Properties

Label 2151.2.a.e.1.1
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.1758214748\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.1767625.1
Defining polynomial: \(x^{6} - 7 x^{4} - x^{3} + 11 x^{2} + x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 717)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(0.267500\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.12029 q^{2} +2.49562 q^{4} +2.47082 q^{5} -2.45271 q^{7} -1.05086 q^{8} +O(q^{10})\) \(q-2.12029 q^{2} +2.49562 q^{4} +2.47082 q^{5} -2.45271 q^{7} -1.05086 q^{8} -5.23885 q^{10} +4.01073 q^{11} +1.62598 q^{13} +5.20045 q^{14} -2.76312 q^{16} -4.89570 q^{17} -6.28561 q^{19} +6.16623 q^{20} -8.50391 q^{22} +3.83124 q^{23} +1.10496 q^{25} -3.44755 q^{26} -6.12103 q^{28} +0.357168 q^{29} -8.44143 q^{31} +7.96032 q^{32} +10.3803 q^{34} -6.06021 q^{35} -1.81610 q^{37} +13.3273 q^{38} -2.59648 q^{40} -10.7403 q^{41} +9.09255 q^{43} +10.0093 q^{44} -8.12332 q^{46} -11.0197 q^{47} -0.984213 q^{49} -2.34284 q^{50} +4.05783 q^{52} -5.78427 q^{53} +9.90981 q^{55} +2.57745 q^{56} -0.757299 q^{58} +1.43742 q^{59} +2.21862 q^{61} +17.8983 q^{62} -11.3519 q^{64} +4.01751 q^{65} -14.2454 q^{67} -12.2178 q^{68} +12.8494 q^{70} +4.12013 q^{71} +2.78648 q^{73} +3.85066 q^{74} -15.6865 q^{76} -9.83717 q^{77} +7.84263 q^{79} -6.82718 q^{80} +22.7725 q^{82} +9.91715 q^{83} -12.0964 q^{85} -19.2788 q^{86} -4.21471 q^{88} -0.978993 q^{89} -3.98806 q^{91} +9.56131 q^{92} +23.3649 q^{94} -15.5306 q^{95} -16.8523 q^{97} +2.08682 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{2} + 4q^{4} - 5q^{5} - 9q^{7} + 3q^{8} + O(q^{10}) \) \( 6q + 2q^{2} + 4q^{4} - 5q^{5} - 9q^{7} + 3q^{8} - 11q^{10} + 13q^{11} - q^{13} - 4q^{16} - 11q^{17} - 22q^{19} + q^{20} - 2q^{22} + 12q^{23} - q^{25} - 12q^{26} - 16q^{28} - 18q^{31} - 7q^{32} - 3q^{34} + 9q^{35} - 8q^{37} + 5q^{38} - 11q^{40} - 10q^{41} - 14q^{43} + 4q^{44} - 18q^{46} + 9q^{47} + 5q^{49} - 4q^{50} - 16q^{52} + 8q^{53} - 20q^{55} - 11q^{56} - 15q^{58} + 10q^{59} - 12q^{61} + 13q^{62} - 31q^{64} + 11q^{65} - 36q^{67} - 22q^{68} + q^{70} + 3q^{71} - 32q^{73} - 9q^{74} - 4q^{76} - 6q^{77} - q^{79} + 7q^{80} + 7q^{82} + 7q^{83} - 14q^{85} - 45q^{86} - 15q^{88} - 17q^{89} - 23q^{91} + 12q^{92} + 50q^{94} - 28q^{97} - 13q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.12029 −1.49927 −0.749635 0.661852i \(-0.769771\pi\)
−0.749635 + 0.661852i \(0.769771\pi\)
\(3\) 0 0
\(4\) 2.49562 1.24781
\(5\) 2.47082 1.10499 0.552493 0.833518i \(-0.313677\pi\)
0.552493 + 0.833518i \(0.313677\pi\)
\(6\) 0 0
\(7\) −2.45271 −0.927037 −0.463519 0.886087i \(-0.653413\pi\)
−0.463519 + 0.886087i \(0.653413\pi\)
\(8\) −1.05086 −0.371534
\(9\) 0 0
\(10\) −5.23885 −1.65667
\(11\) 4.01073 1.20928 0.604641 0.796498i \(-0.293317\pi\)
0.604641 + 0.796498i \(0.293317\pi\)
\(12\) 0 0
\(13\) 1.62598 0.450966 0.225483 0.974247i \(-0.427604\pi\)
0.225483 + 0.974247i \(0.427604\pi\)
\(14\) 5.20045 1.38988
\(15\) 0 0
\(16\) −2.76312 −0.690780
\(17\) −4.89570 −1.18738 −0.593691 0.804693i \(-0.702330\pi\)
−0.593691 + 0.804693i \(0.702330\pi\)
\(18\) 0 0
\(19\) −6.28561 −1.44202 −0.721009 0.692926i \(-0.756321\pi\)
−0.721009 + 0.692926i \(0.756321\pi\)
\(20\) 6.16623 1.37881
\(21\) 0 0
\(22\) −8.50391 −1.81304
\(23\) 3.83124 0.798868 0.399434 0.916762i \(-0.369207\pi\)
0.399434 + 0.916762i \(0.369207\pi\)
\(24\) 0 0
\(25\) 1.10496 0.220992
\(26\) −3.44755 −0.676120
\(27\) 0 0
\(28\) −6.12103 −1.15677
\(29\) 0.357168 0.0663244 0.0331622 0.999450i \(-0.489442\pi\)
0.0331622 + 0.999450i \(0.489442\pi\)
\(30\) 0 0
\(31\) −8.44143 −1.51613 −0.758063 0.652181i \(-0.773854\pi\)
−0.758063 + 0.652181i \(0.773854\pi\)
\(32\) 7.96032 1.40720
\(33\) 0 0
\(34\) 10.3803 1.78020
\(35\) −6.06021 −1.02436
\(36\) 0 0
\(37\) −1.81610 −0.298566 −0.149283 0.988795i \(-0.547696\pi\)
−0.149283 + 0.988795i \(0.547696\pi\)
\(38\) 13.3273 2.16197
\(39\) 0 0
\(40\) −2.59648 −0.410540
\(41\) −10.7403 −1.67735 −0.838676 0.544630i \(-0.816670\pi\)
−0.838676 + 0.544630i \(0.816670\pi\)
\(42\) 0 0
\(43\) 9.09255 1.38660 0.693301 0.720649i \(-0.256156\pi\)
0.693301 + 0.720649i \(0.256156\pi\)
\(44\) 10.0093 1.50895
\(45\) 0 0
\(46\) −8.12332 −1.19772
\(47\) −11.0197 −1.60739 −0.803693 0.595044i \(-0.797134\pi\)
−0.803693 + 0.595044i \(0.797134\pi\)
\(48\) 0 0
\(49\) −0.984213 −0.140602
\(50\) −2.34284 −0.331327
\(51\) 0 0
\(52\) 4.05783 0.562720
\(53\) −5.78427 −0.794530 −0.397265 0.917704i \(-0.630041\pi\)
−0.397265 + 0.917704i \(0.630041\pi\)
\(54\) 0 0
