Properties

Label 1856.4.a.bg.1.1
Level $1856$
Weight $4$
Character 1856.1
Self dual yes
Analytic conductor $109.508$
Analytic rank $1$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1856 = 2^{6} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1856.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(109.507544971\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \( x^{9} - 4x^{8} - 138x^{7} + 394x^{6} + 5872x^{5} - 10822x^{4} - 85158x^{3} + 30654x^{2} + 439999x + 396802 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 928)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-7.63776\) of defining polynomial
Character \(\chi\) \(=\) 1856.1

$q$-expansion

\(f(q)\) \(=\) \(q-7.63776 q^{3} +7.65015 q^{5} -5.64550 q^{7} +31.3353 q^{9} +O(q^{10})\) \(q-7.63776 q^{3} +7.65015 q^{5} -5.64550 q^{7} +31.3353 q^{9} +64.9460 q^{11} -54.0253 q^{13} -58.4300 q^{15} -20.0898 q^{17} -52.3646 q^{19} +43.1190 q^{21} +132.044 q^{23} -66.4752 q^{25} -33.1122 q^{27} +29.0000 q^{29} -135.023 q^{31} -496.042 q^{33} -43.1889 q^{35} +137.761 q^{37} +412.632 q^{39} -229.496 q^{41} +46.2145 q^{43} +239.720 q^{45} -327.225 q^{47} -311.128 q^{49} +153.441 q^{51} -32.9249 q^{53} +496.846 q^{55} +399.948 q^{57} +378.832 q^{59} +18.5312 q^{61} -176.904 q^{63} -413.301 q^{65} +998.295 q^{67} -1008.52 q^{69} -570.499 q^{71} -224.392 q^{73} +507.722 q^{75} -366.653 q^{77} +430.785 q^{79} -593.151 q^{81} +1390.04 q^{83} -153.690 q^{85} -221.495 q^{87} -131.569 q^{89} +305.000 q^{91} +1031.27 q^{93} -400.597 q^{95} +837.766 q^{97} +2035.10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 4 q^{3} - 10 q^{5} - 12 q^{7} + 49 q^{9} + 64 q^{11} - 70 q^{13} - 170 q^{15} - 66 q^{17} + 42 q^{19} - 76 q^{21} - 40 q^{23} + 111 q^{25} + 322 q^{27} + 261 q^{29} + 64 q^{31} - 52 q^{33} + 496 q^{35} + 54 q^{37} - 590 q^{39} - 378 q^{41} - 32 q^{43} - 1046 q^{45} - 1164 q^{47} - 351 q^{49} + 376 q^{51} - 278 q^{53} - 614 q^{55} + 28 q^{57} + 640 q^{59} - 1054 q^{61} - 1660 q^{63} - 708 q^{65} + 1184 q^{67} - 188 q^{69} - 1988 q^{71} - 750 q^{73} + 3126 q^{75} - 1260 q^{77} - 2916 q^{79} + 293 q^{81} + 2832 q^{83} - 56 q^{85} + 116 q^{87} - 370 q^{89} + 3016 q^{91} + 1696 q^{93} - 4412 q^{95} - 2234 q^{97} + 4118 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −7.63776 −1.46989 −0.734943 0.678128i \(-0.762791\pi\)
−0.734943 + 0.678128i \(0.762791\pi\)
\(4\) 0 0
\(5\) 7.65015 0.684250 0.342125 0.939654i \(-0.388853\pi\)
0.342125 + 0.939654i \(0.388853\pi\)
\(6\) 0 0
\(7\) −5.64550 −0.304829 −0.152414 0.988317i \(-0.548705\pi\)
−0.152414 + 0.988317i \(0.548705\pi\)
\(8\) 0 0
\(9\) 31.3353 1.16057
\(10\) 0 0
\(11\) 64.9460 1.78018 0.890089 0.455787i \(-0.150642\pi\)
0.890089 + 0.455787i \(0.150642\pi\)
\(12\) 0 0
\(13\) −54.0253 −1.15261 −0.576304 0.817235i \(-0.695506\pi\)
−0.576304 + 0.817235i \(0.695506\pi\)
\(14\) 0 0
\(15\) −58.4300 −1.00577
\(16\) 0 0
\(17\) −20.0898 −0.286618 −0.143309 0.989678i \(-0.545774\pi\)
−0.143309 + 0.989678i \(0.545774\pi\)
\(18\) 0 0
\(19\) −52.3646 −0.632277 −0.316139 0.948713i \(-0.602386\pi\)
−0.316139 + 0.948713i \(0.602386\pi\)
\(20\) 0 0
\(21\) 43.1190 0.448064
\(22\) 0 0
\(23\) 132.044 1.19709 0.598547 0.801088i \(-0.295745\pi\)
0.598547 + 0.801088i \(0.295745\pi\)
\(24\) 0 0
\(25\) −66.4752 −0.531802
\(26\) 0 0
\(27\) −33.1122 −0.236016
\(28\) 0 0
\(29\) 29.0000 0.185695
\(30\) 0 0
\(31\) −135.023 −0.782284 −0.391142 0.920330i \(-0.627920\pi\)
−0.391142 + 0.920330i \(0.627920\pi\)
\(32\) 0 0
\(33\) −496.042 −2.61666
\(34\) 0 0
\(35\) −43.1889 −0.208579
\(36\) 0 0
\(37\) 137.761 0.612101 0.306050 0.952015i \(-0.400992\pi\)
0.306050 + 0.952015i \(0.400992\pi\)
\(38\) 0 0
\(39\) 412.632 1.69420
\(40\) 0 0
\(41\) −229.496 −0.874176 −0.437088 0.899419i \(-0.643990\pi\)
−0.437088 + 0.899419i \(0.643990\pi\)
\(42\) 0 0
\(43\) 46.2145 0.163899 0.0819493 0.996637i \(-0.473885\pi\)
0.0819493 + 0.996637i \(0.473885\pi\)
\(44\) 0 0
\(45\) 239.720 0.794118
\(46\) 0 0
\(47\) −327.225 −1.01555 −0.507773 0.861491i \(-0.669531\pi\)
−0.507773 + 0.861491i \(0.669531\pi\)
\(48\) 0 0
\(49\) −311.128 −0.907080
\(50\) 0 0
\(51\) 153.441 0.421295
\(52\) 0 0
\(53\) −32.9249 −0.0853318 −0.0426659 0.999089i \(-0.513585\pi\)
−0.0426659 + 0.999089i \(0.513585\pi\)
