Properties

Label 7098.2.a.bs.1.2
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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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 1.2
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 \(-0.447660\pi\)
0.163692 + 0.986512i \(0.447660\pi\)
\(6\) −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 \(-0.690210\pi\)
−0.562628 + 0.826710i \(0.690210\pi\)
\(12\) −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 \(-0.328880\pi\)
0.512068 + 0.858945i \(0.328880\pi\)
\(20\) 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 \(-0.741472\pi\)
−0.687911 + 0.725795i \(0.741472\pi\)
\(32\) 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 \(-0.466764\pi\)
0.104225 + 0.994554i \(0.466764\pi\)
\(38\) 4.46410 0.724173
\(39\) 0 0
\(40\) 0.732051 0.115747
\(41\) 7.00000 1.09322 0.546608 0.837389i \(-0.315919\pi\)
0.546608 + 0.837389i \(0.315919\pi\)
\(42\) 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 \(-0.605559\pi\)
−0.325578 + 0.945515i \(0.605559\pi\)
\(48\) −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 \(-0.519244\pi\)
−0.0604209 + 0.998173i \(0.519244\pi\)
\(60\) −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 \(-0.210228\pi\)
0.789716 + 0.613473i \(0.210228\pi\)
\(68\) 0.267949 0.0324936
\(69\) −3.46410 −0.417029
\(70\) −0.732051 −0.0874968
\(71\) 2.19615 0.260635 0.130318 0.991472i \(-0.458400\pi\)
0.130318 + 0.991472i \(0.458400\pi\)
\(72\) 1.00000 0.117851
\(73\) −6.53590 −0.764969 −0.382485 0.923962i \(-0.624931\pi\)
−0.382485 + 0.923962i \(0.624931\pi\)
\(74\) 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 \(-0.600546\pi\)
−0.310647 + 0.950525i \(0.600546\pi\)
\(84\) 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 \(-0.388694\pi\)
0.342597 + 0.939483i \(0.388694\pi\)
\(90\) 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 \(-0.488168\pi\)
0.0371642 + 0.999309i \(0.488168\pi\)
\(98\) 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 \(-0.556088\pi\)
−0.175294 + 0.984516i \(0.556088\pi\)
\(110\) −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 \(-0.729695\pi\)
−0.660594 + 0.750744i \(0.729695\pi\)
\(138\) −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 \(-0.782416\pi\)
−0.775329 + 0.631557i \(0.782416\pi\)
\(150\) 4.46410 0.364492
\(151\) −13.1962 −1.07389 −0.536944 0.843618i \(-0.680421\pi\)
−0.536944 + 0.843618i \(0.680421\pi\)
\(152\) 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 \(-0.401136\pi\)
0.305622 + 0.952153i \(0.401136\pi\)
\(164\) 7.00000 0.546608
\(165\) 2.73205 0.212690
\(166\) −5.66025 −0.439321
\(167\) 5.85641 0.453182 0.226591 0.973990i \(-0.427242\pi\)
0.226591 + 0.973990i \(0.427242\pi\)
\(168\) 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 \(-0.712423\pi\)
−0.618903 + 0.785467i \(0.712423\pi\)
\(194\) 0.732051 0.0525582
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −2.26795 −0.161585 −0.0807923 0.996731i \(-0.525745\pi\)
−0.0807923 + 0.996731i \(0.525745\pi\)
\(198\) −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 \(-0.590664\pi\)
−0.280995 + 0.959709i \(0.590664\pi\)
\(224\) −1.00000 −0.0668153
\(225\) −4.46410 −0.297607
\(226\) −10.1962 −0.678238
\(227\) 8.53590 0.566547 0.283274 0.959039i \(-0.408580\pi\)
0.283274 + 0.959039i \(0.408580\pi\)
\(228\) −4.46410 −0.295642
\(229\) 10.3205 0.681998 0.340999 0.940064i \(-0.389235\pi\)
0.340999 + 0.940064i \(0.389235\pi\)
\(230\) 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 \(-0.647311\pi\)
−0.446448 + 0.894809i \(0.647311\pi\)
\(240\) −0.732051 −0.0472537
\(241\) 12.7846 0.823529 0.411765 0.911290i \(-0.364913\pi\)
0.411765 + 0.911290i \(0.364913\pi\)
\(242\) 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 \(-0.273291\pi\)
