Properties

Label 2475.2.c.e.199.2
Level $2475$
Weight $2$
Character 2475.199
Analytic conductor $19.763$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 2475.199
Dual form 2475.2.c.e.199.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{7} +3.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000 q^{4} +3.00000i q^{7} +3.00000i q^{8} +1.00000 q^{11} +2.00000i q^{13} -3.00000 q^{14} -1.00000 q^{16} +3.00000i q^{17} +1.00000 q^{19} +1.00000i q^{22} -1.00000i q^{23} -2.00000 q^{26} +3.00000i q^{28} -6.00000 q^{29} +4.00000 q^{31} +5.00000i q^{32} -3.00000 q^{34} -1.00000i q^{37} +1.00000i q^{38} -5.00000 q^{41} +4.00000i q^{43} +1.00000 q^{44} +1.00000 q^{46} +3.00000i q^{47} -2.00000 q^{49} +2.00000i q^{52} -10.0000i q^{53} -9.00000 q^{56} -6.00000i q^{58} -11.0000 q^{59} +14.0000 q^{61} +4.00000i q^{62} -7.00000 q^{64} -2.00000i q^{67} +3.00000i q^{68} -5.00000 q^{71} +2.00000i q^{73} +1.00000 q^{74} +1.00000 q^{76} +3.00000i q^{77} -5.00000 q^{79} -5.00000i q^{82} -8.00000i q^{83} -4.00000 q^{86} +3.00000i q^{88} +10.0000 q^{89} -6.00000 q^{91} -1.00000i q^{92} -3.00000 q^{94} +17.0000i q^{97} -2.00000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + O(q^{10}) \) \( 2 q + 2 q^{4} + 2 q^{11} - 6 q^{14} - 2 q^{16} + 2 q^{19} - 4 q^{26} - 12 q^{29} + 8 q^{31} - 6 q^{34} - 10 q^{41} + 2 q^{44} + 2 q^{46} - 4 q^{49} - 18 q^{56} - 22 q^{59} + 28 q^{61} - 14 q^{64} - 10 q^{71} + 2 q^{74} + 2 q^{76} - 10 q^{79} - 8 q^{86} + 20 q^{89} - 12 q^{91} - 6 q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(2026\) \(2377\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i 0.935414 + 0.353553i \(0.115027\pi\)
−0.935414 + 0.353553i \(0.884973\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 3.00000i 1.13389i 0.823754 + 0.566947i \(0.191875\pi\)
−0.823754 + 0.566947i \(0.808125\pi\)
\(8\) 3.00000i 1.06066i
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) −3.00000 −0.801784
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) 3.00000i 0.727607i 0.931476 + 0.363803i \(0.118522\pi\)
−0.931476 + 0.363803i \(0.881478\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 1.00000i 0.213201i
\(23\) − 1.00000i − 0.208514i −0.994550 0.104257i \(-0.966753\pi\)
0.994550 0.104257i \(-0.0332465\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 3.00000i 0.566947i
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 4.00000 0.718421 0.359211 0.933257i \(-0.383046\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 5.00000i 0.883883i
\(33\) 0 0
\(34\) −3.00000 −0.514496
\(35\) 0 0
\(36\) 0 0
\(37\) − 1.00000i − 0.164399i −0.996616 0.0821995i \(-0.973806\pi\)
0.996616 0.0821995i \(-0.0261945\pi\)
\(38\) 1.00000i 0.162221i
\(39\) 0 0
\(40\) 0 0
\(41\) −5.00000 −0.780869 −0.390434 0.920631i \(-0.627675\pi\)
−0.390434 + 0.920631i \(0.627675\pi\)
\(42\) 0 0
\(43\) 4.00000i 0.609994i 0.952353 + 0.304997i \(0.0986555\pi\)
−0.952353 + 0.304997i \(0.901344\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 1.00000 0.147442
\(47\) 3.00000i 0.437595i 0.975770 + 0.218797i \(0.0702134\pi\)
−0.975770 + 0.218797i \(0.929787\pi\)
\(48\) 0 0
\(49\) −2.00000 −0.285714
\(50\) 0 0
\(51\) 0 0
\(52\) 2.00000i 0.277350i
\(53\) − 10.0000i − 1.37361i −0.726844 0.686803i \(-0.759014\pi\)
0.726844 0.686803i \(-0.240986\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −9.00000 −1.20268
\(57\) 0 0
\(58\) − 6.00000i − 0.787839i
\(59\) −11.0000 −1.43208 −0.716039 0.698060i \(-0.754047\pi\)
−0.716039 + 0.698060i \(0.754047\pi\)
\(60\) 0 0
\(61\) 14.0000 1.79252 0.896258 0.443533i \(-0.146275\pi\)
0.896258 + 0.443533i \(0.146275\pi\)
\(62\) 4.00000i 0.508001i
\(63\) 0 0
\(64\) −7.00000 −0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) − 2.00000i − 0.244339i −0.992509 0.122169i \(-0.961015\pi\)
0.992509 0.122169i \(-0.0389851\pi\)
\(68\) 3.00000i 0.363803i
\(69\) 0 0
\(70\) 0 0
\(71\) −5.00000 −0.593391 −0.296695 0.954972i \(-0.595885\pi\)
−0.296695 + 0.954972i \(0.595885\pi\)
\(72\) 0 0
\(73\) 2.00000i 0.234082i 0.993127 + 0.117041i \(0.0373409\pi\)
−0.993127 + 0.117041i \(0.962659\pi\)
\(74\) 1.00000 0.116248
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) 3.00000i 0.341882i
\(78\) 0 0
