Properties

Label 1815.2.a.w.1.4
Level $1815$
Weight $2$
Character 1815.1
Self dual yes
Analytic conductor $14.493$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4928479669\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
Defining polynomial: \(x^{4} - x^{3} - 3 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.35567\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.35567 q^{2} +1.00000 q^{3} +3.54920 q^{4} +1.00000 q^{5} +2.35567 q^{6} -0.193527 q^{7} +3.64941 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.35567 q^{2} +1.00000 q^{3} +3.54920 q^{4} +1.00000 q^{5} +2.35567 q^{6} -0.193527 q^{7} +3.64941 q^{8} +1.00000 q^{9} +2.35567 q^{10} +3.54920 q^{12} -0.973708 q^{13} -0.455887 q^{14} +1.00000 q^{15} +1.49843 q^{16} +2.67571 q^{17} +2.35567 q^{18} +5.54920 q^{19} +3.54920 q^{20} -0.193527 q^{21} -4.80040 q^{23} +3.64941 q^{24} +1.00000 q^{25} -2.29374 q^{26} +1.00000 q^{27} -0.686867 q^{28} +10.1178 q^{29} +2.35567 q^{30} -2.50533 q^{31} -3.76902 q^{32} +6.30309 q^{34} -0.193527 q^{35} +3.54920 q^{36} +5.71217 q^{37} +13.0721 q^{38} -0.973708 q^{39} +3.64941 q^{40} +8.27861 q^{41} -0.455887 q^{42} -5.11353 q^{43} +1.00000 q^{45} -11.3082 q^{46} -10.8523 q^{47} +1.49843 q^{48} -6.96255 q^{49} +2.35567 q^{50} +2.67571 q^{51} -3.45589 q^{52} -9.28879 q^{53} +2.35567 q^{54} -0.706260 q^{56} +5.54920 q^{57} +23.8342 q^{58} -11.2180 q^{59} +3.54920 q^{60} -1.20602 q^{61} -5.90173 q^{62} -0.193527 q^{63} -11.8754 q^{64} -0.973708 q^{65} +3.25922 q^{67} +9.49662 q^{68} -4.80040 q^{69} -0.455887 q^{70} +5.99078 q^{71} +3.64941 q^{72} -3.30624 q^{73} +13.4560 q^{74} +1.00000 q^{75} +19.6952 q^{76} -2.29374 q^{78} -10.1529 q^{79} +1.49843 q^{80} +1.00000 q^{81} +19.5017 q^{82} -8.31508 q^{83} -0.686867 q^{84} +2.67571 q^{85} -12.0458 q^{86} +10.1178 q^{87} +7.34270 q^{89} +2.35567 q^{90} +0.188439 q^{91} -17.0376 q^{92} -2.50533 q^{93} -25.5645 q^{94} +5.54920 q^{95} -3.76902 q^{96} -15.8523 q^{97} -16.4015 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 3q^{2} + 4q^{3} + q^{4} + 4q^{5} + 3q^{6} + 6q^{7} + 3q^{8} + 4q^{9} + O(q^{10}) \) \( 4q + 3q^{2} + 4q^{3} + q^{4} + 4q^{5} + 3q^{6} + 6q^{7} + 3q^{8} + 4q^{9} + 3q^{10} + q^{12} + 7q^{13} + 3q^{14} + 4q^{15} - q^{16} + 10q^{17} + 3q^{18} + 9q^{19} + q^{20} + 6q^{21} - 3q^{23} + 3q^{24} + 4q^{25} - 4q^{26} + 4q^{27} - 7q^{28} + 15q^{29} + 3q^{30} - 13q^{31} - 6q^{32} - 3q^{34} + 6q^{35} + q^{36} - 3q^{37} + 15q^{38} + 7q^{39} + 3q^{40} + 22q^{41} + 3q^{42} + 4q^{45} + q^{46} - 2q^{47} - q^{48} - 12q^{49} + 3q^{50} + 10q^{51} - 9q^{52} + 10q^{53} + 3q^{54} - 8q^{56} + 9q^{57} + 39q^{58} - 21q^{59} + q^{60} + 11q^{61} + 10q^{62} + 6q^{63} - 3q^{64} + 7q^{65} + q^{67} + 3q^{68} - 3q^{69} + 3q^{70} - 13q^{71} + 3q^{72} + q^{73} - 11q^{74} + 4q^{75} + 19q^{76} - 4q^{78} - 4q^{79} - q^{80} + 4q^{81} + 25q^{82} + 3q^{83} - 7q^{84} + 10q^{85} + 15q^{87} - 10q^{89} + 3q^{90} + 12q^{91} - 24q^{92} - 13q^{93} - 35q^{94} + 9q^{95} - 6q^{96} - 22q^{97} - 11q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.35567 1.66571 0.832857 0.553489i \(-0.186704\pi\)
0.832857 + 0.553489i \(0.186704\pi\)
\(3\) 1.00000 0.577350
\(4\) 3.54920 1.77460
\(5\) 1.00000 0.447214
\(6\) 2.35567 0.961700
\(7\) −0.193527 −0.0731464 −0.0365732 0.999331i \(-0.511644\pi\)
−0.0365732 + 0.999331i \(0.511644\pi\)
\(8\) 3.64941 1.29026
\(9\) 1.00000 0.333333
\(10\) 2.35567 0.744930
\(11\) 0 0
\(12\) 3.54920 1.02457
\(13\) −0.973708 −0.270058 −0.135029 0.990842i \(-0.543113\pi\)
−0.135029 + 0.990842i \(0.543113\pi\)
\(14\) −0.455887 −0.121841
\(15\) 1.00000 0.258199
\(16\) 1.49843 0.374607
\(17\) 2.67571 0.648954 0.324477 0.945894i \(-0.394812\pi\)
0.324477 + 0.945894i \(0.394812\pi\)
\(18\) 2.35567 0.555238
\(19\) 5.54920 1.27307 0.636537 0.771246i \(-0.280366\pi\)
0.636537 + 0.771246i \(0.280366\pi\)
\(20\) 3.54920 0.793626
\(21\) −0.193527 −0.0422311
\(22\) 0 0
\(23\) −4.80040 −1.00095 −0.500476 0.865750i \(-0.666842\pi\)
−0.500476 + 0.865750i \(0.666842\pi\)
\(24\) 3.64941 0.744933
\(25\) 1.00000 0.200000
\(26\) −2.29374 −0.449839
\(27\) 1.00000 0.192450
\(28\) −0.686867 −0.129806
\(29\) 10.1178 1.87883 0.939414 0.342785i \(-0.111370\pi\)
0.939414 + 0.342785i \(0.111370\pi\)
\(30\) 2.35567 0.430085
\(31\) −2.50533 −0.449970 −0.224985 0.974362i \(-0.572233\pi\)
−0.224985 + 0.974362i \(0.572233\pi\)
\(32\) −3.76902 −0.666275
\(33\) 0 0
\(34\) 6.30309 1.08097
\(35\) −0.193527 −0.0327120
\(36\) 3.54920 0.591534
\(37\) 5.71217 0.939076 0.469538 0.882912i \(-0.344421\pi\)
0.469538 + 0.882912i \(0.344421\pi\)
\(38\) 13.0721 2.12058
\(39\) −0.973708 −0.155918
\(40\) 3.64941 0.577023
\(41\) 8.27861 1.29290 0.646451 0.762956i \(-0.276253\pi\)
0.646451 + 0.762956i \(0.276253\pi\)
\(42\) −0.455887 −0.0703449
\(43\) −5.11353 −0.779807 −0.389903 0.920856i \(-0.627492\pi\)
−0.389903 + 0.920856i \(0.627492\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) −11.3082 −1.66730
\(47\) −10.8523 −1.58297 −0.791485 0.611189i \(-0.790692\pi\)
−0.791485 + 0.611189i \(0.790692\pi\)
\(48\) 1.49843 0.216279
\(49\) −6.96255 −0.994650
\(50\) 2.35567 0.333143
\(51\) 2.67571 0.374674
\(52\) −3.45589 −0.479245
