-0.922252\pi\)
0.970319 0.241830i \(-0.0777477\pi\)
\(44\) −1.82843 −0.275646
\(45\) 0 0
\(46\) −1.65685 −0.244290
\(47\) 4.00000i 0.583460i 0.956501 + 0.291730i \(0.0942309\pi\)
−0.956501 + 0.291730i \(0.905769\pi\)
\(48\) − 3.00000i − 0.433013i
\(49\) −16.3137 −2.33053
\(50\) 0 0
\(51\) 6.82843 0.956171
\(52\) 10.3431i 1.43434i
\(53\) − 13.3137i − 1.82878i −0.404836 0.914389i \(-0.632671\pi\)
0.404836 0.914389i \(-0.367329\pi\)
\(54\) 0.414214 0.0563673
\(55\) 0 0
\(56\) 7.65685 1.02319
\(57\) − 1.17157i − 0.155179i
\(58\) 0.343146i 0.0450572i
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −0.343146 −0.0439353 −0.0219677 0.999759i \(-0.506993\pi\)
−0.0219677 + 0.999759i \(0.506993\pi\)
\(62\) 0 0
\(63\) − 4.82843i − 0.608325i
\(64\) 4.17157 0.521447
\(65\) 0 0
\(66\) 0.414214 0.0509862
\(67\) − 5.65685i − 0.691095i −0.938401 0.345547i \(-0.887693\pi\)
0.938401 0.345547i \(-0.112307\pi\)
\(68\) 12.4853i 1.51406i
\(69\) −4.00000 −0.481543
\(70\) 0 0
\(71\) 13.6569 1.62077 0.810385 0.585897i \(-0.199258\pi\)
0.810385 + 0.585897i \(0.199258\pi\)
\(72\) 1.58579i 0.186887i
\(73\) − 11.3137i − 1.32417i −0.749429 0.662085i \(-0.769672\pi\)
0.749429 0.662085i \(-0.230328\pi\)
\(74\) −0.142136 −0.0165229
\(75\) 0 0
\(76\) 2.14214 0.245720
\(77\) − 4.82843i − 0.550250i
\(78\) − 2.34315i − 0.265309i
\(79\) 8.48528 0.954669 0.477334 0.878722i \(-0.341603\pi\)
0.477334 + 0.878722i \(0.341603\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0.343146i 0.0378941i
\(83\) − 10.0000i − 1.09764i −0.835940 0.548821i \(-0.815077\pi\)
0.835940 0.548821i \(-0.184923\pi\)
\(84\) 8.82843 0.963260
\(85\) 0 0
\(86\) −1.31371 −0.141661
\(87\) 0.828427i 0.0888167i
\(88\) 1.58579i 0.169045i
\(89\) 7.65685 0.811625 0.405812 0.913956i \(-0.366989\pi\)
0.405812 + 0.913956i \(0.366989\pi\)
\(90\) 0 0
\(91\) −27.3137 −2.86325
\(92\) − 7.31371i − 0.762507i
\(93\) 0 0
\(94\) 1.65685 0.170891
\(95\) 0 0
\(96\) −4.41421 −0.450524
\(97\) − 0.343146i − 0.0348412i −0.999848 0.0174206i \(-0.994455\pi\)
0.999848 0.0174206i \(-0.00554543\pi\)
\(98\) 6.75736i 0.682596i
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 4.82843 0.480446 0.240223 0.970718i \(-0.422779\pi\)
0.240223 + 0.970718i \(0.422779\pi\)
\(102\) − 2.82843i − 0.280056i
\(103\) 19.3137i 1.90304i 0.307593 + 0.951518i \(0.400477\pi\)
−0.307593 + 0.951518i \(0.599523\pi\)
\(104\) 8.97056 0.879636
\(105\) 0 0
\(106\) −5.51472 −0.535637
\(107\) 5.31371i 0.513696i 0.966452 + 0.256848i \(0.0826839\pi\)
−0.966452 + 0.256848i \(0.917316\pi\)
\(108\) 1.82843i 0.175940i
\(109\) 5.31371 0.508961 0.254480 0.967078i \(-0.418096\pi\)
0.254480 + 0.967078i \(0.418096\pi\)
\(110\) 0 0
\(111\) −0.343146 −0.0325700
\(112\) 14.4853i 1.36873i
\(113\) 14.9706i 1.40831i 0.710045 + 0.704156i \(0.248674\pi\)
−0.710045 + 0.704156i \(0.751326\pi\)
\(114\) −0.485281 −0.0454508
\(115\) 0 0
\(116\) −1.51472 −0.140638
\(117\) − 5.65685i − 0.522976i
\(118\) − 1.65685i − 0.152526i
\(119\) −32.9706 −3.02241
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0.142136i 0.0128684i
\(123\) 0.828427i 0.0746968i
\(124\) 0 0
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) − 2.48528i − 0.220533i −0.993902 0.110267i \(-0.964830\pi\)
0.993902 0.110267i \(-0.0351704\pi\)
\(128\) − 10.5563i − 0.933058i
\(129\) −3.17157 −0.279241
\(130\) 0 0
\(131\) −19.3137 −1.68745 −0.843723 0.536778i \(-0.819641\pi\)
−0.843723 + 0.536778i \(0.819641\pi\)
\(132\) 1.82843i 0.159144i
\(133\) 5.65685i 0.490511i
\(134\) −2.34315 −0.202417
\(135\) 0 0
\(136\) 10.8284 0.928530
\(137\) − 9.31371i − 0.795724i −0.917445 0.397862i \(-0.869752\pi\)
0.917445 0.397862i \(-0.130248\pi\)
\(138\) 1.65685i 0.141041i
\(139\) 16.4853 1.39826 0.699132 0.714993i \(-0.253570\pi\)
0.699132 + 0.714993i \(0.253570\pi\)
\(140\) 0 0
\(141\) 4.00000 0.336861
\(142\) − 5.65685i − 0.474713i
\(143\) − 5.65685i − 0.473050i
\(144\) −3.00000 −0.250000
\(145\) 0 0
\(146\) −4.68629 −0.387840
\(147\) 16.3137i 1.34553i
\(148\) − 0.627417i − 0.0515734i
\(149\) −18.4853 −1.51437 −0.757187 0.653199i \(-0.773427\pi\)
−0.757187 + 0.653199i \(0.773427\pi\)
\(150\) 0 0
\(151\) −0.485281 −0.0394916 −0.0197458 0.999805i \(-0.506286\pi\)
−0.0197458 + 0.999805i \(0.506286\pi\)
\(152\) − 1.85786i − 0.150693i
\(153\) − 6.82843i − 0.552046i
\(154\) −2.00000 −0.161165
\(155\) 0 0
\(156\) 10.3431 0.828114
\(157\) − 18.0000i − 1.43656i −0.695756 0.718278i \(-0.744931\pi\)
0.695756 0.718278i \(-0.255069\pi\)