\(79\) −5.00000 −0.562544 −0.281272 0.959628i \(-0.590756\pi\)
−0.281272 + 0.959628i \(0.590756\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 5.00000i − 0.552158i
\(83\) − 8.00000i − 0.878114i −0.898459 0.439057i \(-0.855313\pi\)
0.898459 0.439057i \(-0.144687\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 3.00000i 0.319801i
\(89\) 10.0000 1.06000 0.529999 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) − 1.00000i − 0.104257i
\(93\) 0 0
\(94\) −3.00000 −0.309426
\(95\) 0 0
\(96\) 0 0
\(97\) 17.0000i 1.72609i 0.505128 + 0.863044i \(0.331445\pi\)
−0.505128 + 0.863044i \(0.668555\pi\)
\(98\) − 2.00000i − 0.202031i
\(99\) 0 0
\(100\) 0 0
\(101\) −11.0000 −1.09454 −0.547270 0.836956i \(-0.684333\pi\)
−0.547270 + 0.836956i \(0.684333\pi\)
\(102\) 0 0
\(103\) 2.00000i 0.197066i 0.995134 + 0.0985329i \(0.0314150\pi\)
−0.995134 + 0.0985329i \(0.968585\pi\)
\(104\) −6.00000 −0.588348
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) 18.0000i 1.74013i 0.492941 + 0.870063i \(0.335922\pi\)
−0.492941 + 0.870063i \(0.664078\pi\)
\(108\) 0 0
\(109\) 12.0000 1.14939 0.574696 0.818367i \(-0.305120\pi\)
0.574696 + 0.818367i \(0.305120\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) − 3.00000i − 0.283473i
\(113\) − 18.0000i − 1.69330i −0.532152 0.846649i \(-0.678617\pi\)
0.532152 0.846649i \(-0.321383\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −6.00000 −0.557086
\(117\) 0 0
\(118\) − 11.0000i − 1.01263i
\(119\) −9.00000 −0.825029
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 14.0000i 1.26750i
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) 5.00000i 0.443678i 0.975083 + 0.221839i \(0.0712060\pi\)
−0.975083 + 0.221839i \(0.928794\pi\)
\(128\) 3.00000i 0.265165i
\(129\) 0 0
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 3.00000i 0.260133i
\(134\) 2.00000 0.172774
\(135\) 0 0
\(136\) −9.00000 −0.771744
\(137\) 6.00000i 0.512615i 0.966595 + 0.256307i \(0.0825059\pi\)
−0.966595 + 0.256307i \(0.917494\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 5.00000i − 0.419591i
\(143\) 2.00000i 0.167248i
\(144\) 0 0
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) − 1.00000i − 0.0821995i
\(149\) −7.00000 −0.573462 −0.286731 0.958011i \(-0.592569\pi\)
−0.286731 + 0.958011i \(0.592569\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 3.00000i 0.243332i
\(153\) 0 0
\(154\) −3.00000 −0.241747
\(155\) 0 0
\(156\) 0 0
\(157\) − 10.0000i − 0.798087i −0.916932 0.399043i \(-0.869342\pi\)
0.916932 0.399043i \(-0.130658\pi\)
\(158\) − 5.00000i − 0.397779i
\(159\) 0 0
\(160\) 0 0
\(161\) 3.00000 0.236433
\(162\) 0 0
\(163\) 10.0000i 0.783260i 0.920123 + 0.391630i \(0.128089\pi\)
−0.920123 + 0.391630i \(0.871911\pi\)
\(164\) −5.00000 −0.390434
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) 10.0000i 0.773823i 0.922117 + 0.386912i \(0.126458\pi\)
−0.922117 + 0.386912i \(0.873542\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 4.00000i 0.304997i
\(173\) 9.00000i 0.684257i 0.939653 + 0.342129i \(0.111148\pi\)
−0.939653 + 0.342129i \(0.888852\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 0 0
\(178\) 10.0000i 0.749532i
\(179\) 1.00000 0.0747435 0.0373718 0.999301i \(-0.488101\pi\)
0.0373718 + 0.999301i \(0.488101\pi\)
\(180\) 0 0
\(181\) −25.0000 −1.85824 −0.929118 0.369784i \(-0.879432\pi\)
−0.929118 + 0.369784i \(0.879432\pi\)
\(182\) − 6.00000i − 0.444750i
\(183\) 0 0
\(184\) 3.00000 0.221163
\(185\) 0 0
\(186\) 0 0
\(187\) 3.00000i 0.219382i
\(188\) 3.00000i 0.218797i
\(189\) 0 0
\(190\) 0 0
\(191\) 19.0000 1.37479 0.687396 0.726283i \(-0.258754\pi\)
0.687396 + 0.726283i \(0.258754\pi\)
\(192\) 0 0
\(193\) 4.00000i 0.287926i 0.989583 + 0.143963i \(0.0459847\pi\)
−0.989583 + 0.143963i \(0.954015\pi\)
\(194\) −17.0000 −1.22053
\(195\) 0 0
\(196\) −2.00000 −0.142857
\(197\) 23.0000i 1.63868i 0.573306 + 0.819341i \(0.305660\pi\)
−0.573306 + 0.819341i \(0.694340\pi\)
\(198\) 0 0
\(199\) 18.0000 1.27599 0.637993 0.770042i \(-0.279765\pi\)
0.637993 + 0.770042i \(0.279765\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 11.0000i − 0.773957i
\(203\) − 18.0000i − 1.26335i
\(204\) 0 0
\(205\) 0 0
\(206\) −2.00000 −0.139347
\(207\) 0 0
