Properties

Label 7098.2.a.cg.1.1
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
Defining polynomial: \(x^{3} - x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.80194\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.19806 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.19806 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.19806 q^{10} +1.55496 q^{11} +1.00000 q^{12} +1.00000 q^{14} +1.19806 q^{15} +1.00000 q^{16} -5.74094 q^{17} -1.00000 q^{18} +4.93900 q^{19} +1.19806 q^{20} -1.00000 q^{21} -1.55496 q^{22} -0.417895 q^{23} -1.00000 q^{24} -3.56465 q^{25} +1.00000 q^{27} -1.00000 q^{28} -2.46681 q^{29} -1.19806 q^{30} -10.2567 q^{31} -1.00000 q^{32} +1.55496 q^{33} +5.74094 q^{34} -1.19806 q^{35} +1.00000 q^{36} -2.58211 q^{37} -4.93900 q^{38} -1.19806 q^{40} +11.0761 q^{41} +1.00000 q^{42} +2.91185 q^{43} +1.55496 q^{44} +1.19806 q^{45} +0.417895 q^{46} +7.89008 q^{47} +1.00000 q^{48} +1.00000 q^{49} +3.56465 q^{50} -5.74094 q^{51} -1.62133 q^{53} -1.00000 q^{54} +1.86294 q^{55} +1.00000 q^{56} +4.93900 q^{57} +2.46681 q^{58} +6.13169 q^{59} +1.19806 q^{60} +14.7017 q^{61} +10.2567 q^{62} -1.00000 q^{63} +1.00000 q^{64} -1.55496 q^{66} +10.5646 q^{67} -5.74094 q^{68} -0.417895 q^{69} +1.19806 q^{70} +1.20775 q^{71} -1.00000 q^{72} +5.86294 q^{73} +2.58211 q^{74} -3.56465 q^{75} +4.93900 q^{76} -1.55496 q^{77} -6.67994 q^{79} +1.19806 q^{80} +1.00000 q^{81} -11.0761 q^{82} +9.06100 q^{83} -1.00000 q^{84} -6.87800 q^{85} -2.91185 q^{86} -2.46681 q^{87} -1.55496 q^{88} +3.67456 q^{89} -1.19806 q^{90} -0.417895 q^{92} -10.2567 q^{93} -7.89008 q^{94} +5.91723 q^{95} -1.00000 q^{96} +17.3937 q^{97} -1.00000 q^{98} +1.55496 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} + 8q^{5} - 3q^{6} - 3q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} + 8q^{5} - 3q^{6} - 3q^{7} - 3q^{8} + 3q^{9} - 8q^{10} + 5q^{11} + 3q^{12} + 3q^{14} + 8q^{15} + 3q^{16} - 3q^{17} - 3q^{18} + 5q^{19} + 8q^{20} - 3q^{21} - 5q^{22} - 7q^{23} - 3q^{24} + 11q^{25} + 3q^{27} - 3q^{28} - 4q^{29} - 8q^{30} - 4q^{31} - 3q^{32} + 5q^{33} + 3q^{34} - 8q^{35} + 3q^{36} - 2q^{37} - 5q^{38} - 8q^{40} + 18q^{41} + 3q^{42} + 5q^{43} + 5q^{44} + 8q^{45} + 7q^{46} + 23q^{47} + 3q^{48} + 3q^{49} - 11q^{50} - 3q^{51} - 12q^{53} - 3q^{54} + 11q^{55} + 3q^{56} + 5q^{57} + 4q^{58} + 16q^{59} + 8q^{60} + 17q^{61} + 4q^{62} - 3q^{63} + 3q^{64} - 5q^{66} + 10q^{67} - 3q^{68} - 7q^{69} + 8q^{70} - 14q^{71} - 3q^{72} + 23q^{73} + 2q^{74} + 11q^{75} + 5q^{76} - 5q^{77} + 4q^{79} + 8q^{80} + 3q^{81} - 18q^{82} + 37q^{83} - 3q^{84} - q^{85} - 5q^{86} - 4q^{87} - 5q^{88} - 10q^{89} - 8q^{90} - 7q^{92} - 4q^{93} - 23q^{94} + 11q^{95} - 3q^{96} + 20q^{97} - 3q^{98} + 5q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 1.19806 0.535790 0.267895 0.963448i \(-0.413672\pi\)
0.267895 + 0.963448i \(0.413672\pi\)
\(6\) −1.00000 −0.408248
\(7\) −1.00000 −0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.19806 −0.378861
\(11\) 1.55496 0.468838 0.234419 0.972136i \(-0.424681\pi\)
0.234419 + 0.972136i \(0.424681\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) 1.00000 0.267261
\(15\) 1.19806 0.309338
\(16\) 1.00000 0.250000
\(17\) −5.74094 −1.39238 −0.696191 0.717857i \(-0.745123\pi\)
−0.696191 + 0.717857i \(0.745123\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.93900 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(20\) 1.19806 0.267895
\(21\) −1.00000 −0.218218
\(22\) −1.55496 −0.331518
\(23\) −0.417895 −0.0871371 −0.0435685 0.999050i \(-0.513873\pi\)
−0.0435685 + 0.999050i \(0.513873\pi\)
\(24\) −1.00000 −0.204124
\(25\) −3.56465 −0.712929
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −1.00000 −0.188982
\(29\) −2.46681 −0.458076 −0.229038 0.973418i \(-0.573558\pi\)
−0.229038 + 0.973418i \(0.573558\pi\)
\(30\) −1.19806 −0.218735
\(31\) −10.2567 −1.84215 −0.921076 0.389383i \(-0.872688\pi\)
−0.921076 + 0.389383i \(0.872688\pi\)
\(32\) −1.00000 −0.176777
\(33\) 1.55496 0.270683
\(34\) 5.74094 0.984563
\(35\) −1.19806 −0.202509
\(36\) 1.00000 0.166667
\(37\) −2.58211 −0.424495 −0.212248 0.977216i \(-0.568078\pi\)
−0.212248 + 0.977216i \(0.568078\pi\)
\(38\) −4.93900 −0.801212
\(39\) 0 0
\(40\) −1.19806 −0.189430
\(41\) 11.0761 1.72979 0.864895 0.501952i \(-0.167385\pi\)
0.864895 + 0.501952i \(0.167385\pi\)
\(42\) 1.00000 0.154303
\(43\) 2.91185 0.444054 0.222027 0.975041i \(-0.428733\pi\)
0.222027 + 0.975041i \(0.428733\pi\)
\(44\) 1.55496 0.234419
\(45\) 1.19806 0.178597
\(46\) 0.417895 0.0616152
\(47\) 7.89008 1.15089 0.575443 0.817842i \(-0.304829\pi\)
0.575443 + 0.817842i \(0.304829\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) 3.56465 0.504117
\(51\) −5.74094 −0.803892
\(52\) 0 0
\(53\) −1.62133 −0.222707 −0.111354 0.993781i \(-0.535519\pi\)
−0.111354 + 0.993781i \(0.535519\pi\)
\(54\) −1.00000 −0.136083
\(55\) 1.86294 0.251198
\(56\) 1.00000 0.133631
\(57\) 4.93900 0.654187
\(58\) 2.46681 0.323908
\(59\) 6.13169 0.798278 0.399139 0.916891i \(-0.369309\pi\)
0.399139 + 0.916891i \(0.369309\pi\)
