Properties

Label 5070.2.a.bo.1.2
Level $5070$
Weight $2$
Character 5070.1
Self dual yes
Analytic conductor $40.484$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.445042\) of defining polynomial
Character \(\chi\) \(=\) 5070.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.44504 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} +3.44504 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +4.24698 q^{11} +1.00000 q^{12} -3.44504 q^{14} -1.00000 q^{15} +1.00000 q^{16} +6.78986 q^{17} -1.00000 q^{18} +6.26875 q^{19} -1.00000 q^{20} +3.44504 q^{21} -4.24698 q^{22} -1.30798 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} +3.44504 q^{28} -9.14675 q^{29} +1.00000 q^{30} +3.75302 q^{31} -1.00000 q^{32} +4.24698 q^{33} -6.78986 q^{34} -3.44504 q^{35} +1.00000 q^{36} +6.82908 q^{37} -6.26875 q^{38} +1.00000 q^{40} +4.26875 q^{41} -3.44504 q^{42} +3.07069 q^{43} +4.24698 q^{44} -1.00000 q^{45} +1.30798 q^{46} +7.76809 q^{47} +1.00000 q^{48} +4.86831 q^{49} -1.00000 q^{50} +6.78986 q^{51} -8.93900 q^{53} -1.00000 q^{54} -4.24698 q^{55} -3.44504 q^{56} +6.26875 q^{57} +9.14675 q^{58} -10.3327 q^{59} -1.00000 q^{60} +2.53319 q^{61} -3.75302 q^{62} +3.44504 q^{63} +1.00000 q^{64} -4.24698 q^{66} -0.0760644 q^{67} +6.78986 q^{68} -1.30798 q^{69} +3.44504 q^{70} +0.374354 q^{71} -1.00000 q^{72} -16.7114 q^{73} -6.82908 q^{74} +1.00000 q^{75} +6.26875 q^{76} +14.6310 q^{77} -1.33513 q^{79} -1.00000 q^{80} +1.00000 q^{81} -4.26875 q^{82} -0.740939 q^{83} +3.44504 q^{84} -6.78986 q^{85} -3.07069 q^{86} -9.14675 q^{87} -4.24698 q^{88} +13.3274 q^{89} +1.00000 q^{90} -1.30798 q^{92} +3.75302 q^{93} -7.76809 q^{94} -6.26875 q^{95} -1.00000 q^{96} -13.1903 q^{97} -4.86831 q^{98} +4.24698 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} - 3q^{6} + 10q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} - 3q^{6} + 10q^{7} - 3q^{8} + 3q^{9} + 3q^{10} + 8q^{11} + 3q^{12} - 10q^{14} - 3q^{15} + 3q^{16} - 3q^{17} - 3q^{18} + 11q^{19} - 3q^{20} + 10q^{21} - 8q^{22} - 9q^{23} - 3q^{24} + 3q^{25} + 3q^{27} + 10q^{28} + 3q^{30} + 16q^{31} - 3q^{32} + 8q^{33} + 3q^{34} - 10q^{35} + 3q^{36} + 10q^{37} - 11q^{38} + 3q^{40} + 5q^{41} - 10q^{42} - 3q^{43} + 8q^{44} - 3q^{45} + 9q^{46} + 3q^{47} + 3q^{48} + 17q^{49} - 3q^{50} - 3q^{51} - 17q^{53} - 3q^{54} - 8q^{55} - 10q^{56} + 11q^{57} + 11q^{59} - 3q^{60} + 11q^{61} - 16q^{62} + 10q^{63} + 3q^{64} - 8q^{66} + 15q^{67} - 3q^{68} - 9q^{69} + 10q^{70} + 13q^{71} - 3q^{72} - q^{73} - 10q^{74} + 3q^{75} + 11q^{76} + 29q^{77} - 3q^{79} - 3q^{80} + 3q^{81} - 5q^{82} + 12q^{83} + 10q^{84} + 3q^{85} + 3q^{86} - 8q^{88} + q^{89} + 3q^{90} - 9q^{92} + 16q^{93} - 3q^{94} - 11q^{95} - 3q^{96} - 6q^{97} - 17q^{98} + 8q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) −1.00000 −0.408248
\(7\) 3.44504 1.30210 0.651052 0.759033i \(-0.274328\pi\)
0.651052 + 0.759033i \(0.274328\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 4.24698 1.28051 0.640256 0.768161i \(-0.278828\pi\)
0.640256 + 0.768161i \(0.278828\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −3.44504 −0.920726
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 6.78986 1.64678 0.823391 0.567474i \(-0.192079\pi\)
0.823391 + 0.567474i \(0.192079\pi\)
\(18\) −1.00000 −0.235702
\(19\) 6.26875 1.43815 0.719075 0.694933i \(-0.244566\pi\)
0.719075 + 0.694933i \(0.244566\pi\)
\(20\) −1.00000 −0.223607
\(21\) 3.44504 0.751770
\(22\) −4.24698 −0.905459
\(23\) −1.30798 −0.272732 −0.136366 0.990658i \(-0.543542\pi\)
−0.136366 + 0.990658i \(0.543542\pi\)
\(24\) −1.00000 −0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 3.44504 0.651052
\(29\) −9.14675 −1.69851 −0.849255 0.527984i \(-0.822948\pi\)
−0.849255 + 0.527984i \(0.822948\pi\)
\(30\) 1.00000 0.182574
\(31\) 3.75302 0.674062 0.337031 0.941493i \(-0.390577\pi\)
0.337031 + 0.941493i \(0.390577\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.24698 0.739304
\(34\) −6.78986 −1.16445
\(35\) −3.44504 −0.582318
\(36\) 1.00000 0.166667
\(37\) 6.82908 1.12269 0.561347 0.827580i \(-0.310283\pi\)
0.561347 + 0.827580i \(0.310283\pi\)
\(38\) −6.26875 −1.01693
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 4.26875 0.666667 0.333333 0.942809i \(-0.391827\pi\)
0.333333 + 0.942809i \(0.391827\pi\)
\(42\) −3.44504 −0.531582
\(43\) 3.07069 0.468275 0.234138 0.972203i \(-0.424773\pi\)
0.234138 + 0.972203i \(0.424773\pi\)
\(44\) 4.24698 0.640256
\(45\) −1.00000 −0.149071
\(46\) 1.30798 0.192851
\(47\) 7.76809 1.13309 0.566546 0.824030i \(-0.308279\pi\)
0.566546 + 0.824030i \(0.308279\pi\)
\(48\) 1.00000 0.144338
\(49\) 4.86831 0.695473
\(50\) −1.00000 −0.141421
\(51\) 6.78986 0.950770
\(52\) 0 0
\(53\) −8.93900 −1.22787 −0.613933 0.789358i \(-0.710414\pi\)
−0.613933 + 0.789358i \(0.710414\pi\)
\(54\) −1.00000 −0.136083
\(55\) −4.24698 −0.572663
\(56\) −3.44504 −0.460363
\(57\) 6.26875 0.830316
\(58\) 9.14675 1.20103
\(59\) −10.3327 −1.34521 −0.672604 0.740003i \(-0.734824\pi\)
−0.672604 + 0.740003i \(0.734824\pi\)
\(60\) −1.00000 −0.129099
\(61\) 2.53319 0.324341 0.162171 0.986763i \(-0.448150\pi\)
