Properties

Label 3381.2.a.bc.1.5
Level $3381$
Weight $2$
Character 3381.1
Self dual yes
Analytic conductor $26.997$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9974209234\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.7997584.1
Defining polynomial: \(x^{6} - x^{5} - 7 x^{4} + 5 x^{3} + 12 x^{2} - 4 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.36549\) of defining polynomial
Character \(\chi\) \(=\) 3381.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.36549 q^{2} -1.00000 q^{3} -0.135449 q^{4} -1.32621 q^{5} -1.36549 q^{6} -2.91593 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.36549 q^{2} -1.00000 q^{3} -0.135449 q^{4} -1.32621 q^{5} -1.36549 q^{6} -2.91593 q^{8} +1.00000 q^{9} -1.81092 q^{10} -1.42009 q^{11} +0.135449 q^{12} +3.16661 q^{13} +1.32621 q^{15} -3.71076 q^{16} +1.15960 q^{17} +1.36549 q^{18} +8.57886 q^{19} +0.179634 q^{20} -1.93911 q^{22} -1.00000 q^{23} +2.91593 q^{24} -3.24116 q^{25} +4.32396 q^{26} -1.00000 q^{27} +7.45480 q^{29} +1.81092 q^{30} +1.86642 q^{31} +0.764868 q^{32} +1.42009 q^{33} +1.58342 q^{34} -0.135449 q^{36} -7.79838 q^{37} +11.7143 q^{38} -3.16661 q^{39} +3.86713 q^{40} -11.2258 q^{41} -11.8479 q^{43} +0.192350 q^{44} -1.32621 q^{45} -1.36549 q^{46} +1.73650 q^{47} +3.71076 q^{48} -4.42576 q^{50} -1.15960 q^{51} -0.428914 q^{52} -4.14898 q^{53} -1.36549 q^{54} +1.88333 q^{55} -8.57886 q^{57} +10.1794 q^{58} +12.6871 q^{59} -0.179634 q^{60} -13.8148 q^{61} +2.54857 q^{62} +8.46593 q^{64} -4.19959 q^{65} +1.93911 q^{66} -8.09295 q^{67} -0.157067 q^{68} +1.00000 q^{69} -5.18979 q^{71} -2.91593 q^{72} -11.3686 q^{73} -10.6486 q^{74} +3.24116 q^{75} -1.16200 q^{76} -4.32396 q^{78} -11.7365 q^{79} +4.92124 q^{80} +1.00000 q^{81} -15.3286 q^{82} +5.70566 q^{83} -1.53788 q^{85} -16.1781 q^{86} -7.45480 q^{87} +4.14087 q^{88} -6.35511 q^{89} -1.81092 q^{90} +0.135449 q^{92} -1.86642 q^{93} +2.37116 q^{94} -11.3774 q^{95} -0.764868 q^{96} +0.408396 q^{97} -1.42009 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - q^{2} - 6q^{3} + 3q^{4} + 3q^{5} + q^{6} - 3q^{8} + 6q^{9} + O(q^{10}) \) \( 6q - q^{2} - 6q^{3} + 3q^{4} + 3q^{5} + q^{6} - 3q^{8} + 6q^{9} - 3q^{10} - 14q^{11} - 3q^{12} - 3q^{15} - 7q^{16} + 15q^{17} - q^{18} - q^{19} + 17q^{20} - 6q^{22} - 6q^{23} + 3q^{24} + 9q^{25} - 15q^{26} - 6q^{27} - 6q^{29} + 3q^{30} - 11q^{31} + 3q^{32} + 14q^{33} + 15q^{34} + 3q^{36} - 5q^{37} + 14q^{38} - 17q^{40} + 18q^{41} - 37q^{43} - 10q^{44} + 3q^{45} + q^{46} + 3q^{47} + 7q^{48} - 30q^{50} - 15q^{51} - 7q^{52} - 15q^{53} + q^{54} - 2q^{55} + q^{57} + 4q^{58} - 2q^{59} - 17q^{60} - 12q^{61} + 36q^{62} - 23q^{64} - 17q^{65} + 6q^{66} - 10q^{67} - q^{68} + 6q^{69} - 21q^{71} - 3q^{72} - 8q^{73} - 16q^{74} - 9q^{75} + 18q^{76} + 15q^{78} - 17q^{79} + 3q^{80} + 6q^{81} - 48q^{82} + 12q^{83} - 13q^{85} + 22q^{86} + 6q^{87} - 2q^{88} + 18q^{89} - 3q^{90} - 3q^{92} + 11q^{93} + 3q^{94} - 16q^{95} - 3q^{96} - 2q^{97} - 14q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.36549 0.965544 0.482772 0.875746i \(-0.339630\pi\)
0.482772 + 0.875746i \(0.339630\pi\)
\(3\) −1.00000 −0.577350
\(4\) −0.135449 −0.0677246
\(5\) −1.32621 −0.593100 −0.296550 0.955017i \(-0.595836\pi\)
−0.296550 + 0.955017i \(0.595836\pi\)
\(6\) −1.36549 −0.557457
\(7\) 0 0
\(8\) −2.91593 −1.03094
\(9\) 1.00000 0.333333
\(10\) −1.81092 −0.572664
\(11\) −1.42009 −0.428172 −0.214086 0.976815i \(-0.568677\pi\)
−0.214086 + 0.976815i \(0.568677\pi\)
\(12\) 0.135449 0.0391008
\(13\) 3.16661 0.878259 0.439129 0.898424i \(-0.355287\pi\)
0.439129 + 0.898424i \(0.355287\pi\)
\(14\) 0 0
\(15\) 1.32621 0.342426
\(16\) −3.71076 −0.927689
\(17\) 1.15960 0.281245 0.140623 0.990063i \(-0.455090\pi\)
0.140623 + 0.990063i \(0.455090\pi\)
\(18\) 1.36549 0.321848
\(19\) 8.57886 1.96813 0.984063 0.177822i \(-0.0569051\pi\)
0.984063 + 0.177822i \(0.0569051\pi\)
\(20\) 0.179634 0.0401674
\(21\) 0 0
\(22\) −1.93911 −0.413419
\(23\) −1.00000 −0.208514
\(24\) 2.91593 0.595211
\(25\) −3.24116 −0.648233
\(26\) 4.32396 0.847998
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 7.45480 1.38432 0.692161 0.721743i \(-0.256659\pi\)
0.692161 + 0.721743i \(0.256659\pi\)
\(30\) 1.81092 0.330628
\(31\) 1.86642 0.335219 0.167609 0.985853i \(-0.446395\pi\)
0.167609 + 0.985853i \(0.446395\pi\)
\(32\) 0.764868 0.135211
\(33\) 1.42009 0.247205
\(34\) 1.58342 0.271555
\(35\) 0 0
\(36\) −0.135449 −0.0225749
\(37\) −7.79838 −1.28205 −0.641023 0.767522i \(-0.721490\pi\)
−0.641023 + 0.767522i \(0.721490\pi\)
\(38\) 11.7143 1.90031
\(39\) −3.16661 −0.507063
\(40\) 3.86713 0.611447
\(41\) −11.2258 −1.75317 −0.876584 0.481248i \(-0.840183\pi\)
−0.876584 + 0.481248i \(0.840183\pi\)
\(42\) 0 0
\(43\) −11.8479 −1.80679 −0.903393 0.428814i \(-0.858932\pi\)
−0.903393 + 0.428814i \(0.858932\pi\)
\(44\) 0.192350 0.0289978
\(45\) −1.32621 −0.197700
\(46\) −1.36549 −0.201330
\(47\) 1.73650 0.253294 0.126647 0.991948i \(-0.459578\pi\)
0.126647 + 0.991948i \(0.459578\pi\)
\(48\) 3.71076 0.535601
