Properties

Label 405.4.a.c.1.2
Level $405$
Weight $4$
Character 405.1
Self dual yes
Analytic conductor $23.896$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.8957735523\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 405.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.732051 q^{2} -7.46410 q^{4} -5.00000 q^{5} -6.92820 q^{7} -11.3205 q^{8} +O(q^{10})\) \(q+0.732051 q^{2} -7.46410 q^{4} -5.00000 q^{5} -6.92820 q^{7} -11.3205 q^{8} -3.66025 q^{10} -37.4974 q^{11} -38.9282 q^{13} -5.07180 q^{14} +51.4256 q^{16} +80.9948 q^{17} +112.779 q^{19} +37.3205 q^{20} -27.4500 q^{22} -13.4256 q^{23} +25.0000 q^{25} -28.4974 q^{26} +51.7128 q^{28} +43.4308 q^{29} -149.785 q^{31} +128.210 q^{32} +59.2923 q^{34} +34.6410 q^{35} -218.344 q^{37} +82.5603 q^{38} +56.6025 q^{40} +372.138 q^{41} +460.200 q^{43} +279.885 q^{44} -9.82824 q^{46} +214.287 q^{47} -295.000 q^{49} +18.3013 q^{50} +290.564 q^{52} +445.205 q^{53} +187.487 q^{55} +78.4308 q^{56} +31.7935 q^{58} +401.497 q^{59} +1.44624 q^{61} -109.650 q^{62} -317.549 q^{64} +194.641 q^{65} -816.067 q^{67} -604.554 q^{68} +25.3590 q^{70} +147.518 q^{71} +432.651 q^{73} -159.839 q^{74} -841.797 q^{76} +259.790 q^{77} +384.133 q^{79} -257.128 q^{80} +272.424 q^{82} +1261.23 q^{83} -404.974 q^{85} +336.890 q^{86} +424.490 q^{88} +513.000 q^{89} +269.703 q^{91} +100.210 q^{92} +156.869 q^{94} -563.897 q^{95} -1096.10 q^{97} -215.955 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 8 q^{4} - 10 q^{5} + 12 q^{8} + O(q^{10}) \) \( 2 q - 2 q^{2} - 8 q^{4} - 10 q^{5} + 12 q^{8} + 10 q^{10} + 22 q^{11} - 64 q^{13} - 24 q^{14} - 8 q^{16} - 32 q^{17} - 10 q^{19} + 40 q^{20} - 190 q^{22} + 84 q^{23} + 50 q^{25} + 40 q^{26} + 48 q^{28} + 170 q^{29} - 258 q^{31} + 104 q^{32} + 368 q^{34} + 76 q^{37} + 418 q^{38} - 60 q^{40} + 578 q^{41} + 380 q^{43} + 248 q^{44} - 276 q^{46} + 484 q^{47} - 590 q^{49} - 50 q^{50} + 304 q^{52} + 544 q^{53} - 110 q^{55} + 240 q^{56} - 314 q^{58} + 706 q^{59} + 668 q^{61} + 186 q^{62} + 224 q^{64} + 320 q^{65} - 1452 q^{67} - 544 q^{68} + 120 q^{70} + 974 q^{71} + 1184 q^{73} - 964 q^{74} - 776 q^{76} + 672 q^{77} + 408 q^{79} + 40 q^{80} - 290 q^{82} + 444 q^{83} + 160 q^{85} + 556 q^{86} + 1812 q^{88} + 1026 q^{89} + 96 q^{91} + 48 q^{92} - 580 q^{94} + 50 q^{95} - 668 q^{97} + 590 q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.732051 0.258819 0.129410 0.991591i \(-0.458692\pi\)
0.129410 + 0.991591i \(0.458692\pi\)
\(3\) 0 0
\(4\) −7.46410 −0.933013
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −6.92820 −0.374088 −0.187044 0.982352i \(-0.559891\pi\)
−0.187044 + 0.982352i \(0.559891\pi\)
\(8\) −11.3205 −0.500301
\(9\) 0 0
\(10\) −3.66025 −0.115747
\(11\) −37.4974 −1.02781 −0.513904 0.857847i \(-0.671801\pi\)
−0.513904 + 0.857847i \(0.671801\pi\)
\(12\) 0 0
\(13\) −38.9282 −0.830519 −0.415259 0.909703i \(-0.636309\pi\)
−0.415259 + 0.909703i \(0.636309\pi\)
\(14\) −5.07180 −0.0968211
\(15\) 0 0
\(16\) 51.4256 0.803525
\(17\) 80.9948 1.15554 0.577769 0.816201i \(-0.303924\pi\)
0.577769 + 0.816201i \(0.303924\pi\)
\(18\) 0 0
\(19\) 112.779 1.36176 0.680878 0.732396i \(-0.261598\pi\)
0.680878 + 0.732396i \(0.261598\pi\)
\(20\) 37.3205 0.417256
\(21\) 0 0
\(22\) −27.4500 −0.266017
\(23\) −13.4256 −0.121715 −0.0608573 0.998146i \(-0.519383\pi\)
−0.0608573 + 0.998146i \(0.519383\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −28.4974 −0.214954
\(27\) 0 0
\(28\) 51.7128 0.349029
\(29\) 43.4308 0.278100 0.139050 0.990285i \(-0.455595\pi\)
0.139050 + 0.990285i \(0.455595\pi\)
\(30\) 0 0
\(31\) −149.785 −0.867810 −0.433905 0.900959i \(-0.642865\pi\)
−0.433905 + 0.900959i \(0.642865\pi\)
\(32\) 128.210 0.708268
\(33\) 0 0
\(34\) 59.2923 0.299075
\(35\) 34.6410 0.167297
\(36\) 0 0
\(37\) −218.344 −0.970147 −0.485074 0.874473i \(-0.661207\pi\)
−0.485074 + 0.874473i \(0.661207\pi\)
\(38\) 82.5603 0.352449
\(39\) 0 0
\(40\) 56.6025 0.223741
\(41\) 372.138 1.41752 0.708759 0.705450i \(-0.249255\pi\)
0.708759 + 0.705450i \(0.249255\pi\)
\(42\) 0 0
\(43\) 460.200 1.63209 0.816045 0.577989i \(-0.196162\pi\)
0.816045 + 0.577989i \(0.196162\pi\)
\(44\) 279.885 0.958959
\(45\) 0 0
\(46\) −9.82824 −0.0315021
\(47\) 214.287 0.665043 0.332521 0.943096i \(-0.392101\pi\)
0.332521 + 0.943096i \(0.392101\pi\)
\(48\) 0 0
\(49\) −295.000 −0.860058
\(50\) 18.3013 0.0517638
\(51\) 0 0
\(52\) 290.564 0.774884
\(53\) 445.205 1.15384 0.576921 0.816800i \(-0.304254\pi\)
0.576921 + 0.816800i \(0.304254\pi\)
\(54\) 0 0
\(55\) 187.487 0.459650
\(56\) 78.4308 0.187156
\(57\) 0 0
\(58\) 31.7935 0.0719775
