Properties

Label 825.2.c.d.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: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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.d.199.3

$q$-expansion

\(f(q)\) \(=\) \(q-0.414214i q^{2} -1.00000i q^{3} +1.82843 q^{4} -0.414214 q^{6} -2.41421i q^{7} -1.58579i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-0.414214i q^{2} -1.00000i q^{3} +1.82843 q^{4} -0.414214 q^{6} -2.41421i q^{7} -1.58579i q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.82843i q^{12} -2.82843i q^{13} -1.00000 q^{14} +3.00000 q^{16} -0.414214i q^{17} +0.414214i q^{18} -3.58579 q^{19} -2.41421 q^{21} +0.414214i q^{22} -1.00000i q^{23} -1.58579 q^{24} -1.17157 q^{26} +1.00000i q^{27} -4.41421i q^{28} -6.82843 q^{29} +8.48528 q^{31} -4.41421i q^{32} +1.00000i q^{33} -0.171573 q^{34} -1.82843 q^{36} -5.82843i q^{37} +1.48528i q^{38} -2.82843 q^{39} +8.89949 q^{41} +1.00000i q^{42} +0.343146i q^{43} -1.82843 q^{44} -0.414214 q^{46} +9.48528i q^{47} -3.00000i q^{48} +1.17157 q^{49} -0.414214 q^{51} -5.17157i q^{52} +3.65685i q^{53} +0.414214 q^{54} -3.82843 q^{56} +3.58579i q^{57} +2.82843i q^{58} -11.0000 q^{59} +3.17157 q^{61} -3.51472i q^{62} +2.41421i q^{63} +4.17157 q^{64} +0.414214 q^{66} -11.6569i q^{67} -0.757359i q^{68} -1.00000 q^{69} +2.17157 q^{71} +1.58579i q^{72} +3.17157i q^{73} -2.41421 q^{74} -6.55635 q^{76} +2.41421i q^{77} +1.17157i q^{78} -4.75736 q^{79} +1.00000 q^{81} -3.68629i q^{82} -12.4853i q^{83} -4.41421 q^{84} +0.142136 q^{86} +6.82843i q^{87} +1.58579i q^{88} +7.65685 q^{89} -6.82843 q^{91} -1.82843i q^{92} -8.48528i q^{93} +3.92893 q^{94} -4.41421 q^{96} +0.171573i q^{97} -0.485281i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} - 4 q^{11} - 4 q^{14} + 12 q^{16} - 20 q^{19} - 4 q^{21} - 12 q^{24} - 16 q^{26} - 16 q^{29} - 12 q^{34} + 4 q^{36} - 4 q^{41} + 4 q^{44} + 4 q^{46} + 16 q^{49} + 4 q^{51} - 4 q^{54} - 4 q^{56} - 44 q^{59} + 24 q^{61} + 28 q^{64} - 4 q^{66} - 4 q^{69} + 20 q^{71} - 4 q^{74} + 36 q^{76} - 36 q^{79} + 4 q^{81} - 12 q^{84} - 56 q^{86} + 8 q^{89} - 16 q^{91} + 44 q^{94} - 12 q^{96} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(7\) − 2.41421i − 0.912487i −0.889855 0.456243i \(-0.849195\pi\)
0.889855 0.456243i \(-0.150805\pi\)
\(8\) − 1.58579i − 0.560660i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) − 1.82843i − 0.527821i
\(13\) − 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 \(-0.984004\pi\)
0.998738 0.0502308i \(-0.0159957\pi\)
\(18\) 0.414214i 0.0976311i
\(19\) −3.58579 −0.822636 −0.411318 0.911492i \(-0.634931\pi\)
−0.411318 + 0.911492i \(0.634931\pi\)
\(20\) 0 0
\(21\) −2.41421 −0.526825
\(22\) 0.414214i 0.0883106i
\(23\) − 1.00000i − 0.208514i −0.994550 0.104257i \(-0.966753\pi\)
0.994550 0.104257i \(-0.0332465\pi\)
\(24\) −1.58579 −0.323697
\(25\) 0 0
\(26\) −1.17157 −0.229764
\(27\) 1.00000i 0.192450i
\(28\) − 4.41421i − 0.834208i
\(29\) −6.82843 −1.26801 −0.634004 0.773330i \(-0.718590\pi\)
−0.634004 + 0.773330i \(0.718590\pi\)
\(30\) 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
\(33\) 1.00000i 0.174078i
\(34\) −0.171573 −0.0294245
\(35\) 0 0
\(36\) −1.82843 −0.304738
\(37\) − 5.82843i − 0.958188i −0.877764 0.479094i \(-0.840965\pi\)
0.877764 0.479094i \(-0.159035\pi\)
\(38\) 1.48528i 0.240944i
\(39\) −2.82843 −0.452911
\(40\) 0 0
\(41\) 8.89949 1.38987 0.694934 0.719074i \(-0.255434\pi\)
0.694934 + 0.719074i \(0.255434\pi\)
\(42\) 1.00000i 0.154303i
\(43\) 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 \(0.243176\pi\)
−0.722103 + 0.691785i \(0.756824\pi\)
\(48\) − 3.00000i − 0.433013i
\(49\) 1.17157 0.167368
\(50\) 0 0
\(51\) −0.414214 −0.0580015
\(52\) − 5.17157i − 0.717168i
\(53\) 3.65685i 0.502308i 0.967947 + 0.251154i \(0.0808100\pi\)
−0.967947 + 0.251154i \(0.919190\pi\)
\(54\) 0.414214 0.0563673
\(55\) 0 0
\(56\) −3.82843 −0.511595
\(57\) 3.58579i 0.474949i
\(58\) 2.82843i 0.371391i
\(59\) −11.0000 −1.43208 −0.716039 0.698060i \(-0.754047\pi\)
−0.716039 + 0.698060i \(0.754047\pi\)
\(60\) 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
\(63\) 2.41421i 0.304162i
\(64\) 4.17157 0.521447
\(65\) 0 0
\(66\) 0.414214 0.0509862
\(67\) − 11.6569i − 1.42411i −0.702123 0.712056i \(-0.747764\pi\)
0.702123 0.712056i \(-0.252236\pi\)