\(60\) 1.19806 0.154669
\(61\) 14.7017 1.88236 0.941181 0.337904i \(-0.109718\pi\)
0.941181 + 0.337904i \(0.109718\pi\)
\(62\) 10.2567 1.30260
\(63\) −1.00000 −0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −1.55496 −0.191402
\(67\) 10.5646 1.29068 0.645339 0.763897i \(-0.276716\pi\)
0.645339 + 0.763897i \(0.276716\pi\)
\(68\) −5.74094 −0.696191
\(69\) −0.417895 −0.0503086
\(70\) 1.19806 0.143196
\(71\) 1.20775 0.143334 0.0716668 0.997429i \(-0.477168\pi\)
0.0716668 + 0.997429i \(0.477168\pi\)
\(72\) −1.00000 −0.117851
\(73\) 5.86294 0.686205 0.343102 0.939298i \(-0.388522\pi\)
0.343102 + 0.939298i \(0.388522\pi\)
\(74\) 2.58211 0.300164
\(75\) −3.56465 −0.411610
\(76\) 4.93900 0.566542
\(77\) −1.55496 −0.177204
\(78\) 0 0
\(79\) −6.67994 −0.751552 −0.375776 0.926711i \(-0.622624\pi\)
−0.375776 + 0.926711i \(0.622624\pi\)
\(80\) 1.19806 0.133947
\(81\) 1.00000 0.111111
\(82\) −11.0761 −1.22315
\(83\) 9.06100 0.994574 0.497287 0.867586i \(-0.334330\pi\)
0.497287 + 0.867586i \(0.334330\pi\)
\(84\) −1.00000 −0.109109
\(85\) −6.87800 −0.746024
\(86\) −2.91185 −0.313993
\(87\) −2.46681 −0.264470
\(88\) −1.55496 −0.165759
\(89\) 3.67456 0.389503 0.194751 0.980853i \(-0.437610\pi\)
0.194751 + 0.980853i \(0.437610\pi\)
\(90\) −1.19806 −0.126287
\(91\) 0 0
\(92\) −0.417895 −0.0435685
\(93\) −10.2567 −1.06357
\(94\) −7.89008 −0.813800
\(95\) 5.91723 0.607095
\(96\) −1.00000 −0.102062
\(97\) 17.3937 1.76607 0.883033 0.469311i \(-0.155498\pi\)
0.883033 + 0.469311i \(0.155498\pi\)
\(98\) −1.00000 −0.101015
\(99\) 1.55496 0.156279
\(100\) −3.56465 −0.356465
\(101\) 0.674563 0.0671215 0.0335608 0.999437i \(-0.489315\pi\)
0.0335608 + 0.999437i \(0.489315\pi\)
\(102\) 5.74094 0.568438
\(103\) −7.75063 −0.763692 −0.381846 0.924226i \(-0.624712\pi\)
−0.381846 + 0.924226i \(0.624712\pi\)
\(104\) 0 0
\(105\) −1.19806 −0.116919
\(106\) 1.62133 0.157478
\(107\) −6.20775 −0.600126 −0.300063 0.953919i \(-0.597008\pi\)
−0.300063 + 0.953919i \(0.597008\pi\)
\(108\) 1.00000 0.0962250
\(109\) 4.91185 0.470470 0.235235 0.971938i \(-0.424414\pi\)
0.235235 + 0.971938i \(0.424414\pi\)
\(110\) −1.86294 −0.177624
\(111\) −2.58211 −0.245083
\(112\) −1.00000 −0.0944911
\(113\) 19.2054 1.80669 0.903344 0.428917i \(-0.141105\pi\)
0.903344 + 0.428917i \(0.141105\pi\)
\(114\) −4.93900 −0.462580
\(115\) −0.500664 −0.0466872
\(116\) −2.46681 −0.229038
\(117\) 0 0
\(118\) −6.13169 −0.564467
\(119\) 5.74094 0.526271
\(120\) −1.19806 −0.109368
\(121\) −8.58211 −0.780191
\(122\) −14.7017 −1.33103
\(123\) 11.0761 0.998695
\(124\) −10.2567 −0.921076
\(125\) −10.2610 −0.917770
\(126\) 1.00000 0.0890871
\(127\) 6.72886 0.597090 0.298545 0.954396i \(-0.403499\pi\)
0.298545 + 0.954396i \(0.403499\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 2.91185 0.256374
\(130\) 0 0
\(131\) 3.19269 0.278946 0.139473 0.990226i \(-0.455459\pi\)
0.139473 + 0.990226i \(0.455459\pi\)
\(132\) 1.55496 0.135342
\(133\) −4.93900 −0.428266
\(134\) −10.5646 −0.912646
\(135\) 1.19806 0.103113
\(136\) 5.74094 0.492281
\(137\) 3.21014 0.274261 0.137131 0.990553i \(-0.456212\pi\)
0.137131 + 0.990553i \(0.456212\pi\)
\(138\) 0.417895 0.0355736
\(139\) −10.5308 −0.893210 −0.446605 0.894731i \(-0.647367\pi\)
−0.446605 + 0.894731i \(0.647367\pi\)
\(140\) −1.19806 −0.101255
\(141\) 7.89008 0.664465
\(142\) −1.20775 −0.101352
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −2.95539 −0.245432
\(146\) −5.86294 −0.485220
\(147\) 1.00000 0.0824786
\(148\) −2.58211 −0.212248
\(149\) 2.44504 0.200306 0.100153 0.994972i \(-0.468067\pi\)
0.100153 + 0.994972i \(0.468067\pi\)
\(150\) 3.56465 0.291052
\(151\) −21.8364 −1.77702 −0.888510 0.458858i \(-0.848259\pi\)
−0.888510 + 0.458858i \(0.848259\pi\)
\(152\) −4.93900 −0.400606
\(153\) −5.74094 −0.464127
\(154\) 1.55496 0.125302
\(155\) −12.2881 −0.987006
\(156\) 0 0
\(157\) 15.3599 1.22585 0.612926 0.790140i \(-0.289992\pi\)
0.612926 + 0.790140i \(0.289992\pi\)
\(158\) 6.67994 0.531427
\(159\) −1.62133 −0.128580
\(160\) −1.19806 −0.0947151
\(161\) 0.417895 0.0329347
\(162\) −1.00000 −0.0785674
\(163\) 16.0911 1.26035 0.630177 0.776451i \(-0.282982\pi\)
0.630177 + 0.776451i \(0.282982\pi\)
\(164\) 11.0761 0.864895
\(165\) 1.86294 0.145029
\(166\) −9.06100 −0.703270
\(167\) 6.65519 0.514994 0.257497 0.966279i \(-0.417102\pi\)
0.257497 + 0.966279i \(0.417102\pi\)
\(168\) 1.00000 0.0771517
\(169\) 0 0
\(170\) 6.87800 0.527519
\(171\) 4.93900 0.377695
\(172\) 2.91185 0.222027
\(173\) 3.83446 0.291529 0.145764 0.989319i \(-0.453436\pi\)
0.145764 + 0.989319i \(0.453436\pi\)
\(174\) 2.46681 0.187009
\(175\) 3.56465 0.269462
\(176\) 1.55496 0.117209
\(177\) 6.13169 0.460886
\(178\) −3.67456 −0.275420
\(179\) −6.48188 −0.484478 −0.242239 0.970217i \(-0.577882\pi\)
−0.242239 + 0.970217i \(0.577882\pi\)
\(180\) 1.19806 0.0892983
\(181\) 10.1075 0.751286 0.375643 0.926764i \(-0.377422\pi\)
0.375643 + 0.926764i \(0.377422\pi\)
\(182\) 0 0
\(183\) 14.7017 1.08678
\(184\) 0.417895 0.0308076
\(185\) −3.09352 −0.227440
\(186\) 10.2567 0.752055
