Properties

Label 8470.2.a.cy.1.1
Level $8470$
Weight $2$
Character 8470.1
Self dual yes
Analytic conductor $67.633$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8470 = 2 \cdot 5 \cdot 7 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8470.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(67.6332905120\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.13298000.1
Defining polynomial: \(x^{6} - x^{5} - 10 x^{4} + 3 x^{3} + 26 x^{2} + 13 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.27063\) of defining polynomial
Character \(\chi\) \(=\) 8470.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -2.80837 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.80837 q^{6} -1.00000 q^{7} -1.00000 q^{8} +4.88695 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -2.80837 q^{3} +1.00000 q^{4} +1.00000 q^{5} +2.80837 q^{6} -1.00000 q^{7} -1.00000 q^{8} +4.88695 q^{9} -1.00000 q^{10} -2.80837 q^{12} +3.05764 q^{13} +1.00000 q^{14} -2.80837 q^{15} +1.00000 q^{16} -6.94459 q^{17} -4.88695 q^{18} +4.69704 q^{19} +1.00000 q^{20} +2.80837 q^{21} -1.73118 q^{23} +2.80837 q^{24} +1.00000 q^{25} -3.05764 q^{26} -5.29926 q^{27} -1.00000 q^{28} +6.74554 q^{29} +2.80837 q^{30} +3.34334 q^{31} -1.00000 q^{32} +6.94459 q^{34} -1.00000 q^{35} +4.88695 q^{36} -7.32736 q^{37} -4.69704 q^{38} -8.58700 q^{39} -1.00000 q^{40} -7.58977 q^{41} -2.80837 q^{42} -7.27592 q^{43} +4.88695 q^{45} +1.73118 q^{46} +4.00555 q^{47} -2.80837 q^{48} +1.00000 q^{49} -1.00000 q^{50} +19.5030 q^{51} +3.05764 q^{52} -1.70116 q^{53} +5.29926 q^{54} +1.00000 q^{56} -13.1910 q^{57} -6.74554 q^{58} +7.64741 q^{59} -2.80837 q^{60} -8.93050 q^{61} -3.34334 q^{62} -4.88695 q^{63} +1.00000 q^{64} +3.05764 q^{65} +10.5718 q^{67} -6.94459 q^{68} +4.86179 q^{69} +1.00000 q^{70} -2.30414 q^{71} -4.88695 q^{72} +13.1113 q^{73} +7.32736 q^{74} -2.80837 q^{75} +4.69704 q^{76} +8.58700 q^{78} -4.25829 q^{79} +1.00000 q^{80} +0.221440 q^{81} +7.58977 q^{82} -3.86073 q^{83} +2.80837 q^{84} -6.94459 q^{85} +7.27592 q^{86} -18.9440 q^{87} -12.3140 q^{89} -4.88695 q^{90} -3.05764 q^{91} -1.73118 q^{92} -9.38933 q^{93} -4.00555 q^{94} +4.69704 q^{95} +2.80837 q^{96} -6.94282 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{2} + q^{3} + 6q^{4} + 6q^{5} - q^{6} - 6q^{7} - 6q^{8} + 15q^{9} + O(q^{10}) \) \( 6q - 6q^{2} + q^{3} + 6q^{4} + 6q^{5} - q^{6} - 6q^{7} - 6q^{8} + 15q^{9} - 6q^{10} + q^{12} - 2q^{13} + 6q^{14} + q^{15} + 6q^{16} - 7q^{17} - 15q^{18} - 11q^{19} + 6q^{20} - q^{21} - 6q^{23} - q^{24} + 6q^{25} + 2q^{26} + 4q^{27} - 6q^{28} - 2q^{29} - q^{30} - 6q^{32} + 7q^{34} - 6q^{35} + 15q^{36} - 14q^{37} + 11q^{38} - 20q^{39} - 6q^{40} - 13q^{41} + q^{42} - 19q^{43} + 15q^{45} + 6q^{46} + 22q^{47} + q^{48} + 6q^{49} - 6q^{50} + 14q^{51} - 2q^{52} - 10q^{53} - 4q^{54} + 6q^{56} - 32q^{57} + 2q^{58} - 7q^{59} + q^{60} - 22q^{61} - 15q^{63} + 6q^{64} - 2q^{65} + 5q^{67} - 7q^{68} - 36q^{69} + 6q^{70} + 8q^{71} - 15q^{72} - 13q^{73} + 14q^{74} + q^{75} - 11q^{76} + 20q^{78} + 16q^{79} + 6q^{80} + 18q^{81} + 13q^{82} + 5q^{83} - q^{84} - 7q^{85} + 19q^{86} - 14q^{87} + q^{89} - 15q^{90} + 2q^{91} - 6q^{92} - 42q^{93} - 22q^{94} - 11q^{95} - q^{96} - 3q^{97} - 6q^{98} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −2.80837 −1.62141 −0.810707 0.585452i \(-0.800917\pi\)
−0.810707 + 0.585452i \(0.800917\pi\)
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 2.80837 1.14651
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 4.88695 1.62898
\(10\) −1.00000 −0.316228
\(11\) 0 0
\(12\) −2.80837 −0.810707
\(13\) 3.05764 0.848037 0.424019 0.905653i \(-0.360619\pi\)
0.424019 + 0.905653i \(0.360619\pi\)
\(14\) 1.00000 0.267261
\(15\) −2.80837 −0.725118
\(16\) 1.00000 0.250000
\(17\) −6.94459 −1.68431 −0.842156 0.539234i \(-0.818714\pi\)
−0.842156 + 0.539234i \(0.818714\pi\)
\(18\) −4.88695 −1.15187
\(19\) 4.69704 1.07757 0.538787 0.842442i \(-0.318883\pi\)
0.538787 + 0.842442i \(0.318883\pi\)
\(20\) 1.00000 0.223607
\(21\) 2.80837 0.612837
\(22\) 0 0
\(23\) −1.73118 −0.360976 −0.180488 0.983577i \(-0.557768\pi\)
−0.180488 + 0.983577i \(0.557768\pi\)
\(24\) 2.80837 0.573256
\(25\) 1.00000 0.200000
\(26\) −3.05764 −0.599653
\(27\) −5.29926 −1.01984
\(28\) −1.00000 −0.188982
\(29\) 6.74554 1.25262 0.626308 0.779576i \(-0.284565\pi\)
0.626308 + 0.779576i \(0.284565\pi\)
\(30\) 2.80837 0.512736
\(31\) 3.34334 0.600481 0.300240 0.953864i \(-0.402933\pi\)
0.300240 + 0.953864i \(0.402933\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 6.94459 1.19099
\(35\) −1.00000 −0.169031
\(36\) 4.88695 0.814492
\(37\) −7.32736 −1.20461 −0.602305 0.798266i \(-0.705751\pi\)
−0.602305 + 0.798266i \(0.705751\pi\)
\(38\) −4.69704 −0.761960
\(39\) −8.58700 −1.37502
\(40\) −1.00000 −0.158114
\(41\) −7.58977 −1.18532 −0.592662 0.805452i \(-0.701923\pi\)
−0.592662 + 0.805452i \(0.701923\pi\)
\(42\) −2.80837 −0.433341
\(43\) −7.27592 −1.10957 −0.554784 0.831995i \(-0.687199\pi\)
−0.554784 + 0.831995i \(0.687199\pi\)
\(44\) 0 0
\(45\) 4.88695 0.728504
\(46\) 1.73118 0.255248
