Properties

Label 825.2.c.g.199.3
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: 6.0.350464.1
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.3
Root \(0.403032 - 0.403032i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.2.c.g.199.4

$q$-expansion

\(f(q)\) \(=\) \(q-0.193937i q^{2} -1.00000i q^{3} +1.96239 q^{4} -0.193937 q^{6} +3.35026i q^{7} -0.768452i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-0.193937i q^{2} -1.00000i q^{3} +1.96239 q^{4} -0.193937 q^{6} +3.35026i q^{7} -0.768452i q^{8} -1.00000 q^{9} +1.00000 q^{11} -1.96239i q^{12} -2.96239i q^{13} +0.649738 q^{14} +3.77575 q^{16} -4.57452i q^{17} +0.193937i q^{18} +4.31265 q^{19} +3.35026 q^{21} -0.193937i q^{22} +6.70052i q^{23} -0.768452 q^{24} -0.574515 q^{26} +1.00000i q^{27} +6.57452i q^{28} +3.61213 q^{29} +9.92478 q^{31} -2.26916i q^{32} -1.00000i q^{33} -0.887166 q^{34} -1.96239 q^{36} -2.00000i q^{37} -0.836381i q^{38} -2.96239 q^{39} -4.38787 q^{41} -0.649738i q^{42} +9.27504i q^{43} +1.96239 q^{44} +1.29948 q^{46} -9.92478i q^{47} -3.77575i q^{48} -4.22425 q^{49} -4.57452 q^{51} -5.81336i q^{52} -4.70052i q^{53} +0.193937 q^{54} +2.57452 q^{56} -4.31265i q^{57} -0.700523i q^{58} -10.7005 q^{59} -8.70052 q^{61} -1.92478i q^{62} -3.35026i q^{63} +7.11142 q^{64} -0.193937 q^{66} +5.92478i q^{67} -8.97698i q^{68} +6.70052 q^{69} +9.92478 q^{71} +0.768452i q^{72} +7.73813i q^{73} -0.387873 q^{74} +8.46310 q^{76} +3.35026i q^{77} +0.574515i q^{78} -11.5369 q^{79} +1.00000 q^{81} +0.850969i q^{82} -10.8872i q^{83} +6.57452 q^{84} +1.79877 q^{86} -3.61213i q^{87} -0.768452i q^{88} +2.77575 q^{89} +9.92478 q^{91} +13.1490i q^{92} -9.92478i q^{93} -1.92478 q^{94} -2.26916 q^{96} +0.0752228i q^{97} +0.819237i q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{4} - 2 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 10 q^{4} - 2 q^{6} - 6 q^{9} + 6 q^{11} + 24 q^{14} + 26 q^{16} - 16 q^{19} + 18 q^{24} + 20 q^{26} + 20 q^{29} + 16 q^{31} + 60 q^{34} + 10 q^{36} + 4 q^{39} - 28 q^{41} - 10 q^{44} + 48 q^{46} - 22 q^{49} - 4 q^{51} + 2 q^{54} - 8 q^{56} - 24 q^{59} - 12 q^{61} - 26 q^{64} - 2 q^{66} + 16 q^{71} - 4 q^{74} + 96 q^{76} - 24 q^{79} + 6 q^{81} + 16 q^{84} - 16 q^{86} + 20 q^{89} + 16 q^{91} + 32 q^{94} - 58 q^{96} - 6 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.193937i − 0.137134i −0.997647 0.0685669i \(-0.978157\pi\)
0.997647 0.0685669i \(-0.0218427\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) 1.96239 0.981194
\(5\) 0 0
\(6\) −0.193937 −0.0791743
\(7\) 3.35026i 1.26628i 0.774037 + 0.633140i \(0.218234\pi\)
−0.774037 + 0.633140i \(0.781766\pi\)
\(8\) − 0.768452i − 0.271689i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) − 1.96239i − 0.566493i
\(13\) − 2.96239i − 0.821619i −0.911721 0.410809i \(-0.865246\pi\)
0.911721 0.410809i \(-0.134754\pi\)
\(14\) 0.649738 0.173650
\(15\) 0 0
\(16\) 3.77575 0.943937
\(17\) − 4.57452i − 1.10948i −0.832023 0.554741i \(-0.812817\pi\)
0.832023 0.554741i \(-0.187183\pi\)
\(18\) 0.193937i 0.0457113i
\(19\) 4.31265 0.989390 0.494695 0.869067i \(-0.335280\pi\)
0.494695 + 0.869067i \(0.335280\pi\)
\(20\) 0 0
\(21\) 3.35026 0.731087
\(22\) − 0.193937i − 0.0413474i
\(23\) 6.70052i 1.39716i 0.715534 + 0.698578i \(0.246183\pi\)
−0.715534 + 0.698578i \(0.753817\pi\)
\(24\) −0.768452 −0.156860
\(25\) 0 0
\(26\) −0.574515 −0.112672
\(27\) 1.00000i 0.192450i
\(28\) 6.57452i 1.24247i
\(29\) 3.61213 0.670755 0.335378 0.942084i \(-0.391136\pi\)
0.335378 + 0.942084i \(0.391136\pi\)
\(30\) 0 0
\(31\) 9.92478 1.78254 0.891271 0.453470i \(-0.149814\pi\)
0.891271 + 0.453470i \(0.149814\pi\)
\(32\) − 2.26916i − 0.401134i
\(33\) − 1.00000i − 0.174078i
\(34\) −0.887166 −0.152148
\(35\) 0 0
\(36\) −1.96239 −0.327065
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) − 0.836381i − 0.135679i
\(39\) −2.96239 −0.474362
\(40\) 0 0
\(41\) −4.38787 −0.685271 −0.342635 0.939468i \(-0.611320\pi\)
−0.342635 + 0.939468i \(0.611320\pi\)
\(42\) − 0.649738i − 0.100257i
\(43\) 9.27504i 1.41443i 0.706998 + 0.707215i \(0.250049\pi\)
−0.706998 + 0.707215i \(0.749951\pi\)
\(44\) 1.96239 0.295841
\(45\) 0 0
\(46\) 1.29948 0.191597
\(47\) − 9.92478i − 1.44768i −0.689969 0.723839i \(-0.742376\pi\)
0.689969 0.723839i \(-0.257624\pi\)
\(48\) − 3.77575i − 0.544982i
\(49\) −4.22425 −0.603465
\(50\) 0 0
\(51\) −4.57452 −0.640560
\(52\) − 5.81336i − 0.806168i
\(53\) − 4.70052i − 0.645667i −0.946456 0.322833i \(-0.895365\pi\)
0.946456 0.322833i \(-0.104635\pi\)
\(54\) 0.193937 0.0263914
\(55\) 0 0
\(56\) 2.57452 0.344034
\(57\) − 4.31265i − 0.571224i
\(58\) − 0.700523i − 0.0919832i
\(59\) −10.7005 −1.39309 −0.696545 0.717513i \(-0.745280\pi\)
−0.696545 + 0.717513i \(0.745280\pi\)
\(60\) 0 0
\(61\) −8.70052 −1.11399 −0.556994 0.830517i \(-0.688045\pi\)
−0.556994 + 0.830517i \(0.688045\pi\)
\(62\) − 1.92478i − 0.244447i
\(63\) − 3.35026i − 0.422093i
\(64\) 7.11142 0.888927
\(65\) 0 0
\(66\) −0.193937 −0.0238719
\(67\) 5.92478i 0.723827i 0.932212 + 0.361913i \(0.117876\pi\)