\(55\) 9.90981 1.33624
\(56\) 2.57745 0.344426
\(57\) 0 0
\(58\) −0.757299 −0.0994382
\(59\) 1.43742 0.187136 0.0935678 0.995613i \(-0.470173\pi\)
0.0935678 + 0.995613i \(0.470173\pi\)
\(60\) 0 0
\(61\) 2.21862 0.284065 0.142032 0.989862i \(-0.454636\pi\)
0.142032 + 0.989862i \(0.454636\pi\)
\(62\) 17.8983 2.27308
\(63\) 0 0
\(64\) −11.3519 −1.41899
\(65\) 4.01751 0.498311
\(66\) 0 0
\(67\) −14.2454 −1.74036 −0.870178 0.492738i \(-0.835996\pi\)
−0.870178 + 0.492738i \(0.835996\pi\)
\(68\) −12.2178 −1.48163
\(69\) 0 0
\(70\) 12.8494 1.53580
\(71\) 4.12013 0.488969 0.244485 0.969653i \(-0.421381\pi\)
0.244485 + 0.969653i \(0.421381\pi\)
\(72\) 0 0
\(73\) 2.78648 0.326133 0.163067 0.986615i \(-0.447861\pi\)
0.163067 + 0.986615i \(0.447861\pi\)
\(74\) 3.85066 0.447630
\(75\) 0 0
\(76\) −15.6865 −1.79936
\(77\) −9.83717 −1.12105
\(78\) 0 0
\(79\) 7.84263 0.882365 0.441182 0.897417i \(-0.354559\pi\)
0.441182 + 0.897417i \(0.354559\pi\)
\(80\) −6.82718 −0.763302
\(81\) 0 0
\(82\) 22.7725 2.51480
\(83\) 9.91715 1.08855 0.544274 0.838907i \(-0.316805\pi\)
0.544274 + 0.838907i \(0.316805\pi\)
\(84\) 0 0
\(85\) −12.0964 −1.31204
\(86\) −19.2788 −2.07889
\(87\) 0 0
\(88\) −4.21471 −0.449290
\(89\) −0.978993 −0.103773 −0.0518865 0.998653i \(-0.516523\pi\)
−0.0518865 + 0.998653i \(0.516523\pi\)
\(90\) 0 0
\(91\) −3.98806 −0.418062
\(92\) 9.56131 0.996836
\(93\) 0 0
\(94\) 23.3649 2.40991
\(95\) −15.5306 −1.59341
\(96\) 0 0
\(97\) −16.8523 −1.71109 −0.855545 0.517728i \(-0.826778\pi\)
−0.855545 + 0.517728i \(0.826778\pi\)
\(98\) 2.08682 0.210800
\(99\) 0 0
\(100\) 2.75756 0.275756
\(101\) −3.80431 −0.378543 −0.189271 0.981925i \(-0.560613\pi\)
−0.189271 + 0.981925i \(0.560613\pi\)
\(102\) 0 0
\(103\) 8.01017 0.789266 0.394633 0.918839i \(-0.370872\pi\)
0.394633 + 0.918839i \(0.370872\pi\)
\(104\) −1.70867 −0.167549
\(105\) 0 0
\(106\) 12.2643 1.19122
\(107\) 12.6857 1.22638 0.613188 0.789937i \(-0.289887\pi\)
0.613188 + 0.789937i \(0.289887\pi\)
\(108\) 0 0
\(109\) −18.7766 −1.79847 −0.899237 0.437461i \(-0.855878\pi\)
−0.899237 + 0.437461i \(0.855878\pi\)
\(110\) −21.0116 −2.00338
\(111\) 0 0
\(112\) 6.77713 0.640379
\(113\) 7.61332 0.716201 0.358101 0.933683i \(-0.383425\pi\)
0.358101 + 0.933683i \(0.383425\pi\)
\(114\) 0 0
\(115\) 9.46630 0.882737
\(116\) 0.891355 0.0827603
\(117\) 0 0
\(118\) −3.04774 −0.280567
\(119\) 12.0077 1.10075
\(120\) 0 0
\(121\) 5.08598 0.462362
\(122\) −4.70410 −0.425890
\(123\) 0 0
\(124\) −21.0666 −1.89184
\(125\) −9.62395 −0.860792
\(126\) 0 0
\(127\) 14.0067 1.24290 0.621448 0.783455i \(-0.286545\pi\)
0.621448 + 0.783455i \(0.286545\pi\)
\(128\) 8.14873 0.720253
\(129\) 0 0
\(130\) −8.51828 −0.747102
\(131\) 4.05565 0.354344 0.177172 0.984180i \(-0.443305\pi\)
0.177172 + 0.984180i \(0.443305\pi\)
\(132\) 0 0
\(133\) 15.4168 1.33680
\(134\) 30.2044 2.60926
\(135\) 0 0
\(136\) 5.14468 0.441153
\(137\) 15.5346 1.32721 0.663604 0.748084i \(-0.269026\pi\)
0.663604 + 0.748084i \(0.269026\pi\)
\(138\) 0 0
\(139\) 1.09201 0.0926232 0.0463116 0.998927i \(-0.485253\pi\)
0.0463116 + 0.998927i \(0.485253\pi\)
\(140\) −15.1240 −1.27821
\(141\) 0 0
\(142\) −8.73586 −0.733097
\(143\) 6.52138 0.545345
\(144\) 0 0
\(145\) 0.882498 0.0732875
\(146\) −5.90815 −0.488962
\(147\) 0 0
\(148\) −4.53230 −0.372553
\(149\) 15.9702 1.30833 0.654167 0.756350i \(-0.273019\pi\)
0.654167 + 0.756350i \(0.273019\pi\)
\(150\) 0 0
\(151\) −8.44617 −0.687339 −0.343670 0.939091i \(-0.611670\pi\)
−0.343670 + 0.939091i \(0.611670\pi\)
\(152\) 6.60528 0.535759
\(153\) 0 0
\(154\) 20.8576 1.68076
\(155\) −20.8573 −1.67530
\(156\) 0 0
\(157\) 8.21810 0.655876 0.327938 0.944699i \(-0.393646\pi\)
0.327938 + 0.944699i \(0.393646\pi\)
\(158\) −16.6286 −1.32290
\(159\) 0 0
\(160\) 19.6685 1.55494
\(161\) −9.39691 −0.740580
\(162\) 0 0
\(163\) 3.58850 0.281073 0.140537 0.990075i \(-0.455117\pi\)
0.140537 + 0.990075i \(0.455117\pi\)
\(164\) −26.8037 −2.09302
\(165\) 0 0
\(166\) −21.0272 −1.63203
\(167\) −14.0773 −1.08934 −0.544668 0.838652i \(-0.683344\pi\)
−0.544668 + 0.838652i \(0.683344\pi\)
\(168\) 0 0
\(169\) −10.3562 −0.796630
\(170\) 25.6478 1.96710
\(171\) 0 0
\(172\) 22.6916 1.73021
\(173\) −5.90842 −0.449208 −0.224604 0.974450i \(-0.572109\pi\)
−0.224604 + 0.974450i \(0.572109\pi\)
\(174\) 0 0
\(175\) −2.71015 −0.204868
\(176\) −11.0821 −0.835348
\(177\) 0 0
\(178\) 2.07575 0.155584
\(179\) −18.5491 −1.38642 −0.693211 0.720735i \(-0.743805\pi\)
−0.693211 + 0.720735i \(0.743805\pi\)
\(180\) 0 0