\(54\) 0 0
\(55\) 496.846 1.21809
\(56\) 0 0
\(57\) 399.948 0.929376
\(58\) 0 0
\(59\) 378.832 0.835929 0.417964 0.908463i \(-0.362744\pi\)
0.417964 + 0.908463i \(0.362744\pi\)
\(60\) 0 0
\(61\) 18.5312 0.0388964 0.0194482 0.999811i \(-0.493809\pi\)
0.0194482 + 0.999811i \(0.493809\pi\)
\(62\) 0 0
\(63\) −176.904 −0.353774
\(64\) 0 0
\(65\) −413.301 −0.788673
\(66\) 0 0
\(67\) 998.295 1.82032 0.910158 0.414261i \(-0.135960\pi\)
0.910158 + 0.414261i \(0.135960\pi\)
\(68\) 0 0
\(69\) −1008.52 −1.75959
\(70\) 0 0
\(71\) −570.499 −0.953602 −0.476801 0.879011i \(-0.658204\pi\)
−0.476801 + 0.879011i \(0.658204\pi\)
\(72\) 0 0
\(73\) −224.392 −0.359768 −0.179884 0.983688i \(-0.557572\pi\)
−0.179884 + 0.983688i \(0.557572\pi\)
\(74\) 0 0
\(75\) 507.722 0.781689
\(76\) 0 0
\(77\) −366.653 −0.542649
\(78\) 0 0
\(79\) 430.785 0.613507 0.306754 0.951789i \(-0.400757\pi\)
0.306754 + 0.951789i \(0.400757\pi\)
\(80\) 0 0
\(81\) −593.151 −0.813650
\(82\) 0 0
\(83\) 1390.04 1.83828 0.919138 0.393935i \(-0.128886\pi\)
0.919138 + 0.393935i \(0.128886\pi\)
\(84\) 0 0
\(85\) −153.690 −0.196118
\(86\) 0 0
\(87\) −221.495 −0.272951
\(88\) 0 0
\(89\) −131.569 −0.156700 −0.0783501 0.996926i \(-0.524965\pi\)
−0.0783501 + 0.996926i \(0.524965\pi\)
\(90\) 0 0
\(91\) 305.000 0.351348
\(92\) 0 0
\(93\) 1031.27 1.14987
\(94\) 0 0
\(95\) −400.597 −0.432636
\(96\) 0 0
\(97\) 837.766 0.876930 0.438465 0.898748i \(-0.355522\pi\)
0.438465 + 0.898748i \(0.355522\pi\)
\(98\) 0 0
\(99\) 2035.10 2.06602
\(100\) 0 0
\(101\) 1595.03 1.57140 0.785702 0.618605i \(-0.212302\pi\)
0.785702 + 0.618605i \(0.212302\pi\)
\(102\) 0 0
\(103\) 1565.54 1.49765 0.748824 0.662769i \(-0.230619\pi\)
0.748824 + 0.662769i \(0.230619\pi\)
\(104\) 0 0
\(105\) 329.867 0.306587
\(106\) 0 0
\(107\) −1243.19 −1.12321 −0.561604 0.827406i \(-0.689816\pi\)
−0.561604 + 0.827406i \(0.689816\pi\)
\(108\) 0 0
\(109\) −1158.97 −1.01843 −0.509217 0.860638i \(-0.670065\pi\)
−0.509217 + 0.860638i \(0.670065\pi\)
\(110\) 0 0
\(111\) −1052.18 −0.899719
\(112\) 0 0
\(113\) −341.404 −0.284217 −0.142109 0.989851i \(-0.545388\pi\)
−0.142109 + 0.989851i \(0.545388\pi\)
\(114\) 0 0
\(115\) 1010.16 0.819111
\(116\) 0 0
\(117\) −1692.90 −1.33768
\(118\) 0 0
\(119\) 113.417 0.0873692
\(120\) 0 0
\(121\) 2886.98 2.16903
\(122\) 0 0
\(123\) 1752.83 1.28494
\(124\) 0 0
\(125\) −1464.81 −1.04814
\(126\) 0 0
\(127\) 258.090 0.180329 0.0901644 0.995927i \(-0.471261\pi\)
0.0901644 + 0.995927i \(0.471261\pi\)
\(128\) 0 0
\(129\) −352.975 −0.240912
\(130\) 0 0
\(131\) −728.188 −0.485665 −0.242832 0.970068i \(-0.578077\pi\)
−0.242832 + 0.970068i \(0.578077\pi\)
\(132\) 0 0
\(133\) 295.625 0.192736
\(134\) 0 0
\(135\) −253.313 −0.161494
\(136\) 0 0
\(137\) 82.1655 0.0512400 0.0256200 0.999672i \(-0.491844\pi\)
0.0256200 + 0.999672i \(0.491844\pi\)
\(138\) 0 0
\(139\) −593.000 −0.361853 −0.180927 0.983497i \(-0.557910\pi\)
−0.180927 + 0.983497i \(0.557910\pi\)
\(140\) 0 0
\(141\) 2499.27 1.49274
\(142\) 0 0
\(143\) −3508.73 −2.05185
\(144\) 0 0
\(145\) 221.854 0.127062
\(146\) 0 0
\(147\) 2376.32 1.33330
\(148\) 0 0
\(149\) −728.700 −0.400654 −0.200327 0.979729i \(-0.564200\pi\)
−0.200327 + 0.979729i \(0.564200\pi\)
\(150\) 0 0
\(151\) 1203.47 0.648590 0.324295 0.945956i \(-0.394873\pi\)
0.324295 + 0.945956i \(0.394873\pi\)
\(152\) 0 0
\(153\) −629.521 −0.332639
\(154\) 0 0
\(155\) −1032.94 −0.535278
\(156\) 0 0
\(157\) −2704.74 −1.37492 −0.687458 0.726224i \(-0.741273\pi\)
−0.687458 + 0.726224i \(0.741273\pi\)
\(158\) 0 0
\(159\) 251.473 0.125428
\(160\) 0 0
\(161\) −745.457 −0.364908
\(162\) 0 0
\(163\) −1310.50 −0.629732 −0.314866 0.949136i \(-0.601960\pi\)
−0.314866 + 0.949136i \(0.601960\pi\)
\(164\) 0 0
\(165\) −3794.79 −1.79045
\(166\) 0 0
\(167\) −2891.17 −1.33967 −0.669837 0.742508i \(-0.733636\pi\)
−0.669837 + 0.742508i \(0.733636\pi\)
\(168\) 0 0
\(169\) 721.730 0.328507
\(170\) 0 0
\(171\) −1640.86 −0.733800
\(172\) 0 0
\(173\) −2915.04 −1.28108 −0.640539 0.767925i \(-0.721289\pi\)
−0.640539 + 0.767925i \(0.721289\pi\)
\(174\) 0 0
\(175\) 375.286 0.162108
\(176\) 0 0
\(177\) −2893.43 −1.22872
\(178\) 0 0
\(179\) −2414.92 −1.00838 −0.504188 0.863594i \(-0.668208\pi\)
−0.504188 + 0.863594i \(0.668208\pi\)
