Properties

Label 1617.2.a.l.1.2
Level $1617$
Weight $2$
Character 1617.1
Self dual yes
Analytic conductor $12.912$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1617.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.9118100068\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 1617.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.381966 q^{2} +1.00000 q^{3} -1.85410 q^{4} -2.23607 q^{5} -0.381966 q^{6} +1.47214 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.381966 q^{2} +1.00000 q^{3} -1.85410 q^{4} -2.23607 q^{5} -0.381966 q^{6} +1.47214 q^{8} +1.00000 q^{9} +0.854102 q^{10} -1.00000 q^{11} -1.85410 q^{12} +5.47214 q^{13} -2.23607 q^{15} +3.14590 q^{16} -6.00000 q^{17} -0.381966 q^{18} -0.236068 q^{19} +4.14590 q^{20} +0.381966 q^{22} +6.47214 q^{23} +1.47214 q^{24} -2.09017 q^{26} +1.00000 q^{27} -5.76393 q^{29} +0.854102 q^{30} +0.472136 q^{31} -4.14590 q^{32} -1.00000 q^{33} +2.29180 q^{34} -1.85410 q^{36} -9.47214 q^{37} +0.0901699 q^{38} +5.47214 q^{39} -3.29180 q^{40} -6.00000 q^{41} +8.47214 q^{43} +1.85410 q^{44} -2.23607 q^{45} -2.47214 q^{46} -2.52786 q^{47} +3.14590 q^{48} -6.00000 q^{51} -10.1459 q^{52} +4.94427 q^{53} -0.381966 q^{54} +2.23607 q^{55} -0.236068 q^{57} +2.20163 q^{58} -5.94427 q^{59} +4.14590 q^{60} -10.9443 q^{61} -0.180340 q^{62} -4.70820 q^{64} -12.2361 q^{65} +0.381966 q^{66} -12.7082 q^{67} +11.1246 q^{68} +6.47214 q^{69} -4.47214 q^{71} +1.47214 q^{72} +1.00000 q^{73} +3.61803 q^{74} +0.437694 q^{76} -2.09017 q^{78} +6.47214 q^{79} -7.03444 q^{80} +1.00000 q^{81} +2.29180 q^{82} -3.52786 q^{83} +13.4164 q^{85} -3.23607 q^{86} -5.76393 q^{87} -1.47214 q^{88} -13.4164 q^{89} +0.854102 q^{90} -12.0000 q^{92} +0.472136 q^{93} +0.965558 q^{94} +0.527864 q^{95} -4.14590 q^{96} -10.9443 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 3 q^{6} - 6 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} + 2 q^{3} + 3 q^{4} - 3 q^{6} - 6 q^{8} + 2 q^{9} - 5 q^{10} - 2 q^{11} + 3 q^{12} + 2 q^{13} + 13 q^{16} - 12 q^{17} - 3 q^{18} + 4 q^{19} + 15 q^{20} + 3 q^{22} + 4 q^{23} - 6 q^{24} + 7 q^{26} + 2 q^{27} - 16 q^{29} - 5 q^{30} - 8 q^{31} - 15 q^{32} - 2 q^{33} + 18 q^{34} + 3 q^{36} - 10 q^{37} - 11 q^{38} + 2 q^{39} - 20 q^{40} - 12 q^{41} + 8 q^{43} - 3 q^{44} + 4 q^{46} - 14 q^{47} + 13 q^{48} - 12 q^{51} - 27 q^{52} - 8 q^{53} - 3 q^{54} + 4 q^{57} + 29 q^{58} + 6 q^{59} + 15 q^{60} - 4 q^{61} + 22 q^{62} + 4 q^{64} - 20 q^{65} + 3 q^{66} - 12 q^{67} - 18 q^{68} + 4 q^{69} - 6 q^{72} + 2 q^{73} + 5 q^{74} + 21 q^{76} + 7 q^{78} + 4 q^{79} + 15 q^{80} + 2 q^{81} + 18 q^{82} - 16 q^{83} - 2 q^{86} - 16 q^{87} + 6 q^{88} - 5 q^{90} - 24 q^{92} - 8 q^{93} + 31 q^{94} + 10 q^{95} - 15 q^{96} - 4 q^{97} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.381966 −0.270091 −0.135045 0.990839i \(-0.543118\pi\)
−0.135045 + 0.990839i \(0.543118\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.85410 −0.927051
\(5\) −2.23607 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) −0.381966 −0.155937
\(7\) 0 0
\(8\) 1.47214 0.520479
\(9\) 1.00000 0.333333
\(10\) 0.854102 0.270091
\(11\) −1.00000 −0.301511
\(12\) −1.85410 −0.535233
\(13\) 5.47214 1.51770 0.758849 0.651267i \(-0.225762\pi\)
0.758849 + 0.651267i \(0.225762\pi\)
\(14\) 0 0
\(15\) −2.23607 −0.577350
\(16\) 3.14590 0.786475
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) −0.381966 −0.0900303
\(19\) −0.236068 −0.0541577 −0.0270789 0.999633i \(-0.508621\pi\)
−0.0270789 + 0.999633i \(0.508621\pi\)
\(20\) 4.14590 0.927051
\(21\) 0 0
\(22\) 0.381966 0.0814354
\(23\) 6.47214 1.34953 0.674767 0.738031i \(-0.264244\pi\)
0.674767 + 0.738031i \(0.264244\pi\)
\(24\) 1.47214 0.300498
\(25\) 0 0
\(26\) −2.09017 −0.409916
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −5.76393 −1.07034 −0.535168 0.844746i \(-0.679752\pi\)
−0.535168 + 0.844746i \(0.679752\pi\)
\(30\) 0.854102 0.155937
\(31\) 0.472136 0.0847981 0.0423991 0.999101i \(-0.486500\pi\)
0.0423991 + 0.999101i \(0.486500\pi\)
\(32\) −4.14590 −0.732898
\(33\) −1.00000 −0.174078
\(34\) 2.29180 0.393040
\(35\) 0 0
\(36\) −1.85410 −0.309017
\(37\) −9.47214 −1.55721 −0.778605 0.627515i \(-0.784072\pi\)
−0.778605 + 0.627515i \(0.784072\pi\)
\(38\) 0.0901699 0.0146275
\(39\) 5.47214 0.876243
\(40\) −3.29180 −0.520479
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 8.47214 1.29199 0.645994 0.763342i \(-0.276443\pi\)
0.645994 + 0.763342i \(0.276443\pi\)
\(44\) 1.85410 0.279516
\(45\) −2.23607 −0.333333
\(46\) −2.47214 −0.364497
\(47\) −2.52786 −0.368727 −0.184363 0.982858i \(-0.559022\pi\)
−0.184363 + 0.982858i \(0.559022\pi\)
\(48\) 3.14590 0.454071
\(49\) 0 0
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) −10.1459 −1.40698
\(53\) 4.94427 0.679148 0.339574 0.940579i \(-0.389717\pi\)
0.339574 + 0.940579i \(0.389717\pi\)
\(54\) −0.381966 −0.0519790
