Properties

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

Related objects

Downloads

Learn more

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: \(1\)
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 \(-0.725672\pi\)
−0.651052 + 0.759033i \(0.725672\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) −4.24698 −1.28051 −0.640256 0.768161i \(-0.721172\pi\)
−0.640256 + 0.768161i \(0.721172\pi\)
\(12\) 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 \(-0.755434\pi\)
−0.719075 + 0.694933i \(0.755434\pi\)
\(20\) 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 \(-0.609423\pi\)
−0.337031 + 0.941493i \(0.609423\pi\)
\(32\) 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 \(-0.689717\pi\)
−0.561347 + 0.827580i \(0.689717\pi\)
\(38\) −6.26875 −1.01693
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) −4.26875 −0.666667 −0.333333 0.942809i \(-0.608173\pi\)
−0.333333 + 0.942809i \(0.608173\pi\)
\(42\) −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 \(-0.691721\pi\)
−0.566546 + 0.824030i \(0.691721\pi\)
\(48\) 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 \(-0.265176\pi\)
0.672604 + 0.740003i \(0.265176\pi\)
\(60\) 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 \(-0.498521\pi\)
0.00464637 + 0.999989i \(0.498521\pi\)
\(68\) 6.78986 0.823391
\(69\) −1.30798 −0.157462
\(70\) −3.44504 −0.411761
\(71\) −0.374354 −0.0444277 −0.0222138 0.999753i \(-0.507071\pi\)
−0.0222138 + 0.999753i \(0.507071\pi\)
\(72\) 1.00000 0.117851
\(73\) 16.7114 1.95592 0.977961 0.208789i \(-0.0669523\pi\)
0.977961 + 0.208789i \(0.0669523\pi\)
\(74\) −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 \(-0.487053\pi\)
0.0406643 + 0.999173i \(0.487053\pi\)
\(84\) −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 \(-0.749659\pi\)
−0.706348 + 0.707864i \(0.749659\pi\)
\(90\) 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 \(-0.266450\pi\)
0.669636 + 0.742690i \(0.266450\pi\)
\(98\) 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 \(-0.653229\pi\)
−0.463006 + 0.886355i \(0.653229\pi\)
\(110\) −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 \(-0.206792\pi\)
0.796292 + 0.604913i \(0.206792\pi\)
\(138\) −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 \(-0.260495\pi\)
0.683414 + 0.730031i \(0.260495\pi\)
\(150\) 1.00000 0.0816497
\(151\) −18.3327 −1.49190 −0.745948 0.666004i \(-0.768003\pi\)
−0.745948 + 0.666004i \(0.768003\pi\)
\(152\) −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 \(-0.765156\pi\)
−0.739962 + 0.672649i \(0.765156\pi\)
\(164\) −4.26875 −0.333333
\(165\) −4.24698 −0.330627
\(166\) 0.740939 0.0575080
\(167\) 4.24698 0.328641 0.164321 0.986407i \(-0.447457\pi\)
0.164321 + 0.986407i \(0.447457\pi\)
\(168\) −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 \(-0.808756\pi\)
−0.824878 + 0.565310i \(0.808756\pi\)
\(194\) 13.1903 0.947008
\(195\) 0 0
\(196\) 4.86831 0.347737
\(197\) 14.3502 1.02241 0.511204 0.859459i \(-0.329200\pi\)
0.511204 + 0.859459i \(0.329200\pi\)
\(198\) −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 \(-0.515955\pi\)
−0.0501016 + 0.998744i \(0.515955\pi\)
\(224\) −3.44504 −0.230182
\(225\) 1.00000 0.0666667
\(226\) −11.3448 −0.754646
\(227\) −25.2150 −1.67358 −0.836791 0.547523i \(-0.815571\pi\)
−0.836791 + 0.547523i \(0.815571\pi\)
\(228\) −6.26875 −0.415158
\(229\) 22.3666 1.47803 0.739013 0.673691i \(-0.235292\pi\)
0.739013 + 0.673691i \(0.235292\pi\)
\(230\) −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 \(-0.555681\pi\)
−0.174036 + 0.984739i \(0.555681\pi\)
\(240\) 1.00000 0.0645497
\(241\) 29.9420 1.92873 0.964366 0.264570i \(-0.0852301\pi\)
0.964366 + 0.264570i \(0.0852301\pi\)
\(242\) 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 \(-0.506925\pi\)