0.653522 + 0.756908i \(0.273291\pi\)
\(272\) 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 \(-0.682615\pi\)
−0.542746 + 0.839897i \(0.682615\pi\)
\(282\) 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 \(-0.378177\pi\)
0.373442 + 0.927653i \(0.378177\pi\)
\(294\) −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 \(-0.0855563\pi\)
0.964095 + 0.265558i \(0.0855563\pi\)
\(308\) 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 \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 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 \(-0.654028\pi\)
−0.465229 + 0.885190i \(0.654028\pi\)
\(332\) −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 \(-0.281141\pi\)
0.634659 + 0.772792i \(0.281141\pi\)
\(350\) 4.46410 0.238616
\(351\) 0 0
\(352\) −3.73205 −0.198919
\(353\) 1.46410 0.0779263 0.0389631 0.999241i \(-0.487595\pi\)
0.0389631 + 0.999241i \(0.487595\pi\)
\(354\) 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 \(-0.636935\pi\)
−0.417048 + 0.908885i \(0.636935\pi\)
\(360\) 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 \(-0.377753\pi\)
0.374679 + 0.927154i \(0.377753\pi\)
\(380\) 3.26795 0.167642
\(381\) 12.5359 0.642234
\(382\) −4.19615 −0.214694
\(383\) 14.6077 0.746418 0.373209 0.927747i \(-0.378257\pi\)
0.373209 + 0.927747i \(0.378257\pi\)
\(384\) −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 \(-0.364427\pi\)
0.413155 + 0.910661i \(0.364427\pi\)
\(398\) −4.19615 −0.210334
\(399\) 4.46410 0.223485
\(400\) −4.46410 −0.223205
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) −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 \(-0.393616\pi\)
0.328029 + 0.944668i \(0.393616\pi\)
\(410\) 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 \(-0.614063\pi\)
−0.350719 + 0.936481i \(0.614063\pi\)
\(422\) −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 \(-0.669897\pi\)
−0.508762 + 0.860907i \(0.669897\pi\)
\(432\) −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 \(-0.418504\pi\)
0.253238 + 0.967404i \(0.418504\pi\)
\(450\) −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 \(-0.550726\pi\)
−0.158685 + 0.987329i \(0.550726\pi\)
\(458\) 10.3205 0.482246
\(459\) −0.267949 −0.0125068
\(460\) 2.53590 0.118237
\(461\) −27.7128 −1.29071 −0.645357 0.763881i \(-0.723291\pi\)
−0.645357 + 0.763881i \(0.723291\pi\)
\(462\) −3.73205 −0.173631
\(463\) 1.19615 0.0555899 0.0277950 0.999614i \(-0.491151\pi\)
0.0277950 + 0.999614i \(0.491151\pi\)
\(464\) −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 \(-0.245435\pi\)
0.717176 + 0.696893i \(0.245435\pi\)
\(480\) −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 \(-0.340548\pi\)
0.480245 + 0.877135i \(0.340548\pi\)
\(488\) −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 \(-0.593838\pi\)
−0.290549 + 0.956860i \(0.593838\pi\)
\(500\) −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 \(-0.304075\pi\)
0.577380 + 0.816476i \(0.304075\pi\)
\(510\) −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 \(-0.724861\pi\)
−0.649117 + 0.760689i \(0.724861\pi\)
\(542\) 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 \(-0.216906\pi\)
0.776672 + 0.629905i \(0.216906\pi\)
\(558\) −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 \(-0.589067\pi\)
−0.276176 + 0.961107i \(0.589067\pi\)
\(578\) −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 \(-0.275834\pi\)
0.647453 + 0.762106i \(0.275834\pi\)
\(588\) −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 \(-0.590625\pi\)
−0.280875 + 0.959744i \(0.590625\pi\)
\(594\) 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 \(-0.820568\pi\)
−0.845282 + 0.534320i \(0.820568\pi\)
\(614\) 33.7846 1.36344
\(615\) −5.12436 −0.206634
\(616\) 3.73205 0.150369
\(617\) 41.5167 1.67140 0.835699 0.549188i \(-0.185063\pi\)
