Properties

Label 1575.2.d.i.1324.4
Level $1575$
Weight $2$
Character 1575.1324
Analytic conductor $12.576$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1575.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.5764383184\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1324.4
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1575.1324
Dual form 1575.2.d.i.1324.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.73205i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.73205i q^{8} +O(q^{10})\) \(q+1.73205i q^{2} -1.00000 q^{4} +1.00000i q^{7} +1.73205i q^{8} +3.46410 q^{11} -2.00000i q^{13} -1.73205 q^{14} -5.00000 q^{16} +3.46410i q^{17} +4.00000 q^{19} +6.00000i q^{22} +3.46410i q^{23} +3.46410 q^{26} -1.00000i q^{28} -4.00000 q^{31} -5.19615i q^{32} -6.00000 q^{34} +2.00000i q^{37} +6.92820i q^{38} +10.3923 q^{41} +4.00000i q^{43} -3.46410 q^{44} -6.00000 q^{46} +6.92820i q^{47} -1.00000 q^{49} +2.00000i q^{52} +6.92820i q^{53} -1.73205 q^{56} +6.92820 q^{59} -10.0000 q^{61} -6.92820i q^{62} -1.00000 q^{64} -4.00000i q^{67} -3.46410i q^{68} -10.3923 q^{71} -14.0000i q^{73} -3.46410 q^{74} -4.00000 q^{76} +3.46410i q^{77} -8.00000 q^{79} +18.0000i q^{82} -6.92820 q^{86} +6.00000i q^{88} +3.46410 q^{89} +2.00000 q^{91} -3.46410i q^{92} -12.0000 q^{94} +14.0000i q^{97} -1.73205i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + O(q^{10}) \) \( 4 q - 4 q^{4} - 20 q^{16} + 16 q^{19} - 16 q^{31} - 24 q^{34} - 24 q^{46} - 4 q^{49} - 40 q^{61} - 4 q^{64} - 16 q^{76} - 32 q^{79} + 8 q^{91} - 48 q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1575\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(1226\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.73205i 1.22474i 0.790569 + 0.612372i \(0.209785\pi\)
−0.790569 + 0.612372i \(0.790215\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 1.73205i 0.612372i
\(9\) 0 0
\(10\) 0 0
\(11\) 3.46410 1.04447 0.522233 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) −1.73205 −0.462910
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 3.46410i 0.840168i 0.907485 + 0.420084i \(0.137999\pi\)
−0.907485 + 0.420084i \(0.862001\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 6.00000i 1.27920i
\(23\) 3.46410i 0.722315i 0.932505 + 0.361158i \(0.117618\pi\)
−0.932505 + 0.361158i \(0.882382\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 3.46410 0.679366
\(27\) 0 0
\(28\) − 1.00000i − 0.188982i
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) − 5.19615i − 0.918559i
\(33\) 0 0
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) 2.00000i 0.328798i 0.986394 + 0.164399i \(0.0525685\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 6.92820i 1.12390i
\(39\) 0 0
\(40\) 0 0
\(41\) 10.3923 1.62301 0.811503 0.584349i \(-0.198650\pi\)
0.811503 + 0.584349i \(0.198650\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) −3.46410 −0.522233
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) 6.92820i 1.01058i 0.862949 + 0.505291i \(0.168615\pi\)
−0.862949 + 0.505291i \(0.831385\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 2.00000i 0.277350i
\(53\) 6.92820i 0.951662i 0.879537 + 0.475831i \(0.157853\pi\)
−0.879537 + 0.475831i \(0.842147\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.73205 −0.231455
\(57\) 0 0
\(58\) 0 0
\(59\) 6.92820 0.901975 0.450988 0.892530i \(-0.351072\pi\)
0.450988 + 0.892530i \(0.351072\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) − 6.92820i − 0.879883i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 4.00000i − 0.488678i −0.969690 0.244339i \(-0.921429\pi\)
0.969690 0.244339i \(-0.0785709\pi\)
\(68\) − 3.46410i − 0.420084i
\(69\) 0 0
\(70\) 0 0
\(71\) −10.3923 −1.23334 −0.616670 0.787222i \(-0.711519\pi\)
−0.616670 + 0.787222i \(0.711519\pi\)
\(72\) 0 0
\(73\) − 14.0000i − 1.63858i −0.573382 0.819288i \(-0.694369\pi\)
0.573382 0.819288i \(-0.305631\pi\)
\(74\) −3.46410 −0.402694
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 3.46410i 0.394771i
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 18.0000i 1.98777i
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −6.92820 −0.747087
\(87\) 0 0
\(88\) 6.00000i 0.639602i
\(89\) 3.46410 0.367194 0.183597 0.983002i \(-0.441226\pi\)
