Properties

Label 2475.2.c.o.199.3
Level $2475$
Weight $2$
Character 2475.199
Analytic conductor $19.763$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 825)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.3
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2475.199
Dual form 2475.2.c.o.199.2

$q$-expansion

\(f(q)\) \(=\) \(q+0.414214i q^{2} +1.82843 q^{4} -2.41421i q^{7} +1.58579i q^{8} +O(q^{10})\) \(q+0.414214i q^{2} +1.82843 q^{4} -2.41421i q^{7} +1.58579i q^{8} +1.00000 q^{11} -2.82843i q^{13} +1.00000 q^{14} +3.00000 q^{16} +0.414214i q^{17} -3.58579 q^{19} +0.414214i q^{22} +1.00000i q^{23} +1.17157 q^{26} -4.41421i q^{28} +6.82843 q^{29} +8.48528 q^{31} +4.41421i q^{32} -0.171573 q^{34} -5.82843i q^{37} -1.48528i q^{38} -8.89949 q^{41} +0.343146i q^{43} +1.82843 q^{44} -0.414214 q^{46} -9.48528i q^{47} +1.17157 q^{49} -5.17157i q^{52} -3.65685i q^{53} +3.82843 q^{56} +2.82843i q^{58} +11.0000 q^{59} +3.17157 q^{61} +3.51472i q^{62} +4.17157 q^{64} -11.6569i q^{67} +0.757359i q^{68} -2.17157 q^{71} +3.17157i q^{73} +2.41421 q^{74} -6.55635 q^{76} -2.41421i q^{77} -4.75736 q^{79} -3.68629i q^{82} +12.4853i q^{83} -0.142136 q^{86} +1.58579i q^{88} -7.65685 q^{89} -6.82843 q^{91} +1.82843i q^{92} +3.92893 q^{94} +0.171573i q^{97} +0.485281i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + O(q^{10}) \) \( 4 q - 4 q^{4} + 4 q^{11} + 4 q^{14} + 12 q^{16} - 20 q^{19} + 16 q^{26} + 16 q^{29} - 12 q^{34} + 4 q^{41} - 4 q^{44} + 4 q^{46} + 16 q^{49} + 4 q^{56} + 44 q^{59} + 24 q^{61} + 28 q^{64} - 20 q^{71} + 4 q^{74} + 36 q^{76} - 36 q^{79} + 56 q^{86} - 8 q^{89} - 16 q^{91} + 44 q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(2026\) \(2377\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(0.0467837\pi\)
−0.989219 + 0.146447i \(0.953216\pi\)
\(3\) 0 0
\(4\) 1.82843 0.914214
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.41421i − 0.912487i −0.889855 0.456243i \(-0.849195\pi\)
0.889855 0.456243i \(-0.150805\pi\)
\(8\) 1.58579i 0.560660i
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) − 2.82843i − 0.784465i −0.919866 0.392232i \(-0.871703\pi\)
0.919866 0.392232i \(-0.128297\pi\)
\(14\) 1.00000 0.267261
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 0.414214i 0.100462i 0.998738 + 0.0502308i \(0.0159957\pi\)
−0.998738 + 0.0502308i \(0.984004\pi\)
\(18\) 0 0
\(19\) −3.58579 −0.822636 −0.411318 0.911492i \(-0.634931\pi\)
−0.411318 + 0.911492i \(0.634931\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.414214i 0.0883106i
\(23\) 1.00000i 0.208514i 0.994550 + 0.104257i \(0.0332465\pi\)
−0.994550 + 0.104257i \(0.966753\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.17157 0.229764
\(27\) 0 0
\(28\) − 4.41421i − 0.834208i
\(29\) 6.82843 1.26801 0.634004 0.773330i \(-0.281410\pi\)
0.634004 + 0.773330i \(0.281410\pi\)
\(30\) 0 0
\(31\) 8.48528 1.52400 0.762001 0.647576i \(-0.224217\pi\)
0.762001 + 0.647576i \(0.224217\pi\)
\(32\) 4.41421i 0.780330i
\(33\) 0 0
\(34\) −0.171573 −0.0294245
\(35\) 0 0
\(36\) 0 0
\(37\) − 5.82843i − 0.958188i −0.877764 0.479094i \(-0.840965\pi\)
0.877764 0.479094i \(-0.159035\pi\)
\(38\) − 1.48528i − 0.240944i
\(39\) 0 0
\(40\) 0 0
\(41\) −8.89949 −1.38987 −0.694934 0.719074i \(-0.744566\pi\)
−0.694934 + 0.719074i \(0.744566\pi\)
\(42\) 0 0
\(43\) 0.343146i 0.0523292i 0.999658 + 0.0261646i \(0.00832941\pi\)
−0.999658 + 0.0261646i \(0.991671\pi\)
\(44\) 1.82843 0.275646
\(45\) 0 0
\(46\) −0.414214 −0.0610725
\(47\) − 9.48528i − 1.38357i −0.722103 0.691785i \(-0.756824\pi\)
0.722103 0.691785i \(-0.243176\pi\)
\(48\) 0 0
\(49\) 1.17157 0.167368
\(50\) 0 0
\(51\) 0 0
\(52\) − 5.17157i − 0.717168i
\(53\) − 3.65685i − 0.502308i −0.967947 0.251154i \(-0.919190\pi\)
0.967947 0.251154i \(-0.0808100\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.82843 0.511595
\(57\) 0 0
\(58\) 2.82843i 0.371391i
\(59\) 11.0000 1.43208 0.716039 0.698060i \(-0.245953\pi\)
0.716039 + 0.698060i \(0.245953\pi\)
\(60\) 0 0
\(61\) 3.17157 0.406078 0.203039 0.979171i \(-0.434918\pi\)
0.203039 + 0.979171i \(0.434918\pi\)
\(62\) 3.51472i 0.446370i
\(63\) 0 0
\(64\) 4.17157 0.521447
\(65\) 0 0
\(66\) 0 0
\(67\) − 11.6569i − 1.42411i −0.702123 0.712056i \(-0.747764\pi\)
0.702123 0.712056i \(-0.252236\pi\)
\(68\) 0.757359i 0.0918433i
\(69\) 0 0
\(70\) 0 0
\(71\) −2.17157 −0.257718 −0.128859 0.991663i \(-0.541132\pi\)
−0.128859 + 0.991663i \(0.541132\pi\)
\(72\) 0 0
\(73\) 3.17157i 0.371205i 0.982625 + 0.185602i \(0.0594236\pi\)