−0.932212 + 0.361913i \(0.882124\pi\)
\(68\) − 8.97698i − 1.08862i
\(69\) 6.70052 0.806648
\(70\) 0 0
\(71\) 9.92478 1.17785 0.588927 0.808186i \(-0.299550\pi\)
0.588927 + 0.808186i \(0.299550\pi\)
\(72\) 0.768452i 0.0905629i
\(73\) 7.73813i 0.905680i 0.891592 + 0.452840i \(0.149589\pi\)
−0.891592 + 0.452840i \(0.850411\pi\)
\(74\) −0.387873 −0.0450893
\(75\) 0 0
\(76\) 8.46310 0.970784
\(77\) 3.35026i 0.381798i
\(78\) 0.574515i 0.0650511i
\(79\) −11.5369 −1.29800 −0.649002 0.760787i \(-0.724813\pi\)
−0.649002 + 0.760787i \(0.724813\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0.850969i 0.0939738i
\(83\) − 10.8872i − 1.19502i −0.801861 0.597511i \(-0.796156\pi\)
0.801861 0.597511i \(-0.203844\pi\)
\(84\) 6.57452 0.717338
\(85\) 0 0
\(86\) 1.79877 0.193966
\(87\) − 3.61213i − 0.387261i
\(88\) − 0.768452i − 0.0819173i
\(89\) 2.77575 0.294229 0.147114 0.989120i \(-0.453001\pi\)
0.147114 + 0.989120i \(0.453001\pi\)
\(90\) 0 0
\(91\) 9.92478 1.04040
\(92\) 13.1490i 1.37088i
\(93\) − 9.92478i − 1.02915i
\(94\) −1.92478 −0.198526
\(95\) 0 0
\(96\) −2.26916 −0.231595
\(97\) 0.0752228i 0.00763772i 0.999993 + 0.00381886i \(0.00121558\pi\)
−0.999993 + 0.00381886i \(0.998784\pi\)
\(98\) 0.819237i 0.0827555i
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −15.0884 −1.50135 −0.750676 0.660671i \(-0.770272\pi\)
−0.750676 + 0.660671i \(0.770272\pi\)
\(102\) 0.887166i 0.0878425i
\(103\) 3.22425i 0.317695i 0.987303 + 0.158848i \(0.0507778\pi\)
−0.987303 + 0.158848i \(0.949222\pi\)
\(104\) −2.27645 −0.223225
\(105\) 0 0
\(106\) −0.911603 −0.0885427
\(107\) − 0.962389i − 0.0930376i −0.998917 0.0465188i \(-0.985187\pi\)
0.998917 0.0465188i \(-0.0148127\pi\)
\(108\) 1.96239i 0.188831i
\(109\) −11.4010 −1.09202 −0.546011 0.837778i \(-0.683854\pi\)
−0.546011 + 0.837778i \(0.683854\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 12.6497i 1.19529i
\(113\) 6.00000i 0.564433i 0.959351 + 0.282216i \(0.0910696\pi\)
−0.959351 + 0.282216i \(0.908930\pi\)
\(114\) −0.836381 −0.0783342
\(115\) 0 0
\(116\) 7.08840 0.658141
\(117\) 2.96239i 0.273873i
\(118\) 2.07522i 0.191040i
\(119\) 15.3258 1.40492
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 1.68735i 0.152765i
\(123\) 4.38787i 0.395641i
\(124\) 19.4763 1.74902
\(125\) 0 0
\(126\) −0.649738 −0.0578833
\(127\) − 14.5745i − 1.29328i −0.762796 0.646640i \(-0.776174\pi\)
0.762796 0.646640i \(-0.223826\pi\)
\(128\) − 5.91748i − 0.523037i
\(129\) 9.27504 0.816622
\(130\) 0 0
\(131\) −5.92478 −0.517650 −0.258825 0.965924i \(-0.583335\pi\)
−0.258825 + 0.965924i \(0.583335\pi\)
\(132\) − 1.96239i − 0.170804i
\(133\) 14.4485i 1.25284i
\(134\) 1.14903 0.0992612
\(135\) 0 0
\(136\) −3.51530 −0.301434
\(137\) 13.8496i 1.18325i 0.806214 + 0.591624i \(0.201513\pi\)
−0.806214 + 0.591624i \(0.798487\pi\)
\(138\) − 1.29948i − 0.110619i
\(139\) −13.6121 −1.15457 −0.577283 0.816544i \(-0.695887\pi\)
−0.577283 + 0.816544i \(0.695887\pi\)
\(140\) 0 0
\(141\) −9.92478 −0.835817
\(142\) − 1.92478i − 0.161524i
\(143\) − 2.96239i − 0.247727i
\(144\) −3.77575 −0.314646
\(145\) 0 0
\(146\) 1.50071 0.124199
\(147\) 4.22425i 0.348411i
\(148\) − 3.92478i − 0.322615i
\(149\) −1.53690 −0.125908 −0.0629540 0.998016i \(-0.520052\pi\)
−0.0629540 + 0.998016i \(0.520052\pi\)
\(150\) 0 0
\(151\) −6.76116 −0.550215 −0.275108 0.961413i \(-0.588713\pi\)
−0.275108 + 0.961413i \(0.588713\pi\)
\(152\) − 3.31406i − 0.268806i
\(153\) 4.57452i 0.369828i
\(154\) 0.649738 0.0523574
\(155\) 0 0
\(156\) −5.81336 −0.465441
\(157\) − 5.47627i − 0.437054i −0.975831 0.218527i \(-0.929875\pi\)
0.975831 0.218527i \(-0.0701252\pi\)
\(158\) 2.23743i 0.178000i
\(159\) −4.70052 −0.372776
\(160\) 0 0
\(161\) −22.4485 −1.76919
\(162\) − 0.193937i − 0.0152371i
\(163\) − 12.6253i − 0.988890i −0.869209 0.494445i \(-0.835371\pi\)
0.869209 0.494445i \(-0.164629\pi\)
\(164\) −8.61071 −0.672384
\(165\) 0 0
\(166\) −2.11142 −0.163878
\(167\) 18.3634i 1.42101i 0.703695 + 0.710503i \(0.251532\pi\)
−0.703695 + 0.710503i \(0.748468\pi\)
\(168\) − 2.57452i − 0.198628i
\(169\) 4.22425 0.324943
\(170\) 0 0
\(171\) −4.31265 −0.329797
\(172\) 18.2012i 1.38783i
\(173\) 8.57452i 0.651908i 0.945386 + 0.325954i \(0.105686\pi\)
−0.945386 + 0.325954i \(0.894314\pi\)
\(174\) −0.700523 −0.0531065
\(175\) 0 0
\(176\) 3.77575 0.284608
\(177\) 10.7005i 0.804301i
\(178\) − 0.538319i − 0.0403487i
\(179\) −14.1768 −1.05962 −0.529812 0.848115i \(-0.677737\pi\)
−0.529812 + 0.848115i \(0.677737\pi\)
\(180\) 0 0
\(181\) −5.22425 −0.388316 −0.194158 0.980970i \(-0.562197\pi\)
−0.194158 + 0.980970i \(0.562197\pi\)
\(182\) − 1.92478i − 0.142674i
\(183\) 8.70052i 0.643161i
\(184\) 5.14903 0.379592
\(185\) 0 0
\(186\) −1.92478 −0.141132
\(187\) − 4.57452i − 0.334522i
\(188\) − 19.4763i − 1.42045i
\(189\) −3.35026 −0.243696
\(190\) 0 0
\(191\) −16.6253 −1.20296 −0.601482 0.798886i \(-0.705423\pi\)
−0.601482 + 0.798886i \(0.705423\pi\)
\(192\) − 7.11142i − 0.513222i
\(193\) 16.3634i 1.17787i 0.808182 + 0.588933i \(0.200452\pi\)