\(68\) − 0.757359i − 0.0918433i
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 2.17157 0.257718 0.128859 0.991663i \(-0.458868\pi\)
0.128859 + 0.991663i \(0.458868\pi\)
\(72\) 1.58579i 0.186887i
\(73\) 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
\(78\) 1.17157i 0.132655i
\(79\) −4.75736 −0.535245 −0.267622 0.963524i \(-0.586238\pi\)
−0.267622 + 0.963524i \(0.586238\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 3.68629i − 0.407083i
\(83\) − 12.4853i − 1.37044i −0.728337 0.685219i \(-0.759707\pi\)
0.728337 0.685219i \(-0.240293\pi\)
\(84\) −4.41421 −0.481630
\(85\) 0 0
\(86\) 0.142136 0.0153269
\(87\) 6.82843i 0.732084i
\(88\) 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
\(91\) −6.82843 −0.715814
\(92\) − 1.82843i − 0.190627i
\(93\) − 8.48528i − 0.879883i
\(94\) 3.92893 0.405238
\(95\) 0 0
\(96\) −4.41421 −0.450524
\(97\) 0.171573i 0.0174206i 0.999962 + 0.00871029i \(0.00277261\pi\)
−0.999962 + 0.00871029i \(0.997227\pi\)
\(98\) − 0.485281i − 0.0490208i
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) −4.89949 −0.487518 −0.243759 0.969836i \(-0.578381\pi\)
−0.243759 + 0.969836i \(0.578381\pi\)
\(102\) 0.171573i 0.0169882i
\(103\) 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 \(0.315628\pi\)
−0.547372 + 0.836890i \(0.684372\pi\)
\(108\) 1.82843i 0.175940i
\(109\) 17.3137 1.65835 0.829176 0.558987i \(-0.188810\pi\)
0.829176 + 0.558987i \(0.188810\pi\)
\(110\) 0 0
\(111\) −5.82843 −0.553210
\(112\) − 7.24264i − 0.684365i
\(113\) 10.0000i 0.940721i 0.882474 + 0.470360i \(0.155876\pi\)
−0.882474 + 0.470360i \(0.844124\pi\)
\(114\) 1.48528 0.139109
\(115\) 0 0
\(116\) −12.4853 −1.15923
\(117\) 2.82843i 0.261488i
\(118\) 4.55635i 0.419446i
\(119\) −1.00000 −0.0916698
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 1.31371i − 0.118938i
\(123\) − 8.89949i − 0.802440i
\(124\) 15.5147 1.39326
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) − 1.24264i − 0.110267i −0.998479 0.0551333i \(-0.982442\pi\)
0.998479 0.0551333i \(-0.0175584\pi\)
\(128\) − 10.5563i − 0.933058i
\(129\) 0.343146 0.0302123
\(130\) 0 0
\(131\) 1.17157 0.102361 0.0511804 0.998689i \(-0.483702\pi\)
0.0511804 + 0.998689i \(0.483702\pi\)
\(132\) 1.82843i 0.159144i
\(133\) 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 \(0.242195\pi\)
−0.724231 + 0.689558i \(0.757805\pi\)
\(138\) 0.414214i 0.0352602i
\(139\) −14.9706 −1.26979 −0.634893 0.772600i \(-0.718956\pi\)
−0.634893 + 0.772600i \(0.718956\pi\)
\(140\) 0 0
\(141\) 9.48528 0.798805
\(142\) − 0.899495i − 0.0754839i
\(143\) 2.82843i 0.236525i
\(144\) −3.00000 −0.250000
\(145\) 0 0
\(146\) 1.31371 0.108723
\(147\) − 1.17157i − 0.0966297i
\(148\) − 10.6569i − 0.875988i
\(149\) 17.7279 1.45233 0.726164 0.687522i \(-0.241301\pi\)
0.726164 + 0.687522i \(0.241301\pi\)
\(150\) 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
\(153\) 0.414214i 0.0334872i
\(154\) 1.00000 0.0805823
\(155\) 0 0
\(156\) −5.17157 −0.414057
\(157\) 6.00000i 0.478852i 0.970915 + 0.239426i \(0.0769593\pi\)
−0.970915 + 0.239426i \(0.923041\pi\)
\(158\) 1.97056i 0.156770i
\(159\) 3.65685 0.290007
\(160\) 0 0
\(161\) −2.41421 −0.190267
\(162\) − 0.414214i − 0.0325437i
\(163\) 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 \(-0.758198\pi\)
0.725081 0.688664i \(-0.241802\pi\)
\(168\) 3.82843i 0.295370i
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 3.58579 0.274212
\(172\) 0.627417i 0.0478401i
\(173\) − 18.5563i − 1.41081i −0.708803 0.705407i \(-0.750764\pi\)
0.708803 0.705407i \(-0.249236\pi\)
\(174\) 2.82843 0.214423
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 11.0000i 0.826811i
\(178\) − 3.17157i − 0.237719i
\(179\) 22.7990 1.70408 0.852038 0.523480i \(-0.175366\pi\)
0.852038 + 0.523480i \(0.175366\pi\)
\(180\) 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
\(183\) − 3.17157i − 0.234449i
\(184\) −1.58579 −0.116906
\(185\) 0 0
\(186\) −3.51472 −0.257712
\(187\) 0.414214i 0.0302903i
\(188\) 17.3431i 1.26488i
\(189\) 2.41421 0.175608
\(190\) 0 0
\(191\) 11.8284 0.855875 0.427937 0.903808i \(-0.359240\pi\)
0.427937 + 0.903808i \(0.359240\pi\)
\(192\) − 4.17157i − 0.301057i
\(193\) 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 \(0.156377\pi\)
−0.881733 + 0.471750i \(0.843623\pi\)
