Properties

Label 5070.2.b.w.1351.2
Level $5070$
Weight $2$
Character 5070.1351
Analytic conductor $40.484$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5070 = 2 \cdot 3 \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5070.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(40.4841538248\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
Defining polynomial: \(x^{6} + 5 x^{4} + 6 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.2
Root \(-0.445042i\) of defining polynomial
Character \(\chi\) \(=\) 5070.1351
Dual form 5070.2.b.w.1351.5

$q$-expansion

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5070\mathbb{Z}\right)^\times\).

\(n\) \(1691\) \(1861\) \(4057\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.00000i − 0.707107i
\(3\) 1.00000 0.577350
\(4\) −1.00000 −0.500000
\(5\) − 1.00000i − 0.447214i
\(6\) − 1.00000i − 0.408248i
\(7\) − 3.44504i − 1.30210i −0.759033 0.651052i \(-0.774328\pi\)
0.759033 0.651052i \(-0.225672\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) − 4.24698i − 1.28051i −0.768161 0.640256i \(-0.778828\pi\)
0.768161 0.640256i \(-0.221172\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −3.44504 −0.920726
\(15\) − 1.00000i − 0.258199i
\(16\) 1.00000 0.250000
\(17\) −6.78986 −1.64678 −0.823391 0.567474i \(-0.807921\pi\)
−0.823391 + 0.567474i \(0.807921\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) 6.26875i 1.43815i 0.694933 + 0.719075i \(0.255434\pi\)
−0.694933 + 0.719075i \(0.744566\pi\)
\(20\) 1.00000i 0.223607i
\(21\) − 3.44504i − 0.751770i
\(22\) −4.24698 −0.905459
\(23\) 1.30798 0.272732 0.136366 0.990658i \(-0.456458\pi\)
0.136366 + 0.990658i \(0.456458\pi\)
\(24\) 1.00000i 0.204124i
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 3.44504i 0.651052i
\(29\) −9.14675 −1.69851 −0.849255 0.527984i \(-0.822948\pi\)
−0.849255 + 0.527984i \(0.822948\pi\)
\(30\) −1.00000 −0.182574
\(31\) 3.75302i 0.674062i 0.941493 + 0.337031i \(0.109423\pi\)
−0.941493 + 0.337031i \(0.890577\pi\)
\(32\) − 1.00000i − 0.176777i
\(33\) − 4.24698i − 0.739304i
\(34\) 6.78986i 1.16445i
\(35\) −3.44504 −0.582318
\(36\) −1.00000 −0.166667
\(37\) − 6.82908i − 1.12269i −0.827580 0.561347i \(-0.810283\pi\)
0.827580 0.561347i \(-0.189717\pi\)
\(38\) 6.26875 1.01693
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 4.26875i 0.666667i 0.942809 + 0.333333i \(0.108173\pi\)
−0.942809 + 0.333333i \(0.891827\pi\)
\(42\) −3.44504 −0.531582
\(43\) −3.07069 −0.468275 −0.234138 0.972203i \(-0.575227\pi\)
−0.234138 + 0.972203i \(0.575227\pi\)
\(44\) 4.24698i 0.640256i
\(45\) − 1.00000i − 0.149071i
\(46\) − 1.30798i − 0.192851i
\(47\) − 7.76809i − 1.13309i −0.824030 0.566546i \(-0.808279\pi\)
0.824030 0.566546i \(-0.191721\pi\)
\(48\) 1.00000 0.144338
\(49\) −4.86831 −0.695473
\(50\) 1.00000i 0.141421i
\(51\) −6.78986 −0.950770
\(52\) 0 0
\(53\) −8.93900 −1.22787 −0.613933 0.789358i \(-0.710414\pi\)
−0.613933 + 0.789358i \(0.710414\pi\)
\(54\) − 1.00000i − 0.136083i
\(55\) −4.24698 −0.572663
\(56\) 3.44504 0.460363
\(57\) 6.26875i 0.830316i
\(58\) 9.14675i 1.20103i
\(59\) 10.3327i 1.34521i 0.740003 + 0.672604i \(0.234824\pi\)
−0.740003 + 0.672604i \(0.765176\pi\)
\(60\) 1.00000i 0.129099i
\(61\) 2.53319 0.324341 0.162171 0.986763i \(-0.448150\pi\)
0.162171 + 0.986763i \(0.448150\pi\)
\(62\) 3.75302 0.476634
\(63\) − 3.44504i − 0.434034i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −4.24698 −0.522767
\(67\) − 0.0760644i − 0.00929275i −0.999989 0.00464637i \(-0.998521\pi\)
0.999989 0.00464637i \(-0.00147899\pi\)
\(68\) 6.78986 0.823391
\(69\) 1.30798 0.157462
\(70\) 3.44504i 0.411761i
\(71\) 0.374354i 0.0444277i 0.999753 + 0.0222138i \(0.00707147\pi\)
−0.999753 + 0.0222138i \(0.992929\pi\)
\(72\) 1.00000i 0.117851i
\(73\) 16.7114i 1.95592i 0.208789 + 0.977961i \(0.433048\pi\)
−0.208789 + 0.977961i \(0.566952\pi\)
\(74\) −6.82908 −0.793865
\(75\) −1.00000 −0.115470
\(76\) − 6.26875i − 0.719075i
\(77\) −14.6310 −1.66736
\(78\) 0 0
\(79\) −1.33513 −0.150213 −0.0751067 0.997176i \(-0.523930\pi\)
−0.0751067 + 0.997176i \(0.523930\pi\)
\(80\) − 1.00000i − 0.111803i
\(81\) 1.00000 0.111111
\(82\) 4.26875 0.471405
\(83\) − 0.740939i − 0.0813286i −0.999173 0.0406643i \(-0.987053\pi\)
0.999173 0.0406643i \(-0.0129474\pi\)
\(84\) 3.44504i 0.375885i
\(85\) 6.78986i 0.736463i
\(86\) 3.07069i 0.331121i
\(87\) −9.14675 −0.980635
\(88\) 4.24698 0.452730
\(89\) − 13.3274i − 1.41270i −0.707864 0.706348i \(-0.750341\pi\)
0.707864 0.706348i \(-0.249659\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) −1.30798 −0.136366
\(93\) 3.75302i 0.389170i
\(94\) −7.76809 −0.801217
\(95\) 6.26875 0.643160
\(96\) − 1.00000i − 0.102062i
\(97\) − 13.1903i − 1.33927i −0.742690 0.669636i \(-0.766450\pi\)
0.742690 0.669636i \(-0.233550\pi\)
\(98\) 4.86831i 0.491774i
\(99\) − 4.24698i − 0.426838i
\(100\) 1.00000 0.100000
\(101\) 15.0422 1.49676 0.748378 0.663272i \(-0.230833\pi\)
0.748378 + 0.663272i \(0.230833\pi\)
\(102\) 6.78986i 0.672296i