\(208\) − 2.00000i − 0.138675i
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) − 10.0000i − 0.686803i
\(213\) 0 0
\(214\) −18.0000 −1.23045
\(215\) 0 0
\(216\) 0 0
\(217\) 12.0000i 0.814613i
\(218\) 12.0000i 0.812743i
\(219\) 0 0
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) 0 0
\(223\) − 22.0000i − 1.47323i −0.676313 0.736614i \(-0.736423\pi\)
0.676313 0.736614i \(-0.263577\pi\)
\(224\) −15.0000 −1.00223
\(225\) 0 0
\(226\) 18.0000 1.19734
\(227\) − 8.00000i − 0.530979i −0.964114 0.265489i \(-0.914466\pi\)
0.964114 0.265489i \(-0.0855335\pi\)
\(228\) 0 0
\(229\) −11.0000 −0.726900 −0.363450 0.931614i \(-0.618401\pi\)
−0.363450 + 0.931614i \(0.618401\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 18.0000i − 1.18176i
\(233\) − 9.00000i − 0.589610i −0.955557 0.294805i \(-0.904745\pi\)
0.955557 0.294805i \(-0.0952546\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −11.0000 −0.716039
\(237\) 0 0
\(238\) − 9.00000i − 0.583383i
\(239\) 20.0000 1.29369 0.646846 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 0 0
\(241\) 4.00000 0.257663 0.128831 0.991667i \(-0.458877\pi\)
0.128831 + 0.991667i \(0.458877\pi\)
\(242\) 1.00000i 0.0642824i
\(243\) 0 0
\(244\) 14.0000 0.896258
\(245\) 0 0
\(246\) 0 0
\(247\) 2.00000i 0.127257i
\(248\) 12.0000i 0.762001i
\(249\) 0 0
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) − 1.00000i − 0.0628695i
\(254\) −5.00000 −0.313728
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 12.0000i 0.748539i 0.927320 + 0.374270i \(0.122107\pi\)
−0.927320 + 0.374270i \(0.877893\pi\)
\(258\) 0 0
\(259\) 3.00000 0.186411
\(260\) 0 0
\(261\) 0 0
\(262\) 12.0000i 0.741362i
\(263\) 18.0000i 1.10993i 0.831875 + 0.554964i \(0.187268\pi\)
−0.831875 + 0.554964i \(0.812732\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −3.00000 −0.183942
\(267\) 0 0
\(268\) − 2.00000i − 0.122169i
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) −3.00000 −0.182237 −0.0911185 0.995840i \(-0.529044\pi\)
−0.0911185 + 0.995840i \(0.529044\pi\)
\(272\) − 3.00000i − 0.181902i
\(273\) 0 0
\(274\) −6.00000 −0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) − 22.0000i − 1.32185i −0.750451 0.660926i \(-0.770164\pi\)
0.750451 0.660926i \(-0.229836\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −21.0000 −1.25275 −0.626377 0.779520i \(-0.715463\pi\)
−0.626377 + 0.779520i \(0.715463\pi\)
\(282\) 0 0
\(283\) − 23.0000i − 1.36721i −0.729853 0.683604i \(-0.760412\pi\)
0.729853 0.683604i \(-0.239588\pi\)
\(284\) −5.00000 −0.296695
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) − 15.0000i − 0.885422i
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) 0 0
\(292\) 2.00000i 0.117041i
\(293\) − 13.0000i − 0.759468i −0.925096 0.379734i \(-0.876015\pi\)
0.925096 0.379734i \(-0.123985\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 3.00000 0.174371
\(297\) 0 0
\(298\) − 7.00000i − 0.405499i
\(299\) 2.00000 0.115663
\(300\) 0 0
\(301\) −12.0000 −0.691669
\(302\) − 16.0000i − 0.920697i
\(303\) 0 0
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 0 0
\(307\) − 32.0000i − 1.82634i −0.407583 0.913168i \(-0.633628\pi\)
0.407583 0.913168i \(-0.366372\pi\)
\(308\) 3.00000i 0.170941i
\(309\) 0 0
\(310\) 0 0
\(311\) −12.0000 −0.680458 −0.340229 0.940343i \(-0.610505\pi\)
−0.340229 + 0.940343i \(0.610505\pi\)
\(312\) 0 0
\(313\) − 1.00000i − 0.0565233i −0.999601 0.0282617i \(-0.991003\pi\)
0.999601 0.0282617i \(-0.00899717\pi\)
\(314\) 10.0000 0.564333
\(315\) 0 0
\(316\) −5.00000 −0.281272
\(317\) − 6.00000i − 0.336994i −0.985702 0.168497i \(-0.946109\pi\)
0.985702 0.168497i \(-0.0538913\pi\)
\(318\) 0 0
\(319\) −6.00000 −0.335936
\(320\) 0 0
\(321\) 0 0
\(322\) 3.00000i 0.167183i
\(323\) 3.00000i 0.166924i
\(324\) 0 0
\(325\) 0 0
\(326\) −10.0000 −0.553849
\(327\) 0 0
\(328\) − 15.0000i − 0.828236i
\(329\) −9.00000 −0.496186
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) − 8.00000i − 0.439057i
\(333\) 0 0
\(334\) −10.0000 −0.547176
\(335\) 0 0
\(336\) 0 0
\(337\) − 6.00000i − 0.326841i −0.986557 0.163420i \(-0.947747\pi\)
0.986557 0.163420i \(-0.0522527\pi\)
\(338\) 9.00000i 0.489535i
\(339\) 0 0
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) 15.0000i 0.809924i
