Properties

Label 2790.2.a.bk.1.2
Level $2790$
Weight $2$
Character 2790.1
Self dual yes
Analytic conductor $22.278$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2790 = 2 \cdot 3^{2} \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2790.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.2782621639\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.17428.1
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 4x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.787711\) of defining polynomial
Character \(\chi\) \(=\) 2790.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -0.662077 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -0.662077 q^{7} +1.00000 q^{8} +1.00000 q^{10} +2.66208 q^{11} +3.57542 q^{13} -0.662077 q^{14} +1.00000 q^{16} -3.18360 q^{17} -2.23750 q^{19} +1.00000 q^{20} +2.66208 q^{22} +7.42110 q^{23} +1.00000 q^{25} +3.57542 q^{26} -0.662077 q^{28} -3.15084 q^{29} +1.00000 q^{31} +1.00000 q^{32} -3.18360 q^{34} -0.662077 q^{35} +9.94263 q^{37} -2.23750 q^{38} +1.00000 q^{40} -3.15084 q^{41} -3.28055 q^{43} +2.66208 q^{44} +7.42110 q^{46} +9.51805 q^{47} -6.56165 q^{49} +1.00000 q^{50} +3.57542 q^{52} -1.08666 q^{53} +2.66208 q^{55} -0.662077 q^{56} -3.15084 q^{58} -2.61847 q^{59} +9.83191 q^{61} +1.00000 q^{62} +1.00000 q^{64} +3.57542 q^{65} -14.4176 q^{67} -3.18360 q^{68} -0.662077 q^{70} +5.81292 q^{71} +11.5617 q^{73} +9.94263 q^{74} -2.23750 q^{76} -1.76250 q^{77} +15.7555 q^{79} +1.00000 q^{80} -3.15084 q^{82} -14.8422 q^{83} -3.18360 q^{85} -3.28055 q^{86} +2.66208 q^{88} -1.61903 q^{89} -2.36721 q^{91} +7.42110 q^{92} +9.51805 q^{94} -2.23750 q^{95} +10.4750 q^{97} -6.56165 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 5 q^{7} + 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 4 q^{4} + 4 q^{5} + 5 q^{7} + 4 q^{8} + 4 q^{10} + 3 q^{11} + 6 q^{13} + 5 q^{14} + 4 q^{16} + 7 q^{19} + 4 q^{20} + 3 q^{22} + q^{23} + 4 q^{25} + 6 q^{26} + 5 q^{28} + 4 q^{29} + 4 q^{31} + 4 q^{32} + 5 q^{35} + 6 q^{37} + 7 q^{38} + 4 q^{40} + 4 q^{41} + 13 q^{43} + 3 q^{44} + q^{46} - 4 q^{47} + 5 q^{49} + 4 q^{50} + 6 q^{52} - 5 q^{53} + 3 q^{55} + 5 q^{56} + 4 q^{58} + 8 q^{59} - 4 q^{61} + 4 q^{62} + 4 q^{64} + 6 q^{65} + 8 q^{67} + 5 q^{70} - q^{71} + 15 q^{73} + 6 q^{74} + 7 q^{76} - 23 q^{77} + 5 q^{79} + 4 q^{80} + 4 q^{82} - 2 q^{83} + 13 q^{86} + 3 q^{88} - 9 q^{89} + 16 q^{91} + q^{92} - 4 q^{94} + 7 q^{95} + 10 q^{97} + 5 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −0.662077 −0.250242 −0.125121 0.992142i \(-0.539932\pi\)
−0.125121 + 0.992142i \(0.539932\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 2.66208 0.802647 0.401323 0.915936i \(-0.368550\pi\)
0.401323 + 0.915936i \(0.368550\pi\)
\(12\) 0 0
\(13\) 3.57542 0.991643 0.495822 0.868424i \(-0.334867\pi\)
0.495822 + 0.868424i \(0.334867\pi\)
\(14\) −0.662077 −0.176948
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.18360 −0.772137 −0.386069 0.922470i \(-0.626167\pi\)
−0.386069 + 0.922470i \(0.626167\pi\)
\(18\) 0 0
\(19\) −2.23750 −0.513317 −0.256659 0.966502i \(-0.582622\pi\)
−0.256659 + 0.966502i \(0.582622\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) 2.66208 0.567557
\(23\) 7.42110 1.54741 0.773703 0.633548i \(-0.218402\pi\)
0.773703 + 0.633548i \(0.218402\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 3.57542 0.701198
\(27\) 0 0
\(28\) −0.662077 −0.125121
\(29\) −3.15084 −0.585097 −0.292548 0.956251i \(-0.594503\pi\)
−0.292548 + 0.956251i \(0.594503\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −3.18360 −0.545983
\(35\) −0.662077 −0.111912
\(36\) 0 0
\(37\) 9.94263 1.63456 0.817279 0.576242i \(-0.195482\pi\)
0.817279 + 0.576242i \(0.195482\pi\)
\(38\) −2.23750 −0.362970
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −3.15084 −0.492079 −0.246039 0.969260i \(-0.579129\pi\)
−0.246039 + 0.969260i \(0.579129\pi\)
\(42\) 0 0
\(43\) −3.28055 −0.500279 −0.250140 0.968210i \(-0.580477\pi\)
−0.250140 + 0.968210i \(0.580477\pi\)
\(44\) 2.66208 0.401323
\(45\) 0 0
\(46\) 7.42110 1.09418
\(47\) 9.51805 1.38835 0.694175 0.719806i \(-0.255769\pi\)
0.694175 + 0.719806i \(0.255769\pi\)
\(48\) 0 0
\(49\) −6.56165 −0.937379
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 3.57542 0.495822
\(53\) −1.08666 −0.149264 −0.0746319 0.997211i \(-0.523778\pi\)
−0.0746319 + 0.997211i \(0.523778\pi\)
\(54\) 0 0
\(55\) 2.66208 0.358954
\(56\) −0.662077 −0.0884738
\(57\) 0 0
\(58\) −3.15084 −0.413726
\(59\) −2.61847 −0.340896 −0.170448 0.985367i \(-0.554521\pi\)
−0.170448 + 0.985367i \(0.554521\pi\)
