Properties

Label 1521.4.a.v.1.4
Level $1521$
Weight $4$
Character 1521.1
Self dual yes
Analytic conductor $89.742$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(89.7419051187\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \( x^{4} - 2x^{3} - 25x^{2} + 24x + 78 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-4.22605\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.22605 q^{2} +9.85953 q^{4} +5.85953 q^{5} -24.1254 q^{7} +7.85849 q^{8} +O(q^{10})\) \(q+4.22605 q^{2} +9.85953 q^{4} +5.85953 q^{5} -24.1254 q^{7} +7.85849 q^{8} +24.7627 q^{10} +33.8892 q^{11} -101.955 q^{14} -45.6659 q^{16} +49.3956 q^{17} -76.8548 q^{19} +57.7723 q^{20} +143.218 q^{22} -6.29163 q^{23} -90.6659 q^{25} -237.866 q^{28} -100.995 q^{29} -307.580 q^{31} -255.854 q^{32} +208.749 q^{34} -141.364 q^{35} -76.0189 q^{37} -324.793 q^{38} +46.0471 q^{40} +514.418 q^{41} -268.184 q^{43} +334.132 q^{44} -26.5888 q^{46} +460.912 q^{47} +239.037 q^{49} -383.159 q^{50} -67.8057 q^{53} +198.575 q^{55} -189.589 q^{56} -426.812 q^{58} -25.2021 q^{59} -588.832 q^{61} -1299.85 q^{62} -715.927 q^{64} -1004.46 q^{67} +487.018 q^{68} -597.411 q^{70} -895.481 q^{71} +968.599 q^{73} -321.260 q^{74} -757.753 q^{76} -817.592 q^{77} -119.053 q^{79} -267.581 q^{80} +2173.96 q^{82} -480.784 q^{83} +289.435 q^{85} -1133.36 q^{86} +266.318 q^{88} -1085.91 q^{89} -62.0325 q^{92} +1947.84 q^{94} -450.333 q^{95} -16.6552 q^{97} +1010.18 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} + 22 q^{4} + 6 q^{5} - 14 q^{7} - 54 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} + 22 q^{4} + 6 q^{5} - 14 q^{7} - 54 q^{8} - 62 q^{10} - 40 q^{11} - 40 q^{14} + 122 q^{16} - 98 q^{17} + 124 q^{19} + 466 q^{20} + 220 q^{22} - 104 q^{23} - 58 q^{25} - 144 q^{28} - 194 q^{29} + 26 q^{31} - 654 q^{32} + 1062 q^{34} - 88 q^{35} + 102 q^{37} - 332 q^{38} - 998 q^{40} + 1054 q^{41} + 450 q^{43} + 44 q^{44} - 172 q^{46} + 96 q^{47} + 1070 q^{49} - 996 q^{50} - 262 q^{53} + 204 q^{55} - 2164 q^{56} + 722 q^{58} - 308 q^{59} - 928 q^{61} - 2780 q^{62} + 1026 q^{64} - 1134 q^{67} - 1786 q^{68} - 2324 q^{70} - 1064 q^{71} + 952 q^{73} - 1158 q^{74} - 1708 q^{76} - 2508 q^{77} - 746 q^{79} + 2922 q^{80} + 1734 q^{82} + 404 q^{83} - 1394 q^{85} - 3168 q^{86} + 3060 q^{88} - 1620 q^{89} - 332 q^{92} - 772 q^{94} - 2204 q^{95} + 2166 q^{97} + 1906 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.22605 1.49414 0.747068 0.664748i \(-0.231461\pi\)
0.747068 + 0.664748i \(0.231461\pi\)
\(3\) 0 0
\(4\) 9.85953 1.23244
\(5\) 5.85953 0.524093 0.262046 0.965055i \(-0.415603\pi\)
0.262046 + 0.965055i \(0.415603\pi\)
\(6\) 0 0
\(7\) −24.1254 −1.30265 −0.651326 0.758798i \(-0.725787\pi\)
−0.651326 + 0.758798i \(0.725787\pi\)
\(8\) 7.85849 0.347299
\(9\) 0 0
\(10\) 24.7627 0.783065
\(11\) 33.8892 0.928907 0.464453 0.885598i \(-0.346251\pi\)
0.464453 + 0.885598i \(0.346251\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −101.955 −1.94634
\(15\) 0 0
\(16\) −45.6659 −0.713529
\(17\) 49.3956 0.704718 0.352359 0.935865i \(-0.385380\pi\)
0.352359 + 0.935865i \(0.385380\pi\)
\(18\) 0 0
\(19\) −76.8548 −0.927985 −0.463992 0.885839i \(-0.653583\pi\)
−0.463992 + 0.885839i \(0.653583\pi\)
\(20\) 57.7723 0.645913
\(21\) 0 0
\(22\) 143.218 1.38791
\(23\) −6.29163 −0.0570389 −0.0285195 0.999593i \(-0.509079\pi\)
−0.0285195 + 0.999593i \(0.509079\pi\)
\(24\) 0 0
\(25\) −90.6659 −0.725327
\(26\) 0 0
\(27\) 0 0
\(28\) −237.866 −1.60544
\(29\) −100.995 −0.646702 −0.323351 0.946279i \(-0.604809\pi\)
−0.323351 + 0.946279i \(0.604809\pi\)
\(30\) 0 0
\(31\) −307.580 −1.78203 −0.891016 0.453972i \(-0.850007\pi\)
−0.891016 + 0.453972i \(0.850007\pi\)
\(32\) −255.854 −1.41341
\(33\) 0 0
\(34\) 208.749 1.05294
\(35\) −141.364 −0.682710
\(36\) 0 0
\(37\) −76.0189 −0.337768 −0.168884 0.985636i \(-0.554016\pi\)
−0.168884 + 0.985636i \(0.554016\pi\)
\(38\) −324.793 −1.38653
\(39\) 0 0
\(40\) 46.0471 0.182017
\(41\) 514.418 1.95948 0.979740 0.200274i \(-0.0641833\pi\)
0.979740 + 0.200274i \(0.0641833\pi\)
\(42\) 0 0
\(43\) −268.184 −0.951108 −0.475554 0.879686i \(-0.657752\pi\)
−0.475554 + 0.879686i \(0.657752\pi\)
\(44\) 334.132 1.14482
\(45\) 0 0
\(46\) −26.5888 −0.0852239
\(47\) 460.912 1.43045 0.715223 0.698896i \(-0.246325\pi\)
0.715223 + 0.698896i \(0.246325\pi\)
\(48\) 0 0
\(49\) 239.037 0.696901
\(50\) −383.159 −1.08374
\(51\) 0 0
\(52\) 0 0
\(53\) −67.8057 −0.175733 −0.0878663 0.996132i \(-0.528005\pi\)
−0.0878663 + 0.996132i \(0.528005\pi\)
\(54\) 0 0
\(55\) 198.575 0.486833
\(56\) −189.589 −0.452410
\(57\) 0 0