−0.808182 + 0.588933i \(0.799548\pi\)
\(194\) 0.0145884 0.00104739
\(195\) 0 0
\(196\) −8.28963 −0.592116
\(197\) − 20.4241i − 1.45515i −0.686026 0.727577i \(-0.740646\pi\)
0.686026 0.727577i \(-0.259354\pi\)
\(198\) 0.193937i 0.0137825i
\(199\) 8.62530 0.611431 0.305716 0.952123i \(-0.401104\pi\)
0.305716 + 0.952123i \(0.401104\pi\)
\(200\) 0 0
\(201\) 5.92478 0.417902
\(202\) 2.92619i 0.205886i
\(203\) 12.1016i 0.849364i
\(204\) −8.97698 −0.628514
\(205\) 0 0
\(206\) 0.625301 0.0435668
\(207\) − 6.70052i − 0.465719i
\(208\) − 11.1852i − 0.775556i
\(209\) 4.31265 0.298312
\(210\) 0 0
\(211\) 9.08840 0.625671 0.312836 0.949807i \(-0.398721\pi\)
0.312836 + 0.949807i \(0.398721\pi\)
\(212\) − 9.22425i − 0.633524i
\(213\) − 9.92478i − 0.680035i
\(214\) −0.186642 −0.0127586
\(215\) 0 0
\(216\) 0.768452 0.0522865
\(217\) 33.2506i 2.25720i
\(218\) 2.21108i 0.149753i
\(219\) 7.73813 0.522895
\(220\) 0 0
\(221\) −13.5515 −0.911572
\(222\) 0.387873i 0.0260323i
\(223\) 6.70052i 0.448700i 0.974509 + 0.224350i \(0.0720259\pi\)
−0.974509 + 0.224350i \(0.927974\pi\)
\(224\) 7.60228 0.507949
\(225\) 0 0
\(226\) 1.16362 0.0774028
\(227\) 16.9624i 1.12583i 0.826514 + 0.562917i \(0.190321\pi\)
−0.826514 + 0.562917i \(0.809679\pi\)
\(228\) − 8.46310i − 0.560482i
\(229\) −25.8496 −1.70819 −0.854093 0.520120i \(-0.825887\pi\)
−0.854093 + 0.520120i \(0.825887\pi\)
\(230\) 0 0
\(231\) 3.35026 0.220431
\(232\) − 2.77575i − 0.182237i
\(233\) 19.2750i 1.26275i 0.775478 + 0.631375i \(0.217509\pi\)
−0.775478 + 0.631375i \(0.782491\pi\)
\(234\) 0.574515 0.0375573
\(235\) 0 0
\(236\) −20.9986 −1.36689
\(237\) 11.5369i 0.749402i
\(238\) − 2.97224i − 0.192662i
\(239\) −26.5501 −1.71738 −0.858691 0.512494i \(-0.828722\pi\)
−0.858691 + 0.512494i \(0.828722\pi\)
\(240\) 0 0
\(241\) 28.5501 1.83907 0.919536 0.393006i \(-0.128565\pi\)
0.919536 + 0.393006i \(0.128565\pi\)
\(242\) − 0.193937i − 0.0124667i
\(243\) − 1.00000i − 0.0641500i
\(244\) −17.0738 −1.09304
\(245\) 0 0
\(246\) 0.850969 0.0542558
\(247\) − 12.7757i − 0.812901i
\(248\) − 7.62672i − 0.484297i
\(249\) −10.8872 −0.689946
\(250\) 0 0
\(251\) 29.9248 1.88884 0.944418 0.328748i \(-0.106627\pi\)
0.944418 + 0.328748i \(0.106627\pi\)
\(252\) − 6.57452i − 0.414156i
\(253\) 6.70052i 0.421258i
\(254\) −2.82653 −0.177352
\(255\) 0 0
\(256\) 13.0752 0.817201
\(257\) 8.70052i 0.542724i 0.962477 + 0.271362i \(0.0874740\pi\)
−0.962477 + 0.271362i \(0.912526\pi\)
\(258\) − 1.79877i − 0.111986i
\(259\) 6.70052 0.416350
\(260\) 0 0
\(261\) −3.61213 −0.223585
\(262\) 1.14903i 0.0709874i
\(263\) − 12.2882i − 0.757724i −0.925453 0.378862i \(-0.876316\pi\)
0.925453 0.378862i \(-0.123684\pi\)
\(264\) −0.768452 −0.0472950
\(265\) 0 0
\(266\) 2.80209 0.171807
\(267\) − 2.77575i − 0.169873i
\(268\) 11.6267i 0.710215i
\(269\) 5.84955 0.356654 0.178327 0.983971i \(-0.442932\pi\)
0.178327 + 0.983971i \(0.442932\pi\)
\(270\) 0 0
\(271\) −5.08840 −0.309098 −0.154549 0.987985i \(-0.549392\pi\)
−0.154549 + 0.987985i \(0.549392\pi\)
\(272\) − 17.2722i − 1.04728i
\(273\) − 9.92478i − 0.600675i
\(274\) 2.68594 0.162263
\(275\) 0 0
\(276\) 13.1490 0.791479
\(277\) 1.41090i 0.0847725i 0.999101 + 0.0423863i \(0.0134960\pi\)
−0.999101 + 0.0423863i \(0.986504\pi\)
\(278\) 2.63989i 0.158330i
\(279\) −9.92478 −0.594181
\(280\) 0 0
\(281\) −4.38787 −0.261759 −0.130879 0.991398i \(-0.541780\pi\)
−0.130879 + 0.991398i \(0.541780\pi\)
\(282\) 1.92478i 0.114619i
\(283\) − 26.5745i − 1.57969i −0.613306 0.789845i \(-0.710161\pi\)
0.613306 0.789845i \(-0.289839\pi\)
\(284\) 19.4763 1.15570
\(285\) 0 0
\(286\) −0.574515 −0.0339718
\(287\) − 14.7005i − 0.867744i
\(288\) 2.26916i 0.133711i
\(289\) −3.92619 −0.230952
\(290\) 0 0
\(291\) 0.0752228 0.00440964
\(292\) 15.1852i 0.888648i
\(293\) 3.42548i 0.200119i 0.994981 + 0.100059i \(0.0319033\pi\)
−0.994981 + 0.100059i \(0.968097\pi\)
\(294\) 0.819237 0.0477789
\(295\) 0 0
\(296\) −1.53690 −0.0893307
\(297\) 1.00000i 0.0580259i
\(298\) 0.298062i 0.0172663i
\(299\) 19.8496 1.14793
\(300\) 0 0
\(301\) −31.0738 −1.79106
\(302\) 1.31124i 0.0754531i
\(303\) 15.0884i 0.866806i
\(304\) 16.2835 0.933921
\(305\) 0 0
\(306\) 0.887166 0.0507159
\(307\) − 16.6497i − 0.950251i −0.879918 0.475125i \(-0.842403\pi\)
0.879918 0.475125i \(-0.157597\pi\)
\(308\) 6.57452i 0.374618i
\(309\) 3.22425 0.183421
\(310\) 0 0
\(311\) 32.9986 1.87118 0.935589 0.353091i \(-0.114869\pi\)
0.935589 + 0.353091i \(0.114869\pi\)
\(312\) 2.27645i 0.128879i
\(313\) − 15.4010i − 0.870519i −0.900305 0.435259i \(-0.856657\pi\)
0.900305 0.435259i \(-0.143343\pi\)
\(314\) −1.06205 −0.0599349
\(315\) 0 0
\(316\) −22.6399 −1.27359
\(317\) 2.15045i 0.120781i 0.998175 + 0.0603905i \(0.0192346\pi\)
−0.998175 + 0.0603905i \(0.980765\pi\)
\(318\) 0.911603i 0.0511202i
\(319\) 3.61213 0.202240
\(320\) 0 0
\(321\) −0.962389 −0.0537153
\(322\) 4.35359i 0.242616i
\(323\) − 19.7283i − 1.09771i
\(324\) 1.96239 0.109022
\(325\) 0 0
\(326\) −2.44851 −0.135610
\(327\) 11.4010i 0.630479i
\(328\) 3.37187i 0.186180i