\(59\) 401.497 0.885941 0.442970 0.896536i \(-0.353925\pi\)
0.442970 + 0.896536i \(0.353925\pi\)
\(60\) 0 0
\(61\) 1.44624 0.00303562 0.00151781 0.999999i \(-0.499517\pi\)
0.00151781 + 0.999999i \(0.499517\pi\)
\(62\) −109.650 −0.224606
\(63\) 0 0
\(64\) −317.549 −0.620212
\(65\) 194.641 0.371419
\(66\) 0 0
\(67\) −816.067 −1.48804 −0.744018 0.668160i \(-0.767082\pi\)
−0.744018 + 0.668160i \(0.767082\pi\)
\(68\) −604.554 −1.07813
\(69\) 0 0
\(70\) 25.3590 0.0432997
\(71\) 147.518 0.246580 0.123290 0.992371i \(-0.460655\pi\)
0.123290 + 0.992371i \(0.460655\pi\)
\(72\) 0 0
\(73\) 432.651 0.693671 0.346836 0.937926i \(-0.387256\pi\)
0.346836 + 0.937926i \(0.387256\pi\)
\(74\) −159.839 −0.251093
\(75\) 0 0
\(76\) −841.797 −1.27054
\(77\) 259.790 0.384491
\(78\) 0 0
\(79\) 384.133 0.547068 0.273534 0.961862i \(-0.411807\pi\)
0.273534 + 0.961862i \(0.411807\pi\)
\(80\) −257.128 −0.359347
\(81\) 0 0
\(82\) 272.424 0.366881
\(83\) 1261.23 1.66793 0.833964 0.551819i \(-0.186066\pi\)
0.833964 + 0.551819i \(0.186066\pi\)
\(84\) 0 0
\(85\) −404.974 −0.516772
\(86\) 336.890 0.422416
\(87\) 0 0
\(88\) 424.490 0.514213
\(89\) 513.000 0.610988 0.305494 0.952194i \(-0.401178\pi\)
0.305494 + 0.952194i \(0.401178\pi\)
\(90\) 0 0
\(91\) 269.703 0.310687
\(92\) 100.210 0.113561
\(93\) 0 0
\(94\) 156.869 0.172126
\(95\) −563.897 −0.608996
\(96\) 0 0
\(97\) −1096.10 −1.14734 −0.573672 0.819085i \(-0.694482\pi\)
−0.573672 + 0.819085i \(0.694482\pi\)
\(98\) −215.955 −0.222599
\(99\) 0 0
\(100\) −186.603 −0.186603
\(101\) −1194.09 −1.17640 −0.588198 0.808717i \(-0.700163\pi\)
−0.588198 + 0.808717i \(0.700163\pi\)
\(102\) 0 0
\(103\) 181.503 0.173631 0.0868154 0.996224i \(-0.472331\pi\)
0.0868154 + 0.996224i \(0.472331\pi\)
\(104\) 440.687 0.415509
\(105\) 0 0
\(106\) 325.913 0.298636
\(107\) −611.667 −0.552636 −0.276318 0.961066i \(-0.589114\pi\)
−0.276318 + 0.961066i \(0.589114\pi\)
\(108\) 0 0
\(109\) 550.467 0.483717 0.241859 0.970312i \(-0.422243\pi\)
0.241859 + 0.970312i \(0.422243\pi\)
\(110\) 137.250 0.118966
\(111\) 0 0
\(112\) −356.287 −0.300589
\(113\) 475.902 0.396187 0.198094 0.980183i \(-0.436525\pi\)
0.198094 + 0.980183i \(0.436525\pi\)
\(114\) 0 0
\(115\) 67.1281 0.0544324
\(116\) −324.172 −0.259471
\(117\) 0 0
\(118\) 293.917 0.229298
\(119\) −561.149 −0.432272
\(120\) 0 0
\(121\) 75.0567 0.0563912
\(122\) 1.05872 0.000785676 0
\(123\) 0 0
\(124\) 1118.01 0.809678
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −1645.91 −1.15001 −0.575003 0.818152i \(-0.694999\pi\)
−0.575003 + 0.818152i \(0.694999\pi\)
\(128\) −1258.14 −0.868791
\(129\) 0 0
\(130\) 142.487 0.0961304
\(131\) −386.707 −0.257914 −0.128957 0.991650i \(-0.541163\pi\)
−0.128957 + 0.991650i \(0.541163\pi\)
\(132\) 0 0
\(133\) −781.359 −0.509417
\(134\) −597.402 −0.385132
\(135\) 0 0
\(136\) −916.903 −0.578116
\(137\) 2973.22 1.85416 0.927078 0.374869i \(-0.122312\pi\)
0.927078 + 0.374869i \(0.122312\pi\)
\(138\) 0 0
\(139\) 2347.35 1.43237 0.716187 0.697909i \(-0.245886\pi\)
0.716187 + 0.697909i \(0.245886\pi\)
\(140\) −258.564 −0.156090
\(141\) 0 0
\(142\) 107.991 0.0638196
\(143\) 1459.71 0.853614
\(144\) 0 0
\(145\) −217.154 −0.124370
\(146\) 316.723 0.179535
\(147\) 0 0
\(148\) 1629.74 0.905160
\(149\) −1402.09 −0.770898 −0.385449 0.922729i \(-0.625953\pi\)
−0.385449 + 0.922729i \(0.625953\pi\)
\(150\) 0 0
\(151\) −2941.26 −1.58514 −0.792571 0.609780i \(-0.791258\pi\)
−0.792571 + 0.609780i \(0.791258\pi\)
\(152\) −1276.72 −0.681288
\(153\) 0 0
\(154\) 190.179 0.0995135
\(155\) 748.923 0.388096
\(156\) 0 0
\(157\) 2551.98 1.29726 0.648631 0.761103i \(-0.275342\pi\)
0.648631 + 0.761103i \(0.275342\pi\)
\(158\) 281.205 0.141592
\(159\) 0 0
\(160\) −641.051 −0.316747
\(161\) 93.0155 0.0455320
\(162\) 0 0
\(163\) 351.559 0.168934 0.0844669 0.996426i \(-0.473081\pi\)
0.0844669 + 0.996426i \(0.473081\pi\)
\(164\) −2777.68 −1.32256
\(165\) 0 0
\(166\) 923.285 0.431692
\(167\) 20.9893 0.00972576 0.00486288 0.999988i \(-0.498452\pi\)
0.00486288 + 0.999988i \(0.498452\pi\)
\(168\) 0 0
\(169\) −681.595 −0.310239
\(170\) −296.462 −0.133750
\(171\) 0 0
\(172\) −3434.98 −1.52276
\(173\) −4524.61 −1.98844 −0.994219 0.107374i \(-0.965756\pi\)
−0.994219 + 0.107374i \(0.965756\pi\)
\(174\) 0 0
\(175\) −173.205 −0.0748176
\(176\) −1928.33 −0.825871
\(177\) 0 0
\(178\) 375.542 0.158135
\(179\) 3626.79 1.51441 0.757204 0.653179i \(-0.226565\pi\)
0.757204 + 0.653179i \(0.226565\pi\)
\(180\) 0 0
\(181\) 3551.41 1.45842 0.729210 0.684289i \(-0.239888\pi\)
0.729210 + 0.684289i \(0.239888\pi\)
\(182\) 197.436 0.0804117
\(183\) 0 0
\(184\) 151.985 0.0608939