\(53\) −9.28879 −1.27591 −0.637956 0.770072i \(-0.720220\pi\)
−0.637956 + 0.770072i \(0.720220\pi\)
\(54\) 2.35567 0.320567
\(55\) 0 0
\(56\) −0.706260 −0.0943780
\(57\) 5.54920 0.735010
\(58\) 23.8342 3.12959
\(59\) −11.2180 −1.46046 −0.730230 0.683201i \(-0.760587\pi\)
−0.730230 + 0.683201i \(0.760587\pi\)
\(60\) 3.54920 0.458200
\(61\) −1.20602 −0.154415 −0.0772077 0.997015i \(-0.524600\pi\)
−0.0772077 + 0.997015i \(0.524600\pi\)
\(62\) −5.90173 −0.749521
\(63\) −0.193527 −0.0243821
\(64\) −11.8754 −1.48443
\(65\) −0.973708 −0.120774
\(66\) 0 0
\(67\) 3.25922 0.398176 0.199088 0.979982i \(-0.436202\pi\)
0.199088 + 0.979982i \(0.436202\pi\)
\(68\) 9.49662 1.15163
\(69\) −4.80040 −0.577900
\(70\) −0.455887 −0.0544889
\(71\) 5.99078 0.710975 0.355488 0.934681i \(-0.384315\pi\)
0.355488 + 0.934681i \(0.384315\pi\)
\(72\) 3.64941 0.430088
\(73\) −3.30624 −0.386966 −0.193483 0.981104i \(-0.561978\pi\)
−0.193483 + 0.981104i \(0.561978\pi\)
\(74\) 13.4560 1.56423
\(75\) 1.00000 0.115470
\(76\) 19.6952 2.25920
\(77\) 0 0
\(78\) −2.29374 −0.259715
\(79\) −10.1529 −1.14229 −0.571147 0.820848i \(-0.693501\pi\)
−0.571147 + 0.820848i \(0.693501\pi\)
\(80\) 1.49843 0.167529
\(81\) 1.00000 0.111111
\(82\) 19.5017 2.15360
\(83\) −8.31508 −0.912698 −0.456349 0.889801i \(-0.650843\pi\)
−0.456349 + 0.889801i \(0.650843\pi\)
\(84\) −0.686867 −0.0749433
\(85\) 2.67571 0.290221
\(86\) −12.0458 −1.29893
\(87\) 10.1178 1.08474
\(88\) 0 0
\(89\) 7.34270 0.778325 0.389163 0.921169i \(-0.372764\pi\)
0.389163 + 0.921169i \(0.372764\pi\)
\(90\) 2.35567 0.248310
\(91\) 0.188439 0.0197538
\(92\) −17.0376 −1.77629
\(93\) −2.50533 −0.259790
\(94\) −25.5645 −2.63677
\(95\) 5.54920 0.569336
\(96\) −3.76902 −0.384674
\(97\) −15.8523 −1.60956 −0.804778 0.593576i \(-0.797716\pi\)
−0.804778 + 0.593576i \(0.797716\pi\)
\(98\) −16.4015 −1.65680
\(99\) 0 0
\(100\) 3.54920 0.354920
\(101\) −13.1591 −1.30938 −0.654692 0.755896i \(-0.727201\pi\)
−0.654692 + 0.755896i \(0.727201\pi\)
\(102\) 6.30309 0.624099
\(103\) 3.99737 0.393872 0.196936 0.980416i \(-0.436901\pi\)
0.196936 + 0.980416i \(0.436901\pi\)
\(104\) −3.55346 −0.348446
\(105\) −0.193527 −0.0188863
\(106\) −21.8814 −2.12530
\(107\) −4.88682 −0.472426 −0.236213 0.971701i \(-0.575906\pi\)
−0.236213 + 0.971701i \(0.575906\pi\)
\(108\) 3.54920 0.341522
\(109\) 7.51977 0.720263 0.360131 0.932902i \(-0.382732\pi\)
0.360131 + 0.932902i \(0.382732\pi\)
\(110\) 0 0
\(111\) 5.71217 0.542176
\(112\) −0.289986 −0.0274011
\(113\) 20.2237 1.90249 0.951243 0.308442i \(-0.0998076\pi\)
0.951243 + 0.308442i \(0.0998076\pi\)
\(114\) 13.0721 1.22432
\(115\) −4.80040 −0.447640
\(116\) 35.9101 3.33417
\(117\) −0.973708 −0.0900194
\(118\) −26.4260 −2.43271
\(119\) −0.517822 −0.0474686
\(120\) 3.64941 0.333144
\(121\) 0 0
\(122\) −2.84100 −0.257212
\(123\) 8.27861 0.746457
\(124\) −8.89190 −0.798517
\(125\) 1.00000 0.0894427
\(126\) −0.455887 −0.0406136
\(127\) −7.10021 −0.630042 −0.315021 0.949085i \(-0.602012\pi\)
−0.315021 + 0.949085i \(0.602012\pi\)
\(128\) −20.4366 −1.80636
\(129\) −5.11353 −0.450222
\(130\) −2.29374 −0.201174
\(131\) 2.50024 0.218447 0.109223 0.994017i \(-0.465164\pi\)
0.109223 + 0.994017i \(0.465164\pi\)
\(132\) 0 0
\(133\) −1.07392 −0.0931207
\(134\) 7.67765 0.663248
\(135\) 1.00000 0.0860663
\(136\) 9.76476 0.837321
\(137\) −15.2944 −1.30669 −0.653344 0.757061i \(-0.726634\pi\)
−0.653344 + 0.757061i \(0.726634\pi\)
\(138\) −11.3082 −0.962616
\(139\) 19.6586 1.66742 0.833711 0.552202i \(-0.186212\pi\)
0.833711 + 0.552202i \(0.186212\pi\)
\(140\) −0.686867 −0.0580508
\(141\) −10.8523 −0.913928
\(142\) 14.1123 1.18428
\(143\) 0 0
\(144\) 1.49843 0.124869
\(145\) 10.1178 0.840237
\(146\) −7.78841 −0.644574
\(147\) −6.96255 −0.574261
\(148\) 20.2737 1.66648
\(149\) 10.1329 0.830122 0.415061 0.909794i \(-0.363760\pi\)
0.415061 + 0.909794i \(0.363760\pi\)
\(150\) 2.35567 0.192340
\(151\) −11.9632 −0.973548 −0.486774 0.873528i \(-0.661826\pi\)
−0.486774 + 0.873528i \(0.661826\pi\)
\(152\) 20.2513 1.64260
\(153\) 2.67571 0.216318
\(154\) 0 0
\(155\) −2.50533 −0.201233
\(156\) −3.45589 −0.276692
\(157\) −2.62605 −0.209582 −0.104791 0.994494i \(-0.533417\pi\)
−0.104791 + 0.994494i \(0.533417\pi\)
\(158\) −23.9170 −1.90273
\(159\) −9.28879 −0.736649
\(160\) −3.76902 −0.297967
\(161\) 0.929007 0.0732160
\(162\) 2.35567 0.185079
\(163\) 23.7235 1.85817 0.929083 0.369872i \(-0.120598\pi\)
0.929083 + 0.369872i \(0.120598\pi\)
\(164\) 29.3825 2.29438
\(165\) 0 0
\(166\) −19.5876 −1.52029
\(167\) −3.49175 −0.270199 −0.135100 0.990832i \(-0.543135\pi\)
−0.135100 + 0.990832i \(0.543135\pi\)
\(168\) −0.706260 −0.0544892
\(169\) −12.0519 −0.927069
\(170\) 6.30309 0.483425
\(171\) 5.54920 0.424358
\(172\) −18.1490 −1.38385
\(173\) 20.3596 1.54791 0.773954 0.633241i \(-0.218276\pi\)
0.773954 + 0.633241i \(0.218276\pi\)
\(174\) 23.8342 1.80687
\(175\) −0.193527 −0.0146293
\(176\) 0 0
\(177\) −11.2180 −0.843197
\(178\) 17.2970 1.29647
\(179\) 3.85117 0.287850 0.143925 0.989589i \(-0.454028\pi\)
0.143925 + 0.989589i \(0.454028\pi\)
\(180\) 3.54920 0.264542
\(181\) −17.6151 −1.30932 −0.654660 0.755923i \(-0.727188\pi\)
−0.654660 + 0.755923i \(0.727188\pi\)