\(158\) − 3.51472i − 0.279616i
\(159\) −13.3137 −1.05585
\(160\) 0 0
\(161\) 19.3137 1.52213
\(162\) − 0.414214i − 0.0325437i
\(163\) 15.3137i 1.19946i 0.800202 + 0.599731i \(0.204726\pi\)
−0.800202 + 0.599731i \(0.795274\pi\)
\(164\) −1.51472 −0.118280
\(165\) 0 0
\(166\) −4.14214 −0.321492
\(167\) − 9.31371i − 0.720716i −0.932814 0.360358i \(-0.882654\pi\)
0.932814 0.360358i \(-0.117346\pi\)
\(168\) − 7.65685i − 0.590739i
\(169\) −19.0000 −1.46154
\(170\) 0 0
\(171\) −1.17157 −0.0895924
\(172\) − 5.79899i − 0.442169i
\(173\) − 2.82843i − 0.215041i −0.994203 0.107521i \(-0.965709\pi\)
0.994203 0.107521i \(-0.0342912\pi\)
\(174\) 0.343146 0.0260138
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) − 4.00000i − 0.300658i
\(178\) − 3.17157i − 0.237719i
\(179\) 6.34315 0.474109 0.237054 0.971496i \(-0.423818\pi\)
0.237054 + 0.971496i \(0.423818\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) 11.3137i 0.838628i
\(183\) 0.343146i 0.0253661i
\(184\) −6.34315 −0.467623
\(185\) 0 0
\(186\) 0 0
\(187\) − 6.82843i − 0.499344i
\(188\) 7.31371i 0.533407i
\(189\) −4.82843 −0.351216
\(190\) 0 0
\(191\) −5.65685 −0.409316 −0.204658 0.978834i \(-0.565608\pi\)
−0.204658 + 0.978834i \(0.565608\pi\)
\(192\) − 4.17157i − 0.301057i
\(193\) 2.34315i 0.168663i 0.996438 + 0.0843317i \(0.0268755\pi\)
−0.996438 + 0.0843317i \(0.973124\pi\)
\(194\) −0.142136 −0.0102047
\(195\) 0 0
\(196\) −29.8284 −2.13060
\(197\) − 8.48528i − 0.604551i −0.953221 0.302276i \(-0.902254\pi\)
0.953221 0.302276i \(-0.0977463\pi\)
\(198\) − 0.414214i − 0.0294369i
\(199\) 10.3431 0.733206 0.366603 0.930377i \(-0.380521\pi\)
0.366603 + 0.930377i \(0.380521\pi\)
\(200\) 0 0
\(201\) −5.65685 −0.399004
\(202\) − 2.00000i − 0.140720i
\(203\) − 4.00000i − 0.280745i
\(204\) 12.4853 0.874145
\(205\) 0 0
\(206\) 8.00000 0.557386
\(207\) 4.00000i 0.278019i
\(208\) 16.9706i 1.17670i
\(209\) −1.17157 −0.0810394
\(210\) 0 0
\(211\) 6.82843 0.470088 0.235044 0.971985i \(-0.424477\pi\)
0.235044 + 0.971985i \(0.424477\pi\)
\(212\) − 24.3431i − 1.67189i
\(213\) − 13.6569i − 0.935752i
\(214\) 2.20101 0.150458
\(215\) 0 0
\(216\) 1.58579 0.107899
\(217\) 0 0
\(218\) − 2.20101i − 0.149071i
\(219\) −11.3137 −0.764510
\(220\) 0 0
\(221\) −38.6274 −2.59836
\(222\) 0.142136i 0.00953952i
\(223\) − 17.6569i − 1.18239i −0.806529 0.591195i \(-0.798656\pi\)
0.806529 0.591195i \(-0.201344\pi\)
\(224\) 21.3137 1.42408
\(225\) 0 0
\(226\) 6.20101 0.412485
\(227\) 14.0000i 0.929213i 0.885517 + 0.464606i \(0.153804\pi\)
−0.885517 + 0.464606i \(0.846196\pi\)
\(228\) − 2.14214i − 0.141866i
\(229\) 2.00000 0.132164 0.0660819 0.997814i \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 0 0
\(231\) −4.82843 −0.317687
\(232\) 1.31371i 0.0862492i
\(233\) − 13.1716i − 0.862898i −0.902137 0.431449i \(-0.858002\pi\)
0.902137 0.431449i \(-0.141998\pi\)
\(234\) −2.34315 −0.153176
\(235\) 0 0
\(236\) 7.31371 0.476082
\(237\) − 8.48528i − 0.551178i
\(238\) 13.6569i 0.885242i
\(239\) −6.34315 −0.410304 −0.205152 0.978730i \(-0.565769\pi\)
−0.205152 + 0.978730i \(0.565769\pi\)
\(240\) 0 0
\(241\) −23.6569 −1.52387 −0.761936 0.647652i \(-0.775751\pi\)
−0.761936 + 0.647652i \(0.775751\pi\)
\(242\) − 0.414214i − 0.0266267i
\(243\) − 1.00000i − 0.0641500i
\(244\) −0.627417 −0.0401663
\(245\) 0 0
\(246\) 0.343146 0.0218782
\(247\) 6.62742i 0.421692i
\(248\) 0 0
\(249\) −10.0000 −0.633724
\(250\) 0 0
\(251\) −12.9706 −0.818695 −0.409347 0.912379i \(-0.634244\pi\)
−0.409347 + 0.912379i \(0.634244\pi\)
\(252\) − 8.82843i − 0.556139i
\(253\) 4.00000i 0.251478i
\(254\) −1.02944 −0.0645926
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) 27.6569i 1.72519i 0.505898 + 0.862594i \(0.331161\pi\)
−0.505898 + 0.862594i \(0.668839\pi\)
\(258\) 1.31371i 0.0817879i
\(259\) 1.65685 0.102952
\(260\) 0 0
\(261\) 0.828427 0.0512784
\(262\) 8.00000i 0.494242i
\(263\) 18.0000i 1.10993i 0.831875 + 0.554964i \(0.187268\pi\)
−0.831875 + 0.554964i \(0.812732\pi\)
\(264\) 1.58579 0.0975984
\(265\) 0 0
\(266\) 2.34315 0.143667
\(267\) − 7.65685i − 0.468592i
\(268\) − 10.3431i − 0.631808i
\(269\) 24.6274 1.50156 0.750780 0.660552i \(-0.229678\pi\)
0.750780 + 0.660552i \(0.229678\pi\)
\(270\) 0 0
\(271\) 27.7990 1.68867 0.844334 0.535817i \(-0.179996\pi\)
0.844334 + 0.535817i \(0.179996\pi\)
\(272\) 20.4853i 1.24210i
\(273\) 27.3137i 1.65310i
\(274\) −3.85786 −0.233062
\(275\) 0 0
\(276\) −7.31371 −0.440234