\(329\) 33.2506 1.83316
\(330\) 0 0
\(331\) −14.5501 −0.799745 −0.399872 0.916571i \(-0.630946\pi\)
−0.399872 + 0.916571i \(0.630946\pi\)
\(332\) − 21.3649i − 1.17255i
\(333\) 2.00000i 0.109599i
\(334\) 3.56134 0.194868
\(335\) 0 0
\(336\) 12.6497 0.690100
\(337\) 16.2619i 0.885840i 0.896561 + 0.442920i \(0.146057\pi\)
−0.896561 + 0.442920i \(0.853943\pi\)
\(338\) − 0.819237i − 0.0445606i
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) 9.92478 0.537457
\(342\) 0.836381i 0.0452263i
\(343\) 9.29948i 0.502125i
\(344\) 7.12742 0.384285
\(345\) 0 0
\(346\) 1.66291 0.0893987
\(347\) − 0.962389i − 0.0516637i −0.999666 0.0258319i \(-0.991777\pi\)
0.999666 0.0258319i \(-0.00822345\pi\)
\(348\) − 7.08840i − 0.379978i
\(349\) −20.7005 −1.10807 −0.554037 0.832492i \(-0.686913\pi\)
−0.554037 + 0.832492i \(0.686913\pi\)
\(350\) 0 0
\(351\) 2.96239 0.158121
\(352\) − 2.26916i − 0.120947i
\(353\) − 20.5501i − 1.09377i −0.837208 0.546885i \(-0.815813\pi\)
0.837208 0.546885i \(-0.184187\pi\)
\(354\) 2.07522 0.110297
\(355\) 0 0
\(356\) 5.44709 0.288695
\(357\) − 15.3258i − 0.811129i
\(358\) 2.74940i 0.145310i
\(359\) −17.9248 −0.946034 −0.473017 0.881053i \(-0.656835\pi\)
−0.473017 + 0.881053i \(0.656835\pi\)
\(360\) 0 0
\(361\) −0.401047 −0.0211077
\(362\) 1.01317i 0.0532512i
\(363\) − 1.00000i − 0.0524864i
\(364\) 19.4763 1.02083
\(365\) 0 0
\(366\) 1.68735 0.0881992
\(367\) − 29.6531i − 1.54788i −0.633261 0.773939i \(-0.718284\pi\)
0.633261 0.773939i \(-0.281716\pi\)
\(368\) 25.2995i 1.31883i
\(369\) 4.38787 0.228424
\(370\) 0 0
\(371\) 15.7480 0.817595
\(372\) − 19.4763i − 1.00980i
\(373\) 9.13918i 0.473209i 0.971606 + 0.236604i \(0.0760345\pi\)
−0.971606 + 0.236604i \(0.923965\pi\)
\(374\) −0.887166 −0.0458743
\(375\) 0 0
\(376\) −7.62672 −0.393318
\(377\) − 10.7005i − 0.551105i
\(378\) 0.649738i 0.0334189i
\(379\) 20.0000 1.02733 0.513665 0.857991i \(-0.328287\pi\)
0.513665 + 0.857991i \(0.328287\pi\)
\(380\) 0 0
\(381\) −14.5745 −0.746675
\(382\) 3.22425i 0.164967i
\(383\) 34.9234i 1.78450i 0.451541 + 0.892250i \(0.350874\pi\)
−0.451541 + 0.892250i \(0.649126\pi\)
\(384\) −5.91748 −0.301975
\(385\) 0 0
\(386\) 3.17347 0.161525
\(387\) − 9.27504i − 0.471477i
\(388\) 0.147616i 0.00749408i
\(389\) −2.77575 −0.140736 −0.0703680 0.997521i \(-0.522417\pi\)
−0.0703680 + 0.997521i \(0.522417\pi\)
\(390\) 0 0
\(391\) 30.6516 1.55012
\(392\) 3.24614i 0.163955i
\(393\) 5.92478i 0.298865i
\(394\) −3.96097 −0.199551
\(395\) 0 0
\(396\) −1.96239 −0.0986137
\(397\) − 19.9248i − 0.999996i −0.866027 0.499998i \(-0.833334\pi\)
0.866027 0.499998i \(-0.166666\pi\)
\(398\) − 1.67276i − 0.0838479i
\(399\) 14.4485 0.723330
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) − 1.14903i − 0.0573085i
\(403\) − 29.4010i − 1.46457i
\(404\) −29.6093 −1.47312
\(405\) 0 0
\(406\) 2.34694 0.116477
\(407\) − 2.00000i − 0.0991363i
\(408\) 3.51530i 0.174033i
\(409\) 13.0738 0.646458 0.323229 0.946321i \(-0.395232\pi\)
0.323229 + 0.946321i \(0.395232\pi\)
\(410\) 0 0
\(411\) 13.8496 0.683148
\(412\) 6.32724i 0.311721i
\(413\) − 35.8496i − 1.76404i
\(414\) −1.29948 −0.0638658
\(415\) 0 0
\(416\) −6.72213 −0.329580
\(417\) 13.6121i 0.666589i
\(418\) − 0.836381i − 0.0409087i
\(419\) −7.22425 −0.352928 −0.176464 0.984307i \(-0.556466\pi\)
−0.176464 + 0.984307i \(0.556466\pi\)
\(420\) 0 0
\(421\) 30.6253 1.49259 0.746293 0.665618i \(-0.231832\pi\)
0.746293 + 0.665618i \(0.231832\pi\)
\(422\) − 1.76257i − 0.0858007i
\(423\) 9.92478i 0.482559i
\(424\) −3.61213 −0.175420
\(425\) 0 0
\(426\) −1.92478 −0.0932558
\(427\) − 29.1490i − 1.41062i
\(428\) − 1.88858i − 0.0912880i
\(429\) −2.96239 −0.143025
\(430\) 0 0
\(431\) −33.8759 −1.63174 −0.815872 0.578232i \(-0.803743\pi\)
−0.815872 + 0.578232i \(0.803743\pi\)
\(432\) 3.77575i 0.181661i
\(433\) 9.47627i 0.455400i 0.973731 + 0.227700i \(0.0731206\pi\)
−0.973731 + 0.227700i \(0.926879\pi\)
\(434\) 6.44851 0.309538
\(435\) 0 0
\(436\) −22.3733 −1.07149
\(437\) 28.8970i 1.38233i
\(438\) − 1.50071i − 0.0717066i
\(439\) 29.4617 1.40613 0.703065 0.711126i \(-0.251814\pi\)
0.703065 + 0.711126i \(0.251814\pi\)
\(440\) 0 0
\(441\) 4.22425 0.201155
\(442\) 2.62813i 0.125007i
\(443\) 19.0738i 0.906224i 0.891454 + 0.453112i \(0.149686\pi\)
−0.891454 + 0.453112i \(0.850314\pi\)
\(444\) −3.92478 −0.186262
\(445\) 0 0
\(446\) 1.29948 0.0615320
\(447\) 1.53690i 0.0726931i
\(448\) 23.8251i 1.12563i
\(449\) −35.8759 −1.69309 −0.846544 0.532318i \(-0.821321\pi\)
−0.846544 + 0.532318i \(0.821321\pi\)
\(450\) 0 0
\(451\) −4.38787 −0.206617
\(452\) 11.7743i 0.553818i
\(453\) 6.76116i 0.317667i
\(454\) 3.28963 0.154390
\(455\) 0 0
\(456\) −3.31406 −0.155195
\(457\) 5.28963i 0.247438i 0.992317 + 0.123719i \(0.0394822\pi\)
−0.992317 + 0.123719i \(0.960518\pi\)
\(458\) 5.01317i 0.234250i
\(459\) 4.57452 0.213520
\(460\) 0 0
\(461\) 36.3390 1.69248 0.846238 0.532805i \(-0.178862\pi\)
0.846238 + 0.532805i \(0.178862\pi\)
\(462\) − 0.649738i − 0.0302286i
\(463\) − 10.5501i − 0.490304i −0.969485 0.245152i \(-0.921162\pi\)
