Properties

Label 3450.2.a.bo.1.1
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.5483886973\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Defining polynomial: \(x^{3} - x^{2} - 3 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 690)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.34017 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -4.34017 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.07838 q^{11} +1.00000 q^{12} +2.34017 q^{13} +4.34017 q^{14} +1.00000 q^{16} -0.921622 q^{17} -1.00000 q^{18} +2.34017 q^{19} -4.34017 q^{21} +1.07838 q^{22} -1.00000 q^{23} -1.00000 q^{24} -2.34017 q^{26} +1.00000 q^{27} -4.34017 q^{28} +10.4969 q^{29} -4.00000 q^{31} -1.00000 q^{32} -1.07838 q^{33} +0.921622 q^{34} +1.00000 q^{36} -2.58145 q^{37} -2.34017 q^{38} +2.34017 q^{39} +0.156755 q^{41} +4.34017 q^{42} -0.738205 q^{43} -1.07838 q^{44} +1.00000 q^{46} -6.83710 q^{47} +1.00000 q^{48} +11.8371 q^{49} -0.921622 q^{51} +2.34017 q^{52} +0.340173 q^{53} -1.00000 q^{54} +4.34017 q^{56} +2.34017 q^{57} -10.4969 q^{58} -8.83710 q^{59} -11.5753 q^{61} +4.00000 q^{62} -4.34017 q^{63} +1.00000 q^{64} +1.07838 q^{66} -2.58145 q^{67} -0.921622 q^{68} -1.00000 q^{69} +15.1773 q^{71} -1.00000 q^{72} +4.68035 q^{73} +2.58145 q^{74} +2.34017 q^{76} +4.68035 q^{77} -2.34017 q^{78} -11.9155 q^{79} +1.00000 q^{81} -0.156755 q^{82} -11.4186 q^{83} -4.34017 q^{84} +0.738205 q^{86} +10.4969 q^{87} +1.07838 q^{88} -4.68035 q^{89} -10.1568 q^{91} -1.00000 q^{92} -4.00000 q^{93} +6.83710 q^{94} -1.00000 q^{96} -9.02052 q^{97} -11.8371 q^{98} -1.07838 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{6} - 2q^{7} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{6} - 2q^{7} - 3q^{8} + 3q^{9} + 3q^{12} - 4q^{13} + 2q^{14} + 3q^{16} - 6q^{17} - 3q^{18} - 4q^{19} - 2q^{21} - 3q^{23} - 3q^{24} + 4q^{26} + 3q^{27} - 2q^{28} + 14q^{29} - 12q^{31} - 3q^{32} + 6q^{34} + 3q^{36} - 22q^{37} + 4q^{38} - 4q^{39} - 6q^{41} + 2q^{42} - 10q^{43} + 3q^{46} + 8q^{47} + 3q^{48} + 7q^{49} - 6q^{51} - 4q^{52} - 10q^{53} - 3q^{54} + 2q^{56} - 4q^{57} - 14q^{58} + 2q^{59} - 14q^{61} + 12q^{62} - 2q^{63} + 3q^{64} - 22q^{67} - 6q^{68} - 3q^{69} + 6q^{71} - 3q^{72} - 8q^{73} + 22q^{74} - 4q^{76} - 8q^{77} + 4q^{78} - 4q^{79} + 3q^{81} + 6q^{82} - 20q^{83} - 2q^{84} + 10q^{86} + 14q^{87} + 8q^{89} - 24q^{91} - 3q^{92} - 12q^{93} - 8q^{94} - 3q^{96} + 6q^{97} - 7q^{98} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) −4.34017 −1.64043 −0.820216 0.572055i \(-0.806147\pi\)
−0.820216 + 0.572055i \(0.806147\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.07838 −0.325143 −0.162572 0.986697i \(-0.551979\pi\)
−0.162572 + 0.986697i \(0.551979\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.34017 0.649047 0.324524 0.945878i \(-0.394796\pi\)
0.324524 + 0.945878i \(0.394796\pi\)
\(14\) 4.34017 1.15996
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −0.921622 −0.223526 −0.111763 0.993735i \(-0.535650\pi\)
−0.111763 + 0.993735i \(0.535650\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.34017 0.536873 0.268436 0.963297i \(-0.413493\pi\)
0.268436 + 0.963297i \(0.413493\pi\)
\(20\) 0 0
\(21\) −4.34017 −0.947103
\(22\) 1.07838 0.229911
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −2.34017 −0.458946
\(27\) 1.00000 0.192450
\(28\) −4.34017 −0.820216
\(29\) 10.4969 1.94923 0.974615 0.223886i \(-0.0718743\pi\)
0.974615 + 0.223886i \(0.0718743\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.07838 −0.187721
\(34\) 0.921622 0.158057
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −2.58145 −0.424388 −0.212194 0.977228i \(-0.568061\pi\)
−0.212194 + 0.977228i \(0.568061\pi\)
\(38\) −2.34017 −0.379626
\(39\) 2.34017 0.374728
\(40\) 0 0
\(41\) 0.156755 0.0244811 0.0122405 0.999925i \(-0.496104\pi\)
0.0122405 + 0.999925i \(0.496104\pi\)
\(42\) 4.34017 0.669703
\(43\) −0.738205 −0.112575 −0.0562876 0.998415i \(-0.517926\pi\)
−0.0562876 + 0.998415i \(0.517926\pi\)
\(44\) −1.07838 −0.162572
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) −6.83710 −0.997294 −0.498647 0.866805i \(-0.666170\pi\)
−0.498647 + 0.866805i \(0.666170\pi\)
\(48\) 1.00000 0.144338
\(49\) 11.8371 1.69101
\(50\) 0 0
\(51\) −0.921622 −0.129053
\(52\) 2.34017 0.324524
\(53\) 0.340173 0.0467264 0.0233632 0.999727i \(-0.492563\pi\)
0.0233632 + 0.999727i \(0.492563\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 4.34017 0.579980
\(57\) 2.34017 0.309963
\(58\) −10.4969 −1.37831
\(59\) −8.83710 −1.15049 −0.575246 0.817980i \(-0.695094\pi\)
−0.575246 + 0.817980i \(0.695094\pi\)
\(60\) 0 0
\(61\) −11.5753 −1.48207 −0.741033 0.671469i \(-0.765664\pi\)
−0.741033 + 0.671469i \(0.765664\pi\)
\(62\) 4.00000 0.508001
\(63\) −4.34017 −0.546810
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.07838 0.132739