\(60\) 0 0
\(61\) 9.83191 1.25885 0.629424 0.777062i \(-0.283291\pi\)
0.629424 + 0.777062i \(0.283291\pi\)
\(62\) 1.00000 0.127000
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.57542 0.443476
\(66\) 0 0
\(67\) −14.4176 −1.76139 −0.880697 0.473681i \(-0.842925\pi\)
−0.880697 + 0.473681i \(0.842925\pi\)
\(68\) −3.18360 −0.386069
\(69\) 0 0
\(70\) −0.662077 −0.0791334
\(71\) 5.81292 0.689867 0.344933 0.938627i \(-0.387902\pi\)
0.344933 + 0.938627i \(0.387902\pi\)
\(72\) 0 0
\(73\) 11.5617 1.35319 0.676595 0.736356i \(-0.263455\pi\)
0.676595 + 0.736356i \(0.263455\pi\)
\(74\) 9.94263 1.15581
\(75\) 0 0
\(76\) −2.23750 −0.256659
\(77\) −1.76250 −0.200856
\(78\) 0 0
\(79\) 15.7555 1.77264 0.886319 0.463076i \(-0.153254\pi\)
0.886319 + 0.463076i \(0.153254\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −3.15084 −0.347952
\(83\) −14.8422 −1.62914 −0.814572 0.580063i \(-0.803028\pi\)
−0.814572 + 0.580063i \(0.803028\pi\)
\(84\) 0 0
\(85\) −3.18360 −0.345310
\(86\) −3.28055 −0.353751
\(87\) 0 0
\(88\) 2.66208 0.283778
\(89\) −1.61903 −0.171616 −0.0858082 0.996312i \(-0.527347\pi\)
−0.0858082 + 0.996312i \(0.527347\pi\)
\(90\) 0 0
\(91\) −2.36721 −0.248151
\(92\) 7.42110 0.773703
\(93\) 0 0
\(94\) 9.51805 0.981712
\(95\) −2.23750 −0.229563
\(96\) 0 0
\(97\) 10.4750 1.06357 0.531787 0.846878i \(-0.321521\pi\)
0.531787 + 0.846878i \(0.321521\pi\)
\(98\) −6.56165 −0.662827
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 10.6047 1.05521 0.527604 0.849491i \(-0.323091\pi\)
0.527604 + 0.849491i \(0.323091\pi\)
\(102\) 0 0
\(103\) 10.6185 1.04627 0.523135 0.852250i \(-0.324763\pi\)
0.523135 + 0.852250i \(0.324763\pi\)
\(104\) 3.57542 0.350599
\(105\) 0 0
\(106\) −1.08666 −0.105545
\(107\) −11.2805 −1.09053 −0.545266 0.838263i \(-0.683571\pi\)
−0.545266 + 0.838263i \(0.683571\pi\)
\(108\) 0 0
\(109\) −2.19389 −0.210137 −0.105068 0.994465i \(-0.533506\pi\)
−0.105068 + 0.994465i \(0.533506\pi\)
\(110\) 2.66208 0.253819
\(111\) 0 0
\(112\) −0.662077 −0.0625604
\(113\) −19.0797 −1.79487 −0.897434 0.441149i \(-0.854571\pi\)
−0.897434 + 0.441149i \(0.854571\pi\)
\(114\) 0 0
\(115\) 7.42110 0.692021
\(116\) −3.15084 −0.292548
\(117\) 0 0
\(118\) −2.61847 −0.241050
\(119\) 2.10779 0.193221
\(120\) 0 0
\(121\) −3.91334 −0.355759
\(122\) 9.83191 0.890140
\(123\) 0 0
\(124\) 1.00000 0.0898027
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 6.72626 0.596859 0.298430 0.954432i \(-0.403537\pi\)
0.298430 + 0.954432i \(0.403537\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 3.57542 0.313585
\(131\) 18.5909 1.62430 0.812149 0.583450i \(-0.198298\pi\)
0.812149 + 0.583450i \(0.198298\pi\)
\(132\) 0 0
\(133\) 1.48140 0.128453
\(134\) −14.4176 −1.24549
\(135\) 0 0
\(136\) −3.18360 −0.272992
\(137\) −22.3069 −1.90581 −0.952904 0.303272i \(-0.901921\pi\)
−0.952904 + 0.303272i \(0.901921\pi\)
\(138\) 0 0
\(139\) 4.14055 0.351197 0.175599 0.984462i \(-0.443814\pi\)
0.175599 + 0.984462i \(0.443814\pi\)
\(140\) −0.662077 −0.0559558
\(141\) 0 0
\(142\) 5.81292 0.487809
\(143\) 9.51805 0.795939
\(144\) 0 0
\(145\) −3.15084 −0.261663
\(146\) 11.5617 0.956849
\(147\) 0 0
\(148\) 9.94263 0.817279
\(149\) 8.43139 0.690727 0.345363 0.938469i \(-0.387756\pi\)
0.345363 + 0.938469i \(0.387756\pi\)
\(150\) 0 0
\(151\) 2.36721 0.192640 0.0963202 0.995350i \(-0.469293\pi\)
0.0963202 + 0.995350i \(0.469293\pi\)
\(152\) −2.23750 −0.181485
\(153\) 0 0
\(154\) −1.76250 −0.142026
\(155\) 1.00000 0.0803219
\(156\) 0 0
\(157\) −17.0797 −1.36311 −0.681554 0.731768i \(-0.738696\pi\)
−0.681554 + 0.731768i \(0.738696\pi\)
\(158\) 15.7555 1.25344
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −4.91334 −0.387226
\(162\) 0 0
\(163\) 25.0935 1.96547 0.982736 0.185013i \(-0.0592327\pi\)
0.982736 + 0.185013i \(0.0592327\pi\)
\(164\) −3.15084 −0.246039
\(165\) 0 0
\(166\) −14.8422 −1.15198
\(167\) −6.59948 −0.510683 −0.255342 0.966851i \(-0.582188\pi\)
−0.255342 + 0.966851i \(0.582188\pi\)
\(168\) 0 0
\(169\) −0.216363 −0.0166433
\(170\) −3.18360 −0.244171
\(171\) 0 0
\(172\) −3.28055 −0.250140
\(173\) 11.9930 0.911814 0.455907 0.890027i \(-0.349315\pi\)
0.455907 + 0.890027i \(0.349315\pi\)
\(174\) 0 0
\(175\) −0.662077 −0.0500483
\(176\) 2.66208 0.200662
\(177\) 0 0
\(178\) −1.61903 −0.121351
\(179\) 15.7418 1.17660 0.588298 0.808644i \(-0.299798\pi\)
0.588298 + 0.808644i \(0.299798\pi\)
\(180\) 0 0
\(181\) −8.57194 −0.637148 −0.318574 0.947898i \(-0.603204\pi\)
−0.318574 + 0.947898i \(0.603204\pi\)
\(182\) −2.36721 −0.175469
\(183\) 0 0
\(184\) 7.42110 0.547091