0.969485 0.245152i \(-0.0788378\pi\)
\(464\) 13.6385 0.633150
\(465\) 0 0
\(466\) 3.73813 0.173166
\(467\) 18.7005i 0.865357i 0.901548 + 0.432679i \(0.142431\pi\)
−0.901548 + 0.432679i \(0.857569\pi\)
\(468\) 5.81336i 0.268723i
\(469\) −19.8496 −0.916567
\(470\) 0 0
\(471\) −5.47627 −0.252333
\(472\) 8.22284i 0.378487i
\(473\) 9.27504i 0.426467i
\(474\) 2.23743 0.102768
\(475\) 0 0
\(476\) 30.0752 1.37850
\(477\) 4.70052i 0.215222i
\(478\) 5.14903i 0.235511i
\(479\) 9.29948 0.424904 0.212452 0.977172i \(-0.431855\pi\)
0.212452 + 0.977172i \(0.431855\pi\)
\(480\) 0 0
\(481\) −5.92478 −0.270147
\(482\) − 5.53690i − 0.252199i
\(483\) 22.4485i 1.02144i
\(484\) 1.96239 0.0891995
\(485\) 0 0
\(486\) −0.193937 −0.00879714
\(487\) − 35.4763i − 1.60758i −0.594911 0.803792i \(-0.702813\pi\)
0.594911 0.803792i \(-0.297187\pi\)
\(488\) 6.68594i 0.302658i
\(489\) −12.6253 −0.570936
\(490\) 0 0
\(491\) 24.7757 1.11811 0.559057 0.829129i \(-0.311163\pi\)
0.559057 + 0.829129i \(0.311163\pi\)
\(492\) 8.61071i 0.388201i
\(493\) − 16.5237i − 0.744191i
\(494\) −2.47768 −0.111476
\(495\) 0 0
\(496\) 37.4734 1.68261
\(497\) 33.2506i 1.49149i
\(498\) 2.11142i 0.0946150i
\(499\) −14.1768 −0.634640 −0.317320 0.948318i \(-0.602783\pi\)
−0.317320 + 0.948318i \(0.602783\pi\)
\(500\) 0 0
\(501\) 18.3634 0.820418
\(502\) − 5.80351i − 0.259023i
\(503\) 8.43866i 0.376261i 0.982144 + 0.188131i \(0.0602428\pi\)
−0.982144 + 0.188131i \(0.939757\pi\)
\(504\) −2.57452 −0.114678
\(505\) 0 0
\(506\) 1.29948 0.0577688
\(507\) − 4.22425i − 0.187606i
\(508\) − 28.6009i − 1.26896i
\(509\) −1.10299 −0.0488890 −0.0244445 0.999701i \(-0.507782\pi\)
−0.0244445 + 0.999701i \(0.507782\pi\)
\(510\) 0 0
\(511\) −25.9248 −1.14684
\(512\) − 14.3707i − 0.635103i
\(513\) 4.31265i 0.190408i
\(514\) 1.68735 0.0744258
\(515\) 0 0
\(516\) 18.2012 0.801265
\(517\) − 9.92478i − 0.436491i
\(518\) − 1.29948i − 0.0570957i
\(519\) 8.57452 0.376379
\(520\) 0 0
\(521\) −12.4485 −0.545379 −0.272690 0.962102i \(-0.587913\pi\)
−0.272690 + 0.962102i \(0.587913\pi\)
\(522\) 0.700523i 0.0306611i
\(523\) − 30.0508i − 1.31403i −0.753878 0.657015i \(-0.771819\pi\)
0.753878 0.657015i \(-0.228181\pi\)
\(524\) −11.6267 −0.507915
\(525\) 0 0
\(526\) −2.38313 −0.103910
\(527\) − 45.4010i − 1.97770i
\(528\) − 3.77575i − 0.164318i
\(529\) −21.8970 −0.952044
\(530\) 0 0
\(531\) 10.7005 0.464363
\(532\) 28.3536i 1.22928i
\(533\) 12.9986i 0.563031i
\(534\) −0.538319 −0.0232953
\(535\) 0 0
\(536\) 4.55291 0.196656
\(537\) 14.1768i 0.611774i
\(538\) − 1.13444i − 0.0489093i
\(539\) −4.22425 −0.181951
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 0.986826i 0.0423878i
\(543\) 5.22425i 0.224194i
\(544\) −10.3803 −0.445052
\(545\) 0 0
\(546\) −1.92478 −0.0823729
\(547\) − 14.3028i − 0.611544i −0.952105 0.305772i \(-0.901086\pi\)
0.952105 0.305772i \(-0.0989145\pi\)
\(548\) 27.1782i 1.16100i
\(549\) 8.70052 0.371329
\(550\) 0 0
\(551\) 15.5778 0.663638
\(552\) − 5.14903i − 0.219157i
\(553\) − 38.6516i − 1.64364i
\(554\) 0.273624 0.0116252
\(555\) 0 0
\(556\) −26.7123 −1.13285
\(557\) − 11.7988i − 0.499930i −0.968255 0.249965i \(-0.919581\pi\)
0.968255 0.249965i \(-0.0804191\pi\)
\(558\) 1.92478i 0.0814823i
\(559\) 27.4763 1.16212
\(560\) 0 0
\(561\) −4.57452 −0.193136
\(562\) 0.850969i 0.0358960i
\(563\) − 30.4847i − 1.28478i −0.766379 0.642389i \(-0.777944\pi\)
0.766379 0.642389i \(-0.222056\pi\)
\(564\) −19.4763 −0.820099
\(565\) 0 0
\(566\) −5.15377 −0.216629
\(567\) 3.35026i 0.140698i
\(568\) − 7.62672i − 0.320010i
\(569\) 27.0884 1.13560 0.567802 0.823165i \(-0.307794\pi\)
0.567802 + 0.823165i \(0.307794\pi\)
\(570\) 0 0
\(571\) 7.28489 0.304863 0.152432 0.988314i \(-0.451290\pi\)
0.152432 + 0.988314i \(0.451290\pi\)
\(572\) − 5.81336i − 0.243069i
\(573\) 16.6253i 0.694532i
\(574\) −2.85097 −0.118997
\(575\) 0 0
\(576\) −7.11142 −0.296309
\(577\) − 31.6239i − 1.31652i −0.752791 0.658260i \(-0.771293\pi\)
0.752791 0.658260i \(-0.228707\pi\)
\(578\) 0.761432i 0.0316714i
\(579\) 16.3634 0.680041
\(580\) 0 0
\(581\) 36.4749 1.51323
\(582\) − 0.0145884i 0 0.000604711i
\(583\) − 4.70052i − 0.194676i
\(584\) 5.94639 0.246063
\(585\) 0 0
\(586\) 0.664327 0.0274431
\(587\) 33.1490i 1.36821i 0.729385 + 0.684103i \(0.239806\pi\)
−0.729385 + 0.684103i \(0.760194\pi\)
\(588\) 8.28963i 0.341858i
\(589\) 42.8021 1.76363
\(590\) 0 0
\(591\) −20.4241 −0.840134
\(592\) − 7.55149i − 0.310364i
\(593\) − 34.4993i − 1.41672i −0.705853 0.708358i \(-0.749436\pi\)
0.705853 0.708358i \(-0.250564\pi\)
\(594\) 0.193937 0.00795731
\(595\) 0 0
\(596\) −3.01600 −0.123540
\(597\) − 8.62530i − 0.353010i
\(598\) − 3.84955i − 0.157420i
\(599\) 14.4485 0.590350 0.295175 0.955443i \(-0.404622\pi\)
0.295175 + 0.955443i \(0.404622\pi\)
\(600\) 0 0
\(601\) −15.9248 −0.649585 −0.324793 0.945785i \(-0.605295\pi\)
−0.324793 + 0.945785i \(0.605295\pi\)
\(602\) 6.02635i 0.245616i
\(603\) − 5.92478i − 0.241276i
\(604\) −13.2680 −0.539868
\(605\) 0 0
\(606\) 2.92619 0.118868