\(47\) 4.00555 0.584270 0.292135 0.956377i \(-0.405634\pi\)
0.292135 + 0.956377i \(0.405634\pi\)
\(48\) −2.80837 −0.405354
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 19.5030 2.73097
\(52\) 3.05764 0.424019
\(53\) −1.70116 −0.233673 −0.116836 0.993151i \(-0.537275\pi\)
−0.116836 + 0.993151i \(0.537275\pi\)
\(54\) 5.29926 0.721138
\(55\) 0 0
\(56\) 1.00000 0.133631
\(57\) −13.1910 −1.74719
\(58\) −6.74554 −0.885733
\(59\) 7.64741 0.995608 0.497804 0.867289i \(-0.334140\pi\)
0.497804 + 0.867289i \(0.334140\pi\)
\(60\) −2.80837 −0.362559
\(61\) −8.93050 −1.14343 −0.571717 0.820451i \(-0.693722\pi\)
−0.571717 + 0.820451i \(0.693722\pi\)
\(62\) −3.34334 −0.424604
\(63\) −4.88695 −0.615698
\(64\) 1.00000 0.125000
\(65\) 3.05764 0.379254
\(66\) 0 0
\(67\) 10.5718 1.29155 0.645773 0.763529i \(-0.276535\pi\)
0.645773 + 0.763529i \(0.276535\pi\)
\(68\) −6.94459 −0.842156
\(69\) 4.86179 0.585291
\(70\) 1.00000 0.119523
\(71\) −2.30414 −0.273451 −0.136725 0.990609i \(-0.543658\pi\)
−0.136725 + 0.990609i \(0.543658\pi\)
\(72\) −4.88695 −0.575933
\(73\) 13.1113 1.53456 0.767278 0.641314i \(-0.221611\pi\)
0.767278 + 0.641314i \(0.221611\pi\)
\(74\) 7.32736 0.851788
\(75\) −2.80837 −0.324283
\(76\) 4.69704 0.538787
\(77\) 0 0
\(78\) 8.58700 0.972286
\(79\) −4.25829 −0.479095 −0.239547 0.970885i \(-0.576999\pi\)
−0.239547 + 0.970885i \(0.576999\pi\)
\(80\) 1.00000 0.111803
\(81\) 0.221440 0.0246044
\(82\) 7.58977 0.838150
\(83\) −3.86073 −0.423770 −0.211885 0.977295i \(-0.567960\pi\)
−0.211885 + 0.977295i \(0.567960\pi\)
\(84\) 2.80837 0.306418
\(85\) −6.94459 −0.753247
\(86\) 7.27592 0.784583
\(87\) −18.9440 −2.03101
\(88\) 0 0
\(89\) −12.3140 −1.30529 −0.652643 0.757665i \(-0.726340\pi\)
−0.652643 + 0.757665i \(0.726340\pi\)
\(90\) −4.88695 −0.515130
\(91\) −3.05764 −0.320528
\(92\) −1.73118 −0.180488
\(93\) −9.38933 −0.973628
\(94\) −4.00555 −0.413141
\(95\) 4.69704 0.481906
\(96\) 2.80837 0.286628
\(97\) −6.94282 −0.704936 −0.352468 0.935824i \(-0.614658\pi\)
−0.352468 + 0.935824i \(0.614658\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −13.5915 −1.35240 −0.676202 0.736717i \(-0.736375\pi\)
−0.676202 + 0.736717i \(0.736375\pi\)
\(102\) −19.5030 −1.93108
\(103\) 15.8591 1.56264 0.781320 0.624131i \(-0.214547\pi\)
0.781320 + 0.624131i \(0.214547\pi\)
\(104\) −3.05764 −0.299826
\(105\) 2.80837 0.274069
\(106\) 1.70116 0.165232
\(107\) 4.39685 0.425060 0.212530 0.977155i \(-0.431830\pi\)
0.212530 + 0.977155i \(0.431830\pi\)
\(108\) −5.29926 −0.509922
\(109\) 14.4931 1.38819 0.694095 0.719884i \(-0.255805\pi\)
0.694095 + 0.719884i \(0.255805\pi\)
\(110\) 0 0
\(111\) 20.5779 1.95317
\(112\) −1.00000 −0.0944911
\(113\) −19.3326 −1.81866 −0.909328 0.416081i \(-0.863403\pi\)
−0.909328 + 0.416081i \(0.863403\pi\)
\(114\) 13.1910 1.23545
\(115\) −1.73118 −0.161433
\(116\) 6.74554 0.626308
\(117\) 14.9425 1.38144
\(118\) −7.64741 −0.704001
\(119\) 6.94459 0.636610
\(120\) 2.80837 0.256368
\(121\) 0 0
\(122\) 8.93050 0.808529
\(123\) 21.3149 1.92190
\(124\) 3.34334 0.300240
\(125\) 1.00000 0.0894427
\(126\) 4.88695 0.435364
\(127\) −21.5848 −1.91534 −0.957669 0.287870i \(-0.907053\pi\)
−0.957669 + 0.287870i \(0.907053\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 20.4335 1.79907
\(130\) −3.05764 −0.268173
\(131\) −5.23468 −0.457356 −0.228678 0.973502i \(-0.573440\pi\)
−0.228678 + 0.973502i \(0.573440\pi\)
\(132\) 0 0
\(133\) −4.69704 −0.407285
\(134\) −10.5718 −0.913261
\(135\) −5.29926 −0.456088
\(136\) 6.94459 0.595494
\(137\) 5.00039 0.427212 0.213606 0.976920i \(-0.431479\pi\)
0.213606 + 0.976920i \(0.431479\pi\)
\(138\) −4.86179 −0.413863
\(139\) 3.08024 0.261262 0.130631 0.991431i \(-0.458300\pi\)
0.130631 + 0.991431i \(0.458300\pi\)
\(140\) −1.00000 −0.0845154
\(141\) −11.2491 −0.947343
\(142\) 2.30414 0.193359
\(143\) 0 0
\(144\) 4.88695 0.407246
\(145\) 6.74554 0.560187
\(146\) −13.1113 −1.08510
\(147\) −2.80837 −0.231631
\(148\) −7.32736 −0.602305
\(149\) 15.8858 1.30141 0.650706 0.759330i \(-0.274473\pi\)
0.650706 + 0.759330i \(0.274473\pi\)
\(150\) 2.80837 0.229303
\(151\) 5.45916 0.444260 0.222130 0.975017i \(-0.428699\pi\)
0.222130 + 0.975017i \(0.428699\pi\)
\(152\) −4.69704 −0.380980
\(153\) −33.9379 −2.74372
\(154\) 0 0
\(155\) 3.34334 0.268543
\(156\) −8.58700 −0.687510
\(157\) 8.44070 0.673641 0.336820 0.941569i \(-0.390648\pi\)
0.336820 + 0.941569i \(0.390648\pi\)
\(158\) 4.25829 0.338771
\(159\) 4.77750 0.378880
\(160\) −1.00000 −0.0790569
\(161\) 1.73118 0.136436
\(162\) −0.221440 −0.0173980
\(163\) 9.74355 0.763174 0.381587 0.924333i \(-0.375378\pi\)
0.381587 + 0.924333i \(0.375378\pi\)
\(164\) −7.58977 −0.592662
\(165\) 0 0
\(166\) 3.86073 0.299650
\(167\) −20.2081 −1.56375 −0.781873 0.623437i \(-0.785736\pi\)
−0.781873 + 0.623437i \(0.785736\pi\)
\(168\) −2.80837 −0.216671
\(169\) −3.65082 −0.280833
\(170\) 6.94459 0.532626
\(171\) 22.9542 1.75535
\(172\) −7.27592 −0.554784
\(173\) −1.06417 −0.0809073 −0.0404537 0.999181i \(-0.512880\pi\)
−0.0404537 + 0.999181i \(0.512880\pi\)
\(174\) 18.9440 1.43614