\(277\) 13.6569i 0.820561i 0.911959 + 0.410280i \(0.134569\pi\)
−0.911959 + 0.410280i \(0.865431\pi\)
\(278\) − 6.82843i − 0.409542i
\(279\) 0 0
\(280\) 0 0
\(281\) −16.8284 −1.00390 −0.501950 0.864897i \(-0.667384\pi\)
−0.501950 + 0.864897i \(0.667384\pi\)
\(282\) − 1.65685i − 0.0986642i
\(283\) − 3.17157i − 0.188530i −0.995547 0.0942652i \(-0.969950\pi\)
0.995547 0.0942652i \(-0.0300502\pi\)
\(284\) 24.9706 1.48173
\(285\) 0 0
\(286\) −2.34315 −0.138553
\(287\) − 4.00000i − 0.236113i
\(288\) 4.41421i 0.260110i
\(289\) −29.6274 −1.74279
\(290\) 0 0
\(291\) −0.343146 −0.0201156
\(292\) − 20.6863i − 1.21057i
\(293\) − 1.17157i − 0.0684440i −0.999414 0.0342220i \(-0.989105\pi\)
0.999414 0.0342220i \(-0.0108953\pi\)
\(294\) 6.75736 0.394097
\(295\) 0 0
\(296\) −0.544156 −0.0316284
\(297\) − 1.00000i − 0.0580259i
\(298\) 7.65685i 0.443550i
\(299\) 22.6274 1.30858
\(300\) 0 0
\(301\) 15.3137 0.882667
\(302\) 0.201010i 0.0115668i
\(303\) − 4.82843i − 0.277386i
\(304\) 3.51472 0.201583
\(305\) 0 0
\(306\) −2.82843 −0.161690
\(307\) 8.82843i 0.503865i 0.967745 + 0.251932i \(0.0810661\pi\)
−0.967745 + 0.251932i \(0.918934\pi\)
\(308\) − 8.82843i − 0.503046i
\(309\) 19.3137 1.09872
\(310\) 0 0
\(311\) 19.3137 1.09518 0.547590 0.836747i \(-0.315545\pi\)
0.547590 + 0.836747i \(0.315545\pi\)
\(312\) − 8.97056i − 0.507858i
\(313\) 4.34315i 0.245489i 0.992438 + 0.122745i \(0.0391696\pi\)
−0.992438 + 0.122745i \(0.960830\pi\)
\(314\) −7.45584 −0.420758
\(315\) 0 0
\(316\) 15.5147 0.872771
\(317\) − 30.2843i − 1.70093i −0.526028 0.850467i \(-0.676319\pi\)
0.526028 0.850467i \(-0.323681\pi\)
\(318\) 5.51472i 0.309250i
\(319\) 0.828427 0.0463830
\(320\) 0 0
\(321\) 5.31371 0.296582
\(322\) − 8.00000i − 0.445823i
\(323\) 8.00000i 0.445132i
\(324\) 1.82843 0.101579
\(325\) 0 0
\(326\) 6.34315 0.351314
\(327\) − 5.31371i − 0.293849i
\(328\) 1.31371i 0.0725374i
\(329\) −19.3137 −1.06480
\(330\) 0 0
\(331\) 17.6569 0.970508 0.485254 0.874373i \(-0.338727\pi\)
0.485254 + 0.874373i \(0.338727\pi\)
\(332\) − 18.2843i − 1.00348i
\(333\) 0.343146i 0.0188043i
\(334\) −3.85786 −0.211093
\(335\) 0 0
\(336\) 14.4853 0.790237
\(337\) 19.3137i 1.05208i 0.850458 + 0.526042i \(0.176325\pi\)
−0.850458 + 0.526042i \(0.823675\pi\)
\(338\) 7.87006i 0.428075i
\(339\) 14.9706 0.813089
\(340\) 0 0
\(341\) 0 0
\(342\) 0.485281i 0.0262410i
\(343\) − 44.9706i − 2.42818i
\(344\) −5.02944 −0.271169
\(345\) 0 0
\(346\) −1.17157 −0.0629841
\(347\) 6.68629i 0.358939i 0.983764 + 0.179469i \(0.0574381\pi\)
−0.983764 + 0.179469i \(0.942562\pi\)
\(348\) 1.51472i 0.0811974i
\(349\) 22.9706 1.22959 0.614793 0.788688i \(-0.289240\pi\)
0.614793 + 0.788688i \(0.289240\pi\)
\(350\) 0 0
\(351\) −5.65685 −0.301941
\(352\) 4.41421i 0.235278i
\(353\) 26.0000i 1.38384i 0.721974 + 0.691920i \(0.243235\pi\)
−0.721974 + 0.691920i \(0.756765\pi\)
\(354\) −1.65685 −0.0880608
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) 32.9706i 1.74499i
\(358\) − 2.62742i − 0.138863i
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) −17.6274 −0.927759
\(362\) 5.79899i 0.304788i
\(363\) − 1.00000i − 0.0524864i
\(364\) −49.9411 −2.61763
\(365\) 0 0
\(366\) 0.142136 0.00742955
\(367\) 1.65685i 0.0864871i 0.999065 + 0.0432435i \(0.0137691\pi\)
−0.999065 + 0.0432435i \(0.986231\pi\)
\(368\) − 12.0000i − 0.625543i
\(369\) 0.828427 0.0431262
\(370\) 0 0
\(371\) 64.2843 3.33747
\(372\) 0 0
\(373\) − 34.6274i − 1.79294i −0.443105 0.896470i \(-0.646123\pi\)
0.443105 0.896470i \(-0.353877\pi\)
\(374\) −2.82843 −0.146254
\(375\) 0 0
\(376\) 6.34315 0.327123
\(377\) − 4.68629i − 0.241356i
\(378\) 2.00000i 0.102869i
\(379\) 0.686292 0.0352524 0.0176262 0.999845i \(-0.494389\pi\)
0.0176262 + 0.999845i \(0.494389\pi\)
\(380\) 0 0
\(381\) −2.48528 −0.127325
\(382\) 2.34315i 0.119886i
\(383\) − 8.00000i − 0.408781i −0.978889 0.204390i \(-0.934479\pi\)
0.978889 0.204390i \(-0.0655212\pi\)
\(384\) −10.5563 −0.538701
\(385\) 0 0
\(386\) 0.970563 0.0494003
\(387\) 3.17157i 0.161220i
\(388\) − 0.627417i − 0.0318523i
\(389\) 12.3431 0.625822 0.312911 0.949782i \(-0.398696\pi\)
0.312911 + 0.949782i \(0.398696\pi\)
\(390\) 0 0
\(391\) 27.3137 1.38131
\(392\) 25.8701i 1.30664i
\(393\) 19.3137i 0.974248i
\(394\) −3.51472 −0.177069
\(395\) 0 0
\(396\) 1.82843 0.0918819
\(397\) − 18.9706i − 0.952105i −0.879417 0.476053i \(-0.842067\pi\)
0.879417 0.476053i \(-0.157933\pi\)
\(398\) − 4.28427i − 0.214751i
\(399\) 5.65685 0.283197
\(400\) 0 0