\(55\) 2.23607 0.301511
\(56\) 0 0
\(57\) −0.236068 −0.0312680
\(58\) 2.20163 0.289088
\(59\) −5.94427 −0.773878 −0.386939 0.922105i \(-0.626468\pi\)
−0.386939 + 0.922105i \(0.626468\pi\)
\(60\) 4.14590 0.535233
\(61\) −10.9443 −1.40127 −0.700635 0.713520i \(-0.747100\pi\)
−0.700635 + 0.713520i \(0.747100\pi\)
\(62\) −0.180340 −0.0229032
\(63\) 0 0
\(64\) −4.70820 −0.588525
\(65\) −12.2361 −1.51770
\(66\) 0.381966 0.0470168
\(67\) −12.7082 −1.55255 −0.776277 0.630392i \(-0.782894\pi\)
−0.776277 + 0.630392i \(0.782894\pi\)
\(68\) 11.1246 1.34906
\(69\) 6.47214 0.779154
\(70\) 0 0
\(71\) −4.47214 −0.530745 −0.265372 0.964146i \(-0.585495\pi\)
−0.265372 + 0.964146i \(0.585495\pi\)
\(72\) 1.47214 0.173493
\(73\) 1.00000 0.117041 0.0585206 0.998286i \(-0.481362\pi\)
0.0585206 + 0.998286i \(0.481362\pi\)
\(74\) 3.61803 0.420588
\(75\) 0 0
\(76\) 0.437694 0.0502070
\(77\) 0 0
\(78\) −2.09017 −0.236665
\(79\) 6.47214 0.728172 0.364086 0.931365i \(-0.381381\pi\)
0.364086 + 0.931365i \(0.381381\pi\)
\(80\) −7.03444 −0.786475
\(81\) 1.00000 0.111111
\(82\) 2.29180 0.253087
\(83\) −3.52786 −0.387233 −0.193617 0.981077i \(-0.562022\pi\)
−0.193617 + 0.981077i \(0.562022\pi\)
\(84\) 0 0
\(85\) 13.4164 1.45521
\(86\) −3.23607 −0.348954
\(87\) −5.76393 −0.617958
\(88\) −1.47214 −0.156930
\(89\) −13.4164 −1.42214 −0.711068 0.703123i \(-0.751788\pi\)
−0.711068 + 0.703123i \(0.751788\pi\)
\(90\) 0.854102 0.0900303
\(91\) 0 0
\(92\) −12.0000 −1.25109
\(93\) 0.472136 0.0489582
\(94\) 0.965558 0.0995897
\(95\) 0.527864 0.0541577
\(96\) −4.14590 −0.423139
\(97\) −10.9443 −1.11122 −0.555611 0.831442i \(-0.687516\pi\)
−0.555611 + 0.831442i \(0.687516\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) −12.4721 −1.24102 −0.620512 0.784197i \(-0.713075\pi\)
−0.620512 + 0.784197i \(0.713075\pi\)
\(102\) 2.29180 0.226922
\(103\) −19.4164 −1.91316 −0.956578 0.291477i \(-0.905853\pi\)
−0.956578 + 0.291477i \(0.905853\pi\)
\(104\) 8.05573 0.789929
\(105\) 0 0
\(106\) −1.88854 −0.183432
\(107\) 2.05573 0.198735 0.0993674 0.995051i \(-0.468318\pi\)
0.0993674 + 0.995051i \(0.468318\pi\)
\(108\) −1.85410 −0.178411
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) −0.854102 −0.0814354
\(111\) −9.47214 −0.899055
\(112\) 0 0
\(113\) −2.47214 −0.232559 −0.116279 0.993217i \(-0.537097\pi\)
−0.116279 + 0.993217i \(0.537097\pi\)
\(114\) 0.0901699 0.00844519
\(115\) −14.4721 −1.34953
\(116\) 10.6869 0.992255
\(117\) 5.47214 0.505899
\(118\) 2.27051 0.209017
\(119\) 0 0
\(120\) −3.29180 −0.300498
\(121\) 1.00000 0.0909091
\(122\) 4.18034 0.378470
\(123\) −6.00000 −0.541002
\(124\) −0.875388 −0.0786122
\(125\) 11.1803 1.00000
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 10.0902 0.891853
\(129\) 8.47214 0.745930
\(130\) 4.67376 0.409916
\(131\) −4.47214 −0.390732 −0.195366 0.980730i \(-0.562590\pi\)
−0.195366 + 0.980730i \(0.562590\pi\)
\(132\) 1.85410 0.161379
\(133\) 0 0
\(134\) 4.85410 0.419331
\(135\) −2.23607 −0.192450
\(136\) −8.83282 −0.757408
\(137\) 6.00000 0.512615 0.256307 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) −2.47214 −0.210442
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) −2.52786 −0.212885
\(142\) 1.70820 0.143349
\(143\) −5.47214 −0.457603
\(144\) 3.14590 0.262158
\(145\) 12.8885 1.07034
\(146\) −0.381966 −0.0316117
\(147\) 0 0
\(148\) 17.5623 1.44361
\(149\) −15.6525 −1.28230 −0.641150 0.767415i \(-0.721543\pi\)
−0.641150 + 0.767415i \(0.721543\pi\)
\(150\) 0 0
\(151\) 10.9443 0.890632 0.445316 0.895373i \(-0.353091\pi\)
0.445316 + 0.895373i \(0.353091\pi\)
\(152\) −0.347524 −0.0281879
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) −1.05573 −0.0847981
\(156\) −10.1459 −0.812322
\(157\) 23.4164 1.86883 0.934416 0.356183i \(-0.115922\pi\)
0.934416 + 0.356183i \(0.115922\pi\)
\(158\) −2.47214 −0.196673
\(159\) 4.94427 0.392106
\(160\) 9.27051 0.732898
\(161\) 0 0
\(162\) −0.381966 −0.0300101
\(163\) 22.1246 1.73293 0.866467 0.499235i \(-0.166386\pi\)
0.866467 + 0.499235i \(0.166386\pi\)
\(164\) 11.1246 0.868686
\(165\) 2.23607 0.174078
\(166\) 1.34752 0.104588
\(167\) −9.52786 −0.737288 −0.368644 0.929571i \(-0.620178\pi\)
−0.368644 + 0.929571i \(0.620178\pi\)
\(168\) 0 0
\(169\) 16.9443 1.30341
\(170\) −5.12461 −0.393040
\(171\) −0.236068 −0.0180526
\(172\) −15.7082 −1.19774
\(173\) −14.4721 −1.10030 −0.550148 0.835067i \(-0.685429\pi\)
−0.550148 + 0.835067i \(0.685429\pi\)
\(174\) 2.20163 0.166905
\(175\) 0 0
\(176\) −3.14590 −0.237131
\(177\) −5.94427 −0.446799
\(178\) 5.12461 0.384106
\(179\) −14.4721 −1.08170 −0.540849 0.841120i \(-0.681897\pi\)
−0.540849 + 0.841120i \(0.681897\pi\)
\(180\) 4.14590 0.309017
\(181\) 8.47214 0.629729 0.314864 0.949137i \(-0.398041\pi\)
0.314864 + 0.949137i \(0.398041\pi\)
\(182\) 0 0
\(183\) −10.9443 −0.809024
\(184\) 9.52786 0.702403
\(185\) 21.1803 1.55721
\(186\) −0.180340 −0.0132232
