Properties

Label 825.2.c.f.199.5
Level $825$
Weight $2$
Character 825.199
Analytic conductor $6.588$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.5161984.1
Defining polynomial: \( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \) 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 199.5
Root \(1.32001 + 1.32001i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.2.c.f.199.2

$q$-expansion

\(f(q)\) \(=\) \(q+2.12489i q^{2} +1.00000i q^{3} -2.51514 q^{4} -2.12489 q^{6} +3.64002i q^{7} -1.09461i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+2.12489i q^{2} +1.00000i q^{3} -2.51514 q^{4} -2.12489 q^{6} +3.64002i q^{7} -1.09461i q^{8} -1.00000 q^{9} +1.00000 q^{11} -2.51514i q^{12} -1.51514i q^{13} -7.73463 q^{14} -2.70436 q^{16} +1.15516i q^{17} -2.12489i q^{18} -2.60975 q^{19} -3.64002 q^{21} +2.12489i q^{22} +5.73463i q^{23} +1.09461 q^{24} +3.21949 q^{26} -1.00000i q^{27} -9.15516i q^{28} -6.24977 q^{29} +5.51514 q^{31} -7.93567i q^{32} +1.00000i q^{33} -2.45459 q^{34} +2.51514 q^{36} +0.454586i q^{37} -5.54541i q^{38} +1.51514 q^{39} +4.12489 q^{41} -7.73463i q^{42} -11.7044i q^{43} -2.51514 q^{44} -12.1854 q^{46} -3.48486i q^{47} -2.70436i q^{48} -6.24977 q^{49} -1.15516 q^{51} +3.81078i q^{52} +12.5601i q^{53} +2.12489 q^{54} +3.98440 q^{56} -2.60975i q^{57} -13.2800i q^{58} +7.73463 q^{59} -12.0147 q^{61} +11.7190i q^{62} -3.64002i q^{63} +11.4537 q^{64} -2.12489 q^{66} -14.2645i q^{67} -2.90539i q^{68} -5.73463 q^{69} +8.51514 q^{71} +1.09461i q^{72} +9.21949i q^{73} -0.965943 q^{74} +6.56387 q^{76} +3.64002i q^{77} +3.21949i q^{78} -5.09461 q^{79} +1.00000 q^{81} +8.76491i q^{82} +14.7493i q^{83} +9.15516 q^{84} +24.8704 q^{86} -6.24977i q^{87} -1.09461i q^{88} +10.4995 q^{89} +5.51514 q^{91} -14.4234i q^{92} +5.51514i q^{93} +7.40493 q^{94} +7.93567 q^{96} -6.77959i q^{97} -13.2800i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 16 q^{4} + 4 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 16 q^{4} + 4 q^{6} - 6 q^{9} + 6 q^{11} - 12 q^{14} + 20 q^{16} + 2 q^{19} - 6 q^{21} - 12 q^{24} - 16 q^{26} - 4 q^{29} + 34 q^{31} - 12 q^{34} + 16 q^{36} + 10 q^{39} + 8 q^{41} - 16 q^{44} - 60 q^{46} - 4 q^{49} + 8 q^{51} - 4 q^{54} - 44 q^{56} + 12 q^{59} - 6 q^{61} - 68 q^{64} + 4 q^{66} + 52 q^{71} - 28 q^{74} - 48 q^{76} - 12 q^{79} + 6 q^{81} + 40 q^{84} + 56 q^{86} - 4 q^{89} + 34 q^{91} - 4 q^{94} + 68 q^{96} - 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.12489i 1.50252i 0.660006 + 0.751260i \(0.270554\pi\)
−0.660006 + 0.751260i \(0.729446\pi\)
\(3\) 1.00000i 0.577350i
\(4\) −2.51514 −1.25757
\(5\) 0 0
\(6\) −2.12489 −0.867481
\(7\) 3.64002i 1.37580i 0.725806 + 0.687900i \(0.241467\pi\)
−0.725806 + 0.687900i \(0.758533\pi\)
\(8\) − 1.09461i − 0.387003i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) − 2.51514i − 0.726058i
\(13\) − 1.51514i − 0.420224i −0.977677 0.210112i \(-0.932617\pi\)
0.977677 0.210112i \(-0.0673828\pi\)
\(14\) −7.73463 −2.06717
\(15\) 0 0
\(16\) −2.70436 −0.676089
\(17\) 1.15516i 0.280168i 0.990140 + 0.140084i \(0.0447372\pi\)
−0.990140 + 0.140084i \(0.955263\pi\)
\(18\) − 2.12489i − 0.500840i
\(19\) −2.60975 −0.598717 −0.299359 0.954141i \(-0.596773\pi\)
−0.299359 + 0.954141i \(0.596773\pi\)
\(20\) 0 0
\(21\) −3.64002 −0.794318
\(22\) 2.12489i 0.453027i
\(23\) 5.73463i 1.19575i 0.801588 + 0.597877i \(0.203989\pi\)
−0.801588 + 0.597877i \(0.796011\pi\)
\(24\) 1.09461 0.223436
\(25\) 0 0
\(26\) 3.21949 0.631395
\(27\) − 1.00000i − 0.192450i
\(28\) − 9.15516i − 1.73016i
\(29\) −6.24977 −1.16055 −0.580277 0.814419i \(-0.697056\pi\)
−0.580277 + 0.814419i \(0.697056\pi\)
\(30\) 0 0
\(31\) 5.51514 0.990548 0.495274 0.868737i \(-0.335068\pi\)
0.495274 + 0.868737i \(0.335068\pi\)
\(32\) − 7.93567i − 1.40284i
\(33\) 1.00000i 0.174078i
\(34\) −2.45459 −0.420958
\(35\) 0 0
\(36\) 2.51514 0.419190
\(37\) 0.454586i 0.0747335i 0.999302 + 0.0373667i \(0.0118970\pi\)
−0.999302 + 0.0373667i \(0.988103\pi\)
\(38\) − 5.54541i − 0.899585i
\(39\) 1.51514 0.242616
\(40\) 0 0
\(41\) 4.12489 0.644199 0.322099 0.946706i \(-0.395611\pi\)
0.322099 + 0.946706i \(0.395611\pi\)
\(42\) − 7.73463i − 1.19348i
\(43\) − 11.7044i − 1.78490i −0.451149 0.892449i \(-0.648986\pi\)
0.451149 0.892449i \(-0.351014\pi\)
\(44\) −2.51514 −0.379171
\(45\) 0 0
\(46\) −12.1854 −1.79664
\(47\) − 3.48486i − 0.508319i −0.967162 0.254160i \(-0.918201\pi\)
0.967162 0.254160i \(-0.0817989\pi\)
\(48\) − 2.70436i − 0.390340i
\(49\) −6.24977 −0.892824
\(50\) 0 0
\(51\) −1.15516 −0.161755
\(52\) 3.81078i 0.528460i
\(53\) 12.5601i 1.72526i 0.505834 + 0.862631i \(0.331185\pi\)
−0.505834 + 0.862631i \(0.668815\pi\)
\(54\) 2.12489 0.289160
\(55\) 0 0
\(56\) 3.98440 0.532438
\(57\) − 2.60975i − 0.345669i
\(58\) − 13.2800i − 1.74376i
\(59\) 7.73463 1.00696 0.503482 0.864006i \(-0.332052\pi\)
0.503482 + 0.864006i \(0.332052\pi\)
\(60\) 0 0
\(61\) −12.0147 −1.53832 −0.769161 0.639055i \(-0.779326\pi\)
−0.769161 + 0.639055i \(0.779326\pi\)