−0.982625 + 0.185602i \(0.940576\pi\)
\(74\) 2.41421 0.280647
\(75\) 0 0
\(76\) −6.55635 −0.752065
\(77\) − 2.41421i − 0.275125i
\(78\) 0 0
\(79\) −4.75736 −0.535245 −0.267622 0.963524i \(-0.586238\pi\)
−0.267622 + 0.963524i \(0.586238\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 3.68629i − 0.407083i
\(83\) 12.4853i 1.37044i 0.728337 + 0.685219i \(0.240293\pi\)
−0.728337 + 0.685219i \(0.759707\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.142136 −0.0153269
\(87\) 0 0
\(88\) 1.58579i 0.169045i
\(89\) −7.65685 −0.811625 −0.405812 0.913956i \(-0.633011\pi\)
−0.405812 + 0.913956i \(0.633011\pi\)
\(90\) 0 0
\(91\) −6.82843 −0.715814
\(92\) 1.82843i 0.190627i
\(93\) 0 0
\(94\) 3.92893 0.405238
\(95\) 0 0
\(96\) 0 0
\(97\) 0.171573i 0.0174206i 0.999962 + 0.00871029i \(0.00277261\pi\)
−0.999962 + 0.00871029i \(0.997227\pi\)
\(98\) 0.485281i 0.0490208i
\(99\) 0 0
\(100\) 0 0
\(101\) 4.89949 0.487518 0.243759 0.969836i \(-0.421619\pi\)
0.243759 + 0.969836i \(0.421619\pi\)
\(102\) 0 0
\(103\) 2.34315i 0.230877i 0.993315 + 0.115439i \(0.0368273\pi\)
−0.993315 + 0.115439i \(0.963173\pi\)
\(104\) 4.48528 0.439818
\(105\) 0 0
\(106\) 1.51472 0.147122
\(107\) − 17.3137i − 1.67378i −0.547372 0.836890i \(-0.684372\pi\)
0.547372 0.836890i \(-0.315628\pi\)
\(108\) 0 0
\(109\) 17.3137 1.65835 0.829176 0.558987i \(-0.188810\pi\)
0.829176 + 0.558987i \(0.188810\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 7.24264i − 0.684365i
\(113\) − 10.0000i − 0.940721i −0.882474 0.470360i \(-0.844124\pi\)
0.882474 0.470360i \(-0.155876\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 12.4853 1.15923
\(117\) 0 0
\(118\) 4.55635i 0.419446i
\(119\) 1.00000 0.0916698
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 1.31371i 0.118938i
\(123\) 0 0
\(124\) 15.5147 1.39326
\(125\) 0 0
\(126\) 0 0
\(127\) − 1.24264i − 0.110267i −0.998479 0.0551333i \(-0.982442\pi\)
0.998479 0.0551333i \(-0.0175584\pi\)
\(128\) 10.5563i 0.933058i
\(129\) 0 0
\(130\) 0 0
\(131\) −1.17157 −0.102361 −0.0511804 0.998689i \(-0.516298\pi\)
−0.0511804 + 0.998689i \(0.516298\pi\)
\(132\) 0 0
\(133\) 8.65685i 0.750644i
\(134\) 4.82843 0.417113
\(135\) 0 0
\(136\) −0.656854 −0.0563248
\(137\) − 16.1421i − 1.37912i −0.724231 0.689558i \(-0.757805\pi\)
0.724231 0.689558i \(-0.242195\pi\)
\(138\) 0 0
\(139\) −14.9706 −1.26979 −0.634893 0.772600i \(-0.718956\pi\)
−0.634893 + 0.772600i \(0.718956\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 0.899495i − 0.0754839i
\(143\) − 2.82843i − 0.236525i
\(144\) 0 0
\(145\) 0 0
\(146\) −1.31371 −0.108723
\(147\) 0 0
\(148\) − 10.6569i − 0.875988i
\(149\) −17.7279 −1.45233 −0.726164 0.687522i \(-0.758699\pi\)
−0.726164 + 0.687522i \(0.758699\pi\)
\(150\) 0 0
\(151\) 14.0000 1.13930 0.569652 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) − 5.68629i − 0.461219i
\(153\) 0 0
\(154\) 1.00000 0.0805823
\(155\) 0 0
\(156\) 0 0
\(157\) 6.00000i 0.478852i 0.970915 + 0.239426i \(0.0769593\pi\)
−0.970915 + 0.239426i \(0.923041\pi\)
\(158\) − 1.97056i − 0.156770i
\(159\) 0 0
\(160\) 0 0
\(161\) 2.41421 0.190267
\(162\) 0 0
\(163\) 23.7990i 1.86408i 0.362354 + 0.932040i \(0.381973\pi\)
−0.362354 + 0.932040i \(0.618027\pi\)
\(164\) −16.2721 −1.27064
\(165\) 0 0
\(166\) −5.17157 −0.401392
\(167\) 17.7990i 1.37733i 0.725081 + 0.688664i \(0.241802\pi\)
−0.725081 + 0.688664i \(0.758198\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 0.627417i 0.0478401i
\(173\) 18.5563i 1.41081i 0.708803 + 0.705407i \(0.249236\pi\)
−0.708803 + 0.705407i \(0.750764\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) − 3.17157i − 0.237719i
\(179\) −22.7990 −1.70408 −0.852038 0.523480i \(-0.824634\pi\)
−0.852038 + 0.523480i \(0.824634\pi\)
\(180\) 0 0
\(181\) 11.9706 0.889765 0.444882 0.895589i \(-0.353245\pi\)
0.444882 + 0.895589i \(0.353245\pi\)
\(182\) − 2.82843i − 0.209657i
\(183\) 0 0
\(184\) −1.58579 −0.116906
\(185\) 0 0
\(186\) 0 0
\(187\) 0.414214i 0.0302903i
\(188\) − 17.3431i − 1.26488i
\(189\) 0 0
\(190\) 0 0
\(191\) −11.8284 −0.855875 −0.427937 0.903808i \(-0.640760\pi\)
−0.427937 + 0.903808i \(0.640760\pi\)
\(192\) 0 0
\(193\) 19.3137i 1.39023i 0.718898 + 0.695116i \(0.244647\pi\)
−0.718898 + 0.695116i \(0.755353\pi\)
\(194\) −0.0710678 −0.00510237
\(195\) 0 0
\(196\) 2.14214 0.153010
\(197\) − 13.2426i − 0.943499i −0.881733 0.471750i \(-0.843623\pi\)
0.881733 0.471750i \(-0.156377\pi\)
\(198\) 0 0
