Properties

Label 3549.2.a.bg.1.11
Level $3549$
Weight $2$
Character 3549.1
Self dual yes
Analytic conductor $28.339$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3549.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.3389076774\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Defining polynomial: \(x^{15} - 2 x^{14} - 27 x^{13} + 51 x^{12} + 290 x^{11} - 510 x^{10} - 1575 x^{9} + 2522 x^{8} + 4553 x^{7} - 6393 x^{6} - 6785 x^{5} + 7806 x^{4} + 4632 x^{3} - 3811 x^{2} - 1041 x + 281\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(-1.29981\) of defining polynomial
Character \(\chi\) \(=\) 3549.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.29981 q^{2} +1.00000 q^{3} -0.310501 q^{4} -1.99349 q^{5} +1.29981 q^{6} +1.00000 q^{7} -3.00321 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.29981 q^{2} +1.00000 q^{3} -0.310501 q^{4} -1.99349 q^{5} +1.29981 q^{6} +1.00000 q^{7} -3.00321 q^{8} +1.00000 q^{9} -2.59115 q^{10} -2.23452 q^{11} -0.310501 q^{12} +1.29981 q^{14} -1.99349 q^{15} -3.28259 q^{16} -0.168513 q^{17} +1.29981 q^{18} +6.89127 q^{19} +0.618982 q^{20} +1.00000 q^{21} -2.90445 q^{22} +1.53967 q^{23} -3.00321 q^{24} -1.02599 q^{25} +1.00000 q^{27} -0.310501 q^{28} +8.81373 q^{29} -2.59115 q^{30} +0.0996388 q^{31} +1.73968 q^{32} -2.23452 q^{33} -0.219034 q^{34} -1.99349 q^{35} -0.310501 q^{36} -4.91920 q^{37} +8.95732 q^{38} +5.98686 q^{40} -0.944193 q^{41} +1.29981 q^{42} +7.75243 q^{43} +0.693822 q^{44} -1.99349 q^{45} +2.00128 q^{46} +1.08393 q^{47} -3.28259 q^{48} +1.00000 q^{49} -1.33359 q^{50} -0.168513 q^{51} +12.0106 q^{53} +1.29981 q^{54} +4.45450 q^{55} -3.00321 q^{56} +6.89127 q^{57} +11.4562 q^{58} -9.17434 q^{59} +0.618982 q^{60} -3.27054 q^{61} +0.129511 q^{62} +1.00000 q^{63} +8.82643 q^{64} -2.90445 q^{66} +8.51847 q^{67} +0.0523235 q^{68} +1.53967 q^{69} -2.59115 q^{70} -6.19104 q^{71} -3.00321 q^{72} +13.2207 q^{73} -6.39401 q^{74} -1.02599 q^{75} -2.13975 q^{76} -2.23452 q^{77} +10.1866 q^{79} +6.54381 q^{80} +1.00000 q^{81} -1.22727 q^{82} +0.338076 q^{83} -0.310501 q^{84} +0.335929 q^{85} +10.0767 q^{86} +8.81373 q^{87} +6.71073 q^{88} +7.96124 q^{89} -2.59115 q^{90} -0.478070 q^{92} +0.0996388 q^{93} +1.40890 q^{94} -13.7377 q^{95} +1.73968 q^{96} +14.8575 q^{97} +1.29981 q^{98} -2.23452 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q - 2q^{2} + 15q^{3} + 28q^{4} + 9q^{5} - 2q^{6} + 15q^{7} - 9q^{8} + 15q^{9} + O(q^{10}) \) \( 15q - 2q^{2} + 15q^{3} + 28q^{4} + 9q^{5} - 2q^{6} + 15q^{7} - 9q^{8} + 15q^{9} + 21q^{10} - 5q^{11} + 28q^{12} - 2q^{14} + 9q^{15} + 50q^{16} - q^{17} - 2q^{18} - 3q^{19} + 23q^{20} + 15q^{21} + 21q^{22} + 4q^{23} - 9q^{24} + 50q^{25} + 15q^{27} + 28q^{28} + 9q^{29} + 21q^{30} - 7q^{31} - 35q^{32} - 5q^{33} + 2q^{34} + 9q^{35} + 28q^{36} - 17q^{37} - 12q^{38} + 46q^{40} + 22q^{41} - 2q^{42} + 36q^{43} - 29q^{44} + 9q^{45} + q^{46} + 12q^{47} + 50q^{48} + 15q^{49} - 53q^{50} - q^{51} - 5q^{53} - 2q^{54} + 43q^{55} - 9q^{56} - 3q^{57} - 29q^{58} + 29q^{59} + 23q^{60} + 12q^{61} + 14q^{62} + 15q^{63} + 95q^{64} + 21q^{66} + 12q^{67} - 16q^{68} + 4q^{69} + 21q^{70} - 36q^{71} - 9q^{72} + 29q^{73} - 5q^{74} + 50q^{75} - 25q^{76} - 5q^{77} + 35q^{79} + 89q^{80} + 15q^{81} + 51q^{82} + 10q^{83} + 28q^{84} - 23q^{85} - 19q^{86} + 9q^{87} + 73q^{88} - 25q^{89} + 21q^{90} - 31q^{92} - 7q^{93} - 19q^{94} - 7q^{95} - 35q^{96} + 26q^{97} - 2q^{98} - 5q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.29981 0.919102 0.459551 0.888151i \(-0.348010\pi\)
0.459551 + 0.888151i \(0.348010\pi\)
\(3\) 1.00000 0.577350
\(4\) −0.310501 −0.155251
\(5\) −1.99349 −0.891516 −0.445758 0.895153i \(-0.647066\pi\)
−0.445758 + 0.895153i \(0.647066\pi\)
\(6\) 1.29981 0.530644
\(7\) 1.00000 0.377964
\(8\) −3.00321 −1.06179
\(9\) 1.00000 0.333333
\(10\) −2.59115 −0.819395
\(11\) −2.23452 −0.673734 −0.336867 0.941552i \(-0.609367\pi\)
−0.336867 + 0.941552i \(0.609367\pi\)
\(12\) −0.310501 −0.0896340
\(13\) 0 0
\(14\) 1.29981 0.347388
\(15\) −1.99349 −0.514717
\(16\) −3.28259 −0.820647
\(17\) −0.168513 −0.0408704 −0.0204352 0.999791i \(-0.506505\pi\)
−0.0204352 + 0.999791i \(0.506505\pi\)
\(18\) 1.29981 0.306367
\(19\) 6.89127 1.58097 0.790483 0.612484i \(-0.209830\pi\)
0.790483 + 0.612484i \(0.209830\pi\)
\(20\) 0.618982 0.138408
\(21\) 1.00000 0.218218
\(22\) −2.90445 −0.619230
\(23\) 1.53967 0.321044 0.160522 0.987032i \(-0.448682\pi\)
0.160522 + 0.987032i \(0.448682\pi\)
\(24\) −3.00321 −0.613027
\(25\) −1.02599 −0.205199
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −0.310501 −0.0586792
\(29\) 8.81373 1.63667 0.818335 0.574742i \(-0.194898\pi\)
0.818335 + 0.574742i \(0.194898\pi\)
\(30\) −2.59115 −0.473078
\(31\) 0.0996388 0.0178956 0.00894782 0.999960i \(-0.497152\pi\)
0.00894782 + 0.999960i \(0.497152\pi\)
\(32\) 1.73968 0.307535
\(33\) −2.23452 −0.388980
\(34\) −0.219034 −0.0375641
\(35\) −1.99349 −0.336961
\(36\) −0.310501 −0.0517502
\(37\) −4.91920 −0.808712 −0.404356 0.914602i \(-0.632504\pi\)
−0.404356 + 0.914602i \(0.632504\pi\)
\(38\) 8.95732 1.45307
\(39\) 0 0
\(40\) 5.98686 0.946606
\(41\) −0.944193 −0.147458 −0.0737291 0.997278i \(-0.523490\pi\)