0.162171 + 0.986763i \(0.448150\pi\)
\(62\) −3.75302 −0.476634
\(63\) 3.44504 0.434034
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.24698 −0.522767
\(67\) −0.0760644 −0.00929275 −0.00464637 0.999989i \(-0.501479\pi\)
−0.00464637 + 0.999989i \(0.501479\pi\)
\(68\) 6.78986 0.823391
\(69\) −1.30798 −0.157462
\(70\) 3.44504 0.411761
\(71\) 0.374354 0.0444277 0.0222138 0.999753i \(-0.492929\pi\)
0.0222138 + 0.999753i \(0.492929\pi\)
\(72\) −1.00000 −0.117851
\(73\) −16.7114 −1.95592 −0.977961 0.208789i \(-0.933048\pi\)
−0.977961 + 0.208789i \(0.933048\pi\)
\(74\) −6.82908 −0.793865
\(75\) 1.00000 0.115470
\(76\) 6.26875 0.719075
\(77\) 14.6310 1.66736
\(78\) 0 0
\(79\) −1.33513 −0.150213 −0.0751067 0.997176i \(-0.523930\pi\)
−0.0751067 + 0.997176i \(0.523930\pi\)
\(80\) −1.00000 −0.111803
\(81\) 1.00000 0.111111
\(82\) −4.26875 −0.471405
\(83\) −0.740939 −0.0813286 −0.0406643 0.999173i \(-0.512947\pi\)
−0.0406643 + 0.999173i \(0.512947\pi\)
\(84\) 3.44504 0.375885
\(85\) −6.78986 −0.736463
\(86\) −3.07069 −0.331121
\(87\) −9.14675 −0.980635
\(88\) −4.24698 −0.452730
\(89\) 13.3274 1.41270 0.706348 0.707864i \(-0.250341\pi\)
0.706348 + 0.707864i \(0.250341\pi\)
\(90\) 1.00000 0.105409
\(91\) 0 0
\(92\) −1.30798 −0.136366
\(93\) 3.75302 0.389170
\(94\) −7.76809 −0.801217
\(95\) −6.26875 −0.643160
\(96\) −1.00000 −0.102062
\(97\) −13.1903 −1.33927 −0.669636 0.742690i \(-0.733550\pi\)
−0.669636 + 0.742690i \(0.733550\pi\)
\(98\) −4.86831 −0.491774
\(99\) 4.24698 0.426838
\(100\) 1.00000 0.100000
\(101\) −15.0422 −1.49676 −0.748378 0.663272i \(-0.769167\pi\)
−0.748378 + 0.663272i \(0.769167\pi\)
\(102\) −6.78986 −0.672296
\(103\) −8.92154 −0.879066 −0.439533 0.898227i \(-0.644856\pi\)
−0.439533 + 0.898227i \(0.644856\pi\)
\(104\) 0 0
\(105\) −3.44504 −0.336202
\(106\) 8.93900 0.868233
\(107\) 14.6746 1.41864 0.709322 0.704885i \(-0.249001\pi\)
0.709322 + 0.704885i \(0.249001\pi\)
\(108\) 1.00000 0.0962250
\(109\) 9.66786 0.926013 0.463006 0.886355i \(-0.346771\pi\)
0.463006 + 0.886355i \(0.346771\pi\)
\(110\) 4.24698 0.404934
\(111\) 6.82908 0.648188
\(112\) 3.44504 0.325526
\(113\) −11.3448 −1.06723 −0.533615 0.845727i \(-0.679167\pi\)
−0.533615 + 0.845727i \(0.679167\pi\)
\(114\) −6.26875 −0.587122
\(115\) 1.30798 0.121970
\(116\) −9.14675 −0.849255
\(117\) 0 0
\(118\) 10.3327 0.951205
\(119\) 23.3913 2.14428
\(120\) 1.00000 0.0912871
\(121\) 7.03684 0.639712
\(122\) −2.53319 −0.229344
\(123\) 4.26875 0.384900
\(124\) 3.75302 0.337031
\(125\) −1.00000 −0.0894427
\(126\) −3.44504 −0.306909
\(127\) −2.54288 −0.225644 −0.112822 0.993615i \(-0.535989\pi\)
−0.112822 + 0.993615i \(0.535989\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 3.07069 0.270359
\(130\) 0 0
\(131\) −10.8509 −0.948044 −0.474022 0.880513i \(-0.657198\pi\)
−0.474022 + 0.880513i \(0.657198\pi\)
\(132\) 4.24698 0.369652
\(133\) 21.5961 1.87262
\(134\) 0.0760644 0.00657096
\(135\) −1.00000 −0.0860663
\(136\) −6.78986 −0.582225
\(137\) −18.6407 −1.59258 −0.796292 0.604913i \(-0.793208\pi\)
−0.796292 + 0.604913i \(0.793208\pi\)
\(138\) 1.30798 0.111343
\(139\) 3.53750 0.300047 0.150023 0.988682i \(-0.452065\pi\)
0.150023 + 0.988682i \(0.452065\pi\)
\(140\) −3.44504 −0.291159
\(141\) 7.76809 0.654191
\(142\) −0.374354 −0.0314151
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 9.14675 0.759596
\(146\) 16.7114 1.38305
\(147\) 4.86831 0.401532
\(148\) 6.82908 0.561347
\(149\) −16.6843 −1.36683 −0.683414 0.730031i \(-0.739505\pi\)
−0.683414 + 0.730031i \(0.739505\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 18.3327 1.49190 0.745948 0.666004i \(-0.231997\pi\)
0.745948 + 0.666004i \(0.231997\pi\)
\(152\) −6.26875 −0.508463
\(153\) 6.78986 0.548927
\(154\) −14.6310 −1.17900
\(155\) −3.75302 −0.301450
\(156\) 0 0
\(157\) −18.6015 −1.48456 −0.742280 0.670090i \(-0.766256\pi\)
−0.742280 + 0.670090i \(0.766256\pi\)
\(158\) 1.33513 0.106217
\(159\) −8.93900 −0.708909
\(160\) 1.00000 0.0790569
\(161\) −4.50604 −0.355126
\(162\) −1.00000 −0.0785674
\(163\) 18.8944 1.47992 0.739962 0.672649i \(-0.234844\pi\)
0.739962 + 0.672649i \(0.234844\pi\)
\(164\) 4.26875 0.333333
\(165\) −4.24698 −0.330627
\(166\) 0.740939 0.0575080
\(167\) −4.24698 −0.328641 −0.164321 0.986407i \(-0.552543\pi\)
−0.164321 + 0.986407i \(0.552543\pi\)
\(168\) −3.44504 −0.265791
\(169\) 0 0
\(170\) 6.78986 0.520758
\(171\) 6.26875 0.479383
\(172\) 3.07069 0.234138
\(173\) 1.16852 0.0888411 0.0444206 0.999013i \(-0.485856\pi\)
0.0444206 + 0.999013i \(0.485856\pi\)
\(174\) 9.14675 0.693413
\(175\) 3.44504 0.260421
\(176\) 4.24698 0.320128
\(177\) −10.3327 −0.776656
\(178\) −13.3274 −0.998928
\(179\) −10.1836 −0.761157 −0.380579 0.924749i \(-0.624275\pi\)
−0.380579 + 0.924749i \(0.624275\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 10.3351 0.768204 0.384102 0.923291i \(-0.374511\pi\)
0.384102 + 0.923291i \(0.374511\pi\)
\(182\) 0 0
\(183\) 2.53319 0.187259
\(184\) 1.30798 0.0964255
\(185\) −6.82908 −0.502084
\(186\) −3.75302 −0.275185
\(187\) 28.8364 2.10872