\(401\) −29.3137 −1.46386 −0.731928 0.681382i \(-0.761379\pi\)
−0.731928 + 0.681382i \(0.761379\pi\)
\(402\) 2.34315i 0.116865i
\(403\) 0 0
\(404\) 8.82843 0.439231
\(405\) 0 0
\(406\) −1.65685 −0.0822283
\(407\) 0.343146i 0.0170091i
\(408\) − 10.8284i − 0.536087i
\(409\) 8.34315 0.412542 0.206271 0.978495i \(-0.433867\pi\)
0.206271 + 0.978495i \(0.433867\pi\)
\(410\) 0 0
\(411\) −9.31371 −0.459411
\(412\) 35.3137i 1.73978i
\(413\) 19.3137i 0.950365i
\(414\) 1.65685 0.0814299
\(415\) 0 0
\(416\) 24.9706 1.22428
\(417\) − 16.4853i − 0.807288i
\(418\) 0.485281i 0.0237359i
\(419\) 3.02944 0.147998 0.0739988 0.997258i \(-0.476424\pi\)
0.0739988 + 0.997258i \(0.476424\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) − 2.82843i − 0.137686i
\(423\) − 4.00000i − 0.194487i
\(424\) −21.1127 −1.02532
\(425\) 0 0
\(426\) −5.65685 −0.274075
\(427\) − 1.65685i − 0.0801808i
\(428\) 9.71573i 0.469627i
\(429\) −5.65685 −0.273115
\(430\) 0 0
\(431\) 10.3431 0.498212 0.249106 0.968476i \(-0.419863\pi\)
0.249106 + 0.968476i \(0.419863\pi\)
\(432\) 3.00000i 0.144338i
\(433\) − 4.34315i − 0.208718i −0.994540 0.104359i \(-0.966721\pi\)
0.994540 0.104359i \(-0.0332791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 9.71573 0.465299
\(437\) − 4.68629i − 0.224176i
\(438\) 4.68629i 0.223920i
\(439\) 3.51472 0.167748 0.0838742 0.996476i \(-0.473271\pi\)
0.0838742 + 0.996476i \(0.473271\pi\)
\(440\) 0 0
\(441\) 16.3137 0.776843
\(442\) 16.0000i 0.761042i
\(443\) 12.0000i 0.570137i 0.958507 + 0.285069i \(0.0920164\pi\)
−0.958507 + 0.285069i \(0.907984\pi\)
\(444\) −0.627417 −0.0297759
\(445\) 0 0
\(446\) −7.31371 −0.346314
\(447\) 18.4853i 0.874324i
\(448\) 20.1421i 0.951626i
\(449\) 2.97056 0.140190 0.0700948 0.997540i \(-0.477670\pi\)
0.0700948 + 0.997540i \(0.477670\pi\)
\(450\) 0 0
\(451\) 0.828427 0.0390091
\(452\) 27.3726i 1.28750i
\(453\) 0.485281i 0.0228005i
\(454\) 5.79899 0.272160
\(455\) 0 0
\(456\) −1.85786 −0.0870025
\(457\) 0.686292i 0.0321034i 0.999871 + 0.0160517i \(0.00510963\pi\)
−0.999871 + 0.0160517i \(0.994890\pi\)
\(458\) − 0.828427i − 0.0387099i
\(459\) −6.82843 −0.318724
\(460\) 0 0
\(461\) −28.1421 −1.31071 −0.655355 0.755321i \(-0.727481\pi\)
−0.655355 + 0.755321i \(0.727481\pi\)
\(462\) 2.00000i 0.0930484i
\(463\) − 28.9706i − 1.34638i −0.739471 0.673188i \(-0.764924\pi\)
0.739471 0.673188i \(-0.235076\pi\)
\(464\) −2.48528 −0.115376
\(465\) 0 0
\(466\) −5.45584 −0.252737
\(467\) − 22.6274i − 1.04707i −0.852004 0.523536i \(-0.824613\pi\)
0.852004 0.523536i \(-0.175387\pi\)
\(468\) − 10.3431i − 0.478112i
\(469\) 27.3137 1.26123
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) − 6.34315i − 0.291967i
\(473\) 3.17157i 0.145829i
\(474\) −3.51472 −0.161436
\(475\) 0 0
\(476\) −60.2843 −2.76313
\(477\) 13.3137i 0.609593i
\(478\) 2.62742i 0.120175i
\(479\) −3.02944 −0.138419 −0.0692093 0.997602i \(-0.522048\pi\)
−0.0692093 + 0.997602i \(0.522048\pi\)
\(480\) 0 0
\(481\) 1.94113 0.0885077
\(482\) 9.79899i 0.446332i
\(483\) − 19.3137i − 0.878804i
\(484\) 1.82843 0.0831103
\(485\) 0 0
\(486\) −0.414214 −0.0187891
\(487\) − 20.9706i − 0.950267i −0.879914 0.475133i \(-0.842400\pi\)
0.879914 0.475133i \(-0.157600\pi\)
\(488\) 0.544156i 0.0246328i
\(489\) 15.3137 0.692510
\(490\) 0 0
\(491\) 25.6569 1.15788 0.578939 0.815371i \(-0.303467\pi\)
0.578939 + 0.815371i \(0.303467\pi\)
\(492\) 1.51472i 0.0682888i
\(493\) − 5.65685i − 0.254772i
\(494\) 2.74517 0.123511
\(495\) 0 0
\(496\) 0 0
\(497\) 65.9411i 2.95786i
\(498\) 4.14214i 0.185614i
\(499\) 33.6569 1.50669 0.753344 0.657627i \(-0.228440\pi\)
0.753344 + 0.657627i \(0.228440\pi\)
\(500\) 0 0
\(501\) −9.31371 −0.416106
\(502\) 5.37258i 0.239790i
\(503\) − 5.31371i − 0.236927i −0.992958 0.118463i \(-0.962203\pi\)
0.992958 0.118463i \(-0.0377968\pi\)
\(504\) −7.65685 −0.341063
\(505\) 0 0
\(506\) 1.65685 0.0736562
\(507\) 19.0000i 0.843820i
\(508\) − 4.54416i − 0.201614i
\(509\) −41.3137 −1.83120 −0.915599 0.402093i \(-0.868283\pi\)
−0.915599 + 0.402093i \(0.868283\pi\)
\(510\) 0 0
\(511\) 54.6274 2.41657
\(512\) − 22.7574i − 1.00574i
\(513\) 1.17157i 0.0517262i
\(514\) 11.4558 0.505296
\(515\) 0 0
\(516\) −5.79899 −0.255286
\(517\) − 4.00000i − 0.175920i
\(518\) − 0.686292i − 0.0301539i
\(519\) −2.82843 −0.124154
\(520\) 0 0
\(521\) 12.6274 0.553217 0.276609 0.960983i \(-0.410789\pi\)
0.276609 + 0.960983i \(0.410789\pi\)
\(522\) − 0.343146i − 0.0150191i
\(523\) − 26.4853i − 1.15812i −0.815285 0.579060i \(-0.803420\pi\)