\(344\) −12.0000 −0.646997
\(345\) 0 0
\(346\) −9.00000 −0.483843
\(347\) − 10.0000i − 0.536828i −0.963304 0.268414i \(-0.913500\pi\)
0.963304 0.268414i \(-0.0864995\pi\)
\(348\) 0 0
\(349\) −8.00000 −0.428230 −0.214115 0.976808i \(-0.568687\pi\)
−0.214115 + 0.976808i \(0.568687\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.00000i 0.266501i
\(353\) − 6.00000i − 0.319348i −0.987170 0.159674i \(-0.948956\pi\)
0.987170 0.159674i \(-0.0510443\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 10.0000 0.529999
\(357\) 0 0
\(358\) 1.00000i 0.0528516i
\(359\) 28.0000 1.47778 0.738892 0.673824i \(-0.235349\pi\)
0.738892 + 0.673824i \(0.235349\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) − 25.0000i − 1.31397i
\(363\) 0 0
\(364\) −6.00000 −0.314485
\(365\) 0 0
\(366\) 0 0
\(367\) − 4.00000i − 0.208798i −0.994535 0.104399i \(-0.966708\pi\)
0.994535 0.104399i \(-0.0332919\pi\)
\(368\) 1.00000i 0.0521286i
\(369\) 0 0
\(370\) 0 0
\(371\) 30.0000 1.55752
\(372\) 0 0
\(373\) 6.00000i 0.310668i 0.987862 + 0.155334i \(0.0496454\pi\)
−0.987862 + 0.155334i \(0.950355\pi\)
\(374\) −3.00000 −0.155126
\(375\) 0 0
\(376\) −9.00000 −0.464140
\(377\) − 12.0000i − 0.618031i
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 19.0000i 0.972125i
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −4.00000 −0.203595
\(387\) 0 0
\(388\) 17.0000i 0.863044i
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 3.00000 0.151717
\(392\) − 6.00000i − 0.303046i
\(393\) 0 0
\(394\) −23.0000 −1.15872
\(395\) 0 0
\(396\) 0 0
\(397\) 6.00000i 0.301131i 0.988600 + 0.150566i \(0.0481095\pi\)
−0.988600 + 0.150566i \(0.951890\pi\)
\(398\) 18.0000i 0.902258i
\(399\) 0 0
\(400\) 0 0
\(401\) −38.0000 −1.89763 −0.948815 0.315833i \(-0.897716\pi\)
−0.948815 + 0.315833i \(0.897716\pi\)
\(402\) 0 0
\(403\) 8.00000i 0.398508i
\(404\) −11.0000 −0.547270
\(405\) 0 0
\(406\) 18.0000 0.893325
\(407\) − 1.00000i − 0.0495682i
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 2.00000i 0.0985329i
\(413\) − 33.0000i − 1.62382i
\(414\) 0 0
\(415\) 0 0
\(416\) −10.0000 −0.490290
\(417\) 0 0
\(418\) 1.00000i 0.0489116i
\(419\) 23.0000 1.12362 0.561812 0.827265i \(-0.310105\pi\)
0.561812 + 0.827265i \(0.310105\pi\)
\(420\) 0 0
\(421\) 9.00000 0.438633 0.219317 0.975654i \(-0.429617\pi\)
0.219317 + 0.975654i \(0.429617\pi\)
\(422\) 4.00000i 0.194717i
\(423\) 0 0
\(424\) 30.0000 1.45693
\(425\) 0 0
\(426\) 0 0
\(427\) 42.0000i 2.03252i
\(428\) 18.0000i 0.870063i
\(429\) 0 0
\(430\) 0 0
\(431\) 36.0000 1.73406 0.867029 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) 0 0
\(433\) 2.00000i 0.0961139i 0.998845 + 0.0480569i \(0.0153029\pi\)
−0.998845 + 0.0480569i \(0.984697\pi\)
\(434\) −12.0000 −0.576018
\(435\) 0 0
\(436\) 12.0000 0.574696
\(437\) − 1.00000i − 0.0478365i
\(438\) 0 0
\(439\) 35.0000 1.67046 0.835229 0.549902i \(-0.185335\pi\)
0.835229 + 0.549902i \(0.185335\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 6.00000i − 0.285391i
\(443\) 31.0000i 1.47285i 0.676517 + 0.736427i \(0.263489\pi\)
−0.676517 + 0.736427i \(0.736511\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 22.0000 1.04173
\(447\) 0 0
\(448\) − 21.0000i − 0.992157i
\(449\) 8.00000 0.377543 0.188772 0.982021i \(-0.439549\pi\)
0.188772 + 0.982021i \(0.439549\pi\)
\(450\) 0 0
\(451\) −5.00000 −0.235441
\(452\) − 18.0000i − 0.846649i
\(453\) 0 0
\(454\) 8.00000 0.375459
\(455\) 0 0
\(456\) 0 0
\(457\) − 28.0000i − 1.30978i −0.755722 0.654892i \(-0.772714\pi\)
0.755722 0.654892i \(-0.227286\pi\)
\(458\) − 11.0000i − 0.513996i
\(459\) 0 0
\(460\) 0 0
\(461\) 34.0000 1.58354 0.791769 0.610821i \(-0.209160\pi\)
0.791769 + 0.610821i \(0.209160\pi\)
\(462\) 0 0
\(463\) 24.0000i 1.11537i 0.830051 + 0.557687i \(0.188311\pi\)
−0.830051 + 0.557687i \(0.811689\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 9.00000 0.416917
\(467\) − 36.0000i − 1.66588i −0.553362 0.832941i \(-0.686655\pi\)
0.553362 0.832941i \(-0.313345\pi\)
\(468\) 0 0
\(469\) 6.00000 0.277054
\(470\) 0 0
\(471\) 0 0
\(472\) − 33.0000i − 1.51895i
\(473\) 4.00000i 0.183920i
\(474\) 0 0
\(475\) 0 0
\(476\) −9.00000 −0.412514
\(477\) 0 0
\(478\) 20.0000i 0.914779i
