Properties

Label 2300.2.c.h.1749.4
Level $2300$
Weight $2$
Character 2300.1749
Analytic conductor $18.366$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2300.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.3655924649\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 460)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1749.4
Root \(2.56155i\) of defining polynomial
Character \(\chi\) \(=\) 2300.1749
Dual form 2300.2.c.h.1749.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.56155i q^{3} +1.56155i q^{7} -3.56155 q^{9} +O(q^{10})\) \(q+2.56155i q^{3} +1.56155i q^{7} -3.56155 q^{9} +2.00000 q^{11} +0.561553i q^{13} +1.56155i q^{17} -6.00000 q^{19} -4.00000 q^{21} +1.00000i q^{23} -1.43845i q^{27} +2.12311 q^{29} -9.24621 q^{31} +5.12311i q^{33} +0.438447i q^{37} -1.43845 q^{39} -4.12311 q^{41} +7.68466i q^{47} +4.56155 q^{49} -4.00000 q^{51} -0.438447i q^{53} -15.3693i q^{57} -8.68466 q^{59} +1.12311 q^{61} -5.56155i q^{63} +4.43845i q^{67} -2.56155 q^{69} +1.87689 q^{71} -8.56155i q^{73} +3.12311i q^{77} -13.1231 q^{79} -7.00000 q^{81} +14.9309i q^{83} +5.43845i q^{87} +2.24621 q^{89} -0.876894 q^{91} -23.6847i q^{93} +4.87689i q^{97} -7.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 6q^{9} + O(q^{10}) \) \( 4q - 6q^{9} + 8q^{11} - 24q^{19} - 16q^{21} - 8q^{29} - 4q^{31} - 14q^{39} + 10q^{49} - 16q^{51} - 10q^{59} - 12q^{61} - 2q^{69} + 24q^{71} - 36q^{79} - 28q^{81} - 24q^{89} - 20q^{91} - 12q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2300\mathbb{Z}\right)^\times\).

\(n\) \(277\) \(1151\) \(1201\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 2.56155i 1.47891i 0.673204 + 0.739457i \(0.264917\pi\)
−0.673204 + 0.739457i \(0.735083\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.56155i 0.590211i 0.955465 + 0.295106i \(0.0953549\pi\)
−0.955465 + 0.295106i \(0.904645\pi\)
\(8\) 0 0
\(9\) −3.56155 −1.18718
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 0.561553i 0.155747i 0.996963 + 0.0778734i \(0.0248130\pi\)
−0.996963 + 0.0778734i \(0.975187\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.56155i 0.378732i 0.981907 + 0.189366i \(0.0606433\pi\)
−0.981907 + 0.189366i \(0.939357\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) −4.00000 −0.872872
\(22\) 0 0
\(23\) 1.00000i 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 1.43845i − 0.276829i
\(28\) 0 0
\(29\) 2.12311 0.394251 0.197125 0.980378i \(-0.436839\pi\)
0.197125 + 0.980378i \(0.436839\pi\)
\(30\) 0 0
\(31\) −9.24621 −1.66067 −0.830334 0.557266i \(-0.811851\pi\)
−0.830334 + 0.557266i \(0.811851\pi\)
\(32\) 0 0
\(33\) 5.12311i 0.891818i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.438447i 0.0720803i 0.999350 + 0.0360401i \(0.0114744\pi\)
−0.999350 + 0.0360401i \(0.988526\pi\)
\(38\) 0 0
\(39\) −1.43845 −0.230336
\(40\) 0 0
\(41\) −4.12311 −0.643921 −0.321960 0.946753i \(-0.604342\pi\)
−0.321960 + 0.946753i \(0.604342\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.68466i 1.12092i 0.828181 + 0.560461i \(0.189376\pi\)
−0.828181 + 0.560461i \(0.810624\pi\)
\(48\) 0 0
\(49\) 4.56155 0.651650
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) − 0.438447i − 0.0602254i −0.999547 0.0301127i \(-0.990413\pi\)
0.999547 0.0301127i \(-0.00958661\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 15.3693i − 2.03572i
\(58\) 0 0
\(59\) −8.68466 −1.13065 −0.565323 0.824870i \(-0.691249\pi\)
−0.565323 + 0.824870i \(0.691249\pi\)
\(60\) 0 0
\(61\) 1.12311 0.143799 0.0718995 0.997412i \(-0.477094\pi\)
0.0718995 + 0.997412i \(0.477094\pi\)
\(62\) 0 0
\(63\) − 5.56155i − 0.700690i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.43845i 0.542243i 0.962545 + 0.271121i \(0.0873945\pi\)
−0.962545 + 0.271121i \(0.912606\pi\)
\(68\) 0 0
\(69\) −2.56155 −0.308375
\(70\) 0 0
\(71\) 1.87689 0.222746 0.111373 0.993779i \(-0.464475\pi\)
0.111373 + 0.993779i \(0.464475\pi\)
\(72\) 0 0
\(73\) − 8.56155i − 1.00205i −0.865432 0.501027i \(-0.832956\pi\)
0.865432 0.501027i \(-0.167044\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 3.12311i 0.355911i
\(78\) 0 0