0.835699 + 0.549188i \(0.185063\pi\)
\(618\) 12.1962 0.490601
\(619\) −2.32051 −0.0932691 −0.0466345 0.998912i \(-0.514850\pi\)
−0.0466345 + 0.998912i \(0.514850\pi\)
\(620\) −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 \(-0.396958\pi\)
0.318092 + 0.948060i \(0.396958\pi\)
\(632\) 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 \(-0.489457\pi\)
0.0331163 + 0.999452i \(0.489457\pi\)
\(644\) −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 \(-0.319592\pi\)
0.536907 + 0.843641i \(0.319592\pi\)
\(662\) −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 \(-0.331516\pi\)
0.504937 + 0.863156i \(0.331516\pi\)
\(684\) 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 \(-0.782633\pi\)
−0.775760 + 0.631028i \(0.782633\pi\)
\(692\) 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 \(-0.649167\pi\)
−0.451656 + 0.892192i \(0.649167\pi\)
\(710\) 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 \(-0.540414\pi\)
−0.126624 + 0.991951i \(0.540414\pi\)
\(734\) 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 \(-0.462489\pi\)
0.117572 + 0.993064i \(0.462489\pi\)
\(740\) 0.928203 0.0341214
\(741\) 0 0
\(742\) 10.4641 0.384149
\(743\) 19.5167 0.715997 0.357998 0.933722i \(-0.383459\pi\)
0.357998 + 0.933722i \(0.383459\pi\)
\(744\) 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 \(-0.496080\pi\)
0.0123158 + 0.999924i \(0.496080\pi\)
\(762\) 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 \(-0.645288\pi\)
−0.440752 + 0.897629i \(0.645288\pi\)
\(770\) 2.73205 0.0984563
\(771\) −11.0526 −0.398048
\(772\) −17.1962 −0.618903
\(773\) 42.3923 1.52475 0.762373 0.647138i \(-0.224034\pi\)
0.762373 + 0.647138i \(0.224034\pi\)
\(774\) 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 \(-0.0341415\pi\)
0.994253 + 0.107053i \(0.0341415\pi\)
\(788\) −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 \(-0.587523\pi\)
−0.271509 + 0.962436i \(0.587523\pi\)
\(812\) 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 \(-0.207196\pi\)
0.795524 + 0.605922i \(0.207196\pi\)
\(822\) 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 \(-0.389264\pi\)
0.340913 + 0.940095i \(0.389264\pi\)
\(828\) 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 \(-0.535195\pi\)
−0.110343 + 0.993894i \(0.535195\pi\)
\(840\) 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 \(-0.569249\pi\)
−0.215840 + 0.976429i \(0.569249\pi\)
\(854\) 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 \(-0.256854\pi\)
0.691718 + 0.722168i \(0.256854\pi\)
\(864\) −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 \(-0.177038\pi\)
0.849277 + 0.527948i \(0.177038\pi\)
\(878\) −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 \(-0.189571\pi\)
0.827837 + 0.560969i \(0.189571\pi\)
\(930\) 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 \(-0.431097\pi\)
0.214778 + 0.976663i \(0.431097\pi\)
\(942\) 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 \(-0.811370\pi\)
−0.829493 + 0.558518i \(0.811370\pi\)
\(948\) −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 \(-0.303505\pi\)
0.578841 + 0.815440i \(0.303505\pi\)
\(968\) 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 \(-0.594292\pi\)
−0.291914 + 0.956445i \(0.594292\pi\)
\(978\) −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 \(-0.351844\pi\)
0.448821 + 0.893622i \(0.351844\pi\)
\(984\) −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.bs.1.2 2
13.6 odd 12 546.2.s.d.127.1 yes 4
13.11 odd 12 546.2.s.d.43.1 4
13.12 even 2 7098.2.a.bj.1.1 2
39.11 even 12 1638.2.bj.d.1135.2 4
39.32 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 13.11 odd 12
546.2.s.d.127.1 yes 4 13.6 odd 12
1638.2.bj.d.127.2 4 39.32 even 12
1638.2.bj.d.1135.2 4 39.11 even 12
7098.2.a.bj.1.1 2 13.12 even 2
7098.2.a.bs.1.2 2 1.1 even 1 trivial