Properties

Label 8281.2.a.cp.1.10
Level $8281$
Weight $2$
Character 8281.1
Self dual yes
Analytic conductor $66.124$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 8281 = 7^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8281.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(66.1241179138\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 16 x^{10} + 88 x^{8} - 197 x^{6} + 172 x^{4} - 36 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(1.37905\) of defining polynomial
Character \(\chi\) \(=\) 8281.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.37905 q^{2} +2.88120 q^{3} -0.0982074 q^{4} +0.805948 q^{5} +3.97334 q^{6} -2.89354 q^{8} +5.30133 q^{9} +O(q^{10})\) \(q+1.37905 q^{2} +2.88120 q^{3} -0.0982074 q^{4} +0.805948 q^{5} +3.97334 q^{6} -2.89354 q^{8} +5.30133 q^{9} +1.11145 q^{10} +5.27158 q^{11} -0.282955 q^{12} +2.32210 q^{15} -3.79394 q^{16} -0.560102 q^{17} +7.31083 q^{18} +5.84469 q^{19} -0.0791501 q^{20} +7.26980 q^{22} -1.60488 q^{23} -8.33689 q^{24} -4.35045 q^{25} +6.63060 q^{27} +2.28015 q^{29} +3.20230 q^{30} +3.47590 q^{31} +0.555034 q^{32} +15.1885 q^{33} -0.772411 q^{34} -0.520630 q^{36} +1.24196 q^{37} +8.06014 q^{38} -2.33205 q^{40} +0.927702 q^{41} +4.44711 q^{43} -0.517708 q^{44} +4.27260 q^{45} -2.21321 q^{46} -3.84418 q^{47} -10.9311 q^{48} -5.99951 q^{50} -1.61377 q^{51} +5.45454 q^{53} +9.14396 q^{54} +4.24862 q^{55} +16.8397 q^{57} +3.14446 q^{58} -10.9940 q^{59} -0.228047 q^{60} +7.30215 q^{61} +4.79346 q^{62} +8.35330 q^{64} +20.9458 q^{66} -7.34556 q^{67} +0.0550061 q^{68} -4.62397 q^{69} +9.31460 q^{71} -15.3396 q^{72} -5.00146 q^{73} +1.71273 q^{74} -12.5345 q^{75} -0.573991 q^{76} +11.3687 q^{79} -3.05772 q^{80} +3.20012 q^{81} +1.27935 q^{82} -5.81962 q^{83} -0.451413 q^{85} +6.13281 q^{86} +6.56959 q^{87} -15.2535 q^{88} +5.00946 q^{89} +5.89215 q^{90} +0.157611 q^{92} +10.0148 q^{93} -5.30133 q^{94} +4.71051 q^{95} +1.59917 q^{96} -10.6483 q^{97} +27.9464 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 6q^{3} + 8q^{4} + 2q^{9} + O(q^{10}) \) \( 12q + 6q^{3} + 8q^{4} + 2q^{9} + 24q^{10} - 2q^{12} + 16q^{16} + 34q^{17} + 30q^{22} + 6q^{23} - 10q^{25} + 12q^{27} + 2q^{29} + 22q^{30} - 26q^{36} + 38q^{38} + 2q^{40} + 22q^{43} - 38q^{48} + 8q^{51} + 16q^{53} + 30q^{55} - 10q^{61} + 82q^{62} - 2q^{64} + 68q^{66} + 22q^{68} + 14q^{69} + 66q^{74} + 2q^{75} + 70q^{79} - 28q^{81} + 10q^{82} - 20q^{87} - 28q^{88} - 66q^{92} - 2q^{94} + 4q^{95} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.37905 0.975139 0.487570 0.873084i \(-0.337884\pi\)
0.487570 + 0.873084i \(0.337884\pi\)
\(3\) 2.88120 1.66346 0.831732 0.555178i \(-0.187350\pi\)
0.831732 + 0.555178i \(0.187350\pi\)
\(4\) −0.0982074 −0.0491037
\(5\) 0.805948 0.360431 0.180216 0.983627i \(-0.442320\pi\)
0.180216 + 0.983627i \(0.442320\pi\)
\(6\) 3.97334 1.62211
\(7\) 0 0
\(8\) −2.89354 −1.02302
\(9\) 5.30133 1.76711
\(10\) 1.11145 0.351470
\(11\) 5.27158 1.58944 0.794720 0.606976i \(-0.207618\pi\)
0.794720 + 0.606976i \(0.207618\pi\)
\(12\) −0.282955 −0.0816822
\(13\) 0 0
\(14\) 0 0
\(15\) 2.32210 0.599564
\(16\) −3.79394 −0.948485
\(17\) −0.560102 −0.135845 −0.0679223 0.997691i \(-0.521637\pi\)
−0.0679223 + 0.997691i \(0.521637\pi\)
\(18\) 7.31083 1.72318
\(19\) 5.84469 1.34086 0.670431 0.741972i \(-0.266109\pi\)
0.670431 + 0.741972i \(0.266109\pi\)
\(20\) −0.0791501 −0.0176985
\(21\) 0 0
\(22\) 7.26980 1.54993
\(23\) −1.60488 −0.334640 −0.167320 0.985903i \(-0.553511\pi\)
−0.167320 + 0.985903i \(0.553511\pi\)
\(24\) −8.33689 −1.70176
\(25\) −4.35045 −0.870089
\(26\) 0 0
\(27\) 6.63060 1.27606
\(28\) 0 0
\(29\) 2.28015 0.423414 0.211707 0.977333i \(-0.432098\pi\)
0.211707 + 0.977333i \(0.432098\pi\)
\(30\) 3.20230 0.584658
\(31\) 3.47590 0.624290 0.312145 0.950034i \(-0.398952\pi\)
0.312145 + 0.950034i \(0.398952\pi\)
\(32\) 0.555034 0.0981171
\(33\) 15.1885 2.64398
\(34\) −0.772411 −0.132467
\(35\) 0 0
\(36\) −0.520630 −0.0867716
\(37\) 1.24196 0.204177 0.102088 0.994775i \(-0.467448\pi\)
0.102088 + 0.994775i \(0.467448\pi\)
\(38\) 8.06014 1.30753
\(39\) 0 0
\(40\) −2.33205 −0.368729
\(41\) 0.927702 0.144883 0.0724413 0.997373i \(-0.476921\pi\)
0.0724413 + 0.997373i \(0.476921\pi\)
\(42\) 0 0
\(43\) 4.44711 0.678179 0.339089 0.940754i \(-0.389881\pi\)
0.339089 + 0.940754i \(0.389881\pi\)
\(44\) −0.517708 −0.0780474
\(45\) 4.27260 0.636921
\(46\) −2.21321 −0.326320
\(47\) −3.84418 −0.560731 −0.280365 0.959893i \(-0.590456\pi\)
−0.280365 + 0.959893i \(0.590456\pi\)
\(48\) −10.9311 −1.57777
\(49\) 0 0
\(50\) −5.99951 −0.848458
\(51\) −1.61377 −0.225973
\(52\) 0 0
\(53\) 5.45454 0.749239 0.374620 0.927179i \(-0.377773\pi\)
0.374620 + 0.927179i \(0.377773\pi\)
\(54\) 9.14396 1.24434
\(55\) 4.24862 0.572884
\(56\) 0 0
\(57\) 16.8397 2.23048
\(58\) 3.14446 0.412887
\(59\) −10.9940 −1.43129 −0.715646 0.698463i \(-0.753868\pi\)
−0.715646 + 0.698463i \(0.753868\pi\)
\(60\) −0.228047 −0.0294408
\(61\) 7.30215 0.934944 0.467472 0.884008i \(-0.345165\pi\)
