Properties

Label 3330.2.a.bl.1.5
Level $3330$
Weight $2$
Character 3330.1
Self dual yes
Analytic conductor $26.590$
Analytic rank $0$
Dimension $5$
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: \(5\)
Coefficient field: 5.5.23544108.1
Defining polynomial: \(x^{5} - x^{4} - 20 x^{3} + 39 x^{2} + 9 x - 36\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.65043\) of defining polynomial
Character \(\chi\) \(=\) 3330.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.56812 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +4.56812 q^{7} +1.00000 q^{8} +1.00000 q^{10} -3.41708 q^{11} +0.971229 q^{13} +4.56812 q^{14} +1.00000 q^{16} -2.56812 q^{17} +4.86416 q^{19} +1.00000 q^{20} -3.41708 q^{22} +1.02877 q^{23} +1.00000 q^{25} +0.971229 q^{26} +4.56812 q^{28} -3.59689 q^{29} +2.55292 q^{31} +1.00000 q^{32} -2.56812 q^{34} +4.56812 q^{35} +1.00000 q^{37} +4.86416 q^{38} +1.00000 q^{40} +7.74670 q^{41} +9.74670 q^{43} -3.41708 q^{44} +1.02877 q^{46} +4.99877 q^{47} +13.8677 q^{49} +1.00000 q^{50} +0.971229 q^{52} -6.71793 q^{53} -3.41708 q^{55} +4.56812 q^{56} -3.59689 q^{58} +6.83416 q^{59} -6.99001 q^{61} +2.55292 q^{62} +1.00000 q^{64} +0.971229 q^{65} -15.9704 q^{67} -2.56812 q^{68} +4.56812 q^{70} -8.29962 q^{71} +1.56331 q^{73} +1.00000 q^{74} +4.86416 q^{76} -15.6096 q^{77} -6.29962 q^{79} +1.00000 q^{80} +7.74670 q^{82} +7.86293 q^{83} -2.56812 q^{85} +9.74670 q^{86} -3.41708 q^{88} -17.3013 q^{89} +4.43669 q^{91} +1.02877 q^{92} +4.99877 q^{94} +4.86416 q^{95} -0.747938 q^{97} +13.8677 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{2} + 5q^{4} + 5q^{5} + 3q^{7} + 5q^{8} + O(q^{10}) \) \( 5q + 5q^{2} + 5q^{4} + 5q^{5} + 3q^{7} + 5q^{8} + 5q^{10} + 5q^{11} + 5q^{13} + 3q^{14} + 5q^{16} + 7q^{17} - q^{19} + 5q^{20} + 5q^{22} + 5q^{23} + 5q^{25} + 5q^{26} + 3q^{28} + 2q^{29} + 16q^{31} + 5q^{32} + 7q^{34} + 3q^{35} + 5q^{37} - q^{38} + 5q^{40} + 2q^{41} + 12q^{43} + 5q^{44} + 5q^{46} + 6q^{47} + 16q^{49} + 5q^{50} + 5q^{52} + 3q^{53} + 5q^{55} + 3q^{56} + 2q^{58} - 10q^{59} + 16q^{61} + 16q^{62} + 5q^{64} + 5q^{65} + 4q^{67} + 7q^{68} + 3q^{70} - 8q^{71} - 3q^{73} + 5q^{74} - q^{76} + 3q^{77} + 2q^{79} + 5q^{80} + 2q^{82} - 5q^{83} + 7q^{85} + 12q^{86} + 5q^{88} - 7q^{89} + 33q^{91} + 5q^{92} + 6q^{94} - q^{95} + 14q^{97} + 16q^{98} + O(q^{100}) \)

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
\(7\) 4.56812 1.72659 0.863294 0.504701i \(-0.168397\pi\)
0.863294 + 0.504701i \(0.168397\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −3.41708 −1.03029 −0.515144 0.857104i \(-0.672262\pi\)
−0.515144 + 0.857104i \(0.672262\pi\)
\(12\) 0 0
\(13\) 0.971229 0.269370 0.134685 0.990888i \(-0.456998\pi\)
0.134685 + 0.990888i \(0.456998\pi\)
\(14\) 4.56812 1.22088
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −2.56812 −0.622861 −0.311431 0.950269i \(-0.600808\pi\)
−0.311431 + 0.950269i \(0.600808\pi\)
\(18\) 0 0
\(19\) 4.86416 1.11592 0.557958 0.829869i \(-0.311585\pi\)
0.557958 + 0.829869i \(0.311585\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −3.41708 −0.728524
\(23\) 1.02877 0.214514 0.107257 0.994231i \(-0.465793\pi\)
0.107257 + 0.994231i \(0.465793\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0.971229 0.190474
\(27\) 0 0
\(28\) 4.56812 0.863294
\(29\) −3.59689 −0.667926 −0.333963 0.942586i \(-0.608386\pi\)
−0.333963 + 0.942586i \(0.608386\pi\)
\(30\) 0 0
\(31\) 2.55292 0.458517 0.229259 0.973366i \(-0.426370\pi\)
0.229259 + 0.973366i \(0.426370\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −2.56812 −0.440429
\(35\) 4.56812 0.772154
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) 4.86416 0.789071
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 7.74670 1.20983 0.604916 0.796289i \(-0.293207\pi\)
0.604916 + 0.796289i \(0.293207\pi\)
\(42\) 0 0
\(43\) 9.74670 1.48636 0.743179 0.669093i \(-0.233317\pi\)
0.743179 + 0.669093i \(0.233317\pi\)
\(44\) −3.41708 −0.515144
\(45\) 0 0
\(46\) 1.02877 0.151684
\(47\) 4.99877 0.729145 0.364572 0.931175i \(-0.381215\pi\)
0.364572 + 0.931175i \(0.381215\pi\)
\(48\) 0 0
\(49\) 13.8677 1.98111
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 0.971229 0.134685
\(53\) −6.71793 −0.922779 −0.461389 0.887198i \(-0.652649\pi\)
−0.461389 + 0.887198i \(0.652649\pi\)
\(54\) 0 0
\(55\) −3.41708 −0.460759
\(56\) 4.56812 0.610441
\(57\) 0 0
\(58\) −3.59689 −0.472295
\(59\) 6.83416 0.889731 0.444866 0.895597i \(-0.353251\pi\)
0.444866 + 0.895597i \(0.353251\pi\)
\(60\) 0 0
\(61\) −6.99001 −0.894980 −0.447490 0.894289i \(-0.647682\pi\)