\(199\) −5.17157 −0.366603 −0.183302 0.983057i \(-0.558678\pi\)
−0.183302 + 0.983057i \(0.558678\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 2.02944i 0.142791i
\(203\) − 16.4853i − 1.15704i
\(204\) 0 0
\(205\) 0 0
\(206\) −0.970563 −0.0676223
\(207\) 0 0
\(208\) − 8.48528i − 0.588348i
\(209\) −3.58579 −0.248034
\(210\) 0 0
\(211\) 9.31371 0.641182 0.320591 0.947218i \(-0.396118\pi\)
0.320591 + 0.947218i \(0.396118\pi\)
\(212\) − 6.68629i − 0.459216i
\(213\) 0 0
\(214\) 7.17157 0.490239
\(215\) 0 0
\(216\) 0 0
\(217\) − 20.4853i − 1.39063i
\(218\) 7.17157i 0.485720i
\(219\) 0 0
\(220\) 0 0
\(221\) 1.17157 0.0788085
\(222\) 0 0
\(223\) 26.8284i 1.79656i 0.439419 + 0.898282i \(0.355184\pi\)
−0.439419 + 0.898282i \(0.644816\pi\)
\(224\) 10.6569 0.712041
\(225\) 0 0
\(226\) 4.14214 0.275531
\(227\) 1.51472i 0.100535i 0.998736 + 0.0502677i \(0.0160075\pi\)
−0.998736 + 0.0502677i \(0.983993\pi\)
\(228\) 0 0
\(229\) 19.4853 1.28762 0.643812 0.765184i \(-0.277352\pi\)
0.643812 + 0.765184i \(0.277352\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 10.8284i 0.710921i
\(233\) − 14.5563i − 0.953618i −0.879007 0.476809i \(-0.841793\pi\)
0.879007 0.476809i \(-0.158207\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 20.1127 1.30923
\(237\) 0 0
\(238\) 0.414214i 0.0268495i
\(239\) 12.3431 0.798412 0.399206 0.916861i \(-0.369286\pi\)
0.399206 + 0.916861i \(0.369286\pi\)
\(240\) 0 0
\(241\) −14.1421 −0.910975 −0.455488 0.890242i \(-0.650535\pi\)
−0.455488 + 0.890242i \(0.650535\pi\)
\(242\) 0.414214i 0.0266267i
\(243\) 0 0
\(244\) 5.79899 0.371242
\(245\) 0 0
\(246\) 0 0
\(247\) 10.1421i 0.645329i
\(248\) 13.4558i 0.854447i
\(249\) 0 0
\(250\) 0 0
\(251\) 24.9706 1.57613 0.788064 0.615593i \(-0.211084\pi\)
0.788064 + 0.615593i \(0.211084\pi\)
\(252\) 0 0
\(253\) 1.00000i 0.0628695i
\(254\) 0.514719 0.0322963
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) 13.3137i 0.830486i 0.909710 + 0.415243i \(0.136304\pi\)
−0.909710 + 0.415243i \(0.863696\pi\)
\(258\) 0 0
\(259\) −14.0711 −0.874334
\(260\) 0 0
\(261\) 0 0
\(262\) − 0.485281i − 0.0299808i
\(263\) − 10.9706i − 0.676474i −0.941061 0.338237i \(-0.890169\pi\)
0.941061 0.338237i \(-0.109831\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −3.58579 −0.219859
\(267\) 0 0
\(268\) − 21.3137i − 1.30194i
\(269\) 23.7990 1.45105 0.725525 0.688196i \(-0.241597\pi\)
0.725525 + 0.688196i \(0.241597\pi\)
\(270\) 0 0
\(271\) −10.8995 −0.662097 −0.331049 0.943614i \(-0.607402\pi\)
−0.331049 + 0.943614i \(0.607402\pi\)
\(272\) 1.24264i 0.0753462i
\(273\) 0 0
\(274\) 6.68629 0.403934
\(275\) 0 0
\(276\) 0 0
\(277\) − 0.828427i − 0.0497754i −0.999690 0.0248877i \(-0.992077\pi\)
0.999690 0.0248877i \(-0.00792281\pi\)
\(278\) − 6.20101i − 0.371912i
\(279\) 0 0
\(280\) 0 0
\(281\) −17.9289 −1.06955 −0.534775 0.844994i \(-0.679604\pi\)
−0.534775 + 0.844994i \(0.679604\pi\)
\(282\) 0 0
\(283\) − 18.8995i − 1.12346i −0.827321 0.561729i \(-0.810136\pi\)
0.827321 0.561729i \(-0.189864\pi\)
\(284\) −3.97056 −0.235610
\(285\) 0 0
\(286\) 1.17157 0.0692766
\(287\) 21.4853i 1.26824i
\(288\) 0 0
\(289\) 16.8284 0.989907
\(290\) 0 0
\(291\) 0 0
\(292\) 5.79899i 0.339360i
\(293\) 20.4142i 1.19261i 0.802758 + 0.596306i \(0.203365\pi\)
−0.802758 + 0.596306i \(0.796635\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 9.24264 0.537218
\(297\) 0 0
\(298\) − 7.34315i − 0.425377i
\(299\) 2.82843 0.163572
\(300\) 0 0
\(301\) 0.828427 0.0477497
\(302\) 5.79899i 0.333694i
\(303\) 0 0
\(304\) −10.7574 −0.616977
\(305\) 0 0
\(306\) 0 0
\(307\) 29.3137i 1.67302i 0.547950 + 0.836511i \(0.315408\pi\)
−0.547950 + 0.836511i \(0.684592\pi\)
\(308\) − 4.41421i − 0.251523i
\(309\) 0 0
\(310\) 0 0
\(311\) −2.34315 −0.132868 −0.0664338 0.997791i \(-0.521162\pi\)
−0.0664338 + 0.997791i \(0.521162\pi\)
\(312\) 0 0
\(313\) − 1.14214i − 0.0645573i −0.999479 0.0322787i \(-0.989724\pi\)
0.999479 0.0322787i \(-0.0102764\pi\)
\(314\) −2.48528 −0.140253
\(315\) 0 0
\(316\) −8.69848 −0.489328
\(317\) − 25.1716i − 1.41378i −0.707325 0.706888i \(-0.750098\pi\)
0.707325 0.706888i \(-0.249902\pi\)
\(318\) 0 0
\(319\) 6.82843 0.382319
\(320\) 0 0
\(321\) 0 0
\(322\) 1.00000i 0.0557278i
\(323\) − 1.48528i − 0.0826433i
\(324\) 0 0
\(325\) 0 0
\(326\) −9.85786 −0.545977
\(327\) 0 0
\(328\) − 14.1127i − 0.779243i
\(329\) −22.8995 −1.26249
\(330\) 0 0
\(331\) −3.85786 −0.212047 −0.106024 0.994364i \(-0.533812\pi\)
−0.106024 + 0.994364i \(0.533812\pi\)
\(332\) 22.8284i 1.25287i
