Properties

Label 547.2.a.b.1.12
Level $547$
Weight $2$
Character 547.1
Self dual yes
Analytic conductor $4.368$
Analytic rank $1$
Dimension $18$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 547 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 547.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.36781699056\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} - 137 x^{10} - 7703 x^{9} + 2068 x^{8} + 11068 x^{7} - 4274 x^{6} - 9021 x^{5} + 4048 x^{4} + 3834 x^{3} - 1851 x^{2} - 654 x + 328\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(-0.924759\) of defining polynomial
Character \(\chi\) \(=\) 547.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.924759 q^{2} +2.22306 q^{3} -1.14482 q^{4} -3.95421 q^{5} +2.05580 q^{6} -3.08028 q^{7} -2.90820 q^{8} +1.94201 q^{9} +O(q^{10})\) \(q+0.924759 q^{2} +2.22306 q^{3} -1.14482 q^{4} -3.95421 q^{5} +2.05580 q^{6} -3.08028 q^{7} -2.90820 q^{8} +1.94201 q^{9} -3.65669 q^{10} -3.50276 q^{11} -2.54501 q^{12} +3.47242 q^{13} -2.84852 q^{14} -8.79046 q^{15} -0.399740 q^{16} -3.04988 q^{17} +1.79589 q^{18} +7.32400 q^{19} +4.52687 q^{20} -6.84766 q^{21} -3.23921 q^{22} -6.16987 q^{23} -6.46511 q^{24} +10.6358 q^{25} +3.21115 q^{26} -2.35198 q^{27} +3.52637 q^{28} -3.33808 q^{29} -8.12905 q^{30} +5.01284 q^{31} +5.44674 q^{32} -7.78685 q^{33} -2.82040 q^{34} +12.1801 q^{35} -2.22325 q^{36} -9.39929 q^{37} +6.77293 q^{38} +7.71940 q^{39} +11.4996 q^{40} -2.63973 q^{41} -6.33243 q^{42} +0.457015 q^{43} +4.01003 q^{44} -7.67911 q^{45} -5.70564 q^{46} -4.87863 q^{47} -0.888648 q^{48} +2.48812 q^{49} +9.83553 q^{50} -6.78008 q^{51} -3.97530 q^{52} +0.519948 q^{53} -2.17502 q^{54} +13.8506 q^{55} +8.95807 q^{56} +16.2817 q^{57} -3.08692 q^{58} +10.3773 q^{59} +10.0635 q^{60} +2.71700 q^{61} +4.63567 q^{62} -5.98193 q^{63} +5.83640 q^{64} -13.7307 q^{65} -7.20096 q^{66} -11.2680 q^{67} +3.49157 q^{68} -13.7160 q^{69} +11.2636 q^{70} +6.36651 q^{71} -5.64775 q^{72} -1.19812 q^{73} -8.69207 q^{74} +23.6440 q^{75} -8.38468 q^{76} +10.7895 q^{77} +7.13858 q^{78} -11.6933 q^{79} +1.58066 q^{80} -11.0546 q^{81} -2.44111 q^{82} -9.73676 q^{83} +7.83934 q^{84} +12.0599 q^{85} +0.422629 q^{86} -7.42077 q^{87} +10.1867 q^{88} +11.4691 q^{89} -7.10132 q^{90} -10.6960 q^{91} +7.06340 q^{92} +11.1439 q^{93} -4.51155 q^{94} -28.9607 q^{95} +12.1084 q^{96} +7.25071 q^{97} +2.30091 q^{98} -6.80238 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q - 4q^{2} - 10q^{3} + 16q^{4} - 27q^{5} - 3q^{6} - 11q^{7} - 12q^{8} + 14q^{9} + O(q^{10}) \) \( 18q - 4q^{2} - 10q^{3} + 16q^{4} - 27q^{5} - 3q^{6} - 11q^{7} - 12q^{8} + 14q^{9} - 5q^{10} + 2q^{11} - 32q^{12} - 25q^{13} - 7q^{14} + 9q^{15} + 8q^{16} - 30q^{17} - 10q^{18} + 4q^{19} - 41q^{20} - 16q^{21} - 24q^{22} - 26q^{23} - 12q^{24} + 31q^{25} - 18q^{26} - 37q^{27} - 16q^{28} - 18q^{29} + 8q^{30} - 5q^{31} - 28q^{32} - 10q^{33} + 5q^{34} - 9q^{35} + 31q^{36} - 18q^{37} - 45q^{38} + 7q^{39} + 7q^{40} - 17q^{41} + 4q^{42} + 8q^{43} + 12q^{44} - 44q^{45} + 30q^{46} - 52q^{47} - 7q^{48} + 29q^{49} + 13q^{50} + 19q^{51} - 14q^{52} - 60q^{53} + 11q^{54} + 11q^{55} + 7q^{56} + 4q^{57} + 14q^{58} - 8q^{59} + 86q^{60} - 26q^{61} + 4q^{62} - q^{63} + 44q^{64} - 6q^{65} + 18q^{66} + 12q^{67} - 61q^{68} - 38q^{69} + 35q^{70} - q^{71} + 28q^{72} - 2q^{73} + 16q^{74} - 17q^{75} + 66q^{76} - 73q^{77} + 115q^{78} + 18q^{79} - 32q^{80} + 18q^{81} + 44q^{82} - 43q^{83} + 41q^{84} + 51q^{85} + 4q^{86} + 3q^{87} - 17q^{88} - 28q^{89} + 58q^{90} - q^{91} - 68q^{92} - 60q^{93} + 78q^{94} - 18q^{95} + 29q^{96} - 34q^{97} + 34q^{98} + 15q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.924759 0.653903 0.326952 0.945041i \(-0.393979\pi\)
0.326952 + 0.945041i \(0.393979\pi\)
\(3\) 2.22306 1.28349 0.641743 0.766920i \(-0.278212\pi\)
0.641743 + 0.766920i \(0.278212\pi\)
\(4\) −1.14482 −0.572411
\(5\) −3.95421 −1.76838 −0.884188 0.467131i \(-0.845288\pi\)
−0.884188 + 0.467131i \(0.845288\pi\)
\(6\) 2.05580 0.839275
\(7\) −3.08028 −1.16424 −0.582118 0.813104i \(-0.697776\pi\)
−0.582118 + 0.813104i \(0.697776\pi\)
\(8\) −2.90820 −1.02820
\(9\) 1.94201 0.647336
\(10\) −3.65669 −1.15635
\(11\) −3.50276 −1.05612 −0.528061 0.849207i \(-0.677081\pi\)
−0.528061 + 0.849207i \(0.677081\pi\)
\(12\) −2.54501 −0.734681
\(13\) 3.47242 0.963075 0.481538 0.876425i \(-0.340079\pi\)
0.481538 + 0.876425i \(0.340079\pi\)
\(14\) −2.84852 −0.761298
\(15\) −8.79046 −2.26969
\(16\) −0.399740 −0.0999351
\(17\) −3.04988 −0.739705 −0.369852 0.929091i \(-0.620592\pi\)
−0.369852 + 0.929091i \(0.620592\pi\)
\(18\) 1.79589 0.423295
\(19\) 7.32400 1.68024 0.840121 0.542399i \(-0.182484\pi\)
0.840121 + 0.542399i \(0.182484\pi\)
\(20\) 4.52687 1.01224
\(21\) −6.84766 −1.49428
\(22\) −3.23921 −0.690601
\(23\) −6.16987 −1.28651 −0.643253 0.765653i \(-0.722416\pi\)
−0.643253 + 0.765653i \(0.722416\pi\)
\(24\) −6.46511 −1.31969
\(25\) 10.6358 2.12716
\(26\) 3.21115 0.629758
\(27\) −2.35198 −0.452639
\(28\) 3.52637 0.666421
\(29\) −3.33808 −0.619867 −0.309933 0.950758i \(-0.600307\pi\)
−0.309933 + 0.950758i \(0.600307\pi\)
\(30\) −8.12905 −1.48415
\(31\) 5.01284 0.900333 0.450167 0.892945i \(-0.351365\pi\)
0.450167 + 0.892945i \(0.351365\pi\)
\(32\) 5.44674 0.962856
\(33\) −7.78685 −1.35552
\(34\) −2.82040 −0.483695