0.467472 + 0.884008i \(0.345165\pi\)
\(62\) 4.79346 0.608770
\(63\) 0 0
\(64\) 8.35330 1.04416
\(65\) 0 0
\(66\) 20.9458 2.57824
\(67\) −7.34556 −0.897403 −0.448701 0.893682i \(-0.648113\pi\)
−0.448701 + 0.893682i \(0.648113\pi\)
\(68\) 0.0550061 0.00667047
\(69\) −4.62397 −0.556661
\(70\) 0 0
\(71\) 9.31460 1.10544 0.552720 0.833367i \(-0.313590\pi\)
0.552720 + 0.833367i \(0.313590\pi\)
\(72\) −15.3396 −1.80779
\(73\) −5.00146 −0.585376 −0.292688 0.956208i \(-0.594550\pi\)
−0.292688 + 0.956208i \(0.594550\pi\)
\(74\) 1.71273 0.199101
\(75\) −12.5345 −1.44736
\(76\) −0.573991 −0.0658413
\(77\) 0 0
\(78\) 0 0
\(79\) 11.3687 1.27908 0.639542 0.768756i \(-0.279124\pi\)
0.639542 + 0.768756i \(0.279124\pi\)
\(80\) −3.05772 −0.341863
\(81\) 3.20012 0.355568
\(82\) 1.27935 0.141281
\(83\) −5.81962 −0.638786 −0.319393 0.947622i \(-0.603479\pi\)
−0.319393 + 0.947622i \(0.603479\pi\)
\(84\) 0 0
\(85\) −0.451413 −0.0489626
\(86\) 6.13281 0.661318
\(87\) 6.56959 0.704334
\(88\) −15.2535 −1.62603
\(89\) 5.00946 0.531001 0.265501 0.964111i \(-0.414463\pi\)
0.265501 + 0.964111i \(0.414463\pi\)
\(90\) 5.89215 0.621087
\(91\) 0 0
\(92\) 0.157611 0.0164320
\(93\) 10.0148 1.03848
\(94\) −5.30133 −0.546791
\(95\) 4.71051 0.483289
\(96\) 1.59917 0.163214
\(97\) −10.6483 −1.08117 −0.540586 0.841289i \(-0.681798\pi\)
−0.540586 + 0.841289i \(0.681798\pi\)
\(98\) 0 0
\(99\) 27.9464 2.80872
\(100\) 0.427246 0.0427246
\(101\) 3.91554 0.389611 0.194805 0.980842i \(-0.437592\pi\)
0.194805 + 0.980842i \(0.437592\pi\)
\(102\) −2.22547 −0.220355
\(103\) 8.45379 0.832977 0.416488 0.909141i \(-0.363261\pi\)
0.416488 + 0.909141i \(0.363261\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 7.52212 0.730613
\(107\) −9.67522 −0.935339 −0.467670 0.883903i \(-0.654906\pi\)
−0.467670 + 0.883903i \(0.654906\pi\)
\(108\) −0.651174 −0.0626592
\(109\) −14.5638 −1.39496 −0.697478 0.716606i \(-0.745695\pi\)
−0.697478 + 0.716606i \(0.745695\pi\)
\(110\) 5.85908 0.558641
\(111\) 3.57833 0.339640
\(112\) 0 0
\(113\) 19.5114 1.83548 0.917741 0.397180i \(-0.130011\pi\)
0.917741 + 0.397180i \(0.130011\pi\)
\(114\) 23.2229 2.17502
\(115\) −1.29345 −0.120615
\(116\) −0.223928 −0.0207912
\(117\) 0 0
\(118\) −15.1613 −1.39571
\(119\) 0 0
\(120\) −6.71910 −0.613367
\(121\) 16.7895 1.52632
\(122\) 10.0701 0.911701
\(123\) 2.67290 0.241007
\(124\) −0.341359 −0.0306550
\(125\) −7.53598 −0.674038
\(126\) 0 0
\(127\) −1.91731 −0.170134 −0.0850670 0.996375i \(-0.527110\pi\)
−0.0850670 + 0.996375i \(0.527110\pi\)
\(128\) 10.4096 0.920087
\(129\) 12.8130 1.12813
\(130\) 0 0
\(131\) 15.5816 1.36137 0.680684 0.732577i \(-0.261683\pi\)
0.680684 + 0.732577i \(0.261683\pi\)
\(132\) −1.49162 −0.129829
\(133\) 0 0
\(134\) −10.1299 −0.875093
\(135\) 5.34392 0.459932
\(136\) 1.62068 0.138972
\(137\) −7.85105 −0.670761 −0.335380 0.942083i \(-0.608865\pi\)
−0.335380 + 0.942083i \(0.608865\pi\)
\(138\) −6.37671 −0.542822
\(139\) 9.92481 0.841812 0.420906 0.907104i \(-0.361712\pi\)
0.420906 + 0.907104i \(0.361712\pi\)
\(140\) 0 0
\(141\) −11.0759 −0.932755
\(142\) 12.8453 1.07796
\(143\) 0 0
\(144\) −20.1129 −1.67608
\(145\) 1.83769 0.152612
\(146\) −6.89728 −0.570823
\(147\) 0 0
\(148\) −0.121969 −0.0100258
\(149\) 7.91925 0.648770 0.324385 0.945925i \(-0.394843\pi\)
0.324385 + 0.945925i \(0.394843\pi\)
\(150\) −17.2858 −1.41138
\(151\) −1.50116 −0.122163 −0.0610815 0.998133i \(-0.519455\pi\)
−0.0610815 + 0.998133i \(0.519455\pi\)
\(152\) −16.9118 −1.37173
\(153\) −2.96928 −0.240052
\(154\) 0 0
\(155\) 2.80140 0.225014
\(156\) 0 0
\(157\) 3.85692 0.307816 0.153908 0.988085i \(-0.450814\pi\)
0.153908 + 0.988085i \(0.450814\pi\)
\(158\) 15.6781 1.24728
\(159\) 15.7156 1.24633
\(160\) 0.447329 0.0353644
\(161\) 0 0
\(162\) 4.41314 0.346729
\(163\) −14.3608 −1.12483 −0.562414 0.826856i \(-0.690127\pi\)
−0.562414 + 0.826856i \(0.690127\pi\)
\(164\) −0.0911072 −0.00711427
\(165\) 12.2411 0.952971
\(166\) −8.02557 −0.622905
\(167\) 4.52138 0.349875 0.174937 0.984580i \(-0.444028\pi\)
0.174937 + 0.984580i \(0.444028\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −0.622523 −0.0477454
\(171\) 30.9846 2.36945
\(172\) −0.436739 −0.0333011
\(173\) −19.5179 −1.48392 −0.741960 0.670444i \(-0.766104\pi\)
−0.741960 + 0.670444i \(0.766104\pi\)
\(174\) 9.05982 0.686823
\(175\) 0 0
\(176\) −20.0001 −1.50756
\(177\) −31.6759 −2.38090
\(178\) 6.90832 0.517800
\(179\) 20.8196 1.55613 0.778065 0.628183i \(-0.216201\pi\)
0.778065 + 0.628183i \(0.216201\pi\)
\(180\) −0.419601 −0.0312752
\(181\) −16.5522 −1.23031 −0.615157 0.788405i \(-0.710907\pi\)
−0.615157 + 0.788405i \(0.710907\pi\)
\(182\) 0 0
\(183\) 21.0390 1.55525
\(184\) 4.64378 0.342344
\(185\) 1.00095 0.0735916
\(186\) 13.8109 1.01267
\(187\) −2.95262 −0.215917
\(188\) 0.377527 0.0275340
\(189\) 0 0
\(190\) 6.49606 0.471274
\(191\) −4.25008 −0.307525 −0.153762 0.988108i \(-0.549139\pi\)
−0.153762 + 0.988108i \(0.549139\pi\)
\(192\) 24.0676 1.73693