\(333\) 0 0
\(334\) −7.37258 −0.403410
\(335\) 0 0
\(336\) 0 0
\(337\) − 24.1421i − 1.31511i −0.753408 0.657553i \(-0.771592\pi\)
0.753408 0.657553i \(-0.228408\pi\)
\(338\) 2.07107i 0.112651i
\(339\) 0 0
\(340\) 0 0
\(341\) 8.48528 0.459504
\(342\) 0 0
\(343\) − 19.7279i − 1.06521i
\(344\) −0.544156 −0.0293389
\(345\) 0 0
\(346\) −7.68629 −0.413218
\(347\) 26.8284i 1.44023i 0.693857 + 0.720113i \(0.255910\pi\)
−0.693857 + 0.720113i \(0.744090\pi\)
\(348\) 0 0
\(349\) −14.4853 −0.775379 −0.387690 0.921790i \(-0.626727\pi\)
−0.387690 + 0.921790i \(0.626727\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4.41421i 0.235278i
\(353\) 12.4853i 0.664524i 0.943187 + 0.332262i \(0.107812\pi\)
−0.943187 + 0.332262i \(0.892188\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) − 9.44365i − 0.499112i
\(359\) 32.4853 1.71451 0.857254 0.514894i \(-0.172169\pi\)
0.857254 + 0.514894i \(0.172169\pi\)
\(360\) 0 0
\(361\) −6.14214 −0.323270
\(362\) 4.95837i 0.260606i
\(363\) 0 0
\(364\) −12.4853 −0.654407
\(365\) 0 0
\(366\) 0 0
\(367\) − 21.3137i − 1.11257i −0.830993 0.556283i \(-0.812227\pi\)
0.830993 0.556283i \(-0.187773\pi\)
\(368\) 3.00000i 0.156386i
\(369\) 0 0
\(370\) 0 0
\(371\) −8.82843 −0.458349
\(372\) 0 0
\(373\) 12.3431i 0.639104i 0.947569 + 0.319552i \(0.103532\pi\)
−0.947569 + 0.319552i \(0.896468\pi\)
\(374\) −0.171573 −0.00887182
\(375\) 0 0
\(376\) 15.0416 0.775713
\(377\) − 19.3137i − 0.994707i
\(378\) 0 0
\(379\) −14.8284 −0.761685 −0.380843 0.924640i \(-0.624366\pi\)
−0.380843 + 0.924640i \(0.624366\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 4.89949i − 0.250680i
\(383\) 20.0000i 1.02195i 0.859595 + 0.510976i \(0.170716\pi\)
−0.859595 + 0.510976i \(0.829284\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −8.00000 −0.407189
\(387\) 0 0
\(388\) 0.313708i 0.0159261i
\(389\) −6.34315 −0.321610 −0.160805 0.986986i \(-0.551409\pi\)
−0.160805 + 0.986986i \(0.551409\pi\)
\(390\) 0 0
\(391\) −0.414214 −0.0209477
\(392\) 1.85786i 0.0938363i
\(393\) 0 0
\(394\) 5.48528 0.276344
\(395\) 0 0
\(396\) 0 0
\(397\) 31.9411i 1.60308i 0.597942 + 0.801540i \(0.295985\pi\)
−0.597942 + 0.801540i \(0.704015\pi\)
\(398\) − 2.14214i − 0.107376i
\(399\) 0 0
\(400\) 0 0
\(401\) 7.79899 0.389463 0.194731 0.980857i \(-0.437616\pi\)
0.194731 + 0.980857i \(0.437616\pi\)
\(402\) 0 0
\(403\) − 24.0000i − 1.19553i
\(404\) 8.95837 0.445696
\(405\) 0 0
\(406\) 6.82843 0.338889
\(407\) − 5.82843i − 0.288904i
\(408\) 0 0
\(409\) −24.1421 −1.19375 −0.596876 0.802334i \(-0.703592\pi\)
−0.596876 + 0.802334i \(0.703592\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 4.28427i 0.211071i
\(413\) − 26.5563i − 1.30675i
\(414\) 0 0
\(415\) 0 0
\(416\) 12.4853 0.612141
\(417\) 0 0
\(418\) − 1.48528i − 0.0726475i
\(419\) −8.51472 −0.415971 −0.207986 0.978132i \(-0.566691\pi\)
−0.207986 + 0.978132i \(0.566691\pi\)
\(420\) 0 0
\(421\) −27.0000 −1.31590 −0.657950 0.753062i \(-0.728576\pi\)
−0.657950 + 0.753062i \(0.728576\pi\)
\(422\) 3.85786i 0.187798i
\(423\) 0 0
\(424\) 5.79899 0.281624
\(425\) 0 0
\(426\) 0 0
\(427\) − 7.65685i − 0.370541i
\(428\) − 31.6569i − 1.53019i
\(429\) 0 0
\(430\) 0 0
\(431\) −6.82843 −0.328914 −0.164457 0.986384i \(-0.552587\pi\)
−0.164457 + 0.986384i \(0.552587\pi\)
\(432\) 0 0
\(433\) − 9.31371i − 0.447588i −0.974636 0.223794i \(-0.928156\pi\)
0.974636 0.223794i \(-0.0718443\pi\)
\(434\) 8.48528 0.407307
\(435\) 0 0
\(436\) 31.6569 1.51609
\(437\) − 3.58579i − 0.171531i
\(438\) 0 0
\(439\) −27.7279 −1.32338 −0.661691 0.749777i \(-0.730161\pi\)
−0.661691 + 0.749777i \(0.730161\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0.485281i 0.0230825i
\(443\) 25.9706i 1.23390i 0.787003 + 0.616949i \(0.211632\pi\)
−0.787003 + 0.616949i \(0.788368\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −11.1127 −0.526202
\(447\) 0 0
\(448\) − 10.0711i − 0.475813i
\(449\) 10.4853 0.494831 0.247416 0.968909i \(-0.420419\pi\)
0.247416 + 0.968909i \(0.420419\pi\)
\(450\) 0 0
\(451\) −8.89949 −0.419061
\(452\) − 18.2843i − 0.860020i
\(453\) 0 0
\(454\) −0.627417 −0.0294461
\(455\) 0 0
\(456\) 0 0
\(457\) 32.1421i 1.50355i 0.659422 + 0.751773i \(0.270801\pi\)
−0.659422 + 0.751773i \(0.729199\pi\)
\(458\) 8.07107i 0.377136i
\(459\) 0 0
\(460\) 0 0
\(461\) −40.7696 −1.89883 −0.949414 0.314028i \(-0.898321\pi\)
−0.949414 + 0.314028i \(0.898321\pi\)
\(462\) 0 0
\(463\) − 1.02944i − 0.0478420i −0.999714 0.0239210i \(-0.992385\pi\)