\(62\) 11.7190i 1.48832i
\(63\) − 3.64002i − 0.458600i
\(64\) 11.4537 1.43171
\(65\) 0 0
\(66\) −2.12489 −0.261555
\(67\) − 14.2645i − 1.74268i −0.490680 0.871340i \(-0.663251\pi\)
0.490680 0.871340i \(-0.336749\pi\)
\(68\) − 2.90539i − 0.352330i
\(69\) −5.73463 −0.690369
\(70\) 0 0
\(71\) 8.51514 1.01056 0.505280 0.862955i \(-0.331389\pi\)
0.505280 + 0.862955i \(0.331389\pi\)
\(72\) 1.09461i 0.129001i
\(73\) 9.21949i 1.07906i 0.841966 + 0.539530i \(0.181398\pi\)
−0.841966 + 0.539530i \(0.818602\pi\)
\(74\) −0.965943 −0.112289
\(75\) 0 0
\(76\) 6.56387 0.752928
\(77\) 3.64002i 0.414819i
\(78\) 3.21949i 0.364536i
\(79\) −5.09461 −0.573188 −0.286594 0.958052i \(-0.592523\pi\)
−0.286594 + 0.958052i \(0.592523\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 8.76491i 0.967922i
\(83\) 14.7493i 1.61895i 0.587156 + 0.809474i \(0.300247\pi\)
−0.587156 + 0.809474i \(0.699753\pi\)
\(84\) 9.15516 0.998910
\(85\) 0 0
\(86\) 24.8704 2.68185
\(87\) − 6.24977i − 0.670046i
\(88\) − 1.09461i − 0.116686i
\(89\) 10.4995 1.11295 0.556475 0.830865i \(-0.312154\pi\)
0.556475 + 0.830865i \(0.312154\pi\)
\(90\) 0 0
\(91\) 5.51514 0.578144
\(92\) − 14.4234i − 1.50374i
\(93\) 5.51514i 0.571893i
\(94\) 7.40493 0.763760
\(95\) 0 0
\(96\) 7.93567 0.809931
\(97\) − 6.77959i − 0.688363i −0.938903 0.344181i \(-0.888156\pi\)
0.938903 0.344181i \(-0.111844\pi\)
\(98\) − 13.2800i − 1.34149i
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 7.40493 0.736818 0.368409 0.929664i \(-0.379903\pi\)
0.368409 + 0.929664i \(0.379903\pi\)
\(102\) − 2.45459i − 0.243040i
\(103\) 16.4995i 1.62575i 0.582439 + 0.812874i \(0.302098\pi\)
−0.582439 + 0.812874i \(0.697902\pi\)
\(104\) −1.65848 −0.162628
\(105\) 0 0
\(106\) −26.6888 −2.59224
\(107\) 3.93945i 0.380841i 0.981703 + 0.190420i \(0.0609851\pi\)
−0.981703 + 0.190420i \(0.939015\pi\)
\(108\) 2.51514i 0.242019i
\(109\) −6.73463 −0.645061 −0.322530 0.946559i \(-0.604533\pi\)
−0.322530 + 0.946559i \(0.604533\pi\)
\(110\) 0 0
\(111\) −0.454586 −0.0431474
\(112\) − 9.84392i − 0.930163i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) 5.54541 0.519376
\(115\) 0 0
\(116\) 15.7190 1.45948
\(117\) 1.51514i 0.140075i
\(118\) 16.4352i 1.51298i
\(119\) −4.20482 −0.385455
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 25.5298i − 2.31136i
\(123\) 4.12489i 0.371928i
\(124\) −13.8713 −1.24568
\(125\) 0 0
\(126\) 7.73463 0.689056
\(127\) 8.06433i 0.715594i 0.933799 + 0.357797i \(0.116472\pi\)
−0.933799 + 0.357797i \(0.883528\pi\)
\(128\) 8.46640i 0.748331i
\(129\) 11.7044 1.03051
\(130\) 0 0
\(131\) −12.8099 −1.11920 −0.559602 0.828762i \(-0.689046\pi\)
−0.559602 + 0.828762i \(0.689046\pi\)
\(132\) − 2.51514i − 0.218915i
\(133\) − 9.49954i − 0.823715i
\(134\) 30.3103 2.61841
\(135\) 0 0
\(136\) 1.26445 0.108426
\(137\) 22.8099i 1.94878i 0.224868 + 0.974389i \(0.427805\pi\)
−0.224868 + 0.974389i \(0.572195\pi\)
\(138\) − 12.1854i − 1.03729i
\(139\) −7.59037 −0.643807 −0.321903 0.946773i \(-0.604323\pi\)
−0.321903 + 0.946773i \(0.604323\pi\)
\(140\) 0 0
\(141\) 3.48486 0.293478
\(142\) 18.0937i 1.51839i
\(143\) − 1.51514i − 0.126702i
\(144\) 2.70436 0.225363
\(145\) 0 0
\(146\) −19.5904 −1.62131
\(147\) − 6.24977i − 0.515472i
\(148\) − 1.14335i − 0.0939825i
\(149\) 1.81456 0.148655 0.0743274 0.997234i \(-0.476319\pi\)
0.0743274 + 0.997234i \(0.476319\pi\)
\(150\) 0 0
\(151\) 24.3250 1.97954 0.989770 0.142670i \(-0.0455687\pi\)
0.989770 + 0.142670i \(0.0455687\pi\)
\(152\) 2.85665i 0.231705i
\(153\) − 1.15516i − 0.0933893i
\(154\) −7.73463 −0.623274
\(155\) 0 0
\(156\) −3.81078 −0.305107
\(157\) 9.76491i 0.779325i 0.920958 + 0.389662i \(0.127408\pi\)
−0.920958 + 0.389662i \(0.872592\pi\)
\(158\) − 10.8255i − 0.861227i
\(159\) −12.5601 −0.996080
\(160\) 0 0
\(161\) −20.8742 −1.64512
\(162\) 2.12489i 0.166947i
\(163\) 6.98440i 0.547061i 0.961863 + 0.273530i \(0.0881914\pi\)
−0.961863 + 0.273530i \(0.911809\pi\)
\(164\) −10.3747 −0.810125
\(165\) 0 0
\(166\) −31.3406 −2.43250
\(167\) − 6.31032i − 0.488307i −0.969737 0.244154i \(-0.921490\pi\)
0.969737 0.244154i \(-0.0785102\pi\)
\(168\) 3.98440i 0.307403i
\(169\) 10.7044 0.823412
\(170\) 0 0
\(171\) 2.60975 0.199572
\(172\) 29.4381i 2.24463i
\(173\) 12.8448i 0.976575i 0.872683 + 0.488287i \(0.162378\pi\)
−0.872683 + 0.488287i \(0.837622\pi\)
\(174\) 13.2800 1.00676
\(175\) 0 0
\(176\) −2.70436 −0.203849
\(177\) 7.73463i 0.581371i
\(178\) 22.3103i 1.67223i
\(179\) −13.4849 −1.00791 −0.503953 0.863731i \(-0.668122\pi\)
−0.503953 + 0.863731i \(0.668122\pi\)
\(180\) 0 0
\(181\) −23.0899 −1.71626 −0.858130 0.513433i \(-0.828374\pi\)
−0.858130 + 0.513433i \(0.828374\pi\)
\(182\) 11.7190i 0.868673i
\(183\) − 12.0147i − 0.888151i
\(184\) 6.27718 0.462760
\(185\) 0 0
\(186\) −11.7190 −0.859281
\(187\) 1.15516i 0.0844738i
\(188\) 8.76491i 0.639247i
\(189\) 3.64002 0.264773
\(190\) 0 0
\(191\) −7.98440 −0.577731 −0.288866 0.957370i \(-0.593278\pi\)
−0.288866 + 0.957370i \(0.593278\pi\)