\(79\) −13.1231 −1.47646 −0.738232 0.674546i \(-0.764339\pi\)
−0.738232 + 0.674546i \(0.764339\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) 14.9309i 1.63888i 0.573168 + 0.819438i \(0.305714\pi\)
−0.573168 + 0.819438i \(0.694286\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 5.43845i 0.583063i
\(88\) 0 0
\(89\) 2.24621 0.238098 0.119049 0.992888i \(-0.462015\pi\)
0.119049 + 0.992888i \(0.462015\pi\)
\(90\) 0 0
\(91\) −0.876894 −0.0919235
\(92\) 0 0
\(93\) − 23.6847i − 2.45598i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.87689i 0.495174i 0.968866 + 0.247587i \(0.0796375\pi\)
−0.968866 + 0.247587i \(0.920362\pi\)
\(98\) 0 0
\(99\) −7.12311 −0.715899
\(100\) 0 0
\(101\) −5.31534 −0.528896 −0.264448 0.964400i \(-0.585190\pi\)
−0.264448 + 0.964400i \(0.585190\pi\)
\(102\) 0 0
\(103\) − 16.0000i − 1.57653i −0.615338 0.788263i \(-0.710980\pi\)
0.615338 0.788263i \(-0.289020\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 7.56155i − 0.731003i −0.930811 0.365501i \(-0.880898\pi\)
0.930811 0.365501i \(-0.119102\pi\)
\(108\) 0 0
\(109\) 9.36932 0.897418 0.448709 0.893678i \(-0.351884\pi\)
0.448709 + 0.893678i \(0.351884\pi\)
\(110\) 0 0
\(111\) −1.12311 −0.106600
\(112\) 0 0
\(113\) − 7.80776i − 0.734493i −0.930124 0.367246i \(-0.880301\pi\)
0.930124 0.367246i \(-0.119699\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 2.00000i − 0.184900i
\(118\) 0 0
\(119\) −2.43845 −0.223532
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) − 10.5616i − 0.952303i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 12.8078i 1.13651i 0.822854 + 0.568253i \(0.192380\pi\)
−0.822854 + 0.568253i \(0.807620\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 20.8078 1.81798 0.908991 0.416815i \(-0.136854\pi\)
0.908991 + 0.416815i \(0.136854\pi\)
\(132\) 0 0
\(133\) − 9.36932i − 0.812423i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 15.1231i − 1.29205i −0.763315 0.646027i \(-0.776429\pi\)
0.763315 0.646027i \(-0.223571\pi\)
\(138\) 0 0
\(139\) −5.24621 −0.444978 −0.222489 0.974935i \(-0.571418\pi\)
−0.222489 + 0.974935i \(0.571418\pi\)
\(140\) 0 0
\(141\) −19.6847 −1.65775
\(142\) 0 0
\(143\) 1.12311i 0.0939188i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 11.6847i 0.963734i
\(148\) 0 0
\(149\) −15.3693 −1.25910 −0.629552 0.776959i \(-0.716761\pi\)
−0.629552 + 0.776959i \(0.716761\pi\)
\(150\) 0 0
\(151\) 23.0540 1.87611 0.938053 0.346492i \(-0.112627\pi\)
0.938053 + 0.346492i \(0.112627\pi\)
\(152\) 0 0
\(153\) − 5.56155i − 0.449625i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 16.6847i 1.33158i 0.746139 + 0.665790i \(0.231905\pi\)
−0.746139 + 0.665790i \(0.768095\pi\)
\(158\) 0 0
\(159\) 1.12311 0.0890681
\(160\) 0 0
\(161\) −1.56155 −0.123068
\(162\) 0 0
\(163\) − 11.9309i − 0.934498i −0.884126 0.467249i \(-0.845245\pi\)
0.884126 0.467249i \(-0.154755\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.00000i 0.619059i 0.950890 + 0.309529i \(0.100171\pi\)
−0.950890 + 0.309529i \(0.899829\pi\)
\(168\) 0 0
\(169\) 12.6847 0.975743
\(170\) 0 0
\(171\) 21.3693 1.63415
\(172\) 0 0
\(173\) − 19.3693i − 1.47262i −0.676643 0.736311i \(-0.736566\pi\)
0.676643 0.736311i \(-0.263434\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 22.2462i − 1.67213i
\(178\) 0 0
\(179\) −21.9309 −1.63919 −0.819595 0.572943i \(-0.805802\pi\)
−0.819595 + 0.572943i \(0.805802\pi\)
\(180\) 0 0
\(181\) 19.3693 1.43971 0.719855 0.694124i \(-0.244208\pi\)
0.719855 + 0.694124i \(0.244208\pi\)
\(182\) 0 0
\(183\) 2.87689i 0.212666i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 3.12311i 0.228384i
\(188\) 0 0
\(189\) 2.24621 0.163388
\(190\) 0 0
\(191\) −7.36932 −0.533225 −0.266613 0.963804i \(-0.585904\pi\)
−0.266613 + 0.963804i \(0.585904\pi\)
\(192\) 0 0
\(193\) 6.56155i 0.472311i 0.971715 + 0.236155i \(0.0758875\pi\)
−0.971715 + 0.236155i \(0.924113\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 6.31534i 0.449949i 0.974365 + 0.224975i \(0.0722300\pi\)
−0.974365 + 0.224975i \(0.927770\pi\)
\(198\) 0 0
\(199\) 10.0000 0.708881 0.354441 0.935079i \(-0.384671\pi\)
0.354441 + 0.935079i \(0.384671\pi\)