0.999714 0.0239210i \(-0.00761502\pi\)
\(464\) 20.4853 0.951005
\(465\) 0 0
\(466\) 6.02944 0.279308
\(467\) 34.6274i 1.60237i 0.598420 + 0.801183i \(0.295796\pi\)
−0.598420 + 0.801183i \(0.704204\pi\)
\(468\) 0 0
\(469\) −28.1421 −1.29948
\(470\) 0 0
\(471\) 0 0
\(472\) 17.4437i 0.802909i
\(473\) 0.343146i 0.0157779i
\(474\) 0 0
\(475\) 0 0
\(476\) 1.82843 0.0838058
\(477\) 0 0
\(478\) 5.11270i 0.233849i
\(479\) −7.51472 −0.343356 −0.171678 0.985153i \(-0.554919\pi\)
−0.171678 + 0.985153i \(0.554919\pi\)
\(480\) 0 0
\(481\) −16.4853 −0.751664
\(482\) − 5.85786i − 0.266818i
\(483\) 0 0
\(484\) 1.82843 0.0831103
\(485\) 0 0
\(486\) 0 0
\(487\) 10.4853i 0.475133i 0.971371 + 0.237567i \(0.0763498\pi\)
−0.971371 + 0.237567i \(0.923650\pi\)
\(488\) 5.02944i 0.227672i
\(489\) 0 0
\(490\) 0 0
\(491\) −4.14214 −0.186932 −0.0934660 0.995622i \(-0.529795\pi\)
−0.0934660 + 0.995622i \(0.529795\pi\)
\(492\) 0 0
\(493\) 2.82843i 0.127386i
\(494\) −4.20101 −0.189012
\(495\) 0 0
\(496\) 25.4558 1.14300
\(497\) 5.24264i 0.235165i
\(498\) 0 0
\(499\) −40.8284 −1.82773 −0.913866 0.406017i \(-0.866918\pi\)
−0.913866 + 0.406017i \(0.866918\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 10.3431i 0.461637i
\(503\) 22.2843i 0.993607i 0.867863 + 0.496803i \(0.165493\pi\)
−0.867863 + 0.496803i \(0.834507\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −0.414214 −0.0184140
\(507\) 0 0
\(508\) − 2.27208i − 0.100807i
\(509\) −40.6274 −1.80078 −0.900389 0.435085i \(-0.856718\pi\)
−0.900389 + 0.435085i \(0.856718\pi\)
\(510\) 0 0
\(511\) 7.65685 0.338719
\(512\) 22.7574i 1.00574i
\(513\) 0 0
\(514\) −5.51472 −0.243244
\(515\) 0 0
\(516\) 0 0
\(517\) − 9.48528i − 0.417162i
\(518\) − 5.82843i − 0.256086i
\(519\) 0 0
\(520\) 0 0
\(521\) 7.85786 0.344259 0.172130 0.985074i \(-0.444935\pi\)
0.172130 + 0.985074i \(0.444935\pi\)
\(522\) 0 0
\(523\) 0.213203i 0.00932274i 0.999989 + 0.00466137i \(0.00148376\pi\)
−0.999989 + 0.00466137i \(0.998516\pi\)
\(524\) −2.14214 −0.0935796
\(525\) 0 0
\(526\) 4.54416 0.198135
\(527\) 3.51472i 0.153104i
\(528\) 0 0
\(529\) 22.0000 0.956522
\(530\) 0 0
\(531\) 0 0
\(532\) 15.8284i 0.686249i
\(533\) 25.1716i 1.09030i
\(534\) 0 0
\(535\) 0 0
\(536\) 18.4853 0.798443
\(537\) 0 0
\(538\) 9.85786i 0.425003i
\(539\) 1.17157 0.0504632
\(540\) 0 0
\(541\) −11.3137 −0.486414 −0.243207 0.969974i \(-0.578199\pi\)
−0.243207 + 0.969974i \(0.578199\pi\)
\(542\) − 4.51472i − 0.193924i
\(543\) 0 0
\(544\) −1.82843 −0.0783932
\(545\) 0 0
\(546\) 0 0
\(547\) − 17.8701i − 0.764068i −0.924148 0.382034i \(-0.875224\pi\)
0.924148 0.382034i \(-0.124776\pi\)
\(548\) − 29.5147i − 1.26081i
\(549\) 0 0
\(550\) 0 0
\(551\) −24.4853 −1.04311
\(552\) 0 0
\(553\) 11.4853i 0.488404i
\(554\) 0.343146 0.0145789
\(555\) 0 0
\(556\) −27.3726 −1.16086
\(557\) 10.8284i 0.458815i 0.973330 + 0.229408i \(0.0736789\pi\)
−0.973330 + 0.229408i \(0.926321\pi\)
\(558\) 0 0
\(559\) 0.970563 0.0410504
\(560\) 0 0
\(561\) 0 0
\(562\) − 7.42641i − 0.313264i
\(563\) − 7.31371i − 0.308236i −0.988052 0.154118i \(-0.950746\pi\)
0.988052 0.154118i \(-0.0492536\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 7.82843 0.329053
\(567\) 0 0
\(568\) − 3.44365i − 0.144492i
\(569\) 6.75736 0.283283 0.141642 0.989918i \(-0.454762\pi\)
0.141642 + 0.989918i \(0.454762\pi\)
\(570\) 0 0
\(571\) −42.9706 −1.79826 −0.899131 0.437680i \(-0.855800\pi\)
−0.899131 + 0.437680i \(0.855800\pi\)
\(572\) − 5.17157i − 0.216234i
\(573\) 0 0
\(574\) −8.89949 −0.371458
\(575\) 0 0
\(576\) 0 0
\(577\) − 9.97056i − 0.415080i −0.978227 0.207540i \(-0.933454\pi\)
0.978227 0.207540i \(-0.0665457\pi\)
\(578\) 6.97056i 0.289937i
\(579\) 0 0
\(580\) 0 0
\(581\) 30.1421 1.25051
\(582\) 0 0
\(583\) − 3.65685i − 0.151451i
\(584\) −5.02944 −0.208120
\(585\) 0 0
\(586\) −8.45584 −0.349308
\(587\) 25.3431i 1.04602i 0.852325 + 0.523012i \(0.175192\pi\)
−0.852325 + 0.523012i \(0.824808\pi\)
\(588\) 0 0
\(589\) −30.4264 −1.25370
\(590\) 0 0
\(591\) 0 0
\(592\) − 17.4853i − 0.718641i
\(593\) − 35.7990i − 1.47009i −0.678019 0.735044i \(-0.737161\pi\)
0.678019 0.735044i \(-0.262839\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −32.4142 −1.32774
\(597\) 0 0
\(598\) 1.17157i 0.0479092i
\(599\) 13.6863 0.559207 0.279603 0.960116i \(-0.409797\pi\)
0.279603 + 0.960116i \(0.409797\pi\)
\(600\) 0 0
\(601\) −9.17157 −0.374116 −0.187058 0.982349i \(-0.559895\pi\)
−0.187058 + 0.982349i \(0.559895\pi\)
\(602\) 0.343146i 0.0139856i
\(603\) 0 0