\(185\) 9.94263 0.730996
\(186\) 0 0
\(187\) −8.47500 −0.619753
\(188\) 9.51805 0.694175
\(189\) 0 0
\(190\) −2.23750 −0.162325
\(191\) −12.5909 −0.911048 −0.455524 0.890223i \(-0.650548\pi\)
−0.455524 + 0.890223i \(0.650548\pi\)
\(192\) 0 0
\(193\) 12.6483 0.910445 0.455223 0.890378i \(-0.349560\pi\)
0.455223 + 0.890378i \(0.349560\pi\)
\(194\) 10.4750 0.752061
\(195\) 0 0
\(196\) −6.56165 −0.468690
\(197\) −21.1508 −1.50694 −0.753468 0.657485i \(-0.771620\pi\)
−0.753468 + 0.657485i \(0.771620\pi\)
\(198\) 0 0
\(199\) −0.0641862 −0.00455004 −0.00227502 0.999997i \(-0.500724\pi\)
−0.00227502 + 0.999997i \(0.500724\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 10.6047 0.746144
\(203\) 2.08610 0.146416
\(204\) 0 0
\(205\) −3.15084 −0.220064
\(206\) 10.6185 0.739824
\(207\) 0 0
\(208\) 3.57542 0.247911
\(209\) −5.95639 −0.412012
\(210\) 0 0
\(211\) 13.5822 0.935040 0.467520 0.883983i \(-0.345148\pi\)
0.467520 + 0.883983i \(0.345148\pi\)
\(212\) −1.08666 −0.0746319
\(213\) 0 0
\(214\) −11.2805 −0.771122
\(215\) −3.28055 −0.223732
\(216\) 0 0
\(217\) −0.662077 −0.0449447
\(218\) −2.19389 −0.148589
\(219\) 0 0
\(220\) 2.66208 0.179477
\(221\) −11.3827 −0.765685
\(222\) 0 0
\(223\) −17.2598 −1.15580 −0.577902 0.816106i \(-0.696128\pi\)
−0.577902 + 0.816106i \(0.696128\pi\)
\(224\) −0.662077 −0.0442369
\(225\) 0 0
\(226\) −19.0797 −1.26916
\(227\) −9.19445 −0.610257 −0.305128 0.952311i \(-0.598699\pi\)
−0.305128 + 0.952311i \(0.598699\pi\)
\(228\) 0 0
\(229\) 16.8314 1.11225 0.556124 0.831100i \(-0.312288\pi\)
0.556124 + 0.831100i \(0.312288\pi\)
\(230\) 7.42110 0.489333
\(231\) 0 0
\(232\) −3.15084 −0.206863
\(233\) −24.6253 −1.61326 −0.806628 0.591059i \(-0.798710\pi\)
−0.806628 + 0.591059i \(0.798710\pi\)
\(234\) 0 0
\(235\) 9.51805 0.620889
\(236\) −2.61847 −0.170448
\(237\) 0 0
\(238\) 2.10779 0.136628
\(239\) −25.4905 −1.64884 −0.824422 0.565975i \(-0.808500\pi\)
−0.824422 + 0.565975i \(0.808500\pi\)
\(240\) 0 0
\(241\) −16.3672 −1.05430 −0.527152 0.849771i \(-0.676740\pi\)
−0.527152 + 0.849771i \(0.676740\pi\)
\(242\) −3.91334 −0.251559
\(243\) 0 0
\(244\) 9.83191 0.629424
\(245\) −6.56165 −0.419209
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) 1.00000 0.0635001
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −15.7418 −0.993612 −0.496806 0.867862i \(-0.665494\pi\)
−0.496806 + 0.867862i \(0.665494\pi\)
\(252\) 0 0
\(253\) 19.7555 1.24202
\(254\) 6.72626 0.422043
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −1.46082 −0.0911232 −0.0455616 0.998962i \(-0.514508\pi\)
−0.0455616 + 0.998962i \(0.514508\pi\)
\(258\) 0 0
\(259\) −6.58279 −0.409035
\(260\) 3.57542 0.221738
\(261\) 0 0
\(262\) 18.5909 1.14855
\(263\) −26.1336 −1.61147 −0.805733 0.592278i \(-0.798229\pi\)
−0.805733 + 0.592278i \(0.798229\pi\)
\(264\) 0 0
\(265\) −1.08666 −0.0667528
\(266\) 1.48140 0.0908303
\(267\) 0 0
\(268\) −14.4176 −0.880697
\(269\) −3.52500 −0.214923 −0.107462 0.994209i \(-0.534272\pi\)
−0.107462 + 0.994209i \(0.534272\pi\)
\(270\) 0 0
\(271\) 7.21503 0.438282 0.219141 0.975693i \(-0.429675\pi\)
0.219141 + 0.975693i \(0.429675\pi\)
\(272\) −3.18360 −0.193034
\(273\) 0 0
\(274\) −22.3069 −1.34761
\(275\) 2.66208 0.160529
\(276\) 0 0
\(277\) −7.07289 −0.424969 −0.212484 0.977164i \(-0.568155\pi\)
−0.212484 + 0.977164i \(0.568155\pi\)
\(278\) 4.14055 0.248334
\(279\) 0 0
\(280\) −0.662077 −0.0395667
\(281\) 6.30168 0.375927 0.187963 0.982176i \(-0.439811\pi\)
0.187963 + 0.982176i \(0.439811\pi\)
\(282\) 0 0
\(283\) 6.05737 0.360073 0.180037 0.983660i \(-0.442378\pi\)
0.180037 + 0.983660i \(0.442378\pi\)
\(284\) 5.81292 0.344933
\(285\) 0 0
\(286\) 9.51805 0.562814
\(287\) 2.08610 0.123139
\(288\) 0 0
\(289\) −6.86467 −0.403804
\(290\) −3.15084 −0.185024
\(291\) 0 0
\(292\) 11.5617 0.676595
\(293\) −17.6914 −1.03354 −0.516770 0.856124i \(-0.672866\pi\)
−0.516770 + 0.856124i \(0.672866\pi\)
\(294\) 0 0
\(295\) −2.61847 −0.152453
\(296\) 9.94263 0.577903
\(297\) 0 0
\(298\) 8.43139 0.488417
\(299\) 26.5336 1.53448
\(300\) 0 0
\(301\) 2.17198 0.125191
\(302\) 2.36721 0.136217
\(303\) 0 0
\(304\) −2.23750 −0.128329
\(305\) 9.83191 0.562974
\(306\) 0 0
\(307\) 19.2392 1.09804 0.549021 0.835809i \(-0.315001\pi\)
0.549021 + 0.835809i \(0.315001\pi\)
\(308\) −1.76250 −0.100428
\(309\) 0 0
\(310\) 1.00000 0.0567962
\(311\) 21.4401 1.21576 0.607878 0.794030i \(-0.292021\pi\)
0.607878 + 0.794030i \(0.292021\pi\)
\(312\) 0 0
\(313\) −12.5405 −0.708832 −0.354416 0.935088i \(-0.615320\pi\)
−0.354416 + 0.935088i \(0.615320\pi\)
\(314\) −17.0797 −0.963863