\(49\) 0 0
\(50\) −4.42576 −0.625897
\(51\) −1.15960 −0.162377
\(52\) −0.428914 −0.0594797
\(53\) −4.14898 −0.569906 −0.284953 0.958541i \(-0.591978\pi\)
−0.284953 + 0.958541i \(0.591978\pi\)
\(54\) −1.36549 −0.185819
\(55\) 1.88333 0.253949
\(56\) 0 0
\(57\) −8.57886 −1.13630
\(58\) 10.1794 1.33662
\(59\) 12.6871 1.65172 0.825862 0.563872i \(-0.190689\pi\)
0.825862 + 0.563872i \(0.190689\pi\)
\(60\) −0.179634 −0.0231907
\(61\) −13.8148 −1.76880 −0.884399 0.466731i \(-0.845432\pi\)
−0.884399 + 0.466731i \(0.845432\pi\)
\(62\) 2.54857 0.323669
\(63\) 0 0
\(64\) 8.46593 1.05824
\(65\) −4.19959 −0.520895
\(66\) 1.93911 0.238688
\(67\) −8.09295 −0.988711 −0.494355 0.869260i \(-0.664596\pi\)
−0.494355 + 0.869260i \(0.664596\pi\)
\(68\) −0.157067 −0.0190472
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −5.18979 −0.615915 −0.307957 0.951400i \(-0.599645\pi\)
−0.307957 + 0.951400i \(0.599645\pi\)
\(72\) −2.91593 −0.343645
\(73\) −11.3686 −1.33059 −0.665297 0.746579i \(-0.731695\pi\)
−0.665297 + 0.746579i \(0.731695\pi\)
\(74\) −10.6486 −1.23787
\(75\) 3.24116 0.374257
\(76\) −1.16200 −0.133291
\(77\) 0 0
\(78\) −4.32396 −0.489592
\(79\) −11.7365 −1.32046 −0.660228 0.751065i \(-0.729540\pi\)
−0.660228 + 0.751065i \(0.729540\pi\)
\(80\) 4.92124 0.550212
\(81\) 1.00000 0.111111
\(82\) −15.3286 −1.69276
\(83\) 5.70566 0.626277 0.313139 0.949707i \(-0.398620\pi\)
0.313139 + 0.949707i \(0.398620\pi\)
\(84\) 0 0
\(85\) −1.53788 −0.166806
\(86\) −16.1781 −1.74453
\(87\) −7.45480 −0.799238
\(88\) 4.14087 0.441418
\(89\) −6.35511 −0.673641 −0.336820 0.941569i \(-0.609352\pi\)
−0.336820 + 0.941569i \(0.609352\pi\)
\(90\) −1.81092 −0.190888
\(91\) 0 0
\(92\) 0.135449 0.0141216
\(93\) −1.86642 −0.193539
\(94\) 2.37116 0.244567
\(95\) −11.3774 −1.16729
\(96\) −0.764868 −0.0780640
\(97\) 0.408396 0.0414663 0.0207331 0.999785i \(-0.493400\pi\)
0.0207331 + 0.999785i \(0.493400\pi\)
\(98\) 0 0
\(99\) −1.42009 −0.142724
\(100\) 0.439013 0.0439013
\(101\) −0.723827 −0.0720235 −0.0360117 0.999351i \(-0.511465\pi\)
−0.0360117 + 0.999351i \(0.511465\pi\)
\(102\) −1.58342 −0.156782
\(103\) −14.2739 −1.40645 −0.703225 0.710967i \(-0.748257\pi\)
−0.703225 + 0.710967i \(0.748257\pi\)
\(104\) −9.23359 −0.905428
\(105\) 0 0
\(106\) −5.66537 −0.550270
\(107\) 13.9959 1.35304 0.676520 0.736425i \(-0.263487\pi\)
0.676520 + 0.736425i \(0.263487\pi\)
\(108\) 0.135449 0.0130336
\(109\) 2.90945 0.278675 0.139338 0.990245i \(-0.455503\pi\)
0.139338 + 0.990245i \(0.455503\pi\)
\(110\) 2.57167 0.245199
\(111\) 7.79838 0.740190
\(112\) 0 0
\(113\) 18.6624 1.75561 0.877804 0.479020i \(-0.159008\pi\)
0.877804 + 0.479020i \(0.159008\pi\)
\(114\) −11.7143 −1.09715
\(115\) 1.32621 0.123670
\(116\) −1.00975 −0.0937526
\(117\) 3.16661 0.292753
\(118\) 17.3241 1.59481
\(119\) 0 0
\(120\) −3.86713 −0.353019
\(121\) −8.98335 −0.816669
\(122\) −18.8639 −1.70785
\(123\) 11.2258 1.01219
\(124\) −0.252805 −0.0227026
\(125\) 10.9295 0.977566
\(126\) 0 0
\(127\) 1.06631 0.0946196 0.0473098 0.998880i \(-0.484935\pi\)
0.0473098 + 0.998880i \(0.484935\pi\)
\(128\) 10.0304 0.886567
\(129\) 11.8479 1.04315
\(130\) −5.73448 −0.502947
\(131\) −16.0144 −1.39919 −0.699593 0.714541i \(-0.746635\pi\)
−0.699593 + 0.714541i \(0.746635\pi\)
\(132\) −0.192350 −0.0167419
\(133\) 0 0
\(134\) −11.0508 −0.954644
\(135\) 1.32621 0.114142
\(136\) −3.38132 −0.289946
\(137\) 0.927579 0.0792484 0.0396242 0.999215i \(-0.487384\pi\)
0.0396242 + 0.999215i \(0.487384\pi\)
\(138\) 1.36549 0.116238
\(139\) −4.22569 −0.358418 −0.179209 0.983811i \(-0.557354\pi\)
−0.179209 + 0.983811i \(0.557354\pi\)
\(140\) 0 0
\(141\) −1.73650 −0.146239
\(142\) −7.08658 −0.594693
\(143\) −4.49686 −0.376046
\(144\) −3.71076 −0.309230
\(145\) −9.88664 −0.821040
\(146\) −15.5237 −1.28475
\(147\) 0 0
\(148\) 1.05628 0.0868261
\(149\) −16.4464 −1.34734 −0.673670 0.739032i \(-0.735283\pi\)
−0.673670 + 0.739032i \(0.735283\pi\)
\(150\) 4.42576 0.361362
\(151\) −8.18838 −0.666361 −0.333180 0.942863i \(-0.608122\pi\)
−0.333180 + 0.942863i \(0.608122\pi\)
\(152\) −25.0153 −2.02901
\(153\) 1.15960 0.0937484
\(154\) 0 0
\(155\) −2.47527 −0.198818
\(156\) 0.428914 0.0343406
\(157\) 22.5151 1.79690 0.898452 0.439072i \(-0.144693\pi\)
0.898452 + 0.439072i \(0.144693\pi\)
\(158\) −16.0260 −1.27496
\(159\) 4.14898 0.329036
\(160\) −1.01438 −0.0801935
\(161\) 0 0
\(162\) 1.36549 0.107283
\(163\) −1.46151 −0.114474 −0.0572370 0.998361i \(-0.518229\pi\)
−0.0572370 + 0.998361i \(0.518229\pi\)
\(164\) 1.52052 0.118733
\(165\) −1.88333 −0.146617
\(166\) 7.79099 0.604698
\(167\) 2.76301 0.213808 0.106904 0.994269i \(-0.465906\pi\)
0.106904 + 0.994269i \(0.465906\pi\)
\(168\) 0 0
\(169\) −2.97260 −0.228661
\(170\) −2.09995 −0.161059
\(171\) 8.57886 0.656042
\(172\) 1.60479 0.122364
\(173\) 18.8661 1.43436 0.717182 0.696886i \(-0.245432\pi\)
0.717182 + 0.696886i \(0.245432\pi\)
\(174\) −10.1794 −0.771700
\(175\) 0 0
\(176\) 5.26959 0.397210
\(177\) −12.6871 −0.953623