\(185\) 1091.72 0.433863
\(186\) 0 0
\(187\) −3037.10 −1.18767
\(188\) −1599.46 −0.620493
\(189\) 0 0
\(190\) −412.801 −0.157620
\(191\) −279.651 −0.105941 −0.0529707 0.998596i \(-0.516869\pi\)
−0.0529707 + 0.998596i \(0.516869\pi\)
\(192\) 0 0
\(193\) 266.159 0.0992672 0.0496336 0.998767i \(-0.484195\pi\)
0.0496336 + 0.998767i \(0.484195\pi\)
\(194\) −802.403 −0.296954
\(195\) 0 0
\(196\) 2201.91 0.802445
\(197\) −4731.12 −1.71106 −0.855528 0.517756i \(-0.826768\pi\)
−0.855528 + 0.517756i \(0.826768\pi\)
\(198\) 0 0
\(199\) 2879.69 1.02581 0.512904 0.858446i \(-0.328570\pi\)
0.512904 + 0.858446i \(0.328570\pi\)
\(200\) −283.013 −0.100060
\(201\) 0 0
\(202\) −874.132 −0.304474
\(203\) −300.897 −0.104034
\(204\) 0 0
\(205\) −1860.69 −0.633934
\(206\) 132.869 0.0449390
\(207\) 0 0
\(208\) −2001.91 −0.667343
\(209\) −4228.94 −1.39963
\(210\) 0 0
\(211\) 2728.21 0.890130 0.445065 0.895498i \(-0.353181\pi\)
0.445065 + 0.895498i \(0.353181\pi\)
\(212\) −3323.06 −1.07655
\(213\) 0 0
\(214\) −447.771 −0.143033
\(215\) −2301.00 −0.729892
\(216\) 0 0
\(217\) 1037.74 0.324637
\(218\) 402.970 0.125195
\(219\) 0 0
\(220\) −1399.42 −0.428859
\(221\) −3152.98 −0.959695
\(222\) 0 0
\(223\) −3674.85 −1.10352 −0.551762 0.834002i \(-0.686044\pi\)
−0.551762 + 0.834002i \(0.686044\pi\)
\(224\) −888.267 −0.264954
\(225\) 0 0
\(226\) 348.385 0.102541
\(227\) 5773.91 1.68823 0.844115 0.536163i \(-0.180127\pi\)
0.844115 + 0.536163i \(0.180127\pi\)
\(228\) 0 0
\(229\) 4818.28 1.39040 0.695198 0.718818i \(-0.255317\pi\)
0.695198 + 0.718818i \(0.255317\pi\)
\(230\) 49.1412 0.0140882
\(231\) 0 0
\(232\) −491.659 −0.139133
\(233\) 91.0828 0.0256096 0.0128048 0.999918i \(-0.495924\pi\)
0.0128048 + 0.999918i \(0.495924\pi\)
\(234\) 0 0
\(235\) −1071.44 −0.297416
\(236\) −2996.82 −0.826594
\(237\) 0 0
\(238\) −410.789 −0.111880
\(239\) 5486.84 1.48500 0.742498 0.669848i \(-0.233641\pi\)
0.742498 + 0.669848i \(0.233641\pi\)
\(240\) 0 0
\(241\) 3017.19 0.806450 0.403225 0.915101i \(-0.367889\pi\)
0.403225 + 0.915101i \(0.367889\pi\)
\(242\) 54.9453 0.0145951
\(243\) 0 0
\(244\) −10.7949 −0.00283227
\(245\) 1475.00 0.384630
\(246\) 0 0
\(247\) −4390.30 −1.13096
\(248\) 1695.64 0.434166
\(249\) 0 0
\(250\) −91.5064 −0.0231495
\(251\) −2740.98 −0.689280 −0.344640 0.938735i \(-0.611999\pi\)
−0.344640 + 0.938735i \(0.611999\pi\)
\(252\) 0 0
\(253\) 503.426 0.125099
\(254\) −1204.89 −0.297643
\(255\) 0 0
\(256\) 1619.36 0.395352
\(257\) −2521.40 −0.611987 −0.305993 0.952034i \(-0.598989\pi\)
−0.305993 + 0.952034i \(0.598989\pi\)
\(258\) 0 0
\(259\) 1512.73 0.362920
\(260\) −1452.82 −0.346539
\(261\) 0 0
\(262\) −283.089 −0.0667531
\(263\) −4487.72 −1.05218 −0.526092 0.850428i \(-0.676343\pi\)
−0.526092 + 0.850428i \(0.676343\pi\)
\(264\) 0 0
\(265\) −2226.03 −0.516014
\(266\) −571.994 −0.131847
\(267\) 0 0
\(268\) 6091.20 1.38836
\(269\) 8531.47 1.93373 0.966864 0.255291i \(-0.0821712\pi\)
0.966864 + 0.255291i \(0.0821712\pi\)
\(270\) 0 0
\(271\) −123.097 −0.0275927 −0.0137964 0.999905i \(-0.504392\pi\)
−0.0137964 + 0.999905i \(0.504392\pi\)
\(272\) 4165.21 0.928504
\(273\) 0 0
\(274\) 2176.55 0.479891
\(275\) −937.436 −0.205562
\(276\) 0 0
\(277\) −6557.58 −1.42241 −0.711204 0.702986i \(-0.751850\pi\)
−0.711204 + 0.702986i \(0.751850\pi\)
\(278\) 1718.38 0.370726
\(279\) 0 0
\(280\) −392.154 −0.0836989
\(281\) 7084.04 1.50391 0.751954 0.659215i \(-0.229111\pi\)
0.751954 + 0.659215i \(0.229111\pi\)
\(282\) 0 0
\(283\) 2072.93 0.435417 0.217709 0.976014i \(-0.430142\pi\)
0.217709 + 0.976014i \(0.430142\pi\)
\(284\) −1101.09 −0.230062
\(285\) 0 0
\(286\) 1068.58 0.220932
\(287\) −2578.25 −0.530276
\(288\) 0 0
\(289\) 1647.16 0.335267
\(290\) −158.968 −0.0321893
\(291\) 0 0
\(292\) −3229.35 −0.647204
\(293\) 7128.40 1.42132 0.710658 0.703537i \(-0.248397\pi\)
0.710658 + 0.703537i \(0.248397\pi\)
\(294\) 0 0
\(295\) −2007.49 −0.396205
\(296\) 2471.76 0.485365
\(297\) 0 0
\(298\) −1026.40 −0.199523
\(299\) 522.635 0.101086
\(300\) 0 0
\(301\) −3188.36 −0.610545
\(302\) −2153.15 −0.410265
\(303\) 0 0
\(304\) 5799.75 1.09421
\(305\) −7.23122 −0.00135757
\(306\) 0 0
\(307\) −6334.70 −1.17766 −0.588828 0.808259i \(-0.700410\pi\)
−0.588828 + 0.808259i \(0.700410\pi\)
\(308\) −1939.10 −0.358735
\(309\) 0 0
\(310\) 548.250 0.100447
\(311\) 7393.90 1.34813 0.674067 0.738671i \(-0.264546\pi\)
0.674067 + 0.738671i \(0.264546\pi\)
\(312\) 0 0
\(313\) −933.055 −0.168496 −0.0842482 0.996445i \(-0.526849\pi\)
−0.0842482 + 0.996445i \(0.526849\pi\)
\(314\) 1868.18 0.335756
\(315\) 0 0
\(316\) −2867.21 −0.510421