\(181\) −0.921544 −0.0684978 −0.0342489 0.999413i \(-0.510904\pi\)
−0.0342489 + 0.999413i \(0.510904\pi\)
\(182\) 8.45584 0.626788
\(183\) 0 0
\(184\) −4.02608 −0.296807
\(185\) −4.48727 −0.329911
\(186\) 0 0
\(187\) −19.6353 −1.43588
\(188\) −27.5009 −2.00571
\(189\) 0 0
\(190\) 32.9294 2.38895
\(191\) −15.1285 −1.09466 −0.547329 0.836918i \(-0.684355\pi\)
−0.547329 + 0.836918i \(0.684355\pi\)
\(192\) 0 0
\(193\) −22.7285 −1.63603 −0.818015 0.575197i \(-0.804925\pi\)
−0.818015 + 0.575197i \(0.804925\pi\)
\(194\) 35.7317 2.56539
\(195\) 0 0
\(196\) −2.45622 −0.175444
\(197\) 2.47078 0.176036 0.0880179 0.996119i \(-0.471947\pi\)
0.0880179 + 0.996119i \(0.471947\pi\)
\(198\) 0 0
\(199\) −5.34461 −0.378869 −0.189435 0.981893i \(-0.560666\pi\)
−0.189435 + 0.981893i \(0.560666\pi\)
\(200\) −1.16116 −0.0821062
\(201\) 0 0
\(202\) 8.06622 0.567538
\(203\) −0.876029 −0.0614852
\(204\) 0 0
\(205\) −26.5374 −1.85345
\(206\) −16.9839 −1.18332
\(207\) 0 0
\(208\) −4.49278 −0.311518
\(209\) −25.2099 −1.74381
\(210\) 0 0
\(211\) −10.3377 −0.711676 −0.355838 0.934548i \(-0.615804\pi\)
−0.355838 + 0.934548i \(0.615804\pi\)
\(212\) −14.4353 −0.991423
\(213\) 0 0
\(214\) −26.8974 −1.83867
\(215\) 22.4661 1.53217
\(216\) 0 0
\(217\) 20.7044 1.40551
\(218\) 39.8119 2.69640
\(219\) 0 0
\(220\) 24.7311 1.66737
\(221\) −7.96031 −0.535469
\(222\) 0 0
\(223\) −10.2702 −0.687745 −0.343873 0.939016i \(-0.611739\pi\)
−0.343873 + 0.939016i \(0.611739\pi\)
\(224\) −19.5244 −1.30453
\(225\) 0 0
\(226\) −16.1424 −1.07378
\(227\) 16.9467 1.12479 0.562396 0.826868i \(-0.309880\pi\)
0.562396 + 0.826868i \(0.309880\pi\)
\(228\) 0 0
\(229\) −29.0495 −1.91964 −0.959822 0.280609i \(-0.909464\pi\)
−0.959822 + 0.280609i \(0.909464\pi\)
\(230\) −20.0713 −1.32346
\(231\) 0 0
\(232\) −0.375332 −0.0246418
\(233\) 19.7368 1.29300 0.646500 0.762914i \(-0.276232\pi\)
0.646500 + 0.762914i \(0.276232\pi\)
\(234\) 0 0
\(235\) −27.2277 −1.77614
\(236\) 3.58725 0.233510
\(237\) 0 0
\(238\) −25.4598 −1.65032
\(239\) 1.00000 0.0646846
\(240\) 0 0
\(241\) 10.4837 0.675311 0.337656 0.941270i \(-0.390366\pi\)
0.337656 + 0.941270i \(0.390366\pi\)
\(242\) −10.7837 −0.693205
\(243\) 0 0
\(244\) 5.53682 0.354459
\(245\) −2.43182 −0.155363
\(246\) 0 0
\(247\) −10.2203 −0.650301
\(248\) 8.87074 0.563293
\(249\) 0 0
\(250\) 20.4055 1.29056
\(251\) −12.1227 −0.765177 −0.382588 0.923919i \(-0.624967\pi\)
−0.382588 + 0.923919i \(0.624967\pi\)
\(252\) 0 0
\(253\) 15.3661 0.966056
\(254\) −29.6983 −1.86344
\(255\) 0 0
\(256\) 5.42623 0.339139
\(257\) −28.3873 −1.77075 −0.885377 0.464874i \(-0.846100\pi\)
−0.885377 + 0.464874i \(0.846100\pi\)
\(258\) 0 0
\(259\) 4.45438 0.276781
\(260\) 10.0262 0.621797
\(261\) 0 0
\(262\) −8.59914 −0.531257
\(263\) 14.0844 0.868482 0.434241 0.900797i \(-0.357017\pi\)
0.434241 + 0.900797i \(0.357017\pi\)
\(264\) 0 0
\(265\) −14.2919 −0.877944
\(266\) −32.6880 −2.00423
\(267\) 0 0
\(268\) −35.5512 −2.17163
\(269\) −22.1224 −1.34883 −0.674415 0.738353i \(-0.735604\pi\)
−0.674415 + 0.738353i \(0.735604\pi\)
\(270\) 0 0
\(271\) −0.185181 −0.0112490 −0.00562449 0.999984i \(-0.501790\pi\)
−0.00562449 + 0.999984i \(0.501790\pi\)
\(272\) 13.5274 0.820219
\(273\) 0 0
\(274\) −32.9377 −1.98984
\(275\) 4.43171 0.267242
\(276\) 0 0
\(277\) −15.8869 −0.954549 −0.477274 0.878754i \(-0.658375\pi\)
−0.477274 + 0.878754i \(0.658375\pi\)
\(278\) −2.31538 −0.138867
\(279\) 0 0
\(280\) 6.36842 0.380586
\(281\) −19.9963 −1.19288 −0.596440 0.802658i \(-0.703419\pi\)
−0.596440 + 0.802658i \(0.703419\pi\)
\(282\) 0 0
\(283\) −30.1737 −1.79364 −0.896819 0.442398i \(-0.854128\pi\)
−0.896819 + 0.442398i \(0.854128\pi\)
\(284\) 10.2823 0.610141
\(285\) 0 0
\(286\) −13.8272 −0.817619
\(287\) 26.3428 1.55497
\(288\) 0 0
\(289\) 6.96786 0.409874
\(290\) −1.87115 −0.109878
\(291\) 0 0
\(292\) 6.95401 0.406953
\(293\) 0.459157 0.0268242 0.0134121 0.999910i \(-0.495731\pi\)
0.0134121 + 0.999910i \(0.495731\pi\)
\(294\) 0 0
\(295\) 3.55160 0.206782
\(296\) 1.90847 0.110927
\(297\) 0 0
\(298\) −33.8615 −1.96155
\(299\) 6.22952 0.360262
\(300\) 0 0
\(301\) −22.3014 −1.28543
\(302\) 17.9083 1.03051
\(303\) 0 0
\(304\) 17.3679 0.996117
\(305\) 5.48181 0.313887
\(306\) 0 0
\(307\) −11.2133 −0.639978 −0.319989 0.947421i \(-0.603679\pi\)
−0.319989 + 0.947421i \(0.603679\pi\)
\(308\) −24.5498 −1.39886
\(309\) 0 0
\(310\) 44.2234 2.51172
\(311\) −12.1228 −0.687423 −0.343712 0.939075i \(-0.611684\pi\)
−0.343712 + 0.939075i \(0.611684\pi\)
\(312\) 0 0
\(313\) 12.8050 0.723782 0.361891 0.932220i \(-0.382131\pi\)