−0.0737291 + 0.997278i \(0.523490\pi\)
\(42\) 1.29981 0.200565
\(43\) 7.75243 1.18223 0.591117 0.806586i \(-0.298687\pi\)
0.591117 + 0.806586i \(0.298687\pi\)
\(44\) 0.693822 0.104598
\(45\) −1.99349 −0.297172
\(46\) 2.00128 0.295072
\(47\) 1.08393 0.158107 0.0790536 0.996870i \(-0.474810\pi\)
0.0790536 + 0.996870i \(0.474810\pi\)
\(48\) −3.28259 −0.473801
\(49\) 1.00000 0.142857
\(50\) −1.33359 −0.188599
\(51\) −0.168513 −0.0235965
\(52\) 0 0
\(53\) 12.0106 1.64978 0.824890 0.565294i \(-0.191237\pi\)
0.824890 + 0.565294i \(0.191237\pi\)
\(54\) 1.29981 0.176881
\(55\) 4.45450 0.600645
\(56\) −3.00321 −0.401320
\(57\) 6.89127 0.912771
\(58\) 11.4562 1.50427
\(59\) −9.17434 −1.19440 −0.597199 0.802093i \(-0.703720\pi\)
−0.597199 + 0.802093i \(0.703720\pi\)
\(60\) 0.618982 0.0799102
\(61\) −3.27054 −0.418750 −0.209375 0.977835i \(-0.567143\pi\)
−0.209375 + 0.977835i \(0.567143\pi\)
\(62\) 0.129511 0.0164479
\(63\) 1.00000 0.125988
\(64\) 8.82643 1.10330
\(65\) 0 0
\(66\) −2.90445 −0.357513
\(67\) 8.51847 1.04070 0.520348 0.853954i \(-0.325802\pi\)
0.520348 + 0.853954i \(0.325802\pi\)
\(68\) 0.0523235 0.00634515
\(69\) 1.53967 0.185355
\(70\) −2.59115 −0.309702
\(71\) −6.19104 −0.734741 −0.367371 0.930075i \(-0.619742\pi\)
−0.367371 + 0.930075i \(0.619742\pi\)
\(72\) −3.00321 −0.353931
\(73\) 13.2207 1.54736 0.773682 0.633574i \(-0.218413\pi\)
0.773682 + 0.633574i \(0.218413\pi\)
\(74\) −6.39401 −0.743289
\(75\) −1.02599 −0.118472
\(76\) −2.13975 −0.245446
\(77\) −2.23452 −0.254647
\(78\) 0 0
\(79\) 10.1866 1.14608 0.573039 0.819528i \(-0.305764\pi\)
0.573039 + 0.819528i \(0.305764\pi\)
\(80\) 6.54381 0.731620
\(81\) 1.00000 0.111111
\(82\) −1.22727 −0.135529
\(83\) 0.338076 0.0371087 0.0185543 0.999828i \(-0.494094\pi\)
0.0185543 + 0.999828i \(0.494094\pi\)
\(84\) −0.310501 −0.0338785
\(85\) 0.335929 0.0364366
\(86\) 10.0767 1.08659
\(87\) 8.81373 0.944931
\(88\) 6.71073 0.715366
\(89\) 7.96124 0.843890 0.421945 0.906621i \(-0.361348\pi\)
0.421945 + 0.906621i \(0.361348\pi\)
\(90\) −2.59115 −0.273132
\(91\) 0 0
\(92\) −0.478070 −0.0498423
\(93\) 0.0996388 0.0103321
\(94\) 1.40890 0.145317
\(95\) −13.7377 −1.40946
\(96\) 1.73968 0.177556
\(97\) 14.8575 1.50855 0.754274 0.656560i \(-0.227989\pi\)
0.754274 + 0.656560i \(0.227989\pi\)
\(98\) 1.29981 0.131300
\(99\) −2.23452 −0.224578
\(100\) 0.318572 0.0318572
\(101\) −6.98055 −0.694590 −0.347295 0.937756i \(-0.612900\pi\)
−0.347295 + 0.937756i \(0.612900\pi\)
\(102\) −0.219034 −0.0216876
\(103\) 8.36180 0.823913 0.411956 0.911204i \(-0.364846\pi\)
0.411956 + 0.911204i \(0.364846\pi\)
\(104\) 0 0
\(105\) −1.99349 −0.194545
\(106\) 15.6114 1.51632
\(107\) 16.7289 1.61725 0.808624 0.588326i \(-0.200213\pi\)
0.808624 + 0.588326i \(0.200213\pi\)
\(108\) −0.310501 −0.0298780
\(109\) −8.00198 −0.766451 −0.383226 0.923655i \(-0.625187\pi\)
−0.383226 + 0.923655i \(0.625187\pi\)
\(110\) 5.78999 0.552054
\(111\) −4.91920 −0.466910
\(112\) −3.28259 −0.310175
\(113\) 1.04525 0.0983287 0.0491643 0.998791i \(-0.484344\pi\)
0.0491643 + 0.998791i \(0.484344\pi\)
\(114\) 8.95732 0.838930
\(115\) −3.06932 −0.286216
\(116\) −2.73668 −0.254094
\(117\) 0 0
\(118\) −11.9249 −1.09777
\(119\) −0.168513 −0.0154475
\(120\) 5.98686 0.546523
\(121\) −6.00691 −0.546083
\(122\) −4.25107 −0.384874
\(123\) −0.944193 −0.0851350
\(124\) −0.0309380 −0.00277831
\(125\) 12.0128 1.07445
\(126\) 1.29981 0.115796
\(127\) 14.7776 1.31130 0.655650 0.755065i \(-0.272395\pi\)
0.655650 + 0.755065i \(0.272395\pi\)
\(128\) 7.99328 0.706513
\(129\) 7.75243 0.682563
\(130\) 0 0
\(131\) −13.6465 −1.19230 −0.596151 0.802872i \(-0.703304\pi\)
−0.596151 + 0.802872i \(0.703304\pi\)
\(132\) 0.693822 0.0603895
\(133\) 6.89127 0.597549
\(134\) 11.0724 0.956507
\(135\) −1.99349 −0.171572
\(136\) 0.506079 0.0433959
\(137\) −8.57759 −0.732833 −0.366417 0.930451i \(-0.619415\pi\)
−0.366417 + 0.930451i \(0.619415\pi\)
\(138\) 2.00128 0.170360
\(139\) 2.79628 0.237177 0.118589 0.992943i \(-0.462163\pi\)
0.118589 + 0.992943i \(0.462163\pi\)
\(140\) 0.618982 0.0523135
\(141\) 1.08393 0.0912832
\(142\) −8.04716 −0.675302
\(143\) 0 0
\(144\) −3.28259 −0.273549
\(145\) −17.5701 −1.45912
\(146\) 17.1843 1.42219
\(147\) 1.00000 0.0824786
\(148\) 1.52742 0.125553
\(149\) −0.948225 −0.0776816 −0.0388408 0.999245i \(-0.512367\pi\)
−0.0388408 + 0.999245i \(0.512367\pi\)
\(150\) −1.33359 −0.108887
\(151\) −16.9937 −1.38293 −0.691463 0.722412i \(-0.743033\pi\)
−0.691463 + 0.722412i \(0.743033\pi\)
\(152\) −20.6959 −1.67866
\(153\) −0.168513 −0.0136235
\(154\) −2.90445 −0.234047
\(155\) −0.198629 −0.0159543
\(156\) 0 0
\(157\) 16.5425 1.32023 0.660117 0.751163i \(-0.270507\pi\)
0.660117 + 0.751163i \(0.270507\pi\)
\(158\) 13.2406 1.05336
\(159\) 12.0106 0.952500
\(160\) −3.46804 −0.274173
\(161\) 1.53967 0.121343
\(162\) 1.29981 0.102122
\(163\) −13.0285 −1.02047 −0.510234 0.860036i \(-0.670441\pi\)
−0.510234 + 0.860036i \(0.670441\pi\)
\(164\) 0.293173 0.0228930
\(165\) 4.45450 0.346782
\(166\) 0.439434 0.0341067
\(167\) −9.63809 −0.745818 −0.372909 0.927868i \(-0.621640\pi\)
−0.372909 + 0.927868i \(0.621640\pi\)
\(168\) −3.00321 −0.231702
\(169\) 0 0