\(198\) − 0.414214i − 0.0294369i
\(199\) −5.17157 −0.366603 −0.183302 0.983057i \(-0.558678\pi\)
−0.183302 + 0.983057i \(0.558678\pi\)
\(200\) 0 0
\(201\) −11.6569 −0.822211
\(202\) 2.02944i 0.142791i
\(203\) 16.4853i 1.15704i
\(204\) −0.757359 −0.0530258
\(205\) 0 0
\(206\) 0.970563 0.0676223
\(207\) 1.00000i 0.0695048i
\(208\) − 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
\(213\) − 2.17157i − 0.148794i
\(214\) 7.17157 0.490239
\(215\) 0 0
\(216\) 1.58579 0.107899
\(217\) − 20.4853i − 1.39063i
\(218\) − 7.17157i − 0.485720i
\(219\) 3.17157 0.214315
\(220\) 0 0
\(221\) −1.17157 −0.0788085
\(222\) 2.41421i 0.162031i
\(223\) 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 \(-0.983993\pi\)
0.998736 0.0502677i \(-0.0160075\pi\)
\(228\) 6.55635i 0.434205i
\(229\) 19.4853 1.28762 0.643812 0.765184i \(-0.277352\pi\)
0.643812 + 0.765184i \(0.277352\pi\)
\(230\) 0 0
\(231\) 2.41421 0.158844
\(232\) 10.8284i 0.710921i
\(233\) 14.5563i 0.953618i 0.879007 + 0.476809i \(0.158207\pi\)
−0.879007 + 0.476809i \(0.841793\pi\)
\(234\) 1.17157 0.0765881
\(235\) 0 0
\(236\) −20.1127 −1.30923
\(237\) 4.75736i 0.309024i
\(238\) 0.414214i 0.0268495i
\(239\) −12.3431 −0.798412 −0.399206 0.916861i \(-0.630714\pi\)
−0.399206 + 0.916861i \(0.630714\pi\)
\(240\) 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
\(243\) − 1.00000i − 0.0641500i
\(244\) 5.79899 0.371242
\(245\) 0 0
\(246\) −3.68629 −0.235029
\(247\) 10.1421i 0.645329i
\(248\) − 13.4558i − 0.854447i
\(249\) −12.4853 −0.791223
\(250\) 0 0
\(251\) −24.9706 −1.57613 −0.788064 0.615593i \(-0.788916\pi\)
−0.788064 + 0.615593i \(0.788916\pi\)
\(252\) 4.41421i 0.278069i
\(253\) 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 \(-0.863696\pi\)
0.909710 0.415243i \(-0.136304\pi\)
\(258\) − 0.142136i − 0.00884898i
\(259\) −14.0711 −0.874334
\(260\) 0 0
\(261\) 6.82843 0.422669
\(262\) − 0.485281i − 0.0299808i
\(263\) 10.9706i 0.676474i 0.941061 + 0.338237i \(0.109831\pi\)
−0.941061 + 0.338237i \(0.890169\pi\)
\(264\) 1.58579 0.0975984
\(265\) 0 0
\(266\) 3.58579 0.219859
\(267\) − 7.65685i − 0.468592i
\(268\) − 21.3137i − 1.30194i
\(269\) −23.7990 −1.45105 −0.725525 0.688196i \(-0.758403\pi\)
−0.725525 + 0.688196i \(0.758403\pi\)
\(270\) 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
\(273\) 6.82843i 0.413275i
\(274\) 6.68629 0.403934
\(275\) 0 0
\(276\) −1.82843 −0.110058
\(277\) − 0.828427i − 0.0497754i −0.999690 0.0248877i \(-0.992077\pi\)
0.999690 0.0248877i \(-0.00792281\pi\)
\(278\) 6.20101i 0.371912i
\(279\) −8.48528 −0.508001
\(280\) 0 0
\(281\) 17.9289 1.06955 0.534775 0.844994i \(-0.320396\pi\)
0.534775 + 0.844994i \(0.320396\pi\)
\(282\) − 3.92893i − 0.233965i
\(283\) − 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
\(288\) 4.41421i 0.260110i
\(289\) 16.8284 0.989907
\(290\) 0 0
\(291\) 0.171573 0.0100578
\(292\) 5.79899i 0.339360i
\(293\) − 20.4142i − 1.19261i −0.802758 0.596306i \(-0.796635\pi\)
0.802758 0.596306i \(-0.203365\pi\)
\(294\) −0.485281 −0.0283022
\(295\) 0 0
\(296\) −9.24264 −0.537218
\(297\) − 1.00000i − 0.0580259i
\(298\) − 7.34315i − 0.425377i
\(299\) −2.82843 −0.163572
\(300\) 0 0
\(301\) 0.828427 0.0477497
\(302\) − 5.79899i − 0.333694i
\(303\) 4.89949i 0.281469i
\(304\) −10.7574 −0.616977
\(305\) 0 0
\(306\) 0.171573 0.00980817
\(307\) 29.3137i 1.67302i 0.547950 + 0.836511i \(0.315408\pi\)
−0.547950 + 0.836511i \(0.684592\pi\)
\(308\) 4.41421i 0.251523i
\(309\) 2.34315 0.133297
\(310\) 0 0
\(311\) 2.34315 0.132868 0.0664338 0.997791i \(-0.478838\pi\)
0.0664338 + 0.997791i \(0.478838\pi\)
\(312\) 4.48528i 0.253929i
\(313\) − 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 \(0.249902\pi\)
−0.707325 + 0.706888i \(0.750098\pi\)
\(318\) − 1.51472i − 0.0849412i
\(319\) 6.82843 0.382319
\(320\) 0 0
\(321\) 17.3137 0.966357
\(322\) 1.00000i 0.0557278i
\(323\) 1.48528i 0.0826433i
\(324\) 1.82843 0.101579
\(325\) 0 0
\(326\) 9.85786 0.545977
\(327\) − 17.3137i − 0.957450i
\(328\) − 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
\(333\) 5.82843i 0.319396i
\(334\) −7.37258 −0.403410
\(335\) 0 0
\(336\) −7.24264 −0.395118
\(337\) − 24.1421i − 1.31511i −0.753408 0.657553i \(-0.771592\pi\)
0.753408 0.657553i \(-0.228408\pi\)
\(338\) − 2.07107i − 0.112651i