\(315\) 0 0
\(316\) 15.7555 0.886319
\(317\) −16.6483 −0.935062 −0.467531 0.883977i \(-0.654856\pi\)
−0.467531 + 0.883977i \(0.654856\pi\)
\(318\) 0 0
\(319\) −8.38779 −0.469626
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −4.91334 −0.273810
\(323\) 7.12331 0.396351
\(324\) 0 0
\(325\) 3.57542 0.198329
\(326\) 25.0935 1.38980
\(327\) 0 0
\(328\) −3.15084 −0.173976
\(329\) −6.30168 −0.347423
\(330\) 0 0
\(331\) −7.79393 −0.428393 −0.214196 0.976791i \(-0.568713\pi\)
−0.214196 + 0.976791i \(0.568713\pi\)
\(332\) −14.8422 −0.814572
\(333\) 0 0
\(334\) −6.59948 −0.361107
\(335\) −14.4176 −0.787719
\(336\) 0 0
\(337\) −20.1939 −1.10003 −0.550016 0.835154i \(-0.685378\pi\)
−0.550016 + 0.835154i \(0.685378\pi\)
\(338\) −0.216363 −0.0117686
\(339\) 0 0
\(340\) −3.18360 −0.172655
\(341\) 2.66208 0.144160
\(342\) 0 0
\(343\) 8.97886 0.484813
\(344\) −3.28055 −0.176875
\(345\) 0 0
\(346\) 11.9930 0.644750
\(347\) −7.71890 −0.414372 −0.207186 0.978302i \(-0.566431\pi\)
−0.207186 + 0.978302i \(0.566431\pi\)
\(348\) 0 0
\(349\) 29.5983 1.58436 0.792180 0.610287i \(-0.208946\pi\)
0.792180 + 0.610287i \(0.208946\pi\)
\(350\) −0.662077 −0.0353895
\(351\) 0 0
\(352\) 2.66208 0.141889
\(353\) 21.9983 1.17085 0.585425 0.810727i \(-0.300928\pi\)
0.585425 + 0.810727i \(0.300928\pi\)
\(354\) 0 0
\(355\) 5.81292 0.308518
\(356\) −1.61903 −0.0858082
\(357\) 0 0
\(358\) 15.7418 0.831979
\(359\) −18.6345 −0.983494 −0.491747 0.870738i \(-0.663641\pi\)
−0.491747 + 0.870738i \(0.663641\pi\)
\(360\) 0 0
\(361\) −13.9936 −0.736505
\(362\) −8.57194 −0.450531
\(363\) 0 0
\(364\) −2.36721 −0.124075
\(365\) 11.5617 0.605165
\(366\) 0 0
\(367\) 24.4107 1.27423 0.637113 0.770770i \(-0.280128\pi\)
0.637113 + 0.770770i \(0.280128\pi\)
\(368\) 7.42110 0.386852
\(369\) 0 0
\(370\) 9.94263 0.516893
\(371\) 0.719451 0.0373520
\(372\) 0 0
\(373\) −9.58223 −0.496149 −0.248075 0.968741i \(-0.579798\pi\)
−0.248075 + 0.968741i \(0.579798\pi\)
\(374\) −8.47500 −0.438232
\(375\) 0 0
\(376\) 9.51805 0.490856
\(377\) −11.2656 −0.580207
\(378\) 0 0
\(379\) 20.0367 1.02921 0.514607 0.857426i \(-0.327938\pi\)
0.514607 + 0.857426i \(0.327938\pi\)
\(380\) −2.23750 −0.114781
\(381\) 0 0
\(382\) −12.5909 −0.644208
\(383\) 7.76639 0.396844 0.198422 0.980117i \(-0.436418\pi\)
0.198422 + 0.980117i \(0.436418\pi\)
\(384\) 0 0
\(385\) −1.76250 −0.0898254
\(386\) 12.6483 0.643782
\(387\) 0 0
\(388\) 10.4750 0.531787
\(389\) −12.0000 −0.608424 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) −23.6258 −1.19481
\(392\) −6.56165 −0.331414
\(393\) 0 0
\(394\) −21.1508 −1.06556
\(395\) 15.7555 0.792748
\(396\) 0 0
\(397\) 9.45386 0.474476 0.237238 0.971452i \(-0.423758\pi\)
0.237238 + 0.971452i \(0.423758\pi\)
\(398\) −0.0641862 −0.00321736
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 32.5679 1.62636 0.813182 0.582010i \(-0.197733\pi\)
0.813182 + 0.582010i \(0.197733\pi\)
\(402\) 0 0
\(403\) 3.57542 0.178104
\(404\) 10.6047 0.527604
\(405\) 0 0
\(406\) 2.08610 0.103531
\(407\) 26.4680 1.31197
\(408\) 0 0
\(409\) −0.994935 −0.0491964 −0.0245982 0.999697i \(-0.507831\pi\)
−0.0245982 + 0.999697i \(0.507831\pi\)
\(410\) −3.15084 −0.155609
\(411\) 0 0
\(412\) 10.6185 0.523135
\(413\) 1.73363 0.0853064
\(414\) 0 0
\(415\) −14.8422 −0.728575
\(416\) 3.57542 0.175299
\(417\) 0 0
\(418\) −5.95639 −0.291337
\(419\) −3.68321 −0.179937 −0.0899684 0.995945i \(-0.528677\pi\)
−0.0899684 + 0.995945i \(0.528677\pi\)
\(420\) 0 0
\(421\) −38.8628 −1.89406 −0.947028 0.321151i \(-0.895930\pi\)
−0.947028 + 0.321151i \(0.895930\pi\)
\(422\) 13.5822 0.661173
\(423\) 0 0
\(424\) −1.08666 −0.0527727
\(425\) −3.18360 −0.154427
\(426\) 0 0
\(427\) −6.50949 −0.315016
\(428\) −11.2805 −0.545266
\(429\) 0 0
\(430\) −3.28055 −0.158202
\(431\) 10.0574 0.484447 0.242223 0.970221i \(-0.422123\pi\)
0.242223 + 0.970221i \(0.422123\pi\)
\(432\) 0 0
\(433\) −17.3011 −0.831439 −0.415720 0.909493i \(-0.636470\pi\)
−0.415720 + 0.909493i \(0.636470\pi\)
\(434\) −0.662077 −0.0317807
\(435\) 0 0
\(436\) −2.19389 −0.105068
\(437\) −16.6047 −0.794311
\(438\) 0 0
\(439\) −15.6258 −0.745781 −0.372890 0.927875i \(-0.621633\pi\)
−0.372890 + 0.927875i \(0.621633\pi\)
\(440\) 2.66208 0.126910
\(441\) 0 0
\(442\) −11.3827 −0.541421
\(443\) −10.4589 −0.496919 −0.248459 0.968642i \(-0.579924\pi\)
−0.248459 + 0.968642i \(0.579924\pi\)
\(444\) 0 0
\(445\) −1.61903 −0.0767492
\(446\) −17.2598 −0.817276
\(447\) 0 0
\(448\) −0.662077 −0.0312802
\(449\) −26.2513 −1.23887 −0.619437 0.785046i \(-0.712639\pi\)
−0.619437 + 0.785046i \(0.712639\pi\)