\(170\) 0.436643 0.0334890
\(171\) 6.89127 0.526988
\(172\) −2.40714 −0.183543
\(173\) −21.7199 −1.65133 −0.825667 0.564158i \(-0.809201\pi\)
−0.825667 + 0.564158i \(0.809201\pi\)
\(174\) 11.4562 0.868489
\(175\) −1.02599 −0.0775578
\(176\) 7.33501 0.552897
\(177\) −9.17434 −0.689586
\(178\) 10.3481 0.775622
\(179\) −14.4043 −1.07663 −0.538315 0.842743i \(-0.680939\pi\)
−0.538315 + 0.842743i \(0.680939\pi\)
\(180\) 0.618982 0.0461362
\(181\) 0.246871 0.0183498 0.00917488 0.999958i \(-0.497080\pi\)
0.00917488 + 0.999958i \(0.497080\pi\)
\(182\) 0 0
\(183\) −3.27054 −0.241765
\(184\) −4.62395 −0.340882
\(185\) 9.80638 0.720980
\(186\) 0.129511 0.00949622
\(187\) 0.376546 0.0275357
\(188\) −0.336561 −0.0245462
\(189\) 1.00000 0.0727393
\(190\) −17.8563 −1.29543
\(191\) 8.32325 0.602249 0.301124 0.953585i \(-0.402638\pi\)
0.301124 + 0.953585i \(0.402638\pi\)
\(192\) 8.82643 0.636992
\(193\) −17.2173 −1.23933 −0.619664 0.784868i \(-0.712731\pi\)
−0.619664 + 0.784868i \(0.712731\pi\)
\(194\) 19.3118 1.38651
\(195\) 0 0
\(196\) −0.310501 −0.0221787
\(197\) 13.8565 0.987231 0.493616 0.869680i \(-0.335675\pi\)
0.493616 + 0.869680i \(0.335675\pi\)
\(198\) −2.90445 −0.206410
\(199\) 13.0628 0.925995 0.462997 0.886360i \(-0.346774\pi\)
0.462997 + 0.886360i \(0.346774\pi\)
\(200\) 3.08127 0.217879
\(201\) 8.51847 0.600846
\(202\) −9.07337 −0.638400
\(203\) 8.81373 0.618603
\(204\) 0.0523235 0.00366337
\(205\) 1.88224 0.131461
\(206\) 10.8687 0.757260
\(207\) 1.53967 0.107015
\(208\) 0 0
\(209\) −15.3987 −1.06515
\(210\) −2.59115 −0.178807
\(211\) 1.40177 0.0965017 0.0482508 0.998835i \(-0.484635\pi\)
0.0482508 + 0.998835i \(0.484635\pi\)
\(212\) −3.72930 −0.256129
\(213\) −6.19104 −0.424203
\(214\) 21.7444 1.48642
\(215\) −15.4544 −1.05398
\(216\) −3.00321 −0.204342
\(217\) 0.0996388 0.00676392
\(218\) −10.4010 −0.704447
\(219\) 13.2207 0.893371
\(220\) −1.38313 −0.0932505
\(221\) 0 0
\(222\) −6.39401 −0.429138
\(223\) −7.12243 −0.476953 −0.238476 0.971148i \(-0.576648\pi\)
−0.238476 + 0.971148i \(0.576648\pi\)
\(224\) 1.73968 0.116237
\(225\) −1.02599 −0.0683996
\(226\) 1.35862 0.0903741
\(227\) −13.5583 −0.899896 −0.449948 0.893055i \(-0.648557\pi\)
−0.449948 + 0.893055i \(0.648557\pi\)
\(228\) −2.13975 −0.141708
\(229\) 12.6022 0.832776 0.416388 0.909187i \(-0.363296\pi\)
0.416388 + 0.909187i \(0.363296\pi\)
\(230\) −3.98953 −0.263062
\(231\) −2.23452 −0.147021
\(232\) −26.4695 −1.73781
\(233\) 14.9463 0.979167 0.489584 0.871956i \(-0.337149\pi\)
0.489584 + 0.871956i \(0.337149\pi\)
\(234\) 0 0
\(235\) −2.16080 −0.140955
\(236\) 2.84865 0.185431
\(237\) 10.1866 0.661688
\(238\) −0.219034 −0.0141979
\(239\) −28.5654 −1.84774 −0.923870 0.382707i \(-0.874992\pi\)
−0.923870 + 0.382707i \(0.874992\pi\)
\(240\) 6.54381 0.422401
\(241\) 24.4026 1.57191 0.785954 0.618285i \(-0.212172\pi\)
0.785954 + 0.618285i \(0.212172\pi\)
\(242\) −7.80783 −0.501906
\(243\) 1.00000 0.0641500
\(244\) 1.01551 0.0650111
\(245\) −1.99349 −0.127359
\(246\) −1.22727 −0.0782478
\(247\) 0 0
\(248\) −0.299236 −0.0190015
\(249\) 0.338076 0.0214247
\(250\) 15.6143 0.987534
\(251\) 2.73250 0.172474 0.0862370 0.996275i \(-0.472516\pi\)
0.0862370 + 0.996275i \(0.472516\pi\)
\(252\) −0.310501 −0.0195597
\(253\) −3.44043 −0.216298
\(254\) 19.2080 1.20522
\(255\) 0.335929 0.0210367
\(256\) −7.26312 −0.453945
\(257\) 4.38194 0.273338 0.136669 0.990617i \(-0.456360\pi\)
0.136669 + 0.990617i \(0.456360\pi\)
\(258\) 10.0767 0.627345
\(259\) −4.91920 −0.305664
\(260\) 0 0
\(261\) 8.81373 0.545556
\(262\) −17.7378 −1.09585
\(263\) −20.7701 −1.28074 −0.640369 0.768067i \(-0.721219\pi\)
−0.640369 + 0.768067i \(0.721219\pi\)
\(264\) 6.71073 0.413017
\(265\) −23.9430 −1.47080
\(266\) 8.95732 0.549209
\(267\) 7.96124 0.487220
\(268\) −2.64500 −0.161569
\(269\) 20.1473 1.22840 0.614201 0.789150i \(-0.289478\pi\)
0.614201 + 0.789150i \(0.289478\pi\)
\(270\) −2.59115 −0.157693
\(271\) 16.1869 0.983284 0.491642 0.870797i \(-0.336397\pi\)
0.491642 + 0.870797i \(0.336397\pi\)
\(272\) 0.553158 0.0335401
\(273\) 0 0
\(274\) −11.1492 −0.673549
\(275\) 2.29261 0.138249
\(276\) −0.478070 −0.0287764
\(277\) −21.9043 −1.31610 −0.658052 0.752972i \(-0.728619\pi\)
−0.658052 + 0.752972i \(0.728619\pi\)
\(278\) 3.63463 0.217990
\(279\) 0.0996388 0.00596522
\(280\) 5.98686 0.357784
\(281\) −19.5379 −1.16553 −0.582765 0.812640i \(-0.698029\pi\)
−0.582765 + 0.812640i \(0.698029\pi\)
\(282\) 1.40890 0.0838987
\(283\) 23.9022 1.42084 0.710418 0.703780i \(-0.248506\pi\)
0.710418 + 0.703780i \(0.248506\pi\)
\(284\) 1.92233 0.114069
\(285\) −13.7377 −0.813750
\(286\) 0 0
\(287\) −0.944193 −0.0557339
\(288\) 1.73968 0.102512
\(289\) −16.9716 −0.998330
\(290\) −22.8377 −1.34108
\(291\) 14.8575 0.870960
\(292\) −4.10504 −0.240229
\(293\) 17.9823 1.05054 0.525268 0.850937i \(-0.323965\pi\)
0.525268 + 0.850937i \(0.323965\pi\)
\(294\) 1.29981 0.0758063
\(295\) 18.2890 1.06483
\(296\) 14.7734 0.858685
\(297\) −2.23452 −0.129660
\(298\) −1.23251 −0.0713974
\(299\) 0 0
\(300\) 0.318572 0.0183928
\(301\) 7.75243 0.446842
\(302\) −22.0885 −1.27105
\(303\) −6.98055 −0.401022
\(304\) −22.6212 −1.29741
\(305\) 6.51979 0.373322