\(339\) 10.0000 0.543125
\(340\) 0 0
\(341\) −8.48528 −0.459504
\(342\) − 1.48528i − 0.0803148i
\(343\) − 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 \(-0.744090\pi\)
0.693857 0.720113i \(-0.255910\pi\)
\(348\) 12.4853i 0.669281i
\(349\) −14.4853 −0.775379 −0.387690 0.921790i \(-0.626727\pi\)
−0.387690 + 0.921790i \(0.626727\pi\)
\(350\) 0 0
\(351\) 2.82843 0.150970
\(352\) 4.41421i 0.235278i
\(353\) − 12.4853i − 0.664524i −0.943187 0.332262i \(-0.892188\pi\)
0.943187 0.332262i \(-0.107812\pi\)
\(354\) 4.55635 0.242167
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) 1.00000i 0.0529256i
\(358\) − 9.44365i − 0.499112i
\(359\) −32.4853 −1.71451 −0.857254 0.514894i \(-0.827831\pi\)
−0.857254 + 0.514894i \(0.827831\pi\)
\(360\) 0 0
\(361\) −6.14214 −0.323270
\(362\) − 4.95837i − 0.260606i
\(363\) − 1.00000i − 0.0524864i
\(364\) −12.4853 −0.654407
\(365\) 0 0
\(366\) −1.31371 −0.0686686
\(367\) − 21.3137i − 1.11257i −0.830993 0.556283i \(-0.812227\pi\)
0.830993 0.556283i \(-0.187773\pi\)
\(368\) − 3.00000i − 0.156386i
\(369\) −8.89949 −0.463289
\(370\) 0 0
\(371\) 8.82843 0.458349
\(372\) − 15.5147i − 0.804401i
\(373\) 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
\(378\) − 1.00000i − 0.0514344i
\(379\) −14.8284 −0.761685 −0.380843 0.924640i \(-0.624366\pi\)
−0.380843 + 0.924640i \(0.624366\pi\)
\(380\) 0 0
\(381\) −1.24264 −0.0636624
\(382\) − 4.89949i − 0.250680i
\(383\) − 20.0000i − 1.02195i −0.859595 0.510976i \(-0.829284\pi\)
0.859595 0.510976i \(-0.170716\pi\)
\(384\) −10.5563 −0.538701
\(385\) 0 0
\(386\) 8.00000 0.407189
\(387\) − 0.343146i − 0.0174431i
\(388\) 0.313708i 0.0159261i
\(389\) 6.34315 0.321610 0.160805 0.986986i \(-0.448591\pi\)
0.160805 + 0.986986i \(0.448591\pi\)
\(390\) 0 0
\(391\) −0.414214 −0.0209477
\(392\) − 1.85786i − 0.0938363i
\(393\) − 1.17157i − 0.0590980i
\(394\) 5.48528 0.276344
\(395\) 0 0
\(396\) 1.82843 0.0918819
\(397\) 31.9411i 1.60308i 0.597942 + 0.801540i \(0.295985\pi\)
−0.597942 + 0.801540i \(0.704015\pi\)
\(398\) 2.14214i 0.107376i
\(399\) 8.65685 0.433385
\(400\) 0 0
\(401\) −7.79899 −0.389463 −0.194731 0.980857i \(-0.562384\pi\)
−0.194731 + 0.980857i \(0.562384\pi\)
\(402\) 4.82843i 0.240820i
\(403\) − 24.0000i − 1.19553i
\(404\) −8.95837 −0.445696
\(405\) 0 0
\(406\) 6.82843 0.338889
\(407\) 5.82843i 0.288904i
\(408\) 0.656854i 0.0325191i
\(409\) −24.1421 −1.19375 −0.596876 0.802334i \(-0.703592\pi\)
−0.596876 + 0.802334i \(0.703592\pi\)
\(410\) 0 0
\(411\) 16.1421 0.796233
\(412\) 4.28427i 0.211071i
\(413\) 26.5563i 1.30675i
\(414\) 0.414214 0.0203575
\(415\) 0 0
\(416\) −12.4853 −0.612141
\(417\) 14.9706i 0.733112i
\(418\) − 1.48528i − 0.0726475i
\(419\) 8.51472 0.415971 0.207986 0.978132i \(-0.433309\pi\)
0.207986 + 0.978132i \(0.433309\pi\)
\(420\) 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
\(423\) − 9.48528i − 0.461190i
\(424\) 5.79899 0.281624
\(425\) 0 0
\(426\) −0.899495 −0.0435807
\(427\) − 7.65685i − 0.370541i
\(428\) 31.6569i 1.53019i
\(429\) 2.82843 0.136558
\(430\) 0 0
\(431\) 6.82843 0.328914 0.164457 0.986384i \(-0.447413\pi\)
0.164457 + 0.986384i \(0.447413\pi\)
\(432\) 3.00000i 0.144338i
\(433\) − 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
\(438\) − 1.31371i − 0.0627714i
\(439\) −27.7279 −1.32338 −0.661691 0.749777i \(-0.730161\pi\)
−0.661691 + 0.749777i \(0.730161\pi\)
\(440\) 0 0
\(441\) −1.17157 −0.0557892
\(442\) 0.485281i 0.0230825i
\(443\) − 25.9706i − 1.23390i −0.787003 0.616949i \(-0.788368\pi\)
0.787003 0.616949i \(-0.211632\pi\)
\(444\) −10.6569 −0.505752
\(445\) 0 0
\(446\) 11.1127 0.526202
\(447\) − 17.7279i − 0.838502i
\(448\) − 10.0711i − 0.475813i
\(449\) −10.4853 −0.494831 −0.247416 0.968909i \(-0.579581\pi\)
−0.247416 + 0.968909i \(0.579581\pi\)
\(450\) 0 0
\(451\) −8.89949 −0.419061
\(452\) 18.2843i 0.860020i
\(453\) − 14.0000i − 0.657777i
\(454\) −0.627417 −0.0294461
\(455\) 0 0
\(456\) 5.68629 0.266285
\(457\) 32.1421i 1.50355i 0.659422 + 0.751773i \(0.270801\pi\)
−0.659422 + 0.751773i \(0.729199\pi\)
\(458\) − 8.07107i − 0.377136i
\(459\) 0.414214 0.0193338
\(460\) 0 0
\(461\) 40.7696 1.89883 0.949414 0.314028i \(-0.101679\pi\)
0.949414 + 0.314028i \(0.101679\pi\)
\(462\) − 1.00000i − 0.0465242i
\(463\) − 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 \(-0.704204\pi\)