\(604\) 25.5980 1.04157
\(605\) 0 0
\(606\) 0 0
\(607\) 30.9706i 1.25706i 0.777787 + 0.628528i \(0.216342\pi\)
−0.777787 + 0.628528i \(0.783658\pi\)
\(608\) − 15.8284i − 0.641927i
\(609\) 0 0
\(610\) 0 0
\(611\) −26.8284 −1.08536
\(612\) 0 0
\(613\) 42.0000i 1.69636i 0.529705 + 0.848182i \(0.322303\pi\)
−0.529705 + 0.848182i \(0.677697\pi\)
\(614\) −12.1421 −0.490017
\(615\) 0 0
\(616\) 3.82843 0.154252
\(617\) 38.1421i 1.53554i 0.640723 + 0.767772i \(0.278635\pi\)
−0.640723 + 0.767772i \(0.721365\pi\)
\(618\) 0 0
\(619\) 6.62742 0.266378 0.133189 0.991091i \(-0.457478\pi\)
0.133189 + 0.991091i \(0.457478\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 0.970563i − 0.0389160i
\(623\) 18.4853i 0.740597i
\(624\) 0 0
\(625\) 0 0
\(626\) 0.473088 0.0189084
\(627\) 0 0
\(628\) 10.9706i 0.437773i
\(629\) 2.41421 0.0962610
\(630\) 0 0
\(631\) −18.6274 −0.741546 −0.370773 0.928724i \(-0.620907\pi\)
−0.370773 + 0.928724i \(0.620907\pi\)
\(632\) − 7.54416i − 0.300090i
\(633\) 0 0
\(634\) 10.4264 0.414086
\(635\) 0 0
\(636\) 0 0
\(637\) − 3.31371i − 0.131294i
\(638\) 2.82843i 0.111979i
\(639\) 0 0
\(640\) 0 0
\(641\) 25.5147 1.00777 0.503885 0.863771i \(-0.331903\pi\)
0.503885 + 0.863771i \(0.331903\pi\)
\(642\) 0 0
\(643\) − 0.970563i − 0.0382753i −0.999817 0.0191376i \(-0.993908\pi\)
0.999817 0.0191376i \(-0.00609207\pi\)
\(644\) 4.41421 0.173944
\(645\) 0 0
\(646\) 0.615224 0.0242057
\(647\) − 28.6569i − 1.12662i −0.826247 0.563309i \(-0.809528\pi\)
0.826247 0.563309i \(-0.190472\pi\)
\(648\) 0 0
\(649\) 11.0000 0.431788
\(650\) 0 0
\(651\) 0 0
\(652\) 43.5147i 1.70417i
\(653\) − 23.5147i − 0.920202i −0.887867 0.460101i \(-0.847813\pi\)
0.887867 0.460101i \(-0.152187\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −26.6985 −1.04240
\(657\) 0 0
\(658\) − 9.48528i − 0.369775i
\(659\) −47.1127 −1.83525 −0.917625 0.397447i \(-0.869896\pi\)
−0.917625 + 0.397447i \(0.869896\pi\)
\(660\) 0 0
\(661\) −23.3431 −0.907943 −0.453972 0.891016i \(-0.649993\pi\)
−0.453972 + 0.891016i \(0.649993\pi\)
\(662\) − 1.59798i − 0.0621072i
\(663\) 0 0
\(664\) −19.7990 −0.768350
\(665\) 0 0
\(666\) 0 0
\(667\) 6.82843i 0.264398i
\(668\) 32.5442i 1.25917i
\(669\) 0 0
\(670\) 0 0
\(671\) 3.17157 0.122437
\(672\) 0 0
\(673\) − 0.343146i − 0.0132273i −0.999978 0.00661365i \(-0.997895\pi\)
0.999978 0.00661365i \(-0.00210520\pi\)
\(674\) 10.0000 0.385186
\(675\) 0 0
\(676\) 9.14214 0.351621
\(677\) − 23.5147i − 0.903744i −0.892083 0.451872i \(-0.850756\pi\)
0.892083 0.451872i \(-0.149244\pi\)
\(678\) 0 0
\(679\) 0.414214 0.0158961
\(680\) 0 0
\(681\) 0 0
\(682\) 3.51472i 0.134586i
\(683\) 12.5147i 0.478862i 0.970913 + 0.239431i \(0.0769610\pi\)
−0.970913 + 0.239431i \(0.923039\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 8.17157 0.311992
\(687\) 0 0
\(688\) 1.02944i 0.0392469i
\(689\) −10.3431 −0.394042
\(690\) 0 0
\(691\) −35.4558 −1.34880 −0.674402 0.738364i \(-0.735598\pi\)
−0.674402 + 0.738364i \(0.735598\pi\)
\(692\) 33.9289i 1.28978i
\(693\) 0 0
\(694\) −11.1127 −0.421832
\(695\) 0 0
\(696\) 0 0
\(697\) − 3.68629i − 0.139628i
\(698\) − 6.00000i − 0.227103i
\(699\) 0 0
\(700\) 0 0
\(701\) 41.3848 1.56308 0.781541 0.623854i \(-0.214434\pi\)
0.781541 + 0.623854i \(0.214434\pi\)
\(702\) 0 0
\(703\) 20.8995i 0.788239i
\(704\) 4.17157 0.157222
\(705\) 0 0
\(706\) −5.17157 −0.194635
\(707\) − 11.8284i − 0.444854i
\(708\) 0 0
\(709\) 29.1421 1.09446 0.547228 0.836984i \(-0.315683\pi\)
0.547228 + 0.836984i \(0.315683\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 12.1421i − 0.455046i
\(713\) 8.48528i 0.317776i
\(714\) 0 0
\(715\) 0 0
\(716\) −41.6863 −1.55789
\(717\) 0 0
\(718\) 13.4558i 0.502168i
\(719\) 9.65685 0.360140 0.180070 0.983654i \(-0.442368\pi\)
0.180070 + 0.983654i \(0.442368\pi\)
\(720\) 0 0
\(721\) 5.65685 0.210672
\(722\) − 2.54416i − 0.0946837i
\(723\) 0 0
\(724\) 21.8873 0.813435
\(725\) 0 0
\(726\) 0 0
\(727\) 16.9706i 0.629403i 0.949191 + 0.314702i \(0.101904\pi\)
−0.949191 + 0.314702i \(0.898096\pi\)
\(728\) − 10.8284i − 0.401328i
\(729\) 0 0
\(730\) 0 0
\(731\) −0.142136 −0.00525708
\(732\) 0 0
\(733\) − 32.1421i − 1.18720i −0.804761 0.593598i \(-0.797707\pi\)
0.804761 0.593598i \(-0.202293\pi\)
\(734\) 8.82843 0.325863
\(735\) 0 0
\(736\) −4.41421 −0.162710
\(737\) − 11.6569i − 0.429386i
\(738\) 0 0
\(739\) −20.4142 −0.750949 −0.375474 0.926833i \(-0.622520\pi\)
−0.375474 + 0.926833i \(0.622520\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 3.65685i − 0.134247i