\(306\) −0.219034 −0.0125214
\(307\) 28.7026 1.63814 0.819071 0.573692i \(-0.194489\pi\)
0.819071 + 0.573692i \(0.194489\pi\)
\(308\) 0.693822 0.0395342
\(309\) 8.36180 0.475686
\(310\) −0.258179 −0.0146636
\(311\) 22.0690 1.25142 0.625710 0.780056i \(-0.284809\pi\)
0.625710 + 0.780056i \(0.284809\pi\)
\(312\) 0 0
\(313\) 20.6739 1.16856 0.584280 0.811552i \(-0.301377\pi\)
0.584280 + 0.811552i \(0.301377\pi\)
\(314\) 21.5020 1.21343
\(315\) −1.99349 −0.112320
\(316\) −3.16294 −0.177929
\(317\) −22.1697 −1.24518 −0.622588 0.782550i \(-0.713919\pi\)
−0.622588 + 0.782550i \(0.713919\pi\)
\(318\) 15.6114 0.875445
\(319\) −19.6945 −1.10268
\(320\) −17.5954 −0.983613
\(321\) 16.7289 0.933718
\(322\) 2.00128 0.111527
\(323\) −1.16127 −0.0646146
\(324\) −0.310501 −0.0172501
\(325\) 0 0
\(326\) −16.9345 −0.937914
\(327\) −8.00198 −0.442511
\(328\) 2.83561 0.156570
\(329\) 1.08393 0.0597589
\(330\) 5.78999 0.318728
\(331\) −1.69293 −0.0930516 −0.0465258 0.998917i \(-0.514815\pi\)
−0.0465258 + 0.998917i \(0.514815\pi\)
\(332\) −0.104973 −0.00576115
\(333\) −4.91920 −0.269571
\(334\) −12.5277 −0.685483
\(335\) −16.9815 −0.927798
\(336\) −3.28259 −0.179080
\(337\) 21.8236 1.18881 0.594404 0.804167i \(-0.297388\pi\)
0.594404 + 0.804167i \(0.297388\pi\)
\(338\) 0 0
\(339\) 1.04525 0.0567701
\(340\) −0.104306 −0.00565681
\(341\) −0.222645 −0.0120569
\(342\) 8.95732 0.484356
\(343\) 1.00000 0.0539949
\(344\) −23.2821 −1.25529
\(345\) −3.06932 −0.165247
\(346\) −28.2317 −1.51774
\(347\) −15.6894 −0.842251 −0.421125 0.907002i \(-0.638365\pi\)
−0.421125 + 0.907002i \(0.638365\pi\)
\(348\) −2.73668 −0.146701
\(349\) −26.8967 −1.43975 −0.719873 0.694106i \(-0.755800\pi\)
−0.719873 + 0.694106i \(0.755800\pi\)
\(350\) −1.33359 −0.0712836
\(351\) 0 0
\(352\) −3.88736 −0.207197
\(353\) 19.0891 1.01601 0.508005 0.861354i \(-0.330383\pi\)
0.508005 + 0.861354i \(0.330383\pi\)
\(354\) −11.9249 −0.633800
\(355\) 12.3418 0.655034
\(356\) −2.47198 −0.131014
\(357\) −0.168513 −0.00891864
\(358\) −18.7229 −0.989534
\(359\) 24.0829 1.27105 0.635523 0.772082i \(-0.280784\pi\)
0.635523 + 0.772082i \(0.280784\pi\)
\(360\) 5.98686 0.315535
\(361\) 28.4896 1.49945
\(362\) 0.320884 0.0168653
\(363\) −6.00691 −0.315281
\(364\) 0 0
\(365\) −26.3553 −1.37950
\(366\) −4.25107 −0.222207
\(367\) −33.8964 −1.76938 −0.884688 0.466184i \(-0.845628\pi\)
−0.884688 + 0.466184i \(0.845628\pi\)
\(368\) −5.05411 −0.263463
\(369\) −0.944193 −0.0491527
\(370\) 12.7464 0.662654
\(371\) 12.0106 0.623558
\(372\) −0.0309380 −0.00160406
\(373\) −5.12346 −0.265283 −0.132641 0.991164i \(-0.542346\pi\)
−0.132641 + 0.991164i \(0.542346\pi\)
\(374\) 0.489437 0.0253082
\(375\) 12.0128 0.620336
\(376\) −3.25526 −0.167877
\(377\) 0 0
\(378\) 1.29981 0.0668549
\(379\) −13.3411 −0.685285 −0.342643 0.939466i \(-0.611322\pi\)
−0.342643 + 0.939466i \(0.611322\pi\)
\(380\) 4.26557 0.218819
\(381\) 14.7776 0.757079
\(382\) 10.8186 0.553528
\(383\) −18.3151 −0.935856 −0.467928 0.883767i \(-0.654999\pi\)
−0.467928 + 0.883767i \(0.654999\pi\)
\(384\) 7.99328 0.407906
\(385\) 4.45450 0.227022
\(386\) −22.3791 −1.13907
\(387\) 7.75243 0.394078
\(388\) −4.61326 −0.234203
\(389\) 7.76659 0.393782 0.196891 0.980425i \(-0.436915\pi\)
0.196891 + 0.980425i \(0.436915\pi\)
\(390\) 0 0
\(391\) −0.259454 −0.0131212
\(392\) −3.00321 −0.151685
\(393\) −13.6465 −0.688376
\(394\) 18.0107 0.907367
\(395\) −20.3068 −1.02175
\(396\) 0.693822 0.0348659
\(397\) −34.9079 −1.75198 −0.875989 0.482332i \(-0.839790\pi\)
−0.875989 + 0.482332i \(0.839790\pi\)
\(398\) 16.9791 0.851084
\(399\) 6.89127 0.344995
\(400\) 3.36791 0.168396
\(401\) 13.5850 0.678403 0.339201 0.940714i \(-0.389843\pi\)
0.339201 + 0.940714i \(0.389843\pi\)
\(402\) 11.0724 0.552239
\(403\) 0 0
\(404\) 2.16747 0.107836
\(405\) −1.99349 −0.0990574
\(406\) 11.4562 0.568559
\(407\) 10.9921 0.544856
\(408\) 0.506079 0.0250546
\(409\) −32.2850 −1.59639 −0.798195 0.602400i \(-0.794211\pi\)
−0.798195 + 0.602400i \(0.794211\pi\)
\(410\) 2.44655 0.120826
\(411\) −8.57759 −0.423101
\(412\) −2.59635 −0.127913
\(413\) −9.17434 −0.451440
\(414\) 2.00128 0.0983574
\(415\) −0.673952 −0.0330830
\(416\) 0 0
\(417\) 2.79628 0.136934
\(418\) −20.0153 −0.978982
\(419\) −7.57878 −0.370247 −0.185124 0.982715i \(-0.559269\pi\)
−0.185124 + 0.982715i \(0.559269\pi\)
\(420\) 0.618982 0.0302032
\(421\) −8.75402 −0.426645 −0.213322 0.976982i \(-0.568429\pi\)
−0.213322 + 0.976982i \(0.568429\pi\)
\(422\) 1.82203 0.0886949
\(423\) 1.08393 0.0527024
\(424\) −36.0702 −1.75173
\(425\) 0.172893 0.00838655
\(426\) −8.04716 −0.389886
\(427\) −3.27054 −0.158272
\(428\) −5.19436 −0.251079
\(429\) 0 0
\(430\) −20.0877 −0.968716
\(431\) 6.43018 0.309731 0.154866 0.987936i \(-0.450506\pi\)
0.154866 + 0.987936i \(0.450506\pi\)
\(432\) −3.28259 −0.157934
\(433\) 14.6413 0.703615 0.351808 0.936072i \(-0.385567\pi\)
0.351808 + 0.936072i \(0.385567\pi\)
\(434\) 0.129511 0.00621674
\(435\) −17.5701 −0.842422
\(436\) 2.48463 0.118992
\(437\) 10.6103 0.507559
\(438\) 17.1843 0.821099
\(439\) 27.4546 1.31034 0.655168 0.755484i \(-0.272598\pi\)
0.655168 + 0.755484i \(0.272598\pi\)
\(440\) −13.3778 −0.637761