\(35\) 12.1801 2.05881
\(36\) −2.22325 −0.370542
\(37\) −9.39929 −1.54523 −0.772617 0.634873i \(-0.781053\pi\)
−0.772617 + 0.634873i \(0.781053\pi\)
\(38\) 6.77293 1.09872
\(39\) 7.71940 1.23609
\(40\) 11.4996 1.81825
\(41\) −2.63973 −0.412256 −0.206128 0.978525i \(-0.566086\pi\)
−0.206128 + 0.978525i \(0.566086\pi\)
\(42\) −6.33243 −0.977115
\(43\) 0.457015 0.0696942 0.0348471 0.999393i \(-0.488906\pi\)
0.0348471 + 0.999393i \(0.488906\pi\)
\(44\) 4.01003 0.604535
\(45\) −7.67911 −1.14473
\(46\) −5.70564 −0.841251
\(47\) −4.87863 −0.711621 −0.355811 0.934558i \(-0.615795\pi\)
−0.355811 + 0.934558i \(0.615795\pi\)
\(48\) −0.888648 −0.128265
\(49\) 2.48812 0.355446
\(50\) 9.83553 1.39095
\(51\) −6.78008 −0.949401
\(52\) −3.97530 −0.551275
\(53\) 0.519948 0.0714203 0.0357101 0.999362i \(-0.488631\pi\)
0.0357101 + 0.999362i \(0.488631\pi\)
\(54\) −2.17502 −0.295982
\(55\) 13.8506 1.86762
\(56\) 8.95807 1.19707
\(57\) 16.2817 2.15657
\(58\) −3.08692 −0.405333
\(59\) 10.3773 1.35101 0.675503 0.737357i \(-0.263927\pi\)
0.675503 + 0.737357i \(0.263927\pi\)
\(60\) 10.0635 1.29919
\(61\) 2.71700 0.347877 0.173938 0.984757i \(-0.444351\pi\)
0.173938 + 0.984757i \(0.444351\pi\)
\(62\) 4.63567 0.588731
\(63\) −5.98193 −0.753652
\(64\) 5.83640 0.729550
\(65\) −13.7307 −1.70308
\(66\) −7.20096 −0.886376
\(67\) −11.2680 −1.37661 −0.688304 0.725423i \(-0.741644\pi\)
−0.688304 + 0.725423i \(0.741644\pi\)
\(68\) 3.49157 0.423415
\(69\) −13.7160 −1.65121
\(70\) 11.2636 1.34626
\(71\) 6.36651 0.755566 0.377783 0.925894i \(-0.376687\pi\)
0.377783 + 0.925894i \(0.376687\pi\)
\(72\) −5.64775 −0.665594
\(73\) −1.19812 −0.140229 −0.0701145 0.997539i \(-0.522336\pi\)
−0.0701145 + 0.997539i \(0.522336\pi\)
\(74\) −8.69207 −1.01043
\(75\) 23.6440 2.73017
\(76\) −8.38468 −0.961788
\(77\) 10.7895 1.22957
\(78\) 7.13858 0.808285
\(79\) −11.6933 −1.31560 −0.657802 0.753191i \(-0.728513\pi\)
−0.657802 + 0.753191i \(0.728513\pi\)
\(80\) 1.58066 0.176723
\(81\) −11.0546 −1.22829
\(82\) −2.44111 −0.269576
\(83\) −9.73676 −1.06875 −0.534374 0.845248i \(-0.679453\pi\)
−0.534374 + 0.845248i \(0.679453\pi\)
\(84\) 7.83934 0.855342
\(85\) 12.0599 1.30808
\(86\) 0.422629 0.0455732
\(87\) −7.42077 −0.795590
\(88\) 10.1867 1.08591
\(89\) 11.4691 1.21572 0.607859 0.794045i \(-0.292028\pi\)
0.607859 + 0.794045i \(0.292028\pi\)
\(90\) −7.10132 −0.748545
\(91\) −10.6960 −1.12125
\(92\) 7.06340 0.736410
\(93\) 11.1439 1.15556
\(94\) −4.51155 −0.465331
\(95\) −28.9607 −2.97130
\(96\) 12.1084 1.23581
\(97\) 7.25071 0.736198 0.368099 0.929787i \(-0.380009\pi\)
0.368099 + 0.929787i \(0.380009\pi\)
\(98\) 2.30091 0.232427
\(99\) −6.80238 −0.683665
\(100\) −12.1761 −1.21761
\(101\) −15.4716 −1.53948 −0.769741 0.638356i \(-0.779615\pi\)
−0.769741 + 0.638356i \(0.779615\pi\)
\(102\) −6.26993 −0.620816
\(103\) −11.1594 −1.09957 −0.549785 0.835306i \(-0.685290\pi\)
−0.549785 + 0.835306i \(0.685290\pi\)
\(104\) −10.0985 −0.990238
\(105\) 27.0771 2.64245
\(106\) 0.480826 0.0467019
\(107\) −15.8859 −1.53574 −0.767872 0.640603i \(-0.778684\pi\)
−0.767872 + 0.640603i \(0.778684\pi\)
\(108\) 2.69260 0.259096
\(109\) 2.66177 0.254951 0.127476 0.991842i \(-0.459313\pi\)
0.127476 + 0.991842i \(0.459313\pi\)
\(110\) 12.8085 1.22124
\(111\) −20.8952 −1.98329
\(112\) 1.23131 0.116348
\(113\) −3.96953 −0.373422 −0.186711 0.982415i \(-0.559783\pi\)
−0.186711 + 0.982415i \(0.559783\pi\)
\(114\) 15.0567 1.41019
\(115\) 24.3970 2.27503
\(116\) 3.82151 0.354818
\(117\) 6.74346 0.623433
\(118\) 9.59647 0.883427
\(119\) 9.39449 0.861191
\(120\) 25.5644 2.33370
\(121\) 1.26931 0.115392
\(122\) 2.51257 0.227478
\(123\) −5.86828 −0.529125
\(124\) −5.73881 −0.515360
\(125\) −22.2851 −1.99324
\(126\) −5.53184 −0.492815
\(127\) −14.0151 −1.24364 −0.621822 0.783159i \(-0.713607\pi\)
−0.621822 + 0.783159i \(0.713607\pi\)
\(128\) −5.49622 −0.485801
\(129\) 1.01597 0.0894515
\(130\) −12.6976 −1.11365
\(131\) 21.9251 1.91560 0.957802 0.287430i \(-0.0928009\pi\)
0.957802 + 0.287430i \(0.0928009\pi\)
\(132\) 8.91455 0.775912
\(133\) −22.5600 −1.95620
\(134\) −10.4202 −0.900168
\(135\) 9.30024 0.800437
\(136\) 8.86967 0.760568
\(137\) 0.725655 0.0619969 0.0309984 0.999519i \(-0.490131\pi\)
0.0309984 + 0.999519i \(0.490131\pi\)
\(138\) −12.6840 −1.07973
\(139\) 2.05950 0.174685 0.0873423 0.996178i \(-0.472163\pi\)
0.0873423 + 0.996178i \(0.472163\pi\)
\(140\) −13.9440 −1.17848
\(141\) −10.8455 −0.913356
\(142\) 5.88749 0.494067
\(143\) −12.1630 −1.01712
\(144\) −0.776299 −0.0646916
\(145\) 13.1995 1.09616
\(146\) −1.10797 −0.0916961
\(147\) 5.53126 0.456210
\(148\) 10.7605 0.884508
\(149\) 13.4893 1.10509 0.552544 0.833484i \(-0.313657\pi\)
0.552544 + 0.833484i \(0.313657\pi\)
\(150\) 21.8650 1.78527
\(151\) 0.695017 0.0565597 0.0282798 0.999600i \(-0.490997\pi\)
0.0282798 + 0.999600i \(0.490997\pi\)
\(152\) −21.2997 −1.72763
\(153\) −5.92289 −0.478837
\(154\) 9.97766 0.804023
\(155\) −19.8218 −1.59213
\(156\) −8.83733 −0.707553
\(157\) 10.9574 0.874499 0.437249 0.899340i \(-0.355953\pi\)
0.437249 + 0.899340i \(0.355953\pi\)
\(158\) −10.8135 −0.860277
\(159\) 1.15588 0.0916669
\(160\) −21.5375 −1.70269
\(161\) 19.0049 1.49780
\(162\) −10.2229 −0.803184
\(163\) 15.9789 1.25156 0.625780 0.779999i \(-0.284781\pi\)
