Properties

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

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Newspace parameters

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

Newform invariants

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

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +2.55496 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +2.55496 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.24698 q^{11} -1.00000 q^{12} -2.55496 q^{14} -1.00000 q^{15} +1.00000 q^{16} +2.02177 q^{17} -1.00000 q^{18} -1.33513 q^{19} +1.00000 q^{20} -2.55496 q^{21} -2.24698 q^{22} +5.58211 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +2.55496 q^{28} +3.26875 q^{29} +1.00000 q^{30} -1.35690 q^{31} -1.00000 q^{32} -2.24698 q^{33} -2.02177 q^{34} +2.55496 q^{35} +1.00000 q^{36} +9.64071 q^{37} +1.33513 q^{38} -1.00000 q^{40} +1.06100 q^{41} +2.55496 q^{42} +0.137063 q^{43} +2.24698 q^{44} +1.00000 q^{45} -5.58211 q^{46} -12.7995 q^{47} -1.00000 q^{48} -0.472189 q^{49} -1.00000 q^{50} -2.02177 q^{51} -3.03684 q^{53} +1.00000 q^{54} +2.24698 q^{55} -2.55496 q^{56} +1.33513 q^{57} -3.26875 q^{58} +11.9608 q^{59} -1.00000 q^{60} -0.972853 q^{61} +1.35690 q^{62} +2.55496 q^{63} +1.00000 q^{64} +2.24698 q^{66} +8.51573 q^{67} +2.02177 q^{68} -5.58211 q^{69} -2.55496 q^{70} +2.69202 q^{71} -1.00000 q^{72} +1.48427 q^{73} -9.64071 q^{74} -1.00000 q^{75} -1.33513 q^{76} +5.74094 q^{77} -7.77479 q^{79} +1.00000 q^{80} +1.00000 q^{81} -1.06100 q^{82} +17.7289 q^{83} -2.55496 q^{84} +2.02177 q^{85} -0.137063 q^{86} -3.26875 q^{87} -2.24698 q^{88} +15.4601 q^{89} -1.00000 q^{90} +5.58211 q^{92} +1.35690 q^{93} +12.7995 q^{94} -1.33513 q^{95} +1.00000 q^{96} -7.14675 q^{97} +0.472189 q^{98} +2.24698 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} + 3q^{6} + 8q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{2} - 3q^{3} + 3q^{4} + 3q^{5} + 3q^{6} + 8q^{7} - 3q^{8} + 3q^{9} - 3q^{10} + 2q^{11} - 3q^{12} - 8q^{14} - 3q^{15} + 3q^{16} + 3q^{17} - 3q^{18} - 3q^{19} + 3q^{20} - 8q^{21} - 2q^{22} + 11q^{23} + 3q^{24} + 3q^{25} - 3q^{27} + 8q^{28} + 2q^{29} + 3q^{30} - 3q^{32} - 2q^{33} - 3q^{34} + 8q^{35} + 3q^{36} - 8q^{37} + 3q^{38} - 3q^{40} + 13q^{41} + 8q^{42} - 5q^{43} + 2q^{44} + 3q^{45} - 11q^{46} + 7q^{47} - 3q^{48} + 5q^{49} - 3q^{50} - 3q^{51} + 19q^{53} + 3q^{54} + 2q^{55} - 8q^{56} + 3q^{57} - 2q^{58} + 23q^{59} - 3q^{60} - 9q^{61} + 8q^{63} + 3q^{64} + 2q^{66} + 13q^{67} + 3q^{68} - 11q^{69} - 8q^{70} + 3q^{71} - 3q^{72} + 17q^{73} + 8q^{74} - 3q^{75} - 3q^{76} + 3q^{77} - 25q^{79} + 3q^{80} + 3q^{81} - 13q^{82} + 20q^{83} - 8q^{84} + 3q^{85} + 5q^{86} - 2q^{87} - 2q^{88} + 21q^{89} - 3q^{90} + 11q^{92} - 7q^{94} - 3q^{95} + 3q^{96} + 6q^{97} - 5q^{98} + 2q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) 2.55496 0.965683 0.482842 0.875708i \(-0.339605\pi\)
0.482842 + 0.875708i \(0.339605\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.00000 −0.316228
\(11\) 2.24698 0.677490 0.338745 0.940878i \(-0.389998\pi\)
0.338745 + 0.940878i \(0.389998\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −2.55496 −0.682841
\(15\) −1.00000 −0.258199
\(16\) 1.00000 0.250000
\(17\) 2.02177 0.490351 0.245176 0.969479i \(-0.421154\pi\)
0.245176 + 0.969479i \(0.421154\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.33513 −0.306299 −0.153149 0.988203i \(-0.548942\pi\)
−0.153149 + 0.988203i \(0.548942\pi\)
\(20\) 1.00000 0.223607
\(21\) −2.55496 −0.557538
\(22\) −2.24698 −0.479058
\(23\) 5.58211 1.16395 0.581975 0.813207i \(-0.302280\pi\)
0.581975 + 0.813207i \(0.302280\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 2.55496 0.482842
\(29\) 3.26875 0.606992 0.303496 0.952833i \(-0.401846\pi\)
0.303496 + 0.952833i \(0.401846\pi\)
\(30\) 1.00000 0.182574
\(31\) −1.35690 −0.243706 −0.121853 0.992548i \(-0.538884\pi\)
−0.121853 + 0.992548i \(0.538884\pi\)
\(32\) −1.00000 −0.176777
\(33\) −2.24698 −0.391149
\(34\) −2.02177 −0.346731
\(35\) 2.55496 0.431867
\(36\) 1.00000 0.166667
\(37\) 9.64071 1.58492 0.792462 0.609922i \(-0.208799\pi\)
0.792462 + 0.609922i \(0.208799\pi\)
\(38\) 1.33513 0.216586
\(39\) 0 0
\(40\) −1.00000 −0.158114
\(41\) 1.06100 0.165700 0.0828501 0.996562i \(-0.473598\pi\)
0.0828501 + 0.996562i \(0.473598\pi\)
\(42\) 2.55496 0.394239
\(43\) 0.137063 0.0209020 0.0104510 0.999945i \(-0.496673\pi\)
0.0104510 + 0.999945i \(0.496673\pi\)
\(44\) 2.24698 0.338745
\(45\) 1.00000 0.149071
\(46\) −5.58211 −0.823037
\(47\) −12.7995 −1.86701 −0.933503 0.358570i \(-0.883264\pi\)
−0.933503 + 0.358570i \(0.883264\pi\)
\(48\) −1.00000 −0.144338
\(49\) −0.472189 −0.0674556
\(50\) −1.00000 −0.141421
\(51\) −2.02177 −0.283104
\(52\) 0 0
\(53\) −3.03684 −0.417141 −0.208571 0.978007i \(-0.566881\pi\)
−0.208571 + 0.978007i \(0.566881\pi\)
\(54\) 1.00000 0.136083
\(55\) 2.24698 0.302983
\(56\) −2.55496 −0.341421
\(57\) 1.33513 0.176842
\(58\) −3.26875 −0.429208
\(59\) 11.9608 1.55716 0.778580 0.627545i \(-0.215940\pi\)
0.778580 + 0.627545i \(0.215940\pi\)
\(60\) −1.00000 −0.129099
