Properties

Label 5054.2.a.g.1.1
Level $5054$
Weight $2$
Character 5054.1
Self dual yes
Analytic conductor $40.356$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5054 = 2 \cdot 7 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5054.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.3563931816\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
Defining polynomial: \(x^{2} - x - 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 5054.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} -0.381966 q^{5} +0.618034 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.61803 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -0.618034 q^{3} +1.00000 q^{4} -0.381966 q^{5} +0.618034 q^{6} +1.00000 q^{7} -1.00000 q^{8} -2.61803 q^{9} +0.381966 q^{10} +1.85410 q^{11} -0.618034 q^{12} +4.47214 q^{13} -1.00000 q^{14} +0.236068 q^{15} +1.00000 q^{16} +5.23607 q^{17} +2.61803 q^{18} -0.381966 q^{20} -0.618034 q^{21} -1.85410 q^{22} -8.94427 q^{23} +0.618034 q^{24} -4.85410 q^{25} -4.47214 q^{26} +3.47214 q^{27} +1.00000 q^{28} +6.85410 q^{29} -0.236068 q^{30} +6.00000 q^{31} -1.00000 q^{32} -1.14590 q^{33} -5.23607 q^{34} -0.381966 q^{35} -2.61803 q^{36} +1.14590 q^{37} -2.76393 q^{39} +0.381966 q^{40} +10.5623 q^{41} +0.618034 q^{42} -2.85410 q^{43} +1.85410 q^{44} +1.00000 q^{45} +8.94427 q^{46} -4.14590 q^{47} -0.618034 q^{48} +1.00000 q^{49} +4.85410 q^{50} -3.23607 q^{51} +4.47214 q^{52} -11.8541 q^{53} -3.47214 q^{54} -0.708204 q^{55} -1.00000 q^{56} -6.85410 q^{58} +8.56231 q^{59} +0.236068 q^{60} +11.5623 q^{61} -6.00000 q^{62} -2.61803 q^{63} +1.00000 q^{64} -1.70820 q^{65} +1.14590 q^{66} +0.763932 q^{67} +5.23607 q^{68} +5.52786 q^{69} +0.381966 q^{70} -11.5623 q^{71} +2.61803 q^{72} -1.14590 q^{74} +3.00000 q^{75} +1.85410 q^{77} +2.76393 q^{78} -15.3262 q^{79} -0.381966 q^{80} +5.70820 q^{81} -10.5623 q^{82} +6.76393 q^{83} -0.618034 q^{84} -2.00000 q^{85} +2.85410 q^{86} -4.23607 q^{87} -1.85410 q^{88} -9.85410 q^{89} -1.00000 q^{90} +4.47214 q^{91} -8.94427 q^{92} -3.70820 q^{93} +4.14590 q^{94} +0.618034 q^{96} +13.8541 q^{97} -1.00000 q^{98} -4.85410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + q^{3} + 2 q^{4} - 3 q^{5} - q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9} + O(q^{10}) \) \( 2 q - 2 q^{2} + q^{3} + 2 q^{4} - 3 q^{5} - q^{6} + 2 q^{7} - 2 q^{8} - 3 q^{9} + 3 q^{10} - 3 q^{11} + q^{12} - 2 q^{14} - 4 q^{15} + 2 q^{16} + 6 q^{17} + 3 q^{18} - 3 q^{20} + q^{21} + 3 q^{22} - q^{24} - 3 q^{25} - 2 q^{27} + 2 q^{28} + 7 q^{29} + 4 q^{30} + 12 q^{31} - 2 q^{32} - 9 q^{33} - 6 q^{34} - 3 q^{35} - 3 q^{36} + 9 q^{37} - 10 q^{39} + 3 q^{40} + q^{41} - q^{42} + q^{43} - 3 q^{44} + 2 q^{45} - 15 q^{47} + q^{48} + 2 q^{49} + 3 q^{50} - 2 q^{51} - 17 q^{53} + 2 q^{54} + 12 q^{55} - 2 q^{56} - 7 q^{58} - 3 q^{59} - 4 q^{60} + 3 q^{61} - 12 q^{62} - 3 q^{63} + 2 q^{64} + 10 q^{65} + 9 q^{66} + 6 q^{67} + 6 q^{68} + 20 q^{69} + 3 q^{70} - 3 q^{71} + 3 q^{72} - 9 q^{74} + 6 q^{75} - 3 q^{77} + 10 q^{78} - 15 q^{79} - 3 q^{80} - 2 q^{81} - q^{82} + 18 q^{83} + q^{84} - 4 q^{85} - q^{86} - 4 q^{87} + 3 q^{88} - 13 q^{89} - 2 q^{90} + 6 q^{93} + 15 q^{94} - q^{96} + 21 q^{97} - 2 q^{98} - 3 q^{99} + O(q^{100}) \)

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
\(3\) −0.618034 −0.356822 −0.178411 0.983956i \(-0.557096\pi\)
−0.178411 + 0.983956i \(0.557096\pi\)
\(4\) 1.00000 0.500000
\(5\) −0.381966 −0.170820 −0.0854102 0.996346i \(-0.527220\pi\)
−0.0854102 + 0.996346i \(0.527220\pi\)
\(6\) 0.618034 0.252311
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) −2.61803 −0.872678
\(10\) 0.381966 0.120788
\(11\) 1.85410 0.559033 0.279516 0.960141i \(-0.409826\pi\)
0.279516 + 0.960141i \(0.409826\pi\)
\(12\) −0.618034 −0.178411
\(13\) 4.47214 1.24035 0.620174 0.784465i \(-0.287062\pi\)
0.620174 + 0.784465i \(0.287062\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0.236068 0.0609525
\(16\) 1.00000 0.250000
\(17\) 5.23607 1.26993 0.634967 0.772540i \(-0.281014\pi\)
0.634967 + 0.772540i \(0.281014\pi\)
\(18\) 2.61803 0.617077
\(19\) 0 0
\(20\) −0.381966 −0.0854102
\(21\) −0.618034 −0.134866
\(22\) −1.85410 −0.395296
\(23\) −8.94427 −1.86501 −0.932505 0.361158i \(-0.882382\pi\)
−0.932505 + 0.361158i \(0.882382\pi\)
\(24\) 0.618034 0.126156
\(25\) −4.85410 −0.970820
\(26\) −4.47214 −0.877058
\(27\) 3.47214 0.668213
\(28\) 1.00000 0.188982
\(29\) 6.85410 1.27277 0.636387 0.771370i \(-0.280428\pi\)
0.636387 + 0.771370i \(0.280428\pi\)
\(30\) −0.236068 −0.0430999
\(31\) 6.00000 1.07763 0.538816 0.842424i \(-0.318872\pi\)
0.538816 + 0.842424i \(0.318872\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.14590 −0.199475
\(34\) −5.23607 −0.897978
\(35\) −0.381966 −0.0645640
\(36\) −2.61803 −0.436339
\(37\) 1.14590 0.188384 0.0941922 0.995554i \(-0.469973\pi\)
0.0941922 + 0.995554i \(0.469973\pi\)
\(38\) 0 0
\(39\) −2.76393 −0.442583
\(40\) 0.381966 0.0603941
\(41\) 10.5623 1.64956 0.824778 0.565457i \(-0.191300\pi\)
0.824778 + 0.565457i \(0.191300\pi\)
\(42\) 0.618034 0.0953647
\(43\) −2.85410 −0.435246 −0.217623 0.976033i \(-0.569830\pi\)