−0.447490 + 0.894289i \(0.647682\pi\)
\(62\) 2.55292 0.324221
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0.971229 0.120466
\(66\) 0 0
\(67\) −15.9704 −1.95109 −0.975547 0.219789i \(-0.929463\pi\)
−0.975547 + 0.219789i \(0.929463\pi\)
\(68\) −2.56812 −0.311431
\(69\) 0 0
\(70\) 4.56812 0.545995
\(71\) −8.29962 −0.984984 −0.492492 0.870317i \(-0.663914\pi\)
−0.492492 + 0.870317i \(0.663914\pi\)
\(72\) 0 0
\(73\) 1.56331 0.182972 0.0914858 0.995806i \(-0.470838\pi\)
0.0914858 + 0.995806i \(0.470838\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 4.86416 0.557958
\(77\) −15.6096 −1.77888
\(78\) 0 0
\(79\) −6.29962 −0.708762 −0.354381 0.935101i \(-0.615308\pi\)
−0.354381 + 0.935101i \(0.615308\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 7.74670 0.855480
\(83\) 7.86293 0.863068 0.431534 0.902097i \(-0.357972\pi\)
0.431534 + 0.902097i \(0.357972\pi\)
\(84\) 0 0
\(85\) −2.56812 −0.278552
\(86\) 9.74670 1.05101
\(87\) 0 0
\(88\) −3.41708 −0.364262
\(89\) −17.3013 −1.83393 −0.916965 0.398968i \(-0.869369\pi\)
−0.916965 + 0.398968i \(0.869369\pi\)
\(90\) 0 0
\(91\) 4.43669 0.465092
\(92\) 1.02877 0.107257
\(93\) 0 0
\(94\) 4.99877 0.515583
\(95\) 4.86416 0.499053
\(96\) 0 0
\(97\) −0.747938 −0.0759416 −0.0379708 0.999279i \(-0.512089\pi\)
−0.0379708 + 0.999279i \(0.512089\pi\)
\(98\) 13.8677 1.40085
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 18.4346 1.83431 0.917157 0.398526i \(-0.130478\pi\)
0.917157 + 0.398526i \(0.130478\pi\)
\(102\) 0 0
\(103\) −2.99877 −0.295477 −0.147739 0.989026i \(-0.547199\pi\)
−0.147739 + 0.989026i \(0.547199\pi\)
\(104\) 0.971229 0.0952368
\(105\) 0 0
\(106\) −6.71793 −0.652503
\(107\) 8.22256 0.794905 0.397452 0.917623i \(-0.369894\pi\)
0.397452 + 0.917623i \(0.369894\pi\)
\(108\) 0 0
\(109\) 13.7167 1.31382 0.656911 0.753968i \(-0.271863\pi\)
0.656911 + 0.753968i \(0.271863\pi\)
\(110\) −3.41708 −0.325806
\(111\) 0 0
\(112\) 4.56812 0.431647
\(113\) −3.23726 −0.304536 −0.152268 0.988339i \(-0.548658\pi\)
−0.152268 + 0.988339i \(0.548658\pi\)
\(114\) 0 0
\(115\) 1.02877 0.0959334
\(116\) −3.59689 −0.333963
\(117\) 0 0
\(118\) 6.83416 0.629135
\(119\) −11.7315 −1.07542
\(120\) 0 0
\(121\) 0.676423 0.0614930
\(122\) −6.99001 −0.632846
\(123\) 0 0
\(124\) 2.55292 0.229259
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −9.21330 −0.817549 −0.408774 0.912636i \(-0.634044\pi\)
−0.408774 + 0.912636i \(0.634044\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0.971229 0.0851824
\(131\) 3.43833 0.300409 0.150204 0.988655i \(-0.452007\pi\)
0.150204 + 0.988655i \(0.452007\pi\)
\(132\) 0 0
\(133\) 22.2201 1.92673
\(134\) −15.9704 −1.37963
\(135\) 0 0
\(136\) −2.56812 −0.220215
\(137\) −3.30085 −0.282011 −0.141005 0.990009i \(-0.545034\pi\)
−0.141005 + 0.990009i \(0.545034\pi\)
\(138\) 0 0
\(139\) −3.06236 −0.259746 −0.129873 0.991531i \(-0.541457\pi\)
−0.129873 + 0.991531i \(0.541457\pi\)
\(140\) 4.56812 0.386077
\(141\) 0 0
\(142\) −8.29962 −0.696489
\(143\) −3.31876 −0.277529
\(144\) 0 0
\(145\) −3.59689 −0.298706
\(146\) 1.56331 0.129380
\(147\) 0 0
\(148\) 1.00000 0.0821995
\(149\) 9.54293 0.781787 0.390894 0.920436i \(-0.372166\pi\)
0.390894 + 0.920436i \(0.372166\pi\)
\(150\) 0 0
\(151\) 8.07706 0.657302 0.328651 0.944451i \(-0.393406\pi\)
0.328651 + 0.944451i \(0.393406\pi\)
\(152\) 4.86416 0.394536
\(153\) 0 0
\(154\) −15.6096 −1.25786
\(155\) 2.55292 0.205055
\(156\) 0 0
\(157\) −21.6248 −1.72585 −0.862925 0.505332i \(-0.831370\pi\)
−0.862925 + 0.505332i \(0.831370\pi\)
\(158\) −6.29962 −0.501171
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) 4.69955 0.370377
\(162\) 0 0
\(163\) 8.12585 0.636466 0.318233 0.948013i \(-0.396911\pi\)
0.318233 + 0.948013i \(0.396911\pi\)
\(164\) 7.74670 0.604916
\(165\) 0 0
\(166\) 7.86293 0.610282
\(167\) −1.02877 −0.0796087 −0.0398044 0.999207i \(-0.512673\pi\)
−0.0398044 + 0.999207i \(0.512673\pi\)
\(168\) 0 0
\(169\) −12.0567 −0.927440
\(170\) −2.56812 −0.196966
\(171\) 0 0
\(172\) 9.74670 0.743179
\(173\) −7.66964 −0.583112 −0.291556 0.956554i \(-0.594173\pi\)
−0.291556 + 0.956554i \(0.594173\pi\)
\(174\) 0 0
\(175\) 4.56812 0.345318
\(176\) −3.41708 −0.257572
\(177\) 0 0
\(178\) −17.3013 −1.29678
\(179\) −1.64284 −0.122792 −0.0613958 0.998114i \(-0.519555\pi\)
−0.0613958 + 0.998114i \(0.519555\pi\)
\(180\) 0 0
\(181\) 25.9704 1.93036 0.965182 0.261578i \(-0.0842429\pi\)
0.965182 + 0.261578i \(0.0842429\pi\)
\(182\) 4.43669 0.328869
\(183\) 0 0
\(184\) 1.02877 0.0758420
\(185\) 1.00000 0.0735215
\(186\) 0 0