\(200\) 0 0
\(201\) −11.3693 −0.801930
\(202\) 0 0
\(203\) 3.31534i 0.232691i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 3.56155i − 0.247545i
\(208\) 0 0
\(209\) −12.0000 −0.830057
\(210\) 0 0
\(211\) 12.6847 0.873248 0.436624 0.899644i \(-0.356174\pi\)
0.436624 + 0.899644i \(0.356174\pi\)
\(212\) 0 0
\(213\) 4.80776i 0.329423i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 14.4384i − 0.980146i
\(218\) 0 0
\(219\) 21.9309 1.48195
\(220\) 0 0
\(221\) −0.876894 −0.0589863
\(222\) 0 0
\(223\) 23.6155i 1.58141i 0.612196 + 0.790706i \(0.290287\pi\)
−0.612196 + 0.790706i \(0.709713\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 9.75379i − 0.647382i −0.946163 0.323691i \(-0.895076\pi\)
0.946163 0.323691i \(-0.104924\pi\)
\(228\) 0 0
\(229\) −22.7386 −1.50261 −0.751306 0.659954i \(-0.770576\pi\)
−0.751306 + 0.659954i \(0.770576\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) 0 0
\(233\) 11.6847i 0.765487i 0.923855 + 0.382744i \(0.125021\pi\)
−0.923855 + 0.382744i \(0.874979\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 33.6155i − 2.18356i
\(238\) 0 0
\(239\) −19.2462 −1.24493 −0.622467 0.782646i \(-0.713869\pi\)
−0.622467 + 0.782646i \(0.713869\pi\)
\(240\) 0 0
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) 0 0
\(243\) − 22.2462i − 1.42710i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 3.36932i − 0.214384i
\(248\) 0 0
\(249\) −38.2462 −2.42376
\(250\) 0 0
\(251\) 17.3693 1.09634 0.548171 0.836366i \(-0.315324\pi\)
0.548171 + 0.836366i \(0.315324\pi\)
\(252\) 0 0
\(253\) 2.00000i 0.125739i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 11.1922i − 0.698152i −0.937094 0.349076i \(-0.886495\pi\)
0.937094 0.349076i \(-0.113505\pi\)
\(258\) 0 0
\(259\) −0.684658 −0.0425426
\(260\) 0 0
\(261\) −7.56155 −0.468048
\(262\) 0 0
\(263\) 28.9309i 1.78395i 0.452081 + 0.891977i \(0.350682\pi\)
−0.452081 + 0.891977i \(0.649318\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 5.75379i 0.352126i
\(268\) 0 0
\(269\) −6.75379 −0.411786 −0.205893 0.978575i \(-0.566010\pi\)
−0.205893 + 0.978575i \(0.566010\pi\)
\(270\) 0 0
\(271\) −6.93087 −0.421020 −0.210510 0.977592i \(-0.567513\pi\)
−0.210510 + 0.977592i \(0.567513\pi\)
\(272\) 0 0
\(273\) − 2.24621i − 0.135947i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 5.68466i − 0.341558i −0.985309 0.170779i \(-0.945372\pi\)
0.985309 0.170779i \(-0.0546284\pi\)
\(278\) 0 0
\(279\) 32.9309 1.97152
\(280\) 0 0
\(281\) −15.1231 −0.902169 −0.451084 0.892481i \(-0.648963\pi\)
−0.451084 + 0.892481i \(0.648963\pi\)
\(282\) 0 0
\(283\) − 5.80776i − 0.345236i −0.984989 0.172618i \(-0.944777\pi\)
0.984989 0.172618i \(-0.0552226\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 6.43845i − 0.380050i
\(288\) 0 0
\(289\) 14.5616 0.856562
\(290\) 0 0
\(291\) −12.4924 −0.732319
\(292\) 0 0
\(293\) 3.80776i 0.222452i 0.993795 + 0.111226i \(0.0354777\pi\)
−0.993795 + 0.111226i \(0.964522\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 2.87689i − 0.166934i
\(298\) 0 0
\(299\) −0.561553 −0.0324754
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) − 13.6155i − 0.782192i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 12.8769i 0.734923i 0.930039 + 0.367462i \(0.119773\pi\)
−0.930039 + 0.367462i \(0.880227\pi\)
\(308\) 0 0
\(309\) 40.9848 2.33155
\(310\) 0 0
\(311\) −3.68466 −0.208938 −0.104469 0.994528i \(-0.533314\pi\)
−0.104469 + 0.994528i \(0.533314\pi\)
\(312\) 0 0
\(313\) 19.8078i 1.11960i 0.828627 + 0.559801i \(0.189122\pi\)
−0.828627 + 0.559801i \(0.810878\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 6.87689i − 0.386245i −0.981175 0.193122i \(-0.938139\pi\)
0.981175 0.193122i \(-0.0618615\pi\)
\(318\) 0 0
\(319\) 4.24621 0.237742
\(320\) 0 0
\(321\) 19.3693 1.08109
\(322\) 0 0
\(323\) − 9.36932i − 0.521323i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 24.0000i 1.32720i
\(328\) 0 0
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) −12.6155 −0.693412 −0.346706 0.937974i \(-0.612700\pi\)
−0.346706 + 0.937974i \(0.612700\pi\)
\(332\) 0 0