−0.217623 + 0.976033i \(0.569830\pi\)
\(44\) 1.85410 0.279516
\(45\) 1.00000 0.149071
\(46\) 8.94427 1.31876
\(47\) −4.14590 −0.604741 −0.302371 0.953190i \(-0.597778\pi\)
−0.302371 + 0.953190i \(0.597778\pi\)
\(48\) −0.618034 −0.0892055
\(49\) 1.00000 0.142857
\(50\) 4.85410 0.686474
\(51\) −3.23607 −0.453140
\(52\) 4.47214 0.620174
\(53\) −11.8541 −1.62829 −0.814143 0.580664i \(-0.802793\pi\)
−0.814143 + 0.580664i \(0.802793\pi\)
\(54\) −3.47214 −0.472498
\(55\) −0.708204 −0.0954942
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −6.85410 −0.899988
\(59\) 8.56231 1.11472 0.557359 0.830272i \(-0.311815\pi\)
0.557359 + 0.830272i \(0.311815\pi\)
\(60\) 0.236068 0.0304762
\(61\) 11.5623 1.48040 0.740201 0.672386i \(-0.234730\pi\)
0.740201 + 0.672386i \(0.234730\pi\)
\(62\) −6.00000 −0.762001
\(63\) −2.61803 −0.329841
\(64\) 1.00000 0.125000
\(65\) −1.70820 −0.211877
\(66\) 1.14590 0.141050
\(67\) 0.763932 0.0933292 0.0466646 0.998911i \(-0.485141\pi\)
0.0466646 + 0.998911i \(0.485141\pi\)
\(68\) 5.23607 0.634967
\(69\) 5.52786 0.665477
\(70\) 0.381966 0.0456537
\(71\) −11.5623 −1.37219 −0.686097 0.727510i \(-0.740677\pi\)
−0.686097 + 0.727510i \(0.740677\pi\)
\(72\) 2.61803 0.308538
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −1.14590 −0.133208
\(75\) 3.00000 0.346410
\(76\) 0 0
\(77\) 1.85410 0.211295
\(78\) 2.76393 0.312954
\(79\) −15.3262 −1.72434 −0.862168 0.506622i \(-0.830894\pi\)
−0.862168 + 0.506622i \(0.830894\pi\)
\(80\) −0.381966 −0.0427051
\(81\) 5.70820 0.634245
\(82\) −10.5623 −1.16641
\(83\) 6.76393 0.742438 0.371219 0.928545i \(-0.378940\pi\)
0.371219 + 0.928545i \(0.378940\pi\)
\(84\) −0.618034 −0.0674330
\(85\) −2.00000 −0.216930
\(86\) 2.85410 0.307766
\(87\) −4.23607 −0.454154
\(88\) −1.85410 −0.197648
\(89\) −9.85410 −1.04453 −0.522266 0.852782i \(-0.674913\pi\)
−0.522266 + 0.852782i \(0.674913\pi\)
\(90\) −1.00000 −0.105409
\(91\) 4.47214 0.468807
\(92\) −8.94427 −0.932505
\(93\) −3.70820 −0.384523
\(94\) 4.14590 0.427617
\(95\) 0 0
\(96\) 0.618034 0.0630778
\(97\) 13.8541 1.40667 0.703335 0.710858i \(-0.251693\pi\)
0.703335 + 0.710858i \(0.251693\pi\)
\(98\) −1.00000 −0.101015
\(99\) −4.85410 −0.487856
\(100\) −4.85410 −0.485410
\(101\) −14.9443 −1.48701 −0.743505 0.668730i \(-0.766838\pi\)
−0.743505 + 0.668730i \(0.766838\pi\)
\(102\) 3.23607 0.320418
\(103\) −8.94427 −0.881305 −0.440653 0.897678i \(-0.645253\pi\)
−0.440653 + 0.897678i \(0.645253\pi\)
\(104\) −4.47214 −0.438529
\(105\) 0.236068 0.0230379
\(106\) 11.8541 1.15137
\(107\) 6.29180 0.608251 0.304125 0.952632i \(-0.401636\pi\)
0.304125 + 0.952632i \(0.401636\pi\)
\(108\) 3.47214 0.334106
\(109\) 4.14590 0.397105 0.198553 0.980090i \(-0.436376\pi\)
0.198553 + 0.980090i \(0.436376\pi\)
\(110\) 0.708204 0.0675246
\(111\) −0.708204 −0.0672197
\(112\) 1.00000 0.0944911
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) 3.41641 0.318582
\(116\) 6.85410 0.636387
\(117\) −11.7082 −1.08242
\(118\) −8.56231 −0.788224
\(119\) 5.23607 0.479990
\(120\) −0.236068 −0.0215500
\(121\) −7.56231 −0.687482
\(122\) −11.5623 −1.04680
\(123\) −6.52786 −0.588598
\(124\) 6.00000 0.538816
\(125\) 3.76393 0.336656
\(126\) 2.61803 0.233233
\(127\) 10.9098 0.968091 0.484045 0.875043i \(-0.339167\pi\)
0.484045 + 0.875043i \(0.339167\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 1.76393 0.155306
\(130\) 1.70820 0.149819
\(131\) 4.47214 0.390732 0.195366 0.980730i \(-0.437410\pi\)
0.195366 + 0.980730i \(0.437410\pi\)
\(132\) −1.14590 −0.0997376
\(133\) 0 0
\(134\) −0.763932 −0.0659937
\(135\) −1.32624 −0.114144
\(136\) −5.23607 −0.448989
\(137\) 5.61803 0.479981 0.239991 0.970775i \(-0.422856\pi\)
0.239991 + 0.970775i \(0.422856\pi\)
\(138\) −5.52786 −0.470563
\(139\) 6.00000 0.508913 0.254457 0.967084i \(-0.418103\pi\)
0.254457 + 0.967084i \(0.418103\pi\)
\(140\) −0.381966 −0.0322820
\(141\) 2.56231 0.215785
\(142\) 11.5623 0.970287
\(143\) 8.29180 0.693395
\(144\) −2.61803 −0.218169
\(145\) −2.61803 −0.217416
\(146\) 0 0
\(147\) −0.618034 −0.0509746
\(148\) 1.14590 0.0941922
\(149\) −12.7639 −1.04566 −0.522831 0.852436i \(-0.675124\pi\)
−0.522831 + 0.852436i \(0.675124\pi\)
\(150\) −3.00000 −0.244949
\(151\) 20.9443 1.70442 0.852210 0.523199i \(-0.175262\pi\)
0.852210 + 0.523199i \(0.175262\pi\)
\(152\) 0 0
\(153\) −13.7082 −1.10824
\(154\) −1.85410 −0.149408
\(155\) −2.29180 −0.184081
\(156\) −2.76393 −0.221292
\(157\) −3.85410 −0.307591 −0.153795 0.988103i \(-0.549150\pi\)
−0.153795 + 0.988103i \(0.549150\pi\)
\(158\) 15.3262 1.21929
\(159\) 7.32624 0.581008
\(160\) 0.381966 0.0301971
\(161\) −8.94427 −0.704907
\(162\) −5.70820 −0.448479
\(163\) 13.2705 1.03943 0.519713 0.854341i \(-0.326039\pi\)
0.519713 + 0.854341i \(0.326039\pi\)
\(164\) 10.5623 0.824778
\(165\) 0.437694 0.0340744
\(166\) −6.76393 −0.524983
\(167\) −10.0000 −0.773823 −0.386912 0.922117i \(-0.626458\pi\)
−0.386912 + 0.922117i \(0.626458\pi\)
\(168\) 0.618034 0.0476824
\(169\) 7.00000 0.538462
\(170\) 2.00000 0.153393
\(171\) 0 0
\(172\) −2.85410 −0.217623
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 4.23607 0.321135
\(175\) −4.85410 −0.366936