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) −21.4768 −1.61429
\(178\) 12.3140 0.922977
\(179\) −5.73780 −0.428863 −0.214432 0.976739i \(-0.568790\pi\)
−0.214432 + 0.976739i \(0.568790\pi\)
\(180\) 4.88695 0.364252
\(181\) 3.77619 0.280682 0.140341 0.990103i \(-0.455180\pi\)
0.140341 + 0.990103i \(0.455180\pi\)
\(182\) 3.05764 0.226648
\(183\) 25.0802 1.85398
\(184\) 1.73118 0.127624
\(185\) −7.32736 −0.538718
\(186\) 9.38933 0.688459
\(187\) 0 0
\(188\) 4.00555 0.292135
\(189\) 5.29926 0.385465
\(190\) −4.69704 −0.340759
\(191\) 11.7298 0.848736 0.424368 0.905490i \(-0.360496\pi\)
0.424368 + 0.905490i \(0.360496\pi\)
\(192\) −2.80837 −0.202677
\(193\) −18.9879 −1.36678 −0.683391 0.730053i \(-0.739495\pi\)
−0.683391 + 0.730053i \(0.739495\pi\)
\(194\) 6.94282 0.498465
\(195\) −8.58700 −0.614928
\(196\) 1.00000 0.0714286
\(197\) 3.27508 0.233340 0.116670 0.993171i \(-0.462778\pi\)
0.116670 + 0.993171i \(0.462778\pi\)
\(198\) 0 0
\(199\) 14.7438 1.04516 0.522580 0.852590i \(-0.324970\pi\)
0.522580 + 0.852590i \(0.324970\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −29.6894 −2.09413
\(202\) 13.5915 0.956294
\(203\) −6.74554 −0.473444
\(204\) 19.5030 1.36548
\(205\) −7.58977 −0.530093
\(206\) −15.8591 −1.10495
\(207\) −8.46019 −0.588024
\(208\) 3.05764 0.212009
\(209\) 0 0
\(210\) −2.80837 −0.193796
\(211\) 13.6709 0.941143 0.470572 0.882362i \(-0.344048\pi\)
0.470572 + 0.882362i \(0.344048\pi\)
\(212\) −1.70116 −0.116836
\(213\) 6.47087 0.443377
\(214\) −4.39685 −0.300563
\(215\) −7.27592 −0.496214
\(216\) 5.29926 0.360569
\(217\) −3.34334 −0.226960
\(218\) −14.4931 −0.981598
\(219\) −36.8213 −2.48815
\(220\) 0 0
\(221\) −21.2341 −1.42836
\(222\) −20.5779 −1.38110
\(223\) −5.23577 −0.350613 −0.175307 0.984514i \(-0.556092\pi\)
−0.175307 + 0.984514i \(0.556092\pi\)
\(224\) 1.00000 0.0668153
\(225\) 4.88695 0.325797
\(226\) 19.3326 1.28598
\(227\) 22.5392 1.49598 0.747991 0.663709i \(-0.231019\pi\)
0.747991 + 0.663709i \(0.231019\pi\)
\(228\) −13.1910 −0.873597
\(229\) 1.54850 0.102328 0.0511639 0.998690i \(-0.483707\pi\)
0.0511639 + 0.998690i \(0.483707\pi\)
\(230\) 1.73118 0.114151
\(231\) 0 0
\(232\) −6.74554 −0.442867
\(233\) 6.59831 0.432270 0.216135 0.976364i \(-0.430655\pi\)
0.216135 + 0.976364i \(0.430655\pi\)
\(234\) −14.9425 −0.976825
\(235\) 4.00555 0.261293
\(236\) 7.64741 0.497804
\(237\) 11.9589 0.776811
\(238\) −6.94459 −0.450151
\(239\) −9.32864 −0.603420 −0.301710 0.953400i \(-0.597557\pi\)
−0.301710 + 0.953400i \(0.597557\pi\)
\(240\) −2.80837 −0.181280
\(241\) 4.69741 0.302587 0.151293 0.988489i \(-0.451656\pi\)
0.151293 + 0.988489i \(0.451656\pi\)
\(242\) 0 0
\(243\) 15.2759 0.979949
\(244\) −8.93050 −0.571717
\(245\) 1.00000 0.0638877
\(246\) −21.3149 −1.35899
\(247\) 14.3619 0.913823
\(248\) −3.34334 −0.212302
\(249\) 10.8424 0.687106
\(250\) −1.00000 −0.0632456
\(251\) 25.1086 1.58484 0.792420 0.609976i \(-0.208821\pi\)
0.792420 + 0.609976i \(0.208821\pi\)
\(252\) −4.88695 −0.307849
\(253\) 0 0
\(254\) 21.5848 1.35435
\(255\) 19.5030 1.22133
\(256\) 1.00000 0.0625000
\(257\) 22.2671 1.38898 0.694491 0.719502i \(-0.255630\pi\)
0.694491 + 0.719502i \(0.255630\pi\)
\(258\) −20.4335 −1.27213
\(259\) 7.32736 0.455300
\(260\) 3.05764 0.189627
\(261\) 32.9651 2.04049
\(262\) 5.23468 0.323400
\(263\) 12.5549 0.774170 0.387085 0.922044i \(-0.373482\pi\)
0.387085 + 0.922044i \(0.373482\pi\)
\(264\) 0 0
\(265\) −1.70116 −0.104502
\(266\) 4.69704 0.287994
\(267\) 34.5824 2.11641
\(268\) 10.5718 0.645773
\(269\) −7.54481 −0.460015 −0.230007 0.973189i \(-0.573875\pi\)
−0.230007 + 0.973189i \(0.573875\pi\)
\(270\) 5.29926 0.322503
\(271\) −27.7828 −1.68768 −0.843842 0.536592i \(-0.819711\pi\)
−0.843842 + 0.536592i \(0.819711\pi\)
\(272\) −6.94459 −0.421078
\(273\) 8.58700 0.519709
\(274\) −5.00039 −0.302085
\(275\) 0 0
\(276\) 4.86179 0.292646
\(277\) −18.8691 −1.13374 −0.566868 0.823809i \(-0.691845\pi\)
−0.566868 + 0.823809i \(0.691845\pi\)
\(278\) −3.08024 −0.184740
\(279\) 16.3387 0.978173
\(280\) 1.00000 0.0597614
\(281\) −6.86665 −0.409630 −0.204815 0.978801i \(-0.565659\pi\)
−0.204815 + 0.978801i \(0.565659\pi\)
\(282\) 11.2491 0.669873
\(283\) 25.1097 1.49261 0.746307 0.665601i \(-0.231825\pi\)
0.746307 + 0.665601i \(0.231825\pi\)
\(284\) −2.30414 −0.136725
\(285\) −13.1910 −0.781369
\(286\) 0 0
\(287\) 7.58977 0.448010
\(288\) −4.88695 −0.287966
\(289\) 31.2274 1.83690
\(290\) −6.74554 −0.396112
\(291\) 19.4980 1.14299
\(292\) 13.1113 0.767278
\(293\) −16.5950 −0.969491 −0.484746 0.874655i \(-0.661088\pi\)
−0.484746 + 0.874655i \(0.661088\pi\)
\(294\) 2.80837 0.163788
\(295\) 7.64741 0.445250
\(296\) 7.32736 0.425894
\(297\) 0 0
\(298\) −15.8858 −0.920237
\(299\) −5.29332 −0.306121
\(300\) −2.80837 −0.162141
\(301\) 7.27592 0.419377
\(302\) −5.45916 −0.314139
\(303\) 38.1699 2.19281
\(304\) 4.69704 0.269394
\(305\) −8.93050 −0.511359
\(306\) 33.9379 1.94010
\(307\) 31.2886 1.78573 0.892867 0.450321i \(-0.148691\pi\)
0.892867 + 0.450321i \(0.148691\pi\)
\(308\) 0 0
\(309\) −44.5381 −2.53368
\(310\) −3.34334 −0.189889