\(61\) −0.972853 −0.124561 −0.0622805 0.998059i \(-0.519837\pi\)
−0.0622805 + 0.998059i \(0.519837\pi\)
\(62\) 1.35690 0.172326
\(63\) 2.55496 0.321894
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 2.24698 0.276584
\(67\) 8.51573 1.04036 0.520181 0.854056i \(-0.325864\pi\)
0.520181 + 0.854056i \(0.325864\pi\)
\(68\) 2.02177 0.245176
\(69\) −5.58211 −0.672006
\(70\) −2.55496 −0.305376
\(71\) 2.69202 0.319484 0.159742 0.987159i \(-0.448934\pi\)
0.159742 + 0.987159i \(0.448934\pi\)
\(72\) −1.00000 −0.117851
\(73\) 1.48427 0.173721 0.0868604 0.996220i \(-0.472317\pi\)
0.0868604 + 0.996220i \(0.472317\pi\)
\(74\) −9.64071 −1.12071
\(75\) −1.00000 −0.115470
\(76\) −1.33513 −0.153149
\(77\) 5.74094 0.654241
\(78\) 0 0
\(79\) −7.77479 −0.874732 −0.437366 0.899284i \(-0.644089\pi\)
−0.437366 + 0.899284i \(0.644089\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −1.06100 −0.117168
\(83\) 17.7289 1.94599 0.972997 0.230816i \(-0.0741395\pi\)
0.972997 + 0.230816i \(0.0741395\pi\)
\(84\) −2.55496 −0.278769
\(85\) 2.02177 0.219292
\(86\) −0.137063 −0.0147799
\(87\) −3.26875 −0.350447
\(88\) −2.24698 −0.239529
\(89\) 15.4601 1.63877 0.819384 0.573245i \(-0.194316\pi\)
0.819384 + 0.573245i \(0.194316\pi\)
\(90\) −1.00000 −0.105409
\(91\) 0 0
\(92\) 5.58211 0.581975
\(93\) 1.35690 0.140704
\(94\) 12.7995 1.32017
\(95\) −1.33513 −0.136981
\(96\) 1.00000 0.102062
\(97\) −7.14675 −0.725643 −0.362821 0.931859i \(-0.618186\pi\)
−0.362821 + 0.931859i \(0.618186\pi\)
\(98\) 0.472189 0.0476983
\(99\) 2.24698 0.225830
\(100\) 1.00000 0.100000
\(101\) −4.71379 −0.469040 −0.234520 0.972111i \(-0.575352\pi\)
−0.234520 + 0.972111i \(0.575352\pi\)
\(102\) 2.02177 0.200185
\(103\) 12.0858 1.19084 0.595422 0.803413i \(-0.296985\pi\)
0.595422 + 0.803413i \(0.296985\pi\)
\(104\) 0 0
\(105\) −2.55496 −0.249338
\(106\) 3.03684 0.294964
\(107\) −8.01507 −0.774846 −0.387423 0.921902i \(-0.626635\pi\)
−0.387423 + 0.921902i \(0.626635\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −9.17390 −0.878700 −0.439350 0.898316i \(-0.644791\pi\)
−0.439350 + 0.898316i \(0.644791\pi\)
\(110\) −2.24698 −0.214241
\(111\) −9.64071 −0.915056
\(112\) 2.55496 0.241421
\(113\) 5.24698 0.493594 0.246797 0.969067i \(-0.420622\pi\)
0.246797 + 0.969067i \(0.420622\pi\)
\(114\) −1.33513 −0.125046
\(115\) 5.58211 0.520534
\(116\) 3.26875 0.303496
\(117\) 0 0
\(118\) −11.9608 −1.10108
\(119\) 5.16554 0.473524
\(120\) 1.00000 0.0912871
\(121\) −5.95108 −0.541008
\(122\) 0.972853 0.0880780
\(123\) −1.06100 −0.0956671
\(124\) −1.35690 −0.121853
\(125\) 1.00000 0.0894427
\(126\) −2.55496 −0.227614
\(127\) −12.6843 −1.12555 −0.562773 0.826612i \(-0.690265\pi\)
−0.562773 + 0.826612i \(0.690265\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −0.137063 −0.0120678
\(130\) 0 0
\(131\) −14.5332 −1.26977 −0.634885 0.772606i \(-0.718953\pi\)
−0.634885 + 0.772606i \(0.718953\pi\)
\(132\) −2.24698 −0.195574
\(133\) −3.41119 −0.295788
\(134\) −8.51573 −0.735647
\(135\) −1.00000 −0.0860663
\(136\) −2.02177 −0.173365
\(137\) 5.06100 0.432390 0.216195 0.976350i \(-0.430635\pi\)
0.216195 + 0.976350i \(0.430635\pi\)
\(138\) 5.58211 0.475180
\(139\) −19.9638 −1.69330 −0.846652 0.532147i \(-0.821385\pi\)
−0.846652 + 0.532147i \(0.821385\pi\)
\(140\) 2.55496 0.215933
\(141\) 12.7995 1.07792
\(142\) −2.69202 −0.225909
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 3.26875 0.271455
\(146\) −1.48427 −0.122839
\(147\) 0.472189 0.0389455
\(148\) 9.64071 0.792462
\(149\) 3.33513 0.273224 0.136612 0.990625i \(-0.456379\pi\)
0.136612 + 0.990625i \(0.456379\pi\)
\(150\) 1.00000 0.0816497
\(151\) −15.6625 −1.27459 −0.637297 0.770618i \(-0.719948\pi\)
−0.637297 + 0.770618i \(0.719948\pi\)
\(152\) 1.33513 0.108293
\(153\) 2.02177 0.163450
\(154\) −5.74094 −0.462618
\(155\) −1.35690 −0.108988
\(156\) 0 0
\(157\) 2.99761 0.239235 0.119618 0.992820i \(-0.461833\pi\)
0.119618 + 0.992820i \(0.461833\pi\)
\(158\) 7.77479 0.618529
\(159\) 3.03684 0.240837
\(160\) −1.00000 −0.0790569
\(161\) 14.2620 1.12401
\(162\) −1.00000 −0.0785674
\(163\) −1.46681 −0.114890 −0.0574448 0.998349i \(-0.518295\pi\)
−0.0574448 + 0.998349i \(0.518295\pi\)
\(164\) 1.06100 0.0828501
\(165\) −2.24698 −0.174927
\(166\) −17.7289 −1.37603
\(167\) 1.13706 0.0879886 0.0439943 0.999032i \(-0.485992\pi\)
0.0439943 + 0.999032i \(0.485992\pi\)
\(168\) 2.55496 0.197119
\(169\) 0 0
\(170\) −2.02177 −0.155063
\(171\) −1.33513 −0.102100
\(172\) 0.137063 0.0104510
\(173\) 0.983607 0.0747822 0.0373911 0.999301i \(-0.488095\pi\)
0.0373911 + 0.999301i \(0.488095\pi\)
\(174\) 3.26875 0.247803
\(175\) 2.55496 0.193137
\(176\) 2.24698 0.169372
\(177\) −11.9608 −0.899027
\(178\) −15.4601 −1.15878
\(179\) 9.86592 0.737414 0.368707 0.929546i \(-0.379801\pi\)
0.368707 + 0.929546i \(0.379801\pi\)
\(180\) 1.00000 0.0745356
\(181\) −3.66487 −0.272408 −0.136204 0.990681i \(-0.543490\pi\)
−0.136204 + 0.990681i \(0.543490\pi\)
\(182\) 0 0
\(183\) 0.972853 0.0719154
\(184\) −5.58211 −0.411518
\(185\) 9.64071 0.708799
\(186\) −1.35690 −0.0994924
\(187\) 4.54288 0.332208
\(188\) −12.7995 −0.933503