0.361891 + 0.932220i \(0.382131\pi\)
\(314\) −17.4247 −0.983335
\(315\) 0 0
\(316\) 19.5722 1.10102
\(317\) 2.96789 0.166693 0.0833467 0.996521i \(-0.473439\pi\)
0.0833467 + 0.996521i \(0.473439\pi\)
\(318\) 0 0
\(319\) 1.43250 0.0802049
\(320\) −28.0486 −1.56797
\(321\) 0 0
\(322\) 19.9242 1.11033
\(323\) 30.7725 1.71223
\(324\) 0 0
\(325\) 1.79665 0.0996600
\(326\) −7.60865 −0.421404
\(327\) 0 0
\(328\) 11.2865 0.623194
\(329\) 27.0281 1.49011
\(330\) 0 0
\(331\) 35.0721 1.92773 0.963867 0.266383i \(-0.0858286\pi\)
0.963867 + 0.266383i \(0.0858286\pi\)
\(332\) 24.7494 1.35830
\(333\) 0 0
\(334\) 29.8480 1.63321
\(335\) −35.1979 −1.92307
\(336\) 0 0
\(337\) 12.4068 0.675840 0.337920 0.941175i \(-0.390277\pi\)
0.337920 + 0.941175i \(0.390277\pi\)
\(338\) 21.9581 1.19436
\(339\) 0 0
\(340\) −30.1880 −1.63718
\(341\) −33.8563 −1.83342
\(342\) 0 0
\(343\) 19.5830 1.05738
\(344\) −9.55498 −0.515170
\(345\) 0 0
\(346\) 12.5275 0.673485
\(347\) 25.6920 1.37922 0.689610 0.724181i \(-0.257782\pi\)
0.689610 + 0.724181i \(0.257782\pi\)
\(348\) 0 0
\(349\) −16.4397 −0.879996 −0.439998 0.897999i \(-0.645021\pi\)
−0.439998 + 0.897999i \(0.645021\pi\)
\(350\) 5.74630 0.307153
\(351\) 0 0
\(352\) 31.9267 1.70170
\(353\) −9.96346 −0.530302 −0.265151 0.964207i \(-0.585422\pi\)
−0.265151 + 0.964207i \(0.585422\pi\)
\(354\) 0 0
\(355\) 10.1801 0.540304
\(356\) −2.44319 −0.129489
\(357\) 0 0
\(358\) 39.3293 2.07862
\(359\) −5.08801 −0.268535 −0.134267 0.990945i \(-0.542868\pi\)
−0.134267 + 0.990945i \(0.542868\pi\)
\(360\) 0 0
\(361\) 20.5089 1.07942
\(362\) 1.95394 0.102697
\(363\) 0 0
\(364\) −9.95269 −0.521662
\(365\) 6.88491 0.360373
\(366\) 0 0
\(367\) 15.5615 0.812302 0.406151 0.913806i \(-0.366871\pi\)
0.406151 + 0.913806i \(0.366871\pi\)
\(368\) −10.5862 −0.551842
\(369\) 0 0
\(370\) 9.51430 0.494625
\(371\) 14.1871 0.736559
\(372\) 0 0
\(373\) −17.8554 −0.924519 −0.462259 0.886745i \(-0.652961\pi\)
−0.462259 + 0.886745i \(0.652961\pi\)
\(374\) 41.6326 2.15277
\(375\) 0 0
\(376\) 11.5801 0.597199
\(377\) 0.580748 0.0299101
\(378\) 0 0
\(379\) 25.1729 1.29304 0.646522 0.762896i \(-0.276223\pi\)
0.646522 + 0.762896i \(0.276223\pi\)
\(380\) −38.7586 −1.98827
\(381\) 0 0
\(382\) 32.0767 1.64119
\(383\) 21.4662 1.09687 0.548435 0.836194i \(-0.315224\pi\)
0.548435 + 0.836194i \(0.315224\pi\)
\(384\) 0 0
\(385\) −24.3059 −1.23874
\(386\) 48.1909 2.45285
\(387\) 0 0
\(388\) −42.0569 −2.13512
\(389\) −14.1828 −0.719096 −0.359548 0.933127i \(-0.617069\pi\)
−0.359548 + 0.933127i \(0.617069\pi\)
\(390\) 0 0
\(391\) −18.7566 −0.948561
\(392\) 1.03427 0.0522384
\(393\) 0 0
\(394\) −5.23877 −0.263925
\(395\) 19.3777 0.975000
\(396\) 0 0
\(397\) −8.60119 −0.431681 −0.215841 0.976429i \(-0.569249\pi\)
−0.215841 + 0.976429i \(0.569249\pi\)
\(398\) 11.3321 0.568027
\(399\) 0 0
\(400\) −3.05314 −0.152657
\(401\) 13.5162 0.674969 0.337485 0.941331i \(-0.390424\pi\)
0.337485 + 0.941331i \(0.390424\pi\)
\(402\) 0 0
\(403\) −13.7256 −0.683721
\(404\) −9.49410 −0.472349
\(405\) 0 0
\(406\) 1.85743 0.0921829
\(407\) −7.28391 −0.361050
\(408\) 0 0
\(409\) 26.6663 1.31856 0.659281 0.751896i \(-0.270861\pi\)
0.659281 + 0.751896i \(0.270861\pi\)
\(410\) 56.2668 2.77882
\(411\) 0 0
\(412\) 19.9904 0.984854
\(413\) −3.52557 −0.173482
\(414\) 0 0
\(415\) 24.5035 1.20283
\(416\) 12.9433 0.634599
\(417\) 0 0
\(418\) 53.4523 2.61444
\(419\) −7.37267 −0.360179 −0.180089 0.983650i \(-0.557639\pi\)
−0.180089 + 0.983650i \(0.557639\pi\)
\(420\) 0 0
\(421\) 36.1603 1.76235 0.881174 0.472793i \(-0.156754\pi\)
0.881174 + 0.472793i \(0.156754\pi\)
\(422\) 21.9189 1.06699
\(423\) 0 0
\(424\) 6.07844 0.295195
\(425\) −5.40956 −0.262402
\(426\) 0 0
\(427\) −5.44162 −0.263339
\(428\) 31.6588 1.53028
\(429\) 0 0
\(430\) −47.6345 −2.29714
\(431\) 13.4750 0.649066 0.324533 0.945874i \(-0.394793\pi\)
0.324533 + 0.945874i \(0.394793\pi\)
\(432\) 0 0
\(433\) −8.22943 −0.395481 −0.197741 0.980254i \(-0.563360\pi\)
−0.197741 + 0.980254i \(0.563360\pi\)
\(434\) −43.8992 −2.10723
\(435\) 0 0
\(436\) −46.8593 −2.24416
\(437\) −24.0817 −1.15198
\(438\) 0 0
\(439\) −1.65529 −0.0790028 −0.0395014 0.999220i \(-0.512577\pi\)
−0.0395014 + 0.999220i \(0.512577\pi\)
\(440\) −10.4138 −0.496458
\(441\) 0 0
\(442\) 16.8782 0.802812
\(443\) 15.9770 0.759092 0.379546 0.925173i \(-0.376080\pi\)
0.379546 + 0.925173i \(0.376080\pi\)
\(444\) 0 0
\(445\) −2.41892 −0.114668
\(446\) 21.7758 1.03112
\(447\) 0 0
\(448\) 27.8430 1.31546