\(311\) −5.69208 −0.322768 −0.161384 0.986892i \(-0.551596\pi\)
−0.161384 + 0.986892i \(0.551596\pi\)
\(312\) 8.58700 0.486143
\(313\) 17.4320 0.985315 0.492658 0.870223i \(-0.336026\pi\)
0.492658 + 0.870223i \(0.336026\pi\)
\(314\) −8.44070 −0.476336
\(315\) −4.88695 −0.275349
\(316\) −4.25829 −0.239547
\(317\) 21.2720 1.19475 0.597376 0.801961i \(-0.296210\pi\)
0.597376 + 0.801961i \(0.296210\pi\)
\(318\) −4.77750 −0.267909
\(319\) 0 0
\(320\) 1.00000 0.0559017
\(321\) −12.3480 −0.689198
\(322\) −1.73118 −0.0964748
\(323\) −32.6190 −1.81497
\(324\) 0.221440 0.0123022
\(325\) 3.05764 0.169607
\(326\) −9.74355 −0.539645
\(327\) −40.7021 −2.25083
\(328\) 7.58977 0.419075
\(329\) −4.00555 −0.220833
\(330\) 0 0
\(331\) 0.712600 0.0391680 0.0195840 0.999808i \(-0.493766\pi\)
0.0195840 + 0.999808i \(0.493766\pi\)
\(332\) −3.86073 −0.211885
\(333\) −35.8084 −1.96229
\(334\) 20.2081 1.10574
\(335\) 10.5718 0.577597
\(336\) 2.80837 0.153209
\(337\) −28.7577 −1.56653 −0.783267 0.621686i \(-0.786448\pi\)
−0.783267 + 0.621686i \(0.786448\pi\)
\(338\) 3.65082 0.198579
\(339\) 54.2930 2.94879
\(340\) −6.94459 −0.376623
\(341\) 0 0
\(342\) −22.9542 −1.24122
\(343\) −1.00000 −0.0539949
\(344\) 7.27592 0.392291
\(345\) 4.86179 0.261750
\(346\) 1.06417 0.0572101
\(347\) −34.5981 −1.85732 −0.928662 0.370927i \(-0.879040\pi\)
−0.928662 + 0.370927i \(0.879040\pi\)
\(348\) −18.9440 −1.01550
\(349\) −15.4593 −0.827518 −0.413759 0.910386i \(-0.635784\pi\)
−0.413759 + 0.910386i \(0.635784\pi\)
\(350\) 1.00000 0.0534522
\(351\) −16.2032 −0.864865
\(352\) 0 0
\(353\) 2.31479 0.123204 0.0616019 0.998101i \(-0.480379\pi\)
0.0616019 + 0.998101i \(0.480379\pi\)
\(354\) 21.4768 1.14148
\(355\) −2.30414 −0.122291
\(356\) −12.3140 −0.652643
\(357\) −19.5030 −1.03221
\(358\) 5.73780 0.303252
\(359\) −5.56161 −0.293530 −0.146765 0.989171i \(-0.546886\pi\)
−0.146765 + 0.989171i \(0.546886\pi\)
\(360\) −4.88695 −0.257565
\(361\) 3.06217 0.161167
\(362\) −3.77619 −0.198472
\(363\) 0 0
\(364\) −3.05764 −0.160264
\(365\) 13.1113 0.686274
\(366\) −25.0802 −1.31096
\(367\) 26.0861 1.36168 0.680842 0.732430i \(-0.261614\pi\)
0.680842 + 0.732430i \(0.261614\pi\)
\(368\) −1.73118 −0.0902439
\(369\) −37.0908 −1.93087
\(370\) 7.32736 0.380931
\(371\) 1.70116 0.0883200
\(372\) −9.38933 −0.486814
\(373\) −28.3183 −1.46627 −0.733134 0.680085i \(-0.761943\pi\)
−0.733134 + 0.680085i \(0.761943\pi\)
\(374\) 0 0
\(375\) −2.80837 −0.145024
\(376\) −4.00555 −0.206571
\(377\) 20.6255 1.06227
\(378\) −5.29926 −0.272565
\(379\) −32.5648 −1.67274 −0.836371 0.548164i \(-0.815327\pi\)
−0.836371 + 0.548164i \(0.815327\pi\)
\(380\) 4.69704 0.240953
\(381\) 60.6181 3.10556
\(382\) −11.7298 −0.600147
\(383\) −12.3943 −0.633319 −0.316660 0.948539i \(-0.602561\pi\)
−0.316660 + 0.948539i \(0.602561\pi\)
\(384\) 2.80837 0.143314
\(385\) 0 0
\(386\) 18.9879 0.966461
\(387\) −35.5571 −1.80747
\(388\) −6.94282 −0.352468
\(389\) −19.8793 −1.00792 −0.503960 0.863727i \(-0.668124\pi\)
−0.503960 + 0.863727i \(0.668124\pi\)
\(390\) 8.58700 0.434819
\(391\) 12.0223 0.607995
\(392\) −1.00000 −0.0505076
\(393\) 14.7009 0.741564
\(394\) −3.27508 −0.164996
\(395\) −4.25829 −0.214258
\(396\) 0 0
\(397\) 27.6288 1.38665 0.693325 0.720625i \(-0.256145\pi\)
0.693325 + 0.720625i \(0.256145\pi\)
\(398\) −14.7438 −0.739040
\(399\) 13.1910 0.660377
\(400\) 1.00000 0.0500000
\(401\) 30.1562 1.50593 0.752965 0.658061i \(-0.228623\pi\)
0.752965 + 0.658061i \(0.228623\pi\)
\(402\) 29.6894 1.48077
\(403\) 10.2227 0.509230
\(404\) −13.5915 −0.676202
\(405\) 0.221440 0.0110034
\(406\) 6.74554 0.334776
\(407\) 0 0
\(408\) −19.5030 −0.965542
\(409\) −18.8749 −0.933302 −0.466651 0.884442i \(-0.654540\pi\)
−0.466651 + 0.884442i \(0.654540\pi\)
\(410\) 7.58977 0.374832
\(411\) −14.0430 −0.692688
\(412\) 15.8591 0.781320
\(413\) −7.64741 −0.376305
\(414\) 8.46019 0.415795
\(415\) −3.86073 −0.189516
\(416\) −3.05764 −0.149913
\(417\) −8.65045 −0.423614
\(418\) 0 0
\(419\) 8.51801 0.416132 0.208066 0.978115i \(-0.433283\pi\)
0.208066 + 0.978115i \(0.433283\pi\)
\(420\) 2.80837 0.137035
\(421\) −13.4733 −0.656647 −0.328323 0.944565i \(-0.606484\pi\)
−0.328323 + 0.944565i \(0.606484\pi\)
\(422\) −13.6709 −0.665489
\(423\) 19.5749 0.951766
\(424\) 1.70116 0.0826158
\(425\) −6.94459 −0.336862
\(426\) −6.47087 −0.313515
\(427\) 8.93050 0.432177
\(428\) 4.39685 0.212530
\(429\) 0 0
\(430\) 7.27592 0.350876
\(431\) 27.8105 1.33959 0.669793 0.742548i \(-0.266383\pi\)
0.669793 + 0.742548i \(0.266383\pi\)
\(432\) −5.29926 −0.254961
\(433\) −19.9578 −0.959111 −0.479555 0.877512i \(-0.659202\pi\)
−0.479555 + 0.877512i \(0.659202\pi\)
\(434\) 3.34334 0.160485
\(435\) −18.9440 −0.908295
\(436\) 14.4931 0.694095
\(437\) −8.13141 −0.388978
\(438\) 36.8213 1.75939
\(439\) 10.7367 0.512436 0.256218 0.966619i \(-0.417524\pi\)
0.256218 + 0.966619i \(0.417524\pi\)
\(440\) 0 0
\(441\) 4.88695 0.232712
\(442\) 21.2341 1.01000
\(443\) −10.0787 −0.478854 −0.239427 0.970914i \(-0.576960\pi\)
−0.239427 + 0.970914i \(0.576960\pi\)
\(444\) 20.5779 0.976586
\(445\) −12.3140 −0.583742