\(189\) −2.55496 −0.185846
\(190\) 1.33513 0.0968602
\(191\) −12.2252 −0.884585 −0.442293 0.896871i \(-0.645835\pi\)
−0.442293 + 0.896871i \(0.645835\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −10.5579 −0.759977 −0.379989 0.924991i \(-0.624072\pi\)
−0.379989 + 0.924991i \(0.624072\pi\)
\(194\) 7.14675 0.513107
\(195\) 0 0
\(196\) −0.472189 −0.0337278
\(197\) 5.46011 0.389017 0.194508 0.980901i \(-0.437689\pi\)
0.194508 + 0.980901i \(0.437689\pi\)
\(198\) −2.24698 −0.159686
\(199\) 5.00969 0.355127 0.177564 0.984109i \(-0.443178\pi\)
0.177564 + 0.984109i \(0.443178\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −8.51573 −0.600653
\(202\) 4.71379 0.331661
\(203\) 8.35152 0.586162
\(204\) −2.02177 −0.141552
\(205\) 1.06100 0.0741034
\(206\) −12.0858 −0.842054
\(207\) 5.58211 0.387983
\(208\) 0 0
\(209\) −3.00000 −0.207514
\(210\) 2.55496 0.176309
\(211\) −9.38404 −0.646024 −0.323012 0.946395i \(-0.604695\pi\)
−0.323012 + 0.946395i \(0.604695\pi\)
\(212\) −3.03684 −0.208571
\(213\) −2.69202 −0.184454
\(214\) 8.01507 0.547899
\(215\) 0.137063 0.00934764
\(216\) 1.00000 0.0680414
\(217\) −3.46681 −0.235343
\(218\) 9.17390 0.621335
\(219\) −1.48427 −0.100298
\(220\) 2.24698 0.151491
\(221\) 0 0
\(222\) 9.64071 0.647042
\(223\) 26.5773 1.77975 0.889874 0.456205i \(-0.150792\pi\)
0.889874 + 0.456205i \(0.150792\pi\)
\(224\) −2.55496 −0.170710
\(225\) 1.00000 0.0666667
\(226\) −5.24698 −0.349024
\(227\) 22.2271 1.47527 0.737633 0.675202i \(-0.235943\pi\)
0.737633 + 0.675202i \(0.235943\pi\)
\(228\) 1.33513 0.0884209
\(229\) −19.1782 −1.26733 −0.633666 0.773607i \(-0.718451\pi\)
−0.633666 + 0.773607i \(0.718451\pi\)
\(230\) −5.58211 −0.368073
\(231\) −5.74094 −0.377726
\(232\) −3.26875 −0.214604
\(233\) 19.5483 1.28065 0.640324 0.768105i \(-0.278800\pi\)
0.640324 + 0.768105i \(0.278800\pi\)
\(234\) 0 0
\(235\) −12.7995 −0.834950
\(236\) 11.9608 0.778580
\(237\) 7.77479 0.505027
\(238\) −5.16554 −0.334832
\(239\) 12.8194 0.829218 0.414609 0.910000i \(-0.363918\pi\)
0.414609 + 0.910000i \(0.363918\pi\)
\(240\) −1.00000 −0.0645497
\(241\) −8.95407 −0.576782 −0.288391 0.957513i \(-0.593120\pi\)
−0.288391 + 0.957513i \(0.593120\pi\)
\(242\) 5.95108 0.382550
\(243\) −1.00000 −0.0641500
\(244\) −0.972853 −0.0622805
\(245\) −0.472189 −0.0301670
\(246\) 1.06100 0.0676468
\(247\) 0 0
\(248\) 1.35690 0.0861630
\(249\) −17.7289 −1.12352
\(250\) −1.00000 −0.0632456
\(251\) −0.276520 −0.0174538 −0.00872688 0.999962i \(-0.502778\pi\)
−0.00872688 + 0.999962i \(0.502778\pi\)
\(252\) 2.55496 0.160947
\(253\) 12.5429 0.788564
\(254\) 12.6843 0.795881
\(255\) −2.02177 −0.126608
\(256\) 1.00000 0.0625000
\(257\) 9.77048 0.609466 0.304733 0.952438i \(-0.401433\pi\)
0.304733 + 0.952438i \(0.401433\pi\)
\(258\) 0.137063 0.00853319
\(259\) 24.6316 1.53053
\(260\) 0 0
\(261\) 3.26875 0.202331
\(262\) 14.5332 0.897863
\(263\) −12.7071 −0.783553 −0.391776 0.920061i \(-0.628139\pi\)
−0.391776 + 0.920061i \(0.628139\pi\)
\(264\) 2.24698 0.138292
\(265\) −3.03684 −0.186551
\(266\) 3.41119 0.209153
\(267\) −15.4601 −0.946143
\(268\) 8.51573 0.520181
\(269\) 9.47757 0.577857 0.288929 0.957351i \(-0.406701\pi\)
0.288929 + 0.957351i \(0.406701\pi\)
\(270\) 1.00000 0.0608581
\(271\) 20.7657 1.26143 0.630713 0.776016i \(-0.282763\pi\)
0.630713 + 0.776016i \(0.282763\pi\)
\(272\) 2.02177 0.122588
\(273\) 0 0
\(274\) −5.06100 −0.305746
\(275\) 2.24698 0.135498
\(276\) −5.58211 −0.336003
\(277\) 15.0911 0.906738 0.453369 0.891323i \(-0.350222\pi\)
0.453369 + 0.891323i \(0.350222\pi\)
\(278\) 19.9638 1.19735
\(279\) −1.35690 −0.0812352
\(280\) −2.55496 −0.152688
\(281\) −17.6015 −1.05002 −0.525008 0.851097i \(-0.675938\pi\)
−0.525008 + 0.851097i \(0.675938\pi\)
\(282\) −12.7995 −0.762202
\(283\) −2.07606 −0.123409 −0.0617046 0.998094i \(-0.519654\pi\)
−0.0617046 + 0.998094i \(0.519654\pi\)
\(284\) 2.69202 0.159742
\(285\) 1.33513 0.0790860
\(286\) 0 0
\(287\) 2.71081 0.160014
\(288\) −1.00000 −0.0589256
\(289\) −12.9124 −0.759556
\(290\) −3.26875 −0.191948
\(291\) 7.14675 0.418950
\(292\) 1.48427 0.0868604
\(293\) −26.0930 −1.52437 −0.762186 0.647358i \(-0.775874\pi\)
−0.762186 + 0.647358i \(0.775874\pi\)
\(294\) −0.472189 −0.0275386
\(295\) 11.9608 0.696383
\(296\) −9.64071 −0.560355
\(297\) −2.24698 −0.130383
\(298\) −3.33513 −0.193199
\(299\) 0 0
\(300\) −1.00000 −0.0577350
\(301\) 0.350191 0.0201847
\(302\) 15.6625 0.901275
\(303\) 4.71379 0.270800
\(304\) −1.33513 −0.0765747
\(305\) −0.972853 −0.0557054
\(306\) −2.02177 −0.115577
\(307\) −15.8092 −0.902281 −0.451140 0.892453i \(-0.648983\pi\)
−0.451140 + 0.892453i \(0.648983\pi\)
\(308\) 5.74094 0.327120
\(309\) −12.0858 −0.687534
\(310\) 1.35690 0.0770665
\(311\) −23.2121 −1.31624 −0.658118 0.752915i \(-0.728647\pi\)
−0.658118 + 0.752915i \(0.728647\pi\)
\(312\) 0 0
\(313\) 28.5297 1.61260 0.806298 0.591510i \(-0.201468\pi\)
0.806298 + 0.591510i \(0.201468\pi\)
\(314\) −2.99761 −0.169165
\(315\) 2.55496 0.143956
\(316\) −7.77479 −0.437366
\(317\) 0.709480 0.0398484 0.0199242 0.999801i \(-0.493658\pi\)