0.815285 0.579060i \(-0.196580\pi\)
\(524\) −35.3137 −1.54269
\(525\) 0 0
\(526\) 7.45584 0.325090
\(527\) 0 0
\(528\) 3.00000i 0.130558i
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) −4.00000 −0.173585
\(532\) 10.3431i 0.448432i
\(533\) − 4.68629i − 0.202986i
\(534\) −3.17157 −0.137247
\(535\) 0 0
\(536\) −8.97056 −0.387469
\(537\) − 6.34315i − 0.273727i
\(538\) − 10.2010i − 0.439797i
\(539\) 16.3137 0.702681
\(540\) 0 0
\(541\) −5.31371 −0.228454 −0.114227 0.993455i \(-0.536439\pi\)
−0.114227 + 0.993455i \(0.536439\pi\)
\(542\) − 11.5147i − 0.494600i
\(543\) 14.0000i 0.600798i
\(544\) 30.1421 1.29233
\(545\) 0 0
\(546\) 11.3137 0.484182
\(547\) − 20.1421i − 0.861216i −0.902539 0.430608i \(-0.858299\pi\)
0.902539 0.430608i \(-0.141701\pi\)
\(548\) − 17.0294i − 0.727462i
\(549\) 0.343146 0.0146451
\(550\) 0 0
\(551\) −0.970563 −0.0413474
\(552\) 6.34315i 0.269982i
\(553\) 40.9706i 1.74225i
\(554\) 5.65685 0.240337
\(555\) 0 0
\(556\) 30.1421 1.27831
\(557\) − 10.8284i − 0.458815i −0.973330 0.229408i \(-0.926321\pi\)
0.973330 0.229408i \(-0.0736789\pi\)
\(558\) 0 0
\(559\) 17.9411 0.758829
\(560\) 0 0
\(561\) −6.82843 −0.288296
\(562\) 6.97056i 0.294035i
\(563\) 20.3431i 0.857361i 0.903456 + 0.428681i \(0.141021\pi\)
−0.903456 + 0.428681i \(0.858979\pi\)
\(564\) 7.31371 0.307963
\(565\) 0 0
\(566\) −1.31371 −0.0552193
\(567\) 4.82843i 0.202775i
\(568\) − 21.6569i − 0.908701i
\(569\) −15.4558 −0.647943 −0.323971 0.946067i \(-0.605018\pi\)
−0.323971 + 0.946067i \(0.605018\pi\)
\(570\) 0 0
\(571\) 0.485281 0.0203084 0.0101542 0.999948i \(-0.496768\pi\)
0.0101542 + 0.999948i \(0.496768\pi\)
\(572\) − 10.3431i − 0.432469i
\(573\) 5.65685i 0.236318i
\(574\) −1.65685 −0.0691558
\(575\) 0 0
\(576\) −4.17157 −0.173816
\(577\) − 14.0000i − 0.582828i −0.956597 0.291414i \(-0.905874\pi\)
0.956597 0.291414i \(-0.0941257\pi\)
\(578\) 12.2721i 0.510451i
\(579\) 2.34315 0.0973778
\(580\) 0 0
\(581\) 48.2843 2.00317
\(582\) 0.142136i 0.00589171i
\(583\) 13.3137i 0.551397i
\(584\) −17.9411 −0.742409
\(585\) 0 0
\(586\) −0.485281 −0.0200468
\(587\) 30.6274i 1.26413i 0.774916 + 0.632064i \(0.217792\pi\)
−0.774916 + 0.632064i \(0.782208\pi\)
\(588\) 29.8284i 1.23010i
\(589\) 0 0
\(590\) 0 0
\(591\) −8.48528 −0.349038
\(592\) − 1.02944i − 0.0423096i
\(593\) − 17.1716i − 0.705152i −0.935783 0.352576i \(-0.885306\pi\)
0.935783 0.352576i \(-0.114694\pi\)
\(594\) −0.414214 −0.0169954
\(595\) 0 0
\(596\) −33.7990 −1.38446
\(597\) − 10.3431i − 0.423317i
\(598\) − 9.37258i − 0.383273i
\(599\) −4.68629 −0.191477 −0.0957383 0.995407i \(-0.530521\pi\)
−0.0957383 + 0.995407i \(0.530521\pi\)
\(600\) 0 0
\(601\) 17.3137 0.706241 0.353120 0.935578i \(-0.385121\pi\)
0.353120 + 0.935578i \(0.385121\pi\)
\(602\) − 6.34315i − 0.258527i
\(603\) 5.65685i 0.230365i
\(604\) −0.887302 −0.0361038
\(605\) 0 0
\(606\) −2.00000 −0.0812444
\(607\) − 18.4853i − 0.750294i −0.926965 0.375147i \(-0.877592\pi\)
0.926965 0.375147i \(-0.122408\pi\)
\(608\) − 5.17157i − 0.209735i
\(609\) −4.00000 −0.162088
\(610\) 0 0
\(611\) −22.6274 −0.915407
\(612\) − 12.4853i − 0.504688i
\(613\) 21.9411i 0.886194i 0.896474 + 0.443097i \(0.146120\pi\)
−0.896474 + 0.443097i \(0.853880\pi\)
\(614\) 3.65685 0.147579
\(615\) 0 0
\(616\) −7.65685 −0.308503
\(617\) − 11.6569i − 0.469287i −0.972081 0.234644i \(-0.924608\pi\)
0.972081 0.234644i \(-0.0753923\pi\)
\(618\) − 8.00000i − 0.321807i
\(619\) 25.6569 1.03124 0.515618 0.856819i \(-0.327562\pi\)
0.515618 + 0.856819i \(0.327562\pi\)
\(620\) 0 0
\(621\) 4.00000 0.160514
\(622\) − 8.00000i − 0.320771i
\(623\) 36.9706i 1.48119i
\(624\) 16.9706 0.679366
\(625\) 0 0
\(626\) 1.79899 0.0719021
\(627\) 1.17157i 0.0467881i
\(628\) − 32.9117i − 1.31332i
\(629\) 2.34315 0.0934273
\(630\) 0 0
\(631\) 34.3431 1.36718 0.683590 0.729867i \(-0.260418\pi\)
0.683590 + 0.729867i \(0.260418\pi\)
\(632\) − 13.4558i − 0.535245i
\(633\) − 6.82843i − 0.271406i
\(634\) −12.5442 −0.498192
\(635\) 0 0
\(636\) −24.3431 −0.965269
\(637\) − 92.2843i − 3.65644i
\(638\) − 0.343146i − 0.0135853i
\(639\) −13.6569 −0.540257
\(640\) 0 0
\(641\) −26.9706 −1.06527 −0.532637 0.846344i \(-0.678799\pi\)
−0.532637 + 0.846344i \(0.678799\pi\)
\(642\) − 2.20101i − 0.0868669i
\(643\) − 29.9411i − 1.18076i −0.807124 0.590381i \(-0.798977\pi\)
0.807124 0.590381i \(-0.201023\pi\)
\(644\) 35.3137 1.39156
\(645\) 0 0
\(646\) 3.31371 0.130376
\(647\) − 27.3137i − 1.07381i −0.843642 0.536906i \(-0.819593\pi\)
