Properties

Label 3330.2.a.bf.1.2
Level $3330$
Weight $2$
Character 3330.1
Self dual yes
Analytic conductor $26.590$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.5901838731\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{113}) \)
Defining polynomial: \(x^{2} - x - 28\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1110)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-4.81507\) of defining polynomial
Character \(\chi\) \(=\) 3330.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +3.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +1.00000 q^{11} +6.81507 q^{13} +3.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} -2.81507 q^{19} -1.00000 q^{20} +1.00000 q^{22} +4.81507 q^{23} +1.00000 q^{25} +6.81507 q^{26} +3.00000 q^{28} -3.81507 q^{29} -3.81507 q^{31} +1.00000 q^{32} -1.00000 q^{34} -3.00000 q^{35} +1.00000 q^{37} -2.81507 q^{38} -1.00000 q^{40} -5.81507 q^{41} +5.81507 q^{43} +1.00000 q^{44} +4.81507 q^{46} +8.00000 q^{47} +2.00000 q^{49} +1.00000 q^{50} +6.81507 q^{52} +10.6301 q^{53} -1.00000 q^{55} +3.00000 q^{56} -3.81507 q^{58} -2.00000 q^{59} -3.81507 q^{61} -3.81507 q^{62} +1.00000 q^{64} -6.81507 q^{65} -13.6301 q^{67} -1.00000 q^{68} -3.00000 q^{70} -9.63015 q^{71} +4.81507 q^{73} +1.00000 q^{74} -2.81507 q^{76} +3.00000 q^{77} +12.0000 q^{79} -1.00000 q^{80} -5.81507 q^{82} -6.81507 q^{83} +1.00000 q^{85} +5.81507 q^{86} +1.00000 q^{88} -1.18493 q^{89} +20.4452 q^{91} +4.81507 q^{92} +8.00000 q^{94} +2.81507 q^{95} +5.81507 q^{97} +2.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} - 2q^{5} + 6q^{7} + 2q^{8} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} - 2q^{5} + 6q^{7} + 2q^{8} - 2q^{10} + 2q^{11} + 3q^{13} + 6q^{14} + 2q^{16} - 2q^{17} + 5q^{19} - 2q^{20} + 2q^{22} - q^{23} + 2q^{25} + 3q^{26} + 6q^{28} + 3q^{29} + 3q^{31} + 2q^{32} - 2q^{34} - 6q^{35} + 2q^{37} + 5q^{38} - 2q^{40} - q^{41} + q^{43} + 2q^{44} - q^{46} + 16q^{47} + 4q^{49} + 2q^{50} + 3q^{52} - 2q^{55} + 6q^{56} + 3q^{58} - 4q^{59} + 3q^{61} + 3q^{62} + 2q^{64} - 3q^{65} - 6q^{67} - 2q^{68} - 6q^{70} + 2q^{71} - q^{73} + 2q^{74} + 5q^{76} + 6q^{77} + 24q^{79} - 2q^{80} - q^{82} - 3q^{83} + 2q^{85} + q^{86} + 2q^{88} - 13q^{89} + 9q^{91} - q^{92} + 16q^{94} - 5q^{95} + q^{97} + 4q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.00000 1.13389 0.566947 0.823754i \(-0.308125\pi\)
0.566947 + 0.823754i \(0.308125\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.00000 −0.316228
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) 6.81507 1.89016 0.945081 0.326837i \(-0.105983\pi\)
0.945081 + 0.326837i \(0.105983\pi\)
\(14\) 3.00000 0.801784
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 0 0
\(19\) −2.81507 −0.645822 −0.322911 0.946429i \(-0.604661\pi\)
−0.322911 + 0.946429i \(0.604661\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 1.00000 0.213201
\(23\) 4.81507 1.00401 0.502006 0.864864i \(-0.332596\pi\)
0.502006 + 0.864864i \(0.332596\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 6.81507 1.33655
\(27\) 0 0
\(28\) 3.00000 0.566947
\(29\) −3.81507 −0.708441 −0.354221 0.935162i \(-0.615254\pi\)
−0.354221 + 0.935162i \(0.615254\pi\)
\(30\) 0 0
\(31\) −3.81507 −0.685207 −0.342604 0.939480i \(-0.611309\pi\)
−0.342604 + 0.939480i \(0.611309\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −1.00000 −0.171499
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) −2.81507 −0.456665
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) −5.81507 −0.908162 −0.454081 0.890960i \(-0.650032\pi\)
−0.454081 + 0.890960i \(0.650032\pi\)
\(42\) 0 0
\(43\) 5.81507 0.886790 0.443395 0.896326i \(-0.353774\pi\)
0.443395 + 0.896326i \(0.353774\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 4.81507 0.709944
\(47\) 8.00000 1.16692 0.583460 0.812142i \(-0.301699\pi\)
0.583460 + 0.812142i \(0.301699\pi\)
\(48\) 0 0
\(49\) 2.00000 0.285714
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 6.81507 0.945081
\(53\) 10.6301 1.46016 0.730081 0.683360i \(-0.239482\pi\)
0.730081 + 0.683360i \(0.239482\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 3.00000 0.400892
\(57\) 0 0
\(58\) −3.81507 −0.500944
\(59\) −2.00000 −0.260378 −0.130189 0.991489i \(-0.541558\pi\)
−0.130189 + 0.991489i \(0.541558\pi\)
\(60\) 0 0
\(61\) −3.81507 −0.488470 −0.244235 0.969716i \(-0.578537\pi\)
−0.244235 + 0.969716i \(0.578537\pi\)
\(62\) −3.81507 −0.484515