\(182\) 0.443901 0.0329041
\(183\) −1.20602 −0.0891518
\(184\) −17.5186 −1.29149
\(185\) 5.71217 0.419967
\(186\) −5.90173 −0.432736
\(187\) 0 0
\(188\) −38.5170 −2.80914
\(189\) −0.193527 −0.0140770
\(190\) 13.0721 0.948351
\(191\) 0.417099 0.0301802 0.0150901 0.999886i \(-0.495196\pi\)
0.0150901 + 0.999886i \(0.495196\pi\)
\(192\) −11.8754 −0.857036
\(193\) −11.6965 −0.841935 −0.420967 0.907076i \(-0.638309\pi\)
−0.420967 + 0.907076i \(0.638309\pi\)
\(194\) −37.3428 −2.68106
\(195\) −0.973708 −0.0697287
\(196\) −24.7115 −1.76511
\(197\) −21.5958 −1.53864 −0.769320 0.638863i \(-0.779405\pi\)
−0.769320 + 0.638863i \(0.779405\pi\)
\(198\) 0 0
\(199\) 7.76028 0.550111 0.275056 0.961428i \(-0.411304\pi\)
0.275056 + 0.961428i \(0.411304\pi\)
\(200\) 3.64941 0.258053
\(201\) 3.25922 0.229887
\(202\) −30.9986 −2.18106
\(203\) −1.95807 −0.137429
\(204\) 9.49662 0.664896
\(205\) 8.27861 0.578203
\(206\) 9.41649 0.656078
\(207\) −4.80040 −0.333651
\(208\) −1.45903 −0.101166
\(209\) 0 0
\(210\) −0.455887 −0.0314592
\(211\) 12.5057 0.860926 0.430463 0.902608i \(-0.358350\pi\)
0.430463 + 0.902608i \(0.358350\pi\)
\(212\) −32.9678 −2.26424
\(213\) 5.99078 0.410482
\(214\) −11.5117 −0.786927
\(215\) −5.11353 −0.348740
\(216\) 3.64941 0.248311
\(217\) 0.484848 0.0329137
\(218\) 17.7141 1.19975
\(219\) −3.30624 −0.223415
\(220\) 0 0
\(221\) −2.60536 −0.175255
\(222\) 13.4560 0.903109
\(223\) −5.37568 −0.359982 −0.179991 0.983668i \(-0.557607\pi\)
−0.179991 + 0.983668i \(0.557607\pi\)
\(224\) 0.729407 0.0487356
\(225\) 1.00000 0.0666667
\(226\) 47.6405 3.16900
\(227\) 21.7529 1.44379 0.721895 0.692002i \(-0.243271\pi\)
0.721895 + 0.692002i \(0.243271\pi\)
\(228\) 19.6952 1.30435
\(229\) 2.67075 0.176488 0.0882441 0.996099i \(-0.471874\pi\)
0.0882441 + 0.996099i \(0.471874\pi\)
\(230\) −11.3082 −0.745639
\(231\) 0 0
\(232\) 36.9240 2.42418
\(233\) 5.72699 0.375188 0.187594 0.982247i \(-0.439931\pi\)
0.187594 + 0.982247i \(0.439931\pi\)
\(234\) −2.29374 −0.149946
\(235\) −10.8523 −0.707925
\(236\) −39.8150 −2.59173
\(237\) −10.1529 −0.659504
\(238\) −1.21982 −0.0790691
\(239\) 24.3680 1.57624 0.788118 0.615524i \(-0.211056\pi\)
0.788118 + 0.615524i \(0.211056\pi\)
\(240\) 1.49843 0.0967231
\(241\) 16.7082 1.07627 0.538135 0.842859i \(-0.319129\pi\)
0.538135 + 0.842859i \(0.319129\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −4.28042 −0.274026
\(245\) −6.96255 −0.444821
\(246\) 19.5017 1.24338
\(247\) −5.40330 −0.343804
\(248\) −9.14297 −0.580579
\(249\) −8.31508 −0.526947
\(250\) 2.35567 0.148986
\(251\) −16.0984 −1.01612 −0.508061 0.861321i \(-0.669638\pi\)
−0.508061 + 0.861321i \(0.669638\pi\)
\(252\) −0.686867 −0.0432685
\(253\) 0 0
\(254\) −16.7258 −1.04947
\(255\) 2.67571 0.167559
\(256\) −24.3912 −1.52445
\(257\) 5.96875 0.372321 0.186160 0.982519i \(-0.440396\pi\)
0.186160 + 0.982519i \(0.440396\pi\)
\(258\) −12.0458 −0.749940
\(259\) −1.10546 −0.0686900
\(260\) −3.45589 −0.214325
\(261\) 10.1178 0.626276
\(262\) 5.88974 0.363870
\(263\) 0.451149 0.0278190 0.0139095 0.999903i \(-0.495572\pi\)
0.0139095 + 0.999903i \(0.495572\pi\)
\(264\) 0 0
\(265\) −9.28879 −0.570606
\(266\) −2.52981 −0.155112
\(267\) 7.34270 0.449366
\(268\) 11.5676 0.706604
\(269\) 14.7197 0.897477 0.448738 0.893663i \(-0.351873\pi\)
0.448738 + 0.893663i \(0.351873\pi\)
\(270\) 2.35567 0.143362
\(271\) −4.42787 −0.268974 −0.134487 0.990915i \(-0.542939\pi\)
−0.134487 + 0.990915i \(0.542939\pi\)
\(272\) 4.00935 0.243103
\(273\) 0.188439 0.0114048
\(274\) −36.0286 −2.17657
\(275\) 0 0
\(276\) −17.0376 −1.02554
\(277\) 0.0923299 0.00554757 0.00277378 0.999996i \(-0.499117\pi\)
0.00277378 + 0.999996i \(0.499117\pi\)
\(278\) 46.3093 2.77745
\(279\) −2.50533 −0.149990
\(280\) −0.706260 −0.0422071
\(281\) −5.18301 −0.309192 −0.154596 0.987978i \(-0.549408\pi\)
−0.154596 + 0.987978i \(0.549408\pi\)
\(282\) −25.5645 −1.52234
\(283\) −1.56463 −0.0930073 −0.0465037 0.998918i \(-0.514808\pi\)
−0.0465037 + 0.998918i \(0.514808\pi\)
\(284\) 21.2625 1.26170
\(285\) 5.54920 0.328706
\(286\) 0 0
\(287\) −1.60214 −0.0945710
\(288\) −3.76902 −0.222092
\(289\) −9.84060 −0.578859
\(290\) 23.8342 1.39959
\(291\) −15.8523 −0.929278
\(292\) −11.7345 −0.686709
\(293\) 23.1570 1.35285 0.676424 0.736512i \(-0.263529\pi\)
0.676424 + 0.736512i \(0.263529\pi\)
\(294\) −16.4015 −0.956555
\(295\) −11.2180 −0.653138
\(296\) 20.8461 1.21165
\(297\) 0 0
\(298\) 23.8699 1.38274
\(299\) 4.67419 0.270315
\(300\) 3.54920 0.204913
\(301\) 0.989607 0.0570400
\(302\) −28.1813 −1.62165
\(303\) −13.1591 −0.755973
\(304\) 8.31508 0.476902
\(305\) −1.20602 −0.0690567
\(306\) 6.30309 0.360324
\(307\) 7.72480 0.440878 0.220439 0.975401i \(-0.429251\pi\)
0.220439 + 0.975401i \(0.429251\pi\)
\(308\) 0 0
\(309\) 3.99737 0.227402
\(310\) −5.90173 −0.335196
\(311\) −19.1209 −1.08424 −0.542122 0.840300i \(-0.682379\pi\)
−0.542122 + 0.840300i \(0.682379\pi\)
\(312\) −3.55346 −0.201175
\(313\) 2.90307 0.164091 0.0820455 0.996629i \(-0.473855\pi\)
0.0820455 + 0.996629i \(0.473855\pi\)
\(314\) −6.18612 −0.349103
\(315\) −0.193527 −0.0109040
\(316\) −36.0348 −2.02712
\(317\) 2.26153 0.127020 0.0635102 0.997981i \(-0.479770\pi\)
