Properties

Label 2925.2.c.u.2224.2
Level $2925$
Weight $2$
Character 2925.2224
Analytic conductor $23.356$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2925.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(23.3562425912\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2224.2
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2925.2224
Dual form 2925.2.c.u.2224.3

$q$-expansion

\(f(q)\) \(=\) \(q-0.414214i q^{2} +1.82843 q^{4} +2.82843i q^{7} -1.58579i q^{8} +O(q^{10})\) \(q-0.414214i q^{2} +1.82843 q^{4} +2.82843i q^{7} -1.58579i q^{8} +2.00000 q^{11} +1.00000i q^{13} +1.17157 q^{14} +3.00000 q^{16} -7.65685i q^{17} +2.82843 q^{19} -0.828427i q^{22} -4.00000i q^{23} +0.414214 q^{26} +5.17157i q^{28} +2.00000 q^{29} -1.17157 q^{31} -4.41421i q^{32} -3.17157 q^{34} -7.65685i q^{37} -1.17157i q^{38} -5.17157 q^{41} +1.65685i q^{43} +3.65685 q^{44} -1.65685 q^{46} +11.6569i q^{47} -1.00000 q^{49} +1.82843i q^{52} -2.00000i q^{53} +4.48528 q^{56} -0.828427i q^{58} +7.65685 q^{59} +13.3137 q^{61} +0.485281i q^{62} +4.17157 q^{64} +6.82843i q^{67} -14.0000i q^{68} -2.00000 q^{71} -0.343146i q^{73} -3.17157 q^{74} +5.17157 q^{76} +5.65685i q^{77} +11.3137 q^{79} +2.14214i q^{82} +3.65685i q^{83} +0.686292 q^{86} -3.17157i q^{88} +14.8284 q^{89} -2.82843 q^{91} -7.31371i q^{92} +4.82843 q^{94} +3.65685i q^{97} +0.414214i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + O(q^{10}) \) \( 4q - 4q^{4} + 8q^{11} + 16q^{14} + 12q^{16} - 4q^{26} + 8q^{29} - 16q^{31} - 24q^{34} - 32q^{41} - 8q^{44} + 16q^{46} - 4q^{49} - 16q^{56} + 8q^{59} + 8q^{61} + 28q^{64} - 8q^{71} - 24q^{74} + 32q^{76} + 48q^{86} + 48q^{89} + 8q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2925\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\) \(2251\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.414214i − 0.292893i −0.989219 0.146447i \(-0.953216\pi\)
0.989219 0.146447i \(-0.0467837\pi\)
\(3\) 0 0
\(4\) 1.82843 0.914214
\(5\) 0 0
\(6\) 0 0
\(7\) 2.82843i 1.06904i 0.845154 + 0.534522i \(0.179509\pi\)
−0.845154 + 0.534522i \(0.820491\pi\)
\(8\) − 1.58579i − 0.560660i
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 1.00000i 0.277350i
\(14\) 1.17157 0.313116
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) − 7.65685i − 1.85706i −0.371257 0.928530i \(-0.621073\pi\)
0.371257 0.928530i \(-0.378927\pi\)
\(18\) 0 0
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 0.828427i − 0.176621i
\(23\) − 4.00000i − 0.834058i −0.908893 0.417029i \(-0.863071\pi\)
0.908893 0.417029i \(-0.136929\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0.414214 0.0812340
\(27\) 0 0
\(28\) 5.17157i 0.977335i
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −1.17157 −0.210421 −0.105210 0.994450i \(-0.533552\pi\)
−0.105210 + 0.994450i \(0.533552\pi\)
\(32\) − 4.41421i − 0.780330i
\(33\) 0 0
\(34\) −3.17157 −0.543920
\(35\) 0 0
\(36\) 0 0
\(37\) − 7.65685i − 1.25878i −0.777090 0.629390i \(-0.783305\pi\)
0.777090 0.629390i \(-0.216695\pi\)
\(38\) − 1.17157i − 0.190054i
\(39\) 0 0
\(40\) 0 0
\(41\) −5.17157 −0.807664 −0.403832 0.914833i \(-0.632322\pi\)
−0.403832 + 0.914833i \(0.632322\pi\)
\(42\) 0 0
\(43\) 1.65685i 0.252668i 0.991988 + 0.126334i \(0.0403211\pi\)