\(58\) −426.812 −0.966261
\(59\) −25.2021 −0.0556107 −0.0278053 0.999613i \(-0.508852\pi\)
−0.0278053 + 0.999613i \(0.508852\pi\)
\(60\) 0 0
\(61\) −588.832 −1.23594 −0.617969 0.786202i \(-0.712044\pi\)
−0.617969 + 0.786202i \(0.712044\pi\)
\(62\) −1299.85 −2.66260
\(63\) 0 0
\(64\) −715.927 −1.39830
\(65\) 0 0
\(66\) 0 0
\(67\) −1004.46 −1.83156 −0.915778 0.401684i \(-0.868425\pi\)
−0.915778 + 0.401684i \(0.868425\pi\)
\(68\) 487.018 0.868523
\(69\) 0 0
\(70\) −597.411 −1.02006
\(71\) −895.481 −1.49682 −0.748408 0.663238i \(-0.769182\pi\)
−0.748408 + 0.663238i \(0.769182\pi\)
\(72\) 0 0
\(73\) 968.599 1.55296 0.776479 0.630143i \(-0.217004\pi\)
0.776479 + 0.630143i \(0.217004\pi\)
\(74\) −321.260 −0.504672
\(75\) 0 0
\(76\) −757.753 −1.14369
\(77\) −817.592 −1.21004
\(78\) 0 0
\(79\) −119.053 −0.169551 −0.0847755 0.996400i \(-0.527017\pi\)
−0.0847755 + 0.996400i \(0.527017\pi\)
\(80\) −267.581 −0.373955
\(81\) 0 0
\(82\) 2173.96 2.92773
\(83\) −480.784 −0.635818 −0.317909 0.948121i \(-0.602981\pi\)
−0.317909 + 0.948121i \(0.602981\pi\)
\(84\) 0 0
\(85\) 289.435 0.369337
\(86\) −1133.36 −1.42109
\(87\) 0 0
\(88\) 266.318 0.322609
\(89\) −1085.91 −1.29333 −0.646663 0.762776i \(-0.723836\pi\)
−0.646663 + 0.762776i \(0.723836\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −62.0325 −0.0702972
\(93\) 0 0
\(94\) 1947.84 2.13728
\(95\) −450.333 −0.486350
\(96\) 0 0
\(97\) −16.6552 −0.0174338 −0.00871692 0.999962i \(-0.502775\pi\)
−0.00871692 + 0.999962i \(0.502775\pi\)
\(98\) 1010.18 1.04126
\(99\) 0 0
\(100\) −893.923 −0.893923
\(101\) 958.004 0.943811 0.471906 0.881649i \(-0.343566\pi\)
0.471906 + 0.881649i \(0.343566\pi\)
\(102\) 0 0
\(103\) −2.70560 −0.00258826 −0.00129413 0.999999i \(-0.500412\pi\)
−0.00129413 + 0.999999i \(0.500412\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −286.551 −0.262568
\(107\) −1351.28 −1.22087 −0.610437 0.792065i \(-0.709006\pi\)
−0.610437 + 0.792065i \(0.709006\pi\)
\(108\) 0 0
\(109\) 448.455 0.394075 0.197037 0.980396i \(-0.436868\pi\)
0.197037 + 0.980396i \(0.436868\pi\)
\(110\) 839.188 0.727395
\(111\) 0 0
\(112\) 1101.71 0.929480
\(113\) −1398.85 −1.16453 −0.582267 0.812997i \(-0.697834\pi\)
−0.582267 + 0.812997i \(0.697834\pi\)
\(114\) 0 0
\(115\) −36.8660 −0.0298937
\(116\) −995.767 −0.797023
\(117\) 0 0
\(118\) −106.505 −0.0830899
\(119\) −1191.69 −0.918002
\(120\) 0 0
\(121\) −182.523 −0.137132
\(122\) −2488.44 −1.84666
\(123\) 0 0
\(124\) −3032.59 −2.19625
\(125\) −1263.70 −0.904231
\(126\) 0 0
\(127\) −119.504 −0.0834985 −0.0417492 0.999128i \(-0.513293\pi\)
−0.0417492 + 0.999128i \(0.513293\pi\)
\(128\) −978.713 −0.675834
\(129\) 0 0
\(130\) 0 0
\(131\) 2251.70 1.50177 0.750886 0.660432i \(-0.229627\pi\)
0.750886 + 0.660432i \(0.229627\pi\)
\(132\) 0 0
\(133\) 1854.16 1.20884
\(134\) −4244.90 −2.73659
\(135\) 0 0
\(136\) 388.175 0.244748
\(137\) −1130.62 −0.705076 −0.352538 0.935797i \(-0.614681\pi\)
−0.352538 + 0.935797i \(0.614681\pi\)
\(138\) 0 0
\(139\) −595.287 −0.363249 −0.181624 0.983368i \(-0.558136\pi\)
−0.181624 + 0.983368i \(0.558136\pi\)
\(140\) −1393.78 −0.841400
\(141\) 0 0
\(142\) −3784.35 −2.23645
\(143\) 0 0
\(144\) 0 0
\(145\) −591.786 −0.338932
\(146\) 4093.35 2.32033
\(147\) 0 0
\(148\) −749.511 −0.416280
\(149\) 793.175 0.436103 0.218052 0.975937i \(-0.430030\pi\)
0.218052 + 0.975937i \(0.430030\pi\)
\(150\) 0 0
\(151\) −134.213 −0.0723317 −0.0361659 0.999346i \(-0.511514\pi\)
−0.0361659 + 0.999346i \(0.511514\pi\)
\(152\) −603.963 −0.322288
\(153\) 0 0
\(154\) −3455.19 −1.80797
\(155\) −1802.28 −0.933950
\(156\) 0 0
\(157\) 1509.07 0.767114 0.383557 0.923517i \(-0.374699\pi\)
0.383557 + 0.923517i \(0.374699\pi\)
\(158\) −503.125 −0.253332
\(159\) 0 0
\(160\) −1499.19 −0.740757
\(161\) 151.788 0.0743019
\(162\) 0 0
\(163\) 1175.08 0.564658 0.282329 0.959318i \(-0.408893\pi\)
0.282329 + 0.959318i \(0.408893\pi\)
\(164\) 5071.93 2.41494
\(165\) 0 0
\(166\) −2031.82 −0.949999
\(167\) −1474.01 −0.683010 −0.341505 0.939880i \(-0.610937\pi\)
−0.341505 + 0.939880i \(0.610937\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 1223.17 0.551840
\(171\) 0 0
\(172\) −2644.17 −1.17219
\(173\) 2328.31 1.02322 0.511612 0.859216i \(-0.329048\pi\)
0.511612 + 0.859216i \(0.329048\pi\)
\(174\) 0 0
\(175\) 2187.35 0.944848
\(176\) −1547.58 −0.662802
\(177\) 0 0
\(178\) −4589.10 −1.93240
\(179\) 2133.85 0.891015 0.445508 0.895278i \(-0.353023\pi\)
0.445508 + 0.895278i \(0.353023\pi\)
\(180\) 0 0
\(181\) −2485.41 −1.02066 −0.510329 0.859979i \(-0.670476\pi\)