0.0199242 + 0.999801i \(0.493658\pi\)
\(318\) −3.03684 −0.170297
\(319\) 7.34481 0.411231
\(320\) 1.00000 0.0559017
\(321\) 8.01507 0.447357
\(322\) −14.2620 −0.794793
\(323\) −2.69932 −0.150194
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 1.46681 0.0812392
\(327\) 9.17390 0.507318
\(328\) −1.06100 −0.0585839
\(329\) −32.7023 −1.80294
\(330\) 2.24698 0.123692
\(331\) 22.8605 1.25653 0.628265 0.778000i \(-0.283766\pi\)
0.628265 + 0.778000i \(0.283766\pi\)
\(332\) 17.7289 0.972997
\(333\) 9.64071 0.528308
\(334\) −1.13706 −0.0622173
\(335\) 8.51573 0.465264
\(336\) −2.55496 −0.139384
\(337\) 33.0629 1.80105 0.900526 0.434802i \(-0.143182\pi\)
0.900526 + 0.434802i \(0.143182\pi\)
\(338\) 0 0
\(339\) −5.24698 −0.284977
\(340\) 2.02177 0.109646
\(341\) −3.04892 −0.165108
\(342\) 1.33513 0.0721953
\(343\) −19.0911 −1.03082
\(344\) −0.137063 −0.00738996
\(345\) −5.58211 −0.300530
\(346\) −0.983607 −0.0528790
\(347\) 23.4808 1.26052 0.630258 0.776386i \(-0.282949\pi\)
0.630258 + 0.776386i \(0.282949\pi\)
\(348\) −3.26875 −0.175223
\(349\) 35.9004 1.92170 0.960851 0.277065i \(-0.0893617\pi\)
0.960851 + 0.277065i \(0.0893617\pi\)
\(350\) −2.55496 −0.136568
\(351\) 0 0
\(352\) −2.24698 −0.119764
\(353\) 31.0562 1.65296 0.826478 0.562969i \(-0.190341\pi\)
0.826478 + 0.562969i \(0.190341\pi\)
\(354\) 11.9608 0.635708
\(355\) 2.69202 0.142878
\(356\) 15.4601 0.819384
\(357\) −5.16554 −0.273389
\(358\) −9.86592 −0.521430
\(359\) −6.14675 −0.324413 −0.162207 0.986757i \(-0.551861\pi\)
−0.162207 + 0.986757i \(0.551861\pi\)
\(360\) −1.00000 −0.0527046
\(361\) −17.2174 −0.906181
\(362\) 3.66487 0.192622
\(363\) 5.95108 0.312351
\(364\) 0 0
\(365\) 1.48427 0.0776903
\(366\) −0.972853 −0.0508518
\(367\) −26.5163 −1.38414 −0.692070 0.721830i \(-0.743301\pi\)
−0.692070 + 0.721830i \(0.743301\pi\)
\(368\) 5.58211 0.290987
\(369\) 1.06100 0.0552334
\(370\) −9.64071 −0.501197
\(371\) −7.75899 −0.402827
\(372\) 1.35690 0.0703518
\(373\) −27.0616 −1.40120 −0.700598 0.713556i \(-0.747083\pi\)
−0.700598 + 0.713556i \(0.747083\pi\)
\(374\) −4.54288 −0.234907
\(375\) −1.00000 −0.0516398
\(376\) 12.7995 0.660086
\(377\) 0 0
\(378\) 2.55496 0.131413
\(379\) −16.7313 −0.859427 −0.429713 0.902965i \(-0.641385\pi\)
−0.429713 + 0.902965i \(0.641385\pi\)
\(380\) −1.33513 −0.0684905
\(381\) 12.6843 0.649834
\(382\) 12.2252 0.625496
\(383\) 34.3569 1.75556 0.877778 0.479068i \(-0.159025\pi\)
0.877778 + 0.479068i \(0.159025\pi\)
\(384\) 1.00000 0.0510310
\(385\) 5.74094 0.292585
\(386\) 10.5579 0.537385
\(387\) 0.137063 0.00696732
\(388\) −7.14675 −0.362821
\(389\) 11.6732 0.591857 0.295928 0.955210i \(-0.404371\pi\)
0.295928 + 0.955210i \(0.404371\pi\)
\(390\) 0 0
\(391\) 11.2857 0.570744
\(392\) 0.472189 0.0238491
\(393\) 14.5332 0.733102
\(394\) −5.46011 −0.275076
\(395\) −7.77479 −0.391192
\(396\) 2.24698 0.112915
\(397\) 12.6431 0.634539 0.317270 0.948335i \(-0.397234\pi\)
0.317270 + 0.948335i \(0.397234\pi\)
\(398\) −5.00969 −0.251113
\(399\) 3.41119 0.170773
\(400\) 1.00000 0.0500000
\(401\) 29.7724 1.48676 0.743381 0.668868i \(-0.233221\pi\)
0.743381 + 0.668868i \(0.233221\pi\)
\(402\) 8.51573 0.424726
\(403\) 0 0
\(404\) −4.71379 −0.234520
\(405\) 1.00000 0.0496904
\(406\) −8.35152 −0.414479
\(407\) 21.6625 1.07377
\(408\) 2.02177 0.100093
\(409\) 5.61117 0.277455 0.138727 0.990331i \(-0.455699\pi\)
0.138727 + 0.990331i \(0.455699\pi\)
\(410\) −1.06100 −0.0523990
\(411\) −5.06100 −0.249641
\(412\) 12.0858 0.595422
\(413\) 30.5593 1.50372
\(414\) −5.58211 −0.274346
\(415\) 17.7289 0.870275
\(416\) 0 0
\(417\) 19.9638 0.977629
\(418\) 3.00000 0.146735
\(419\) −1.24937 −0.0610358 −0.0305179 0.999534i \(-0.509716\pi\)
−0.0305179 + 0.999534i \(0.509716\pi\)
\(420\) −2.55496 −0.124669
\(421\) 2.76749 0.134879 0.0674397 0.997723i \(-0.478517\pi\)
0.0674397 + 0.997723i \(0.478517\pi\)
\(422\) 9.38404 0.456808
\(423\) −12.7995 −0.622335
\(424\) 3.03684 0.147482
\(425\) 2.02177 0.0980703
\(426\) 2.69202 0.130429
\(427\) −2.48560 −0.120287
\(428\) −8.01507 −0.387423
\(429\) 0 0
\(430\) −0.137063 −0.00660978
\(431\) 19.1793 0.923833 0.461917 0.886923i \(-0.347162\pi\)
0.461917 + 0.886923i \(0.347162\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 36.9051 1.77355 0.886774 0.462203i \(-0.152941\pi\)
0.886774 + 0.462203i \(0.152941\pi\)
\(434\) 3.46681 0.166412
\(435\) −3.26875 −0.156725
\(436\) −9.17390 −0.439350
\(437\) −7.45281 −0.356516
\(438\) 1.48427 0.0709212
\(439\) 14.6213 0.697838 0.348919 0.937153i \(-0.386549\pi\)
0.348919 + 0.937153i \(0.386549\pi\)
\(440\) −2.24698 −0.107121
\(441\) −0.472189 −0.0224852
\(442\) 0 0
\(443\) −35.1377 −1.66944 −0.834720 0.550674i \(-0.814371\pi\)
−0.834720 + 0.550674i \(0.814371\pi\)
\(444\) −9.64071 −0.457528
\(445\) 15.4601 0.732879
\(446\) −26.5773 −1.25847
\(447\) −3.33513 −0.157746
\(448\) 2.55496 0.120710
\(449\) −6.67994 −0.315246 −0.157623 0.987499i \(-0.550383\pi\)
−0.157623 + 0.987499i \(0.550383\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 2.38404 0.112260
\(452\) 5.24698 0.246797
\(453\) 15.6625 0.735888
\(454\) −22.2271 −1.04317