0.625780 + 0.779999i \(0.284781\pi\)
\(164\) 3.02202 0.235980
\(165\) 30.7908 2.39706
\(166\) −9.00415 −0.698858
\(167\) 13.3872 1.03594 0.517968 0.855400i \(-0.326689\pi\)
0.517968 + 0.855400i \(0.326689\pi\)
\(168\) 19.9144 1.53643
\(169\) −0.942324 −0.0724865
\(170\) 11.1525 0.855355
\(171\) 14.2233 1.08768
\(172\) −0.523201 −0.0398937
\(173\) −17.9016 −1.36103 −0.680515 0.732734i \(-0.738244\pi\)
−0.680515 + 0.732734i \(0.738244\pi\)
\(174\) −6.86242 −0.520239
\(175\) −32.7612 −2.47651
\(176\) 1.40019 0.105544
\(177\) 23.0693 1.73400
\(178\) 10.6061 0.794962
\(179\) 24.0861 1.80028 0.900139 0.435602i \(-0.143464\pi\)
0.900139 + 0.435602i \(0.143464\pi\)
\(180\) 8.79121 0.655258
\(181\) 4.37140 0.324924 0.162462 0.986715i \(-0.448057\pi\)
0.162462 + 0.986715i \(0.448057\pi\)
\(182\) −9.89123 −0.733187
\(183\) 6.04007 0.446495
\(184\) 17.9432 1.32279
\(185\) 37.1668 2.73255
\(186\) 10.3054 0.755627
\(187\) 10.6830 0.781218
\(188\) 5.58516 0.407340
\(189\) 7.24477 0.526979
\(190\) −26.7816 −1.94294
\(191\) −14.3169 −1.03593 −0.517967 0.855401i \(-0.673311\pi\)
−0.517967 + 0.855401i \(0.673311\pi\)
\(192\) 12.9747 0.936367
\(193\) 12.7297 0.916305 0.458152 0.888874i \(-0.348511\pi\)
0.458152 + 0.888874i \(0.348511\pi\)
\(194\) 6.70516 0.481402
\(195\) −30.5241 −2.18588
\(196\) −2.84846 −0.203461
\(197\) 13.4238 0.956407 0.478203 0.878249i \(-0.341288\pi\)
0.478203 + 0.878249i \(0.341288\pi\)
\(198\) −6.29056 −0.447051
\(199\) −1.56808 −0.111159 −0.0555793 0.998454i \(-0.517701\pi\)
−0.0555793 + 0.998454i \(0.517701\pi\)
\(200\) −30.9310 −2.18715
\(201\) −25.0495 −1.76686
\(202\) −14.3075 −1.00667
\(203\) 10.2822 0.721671
\(204\) 7.76198 0.543447
\(205\) 10.4380 0.729024
\(206\) −10.3198 −0.719012
\(207\) −11.9819 −0.832802
\(208\) −1.38807 −0.0962450
\(209\) −25.6542 −1.77454
\(210\) 25.0398 1.72791
\(211\) −16.3089 −1.12275 −0.561376 0.827561i \(-0.689728\pi\)
−0.561376 + 0.827561i \(0.689728\pi\)
\(212\) −0.595247 −0.0408817
\(213\) 14.1532 0.969758
\(214\) −14.6906 −1.00423
\(215\) −1.80714 −0.123246
\(216\) 6.84004 0.465406
\(217\) −15.4410 −1.04820
\(218\) 2.46149 0.166713
\(219\) −2.66349 −0.179982
\(220\) −15.8565 −1.06905
\(221\) −10.5905 −0.712391
\(222\) −19.3230 −1.29688
\(223\) −6.07966 −0.407124 −0.203562 0.979062i \(-0.565252\pi\)
−0.203562 + 0.979062i \(0.565252\pi\)
\(224\) −16.7775 −1.12099
\(225\) 20.6548 1.37698
\(226\) −3.67086 −0.244182
\(227\) 19.1850 1.27336 0.636678 0.771130i \(-0.280308\pi\)
0.636678 + 0.771130i \(0.280308\pi\)
\(228\) −18.6397 −1.23444
\(229\) 9.82576 0.649305 0.324652 0.945833i \(-0.394753\pi\)
0.324652 + 0.945833i \(0.394753\pi\)
\(230\) 22.5613 1.48765
\(231\) 23.9857 1.57814
\(232\) 9.70782 0.637350
\(233\) −30.2087 −1.97904 −0.989519 0.144406i \(-0.953873\pi\)
−0.989519 + 0.144406i \(0.953873\pi\)
\(234\) 6.23607 0.407665
\(235\) 19.2911 1.25841
\(236\) −11.8801 −0.773330
\(237\) −25.9950 −1.68856
\(238\) 8.68763 0.563136
\(239\) 20.0414 1.29637 0.648186 0.761482i \(-0.275528\pi\)
0.648186 + 0.761482i \(0.275528\pi\)
\(240\) 3.51390 0.226821
\(241\) −22.2442 −1.43287 −0.716436 0.697653i \(-0.754228\pi\)
−0.716436 + 0.697653i \(0.754228\pi\)
\(242\) 1.17381 0.0754551
\(243\) −17.5192 −1.12386
\(244\) −3.11048 −0.199128
\(245\) −9.83857 −0.628563
\(246\) −5.42674 −0.345997
\(247\) 25.4320 1.61820
\(248\) −14.5784 −0.925726
\(249\) −21.6454 −1.37172
\(250\) −20.6083 −1.30338
\(251\) −12.5368 −0.791313 −0.395656 0.918399i \(-0.629483\pi\)
−0.395656 + 0.918399i \(0.629483\pi\)
\(252\) 6.84824 0.431399
\(253\) 21.6116 1.35871
\(254\) −12.9606 −0.813222
\(255\) 26.8099 1.67890
\(256\) −16.7555 −1.04722
\(257\) 13.1786 0.822055 0.411028 0.911623i \(-0.365170\pi\)
0.411028 + 0.911623i \(0.365170\pi\)
\(258\) 0.939531 0.0584926
\(259\) 28.9524 1.79902
\(260\) 15.7192 0.974861
\(261\) −6.48259 −0.401262
\(262\) 20.2754 1.25262
\(263\) 1.98374 0.122322 0.0611612 0.998128i \(-0.480520\pi\)
0.0611612 + 0.998128i \(0.480520\pi\)
\(264\) 22.6457 1.39375
\(265\) −2.05598 −0.126298
\(266\) −20.8625 −1.27916
\(267\) 25.4965 1.56036
\(268\) 12.8999 0.787985
\(269\) −0.592176 −0.0361056 −0.0180528 0.999837i \(-0.505747\pi\)
−0.0180528 + 0.999837i \(0.505747\pi\)
\(270\) 8.60047 0.523408
\(271\) −21.4167 −1.30097 −0.650486 0.759518i \(-0.725435\pi\)
−0.650486 + 0.759518i \(0.725435\pi\)
\(272\) 1.21916 0.0739225
\(273\) −23.7779 −1.43910
\(274\) 0.671056 0.0405400
\(275\) −37.2546 −2.24653
\(276\) 15.7024 0.945172
\(277\) −9.10027 −0.546783 −0.273391 0.961903i \(-0.588145\pi\)
−0.273391 + 0.961903i \(0.588145\pi\)
\(278\) 1.90454 0.114227
\(279\) 9.73498 0.582818
\(280\) −35.4221 −2.11688
\(281\) −2.45423 −0.146407 −0.0732036 0.997317i \(-0.523322\pi\)
−0.0732036 + 0.997317i \(0.523322\pi\)
\(282\) −10.0295 −0.597246
\(283\) −7.57111 −0.450056 −0.225028 0.974352i \(-0.572247\pi\)
−0.225028 + 0.974352i \(0.572247\pi\)
\(284\) −7.28852 −0.432494
\(285\) −64.3813 −3.81362
\(286\) −11.2479 −0.665100
\(287\) 8.13110 0.479964
\(288\) 10.5776 0.623291
\(289\) −7.69823 −0.452837
\(290\) 12.2063 0.716781
\(291\) 16.1188 0.944900
\(292\) 1.37163 0.0802685
\(293\) −15.9522 −0.931935 −0.465967 0.884802i \(-0.654294\pi\)
−0.465967 + 0.884802i \(0.654294\pi\)
\(294\) 5.11508 0.298317
\(295\) −41.0339 −2.38909
\(296\) 27.3350 1.58882
\(297\) 8.23843 0.478042