\(192\) 11.4537i 0.826597i
\(193\) − 11.7649i − 0.846857i −0.905930 0.423428i \(-0.860827\pi\)
0.905930 0.423428i \(-0.139173\pi\)
\(194\) 14.4058 1.03428
\(195\) 0 0
\(196\) 15.7190 1.12279
\(197\) − 3.81456i − 0.271776i −0.990724 0.135888i \(-0.956611\pi\)
0.990724 0.135888i \(-0.0433888\pi\)
\(198\) − 2.12489i − 0.151009i
\(199\) 12.0752 0.855990 0.427995 0.903781i \(-0.359220\pi\)
0.427995 + 0.903781i \(0.359220\pi\)
\(200\) 0 0
\(201\) 14.2645 1.00614
\(202\) 15.7346i 1.10708i
\(203\) − 22.7493i − 1.59669i
\(204\) 2.90539 0.203418
\(205\) 0 0
\(206\) −35.0596 −2.44272
\(207\) − 5.73463i − 0.398585i
\(208\) 4.09747i 0.284109i
\(209\) −2.60975 −0.180520
\(210\) 0 0
\(211\) 10.2645 0.706634 0.353317 0.935504i \(-0.385054\pi\)
0.353317 + 0.935504i \(0.385054\pi\)
\(212\) − 31.5904i − 2.16964i
\(213\) 8.51514i 0.583448i
\(214\) −8.37088 −0.572221
\(215\) 0 0
\(216\) −1.09461 −0.0744787
\(217\) 20.0752i 1.36280i
\(218\) − 14.3103i − 0.969217i
\(219\) −9.21949 −0.622996
\(220\) 0 0
\(221\) 1.75023 0.117733
\(222\) − 0.965943i − 0.0648298i
\(223\) − 12.9239i − 0.865445i −0.901527 0.432723i \(-0.857553\pi\)
0.901527 0.432723i \(-0.142447\pi\)
\(224\) 28.8860 1.93003
\(225\) 0 0
\(226\) −12.7493 −0.848072
\(227\) 22.8099i 1.51394i 0.653447 + 0.756972i \(0.273322\pi\)
−0.653447 + 0.756972i \(0.726678\pi\)
\(228\) 6.56387i 0.434703i
\(229\) −14.7796 −0.976663 −0.488331 0.872658i \(-0.662394\pi\)
−0.488331 + 0.872658i \(0.662394\pi\)
\(230\) 0 0
\(231\) −3.64002 −0.239496
\(232\) 6.84106i 0.449137i
\(233\) − 4.96594i − 0.325330i −0.986681 0.162665i \(-0.947991\pi\)
0.986681 0.162665i \(-0.0520089\pi\)
\(234\) −3.21949 −0.210465
\(235\) 0 0
\(236\) −19.4537 −1.26633
\(237\) − 5.09461i − 0.330930i
\(238\) − 8.93475i − 0.579154i
\(239\) 14.9991 0.970210 0.485105 0.874456i \(-0.338781\pi\)
0.485105 + 0.874456i \(0.338781\pi\)
\(240\) 0 0
\(241\) 5.04496 0.324974 0.162487 0.986711i \(-0.448048\pi\)
0.162487 + 0.986711i \(0.448048\pi\)
\(242\) 2.12489i 0.136593i
\(243\) 1.00000i 0.0641500i
\(244\) 30.2186 1.93455
\(245\) 0 0
\(246\) −8.76491 −0.558830
\(247\) 3.95413i 0.251595i
\(248\) − 6.03692i − 0.383345i
\(249\) −14.7493 −0.934700
\(250\) 0 0
\(251\) −3.03028 −0.191269 −0.0956347 0.995417i \(-0.530488\pi\)
−0.0956347 + 0.995417i \(0.530488\pi\)
\(252\) 9.15516i 0.576721i
\(253\) 5.73463i 0.360533i
\(254\) −17.1358 −1.07519
\(255\) 0 0
\(256\) 4.91721 0.307325
\(257\) − 13.6509i − 0.851521i −0.904836 0.425761i \(-0.860007\pi\)
0.904836 0.425761i \(-0.139993\pi\)
\(258\) 24.8704i 1.54836i
\(259\) −1.65470 −0.102818
\(260\) 0 0
\(261\) 6.24977 0.386851
\(262\) − 27.2195i − 1.68163i
\(263\) − 12.5601i − 0.774489i −0.921977 0.387244i \(-0.873427\pi\)
0.921977 0.387244i \(-0.126573\pi\)
\(264\) 1.09461 0.0673685
\(265\) 0 0
\(266\) 20.1854 1.23765
\(267\) 10.4995i 0.642562i
\(268\) 35.8771i 2.19154i
\(269\) 24.6888 1.50530 0.752650 0.658421i \(-0.228775\pi\)
0.752650 + 0.658421i \(0.228775\pi\)
\(270\) 0 0
\(271\) 7.56479 0.459528 0.229764 0.973246i \(-0.426205\pi\)
0.229764 + 0.973246i \(0.426205\pi\)
\(272\) − 3.12397i − 0.189418i
\(273\) 5.51514i 0.333791i
\(274\) −48.4683 −2.92808
\(275\) 0 0
\(276\) 14.4234 0.868186
\(277\) 1.92477i 0.115648i 0.998327 + 0.0578241i \(0.0184162\pi\)
−0.998327 + 0.0578241i \(0.981584\pi\)
\(278\) − 16.1287i − 0.967333i
\(279\) −5.51514 −0.330183
\(280\) 0 0
\(281\) −1.87511 −0.111860 −0.0559300 0.998435i \(-0.517812\pi\)
−0.0559300 + 0.998435i \(0.517812\pi\)
\(282\) 7.40493i 0.440957i
\(283\) − 30.1396i − 1.79161i −0.444446 0.895806i \(-0.646599\pi\)
0.444446 0.895806i \(-0.353401\pi\)
\(284\) −21.4167 −1.27085
\(285\) 0 0
\(286\) 3.21949 0.190373
\(287\) 15.0147i 0.886289i
\(288\) 7.93567i 0.467614i
\(289\) 15.6656 0.921506
\(290\) 0 0
\(291\) 6.77959 0.397427
\(292\) − 23.1883i − 1.35699i
\(293\) 29.1552i 1.70326i 0.524141 + 0.851631i \(0.324386\pi\)
−0.524141 + 0.851631i \(0.675614\pi\)
\(294\) 13.2800 0.774508
\(295\) 0 0
\(296\) 0.497594 0.0289221
\(297\) − 1.00000i − 0.0580259i
\(298\) 3.85574i 0.223357i
\(299\) 8.68876 0.502484
\(300\) 0 0
\(301\) 42.6041 2.45566
\(302\) 51.6878i 2.97430i
\(303\) 7.40493i 0.425402i
\(304\) 7.05769 0.404786
\(305\) 0 0
\(306\) 2.45459 0.140319
\(307\) 27.8548i 1.58976i 0.606768 + 0.794879i \(0.292466\pi\)
−0.606768 + 0.794879i \(0.707534\pi\)
\(308\) − 9.15516i − 0.521664i
\(309\) −16.4995 −0.938626
\(310\) 0 0
\(311\) 23.9083 1.35571 0.677856 0.735194i \(-0.262909\pi\)
0.677856 + 0.735194i \(0.262909\pi\)
\(312\) − 1.65848i − 0.0938932i
\(313\) − 28.3094i − 1.60014i −0.599905 0.800071i \(-0.704795\pi\)
0.599905 0.800071i \(-0.295205\pi\)
\(314\) −20.7493 −1.17095
\(315\) 0 0
\(316\) 12.8136 0.720824
\(317\) − 8.80986i − 0.494811i −0.968912 0.247406i \(-0.920422\pi\)
0.968912 0.247406i \(-0.0795780\pi\)
\(318\) − 26.6888i − 1.49663i
\(319\) −6.24977 −0.349920
\(320\) 0 0
\(321\) −3.93945 −0.219879
\(322\) − 44.3553i − 2.47182i
\(323\) − 3.01468i − 0.167741i