−0.510329 + 0.859979i \(0.670476\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −49.4427 −0.0198096
\(185\) −445.435 −0.177022
\(186\) 0 0
\(187\) 1673.98 0.654617
\(188\) 4544.38 1.76294
\(189\) 0 0
\(190\) −1903.13 −0.726673
\(191\) 2325.07 0.880816 0.440408 0.897798i \(-0.354834\pi\)
0.440408 + 0.897798i \(0.354834\pi\)
\(192\) 0 0
\(193\) 3350.12 1.24946 0.624732 0.780839i \(-0.285208\pi\)
0.624732 + 0.780839i \(0.285208\pi\)
\(194\) −70.3859 −0.0260485
\(195\) 0 0
\(196\) 2356.79 0.858890
\(197\) 3859.30 1.39576 0.697878 0.716217i \(-0.254128\pi\)
0.697878 + 0.716217i \(0.254128\pi\)
\(198\) 0 0
\(199\) −4083.60 −1.45467 −0.727333 0.686284i \(-0.759241\pi\)
−0.727333 + 0.686284i \(0.759241\pi\)
\(200\) −712.497 −0.251906
\(201\) 0 0
\(202\) 4048.58 1.41018
\(203\) 2436.56 0.842428
\(204\) 0 0
\(205\) 3014.25 1.02695
\(206\) −11.4340 −0.00386721
\(207\) 0 0
\(208\) 0 0
\(209\) −2604.55 −0.862011
\(210\) 0 0
\(211\) 3027.30 0.987714 0.493857 0.869543i \(-0.335587\pi\)
0.493857 + 0.869543i \(0.335587\pi\)
\(212\) −668.533 −0.216580
\(213\) 0 0
\(214\) −5710.60 −1.82415
\(215\) −1571.43 −0.498469
\(216\) 0 0
\(217\) 7420.50 2.32137
\(218\) 1895.19 0.588801
\(219\) 0 0
\(220\) 1957.86 0.599994
\(221\) 0 0
\(222\) 0 0
\(223\) −1724.76 −0.517930 −0.258965 0.965887i \(-0.583381\pi\)
−0.258965 + 0.965887i \(0.583381\pi\)
\(224\) 6172.60 1.84118
\(225\) 0 0
\(226\) −5911.60 −1.73997
\(227\) 1923.27 0.562344 0.281172 0.959657i \(-0.409277\pi\)
0.281172 + 0.959657i \(0.409277\pi\)
\(228\) 0 0
\(229\) 373.993 0.107922 0.0539610 0.998543i \(-0.482815\pi\)
0.0539610 + 0.998543i \(0.482815\pi\)
\(230\) −155.798 −0.0446652
\(231\) 0 0
\(232\) −793.671 −0.224599
\(233\) −3094.49 −0.870073 −0.435036 0.900413i \(-0.643265\pi\)
−0.435036 + 0.900413i \(0.643265\pi\)
\(234\) 0 0
\(235\) 2700.73 0.749686
\(236\) −248.481 −0.0685369
\(237\) 0 0
\(238\) −5036.15 −1.37162
\(239\) 1221.18 0.330510 0.165255 0.986251i \(-0.447155\pi\)
0.165255 + 0.986251i \(0.447155\pi\)
\(240\) 0 0
\(241\) 145.401 0.0388634 0.0194317 0.999811i \(-0.493814\pi\)
0.0194317 + 0.999811i \(0.493814\pi\)
\(242\) −771.350 −0.204894
\(243\) 0 0
\(244\) −5805.61 −1.52322
\(245\) 1400.65 0.365241
\(246\) 0 0
\(247\) 0 0
\(248\) −2417.11 −0.618899
\(249\) 0 0
\(250\) −5340.47 −1.35104
\(251\) 985.670 0.247868 0.123934 0.992290i \(-0.460449\pi\)
0.123934 + 0.992290i \(0.460449\pi\)
\(252\) 0 0
\(253\) −213.218 −0.0529839
\(254\) −505.032 −0.124758
\(255\) 0 0
\(256\) 1591.33 0.388507
\(257\) −2929.32 −0.710995 −0.355498 0.934677i \(-0.615689\pi\)
−0.355498 + 0.934677i \(0.615689\pi\)
\(258\) 0 0
\(259\) 1833.99 0.439995
\(260\) 0 0
\(261\) 0 0
\(262\) 9515.82 2.24385
\(263\) −2238.00 −0.524719 −0.262360 0.964970i \(-0.584501\pi\)
−0.262360 + 0.964970i \(0.584501\pi\)
\(264\) 0 0
\(265\) −397.310 −0.0921002
\(266\) 7835.77 1.80617
\(267\) 0 0
\(268\) −9903.50 −2.25729
\(269\) −1925.98 −0.436540 −0.218270 0.975888i \(-0.570041\pi\)
−0.218270 + 0.975888i \(0.570041\pi\)
\(270\) 0 0
\(271\) −3562.29 −0.798500 −0.399250 0.916842i \(-0.630729\pi\)
−0.399250 + 0.916842i \(0.630729\pi\)
\(272\) −2255.69 −0.502837
\(273\) 0 0
\(274\) −4778.06 −1.05348
\(275\) −3072.59 −0.673761
\(276\) 0 0
\(277\) −1437.42 −0.311792 −0.155896 0.987773i \(-0.549827\pi\)
−0.155896 + 0.987773i \(0.549827\pi\)
\(278\) −2515.72 −0.542743
\(279\) 0 0
\(280\) −1110.91 −0.237105
\(281\) 3913.51 0.830820 0.415410 0.909634i \(-0.363638\pi\)
0.415410 + 0.909634i \(0.363638\pi\)
\(282\) 0 0
\(283\) −3212.31 −0.674743 −0.337371 0.941372i \(-0.609538\pi\)
−0.337371 + 0.941372i \(0.609538\pi\)
\(284\) −8829.02 −1.84474
\(285\) 0 0
\(286\) 0 0
\(287\) −12410.6 −2.55252
\(288\) 0 0
\(289\) −2473.07 −0.503373
\(290\) −2500.92 −0.506410
\(291\) 0 0
\(292\) 9549.94 1.91393
\(293\) 4901.77 0.977353 0.488677 0.872465i \(-0.337480\pi\)
0.488677 + 0.872465i \(0.337480\pi\)
\(294\) 0 0
\(295\) −147.672 −0.0291452
\(296\) −597.394 −0.117307
\(297\) 0 0
\(298\) 3352.00 0.651598
\(299\) 0 0
\(300\) 0 0
\(301\) 6470.06 1.23896
\(302\) −567.191 −0.108073
\(303\) 0 0
\(304\) 3509.64 0.662144
\(305\) −3450.28 −0.647746
\(306\) 0 0
\(307\) 5800.63 1.07837 0.539185 0.842188i \(-0.318733\pi\)
0.539185 + 0.842188i \(0.318733\pi\)
\(308\) −8061.07 −1.49131
\(309\) 0 0
\(310\) −7616.51 −1.39545
\(311\) 4913.51 0.895884 0.447942 0.894063i \(-0.352157\pi\)
0.447942 + 0.894063i \(0.352157\pi\)
\(312\) 0 0
\(313\) −8104.97 −1.46364 −0.731822 0.681496i \(-0.761330\pi\)
−0.731822 + 0.681496i \(0.761330\pi\)
\(314\) 6377.41 1.14617
\(315\) 0 0
\(316\) −1173.81 −0.208962