\(743\) − 31.1127i − 1.14141i −0.821154 0.570707i \(-0.806669\pi\)
0.821154 0.570707i \(-0.193331\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −5.11270 −0.187189
\(747\) 0 0
\(748\) 0.757359i 0.0276918i
\(749\) −41.7990 −1.52730
\(750\) 0 0
\(751\) 31.5980 1.15303 0.576513 0.817088i \(-0.304413\pi\)
0.576513 + 0.817088i \(0.304413\pi\)
\(752\) − 28.4558i − 1.03768i
\(753\) 0 0
\(754\) 8.00000 0.291343
\(755\) 0 0
\(756\) 0 0
\(757\) − 10.6863i − 0.388400i −0.980962 0.194200i \(-0.937789\pi\)
0.980962 0.194200i \(-0.0622111\pi\)
\(758\) − 6.14214i − 0.223092i
\(759\) 0 0
\(760\) 0 0
\(761\) 5.17157 0.187469 0.0937347 0.995597i \(-0.470119\pi\)
0.0937347 + 0.995597i \(0.470119\pi\)
\(762\) 0 0
\(763\) − 41.7990i − 1.51323i
\(764\) −21.6274 −0.782452
\(765\) 0 0
\(766\) −8.28427 −0.299323
\(767\) − 31.1127i − 1.12341i
\(768\) 0 0
\(769\) 7.65685 0.276113 0.138057 0.990424i \(-0.455914\pi\)
0.138057 + 0.990424i \(0.455914\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 35.3137i 1.27097i
\(773\) − 0.828427i − 0.0297965i −0.999889 0.0148982i \(-0.995258\pi\)
0.999889 0.0148982i \(-0.00474243\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −0.272078 −0.00976703
\(777\) 0 0
\(778\) − 2.62742i − 0.0941975i
\(779\) 31.9117 1.14335
\(780\) 0 0
\(781\) −2.17157 −0.0777050
\(782\) − 0.171573i − 0.00613543i
\(783\) 0 0
\(784\) 3.51472 0.125526
\(785\) 0 0
\(786\) 0 0
\(787\) − 7.92893i − 0.282636i −0.989964 0.141318i \(-0.954866\pi\)
0.989964 0.141318i \(-0.0451340\pi\)
\(788\) − 24.2132i − 0.862560i
\(789\) 0 0
\(790\) 0 0
\(791\) −24.1421 −0.858396
\(792\) 0 0
\(793\) − 8.97056i − 0.318554i
\(794\) −13.2304 −0.469531
\(795\) 0 0
\(796\) −9.45584 −0.335154
\(797\) − 40.9706i − 1.45125i −0.688089 0.725626i \(-0.741550\pi\)
0.688089 0.725626i \(-0.258450\pi\)
\(798\) 0 0
\(799\) 3.92893 0.138996
\(800\) 0 0
\(801\) 0 0
\(802\) 3.23045i 0.114071i
\(803\) 3.17157i 0.111922i
\(804\) 0 0
\(805\) 0 0
\(806\) 9.94113 0.350161
\(807\) 0 0
\(808\) 7.76955i 0.273332i
\(809\) 7.72792 0.271699 0.135850 0.990729i \(-0.456624\pi\)
0.135850 + 0.990729i \(0.456624\pi\)
\(810\) 0 0
\(811\) 10.2132 0.358634 0.179317 0.983791i \(-0.442611\pi\)
0.179317 + 0.983791i \(0.442611\pi\)
\(812\) − 30.1421i − 1.05778i
\(813\) 0 0
\(814\) 2.41421 0.0846181
\(815\) 0 0
\(816\) 0 0
\(817\) − 1.23045i − 0.0430479i
\(818\) − 10.0000i − 0.349642i
\(819\) 0 0
\(820\) 0 0
\(821\) −15.7990 −0.551389 −0.275694 0.961245i \(-0.588908\pi\)
−0.275694 + 0.961245i \(0.588908\pi\)
\(822\) 0 0
\(823\) 14.9706i 0.521841i 0.965360 + 0.260921i \(0.0840260\pi\)
−0.965360 + 0.260921i \(0.915974\pi\)
\(824\) −3.71573 −0.129444
\(825\) 0 0
\(826\) 11.0000 0.382739
\(827\) 12.6863i 0.441146i 0.975371 + 0.220573i \(0.0707927\pi\)
−0.975371 + 0.220573i \(0.929207\pi\)
\(828\) 0 0
\(829\) 47.9411 1.66506 0.832532 0.553977i \(-0.186890\pi\)
0.832532 + 0.553977i \(0.186890\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 11.7990i − 0.409056i
\(833\) 0.485281i 0.0168140i
\(834\) 0 0
\(835\) 0 0
\(836\) −6.55635 −0.226756
\(837\) 0 0
\(838\) − 3.52691i − 0.121835i
\(839\) −47.3137 −1.63345 −0.816725 0.577027i \(-0.804213\pi\)
−0.816725 + 0.577027i \(0.804213\pi\)
\(840\) 0 0
\(841\) 17.6274 0.607842
\(842\) − 11.1838i − 0.385418i
\(843\) 0 0
\(844\) 17.0294 0.586177
\(845\) 0 0
\(846\) 0 0
\(847\) − 2.41421i − 0.0829534i
\(848\) − 10.9706i − 0.376731i
\(849\) 0 0
\(850\) 0 0
\(851\) 5.82843 0.199796
\(852\) 0 0
\(853\) 19.1716i 0.656422i 0.944604 + 0.328211i \(0.106446\pi\)
−0.944604 + 0.328211i \(0.893554\pi\)
\(854\) 3.17157 0.108529
\(855\) 0 0
\(856\) 27.4558 0.938421
\(857\) 36.6985i 1.25360i 0.779182 + 0.626798i \(0.215635\pi\)
−0.779182 + 0.626798i \(0.784365\pi\)
\(858\) 0 0
\(859\) 20.4853 0.698949 0.349474 0.936946i \(-0.386360\pi\)
0.349474 + 0.936946i \(0.386360\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 2.82843i − 0.0963366i
\(863\) − 28.6863i − 0.976493i −0.872706 0.488246i \(-0.837637\pi\)
0.872706 0.488246i \(-0.162363\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 3.85786 0.131096
\(867\) 0 0
\(868\) − 37.4558i − 1.27133i
\(869\) −4.75736 −0.161382
\(870\) 0 0
\(871\) −32.9706 −1.11716
\(872\) 27.4558i 0.929772i
\(873\) 0 0
\(874\) 1.48528 0.0502404
\(875\) 0 0
\(876\) 0 0
\(877\) 15.1127i 0.510320i 0.966899 + 0.255160i \(0.0821281\pi\)
−0.966899 + 0.255160i \(0.917872\pi\)
\(878\) − 11.4853i − 0.387609i
\(879\) 0 0
\(880\) 0 0