\(324\) −2.51514 −0.139730
\(325\) 0 0
\(326\) −14.8411 −0.821970
\(327\) − 6.73463i − 0.372426i
\(328\) − 4.51514i − 0.249307i
\(329\) 12.6850 0.699346
\(330\) 0 0
\(331\) 32.2498 1.77261 0.886304 0.463104i \(-0.153264\pi\)
0.886304 + 0.463104i \(0.153264\pi\)
\(332\) − 37.0966i − 2.03594i
\(333\) − 0.454586i − 0.0249112i
\(334\) 13.4087 0.733692
\(335\) 0 0
\(336\) 9.84392 0.537030
\(337\) 28.9844i 1.57888i 0.613827 + 0.789441i \(0.289629\pi\)
−0.613827 + 0.789441i \(0.710371\pi\)
\(338\) 22.7455i 1.23719i
\(339\) −6.00000 −0.325875
\(340\) 0 0
\(341\) 5.51514 0.298661
\(342\) 5.54541i 0.299862i
\(343\) 2.73085i 0.147452i
\(344\) −12.8117 −0.690760
\(345\) 0 0
\(346\) −27.2938 −1.46732
\(347\) − 35.7190i − 1.91750i −0.284253 0.958749i \(-0.591746\pi\)
0.284253 0.958749i \(-0.408254\pi\)
\(348\) 15.7190i 0.842629i
\(349\) 23.2800 1.24615 0.623076 0.782161i \(-0.285883\pi\)
0.623076 + 0.782161i \(0.285883\pi\)
\(350\) 0 0
\(351\) −1.51514 −0.0808721
\(352\) − 7.93567i − 0.422972i
\(353\) 9.75023i 0.518952i 0.965750 + 0.259476i \(0.0835499\pi\)
−0.965750 + 0.259476i \(0.916450\pi\)
\(354\) −16.4352 −0.873521
\(355\) 0 0
\(356\) −26.4078 −1.39961
\(357\) − 4.20482i − 0.222542i
\(358\) − 28.6538i − 1.51440i
\(359\) −33.9007 −1.78921 −0.894605 0.446858i \(-0.852543\pi\)
−0.894605 + 0.446858i \(0.852543\pi\)
\(360\) 0 0
\(361\) −12.1892 −0.641538
\(362\) − 49.0634i − 2.57872i
\(363\) 1.00000i 0.0524864i
\(364\) −13.8713 −0.727055
\(365\) 0 0
\(366\) 25.5298 1.33446
\(367\) 1.88601i 0.0984491i 0.998788 + 0.0492245i \(0.0156750\pi\)
−0.998788 + 0.0492245i \(0.984325\pi\)
\(368\) − 15.5085i − 0.808436i
\(369\) −4.12489 −0.214733
\(370\) 0 0
\(371\) −45.7190 −2.37361
\(372\) − 13.8713i − 0.719195i
\(373\) 16.3250i 0.845277i 0.906298 + 0.422638i \(0.138896\pi\)
−0.906298 + 0.422638i \(0.861104\pi\)
\(374\) −2.45459 −0.126924
\(375\) 0 0
\(376\) −3.81456 −0.196721
\(377\) 9.46927i 0.487692i
\(378\) 7.73463i 0.397827i
\(379\) 26.0440 1.33779 0.668896 0.743356i \(-0.266767\pi\)
0.668896 + 0.743356i \(0.266767\pi\)
\(380\) 0 0
\(381\) −8.06433 −0.413148
\(382\) − 16.9659i − 0.868053i
\(383\) − 12.4702i − 0.637197i −0.947890 0.318598i \(-0.896788\pi\)
0.947890 0.318598i \(-0.103212\pi\)
\(384\) −8.46640 −0.432049
\(385\) 0 0
\(386\) 24.9991 1.27242
\(387\) 11.7044i 0.594966i
\(388\) 17.0516i 0.865664i
\(389\) 18.0899 0.917195 0.458597 0.888644i \(-0.348352\pi\)
0.458597 + 0.888644i \(0.348352\pi\)
\(390\) 0 0
\(391\) −6.62443 −0.335012
\(392\) 6.84106i 0.345526i
\(393\) − 12.8099i − 0.646172i
\(394\) 8.10551 0.408350
\(395\) 0 0
\(396\) 2.51514 0.126390
\(397\) − 15.2342i − 0.764581i −0.924042 0.382291i \(-0.875135\pi\)
0.924042 0.382291i \(-0.124865\pi\)
\(398\) 25.6585i 1.28614i
\(399\) 9.49954 0.475572
\(400\) 0 0
\(401\) 2.74931 0.137294 0.0686471 0.997641i \(-0.478132\pi\)
0.0686471 + 0.997641i \(0.478132\pi\)
\(402\) 30.3103i 1.51174i
\(403\) − 8.35620i − 0.416252i
\(404\) −18.6244 −0.926600
\(405\) 0 0
\(406\) 48.3397 2.39906
\(407\) 0.454586i 0.0225330i
\(408\) 1.26445i 0.0625996i
\(409\) 3.98532 0.197061 0.0985307 0.995134i \(-0.468586\pi\)
0.0985307 + 0.995134i \(0.468586\pi\)
\(410\) 0 0
\(411\) −22.8099 −1.12513
\(412\) − 41.4986i − 2.04449i
\(413\) 28.1542i 1.38538i
\(414\) 12.1854 0.598882
\(415\) 0 0
\(416\) −12.0236 −0.589507
\(417\) − 7.59037i − 0.371702i
\(418\) − 5.54541i − 0.271235i
\(419\) 5.13578 0.250899 0.125450 0.992100i \(-0.459963\pi\)
0.125450 + 0.992100i \(0.459963\pi\)
\(420\) 0 0
\(421\) −8.94657 −0.436029 −0.218014 0.975946i \(-0.569958\pi\)
−0.218014 + 0.975946i \(0.569958\pi\)
\(422\) 21.8108i 1.06173i
\(423\) 3.48486i 0.169440i
\(424\) 13.7484 0.667681
\(425\) 0 0
\(426\) −18.0937 −0.876642
\(427\) − 43.7337i − 2.11642i
\(428\) − 9.90826i − 0.478934i
\(429\) 1.51514 0.0731516
\(430\) 0 0
\(431\) 22.7493 1.09580 0.547898 0.836545i \(-0.315428\pi\)
0.547898 + 0.836545i \(0.315428\pi\)
\(432\) 2.70436i 0.130113i
\(433\) − 7.58325i − 0.364428i −0.983259 0.182214i \(-0.941674\pi\)
0.983259 0.182214i \(-0.0583263\pi\)
\(434\) −42.6576 −2.04763
\(435\) 0 0
\(436\) 16.9385 0.811209
\(437\) − 14.9659i − 0.715918i
\(438\) − 19.5904i − 0.936064i
\(439\) −17.3903 −0.829991 −0.414996 0.909823i \(-0.636217\pi\)
−0.414996 + 0.909823i \(0.636217\pi\)
\(440\) 0 0
\(441\) 6.24977 0.297608
\(442\) 3.71904i 0.176897i
\(443\) − 10.6438i − 0.505702i −0.967505 0.252851i \(-0.918632\pi\)
0.967505 0.252851i \(-0.0813683\pi\)
\(444\) 1.14335 0.0542608
\(445\) 0 0
\(446\) 27.4617 1.30035
\(447\) 1.81456i 0.0858259i
\(448\) 41.6916i 1.96974i
\(449\) 26.9310 1.27095 0.635476 0.772121i \(-0.280804\pi\)
0.635476 + 0.772121i \(0.280804\pi\)
\(450\) 0 0
\(451\) 4.12489 0.194233
\(452\) − 15.0908i − 0.709813i
\(453\) 24.3250i 1.14289i
\(454\) −48.4683 −2.27473
\(455\) 0 0
\(456\) −2.85665 −0.133775
\(457\) − 15.7796i − 0.738138i −0.929402 0.369069i \(-0.879677\pi\)
0.929402 0.369069i \(-0.120323\pi\)
\(458\) − 31.4049i − 1.46746i
\(459\) 1.15516 0.0539183