\(317\) −5149.92 −0.912455 −0.456227 0.889863i \(-0.650800\pi\)
−0.456227 + 0.889863i \(0.650800\pi\)
\(318\) 0 0
\(319\) −3422.65 −0.600726
\(320\) −4195.00 −0.732836
\(321\) 0 0
\(322\) 641.466 0.111017
\(323\) −3796.29 −0.653967
\(324\) 0 0
\(325\) 0 0
\(326\) 4965.95 0.843676
\(327\) 0 0
\(328\) 4042.55 0.680526
\(329\) −11119.7 −1.86337
\(330\) 0 0
\(331\) 6061.98 1.00664 0.503318 0.864101i \(-0.332112\pi\)
0.503318 + 0.864101i \(0.332112\pi\)
\(332\) −4740.31 −0.783609
\(333\) 0 0
\(334\) −6229.26 −1.02051
\(335\) −5885.67 −0.959905
\(336\) 0 0
\(337\) 3743.50 0.605108 0.302554 0.953132i \(-0.402161\pi\)
0.302554 + 0.953132i \(0.402161\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 2853.70 0.455187
\(341\) −10423.6 −1.65534
\(342\) 0 0
\(343\) 2508.15 0.394832
\(344\) −2107.52 −0.330319
\(345\) 0 0
\(346\) 9839.55 1.52884
\(347\) 2520.41 0.389921 0.194961 0.980811i \(-0.437542\pi\)
0.194961 + 0.980811i \(0.437542\pi\)
\(348\) 0 0
\(349\) 10650.7 1.63359 0.816793 0.576931i \(-0.195750\pi\)
0.816793 + 0.576931i \(0.195750\pi\)
\(350\) 9243.88 1.41173
\(351\) 0 0
\(352\) −8670.70 −1.31293
\(353\) 9002.82 1.35743 0.678714 0.734403i \(-0.262538\pi\)
0.678714 + 0.734403i \(0.262538\pi\)
\(354\) 0 0
\(355\) −5247.10 −0.784471
\(356\) −10706.5 −1.59395
\(357\) 0 0
\(358\) 9017.78 1.33130
\(359\) 11360.9 1.67021 0.835106 0.550089i \(-0.185406\pi\)
0.835106 + 0.550089i \(0.185406\pi\)
\(360\) 0 0
\(361\) −952.336 −0.138845
\(362\) −10503.5 −1.52500
\(363\) 0 0
\(364\) 0 0
\(365\) 5675.54 0.813894
\(366\) 0 0
\(367\) −13938.8 −1.98257 −0.991283 0.131753i \(-0.957939\pi\)
−0.991283 + 0.131753i \(0.957939\pi\)
\(368\) 287.313 0.0406990
\(369\) 0 0
\(370\) −1882.43 −0.264495
\(371\) 1635.84 0.228918
\(372\) 0 0
\(373\) 1593.07 0.221142 0.110571 0.993868i \(-0.464732\pi\)
0.110571 + 0.993868i \(0.464732\pi\)
\(374\) 7074.32 0.978087
\(375\) 0 0
\(376\) 3622.07 0.496793
\(377\) 0 0
\(378\) 0 0
\(379\) 9137.56 1.23843 0.619215 0.785221i \(-0.287451\pi\)
0.619215 + 0.785221i \(0.287451\pi\)
\(380\) −4440.08 −0.599398
\(381\) 0 0
\(382\) 9825.86 1.31606
\(383\) −9551.07 −1.27425 −0.637124 0.770761i \(-0.719876\pi\)
−0.637124 + 0.770761i \(0.719876\pi\)
\(384\) 0 0
\(385\) −4790.71 −0.634174
\(386\) 14157.8 1.86687
\(387\) 0 0
\(388\) −164.213 −0.0214862
\(389\) −7366.50 −0.960145 −0.480072 0.877229i \(-0.659390\pi\)
−0.480072 + 0.877229i \(0.659390\pi\)
\(390\) 0 0
\(391\) −310.779 −0.0401964
\(392\) 1878.47 0.242033
\(393\) 0 0
\(394\) 16309.6 2.08545
\(395\) −697.596 −0.0888604
\(396\) 0 0
\(397\) −11696.5 −1.47866 −0.739332 0.673342i \(-0.764858\pi\)
−0.739332 + 0.673342i \(0.764858\pi\)
\(398\) −17257.5 −2.17347
\(399\) 0 0
\(400\) 4140.34 0.517542
\(401\) 14167.6 1.76433 0.882167 0.470937i \(-0.156084\pi\)
0.882167 + 0.470937i \(0.156084\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 9445.47 1.16319
\(405\) 0 0
\(406\) 10297.0 1.25870
\(407\) −2576.22 −0.313755
\(408\) 0 0
\(409\) 2703.69 0.326868 0.163434 0.986554i \(-0.447743\pi\)
0.163434 + 0.986554i \(0.447743\pi\)
\(410\) 12738.4 1.53440
\(411\) 0 0
\(412\) −26.6759 −0.00318988
\(413\) 608.011 0.0724414
\(414\) 0 0
\(415\) −2817.17 −0.333228
\(416\) 0 0
\(417\) 0 0
\(418\) −11007.0 −1.28796
\(419\) −7142.52 −0.832781 −0.416390 0.909186i \(-0.636705\pi\)
−0.416390 + 0.909186i \(0.636705\pi\)
\(420\) 0 0
\(421\) −3406.45 −0.394347 −0.197174 0.980369i \(-0.563176\pi\)
−0.197174 + 0.980369i \(0.563176\pi\)
\(422\) 12793.5 1.47578
\(423\) 0 0
\(424\) −532.850 −0.0610318
\(425\) −4478.50 −0.511151
\(426\) 0 0
\(427\) 14205.8 1.61000
\(428\) −13323.0 −1.50466
\(429\) 0 0
\(430\) −6640.96 −0.744780
\(431\) −5172.97 −0.578128 −0.289064 0.957310i \(-0.593344\pi\)
−0.289064 + 0.957310i \(0.593344\pi\)
\(432\) 0 0
\(433\) 10955.0 1.21585 0.607924 0.793995i \(-0.292002\pi\)
0.607924 + 0.793995i \(0.292002\pi\)
\(434\) 31359.4 3.46844
\(435\) 0 0
\(436\) 4421.55 0.485674
\(437\) 483.542 0.0529313
\(438\) 0 0
\(439\) 11832.4 1.28640 0.643202 0.765696i \(-0.277605\pi\)
0.643202 + 0.765696i \(0.277605\pi\)
\(440\) 1560.50 0.169077
\(441\) 0 0
\(442\) 0 0
\(443\) −13479.8 −1.44570 −0.722852 0.691003i \(-0.757169\pi\)
−0.722852 + 0.691003i \(0.757169\pi\)
\(444\) 0 0
\(445\) −6362.91 −0.677822
\(446\) −7288.92 −0.773857
\(447\) 0 0
\(448\) 17272.1 1.82149
\(449\) 6774.34 0.712028 0.356014 0.934481i \(-0.384135\pi\)
0.356014 + 0.934481i \(0.384135\pi\)
\(450\) 0 0
\(451\) 17433.2 1.82017
\(452\) −13792.0 −1.43522
\(453\) 0 0
\(454\) 8127.86 0.840219
\(455\) 0 0
\(456\) 0 0