\(455\) 0 0
\(456\) −1.33513 −0.0625230
\(457\) −22.9963 −1.07572 −0.537860 0.843034i \(-0.680767\pi\)
−0.537860 + 0.843034i \(0.680767\pi\)
\(458\) 19.1782 0.896139
\(459\) −2.02177 −0.0943682
\(460\) 5.58211 0.260267
\(461\) 33.9705 1.58216 0.791081 0.611711i \(-0.209519\pi\)
0.791081 + 0.611711i \(0.209519\pi\)
\(462\) 5.74094 0.267093
\(463\) 26.9594 1.25291 0.626456 0.779457i \(-0.284505\pi\)
0.626456 + 0.779457i \(0.284505\pi\)
\(464\) 3.26875 0.151748
\(465\) 1.35690 0.0629245
\(466\) −19.5483 −0.905555
\(467\) −25.0411 −1.15877 −0.579383 0.815055i \(-0.696706\pi\)
−0.579383 + 0.815055i \(0.696706\pi\)
\(468\) 0 0
\(469\) 21.7573 1.00466
\(470\) 12.7995 0.590399
\(471\) −2.99761 −0.138122
\(472\) −11.9608 −0.550539
\(473\) 0.307979 0.0141609
\(474\) −7.77479 −0.357108
\(475\) −1.33513 −0.0612598
\(476\) 5.16554 0.236762
\(477\) −3.03684 −0.139047
\(478\) −12.8194 −0.586346
\(479\) 7.21983 0.329883 0.164941 0.986303i \(-0.447257\pi\)
0.164941 + 0.986303i \(0.447257\pi\)
\(480\) 1.00000 0.0456435
\(481\) 0 0
\(482\) 8.95407 0.407847
\(483\) −14.2620 −0.648946
\(484\) −5.95108 −0.270504
\(485\) −7.14675 −0.324517
\(486\) 1.00000 0.0453609
\(487\) −24.1497 −1.09433 −0.547164 0.837025i \(-0.684293\pi\)
−0.547164 + 0.837025i \(0.684293\pi\)
\(488\) 0.972853 0.0440390
\(489\) 1.46681 0.0663315
\(490\) 0.472189 0.0213313
\(491\) −19.7259 −0.890216 −0.445108 0.895477i \(-0.646835\pi\)
−0.445108 + 0.895477i \(0.646835\pi\)
\(492\) −1.06100 −0.0478335
\(493\) 6.60866 0.297639
\(494\) 0 0
\(495\) 2.24698 0.100994
\(496\) −1.35690 −0.0609264
\(497\) 6.87800 0.308521
\(498\) 17.7289 0.794449
\(499\) −35.3599 −1.58293 −0.791463 0.611217i \(-0.790680\pi\)
−0.791463 + 0.611217i \(0.790680\pi\)
\(500\) 1.00000 0.0447214
\(501\) −1.13706 −0.0508002
\(502\) 0.276520 0.0123417
\(503\) 14.3558 0.640095 0.320047 0.947402i \(-0.396301\pi\)
0.320047 + 0.947402i \(0.396301\pi\)
\(504\) −2.55496 −0.113807
\(505\) −4.71379 −0.209761
\(506\) −12.5429 −0.557599
\(507\) 0 0
\(508\) −12.6843 −0.562773
\(509\) 21.4426 0.950429 0.475214 0.879870i \(-0.342371\pi\)
0.475214 + 0.879870i \(0.342371\pi\)
\(510\) 2.02177 0.0895255
\(511\) 3.79225 0.167759
\(512\) −1.00000 −0.0441942
\(513\) 1.33513 0.0589472
\(514\) −9.77048 −0.430957
\(515\) 12.0858 0.532562
\(516\) −0.137063 −0.00603388
\(517\) −28.7603 −1.26488
\(518\) −24.6316 −1.08225
\(519\) −0.983607 −0.0431755
\(520\) 0 0
\(521\) 9.58748 0.420035 0.210018 0.977698i \(-0.432648\pi\)
0.210018 + 0.977698i \(0.432648\pi\)
\(522\) −3.26875 −0.143069
\(523\) 32.5241 1.42218 0.711090 0.703101i \(-0.248202\pi\)
0.711090 + 0.703101i \(0.248202\pi\)
\(524\) −14.5332 −0.634885
\(525\) −2.55496 −0.111508
\(526\) 12.7071 0.554055
\(527\) −2.74333 −0.119501
\(528\) −2.24698 −0.0977872
\(529\) 8.15990 0.354778
\(530\) 3.03684 0.131912
\(531\) 11.9608 0.519053
\(532\) −3.41119 −0.147894
\(533\) 0 0
\(534\) 15.4601 0.669024
\(535\) −8.01507 −0.346521
\(536\) −8.51573 −0.367823
\(537\) −9.86592 −0.425746
\(538\) −9.47757 −0.408607
\(539\) −1.06100 −0.0457005
\(540\) −1.00000 −0.0430331
\(541\) 27.5381 1.18395 0.591977 0.805954i \(-0.298347\pi\)
0.591977 + 0.805954i \(0.298347\pi\)
\(542\) −20.7657 −0.891963
\(543\) 3.66487 0.157275
\(544\) −2.02177 −0.0866827
\(545\) −9.17390 −0.392967
\(546\) 0 0
\(547\) −18.0670 −0.772488 −0.386244 0.922397i \(-0.626228\pi\)
−0.386244 + 0.922397i \(0.626228\pi\)
\(548\) 5.06100 0.216195
\(549\) −0.972853 −0.0415204
\(550\) −2.24698 −0.0958115
\(551\) −4.36419 −0.185921
\(552\) 5.58211 0.237590
\(553\) −19.8643 −0.844714
\(554\) −15.0911 −0.641161
\(555\) −9.64071 −0.409225
\(556\) −19.9638 −0.846652
\(557\) −33.5609 −1.42202 −0.711011 0.703181i \(-0.751762\pi\)
−0.711011 + 0.703181i \(0.751762\pi\)
\(558\) 1.35690 0.0574420
\(559\) 0 0
\(560\) 2.55496 0.107967
\(561\) −4.54288 −0.191800
\(562\) 17.6015 0.742474
\(563\) 12.6974 0.535132 0.267566 0.963540i \(-0.413781\pi\)
0.267566 + 0.963540i \(0.413781\pi\)
\(564\) 12.7995 0.538958
\(565\) 5.24698 0.220742
\(566\) 2.07606 0.0872635
\(567\) 2.55496 0.107298
\(568\) −2.69202 −0.112955
\(569\) 30.1317 1.26319 0.631593 0.775300i \(-0.282401\pi\)
0.631593 + 0.775300i \(0.282401\pi\)
\(570\) −1.33513 −0.0559223
\(571\) 0.693349 0.0290158 0.0145079 0.999895i \(-0.495382\pi\)
0.0145079 + 0.999895i \(0.495382\pi\)
\(572\) 0 0
\(573\) 12.2252 0.510715
\(574\) −2.71081 −0.113147
\(575\) 5.58211 0.232790
\(576\) 1.00000 0.0416667
\(577\) 34.6983 1.44451 0.722254 0.691628i \(-0.243106\pi\)
0.722254 + 0.691628i \(0.243106\pi\)
\(578\) 12.9124 0.537087
\(579\) 10.5579 0.438773
\(580\) 3.26875 0.135727
\(581\) 45.2965 1.87921
\(582\) −7.14675 −0.296242
\(583\) −6.82371 −0.282609
\(584\) −1.48427 −0.0614196
\(585\) 0 0
\(586\) 26.0930 1.07789
\(587\) 0.891149 0.0367816 0.0183908 0.999831i \(-0.494146\pi\)
0.0183908 + 0.999831i \(0.494146\pi\)
\(588\) 0.472189 0.0194727
\(589\) 1.81163 0.0746468
\(590\) −11.9608 −0.492417
\(591\) −5.46011 −0.224599
\(592\) 9.64071 0.396231
\(593\) 34.7439 1.42676 0.713381 0.700776i \(-0.247163\pi\)
0.713381 + 0.700776i \(0.247163\pi\)