\(188\) 7.76809 0.566546
\(189\) 3.44504 0.250590
\(190\) 6.26875 0.454783
\(191\) −9.43296 −0.682545 −0.341273 0.939964i \(-0.610858\pi\)
−0.341273 + 0.939964i \(0.610858\pi\)
\(192\) 1.00000 0.0721688
\(193\) 22.9191 1.64976 0.824878 0.565310i \(-0.191244\pi\)
0.824878 + 0.565310i \(0.191244\pi\)
\(194\) 13.1903 0.947008
\(195\) 0 0
\(196\) 4.86831 0.347737
\(197\) −14.3502 −1.02241 −0.511204 0.859459i \(-0.670800\pi\)
−0.511204 + 0.859459i \(0.670800\pi\)
\(198\) −4.24698 −0.301820
\(199\) 14.2524 1.01032 0.505161 0.863025i \(-0.331433\pi\)
0.505161 + 0.863025i \(0.331433\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −0.0760644 −0.00536517
\(202\) 15.0422 1.05837
\(203\) −31.5109 −2.21163
\(204\) 6.78986 0.475385
\(205\) −4.26875 −0.298142
\(206\) 8.92154 0.621593
\(207\) −1.30798 −0.0909108
\(208\) 0 0
\(209\) 26.6233 1.84157
\(210\) 3.44504 0.237730
\(211\) 0.396125 0.0272703 0.0136352 0.999907i \(-0.495660\pi\)
0.0136352 + 0.999907i \(0.495660\pi\)
\(212\) −8.93900 −0.613933
\(213\) 0.374354 0.0256503
\(214\) −14.6746 −1.00313
\(215\) −3.07069 −0.209419
\(216\) −1.00000 −0.0680414
\(217\) 12.9293 0.877699
\(218\) −9.66786 −0.654790
\(219\) −16.7114 −1.12925
\(220\) −4.24698 −0.286331
\(221\) 0 0
\(222\) −6.82908 −0.458338
\(223\) 1.49635 0.100203 0.0501016 0.998744i \(-0.484045\pi\)
0.0501016 + 0.998744i \(0.484045\pi\)
\(224\) −3.44504 −0.230182
\(225\) 1.00000 0.0666667
\(226\) 11.3448 0.754646
\(227\) 25.2150 1.67358 0.836791 0.547523i \(-0.184429\pi\)
0.836791 + 0.547523i \(0.184429\pi\)
\(228\) 6.26875 0.415158
\(229\) −22.3666 −1.47803 −0.739013 0.673691i \(-0.764708\pi\)
−0.739013 + 0.673691i \(0.764708\pi\)
\(230\) −1.30798 −0.0862456
\(231\) 14.6310 0.962651
\(232\) 9.14675 0.600514
\(233\) 14.0858 0.922788 0.461394 0.887195i \(-0.347349\pi\)
0.461394 + 0.887195i \(0.347349\pi\)
\(234\) 0 0
\(235\) −7.76809 −0.506734
\(236\) −10.3327 −0.672604
\(237\) −1.33513 −0.0867257
\(238\) −23.3913 −1.51624
\(239\) 5.38106 0.348072 0.174036 0.984739i \(-0.444319\pi\)
0.174036 + 0.984739i \(0.444319\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −29.9420 −1.92873 −0.964366 0.264570i \(-0.914770\pi\)
−0.964366 + 0.264570i \(0.914770\pi\)
\(242\) −7.03684 −0.452345
\(243\) 1.00000 0.0641500
\(244\) 2.53319 0.162171
\(245\) −4.86831 −0.311025
\(246\) −4.26875 −0.272166
\(247\) 0 0
\(248\) −3.75302 −0.238317
\(249\) −0.740939 −0.0469551
\(250\) 1.00000 0.0632456
\(251\) −8.73556 −0.551384 −0.275692 0.961246i \(-0.588907\pi\)
−0.275692 + 0.961246i \(0.588907\pi\)
\(252\) 3.44504 0.217017
\(253\) −5.55496 −0.349237
\(254\) 2.54288 0.159554
\(255\) −6.78986 −0.425197
\(256\) 1.00000 0.0625000
\(257\) 21.2717 1.32689 0.663447 0.748223i \(-0.269093\pi\)
0.663447 + 0.748223i \(0.269093\pi\)
\(258\) −3.07069 −0.191173
\(259\) 23.5265 1.46186
\(260\) 0 0
\(261\) −9.14675 −0.566170
\(262\) 10.8509 0.670368
\(263\) −22.2325 −1.37091 −0.685457 0.728113i \(-0.740398\pi\)
−0.685457 + 0.728113i \(0.740398\pi\)
\(264\) −4.24698 −0.261384
\(265\) 8.93900 0.549118
\(266\) −21.5961 −1.32414
\(267\) 13.3274 0.815621
\(268\) −0.0760644 −0.00464637
\(269\) −10.2241 −0.623377 −0.311689 0.950184i \(-0.600895\pi\)
−0.311689 + 0.950184i \(0.600895\pi\)
\(270\) 1.00000 0.0608581
\(271\) 0.716185 0.0435051 0.0217526 0.999763i \(-0.493075\pi\)
0.0217526 + 0.999763i \(0.493075\pi\)
\(272\) 6.78986 0.411696
\(273\) 0 0
\(274\) 18.6407 1.12613
\(275\) 4.24698 0.256103
\(276\) −1.30798 −0.0787311
\(277\) −22.8713 −1.37420 −0.687102 0.726561i \(-0.741117\pi\)
−0.687102 + 0.726561i \(0.741117\pi\)
\(278\) −3.53750 −0.212165
\(279\) 3.75302 0.224687
\(280\) 3.44504 0.205881
\(281\) 4.13036 0.246397 0.123198 0.992382i \(-0.460685\pi\)
0.123198 + 0.992382i \(0.460685\pi\)
\(282\) −7.76809 −0.462583
\(283\) 16.2959 0.968691 0.484345 0.874877i \(-0.339058\pi\)
0.484345 + 0.874877i \(0.339058\pi\)
\(284\) 0.374354 0.0222138
\(285\) −6.26875 −0.371329
\(286\) 0 0
\(287\) 14.7060 0.868069
\(288\) −1.00000 −0.0589256
\(289\) 29.1021 1.71189
\(290\) −9.14675 −0.537116
\(291\) −13.1903 −0.773229
\(292\) −16.7114 −0.977961
\(293\) 7.72587 0.451350 0.225675 0.974203i \(-0.427541\pi\)
0.225675 + 0.974203i \(0.427541\pi\)
\(294\) −4.86831 −0.283926
\(295\) 10.3327 0.601595
\(296\) −6.82908 −0.396932
\(297\) 4.24698 0.246435
\(298\) 16.6843 0.966493
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 10.5786 0.609743
\(302\) −18.3327 −1.05493
\(303\) −15.0422 −0.864153
\(304\) 6.26875 0.359537
\(305\) −2.53319 −0.145050
\(306\) −6.78986 −0.388150
\(307\) −5.13036 −0.292805 −0.146403 0.989225i \(-0.546769\pi\)
−0.146403 + 0.989225i \(0.546769\pi\)
\(308\) 14.6310 0.833680
\(309\) −8.92154 −0.507529
\(310\) 3.75302 0.213157
\(311\) 33.1987 1.88252 0.941261 0.337679i \(-0.109642\pi\)
0.941261 + 0.337679i \(0.109642\pi\)
\(312\) 0 0
\(313\) 0.0814412 0.00460333 0.00230167 0.999997i \(-0.499267\pi\)
0.00230167 + 0.999997i \(0.499267\pi\)
\(314\) 18.6015 1.04974
\(315\) −3.44504 −0.194106
\(316\) −1.33513 −0.0751067
\(317\) −19.4969 −1.09506 −0.547529 0.836787i \(-0.684431\pi\)