\(103\) 8.92154 0.879066 0.439533 0.898227i \(-0.355144\pi\)
0.439533 + 0.898227i \(0.355144\pi\)
\(104\) 0 0
\(105\) −3.44504 −0.336202
\(106\) 8.93900i 0.868233i
\(107\) 14.6746 1.41864 0.709322 0.704885i \(-0.249001\pi\)
0.709322 + 0.704885i \(0.249001\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 9.66786i 0.926013i 0.886355 + 0.463006i \(0.153229\pi\)
−0.886355 + 0.463006i \(0.846771\pi\)
\(110\) 4.24698i 0.404934i
\(111\) − 6.82908i − 0.648188i
\(112\) − 3.44504i − 0.325526i
\(113\) −11.3448 −1.06723 −0.533615 0.845727i \(-0.679167\pi\)
−0.533615 + 0.845727i \(0.679167\pi\)
\(114\) 6.26875 0.587122
\(115\) − 1.30798i − 0.121970i
\(116\) 9.14675 0.849255
\(117\) 0 0
\(118\) 10.3327 0.951205
\(119\) 23.3913i 2.14428i
\(120\) 1.00000 0.0912871
\(121\) −7.03684 −0.639712
\(122\) − 2.53319i − 0.229344i
\(123\) 4.26875i 0.384900i
\(124\) − 3.75302i − 0.337031i
\(125\) 1.00000i 0.0894427i
\(126\) −3.44504 −0.306909
\(127\) 2.54288 0.225644 0.112822 0.993615i \(-0.464011\pi\)
0.112822 + 0.993615i \(0.464011\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −3.07069 −0.270359
\(130\) 0 0
\(131\) −10.8509 −0.948044 −0.474022 0.880513i \(-0.657198\pi\)
−0.474022 + 0.880513i \(0.657198\pi\)
\(132\) 4.24698i 0.369652i
\(133\) 21.5961 1.87262
\(134\) −0.0760644 −0.00657096
\(135\) − 1.00000i − 0.0860663i
\(136\) − 6.78986i − 0.582225i
\(137\) 18.6407i 1.59258i 0.604913 + 0.796292i \(0.293208\pi\)
−0.604913 + 0.796292i \(0.706792\pi\)
\(138\) − 1.30798i − 0.111343i
\(139\) 3.53750 0.300047 0.150023 0.988682i \(-0.452065\pi\)
0.150023 + 0.988682i \(0.452065\pi\)
\(140\) 3.44504 0.291159
\(141\) − 7.76809i − 0.654191i
\(142\) 0.374354 0.0314151
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 9.14675i 0.759596i
\(146\) 16.7114 1.38305
\(147\) −4.86831 −0.401532
\(148\) 6.82908i 0.561347i
\(149\) − 16.6843i − 1.36683i −0.730031 0.683414i \(-0.760495\pi\)
0.730031 0.683414i \(-0.239505\pi\)
\(150\) 1.00000i 0.0816497i
\(151\) − 18.3327i − 1.49190i −0.666004 0.745948i \(-0.731997\pi\)
0.666004 0.745948i \(-0.268003\pi\)
\(152\) −6.26875 −0.508463
\(153\) −6.78986 −0.548927
\(154\) 14.6310i 1.17900i
\(155\) 3.75302 0.301450
\(156\) 0 0
\(157\) −18.6015 −1.48456 −0.742280 0.670090i \(-0.766256\pi\)
−0.742280 + 0.670090i \(0.766256\pi\)
\(158\) 1.33513i 0.106217i
\(159\) −8.93900 −0.708909
\(160\) −1.00000 −0.0790569
\(161\) − 4.50604i − 0.355126i
\(162\) − 1.00000i − 0.0785674i
\(163\) − 18.8944i − 1.47992i −0.672649 0.739962i \(-0.734844\pi\)
0.672649 0.739962i \(-0.265156\pi\)
\(164\) − 4.26875i − 0.333333i
\(165\) −4.24698 −0.330627
\(166\) −0.740939 −0.0575080
\(167\) 4.24698i 0.328641i 0.986407 + 0.164321i \(0.0525432\pi\)
−0.986407 + 0.164321i \(0.947457\pi\)
\(168\) 3.44504 0.265791
\(169\) 0 0
\(170\) 6.78986 0.520758
\(171\) 6.26875i 0.479383i
\(172\) 3.07069 0.234138
\(173\) −1.16852 −0.0888411 −0.0444206 0.999013i \(-0.514144\pi\)
−0.0444206 + 0.999013i \(0.514144\pi\)
\(174\) 9.14675i 0.693413i
\(175\) 3.44504i 0.260421i
\(176\) − 4.24698i − 0.320128i
\(177\) 10.3327i 0.776656i
\(178\) −13.3274 −0.998928
\(179\) 10.1836 0.761157 0.380579 0.924749i \(-0.375725\pi\)
0.380579 + 0.924749i \(0.375725\pi\)
\(180\) 1.00000i 0.0745356i
\(181\) −10.3351 −0.768204 −0.384102 0.923291i \(-0.625489\pi\)
−0.384102 + 0.923291i \(0.625489\pi\)
\(182\) 0 0
\(183\) 2.53319 0.187259
\(184\) 1.30798i 0.0964255i
\(185\) −6.82908 −0.502084
\(186\) 3.75302 0.275185
\(187\) 28.8364i 2.10872i
\(188\) 7.76809i 0.566546i
\(189\) − 3.44504i − 0.250590i
\(190\) − 6.26875i − 0.454783i
\(191\) −9.43296 −0.682545 −0.341273 0.939964i \(-0.610858\pi\)
−0.341273 + 0.939964i \(0.610858\pi\)
\(192\) −1.00000 −0.0721688
\(193\) − 22.9191i − 1.64976i −0.565310 0.824878i \(-0.691244\pi\)
0.565310 0.824878i \(-0.308756\pi\)
\(194\) −13.1903 −0.947008
\(195\) 0 0
\(196\) 4.86831 0.347737
\(197\) − 14.3502i − 1.02241i −0.859459 0.511204i \(-0.829200\pi\)
0.859459 0.511204i \(-0.170800\pi\)
\(198\) −4.24698 −0.301820
\(199\) −14.2524 −1.01032 −0.505161 0.863025i \(-0.668567\pi\)
−0.505161 + 0.863025i \(0.668567\pi\)
\(200\) − 1.00000i − 0.0707107i
\(201\) − 0.0760644i − 0.00536517i
\(202\) − 15.0422i − 1.05837i
\(203\) 31.5109i 2.21163i
\(204\) 6.78986 0.475385
\(205\) 4.26875 0.298142
\(206\) − 8.92154i − 0.621593i
\(207\) 1.30798 0.0909108
\(208\) 0 0
\(209\) 26.6233 1.84157
\(210\) 3.44504i 0.237730i
\(211\) 0.396125 0.0272703 0.0136352 0.999907i \(-0.495660\pi\)
0.0136352 + 0.999907i \(0.495660\pi\)
\(212\) 8.93900 0.613933
\(213\) 0.374354i 0.0256503i
\(214\) − 14.6746i − 1.00313i
\(215\) 3.07069i 0.209419i
\(216\) 1.00000i 0.0680414i
\(217\) 12.9293 0.877699
\(218\) 9.66786 0.654790
\(219\) 16.7114i 1.12925i
\(220\) 4.24698 0.286331
\(221\) 0 0
\(222\) −6.82908 −0.458338
\(223\) 1.49635i 0.100203i 0.998744 + 0.0501016i \(0.0159545\pi\)
−0.998744 + 0.0501016i \(0.984045\pi\)
\(224\) −3.44504 −0.230182
\(225\) −1.00000 −0.0666667
\(226\) 11.3448i 0.754646i
\(227\) 25.2150i 1.67358i 0.547523 + 0.836791i \(0.315571\pi\)