\(446\) 5.23577 0.247921
\(447\) −44.6131 −2.11013
\(448\) −1.00000 −0.0472456
\(449\) −31.1076 −1.46806 −0.734030 0.679117i \(-0.762363\pi\)
−0.734030 + 0.679117i \(0.762363\pi\)
\(450\) −4.88695 −0.230373
\(451\) 0 0
\(452\) −19.3326 −0.909328
\(453\) −15.3313 −0.720330
\(454\) −22.5392 −1.05782
\(455\) −3.05764 −0.143344
\(456\) 13.1910 0.617727
\(457\) −26.2062 −1.22587 −0.612936 0.790132i \(-0.710012\pi\)
−0.612936 + 0.790132i \(0.710012\pi\)
\(458\) −1.54850 −0.0723566
\(459\) 36.8012 1.71773
\(460\) −1.73118 −0.0807166
\(461\) 5.33572 0.248509 0.124255 0.992250i \(-0.460346\pi\)
0.124255 + 0.992250i \(0.460346\pi\)
\(462\) 0 0
\(463\) 28.8817 1.34224 0.671122 0.741347i \(-0.265813\pi\)
0.671122 + 0.741347i \(0.265813\pi\)
\(464\) 6.74554 0.313154
\(465\) −9.38933 −0.435420
\(466\) −6.59831 −0.305661
\(467\) −37.6575 −1.74258 −0.871291 0.490766i \(-0.836717\pi\)
−0.871291 + 0.490766i \(0.836717\pi\)
\(468\) 14.9425 0.690720
\(469\) −10.5718 −0.488159
\(470\) −4.00555 −0.184762
\(471\) −23.7046 −1.09225
\(472\) −7.64741 −0.352001
\(473\) 0 0
\(474\) −11.9589 −0.549288
\(475\) 4.69704 0.215515
\(476\) 6.94459 0.318305
\(477\) −8.31350 −0.380649
\(478\) 9.32864 0.426682
\(479\) −25.5852 −1.16902 −0.584509 0.811387i \(-0.698713\pi\)
−0.584509 + 0.811387i \(0.698713\pi\)
\(480\) 2.80837 0.128184
\(481\) −22.4044 −1.02155
\(482\) −4.69741 −0.213961
\(483\) −4.86179 −0.221219
\(484\) 0 0
\(485\) −6.94282 −0.315257
\(486\) −15.2759 −0.692929
\(487\) 20.1951 0.915125 0.457563 0.889177i \(-0.348723\pi\)
0.457563 + 0.889177i \(0.348723\pi\)
\(488\) 8.93050 0.404265
\(489\) −27.3635 −1.23742
\(490\) −1.00000 −0.0451754
\(491\) −18.2166 −0.822104 −0.411052 0.911612i \(-0.634839\pi\)
−0.411052 + 0.911612i \(0.634839\pi\)
\(492\) 21.3149 0.960950
\(493\) −46.8451 −2.10980
\(494\) −14.3619 −0.646171
\(495\) 0 0
\(496\) 3.34334 0.150120
\(497\) 2.30414 0.103355
\(498\) −10.8424 −0.485857
\(499\) −21.4887 −0.961966 −0.480983 0.876730i \(-0.659720\pi\)
−0.480983 + 0.876730i \(0.659720\pi\)
\(500\) 1.00000 0.0447214
\(501\) 56.7518 2.53548
\(502\) −25.1086 −1.12065
\(503\) 5.56623 0.248186 0.124093 0.992271i \(-0.460398\pi\)
0.124093 + 0.992271i \(0.460398\pi\)
\(504\) 4.88695 0.217682
\(505\) −13.5915 −0.604813
\(506\) 0 0
\(507\) 10.2529 0.455346
\(508\) −21.5848 −0.957669
\(509\) −7.22200 −0.320110 −0.160055 0.987108i \(-0.551167\pi\)
−0.160055 + 0.987108i \(0.551167\pi\)
\(510\) −19.5030 −0.863607
\(511\) −13.1113 −0.580008
\(512\) −1.00000 −0.0441942
\(513\) −24.8908 −1.09896
\(514\) −22.2671 −0.982158
\(515\) 15.8591 0.698833
\(516\) 20.4335 0.899534
\(517\) 0 0
\(518\) −7.32736 −0.321946
\(519\) 2.98858 0.131184
\(520\) −3.05764 −0.134086
\(521\) −6.64589 −0.291162 −0.145581 0.989346i \(-0.546505\pi\)
−0.145581 + 0.989346i \(0.546505\pi\)
\(522\) −32.9651 −1.44285
\(523\) −24.8014 −1.08449 −0.542246 0.840220i \(-0.682426\pi\)
−0.542246 + 0.840220i \(0.682426\pi\)
\(524\) −5.23468 −0.228678
\(525\) 2.80837 0.122567
\(526\) −12.5549 −0.547421
\(527\) −23.2181 −1.01140
\(528\) 0 0
\(529\) −20.0030 −0.869697
\(530\) 1.70116 0.0738938
\(531\) 37.3725 1.62183
\(532\) −4.69704 −0.203642
\(533\) −23.2068 −1.00520
\(534\) −34.5824 −1.49653
\(535\) 4.39685 0.190092
\(536\) −10.5718 −0.456631
\(537\) 16.1139 0.695365
\(538\) 7.54481 0.325280
\(539\) 0 0
\(540\) −5.29926 −0.228044
\(541\) 34.8421 1.49798 0.748990 0.662581i \(-0.230539\pi\)
0.748990 + 0.662581i \(0.230539\pi\)
\(542\) 27.7828 1.19337
\(543\) −10.6049 −0.455101
\(544\) 6.94459 0.297747
\(545\) 14.4931 0.620817
\(546\) −8.58700 −0.367489
\(547\) −22.5413 −0.963798 −0.481899 0.876227i \(-0.660053\pi\)
−0.481899 + 0.876227i \(0.660053\pi\)
\(548\) 5.00039 0.213606
\(549\) −43.6429 −1.86263
\(550\) 0 0
\(551\) 31.6841 1.34979
\(552\) −4.86179 −0.206932
\(553\) 4.25829 0.181081
\(554\) 18.8691 0.801672
\(555\) 20.5779 0.873485
\(556\) 3.08024 0.130631
\(557\) −0.716288 −0.0303501 −0.0151750 0.999885i \(-0.504831\pi\)
−0.0151750 + 0.999885i \(0.504831\pi\)
\(558\) −16.3387 −0.691673
\(559\) −22.2472 −0.940955
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 6.86665 0.289652
\(563\) −44.5640 −1.87815 −0.939075 0.343713i \(-0.888315\pi\)
−0.939075 + 0.343713i \(0.888315\pi\)
\(564\) −11.2491 −0.473672
\(565\) −19.3326 −0.813327
\(566\) −25.1097 −1.05544
\(567\) −0.221440 −0.00929961
\(568\) 2.30414 0.0966795
\(569\) −9.43478 −0.395526 −0.197763 0.980250i \(-0.563368\pi\)
−0.197763 + 0.980250i \(0.563368\pi\)
\(570\) 13.1910 0.552511
\(571\) −31.2693 −1.30858 −0.654289 0.756244i \(-0.727032\pi\)
−0.654289 + 0.756244i \(0.727032\pi\)
\(572\) 0 0
\(573\) −32.9415 −1.37615
\(574\) −7.58977 −0.316791
\(575\) −1.73118 −0.0721951
\(576\) 4.88695 0.203623
\(577\) 33.7440 1.40478 0.702390 0.711792i \(-0.252116\pi\)
0.702390 + 0.711792i \(0.252116\pi\)
\(578\) −31.2274 −1.29889
\(579\) 53.3252 2.21612
\(580\) 6.74554 0.280093
\(581\) 3.86073 0.160170
\(582\) −19.4980 −0.808219
\(583\) 0 0
\(584\) −13.1113 −0.542548
\(585\) 14.9425 0.617798
\(586\) 16.5950 0.685534
\(587\) 26.2246 1.08240 0.541202 0.840892i \(-0.317969\pi\)