\(449\) −13.7005 −0.646566 −0.323283 0.946302i \(-0.604787\pi\)
−0.323283 + 0.946302i \(0.604787\pi\)
\(450\) 0 0
\(451\) −43.0765 −2.02839
\(452\) 19.0000 0.893683
\(453\) 0 0
\(454\) −35.9319 −1.68637
\(455\) −9.85379 −0.461953
\(456\) 0 0
\(457\) 34.4962 1.61367 0.806833 0.590779i \(-0.201180\pi\)
0.806833 + 0.590779i \(0.201180\pi\)
\(458\) 61.5933 2.87806
\(459\) 0 0
\(460\) 23.6243 1.10149
\(461\) 30.3840 1.41513 0.707563 0.706651i \(-0.249795\pi\)
0.707563 + 0.706651i \(0.249795\pi\)
\(462\) 0 0
\(463\) −19.9606 −0.927648 −0.463824 0.885927i \(-0.653523\pi\)
−0.463824 + 0.885927i \(0.653523\pi\)
\(464\) −0.986897 −0.0458156
\(465\) 0 0
\(466\) −41.8477 −1.93856
\(467\) −8.39058 −0.388270 −0.194135 0.980975i \(-0.562190\pi\)
−0.194135 + 0.980975i \(0.562190\pi\)
\(468\) 0 0
\(469\) 34.9399 1.61337
\(470\) 57.7305 2.66291
\(471\) 0 0
\(472\) −1.51052 −0.0695273
\(473\) 36.4678 1.67679
\(474\) 0 0
\(475\) −6.94536 −0.318675
\(476\) 29.9667 1.37352
\(477\) 0 0
\(478\) −2.12029 −0.0969797
\(479\) −21.2295 −0.969999 −0.485000 0.874514i \(-0.661180\pi\)
−0.485000 + 0.874514i \(0.661180\pi\)
\(480\) 0 0
\(481\) −2.95295 −0.134643
\(482\) −22.2284 −1.01247
\(483\) 0 0
\(484\) 12.6927 0.576940
\(485\) −41.6390 −1.89073
\(486\) 0 0
\(487\) 5.32666 0.241374 0.120687 0.992691i \(-0.461490\pi\)
0.120687 + 0.992691i \(0.461490\pi\)
\(488\) −2.33145 −0.105540
\(489\) 0 0
\(490\) 5.15615 0.232931
\(491\) −24.0997 −1.08760 −0.543802 0.839213i \(-0.683016\pi\)
−0.543802 + 0.839213i \(0.683016\pi\)
\(492\) 0 0
\(493\) −1.74859 −0.0787523
\(494\) 21.6700 0.974977
\(495\) 0 0
\(496\) 23.3247 1.04731
\(497\) −10.1055 −0.453293
\(498\) 0 0
\(499\) −34.1034 −1.52668 −0.763339 0.645998i \(-0.776441\pi\)
−0.763339 + 0.645998i \(0.776441\pi\)
\(500\) −24.0177 −1.07410
\(501\) 0 0
\(502\) 25.7036 1.14721
\(503\) 10.6016 0.472704 0.236352 0.971667i \(-0.424048\pi\)
0.236352 + 0.971667i \(0.424048\pi\)
\(504\) 0 0
\(505\) −9.39976 −0.418284
\(506\) −32.5805 −1.44838
\(507\) 0 0
\(508\) 34.9555 1.55090
\(509\) −20.2990 −0.899736 −0.449868 0.893095i \(-0.648529\pi\)
−0.449868 + 0.893095i \(0.648529\pi\)
\(510\) 0 0
\(511\) −6.83444 −0.302338
\(512\) −27.8026 −1.22871
\(513\) 0 0
\(514\) 60.1893 2.65484
\(515\) 19.7917 0.872127
\(516\) 0 0
\(517\) −44.1970 −1.94378
\(518\) −9.44456 −0.414970
\(519\) 0 0
\(520\) −4.22183 −0.185140
\(521\) 1.16040 0.0508378 0.0254189 0.999677i \(-0.491908\pi\)
0.0254189 + 0.999677i \(0.491908\pi\)
\(522\) 0 0
\(523\) −19.3796 −0.847412 −0.423706 0.905800i \(-0.639271\pi\)
−0.423706 + 0.905800i \(0.639271\pi\)
\(524\) 10.1214 0.442154
\(525\) 0 0
\(526\) −29.8630 −1.30209
\(527\) 41.3267 1.80022
\(528\) 0 0
\(529\) −8.32163 −0.361810
\(530\) 30.3029 1.31628
\(531\) 0 0
\(532\) 38.4744 1.66808
\(533\) −17.4635 −0.756429
\(534\) 0 0
\(535\) 31.3442 1.35513
\(536\) 14.9699 0.646602
\(537\) 0 0
\(538\) 46.9059 2.02226
\(539\) −3.94742 −0.170027
\(540\) 0 0
\(541\) −4.77487 −0.205288 −0.102644 0.994718i \(-0.532730\pi\)
−0.102644 + 0.994718i \(0.532730\pi\)
\(542\) 0.392638 0.0168652
\(543\) 0 0
\(544\) −38.9713 −1.67088
\(545\) −46.3937 −1.98729
\(546\) 0 0
\(547\) −12.9501 −0.553704 −0.276852 0.960913i \(-0.589291\pi\)
−0.276852 + 0.960913i \(0.589291\pi\)
\(548\) 38.7684 1.65610
\(549\) 0 0
\(550\) −9.39649 −0.400668
\(551\) −2.24502 −0.0956410
\(552\) 0 0
\(553\) −19.2357 −0.817985
\(554\) 33.6847 1.43113
\(555\) 0 0
\(556\) 2.72525 0.115576
\(557\) −15.2371 −0.645618 −0.322809 0.946464i \(-0.604627\pi\)
−0.322809 + 0.946464i \(0.604627\pi\)
\(558\) 0 0
\(559\) 14.7843 0.625310
\(560\) 16.7451 0.707609
\(561\) 0 0
\(562\) 42.3979 1.78845
\(563\) 44.4294 1.87247 0.936237 0.351368i \(-0.114284\pi\)
0.936237 + 0.351368i \(0.114284\pi\)
\(564\) 0 0
\(565\) 18.8112 0.791392
\(566\) 63.9768 2.68915
\(567\) 0 0
\(568\) −4.32967 −0.181669
\(569\) 32.3049 1.35429 0.677146 0.735848i \(-0.263216\pi\)
0.677146 + 0.735848i \(0.263216\pi\)
\(570\) 0 0
\(571\) −17.6597 −0.739036 −0.369518 0.929224i \(-0.620477\pi\)
−0.369518 + 0.929224i \(0.620477\pi\)
\(572\) 16.2749 0.680487
\(573\) 0 0
\(574\) −55.8544 −2.33132
\(575\) 4.23337 0.176544
\(576\) 0 0
\(577\) −5.92655 −0.246726 −0.123363 0.992362i \(-0.539368\pi\)
−0.123363 + 0.992362i \(0.539368\pi\)
\(578\) −14.7739 −0.614512
\(579\) 0 0
\(580\) 2.20238 0.0914489
\(581\) −24.3239 −1.00913
\(582\) 0 0
\(583\) −23.1992 −0.960811
\(584\) −2.92820 −0.121170
\(585\) 0 0
\(586\) −0.973544 −0.0402167
\(587\) −25.8268 −1.06599 −0.532993 0.846120i \(-0.678933\pi\)
−0.532993 + 0.846120i \(0.678933\pi\)