0.0635102 + 0.997981i \(0.479770\pi\)
\(318\) −21.8814 −1.22705
\(319\) 0 0
\(320\) −11.8754 −0.663857
\(321\) −4.88682 −0.272756
\(322\) 2.18844 0.121957
\(323\) 14.8480 0.826166
\(324\) 3.54920 0.197178
\(325\) −0.973708 −0.0540116
\(326\) 55.8848 3.09517
\(327\) 7.51977 0.415844
\(328\) 30.2121 1.66818
\(329\) 2.10021 0.115788
\(330\) 0 0
\(331\) 6.02336 0.331074 0.165537 0.986204i \(-0.447064\pi\)
0.165537 + 0.986204i \(0.447064\pi\)
\(332\) −29.5119 −1.61967
\(333\) 5.71217 0.313025
\(334\) −8.22542 −0.450075
\(335\) 3.25922 0.178070
\(336\) −0.289986 −0.0158201
\(337\) 14.5648 0.793393 0.396696 0.917950i \(-0.370157\pi\)
0.396696 + 0.917950i \(0.370157\pi\)
\(338\) −28.3903 −1.54423
\(339\) 20.2237 1.09840
\(340\) 9.49662 0.515026
\(341\) 0 0
\(342\) 13.0721 0.706859
\(343\) 2.70213 0.145901
\(344\) −18.6614 −1.00616
\(345\) −4.80040 −0.258445
\(346\) 47.9605 2.57837
\(347\) −28.8351 −1.54795 −0.773976 0.633215i \(-0.781735\pi\)
−0.773976 + 0.633215i \(0.781735\pi\)
\(348\) 35.9101 1.92498
\(349\) 1.63234 0.0873771 0.0436886 0.999045i \(-0.486089\pi\)
0.0436886 + 0.999045i \(0.486089\pi\)
\(350\) −0.455887 −0.0243682
\(351\) −0.973708 −0.0519727
\(352\) 0 0
\(353\) 24.9297 1.32687 0.663437 0.748232i \(-0.269097\pi\)
0.663437 + 0.748232i \(0.269097\pi\)
\(354\) −26.4260 −1.40452
\(355\) 5.99078 0.317958
\(356\) 26.0607 1.38122
\(357\) −0.517822 −0.0274060
\(358\) 9.07211 0.479476
\(359\) −28.4409 −1.50106 −0.750528 0.660839i \(-0.770201\pi\)
−0.750528 + 0.660839i \(0.770201\pi\)
\(360\) 3.64941 0.192341
\(361\) 11.7936 0.620718
\(362\) −41.4955 −2.18095
\(363\) 0 0
\(364\) 0.668808 0.0350550
\(365\) −3.30624 −0.173056
\(366\) −2.84100 −0.148501
\(367\) −10.4413 −0.545030 −0.272515 0.962152i \(-0.587855\pi\)
−0.272515 + 0.962152i \(0.587855\pi\)
\(368\) −7.19305 −0.374964
\(369\) 8.27861 0.430967
\(370\) 13.4560 0.699545
\(371\) 1.79763 0.0933284
\(372\) −8.89190 −0.461024
\(373\) 2.81747 0.145883 0.0729416 0.997336i \(-0.476761\pi\)
0.0729416 + 0.997336i \(0.476761\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) −39.6045 −2.04245
\(377\) −9.85178 −0.507393
\(378\) −0.455887 −0.0234483
\(379\) 8.93319 0.458867 0.229434 0.973324i \(-0.426313\pi\)
0.229434 + 0.973324i \(0.426313\pi\)
\(380\) 19.6952 1.01034
\(381\) −7.10021 −0.363755
\(382\) 0.982550 0.0502716
\(383\) −8.53279 −0.436005 −0.218003 0.975948i \(-0.569954\pi\)
−0.218003 + 0.975948i \(0.569954\pi\)
\(384\) −20.4366 −1.04290
\(385\) 0 0
\(386\) −27.5532 −1.40242
\(387\) −5.11353 −0.259936
\(388\) −56.2630 −2.85632
\(389\) 11.7757 0.597054 0.298527 0.954401i \(-0.403505\pi\)
0.298527 + 0.954401i \(0.403505\pi\)
\(390\) −2.29374 −0.116148
\(391\) −12.8445 −0.649572
\(392\) −25.4092 −1.28336
\(393\) 2.50024 0.126120
\(394\) −50.8728 −2.56293
\(395\) −10.1529 −0.510849
\(396\) 0 0
\(397\) −1.41214 −0.0708735 −0.0354368 0.999372i \(-0.511282\pi\)
−0.0354368 + 0.999372i \(0.511282\pi\)
\(398\) 18.2807 0.916328
\(399\) −1.07392 −0.0537633
\(400\) 1.49843 0.0749214
\(401\) 8.45917 0.422431 0.211215 0.977440i \(-0.432258\pi\)
0.211215 + 0.977440i \(0.432258\pi\)
\(402\) 7.67765 0.382926
\(403\) 2.43946 0.121518
\(404\) −46.7044 −2.32363
\(405\) 1.00000 0.0496904
\(406\) −4.61257 −0.228918
\(407\) 0 0
\(408\) 9.76476 0.483428
\(409\) −10.1255 −0.500675 −0.250337 0.968159i \(-0.580542\pi\)
−0.250337 + 0.968159i \(0.580542\pi\)
\(410\) 19.5017 0.963121
\(411\) −15.2944 −0.754416
\(412\) 14.1875 0.698966
\(413\) 2.17099 0.106827
\(414\) −11.3082 −0.555767
\(415\) −8.31508 −0.408171
\(416\) 3.66993 0.179933
\(417\) 19.6586 0.962686
\(418\) 0 0
\(419\) 32.8019 1.60248 0.801240 0.598343i \(-0.204174\pi\)
0.801240 + 0.598343i \(0.204174\pi\)
\(420\) −0.686867 −0.0335157
\(421\) −14.1293 −0.688619 −0.344309 0.938856i \(-0.611887\pi\)
−0.344309 + 0.938856i \(0.611887\pi\)
\(422\) 29.4593 1.43406
\(423\) −10.8523 −0.527657
\(424\) −33.8986 −1.64626
\(425\) 2.67571 0.129791
\(426\) 14.1123 0.683745
\(427\) 0.233398 0.0112949
\(428\) −17.3443 −0.838368
\(429\) 0 0
\(430\) −12.0458 −0.580901
\(431\) −19.1057 −0.920291 −0.460145 0.887844i \(-0.652203\pi\)
−0.460145 + 0.887844i \(0.652203\pi\)
\(432\) 1.49843 0.0720931
\(433\) 15.1200 0.726622 0.363311 0.931668i \(-0.381646\pi\)
0.363311 + 0.931668i \(0.381646\pi\)
\(434\) 1.14214 0.0548247
\(435\) 10.1178 0.485111
\(436\) 26.6892 1.27818
\(437\) −26.6384 −1.27429
\(438\) −7.78841 −0.372145
\(439\) −37.1642 −1.77375 −0.886876 0.462008i \(-0.847129\pi\)
−0.886876 + 0.462008i \(0.847129\pi\)
\(440\) 0 0
\(441\) −6.96255 −0.331550
\(442\) −6.13737 −0.291925
\(443\) −2.76049 −0.131155 −0.0655775 0.997847i \(-0.520889\pi\)
−0.0655775 + 0.997847i \(0.520889\pi\)
\(444\) 20.2737 0.962145
\(445\) 7.34270 0.348078
\(446\) −12.6633 −0.599627
\(447\) 10.1329 0.479271
\(448\) 2.29822 0.108581
\(449\) −17.8661 −0.843156 −0.421578 0.906792i \(-0.638524\pi\)
−0.421578 + 0.906792i \(0.638524\pi\)
\(450\) 2.35567 0.111048
\(451\) 0 0
\(452\) 71.7780 3.37615
\(453\) −11.9632 −0.562078
\(454\) 51.2428 2.40494
\(455\) 0.188439 0.00883415
\(456\) 20.2513 0.948356
\(457\) 21.1621 0.989922 0.494961 0.868915i \(-0.335182\pi\)