−0.547523 + 0.836791i \(0.684429\pi\)
\(228\) − 6.26875i − 0.415158i
\(229\) 22.3666i 1.47803i 0.673691 + 0.739013i \(0.264708\pi\)
−0.673691 + 0.739013i \(0.735292\pi\)
\(230\) −1.30798 −0.0862456
\(231\) −14.6310 −0.962651
\(232\) − 9.14675i − 0.600514i
\(233\) −14.0858 −0.922788 −0.461394 0.887195i \(-0.652651\pi\)
−0.461394 + 0.887195i \(0.652651\pi\)
\(234\) 0 0
\(235\) −7.76809 −0.506734
\(236\) − 10.3327i − 0.672604i
\(237\) −1.33513 −0.0867257
\(238\) 23.3913 1.51624
\(239\) 5.38106i 0.348072i 0.984739 + 0.174036i \(0.0556809\pi\)
−0.984739 + 0.174036i \(0.944319\pi\)
\(240\) − 1.00000i − 0.0645497i
\(241\) 29.9420i 1.92873i 0.264570 + 0.964366i \(0.414770\pi\)
−0.264570 + 0.964366i \(0.585230\pi\)
\(242\) 7.03684i 0.452345i
\(243\) 1.00000 0.0641500
\(244\) −2.53319 −0.162171
\(245\) 4.86831i 0.311025i
\(246\) 4.26875 0.272166
\(247\) 0 0
\(248\) −3.75302 −0.238317
\(249\) − 0.740939i − 0.0469551i
\(250\) 1.00000 0.0632456
\(251\) 8.73556 0.551384 0.275692 0.961246i \(-0.411093\pi\)
0.275692 + 0.961246i \(0.411093\pi\)
\(252\) 3.44504i 0.217017i
\(253\) − 5.55496i − 0.349237i
\(254\) − 2.54288i − 0.159554i
\(255\) 6.78986i 0.425197i
\(256\) 1.00000 0.0625000
\(257\) −21.2717 −1.32689 −0.663447 0.748223i \(-0.730907\pi\)
−0.663447 + 0.748223i \(0.730907\pi\)
\(258\) 3.07069i 0.191173i
\(259\) −23.5265 −1.46186
\(260\) 0 0
\(261\) −9.14675 −0.566170
\(262\) 10.8509i 0.670368i
\(263\) −22.2325 −1.37091 −0.685457 0.728113i \(-0.740398\pi\)
−0.685457 + 0.728113i \(0.740398\pi\)
\(264\) 4.24698 0.261384
\(265\) 8.93900i 0.549118i
\(266\) − 21.5961i − 1.32414i
\(267\) − 13.3274i − 0.815621i
\(268\) 0.0760644i 0.00464637i
\(269\) −10.2241 −0.623377 −0.311689 0.950184i \(-0.600895\pi\)
−0.311689 + 0.950184i \(0.600895\pi\)
\(270\) −1.00000 −0.0608581
\(271\) − 0.716185i − 0.0435051i −0.999763 0.0217526i \(-0.993075\pi\)
0.999763 0.0217526i \(-0.00692460\pi\)
\(272\) −6.78986 −0.411696
\(273\) 0 0
\(274\) 18.6407 1.12613
\(275\) 4.24698i 0.256103i
\(276\) −1.30798 −0.0787311
\(277\) 22.8713 1.37420 0.687102 0.726561i \(-0.258883\pi\)
0.687102 + 0.726561i \(0.258883\pi\)
\(278\) − 3.53750i − 0.212165i
\(279\) 3.75302i 0.224687i
\(280\) − 3.44504i − 0.205881i
\(281\) − 4.13036i − 0.246397i −0.992382 0.123198i \(-0.960685\pi\)
0.992382 0.123198i \(-0.0393151\pi\)
\(282\) −7.76809 −0.462583
\(283\) −16.2959 −0.968691 −0.484345 0.874877i \(-0.660942\pi\)
−0.484345 + 0.874877i \(0.660942\pi\)
\(284\) − 0.374354i − 0.0222138i
\(285\) 6.26875 0.371329
\(286\) 0 0
\(287\) 14.7060 0.868069
\(288\) − 1.00000i − 0.0589256i
\(289\) 29.1021 1.71189
\(290\) 9.14675 0.537116
\(291\) − 13.1903i − 0.773229i
\(292\) − 16.7114i − 0.977961i
\(293\) − 7.72587i − 0.451350i −0.974203 0.225675i \(-0.927541\pi\)
0.974203 0.225675i \(-0.0724588\pi\)
\(294\) 4.86831i 0.283926i
\(295\) 10.3327 0.601595
\(296\) 6.82908 0.396932
\(297\) − 4.24698i − 0.246435i
\(298\) −16.6843 −0.966493
\(299\) 0 0
\(300\) 1.00000 0.0577350
\(301\) 10.5786i 0.609743i
\(302\) −18.3327 −1.05493
\(303\) 15.0422 0.864153
\(304\) 6.26875i 0.359537i
\(305\) − 2.53319i − 0.145050i
\(306\) 6.78986i 0.388150i
\(307\) 5.13036i 0.292805i 0.989225 + 0.146403i \(0.0467695\pi\)
−0.989225 + 0.146403i \(0.953231\pi\)
\(308\) 14.6310 0.833680
\(309\) 8.92154 0.507529
\(310\) − 3.75302i − 0.213157i
\(311\) −33.1987 −1.88252 −0.941261 0.337679i \(-0.890358\pi\)
−0.941261 + 0.337679i \(0.890358\pi\)
\(312\) 0 0
\(313\) 0.0814412 0.00460333 0.00230167 0.999997i \(-0.499267\pi\)
0.00230167 + 0.999997i \(0.499267\pi\)
\(314\) 18.6015i 1.04974i
\(315\) −3.44504 −0.194106
\(316\) 1.33513 0.0751067
\(317\) − 19.4969i − 1.09506i −0.836787 0.547529i \(-0.815569\pi\)
0.836787 0.547529i \(-0.184431\pi\)
\(318\) 8.93900i 0.501274i
\(319\) 38.8461i 2.17496i
\(320\) 1.00000i 0.0559017i
\(321\) 14.6746 0.819054
\(322\) −4.50604 −0.251112
\(323\) − 42.5639i − 2.36832i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −18.8944 −1.04646
\(327\) 9.66786i 0.534634i
\(328\) −4.26875 −0.235702
\(329\) −26.7614 −1.47540
\(330\) 4.24698i 0.233789i
\(331\) 23.8049i 1.30844i 0.756306 + 0.654218i \(0.227002\pi\)
−0.756306 + 0.654218i \(0.772998\pi\)
\(332\) 0.740939i 0.0406643i
\(333\) − 6.82908i − 0.374232i
\(334\) 4.24698 0.232384
\(335\) −0.0760644 −0.00415584
\(336\) − 3.44504i − 0.187942i
\(337\) −21.2064 −1.15519 −0.577594 0.816324i \(-0.696008\pi\)
−0.577594 + 0.816324i \(0.696008\pi\)
\(338\) 0 0
\(339\) −11.3448 −0.616166
\(340\) − 6.78986i − 0.368232i
\(341\) 15.9390 0.863145
\(342\) 6.26875 0.338975
\(343\) − 7.34375i − 0.396525i
\(344\) − 3.07069i − 0.165560i
\(345\) − 1.30798i − 0.0704192i
\(346\) 1.16852i 0.0628201i
\(347\) −16.5773 −0.889917 −0.444959 0.895551i \(-0.646782\pi\)
−0.444959 + 0.895551i \(0.646782\pi\)
\(348\) 9.14675 0.490317
\(349\) − 2.51035i − 0.134376i −0.997740 0.0671880i \(-0.978597\pi\)
0.997740 0.0671880i \(-0.0214027\pi\)
\(350\) 3.44504 0.184145
\(351\) 0 0
\(352\) −4.24698 −0.226365