0.183597 + 0.983002i \(0.441226\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) − 3.46410i − 0.361158i
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) 0 0
\(97\) 14.0000i 1.42148i 0.703452 + 0.710742i \(0.251641\pi\)
−0.703452 + 0.710742i \(0.748359\pi\)
\(98\) − 1.73205i − 0.174964i
\(99\) 0 0
\(100\) 0 0
\(101\) 3.46410 0.344691 0.172345 0.985037i \(-0.444865\pi\)
0.172345 + 0.985037i \(0.444865\pi\)
\(102\) 0 0
\(103\) 4.00000i 0.394132i 0.980390 + 0.197066i \(0.0631413\pi\)
−0.980390 + 0.197066i \(0.936859\pi\)
\(104\) 3.46410 0.339683
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) 17.3205i 1.67444i 0.546869 + 0.837218i \(0.315820\pi\)
−0.546869 + 0.837218i \(0.684180\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 5.00000i − 0.472456i
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 12.0000i 1.10469i
\(119\) −3.46410 −0.317554
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 17.3205i − 1.56813i
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) 8.00000i 0.709885i 0.934888 + 0.354943i \(0.115500\pi\)
−0.934888 + 0.354943i \(0.884500\pi\)
\(128\) − 12.1244i − 1.07165i
\(129\) 0 0
\(130\) 0 0
\(131\) −13.8564 −1.21064 −0.605320 0.795982i \(-0.706955\pi\)
−0.605320 + 0.795982i \(0.706955\pi\)
\(132\) 0 0
\(133\) 4.00000i 0.346844i
\(134\) 6.92820 0.598506
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) − 6.92820i − 0.591916i −0.955201 0.295958i \(-0.904361\pi\)
0.955201 0.295958i \(-0.0956389\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 18.0000i − 1.51053i
\(143\) − 6.92820i − 0.579365i
\(144\) 0 0
\(145\) 0 0
\(146\) 24.2487 2.00684
\(147\) 0 0
\(148\) − 2.00000i − 0.164399i
\(149\) −6.92820 −0.567581 −0.283790 0.958886i \(-0.591592\pi\)
−0.283790 + 0.958886i \(0.591592\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 6.92820i 0.561951i
\(153\) 0 0
\(154\) −6.00000 −0.483494
\(155\) 0 0
\(156\) 0 0
\(157\) − 10.0000i − 0.798087i −0.916932 0.399043i \(-0.869342\pi\)
0.916932 0.399043i \(-0.130658\pi\)
\(158\) − 13.8564i − 1.10236i
\(159\) 0 0
\(160\) 0 0
\(161\) −3.46410 −0.273009
\(162\) 0 0
\(163\) − 20.0000i − 1.56652i −0.621694 0.783260i \(-0.713555\pi\)
0.621694 0.783260i \(-0.286445\pi\)
\(164\) −10.3923 −0.811503
\(165\) 0 0
\(166\) 0 0
\(167\) − 20.7846i − 1.60836i −0.594385 0.804181i \(-0.702604\pi\)
0.594385 0.804181i \(-0.297396\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) − 4.00000i − 0.304997i
\(173\) − 17.3205i − 1.31685i −0.752645 0.658427i \(-0.771222\pi\)
0.752645 0.658427i \(-0.228778\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −17.3205 −1.30558
\(177\) 0 0
\(178\) 6.00000i 0.449719i
\(179\) 17.3205 1.29460 0.647298 0.762237i \(-0.275899\pi\)
0.647298 + 0.762237i \(0.275899\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 3.46410i 0.256776i
\(183\) 0 0
\(184\) −6.00000 −0.442326
\(185\) 0 0
\(186\) 0 0
\(187\) 12.0000i 0.877527i
\(188\) − 6.92820i − 0.505291i
\(189\) 0 0
\(190\) 0 0
\(191\) −24.2487 −1.75458 −0.877288 0.479965i \(-0.840649\pi\)
−0.877288 + 0.479965i \(0.840649\pi\)
\(192\) 0 0
\(193\) − 14.0000i − 1.00774i −0.863779 0.503871i \(-0.831909\pi\)
0.863779 0.503871i \(-0.168091\pi\)
\(194\) −24.2487 −1.74096
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 20.7846i 1.48084i 0.672143 + 0.740421i \(0.265374\pi\)
−0.672143 + 0.740421i \(0.734626\pi\)
\(198\) 0 0
\(199\) 16.0000 1.13421 0.567105 0.823646i \(-0.308063\pi\)
0.567105 + 0.823646i \(0.308063\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 6.00000i 0.422159i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) −6.92820 −0.482711
\(207\) 0 0
\(208\) 10.0000i 0.693375i
\(209\) 13.8564 0.958468
\(210\) 0 0
\(211\) 20.0000 1.37686 0.688428 0.725304i \(-0.258301\pi\)
0.688428 + 0.725304i \(0.258301\pi\)
\(212\) − 6.92820i − 0.475831i
\(213\) 0 0
\(214\) −30.0000 −2.05076
\(215\) 0 0
\(216\) 0 0
\(217\) − 4.00000i − 0.271538i
\(218\) − 3.46410i − 0.234619i
\(219\) 0 0
\(220\) 0 0
\(221\) 6.92820 0.466041
\(222\) 0 0
\(223\) − 8.00000i − 0.535720i −0.963458 0.267860i \(-0.913684\pi\)
0.963458 0.267860i \(-0.0863164\pi\)
\(224\) 5.19615 0.347183
\(225\) 0 0
\(226\) 0 0
\(227\) − 6.92820i − 0.459841i −0.973209 0.229920i \(-0.926153\pi\)
0.973209 0.229920i \(-0.0738466\pi\)
\(228\) 0 0