\(187\) 8.77547 0.641726
\(188\) 4.99877 0.364572
\(189\) 0 0
\(190\) 4.86416 0.352883
\(191\) −18.9253 −1.36939 −0.684693 0.728832i \(-0.740064\pi\)
−0.684693 + 0.728832i \(0.740064\pi\)
\(192\) 0 0
\(193\) −2.63913 −0.189969 −0.0949845 0.995479i \(-0.530280\pi\)
−0.0949845 + 0.995479i \(0.530280\pi\)
\(194\) −0.747938 −0.0536988
\(195\) 0 0
\(196\) 13.8677 0.990553
\(197\) −10.9416 −0.779559 −0.389779 0.920908i \(-0.627449\pi\)
−0.389779 + 0.920908i \(0.627449\pi\)
\(198\) 0 0
\(199\) −1.75545 −0.124441 −0.0622204 0.998062i \(-0.519818\pi\)
−0.0622204 + 0.998062i \(0.519818\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 18.4346 1.29706
\(203\) −16.4310 −1.15323
\(204\) 0 0
\(205\) 7.74670 0.541053
\(206\) −2.99877 −0.208934
\(207\) 0 0
\(208\) 0.971229 0.0673426
\(209\) −16.6212 −1.14971
\(210\) 0 0
\(211\) 11.3252 0.779660 0.389830 0.920887i \(-0.372534\pi\)
0.389830 + 0.920887i \(0.372534\pi\)
\(212\) −6.71793 −0.461389
\(213\) 0 0
\(214\) 8.22256 0.562083
\(215\) 9.74670 0.664720
\(216\) 0 0
\(217\) 11.6620 0.791670
\(218\) 13.7167 0.929012
\(219\) 0 0
\(220\) −3.41708 −0.230379
\(221\) −2.49423 −0.167780
\(222\) 0 0
\(223\) −15.0328 −1.00667 −0.503334 0.864092i \(-0.667893\pi\)
−0.503334 + 0.864092i \(0.667893\pi\)
\(224\) 4.56812 0.305221
\(225\) 0 0
\(226\) −3.23726 −0.215340
\(227\) −14.3735 −0.954003 −0.477002 0.878902i \(-0.658276\pi\)
−0.477002 + 0.878902i \(0.658276\pi\)
\(228\) 0 0
\(229\) 13.2242 0.873880 0.436940 0.899491i \(-0.356062\pi\)
0.436940 + 0.899491i \(0.356062\pi\)
\(230\) 1.02877 0.0678352
\(231\) 0 0
\(232\) −3.59689 −0.236148
\(233\) 0.222152 0.0145536 0.00727682 0.999974i \(-0.497684\pi\)
0.00727682 + 0.999974i \(0.497684\pi\)
\(234\) 0 0
\(235\) 4.99877 0.326083
\(236\) 6.83416 0.444866
\(237\) 0 0
\(238\) −11.7315 −0.760440
\(239\) 2.14106 0.138493 0.0692467 0.997600i \(-0.477940\pi\)
0.0692467 + 0.997600i \(0.477940\pi\)
\(240\) 0 0
\(241\) 28.3876 1.82860 0.914302 0.405033i \(-0.132740\pi\)
0.914302 + 0.405033i \(0.132740\pi\)
\(242\) 0.676423 0.0434821
\(243\) 0 0
\(244\) −6.99001 −0.447490
\(245\) 13.8677 0.885977
\(246\) 0 0
\(247\) 4.72421 0.300595
\(248\) 2.55292 0.162110
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 22.3276 1.40930 0.704652 0.709553i \(-0.251103\pi\)
0.704652 + 0.709553i \(0.251103\pi\)
\(252\) 0 0
\(253\) −3.51539 −0.221011
\(254\) −9.21330 −0.578094
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 26.9392 1.68042 0.840209 0.542262i \(-0.182432\pi\)
0.840209 + 0.542262i \(0.182432\pi\)
\(258\) 0 0
\(259\) 4.56812 0.283849
\(260\) 0.971229 0.0602330
\(261\) 0 0
\(262\) 3.43833 0.212421
\(263\) −16.2134 −0.999761 −0.499881 0.866094i \(-0.666623\pi\)
−0.499881 + 0.866094i \(0.666623\pi\)
\(264\) 0 0
\(265\) −6.71793 −0.412679
\(266\) 22.2201 1.36240
\(267\) 0 0
\(268\) −15.9704 −0.975547
\(269\) −19.0487 −1.16142 −0.580710 0.814111i \(-0.697225\pi\)
−0.580710 + 0.814111i \(0.697225\pi\)
\(270\) 0 0
\(271\) −1.99753 −0.121341 −0.0606707 0.998158i \(-0.519324\pi\)
−0.0606707 + 0.998158i \(0.519324\pi\)
\(272\) −2.56812 −0.155715
\(273\) 0 0
\(274\) −3.30085 −0.199412
\(275\) −3.41708 −0.206058
\(276\) 0 0
\(277\) −2.46710 −0.148234 −0.0741170 0.997250i \(-0.523614\pi\)
−0.0741170 + 0.997250i \(0.523614\pi\)
\(278\) −3.06236 −0.183668
\(279\) 0 0
\(280\) 4.56812 0.272998
\(281\) −15.3284 −0.914415 −0.457208 0.889360i \(-0.651150\pi\)
−0.457208 + 0.889360i \(0.651150\pi\)
\(282\) 0 0
\(283\) −4.75710 −0.282780 −0.141390 0.989954i \(-0.545157\pi\)
−0.141390 + 0.989954i \(0.545157\pi\)
\(284\) −8.29962 −0.492492
\(285\) 0 0
\(286\) −3.31876 −0.196243
\(287\) 35.3879 2.08888
\(288\) 0 0
\(289\) −10.4047 −0.612044
\(290\) −3.59689 −0.211217
\(291\) 0 0
\(292\) 1.56331 0.0914858
\(293\) 2.17377 0.126993 0.0634964 0.997982i \(-0.479775\pi\)
0.0634964 + 0.997982i \(0.479775\pi\)
\(294\) 0 0
\(295\) 6.83416 0.397900
\(296\) 1.00000 0.0581238
\(297\) 0 0
\(298\) 9.54293 0.552807
\(299\) 0.999172 0.0577836
\(300\) 0 0
\(301\) 44.5241 2.56633
\(302\) 8.07706 0.464783
\(303\) 0 0
\(304\) 4.86416 0.278979
\(305\) −6.99001 −0.400247
\(306\) 0 0
\(307\) 2.83416 0.161754 0.0808769 0.996724i \(-0.474228\pi\)
0.0808769 + 0.996724i \(0.474228\pi\)
\(308\) −15.6096 −0.889441
\(309\) 0 0
\(310\) 2.55292 0.144996
\(311\) −30.4039 −1.72405 −0.862024 0.506867i \(-0.830803\pi\)
−0.862024 + 0.506867i \(0.830803\pi\)
\(312\) 0 0
\(313\) −27.2992 −1.54304 −0.771521 0.636204i \(-0.780504\pi\)
−0.771521 + 0.636204i \(0.780504\pi\)
\(314\) −21.6248 −1.22036
\(315\) 0 0