\(193\) −11.5972 −0.834787 −0.417393 0.908726i \(-0.637056\pi\)
−0.417393 + 0.908726i \(0.637056\pi\)
\(194\) −14.6846 −1.05429
\(195\) 0 0
\(196\) 0 0
\(197\) −14.4213 −1.02748 −0.513738 0.857947i \(-0.671740\pi\)
−0.513738 + 0.857947i \(0.671740\pi\)
\(198\) 38.5396 2.73889
\(199\) 7.05924 0.500416 0.250208 0.968192i \(-0.419501\pi\)
0.250208 + 0.968192i \(0.419501\pi\)
\(200\) 12.5882 0.890121
\(201\) −21.1640 −1.49280
\(202\) 5.39974 0.379925
\(203\) 0 0
\(204\) 0.158484 0.0110961
\(205\) 0.747680 0.0522202
\(206\) 11.6582 0.812268
\(207\) −8.50798 −0.591345
\(208\) 0 0
\(209\) 30.8107 2.13122
\(210\) 0 0
\(211\) −26.4226 −1.81901 −0.909505 0.415693i \(-0.863539\pi\)
−0.909505 + 0.415693i \(0.863539\pi\)
\(212\) −0.535677 −0.0367904
\(213\) 26.8372 1.83886
\(214\) −13.3427 −0.912086
\(215\) 3.58414 0.244437
\(216\) −19.1859 −1.30544
\(217\) 0 0
\(218\) −20.0842 −1.36028
\(219\) −14.4102 −0.973752
\(220\) −0.417246 −0.0281307
\(221\) 0 0
\(222\) 4.93472 0.331197
\(223\) 23.0005 1.54023 0.770115 0.637905i \(-0.220199\pi\)
0.770115 + 0.637905i \(0.220199\pi\)
\(224\) 0 0
\(225\) −23.0632 −1.53754
\(226\) 26.9073 1.78985
\(227\) −0.453367 −0.0300911 −0.0150455 0.999887i \(-0.504789\pi\)
−0.0150455 + 0.999887i \(0.504789\pi\)
\(228\) −1.65379 −0.109525
\(229\) 17.3335 1.14543 0.572714 0.819755i \(-0.305890\pi\)
0.572714 + 0.819755i \(0.305890\pi\)
\(230\) −1.78373 −0.117616
\(231\) 0 0
\(232\) −6.59772 −0.433162
\(233\) 7.81511 0.511985 0.255992 0.966679i \(-0.417598\pi\)
0.255992 + 0.966679i \(0.417598\pi\)
\(234\) 0 0
\(235\) −3.09821 −0.202105
\(236\) 1.07969 0.0702818
\(237\) 32.7557 2.12771
\(238\) 0 0
\(239\) −13.5314 −0.875276 −0.437638 0.899151i \(-0.644185\pi\)
−0.437638 + 0.899151i \(0.644185\pi\)
\(240\) −8.80991 −0.568677
\(241\) 22.5592 1.45317 0.726583 0.687078i \(-0.241107\pi\)
0.726583 + 0.687078i \(0.241107\pi\)
\(242\) 23.1537 1.48838
\(243\) −10.6716 −0.684585
\(244\) −0.717125 −0.0459092
\(245\) 0 0
\(246\) 3.68607 0.235015
\(247\) 0 0
\(248\) −10.0577 −0.638663
\(249\) −16.7675 −1.06260
\(250\) −10.3925 −0.657281
\(251\) −6.73236 −0.424943 −0.212471 0.977167i \(-0.568151\pi\)
−0.212471 + 0.977167i \(0.568151\pi\)
\(252\) 0 0
\(253\) −8.46023 −0.531890
\(254\) −2.64408 −0.165904
\(255\) −1.30061 −0.0814475
\(256\) −2.35120 −0.146950
\(257\) 16.5381 1.03162 0.515811 0.856703i \(-0.327491\pi\)
0.515811 + 0.856703i \(0.327491\pi\)
\(258\) 17.6699 1.10008
\(259\) 0 0
\(260\) 0 0
\(261\) 12.0878 0.748219
\(262\) 21.4878 1.32752
\(263\) −10.0227 −0.618028 −0.309014 0.951057i \(-0.599999\pi\)
−0.309014 + 0.951057i \(0.599999\pi\)
\(264\) −43.9485 −2.70485
\(265\) 4.39608 0.270049
\(266\) 0 0
\(267\) 14.4333 0.883302
\(268\) 0.721388 0.0440658
\(269\) −15.7230 −0.958647 −0.479323 0.877638i \(-0.659118\pi\)
−0.479323 + 0.877638i \(0.659118\pi\)
\(270\) 7.36956 0.448497
\(271\) −5.21618 −0.316860 −0.158430 0.987370i \(-0.550643\pi\)
−0.158430 + 0.987370i \(0.550643\pi\)
\(272\) 2.12499 0.128847
\(273\) 0 0
\(274\) −10.8270 −0.654085
\(275\) −22.9337 −1.38296
\(276\) 0.454108 0.0273341
\(277\) −19.2724 −1.15797 −0.578983 0.815340i \(-0.696550\pi\)
−0.578983 + 0.815340i \(0.696550\pi\)
\(278\) 13.6869 0.820884
\(279\) 18.4269 1.10319
\(280\) 0 0
\(281\) 2.14283 0.127831 0.0639153 0.997955i \(-0.479641\pi\)
0.0639153 + 0.997955i \(0.479641\pi\)
\(282\) −15.2742 −0.909566
\(283\) −15.7502 −0.936255 −0.468127 0.883661i \(-0.655071\pi\)
−0.468127 + 0.883661i \(0.655071\pi\)
\(284\) −0.914762 −0.0542812
\(285\) 13.5719 0.803933
\(286\) 0 0
\(287\) 0 0
\(288\) 2.94242 0.173384
\(289\) −16.6863 −0.981546
\(290\) 2.53427 0.148817
\(291\) −30.6800 −1.79849
\(292\) 0.491180 0.0287441
\(293\) −23.1487 −1.35236 −0.676182 0.736735i \(-0.736367\pi\)
−0.676182 + 0.736735i \(0.736367\pi\)
\(294\) 0 0
\(295\) −8.86057 −0.515882
\(296\) −3.59366 −0.208877
\(297\) 34.9537 2.02822
\(298\) 10.9211 0.632641
\(299\) 0 0
\(300\) 1.23098 0.0710708
\(301\) 0 0
\(302\) −2.07019 −0.119126
\(303\) 11.2815 0.648103
\(304\) −22.1744 −1.27179
\(305\) 5.88515 0.336983
\(306\) −4.09481 −0.234085
\(307\) −4.23590 −0.241756 −0.120878 0.992667i \(-0.538571\pi\)
−0.120878 + 0.992667i \(0.538571\pi\)
\(308\) 0 0
\(309\) 24.3571 1.38563
\(310\) 3.86328 0.219420
\(311\) 27.2501 1.54521 0.772606 0.634885i \(-0.218953\pi\)
0.772606 + 0.634885i \(0.218953\pi\)
\(312\) 0 0
\(313\) 2.69697 0.152442 0.0762209 0.997091i \(-0.475715\pi\)
0.0762209 + 0.997091i \(0.475715\pi\)
\(314\) 5.31891 0.300163
\(315\) 0 0
\(316\) −1.11649 −0.0628077
\(317\) −24.0705 −1.35193 −0.675966 0.736933i \(-0.736273\pi\)
−0.675966 + 0.736933i \(0.736273\pi\)
\(318\) 21.6727 1.21535
\(319\) 12.0200 0.672991
\(320\) 6.73233 0.376349
\(321\) −27.8763 −1.55590
\(322\) 0 0
\(323\) −3.27362 −0.182149
\(324\) −0.314275 −0.0174597
\(325\) 0 0
\(326\) −19.8044 −1.09686
\(327\) −41.9612 −2.32046
\(328\) −2.68434 −0.148218
\(329\) 0 0
\(330\) 16.8812 0.929279
\(331\) 0.619723 0.0340631 0.0170315 0.999855i \(-0.494578\pi\)