\(460\) 0 0
\(461\) 8.18922 0.381410 0.190705 0.981647i \(-0.438923\pi\)
0.190705 + 0.981647i \(0.438923\pi\)
\(462\) − 7.73463i − 0.359848i
\(463\) − 16.0899i − 0.747762i −0.927477 0.373881i \(-0.878027\pi\)
0.927477 0.373881i \(-0.121973\pi\)
\(464\) 16.9016 0.784638
\(465\) 0 0
\(466\) 10.5521 0.488815
\(467\) − 29.4693i − 1.36367i −0.731504 0.681837i \(-0.761181\pi\)
0.731504 0.681837i \(-0.238819\pi\)
\(468\) − 3.81078i − 0.176153i
\(469\) 51.9229 2.39758
\(470\) 0 0
\(471\) −9.76491 −0.449943
\(472\) − 8.46640i − 0.389698i
\(473\) − 11.7044i − 0.538167i
\(474\) 10.8255 0.497230
\(475\) 0 0
\(476\) 10.5757 0.484736
\(477\) − 12.5601i − 0.575087i
\(478\) 31.8713i 1.45776i
\(479\) −32.2186 −1.47210 −0.736052 0.676925i \(-0.763312\pi\)
−0.736052 + 0.676925i \(0.763312\pi\)
\(480\) 0 0
\(481\) 0.688760 0.0314048
\(482\) 10.7200i 0.488280i
\(483\) − 20.8742i − 0.949809i
\(484\) −2.51514 −0.114324
\(485\) 0 0
\(486\) −2.12489 −0.0963868
\(487\) − 35.8936i − 1.62649i −0.581919 0.813247i \(-0.697698\pi\)
0.581919 0.813247i \(-0.302302\pi\)
\(488\) 13.1514i 0.595335i
\(489\) −6.98440 −0.315846
\(490\) 0 0
\(491\) −7.15894 −0.323079 −0.161539 0.986866i \(-0.551646\pi\)
−0.161539 + 0.986866i \(0.551646\pi\)
\(492\) − 10.3747i − 0.467726i
\(493\) − 7.21949i − 0.325150i
\(494\) −8.40207 −0.378027
\(495\) 0 0
\(496\) −14.9149 −0.669699
\(497\) 30.9953i 1.39033i
\(498\) − 31.3406i − 1.40441i
\(499\) −27.0743 −1.21201 −0.606006 0.795460i \(-0.707229\pi\)
−0.606006 + 0.795460i \(0.707229\pi\)
\(500\) 0 0
\(501\) 6.31032 0.281924
\(502\) − 6.43899i − 0.287386i
\(503\) − 26.9991i − 1.20383i −0.798560 0.601915i \(-0.794405\pi\)
0.798560 0.601915i \(-0.205595\pi\)
\(504\) −3.98440 −0.177479
\(505\) 0 0
\(506\) −12.1854 −0.541709
\(507\) 10.7044i 0.475397i
\(508\) − 20.2829i − 0.899909i
\(509\) −15.5904 −0.691031 −0.345515 0.938413i \(-0.612296\pi\)
−0.345515 + 0.938413i \(0.612296\pi\)
\(510\) 0 0
\(511\) −33.5592 −1.48457
\(512\) 27.3813i 1.21009i
\(513\) 2.60975i 0.115223i
\(514\) 29.0066 1.27943
\(515\) 0 0
\(516\) −29.4381 −1.29594
\(517\) − 3.48486i − 0.153264i
\(518\) − 3.51605i − 0.154487i
\(519\) −12.8448 −0.563826
\(520\) 0 0
\(521\) 11.1589 0.488882 0.244441 0.969664i \(-0.421396\pi\)
0.244441 + 0.969664i \(0.421396\pi\)
\(522\) 13.2800i 0.581252i
\(523\) 10.5786i 0.462568i 0.972886 + 0.231284i \(0.0742926\pi\)
−0.972886 + 0.231284i \(0.925707\pi\)
\(524\) 32.2186 1.40748
\(525\) 0 0
\(526\) 26.6888 1.16369
\(527\) 6.37088i 0.277520i
\(528\) − 2.70436i − 0.117692i
\(529\) −9.88601 −0.429827
\(530\) 0 0
\(531\) −7.73463 −0.335654
\(532\) 23.8927i 1.03588i
\(533\) − 6.24977i − 0.270708i
\(534\) −22.3103 −0.965462
\(535\) 0 0
\(536\) −15.6140 −0.674422
\(537\) − 13.4849i − 0.581915i
\(538\) 52.4608i 2.26175i
\(539\) −6.24977 −0.269197
\(540\) 0 0
\(541\) −11.2947 −0.485598 −0.242799 0.970077i \(-0.578066\pi\)
−0.242799 + 0.970077i \(0.578066\pi\)
\(542\) 16.0743i 0.690451i
\(543\) − 23.0899i − 0.990883i
\(544\) 9.16698 0.393031
\(545\) 0 0
\(546\) −11.7190 −0.501528
\(547\) 6.09369i 0.260547i 0.991478 + 0.130274i \(0.0415856\pi\)
−0.991478 + 0.130274i \(0.958414\pi\)
\(548\) − 57.3700i − 2.45072i
\(549\) 12.0147 0.512774
\(550\) 0 0
\(551\) 16.3103 0.694843
\(552\) 6.27718i 0.267175i
\(553\) − 18.5445i − 0.788592i
\(554\) −4.08991 −0.173764
\(555\) 0 0
\(556\) 19.0908 0.809631
\(557\) − 5.90069i − 0.250020i −0.992155 0.125010i \(-0.960104\pi\)
0.992155 0.125010i \(-0.0398964\pi\)
\(558\) − 11.7190i − 0.496106i
\(559\) −17.7337 −0.750056
\(560\) 0 0
\(561\) −1.15516 −0.0487710
\(562\) − 3.98440i − 0.168072i
\(563\) − 3.03028i − 0.127711i −0.997959 0.0638555i \(-0.979660\pi\)
0.997959 0.0638555i \(-0.0203397\pi\)
\(564\) −8.76491 −0.369069
\(565\) 0 0
\(566\) 64.0431 2.69193
\(567\) 3.64002i 0.152867i
\(568\) − 9.32075i − 0.391090i
\(569\) −13.4049 −0.561964 −0.280982 0.959713i \(-0.590660\pi\)
−0.280982 + 0.959713i \(0.590660\pi\)
\(570\) 0 0
\(571\) 26.8851 1.12511 0.562553 0.826761i \(-0.309819\pi\)
0.562553 + 0.826761i \(0.309819\pi\)
\(572\) 3.81078i 0.159337i
\(573\) − 7.98440i − 0.333553i
\(574\) −31.9045 −1.33167
\(575\) 0 0
\(576\) −11.4537 −0.477236
\(577\) 2.03028i 0.0845215i 0.999107 + 0.0422607i \(0.0134560\pi\)
−0.999107 + 0.0422607i \(0.986544\pi\)
\(578\) 33.2876i 1.38458i
\(579\) 11.7649 0.488933
\(580\) 0 0
\(581\) −53.6878 −2.22735
\(582\) 14.4058i 0.597142i
\(583\) 12.5601i 0.520186i
\(584\) 10.0917 0.417599
\(585\) 0 0
\(586\) −61.9514 −2.55919
\(587\) 21.8245i 0.900795i 0.892828 + 0.450398i \(0.148718\pi\)
−0.892828 + 0.450398i \(0.851282\pi\)
\(588\) 15.7190i 0.648242i
\(589\) −14.3931 −0.593058
\(590\) 0 0
\(591\) 3.81456 0.156910
\(592\) − 1.22936i − 0.0505265i
\(593\) − 8.06811i − 0.331318i −0.986183 0.165659i \(-0.947025\pi\)
0.986183 0.165659i \(-0.0529751\pi\)
\(594\) 2.12489 0.0871851
\(595\) 0 0
\(596\) −4.56387 −0.186944
\(597\) 12.0752i 0.494206i
\(598\) 18.4626i 0.754993i
\(599\) −7.61353 −0.311080 −0.155540 0.987830i \(-0.549712\pi\)