\(176\) 1.85410 0.139758
\(177\) −5.29180 −0.397756
\(178\) 9.85410 0.738596
\(179\) 4.29180 0.320784 0.160392 0.987053i \(-0.448724\pi\)
0.160392 + 0.987053i \(0.448724\pi\)
\(180\) 1.00000 0.0745356
\(181\) 19.4164 1.44321 0.721605 0.692305i \(-0.243405\pi\)
0.721605 + 0.692305i \(0.243405\pi\)
\(182\) −4.47214 −0.331497
\(183\) −7.14590 −0.528240
\(184\) 8.94427 0.659380
\(185\) −0.437694 −0.0321799
\(186\) 3.70820 0.271899
\(187\) 9.70820 0.709934
\(188\) −4.14590 −0.302371
\(189\) 3.47214 0.252561
\(190\) 0 0
\(191\) −2.94427 −0.213040 −0.106520 0.994311i \(-0.533971\pi\)
−0.106520 + 0.994311i \(0.533971\pi\)
\(192\) −0.618034 −0.0446028
\(193\) −3.05573 −0.219956 −0.109978 0.993934i \(-0.535078\pi\)
−0.109978 + 0.993934i \(0.535078\pi\)
\(194\) −13.8541 −0.994667
\(195\) 1.05573 0.0756023
\(196\) 1.00000 0.0714286
\(197\) −0.763932 −0.0544279 −0.0272140 0.999630i \(-0.508664\pi\)
−0.0272140 + 0.999630i \(0.508664\pi\)
\(198\) 4.85410 0.344966
\(199\) −18.5623 −1.31585 −0.657923 0.753085i \(-0.728565\pi\)
−0.657923 + 0.753085i \(0.728565\pi\)
\(200\) 4.85410 0.343237
\(201\) −0.472136 −0.0333019
\(202\) 14.9443 1.05148
\(203\) 6.85410 0.481064
\(204\) −3.23607 −0.226570
\(205\) −4.03444 −0.281778
\(206\) 8.94427 0.623177
\(207\) 23.4164 1.62755
\(208\) 4.47214 0.310087
\(209\) 0 0
\(210\) −0.236068 −0.0162902
\(211\) 12.7639 0.878705 0.439353 0.898315i \(-0.355208\pi\)
0.439353 + 0.898315i \(0.355208\pi\)
\(212\) −11.8541 −0.814143
\(213\) 7.14590 0.489629
\(214\) −6.29180 −0.430098
\(215\) 1.09017 0.0743490
\(216\) −3.47214 −0.236249
\(217\) 6.00000 0.407307
\(218\) −4.14590 −0.280796
\(219\) 0 0
\(220\) −0.708204 −0.0477471
\(221\) 23.4164 1.57516
\(222\) 0.708204 0.0475315
\(223\) −0.763932 −0.0511567 −0.0255783 0.999673i \(-0.508143\pi\)
−0.0255783 + 0.999673i \(0.508143\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 12.7082 0.847214
\(226\) −10.0000 −0.665190
\(227\) −7.41641 −0.492244 −0.246122 0.969239i \(-0.579156\pi\)
−0.246122 + 0.969239i \(0.579156\pi\)
\(228\) 0 0
\(229\) 17.1459 1.13303 0.566516 0.824050i \(-0.308291\pi\)
0.566516 + 0.824050i \(0.308291\pi\)
\(230\) −3.41641 −0.225271
\(231\) −1.14590 −0.0753946
\(232\) −6.85410 −0.449994
\(233\) −7.09017 −0.464492 −0.232246 0.972657i \(-0.574608\pi\)
−0.232246 + 0.972657i \(0.574608\pi\)
\(234\) 11.7082 0.765389
\(235\) 1.58359 0.103302
\(236\) 8.56231 0.557359
\(237\) 9.47214 0.615281
\(238\) −5.23607 −0.339404
\(239\) −8.18034 −0.529142 −0.264571 0.964366i \(-0.585230\pi\)
−0.264571 + 0.964366i \(0.585230\pi\)
\(240\) 0.236068 0.0152381
\(241\) −2.56231 −0.165053 −0.0825263 0.996589i \(-0.526299\pi\)
−0.0825263 + 0.996589i \(0.526299\pi\)
\(242\) 7.56231 0.486123
\(243\) −13.9443 −0.894525
\(244\) 11.5623 0.740201
\(245\) −0.381966 −0.0244029
\(246\) 6.52786 0.416201
\(247\) 0 0
\(248\) −6.00000 −0.381000
\(249\) −4.18034 −0.264918
\(250\) −3.76393 −0.238052
\(251\) 26.9443 1.70071 0.850354 0.526212i \(-0.176388\pi\)
0.850354 + 0.526212i \(0.176388\pi\)
\(252\) −2.61803 −0.164921
\(253\) −16.5836 −1.04260
\(254\) −10.9098 −0.684544
\(255\) 1.23607 0.0774056
\(256\) 1.00000 0.0625000
\(257\) 3.14590 0.196236 0.0981179 0.995175i \(-0.468718\pi\)
0.0981179 + 0.995175i \(0.468718\pi\)
\(258\) −1.76393 −0.109818
\(259\) 1.14590 0.0712026
\(260\) −1.70820 −0.105938
\(261\) −17.9443 −1.11072
\(262\) −4.47214 −0.276289
\(263\) −18.6525 −1.15016 −0.575080 0.818097i \(-0.695029\pi\)
−0.575080 + 0.818097i \(0.695029\pi\)
\(264\) 1.14590 0.0705251
\(265\) 4.52786 0.278144
\(266\) 0 0
\(267\) 6.09017 0.372712
\(268\) 0.763932 0.0466646
\(269\) −31.4164 −1.91549 −0.957746 0.287615i \(-0.907138\pi\)
−0.957746 + 0.287615i \(0.907138\pi\)
\(270\) 1.32624 0.0807123
\(271\) 22.5623 1.37056 0.685281 0.728279i \(-0.259679\pi\)
0.685281 + 0.728279i \(0.259679\pi\)
\(272\) 5.23607 0.317483
\(273\) −2.76393 −0.167281
\(274\) −5.61803 −0.339398
\(275\) −9.00000 −0.542720
\(276\) 5.52786 0.332738
\(277\) 1.41641 0.0851037 0.0425519 0.999094i \(-0.486451\pi\)
0.0425519 + 0.999094i \(0.486451\pi\)
\(278\) −6.00000 −0.359856
\(279\) −15.7082 −0.940426
\(280\) 0.381966 0.0228268
\(281\) 3.41641 0.203806 0.101903 0.994794i \(-0.467507\pi\)
0.101903 + 0.994794i \(0.467507\pi\)
\(282\) −2.56231 −0.152583
\(283\) 8.00000 0.475551 0.237775 0.971320i \(-0.423582\pi\)
0.237775 + 0.971320i \(0.423582\pi\)
\(284\) −11.5623 −0.686097
\(285\) 0 0
\(286\) −8.29180 −0.490304
\(287\) 10.5623 0.623473
\(288\) 2.61803 0.154269
\(289\) 10.4164 0.612730
\(290\) 2.61803 0.153736
\(291\) −8.56231 −0.501931
\(292\) 0 0
\(293\) 4.58359 0.267776 0.133888 0.990996i \(-0.457254\pi\)
0.133888 + 0.990996i \(0.457254\pi\)
\(294\) 0.618034 0.0360445
\(295\) −3.27051 −0.190416
\(296\) −1.14590 −0.0666040
\(297\) 6.43769 0.373553
\(298\) 12.7639 0.739395
\(299\) −40.0000 −2.31326
\(300\) 3.00000 0.173205
\(301\) −2.85410 −0.164508
\(302\) −20.9443 −1.20521
\(303\) 9.23607 0.530598
\(304\) 0 0
\(305\) −4.41641 −0.252883
\(306\) 13.7082 0.783646
\(307\) −27.9787 −1.59683 −0.798415 0.602108i \(-0.794328\pi\)
−0.798415 + 0.602108i \(0.794328\pi\)
\(308\) 1.85410 0.105647