\(316\) −6.29962 −0.354381
\(317\) 29.4271 1.65279 0.826396 0.563090i \(-0.190388\pi\)
0.826396 + 0.563090i \(0.190388\pi\)
\(318\) 0 0
\(319\) 12.2909 0.688156
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 4.69955 0.261896
\(323\) −12.4918 −0.695060
\(324\) 0 0
\(325\) 0.971229 0.0538741
\(326\) 8.12585 0.450049
\(327\) 0 0
\(328\) 7.74670 0.427740
\(329\) 22.8350 1.25893
\(330\) 0 0
\(331\) −21.9305 −1.20541 −0.602705 0.797964i \(-0.705910\pi\)
−0.602705 + 0.797964i \(0.705910\pi\)
\(332\) 7.86293 0.431534
\(333\) 0 0
\(334\) −1.02877 −0.0562919
\(335\) −15.9704 −0.872556
\(336\) 0 0
\(337\) 29.0271 1.58121 0.790604 0.612328i \(-0.209767\pi\)
0.790604 + 0.612328i \(0.209767\pi\)
\(338\) −12.0567 −0.655799
\(339\) 0 0
\(340\) −2.56812 −0.139276
\(341\) −8.72351 −0.472405
\(342\) 0 0
\(343\) 31.3727 1.69397
\(344\) 9.74670 0.525507
\(345\) 0 0
\(346\) −7.66964 −0.412323
\(347\) −12.3300 −0.661911 −0.330955 0.943646i \(-0.607371\pi\)
−0.330955 + 0.943646i \(0.607371\pi\)
\(348\) 0 0
\(349\) 1.82262 0.0975628 0.0487814 0.998809i \(-0.484466\pi\)
0.0487814 + 0.998809i \(0.484466\pi\)
\(350\) 4.56812 0.244176
\(351\) 0 0
\(352\) −3.41708 −0.182131
\(353\) −8.04348 −0.428111 −0.214055 0.976822i \(-0.568667\pi\)
−0.214055 + 0.976822i \(0.568667\pi\)
\(354\) 0 0
\(355\) −8.29962 −0.440498
\(356\) −17.3013 −0.916965
\(357\) 0 0
\(358\) −1.64284 −0.0868267
\(359\) −5.52547 −0.291623 −0.145812 0.989312i \(-0.546579\pi\)
−0.145812 + 0.989312i \(0.546579\pi\)
\(360\) 0 0
\(361\) 4.66008 0.245267
\(362\) 25.9704 1.36497
\(363\) 0 0
\(364\) 4.43669 0.232546
\(365\) 1.56331 0.0818273
\(366\) 0 0
\(367\) −10.4115 −0.543478 −0.271739 0.962371i \(-0.587599\pi\)
−0.271739 + 0.962371i \(0.587599\pi\)
\(368\) 1.02877 0.0536284
\(369\) 0 0
\(370\) 1.00000 0.0519875
\(371\) −30.6883 −1.59326
\(372\) 0 0
\(373\) 20.2125 1.04656 0.523281 0.852160i \(-0.324708\pi\)
0.523281 + 0.852160i \(0.324708\pi\)
\(374\) 8.77547 0.453769
\(375\) 0 0
\(376\) 4.99877 0.257792
\(377\) −3.49341 −0.179920
\(378\) 0 0
\(379\) −15.5557 −0.799042 −0.399521 0.916724i \(-0.630824\pi\)
−0.399521 + 0.916724i \(0.630824\pi\)
\(380\) 4.86416 0.249526
\(381\) 0 0
\(382\) −18.9253 −0.968302
\(383\) 29.5137 1.50808 0.754041 0.656828i \(-0.228102\pi\)
0.754041 + 0.656828i \(0.228102\pi\)
\(384\) 0 0
\(385\) −15.6096 −0.795540
\(386\) −2.63913 −0.134328
\(387\) 0 0
\(388\) −0.747938 −0.0379708
\(389\) −14.7235 −0.746512 −0.373256 0.927728i \(-0.621759\pi\)
−0.373256 + 0.927728i \(0.621759\pi\)
\(390\) 0 0
\(391\) −2.64201 −0.133612
\(392\) 13.8677 0.700427
\(393\) 0 0
\(394\) −10.9416 −0.551231
\(395\) −6.29962 −0.316968
\(396\) 0 0
\(397\) −17.7859 −0.892647 −0.446324 0.894872i \(-0.647267\pi\)
−0.446324 + 0.894872i \(0.647267\pi\)
\(398\) −1.75545 −0.0879930
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −33.4858 −1.67220 −0.836100 0.548577i \(-0.815170\pi\)
−0.836100 + 0.548577i \(0.815170\pi\)
\(402\) 0 0
\(403\) 2.47946 0.123511
\(404\) 18.4346 0.917157
\(405\) 0 0
\(406\) −16.4310 −0.815459
\(407\) −3.41708 −0.169378
\(408\) 0 0
\(409\) −30.2700 −1.49676 −0.748378 0.663272i \(-0.769167\pi\)
−0.748378 + 0.663272i \(0.769167\pi\)
\(410\) 7.74670 0.382582
\(411\) 0 0
\(412\) −2.99877 −0.147739
\(413\) 31.2193 1.53620
\(414\) 0 0
\(415\) 7.86293 0.385976
\(416\) 0.971229 0.0476184
\(417\) 0 0
\(418\) −16.6212 −0.812971
\(419\) −34.5421 −1.68749 −0.843746 0.536743i \(-0.819655\pi\)
−0.843746 + 0.536743i \(0.819655\pi\)
\(420\) 0 0
\(421\) 10.6991 0.521445 0.260722 0.965414i \(-0.416039\pi\)
0.260722 + 0.965414i \(0.416039\pi\)
\(422\) 11.3252 0.551303
\(423\) 0 0
\(424\) −6.71793 −0.326252
\(425\) −2.56812 −0.124572
\(426\) 0 0
\(427\) −31.9312 −1.54526
\(428\) 8.22256 0.397452
\(429\) 0 0
\(430\) 9.74670 0.470028
\(431\) 5.74112 0.276540 0.138270 0.990395i \(-0.455846\pi\)
0.138270 + 0.990395i \(0.455846\pi\)
\(432\) 0 0
\(433\) −16.1650 −0.776841 −0.388421 0.921482i \(-0.626979\pi\)
−0.388421 + 0.921482i \(0.626979\pi\)
\(434\) 11.6620 0.559795
\(435\) 0 0
\(436\) 13.7167 0.656911
\(437\) 5.00411 0.239379
\(438\) 0 0
\(439\) 33.3279 1.59065 0.795326 0.606182i \(-0.207300\pi\)
0.795326 + 0.606182i \(0.207300\pi\)
\(440\) −3.41708 −0.162903
\(441\) 0 0
\(442\) −2.49423 −0.118639
\(443\) 11.7930 0.560303 0.280152 0.959956i \(-0.409615\pi\)
0.280152 + 0.959956i \(0.409615\pi\)
\(444\) 0 0
\(445\) −17.3013 −0.820158
\(446\) −15.0328 −0.711822
\(447\) 0 0
\(448\) 4.56812 0.215823
\(449\) −14.8917 −0.702783 −0.351391 0.936229i \(-0.614291\pi\)
−0.351391 + 0.936229i \(0.614291\pi\)