\(180\) 0 0
\(181\) −100.467 −0.0412576 −0.0206288 0.999787i \(-0.506567\pi\)
−0.0206288 + 0.999787i \(0.506567\pi\)
\(182\) 0 0
\(183\) −141.537 −0.0571733
\(184\) 0 0
\(185\) 1053.89 0.418830
\(186\) 0 0
\(187\) −1304.75 −0.510230
\(188\) 0 0
\(189\) 186.935 0.0719445
\(190\) 0 0
\(191\) −4186.06 −1.58583 −0.792914 0.609334i \(-0.791437\pi\)
−0.792914 + 0.609334i \(0.791437\pi\)
\(192\) 0 0
\(193\) 2681.27 1.00001 0.500006 0.866022i \(-0.333331\pi\)
0.500006 + 0.866022i \(0.333331\pi\)
\(194\) 0 0
\(195\) 3156.69 1.15926
\(196\) 0 0
\(197\) −2684.48 −0.970869 −0.485435 0.874273i \(-0.661339\pi\)
−0.485435 + 0.874273i \(0.661339\pi\)
\(198\) 0 0
\(199\) −3671.70 −1.30794 −0.653970 0.756520i \(-0.726898\pi\)
−0.653970 + 0.756520i \(0.726898\pi\)
\(200\) 0 0
\(201\) −7624.74 −2.67566
\(202\) 0 0
\(203\) −163.720 −0.0566052
\(204\) 0 0
\(205\) −1755.68 −0.598155
\(206\) 0 0
\(207\) 4137.65 1.38931
\(208\) 0 0
\(209\) −3400.87 −1.12557
\(210\) 0 0
\(211\) 4641.03 1.51423 0.757113 0.653284i \(-0.226609\pi\)
0.757113 + 0.653284i \(0.226609\pi\)
\(212\) 0 0
\(213\) 4357.33 1.40169
\(214\) 0 0
\(215\) 353.547 0.112148
\(216\) 0 0
\(217\) 762.272 0.238462
\(218\) 0 0
\(219\) 1713.85 0.528819
\(220\) 0 0
\(221\) 1085.36 0.330358
\(222\) 0 0
\(223\) −1275.71 −0.383086 −0.191543 0.981484i \(-0.561349\pi\)
−0.191543 + 0.981484i \(0.561349\pi\)
\(224\) 0 0
\(225\) −2083.02 −0.617192
\(226\) 0 0
\(227\) 876.500 0.256279 0.128140 0.991756i \(-0.459099\pi\)
0.128140 + 0.991756i \(0.459099\pi\)
\(228\) 0 0
\(229\) 1083.31 0.312606 0.156303 0.987709i \(-0.450042\pi\)
0.156303 + 0.987709i \(0.450042\pi\)
\(230\) 0 0
\(231\) 2800.41 0.797633
\(232\) 0 0
\(233\) 5786.21 1.62690 0.813448 0.581637i \(-0.197588\pi\)
0.813448 + 0.581637i \(0.197588\pi\)
\(234\) 0 0
\(235\) −2503.32 −0.694888
\(236\) 0 0
\(237\) −3290.23 −0.901786
\(238\) 0 0
\(239\) 1971.04 0.533456 0.266728 0.963772i \(-0.414057\pi\)
0.266728 + 0.963772i \(0.414057\pi\)
\(240\) 0 0
\(241\) −4272.70 −1.14203 −0.571014 0.820941i \(-0.693450\pi\)
−0.571014 + 0.820941i \(0.693450\pi\)
\(242\) 0 0
\(243\) 5424.37 1.43199
\(244\) 0 0
\(245\) −2380.18 −0.620669
\(246\) 0 0
\(247\) 2829.01 0.728768
\(248\) 0 0
\(249\) −10616.8 −2.70206
\(250\) 0 0
\(251\) 2804.44 0.705238 0.352619 0.935767i \(-0.385291\pi\)
0.352619 + 0.935767i \(0.385291\pi\)
\(252\) 0 0
\(253\) 8575.75 2.13104
\(254\) 0 0
\(255\) 1173.85 0.288271
\(256\) 0 0
\(257\) 3591.88 0.871811 0.435906 0.899992i \(-0.356428\pi\)
0.435906 + 0.899992i \(0.356428\pi\)
\(258\) 0 0
\(259\) −777.729 −0.186586
\(260\) 0 0
\(261\) 908.724 0.215512
\(262\) 0 0
\(263\) −4143.91 −0.971576 −0.485788 0.874077i \(-0.661467\pi\)
−0.485788 + 0.874077i \(0.661467\pi\)
\(264\) 0 0
\(265\) −251.880 −0.0583883
\(266\) 0 0
\(267\) 1004.89 0.230331
\(268\) 0 0
\(269\) −4024.31 −0.912143 −0.456072 0.889943i \(-0.650744\pi\)
−0.456072 + 0.889943i \(0.650744\pi\)
\(270\) 0 0
\(271\) −6233.96 −1.39737 −0.698683 0.715431i \(-0.746230\pi\)
−0.698683 + 0.715431i \(0.746230\pi\)
\(272\) 0 0
\(273\) −2329.52 −0.516442
\(274\) 0 0
\(275\) −4317.30 −0.946702
\(276\) 0 0
\(277\) 435.106 0.0943789 0.0471895 0.998886i \(-0.484974\pi\)
0.0471895 + 0.998886i \(0.484974\pi\)
\(278\) 0 0
\(279\) −4230.98 −0.907893
\(280\) 0 0
\(281\) −3805.46 −0.807882 −0.403941 0.914785i \(-0.632360\pi\)
−0.403941 + 0.914785i \(0.632360\pi\)
\(282\) 0 0
\(283\) −239.164 −0.0502361 −0.0251180 0.999684i \(-0.507996\pi\)
−0.0251180 + 0.999684i \(0.507996\pi\)
\(284\) 0 0
\(285\) 3059.66 0.635926
\(286\) 0 0
\(287\) 1295.62 0.266474
\(288\) 0 0
\(289\) −4509.40 −0.917850
\(290\) 0 0
\(291\) −6398.65 −1.28899
\(292\) 0 0
\(293\) −719.126 −0.143385 −0.0716925 0.997427i \(-0.522840\pi\)
−0.0716925 + 0.997427i \(0.522840\pi\)
\(294\) 0 0
\(295\) 2898.12 0.571984
\(296\) 0 0
\(297\) −2150.50 −0.420151
\(298\) 0 0
\(299\) −7133.73 −1.37978
\(300\) 0 0
\(301\) −260.904 −0.0499610
\(302\) 0 0
\(303\) −12182.5 −2.30979
\(304\) 0 0
\(305\) 141.767 0.0266148
\(306\) 0 0
\(307\) −9702.09 −1.80367 −0.901836 0.432078i \(-0.857780\pi\)
−0.901836 + 0.432078i \(0.857780\pi\)
\(308\) 0 0
\(309\) −11957.2 −2.20137
\(310\) 0 0
\(311\) −3369.02 −0.614275 −0.307137 0.951665i \(-0.599371\pi\)
−0.307137 + 0.951665i \(0.599371\pi\)