\(588\) 0 0
\(589\) 53.0596 2.18628
\(590\) −7.53041 −0.310022
\(591\) 0 0
\(592\) 5.01811 0.206243
\(593\) 9.21060 0.378234 0.189117 0.981955i \(-0.439437\pi\)
0.189117 + 0.981955i \(0.439437\pi\)
\(594\) 0 0
\(595\) 29.6690 1.21631
\(596\) 39.8557 1.63255
\(597\) 0 0
\(598\) −13.2084 −0.540131
\(599\) 43.4301 1.77450 0.887252 0.461285i \(-0.152611\pi\)
0.887252 + 0.461285i \(0.152611\pi\)
\(600\) 0 0
\(601\) −10.2416 −0.417765 −0.208883 0.977941i \(-0.566983\pi\)
−0.208883 + 0.977941i \(0.566983\pi\)
\(602\) 47.2854 1.92721
\(603\) 0 0
\(604\) −21.0784 −0.857669
\(605\) 12.5666 0.510903
\(606\) 0 0
\(607\) 45.2285 1.83577 0.917884 0.396849i \(-0.129896\pi\)
0.917884 + 0.396849i \(0.129896\pi\)
\(608\) −50.0355 −2.02921
\(609\) 0 0
\(610\) −11.6230 −0.470602
\(611\) −17.9178 −0.724876
\(612\) 0 0
\(613\) 20.4029 0.824064 0.412032 0.911169i \(-0.364819\pi\)
0.412032 + 0.911169i \(0.364819\pi\)
\(614\) 23.7755 0.959500
\(615\) 0 0
\(616\) 10.3375 0.416508
\(617\) −4.40978 −0.177531 −0.0887655 0.996053i \(-0.528292\pi\)
−0.0887655 + 0.996053i \(0.528292\pi\)
\(618\) 0 0
\(619\) 20.8034 0.836160 0.418080 0.908410i \(-0.362703\pi\)
0.418080 + 0.908410i \(0.362703\pi\)
\(620\) −52.0518 −2.09045
\(621\) 0 0
\(622\) 25.7039 1.03063
\(623\) 2.40119 0.0962015
\(624\) 0 0
\(625\) −29.3039 −1.17215
\(626\) −27.1503 −1.08514
\(627\) 0 0
\(628\) 20.5093 0.818409
\(629\) 8.89109 0.354511
\(630\) 0 0
\(631\) 33.5580 1.33592 0.667962 0.744196i \(-0.267167\pi\)
0.667962 + 0.744196i \(0.267167\pi\)
\(632\) −8.24149 −0.327829
\(633\) 0 0
\(634\) −6.29278 −0.249918
\(635\) 34.6081 1.37338
\(636\) 0 0
\(637\) −1.60031 −0.0634067
\(638\) −3.03732 −0.120249
\(639\) 0 0
\(640\) 20.1341 0.795869
\(641\) −24.5383 −0.969206 −0.484603 0.874734i \(-0.661036\pi\)
−0.484603 + 0.874734i \(0.661036\pi\)
\(642\) 0 0
\(643\) 9.18123 0.362073 0.181036 0.983476i \(-0.442055\pi\)
0.181036 + 0.983476i \(0.442055\pi\)
\(644\) −23.4511 −0.924104
\(645\) 0 0
\(646\) −65.2465 −2.56709
\(647\) −33.3174 −1.30984 −0.654922 0.755697i \(-0.727298\pi\)
−0.654922 + 0.755697i \(0.727298\pi\)
\(648\) 0 0
\(649\) 5.76509 0.226300
\(650\) −3.80941 −0.149417
\(651\) 0 0
\(652\) 8.95553 0.350726
\(653\) 8.78496 0.343782 0.171891 0.985116i \(-0.445012\pi\)
0.171891 + 0.985116i \(0.445012\pi\)
\(654\) 0 0
\(655\) 10.0208 0.391545
\(656\) 29.6767 1.15868
\(657\) 0 0
\(658\) −57.3073 −2.23407
\(659\) 27.9953 1.09054 0.545270 0.838260i \(-0.316427\pi\)
0.545270 + 0.838260i \(0.316427\pi\)
\(660\) 0 0
\(661\) −31.7412 −1.23459 −0.617294 0.786732i \(-0.711771\pi\)
−0.617294 + 0.786732i \(0.711771\pi\)
\(662\) −74.3628 −2.89019
\(663\) 0 0
\(664\) −10.4215 −0.404433
\(665\) 38.0921 1.47715
\(666\) 0 0
\(667\) 1.36839 0.0529844
\(668\) −35.1316 −1.35928
\(669\) 0 0
\(670\) 74.6297 2.88320
\(671\) 8.89828 0.343514
\(672\) 0 0
\(673\) −0.521763 −0.0201125 −0.0100562 0.999949i \(-0.503201\pi\)
−0.0100562 + 0.999949i \(0.503201\pi\)
\(674\) −26.3059 −1.01327
\(675\) 0 0
\(676\) −25.8451 −0.994042
\(677\) 2.22663 0.0855765 0.0427882 0.999084i \(-0.486376\pi\)
0.0427882 + 0.999084i \(0.486376\pi\)
\(678\) 0 0
\(679\) 41.3338 1.58624
\(680\) 12.7116 0.487467
\(681\) 0 0
\(682\) 71.7852 2.74880
\(683\) 18.9101 0.723577 0.361788 0.932260i \(-0.382166\pi\)
0.361788 + 0.932260i \(0.382166\pi\)
\(684\) 0 0
\(685\) 38.3831 1.46654
\(686\) −41.5215 −1.58530
\(687\) 0 0
\(688\) −25.1238 −0.957836
\(689\) −9.40511 −0.358306
\(690\) 0 0
\(691\) −18.7431 −0.713022 −0.356511 0.934291i \(-0.616034\pi\)
−0.356511 + 0.934291i \(0.616034\pi\)
\(692\) −14.7452 −0.560527
\(693\) 0 0
\(694\) −54.4745 −2.06782
\(695\) 2.69817 0.102347
\(696\) 0 0
\(697\) 52.5812 1.99166
\(698\) 34.8568 1.31935
\(699\) 0 0
\(700\) −6.76351 −0.255637
\(701\) −16.1798 −0.611102 −0.305551 0.952176i \(-0.598841\pi\)
−0.305551 + 0.952176i \(0.598841\pi\)
\(702\) 0 0
\(703\) 11.4153 0.430537
\(704\) −45.5296 −1.71596
\(705\) 0 0
\(706\) 21.1254 0.795065
\(707\) 9.33086 0.350923
\(708\) 0 0
\(709\) −8.57907 −0.322194 −0.161097 0.986939i \(-0.551503\pi\)
−0.161097 + 0.986939i \(0.551503\pi\)
\(710\) −21.5848 −0.810062
\(711\) 0 0
\(712\) 1.02878 0.0385552
\(713\) −32.3411 −1.21118
\(714\) 0 0
\(715\) 16.1132 0.602598
\(716\) −46.2914 −1.72999
\(717\) 0 0
\(718\) 10.7880 0.402606
\(719\) 32.5343 1.21333 0.606663 0.794959i \(-0.292508\pi\)
0.606663 + 0.794959i \(0.292508\pi\)
\(720\) 0 0
\(721\) −19.6466 −0.731679
\(722\) −43.4848 −1.61834
\(723\) 0 0
\(724\) −2.29982 −0.0854723
\(725\) 0.394657 0.0146572