\(450\) 0 0
\(451\) −26.4711 −1.24647
\(452\) −3.23726 −0.152268
\(453\) 0 0
\(454\) −14.3735 −0.674582
\(455\) 4.43669 0.207995
\(456\) 0 0
\(457\) −3.03793 −0.142108 −0.0710542 0.997472i \(-0.522636\pi\)
−0.0710542 + 0.997472i \(0.522636\pi\)
\(458\) 13.2242 0.617926
\(459\) 0 0
\(460\) 1.02877 0.0479667
\(461\) −17.5465 −0.817222 −0.408611 0.912709i \(-0.633987\pi\)
−0.408611 + 0.912709i \(0.633987\pi\)
\(462\) 0 0
\(463\) 14.5522 0.676297 0.338149 0.941093i \(-0.390199\pi\)
0.338149 + 0.941093i \(0.390199\pi\)
\(464\) −3.59689 −0.166982
\(465\) 0 0
\(466\) 0.222152 0.0102910
\(467\) 16.9681 0.785188 0.392594 0.919712i \(-0.371578\pi\)
0.392594 + 0.919712i \(0.371578\pi\)
\(468\) 0 0
\(469\) −72.9547 −3.36874
\(470\) 4.99877 0.230576
\(471\) 0 0
\(472\) 6.83416 0.314568
\(473\) −33.3052 −1.53138
\(474\) 0 0
\(475\) 4.86416 0.223183
\(476\) −11.7315 −0.537712
\(477\) 0 0
\(478\) 2.14106 0.0979297
\(479\) −26.7267 −1.22117 −0.610587 0.791949i \(-0.709066\pi\)
−0.610587 + 0.791949i \(0.709066\pi\)
\(480\) 0 0
\(481\) 0.971229 0.0442842
\(482\) 28.3876 1.29302
\(483\) 0 0
\(484\) 0.676423 0.0307465
\(485\) −0.747938 −0.0339621
\(486\) 0 0
\(487\) −32.1054 −1.45484 −0.727418 0.686195i \(-0.759280\pi\)
−0.727418 + 0.686195i \(0.759280\pi\)
\(488\) −6.99001 −0.316423
\(489\) 0 0
\(490\) 13.8677 0.626481
\(491\) −38.0588 −1.71757 −0.858784 0.512338i \(-0.828780\pi\)
−0.858784 + 0.512338i \(0.828780\pi\)
\(492\) 0 0
\(493\) 9.23726 0.416025
\(494\) 4.72421 0.212552
\(495\) 0 0
\(496\) 2.55292 0.114629
\(497\) −37.9137 −1.70066
\(498\) 0 0
\(499\) −32.5421 −1.45678 −0.728392 0.685160i \(-0.759732\pi\)
−0.728392 + 0.685160i \(0.759732\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 22.3276 0.996528
\(503\) 16.1549 0.720313 0.360156 0.932892i \(-0.382723\pi\)
0.360156 + 0.932892i \(0.382723\pi\)
\(504\) 0 0
\(505\) 18.4346 0.820330
\(506\) −3.51539 −0.156278
\(507\) 0 0
\(508\) −9.21330 −0.408774
\(509\) −12.5021 −0.554144 −0.277072 0.960849i \(-0.589364\pi\)
−0.277072 + 0.960849i \(0.589364\pi\)
\(510\) 0 0
\(511\) 7.14139 0.315916
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 26.9392 1.18824
\(515\) −2.99877 −0.132141
\(516\) 0 0
\(517\) −17.0812 −0.751229
\(518\) 4.56812 0.200712
\(519\) 0 0
\(520\) 0.971229 0.0425912
\(521\) 1.29912 0.0569153 0.0284576 0.999595i \(-0.490940\pi\)
0.0284576 + 0.999595i \(0.490940\pi\)
\(522\) 0 0
\(523\) 26.3995 1.15437 0.577184 0.816614i \(-0.304151\pi\)
0.577184 + 0.816614i \(0.304151\pi\)
\(524\) 3.43833 0.150204
\(525\) 0 0
\(526\) −16.2134 −0.706938
\(527\) −6.55620 −0.285593
\(528\) 0 0
\(529\) −21.9416 −0.953984
\(530\) −6.71793 −0.291808
\(531\) 0 0
\(532\) 22.2201 0.963363
\(533\) 7.52382 0.325893
\(534\) 0 0
\(535\) 8.22256 0.355492
\(536\) −15.9704 −0.689816
\(537\) 0 0
\(538\) −19.0487 −0.821248
\(539\) −47.3872 −2.04111
\(540\) 0 0
\(541\) 10.3600 0.445413 0.222706 0.974886i \(-0.428511\pi\)
0.222706 + 0.974886i \(0.428511\pi\)
\(542\) −1.99753 −0.0858013
\(543\) 0 0
\(544\) −2.56812 −0.110107
\(545\) 13.7167 0.587559
\(546\) 0 0
\(547\) 31.4051 1.34279 0.671393 0.741101i \(-0.265696\pi\)
0.671393 + 0.741101i \(0.265696\pi\)
\(548\) −3.30085 −0.141005
\(549\) 0 0
\(550\) −3.41708 −0.145705
\(551\) −17.4959 −0.745349
\(552\) 0 0
\(553\) −28.7774 −1.22374
\(554\) −2.46710 −0.104817
\(555\) 0 0
\(556\) −3.06236 −0.129873
\(557\) 40.6872 1.72397 0.861986 0.506932i \(-0.169221\pi\)
0.861986 + 0.506932i \(0.169221\pi\)
\(558\) 0 0
\(559\) 9.46628 0.400381
\(560\) 4.56812 0.193038
\(561\) 0 0
\(562\) −15.3284 −0.646589
\(563\) −22.9777 −0.968394 −0.484197 0.874959i \(-0.660888\pi\)
−0.484197 + 0.874959i \(0.660888\pi\)
\(564\) 0 0
\(565\) −3.23726 −0.136193
\(566\) −4.75710 −0.199956
\(567\) 0 0
\(568\) −8.29962 −0.348244
\(569\) 19.5129 0.818024 0.409012 0.912529i \(-0.365873\pi\)
0.409012 + 0.912529i \(0.365873\pi\)
\(570\) 0 0
\(571\) −44.0843 −1.84487 −0.922436 0.386150i \(-0.873805\pi\)
−0.922436 + 0.386150i \(0.873805\pi\)
\(572\) −3.31876 −0.138764
\(573\) 0 0
\(574\) 35.3879 1.47706
\(575\) 1.02877 0.0429027
\(576\) 0 0
\(577\) 1.03165 0.0429481 0.0214740 0.999769i \(-0.493164\pi\)
0.0214740 + 0.999769i \(0.493164\pi\)
\(578\) −10.4047 −0.432780
\(579\) 0 0
\(580\) −3.59689 −0.149353
\(581\) 35.9188 1.49016
\(582\) 0 0
\(583\) 22.9557 0.950728
\(584\) 1.56331 0.0646902
\(585\) 0 0
\(586\) 2.17377 0.0897975
\(587\) 21.7214 0.896539 0.448269 0.893899i \(-0.352041\pi\)
0.448269 + 0.893899i \(0.352041\pi\)
\(588\) 0 0
\(589\) 12.4178 0.511666
\(590\) 6.83416 0.281358