\(607\) − 14.5745i − 0.591561i −0.955256 0.295781i \(-0.904420\pi\)
0.955256 0.295781i \(-0.0955798\pi\)
\(608\) − 9.78609i − 0.396878i
\(609\) 12.1016 0.490380
\(610\) 0 0
\(611\) −29.4010 −1.18944
\(612\) 8.97698i 0.362873i
\(613\) − 16.4123i − 0.662887i −0.943475 0.331443i \(-0.892464\pi\)
0.943475 0.331443i \(-0.107536\pi\)
\(614\) −3.22899 −0.130312
\(615\) 0 0
\(616\) 2.57452 0.103730
\(617\) − 17.8496i − 0.718596i −0.933223 0.359298i \(-0.883016\pi\)
0.933223 0.359298i \(-0.116984\pi\)
\(618\) − 0.625301i − 0.0251533i
\(619\) 0.402462 0.0161763 0.00808815 0.999967i \(-0.497425\pi\)
0.00808815 + 0.999967i \(0.497425\pi\)
\(620\) 0 0
\(621\) −6.70052 −0.268883
\(622\) − 6.39963i − 0.256602i
\(623\) 9.29948i 0.372576i
\(624\) −11.1852 −0.447767
\(625\) 0 0
\(626\) −2.98683 −0.119378
\(627\) − 4.31265i − 0.172231i
\(628\) − 10.7466i − 0.428835i
\(629\) −9.14903 −0.364796
\(630\) 0 0
\(631\) −38.0263 −1.51380 −0.756902 0.653528i \(-0.773288\pi\)
−0.756902 + 0.653528i \(0.773288\pi\)
\(632\) 8.86556i 0.352653i
\(633\) − 9.08840i − 0.361231i
\(634\) 0.417050 0.0165632
\(635\) 0 0
\(636\) −9.22425 −0.365765
\(637\) 12.5139i 0.495818i
\(638\) − 0.700523i − 0.0277340i
\(639\) −9.92478 −0.392618
\(640\) 0 0
\(641\) −28.0263 −1.10697 −0.553487 0.832858i \(-0.686703\pi\)
−0.553487 + 0.832858i \(0.686703\pi\)
\(642\) 0.186642i 0.00736619i
\(643\) 4.62530i 0.182404i 0.995832 + 0.0912020i \(0.0290709\pi\)
−0.995832 + 0.0912020i \(0.970929\pi\)
\(644\) −44.0527 −1.73592
\(645\) 0 0
\(646\) −3.82604 −0.150533
\(647\) 23.5778i 0.926941i 0.886112 + 0.463470i \(0.153396\pi\)
−0.886112 + 0.463470i \(0.846604\pi\)
\(648\) − 0.768452i − 0.0301876i
\(649\) −10.7005 −0.420032
\(650\) 0 0
\(651\) 33.2506 1.30319
\(652\) − 24.7757i − 0.970293i
\(653\) 2.25202i 0.0881282i 0.999029 + 0.0440641i \(0.0140306\pi\)
−0.999029 + 0.0440641i \(0.985969\pi\)
\(654\) 2.21108 0.0864601
\(655\) 0 0
\(656\) −16.5675 −0.646852
\(657\) − 7.73813i − 0.301893i
\(658\) − 6.44851i − 0.251389i
\(659\) 41.4010 1.61276 0.806378 0.591401i \(-0.201425\pi\)
0.806378 + 0.591401i \(0.201425\pi\)
\(660\) 0 0
\(661\) 3.40105 0.132285 0.0661427 0.997810i \(-0.478931\pi\)
0.0661427 + 0.997810i \(0.478931\pi\)
\(662\) 2.82179i 0.109672i
\(663\) 13.5515i 0.526296i
\(664\) −8.36626 −0.324674
\(665\) 0 0
\(666\) 0.387873 0.0150298
\(667\) 24.2031i 0.937149i
\(668\) 36.0362i 1.39428i
\(669\) 6.70052 0.259057
\(670\) 0 0
\(671\) −8.70052 −0.335880
\(672\) − 7.60228i − 0.293264i
\(673\) − 0.887166i − 0.0341977i −0.999854 0.0170989i \(-0.994557\pi\)
0.999854 0.0170989i \(-0.00544300\pi\)
\(674\) 3.15377 0.121479
\(675\) 0 0
\(676\) 8.28963 0.318832
\(677\) 18.9018i 0.726453i 0.931701 + 0.363227i \(0.118325\pi\)
−0.931701 + 0.363227i \(0.881675\pi\)
\(678\) − 1.16362i − 0.0446885i
\(679\) −0.252016 −0.00967149
\(680\) 0 0
\(681\) 16.9624 0.650000
\(682\) − 1.92478i − 0.0737035i
\(683\) 20.8773i 0.798848i 0.916766 + 0.399424i \(0.130790\pi\)
−0.916766 + 0.399424i \(0.869210\pi\)
\(684\) −8.46310 −0.323595
\(685\) 0 0
\(686\) 1.80351 0.0688583
\(687\) 25.8496i 0.986222i
\(688\) 35.0202i 1.33513i
\(689\) −13.9248 −0.530492
\(690\) 0 0
\(691\) −2.44851 −0.0931456 −0.0465728 0.998915i \(-0.514830\pi\)
−0.0465728 + 0.998915i \(0.514830\pi\)
\(692\) 16.8265i 0.639649i
\(693\) − 3.35026i − 0.127266i
\(694\) −0.186642 −0.00708485
\(695\) 0 0
\(696\) −2.77575 −0.105214
\(697\) 20.0724i 0.760296i
\(698\) 4.01459i 0.151954i
\(699\) 19.2750 0.729049
\(700\) 0 0
\(701\) −2.98683 −0.112811 −0.0564054 0.998408i \(-0.517964\pi\)
−0.0564054 + 0.998408i \(0.517964\pi\)
\(702\) − 0.574515i − 0.0216837i
\(703\) − 8.62530i − 0.325309i
\(704\) 7.11142 0.268022
\(705\) 0 0
\(706\) −3.98541 −0.149993
\(707\) − 50.5501i − 1.90113i
\(708\) 20.9986i 0.789175i
\(709\) −24.1768 −0.907979 −0.453989 0.891007i \(-0.650000\pi\)
−0.453989 + 0.891007i \(0.650000\pi\)
\(710\) 0 0
\(711\) 11.5369 0.432668
\(712\) − 2.13303i − 0.0799386i
\(713\) 66.5012i 2.49049i
\(714\) −2.97224 −0.111233
\(715\) 0 0
\(716\) −27.8204 −1.03970
\(717\) 26.5501i 0.991531i
\(718\) 3.47627i 0.129733i
\(719\) 30.0263 1.11979 0.559897 0.828562i \(-0.310841\pi\)
0.559897 + 0.828562i \(0.310841\pi\)
\(720\) 0 0
\(721\) −10.8021 −0.402291
\(722\) 0.0777777i 0.00289459i
\(723\) − 28.5501i − 1.06179i
\(724\) −10.2520 −0.381013
\(725\) 0 0
\(726\) −0.193937 −0.00719766
\(727\) 14.9525i 0.554559i 0.960789 + 0.277279i \(0.0894328\pi\)
−0.960789 + 0.277279i \(0.910567\pi\)
\(728\) − 7.62672i − 0.282665i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 42.4288 1.56929
\(732\) 17.0738i 0.631066i
\(733\) 19.1128i 0.705949i 0.935633 + 0.352974i \(0.114830\pi\)
−0.935633 + 0.352974i \(0.885170\pi\)
\(734\) −5.75081 −0.212266
\(735\) 0 0
\(736\) 15.2046 0.560447
\(737\) 5.92478i 0.218242i
\(738\) − 0.850969i − 0.0313246i
\(739\) 3.31406 0.121910 0.0609549 0.998141i \(-0.480585\pi\)
0.0609549 + 0.998141i \(0.480585\pi\)
\(740\) 0 0
\(741\) −12.7757 −0.469329
\(742\) − 3.05411i − 0.112120i
\(743\) 34.9887i 1.28361i 0.766867 + 0.641806i \(0.221815\pi\)