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −6.81507 −0.845306
\(66\) 0 0
\(67\) −13.6301 −1.66519 −0.832594 0.553884i \(-0.813145\pi\)
−0.832594 + 0.553884i \(0.813145\pi\)
\(68\) −1.00000 −0.121268
\(69\) 0 0
\(70\) −3.00000 −0.358569
\(71\) −9.63015 −1.14289 −0.571444 0.820641i \(-0.693617\pi\)
−0.571444 + 0.820641i \(0.693617\pi\)
\(72\) 0 0
\(73\) 4.81507 0.563562 0.281781 0.959479i \(-0.409075\pi\)
0.281781 + 0.959479i \(0.409075\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) −2.81507 −0.322911
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −5.81507 −0.642167
\(83\) −6.81507 −0.748051 −0.374026 0.927418i \(-0.622023\pi\)
−0.374026 + 0.927418i \(0.622023\pi\)
\(84\) 0 0
\(85\) 1.00000 0.108465
\(86\) 5.81507 0.627055
\(87\) 0 0
\(88\) 1.00000 0.106600
\(89\) −1.18493 −0.125602 −0.0628010 0.998026i \(-0.520003\pi\)
−0.0628010 + 0.998026i \(0.520003\pi\)
\(90\) 0 0
\(91\) 20.4452 2.14324
\(92\) 4.81507 0.502006
\(93\) 0 0
\(94\) 8.00000 0.825137
\(95\) 2.81507 0.288820
\(96\) 0 0
\(97\) 5.81507 0.590431 0.295216 0.955431i \(-0.404609\pi\)
0.295216 + 0.955431i \(0.404609\pi\)
\(98\) 2.00000 0.202031
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 9.63015 0.958235 0.479118 0.877751i \(-0.340957\pi\)
0.479118 + 0.877751i \(0.340957\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 6.81507 0.668273
\(105\) 0 0
\(106\) 10.6301 1.03249
\(107\) 18.8151 1.81892 0.909461 0.415790i \(-0.136495\pi\)
0.909461 + 0.415790i \(0.136495\pi\)
\(108\) 0 0
\(109\) 13.0000 1.24517 0.622587 0.782551i \(-0.286082\pi\)
0.622587 + 0.782551i \(0.286082\pi\)
\(110\) −1.00000 −0.0953463
\(111\) 0 0
\(112\) 3.00000 0.283473
\(113\) −3.81507 −0.358892 −0.179446 0.983768i \(-0.557430\pi\)
−0.179446 + 0.983768i \(0.557430\pi\)
\(114\) 0 0
\(115\) −4.81507 −0.449008
\(116\) −3.81507 −0.354221
\(117\) 0 0
\(118\) −2.00000 −0.184115
\(119\) −3.00000 −0.275010
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) −3.81507 −0.345400
\(123\) 0 0
\(124\) −3.81507 −0.342604
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 10.4452 0.926863 0.463432 0.886133i \(-0.346618\pi\)
0.463432 + 0.886133i \(0.346618\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −6.81507 −0.597721
\(131\) −11.6301 −1.01613 −0.508065 0.861319i \(-0.669639\pi\)
−0.508065 + 0.861319i \(0.669639\pi\)
\(132\) 0 0
\(133\) −8.44522 −0.732293
\(134\) −13.6301 −1.17747
\(135\) 0 0
\(136\) −1.00000 −0.0857493
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) 12.1849 1.03351 0.516756 0.856133i \(-0.327139\pi\)
0.516756 + 0.856133i \(0.327139\pi\)
\(140\) −3.00000 −0.253546
\(141\) 0 0
\(142\) −9.63015 −0.808144
\(143\) 6.81507 0.569905
\(144\) 0 0
\(145\) 3.81507 0.316825
\(146\) 4.81507 0.398498
\(147\) 0 0
\(148\) 1.00000 0.0821995
\(149\) −2.00000 −0.163846 −0.0819232 0.996639i \(-0.526106\pi\)
−0.0819232 + 0.996639i \(0.526106\pi\)
\(150\) 0 0
\(151\) −6.81507 −0.554603 −0.277301 0.960783i \(-0.589440\pi\)
−0.277301 + 0.960783i \(0.589440\pi\)
\(152\) −2.81507 −0.228333
\(153\) 0 0
\(154\) 3.00000 0.241747
\(155\) 3.81507 0.306434
\(156\) 0 0
\(157\) −2.18493 −0.174376 −0.0871881 0.996192i \(-0.527788\pi\)
−0.0871881 + 0.996192i \(0.527788\pi\)
\(158\) 12.0000 0.954669
\(159\) 0 0
\(160\) −1.00000 −0.0790569
\(161\) 14.4452 1.13844
\(162\) 0 0
\(163\) −4.63015 −0.362661 −0.181331 0.983422i \(-0.558040\pi\)
−0.181331 + 0.983422i \(0.558040\pi\)
\(164\) −5.81507 −0.454081
\(165\) 0 0
\(166\) −6.81507 −0.528952
\(167\) 11.1849 0.865516 0.432758 0.901510i \(-0.357541\pi\)
0.432758 + 0.901510i \(0.357541\pi\)
\(168\) 0 0
\(169\) 33.4452 2.57271
\(170\) 1.00000 0.0766965
\(171\) 0 0
\(172\) 5.81507 0.443395
\(173\) 1.00000 0.0760286 0.0380143 0.999277i \(-0.487897\pi\)
0.0380143 + 0.999277i \(0.487897\pi\)
\(174\) 0 0
\(175\) 3.00000 0.226779
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −1.18493 −0.0888140
\(179\) −16.0000 −1.19590 −0.597948 0.801535i \(-0.704017\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 20.4452 1.51550
\(183\) 0 0
\(184\) 4.81507 0.354972
\(185\) −1.00000 −0.0735215
\(186\) 0 0
\(187\) −1.00000 −0.0731272
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) 2.81507 0.204227
\(191\) 16.6301 1.20332 0.601658 0.798754i \(-0.294507\pi\)
0.601658 + 0.798754i \(0.294507\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 5.81507 0.417498
\(195\) 0 0
\(196\) 2.00000 0.142857