0.541202 + 0.840892i \(0.317969\pi\)
\(588\) −2.80837 −0.115815
\(589\) 15.7038 0.647063
\(590\) −7.64741 −0.314839
\(591\) −9.19763 −0.378340
\(592\) −7.32736 −0.301153
\(593\) −23.6431 −0.970906 −0.485453 0.874263i \(-0.661345\pi\)
−0.485453 + 0.874263i \(0.661345\pi\)
\(594\) 0 0
\(595\) 6.94459 0.284701
\(596\) 15.8858 0.650706
\(597\) −41.4060 −1.69464
\(598\) 5.29332 0.216460
\(599\) −35.9599 −1.46928 −0.734640 0.678457i \(-0.762649\pi\)
−0.734640 + 0.678457i \(0.762649\pi\)
\(600\) 2.80837 0.114651
\(601\) 31.4056 1.28106 0.640530 0.767933i \(-0.278715\pi\)
0.640530 + 0.767933i \(0.278715\pi\)
\(602\) −7.27592 −0.296544
\(603\) 51.6637 2.10391
\(604\) 5.45916 0.222130
\(605\) 0 0
\(606\) −38.1699 −1.55055
\(607\) −35.9288 −1.45830 −0.729152 0.684352i \(-0.760085\pi\)
−0.729152 + 0.684352i \(0.760085\pi\)
\(608\) −4.69704 −0.190490
\(609\) 18.9440 0.767649
\(610\) 8.93050 0.361585
\(611\) 12.2475 0.495482
\(612\) −33.9379 −1.37186
\(613\) −47.6086 −1.92289 −0.961446 0.274992i \(-0.911325\pi\)
−0.961446 + 0.274992i \(0.911325\pi\)
\(614\) −31.2886 −1.26270
\(615\) 21.3149 0.859500
\(616\) 0 0
\(617\) −5.44411 −0.219172 −0.109586 0.993977i \(-0.534952\pi\)
−0.109586 + 0.993977i \(0.534952\pi\)
\(618\) 44.5381 1.79159
\(619\) 24.6723 0.991665 0.495833 0.868418i \(-0.334863\pi\)
0.495833 + 0.868418i \(0.334863\pi\)
\(620\) 3.34334 0.134272
\(621\) 9.17397 0.368139
\(622\) 5.69208 0.228231
\(623\) 12.3140 0.493352
\(624\) −8.58700 −0.343755
\(625\) 1.00000 0.0400000
\(626\) −17.4320 −0.696723
\(627\) 0 0
\(628\) 8.44070 0.336820
\(629\) 50.8855 2.02894
\(630\) 4.88695 0.194701
\(631\) −15.8756 −0.631997 −0.315998 0.948760i \(-0.602339\pi\)
−0.315998 + 0.948760i \(0.602339\pi\)
\(632\) 4.25829 0.169386
\(633\) −38.3930 −1.52598
\(634\) −21.2720 −0.844818
\(635\) −21.5848 −0.856566
\(636\) 4.77750 0.189440
\(637\) 3.05764 0.121148
\(638\) 0 0
\(639\) −11.2602 −0.445447
\(640\) −1.00000 −0.0395285
\(641\) −3.37605 −0.133346 −0.0666730 0.997775i \(-0.521238\pi\)
−0.0666730 + 0.997775i \(0.521238\pi\)
\(642\) 12.3480 0.487336
\(643\) 20.7120 0.816803 0.408402 0.912802i \(-0.366086\pi\)
0.408402 + 0.912802i \(0.366086\pi\)
\(644\) 1.73118 0.0682180
\(645\) 20.4335 0.804568
\(646\) 32.6190 1.28338
\(647\) −6.82833 −0.268449 −0.134225 0.990951i \(-0.542854\pi\)
−0.134225 + 0.990951i \(0.542854\pi\)
\(648\) −0.221440 −0.00869898
\(649\) 0 0
\(650\) −3.05764 −0.119931
\(651\) 9.38933 0.367997
\(652\) 9.74355 0.381587
\(653\) 30.9046 1.20939 0.604696 0.796456i \(-0.293295\pi\)
0.604696 + 0.796456i \(0.293295\pi\)
\(654\) 40.7021 1.59158
\(655\) −5.23468 −0.204536
\(656\) −7.58977 −0.296331
\(657\) 64.0741 2.49977
\(658\) 4.00555 0.156153
\(659\) −17.6185 −0.686318 −0.343159 0.939277i \(-0.611497\pi\)
−0.343159 + 0.939277i \(0.611497\pi\)
\(660\) 0 0
\(661\) 11.7396 0.456617 0.228309 0.973589i \(-0.426680\pi\)
0.228309 + 0.973589i \(0.426680\pi\)
\(662\) −0.712600 −0.0276960
\(663\) 59.6332 2.31596
\(664\) 3.86073 0.149825
\(665\) −4.69704 −0.182143
\(666\) 35.8084 1.38755
\(667\) −11.6777 −0.452164
\(668\) −20.2081 −0.781873
\(669\) 14.7040 0.568489
\(670\) −10.5718 −0.408423
\(671\) 0 0
\(672\) −2.80837 −0.108335
\(673\) −17.0012 −0.655347 −0.327673 0.944791i \(-0.606265\pi\)
−0.327673 + 0.944791i \(0.606265\pi\)
\(674\) 28.7577 1.10771
\(675\) −5.29926 −0.203969
\(676\) −3.65082 −0.140416
\(677\) 3.84431 0.147749 0.0738744 0.997268i \(-0.476464\pi\)
0.0738744 + 0.997268i \(0.476464\pi\)
\(678\) −54.2930 −2.08511
\(679\) 6.94282 0.266441
\(680\) 6.94459 0.266313
\(681\) −63.2986 −2.42561
\(682\) 0 0
\(683\) −2.04787 −0.0783595 −0.0391798 0.999232i \(-0.512474\pi\)
−0.0391798 + 0.999232i \(0.512474\pi\)
\(684\) 22.9542 0.877676
\(685\) 5.00039 0.191055
\(686\) 1.00000 0.0381802
\(687\) −4.34876 −0.165916
\(688\) −7.27592 −0.277392
\(689\) −5.20155 −0.198163
\(690\) −4.86179 −0.185085
\(691\) −24.0616 −0.915348 −0.457674 0.889120i \(-0.651317\pi\)
−0.457674 + 0.889120i \(0.651317\pi\)
\(692\) −1.06417 −0.0404537
\(693\) 0 0
\(694\) 34.5981 1.31333
\(695\) 3.08024 0.116840
\(696\) 18.9440 0.718070
\(697\) 52.7079 1.99645
\(698\) 15.4593 0.585144
\(699\) −18.5305 −0.700888
\(700\) −1.00000 −0.0377964
\(701\) −7.99724 −0.302052 −0.151026 0.988530i \(-0.548258\pi\)
−0.151026 + 0.988530i \(0.548258\pi\)
\(702\) 16.2032 0.611552
\(703\) −34.4169 −1.29806
\(704\) 0 0
\(705\) −11.2491 −0.423665
\(706\) −2.31479 −0.0871183
\(707\) 13.5915 0.511160
\(708\) −21.4768 −0.807147
\(709\) 4.28038 0.160753 0.0803765 0.996765i \(-0.474388\pi\)
0.0803765 + 0.996765i \(0.474388\pi\)
\(710\) 2.30414 0.0864727
\(711\) −20.8100 −0.780438
\(712\) 12.3140 0.461488
\(713\) −5.78791 −0.216759
\(714\) 19.5030 0.729881
\(715\) 0 0
\(716\) −5.73780 −0.214432
\(717\) 26.1983 0.978393
\(718\) 5.56161 0.207557
\(719\) 36.8044 1.37257 0.686286 0.727332i \(-0.259240\pi\)
0.686286 + 0.727332i \(0.259240\pi\)
\(720\) 4.88695 0.182126
\(721\) −15.8591 −0.590622
\(722\) −3.06217 −0.113962
\(723\) −13.1921 −0.490618
\(724\) 3.77619 0.140341
\(725\) 6.74554 0.250523