0.598420 0.801183i \(-0.295796\pi\)
\(468\) 5.17157i 0.239056i
\(469\) −28.1421 −1.29948
\(470\) 0 0
\(471\) 6.00000 0.276465
\(472\) 17.4437i 0.802909i
\(473\) − 0.343146i − 0.0157779i
\(474\) 1.97056 0.0905109
\(475\) 0 0
\(476\) −1.82843 −0.0838058
\(477\) − 3.65685i − 0.167436i
\(478\) 5.11270i 0.233849i
\(479\) 7.51472 0.343356 0.171678 0.985153i \(-0.445081\pi\)
0.171678 + 0.985153i \(0.445081\pi\)
\(480\) 0 0
\(481\) −16.4853 −0.751664
\(482\) 5.85786i 0.266818i
\(483\) 2.41421i 0.109851i
\(484\) 1.82843 0.0831103
\(485\) 0 0
\(486\) −0.414214 −0.0187891
\(487\) 10.4853i 0.475133i 0.971371 + 0.237567i \(0.0763498\pi\)
−0.971371 + 0.237567i \(0.923650\pi\)
\(488\) − 5.02944i − 0.227672i
\(489\) 23.7990 1.07623
\(490\) 0 0
\(491\) 4.14214 0.186932 0.0934660 0.995622i \(-0.470205\pi\)
0.0934660 + 0.995622i \(0.470205\pi\)
\(492\) − 16.2721i − 0.733602i
\(493\) 2.82843i 0.127386i
\(494\) 4.20101 0.189012
\(495\) 0 0
\(496\) 25.4558 1.14300
\(497\) − 5.24264i − 0.235165i
\(498\) 5.17157i 0.231744i
\(499\) −40.8284 −1.82773 −0.913866 0.406017i \(-0.866918\pi\)
−0.913866 + 0.406017i \(0.866918\pi\)
\(500\) 0 0
\(501\) −17.7990 −0.795200
\(502\) 10.3431i 0.461637i
\(503\) − 22.2843i − 0.993607i −0.867863 0.496803i \(-0.834507\pi\)
0.867863 0.496803i \(-0.165493\pi\)
\(504\) 3.82843 0.170532
\(505\) 0 0
\(506\) 0.414214 0.0184140
\(507\) − 5.00000i − 0.222058i
\(508\) − 2.27208i − 0.100807i
\(509\) 40.6274 1.80078 0.900389 0.435085i \(-0.143282\pi\)
0.900389 + 0.435085i \(0.143282\pi\)
\(510\) 0 0
\(511\) 7.65685 0.338719
\(512\) − 22.7574i − 1.00574i
\(513\) − 3.58579i − 0.158316i
\(514\) −5.51472 −0.243244
\(515\) 0 0
\(516\) 0.627417 0.0276205
\(517\) − 9.48528i − 0.417162i
\(518\) 5.82843i 0.256086i
\(519\) −18.5563 −0.814533
\(520\) 0 0
\(521\) −7.85786 −0.344259 −0.172130 0.985074i \(-0.555065\pi\)
−0.172130 + 0.985074i \(0.555065\pi\)
\(522\) − 2.82843i − 0.123797i
\(523\) 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
\(528\) 3.00000i 0.130558i
\(529\) 22.0000 0.956522
\(530\) 0 0
\(531\) 11.0000 0.477359
\(532\) 15.8284i 0.686249i
\(533\) − 25.1716i − 1.09030i
\(534\) −3.17157 −0.137247
\(535\) 0 0
\(536\) −18.4853 −0.798443
\(537\) − 22.7990i − 0.983849i
\(538\) 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
\(543\) − 11.9706i − 0.513706i
\(544\) −1.82843 −0.0783932
\(545\) 0 0
\(546\) 2.82843 0.121046
\(547\) − 17.8701i − 0.764068i −0.924148 0.382034i \(-0.875224\pi\)
0.924148 0.382034i \(-0.124776\pi\)
\(548\) 29.5147i 1.26081i
\(549\) −3.17157 −0.135359
\(550\) 0 0
\(551\) 24.4853 1.04311
\(552\) 1.58579i 0.0674956i
\(553\) 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 \(-0.926321\pi\)
0.973330 0.229408i \(-0.0736789\pi\)
\(558\) 3.51472i 0.148790i
\(559\) 0.970563 0.0410504
\(560\) 0 0
\(561\) 0.414214 0.0174881
\(562\) − 7.42641i − 0.313264i
\(563\) 7.31371i 0.308236i 0.988052 + 0.154118i \(0.0492536\pi\)
−0.988052 + 0.154118i \(0.950746\pi\)
\(564\) 17.3431 0.730278
\(565\) 0 0
\(566\) −7.82843 −0.329053
\(567\) − 2.41421i − 0.101387i
\(568\) − 3.44365i − 0.144492i
\(569\) −6.75736 −0.283283 −0.141642 0.989918i \(-0.545238\pi\)
−0.141642 + 0.989918i \(0.545238\pi\)
\(570\) 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
\(573\) − 11.8284i − 0.494140i
\(574\) −8.89949 −0.371458
\(575\) 0 0
\(576\) −4.17157 −0.173816
\(577\) − 9.97056i − 0.415080i −0.978227 0.207540i \(-0.933454\pi\)
0.978227 0.207540i \(-0.0665457\pi\)
\(578\) − 6.97056i − 0.289937i
\(579\) 19.3137 0.802650
\(580\) 0 0
\(581\) −30.1421 −1.25051
\(582\) − 0.0710678i − 0.00294586i
\(583\) − 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 \(-0.824808\pi\)
0.852325 0.523012i \(-0.175192\pi\)
\(588\) − 2.14214i − 0.0883402i
\(589\) −30.4264 −1.25370
\(590\) 0 0
\(591\) 13.2426 0.544729
\(592\) − 17.4853i − 0.718641i
\(593\) 35.7990i 1.47009i 0.678019 + 0.735044i \(0.262839\pi\)
−0.678019 + 0.735044i \(0.737161\pi\)
\(594\) −0.414214 −0.0169954
\(595\) 0 0
\(596\) 32.4142 1.32774
\(597\) 5.17157i 0.211658i
\(598\) 1.17157i 0.0479092i
\(599\) −13.6863 −0.559207 −0.279603 0.960116i \(-0.590203\pi\)
−0.279603 + 0.960116i \(0.590203\pi\)
\(600\) 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
\(603\) 11.6569i 0.474704i