\(317\) 8734.68 1.54760 0.773799 0.633432i \(-0.218354\pi\)
0.773799 + 0.633432i \(0.218354\pi\)
\(318\) 0 0
\(319\) −1628.54 −0.285833
\(320\) 1587.74 0.277367
\(321\) 0 0
\(322\) 68.0920 0.0117845
\(323\) 9134.55 1.57356
\(324\) 0 0
\(325\) −973.205 −0.166104
\(326\) 257.359 0.0437233
\(327\) 0 0
\(328\) −4212.80 −0.709185
\(329\) −1484.63 −0.248784
\(330\) 0 0
\(331\) −1290.53 −0.214302 −0.107151 0.994243i \(-0.534173\pi\)
−0.107151 + 0.994243i \(0.534173\pi\)
\(332\) −9413.95 −1.55620
\(333\) 0 0
\(334\) 15.3652 0.00251721
\(335\) 4080.33 0.665470
\(336\) 0 0
\(337\) −10001.3 −1.61664 −0.808318 0.588746i \(-0.799622\pi\)
−0.808318 + 0.588746i \(0.799622\pi\)
\(338\) −498.962 −0.0802958
\(339\) 0 0
\(340\) 3022.77 0.482155
\(341\) 5616.54 0.891943
\(342\) 0 0
\(343\) 4420.19 0.695825
\(344\) −5209.70 −0.816535
\(345\) 0 0
\(346\) −3312.24 −0.514645
\(347\) 6468.37 1.00069 0.500346 0.865825i \(-0.333206\pi\)
0.500346 + 0.865825i \(0.333206\pi\)
\(348\) 0 0
\(349\) 11169.8 1.71319 0.856597 0.515985i \(-0.172574\pi\)
0.856597 + 0.515985i \(0.172574\pi\)
\(350\) −126.795 −0.0193642
\(351\) 0 0
\(352\) −4807.55 −0.727964
\(353\) 1816.77 0.273930 0.136965 0.990576i \(-0.456265\pi\)
0.136965 + 0.990576i \(0.456265\pi\)
\(354\) 0 0
\(355\) −737.590 −0.110274
\(356\) −3829.08 −0.570059
\(357\) 0 0
\(358\) 2654.99 0.391957
\(359\) 918.073 0.134969 0.0674847 0.997720i \(-0.478503\pi\)
0.0674847 + 0.997720i \(0.478503\pi\)
\(360\) 0 0
\(361\) 5860.21 0.854382
\(362\) 2599.81 0.377467
\(363\) 0 0
\(364\) −2013.09 −0.289875
\(365\) −2163.26 −0.310219
\(366\) 0 0
\(367\) 7354.27 1.04602 0.523011 0.852326i \(-0.324809\pi\)
0.523011 + 0.852326i \(0.324809\pi\)
\(368\) −690.421 −0.0978008
\(369\) 0 0
\(370\) 799.193 0.112292
\(371\) −3084.47 −0.431638
\(372\) 0 0
\(373\) −602.607 −0.0836509 −0.0418255 0.999125i \(-0.513317\pi\)
−0.0418255 + 0.999125i \(0.513317\pi\)
\(374\) −2223.31 −0.307392
\(375\) 0 0
\(376\) −2425.84 −0.332721
\(377\) −1690.68 −0.230967
\(378\) 0 0
\(379\) 13319.0 1.80515 0.902575 0.430533i \(-0.141674\pi\)
0.902575 + 0.430533i \(0.141674\pi\)
\(380\) 4208.99 0.568201
\(381\) 0 0
\(382\) −204.718 −0.0274197
\(383\) −8537.75 −1.13906 −0.569528 0.821972i \(-0.692874\pi\)
−0.569528 + 0.821972i \(0.692874\pi\)
\(384\) 0 0
\(385\) −1298.95 −0.171950
\(386\) 194.842 0.0256922
\(387\) 0 0
\(388\) 8181.42 1.07049
\(389\) −12640.0 −1.64749 −0.823744 0.566962i \(-0.808119\pi\)
−0.823744 + 0.566962i \(0.808119\pi\)
\(390\) 0 0
\(391\) −1087.41 −0.140646
\(392\) 3339.55 0.430288
\(393\) 0 0
\(394\) −3463.42 −0.442854
\(395\) −1920.67 −0.244656
\(396\) 0 0
\(397\) 5473.94 0.692013 0.346006 0.938232i \(-0.387538\pi\)
0.346006 + 0.938232i \(0.387538\pi\)
\(398\) 2108.08 0.265499
\(399\) 0 0
\(400\) 1285.64 0.160705
\(401\) 13045.6 1.62461 0.812305 0.583233i \(-0.198212\pi\)
0.812305 + 0.583233i \(0.198212\pi\)
\(402\) 0 0
\(403\) 5830.85 0.720732
\(404\) 8912.79 1.09759
\(405\) 0 0
\(406\) −220.272 −0.0269259
\(407\) 8187.32 0.997126
\(408\) 0 0
\(409\) −7182.72 −0.868368 −0.434184 0.900824i \(-0.642963\pi\)
−0.434184 + 0.900824i \(0.642963\pi\)
\(410\) −1362.12 −0.164074
\(411\) 0 0
\(412\) −1354.75 −0.162000
\(413\) −2781.66 −0.331420
\(414\) 0 0
\(415\) −6306.15 −0.745920
\(416\) −4990.99 −0.588230
\(417\) 0 0
\(418\) −3095.80 −0.362250
\(419\) 8720.40 1.01675 0.508376 0.861135i \(-0.330246\pi\)
0.508376 + 0.861135i \(0.330246\pi\)
\(420\) 0 0
\(421\) 6553.03 0.758611 0.379305 0.925272i \(-0.376163\pi\)
0.379305 + 0.925272i \(0.376163\pi\)
\(422\) 1997.18 0.230383
\(423\) 0 0
\(424\) −5039.95 −0.577268
\(425\) 2024.87 0.231107
\(426\) 0 0
\(427\) −10.0199 −0.00113559
\(428\) 4565.54 0.515616
\(429\) 0 0
\(430\) −1684.45 −0.188910
\(431\) 5217.70 0.583127 0.291564 0.956551i \(-0.405824\pi\)
0.291564 + 0.956551i \(0.405824\pi\)
\(432\) 0 0
\(433\) 3377.56 0.374862 0.187431 0.982278i \(-0.439984\pi\)
0.187431 + 0.982278i \(0.439984\pi\)
\(434\) 759.677 0.0840223
\(435\) 0 0
\(436\) −4108.74 −0.451314
\(437\) −1514.13 −0.165746
\(438\) 0 0
\(439\) 2138.92 0.232540 0.116270 0.993218i \(-0.462906\pi\)
0.116270 + 0.993218i \(0.462906\pi\)
\(440\) −2122.45 −0.229963
\(441\) 0 0
\(442\) −2308.14 −0.248387
\(443\) 8512.62 0.912973 0.456486 0.889730i \(-0.349108\pi\)
0.456486 + 0.889730i \(0.349108\pi\)
\(444\) 0 0
\(445\) −2565.00 −0.273242
\(446\) −2690.17 −0.285613
\(447\) 0 0
\(448\) 2200.04 0.232014
\(449\) −4542.67 −0.477465 −0.238733 0.971085i \(-0.576732\pi\)
−0.238733 + 0.971085i \(0.576732\pi\)
\(450\) 0 0
\(451\) −13954.2 −1.45694
\(452\) −3552.18 −0.369648
\(453\) 0 0
\(454\) 4226.80 0.436946