\(726\) 0 0
\(727\) −46.5408 −1.72610 −0.863051 0.505116i \(-0.831450\pi\)
−0.863051 + 0.505116i \(0.831450\pi\)
\(728\) 3.05764 0.113324
\(729\) −43.5647 −1.61351
\(730\) −13.1113 −0.485269
\(731\) 50.5283 1.86886
\(732\) 25.0802 0.926989
\(733\) −15.6659 −0.578633 −0.289316 0.957234i \(-0.593428\pi\)
−0.289316 + 0.957234i \(0.593428\pi\)
\(734\) −26.0861 −0.962857
\(735\) −2.80837 −0.103588
\(736\) 1.73118 0.0638121
\(737\) 0 0
\(738\) 37.0908 1.36533
\(739\) −6.15005 −0.226233 −0.113117 0.993582i \(-0.536083\pi\)
−0.113117 + 0.993582i \(0.536083\pi\)
\(740\) −7.32736 −0.269359
\(741\) −40.3334 −1.48169
\(742\) −1.70116 −0.0624517
\(743\) −27.2721 −1.00052 −0.500259 0.865876i \(-0.666762\pi\)
−0.500259 + 0.865876i \(0.666762\pi\)
\(744\) 9.38933 0.344229
\(745\) 15.8858 0.582009
\(746\) 28.3183 1.03681
\(747\) −18.8672 −0.690314
\(748\) 0 0
\(749\) −4.39685 −0.160657
\(750\) 2.80837 0.102547
\(751\) 7.95258 0.290194 0.145097 0.989417i \(-0.453651\pi\)
0.145097 + 0.989417i \(0.453651\pi\)
\(752\) 4.00555 0.146067
\(753\) −70.5142 −2.56968
\(754\) −20.6255 −0.751135
\(755\) 5.45916 0.198679
\(756\) 5.29926 0.192732
\(757\) 24.8369 0.902713 0.451356 0.892344i \(-0.350940\pi\)
0.451356 + 0.892344i \(0.350940\pi\)
\(758\) 32.5648 1.18281
\(759\) 0 0
\(760\) −4.69704 −0.170379
\(761\) −34.0673 −1.23494 −0.617469 0.786595i \(-0.711842\pi\)
−0.617469 + 0.786595i \(0.711842\pi\)
\(762\) −60.6181 −2.19596
\(763\) −14.4931 −0.524686
\(764\) 11.7298 0.424368
\(765\) −33.9379 −1.22703
\(766\) 12.3943 0.447824
\(767\) 23.3831 0.844313
\(768\) −2.80837 −0.101338
\(769\) −10.7430 −0.387404 −0.193702 0.981060i \(-0.562050\pi\)
−0.193702 + 0.981060i \(0.562050\pi\)
\(770\) 0 0
\(771\) −62.5342 −2.25211
\(772\) −18.9879 −0.683391
\(773\) −35.4511 −1.27509 −0.637544 0.770414i \(-0.720050\pi\)
−0.637544 + 0.770414i \(0.720050\pi\)
\(774\) 35.5571 1.27807
\(775\) 3.34334 0.120096
\(776\) 6.94282 0.249233
\(777\) −20.5779 −0.738230
\(778\) 19.8793 0.712708
\(779\) −35.6494 −1.27727
\(780\) −8.58700 −0.307464
\(781\) 0 0
\(782\) −12.0223 −0.429918
\(783\) −35.7464 −1.27747
\(784\) 1.00000 0.0357143
\(785\) 8.44070 0.301261
\(786\) −14.7009 −0.524365
\(787\) 22.9148 0.816825 0.408413 0.912797i \(-0.366082\pi\)
0.408413 + 0.912797i \(0.366082\pi\)
\(788\) 3.27508 0.116670
\(789\) −35.2589 −1.25525
\(790\) 4.25829 0.151503
\(791\) 19.3326 0.687387
\(792\) 0 0
\(793\) −27.3063 −0.969674
\(794\) −27.6288 −0.980510
\(795\) 4.77750 0.169440
\(796\) 14.7438 0.522580
\(797\) −42.9156 −1.52015 −0.760074 0.649836i \(-0.774837\pi\)
−0.760074 + 0.649836i \(0.774837\pi\)
\(798\) −13.1910 −0.466957
\(799\) −27.8169 −0.984092
\(800\) −1.00000 −0.0353553
\(801\) −60.1782 −2.12629
\(802\) −30.1562 −1.06485
\(803\) 0 0
\(804\) −29.6894 −1.04707
\(805\) 1.73118 0.0610160
\(806\) −10.2227 −0.360080
\(807\) 21.1886 0.745875
\(808\) 13.5915 0.478147
\(809\) −16.9974 −0.597598 −0.298799 0.954316i \(-0.596586\pi\)
−0.298799 + 0.954316i \(0.596586\pi\)
\(810\) −0.221440 −0.00778061
\(811\) 10.6859 0.375231 0.187616 0.982243i \(-0.439924\pi\)
0.187616 + 0.982243i \(0.439924\pi\)
\(812\) −6.74554 −0.236722
\(813\) 78.0243 2.73643
\(814\) 0 0
\(815\) 9.74355 0.341302
\(816\) 19.5030 0.682742
\(817\) −34.1753 −1.19564
\(818\) 18.8749 0.659944
\(819\) −14.9425 −0.522135
\(820\) −7.58977 −0.265046
\(821\) 20.3259 0.709379 0.354689 0.934984i \(-0.384587\pi\)
0.354689 + 0.934984i \(0.384587\pi\)
\(822\) 14.0430 0.489805
\(823\) −29.8555 −1.04070 −0.520348 0.853954i \(-0.674198\pi\)
−0.520348 + 0.853954i \(0.674198\pi\)
\(824\) −15.8591 −0.552476
\(825\) 0 0
\(826\) 7.64741 0.266088
\(827\) 4.53541 0.157712 0.0788558 0.996886i \(-0.474873\pi\)
0.0788558 + 0.996886i \(0.474873\pi\)
\(828\) −8.46019 −0.294012
\(829\) −29.0225 −1.00799 −0.503997 0.863705i \(-0.668138\pi\)
−0.503997 + 0.863705i \(0.668138\pi\)
\(830\) 3.86073 0.134008
\(831\) 52.9915 1.83826
\(832\) 3.05764 0.106005
\(833\) −6.94459 −0.240616
\(834\) 8.65045 0.299540
\(835\) −20.2081 −0.699329
\(836\) 0 0
\(837\) −17.7172 −0.612396
\(838\) −8.51801 −0.294250
\(839\) −3.85780 −0.133186 −0.0665930 0.997780i \(-0.521213\pi\)
−0.0665930 + 0.997780i \(0.521213\pi\)
\(840\) −2.80837 −0.0968980
\(841\) 16.5024 0.569047
\(842\) 13.4733 0.464319
\(843\) 19.2841 0.664180
\(844\) 13.6709 0.470572
\(845\) −3.65082 −0.125592
\(846\) −19.5749 −0.673000
\(847\) 0 0
\(848\) −1.70116 −0.0584182
\(849\) −70.5173 −2.42015
\(850\) 6.94459 0.238198
\(851\) 12.6850 0.434835
\(852\) 6.47087 0.221688
\(853\) 14.8560 0.508659 0.254329 0.967118i \(-0.418145\pi\)
0.254329 + 0.967118i \(0.418145\pi\)
\(854\) −8.93050 −0.305595
\(855\) 22.9542 0.785017
\(856\) −4.39685 −0.150281
\(857\) 21.4567 0.732948 0.366474 0.930428i \(-0.380565\pi\)
0.366474 + 0.930428i \(0.380565\pi\)
\(858\) 0 0
\(859\) −50.7945 −1.73309 −0.866543 0.499103i \(-0.833663\pi\)
−0.866543 + 0.499103i \(0.833663\pi\)
\(860\) −7.27592 −0.248107
\(861\) −21.3149 −0.726410
\(862\) −27.8105 −0.947231
\(863\) −13.6152 −0.463466 −0.231733 0.972779i \(-0.574440\pi\)
−0.231733 + 0.972779i \(0.574440\pi\)