\(312\) 0 0
\(313\) −5269.30 −0.951561 −0.475781 0.879564i \(-0.657834\pi\)
−0.475781 + 0.879564i \(0.657834\pi\)
\(314\) 0 0
\(315\) −1353.34 −0.242070
\(316\) 0 0
\(317\) 7664.88 1.35805 0.679026 0.734114i \(-0.262402\pi\)
0.679026 + 0.734114i \(0.262402\pi\)
\(318\) 0 0
\(319\) 1883.43 0.330571
\(320\) 0 0
\(321\) 9495.15 1.65099
\(322\) 0 0
\(323\) 1052.00 0.181222
\(324\) 0 0
\(325\) 3591.34 0.612960
\(326\) 0 0
\(327\) 8851.94 1.49698
\(328\) 0 0
\(329\) 1847.35 0.309568
\(330\) 0 0
\(331\) −6587.05 −1.09383 −0.546914 0.837189i \(-0.684198\pi\)
−0.546914 + 0.837189i \(0.684198\pi\)
\(332\) 0 0
\(333\) 4316.78 0.710385
\(334\) 0 0
\(335\) 7637.11 1.24555
\(336\) 0 0
\(337\) 11419.9 1.84595 0.922973 0.384865i \(-0.125752\pi\)
0.922973 + 0.384865i \(0.125752\pi\)
\(338\) 0 0
\(339\) 2607.56 0.417767
\(340\) 0 0
\(341\) −8769.19 −1.39260
\(342\) 0 0
\(343\) 3692.88 0.581332
\(344\) 0 0
\(345\) −7715.35 −1.20400
\(346\) 0 0
\(347\) −2175.03 −0.336488 −0.168244 0.985745i \(-0.553810\pi\)
−0.168244 + 0.985745i \(0.553810\pi\)
\(348\) 0 0
\(349\) −5659.99 −0.868115 −0.434058 0.900885i \(-0.642919\pi\)
−0.434058 + 0.900885i \(0.642919\pi\)
\(350\) 0 0
\(351\) 1788.89 0.272034
\(352\) 0 0
\(353\) −2506.30 −0.377895 −0.188948 0.981987i \(-0.560508\pi\)
−0.188948 + 0.981987i \(0.560508\pi\)
\(354\) 0 0
\(355\) −4364.40 −0.652502
\(356\) 0 0
\(357\) −866.253 −0.128423
\(358\) 0 0
\(359\) −4848.73 −0.712831 −0.356415 0.934328i \(-0.616001\pi\)
−0.356415 + 0.934328i \(0.616001\pi\)
\(360\) 0 0
\(361\) −4116.95 −0.600226
\(362\) 0 0
\(363\) −22050.1 −3.18823
\(364\) 0 0
\(365\) −1716.63 −0.246171
\(366\) 0 0
\(367\) −9034.96 −1.28507 −0.642535 0.766256i \(-0.722117\pi\)
−0.642535 + 0.766256i \(0.722117\pi\)
\(368\) 0 0
\(369\) −7191.32 −1.01454
\(370\) 0 0
\(371\) 185.878 0.0260116
\(372\) 0 0
\(373\) −4060.90 −0.563715 −0.281857 0.959456i \(-0.590950\pi\)
−0.281857 + 0.959456i \(0.590950\pi\)
\(374\) 0 0
\(375\) 11187.9 1.54064
\(376\) 0 0
\(377\) −1566.73 −0.214034
\(378\) 0 0
\(379\) 6094.76 0.826034 0.413017 0.910723i \(-0.364475\pi\)
0.413017 + 0.910723i \(0.364475\pi\)
\(380\) 0 0
\(381\) −1971.23 −0.265063
\(382\) 0 0
\(383\) −4602.40 −0.614025 −0.307013 0.951705i \(-0.599329\pi\)
−0.307013 + 0.951705i \(0.599329\pi\)
\(384\) 0 0
\(385\) −2804.95 −0.371308
\(386\) 0 0
\(387\) 1448.15 0.190215
\(388\) 0 0
\(389\) −9393.43 −1.22433 −0.612167 0.790729i \(-0.709702\pi\)
−0.612167 + 0.790729i \(0.709702\pi\)
\(390\) 0 0
\(391\) −2652.75 −0.343108
\(392\) 0 0
\(393\) 5561.72 0.713872
\(394\) 0 0
\(395\) 3295.57 0.419792
\(396\) 0 0
\(397\) −13398.1 −1.69378 −0.846889 0.531769i \(-0.821527\pi\)
−0.846889 + 0.531769i \(0.821527\pi\)
\(398\) 0 0
\(399\) −2257.91 −0.283300
\(400\) 0 0
\(401\) −12917.9 −1.60870 −0.804348 0.594159i \(-0.797485\pi\)
−0.804348 + 0.594159i \(0.797485\pi\)
\(402\) 0 0
\(403\) 7294.64 0.901667
\(404\) 0 0
\(405\) −4537.69 −0.556740
\(406\) 0 0
\(407\) 8947.01 1.08965
\(408\) 0 0
\(409\) −5221.70 −0.631288 −0.315644 0.948878i \(-0.602220\pi\)
−0.315644 + 0.948878i \(0.602220\pi\)
\(410\) 0 0
\(411\) −627.560 −0.0753169
\(412\) 0 0
\(413\) −2138.70 −0.254815
\(414\) 0 0
\(415\) 10634.0 1.25784
\(416\) 0 0
\(417\) 4529.19 0.531883
\(418\) 0 0
\(419\) −6246.54 −0.728314 −0.364157 0.931338i \(-0.618643\pi\)
−0.364157 + 0.931338i \(0.618643\pi\)
\(420\) 0 0
\(421\) −583.420 −0.0675396 −0.0337698 0.999430i \(-0.510751\pi\)
−0.0337698 + 0.999430i \(0.510751\pi\)
\(422\) 0 0
\(423\) −10253.7 −1.17861
\(424\) 0 0
\(425\) 1335.48 0.152424
\(426\) 0 0
\(427\) −104.618 −0.0118567
\(428\) 0 0
\(429\) 26798.8 3.01599
\(430\) 0 0
\(431\) −4845.37 −0.541516 −0.270758 0.962647i \(-0.587274\pi\)
−0.270758 + 0.962647i \(0.587274\pi\)
\(432\) 0 0
\(433\) 2159.85 0.239714 0.119857 0.992791i \(-0.461756\pi\)
0.119857 + 0.992791i \(0.461756\pi\)
\(434\) 0 0
\(435\) −1694.47 −0.186767
\(436\) 0 0
\(437\) −6914.45 −0.756895
\(438\) 0 0
\(439\) 17016.0 1.84995 0.924976 0.380026i \(-0.124085\pi\)
0.924976 + 0.380026i \(0.124085\pi\)
\(440\) 0 0
\(441\) −9749.31 −1.05273
\(442\) 0 0
\(443\) −3676.92 −0.394347 −0.197173 0.980369i \(-0.563176\pi\)
−0.197173 + 0.980369i \(0.563176\pi\)
\(444\) 0 0
\(445\) −1006.52 −0.107222
\(446\) 0 0