\(881\) 24.0000 0.808581 0.404290 0.914631i \(-0.367519\pi\)
0.404290 + 0.914631i \(0.367519\pi\)
\(882\) 0 0
\(883\) − 38.6274i − 1.29992i −0.759970 0.649958i \(-0.774786\pi\)
0.759970 0.649958i \(-0.225214\pi\)
\(884\) 2.14214 0.0720478
\(885\) 0 0
\(886\) −10.7574 −0.361401
\(887\) 22.1421i 0.743460i 0.928341 + 0.371730i \(0.121235\pi\)
−0.928341 + 0.371730i \(0.878765\pi\)
\(888\) 0 0
\(889\) −3.00000 −0.100617
\(890\) 0 0
\(891\) 0 0
\(892\) 49.0538i 1.64244i
\(893\) 34.0122i 1.13817i
\(894\) 0 0
\(895\) 0 0
\(896\) 25.4853 0.851403
\(897\) 0 0
\(898\) 4.34315i 0.144933i
\(899\) 57.9411 1.93244
\(900\) 0 0
\(901\) 1.51472 0.0504626
\(902\) − 3.68629i − 0.122740i
\(903\) 0 0
\(904\) 15.8579 0.527425
\(905\) 0 0
\(906\) 0 0
\(907\) − 2.48528i − 0.0825224i −0.999148 0.0412612i \(-0.986862\pi\)
0.999148 0.0412612i \(-0.0131376\pi\)
\(908\) 2.76955i 0.0919108i
\(909\) 0 0
\(910\) 0 0
\(911\) −28.5147 −0.944735 −0.472367 0.881402i \(-0.656600\pi\)
−0.472367 + 0.881402i \(0.656600\pi\)
\(912\) 0 0
\(913\) 12.4853i 0.413203i
\(914\) −13.3137 −0.440378
\(915\) 0 0
\(916\) 35.6274 1.17716
\(917\) 2.82843i 0.0934029i
\(918\) 0 0
\(919\) −1.78680 −0.0589410 −0.0294705 0.999566i \(-0.509382\pi\)
−0.0294705 + 0.999566i \(0.509382\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 16.8873i − 0.556154i
\(923\) 6.14214i 0.202171i
\(924\) 0 0
\(925\) 0 0
\(926\) 0.426407 0.0140126
\(927\) 0 0
\(928\) 30.1421i 0.989464i
\(929\) −13.7990 −0.452730 −0.226365 0.974043i \(-0.572684\pi\)
−0.226365 + 0.974043i \(0.572684\pi\)
\(930\) 0 0
\(931\) −4.20101 −0.137683
\(932\) − 26.6152i − 0.871811i
\(933\) 0 0
\(934\) −14.3431 −0.469322
\(935\) 0 0
\(936\) 0 0
\(937\) − 16.0000i − 0.522697i −0.965244 0.261349i \(-0.915833\pi\)
0.965244 0.261349i \(-0.0841672\pi\)
\(938\) − 11.6569i − 0.380610i
\(939\) 0 0
\(940\) 0 0
\(941\) 22.8995 0.746502 0.373251 0.927730i \(-0.378243\pi\)
0.373251 + 0.927730i \(0.378243\pi\)
\(942\) 0 0
\(943\) − 8.89949i − 0.289807i
\(944\) 33.0000 1.07406
\(945\) 0 0
\(946\) −0.142136 −0.00462123
\(947\) 36.7990i 1.19581i 0.801568 + 0.597903i \(0.203999\pi\)
−0.801568 + 0.597903i \(0.796001\pi\)
\(948\) 0 0
\(949\) 8.97056 0.291197
\(950\) 0 0
\(951\) 0 0
\(952\) 1.58579i 0.0513956i
\(953\) − 29.0416i − 0.940751i −0.882466 0.470375i \(-0.844119\pi\)
0.882466 0.470375i \(-0.155881\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 22.5685 0.729919
\(957\) 0 0
\(958\) − 3.11270i − 0.100567i
\(959\) −38.9706 −1.25843
\(960\) 0 0
\(961\) 41.0000 1.32258
\(962\) − 6.82843i − 0.220157i
\(963\) 0 0
\(964\) −25.8579 −0.832826
\(965\) 0 0
\(966\) 0 0
\(967\) 38.0000i 1.22200i 0.791632 + 0.610999i \(0.209232\pi\)
−0.791632 + 0.610999i \(0.790768\pi\)
\(968\) 1.58579i 0.0509691i
\(969\) 0 0
\(970\) 0 0
\(971\) 50.3137 1.61464 0.807322 0.590111i \(-0.200916\pi\)
0.807322 + 0.590111i \(0.200916\pi\)
\(972\) 0 0
\(973\) 36.1421i 1.15866i
\(974\) −4.34315 −0.139163
\(975\) 0 0
\(976\) 9.51472 0.304559
\(977\) − 60.5685i − 1.93776i −0.247532 0.968880i \(-0.579620\pi\)
0.247532 0.968880i \(-0.420380\pi\)
\(978\) 0 0
\(979\) −7.65685 −0.244714
\(980\) 0 0
\(981\) 0 0
\(982\) − 1.71573i − 0.0547511i
\(983\) 51.2843i 1.63571i 0.575421 + 0.817857i \(0.304838\pi\)
−0.575421 + 0.817857i \(0.695162\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −1.17157 −0.0373105
\(987\) 0 0
\(988\) 18.5442i 0.589968i
\(989\) −0.343146 −0.0109114
\(990\) 0 0
\(991\) −42.2843 −1.34320 −0.671602 0.740912i \(-0.734394\pi\)
−0.671602 + 0.740912i \(0.734394\pi\)
\(992\) 37.4558i 1.18922i
\(993\) 0 0
\(994\) −2.17157 −0.0688781
\(995\) 0 0
\(996\) 0 0
\(997\) − 47.2548i − 1.49658i −0.663374 0.748288i \(-0.730876\pi\)
0.663374 0.748288i \(-0.269124\pi\)
\(998\) − 16.9117i − 0.535330i
\(999\) 0 0
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 2475.2.c.o.199.3 4
3.2 odd 2 825.2.c.d.199.2 4
5.2 odd 4 2475.2.a.w.1.1 2
5.3 odd 4 2475.2.a.l.1.2 2
5.4 even 2 inner 2475.2.c.o.199.2 4
15.2 even 4 825.2.a.d.1.2 2
15.8 even 4 825.2.a.f.1.1 yes 2
15.14 odd 2 825.2.c.d.199.3 4
165.32 odd 4 9075.2.a.ca.1.1 2
165.98 odd 4 9075.2.a.w.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.a.d.1.2 2 15.2 even 4
825.2.a.f.1.1 yes 2 15.8 even 4
825.2.c.d.199.2 4 3.2 odd 2
825.2.c.d.199.3 4 15.14 odd 2
2475.2.a.l.1.2 2 5.3 odd 4
2475.2.a.w.1.1 2 5.2 odd 4
2475.2.c.o.199.2 4 5.4 even 2 inner
2475.2.c.o.199.3 4 1.1 even 1 trivial
9075.2.a.w.1.2 2 165.98 odd 4
9075.2.a.ca.1.1 2 165.32 odd 4