\(197\) 14.8151 1.05553 0.527765 0.849390i \(-0.323030\pi\)
0.527765 + 0.849390i \(0.323030\pi\)
\(198\) 0 0
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 9.63015 0.677575
\(203\) −11.4452 −0.803297
\(204\) 0 0
\(205\) 5.81507 0.406142
\(206\) 0 0
\(207\) 0 0
\(208\) 6.81507 0.472540
\(209\) −2.81507 −0.194723
\(210\) 0 0
\(211\) −23.4452 −1.61404 −0.807018 0.590527i \(-0.798920\pi\)
−0.807018 + 0.590527i \(0.798920\pi\)
\(212\) 10.6301 0.730081
\(213\) 0 0
\(214\) 18.8151 1.28617
\(215\) −5.81507 −0.396585
\(216\) 0 0
\(217\) −11.4452 −0.776952
\(218\) 13.0000 0.880471
\(219\) 0 0
\(220\) −1.00000 −0.0674200
\(221\) −6.81507 −0.458431
\(222\) 0 0
\(223\) −6.18493 −0.414173 −0.207087 0.978323i \(-0.566398\pi\)
−0.207087 + 0.978323i \(0.566398\pi\)
\(224\) 3.00000 0.200446
\(225\) 0 0
\(226\) −3.81507 −0.253775
\(227\) 14.1849 0.941487 0.470743 0.882270i \(-0.343986\pi\)
0.470743 + 0.882270i \(0.343986\pi\)
\(228\) 0 0
\(229\) 21.6301 1.42936 0.714680 0.699451i \(-0.246572\pi\)
0.714680 + 0.699451i \(0.246572\pi\)
\(230\) −4.81507 −0.317497
\(231\) 0 0
\(232\) −3.81507 −0.250472
\(233\) −13.6301 −0.892941 −0.446470 0.894798i \(-0.647319\pi\)
−0.446470 + 0.894798i \(0.647319\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) −2.00000 −0.130189
\(237\) 0 0
\(238\) −3.00000 −0.194461
\(239\) 25.4452 1.64591 0.822957 0.568103i \(-0.192323\pi\)
0.822957 + 0.568103i \(0.192323\pi\)
\(240\) 0 0
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) −10.0000 −0.642824
\(243\) 0 0
\(244\) −3.81507 −0.244235
\(245\) −2.00000 −0.127775
\(246\) 0 0
\(247\) −19.1849 −1.22071
\(248\) −3.81507 −0.242257
\(249\) 0 0
\(250\) −1.00000 −0.0632456
\(251\) −9.63015 −0.607849 −0.303925 0.952696i \(-0.598297\pi\)
−0.303925 + 0.952696i \(0.598297\pi\)
\(252\) 0 0
\(253\) 4.81507 0.302721
\(254\) 10.4452 0.655391
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −6.81507 −0.425113 −0.212556 0.977149i \(-0.568179\pi\)
−0.212556 + 0.977149i \(0.568179\pi\)
\(258\) 0 0
\(259\) 3.00000 0.186411
\(260\) −6.81507 −0.422653
\(261\) 0 0
\(262\) −11.6301 −0.718513
\(263\) −1.44522 −0.0891160 −0.0445580 0.999007i \(-0.514188\pi\)
−0.0445580 + 0.999007i \(0.514188\pi\)
\(264\) 0 0
\(265\) −10.6301 −0.653005
\(266\) −8.44522 −0.517810
\(267\) 0 0
\(268\) −13.6301 −0.832594
\(269\) 0.815073 0.0496959 0.0248479 0.999691i \(-0.492090\pi\)
0.0248479 + 0.999691i \(0.492090\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) −1.00000 −0.0606339
\(273\) 0 0
\(274\) −2.00000 −0.120824
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −10.4452 −0.627592 −0.313796 0.949490i \(-0.601601\pi\)
−0.313796 + 0.949490i \(0.601601\pi\)
\(278\) 12.1849 0.730803
\(279\) 0 0
\(280\) −3.00000 −0.179284
\(281\) −12.4452 −0.742420 −0.371210 0.928549i \(-0.621057\pi\)
−0.371210 + 0.928549i \(0.621057\pi\)
\(282\) 0 0
\(283\) −11.1849 −0.664875 −0.332437 0.943125i \(-0.607871\pi\)
−0.332437 + 0.943125i \(0.607871\pi\)
\(284\) −9.63015 −0.571444
\(285\) 0 0
\(286\) 6.81507 0.402984
\(287\) −17.4452 −1.02976
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 3.81507 0.224029
\(291\) 0 0
\(292\) 4.81507 0.281781
\(293\) 14.2603 0.833095 0.416548 0.909114i \(-0.363240\pi\)
0.416548 + 0.909114i \(0.363240\pi\)
\(294\) 0 0
\(295\) 2.00000 0.116445
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) −2.00000 −0.115857
\(299\) 32.8151 1.89774
\(300\) 0 0
\(301\) 17.4452 1.00553
\(302\) −6.81507 −0.392163
\(303\) 0 0
\(304\) −2.81507 −0.161456
\(305\) 3.81507 0.218450
\(306\) 0 0
\(307\) −18.0000 −1.02731 −0.513657 0.857996i \(-0.671710\pi\)
−0.513657 + 0.857996i \(0.671710\pi\)
\(308\) 3.00000 0.170941
\(309\) 0 0
\(310\) 3.81507 0.216682
\(311\) 21.4452 1.21605 0.608023 0.793919i \(-0.291963\pi\)
0.608023 + 0.793919i \(0.291963\pi\)
\(312\) 0 0
\(313\) −30.8904 −1.74603 −0.873015 0.487693i \(-0.837839\pi\)
−0.873015 + 0.487693i \(0.837839\pi\)
\(314\) −2.18493 −0.123303
\(315\) 0 0
\(316\) 12.0000 0.675053
\(317\) −7.81507 −0.438938 −0.219469 0.975619i \(-0.570433\pi\)
−0.219469 + 0.975619i \(0.570433\pi\)
\(318\) 0 0
\(319\) −3.81507 −0.213603
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 14.4452 0.805001
\(323\) 2.81507 0.156635
\(324\) 0 0
\(325\) 6.81507 0.378032
\(326\) −4.63015 −0.256440
\(327\) 0 0
\(328\) −5.81507 −0.321084
\(329\) 24.0000 1.32316
\(330\) 0 0
\(331\) 19.2603 1.05864 0.529321 0.848422i \(-0.322447\pi\)