\(447\) 5565.63 0.588916
\(448\) 0 0
\(449\) 11074.1 1.16396 0.581979 0.813204i \(-0.302279\pi\)
0.581979 + 0.813204i \(0.302279\pi\)
\(450\) 0 0
\(451\) −14904.8 −1.55619
\(452\) 0 0
\(453\) −9191.82 −0.953354
\(454\) 0 0
\(455\) 2333.29 0.240410
\(456\) 0 0
\(457\) −1146.80 −0.117386 −0.0586928 0.998276i \(-0.518693\pi\)
−0.0586928 + 0.998276i \(0.518693\pi\)
\(458\) 0 0
\(459\) 665.217 0.0676464
\(460\) 0 0
\(461\) −15788.0 −1.59505 −0.797526 0.603284i \(-0.793858\pi\)
−0.797526 + 0.603284i \(0.793858\pi\)
\(462\) 0 0
\(463\) 2709.43 0.271961 0.135981 0.990711i \(-0.456582\pi\)
0.135981 + 0.990711i \(0.456582\pi\)
\(464\) 0 0
\(465\) 7889.37 0.786798
\(466\) 0 0
\(467\) 16216.0 1.60682 0.803411 0.595425i \(-0.203016\pi\)
0.803411 + 0.595425i \(0.203016\pi\)
\(468\) 0 0
\(469\) −5635.88 −0.554884
\(470\) 0 0
\(471\) 20658.2 2.02097
\(472\) 0 0
\(473\) 3001.44 0.291769
\(474\) 0 0
\(475\) 3480.95 0.336246
\(476\) 0 0
\(477\) −1031.71 −0.0990333
\(478\) 0 0
\(479\) −12106.9 −1.15487 −0.577433 0.816438i \(-0.695945\pi\)
−0.577433 + 0.816438i \(0.695945\pi\)
\(480\) 0 0
\(481\) −7442.56 −0.705513
\(482\) 0 0
\(483\) 5693.62 0.536374
\(484\) 0 0
\(485\) 6409.03 0.600039
\(486\) 0 0
\(487\) 14106.0 1.31253 0.656267 0.754529i \(-0.272135\pi\)
0.656267 + 0.754529i \(0.272135\pi\)
\(488\) 0 0
\(489\) 10009.3 0.925635
\(490\) 0 0
\(491\) 16587.4 1.52460 0.762299 0.647225i \(-0.224070\pi\)
0.762299 + 0.647225i \(0.224070\pi\)
\(492\) 0 0
\(493\) −582.605 −0.0532235
\(494\) 0 0
\(495\) 15568.8 1.41367
\(496\) 0 0
\(497\) 3220.75 0.290685
\(498\) 0 0
\(499\) 5294.94 0.475018 0.237509 0.971385i \(-0.423669\pi\)
0.237509 + 0.971385i \(0.423669\pi\)
\(500\) 0 0
\(501\) 22082.1 1.96917
\(502\) 0 0
\(503\) 5856.03 0.519101 0.259550 0.965730i \(-0.416426\pi\)
0.259550 + 0.965730i \(0.416426\pi\)
\(504\) 0 0
\(505\) 12202.2 1.07523
\(506\) 0 0
\(507\) −5512.40 −0.482868
\(508\) 0 0
\(509\) −3680.23 −0.320478 −0.160239 0.987078i \(-0.551227\pi\)
−0.160239 + 0.987078i \(0.551227\pi\)
\(510\) 0 0
\(511\) 1266.81 0.109668
\(512\) 0 0
\(513\) 1733.91 0.149228
\(514\) 0 0
\(515\) 11976.6 1.02477
\(516\) 0 0
\(517\) −21252.0 −1.80785
\(518\) 0 0
\(519\) 22264.4 1.88304
\(520\) 0 0
\(521\) 15679.1 1.31845 0.659227 0.751944i \(-0.270884\pi\)
0.659227 + 0.751944i \(0.270884\pi\)
\(522\) 0 0
\(523\) −17311.4 −1.44737 −0.723686 0.690130i \(-0.757553\pi\)
−0.723686 + 0.690130i \(0.757553\pi\)
\(524\) 0 0
\(525\) −2866.35 −0.238281
\(526\) 0 0
\(527\) 2712.58 0.224216
\(528\) 0 0
\(529\) 5268.72 0.433034
\(530\) 0 0
\(531\) 11870.8 0.970152
\(532\) 0 0
\(533\) 12398.6 1.00758
\(534\) 0 0
\(535\) −9510.55 −0.768555
\(536\) 0 0
\(537\) 18444.6 1.48220
\(538\) 0 0
\(539\) −20206.5 −1.61476
\(540\) 0 0
\(541\) −7300.99 −0.580211 −0.290105 0.956995i \(-0.593690\pi\)
−0.290105 + 0.956995i \(0.593690\pi\)
\(542\) 0 0
\(543\) 767.340 0.0606441
\(544\) 0 0
\(545\) −8866.30 −0.696863
\(546\) 0 0
\(547\) 12610.1 0.985684 0.492842 0.870119i \(-0.335958\pi\)
0.492842 + 0.870119i \(0.335958\pi\)
\(548\) 0 0
\(549\) 580.682 0.0451419
\(550\) 0 0
\(551\) −1518.57 −0.117411
\(552\) 0 0
\(553\) −2432.00 −0.187015
\(554\) 0 0
\(555\) −8049.36 −0.615633
\(556\) 0 0
\(557\) 2845.32 0.216446 0.108223 0.994127i \(-0.465484\pi\)
0.108223 + 0.994127i \(0.465484\pi\)
\(558\) 0 0
\(559\) −2496.75 −0.188911
\(560\) 0 0
\(561\) 9965.39 0.749981
\(562\) 0 0
\(563\) 3877.50 0.290261 0.145131 0.989412i \(-0.453640\pi\)
0.145131 + 0.989412i \(0.453640\pi\)
\(564\) 0 0
\(565\) −2611.79 −0.194476
\(566\) 0 0
\(567\) 3348.64 0.248024
\(568\) 0 0
\(569\) −22052.8 −1.62478 −0.812391 0.583113i \(-0.801835\pi\)
−0.812391 + 0.583113i \(0.801835\pi\)
\(570\) 0 0
\(571\) 3017.56 0.221158 0.110579 0.993867i \(-0.464730\pi\)
0.110579 + 0.993867i \(0.464730\pi\)
\(572\) 0 0
\(573\) 31972.1 2.33099
\(574\) 0 0
\(575\) −8777.68 −0.636617
\(576\) 0 0
\(577\) 9766.04 0.704620 0.352310 0.935883i \(-0.385396\pi\)
0.352310 + 0.935883i \(0.385396\pi\)
\(578\) 0 0
\(579\) −20478.9 −1.46990
\(580\) 0 0
\(581\) −7847.49 −0.560359
\(582\) 0 0
\(583\) −2138.34 −0.151906
\(584\) 0 0
\(585\) −12950.9 −0.915308
\(586\) 0 0
\(587\) 20684.2 1.45439 0.727195 0.686431i \(-0.240824\pi\)
0.727195 + 0.686431i \(0.240824\pi\)
\(588\) 0 0