−0.991988 + 0.126334i \(0.959679\pi\)
\(44\) 3.65685 0.551292
\(45\) 0 0
\(46\) −1.65685 −0.244290
\(47\) 11.6569i 1.70033i 0.526519 + 0.850163i \(0.323497\pi\)
−0.526519 + 0.850163i \(0.676503\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 1.82843i 0.253557i
\(53\) − 2.00000i − 0.274721i −0.990521 0.137361i \(-0.956138\pi\)
0.990521 0.137361i \(-0.0438619\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.48528 0.599371
\(57\) 0 0
\(58\) − 0.828427i − 0.108778i
\(59\) 7.65685 0.996838 0.498419 0.866936i \(-0.333914\pi\)
0.498419 + 0.866936i \(0.333914\pi\)
\(60\) 0 0
\(61\) 13.3137 1.70465 0.852323 0.523016i \(-0.175193\pi\)
0.852323 + 0.523016i \(0.175193\pi\)
\(62\) 0.485281i 0.0616308i
\(63\) 0 0
\(64\) 4.17157 0.521447
\(65\) 0 0
\(66\) 0 0
\(67\) 6.82843i 0.834225i 0.908855 + 0.417113i \(0.136958\pi\)
−0.908855 + 0.417113i \(0.863042\pi\)
\(68\) − 14.0000i − 1.69775i
\(69\) 0 0
\(70\) 0 0
\(71\) −2.00000 −0.237356 −0.118678 0.992933i \(-0.537866\pi\)
−0.118678 + 0.992933i \(0.537866\pi\)
\(72\) 0 0
\(73\) − 0.343146i − 0.0401622i −0.999798 0.0200811i \(-0.993608\pi\)
0.999798 0.0200811i \(-0.00639244\pi\)
\(74\) −3.17157 −0.368688
\(75\) 0 0
\(76\) 5.17157 0.593220
\(77\) 5.65685i 0.644658i
\(78\) 0 0
\(79\) 11.3137 1.27289 0.636446 0.771321i \(-0.280404\pi\)
0.636446 + 0.771321i \(0.280404\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 2.14214i 0.236559i
\(83\) 3.65685i 0.401392i 0.979654 + 0.200696i \(0.0643203\pi\)
−0.979654 + 0.200696i \(0.935680\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.686292 0.0740047
\(87\) 0 0
\(88\) − 3.17157i − 0.338091i
\(89\) 14.8284 1.57181 0.785905 0.618347i \(-0.212197\pi\)
0.785905 + 0.618347i \(0.212197\pi\)
\(90\) 0 0
\(91\) −2.82843 −0.296500
\(92\) − 7.31371i − 0.762507i
\(93\) 0 0
\(94\) 4.82843 0.498014
\(95\) 0 0
\(96\) 0 0
\(97\) 3.65685i 0.371297i 0.982616 + 0.185649i \(0.0594386\pi\)
−0.982616 + 0.185649i \(0.940561\pi\)
\(98\) 0.414214i 0.0418419i
\(99\) 0 0
\(100\) 0 0
\(101\) −7.65685 −0.761885 −0.380943 0.924599i \(-0.624401\pi\)
−0.380943 + 0.924599i \(0.624401\pi\)
\(102\) 0 0
\(103\) − 2.34315i − 0.230877i −0.993315 0.115439i \(-0.963173\pi\)
0.993315 0.115439i \(-0.0368273\pi\)
\(104\) 1.58579 0.155499
\(105\) 0 0
\(106\) −0.828427 −0.0804640
\(107\) 11.3137i 1.09374i 0.837218 + 0.546869i \(0.184180\pi\)
−0.837218 + 0.546869i \(0.815820\pi\)
\(108\) 0 0
\(109\) −5.31371 −0.508961 −0.254480 0.967078i \(-0.581904\pi\)
−0.254480 + 0.967078i \(0.581904\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 8.48528i 0.801784i
\(113\) − 5.31371i − 0.499872i −0.968262 0.249936i \(-0.919590\pi\)
0.968262 0.249936i \(-0.0804095\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 3.65685 0.339530
\(117\) 0 0
\(118\) − 3.17157i − 0.291967i
\(119\) 21.6569 1.98528
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) − 5.51472i − 0.499279i
\(123\) 0 0
\(124\) −2.14214 −0.192369
\(125\) 0 0
\(126\) 0 0
\(127\) 5.65685i 0.501965i 0.967992 + 0.250982i \(0.0807536\pi\)
−0.967992 + 0.250982i \(0.919246\pi\)
\(128\) − 10.5563i − 0.933058i
\(129\) 0 0
\(130\) 0 0
\(131\) 8.00000 0.698963 0.349482 0.936943i \(-0.386358\pi\)
0.349482 + 0.936943i \(0.386358\pi\)