0.494961 + 0.868915i \(0.335182\pi\)
\(458\) 6.29142 0.293979
\(459\) 2.67571 0.124891
\(460\) −17.0376 −0.794382
\(461\) 34.3476 1.59973 0.799864 0.600181i \(-0.204905\pi\)
0.799864 + 0.600181i \(0.204905\pi\)
\(462\) 0 0
\(463\) 10.4516 0.485727 0.242864 0.970060i \(-0.421913\pi\)
0.242864 + 0.970060i \(0.421913\pi\)
\(464\) 15.1608 0.703822
\(465\) −2.50533 −0.116182
\(466\) 13.4909 0.624955
\(467\) 29.6434 1.37173 0.685866 0.727728i \(-0.259424\pi\)
0.685866 + 0.727728i \(0.259424\pi\)
\(468\) −3.45589 −0.159748
\(469\) −0.630746 −0.0291252
\(470\) −25.5645 −1.17920
\(471\) −2.62605 −0.121002
\(472\) −40.9392 −1.88438
\(473\) 0 0
\(474\) −23.9170 −1.09854
\(475\) 5.54920 0.254615
\(476\) −1.83785 −0.0842378
\(477\) −9.28879 −0.425304
\(478\) 57.4031 2.62556
\(479\) 13.2186 0.603974 0.301987 0.953312i \(-0.402350\pi\)
0.301987 + 0.953312i \(0.402350\pi\)
\(480\) −3.76902 −0.172031
\(481\) −5.56199 −0.253605
\(482\) 39.3591 1.79276
\(483\) 0.929007 0.0422713
\(484\) 0 0
\(485\) −15.8523 −0.719816
\(486\) 2.35567 0.106856
\(487\) −42.4976 −1.92575 −0.962875 0.269949i \(-0.912993\pi\)
−0.962875 + 0.269949i \(0.912993\pi\)
\(488\) −4.40128 −0.199236
\(489\) 23.7235 1.07281
\(490\) −16.4015 −0.740944
\(491\) 15.7457 0.710592 0.355296 0.934754i \(-0.384380\pi\)
0.355296 + 0.934754i \(0.384380\pi\)
\(492\) 29.3825 1.32466
\(493\) 27.0722 1.21927
\(494\) −12.7284 −0.572679
\(495\) 0 0
\(496\) −3.75405 −0.168562
\(497\) −1.15938 −0.0520052
\(498\) −19.5876 −0.877742
\(499\) −33.5279 −1.50092 −0.750458 0.660918i \(-0.770167\pi\)
−0.750458 + 0.660918i \(0.770167\pi\)
\(500\) 3.54920 0.158725
\(501\) −3.49175 −0.156000
\(502\) −37.9226 −1.69257
\(503\) 22.8163 1.01733 0.508664 0.860965i \(-0.330139\pi\)
0.508664 + 0.860965i \(0.330139\pi\)
\(504\) −0.706260 −0.0314593
\(505\) −13.1591 −0.585574
\(506\) 0 0
\(507\) −12.0519 −0.535243
\(508\) −25.2001 −1.11807
\(509\) 39.7166 1.76041 0.880204 0.474596i \(-0.157406\pi\)
0.880204 + 0.474596i \(0.157406\pi\)
\(510\) 6.30309 0.279106
\(511\) 0.639846 0.0283051
\(512\) −16.5844 −0.732933
\(513\) 5.54920 0.245003
\(514\) 14.0604 0.620179
\(515\) 3.99737 0.176145
\(516\) −18.1490 −0.798963
\(517\) 0 0
\(518\) −2.60410 −0.114418
\(519\) 20.3596 0.893686
\(520\) −3.55346 −0.155830
\(521\) 25.5004 1.11719 0.558596 0.829440i \(-0.311340\pi\)
0.558596 + 0.829440i \(0.311340\pi\)
\(522\) 23.8342 1.04320
\(523\) 15.4304 0.674725 0.337363 0.941375i \(-0.390465\pi\)
0.337363 + 0.941375i \(0.390465\pi\)
\(524\) 8.87385 0.387656
\(525\) −0.193527 −0.00844621
\(526\) 1.06276 0.0463385
\(527\) −6.70351 −0.292010
\(528\) 0 0
\(529\) 0.0438407 0.00190612
\(530\) −21.8814 −0.950465
\(531\) −11.2180 −0.486820
\(532\) −3.81156 −0.165252
\(533\) −8.06095 −0.349159
\(534\) 17.2970 0.748515
\(535\) −4.88682 −0.211276
\(536\) 11.8942 0.513752
\(537\) 3.85117 0.166190
\(538\) 34.6749 1.49494
\(539\) 0 0
\(540\) 3.54920 0.152733
\(541\) −27.6007 −1.18665 −0.593324 0.804964i \(-0.702185\pi\)
−0.593324 + 0.804964i \(0.702185\pi\)
\(542\) −10.4306 −0.448033
\(543\) −17.6151 −0.755937
\(544\) −10.0848 −0.432382
\(545\) 7.51977 0.322111
\(546\) 0.443901 0.0189972
\(547\) −30.7685 −1.31557 −0.657784 0.753207i \(-0.728506\pi\)
−0.657784 + 0.753207i \(0.728506\pi\)
\(548\) −54.2828 −2.31885
\(549\) −1.20602 −0.0514718
\(550\) 0 0
\(551\) 56.1457 2.39189
\(552\) −17.5186 −0.745643
\(553\) 1.96487 0.0835546
\(554\) 0.217499 0.00924065
\(555\) 5.71217 0.242468
\(556\) 69.7723 2.95901
\(557\) 13.9359 0.590482 0.295241 0.955423i \(-0.404600\pi\)
0.295241 + 0.955423i \(0.404600\pi\)
\(558\) −5.90173 −0.249840
\(559\) 4.97909 0.210593
\(560\) −0.289986 −0.0122542
\(561\) 0 0
\(562\) −12.2095 −0.515026
\(563\) −5.83988 −0.246122 −0.123061 0.992399i \(-0.539271\pi\)
−0.123061 + 0.992399i \(0.539271\pi\)
\(564\) −38.5170 −1.62186
\(565\) 20.2237 0.850818
\(566\) −3.68575 −0.154924
\(567\) −0.193527 −0.00812737
\(568\) 21.8628 0.917345
\(569\) 5.45394 0.228641 0.114321 0.993444i \(-0.463531\pi\)
0.114321 + 0.993444i \(0.463531\pi\)
\(570\) 13.0721 0.547530
\(571\) −40.2894 −1.68606 −0.843030 0.537866i \(-0.819230\pi\)
−0.843030 + 0.537866i \(0.819230\pi\)
\(572\) 0 0
\(573\) 0.417099 0.0174246
\(574\) −3.77411 −0.157528
\(575\) −4.80040 −0.200191
\(576\) −11.8754 −0.494810
\(577\) −23.8236 −0.991789 −0.495894 0.868383i \(-0.665160\pi\)
−0.495894 + 0.868383i \(0.665160\pi\)
\(578\) −23.1812 −0.964213
\(579\) −11.6965 −0.486091
\(580\) 35.9101 1.49109
\(581\) 1.60919 0.0667606
\(582\) −37.3428 −1.54791
\(583\) 0 0
\(584\) −12.0658 −0.499287
\(585\) −0.973708 −0.0402579
\(586\) 54.5504 2.25346
\(587\) 19.6080 0.809307 0.404654 0.914470i \(-0.367392\pi\)
0.404654 + 0.914470i \(0.367392\pi\)
\(588\) −24.7115 −1.01908
\(589\) −13.9026 −0.572845
\(590\) −26.4260 −1.08794
\(591\) −21.5958 −0.888334
\(592\) 8.55928 0.351784
\(593\) −3.31095 −0.135964 −0.0679822 0.997687i \(-0.521656\pi\)
−0.0679822 + 0.997687i \(0.521656\pi\)
\(594\) 0 0
\(595\) −0.517822 −0.0212286
\(596\) 35.9638 1.47313
\(597\) 7.76028 0.317607
\(598\) 11.0109 0.450268
\(599\) 32.0710 1.31039 0.655193 0.755462i \(-0.272587\pi\)
0.655193 + 0.755462i \(0.272587\pi\)