\(229\) 22.0000 1.45380 0.726900 0.686743i \(-0.240960\pi\)
0.726900 + 0.686743i \(0.240960\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 6.92820i − 0.453882i −0.973909 0.226941i \(-0.927128\pi\)
0.973909 0.226941i \(-0.0728724\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −6.92820 −0.450988
\(237\) 0 0
\(238\) − 6.00000i − 0.388922i
\(239\) −10.3923 −0.672222 −0.336111 0.941822i \(-0.609112\pi\)
−0.336111 + 0.941822i \(0.609112\pi\)
\(240\) 0 0
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 1.73205i 0.111340i
\(243\) 0 0
\(244\) 10.0000 0.640184
\(245\) 0 0
\(246\) 0 0
\(247\) − 8.00000i − 0.509028i
\(248\) − 6.92820i − 0.439941i
\(249\) 0 0
\(250\) 0 0
\(251\) 20.7846 1.31191 0.655956 0.754799i \(-0.272265\pi\)
0.655956 + 0.754799i \(0.272265\pi\)
\(252\) 0 0
\(253\) 12.0000i 0.754434i
\(254\) −13.8564 −0.869428
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) − 3.46410i − 0.216085i −0.994146 0.108042i \(-0.965542\pi\)
0.994146 0.108042i \(-0.0344582\pi\)
\(258\) 0 0
\(259\) −2.00000 −0.124274
\(260\) 0 0
\(261\) 0 0
\(262\) − 24.0000i − 1.48272i
\(263\) 17.3205i 1.06803i 0.845476 + 0.534014i \(0.179317\pi\)
−0.845476 + 0.534014i \(0.820683\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −6.92820 −0.424795
\(267\) 0 0
\(268\) 4.00000i 0.244339i
\(269\) 17.3205 1.05605 0.528025 0.849229i \(-0.322933\pi\)
0.528025 + 0.849229i \(0.322933\pi\)
\(270\) 0 0
\(271\) 20.0000 1.21491 0.607457 0.794353i \(-0.292190\pi\)
0.607457 + 0.794353i \(0.292190\pi\)
\(272\) − 17.3205i − 1.05021i
\(273\) 0 0
\(274\) 12.0000 0.724947
\(275\) 0 0
\(276\) 0 0
\(277\) − 10.0000i − 0.600842i −0.953807 0.300421i \(-0.902873\pi\)
0.953807 0.300421i \(-0.0971271\pi\)
\(278\) 27.7128i 1.66210i
\(279\) 0 0
\(280\) 0 0
\(281\) −20.7846 −1.23991 −0.619953 0.784639i \(-0.712848\pi\)
−0.619953 + 0.784639i \(0.712848\pi\)
\(282\) 0 0
\(283\) 4.00000i 0.237775i 0.992908 + 0.118888i \(0.0379328\pi\)
−0.992908 + 0.118888i \(0.962067\pi\)
\(284\) 10.3923 0.616670
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) 10.3923i 0.613438i
\(288\) 0 0
\(289\) 5.00000 0.294118
\(290\) 0 0
\(291\) 0 0
\(292\) 14.0000i 0.819288i
\(293\) 10.3923i 0.607125i 0.952812 + 0.303562i \(0.0981761\pi\)
−0.952812 + 0.303562i \(0.901824\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −3.46410 −0.201347
\(297\) 0 0
\(298\) − 12.0000i − 0.695141i
\(299\) 6.92820 0.400668
\(300\) 0 0
\(301\) −4.00000 −0.230556
\(302\) 13.8564i 0.797347i
\(303\) 0 0
\(304\) −20.0000 −1.14708
\(305\) 0 0
\(306\) 0 0
\(307\) − 28.0000i − 1.59804i −0.601302 0.799022i \(-0.705351\pi\)
0.601302 0.799022i \(-0.294649\pi\)
\(308\) − 3.46410i − 0.197386i
\(309\) 0 0
\(310\) 0 0
\(311\) 34.6410 1.96431 0.982156 0.188069i \(-0.0602227\pi\)
0.982156 + 0.188069i \(0.0602227\pi\)
\(312\) 0 0
\(313\) − 2.00000i − 0.113047i −0.998401 0.0565233i \(-0.981998\pi\)
0.998401 0.0565233i \(-0.0180015\pi\)
\(314\) 17.3205 0.977453
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) 6.92820i 0.389127i 0.980890 + 0.194563i \(0.0623290\pi\)
−0.980890 + 0.194563i \(0.937671\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) − 6.00000i − 0.334367i
\(323\) 13.8564i 0.770991i
\(324\) 0 0
\(325\) 0 0
\(326\) 34.6410 1.91859
\(327\) 0 0
\(328\) 18.0000i 0.993884i
\(329\) −6.92820 −0.381964
\(330\) 0 0
\(331\) 20.0000 1.09930 0.549650 0.835395i \(-0.314761\pi\)
0.549650 + 0.835395i \(0.314761\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 36.0000 1.96983
\(335\) 0 0
\(336\) 0 0
\(337\) 14.0000i 0.762629i 0.924445 + 0.381314i \(0.124528\pi\)
−0.924445 + 0.381314i \(0.875472\pi\)
\(338\) 15.5885i 0.847900i
\(339\) 0 0
\(340\) 0 0
\(341\) −13.8564 −0.750366
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) −6.92820 −0.373544
\(345\) 0 0
\(346\) 30.0000 1.61281
\(347\) − 17.3205i − 0.929814i −0.885360 0.464907i \(-0.846088\pi\)
0.885360 0.464907i \(-0.153912\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 18.0000i − 0.959403i
\(353\) − 3.46410i − 0.184376i −0.995742 0.0921878i \(-0.970614\pi\)
0.995742 0.0921878i \(-0.0293860\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −3.46410 −0.183597
\(357\) 0 0
\(358\) 30.0000i 1.58555i
\(359\) 24.2487 1.27980 0.639899 0.768459i \(-0.278976\pi\)