0.0170315 + 0.999855i \(0.494578\pi\)
\(332\) 0.571530 0.0313668
\(333\) 6.58403 0.360803
\(334\) 6.23523 0.341177
\(335\) −5.92014 −0.323452
\(336\) 0 0
\(337\) 5.72118 0.311652 0.155826 0.987784i \(-0.450196\pi\)
0.155826 + 0.987784i \(0.450196\pi\)
\(338\) 0 0
\(339\) 56.2164 3.05326
\(340\) 0.0443321 0.00240425
\(341\) 18.3235 0.992272
\(342\) 42.7295 2.31055
\(343\) 0 0
\(344\) −12.8679 −0.693792
\(345\) −3.72668 −0.200638
\(346\) −26.9163 −1.44703
\(347\) −1.86486 −0.100111 −0.0500554 0.998746i \(-0.515940\pi\)
−0.0500554 + 0.998746i \(0.515940\pi\)
\(348\) −0.645182 −0.0345854
\(349\) 22.3172 1.19461 0.597307 0.802012i \(-0.296237\pi\)
0.597307 + 0.802012i \(0.296237\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.92591 0.155951
\(353\) −2.33199 −0.124119 −0.0620597 0.998072i \(-0.519767\pi\)
−0.0620597 + 0.998072i \(0.519767\pi\)
\(354\) −43.6827 −2.32171
\(355\) 7.50708 0.398435
\(356\) −0.491966 −0.0260741
\(357\) 0 0
\(358\) 28.7114 1.51744
\(359\) −3.27105 −0.172639 −0.0863197 0.996267i \(-0.527511\pi\)
−0.0863197 + 0.996267i \(0.527511\pi\)
\(360\) −12.3629 −0.651585
\(361\) 15.1603 0.797913
\(362\) −22.8264 −1.19973
\(363\) 48.3741 2.53898
\(364\) 0 0
\(365\) −4.03092 −0.210988
\(366\) 29.0139 1.51658
\(367\) 4.15290 0.216780 0.108390 0.994108i \(-0.465431\pi\)
0.108390 + 0.994108i \(0.465431\pi\)
\(368\) 6.08880 0.317401
\(369\) 4.91805 0.256024
\(370\) 1.38037 0.0717620
\(371\) 0 0
\(372\) −0.983525 −0.0509934
\(373\) −11.1089 −0.575198 −0.287599 0.957751i \(-0.592857\pi\)
−0.287599 + 0.957751i \(0.592857\pi\)
\(374\) −4.07183 −0.210549
\(375\) −21.7127 −1.12124
\(376\) 11.1233 0.573640
\(377\) 0 0
\(378\) 0 0
\(379\) −4.64030 −0.238356 −0.119178 0.992873i \(-0.538026\pi\)
−0.119178 + 0.992873i \(0.538026\pi\)
\(380\) −0.462607 −0.0237312
\(381\) −5.52416 −0.283012
\(382\) −5.86109 −0.299880
\(383\) −3.66933 −0.187494 −0.0937469 0.995596i \(-0.529884\pi\)
−0.0937469 + 0.995596i \(0.529884\pi\)
\(384\) 29.9922 1.53053
\(385\) 0 0
\(386\) −15.9932 −0.814033
\(387\) 23.5756 1.19842
\(388\) 1.04574 0.0530896
\(389\) 16.8831 0.856008 0.428004 0.903777i \(-0.359217\pi\)
0.428004 + 0.903777i \(0.359217\pi\)
\(390\) 0 0
\(391\) 0.898894 0.0454590
\(392\) 0 0
\(393\) 44.8937 2.26459
\(394\) −19.8878 −1.00193
\(395\) 9.16262 0.461021
\(396\) −2.74454 −0.137918
\(397\) 16.7086 0.838578 0.419289 0.907853i \(-0.362279\pi\)
0.419289 + 0.907853i \(0.362279\pi\)
\(398\) 9.73508 0.487976
\(399\) 0 0
\(400\) 16.5053 0.825267
\(401\) 25.3134 1.26409 0.632046 0.774931i \(-0.282215\pi\)
0.632046 + 0.774931i \(0.282215\pi\)
\(402\) −29.1864 −1.45568
\(403\) 0 0
\(404\) −0.384535 −0.0191313
\(405\) 2.57913 0.128158
\(406\) 0 0
\(407\) 6.54708 0.324527
\(408\) 4.66950 0.231175
\(409\) 5.73343 0.283500 0.141750 0.989903i \(-0.454727\pi\)
0.141750 + 0.989903i \(0.454727\pi\)
\(410\) 1.03109 0.0509220
\(411\) −22.6205 −1.11579
\(412\) −0.830225 −0.0409022
\(413\) 0 0
\(414\) −11.7330 −0.576644
\(415\) −4.69031 −0.230238
\(416\) 0 0
\(417\) 28.5954 1.40032
\(418\) 42.4897 2.07824
\(419\) −34.3458 −1.67790 −0.838950 0.544208i \(-0.816830\pi\)
−0.838950 + 0.544208i \(0.816830\pi\)
\(420\) 0 0
\(421\) −2.94167 −0.143368 −0.0716842 0.997427i \(-0.522837\pi\)
−0.0716842 + 0.997427i \(0.522837\pi\)
\(422\) −36.4383 −1.77379
\(423\) −20.3793 −0.990873
\(424\) −15.7830 −0.766488
\(425\) 2.43669 0.118197
\(426\) 37.0100 1.79314
\(427\) 0 0
\(428\) 0.950178 0.0459286
\(429\) 0 0
\(430\) 4.94273 0.238360
\(431\) −39.6955 −1.91207 −0.956033 0.293258i \(-0.905261\pi\)
−0.956033 + 0.293258i \(0.905261\pi\)
\(432\) −25.1561 −1.21032
\(433\) −9.83653 −0.472714 −0.236357 0.971666i \(-0.575953\pi\)
−0.236357 + 0.971666i \(0.575953\pi\)
\(434\) 0 0
\(435\) 5.29475 0.253864
\(436\) 1.43027 0.0684975
\(437\) −9.37999 −0.448706
\(438\) −19.8725 −0.949544
\(439\) 28.5465 1.36245 0.681226 0.732073i \(-0.261447\pi\)
0.681226 + 0.732073i \(0.261447\pi\)
\(440\) −12.2936 −0.586073
\(441\) 0 0
\(442\) 0 0
\(443\) 3.33901 0.158641 0.0793207 0.996849i \(-0.474725\pi\)
0.0793207 + 0.996849i \(0.474725\pi\)
\(444\) −0.351419 −0.0166776
\(445\) 4.03736 0.191389
\(446\) 31.7190 1.50194
\(447\) 22.8170 1.07921
\(448\) 0 0
\(449\) 18.1851 0.858206 0.429103 0.903256i \(-0.358830\pi\)
0.429103 + 0.903256i \(0.358830\pi\)
\(450\) −31.8054 −1.49932
\(451\) 4.89045 0.230282
\(452\) −1.91617 −0.0901289
\(453\) −4.32516 −0.203214
\(454\) −0.625219 −0.0293430
\(455\) 0 0
\(456\) −48.7265 −2.28183
\(457\) −8.72932 −0.408341 −0.204170 0.978935i \(-0.565450\pi\)
−0.204170 + 0.978935i \(0.565450\pi\)
\(458\) 23.9038 1.11695
\(459\) −3.71381 −0.173346
\(460\) 0.127026 0.00592262
\(461\) 2.27124 0.105782 0.0528910 0.998600i \(-0.483156\pi\)
0.0528910 + 0.998600i \(0.483156\pi\)
\(462\) 0 0
\(463\) 5.48326 0.254829 0.127414 0.991850i \(-0.459332\pi\)
0.127414 + 0.991850i \(0.459332\pi\)
\(464\) −8.65077 −0.401602
\(465\) 8.07139 0.374302
\(466\) 10.7775 0.499257
\(467\) 18.8819 0.873750 0.436875 0.899522i \(-0.356085\pi\)