\(591\) 0 0
\(592\) 1.00000 0.0410997
\(593\) −9.95763 −0.408911 −0.204455 0.978876i \(-0.565542\pi\)
−0.204455 + 0.978876i \(0.565542\pi\)
\(594\) 0 0
\(595\) −11.7315 −0.480944
\(596\) 9.54293 0.390894
\(597\) 0 0
\(598\) 0.999172 0.0408592
\(599\) 11.9704 0.489097 0.244549 0.969637i \(-0.421360\pi\)
0.244549 + 0.969637i \(0.421360\pi\)
\(600\) 0 0
\(601\) 15.3447 0.625925 0.312962 0.949766i \(-0.398679\pi\)
0.312962 + 0.949766i \(0.398679\pi\)
\(602\) 44.5241 1.81467
\(603\) 0 0
\(604\) 8.07706 0.328651
\(605\) 0.676423 0.0275005
\(606\) 0 0
\(607\) −26.0471 −1.05722 −0.528609 0.848866i \(-0.677286\pi\)
−0.528609 + 0.848866i \(0.677286\pi\)
\(608\) 4.86416 0.197268
\(609\) 0 0
\(610\) −6.99001 −0.283017
\(611\) 4.85494 0.196410
\(612\) 0 0
\(613\) −1.83897 −0.0742753 −0.0371376 0.999310i \(-0.511824\pi\)
−0.0371376 + 0.999310i \(0.511824\pi\)
\(614\) 2.83416 0.114377
\(615\) 0 0
\(616\) −15.6096 −0.628930
\(617\) 46.2326 1.86125 0.930627 0.365969i \(-0.119262\pi\)
0.930627 + 0.365969i \(0.119262\pi\)
\(618\) 0 0
\(619\) −9.17991 −0.368972 −0.184486 0.982835i \(-0.559062\pi\)
−0.184486 + 0.982835i \(0.559062\pi\)
\(620\) 2.55292 0.102528
\(621\) 0 0
\(622\) −30.4039 −1.21909
\(623\) −79.0343 −3.16644
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −27.2992 −1.09110
\(627\) 0 0
\(628\) −21.6248 −0.862925
\(629\) −2.56812 −0.102398
\(630\) 0 0
\(631\) 43.0379 1.71331 0.856655 0.515889i \(-0.172538\pi\)
0.856655 + 0.515889i \(0.172538\pi\)
\(632\) −6.29962 −0.250585
\(633\) 0 0
\(634\) 29.4271 1.16870
\(635\) −9.21330 −0.365619
\(636\) 0 0
\(637\) 13.4687 0.533651
\(638\) 12.2909 0.486600
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −4.81578 −0.190212 −0.0951059 0.995467i \(-0.530319\pi\)
−0.0951059 + 0.995467i \(0.530319\pi\)
\(642\) 0 0
\(643\) −9.60247 −0.378685 −0.189342 0.981911i \(-0.560636\pi\)
−0.189342 + 0.981911i \(0.560636\pi\)
\(644\) 4.69955 0.185188
\(645\) 0 0
\(646\) −12.4918 −0.491482
\(647\) −33.6288 −1.32209 −0.661043 0.750348i \(-0.729886\pi\)
−0.661043 + 0.750348i \(0.729886\pi\)
\(648\) 0 0
\(649\) −23.3528 −0.916679
\(650\) 0.971229 0.0380947
\(651\) 0 0
\(652\) 8.12585 0.318233
\(653\) −13.6085 −0.532541 −0.266271 0.963898i \(-0.585792\pi\)
−0.266271 + 0.963898i \(0.585792\pi\)
\(654\) 0 0
\(655\) 3.43833 0.134347
\(656\) 7.74670 0.302458
\(657\) 0 0
\(658\) 22.8350 0.890200
\(659\) −9.21290 −0.358884 −0.179442 0.983769i \(-0.557429\pi\)
−0.179442 + 0.983769i \(0.557429\pi\)
\(660\) 0 0
\(661\) −12.9033 −0.501881 −0.250941 0.968002i \(-0.580740\pi\)
−0.250941 + 0.968002i \(0.580740\pi\)
\(662\) −21.9305 −0.852353
\(663\) 0 0
\(664\) 7.86293 0.305141
\(665\) 22.2201 0.861658
\(666\) 0 0
\(667\) −3.70038 −0.143279
\(668\) −1.02877 −0.0398044
\(669\) 0 0
\(670\) −15.9704 −0.616990
\(671\) 23.8854 0.922086
\(672\) 0 0
\(673\) 4.76919 0.183839 0.0919194 0.995766i \(-0.470700\pi\)
0.0919194 + 0.995766i \(0.470700\pi\)
\(674\) 29.0271 1.11808
\(675\) 0 0
\(676\) −12.0567 −0.463720
\(677\) 42.6724 1.64003 0.820017 0.572339i \(-0.193964\pi\)
0.820017 + 0.572339i \(0.193964\pi\)
\(678\) 0 0
\(679\) −3.41667 −0.131120
\(680\) −2.56812 −0.0984830
\(681\) 0 0
\(682\) −8.72351 −0.334041
\(683\) 12.0141 0.459706 0.229853 0.973225i \(-0.426176\pi\)
0.229853 + 0.973225i \(0.426176\pi\)
\(684\) 0 0
\(685\) −3.30085 −0.126119
\(686\) 31.3727 1.19781
\(687\) 0 0
\(688\) 9.74670 0.371590
\(689\) −6.52465 −0.248569
\(690\) 0 0
\(691\) −32.3464 −1.23051 −0.615257 0.788327i \(-0.710948\pi\)
−0.615257 + 0.788327i \(0.710948\pi\)
\(692\) −7.66964 −0.291556
\(693\) 0 0
\(694\) −12.3300 −0.468042
\(695\) −3.06236 −0.116162
\(696\) 0 0
\(697\) −19.8945 −0.753557
\(698\) 1.82262 0.0689873
\(699\) 0 0
\(700\) 4.56812 0.172659
\(701\) 43.9983 1.66179 0.830897 0.556426i \(-0.187828\pi\)
0.830897 + 0.556426i \(0.187828\pi\)
\(702\) 0 0
\(703\) 4.86416 0.183455
\(704\) −3.41708 −0.128786
\(705\) 0 0
\(706\) −8.04348 −0.302720
\(707\) 84.2116 3.16710
\(708\) 0 0
\(709\) −7.23008 −0.271531 −0.135766 0.990741i \(-0.543349\pi\)
−0.135766 + 0.990741i \(0.543349\pi\)
\(710\) −8.29962 −0.311479
\(711\) 0 0
\(712\) −17.3013 −0.648392
\(713\) 2.62637 0.0983582
\(714\) 0 0
\(715\) −3.31876 −0.124115
\(716\) −1.64284 −0.0613958
\(717\) 0 0
\(718\) −5.52547 −0.206209
\(719\) −18.0454 −0.672982 −0.336491 0.941687i \(-0.609240\pi\)
−0.336491 + 0.941687i \(0.609240\pi\)
\(720\) 0 0
\(721\) −13.6987 −0.510167
\(722\) 4.66008 0.173430
\(723\) 0 0
\(724\) 25.9704 0.965182
\(725\) −3.59689 −0.133585
\(726\) 0 0