\(187\) −8.92692 −0.652801
\(188\) 7.89008 0.575443
\(189\) −1.00000 −0.0727393
\(190\) −5.91723 −0.429281
\(191\) 22.9681 1.66191 0.830956 0.556339i \(-0.187794\pi\)
0.830956 + 0.556339i \(0.187794\pi\)
\(192\) 1.00000 0.0721688
\(193\) 4.37435 0.314873 0.157436 0.987529i \(-0.449677\pi\)
0.157436 + 0.987529i \(0.449677\pi\)
\(194\) −17.3937 −1.24880
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 4.90515 0.349477 0.174739 0.984615i \(-0.444092\pi\)
0.174739 + 0.984615i \(0.444092\pi\)
\(198\) −1.55496 −0.110506
\(199\) −7.73125 −0.548054 −0.274027 0.961722i \(-0.588356\pi\)
−0.274027 + 0.961722i \(0.588356\pi\)
\(200\) 3.56465 0.252059
\(201\) 10.5646 0.745173
\(202\) −0.674563 −0.0474621
\(203\) 2.46681 0.173136
\(204\) −5.74094 −0.401946
\(205\) 13.2698 0.926804
\(206\) 7.75063 0.540012
\(207\) −0.417895 −0.0290457
\(208\) 0 0
\(209\) 7.67994 0.531233
\(210\) 1.19806 0.0826742
\(211\) 6.35690 0.437627 0.218813 0.975767i \(-0.429781\pi\)
0.218813 + 0.975767i \(0.429781\pi\)
\(212\) −1.62133 −0.111354
\(213\) 1.20775 0.0827537
\(214\) 6.20775 0.424353
\(215\) 3.48858 0.237919
\(216\) −1.00000 −0.0680414
\(217\) 10.2567 0.696268
\(218\) −4.91185 −0.332673
\(219\) 5.86294 0.396181
\(220\) 1.86294 0.125599
\(221\) 0 0
\(222\) 2.58211 0.173300
\(223\) −6.49157 −0.434707 −0.217354 0.976093i \(-0.569743\pi\)
−0.217354 + 0.976093i \(0.569743\pi\)
\(224\) 1.00000 0.0668153
\(225\) −3.56465 −0.237643
\(226\) −19.2054 −1.27752
\(227\) 7.92931 0.526287 0.263143 0.964757i \(-0.415241\pi\)
0.263143 + 0.964757i \(0.415241\pi\)
\(228\) 4.93900 0.327093
\(229\) −21.3502 −1.41086 −0.705430 0.708779i \(-0.749246\pi\)
−0.705430 + 0.708779i \(0.749246\pi\)
\(230\) 0.500664 0.0330128
\(231\) −1.55496 −0.102309
\(232\) 2.46681 0.161954
\(233\) 10.3002 0.674789 0.337395 0.941363i \(-0.390454\pi\)
0.337395 + 0.941363i \(0.390454\pi\)
\(234\) 0 0
\(235\) 9.45281 0.616633
\(236\) 6.13169 0.399139
\(237\) −6.67994 −0.433909
\(238\) −5.74094 −0.372130
\(239\) −19.2814 −1.24721 −0.623606 0.781739i \(-0.714333\pi\)
−0.623606 + 0.781739i \(0.714333\pi\)
\(240\) 1.19806 0.0773346
\(241\) −10.6920 −0.688734 −0.344367 0.938835i \(-0.611906\pi\)
−0.344367 + 0.938835i \(0.611906\pi\)
\(242\) 8.58211 0.551679
\(243\) 1.00000 0.0641500
\(244\) 14.7017 0.941181
\(245\) 1.19806 0.0765414
\(246\) −11.0761 −0.706184
\(247\) 0 0
\(248\) 10.2567 0.651299
\(249\) 9.06100 0.574217
\(250\) 10.2610 0.648961
\(251\) −6.49827 −0.410167 −0.205084 0.978744i \(-0.565747\pi\)
−0.205084 + 0.978744i \(0.565747\pi\)
\(252\) −1.00000 −0.0629941
\(253\) −0.649809 −0.0408531
\(254\) −6.72886 −0.422206
\(255\) −6.87800 −0.430717
\(256\) 1.00000 0.0625000
\(257\) 0.907542 0.0566109 0.0283055 0.999599i \(-0.490989\pi\)
0.0283055 + 0.999599i \(0.490989\pi\)
\(258\) −2.91185 −0.181284
\(259\) 2.58211 0.160444
\(260\) 0 0
\(261\) −2.46681 −0.152692
\(262\) −3.19269 −0.197245
\(263\) 17.1564 1.05791 0.528956 0.848649i \(-0.322584\pi\)
0.528956 + 0.848649i \(0.322584\pi\)
\(264\) −1.55496 −0.0957011
\(265\) −1.94246 −0.119324
\(266\) 4.93900 0.302830
\(267\) 3.67456 0.224880
\(268\) 10.5646 0.645339
\(269\) −19.7047 −1.20142 −0.600708 0.799468i \(-0.705115\pi\)
−0.600708 + 0.799468i \(0.705115\pi\)
\(270\) −1.19806 −0.0729117
\(271\) 16.0194 0.973108 0.486554 0.873651i \(-0.338254\pi\)
0.486554 + 0.873651i \(0.338254\pi\)
\(272\) −5.74094 −0.348096
\(273\) 0 0
\(274\) −3.21014 −0.193932
\(275\) −5.54288 −0.334248
\(276\) −0.417895 −0.0251543
\(277\) −10.6571 −0.640323 −0.320162 0.947363i \(-0.603737\pi\)
−0.320162 + 0.947363i \(0.603737\pi\)
\(278\) 10.5308 0.631595
\(279\) −10.2567 −0.614051
\(280\) 1.19806 0.0715979
\(281\) −28.6286 −1.70784 −0.853920 0.520404i \(-0.825782\pi\)
−0.853920 + 0.520404i \(0.825782\pi\)
\(282\) −7.89008 −0.469848
\(283\) −7.11960 −0.423217 −0.211608 0.977355i \(-0.567870\pi\)
−0.211608 + 0.977355i \(0.567870\pi\)
\(284\) 1.20775 0.0716668
\(285\) 5.91723 0.350507
\(286\) 0 0
\(287\) −11.0761 −0.653799
\(288\) −1.00000 −0.0589256
\(289\) 15.9584 0.938728
\(290\) 2.95539 0.173547
\(291\) 17.3937 1.01964
\(292\) 5.86294 0.343102
\(293\) 25.0978 1.46623 0.733116 0.680104i \(-0.238065\pi\)
0.733116 + 0.680104i \(0.238065\pi\)
\(294\) −1.00000 −0.0583212
\(295\) 7.34614 0.427709
\(296\) 2.58211 0.150082
\(297\) 1.55496 0.0902278
\(298\) −2.44504 −0.141637
\(299\) 0 0
\(300\) −3.56465 −0.205805
\(301\) −2.91185 −0.167836
\(302\) 21.8364 1.25654
\(303\) 0.674563 0.0387526
\(304\) 4.93900 0.283271
\(305\) 17.6136 1.00855
\(306\) 5.74094 0.328188
\(307\) −8.36658 −0.477506 −0.238753 0.971080i \(-0.576739\pi\)
−0.238753 + 0.971080i \(0.576739\pi\)
\(308\) −1.55496 −0.0886020
\(309\) −7.75063 −0.440918
\(310\) 12.2881 0.697919
\(311\) −21.6668 −1.22861 −0.614306 0.789068i \(-0.710564\pi\)
−0.614306 + 0.789068i \(0.710564\pi\)
\(312\) 0 0
\(313\) 14.6950 0.830611 0.415305 0.909682i \(-0.363675\pi\)
0.415305 + 0.909682i \(0.363675\pi\)
\(314\) −15.3599 −0.866808
\(315\) −1.19806 −0.0675032
\(316\) −6.67994 −0.375776
\(317\) 27.1672 1.52586 0.762931 0.646480i \(-0.223760\pi\)