\(604\) 25.5980 1.04157
\(605\) 0 0
\(606\) 2.02944 0.0824403
\(607\) 30.9706i 1.25706i 0.777787 + 0.628528i \(0.216342\pi\)
−0.777787 + 0.628528i \(0.783658\pi\)
\(608\) 15.8284i 0.641927i
\(609\) 16.4853 0.668017
\(610\) 0 0
\(611\) 26.8284 1.08536
\(612\) 0.757359i 0.0306144i
\(613\) 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 \(-0.721365\pi\)
0.640723 0.767772i \(-0.278635\pi\)
\(618\) − 0.970563i − 0.0390418i
\(619\) 6.62742 0.266378 0.133189 0.991091i \(-0.457478\pi\)
0.133189 + 0.991091i \(0.457478\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) − 0.970563i − 0.0389160i
\(623\) − 18.4853i − 0.740597i
\(624\) −8.48528 −0.339683
\(625\) 0 0
\(626\) −0.473088 −0.0189084
\(627\) − 3.58579i − 0.143203i
\(628\) 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
\(633\) − 9.31371i − 0.370187i
\(634\) 10.4264 0.414086
\(635\) 0 0
\(636\) 6.68629 0.265129
\(637\) − 3.31371i − 0.131294i
\(638\) − 2.82843i − 0.111979i
\(639\) −2.17157 −0.0859061
\(640\) 0 0
\(641\) −25.5147 −1.00777 −0.503885 0.863771i \(-0.668097\pi\)
−0.503885 + 0.863771i \(0.668097\pi\)
\(642\) − 7.17157i − 0.283039i
\(643\) − 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 \(0.190472\pi\)
−0.826247 + 0.563309i \(0.809528\pi\)
\(648\) − 1.58579i − 0.0622956i
\(649\) 11.0000 0.431788
\(650\) 0 0
\(651\) −20.4853 −0.802881
\(652\) 43.5147i 1.70417i
\(653\) 23.5147i 0.920202i 0.887867 + 0.460101i \(0.152187\pi\)
−0.887867 + 0.460101i \(0.847813\pi\)
\(654\) −7.17157 −0.280431
\(655\) 0 0
\(656\) 26.6985 1.04240
\(657\) − 3.17157i − 0.123735i
\(658\) − 9.48528i − 0.369775i
\(659\) 47.1127 1.83525 0.917625 0.397447i \(-0.130104\pi\)
0.917625 + 0.397447i \(0.130104\pi\)
\(660\) 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
\(663\) 1.17157i 0.0455001i
\(664\) −19.7990 −0.768350
\(665\) 0 0
\(666\) 2.41421 0.0935489
\(667\) 6.82843i 0.264398i
\(668\) − 32.5442i − 1.25917i
\(669\) 26.8284 1.03725
\(670\) 0 0
\(671\) −3.17157 −0.122437
\(672\) 10.6569i 0.411097i
\(673\) − 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 \(0.149244\pi\)
−0.892083 + 0.451872i \(0.850756\pi\)
\(678\) − 4.14214i − 0.159078i
\(679\) 0.414214 0.0158961
\(680\) 0 0
\(681\) −1.51472 −0.0580441
\(682\) 3.51472i 0.134586i
\(683\) − 12.5147i − 0.478862i −0.970913 0.239431i \(-0.923039\pi\)
0.970913 0.239431i \(-0.0769610\pi\)
\(684\) 6.55635 0.250688
\(685\) 0 0
\(686\) −8.17157 −0.311992
\(687\) − 19.4853i − 0.743410i
\(688\) 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
\(693\) − 2.41421i − 0.0917084i
\(694\) −11.1127 −0.421832
\(695\) 0 0
\(696\) 10.8284 0.410450
\(697\) − 3.68629i − 0.139628i
\(698\) 6.00000i 0.227103i
\(699\) 14.5563 0.550572
\(700\) 0 0
\(701\) −41.3848 −1.56308 −0.781541 0.623854i \(-0.785566\pi\)
−0.781541 + 0.623854i \(0.785566\pi\)
\(702\) − 1.17157i − 0.0442182i
\(703\) 20.8995i 0.788239i
\(704\) −4.17157 −0.157222
\(705\) 0 0
\(706\) −5.17157 −0.194635
\(707\) 11.8284i 0.444854i
\(708\) 20.1127i 0.755881i
\(709\) 29.1421 1.09446 0.547228 0.836984i \(-0.315683\pi\)
0.547228 + 0.836984i \(0.315683\pi\)
\(710\) 0 0
\(711\) 4.75736 0.178415
\(712\) − 12.1421i − 0.455046i
\(713\) − 8.48528i − 0.317776i
\(714\) 0.414214 0.0155016
\(715\) 0 0
\(716\) 41.6863 1.55789
\(717\) 12.3431i 0.460963i
\(718\) 13.4558i 0.502168i
\(719\) −9.65685 −0.360140 −0.180070 0.983654i \(-0.557632\pi\)
−0.180070 + 0.983654i \(0.557632\pi\)
\(720\) 0 0
\(721\) 5.65685 0.210672
\(722\) 2.54416i 0.0946837i
\(723\) 14.1421i 0.525952i
\(724\) 21.8873 0.813435
\(725\) 0 0
\(726\) −0.414214 −0.0153729
\(727\) 16.9706i 0.629403i 0.949191 + 0.314702i \(0.101904\pi\)
−0.949191 + 0.314702i \(0.898096\pi\)
\(728\) 10.8284i 0.401328i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0.142136 0.00525708
\(732\) − 5.79899i − 0.214337i
\(733\) − 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
\(738\) 3.68629i 0.135694i
\(739\) −20.4142 −0.750949 −0.375474 0.926833i \(-0.622520\pi\)
−0.375474 + 0.926833i \(0.622520\pi\)
\(740\) 0 0
\(741\) 10.1421 0.372581
\(742\) − 3.65685i − 0.134247i
\(743\) 31.1127i 1.14141i 0.821154 + 0.570707i \(0.193331\pi\)
−0.821154 + 0.570707i \(0.806669\pi\)
\(744\) −13.4558 −0.493315
\(745\) 0 0
\(746\) 5.11270 0.187189
\(747\) 12.4853i 0.456813i