\(298\) 12.4744 0.722620
\(299\) −21.4244 −1.23900
\(300\) −27.0682 −1.56278
\(301\) −1.40774 −0.0811405
\(302\) 0.642723 0.0369846
\(303\) −34.3943 −1.97590
\(304\) −2.92770 −0.167915
\(305\) −10.7436 −0.615177
\(306\) −5.47725 −0.313113
\(307\) −10.0561 −0.573931 −0.286965 0.957941i \(-0.592646\pi\)
−0.286965 + 0.957941i \(0.592646\pi\)
\(308\) −12.3520 −0.703822
\(309\) −24.8081 −1.41128
\(310\) −18.3304 −1.04110
\(311\) −17.8541 −1.01241 −0.506207 0.862412i \(-0.668953\pi\)
−0.506207 + 0.862412i \(0.668953\pi\)
\(312\) −22.4496 −1.27096
\(313\) −11.0835 −0.626475 −0.313238 0.949675i \(-0.601414\pi\)
−0.313238 + 0.949675i \(0.601414\pi\)
\(314\) 10.1330 0.571837
\(315\) 23.6538 1.33274
\(316\) 13.3868 0.753065
\(317\) 4.47220 0.251184 0.125592 0.992082i \(-0.459917\pi\)
0.125592 + 0.992082i \(0.459917\pi\)
\(318\) 1.06891 0.0599413
\(319\) 11.6925 0.654654
\(320\) −23.0783 −1.29012
\(321\) −35.3153 −1.97111
\(322\) 17.5750 0.979415
\(323\) −22.3373 −1.24288
\(324\) 12.6556 0.703088
\(325\) 36.9319 2.04861
\(326\) 14.7766 0.818399
\(327\) 5.91728 0.327226
\(328\) 7.67686 0.423884
\(329\) 15.0275 0.828495
\(330\) 28.4741 1.56745
\(331\) 28.5978 1.57187 0.785937 0.618306i \(-0.212181\pi\)
0.785937 + 0.618306i \(0.212181\pi\)
\(332\) 11.1469 0.611763
\(333\) −18.2535 −1.00029
\(334\) 12.3800 0.677402
\(335\) 44.5561 2.43436
\(336\) 2.73729 0.149331
\(337\) 34.0126 1.85278 0.926392 0.376561i \(-0.122893\pi\)
0.926392 + 0.376561i \(0.122893\pi\)
\(338\) −0.871422 −0.0473991
\(339\) −8.82452 −0.479282
\(340\) −13.8064 −0.748757
\(341\) −17.5588 −0.950861
\(342\) 13.1531 0.711238
\(343\) 13.8978 0.750413
\(344\) −1.32909 −0.0716599
\(345\) 54.2360 2.91997
\(346\) −16.5546 −0.889982
\(347\) −5.18016 −0.278086 −0.139043 0.990286i \(-0.544403\pi\)
−0.139043 + 0.990286i \(0.544403\pi\)
\(348\) 8.49546 0.455404
\(349\) 4.21519 0.225634 0.112817 0.993616i \(-0.464013\pi\)
0.112817 + 0.993616i \(0.464013\pi\)
\(350\) −30.2962 −1.61940
\(351\) −8.16706 −0.435926
\(352\) −19.0786 −1.01689
\(353\) 22.6574 1.20593 0.602967 0.797766i \(-0.293985\pi\)
0.602967 + 0.797766i \(0.293985\pi\)
\(354\) 21.3336 1.13387
\(355\) −25.1745 −1.33613
\(356\) −13.1300 −0.695891
\(357\) 20.8845 1.10533
\(358\) 22.2738 1.17721
\(359\) −5.93624 −0.313303 −0.156651 0.987654i \(-0.550070\pi\)
−0.156651 + 0.987654i \(0.550070\pi\)
\(360\) 22.3324 1.17702
\(361\) 34.6410 1.82321
\(362\) 4.04249 0.212469
\(363\) 2.82176 0.148104
\(364\) 12.2450 0.641814
\(365\) 4.73760 0.247978
\(366\) 5.58560 0.291964
\(367\) 16.1865 0.844930 0.422465 0.906379i \(-0.361165\pi\)
0.422465 + 0.906379i \(0.361165\pi\)
\(368\) 2.46635 0.128567
\(369\) −5.12637 −0.266868
\(370\) 34.3703 1.78683
\(371\) −1.60158 −0.0831501
\(372\) −12.7577 −0.661458
\(373\) −8.96744 −0.464316 −0.232158 0.972678i \(-0.574579\pi\)
−0.232158 + 0.972678i \(0.574579\pi\)
\(374\) 9.87919 0.510841
\(375\) −49.5411 −2.55829
\(376\) 14.1880 0.731692
\(377\) −11.5912 −0.596978
\(378\) 6.69966 0.344593
\(379\) −32.2073 −1.65438 −0.827189 0.561924i \(-0.810061\pi\)
−0.827189 + 0.561924i \(0.810061\pi\)
\(380\) 33.1548 1.70080
\(381\) −31.1566 −1.59620
\(382\) −13.2397 −0.677400
\(383\) 32.7239 1.67211 0.836056 0.548644i \(-0.184856\pi\)
0.836056 + 0.548644i \(0.184856\pi\)
\(384\) −12.2184 −0.623519
\(385\) −42.6639 −2.17435
\(386\) 11.7719 0.599175
\(387\) 0.887527 0.0451156
\(388\) −8.30077 −0.421408
\(389\) −15.2384 −0.772616 −0.386308 0.922370i \(-0.626250\pi\)
−0.386308 + 0.922370i \(0.626250\pi\)
\(390\) −28.2275 −1.42935
\(391\) 18.8174 0.951635
\(392\) −7.23596 −0.365471
\(393\) 48.7408 2.45865
\(394\) 12.4138 0.625397
\(395\) 46.2379 2.32648
\(396\) 7.78751 0.391337
\(397\) −31.8678 −1.59940 −0.799699 0.600401i \(-0.795008\pi\)
−0.799699 + 0.600401i \(0.795008\pi\)
\(398\) −1.45010 −0.0726869
\(399\) −50.1522 −2.51075
\(400\) −4.25155 −0.212578
\(401\) −7.87598 −0.393307 −0.196654 0.980473i \(-0.563007\pi\)
−0.196654 + 0.980473i \(0.563007\pi\)
\(402\) −23.1647 −1.15535
\(403\) 17.4067 0.867088
\(404\) 17.7122 0.881216
\(405\) 43.7123 2.17208
\(406\) 9.50858 0.471903
\(407\) 32.9234 1.63195
\(408\) 19.7178 0.976178
\(409\) −21.2120 −1.04886 −0.524432 0.851452i \(-0.675722\pi\)
−0.524432 + 0.851452i \(0.675722\pi\)
\(410\) 9.65267 0.476711
\(411\) 1.61318 0.0795721
\(412\) 12.7755 0.629405
\(413\) −31.9649 −1.57289
\(414\) −11.0804 −0.544572
\(415\) 38.5012 1.88995
\(416\) 18.9133 0.927303
\(417\) 4.57840 0.224205
\(418\) −23.7239 −1.16038
\(419\) −17.3991 −0.850003 −0.425001 0.905193i \(-0.639726\pi\)
−0.425001 + 0.905193i \(0.639726\pi\)
\(420\) −30.9984 −1.51257
\(421\) −13.3962 −0.652890 −0.326445 0.945216i \(-0.605851\pi\)
−0.326445 + 0.945216i \(0.605851\pi\)
\(422\) −15.0818 −0.734171
\(423\) −9.47433 −0.460658
\(424\) −1.51211 −0.0734346
\(425\) −32.4379 −1.57347
\(426\) 13.0883 0.634128
\(427\) −8.36913 −0.405011
\(428\) 18.1865 0.879076
\(429\) −27.0392 −1.30546
\(430\) −1.67116 −0.0805907
\(431\) 32.3565 1.55856 0.779279 0.626677i \(-0.215586\pi\)
0.779279 + 0.626677i \(0.215586\pi\)
\(432\) 0.940183 0.0452346
\(433\) 8.08088 0.388342 0.194171 0.980968i \(-0.437798\pi\)
0.194171 + 0.980968i \(0.437798\pi\)
\(434\) −14.2792 −0.685422
\(435\) 29.3433 1.40690
\(436\) −3.04725 −0.145937
\(437\) −45.1881 −2.16164
