Properties

Label 7098.2.a.bj.1.1
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.732051 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -0.732051 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +0.732051 q^{10} +3.73205 q^{11} -1.00000 q^{12} -1.00000 q^{14} +0.732051 q^{15} +1.00000 q^{16} +0.267949 q^{17} -1.00000 q^{18} -4.46410 q^{19} -0.732051 q^{20} -1.00000 q^{21} -3.73205 q^{22} +3.46410 q^{23} +1.00000 q^{24} -4.46410 q^{25} -1.00000 q^{27} +1.00000 q^{28} -3.00000 q^{29} -0.732051 q^{30} +7.66025 q^{31} -1.00000 q^{32} -3.73205 q^{33} -0.267949 q^{34} -0.732051 q^{35} +1.00000 q^{36} -1.26795 q^{37} +4.46410 q^{38} +0.732051 q^{40} -7.00000 q^{41} +1.00000 q^{42} +0.732051 q^{43} +3.73205 q^{44} -0.732051 q^{45} -3.46410 q^{46} +4.46410 q^{47} -1.00000 q^{48} +1.00000 q^{49} +4.46410 q^{50} -0.267949 q^{51} -10.4641 q^{53} +1.00000 q^{54} -2.73205 q^{55} -1.00000 q^{56} +4.46410 q^{57} +3.00000 q^{58} +0.928203 q^{59} +0.732051 q^{60} -11.7321 q^{61} -7.66025 q^{62} +1.00000 q^{63} +1.00000 q^{64} +3.73205 q^{66} -12.9282 q^{67} +0.267949 q^{68} -3.46410 q^{69} +0.732051 q^{70} -2.19615 q^{71} -1.00000 q^{72} +6.53590 q^{73} +1.26795 q^{74} +4.46410 q^{75} -4.46410 q^{76} +3.73205 q^{77} +10.8564 q^{79} -0.732051 q^{80} +1.00000 q^{81} +7.00000 q^{82} +5.66025 q^{83} -1.00000 q^{84} -0.196152 q^{85} -0.732051 q^{86} +3.00000 q^{87} -3.73205 q^{88} -6.46410 q^{89} +0.732051 q^{90} +3.46410 q^{92} -7.66025 q^{93} -4.46410 q^{94} +3.26795 q^{95} +1.00000 q^{96} -0.732051 q^{97} -1.00000 q^{98} +3.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} - 2q^{10} + 4q^{11} - 2q^{12} - 2q^{14} - 2q^{15} + 2q^{16} + 4q^{17} - 2q^{18} - 2q^{19} + 2q^{20} - 2q^{21} - 4q^{22} + 2q^{24} - 2q^{25} - 2q^{27} + 2q^{28} - 6q^{29} + 2q^{30} - 2q^{31} - 2q^{32} - 4q^{33} - 4q^{34} + 2q^{35} + 2q^{36} - 6q^{37} + 2q^{38} - 2q^{40} - 14q^{41} + 2q^{42} - 2q^{43} + 4q^{44} + 2q^{45} + 2q^{47} - 2q^{48} + 2q^{49} + 2q^{50} - 4q^{51} - 14q^{53} + 2q^{54} - 2q^{55} - 2q^{56} + 2q^{57} + 6q^{58} - 12q^{59} - 2q^{60} - 20q^{61} + 2q^{62} + 2q^{63} + 2q^{64} + 4q^{66} - 12q^{67} + 4q^{68} - 2q^{70} + 6q^{71} - 2q^{72} + 20q^{73} + 6q^{74} + 2q^{75} - 2q^{76} + 4q^{77} - 6q^{79} + 2q^{80} + 2q^{81} + 14q^{82} - 6q^{83} - 2q^{84} + 10q^{85} + 2q^{86} + 6q^{87} - 4q^{88} - 6q^{89} - 2q^{90} + 2q^{93} - 2q^{94} + 10q^{95} + 2q^{96} + 2q^{97} - 2q^{98} + 4q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −0.732051 −0.327383 −0.163692 0.986512i \(-0.552340\pi\)
−0.163692 + 0.986512i \(0.552340\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0.732051 0.231495
\(11\) 3.73205 1.12526 0.562628 0.826710i \(-0.309790\pi\)
0.562628 + 0.826710i \(0.309790\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) 0.732051 0.189015
\(16\) 1.00000 0.250000
\(17\) 0.267949 0.0649872 0.0324936 0.999472i \(-0.489655\pi\)
0.0324936 + 0.999472i \(0.489655\pi\)
\(18\) −1.00000 −0.235702
\(19\) −4.46410 −1.02414 −0.512068 0.858945i \(-0.671120\pi\)
−0.512068 + 0.858945i \(0.671120\pi\)
\(20\) −0.732051 −0.163692
\(21\) −1.00000 −0.218218
\(22\) −3.73205 −0.795676
\(23\) 3.46410 0.722315 0.361158 0.932505i \(-0.382382\pi\)
0.361158 + 0.932505i \(0.382382\pi\)
\(24\) 1.00000 0.204124
\(25\) −4.46410 −0.892820
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) −0.732051 −0.133654
\(31\) 7.66025 1.37582 0.687911 0.725795i \(-0.258528\pi\)
0.687911 + 0.725795i \(0.258528\pi\)
\(32\) −1.00000 −0.176777
\(33\) −3.73205 −0.649667
\(34\) −0.267949 −0.0459529
\(35\) −0.732051 −0.123739
\(36\) 1.00000 0.166667
\(37\) −1.26795 −0.208450 −0.104225 0.994554i \(-0.533236\pi\)
−0.104225 + 0.994554i \(0.533236\pi\)
\(38\) 4.46410 0.724173
\(39\) 0 0
\(40\) 0.732051 0.115747
\(41\) −7.00000 −1.09322 −0.546608 0.837389i \(-0.684081\pi\)
−0.546608 + 0.837389i \(0.684081\pi\)
\(42\) 1.00000 0.154303
\(43\) 0.732051 0.111637 0.0558184 0.998441i \(-0.482223\pi\)
0.0558184 + 0.998441i \(0.482223\pi\)
\(44\) 3.73205 0.562628
\(45\) −0.732051 −0.109128
\(46\) −3.46410 −0.510754
\(47\) 4.46410 0.651156 0.325578 0.945515i \(-0.394441\pi\)
0.325578 + 0.945515i \(0.394441\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) 4.46410 0.631319
\(51\) −0.267949 −0.0375204
\(52\) 0 0
\(53\) −10.4641 −1.43735 −0.718677 0.695344i \(-0.755252\pi\)
−0.718677 + 0.695344i \(0.755252\pi\)
\(54\) 1.00000 0.136083
\(55\) −2.73205 −0.368390
\(56\) −1.00000 −0.133631
\(57\) 4.46410 0.591285
\(58\) 3.00000 0.393919
\(59\) 0.928203 0.120842 0.0604209 0.998173i \(-0.480756\pi\)
0.0604209 + 0.998173i \(0.480756\pi\)
\(60\) 0.732051 0.0945074
\(61\) −11.7321 −1.50214 −0.751068 0.660225i \(-0.770461\pi\)
−0.751068 + 0.660225i \(0.770461\pi\)
\(62\) −7.66025 −0.972853
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.73205 0.459384
\(67\) −12.9282 −1.57943 −0.789716 0.613473i \(-0.789772\pi\)
−0.789716 + 0.613473i \(0.789772\pi\)
\(68\) 0.267949 0.0324936
\(69\) −3.46410 −0.417029
\(70\) 0.732051 0.0874968
\(71\) −2.19615 −0.260635 −0.130318 0.991472i \(-0.541600\pi\)