0.762931 + 0.646480i \(0.223760\pi\)
\(318\) 1.62133 0.0909199
\(319\) −3.83579 −0.214763
\(320\) 1.19806 0.0669737
\(321\) −6.20775 −0.346483
\(322\) −0.417895 −0.0232884
\(323\) −28.3545 −1.57769
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −16.0911 −0.891205
\(327\) 4.91185 0.271626
\(328\) −11.0761 −0.611573
\(329\) −7.89008 −0.434994
\(330\) −1.86294 −0.102551
\(331\) 30.9885 1.70328 0.851641 0.524125i \(-0.175608\pi\)
0.851641 + 0.524125i \(0.175608\pi\)
\(332\) 9.06100 0.497287
\(333\) −2.58211 −0.141498
\(334\) −6.65519 −0.364156
\(335\) 12.6571 0.691532
\(336\) −1.00000 −0.0545545
\(337\) −1.95539 −0.106517 −0.0532586 0.998581i \(-0.516961\pi\)
−0.0532586 + 0.998581i \(0.516961\pi\)
\(338\) 0 0
\(339\) 19.2054 1.04309
\(340\) −6.87800 −0.373012
\(341\) −15.9487 −0.863670
\(342\) −4.93900 −0.267071
\(343\) −1.00000 −0.0539949
\(344\) −2.91185 −0.156997
\(345\) −0.500664 −0.0269548
\(346\) −3.83446 −0.206142
\(347\) 22.9259 1.23072 0.615362 0.788244i \(-0.289010\pi\)
0.615362 + 0.788244i \(0.289010\pi\)
\(348\) −2.46681 −0.132235
\(349\) 3.67994 0.196983 0.0984913 0.995138i \(-0.468598\pi\)
0.0984913 + 0.995138i \(0.468598\pi\)
\(350\) −3.56465 −0.190538
\(351\) 0 0
\(352\) −1.55496 −0.0828795
\(353\) 26.6353 1.41766 0.708828 0.705381i \(-0.249224\pi\)
0.708828 + 0.705381i \(0.249224\pi\)
\(354\) −6.13169 −0.325895
\(355\) 1.44696 0.0767967
\(356\) 3.67456 0.194751
\(357\) 5.74094 0.303843
\(358\) 6.48188 0.342578
\(359\) −3.05967 −0.161483 −0.0807416 0.996735i \(-0.525729\pi\)
−0.0807416 + 0.996735i \(0.525729\pi\)
\(360\) −1.19806 −0.0631434
\(361\) 5.39373 0.283881
\(362\) −10.1075 −0.531240
\(363\) −8.58211 −0.450444
\(364\) 0 0
\(365\) 7.02416 0.367662
\(366\) −14.7017 −0.768471
\(367\) −25.6582 −1.33935 −0.669673 0.742656i \(-0.733566\pi\)
−0.669673 + 0.742656i \(0.733566\pi\)
\(368\) −0.417895 −0.0217843
\(369\) 11.0761 0.576597
\(370\) 3.09352 0.160825
\(371\) 1.62133 0.0841755
\(372\) −10.2567 −0.531783
\(373\) −21.6612 −1.12157 −0.560786 0.827961i \(-0.689501\pi\)
−0.560786 + 0.827961i \(0.689501\pi\)
\(374\) 8.92692 0.461600
\(375\) −10.2610 −0.529875
\(376\) −7.89008 −0.406900
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) −6.93602 −0.356279 −0.178140 0.984005i \(-0.557008\pi\)
−0.178140 + 0.984005i \(0.557008\pi\)
\(380\) 5.91723 0.303548
\(381\) 6.72886 0.344730
\(382\) −22.9681 −1.17515
\(383\) 19.9734 1.02060 0.510298 0.859998i \(-0.329535\pi\)
0.510298 + 0.859998i \(0.329535\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −1.86294 −0.0949440
\(386\) −4.37435 −0.222649
\(387\) 2.91185 0.148018
\(388\) 17.3937 0.883033
\(389\) −30.7289 −1.55801 −0.779007 0.627015i \(-0.784277\pi\)
−0.779007 + 0.627015i \(0.784277\pi\)
\(390\) 0 0
\(391\) 2.39911 0.121328
\(392\) −1.00000 −0.0505076
\(393\) 3.19269 0.161050
\(394\) −4.90515 −0.247118
\(395\) −8.00298 −0.402674
\(396\) 1.55496 0.0781396
\(397\) −12.3405 −0.619352 −0.309676 0.950842i \(-0.600221\pi\)
−0.309676 + 0.950842i \(0.600221\pi\)
\(398\) 7.73125 0.387533
\(399\) −4.93900 −0.247259
\(400\) −3.56465 −0.178232
\(401\) 10.2241 0.510569 0.255285 0.966866i \(-0.417831\pi\)
0.255285 + 0.966866i \(0.417831\pi\)
\(402\) −10.5646 −0.526917
\(403\) 0 0
\(404\) 0.674563 0.0335608
\(405\) 1.19806 0.0595322
\(406\) −2.46681 −0.122426
\(407\) −4.01507 −0.199019
\(408\) 5.74094 0.284219
\(409\) 14.0127 0.692882 0.346441 0.938072i \(-0.387390\pi\)
0.346441 + 0.938072i \(0.387390\pi\)
\(410\) −13.2698 −0.655349
\(411\) 3.21014 0.158345
\(412\) −7.75063 −0.381846
\(413\) −6.13169 −0.301721
\(414\) 0.417895 0.0205384
\(415\) 10.8556 0.532882
\(416\) 0 0
\(417\) −10.5308 −0.515695
\(418\) −7.67994 −0.375638
\(419\) 1.10859 0.0541581 0.0270790 0.999633i \(-0.491379\pi\)
0.0270790 + 0.999633i \(0.491379\pi\)
\(420\) −1.19806 −0.0584595
\(421\) −5.63342 −0.274556 −0.137278 0.990533i \(-0.543835\pi\)
−0.137278 + 0.990533i \(0.543835\pi\)
\(422\) −6.35690 −0.309449
\(423\) 7.89008 0.383629
\(424\) 1.62133 0.0787389
\(425\) 20.4644 0.992670
\(426\) −1.20775 −0.0585157
\(427\) −14.7017 −0.711466
\(428\) −6.20775 −0.300063
\(429\) 0 0
\(430\) −3.48858 −0.168234
\(431\) 18.6907 0.900299 0.450150 0.892953i \(-0.351371\pi\)
0.450150 + 0.892953i \(0.351371\pi\)
\(432\) 1.00000 0.0481125
\(433\) −26.2640 −1.26217 −0.631083 0.775715i \(-0.717389\pi\)
−0.631083 + 0.775715i \(0.717389\pi\)
\(434\) −10.2567 −0.492336
\(435\) −2.95539 −0.141700
\(436\) 4.91185 0.235235
\(437\) −2.06398 −0.0987337
\(438\) −5.86294 −0.280142
\(439\) 11.8092 0.563624 0.281812 0.959470i \(-0.409065\pi\)
0.281812 + 0.959470i \(0.409065\pi\)
\(440\) −1.86294 −0.0888120
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 7.16660 0.340496 0.170248 0.985401i \(-0.445543\pi\)
0.170248 + 0.985401i \(0.445543\pi\)
\(444\) −2.58211 −0.122541
\(445\) 4.40236 0.208692
\(446\) 6.49157 0.307385
\(447\) 2.44504 0.115647
\(448\) −1.00000 −0.0472456
\(449\) 31.3357 1.47882 0.739412 0.673253i \(-0.235104\pi\)
0.739412 + 0.673253i \(0.235104\pi\)
\(450\) 3.56465 0.168039