\(67\) −2.58145 −0.315374 −0.157687 0.987489i \(-0.550404\pi\)
−0.157687 + 0.987489i \(0.550404\pi\)
\(68\) −0.921622 −0.111763
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 15.1773 1.80121 0.900606 0.434637i \(-0.143123\pi\)
0.900606 + 0.434637i \(0.143123\pi\)
\(72\) −1.00000 −0.117851
\(73\) 4.68035 0.547793 0.273897 0.961759i \(-0.411687\pi\)
0.273897 + 0.961759i \(0.411687\pi\)
\(74\) 2.58145 0.300087
\(75\) 0 0
\(76\) 2.34017 0.268436
\(77\) 4.68035 0.533375
\(78\) −2.34017 −0.264972
\(79\) −11.9155 −1.34060 −0.670298 0.742092i \(-0.733834\pi\)
−0.670298 + 0.742092i \(0.733834\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −0.156755 −0.0173107
\(83\) −11.4186 −1.25335 −0.626674 0.779281i \(-0.715584\pi\)
−0.626674 + 0.779281i \(0.715584\pi\)
\(84\) −4.34017 −0.473552
\(85\) 0 0
\(86\) 0.738205 0.0796027
\(87\) 10.4969 1.12539
\(88\) 1.07838 0.114955
\(89\) −4.68035 −0.496116 −0.248058 0.968745i \(-0.579792\pi\)
−0.248058 + 0.968745i \(0.579792\pi\)
\(90\) 0 0
\(91\) −10.1568 −1.06472
\(92\) −1.00000 −0.104257
\(93\) −4.00000 −0.414781
\(94\) 6.83710 0.705193
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −9.02052 −0.915895 −0.457947 0.888979i \(-0.651415\pi\)
−0.457947 + 0.888979i \(0.651415\pi\)
\(98\) −11.8371 −1.19573
\(99\) −1.07838 −0.108381
\(100\) 0 0
\(101\) 1.50307 0.149561 0.0747806 0.997200i \(-0.476174\pi\)
0.0747806 + 0.997200i \(0.476174\pi\)
\(102\) 0.921622 0.0912542
\(103\) 2.49693 0.246030 0.123015 0.992405i \(-0.460744\pi\)
0.123015 + 0.992405i \(0.460744\pi\)
\(104\) −2.34017 −0.229473
\(105\) 0 0
\(106\) −0.340173 −0.0330405
\(107\) −9.57531 −0.925680 −0.462840 0.886442i \(-0.653170\pi\)
−0.462840 + 0.886442i \(0.653170\pi\)
\(108\) 1.00000 0.0962250
\(109\) −9.78539 −0.937270 −0.468635 0.883392i \(-0.655254\pi\)
−0.468635 + 0.883392i \(0.655254\pi\)
\(110\) 0 0
\(111\) −2.58145 −0.245020
\(112\) −4.34017 −0.410108
\(113\) −8.92162 −0.839276 −0.419638 0.907692i \(-0.637843\pi\)
−0.419638 + 0.907692i \(0.637843\pi\)
\(114\) −2.34017 −0.219177
\(115\) 0 0
\(116\) 10.4969 0.974615
\(117\) 2.34017 0.216349
\(118\) 8.83710 0.813521
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) −9.83710 −0.894282
\(122\) 11.5753 1.04798
\(123\) 0.156755 0.0141342
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 4.34017 0.386653
\(127\) −17.1773 −1.52424 −0.762118 0.647438i \(-0.775841\pi\)
−0.762118 + 0.647438i \(0.775841\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −0.738205 −0.0649953
\(130\) 0 0
\(131\) −6.68035 −0.583665 −0.291832 0.956470i \(-0.594265\pi\)
−0.291832 + 0.956470i \(0.594265\pi\)
\(132\) −1.07838 −0.0938607
\(133\) −10.1568 −0.880702
\(134\) 2.58145 0.223003
\(135\) 0 0
\(136\) 0.921622 0.0790285
\(137\) −22.2823 −1.90371 −0.951853 0.306554i \(-0.900824\pi\)
−0.951853 + 0.306554i \(0.900824\pi\)
\(138\) 1.00000 0.0851257
\(139\) −10.5236 −0.892599 −0.446300 0.894884i \(-0.647258\pi\)
−0.446300 + 0.894884i \(0.647258\pi\)
\(140\) 0 0
\(141\) −6.83710 −0.575788
\(142\) −15.1773 −1.27365
\(143\) −2.52359 −0.211033
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −4.68035 −0.387348
\(147\) 11.8371 0.976308
\(148\) −2.58145 −0.212194
\(149\) 6.92162 0.567041 0.283521 0.958966i \(-0.408498\pi\)
0.283521 + 0.958966i \(0.408498\pi\)
\(150\) 0 0
\(151\) 6.15676 0.501030 0.250515 0.968113i \(-0.419400\pi\)
0.250515 + 0.968113i \(0.419400\pi\)
\(152\) −2.34017 −0.189813
\(153\) −0.921622 −0.0745087
\(154\) −4.68035 −0.377153
\(155\) 0 0
\(156\) 2.34017 0.187364
\(157\) −2.89496 −0.231043 −0.115521 0.993305i \(-0.536854\pi\)
−0.115521 + 0.993305i \(0.536854\pi\)
\(158\) 11.9155 0.947945
\(159\) 0.340173 0.0269775
\(160\) 0 0
\(161\) 4.34017 0.342054
\(162\) −1.00000 −0.0785674
\(163\) −6.52359 −0.510967 −0.255484 0.966813i \(-0.582235\pi\)
−0.255484 + 0.966813i \(0.582235\pi\)
\(164\) 0.156755 0.0122405
\(165\) 0 0
\(166\) 11.4186 0.886251
\(167\) −5.47641 −0.423777 −0.211889 0.977294i \(-0.567961\pi\)
−0.211889 + 0.977294i \(0.567961\pi\)
\(168\) 4.34017 0.334852
\(169\) −7.52359 −0.578738
\(170\) 0 0
\(171\) 2.34017 0.178958
\(172\) −0.738205 −0.0562876
\(173\) 15.6742 1.19169 0.595844 0.803100i \(-0.296818\pi\)
0.595844 + 0.803100i \(0.296818\pi\)
\(174\) −10.4969 −0.795770
\(175\) 0 0
\(176\) −1.07838 −0.0812858
\(177\) −8.83710 −0.664237
\(178\) 4.68035 0.350807
\(179\) 15.3607 1.14811 0.574056 0.818816i \(-0.305369\pi\)
0.574056 + 0.818816i \(0.305369\pi\)
\(180\) 0 0
\(181\) 21.6163 1.60673 0.803365 0.595487i \(-0.203041\pi\)
0.803365 + 0.595487i \(0.203041\pi\)
\(182\) 10.1568 0.752869
\(183\) −11.5753 −0.855671
\(184\) 1.00000 0.0737210
\(185\) 0 0
\(186\) 4.00000 0.293294
\(187\) 0.993857 0.0726780
\(188\) −6.83710 −0.498647
\(189\) −4.34017 −0.315701
\(190\) 0 0
\(191\) 4.36683 0.315973 0.157987 0.987441i \(-0.449500\pi\)
0.157987 + 0.987441i \(0.449500\pi\)