\(333\) − 1.56155i − 0.0855726i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 27.6155i 1.50431i 0.658984 + 0.752157i \(0.270986\pi\)
−0.658984 + 0.752157i \(0.729014\pi\)
\(338\) 0 0
\(339\) 20.0000 1.08625
\(340\) 0 0
\(341\) −18.4924 −1.00142
\(342\) 0 0
\(343\) 18.0540i 0.974823i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 16.4924i 0.885360i 0.896680 + 0.442680i \(0.145972\pi\)
−0.896680 + 0.442680i \(0.854028\pi\)
\(348\) 0 0
\(349\) 1.63068 0.0872885 0.0436442 0.999047i \(-0.486103\pi\)
0.0436442 + 0.999047i \(0.486103\pi\)
\(350\) 0 0
\(351\) 0.807764 0.0431153
\(352\) 0 0
\(353\) 35.0540i 1.86573i 0.360220 + 0.932867i \(0.382702\pi\)
−0.360220 + 0.932867i \(0.617298\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 6.24621i − 0.330585i
\(358\) 0 0
\(359\) 25.6155 1.35194 0.675968 0.736931i \(-0.263726\pi\)
0.675968 + 0.736931i \(0.263726\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) − 17.9309i − 0.941127i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 18.4384i 0.962479i 0.876589 + 0.481240i \(0.159813\pi\)
−0.876589 + 0.481240i \(0.840187\pi\)
\(368\) 0 0
\(369\) 14.6847 0.764453
\(370\) 0 0
\(371\) 0.684658 0.0355457
\(372\) 0 0
\(373\) − 3.75379i − 0.194364i −0.995267 0.0971819i \(-0.969017\pi\)
0.995267 0.0971819i \(-0.0309829\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.19224i 0.0614033i
\(378\) 0 0
\(379\) 3.50758 0.180172 0.0900861 0.995934i \(-0.471286\pi\)
0.0900861 + 0.995934i \(0.471286\pi\)
\(380\) 0 0
\(381\) −32.8078 −1.68079
\(382\) 0 0
\(383\) 23.8078i 1.21652i 0.793738 + 0.608260i \(0.208132\pi\)
−0.793738 + 0.608260i \(0.791868\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 19.1231 0.969580 0.484790 0.874631i \(-0.338896\pi\)
0.484790 + 0.874631i \(0.338896\pi\)
\(390\) 0 0
\(391\) −1.56155 −0.0789711
\(392\) 0 0
\(393\) 53.3002i 2.68864i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 35.5464i 1.78402i 0.452013 + 0.892011i \(0.350706\pi\)
−0.452013 + 0.892011i \(0.649294\pi\)
\(398\) 0 0
\(399\) 24.0000 1.20150
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 0 0
\(403\) − 5.19224i − 0.258644i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.876894i 0.0434660i
\(408\) 0 0
\(409\) −29.9848 −1.48266 −0.741328 0.671143i \(-0.765803\pi\)
−0.741328 + 0.671143i \(0.765803\pi\)
\(410\) 0 0
\(411\) 38.7386 1.91084
\(412\) 0 0
\(413\) − 13.5616i − 0.667320i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 13.4384i − 0.658084i
\(418\) 0 0
\(419\) 27.1231 1.32505 0.662525 0.749040i \(-0.269485\pi\)
0.662525 + 0.749040i \(0.269485\pi\)
\(420\) 0 0
\(421\) 16.8769 0.822530 0.411265 0.911516i \(-0.365087\pi\)
0.411265 + 0.911516i \(0.365087\pi\)
\(422\) 0 0
\(423\) − 27.3693i − 1.33074i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.75379i 0.0848718i
\(428\) 0 0
\(429\) −2.87689 −0.138898
\(430\) 0 0
\(431\) 6.24621 0.300869 0.150435 0.988620i \(-0.451933\pi\)
0.150435 + 0.988620i \(0.451933\pi\)
\(432\) 0 0
\(433\) 11.0691i 0.531948i 0.963980 + 0.265974i \(0.0856936\pi\)
−0.963980 + 0.265974i \(0.914306\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 6.00000i − 0.287019i
\(438\) 0 0
\(439\) 0.807764 0.0385525 0.0192762 0.999814i \(-0.493864\pi\)
0.0192762 + 0.999814i \(0.493864\pi\)
\(440\) 0 0
\(441\) −16.2462 −0.773629
\(442\) 0 0
\(443\) 8.80776i 0.418469i 0.977865 + 0.209235i \(0.0670973\pi\)
−0.977865 + 0.209235i \(0.932903\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 39.3693i − 1.86210i
\(448\) 0 0
\(449\) 9.31534 0.439618 0.219809 0.975543i \(-0.429457\pi\)
0.219809 + 0.975543i \(0.429457\pi\)
\(450\) 0 0
\(451\) −8.24621 −0.388299
\(452\) 0 0
\(453\) 59.0540i 2.77460i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 13.5616i − 0.634383i −0.948362 0.317191i \(-0.897260\pi\)
0.948362 0.317191i \(-0.102740\pi\)
\(458\) 0 0
\(459\) 2.24621 0.104844
\(460\) 0 0
\(461\) −12.0691 −0.562115 −0.281058 0.959691i \(-0.590685\pi\)
−0.281058 + 0.959691i \(0.590685\pi\)
\(462\) 0 0
\(463\) − 6.63068i − 0.308154i −0.988059 0.154077i \(-0.950760\pi\)
0.988059 0.154077i \(-0.0492404\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 34.6847i 1.60501i 0.596642 + 0.802507i \(0.296501\pi\)