\(309\) 5.52786 0.314469
\(310\) 2.29180 0.130165
\(311\) −5.67376 −0.321730 −0.160865 0.986976i \(-0.551428\pi\)
−0.160865 + 0.986976i \(0.551428\pi\)
\(312\) 2.76393 0.156477
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 3.85410 0.217500
\(315\) 1.00000 0.0563436
\(316\) −15.3262 −0.862168
\(317\) 34.6869 1.94821 0.974106 0.226093i \(-0.0725954\pi\)
0.974106 + 0.226093i \(0.0725954\pi\)
\(318\) −7.32624 −0.410835
\(319\) 12.7082 0.711523
\(320\) −0.381966 −0.0213525
\(321\) −3.88854 −0.217037
\(322\) 8.94427 0.498445
\(323\) 0 0
\(324\) 5.70820 0.317122
\(325\) −21.7082 −1.20415
\(326\) −13.2705 −0.734986
\(327\) −2.56231 −0.141696
\(328\) −10.5623 −0.583206
\(329\) −4.14590 −0.228571
\(330\) −0.437694 −0.0240943
\(331\) 9.70820 0.533611 0.266806 0.963750i \(-0.414032\pi\)
0.266806 + 0.963750i \(0.414032\pi\)
\(332\) 6.76393 0.371219
\(333\) −3.00000 −0.164399
\(334\) 10.0000 0.547176
\(335\) −0.291796 −0.0159425
\(336\) −0.618034 −0.0337165
\(337\) −3.70820 −0.201999 −0.100999 0.994886i \(-0.532204\pi\)
−0.100999 + 0.994886i \(0.532204\pi\)
\(338\) −7.00000 −0.380750
\(339\) −6.18034 −0.335670
\(340\) −2.00000 −0.108465
\(341\) 11.1246 0.602432
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 2.85410 0.153883
\(345\) −2.11146 −0.113677
\(346\) −18.0000 −0.967686
\(347\) −8.94427 −0.480154 −0.240077 0.970754i \(-0.577173\pi\)
−0.240077 + 0.970754i \(0.577173\pi\)
\(348\) −4.23607 −0.227077
\(349\) 36.8328 1.97162 0.985808 0.167878i \(-0.0536916\pi\)
0.985808 + 0.167878i \(0.0536916\pi\)
\(350\) 4.85410 0.259463
\(351\) 15.5279 0.828816
\(352\) −1.85410 −0.0988240
\(353\) −13.4164 −0.714083 −0.357042 0.934088i \(-0.616215\pi\)
−0.357042 + 0.934088i \(0.616215\pi\)
\(354\) 5.29180 0.281256
\(355\) 4.41641 0.234399
\(356\) −9.85410 −0.522266
\(357\) −3.23607 −0.171271
\(358\) −4.29180 −0.226828
\(359\) 15.7082 0.829047 0.414524 0.910039i \(-0.363948\pi\)
0.414524 + 0.910039i \(0.363948\pi\)
\(360\) −1.00000 −0.0527046
\(361\) 0 0
\(362\) −19.4164 −1.02050
\(363\) 4.67376 0.245309
\(364\) 4.47214 0.234404
\(365\) 0 0
\(366\) 7.14590 0.373522
\(367\) 27.2705 1.42351 0.711755 0.702428i \(-0.247901\pi\)
0.711755 + 0.702428i \(0.247901\pi\)
\(368\) −8.94427 −0.466252
\(369\) −27.6525 −1.43953
\(370\) 0.437694 0.0227546
\(371\) −11.8541 −0.615434
\(372\) −3.70820 −0.192261
\(373\) 12.3820 0.641114 0.320557 0.947229i \(-0.396130\pi\)
0.320557 + 0.947229i \(0.396130\pi\)
\(374\) −9.70820 −0.501999
\(375\) −2.32624 −0.120126
\(376\) 4.14590 0.213808
\(377\) 30.6525 1.57868
\(378\) −3.47214 −0.178587
\(379\) 9.81966 0.504402 0.252201 0.967675i \(-0.418846\pi\)
0.252201 + 0.967675i \(0.418846\pi\)
\(380\) 0 0
\(381\) −6.74265 −0.345436
\(382\) 2.94427 0.150642
\(383\) 27.1246 1.38600 0.693001 0.720936i \(-0.256288\pi\)
0.693001 + 0.720936i \(0.256288\pi\)
\(384\) 0.618034 0.0315389
\(385\) −0.708204 −0.0360934
\(386\) 3.05573 0.155532
\(387\) 7.47214 0.379830
\(388\) 13.8541 0.703335
\(389\) 25.3050 1.28301 0.641506 0.767118i \(-0.278310\pi\)
0.641506 + 0.767118i \(0.278310\pi\)
\(390\) −1.05573 −0.0534589
\(391\) −46.8328 −2.36844
\(392\) −1.00000 −0.0505076
\(393\) −2.76393 −0.139422
\(394\) 0.763932 0.0384863
\(395\) 5.85410 0.294552
\(396\) −4.85410 −0.243928
\(397\) −8.43769 −0.423476 −0.211738 0.977326i \(-0.567912\pi\)
−0.211738 + 0.977326i \(0.567912\pi\)
\(398\) 18.5623 0.930444
\(399\) 0 0
\(400\) −4.85410 −0.242705
\(401\) 1.41641 0.0707320 0.0353660 0.999374i \(-0.488740\pi\)
0.0353660 + 0.999374i \(0.488740\pi\)
\(402\) 0.472136 0.0235480
\(403\) 26.8328 1.33664
\(404\) −14.9443 −0.743505
\(405\) −2.18034 −0.108342
\(406\) −6.85410 −0.340163
\(407\) 2.12461 0.105313
\(408\) 3.23607 0.160209
\(409\) 22.7984 1.12731 0.563654 0.826011i \(-0.309395\pi\)
0.563654 + 0.826011i \(0.309395\pi\)
\(410\) 4.03444 0.199247
\(411\) −3.47214 −0.171268
\(412\) −8.94427 −0.440653
\(413\) 8.56231 0.421324
\(414\) −23.4164 −1.15085
\(415\) −2.58359 −0.126824
\(416\) −4.47214 −0.219265
\(417\) −3.70820 −0.181592
\(418\) 0 0
\(419\) 18.6525 0.911233 0.455617 0.890176i \(-0.349419\pi\)
0.455617 + 0.890176i \(0.349419\pi\)
\(420\) 0.236068 0.0115189
\(421\) 28.4721 1.38765 0.693823 0.720145i \(-0.255925\pi\)
0.693823 + 0.720145i \(0.255925\pi\)
\(422\) −12.7639 −0.621338
\(423\) 10.8541 0.527744
\(424\) 11.8541 0.575686
\(425\) −25.4164 −1.23288
\(426\) −7.14590 −0.346220
\(427\) 11.5623 0.559539
\(428\) 6.29180 0.304125
\(429\) −5.12461 −0.247419
\(430\) −1.09017 −0.0525727
\(431\) 16.1459 0.777721 0.388860 0.921297i \(-0.372869\pi\)
0.388860 + 0.921297i \(0.372869\pi\)
\(432\) 3.47214 0.167053
\(433\) −5.61803 −0.269985 −0.134993 0.990847i \(-0.543101\pi\)
−0.134993 + 0.990847i \(0.543101\pi\)
\(434\) −6.00000 −0.288009
\(435\) 1.61803 0.0775788
\(436\) 4.14590 0.198553
\(437\) 0 0
\(438\) 0 0
\(439\) 20.1803 0.963155 0.481578 0.876403i \(-0.340064\pi\)
0.481578 + 0.876403i \(0.340064\pi\)
\(440\) 0.708204 0.0337623
\(441\) −2.61803 −0.124668
\(442\) −23.4164 −1.11380
\(443\) −10.0902 −0.479398 −0.239699 0.970847i \(-0.577049\pi\)
−0.239699 + 0.970847i \(0.577049\pi\)