0.639899 + 0.768459i \(0.278976\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 3.46410i 0.182069i
\(363\) 0 0
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) 0 0
\(367\) − 16.0000i − 0.835193i −0.908633 0.417597i \(-0.862873\pi\)
0.908633 0.417597i \(-0.137127\pi\)
\(368\) − 17.3205i − 0.902894i
\(369\) 0 0
\(370\) 0 0
\(371\) −6.92820 −0.359694
\(372\) 0 0
\(373\) 10.0000i 0.517780i 0.965907 + 0.258890i \(0.0833568\pi\)
−0.965907 + 0.258890i \(0.916643\pi\)
\(374\) −20.7846 −1.07475
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 0 0
\(378\) 0 0
\(379\) 28.0000 1.43826 0.719132 0.694874i \(-0.244540\pi\)
0.719132 + 0.694874i \(0.244540\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 42.0000i − 2.14891i
\(383\) 13.8564i 0.708029i 0.935240 + 0.354015i \(0.115184\pi\)
−0.935240 + 0.354015i \(0.884816\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 24.2487 1.23423
\(387\) 0 0
\(388\) − 14.0000i − 0.710742i
\(389\) −13.8564 −0.702548 −0.351274 0.936273i \(-0.614251\pi\)
−0.351274 + 0.936273i \(0.614251\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) − 1.73205i − 0.0874818i
\(393\) 0 0
\(394\) −36.0000 −1.81365
\(395\) 0 0
\(396\) 0 0
\(397\) 38.0000i 1.90717i 0.301131 + 0.953583i \(0.402636\pi\)
−0.301131 + 0.953583i \(0.597364\pi\)
\(398\) 27.7128i 1.38912i
\(399\) 0 0
\(400\) 0 0
\(401\) 6.92820 0.345978 0.172989 0.984924i \(-0.444657\pi\)
0.172989 + 0.984924i \(0.444657\pi\)
\(402\) 0 0
\(403\) 8.00000i 0.398508i
\(404\) −3.46410 −0.172345
\(405\) 0 0
\(406\) 0 0
\(407\) 6.92820i 0.343418i
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 4.00000i − 0.197066i
\(413\) 6.92820i 0.340915i
\(414\) 0 0
\(415\) 0 0
\(416\) −10.3923 −0.509525
\(417\) 0 0
\(418\) 24.0000i 1.17388i
\(419\) −20.7846 −1.01539 −0.507697 0.861536i \(-0.669503\pi\)
−0.507697 + 0.861536i \(0.669503\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 34.6410i 1.68630i
\(423\) 0 0
\(424\) −12.0000 −0.582772
\(425\) 0 0
\(426\) 0 0
\(427\) − 10.0000i − 0.483934i
\(428\) − 17.3205i − 0.837218i
\(429\) 0 0
\(430\) 0 0
\(431\) 3.46410 0.166860 0.0834300 0.996514i \(-0.473413\pi\)
0.0834300 + 0.996514i \(0.473413\pi\)
\(432\) 0 0
\(433\) − 26.0000i − 1.24948i −0.780833 0.624740i \(-0.785205\pi\)
0.780833 0.624740i \(-0.214795\pi\)
\(434\) 6.92820 0.332564
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 13.8564i 0.662842i
\(438\) 0 0
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 12.0000i 0.570782i
\(443\) 3.46410i 0.164584i 0.996608 + 0.0822922i \(0.0262241\pi\)
−0.996608 + 0.0822922i \(0.973776\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 13.8564 0.656120
\(447\) 0 0
\(448\) − 1.00000i − 0.0472456i
\(449\) −41.5692 −1.96177 −0.980886 0.194581i \(-0.937665\pi\)
−0.980886 + 0.194581i \(0.937665\pi\)
\(450\) 0 0
\(451\) 36.0000 1.69517
\(452\) 0 0
\(453\) 0 0
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) 0 0
\(457\) − 10.0000i − 0.467780i −0.972263 0.233890i \(-0.924854\pi\)
0.972263 0.233890i \(-0.0751456\pi\)
\(458\) 38.1051i 1.78054i
\(459\) 0 0
\(460\) 0 0
\(461\) 31.1769 1.45205 0.726027 0.687666i \(-0.241365\pi\)
0.726027 + 0.687666i \(0.241365\pi\)
\(462\) 0 0
\(463\) − 32.0000i − 1.48717i −0.668644 0.743583i \(-0.733125\pi\)
0.668644 0.743583i \(-0.266875\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 12.0000 0.555889
\(467\) 6.92820i 0.320599i 0.987068 + 0.160300i \(0.0512460\pi\)
−0.987068 + 0.160300i \(0.948754\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) 12.0000i 0.552345i
\(473\) 13.8564i 0.637118i
\(474\) 0 0
\(475\) 0 0
\(476\) 3.46410 0.158777
\(477\) 0 0
\(478\) − 18.0000i − 0.823301i
\(479\) 6.92820 0.316558 0.158279 0.987394i \(-0.449406\pi\)
0.158279 + 0.987394i \(0.449406\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) − 17.3205i − 0.788928i
\(483\) 0 0
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) 0 0
\(487\) − 40.0000i − 1.81257i −0.422664 0.906287i \(-0.638905\pi\)
0.422664 0.906287i \(-0.361095\pi\)
\(488\) − 17.3205i − 0.784063i
\(489\) 0 0
\(490\) 0 0
\(491\) 10.3923 0.468998 0.234499 0.972116i \(-0.424655\pi\)
0.234499 + 0.972116i \(0.424655\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 13.8564 0.623429
\(495\) 0 0
\(496\) 20.0000 0.898027
\(497\) − 10.3923i − 0.466159i