\(187\) 6.00000 0.438763
\(188\) 4.68692 0.341829
\(189\) 0 0
\(190\) −0.201626 −0.0146275
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) −4.70820 −0.339785
\(193\) −9.88854 −0.711793 −0.355896 0.934525i \(-0.615824\pi\)
−0.355896 + 0.934525i \(0.615824\pi\)
\(194\) 4.18034 0.300131
\(195\) −12.2361 −0.876243
\(196\) 0 0
\(197\) 0.472136 0.0336383 0.0168191 0.999859i \(-0.494646\pi\)
0.0168191 + 0.999859i \(0.494646\pi\)
\(198\) 0.381966 0.0271451
\(199\) −15.5279 −1.10074 −0.550371 0.834921i \(-0.685514\pi\)
−0.550371 + 0.834921i \(0.685514\pi\)
\(200\) 0 0
\(201\) −12.7082 −0.896368
\(202\) 4.76393 0.335189
\(203\) 0 0
\(204\) 11.1246 0.778879
\(205\) 13.4164 0.937043
\(206\) 7.41641 0.516726
\(207\) 6.47214 0.449845
\(208\) 17.2148 1.19363
\(209\) 0.236068 0.0163292
\(210\) 0 0
\(211\) 1.05573 0.0726793 0.0363397 0.999339i \(-0.488430\pi\)
0.0363397 + 0.999339i \(0.488430\pi\)
\(212\) −9.16718 −0.629605
\(213\) −4.47214 −0.306426
\(214\) −0.785218 −0.0536764
\(215\) −18.9443 −1.29199
\(216\) 1.47214 0.100166
\(217\) 0 0
\(218\) −2.29180 −0.155220
\(219\) 1.00000 0.0675737
\(220\) −4.14590 −0.279516
\(221\) −32.8328 −2.20857
\(222\) 3.61803 0.242827
\(223\) −18.0000 −1.20537 −0.602685 0.797980i \(-0.705902\pi\)
−0.602685 + 0.797980i \(0.705902\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0.944272 0.0628120
\(227\) 26.8328 1.78096 0.890478 0.455026i \(-0.150370\pi\)
0.890478 + 0.455026i \(0.150370\pi\)
\(228\) 0.437694 0.0289870
\(229\) 11.4164 0.754417 0.377209 0.926128i \(-0.376884\pi\)
0.377209 + 0.926128i \(0.376884\pi\)
\(230\) 5.52786 0.364497
\(231\) 0 0
\(232\) −8.48529 −0.557087
\(233\) −13.4164 −0.878938 −0.439469 0.898258i \(-0.644833\pi\)
−0.439469 + 0.898258i \(0.644833\pi\)
\(234\) −2.09017 −0.136639
\(235\) 5.65248 0.368727
\(236\) 11.0213 0.717425
\(237\) 6.47214 0.420410
\(238\) 0 0
\(239\) −27.3607 −1.76982 −0.884908 0.465767i \(-0.845779\pi\)
−0.884908 + 0.465767i \(0.845779\pi\)
\(240\) −7.03444 −0.454071
\(241\) 19.0000 1.22390 0.611949 0.790897i \(-0.290386\pi\)
0.611949 + 0.790897i \(0.290386\pi\)
\(242\) −0.381966 −0.0245537
\(243\) 1.00000 0.0641500
\(244\) 20.2918 1.29905
\(245\) 0 0
\(246\) 2.29180 0.146120
\(247\) −1.29180 −0.0821950
\(248\) 0.695048 0.0441356
\(249\) −3.52786 −0.223569
\(250\) −4.27051 −0.270091
\(251\) 13.9443 0.880155 0.440077 0.897960i \(-0.354951\pi\)
0.440077 + 0.897960i \(0.354951\pi\)
\(252\) 0 0
\(253\) −6.47214 −0.406900
\(254\) −6.11146 −0.383467
\(255\) 13.4164 0.840168
\(256\) 5.56231 0.347644
\(257\) 12.2361 0.763265 0.381632 0.924314i \(-0.375362\pi\)
0.381632 + 0.924314i \(0.375362\pi\)
\(258\) −3.23607 −0.201469
\(259\) 0 0
\(260\) 22.6869 1.40698
\(261\) −5.76393 −0.356778
\(262\) 1.70820 0.105533
\(263\) 22.4164 1.38225 0.691127 0.722733i \(-0.257114\pi\)
0.691127 + 0.722733i \(0.257114\pi\)
\(264\) −1.47214 −0.0906037
\(265\) −11.0557 −0.679148
\(266\) 0 0
\(267\) −13.4164 −0.821071
\(268\) 23.5623 1.43930
\(269\) 7.52786 0.458982 0.229491 0.973311i \(-0.426294\pi\)
0.229491 + 0.973311i \(0.426294\pi\)
\(270\) 0.854102 0.0519790
\(271\) −18.7082 −1.13644 −0.568221 0.822876i \(-0.692368\pi\)
−0.568221 + 0.822876i \(0.692368\pi\)
\(272\) −18.8754 −1.14449
\(273\) 0 0
\(274\) −2.29180 −0.138452
\(275\) 0 0
\(276\) −12.0000 −0.722315
\(277\) −1.05573 −0.0634326 −0.0317163 0.999497i \(-0.510097\pi\)
−0.0317163 + 0.999497i \(0.510097\pi\)
\(278\) 3.05573 0.183270
\(279\) 0.472136 0.0282660
\(280\) 0 0
\(281\) −17.1803 −1.02489 −0.512447 0.858719i \(-0.671261\pi\)
−0.512447 + 0.858719i \(0.671261\pi\)
\(282\) 0.965558 0.0574982
\(283\) 8.70820 0.517649 0.258824 0.965924i \(-0.416665\pi\)
0.258824 + 0.965924i \(0.416665\pi\)
\(284\) 8.29180 0.492028
\(285\) 0.527864 0.0312680
\(286\) 2.09017 0.123594
\(287\) 0 0
\(288\) −4.14590 −0.244299
\(289\) 19.0000 1.11765
\(290\) −4.92299 −0.289088
\(291\) −10.9443 −0.641565
\(292\) −1.85410 −0.108503
\(293\) −25.8885 −1.51242 −0.756212 0.654326i \(-0.772952\pi\)
−0.756212 + 0.654326i \(0.772952\pi\)
\(294\) 0 0
\(295\) 13.2918 0.773878
\(296\) −13.9443 −0.810494
\(297\) −1.00000 −0.0580259
\(298\) 5.97871 0.346338
\(299\) 35.4164 2.04818
\(300\) 0 0
\(301\) 0 0
\(302\) −4.18034 −0.240552
\(303\) −12.4721 −0.716505
\(304\) −0.742646 −0.0425937
\(305\) 24.4721 1.40127
\(306\) 2.29180 0.131013
\(307\) −16.0000 −0.913168 −0.456584 0.889680i \(-0.650927\pi\)
−0.456584 + 0.889680i \(0.650927\pi\)
\(308\) 0 0
\(309\) −19.4164 −1.10456
\(310\) 0.403252 0.0229032
\(311\) −4.94427 −0.280364 −0.140182 0.990126i \(-0.544769\pi\)
−0.140182 + 0.990126i \(0.544769\pi\)
\(312\) 8.05573 0.456066
\(313\) 18.3607 1.03781 0.518903 0.854833i \(-0.326340\pi\)
0.518903 + 0.854833i \(0.326340\pi\)
\(314\) −8.94427 −0.504754
\(315\) 0 0
\(316\) −12.0000 −0.675053
\(317\) 9.05573 0.508620 0.254310 0.967123i \(-0.418152\pi\)
0.254310 + 0.967123i \(0.418152\pi\)