\(594\) 2.24698 0.0921947
\(595\) 5.16554 0.211766
\(596\) 3.33513 0.136612
\(597\) −5.00969 −0.205033
\(598\) 0 0
\(599\) 10.4843 0.428376 0.214188 0.976792i \(-0.431290\pi\)
0.214188 + 0.976792i \(0.431290\pi\)
\(600\) 1.00000 0.0408248
\(601\) −2.28813 −0.0933347 −0.0466673 0.998910i \(-0.514860\pi\)
−0.0466673 + 0.998910i \(0.514860\pi\)
\(602\) −0.350191 −0.0142727
\(603\) 8.51573 0.346787
\(604\) −15.6625 −0.637297
\(605\) −5.95108 −0.241946
\(606\) −4.71379 −0.191485
\(607\) 0.0499823 0.00202872 0.00101436 0.999999i \(-0.499677\pi\)
0.00101436 + 0.999999i \(0.499677\pi\)
\(608\) 1.33513 0.0541465
\(609\) −8.35152 −0.338421
\(610\) 0.972853 0.0393897
\(611\) 0 0
\(612\) 2.02177 0.0817252
\(613\) −41.2737 −1.66703 −0.833514 0.552499i \(-0.813674\pi\)
−0.833514 + 0.552499i \(0.813674\pi\)
\(614\) 15.8092 0.638009
\(615\) −1.06100 −0.0427836
\(616\) −5.74094 −0.231309
\(617\) 2.35450 0.0947887 0.0473944 0.998876i \(-0.484908\pi\)
0.0473944 + 0.998876i \(0.484908\pi\)
\(618\) 12.0858 0.486160
\(619\) −30.0097 −1.20619 −0.603096 0.797669i \(-0.706066\pi\)
−0.603096 + 0.797669i \(0.706066\pi\)
\(620\) −1.35690 −0.0544942
\(621\) −5.58211 −0.224002
\(622\) 23.2121 0.930719
\(623\) 39.4999 1.58253
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −28.5297 −1.14028
\(627\) 3.00000 0.119808
\(628\) 2.99761 0.119618
\(629\) 19.4913 0.777169
\(630\) −2.55496 −0.101792
\(631\) 35.6209 1.41804 0.709022 0.705186i \(-0.249137\pi\)
0.709022 + 0.705186i \(0.249137\pi\)
\(632\) 7.77479 0.309265
\(633\) 9.38404 0.372982
\(634\) −0.709480 −0.0281770
\(635\) −12.6843 −0.503359
\(636\) 3.03684 0.120418
\(637\) 0 0
\(638\) −7.34481 −0.290784
\(639\) 2.69202 0.106495
\(640\) −1.00000 −0.0395285
\(641\) −1.93230 −0.0763211 −0.0381606 0.999272i \(-0.512150\pi\)
−0.0381606 + 0.999272i \(0.512150\pi\)
\(642\) −8.01507 −0.316329
\(643\) −31.8812 −1.25727 −0.628637 0.777699i \(-0.716387\pi\)
−0.628637 + 0.777699i \(0.716387\pi\)
\(644\) 14.2620 0.562003
\(645\) −0.137063 −0.00539686
\(646\) 2.69932 0.106203
\(647\) 34.1527 1.34268 0.671341 0.741149i \(-0.265719\pi\)
0.671341 + 0.741149i \(0.265719\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 26.8756 1.05496
\(650\) 0 0
\(651\) 3.46681 0.135875
\(652\) −1.46681 −0.0574448
\(653\) 11.7909 0.461414 0.230707 0.973023i \(-0.425896\pi\)
0.230707 + 0.973023i \(0.425896\pi\)
\(654\) −9.17390 −0.358728
\(655\) −14.5332 −0.567859
\(656\) 1.06100 0.0414250
\(657\) 1.48427 0.0579069
\(658\) 32.7023 1.27487
\(659\) −1.19806 −0.0466699 −0.0233349 0.999728i \(-0.507428\pi\)
−0.0233349 + 0.999728i \(0.507428\pi\)
\(660\) −2.24698 −0.0874636
\(661\) 2.95348 0.114877 0.0574384 0.998349i \(-0.481707\pi\)
0.0574384 + 0.998349i \(0.481707\pi\)
\(662\) −22.8605 −0.888500
\(663\) 0 0
\(664\) −17.7289 −0.688013
\(665\) −3.41119 −0.132280
\(666\) −9.64071 −0.373570
\(667\) 18.2465 0.706508
\(668\) 1.13706 0.0439943
\(669\) −26.5773 −1.02754
\(670\) −8.51573 −0.328991
\(671\) −2.18598 −0.0843888
\(672\) 2.55496 0.0985596
\(673\) 32.4771 1.25190 0.625950 0.779863i \(-0.284711\pi\)
0.625950 + 0.779863i \(0.284711\pi\)
\(674\) −33.0629 −1.27354
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 11.8418 0.455116 0.227558 0.973765i \(-0.426926\pi\)
0.227558 + 0.973765i \(0.426926\pi\)
\(678\) 5.24698 0.201509
\(679\) −18.2597 −0.700741
\(680\) −2.02177 −0.0775314
\(681\) −22.2271 −0.851745
\(682\) 3.04892 0.116749
\(683\) 7.34481 0.281042 0.140521 0.990078i \(-0.455122\pi\)
0.140521 + 0.990078i \(0.455122\pi\)
\(684\) −1.33513 −0.0510498
\(685\) 5.06100 0.193371
\(686\) 19.0911 0.728903
\(687\) 19.1782 0.731694
\(688\) 0.137063 0.00522549
\(689\) 0 0
\(690\) 5.58211 0.212507
\(691\) −32.8678 −1.25035 −0.625176 0.780484i \(-0.714973\pi\)
−0.625176 + 0.780484i \(0.714973\pi\)
\(692\) 0.983607 0.0373911
\(693\) 5.74094 0.218080
\(694\) −23.4808 −0.891319
\(695\) −19.9638 −0.757268
\(696\) 3.26875 0.123902
\(697\) 2.14510 0.0812513
\(698\) −35.9004 −1.35885
\(699\) −19.5483 −0.739383
\(700\) 2.55496 0.0965683
\(701\) −48.7133 −1.83988 −0.919938 0.392063i \(-0.871761\pi\)
−0.919938 + 0.392063i \(0.871761\pi\)
\(702\) 0 0
\(703\) −12.8716 −0.485460
\(704\) 2.24698 0.0846862
\(705\) 12.7995 0.482059
\(706\) −31.0562 −1.16882
\(707\) −12.0435 −0.452944
\(708\) −11.9608 −0.449513
\(709\) −29.6209 −1.11243 −0.556217 0.831037i \(-0.687748\pi\)
−0.556217 + 0.831037i \(0.687748\pi\)
\(710\) −2.69202 −0.101030
\(711\) −7.77479 −0.291577
\(712\) −15.4601 −0.579392
\(713\) −7.57434 −0.283661
\(714\) 5.16554 0.193315
\(715\) 0 0
\(716\) 9.86592 0.368707
\(717\) −12.8194 −0.478749
\(718\) 6.14675 0.229395
\(719\) −33.5599 −1.25157 −0.625786 0.779995i \(-0.715222\pi\)
−0.625786 + 0.779995i \(0.715222\pi\)
\(720\) 1.00000 0.0372678
\(721\) 30.8786 1.14998
\(722\) 17.2174 0.640767
\(723\) 8.95407 0.333005
\(724\) −3.66487 −0.136204
\(725\) 3.26875 0.121398
\(726\) −5.95108 −0.220865
\(727\) 29.8950 1.10874 0.554372 0.832269i \(-0.312959\pi\)
0.554372 + 0.832269i \(0.312959\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −1.48427 −0.0549353