\(444\) −0.708204 −0.0336099
\(445\) 3.76393 0.178427
\(446\) 0.763932 0.0361732
\(447\) 7.88854 0.373115
\(448\) 1.00000 0.0472456
\(449\) −13.7082 −0.646930 −0.323465 0.946240i \(-0.604848\pi\)
−0.323465 + 0.946240i \(0.604848\pi\)
\(450\) −12.7082 −0.599070
\(451\) 19.5836 0.922155
\(452\) 10.0000 0.470360
\(453\) −12.9443 −0.608175
\(454\) 7.41641 0.348069
\(455\) −1.70820 −0.0800818
\(456\) 0 0
\(457\) 37.8541 1.77074 0.885370 0.464887i \(-0.153905\pi\)
0.885370 + 0.464887i \(0.153905\pi\)
\(458\) −17.1459 −0.801175
\(459\) 18.1803 0.848586
\(460\) 3.41641 0.159291
\(461\) −3.49342 −0.162705 −0.0813524 0.996685i \(-0.525924\pi\)
−0.0813524 + 0.996685i \(0.525924\pi\)
\(462\) 1.14590 0.0533120
\(463\) −12.0000 −0.557687 −0.278844 0.960337i \(-0.589951\pi\)
−0.278844 + 0.960337i \(0.589951\pi\)
\(464\) 6.85410 0.318194
\(465\) 1.41641 0.0656843
\(466\) 7.09017 0.328446
\(467\) 4.36068 0.201788 0.100894 0.994897i \(-0.467830\pi\)
0.100894 + 0.994897i \(0.467830\pi\)
\(468\) −11.7082 −0.541212
\(469\) 0.763932 0.0352751
\(470\) −1.58359 −0.0730457
\(471\) 2.38197 0.109755
\(472\) −8.56231 −0.394112
\(473\) −5.29180 −0.243317
\(474\) −9.47214 −0.435070
\(475\) 0 0
\(476\) 5.23607 0.239995
\(477\) 31.0344 1.42097
\(478\) 8.18034 0.374160
\(479\) 20.6738 0.944608 0.472304 0.881436i \(-0.343422\pi\)
0.472304 + 0.881436i \(0.343422\pi\)
\(480\) −0.236068 −0.0107750
\(481\) 5.12461 0.233662
\(482\) 2.56231 0.116710
\(483\) 5.52786 0.251527
\(484\) −7.56231 −0.343741
\(485\) −5.29180 −0.240288
\(486\) 13.9443 0.632525
\(487\) 28.0902 1.27289 0.636444 0.771323i \(-0.280405\pi\)
0.636444 + 0.771323i \(0.280405\pi\)
\(488\) −11.5623 −0.523401
\(489\) −8.20163 −0.370890
\(490\) 0.381966 0.0172555
\(491\) 4.36068 0.196795 0.0983974 0.995147i \(-0.468628\pi\)
0.0983974 + 0.995147i \(0.468628\pi\)
\(492\) −6.52786 −0.294299
\(493\) 35.8885 1.61634
\(494\) 0 0
\(495\) 1.85410 0.0833357
\(496\) 6.00000 0.269408
\(497\) −11.5623 −0.518640
\(498\) 4.18034 0.187326
\(499\) 41.6869 1.86616 0.933081 0.359665i \(-0.117109\pi\)
0.933081 + 0.359665i \(0.117109\pi\)
\(500\) 3.76393 0.168328
\(501\) 6.18034 0.276117
\(502\) −26.9443 −1.20258
\(503\) −21.2148 −0.945920 −0.472960 0.881084i \(-0.656815\pi\)
−0.472960 + 0.881084i \(0.656815\pi\)
\(504\) 2.61803 0.116617
\(505\) 5.70820 0.254012
\(506\) 16.5836 0.737231
\(507\) −4.32624 −0.192135
\(508\) 10.9098 0.484045
\(509\) 5.12461 0.227144 0.113572 0.993530i \(-0.463771\pi\)
0.113572 + 0.993530i \(0.463771\pi\)
\(510\) −1.23607 −0.0547340
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −3.14590 −0.138760
\(515\) 3.41641 0.150545
\(516\) 1.76393 0.0776528
\(517\) −7.68692 −0.338070
\(518\) −1.14590 −0.0503479
\(519\) −11.1246 −0.488316
\(520\) 1.70820 0.0749097
\(521\) 5.41641 0.237297 0.118649 0.992936i \(-0.462144\pi\)
0.118649 + 0.992936i \(0.462144\pi\)
\(522\) 17.9443 0.785399
\(523\) 24.0000 1.04945 0.524723 0.851273i \(-0.324169\pi\)
0.524723 + 0.851273i \(0.324169\pi\)
\(524\) 4.47214 0.195366
\(525\) 3.00000 0.130931
\(526\) 18.6525 0.813287
\(527\) 31.4164 1.36852
\(528\) −1.14590 −0.0498688
\(529\) 57.0000 2.47826
\(530\) −4.52786 −0.196678
\(531\) −22.4164 −0.972789
\(532\) 0 0
\(533\) 47.2361 2.04602
\(534\) −6.09017 −0.263547
\(535\) −2.40325 −0.103902
\(536\) −0.763932 −0.0329968
\(537\) −2.65248 −0.114463
\(538\) 31.4164 1.35446
\(539\) 1.85410 0.0798618
\(540\) −1.32624 −0.0570722
\(541\) −21.1246 −0.908218 −0.454109 0.890946i \(-0.650042\pi\)
−0.454109 + 0.890946i \(0.650042\pi\)
\(542\) −22.5623 −0.969134
\(543\) −12.0000 −0.514969
\(544\) −5.23607 −0.224495
\(545\) −1.58359 −0.0678336
\(546\) 2.76393 0.118285
\(547\) −18.6525 −0.797522 −0.398761 0.917055i \(-0.630560\pi\)
−0.398761 + 0.917055i \(0.630560\pi\)
\(548\) 5.61803 0.239991
\(549\) −30.2705 −1.29191
\(550\) 9.00000 0.383761
\(551\) 0 0
\(552\) −5.52786 −0.235282
\(553\) −15.3262 −0.651738
\(554\) −1.41641 −0.0601774
\(555\) 0.270510 0.0114825
\(556\) 6.00000 0.254457
\(557\) 26.0689 1.10457 0.552287 0.833654i \(-0.313755\pi\)
0.552287 + 0.833654i \(0.313755\pi\)
\(558\) 15.7082 0.664981
\(559\) −12.7639 −0.539857
\(560\) −0.381966 −0.0161410
\(561\) −6.00000 −0.253320
\(562\) −3.41641 −0.144112
\(563\) 13.5623 0.571583 0.285792 0.958292i \(-0.407744\pi\)
0.285792 + 0.958292i \(0.407744\pi\)
\(564\) 2.56231 0.107893
\(565\) −3.81966 −0.160694
\(566\) −8.00000 −0.336265
\(567\) 5.70820 0.239722
\(568\) 11.5623 0.485144
\(569\) −12.2918 −0.515299 −0.257649 0.966238i \(-0.582948\pi\)
−0.257649 + 0.966238i \(0.582948\pi\)
\(570\) 0 0
\(571\) −44.6869 −1.87009 −0.935045 0.354530i \(-0.884641\pi\)
−0.935045 + 0.354530i \(0.884641\pi\)
\(572\) 8.29180 0.346697
\(573\) 1.81966 0.0760174
\(574\) −10.5623 −0.440862
\(575\) 43.4164 1.81059
\(576\) −2.61803 −0.109085
\(577\) 27.1246 1.12921 0.564606 0.825360i \(-0.309028\pi\)
0.564606 + 0.825360i \(0.309028\pi\)
\(578\) −10.4164 −0.433265
\(579\) 1.88854 0.0784852
\(580\) −2.61803 −0.108708
\(581\) 6.76393 0.280615
\(582\) 8.56231 0.354919
\(583\) −21.9787 −0.910265
\(584\) 0 0
\(585\) 4.47214 0.184900
\(586\) −4.58359 −0.189346