\(451\) 17.2228 0.810991
\(452\) 19.2054 0.903344
\(453\) −21.8364 −1.02596
\(454\) −7.92931 −0.372141
\(455\) 0 0
\(456\) −4.93900 −0.231290
\(457\) −25.4058 −1.18843 −0.594217 0.804305i \(-0.702538\pi\)
−0.594217 + 0.804305i \(0.702538\pi\)
\(458\) 21.3502 0.997629
\(459\) −5.74094 −0.267964
\(460\) −0.500664 −0.0233436
\(461\) −37.0417 −1.72521 −0.862603 0.505882i \(-0.831167\pi\)
−0.862603 + 0.505882i \(0.831167\pi\)
\(462\) 1.55496 0.0723432
\(463\) −10.2929 −0.478352 −0.239176 0.970976i \(-0.576877\pi\)
−0.239176 + 0.970976i \(0.576877\pi\)
\(464\) −2.46681 −0.114519
\(465\) −12.2881 −0.569848
\(466\) −10.3002 −0.477148
\(467\) −12.1890 −0.564038 −0.282019 0.959409i \(-0.591004\pi\)
−0.282019 + 0.959409i \(0.591004\pi\)
\(468\) 0 0
\(469\) −10.5646 −0.487830
\(470\) −9.45281 −0.436026
\(471\) 15.3599 0.707746
\(472\) −6.13169 −0.282234
\(473\) 4.52781 0.208189
\(474\) 6.67994 0.306820
\(475\) −17.6058 −0.807809
\(476\) 5.74094 0.263135
\(477\) −1.62133 −0.0742358
\(478\) 19.2814 0.881912
\(479\) 42.4413 1.93919 0.969597 0.244708i \(-0.0786922\pi\)
0.969597 + 0.244708i \(0.0786922\pi\)
\(480\) −1.19806 −0.0546838
\(481\) 0 0
\(482\) 10.6920 0.487008
\(483\) 0.417895 0.0190149
\(484\) −8.58211 −0.390096
\(485\) 20.8388 0.946240
\(486\) −1.00000 −0.0453609
\(487\) −7.26742 −0.329318 −0.164659 0.986351i \(-0.552652\pi\)
−0.164659 + 0.986351i \(0.552652\pi\)
\(488\) −14.7017 −0.665515
\(489\) 16.0911 0.727666
\(490\) −1.19806 −0.0541229
\(491\) 14.3763 0.648792 0.324396 0.945921i \(-0.394839\pi\)
0.324396 + 0.945921i \(0.394839\pi\)
\(492\) 11.0761 0.499348
\(493\) 14.1618 0.637816
\(494\) 0 0
\(495\) 1.86294 0.0837328
\(496\) −10.2567 −0.460538
\(497\) −1.20775 −0.0541750
\(498\) −9.06100 −0.406033
\(499\) −5.55602 −0.248722 −0.124361 0.992237i \(-0.539688\pi\)
−0.124361 + 0.992237i \(0.539688\pi\)
\(500\) −10.2610 −0.458885
\(501\) 6.65519 0.297332
\(502\) 6.49827 0.290032
\(503\) 38.2747 1.70659 0.853293 0.521432i \(-0.174602\pi\)
0.853293 + 0.521432i \(0.174602\pi\)
\(504\) 1.00000 0.0445435
\(505\) 0.808169 0.0359630
\(506\) 0.649809 0.0288875
\(507\) 0 0
\(508\) 6.72886 0.298545
\(509\) −33.3685 −1.47903 −0.739516 0.673138i \(-0.764946\pi\)
−0.739516 + 0.673138i \(0.764946\pi\)
\(510\) 6.87800 0.304563
\(511\) −5.86294 −0.259361
\(512\) −1.00000 −0.0441942
\(513\) 4.93900 0.218062
\(514\) −0.907542 −0.0400300
\(515\) −9.28573 −0.409178
\(516\) 2.91185 0.128187
\(517\) 12.2687 0.539579
\(518\) −2.58211 −0.113451
\(519\) 3.83446 0.168314
\(520\) 0 0
\(521\) 14.8291 0.649674 0.324837 0.945770i \(-0.394691\pi\)
0.324837 + 0.945770i \(0.394691\pi\)
\(522\) 2.46681 0.107969
\(523\) −27.8625 −1.21834 −0.609170 0.793040i \(-0.708497\pi\)
−0.609170 + 0.793040i \(0.708497\pi\)
\(524\) 3.19269 0.139473
\(525\) 3.56465 0.155574
\(526\) −17.1564 −0.748056
\(527\) 58.8829 2.56498
\(528\) 1.55496 0.0676709
\(529\) −22.8254 −0.992407
\(530\) 1.94246 0.0843750
\(531\) 6.13169 0.266093
\(532\) −4.93900 −0.214133
\(533\) 0 0
\(534\) −3.67456 −0.159014
\(535\) −7.43727 −0.321541
\(536\) −10.5646 −0.456323
\(537\) −6.48188 −0.279714
\(538\) 19.7047 0.849530
\(539\) 1.55496 0.0669768
\(540\) 1.19806 0.0515564
\(541\) −27.8019 −1.19530 −0.597649 0.801758i \(-0.703898\pi\)
−0.597649 + 0.801758i \(0.703898\pi\)
\(542\) −16.0194 −0.688091
\(543\) 10.1075 0.433755
\(544\) 5.74094 0.246141
\(545\) 5.88471 0.252073
\(546\) 0 0
\(547\) −11.2500 −0.481014 −0.240507 0.970647i \(-0.577314\pi\)
−0.240507 + 0.970647i \(0.577314\pi\)
\(548\) 3.21014 0.137131
\(549\) 14.7017 0.627454
\(550\) 5.54288 0.236349
\(551\) −12.1836 −0.519038
\(552\) 0.417895 0.0177868
\(553\) 6.67994 0.284060
\(554\) 10.6571 0.452777
\(555\) −3.09352 −0.131313
\(556\) −10.5308 −0.446605
\(557\) 30.0062 1.27140 0.635702 0.771934i \(-0.280711\pi\)
0.635702 + 0.771934i \(0.280711\pi\)
\(558\) 10.2567 0.434199
\(559\) 0 0
\(560\) −1.19806 −0.0506274
\(561\) −8.92692 −0.376895
\(562\) 28.6286 1.20763
\(563\) 39.3491 1.65837 0.829184 0.558976i \(-0.188806\pi\)
0.829184 + 0.558976i \(0.188806\pi\)
\(564\) 7.89008 0.332232
\(565\) 23.0092 0.968005
\(566\) 7.11960 0.299259
\(567\) −1.00000 −0.0419961
\(568\) −1.20775 −0.0506761
\(569\) 9.36360 0.392543 0.196271 0.980550i \(-0.437117\pi\)
0.196271 + 0.980550i \(0.437117\pi\)
\(570\) −5.91723 −0.247846
\(571\) 24.1540 1.01082 0.505408 0.862881i \(-0.331342\pi\)
0.505408 + 0.862881i \(0.331342\pi\)
\(572\) 0 0
\(573\) 22.9681 0.959505
\(574\) 11.0761 0.462306
\(575\) 1.48965 0.0621226
\(576\) 1.00000 0.0416667
\(577\) −28.7885 −1.19848 −0.599241 0.800569i \(-0.704531\pi\)
−0.599241 + 0.800569i \(0.704531\pi\)
\(578\) −15.9584 −0.663781
\(579\) 4.37435 0.181792
\(580\) −2.95539 −0.122716
\(581\) −9.06100 −0.375914
\(582\) −17.3937 −0.720993
\(583\) −2.52111 −0.104414
\(584\) −5.86294 −0.242610
\(585\) 0 0
\(586\) −25.0978 −1.03678
\(587\) 6.58211 0.271673 0.135836 0.990731i \(-0.456628\pi\)
0.135836 + 0.990731i \(0.456628\pi\)
\(588\) 1.00000 0.0412393
\(589\) −50.6577 −2.08731
\(590\) −7.34614 −0.302436
\(591\) 4.90515 0.201771