\(353\) − 20.8412i − 1.10926i −0.832096 0.554632i \(-0.812859\pi\)
0.832096 0.554632i \(-0.187141\pi\)
\(354\) 10.3327 0.549179
\(355\) 0.374354 0.0198687
\(356\) 13.3274i 0.706348i
\(357\) 23.3913i 1.23800i
\(358\) − 10.1836i − 0.538219i
\(359\) − 2.42566i − 0.128022i −0.997949 0.0640108i \(-0.979611\pi\)
0.997949 0.0640108i \(-0.0203892\pi\)
\(360\) 1.00000 0.0527046
\(361\) −20.2972 −1.06827
\(362\) 10.3351i 0.543202i
\(363\) −7.03684 −0.369338
\(364\) 0 0
\(365\) 16.7114 0.874715
\(366\) − 2.53319i − 0.132412i
\(367\) 2.66248 0.138980 0.0694902 0.997583i \(-0.477863\pi\)
0.0694902 + 0.997583i \(0.477863\pi\)
\(368\) 1.30798 0.0681831
\(369\) 4.26875i 0.222222i
\(370\) 6.82908i 0.355027i
\(371\) 30.7952i 1.59881i
\(372\) − 3.75302i − 0.194585i
\(373\) 5.87800 0.304351 0.152176 0.988353i \(-0.451372\pi\)
0.152176 + 0.988353i \(0.451372\pi\)
\(374\) 28.8364 1.49109
\(375\) 1.00000i 0.0516398i
\(376\) 7.76809 0.400608
\(377\) 0 0
\(378\) −3.44504 −0.177194
\(379\) 36.1540i 1.85711i 0.371196 + 0.928554i \(0.378948\pi\)
−0.371196 + 0.928554i \(0.621052\pi\)
\(380\) −6.26875 −0.321580
\(381\) 2.54288 0.130276
\(382\) 9.43296i 0.482632i
\(383\) − 8.98361i − 0.459041i −0.973304 0.229520i \(-0.926284\pi\)
0.973304 0.229520i \(-0.0737158\pi\)
\(384\) 1.00000i 0.0510310i
\(385\) 14.6310i 0.745666i
\(386\) −22.9191 −1.16655
\(387\) −3.07069 −0.156092
\(388\) 13.1903i 0.669636i
\(389\) −0.655186 −0.0332192 −0.0166096 0.999862i \(-0.505287\pi\)
−0.0166096 + 0.999862i \(0.505287\pi\)
\(390\) 0 0
\(391\) −8.88099 −0.449131
\(392\) − 4.86831i − 0.245887i
\(393\) −10.8509 −0.547353
\(394\) −14.3502 −0.722952
\(395\) 1.33513i 0.0671775i
\(396\) 4.24698i 0.213419i
\(397\) − 0.271143i − 0.0136083i −0.999977 0.00680413i \(-0.997834\pi\)
0.999977 0.00680413i \(-0.00216584\pi\)
\(398\) 14.2524i 0.714406i
\(399\) 21.5961 1.08116
\(400\) −1.00000 −0.0500000
\(401\) − 0.344814i − 0.0172192i −0.999963 0.00860960i \(-0.997259\pi\)
0.999963 0.00860960i \(-0.00274056\pi\)
\(402\) −0.0760644 −0.00379375
\(403\) 0 0
\(404\) −15.0422 −0.748378
\(405\) − 1.00000i − 0.0496904i
\(406\) 31.5109 1.56386
\(407\) −29.0030 −1.43762
\(408\) − 6.78986i − 0.336148i
\(409\) 27.0388i 1.33698i 0.743721 + 0.668490i \(0.233059\pi\)
−0.743721 + 0.668490i \(0.766941\pi\)
\(410\) − 4.26875i − 0.210819i
\(411\) 18.6407i 0.919478i
\(412\) −8.92154 −0.439533
\(413\) 35.5967 1.75160
\(414\) − 1.30798i − 0.0642836i
\(415\) −0.740939 −0.0363713
\(416\) 0 0
\(417\) 3.53750 0.173232
\(418\) − 26.6233i − 1.30219i
\(419\) 5.86533 0.286540 0.143270 0.989684i \(-0.454238\pi\)
0.143270 + 0.989684i \(0.454238\pi\)
\(420\) 3.44504 0.168101
\(421\) − 28.9476i − 1.41082i −0.708799 0.705410i \(-0.750763\pi\)
0.708799 0.705410i \(-0.249237\pi\)
\(422\) − 0.396125i − 0.0192830i
\(423\) − 7.76809i − 0.377697i
\(424\) − 8.93900i − 0.434116i
\(425\) 6.78986 0.329356
\(426\) 0.374354 0.0181375
\(427\) − 8.72694i − 0.422326i
\(428\) −14.6746 −0.709322
\(429\) 0 0
\(430\) 3.07069 0.148082
\(431\) 4.87023i 0.234591i 0.993097 + 0.117295i \(0.0374224\pi\)
−0.993097 + 0.117295i \(0.962578\pi\)
\(432\) 1.00000 0.0481125
\(433\) 36.9788 1.77709 0.888544 0.458791i \(-0.151717\pi\)
0.888544 + 0.458791i \(0.151717\pi\)
\(434\) − 12.9293i − 0.620627i
\(435\) 9.14675i 0.438553i
\(436\) − 9.66786i − 0.463006i
\(437\) 8.19939i 0.392230i
\(438\) 16.7114 0.798502
\(439\) −22.8799 −1.09200 −0.546000 0.837785i \(-0.683850\pi\)
−0.546000 + 0.837785i \(0.683850\pi\)
\(440\) − 4.24698i − 0.202467i
\(441\) −4.86831 −0.231824
\(442\) 0 0
\(443\) −10.5942 −0.503345 −0.251673 0.967812i \(-0.580981\pi\)
−0.251673 + 0.967812i \(0.580981\pi\)
\(444\) 6.82908i 0.324094i
\(445\) −13.3274 −0.631777
\(446\) 1.49635 0.0708543
\(447\) − 16.6843i − 0.789138i
\(448\) 3.44504i 0.162763i
\(449\) − 19.6485i − 0.927269i −0.886027 0.463635i \(-0.846545\pi\)
0.886027 0.463635i \(-0.153455\pi\)
\(450\) 1.00000i 0.0471405i
\(451\) 18.1293 0.853675
\(452\) 11.3448 0.533615
\(453\) − 18.3327i − 0.861347i
\(454\) 25.2150 1.18340
\(455\) 0 0
\(456\) −6.26875 −0.293561
\(457\) − 18.4239i − 0.861832i −0.902392 0.430916i \(-0.858191\pi\)
0.902392 0.430916i \(-0.141809\pi\)
\(458\) 22.3666 1.04512
\(459\) −6.78986 −0.316923
\(460\) 1.30798i 0.0609848i
\(461\) − 22.2446i − 1.03603i −0.855370 0.518017i \(-0.826670\pi\)
0.855370 0.518017i \(-0.173330\pi\)
\(462\) 14.6310i 0.680697i
\(463\) 8.68532i 0.403641i 0.979423 + 0.201820i \(0.0646857\pi\)
−0.979423 + 0.201820i \(0.935314\pi\)
\(464\) −9.14675 −0.424627
\(465\) 3.75302 0.174042
\(466\) 14.0858i 0.652510i
\(467\) −19.6136 −0.907608 −0.453804 0.891102i \(-0.649933\pi\)
−0.453804 + 0.891102i \(0.649933\pi\)
\(468\) 0 0
\(469\) −0.262045 −0.0121001
\(470\) 7.76809i 0.358315i
\(471\) −18.6015 −0.857111
\(472\) −10.3327 −0.475603
\(473\) 13.0411i 0.599633i
\(474\) 1.33513i 0.0613243i
\(475\) − 6.26875i − 0.287630i
\(476\) − 23.3913i − 1.07214i
\(477\) −8.93900 −0.409289
\(478\) 5.38106 0.246124
\(479\) − 31.4698i − 1.43789i −0.695066 0.718946i \(-0.744625\pi\)