\(587\) 28.4721 1.17517 0.587585 0.809162i \(-0.300079\pi\)
0.587585 + 0.809162i \(0.300079\pi\)
\(588\) −0.618034 −0.0254873
\(589\) 0 0
\(590\) 3.27051 0.134645
\(591\) 0.472136 0.0194211
\(592\) 1.14590 0.0470961
\(593\) 7.52786 0.309132 0.154566 0.987982i \(-0.450602\pi\)
0.154566 + 0.987982i \(0.450602\pi\)
\(594\) −6.43769 −0.264142
\(595\) −2.00000 −0.0819920
\(596\) −12.7639 −0.522831
\(597\) 11.4721 0.469523
\(598\) 40.0000 1.63572
\(599\) −34.9787 −1.42919 −0.714596 0.699538i \(-0.753389\pi\)
−0.714596 + 0.699538i \(0.753389\pi\)
\(600\) −3.00000 −0.122474
\(601\) −34.3607 −1.40160 −0.700801 0.713357i \(-0.747174\pi\)
−0.700801 + 0.713357i \(0.747174\pi\)
\(602\) 2.85410 0.116325
\(603\) −2.00000 −0.0814463
\(604\) 20.9443 0.852210
\(605\) 2.88854 0.117436
\(606\) −9.23607 −0.375190
\(607\) −28.3607 −1.15112 −0.575562 0.817758i \(-0.695217\pi\)
−0.575562 + 0.817758i \(0.695217\pi\)
\(608\) 0 0
\(609\) −4.23607 −0.171654
\(610\) 4.41641 0.178815
\(611\) −18.5410 −0.750089
\(612\) −13.7082 −0.554121
\(613\) 25.7082 1.03834 0.519172 0.854670i \(-0.326240\pi\)
0.519172 + 0.854670i \(0.326240\pi\)
\(614\) 27.9787 1.12913
\(615\) 2.49342 0.100544
\(616\) −1.85410 −0.0747039
\(617\) −30.9230 −1.24491 −0.622456 0.782655i \(-0.713865\pi\)
−0.622456 + 0.782655i \(0.713865\pi\)
\(618\) −5.52786 −0.222363
\(619\) −8.58359 −0.345004 −0.172502 0.985009i \(-0.555185\pi\)
−0.172502 + 0.985009i \(0.555185\pi\)
\(620\) −2.29180 −0.0920407
\(621\) −31.0557 −1.24622
\(622\) 5.67376 0.227497
\(623\) −9.85410 −0.394796
\(624\) −2.76393 −0.110646
\(625\) 22.8328 0.913313
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) −3.85410 −0.153795
\(629\) 6.00000 0.239236
\(630\) −1.00000 −0.0398410
\(631\) −30.8328 −1.22744 −0.613718 0.789526i \(-0.710327\pi\)
−0.613718 + 0.789526i \(0.710327\pi\)
\(632\) 15.3262 0.609645
\(633\) −7.88854 −0.313541
\(634\) −34.6869 −1.37759
\(635\) −4.16718 −0.165370
\(636\) 7.32624 0.290504
\(637\) 4.47214 0.177192
\(638\) −12.7082 −0.503123
\(639\) 30.2705 1.19748
\(640\) 0.381966 0.0150985
\(641\) 19.4164 0.766902 0.383451 0.923561i \(-0.374735\pi\)
0.383451 + 0.923561i \(0.374735\pi\)
\(642\) 3.88854 0.153469
\(643\) −5.41641 −0.213602 −0.106801 0.994280i \(-0.534061\pi\)
−0.106801 + 0.994280i \(0.534061\pi\)
\(644\) −8.94427 −0.352454
\(645\) −0.673762 −0.0265294
\(646\) 0 0
\(647\) 15.3262 0.602537 0.301268 0.953539i \(-0.402590\pi\)
0.301268 + 0.953539i \(0.402590\pi\)
\(648\) −5.70820 −0.224239
\(649\) 15.8754 0.623163
\(650\) 21.7082 0.851466
\(651\) −3.70820 −0.145336
\(652\) 13.2705 0.519713
\(653\) −22.4721 −0.879403 −0.439701 0.898144i \(-0.644916\pi\)
−0.439701 + 0.898144i \(0.644916\pi\)
\(654\) 2.56231 0.100194
\(655\) −1.70820 −0.0667451
\(656\) 10.5623 0.412389
\(657\) 0 0
\(658\) 4.14590 0.161624
\(659\) −41.1246 −1.60199 −0.800994 0.598673i \(-0.795695\pi\)
−0.800994 + 0.598673i \(0.795695\pi\)
\(660\) 0.437694 0.0170372
\(661\) 26.8328 1.04368 0.521838 0.853045i \(-0.325247\pi\)
0.521838 + 0.853045i \(0.325247\pi\)
\(662\) −9.70820 −0.377320
\(663\) −14.4721 −0.562051
\(664\) −6.76393 −0.262491
\(665\) 0 0
\(666\) 3.00000 0.116248
\(667\) −61.3050 −2.37374
\(668\) −10.0000 −0.386912
\(669\) 0.472136 0.0182538
\(670\) 0.291796 0.0112731
\(671\) 21.4377 0.827593
\(672\) 0.618034 0.0238412
\(673\) −20.9443 −0.807342 −0.403671 0.914904i \(-0.632266\pi\)
−0.403671 + 0.914904i \(0.632266\pi\)
\(674\) 3.70820 0.142835
\(675\) −16.8541 −0.648715
\(676\) 7.00000 0.269231
\(677\) −28.2918 −1.08734 −0.543671 0.839298i \(-0.682966\pi\)
−0.543671 + 0.839298i \(0.682966\pi\)
\(678\) 6.18034 0.237355
\(679\) 13.8541 0.531672
\(680\) 2.00000 0.0766965
\(681\) 4.58359 0.175644
\(682\) −11.1246 −0.425983
\(683\) −15.7082 −0.601058 −0.300529 0.953773i \(-0.597163\pi\)
−0.300529 + 0.953773i \(0.597163\pi\)
\(684\) 0 0
\(685\) −2.14590 −0.0819905
\(686\) −1.00000 −0.0381802
\(687\) −10.5967 −0.404291
\(688\) −2.85410 −0.108812
\(689\) −53.0132 −2.01964
\(690\) 2.11146 0.0803818
\(691\) −46.5410 −1.77050 −0.885252 0.465112i \(-0.846014\pi\)
−0.885252 + 0.465112i \(0.846014\pi\)
\(692\) 18.0000 0.684257
\(693\) −4.85410 −0.184392
\(694\) 8.94427 0.339520
\(695\) −2.29180 −0.0869328
\(696\) 4.23607 0.160568
\(697\) 55.3050 2.09482
\(698\) −36.8328 −1.39414
\(699\) 4.38197 0.165741
\(700\) −4.85410 −0.183468
\(701\) −0.763932 −0.0288533 −0.0144267 0.999896i \(-0.504592\pi\)
−0.0144267 + 0.999896i \(0.504592\pi\)
\(702\) −15.5279 −0.586061
\(703\) 0 0
\(704\) 1.85410 0.0698791
\(705\) −0.978714 −0.0368605
\(706\) 13.4164 0.504933
\(707\) −14.9443 −0.562037
\(708\) −5.29180 −0.198878
\(709\) −31.1246 −1.16891 −0.584455 0.811426i \(-0.698692\pi\)
−0.584455 + 0.811426i \(0.698692\pi\)
\(710\) −4.41641 −0.165745
\(711\) 40.1246 1.50479
\(712\) 9.85410 0.369298
\(713\) −53.6656 −2.00979
\(714\) 3.23607 0.121107
\(715\) −3.16718 −0.118446
\(716\) 4.29180 0.160392
\(717\) 5.05573 0.188810
\(718\) −15.7082 −0.586225
\(719\) −28.3607 −1.05767 −0.528837 0.848723i \(-0.677372\pi\)
−0.528837 + 0.848723i \(0.677372\pi\)
\(720\) 1.00000 0.0372678
\(721\) −8.94427 −0.333102
\(722\) 0 0
\(723\) 1.58359 0.0588944