\(192\) 1.00000 0.0721688
\(193\) 8.68035 0.624825 0.312412 0.949947i \(-0.398863\pi\)
0.312412 + 0.949947i \(0.398863\pi\)
\(194\) 9.02052 0.647636
\(195\) 0 0
\(196\) 11.8371 0.845507
\(197\) 22.6803 1.61591 0.807954 0.589246i \(-0.200575\pi\)
0.807954 + 0.589246i \(0.200575\pi\)
\(198\) 1.07838 0.0766370
\(199\) −5.44521 −0.386001 −0.193000 0.981199i \(-0.561822\pi\)
−0.193000 + 0.981199i \(0.561822\pi\)
\(200\) 0 0
\(201\) −2.58145 −0.182081
\(202\) −1.50307 −0.105756
\(203\) −45.5585 −3.19758
\(204\) −0.921622 −0.0645265
\(205\) 0 0
\(206\) −2.49693 −0.173969
\(207\) −1.00000 −0.0695048
\(208\) 2.34017 0.162262
\(209\) −2.52359 −0.174560
\(210\) 0 0
\(211\) −22.0410 −1.51737 −0.758684 0.651459i \(-0.774157\pi\)
−0.758684 + 0.651459i \(0.774157\pi\)
\(212\) 0.340173 0.0233632
\(213\) 15.1773 1.03993
\(214\) 9.57531 0.654554
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 17.3607 1.17852
\(218\) 9.78539 0.662750
\(219\) 4.68035 0.316268
\(220\) 0 0
\(221\) −2.15676 −0.145079
\(222\) 2.58145 0.173256
\(223\) −21.1773 −1.41814 −0.709068 0.705141i \(-0.750884\pi\)
−0.709068 + 0.705141i \(0.750884\pi\)
\(224\) 4.34017 0.289990
\(225\) 0 0
\(226\) 8.92162 0.593457
\(227\) 11.7854 0.782224 0.391112 0.920343i \(-0.372091\pi\)
0.391112 + 0.920343i \(0.372091\pi\)
\(228\) 2.34017 0.154982
\(229\) 1.78539 0.117982 0.0589908 0.998259i \(-0.481212\pi\)
0.0589908 + 0.998259i \(0.481212\pi\)
\(230\) 0 0
\(231\) 4.68035 0.307944
\(232\) −10.4969 −0.689157
\(233\) −18.3135 −1.19976 −0.599879 0.800091i \(-0.704785\pi\)
−0.599879 + 0.800091i \(0.704785\pi\)
\(234\) −2.34017 −0.152982
\(235\) 0 0
\(236\) −8.83710 −0.575246
\(237\) −11.9155 −0.773994
\(238\) −4.00000 −0.259281
\(239\) −5.33403 −0.345030 −0.172515 0.985007i \(-0.555189\pi\)
−0.172515 + 0.985007i \(0.555189\pi\)
\(240\) 0 0
\(241\) 26.1978 1.68755 0.843774 0.536698i \(-0.180329\pi\)
0.843774 + 0.536698i \(0.180329\pi\)
\(242\) 9.83710 0.632353
\(243\) 1.00000 0.0641500
\(244\) −11.5753 −0.741033
\(245\) 0 0
\(246\) −0.156755 −0.00999437
\(247\) 5.47641 0.348456
\(248\) 4.00000 0.254000
\(249\) −11.4186 −0.723621
\(250\) 0 0
\(251\) 27.4329 1.73155 0.865775 0.500433i \(-0.166826\pi\)
0.865775 + 0.500433i \(0.166826\pi\)
\(252\) −4.34017 −0.273405
\(253\) 1.07838 0.0677970
\(254\) 17.1773 1.07780
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 10.3135 0.643339 0.321670 0.946852i \(-0.395756\pi\)
0.321670 + 0.946852i \(0.395756\pi\)
\(258\) 0.738205 0.0459586
\(259\) 11.2039 0.696179
\(260\) 0 0
\(261\) 10.4969 0.649744
\(262\) 6.68035 0.412713
\(263\) 20.9939 1.29454 0.647268 0.762262i \(-0.275911\pi\)
0.647268 + 0.762262i \(0.275911\pi\)
\(264\) 1.07838 0.0663696
\(265\) 0 0
\(266\) 10.1568 0.622751
\(267\) −4.68035 −0.286433
\(268\) −2.58145 −0.157687
\(269\) 25.3874 1.54789 0.773947 0.633250i \(-0.218280\pi\)
0.773947 + 0.633250i \(0.218280\pi\)
\(270\) 0 0
\(271\) −32.5646 −1.97816 −0.989080 0.147379i \(-0.952916\pi\)
−0.989080 + 0.147379i \(0.952916\pi\)
\(272\) −0.921622 −0.0558816
\(273\) −10.1568 −0.614715
\(274\) 22.2823 1.34612
\(275\) 0 0
\(276\) −1.00000 −0.0601929
\(277\) −15.7009 −0.943374 −0.471687 0.881766i \(-0.656355\pi\)
−0.471687 + 0.881766i \(0.656355\pi\)
\(278\) 10.5236 0.631163
\(279\) −4.00000 −0.239474
\(280\) 0 0
\(281\) −9.36069 −0.558412 −0.279206 0.960231i \(-0.590071\pi\)
−0.279206 + 0.960231i \(0.590071\pi\)
\(282\) 6.83710 0.407143
\(283\) 17.4186 1.03543 0.517713 0.855555i \(-0.326784\pi\)
0.517713 + 0.855555i \(0.326784\pi\)
\(284\) 15.1773 0.900606
\(285\) 0 0
\(286\) 2.52359 0.149223
\(287\) −0.680346 −0.0401596
\(288\) −1.00000 −0.0589256
\(289\) −16.1506 −0.950036
\(290\) 0 0
\(291\) −9.02052 −0.528792
\(292\) 4.68035 0.273897
\(293\) 16.3402 0.954603 0.477302 0.878740i \(-0.341615\pi\)
0.477302 + 0.878740i \(0.341615\pi\)
\(294\) −11.8371 −0.690354
\(295\) 0 0
\(296\) 2.58145 0.150044
\(297\) −1.07838 −0.0625738
\(298\) −6.92162 −0.400959
\(299\) −2.34017 −0.135336
\(300\) 0 0
\(301\) 3.20394 0.184672
\(302\) −6.15676 −0.354281
\(303\) 1.50307 0.0863492
\(304\) 2.34017 0.134218
\(305\) 0 0
\(306\) 0.921622 0.0526856
\(307\) 3.20394 0.182858 0.0914292 0.995812i \(-0.470856\pi\)
0.0914292 + 0.995812i \(0.470856\pi\)
\(308\) 4.68035 0.266687
\(309\) 2.49693 0.142045
\(310\) 0 0
\(311\) 12.6537 0.717525 0.358762 0.933429i \(-0.383199\pi\)
0.358762 + 0.933429i \(0.383199\pi\)
\(312\) −2.34017 −0.132486
\(313\) −33.2183 −1.87761 −0.938805 0.344449i \(-0.888066\pi\)
−0.938805 + 0.344449i \(0.888066\pi\)
\(314\) 2.89496 0.163372
\(315\) 0 0
\(316\) −11.9155 −0.670298
\(317\) 6.99386 0.392814 0.196407 0.980522i \(-0.437073\pi\)
0.196407 + 0.980522i \(0.437073\pi\)
\(318\) −0.340173 −0.0190760
\(319\) −11.3197 −0.633779
\(320\) 0 0
\(321\) −9.57531 −0.534441
\(322\) −4.34017 −0.241868