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 25.4047 1.20701 0.603506 0.797358i \(-0.293770\pi\)
0.603506 + 0.797358i \(0.293770\pi\)
\(444\) 1.52742 0.0724881
\(445\) −15.8707 −0.752342
\(446\) −9.25778 −0.438369
\(447\) −0.948225 −0.0448495
\(448\) 8.82643 0.417009
\(449\) −8.66570 −0.408960 −0.204480 0.978871i \(-0.565550\pi\)
−0.204480 + 0.978871i \(0.565550\pi\)
\(450\) −1.33359 −0.0628662
\(451\) 2.10982 0.0993475
\(452\) −0.324551 −0.0152656
\(453\) −16.9937 −0.798433
\(454\) −17.6232 −0.827096
\(455\) 0 0
\(456\) −20.6959 −0.969174
\(457\) 31.5437 1.47555 0.737775 0.675047i \(-0.235876\pi\)
0.737775 + 0.675047i \(0.235876\pi\)
\(458\) 16.3804 0.765406
\(459\) −0.168513 −0.00786550
\(460\) 0.953028 0.0444352
\(461\) −18.5257 −0.862829 −0.431415 0.902154i \(-0.641985\pi\)
−0.431415 + 0.902154i \(0.641985\pi\)
\(462\) −2.90445 −0.135127
\(463\) 3.91466 0.181930 0.0909649 0.995854i \(-0.471005\pi\)
0.0909649 + 0.995854i \(0.471005\pi\)
\(464\) −28.9318 −1.34313
\(465\) −0.198629 −0.00921120
\(466\) 19.4274 0.899955
\(467\) −22.9029 −1.05982 −0.529909 0.848055i \(-0.677774\pi\)
−0.529909 + 0.848055i \(0.677774\pi\)
\(468\) 0 0
\(469\) 8.51847 0.393346
\(470\) −2.80863 −0.129552
\(471\) 16.5425 0.762237
\(472\) 27.5524 1.26820
\(473\) −17.3230 −0.796511
\(474\) 13.2406 0.608159
\(475\) −7.07040 −0.324412
\(476\) 0.0523235 0.00239824
\(477\) 12.0106 0.549926
\(478\) −37.1295 −1.69826
\(479\) 10.5619 0.482588 0.241294 0.970452i \(-0.422428\pi\)
0.241294 + 0.970452i \(0.422428\pi\)
\(480\) −3.46804 −0.158294
\(481\) 0 0
\(482\) 31.7186 1.44474
\(483\) 1.53967 0.0700575
\(484\) 1.86515 0.0847797
\(485\) −29.6182 −1.34489
\(486\) 1.29981 0.0589605
\(487\) −10.8592 −0.492076 −0.246038 0.969260i \(-0.579129\pi\)
−0.246038 + 0.969260i \(0.579129\pi\)
\(488\) 9.82210 0.444626
\(489\) −13.0285 −0.589167
\(490\) −2.59115 −0.117056
\(491\) 19.6903 0.888613 0.444306 0.895875i \(-0.353450\pi\)
0.444306 + 0.895875i \(0.353450\pi\)
\(492\) 0.293173 0.0132173
\(493\) −1.48523 −0.0668913
\(494\) 0 0
\(495\) 4.45450 0.200215
\(496\) −0.327073 −0.0146860
\(497\) −6.19104 −0.277706
\(498\) 0.439434 0.0196915
\(499\) 23.1892 1.03809 0.519045 0.854747i \(-0.326288\pi\)
0.519045 + 0.854747i \(0.326288\pi\)
\(500\) −3.72998 −0.166810
\(501\) −9.63809 −0.430598
\(502\) 3.55172 0.158521
\(503\) −17.2440 −0.768871 −0.384435 0.923152i \(-0.625604\pi\)
−0.384435 + 0.923152i \(0.625604\pi\)
\(504\) −3.00321 −0.133773
\(505\) 13.9157 0.619239
\(506\) −4.47190 −0.198800
\(507\) 0 0
\(508\) −4.58846 −0.203580
\(509\) 8.87248 0.393266 0.196633 0.980477i \(-0.436999\pi\)
0.196633 + 0.980477i \(0.436999\pi\)
\(510\) 0.436643 0.0193349
\(511\) 13.2207 0.584849
\(512\) −25.4272 −1.12374
\(513\) 6.89127 0.304257
\(514\) 5.69568 0.251226
\(515\) −16.6692 −0.734531
\(516\) −2.40714 −0.105968
\(517\) −2.42206 −0.106522
\(518\) −6.39401 −0.280937
\(519\) −21.7199 −0.953398
\(520\) 0 0
\(521\) −16.5018 −0.722956 −0.361478 0.932381i \(-0.617728\pi\)
−0.361478 + 0.932381i \(0.617728\pi\)
\(522\) 11.4562 0.501422
\(523\) −26.6069 −1.16344 −0.581720 0.813389i \(-0.697620\pi\)
−0.581720 + 0.813389i \(0.697620\pi\)
\(524\) 4.23726 0.185106
\(525\) −1.02599 −0.0447780
\(526\) −26.9971 −1.17713
\(527\) −0.0167904 −0.000731402 0
\(528\) 7.33501 0.319215
\(529\) −20.6294 −0.896931
\(530\) −31.1212 −1.35182
\(531\) −9.17434 −0.398133
\(532\) −2.13975 −0.0927698
\(533\) 0 0
\(534\) 10.3481 0.447805
\(535\) −33.3490 −1.44180
\(536\) −25.5827 −1.10500
\(537\) −14.4043 −0.621593
\(538\) 26.1876 1.12903
\(539\) −2.23452 −0.0962477
\(540\) 0.618982 0.0266367
\(541\) 8.60187 0.369823 0.184912 0.982755i \(-0.440800\pi\)
0.184912 + 0.982755i \(0.440800\pi\)
\(542\) 21.0398 0.903739
\(543\) 0.246871 0.0105942
\(544\) −0.293159 −0.0125691
\(545\) 15.9519 0.683304
\(546\) 0 0
\(547\) −35.2695 −1.50801 −0.754007 0.656867i \(-0.771881\pi\)
−0.754007 + 0.656867i \(0.771881\pi\)
\(548\) 2.66335 0.113773
\(549\) −3.27054 −0.139583
\(550\) 2.97994 0.127065
\(551\) 60.7378 2.58752
\(552\) −4.62395 −0.196808
\(553\) 10.1866 0.433177
\(554\) −28.4714 −1.20963
\(555\) 9.80638 0.416258
\(556\) −0.868249 −0.0368219
\(557\) 29.0820 1.23225 0.616123 0.787650i \(-0.288703\pi\)
0.616123 + 0.787650i \(0.288703\pi\)
\(558\) 0.129511 0.00548265
\(559\) 0 0
\(560\) 6.54381 0.276526
\(561\) 0.376546 0.0158978
\(562\) −25.3954 −1.07124
\(563\) 21.7125 0.915074 0.457537 0.889191i \(-0.348732\pi\)
0.457537 + 0.889191i \(0.348732\pi\)
\(564\) −0.336561 −0.0141718
\(565\) −2.08369 −0.0876616
\(566\) 31.0682 1.30589
\(567\) 1.00000 0.0419961
\(568\) 18.5930 0.780144
\(569\) −42.5801 −1.78505 −0.892525 0.450998i \(-0.851068\pi\)
−0.892525 + 0.450998i \(0.851068\pi\)
\(570\) −17.8563 −0.747920
\(571\) −14.4690 −0.605509 −0.302754 0.953069i \(-0.597906\pi\)
−0.302754 + 0.953069i \(0.597906\pi\)
\(572\) 0 0
\(573\) 8.32325 0.347709
\(574\) −1.22727 −0.0512252
\(575\) −1.57969 −0.0658778
\(576\) 8.82643 0.367768
\(577\) 12.3774 0.515277 0.257639 0.966241i \(-0.417056\pi\)
0.257639 + 0.966241i \(0.417056\pi\)
\(578\) −22.0598 −0.917567
\(579\) −17.2173 −0.715526
\(580\) 5.45554 0.226529
\(581\) 0.338076 0.0140258