0.695066 0.718946i \(-0.255375\pi\)
\(480\) −1.00000 −0.0456435
\(481\) 0 0
\(482\) 29.9420 1.36382
\(483\) − 4.50604i − 0.205032i
\(484\) 7.03684 0.319856
\(485\) −13.1903 −0.598940
\(486\) − 1.00000i − 0.0453609i
\(487\) 13.2644i 0.601069i 0.953771 + 0.300535i \(0.0971651\pi\)
−0.953771 + 0.300535i \(0.902835\pi\)
\(488\) 2.53319i 0.114672i
\(489\) − 18.8944i − 0.854434i
\(490\) 4.86831 0.219928
\(491\) −11.7259 −0.529181 −0.264591 0.964361i \(-0.585237\pi\)
−0.264591 + 0.964361i \(0.585237\pi\)
\(492\) − 4.26875i − 0.192450i
\(493\) 62.1051 2.79707
\(494\) 0 0
\(495\) −4.24698 −0.190888
\(496\) 3.75302i 0.168516i
\(497\) 1.28967 0.0578494
\(498\) −0.740939 −0.0332023
\(499\) − 26.7633i − 1.19809i −0.800715 0.599045i \(-0.795547\pi\)
0.800715 0.599045i \(-0.204453\pi\)
\(500\) − 1.00000i − 0.0447214i
\(501\) 4.24698i 0.189741i
\(502\) − 8.73556i − 0.389887i
\(503\) 17.2239 0.767975 0.383987 0.923338i \(-0.374551\pi\)
0.383987 + 0.923338i \(0.374551\pi\)
\(504\) 3.44504 0.153454
\(505\) − 15.0422i − 0.669370i
\(506\) −5.55496 −0.246948
\(507\) 0 0
\(508\) −2.54288 −0.112822
\(509\) − 28.2737i − 1.25321i −0.779338 0.626604i \(-0.784444\pi\)
0.779338 0.626604i \(-0.215556\pi\)
\(510\) 6.78986 0.300660
\(511\) 57.5715 2.54681
\(512\) − 1.00000i − 0.0441942i
\(513\) 6.26875i 0.276772i
\(514\) 21.2717i 0.938256i
\(515\) − 8.92154i − 0.393130i
\(516\) 3.07069 0.135179
\(517\) −32.9909 −1.45094
\(518\) 23.5265i 1.03369i
\(519\) −1.16852 −0.0512924
\(520\) 0 0
\(521\) 1.47889 0.0647915 0.0323958 0.999475i \(-0.489686\pi\)
0.0323958 + 0.999475i \(0.489686\pi\)
\(522\) 9.14675i 0.400342i
\(523\) −28.2198 −1.23397 −0.616984 0.786976i \(-0.711646\pi\)
−0.616984 + 0.786976i \(0.711646\pi\)
\(524\) 10.8509 0.474022
\(525\) 3.44504i 0.150354i
\(526\) 22.2325i 0.969383i
\(527\) − 25.4825i − 1.11003i
\(528\) − 4.24698i − 0.184826i
\(529\) −21.2892 −0.925617
\(530\) 8.93900 0.388285
\(531\) 10.3327i 0.448402i
\(532\) −21.5961 −0.936310
\(533\) 0 0
\(534\) −13.3274 −0.576731
\(535\) − 14.6746i − 0.634437i
\(536\) 0.0760644 0.00328548
\(537\) 10.1836 0.439454
\(538\) 10.2241i 0.440794i
\(539\) 20.6756i 0.890562i
\(540\) 1.00000i 0.0430331i
\(541\) − 0.245915i − 0.0105727i −0.999986 0.00528635i \(-0.998317\pi\)
0.999986 0.00528635i \(-0.00168270\pi\)
\(542\) −0.716185 −0.0307628
\(543\) −10.3351 −0.443523
\(544\) 6.78986i 0.291113i
\(545\) 9.66786 0.414126
\(546\) 0 0
\(547\) −13.6407 −0.583235 −0.291617 0.956535i \(-0.594193\pi\)
−0.291617 + 0.956535i \(0.594193\pi\)
\(548\) − 18.6407i − 0.796292i
\(549\) 2.53319 0.108114
\(550\) 4.24698 0.181092
\(551\) − 57.3387i − 2.44271i
\(552\) 1.30798i 0.0556713i
\(553\) 4.59956i 0.195593i
\(554\) − 22.8713i − 0.971708i
\(555\) −6.82908 −0.289879
\(556\) −3.53750 −0.150023
\(557\) − 14.0248i − 0.594248i −0.954839 0.297124i \(-0.903973\pi\)
0.954839 0.297124i \(-0.0960275\pi\)
\(558\) 3.75302 0.158878
\(559\) 0 0
\(560\) −3.44504 −0.145580
\(561\) 28.8364i 1.21747i
\(562\) −4.13036 −0.174229
\(563\) 3.26981 0.137806 0.0689031 0.997623i \(-0.478050\pi\)
0.0689031 + 0.997623i \(0.478050\pi\)
\(564\) 7.76809i 0.327095i
\(565\) 11.3448i 0.477280i
\(566\) 16.2959i 0.684968i
\(567\) − 3.44504i − 0.144678i
\(568\) −0.374354 −0.0157076
\(569\) −4.10752 −0.172196 −0.0860982 0.996287i \(-0.527440\pi\)
−0.0860982 + 0.996287i \(0.527440\pi\)
\(570\) − 6.26875i − 0.262569i
\(571\) −9.18465 −0.384366 −0.192183 0.981359i \(-0.561557\pi\)
−0.192183 + 0.981359i \(0.561557\pi\)
\(572\) 0 0
\(573\) −9.43296 −0.394068
\(574\) − 14.7060i − 0.613817i
\(575\) −1.30798 −0.0545465
\(576\) −1.00000 −0.0416667
\(577\) 28.0000i 1.16566i 0.812596 + 0.582828i \(0.198054\pi\)
−0.812596 + 0.582828i \(0.801946\pi\)
\(578\) − 29.1021i − 1.21049i
\(579\) − 22.9191i − 0.952487i
\(580\) − 9.14675i − 0.379798i
\(581\) −2.55257 −0.105898
\(582\) −13.1903 −0.546755
\(583\) 37.9638i 1.57230i
\(584\) −16.7114 −0.691523
\(585\) 0 0
\(586\) −7.72587 −0.319153
\(587\) − 47.6765i − 1.96782i −0.178668 0.983910i \(-0.557179\pi\)
0.178668 0.983910i \(-0.442821\pi\)
\(588\) 4.86831 0.200766
\(589\) −23.5267 −0.969403
\(590\) − 10.3327i − 0.425392i
\(591\) − 14.3502i − 0.590288i
\(592\) − 6.82908i − 0.280674i
\(593\) 16.7681i 0.688583i 0.938863 + 0.344291i \(0.111881\pi\)
−0.938863 + 0.344291i \(0.888119\pi\)
\(594\) −4.24698 −0.174256
\(595\) 23.3913 0.958951
\(596\) 16.6843i 0.683414i
\(597\) −14.2524 −0.583310
\(598\) 0 0
\(599\) 44.0157 1.79843 0.899215 0.437506i \(-0.144138\pi\)
0.899215 + 0.437506i \(0.144138\pi\)
\(600\) − 1.00000i − 0.0408248i
\(601\) 39.1299 1.59614 0.798071 0.602564i \(-0.205854\pi\)
0.798071 + 0.602564i \(0.205854\pi\)
\(602\) 10.5786 0.431153
\(603\) − 0.0760644i − 0.00309758i
\(604\) 18.3327i 0.745948i
\(605\) 7.03684i 0.286088i
\(606\) − 15.0422i − 0.611048i
\(607\) 6.33273 0.257038 0.128519 0.991707i \(-0.458978\pi\)
0.128519 + 0.991707i \(0.458978\pi\)
\(608\) 6.26875 0.254231
\(609\) 31.5109i 1.27689i
\(610\) −2.53319 −0.102566
\(611\) 0 0