\(455\) −1348.51 −0.138943
\(456\) 0 0
\(457\) −17942.2 −1.83654 −0.918272 0.395951i \(-0.870415\pi\)
−0.918272 + 0.395951i \(0.870415\pi\)
\(458\) 3527.22 0.359861
\(459\) 0 0
\(460\) −501.051 −0.0507862
\(461\) −5180.89 −0.523423 −0.261711 0.965146i \(-0.584287\pi\)
−0.261711 + 0.965146i \(0.584287\pi\)
\(462\) 0 0
\(463\) 14192.9 1.42462 0.712310 0.701864i \(-0.247649\pi\)
0.712310 + 0.701864i \(0.247649\pi\)
\(464\) 2233.46 0.223460
\(465\) 0 0
\(466\) 66.6773 0.00662825
\(467\) −11886.3 −1.17780 −0.588901 0.808205i \(-0.700439\pi\)
−0.588901 + 0.808205i \(0.700439\pi\)
\(468\) 0 0
\(469\) 5653.88 0.556656
\(470\) −784.346 −0.0769769
\(471\) 0 0
\(472\) −4545.15 −0.443237
\(473\) −17256.3 −1.67748
\(474\) 0 0
\(475\) 2819.49 0.272351
\(476\) 4188.47 0.403316
\(477\) 0 0
\(478\) 4016.65 0.384345
\(479\) −4319.46 −0.412027 −0.206014 0.978549i \(-0.566049\pi\)
−0.206014 + 0.978549i \(0.566049\pi\)
\(480\) 0 0
\(481\) 8499.72 0.805725
\(482\) 2208.74 0.208725
\(483\) 0 0
\(484\) −560.231 −0.0526137
\(485\) 5480.51 0.513108
\(486\) 0 0
\(487\) −6436.94 −0.598944 −0.299472 0.954105i \(-0.596810\pi\)
−0.299472 + 0.954105i \(0.596810\pi\)
\(488\) −16.3722 −0.00151872
\(489\) 0 0
\(490\) 1079.77 0.0995495
\(491\) −2897.73 −0.266340 −0.133170 0.991093i \(-0.542516\pi\)
−0.133170 + 0.991093i \(0.542516\pi\)
\(492\) 0 0
\(493\) 3517.67 0.321355
\(494\) −3213.92 −0.292715
\(495\) 0 0
\(496\) −7702.77 −0.697307
\(497\) −1022.03 −0.0922425
\(498\) 0 0
\(499\) 15054.6 1.35057 0.675286 0.737556i \(-0.264020\pi\)
0.675286 + 0.737556i \(0.264020\pi\)
\(500\) 933.013 0.0834512
\(501\) 0 0
\(502\) −2006.54 −0.178399
\(503\) 15582.5 1.38129 0.690646 0.723193i \(-0.257326\pi\)
0.690646 + 0.723193i \(0.257326\pi\)
\(504\) 0 0
\(505\) 5970.43 0.526101
\(506\) 368.534 0.0323781
\(507\) 0 0
\(508\) 12285.2 1.07297
\(509\) 4071.01 0.354508 0.177254 0.984165i \(-0.443279\pi\)
0.177254 + 0.984165i \(0.443279\pi\)
\(510\) 0 0
\(511\) −2997.50 −0.259494
\(512\) 11250.6 0.971116
\(513\) 0 0
\(514\) −1845.79 −0.158394
\(515\) −907.513 −0.0776501
\(516\) 0 0
\(517\) −8035.22 −0.683537
\(518\) 1107.39 0.0939307
\(519\) 0 0
\(520\) −2203.44 −0.185821
\(521\) −18597.9 −1.56389 −0.781947 0.623345i \(-0.785773\pi\)
−0.781947 + 0.623345i \(0.785773\pi\)
\(522\) 0 0
\(523\) −12073.5 −1.00944 −0.504722 0.863282i \(-0.668405\pi\)
−0.504722 + 0.863282i \(0.668405\pi\)
\(524\) 2886.42 0.240637
\(525\) 0 0
\(526\) −3285.24 −0.272325
\(527\) −12131.8 −1.00279
\(528\) 0 0
\(529\) −11986.8 −0.985186
\(530\) −1629.56 −0.133554
\(531\) 0 0
\(532\) 5832.14 0.475292
\(533\) −14486.7 −1.17728
\(534\) 0 0
\(535\) 3058.33 0.247146
\(536\) 9238.29 0.744465
\(537\) 0 0
\(538\) 6245.47 0.500486
\(539\) 11061.7 0.883976
\(540\) 0 0
\(541\) 1802.04 0.143208 0.0716041 0.997433i \(-0.477188\pi\)
0.0716041 + 0.997433i \(0.477188\pi\)
\(542\) −90.1134 −0.00714152
\(543\) 0 0
\(544\) 10384.4 0.818430
\(545\) −2752.33 −0.216325
\(546\) 0 0
\(547\) 6589.73 0.515094 0.257547 0.966266i \(-0.417086\pi\)
0.257547 + 0.966266i \(0.417086\pi\)
\(548\) −22192.4 −1.72995
\(549\) 0 0
\(550\) −686.250 −0.0532033
\(551\) 4898.10 0.378704
\(552\) 0 0
\(553\) −2661.35 −0.204651
\(554\) −4800.48 −0.368146
\(555\) 0 0
\(556\) −17520.9 −1.33642
\(557\) −21128.6 −1.60727 −0.803633 0.595125i \(-0.797103\pi\)
−0.803633 + 0.595125i \(0.797103\pi\)
\(558\) 0 0
\(559\) −17914.8 −1.35548
\(560\) 1781.44 0.134428
\(561\) 0 0
\(562\) 5185.88 0.389240
\(563\) 110.427 0.00826636 0.00413318 0.999991i \(-0.498684\pi\)
0.00413318 + 0.999991i \(0.498684\pi\)
\(564\) 0 0
\(565\) −2379.51 −0.177180
\(566\) 1517.49 0.112694
\(567\) 0 0
\(568\) −1669.98 −0.123364
\(569\) 6930.08 0.510587 0.255294 0.966864i \(-0.417828\pi\)
0.255294 + 0.966864i \(0.417828\pi\)
\(570\) 0 0
\(571\) 21264.6 1.55849 0.779244 0.626721i \(-0.215603\pi\)
0.779244 + 0.626721i \(0.215603\pi\)
\(572\) −10895.4 −0.796433
\(573\) 0 0
\(574\) −1887.41 −0.137246
\(575\) −335.641 −0.0243429
\(576\) 0 0
\(577\) 2927.39 0.211211 0.105606 0.994408i \(-0.466322\pi\)
0.105606 + 0.994408i \(0.466322\pi\)
\(578\) 1205.81 0.0867734
\(579\) 0 0
\(580\) 1620.86 0.116039
\(581\) −8738.06 −0.623952
\(582\) 0 0
\(583\) −16694.0 −1.18593
\(584\) −4897.83 −0.347044
\(585\) 0 0
\(586\) 5218.35 0.367864
\(587\) −21750.5 −1.52937 −0.764683 0.644407i \(-0.777104\pi\)
−0.764683 + 0.644407i \(0.777104\pi\)
\(588\) 0 0
\(589\) −16892.6 −1.18175
\(590\) −1469.58 −0.102545
\(591\) 0 0
\(592\) −11228.5 −0.779538
\(593\) −5635.88 −0.390283 −0.195142 0.980775i \(-0.562517\pi\)
−0.195142 + 0.980775i \(0.562517\pi\)
\(594\) 0 0
\(595\) 2805.74 0.193318