\(323\) −2.15676 −0.120005
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 6.52359 0.361308
\(327\) −9.78539 −0.541133
\(328\) −0.156755 −0.00865537
\(329\) 29.6742 1.63599
\(330\) 0 0
\(331\) −29.7275 −1.63397 −0.816986 0.576657i \(-0.804357\pi\)
−0.816986 + 0.576657i \(0.804357\pi\)
\(332\) −11.4186 −0.626674
\(333\) −2.58145 −0.141463
\(334\) 5.47641 0.299656
\(335\) 0 0
\(336\) −4.34017 −0.236776
\(337\) −28.3402 −1.54379 −0.771894 0.635751i \(-0.780690\pi\)
−0.771894 + 0.635751i \(0.780690\pi\)
\(338\) 7.52359 0.409229
\(339\) −8.92162 −0.484556
\(340\) 0 0
\(341\) 4.31351 0.233590
\(342\) −2.34017 −0.126542
\(343\) −20.9939 −1.13356
\(344\) 0.738205 0.0398013
\(345\) 0 0
\(346\) −15.6742 −0.842650
\(347\) 22.5236 1.20913 0.604565 0.796556i \(-0.293347\pi\)
0.604565 + 0.796556i \(0.293347\pi\)
\(348\) 10.4969 0.562694
\(349\) 15.6742 0.839021 0.419510 0.907751i \(-0.362202\pi\)
0.419510 + 0.907751i \(0.362202\pi\)
\(350\) 0 0
\(351\) 2.34017 0.124909
\(352\) 1.07838 0.0574777
\(353\) −26.3135 −1.40053 −0.700263 0.713885i \(-0.746934\pi\)
−0.700263 + 0.713885i \(0.746934\pi\)
\(354\) 8.83710 0.469687
\(355\) 0 0
\(356\) −4.68035 −0.248058
\(357\) 4.00000 0.211702
\(358\) −15.3607 −0.811838
\(359\) −0.796064 −0.0420146 −0.0210073 0.999779i \(-0.506687\pi\)
−0.0210073 + 0.999779i \(0.506687\pi\)
\(360\) 0 0
\(361\) −13.5236 −0.711768
\(362\) −21.6163 −1.13613
\(363\) −9.83710 −0.516314
\(364\) −10.1568 −0.532359
\(365\) 0 0
\(366\) 11.5753 0.605051
\(367\) −13.7009 −0.715179 −0.357590 0.933879i \(-0.616401\pi\)
−0.357590 + 0.933879i \(0.616401\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0.156755 0.00816036
\(370\) 0 0
\(371\) −1.47641 −0.0766514
\(372\) −4.00000 −0.207390
\(373\) −4.25565 −0.220349 −0.110175 0.993912i \(-0.535141\pi\)
−0.110175 + 0.993912i \(0.535141\pi\)
\(374\) −0.993857 −0.0513911
\(375\) 0 0
\(376\) 6.83710 0.352597
\(377\) 24.5646 1.26514
\(378\) 4.34017 0.223234
\(379\) −30.6537 −1.57457 −0.787287 0.616587i \(-0.788515\pi\)
−0.787287 + 0.616587i \(0.788515\pi\)
\(380\) 0 0
\(381\) −17.1773 −0.880018
\(382\) −4.36683 −0.223427
\(383\) 20.8781 1.06682 0.533412 0.845856i \(-0.320910\pi\)
0.533412 + 0.845856i \(0.320910\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −8.68035 −0.441818
\(387\) −0.738205 −0.0375251
\(388\) −9.02052 −0.457947
\(389\) −16.9627 −0.860041 −0.430021 0.902819i \(-0.641494\pi\)
−0.430021 + 0.902819i \(0.641494\pi\)
\(390\) 0 0
\(391\) 0.921622 0.0466084
\(392\) −11.8371 −0.597864
\(393\) −6.68035 −0.336979
\(394\) −22.6803 −1.14262
\(395\) 0 0
\(396\) −1.07838 −0.0541905
\(397\) 16.4969 0.827957 0.413979 0.910287i \(-0.364139\pi\)
0.413979 + 0.910287i \(0.364139\pi\)
\(398\) 5.44521 0.272944
\(399\) −10.1568 −0.508474
\(400\) 0 0
\(401\) −35.2039 −1.75800 −0.879000 0.476821i \(-0.841789\pi\)
−0.879000 + 0.476821i \(0.841789\pi\)
\(402\) 2.58145 0.128751
\(403\) −9.36069 −0.466289
\(404\) 1.50307 0.0747806
\(405\) 0 0
\(406\) 45.5585 2.26103
\(407\) 2.78378 0.137987
\(408\) 0.921622 0.0456271
\(409\) 11.3607 0.561750 0.280875 0.959744i \(-0.409375\pi\)
0.280875 + 0.959744i \(0.409375\pi\)
\(410\) 0 0
\(411\) −22.2823 −1.09911
\(412\) 2.49693 0.123015
\(413\) 38.3545 1.88730
\(414\) 1.00000 0.0491473
\(415\) 0 0
\(416\) −2.34017 −0.114736
\(417\) −10.5236 −0.515342
\(418\) 2.52359 0.123433
\(419\) −18.1256 −0.885491 −0.442746 0.896647i \(-0.645996\pi\)
−0.442746 + 0.896647i \(0.645996\pi\)
\(420\) 0 0
\(421\) 2.58145 0.125812 0.0629061 0.998019i \(-0.479963\pi\)
0.0629061 + 0.998019i \(0.479963\pi\)
\(422\) 22.0410 1.07294
\(423\) −6.83710 −0.332431
\(424\) −0.340173 −0.0165203
\(425\) 0 0
\(426\) −15.1773 −0.735341
\(427\) 50.2388 2.43123
\(428\) −9.57531 −0.462840
\(429\) −2.52359 −0.121840
\(430\) 0 0
\(431\) −33.9877 −1.63713 −0.818565 0.574414i \(-0.805230\pi\)
−0.818565 + 0.574414i \(0.805230\pi\)
\(432\) 1.00000 0.0481125
\(433\) 8.28685 0.398241 0.199120 0.979975i \(-0.436192\pi\)
0.199120 + 0.979975i \(0.436192\pi\)
\(434\) −17.3607 −0.833340
\(435\) 0 0
\(436\) −9.78539 −0.468635
\(437\) −2.34017 −0.111946
\(438\) −4.68035 −0.223636
\(439\) −28.9939 −1.38380 −0.691901 0.721993i \(-0.743226\pi\)
−0.691901 + 0.721993i \(0.743226\pi\)
\(440\) 0 0
\(441\) 11.8371 0.563671
\(442\) 2.15676 0.102586
\(443\) 3.20394 0.152224 0.0761118 0.997099i \(-0.475749\pi\)
0.0761118 + 0.997099i \(0.475749\pi\)
\(444\) −2.58145 −0.122510
\(445\) 0 0
\(446\) 21.1773 1.00277
\(447\) 6.92162 0.327381
\(448\) −4.34017 −0.205054
\(449\) 28.1568 1.32880 0.664400 0.747377i \(-0.268687\pi\)
0.664400 + 0.747377i \(0.268687\pi\)
\(450\) 0 0
\(451\) −0.169042 −0.00795986
\(452\) −8.92162 −0.419638
\(453\) 6.15676 0.289270
\(454\) −11.7854 −0.553116
\(455\) 0 0
\(456\) −2.34017 −0.109589
\(457\) 23.1773 1.08419 0.542094 0.840318i \(-0.317632\pi\)