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 36.0000i 1.60676i
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −20.7846 −0.923989
\(507\) 0 0
\(508\) − 8.00000i − 0.354943i
\(509\) 3.46410 0.153544 0.0767718 0.997049i \(-0.475539\pi\)
0.0767718 + 0.997049i \(0.475539\pi\)
\(510\) 0 0
\(511\) 14.0000 0.619324
\(512\) 8.66025i 0.382733i
\(513\) 0 0
\(514\) 6.00000 0.264649
\(515\) 0 0
\(516\) 0 0
\(517\) 24.0000i 1.05552i
\(518\) − 3.46410i − 0.152204i
\(519\) 0 0
\(520\) 0 0
\(521\) 3.46410 0.151765 0.0758825 0.997117i \(-0.475823\pi\)
0.0758825 + 0.997117i \(0.475823\pi\)
\(522\) 0 0
\(523\) 16.0000i 0.699631i 0.936819 + 0.349816i \(0.113756\pi\)
−0.936819 + 0.349816i \(0.886244\pi\)
\(524\) 13.8564 0.605320
\(525\) 0 0
\(526\) −30.0000 −1.30806
\(527\) − 13.8564i − 0.603595i
\(528\) 0 0
\(529\) 11.0000 0.478261
\(530\) 0 0
\(531\) 0 0
\(532\) − 4.00000i − 0.173422i
\(533\) − 20.7846i − 0.900281i
\(534\) 0 0
\(535\) 0 0
\(536\) 6.92820 0.299253
\(537\) 0 0
\(538\) 30.0000i 1.29339i
\(539\) −3.46410 −0.149209
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 34.6410i 1.48796i
\(543\) 0 0
\(544\) 18.0000 0.771744
\(545\) 0 0
\(546\) 0 0
\(547\) − 4.00000i − 0.171028i −0.996337 0.0855138i \(-0.972747\pi\)
0.996337 0.0855138i \(-0.0272532\pi\)
\(548\) 6.92820i 0.295958i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) − 8.00000i − 0.340195i
\(554\) 17.3205 0.735878
\(555\) 0 0
\(556\) −16.0000 −0.678551
\(557\) − 6.92820i − 0.293557i −0.989169 0.146779i \(-0.953109\pi\)
0.989169 0.146779i \(-0.0468905\pi\)
\(558\) 0 0
\(559\) 8.00000 0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) − 36.0000i − 1.51857i
\(563\) − 34.6410i − 1.45994i −0.683477 0.729972i \(-0.739533\pi\)
0.683477 0.729972i \(-0.260467\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −6.92820 −0.291214
\(567\) 0 0
\(568\) − 18.0000i − 0.755263i
\(569\) −6.92820 −0.290445 −0.145223 0.989399i \(-0.546390\pi\)
−0.145223 + 0.989399i \(0.546390\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 6.92820i 0.289683i
\(573\) 0 0
\(574\) −18.0000 −0.751305
\(575\) 0 0
\(576\) 0 0
\(577\) 2.00000i 0.0832611i 0.999133 + 0.0416305i \(0.0132552\pi\)
−0.999133 + 0.0416305i \(0.986745\pi\)
\(578\) 8.66025i 0.360219i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 24.0000i 0.993978i
\(584\) 24.2487 1.00342
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) − 20.7846i − 0.857873i −0.903335 0.428936i \(-0.858888\pi\)
0.903335 0.428936i \(-0.141112\pi\)
\(588\) 0 0
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) 0 0
\(592\) − 10.0000i − 0.410997i
\(593\) 24.2487i 0.995775i 0.867242 + 0.497888i \(0.165891\pi\)
−0.867242 + 0.497888i \(0.834109\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.92820 0.283790
\(597\) 0 0
\(598\) 12.0000i 0.490716i
\(599\) −45.0333 −1.84001 −0.920006 0.391905i \(-0.871816\pi\)
−0.920006 + 0.391905i \(0.871816\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) − 6.92820i − 0.282372i
\(603\) 0 0
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) 0 0
\(607\) 8.00000i 0.324710i 0.986732 + 0.162355i \(0.0519090\pi\)
−0.986732 + 0.162355i \(0.948091\pi\)
\(608\) − 20.7846i − 0.842927i
\(609\) 0 0
\(610\) 0 0
\(611\) 13.8564 0.560570
\(612\) 0 0
\(613\) − 38.0000i − 1.53481i −0.641165 0.767403i \(-0.721549\pi\)
0.641165 0.767403i \(-0.278451\pi\)
\(614\) 48.4974 1.95720
\(615\) 0 0
\(616\) −6.00000 −0.241747
\(617\) − 20.7846i − 0.836757i −0.908273 0.418378i \(-0.862599\pi\)
0.908273 0.418378i \(-0.137401\pi\)
\(618\) 0 0
\(619\) −8.00000 −0.321547 −0.160774 0.986991i \(-0.551399\pi\)
−0.160774 + 0.986991i \(0.551399\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 60.0000i 2.40578i
\(623\) 3.46410i 0.138786i
\(624\) 0 0
\(625\) 0 0
\(626\) 3.46410 0.138453
\(627\) 0 0
\(628\) 10.0000i 0.399043i
\(629\) −6.92820 −0.276246
\(630\) 0 0
\(631\) 8.00000 0.318475 0.159237 0.987240i \(-0.449096\pi\)
0.159237 + 0.987240i \(0.449096\pi\)
\(632\) − 13.8564i − 0.551178i
\(633\) 0 0
\(634\) −12.0000 −0.476581
\(635\) 0 0
\(636\) 0 0
\(637\) 2.00000i 0.0792429i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −48.4974 −1.91553 −0.957767 0.287547i \(-0.907160\pi\)
−0.957767 + 0.287547i \(0.907160\pi\)
\(642\) 0 0