−0.155540 + 0.987830i \(0.549712\pi\)
\(600\) 0 0
\(601\) 3.57569 0.145855 0.0729277 0.997337i \(-0.476766\pi\)
0.0729277 + 0.997337i \(0.476766\pi\)
\(602\) 90.5289i 3.68968i
\(603\) 14.2645i 0.580893i
\(604\) −61.1807 −2.48941
\(605\) 0 0
\(606\) −15.7346 −0.639176
\(607\) 17.5298i 0.711513i 0.934579 + 0.355757i \(0.115777\pi\)
−0.934579 + 0.355757i \(0.884223\pi\)
\(608\) 20.7101i 0.839905i
\(609\) 22.7493 0.921849
\(610\) 0 0
\(611\) −5.28005 −0.213608
\(612\) 2.90539i 0.117443i
\(613\) − 12.5601i − 0.507297i −0.967296 0.253649i \(-0.918369\pi\)
0.967296 0.253649i \(-0.0816307\pi\)
\(614\) −59.1883 −2.38865
\(615\) 0 0
\(616\) 3.98440 0.160536
\(617\) 15.9612i 0.642576i 0.946982 + 0.321288i \(0.104116\pi\)
−0.946982 + 0.321288i \(0.895884\pi\)
\(618\) − 35.0596i − 1.41031i
\(619\) 9.23417 0.371153 0.185576 0.982630i \(-0.440585\pi\)
0.185576 + 0.982630i \(0.440585\pi\)
\(620\) 0 0
\(621\) 5.73463 0.230123
\(622\) 50.8023i 2.03699i
\(623\) 38.2186i 1.53119i
\(624\) −4.09747 −0.164030
\(625\) 0 0
\(626\) 60.1542 2.40425
\(627\) − 2.60975i − 0.104223i
\(628\) − 24.5601i − 0.980054i
\(629\) −0.525120 −0.0209379
\(630\) 0 0
\(631\) 29.2342 1.16379 0.581897 0.813262i \(-0.302311\pi\)
0.581897 + 0.813262i \(0.302311\pi\)
\(632\) 5.57661i 0.221826i
\(633\) 10.2645i 0.407975i
\(634\) 18.7200 0.743464
\(635\) 0 0
\(636\) 31.5904 1.25264
\(637\) 9.46927i 0.375186i
\(638\) − 13.2800i − 0.525762i
\(639\) −8.51514 −0.336854
\(640\) 0 0
\(641\) 13.9612 0.551436 0.275718 0.961239i \(-0.411084\pi\)
0.275718 + 0.961239i \(0.411084\pi\)
\(642\) − 8.37088i − 0.330372i
\(643\) − 12.6206i − 0.497710i −0.968541 0.248855i \(-0.919946\pi\)
0.968541 0.248855i \(-0.0800542\pi\)
\(644\) 52.5015 2.06885
\(645\) 0 0
\(646\) 6.40585 0.252035
\(647\) 29.6429i 1.16538i 0.812694 + 0.582691i \(0.198000\pi\)
−0.812694 + 0.582691i \(0.802000\pi\)
\(648\) − 1.09461i − 0.0430003i
\(649\) 7.73463 0.303611
\(650\) 0 0
\(651\) −20.0752 −0.786810
\(652\) − 17.5667i − 0.687967i
\(653\) − 9.90069i − 0.387444i −0.981056 0.193722i \(-0.937944\pi\)
0.981056 0.193722i \(-0.0620560\pi\)
\(654\) 14.3103 0.559578
\(655\) 0 0
\(656\) −11.1552 −0.435536
\(657\) − 9.21949i − 0.359687i
\(658\) 26.9541i 1.05078i
\(659\) 5.28005 0.205681 0.102841 0.994698i \(-0.467207\pi\)
0.102841 + 0.994698i \(0.467207\pi\)
\(660\) 0 0
\(661\) −26.8548 −1.04453 −0.522266 0.852783i \(-0.674913\pi\)
−0.522266 + 0.852783i \(0.674913\pi\)
\(662\) 68.5271i 2.66338i
\(663\) 1.75023i 0.0679733i
\(664\) 16.1447 0.626537
\(665\) 0 0
\(666\) 0.965943 0.0374295
\(667\) − 35.8401i − 1.38774i
\(668\) 15.8713i 0.614080i
\(669\) 12.9239 0.499665
\(670\) 0 0
\(671\) −12.0147 −0.463822
\(672\) 28.8860i 1.11430i
\(673\) 3.81834i 0.147186i 0.997288 + 0.0735932i \(0.0234466\pi\)
−0.997288 + 0.0735932i \(0.976553\pi\)
\(674\) −61.5885 −2.37230
\(675\) 0 0
\(676\) −26.9229 −1.03550
\(677\) − 15.6897i − 0.603003i −0.953466 0.301502i \(-0.902512\pi\)
0.953466 0.301502i \(-0.0974879\pi\)
\(678\) − 12.7493i − 0.489634i
\(679\) 24.6779 0.947049
\(680\) 0 0
\(681\) −22.8099 −0.874076
\(682\) 11.7190i 0.448745i
\(683\) 15.6353i 0.598269i 0.954211 + 0.299135i \(0.0966979\pi\)
−0.954211 + 0.299135i \(0.903302\pi\)
\(684\) −6.56387 −0.250976
\(685\) 0 0
\(686\) −5.80275 −0.221550
\(687\) − 14.7796i − 0.563876i
\(688\) 31.6528i 1.20675i
\(689\) 19.0303 0.724996
\(690\) 0 0
\(691\) −31.4305 −1.19567 −0.597836 0.801618i \(-0.703973\pi\)
−0.597836 + 0.801618i \(0.703973\pi\)
\(692\) − 32.3065i − 1.22811i
\(693\) − 3.64002i − 0.138273i
\(694\) 75.8989 2.88108
\(695\) 0 0
\(696\) −6.84106 −0.259310
\(697\) 4.76491i 0.180484i
\(698\) 49.4674i 1.87237i
\(699\) 4.96594 0.187829
\(700\) 0 0
\(701\) 3.24507 0.122565 0.0612824 0.998120i \(-0.480481\pi\)
0.0612824 + 0.998120i \(0.480481\pi\)
\(702\) − 3.21949i − 0.121512i
\(703\) − 1.18635i − 0.0447442i
\(704\) 11.4537 0.431676
\(705\) 0 0
\(706\) −20.7181 −0.779737
\(707\) 26.9541i 1.01371i
\(708\) − 19.4537i − 0.731114i
\(709\) 26.7190 1.00345 0.501727 0.865026i \(-0.332698\pi\)
0.501727 + 0.865026i \(0.332698\pi\)
\(710\) 0 0
\(711\) 5.09461 0.191063
\(712\) − 11.4929i − 0.430714i
\(713\) 31.6273i 1.18445i
\(714\) 8.93475 0.334375
\(715\) 0 0
\(716\) 33.9163 1.26751
\(717\) 14.9991i 0.560151i
\(718\) − 72.0351i − 2.68833i
\(719\) 6.78807 0.253152 0.126576 0.991957i \(-0.459601\pi\)
0.126576 + 0.991957i \(0.459601\pi\)
\(720\) 0 0
\(721\) −60.0587 −2.23670
\(722\) − 25.9007i − 0.963924i
\(723\) 5.04496i 0.187624i
\(724\) 58.0743 2.15831
\(725\) 0 0
\(726\) −2.12489 −0.0788619
\(727\) − 19.9154i − 0.738620i −0.929306 0.369310i \(-0.879594\pi\)
0.929306 0.369310i \(-0.120406\pi\)
\(728\) − 6.03692i − 0.223743i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 13.5204 0.500071
\(732\) 30.2186i 1.11691i
\(733\) 31.3388i 1.15752i 0.815497 + 0.578762i \(0.196464\pi\)
−0.815497 + 0.578762i \(0.803536\pi\)
\(734\) −4.00756 −0.147922
\(735\) 0 0
\(736\) 45.5081 1.67745