0.529321 + 0.848422i \(0.322447\pi\)
\(332\) −6.81507 −0.374026
\(333\) 0 0
\(334\) 11.1849 0.612012
\(335\) 13.6301 0.744694
\(336\) 0 0
\(337\) −14.8151 −0.807028 −0.403514 0.914973i \(-0.632211\pi\)
−0.403514 + 0.914973i \(0.632211\pi\)
\(338\) 33.4452 1.81918
\(339\) 0 0
\(340\) 1.00000 0.0542326
\(341\) −3.81507 −0.206598
\(342\) 0 0
\(343\) −15.0000 −0.809924
\(344\) 5.81507 0.313528
\(345\) 0 0
\(346\) 1.00000 0.0537603
\(347\) 31.2603 1.67814 0.839070 0.544023i \(-0.183100\pi\)
0.839070 + 0.544023i \(0.183100\pi\)
\(348\) 0 0
\(349\) −35.2603 −1.88744 −0.943720 0.330745i \(-0.892700\pi\)
−0.943720 + 0.330745i \(0.892700\pi\)
\(350\) 3.00000 0.160357
\(351\) 0 0
\(352\) 1.00000 0.0533002
\(353\) 23.8151 1.26755 0.633774 0.773518i \(-0.281505\pi\)
0.633774 + 0.773518i \(0.281505\pi\)
\(354\) 0 0
\(355\) 9.63015 0.511115
\(356\) −1.18493 −0.0628010
\(357\) 0 0
\(358\) −16.0000 −0.845626
\(359\) −27.6301 −1.45826 −0.729132 0.684373i \(-0.760076\pi\)
−0.729132 + 0.684373i \(0.760076\pi\)
\(360\) 0 0
\(361\) −11.0754 −0.582914
\(362\) −10.0000 −0.525588
\(363\) 0 0
\(364\) 20.4452 1.07162
\(365\) −4.81507 −0.252032
\(366\) 0 0
\(367\) −21.0000 −1.09619 −0.548096 0.836416i \(-0.684647\pi\)
−0.548096 + 0.836416i \(0.684647\pi\)
\(368\) 4.81507 0.251003
\(369\) 0 0
\(370\) −1.00000 −0.0519875
\(371\) 31.8904 1.65567
\(372\) 0 0
\(373\) 25.2603 1.30793 0.653964 0.756526i \(-0.273105\pi\)
0.653964 + 0.756526i \(0.273105\pi\)
\(374\) −1.00000 −0.0517088
\(375\) 0 0
\(376\) 8.00000 0.412568
\(377\) −26.0000 −1.33907
\(378\) 0 0
\(379\) 1.63015 0.0837350 0.0418675 0.999123i \(-0.486669\pi\)
0.0418675 + 0.999123i \(0.486669\pi\)
\(380\) 2.81507 0.144410
\(381\) 0 0
\(382\) 16.6301 0.850872
\(383\) −1.55478 −0.0794456 −0.0397228 0.999211i \(-0.512647\pi\)
−0.0397228 + 0.999211i \(0.512647\pi\)
\(384\) 0 0
\(385\) −3.00000 −0.152894
\(386\) −2.00000 −0.101797
\(387\) 0 0
\(388\) 5.81507 0.295216
\(389\) −38.7055 −1.96245 −0.981224 0.192873i \(-0.938219\pi\)
−0.981224 + 0.192873i \(0.938219\pi\)
\(390\) 0 0
\(391\) −4.81507 −0.243509
\(392\) 2.00000 0.101015
\(393\) 0 0
\(394\) 14.8151 0.746373
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) 15.6301 0.784455 0.392227 0.919868i \(-0.371705\pi\)
0.392227 + 0.919868i \(0.371705\pi\)
\(398\) −4.00000 −0.200502
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −32.4452 −1.62024 −0.810118 0.586266i \(-0.800597\pi\)
−0.810118 + 0.586266i \(0.800597\pi\)
\(402\) 0 0
\(403\) −26.0000 −1.29515
\(404\) 9.63015 0.479118
\(405\) 0 0
\(406\) −11.4452 −0.568017
\(407\) 1.00000 0.0495682
\(408\) 0 0
\(409\) 27.6301 1.36622 0.683111 0.730314i \(-0.260626\pi\)
0.683111 + 0.730314i \(0.260626\pi\)
\(410\) 5.81507 0.287186
\(411\) 0 0
\(412\) 0 0
\(413\) −6.00000 −0.295241
\(414\) 0 0
\(415\) 6.81507 0.334539
\(416\) 6.81507 0.334136
\(417\) 0 0
\(418\) −2.81507 −0.137690
\(419\) −6.44522 −0.314870 −0.157435 0.987529i \(-0.550322\pi\)
−0.157435 + 0.987529i \(0.550322\pi\)
\(420\) 0 0
\(421\) 13.2603 0.646267 0.323134 0.946353i \(-0.395264\pi\)
0.323134 + 0.946353i \(0.395264\pi\)
\(422\) −23.4452 −1.14130
\(423\) 0 0
\(424\) 10.6301 0.516246
\(425\) −1.00000 −0.0485071
\(426\) 0 0
\(427\) −11.4452 −0.553873
\(428\) 18.8151 0.909461
\(429\) 0 0
\(430\) −5.81507 −0.280428
\(431\) −21.0000 −1.01153 −0.505767 0.862670i \(-0.668791\pi\)
−0.505767 + 0.862670i \(0.668791\pi\)
\(432\) 0 0
\(433\) −8.81507 −0.423625 −0.211813 0.977310i \(-0.567937\pi\)
−0.211813 + 0.977310i \(0.567937\pi\)
\(434\) −11.4452 −0.549388
\(435\) 0 0
\(436\) 13.0000 0.622587
\(437\) −13.5548 −0.648413
\(438\) 0 0
\(439\) 13.4452 0.641705 0.320853 0.947129i \(-0.396031\pi\)
0.320853 + 0.947129i \(0.396031\pi\)
\(440\) −1.00000 −0.0476731
\(441\) 0 0
\(442\) −6.81507 −0.324160
\(443\) 7.63015 0.362519 0.181260 0.983435i \(-0.441983\pi\)
0.181260 + 0.983435i \(0.441983\pi\)
\(444\) 0 0
\(445\) 1.18493 0.0561709
\(446\) −6.18493 −0.292865
\(447\) 0 0
\(448\) 3.00000 0.141737
\(449\) 32.8904 1.55220 0.776098 0.630612i \(-0.217196\pi\)
0.776098 + 0.630612i \(0.217196\pi\)
\(450\) 0 0
\(451\) −5.81507 −0.273821
\(452\) −3.81507 −0.179446
\(453\) 0 0
\(454\) 14.1849 0.665732
\(455\) −20.4452 −0.958487
\(456\) 0 0
\(457\) 25.0754 1.17298 0.586488 0.809958i \(-0.300510\pi\)
0.586488 + 0.809958i \(0.300510\pi\)
\(458\) 21.6301 1.01071
\(459\) 0 0
\(460\) −4.81507 −0.224504
\(461\) −24.1849 −1.12640 −0.563202 0.826319i \(-0.690431\pi\)