\(643\) − 20.0000i − 0.788723i −0.918955 0.394362i \(-0.870966\pi\)
0.918955 0.394362i \(-0.129034\pi\)
\(644\) 3.46410 0.136505
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) − 6.92820i − 0.272376i −0.990683 0.136188i \(-0.956515\pi\)
0.990683 0.136188i \(-0.0434851\pi\)
\(648\) 0 0
\(649\) 24.0000 0.942082
\(650\) 0 0
\(651\) 0 0
\(652\) 20.0000i 0.783260i
\(653\) − 27.7128i − 1.08449i −0.840222 0.542243i \(-0.817575\pi\)
0.840222 0.542243i \(-0.182425\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −51.9615 −2.02876
\(657\) 0 0
\(658\) − 12.0000i − 0.467809i
\(659\) 10.3923 0.404827 0.202413 0.979300i \(-0.435122\pi\)
0.202413 + 0.979300i \(0.435122\pi\)
\(660\) 0 0
\(661\) −10.0000 −0.388955 −0.194477 0.980907i \(-0.562301\pi\)
−0.194477 + 0.980907i \(0.562301\pi\)
\(662\) 34.6410i 1.34636i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 20.7846i 0.804181i
\(669\) 0 0
\(670\) 0 0
\(671\) −34.6410 −1.33730
\(672\) 0 0
\(673\) 10.0000i 0.385472i 0.981251 + 0.192736i \(0.0617360\pi\)
−0.981251 + 0.192736i \(0.938264\pi\)
\(674\) −24.2487 −0.934025
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) − 24.2487i − 0.931954i −0.884797 0.465977i \(-0.845703\pi\)
0.884797 0.465977i \(-0.154297\pi\)
\(678\) 0 0
\(679\) −14.0000 −0.537271
\(680\) 0 0
\(681\) 0 0
\(682\) − 24.0000i − 0.919007i
\(683\) − 24.2487i − 0.927851i −0.885874 0.463926i \(-0.846441\pi\)
0.885874 0.463926i \(-0.153559\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 1.73205 0.0661300
\(687\) 0 0
\(688\) − 20.0000i − 0.762493i
\(689\) 13.8564 0.527887
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 17.3205i 0.658427i
\(693\) 0 0
\(694\) 30.0000 1.13878
\(695\) 0 0
\(696\) 0 0
\(697\) 36.0000i 1.36360i
\(698\) − 24.2487i − 0.917827i
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 8.00000i 0.301726i
\(704\) −3.46410 −0.130558
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 3.46410i 0.130281i
\(708\) 0 0
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 6.00000i 0.224860i
\(713\) − 13.8564i − 0.518927i
\(714\) 0 0
\(715\) 0 0
\(716\) −17.3205 −0.647298
\(717\) 0 0
\(718\) 42.0000i 1.56743i
\(719\) −27.7128 −1.03351 −0.516757 0.856132i \(-0.672861\pi\)
−0.516757 + 0.856132i \(0.672861\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) − 5.19615i − 0.193381i
\(723\) 0 0
\(724\) −2.00000 −0.0743294
\(725\) 0 0
\(726\) 0 0
\(727\) − 4.00000i − 0.148352i −0.997245 0.0741759i \(-0.976367\pi\)
0.997245 0.0741759i \(-0.0236326\pi\)
\(728\) 3.46410i 0.128388i
\(729\) 0 0
\(730\) 0 0
\(731\) −13.8564 −0.512498
\(732\) 0 0
\(733\) 22.0000i 0.812589i 0.913742 + 0.406294i \(0.133179\pi\)
−0.913742 + 0.406294i \(0.866821\pi\)
\(734\) 27.7128 1.02290
\(735\) 0 0
\(736\) 18.0000 0.663489
\(737\) − 13.8564i − 0.510407i
\(738\) 0 0
\(739\) −20.0000 −0.735712 −0.367856 0.929883i \(-0.619908\pi\)
−0.367856 + 0.929883i \(0.619908\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 12.0000i − 0.440534i
\(743\) 10.3923i 0.381257i 0.981662 + 0.190628i \(0.0610525\pi\)
−0.981662 + 0.190628i \(0.938947\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −17.3205 −0.634149
\(747\) 0 0
\(748\) − 12.0000i − 0.438763i
\(749\) −17.3205 −0.632878
\(750\) 0 0
\(751\) 8.00000 0.291924 0.145962 0.989290i \(-0.453372\pi\)
0.145962 + 0.989290i \(0.453372\pi\)
\(752\) − 34.6410i − 1.26323i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 26.0000i 0.944986i 0.881334 + 0.472493i \(0.156646\pi\)
−0.881334 + 0.472493i \(0.843354\pi\)
\(758\) 48.4974i 1.76151i
\(759\) 0 0
\(760\) 0 0
\(761\) 38.1051 1.38131 0.690655 0.723185i \(-0.257322\pi\)
0.690655 + 0.723185i \(0.257322\pi\)
\(762\) 0 0
\(763\) − 2.00000i − 0.0724049i
\(764\) 24.2487 0.877288
\(765\) 0 0
\(766\) −24.0000 −0.867155
\(767\) − 13.8564i − 0.500326i
\(768\) 0 0
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 14.0000i 0.503871i
\(773\) − 45.0333i − 1.61974i −0.586612 0.809868i \(-0.699539\pi\)
0.586612 0.809868i \(-0.300461\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −24.2487 −0.870478
\(777\) 0 0
\(778\) − 24.0000i − 0.860442i
\(779\) 41.5692 1.48937
\(780\) 0 0
\(781\) −36.0000 −1.28818
\(782\) − 20.7846i − 0.743256i
\(783\) 0 0
\(784\) 5.00000 0.178571
\(785\) 0 0
\(786\) 0 0
\(787\) 32.0000i 1.14068i 0.821410 + 0.570338i \(0.193188\pi\)