\(726\) 0 0
\(727\) −4.65810 −0.172759 −0.0863797 0.996262i \(-0.527530\pi\)
−0.0863797 + 0.996262i \(0.527530\pi\)
\(728\) 4.19088 0.155325
\(729\) 0 0
\(730\) −14.5980 −0.540296
\(731\) −44.5144 −1.64642
\(732\) 0 0
\(733\) 8.64374 0.319264 0.159632 0.987177i \(-0.448969\pi\)
0.159632 + 0.987177i \(0.448969\pi\)
\(734\) −32.9948 −1.21786
\(735\) 0 0
\(736\) 30.4979 1.12417
\(737\) −57.1346 −2.10458
\(738\) 0 0
\(739\) 42.9217 1.57890 0.789450 0.613815i \(-0.210366\pi\)
0.789450 + 0.613815i \(0.210366\pi\)
\(740\) −11.1985 −0.411666
\(741\) 0 0
\(742\) −30.0808 −1.10430
\(743\) −39.0878 −1.43399 −0.716997 0.697076i \(-0.754484\pi\)
−0.716997 + 0.697076i \(0.754484\pi\)
\(744\) 0 0
\(745\) 39.4596 1.44569
\(746\) 37.8586 1.38610
\(747\) 0 0
\(748\) −49.0024 −1.79170
\(749\) −31.1144 −1.13690
\(750\) 0 0
\(751\) 33.4998 1.22243 0.611213 0.791466i \(-0.290682\pi\)
0.611213 + 0.791466i \(0.290682\pi\)
\(752\) 30.4487 1.11035
\(753\) 0 0
\(754\) −1.23135 −0.0448432
\(755\) −20.8690 −0.759500
\(756\) 0 0
\(757\) 42.7926 1.55533 0.777663 0.628682i \(-0.216405\pi\)
0.777663 + 0.628682i \(0.216405\pi\)
\(758\) −53.3737 −1.93862
\(759\) 0 0
\(760\) 16.3205 0.592006
\(761\) −3.50389 −0.127016 −0.0635079 0.997981i \(-0.520229\pi\)
−0.0635079 + 0.997981i \(0.520229\pi\)
\(762\) 0 0
\(763\) 46.0536 1.66725
\(764\) −37.7549 −1.36592
\(765\) 0 0
\(766\) −45.5144 −1.64450
\(767\) 2.33721 0.0843919
\(768\) 0 0
\(769\) −33.9827 −1.22545 −0.612724 0.790297i \(-0.709926\pi\)
−0.612724 + 0.790297i \(0.709926\pi\)
\(770\) 51.5355 1.85721
\(771\) 0 0
\(772\) −56.7216 −2.04146
\(773\) −16.2523 −0.584553 −0.292277 0.956334i \(-0.594413\pi\)
−0.292277 + 0.956334i \(0.594413\pi\)
\(774\) 0 0
\(775\) −9.32746 −0.335052
\(776\) 17.7094 0.635729
\(777\) 0 0
\(778\) 30.0716 1.07812
\(779\) 67.5093 2.41877
\(780\) 0 0
\(781\) 16.5247 0.591302
\(782\) 39.7693 1.42215
\(783\) 0 0
\(784\) 2.71950 0.0971250
\(785\) 20.3055 0.724733
\(786\) 0 0
\(787\) −10.9298 −0.389606 −0.194803 0.980842i \(-0.562407\pi\)
−0.194803 + 0.980842i \(0.562407\pi\)
\(788\) 6.16613 0.219659
\(789\) 0 0
\(790\) −41.0864 −1.46179
\(791\) −18.6733 −0.663945
\(792\) 0 0
\(793\) 3.60743 0.128104
\(794\) 18.2370 0.647207
\(795\) 0 0
\(796\) −13.3381 −0.472757
\(797\) 17.4122 0.616772 0.308386 0.951261i \(-0.400211\pi\)
0.308386 + 0.951261i \(0.400211\pi\)
\(798\) 0 0
\(799\) 53.9490 1.90858
\(800\) 8.79585 0.310980
\(801\) 0 0
\(802\) −28.6583 −1.01196
\(803\) 11.1758 0.394387
\(804\) 0 0
\(805\) −23.2181 −0.818330
\(806\) 29.1022 1.02508
\(807\) 0 0
\(808\) 3.99778 0.140642
\(809\) 21.8841 0.769403 0.384702 0.923041i \(-0.374304\pi\)
0.384702 + 0.923041i \(0.374304\pi\)
\(810\) 0 0
\(811\) 41.8415 1.46925 0.734627 0.678471i \(-0.237357\pi\)
0.734627 + 0.678471i \(0.237357\pi\)
\(812\) −2.18624 −0.0767218
\(813\) 0 0
\(814\) 15.4440 0.541311
\(815\) 8.86655 0.310582
\(816\) 0 0
\(817\) −57.1522 −1.99950
\(818\) −56.5402 −1.97688
\(819\) 0 0
\(820\) −66.2272 −2.31275
\(821\) −36.7094 −1.28117 −0.640584 0.767888i \(-0.721308\pi\)
−0.640584 + 0.767888i \(0.721308\pi\)
\(822\) 0 0
\(823\) −31.0617 −1.08274 −0.541371 0.840784i \(-0.682095\pi\)
−0.541371 + 0.840784i \(0.682095\pi\)
\(824\) −8.41755 −0.293239
\(825\) 0 0
\(826\) 7.47521 0.260096
\(827\) −55.1422 −1.91748 −0.958741 0.284281i \(-0.908245\pi\)
−0.958741 + 0.284281i \(0.908245\pi\)
\(828\) 0 0
\(829\) 11.6426 0.404366 0.202183 0.979348i \(-0.435196\pi\)
0.202183 + 0.979348i \(0.435196\pi\)
\(830\) −51.9545 −1.80337
\(831\) 0 0
\(832\) −18.4580 −0.639917
\(833\) 4.81841 0.166948
\(834\) 0 0
\(835\) −34.7825 −1.20370
\(836\) −62.9144 −2.17594
\(837\) 0 0
\(838\) 15.6322 0.540005
\(839\) −9.99713 −0.345139 −0.172570 0.984997i \(-0.555207\pi\)
−0.172570 + 0.984997i \(0.555207\pi\)
\(840\) 0 0
\(841\) −28.8724 −0.995601
\(842\) −76.6703 −2.64223
\(843\) 0 0
\(844\) −25.7989 −0.888036
\(845\) −25.5883 −0.880264
\(846\) 0 0
\(847\) −12.4744 −0.428627
\(848\) 15.9826 0.548846
\(849\) 0 0
\(850\) 11.4698 0.393412
\(851\) −6.95792 −0.238514
\(852\) 0 0
\(853\) 41.0682 1.40615 0.703073 0.711117i \(-0.251811\pi\)
0.703073 + 0.711117i \(0.251811\pi\)
\(854\) 11.5378 0.394816
\(855\) 0 0
\(856\) −13.3309 −0.455641
\(857\) 22.0503 0.753224 0.376612 0.926371i \(-0.377089\pi\)
0.376612 + 0.926371i \(0.377089\pi\)
\(858\) 0 0
\(859\) 5.68153 0.193851 0.0969256 0.995292i \(-0.469099\pi\)
0.0969256 + 0.995292i \(0.469099\pi\)
\(860\) 56.0668 1.91186
\(861\) 0 0
\(862\) −28.5708 −0.973125