−0.130318 + 0.991472i \(0.541600\pi\)
\(72\) −1.00000 −0.117851
\(73\) 6.53590 0.764969 0.382485 0.923962i \(-0.375069\pi\)
0.382485 + 0.923962i \(0.375069\pi\)
\(74\) 1.26795 0.147396
\(75\) 4.46410 0.515470
\(76\) −4.46410 −0.512068
\(77\) 3.73205 0.425307
\(78\) 0 0
\(79\) 10.8564 1.22144 0.610721 0.791846i \(-0.290880\pi\)
0.610721 + 0.791846i \(0.290880\pi\)
\(80\) −0.732051 −0.0818458
\(81\) 1.00000 0.111111
\(82\) 7.00000 0.773021
\(83\) 5.66025 0.621294 0.310647 0.950525i \(-0.399454\pi\)
0.310647 + 0.950525i \(0.399454\pi\)
\(84\) −1.00000 −0.109109
\(85\) −0.196152 −0.0212757
\(86\) −0.732051 −0.0789391
\(87\) 3.00000 0.321634
\(88\) −3.73205 −0.397838
\(89\) −6.46410 −0.685193 −0.342597 0.939483i \(-0.611306\pi\)
−0.342597 + 0.939483i \(0.611306\pi\)
\(90\) 0.732051 0.0771649
\(91\) 0 0
\(92\) 3.46410 0.361158
\(93\) −7.66025 −0.794331
\(94\) −4.46410 −0.460437
\(95\) 3.26795 0.335285
\(96\) 1.00000 0.102062
\(97\) −0.732051 −0.0743285 −0.0371642 0.999309i \(-0.511832\pi\)
−0.0371642 + 0.999309i \(0.511832\pi\)
\(98\) −1.00000 −0.101015
\(99\) 3.73205 0.375085
\(100\) −4.46410 −0.446410
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0.267949 0.0265309
\(103\) −12.1962 −1.20172 −0.600861 0.799353i \(-0.705176\pi\)
−0.600861 + 0.799353i \(0.705176\pi\)
\(104\) 0 0
\(105\) 0.732051 0.0714408
\(106\) 10.4641 1.01636
\(107\) −1.53590 −0.148481 −0.0742405 0.997240i \(-0.523653\pi\)
−0.0742405 + 0.997240i \(0.523653\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 3.66025 0.350589 0.175294 0.984516i \(-0.443912\pi\)
0.175294 + 0.984516i \(0.443912\pi\)
\(110\) 2.73205 0.260491
\(111\) 1.26795 0.120348
\(112\) 1.00000 0.0944911
\(113\) −10.1962 −0.959173 −0.479587 0.877495i \(-0.659213\pi\)
−0.479587 + 0.877495i \(0.659213\pi\)
\(114\) −4.46410 −0.418101
\(115\) −2.53590 −0.236474
\(116\) −3.00000 −0.278543
\(117\) 0 0
\(118\) −0.928203 −0.0854480
\(119\) 0.267949 0.0245629
\(120\) −0.732051 −0.0668268
\(121\) 2.92820 0.266200
\(122\) 11.7321 1.06217
\(123\) 7.00000 0.631169
\(124\) 7.66025 0.687911
\(125\) 6.92820 0.619677
\(126\) −1.00000 −0.0890871
\(127\) −12.5359 −1.11238 −0.556191 0.831055i \(-0.687738\pi\)
−0.556191 + 0.831055i \(0.687738\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −0.732051 −0.0644535
\(130\) 0 0
\(131\) 18.9282 1.65376 0.826882 0.562375i \(-0.190112\pi\)
0.826882 + 0.562375i \(0.190112\pi\)
\(132\) −3.73205 −0.324833
\(133\) −4.46410 −0.387087
\(134\) 12.9282 1.11683
\(135\) 0.732051 0.0630049
\(136\) −0.267949 −0.0229765
\(137\) 15.4641 1.32119 0.660594 0.750744i \(-0.270305\pi\)
0.660594 + 0.750744i \(0.270305\pi\)
\(138\) 3.46410 0.294884
\(139\) −18.1244 −1.53729 −0.768644 0.639677i \(-0.779068\pi\)
−0.768644 + 0.639677i \(0.779068\pi\)
\(140\) −0.732051 −0.0618696
\(141\) −4.46410 −0.375945
\(142\) 2.19615 0.184297
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 2.19615 0.182381
\(146\) −6.53590 −0.540915
\(147\) −1.00000 −0.0824786
\(148\) −1.26795 −0.104225
\(149\) 18.9282 1.55066 0.775329 0.631557i \(-0.217584\pi\)
0.775329 + 0.631557i \(0.217584\pi\)
\(150\) −4.46410 −0.364492
\(151\) 13.1962 1.07389 0.536944 0.843618i \(-0.319579\pi\)
0.536944 + 0.843618i \(0.319579\pi\)
\(152\) 4.46410 0.362086
\(153\) 0.267949 0.0216624
\(154\) −3.73205 −0.300737
\(155\) −5.60770 −0.450421
\(156\) 0 0
\(157\) −15.4641 −1.23417 −0.617085 0.786897i \(-0.711686\pi\)
−0.617085 + 0.786897i \(0.711686\pi\)
\(158\) −10.8564 −0.863689
\(159\) 10.4641 0.829857
\(160\) 0.732051 0.0578737
\(161\) 3.46410 0.273009
\(162\) −1.00000 −0.0785674
\(163\) −7.80385 −0.611245 −0.305622 0.952153i \(-0.598864\pi\)
−0.305622 + 0.952153i \(0.598864\pi\)
\(164\) −7.00000 −0.546608
\(165\) 2.73205 0.212690
\(166\) −5.66025 −0.439321
\(167\) −5.85641 −0.453182 −0.226591 0.973990i \(-0.572758\pi\)
−0.226591 + 0.973990i \(0.572758\pi\)
\(168\) 1.00000 0.0771517
\(169\) 0 0
\(170\) 0.196152 0.0150442
\(171\) −4.46410 −0.341378
\(172\) 0.732051 0.0558184
\(173\) 20.7321 1.57623 0.788114 0.615529i \(-0.211058\pi\)
0.788114 + 0.615529i \(0.211058\pi\)
\(174\) −3.00000 −0.227429
\(175\) −4.46410 −0.337454
\(176\) 3.73205 0.281314
\(177\) −0.928203 −0.0697680
\(178\) 6.46410 0.484505
\(179\) 22.3923 1.67368 0.836840 0.547448i \(-0.184401\pi\)
0.836840 + 0.547448i \(0.184401\pi\)
\(180\) −0.732051 −0.0545638
\(181\) 1.19615 0.0889093 0.0444547 0.999011i \(-0.485845\pi\)
0.0444547 + 0.999011i \(0.485845\pi\)
\(182\) 0 0
\(183\) 11.7321 0.867258
\(184\) −3.46410 −0.255377
\(185\) 0.928203 0.0682429
\(186\) 7.66025 0.561677
\(187\) 1.00000 0.0731272
\(188\) 4.46410 0.325578
\(189\) −1.00000 −0.0727393
\(190\) −3.26795 −0.237082
\(191\) −4.19615 −0.303623 −0.151811 0.988409i \(-0.548511\pi\)
−0.151811 + 0.988409i \(0.548511\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 17.1962 1.23781 0.618903 0.785467i \(-0.287577\pi\)
0.618903 + 0.785467i \(0.287577\pi\)
\(194\) 0.732051 0.0525582
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 2.26795 0.161585 0.0807923 0.996731i \(-0.474255\pi\)