\(438\) −2.46308 −0.117691
\(439\) 26.4717 1.26342 0.631712 0.775203i \(-0.282353\pi\)
0.631712 + 0.775203i \(0.282353\pi\)
\(440\) −40.2804 −1.92029
\(441\) 4.83196 0.230093
\(442\) −9.79362 −0.465835
\(443\) −38.7060 −1.83898 −0.919488 0.393118i \(-0.871396\pi\)
−0.919488 + 0.393118i \(0.871396\pi\)
\(444\) 23.9213 1.13525
\(445\) −45.3511 −2.14985
\(446\) −5.62222 −0.266220
\(447\) 29.9876 1.41836
\(448\) −17.9777 −0.849368
\(449\) −4.41373 −0.208297 −0.104148 0.994562i \(-0.533212\pi\)
−0.104148 + 0.994562i \(0.533212\pi\)
\(450\) 19.1007 0.900414
\(451\) 9.24633 0.435393
\(452\) 4.54440 0.213751
\(453\) 1.54507 0.0725936
\(454\) 17.7415 0.832651
\(455\) 42.2943 1.98279
\(456\) −47.3505 −2.21739
\(457\) 0.171589 0.00802659 0.00401329 0.999992i \(-0.498723\pi\)
0.00401329 + 0.999992i \(0.498723\pi\)
\(458\) 9.08646 0.424582
\(459\) 7.17327 0.334819
\(460\) −27.9302 −1.30225
\(461\) −32.6067 −1.51864 −0.759322 0.650715i \(-0.774469\pi\)
−0.759322 + 0.650715i \(0.774469\pi\)
\(462\) 22.1810 1.03195
\(463\) 25.5148 1.18577 0.592887 0.805285i \(-0.297988\pi\)
0.592887 + 0.805285i \(0.297988\pi\)
\(464\) 1.33437 0.0619465
\(465\) −44.0652 −2.04347
\(466\) −27.9357 −1.29410
\(467\) 3.58271 0.165788 0.0828939 0.996558i \(-0.473584\pi\)
0.0828939 + 0.996558i \(0.473584\pi\)
\(468\) −7.72006 −0.356860
\(469\) 34.7086 1.60270
\(470\) 17.8396 0.822881
\(471\) 24.3591 1.12241
\(472\) −30.1792 −1.38911
\(473\) −1.60081 −0.0736055
\(474\) −24.0391 −1.10415
\(475\) 77.8965 3.57414
\(476\) −10.7550 −0.492955
\(477\) 1.00974 0.0462329
\(478\) 18.5335 0.847702
\(479\) −27.2764 −1.24629 −0.623145 0.782106i \(-0.714146\pi\)
−0.623145 + 0.782106i \(0.714146\pi\)
\(480\) −47.8793 −2.18538
\(481\) −32.6382 −1.48818
\(482\) −20.5705 −0.936959
\(483\) 42.2491 1.92240
\(484\) −1.45313 −0.0660515
\(485\) −28.6709 −1.30188
\(486\) −16.2010 −0.734893
\(487\) −32.4550 −1.47068 −0.735339 0.677700i \(-0.762977\pi\)
−0.735339 + 0.677700i \(0.762977\pi\)
\(488\) −7.90159 −0.357688
\(489\) 35.5220 1.60636
\(490\) −9.09830 −0.411019
\(491\) 8.45330 0.381492 0.190746 0.981639i \(-0.438909\pi\)
0.190746 + 0.981639i \(0.438909\pi\)
\(492\) 6.71813 0.302877
\(493\) 10.1808 0.458518
\(494\) 23.5185 1.05815
\(495\) 26.8981 1.20898
\(496\) −2.00384 −0.0899749
\(497\) −19.6106 −0.879657
\(498\) −20.0168 −0.896974
\(499\) −28.5526 −1.27819 −0.639094 0.769128i \(-0.720691\pi\)
−0.639094 + 0.769128i \(0.720691\pi\)
\(500\) 25.5124 1.14095
\(501\) 29.7607 1.32961
\(502\) −11.5935 −0.517442
\(503\) 14.5826 0.650208 0.325104 0.945678i \(-0.394601\pi\)
0.325104 + 0.945678i \(0.394601\pi\)
\(504\) 17.3966 0.774908
\(505\) 61.1780 2.72238
\(506\) 19.9855 0.888463
\(507\) −2.09485 −0.0930354
\(508\) 16.0448 0.711875
\(509\) −16.1161 −0.714331 −0.357166 0.934041i \(-0.616257\pi\)
−0.357166 + 0.934041i \(0.616257\pi\)
\(510\) 24.7926 1.09784
\(511\) 3.69053 0.163260
\(512\) −4.50233 −0.198977
\(513\) −17.2259 −0.760543
\(514\) 12.1870 0.537545
\(515\) 44.1267 1.94445
\(516\) −1.16311 −0.0512030
\(517\) 17.0887 0.751558
\(518\) 26.7740 1.17638
\(519\) −39.7963 −1.74686
\(520\) 39.9315 1.75111
\(521\) −7.13196 −0.312457 −0.156228 0.987721i \(-0.549934\pi\)
−0.156228 + 0.987721i \(0.549934\pi\)
\(522\) −5.99483 −0.262386
\(523\) −4.52473 −0.197853 −0.0989264 0.995095i \(-0.531541\pi\)
−0.0989264 + 0.995095i \(0.531541\pi\)
\(524\) −25.1003 −1.09651
\(525\) −72.8302 −3.17857
\(526\) 1.83448 0.0799870
\(527\) −15.2886 −0.665981
\(528\) 3.11272 0.135464
\(529\) 15.0673 0.655099
\(530\) −1.90129 −0.0825866
\(531\) 20.1527 0.874555
\(532\) 25.8272 1.11975
\(533\) −9.16623 −0.397034
\(534\) 23.5781 1.02032
\(535\) 62.8160 2.71577
\(536\) 32.7697 1.41543
\(537\) 53.5449 2.31063
\(538\) −0.547619 −0.0236096
\(539\) −8.71529 −0.375394
\(540\) −10.6471 −0.458179
\(541\) −6.61816 −0.284537 −0.142268 0.989828i \(-0.545440\pi\)
−0.142268 + 0.989828i \(0.545440\pi\)
\(542\) −19.8053 −0.850709
\(543\) 9.71790 0.417035
\(544\) −16.6119 −0.712229
\(545\) −10.5252 −0.450849
\(546\) −21.9888 −0.941035
\(547\) −1.00000 −0.0427569
\(548\) −0.830745 −0.0354877
\(549\) 5.27644 0.225193
\(550\) −34.4515 −1.46902
\(551\) −24.4481 −1.04153
\(552\) 39.8889 1.69778
\(553\) 36.0188 1.53167
\(554\) −8.41556 −0.357543
\(555\) 82.6240 3.50720
\(556\) −2.35776 −0.0999914
\(557\) −9.58172 −0.405991 −0.202995 0.979180i \(-0.565068\pi\)
−0.202995 + 0.979180i \(0.565068\pi\)
\(558\) 9.00251 0.381106
\(559\) 1.58695 0.0671207
\(560\) −4.86887 −0.205747
\(561\) 23.7490 1.00268
\(562\) −2.26957 −0.0957362
\(563\) 44.4878 1.87494 0.937468 0.348072i \(-0.113164\pi\)
0.937468 + 0.348072i \(0.113164\pi\)
\(564\) 12.4162 0.522815
\(565\) 15.6964 0.660351
\(566\) −7.00145 −0.294293
\(567\) 34.0514 1.43002
\(568\) −18.5151 −0.776876
\(569\) −14.0592 −0.589394 −0.294697 0.955591i \(-0.595219\pi\)
−0.294697 + 0.955591i \(0.595219\pi\)
\(570\) −59.5372 −2.49374
\(571\) 38.0971 1.59431 0.797156 0.603773i \(-0.206337\pi\)
0.797156 + 0.603773i \(0.206337\pi\)
\(572\) 13.9245 0.582213
\(573\) −31.8273 −1.32961
\(574\) 7.51931 0.313850
\(575\) −65.6214 −2.73660
\(576\) 11.3343 0.472264
\(577\) 7.80242 0.324819 0.162410 0.986723i \(-0.448073\pi\)
0.162410 + 0.986723i \(0.448073\pi\)
\(578\) −7.11900 −0.296111
\(579\) 28.2990 1.17606