\(450\) 0 0
\(451\) −8.38779 −0.394965
\(452\) −19.0797 −0.897434
\(453\) 0 0
\(454\) −9.19445 −0.431517
\(455\) −2.36721 −0.110976
\(456\) 0 0
\(457\) −41.4905 −1.94084 −0.970422 0.241414i \(-0.922389\pi\)
−0.970422 + 0.241414i \(0.922389\pi\)
\(458\) 16.8314 0.786478
\(459\) 0 0
\(460\) 7.42110 0.346011
\(461\) −29.6844 −1.38254 −0.691270 0.722596i \(-0.742949\pi\)
−0.691270 + 0.722596i \(0.742949\pi\)
\(462\) 0 0
\(463\) 23.9632 1.11366 0.556832 0.830625i \(-0.312017\pi\)
0.556832 + 0.830625i \(0.312017\pi\)
\(464\) −3.15084 −0.146274
\(465\) 0 0
\(466\) −24.6253 −1.14074
\(467\) −23.5386 −1.08924 −0.544619 0.838684i \(-0.683326\pi\)
−0.544619 + 0.838684i \(0.683326\pi\)
\(468\) 0 0
\(469\) 9.54558 0.440774
\(470\) 9.51805 0.439035
\(471\) 0 0
\(472\) −2.61847 −0.120525
\(473\) −8.73308 −0.401547
\(474\) 0 0
\(475\) −2.23750 −0.102663
\(476\) 2.10779 0.0966105
\(477\) 0 0
\(478\) −25.4905 −1.16591
\(479\) −11.1371 −0.508866 −0.254433 0.967090i \(-0.581889\pi\)
−0.254433 + 0.967090i \(0.581889\pi\)
\(480\) 0 0
\(481\) 35.5491 1.62090
\(482\) −16.3672 −0.745506
\(483\) 0 0
\(484\) −3.91334 −0.177879
\(485\) 10.4750 0.475645
\(486\) 0 0
\(487\) −21.0279 −0.952867 −0.476434 0.879210i \(-0.658071\pi\)
−0.476434 + 0.879210i \(0.658071\pi\)
\(488\) 9.83191 0.445070
\(489\) 0 0
\(490\) −6.56165 −0.296425
\(491\) 12.8629 0.580496 0.290248 0.956952i \(-0.406262\pi\)
0.290248 + 0.956952i \(0.406262\pi\)
\(492\) 0 0
\(493\) 10.0310 0.451775
\(494\) −8.00000 −0.359937
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) −3.84860 −0.172633
\(498\) 0 0
\(499\) 2.70165 0.120943 0.0604713 0.998170i \(-0.480740\pi\)
0.0604713 + 0.998170i \(0.480740\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −15.7418 −0.702590
\(503\) 21.4525 0.956521 0.478260 0.878218i \(-0.341267\pi\)
0.478260 + 0.878218i \(0.341267\pi\)
\(504\) 0 0
\(505\) 10.6047 0.471903
\(506\) 19.7555 0.878241
\(507\) 0 0
\(508\) 6.72626 0.298430
\(509\) −19.1233 −0.847626 −0.423813 0.905750i \(-0.639309\pi\)
−0.423813 + 0.905750i \(0.639309\pi\)
\(510\) 0 0
\(511\) −7.65471 −0.338624
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −1.46082 −0.0644339
\(515\) 10.6185 0.467906
\(516\) 0 0
\(517\) 25.3378 1.11435
\(518\) −6.58279 −0.289231
\(519\) 0 0
\(520\) 3.57542 0.156793
\(521\) 4.50253 0.197260 0.0986298 0.995124i \(-0.468554\pi\)
0.0986298 + 0.995124i \(0.468554\pi\)
\(522\) 0 0
\(523\) −23.4469 −1.02526 −0.512631 0.858609i \(-0.671329\pi\)
−0.512631 + 0.858609i \(0.671329\pi\)
\(524\) 18.5909 0.812149
\(525\) 0 0
\(526\) −26.1336 −1.13948
\(527\) −3.18360 −0.138680
\(528\) 0 0
\(529\) 32.0727 1.39447
\(530\) −1.08666 −0.0472013
\(531\) 0 0
\(532\) 1.48140 0.0642267
\(533\) −11.2656 −0.487967
\(534\) 0 0
\(535\) −11.2805 −0.487701
\(536\) −14.4176 −0.622747
\(537\) 0 0
\(538\) −3.52500 −0.151974
\(539\) −17.4676 −0.752384
\(540\) 0 0
\(541\) 10.5405 0.453172 0.226586 0.973991i \(-0.427244\pi\)
0.226586 + 0.973991i \(0.427244\pi\)
\(542\) 7.21503 0.309912
\(543\) 0 0
\(544\) −3.18360 −0.136496
\(545\) −2.19389 −0.0939761
\(546\) 0 0
\(547\) −26.4176 −1.12954 −0.564768 0.825250i \(-0.691034\pi\)
−0.564768 + 0.825250i \(0.691034\pi\)
\(548\) −22.3069 −0.952904
\(549\) 0 0
\(550\) 2.66208 0.113511
\(551\) 7.05001 0.300340
\(552\) 0 0
\(553\) −10.4314 −0.443588
\(554\) −7.07289 −0.300498
\(555\) 0 0
\(556\) 4.14055 0.175599
\(557\) −6.71945 −0.284712 −0.142356 0.989816i \(-0.545468\pi\)
−0.142356 + 0.989816i \(0.545468\pi\)
\(558\) 0 0
\(559\) −11.7293 −0.496098
\(560\) −0.662077 −0.0279779
\(561\) 0 0
\(562\) 6.30168 0.265821
\(563\) −9.29662 −0.391806 −0.195903 0.980623i \(-0.562764\pi\)
−0.195903 + 0.980623i \(0.562764\pi\)
\(564\) 0 0
\(565\) −19.0797 −0.802689
\(566\) 6.05737 0.254610
\(567\) 0 0
\(568\) 5.81292 0.243905
\(569\) −18.6345 −0.781201 −0.390600 0.920560i \(-0.627733\pi\)
−0.390600 + 0.920560i \(0.627733\pi\)
\(570\) 0 0
\(571\) −15.0700 −0.630658 −0.315329 0.948982i \(-0.602115\pi\)
−0.315329 + 0.948982i \(0.602115\pi\)
\(572\) 9.51805 0.397970
\(573\) 0 0
\(574\) 2.08610 0.0870722
\(575\) 7.42110 0.309481
\(576\) 0 0
\(577\) −41.8577 −1.74256 −0.871280 0.490787i \(-0.836709\pi\)
−0.871280 + 0.490787i \(0.836709\pi\)
\(578\) −6.86467 −0.285533
\(579\) 0 0
\(580\) −3.15084 −0.130832
\(581\) 9.82669 0.407680
\(582\) 0 0
\(583\) −2.89276 −0.119806
\(584\) 11.5617 0.478425
\(585\) 0 0
\(586\) −17.6914 −0.730823
\(587\) 21.4525 0.885441 0.442720 0.896660i \(-0.354013\pi\)
0.442720 + 0.896660i \(0.354013\pi\)
\(588\) 0 0
\(589\) −2.23750 −0.0921945