\(178\) −8.67782 −0.650430
\(179\) −20.7182 −1.54855 −0.774276 0.632847i \(-0.781886\pi\)
−0.774276 + 0.632847i \(0.781886\pi\)
\(180\) 0.179634 0.0133891
\(181\) 0.686865 0.0510543 0.0255271 0.999674i \(-0.491874\pi\)
0.0255271 + 0.999674i \(0.491874\pi\)
\(182\) 0 0
\(183\) 13.8148 1.02122
\(184\) 2.91593 0.214965
\(185\) 10.3423 0.760381
\(186\) −2.54857 −0.186870
\(187\) −1.64674 −0.120421
\(188\) −0.235207 −0.0171542
\(189\) 0 0
\(190\) −15.5356 −1.12707
\(191\) 1.09300 0.0790868 0.0395434 0.999218i \(-0.487410\pi\)
0.0395434 + 0.999218i \(0.487410\pi\)
\(192\) −8.46593 −0.610976
\(193\) 7.41394 0.533667 0.266833 0.963743i \(-0.414023\pi\)
0.266833 + 0.963743i \(0.414023\pi\)
\(194\) 0.557658 0.0400375
\(195\) 4.19959 0.300739
\(196\) 0 0
\(197\) 3.16080 0.225198 0.112599 0.993641i \(-0.464082\pi\)
0.112599 + 0.993641i \(0.464082\pi\)
\(198\) −1.93911 −0.137806
\(199\) −1.80928 −0.128257 −0.0641283 0.997942i \(-0.520427\pi\)
−0.0641283 + 0.997942i \(0.520427\pi\)
\(200\) 9.45099 0.668286
\(201\) 8.09295 0.570833
\(202\) −0.988375 −0.0695419
\(203\) 0 0
\(204\) 0.157067 0.0109969
\(205\) 14.8877 1.03980
\(206\) −19.4908 −1.35799
\(207\) −1.00000 −0.0695048
\(208\) −11.7505 −0.814751
\(209\) −12.1827 −0.842696
\(210\) 0 0
\(211\) −22.5516 −1.55252 −0.776260 0.630413i \(-0.782885\pi\)
−0.776260 + 0.630413i \(0.782885\pi\)
\(212\) 0.561976 0.0385967
\(213\) 5.18979 0.355599
\(214\) 19.1113 1.30642
\(215\) 15.7128 1.07160
\(216\) 2.91593 0.198404
\(217\) 0 0
\(218\) 3.97282 0.269073
\(219\) 11.3686 0.768218
\(220\) −0.255096 −0.0171986
\(221\) 3.67201 0.247006
\(222\) 10.6486 0.714686
\(223\) −14.4484 −0.967535 −0.483767 0.875197i \(-0.660732\pi\)
−0.483767 + 0.875197i \(0.660732\pi\)
\(224\) 0 0
\(225\) −3.24116 −0.216078
\(226\) 25.4832 1.69512
\(227\) −3.27881 −0.217622 −0.108811 0.994062i \(-0.534704\pi\)
−0.108811 + 0.994062i \(0.534704\pi\)
\(228\) 1.16200 0.0769553
\(229\) −0.263933 −0.0174412 −0.00872059 0.999962i \(-0.502776\pi\)
−0.00872059 + 0.999962i \(0.502776\pi\)
\(230\) 1.81092 0.119409
\(231\) 0 0
\(232\) −21.7376 −1.42715
\(233\) 17.8509 1.16945 0.584727 0.811230i \(-0.301202\pi\)
0.584727 + 0.811230i \(0.301202\pi\)
\(234\) 4.32396 0.282666
\(235\) −2.30296 −0.150229
\(236\) −1.71846 −0.111862
\(237\) 11.7365 0.762366
\(238\) 0 0
\(239\) 1.68846 0.109217 0.0546086 0.998508i \(-0.482609\pi\)
0.0546086 + 0.998508i \(0.482609\pi\)
\(240\) −4.92124 −0.317665
\(241\) 19.0420 1.22660 0.613300 0.789850i \(-0.289842\pi\)
0.613300 + 0.789850i \(0.289842\pi\)
\(242\) −12.2666 −0.788530
\(243\) −1.00000 −0.0641500
\(244\) 1.87120 0.119791
\(245\) 0 0
\(246\) 15.3286 0.977316
\(247\) 27.1659 1.72852
\(248\) −5.44234 −0.345589
\(249\) −5.70566 −0.361581
\(250\) 14.9241 0.943883
\(251\) −20.3390 −1.28379 −0.641895 0.766793i \(-0.721851\pi\)
−0.641895 + 0.766793i \(0.721851\pi\)
\(252\) 0 0
\(253\) 1.42009 0.0892801
\(254\) 1.45603 0.0913594
\(255\) 1.53788 0.0963057
\(256\) −3.23553 −0.202221
\(257\) −9.62955 −0.600675 −0.300337 0.953833i \(-0.597099\pi\)
−0.300337 + 0.953833i \(0.597099\pi\)
\(258\) 16.1781 1.00721
\(259\) 0 0
\(260\) 0.568831 0.0352774
\(261\) 7.45480 0.461440
\(262\) −21.8675 −1.35098
\(263\) −4.85507 −0.299377 −0.149688 0.988733i \(-0.547827\pi\)
−0.149688 + 0.988733i \(0.547827\pi\)
\(264\) −4.14087 −0.254853
\(265\) 5.50243 0.338011
\(266\) 0 0
\(267\) 6.35511 0.388927
\(268\) 1.09618 0.0669601
\(269\) 12.7386 0.776684 0.388342 0.921515i \(-0.373048\pi\)
0.388342 + 0.921515i \(0.373048\pi\)
\(270\) 1.81092 0.110209
\(271\) −22.0021 −1.33653 −0.668265 0.743923i \(-0.732963\pi\)
−0.668265 + 0.743923i \(0.732963\pi\)
\(272\) −4.30301 −0.260908
\(273\) 0 0
\(274\) 1.26660 0.0765179
\(275\) 4.60273 0.277555
\(276\) −0.135449 −0.00815308
\(277\) 0.165444 0.00994059 0.00497030 0.999988i \(-0.498418\pi\)
0.00497030 + 0.999988i \(0.498418\pi\)
\(278\) −5.77012 −0.346069
\(279\) 1.86642 0.111740
\(280\) 0 0
\(281\) −20.9797 −1.25155 −0.625773 0.780006i \(-0.715216\pi\)
−0.625773 + 0.780006i \(0.715216\pi\)
\(282\) −2.37116 −0.141201
\(283\) −0.814822 −0.0484362 −0.0242181 0.999707i \(-0.507710\pi\)
−0.0242181 + 0.999707i \(0.507710\pi\)
\(284\) 0.702953 0.0417126
\(285\) 11.3774 0.673938
\(286\) −6.14039 −0.363089
\(287\) 0 0
\(288\) 0.764868 0.0450703
\(289\) −15.6553 −0.920901
\(290\) −13.5001 −0.792751
\(291\) −0.408396 −0.0239406
\(292\) 1.53987 0.0901139
\(293\) −12.6589 −0.739543 −0.369771 0.929123i \(-0.620564\pi\)
−0.369771 + 0.929123i \(0.620564\pi\)
\(294\) 0 0
\(295\) −16.8258 −0.979637
\(296\) 22.7395 1.32171
\(297\) 1.42009 0.0824018
\(298\) −22.4573 −1.30092
\(299\) −3.16661 −0.183130
\(300\) −0.439013 −0.0253464
\(301\) 0 0
\(302\) −11.1811 −0.643401
\(303\) 0.723827 0.0415828
\(304\) −31.8340 −1.82581
\(305\) 18.3213 1.04907
\(306\) 1.58342 0.0905182
\(307\) −3.03467 −0.173198 −0.0865989 0.996243i \(-0.527600\pi\)
−0.0865989 + 0.996243i \(0.527600\pi\)
\(308\) 0 0
\(309\) 14.2739 0.812014
\(310\) −3.37994 −0.191968