−0.821410 + 0.570338i \(0.806812\pi\)
\(788\) − 20.7846i − 0.740421i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 20.0000i 0.710221i
\(794\) −65.8179 −2.33579
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 31.1769i 1.10434i 0.833731 + 0.552171i \(0.186201\pi\)
−0.833731 + 0.552171i \(0.813799\pi\)
\(798\) 0 0
\(799\) −24.0000 −0.849059
\(800\) 0 0
\(801\) 0 0
\(802\) 12.0000i 0.423735i
\(803\) − 48.4974i − 1.71144i
\(804\) 0 0
\(805\) 0 0
\(806\) −13.8564 −0.488071
\(807\) 0 0
\(808\) 6.00000i 0.211079i
\(809\) 27.7128 0.974331 0.487165 0.873310i \(-0.338031\pi\)
0.487165 + 0.873310i \(0.338031\pi\)
\(810\) 0 0
\(811\) −16.0000 −0.561836 −0.280918 0.959732i \(-0.590639\pi\)
−0.280918 + 0.959732i \(0.590639\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −12.0000 −0.420600
\(815\) 0 0
\(816\) 0 0
\(817\) 16.0000i 0.559769i
\(818\) − 24.2487i − 0.847836i
\(819\) 0 0
\(820\) 0 0
\(821\) 6.92820 0.241796 0.120898 0.992665i \(-0.461423\pi\)
0.120898 + 0.992665i \(0.461423\pi\)
\(822\) 0 0
\(823\) − 32.0000i − 1.11545i −0.830026 0.557725i \(-0.811674\pi\)
0.830026 0.557725i \(-0.188326\pi\)
\(824\) −6.92820 −0.241355
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) − 10.3923i − 0.361376i −0.983540 0.180688i \(-0.942168\pi\)
0.983540 0.180688i \(-0.0578324\pi\)
\(828\) 0 0
\(829\) −50.0000 −1.73657 −0.868286 0.496064i \(-0.834778\pi\)
−0.868286 + 0.496064i \(0.834778\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 2.00000i 0.0693375i
\(833\) − 3.46410i − 0.120024i
\(834\) 0 0
\(835\) 0 0
\(836\) −13.8564 −0.479234
\(837\) 0 0
\(838\) − 36.0000i − 1.24360i
\(839\) −20.7846 −0.717564 −0.358782 0.933421i \(-0.616808\pi\)
−0.358782 + 0.933421i \(0.616808\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) − 17.3205i − 0.596904i
\(843\) 0 0
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) 0 0
\(847\) 1.00000i 0.0343604i
\(848\) − 34.6410i − 1.18958i
\(849\) 0 0
\(850\) 0 0
\(851\) −6.92820 −0.237496
\(852\) 0 0
\(853\) 10.0000i 0.342393i 0.985237 + 0.171197i \(0.0547634\pi\)
−0.985237 + 0.171197i \(0.945237\pi\)
\(854\) 17.3205 0.592696
\(855\) 0 0
\(856\) −30.0000 −1.02538
\(857\) − 17.3205i − 0.591657i −0.955241 0.295829i \(-0.904404\pi\)
0.955241 0.295829i \(-0.0955957\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 6.00000i 0.204361i
\(863\) − 38.1051i − 1.29711i −0.761166 0.648557i \(-0.775373\pi\)
0.761166 0.648557i \(-0.224627\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 45.0333 1.53029
\(867\) 0 0
\(868\) 4.00000i 0.135769i
\(869\) −27.7128 −0.940093
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) − 3.46410i − 0.117309i
\(873\) 0 0
\(874\) −24.0000 −0.811812
\(875\) 0 0
\(876\) 0 0
\(877\) 14.0000i 0.472746i 0.971662 + 0.236373i \(0.0759588\pi\)
−0.971662 + 0.236373i \(0.924041\pi\)
\(878\) 27.7128i 0.935262i
\(879\) 0 0
\(880\) 0 0
\(881\) −51.9615 −1.75063 −0.875314 0.483555i \(-0.839345\pi\)
−0.875314 + 0.483555i \(0.839345\pi\)
\(882\) 0 0
\(883\) 52.0000i 1.74994i 0.484178 + 0.874970i \(0.339119\pi\)
−0.484178 + 0.874970i \(0.660881\pi\)
\(884\) −6.92820 −0.233021
\(885\) 0 0
\(886\) −6.00000 −0.201574
\(887\) 6.92820i 0.232626i 0.993213 + 0.116313i \(0.0371076\pi\)
−0.993213 + 0.116313i \(0.962892\pi\)
\(888\) 0 0
\(889\) −8.00000 −0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) 8.00000i 0.267860i
\(893\) 27.7128i 0.927374i
\(894\) 0 0
\(895\) 0 0
\(896\) 12.1244 0.405046
\(897\) 0 0
\(898\) − 72.0000i − 2.40267i
\(899\) 0 0
\(900\) 0 0
\(901\) −24.0000 −0.799556
\(902\) 62.3538i 2.07616i
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 28.0000i − 0.929725i −0.885383 0.464862i \(-0.846104\pi\)
0.885383 0.464862i \(-0.153896\pi\)
\(908\) 6.92820i 0.229920i
\(909\) 0 0
\(910\) 0 0
\(911\) 31.1769 1.03294 0.516469 0.856306i \(-0.327246\pi\)
0.516469 + 0.856306i \(0.327246\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 17.3205 0.572911
\(915\) 0 0
\(916\) −22.0000 −0.726900
\(917\) − 13.8564i − 0.457579i
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 54.0000i 1.77840i
\(923\) 20.7846i 0.684134i
\(924\) 0 0
\(925\) 0 0
\(926\) 55.4256 1.82140
\(927\) 0 0
\(928\) 0 0
\(929\) 45.0333 1.47750 0.738748 0.673982i \(-0.235418\pi\)
0.738748 + 0.673982i \(0.235418\pi\)