0.436875 + 0.899522i \(0.356085\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −4.27260 −0.197080
\(471\) 11.1126 0.512040
\(472\) 31.8115 1.46424
\(473\) 23.4433 1.07792
\(474\) 45.1718 2.07481
\(475\) −25.4270 −1.16667
\(476\) 0 0
\(477\) 28.9163 1.32399
\(478\) −18.6606 −0.853516
\(479\) −33.1354 −1.51399 −0.756997 0.653418i \(-0.773334\pi\)
−0.756997 + 0.653418i \(0.773334\pi\)
\(480\) 1.28884 0.0588275
\(481\) 0 0
\(482\) 31.1104 1.41704
\(483\) 0 0
\(484\) −1.64886 −0.0749480
\(485\) −8.58199 −0.389688
\(486\) −14.7168 −0.667565
\(487\) −15.9563 −0.723048 −0.361524 0.932363i \(-0.617743\pi\)
−0.361524 + 0.932363i \(0.617743\pi\)
\(488\) −21.1291 −0.956469
\(489\) −41.3765 −1.87111
\(490\) 0 0
\(491\) −31.6928 −1.43028 −0.715138 0.698983i \(-0.753636\pi\)
−0.715138 + 0.698983i \(0.753636\pi\)
\(492\) −0.262498 −0.0118343
\(493\) −1.27712 −0.0575185
\(494\) 0 0
\(495\) 22.5233 1.01235
\(496\) −13.1874 −0.592130
\(497\) 0 0
\(498\) −23.1233 −1.03618
\(499\) −24.2184 −1.08417 −0.542083 0.840325i \(-0.682364\pi\)
−0.542083 + 0.840325i \(0.682364\pi\)
\(500\) 0.740089 0.0330978
\(501\) 13.0270 0.582004
\(502\) −9.28429 −0.414378
\(503\) 0.854498 0.0381002 0.0190501 0.999819i \(-0.493936\pi\)
0.0190501 + 0.999819i \(0.493936\pi\)
\(504\) 0 0
\(505\) 3.15572 0.140428
\(506\) −11.6671 −0.518667
\(507\) 0 0
\(508\) 0.188294 0.00835420
\(509\) 1.30000 0.0576215 0.0288108 0.999585i \(-0.490828\pi\)
0.0288108 + 0.999585i \(0.490828\pi\)
\(510\) −1.79362 −0.0794227
\(511\) 0 0
\(512\) −24.0616 −1.06338
\(513\) 38.7538 1.71102
\(514\) 22.8070 1.00597
\(515\) 6.81332 0.300231
\(516\) −1.25833 −0.0553951
\(517\) −20.2649 −0.891248
\(518\) 0 0
\(519\) −56.2351 −2.46845
\(520\) 0 0
\(521\) −25.0455 −1.09726 −0.548632 0.836064i \(-0.684851\pi\)
−0.548632 + 0.836064i \(0.684851\pi\)
\(522\) 16.6698 0.729618
\(523\) 12.8239 0.560752 0.280376 0.959890i \(-0.409541\pi\)
0.280376 + 0.959890i \(0.409541\pi\)
\(524\) −1.53022 −0.0668482
\(525\) 0 0
\(526\) −13.8219 −0.602664
\(527\) −1.94686 −0.0848065
\(528\) −57.6242 −2.50777
\(529\) −20.4244 −0.888016
\(530\) 6.06244 0.263335
\(531\) −58.2827 −2.52925
\(532\) 0 0
\(533\) 0 0
\(534\) 19.9043 0.861342
\(535\) −7.79773 −0.337125
\(536\) 21.2547 0.918063
\(537\) 59.9855 2.58857
\(538\) −21.6828 −0.934814
\(539\) 0 0
\(540\) −0.524813 −0.0225843
\(541\) −28.7449 −1.23584 −0.617920 0.786241i \(-0.712025\pi\)
−0.617920 + 0.786241i \(0.712025\pi\)
\(542\) −7.19340 −0.308983
\(543\) −47.6902 −2.04658
\(544\) −0.310876 −0.0133287
\(545\) −11.7376 −0.502785
\(546\) 0 0
\(547\) −8.88085 −0.379718 −0.189859 0.981811i \(-0.560803\pi\)
−0.189859 + 0.981811i \(0.560803\pi\)
\(548\) 0.771031 0.0329368
\(549\) 38.7111 1.65215
\(550\) −31.6269 −1.34857
\(551\) 13.3268 0.567740
\(552\) 13.3797 0.569476
\(553\) 0 0
\(554\) −26.5777 −1.12918
\(555\) 2.88395 0.122417
\(556\) −0.974690 −0.0413361
\(557\) 38.7273 1.64093 0.820465 0.571696i \(-0.193714\pi\)
0.820465 + 0.571696i \(0.193714\pi\)
\(558\) 25.4117 1.07576
\(559\) 0 0
\(560\) 0 0
\(561\) −8.50710 −0.359170
\(562\) 2.95508 0.124653
\(563\) 6.90882 0.291172 0.145586 0.989346i \(-0.453493\pi\)
0.145586 + 0.989346i \(0.453493\pi\)
\(564\) 1.08773 0.0458017
\(565\) 15.7252 0.661565
\(566\) −21.7205 −0.912979
\(567\) 0 0
\(568\) −26.9522 −1.13089
\(569\) −2.83745 −0.118952 −0.0594759 0.998230i \(-0.518943\pi\)
−0.0594759 + 0.998230i \(0.518943\pi\)
\(570\) 18.7165 0.783946
\(571\) −46.6724 −1.95318 −0.976589 0.215113i \(-0.930988\pi\)
−0.976589 + 0.215113i \(0.930988\pi\)
\(572\) 0 0
\(573\) −12.2453 −0.511557
\(574\) 0 0
\(575\) 6.98193 0.291167
\(576\) 44.2836 1.84515
\(577\) −11.4088 −0.474955 −0.237478 0.971393i \(-0.576321\pi\)
−0.237478 + 0.971393i \(0.576321\pi\)
\(578\) −23.0113 −0.957144
\(579\) −33.4140 −1.38864
\(580\) −0.180474 −0.00749379
\(581\) 0 0
\(582\) −42.3094 −1.75378
\(583\) 28.7541 1.19087
\(584\) 14.4719 0.598853
\(585\) 0 0
\(586\) −31.9234 −1.31874
\(587\) −46.4410 −1.91683 −0.958413 0.285384i \(-0.907879\pi\)
−0.958413 + 0.285384i \(0.907879\pi\)
\(588\) 0 0
\(589\) 20.3156 0.837088
\(590\) −12.2192 −0.503057
\(591\) −41.5508 −1.70917
\(592\) −4.71191 −0.193658
\(593\) 20.2606 0.832002 0.416001 0.909364i \(-0.363431\pi\)
0.416001 + 0.909364i \(0.363431\pi\)
\(594\) 48.2031 1.97780
\(595\) 0 0
\(596\) −0.777728 −0.0318570
\(597\) 20.3391 0.832424
\(598\) 0 0
\(599\) −38.9876 −1.59299 −0.796494 0.604646i \(-0.793315\pi\)
−0.796494 + 0.604646i \(0.793315\pi\)
\(600\) 36.2692 1.48068
\(601\) 19.1390 0.780697 0.390348 0.920667i \(-0.372355\pi\)
0.390348 + 0.920667i \(0.372355\pi\)
\(602\) 0 0
\(603\) −38.9412 −1.58581
\(604\) 0.147425 0.00599865
\(605\) 13.5315 0.550134
\(606\) 15.5578 0.631991
\(607\) 43.3336 1.75886 0.879428 0.476033i \(-0.157926\pi\)
0.879428 + 0.476033i \(0.157926\pi\)
\(608\) 3.24400 0.131562
\(609\) 0 0
\(610\) 8.11595 0.328605
\(611\) 0 0
\(612\) 0.291606 0.0117875
\(613\) −10.3096 −0.416399 −0.208200 0.978086i \(-0.566760\pi\)