\(311\) 27.4766 1.55806 0.779029 0.626988i \(-0.215713\pi\)
0.779029 + 0.626988i \(0.215713\pi\)
\(312\) 9.23359 0.522749
\(313\) 1.34203 0.0758558 0.0379279 0.999280i \(-0.487924\pi\)
0.0379279 + 0.999280i \(0.487924\pi\)
\(314\) 30.7441 1.73499
\(315\) 0 0
\(316\) 1.58970 0.0894274
\(317\) 0.508616 0.0285667 0.0142834 0.999898i \(-0.495453\pi\)
0.0142834 + 0.999898i \(0.495453\pi\)
\(318\) 5.66537 0.317698
\(319\) −10.5865 −0.592728
\(320\) −11.2276 −0.627642
\(321\) −13.9959 −0.781178
\(322\) 0 0
\(323\) 9.94808 0.553526
\(324\) −0.135449 −0.00752496
\(325\) −10.2635 −0.569316
\(326\) −1.99566 −0.110530
\(327\) −2.90945 −0.160893
\(328\) 32.7335 1.80740
\(329\) 0 0
\(330\) −2.57167 −0.141566
\(331\) 6.19033 0.340252 0.170126 0.985422i \(-0.445583\pi\)
0.170126 + 0.985422i \(0.445583\pi\)
\(332\) −0.772827 −0.0424144
\(333\) −7.79838 −0.427349
\(334\) 3.77284 0.206441
\(335\) 10.7330 0.586404
\(336\) 0 0
\(337\) −5.07603 −0.276509 −0.138254 0.990397i \(-0.544149\pi\)
−0.138254 + 0.990397i \(0.544149\pi\)
\(338\) −4.05904 −0.220783
\(339\) −18.6624 −1.01360
\(340\) 0.208305 0.0112969
\(341\) −2.65048 −0.143531
\(342\) 11.7143 0.633437
\(343\) 0 0
\(344\) 34.5476 1.86268
\(345\) −1.32621 −0.0714008
\(346\) 25.7614 1.38494
\(347\) −25.4952 −1.36865 −0.684326 0.729176i \(-0.739903\pi\)
−0.684326 + 0.729176i \(0.739903\pi\)
\(348\) 1.00975 0.0541281
\(349\) −11.0135 −0.589541 −0.294771 0.955568i \(-0.595243\pi\)
−0.294771 + 0.955568i \(0.595243\pi\)
\(350\) 0 0
\(351\) −3.16661 −0.169021
\(352\) −1.08618 −0.0578935
\(353\) −6.64781 −0.353827 −0.176914 0.984226i \(-0.556611\pi\)
−0.176914 + 0.984226i \(0.556611\pi\)
\(354\) −17.3241 −0.920765
\(355\) 6.88276 0.365299
\(356\) 0.860795 0.0456221
\(357\) 0 0
\(358\) −28.2904 −1.49520
\(359\) −7.85473 −0.414557 −0.207278 0.978282i \(-0.566461\pi\)
−0.207278 + 0.978282i \(0.566461\pi\)
\(360\) 3.86713 0.203816
\(361\) 54.5968 2.87352
\(362\) 0.937905 0.0492952
\(363\) 8.98335 0.471504
\(364\) 0 0
\(365\) 15.0772 0.789174
\(366\) 18.8639 0.986029
\(367\) 1.87237 0.0977369 0.0488684 0.998805i \(-0.484438\pi\)
0.0488684 + 0.998805i \(0.484438\pi\)
\(368\) 3.71076 0.193436
\(369\) −11.2258 −0.584389
\(370\) 14.1223 0.734182
\(371\) 0 0
\(372\) 0.252805 0.0131073
\(373\) −23.3775 −1.21044 −0.605220 0.796058i \(-0.706915\pi\)
−0.605220 + 0.796058i \(0.706915\pi\)
\(374\) −2.24860 −0.116272
\(375\) −10.9295 −0.564398
\(376\) −5.06350 −0.261130
\(377\) 23.6064 1.21579
\(378\) 0 0
\(379\) 20.9257 1.07488 0.537441 0.843301i \(-0.319391\pi\)
0.537441 + 0.843301i \(0.319391\pi\)
\(380\) 1.54106 0.0790546
\(381\) −1.06631 −0.0546287
\(382\) 1.49248 0.0763618
\(383\) −26.8189 −1.37038 −0.685191 0.728363i \(-0.740281\pi\)
−0.685191 + 0.728363i \(0.740281\pi\)
\(384\) −10.0304 −0.511860
\(385\) 0 0
\(386\) 10.1236 0.515279
\(387\) −11.8479 −0.602262
\(388\) −0.0553169 −0.00280829
\(389\) −25.6338 −1.29968 −0.649842 0.760070i \(-0.725165\pi\)
−0.649842 + 0.760070i \(0.725165\pi\)
\(390\) 5.73448 0.290377
\(391\) −1.15960 −0.0586437
\(392\) 0 0
\(393\) 16.0144 0.807821
\(394\) 4.31603 0.217439
\(395\) 15.5650 0.783162
\(396\) 0.192350 0.00966593
\(397\) 10.1904 0.511444 0.255722 0.966750i \(-0.417687\pi\)
0.255722 + 0.966750i \(0.417687\pi\)
\(398\) −2.47055 −0.123837
\(399\) 0 0
\(400\) 12.0272 0.601358
\(401\) 23.7815 1.18759 0.593796 0.804615i \(-0.297629\pi\)
0.593796 + 0.804615i \(0.297629\pi\)
\(402\) 11.0508 0.551164
\(403\) 5.91022 0.294409
\(404\) 0.0980418 0.00487776
\(405\) −1.32621 −0.0659000
\(406\) 0 0
\(407\) 11.0744 0.548937
\(408\) 3.38132 0.167400
\(409\) −14.4367 −0.713849 −0.356925 0.934133i \(-0.616175\pi\)
−0.356925 + 0.934133i \(0.616175\pi\)
\(410\) 20.3290 1.00398
\(411\) −0.927579 −0.0457541
\(412\) 1.93339 0.0952513
\(413\) 0 0
\(414\) −1.36549 −0.0671100
\(415\) −7.56691 −0.371445
\(416\) 2.42204 0.118750
\(417\) 4.22569 0.206933
\(418\) −16.6353 −0.813661
\(419\) 18.5772 0.907558 0.453779 0.891114i \(-0.350076\pi\)
0.453779 + 0.891114i \(0.350076\pi\)
\(420\) 0 0
\(421\) −7.98595 −0.389211 −0.194606 0.980882i \(-0.562343\pi\)
−0.194606 + 0.980882i \(0.562343\pi\)
\(422\) −30.7939 −1.49903
\(423\) 1.73650 0.0844314
\(424\) 12.0981 0.587537
\(425\) −3.75847 −0.182312
\(426\) 7.08658 0.343346
\(427\) 0 0
\(428\) −1.89574 −0.0916340
\(429\) 4.49686 0.217110
\(430\) 21.4556 1.03468
\(431\) −18.0343 −0.868680 −0.434340 0.900749i \(-0.643018\pi\)
−0.434340 + 0.900749i \(0.643018\pi\)
\(432\) 3.71076 0.178534
\(433\) −13.6502 −0.655989 −0.327995 0.944680i \(-0.606373\pi\)
−0.327995 + 0.944680i \(0.606373\pi\)
\(434\) 0 0
\(435\) 9.88664 0.474028
\(436\) −0.394083 −0.0188732
\(437\) −8.57886 −0.410383
\(438\) 15.5237 0.741749
\(439\) 10.8654 0.518578 0.259289 0.965800i \(-0.416512\pi\)
0.259289 + 0.965800i \(0.416512\pi\)
\(440\) −5.49166 −0.261805
\(441\) 0 0
\(442\) 5.01408 0.238495
\(443\) 15.1123 0.718006 0.359003 0.933336i \(-0.383117\pi\)
0.359003 + 0.933336i \(0.383117\pi\)
\(444\) −1.05628 −0.0501291