−0.766867 + 0.641806i \(0.778185\pi\)
\(744\) −7.62672 −0.279609
\(745\) 0 0
\(746\) 1.77242 0.0648930
\(747\) 10.8872i 0.398341i
\(748\) − 8.97698i − 0.328231i
\(749\) 3.22425 0.117812
\(750\) 0 0
\(751\) −26.9234 −0.982447 −0.491224 0.871033i \(-0.663450\pi\)
−0.491224 + 0.871033i \(0.663450\pi\)
\(752\) − 37.4734i − 1.36652i
\(753\) − 29.9248i − 1.09052i
\(754\) −2.07522 −0.0755752
\(755\) 0 0
\(756\) −6.57452 −0.239113
\(757\) 15.9248i 0.578796i 0.957209 + 0.289398i \(0.0934551\pi\)
−0.957209 + 0.289398i \(0.906545\pi\)
\(758\) − 3.87873i − 0.140882i
\(759\) 6.70052 0.243214
\(760\) 0 0
\(761\) −30.9380 −1.12150 −0.560750 0.827985i \(-0.689487\pi\)
−0.560750 + 0.827985i \(0.689487\pi\)
\(762\) 2.82653i 0.102394i
\(763\) − 38.1965i − 1.38281i
\(764\) −32.6253 −1.18034
\(765\) 0 0
\(766\) 6.77292 0.244715
\(767\) 31.6991i 1.14459i
\(768\) − 13.0752i − 0.471811i
\(769\) −9.32582 −0.336298 −0.168149 0.985762i \(-0.553779\pi\)
−0.168149 + 0.985762i \(0.553779\pi\)
\(770\) 0 0
\(771\) 8.70052 0.313342
\(772\) 32.1114i 1.15572i
\(773\) − 44.7005i − 1.60777i −0.594787 0.803883i \(-0.702764\pi\)
0.594787 0.803883i \(-0.297236\pi\)
\(774\) −1.79877 −0.0646554
\(775\) 0 0
\(776\) 0.0578051 0.00207508
\(777\) − 6.70052i − 0.240380i
\(778\) 0.538319i 0.0192997i
\(779\) −18.9234 −0.678000
\(780\) 0 0
\(781\) 9.92478 0.355136
\(782\) − 5.94448i − 0.212574i
\(783\) 3.61213i 0.129087i
\(784\) −15.9497 −0.569633
\(785\) 0 0
\(786\) 1.14903 0.0409846
\(787\) 21.6775i 0.772719i 0.922348 + 0.386360i \(0.126268\pi\)
−0.922348 + 0.386360i \(0.873732\pi\)
\(788\) − 40.0800i − 1.42779i
\(789\) −12.2882 −0.437472
\(790\) 0 0
\(791\) −20.1016 −0.714730
\(792\) 0.768452i 0.0273058i
\(793\) 25.7743i 0.915273i
\(794\) −3.86414 −0.137133
\(795\) 0 0
\(796\) 16.9262 0.599933
\(797\) 22.7466i 0.805725i 0.915261 + 0.402862i \(0.131985\pi\)
−0.915261 + 0.402862i \(0.868015\pi\)
\(798\) − 2.80209i − 0.0991930i
\(799\) −45.4010 −1.60617
\(800\) 0 0
\(801\) −2.77575 −0.0980762
\(802\) − 0.387873i − 0.0136963i
\(803\) 7.73813i 0.273073i
\(804\) 11.6267 0.410043
\(805\) 0 0
\(806\) −5.70194 −0.200842
\(807\) − 5.84955i − 0.205914i
\(808\) 11.5947i 0.407900i
\(809\) 23.6121 0.830158 0.415079 0.909785i \(-0.363754\pi\)
0.415079 + 0.909785i \(0.363754\pi\)
\(810\) 0 0
\(811\) −26.0870 −0.916038 −0.458019 0.888942i \(-0.651441\pi\)
−0.458019 + 0.888942i \(0.651441\pi\)
\(812\) 23.7480i 0.833391i
\(813\) 5.08840i 0.178458i
\(814\) −0.387873 −0.0135949
\(815\) 0 0
\(816\) −17.2722 −0.604648
\(817\) 40.0000i 1.39942i
\(818\) − 2.53549i − 0.0886513i
\(819\) −9.92478 −0.346800
\(820\) 0 0
\(821\) −54.4142 −1.89907 −0.949535 0.313662i \(-0.898444\pi\)
−0.949535 + 0.313662i \(0.898444\pi\)
\(822\) − 2.68594i − 0.0936827i
\(823\) − 0.121269i − 0.00422716i −0.999998 0.00211358i \(-0.999327\pi\)
0.999998 0.00211358i \(-0.000672774\pi\)
\(824\) 2.47768 0.0863142
\(825\) 0 0
\(826\) −6.95254 −0.241910
\(827\) − 18.2130i − 0.633328i −0.948538 0.316664i \(-0.897437\pi\)
0.948538 0.316664i \(-0.102563\pi\)
\(828\) − 13.1490i − 0.456960i
\(829\) −13.0738 −0.454072 −0.227036 0.973886i \(-0.572904\pi\)
−0.227036 + 0.973886i \(0.572904\pi\)
\(830\) 0 0
\(831\) 1.41090 0.0489434
\(832\) − 21.0668i − 0.730359i
\(833\) 19.3239i 0.669534i
\(834\) 2.63989 0.0914119
\(835\) 0 0
\(836\) 8.46310 0.292702
\(837\) 9.92478i 0.343050i
\(838\) 1.40105i 0.0483984i
\(839\) 26.5501 0.916610 0.458305 0.888795i \(-0.348457\pi\)
0.458305 + 0.888795i \(0.348457\pi\)
\(840\) 0 0
\(841\) −15.9525 −0.550088
\(842\) − 5.93937i − 0.204684i
\(843\) 4.38787i 0.151126i
\(844\) 17.8350 0.613905
\(845\) 0 0
\(846\) 1.92478 0.0661752
\(847\) 3.35026i 0.115116i
\(848\) − 17.7480i − 0.609468i
\(849\) −26.5745 −0.912035
\(850\) 0 0
\(851\) 13.4010 0.459382
\(852\) − 19.4763i − 0.667246i
\(853\) − 40.6155i − 1.39065i −0.718697 0.695323i \(-0.755261\pi\)
0.718697 0.695323i \(-0.244739\pi\)
\(854\) −5.65306 −0.193444
\(855\) 0 0
\(856\) −0.739549 −0.0252773
\(857\) 20.1721i 0.689064i 0.938775 + 0.344532i \(0.111962\pi\)
−0.938775 + 0.344532i \(0.888038\pi\)
\(858\) 0.574515i 0.0196136i
\(859\) −21.8035 −0.743926 −0.371963 0.928248i \(-0.621315\pi\)
−0.371963 + 0.928248i \(0.621315\pi\)
\(860\) 0 0
\(861\) −14.7005 −0.500993
\(862\) 6.56978i 0.223767i
\(863\) − 35.4274i − 1.20596i −0.797755 0.602981i \(-0.793979\pi\)
0.797755 0.602981i \(-0.206021\pi\)
\(864\) 2.26916 0.0771984
\(865\) 0 0
\(866\) 1.83780 0.0624508
\(867\) 3.92619i 0.133340i
\(868\) 65.2506i 2.21475i
\(869\) −11.5369 −0.391363
\(870\) 0 0
\(871\) 17.5515 0.594710
\(872\) 8.76116i 0.296690i
\(873\) − 0.0752228i − 0.00254591i
\(874\) 5.60419 0.189564
\(875\) 0 0
\(876\) 15.1852 0.513061
\(877\) − 14.0362i − 0.473969i −0.971513 0.236984i \(-0.923841\pi\)
0.971513 0.236984i \(-0.0761590\pi\)
\(878\) − 5.71370i − 0.192828i
\(879\) 3.42548 0.115539
\(880\) 0 0
\(881\) −21.0738 −0.709995 −0.354997 0.934867i \(-0.615518\pi\)
−0.354997 + 0.934867i \(0.615518\pi\)
\(882\) − 0.819237i − 0.0275852i