\(612\) 6.78986 0.274464
\(613\) 29.1129i 1.17586i 0.808912 + 0.587929i \(0.200057\pi\)
−0.808912 + 0.587929i \(0.799943\pi\)
\(614\) 5.13036 0.207044
\(615\) 4.26875 0.172133
\(616\) − 14.6310i − 0.589501i
\(617\) − 27.9933i − 1.12697i −0.826127 0.563484i \(-0.809461\pi\)
0.826127 0.563484i \(-0.190539\pi\)
\(618\) − 8.92154i − 0.358877i
\(619\) − 23.0084i − 0.924784i −0.886676 0.462392i \(-0.846991\pi\)
0.886676 0.462392i \(-0.153009\pi\)
\(620\) −3.75302 −0.150725
\(621\) 1.30798 0.0524874
\(622\) 33.1987i 1.33114i
\(623\) −45.9133 −1.83948
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) − 0.0814412i − 0.00325505i
\(627\) 26.6233 1.06323
\(628\) 18.6015 0.742280
\(629\) 46.3685i 1.84883i
\(630\) 3.44504i 0.137254i
\(631\) 6.18731i 0.246313i 0.992387 + 0.123156i \(0.0393017\pi\)
−0.992387 + 0.123156i \(0.960698\pi\)
\(632\) − 1.33513i − 0.0531084i
\(633\) 0.396125 0.0157445
\(634\) −19.4969 −0.774323
\(635\) − 2.54288i − 0.100911i
\(636\) 8.93900 0.354454
\(637\) 0 0
\(638\) 38.8461 1.53793
\(639\) 0.374354i 0.0148092i
\(640\) 1.00000 0.0395285
\(641\) −39.1159 −1.54498 −0.772492 0.635024i \(-0.780990\pi\)
−0.772492 + 0.635024i \(0.780990\pi\)
\(642\) − 14.6746i − 0.579159i
\(643\) 8.78209i 0.346332i 0.984893 + 0.173166i \(0.0553997\pi\)
−0.984893 + 0.173166i \(0.944600\pi\)
\(644\) 4.50604i 0.177563i
\(645\) 3.07069i 0.120908i
\(646\) −42.5639 −1.67465
\(647\) 30.1333 1.18466 0.592332 0.805694i \(-0.298207\pi\)
0.592332 + 0.805694i \(0.298207\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 43.8829 1.72255
\(650\) 0 0
\(651\) 12.9293 0.506740
\(652\) 18.8944i 0.739962i
\(653\) −27.0858 −1.05995 −0.529974 0.848014i \(-0.677798\pi\)
−0.529974 + 0.848014i \(0.677798\pi\)
\(654\) 9.66786 0.378043
\(655\) 10.8509i 0.423978i
\(656\) 4.26875i 0.166667i
\(657\) 16.7114i 0.651974i
\(658\) 26.7614i 1.04327i
\(659\) 5.69681 0.221916 0.110958 0.993825i \(-0.464608\pi\)
0.110958 + 0.993825i \(0.464608\pi\)
\(660\) 4.24698 0.165313
\(661\) 17.1008i 0.665145i 0.943078 + 0.332572i \(0.107917\pi\)
−0.943078 + 0.332572i \(0.892083\pi\)
\(662\) 23.8049 0.925205
\(663\) 0 0
\(664\) 0.740939 0.0287540
\(665\) − 21.5961i − 0.837461i
\(666\) −6.82908 −0.264622
\(667\) −11.9638 −0.463238
\(668\) − 4.24698i − 0.164321i
\(669\) 1.49635i 0.0578523i
\(670\) 0.0760644i 0.00293862i
\(671\) − 10.7584i − 0.415323i
\(672\) −3.44504 −0.132895
\(673\) 43.1957 1.66507 0.832535 0.553972i \(-0.186889\pi\)
0.832535 + 0.553972i \(0.186889\pi\)
\(674\) 21.2064i 0.816841i
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) −7.78746 −0.299297 −0.149648 0.988739i \(-0.547814\pi\)
−0.149648 + 0.988739i \(0.547814\pi\)
\(678\) 11.3448i 0.435695i
\(679\) −45.4411 −1.74387
\(680\) −6.78986 −0.260379
\(681\) 25.2150i 0.966243i
\(682\) − 15.9390i − 0.610336i
\(683\) − 30.9138i − 1.18288i −0.806348 0.591441i \(-0.798559\pi\)
0.806348 0.591441i \(-0.201441\pi\)
\(684\) − 6.26875i − 0.239692i
\(685\) 18.6407 0.712225
\(686\) −7.34375 −0.280386
\(687\) 22.3666i 0.853338i
\(688\) −3.07069 −0.117069
\(689\) 0 0
\(690\) −1.30798 −0.0497939
\(691\) − 6.20237i − 0.235949i −0.993017 0.117975i \(-0.962360\pi\)
0.993017 0.117975i \(-0.0376402\pi\)
\(692\) 1.16852 0.0444206
\(693\) −14.6310 −0.555787
\(694\) 16.5773i 0.629266i
\(695\) − 3.53750i − 0.134185i
\(696\) − 9.14675i − 0.346707i
\(697\) − 28.9842i − 1.09785i
\(698\) −2.51035 −0.0950182
\(699\) −14.0858 −0.532772
\(700\) − 3.44504i − 0.130210i
\(701\) −26.3220 −0.994167 −0.497084 0.867703i \(-0.665596\pi\)
−0.497084 + 0.867703i \(0.665596\pi\)
\(702\) 0 0
\(703\) 42.8098 1.61460
\(704\) 4.24698i 0.160064i
\(705\) −7.76809 −0.292563
\(706\) −20.8412 −0.784368
\(707\) − 51.8211i − 1.94893i
\(708\) − 10.3327i − 0.388328i
\(709\) − 33.7429i − 1.26724i −0.773645 0.633620i \(-0.781568\pi\)
0.773645 0.633620i \(-0.218432\pi\)
\(710\) − 0.374354i − 0.0140493i
\(711\) −1.33513 −0.0500711
\(712\) 13.3274 0.499464
\(713\) 4.90887i 0.183839i
\(714\) 23.3913 0.875399
\(715\) 0 0
\(716\) −10.1836 −0.380579
\(717\) 5.38106i 0.200959i
\(718\) −2.42566 −0.0905250
\(719\) 9.50173 0.354355 0.177177 0.984179i \(-0.443303\pi\)
0.177177 + 0.984179i \(0.443303\pi\)
\(720\) − 1.00000i − 0.0372678i
\(721\) − 30.7351i − 1.14463i
\(722\) 20.2972i 0.755384i
\(723\) 29.9420i 1.11355i
\(724\) 10.3351 0.384102
\(725\) 9.14675 0.339702
\(726\) 7.03684i 0.261161i
\(727\) 12.2631 0.454814 0.227407 0.973800i \(-0.426975\pi\)
0.227407 + 0.973800i \(0.426975\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) − 16.7114i − 0.618517i
\(731\) 20.8495 0.771148
\(732\) −2.53319 −0.0936293
\(733\) 1.52217i 0.0562227i 0.999605 + 0.0281113i \(0.00894930\pi\)
−0.999605 + 0.0281113i \(0.991051\pi\)
\(734\) − 2.66248i − 0.0982740i
\(735\) 4.86831i 0.179570i
\(736\) − 1.30798i − 0.0482127i
\(737\) −0.323044 −0.0118995
\(738\) 4.26875 0.157135
\(739\) − 51.1075i − 1.88002i −0.341146 0.940010i \(-0.610815\pi\)
0.341146 0.940010i \(-0.389185\pi\)
\(740\) 6.82908 0.251042
\(741\) 0 0