−0.596642 + 0.802507i \(0.703499\pi\)
\(468\) 0 0
\(469\) −6.93087 −0.320038
\(470\) 0 0
\(471\) −42.7386 −1.96929
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.56155i 0.0714986i
\(478\) 0 0
\(479\) −39.2311 −1.79251 −0.896256 0.443536i \(-0.853724\pi\)
−0.896256 + 0.443536i \(0.853724\pi\)
\(480\) 0 0
\(481\) −0.246211 −0.0112263
\(482\) 0 0
\(483\) − 4.00000i − 0.182006i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 12.8078i 0.580375i 0.956970 + 0.290188i \(0.0937177\pi\)
−0.956970 + 0.290188i \(0.906282\pi\)
\(488\) 0 0
\(489\) 30.5616 1.38204
\(490\) 0 0
\(491\) −21.4924 −0.969939 −0.484970 0.874531i \(-0.661169\pi\)
−0.484970 + 0.874531i \(0.661169\pi\)
\(492\) 0 0
\(493\) 3.31534i 0.149315i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.93087i 0.131467i
\(498\) 0 0
\(499\) 18.6155 0.833345 0.416673 0.909057i \(-0.363196\pi\)
0.416673 + 0.909057i \(0.363196\pi\)
\(500\) 0 0
\(501\) −20.4924 −0.915534
\(502\) 0 0
\(503\) 24.9309i 1.11161i 0.831312 + 0.555806i \(0.187590\pi\)
−0.831312 + 0.555806i \(0.812410\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 32.4924i 1.44304i
\(508\) 0 0
\(509\) −18.1771 −0.805685 −0.402842 0.915269i \(-0.631978\pi\)
−0.402842 + 0.915269i \(0.631978\pi\)
\(510\) 0 0
\(511\) 13.3693 0.591424
\(512\) 0 0
\(513\) 8.63068i 0.381054i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 15.3693i 0.675942i
\(518\) 0 0
\(519\) 49.6155 2.17788
\(520\) 0 0
\(521\) 27.6155 1.20986 0.604929 0.796279i \(-0.293201\pi\)
0.604929 + 0.796279i \(0.293201\pi\)
\(522\) 0 0
\(523\) 34.7386i 1.51901i 0.650499 + 0.759507i \(0.274560\pi\)
−0.650499 + 0.759507i \(0.725440\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 14.4384i − 0.628949i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 30.9309 1.34229
\(532\) 0 0
\(533\) − 2.31534i − 0.100289i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 56.1771i − 2.42422i
\(538\) 0 0
\(539\) 9.12311 0.392960
\(540\) 0 0
\(541\) 9.68466 0.416376 0.208188 0.978089i \(-0.433243\pi\)
0.208188 + 0.978089i \(0.433243\pi\)
\(542\) 0 0
\(543\) 49.6155i 2.12921i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 24.1771i − 1.03374i −0.856065 0.516869i \(-0.827098\pi\)
0.856065 0.516869i \(-0.172902\pi\)
\(548\) 0 0
\(549\) −4.00000 −0.170716
\(550\) 0 0
\(551\) −12.7386 −0.542684
\(552\) 0 0
\(553\) − 20.4924i − 0.871426i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 3.31534i 0.140476i 0.997530 + 0.0702378i \(0.0223758\pi\)
−0.997530 + 0.0702378i \(0.977624\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −8.00000 −0.337760
\(562\) 0 0
\(563\) − 45.6695i − 1.92474i −0.271742 0.962370i \(-0.587600\pi\)
0.271742 0.962370i \(-0.412400\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 10.9309i − 0.459053i
\(568\) 0 0
\(569\) 32.7386 1.37247 0.686237 0.727378i \(-0.259261\pi\)
0.686237 + 0.727378i \(0.259261\pi\)
\(570\) 0 0
\(571\) 36.0000 1.50655 0.753277 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) 0 0
\(573\) − 18.8769i − 0.788594i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 25.4384i 1.05902i 0.848305 + 0.529508i \(0.177624\pi\)
−0.848305 + 0.529508i \(0.822376\pi\)
\(578\) 0 0
\(579\) −16.8078 −0.698507
\(580\) 0 0
\(581\) −23.3153 −0.967283
\(582\) 0 0
\(583\) − 0.876894i − 0.0363173i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 35.9309i − 1.48303i −0.670939 0.741513i \(-0.734109\pi\)
0.670939 0.741513i \(-0.265891\pi\)
\(588\) 0 0
\(589\) 55.4773 2.28590
\(590\) 0 0
\(591\) −16.1771 −0.665436
\(592\) 0 0
\(593\) − 25.6155i − 1.05190i −0.850514 0.525952i \(-0.823709\pi\)
0.850514 0.525952i \(-0.176291\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 25.6155i 1.04837i
\(598\) 0 0
\(599\) −32.9848 −1.34772 −0.673862 0.738857i \(-0.735366\pi\)
−0.673862 + 0.738857i \(0.735366\pi\)
\(600\) 0 0
\(601\) 11.6307 0.474425 0.237213 0.971458i \(-0.423766\pi\)
0.237213 + 0.971458i \(0.423766\pi\)
\(602\) 0 0
\(603\) − 15.8078i − 0.643742i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 24.4924i 0.994117i 0.867717 + 0.497058i \(0.165587\pi\)