\(864\) 5.29926 0.180285
\(865\) −1.06417 −0.0361829
\(866\) 19.9578 0.678194
\(867\) −87.6981 −2.97838
\(868\) −3.34334 −0.113480
\(869\) 0 0
\(870\) 18.9440 0.642262
\(871\) 32.3247 1.09528
\(872\) −14.4931 −0.490799
\(873\) −33.9292 −1.14833
\(874\) 8.13141 0.275049
\(875\) −1.00000 −0.0338062
\(876\) −36.8213 −1.24408
\(877\) −8.90340 −0.300647 −0.150323 0.988637i \(-0.548031\pi\)
−0.150323 + 0.988637i \(0.548031\pi\)
\(878\) −10.7367 −0.362347
\(879\) 46.6050 1.57195
\(880\) 0 0
\(881\) −2.11899 −0.0713904 −0.0356952 0.999363i \(-0.511365\pi\)
−0.0356952 + 0.999363i \(0.511365\pi\)
\(882\) −4.88695 −0.164552
\(883\) −10.2827 −0.346041 −0.173021 0.984918i \(-0.555353\pi\)
−0.173021 + 0.984918i \(0.555353\pi\)
\(884\) −21.2341 −0.714179
\(885\) −21.4768 −0.721934
\(886\) 10.0787 0.338601
\(887\) 5.63590 0.189235 0.0946175 0.995514i \(-0.469837\pi\)
0.0946175 + 0.995514i \(0.469837\pi\)
\(888\) −20.5779 −0.690551
\(889\) 21.5848 0.723930
\(890\) 12.3140 0.412768
\(891\) 0 0
\(892\) −5.23577 −0.175307
\(893\) 18.8142 0.629594
\(894\) 44.6131 1.49209
\(895\) −5.73780 −0.191793
\(896\) 1.00000 0.0334077
\(897\) 14.8656 0.496349
\(898\) 31.1076 1.03808
\(899\) 22.5526 0.752172
\(900\) 4.88695 0.162898
\(901\) 11.8139 0.393578
\(902\) 0 0
\(903\) −20.4335 −0.679984
\(904\) 19.3326 0.642992
\(905\) 3.77619 0.125525
\(906\) 15.3313 0.509350
\(907\) −39.0697 −1.29729 −0.648643 0.761092i \(-0.724663\pi\)
−0.648643 + 0.761092i \(0.724663\pi\)
\(908\) 22.5392 0.747991
\(909\) −66.4209 −2.20304
\(910\) 3.05764 0.101360
\(911\) 22.8604 0.757398 0.378699 0.925520i \(-0.376372\pi\)
0.378699 + 0.925520i \(0.376372\pi\)
\(912\) −13.1910 −0.436799
\(913\) 0 0
\(914\) 26.2062 0.866823
\(915\) 25.0802 0.829124
\(916\) 1.54850 0.0511639
\(917\) 5.23468 0.172864
\(918\) −36.8012 −1.21462
\(919\) 13.0335 0.429935 0.214968 0.976621i \(-0.431035\pi\)
0.214968 + 0.976621i \(0.431035\pi\)
\(920\) 1.73118 0.0570753
\(921\) −87.8699 −2.89541
\(922\) −5.33572 −0.175723
\(923\) −7.04523 −0.231896
\(924\) 0 0
\(925\) −7.32736 −0.240922
\(926\) −28.8817 −0.949110
\(927\) 77.5024 2.54551
\(928\) −6.74554 −0.221433
\(929\) 27.3966 0.898853 0.449427 0.893317i \(-0.351628\pi\)
0.449427 + 0.893317i \(0.351628\pi\)
\(930\) 9.38933 0.307888
\(931\) 4.69704 0.153939
\(932\) 6.59831 0.216135
\(933\) 15.9855 0.523341
\(934\) 37.6575 1.23219
\(935\) 0 0
\(936\) −14.9425 −0.488412
\(937\) −22.1523 −0.723684 −0.361842 0.932240i \(-0.617852\pi\)
−0.361842 + 0.932240i \(0.617852\pi\)
\(938\) 10.5718 0.345180
\(939\) −48.9556 −1.59760
\(940\) 4.00555 0.130647
\(941\) 12.0202 0.391847 0.195924 0.980619i \(-0.437229\pi\)
0.195924 + 0.980619i \(0.437229\pi\)
\(942\) 23.7046 0.772338
\(943\) 13.1392 0.427873
\(944\) 7.64741 0.248902
\(945\) 5.29926 0.172385
\(946\) 0 0
\(947\) −29.5380 −0.959857 −0.479929 0.877307i \(-0.659337\pi\)
−0.479929 + 0.877307i \(0.659337\pi\)
\(948\) 11.9589 0.388406
\(949\) 40.0895 1.30136
\(950\) −4.69704 −0.152392
\(951\) −59.7396 −1.93719
\(952\) −6.94459 −0.225076
\(953\) −38.2985 −1.24061 −0.620306 0.784360i \(-0.712991\pi\)
−0.620306 + 0.784360i \(0.712991\pi\)
\(954\) 8.31350 0.269160
\(955\) 11.7298 0.379566
\(956\) −9.32864 −0.301710
\(957\) 0 0
\(958\) 25.5852 0.826621
\(959\) −5.00039 −0.161471
\(960\) −2.80837 −0.0906398
\(961\) −19.8221 −0.639423
\(962\) 22.4044 0.722348
\(963\) 21.4872 0.692415
\(964\) 4.69741 0.151293
\(965\) −18.9879 −0.611243
\(966\) 4.86179 0.156426
\(967\) −2.04155 −0.0656517 −0.0328259 0.999461i \(-0.510451\pi\)
−0.0328259 + 0.999461i \(0.510451\pi\)
\(968\) 0 0
\(969\) 91.6063 2.94282
\(970\) 6.94282 0.222920
\(971\) −5.03293 −0.161514 −0.0807572 0.996734i \(-0.525734\pi\)
−0.0807572 + 0.996734i \(0.525734\pi\)
\(972\) 15.2759 0.489975
\(973\) −3.08024 −0.0987478
\(974\) −20.1951 −0.647091
\(975\) −8.58700 −0.275004
\(976\) −8.93050 −0.285858
\(977\) −10.1256 −0.323945 −0.161973 0.986795i \(-0.551786\pi\)
−0.161973 + 0.986795i \(0.551786\pi\)
\(978\) 27.3635 0.874989
\(979\) 0 0
\(980\) 1.00000 0.0319438
\(981\) 70.8272 2.26134
\(982\) 18.2166 0.581315
\(983\) 33.8126 1.07846 0.539228 0.842160i \(-0.318716\pi\)
0.539228 + 0.842160i \(0.318716\pi\)
\(984\) −21.3149 −0.679494
\(985\) 3.27508 0.104353
\(986\) 46.8451 1.49185
\(987\) 11.2491 0.358062
\(988\) 14.3619 0.456912
\(989\) 12.5959 0.400527
\(990\) 0 0
\(991\) 13.8867 0.441127 0.220563 0.975373i \(-0.429210\pi\)
0.220563 + 0.975373i \(0.429210\pi\)
\(992\) −3.34334 −0.106151
\(993\) −2.00124 −0.0635076
\(994\) −2.30414 −0.0730828
\(995\) 14.7438 0.467410
\(996\) 10.8424 0.343553
\(997\) −5.72382 −0.181275 −0.0906375 0.995884i \(-0.528890\pi\)
−0.0906375 + 0.995884i \(0.528890\pi\)
\(998\) 21.4887 0.680213
\(999\) 38.8296 1.22851
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 8470.2.a.cy.1.1 6
11.2 odd 10 770.2.n.i.631.1 yes 12
11.6 odd 10 770.2.n.i.421.1 12
11.10 odd 2 8470.2.a.de.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.n.i.421.1 12 11.6 odd 10
770.2.n.i.631.1 yes 12 11.2 odd 10
8470.2.a.cy.1.1 6 1.1 even 1 trivial
8470.2.a.de.1.1 6 11.10 odd 2