\(589\) 7070.41 0.494620
\(590\) 0 0
\(591\) 20503.4 1.42707
\(592\) 0 0
\(593\) 20556.7 1.42354 0.711772 0.702410i \(-0.247893\pi\)
0.711772 + 0.702410i \(0.247893\pi\)
\(594\) 0 0
\(595\) 867.658 0.0597824
\(596\) 0 0
\(597\) 28043.6 1.92252
\(598\) 0 0
\(599\) −11055.7 −0.754131 −0.377066 0.926186i \(-0.623067\pi\)
−0.377066 + 0.926186i \(0.623067\pi\)
\(600\) 0 0
\(601\) −8326.30 −0.565119 −0.282560 0.959250i \(-0.591184\pi\)
−0.282560 + 0.959250i \(0.591184\pi\)
\(602\) 0 0
\(603\) 31281.9 2.11260
\(604\) 0 0
\(605\) 22085.8 1.48416
\(606\) 0 0
\(607\) −22922.1 −1.53275 −0.766374 0.642395i \(-0.777941\pi\)
−0.766374 + 0.642395i \(0.777941\pi\)
\(608\) 0 0
\(609\) 1250.45 0.0832033
\(610\) 0 0
\(611\) 17678.4 1.17053
\(612\) 0 0
\(613\) 8197.48 0.540119 0.270060 0.962844i \(-0.412957\pi\)
0.270060 + 0.962844i \(0.412957\pi\)
\(614\) 0 0
\(615\) 13409.4 0.879220
\(616\) 0 0
\(617\) −27147.0 −1.77131 −0.885656 0.464343i \(-0.846291\pi\)
−0.885656 + 0.464343i \(0.846291\pi\)
\(618\) 0 0
\(619\) 472.406 0.0306747 0.0153373 0.999882i \(-0.495118\pi\)
0.0153373 + 0.999882i \(0.495118\pi\)
\(620\) 0 0
\(621\) −4372.27 −0.282534
\(622\) 0 0
\(623\) 742.775 0.0477667
\(624\) 0 0
\(625\) −2896.64 −0.185385
\(626\) 0 0
\(627\) 25975.0 1.65445
\(628\) 0 0
\(629\) −2767.59 −0.175439
\(630\) 0 0
\(631\) −2414.00 −0.152297 −0.0761487 0.997096i \(-0.524262\pi\)
−0.0761487 + 0.997096i \(0.524262\pi\)
\(632\) 0 0
\(633\) −35447.1 −2.22574
\(634\) 0 0
\(635\) 1974.42 0.123390
\(636\) 0 0
\(637\) 16808.8 1.04551
\(638\) 0 0
\(639\) −17876.8 −1.10672
\(640\) 0 0
\(641\) −10379.6 −0.639575 −0.319788 0.947489i \(-0.603612\pi\)
−0.319788 + 0.947489i \(0.603612\pi\)
\(642\) 0 0
\(643\) −19509.5 −1.19655 −0.598275 0.801291i \(-0.704147\pi\)
−0.598275 + 0.801291i \(0.704147\pi\)
\(644\) 0 0
\(645\) −2700.31 −0.164844
\(646\) 0 0
\(647\) −15125.8 −0.919096 −0.459548 0.888153i \(-0.651989\pi\)
−0.459548 + 0.888153i \(0.651989\pi\)
\(648\) 0 0
\(649\) 24603.7 1.48810
\(650\) 0 0
\(651\) −5822.04 −0.350513
\(652\) 0 0
\(653\) 22790.8 1.36581 0.682905 0.730507i \(-0.260716\pi\)
0.682905 + 0.730507i \(0.260716\pi\)
\(654\) 0 0
\(655\) −5570.75 −0.332316
\(656\) 0 0
\(657\) −7031.39 −0.417535
\(658\) 0 0
\(659\) −13542.4 −0.800514 −0.400257 0.916403i \(-0.631079\pi\)
−0.400257 + 0.916403i \(0.631079\pi\)
\(660\) 0 0
\(661\) 23999.0 1.41218 0.706092 0.708120i \(-0.250456\pi\)
0.706092 + 0.708120i \(0.250456\pi\)
\(662\) 0 0
\(663\) −8289.70 −0.485589
\(664\) 0 0
\(665\) 2261.57 0.131880
\(666\) 0 0
\(667\) 3829.29 0.222295
\(668\) 0 0
\(669\) 9743.59 0.563092
\(670\) 0 0
\(671\) 1203.53 0.0692425
\(672\) 0 0
\(673\) 34631.5 1.98357 0.991787 0.127901i \(-0.0408240\pi\)
0.991787 + 0.127901i \(0.0408240\pi\)
\(674\) 0 0
\(675\) 2201.14 0.125514
\(676\) 0 0
\(677\) −5200.08 −0.295207 −0.147604 0.989047i \(-0.547156\pi\)
−0.147604 + 0.989047i \(0.547156\pi\)
\(678\) 0 0
\(679\) −4729.61 −0.267313
\(680\) 0 0
\(681\) −6694.50 −0.376701
\(682\) 0 0
\(683\) −11690.2 −0.654926 −0.327463 0.944864i \(-0.606194\pi\)
−0.327463 + 0.944864i \(0.606194\pi\)
\(684\) 0 0
\(685\) 628.578 0.0350609
\(686\) 0 0
\(687\) −8274.03 −0.459496
\(688\) 0 0
\(689\) 1778.78 0.0983542
\(690\) 0 0
\(691\) 13188.4 0.726064 0.363032 0.931777i \(-0.381742\pi\)
0.363032 + 0.931777i \(0.381742\pi\)
\(692\) 0 0
\(693\) −11489.2 −0.629781
\(694\) 0 0
\(695\) −4536.54 −0.247598
\(696\) 0 0
\(697\) 4610.53 0.250554
\(698\) 0 0
\(699\) −44193.6 −2.39135
\(700\) 0 0
\(701\) 19761.3 1.06473 0.532364 0.846516i \(-0.321304\pi\)
0.532364 + 0.846516i \(0.321304\pi\)
\(702\) 0 0
\(703\) −7213.79 −0.387017
\(704\) 0 0
\(705\) 19119.8 1.02141
\(706\) 0 0
\(707\) −9004.77 −0.479009
\(708\) 0 0
\(709\) 16955.0 0.898106 0.449053 0.893505i \(-0.351761\pi\)
0.449053 + 0.893505i \(0.351761\pi\)
\(710\) 0 0
\(711\) 13498.8 0.712017
\(712\) 0 0
\(713\) −17829.0 −0.936467
\(714\) 0 0
\(715\) −26842.3 −1.40398
\(716\) 0 0
\(717\) −15054.3 −0.784120
\(718\) 0 0
\(719\) 8924.11 0.462884 0.231442 0.972849i \(-0.425656\pi\)
0.231442 + 0.972849i \(0.425656\pi\)
\(720\) 0 0
\(721\) −8838.29 −0.456526
\(722\) 0 0
\(723\) 32633.8 1.67865
\(724\) 0 0
\(725\) −1927.78 −0.0987531
\(726\) 0 0
\(727\) 14644.9 0.747112 0.373556 0.927608i \(-0.378138\pi\)