\(318\) −1.88854 −0.105904
\(319\) 5.76393 0.322718
\(320\) 10.5279 0.588525
\(321\) 2.05573 0.114740
\(322\) 0 0
\(323\) 1.41641 0.0788110
\(324\) −1.85410 −0.103006
\(325\) 0 0
\(326\) −8.45085 −0.468049
\(327\) 6.00000 0.331801
\(328\) −8.83282 −0.487711
\(329\) 0 0
\(330\) −0.854102 −0.0470168
\(331\) −16.0000 −0.879440 −0.439720 0.898135i \(-0.644922\pi\)
−0.439720 + 0.898135i \(0.644922\pi\)
\(332\) 6.54102 0.358985
\(333\) −9.47214 −0.519070
\(334\) 3.63932 0.199135
\(335\) 28.4164 1.55255
\(336\) 0 0
\(337\) −15.4164 −0.839785 −0.419893 0.907574i \(-0.637932\pi\)
−0.419893 + 0.907574i \(0.637932\pi\)
\(338\) −6.47214 −0.352038
\(339\) −2.47214 −0.134268
\(340\) −24.8754 −1.34906
\(341\) −0.472136 −0.0255676
\(342\) 0.0901699 0.00487583
\(343\) 0 0
\(344\) 12.4721 0.672453
\(345\) −14.4721 −0.779154
\(346\) 5.52786 0.297180
\(347\) −16.9443 −0.909616 −0.454808 0.890589i \(-0.650292\pi\)
−0.454808 + 0.890589i \(0.650292\pi\)
\(348\) 10.6869 0.572879
\(349\) −24.4164 −1.30698 −0.653490 0.756935i \(-0.726696\pi\)
−0.653490 + 0.756935i \(0.726696\pi\)
\(350\) 0 0
\(351\) 5.47214 0.292081
\(352\) 4.14590 0.220977
\(353\) −0.236068 −0.0125646 −0.00628232 0.999980i \(-0.502000\pi\)
−0.00628232 + 0.999980i \(0.502000\pi\)
\(354\) 2.27051 0.120676
\(355\) 10.0000 0.530745
\(356\) 24.8754 1.31839
\(357\) 0 0
\(358\) 5.52786 0.292157
\(359\) −3.05573 −0.161275 −0.0806376 0.996743i \(-0.525696\pi\)
−0.0806376 + 0.996743i \(0.525696\pi\)
\(360\) −3.29180 −0.173493
\(361\) −18.9443 −0.997067
\(362\) −3.23607 −0.170084
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) −2.23607 −0.117041
\(366\) 4.18034 0.218510
\(367\) −1.41641 −0.0739359 −0.0369679 0.999316i \(-0.511770\pi\)
−0.0369679 + 0.999316i \(0.511770\pi\)
\(368\) 20.3607 1.06137
\(369\) −6.00000 −0.312348
\(370\) −8.09017 −0.420588
\(371\) 0 0
\(372\) −0.875388 −0.0453868
\(373\) 0.944272 0.0488925 0.0244463 0.999701i \(-0.492218\pi\)
0.0244463 + 0.999701i \(0.492218\pi\)
\(374\) −2.29180 −0.118506
\(375\) 11.1803 0.577350
\(376\) −3.72136 −0.191914
\(377\) −31.5410 −1.62445
\(378\) 0 0
\(379\) 5.65248 0.290348 0.145174 0.989406i \(-0.453626\pi\)
0.145174 + 0.989406i \(0.453626\pi\)
\(380\) −0.978714 −0.0502070
\(381\) 16.0000 0.819705
\(382\) −4.58359 −0.234517
\(383\) 8.94427 0.457031 0.228515 0.973540i \(-0.426613\pi\)
0.228515 + 0.973540i \(0.426613\pi\)
\(384\) 10.0902 0.514912
\(385\) 0 0
\(386\) 3.77709 0.192249
\(387\) 8.47214 0.430663
\(388\) 20.2918 1.03016
\(389\) −32.8328 −1.66469 −0.832345 0.554258i \(-0.813002\pi\)
−0.832345 + 0.554258i \(0.813002\pi\)
\(390\) 4.67376 0.236665
\(391\) −38.8328 −1.96386
\(392\) 0 0
\(393\) −4.47214 −0.225589
\(394\) −0.180340 −0.00908539
\(395\) −14.4721 −0.728172
\(396\) 1.85410 0.0931721
\(397\) 17.4164 0.874104 0.437052 0.899436i \(-0.356022\pi\)
0.437052 + 0.899436i \(0.356022\pi\)
\(398\) 5.93112 0.297300
\(399\) 0 0
\(400\) 0 0
\(401\) 0.472136 0.0235773 0.0117887 0.999931i \(-0.496247\pi\)
0.0117887 + 0.999931i \(0.496247\pi\)
\(402\) 4.85410 0.242101
\(403\) 2.58359 0.128698
\(404\) 23.1246 1.15049
\(405\) −2.23607 −0.111111
\(406\) 0 0
\(407\) 9.47214 0.469516
\(408\) −8.83282 −0.437290
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) −5.12461 −0.253087
\(411\) 6.00000 0.295958
\(412\) 36.0000 1.77359
\(413\) 0 0
\(414\) −2.47214 −0.121499
\(415\) 7.88854 0.387233
\(416\) −22.6869 −1.11232
\(417\) −8.00000 −0.391762
\(418\) −0.0901699 −0.00441036
\(419\) −3.00000 −0.146560 −0.0732798 0.997311i \(-0.523347\pi\)
−0.0732798 + 0.997311i \(0.523347\pi\)
\(420\) 0 0
\(421\) 20.5279 1.00047 0.500233 0.865891i \(-0.333248\pi\)
0.500233 + 0.865891i \(0.333248\pi\)
\(422\) −0.403252 −0.0196300
\(423\) −2.52786 −0.122909
\(424\) 7.27864 0.353482
\(425\) 0 0
\(426\) 1.70820 0.0827628
\(427\) 0 0
\(428\) −3.81153 −0.184237
\(429\) −5.47214 −0.264197
\(430\) 7.23607 0.348954
\(431\) 27.3607 1.31792 0.658959 0.752179i \(-0.270997\pi\)
0.658959 + 0.752179i \(0.270997\pi\)
\(432\) 3.14590 0.151357
\(433\) 39.4164 1.89423 0.947116 0.320892i \(-0.103983\pi\)
0.947116 + 0.320892i \(0.103983\pi\)
\(434\) 0 0
\(435\) 12.8885 0.617958
\(436\) −11.1246 −0.532772
\(437\) −1.52786 −0.0730876
\(438\) −0.381966 −0.0182510
\(439\) 32.1246 1.53322 0.766612 0.642111i \(-0.221941\pi\)
0.766612 + 0.642111i \(0.221941\pi\)
\(440\) 3.29180 0.156930
\(441\) 0 0
\(442\) 12.5410 0.596515
\(443\) 26.9443 1.28016 0.640080 0.768308i \(-0.278901\pi\)
0.640080 + 0.768308i \(0.278901\pi\)
\(444\) 17.5623 0.833470
\(445\) 30.0000 1.42214
\(446\) 6.87539 0.325559
\(447\) −15.6525 −0.740337
\(448\) 0 0
\(449\) 1.41641 0.0668444 0.0334222 0.999441i \(-0.489359\pi\)
0.0334222 + 0.999441i \(0.489359\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) 4.58359 0.215594
\(453\) 10.9443 0.514207
\(454\) −10.2492 −0.481020
\(455\) 0 0
\(456\) −0.347524 −0.0162743