\(724\) 19.4164 0.721605
\(725\) −33.2705 −1.23564
\(726\) −4.67376 −0.173460
\(727\) 8.43769 0.312937 0.156468 0.987683i \(-0.449989\pi\)
0.156468 + 0.987683i \(0.449989\pi\)
\(728\) −4.47214 −0.165748
\(729\) −8.50658 −0.315058
\(730\) 0 0
\(731\) −14.9443 −0.552734
\(732\) −7.14590 −0.264120
\(733\) 0.562306 0.0207692 0.0103846 0.999946i \(-0.496694\pi\)
0.0103846 + 0.999946i \(0.496694\pi\)
\(734\) −27.2705 −1.00657
\(735\) 0.236068 0.00870750
\(736\) 8.94427 0.329690
\(737\) 1.41641 0.0521741
\(738\) 27.6525 1.01790
\(739\) 1.72949 0.0636203 0.0318102 0.999494i \(-0.489873\pi\)
0.0318102 + 0.999494i \(0.489873\pi\)
\(740\) −0.437694 −0.0160900
\(741\) 0 0
\(742\) 11.8541 0.435178
\(743\) −21.9787 −0.806321 −0.403160 0.915129i \(-0.632088\pi\)
−0.403160 + 0.915129i \(0.632088\pi\)
\(744\) 3.70820 0.135949
\(745\) 4.87539 0.178620
\(746\) −12.3820 −0.453336
\(747\) −17.7082 −0.647909
\(748\) 9.70820 0.354967
\(749\) 6.29180 0.229897
\(750\) 2.32624 0.0849422
\(751\) 15.3262 0.559262 0.279631 0.960107i \(-0.409788\pi\)
0.279631 + 0.960107i \(0.409788\pi\)
\(752\) −4.14590 −0.151185
\(753\) −16.6525 −0.606850
\(754\) −30.6525 −1.11630
\(755\) −8.00000 −0.291150
\(756\) 3.47214 0.126280
\(757\) −34.2492 −1.24481 −0.622405 0.782696i \(-0.713844\pi\)
−0.622405 + 0.782696i \(0.713844\pi\)
\(758\) −9.81966 −0.356666
\(759\) 10.2492 0.372023
\(760\) 0 0
\(761\) 53.7771 1.94942 0.974709 0.223478i \(-0.0717411\pi\)
0.974709 + 0.223478i \(0.0717411\pi\)
\(762\) 6.74265 0.244260
\(763\) 4.14590 0.150092
\(764\) −2.94427 −0.106520
\(765\) 5.23607 0.189310
\(766\) −27.1246 −0.980052
\(767\) 38.2918 1.38264
\(768\) −0.618034 −0.0223014
\(769\) 25.7082 0.927062 0.463531 0.886081i \(-0.346582\pi\)
0.463531 + 0.886081i \(0.346582\pi\)
\(770\) 0.708204 0.0255219
\(771\) −1.94427 −0.0700212
\(772\) −3.05573 −0.109978
\(773\) −11.7082 −0.421115 −0.210557 0.977581i \(-0.567528\pi\)
−0.210557 + 0.977581i \(0.567528\pi\)
\(774\) −7.47214 −0.268580
\(775\) −29.1246 −1.04619
\(776\) −13.8541 −0.497333
\(777\) −0.708204 −0.0254067
\(778\) −25.3050 −0.907226
\(779\) 0 0
\(780\) 1.05573 0.0378011
\(781\) −21.4377 −0.767101
\(782\) 46.8328 1.67474
\(783\) 23.7984 0.850484
\(784\) 1.00000 0.0357143
\(785\) 1.47214 0.0525428
\(786\) 2.76393 0.0985862
\(787\) 6.27051 0.223520 0.111760 0.993735i \(-0.464351\pi\)
0.111760 + 0.993735i \(0.464351\pi\)
\(788\) −0.763932 −0.0272140
\(789\) 11.5279 0.410403
\(790\) −5.85410 −0.208280
\(791\) 10.0000 0.355559
\(792\) 4.85410 0.172483
\(793\) 51.7082 1.83621
\(794\) 8.43769 0.299443
\(795\) −2.79837 −0.0992481
\(796\) −18.5623 −0.657923
\(797\) 8.83282 0.312874 0.156437 0.987688i \(-0.449999\pi\)
0.156437 + 0.987688i \(0.449999\pi\)
\(798\) 0 0
\(799\) −21.7082 −0.767981
\(800\) 4.85410 0.171618
\(801\) 25.7984 0.911541
\(802\) −1.41641 −0.0500151
\(803\) 0 0
\(804\) −0.472136 −0.0166510
\(805\) 3.41641 0.120413
\(806\) −26.8328 −0.945146
\(807\) 19.4164 0.683490
\(808\) 14.9443 0.525738
\(809\) 13.0344 0.458267 0.229133 0.973395i \(-0.426411\pi\)
0.229133 + 0.973395i \(0.426411\pi\)
\(810\) 2.18034 0.0766093
\(811\) 48.9787 1.71988 0.859938 0.510399i \(-0.170502\pi\)
0.859938 + 0.510399i \(0.170502\pi\)
\(812\) 6.85410 0.240532
\(813\) −13.9443 −0.489047
\(814\) −2.12461 −0.0744676
\(815\) −5.06888 −0.177555
\(816\) −3.23607 −0.113285
\(817\) 0 0
\(818\) −22.7984 −0.797126
\(819\) −11.7082 −0.409118
\(820\) −4.03444 −0.140889
\(821\) 47.1246 1.64466 0.822330 0.569011i \(-0.192674\pi\)
0.822330 + 0.569011i \(0.192674\pi\)
\(822\) 3.47214 0.121105
\(823\) −15.7082 −0.547554 −0.273777 0.961793i \(-0.588273\pi\)
−0.273777 + 0.961793i \(0.588273\pi\)
\(824\) 8.94427 0.311588
\(825\) 5.56231 0.193655
\(826\) −8.56231 −0.297921
\(827\) 49.9574 1.73719 0.868595 0.495523i \(-0.165023\pi\)
0.868595 + 0.495523i \(0.165023\pi\)
\(828\) 23.4164 0.813776
\(829\) 5.12461 0.177985 0.0889926 0.996032i \(-0.471635\pi\)
0.0889926 + 0.996032i \(0.471635\pi\)
\(830\) 2.58359 0.0896778
\(831\) −0.875388 −0.0303669
\(832\) 4.47214 0.155043
\(833\) 5.23607 0.181419
\(834\) 3.70820 0.128405
\(835\) 3.81966 0.132185
\(836\) 0 0
\(837\) 20.8328 0.720087
\(838\) −18.6525 −0.644339
\(839\) 48.8328 1.68590 0.842948 0.537995i \(-0.180818\pi\)
0.842948 + 0.537995i \(0.180818\pi\)
\(840\) −0.236068 −0.00814512
\(841\) 17.9787 0.619956
\(842\) −28.4721 −0.981215
\(843\) −2.11146 −0.0727224
\(844\) 12.7639 0.439353
\(845\) −2.67376 −0.0919802
\(846\) −10.8541 −0.373172
\(847\) −7.56231 −0.259844
\(848\) −11.8541 −0.407072
\(849\) −4.94427 −0.169687
\(850\) 25.4164 0.871776
\(851\) −10.2492 −0.351339
\(852\) 7.14590 0.244814
\(853\) 10.2705 0.351656 0.175828 0.984421i \(-0.443740\pi\)
0.175828 + 0.984421i \(0.443740\pi\)
\(854\) −11.5623 −0.395654
\(855\) 0 0
\(856\) −6.29180 −0.215049
\(857\) −48.2492 −1.64816 −0.824081 0.566472i \(-0.808308\pi\)
−0.824081 + 0.566472i \(0.808308\pi\)
\(858\) 5.12461 0.174951
\(859\) −34.0000 −1.16007 −0.580033 0.814593i \(-0.696960\pi\)
−0.580033 + 0.814593i \(0.696960\pi\)
\(860\) 1.09017 0.0371745
\(861\) −6.52786 −0.222469
\(862\) −16.1459 −0.549931