−0.547529 + 0.836787i \(0.684431\pi\)
\(318\) 8.93900 0.501274
\(319\) −38.8461 −2.17496
\(320\) −1.00000 −0.0559017
\(321\) 14.6746 0.819054
\(322\) 4.50604 0.251112
\(323\) 42.5639 2.36832
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −18.8944 −1.04646
\(327\) 9.66786 0.534634
\(328\) −4.26875 −0.235702
\(329\) 26.7614 1.47540
\(330\) 4.24698 0.233789
\(331\) 23.8049 1.30844 0.654218 0.756306i \(-0.272998\pi\)
0.654218 + 0.756306i \(0.272998\pi\)
\(332\) −0.740939 −0.0406643
\(333\) 6.82908 0.374232
\(334\) 4.24698 0.232384
\(335\) 0.0760644 0.00415584
\(336\) 3.44504 0.187942
\(337\) 21.2064 1.15519 0.577594 0.816324i \(-0.303992\pi\)
0.577594 + 0.816324i \(0.303992\pi\)
\(338\) 0 0
\(339\) −11.3448 −0.616166
\(340\) −6.78986 −0.368232
\(341\) 15.9390 0.863145
\(342\) −6.26875 −0.338975
\(343\) −7.34375 −0.396525
\(344\) −3.07069 −0.165560
\(345\) 1.30798 0.0704192
\(346\) −1.16852 −0.0628201
\(347\) −16.5773 −0.889917 −0.444959 0.895551i \(-0.646782\pi\)
−0.444959 + 0.895551i \(0.646782\pi\)
\(348\) −9.14675 −0.490317
\(349\) 2.51035 0.134376 0.0671880 0.997740i \(-0.478597\pi\)
0.0671880 + 0.997740i \(0.478597\pi\)
\(350\) −3.44504 −0.184145
\(351\) 0 0
\(352\) −4.24698 −0.226365
\(353\) −20.8412 −1.10926 −0.554632 0.832096i \(-0.687141\pi\)
−0.554632 + 0.832096i \(0.687141\pi\)
\(354\) 10.3327 0.549179
\(355\) −0.374354 −0.0198687
\(356\) 13.3274 0.706348
\(357\) 23.3913 1.23800
\(358\) 10.1836 0.538219
\(359\) 2.42566 0.128022 0.0640108 0.997949i \(-0.479611\pi\)
0.0640108 + 0.997949i \(0.479611\pi\)
\(360\) 1.00000 0.0527046
\(361\) 20.2972 1.06827
\(362\) −10.3351 −0.543202
\(363\) 7.03684 0.369338
\(364\) 0 0
\(365\) 16.7114 0.874715
\(366\) −2.53319 −0.132412
\(367\) 2.66248 0.138980 0.0694902 0.997583i \(-0.477863\pi\)
0.0694902 + 0.997583i \(0.477863\pi\)
\(368\) −1.30798 −0.0681831
\(369\) 4.26875 0.222222
\(370\) 6.82908 0.355027
\(371\) −30.7952 −1.59881
\(372\) 3.75302 0.194585
\(373\) 5.87800 0.304351 0.152176 0.988353i \(-0.451372\pi\)
0.152176 + 0.988353i \(0.451372\pi\)
\(374\) −28.8364 −1.49109
\(375\) −1.00000 −0.0516398
\(376\) −7.76809 −0.400608
\(377\) 0 0
\(378\) −3.44504 −0.177194
\(379\) 36.1540 1.85711 0.928554 0.371196i \(-0.121052\pi\)
0.928554 + 0.371196i \(0.121052\pi\)
\(380\) −6.26875 −0.321580
\(381\) −2.54288 −0.130276
\(382\) 9.43296 0.482632
\(383\) −8.98361 −0.459041 −0.229520 0.973304i \(-0.573716\pi\)
−0.229520 + 0.973304i \(0.573716\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −14.6310 −0.745666
\(386\) −22.9191 −1.16655
\(387\) 3.07069 0.156092
\(388\) −13.1903 −0.669636
\(389\) 0.655186 0.0332192 0.0166096 0.999862i \(-0.494713\pi\)
0.0166096 + 0.999862i \(0.494713\pi\)
\(390\) 0 0
\(391\) −8.88099 −0.449131
\(392\) −4.86831 −0.245887
\(393\) −10.8509 −0.547353
\(394\) 14.3502 0.722952
\(395\) 1.33513 0.0671775
\(396\) 4.24698 0.213419
\(397\) 0.271143 0.0136083 0.00680413 0.999977i \(-0.497834\pi\)
0.00680413 + 0.999977i \(0.497834\pi\)
\(398\) −14.2524 −0.714406
\(399\) 21.5961 1.08116
\(400\) 1.00000 0.0500000
\(401\) 0.344814 0.0172192 0.00860960 0.999963i \(-0.497259\pi\)
0.00860960 + 0.999963i \(0.497259\pi\)
\(402\) 0.0760644 0.00379375
\(403\) 0 0
\(404\) −15.0422 −0.748378
\(405\) −1.00000 −0.0496904
\(406\) 31.5109 1.56386
\(407\) 29.0030 1.43762
\(408\) −6.78986 −0.336148
\(409\) 27.0388 1.33698 0.668490 0.743721i \(-0.266941\pi\)
0.668490 + 0.743721i \(0.266941\pi\)
\(410\) 4.26875 0.210819
\(411\) −18.6407 −0.919478
\(412\) −8.92154 −0.439533
\(413\) −35.5967 −1.75160
\(414\) 1.30798 0.0642836
\(415\) 0.740939 0.0363713
\(416\) 0 0
\(417\) 3.53750 0.173232
\(418\) −26.6233 −1.30219
\(419\) 5.86533 0.286540 0.143270 0.989684i \(-0.454238\pi\)
0.143270 + 0.989684i \(0.454238\pi\)
\(420\) −3.44504 −0.168101
\(421\) −28.9476 −1.41082 −0.705410 0.708799i \(-0.749237\pi\)
−0.705410 + 0.708799i \(0.749237\pi\)
\(422\) −0.396125 −0.0192830
\(423\) 7.76809 0.377697
\(424\) 8.93900 0.434116
\(425\) 6.78986 0.329356
\(426\) −0.374354 −0.0181375
\(427\) 8.72694 0.422326
\(428\) 14.6746 0.709322
\(429\) 0 0
\(430\) 3.07069 0.148082
\(431\) 4.87023 0.234591 0.117295 0.993097i \(-0.462578\pi\)
0.117295 + 0.993097i \(0.462578\pi\)
\(432\) 1.00000 0.0481125
\(433\) −36.9788 −1.77709 −0.888544 0.458791i \(-0.848283\pi\)
−0.888544 + 0.458791i \(0.848283\pi\)
\(434\) −12.9293 −0.620627
\(435\) 9.14675 0.438553
\(436\) 9.66786 0.463006
\(437\) −8.19939 −0.392230
\(438\) 16.7114 0.798502
\(439\) 22.8799 1.09200 0.546000 0.837785i \(-0.316150\pi\)
0.546000 + 0.837785i \(0.316150\pi\)
\(440\) 4.24698 0.202467
\(441\) 4.86831 0.231824
\(442\) 0 0
\(443\) −10.5942 −0.503345 −0.251673 0.967812i \(-0.580981\pi\)
−0.251673 + 0.967812i \(0.580981\pi\)
\(444\) 6.82908 0.324094
\(445\) −13.3274 −0.631777
\(446\) −1.49635 −0.0708543
\(447\) −16.6843 −0.789138
\(448\) 3.44504 0.162763
\(449\) 19.6485 0.927269 0.463635 0.886027i \(-0.346545\pi\)
0.463635 + 0.886027i \(0.346545\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 18.1293 0.853675
\(452\) −11.3448 −0.533615