0.542094 + 0.840318i \(0.317632\pi\)
\(458\) −1.78539 −0.0834256
\(459\) −0.921622 −0.0430176
\(460\) 0 0
\(461\) −42.3279 −1.97141 −0.985703 0.168491i \(-0.946110\pi\)
−0.985703 + 0.168491i \(0.946110\pi\)
\(462\) −4.68035 −0.217749
\(463\) 34.9048 1.62216 0.811082 0.584933i \(-0.198879\pi\)
0.811082 + 0.584933i \(0.198879\pi\)
\(464\) 10.4969 0.487308
\(465\) 0 0
\(466\) 18.3135 0.848357
\(467\) 37.7731 1.74793 0.873965 0.485988i \(-0.161540\pi\)
0.873965 + 0.485988i \(0.161540\pi\)
\(468\) 2.34017 0.108175
\(469\) 11.2039 0.517350
\(470\) 0 0
\(471\) −2.89496 −0.133393
\(472\) 8.83710 0.406761
\(473\) 0.796064 0.0366030
\(474\) 11.9155 0.547296
\(475\) 0 0
\(476\) 4.00000 0.183340
\(477\) 0.340173 0.0155755
\(478\) 5.33403 0.243973
\(479\) −28.1978 −1.28839 −0.644195 0.764861i \(-0.722807\pi\)
−0.644195 + 0.764861i \(0.722807\pi\)
\(480\) 0 0
\(481\) −6.04104 −0.275448
\(482\) −26.1978 −1.19328
\(483\) 4.34017 0.197485
\(484\) −9.83710 −0.447141
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −38.2245 −1.73212 −0.866058 0.499944i \(-0.833354\pi\)
−0.866058 + 0.499944i \(0.833354\pi\)
\(488\) 11.5753 0.523989
\(489\) −6.52359 −0.295007
\(490\) 0 0
\(491\) 11.3074 0.510294 0.255147 0.966902i \(-0.417876\pi\)
0.255147 + 0.966902i \(0.417876\pi\)
\(492\) 0.156755 0.00706708
\(493\) −9.67420 −0.435704
\(494\) −5.47641 −0.246395
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) −65.8720 −2.95476
\(498\) 11.4186 0.511677
\(499\) −2.47027 −0.110584 −0.0552922 0.998470i \(-0.517609\pi\)
−0.0552922 + 0.998470i \(0.517609\pi\)
\(500\) 0 0
\(501\) −5.47641 −0.244668
\(502\) −27.4329 −1.22439
\(503\) 20.6270 0.919713 0.459857 0.887993i \(-0.347901\pi\)
0.459857 + 0.887993i \(0.347901\pi\)
\(504\) 4.34017 0.193327
\(505\) 0 0
\(506\) −1.07838 −0.0479397
\(507\) −7.52359 −0.334134
\(508\) −17.1773 −0.762118
\(509\) −32.2245 −1.42832 −0.714162 0.699981i \(-0.753192\pi\)
−0.714162 + 0.699981i \(0.753192\pi\)
\(510\) 0 0
\(511\) −20.3135 −0.898617
\(512\) −1.00000 −0.0441942
\(513\) 2.34017 0.103321
\(514\) −10.3135 −0.454909
\(515\) 0 0
\(516\) −0.738205 −0.0324977
\(517\) 7.37298 0.324263
\(518\) −11.2039 −0.492273
\(519\) 15.6742 0.688021
\(520\) 0 0
\(521\) −3.51745 −0.154102 −0.0770511 0.997027i \(-0.524550\pi\)
−0.0770511 + 0.997027i \(0.524550\pi\)
\(522\) −10.4969 −0.459438
\(523\) 35.9421 1.57164 0.785820 0.618455i \(-0.212241\pi\)
0.785820 + 0.618455i \(0.212241\pi\)
\(524\) −6.68035 −0.291832
\(525\) 0 0
\(526\) −20.9939 −0.915376
\(527\) 3.68649 0.160586
\(528\) −1.07838 −0.0469304
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −8.83710 −0.383498
\(532\) −10.1568 −0.440351
\(533\) 0.366835 0.0158894
\(534\) 4.68035 0.202538
\(535\) 0 0
\(536\) 2.58145 0.111502
\(537\) 15.3607 0.662863
\(538\) −25.3874 −1.09453
\(539\) −12.7649 −0.549822
\(540\) 0 0
\(541\) 31.7275 1.36407 0.682036 0.731318i \(-0.261095\pi\)
0.682036 + 0.731318i \(0.261095\pi\)
\(542\) 32.5646 1.39877
\(543\) 21.6163 0.927646
\(544\) 0.921622 0.0395142
\(545\) 0 0
\(546\) 10.1568 0.434669
\(547\) 9.84324 0.420867 0.210433 0.977608i \(-0.432512\pi\)
0.210433 + 0.977608i \(0.432512\pi\)
\(548\) −22.2823 −0.951853
\(549\) −11.5753 −0.494022
\(550\) 0 0
\(551\) 24.5646 1.04649
\(552\) 1.00000 0.0425628
\(553\) 51.7152 2.19916
\(554\) 15.7009 0.667066
\(555\) 0 0
\(556\) −10.5236 −0.446300
\(557\) −26.3279 −1.11555 −0.557774 0.829993i \(-0.688344\pi\)
−0.557774 + 0.829993i \(0.688344\pi\)
\(558\) 4.00000 0.169334
\(559\) −1.72753 −0.0730666
\(560\) 0 0
\(561\) 0.993857 0.0419607
\(562\) 9.36069 0.394857
\(563\) 30.2557 1.27512 0.637562 0.770399i \(-0.279943\pi\)
0.637562 + 0.770399i \(0.279943\pi\)
\(564\) −6.83710 −0.287894
\(565\) 0 0
\(566\) −17.4186 −0.732156
\(567\) −4.34017 −0.182270
\(568\) −15.1773 −0.636824
\(569\) 33.8720 1.41999 0.709994 0.704208i \(-0.248698\pi\)
0.709994 + 0.704208i \(0.248698\pi\)
\(570\) 0 0
\(571\) 35.1650 1.47161 0.735804 0.677194i \(-0.236804\pi\)
0.735804 + 0.677194i \(0.236804\pi\)
\(572\) −2.52359 −0.105517
\(573\) 4.36683 0.182427
\(574\) 0.680346 0.0283971
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 4.00000 0.166522 0.0832611 0.996528i \(-0.473466\pi\)
0.0832611 + 0.996528i \(0.473466\pi\)
\(578\) 16.1506 0.671777
\(579\) 8.68035 0.360743
\(580\) 0 0
\(581\) 49.5585 2.05603
\(582\) 9.02052 0.373913
\(583\) −0.366835 −0.0151928
\(584\) −4.68035 −0.193674
\(585\) 0 0
\(586\) −16.3402 −0.675006
\(587\) −9.30737 −0.384156 −0.192078 0.981380i \(-0.561523\pi\)
−0.192078 + 0.981380i \(0.561523\pi\)
\(588\) 11.8371 0.488154
\(589\) −9.36069 −0.385701
\(590\) 0 0
\(591\) 22.6803 0.932945
\(592\) −2.58145 −0.106097
\(593\) −33.0349 −1.35658 −0.678290 0.734794i \(-0.737279\pi\)
−0.678290 + 0.734794i \(0.737279\pi\)
\(594\) 1.07838 0.0442464
\(595\) 0 0
\(596\) 6.92162 0.283521
\(597\) −5.44521 −0.222858