−0.563202 + 0.826319i \(0.690431\pi\)
\(462\) 0 0
\(463\) 3.63015 0.168707 0.0843536 0.996436i \(-0.473117\pi\)
0.0843536 + 0.996436i \(0.473117\pi\)
\(464\) −3.81507 −0.177110
\(465\) 0 0
\(466\) −13.6301 −0.631404
\(467\) −18.1849 −0.841498 −0.420749 0.907177i \(-0.638233\pi\)
−0.420749 + 0.907177i \(0.638233\pi\)
\(468\) 0 0
\(469\) −40.8904 −1.88814
\(470\) −8.00000 −0.369012
\(471\) 0 0
\(472\) −2.00000 −0.0920575
\(473\) 5.81507 0.267377
\(474\) 0 0
\(475\) −2.81507 −0.129164
\(476\) −3.00000 −0.137505
\(477\) 0 0
\(478\) 25.4452 1.16384
\(479\) 0.815073 0.0372416 0.0186208 0.999827i \(-0.494072\pi\)
0.0186208 + 0.999827i \(0.494072\pi\)
\(480\) 0 0
\(481\) 6.81507 0.310741
\(482\) −26.0000 −1.18427
\(483\) 0 0
\(484\) −10.0000 −0.454545
\(485\) −5.81507 −0.264049
\(486\) 0 0
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) −3.81507 −0.172700
\(489\) 0 0
\(490\) −2.00000 −0.0903508
\(491\) −12.8151 −0.578336 −0.289168 0.957278i \(-0.593379\pi\)
−0.289168 + 0.957278i \(0.593379\pi\)
\(492\) 0 0
\(493\) 3.81507 0.171822
\(494\) −19.1849 −0.863171
\(495\) 0 0
\(496\) −3.81507 −0.171302
\(497\) −28.8904 −1.29591
\(498\) 0 0
\(499\) 14.4452 0.646657 0.323328 0.946287i \(-0.395198\pi\)
0.323328 + 0.946287i \(0.395198\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −9.63015 −0.429814
\(503\) −43.2603 −1.92888 −0.964441 0.264300i \(-0.914859\pi\)
−0.964441 + 0.264300i \(0.914859\pi\)
\(504\) 0 0
\(505\) −9.63015 −0.428536
\(506\) 4.81507 0.214056
\(507\) 0 0
\(508\) 10.4452 0.463432
\(509\) 14.4452 0.640273 0.320137 0.947371i \(-0.396271\pi\)
0.320137 + 0.947371i \(0.396271\pi\)
\(510\) 0 0
\(511\) 14.4452 0.639019
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −6.81507 −0.300600
\(515\) 0 0
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) 3.00000 0.131812
\(519\) 0 0
\(520\) −6.81507 −0.298861
\(521\) 9.81507 0.430006 0.215003 0.976613i \(-0.431024\pi\)
0.215003 + 0.976613i \(0.431024\pi\)
\(522\) 0 0
\(523\) −43.2603 −1.89164 −0.945820 0.324691i \(-0.894740\pi\)
−0.945820 + 0.324691i \(0.894740\pi\)
\(524\) −11.6301 −0.508065
\(525\) 0 0
\(526\) −1.44522 −0.0630145
\(527\) 3.81507 0.166187
\(528\) 0 0
\(529\) 0.184927 0.00804031
\(530\) −10.6301 −0.461744
\(531\) 0 0
\(532\) −8.44522 −0.366147
\(533\) −39.6301 −1.71657
\(534\) 0 0
\(535\) −18.8151 −0.813447
\(536\) −13.6301 −0.588733
\(537\) 0 0
\(538\) 0.815073 0.0351403
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) 17.1849 0.738838 0.369419 0.929263i \(-0.379557\pi\)
0.369419 + 0.929263i \(0.379557\pi\)
\(542\) −20.0000 −0.859074
\(543\) 0 0
\(544\) −1.00000 −0.0428746
\(545\) −13.0000 −0.556859
\(546\) 0 0
\(547\) −39.8904 −1.70559 −0.852796 0.522244i \(-0.825095\pi\)
−0.852796 + 0.522244i \(0.825095\pi\)
\(548\) −2.00000 −0.0854358
\(549\) 0 0
\(550\) 1.00000 0.0426401
\(551\) 10.7397 0.457527
\(552\) 0 0
\(553\) 36.0000 1.53088
\(554\) −10.4452 −0.443775
\(555\) 0 0
\(556\) 12.1849 0.516756
\(557\) 0.369854 0.0156712 0.00783561 0.999969i \(-0.497506\pi\)
0.00783561 + 0.999969i \(0.497506\pi\)
\(558\) 0 0
\(559\) 39.6301 1.67618
\(560\) −3.00000 −0.126773
\(561\) 0 0
\(562\) −12.4452 −0.524970
\(563\) −23.4452 −0.988098 −0.494049 0.869434i \(-0.664484\pi\)
−0.494049 + 0.869434i \(0.664484\pi\)
\(564\) 0 0
\(565\) 3.81507 0.160501
\(566\) −11.1849 −0.470138
\(567\) 0 0
\(568\) −9.63015 −0.404072
\(569\) 25.1849 1.05581 0.527904 0.849304i \(-0.322978\pi\)
0.527904 + 0.849304i \(0.322978\pi\)
\(570\) 0 0
\(571\) −4.18493 −0.175134 −0.0875669 0.996159i \(-0.527909\pi\)
−0.0875669 + 0.996159i \(0.527909\pi\)
\(572\) 6.81507 0.284953
\(573\) 0 0
\(574\) −17.4452 −0.728149
\(575\) 4.81507 0.200802
\(576\) 0 0
\(577\) −22.0000 −0.915872 −0.457936 0.888985i \(-0.651411\pi\)
−0.457936 + 0.888985i \(0.651411\pi\)
\(578\) −16.0000 −0.665512
\(579\) 0 0
\(580\) 3.81507 0.158412
\(581\) −20.4452 −0.848211
\(582\) 0 0
\(583\) 10.6301 0.440256
\(584\) 4.81507 0.199249
\(585\) 0 0
\(586\) 14.2603 0.589087
\(587\) −10.1849 −0.420377 −0.210188 0.977661i \(-0.567408\pi\)
−0.210188 + 0.977661i \(0.567408\pi\)
\(588\) 0 0
\(589\) 10.7397 0.442522
\(590\) 2.00000 0.0823387
\(591\) 0 0
\(592\) 1.00000 0.0410997
\(593\) −7.63015 −0.313333 −0.156666 0.987652i \(-0.550075\pi\)
−0.156666 + 0.987652i \(0.550075\pi\)
\(594\) 0 0
\(595\) 3.00000 0.122988
\(596\) −2.00000 −0.0819232
\(597\) 0 0
\(598\) 32.8151 1.34191