\(582\) 19.3118 0.800502
\(583\) −26.8379 −1.11151
\(584\) −39.7044 −1.64298
\(585\) 0 0
\(586\) 23.3735 0.965550
\(587\) −46.2313 −1.90817 −0.954086 0.299534i \(-0.903169\pi\)
−0.954086 + 0.299534i \(0.903169\pi\)
\(588\) −0.310501 −0.0128049
\(589\) 0.686637 0.0282924
\(590\) 23.7721 0.978683
\(591\) 13.8565 0.569978
\(592\) 16.1477 0.663666
\(593\) 45.4459 1.86624 0.933120 0.359565i \(-0.117075\pi\)
0.933120 + 0.359565i \(0.117075\pi\)
\(594\) −2.90445 −0.119171
\(595\) 0.335929 0.0137717
\(596\) 0.294425 0.0120601
\(597\) 13.0628 0.534623
\(598\) 0 0
\(599\) 27.7068 1.13207 0.566035 0.824381i \(-0.308477\pi\)
0.566035 + 0.824381i \(0.308477\pi\)
\(600\) 3.08127 0.125792
\(601\) 34.5036 1.40743 0.703716 0.710481i \(-0.251523\pi\)
0.703716 + 0.710481i \(0.251523\pi\)
\(602\) 10.0767 0.410694
\(603\) 8.51847 0.346899
\(604\) 5.27656 0.214700
\(605\) 11.9747 0.486842
\(606\) −9.07337 −0.368580
\(607\) 22.1561 0.899290 0.449645 0.893207i \(-0.351551\pi\)
0.449645 + 0.893207i \(0.351551\pi\)
\(608\) 11.9886 0.486203
\(609\) 8.81373 0.357150
\(610\) 8.47447 0.343121
\(611\) 0 0
\(612\) 0.0523235 0.00211505
\(613\) 8.13812 0.328696 0.164348 0.986402i \(-0.447448\pi\)
0.164348 + 0.986402i \(0.447448\pi\)
\(614\) 37.3078 1.50562
\(615\) 1.88224 0.0758992
\(616\) 6.71073 0.270383
\(617\) −34.3640 −1.38344 −0.691722 0.722164i \(-0.743148\pi\)
−0.691722 + 0.722164i \(0.743148\pi\)
\(618\) 10.8687 0.437204
\(619\) −22.8723 −0.919317 −0.459658 0.888096i \(-0.652028\pi\)
−0.459658 + 0.888096i \(0.652028\pi\)
\(620\) 0.0616746 0.00247691
\(621\) 1.53967 0.0617849
\(622\) 28.6855 1.15018
\(623\) 7.96124 0.318960
\(624\) 0 0
\(625\) −18.8174 −0.752695
\(626\) 26.8721 1.07403
\(627\) −15.3987 −0.614964
\(628\) −5.13646 −0.204967
\(629\) 0.828948 0.0330523
\(630\) −2.59115 −0.103234
\(631\) 16.5005 0.656874 0.328437 0.944526i \(-0.393478\pi\)
0.328437 + 0.944526i \(0.393478\pi\)
\(632\) −30.5923 −1.21690
\(633\) 1.40177 0.0557153
\(634\) −28.8164 −1.14444
\(635\) −29.4590 −1.16904
\(636\) −3.72930 −0.147876
\(637\) 0 0
\(638\) −25.5990 −1.01348
\(639\) −6.19104 −0.244914
\(640\) −15.9345 −0.629868
\(641\) −35.2485 −1.39223 −0.696115 0.717930i \(-0.745090\pi\)
−0.696115 + 0.717930i \(0.745090\pi\)
\(642\) 21.7444 0.858183
\(643\) 4.35068 0.171574 0.0857871 0.996313i \(-0.472660\pi\)
0.0857871 + 0.996313i \(0.472660\pi\)
\(644\) −0.478070 −0.0188386
\(645\) −15.4544 −0.608516
\(646\) −1.50942 −0.0593875
\(647\) −36.0405 −1.41690 −0.708450 0.705761i \(-0.750605\pi\)
−0.708450 + 0.705761i \(0.750605\pi\)
\(648\) −3.00321 −0.117977
\(649\) 20.5003 0.804706
\(650\) 0 0
\(651\) 0.0996388 0.00390515
\(652\) 4.04535 0.158428
\(653\) −19.4384 −0.760683 −0.380342 0.924846i \(-0.624194\pi\)
−0.380342 + 0.924846i \(0.624194\pi\)
\(654\) −10.4010 −0.406713
\(655\) 27.2042 1.06296
\(656\) 3.09939 0.121011
\(657\) 13.2207 0.515788
\(658\) 1.40890 0.0549246
\(659\) 19.6182 0.764217 0.382109 0.924117i \(-0.375198\pi\)
0.382109 + 0.924117i \(0.375198\pi\)
\(660\) −1.38313 −0.0538382
\(661\) 6.96154 0.270773 0.135386 0.990793i \(-0.456772\pi\)
0.135386 + 0.990793i \(0.456772\pi\)
\(662\) −2.20048 −0.0855240
\(663\) 0 0
\(664\) −1.01531 −0.0394018
\(665\) −13.7377 −0.532724
\(666\) −6.39401 −0.247763
\(667\) 13.5703 0.525442
\(668\) 2.99264 0.115789
\(669\) −7.12243 −0.275369
\(670\) −22.0727 −0.852741
\(671\) 7.30809 0.282126
\(672\) 1.73968 0.0671097
\(673\) 23.6004 0.909728 0.454864 0.890561i \(-0.349688\pi\)
0.454864 + 0.890561i \(0.349688\pi\)
\(674\) 28.3665 1.09264
\(675\) −1.02599 −0.0394905
\(676\) 0 0
\(677\) 19.0748 0.733105 0.366552 0.930397i \(-0.380538\pi\)
0.366552 + 0.930397i \(0.380538\pi\)
\(678\) 1.35862 0.0521775
\(679\) 14.8575 0.570177
\(680\) −1.00886 −0.0386881
\(681\) −13.5583 −0.519555
\(682\) −0.289396 −0.0110815
\(683\) −17.7879 −0.680633 −0.340317 0.940311i \(-0.610534\pi\)
−0.340317 + 0.940311i \(0.610534\pi\)
\(684\) −2.13975 −0.0818153
\(685\) 17.0994 0.653333
\(686\) 1.29981 0.0496269
\(687\) 12.6022 0.480803
\(688\) −25.4480 −0.970196
\(689\) 0 0
\(690\) −3.98953 −0.151879
\(691\) −22.9906 −0.874603 −0.437302 0.899315i \(-0.644066\pi\)
−0.437302 + 0.899315i \(0.644066\pi\)
\(692\) 6.74406 0.256371
\(693\) −2.23452 −0.0848825
\(694\) −20.3932 −0.774115
\(695\) −5.57436 −0.211448
\(696\) −26.4695 −1.00332
\(697\) 0.159109 0.00602667
\(698\) −34.9605 −1.32327
\(699\) 14.9463 0.565322
\(700\) 0.318572 0.0120409
\(701\) 6.30757 0.238234 0.119117 0.992880i \(-0.461994\pi\)
0.119117 + 0.992880i \(0.461994\pi\)
\(702\) 0 0
\(703\) −33.8995 −1.27855
\(704\) −19.7228 −0.743333
\(705\) −2.16080 −0.0813805
\(706\) 24.8121 0.933817
\(707\) −6.98055 −0.262531
\(708\) 2.84865 0.107059
\(709\) −10.9014 −0.409412 −0.204706 0.978823i \(-0.565624\pi\)
−0.204706 + 0.978823i \(0.565624\pi\)
\(710\) 16.0419 0.602043
\(711\) 10.1866 0.382026
\(712\) −23.9093 −0.896037
\(713\) 0.153411 0.00574529
\(714\) −0.219034 −0.00819715
\(715\) 0 0
\(716\) 4.47257 0.167148
\(717\) −28.5654 −1.06679
\(718\) 31.3031 1.16822
\(719\) 5.44927 0.203223 0.101612 0.994824i \(-0.467600\pi\)
0.101612 + 0.994824i \(0.467600\pi\)
\(720\) 6.54381 0.243873