\(479\) 18.0000 0.822441 0.411220 0.911536i \(-0.365103\pi\)
0.411220 + 0.911536i \(0.365103\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 4.00000i 0.182195i
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) 0 0
\(487\) 26.0000i 1.17817i 0.808070 + 0.589086i \(0.200512\pi\)
−0.808070 + 0.589086i \(0.799488\pi\)
\(488\) 42.0000i 1.90125i
\(489\) 0 0
\(490\) 0 0
\(491\) 28.0000 1.26362 0.631811 0.775122i \(-0.282312\pi\)
0.631811 + 0.775122i \(0.282312\pi\)
\(492\) 0 0
\(493\) − 18.0000i − 0.810679i
\(494\) −2.00000 −0.0899843
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) − 15.0000i − 0.672842i
\(498\) 0 0
\(499\) −24.0000 −1.07439 −0.537194 0.843459i \(-0.680516\pi\)
−0.537194 + 0.843459i \(0.680516\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 12.0000i 0.535586i
\(503\) − 20.0000i − 0.891756i −0.895094 0.445878i \(-0.852892\pi\)
0.895094 0.445878i \(-0.147108\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.00000 0.0444554
\(507\) 0 0
\(508\) 5.00000i 0.221839i
\(509\) 26.0000 1.15243 0.576215 0.817298i \(-0.304529\pi\)
0.576215 + 0.817298i \(0.304529\pi\)
\(510\) 0 0
\(511\) −6.00000 −0.265424
\(512\) − 11.0000i − 0.486136i
\(513\) 0 0
\(514\) −12.0000 −0.529297
\(515\) 0 0
\(516\) 0 0
\(517\) 3.00000i 0.131940i
\(518\) 3.00000i 0.131812i
\(519\) 0 0
\(520\) 0 0
\(521\) 42.0000 1.84005 0.920027 0.391856i \(-0.128167\pi\)
0.920027 + 0.391856i \(0.128167\pi\)
\(522\) 0 0
\(523\) 7.00000i 0.306089i 0.988219 + 0.153044i \(0.0489077\pi\)
−0.988219 + 0.153044i \(0.951092\pi\)
\(524\) 12.0000 0.524222
\(525\) 0 0
\(526\) −18.0000 −0.784837
\(527\) 12.0000i 0.522728i
\(528\) 0 0
\(529\) 22.0000 0.956522
\(530\) 0 0
\(531\) 0 0
\(532\) 3.00000i 0.130066i
\(533\) − 10.0000i − 0.433148i
\(534\) 0 0
\(535\) 0 0
\(536\) 6.00000 0.259161
\(537\) 0 0
\(538\) 24.0000i 1.03471i
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) 32.0000 1.37579 0.687894 0.725811i \(-0.258536\pi\)
0.687894 + 0.725811i \(0.258536\pi\)
\(542\) − 3.00000i − 0.128861i
\(543\) 0 0
\(544\) −15.0000 −0.643120
\(545\) 0 0
\(546\) 0 0
\(547\) 33.0000i 1.41098i 0.708721 + 0.705489i \(0.249273\pi\)
−0.708721 + 0.705489i \(0.750727\pi\)
\(548\) 6.00000i 0.256307i
\(549\) 0 0
\(550\) 0 0
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) − 15.0000i − 0.637865i
\(554\) 22.0000 0.934690
\(555\) 0 0
\(556\) 0 0
\(557\) 22.0000i 0.932170i 0.884740 + 0.466085i \(0.154336\pi\)
−0.884740 + 0.466085i \(0.845664\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) − 21.0000i − 0.885832i
\(563\) − 18.0000i − 0.758610i −0.925272 0.379305i \(-0.876163\pi\)
0.925272 0.379305i \(-0.123837\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 23.0000 0.966762
\(567\) 0 0
\(568\) − 15.0000i − 0.629386i
\(569\) −37.0000 −1.55112 −0.775560 0.631273i \(-0.782533\pi\)
−0.775560 + 0.631273i \(0.782533\pi\)
\(570\) 0 0
\(571\) 36.0000 1.50655 0.753277 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) 2.00000i 0.0836242i
\(573\) 0 0
\(574\) 15.0000 0.626088
\(575\) 0 0
\(576\) 0 0
\(577\) 43.0000i 1.79011i 0.445952 + 0.895057i \(0.352865\pi\)
−0.445952 + 0.895057i \(0.647135\pi\)
\(578\) 8.00000i 0.332756i
\(579\) 0 0
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) 0 0
\(583\) − 10.0000i − 0.414158i
\(584\) −6.00000 −0.248282
\(585\) 0 0
\(586\) 13.0000 0.537025
\(587\) 9.00000i 0.371470i 0.982600 + 0.185735i \(0.0594666\pi\)
−0.982600 + 0.185735i \(0.940533\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) 1.00000i 0.0410997i
\(593\) 14.0000i 0.574911i 0.957794 + 0.287456i \(0.0928094\pi\)
−0.957794 + 0.287456i \(0.907191\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −7.00000 −0.286731
\(597\) 0 0
\(598\) 2.00000i 0.0817861i
\(599\) 27.0000 1.10319 0.551595 0.834112i \(-0.314019\pi\)
0.551595 + 0.834112i \(0.314019\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) − 12.0000i − 0.489083i
\(603\) 0 0
\(604\) −16.0000 −0.651031
\(605\) 0 0
\(606\) 0 0
\(607\) − 40.0000i − 1.62355i −0.583970 0.811775i \(-0.698502\pi\)
0.583970 0.811775i \(-0.301498\pi\)
\(608\) 5.00000i 0.202777i
\(609\) 0 0
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) 16.0000i 0.646234i 0.946359 + 0.323117i \(0.104731\pi\)