\(930\) 0 0
\(931\) −4.00000 −0.131095
\(932\) 6.92820i 0.226941i
\(933\) 0 0
\(934\) −12.0000 −0.392652
\(935\) 0 0
\(936\) 0 0
\(937\) − 46.0000i − 1.50275i −0.659873 0.751377i \(-0.729390\pi\)
0.659873 0.751377i \(-0.270610\pi\)
\(938\) 6.92820i 0.226214i
\(939\) 0 0
\(940\) 0 0
\(941\) −17.3205 −0.564632 −0.282316 0.959321i \(-0.591103\pi\)
−0.282316 + 0.959321i \(0.591103\pi\)
\(942\) 0 0
\(943\) 36.0000i 1.17232i
\(944\) −34.6410 −1.12747
\(945\) 0 0
\(946\) −24.0000 −0.780307
\(947\) − 24.2487i − 0.787977i −0.919115 0.393989i \(-0.871095\pi\)
0.919115 0.393989i \(-0.128905\pi\)
\(948\) 0 0
\(949\) −28.0000 −0.908918
\(950\) 0 0
\(951\) 0 0
\(952\) − 6.00000i − 0.194461i
\(953\) 41.5692i 1.34656i 0.739388 + 0.673280i \(0.235115\pi\)
−0.739388 + 0.673280i \(0.764885\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 10.3923 0.336111
\(957\) 0 0
\(958\) 12.0000i 0.387702i
\(959\) 6.92820 0.223723
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 6.92820i 0.223374i
\(963\) 0 0
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) − 16.0000i − 0.514525i −0.966342 0.257263i \(-0.917179\pi\)
0.966342 0.257263i \(-0.0828206\pi\)
\(968\) 1.73205i 0.0556702i
\(969\) 0 0
\(970\) 0 0
\(971\) 27.7128 0.889346 0.444673 0.895693i \(-0.353320\pi\)
0.444673 + 0.895693i \(0.353320\pi\)
\(972\) 0 0
\(973\) 16.0000i 0.512936i
\(974\) 69.2820 2.21994
\(975\) 0 0
\(976\) 50.0000 1.60046
\(977\) 34.6410i 1.10826i 0.832429 + 0.554132i \(0.186950\pi\)
−0.832429 + 0.554132i \(0.813050\pi\)
\(978\) 0 0
\(979\) 12.0000 0.383522
\(980\) 0 0
\(981\) 0 0
\(982\) 18.0000i 0.574403i
\(983\) − 13.8564i − 0.441951i −0.975279 0.220975i \(-0.929076\pi\)
0.975279 0.220975i \(-0.0709240\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 8.00000i 0.254514i
\(989\) −13.8564 −0.440608
\(990\) 0 0
\(991\) 56.0000 1.77890 0.889449 0.457034i \(-0.151088\pi\)
0.889449 + 0.457034i \(0.151088\pi\)
\(992\) 20.7846i 0.659912i
\(993\) 0 0
\(994\) 18.0000 0.570925
\(995\) 0 0
\(996\) 0 0
\(997\) − 10.0000i − 0.316703i −0.987383 0.158352i \(-0.949382\pi\)
0.987383 0.158352i \(-0.0506179\pi\)
\(998\) 6.92820i 0.219308i
\(999\) 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 1575.2.d.i.1324.4 4
3.2 odd 2 inner 1575.2.d.i.1324.2 4
5.2 odd 4 1575.2.a.q.1.1 2
5.3 odd 4 63.2.a.b.1.2 yes 2
5.4 even 2 inner 1575.2.d.i.1324.1 4
15.2 even 4 1575.2.a.q.1.2 2
15.8 even 4 63.2.a.b.1.1 2
15.14 odd 2 inner 1575.2.d.i.1324.3 4
20.3 even 4 1008.2.a.n.1.1 2
35.3 even 12 441.2.e.i.226.1 4
35.13 even 4 441.2.a.g.1.2 2
35.18 odd 12 441.2.e.j.226.1 4
35.23 odd 12 441.2.e.j.361.1 4
35.33 even 12 441.2.e.i.361.1 4
40.3 even 4 4032.2.a.bq.1.2 2
40.13 odd 4 4032.2.a.bt.1.2 2
45.13 odd 12 567.2.f.j.379.1 4
45.23 even 12 567.2.f.j.379.2 4
45.38 even 12 567.2.f.j.190.2 4
45.43 odd 12 567.2.f.j.190.1 4
55.43 even 4 7623.2.a.bi.1.1 2
60.23 odd 4 1008.2.a.n.1.2 2
105.23 even 12 441.2.e.j.361.2 4
105.38 odd 12 441.2.e.i.226.2 4
105.53 even 12 441.2.e.j.226.2 4
105.68 odd 12 441.2.e.i.361.2 4
105.83 odd 4 441.2.a.g.1.1 2
120.53 even 4 4032.2.a.bt.1.1 2
120.83 odd 4 4032.2.a.bq.1.1 2
140.83 odd 4 7056.2.a.cm.1.2 2
165.98 odd 4 7623.2.a.bi.1.2 2
420.83 even 4 7056.2.a.cm.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.a.b.1.1 2 15.8 even 4
63.2.a.b.1.2 yes 2 5.3 odd 4
441.2.a.g.1.1 2 105.83 odd 4
441.2.a.g.1.2 2 35.13 even 4
441.2.e.i.226.1 4 35.3 even 12
441.2.e.i.226.2 4 105.38 odd 12
441.2.e.i.361.1 4 35.33 even 12
441.2.e.i.361.2 4 105.68 odd 12
441.2.e.j.226.1 4 35.18 odd 12
441.2.e.j.226.2 4 105.53 even 12
441.2.e.j.361.1 4 35.23 odd 12
441.2.e.j.361.2 4 105.23 even 12
567.2.f.j.190.1 4 45.43 odd 12
567.2.f.j.190.2 4 45.38 even 12
567.2.f.j.379.1 4 45.13 odd 12
567.2.f.j.379.2 4 45.23 even 12
1008.2.a.n.1.1 2 20.3 even 4
1008.2.a.n.1.2 2 60.23 odd 4
1575.2.a.q.1.1 2 5.2 odd 4
1575.2.a.q.1.2 2 15.2 even 4
1575.2.d.i.1324.1 4 5.4 even 2 inner
1575.2.d.i.1324.2 4 3.2 odd 2 inner
1575.2.d.i.1324.3 4 15.14 odd 2 inner
1575.2.d.i.1324.4 4 1.1 even 1 trivial
4032.2.a.bq.1.1 2 120.83 odd 4
4032.2.a.bq.1.2 2 40.3 even 4
4032.2.a.bt.1.1 2 120.53 even 4
4032.2.a.bt.1.2 2 40.13 odd 4
7056.2.a.cm.1.1 2 420.83 even 4
7056.2.a.cm.1.2 2 140.83 odd 4
7623.2.a.bi.1.1 2 55.43 even 4
7623.2.a.bi.1.2 2 165.98 odd 4