−0.208200 + 0.978086i \(0.566760\pi\)
\(614\) −5.84154 −0.235745
\(615\) 2.15422 0.0868664
\(616\) 0 0
\(617\) 11.0699 0.445659 0.222829 0.974857i \(-0.428471\pi\)
0.222829 + 0.974857i \(0.428471\pi\)
\(618\) 33.5898 1.35118
\(619\) −33.7616 −1.35700 −0.678498 0.734603i \(-0.737369\pi\)
−0.678498 + 0.734603i \(0.737369\pi\)
\(620\) −0.275118 −0.0110490
\(621\) −10.6413 −0.427020
\(622\) 37.5794 1.50680
\(623\) 0 0
\(624\) 0 0
\(625\) 15.6786 0.627145
\(626\) 3.71927 0.148652
\(627\) 88.7719 3.54521
\(628\) −0.378778 −0.0151149
\(629\) −0.695623 −0.0277363
\(630\) 0 0
\(631\) 38.5975 1.53654 0.768271 0.640125i \(-0.221117\pi\)
0.768271 + 0.640125i \(0.221117\pi\)
\(632\) −32.8959 −1.30853
\(633\) −76.1290 −3.02586
\(634\) −33.1945 −1.31832
\(635\) −1.54525 −0.0613215
\(636\) −1.54339 −0.0611995
\(637\) 0 0
\(638\) 16.5763 0.656260
\(639\) 49.3798 1.95343
\(640\) 8.38960 0.331628
\(641\) 19.5228 0.771105 0.385553 0.922686i \(-0.374011\pi\)
0.385553 + 0.922686i \(0.374011\pi\)
\(642\) −38.4429 −1.51722
\(643\) −12.4718 −0.491839 −0.245920 0.969290i \(-0.579090\pi\)
−0.245920 + 0.969290i \(0.579090\pi\)
\(644\) 0 0
\(645\) 10.3266 0.406611
\(646\) −4.51450 −0.177621
\(647\) 35.9391 1.41291 0.706455 0.707758i \(-0.250293\pi\)
0.706455 + 0.707758i \(0.250293\pi\)
\(648\) −9.25967 −0.363754
\(649\) −57.9556 −2.27496
\(650\) 0 0
\(651\) 0 0
\(652\) 1.41034 0.0552332
\(653\) 4.85888 0.190143 0.0950713 0.995470i \(-0.469692\pi\)
0.0950713 + 0.995470i \(0.469692\pi\)
\(654\) −57.8668 −2.26277
\(655\) 12.5579 0.490679
\(656\) −3.51964 −0.137419
\(657\) −26.5144 −1.03442
\(658\) 0 0
\(659\) −23.6206 −0.920127 −0.460063 0.887886i \(-0.652173\pi\)
−0.460063 + 0.887886i \(0.652173\pi\)
\(660\) −1.20217 −0.0467944
\(661\) 16.3932 0.637623 0.318812 0.947818i \(-0.396716\pi\)
0.318812 + 0.947818i \(0.396716\pi\)
\(662\) 0.854633 0.0332162
\(663\) 0 0
\(664\) 16.8393 0.653492
\(665\) 0 0
\(666\) 9.07974 0.351833
\(667\) −3.65936 −0.141691
\(668\) −0.444033 −0.0171801
\(669\) 66.2692 2.56212
\(670\) −8.16420 −0.315410
\(671\) 38.4939 1.48604
\(672\) 0 0
\(673\) −14.2536 −0.549434 −0.274717 0.961525i \(-0.588584\pi\)
−0.274717 + 0.961525i \(0.588584\pi\)
\(674\) 7.88982 0.303904
\(675\) −28.8461 −1.11029
\(676\) 0 0
\(677\) −10.2715 −0.394765 −0.197383 0.980327i \(-0.563244\pi\)
−0.197383 + 0.980327i \(0.563244\pi\)
\(678\) 77.5255 2.97735
\(679\) 0 0
\(680\) 1.30618 0.0500898
\(681\) −1.30624 −0.0500554
\(682\) 25.2691 0.967604
\(683\) −2.22201 −0.0850230 −0.0425115 0.999096i \(-0.513536\pi\)
−0.0425115 + 0.999096i \(0.513536\pi\)
\(684\) −3.04292 −0.116349
\(685\) −6.32754 −0.241763
\(686\) 0 0
\(687\) 49.9413 1.90538
\(688\) −16.8721 −0.643242
\(689\) 0 0
\(690\) −5.13930 −0.195650
\(691\) −2.64015 −0.100436 −0.0502179 0.998738i \(-0.515992\pi\)
−0.0502179 + 0.998738i \(0.515992\pi\)
\(692\) 1.91680 0.0728659
\(693\) 0 0
\(694\) −2.57174 −0.0976220
\(695\) 7.99889 0.303415
\(696\) −19.0094 −0.720549
\(697\) −0.519607 −0.0196815
\(698\) 30.7767 1.16492
\(699\) 22.5169 0.851668
\(700\) 0 0
\(701\) 8.89991 0.336145 0.168072 0.985775i \(-0.446246\pi\)
0.168072 + 0.985775i \(0.446246\pi\)
\(702\) 0 0
\(703\) 7.25885 0.273773
\(704\) 44.0351 1.65964
\(705\) −8.92656 −0.336194
\(706\) −3.21594 −0.121034
\(707\) 0 0
\(708\) 3.11080 0.116911
\(709\) 40.5944 1.52456 0.762278 0.647250i \(-0.224081\pi\)
0.762278 + 0.647250i \(0.224081\pi\)
\(710\) 10.3527 0.388529
\(711\) 60.2695 2.26028
\(712\) −14.4951 −0.543226
\(713\) −5.57839 −0.208912
\(714\) 0 0
\(715\) 0 0
\(716\) −2.04464 −0.0764118
\(717\) −38.9868 −1.45599
\(718\) −4.51096 −0.168348
\(719\) 14.5135 0.541262 0.270631 0.962683i \(-0.412768\pi\)
0.270631 + 0.962683i \(0.412768\pi\)
\(720\) −16.2100 −0.604110
\(721\) 0 0
\(722\) 20.9069 0.778076
\(723\) 64.9977 2.41729
\(724\) 1.62555 0.0604129
\(725\) −9.91969 −0.368408
\(726\) 66.7105 2.47586
\(727\) −30.6942 −1.13839 −0.569193 0.822204i \(-0.692744\pi\)
−0.569193 + 0.822204i \(0.692744\pi\)
\(728\) 0 0
\(729\) −40.3475 −1.49435
\(730\) −5.55885 −0.205742
\(731\) −2.49084 −0.0921269
\(732\) −2.06618 −0.0763683
\(733\) 13.2644 0.489930 0.244965 0.969532i \(-0.421223\pi\)
0.244965 + 0.969532i \(0.421223\pi\)
\(734\) 5.72708 0.211390
\(735\) 0 0
\(736\) −0.890761 −0.0328339
\(737\) −38.7227 −1.42637
\(738\) 6.78227 0.249659
\(739\) −7.25474 −0.266870 −0.133435 0.991058i \(-0.542601\pi\)
−0.133435 + 0.991058i \(0.542601\pi\)
\(740\) −0.0983010 −0.00361362
\(741\) 0 0
\(742\) 0 0
\(743\) −46.2694 −1.69746 −0.848730 0.528827i \(-0.822632\pi\)
−0.848730 + 0.528827i \(0.822632\pi\)
\(744\) −28.9782 −1.06239
\(745\) 6.38250 0.233837
\(746\) −15.3198 −0.560898
\(747\) −30.8517 −1.12881
\(748\) 0.289969 0.0106023
\(749\) 0 0
\(750\) −29.9430 −1.09336
\(751\) −36.0260 −1.31461 −0.657305 0.753625i \(-0.728303\pi\)
−0.657305 + 0.753625i \(0.728303\pi\)
\(752\) 14.5846 0.531845
\(753\) −19.3973 −0.706877
\(754\) 0 0
\(755\) −1.20986 −0.0440313
\(756\) 0 0