−0.946359 + 0.323117i \(0.895269\pi\)
\(614\) 32.0000 1.29141
\(615\) 0 0
\(616\) −9.00000 −0.362620
\(617\) − 28.0000i − 1.12724i −0.826035 0.563619i \(-0.809409\pi\)
0.826035 0.563619i \(-0.190591\pi\)
\(618\) 0 0
\(619\) −34.0000 −1.36658 −0.683288 0.730149i \(-0.739451\pi\)
−0.683288 + 0.730149i \(0.739451\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 12.0000i − 0.481156i
\(623\) 30.0000i 1.20192i
\(624\) 0 0
\(625\) 0 0
\(626\) 1.00000 0.0399680
\(627\) 0 0
\(628\) − 10.0000i − 0.399043i
\(629\) 3.00000 0.119618
\(630\) 0 0
\(631\) 34.0000 1.35352 0.676759 0.736204i \(-0.263384\pi\)
0.676759 + 0.736204i \(0.263384\pi\)
\(632\) − 15.0000i − 0.596668i
\(633\) 0 0
\(634\) 6.00000 0.238290
\(635\) 0 0
\(636\) 0 0
\(637\) − 4.00000i − 0.158486i
\(638\) − 6.00000i − 0.237542i
\(639\) 0 0
\(640\) 0 0
\(641\) −16.0000 −0.631962 −0.315981 0.948766i \(-0.602334\pi\)
−0.315981 + 0.948766i \(0.602334\pi\)
\(642\) 0 0
\(643\) 14.0000i 0.552106i 0.961142 + 0.276053i \(0.0890266\pi\)
−0.961142 + 0.276053i \(0.910973\pi\)
\(644\) 3.00000 0.118217
\(645\) 0 0
\(646\) −3.00000 −0.118033
\(647\) − 13.0000i − 0.511083i −0.966798 0.255541i \(-0.917746\pi\)
0.966798 0.255541i \(-0.0822537\pi\)
\(648\) 0 0
\(649\) −11.0000 −0.431788
\(650\) 0 0
\(651\) 0 0
\(652\) 10.0000i 0.391630i
\(653\) 4.00000i 0.156532i 0.996933 + 0.0782660i \(0.0249384\pi\)
−0.996933 + 0.0782660i \(0.975062\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 5.00000 0.195217
\(657\) 0 0
\(658\) − 9.00000i − 0.350857i
\(659\) −6.00000 −0.233727 −0.116863 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) 0 0
\(661\) 35.0000 1.36134 0.680671 0.732589i \(-0.261688\pi\)
0.680671 + 0.732589i \(0.261688\pi\)
\(662\) 4.00000i 0.155464i
\(663\) 0 0
\(664\) 24.0000 0.931381
\(665\) 0 0
\(666\) 0 0
\(667\) 6.00000i 0.232321i
\(668\) 10.0000i 0.386912i
\(669\) 0 0
\(670\) 0 0
\(671\) 14.0000 0.540464
\(672\) 0 0
\(673\) 4.00000i 0.154189i 0.997024 + 0.0770943i \(0.0245643\pi\)
−0.997024 + 0.0770943i \(0.975436\pi\)
\(674\) 6.00000 0.231111
\(675\) 0 0
\(676\) 9.00000 0.346154
\(677\) − 14.0000i − 0.538064i −0.963131 0.269032i \(-0.913296\pi\)
0.963131 0.269032i \(-0.0867037\pi\)
\(678\) 0 0
\(679\) −51.0000 −1.95720
\(680\) 0 0
\(681\) 0 0
\(682\) 4.00000i 0.153168i
\(683\) − 47.0000i − 1.79841i −0.437533 0.899203i \(-0.644148\pi\)
0.437533 0.899203i \(-0.355852\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −15.0000 −0.572703
\(687\) 0 0
\(688\) − 4.00000i − 0.152499i
\(689\) 20.0000 0.761939
\(690\) 0 0
\(691\) −40.0000 −1.52167 −0.760836 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) 9.00000i 0.342129i
\(693\) 0 0
\(694\) 10.0000 0.379595
\(695\) 0 0
\(696\) 0 0
\(697\) − 15.0000i − 0.568166i
\(698\) − 8.00000i − 0.302804i
\(699\) 0 0
\(700\) 0 0
\(701\) 23.0000 0.868698 0.434349 0.900745i \(-0.356978\pi\)
0.434349 + 0.900745i \(0.356978\pi\)
\(702\) 0 0
\(703\) − 1.00000i − 0.0377157i
\(704\) −7.00000 −0.263822
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) − 33.0000i − 1.24109i
\(708\) 0 0
\(709\) −35.0000 −1.31445 −0.657226 0.753693i \(-0.728270\pi\)
−0.657226 + 0.753693i \(0.728270\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 30.0000i 1.12430i
\(713\) − 4.00000i − 0.149801i
\(714\) 0 0
\(715\) 0 0
\(716\) 1.00000 0.0373718
\(717\) 0 0
\(718\) 28.0000i 1.04495i
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) −6.00000 −0.223452
\(722\) − 18.0000i − 0.669891i
\(723\) 0 0
\(724\) −25.0000 −0.929118
\(725\) 0 0
\(726\) 0 0
\(727\) − 28.0000i − 1.03846i −0.854634 0.519231i \(-0.826218\pi\)
0.854634 0.519231i \(-0.173782\pi\)
\(728\) − 18.0000i − 0.667124i
\(729\) 0 0
\(730\) 0 0
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) − 6.00000i − 0.221615i −0.993842 0.110808i \(-0.964656\pi\)
0.993842 0.110808i \(-0.0353437\pi\)
\(734\) 4.00000 0.147643
\(735\) 0 0
\(736\) 5.00000 0.184302
\(737\) − 2.00000i − 0.0736709i
\(738\) 0 0
\(739\) −1.00000 −0.0367856 −0.0183928 0.999831i \(-0.505855\pi\)
−0.0183928 + 0.999831i \(0.505855\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 30.0000i 1.10133i
\(743\) 14.0000i 0.513610i 0.966463 + 0.256805i \(0.0826698\pi\)