\(737\) − 14.2645i − 0.525438i
\(738\) − 8.76491i − 0.322641i
\(739\) 2.25355 0.0828982 0.0414491 0.999141i \(-0.486803\pi\)
0.0414491 + 0.999141i \(0.486803\pi\)
\(740\) 0 0
\(741\) −3.95413 −0.145259
\(742\) − 97.1477i − 3.56640i
\(743\) 6.74931i 0.247608i 0.992307 + 0.123804i \(0.0395095\pi\)
−0.992307 + 0.123804i \(0.960491\pi\)
\(744\) 6.03692 0.221324
\(745\) 0 0
\(746\) −34.6888 −1.27005
\(747\) − 14.7493i − 0.539649i
\(748\) − 2.90539i − 0.106232i
\(749\) −14.3397 −0.523961
\(750\) 0 0
\(751\) 22.4390 0.818810 0.409405 0.912353i \(-0.365736\pi\)
0.409405 + 0.912353i \(0.365736\pi\)
\(752\) 9.42431i 0.343669i
\(753\) − 3.03028i − 0.110429i
\(754\) −20.1211 −0.732767
\(755\) 0 0
\(756\) −9.15516 −0.332970
\(757\) 25.4158i 0.923754i 0.886944 + 0.461877i \(0.152824\pi\)
−0.886944 + 0.461877i \(0.847176\pi\)
\(758\) 55.3406i 2.01006i
\(759\) −5.73463 −0.208154
\(760\) 0 0
\(761\) −30.7493 −1.11466 −0.557331 0.830291i \(-0.688174\pi\)
−0.557331 + 0.830291i \(0.688174\pi\)
\(762\) − 17.1358i − 0.620764i
\(763\) − 24.5142i − 0.887474i
\(764\) 20.0819 0.726537
\(765\) 0 0
\(766\) 26.4977 0.957401
\(767\) − 11.7190i − 0.423150i
\(768\) 4.91721i 0.177434i
\(769\) 16.2956 0.587636 0.293818 0.955861i \(-0.405074\pi\)
0.293818 + 0.955861i \(0.405074\pi\)
\(770\) 0 0
\(771\) 13.6509 0.491626
\(772\) 29.5904i 1.06498i
\(773\) − 48.7787i − 1.75445i −0.480082 0.877223i \(-0.659393\pi\)
0.480082 0.877223i \(-0.340607\pi\)
\(774\) −24.8704 −0.893949
\(775\) 0 0
\(776\) −7.42100 −0.266398
\(777\) − 1.65470i − 0.0593621i
\(778\) 38.4390i 1.37810i
\(779\) −10.7649 −0.385693
\(780\) 0 0
\(781\) 8.51514 0.304696
\(782\) − 14.0761i − 0.503362i
\(783\) 6.24977i 0.223349i
\(784\) 16.9016 0.603629
\(785\) 0 0
\(786\) 27.2195 0.970887
\(787\) − 46.2001i − 1.64686i −0.567421 0.823428i \(-0.692059\pi\)
0.567421 0.823428i \(-0.307941\pi\)
\(788\) 9.59415i 0.341777i
\(789\) 12.5601 0.447151
\(790\) 0 0
\(791\) −21.8401 −0.776546
\(792\) 1.09461i 0.0388952i
\(793\) 18.2039i 0.646439i
\(794\) 32.3709 1.14880
\(795\) 0 0
\(796\) −30.3709 −1.07647
\(797\) 36.3784i 1.28859i 0.764777 + 0.644295i \(0.222849\pi\)
−0.764777 + 0.644295i \(0.777151\pi\)
\(798\) 20.1854i 0.714557i
\(799\) 4.02558 0.142415
\(800\) 0 0
\(801\) −10.4995 −0.370983
\(802\) 5.84197i 0.206287i
\(803\) 9.21949i 0.325349i
\(804\) −35.8771 −1.26529
\(805\) 0 0
\(806\) 17.7560 0.625427
\(807\) 24.6888i 0.869086i
\(808\) − 8.10551i − 0.285151i
\(809\) −11.3737 −0.399879 −0.199940 0.979808i \(-0.564075\pi\)
−0.199940 + 0.979808i \(0.564075\pi\)
\(810\) 0 0
\(811\) −13.3903 −0.470195 −0.235098 0.971972i \(-0.575541\pi\)
−0.235098 + 0.971972i \(0.575541\pi\)
\(812\) 57.2177i 2.00795i
\(813\) 7.56479i 0.265309i
\(814\) −0.965943 −0.0338563
\(815\) 0 0
\(816\) 3.12397 0.109361
\(817\) 30.5454i 1.06865i
\(818\) 8.46835i 0.296089i
\(819\) −5.51514 −0.192715
\(820\) 0 0
\(821\) 32.0975 1.12021 0.560105 0.828422i \(-0.310761\pi\)
0.560105 + 0.828422i \(0.310761\pi\)
\(822\) − 48.4683i − 1.69053i
\(823\) − 16.7952i − 0.585443i −0.956198 0.292722i \(-0.905439\pi\)
0.956198 0.292722i \(-0.0945609\pi\)
\(824\) 18.0606 0.629169
\(825\) 0 0
\(826\) −59.8245 −2.08156
\(827\) − 45.5904i − 1.58533i −0.609656 0.792666i \(-0.708692\pi\)
0.609656 0.792666i \(-0.291308\pi\)
\(828\) 14.4234i 0.501248i
\(829\) −12.9385 −0.449374 −0.224687 0.974431i \(-0.572136\pi\)
−0.224687 + 0.974431i \(0.572136\pi\)
\(830\) 0 0
\(831\) −1.92477 −0.0667695
\(832\) − 17.3539i − 0.601638i
\(833\) − 7.21949i − 0.250141i
\(834\) 16.1287 0.558490
\(835\) 0 0
\(836\) 6.56387 0.227016
\(837\) − 5.51514i − 0.190631i
\(838\) 10.9130i 0.376982i
\(839\) 1.59037 0.0549057 0.0274528 0.999623i \(-0.491260\pi\)
0.0274528 + 0.999623i \(0.491260\pi\)
\(840\) 0 0
\(841\) 10.0596 0.346884
\(842\) − 19.0104i − 0.655143i
\(843\) − 1.87511i − 0.0645824i
\(844\) −25.8165 −0.888641
\(845\) 0 0
\(846\) −7.40493 −0.254587
\(847\) 3.64002i 0.125073i
\(848\) − 33.9670i − 1.16643i
\(849\) 30.1396 1.03439
\(850\) 0 0
\(851\) −2.60688 −0.0893628
\(852\) − 21.4167i − 0.733726i
\(853\) − 10.5161i − 0.360063i −0.983661 0.180031i \(-0.942380\pi\)
0.983661 0.180031i \(-0.0576199\pi\)
\(854\) 92.9291 3.17997
\(855\) 0 0
\(856\) 4.31216 0.147386
\(857\) − 57.4637i − 1.96292i −0.191665 0.981460i \(-0.561389\pi\)
0.191665 0.981460i \(-0.438611\pi\)
\(858\) 3.21949i 0.109912i
\(859\) −32.7181 −1.11633 −0.558164 0.829731i \(-0.688494\pi\)
−0.558164 + 0.829731i \(0.688494\pi\)
\(860\) 0 0
\(861\) −15.0147 −0.511699
\(862\) 48.3397i 1.64646i
\(863\) 43.1807i 1.46989i 0.678127 + 0.734945i \(0.262792\pi\)
−0.678127 + 0.734945i \(0.737208\pi\)
\(864\) −7.93567 −0.269977
\(865\) 0 0
\(866\) 16.1135 0.547560
\(867\) 15.6656i 0.532032i
\(868\) − 50.4920i − 1.71381i
\(869\) −5.09461 −0.172823
\(870\) 0 0
\(871\) −21.6126 −0.732315
\(872\) 7.37179i 0.249640i
\(873\) 6.77959i 0.229454i
\(874\) 31.8009 1.07568
\(875\) 0 0
\(876\) 23.1883 0.783460
\(877\) − 16.0752i − 0.542822i −0.962464 0.271411i \(-0.912510\pi\)