\(590\) −2.61847 −0.107801
\(591\) 0 0
\(592\) 9.94263 0.408639
\(593\) 25.1016 1.03080 0.515400 0.856950i \(-0.327643\pi\)
0.515400 + 0.856950i \(0.327643\pi\)
\(594\) 0 0
\(595\) 2.10779 0.0864110
\(596\) 8.43139 0.345363
\(597\) 0 0
\(598\) 26.5336 1.08504
\(599\) 29.4388 1.20284 0.601418 0.798935i \(-0.294603\pi\)
0.601418 + 0.798935i \(0.294603\pi\)
\(600\) 0 0
\(601\) 3.40330 0.138824 0.0694118 0.997588i \(-0.477888\pi\)
0.0694118 + 0.997588i \(0.477888\pi\)
\(602\) 2.17198 0.0885232
\(603\) 0 0
\(604\) 2.36721 0.0963202
\(605\) −3.91334 −0.159100
\(606\) 0 0
\(607\) −48.0448 −1.95008 −0.975039 0.222033i \(-0.928731\pi\)
−0.975039 + 0.222033i \(0.928731\pi\)
\(608\) −2.23750 −0.0907426
\(609\) 0 0
\(610\) 9.83191 0.398083
\(611\) 34.0310 1.37675
\(612\) 0 0
\(613\) −23.3323 −0.942383 −0.471191 0.882031i \(-0.656176\pi\)
−0.471191 + 0.882031i \(0.656176\pi\)
\(614\) 19.2392 0.776433
\(615\) 0 0
\(616\) −1.76250 −0.0710132
\(617\) 23.2736 0.936960 0.468480 0.883474i \(-0.344802\pi\)
0.468480 + 0.883474i \(0.344802\pi\)
\(618\) 0 0
\(619\) 9.82496 0.394898 0.197449 0.980313i \(-0.436734\pi\)
0.197449 + 0.980313i \(0.436734\pi\)
\(620\) 1.00000 0.0401610
\(621\) 0 0
\(622\) 21.4401 0.859669
\(623\) 1.07192 0.0429456
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −12.5405 −0.501220
\(627\) 0 0
\(628\) −17.0797 −0.681554
\(629\) −31.6534 −1.26210
\(630\) 0 0
\(631\) 38.0297 1.51394 0.756969 0.653451i \(-0.226679\pi\)
0.756969 + 0.653451i \(0.226679\pi\)
\(632\) 15.7555 0.626722
\(633\) 0 0
\(634\) −16.6483 −0.661189
\(635\) 6.72626 0.266924
\(636\) 0 0
\(637\) −23.4607 −0.929546
\(638\) −8.38779 −0.332076
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 47.1520 1.86239 0.931197 0.364517i \(-0.118766\pi\)
0.931197 + 0.364517i \(0.118766\pi\)
\(642\) 0 0
\(643\) 19.7935 0.780581 0.390290 0.920692i \(-0.372375\pi\)
0.390290 + 0.920692i \(0.372375\pi\)
\(644\) −4.91334 −0.193613
\(645\) 0 0
\(646\) 7.12331 0.280263
\(647\) −10.3505 −0.406921 −0.203460 0.979083i \(-0.565219\pi\)
−0.203460 + 0.979083i \(0.565219\pi\)
\(648\) 0 0
\(649\) −6.97058 −0.273619
\(650\) 3.57542 0.140240
\(651\) 0 0
\(652\) 25.0935 0.982736
\(653\) 15.5180 0.607268 0.303634 0.952789i \(-0.401800\pi\)
0.303634 + 0.952789i \(0.401800\pi\)
\(654\) 0 0
\(655\) 18.5909 0.726408
\(656\) −3.15084 −0.123020
\(657\) 0 0
\(658\) −6.30168 −0.245665
\(659\) −12.2020 −0.475324 −0.237662 0.971348i \(-0.576381\pi\)
−0.237662 + 0.971348i \(0.576381\pi\)
\(660\) 0 0
\(661\) −14.2594 −0.554627 −0.277313 0.960779i \(-0.589444\pi\)
−0.277313 + 0.960779i \(0.589444\pi\)
\(662\) −7.79393 −0.302920
\(663\) 0 0
\(664\) −14.8422 −0.575989
\(665\) 1.48140 0.0574461
\(666\) 0 0
\(667\) −23.3827 −0.905383
\(668\) −6.59948 −0.255342
\(669\) 0 0
\(670\) −14.4176 −0.557001
\(671\) 26.1733 1.01041
\(672\) 0 0
\(673\) 1.99305 0.0768263 0.0384131 0.999262i \(-0.487770\pi\)
0.0384131 + 0.999262i \(0.487770\pi\)
\(674\) −20.1939 −0.777840
\(675\) 0 0
\(676\) −0.216363 −0.00832166
\(677\) −19.9917 −0.768344 −0.384172 0.923262i \(-0.625513\pi\)
−0.384172 + 0.923262i \(0.625513\pi\)
\(678\) 0 0
\(679\) −6.93526 −0.266151
\(680\) −3.18360 −0.122086
\(681\) 0 0
\(682\) 2.66208 0.101936
\(683\) −19.2255 −0.735642 −0.367821 0.929897i \(-0.619896\pi\)
−0.367821 + 0.929897i \(0.619896\pi\)
\(684\) 0 0
\(685\) −22.3069 −0.852303
\(686\) 8.97886 0.342815
\(687\) 0 0
\(688\) −3.28055 −0.125070
\(689\) −3.88525 −0.148016
\(690\) 0 0
\(691\) −9.80049 −0.372828 −0.186414 0.982471i \(-0.559687\pi\)
−0.186414 + 0.982471i \(0.559687\pi\)
\(692\) 11.9930 0.455907
\(693\) 0 0
\(694\) −7.71890 −0.293005
\(695\) 4.14055 0.157060
\(696\) 0 0
\(697\) 10.0310 0.379952
\(698\) 29.5983 1.12031
\(699\) 0 0
\(700\) −0.662077 −0.0250242
\(701\) 0.865228 0.0326792 0.0163396 0.999866i \(-0.494799\pi\)
0.0163396 + 0.999866i \(0.494799\pi\)
\(702\) 0 0
\(703\) −22.2466 −0.839047
\(704\) 2.66208 0.100331
\(705\) 0 0
\(706\) 21.9983 0.827916
\(707\) −7.02114 −0.264057
\(708\) 0 0
\(709\) −50.5650 −1.89901 −0.949504 0.313755i \(-0.898413\pi\)
−0.949504 + 0.313755i \(0.898413\pi\)
\(710\) 5.81292 0.218155
\(711\) 0 0
\(712\) −1.61903 −0.0606756
\(713\) 7.42110 0.277922
\(714\) 0 0
\(715\) 9.51805 0.355955
\(716\) 15.7418 0.588298
\(717\) 0 0
\(718\) −18.6345 −0.695435
\(719\) −30.9706 −1.15501 −0.577504 0.816388i \(-0.695973\pi\)
−0.577504 + 0.816388i \(0.695973\pi\)
\(720\) 0 0
\(721\) −7.03025 −0.261820
\(722\) −13.9936 −0.520788
\(723\) 0 0
\(724\) −8.57194 −0.318574
\(725\) −3.15084 −0.117019
\(726\) 0 0