\(445\) 8.42822 0.399536
\(446\) −19.7290 −0.934197
\(447\) 16.4464 0.777887
\(448\) 0 0
\(449\) 6.14432 0.289968 0.144984 0.989434i \(-0.453687\pi\)
0.144984 + 0.989434i \(0.453687\pi\)
\(450\) −4.42576 −0.208632
\(451\) 15.9415 0.750658
\(452\) −2.52780 −0.118898
\(453\) 8.18838 0.384724
\(454\) −4.47717 −0.210124
\(455\) 0 0
\(456\) 25.0153 1.17145
\(457\) 29.5848 1.38392 0.691960 0.721936i \(-0.256747\pi\)
0.691960 + 0.721936i \(0.256747\pi\)
\(458\) −0.360396 −0.0168402
\(459\) −1.15960 −0.0541257
\(460\) −0.179634 −0.00837549
\(461\) −34.7997 −1.62078 −0.810391 0.585890i \(-0.800745\pi\)
−0.810391 + 0.585890i \(0.800745\pi\)
\(462\) 0 0
\(463\) 15.8969 0.738793 0.369397 0.929272i \(-0.379564\pi\)
0.369397 + 0.929272i \(0.379564\pi\)
\(464\) −27.6629 −1.28422
\(465\) 2.47527 0.114788
\(466\) 24.3752 1.12916
\(467\) 7.28616 0.337163 0.168582 0.985688i \(-0.446081\pi\)
0.168582 + 0.985688i \(0.446081\pi\)
\(468\) −0.428914 −0.0198266
\(469\) 0 0
\(470\) −3.14466 −0.145052
\(471\) −22.5151 −1.03744
\(472\) −36.9947 −1.70282
\(473\) 16.8250 0.773615
\(474\) 16.0260 0.736098
\(475\) −27.8055 −1.27580
\(476\) 0 0
\(477\) −4.14898 −0.189969
\(478\) 2.30556 0.105454
\(479\) −18.6156 −0.850569 −0.425285 0.905060i \(-0.639826\pi\)
−0.425285 + 0.905060i \(0.639826\pi\)
\(480\) 1.01438 0.0462997
\(481\) −24.6944 −1.12597
\(482\) 26.0015 1.18434
\(483\) 0 0
\(484\) 1.21679 0.0553086
\(485\) −0.541619 −0.0245936
\(486\) −1.36549 −0.0619397
\(487\) 9.62891 0.436328 0.218164 0.975912i \(-0.429993\pi\)
0.218164 + 0.975912i \(0.429993\pi\)
\(488\) 40.2828 1.82352
\(489\) 1.46151 0.0660915
\(490\) 0 0
\(491\) −4.61117 −0.208099 −0.104050 0.994572i \(-0.533180\pi\)
−0.104050 + 0.994572i \(0.533180\pi\)
\(492\) −1.52052 −0.0685503
\(493\) 8.64461 0.389334
\(494\) 37.0946 1.66897
\(495\) 1.88333 0.0846496
\(496\) −6.92583 −0.310979
\(497\) 0 0
\(498\) −7.79099 −0.349123
\(499\) 12.8567 0.575543 0.287771 0.957699i \(-0.407086\pi\)
0.287771 + 0.957699i \(0.407086\pi\)
\(500\) −1.48040 −0.0662053
\(501\) −2.76301 −0.123442
\(502\) −27.7727 −1.23955
\(503\) 15.4383 0.688360 0.344180 0.938904i \(-0.388157\pi\)
0.344180 + 0.938904i \(0.388157\pi\)
\(504\) 0 0
\(505\) 0.959948 0.0427171
\(506\) 1.93911 0.0862038
\(507\) 2.97260 0.132018
\(508\) −0.144431 −0.00640807
\(509\) 11.4032 0.505439 0.252720 0.967540i \(-0.418675\pi\)
0.252720 + 0.967540i \(0.418675\pi\)
\(510\) 2.09995 0.0929874
\(511\) 0 0
\(512\) −24.4788 −1.08182
\(513\) −8.57886 −0.378766
\(514\) −13.1490 −0.579978
\(515\) 18.9302 0.834165
\(516\) −1.60479 −0.0706468
\(517\) −2.46598 −0.108454
\(518\) 0 0
\(519\) −18.8661 −0.828131
\(520\) 12.2457 0.537009
\(521\) 14.9757 0.656097 0.328049 0.944661i \(-0.393609\pi\)
0.328049 + 0.944661i \(0.393609\pi\)
\(522\) 10.1794 0.445541
\(523\) −11.4151 −0.499147 −0.249573 0.968356i \(-0.580290\pi\)
−0.249573 + 0.968356i \(0.580290\pi\)
\(524\) 2.16914 0.0947593
\(525\) 0 0
\(526\) −6.62953 −0.289061
\(527\) 2.16431 0.0942787
\(528\) −5.26959 −0.229330
\(529\) 1.00000 0.0434783
\(530\) 7.51348 0.326365
\(531\) 12.6871 0.550575
\(532\) 0 0
\(533\) −35.5476 −1.53974
\(534\) 8.67782 0.375526
\(535\) −18.5616 −0.802487
\(536\) 23.5984 1.01930
\(537\) 20.7182 0.894057
\(538\) 17.3943 0.749923
\(539\) 0 0
\(540\) −0.179634 −0.00773023
\(541\) 18.1297 0.779455 0.389727 0.920930i \(-0.372569\pi\)
0.389727 + 0.920930i \(0.372569\pi\)
\(542\) −30.0435 −1.29048
\(543\) −0.686865 −0.0294762
\(544\) 0.886943 0.0380274
\(545\) −3.85855 −0.165282
\(546\) 0 0
\(547\) −13.0632 −0.558543 −0.279271 0.960212i \(-0.590093\pi\)
−0.279271 + 0.960212i \(0.590093\pi\)
\(548\) −0.125640 −0.00536707
\(549\) −13.8148 −0.589599
\(550\) 6.28497 0.267992
\(551\) 63.9537 2.72452
\(552\) −2.91593 −0.124110
\(553\) 0 0
\(554\) 0.225912 0.00959808
\(555\) −10.3423 −0.439006
\(556\) 0.572366 0.0242737
\(557\) −26.8913 −1.13942 −0.569711 0.821845i \(-0.692945\pi\)
−0.569711 + 0.821845i \(0.692945\pi\)
\(558\) 2.54857 0.107890
\(559\) −37.5176 −1.58683
\(560\) 0 0
\(561\) 1.64674 0.0695253
\(562\) −28.6475 −1.20842
\(563\) 9.77405 0.411927 0.205963 0.978560i \(-0.433967\pi\)
0.205963 + 0.978560i \(0.433967\pi\)
\(564\) 0.235207 0.00990401
\(565\) −24.7502 −1.04125
\(566\) −1.11263 −0.0467673
\(567\) 0 0
\(568\) 15.1330 0.634968
\(569\) −43.9890 −1.84411 −0.922057 0.387054i \(-0.873493\pi\)
−0.922057 + 0.387054i \(0.873493\pi\)
\(570\) 15.5356 0.650717
\(571\) 6.19921 0.259429 0.129714 0.991551i \(-0.458594\pi\)
0.129714 + 0.991551i \(0.458594\pi\)
\(572\) 0.609096 0.0254676
\(573\) −1.09300 −0.0456608
\(574\) 0 0
\(575\) 3.24116 0.135166
\(576\) 8.46593 0.352747
\(577\) 14.5603 0.606154 0.303077 0.952966i \(-0.401986\pi\)
0.303077 + 0.952966i \(0.401986\pi\)
\(578\) −21.3771 −0.889171
\(579\) −7.41394 −0.308113
\(580\) 1.33914 0.0556046
\(581\) 0 0
\(582\) −0.557658 −0.0231157
\(583\) 5.89191 0.244018
\(584\) 33.1500 1.37176
\(585\) −4.19959 −0.173632
\(586\) −17.2856 −0.714061
\(587\) −16.3784 −0.676009 −0.338004 0.941145i \(-0.609752\pi\)