\(580\) −15.1111 −0.627453
\(581\) 29.9919 1.24428
\(582\) 14.9060 0.617873
\(583\) −1.82125 −0.0754285
\(584\) 3.48436 0.144184
\(585\) −26.6651 −1.10246
\(586\) −14.7519 −0.609395
\(587\) −26.8275 −1.10729 −0.553644 0.832753i \(-0.686763\pi\)
−0.553644 + 0.832753i \(0.686763\pi\)
\(588\) −6.33230 −0.261140
\(589\) 36.7141 1.51278
\(590\) −37.9465 −1.56223
\(591\) 29.8420 1.22753
\(592\) 3.75728 0.154423
\(593\) −34.0163 −1.39688 −0.698441 0.715667i \(-0.746123\pi\)
−0.698441 + 0.715667i \(0.746123\pi\)
\(594\) 7.61856 0.312593
\(595\) −37.1478 −1.52291
\(596\) −15.4428 −0.632564
\(597\) −3.48595 −0.142670
\(598\) −19.8124 −0.810187
\(599\) 22.3457 0.913021 0.456510 0.889718i \(-0.349099\pi\)
0.456510 + 0.889718i \(0.349099\pi\)
\(600\) −68.7615 −2.80718
\(601\) 25.2733 1.03092 0.515460 0.856914i \(-0.327621\pi\)
0.515460 + 0.856914i \(0.327621\pi\)
\(602\) −1.30182 −0.0530580
\(603\) −21.8826 −0.891127
\(604\) −0.795671 −0.0323754
\(605\) −5.01912 −0.204056
\(606\) −31.8065 −1.29205
\(607\) 40.1847 1.63105 0.815524 0.578723i \(-0.196449\pi\)
0.815524 + 0.578723i \(0.196449\pi\)
\(608\) 39.8919 1.61783
\(609\) 22.8581 0.926255
\(610\) −9.93524 −0.402266
\(611\) −16.9406 −0.685344
\(612\) 6.78065 0.274092
\(613\) 44.1657 1.78384 0.891919 0.452196i \(-0.149359\pi\)
0.891919 + 0.452196i \(0.149359\pi\)
\(614\) −9.29945 −0.375295
\(615\) 23.2044 0.935692
\(616\) −31.3780 −1.26425
\(617\) −22.9118 −0.922394 −0.461197 0.887298i \(-0.652580\pi\)
−0.461197 + 0.887298i \(0.652580\pi\)
\(618\) −22.9415 −0.922841
\(619\) 26.6090 1.06951 0.534754 0.845008i \(-0.320404\pi\)
0.534754 + 0.845008i \(0.320404\pi\)
\(620\) 22.6925 0.911351
\(621\) 14.5114 0.582324
\(622\) −16.5107 −0.662020
\(623\) −35.3279 −1.41538
\(624\) −3.08576 −0.123529
\(625\) 34.9409 1.39764
\(626\) −10.2495 −0.409654
\(627\) −57.0309 −2.27760
\(628\) −12.5443 −0.500572
\(629\) 28.6667 1.14302
\(630\) 21.8741 0.871483
\(631\) −10.4893 −0.417573 −0.208787 0.977961i \(-0.566951\pi\)
−0.208787 + 0.977961i \(0.566951\pi\)
\(632\) 34.0066 1.35271
\(633\) −36.2557 −1.44104
\(634\) 4.13570 0.164250
\(635\) 55.4189 2.19923
\(636\) −1.32327 −0.0524711
\(637\) 8.63980 0.342321
\(638\) 10.8127 0.428080
\(639\) 12.3638 0.489105
\(640\) 21.7332 0.859080
\(641\) 11.4048 0.450463 0.225232 0.974305i \(-0.427686\pi\)
0.225232 + 0.974305i \(0.427686\pi\)
\(642\) −32.6581 −1.28891
\(643\) −24.0029 −0.946581 −0.473291 0.880906i \(-0.656934\pi\)
−0.473291 + 0.880906i \(0.656934\pi\)
\(644\) −21.7572 −0.857356
\(645\) −4.01737 −0.158184
\(646\) −20.6566 −0.812725
\(647\) 39.8641 1.56722 0.783609 0.621254i \(-0.213377\pi\)
0.783609 + 0.621254i \(0.213377\pi\)
\(648\) 32.1491 1.26294
\(649\) −36.3491 −1.42683
\(650\) 34.1531 1.33959
\(651\) −34.3262 −1.34535
\(652\) −18.2929 −0.716407
\(653\) −35.3964 −1.38517 −0.692584 0.721337i \(-0.743528\pi\)
−0.692584 + 0.721337i \(0.743528\pi\)
\(654\) 5.47205 0.213974
\(655\) −86.6964 −3.38751
\(656\) 1.05521 0.0411989
\(657\) −2.32675 −0.0907752
\(658\) 13.8968 0.541755
\(659\) 24.7728 0.965010 0.482505 0.875893i \(-0.339727\pi\)
0.482505 + 0.875893i \(0.339727\pi\)
\(660\) −35.2500 −1.37211
\(661\) −23.3350 −0.907626 −0.453813 0.891097i \(-0.649937\pi\)
−0.453813 + 0.891097i \(0.649937\pi\)
\(662\) 26.4460 1.02785
\(663\) −23.5432 −0.914344
\(664\) 28.3164 1.09889
\(665\) 89.2069 3.45930
\(666\) −16.8801 −0.654090
\(667\) 20.5955 0.797463
\(668\) −15.3260 −0.592981
\(669\) −13.5155 −0.522538
\(670\) 41.2036 1.59184
\(671\) −9.51700 −0.367400
\(672\) −37.2974 −1.43878
\(673\) −5.98598 −0.230742 −0.115371 0.993322i \(-0.536806\pi\)
−0.115371 + 0.993322i \(0.536806\pi\)
\(674\) 31.4534 1.21154
\(675\) −25.0152 −0.962835
\(676\) 1.07879 0.0414920
\(677\) −27.5508 −1.05886 −0.529431 0.848353i \(-0.677595\pi\)
−0.529431 + 0.848353i \(0.677595\pi\)
\(678\) −8.16055 −0.313404
\(679\) −22.3342 −0.857109
\(680\) −35.0725 −1.34497
\(681\) 42.6495 1.63433
\(682\) −16.2376 −0.621771
\(683\) 41.1849 1.57590 0.787948 0.615741i \(-0.211143\pi\)
0.787948 + 0.615741i \(0.211143\pi\)
\(684\) −16.2831 −0.622600
\(685\) −2.86939 −0.109634
\(686\) 12.8521 0.490697
\(687\) 21.8433 0.833373
\(688\) −0.182688 −0.00696490
\(689\) 1.80547 0.0687831
\(690\) 50.1552 1.90938
\(691\) −6.87493 −0.261535 −0.130767 0.991413i \(-0.541744\pi\)
−0.130767 + 0.991413i \(0.541744\pi\)
\(692\) 20.4941 0.779069
\(693\) 20.9532 0.795948
\(694\) −4.79040 −0.181841
\(695\) −8.14370 −0.308908
\(696\) 21.5811 0.818029
\(697\) 8.05086 0.304948
\(698\) 3.89803 0.147543
\(699\) −67.1558 −2.54007
\(700\) 37.5057 1.41758
\(701\) 15.9268 0.601548 0.300774 0.953695i \(-0.402755\pi\)
0.300774 + 0.953695i \(0.402755\pi\)
\(702\) −7.55256 −0.285053
\(703\) −68.8404 −2.59637
\(704\) −20.4435 −0.770493
\(705\) 42.8854 1.61516
\(706\) 20.9526 0.788563
\(707\) 47.6569 1.79232
\(708\) −26.4103 −0.992559
\(709\) −8.19363 −0.307718 −0.153859 0.988093i \(-0.549170\pi\)
−0.153859 + 0.988093i \(0.549170\pi\)
\(710\) −23.2804 −0.873696
\(711\) −22.7086 −0.851637
\(712\) −33.3544 −1.25001
\(713\) −30.9286 −1.15828
\(714\) 19.3132 0.722777
\(715\) 48.0952 1.79866
\(716\) −27.5743 −1.03050
\(717\) 44.5534 1.66388
\(718\) −5.48959 −0.204870
\(719\) 13.0692 0.487397 0.243699 0.969851i \(-0.421639\pi\)
0.243699 + 0.969851i \(0.421639\pi\)