\(600\) 3.64941 0.148987
\(601\) 29.2905 1.19478 0.597392 0.801950i \(-0.296204\pi\)
0.597392 + 0.801950i \(0.296204\pi\)
\(602\) 2.33119 0.0950123
\(603\) 3.25922 0.132725
\(604\) −42.4596 −1.72766
\(605\) 0 0
\(606\) −30.9986 −1.25923
\(607\) 39.0806 1.58623 0.793117 0.609070i \(-0.208457\pi\)
0.793117 + 0.609070i \(0.208457\pi\)
\(608\) −20.9151 −0.848217
\(609\) −1.95807 −0.0793449
\(610\) −2.84100 −0.115029
\(611\) 10.5670 0.427494
\(612\) 9.49662 0.383878
\(613\) 12.7324 0.514256 0.257128 0.966377i \(-0.417224\pi\)
0.257128 + 0.966377i \(0.417224\pi\)
\(614\) 18.1971 0.734376
\(615\) 8.27861 0.333826
\(616\) 0 0
\(617\) −8.97789 −0.361436 −0.180718 0.983535i \(-0.557842\pi\)
−0.180718 + 0.983535i \(0.557842\pi\)
\(618\) 9.41649 0.378787
\(619\) −22.2121 −0.892780 −0.446390 0.894839i \(-0.647290\pi\)
−0.446390 + 0.894839i \(0.647290\pi\)
\(620\) −8.89190 −0.357107
\(621\) −4.80040 −0.192633
\(622\) −45.0425 −1.80604
\(623\) −1.42101 −0.0569316
\(624\) −1.45903 −0.0584080
\(625\) 1.00000 0.0400000
\(626\) 6.83868 0.273329
\(627\) 0 0
\(628\) −9.32038 −0.371924
\(629\) 15.2841 0.609417
\(630\) −0.455887 −0.0181630
\(631\) 26.7672 1.06559 0.532794 0.846245i \(-0.321142\pi\)
0.532794 + 0.846245i \(0.321142\pi\)
\(632\) −37.0522 −1.47386
\(633\) 12.5057 0.497056
\(634\) 5.32744 0.211580
\(635\) −7.10021 −0.281763
\(636\) −32.9678 −1.30726
\(637\) 6.77949 0.268613
\(638\) 0 0
\(639\) 5.99078 0.236992
\(640\) −20.4366 −0.807829
\(641\) 15.0726 0.595333 0.297666 0.954670i \(-0.403792\pi\)
0.297666 + 0.954670i \(0.403792\pi\)
\(642\) −11.5117 −0.454332
\(643\) −39.2872 −1.54934 −0.774669 0.632367i \(-0.782083\pi\)
−0.774669 + 0.632367i \(0.782083\pi\)
\(644\) 3.29723 0.129929
\(645\) −5.11353 −0.201345
\(646\) 34.9771 1.37616
\(647\) 23.8136 0.936208 0.468104 0.883673i \(-0.344937\pi\)
0.468104 + 0.883673i \(0.344937\pi\)
\(648\) 3.64941 0.143363
\(649\) 0 0
\(650\) −2.29374 −0.0899679
\(651\) 0.484848 0.0190027
\(652\) 84.1994 3.29750
\(653\) 6.90354 0.270156 0.135078 0.990835i \(-0.456871\pi\)
0.135078 + 0.990835i \(0.456871\pi\)
\(654\) 17.7141 0.692677
\(655\) 2.50024 0.0976924
\(656\) 12.4049 0.484330
\(657\) −3.30624 −0.128989
\(658\) 4.94742 0.192870
\(659\) 20.3718 0.793571 0.396786 0.917911i \(-0.370126\pi\)
0.396786 + 0.917911i \(0.370126\pi\)
\(660\) 0 0
\(661\) −19.7451 −0.767994 −0.383997 0.923334i \(-0.625453\pi\)
−0.383997 + 0.923334i \(0.625453\pi\)
\(662\) 14.1891 0.551474
\(663\) −2.60536 −0.101184
\(664\) −30.3452 −1.17762
\(665\) −1.07392 −0.0416449
\(666\) 13.4560 0.521410
\(667\) −48.5695 −1.88062
\(668\) −12.3929 −0.479496
\(669\) −5.37568 −0.207836
\(670\) 7.67765 0.296613
\(671\) 0 0
\(672\) 0.729407 0.0281375
\(673\) −36.6132 −1.41134 −0.705669 0.708542i \(-0.749353\pi\)
−0.705669 + 0.708542i \(0.749353\pi\)
\(674\) 34.3098 1.32157
\(675\) 1.00000 0.0384900
\(676\) −42.7746 −1.64518
\(677\) −7.37138 −0.283305 −0.141653 0.989916i \(-0.545242\pi\)
−0.141653 + 0.989916i \(0.545242\pi\)
\(678\) 47.6405 1.82962
\(679\) 3.06785 0.117733
\(680\) 9.76476 0.374461
\(681\) 21.7529 0.833573
\(682\) 0 0
\(683\) 24.5651 0.939959 0.469979 0.882677i \(-0.344261\pi\)
0.469979 + 0.882677i \(0.344261\pi\)
\(684\) 19.6952 0.753066
\(685\) −15.2944 −0.584368
\(686\) 6.36534 0.243030
\(687\) 2.67075 0.101896
\(688\) −7.66226 −0.292121
\(689\) 9.04457 0.344571
\(690\) −11.3082 −0.430495
\(691\) 29.8435 1.13530 0.567651 0.823269i \(-0.307852\pi\)
0.567651 + 0.823269i \(0.307852\pi\)
\(692\) 72.2602 2.74692
\(693\) 0 0
\(694\) −67.9262 −2.57844
\(695\) 19.6586 0.745693
\(696\) 36.9240 1.39960
\(697\) 22.1511 0.839033
\(698\) 3.84526 0.145545
\(699\) 5.72699 0.216615
\(700\) −0.686867 −0.0259611
\(701\) 14.4189 0.544595 0.272297 0.962213i \(-0.412217\pi\)
0.272297 + 0.962213i \(0.412217\pi\)
\(702\) −2.29374 −0.0865716
\(703\) 31.6980 1.19551
\(704\) 0 0
\(705\) −10.8523 −0.408721
\(706\) 58.7263 2.21019
\(707\) 2.54665 0.0957766
\(708\) −39.8150 −1.49634
\(709\) −44.5088 −1.67156 −0.835781 0.549063i \(-0.814985\pi\)
−0.835781 + 0.549063i \(0.814985\pi\)
\(710\) 14.1123 0.529626
\(711\) −10.1529 −0.380765
\(712\) 26.7966 1.00424
\(713\) 12.0266 0.450398
\(714\) −1.21982 −0.0456506
\(715\) 0 0
\(716\) 13.6686 0.510819
\(717\) 24.3680 0.910040
\(718\) −66.9976 −2.50033
\(719\) 3.59345 0.134013 0.0670066 0.997753i \(-0.478655\pi\)
0.0670066 + 0.997753i \(0.478655\pi\)
\(720\) 1.49843 0.0558431
\(721\) −0.773598 −0.0288103
\(722\) 27.7820 1.03394
\(723\) 16.7082 0.621385
\(724\) −62.5196 −2.32352
\(725\) 10.1178 0.375766
\(726\) 0 0
\(727\) −39.0846 −1.44957 −0.724784 0.688976i \(-0.758060\pi\)
−0.724784 + 0.688976i \(0.758060\pi\)
\(728\) 0.687692 0.0254875
\(729\) 1.00000 0.0370370
\(730\) −7.78841 −0.288262
\(731\) −13.6823 −0.506059
\(732\) −4.28042 −0.158209
\(733\) −37.7065 −1.39272 −0.696360 0.717693i \(-0.745198\pi\)
−0.696360 + 0.717693i \(0.745198\pi\)
\(734\) −24.5962 −0.907863
\(735\) −6.96255 −0.256817
\(736\) 18.0928 0.666910
\(737\) 0 0
\(738\) 19.5017 0.717868
\(739\) −14.9324 −0.549298 −0.274649 0.961545i \(-0.588562\pi\)
−0.274649 + 0.961545i \(0.588562\pi\)
\(740\) 20.2737 0.745274
\(741\) −5.40330 −0.198495