0.0807923 + 0.996731i \(0.474255\pi\)
\(198\) −3.73205 −0.265225
\(199\) −4.19615 −0.297457 −0.148729 0.988878i \(-0.547518\pi\)
−0.148729 + 0.988878i \(0.547518\pi\)
\(200\) 4.46410 0.315660
\(201\) 12.9282 0.911885
\(202\) 10.0000 0.703598
\(203\) −3.00000 −0.210559
\(204\) −0.267949 −0.0187602
\(205\) 5.12436 0.357901
\(206\) 12.1962 0.849746
\(207\) 3.46410 0.240772
\(208\) 0 0
\(209\) −16.6603 −1.15241
\(210\) −0.732051 −0.0505163
\(211\) −15.8564 −1.09160 −0.545800 0.837915i \(-0.683774\pi\)
−0.545800 + 0.837915i \(0.683774\pi\)
\(212\) −10.4641 −0.718677
\(213\) 2.19615 0.150478
\(214\) 1.53590 0.104992
\(215\) −0.535898 −0.0365480
\(216\) 1.00000 0.0680414
\(217\) 7.66025 0.520012
\(218\) −3.66025 −0.247904
\(219\) −6.53590 −0.441655
\(220\) −2.73205 −0.184195
\(221\) 0 0
\(222\) −1.26795 −0.0850992
\(223\) 8.39230 0.561990 0.280995 0.959709i \(-0.409336\pi\)
0.280995 + 0.959709i \(0.409336\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −4.46410 −0.297607
\(226\) 10.1962 0.678238
\(227\) −8.53590 −0.566547 −0.283274 0.959039i \(-0.591420\pi\)
−0.283274 + 0.959039i \(0.591420\pi\)
\(228\) 4.46410 0.295642
\(229\) −10.3205 −0.681998 −0.340999 0.940064i \(-0.610765\pi\)
−0.340999 + 0.940064i \(0.610765\pi\)
\(230\) 2.53590 0.167212
\(231\) −3.73205 −0.245551
\(232\) 3.00000 0.196960
\(233\) 2.19615 0.143875 0.0719374 0.997409i \(-0.477082\pi\)
0.0719374 + 0.997409i \(0.477082\pi\)
\(234\) 0 0
\(235\) −3.26795 −0.213177
\(236\) 0.928203 0.0604209
\(237\) −10.8564 −0.705199
\(238\) −0.267949 −0.0173686
\(239\) 13.8038 0.892897 0.446448 0.894809i \(-0.352689\pi\)
0.446448 + 0.894809i \(0.352689\pi\)
\(240\) 0.732051 0.0472537
\(241\) −12.7846 −0.823529 −0.411765 0.911290i \(-0.635087\pi\)
−0.411765 + 0.911290i \(0.635087\pi\)
\(242\) −2.92820 −0.188232
\(243\) −1.00000 −0.0641500
\(244\) −11.7321 −0.751068
\(245\) −0.732051 −0.0467690
\(246\) −7.00000 −0.446304
\(247\) 0 0
\(248\) −7.66025 −0.486427
\(249\) −5.66025 −0.358704
\(250\) −6.92820 −0.438178
\(251\) 18.1962 1.14853 0.574265 0.818669i \(-0.305288\pi\)
0.574265 + 0.818669i \(0.305288\pi\)
\(252\) 1.00000 0.0629941
\(253\) 12.9282 0.812789
\(254\) 12.5359 0.786572
\(255\) 0.196152 0.0122835
\(256\) 1.00000 0.0625000
\(257\) 11.0526 0.689440 0.344720 0.938706i \(-0.387974\pi\)
0.344720 + 0.938706i \(0.387974\pi\)
\(258\) 0.732051 0.0455755
\(259\) −1.26795 −0.0787865
\(260\) 0 0
\(261\) −3.00000 −0.185695
\(262\) −18.9282 −1.16939
\(263\) −18.7321 −1.15507 −0.577534 0.816367i \(-0.695985\pi\)
−0.577534 + 0.816367i \(0.695985\pi\)
\(264\) 3.73205 0.229692
\(265\) 7.66025 0.470566
\(266\) 4.46410 0.273712
\(267\) 6.46410 0.395597
\(268\) −12.9282 −0.789716
\(269\) −23.7128 −1.44580 −0.722898 0.690955i \(-0.757190\pi\)
−0.722898 + 0.690955i \(0.757190\pi\)
\(270\) −0.732051 −0.0445512
\(271\) −21.5167 −1.30704 −0.653522 0.756908i \(-0.726709\pi\)
−0.653522 + 0.756908i \(0.726709\pi\)
\(272\) 0.267949 0.0162468
\(273\) 0 0
\(274\) −15.4641 −0.934221
\(275\) −16.6603 −1.00465
\(276\) −3.46410 −0.208514
\(277\) −26.3923 −1.58576 −0.792880 0.609378i \(-0.791419\pi\)
−0.792880 + 0.609378i \(0.791419\pi\)
\(278\) 18.1244 1.08703
\(279\) 7.66025 0.458607
\(280\) 0.732051 0.0437484
\(281\) 18.1962 1.08549 0.542746 0.839897i \(-0.317385\pi\)
0.542746 + 0.839897i \(0.317385\pi\)
\(282\) 4.46410 0.265833
\(283\) −22.9282 −1.36294 −0.681470 0.731846i \(-0.738659\pi\)
−0.681470 + 0.731846i \(0.738659\pi\)
\(284\) −2.19615 −0.130318
\(285\) −3.26795 −0.193577
\(286\) 0 0
\(287\) −7.00000 −0.413197
\(288\) −1.00000 −0.0589256
\(289\) −16.9282 −0.995777
\(290\) −2.19615 −0.128963
\(291\) 0.732051 0.0429136
\(292\) 6.53590 0.382485
\(293\) −12.7846 −0.746885 −0.373442 0.927653i \(-0.621823\pi\)
−0.373442 + 0.927653i \(0.621823\pi\)
\(294\) 1.00000 0.0583212
\(295\) −0.679492 −0.0395615
\(296\) 1.26795 0.0736980
\(297\) −3.73205 −0.216556
\(298\) −18.9282 −1.09648
\(299\) 0 0
\(300\) 4.46410 0.257735
\(301\) 0.732051 0.0421947
\(302\) −13.1962 −0.759353
\(303\) 10.0000 0.574485
\(304\) −4.46410 −0.256034
\(305\) 8.58846 0.491774
\(306\) −0.267949 −0.0153176
\(307\) −33.7846 −1.92819 −0.964095 0.265558i \(-0.914444\pi\)
−0.964095 + 0.265558i \(0.914444\pi\)
\(308\) 3.73205 0.212653
\(309\) 12.1962 0.693815
\(310\) 5.60770 0.318496
\(311\) −19.1962 −1.08851 −0.544257 0.838919i \(-0.683188\pi\)
−0.544257 + 0.838919i \(0.683188\pi\)
\(312\) 0 0
\(313\) 1.80385 0.101959 0.0509797 0.998700i \(-0.483766\pi\)
0.0509797 + 0.998700i \(0.483766\pi\)
\(314\) 15.4641 0.872690
\(315\) −0.732051 −0.0412464
\(316\) 10.8564 0.610721
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) −10.4641 −0.586798
\(319\) −11.1962 −0.626864
\(320\) −0.732051 −0.0409229
\(321\) 1.53590 0.0857255
\(322\) −3.46410 −0.193047
\(323\) −1.19615 −0.0665557
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 7.80385 0.432215
\(327\) −3.66025 −0.202413
\(328\) 7.00000 0.386510
\(329\) 4.46410 0.246114
\(330\) −2.73205 −0.150394
\(331\) 16.9282 0.930458 0.465229 0.885190i \(-0.345972\pi\)