−0.338004 + 0.941145i \(0.609752\pi\)
\(588\) 0 0
\(589\) 16.0118 0.659753
\(590\) −22.9754 −0.945883
\(591\) −3.16080 −0.130018
\(592\) 28.9379 1.18934
\(593\) 27.2314 1.11826 0.559129 0.829081i \(-0.311136\pi\)
0.559129 + 0.829081i \(0.311136\pi\)
\(594\) 1.93911 0.0795625
\(595\) 0 0
\(596\) 2.22765 0.0912481
\(597\) 1.80928 0.0740490
\(598\) −4.32396 −0.176820
\(599\) 40.5288 1.65596 0.827981 0.560756i \(-0.189489\pi\)
0.827981 + 0.560756i \(0.189489\pi\)
\(600\) −9.45099 −0.385835
\(601\) 8.69739 0.354774 0.177387 0.984141i \(-0.443236\pi\)
0.177387 + 0.984141i \(0.443236\pi\)
\(602\) 0 0
\(603\) −8.09295 −0.329570
\(604\) 1.10911 0.0451290
\(605\) 11.9138 0.484366
\(606\) 0.988375 0.0401500
\(607\) 20.0430 0.813522 0.406761 0.913535i \(-0.366658\pi\)
0.406761 + 0.913535i \(0.366658\pi\)
\(608\) 6.56169 0.266112
\(609\) 0 0
\(610\) 25.0175 1.01293
\(611\) 5.49881 0.222458
\(612\) −0.157067 −0.00634907
\(613\) −20.0601 −0.810221 −0.405111 0.914268i \(-0.632767\pi\)
−0.405111 + 0.914268i \(0.632767\pi\)
\(614\) −4.14380 −0.167230
\(615\) −14.8877 −0.600331
\(616\) 0 0
\(617\) 35.9585 1.44764 0.723818 0.689991i \(-0.242385\pi\)
0.723818 + 0.689991i \(0.242385\pi\)
\(618\) 19.4908 0.784036
\(619\) −14.8199 −0.595663 −0.297832 0.954618i \(-0.596263\pi\)
−0.297832 + 0.954618i \(0.596263\pi\)
\(620\) 0.335273 0.0134649
\(621\) 1.00000 0.0401286
\(622\) 37.5190 1.50437
\(623\) 0 0
\(624\) 11.7505 0.470397
\(625\) 1.71096 0.0684386
\(626\) 1.83252 0.0732421
\(627\) 12.1827 0.486531
\(628\) −3.04966 −0.121695
\(629\) −9.04303 −0.360569
\(630\) 0 0
\(631\) −25.6933 −1.02284 −0.511418 0.859332i \(-0.670880\pi\)
−0.511418 + 0.859332i \(0.670880\pi\)
\(632\) 34.2227 1.36130
\(633\) 22.5516 0.896347
\(634\) 0.694508 0.0275825
\(635\) −1.41415 −0.0561189
\(636\) −0.561976 −0.0222838
\(637\) 0 0
\(638\) −14.4557 −0.572305
\(639\) −5.18979 −0.205305
\(640\) −13.3024 −0.525823
\(641\) 19.7597 0.780460 0.390230 0.920717i \(-0.372395\pi\)
0.390230 + 0.920717i \(0.372395\pi\)
\(642\) −19.1113 −0.754261
\(643\) −6.27845 −0.247598 −0.123799 0.992307i \(-0.539508\pi\)
−0.123799 + 0.992307i \(0.539508\pi\)
\(644\) 0 0
\(645\) −15.7128 −0.618691
\(646\) 13.5840 0.534454
\(647\) −46.9083 −1.84416 −0.922078 0.387004i \(-0.873510\pi\)
−0.922078 + 0.387004i \(0.873510\pi\)
\(648\) −2.91593 −0.114548
\(649\) −18.0168 −0.707222
\(650\) −14.0147 −0.549700
\(651\) 0 0
\(652\) 0.197960 0.00775270
\(653\) 29.3176 1.14729 0.573643 0.819105i \(-0.305530\pi\)
0.573643 + 0.819105i \(0.305530\pi\)
\(654\) −3.97282 −0.155349
\(655\) 21.2385 0.829857
\(656\) 41.6560 1.62639
\(657\) −11.3686 −0.443531
\(658\) 0 0
\(659\) −8.04283 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(660\) 0.255096 0.00992960
\(661\) 12.8774 0.500872 0.250436 0.968133i \(-0.419426\pi\)
0.250436 + 0.968133i \(0.419426\pi\)
\(662\) 8.45281 0.328528
\(663\) −3.67201 −0.142609
\(664\) −16.6373 −0.645651
\(665\) 0 0
\(666\) −10.6486 −0.412624
\(667\) −7.45480 −0.288651
\(668\) −0.374247 −0.0144800
\(669\) 14.4484 0.558606
\(670\) 14.6557 0.566199
\(671\) 19.6181 0.757350
\(672\) 0 0
\(673\) 42.5924 1.64182 0.820909 0.571059i \(-0.193467\pi\)
0.820909 + 0.571059i \(0.193467\pi\)
\(674\) −6.93124 −0.266981
\(675\) 3.24116 0.124752
\(676\) 0.402636 0.0154860
\(677\) 11.6245 0.446766 0.223383 0.974731i \(-0.428290\pi\)
0.223383 + 0.974731i \(0.428290\pi\)
\(678\) −25.4832 −0.978676
\(679\) 0 0
\(680\) 4.48434 0.171967
\(681\) 3.27881 0.125644
\(682\) −3.61919 −0.138586
\(683\) 8.84158 0.338314 0.169157 0.985589i \(-0.445896\pi\)
0.169157 + 0.985589i \(0.445896\pi\)
\(684\) −1.16200 −0.0444302
\(685\) −1.23017 −0.0470022
\(686\) 0 0
\(687\) 0.263933 0.0100697
\(688\) 43.9646 1.67614
\(689\) −13.1382 −0.500525
\(690\) −1.81092 −0.0689406
\(691\) 25.2677 0.961230 0.480615 0.876932i \(-0.340413\pi\)
0.480615 + 0.876932i \(0.340413\pi\)
\(692\) −2.55540 −0.0971417
\(693\) 0 0
\(694\) −34.8133 −1.32149
\(695\) 5.60416 0.212578
\(696\) 21.7376 0.823963
\(697\) −13.0174 −0.493070
\(698\) −15.0388 −0.569228
\(699\) −17.8509 −0.675184
\(700\) 0 0
\(701\) 17.9512 0.678009 0.339005 0.940785i \(-0.389910\pi\)
0.339005 + 0.940785i \(0.389910\pi\)
\(702\) −4.32396 −0.163197
\(703\) −66.9012 −2.52323
\(704\) −12.0223 −0.453109
\(705\) 2.30296 0.0867346
\(706\) −9.07748 −0.341636
\(707\) 0 0
\(708\) 1.71846 0.0645838
\(709\) −37.4627 −1.40694 −0.703470 0.710725i \(-0.748367\pi\)
−0.703470 + 0.710725i \(0.748367\pi\)
\(710\) 9.39831 0.352712
\(711\) −11.7365 −0.440152
\(712\) 18.5310 0.694480
\(713\) −1.86642 −0.0698980
\(714\) 0 0
\(715\) 5.96378 0.223033
\(716\) 2.80627 0.104875
\(717\) −1.68846 −0.0630566
\(718\) −10.7255 −0.400273
\(719\) 31.6182 1.17916 0.589580 0.807710i \(-0.299293\pi\)
0.589580 + 0.807710i \(0.299293\pi\)
\(720\) 4.92124 0.183404
\(721\) 0 0
\(722\) 74.5512 2.77451
\(723\) −19.0420 −0.708178
\(724\) −0.0930354 −0.00345763
\(725\) −24.1622 −0.897362
\(726\) 12.2666 0.455258