0.843642 0.536906i \(-0.180407\pi\)
\(648\) − 1.58579i − 0.0622956i
\(649\) −4.00000 −0.157014
\(650\) 0 0
\(651\) 0 0
\(652\) 28.0000i 1.09656i
\(653\) − 26.9706i − 1.05544i −0.849418 0.527720i \(-0.823047\pi\)
0.849418 0.527720i \(-0.176953\pi\)
\(654\) −2.20101 −0.0860663
\(655\) 0 0
\(656\) −2.48528 −0.0970339
\(657\) 11.3137i 0.441390i
\(658\) 8.00000i 0.311872i
\(659\) −7.31371 −0.284902 −0.142451 0.989802i \(-0.545498\pi\)
−0.142451 + 0.989802i \(0.545498\pi\)
\(660\) 0 0
\(661\) −13.3137 −0.517843 −0.258922 0.965898i \(-0.583367\pi\)
−0.258922 + 0.965898i \(0.583367\pi\)
\(662\) − 7.31371i − 0.284255i
\(663\) 38.6274i 1.50016i
\(664\) −15.8579 −0.615404
\(665\) 0 0
\(666\) 0.142136 0.00550764
\(667\) 3.31371i 0.128307i
\(668\) − 17.0294i − 0.658889i
\(669\) −17.6569 −0.682653
\(670\) 0 0
\(671\) 0.343146 0.0132470
\(672\) − 21.3137i − 0.822194i
\(673\) 29.6569i 1.14319i 0.820537 + 0.571594i \(0.193675\pi\)
−0.820537 + 0.571594i \(0.806325\pi\)
\(674\) 8.00000 0.308148
\(675\) 0 0
\(676\) −34.7401 −1.33616
\(677\) 21.4558i 0.824615i 0.911045 + 0.412308i \(0.135277\pi\)
−0.911045 + 0.412308i \(0.864723\pi\)
\(678\) − 6.20101i − 0.238148i
\(679\) 1.65685 0.0635842
\(680\) 0 0
\(681\) 14.0000 0.536481
\(682\) 0 0
\(683\) 24.0000i 0.918334i 0.888350 + 0.459167i \(0.151852\pi\)
−0.888350 + 0.459167i \(0.848148\pi\)
\(684\) −2.14214 −0.0819066
\(685\) 0 0
\(686\) −18.6274 −0.711198
\(687\) − 2.00000i − 0.0763048i
\(688\) − 9.51472i − 0.362745i
\(689\) 75.3137 2.86922
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) − 5.17157i − 0.196594i
\(693\) 4.82843i 0.183417i
\(694\) 2.76955 0.105131
\(695\) 0 0
\(696\) 1.31371 0.0497960
\(697\) − 5.65685i − 0.214269i
\(698\) − 9.51472i − 0.360137i
\(699\) −13.1716 −0.498195
\(700\) 0 0
\(701\) 7.85786 0.296787 0.148394 0.988928i \(-0.452590\pi\)
0.148394 + 0.988928i \(0.452590\pi\)
\(702\) 2.34315i 0.0884363i
\(703\) − 0.402020i − 0.0151625i
\(704\) −4.17157 −0.157222
\(705\) 0 0
\(706\) 10.7696 0.405317
\(707\) 23.3137i 0.876802i
\(708\) − 7.31371i − 0.274866i
\(709\) −29.3137 −1.10090 −0.550450 0.834868i \(-0.685544\pi\)
−0.550450 + 0.834868i \(0.685544\pi\)
\(710\) 0 0
\(711\) −8.48528 −0.318223
\(712\) − 12.1421i − 0.455046i
\(713\) 0 0
\(714\) 13.6569 0.511095
\(715\) 0 0
\(716\) 11.5980 0.433437
\(717\) 6.34315i 0.236889i
\(718\) 4.97056i 0.185500i
\(719\) −31.5980 −1.17841 −0.589203 0.807985i \(-0.700558\pi\)
−0.589203 + 0.807985i \(0.700558\pi\)
\(720\) 0 0
\(721\) −93.2548 −3.47299
\(722\) 7.30152i 0.271734i
\(723\) 23.6569i 0.879808i
\(724\) −25.5980 −0.951341
\(725\) 0 0
\(726\) −0.414214 −0.0153729
\(727\) 33.9411i 1.25881i 0.777079 + 0.629403i \(0.216701\pi\)
−0.777079 + 0.629403i \(0.783299\pi\)
\(728\) 43.3137i 1.60531i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 21.6569 0.801008
\(732\) 0.627417i 0.0231900i
\(733\) − 17.6569i − 0.652171i −0.945340 0.326085i \(-0.894270\pi\)
0.945340 0.326085i \(-0.105730\pi\)
\(734\) 0.686292 0.0253315
\(735\) 0 0
\(736\) −17.6569 −0.650840
\(737\) 5.65685i 0.208373i
\(738\) − 0.343146i − 0.0126314i
\(739\) −47.1127 −1.73307 −0.866534 0.499118i \(-0.833658\pi\)
−0.866534 + 0.499118i \(0.833658\pi\)
\(740\) 0 0
\(741\) 6.62742 0.243464
\(742\) − 26.6274i − 0.977523i
\(743\) 47.6569i 1.74836i 0.485602 + 0.874180i \(0.338601\pi\)
−0.485602 + 0.874180i \(0.661399\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −14.3431 −0.525140
\(747\) 10.0000i 0.365881i
\(748\) − 12.4853i − 0.456507i
\(749\) −25.6569 −0.937481
\(750\) 0 0
\(751\) −36.2843 −1.32403 −0.662016 0.749490i \(-0.730299\pi\)
−0.662016 + 0.749490i \(0.730299\pi\)
\(752\) 12.0000i 0.437595i
\(753\) 12.9706i 0.472674i
\(754\) −1.94113 −0.0706916
\(755\) 0 0
\(756\) −8.82843 −0.321087
\(757\) − 8.62742i − 0.313569i −0.987633 0.156784i \(-0.949887\pi\)
0.987633 0.156784i \(-0.0501128\pi\)
\(758\) − 0.284271i − 0.0103252i
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) −23.1716 −0.839969 −0.419984 0.907531i \(-0.637964\pi\)
−0.419984 + 0.907531i \(0.637964\pi\)
\(762\) 1.02944i 0.0372926i
\(763\) 25.6569i 0.928840i
\(764\) −10.3431 −0.374202
\(765\) 0 0
\(766\) −3.31371 −0.119729
\(767\) 22.6274i 0.817029i
\(768\) − 3.97056i − 0.143275i
\(769\) −33.3137 −1.20132 −0.600662 0.799503i \(-0.705096\pi\)
−0.600662 + 0.799503i \(0.705096\pi\)
\(770\) 0 0
\(771\) 27.6569 0.996037
\(772\) 4.28427i 0.154194i
\(773\) − 7.65685i − 0.275398i −0.990474 0.137699i \(-0.956029\pi\)