\(457\) 15.4164 0.721149 0.360575 0.932730i \(-0.382581\pi\)
0.360575 + 0.932730i \(0.382581\pi\)
\(458\) −4.36068 −0.203761
\(459\) −6.00000 −0.280056
\(460\) 26.8328 1.25109
\(461\) −7.52786 −0.350608 −0.175304 0.984514i \(-0.556091\pi\)
−0.175304 + 0.984514i \(0.556091\pi\)
\(462\) 0 0
\(463\) 10.7082 0.497652 0.248826 0.968548i \(-0.419955\pi\)
0.248826 + 0.968548i \(0.419955\pi\)
\(464\) −18.1327 −0.841791
\(465\) −1.05573 −0.0489582
\(466\) 5.12461 0.237393
\(467\) 29.8328 1.38050 0.690249 0.723572i \(-0.257501\pi\)
0.690249 + 0.723572i \(0.257501\pi\)
\(468\) −10.1459 −0.468994
\(469\) 0 0
\(470\) −2.15905 −0.0995897
\(471\) 23.4164 1.07897
\(472\) −8.75078 −0.402787
\(473\) −8.47214 −0.389549
\(474\) −2.47214 −0.113549
\(475\) 0 0
\(476\) 0 0
\(477\) 4.94427 0.226383
\(478\) 10.4508 0.478011
\(479\) 13.8885 0.634584 0.317292 0.948328i \(-0.397227\pi\)
0.317292 + 0.948328i \(0.397227\pi\)
\(480\) 9.27051 0.423139
\(481\) −51.8328 −2.36337
\(482\) −7.25735 −0.330563
\(483\) 0 0
\(484\) −1.85410 −0.0842774
\(485\) 24.4721 1.11122
\(486\) −0.381966 −0.0173263
\(487\) 13.8885 0.629350 0.314675 0.949199i \(-0.398104\pi\)
0.314675 + 0.949199i \(0.398104\pi\)
\(488\) −16.1115 −0.729331
\(489\) 22.1246 1.00051
\(490\) 0 0
\(491\) 24.8885 1.12320 0.561602 0.827407i \(-0.310185\pi\)
0.561602 + 0.827407i \(0.310185\pi\)
\(492\) 11.1246 0.501536
\(493\) 34.5836 1.55757
\(494\) 0.493422 0.0222001
\(495\) 2.23607 0.100504
\(496\) 1.48529 0.0666916
\(497\) 0 0
\(498\) 1.34752 0.0603840
\(499\) −18.1246 −0.811369 −0.405685 0.914013i \(-0.632967\pi\)
−0.405685 + 0.914013i \(0.632967\pi\)
\(500\) −20.7295 −0.927051
\(501\) −9.52786 −0.425674
\(502\) −5.32624 −0.237722
\(503\) −37.4164 −1.66832 −0.834158 0.551526i \(-0.814046\pi\)
−0.834158 + 0.551526i \(0.814046\pi\)
\(504\) 0 0
\(505\) 27.8885 1.24102
\(506\) 2.47214 0.109900
\(507\) 16.9443 0.752522
\(508\) −29.6656 −1.31620
\(509\) −44.4721 −1.97119 −0.985596 0.169115i \(-0.945909\pi\)
−0.985596 + 0.169115i \(0.945909\pi\)
\(510\) −5.12461 −0.226922
\(511\) 0 0
\(512\) −22.3050 −0.985749
\(513\) −0.236068 −0.0104227
\(514\) −4.67376 −0.206151
\(515\) 43.4164 1.91316
\(516\) −15.7082 −0.691515
\(517\) 2.52786 0.111175
\(518\) 0 0
\(519\) −14.4721 −0.635256
\(520\) −18.0132 −0.789929
\(521\) 28.2361 1.23704 0.618522 0.785767i \(-0.287732\pi\)
0.618522 + 0.785767i \(0.287732\pi\)
\(522\) 2.20163 0.0963626
\(523\) −16.7082 −0.730599 −0.365299 0.930890i \(-0.619033\pi\)
−0.365299 + 0.930890i \(0.619033\pi\)
\(524\) 8.29180 0.362229
\(525\) 0 0
\(526\) −8.56231 −0.373334
\(527\) −2.83282 −0.123399
\(528\) −3.14590 −0.136908
\(529\) 18.8885 0.821241
\(530\) 4.22291 0.183432
\(531\) −5.94427 −0.257959
\(532\) 0 0
\(533\) −32.8328 −1.42215
\(534\) 5.12461 0.221764
\(535\) −4.59675 −0.198735
\(536\) −18.7082 −0.808071
\(537\) −14.4721 −0.624519
\(538\) −2.87539 −0.123967
\(539\) 0 0
\(540\) 4.14590 0.178411
\(541\) −5.41641 −0.232870 −0.116435 0.993198i \(-0.537147\pi\)
−0.116435 + 0.993198i \(0.537147\pi\)
\(542\) 7.14590 0.306943
\(543\) 8.47214 0.363574
\(544\) 24.8754 1.06652
\(545\) −13.4164 −0.574696
\(546\) 0 0
\(547\) 13.4164 0.573644 0.286822 0.957984i \(-0.407401\pi\)
0.286822 + 0.957984i \(0.407401\pi\)
\(548\) −11.1246 −0.475220
\(549\) −10.9443 −0.467090
\(550\) 0 0
\(551\) 1.36068 0.0579669
\(552\) 9.52786 0.405533
\(553\) 0 0
\(554\) 0.403252 0.0171325
\(555\) 21.1803 0.899055
\(556\) 14.8328 0.629052
\(557\) 32.1246 1.36116 0.680582 0.732672i \(-0.261727\pi\)
0.680582 + 0.732672i \(0.261727\pi\)
\(558\) −0.180340 −0.00763440
\(559\) 46.3607 1.96085
\(560\) 0 0
\(561\) 6.00000 0.253320
\(562\) 6.56231 0.276814
\(563\) −4.47214 −0.188478 −0.0942390 0.995550i \(-0.530042\pi\)
−0.0942390 + 0.995550i \(0.530042\pi\)
\(564\) 4.68692 0.197355
\(565\) 5.52786 0.232559
\(566\) −3.32624 −0.139812
\(567\) 0 0
\(568\) −6.58359 −0.276241
\(569\) −3.52786 −0.147896 −0.0739479 0.997262i \(-0.523560\pi\)
−0.0739479 + 0.997262i \(0.523560\pi\)
\(570\) −0.201626 −0.00844519
\(571\) −37.4164 −1.56583 −0.782914 0.622130i \(-0.786268\pi\)
−0.782914 + 0.622130i \(0.786268\pi\)
\(572\) 10.1459 0.424221
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) 0 0
\(576\) −4.70820 −0.196175
\(577\) −17.0557 −0.710039 −0.355020 0.934859i \(-0.615526\pi\)
−0.355020 + 0.934859i \(0.615526\pi\)
\(578\) −7.25735 −0.301866
\(579\) −9.88854 −0.410954
\(580\) −23.8967 −0.992255
\(581\) 0 0
\(582\) 4.18034 0.173281
\(583\) −4.94427 −0.204771
\(584\) 1.47214 0.0609174
\(585\) −12.2361 −0.505899
\(586\) 9.88854 0.408492
\(587\) −31.9443 −1.31848 −0.659241 0.751932i \(-0.729122\pi\)
−0.659241 + 0.751932i \(0.729122\pi\)
\(588\) 0 0
\(589\) −0.111456 −0.00459247
\(590\) −5.07701 −0.209017
\(591\) 0.472136 0.0194211
\(592\) −29.7984 −1.22471
\(593\) 10.5836 0.434616 0.217308 0.976103i \(-0.430272\pi\)