−0.966463 + 0.256805i \(0.917330\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −6.00000 −0.219676
\(747\) 0 0
\(748\) 3.00000i 0.109691i
\(749\) −54.0000 −1.97312
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) − 3.00000i − 0.109399i
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) 42.0000i 1.52652i 0.646094 + 0.763258i \(0.276401\pi\)
−0.646094 + 0.763258i \(0.723599\pi\)
\(758\) 16.0000i 0.581146i
\(759\) 0 0
\(760\) 0 0
\(761\) −22.0000 −0.797499 −0.398750 0.917060i \(-0.630556\pi\)
−0.398750 + 0.917060i \(0.630556\pi\)
\(762\) 0 0
\(763\) 36.0000i 1.30329i
\(764\) 19.0000 0.687396
\(765\) 0 0
\(766\) 0 0
\(767\) − 22.0000i − 0.794374i
\(768\) 0 0
\(769\) 28.0000 1.00971 0.504853 0.863205i \(-0.331547\pi\)
0.504853 + 0.863205i \(0.331547\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.00000i 0.143963i
\(773\) − 44.0000i − 1.58257i −0.611448 0.791285i \(-0.709412\pi\)
0.611448 0.791285i \(-0.290588\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −51.0000 −1.83079
\(777\) 0 0
\(778\) 6.00000i 0.215110i
\(779\) −5.00000 −0.179144
\(780\) 0 0
\(781\) −5.00000 −0.178914
\(782\) 3.00000i 0.107280i
\(783\) 0 0
\(784\) 2.00000 0.0714286
\(785\) 0 0
\(786\) 0 0
\(787\) − 43.0000i − 1.53278i −0.642373 0.766392i \(-0.722050\pi\)
0.642373 0.766392i \(-0.277950\pi\)
\(788\) 23.0000i 0.819341i
\(789\) 0 0
\(790\) 0 0
\(791\) 54.0000 1.92002
\(792\) 0 0
\(793\) 28.0000i 0.994309i
\(794\) −6.00000 −0.212932
\(795\) 0 0
\(796\) 18.0000 0.637993
\(797\) 22.0000i 0.779280i 0.920967 + 0.389640i \(0.127401\pi\)
−0.920967 + 0.389640i \(0.872599\pi\)
\(798\) 0 0
\(799\) −9.00000 −0.318397
\(800\) 0 0
\(801\) 0 0
\(802\) − 38.0000i − 1.34183i
\(803\) 2.00000i 0.0705785i
\(804\) 0 0
\(805\) 0 0
\(806\) −8.00000 −0.281788
\(807\) 0 0
\(808\) − 33.0000i − 1.16094i
\(809\) 15.0000 0.527372 0.263686 0.964609i \(-0.415062\pi\)
0.263686 + 0.964609i \(0.415062\pi\)
\(810\) 0 0
\(811\) −21.0000 −0.737410 −0.368705 0.929547i \(-0.620199\pi\)
−0.368705 + 0.929547i \(0.620199\pi\)
\(812\) − 18.0000i − 0.631676i
\(813\) 0 0
\(814\) 1.00000 0.0350500
\(815\) 0 0
\(816\) 0 0
\(817\) 4.00000i 0.139942i
\(818\) − 6.00000i − 0.209785i
\(819\) 0 0
\(820\) 0 0
\(821\) −18.0000 −0.628204 −0.314102 0.949389i \(-0.601703\pi\)
−0.314102 + 0.949389i \(0.601703\pi\)
\(822\) 0 0
\(823\) − 42.0000i − 1.46403i −0.681290 0.732014i \(-0.738581\pi\)
0.681290 0.732014i \(-0.261419\pi\)
\(824\) −6.00000 −0.209020
\(825\) 0 0
\(826\) 33.0000 1.14822
\(827\) − 48.0000i − 1.66912i −0.550914 0.834562i \(-0.685721\pi\)
0.550914 0.834562i \(-0.314279\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 14.0000i − 0.485363i
\(833\) − 6.00000i − 0.207888i
\(834\) 0 0
\(835\) 0 0
\(836\) 1.00000 0.0345857
\(837\) 0 0
\(838\) 23.0000i 0.794522i
\(839\) 40.0000 1.38095 0.690477 0.723355i \(-0.257401\pi\)
0.690477 + 0.723355i \(0.257401\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 9.00000i 0.310160i
\(843\) 0 0
\(844\) 4.00000 0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) 3.00000i 0.103081i
\(848\) 10.0000i 0.343401i
\(849\) 0 0
\(850\) 0 0
\(851\) −1.00000 −0.0342796
\(852\) 0 0
\(853\) − 40.0000i − 1.36957i −0.728743 0.684787i \(-0.759895\pi\)
0.728743 0.684787i \(-0.240105\pi\)
\(854\) −42.0000 −1.43721
\(855\) 0 0
\(856\) −54.0000 −1.84568
\(857\) 39.0000i 1.33221i 0.745856 + 0.666107i \(0.232041\pi\)
−0.745856 + 0.666107i \(0.767959\pi\)
\(858\) 0 0
\(859\) 26.0000 0.887109 0.443554 0.896248i \(-0.353717\pi\)
0.443554 + 0.896248i \(0.353717\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 36.0000i 1.22616i
\(863\) − 4.00000i − 0.136162i −0.997680 0.0680808i \(-0.978312\pi\)
0.997680 0.0680808i \(-0.0216876\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −2.00000 −0.0679628
\(867\) 0 0
\(868\) 12.0000i 0.407307i
\(869\) −5.00000 −0.169613
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 36.0000i 1.21911i
\(873\) 0 0
\(874\) 1.00000 0.0338255
\(875\) 0 0
\(876\) 0 0
\(877\) 52.0000i 1.75592i 0.478738 + 0.877958i \(0.341094\pi\)
−0.478738 + 0.877958i \(0.658906\pi\)
\(878\) 35.0000i 1.18119i
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 2.00000i 0.0673054i 0.999434 + 0.0336527i \(0.0107140\pi\)