\(457\) −4642.36 −0.475187 −0.237594 0.971365i \(-0.576359\pi\)
−0.237594 + 0.971365i \(0.576359\pi\)
\(458\) 1580.51 0.161250
\(459\) 0 0
\(460\) −363.482 −0.0368422
\(461\) −2460.37 −0.248570 −0.124285 0.992247i \(-0.539664\pi\)
−0.124285 + 0.992247i \(0.539664\pi\)
\(462\) 0 0
\(463\) 4290.01 0.430613 0.215306 0.976547i \(-0.430925\pi\)
0.215306 + 0.976547i \(0.430925\pi\)
\(464\) 4612.04 0.461441
\(465\) 0 0
\(466\) −13077.5 −1.30001
\(467\) 8798.99 0.871882 0.435941 0.899975i \(-0.356416\pi\)
0.435941 + 0.899975i \(0.356416\pi\)
\(468\) 0 0
\(469\) 24233.0 2.38588
\(470\) 11413.4 1.12013
\(471\) 0 0
\(472\) −198.050 −0.0193136
\(473\) −9088.54 −0.883491
\(474\) 0 0
\(475\) 6968.11 0.673092
\(476\) −11749.5 −1.13138
\(477\) 0 0
\(478\) 5160.79 0.493826
\(479\) 10973.4 1.04673 0.523367 0.852107i \(-0.324676\pi\)
0.523367 + 0.852107i \(0.324676\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 614.470 0.0580671
\(483\) 0 0
\(484\) −1799.59 −0.169007
\(485\) −97.5919 −0.00913694
\(486\) 0 0
\(487\) −5209.58 −0.484740 −0.242370 0.970184i \(-0.577925\pi\)
−0.242370 + 0.970184i \(0.577925\pi\)
\(488\) −4627.33 −0.429241
\(489\) 0 0
\(490\) 5919.20 0.545719
\(491\) 8779.22 0.806926 0.403463 0.914996i \(-0.367806\pi\)
0.403463 + 0.914996i \(0.367806\pi\)
\(492\) 0 0
\(493\) −4988.73 −0.455742
\(494\) 0 0
\(495\) 0 0
\(496\) 14045.9 1.27153
\(497\) 21603.9 1.94983
\(498\) 0 0
\(499\) −15590.1 −1.39861 −0.699305 0.714823i \(-0.746507\pi\)
−0.699305 + 0.714823i \(0.746507\pi\)
\(500\) −12459.5 −1.11441
\(501\) 0 0
\(502\) 4165.49 0.370349
\(503\) 64.3909 0.00570785 0.00285393 0.999996i \(-0.499092\pi\)
0.00285393 + 0.999996i \(0.499092\pi\)
\(504\) 0 0
\(505\) 5613.45 0.494644
\(506\) −901.072 −0.0791651
\(507\) 0 0
\(508\) −1178.26 −0.102907
\(509\) −3214.15 −0.279891 −0.139946 0.990159i \(-0.544693\pi\)
−0.139946 + 0.990159i \(0.544693\pi\)
\(510\) 0 0
\(511\) −23367.9 −2.02296
\(512\) 14554.7 1.25632
\(513\) 0 0
\(514\) −12379.5 −1.06232
\(515\) −15.8535 −0.00135649
\(516\) 0 0
\(517\) 15619.9 1.32875
\(518\) 7750.54 0.657412
\(519\) 0 0
\(520\) 0 0
\(521\) 3053.01 0.256727 0.128363 0.991727i \(-0.459028\pi\)
0.128363 + 0.991727i \(0.459028\pi\)
\(522\) 0 0
\(523\) 5096.02 0.426067 0.213034 0.977045i \(-0.431666\pi\)
0.213034 + 0.977045i \(0.431666\pi\)
\(524\) 22200.7 1.85085
\(525\) 0 0
\(526\) −9457.92 −0.784002
\(527\) −15193.1 −1.25583
\(528\) 0 0
\(529\) −12127.4 −0.996747
\(530\) −1679.05 −0.137610
\(531\) 0 0
\(532\) 18281.1 1.48983
\(533\) 0 0
\(534\) 0 0
\(535\) −7917.89 −0.639851
\(536\) −7893.53 −0.636098
\(537\) 0 0
\(538\) −8139.31 −0.652250
\(539\) 8100.77 0.647356
\(540\) 0 0
\(541\) 7861.99 0.624793 0.312397 0.949952i \(-0.398868\pi\)
0.312397 + 0.949952i \(0.398868\pi\)
\(542\) −15054.4 −1.19307
\(543\) 0 0
\(544\) −12638.1 −0.996054
\(545\) 2627.73 0.206532
\(546\) 0 0
\(547\) −6317.48 −0.493814 −0.246907 0.969039i \(-0.579414\pi\)
−0.246907 + 0.969039i \(0.579414\pi\)
\(548\) −11147.4 −0.868965
\(549\) 0 0
\(550\) −12984.9 −1.00669
\(551\) 7761.98 0.600130
\(552\) 0 0
\(553\) 2872.21 0.220866
\(554\) −6074.63 −0.465860
\(555\) 0 0
\(556\) −5869.25 −0.447683
\(557\) 971.234 0.0738824 0.0369412 0.999317i \(-0.488239\pi\)
0.0369412 + 0.999317i \(0.488239\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 6455.50 0.487134
\(561\) 0 0
\(562\) 16538.7 1.24136
\(563\) −9328.49 −0.698311 −0.349155 0.937065i \(-0.613531\pi\)
−0.349155 + 0.937065i \(0.613531\pi\)
\(564\) 0 0
\(565\) −8196.59 −0.610324
\(566\) −13575.4 −1.00816
\(567\) 0 0
\(568\) −7037.12 −0.519843
\(569\) −17452.2 −1.28582 −0.642911 0.765941i \(-0.722273\pi\)
−0.642911 + 0.765941i \(0.722273\pi\)
\(570\) 0 0
\(571\) −20181.4 −1.47910 −0.739548 0.673103i \(-0.764961\pi\)
−0.739548 + 0.673103i \(0.764961\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −52447.8 −3.81381
\(575\) 570.436 0.0413719
\(576\) 0 0
\(577\) 6382.72 0.460513 0.230257 0.973130i \(-0.426043\pi\)
0.230257 + 0.973130i \(0.426043\pi\)
\(578\) −10451.3 −0.752107
\(579\) 0 0
\(580\) −5834.73 −0.417714
\(581\) 11599.1 0.828250
\(582\) 0 0
\(583\) −2297.88 −0.163239
\(584\) 7611.72 0.539341
\(585\) 0 0
\(586\) 20715.2 1.46030
\(587\) 775.527 0.0545305 0.0272653 0.999628i \(-0.491320\pi\)
0.0272653 + 0.999628i \(0.491320\pi\)
\(588\) 0 0
\(589\) 23639.0 1.65370
\(590\) −624.072 −0.0435468
\(591\) 0 0
\(592\) 3471.47 0.241008
\(593\) 17843.3 1.23564 0.617821 0.786319i \(-0.288016\pi\)
0.617821 + 0.786319i \(0.288016\pi\)
\(594\) 0 0
\(595\) −6982.76 −0.481118
\(596\) 7820.33 0.537472
\(597\) 0 0