\(731\) 0.277111 0.0102493
\(732\) 0.972853 0.0359577
\(733\) 19.8812 0.734331 0.367165 0.930156i \(-0.380328\pi\)
0.367165 + 0.930156i \(0.380328\pi\)
\(734\) 26.5163 0.978735
\(735\) 0.472189 0.0174170
\(736\) −5.58211 −0.205759
\(737\) 19.1347 0.704835
\(738\) −1.06100 −0.0390559
\(739\) −35.4144 −1.30274 −0.651371 0.758760i \(-0.725806\pi\)
−0.651371 + 0.758760i \(0.725806\pi\)
\(740\) 9.64071 0.354400
\(741\) 0 0
\(742\) 7.75899 0.284841
\(743\) 14.9815 0.549617 0.274809 0.961499i \(-0.411386\pi\)
0.274809 + 0.961499i \(0.411386\pi\)
\(744\) −1.35690 −0.0497462
\(745\) 3.33513 0.122190
\(746\) 27.0616 0.990795
\(747\) 17.7289 0.648665
\(748\) 4.54288 0.166104
\(749\) −20.4782 −0.748256
\(750\) 1.00000 0.0365148
\(751\) −34.3599 −1.25381 −0.626905 0.779096i \(-0.715679\pi\)
−0.626905 + 0.779096i \(0.715679\pi\)
\(752\) −12.7995 −0.466751
\(753\) 0.276520 0.0100769
\(754\) 0 0
\(755\) −15.6625 −0.570016
\(756\) −2.55496 −0.0929229
\(757\) −20.8653 −0.758363 −0.379182 0.925322i \(-0.623794\pi\)
−0.379182 + 0.925322i \(0.623794\pi\)
\(758\) 16.7313 0.607706
\(759\) −12.5429 −0.455278
\(760\) 1.33513 0.0484301
\(761\) 48.0974 1.74353 0.871764 0.489926i \(-0.162976\pi\)
0.871764 + 0.489926i \(0.162976\pi\)
\(762\) −12.6843 −0.459502
\(763\) −23.4389 −0.848546
\(764\) −12.2252 −0.442293
\(765\) 2.02177 0.0730973
\(766\) −34.3569 −1.24137
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −15.5235 −0.559792 −0.279896 0.960030i \(-0.590300\pi\)
−0.279896 + 0.960030i \(0.590300\pi\)
\(770\) −5.74094 −0.206889
\(771\) −9.77048 −0.351875
\(772\) −10.5579 −0.379989
\(773\) 15.0043 0.539668 0.269834 0.962907i \(-0.413031\pi\)
0.269834 + 0.962907i \(0.413031\pi\)
\(774\) −0.137063 −0.00492664
\(775\) −1.35690 −0.0487411
\(776\) 7.14675 0.256553
\(777\) −24.6316 −0.883654
\(778\) −11.6732 −0.418506
\(779\) −1.41657 −0.0507538
\(780\) 0 0
\(781\) 6.04892 0.216447
\(782\) −11.2857 −0.403577
\(783\) −3.26875 −0.116816
\(784\) −0.472189 −0.0168639
\(785\) 2.99761 0.106989
\(786\) −14.5332 −0.518382
\(787\) 18.2024 0.648845 0.324422 0.945912i \(-0.394830\pi\)
0.324422 + 0.945912i \(0.394830\pi\)
\(788\) 5.46011 0.194508
\(789\) 12.7071 0.452384
\(790\) 7.77479 0.276615
\(791\) 13.4058 0.476656
\(792\) −2.24698 −0.0798429
\(793\) 0 0
\(794\) −12.6431 −0.448687
\(795\) 3.03684 0.107705
\(796\) 5.00969 0.177564
\(797\) −36.0079 −1.27546 −0.637732 0.770258i \(-0.720127\pi\)
−0.637732 + 0.770258i \(0.720127\pi\)
\(798\) −3.41119 −0.120755
\(799\) −25.8777 −0.915489
\(800\) −1.00000 −0.0353553
\(801\) 15.4601 0.546256
\(802\) −29.7724 −1.05130
\(803\) 3.33513 0.117694
\(804\) −8.51573 −0.300327
\(805\) 14.2620 0.502671
\(806\) 0 0
\(807\) −9.47757 −0.333626
\(808\) 4.71379 0.165831
\(809\) 21.6832 0.762340 0.381170 0.924505i \(-0.375521\pi\)
0.381170 + 0.924505i \(0.375521\pi\)
\(810\) −1.00000 −0.0351364
\(811\) −16.0301 −0.562894 −0.281447 0.959577i \(-0.590814\pi\)
−0.281447 + 0.959577i \(0.590814\pi\)
\(812\) 8.35152 0.293081
\(813\) −20.7657 −0.728285
\(814\) −21.6625 −0.759270
\(815\) −1.46681 −0.0513802
\(816\) −2.02177 −0.0707761
\(817\) −0.182997 −0.00640225
\(818\) −5.61117 −0.196190
\(819\) 0 0
\(820\) 1.06100 0.0370517
\(821\) −10.2543 −0.357877 −0.178938 0.983860i \(-0.557266\pi\)
−0.178938 + 0.983860i \(0.557266\pi\)
\(822\) 5.06100 0.176523
\(823\) 3.81807 0.133089 0.0665447 0.997783i \(-0.478802\pi\)
0.0665447 + 0.997783i \(0.478802\pi\)
\(824\) −12.0858 −0.421027
\(825\) −2.24698 −0.0782298
\(826\) −30.5593 −1.06329
\(827\) −29.9172 −1.04032 −0.520162 0.854068i \(-0.674128\pi\)
−0.520162 + 0.854068i \(0.674128\pi\)
\(828\) 5.58211 0.193992
\(829\) 48.3105 1.67789 0.838946 0.544214i \(-0.183172\pi\)
0.838946 + 0.544214i \(0.183172\pi\)
\(830\) −17.7289 −0.615378
\(831\) −15.0911 −0.523505
\(832\) 0 0
\(833\) −0.954658 −0.0330769
\(834\) −19.9638 −0.691288
\(835\) 1.13706 0.0393497
\(836\) −3.00000 −0.103757
\(837\) 1.35690 0.0469012
\(838\) 1.24937 0.0431589
\(839\) 28.9275 0.998689 0.499344 0.866404i \(-0.333574\pi\)
0.499344 + 0.866404i \(0.333574\pi\)
\(840\) 2.55496 0.0881544
\(841\) −18.3153 −0.631561
\(842\) −2.76749 −0.0953742
\(843\) 17.6015 0.606227
\(844\) −9.38404 −0.323012
\(845\) 0 0
\(846\) 12.7995 0.440057
\(847\) −15.2048 −0.522442
\(848\) −3.03684 −0.104285
\(849\) 2.07606 0.0712503
\(850\) −2.02177 −0.0693461
\(851\) 53.8155 1.84477
\(852\) −2.69202 −0.0922271
\(853\) 54.7193 1.87355 0.936776 0.349929i \(-0.113794\pi\)
0.936776 + 0.349929i \(0.113794\pi\)
\(854\) 2.48560 0.0850554
\(855\) −1.33513 −0.0456603
\(856\) 8.01507 0.273949
\(857\) −57.7904 −1.97408 −0.987042 0.160462i \(-0.948702\pi\)
−0.987042 + 0.160462i \(0.948702\pi\)
\(858\) 0 0
\(859\) −11.0261 −0.376205 −0.188103 0.982149i \(-0.560234\pi\)
−0.188103 + 0.982149i \(0.560234\pi\)
\(860\) 0.137063 0.00467382
\(861\) −2.71081 −0.0923841
\(862\) −19.1793 −0.653249
\(863\) −47.8501 −1.62884 −0.814418 0.580278i \(-0.802944\pi\)
−0.814418 + 0.580278i \(0.802944\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0.983607 0.0334436
\(866\) −36.9051 −1.25409
\(867\) 12.9124 0.438530