\(727\) −11.0659 −0.410413 −0.205206 0.978719i \(-0.565787\pi\)
−0.205206 + 0.978719i \(0.565787\pi\)
\(728\) 4.43669 0.164435
\(729\) 0 0
\(730\) 1.56331 0.0578607
\(731\) −25.0307 −0.925795
\(732\) 0 0
\(733\) −21.8869 −0.808411 −0.404205 0.914668i \(-0.632452\pi\)
−0.404205 + 0.914668i \(0.632452\pi\)
\(734\) −10.4115 −0.384297
\(735\) 0 0
\(736\) 1.02877 0.0379210
\(737\) 54.5721 2.01019
\(738\) 0 0
\(739\) 7.26521 0.267255 0.133627 0.991032i \(-0.457337\pi\)
0.133627 + 0.991032i \(0.457337\pi\)
\(740\) 1.00000 0.0367607
\(741\) 0 0
\(742\) −30.6883 −1.12660
\(743\) −15.4941 −0.568425 −0.284212 0.958761i \(-0.591732\pi\)
−0.284212 + 0.958761i \(0.591732\pi\)
\(744\) 0 0
\(745\) 9.54293 0.349626
\(746\) 20.2125 0.740031
\(747\) 0 0
\(748\) 8.77547 0.320863
\(749\) 37.5617 1.37247
\(750\) 0 0
\(751\) −30.9013 −1.12761 −0.563803 0.825910i \(-0.690662\pi\)
−0.563803 + 0.825910i \(0.690662\pi\)
\(752\) 4.99877 0.182286
\(753\) 0 0
\(754\) −3.49341 −0.127222
\(755\) 8.07706 0.293954
\(756\) 0 0
\(757\) −38.2601 −1.39059 −0.695294 0.718725i \(-0.744726\pi\)
−0.695294 + 0.718725i \(0.744726\pi\)
\(758\) −15.5557 −0.565008
\(759\) 0 0
\(760\) 4.86416 0.176442
\(761\) 7.92408 0.287248 0.143624 0.989632i \(-0.454124\pi\)
0.143624 + 0.989632i \(0.454124\pi\)
\(762\) 0 0
\(763\) 62.6595 2.26843
\(764\) −18.9253 −0.684693
\(765\) 0 0
\(766\) 29.5137 1.06637
\(767\) 6.63753 0.239667
\(768\) 0 0
\(769\) −33.6108 −1.21204 −0.606018 0.795451i \(-0.707234\pi\)
−0.606018 + 0.795451i \(0.707234\pi\)
\(770\) −15.6096 −0.562532
\(771\) 0 0
\(772\) −2.63913 −0.0949845
\(773\) 25.9812 0.934478 0.467239 0.884131i \(-0.345249\pi\)
0.467239 + 0.884131i \(0.345249\pi\)
\(774\) 0 0
\(775\) 2.55292 0.0917034
\(776\) −0.747938 −0.0268494
\(777\) 0 0
\(778\) −14.7235 −0.527863
\(779\) 37.6812 1.35007
\(780\) 0 0
\(781\) 28.3604 1.01482
\(782\) −2.64201 −0.0944781
\(783\) 0 0
\(784\) 13.8677 0.495276
\(785\) −21.6248 −0.771824
\(786\) 0 0
\(787\) −16.6897 −0.594922 −0.297461 0.954734i \(-0.596140\pi\)
−0.297461 + 0.954734i \(0.596140\pi\)
\(788\) −10.9416 −0.389779
\(789\) 0 0
\(790\) −6.29962 −0.224130
\(791\) −14.7882 −0.525808
\(792\) 0 0
\(793\) −6.78890 −0.241081
\(794\) −17.7859 −0.631197
\(795\) 0 0
\(796\) −1.75545 −0.0622204
\(797\) −0.474525 −0.0168085 −0.00840427 0.999965i \(-0.502675\pi\)
−0.00840427 + 0.999965i \(0.502675\pi\)
\(798\) 0 0
\(799\) −12.8374 −0.454156
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) −33.4858 −1.18242
\(803\) −5.34195 −0.188513
\(804\) 0 0
\(805\) 4.69955 0.165638
\(806\) 2.47946 0.0873354
\(807\) 0 0
\(808\) 18.4346 0.648528
\(809\) −27.1716 −0.955303 −0.477651 0.878549i \(-0.658512\pi\)
−0.477651 + 0.878549i \(0.658512\pi\)
\(810\) 0 0
\(811\) −19.8249 −0.696146 −0.348073 0.937467i \(-0.613164\pi\)
−0.348073 + 0.937467i \(0.613164\pi\)
\(812\) −16.4310 −0.576617
\(813\) 0 0
\(814\) −3.41708 −0.119769
\(815\) 8.12585 0.284636
\(816\) 0 0
\(817\) 47.4095 1.65865
\(818\) −30.2700 −1.05837
\(819\) 0 0
\(820\) 7.74670 0.270527
\(821\) 40.2450 1.40456 0.702279 0.711902i \(-0.252166\pi\)
0.702279 + 0.711902i \(0.252166\pi\)
\(822\) 0 0
\(823\) −9.93257 −0.346227 −0.173114 0.984902i \(-0.555383\pi\)
−0.173114 + 0.984902i \(0.555383\pi\)
\(824\) −2.99877 −0.104467
\(825\) 0 0
\(826\) 31.2193 1.08626
\(827\) −7.50975 −0.261140 −0.130570 0.991439i \(-0.541681\pi\)
−0.130570 + 0.991439i \(0.541681\pi\)
\(828\) 0 0
\(829\) 15.4826 0.537733 0.268866 0.963178i \(-0.413351\pi\)
0.268866 + 0.963178i \(0.413351\pi\)
\(830\) 7.86293 0.272926
\(831\) 0 0
\(832\) 0.971229 0.0336713
\(833\) −35.6141 −1.23395
\(834\) 0 0
\(835\) −1.02877 −0.0356021
\(836\) −16.6212 −0.574857
\(837\) 0 0
\(838\) −34.5421 −1.19324
\(839\) −4.72154 −0.163006 −0.0815028 0.996673i \(-0.525972\pi\)
−0.0815028 + 0.996673i \(0.525972\pi\)
\(840\) 0 0
\(841\) −16.0624 −0.553874
\(842\) 10.6991 0.368717
\(843\) 0 0
\(844\) 11.3252 0.389830
\(845\) −12.0567 −0.414764
\(846\) 0 0
\(847\) 3.08998 0.106173
\(848\) −6.71793 −0.230695
\(849\) 0 0
\(850\) −2.56812 −0.0880859
\(851\) 1.02877 0.0352658
\(852\) 0 0
\(853\) 10.4846 0.358986 0.179493 0.983759i \(-0.442554\pi\)
0.179493 + 0.983759i \(0.442554\pi\)
\(854\) −31.9312 −1.09266
\(855\) 0 0
\(856\) 8.22256 0.281041
\(857\) −16.5665 −0.565900 −0.282950 0.959135i \(-0.591313\pi\)
−0.282950 + 0.959135i \(0.591313\pi\)
\(858\) 0 0
\(859\) −6.70676 −0.228832 −0.114416 0.993433i \(-0.536500\pi\)
−0.114416 + 0.993433i \(0.536500\pi\)
\(860\) 9.74670 0.332360
\(861\) 0 0
\(862\) 5.74112 0.195543
\(863\) 6.65751 0.226624 0.113312 0.993559i \(-0.463854\pi\)