0.217308 + 0.976103i \(0.430272\pi\)
\(594\) 0.381966 0.0156723
\(595\) 0 0
\(596\) 29.0213 1.18876
\(597\) −15.5279 −0.635513
\(598\) −13.5279 −0.553195
\(599\) −37.3050 −1.52424 −0.762120 0.647436i \(-0.775841\pi\)
−0.762120 + 0.647436i \(0.775841\pi\)
\(600\) 0 0
\(601\) 47.8328 1.95114 0.975571 0.219686i \(-0.0705033\pi\)
0.975571 + 0.219686i \(0.0705033\pi\)
\(602\) 0 0
\(603\) −12.7082 −0.517518
\(604\) −20.2918 −0.825661
\(605\) −2.23607 −0.0909091
\(606\) 4.76393 0.193522
\(607\) 9.29180 0.377142 0.188571 0.982060i \(-0.439614\pi\)
0.188571 + 0.982060i \(0.439614\pi\)
\(608\) 0.978714 0.0396921
\(609\) 0 0
\(610\) −9.34752 −0.378470
\(611\) −13.8328 −0.559616
\(612\) 11.1246 0.449686
\(613\) −7.41641 −0.299546 −0.149773 0.988720i \(-0.547854\pi\)
−0.149773 + 0.988720i \(0.547854\pi\)
\(614\) 6.11146 0.246638
\(615\) 13.4164 0.541002
\(616\) 0 0
\(617\) 40.2492 1.62037 0.810186 0.586172i \(-0.199366\pi\)
0.810186 + 0.586172i \(0.199366\pi\)
\(618\) 7.41641 0.298332
\(619\) −0.583592 −0.0234565 −0.0117283 0.999931i \(-0.503733\pi\)
−0.0117283 + 0.999931i \(0.503733\pi\)
\(620\) 1.95743 0.0786122
\(621\) 6.47214 0.259718
\(622\) 1.88854 0.0757237
\(623\) 0 0
\(624\) 17.2148 0.689143
\(625\) −25.0000 −1.00000
\(626\) −7.01316 −0.280302
\(627\) 0.236068 0.00942765
\(628\) −43.4164 −1.73250
\(629\) 56.8328 2.26607
\(630\) 0 0
\(631\) −4.00000 −0.159237 −0.0796187 0.996825i \(-0.525370\pi\)
−0.0796187 + 0.996825i \(0.525370\pi\)
\(632\) 9.52786 0.378998
\(633\) 1.05573 0.0419614
\(634\) −3.45898 −0.137374
\(635\) −35.7771 −1.41977
\(636\) −9.16718 −0.363503
\(637\) 0 0
\(638\) −2.20163 −0.0871632
\(639\) −4.47214 −0.176915
\(640\) −22.5623 −0.891853
\(641\) 40.4721 1.59855 0.799277 0.600963i \(-0.205216\pi\)
0.799277 + 0.600963i \(0.205216\pi\)
\(642\) −0.785218 −0.0309901
\(643\) 21.4164 0.844581 0.422290 0.906461i \(-0.361226\pi\)
0.422290 + 0.906461i \(0.361226\pi\)
\(644\) 0 0
\(645\) −18.9443 −0.745930
\(646\) −0.541020 −0.0212861
\(647\) −22.4164 −0.881280 −0.440640 0.897684i \(-0.645248\pi\)
−0.440640 + 0.897684i \(0.645248\pi\)
\(648\) 1.47214 0.0578310
\(649\) 5.94427 0.233333
\(650\) 0 0
\(651\) 0 0
\(652\) −41.0213 −1.60652
\(653\) −29.8885 −1.16963 −0.584815 0.811167i \(-0.698833\pi\)
−0.584815 + 0.811167i \(0.698833\pi\)
\(654\) −2.29180 −0.0896163
\(655\) 10.0000 0.390732
\(656\) −18.8754 −0.736960
\(657\) 1.00000 0.0390137
\(658\) 0 0
\(659\) −2.88854 −0.112522 −0.0562608 0.998416i \(-0.517918\pi\)
−0.0562608 + 0.998416i \(0.517918\pi\)
\(660\) −4.14590 −0.161379
\(661\) 4.00000 0.155582 0.0777910 0.996970i \(-0.475213\pi\)
0.0777910 + 0.996970i \(0.475213\pi\)
\(662\) 6.11146 0.237528
\(663\) −32.8328 −1.27512
\(664\) −5.19350 −0.201547
\(665\) 0 0
\(666\) 3.61803 0.140196
\(667\) −37.3050 −1.44445
\(668\) 17.6656 0.683504
\(669\) −18.0000 −0.695920
\(670\) −10.8541 −0.419331
\(671\) 10.9443 0.422499
\(672\) 0 0
\(673\) −45.4164 −1.75067 −0.875337 0.483513i \(-0.839360\pi\)
−0.875337 + 0.483513i \(0.839360\pi\)
\(674\) 5.88854 0.226818
\(675\) 0 0
\(676\) −31.4164 −1.20832
\(677\) −16.5836 −0.637359 −0.318680 0.947862i \(-0.603239\pi\)
−0.318680 + 0.947862i \(0.603239\pi\)
\(678\) 0.944272 0.0362645
\(679\) 0 0
\(680\) 19.7508 0.757408
\(681\) 26.8328 1.02824
\(682\) 0.180340 0.00690557
\(683\) 18.4721 0.706817 0.353408 0.935469i \(-0.385023\pi\)
0.353408 + 0.935469i \(0.385023\pi\)
\(684\) 0.437694 0.0167357
\(685\) −13.4164 −0.512615
\(686\) 0 0
\(687\) 11.4164 0.435563
\(688\) 26.6525 1.01612
\(689\) 27.0557 1.03074
\(690\) 5.52786 0.210442
\(691\) −33.4164 −1.27122 −0.635610 0.772010i \(-0.719251\pi\)
−0.635610 + 0.772010i \(0.719251\pi\)
\(692\) 26.8328 1.02003
\(693\) 0 0
\(694\) 6.47214 0.245679
\(695\) 17.8885 0.678551
\(696\) −8.48529 −0.321634
\(697\) 36.0000 1.36360
\(698\) 9.32624 0.353003
\(699\) −13.4164 −0.507455
\(700\) 0 0
\(701\) 13.4164 0.506731 0.253365 0.967371i \(-0.418463\pi\)
0.253365 + 0.967371i \(0.418463\pi\)
\(702\) −2.09017 −0.0788884
\(703\) 2.23607 0.0843349
\(704\) 4.70820 0.177447
\(705\) 5.65248 0.212885
\(706\) 0.0901699 0.00339359
\(707\) 0 0
\(708\) 11.0213 0.414205
\(709\) −4.41641 −0.165862 −0.0829308 0.996555i \(-0.526428\pi\)
−0.0829308 + 0.996555i \(0.526428\pi\)
\(710\) −3.81966 −0.143349
\(711\) 6.47214 0.242724
\(712\) −19.7508 −0.740192
\(713\) 3.05573 0.114438
\(714\) 0 0
\(715\) 12.2361 0.457603
\(716\) 26.8328 1.00279
\(717\) −27.3607 −1.02180
\(718\) 1.16718 0.0435589
\(719\) 49.3607 1.84084 0.920421 0.390928i \(-0.127846\pi\)
0.920421 + 0.390928i \(0.127846\pi\)
\(720\) −7.03444 −0.262158
\(721\) 0 0
\(722\) 7.23607 0.269299
\(723\) 19.0000 0.706618
\(724\) −15.7082 −0.583791
\(725\) 0 0
\(726\) −0.381966 −0.0141761
\(727\) −31.4164 −1.16517 −0.582585 0.812770i \(-0.697959\pi\)
−0.582585 + 0.812770i \(0.697959\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0.854102 0.0316117