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) −40.6301 −1.65734 −0.828669 0.559739i \(-0.810901\pi\)
−0.828669 + 0.559739i \(0.810901\pi\)
\(602\) 17.4452 0.711014
\(603\) 0 0
\(604\) −6.81507 −0.277301
\(605\) 10.0000 0.406558
\(606\) 0 0
\(607\) 5.26029 0.213509 0.106754 0.994285i \(-0.465954\pi\)
0.106754 + 0.994285i \(0.465954\pi\)
\(608\) −2.81507 −0.114166
\(609\) 0 0
\(610\) 3.81507 0.154468
\(611\) 54.5206 2.20567
\(612\) 0 0
\(613\) 47.0754 1.90136 0.950678 0.310179i \(-0.100389\pi\)
0.950678 + 0.310179i \(0.100389\pi\)
\(614\) −18.0000 −0.726421
\(615\) 0 0
\(616\) 3.00000 0.120873
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) −45.4452 −1.82660 −0.913299 0.407290i \(-0.866474\pi\)
−0.913299 + 0.407290i \(0.866474\pi\)
\(620\) 3.81507 0.153217
\(621\) 0 0
\(622\) 21.4452 0.859875
\(623\) −3.55478 −0.142419
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −30.8904 −1.23463
\(627\) 0 0
\(628\) −2.18493 −0.0871881
\(629\) −1.00000 −0.0398726
\(630\) 0 0
\(631\) −40.7055 −1.62046 −0.810230 0.586112i \(-0.800658\pi\)
−0.810230 + 0.586112i \(0.800658\pi\)
\(632\) 12.0000 0.477334
\(633\) 0 0
\(634\) −7.81507 −0.310376
\(635\) −10.4452 −0.414506
\(636\) 0 0
\(637\) 13.6301 0.540046
\(638\) −3.81507 −0.151040
\(639\) 0 0
\(640\) −1.00000 −0.0395285
\(641\) 45.4452 1.79498 0.897489 0.441037i \(-0.145389\pi\)
0.897489 + 0.441037i \(0.145389\pi\)
\(642\) 0 0
\(643\) 17.0000 0.670415 0.335207 0.942144i \(-0.391194\pi\)
0.335207 + 0.942144i \(0.391194\pi\)
\(644\) 14.4452 0.569221
\(645\) 0 0
\(646\) 2.81507 0.110758
\(647\) 28.0754 1.10376 0.551878 0.833925i \(-0.313911\pi\)
0.551878 + 0.833925i \(0.313911\pi\)
\(648\) 0 0
\(649\) −2.00000 −0.0785069
\(650\) 6.81507 0.267309
\(651\) 0 0
\(652\) −4.63015 −0.181331
\(653\) −4.00000 −0.156532 −0.0782660 0.996933i \(-0.524938\pi\)
−0.0782660 + 0.996933i \(0.524938\pi\)
\(654\) 0 0
\(655\) 11.6301 0.454427
\(656\) −5.81507 −0.227040
\(657\) 0 0
\(658\) 24.0000 0.935617
\(659\) −11.2603 −0.438639 −0.219319 0.975653i \(-0.570384\pi\)
−0.219319 + 0.975653i \(0.570384\pi\)
\(660\) 0 0
\(661\) 46.2603 1.79932 0.899658 0.436595i \(-0.143816\pi\)
0.899658 + 0.436595i \(0.143816\pi\)
\(662\) 19.2603 0.748572
\(663\) 0 0
\(664\) −6.81507 −0.264476
\(665\) 8.44522 0.327492
\(666\) 0 0
\(667\) −18.3699 −0.711284
\(668\) 11.1849 0.432758
\(669\) 0 0
\(670\) 13.6301 0.526578
\(671\) −3.81507 −0.147279
\(672\) 0 0
\(673\) 16.8151 0.648173 0.324087 0.946027i \(-0.394943\pi\)
0.324087 + 0.946027i \(0.394943\pi\)
\(674\) −14.8151 −0.570655
\(675\) 0 0
\(676\) 33.4452 1.28635
\(677\) 6.07536 0.233495 0.116748 0.993162i \(-0.462753\pi\)
0.116748 + 0.993162i \(0.462753\pi\)
\(678\) 0 0
\(679\) 17.4452 0.669486
\(680\) 1.00000 0.0383482
\(681\) 0 0
\(682\) −3.81507 −0.146087
\(683\) −48.3357 −1.84951 −0.924756 0.380560i \(-0.875731\pi\)
−0.924756 + 0.380560i \(0.875731\pi\)
\(684\) 0 0
\(685\) 2.00000 0.0764161
\(686\) −15.0000 −0.572703
\(687\) 0 0
\(688\) 5.81507 0.221698
\(689\) 72.4452 2.75994
\(690\) 0 0
\(691\) 19.0754 0.725661 0.362831 0.931855i \(-0.381810\pi\)
0.362831 + 0.931855i \(0.381810\pi\)
\(692\) 1.00000 0.0380143
\(693\) 0 0
\(694\) 31.2603 1.18662
\(695\) −12.1849 −0.462201
\(696\) 0 0
\(697\) 5.81507 0.220262
\(698\) −35.2603 −1.33462
\(699\) 0 0
\(700\) 3.00000 0.113389
\(701\) −12.3699 −0.467203 −0.233601 0.972332i \(-0.575051\pi\)
−0.233601 + 0.972332i \(0.575051\pi\)
\(702\) 0 0
\(703\) −2.81507 −0.106172
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 23.8151 0.896292
\(707\) 28.8904 1.08654
\(708\) 0 0
\(709\) −0.630146 −0.0236656 −0.0118328 0.999930i \(-0.503767\pi\)
−0.0118328 + 0.999930i \(0.503767\pi\)
\(710\) 9.63015 0.361413
\(711\) 0 0
\(712\) −1.18493 −0.0444070
\(713\) −18.3699 −0.687956
\(714\) 0 0
\(715\) −6.81507 −0.254869
\(716\) −16.0000 −0.597948
\(717\) 0 0
\(718\) −27.6301 −1.03115
\(719\) −34.0000 −1.26799 −0.633993 0.773339i \(-0.718585\pi\)
−0.633993 + 0.773339i \(0.718585\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −11.0754 −0.412182
\(723\) 0 0
\(724\) −10.0000 −0.371647
\(725\) −3.81507 −0.141688
\(726\) 0 0
\(727\) −18.0000 −0.667583 −0.333792 0.942647i \(-0.608328\pi\)
−0.333792 + 0.942647i \(0.608328\pi\)
\(728\) 20.4452 0.757750
\(729\) 0 0
\(730\) −4.81507 −0.178214
\(731\) −5.81507 −0.215078
\(732\) 0 0
\(733\) 8.55478 0.315978 0.157989 0.987441i \(-0.449499\pi\)