0.465229 + 0.885190i \(0.345972\pi\)
\(332\) 5.66025 0.310647
\(333\) −1.26795 −0.0694832
\(334\) 5.85641 0.320448
\(335\) 9.46410 0.517079
\(336\) −1.00000 −0.0545545
\(337\) 27.7846 1.51352 0.756762 0.653690i \(-0.226780\pi\)
0.756762 + 0.653690i \(0.226780\pi\)
\(338\) 0 0
\(339\) 10.1962 0.553779
\(340\) −0.196152 −0.0106379
\(341\) 28.5885 1.54815
\(342\) 4.46410 0.241391
\(343\) 1.00000 0.0539949
\(344\) −0.732051 −0.0394695
\(345\) 2.53590 0.136528
\(346\) −20.7321 −1.11456
\(347\) 22.8564 1.22700 0.613498 0.789696i \(-0.289762\pi\)
0.613498 + 0.789696i \(0.289762\pi\)
\(348\) 3.00000 0.160817
\(349\) −23.7128 −1.26932 −0.634659 0.772792i \(-0.718859\pi\)
−0.634659 + 0.772792i \(0.718859\pi\)
\(350\) 4.46410 0.238616
\(351\) 0 0
\(352\) −3.73205 −0.198919
\(353\) −1.46410 −0.0779263 −0.0389631 0.999241i \(-0.512405\pi\)
−0.0389631 + 0.999241i \(0.512405\pi\)
\(354\) 0.928203 0.0493334
\(355\) 1.60770 0.0853276
\(356\) −6.46410 −0.342597
\(357\) −0.267949 −0.0141814
\(358\) −22.3923 −1.18347
\(359\) 15.8038 0.834095 0.417048 0.908885i \(-0.363065\pi\)
0.417048 + 0.908885i \(0.363065\pi\)
\(360\) 0.732051 0.0385825
\(361\) 0.928203 0.0488528
\(362\) −1.19615 −0.0628684
\(363\) −2.92820 −0.153691
\(364\) 0 0
\(365\) −4.78461 −0.250438
\(366\) −11.7321 −0.613244
\(367\) 16.2487 0.848176 0.424088 0.905621i \(-0.360595\pi\)
0.424088 + 0.905621i \(0.360595\pi\)
\(368\) 3.46410 0.180579
\(369\) −7.00000 −0.364405
\(370\) −0.928203 −0.0482550
\(371\) −10.4641 −0.543269
\(372\) −7.66025 −0.397166
\(373\) −20.5885 −1.06603 −0.533015 0.846106i \(-0.678941\pi\)
−0.533015 + 0.846106i \(0.678941\pi\)
\(374\) −1.00000 −0.0517088
\(375\) −6.92820 −0.357771
\(376\) −4.46410 −0.230218
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) −14.5885 −0.749359 −0.374679 0.927154i \(-0.622247\pi\)
−0.374679 + 0.927154i \(0.622247\pi\)
\(380\) 3.26795 0.167642
\(381\) 12.5359 0.642234
\(382\) 4.19615 0.214694
\(383\) −14.6077 −0.746418 −0.373209 0.927747i \(-0.621743\pi\)
−0.373209 + 0.927747i \(0.621743\pi\)
\(384\) 1.00000 0.0510310
\(385\) −2.73205 −0.139238
\(386\) −17.1962 −0.875261
\(387\) 0.732051 0.0372122
\(388\) −0.732051 −0.0371642
\(389\) −33.1769 −1.68214 −0.841068 0.540929i \(-0.818073\pi\)
−0.841068 + 0.540929i \(0.818073\pi\)
\(390\) 0 0
\(391\) 0.928203 0.0469413
\(392\) −1.00000 −0.0505076
\(393\) −18.9282 −0.954802
\(394\) −2.26795 −0.114258
\(395\) −7.94744 −0.399879
\(396\) 3.73205 0.187543
\(397\) −16.4641 −0.826310 −0.413155 0.910661i \(-0.635573\pi\)
−0.413155 + 0.910661i \(0.635573\pi\)
\(398\) 4.19615 0.210334
\(399\) 4.46410 0.223485
\(400\) −4.46410 −0.223205
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) −12.9282 −0.644800
\(403\) 0 0
\(404\) −10.0000 −0.497519
\(405\) −0.732051 −0.0363759
\(406\) 3.00000 0.148888
\(407\) −4.73205 −0.234559
\(408\) 0.267949 0.0132655
\(409\) −13.2679 −0.656058 −0.328029 0.944668i \(-0.606384\pi\)
−0.328029 + 0.944668i \(0.606384\pi\)
\(410\) −5.12436 −0.253074
\(411\) −15.4641 −0.762788
\(412\) −12.1962 −0.600861
\(413\) 0.928203 0.0456739
\(414\) −3.46410 −0.170251
\(415\) −4.14359 −0.203401
\(416\) 0 0
\(417\) 18.1244 0.887554
\(418\) 16.6603 0.814880
\(419\) −6.19615 −0.302702 −0.151351 0.988480i \(-0.548362\pi\)
−0.151351 + 0.988480i \(0.548362\pi\)
\(420\) 0.732051 0.0357204
\(421\) 14.3923 0.701438 0.350719 0.936481i \(-0.385937\pi\)
0.350719 + 0.936481i \(0.385937\pi\)
\(422\) 15.8564 0.771878
\(423\) 4.46410 0.217052
\(424\) 10.4641 0.508182
\(425\) −1.19615 −0.0580219
\(426\) −2.19615 −0.106404
\(427\) −11.7321 −0.567754
\(428\) −1.53590 −0.0742405
\(429\) 0 0
\(430\) 0.535898 0.0258433
\(431\) 21.1244 1.01752 0.508762 0.860907i \(-0.330103\pi\)
0.508762 + 0.860907i \(0.330103\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 8.92820 0.429062 0.214531 0.976717i \(-0.431178\pi\)
0.214531 + 0.976717i \(0.431178\pi\)
\(434\) −7.66025 −0.367704
\(435\) −2.19615 −0.105297
\(436\) 3.66025 0.175294
\(437\) −15.4641 −0.739748
\(438\) 6.53590 0.312297
\(439\) −16.3923 −0.782362 −0.391181 0.920314i \(-0.627933\pi\)
−0.391181 + 0.920314i \(0.627933\pi\)
\(440\) 2.73205 0.130245
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −31.3923 −1.49149 −0.745747 0.666230i \(-0.767907\pi\)
−0.745747 + 0.666230i \(0.767907\pi\)
\(444\) 1.26795 0.0601742
\(445\) 4.73205 0.224321
\(446\) −8.39230 −0.397387
\(447\) −18.9282 −0.895273
\(448\) 1.00000 0.0472456
\(449\) −10.7321 −0.506477 −0.253238 0.967404i \(-0.581496\pi\)
−0.253238 + 0.967404i \(0.581496\pi\)
\(450\) 4.46410 0.210440
\(451\) −26.1244 −1.23015
\(452\) −10.1962 −0.479587
\(453\) −13.1962 −0.620009
\(454\) 8.53590 0.400610
\(455\) 0 0
\(456\) −4.46410 −0.209051
\(457\) 6.78461 0.317371 0.158685 0.987329i \(-0.449274\pi\)
0.158685 + 0.987329i \(0.449274\pi\)
\(458\) 10.3205 0.482246
\(459\) −0.267949 −0.0125068
\(460\) −2.53590 −0.118237
\(461\) 27.7128 1.29071 0.645357 0.763881i \(-0.276709\pi\)
0.645357 + 0.763881i \(0.276709\pi\)
\(462\) 3.73205 0.173631