\(453\) 18.3327 0.861347
\(454\) −25.2150 −1.18340
\(455\) 0 0
\(456\) −6.26875 −0.293561
\(457\) −18.4239 −0.861832 −0.430916 0.902392i \(-0.641809\pi\)
−0.430916 + 0.902392i \(0.641809\pi\)
\(458\) 22.3666 1.04512
\(459\) 6.78986 0.316923
\(460\) 1.30798 0.0609848
\(461\) −22.2446 −1.03603 −0.518017 0.855370i \(-0.673330\pi\)
−0.518017 + 0.855370i \(0.673330\pi\)
\(462\) −14.6310 −0.680697
\(463\) −8.68532 −0.403641 −0.201820 0.979423i \(-0.564686\pi\)
−0.201820 + 0.979423i \(0.564686\pi\)
\(464\) −9.14675 −0.424627
\(465\) −3.75302 −0.174042
\(466\) −14.0858 −0.652510
\(467\) 19.6136 0.907608 0.453804 0.891102i \(-0.350067\pi\)
0.453804 + 0.891102i \(0.350067\pi\)
\(468\) 0 0
\(469\) −0.262045 −0.0121001
\(470\) 7.76809 0.358315
\(471\) −18.6015 −0.857111
\(472\) 10.3327 0.475603
\(473\) 13.0411 0.599633
\(474\) 1.33513 0.0613243
\(475\) 6.26875 0.287630
\(476\) 23.3913 1.07214
\(477\) −8.93900 −0.409289
\(478\) −5.38106 −0.246124
\(479\) 31.4698 1.43789 0.718946 0.695066i \(-0.244625\pi\)
0.718946 + 0.695066i \(0.244625\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) 29.9420 1.36382
\(483\) −4.50604 −0.205032
\(484\) 7.03684 0.319856
\(485\) 13.1903 0.598940
\(486\) −1.00000 −0.0453609
\(487\) 13.2644 0.601069 0.300535 0.953771i \(-0.402835\pi\)
0.300535 + 0.953771i \(0.402835\pi\)
\(488\) −2.53319 −0.114672
\(489\) 18.8944 0.854434
\(490\) 4.86831 0.219928
\(491\) 11.7259 0.529181 0.264591 0.964361i \(-0.414763\pi\)
0.264591 + 0.964361i \(0.414763\pi\)
\(492\) 4.26875 0.192450
\(493\) −62.1051 −2.79707
\(494\) 0 0
\(495\) −4.24698 −0.190888
\(496\) 3.75302 0.168516
\(497\) 1.28967 0.0578494
\(498\) 0.740939 0.0332023
\(499\) −26.7633 −1.19809 −0.599045 0.800715i \(-0.704453\pi\)
−0.599045 + 0.800715i \(0.704453\pi\)
\(500\) −1.00000 −0.0447214
\(501\) −4.24698 −0.189741
\(502\) 8.73556 0.389887
\(503\) 17.2239 0.767975 0.383987 0.923338i \(-0.374551\pi\)
0.383987 + 0.923338i \(0.374551\pi\)
\(504\) −3.44504 −0.153454
\(505\) 15.0422 0.669370
\(506\) 5.55496 0.246948
\(507\) 0 0
\(508\) −2.54288 −0.112822
\(509\) −28.2737 −1.25321 −0.626604 0.779338i \(-0.715556\pi\)
−0.626604 + 0.779338i \(0.715556\pi\)
\(510\) 6.78986 0.300660
\(511\) −57.5715 −2.54681
\(512\) −1.00000 −0.0441942
\(513\) 6.26875 0.276772
\(514\) −21.2717 −0.938256
\(515\) 8.92154 0.393130
\(516\) 3.07069 0.135179
\(517\) 32.9909 1.45094
\(518\) −23.5265 −1.03369
\(519\) 1.16852 0.0512924
\(520\) 0 0
\(521\) 1.47889 0.0647915 0.0323958 0.999475i \(-0.489686\pi\)
0.0323958 + 0.999475i \(0.489686\pi\)
\(522\) 9.14675 0.400342
\(523\) −28.2198 −1.23397 −0.616984 0.786976i \(-0.711646\pi\)
−0.616984 + 0.786976i \(0.711646\pi\)
\(524\) −10.8509 −0.474022
\(525\) 3.44504 0.150354
\(526\) 22.2325 0.969383
\(527\) 25.4825 1.11003
\(528\) 4.24698 0.184826
\(529\) −21.2892 −0.925617
\(530\) −8.93900 −0.388285
\(531\) −10.3327 −0.448402
\(532\) 21.5961 0.936310
\(533\) 0 0
\(534\) −13.3274 −0.576731
\(535\) −14.6746 −0.634437
\(536\) 0.0760644 0.00328548
\(537\) −10.1836 −0.439454
\(538\) 10.2241 0.440794
\(539\) 20.6756 0.890562
\(540\) −1.00000 −0.0430331
\(541\) 0.245915 0.0105727 0.00528635 0.999986i \(-0.498317\pi\)
0.00528635 + 0.999986i \(0.498317\pi\)
\(542\) −0.716185 −0.0307628
\(543\) 10.3351 0.443523
\(544\) −6.78986 −0.291113
\(545\) −9.66786 −0.414126
\(546\) 0 0
\(547\) −13.6407 −0.583235 −0.291617 0.956535i \(-0.594193\pi\)
−0.291617 + 0.956535i \(0.594193\pi\)
\(548\) −18.6407 −0.796292
\(549\) 2.53319 0.108114
\(550\) −4.24698 −0.181092
\(551\) −57.3387 −2.44271
\(552\) 1.30798 0.0556713
\(553\) −4.59956 −0.195593
\(554\) 22.8713 0.971708
\(555\) −6.82908 −0.289879
\(556\) 3.53750 0.150023
\(557\) 14.0248 0.594248 0.297124 0.954839i \(-0.403973\pi\)
0.297124 + 0.954839i \(0.403973\pi\)
\(558\) −3.75302 −0.158878
\(559\) 0 0
\(560\) −3.44504 −0.145580
\(561\) 28.8364 1.21747
\(562\) −4.13036 −0.174229
\(563\) −3.26981 −0.137806 −0.0689031 0.997623i \(-0.521950\pi\)
−0.0689031 + 0.997623i \(0.521950\pi\)
\(564\) 7.76809 0.327095
\(565\) 11.3448 0.477280
\(566\) −16.2959 −0.684968
\(567\) 3.44504 0.144678
\(568\) −0.374354 −0.0157076
\(569\) 4.10752 0.172196 0.0860982 0.996287i \(-0.472560\pi\)
0.0860982 + 0.996287i \(0.472560\pi\)
\(570\) 6.26875 0.262569
\(571\) 9.18465 0.384366 0.192183 0.981359i \(-0.438443\pi\)
0.192183 + 0.981359i \(0.438443\pi\)
\(572\) 0 0
\(573\) −9.43296 −0.394068
\(574\) −14.7060 −0.613817
\(575\) −1.30798 −0.0545465
\(576\) 1.00000 0.0416667
\(577\) 28.0000 1.16566 0.582828 0.812596i \(-0.301946\pi\)
0.582828 + 0.812596i \(0.301946\pi\)
\(578\) −29.1021 −1.21049
\(579\) 22.9191 0.952487
\(580\) 9.14675 0.379798
\(581\) −2.55257 −0.105898
\(582\) 13.1903 0.546755
\(583\) −37.9638 −1.57230
\(584\) 16.7114 0.691523
\(585\) 0 0
\(586\) −7.72587 −0.319153
\(587\) −47.6765 −1.96782 −0.983910 0.178668i \(-0.942821\pi\)
−0.983910 + 0.178668i \(0.942821\pi\)
\(588\) 4.86831 0.200766
\(589\) 23.5267 0.969403
\(590\) −10.3327 −0.425392
\(591\) −14.3502 −0.590288
\(592\) 6.82908 0.280674
\(593\) −16.7681 −0.688583 −0.344291 0.938863i \(-0.611881\pi\)