\(727\) 0.00894509 0.000331755 0 0.000165878 1.00000i \(-0.499947\pi\)
0.000165878 1.00000i \(0.499947\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 20.5876 0.761983
\(731\) −13.7389 −0.508150
\(732\) −1.87120 −0.0691615
\(733\) 25.7110 0.949656 0.474828 0.880079i \(-0.342510\pi\)
0.474828 + 0.880079i \(0.342510\pi\)
\(734\) 2.55669 0.0943693
\(735\) 0 0
\(736\) −0.764868 −0.0281934
\(737\) 11.4927 0.423339
\(738\) −15.3286 −0.564254
\(739\) −16.7658 −0.616739 −0.308370 0.951267i \(-0.599783\pi\)
−0.308370 + 0.951267i \(0.599783\pi\)
\(740\) −1.40086 −0.0514965
\(741\) −27.1659 −0.997964
\(742\) 0 0
\(743\) 13.6484 0.500712 0.250356 0.968154i \(-0.419452\pi\)
0.250356 + 0.968154i \(0.419452\pi\)
\(744\) 5.44234 0.199526
\(745\) 21.8114 0.799107
\(746\) −31.9216 −1.16873
\(747\) 5.70566 0.208759
\(748\) 0.223049 0.00815549
\(749\) 0 0
\(750\) −14.9241 −0.544951
\(751\) −19.5144 −0.712092 −0.356046 0.934468i \(-0.615875\pi\)
−0.356046 + 0.934468i \(0.615875\pi\)
\(752\) −6.44372 −0.234978
\(753\) 20.3390 0.741196
\(754\) 32.2342 1.17390
\(755\) 10.8595 0.395218
\(756\) 0 0
\(757\) 11.1059 0.403652 0.201826 0.979421i \(-0.435312\pi\)
0.201826 + 0.979421i \(0.435312\pi\)
\(758\) 28.5738 1.03785
\(759\) −1.42009 −0.0515459
\(760\) 33.1756 1.20340
\(761\) 29.3210 1.06288 0.531442 0.847095i \(-0.321650\pi\)
0.531442 + 0.847095i \(0.321650\pi\)
\(762\) −1.45603 −0.0527464
\(763\) 0 0
\(764\) −0.148046 −0.00535613
\(765\) −1.53788 −0.0556021
\(766\) −36.6208 −1.32316
\(767\) 40.1752 1.45064
\(768\) 3.23553 0.116752
\(769\) −9.29965 −0.335354 −0.167677 0.985842i \(-0.553627\pi\)
−0.167677 + 0.985842i \(0.553627\pi\)
\(770\) 0 0
\(771\) 9.62955 0.346800
\(772\) −1.00421 −0.0361424
\(773\) 33.5436 1.20648 0.603240 0.797559i \(-0.293876\pi\)
0.603240 + 0.797559i \(0.293876\pi\)
\(774\) −16.1781 −0.581511
\(775\) −6.04937 −0.217300
\(776\) −1.19085 −0.0427491
\(777\) 0 0
\(778\) −35.0025 −1.25490
\(779\) −96.3042 −3.45046
\(780\) −0.568831 −0.0203674
\(781\) 7.36995 0.263718
\(782\) −1.58342 −0.0566231
\(783\) −7.45480 −0.266413
\(784\) 0 0
\(785\) −29.8598 −1.06574
\(786\) 21.8675 0.779986
\(787\) −30.5190 −1.08789 −0.543943 0.839122i \(-0.683069\pi\)
−0.543943 + 0.839122i \(0.683069\pi\)
\(788\) −0.428128 −0.0152514
\(789\) 4.85507 0.172845
\(790\) 21.2538 0.756177
\(791\) 0 0
\(792\) 4.14087 0.147139
\(793\) −43.7459 −1.55346
\(794\) 13.9149 0.493821
\(795\) −5.50243 −0.195151
\(796\) 0.245066 0.00868613
\(797\) −41.6491 −1.47529 −0.737643 0.675191i \(-0.764061\pi\)
−0.737643 + 0.675191i \(0.764061\pi\)
\(798\) 0 0
\(799\) 2.01365 0.0712378
\(800\) −2.47906 −0.0876481
\(801\) −6.35511 −0.224547
\(802\) 32.4733 1.14667
\(803\) 16.1444 0.569723
\(804\) −1.09618 −0.0386594
\(805\) 0 0
\(806\) 8.07032 0.284265
\(807\) −12.7386 −0.448419
\(808\) 2.11063 0.0742515
\(809\) −31.5420 −1.10896 −0.554479 0.832198i \(-0.687082\pi\)
−0.554479 + 0.832198i \(0.687082\pi\)
\(810\) −1.81092 −0.0636293
\(811\) −22.0480 −0.774211 −0.387106 0.922035i \(-0.626525\pi\)
−0.387106 + 0.922035i \(0.626525\pi\)
\(812\) 0 0
\(813\) 22.0021 0.771646
\(814\) 15.1219 0.530022
\(815\) 1.93826 0.0678944
\(816\) 4.30301 0.150635
\(817\) −101.641 −3.55598
\(818\) −19.7131 −0.689253
\(819\) 0 0
\(820\) −2.01653 −0.0704203
\(821\) 52.0995 1.81828 0.909142 0.416487i \(-0.136739\pi\)
0.909142 + 0.416487i \(0.136739\pi\)
\(822\) −1.26660 −0.0441776
\(823\) −38.4976 −1.34194 −0.670970 0.741484i \(-0.734122\pi\)
−0.670970 + 0.741484i \(0.734122\pi\)
\(824\) 41.6217 1.44996
\(825\) −4.60273 −0.160247
\(826\) 0 0
\(827\) −36.5676 −1.27158 −0.635790 0.771862i \(-0.719325\pi\)
−0.635790 + 0.771862i \(0.719325\pi\)
\(828\) 0.135449 0.00470719
\(829\) 27.5191 0.955777 0.477888 0.878421i \(-0.341402\pi\)
0.477888 + 0.878421i \(0.341402\pi\)
\(830\) −10.3325 −0.358646
\(831\) −0.165444 −0.00573921
\(832\) 26.8083 0.929409
\(833\) 0 0
\(834\) 5.77012 0.199803
\(835\) −3.66433 −0.126809
\(836\) 1.65014 0.0570713
\(837\) −1.86642 −0.0645129
\(838\) 25.3670 0.876287
\(839\) 2.02614 0.0699502 0.0349751 0.999388i \(-0.488865\pi\)
0.0349751 + 0.999388i \(0.488865\pi\)
\(840\) 0 0
\(841\) 26.5740 0.916345
\(842\) −10.9047 −0.375801
\(843\) 20.9797 0.722580
\(844\) 3.05460 0.105144
\(845\) 3.94229 0.135619
\(846\) 2.37116 0.0815222
\(847\) 0 0
\(848\) 15.3959 0.528696
\(849\) 0.814822 0.0279646
\(850\) −5.13213 −0.176031
\(851\) 7.79838 0.267325
\(852\) −0.702953 −0.0240828
\(853\) −38.6105 −1.32200 −0.660999 0.750387i \(-0.729867\pi\)
−0.660999 + 0.750387i \(0.729867\pi\)
\(854\) 0 0
\(855\) −11.3774 −0.389098
\(856\) −40.8111 −1.39490
\(857\) 41.9796 1.43399 0.716997 0.697076i \(-0.245516\pi\)
0.716997 + 0.697076i \(0.245516\pi\)
\(858\) 6.14039 0.209630
\(859\) 40.8629 1.39422 0.697111 0.716963i \(-0.254468\pi\)
0.697111 + 0.716963i \(0.254468\pi\)
\(860\) −2.12829 −0.0725740
\(861\) 0 0
\(862\) −24.6255 −0.838748
\(863\) −33.2645 −1.13234 −0.566169 0.824289i \(-0.691575\pi\)
−0.566169 + 0.824289i \(0.691575\pi\)
\(864\) −0.764868 −0.0260213