\(463\) −1.19615 −0.0555899 −0.0277950 0.999614i \(-0.508849\pi\)
−0.0277950 + 0.999614i \(0.508849\pi\)
\(464\) −3.00000 −0.139272
\(465\) 5.60770 0.260051
\(466\) −2.19615 −0.101735
\(467\) −9.85641 −0.456100 −0.228050 0.973649i \(-0.573235\pi\)
−0.228050 + 0.973649i \(0.573235\pi\)
\(468\) 0 0
\(469\) −12.9282 −0.596969
\(470\) 3.26795 0.150739
\(471\) 15.4641 0.712548
\(472\) −0.928203 −0.0427240
\(473\) 2.73205 0.125620
\(474\) 10.8564 0.498651
\(475\) 19.9282 0.914369
\(476\) 0.267949 0.0122814
\(477\) −10.4641 −0.479118
\(478\) −13.8038 −0.631373
\(479\) −31.3923 −1.43435 −0.717176 0.696893i \(-0.754565\pi\)
−0.717176 + 0.696893i \(0.754565\pi\)
\(480\) −0.732051 −0.0334134
\(481\) 0 0
\(482\) 12.7846 0.582323
\(483\) −3.46410 −0.157622
\(484\) 2.92820 0.133100
\(485\) 0.535898 0.0243339
\(486\) 1.00000 0.0453609
\(487\) −21.1962 −0.960489 −0.480245 0.877135i \(-0.659452\pi\)
−0.480245 + 0.877135i \(0.659452\pi\)
\(488\) 11.7321 0.531085
\(489\) 7.80385 0.352902
\(490\) 0.732051 0.0330707
\(491\) 36.2487 1.63588 0.817941 0.575303i \(-0.195116\pi\)
0.817941 + 0.575303i \(0.195116\pi\)
\(492\) 7.00000 0.315584
\(493\) −0.803848 −0.0362035
\(494\) 0 0
\(495\) −2.73205 −0.122797
\(496\) 7.66025 0.343956
\(497\) −2.19615 −0.0985109
\(498\) 5.66025 0.253642
\(499\) 12.9808 0.581099 0.290549 0.956860i \(-0.406162\pi\)
0.290549 + 0.956860i \(0.406162\pi\)
\(500\) 6.92820 0.309839
\(501\) 5.85641 0.261645
\(502\) −18.1962 −0.812134
\(503\) −23.7128 −1.05730 −0.528651 0.848839i \(-0.677302\pi\)
−0.528651 + 0.848839i \(0.677302\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 7.32051 0.325758
\(506\) −12.9282 −0.574729
\(507\) 0 0
\(508\) −12.5359 −0.556191
\(509\) −26.0526 −1.15476 −0.577380 0.816476i \(-0.695925\pi\)
−0.577380 + 0.816476i \(0.695925\pi\)
\(510\) −0.196152 −0.00868578
\(511\) 6.53590 0.289131
\(512\) −1.00000 −0.0441942
\(513\) 4.46410 0.197095
\(514\) −11.0526 −0.487507
\(515\) 8.92820 0.393424
\(516\) −0.732051 −0.0322267
\(517\) 16.6603 0.732717
\(518\) 1.26795 0.0557105
\(519\) −20.7321 −0.910036
\(520\) 0 0
\(521\) −2.26795 −0.0993607 −0.0496803 0.998765i \(-0.515820\pi\)
−0.0496803 + 0.998765i \(0.515820\pi\)
\(522\) 3.00000 0.131306
\(523\) −18.8038 −0.822235 −0.411117 0.911582i \(-0.634861\pi\)
−0.411117 + 0.911582i \(0.634861\pi\)
\(524\) 18.9282 0.826882
\(525\) 4.46410 0.194829
\(526\) 18.7321 0.816756
\(527\) 2.05256 0.0894109
\(528\) −3.73205 −0.162417
\(529\) −11.0000 −0.478261
\(530\) −7.66025 −0.332740
\(531\) 0.928203 0.0402806
\(532\) −4.46410 −0.193543
\(533\) 0 0
\(534\) −6.46410 −0.279729
\(535\) 1.12436 0.0486101
\(536\) 12.9282 0.558413
\(537\) −22.3923 −0.966299
\(538\) 23.7128 1.02233
\(539\) 3.73205 0.160751
\(540\) 0.732051 0.0315025
\(541\) 30.1962 1.29823 0.649117 0.760689i \(-0.275139\pi\)
0.649117 + 0.760689i \(0.275139\pi\)
\(542\) 21.5167 0.924220
\(543\) −1.19615 −0.0513318
\(544\) −0.267949 −0.0114882
\(545\) −2.67949 −0.114777
\(546\) 0 0
\(547\) −12.8756 −0.550523 −0.275261 0.961369i \(-0.588764\pi\)
−0.275261 + 0.961369i \(0.588764\pi\)
\(548\) 15.4641 0.660594
\(549\) −11.7321 −0.500712
\(550\) 16.6603 0.710396
\(551\) 13.3923 0.570531
\(552\) 3.46410 0.147442
\(553\) 10.8564 0.461661
\(554\) 26.3923 1.12130
\(555\) −0.928203 −0.0394000
\(556\) −18.1244 −0.768644
\(557\) −36.6603 −1.55334 −0.776672 0.629905i \(-0.783094\pi\)
−0.776672 + 0.629905i \(0.783094\pi\)
\(558\) −7.66025 −0.324284
\(559\) 0 0
\(560\) −0.732051 −0.0309348
\(561\) −1.00000 −0.0422200
\(562\) −18.1962 −0.767558
\(563\) −33.2679 −1.40208 −0.701038 0.713123i \(-0.747280\pi\)
−0.701038 + 0.713123i \(0.747280\pi\)
\(564\) −4.46410 −0.187973
\(565\) 7.46410 0.314017
\(566\) 22.9282 0.963744
\(567\) 1.00000 0.0419961
\(568\) 2.19615 0.0921485
\(569\) 22.7321 0.952977 0.476489 0.879181i \(-0.341909\pi\)
0.476489 + 0.879181i \(0.341909\pi\)
\(570\) 3.26795 0.136879
\(571\) 19.8038 0.828765 0.414383 0.910103i \(-0.363998\pi\)
0.414383 + 0.910103i \(0.363998\pi\)
\(572\) 0 0
\(573\) 4.19615 0.175297
\(574\) 7.00000 0.292174
\(575\) −15.4641 −0.644898
\(576\) 1.00000 0.0416667
\(577\) 13.2679 0.552352 0.276176 0.961107i \(-0.410933\pi\)
0.276176 + 0.961107i \(0.410933\pi\)
\(578\) 16.9282 0.704120
\(579\) −17.1962 −0.714648
\(580\) 2.19615 0.0911903
\(581\) 5.66025 0.234827
\(582\) −0.732051 −0.0303445
\(583\) −39.0526 −1.61739
\(584\) −6.53590 −0.270457
\(585\) 0 0
\(586\) 12.7846 0.528127
\(587\) −31.3731 −1.29491 −0.647453 0.762106i \(-0.724166\pi\)
−0.647453 + 0.762106i \(0.724166\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −34.1962 −1.40903
\(590\) 0.679492 0.0279742
\(591\) −2.26795 −0.0932910
\(592\) −1.26795 −0.0521124
\(593\) 13.6795 0.561749 0.280875 0.959744i \(-0.409375\pi\)
0.280875 + 0.959744i \(0.409375\pi\)
\(594\) 3.73205 0.153128
\(595\) −0.196152 −0.00804147
\(596\) 18.9282 0.775329
\(597\) 4.19615 0.171737
\(598\) 0 0
\(599\) −21.7128 −0.887161 −0.443581 0.896234i \(-0.646292\pi\)
−0.443581 + 0.896234i \(0.646292\pi\)
\(600\) −4.46410 −0.182246