0.373556 + 0.927608i \(0.378138\pi\)
\(728\) 0 0
\(729\) −25415.0 −1.29121
\(730\) 0 0
\(731\) −928.440 −0.0469762
\(732\) 0 0
\(733\) −31105.6 −1.56741 −0.783704 0.621134i \(-0.786672\pi\)
−0.783704 + 0.621134i \(0.786672\pi\)
\(734\) 0 0
\(735\) 18179.2 0.912313
\(736\) 0 0
\(737\) 64835.3 3.24049
\(738\) 0 0
\(739\) −29071.3 −1.44710 −0.723549 0.690273i \(-0.757490\pi\)
−0.723549 + 0.690273i \(0.757490\pi\)
\(740\) 0 0
\(741\) −21607.3 −1.07121
\(742\) 0 0
\(743\) 8095.53 0.399726 0.199863 0.979824i \(-0.435950\pi\)
0.199863 + 0.979824i \(0.435950\pi\)
\(744\) 0 0
\(745\) −5574.66 −0.274147
\(746\) 0 0
\(747\) 43557.4 2.13344
\(748\) 0 0
\(749\) 7018.41 0.342386
\(750\) 0 0
\(751\) −6109.17 −0.296840 −0.148420 0.988924i \(-0.547419\pi\)
−0.148420 + 0.988924i \(0.547419\pi\)
\(752\) 0 0
\(753\) −21419.6 −1.03662
\(754\) 0 0
\(755\) 9206.73 0.443798
\(756\) 0 0
\(757\) −17269.3 −0.829146 −0.414573 0.910016i \(-0.636069\pi\)
−0.414573 + 0.910016i \(0.636069\pi\)
\(758\) 0 0
\(759\) −65499.5 −3.13239
\(760\) 0 0
\(761\) 39975.0 1.90420 0.952098 0.305792i \(-0.0989212\pi\)
0.952098 + 0.305792i \(0.0989212\pi\)
\(762\) 0 0
\(763\) 6542.97 0.310448
\(764\) 0 0
\(765\) −4815.93 −0.227608
\(766\) 0 0
\(767\) −20466.5 −0.963499
\(768\) 0 0
\(769\) −29413.2 −1.37928 −0.689640 0.724152i \(-0.742231\pi\)
−0.689640 + 0.724152i \(0.742231\pi\)
\(770\) 0 0
\(771\) −27433.9 −1.28146
\(772\) 0 0
\(773\) −29797.9 −1.38649 −0.693245 0.720702i \(-0.743820\pi\)
−0.693245 + 0.720702i \(0.743820\pi\)
\(774\) 0 0
\(775\) 8975.67 0.416020
\(776\) 0 0
\(777\) 5940.11 0.274260
\(778\) 0 0
\(779\) 12017.5 0.552721
\(780\) 0 0
\(781\) −37051.6 −1.69758
\(782\) 0 0
\(783\) −960.253 −0.0438271
\(784\) 0 0
\(785\) −20691.7 −0.940786
\(786\) 0 0
\(787\) 29580.7 1.33982 0.669910 0.742443i \(-0.266333\pi\)
0.669910 + 0.742443i \(0.266333\pi\)
\(788\) 0 0
\(789\) 31650.2 1.42811
\(790\) 0 0
\(791\) 1927.40 0.0866375
\(792\) 0 0
\(793\) −1001.15 −0.0448323
\(794\) 0 0
\(795\) 1923.80 0.0858242
\(796\) 0 0
\(797\) −35669.3 −1.58528 −0.792642 0.609688i \(-0.791295\pi\)
−0.792642 + 0.609688i \(0.791295\pi\)
\(798\) 0 0
\(799\) 6573.90 0.291073
\(800\) 0 0
\(801\) −4122.77 −0.181861
\(802\) 0 0
\(803\) −14573.4 −0.640451
\(804\) 0 0
\(805\) −5702.86 −0.249689
\(806\) 0 0
\(807\) 30736.7 1.34075
\(808\) 0 0
\(809\) −27538.1 −1.19677 −0.598386 0.801208i \(-0.704191\pi\)
−0.598386 + 0.801208i \(0.704191\pi\)
\(810\) 0 0
\(811\) −428.113 −0.0185365 −0.00926824 0.999957i \(-0.502950\pi\)
−0.00926824 + 0.999957i \(0.502950\pi\)
\(812\) 0 0
\(813\) 47613.5 2.05397
\(814\) 0 0
\(815\) −10025.5 −0.430894
\(816\) 0 0
\(817\) −2420.00 −0.103629
\(818\) 0 0
\(819\) 9557.27 0.407763
\(820\) 0 0
\(821\) 22381.3 0.951415 0.475707 0.879604i \(-0.342192\pi\)
0.475707 + 0.879604i \(0.342192\pi\)
\(822\) 0 0
\(823\) 4884.52 0.206882 0.103441 0.994636i \(-0.467015\pi\)
0.103441 + 0.994636i \(0.467015\pi\)
\(824\) 0 0
\(825\) 32974.5 1.39154
\(826\) 0 0
\(827\) 20727.9 0.871558 0.435779 0.900054i \(-0.356473\pi\)
0.435779 + 0.900054i \(0.356473\pi\)
\(828\) 0 0
\(829\) 7720.98 0.323475 0.161738 0.986834i \(-0.448290\pi\)
0.161738 + 0.986834i \(0.448290\pi\)
\(830\) 0 0
\(831\) −3323.23 −0.138726
\(832\) 0 0
\(833\) 6250.51 0.259985
\(834\) 0 0
\(835\) −22117.9 −0.916671
\(836\) 0 0
\(837\) 4470.90 0.184632
\(838\) 0 0
\(839\) −39526.8 −1.62648 −0.813239 0.581929i \(-0.802298\pi\)
−0.813239 + 0.581929i \(0.802298\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) 29065.2 1.18749
\(844\) 0 0
\(845\) 5521.34 0.224781
\(846\) 0 0
\(847\) −16298.5 −0.661183
\(848\) 0 0
\(849\) 1826.67 0.0738414
\(850\) 0 0
\(851\) 18190.5 0.732742
\(852\) 0 0
\(853\) 6398.26 0.256826 0.128413 0.991721i \(-0.459012\pi\)
0.128413 + 0.991721i \(0.459012\pi\)
\(854\) 0 0
\(855\) −12552.8 −0.502103
\(856\) 0 0
\(857\) 18780.2 0.748563 0.374282 0.927315i \(-0.377889\pi\)
0.374282 + 0.927315i \(0.377889\pi\)
\(858\) 0 0
\(859\) 18787.8 0.746253 0.373127 0.927780i \(-0.378286\pi\)
0.373127 + 0.927780i \(0.378286\pi\)
\(860\) 0 0
\(861\) −9895.62 −0.391686
\(862\) 0 0
\(863\) 28372.1 1.11912 0.559559 0.828791i \(-0.310971\pi\)
0.559559 + 0.828791i \(0.310971\pi\)
\(864\) 0 0
\(865\) −22300.5 −0.876578