\(865\) −25.0205 −0.850721
\(866\) −18.6392 −0.633386
\(867\) 15.6553 0.531683
\(868\) 0 0
\(869\) 16.6668 0.565382
\(870\) 13.5001 0.457695
\(871\) −25.6272 −0.868344
\(872\) −8.48375 −0.287296
\(873\) 0.408396 0.0138221
\(874\) −11.7143 −0.396242
\(875\) 0 0
\(876\) −1.53987 −0.0520273
\(877\) 18.2687 0.616892 0.308446 0.951242i \(-0.400191\pi\)
0.308446 + 0.951242i \(0.400191\pi\)
\(878\) 14.8366 0.500710
\(879\) 12.6589 0.426975
\(880\) −6.98859 −0.235585
\(881\) 27.6269 0.930775 0.465387 0.885107i \(-0.345915\pi\)
0.465387 + 0.885107i \(0.345915\pi\)
\(882\) 0 0
\(883\) −46.7892 −1.57458 −0.787290 0.616582i \(-0.788517\pi\)
−0.787290 + 0.616582i \(0.788517\pi\)
\(884\) −0.497371 −0.0167284
\(885\) 16.8258 0.565594
\(886\) 20.6356 0.693267
\(887\) −21.8623 −0.734064 −0.367032 0.930208i \(-0.619626\pi\)
−0.367032 + 0.930208i \(0.619626\pi\)
\(888\) −22.7395 −0.763088
\(889\) 0 0
\(890\) 11.5086 0.385770
\(891\) −1.42009 −0.0475747
\(892\) 1.95702 0.0655259
\(893\) 14.8972 0.498515
\(894\) 22.4573 0.751084
\(895\) 27.4767 0.918446
\(896\) 0 0
\(897\) 3.16661 0.105730
\(898\) 8.38998 0.279977
\(899\) 13.9138 0.464051
\(900\) 0.439013 0.0146338
\(901\) −4.81117 −0.160283
\(902\) 21.7679 0.724793
\(903\) 0 0
\(904\) −54.4181 −1.80992
\(905\) −0.910928 −0.0302803
\(906\) 11.1811 0.371468
\(907\) −51.1525 −1.69849 −0.849246 0.527998i \(-0.822943\pi\)
−0.849246 + 0.527998i \(0.822943\pi\)
\(908\) 0.444112 0.0147384
\(909\) −0.723827 −0.0240078
\(910\) 0 0
\(911\) −26.2401 −0.869373 −0.434686 0.900582i \(-0.643141\pi\)
−0.434686 + 0.900582i \(0.643141\pi\)
\(912\) 31.8340 1.05413
\(913\) −8.10253 −0.268155
\(914\) 40.3977 1.33624
\(915\) −18.3213 −0.605683
\(916\) 0.0357495 0.00118120
\(917\) 0 0
\(918\) −1.58342 −0.0522607
\(919\) 31.9401 1.05361 0.526803 0.849987i \(-0.323390\pi\)
0.526803 + 0.849987i \(0.323390\pi\)
\(920\) −3.86713 −0.127496
\(921\) 3.03467 0.0999957
\(922\) −47.5184 −1.56494
\(923\) −16.4340 −0.540933
\(924\) 0 0
\(925\) 25.2758 0.831065
\(926\) 21.7070 0.713337
\(927\) −14.2739 −0.468817
\(928\) 5.70193 0.187175
\(929\) −13.4153 −0.440140 −0.220070 0.975484i \(-0.570629\pi\)
−0.220070 + 0.975484i \(0.570629\pi\)
\(930\) 3.37994 0.110833
\(931\) 0 0
\(932\) −2.41789 −0.0792008
\(933\) −27.4766 −0.899545
\(934\) 9.94915 0.325546
\(935\) 2.18392 0.0714219
\(936\) −9.23359 −0.301809
\(937\) 21.3033 0.695950 0.347975 0.937504i \(-0.386869\pi\)
0.347975 + 0.937504i \(0.386869\pi\)
\(938\) 0 0
\(939\) −1.34203 −0.0437954
\(940\) 0.311934 0.0101742
\(941\) 5.22573 0.170354 0.0851770 0.996366i \(-0.472854\pi\)
0.0851770 + 0.996366i \(0.472854\pi\)
\(942\) −30.7441 −1.00170
\(943\) 11.2258 0.365561
\(944\) −47.0788 −1.53229
\(945\) 0 0
\(946\) 22.9743 0.746960
\(947\) −14.7609 −0.479664 −0.239832 0.970814i \(-0.577092\pi\)
−0.239832 + 0.970814i \(0.577092\pi\)
\(948\) −1.58970 −0.0516309
\(949\) −35.9999 −1.16861
\(950\) −37.9680 −1.23184
\(951\) −0.508616 −0.0164930
\(952\) 0 0
\(953\) −33.6944 −1.09147 −0.545735 0.837958i \(-0.683749\pi\)
−0.545735 + 0.837958i \(0.683749\pi\)
\(954\) −5.66537 −0.183423
\(955\) −1.44955 −0.0469064
\(956\) −0.228700 −0.00739669
\(957\) 10.5865 0.342212
\(958\) −25.4193 −0.821262
\(959\) 0 0
\(960\) 11.2276 0.362369
\(961\) −27.5165 −0.887628
\(962\) −33.7199 −1.08717
\(963\) 13.9959 0.451013
\(964\) −2.57922 −0.0830710
\(965\) −9.83244 −0.316518
\(966\) 0 0
\(967\) 49.9879 1.60750 0.803751 0.594966i \(-0.202835\pi\)
0.803751 + 0.594966i \(0.202835\pi\)
\(968\) 26.1948 0.841932
\(969\) −9.94808 −0.319578
\(970\) −0.739573 −0.0237462
\(971\) −11.4283 −0.366751 −0.183375 0.983043i \(-0.558702\pi\)
−0.183375 + 0.983043i \(0.558702\pi\)
\(972\) 0.135449 0.00434454
\(973\) 0 0
\(974\) 13.1481 0.421294
\(975\) 10.2635 0.328695
\(976\) 51.2632 1.64089
\(977\) 4.98782 0.159575 0.0797873 0.996812i \(-0.474576\pi\)
0.0797873 + 0.996812i \(0.474576\pi\)
\(978\) 1.99566 0.0638143
\(979\) 9.02481 0.288434
\(980\) 0 0
\(981\) 2.90945 0.0928917
\(982\) −6.29649 −0.200929
\(983\) 45.8555 1.46256 0.731282 0.682075i \(-0.238922\pi\)
0.731282 + 0.682075i \(0.238922\pi\)
\(984\) −32.7335 −1.04350
\(985\) −4.19189 −0.133565
\(986\) 11.8041 0.375919
\(987\) 0 0
\(988\) −3.67960 −0.117064
\(989\) 11.8479 0.376741
\(990\) 2.57167 0.0817329
\(991\) 33.4065 1.06119 0.530596 0.847625i \(-0.321968\pi\)
0.530596 + 0.847625i \(0.321968\pi\)
\(992\) 1.42756 0.0453252
\(993\) −6.19033 −0.196444
\(994\) 0 0
\(995\) 2.39949 0.0760690
\(996\) 0.772827 0.0244880
\(997\) −21.8973 −0.693494 −0.346747 0.937959i \(-0.612714\pi\)
−0.346747 + 0.937959i \(0.612714\pi\)
\(998\) 17.5556 0.555712
\(999\) 7.79838 0.246730
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 3381.2.a.bc.1.5 6
7.2 even 3 483.2.i.f.277.2 12
7.4 even 3 483.2.i.f.415.2 yes 12
7.6 odd 2 3381.2.a.bd.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.i.f.277.2 12 7.2 even 3
483.2.i.f.415.2 yes 12 7.4 even 3
3381.2.a.bc.1.5 6 1.1 even 1 trivial
3381.2.a.bd.1.5 6 7.6 odd 2