\(601\) −12.8756 −0.525208 −0.262604 0.964904i \(-0.584581\pi\)
−0.262604 + 0.964904i \(0.584581\pi\)
\(602\) −0.732051 −0.0298362
\(603\) −12.9282 −0.526477
\(604\) 13.1962 0.536944
\(605\) −2.14359 −0.0871495
\(606\) −10.0000 −0.406222
\(607\) 39.9090 1.61985 0.809927 0.586530i \(-0.199506\pi\)
0.809927 + 0.586530i \(0.199506\pi\)
\(608\) 4.46410 0.181043
\(609\) 3.00000 0.121566
\(610\) −8.58846 −0.347736
\(611\) 0 0
\(612\) 0.267949 0.0108312
\(613\) 41.8564 1.69056 0.845282 0.534320i \(-0.179432\pi\)
0.845282 + 0.534320i \(0.179432\pi\)
\(614\) 33.7846 1.36344
\(615\) −5.12436 −0.206634
\(616\) −3.73205 −0.150369
\(617\) −41.5167 −1.67140 −0.835699 0.549188i \(-0.814937\pi\)
−0.835699 + 0.549188i \(0.814937\pi\)
\(618\) −12.1962 −0.490601
\(619\) 2.32051 0.0932691 0.0466345 0.998912i \(-0.485150\pi\)
0.0466345 + 0.998912i \(0.485150\pi\)
\(620\) −5.60770 −0.225210
\(621\) −3.46410 −0.139010
\(622\) 19.1962 0.769696
\(623\) −6.46410 −0.258979
\(624\) 0 0
\(625\) 17.2487 0.689948
\(626\) −1.80385 −0.0720962
\(627\) 16.6603 0.665346
\(628\) −15.4641 −0.617085
\(629\) −0.339746 −0.0135466
\(630\) 0.732051 0.0291656
\(631\) −15.9808 −0.636184 −0.318092 0.948060i \(-0.603042\pi\)
−0.318092 + 0.948060i \(0.603042\pi\)
\(632\) −10.8564 −0.431845
\(633\) 15.8564 0.630236
\(634\) −18.0000 −0.714871
\(635\) 9.17691 0.364175
\(636\) 10.4641 0.414929
\(637\) 0 0
\(638\) 11.1962 0.443260
\(639\) −2.19615 −0.0868784
\(640\) 0.732051 0.0289368
\(641\) −16.2487 −0.641786 −0.320893 0.947116i \(-0.603983\pi\)
−0.320893 + 0.947116i \(0.603983\pi\)
\(642\) −1.53590 −0.0606171
\(643\) −1.67949 −0.0662327 −0.0331163 0.999452i \(-0.510543\pi\)
−0.0331163 + 0.999452i \(0.510543\pi\)
\(644\) 3.46410 0.136505
\(645\) 0.535898 0.0211010
\(646\) 1.19615 0.0470620
\(647\) 1.73205 0.0680939 0.0340470 0.999420i \(-0.489160\pi\)
0.0340470 + 0.999420i \(0.489160\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 3.46410 0.135978
\(650\) 0 0
\(651\) −7.66025 −0.300229
\(652\) −7.80385 −0.305622
\(653\) 11.6795 0.457054 0.228527 0.973538i \(-0.426609\pi\)
0.228527 + 0.973538i \(0.426609\pi\)
\(654\) 3.66025 0.143127
\(655\) −13.8564 −0.541415
\(656\) −7.00000 −0.273304
\(657\) 6.53590 0.254990
\(658\) −4.46410 −0.174029
\(659\) 33.9282 1.32166 0.660828 0.750538i \(-0.270205\pi\)
0.660828 + 0.750538i \(0.270205\pi\)
\(660\) 2.73205 0.106345
\(661\) −27.6077 −1.07381 −0.536907 0.843641i \(-0.680408\pi\)
−0.536907 + 0.843641i \(0.680408\pi\)
\(662\) −16.9282 −0.657933
\(663\) 0 0
\(664\) −5.66025 −0.219660
\(665\) 3.26795 0.126726
\(666\) 1.26795 0.0491320
\(667\) −10.3923 −0.402392
\(668\) −5.85641 −0.226591
\(669\) −8.39230 −0.324465
\(670\) −9.46410 −0.365630
\(671\) −43.7846 −1.69029
\(672\) 1.00000 0.0385758
\(673\) −47.9282 −1.84750 −0.923748 0.383000i \(-0.874891\pi\)
−0.923748 + 0.383000i \(0.874891\pi\)
\(674\) −27.7846 −1.07022
\(675\) 4.46410 0.171823
\(676\) 0 0
\(677\) −47.7654 −1.83577 −0.917886 0.396844i \(-0.870105\pi\)
−0.917886 + 0.396844i \(0.870105\pi\)
\(678\) −10.1962 −0.391581
\(679\) −0.732051 −0.0280935
\(680\) 0.196152 0.00752210
\(681\) 8.53590 0.327096
\(682\) −28.5885 −1.09471
\(683\) −26.3923 −1.00987 −0.504937 0.863156i \(-0.668484\pi\)
−0.504937 + 0.863156i \(0.668484\pi\)
\(684\) −4.46410 −0.170689
\(685\) −11.3205 −0.432534
\(686\) −1.00000 −0.0381802
\(687\) 10.3205 0.393752
\(688\) 0.732051 0.0279092
\(689\) 0 0
\(690\) −2.53590 −0.0965400
\(691\) 40.7846 1.55152 0.775760 0.631028i \(-0.217367\pi\)
0.775760 + 0.631028i \(0.217367\pi\)
\(692\) 20.7321 0.788114
\(693\) 3.73205 0.141769
\(694\) −22.8564 −0.867617
\(695\) 13.2679 0.503282
\(696\) −3.00000 −0.113715
\(697\) −1.87564 −0.0710451
\(698\) 23.7128 0.897543
\(699\) −2.19615 −0.0830661
\(700\) −4.46410 −0.168727
\(701\) 14.3205 0.540878 0.270439 0.962737i \(-0.412831\pi\)
0.270439 + 0.962737i \(0.412831\pi\)
\(702\) 0 0
\(703\) 5.66025 0.213481
\(704\) 3.73205 0.140657
\(705\) 3.26795 0.123078
\(706\) 1.46410 0.0551022
\(707\) −10.0000 −0.376089
\(708\) −0.928203 −0.0348840
\(709\) 24.0526 0.903313 0.451656 0.892192i \(-0.350833\pi\)
0.451656 + 0.892192i \(0.350833\pi\)
\(710\) −1.60770 −0.0603357
\(711\) 10.8564 0.407147
\(712\) 6.46410 0.242252
\(713\) 26.5359 0.993777
\(714\) 0.267949 0.0100277
\(715\) 0 0
\(716\) 22.3923 0.836840
\(717\) −13.8038 −0.515514
\(718\) −15.8038 −0.589794
\(719\) 3.05256 0.113841 0.0569206 0.998379i \(-0.481872\pi\)
0.0569206 + 0.998379i \(0.481872\pi\)
\(720\) −0.732051 −0.0272819
\(721\) −12.1962 −0.454208
\(722\) −0.928203 −0.0345441
\(723\) 12.7846 0.475465
\(724\) 1.19615 0.0444547
\(725\) 13.3923 0.497378
\(726\) 2.92820 0.108676
\(727\) 1.46410 0.0543005 0.0271503 0.999631i \(-0.491357\pi\)
0.0271503 + 0.999631i \(0.491357\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 4.78461 0.177086
\(731\) 0.196152 0.00725496
\(732\) 11.7321 0.433629
\(733\) 6.85641 0.253247 0.126624 0.991951i \(-0.459586\pi\)
0.126624 + 0.991951i \(0.459586\pi\)
\(734\) −16.2487 −0.599751
\(735\) 0.732051 0.0270021
\(736\) −3.46410 −0.127688