\(731\) −50.8328 −1.88012
\(732\) 20.2918 0.750006
\(733\) −46.9443 −1.73393 −0.866963 0.498372i \(-0.833931\pi\)
−0.866963 + 0.498372i \(0.833931\pi\)
\(734\) 0.541020 0.0199694
\(735\) 0 0
\(736\) −26.8328 −0.989071
\(737\) 12.7082 0.468113
\(738\) 2.29180 0.0843622
\(739\) 16.5836 0.610037 0.305019 0.952346i \(-0.401337\pi\)
0.305019 + 0.952346i \(0.401337\pi\)
\(740\) −39.2705 −1.44361
\(741\) −1.29180 −0.0474553
\(742\) 0 0
\(743\) −50.3050 −1.84551 −0.922755 0.385387i \(-0.874068\pi\)
−0.922755 + 0.385387i \(0.874068\pi\)
\(744\) 0.695048 0.0254817
\(745\) 35.0000 1.28230
\(746\) −0.360680 −0.0132054
\(747\) −3.52786 −0.129078
\(748\) −11.1246 −0.406756
\(749\) 0 0
\(750\) −4.27051 −0.155937
\(751\) −28.1246 −1.02628 −0.513141 0.858304i \(-0.671518\pi\)
−0.513141 + 0.858304i \(0.671518\pi\)
\(752\) −7.95240 −0.289994
\(753\) 13.9443 0.508158
\(754\) 12.0476 0.438748
\(755\) −24.4721 −0.890632
\(756\) 0 0
\(757\) −38.4164 −1.39627 −0.698134 0.715967i \(-0.745986\pi\)
−0.698134 + 0.715967i \(0.745986\pi\)
\(758\) −2.15905 −0.0784204
\(759\) −6.47214 −0.234924
\(760\) 0.777088 0.0281879
\(761\) −22.4721 −0.814614 −0.407307 0.913291i \(-0.633532\pi\)
−0.407307 + 0.913291i \(0.633532\pi\)
\(762\) −6.11146 −0.221395
\(763\) 0 0
\(764\) −22.2492 −0.804949
\(765\) 13.4164 0.485071
\(766\) −3.41641 −0.123440
\(767\) −32.5279 −1.17451
\(768\) 5.56231 0.200712
\(769\) −1.00000 −0.0360609 −0.0180305 0.999837i \(-0.505740\pi\)
−0.0180305 + 0.999837i \(0.505740\pi\)
\(770\) 0 0
\(771\) 12.2361 0.440671
\(772\) 18.3344 0.659868
\(773\) 51.6525 1.85781 0.928905 0.370318i \(-0.120751\pi\)
0.928905 + 0.370318i \(0.120751\pi\)
\(774\) −3.23607 −0.116318
\(775\) 0 0
\(776\) −16.1115 −0.578368
\(777\) 0 0
\(778\) 12.5410 0.449617
\(779\) 1.41641 0.0507481
\(780\) 22.6869 0.812322
\(781\) 4.47214 0.160026
\(782\) 14.8328 0.530420
\(783\) −5.76393 −0.205986
\(784\) 0 0
\(785\) −52.3607 −1.86883
\(786\) 1.70820 0.0609296
\(787\) 2.12461 0.0757342 0.0378671 0.999283i \(-0.487944\pi\)
0.0378671 + 0.999283i \(0.487944\pi\)
\(788\) −0.875388 −0.0311844
\(789\) 22.4164 0.798045
\(790\) 5.52786 0.196673
\(791\) 0 0
\(792\) −1.47214 −0.0523101
\(793\) −59.8885 −2.12670
\(794\) −6.65248 −0.236088
\(795\) −11.0557 −0.392106
\(796\) 28.7902 1.02044
\(797\) 50.2361 1.77945 0.889726 0.456494i \(-0.150895\pi\)
0.889726 + 0.456494i \(0.150895\pi\)
\(798\) 0 0
\(799\) 15.1672 0.536576
\(800\) 0 0
\(801\) −13.4164 −0.474045
\(802\) −0.180340 −0.00636802
\(803\) −1.00000 −0.0352892
\(804\) 23.5623 0.830978
\(805\) 0 0
\(806\) −0.986844 −0.0347601
\(807\) 7.52786 0.264993
\(808\) −18.3607 −0.645926
\(809\) −37.1803 −1.30719 −0.653596 0.756844i \(-0.726740\pi\)
−0.653596 + 0.756844i \(0.726740\pi\)
\(810\) 0.854102 0.0300101
\(811\) −39.5410 −1.38847 −0.694236 0.719747i \(-0.744258\pi\)
−0.694236 + 0.719747i \(0.744258\pi\)
\(812\) 0 0
\(813\) −18.7082 −0.656125
\(814\) −3.61803 −0.126812
\(815\) −49.4721 −1.73293
\(816\) −18.8754 −0.660771
\(817\) −2.00000 −0.0699711
\(818\) 5.34752 0.186972
\(819\) 0 0
\(820\) −24.8754 −0.868686
\(821\) 33.5410 1.17059 0.585295 0.810821i \(-0.300979\pi\)
0.585295 + 0.810821i \(0.300979\pi\)
\(822\) −2.29180 −0.0799356
\(823\) 21.5410 0.750873 0.375436 0.926848i \(-0.377493\pi\)
0.375436 + 0.926848i \(0.377493\pi\)
\(824\) −28.5836 −0.995757
\(825\) 0 0
\(826\) 0 0
\(827\) 53.8328 1.87195 0.935975 0.352066i \(-0.114521\pi\)
0.935975 + 0.352066i \(0.114521\pi\)
\(828\) −12.0000 −0.417029
\(829\) −18.8328 −0.654091 −0.327045 0.945009i \(-0.606053\pi\)
−0.327045 + 0.945009i \(0.606053\pi\)
\(830\) −3.01316 −0.104588
\(831\) −1.05573 −0.0366228
\(832\) −25.7639 −0.893204
\(833\) 0 0
\(834\) 3.05573 0.105811
\(835\) 21.3050 0.737288
\(836\) −0.437694 −0.0151380
\(837\) 0.472136 0.0163194
\(838\) 1.14590 0.0395844
\(839\) 29.4721 1.01749 0.508746 0.860917i \(-0.330109\pi\)
0.508746 + 0.860917i \(0.330109\pi\)
\(840\) 0 0
\(841\) 4.22291 0.145618
\(842\) −7.84095 −0.270217
\(843\) −17.1803 −0.591722
\(844\) −1.95743 −0.0673774
\(845\) −37.8885 −1.30341
\(846\) 0.965558 0.0331966
\(847\) 0 0
\(848\) 15.5542 0.534133
\(849\) 8.70820 0.298865
\(850\) 0 0
\(851\) −61.3050 −2.10151
\(852\) 8.29180 0.284072
\(853\) −34.7214 −1.18884 −0.594418 0.804156i \(-0.702618\pi\)
−0.594418 + 0.804156i \(0.702618\pi\)
\(854\) 0 0
\(855\) 0.527864 0.0180526
\(856\) 3.02631 0.103437
\(857\) −38.7214 −1.32270 −0.661348 0.750079i \(-0.730015\pi\)
−0.661348 + 0.750079i \(0.730015\pi\)
\(858\) 2.09017 0.0713572
\(859\) 11.4164 0.389523 0.194761 0.980851i \(-0.437607\pi\)
0.194761 + 0.980851i \(0.437607\pi\)
\(860\) 35.1246 1.19774
\(861\) 0 0
\(862\) −10.4508 −0.355957
\(863\) −27.3050 −0.929471 −0.464736 0.885449i \(-0.653851\pi\)
−0.464736 + 0.885449i \(0.653851\pi\)
\(864\) −4.14590 −0.141046
\(865\) 32.3607 1.10030
\(866\) −15.0557 −0.511614
\(867\) 19.0000 0.645274