\(742\) 30.7952 1.13053
\(743\) − 9.08144i − 0.333166i −0.986027 0.166583i \(-0.946727\pi\)
0.986027 0.166583i \(-0.0532733\pi\)
\(744\) −3.75302 −0.137592
\(745\) −16.6843 −0.611264
\(746\) − 5.87800i − 0.215209i
\(747\) − 0.740939i − 0.0271095i
\(748\) − 28.8364i − 1.05436i
\(749\) − 50.5545i − 1.84722i
\(750\) 1.00000 0.0365148
\(751\) −31.7090 −1.15708 −0.578539 0.815655i \(-0.696377\pi\)
−0.578539 + 0.815655i \(0.696377\pi\)
\(752\) − 7.76809i − 0.283273i
\(753\) 8.73556 0.318342
\(754\) 0 0
\(755\) −18.3327 −0.667196
\(756\) 3.44504i 0.125295i
\(757\) −8.49934 −0.308914 −0.154457 0.988000i \(-0.549363\pi\)
−0.154457 + 0.988000i \(0.549363\pi\)
\(758\) 36.1540 1.31317
\(759\) − 5.55496i − 0.201632i
\(760\) 6.26875i 0.227391i
\(761\) 16.4198i 0.595218i 0.954688 + 0.297609i \(0.0961891\pi\)
−0.954688 + 0.297609i \(0.903811\pi\)
\(762\) − 2.54288i − 0.0921187i
\(763\) 33.3062 1.20576
\(764\) 9.43296 0.341273
\(765\) 6.78986i 0.245488i
\(766\) −8.98361 −0.324591
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) − 27.1011i − 0.977290i −0.872483 0.488645i \(-0.837491\pi\)
0.872483 0.488645i \(-0.162509\pi\)
\(770\) 14.6310 0.527265
\(771\) −21.2717 −0.766083
\(772\) 22.9191i 0.824878i
\(773\) 12.6297i 0.454259i 0.973865 + 0.227129i \(0.0729340\pi\)
−0.973865 + 0.227129i \(0.927066\pi\)
\(774\) 3.07069i 0.110374i
\(775\) − 3.75302i − 0.134812i
\(776\) 13.1903 0.473504
\(777\) −23.5265 −0.844008
\(778\) 0.655186i 0.0234895i
\(779\) −26.7597 −0.958767
\(780\) 0 0
\(781\) 1.58987 0.0568902
\(782\) 8.88099i 0.317583i
\(783\) −9.14675 −0.326878
\(784\) −4.86831 −0.173868
\(785\) 18.6015i 0.663915i
\(786\) 10.8509i 0.387037i
\(787\) − 3.02954i − 0.107991i −0.998541 0.0539957i \(-0.982804\pi\)
0.998541 0.0539957i \(-0.0171957\pi\)
\(788\) 14.3502i 0.511204i
\(789\) −22.2325 −0.791498
\(790\) 1.33513 0.0475016
\(791\) 39.0834i 1.38964i
\(792\) 4.24698 0.150910
\(793\) 0 0
\(794\) −0.271143 −0.00962250
\(795\) 8.93900i 0.317034i
\(796\) 14.2524 0.505161
\(797\) −40.2218 −1.42473 −0.712364 0.701810i \(-0.752375\pi\)
−0.712364 + 0.701810i \(0.752375\pi\)
\(798\) − 21.5961i − 0.764494i
\(799\) 52.7442i 1.86596i
\(800\) 1.00000i 0.0353553i
\(801\) − 13.3274i − 0.470899i
\(802\) −0.344814 −0.0121758
\(803\) 70.9730 2.50458
\(804\) 0.0760644i 0.00268259i
\(805\) −4.50604 −0.158817
\(806\) 0 0
\(807\) −10.2241 −0.359907
\(808\) 15.0422i 0.529183i
\(809\) 33.7918 1.18806 0.594028 0.804445i \(-0.297537\pi\)
0.594028 + 0.804445i \(0.297537\pi\)
\(810\) −1.00000 −0.0351364
\(811\) 33.6534i 1.18173i 0.806770 + 0.590865i \(0.201213\pi\)
−0.806770 + 0.590865i \(0.798787\pi\)
\(812\) − 31.5109i − 1.10582i
\(813\) − 0.716185i − 0.0251177i
\(814\) 29.0030i 1.01655i
\(815\) −18.8944 −0.661842
\(816\) −6.78986 −0.237693
\(817\) − 19.2494i − 0.673450i
\(818\) 27.0388 0.945388
\(819\) 0 0
\(820\) −4.26875 −0.149071
\(821\) 12.6939i 0.443022i 0.975158 + 0.221511i \(0.0710988\pi\)
−0.975158 + 0.221511i \(0.928901\pi\)
\(822\) 18.6407 0.650169
\(823\) 8.79523 0.306583 0.153291 0.988181i \(-0.451013\pi\)
0.153291 + 0.988181i \(0.451013\pi\)
\(824\) 8.92154i 0.310797i
\(825\) 4.24698i 0.147861i
\(826\) − 35.5967i − 1.23857i
\(827\) − 32.1758i − 1.11886i −0.828877 0.559431i \(-0.811020\pi\)
0.828877 0.559431i \(-0.188980\pi\)
\(828\) −1.30798 −0.0454554
\(829\) −21.1812 −0.735653 −0.367827 0.929894i \(-0.619898\pi\)
−0.367827 + 0.929894i \(0.619898\pi\)
\(830\) 0.740939i 0.0257184i
\(831\) 22.8713 0.793397
\(832\) 0 0
\(833\) 33.0551 1.14529
\(834\) − 3.53750i − 0.122494i
\(835\) 4.24698 0.146973
\(836\) −26.6233 −0.920784
\(837\) 3.75302i 0.129723i
\(838\) − 5.86533i − 0.202614i
\(839\) − 26.5931i − 0.918097i −0.888411 0.459048i \(-0.848190\pi\)
0.888411 0.459048i \(-0.151810\pi\)
\(840\) − 3.44504i − 0.118865i
\(841\) 54.6631 1.88493
\(842\) −28.9476 −0.997601
\(843\) − 4.13036i − 0.142257i
\(844\) −0.396125 −0.0136352
\(845\) 0 0
\(846\) −7.76809 −0.267072
\(847\) 24.2422i 0.832972i
\(848\) −8.93900 −0.306967
\(849\) −16.2959 −0.559274
\(850\) − 6.78986i − 0.232890i
\(851\) − 8.93230i − 0.306195i
\(852\) − 0.374354i − 0.0128252i
\(853\) 32.2457i 1.10407i 0.833821 + 0.552035i \(0.186149\pi\)
−0.833821 + 0.552035i \(0.813851\pi\)
\(854\) −8.72694 −0.298630
\(855\) 6.26875 0.214387
\(856\) 14.6746i 0.501566i
\(857\) 43.0176 1.46945 0.734726 0.678364i \(-0.237311\pi\)
0.734726 + 0.678364i \(0.237311\pi\)
\(858\) 0 0
\(859\) −31.1997 −1.06452 −0.532260 0.846581i \(-0.678657\pi\)
−0.532260 + 0.846581i \(0.678657\pi\)
\(860\) − 3.07069i − 0.104710i
\(861\) 14.7060 0.501180
\(862\) 4.87023 0.165881
\(863\) − 34.5424i − 1.17584i −0.808920 0.587919i \(-0.799948\pi\)
0.808920 0.587919i \(-0.200052\pi\)
\(864\) − 1.00000i − 0.0340207i
\(865\) 1.16852i 0.0397309i
\(866\) − 36.9788i − 1.25659i
\(867\) 29.1021 0.988361
\(868\) −12.9293 −0.438849
\(869\) 5.67025i 0.192350i
\(870\) 9.14675 0.310104
\(871\) 0 0
\(872\) −9.66786 −0.327395
\(873\) − 13.1903i − 0.446424i
\(874\) 8.19939 0.277349
\(875\) 3.44504 0.116464