0.157989 + 0.987441i \(0.449499\pi\)
\(734\) −21.0000 −0.775124
\(735\) 0 0
\(736\) 4.81507 0.177486
\(737\) −13.6301 −0.502073
\(738\) 0 0
\(739\) −19.0754 −0.701699 −0.350849 0.936432i \(-0.614107\pi\)
−0.350849 + 0.936432i \(0.614107\pi\)
\(740\) −1.00000 −0.0367607
\(741\) 0 0
\(742\) 31.8904 1.17073
\(743\) 27.4452 1.00687 0.503434 0.864034i \(-0.332070\pi\)
0.503434 + 0.864034i \(0.332070\pi\)
\(744\) 0 0
\(745\) 2.00000 0.0732743
\(746\) 25.2603 0.924845
\(747\) 0 0
\(748\) −1.00000 −0.0365636
\(749\) 56.4452 2.06246
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) 8.00000 0.291730
\(753\) 0 0
\(754\) −26.0000 −0.946864
\(755\) 6.81507 0.248026
\(756\) 0 0
\(757\) 13.1849 0.479214 0.239607 0.970870i \(-0.422981\pi\)
0.239607 + 0.970870i \(0.422981\pi\)
\(758\) 1.63015 0.0592096
\(759\) 0 0
\(760\) 2.81507 0.102113
\(761\) 19.8151 0.718296 0.359148 0.933281i \(-0.383067\pi\)
0.359148 + 0.933281i \(0.383067\pi\)
\(762\) 0 0
\(763\) 39.0000 1.41189
\(764\) 16.6301 0.601658
\(765\) 0 0
\(766\) −1.55478 −0.0561765
\(767\) −13.6301 −0.492156
\(768\) 0 0
\(769\) 6.73971 0.243040 0.121520 0.992589i \(-0.461223\pi\)
0.121520 + 0.992589i \(0.461223\pi\)
\(770\) −3.00000 −0.108112
\(771\) 0 0
\(772\) −2.00000 −0.0719816
\(773\) 28.6301 1.02975 0.514877 0.857264i \(-0.327837\pi\)
0.514877 + 0.857264i \(0.327837\pi\)
\(774\) 0 0
\(775\) −3.81507 −0.137041
\(776\) 5.81507 0.208749
\(777\) 0 0
\(778\) −38.7055 −1.38766
\(779\) 16.3699 0.586511
\(780\) 0 0
\(781\) −9.63015 −0.344594
\(782\) −4.81507 −0.172187
\(783\) 0 0
\(784\) 2.00000 0.0714286
\(785\) 2.18493 0.0779834
\(786\) 0 0
\(787\) −18.3699 −0.654815 −0.327407 0.944883i \(-0.606175\pi\)
−0.327407 + 0.944883i \(0.606175\pi\)
\(788\) 14.8151 0.527765
\(789\) 0 0
\(790\) −12.0000 −0.426941
\(791\) −11.4452 −0.406945
\(792\) 0 0
\(793\) −26.0000 −0.923287
\(794\) 15.6301 0.554693
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) −28.8904 −1.02335 −0.511676 0.859179i \(-0.670975\pi\)
−0.511676 + 0.859179i \(0.670975\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −32.4452 −1.14568
\(803\) 4.81507 0.169920
\(804\) 0 0
\(805\) −14.4452 −0.509127
\(806\) −26.0000 −0.915811
\(807\) 0 0
\(808\) 9.63015 0.338787
\(809\) −40.0754 −1.40897 −0.704487 0.709717i \(-0.748823\pi\)
−0.704487 + 0.709717i \(0.748823\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) −11.4452 −0.401648
\(813\) 0 0
\(814\) 1.00000 0.0350500
\(815\) 4.63015 0.162187
\(816\) 0 0
\(817\) −16.3699 −0.572709
\(818\) 27.6301 0.966065
\(819\) 0 0
\(820\) 5.81507 0.203071
\(821\) 42.0754 1.46844 0.734220 0.678911i \(-0.237548\pi\)
0.734220 + 0.678911i \(0.237548\pi\)
\(822\) 0 0
\(823\) −12.0754 −0.420921 −0.210460 0.977602i \(-0.567496\pi\)
−0.210460 + 0.977602i \(0.567496\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −6.00000 −0.208767
\(827\) −25.0754 −0.871956 −0.435978 0.899957i \(-0.643597\pi\)
−0.435978 + 0.899957i \(0.643597\pi\)
\(828\) 0 0
\(829\) 31.5206 1.09476 0.547378 0.836886i \(-0.315626\pi\)
0.547378 + 0.836886i \(0.315626\pi\)
\(830\) 6.81507 0.236555
\(831\) 0 0
\(832\) 6.81507 0.236270
\(833\) −2.00000 −0.0692959
\(834\) 0 0
\(835\) −11.1849 −0.387070
\(836\) −2.81507 −0.0973613
\(837\) 0 0
\(838\) −6.44522 −0.222646
\(839\) −18.0000 −0.621429 −0.310715 0.950503i \(-0.600568\pi\)
−0.310715 + 0.950503i \(0.600568\pi\)
\(840\) 0 0
\(841\) −14.4452 −0.498111
\(842\) 13.2603 0.456980
\(843\) 0 0
\(844\) −23.4452 −0.807018
\(845\) −33.4452 −1.15055
\(846\) 0 0
\(847\) −30.0000 −1.03081
\(848\) 10.6301 0.365041
\(849\) 0 0
\(850\) −1.00000 −0.0342997
\(851\) 4.81507 0.165059
\(852\) 0 0
\(853\) 42.0754 1.44063 0.720317 0.693646i \(-0.243997\pi\)
0.720317 + 0.693646i \(0.243997\pi\)
\(854\) −11.4452 −0.391647
\(855\) 0 0
\(856\) 18.8151 0.643086
\(857\) 12.2603 0.418804 0.209402 0.977830i \(-0.432848\pi\)
0.209402 + 0.977830i \(0.432848\pi\)
\(858\) 0 0
\(859\) −0.445219 −0.0151907 −0.00759533 0.999971i \(-0.502418\pi\)
−0.00759533 + 0.999971i \(0.502418\pi\)
\(860\) −5.81507 −0.198292
\(861\) 0 0
\(862\) −21.0000 −0.715263
\(863\) 33.4452 1.13849 0.569244 0.822168i \(-0.307236\pi\)
0.569244 + 0.822168i \(0.307236\pi\)
\(864\) 0 0
\(865\) −1.00000 −0.0340010
\(866\) −8.81507 −0.299548
\(867\) 0 0
\(868\) −11.4452 −0.388476
\(869\) 12.0000 0.407072
\(870\) 0 0
\(871\) −92.8904 −3.14747
\(872\) 13.0000 0.440236
\(873\) 0 0
\(874\) −13.5548 −0.458497