\(748\) 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
\(753\) 24.9706i 0.909978i
\(754\) 8.00000 0.291343
\(755\) 0 0
\(756\) 4.41421 0.160543
\(757\) − 10.6863i − 0.388400i −0.980962 0.194200i \(-0.937789\pi\)
0.980962 0.194200i \(-0.0622111\pi\)
\(758\) 6.14214i 0.223092i
\(759\) 1.00000 0.0362977
\(760\) 0 0
\(761\) −5.17157 −0.187469 −0.0937347 0.995597i \(-0.529881\pi\)
−0.0937347 + 0.995597i \(0.529881\pi\)
\(762\) 0.514719i 0.0186463i
\(763\) − 41.7990i − 1.51323i
\(764\) 21.6274 0.782452
\(765\) 0 0
\(766\) −8.28427 −0.299323
\(767\) 31.1127i 1.12341i
\(768\) − 3.97056i − 0.143275i
\(769\) 7.65685 0.276113 0.138057 0.990424i \(-0.455914\pi\)
0.138057 + 0.990424i \(0.455914\pi\)
\(770\) 0 0
\(771\) −13.3137 −0.479481
\(772\) 35.3137i 1.27097i
\(773\) 0.828427i 0.0297965i 0.999889 + 0.0148982i \(0.00474243\pi\)
−0.999889 + 0.0148982i \(0.995258\pi\)
\(774\) −0.142136 −0.00510896
\(775\) 0 0
\(776\) 0.272078 0.00976703
\(777\) 14.0711i 0.504797i
\(778\) − 2.62742i − 0.0941975i
\(779\) −31.9117 −1.14335
\(780\) 0 0
\(781\) −2.17157 −0.0777050
\(782\) 0.171573i 0.00613543i
\(783\) − 6.82843i − 0.244028i
\(784\) 3.51472 0.125526
\(785\) 0 0
\(786\) −0.485281 −0.0173094
\(787\) − 7.92893i − 0.282636i −0.989964 0.141318i \(-0.954866\pi\)
0.989964 0.141318i \(-0.0451340\pi\)
\(788\) 24.2132i 0.862560i
\(789\) 10.9706 0.390562
\(790\) 0 0
\(791\) 24.1421 0.858396
\(792\) − 1.58579i − 0.0563485i
\(793\) − 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 \(0.258450\pi\)
−0.688089 + 0.725626i \(0.741550\pi\)
\(798\) − 3.58579i − 0.126935i
\(799\) 3.92893 0.138996
\(800\) 0 0
\(801\) −7.65685 −0.270542
\(802\) 3.23045i 0.114071i
\(803\) − 3.17157i − 0.111922i
\(804\) −21.3137 −0.751677
\(805\) 0 0
\(806\) −9.94113 −0.350161
\(807\) 23.7990i 0.837764i
\(808\) 7.76955i 0.273332i
\(809\) −7.72792 −0.271699 −0.135850 0.990729i \(-0.543376\pi\)
−0.135850 + 0.990729i \(0.543376\pi\)
\(810\) 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
\(813\) 10.8995i 0.382262i
\(814\) 2.41421 0.0846181
\(815\) 0 0
\(816\) −1.24264 −0.0435011
\(817\) − 1.23045i − 0.0430479i
\(818\) 10.0000i 0.349642i
\(819\) 6.82843 0.238605
\(820\) 0 0
\(821\) 15.7990 0.551389 0.275694 0.961245i \(-0.411092\pi\)
0.275694 + 0.961245i \(0.411092\pi\)
\(822\) − 6.68629i − 0.233211i
\(823\) 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 \(-0.929207\pi\)
0.975371 0.220573i \(-0.0707927\pi\)
\(828\) 1.82843i 0.0635422i
\(829\) 47.9411 1.66506 0.832532 0.553977i \(-0.186890\pi\)
0.832532 + 0.553977i \(0.186890\pi\)
\(830\) 0 0
\(831\) −0.828427 −0.0287378
\(832\) − 11.7990i − 0.409056i
\(833\) − 0.485281i − 0.0168140i
\(834\) 6.20101 0.214723
\(835\) 0 0
\(836\) 6.55635 0.226756
\(837\) 8.48528i 0.293294i
\(838\) − 3.52691i − 0.121835i
\(839\) 47.3137 1.63345 0.816725 0.577027i \(-0.195787\pi\)
0.816725 + 0.577027i \(0.195787\pi\)
\(840\) 0 0
\(841\) 17.6274 0.607842
\(842\) 11.1838i 0.385418i
\(843\) − 17.9289i − 0.617505i
\(844\) 17.0294 0.586177
\(845\) 0 0
\(846\) −3.92893 −0.135079
\(847\) − 2.41421i − 0.0829534i
\(848\) 10.9706i 0.376731i
\(849\) −18.8995 −0.648629
\(850\) 0 0
\(851\) −5.82843 −0.199796
\(852\) − 3.97056i − 0.136029i
\(853\) 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 \(-0.784365\pi\)
0.779182 0.626798i \(-0.215635\pi\)
\(858\) − 1.17157i − 0.0399968i
\(859\) 20.4853 0.698949 0.349474 0.936946i \(-0.386360\pi\)
0.349474 + 0.936946i \(0.386360\pi\)
\(860\) 0 0
\(861\) −21.4853 −0.732216
\(862\) − 2.82843i − 0.0963366i
\(863\) 28.6863i 0.976493i 0.872706 + 0.488246i \(0.162363\pi\)
−0.872706 + 0.488246i \(0.837637\pi\)
\(864\) 4.41421 0.150175
\(865\) 0 0
\(866\) −3.85786 −0.131096
\(867\) − 16.8284i − 0.571523i
\(868\) − 37.4558i − 1.27133i
\(869\) 4.75736 0.161382
\(870\) 0 0
\(871\) −32.9706 −1.11716
\(872\) − 27.4558i − 0.929772i
\(873\) − 0.171573i − 0.00580686i
\(874\) 1.48528 0.0502404
\(875\) 0 0
\(876\) 5.79899 0.195930
\(877\) 15.1127i 0.510320i 0.966899 + 0.255160i \(0.0821281\pi\)
−0.966899 + 0.255160i \(0.917872\pi\)
\(878\) 11.4853i 0.387609i
\(879\) −20.4142 −0.688554
\(880\) 0 0
\(881\) −24.0000 −0.808581 −0.404290 0.914631i \(-0.632481\pi\)
−0.404290 + 0.914631i \(0.632481\pi\)
\(882\) 0.485281i 0.0163403i
\(883\) − 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 \(-0.878765\pi\)