−0.999434 + 0.0336527i \(0.989286\pi\)
\(884\) −6.00000 −0.201802
\(885\) 0 0
\(886\) −31.0000 −1.04147
\(887\) 14.0000i 0.470074i 0.971986 + 0.235037i \(0.0755211\pi\)
−0.971986 + 0.235037i \(0.924479\pi\)
\(888\) 0 0
\(889\) −15.0000 −0.503084
\(890\) 0 0
\(891\) 0 0
\(892\) − 22.0000i − 0.736614i
\(893\) 3.00000i 0.100391i
\(894\) 0 0
\(895\) 0 0
\(896\) −9.00000 −0.300669
\(897\) 0 0
\(898\) 8.00000i 0.266963i
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) 30.0000 0.999445
\(902\) − 5.00000i − 0.166482i
\(903\) 0 0
\(904\) 54.0000 1.79601
\(905\) 0 0
\(906\) 0 0
\(907\) − 2.00000i − 0.0664089i −0.999449 0.0332045i \(-0.989429\pi\)
0.999449 0.0332045i \(-0.0105712\pi\)
\(908\) − 8.00000i − 0.265489i
\(909\) 0 0
\(910\) 0 0
\(911\) −53.0000 −1.75597 −0.877984 0.478690i \(-0.841112\pi\)
−0.877984 + 0.478690i \(0.841112\pi\)
\(912\) 0 0
\(913\) − 8.00000i − 0.264761i
\(914\) 28.0000 0.926158
\(915\) 0 0
\(916\) −11.0000 −0.363450
\(917\) 36.0000i 1.18882i
\(918\) 0 0
\(919\) 39.0000 1.28649 0.643246 0.765660i \(-0.277587\pi\)
0.643246 + 0.765660i \(0.277587\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 34.0000i 1.11973i
\(923\) − 10.0000i − 0.329154i
\(924\) 0 0
\(925\) 0 0
\(926\) −24.0000 −0.788689
\(927\) 0 0
\(928\) − 30.0000i − 0.984798i
\(929\) −32.0000 −1.04989 −0.524943 0.851137i \(-0.675913\pi\)
−0.524943 + 0.851137i \(0.675913\pi\)
\(930\) 0 0
\(931\) −2.00000 −0.0655474
\(932\) − 9.00000i − 0.294805i
\(933\) 0 0
\(934\) 36.0000 1.17796
\(935\) 0 0
\(936\) 0 0
\(937\) − 18.0000i − 0.588034i −0.955800 0.294017i \(-0.905008\pi\)
0.955800 0.294017i \(-0.0949923\pi\)
\(938\) 6.00000i 0.195907i
\(939\) 0 0
\(940\) 0 0
\(941\) 5.00000 0.162995 0.0814977 0.996674i \(-0.474030\pi\)
0.0814977 + 0.996674i \(0.474030\pi\)
\(942\) 0 0
\(943\) 5.00000i 0.162822i
\(944\) 11.0000 0.358020
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) − 27.0000i − 0.877382i −0.898638 0.438691i \(-0.855442\pi\)
0.898638 0.438691i \(-0.144558\pi\)
\(948\) 0 0
\(949\) −4.00000 −0.129845
\(950\) 0 0
\(951\) 0 0
\(952\) − 27.0000i − 0.875075i
\(953\) 21.0000i 0.680257i 0.940379 + 0.340128i \(0.110471\pi\)
−0.940379 + 0.340128i \(0.889529\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 20.0000 0.646846
\(957\) 0 0
\(958\) 18.0000i 0.581554i
\(959\) −18.0000 −0.581250
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 2.00000i 0.0644826i
\(963\) 0 0
\(964\) 4.00000 0.128831
\(965\) 0 0
\(966\) 0 0
\(967\) − 20.0000i − 0.643157i −0.946883 0.321578i \(-0.895787\pi\)
0.946883 0.321578i \(-0.104213\pi\)
\(968\) 3.00000i 0.0964237i
\(969\) 0 0
\(970\) 0 0
\(971\) 21.0000 0.673922 0.336961 0.941519i \(-0.390601\pi\)
0.336961 + 0.941519i \(0.390601\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −26.0000 −0.833094
\(975\) 0 0
\(976\) −14.0000 −0.448129
\(977\) 46.0000i 1.47167i 0.677161 + 0.735835i \(0.263210\pi\)
−0.677161 + 0.735835i \(0.736790\pi\)
\(978\) 0 0
\(979\) 10.0000 0.319601
\(980\) 0 0
\(981\) 0 0
\(982\) 28.0000i 0.893516i
\(983\) 49.0000i 1.56286i 0.623995 + 0.781429i \(0.285509\pi\)
−0.623995 + 0.781429i \(0.714491\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 18.0000 0.573237
\(987\) 0 0
\(988\) 2.00000i 0.0636285i
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) −34.0000 −1.08005 −0.540023 0.841650i \(-0.681584\pi\)
−0.540023 + 0.841650i \(0.681584\pi\)
\(992\) 20.0000i 0.635001i
\(993\) 0 0
\(994\) 15.0000 0.475771
\(995\) 0 0
\(996\) 0 0
\(997\) − 50.0000i − 1.58352i −0.610835 0.791758i \(-0.709166\pi\)
0.610835 0.791758i \(-0.290834\pi\)
\(998\) − 24.0000i − 0.759707i
\(999\) 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 2475.2.c.e.199.2 2
3.2 odd 2 2475.2.c.c.199.1 2
5.2 odd 4 2475.2.a.b.1.1 1
5.3 odd 4 2475.2.a.k.1.1 yes 1
5.4 even 2 inner 2475.2.c.e.199.1 2
15.2 even 4 2475.2.a.h.1.1 yes 1
15.8 even 4 2475.2.a.d.1.1 yes 1
15.14 odd 2 2475.2.c.c.199.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2475.2.a.b.1.1 1 5.2 odd 4
2475.2.a.d.1.1 yes 1 15.8 even 4
2475.2.a.h.1.1 yes 1 15.2 even 4
2475.2.a.k.1.1 yes 1 5.3 odd 4
2475.2.c.c.199.1 2 3.2 odd 2
2475.2.c.c.199.2 2 15.14 odd 2
2475.2.c.e.199.1 2 5.4 even 2 inner
2475.2.c.e.199.2 2 1.1 even 1 trivial