\(883\) 42.1476i 1.41838i 0.705017 + 0.709190i \(0.250939\pi\)
−0.705017 + 0.709190i \(0.749061\pi\)
\(884\) −26.5933 −0.894429
\(885\) 0 0
\(886\) 3.69911 0.124274
\(887\) 6.93604i 0.232889i 0.993197 + 0.116445i \(0.0371498\pi\)
−0.993197 + 0.116445i \(0.962850\pi\)
\(888\) 1.53690i 0.0515751i
\(889\) 48.8284 1.63765
\(890\) 0 0
\(891\) 1.00000 0.0335013
\(892\) 13.1490i 0.440262i
\(893\) − 42.8021i − 1.43232i
\(894\) 0.298062 0.00996868
\(895\) 0 0
\(896\) 19.8251 0.662311
\(897\) − 19.8496i − 0.662757i
\(898\) 6.95765i 0.232180i
\(899\) 35.8496 1.19565
\(900\) 0 0
\(901\) −21.5026 −0.716356
\(902\) 0.850969i 0.0283342i
\(903\) 31.0738i 1.03407i
\(904\) 4.61071 0.153350
\(905\) 0 0
\(906\) 1.31124 0.0435629
\(907\) 53.2017i 1.76653i 0.468870 + 0.883267i \(0.344661\pi\)
−0.468870 + 0.883267i \(0.655339\pi\)
\(908\) 33.2868i 1.10466i
\(909\) 15.0884 0.500451
\(910\) 0 0
\(911\) 36.4749 1.20847 0.604233 0.796808i \(-0.293480\pi\)
0.604233 + 0.796808i \(0.293480\pi\)
\(912\) − 16.2835i − 0.539200i
\(913\) − 10.8872i − 0.360313i
\(914\) 1.02585 0.0339322
\(915\) 0 0
\(916\) −50.7269 −1.67606
\(917\) − 19.8496i − 0.655490i
\(918\) − 0.887166i − 0.0292808i
\(919\) −9.73340 −0.321075 −0.160538 0.987030i \(-0.551323\pi\)
−0.160538 + 0.987030i \(0.551323\pi\)
\(920\) 0 0
\(921\) −16.6497 −0.548628
\(922\) − 7.04746i − 0.232096i
\(923\) − 29.4010i − 0.967747i
\(924\) 6.57452 0.216286
\(925\) 0 0
\(926\) −2.04605 −0.0672372
\(927\) − 3.22425i − 0.105898i
\(928\) − 8.19649i − 0.269063i
\(929\) 24.1768 0.793215 0.396607 0.917988i \(-0.370187\pi\)
0.396607 + 0.917988i \(0.370187\pi\)
\(930\) 0 0
\(931\) −18.2177 −0.597062
\(932\) 37.8251i 1.23900i
\(933\) − 32.9986i − 1.08033i
\(934\) 3.62672 0.118670
\(935\) 0 0
\(936\) 2.27645 0.0744082
\(937\) − 7.48612i − 0.244561i −0.992496 0.122280i \(-0.960979\pi\)
0.992496 0.122280i \(-0.0390207\pi\)
\(938\) 3.84955i 0.125692i
\(939\) −15.4010 −0.502594
\(940\) 0 0
\(941\) −21.2360 −0.692274 −0.346137 0.938184i \(-0.612507\pi\)
−0.346137 + 0.938184i \(0.612507\pi\)
\(942\) 1.06205i 0.0346034i
\(943\) − 29.4010i − 0.957430i
\(944\) −40.4025 −1.31499
\(945\) 0 0
\(946\) 1.79877 0.0584830
\(947\) − 15.4763i − 0.502911i −0.967869 0.251456i \(-0.919091\pi\)
0.967869 0.251456i \(-0.0809092\pi\)
\(948\) 22.6399i 0.735309i
\(949\) 22.9234 0.744124
\(950\) 0 0
\(951\) 2.15045 0.0697330
\(952\) − 11.7772i − 0.381700i
\(953\) 32.0508i 1.03823i 0.854705 + 0.519113i \(0.173738\pi\)
−0.854705 + 0.519113i \(0.826262\pi\)
\(954\) 0.911603 0.0295142
\(955\) 0 0
\(956\) −52.1016 −1.68509
\(957\) − 3.61213i − 0.116763i
\(958\) − 1.80351i − 0.0582687i
\(959\) −46.3996 −1.49832
\(960\) 0 0
\(961\) 67.5012 2.17746
\(962\) 1.14903i 0.0370462i
\(963\) 0.962389i 0.0310125i
\(964\) 56.0263 1.80449
\(965\) 0 0
\(966\) 4.35359 0.140074
\(967\) − 17.3766i − 0.558794i −0.960176 0.279397i \(-0.909865\pi\)
0.960176 0.279397i \(-0.0901346\pi\)
\(968\) − 0.768452i − 0.0246990i
\(969\) −19.7283 −0.633764
\(970\) 0 0
\(971\) 36.2031 1.16181 0.580907 0.813970i \(-0.302698\pi\)
0.580907 + 0.813970i \(0.302698\pi\)
\(972\) − 1.96239i − 0.0629436i
\(973\) − 45.6042i − 1.46200i
\(974\) −6.88015 −0.220454
\(975\) 0 0
\(976\) −32.8510 −1.05153
\(977\) − 28.1476i − 0.900522i −0.892897 0.450261i \(-0.851331\pi\)
0.892897 0.450261i \(-0.148669\pi\)
\(978\) 2.44851i 0.0782946i
\(979\) 2.77575 0.0887132
\(980\) 0 0
\(981\) 11.4010 0.364007
\(982\) − 4.80492i − 0.153331i
\(983\) − 7.07381i − 0.225619i −0.993617 0.112810i \(-0.964015\pi\)
0.993617 0.112810i \(-0.0359851\pi\)
\(984\) 3.37187 0.107491
\(985\) 0 0
\(986\) −3.20456 −0.102054
\(987\) − 33.2506i − 1.05838i
\(988\) − 25.0710i − 0.797614i
\(989\) −62.1476 −1.97618
\(990\) 0 0
\(991\) 44.4260 1.41124 0.705619 0.708592i \(-0.250669\pi\)
0.705619 + 0.708592i \(0.250669\pi\)
\(992\) − 22.5209i − 0.715039i
\(993\) 14.5501i 0.461733i
\(994\) 6.44851 0.204534
\(995\) 0 0
\(996\) −21.3649 −0.676971
\(997\) − 28.4847i − 0.902120i −0.892494 0.451060i \(-0.851046\pi\)
0.892494 0.451060i \(-0.148954\pi\)
\(998\) 2.74940i 0.0870307i
\(999\) 2.00000 0.0632772
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.g.199.3 6
3.2 odd 2 2475.2.c.r.199.4 6
5.2 odd 4 825.2.a.k.1.2 3
5.3 odd 4 165.2.a.c.1.2 3
5.4 even 2 inner 825.2.c.g.199.4 6
15.2 even 4 2475.2.a.bb.1.2 3
15.8 even 4 495.2.a.e.1.2 3
15.14 odd 2 2475.2.c.r.199.3 6
20.3 even 4 2640.2.a.be.1.1 3
35.13 even 4 8085.2.a.bk.1.2 3
55.32 even 4 9075.2.a.cf.1.2 3
55.43 even 4 1815.2.a.m.1.2 3
60.23 odd 4 7920.2.a.cj.1.1 3
165.98 odd 4 5445.2.a.z.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.c.1.2 3 5.3 odd 4
495.2.a.e.1.2 3 15.8 even 4
825.2.a.k.1.2 3 5.2 odd 4
825.2.c.g.199.3 6 1.1 even 1 trivial
825.2.c.g.199.4 6 5.4 even 2 inner
1815.2.a.m.1.2 3 55.43 even 4
2475.2.a.bb.1.2 3 15.2 even 4
2475.2.c.r.199.3 6 15.14 odd 2
2475.2.c.r.199.4 6 3.2 odd 2
2640.2.a.be.1.1 3 20.3 even 4
5445.2.a.z.1.2 3 165.98 odd 4
7920.2.a.cj.1.1 3 60.23 odd 4
8085.2.a.bk.1.2 3 35.13 even 4
9075.2.a.cf.1.2 3 55.32 even 4