\(721\) 8.36180 0.311410
\(722\) 37.0310 1.37815
\(723\) 24.4026 0.907541
\(724\) −0.0766537 −0.00284881
\(725\) −9.04283 −0.335842
\(726\) −7.80783 −0.289776
\(727\) 33.5071 1.24271 0.621355 0.783529i \(-0.286582\pi\)
0.621355 + 0.783529i \(0.286582\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −34.2568 −1.26790
\(731\) −1.30638 −0.0483183
\(732\) 1.01551 0.0375342
\(733\) 48.1935 1.78007 0.890033 0.455895i \(-0.150681\pi\)
0.890033 + 0.455895i \(0.150681\pi\)
\(734\) −44.0587 −1.62624
\(735\) −1.99349 −0.0735310
\(736\) 2.67854 0.0987323
\(737\) −19.0347 −0.701152
\(738\) −1.22727 −0.0451764
\(739\) 42.2008 1.55238 0.776190 0.630499i \(-0.217150\pi\)
0.776190 + 0.630499i \(0.217150\pi\)
\(740\) −3.04489 −0.111933
\(741\) 0 0
\(742\) 15.6114 0.573114
\(743\) −17.3622 −0.636956 −0.318478 0.947930i \(-0.603172\pi\)
−0.318478 + 0.947930i \(0.603172\pi\)
\(744\) −0.299236 −0.0109705
\(745\) 1.89028 0.0692544
\(746\) −6.65951 −0.243822
\(747\) 0.338076 0.0123696
\(748\) −0.116918 −0.00427494
\(749\) 16.7289 0.611262
\(750\) 15.6143 0.570153
\(751\) −17.2502 −0.629470 −0.314735 0.949180i \(-0.601916\pi\)
−0.314735 + 0.949180i \(0.601916\pi\)
\(752\) −3.55809 −0.129750
\(753\) 2.73250 0.0995779
\(754\) 0 0
\(755\) 33.8767 1.23290
\(756\) −0.310501 −0.0112928
\(757\) −27.7065 −1.00701 −0.503506 0.863992i \(-0.667957\pi\)
−0.503506 + 0.863992i \(0.667957\pi\)
\(758\) −17.3408 −0.629847
\(759\) −3.44043 −0.124880
\(760\) 41.2571 1.49655
\(761\) 6.49989 0.235621 0.117810 0.993036i \(-0.462412\pi\)
0.117810 + 0.993036i \(0.462412\pi\)
\(762\) 19.2080 0.695833
\(763\) −8.00198 −0.289691
\(764\) −2.58438 −0.0934995
\(765\) 0.335929 0.0121455
\(766\) −23.8061 −0.860148
\(767\) 0 0
\(768\) −7.26312 −0.262085
\(769\) −16.2258 −0.585117 −0.292559 0.956248i \(-0.594507\pi\)
−0.292559 + 0.956248i \(0.594507\pi\)
\(770\) 5.78999 0.208657
\(771\) 4.38194 0.157812
\(772\) 5.34599 0.192406
\(773\) −15.1651 −0.545451 −0.272725 0.962092i \(-0.587925\pi\)
−0.272725 + 0.962092i \(0.587925\pi\)
\(774\) 10.0767 0.362198
\(775\) −0.102229 −0.00367216
\(776\) −44.6200 −1.60177
\(777\) −4.91920 −0.176475
\(778\) 10.0951 0.361926
\(779\) −6.50668 −0.233126
\(780\) 0 0
\(781\) 13.8340 0.495020
\(782\) −0.337241 −0.0120597
\(783\) 8.81373 0.314977
\(784\) −3.28259 −0.117235
\(785\) −32.9773 −1.17701
\(786\) −17.7378 −0.632688
\(787\) 35.6231 1.26983 0.634913 0.772584i \(-0.281036\pi\)
0.634913 + 0.772584i \(0.281036\pi\)
\(788\) −4.30245 −0.153268
\(789\) −20.7701 −0.739435
\(790\) −26.3949 −0.939090
\(791\) 1.04525 0.0371647
\(792\) 6.71073 0.238455
\(793\) 0 0
\(794\) −45.3735 −1.61025
\(795\) −23.9430 −0.849170
\(796\) −4.05600 −0.143761
\(797\) 27.7190 0.981856 0.490928 0.871200i \(-0.336658\pi\)
0.490928 + 0.871200i \(0.336658\pi\)
\(798\) 8.95732 0.317086
\(799\) −0.182656 −0.00646190
\(800\) −1.78490 −0.0631059
\(801\) 7.96124 0.281297
\(802\) 17.6579 0.623522
\(803\) −29.5419 −1.04251
\(804\) −2.64500 −0.0932818
\(805\) −3.06932 −0.108179
\(806\) 0 0
\(807\) 20.1473 0.709218
\(808\) 20.9640 0.737512
\(809\) −53.4008 −1.87747 −0.938735 0.344639i \(-0.888001\pi\)
−0.938735 + 0.344639i \(0.888001\pi\)
\(810\) −2.59115 −0.0910439
\(811\) 36.7023 1.28879 0.644397 0.764691i \(-0.277109\pi\)
0.644397 + 0.764691i \(0.277109\pi\)
\(812\) −2.73668 −0.0960385
\(813\) 16.1869 0.567699
\(814\) 14.2876 0.500779
\(815\) 25.9721 0.909763
\(816\) 0.553158 0.0193644
\(817\) 53.4241 1.86907
\(818\) −41.9642 −1.46725
\(819\) 0 0
\(820\) −0.584438 −0.0204095
\(821\) 42.0190 1.46647 0.733236 0.679974i \(-0.238009\pi\)
0.733236 + 0.679974i \(0.238009\pi\)
\(822\) −11.1492 −0.388874
\(823\) −51.9482 −1.81080 −0.905400 0.424559i \(-0.860429\pi\)
−0.905400 + 0.424559i \(0.860429\pi\)
\(824\) −25.1122 −0.874825
\(825\) 2.29261 0.0798183
\(826\) −11.9249 −0.414920
\(827\) 13.0284 0.453042 0.226521 0.974006i \(-0.427265\pi\)
0.226521 + 0.974006i \(0.427265\pi\)
\(828\) −0.478070 −0.0166141
\(829\) −8.22530 −0.285677 −0.142838 0.989746i \(-0.545623\pi\)
−0.142838 + 0.989746i \(0.545623\pi\)
\(830\) −0.876008 −0.0304067
\(831\) −21.9043 −0.759853
\(832\) 0 0
\(833\) −0.168513 −0.00583862
\(834\) 3.63463 0.125857
\(835\) 19.2134 0.664909
\(836\) 4.78131 0.165365
\(837\) 0.0996388 0.00344402
\(838\) −9.85095 −0.340295
\(839\) −39.8317 −1.37514 −0.687572 0.726116i \(-0.741324\pi\)
−0.687572 + 0.726116i \(0.741324\pi\)
\(840\) 5.98686 0.206566
\(841\) 48.6819 1.67869
\(842\) −11.3785 −0.392130
\(843\) −19.5379 −0.672920
\(844\) −0.435251 −0.0149819
\(845\) 0 0
\(846\) 1.40890 0.0484389
\(847\) −6.00691 −0.206400
\(848\) −39.4257 −1.35389
\(849\) 23.9022 0.820320
\(850\) 0.224728 0.00770809
\(851\) −7.57395 −0.259632
\(852\) 1.92233 0.0658578
\(853\) 31.4971 1.07844 0.539220 0.842165i \(-0.318719\pi\)
0.539220 + 0.842165i \(0.318719\pi\)
\(854\) −4.25107 −0.145469
\(855\) −13.7377 −0.469819
\(856\) −50.2405 −1.71718
\(857\) 24.9026 0.850658 0.425329 0.905039i \(-0.360158\pi\)
0.425329 + 0.905039i \(0.360158\pi\)
\(858\) 0 0
\(859\) 57.6219 1.96603 0.983016 0.183519i \(-0.0587490\pi\)
0.983016 + 0.183519i \(0.0587490\pi\)
\(860\) 4.79861 0.163631
\(861\) −0.944193 −0.0321780
\(862\) 8.35800 0.284675