0.990474 0.137699i \(-0.0439706\pi\)
\(774\) 1.31371 0.0472203
\(775\) 0 0
\(776\) −0.544156 −0.0195341
\(777\) − 1.65685i − 0.0594393i
\(778\) − 5.11270i − 0.183299i
\(779\) −0.970563 −0.0347740
\(780\) 0 0
\(781\) −13.6569 −0.488681
\(782\) − 11.3137i − 0.404577i
\(783\) − 0.828427i − 0.0296056i
\(784\) −48.9411 −1.74790
\(785\) 0 0
\(786\) 8.00000 0.285351
\(787\) − 8.14214i − 0.290236i −0.989414 0.145118i \(-0.953644\pi\)
0.989414 0.145118i \(-0.0463561\pi\)
\(788\) − 15.5147i − 0.552689i
\(789\) 18.0000 0.640817
\(790\) 0 0
\(791\) −72.2843 −2.57013
\(792\) − 1.58579i − 0.0563485i
\(793\) − 1.94113i − 0.0689314i
\(794\) −7.85786 −0.278865
\(795\) 0 0
\(796\) 18.9117 0.670307
\(797\) − 1.02944i − 0.0364645i −0.999834 0.0182323i \(-0.994196\pi\)
0.999834 0.0182323i \(-0.00580383\pi\)
\(798\) − 2.34315i − 0.0829465i
\(799\) −27.3137 −0.966290
\(800\) 0 0
\(801\) −7.65685 −0.270542
\(802\) 12.1421i 0.428754i
\(803\) 11.3137i 0.399252i
\(804\) −10.3431 −0.364775
\(805\) 0 0
\(806\) 0 0
\(807\) − 24.6274i − 0.866926i
\(808\) − 7.65685i − 0.269367i
\(809\) 56.4264 1.98385 0.991923 0.126838i \(-0.0404829\pi\)
0.991923 + 0.126838i \(0.0404829\pi\)
\(810\) 0 0
\(811\) −16.4853 −0.578877 −0.289438 0.957197i \(-0.593468\pi\)
−0.289438 + 0.957197i \(0.593468\pi\)
\(812\) − 7.31371i − 0.256661i
\(813\) − 27.7990i − 0.974953i
\(814\) 0.142136 0.00498185
\(815\) 0 0
\(816\) 20.4853 0.717128
\(817\) − 3.71573i − 0.129997i
\(818\) − 3.45584i − 0.120831i
\(819\) 27.3137 0.954418
\(820\) 0 0
\(821\) −7.17157 −0.250290 −0.125145 0.992138i \(-0.539940\pi\)
−0.125145 + 0.992138i \(0.539940\pi\)
\(822\) 3.85786i 0.134558i
\(823\) 16.0000i 0.557725i 0.960331 + 0.278862i \(0.0899574\pi\)
−0.960331 + 0.278862i \(0.910043\pi\)
\(824\) 30.6274 1.06696
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) − 18.6863i − 0.649786i −0.945751 0.324893i \(-0.894672\pi\)
0.945751 0.324893i \(-0.105328\pi\)
\(828\) 7.31371i 0.254169i
\(829\) 38.0000 1.31979 0.659897 0.751356i \(-0.270600\pi\)
0.659897 + 0.751356i \(0.270600\pi\)
\(830\) 0 0
\(831\) 13.6569 0.473751
\(832\) 23.5980i 0.818113i
\(833\) − 111.397i − 3.85968i
\(834\) −6.82843 −0.236449
\(835\) 0 0
\(836\) −2.14214 −0.0740873
\(837\) 0 0
\(838\) − 1.25483i − 0.0433475i
\(839\) −22.6274 −0.781185 −0.390593 0.920564i \(-0.627730\pi\)
−0.390593 + 0.920564i \(0.627730\pi\)
\(840\) 0 0
\(841\) −28.3137 −0.976335
\(842\) 2.48528i 0.0856485i
\(843\) 16.8284i 0.579602i
\(844\) 12.4853 0.429761
\(845\) 0 0
\(846\) −1.65685 −0.0569638
\(847\) 4.82843i 0.165907i
\(848\) − 39.9411i − 1.37158i
\(849\) −3.17157 −0.108848
\(850\) 0 0
\(851\) −1.37258 −0.0470515
\(852\) − 24.9706i − 0.855477i
\(853\) − 31.3137i − 1.07216i −0.844167 0.536080i \(-0.819904\pi\)
0.844167 0.536080i \(-0.180096\pi\)
\(854\) −0.686292 −0.0234844
\(855\) 0 0
\(856\) 8.42641 0.288009
\(857\) 11.5147i 0.393335i 0.980470 + 0.196668i \(0.0630120\pi\)
−0.980470 + 0.196668i \(0.936988\pi\)
\(858\) 2.34315i 0.0799937i
\(859\) 19.0294 0.649276 0.324638 0.945838i \(-0.394758\pi\)
0.324638 + 0.945838i \(0.394758\pi\)
\(860\) 0 0
\(861\) −4.00000 −0.136320
\(862\) − 4.28427i − 0.145923i
\(863\) − 43.3137i − 1.47442i −0.675666 0.737208i \(-0.736144\pi\)
0.675666 0.737208i \(-0.263856\pi\)
\(864\) 4.41421 0.150175
\(865\) 0 0
\(866\) −1.79899 −0.0611322
\(867\) 29.6274i 1.00620i
\(868\) 0 0
\(869\) −8.48528 −0.287843
\(870\) 0 0
\(871\) 32.0000 1.08428
\(872\) − 8.42641i − 0.285354i
\(873\) 0.343146i 0.0116137i
\(874\) −1.94113 −0.0656595
\(875\) 0 0
\(876\) −20.6863 −0.698925
\(877\) 42.6274i 1.43943i 0.694272 + 0.719713i \(0.255727\pi\)
−0.694272 + 0.719713i \(0.744273\pi\)
\(878\) − 1.45584i − 0.0491324i
\(879\) −1.17157 −0.0395162
\(880\) 0 0
\(881\) −13.0294 −0.438973 −0.219486 0.975616i \(-0.570438\pi\)
−0.219486 + 0.975616i \(0.570438\pi\)
\(882\) − 6.75736i − 0.227532i
\(883\) − 50.6274i − 1.70375i −0.523747 0.851874i \(-0.675466\pi\)
0.523747 0.851874i \(-0.324534\pi\)
\(884\) −70.6274 −2.37546
\(885\) 0 0
\(886\) 4.97056 0.166989
\(887\) 4.34315i 0.145829i 0.997338 + 0.0729143i \(0.0232300\pi\)
−0.997338 + 0.0729143i \(0.976770\pi\)
\(888\) 0.544156i 0.0182607i
\(889\) 12.0000 0.402467
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) − 32.2843i − 1.08096i
\(893\) 4.68629i 0.156821i
\(894\) 7.65685 0.256084
\(895\) 0 0
\(896\) 50.9706 1.70281
\(897\) − 22.6274i − 0.755507i
\(898\) − 1.23045i − 0.0410606i
\(899\) 0 0