\(742\) 4.23463 0.155458
\(743\) −13.3680 −0.490423 −0.245211 0.969470i \(-0.578857\pi\)
−0.245211 + 0.969470i \(0.578857\pi\)
\(744\) −9.14297 −0.335198
\(745\) 10.1329 0.371242
\(746\) 6.63705 0.243000
\(747\) −8.31508 −0.304233
\(748\) 0 0
\(749\) 0.945731 0.0345563
\(750\) 2.35567 0.0860171
\(751\) 40.4655 1.47661 0.738303 0.674469i \(-0.235627\pi\)
0.738303 + 0.674469i \(0.235627\pi\)
\(752\) −16.2614 −0.592991
\(753\) −16.0984 −0.586658
\(754\) −23.2076 −0.845171
\(755\) −11.9632 −0.435384
\(756\) −0.686867 −0.0249811
\(757\) −8.53757 −0.310303 −0.155152 0.987891i \(-0.549587\pi\)
−0.155152 + 0.987891i \(0.549587\pi\)
\(758\) 21.0437 0.764341
\(759\) 0 0
\(760\) 20.2513 0.734593
\(761\) 21.2805 0.771416 0.385708 0.922621i \(-0.373957\pi\)
0.385708 + 0.922621i \(0.373957\pi\)
\(762\) −16.7258 −0.605911
\(763\) −1.45528 −0.0526846
\(764\) 1.48037 0.0535579
\(765\) 2.67571 0.0967403
\(766\) −20.1005 −0.726260
\(767\) 10.9231 0.394409
\(768\) −24.3912 −0.880140
\(769\) 24.4717 0.882471 0.441235 0.897391i \(-0.354540\pi\)
0.441235 + 0.897391i \(0.354540\pi\)
\(770\) 0 0
\(771\) 5.96875 0.214959
\(772\) −41.5134 −1.49410
\(773\) 13.6472 0.490855 0.245427 0.969415i \(-0.421072\pi\)
0.245427 + 0.969415i \(0.421072\pi\)
\(774\) −12.0458 −0.432978
\(775\) −2.50533 −0.0899940
\(776\) −57.8516 −2.07675
\(777\) −1.10546 −0.0396582
\(778\) 27.7398 0.994520
\(779\) 45.9397 1.64596
\(780\) −3.45589 −0.123741
\(781\) 0 0
\(782\) −30.2574 −1.08200
\(783\) 10.1178 0.361581
\(784\) −10.4329 −0.372603
\(785\) −2.62605 −0.0937278
\(786\) 5.88974 0.210080
\(787\) 7.45709 0.265816 0.132908 0.991128i \(-0.457568\pi\)
0.132908 + 0.991128i \(0.457568\pi\)
\(788\) −76.6480 −2.73047
\(789\) 0.451149 0.0160613
\(790\) −23.9170 −0.850929
\(791\) −3.91383 −0.139160
\(792\) 0 0
\(793\) 1.17431 0.0417011
\(794\) −3.32655 −0.118055
\(795\) −9.28879 −0.329439
\(796\) 27.5428 0.976228
\(797\) 12.0347 0.426289 0.213145 0.977021i \(-0.431629\pi\)
0.213145 + 0.977021i \(0.431629\pi\)
\(798\) −2.52981 −0.0895542
\(799\) −29.0375 −1.02727
\(800\) −3.76902 −0.133255
\(801\) 7.34270 0.259442
\(802\) 19.9270 0.703648
\(803\) 0 0
\(804\) 11.5676 0.407958
\(805\) 0.929007 0.0327432
\(806\) 5.74656 0.202414
\(807\) 14.7197 0.518159
\(808\) −48.0231 −1.68945
\(809\) 50.0338 1.75909 0.879546 0.475813i \(-0.157846\pi\)
0.879546 + 0.475813i \(0.157846\pi\)
\(810\) 2.35567 0.0827700
\(811\) 11.2188 0.393944 0.196972 0.980409i \(-0.436889\pi\)
0.196972 + 0.980409i \(0.436889\pi\)
\(812\) −6.94958 −0.243882
\(813\) −4.42787 −0.155292
\(814\) 0 0
\(815\) 23.7235 0.830997
\(816\) 4.00935 0.140355
\(817\) −28.3760 −0.992752
\(818\) −23.8524 −0.833981
\(819\) 0.188439 0.00658459
\(820\) 29.3825 1.02608
\(821\) −7.37723 −0.257467 −0.128734 0.991679i \(-0.541091\pi\)
−0.128734 + 0.991679i \(0.541091\pi\)
\(822\) −36.0286 −1.25664
\(823\) 35.6380 1.24226 0.621132 0.783706i \(-0.286673\pi\)
0.621132 + 0.783706i \(0.286673\pi\)
\(824\) 14.5880 0.508198
\(825\) 0 0
\(826\) 5.11414 0.177944
\(827\) 50.5291 1.75707 0.878535 0.477678i \(-0.158522\pi\)
0.878535 + 0.477678i \(0.158522\pi\)
\(828\) −17.0376 −0.592097
\(829\) −6.14491 −0.213422 −0.106711 0.994290i \(-0.534032\pi\)
−0.106711 + 0.994290i \(0.534032\pi\)
\(830\) −19.5876 −0.679896
\(831\) 0.0923299 0.00320289
\(832\) 11.5632 0.400882
\(833\) −18.6297 −0.645482
\(834\) 46.3093 1.60356
\(835\) −3.49175 −0.120837
\(836\) 0 0
\(837\) −2.50533 −0.0865967
\(838\) 77.2707 2.66927
\(839\) −37.5230 −1.29544 −0.647719 0.761879i \(-0.724277\pi\)
−0.647719 + 0.761879i \(0.724277\pi\)
\(840\) −0.706260 −0.0243683
\(841\) 73.3698 2.52999
\(842\) −33.2840 −1.14704
\(843\) −5.18301 −0.178512
\(844\) 44.3852 1.52780
\(845\) −12.0519 −0.414598
\(846\) −25.5645 −0.878924
\(847\) 0 0
\(848\) −13.9186 −0.477966
\(849\) −1.56463 −0.0536978
\(850\) 6.30309 0.216194
\(851\) −27.4207 −0.939970
\(852\) 21.2625 0.728441
\(853\) 27.0122 0.924880 0.462440 0.886651i \(-0.346974\pi\)
0.462440 + 0.886651i \(0.346974\pi\)
\(854\) 0.549810 0.0188141
\(855\) 5.54920 0.189779
\(856\) −17.8340 −0.609554
\(857\) 2.51515 0.0859159 0.0429580 0.999077i \(-0.486322\pi\)
0.0429580 + 0.999077i \(0.486322\pi\)
\(858\) 0 0
\(859\) 6.70885 0.228903 0.114451 0.993429i \(-0.463489\pi\)
0.114451 + 0.993429i \(0.463489\pi\)
\(860\) −18.1490 −0.618874
\(861\) −1.60214 −0.0546006
\(862\) −45.0069 −1.53294
\(863\) −3.30744 −0.112586 −0.0562932 0.998414i \(-0.517928\pi\)
−0.0562932 + 0.998414i \(0.517928\pi\)
\(864\) −3.76902 −0.128225
\(865\) 20.3596 0.692246
\(866\) 35.6179 1.21034
\(867\) −9.84060 −0.334204
\(868\) 1.72082 0.0584086
\(869\) 0 0
\(870\) 23.8342 0.808056
\(871\) −3.17352 −0.107531
\(872\) 27.4427 0.929328
\(873\) −15.8523 −0.536519
\(874\) −62.7514 −2.12260
\(875\) −0.193527 −0.00654241
\(876\) −11.7345 −0.396472
\(877\) −32.1004 −1.08395 −0.541976 0.840394i \(-0.682324\pi\)
−0.541976 + 0.840394i \(0.682324\pi\)
\(878\) −87.5468 −2.95456
\(879\) 23.1570 0.781067
\(880\) 0 0
\(881\) −35.2547 −1.18776 −0.593881 0.804553i \(-0.702405\pi\)
−0.593881 + 0.804553i \(0.702405\pi\)
\(882\) −16.4015 −0.552267
\(883\) −0.455852 −0.0153406 −0.00767032 0.999971i \(-0.502442\pi\)