0.928341 0.371730i \(-0.121235\pi\)
\(888\) 9.24264i 0.310163i
\(889\) −3.00000 −0.100617
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 49.0538i 1.64244i
\(893\) − 34.0122i − 1.13817i
\(894\) −7.34315 −0.245592
\(895\) 0 0
\(896\) −25.4853 −0.851403
\(897\) 2.82843i 0.0944384i
\(898\) 4.34315i 0.144933i
\(899\) −57.9411 −1.93244
\(900\) 0 0
\(901\) 1.51472 0.0504626
\(902\) 3.68629i 0.122740i
\(903\) − 0.828427i − 0.0275683i
\(904\) 15.8579 0.527425
\(905\) 0 0
\(906\) −5.79899 −0.192659
\(907\) − 2.48528i − 0.0825224i −0.999148 0.0412612i \(-0.986862\pi\)
0.999148 0.0412612i \(-0.0131376\pi\)
\(908\) − 2.76955i − 0.0919108i
\(909\) 4.89949 0.162506
\(910\) 0 0
\(911\) 28.5147 0.944735 0.472367 0.881402i \(-0.343400\pi\)
0.472367 + 0.881402i \(0.343400\pi\)
\(912\) 10.7574i 0.356212i
\(913\) 12.4853i 0.413203i
\(914\) 13.3137 0.440378
\(915\) 0 0
\(916\) 35.6274 1.17716
\(917\) − 2.82843i − 0.0934029i
\(918\) − 0.171573i − 0.00566275i
\(919\) −1.78680 −0.0589410 −0.0294705 0.999566i \(-0.509382\pi\)
−0.0294705 + 0.999566i \(0.509382\pi\)
\(920\) 0 0
\(921\) 29.3137 0.965920
\(922\) − 16.8873i − 0.556154i
\(923\) − 6.14214i − 0.202171i
\(924\) 4.41421 0.145217
\(925\) 0 0
\(926\) −0.426407 −0.0140126
\(927\) − 2.34315i − 0.0769590i
\(928\) 30.1421i 0.989464i
\(929\) 13.7990 0.452730 0.226365 0.974043i \(-0.427316\pi\)
0.226365 + 0.974043i \(0.427316\pi\)
\(930\) 0 0
\(931\) −4.20101 −0.137683
\(932\) 26.6152i 0.871811i
\(933\) − 2.34315i − 0.0767111i
\(934\) −14.3431 −0.469322
\(935\) 0 0
\(936\) 4.48528 0.146606
\(937\) − 16.0000i − 0.522697i −0.965244 0.261349i \(-0.915833\pi\)
0.965244 0.261349i \(-0.0841672\pi\)
\(938\) 11.6569i 0.380610i
\(939\) −1.14214 −0.0372722
\(940\) 0 0
\(941\) −22.8995 −0.746502 −0.373251 0.927730i \(-0.621757\pi\)
−0.373251 + 0.927730i \(0.621757\pi\)
\(942\) − 2.48528i − 0.0809748i
\(943\) − 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 \(-0.796001\pi\)
0.801568 0.597903i \(-0.203999\pi\)
\(948\) 8.69848i 0.282514i
\(949\) 8.97056 0.291197
\(950\) 0 0
\(951\) 25.1716 0.816244
\(952\) 1.58579i 0.0513956i
\(953\) 29.0416i 0.940751i 0.882466 + 0.470375i \(0.155881\pi\)
−0.882466 + 0.470375i \(0.844119\pi\)
\(954\) −1.51472 −0.0490408
\(955\) 0 0
\(956\) −22.5685 −0.729919
\(957\) − 6.82843i − 0.220732i
\(958\) − 3.11270i − 0.100567i
\(959\) 38.9706 1.25843
\(960\) 0 0
\(961\) 41.0000 1.32258
\(962\) 6.82843i 0.220157i
\(963\) − 17.3137i − 0.557926i
\(964\) −25.8579 −0.832826
\(965\) 0 0
\(966\) 1.00000 0.0321745
\(967\) 38.0000i 1.22200i 0.791632 + 0.610999i \(0.209232\pi\)
−0.791632 + 0.610999i \(0.790768\pi\)
\(968\) − 1.58579i − 0.0509691i
\(969\) 1.48528 0.0477141
\(970\) 0 0
\(971\) −50.3137 −1.61464 −0.807322 0.590111i \(-0.799084\pi\)
−0.807322 + 0.590111i \(0.799084\pi\)
\(972\) − 1.82843i − 0.0586468i
\(973\) 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 \(0.420380\pi\)
−0.247532 + 0.968880i \(0.579620\pi\)
\(978\) − 9.85786i − 0.315220i
\(979\) −7.65685 −0.244714
\(980\) 0 0
\(981\) −17.3137 −0.552784
\(982\) − 1.71573i − 0.0547511i
\(983\) − 51.2843i − 1.63571i −0.575421 0.817857i \(-0.695162\pi\)
0.575421 0.817857i \(-0.304838\pi\)
\(984\) −14.1127 −0.449896
\(985\) 0 0
\(986\) 1.17157 0.0373105
\(987\) − 22.8995i − 0.728899i
\(988\) 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
\(993\) 3.85786i 0.122426i
\(994\) −2.17157 −0.0688781
\(995\) 0 0
\(996\) −22.8284 −0.723346
\(997\) − 47.2548i − 1.49658i −0.663374 0.748288i \(-0.730876\pi\)
0.663374 0.748288i \(-0.269124\pi\)
\(998\) 16.9117i 0.535330i
\(999\) 5.82843 0.184403
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.d.199.2 4
3.2 odd 2 2475.2.c.o.199.3 4
5.2 odd 4 825.2.a.d.1.2 2
5.3 odd 4 825.2.a.f.1.1 yes 2
5.4 even 2 inner 825.2.c.d.199.3 4
15.2 even 4 2475.2.a.w.1.1 2
15.8 even 4 2475.2.a.l.1.2 2
15.14 odd 2 2475.2.c.o.199.2 4
55.32 even 4 9075.2.a.ca.1.1 2
55.43 even 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 5.2 odd 4
825.2.a.f.1.1 yes 2 5.3 odd 4
825.2.c.d.199.2 4 1.1 even 1 trivial
825.2.c.d.199.3 4 5.4 even 2 inner
2475.2.a.l.1.2 2 15.8 even 4
2475.2.a.w.1.1 2 15.2 even 4
2475.2.c.o.199.2 4 15.14 odd 2
2475.2.c.o.199.3 4 3.2 odd 2
9075.2.a.w.1.2 2 55.43 even 4
9075.2.a.ca.1.1 2 55.32 even 4