\(598\) 2.34017 0.0956968
\(599\) 13.6475 0.557623 0.278812 0.960346i \(-0.410059\pi\)
0.278812 + 0.960346i \(0.410059\pi\)
\(600\) 0 0
\(601\) 1.68649 0.0687933 0.0343967 0.999408i \(-0.489049\pi\)
0.0343967 + 0.999408i \(0.489049\pi\)
\(602\) −3.20394 −0.130583
\(603\) −2.58145 −0.105125
\(604\) 6.15676 0.250515
\(605\) 0 0
\(606\) −1.50307 −0.0610581
\(607\) 47.2183 1.91653 0.958266 0.285878i \(-0.0922851\pi\)
0.958266 + 0.285878i \(0.0922851\pi\)
\(608\) −2.34017 −0.0949065
\(609\) −45.5585 −1.84612
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) −0.921622 −0.0372544
\(613\) 7.45959 0.301290 0.150645 0.988588i \(-0.451865\pi\)
0.150645 + 0.988588i \(0.451865\pi\)
\(614\) −3.20394 −0.129300
\(615\) 0 0
\(616\) −4.68035 −0.188577
\(617\) −15.8120 −0.636569 −0.318285 0.947995i \(-0.603107\pi\)
−0.318285 + 0.947995i \(0.603107\pi\)
\(618\) −2.49693 −0.100441
\(619\) −26.3402 −1.05870 −0.529350 0.848403i \(-0.677564\pi\)
−0.529350 + 0.848403i \(0.677564\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) −12.6537 −0.507367
\(623\) 20.3135 0.813844
\(624\) 2.34017 0.0936819
\(625\) 0 0
\(626\) 33.2183 1.32767
\(627\) −2.52359 −0.100782
\(628\) −2.89496 −0.115521
\(629\) 2.37912 0.0948618
\(630\) 0 0
\(631\) −14.4391 −0.574810 −0.287405 0.957809i \(-0.592793\pi\)
−0.287405 + 0.957809i \(0.592793\pi\)
\(632\) 11.9155 0.473972
\(633\) −22.0410 −0.876053
\(634\) −6.99386 −0.277762
\(635\) 0 0
\(636\) 0.340173 0.0134887
\(637\) 27.7009 1.09755
\(638\) 11.3197 0.448149
\(639\) 15.1773 0.600404
\(640\) 0 0
\(641\) −5.78992 −0.228688 −0.114344 0.993441i \(-0.536477\pi\)
−0.114344 + 0.993441i \(0.536477\pi\)
\(642\) 9.57531 0.377907
\(643\) −23.4063 −0.923053 −0.461526 0.887126i \(-0.652698\pi\)
−0.461526 + 0.887126i \(0.652698\pi\)
\(644\) 4.34017 0.171027
\(645\) 0 0
\(646\) 2.15676 0.0848564
\(647\) −25.9877 −1.02168 −0.510841 0.859675i \(-0.670666\pi\)
−0.510841 + 0.859675i \(0.670666\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 9.52973 0.374075
\(650\) 0 0
\(651\) 17.3607 0.680419
\(652\) −6.52359 −0.255484
\(653\) 0.325797 0.0127494 0.00637471 0.999980i \(-0.497971\pi\)
0.00637471 + 0.999980i \(0.497971\pi\)
\(654\) 9.78539 0.382639
\(655\) 0 0
\(656\) 0.156755 0.00612027
\(657\) 4.68035 0.182598
\(658\) −29.6742 −1.15682
\(659\) −3.11942 −0.121515 −0.0607576 0.998153i \(-0.519352\pi\)
−0.0607576 + 0.998153i \(0.519352\pi\)
\(660\) 0 0
\(661\) 24.1399 0.938935 0.469467 0.882950i \(-0.344446\pi\)
0.469467 + 0.882950i \(0.344446\pi\)
\(662\) 29.7275 1.15539
\(663\) −2.15676 −0.0837614
\(664\) 11.4186 0.443126
\(665\) 0 0
\(666\) 2.58145 0.100029
\(667\) −10.4969 −0.406443
\(668\) −5.47641 −0.211889
\(669\) −21.1773 −0.818761
\(670\) 0 0
\(671\) 12.4826 0.481884
\(672\) 4.34017 0.167426
\(673\) 45.6742 1.76061 0.880306 0.474407i \(-0.157338\pi\)
0.880306 + 0.474407i \(0.157338\pi\)
\(674\) 28.3402 1.09162
\(675\) 0 0
\(676\) −7.52359 −0.289369
\(677\) −9.38735 −0.360785 −0.180393 0.983595i \(-0.557737\pi\)
−0.180393 + 0.983595i \(0.557737\pi\)
\(678\) 8.92162 0.342633
\(679\) 39.1506 1.50246
\(680\) 0 0
\(681\) 11.7854 0.451617
\(682\) −4.31351 −0.165173
\(683\) −20.9939 −0.803308 −0.401654 0.915792i \(-0.631564\pi\)
−0.401654 + 0.915792i \(0.631564\pi\)
\(684\) 2.34017 0.0894788
\(685\) 0 0
\(686\) 20.9939 0.801549
\(687\) 1.78539 0.0681167
\(688\) −0.738205 −0.0281438
\(689\) 0.796064 0.0303276
\(690\) 0 0
\(691\) −44.1445 −1.67933 −0.839667 0.543101i \(-0.817250\pi\)
−0.839667 + 0.543101i \(0.817250\pi\)
\(692\) 15.6742 0.595844
\(693\) 4.68035 0.177792
\(694\) −22.5236 −0.854984
\(695\) 0 0
\(696\) −10.4969 −0.397885
\(697\) −0.144469 −0.00547217
\(698\) −15.6742 −0.593277
\(699\) −18.3135 −0.692681
\(700\) 0 0
\(701\) −27.7998 −1.04998 −0.524991 0.851108i \(-0.675931\pi\)
−0.524991 + 0.851108i \(0.675931\pi\)
\(702\) −2.34017 −0.0883241
\(703\) −6.04104 −0.227842
\(704\) −1.07838 −0.0406429
\(705\) 0 0
\(706\) 26.3135 0.990322
\(707\) −6.52359 −0.245345
\(708\) −8.83710 −0.332119
\(709\) 4.30898 0.161827 0.0809135 0.996721i \(-0.474216\pi\)
0.0809135 + 0.996721i \(0.474216\pi\)
\(710\) 0 0
\(711\) −11.9155 −0.446865
\(712\) 4.68035 0.175403
\(713\) 4.00000 0.149801
\(714\) −4.00000 −0.149696
\(715\) 0 0
\(716\) 15.3607 0.574056
\(717\) −5.33403 −0.199203
\(718\) 0.796064 0.0297088
\(719\) 15.9733 0.595705 0.297852 0.954612i \(-0.403730\pi\)
0.297852 + 0.954612i \(0.403730\pi\)
\(720\) 0 0
\(721\) −10.8371 −0.403595
\(722\) 13.5236 0.503296
\(723\) 26.1978 0.974306
\(724\) 21.6163 0.803365
\(725\) 0 0
\(726\) 9.83710 0.365089
\(727\) 0.340173 0.0126163 0.00630816 0.999980i \(-0.497992\pi\)
0.00630816 + 0.999980i \(0.497992\pi\)
\(728\) 10.1568 0.376434
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0.680346 0.0251635
\(732\) −11.5753 −0.427836