\(863\) 2.02129 0.0688054 0.0344027 0.999408i \(-0.489047\pi\)
0.0344027 + 0.999408i \(0.489047\pi\)
\(864\) −3.47214 −0.118124
\(865\) −6.87539 −0.233770
\(866\) 5.61803 0.190909
\(867\) −6.43769 −0.218636
\(868\) 6.00000 0.203653
\(869\) −28.4164 −0.963961
\(870\) −1.61803 −0.0548565
\(871\) 3.41641 0.115761
\(872\) −4.14590 −0.140398
\(873\) −36.2705 −1.22757
\(874\) 0 0
\(875\) 3.76393 0.127244
\(876\) 0 0
\(877\) 26.3951 0.891300 0.445650 0.895207i \(-0.352973\pi\)
0.445650 + 0.895207i \(0.352973\pi\)
\(878\) −20.1803 −0.681053
\(879\) −2.83282 −0.0955485
\(880\) −0.708204 −0.0238735
\(881\) 32.1803 1.08418 0.542092 0.840319i \(-0.317633\pi\)
0.542092 + 0.840319i \(0.317633\pi\)
\(882\) 2.61803 0.0881538
\(883\) 32.6869 1.10000 0.550001 0.835164i \(-0.314627\pi\)
0.550001 + 0.835164i \(0.314627\pi\)
\(884\) 23.4164 0.787579
\(885\) 2.02129 0.0679448
\(886\) 10.0902 0.338986
\(887\) −50.8328 −1.70680 −0.853399 0.521257i \(-0.825463\pi\)
−0.853399 + 0.521257i \(0.825463\pi\)
\(888\) 0.708204 0.0237658
\(889\) 10.9098 0.365904
\(890\) −3.76393 −0.126167
\(891\) 10.5836 0.354564
\(892\) −0.763932 −0.0255783
\(893\) 0 0
\(894\) −7.88854 −0.263832
\(895\) −1.63932 −0.0547964
\(896\) −1.00000 −0.0334077
\(897\) 24.7214 0.825422
\(898\) 13.7082 0.457449
\(899\) 41.1246 1.37158
\(900\) 12.7082 0.423607
\(901\) −62.0689 −2.06781
\(902\) −19.5836 −0.652062
\(903\) 1.76393 0.0587000
\(904\) −10.0000 −0.332595
\(905\) −7.41641 −0.246530
\(906\) 12.9443 0.430045
\(907\) 48.0000 1.59381 0.796907 0.604102i \(-0.206468\pi\)
0.796907 + 0.604102i \(0.206468\pi\)
\(908\) −7.41641 −0.246122
\(909\) 39.1246 1.29768
\(910\) 1.70820 0.0566264
\(911\) −43.1459 −1.42949 −0.714744 0.699386i \(-0.753457\pi\)
−0.714744 + 0.699386i \(0.753457\pi\)
\(912\) 0 0
\(913\) 12.5410 0.415047
\(914\) −37.8541 −1.25210
\(915\) 2.72949 0.0902342
\(916\) 17.1459 0.566516
\(917\) 4.47214 0.147683
\(918\) −18.1803 −0.600041
\(919\) −25.7082 −0.848035 −0.424018 0.905654i \(-0.639381\pi\)
−0.424018 + 0.905654i \(0.639381\pi\)
\(920\) −3.41641 −0.112636
\(921\) 17.2918 0.569784
\(922\) 3.49342 0.115050
\(923\) −51.7082 −1.70200
\(924\) −1.14590 −0.0376973
\(925\) −5.56231 −0.182887
\(926\) 12.0000 0.394344
\(927\) 23.4164 0.769096
\(928\) −6.85410 −0.224997
\(929\) 47.2361 1.54977 0.774883 0.632105i \(-0.217809\pi\)
0.774883 + 0.632105i \(0.217809\pi\)
\(930\) −1.41641 −0.0464458
\(931\) 0 0
\(932\) −7.09017 −0.232246
\(933\) 3.50658 0.114800
\(934\) −4.36068 −0.142686
\(935\) −3.70820 −0.121271
\(936\) 11.7082 0.382695
\(937\) 0.291796 0.00953256 0.00476628 0.999989i \(-0.498483\pi\)
0.00476628 + 0.999989i \(0.498483\pi\)
\(938\) −0.763932 −0.0249433
\(939\) −6.18034 −0.201688
\(940\) 1.58359 0.0516511
\(941\) −20.2918 −0.661494 −0.330747 0.943720i \(-0.607301\pi\)
−0.330747 + 0.943720i \(0.607301\pi\)
\(942\) −2.38197 −0.0776086
\(943\) −94.4721 −3.07644
\(944\) 8.56231 0.278679
\(945\) −1.32624 −0.0431425
\(946\) 5.29180 0.172051
\(947\) −49.8541 −1.62004 −0.810020 0.586402i \(-0.800544\pi\)
−0.810020 + 0.586402i \(0.800544\pi\)
\(948\) 9.47214 0.307641
\(949\) 0 0
\(950\) 0 0
\(951\) −21.4377 −0.695165
\(952\) −5.23607 −0.169702
\(953\) 15.7082 0.508839 0.254419 0.967094i \(-0.418116\pi\)
0.254419 + 0.967094i \(0.418116\pi\)
\(954\) −31.0344 −1.00478
\(955\) 1.12461 0.0363916
\(956\) −8.18034 −0.264571
\(957\) −7.85410 −0.253887
\(958\) −20.6738 −0.667939
\(959\) 5.61803 0.181416
\(960\) 0.236068 0.00761906
\(961\) 5.00000 0.161290
\(962\) −5.12461 −0.165224
\(963\) −16.4721 −0.530807
\(964\) −2.56231 −0.0825263
\(965\) 1.16718 0.0375730
\(966\) −5.52786 −0.177856
\(967\) −26.2492 −0.844118 −0.422059 0.906568i \(-0.638693\pi\)
−0.422059 + 0.906568i \(0.638693\pi\)
\(968\) 7.56231 0.243062
\(969\) 0 0
\(970\) 5.29180 0.169909
\(971\) 45.2705 1.45280 0.726400 0.687272i \(-0.241192\pi\)
0.726400 + 0.687272i \(0.241192\pi\)
\(972\) −13.9443 −0.447263
\(973\) 6.00000 0.192351
\(974\) −28.0902 −0.900067
\(975\) 13.4164 0.429669
\(976\) 11.5623 0.370100
\(977\) 22.2918 0.713178 0.356589 0.934261i \(-0.383940\pi\)
0.356589 + 0.934261i \(0.383940\pi\)
\(978\) 8.20163 0.262259
\(979\) −18.2705 −0.583928
\(980\) −0.381966 −0.0122015
\(981\) −10.8541 −0.346545
\(982\) −4.36068 −0.139155
\(983\) −21.1246 −0.673770 −0.336885 0.941546i \(-0.609373\pi\)
−0.336885 + 0.941546i \(0.609373\pi\)
\(984\) 6.52786 0.208101
\(985\) 0.291796 0.00929740
\(986\) −35.8885 −1.14292
\(987\) 2.56231 0.0815591
\(988\) 0 0
\(989\) 25.5279 0.811739
\(990\) −1.85410 −0.0589272
\(991\) −26.3951 −0.838469 −0.419234 0.907878i \(-0.637701\pi\)
−0.419234 + 0.907878i \(0.637701\pi\)
\(992\) −6.00000 −0.190500
\(993\) −6.00000 −0.190404
\(994\) 11.5623 0.366734
\(995\) 7.09017 0.224773
\(996\) −4.18034 −0.132459
\(997\) −45.1459 −1.42978 −0.714892 0.699234i \(-0.753524\pi\)
−0.714892 + 0.699234i \(0.753524\pi\)
\(998\) −41.6869 −1.31958
\(999\) 3.97871 0.125881
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 5054.2.a.g.1.1 2
19.18 odd 2 5054.2.a.l.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5054.2.a.g.1.1 2 1.1 even 1 trivial
5054.2.a.l.1.2 yes 2 19.18 odd 2