0.113312 + 0.993559i \(0.463854\pi\)
\(864\) 0 0
\(865\) −7.66964 −0.260776
\(866\) −16.1650 −0.549310
\(867\) 0 0
\(868\) 11.6620 0.395835
\(869\) 21.5263 0.730229
\(870\) 0 0
\(871\) −15.5109 −0.525567
\(872\) 13.7167 0.464506
\(873\) 0 0
\(874\) 5.00411 0.169267
\(875\) 4.56812 0.154431
\(876\) 0 0
\(877\) −35.9444 −1.21376 −0.606879 0.794795i \(-0.707579\pi\)
−0.606879 + 0.794795i \(0.707579\pi\)
\(878\) 33.3279 1.12476
\(879\) 0 0
\(880\) −3.41708 −0.115190
\(881\) 31.8018 1.07143 0.535715 0.844399i \(-0.320042\pi\)
0.535715 + 0.844399i \(0.320042\pi\)
\(882\) 0 0
\(883\) −33.2325 −1.11836 −0.559181 0.829045i \(-0.688884\pi\)
−0.559181 + 0.829045i \(0.688884\pi\)
\(884\) −2.49423 −0.0838902
\(885\) 0 0
\(886\) 11.7930 0.396194
\(887\) −12.0543 −0.404745 −0.202373 0.979309i \(-0.564865\pi\)
−0.202373 + 0.979309i \(0.564865\pi\)
\(888\) 0 0
\(889\) −42.0875 −1.41157
\(890\) −17.3013 −0.579940
\(891\) 0 0
\(892\) −15.0328 −0.503334
\(893\) 24.3148 0.813664
\(894\) 0 0
\(895\) −1.64284 −0.0549140
\(896\) 4.56812 0.152610
\(897\) 0 0
\(898\) −14.8917 −0.496942
\(899\) −9.18257 −0.306256
\(900\) 0 0
\(901\) 17.2525 0.574763
\(902\) −26.4711 −0.881391
\(903\) 0 0
\(904\) −3.23726 −0.107670
\(905\) 25.9704 0.863285
\(906\) 0 0
\(907\) 34.8146 1.15600 0.578001 0.816036i \(-0.303833\pi\)
0.578001 + 0.816036i \(0.303833\pi\)
\(908\) −14.3735 −0.477002
\(909\) 0 0
\(910\) 4.43669 0.147075
\(911\) 5.67325 0.187963 0.0939816 0.995574i \(-0.470041\pi\)
0.0939816 + 0.995574i \(0.470041\pi\)
\(912\) 0 0
\(913\) −26.8682 −0.889209
\(914\) −3.03793 −0.100486
\(915\) 0 0
\(916\) 13.2242 0.436940
\(917\) 15.7067 0.518682
\(918\) 0 0
\(919\) −43.2346 −1.42618 −0.713090 0.701073i \(-0.752705\pi\)
−0.713090 + 0.701073i \(0.752705\pi\)
\(920\) 1.02877 0.0339176
\(921\) 0 0
\(922\) −17.5465 −0.577863
\(923\) −8.06083 −0.265325
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) 14.5522 0.478214
\(927\) 0 0
\(928\) −3.59689 −0.118074
\(929\) −45.4575 −1.49141 −0.745707 0.666275i \(-0.767888\pi\)
−0.745707 + 0.666275i \(0.767888\pi\)
\(930\) 0 0
\(931\) 67.4549 2.21075
\(932\) 0.222152 0.00727682
\(933\) 0 0
\(934\) 16.9681 0.555212
\(935\) 8.77547 0.286989
\(936\) 0 0
\(937\) 47.5222 1.55248 0.776241 0.630437i \(-0.217124\pi\)
0.776241 + 0.630437i \(0.217124\pi\)
\(938\) −72.9547 −2.38206
\(939\) 0 0
\(940\) 4.99877 0.163042
\(941\) 18.5824 0.605769 0.302885 0.953027i \(-0.402050\pi\)
0.302885 + 0.953027i \(0.402050\pi\)
\(942\) 0 0
\(943\) 7.96959 0.259525
\(944\) 6.83416 0.222433
\(945\) 0 0
\(946\) −33.3052 −1.08285
\(947\) 12.0141 0.390405 0.195202 0.980763i \(-0.437464\pi\)
0.195202 + 0.980763i \(0.437464\pi\)
\(948\) 0 0
\(949\) 1.51833 0.0492871
\(950\) 4.86416 0.157814
\(951\) 0 0
\(952\) −11.7315 −0.380220
\(953\) 34.1350 1.10574 0.552871 0.833267i \(-0.313532\pi\)
0.552871 + 0.833267i \(0.313532\pi\)
\(954\) 0 0
\(955\) −18.9253 −0.612408
\(956\) 2.14106 0.0692467
\(957\) 0 0
\(958\) −26.7267 −0.863500
\(959\) −15.0787 −0.486917
\(960\) 0 0
\(961\) −24.4826 −0.789762
\(962\) 0.971229 0.0313137
\(963\) 0 0
\(964\) 28.3876 0.914302
\(965\) −2.63913 −0.0849567
\(966\) 0 0
\(967\) 19.3288 0.621572 0.310786 0.950480i \(-0.399408\pi\)
0.310786 + 0.950480i \(0.399408\pi\)
\(968\) 0.676423 0.0217411
\(969\) 0 0
\(970\) −0.747938 −0.0240148
\(971\) 16.7079 0.536182 0.268091 0.963394i \(-0.413607\pi\)
0.268091 + 0.963394i \(0.413607\pi\)
\(972\) 0 0
\(973\) −13.9892 −0.448474
\(974\) −32.1054 −1.02872
\(975\) 0 0
\(976\) −6.99001 −0.223745
\(977\) 26.4539 0.846334 0.423167 0.906052i \(-0.360918\pi\)
0.423167 + 0.906052i \(0.360918\pi\)
\(978\) 0 0
\(979\) 59.1198 1.88948
\(980\) 13.8677 0.442989
\(981\) 0 0
\(982\) −38.0588 −1.21450
\(983\) −47.7091 −1.52168 −0.760842 0.648937i \(-0.775214\pi\)
−0.760842 + 0.648937i \(0.775214\pi\)
\(984\) 0 0
\(985\) −10.9416 −0.348629
\(986\) 9.23726 0.294174
\(987\) 0 0
\(988\) 4.72421 0.150297
\(989\) 10.0271 0.318844
\(990\) 0 0
\(991\) −15.3846 −0.488708 −0.244354 0.969686i \(-0.578576\pi\)
−0.244354 + 0.969686i \(0.578576\pi\)
\(992\) 2.55292 0.0810552
\(993\) 0 0
\(994\) −37.9137 −1.20255
\(995\) −1.75545 −0.0556517
\(996\) 0 0
\(997\) −25.0966 −0.794816 −0.397408 0.917642i \(-0.630090\pi\)
−0.397408 + 0.917642i \(0.630090\pi\)
\(998\) −32.5421 −1.03010
\(999\) 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.bl.1.5 yes 5
3.2 odd 2 3330.2.a.bk.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3330.2.a.bk.1.5 5 3.2 odd 2
3330.2.a.bl.1.5 yes 5 1.1 even 1 trivial