\(866\) 0 0
\(867\) 34441.7 1.34914
\(868\) 0 0
\(869\) 27977.7 1.09215
\(870\) 0 0
\(871\) −53933.2 −2.09811
\(872\) 0 0
\(873\) 26251.7 1.01774
\(874\) 0 0
\(875\) 8269.61 0.319502
\(876\) 0 0
\(877\) −3003.79 −0.115656 −0.0578282 0.998327i \(-0.518418\pi\)
−0.0578282 + 0.998327i \(0.518418\pi\)
\(878\) 0 0
\(879\) 5492.51 0.210760
\(880\) 0 0
\(881\) 28184.2 1.07781 0.538904 0.842367i \(-0.318838\pi\)
0.538904 + 0.842367i \(0.318838\pi\)
\(882\) 0 0
\(883\) −12092.6 −0.460871 −0.230435 0.973088i \(-0.574015\pi\)
−0.230435 + 0.973088i \(0.574015\pi\)
\(884\) 0 0
\(885\) −22135.2 −0.840752
\(886\) 0 0
\(887\) −41525.1 −1.57190 −0.785951 0.618289i \(-0.787826\pi\)
−0.785951 + 0.618289i \(0.787826\pi\)
\(888\) 0 0
\(889\) −1457.05 −0.0549694
\(890\) 0 0
\(891\) −38522.8 −1.44844
\(892\) 0 0
\(893\) 17135.0 0.642107
\(894\) 0 0
\(895\) −18474.5 −0.689982
\(896\) 0 0
\(897\) 54485.7 2.02812
\(898\) 0 0
\(899\) −3915.66 −0.145266
\(900\) 0 0
\(901\) 661.456 0.0244576
\(902\) 0 0
\(903\) 1992.72 0.0734370
\(904\) 0 0
\(905\) −768.585 −0.0282305
\(906\) 0 0
\(907\) −17464.3 −0.639353 −0.319676 0.947527i \(-0.603574\pi\)
−0.319676 + 0.947527i \(0.603574\pi\)
\(908\) 0 0
\(909\) 49980.9 1.82372
\(910\) 0 0
\(911\) −8847.73 −0.321776 −0.160888 0.986973i \(-0.551436\pi\)
−0.160888 + 0.986973i \(0.551436\pi\)
\(912\) 0 0
\(913\) 90277.7 3.27246
\(914\) 0 0
\(915\) −1082.78 −0.0391208
\(916\) 0 0
\(917\) 4110.99 0.148044
\(918\) 0 0
\(919\) −45221.9 −1.62321 −0.811606 0.584205i \(-0.801406\pi\)
−0.811606 + 0.584205i \(0.801406\pi\)
\(920\) 0 0
\(921\) 74102.2 2.65119
\(922\) 0 0
\(923\) 30821.4 1.09913
\(924\) 0 0
\(925\) −9157.68 −0.325516
\(926\) 0 0
\(927\) 49056.9 1.73812
\(928\) 0 0
\(929\) −5013.65 −0.177064 −0.0885320 0.996073i \(-0.528218\pi\)
−0.0885320 + 0.996073i \(0.528218\pi\)
\(930\) 0 0
\(931\) 16292.1 0.573526
\(932\) 0 0
\(933\) 25731.7 0.902915
\(934\) 0 0
\(935\) −9981.56 −0.349125
\(936\) 0 0
\(937\) −39065.1 −1.36201 −0.681004 0.732280i \(-0.738456\pi\)
−0.681004 + 0.732280i \(0.738456\pi\)
\(938\) 0 0
\(939\) 40245.7 1.39869
\(940\) 0 0
\(941\) −18526.0 −0.641796 −0.320898 0.947114i \(-0.603985\pi\)
−0.320898 + 0.947114i \(0.603985\pi\)
\(942\) 0 0
\(943\) −30303.6 −1.04647
\(944\) 0 0
\(945\) 1430.08 0.0492280
\(946\) 0 0
\(947\) −30773.5 −1.05597 −0.527986 0.849253i \(-0.677053\pi\)
−0.527986 + 0.849253i \(0.677053\pi\)
\(948\) 0 0
\(949\) 12122.8 0.414672
\(950\) 0 0
\(951\) −58542.5 −1.99618
\(952\) 0 0
\(953\) −32135.7 −1.09232 −0.546158 0.837682i \(-0.683910\pi\)
−0.546158 + 0.837682i \(0.683910\pi\)
\(954\) 0 0
\(955\) −32024.0 −1.08510
\(956\) 0 0
\(957\) −14385.2 −0.485902
\(958\) 0 0
\(959\) −463.866 −0.0156194
\(960\) 0 0
\(961\) −11559.9 −0.388032
\(962\) 0 0
\(963\) −38955.6 −1.30356
\(964\) 0 0
\(965\) 20512.1 0.684258
\(966\) 0 0
\(967\) −10300.8 −0.342555 −0.171278 0.985223i \(-0.554789\pi\)
−0.171278 + 0.985223i \(0.554789\pi\)
\(968\) 0 0
\(969\) −8034.89 −0.266375
\(970\) 0 0
\(971\) 26187.1 0.865485 0.432742 0.901518i \(-0.357546\pi\)
0.432742 + 0.901518i \(0.357546\pi\)
\(972\) 0 0
\(973\) 3347.78 0.110303
\(974\) 0 0
\(975\) −27429.8 −0.900981
\(976\) 0 0
\(977\) 21578.3 0.706604 0.353302 0.935509i \(-0.385059\pi\)
0.353302 + 0.935509i \(0.385059\pi\)
\(978\) 0 0
\(979\) −8544.90 −0.278954
\(980\) 0 0
\(981\) −36316.7 −1.18196
\(982\) 0 0
\(983\) 2362.28 0.0766481 0.0383240 0.999265i \(-0.487798\pi\)
0.0383240 + 0.999265i \(0.487798\pi\)
\(984\) 0 0
\(985\) −20536.7 −0.664317
\(986\) 0 0
\(987\) −14109.6 −0.455029
\(988\) 0 0
\(989\) 6102.36 0.196202
\(990\) 0 0
\(991\) 15492.6 0.496609 0.248304 0.968682i \(-0.420127\pi\)
0.248304 + 0.968682i \(0.420127\pi\)
\(992\) 0 0
\(993\) 50310.3 1.60780
\(994\) 0 0
\(995\) −28089.1 −0.894958
\(996\) 0 0
\(997\) −30991.6 −0.984467 −0.492233 0.870463i \(-0.663819\pi\)
−0.492233 + 0.870463i \(0.663819\pi\)
\(998\) 0 0
\(999\) −4561.56 −0.144466
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 1856.4.a.bg.1.1 9
4.3 odd 2 1856.4.a.bf.1.9 9
8.3 odd 2 928.4.a.e.1.1 yes 9
8.5 even 2 928.4.a.d.1.9 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
928.4.a.d.1.9 9 8.5 even 2
928.4.a.e.1.1 yes 9 8.3 odd 2
1856.4.a.bf.1.9 9 4.3 odd 2
1856.4.a.bg.1.1 9 1.1 even 1 trivial