\(598\) 0 0
\(599\) 24373.3 1.66255 0.831274 0.555863i \(-0.187612\pi\)
0.831274 + 0.555863i \(0.187612\pi\)
\(600\) 0 0
\(601\) −3526.99 −0.239383 −0.119691 0.992811i \(-0.538190\pi\)
−0.119691 + 0.992811i \(0.538190\pi\)
\(602\) 27342.8 1.85118
\(603\) 0 0
\(604\) −1323.28 −0.0891446
\(605\) −1069.50 −0.0718698
\(606\) 0 0
\(607\) 7991.55 0.534377 0.267189 0.963644i \(-0.413905\pi\)
0.267189 + 0.963644i \(0.413905\pi\)
\(608\) 19663.6 1.31162
\(609\) 0 0
\(610\) −14581.1 −0.967821
\(611\) 0 0
\(612\) 0 0
\(613\) −16332.2 −1.07610 −0.538051 0.842912i \(-0.680839\pi\)
−0.538051 + 0.842912i \(0.680839\pi\)
\(614\) 24513.8 1.61123
\(615\) 0 0
\(616\) −6425.04 −0.420247
\(617\) 19353.6 1.26280 0.631401 0.775457i \(-0.282480\pi\)
0.631401 + 0.775457i \(0.282480\pi\)
\(618\) 0 0
\(619\) −9982.52 −0.648193 −0.324096 0.946024i \(-0.605060\pi\)
−0.324096 + 0.946024i \(0.605060\pi\)
\(620\) −17769.6 −1.15104
\(621\) 0 0
\(622\) 20764.8 1.33857
\(623\) 26198.0 1.68475
\(624\) 0 0
\(625\) 3928.53 0.251426
\(626\) −34252.1 −2.18688
\(627\) 0 0
\(628\) 14878.7 0.945423
\(629\) −3755.00 −0.238031
\(630\) 0 0
\(631\) −575.775 −0.0363253 −0.0181626 0.999835i \(-0.505782\pi\)
−0.0181626 + 0.999835i \(0.505782\pi\)
\(632\) −935.578 −0.0588850
\(633\) 0 0
\(634\) −21763.8 −1.36333
\(635\) −700.240 −0.0437609
\(636\) 0 0
\(637\) 0 0
\(638\) −14464.3 −0.897566
\(639\) 0 0
\(640\) −5734.80 −0.354200
\(641\) 24521.8 1.51100 0.755500 0.655149i \(-0.227394\pi\)
0.755500 + 0.655149i \(0.227394\pi\)
\(642\) 0 0
\(643\) −22667.0 −1.39020 −0.695099 0.718914i \(-0.744640\pi\)
−0.695099 + 0.718914i \(0.744640\pi\)
\(644\) 1496.56 0.0915727
\(645\) 0 0
\(646\) −16043.3 −0.977116
\(647\) 2397.45 0.145678 0.0728389 0.997344i \(-0.476794\pi\)
0.0728389 + 0.997344i \(0.476794\pi\)
\(648\) 0 0
\(649\) −854.078 −0.0516572
\(650\) 0 0
\(651\) 0 0
\(652\) 11585.7 0.695908
\(653\) −20002.1 −1.19868 −0.599342 0.800493i \(-0.704571\pi\)
−0.599342 + 0.800493i \(0.704571\pi\)
\(654\) 0 0
\(655\) 13193.9 0.787068
\(656\) −23491.4 −1.39815
\(657\) 0 0
\(658\) −46992.5 −2.78413
\(659\) 3517.96 0.207952 0.103976 0.994580i \(-0.466844\pi\)
0.103976 + 0.994580i \(0.466844\pi\)
\(660\) 0 0
\(661\) 13583.4 0.799294 0.399647 0.916669i \(-0.369133\pi\)
0.399647 + 0.916669i \(0.369133\pi\)
\(662\) 25618.2 1.50405
\(663\) 0 0
\(664\) −3778.24 −0.220819
\(665\) 10864.5 0.633544
\(666\) 0 0
\(667\) 635.425 0.0368872
\(668\) −14533.1 −0.841770
\(669\) 0 0
\(670\) −24873.1 −1.43423
\(671\) −19955.1 −1.14807
\(672\) 0 0
\(673\) −10895.8 −0.624077 −0.312038 0.950069i \(-0.601012\pi\)
−0.312038 + 0.950069i \(0.601012\pi\)
\(674\) 15820.2 0.904113
\(675\) 0 0
\(676\) 0 0
\(677\) −1449.03 −0.0822609 −0.0411305 0.999154i \(-0.513096\pi\)
−0.0411305 + 0.999154i \(0.513096\pi\)
\(678\) 0 0
\(679\) 401.815 0.0227102
\(680\) 2274.52 0.128271
\(681\) 0 0
\(682\) −44050.9 −2.47331
\(683\) 15366.4 0.860878 0.430439 0.902620i \(-0.358359\pi\)
0.430439 + 0.902620i \(0.358359\pi\)
\(684\) 0 0
\(685\) −6624.91 −0.369525
\(686\) 10599.6 0.589933
\(687\) 0 0
\(688\) 12246.9 0.678644
\(689\) 0 0
\(690\) 0 0
\(691\) −2019.71 −0.111191 −0.0555957 0.998453i \(-0.517706\pi\)
−0.0555957 + 0.998453i \(0.517706\pi\)
\(692\) 22956.0 1.26106
\(693\) 0 0
\(694\) 10651.4 0.582595
\(695\) −3488.10 −0.190376
\(696\) 0 0
\(697\) 25410.0 1.38088
\(698\) 45010.6 2.44080
\(699\) 0 0
\(700\) 21566.3 1.16447
\(701\) −28031.6 −1.51033 −0.755164 0.655536i \(-0.772443\pi\)
−0.755164 + 0.655536i \(0.772443\pi\)
\(702\) 0 0
\(703\) 5842.42 0.313444
\(704\) −24262.2 −1.29889
\(705\) 0 0
\(706\) 38046.4 2.02818
\(707\) −23112.3 −1.22946
\(708\) 0 0
\(709\) 19604.4 1.03845 0.519224 0.854638i \(-0.326221\pi\)
0.519224 + 0.854638i \(0.326221\pi\)
\(710\) −22174.5 −1.17211
\(711\) 0 0
\(712\) −8533.59 −0.449171
\(713\) 1935.18 0.101645
\(714\) 0 0
\(715\) 0 0
\(716\) 21038.8 1.09812
\(717\) 0 0
\(718\) 48011.8 2.49552
\(719\) 14727.4 0.763894 0.381947 0.924184i \(-0.375254\pi\)
0.381947 + 0.924184i \(0.375254\pi\)
\(720\) 0 0
\(721\) 65.2737 0.00337160
\(722\) −4024.62 −0.207453
\(723\) 0 0
\(724\) −24505.0 −1.25790
\(725\) 9156.83 0.469071
\(726\) 0 0
\(727\) −16890.5 −0.861668 −0.430834 0.902431i \(-0.641781\pi\)
−0.430834 + 0.902431i \(0.641781\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 23985.1 1.21607
\(731\) −13247.1 −0.670263
\(732\) 0 0
\(733\) 12553.6 0.632578 0.316289 0.948663i \(-0.397563\pi\)
0.316289 + 0.948663i \(0.397563\pi\)
\(734\) −58906.3 −2.96222
\(735\) 0 0
\(736\) 1609.74 0.0806194
\(737\) −34040.3 −1.70135
