Properties

Label 3381.2.a.bd.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 \(-0.404164\pi\)
0.296550 + 0.955017i \(0.404164\pi\)
\(6\) 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 \(-0.644713\pi\)
−0.439129 + 0.898424i \(0.644713\pi\)
\(14\) 0 0
\(15\) 1.32621 0.342426
\(16\) −3.71076 −0.927689
\(17\) −1.15960 −0.281245 −0.140623 0.990063i \(-0.544910\pi\)
−0.140623 + 0.990063i \(0.544910\pi\)
\(18\) 1.36549 0.321848
\(19\) −8.57886 −1.96813 −0.984063 0.177822i \(-0.943095\pi\)
−0.984063 + 0.177822i \(0.943095\pi\)
\(20\) −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 \(-0.553605\pi\)
−0.167609 + 0.985853i \(0.553605\pi\)
\(32\) 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 \(-0.159817\pi\)
0.876584 + 0.481248i \(0.159817\pi\)
\(42\) 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 \(-0.540422\pi\)
−0.126647 + 0.991948i \(0.540422\pi\)
\(48\) −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 \(-0.809311\pi\)
−0.825862 + 0.563872i \(0.809311\pi\)
\(60\) −0.179634 −0.0231907
\(61\) 13.8148 1.76880 0.884399 0.466731i \(-0.154568\pi\)
0.884399 + 0.466731i \(0.154568\pi\)
\(62\) −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 \(-0.268305\pi\)
0.665297 + 0.746579i \(0.268305\pi\)
\(74\) −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 \(-0.601380\pi\)
−0.313139 + 0.949707i \(0.601380\pi\)
\(84\) 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 \(-0.390648\pi\)
0.336820 + 0.941569i \(0.390648\pi\)
\(90\) 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 \(-0.506600\pi\)
−0.0207331 + 0.999785i \(0.506600\pi\)
\(98\) 0 0
\(99\) −1.42009 −0.142724
\(100\) 0.439013 0.0439013
\(101\) 0.723827 0.0720235 0.0360117 0.999351i \(-0.488535\pi\)
0.0360117 + 0.999351i \(0.488535\pi\)
\(102\) −1.58342 −0.156782
\(103\) 14.2739 1.40645 0.703225 0.710967i \(-0.251743\pi\)
0.703225 + 0.710967i \(0.251743\pi\)
\(104\) 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 \(-0.253365\pi\)
0.699593 + 0.714541i \(0.253365\pi\)
\(132\) 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 \(-0.442646\pi\)
0.179209 + 0.983811i \(0.442646\pi\)
\(140\) 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 \(-0.855307\pi\)
−0.898452 + 0.439072i \(0.855307\pi\)
\(158\) −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 \(-0.534094\pi\)
−0.106904 + 0.994269i \(0.534094\pi\)
\(168\) 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 \(-0.754568\pi\)
−0.717182 + 0.696886i \(0.754568\pi\)
\(174\) 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 \(-0.508126\pi\)
−0.0255271 + 0.999674i \(0.508126\pi\)
\(182\) 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 \(-0.479573\pi\)
0.0641283 + 0.997942i \(0.479573\pi\)
\(200\) 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 \(-0.339268\pi\)
0.483767 + 0.875197i \(0.339268\pi\)
\(224\) 0 0
\(225\) −3.24116 −0.216078
\(226\) 25.4832 1.69512
\(227\) 3.27881 0.217622 0.108811 0.994062i \(-0.465296\pi\)
0.108811 + 0.994062i \(0.465296\pi\)
\(228\) 1.16200 0.0769553
\(229\) 0.263933 0.0174412 0.00872059 0.999962i \(-0.497224\pi\)
0.00872059 + 0.999962i \(0.497224\pi\)
\(230\) −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 \(-0.710158\pi\)
−0.613300 + 0.789850i \(0.710158\pi\)
\(242\) −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 \(-0.278149\pi\)
0.641895 + 0.766793i \(0.278149\pi\)
\(252\) 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 \(-0.402901\pi\)
0.300337 + 0.953833i \(0.402901\pi\)
\(258\) −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 \(-0.626952\pi\)
−0.388342 + 0.921515i \(0.626952\pi\)
\(270\) 1.81092 0.110209