\(727\) −43.4112 −1.61003 −0.805017 0.593252i \(-0.797844\pi\)
−0.805017 + 0.593252i \(0.797844\pi\)
\(728\) −2.36721 −0.0877345
\(729\) 0 0
\(730\) 11.5617 0.427916
\(731\) 10.4440 0.386284
\(732\) 0 0
\(733\) 19.6258 0.724897 0.362448 0.932004i \(-0.381941\pi\)
0.362448 + 0.932004i \(0.381941\pi\)
\(734\) 24.4107 0.901014
\(735\) 0 0
\(736\) 7.42110 0.273545
\(737\) −38.3808 −1.41378
\(738\) 0 0
\(739\) −17.0033 −0.625478 −0.312739 0.949839i \(-0.601246\pi\)
−0.312739 + 0.949839i \(0.601246\pi\)
\(740\) 9.94263 0.365498
\(741\) 0 0
\(742\) 0.719451 0.0264119
\(743\) 12.2323 0.448759 0.224379 0.974502i \(-0.427965\pi\)
0.224379 + 0.974502i \(0.427965\pi\)
\(744\) 0 0
\(745\) 8.43139 0.308902
\(746\) −9.58223 −0.350831
\(747\) 0 0
\(748\) −8.47500 −0.309877
\(749\) 7.46860 0.272897
\(750\) 0 0
\(751\) −21.2369 −0.774947 −0.387474 0.921881i \(-0.626652\pi\)
−0.387474 + 0.921881i \(0.626652\pi\)
\(752\) 9.51805 0.347087
\(753\) 0 0
\(754\) −11.2656 −0.410269
\(755\) 2.36721 0.0861514
\(756\) 0 0
\(757\) 17.2874 0.628320 0.314160 0.949370i \(-0.398277\pi\)
0.314160 + 0.949370i \(0.398277\pi\)
\(758\) 20.0367 0.727764
\(759\) 0 0
\(760\) −2.23750 −0.0811626
\(761\) −13.8921 −0.503587 −0.251794 0.967781i \(-0.581020\pi\)
−0.251794 + 0.967781i \(0.581020\pi\)
\(762\) 0 0
\(763\) 1.45253 0.0525850
\(764\) −12.5909 −0.455524
\(765\) 0 0
\(766\) 7.76639 0.280611
\(767\) −9.36214 −0.338047
\(768\) 0 0
\(769\) −13.8564 −0.499674 −0.249837 0.968288i \(-0.580377\pi\)
−0.249837 + 0.968288i \(0.580377\pi\)
\(770\) −1.76250 −0.0635161
\(771\) 0 0
\(772\) 12.6483 0.455223
\(773\) −26.4027 −0.949641 −0.474820 0.880083i \(-0.657487\pi\)
−0.474820 + 0.880083i \(0.657487\pi\)
\(774\) 0 0
\(775\) 1.00000 0.0359211
\(776\) 10.4750 0.376030
\(777\) 0 0
\(778\) −12.0000 −0.430221
\(779\) 7.05001 0.252593
\(780\) 0 0
\(781\) 15.4744 0.553719
\(782\) −23.6258 −0.844858
\(783\) 0 0
\(784\) −6.56165 −0.234345
\(785\) −17.0797 −0.609601
\(786\) 0 0
\(787\) −37.9289 −1.35202 −0.676009 0.736893i \(-0.736292\pi\)
−0.676009 + 0.736893i \(0.736292\pi\)
\(788\) −21.1508 −0.753468
\(789\) 0 0
\(790\) 15.7555 0.560557
\(791\) 12.6322 0.449151
\(792\) 0 0
\(793\) 35.1532 1.24833
\(794\) 9.45386 0.335505
\(795\) 0 0
\(796\) −0.0641862 −0.00227502
\(797\) 25.4836 0.902674 0.451337 0.892354i \(-0.350947\pi\)
0.451337 + 0.892354i \(0.350947\pi\)
\(798\) 0 0
\(799\) −30.3017 −1.07200
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 32.5679 1.15001
\(803\) 30.7780 1.08613
\(804\) 0 0
\(805\) −4.91334 −0.173173
\(806\) 3.57542 0.125939
\(807\) 0 0
\(808\) 10.6047 0.373072
\(809\) 21.6327 0.760564 0.380282 0.924871i \(-0.375827\pi\)
0.380282 + 0.924871i \(0.375827\pi\)
\(810\) 0 0
\(811\) 4.52556 0.158914 0.0794569 0.996838i \(-0.474681\pi\)
0.0794569 + 0.996838i \(0.474681\pi\)
\(812\) 2.08610 0.0732078
\(813\) 0 0
\(814\) 26.4680 0.927704
\(815\) 25.0935 0.878986
\(816\) 0 0
\(817\) 7.34022 0.256802
\(818\) −0.994935 −0.0347871
\(819\) 0 0
\(820\) −3.15084 −0.110032
\(821\) −9.82669 −0.342954 −0.171477 0.985188i \(-0.554854\pi\)
−0.171477 + 0.985188i \(0.554854\pi\)
\(822\) 0 0
\(823\) 12.1815 0.424619 0.212310 0.977202i \(-0.431902\pi\)
0.212310 + 0.977202i \(0.431902\pi\)
\(824\) 10.6185 0.369912
\(825\) 0 0
\(826\) 1.73363 0.0603207
\(827\) 11.6258 0.404270 0.202135 0.979358i \(-0.435212\pi\)
0.202135 + 0.979358i \(0.435212\pi\)
\(828\) 0 0
\(829\) 3.03221 0.105313 0.0526564 0.998613i \(-0.483231\pi\)
0.0526564 + 0.998613i \(0.483231\pi\)
\(830\) −14.8422 −0.515180
\(831\) 0 0
\(832\) 3.57542 0.123955
\(833\) 20.8897 0.723785
\(834\) 0 0
\(835\) −6.59948 −0.228384
\(836\) −5.95639 −0.206006
\(837\) 0 0
\(838\) −3.68321 −0.127234
\(839\) 30.0860 1.03868 0.519341 0.854567i \(-0.326177\pi\)
0.519341 + 0.854567i \(0.326177\pi\)
\(840\) 0 0
\(841\) −19.0722 −0.657662
\(842\) −38.8628 −1.33930
\(843\) 0 0
\(844\) 13.5822 0.467520
\(845\) −0.216363 −0.00744312
\(846\) 0 0
\(847\) 2.59094 0.0890256
\(848\) −1.08666 −0.0373159
\(849\) 0 0
\(850\) −3.18360 −0.109197
\(851\) 73.7852 2.52933
\(852\) 0 0
\(853\) −12.8641 −0.440459 −0.220230 0.975448i \(-0.570681\pi\)
−0.220230 + 0.975448i \(0.570681\pi\)
\(854\) −6.50949 −0.222750
\(855\) 0 0
\(856\) −11.2805 −0.385561
\(857\) 39.9000 1.36296 0.681479 0.731838i \(-0.261337\pi\)
0.681479 + 0.731838i \(0.261337\pi\)
\(858\) 0 0
\(859\) 20.6260 0.703750 0.351875 0.936047i \(-0.385544\pi\)
0.351875 + 0.936047i \(0.385544\pi\)
\(860\) −3.28055 −0.111866
\(861\) 0 0
\(862\) 10.0574 0.342555
\(863\) 27.2272 0.926825 0.463412 0.886143i \(-0.346625\pi\)