\(592\) −2.58211 −0.106124
\(593\) 1.66355 0.0683137 0.0341568 0.999416i \(-0.489125\pi\)
0.0341568 + 0.999416i \(0.489125\pi\)
\(594\) −1.55496 −0.0638007
\(595\) 6.87800 0.281971
\(596\) 2.44504 0.100153
\(597\) −7.73125 −0.316419
\(598\) 0 0
\(599\) 32.9788 1.34748 0.673739 0.738969i \(-0.264687\pi\)
0.673739 + 0.738969i \(0.264687\pi\)
\(600\) 3.56465 0.145526
\(601\) −30.2107 −1.23232 −0.616161 0.787620i \(-0.711313\pi\)
−0.616161 + 0.787620i \(0.711313\pi\)
\(602\) 2.91185 0.118678
\(603\) 10.5646 0.430226
\(604\) −21.8364 −0.888510
\(605\) −10.2819 −0.418019
\(606\) −0.674563 −0.0274023
\(607\) 7.50471 0.304607 0.152303 0.988334i \(-0.451331\pi\)
0.152303 + 0.988334i \(0.451331\pi\)
\(608\) −4.93900 −0.200303
\(609\) 2.46681 0.0999603
\(610\) −17.6136 −0.713152
\(611\) 0 0
\(612\) −5.74094 −0.232064
\(613\) 18.5187 0.747964 0.373982 0.927436i \(-0.377992\pi\)
0.373982 + 0.927436i \(0.377992\pi\)
\(614\) 8.36658 0.337648
\(615\) 13.2698 0.535091
\(616\) 1.55496 0.0626510
\(617\) −31.8998 −1.28424 −0.642118 0.766606i \(-0.721944\pi\)
−0.642118 + 0.766606i \(0.721944\pi\)
\(618\) 7.75063 0.311776
\(619\) −48.4566 −1.94764 −0.973819 0.227327i \(-0.927002\pi\)
−0.973819 + 0.227327i \(0.927002\pi\)
\(620\) −12.2881 −0.493503
\(621\) −0.417895 −0.0167695
\(622\) 21.6668 0.868759
\(623\) −3.67456 −0.147218
\(624\) 0 0
\(625\) 5.52994 0.221198
\(626\) −14.6950 −0.587331
\(627\) 7.67994 0.306707
\(628\) 15.3599 0.612926
\(629\) 14.8237 0.591060
\(630\) 1.19806 0.0477319
\(631\) 3.20477 0.127580 0.0637899 0.997963i \(-0.479681\pi\)
0.0637899 + 0.997963i \(0.479681\pi\)
\(632\) 6.67994 0.265714
\(633\) 6.35690 0.252664
\(634\) −27.1672 −1.07895
\(635\) 8.06159 0.319914
\(636\) −1.62133 −0.0642901
\(637\) 0 0
\(638\) 3.83579 0.151860
\(639\) 1.20775 0.0477779
\(640\) −1.19806 −0.0473576
\(641\) −32.5515 −1.28571 −0.642853 0.765989i \(-0.722250\pi\)
−0.642853 + 0.765989i \(0.722250\pi\)
\(642\) 6.20775 0.245000
\(643\) −9.77107 −0.385333 −0.192667 0.981264i \(-0.561714\pi\)
−0.192667 + 0.981264i \(0.561714\pi\)
\(644\) 0.417895 0.0164674
\(645\) 3.48858 0.137363
\(646\) 28.3545 1.11559
\(647\) −12.4829 −0.490755 −0.245378 0.969428i \(-0.578912\pi\)
−0.245378 + 0.969428i \(0.578912\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 9.53452 0.374262
\(650\) 0 0
\(651\) 10.2567 0.401991
\(652\) 16.0911 0.630177
\(653\) 42.5488 1.66506 0.832532 0.553976i \(-0.186890\pi\)
0.832532 + 0.553976i \(0.186890\pi\)
\(654\) −4.91185 −0.192069
\(655\) 3.82504 0.149457
\(656\) 11.0761 0.432448
\(657\) 5.86294 0.228735
\(658\) 7.89008 0.307587
\(659\) −40.8068 −1.58961 −0.794804 0.606866i \(-0.792427\pi\)
−0.794804 + 0.606866i \(0.792427\pi\)
\(660\) 1.86294 0.0725147
\(661\) −39.2597 −1.52702 −0.763512 0.645794i \(-0.776527\pi\)
−0.763512 + 0.645794i \(0.776527\pi\)
\(662\) −30.9885 −1.20440
\(663\) 0 0
\(664\) −9.06100 −0.351635
\(665\) −5.91723 −0.229460
\(666\) 2.58211 0.100055
\(667\) 1.03087 0.0399154
\(668\) 6.65519 0.257497
\(669\) −6.49157 −0.250978
\(670\) −12.6571 −0.488987
\(671\) 22.8605 0.882522
\(672\) 1.00000 0.0385758
\(673\) 40.8353 1.57409 0.787043 0.616898i \(-0.211611\pi\)
0.787043 + 0.616898i \(0.211611\pi\)
\(674\) 1.95539 0.0753190
\(675\) −3.56465 −0.137203
\(676\) 0 0
\(677\) −22.3817 −0.860197 −0.430098 0.902782i \(-0.641521\pi\)
−0.430098 + 0.902782i \(0.641521\pi\)
\(678\) −19.2054 −0.737577
\(679\) −17.3937 −0.667510
\(680\) 6.87800 0.263759
\(681\) 7.92931 0.303852
\(682\) 15.9487 0.610707
\(683\) −9.96482 −0.381293 −0.190647 0.981659i \(-0.561058\pi\)
−0.190647 + 0.981659i \(0.561058\pi\)
\(684\) 4.93900 0.188847
\(685\) 3.84595 0.146946
\(686\) 1.00000 0.0381802
\(687\) −21.3502 −0.814561
\(688\) 2.91185 0.111013
\(689\) 0 0
\(690\) 0.500664 0.0190600
\(691\) 24.9366 0.948633 0.474317 0.880354i \(-0.342695\pi\)
0.474317 + 0.880354i \(0.342695\pi\)
\(692\) 3.83446 0.145764
\(693\) −1.55496 −0.0590680
\(694\) −22.9259 −0.870254
\(695\) −12.6165 −0.478573
\(696\) 2.46681 0.0935043
\(697\) −63.5870 −2.40853
\(698\) −3.67994 −0.139288
\(699\) 10.3002 0.389590
\(700\) 3.56465 0.134731
\(701\) 12.2567 0.462928 0.231464 0.972843i \(-0.425648\pi\)
0.231464 + 0.972843i \(0.425648\pi\)
\(702\) 0 0
\(703\) −12.7530 −0.480989
\(704\) 1.55496 0.0586047
\(705\) 9.45281 0.356013
\(706\) −26.6353 −1.00243
\(707\) −0.674563 −0.0253696
\(708\) 6.13169 0.230443
\(709\) −27.1825 −1.02086 −0.510431 0.859919i \(-0.670514\pi\)
−0.510431 + 0.859919i \(0.670514\pi\)
\(710\) −1.44696 −0.0543035
\(711\) −6.67994 −0.250517
\(712\) −3.67456 −0.137710
\(713\) 4.28621 0.160520
\(714\) −5.74094 −0.214849
\(715\) 0 0
\(716\) −6.48188 −0.242239
\(717\) −19.2814 −0.720078
\(718\) 3.05967 0.114186
\(719\) 24.6625 0.919755 0.459878 0.887982i \(-0.347893\pi\)
0.459878 + 0.887982i \(0.347893\pi\)
\(720\) 1.19806 0.0446491
\(721\) 7.75063 0.288648
\(722\) −5.39373 −0.200734
\(723\) −10.6920 −0.397641
\(724\) 10.1075 0.375643
\(725\) 8.79331 0.326575
\(726\) 8.58211 0.318512
\(727\) 38.9657 1.44516 0.722578 0.691289i \(-0.242957\pi\)