\(271\) 22.0021 1.33653 0.668265 0.743923i \(-0.267037\pi\)
0.668265 + 0.743923i \(0.267037\pi\)
\(272\) 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 \(-0.492290\pi\)
0.0242181 + 0.999707i \(0.492290\pi\)
\(284\) 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 \(-0.379436\pi\)
0.369771 + 0.929123i \(0.379436\pi\)
\(294\) 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 \(-0.472400\pi\)
0.0865989 + 0.996243i \(0.472400\pi\)
\(308\) 0 0
\(309\) 14.2739 0.812014
\(310\) −3.37994 −0.191968
\(311\) −27.4766 −1.55806 −0.779029 0.626988i \(-0.784287\pi\)
−0.779029 + 0.626988i \(0.784287\pi\)
\(312\) 9.23359 0.522749
\(313\) −1.34203 −0.0758558 −0.0379279 0.999280i \(-0.512076\pi\)
−0.0379279 + 0.999280i \(0.512076\pi\)
\(314\) −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 \(-0.404757\pi\)
0.294771 + 0.955568i \(0.404757\pi\)
\(350\) 0 0
\(351\) −3.16661 −0.169021
\(352\) −1.08618 −0.0578935
\(353\) 6.64781 0.353827 0.176914 0.984226i \(-0.443389\pi\)
0.176914 + 0.984226i \(0.443389\pi\)
\(354\) −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 \(-0.515562\pi\)
−0.0488684 + 0.998805i \(0.515562\pi\)
\(368\) 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 \(-0.259719\pi\)
0.685191 + 0.728363i \(0.259719\pi\)
\(384\) 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 \(-0.582313\pi\)
−0.255722 + 0.966750i \(0.582313\pi\)
\(398\) 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 \(-0.383825\pi\)
0.356925 + 0.934133i \(0.383825\pi\)
\(410\) 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 \(-0.649924\pi\)
−0.453779 + 0.891114i \(0.649924\pi\)
\(420\) 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 \(-0.393627\pi\)
0.327995 + 0.944680i \(0.393627\pi\)
\(434\) 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 \(-0.583488\pi\)
−0.259289 + 0.965800i \(0.583488\pi\)
\(440\) 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 \(-0.199255\pi\)
0.810391 + 0.585890i \(0.199255\pi\)
\(462\) 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 \(-0.553919\pi\)
−0.168582 + 0.985688i \(0.553919\pi\)
\(468\) 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 \(-0.360174\pi\)
0.425285 + 0.905060i \(0.360174\pi\)
\(480\) 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 \(-0.611843\pi\)
−0.344180 + 0.938904i \(0.611843\pi\)
\(504\) 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 \(-0.581325\pi\)
−0.252720 + 0.967540i \(0.581325\pi\)
\(510\) −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 \(-0.606391\pi\)
−0.328049 + 0.944661i \(0.606391\pi\)
\(522\) 10.1794 0.445541
\(523\) 11.4151 0.499147 0.249573 0.968356i \(-0.419710\pi\)
0.249573 + 0.968356i \(0.419710\pi\)
\(524\) −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 \(-0.566033\pi\)
−0.205963 + 0.978560i \(0.566033\pi\)
\(564\) 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 \(-0.598014\pi\)
−0.303077 + 0.952966i \(0.598014\pi\)
\(578\) −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 \(-0.390248\pi\)
0.338004 + 0.941145i \(0.390248\pi\)
\(588\) 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 \(-0.688864\pi\)
−0.559129 + 0.829081i \(0.688864\pi\)
\(594\) −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 \(-0.556764\pi\)
−0.177387 + 0.984141i \(0.556764\pi\)
\(602\) 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 \(-0.633342\pi\)
−0.406761 + 0.913535i \(0.633342\pi\)
\(608\) −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 \(-0.403737\pi\)
0.297832 + 0.954618i \(0.403737\pi\)
\(620\) 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 \(-0.460492\pi\)
0.123799 + 0.992307i \(0.460492\pi\)