0.962464 0.271411i \(-0.0874903\pi\)
\(878\) − 36.9523i − 1.24708i
\(879\) −29.1552 −0.983379
\(880\) 0 0
\(881\) 31.2876 1.05411 0.527053 0.849832i \(-0.323297\pi\)
0.527053 + 0.849832i \(0.323297\pi\)
\(882\) 13.2800i 0.447162i
\(883\) − 24.7640i − 0.833375i −0.909050 0.416687i \(-0.863191\pi\)
0.909050 0.416687i \(-0.136809\pi\)
\(884\) −4.40207 −0.148058
\(885\) 0 0
\(886\) 22.6169 0.759828
\(887\) − 26.3085i − 0.883353i −0.897174 0.441676i \(-0.854384\pi\)
0.897174 0.441676i \(-0.145616\pi\)
\(888\) 0.497594i 0.0166982i
\(889\) −29.3544 −0.984514
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 32.5053i 1.08836i
\(893\) 9.09461i 0.304339i
\(894\) −3.85574 −0.128955
\(895\) 0 0
\(896\) −30.8179 −1.02955
\(897\) 8.68876i 0.290109i
\(898\) 57.2252i 1.90963i
\(899\) −34.4683 −1.14958
\(900\) 0 0
\(901\) −14.5089 −0.483363
\(902\) 8.76491i 0.291840i
\(903\) 42.6041i 1.41778i
\(904\) 6.56766 0.218437
\(905\) 0 0
\(906\) −51.6878 −1.71721
\(907\) − 55.9301i − 1.85713i −0.371174 0.928563i \(-0.621045\pi\)
0.371174 0.928563i \(-0.378955\pi\)
\(908\) − 57.3700i − 1.90389i
\(909\) −7.40493 −0.245606
\(910\) 0 0
\(911\) −15.3931 −0.509997 −0.254998 0.966941i \(-0.582075\pi\)
−0.254998 + 0.966941i \(0.582075\pi\)
\(912\) 7.05769i 0.233703i
\(913\) 14.7493i 0.488131i
\(914\) 33.5298 1.10907
\(915\) 0 0
\(916\) 37.1727 1.22822
\(917\) − 46.6282i − 1.53980i
\(918\) 2.45459i 0.0810134i
\(919\) 51.2598 1.69090 0.845452 0.534052i \(-0.179331\pi\)
0.845452 + 0.534052i \(0.179331\pi\)
\(920\) 0 0
\(921\) −27.8548 −0.917848
\(922\) 17.4012i 0.573076i
\(923\) − 12.9016i − 0.424662i
\(924\) 9.15516 0.301183
\(925\) 0 0
\(926\) 34.1892 1.12353
\(927\) − 16.4995i − 0.541916i
\(928\) 49.5961i 1.62807i
\(929\) −14.8099 −0.485896 −0.242948 0.970039i \(-0.578114\pi\)
−0.242948 + 0.970039i \(0.578114\pi\)
\(930\) 0 0
\(931\) 16.3103 0.534549
\(932\) 12.4900i 0.409125i
\(933\) 23.9083i 0.782721i
\(934\) 62.6188 2.04895
\(935\) 0 0
\(936\) 1.65848 0.0542093
\(937\) 27.2654i 0.890721i 0.895351 + 0.445360i \(0.146924\pi\)
−0.895351 + 0.445360i \(0.853076\pi\)
\(938\) 110.330i 3.60241i
\(939\) 28.3094 0.923843
\(940\) 0 0
\(941\) −49.5630 −1.61571 −0.807853 0.589384i \(-0.799371\pi\)
−0.807853 + 0.589384i \(0.799371\pi\)
\(942\) − 20.7493i − 0.676049i
\(943\) 23.6547i 0.770303i
\(944\) −20.9172 −0.680797
\(945\) 0 0
\(946\) 24.8704 0.808607
\(947\) − 13.9844i − 0.454432i −0.973844 0.227216i \(-0.927038\pi\)
0.973844 0.227216i \(-0.0729623\pi\)
\(948\) 12.8136i 0.416168i
\(949\) 13.9688 0.453447
\(950\) 0 0
\(951\) 8.80986 0.285679
\(952\) 4.60263i 0.149172i
\(953\) − 17.1240i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894561\pi\)
\(954\) 26.6888 0.864081
\(955\) 0 0
\(956\) −37.7248 −1.22011
\(957\) − 6.24977i − 0.202026i
\(958\) − 68.4608i − 2.21187i
\(959\) −83.0284 −2.68113
\(960\) 0 0
\(961\) −0.583252 −0.0188146
\(962\) 1.46354i 0.0471863i
\(963\) − 3.93945i − 0.126947i
\(964\) −12.6888 −0.408677
\(965\) 0 0
\(966\) 44.3553 1.42711
\(967\) 1.90826i 0.0613653i 0.999529 + 0.0306827i \(0.00976813\pi\)
−0.999529 + 0.0306827i \(0.990232\pi\)
\(968\) − 1.09461i − 0.0351821i
\(969\) 3.01468 0.0968455
\(970\) 0 0
\(971\) 31.3856 1.00721 0.503605 0.863934i \(-0.332007\pi\)
0.503605 + 0.863934i \(0.332007\pi\)
\(972\) − 2.51514i − 0.0806731i
\(973\) − 27.6291i − 0.885749i
\(974\) 76.2697 2.44384
\(975\) 0 0
\(976\) 32.4920 1.04004
\(977\) 41.4693i 1.32672i 0.748301 + 0.663360i \(0.230870\pi\)
−0.748301 + 0.663360i \(0.769130\pi\)
\(978\) − 14.8411i − 0.474565i
\(979\) 10.4995 0.335567
\(980\) 0 0
\(981\) 6.73463 0.215020
\(982\) − 15.2119i − 0.485432i
\(983\) 6.23601i 0.198898i 0.995043 + 0.0994489i \(0.0317080\pi\)
−0.995043 + 0.0994489i \(0.968292\pi\)
\(984\) 4.51514 0.143937
\(985\) 0 0
\(986\) 15.3406 0.488544
\(987\) 12.6850i 0.403767i
\(988\) − 9.94518i − 0.316398i
\(989\) 67.1202 2.13430
\(990\) 0 0
\(991\) 27.2048 0.864189 0.432095 0.901828i \(-0.357775\pi\)
0.432095 + 0.901828i \(0.357775\pi\)
\(992\) − 43.7663i − 1.38958i
\(993\) 32.2498i 1.02342i
\(994\) −65.8615 −2.08900
\(995\) 0 0
\(996\) 37.0966 1.17545
\(997\) 28.0606i 0.888687i 0.895857 + 0.444343i \(0.146563\pi\)
−0.895857 + 0.444343i \(0.853437\pi\)
\(998\) − 57.5298i − 1.82107i
\(999\) 0.454586 0.0143825
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.f.199.5 6
3.2 odd 2 2475.2.c.q.199.2 6
5.2 odd 4 825.2.a.m.1.1 yes 3
5.3 odd 4 825.2.a.i.1.3 3
5.4 even 2 inner 825.2.c.f.199.2 6
15.2 even 4 2475.2.a.z.1.3 3
15.8 even 4 2475.2.a.bd.1.1 3
15.14 odd 2 2475.2.c.q.199.5 6
55.32 even 4 9075.2.a.cd.1.3 3
55.43 even 4 9075.2.a.cj.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.a.i.1.3 3 5.3 odd 4
825.2.a.m.1.1 yes 3 5.2 odd 4
825.2.c.f.199.2 6 5.4 even 2 inner
825.2.c.f.199.5 6 1.1 even 1 trivial
2475.2.a.z.1.3 3 15.2 even 4
2475.2.a.bd.1.1 3 15.8 even 4
2475.2.c.q.199.2 6 3.2 odd 2
2475.2.c.q.199.5 6 15.14 odd 2
9075.2.a.cd.1.3 3 55.32 even 4
9075.2.a.cj.1.1 3 55.43 even 4