0.722578 + 0.691289i \(0.242957\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −7.02416 −0.259976
\(731\) −16.7168 −0.618292
\(732\) 14.7017 0.543391
\(733\) −8.73855 −0.322766 −0.161383 0.986892i \(-0.551595\pi\)
−0.161383 + 0.986892i \(0.551595\pi\)
\(734\) 25.6582 0.947060
\(735\) 1.19806 0.0441912
\(736\) 0.417895 0.0154038
\(737\) 16.4276 0.605118
\(738\) −11.0761 −0.407716
\(739\) 21.8200 0.802661 0.401331 0.915933i \(-0.368548\pi\)
0.401331 + 0.915933i \(0.368548\pi\)
\(740\) −3.09352 −0.113720
\(741\) 0 0
\(742\) −1.62133 −0.0595210
\(743\) 25.6437 0.940776 0.470388 0.882460i \(-0.344114\pi\)
0.470388 + 0.882460i \(0.344114\pi\)
\(744\) 10.2567 0.376028
\(745\) 2.92931 0.107322
\(746\) 21.6612 0.793071
\(747\) 9.06100 0.331525
\(748\) −8.92692 −0.326401
\(749\) 6.20775 0.226826
\(750\) 10.2610 0.374678
\(751\) 43.4034 1.58381 0.791907 0.610642i \(-0.209089\pi\)
0.791907 + 0.610642i \(0.209089\pi\)
\(752\) 7.89008 0.287722
\(753\) −6.49827 −0.236810
\(754\) 0 0
\(755\) −26.1613 −0.952109
\(756\) −1.00000 −0.0363696
\(757\) 39.9111 1.45059 0.725297 0.688436i \(-0.241702\pi\)
0.725297 + 0.688436i \(0.241702\pi\)
\(758\) 6.93602 0.251927
\(759\) −0.649809 −0.0235866
\(760\) −5.91723 −0.214641
\(761\) 3.46011 0.125429 0.0627144 0.998032i \(-0.480024\pi\)
0.0627144 + 0.998032i \(0.480024\pi\)
\(762\) −6.72886 −0.243761
\(763\) −4.91185 −0.177821
\(764\) 22.9681 0.830956
\(765\) −6.87800 −0.248675
\(766\) −19.9734 −0.721670
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −31.8146 −1.14726 −0.573632 0.819113i \(-0.694466\pi\)
−0.573632 + 0.819113i \(0.694466\pi\)
\(770\) 1.86294 0.0671356
\(771\) 0.907542 0.0326843
\(772\) 4.37435 0.157436
\(773\) −1.18300 −0.0425494 −0.0212747 0.999774i \(-0.506772\pi\)
−0.0212747 + 0.999774i \(0.506772\pi\)
\(774\) −2.91185 −0.104664
\(775\) 36.5614 1.31332
\(776\) −17.3937 −0.624399
\(777\) 2.58211 0.0926325
\(778\) 30.7289 1.10168
\(779\) 54.7047 1.96000
\(780\) 0 0
\(781\) 1.87800 0.0672002
\(782\) −2.39911 −0.0857919
\(783\) −2.46681 −0.0881567
\(784\) 1.00000 0.0357143
\(785\) 18.4021 0.656799
\(786\) −3.19269 −0.113879
\(787\) 15.2553 0.543794 0.271897 0.962326i \(-0.412349\pi\)
0.271897 + 0.962326i \(0.412349\pi\)
\(788\) 4.90515 0.174739
\(789\) 17.1564 0.610785
\(790\) 8.00298 0.284733
\(791\) −19.2054 −0.682864
\(792\) −1.55496 −0.0552530
\(793\) 0 0
\(794\) 12.3405 0.437948
\(795\) −1.94246 −0.0688919
\(796\) −7.73125 −0.274027
\(797\) −20.1575 −0.714015 −0.357008 0.934101i \(-0.616203\pi\)
−0.357008 + 0.934101i \(0.616203\pi\)
\(798\) 4.93900 0.174839
\(799\) −45.2965 −1.60247
\(800\) 3.56465 0.126029
\(801\) 3.67456 0.129834
\(802\) −10.2241 −0.361027
\(803\) 9.11662 0.321719
\(804\) 10.5646 0.372586
\(805\) 0.500664 0.0176461
\(806\) 0 0
\(807\) −19.7047 −0.693638
\(808\) −0.674563 −0.0237310
\(809\) −27.9842 −0.983872 −0.491936 0.870631i \(-0.663711\pi\)
−0.491936 + 0.870631i \(0.663711\pi\)
\(810\) −1.19806 −0.0420956
\(811\) 43.1196 1.51413 0.757067 0.653337i \(-0.226632\pi\)
0.757067 + 0.653337i \(0.226632\pi\)
\(812\) 2.46681 0.0865681
\(813\) 16.0194 0.561824
\(814\) 4.01507 0.140728
\(815\) 19.2782 0.675285
\(816\) −5.74094 −0.200973
\(817\) 14.3817 0.503150
\(818\) −14.0127 −0.489942
\(819\) 0 0
\(820\) 13.2698 0.463402
\(821\) −46.3075 −1.61614 −0.808072 0.589084i \(-0.799489\pi\)
−0.808072 + 0.589084i \(0.799489\pi\)
\(822\) −3.21014 −0.111967
\(823\) 0.214456 0.00747546 0.00373773 0.999993i \(-0.498810\pi\)
0.00373773 + 0.999993i \(0.498810\pi\)
\(824\) 7.75063 0.270006
\(825\) −5.54288 −0.192978
\(826\) 6.13169 0.213349
\(827\) 5.83100 0.202764 0.101382 0.994848i \(-0.467674\pi\)
0.101382 + 0.994848i \(0.467674\pi\)
\(828\) −0.417895 −0.0145228
\(829\) −11.0718 −0.384538 −0.192269 0.981342i \(-0.561585\pi\)
−0.192269 + 0.981342i \(0.561585\pi\)
\(830\) −10.8556 −0.376805
\(831\) −10.6571 −0.369691
\(832\) 0 0
\(833\) −5.74094 −0.198912
\(834\) 10.5308 0.364652
\(835\) 7.97333 0.275928
\(836\) 7.67994 0.265616
\(837\) −10.2567 −0.354522
\(838\) −1.10859 −0.0382955
\(839\) 49.6601 1.71446 0.857228 0.514936i \(-0.172185\pi\)
0.857228 + 0.514936i \(0.172185\pi\)
\(840\) 1.19806 0.0413371
\(841\) −22.9148 −0.790167
\(842\) 5.63342 0.194140
\(843\) −28.6286 −0.986022
\(844\) 6.35690 0.218813
\(845\) 0 0
\(846\) −7.89008 −0.271267
\(847\) 8.58211 0.294885
\(848\) −1.62133 −0.0556768
\(849\) −7.11960 −0.244344
\(850\) −20.4644 −0.701924
\(851\) 1.07905 0.0369893
\(852\) 1.20775 0.0413769
\(853\) 45.7120 1.56515 0.782574 0.622557i \(-0.213906\pi\)
0.782574 + 0.622557i \(0.213906\pi\)
\(854\) 14.7017 0.503082
\(855\) 5.91723 0.202365
\(856\) 6.20775 0.212177
\(857\) 27.3913 0.935670 0.467835 0.883816i \(-0.345034\pi\)
0.467835 + 0.883816i \(0.345034\pi\)
\(858\) 0 0
\(859\) −35.4325 −1.20894 −0.604470 0.796628i \(-0.706615\pi\)
−0.604470 + 0.796628i \(0.706615\pi\)
\(860\) 3.48858 0.118960
\(861\) −11.0761 −0.377471
\(862\) −18.6907 −0.636608
\(863\) 10.0489 0.342069 0.171035 0.985265i \(-0.445289\pi\)