\(733\) 19.2618 0.711451 0.355725 0.934591i \(-0.384234\pi\)
0.355725 + 0.934591i \(0.384234\pi\)
\(734\) 13.7009 0.505708
\(735\) 0 0
\(736\) 1.00000 0.0368605
\(737\) 2.78378 0.102542
\(738\) −0.156755 −0.00577025
\(739\) 25.5585 0.940184 0.470092 0.882617i \(-0.344221\pi\)
0.470092 + 0.882617i \(0.344221\pi\)
\(740\) 0 0
\(741\) 5.47641 0.201181
\(742\) 1.47641 0.0542007
\(743\) 36.8781 1.35293 0.676464 0.736476i \(-0.263511\pi\)
0.676464 + 0.736476i \(0.263511\pi\)
\(744\) 4.00000 0.146647
\(745\) 0 0
\(746\) 4.25565 0.155810
\(747\) −11.4186 −0.417783
\(748\) 0.993857 0.0363390
\(749\) 41.5585 1.51851
\(750\) 0 0
\(751\) 18.1256 0.661411 0.330706 0.943734i \(-0.392713\pi\)
0.330706 + 0.943734i \(0.392713\pi\)
\(752\) −6.83710 −0.249323
\(753\) 27.4329 0.999711
\(754\) −24.5646 −0.894591
\(755\) 0 0
\(756\) −4.34017 −0.157851
\(757\) 17.4186 0.633088 0.316544 0.948578i \(-0.397478\pi\)
0.316544 + 0.948578i \(0.397478\pi\)
\(758\) 30.6537 1.11339
\(759\) 1.07838 0.0391426
\(760\) 0 0
\(761\) −23.4140 −0.848757 −0.424379 0.905485i \(-0.639507\pi\)
−0.424379 + 0.905485i \(0.639507\pi\)
\(762\) 17.1773 0.622267
\(763\) 42.4703 1.53753
\(764\) 4.36683 0.157987
\(765\) 0 0
\(766\) −20.8781 −0.754358
\(767\) −20.6803 −0.746724
\(768\) 1.00000 0.0360844
\(769\) −29.7152 −1.07156 −0.535779 0.844358i \(-0.679982\pi\)
−0.535779 + 0.844358i \(0.679982\pi\)
\(770\) 0 0
\(771\) 10.3135 0.371432
\(772\) 8.68035 0.312412
\(773\) −15.3463 −0.551969 −0.275984 0.961162i \(-0.589004\pi\)
−0.275984 + 0.961162i \(0.589004\pi\)
\(774\) 0.738205 0.0265342
\(775\) 0 0
\(776\) 9.02052 0.323818
\(777\) 11.2039 0.401939
\(778\) 16.9627 0.608141
\(779\) 0.366835 0.0131432
\(780\) 0 0
\(781\) −16.3668 −0.585651
\(782\) −0.921622 −0.0329571
\(783\) 10.4969 0.375130
\(784\) 11.8371 0.422754
\(785\) 0 0
\(786\) 6.68035 0.238280
\(787\) 22.8950 0.816117 0.408059 0.912956i \(-0.366206\pi\)
0.408059 + 0.912956i \(0.366206\pi\)
\(788\) 22.6803 0.807954
\(789\) 20.9939 0.747401
\(790\) 0 0
\(791\) 38.7214 1.37677
\(792\) 1.07838 0.0383185
\(793\) −27.0882 −0.961931
\(794\) −16.4969 −0.585454
\(795\) 0 0
\(796\) −5.44521 −0.193000
\(797\) 7.60650 0.269436 0.134718 0.990884i \(-0.456987\pi\)
0.134718 + 0.990884i \(0.456987\pi\)
\(798\) 10.1568 0.359545
\(799\) 6.30122 0.222921
\(800\) 0 0
\(801\) −4.68035 −0.165372
\(802\) 35.2039 1.24309
\(803\) −5.04718 −0.178111
\(804\) −2.58145 −0.0910407
\(805\) 0 0
\(806\) 9.36069 0.329716
\(807\) 25.3874 0.893677
\(808\) −1.50307 −0.0528779
\(809\) 43.7275 1.53738 0.768689 0.639623i \(-0.220909\pi\)
0.768689 + 0.639623i \(0.220909\pi\)
\(810\) 0 0
\(811\) −4.79606 −0.168413 −0.0842063 0.996448i \(-0.526835\pi\)
−0.0842063 + 0.996448i \(0.526835\pi\)
\(812\) −45.5585 −1.59879
\(813\) −32.5646 −1.14209
\(814\) −2.78378 −0.0975713
\(815\) 0 0
\(816\) −0.921622 −0.0322632
\(817\) −1.72753 −0.0604385
\(818\) −11.3607 −0.397217
\(819\) −10.1568 −0.354906
\(820\) 0 0
\(821\) −12.2245 −0.426636 −0.213318 0.976983i \(-0.568427\pi\)
−0.213318 + 0.976983i \(0.568427\pi\)
\(822\) 22.2823 0.777185
\(823\) 37.5441 1.30871 0.654353 0.756189i \(-0.272941\pi\)
0.654353 + 0.756189i \(0.272941\pi\)
\(824\) −2.49693 −0.0869846
\(825\) 0 0
\(826\) −38.3545 −1.33453
\(827\) 38.3090 1.33213 0.666067 0.745892i \(-0.267977\pi\)
0.666067 + 0.745892i \(0.267977\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −31.4140 −1.09105 −0.545527 0.838093i \(-0.683670\pi\)
−0.545527 + 0.838093i \(0.683670\pi\)
\(830\) 0 0
\(831\) −15.7009 −0.544657
\(832\) 2.34017 0.0811309
\(833\) −10.9093 −0.377986
\(834\) 10.5236 0.364402
\(835\) 0 0
\(836\) −2.52359 −0.0872802
\(837\) −4.00000 −0.138260
\(838\) 18.1256 0.626137
\(839\) 23.1506 0.799248 0.399624 0.916679i \(-0.369141\pi\)
0.399624 + 0.916679i \(0.369141\pi\)
\(840\) 0 0
\(841\) 81.1855 2.79950
\(842\) −2.58145 −0.0889626
\(843\) −9.36069 −0.322399
\(844\) −22.0410 −0.758684
\(845\) 0 0
\(846\) 6.83710 0.235064
\(847\) 42.6947 1.46701
\(848\) 0.340173 0.0116816
\(849\) 17.4186 0.597803
\(850\) 0 0
\(851\) 2.58145 0.0884909
\(852\) 15.1773 0.519965
\(853\) 32.4969 1.11267 0.556337 0.830957i \(-0.312206\pi\)
0.556337 + 0.830957i \(0.312206\pi\)
\(854\) −50.2388 −1.71914
\(855\) 0 0
\(856\) 9.57531 0.327277
\(857\) −37.8310 −1.29228 −0.646140 0.763219i \(-0.723618\pi\)
−0.646140 + 0.763219i \(0.723618\pi\)
\(858\) 2.52359 0.0861540
\(859\) −28.1978 −0.962096 −0.481048 0.876694i \(-0.659744\pi\)
−0.481048 + 0.876694i \(0.659744\pi\)
\(860\) 0 0
\(861\) −0.680346 −0.0231861
\(862\) 33.9877 1.15763
\(863\) −32.0288 −1.09027 −0.545136 0.838348i \(-0.683522\pi\)
−0.545136 + 0.838348i \(0.683522\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −8.28685 −0.281599
\(867\) −16.1506 −0.548504
\(868\) 17.3607 0.589260
\(869\) 12.8494 0.435886
\(870\) 0 0
\(871\) −6.04104 −0.204693
\(872\) 9.78539 0.331375