−0.0217526 + 0.999763i \(0.506925\pi\)
\(272\) 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 \(-0.539315\pi\)
−0.123198 + 0.992382i \(0.539315\pi\)
\(282\) −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 \(-0.572459\pi\)
−0.225675 + 0.974203i \(0.572459\pi\)
\(294\) 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 \(-0.453231\pi\)
0.146403 + 0.989225i \(0.453231\pi\)
\(308\) 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 \(-0.315569\pi\)
0.547529 + 0.836787i \(0.315569\pi\)
\(318\) −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 \(-0.727002\pi\)
−0.654218 + 0.756306i \(0.727002\pi\)
\(332\) 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 \(-0.521403\pi\)
−0.0671880 + 0.997740i \(0.521403\pi\)
\(350\) −3.44504 −0.184145
\(351\) 0 0
\(352\) −4.24698 −0.226365
\(353\) 20.8412 1.10926 0.554632 0.832096i \(-0.312859\pi\)
0.554632 + 0.832096i \(0.312859\pi\)
\(354\) 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 \(-0.520389\pi\)
−0.0640108 + 0.997949i \(0.520389\pi\)
\(360\) 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 \(-0.878948\pi\)
−0.928554 + 0.371196i \(0.878948\pi\)
\(380\) −6.26875 −0.321580
\(381\) −2.54288 −0.130276
\(382\) −9.43296 −0.482632
\(383\) 8.98361 0.459041 0.229520 0.973304i \(-0.426284\pi\)
0.229520 + 0.973304i \(0.426284\pi\)
\(384\) 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 \(-0.502166\pi\)
−0.00680413 + 0.999977i \(0.502166\pi\)
\(398\) 14.2524 0.714406
\(399\) 21.5961 1.08116
\(400\) 1.00000 0.0500000
\(401\) −0.344814 −0.0172192 −0.00860960 0.999963i \(-0.502741\pi\)
−0.00860960 + 0.999963i \(0.502741\pi\)
\(402\) 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 \(-0.733059\pi\)
−0.668490 + 0.743721i \(0.733059\pi\)
\(410\) −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 \(-0.250763\pi\)
0.705410 + 0.708799i \(0.250763\pi\)
\(422\) 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 \(-0.537422\pi\)
−0.117295 + 0.993097i \(0.537422\pi\)
\(432\) 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 \(-0.653455\pi\)
−0.463635 + 0.886027i \(0.653455\pi\)
\(450\) 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 \(-0.358191\pi\)
0.430916 + 0.902392i \(0.358191\pi\)
\(458\) 22.3666 1.04512
\(459\) 6.78986 0.316923
\(460\) −1.30798 −0.0609848
\(461\) 22.2446 1.03603 0.518017 0.855370i \(-0.326670\pi\)
0.518017 + 0.855370i \(0.326670\pi\)
\(462\) 14.6310 0.680697
\(463\) 8.68532 0.403641 0.201820 0.979423i \(-0.435314\pi\)
0.201820 + 0.979423i \(0.435314\pi\)
\(464\) −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 \(-0.755375\pi\)
−0.718946 + 0.695066i \(0.755375\pi\)
\(480\) 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 \(-0.597165\pi\)
−0.300535 + 0.953771i \(0.597165\pi\)
\(488\) 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 \(-0.295547\pi\)
0.599045 + 0.800715i \(0.295547\pi\)
\(500\) 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 \(-0.284444\pi\)
0.626604 + 0.779338i \(0.284444\pi\)
\(510\) 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 \(-0.501683\pi\)
−0.00528635 + 0.999986i \(0.501683\pi\)
\(542\) −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 \(-0.596027\pi\)
−0.297124 + 0.954839i \(0.596027\pi\)
\(558\) −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 \(-0.698054\pi\)
−0.582828 + 0.812596i \(0.698054\pi\)
\(578\) 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 \(-0.0571786\pi\)
0.983910 + 0.178668i \(0.0571786\pi\)
\(588\) 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 \(-0.388119\pi\)
0.344291 + 0.938863i \(0.388119\pi\)
\(594\) −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 \(-0.700057\pi\)
−0.587929 + 0.808912i \(0.700057\pi\)
\(614\) 5.13036 0.207044
\(615\) −4.26875 −0.172133