\(738\) 0 0
\(739\) −37874.8 −1.88531 −0.942656 0.333766i \(-0.891680\pi\)
−0.942656 + 0.333766i \(0.891680\pi\)
\(740\) −4391.78 −0.218169
\(741\) 0 0
\(742\) 6913.16 0.342035
\(743\) −35882.6 −1.77174 −0.885872 0.463929i \(-0.846439\pi\)
−0.885872 + 0.463929i \(0.846439\pi\)
\(744\) 0 0
\(745\) 4647.63 0.228559
\(746\) 6732.40 0.330416
\(747\) 0 0
\(748\) 16504.6 0.806777
\(749\) 32600.3 1.59037
\(750\) 0 0
\(751\) −181.689 −0.00882815 −0.00441407 0.999990i \(-0.501405\pi\)
−0.00441407 + 0.999990i \(0.501405\pi\)
\(752\) −21048.0 −1.02067
\(753\) 0 0
\(754\) 0 0
\(755\) −786.425 −0.0379085
\(756\) 0 0
\(757\) −491.053 −0.0235768 −0.0117884 0.999931i \(-0.503752\pi\)
−0.0117884 + 0.999931i \(0.503752\pi\)
\(758\) 38615.8 1.85038
\(759\) 0 0
\(760\) −3538.94 −0.168909
\(761\) 8113.01 0.386460 0.193230 0.981153i \(-0.438104\pi\)
0.193230 + 0.981153i \(0.438104\pi\)
\(762\) 0 0
\(763\) −10819.2 −0.513342
\(764\) 22924.1 1.08555
\(765\) 0 0
\(766\) −40363.3 −1.90390
\(767\) 0 0
\(768\) 0 0
\(769\) 19864.7 0.931519 0.465759 0.884911i \(-0.345781\pi\)
0.465759 + 0.884911i \(0.345781\pi\)
\(770\) −20245.8 −0.947542
\(771\) 0 0
\(772\) 33030.6 1.53989
\(773\) −9047.42 −0.420974 −0.210487 0.977597i \(-0.567505\pi\)
−0.210487 + 0.977597i \(0.567505\pi\)
\(774\) 0 0
\(775\) 27887.0 1.29256
\(776\) −130.885 −0.00605476
\(777\) 0 0
\(778\) −31131.2 −1.43459
\(779\) −39535.5 −1.81837
\(780\) 0 0
\(781\) −30347.1 −1.39040
\(782\) −1313.37 −0.0600588
\(783\) 0 0
\(784\) −10915.8 −0.497259
\(785\) 8842.45 0.402039
\(786\) 0 0
\(787\) −15018.4 −0.680240 −0.340120 0.940382i \(-0.610468\pi\)
−0.340120 + 0.940382i \(0.610468\pi\)
\(788\) 38050.9 1.72019
\(789\) 0 0
\(790\) −2948.08 −0.132770
\(791\) 33747.8 1.51698
\(792\) 0 0
\(793\) 0 0
\(794\) −49429.9 −2.20932
\(795\) 0 0
\(796\) −40262.4 −1.79279
\(797\) −31941.1 −1.41959 −0.709794 0.704409i \(-0.751212\pi\)
−0.709794 + 0.704409i \(0.751212\pi\)
\(798\) 0 0
\(799\) 22767.1 1.00806
\(800\) 23197.3 1.02518
\(801\) 0 0
\(802\) 59873.2 2.63615
\(803\) 32825.0 1.44255
\(804\) 0 0
\(805\) 889.409 0.0389411
\(806\) 0 0
\(807\) 0 0
\(808\) 7528.46 0.327785
\(809\) −27260.1 −1.18469 −0.592344 0.805685i \(-0.701797\pi\)
−0.592344 + 0.805685i \(0.701797\pi\)
\(810\) 0 0
\(811\) −20707.8 −0.896607 −0.448303 0.893881i \(-0.647972\pi\)
−0.448303 + 0.893881i \(0.647972\pi\)
\(812\) 24023.3 1.03824
\(813\) 0 0
\(814\) −10887.2 −0.468793
\(815\) 6885.42 0.295933
\(816\) 0 0
\(817\) 20611.2 0.882614
\(818\) 11426.0 0.488385
\(819\) 0 0
\(820\) 29719.1 1.26565
\(821\) −15658.6 −0.665636 −0.332818 0.942991i \(-0.608000\pi\)
−0.332818 + 0.942991i \(0.608000\pi\)
\(822\) 0 0
\(823\) −4106.58 −0.173932 −0.0869662 0.996211i \(-0.527717\pi\)
−0.0869662 + 0.996211i \(0.527717\pi\)
\(824\) −21.2619 −0.000898900 0
\(825\) 0 0
\(826\) 2569.49 0.108237
\(827\) −16747.3 −0.704184 −0.352092 0.935965i \(-0.614530\pi\)
−0.352092 + 0.935965i \(0.614530\pi\)
\(828\) 0 0
\(829\) −29157.1 −1.22155 −0.610776 0.791803i \(-0.709142\pi\)
−0.610776 + 0.791803i \(0.709142\pi\)
\(830\) −11905.5 −0.497887
\(831\) 0 0
\(832\) 0 0
\(833\) 11807.4 0.491119
\(834\) 0 0
\(835\) −8637.03 −0.357960
\(836\) −25679.6 −1.06238
\(837\) 0 0
\(838\) −30184.7 −1.24429
\(839\) −45819.4 −1.88541 −0.942707 0.333621i \(-0.891729\pi\)
−0.942707 + 0.333621i \(0.891729\pi\)
\(840\) 0 0
\(841\) −14188.9 −0.581776
\(842\) −14395.8 −0.589209
\(843\) 0 0
\(844\) 29847.7 1.21730
\(845\) 0 0
\(846\) 0 0
\(847\) 4403.44 0.178635
\(848\) 3096.41 0.125390
\(849\) 0 0
\(850\) −18926.4 −0.763729
\(851\) 478.283 0.0192660
\(852\) 0 0
\(853\) 17351.1 0.696471 0.348235 0.937407i \(-0.386781\pi\)
0.348235 + 0.937407i \(0.386781\pi\)
\(854\) 60034.7 2.40556
\(855\) 0 0
\(856\) −10619.0 −0.424009
\(857\) 21768.1 0.867659 0.433829 0.900995i \(-0.357162\pi\)
0.433829 + 0.900995i \(0.357162\pi\)
\(858\) 0 0
\(859\) −29878.4 −1.18677 −0.593387 0.804918i \(-0.702209\pi\)
−0.593387 + 0.804918i \(0.702209\pi\)
\(860\) −15493.6 −0.614334
\(861\) 0 0
\(862\) −21861.2 −0.863801
\(863\) 15067.7 0.594335 0.297168 0.954825i \(-0.403958\pi\)
0.297168 + 0.954825i \(0.403958\pi\)
\(864\) 0 0
\(865\) 13642.8 0.536265
\(866\) 46296.3 1.81664
\(867\) 0 0
\(868\) 73162.7 2.86095
\(869\) −4034.62 −0.157497
\(870\) 0 0
\(871\) 0 0
\(872\) 3524.17 0.136862
\(873\) 0 0
\(874\) 2043.48 0.0790865
\(875\) 30487.4 1.17790
\(876\) 0 0
\(877\) −21119.5 −0.813177 −0.406588 0.913611i \(-0.633282\pi\)
−0.406588 + 0.913611i \(0.633282\pi\)
\(878\) 50004.5 1.92206
\(879\) 0 0
\(880\) −9068.09 −0.347370