\(596\) 10465.4 0.719258
\(597\) 0 0
\(598\) 382.596 0.0261630
\(599\) −18324.4 −1.24994 −0.624970 0.780648i \(-0.714889\pi\)
−0.624970 + 0.780648i \(0.714889\pi\)
\(600\) 0 0
\(601\) −2503.90 −0.169944 −0.0849719 0.996383i \(-0.527080\pi\)
−0.0849719 + 0.996383i \(0.527080\pi\)
\(602\) −2334.04 −0.158021
\(603\) 0 0
\(604\) 21953.9 1.47896
\(605\) −375.284 −0.0252189
\(606\) 0 0
\(607\) 17466.6 1.16796 0.583978 0.811770i \(-0.301495\pi\)
0.583978 + 0.811770i \(0.301495\pi\)
\(608\) 14459.5 0.964489
\(609\) 0 0
\(610\) −5.29362 −0.000351365 0
\(611\) −8341.82 −0.552330
\(612\) 0 0
\(613\) −2244.45 −0.147883 −0.0739417 0.997263i \(-0.523558\pi\)
−0.0739417 + 0.997263i \(0.523558\pi\)
\(614\) −4637.32 −0.304800
\(615\) 0 0
\(616\) −2940.95 −0.192361
\(617\) 5809.84 0.379085 0.189542 0.981873i \(-0.439300\pi\)
0.189542 + 0.981873i \(0.439300\pi\)
\(618\) 0 0
\(619\) −1543.48 −0.100223 −0.0501114 0.998744i \(-0.515958\pi\)
−0.0501114 + 0.998744i \(0.515958\pi\)
\(620\) −5590.04 −0.362099
\(621\) 0 0
\(622\) 5412.71 0.348922
\(623\) −3554.17 −0.228563
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) −683.044 −0.0436101
\(627\) 0 0
\(628\) −19048.2 −1.21036
\(629\) −17684.7 −1.12104
\(630\) 0 0
\(631\) 7992.93 0.504269 0.252134 0.967692i \(-0.418868\pi\)
0.252134 + 0.967692i \(0.418868\pi\)
\(632\) −4348.58 −0.273698
\(633\) 0 0
\(634\) 6394.23 0.400548
\(635\) 8229.54 0.514298
\(636\) 0 0
\(637\) 11483.8 0.714294
\(638\) −1192.18 −0.0739791
\(639\) 0 0
\(640\) 6290.72 0.388535
\(641\) −26129.8 −1.61009 −0.805044 0.593215i \(-0.797858\pi\)
−0.805044 + 0.593215i \(0.797858\pi\)
\(642\) 0 0
\(643\) −28772.1 −1.76464 −0.882319 0.470653i \(-0.844018\pi\)
−0.882319 + 0.470653i \(0.844018\pi\)
\(644\) −694.277 −0.0424819
\(645\) 0 0
\(646\) 6686.96 0.407268
\(647\) 7156.38 0.434847 0.217424 0.976077i \(-0.430235\pi\)
0.217424 + 0.976077i \(0.430235\pi\)
\(648\) 0 0
\(649\) −15055.1 −0.910578
\(650\) −712.436 −0.0429908
\(651\) 0 0
\(652\) −2624.07 −0.157617
\(653\) −15434.9 −0.924981 −0.462491 0.886624i \(-0.653044\pi\)
−0.462491 + 0.886624i \(0.653044\pi\)
\(654\) 0 0
\(655\) 1933.54 0.115343
\(656\) 19137.5 1.13901
\(657\) 0 0
\(658\) −1086.82 −0.0643901
\(659\) 6462.64 0.382016 0.191008 0.981588i \(-0.438824\pi\)
0.191008 + 0.981588i \(0.438824\pi\)
\(660\) 0 0
\(661\) 1747.60 0.102835 0.0514174 0.998677i \(-0.483626\pi\)
0.0514174 + 0.998677i \(0.483626\pi\)
\(662\) −944.735 −0.0554655
\(663\) 0 0
\(664\) −14277.8 −0.834465
\(665\) 3906.79 0.227818
\(666\) 0 0
\(667\) −583.085 −0.0338488
\(668\) −156.666 −0.00907426
\(669\) 0 0
\(670\) 2987.01 0.172236
\(671\) −54.2305 −0.00312004
\(672\) 0 0
\(673\) −5478.38 −0.313783 −0.156892 0.987616i \(-0.550147\pi\)
−0.156892 + 0.987616i \(0.550147\pi\)
\(674\) −7321.47 −0.418416
\(675\) 0 0
\(676\) 5087.49 0.289457
\(677\) 4018.53 0.228131 0.114066 0.993473i \(-0.463613\pi\)
0.114066 + 0.993473i \(0.463613\pi\)
\(678\) 0 0
\(679\) 7594.02 0.429207
\(680\) 4584.51 0.258541
\(681\) 0 0
\(682\) 4111.59 0.230852
\(683\) −9894.95 −0.554348 −0.277174 0.960820i \(-0.589398\pi\)
−0.277174 + 0.960820i \(0.589398\pi\)
\(684\) 0 0
\(685\) −14866.1 −0.829204
\(686\) 3235.81 0.180093
\(687\) 0 0
\(688\) 23666.1 1.31143
\(689\) −17331.0 −0.958287
\(690\) 0 0
\(691\) 21847.9 1.20280 0.601400 0.798948i \(-0.294610\pi\)
0.601400 + 0.798948i \(0.294610\pi\)
\(692\) 33772.1 1.85524
\(693\) 0 0
\(694\) 4735.18 0.258998
\(695\) −11736.8 −0.640577
\(696\) 0 0
\(697\) 30141.3 1.63800
\(698\) 8176.85 0.443407
\(699\) 0 0
\(700\) 1292.82 0.0698057
\(701\) −15506.5 −0.835480 −0.417740 0.908567i \(-0.637178\pi\)
−0.417740 + 0.908567i \(0.637178\pi\)
\(702\) 0 0
\(703\) −24624.7 −1.32110
\(704\) 11907.3 0.637460
\(705\) 0 0
\(706\) 1329.97 0.0708982
\(707\) 8272.88 0.440076
\(708\) 0 0
\(709\) 6399.70 0.338993 0.169496 0.985531i \(-0.445786\pi\)
0.169496 + 0.985531i \(0.445786\pi\)
\(710\) −539.954 −0.0285410
\(711\) 0 0
\(712\) −5807.42 −0.305677
\(713\) 2010.95 0.105625
\(714\) 0 0
\(715\) −7298.54 −0.381748
\(716\) −27070.7 −1.41296
\(717\) 0 0
\(718\) 672.076 0.0349327
\(719\) −1771.66 −0.0918941 −0.0459470 0.998944i \(-0.514631\pi\)
−0.0459470 + 0.998944i \(0.514631\pi\)
\(720\) 0 0
\(721\) −1257.49 −0.0649532
\(722\) 4289.97 0.221130
\(723\) 0 0
\(724\) −26508.1 −1.36073
\(725\) 1085.77 0.0556200
\(726\) 0 0
\(727\) 9575.06 0.488472 0.244236 0.969716i \(-0.421463\pi\)
0.244236 + 0.969716i \(0.421463\pi\)
\(728\) −3053.17 −0.155437
\(729\) 0 0
\(730\) −1583.61 −0.0802907
\(731\) 37273.8 1.88594
\(732\) 0 0
\(733\) −19790.5 −0.997243 −0.498622 0.866820i \(-0.666160\pi\)