\(875\) −3.00000 −0.101419
\(876\) 0 0
\(877\) −24.7055 −0.834246 −0.417123 0.908850i \(-0.636962\pi\)
−0.417123 + 0.908850i \(0.636962\pi\)
\(878\) 13.4452 0.453754
\(879\) 0 0
\(880\) −1.00000 −0.0337100
\(881\) −53.8151 −1.81308 −0.906538 0.422124i \(-0.861285\pi\)
−0.906538 + 0.422124i \(0.861285\pi\)
\(882\) 0 0
\(883\) 27.0000 0.908622 0.454311 0.890843i \(-0.349885\pi\)
0.454311 + 0.890843i \(0.349885\pi\)
\(884\) −6.81507 −0.229216
\(885\) 0 0
\(886\) 7.63015 0.256340
\(887\) −39.0754 −1.31202 −0.656011 0.754751i \(-0.727758\pi\)
−0.656011 + 0.754751i \(0.727758\pi\)
\(888\) 0 0
\(889\) 31.3357 1.05096
\(890\) 1.18493 0.0397188
\(891\) 0 0
\(892\) −6.18493 −0.207087
\(893\) −22.5206 −0.753623
\(894\) 0 0
\(895\) 16.0000 0.534821
\(896\) 3.00000 0.100223
\(897\) 0 0
\(898\) 32.8904 1.09757
\(899\) 14.5548 0.485429
\(900\) 0 0
\(901\) −10.6301 −0.354142
\(902\) −5.81507 −0.193621
\(903\) 0 0
\(904\) −3.81507 −0.126887
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) −35.1849 −1.16830 −0.584148 0.811647i \(-0.698571\pi\)
−0.584148 + 0.811647i \(0.698571\pi\)
\(908\) 14.1849 0.470743
\(909\) 0 0
\(910\) −20.4452 −0.677752
\(911\) −30.5206 −1.01119 −0.505596 0.862770i \(-0.668727\pi\)
−0.505596 + 0.862770i \(0.668727\pi\)
\(912\) 0 0
\(913\) −6.81507 −0.225546
\(914\) 25.0754 0.829419
\(915\) 0 0
\(916\) 21.6301 0.714680
\(917\) −34.8904 −1.15218
\(918\) 0 0
\(919\) 24.0000 0.791687 0.395843 0.918318i \(-0.370452\pi\)
0.395843 + 0.918318i \(0.370452\pi\)
\(920\) −4.81507 −0.158748
\(921\) 0 0
\(922\) −24.1849 −0.796488
\(923\) −65.6301 −2.16024
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) 3.63015 0.119294
\(927\) 0 0
\(928\) −3.81507 −0.125236
\(929\) 7.44522 0.244270 0.122135 0.992514i \(-0.461026\pi\)
0.122135 + 0.992514i \(0.461026\pi\)
\(930\) 0 0
\(931\) −5.63015 −0.184521
\(932\) −13.6301 −0.446470
\(933\) 0 0
\(934\) −18.1849 −0.595029
\(935\) 1.00000 0.0327035
\(936\) 0 0
\(937\) 9.63015 0.314603 0.157302 0.987551i \(-0.449721\pi\)
0.157302 + 0.987551i \(0.449721\pi\)
\(938\) −40.8904 −1.33512
\(939\) 0 0
\(940\) −8.00000 −0.260931
\(941\) −41.2603 −1.34505 −0.672524 0.740076i \(-0.734790\pi\)
−0.672524 + 0.740076i \(0.734790\pi\)
\(942\) 0 0
\(943\) −28.0000 −0.911805
\(944\) −2.00000 −0.0650945
\(945\) 0 0
\(946\) 5.81507 0.189064
\(947\) 57.4452 1.86672 0.933359 0.358943i \(-0.116863\pi\)
0.933359 + 0.358943i \(0.116863\pi\)
\(948\) 0 0
\(949\) 32.8151 1.06522
\(950\) −2.81507 −0.0913330
\(951\) 0 0
\(952\) −3.00000 −0.0972306
\(953\) 48.8904 1.58372 0.791858 0.610705i \(-0.209114\pi\)
0.791858 + 0.610705i \(0.209114\pi\)
\(954\) 0 0
\(955\) −16.6301 −0.538139
\(956\) 25.4452 0.822957
\(957\) 0 0
\(958\) 0.815073 0.0263338
\(959\) −6.00000 −0.193750
\(960\) 0 0
\(961\) −16.4452 −0.530491
\(962\) 6.81507 0.219727
\(963\) 0 0
\(964\) −26.0000 −0.837404
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) −13.6301 −0.438316 −0.219158 0.975689i \(-0.570331\pi\)
−0.219158 + 0.975689i \(0.570331\pi\)
\(968\) −10.0000 −0.321412
\(969\) 0 0
\(970\) −5.81507 −0.186711
\(971\) −5.07536 −0.162876 −0.0814381 0.996678i \(-0.525951\pi\)
−0.0814381 + 0.996678i \(0.525951\pi\)
\(972\) 0 0
\(973\) 36.5548 1.17189
\(974\) 2.00000 0.0640841
\(975\) 0 0
\(976\) −3.81507 −0.122118
\(977\) −21.0000 −0.671850 −0.335925 0.941889i \(-0.609049\pi\)
−0.335925 + 0.941889i \(0.609049\pi\)
\(978\) 0 0
\(979\) −1.18493 −0.0378704
\(980\) −2.00000 −0.0638877
\(981\) 0 0
\(982\) −12.8151 −0.408945
\(983\) 38.3357 1.22272 0.611359 0.791354i \(-0.290623\pi\)
0.611359 + 0.791354i \(0.290623\pi\)
\(984\) 0 0
\(985\) −14.8151 −0.472047
\(986\) 3.81507 0.121497
\(987\) 0 0
\(988\) −19.1849 −0.610354
\(989\) 28.0000 0.890348
\(990\) 0 0
\(991\) −6.55478 −0.208219 −0.104110 0.994566i \(-0.533199\pi\)
−0.104110 + 0.994566i \(0.533199\pi\)
\(992\) −3.81507 −0.121129
\(993\) 0 0
\(994\) −28.8904 −0.916349
\(995\) 4.00000 0.126809
\(996\) 0 0
\(997\) 1.92464 0.0609538 0.0304769 0.999535i \(-0.490297\pi\)
0.0304769 + 0.999535i \(0.490297\pi\)
\(998\) 14.4452 0.457255
\(999\) 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 3330.2.a.bf.1.2 2
3.2 odd 2 1110.2.a.q.1.2 2
12.11 even 2 8880.2.a.bh.1.2 2
15.14 odd 2 5550.2.a.bx.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1110.2.a.q.1.2 2 3.2 odd 2
3330.2.a.bf.1.2 2 1.1 even 1 trivial
5550.2.a.bx.1.1 2 15.14 odd 2
8880.2.a.bh.1.2 2 12.11 even 2