\(720\) 3.06965 0.114399
\(721\) 34.3741 1.28016
\(722\) 32.0346 1.19220
\(723\) −49.4502 −1.83907
\(724\) −5.00447 −0.185990
\(725\) −35.5031 −1.31855
\(726\) 2.60944 0.0968455
\(727\) −18.2107 −0.675397 −0.337699 0.941254i \(-0.609648\pi\)
−0.337699 + 0.941254i \(0.609648\pi\)
\(728\) 31.1062 1.15287
\(729\) −5.78236 −0.214161
\(730\) 4.38114 0.162153
\(731\) −1.39384 −0.0515531
\(732\) −6.91480 −0.255578
\(733\) −7.19358 −0.265701 −0.132851 0.991136i \(-0.542413\pi\)
−0.132851 + 0.991136i \(0.542413\pi\)
\(734\) 14.9686 0.552502
\(735\) −21.8717 −0.806752
\(736\) −33.6057 −1.23872
\(737\) 39.4691 1.45386
\(738\) −4.74066 −0.174506
\(739\) −0.780351 −0.0287057 −0.0143528 0.999897i \(-0.504569\pi\)
−0.0143528 + 0.999897i \(0.504569\pi\)
\(740\) −42.5493 −1.56414
\(741\) 56.5369 2.07694
\(742\) −1.48108 −0.0543721
\(743\) −13.0511 −0.478797 −0.239399 0.970921i \(-0.576950\pi\)
−0.239399 + 0.970921i \(0.576950\pi\)
\(744\) −32.4086 −1.18816
\(745\) −53.3396 −1.95421
\(746\) −8.29272 −0.303618
\(747\) −18.9089 −0.691839
\(748\) −12.2301 −0.447178
\(749\) 48.9329 1.78797
\(750\) −45.8136 −1.67287
\(751\) −28.3735 −1.03536 −0.517681 0.855573i \(-0.673205\pi\)
−0.517681 + 0.855573i \(0.673205\pi\)
\(752\) 1.95019 0.0711159
\(753\) −27.8700 −1.01564
\(754\) −10.7191 −0.390366
\(755\) −2.74824 −0.100019
\(756\) −8.29396 −0.301649
\(757\) −11.4037 −0.414476 −0.207238 0.978291i \(-0.566447\pi\)
−0.207238 + 0.978291i \(0.566447\pi\)
\(758\) −29.7840 −1.08180
\(759\) 48.0438 1.74388
\(760\) 84.2234 3.05510
\(761\) 19.8526 0.719658 0.359829 0.933018i \(-0.382835\pi\)
0.359829 + 0.933018i \(0.382835\pi\)
\(762\) −28.8123 −1.04376
\(763\) −8.19899 −0.296823
\(764\) 16.3903 0.592980
\(765\) 23.4204 0.846765
\(766\) 30.2617 1.09340
\(767\) 36.0342 1.30112
\(768\) −37.2485 −1.34409
\(769\) −50.3300 −1.81495 −0.907474 0.420109i \(-0.861992\pi\)
−0.907474 + 0.420109i \(0.861992\pi\)
\(770\) −39.4538 −1.42181
\(771\) 29.2967 1.05510
\(772\) −14.5733 −0.524503
\(773\) 37.6442 1.35397 0.676984 0.735998i \(-0.263287\pi\)
0.676984 + 0.735998i \(0.263287\pi\)
\(774\) 0.820749 0.0295012
\(775\) 53.3155 1.91515
\(776\) −21.0865 −0.756962
\(777\) 64.3631 2.30901
\(778\) −14.0918 −0.505216
\(779\) −19.3334 −0.692690
\(780\) 34.9447 1.25122
\(781\) −22.3003 −0.797969
\(782\) 17.4015 0.622277
\(783\) 7.85112 0.280576
\(784\) −0.994604 −0.0355216
\(785\) −43.3280 −1.54644
\(786\) 45.0735 1.60772
\(787\) −39.1131 −1.39423 −0.697116 0.716958i \(-0.745534\pi\)
−0.697116 + 0.716958i \(0.745534\pi\)
\(788\) −15.3679 −0.547458
\(789\) 4.40997 0.156999
\(790\) 42.7589 1.52129
\(791\) 12.2273 0.434751
\(792\) 19.7827 0.702947
\(793\) 9.43456 0.335031
\(794\) −29.4700 −1.04585
\(795\) −4.57058 −0.162102
\(796\) 1.79518 0.0636284
\(797\) −41.6279 −1.47454 −0.737269 0.675600i \(-0.763885\pi\)
−0.737269 + 0.675600i \(0.763885\pi\)
\(798\) −46.3787 −1.64179
\(799\) 14.8792 0.526389
\(800\) 57.9303 2.04815
\(801\) 22.2730 0.786979
\(802\) −7.28338 −0.257185
\(803\) 4.19671 0.148099
\(804\) 28.6772 1.01137
\(805\) −75.1495 −2.64867
\(806\) 16.0970 0.566992
\(807\) −1.31644 −0.0463410
\(808\) 44.9945 1.58290
\(809\) −1.32763 −0.0466770 −0.0233385 0.999728i \(-0.507430\pi\)
−0.0233385 + 0.999728i \(0.507430\pi\)
\(810\) 40.4234 1.42033
\(811\) −47.7210 −1.67571 −0.837856 0.545892i \(-0.816191\pi\)
−0.837856 + 0.545892i \(0.816191\pi\)
\(812\) −11.7713 −0.413092
\(813\) −47.6107 −1.66978
\(814\) 30.4462 1.06714
\(815\) −63.1838 −2.21323
\(816\) 2.71027 0.0948785
\(817\) 3.34718 0.117103
\(818\) −19.6159 −0.685855
\(819\) −20.7717 −0.725823
\(820\) −11.9497 −0.417301
\(821\) 18.2322 0.636307 0.318153 0.948039i \(-0.396937\pi\)
0.318153 + 0.948039i \(0.396937\pi\)
\(822\) 1.49180 0.0520325
\(823\) 10.6665 0.371810 0.185905 0.982568i \(-0.440478\pi\)
0.185905 + 0.982568i \(0.440478\pi\)
\(824\) 32.4538 1.13058
\(825\) −82.8192 −2.88340
\(826\) −29.5598 −1.02852
\(827\) −3.91954 −0.136296 −0.0681478 0.997675i \(-0.521709\pi\)
−0.0681478 + 0.997675i \(0.521709\pi\)
\(828\) 13.7172 0.476705
\(829\) 34.1483 1.18602 0.593009 0.805196i \(-0.297940\pi\)
0.593009 + 0.805196i \(0.297940\pi\)
\(830\) 35.6043 1.23584
\(831\) −20.2305 −0.701788
\(832\) 20.2664 0.702611
\(833\) −7.58848 −0.262925
\(834\) 4.23392 0.146609
\(835\) −52.9360 −1.83193
\(836\) 29.3695 1.01577
\(837\) −11.7901 −0.407526
\(838\) −16.0900 −0.555819
\(839\) 18.3406 0.633188 0.316594 0.948561i \(-0.397461\pi\)
0.316594 + 0.948561i \(0.397461\pi\)
\(840\) −78.7456 −2.71698
\(841\) −17.8572 −0.615765
\(842\) −12.3882 −0.426927
\(843\) −5.45591 −0.187912
\(844\) 18.6708 0.642675
\(845\) 3.72615 0.128183
\(846\) −8.76147 −0.301226
\(847\) −3.90983 −0.134343
\(848\) −0.207844 −0.00713740
\(849\) −16.8310 −0.577640
\(850\) −29.9972 −1.02890
\(851\) 57.9924 1.98795
\(852\) −16.2028 −0.555100
\(853\) −24.8775 −0.851790 −0.425895 0.904773i \(-0.640041\pi\)
−0.425895 + 0.904773i \(0.640041\pi\)
\(854\) −7.73942 −0.264838
\(855\) −56.2418 −1.92343
\(856\) 46.1993 1.57906
\(857\) 7.01498 0.239627 0.119814 0.992796i \(-0.461770\pi\)
0.119814 + 0.992796i \(0.461770\pi\)
\(858\) −25.0047 −0.853647
\(859\) 29.1462 0.994456 0.497228 0.867620i \(-0.334351\pi\)
0.497228 + 0.867620i \(0.334351\pi\)
\(860\) 2.06885 0.0705471
\(861\) 18.0759 0.616027