\(863\) −27.0640 −0.921269 −0.460635 0.887590i \(-0.652378\pi\)
−0.460635 + 0.887590i \(0.652378\pi\)
\(864\) 0 0
\(865\) −14.5986 −0.496369
\(866\) 17.4488 0.592933
\(867\) 0 0
\(868\) 51.6703 1.75380
\(869\) 31.4547 1.06703
\(870\) 0 0
\(871\) −23.1628 −0.784841
\(872\) 19.7316 0.668195
\(873\) 0 0
\(874\) 51.0601 1.72713
\(875\) 23.6048 0.797986
\(876\) 0 0
\(877\) −31.5393 −1.06501 −0.532503 0.846428i \(-0.678749\pi\)
−0.532503 + 0.846428i \(0.678749\pi\)
\(878\) 3.50970 0.118447
\(879\) 0 0
\(880\) −27.3820 −0.923047
\(881\) 9.23845 0.311251 0.155626 0.987816i \(-0.450261\pi\)
0.155626 + 0.987816i \(0.450261\pi\)
\(882\) 0 0
\(883\) −22.5049 −0.757350 −0.378675 0.925530i \(-0.623620\pi\)
−0.378675 + 0.925530i \(0.623620\pi\)
\(884\) −19.8659 −0.668163
\(885\) 0 0
\(886\) −33.8759 −1.13808
\(887\) 30.2960 1.01724 0.508621 0.860991i \(-0.330156\pi\)
0.508621 + 0.860991i \(0.330156\pi\)
\(888\) 0 0
\(889\) −34.3545 −1.15221
\(890\) 5.12880 0.171918
\(891\) 0 0
\(892\) −25.6306 −0.858176
\(893\) 69.2654 2.31788
\(894\) 0 0
\(895\) −45.8314 −1.53198
\(896\) −19.9865 −0.667701
\(897\) 0 0
\(898\) 29.0490 0.969377
\(899\) −3.01501 −0.100556
\(900\) 0 0
\(901\) 28.3180 0.943410
\(902\) 91.3345 3.04111
\(903\) 0 0
\(904\) −8.00052 −0.266093
\(905\) −2.27697 −0.0756891
\(906\) 0 0
\(907\) −34.6786 −1.15148 −0.575742 0.817632i \(-0.695287\pi\)
−0.575742 + 0.817632i \(0.695287\pi\)
\(908\) 42.2925 1.40353
\(909\) 0 0
\(910\) 20.8929 0.692592
\(911\) 29.6688 0.982970 0.491485 0.870886i \(-0.336454\pi\)
0.491485 + 0.870886i \(0.336454\pi\)
\(912\) 0 0
\(913\) 39.7750 1.31636
\(914\) −73.1420 −2.41932
\(915\) 0 0
\(916\) −72.4965 −2.39535
\(917\) −9.94733 −0.328490
\(918\) 0 0
\(919\) 3.41218 0.112558 0.0562788 0.998415i \(-0.482076\pi\)
0.0562788 + 0.998415i \(0.482076\pi\)
\(920\) −9.94774 −0.327967
\(921\) 0 0
\(922\) −64.4229 −2.12165
\(923\) 6.69925 0.220509
\(924\) 0 0
\(925\) −2.00672 −0.0659807
\(926\) 42.3222 1.39079
\(927\) 0 0
\(928\) 2.84317 0.0933317
\(929\) 5.27469 0.173057 0.0865285 0.996249i \(-0.472423\pi\)
0.0865285 + 0.996249i \(0.472423\pi\)
\(930\) 0 0
\(931\) 6.18638 0.202750
\(932\) 49.2555 1.61342
\(933\) 0 0
\(934\) 17.7905 0.582121
\(935\) −48.5154 −1.58662
\(936\) 0 0
\(937\) −11.6027 −0.379043 −0.189521 0.981877i \(-0.560694\pi\)
−0.189521 + 0.981877i \(0.560694\pi\)
\(938\) −74.0826 −2.41888
\(939\) 0 0
\(940\) −67.9499 −2.21628
\(941\) 28.2271 0.920178 0.460089 0.887873i \(-0.347818\pi\)
0.460089 + 0.887873i \(0.347818\pi\)
\(942\) 0 0
\(943\) −41.1486 −1.33998
\(944\) −3.97175 −0.129270
\(945\) 0 0
\(946\) −77.3222 −2.51396
\(947\) 41.9537 1.36331 0.681657 0.731672i \(-0.261260\pi\)
0.681657 + 0.731672i \(0.261260\pi\)
\(948\) 0 0
\(949\) 4.53077 0.147075
\(950\) 14.7262 0.477780
\(951\) 0 0
\(952\) −12.6184 −0.408965
\(953\) 51.3228 1.66251 0.831255 0.555892i \(-0.187623\pi\)
0.831255 + 0.555892i \(0.187623\pi\)
\(954\) 0 0
\(955\) −37.3798 −1.20958
\(956\) 2.49562 0.0807141
\(957\) 0 0
\(958\) 45.0126 1.45429
\(959\) −38.1018 −1.23037
\(960\) 0 0
\(961\) 40.2578 1.29864
\(962\) 6.26111 0.201866
\(963\) 0 0
\(964\) 26.1632 0.842660
\(965\) −56.1580 −1.80779
\(966\) 0 0
\(967\) −15.0557 −0.484158 −0.242079 0.970257i \(-0.577829\pi\)
−0.242079 + 0.970257i \(0.577829\pi\)
\(968\) −5.34464 −0.171783
\(969\) 0 0
\(970\) 88.2867 2.83471
\(971\) −18.2675 −0.586232 −0.293116 0.956077i \(-0.594692\pi\)
−0.293116 + 0.956077i \(0.594692\pi\)
\(972\) 0 0
\(973\) −2.67839 −0.0858651
\(974\) −11.2940 −0.361885
\(975\) 0 0
\(976\) −6.13030 −0.196226
\(977\) −11.8112 −0.377873 −0.188936 0.981989i \(-0.560504\pi\)
−0.188936 + 0.981989i \(0.560504\pi\)
\(978\) 0 0
\(979\) −3.92648 −0.125491
\(980\) −6.06889 −0.193864
\(981\) 0 0
\(982\) 51.0983 1.63061
\(983\) 30.4350 0.970725 0.485363 0.874313i \(-0.338688\pi\)
0.485363 + 0.874313i \(0.338688\pi\)
\(984\) 0 0
\(985\) 6.10486 0.194517
\(986\) 3.70750 0.118071
\(987\) 0 0
\(988\) −25.5060 −0.811453
\(989\) 34.8357 1.10771
\(990\) 0 0
\(991\) 54.3952 1.72792 0.863960 0.503561i \(-0.167977\pi\)
0.863960 + 0.503561i \(0.167977\pi\)
\(992\) −67.1965 −2.13349
\(993\) 0 0
\(994\) 21.4265 0.679608
\(995\) −13.2056 −0.418645
\(996\) 0 0
\(997\) 14.6453 0.463821 0.231910 0.972737i \(-0.425502\pi\)
0.231910 + 0.972737i \(0.425502\pi\)
\(998\) 72.3090 2.28890
\(999\) 0 0
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 2151.2.a.e.1.1 6
3.2 odd 2 717.2.a.d.1.6 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
717.2.a.d.1.6 6 3.2 odd 2
2151.2.a.e.1.1 6 1.1 even 1 trivial