\(868\) 0 0
\(869\) −6.47214 −0.219552
\(870\) −4.92299 −0.166905
\(871\) −69.5410 −2.35631
\(872\) 8.83282 0.299117
\(873\) −10.9443 −0.370407
\(874\) 0.583592 0.0197403
\(875\) 0 0
\(876\) −1.85410 −0.0626443
\(877\) 13.8885 0.468983 0.234491 0.972118i \(-0.424658\pi\)
0.234491 + 0.972118i \(0.424658\pi\)
\(878\) −12.2705 −0.414110
\(879\) −25.8885 −0.873199
\(880\) 7.03444 0.237131
\(881\) −26.5967 −0.896067 −0.448034 0.894017i \(-0.647876\pi\)
−0.448034 + 0.894017i \(0.647876\pi\)
\(882\) 0 0
\(883\) −0.236068 −0.00794432 −0.00397216 0.999992i \(-0.501264\pi\)
−0.00397216 + 0.999992i \(0.501264\pi\)
\(884\) 60.8754 2.04746
\(885\) 13.2918 0.446799
\(886\) −10.2918 −0.345760
\(887\) −16.3607 −0.549338 −0.274669 0.961539i \(-0.588568\pi\)
−0.274669 + 0.961539i \(0.588568\pi\)
\(888\) −13.9443 −0.467939
\(889\) 0 0
\(890\) −11.4590 −0.384106
\(891\) −1.00000 −0.0335013
\(892\) 33.3738 1.11744
\(893\) 0.596748 0.0199694
\(894\) 5.97871 0.199958
\(895\) 32.3607 1.08170
\(896\) 0 0
\(897\) 35.4164 1.18252
\(898\) −0.541020 −0.0180541
\(899\) −2.72136 −0.0907624
\(900\) 0 0
\(901\) −29.6656 −0.988305
\(902\) −2.29180 −0.0763085
\(903\) 0 0
\(904\) −3.63932 −0.121042
\(905\) −18.9443 −0.629729
\(906\) −4.18034 −0.138882
\(907\) −49.8885 −1.65652 −0.828261 0.560343i \(-0.810669\pi\)
−0.828261 + 0.560343i \(0.810669\pi\)
\(908\) −49.7508 −1.65104
\(909\) −12.4721 −0.413675
\(910\) 0 0
\(911\) −4.58359 −0.151861 −0.0759306 0.997113i \(-0.524193\pi\)
−0.0759306 + 0.997113i \(0.524193\pi\)
\(912\) −0.742646 −0.0245915
\(913\) 3.52786 0.116755
\(914\) −5.88854 −0.194776
\(915\) 24.4721 0.809024
\(916\) −21.1672 −0.699383
\(917\) 0 0
\(918\) 2.29180 0.0756405
\(919\) −33.8885 −1.11788 −0.558940 0.829208i \(-0.688792\pi\)
−0.558940 + 0.829208i \(0.688792\pi\)
\(920\) −21.3050 −0.702403
\(921\) −16.0000 −0.527218
\(922\) 2.87539 0.0946959
\(923\) −24.4721 −0.805510
\(924\) 0 0
\(925\) 0 0
\(926\) −4.09017 −0.134411
\(927\) −19.4164 −0.637719
\(928\) 23.8967 0.784447
\(929\) 32.4853 1.06581 0.532904 0.846176i \(-0.321101\pi\)
0.532904 + 0.846176i \(0.321101\pi\)
\(930\) 0.403252 0.0132232
\(931\) 0 0
\(932\) 24.8754 0.814820
\(933\) −4.94427 −0.161868
\(934\) −11.3951 −0.372860
\(935\) −13.4164 −0.438763
\(936\) 8.05573 0.263310
\(937\) 46.7214 1.52632 0.763160 0.646209i \(-0.223647\pi\)
0.763160 + 0.646209i \(0.223647\pi\)
\(938\) 0 0
\(939\) 18.3607 0.599178
\(940\) −10.4803 −0.341829
\(941\) 47.1935 1.53846 0.769232 0.638970i \(-0.220639\pi\)
0.769232 + 0.638970i \(0.220639\pi\)
\(942\) −8.94427 −0.291420
\(943\) −38.8328 −1.26457
\(944\) −18.7001 −0.608636
\(945\) 0 0
\(946\) 3.23607 0.105214
\(947\) 35.8885 1.16622 0.583110 0.812393i \(-0.301835\pi\)
0.583110 + 0.812393i \(0.301835\pi\)
\(948\) −12.0000 −0.389742
\(949\) 5.47214 0.177633
\(950\) 0 0
\(951\) 9.05573 0.293652
\(952\) 0 0
\(953\) 56.4853 1.82974 0.914869 0.403751i \(-0.132294\pi\)
0.914869 + 0.403751i \(0.132294\pi\)
\(954\) −1.88854 −0.0611439
\(955\) −26.8328 −0.868290
\(956\) 50.7295 1.64071
\(957\) 5.76393 0.186321
\(958\) −5.30495 −0.171395
\(959\) 0 0
\(960\) 10.5279 0.339785
\(961\) −30.7771 −0.992809
\(962\) 19.7984 0.638325
\(963\) 2.05573 0.0662449
\(964\) −35.2279 −1.13462
\(965\) 22.1115 0.711793
\(966\) 0 0
\(967\) −34.2492 −1.10138 −0.550690 0.834710i \(-0.685635\pi\)
−0.550690 + 0.834710i \(0.685635\pi\)
\(968\) 1.47214 0.0473162
\(969\) 1.41641 0.0455016
\(970\) −9.34752 −0.300131
\(971\) 10.8885 0.349430 0.174715 0.984619i \(-0.444100\pi\)
0.174715 + 0.984619i \(0.444100\pi\)
\(972\) −1.85410 −0.0594703
\(973\) 0 0
\(974\) −5.30495 −0.169982
\(975\) 0 0
\(976\) −34.4296 −1.10206
\(977\) −22.4721 −0.718947 −0.359474 0.933155i \(-0.617044\pi\)
−0.359474 + 0.933155i \(0.617044\pi\)
\(978\) −8.45085 −0.270228
\(979\) 13.4164 0.428790
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) −9.50658 −0.303367
\(983\) 14.8328 0.473093 0.236547 0.971620i \(-0.423984\pi\)
0.236547 + 0.971620i \(0.423984\pi\)
\(984\) −8.83282 −0.281580
\(985\) −1.05573 −0.0336383
\(986\) −13.2098 −0.420684
\(987\) 0 0
\(988\) 2.39512 0.0761990
\(989\) 54.8328 1.74358
\(990\) −0.854102 −0.0271451
\(991\) 26.7082 0.848414 0.424207 0.905565i \(-0.360553\pi\)
0.424207 + 0.905565i \(0.360553\pi\)
\(992\) −1.95743 −0.0621484
\(993\) −16.0000 −0.507745
\(994\) 0 0
\(995\) 34.7214 1.10074
\(996\) 6.54102 0.207260
\(997\) −42.0000 −1.33015 −0.665077 0.746775i \(-0.731601\pi\)
−0.665077 + 0.746775i \(0.731601\pi\)
\(998\) 6.92299 0.219143
\(999\) −9.47214 −0.299685
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 1617.2.a.l.1.2 yes 2
3.2 odd 2 4851.2.a.bg.1.1 2
7.6 odd 2 1617.2.a.k.1.2 2
21.20 even 2 4851.2.a.bh.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1617.2.a.k.1.2 2 7.6 odd 2
1617.2.a.l.1.2 yes 2 1.1 even 1 trivial
4851.2.a.bg.1.1 2 3.2 odd 2
4851.2.a.bh.1.1 2 21.20 even 2