−0.867717 + 0.497058i \(0.834413\pi\)
\(608\) 0 0
\(609\) −8.49242 −0.344130
\(610\) 0 0
\(611\) −4.31534 −0.174580
\(612\) 0 0
\(613\) 35.6155i 1.43850i 0.694753 + 0.719249i \(0.255514\pi\)
−0.694753 + 0.719249i \(0.744486\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.56155i 0.143383i 0.997427 + 0.0716914i \(0.0228397\pi\)
−0.997427 + 0.0716914i \(0.977160\pi\)
\(618\) 0 0
\(619\) 20.4924 0.823660 0.411830 0.911261i \(-0.364890\pi\)
0.411830 + 0.911261i \(0.364890\pi\)
\(620\) 0 0
\(621\) 1.43845 0.0577229
\(622\) 0 0
\(623\) 3.50758i 0.140528i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 30.7386i − 1.22758i
\(628\) 0 0
\(629\) −0.684658 −0.0272991
\(630\) 0 0
\(631\) 13.7538 0.547530 0.273765 0.961797i \(-0.411731\pi\)
0.273765 + 0.961797i \(0.411731\pi\)
\(632\) 0 0
\(633\) 32.4924i 1.29146i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.56155i 0.101492i
\(638\) 0 0
\(639\) −6.68466 −0.264441
\(640\) 0 0
\(641\) 15.1231 0.597327 0.298663 0.954359i \(-0.403459\pi\)
0.298663 + 0.954359i \(0.403459\pi\)
\(642\) 0 0
\(643\) − 41.4233i − 1.63358i −0.576939 0.816788i \(-0.695753\pi\)
0.576939 0.816788i \(-0.304247\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 17.6847i − 0.695256i −0.937633 0.347628i \(-0.886987\pi\)
0.937633 0.347628i \(-0.113013\pi\)
\(648\) 0 0
\(649\) −17.3693 −0.681805
\(650\) 0 0
\(651\) 36.9848 1.44955
\(652\) 0 0
\(653\) − 38.6695i − 1.51325i −0.653846 0.756627i \(-0.726846\pi\)
0.653846 0.756627i \(-0.273154\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 30.4924i 1.18962i
\(658\) 0 0
\(659\) 9.36932 0.364977 0.182488 0.983208i \(-0.441585\pi\)
0.182488 + 0.983208i \(0.441585\pi\)
\(660\) 0 0
\(661\) 26.4924 1.03044 0.515218 0.857059i \(-0.327711\pi\)
0.515218 + 0.857059i \(0.327711\pi\)
\(662\) 0 0
\(663\) − 2.24621i − 0.0872356i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 2.12311i 0.0822070i
\(668\) 0 0
\(669\) −60.4924 −2.33877
\(670\) 0 0
\(671\) 2.24621 0.0867140
\(672\) 0 0
\(673\) − 35.5464i − 1.37021i −0.728443 0.685106i \(-0.759756\pi\)
0.728443 0.685106i \(-0.240244\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 6.19224i − 0.237987i −0.992895 0.118993i \(-0.962033\pi\)
0.992895 0.118993i \(-0.0379668\pi\)
\(678\) 0 0
\(679\) −7.61553 −0.292257
\(680\) 0 0
\(681\) 24.9848 0.957421
\(682\) 0 0
\(683\) 4.94602i 0.189254i 0.995513 + 0.0946272i \(0.0301659\pi\)
−0.995513 + 0.0946272i \(0.969834\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 58.2462i − 2.22223i
\(688\) 0 0
\(689\) 0.246211 0.00937990
\(690\) 0 0
\(691\) −16.4924 −0.627401 −0.313701 0.949522i \(-0.601569\pi\)
−0.313701 + 0.949522i \(0.601569\pi\)
\(692\) 0 0
\(693\) − 11.1231i − 0.422532i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 6.43845i − 0.243874i
\(698\) 0 0
\(699\) −29.9309 −1.13209
\(700\) 0 0
\(701\) −18.2462 −0.689150 −0.344575 0.938759i \(-0.611977\pi\)
−0.344575 + 0.938759i \(0.611977\pi\)
\(702\) 0 0
\(703\) − 2.63068i − 0.0992181i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 8.30019i − 0.312161i
\(708\) 0 0
\(709\) −46.2462 −1.73681 −0.868406 0.495853i \(-0.834855\pi\)
−0.868406 + 0.495853i \(0.834855\pi\)
\(710\) 0 0
\(711\) 46.7386 1.75284
\(712\) 0 0
\(713\) − 9.24621i − 0.346273i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 49.3002i − 1.84115i
\(718\) 0 0
\(719\) 18.0540 0.673300 0.336650 0.941630i \(-0.390706\pi\)
0.336650 + 0.941630i \(0.390706\pi\)
\(720\) 0 0
\(721\) 24.9848 0.930484
\(722\) 0 0
\(723\) 15.3693i 0.571591i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 41.8078i 1.55056i 0.631615 + 0.775282i \(0.282392\pi\)
−0.631615 + 0.775282i \(0.717608\pi\)
\(728\) 0 0
\(729\) 35.9848 1.33277
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 12.6847i 0.468519i 0.972174 + 0.234259i \(0.0752665\pi\)
−0.972174 + 0.234259i \(0.924734\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8.87689i 0.326985i
\(738\) 0 0
\(739\) 40.6155 1.49407 0.747033 0.664787i \(-0.231478\pi\)
0.747033 + 0.664787i \(0.231478\pi\)
\(740\) 0 0
\(741\) 8.63068 0.317056
\(742\) 0 0
\(743\) 36.4924i 1.33878i 0.742912 + 0.669389i \(0.233444\pi\)