−0.00767032 + 0.999971i \(0.502442\pi\)
\(884\) −9.24694 −0.311008
\(885\) −11.2180 −0.377089
\(886\) −6.50282 −0.218467
\(887\) 1.80160 0.0604918 0.0302459 0.999542i \(-0.490371\pi\)
0.0302459 + 0.999542i \(0.490371\pi\)
\(888\) 20.8461 0.699549
\(889\) 1.37408 0.0460853
\(890\) 17.2970 0.579797
\(891\) 0 0
\(892\) −19.0794 −0.638824
\(893\) −60.2216 −2.01524
\(894\) 23.8699 0.798328
\(895\) 3.85117 0.128731
\(896\) 3.95504 0.132129
\(897\) 4.67419 0.156067
\(898\) −42.0868 −1.40446
\(899\) −25.3484 −0.845416
\(900\) 3.54920 0.118307
\(901\) −24.8541 −0.828009
\(902\) 0 0
\(903\) 0.989607 0.0329321
\(904\) 73.8047 2.45471
\(905\) −17.6151 −0.585546
\(906\) −28.1813 −0.936261
\(907\) −35.3469 −1.17367 −0.586837 0.809705i \(-0.699627\pi\)
−0.586837 + 0.809705i \(0.699627\pi\)
\(908\) 77.2054 2.56215
\(909\) −13.1591 −0.436461
\(910\) 0.443901 0.0147152
\(911\) −1.64877 −0.0546262 −0.0273131 0.999627i \(-0.508695\pi\)
−0.0273131 + 0.999627i \(0.508695\pi\)
\(912\) 8.31508 0.275340
\(913\) 0 0
\(914\) 49.8511 1.64893
\(915\) −1.20602 −0.0398699
\(916\) 9.47903 0.313196
\(917\) −0.483864 −0.0159786
\(918\) 6.30309 0.208033
\(919\) 0.318990 0.0105225 0.00526125 0.999986i \(-0.498325\pi\)
0.00526125 + 0.999986i \(0.498325\pi\)
\(920\) −17.5186 −0.577573
\(921\) 7.72480 0.254541
\(922\) 80.9118 2.66469
\(923\) −5.83327 −0.192005
\(924\) 0 0
\(925\) 5.71217 0.187815
\(926\) 24.6206 0.809082
\(927\) 3.99737 0.131291
\(928\) −38.1342 −1.25182
\(929\) 56.1509 1.84225 0.921125 0.389267i \(-0.127272\pi\)
0.921125 + 0.389267i \(0.127272\pi\)
\(930\) −5.90173 −0.193525
\(931\) −38.6366 −1.26626
\(932\) 20.3262 0.665808
\(933\) −19.1209 −0.625989
\(934\) 69.8301 2.28491
\(935\) 0 0
\(936\) −3.55346 −0.116149
\(937\) −36.8424 −1.20359 −0.601795 0.798651i \(-0.705547\pi\)
−0.601795 + 0.798651i \(0.705547\pi\)
\(938\) −1.48583 −0.0485142
\(939\) 2.90307 0.0947380
\(940\) −38.5170 −1.25629
\(941\) 15.3105 0.499108 0.249554 0.968361i \(-0.419716\pi\)
0.249554 + 0.968361i \(0.419716\pi\)
\(942\) −6.18612 −0.201555
\(943\) −39.7406 −1.29413
\(944\) −16.8094 −0.547099
\(945\) −0.193527 −0.00629544
\(946\) 0 0
\(947\) −22.6654 −0.736526 −0.368263 0.929722i \(-0.620047\pi\)
−0.368263 + 0.929722i \(0.620047\pi\)
\(948\) −36.0348 −1.17036
\(949\) 3.21931 0.104503
\(950\) 13.0721 0.424115
\(951\) 2.26153 0.0733353
\(952\) −1.88974 −0.0612470
\(953\) −39.3324 −1.27410 −0.637051 0.770822i \(-0.719846\pi\)
−0.637051 + 0.770822i \(0.719846\pi\)
\(954\) −21.8814 −0.708435
\(955\) 0.417099 0.0134970
\(956\) 86.4870 2.79719
\(957\) 0 0
\(958\) 31.1388 1.00605
\(959\) 2.95988 0.0955794
\(960\) −11.8754 −0.383278
\(961\) −24.7233 −0.797527
\(962\) −13.1022 −0.422433
\(963\) −4.88682 −0.157475
\(964\) 59.3008 1.90995
\(965\) −11.6965 −0.376525
\(966\) 2.18844 0.0704119
\(967\) −3.06103 −0.0984360 −0.0492180 0.998788i \(-0.515673\pi\)
−0.0492180 + 0.998788i \(0.515673\pi\)
\(968\) 0 0
\(969\) 14.8480 0.476987
\(970\) −37.3428 −1.19901
\(971\) 32.0385 1.02816 0.514082 0.857741i \(-0.328133\pi\)
0.514082 + 0.857741i \(0.328133\pi\)
\(972\) 3.54920 0.113841
\(973\) −3.80447 −0.121966
\(974\) −100.110 −3.20775
\(975\) −0.973708 −0.0311836
\(976\) −1.80714 −0.0578451
\(977\) −53.1962 −1.70190 −0.850948 0.525250i \(-0.823972\pi\)
−0.850948 + 0.525250i \(0.823972\pi\)
\(978\) 55.8848 1.78700
\(979\) 0 0
\(980\) −24.7115 −0.789379
\(981\) 7.51977 0.240088
\(982\) 37.0916 1.18364
\(983\) −24.5004 −0.781442 −0.390721 0.920509i \(-0.627774\pi\)
−0.390721 + 0.920509i \(0.627774\pi\)
\(984\) 30.2121 0.963126
\(985\) −21.5958 −0.688101
\(986\) 63.7734 2.03096
\(987\) 2.10021 0.0668505
\(988\) −19.1774 −0.610115
\(989\) 24.5470 0.780549
\(990\) 0 0
\(991\) −6.34819 −0.201657 −0.100828 0.994904i \(-0.532149\pi\)
−0.100828 + 0.994904i \(0.532149\pi\)
\(992\) 9.44262 0.299804
\(993\) 6.02336 0.191146
\(994\) −2.73112 −0.0866258
\(995\) 7.76028 0.246017
\(996\) −29.5119 −0.935120
\(997\) 25.9460 0.821719 0.410860 0.911699i \(-0.365229\pi\)
0.410860 + 0.911699i \(0.365229\pi\)
\(998\) −78.9808 −2.50010
\(999\) 5.71217 0.180725
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 1815.2.a.w.1.4 4
3.2 odd 2 5445.2.a.bf.1.1 4
5.4 even 2 9075.2.a.cm.1.1 4
11.2 odd 10 165.2.m.d.136.2 yes 8
11.6 odd 10 165.2.m.d.91.2 8
11.10 odd 2 1815.2.a.p.1.1 4
33.2 even 10 495.2.n.a.136.1 8
33.17 even 10 495.2.n.a.91.1 8
33.32 even 2 5445.2.a.bt.1.4 4
55.2 even 20 825.2.bx.f.499.1 16
55.13 even 20 825.2.bx.f.499.4 16
55.17 even 20 825.2.bx.f.124.4 16
55.24 odd 10 825.2.n.g.301.1 8
55.28 even 20 825.2.bx.f.124.1 16
55.39 odd 10 825.2.n.g.751.1 8
55.54 odd 2 9075.2.a.di.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.m.d.91.2 8 11.6 odd 10
165.2.m.d.136.2 yes 8 11.2 odd 10
495.2.n.a.91.1 8 33.17 even 10
495.2.n.a.136.1 8 33.2 even 10
825.2.n.g.301.1 8 55.24 odd 10
825.2.n.g.751.1 8 55.39 odd 10
825.2.bx.f.124.1 16 55.28 even 20
825.2.bx.f.124.4 16 55.17 even 20
825.2.bx.f.499.1 16 55.2 even 20
825.2.bx.f.499.4 16 55.13 even 20
1815.2.a.p.1.1 4 11.10 odd 2
1815.2.a.w.1.4 4 1.1 even 1 trivial
5445.2.a.bf.1.1 4 3.2 odd 2
5445.2.a.bt.1.4 4 33.32 even 2
9075.2.a.cm.1.1 4 5.4 even 2
9075.2.a.di.1.4 4 55.54 odd 2