\(876\) − 16.7114i − 0.564626i
\(877\) 2.12093i 0.0716188i 0.999359 + 0.0358094i \(0.0114009\pi\)
−0.999359 + 0.0358094i \(0.988599\pi\)
\(878\) 22.8799i 0.772160i
\(879\) − 7.72587i − 0.260587i
\(880\) −4.24698 −0.143166
\(881\) −27.3489 −0.921407 −0.460703 0.887554i \(-0.652403\pi\)
−0.460703 + 0.887554i \(0.652403\pi\)
\(882\) 4.86831i 0.163925i
\(883\) 48.2959 1.62529 0.812643 0.582762i \(-0.198028\pi\)
0.812643 + 0.582762i \(0.198028\pi\)
\(884\) 0 0
\(885\) 10.3327 0.347331
\(886\) 10.5942i 0.355919i
\(887\) 25.1943 0.845943 0.422972 0.906143i \(-0.360987\pi\)
0.422972 + 0.906143i \(0.360987\pi\)
\(888\) 6.82908 0.229169
\(889\) − 8.76032i − 0.293812i
\(890\) 13.3274i 0.446734i
\(891\) − 4.24698i − 0.142279i
\(892\) − 1.49635i − 0.0501016i
\(893\) 48.6962 1.62956
\(894\) −16.6843 −0.558005
\(895\) − 10.1836i − 0.340400i
\(896\) 3.44504 0.115091
\(897\) 0 0
\(898\) −19.6485 −0.655678
\(899\) − 34.3279i − 1.14490i
\(900\) 1.00000 0.0333333
\(901\) 60.6945 2.02203
\(902\) − 18.1293i − 0.603639i
\(903\) 10.5786i 0.352035i
\(904\) − 11.3448i − 0.377323i
\(905\) 10.3351i 0.343551i
\(906\) −18.3327 −0.609064
\(907\) 20.0949 0.667239 0.333619 0.942708i \(-0.391730\pi\)
0.333619 + 0.942708i \(0.391730\pi\)
\(908\) − 25.2150i − 0.836791i
\(909\) 15.0422 0.498919
\(910\) 0 0
\(911\) −7.44563 −0.246685 −0.123342 0.992364i \(-0.539361\pi\)
−0.123342 + 0.992364i \(0.539361\pi\)
\(912\) 6.26875i 0.207579i
\(913\) −3.14675 −0.104142
\(914\) −18.4239 −0.609407
\(915\) − 2.53319i − 0.0837446i
\(916\) − 22.3666i − 0.739013i
\(917\) 37.3817i 1.23445i
\(918\) 6.78986i 0.224099i
\(919\) −14.2765 −0.470939 −0.235469 0.971882i \(-0.575663\pi\)
−0.235469 + 0.971882i \(0.575663\pi\)
\(920\) 1.30798 0.0431228
\(921\) 5.13036i 0.169051i
\(922\) −22.2446 −0.732586
\(923\) 0 0
\(924\) 14.6310 0.481325
\(925\) 6.82908i 0.224539i
\(926\) 8.68532 0.285417
\(927\) 8.92154 0.293022
\(928\) 9.14675i 0.300257i
\(929\) 30.9148i 1.01428i 0.861863 + 0.507141i \(0.169298\pi\)
−0.861863 + 0.507141i \(0.830702\pi\)
\(930\) − 3.75302i − 0.123066i
\(931\) − 30.5182i − 1.00019i
\(932\) 14.0858 0.461394
\(933\) −33.1987 −1.08688
\(934\) 19.6136i 0.641775i
\(935\) 28.8364 0.943050
\(936\) 0 0
\(937\) −16.7187 −0.546176 −0.273088 0.961989i \(-0.588045\pi\)
−0.273088 + 0.961989i \(0.588045\pi\)
\(938\) 0.262045i 0.00855608i
\(939\) 0.0814412 0.00265773
\(940\) 7.76809 0.253367
\(941\) 36.9033i 1.20301i 0.798867 + 0.601507i \(0.205433\pi\)
−0.798867 + 0.601507i \(0.794567\pi\)
\(942\) 18.6015i 0.606069i
\(943\) 5.58343i 0.181822i
\(944\) 10.3327i 0.336302i
\(945\) −3.44504 −0.112067
\(946\) 13.0411 0.424004
\(947\) 39.1661i 1.27273i 0.771389 + 0.636364i \(0.219562\pi\)
−0.771389 + 0.636364i \(0.780438\pi\)
\(948\) 1.33513 0.0433629
\(949\) 0 0
\(950\) −6.26875 −0.203385
\(951\) − 19.4969i − 0.632232i
\(952\) −23.3913 −0.758118
\(953\) 19.6200 0.635554 0.317777 0.948165i \(-0.397064\pi\)
0.317777 + 0.948165i \(0.397064\pi\)
\(954\) 8.93900i 0.289411i
\(955\) 9.43296i 0.305243i
\(956\) − 5.38106i − 0.174036i
\(957\) 38.8461i 1.25572i
\(958\) −31.4698 −1.01674
\(959\) 64.2180 2.07371
\(960\) 1.00000i 0.0322749i
\(961\) 16.9148 0.545640
\(962\) 0 0
\(963\) 14.6746 0.472881
\(964\) − 29.9420i − 0.964366i
\(965\) −22.9191 −0.737794
\(966\) −4.50604 −0.144979
\(967\) − 43.1564i − 1.38782i −0.720063 0.693909i \(-0.755887\pi\)
0.720063 0.693909i \(-0.244113\pi\)
\(968\) − 7.03684i − 0.226172i
\(969\) − 42.5639i − 1.36735i
\(970\) 13.1903i 0.423515i
\(971\) 39.5032 1.26772 0.633859 0.773449i \(-0.281470\pi\)
0.633859 + 0.773449i \(0.281470\pi\)
\(972\) −1.00000 −0.0320750
\(973\) − 12.1868i − 0.390692i
\(974\) 13.2644 0.425020
\(975\) 0 0
\(976\) 2.53319 0.0810854
\(977\) − 3.84117i − 0.122890i −0.998110 0.0614449i \(-0.980429\pi\)
0.998110 0.0614449i \(-0.0195708\pi\)
\(978\) −18.8944 −0.604176
\(979\) −56.6010 −1.80898
\(980\) − 4.86831i − 0.155513i
\(981\) 9.66786i 0.308671i
\(982\) 11.7259i 0.374188i
\(983\) 32.4300i 1.03436i 0.855878 + 0.517178i \(0.173017\pi\)
−0.855878 + 0.517178i \(0.826983\pi\)
\(984\) −4.26875 −0.136083
\(985\) −14.3502 −0.457235
\(986\) − 62.1051i − 1.97783i
\(987\) −26.7614 −0.851824
\(988\) 0 0
\(989\) −4.01639 −0.127714
\(990\) 4.24698i 0.134978i
\(991\) −61.9807 −1.96888 −0.984442 0.175712i \(-0.943777\pi\)
−0.984442 + 0.175712i \(0.943777\pi\)
\(992\) 3.75302 0.119159
\(993\) 23.8049i 0.755426i
\(994\) − 1.28967i − 0.0409057i
\(995\) 14.2524i 0.451830i
\(996\) 0.740939i 0.0234775i
\(997\) 28.9162 0.915784 0.457892 0.889008i \(-0.348605\pi\)
0.457892 + 0.889008i \(0.348605\pi\)
\(998\) −26.7633 −0.847177
\(999\) − 6.82908i − 0.216063i
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.b.w.1351.2 6
13.5 odd 4 5070.2.a.bo.1.2 3
13.8 odd 4 5070.2.a.bx.1.2 yes 3
13.12 even 2 inner 5070.2.b.w.1351.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bo.1.2 3 13.5 odd 4
5070.2.a.bx.1.2 yes 3 13.8 odd 4
5070.2.b.w.1351.2 6 1.1 even 1 trivial
5070.2.b.w.1351.5 6 13.12 even 2 inner