−0.344291 + 0.938863i \(0.611881\pi\)
\(594\) −4.24698 −0.174256
\(595\) −23.3913 −0.958951
\(596\) −16.6843 −0.683414
\(597\) 14.2524 0.583310
\(598\) 0 0
\(599\) 44.0157 1.79843 0.899215 0.437506i \(-0.144138\pi\)
0.899215 + 0.437506i \(0.144138\pi\)
\(600\) −1.00000 −0.0408248
\(601\) 39.1299 1.59614 0.798071 0.602564i \(-0.205854\pi\)
0.798071 + 0.602564i \(0.205854\pi\)
\(602\) −10.5786 −0.431153
\(603\) −0.0760644 −0.00309758
\(604\) 18.3327 0.745948
\(605\) −7.03684 −0.286088
\(606\) 15.0422 0.611048
\(607\) 6.33273 0.257038 0.128519 0.991707i \(-0.458978\pi\)
0.128519 + 0.991707i \(0.458978\pi\)
\(608\) −6.26875 −0.254231
\(609\) −31.5109 −1.27689
\(610\) 2.53319 0.102566
\(611\) 0 0
\(612\) 6.78986 0.274464
\(613\) 29.1129 1.17586 0.587929 0.808912i \(-0.299943\pi\)
0.587929 + 0.808912i \(0.299943\pi\)
\(614\) 5.13036 0.207044
\(615\) −4.26875 −0.172133
\(616\) −14.6310 −0.589501
\(617\) −27.9933 −1.12697 −0.563484 0.826127i \(-0.690539\pi\)
−0.563484 + 0.826127i \(0.690539\pi\)
\(618\) 8.92154 0.358877
\(619\) 23.0084 0.924784 0.462392 0.886676i \(-0.346991\pi\)
0.462392 + 0.886676i \(0.346991\pi\)
\(620\) −3.75302 −0.150725
\(621\) −1.30798 −0.0524874
\(622\) −33.1987 −1.33114
\(623\) 45.9133 1.83948
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −0.0814412 −0.00325505
\(627\) 26.6233 1.06323
\(628\) −18.6015 −0.742280
\(629\) 46.3685 1.84883
\(630\) 3.44504 0.137254
\(631\) −6.18731 −0.246313 −0.123156 0.992387i \(-0.539302\pi\)
−0.123156 + 0.992387i \(0.539302\pi\)
\(632\) 1.33513 0.0531084
\(633\) 0.396125 0.0157445
\(634\) 19.4969 0.774323
\(635\) 2.54288 0.100911
\(636\) −8.93900 −0.354454
\(637\) 0 0
\(638\) 38.8461 1.53793
\(639\) 0.374354 0.0148092
\(640\) 1.00000 0.0395285
\(641\) 39.1159 1.54498 0.772492 0.635024i \(-0.219010\pi\)
0.772492 + 0.635024i \(0.219010\pi\)
\(642\) −14.6746 −0.579159
\(643\) 8.78209 0.346332 0.173166 0.984893i \(-0.444600\pi\)
0.173166 + 0.984893i \(0.444600\pi\)
\(644\) −4.50604 −0.177563
\(645\) −3.07069 −0.120908
\(646\) −42.5639 −1.67465
\(647\) −30.1333 −1.18466 −0.592332 0.805694i \(-0.701793\pi\)
−0.592332 + 0.805694i \(0.701793\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −43.8829 −1.72255
\(650\) 0 0
\(651\) 12.9293 0.506740
\(652\) 18.8944 0.739962
\(653\) −27.0858 −1.05995 −0.529974 0.848014i \(-0.677798\pi\)
−0.529974 + 0.848014i \(0.677798\pi\)
\(654\) −9.66786 −0.378043
\(655\) 10.8509 0.423978
\(656\) 4.26875 0.166667
\(657\) −16.7114 −0.651974
\(658\) −26.7614 −1.04327
\(659\) 5.69681 0.221916 0.110958 0.993825i \(-0.464608\pi\)
0.110958 + 0.993825i \(0.464608\pi\)
\(660\) −4.24698 −0.165313
\(661\) −17.1008 −0.665145 −0.332572 0.943078i \(-0.607917\pi\)
−0.332572 + 0.943078i \(0.607917\pi\)
\(662\) −23.8049 −0.925205
\(663\) 0 0
\(664\) 0.740939 0.0287540
\(665\) −21.5961 −0.837461
\(666\) −6.82908 −0.264622
\(667\) 11.9638 0.463238
\(668\) −4.24698 −0.164321
\(669\) 1.49635 0.0578523
\(670\) −0.0760644 −0.00293862
\(671\) 10.7584 0.415323
\(672\) −3.44504 −0.132895
\(673\) −43.1957 −1.66507 −0.832535 0.553972i \(-0.813111\pi\)
−0.832535 + 0.553972i \(0.813111\pi\)
\(674\) −21.2064 −0.816841
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −7.78746 −0.299297 −0.149648 0.988739i \(-0.547814\pi\)
−0.149648 + 0.988739i \(0.547814\pi\)
\(678\) 11.3448 0.435695
\(679\) −45.4411 −1.74387
\(680\) 6.78986 0.260379
\(681\) 25.2150 0.966243
\(682\) −15.9390 −0.610336
\(683\) 30.9138 1.18288 0.591441 0.806348i \(-0.298559\pi\)
0.591441 + 0.806348i \(0.298559\pi\)
\(684\) 6.26875 0.239692
\(685\) 18.6407 0.712225
\(686\) 7.34375 0.280386
\(687\) −22.3666 −0.853338
\(688\) 3.07069 0.117069
\(689\) 0 0
\(690\) −1.30798 −0.0497939
\(691\) −6.20237 −0.235949 −0.117975 0.993017i \(-0.537640\pi\)
−0.117975 + 0.993017i \(0.537640\pi\)
\(692\) 1.16852 0.0444206
\(693\) 14.6310 0.555787
\(694\) 16.5773 0.629266
\(695\) −3.53750 −0.134185
\(696\) 9.14675 0.346707
\(697\) 28.9842 1.09785
\(698\) −2.51035 −0.0950182
\(699\) 14.0858 0.532772
\(700\) 3.44504 0.130210
\(701\) 26.3220 0.994167 0.497084 0.867703i \(-0.334404\pi\)
0.497084 + 0.867703i \(0.334404\pi\)
\(702\) 0 0
\(703\) 42.8098 1.61460
\(704\) 4.24698 0.160064
\(705\) −7.76809 −0.292563
\(706\) 20.8412 0.784368
\(707\) −51.8211 −1.94893
\(708\) −10.3327 −0.388328
\(709\) 33.7429 1.26724 0.633620 0.773645i \(-0.281568\pi\)
0.633620 + 0.773645i \(0.281568\pi\)
\(710\) 0.374354 0.0140493
\(711\) −1.33513 −0.0500711
\(712\) −13.3274 −0.499464
\(713\) −4.90887 −0.183839
\(714\) −23.3913 −0.875399
\(715\) 0 0
\(716\) −10.1836 −0.380579
\(717\) 5.38106 0.200959
\(718\) −2.42566 −0.0905250
\(719\) −9.50173 −0.354355 −0.177177 0.984179i \(-0.556697\pi\)
−0.177177 + 0.984179i \(0.556697\pi\)
\(720\) −1.00000 −0.0372678
\(721\) −30.7351 −1.14463
\(722\) −20.2972 −0.755384
\(723\) −29.9420 −1.11355
\(724\) 10.3351 0.384102
\(725\) −9.14675 −0.339702
\(726\) −7.03684 −0.261161
\(727\) −12.2631 −0.454814 −0.227407 0.973800i \(-0.573025\pi\)
−0.227407 + 0.973800i \(0.573025\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −16.7114 −0.618517