\(900\) 0 0
\(901\) 90.9117 3.02871
\(902\) − 0.343146i − 0.0114255i
\(903\) − 15.3137i − 0.509608i
\(904\) 23.7401 0.789584
\(905\) 0 0
\(906\) 0.201010 0.00667811
\(907\) − 7.02944i − 0.233409i −0.993167 0.116704i \(-0.962767\pi\)
0.993167 0.116704i \(-0.0372330\pi\)
\(908\) 25.5980i 0.849499i
\(909\) −4.82843 −0.160149
\(910\) 0 0
\(911\) −15.0294 −0.497947 −0.248974 0.968510i \(-0.580093\pi\)
−0.248974 + 0.968510i \(0.580093\pi\)
\(912\) − 3.51472i − 0.116384i
\(913\) 10.0000i 0.330952i
\(914\) 0.284271 0.00940286
\(915\) 0 0
\(916\) 3.65685 0.120826
\(917\) − 93.2548i − 3.07955i
\(918\) 2.82843i 0.0933520i
\(919\) −28.4853 −0.939643 −0.469821 0.882762i \(-0.655682\pi\)
−0.469821 + 0.882762i \(0.655682\pi\)
\(920\) 0 0
\(921\) 8.82843 0.290907
\(922\) 11.6569i 0.383898i
\(923\) 77.2548i 2.54287i
\(924\) −8.82843 −0.290434
\(925\) 0 0
\(926\) −12.0000 −0.394344
\(927\) − 19.3137i − 0.634345i
\(928\) 3.65685i 0.120042i
\(929\) −33.5980 −1.10231 −0.551157 0.834402i \(-0.685813\pi\)
−0.551157 + 0.834402i \(0.685813\pi\)
\(930\) 0 0
\(931\) −19.1127 −0.626393
\(932\) − 24.0833i − 0.788873i
\(933\) − 19.3137i − 0.632302i
\(934\) −9.37258 −0.306680
\(935\) 0 0
\(936\) −8.97056 −0.293212
\(937\) − 44.9706i − 1.46912i −0.678541 0.734562i \(-0.737388\pi\)
0.678541 0.734562i \(-0.262612\pi\)
\(938\) − 11.3137i − 0.369406i
\(939\) 4.34315 0.141733
\(940\) 0 0
\(941\) 38.7696 1.26385 0.631926 0.775029i \(-0.282265\pi\)
0.631926 + 0.775029i \(0.282265\pi\)
\(942\) 7.45584i 0.242925i
\(943\) 3.31371i 0.107909i
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 1.31371 0.0427123
\(947\) 38.6274i 1.25522i 0.778527 + 0.627611i \(0.215967\pi\)
−0.778527 + 0.627611i \(0.784033\pi\)
\(948\) − 15.5147i − 0.503895i
\(949\) 64.0000 2.07753
\(950\) 0 0
\(951\) −30.2843 −0.982035
\(952\) 52.2843i 1.69454i
\(953\) 27.7990i 0.900498i 0.892903 + 0.450249i \(0.148665\pi\)
−0.892903 + 0.450249i \(0.851335\pi\)
\(954\) 5.51472 0.178546
\(955\) 0 0
\(956\) −11.5980 −0.375105
\(957\) − 0.828427i − 0.0267792i
\(958\) 1.25483i 0.0405418i
\(959\) 44.9706 1.45218
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) − 0.804041i − 0.0259233i
\(963\) − 5.31371i − 0.171232i
\(964\) −43.2548 −1.39314
\(965\) 0 0
\(966\) −8.00000 −0.257396
\(967\) 39.4558i 1.26881i 0.772999 + 0.634407i \(0.218756\pi\)
−0.772999 + 0.634407i \(0.781244\pi\)
\(968\) − 1.58579i − 0.0509691i
\(969\) 8.00000 0.256997
\(970\) 0 0
\(971\) 10.6274 0.341050 0.170525 0.985353i \(-0.445454\pi\)
0.170525 + 0.985353i \(0.445454\pi\)
\(972\) − 1.82843i − 0.0586468i
\(973\) 79.5980i 2.55179i
\(974\) −8.68629 −0.278327
\(975\) 0 0
\(976\) −1.02944 −0.0329515
\(977\) − 25.3137i − 0.809857i −0.914348 0.404929i \(-0.867296\pi\)
0.914348 0.404929i \(-0.132704\pi\)
\(978\) − 6.34315i − 0.202831i
\(979\) −7.65685 −0.244714
\(980\) 0 0
\(981\) −5.31371 −0.169654
\(982\) − 10.6274i − 0.339135i
\(983\) 14.6274i 0.466542i 0.972412 + 0.233271i \(0.0749429\pi\)
−0.972412 + 0.233271i \(0.925057\pi\)
\(984\) 1.31371 0.0418795
\(985\) 0 0
\(986\) −2.34315 −0.0746210
\(987\) 19.3137i 0.614762i
\(988\) 12.1177i 0.385517i
\(989\) −12.6863 −0.403401
\(990\) 0 0
\(991\) 14.6274 0.464655 0.232328 0.972638i \(-0.425366\pi\)
0.232328 + 0.972638i \(0.425366\pi\)
\(992\) 0 0
\(993\) − 17.6569i − 0.560323i
\(994\) 27.3137 0.866338
\(995\) 0 0
\(996\) −18.2843 −0.579359
\(997\) 16.6863i 0.528460i 0.964460 + 0.264230i \(0.0851178\pi\)
−0.964460 + 0.264230i \(0.914882\pi\)
\(998\) − 13.9411i − 0.441299i
\(999\) 0.343146 0.0108567
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 825.2.c.e.199.2 4
3.2 odd 2 2475.2.c.m.199.3 4
5.2 odd 4 165.2.a.a.1.2 2
5.3 odd 4 825.2.a.g.1.1 2
5.4 even 2 inner 825.2.c.e.199.3 4
15.2 even 4 495.2.a.d.1.1 2
15.8 even 4 2475.2.a.m.1.2 2
15.14 odd 2 2475.2.c.m.199.2 4
20.7 even 4 2640.2.a.bb.1.2 2
35.27 even 4 8085.2.a.ba.1.2 2
55.32 even 4 1815.2.a.k.1.1 2
55.43 even 4 9075.2.a.v.1.2 2
60.47 odd 4 7920.2.a.cg.1.2 2
165.32 odd 4 5445.2.a.m.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.a.1.2 2 5.2 odd 4
495.2.a.d.1.1 2 15.2 even 4
825.2.a.g.1.1 2 5.3 odd 4
825.2.c.e.199.2 4 1.1 even 1 trivial
825.2.c.e.199.3 4 5.4 even 2 inner
1815.2.a.k.1.1 2 55.32 even 4
2475.2.a.m.1.2 2 15.8 even 4
2475.2.c.m.199.2 4 15.14 odd 2
2475.2.c.m.199.3 4 3.2 odd 2
2640.2.a.bb.1.2 2 20.7 even 4
5445.2.a.m.1.2 2 165.32 odd 4
7920.2.a.cg.1.2 2 60.47 odd 4
8085.2.a.ba.1.2 2 35.27 even 4
9075.2.a.v.1.2 2 55.43 even 4