\(737\) −48.2487 −1.77726
\(738\) 7.00000 0.257674
\(739\) −6.39230 −0.235145 −0.117572 0.993064i \(-0.537511\pi\)
−0.117572 + 0.993064i \(0.537511\pi\)
\(740\) 0.928203 0.0341214
\(741\) 0 0
\(742\) 10.4641 0.384149
\(743\) −19.5167 −0.715997 −0.357998 0.933722i \(-0.616541\pi\)
−0.357998 + 0.933722i \(0.616541\pi\)
\(744\) 7.66025 0.280839
\(745\) −13.8564 −0.507659
\(746\) 20.5885 0.753797
\(747\) 5.66025 0.207098
\(748\) 1.00000 0.0365636
\(749\) −1.53590 −0.0561205
\(750\) 6.92820 0.252982
\(751\) −6.85641 −0.250194 −0.125097 0.992145i \(-0.539924\pi\)
−0.125097 + 0.992145i \(0.539924\pi\)
\(752\) 4.46410 0.162789
\(753\) −18.1962 −0.663105
\(754\) 0 0
\(755\) −9.66025 −0.351573
\(756\) −1.00000 −0.0363696
\(757\) −38.0526 −1.38304 −0.691522 0.722356i \(-0.743059\pi\)
−0.691522 + 0.722356i \(0.743059\pi\)
\(758\) 14.5885 0.529877
\(759\) −12.9282 −0.469264
\(760\) −3.26795 −0.118541
\(761\) −0.679492 −0.0246316 −0.0123158 0.999924i \(-0.503920\pi\)
−0.0123158 + 0.999924i \(0.503920\pi\)
\(762\) −12.5359 −0.454128
\(763\) 3.66025 0.132510
\(764\) −4.19615 −0.151811
\(765\) −0.196152 −0.00709191
\(766\) 14.6077 0.527797
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) 24.4449 0.881504 0.440752 0.897629i \(-0.354712\pi\)
0.440752 + 0.897629i \(0.354712\pi\)
\(770\) 2.73205 0.0984563
\(771\) −11.0526 −0.398048
\(772\) 17.1962 0.618903
\(773\) −42.3923 −1.52475 −0.762373 0.647138i \(-0.775966\pi\)
−0.762373 + 0.647138i \(0.775966\pi\)
\(774\) −0.732051 −0.0263130
\(775\) −34.1962 −1.22836
\(776\) 0.732051 0.0262791
\(777\) 1.26795 0.0454874
\(778\) 33.1769 1.18945
\(779\) 31.2487 1.11960
\(780\) 0 0
\(781\) −8.19615 −0.293281
\(782\) −0.928203 −0.0331925
\(783\) 3.00000 0.107211
\(784\) 1.00000 0.0357143
\(785\) 11.3205 0.404046
\(786\) 18.9282 0.675147
\(787\) −55.7846 −1.98851 −0.994253 0.107053i \(-0.965858\pi\)
−0.994253 + 0.107053i \(0.965858\pi\)
\(788\) 2.26795 0.0807923
\(789\) 18.7321 0.666879
\(790\) 7.94744 0.282757
\(791\) −10.1962 −0.362533
\(792\) −3.73205 −0.132613
\(793\) 0 0
\(794\) 16.4641 0.584289
\(795\) −7.66025 −0.271681
\(796\) −4.19615 −0.148729
\(797\) −0.732051 −0.0259306 −0.0129653 0.999916i \(-0.504127\pi\)
−0.0129653 + 0.999916i \(0.504127\pi\)
\(798\) −4.46410 −0.158027
\(799\) 1.19615 0.0423168
\(800\) 4.46410 0.157830
\(801\) −6.46410 −0.228398
\(802\) −10.0000 −0.353112
\(803\) 24.3923 0.860786
\(804\) 12.9282 0.455943
\(805\) −2.53590 −0.0893787
\(806\) 0 0
\(807\) 23.7128 0.834731
\(808\) 10.0000 0.351799
\(809\) −0.196152 −0.00689635 −0.00344818 0.999994i \(-0.501098\pi\)
−0.00344818 + 0.999994i \(0.501098\pi\)
\(810\) 0.732051 0.0257216
\(811\) 15.4641 0.543018 0.271509 0.962436i \(-0.412477\pi\)
0.271509 + 0.962436i \(0.412477\pi\)
\(812\) −3.00000 −0.105279
\(813\) 21.5167 0.754622
\(814\) 4.73205 0.165858
\(815\) 5.71281 0.200111
\(816\) −0.267949 −0.00938010
\(817\) −3.26795 −0.114331
\(818\) 13.2679 0.463903
\(819\) 0 0
\(820\) 5.12436 0.178950
\(821\) −45.5885 −1.59105 −0.795524 0.605922i \(-0.792804\pi\)
−0.795524 + 0.605922i \(0.792804\pi\)
\(822\) 15.4641 0.539372
\(823\) 7.07180 0.246507 0.123254 0.992375i \(-0.460667\pi\)
0.123254 + 0.992375i \(0.460667\pi\)
\(824\) 12.1962 0.424873
\(825\) 16.6603 0.580036
\(826\) −0.928203 −0.0322963
\(827\) −19.6077 −0.681826 −0.340913 0.940095i \(-0.610736\pi\)
−0.340913 + 0.940095i \(0.610736\pi\)
\(828\) 3.46410 0.120386
\(829\) 16.6603 0.578635 0.289317 0.957233i \(-0.406572\pi\)
0.289317 + 0.957233i \(0.406572\pi\)
\(830\) 4.14359 0.143826
\(831\) 26.3923 0.915539
\(832\) 0 0
\(833\) 0.267949 0.00928389
\(834\) −18.1244 −0.627595
\(835\) 4.28719 0.148364
\(836\) −16.6603 −0.576207
\(837\) −7.66025 −0.264777
\(838\) 6.19615 0.214043
\(839\) 6.39230 0.220687 0.110343 0.993894i \(-0.464805\pi\)
0.110343 + 0.993894i \(0.464805\pi\)
\(840\) −0.732051 −0.0252582
\(841\) −20.0000 −0.689655
\(842\) −14.3923 −0.495992
\(843\) −18.1962 −0.626709
\(844\) −15.8564 −0.545800
\(845\) 0 0
\(846\) −4.46410 −0.153479
\(847\) 2.92820 0.100614
\(848\) −10.4641 −0.359339
\(849\) 22.9282 0.786894
\(850\) 1.19615 0.0410277
\(851\) −4.39230 −0.150566
\(852\) 2.19615 0.0752389
\(853\) 12.6077 0.431679 0.215840 0.976429i \(-0.430751\pi\)
0.215840 + 0.976429i \(0.430751\pi\)
\(854\) 11.7321 0.401463
\(855\) 3.26795 0.111762
\(856\) 1.53590 0.0524959
\(857\) −42.3923 −1.44809 −0.724047 0.689751i \(-0.757720\pi\)
−0.724047 + 0.689751i \(0.757720\pi\)
\(858\) 0 0
\(859\) −6.94744 −0.237044 −0.118522 0.992951i \(-0.537816\pi\)
−0.118522 + 0.992951i \(0.537816\pi\)
\(860\) −0.535898 −0.0182740
\(861\) 7.00000 0.238559
\(862\) −21.1244 −0.719498
\(863\) −40.6410 −1.38344 −0.691718 0.722168i \(-0.743146\pi\)
−0.691718 + 0.722168i \(0.743146\pi\)
\(864\) 1.00000 0.0340207
\(865\) −15.1769 −0.516031
\(866\) −8.92820 −0.303393
\(867\) 16.9282 0.574912
\(868\) 7.66025 0.260006
\(869\) 40.5167 1.37443
\(870\) 2.19615 0.0744565
\(871\) 0 0
\(872\) −3.66025 −0.123952
\(873\) −0.732051 −0.0247762
\(874\) 15.4641 0.523081
\(875\) 6.92820 0.234216