\(731\) 20.8495 0.771148
\(732\) 2.53319 0.0936293
\(733\) 1.52217 0.0562227 0.0281113 0.999605i \(-0.491051\pi\)
0.0281113 + 0.999605i \(0.491051\pi\)
\(734\) −2.66248 −0.0982740
\(735\) −4.86831 −0.179570
\(736\) 1.30798 0.0482127
\(737\) −0.323044 −0.0118995
\(738\) −4.26875 −0.157135
\(739\) 51.1075 1.88002 0.940010 0.341146i \(-0.110815\pi\)
0.940010 + 0.341146i \(0.110815\pi\)
\(740\) −6.82908 −0.251042
\(741\) 0 0
\(742\) 30.7952 1.13053
\(743\) −9.08144 −0.333166 −0.166583 0.986027i \(-0.553273\pi\)
−0.166583 + 0.986027i \(0.553273\pi\)
\(744\) −3.75302 −0.137592
\(745\) 16.6843 0.611264
\(746\) −5.87800 −0.215209
\(747\) −0.740939 −0.0271095
\(748\) 28.8364 1.05436
\(749\) 50.5545 1.84722
\(750\) 1.00000 0.0365148
\(751\) 31.7090 1.15708 0.578539 0.815655i \(-0.303623\pi\)
0.578539 + 0.815655i \(0.303623\pi\)
\(752\) 7.76809 0.283273
\(753\) −8.73556 −0.318342
\(754\) 0 0
\(755\) −18.3327 −0.667196
\(756\) 3.44504 0.125295
\(757\) −8.49934 −0.308914 −0.154457 0.988000i \(-0.549363\pi\)
−0.154457 + 0.988000i \(0.549363\pi\)
\(758\) −36.1540 −1.31317
\(759\) −5.55496 −0.201632
\(760\) 6.26875 0.227391
\(761\) −16.4198 −0.595218 −0.297609 0.954688i \(-0.596189\pi\)
−0.297609 + 0.954688i \(0.596189\pi\)
\(762\) 2.54288 0.0921187
\(763\) 33.3062 1.20576
\(764\) −9.43296 −0.341273
\(765\) −6.78986 −0.245488
\(766\) 8.98361 0.324591
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −27.1011 −0.977290 −0.488645 0.872483i \(-0.662509\pi\)
−0.488645 + 0.872483i \(0.662509\pi\)
\(770\) 14.6310 0.527265
\(771\) 21.2717 0.766083
\(772\) 22.9191 0.824878
\(773\) 12.6297 0.454259 0.227129 0.973865i \(-0.427066\pi\)
0.227129 + 0.973865i \(0.427066\pi\)
\(774\) −3.07069 −0.110374
\(775\) 3.75302 0.134812
\(776\) 13.1903 0.473504
\(777\) 23.5265 0.844008
\(778\) −0.655186 −0.0234895
\(779\) 26.7597 0.958767
\(780\) 0 0
\(781\) 1.58987 0.0568902
\(782\) 8.88099 0.317583
\(783\) −9.14675 −0.326878
\(784\) 4.86831 0.173868
\(785\) 18.6015 0.663915
\(786\) 10.8509 0.387037
\(787\) 3.02954 0.107991 0.0539957 0.998541i \(-0.482804\pi\)
0.0539957 + 0.998541i \(0.482804\pi\)
\(788\) −14.3502 −0.511204
\(789\) −22.2325 −0.791498
\(790\) −1.33513 −0.0475016
\(791\) −39.0834 −1.38964
\(792\) −4.24698 −0.150910
\(793\) 0 0
\(794\) −0.271143 −0.00962250
\(795\) 8.93900 0.317034
\(796\) 14.2524 0.505161
\(797\) 40.2218 1.42473 0.712364 0.701810i \(-0.247625\pi\)
0.712364 + 0.701810i \(0.247625\pi\)
\(798\) −21.5961 −0.764494
\(799\) 52.7442 1.86596
\(800\) −1.00000 −0.0353553
\(801\) 13.3274 0.470899
\(802\) −0.344814 −0.0121758
\(803\) −70.9730 −2.50458
\(804\) −0.0760644 −0.00268259
\(805\) 4.50604 0.158817
\(806\) 0 0
\(807\) −10.2241 −0.359907
\(808\) 15.0422 0.529183
\(809\) 33.7918 1.18806 0.594028 0.804445i \(-0.297537\pi\)
0.594028 + 0.804445i \(0.297537\pi\)
\(810\) 1.00000 0.0351364
\(811\) 33.6534 1.18173 0.590865 0.806770i \(-0.298787\pi\)
0.590865 + 0.806770i \(0.298787\pi\)
\(812\) −31.5109 −1.10582
\(813\) 0.716185 0.0251177
\(814\) −29.0030 −1.01655
\(815\) −18.8944 −0.661842
\(816\) 6.78986 0.237693
\(817\) 19.2494 0.673450
\(818\) −27.0388 −0.945388
\(819\) 0 0
\(820\) −4.26875 −0.149071
\(821\) 12.6939 0.443022 0.221511 0.975158i \(-0.428901\pi\)
0.221511 + 0.975158i \(0.428901\pi\)
\(822\) 18.6407 0.650169
\(823\) −8.79523 −0.306583 −0.153291 0.988181i \(-0.548987\pi\)
−0.153291 + 0.988181i \(0.548987\pi\)
\(824\) 8.92154 0.310797
\(825\) 4.24698 0.147861
\(826\) 35.5967 1.23857
\(827\) 32.1758 1.11886 0.559431 0.828877i \(-0.311020\pi\)
0.559431 + 0.828877i \(0.311020\pi\)
\(828\) −1.30798 −0.0454554
\(829\) 21.1812 0.735653 0.367827 0.929894i \(-0.380102\pi\)
0.367827 + 0.929894i \(0.380102\pi\)
\(830\) −0.740939 −0.0257184
\(831\) −22.8713 −0.793397
\(832\) 0 0
\(833\) 33.0551 1.14529
\(834\) −3.53750 −0.122494
\(835\) 4.24698 0.146973
\(836\) 26.6233 0.920784
\(837\) 3.75302 0.129723
\(838\) −5.86533 −0.202614
\(839\) 26.5931 0.918097 0.459048 0.888411i \(-0.348190\pi\)
0.459048 + 0.888411i \(0.348190\pi\)
\(840\) 3.44504 0.118865
\(841\) 54.6631 1.88493
\(842\) 28.9476 0.997601
\(843\) 4.13036 0.142257
\(844\) 0.396125 0.0136352
\(845\) 0 0
\(846\) −7.76809 −0.267072
\(847\) 24.2422 0.832972
\(848\) −8.93900 −0.306967
\(849\) 16.2959 0.559274
\(850\) −6.78986 −0.232890
\(851\) −8.93230 −0.306195
\(852\) 0.374354 0.0128252
\(853\) −32.2457 −1.10407 −0.552035 0.833821i \(-0.686149\pi\)
−0.552035 + 0.833821i \(0.686149\pi\)
\(854\) −8.72694 −0.298630
\(855\) −6.26875 −0.214387
\(856\) −14.6746 −0.501566
\(857\) −43.0176 −1.46945 −0.734726 0.678364i \(-0.762689\pi\)
−0.734726 + 0.678364i \(0.762689\pi\)
\(858\) 0 0
\(859\) −31.1997 −1.06452 −0.532260 0.846581i \(-0.678657\pi\)
−0.532260 + 0.846581i \(0.678657\pi\)
\(860\) −3.07069 −0.104710
\(861\) 14.7060 0.501180
\(862\) −4.87023 −0.165881
\(863\) −34.5424 −1.17584 −0.587919 0.808920i \(-0.700052\pi\)
−0.587919 + 0.808920i \(0.700052\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −1.16852 −0.0397309
\(866\) 36.9788 1.25659