\(873\) −9.02052 −0.305298
\(874\) 2.34017 0.0791575
\(875\) 0 0
\(876\) 4.68035 0.158134
\(877\) −27.7009 −0.935392 −0.467696 0.883889i \(-0.654916\pi\)
−0.467696 + 0.883889i \(0.654916\pi\)
\(878\) 28.9939 0.978495
\(879\) 16.3402 0.551140
\(880\) 0 0
\(881\) 12.3135 0.414853 0.207426 0.978251i \(-0.433491\pi\)
0.207426 + 0.978251i \(0.433491\pi\)
\(882\) −11.8371 −0.398576
\(883\) −7.15061 −0.240637 −0.120319 0.992735i \(-0.538392\pi\)
−0.120319 + 0.992735i \(0.538392\pi\)
\(884\) −2.15676 −0.0725395
\(885\) 0 0
\(886\) −3.20394 −0.107638
\(887\) −9.19165 −0.308625 −0.154313 0.988022i \(-0.549316\pi\)
−0.154313 + 0.988022i \(0.549316\pi\)
\(888\) 2.58145 0.0866278
\(889\) 74.5523 2.50041
\(890\) 0 0
\(891\) −1.07838 −0.0361270
\(892\) −21.1773 −0.709068
\(893\) −16.0000 −0.535420
\(894\) −6.92162 −0.231494
\(895\) 0 0
\(896\) 4.34017 0.144995
\(897\) −2.34017 −0.0781361
\(898\) −28.1568 −0.939603
\(899\) −41.9877 −1.40037
\(900\) 0 0
\(901\) −0.313511 −0.0104446
\(902\) 0.169042 0.00562847
\(903\) 3.20394 0.106620
\(904\) 8.92162 0.296729
\(905\) 0 0
\(906\) −6.15676 −0.204545
\(907\) 14.0989 0.468146 0.234073 0.972219i \(-0.424794\pi\)
0.234073 + 0.972219i \(0.424794\pi\)
\(908\) 11.7854 0.391112
\(909\) 1.50307 0.0498537
\(910\) 0 0
\(911\) −20.9939 −0.695558 −0.347779 0.937577i \(-0.613064\pi\)
−0.347779 + 0.937577i \(0.613064\pi\)
\(912\) 2.34017 0.0774909
\(913\) 12.3135 0.407518
\(914\) −23.1773 −0.766636
\(915\) 0 0
\(916\) 1.78539 0.0589908
\(917\) 28.9939 0.957462
\(918\) 0.921622 0.0304181
\(919\) −1.92777 −0.0635911 −0.0317956 0.999494i \(-0.510123\pi\)
−0.0317956 + 0.999494i \(0.510123\pi\)
\(920\) 0 0
\(921\) 3.20394 0.105573
\(922\) 42.3279 1.39399
\(923\) 35.5174 1.16907
\(924\) 4.68035 0.153972
\(925\) 0 0
\(926\) −34.9048 −1.14704
\(927\) 2.49693 0.0820099
\(928\) −10.4969 −0.344579
\(929\) 18.5646 0.609086 0.304543 0.952499i \(-0.401496\pi\)
0.304543 + 0.952499i \(0.401496\pi\)
\(930\) 0 0
\(931\) 27.7009 0.907859
\(932\) −18.3135 −0.599879
\(933\) 12.6537 0.414263
\(934\) −37.7731 −1.23597
\(935\) 0 0
\(936\) −2.34017 −0.0764909
\(937\) −22.3279 −0.729420 −0.364710 0.931121i \(-0.618832\pi\)
−0.364710 + 0.931121i \(0.618832\pi\)
\(938\) −11.2039 −0.365821
\(939\) −33.2183 −1.08404
\(940\) 0 0
\(941\) −11.0661 −0.360744 −0.180372 0.983598i \(-0.557730\pi\)
−0.180372 + 0.983598i \(0.557730\pi\)
\(942\) 2.89496 0.0943229
\(943\) −0.156755 −0.00510466
\(944\) −8.83710 −0.287623
\(945\) 0 0
\(946\) −0.796064 −0.0258823
\(947\) −8.05332 −0.261698 −0.130849 0.991402i \(-0.541770\pi\)
−0.130849 + 0.991402i \(0.541770\pi\)
\(948\) −11.9155 −0.386997
\(949\) 10.9528 0.355544
\(950\) 0 0
\(951\) 6.99386 0.226791
\(952\) −4.00000 −0.129641
\(953\) −39.5897 −1.28244 −0.641218 0.767359i \(-0.721570\pi\)
−0.641218 + 0.767359i \(0.721570\pi\)
\(954\) −0.340173 −0.0110135
\(955\) 0 0
\(956\) −5.33403 −0.172515
\(957\) −11.3197 −0.365912
\(958\) 28.1978 0.911029
\(959\) 96.7091 3.12290
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 6.04104 0.194771
\(963\) −9.57531 −0.308560
\(964\) 26.1978 0.843774
\(965\) 0 0
\(966\) −4.34017 −0.139643
\(967\) −48.2655 −1.55211 −0.776057 0.630663i \(-0.782783\pi\)
−0.776057 + 0.630663i \(0.782783\pi\)
\(968\) 9.83710 0.316176
\(969\) −2.15676 −0.0692850
\(970\) 0 0
\(971\) 18.0722 0.579966 0.289983 0.957032i \(-0.406350\pi\)
0.289983 + 0.957032i \(0.406350\pi\)
\(972\) 1.00000 0.0320750
\(973\) 45.6742 1.46425
\(974\) 38.2245 1.22479
\(975\) 0 0
\(976\) −11.5753 −0.370517
\(977\) 45.9442 1.46989 0.734943 0.678129i \(-0.237209\pi\)
0.734943 + 0.678129i \(0.237209\pi\)
\(978\) 6.52359 0.208601
\(979\) 5.04718 0.161309
\(980\) 0 0
\(981\) −9.78539 −0.312423
\(982\) −11.3074 −0.360833
\(983\) −1.78992 −0.0570896 −0.0285448 0.999593i \(-0.509087\pi\)
−0.0285448 + 0.999593i \(0.509087\pi\)
\(984\) −0.156755 −0.00499718
\(985\) 0 0
\(986\) 9.67420 0.308089
\(987\) 29.6742 0.944540
\(988\) 5.47641 0.174228
\(989\) 0.738205 0.0234735
\(990\) 0 0
\(991\) 7.68649 0.244169 0.122085 0.992520i \(-0.461042\pi\)
0.122085 + 0.992520i \(0.461042\pi\)
\(992\) 4.00000 0.127000
\(993\) −29.7275 −0.943375
\(994\) 65.8720 2.08933
\(995\) 0 0
\(996\) −11.4186 −0.361811
\(997\) 21.4908 0.680620 0.340310 0.940313i \(-0.389468\pi\)
0.340310 + 0.940313i \(0.389468\pi\)
\(998\) 2.47027 0.0781949
\(999\) −2.58145 −0.0816734
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 3450.2.a.bo.1.1 3
5.2 odd 4 690.2.d.c.139.2 6
5.3 odd 4 690.2.d.c.139.5 yes 6
5.4 even 2 3450.2.a.bt.1.3 3
15.2 even 4 2070.2.d.e.829.5 6
15.8 even 4 2070.2.d.e.829.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.d.c.139.2 6 5.2 odd 4
690.2.d.c.139.5 yes 6 5.3 odd 4
2070.2.d.e.829.2 6 15.8 even 4
2070.2.d.e.829.5 6 15.2 even 4
3450.2.a.bo.1.1 3 1.1 even 1 trivial
3450.2.a.bt.1.3 3 5.4 even 2