\(876\) −6.53590 −0.220828
\(877\) −50.3013 −1.69855 −0.849277 0.527948i \(-0.822962\pi\)
−0.849277 + 0.527948i \(0.822962\pi\)
\(878\) 16.3923 0.553213
\(879\) 12.7846 0.431214
\(880\) −2.73205 −0.0920974
\(881\) 14.9282 0.502944 0.251472 0.967865i \(-0.419085\pi\)
0.251472 + 0.967865i \(0.419085\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −24.4449 −0.822635 −0.411318 0.911492i \(-0.634931\pi\)
−0.411318 + 0.911492i \(0.634931\pi\)
\(884\) 0 0
\(885\) 0.679492 0.0228409
\(886\) 31.3923 1.05465
\(887\) 6.94744 0.233272 0.116636 0.993175i \(-0.462789\pi\)
0.116636 + 0.993175i \(0.462789\pi\)
\(888\) −1.26795 −0.0425496
\(889\) −12.5359 −0.420441
\(890\) −4.73205 −0.158619
\(891\) 3.73205 0.125028
\(892\) 8.39230 0.280995
\(893\) −19.9282 −0.666872
\(894\) 18.9282 0.633054
\(895\) −16.3923 −0.547934
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 10.7321 0.358133
\(899\) −22.9808 −0.766451
\(900\) −4.46410 −0.148803
\(901\) −2.80385 −0.0934097
\(902\) 26.1244 0.869846
\(903\) −0.732051 −0.0243611
\(904\) 10.1962 0.339119
\(905\) −0.875644 −0.0291074
\(906\) 13.1962 0.438413
\(907\) −20.5359 −0.681883 −0.340942 0.940084i \(-0.610746\pi\)
−0.340942 + 0.940084i \(0.610746\pi\)
\(908\) −8.53590 −0.283274
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) 38.0526 1.26074 0.630369 0.776296i \(-0.282904\pi\)
0.630369 + 0.776296i \(0.282904\pi\)
\(912\) 4.46410 0.147821
\(913\) 21.1244 0.699114
\(914\) −6.78461 −0.224415
\(915\) −8.58846 −0.283926
\(916\) −10.3205 −0.340999
\(917\) 18.9282 0.625064
\(918\) 0.267949 0.00884364
\(919\) −27.6410 −0.911793 −0.455896 0.890033i \(-0.650681\pi\)
−0.455896 + 0.890033i \(0.650681\pi\)
\(920\) 2.53590 0.0836061
\(921\) 33.7846 1.11324
\(922\) −27.7128 −0.912673
\(923\) 0 0
\(924\) −3.73205 −0.122775
\(925\) 5.66025 0.186108
\(926\) 1.19615 0.0393080
\(927\) −12.1962 −0.400574
\(928\) 3.00000 0.0984798
\(929\) −50.4641 −1.65567 −0.827837 0.560969i \(-0.810429\pi\)
−0.827837 + 0.560969i \(0.810429\pi\)
\(930\) −5.60770 −0.183884
\(931\) −4.46410 −0.146305
\(932\) 2.19615 0.0719374
\(933\) 19.1962 0.628454
\(934\) 9.85641 0.322511
\(935\) −0.732051 −0.0239406
\(936\) 0 0
\(937\) −37.5692 −1.22733 −0.613666 0.789565i \(-0.710306\pi\)
−0.613666 + 0.789565i \(0.710306\pi\)
\(938\) 12.9282 0.422121
\(939\) −1.80385 −0.0588663
\(940\) −3.26795 −0.106589
\(941\) −13.1769 −0.429555 −0.214778 0.976663i \(-0.568903\pi\)
−0.214778 + 0.976663i \(0.568903\pi\)
\(942\) −15.4641 −0.503848
\(943\) −24.2487 −0.789647
\(944\) 0.928203 0.0302104
\(945\) 0.732051 0.0238136
\(946\) −2.73205 −0.0888266
\(947\) 51.0526 1.65899 0.829493 0.558518i \(-0.188630\pi\)
0.829493 + 0.558518i \(0.188630\pi\)
\(948\) −10.8564 −0.352600
\(949\) 0 0
\(950\) −19.9282 −0.646556
\(951\) −18.0000 −0.583690
\(952\) −0.267949 −0.00868428
\(953\) 51.3731 1.66414 0.832068 0.554673i \(-0.187157\pi\)
0.832068 + 0.554673i \(0.187157\pi\)
\(954\) 10.4641 0.338788
\(955\) 3.07180 0.0994010
\(956\) 13.8038 0.446448
\(957\) 11.1962 0.361920
\(958\) 31.3923 1.01424
\(959\) 15.4641 0.499362
\(960\) 0.732051 0.0236268
\(961\) 27.6795 0.892887
\(962\) 0 0
\(963\) −1.53590 −0.0494936
\(964\) −12.7846 −0.411765
\(965\) −12.5885 −0.405237
\(966\) 3.46410 0.111456
\(967\) −36.0000 −1.15768 −0.578841 0.815440i \(-0.696495\pi\)
−0.578841 + 0.815440i \(0.696495\pi\)
\(968\) −2.92820 −0.0941160
\(969\) 1.19615 0.0384260
\(970\) −0.535898 −0.0172067
\(971\) 44.2487 1.42001 0.710004 0.704197i \(-0.248693\pi\)
0.710004 + 0.704197i \(0.248693\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −18.1244 −0.581040
\(974\) 21.1962 0.679169
\(975\) 0 0
\(976\) −11.7321 −0.375534
\(977\) 18.2487 0.583828 0.291914 0.956445i \(-0.405708\pi\)
0.291914 + 0.956445i \(0.405708\pi\)
\(978\) −7.80385 −0.249540
\(979\) −24.1244 −0.771018
\(980\) −0.732051 −0.0233845
\(981\) 3.66025 0.116863
\(982\) −36.2487 −1.15674
\(983\) −28.1436 −0.897641 −0.448821 0.893622i \(-0.648156\pi\)
−0.448821 + 0.893622i \(0.648156\pi\)
\(984\) −7.00000 −0.223152
\(985\) −1.66025 −0.0529001
\(986\) 0.803848 0.0255997
\(987\) −4.46410 −0.142094
\(988\) 0 0
\(989\) 2.53590 0.0806369
\(990\) 2.73205 0.0868303
\(991\) −4.60770 −0.146368 −0.0731841 0.997318i \(-0.523316\pi\)
−0.0731841 + 0.997318i \(0.523316\pi\)
\(992\) −7.66025 −0.243213
\(993\) −16.9282 −0.537200
\(994\) 2.19615 0.0696577
\(995\) 3.07180 0.0973825
\(996\) −5.66025 −0.179352
\(997\) 42.2295 1.33742 0.668710 0.743523i \(-0.266847\pi\)
0.668710 + 0.743523i \(0.266847\pi\)
\(998\) −12.9808 −0.410899
\(999\) 1.26795 0.0401161
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 7098.2.a.bj.1.1 2
13.2 odd 12 546.2.s.d.43.1 4
13.7 odd 12 546.2.s.d.127.1 yes 4
13.12 even 2 7098.2.a.bs.1.2 2
39.2 even 12 1638.2.bj.d.1135.2 4
39.20 even 12 1638.2.bj.d.127.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.s.d.43.1 4 13.2 odd 12
546.2.s.d.127.1 yes 4 13.7 odd 12
1638.2.bj.d.127.2 4 39.20 even 12
1638.2.bj.d.1135.2 4 39.2 even 12
7098.2.a.bj.1.1 2 1.1 even 1 trivial
7098.2.a.bs.1.2 2 13.12 even 2