\(881\) −31652.5 −1.21044 −0.605221 0.796057i \(-0.706915\pi\)
−0.605221 + 0.796057i \(0.706915\pi\)
\(882\) 0 0
\(883\) 11701.6 0.445969 0.222984 0.974822i \(-0.428420\pi\)
0.222984 + 0.974822i \(0.428420\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −56966.6 −2.16008
\(887\) −22507.9 −0.852020 −0.426010 0.904719i \(-0.640081\pi\)
−0.426010 + 0.904719i \(0.640081\pi\)
\(888\) 0 0
\(889\) 2883.10 0.108769
\(890\) −26890.0 −1.01276
\(891\) 0 0
\(892\) −17005.3 −0.638318
\(893\) −35423.3 −1.32743
\(894\) 0 0
\(895\) 12503.4 0.466974
\(896\) 23611.9 0.880377
\(897\) 0 0
\(898\) 28628.7 1.06387
\(899\) 31064.1 1.15244
\(900\) 0 0
\(901\) −3349.31 −0.123842
\(902\) 73673.8 2.71959
\(903\) 0 0
\(904\) −10992.8 −0.404442
\(905\) −14563.3 −0.534919
\(906\) 0 0
\(907\) 18718.6 0.685273 0.342636 0.939468i \(-0.388680\pi\)
0.342636 + 0.939468i \(0.388680\pi\)
\(908\) 18962.6 0.693057
\(909\) 0 0
\(910\) 0 0
\(911\) −18616.7 −0.677057 −0.338529 0.940956i \(-0.609929\pi\)
−0.338529 + 0.940956i \(0.609929\pi\)
\(912\) 0 0
\(913\) −16293.4 −0.590616
\(914\) −19618.9 −0.709994
\(915\) 0 0
\(916\) 3687.39 0.133008
\(917\) −54323.3 −1.95629
\(918\) 0 0
\(919\) −54764.4 −1.96573 −0.982867 0.184316i \(-0.940993\pi\)
−0.982867 + 0.184316i \(0.940993\pi\)
\(920\) −289.711 −0.0103821
\(921\) 0 0
\(922\) −10397.6 −0.371397
\(923\) 0 0
\(924\) 0 0
\(925\) 6892.32 0.244993
\(926\) 18129.8 0.643394
\(927\) 0 0
\(928\) 25840.1 0.914055
\(929\) 31832.1 1.12419 0.562097 0.827071i \(-0.309995\pi\)
0.562097 + 0.827071i \(0.309995\pi\)
\(930\) 0 0
\(931\) −18371.2 −0.646713
\(932\) −30510.3 −1.07231
\(933\) 0 0
\(934\) 37185.0 1.30271
\(935\) 9808.73 0.343080
\(936\) 0 0
\(937\) −27408.7 −0.955607 −0.477803 0.878467i \(-0.658567\pi\)
−0.477803 + 0.878467i \(0.658567\pi\)
\(938\) 102410. 3.56483
\(939\) 0 0
\(940\) 26627.9 0.923945
\(941\) −54837.8 −1.89975 −0.949874 0.312634i \(-0.898789\pi\)
−0.949874 + 0.312634i \(0.898789\pi\)
\(942\) 0 0
\(943\) −3236.53 −0.111767
\(944\) 1150.87 0.0396799
\(945\) 0 0
\(946\) −38408.6 −1.32006
\(947\) 39707.8 1.36255 0.681273 0.732030i \(-0.261427\pi\)
0.681273 + 0.732030i \(0.261427\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 29447.6 1.00569
\(951\) 0 0
\(952\) −9364.89 −0.318821
\(953\) 17106.6 0.581468 0.290734 0.956804i \(-0.406101\pi\)
0.290734 + 0.956804i \(0.406101\pi\)
\(954\) 0 0
\(955\) 13623.8 0.461629
\(956\) 12040.3 0.407334
\(957\) 0 0
\(958\) 46374.0 1.56396
\(959\) 27276.7 0.918469
\(960\) 0 0
\(961\) 64814.4 2.17564
\(962\) 0 0
\(963\) 0 0
\(964\) 1433.58 0.0478968
\(965\) 19630.1 0.654835
\(966\) 0 0
\(967\) −23417.5 −0.778756 −0.389378 0.921078i \(-0.627310\pi\)
−0.389378 + 0.921078i \(0.627310\pi\)
\(968\) −1434.35 −0.0476258
\(969\) 0 0
\(970\) −412.428 −0.0136518
\(971\) −16430.6 −0.543032 −0.271516 0.962434i \(-0.587525\pi\)
−0.271516 + 0.962434i \(0.587525\pi\)
\(972\) 0 0
\(973\) 14361.6 0.473187
\(974\) −22015.9 −0.724267
\(975\) 0 0
\(976\) 26889.5 0.881879
\(977\) 10554.6 0.345622 0.172811 0.984955i \(-0.444715\pi\)
0.172811 + 0.984955i \(0.444715\pi\)
\(978\) 0 0
\(979\) −36800.5 −1.20138
\(980\) 13809.7 0.450138
\(981\) 0 0
\(982\) 37101.5 1.20566
\(983\) −1534.33 −0.0497839 −0.0248919 0.999690i \(-0.507924\pi\)
−0.0248919 + 0.999690i \(0.507924\pi\)
\(984\) 0 0
\(985\) 22613.7 0.731505
\(986\) −21082.6 −0.680941
\(987\) 0 0
\(988\) 0 0
\(989\) 1687.31 0.0542502
\(990\) 0 0
\(991\) −18018.2 −0.577563 −0.288782 0.957395i \(-0.593250\pi\)
−0.288782 + 0.957395i \(0.593250\pi\)
\(992\) 78695.7 2.51874
\(993\) 0 0
\(994\) 91299.1 2.91331
\(995\) −23928.0 −0.762380
\(996\) 0 0
\(997\) 48287.7 1.53389 0.766944 0.641714i \(-0.221776\pi\)
0.766944 + 0.641714i \(0.221776\pi\)
\(998\) −65884.4 −2.08971
\(999\) 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 1521.4.a.v.1.4 4
3.2 odd 2 507.4.a.m.1.1 4
13.3 even 3 117.4.g.e.100.1 8
13.9 even 3 117.4.g.e.55.1 8
13.12 even 2 1521.4.a.bb.1.1 4
39.5 even 4 507.4.b.h.337.7 8
39.8 even 4 507.4.b.h.337.2 8
39.29 odd 6 39.4.e.c.22.4 yes 8
39.35 odd 6 39.4.e.c.16.4 8
39.38 odd 2 507.4.a.i.1.4 4
156.35 even 6 624.4.q.i.289.2 8
156.107 even 6 624.4.q.i.529.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.4.e.c.16.4 8 39.35 odd 6
39.4.e.c.22.4 yes 8 39.29 odd 6
117.4.g.e.55.1 8 13.9 even 3
117.4.g.e.100.1 8 13.3 even 3
507.4.a.i.1.4 4 39.38 odd 2
507.4.a.m.1.1 4 3.2 odd 2
507.4.b.h.337.2 8 39.8 even 4
507.4.b.h.337.7 8 39.5 even 4
624.4.q.i.289.2 8 156.35 even 6
624.4.q.i.529.2 8 156.107 even 6
1521.4.a.v.1.4 4 1.1 even 1 trivial
1521.4.a.bb.1.1 4 13.12 even 2