\(757\) −10.5626 −0.383906 −0.191953 0.981404i \(-0.561482\pi\)
−0.191953 + 0.981404i \(0.561482\pi\)
\(758\) −6.39923 −0.232430
\(759\) −24.3756 −0.884780
\(760\) −13.6301 −0.494415
\(761\) −7.81202 −0.283185 −0.141593 0.989925i \(-0.545222\pi\)
−0.141593 + 0.989925i \(0.545222\pi\)
\(762\) −7.61813 −0.275976
\(763\) 0 0
\(764\) 0.417389 0.0151006
\(765\) −2.39309 −0.0865223
\(766\) −5.06020 −0.182833
\(767\) 0 0
\(768\) −6.77429 −0.244446
\(769\) 25.2915 0.912033 0.456017 0.889971i \(-0.349276\pi\)
0.456017 + 0.889971i \(0.349276\pi\)
\(770\) 0 0
\(771\) 47.6497 1.71606
\(772\) 1.13893 0.0409911
\(773\) −46.6004 −1.67610 −0.838051 0.545592i \(-0.816305\pi\)
−0.838051 + 0.545592i \(0.816305\pi\)
\(774\) 32.5121 1.16862
\(775\) −15.1217 −0.543188
\(776\) 30.8114 1.10606
\(777\) 0 0
\(778\) 23.2827 0.834727
\(779\) 5.42212 0.194268
\(780\) 0 0
\(781\) 49.1026 1.75703
\(782\) 1.23962 0.0443289
\(783\) 15.1188 0.540302
\(784\) 0 0
\(785\) 3.10848 0.110946
\(786\) 61.9108 2.20829
\(787\) −39.7332 −1.41633 −0.708167 0.706045i \(-0.750478\pi\)
−0.708167 + 0.706045i \(0.750478\pi\)
\(788\) 1.41628 0.0504529
\(789\) −28.8775 −1.02807
\(790\) 12.6358 0.449560
\(791\) 0 0
\(792\) −80.8641 −2.87338
\(793\) 0 0
\(794\) 23.0420 0.817730
\(795\) 12.6660 0.449217
\(796\) −0.693270 −0.0245723
\(797\) −2.78598 −0.0986844 −0.0493422 0.998782i \(-0.515712\pi\)
−0.0493422 + 0.998782i \(0.515712\pi\)
\(798\) 0 0
\(799\) 2.15313 0.0761723
\(800\) −2.41465 −0.0853706
\(801\) 26.5568 0.938338
\(802\) 34.9086 1.23267
\(803\) −26.3656 −0.930421
\(804\) 2.07847 0.0733018
\(805\) 0 0
\(806\) 0 0
\(807\) −45.3011 −1.59467
\(808\) −11.3298 −0.398580
\(809\) −41.4586 −1.45761 −0.728803 0.684723i \(-0.759923\pi\)
−0.728803 + 0.684723i \(0.759923\pi\)
\(810\) 3.55676 0.124972
\(811\) 27.8622 0.978375 0.489188 0.872179i \(-0.337293\pi\)
0.489188 + 0.872179i \(0.337293\pi\)
\(812\) 0 0
\(813\) −15.0289 −0.527085
\(814\) 9.02878 0.316459
\(815\) −11.5741 −0.405423
\(816\) 6.12254 0.214332
\(817\) 25.9920 0.909344
\(818\) 7.90671 0.276452
\(819\) 0 0
\(820\) −0.0734276 −0.00256420
\(821\) 22.4202 0.782469 0.391235 0.920291i \(-0.372048\pi\)
0.391235 + 0.920291i \(0.372048\pi\)
\(822\) −31.1949 −1.08805
\(823\) −2.36166 −0.0823221 −0.0411611 0.999153i \(-0.513106\pi\)
−0.0411611 + 0.999153i \(0.513106\pi\)
\(824\) −24.4614 −0.852154
\(825\) −66.0767 −2.30050
\(826\) 0 0
\(827\) −43.3148 −1.50620 −0.753102 0.657904i \(-0.771443\pi\)
−0.753102 + 0.657904i \(0.771443\pi\)
\(828\) 0.835546 0.0290372
\(829\) −54.9280 −1.90773 −0.953864 0.300239i \(-0.902933\pi\)
−0.953864 + 0.300239i \(0.902933\pi\)
\(830\) −6.46820 −0.224514
\(831\) −55.5277 −1.92623
\(832\) 0 0
\(833\) 0 0
\(834\) 39.4346 1.36551
\(835\) 3.64400 0.126106
\(836\) −3.02584 −0.104651
\(837\) 23.0473 0.796632
\(838\) −47.3647 −1.63619
\(839\) −14.7410 −0.508917 −0.254459 0.967084i \(-0.581897\pi\)
−0.254459 + 0.967084i \(0.581897\pi\)
\(840\) 0 0
\(841\) −23.8009 −0.820721
\(842\) −4.05673 −0.139804
\(843\) 6.17393 0.212642
\(844\) 2.59490 0.0893201
\(845\) 0 0
\(846\) −28.1041 −0.966239
\(847\) 0 0
\(848\) −20.6942 −0.710642
\(849\) −45.3796 −1.55743
\(850\) 3.36033 0.115258
\(851\) −1.99319 −0.0683256
\(852\) −2.63562 −0.0902947
\(853\) −24.1038 −0.825297 −0.412649 0.910890i \(-0.635396\pi\)
−0.412649 + 0.910890i \(0.635396\pi\)
\(854\) 0 0
\(855\) 24.9720 0.854024
\(856\) 27.9957 0.956873
\(857\) −18.5850 −0.634851 −0.317425 0.948283i \(-0.602818\pi\)
−0.317425 + 0.948283i \(0.602818\pi\)
\(858\) 0 0
\(859\) 29.4975 1.00644 0.503221 0.864158i \(-0.332148\pi\)
0.503221 + 0.864158i \(0.332148\pi\)
\(860\) −0.351989 −0.0120027
\(861\) 0 0
\(862\) −54.7423 −1.86453
\(863\) 18.6435 0.634631 0.317315 0.948320i \(-0.397219\pi\)
0.317315 + 0.948320i \(0.397219\pi\)
\(864\) 3.68021 0.125203
\(865\) −15.7304 −0.534851
\(866\) −13.5651 −0.460961
\(867\) −48.0766 −1.63277
\(868\) 0 0
\(869\) 59.9312 2.03303
\(870\) 7.30175 0.247552
\(871\) 0 0
\(872\) 42.1409 1.42707
\(873\) −56.4502 −1.91055
\(874\) −12.9355 −0.437551
\(875\) 0 0
\(876\) 1.41519 0.0478148
\(877\) −37.7518 −1.27479 −0.637395 0.770538i \(-0.719988\pi\)
−0.637395 + 0.770538i \(0.719988\pi\)
\(878\) 39.3673 1.32858
\(879\) −66.6962 −2.24961
\(880\) −16.1190 −0.543372
\(881\) 29.8298 1.00499 0.502497 0.864579i \(-0.332415\pi\)
0.502497 + 0.864579i \(0.332415\pi\)
\(882\) 0 0
\(883\) 32.3979 1.09028 0.545138 0.838346i \(-0.316477\pi\)
0.545138 + 0.838346i \(0.316477\pi\)
\(884\) 0 0
\(885\) −25.5291 −0.858151
\(886\) 4.60468 0.154697
\(887\) −25.9198 −0.870302 −0.435151 0.900358i \(-0.643305\pi\)
−0.435151 + 0.900358i \(0.643305\pi\)
\(888\) −10.3541 −0.347459
\(889\) 0 0
\(890\) 5.56775 0.186631
\(891\) 16.8697 0.565155
\(892\) −2.25882 −0.0756310
\(893\) −22.4680 −0.751863
\(894\) 31.4658 1.05238
\(895\) 16.7795 0.560878
\(896\) 0 0
\(897\) 0 0
\(898\) 25.0782 0.836870
\(899\) 7.92559 0.264333
\(900\) 2.26497 0.0754991
\(901\) −3.05510 −0.101780
\(902\) 6.74420 0.224557
\(903\) 0 0
\(904\) −56.4572 −1.87774