−0.742912 + 0.669389i \(0.766556\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 53.1771i − 1.94565i
\(748\) 0 0
\(749\) 11.8078 0.431446
\(750\) 0 0
\(751\) −4.87689 −0.177960 −0.0889802 0.996033i \(-0.528361\pi\)
−0.0889802 + 0.996033i \(0.528361\pi\)
\(752\) 0 0
\(753\) 44.4924i 1.62139i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 8.43845i − 0.306701i −0.988172 0.153350i \(-0.950994\pi\)
0.988172 0.153350i \(-0.0490063\pi\)
\(758\) 0 0
\(759\) −5.12311 −0.185957
\(760\) 0 0
\(761\) 21.9848 0.796950 0.398475 0.917179i \(-0.369540\pi\)
0.398475 + 0.917179i \(0.369540\pi\)
\(762\) 0 0
\(763\) 14.6307i 0.529666i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 4.87689i − 0.176094i
\(768\) 0 0
\(769\) 35.3693 1.27545 0.637725 0.770264i \(-0.279876\pi\)
0.637725 + 0.770264i \(0.279876\pi\)
\(770\) 0 0
\(771\) 28.6695 1.03251
\(772\) 0 0
\(773\) − 32.8769i − 1.18250i −0.806488 0.591250i \(-0.798635\pi\)
0.806488 0.591250i \(-0.201365\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 1.75379i − 0.0629168i
\(778\) 0 0
\(779\) 24.7386 0.886354
\(780\) 0 0
\(781\) 3.75379 0.134321
\(782\) 0 0
\(783\) − 3.05398i − 0.109140i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 27.3153i 0.973687i 0.873489 + 0.486843i \(0.161852\pi\)
−0.873489 + 0.486843i \(0.838148\pi\)
\(788\) 0 0
\(789\) −74.1080 −2.63831
\(790\) 0 0
\(791\) 12.1922 0.433506
\(792\) 0 0
\(793\) 0.630683i 0.0223962i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 9.42329i − 0.333790i −0.985975 0.166895i \(-0.946626\pi\)
0.985975 0.166895i \(-0.0533741\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) −8.00000 −0.282666
\(802\) 0 0
\(803\) − 17.1231i − 0.604261i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 17.3002i − 0.608995i
\(808\) 0 0
\(809\) 8.82292 0.310197 0.155099 0.987899i \(-0.450430\pi\)
0.155099 + 0.987899i \(0.450430\pi\)
\(810\) 0 0
\(811\) 30.7538 1.07991 0.539956 0.841693i \(-0.318441\pi\)
0.539956 + 0.841693i \(0.318441\pi\)
\(812\) 0 0
\(813\) − 17.7538i − 0.622653i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 3.12311 0.109130
\(820\) 0 0
\(821\) −1.50758 −0.0526148 −0.0263074 0.999654i \(-0.508375\pi\)
−0.0263074 + 0.999654i \(0.508375\pi\)
\(822\) 0 0
\(823\) − 0.946025i − 0.0329763i −0.999864 0.0164882i \(-0.994751\pi\)
0.999864 0.0164882i \(-0.00524859\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 7.31534i − 0.254379i −0.991878 0.127190i \(-0.959404\pi\)
0.991878 0.127190i \(-0.0405957\pi\)
\(828\) 0 0
\(829\) −40.5464 −1.40823 −0.704117 0.710084i \(-0.748657\pi\)
−0.704117 + 0.710084i \(0.748657\pi\)
\(830\) 0 0
\(831\) 14.5616 0.505135
\(832\) 0 0
\(833\) 7.12311i 0.246801i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 13.3002i 0.459722i
\(838\) 0 0
\(839\) 51.1231 1.76497 0.882483 0.470345i \(-0.155870\pi\)
0.882483 + 0.470345i \(0.155870\pi\)
\(840\) 0 0
\(841\) −24.4924 −0.844566
\(842\) 0 0
\(843\) − 38.7386i − 1.33423i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 10.9309i − 0.375589i
\(848\) 0 0
\(849\) 14.8769 0.510574
\(850\) 0 0
\(851\) −0.438447 −0.0150298
\(852\) 0 0
\(853\) − 3.75379i − 0.128527i −0.997933 0.0642636i \(-0.979530\pi\)
0.997933 0.0642636i \(-0.0204699\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 45.6847i 1.56056i 0.625431 + 0.780279i \(0.284923\pi\)
−0.625431 + 0.780279i \(0.715077\pi\)
\(858\) 0 0
\(859\) 47.4924 1.62042 0.810210 0.586139i \(-0.199353\pi\)
0.810210 + 0.586139i \(0.199353\pi\)
\(860\) 0 0
\(861\) 16.4924 0.562060
\(862\) 0 0
\(863\) − 3.43845i − 0.117046i −0.998286 0.0585231i \(-0.981361\pi\)
0.998286 0.0585231i \(-0.0186391\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 37.3002i 1.26678i
\(868\) 0 0
\(869\) −26.2462 −0.890342
\(870\) 0 0
\(871\) −2.49242 −0.0844525
\(872\) 0 0
\(873\) − 17.3693i − 0.587862i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 46.9848i − 1.58657i −0.608853 0.793283i \(-0.708370\pi\)
0.608853 0.793283i \(-0.291630\pi\)
\(878\) 0 0
\(879\) −9.75379 −0.328987
\(880\) 0 0
\(881\) −57.4773 −1.93646 −0.968229 0.250065i \(-0.919548\pi\)