0.171035 + 0.985265i \(0.445289\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 4.59392 0.156198
\(866\) 26.2640 0.892486
\(867\) 15.9584 0.541975
\(868\) 10.2567 0.348134
\(869\) −10.3870 −0.352356
\(870\) 2.95539 0.100197
\(871\) 0 0
\(872\) −4.91185 −0.166336
\(873\) 17.3937 0.588689
\(874\) 2.06398 0.0698153
\(875\) 10.2610 0.346884
\(876\) 5.86294 0.198090
\(877\) −43.7676 −1.47793 −0.738964 0.673745i \(-0.764684\pi\)
−0.738964 + 0.673745i \(0.764684\pi\)
\(878\) −11.8092 −0.398542
\(879\) 25.0978 0.846529
\(880\) 1.86294 0.0627996
\(881\) 15.3612 0.517532 0.258766 0.965940i \(-0.416684\pi\)
0.258766 + 0.965940i \(0.416684\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −6.10752 −0.205535 −0.102767 0.994705i \(-0.532770\pi\)
−0.102767 + 0.994705i \(0.532770\pi\)
\(884\) 0 0
\(885\) 7.34614 0.246938
\(886\) −7.16660 −0.240767
\(887\) −7.64801 −0.256795 −0.128397 0.991723i \(-0.540983\pi\)
−0.128397 + 0.991723i \(0.540983\pi\)
\(888\) 2.58211 0.0866498
\(889\) −6.72886 −0.225679
\(890\) −4.40236 −0.147567
\(891\) 1.55496 0.0520931
\(892\) −6.49157 −0.217354
\(893\) 38.9691 1.30405
\(894\) −2.44504 −0.0817744
\(895\) −7.76569 −0.259578
\(896\) 1.00000 0.0334077
\(897\) 0 0
\(898\) −31.3357 −1.04569
\(899\) 25.3013 0.843845
\(900\) −3.56465 −0.118822
\(901\) 9.30798 0.310094
\(902\) −17.2228 −0.573457
\(903\) −2.91185 −0.0969004
\(904\) −19.2054 −0.638761
\(905\) 12.1094 0.402531
\(906\) 21.8364 0.725465
\(907\) 33.3038 1.10583 0.552917 0.833236i \(-0.313515\pi\)
0.552917 + 0.833236i \(0.313515\pi\)
\(908\) 7.92931 0.263143
\(909\) 0.674563 0.0223738
\(910\) 0 0
\(911\) 20.0780 0.665213 0.332607 0.943066i \(-0.392072\pi\)
0.332607 + 0.943066i \(0.392072\pi\)
\(912\) 4.93900 0.163547
\(913\) 14.0895 0.466294
\(914\) 25.4058 0.840350
\(915\) 17.6136 0.582286
\(916\) −21.3502 −0.705430
\(917\) −3.19269 −0.105432
\(918\) 5.74094 0.189479
\(919\) 46.9541 1.54887 0.774436 0.632652i \(-0.218034\pi\)
0.774436 + 0.632652i \(0.218034\pi\)
\(920\) 0.500664 0.0165064
\(921\) −8.36658 −0.275688
\(922\) 37.0417 1.21990
\(923\) 0 0
\(924\) −1.55496 −0.0511544
\(925\) 9.20429 0.302635
\(926\) 10.2929 0.338246
\(927\) −7.75063 −0.254564
\(928\) 2.46681 0.0809771
\(929\) −3.66727 −0.120319 −0.0601596 0.998189i \(-0.519161\pi\)
−0.0601596 + 0.998189i \(0.519161\pi\)
\(930\) 12.2881 0.402944
\(931\) 4.93900 0.161869
\(932\) 10.3002 0.337395
\(933\) −21.6668 −0.709339
\(934\) 12.1890 0.398835
\(935\) −10.6950 −0.349764
\(936\) 0 0
\(937\) −3.10992 −0.101597 −0.0507983 0.998709i \(-0.516177\pi\)
−0.0507983 + 0.998709i \(0.516177\pi\)
\(938\) 10.5646 0.344948
\(939\) 14.6950 0.479553
\(940\) 9.45281 0.308317
\(941\) −7.25475 −0.236498 −0.118249 0.992984i \(-0.537728\pi\)
−0.118249 + 0.992984i \(0.537728\pi\)
\(942\) −15.3599 −0.500452
\(943\) −4.62863 −0.150729
\(944\) 6.13169 0.199569
\(945\) −1.19806 −0.0389730
\(946\) −4.52781 −0.147212
\(947\) −9.91782 −0.322286 −0.161143 0.986931i \(-0.551518\pi\)
−0.161143 + 0.986931i \(0.551518\pi\)
\(948\) −6.67994 −0.216954
\(949\) 0 0
\(950\) 17.6058 0.571207
\(951\) 27.1672 0.880957
\(952\) −5.74094 −0.186065
\(953\) −0.0204423 −0.000662191 0 −0.000331096 1.00000i \(-0.500105\pi\)
−0.000331096 1.00000i \(0.500105\pi\)
\(954\) 1.62133 0.0524926
\(955\) 27.5172 0.890435
\(956\) −19.2814 −0.623606
\(957\) −3.83579 −0.123993
\(958\) −42.4413 −1.37122
\(959\) −3.21014 −0.103661
\(960\) 1.19806 0.0386673
\(961\) 74.1992 2.39352
\(962\) 0 0
\(963\) −6.20775 −0.200042
\(964\) −10.6920 −0.344367
\(965\) 5.24075 0.168706
\(966\) −0.417895 −0.0134455
\(967\) −13.2731 −0.426833 −0.213416 0.976961i \(-0.568459\pi\)
−0.213416 + 0.976961i \(0.568459\pi\)
\(968\) 8.58211 0.275839
\(969\) −28.3545 −0.910878
\(970\) −20.8388 −0.669093
\(971\) 55.2409 1.77276 0.886382 0.462955i \(-0.153211\pi\)
0.886382 + 0.462955i \(0.153211\pi\)
\(972\) 1.00000 0.0320750
\(973\) 10.5308 0.337602
\(974\) 7.26742 0.232863
\(975\) 0 0
\(976\) 14.7017 0.470590
\(977\) −35.9235 −1.14929 −0.574647 0.818402i \(-0.694861\pi\)
−0.574647 + 0.818402i \(0.694861\pi\)
\(978\) −16.0911 −0.514538
\(979\) 5.71379 0.182614
\(980\) 1.19806 0.0382707
\(981\) 4.91185 0.156823
\(982\) −14.3763 −0.458765
\(983\) 31.1360 0.993084 0.496542 0.868013i \(-0.334603\pi\)
0.496542 + 0.868013i \(0.334603\pi\)
\(984\) −11.0761 −0.353092
\(985\) 5.87667 0.187246
\(986\) −14.1618 −0.451004
\(987\) −7.89008 −0.251144
\(988\) 0 0
\(989\) −1.21685 −0.0386935
\(990\) −1.86294 −0.0592080
\(991\) −51.4161 −1.63329 −0.816643 0.577143i \(-0.804167\pi\)
−0.816643 + 0.577143i \(0.804167\pi\)
\(992\) 10.2567 0.325650
\(993\) 30.9885 0.983391
\(994\) 1.20775 0.0383075
\(995\) −9.26252 −0.293642
\(996\) 9.06100 0.287109
\(997\) −32.4993 −1.02926 −0.514632 0.857411i \(-0.672072\pi\)
−0.514632 + 0.857411i \(0.672072\pi\)
\(998\) 5.55602 0.175873
\(999\) −2.58211 −0.0816942
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 7098.2.a.cg.1.1 3
13.12 even 2 7098.2.a.cl.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7098.2.a.cg.1.1 3 1.1 even 1 trivial
7098.2.a.cl.1.3 yes 3 13.12 even 2