\(905\) −13.3402 −0.443443
\(906\) −5.96463 −0.198162
\(907\) 15.5423 0.516072 0.258036 0.966135i \(-0.416925\pi\)
0.258036 + 0.966135i \(0.416925\pi\)
\(908\) 0.0445240 0.00147758
\(909\) 20.7576 0.688485
\(910\) 0 0
\(911\) 23.6358 0.783090 0.391545 0.920159i \(-0.371941\pi\)
0.391545 + 0.920159i \(0.371941\pi\)
\(912\) −63.8889 −2.11557
\(913\) −30.6786 −1.01531
\(914\) −12.0382 −0.398189
\(915\) 16.9563 0.560559
\(916\) −1.70228 −0.0562448
\(917\) 0 0
\(918\) −5.12155 −0.169036
\(919\) 44.4817 1.46732 0.733659 0.679518i \(-0.237811\pi\)
0.733659 + 0.679518i \(0.237811\pi\)
\(920\) 3.74264 0.123391
\(921\) −12.2045 −0.402151
\(922\) 3.13216 0.103152
\(923\) 0 0
\(924\) 0 0
\(925\) −5.40307 −0.177652
\(926\) 7.56171 0.248493
\(927\) 44.8163 1.47196
\(928\) 1.26556 0.0415441
\(929\) −2.94270 −0.0965470 −0.0482735 0.998834i \(-0.515372\pi\)
−0.0482735 + 0.998834i \(0.515372\pi\)
\(930\) 11.1309 0.364996
\(931\) 0 0
\(932\) −0.767502 −0.0251404
\(933\) 78.5131 2.57040
\(934\) 26.0392 0.852028
\(935\) −2.37966 −0.0778232
\(936\) 0 0
\(937\) −0.951020 −0.0310685 −0.0155342 0.999879i \(-0.504945\pi\)
−0.0155342 + 0.999879i \(0.504945\pi\)
\(938\) 0 0
\(939\) 7.77052 0.253581
\(940\) 0.304267 0.00992409
\(941\) −22.0692 −0.719436 −0.359718 0.933061i \(-0.617127\pi\)
−0.359718 + 0.933061i \(0.617127\pi\)
\(942\) 15.3248 0.499311
\(943\) −1.48885 −0.0484835
\(944\) 41.7105 1.35756
\(945\) 0 0
\(946\) 32.3296 1.05113
\(947\) −51.1717 −1.66286 −0.831429 0.555631i \(-0.812477\pi\)
−0.831429 + 0.555631i \(0.812477\pi\)
\(948\) −3.21685 −0.104478
\(949\) 0 0
\(950\) −35.0652 −1.13767
\(951\) −69.3519 −2.24889
\(952\) 0 0
\(953\) 45.8470 1.48513 0.742565 0.669773i \(-0.233609\pi\)
0.742565 + 0.669773i \(0.233609\pi\)
\(954\) 39.8772 1.29107
\(955\) −3.42534 −0.110842
\(956\) 1.32889 0.0429793
\(957\) 34.6321 1.11950
\(958\) −45.6955 −1.47635
\(959\) 0 0
\(960\) 19.3972 0.626042
\(961\) −18.9181 −0.610262
\(962\) 0 0
\(963\) −51.2916 −1.65285
\(964\) −2.21548 −0.0713559
\(965\) −9.34676 −0.300883
\(966\) 0 0
\(967\) 19.2609 0.619387 0.309694 0.950836i \(-0.399773\pi\)
0.309694 + 0.950836i \(0.399773\pi\)
\(968\) −48.5813 −1.56146
\(969\) −9.43196 −0.302998
\(970\) −11.8350 −0.380000
\(971\) 47.3326 1.51897 0.759487 0.650522i \(-0.225450\pi\)
0.759487 + 0.650522i \(0.225450\pi\)
\(972\) 1.04803 0.0336156
\(973\) 0 0
\(974\) −22.0046 −0.705072
\(975\) 0 0
\(976\) −27.7039 −0.886781
\(977\) −47.8571 −1.53108 −0.765541 0.643387i \(-0.777529\pi\)
−0.765541 + 0.643387i \(0.777529\pi\)
\(978\) −57.0605 −1.82459
\(979\) 26.4078 0.843995
\(980\) 0 0
\(981\) −77.2074 −2.46504
\(982\) −43.7061 −1.39472
\(983\) −15.7686 −0.502941 −0.251471 0.967865i \(-0.580914\pi\)
−0.251471 + 0.967865i \(0.580914\pi\)
\(984\) −7.73414 −0.246555
\(985\) −11.6228 −0.370335
\(986\) −1.76122 −0.0560885
\(987\) 0 0
\(988\) 0 0
\(989\) −7.13707 −0.226945
\(990\) 31.0609 0.987181
\(991\) −12.1378 −0.385571 −0.192786 0.981241i \(-0.561752\pi\)
−0.192786 + 0.981241i \(0.561752\pi\)
\(992\) 1.92924 0.0612536
\(993\) 1.78555 0.0566627
\(994\) 0 0
\(995\) 5.68938 0.180366
\(996\) 1.64669 0.0521775
\(997\) 33.7876 1.07006 0.535032 0.844832i \(-0.320299\pi\)
0.535032 + 0.844832i \(0.320299\pi\)
\(998\) −33.3986 −1.05721
\(999\) 8.23493 0.260542
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 8281.2.a.cp.1.10 12
7.2 even 3 1183.2.e.j.508.3 24
7.4 even 3 1183.2.e.j.170.3 24
7.6 odd 2 8281.2.a.co.1.10 12
13.2 odd 12 637.2.q.g.589.5 12
13.7 odd 12 637.2.q.g.491.5 12
13.12 even 2 inner 8281.2.a.cp.1.3 12
91.2 odd 12 91.2.k.b.4.2 12
91.20 even 12 637.2.q.i.491.5 12
91.25 even 6 1183.2.e.j.170.10 24
91.33 even 12 637.2.u.g.361.2 12
91.41 even 12 637.2.q.i.589.5 12
91.46 odd 12 91.2.k.b.23.5 yes 12
91.51 even 6 1183.2.e.j.508.10 24
91.54 even 12 637.2.k.i.459.2 12
91.59 even 12 637.2.k.i.569.5 12
91.67 odd 12 91.2.u.b.30.2 yes 12
91.72 odd 12 91.2.u.b.88.2 yes 12
91.80 even 12 637.2.u.g.30.2 12
91.90 odd 2 8281.2.a.co.1.3 12
273.2 even 12 819.2.bm.f.550.5 12
273.137 even 12 819.2.bm.f.478.2 12
273.158 even 12 819.2.do.e.667.5 12
273.254 even 12 819.2.do.e.361.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.k.b.4.2 12 91.2 odd 12
91.2.k.b.23.5 yes 12 91.46 odd 12
91.2.u.b.30.2 yes 12 91.67 odd 12
91.2.u.b.88.2 yes 12 91.72 odd 12
637.2.k.i.459.2 12 91.54 even 12
637.2.k.i.569.5 12 91.59 even 12
637.2.q.g.491.5 12 13.7 odd 12
637.2.q.g.589.5 12 13.2 odd 12
637.2.q.i.491.5 12 91.20 even 12
637.2.q.i.589.5 12 91.41 even 12
637.2.u.g.30.2 12 91.80 even 12
637.2.u.g.361.2 12 91.33 even 12
819.2.bm.f.478.2 12 273.137 even 12
819.2.bm.f.550.5 12 273.2 even 12
819.2.do.e.361.5 12 273.254 even 12
819.2.do.e.667.5 12 273.158 even 12
1183.2.e.j.170.3 24 7.4 even 3
1183.2.e.j.170.10 24 91.25 even 6
1183.2.e.j.508.3 24 7.2 even 3
1183.2.e.j.508.10 24 91.51 even 6
8281.2.a.co.1.3 12 91.90 odd 2
8281.2.a.co.1.10 12 7.6 odd 2
8281.2.a.cp.1.3 12 13.12 even 2 inner
8281.2.a.cp.1.10 12 1.1 even 1 trivial