\(862\) 29.9220 1.01915
\(863\) −21.2381 −0.722952 −0.361476 0.932381i \(-0.617727\pi\)
−0.361476 + 0.932381i \(0.617727\pi\)
\(864\) −12.8106 −0.435827
\(865\) 70.7866 2.40682
\(866\) 7.47287 0.253938
\(867\) −17.1136 −0.581210
\(868\) 17.6771 0.600001
\(869\) 40.9589 1.38944
\(870\) 27.1355 0.919978
\(871\) −39.1272 −1.32578
\(872\) −7.74095 −0.262142
\(873\) 14.0809 0.476568
\(874\) −41.7881 −1.41350
\(875\) 68.6442 2.32060
\(876\) 3.04922 0.103024
\(877\) −0.350498 −0.0118355 −0.00591773 0.999982i \(-0.501884\pi\)
−0.00591773 + 0.999982i \(0.501884\pi\)
\(878\) 24.4799 0.826157
\(879\) −35.4626 −1.19613
\(880\) −5.53666 −0.186641
\(881\) 20.5547 0.692504 0.346252 0.938142i \(-0.387454\pi\)
0.346252 + 0.938142i \(0.387454\pi\)
\(882\) 4.46839 0.150459
\(883\) 38.4958 1.29549 0.647744 0.761858i \(-0.275713\pi\)
0.647744 + 0.761858i \(0.275713\pi\)
\(884\) 12.1242 0.407780
\(885\) −91.2210 −3.06636
\(886\) −35.7937 −1.20251
\(887\) −32.5923 −1.09434 −0.547171 0.837021i \(-0.684295\pi\)
−0.547171 + 0.837021i \(0.684295\pi\)
\(888\) 60.7675 2.03922
\(889\) 43.1706 1.44789
\(890\) −41.9388 −1.40579
\(891\) 38.7217 1.29723
\(892\) 6.96012 0.233042
\(893\) −35.7311 −1.19570
\(894\) 27.7313 0.927472
\(895\) −95.2415 −3.18357
\(896\) 16.9299 0.565588
\(897\) −47.6277 −1.59024
\(898\) −4.08164 −0.136206
\(899\) −16.7333 −0.558087
\(900\) −23.6460 −0.788201
\(901\) −1.58578 −0.0528299
\(902\) 8.55062 0.284705
\(903\) −3.12948 −0.104143
\(904\) 11.5442 0.383954
\(905\) −17.2854 −0.574588
\(906\) 1.42881 0.0474692
\(907\) −18.8155 −0.624757 −0.312379 0.949958i \(-0.601126\pi\)
−0.312379 + 0.949958i \(0.601126\pi\)
\(908\) −21.9634 −0.728882
\(909\) −30.0460 −0.996562
\(910\) 39.1120 1.29655
\(911\) −6.57999 −0.218005 −0.109002 0.994041i \(-0.534766\pi\)
−0.109002 + 0.994041i \(0.534766\pi\)
\(912\) −6.50846 −0.215517
\(913\) 34.1055 1.12873
\(914\) 0.158678 0.00524861
\(915\) −23.8837 −0.789571
\(916\) −11.2487 −0.371669
\(917\) −67.5354 −2.23021
\(918\) 6.63354 0.218939
\(919\) 26.0037 0.857782 0.428891 0.903356i \(-0.358904\pi\)
0.428891 + 0.903356i \(0.358904\pi\)
\(920\) −70.9513 −2.33919
\(921\) −22.3553 −0.736632
\(922\) −30.1533 −0.993046
\(923\) 22.1072 0.727667
\(924\) −27.4593 −0.903345
\(925\) −99.9688 −3.28695
\(926\) 23.5951 0.775382
\(927\) −21.6717 −0.711791
\(928\) −18.1817 −0.596843
\(929\) 8.06188 0.264502 0.132251 0.991216i \(-0.457780\pi\)
0.132251 + 0.991216i \(0.457780\pi\)
\(930\) −40.7497 −1.33623
\(931\) 18.2230 0.597236
\(932\) 34.5836 1.13282
\(933\) −39.6908 −1.29942
\(934\) 3.31314 0.108409
\(935\) −42.2428 −1.38149
\(936\) −19.6113 −0.641017
\(937\) 23.0454 0.752862 0.376431 0.926445i \(-0.377151\pi\)
0.376431 + 0.926445i \(0.377151\pi\)
\(938\) 32.0971 1.04801
\(939\) −24.6393 −0.804072
\(940\) −22.0849 −0.720330
\(941\) 37.3296 1.21691 0.608454 0.793589i \(-0.291790\pi\)
0.608454 + 0.793589i \(0.291790\pi\)
\(942\) 22.5263 0.733945
\(943\) 16.2868 0.530370
\(944\) −4.14822 −0.135013
\(945\) −28.6473 −0.931898
\(946\) −1.48037 −0.0481309
\(947\) −29.3980 −0.955306 −0.477653 0.878549i \(-0.658512\pi\)
−0.477653 + 0.878549i \(0.658512\pi\)
\(948\) 29.7597 0.966549
\(949\) −4.16036 −0.135051
\(950\) 72.0355 2.33714
\(951\) 9.94198 0.322391
\(952\) −27.3211 −0.885480
\(953\) 17.1464 0.555426 0.277713 0.960664i \(-0.410424\pi\)
0.277713 + 0.960664i \(0.410424\pi\)
\(954\) 0.933768 0.0302318
\(955\) 56.6120 1.83192
\(956\) −22.9439 −0.742058
\(957\) 25.9932 0.840240
\(958\) −25.2241 −0.814953
\(959\) −2.23522 −0.0721790
\(960\) −51.3046 −1.65585
\(961\) −5.87140 −0.189400
\(962\) −30.1825 −0.973123
\(963\) −30.8505 −0.994142
\(964\) 25.4656 0.820191
\(965\) −50.3360 −1.62037
\(966\) 39.0703 1.25706
\(967\) −34.1166 −1.09711 −0.548557 0.836113i \(-0.684823\pi\)
−0.548557 + 0.836113i \(0.684823\pi\)
\(968\) −3.69141 −0.118646
\(969\) −49.6573 −1.59522
\(970\) −26.5136 −0.851301
\(971\) −6.71553 −0.215512 −0.107756 0.994177i \(-0.534366\pi\)
−0.107756 + 0.994177i \(0.534366\pi\)
\(972\) 20.0563 0.643307
\(973\) −6.34384 −0.203374
\(974\) −30.0131 −0.961680
\(975\) 82.1018 2.62936
\(976\) −1.08610 −0.0347651
\(977\) −12.8585 −0.411380 −0.205690 0.978617i \(-0.565944\pi\)
−0.205690 + 0.978617i \(0.565944\pi\)
\(978\) 32.8493 1.05040
\(979\) −40.1734 −1.28395
\(980\) 11.2634 0.359796
\(981\) 5.16917 0.165039
\(982\) 7.81726 0.249459
\(983\) 44.7830 1.42836 0.714178 0.699964i \(-0.246801\pi\)
0.714178 + 0.699964i \(0.246801\pi\)
\(984\) 17.0661 0.544049
\(985\) −53.0806 −1.69129
\(986\) 9.41475 0.299827
\(987\) 33.4072 1.06336
\(988\) −29.1151 −0.926274
\(989\) −2.81973 −0.0896620
\(990\) 24.8742 0.790554
\(991\) 38.3443 1.21805 0.609024 0.793152i \(-0.291561\pi\)
0.609024 + 0.793152i \(0.291561\pi\)
\(992\) 27.3036 0.866892
\(993\) 63.5746 2.01748
\(994\) −18.1351 −0.575211
\(995\) 6.20054 0.196570
\(996\) 24.7801 0.785189
\(997\) −50.7982 −1.60880 −0.804398 0.594090i \(-0.797512\pi\)
−0.804398 + 0.594090i \(0.797512\pi\)
\(998\) −26.4042 −0.835811
\(999\) 22.1070 0.699433
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 547.2.a.b.1.12 18
3.2 odd 2 4923.2.a.l.1.7 18
4.3 odd 2 8752.2.a.s.1.2 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
547.2.a.b.1.12 18 1.1 even 1 trivial
4923.2.a.l.1.7 18 3.2 odd 2
8752.2.a.s.1.2 18 4.3 odd 2