\(644\) 0 0
\(645\) −15.7128 −0.618691
\(646\) 13.5840 0.534454
\(647\) 46.9083 1.84416 0.922078 0.387004i \(-0.126490\pi\)
0.922078 + 0.387004i \(0.126490\pi\)
\(648\) −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 \(-0.580574\pi\)
−0.250436 + 0.968133i \(0.580574\pi\)
\(662\) 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 \(-0.571710\pi\)
−0.223383 + 0.974731i \(0.571710\pi\)
\(678\) 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 \(-0.659587\pi\)
−0.480615 + 0.876932i \(0.659587\pi\)
\(692\) 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 \(-0.700707\pi\)
−0.589580 + 0.807710i \(0.700707\pi\)
\(720\) −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 \(-0.500053\pi\)
−0.000165878 1.00000i \(0.500053\pi\)
\(728\) 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 \(-0.657490\pi\)
−0.474828 + 0.880079i \(0.657490\pi\)
\(734\) −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 \(-0.678350\pi\)
−0.531442 + 0.847095i \(0.678350\pi\)
\(762\) 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 \(-0.446373\pi\)
0.167677 + 0.985842i \(0.446373\pi\)
\(770\) 0 0
\(771\) 9.62955 0.346800
\(772\) −1.00421 −0.0361424
\(773\) −33.5436 −1.20648 −0.603240 0.797559i \(-0.706124\pi\)
−0.603240 + 0.797559i \(0.706124\pi\)
\(774\) −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 \(-0.316931\pi\)
0.543943 + 0.839122i \(0.316931\pi\)
\(788\) −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 \(-0.235939\pi\)
0.737643 + 0.675191i \(0.235939\pi\)
\(798\) 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 \(-0.373475\pi\)
0.387106 + 0.922035i \(0.373475\pi\)
\(812\) 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 \(-0.658598\pi\)
−0.477888 + 0.878421i \(0.658598\pi\)
\(830\) −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 \(-0.511135\pi\)
−0.0349751 + 0.999388i \(0.511135\pi\)
\(840\) 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 \(-0.270133\pi\)
0.660999 + 0.750387i \(0.270133\pi\)
\(854\) 0 0
\(855\) −11.3774 −0.389098
\(856\) −40.8111 −1.39490
\(857\) −41.9796 −1.43399 −0.716997 0.697076i \(-0.754484\pi\)
−0.716997 + 0.697076i \(0.754484\pi\)
\(858\) 6.14039 0.209630
\(859\) −40.8629 −1.39422 −0.697111 0.716963i \(-0.745532\pi\)
−0.697111 + 0.716963i \(0.745532\pi\)
\(860\) 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 \(-0.654085\pi\)
−0.465387 + 0.885107i \(0.654085\pi\)
\(882\) 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 \(-0.380374\pi\)
0.367032 + 0.930208i \(0.380374\pi\)
\(888\) 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 \(-0.429371\pi\)
0.220070 + 0.975484i \(0.429371\pi\)
\(930\) −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 \(-0.613131\pi\)
−0.347975 + 0.937504i \(0.613131\pi\)
\(938\) 0 0
\(939\) −1.34203 −0.0437954
\(940\) 0.311934 0.0101742
\(941\) −5.22573 −0.170354 −0.0851770 0.996366i \(-0.527146\pi\)
−0.0851770 + 0.996366i \(0.527146\pi\)
\(942\) −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 \(-0.441298\pi\)
0.183375 + 0.983043i \(0.441298\pi\)
\(972\) −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 \(-0.761078\pi\)
−0.731282 + 0.682075i \(0.761078\pi\)
\(984\) −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 \(-0.387286\pi\)
0.346747 + 0.937959i \(0.387286\pi\)
\(998\) 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.bd.1.5 6
7.3 odd 6 483.2.i.f.415.2 yes 12
7.5 odd 6 483.2.i.f.277.2 12
7.6 odd 2 3381.2.a.bc.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.i.f.277.2 12 7.5 odd 6
483.2.i.f.415.2 yes 12 7.3 odd 6
3381.2.a.bc.1.5 6 7.6 odd 2
3381.2.a.bd.1.5 6 1.1 even 1 trivial