\(863\) 42.6303 1.45115 0.725576 0.688142i \(-0.241573\pi\)
0.725576 + 0.688142i \(0.241573\pi\)
\(864\) 1.73968 0.0591852
\(865\) 43.2984 1.47219
\(866\) 19.0309 0.646695
\(867\) −16.9716 −0.576386
\(868\) −0.0309380 −0.00105010
\(869\) −22.7621 −0.772151
\(870\) −22.8377 −0.774272
\(871\) 0 0
\(872\) 24.0316 0.813813
\(873\) 14.8575 0.502849
\(874\) 13.7913 0.466499
\(875\) 12.0128 0.406106
\(876\) −4.10504 −0.138696
\(877\) 44.6625 1.50814 0.754072 0.656792i \(-0.228087\pi\)
0.754072 + 0.656792i \(0.228087\pi\)
\(878\) 35.6856 1.20433
\(879\) 17.9823 0.606527
\(880\) −14.6223 −0.492917
\(881\) −9.04872 −0.304859 −0.152430 0.988314i \(-0.548710\pi\)
−0.152430 + 0.988314i \(0.548710\pi\)
\(882\) 1.29981 0.0437668
\(883\) 0.876086 0.0294826 0.0147413 0.999891i \(-0.495308\pi\)
0.0147413 + 0.999891i \(0.495308\pi\)
\(884\) 0 0
\(885\) 18.2890 0.614777
\(886\) 33.0212 1.10937
\(887\) −14.8426 −0.498366 −0.249183 0.968456i \(-0.580162\pi\)
−0.249183 + 0.968456i \(0.580162\pi\)
\(888\) 14.7734 0.495762
\(889\) 14.7776 0.495625
\(890\) −20.6288 −0.691479
\(891\) −2.23452 −0.0748593
\(892\) 2.21152 0.0740473
\(893\) 7.46964 0.249962
\(894\) −1.23251 −0.0412213
\(895\) 28.7149 0.959834
\(896\) 7.99328 0.267037
\(897\) 0 0
\(898\) −11.2637 −0.375876
\(899\) 0.878189 0.0292893
\(900\) 0.318572 0.0106191
\(901\) −2.02394 −0.0674271
\(902\) 2.74236 0.0913105
\(903\) 7.75243 0.257985
\(904\) −3.13910 −0.104405
\(905\) −0.492135 −0.0163591
\(906\) −22.0885 −0.733841
\(907\) 30.2415 1.00415 0.502076 0.864824i \(-0.332570\pi\)
0.502076 + 0.864824i \(0.332570\pi\)
\(908\) 4.20987 0.139709
\(909\) −6.98055 −0.231530
\(910\) 0 0
\(911\) 1.72413 0.0571229 0.0285614 0.999592i \(-0.490907\pi\)
0.0285614 + 0.999592i \(0.490907\pi\)
\(912\) −22.6212 −0.749062
\(913\) −0.755439 −0.0250014
\(914\) 41.0007 1.35618
\(915\) 6.51979 0.215538
\(916\) −3.91299 −0.129289
\(917\) −13.6465 −0.450648
\(918\) −0.219034 −0.00722920
\(919\) −14.1914 −0.468131 −0.234066 0.972221i \(-0.575203\pi\)
−0.234066 + 0.972221i \(0.575203\pi\)
\(920\) 9.21781 0.303902
\(921\) 28.7026 0.945782
\(922\) −24.0799 −0.793029
\(923\) 0 0
\(924\) 0.693822 0.0228251
\(925\) 5.04707 0.165947
\(926\) 5.08831 0.167212
\(927\) 8.36180 0.274638
\(928\) 15.3331 0.503334
\(929\) 46.0189 1.50983 0.754916 0.655821i \(-0.227678\pi\)
0.754916 + 0.655821i \(0.227678\pi\)
\(930\) −0.258179 −0.00846603
\(931\) 6.89127 0.225852
\(932\) −4.64086 −0.152016
\(933\) 22.0690 0.722507
\(934\) −29.7693 −0.974081
\(935\) −0.750640 −0.0245486
\(936\) 0 0
\(937\) 45.9281 1.50041 0.750203 0.661208i \(-0.229956\pi\)
0.750203 + 0.661208i \(0.229956\pi\)
\(938\) 11.0724 0.361525
\(939\) 20.6739 0.674668
\(940\) 0.670932 0.0218834
\(941\) −10.3266 −0.336638 −0.168319 0.985733i \(-0.553834\pi\)
−0.168319 + 0.985733i \(0.553834\pi\)
\(942\) 21.5020 0.700574
\(943\) −1.45375 −0.0473405
\(944\) 30.1156 0.980179
\(945\) −1.99349 −0.0648483
\(946\) −22.5165 −0.732075
\(947\) 38.6597 1.25627 0.628135 0.778104i \(-0.283818\pi\)
0.628135 + 0.778104i \(0.283818\pi\)
\(948\) −3.16294 −0.102728
\(949\) 0 0
\(950\) −9.19015 −0.298168
\(951\) −22.1697 −0.718903
\(952\) 0.506079 0.0164021
\(953\) 3.49088 0.113081 0.0565404 0.998400i \(-0.481993\pi\)
0.0565404 + 0.998400i \(0.481993\pi\)
\(954\) 15.6114 0.505439
\(955\) −16.5923 −0.536915
\(956\) 8.86958 0.286863
\(957\) −19.6945 −0.636632
\(958\) 13.7285 0.443548
\(959\) −8.57759 −0.276985
\(960\) −17.5954 −0.567889
\(961\) −30.9901 −0.999680
\(962\) 0 0
\(963\) 16.7289 0.539082
\(964\) −7.57703 −0.244040
\(965\) 34.3225 1.10488
\(966\) 2.00128 0.0643900
\(967\) −27.3810 −0.880513 −0.440257 0.897872i \(-0.645113\pi\)
−0.440257 + 0.897872i \(0.645113\pi\)
\(968\) 18.0400 0.579827
\(969\) −1.16127 −0.0373053
\(970\) −38.4980 −1.23610
\(971\) −35.1329 −1.12747 −0.563735 0.825956i \(-0.690636\pi\)
−0.563735 + 0.825956i \(0.690636\pi\)
\(972\) −0.310501 −0.00995933
\(973\) 2.79628 0.0896446
\(974\) −14.1148 −0.452268
\(975\) 0 0
\(976\) 10.7358 0.343645
\(977\) −16.4403 −0.525973 −0.262987 0.964799i \(-0.584707\pi\)
−0.262987 + 0.964799i \(0.584707\pi\)
\(978\) −16.9345 −0.541505
\(979\) −17.7896 −0.568557
\(980\) 0.618982 0.0197726
\(981\) −8.00198 −0.255484
\(982\) 25.5936 0.816726
\(983\) −15.8250 −0.504739 −0.252370 0.967631i \(-0.581210\pi\)
−0.252370 + 0.967631i \(0.581210\pi\)
\(984\) 2.83561 0.0903958
\(985\) −27.6227 −0.880133
\(986\) −1.93051 −0.0614799
\(987\) 1.08393 0.0345018
\(988\) 0 0
\(989\) 11.9362 0.379549
\(990\) 5.78999 0.184018
\(991\) −27.8250 −0.883890 −0.441945 0.897042i \(-0.645711\pi\)
−0.441945 + 0.897042i \(0.645711\pi\)
\(992\) 0.173340 0.00550355
\(993\) −1.69293 −0.0537234
\(994\) −8.04716 −0.255240
\(995\) −26.0405 −0.825539
\(996\) −0.104973 −0.00332620
\(997\) −36.7129 −1.16271 −0.581355 0.813650i \(-0.697477\pi\)
−0.581355 + 0.813650i \(0.697477\pi\)
\(998\) 30.1414 0.954110
\(999\) −4.91920 −0.155637
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 3549.2.a.bg.1.11 15
13.12 even 2 3549.2.a.bh.1.5 yes 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3549.2.a.bg.1.11 15 1.1 even 1 trivial
3549.2.a.bh.1.5 yes 15 13.12 even 2