−0.968229 + 0.250065i \(0.919548\pi\)
\(882\) 0 0
\(883\) − 38.7386i − 1.30366i −0.758366 0.651829i \(-0.774002\pi\)
0.758366 0.651829i \(-0.225998\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 43.7926i 1.47041i 0.677844 + 0.735206i \(0.262915\pi\)
−0.677844 + 0.735206i \(0.737085\pi\)
\(888\) 0 0
\(889\) −20.0000 −0.670778
\(890\) 0 0
\(891\) −14.0000 −0.469018
\(892\) 0 0
\(893\) − 46.1080i − 1.54294i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 1.43845i − 0.0480284i
\(898\) 0 0
\(899\) −19.6307 −0.654720
\(900\) 0 0
\(901\) 0.684658 0.0228093
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 9.31534i − 0.309311i −0.987968 0.154655i \(-0.950573\pi\)
0.987968 0.154655i \(-0.0494267\pi\)
\(908\) 0 0
\(909\) 18.9309 0.627897
\(910\) 0 0
\(911\) 5.12311 0.169736 0.0848680 0.996392i \(-0.472953\pi\)
0.0848680 + 0.996392i \(0.472953\pi\)
\(912\) 0 0
\(913\) 29.8617i 0.988279i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 32.4924i 1.07299i
\(918\) 0 0
\(919\) 2.73863 0.0903392 0.0451696 0.998979i \(-0.485617\pi\)
0.0451696 + 0.998979i \(0.485617\pi\)
\(920\) 0 0
\(921\) −32.9848 −1.08689
\(922\) 0 0
\(923\) 1.05398i 0.0346920i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 56.9848i 1.87163i
\(928\) 0 0
\(929\) −58.7235 −1.92665 −0.963327 0.268329i \(-0.913529\pi\)
−0.963327 + 0.268329i \(0.913529\pi\)
\(930\) 0 0
\(931\) −27.3693 −0.896993
\(932\) 0 0
\(933\) − 9.43845i − 0.309001i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 0.246211i − 0.00804337i −0.999992 0.00402169i \(-0.998720\pi\)
0.999992 0.00402169i \(-0.00128015\pi\)
\(938\) 0 0
\(939\) −50.7386 −1.65579
\(940\) 0 0
\(941\) −54.2462 −1.76838 −0.884188 0.467131i \(-0.845288\pi\)
−0.884188 + 0.467131i \(0.845288\pi\)
\(942\) 0 0
\(943\) − 4.12311i − 0.134267i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 44.3153i − 1.44006i −0.693945 0.720028i \(-0.744129\pi\)
0.693945 0.720028i \(-0.255871\pi\)
\(948\) 0 0
\(949\) 4.80776 0.156067
\(950\) 0 0
\(951\) 17.6155 0.571223
\(952\) 0 0
\(953\) − 5.50758i − 0.178408i −0.996013 0.0892040i \(-0.971568\pi\)
0.996013 0.0892040i \(-0.0284323\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 10.8769i 0.351600i
\(958\) 0 0
\(959\) 23.6155 0.762585
\(960\) 0 0
\(961\) 54.4924 1.75782
\(962\) 0 0
\(963\) 26.9309i 0.867835i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 44.3153i 1.42509i 0.701629 + 0.712543i \(0.252457\pi\)
−0.701629 + 0.712543i \(0.747543\pi\)
\(968\) 0 0
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) −47.3693 −1.52015 −0.760077 0.649833i \(-0.774839\pi\)
−0.760077 + 0.649833i \(0.774839\pi\)
\(972\) 0 0
\(973\) − 8.19224i − 0.262631i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 26.7926i 0.857172i 0.903501 + 0.428586i \(0.140988\pi\)
−0.903501 + 0.428586i \(0.859012\pi\)
\(978\) 0 0
\(979\) 4.49242 0.143578
\(980\) 0 0
\(981\) −33.3693 −1.06540
\(982\) 0 0
\(983\) 0.0539753i 0.00172155i 1.00000 0.000860773i \(0.000273992\pi\)
−1.00000 0.000860773i \(0.999726\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 30.7386i − 0.978421i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −0.300187 −0.00953574 −0.00476787 0.999989i \(-0.501518\pi\)
−0.00476787 + 0.999989i \(0.501518\pi\)
\(992\) 0 0
\(993\) − 32.3153i − 1.02550i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 14.3845i − 0.455561i −0.973713 0.227780i \(-0.926853\pi\)
0.973713 0.227780i \(-0.0731468\pi\)
\(998\) 0 0
\(999\) 0.630683 0.0199539
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 2300.2.c.h.1749.4 4
5.2 odd 4 460.2.a.e.1.2 2
5.3 odd 4 2300.2.a.i.1.1 2
5.4 even 2 inner 2300.2.c.h.1749.1 4
15.2 even 4 4140.2.a.m.1.1 2
20.3 even 4 9200.2.a.bv.1.2 2
20.7 even 4 1840.2.a.m.1.1 2
40.27 even 4 7360.2.a.bo.1.2 2
40.37 odd 4 7360.2.a.bi.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
460.2.a.e.1.2 2 5.2 odd 4
1840.2.a.m.1.1 2 20.7 even 4
2300.2.a.i.1.1 2 5.3 odd 4
2300.2.c.h.1749.1 4 5.4 even 2 inner
2300.2.c.h.1749.4 4 1.1 even 1 trivial
4140.2.a.m.1.1 2 15.2 even 4
7360.2.a.bi.1.1 2 40.37 odd 4
7360.2.a.bo.1.2 2 40.27 even 4
9200.2.a.bv.1.2 2 20.3 even 4