−0.498622 + 0.866820i \(0.666160\pi\)
\(734\) 5383.70 0.270730
\(735\) 0 0
\(736\) −1721.30 −0.0862066
\(737\) 30600.4 1.52942
\(738\) 0 0
\(739\) −13106.5 −0.652407 −0.326203 0.945300i \(-0.605769\pi\)
−0.326203 + 0.945300i \(0.605769\pi\)
\(740\) −8148.69 −0.404800
\(741\) 0 0
\(742\) −2257.99 −0.111716
\(743\) 17068.9 0.842795 0.421398 0.906876i \(-0.361540\pi\)
0.421398 + 0.906876i \(0.361540\pi\)
\(744\) 0 0
\(745\) 7010.46 0.344756
\(746\) −441.139 −0.0216505
\(747\) 0 0
\(748\) 22669.2 1.10811
\(749\) 4237.75 0.206734
\(750\) 0 0
\(751\) 4112.93 0.199844 0.0999220 0.994995i \(-0.468141\pi\)
0.0999220 + 0.994995i \(0.468141\pi\)
\(752\) 11019.9 0.534379
\(753\) 0 0
\(754\) −1237.67 −0.0597787
\(755\) 14706.3 0.708897
\(756\) 0 0
\(757\) 17493.6 0.839915 0.419958 0.907544i \(-0.362045\pi\)
0.419958 + 0.907544i \(0.362045\pi\)
\(758\) 9750.19 0.467207
\(759\) 0 0
\(760\) 6383.60 0.304681
\(761\) −5212.04 −0.248274 −0.124137 0.992265i \(-0.539616\pi\)
−0.124137 + 0.992265i \(0.539616\pi\)
\(762\) 0 0
\(763\) −3813.75 −0.180953
\(764\) 2087.34 0.0988447
\(765\) 0 0
\(766\) −6250.07 −0.294809
\(767\) −15629.6 −0.735790
\(768\) 0 0
\(769\) −23324.7 −1.09377 −0.546885 0.837208i \(-0.684186\pi\)
−0.546885 + 0.837208i \(0.684186\pi\)
\(770\) −950.897 −0.0445038
\(771\) 0 0
\(772\) −1986.64 −0.0926176
\(773\) 20455.0 0.951765 0.475883 0.879509i \(-0.342129\pi\)
0.475883 + 0.879509i \(0.342129\pi\)
\(774\) 0 0
\(775\) −3744.62 −0.173562
\(776\) 12408.4 0.574017
\(777\) 0 0
\(778\) −9253.11 −0.426401
\(779\) 41969.6 1.93032
\(780\) 0 0
\(781\) −5531.55 −0.253437
\(782\) −796.037 −0.0364018
\(783\) 0 0
\(784\) −15170.6 −0.691079
\(785\) −12759.9 −0.580153
\(786\) 0 0
\(787\) 9110.97 0.412670 0.206335 0.978481i \(-0.433846\pi\)
0.206335 + 0.978481i \(0.433846\pi\)
\(788\) 35313.5 1.59644
\(789\) 0 0
\(790\) −1406.03 −0.0633217
\(791\) −3297.15 −0.148209
\(792\) 0 0
\(793\) −56.2997 −0.00252114
\(794\) 4007.20 0.179106
\(795\) 0 0
\(796\) −21494.3 −0.957093
\(797\) −7925.09 −0.352222 −0.176111 0.984370i \(-0.556352\pi\)
−0.176111 + 0.984370i \(0.556352\pi\)
\(798\) 0 0
\(799\) 17356.2 0.768481
\(800\) 3205.26 0.141654
\(801\) 0 0
\(802\) 9550.08 0.420480
\(803\) −16223.3 −0.712962
\(804\) 0 0
\(805\) −465.077 −0.0203625
\(806\) 4268.48 0.186539
\(807\) 0 0
\(808\) 13517.7 0.588552
\(809\) −33705.8 −1.46481 −0.732406 0.680868i \(-0.761603\pi\)
−0.732406 + 0.680868i \(0.761603\pi\)
\(810\) 0 0
\(811\) 10424.5 0.451360 0.225680 0.974201i \(-0.427540\pi\)
0.225680 + 0.974201i \(0.427540\pi\)
\(812\) 2245.93 0.0970648
\(813\) 0 0
\(814\) 5993.53 0.258075
\(815\) −1757.79 −0.0755495
\(816\) 0 0
\(817\) 51901.1 2.22251
\(818\) −5258.11 −0.224750
\(819\) 0 0
\(820\) 13888.4 0.591468
\(821\) −9467.66 −0.402465 −0.201232 0.979544i \(-0.564495\pi\)
−0.201232 + 0.979544i \(0.564495\pi\)
\(822\) 0 0
\(823\) 1160.55 0.0491545 0.0245772 0.999698i \(-0.492176\pi\)
0.0245772 + 0.999698i \(0.492176\pi\)
\(824\) −2054.70 −0.0868676
\(825\) 0 0
\(826\) −2036.31 −0.0857777
\(827\) 26499.9 1.11426 0.557128 0.830426i \(-0.311903\pi\)
0.557128 + 0.830426i \(0.311903\pi\)
\(828\) 0 0
\(829\) 653.861 0.0273939 0.0136969 0.999906i \(-0.495640\pi\)
0.0136969 + 0.999906i \(0.495640\pi\)
\(830\) −4616.42 −0.193058
\(831\) 0 0
\(832\) 12361.6 0.515098
\(833\) −23893.5 −0.993830
\(834\) 0 0
\(835\) −104.947 −0.00434949
\(836\) 31565.2 1.30587
\(837\) 0 0
\(838\) 6383.77 0.263155
\(839\) 41269.7 1.69820 0.849099 0.528234i \(-0.177146\pi\)
0.849099 + 0.528234i \(0.177146\pi\)
\(840\) 0 0
\(841\) −22502.8 −0.922661
\(842\) 4797.15 0.196343
\(843\) 0 0
\(844\) −20363.6 −0.830502
\(845\) 3407.97 0.138743
\(846\) 0 0
\(847\) −520.008 −0.0210953
\(848\) 22894.9 0.927141
\(849\) 0 0
\(850\) 1482.31 0.0598150
\(851\) 2931.40 0.118081
\(852\) 0 0
\(853\) 5069.39 0.203485 0.101743 0.994811i \(-0.467558\pi\)
0.101743 + 0.994811i \(0.467558\pi\)
\(854\) −7.33506 −0.000293912 0
\(855\) 0 0
\(856\) 6924.38 0.276484
\(857\) −43639.0 −1.73941 −0.869707 0.493568i \(-0.835693\pi\)
−0.869707 + 0.493568i \(0.835693\pi\)
\(858\) 0 0
\(859\) −12898.1 −0.512315 −0.256157 0.966635i \(-0.582457\pi\)
−0.256157 + 0.966635i \(0.582457\pi\)
\(860\) 17174.9 0.680999
\(861\) 0 0
\(862\) 3819.62 0.150924
\(863\) 8962.98 0.353538 0.176769 0.984252i \(-0.443435\pi\)
0.176769 + 0.984252i \(0.443435\pi\)
\(864\) 0 0
\(865\) 22623.0 0.889256
\(866\) 2472.54 0.0970214
\(867\) 0 0
\(868\) −7745.78 −0.302891
\(869\) −14404.0 −0.562281
\(870\) 0 0
\(871\) 31768.0 1.23584
\(872\) −6231.56 −0.242004
\(873\) 0 0
\(874\) −1108.42 −0.0428982