\(867\) 29.1021 0.988361
\(868\) 12.9293 0.438849
\(869\) −5.67025 −0.192350
\(870\) −9.14675 −0.310104
\(871\) 0 0
\(872\) −9.66786 −0.327395
\(873\) −13.1903 −0.446424
\(874\) 8.19939 0.277349
\(875\) −3.44504 −0.116464
\(876\) −16.7114 −0.564626
\(877\) 2.12093 0.0716188 0.0358094 0.999359i \(-0.488599\pi\)
0.0358094 + 0.999359i \(0.488599\pi\)
\(878\) −22.8799 −0.772160
\(879\) 7.72587 0.260587
\(880\) −4.24698 −0.143166
\(881\) 27.3489 0.921407 0.460703 0.887554i \(-0.347597\pi\)
0.460703 + 0.887554i \(0.347597\pi\)
\(882\) −4.86831 −0.163925
\(883\) −48.2959 −1.62529 −0.812643 0.582762i \(-0.801972\pi\)
−0.812643 + 0.582762i \(0.801972\pi\)
\(884\) 0 0
\(885\) 10.3327 0.347331
\(886\) 10.5942 0.355919
\(887\) 25.1943 0.845943 0.422972 0.906143i \(-0.360987\pi\)
0.422972 + 0.906143i \(0.360987\pi\)
\(888\) −6.82908 −0.229169
\(889\) −8.76032 −0.293812
\(890\) 13.3274 0.446734
\(891\) 4.24698 0.142279
\(892\) 1.49635 0.0501016
\(893\) 48.6962 1.62956
\(894\) 16.6843 0.558005
\(895\) 10.1836 0.340400
\(896\) −3.44504 −0.115091
\(897\) 0 0
\(898\) −19.6485 −0.655678
\(899\) −34.3279 −1.14490
\(900\) 1.00000 0.0333333
\(901\) −60.6945 −2.02203
\(902\) −18.1293 −0.603639
\(903\) 10.5786 0.352035
\(904\) 11.3448 0.377323
\(905\) −10.3351 −0.343551
\(906\) −18.3327 −0.609064
\(907\) −20.0949 −0.667239 −0.333619 0.942708i \(-0.608270\pi\)
−0.333619 + 0.942708i \(0.608270\pi\)
\(908\) 25.2150 0.836791
\(909\) −15.0422 −0.498919
\(910\) 0 0
\(911\) −7.44563 −0.246685 −0.123342 0.992364i \(-0.539361\pi\)
−0.123342 + 0.992364i \(0.539361\pi\)
\(912\) 6.26875 0.207579
\(913\) −3.14675 −0.104142
\(914\) 18.4239 0.609407
\(915\) −2.53319 −0.0837446
\(916\) −22.3666 −0.739013
\(917\) −37.3817 −1.23445
\(918\) −6.78986 −0.224099
\(919\) −14.2765 −0.470939 −0.235469 0.971882i \(-0.575663\pi\)
−0.235469 + 0.971882i \(0.575663\pi\)
\(920\) −1.30798 −0.0431228
\(921\) −5.13036 −0.169051
\(922\) 22.2446 0.732586
\(923\) 0 0
\(924\) 14.6310 0.481325
\(925\) 6.82908 0.224539
\(926\) 8.68532 0.285417
\(927\) −8.92154 −0.293022
\(928\) 9.14675 0.300257
\(929\) 30.9148 1.01428 0.507141 0.861863i \(-0.330702\pi\)
0.507141 + 0.861863i \(0.330702\pi\)
\(930\) 3.75302 0.123066
\(931\) 30.5182 1.00019
\(932\) 14.0858 0.461394
\(933\) 33.1987 1.08688
\(934\) −19.6136 −0.641775
\(935\) −28.8364 −0.943050
\(936\) 0 0
\(937\) −16.7187 −0.546176 −0.273088 0.961989i \(-0.588045\pi\)
−0.273088 + 0.961989i \(0.588045\pi\)
\(938\) 0.262045 0.00855608
\(939\) 0.0814412 0.00265773
\(940\) −7.76809 −0.253367
\(941\) 36.9033 1.20301 0.601507 0.798867i \(-0.294567\pi\)
0.601507 + 0.798867i \(0.294567\pi\)
\(942\) 18.6015 0.606069
\(943\) −5.58343 −0.181822
\(944\) −10.3327 −0.336302
\(945\) −3.44504 −0.112067
\(946\) −13.0411 −0.424004
\(947\) −39.1661 −1.27273 −0.636364 0.771389i \(-0.719562\pi\)
−0.636364 + 0.771389i \(0.719562\pi\)
\(948\) −1.33513 −0.0433629
\(949\) 0 0
\(950\) −6.26875 −0.203385
\(951\) −19.4969 −0.632232
\(952\) −23.3913 −0.758118
\(953\) −19.6200 −0.635554 −0.317777 0.948165i \(-0.602936\pi\)
−0.317777 + 0.948165i \(0.602936\pi\)
\(954\) 8.93900 0.289411
\(955\) 9.43296 0.305243
\(956\) 5.38106 0.174036
\(957\) −38.8461 −1.25572
\(958\) −31.4698 −1.01674
\(959\) −64.2180 −2.07371
\(960\) −1.00000 −0.0322749
\(961\) −16.9148 −0.545640
\(962\) 0 0
\(963\) 14.6746 0.472881
\(964\) −29.9420 −0.964366
\(965\) −22.9191 −0.737794
\(966\) 4.50604 0.144979
\(967\) −43.1564 −1.38782 −0.693909 0.720063i \(-0.744113\pi\)
−0.693909 + 0.720063i \(0.744113\pi\)
\(968\) −7.03684 −0.226172
\(969\) 42.5639 1.36735
\(970\) −13.1903 −0.423515
\(971\) 39.5032 1.26772 0.633859 0.773449i \(-0.281470\pi\)
0.633859 + 0.773449i \(0.281470\pi\)
\(972\) 1.00000 0.0320750
\(973\) 12.1868 0.390692
\(974\) −13.2644 −0.425020
\(975\) 0 0
\(976\) 2.53319 0.0810854
\(977\) −3.84117 −0.122890 −0.0614449 0.998110i \(-0.519571\pi\)
−0.0614449 + 0.998110i \(0.519571\pi\)
\(978\) −18.8944 −0.604176
\(979\) 56.6010 1.80898
\(980\) −4.86831 −0.155513
\(981\) 9.66786 0.308671
\(982\) −11.7259 −0.374188
\(983\) −32.4300 −1.03436 −0.517178 0.855878i \(-0.673017\pi\)
−0.517178 + 0.855878i \(0.673017\pi\)
\(984\) −4.26875 −0.136083
\(985\) 14.3502 0.457235
\(986\) 62.1051 1.97783
\(987\) 26.7614 0.851824
\(988\) 0 0
\(989\) −4.01639 −0.127714
\(990\) 4.24698 0.134978
\(991\) −61.9807 −1.96888 −0.984442 0.175712i \(-0.943777\pi\)
−0.984442 + 0.175712i \(0.943777\pi\)
\(992\) −3.75302 −0.119159
\(993\) 23.8049 0.755426
\(994\) −1.28967 −0.0409057
\(995\) −14.2524 −0.451830
\(996\) −0.740939 −0.0234775
\(997\) 28.9162 0.915784 0.457892 0.889008i \(-0.348605\pi\)
0.457892 + 0.889008i \(0.348605\pi\)
\(998\) 26.7633 0.847177
\(999\) 6.82908 0.216063
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 5070.2.a.bo.1.2 3
13.5 odd 4 5070.2.b.w.1351.5 6
13.8 odd 4 5070.2.b.w.1351.2 6
13.12 even 2 5070.2.a.bx.1.2 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bo.1.2 3 1.1 even 1 trivial
5070.2.a.bx.1.2 yes 3 13.12 even 2
5070.2.b.w.1351.2 6 13.8 odd 4
5070.2.b.w.1351.5 6 13.5 odd 4