0.463412 + 0.886143i \(0.346625\pi\)
\(864\) 0 0
\(865\) 11.9930 0.407776
\(866\) −17.3011 −0.587916
\(867\) 0 0
\(868\) −0.662077 −0.0224724
\(869\) 41.9425 1.42280
\(870\) 0 0
\(871\) −51.5491 −1.74667
\(872\) −2.19389 −0.0742946
\(873\) 0 0
\(874\) −16.6047 −0.561663
\(875\) −0.662077 −0.0223823
\(876\) 0 0
\(877\) 9.88636 0.333839 0.166919 0.985971i \(-0.446618\pi\)
0.166919 + 0.985971i \(0.446618\pi\)
\(878\) −15.6258 −0.527347
\(879\) 0 0
\(880\) 2.66208 0.0897386
\(881\) −3.82899 −0.129002 −0.0645010 0.997918i \(-0.520546\pi\)
−0.0645010 + 0.997918i \(0.520546\pi\)
\(882\) 0 0
\(883\) 39.1600 1.31784 0.658919 0.752214i \(-0.271014\pi\)
0.658919 + 0.752214i \(0.271014\pi\)
\(884\) −11.3827 −0.382842
\(885\) 0 0
\(886\) −10.4589 −0.351375
\(887\) 2.56916 0.0862640 0.0431320 0.999069i \(-0.486266\pi\)
0.0431320 + 0.999069i \(0.486266\pi\)
\(888\) 0 0
\(889\) −4.45331 −0.149359
\(890\) −1.61903 −0.0542699
\(891\) 0 0
\(892\) −17.2598 −0.577902
\(893\) −21.2966 −0.712664
\(894\) 0 0
\(895\) 15.7418 0.526190
\(896\) −0.662077 −0.0221185
\(897\) 0 0
\(898\) −26.2513 −0.876016
\(899\) −3.15084 −0.105086
\(900\) 0 0
\(901\) 3.45948 0.115252
\(902\) −8.38779 −0.279283
\(903\) 0 0
\(904\) −19.0797 −0.634581
\(905\) −8.57194 −0.284941
\(906\) 0 0
\(907\) 22.7642 0.755874 0.377937 0.925831i \(-0.376634\pi\)
0.377937 + 0.925831i \(0.376634\pi\)
\(908\) −9.19445 −0.305128
\(909\) 0 0
\(910\) −2.36721 −0.0784721
\(911\) 45.1955 1.49739 0.748697 0.662913i \(-0.230680\pi\)
0.748697 + 0.662913i \(0.230680\pi\)
\(912\) 0 0
\(913\) −39.5111 −1.30763
\(914\) −41.4905 −1.37238
\(915\) 0 0
\(916\) 16.8314 0.556124
\(917\) −12.3086 −0.406467
\(918\) 0 0
\(919\) 54.6344 1.80222 0.901111 0.433588i \(-0.142753\pi\)
0.901111 + 0.433588i \(0.142753\pi\)
\(920\) 7.42110 0.244666
\(921\) 0 0
\(922\) −29.6844 −0.977604
\(923\) 20.7836 0.684102
\(924\) 0 0
\(925\) 9.94263 0.326912
\(926\) 23.9632 0.787480
\(927\) 0 0
\(928\) −3.15084 −0.103431
\(929\) −55.5982 −1.82412 −0.912058 0.410061i \(-0.865508\pi\)
−0.912058 + 0.410061i \(0.865508\pi\)
\(930\) 0 0
\(931\) 14.6817 0.481173
\(932\) −24.6253 −0.806628
\(933\) 0 0
\(934\) −23.5386 −0.770207
\(935\) −8.47500 −0.277162
\(936\) 0 0
\(937\) 30.9500 1.01109 0.505546 0.862800i \(-0.331291\pi\)
0.505546 + 0.862800i \(0.331291\pi\)
\(938\) 9.54558 0.311674
\(939\) 0 0
\(940\) 9.51805 0.310444
\(941\) −19.1233 −0.623402 −0.311701 0.950180i \(-0.600899\pi\)
−0.311701 + 0.950180i \(0.600899\pi\)
\(942\) 0 0
\(943\) −23.3827 −0.761446
\(944\) −2.61847 −0.0852240
\(945\) 0 0
\(946\) −8.73308 −0.283937
\(947\) −10.3397 −0.335994 −0.167997 0.985787i \(-0.553730\pi\)
−0.167997 + 0.985787i \(0.553730\pi\)
\(948\) 0 0
\(949\) 41.3378 1.34188
\(950\) −2.23750 −0.0725940
\(951\) 0 0
\(952\) 2.10779 0.0683139
\(953\) −25.3922 −0.822535 −0.411268 0.911515i \(-0.634914\pi\)
−0.411268 + 0.911515i \(0.634914\pi\)
\(954\) 0 0
\(955\) −12.5909 −0.407433
\(956\) −25.4905 −0.824422
\(957\) 0 0
\(958\) −11.1371 −0.359823
\(959\) 14.7689 0.476913
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 35.5491 1.14615
\(963\) 0 0
\(964\) −16.3672 −0.527152
\(965\) 12.6483 0.407163
\(966\) 0 0
\(967\) 13.0718 0.420360 0.210180 0.977663i \(-0.432595\pi\)
0.210180 + 0.977663i \(0.432595\pi\)
\(968\) −3.91334 −0.125780
\(969\) 0 0
\(970\) 10.4750 0.336332
\(971\) 47.5409 1.52566 0.762831 0.646598i \(-0.223809\pi\)
0.762831 + 0.646598i \(0.223809\pi\)
\(972\) 0 0
\(973\) −2.74137 −0.0878842
\(974\) −21.0279 −0.673779
\(975\) 0 0
\(976\) 9.83191 0.314712
\(977\) −23.0155 −0.736332 −0.368166 0.929760i \(-0.620014\pi\)
−0.368166 + 0.929760i \(0.620014\pi\)
\(978\) 0 0
\(979\) −4.30997 −0.137747
\(980\) −6.56165 −0.209604
\(981\) 0 0
\(982\) 12.8629 0.410472
\(983\) −24.8474 −0.792510 −0.396255 0.918141i \(-0.629690\pi\)
−0.396255 + 0.918141i \(0.629690\pi\)
\(984\) 0 0
\(985\) −21.1508 −0.673922
\(986\) 10.0310 0.319453
\(987\) 0 0
\(988\) −8.00000 −0.254514
\(989\) −24.3453 −0.774135
\(990\) 0 0
\(991\) 39.1383 1.24327 0.621634 0.783308i \(-0.286469\pi\)
0.621634 + 0.783308i \(0.286469\pi\)
\(992\) 1.00000 0.0317500
\(993\) 0 0
\(994\) −3.84860 −0.122070
\(995\) −0.0641862 −0.00203484
\(996\) 0 0
\(997\) 32.3603 1.02486 0.512430 0.858729i \(-0.328746\pi\)
0.512430 + 0.858729i \(0.328746\pi\)
\(998\) 2.70165 0.0855193
\(999\) 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 2790.2.a.bk.1.2 yes 4
3.2 odd 2 2790.2.a.bj.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2790.2.a.bj.1.2 4 3.2 odd 2
2790.2.a.bk.1.2 yes 4 1.1 even 1 trivial