\(616\) 14.6310 0.589501
\(617\) 27.9933 1.12697 0.563484 0.826127i \(-0.309461\pi\)
0.563484 + 0.826127i \(0.309461\pi\)
\(618\) −8.92154 −0.358877
\(619\) −23.0084 −0.924784 −0.462392 0.886676i \(-0.653009\pi\)
−0.462392 + 0.886676i \(0.653009\pi\)
\(620\) −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 \(-0.460698\pi\)
0.123156 + 0.992387i \(0.460698\pi\)
\(632\) −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 \(-0.555400\pi\)
−0.173166 + 0.984893i \(0.555400\pi\)
\(644\) 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 \(-0.392083\pi\)
0.332572 + 0.943078i \(0.392083\pi\)
\(662\) −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 \(-0.701441\pi\)
−0.591441 + 0.806348i \(0.701441\pi\)
\(684\) −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 \(-0.462360\pi\)
0.117975 + 0.993017i \(0.462360\pi\)
\(692\) 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 \(-0.718432\pi\)
−0.633620 + 0.773645i \(0.718432\pi\)
\(710\) −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 \(-0.508949\pi\)
−0.0281113 + 0.999605i \(0.508949\pi\)
\(734\) 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 \(-0.889185\pi\)
−0.940010 + 0.341146i \(0.889185\pi\)
\(740\) −6.82908 −0.251042
\(741\) 0 0
\(742\) 30.7952 1.13053
\(743\) 9.08144 0.333166 0.166583 0.986027i \(-0.446727\pi\)
0.166583 + 0.986027i \(0.446727\pi\)
\(744\) −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 \(-0.403811\pi\)
0.297609 + 0.954688i \(0.403811\pi\)
\(762\) −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 \(-0.337491\pi\)
0.488645 + 0.872483i \(0.337491\pi\)
\(770\) 14.6310 0.527265
\(771\) 21.2717 0.766083
\(772\) −22.9191 −0.824878
\(773\) −12.6297 −0.454259 −0.227129 0.973865i \(-0.572934\pi\)
−0.227129 + 0.973865i \(0.572934\pi\)
\(774\) 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 \(-0.517196\pi\)
−0.0539957 + 0.998541i \(0.517196\pi\)
\(788\) 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 \(-0.701213\pi\)
−0.590865 + 0.806770i \(0.701213\pi\)
\(812\) 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 \(-0.571099\pi\)
−0.221511 + 0.975158i \(0.571099\pi\)
\(822\) 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 \(-0.688980\pi\)
−0.559431 + 0.828877i \(0.688980\pi\)
\(828\) −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 \(-0.651810\pi\)
−0.459048 + 0.888411i \(0.651810\pi\)
\(840\) −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 \(-0.313851\pi\)
0.552035 + 0.833821i \(0.313851\pi\)
\(854\) −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 \(-0.299948\pi\)
0.587919 + 0.808920i \(0.299948\pi\)
\(864\) 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 \(-0.511401\pi\)
−0.0358094 + 0.999359i \(0.511401\pi\)
\(878\) 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 \(-0.669298\pi\)
−0.507141 + 0.861863i \(0.669298\pi\)
\(930\) −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 \(-0.705433\pi\)
−0.601507 + 0.798867i \(0.705433\pi\)
\(942\) −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 \(-0.280438\pi\)
0.636364 + 0.771389i \(0.280438\pi\)
\(948\) −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 \(-0.255887\pi\)
0.693909 + 0.720063i \(0.255887\pi\)
\(968\) 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 \(-0.480429\pi\)
0.0614449 + 0.998110i \(0.480429\pi\)
\(978\) −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 \(-0.326983\pi\)
0.517178 + 0.855878i \(0.326983\pi\)
\(984\) −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.bx.1.2 yes 3
13.5 odd 4 5070.2.b.w.1351.2 6
13.8 odd 4 5070.2.b.w.1351.5 6
13.12 even 2 5070.2.a.bo.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bo.1.2 3 13.12 even 2
5070.2.a.bx.1.2 yes 3 1.1 even 1 trivial
5070.2.b.w.1351.2 6 13.5 odd 4
5070.2.b.w.1351.5 6 13.8 odd 4