\(868\) −3.46681 −0.117671
\(869\) −17.4698 −0.592622
\(870\) 3.26875 0.110821
\(871\) 0 0
\(872\) 9.17390 0.310667
\(873\) −7.14675 −0.241881
\(874\) 7.45281 0.252095
\(875\) 2.55496 0.0863733
\(876\) −1.48427 −0.0501489
\(877\) 1.76569 0.0596232 0.0298116 0.999556i \(-0.490509\pi\)
0.0298116 + 0.999556i \(0.490509\pi\)
\(878\) −14.6213 −0.493446
\(879\) 26.0930 0.880097
\(880\) 2.24698 0.0757457
\(881\) 46.8394 1.57806 0.789029 0.614356i \(-0.210584\pi\)
0.789029 + 0.614356i \(0.210584\pi\)
\(882\) 0.472189 0.0158994
\(883\) −11.3730 −0.382733 −0.191366 0.981519i \(-0.561292\pi\)
−0.191366 + 0.981519i \(0.561292\pi\)
\(884\) 0 0
\(885\) −11.9608 −0.402057
\(886\) 35.1377 1.18047
\(887\) −53.1245 −1.78375 −0.891873 0.452286i \(-0.850609\pi\)
−0.891873 + 0.452286i \(0.850609\pi\)
\(888\) 9.64071 0.323521
\(889\) −32.4077 −1.08692
\(890\) −15.4601 −0.518224
\(891\) 2.24698 0.0752766
\(892\) 26.5773 0.889874
\(893\) 17.0890 0.571862
\(894\) 3.33513 0.111543
\(895\) 9.86592 0.329781
\(896\) −2.55496 −0.0853552
\(897\) 0 0
\(898\) 6.67994 0.222912
\(899\) −4.43535 −0.147927
\(900\) 1.00000 0.0333333
\(901\) −6.13978 −0.204546
\(902\) −2.38404 −0.0793799
\(903\) −0.350191 −0.0116536
\(904\) −5.24698 −0.174512
\(905\) −3.66487 −0.121825
\(906\) −15.6625 −0.520351
\(907\) −0.871560 −0.0289397 −0.0144698 0.999895i \(-0.504606\pi\)
−0.0144698 + 0.999895i \(0.504606\pi\)
\(908\) 22.2271 0.737633
\(909\) −4.71379 −0.156347
\(910\) 0 0
\(911\) −34.8431 −1.15440 −0.577201 0.816602i \(-0.695855\pi\)
−0.577201 + 0.816602i \(0.695855\pi\)
\(912\) 1.33513 0.0442104
\(913\) 39.8364 1.31839
\(914\) 22.9963 0.760649
\(915\) 0.972853 0.0321615
\(916\) −19.1782 −0.633666
\(917\) −37.1317 −1.22620
\(918\) 2.02177 0.0667284
\(919\) 36.6765 1.20985 0.604923 0.796284i \(-0.293204\pi\)
0.604923 + 0.796284i \(0.293204\pi\)
\(920\) −5.58211 −0.184037
\(921\) 15.8092 0.520932
\(922\) −33.9705 −1.11876
\(923\) 0 0
\(924\) −5.74094 −0.188863
\(925\) 9.64071 0.316985
\(926\) −26.9594 −0.885942
\(927\) 12.0858 0.396948
\(928\) −3.26875 −0.107302
\(929\) −20.2446 −0.664203 −0.332102 0.943244i \(-0.607758\pi\)
−0.332102 + 0.943244i \(0.607758\pi\)
\(930\) −1.35690 −0.0444944
\(931\) 0.630432 0.0206616
\(932\) 19.5483 0.640324
\(933\) 23.2121 0.759929
\(934\) 25.0411 0.819371
\(935\) 4.54288 0.148568
\(936\) 0 0
\(937\) 7.66189 0.250303 0.125152 0.992138i \(-0.460058\pi\)
0.125152 + 0.992138i \(0.460058\pi\)
\(938\) −21.7573 −0.710402
\(939\) −28.5297 −0.931033
\(940\) −12.7995 −0.417475
\(941\) −21.4107 −0.697969 −0.348985 0.937128i \(-0.613473\pi\)
−0.348985 + 0.937128i \(0.613473\pi\)
\(942\) 2.99761 0.0976673
\(943\) 5.92261 0.192867
\(944\) 11.9608 0.389290
\(945\) −2.55496 −0.0831128
\(946\) −0.307979 −0.0100132
\(947\) −36.3631 −1.18164 −0.590821 0.806802i \(-0.701196\pi\)
−0.590821 + 0.806802i \(0.701196\pi\)
\(948\) 7.77479 0.252513
\(949\) 0 0
\(950\) 1.33513 0.0433172
\(951\) −0.709480 −0.0230065
\(952\) −5.16554 −0.167416
\(953\) 48.8256 1.58162 0.790809 0.612064i \(-0.209660\pi\)
0.790809 + 0.612064i \(0.209660\pi\)
\(954\) 3.03684 0.0983212
\(955\) −12.2252 −0.395598
\(956\) 12.8194 0.414609
\(957\) −7.34481 −0.237424
\(958\) −7.21983 −0.233262
\(959\) 12.9306 0.417552
\(960\) −1.00000 −0.0322749
\(961\) −29.1588 −0.940608
\(962\) 0 0
\(963\) −8.01507 −0.258282
\(964\) −8.95407 −0.288391
\(965\) −10.5579 −0.339872
\(966\) 14.2620 0.458874
\(967\) −4.23623 −0.136228 −0.0681139 0.997678i \(-0.521698\pi\)
−0.0681139 + 0.997678i \(0.521698\pi\)
\(968\) 5.95108 0.191275
\(969\) 2.69932 0.0867146
\(970\) 7.14675 0.229468
\(971\) −25.7735 −0.827110 −0.413555 0.910479i \(-0.635713\pi\)
−0.413555 + 0.910479i \(0.635713\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −51.0066 −1.63520
\(974\) 24.1497 0.773807
\(975\) 0 0
\(976\) −0.972853 −0.0311403
\(977\) −33.2881 −1.06498 −0.532491 0.846436i \(-0.678744\pi\)
−0.532491 + 0.846436i \(0.678744\pi\)
\(978\) −1.46681 −0.0469035
\(979\) 34.7385 1.11025
\(980\) −0.472189 −0.0150835
\(981\) −9.17390 −0.292900
\(982\) 19.7259 0.629478
\(983\) 45.1605 1.44040 0.720198 0.693769i \(-0.244051\pi\)
0.720198 + 0.693769i \(0.244051\pi\)
\(984\) 1.06100 0.0338234
\(985\) 5.46011 0.173973
\(986\) −6.60866 −0.210463
\(987\) 32.7023 1.04093
\(988\) 0 0
\(989\) 0.765102 0.0243288
\(990\) −2.24698 −0.0714137
\(991\) −24.5364 −0.779426 −0.389713 0.920936i \(-0.627426\pi\)
−0.389713 + 0.920936i \(0.627426\pi\)
\(992\) 1.35690 0.0430815
\(993\) −22.8605 −0.725457
\(994\) −6.87800 −0.218157
\(995\) 5.00969 0.158818
\(996\) −17.7289 −0.561760
\(997\) 20.6286 0.653315 0.326658 0.945143i \(-0.394078\pi\)
0.326658 + 0.945143i \(0.394078\pi\)
\(998\) 35.3599 1.11930
\(999\) −9.64071 −0.305019
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 5070.2.a.bm.1.2 3
13.5 odd 4 5070.2.b.u.1351.5 6
13.8 odd 4 5070.2.b.u.1351.2 6
13.12 even 2 5070.2.a.br.1.2 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5070.2.a.bm.1.2 3 1.1 even 1 trivial
5070.2.a.br.1.2 yes 3 13.12 even 2
5070.2.b.u.1351.2 6 13.8 odd 4
5070.2.b.u.1351.5 6 13.5 odd 4