\(875\) 866.025 0.0334594
\(876\) 0 0
\(877\) −37720.1 −1.45236 −0.726179 0.687506i \(-0.758705\pi\)
−0.726179 + 0.687506i \(0.758705\pi\)
\(878\) 1565.80 0.0601859
\(879\) 0 0
\(880\) 9641.64 0.369341
\(881\) −7961.44 −0.304458 −0.152229 0.988345i \(-0.548645\pi\)
−0.152229 + 0.988345i \(0.548645\pi\)
\(882\) 0 0
\(883\) −37426.4 −1.42639 −0.713193 0.700968i \(-0.752752\pi\)
−0.713193 + 0.700968i \(0.752752\pi\)
\(884\) 23534.2 0.895408
\(885\) 0 0
\(886\) 6231.67 0.236295
\(887\) 26853.3 1.01651 0.508256 0.861206i \(-0.330290\pi\)
0.508256 + 0.861206i \(0.330290\pi\)
\(888\) 0 0
\(889\) 11403.2 0.430203
\(890\) −1877.71 −0.0707202
\(891\) 0 0
\(892\) 27429.4 1.02960
\(893\) 24167.2 0.905626
\(894\) 0 0
\(895\) −18133.9 −0.677264
\(896\) 8716.67 0.325004
\(897\) 0 0
\(898\) −3325.46 −0.123577
\(899\) −6505.26 −0.241338
\(900\) 0 0
\(901\) 36059.3 1.33331
\(902\) −10215.2 −0.377083
\(903\) 0 0
\(904\) −5387.46 −0.198213
\(905\) −17757.0 −0.652226
\(906\) 0 0
\(907\) 27150.9 0.993969 0.496985 0.867759i \(-0.334441\pi\)
0.496985 + 0.867759i \(0.334441\pi\)
\(908\) −43097.1 −1.57514
\(909\) 0 0
\(910\) −987.180 −0.0359612
\(911\) 15300.5 0.556452 0.278226 0.960516i \(-0.410254\pi\)
0.278226 + 0.960516i \(0.410254\pi\)
\(912\) 0 0
\(913\) −47292.9 −1.71431
\(914\) −13134.6 −0.475332
\(915\) 0 0
\(916\) −35964.1 −1.29726
\(917\) 2679.19 0.0964826
\(918\) 0 0
\(919\) −26325.6 −0.944943 −0.472471 0.881346i \(-0.656638\pi\)
−0.472471 + 0.881346i \(0.656638\pi\)
\(920\) −759.925 −0.0272326
\(921\) 0 0
\(922\) −3792.67 −0.135472
\(923\) −5742.61 −0.204789
\(924\) 0 0
\(925\) −5458.59 −0.194029
\(926\) 10389.9 0.368719
\(927\) 0 0
\(928\) 5568.27 0.196969
\(929\) 35507.1 1.25398 0.626991 0.779027i \(-0.284286\pi\)
0.626991 + 0.779027i \(0.284286\pi\)
\(930\) 0 0
\(931\) −33269.9 −1.17119
\(932\) −679.852 −0.0238941
\(933\) 0 0
\(934\) −8701.39 −0.304838
\(935\) 15185.5 0.531143
\(936\) 0 0
\(937\) 35105.4 1.22395 0.611975 0.790877i \(-0.290375\pi\)
0.611975 + 0.790877i \(0.290375\pi\)
\(938\) 4138.92 0.144073
\(939\) 0 0
\(940\) 7997.31 0.277493
\(941\) 3574.47 0.123830 0.0619151 0.998081i \(-0.480279\pi\)
0.0619151 + 0.998081i \(0.480279\pi\)
\(942\) 0 0
\(943\) −4996.19 −0.172533
\(944\) 20647.3 0.711876
\(945\) 0 0
\(946\) −12632.5 −0.434163
\(947\) −33058.5 −1.13438 −0.567189 0.823588i \(-0.691969\pi\)
−0.567189 + 0.823588i \(0.691969\pi\)
\(948\) 0 0
\(949\) −16842.3 −0.576107
\(950\) 2064.01 0.0704897
\(951\) 0 0
\(952\) 6352.49 0.216266
\(953\) −1355.24 −0.0460656 −0.0230328 0.999735i \(-0.507332\pi\)
−0.0230328 + 0.999735i \(0.507332\pi\)
\(954\) 0 0
\(955\) 1398.25 0.0473784
\(956\) −40954.3 −1.38552
\(957\) 0 0
\(958\) −3162.06 −0.106641
\(959\) −20599.1 −0.693617
\(960\) 0 0
\(961\) −7355.57 −0.246906
\(962\) 6222.23 0.208537
\(963\) 0 0
\(964\) −22520.6 −0.752428
\(965\) −1330.80 −0.0443937
\(966\) 0 0
\(967\) 56118.0 1.86622 0.933110 0.359592i \(-0.117084\pi\)
0.933110 + 0.359592i \(0.117084\pi\)
\(968\) −849.680 −0.0282126
\(969\) 0 0
\(970\) 4012.01 0.132802
\(971\) −32718.5 −1.08135 −0.540673 0.841233i \(-0.681830\pi\)
−0.540673 + 0.841233i \(0.681830\pi\)
\(972\) 0 0
\(973\) −16262.9 −0.535834
\(974\) −4712.17 −0.155018
\(975\) 0 0
\(976\) 74.3741 0.00243920
\(977\) 8850.88 0.289831 0.144915 0.989444i \(-0.453709\pi\)
0.144915 + 0.989444i \(0.453709\pi\)
\(978\) 0 0
\(979\) −19236.2 −0.627978
\(980\) −11009.5 −0.358864
\(981\) 0 0
\(982\) −2121.29 −0.0689338
\(983\) 34107.9 1.10669 0.553343 0.832953i \(-0.313352\pi\)
0.553343 + 0.832953i \(0.313352\pi\)
\(984\) 0 0
\(985\) 23655.6 0.765208
\(986\) 2575.11 0.0831727
\(987\) 0 0
\(988\) 32769.7 1.05520
\(989\) −6178.47 −0.198649
\(990\) 0 0
\(991\) −52154.2 −1.67178 −0.835890 0.548897i \(-0.815048\pi\)
−0.835890 + 0.548897i \(0.815048\pi\)
\(992\) −19203.9 −0.614642
\(993\) 0 0
\(994\) −748.182 −0.0238741
\(995\) −14398.5 −0.458756
\(996\) 0 0
\(997\) 2111.10 0.0670604 0.0335302 0.999438i \(-0.489325\pi\)
0.0335302 + 0.999438i \(0.489325\pi\)
\(998\) 11020.7 0.349554
\(999\) 0 0
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 405.4.a.c.1.2 2
3.2 odd 2 405.4.a.f.1.1 yes 2
5.4 even 2 2025.4.a.m.1.1 2
9.2 odd 6 405.4.e.o.271.2 4
9.4 even 3 405.4.e.p.136.1 4
9.5 odd 6 405.4.e.o.136.2 4
9.7 even 3 405.4.e.p.271.1 4
15.14 odd 2 2025.4.a.i.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
405.4.a.c.1.2 2 1.1 even 1 trivial
405.4.a.f.1.1 yes 2 3.2 odd 2
405.4.e.o.136.2 4 9.5 odd 6
405.4.e.o.271.2 4 9.2 odd 6
405.4.e.p.136.1 4 9.4 even 3
405.4.e.p.271.1 4 9.7 even 3
2025.4.a.i.1.2 2 15.14 odd 2
2025.4.a.m.1.1 2 5.4 even 2