\(132\) 0 0
\(133\) 8.00000i 0.693688i
\(134\) 2.82843 0.244339
\(135\) 0 0
\(136\) −12.1421 −1.04118
\(137\) 10.8284i 0.925135i 0.886584 + 0.462567i \(0.153072\pi\)
−0.886584 + 0.462567i \(0.846928\pi\)
\(138\) 0 0
\(139\) 7.31371 0.620341 0.310170 0.950681i \(-0.399614\pi\)
0.310170 + 0.950681i \(0.399614\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0.828427i 0.0695201i
\(143\) 2.00000i 0.167248i
\(144\) 0 0
\(145\) 0 0
\(146\) −0.142136 −0.0117632
\(147\) 0 0
\(148\) − 14.0000i − 1.15079i
\(149\) −9.17157 −0.751365 −0.375682 0.926749i \(-0.622592\pi\)
−0.375682 + 0.926749i \(0.622592\pi\)
\(150\) 0 0
\(151\) −3.51472 −0.286024 −0.143012 0.989721i \(-0.545679\pi\)
−0.143012 + 0.989721i \(0.545679\pi\)
\(152\) − 4.48528i − 0.363804i
\(153\) 0 0
\(154\) 2.34315 0.188816
\(155\) 0 0
\(156\) 0 0
\(157\) − 10.0000i − 0.798087i −0.916932 0.399043i \(-0.869342\pi\)
0.916932 0.399043i \(-0.130658\pi\)
\(158\) − 4.68629i − 0.372821i
\(159\) 0 0
\(160\) 0 0
\(161\) 11.3137 0.891645
\(162\) 0 0
\(163\) − 18.8284i − 1.47476i −0.675480 0.737378i \(-0.736064\pi\)
0.675480 0.737378i \(-0.263936\pi\)
\(164\) −9.45584 −0.738377
\(165\) 0 0
\(166\) 1.51472 0.117565
\(167\) 3.65685i 0.282976i 0.989940 + 0.141488i \(0.0451886\pi\)
−0.989940 + 0.141488i \(0.954811\pi\)
\(168\) 0 0
\(169\) −1.00000 −0.0769231
\(170\) 0 0
\(171\) 0 0
\(172\) 3.02944i 0.230992i
\(173\) − 11.6569i − 0.886254i −0.896459 0.443127i \(-0.853869\pi\)
0.896459 0.443127i \(-0.146131\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 6.00000 0.452267
\(177\) 0 0
\(178\) − 6.14214i − 0.460373i
\(179\) −23.3137 −1.74255 −0.871274 0.490797i \(-0.836706\pi\)
−0.871274 + 0.490797i \(0.836706\pi\)
\(180\) 0 0
\(181\) 14.0000 1.04061 0.520306 0.853980i \(-0.325818\pi\)
0.520306 + 0.853980i \(0.325818\pi\)
\(182\) 1.17157i 0.0868428i
\(183\) 0 0
\(184\) −6.34315 −0.467623
\(185\) 0 0
\(186\) 0 0
\(187\) − 15.3137i − 1.11985i
\(188\) 21.3137i 1.55446i
\(189\) 0 0
\(190\) 0 0
\(191\) −3.31371 −0.239772 −0.119886 0.992788i \(-0.538253\pi\)
−0.119886 + 0.992788i \(0.538253\pi\)
\(192\) 0 0
\(193\) − 5.31371i − 0.382489i −0.981542 0.191245i \(-0.938748\pi\)
0.981542 0.191245i \(-0.0612524\pi\)
\(194\) 1.51472 0.108750
\(195\) 0 0
\(196\) −1.82843 −0.130602
\(197\) − 0.485281i − 0.0345749i −0.999851 0.0172874i \(-0.994497\pi\)
0.999851 0.0172874i \(-0.00550304\pi\)
\(198\) 0 0
\(199\) −21.6569 −1.53521 −0.767607 0.640921i \(-0.778553\pi\)
−0.767607 + 0.640921i \(0.778553\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 3.17157i 0.223151i
\(203\) 5.65685i 0.397033i
\(204\) 0 0
\(205\) 0 0
\(206\) −0.970563 −0.0676223
\(207\) 0 0
\(208\) 3.00000i 0.208013i
\(209\) 5.65685 0.391293
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) − 3.65685i − 0.251154i
\(213\) 0 0
\(214\) 4.68629 0.320348
\(215\) 0 0
\(216\) 0 0
\(217\) − 3.31371i − 0.224949i
\(218\) 2.20101i 0.149071i
\(219\) 0 0
\(220\) 0 0
\(221\) 7.65685 0.515056
\(222\) 0 0
\(223\) 12.4853i 0.836076i 0.908429 + 0.418038i \(0.137282\pi\)
−0.908429 + 0.418038i \(0.862718\pi\)
\(224\) 12.4853 0.834208
\(225\) 0 0
\(226\) −2.20101 −0.146409
\(227\) 17.3137i 1.14915i 0.818452 + 0.574576i \(0.194833\pi\)
−0.818452 + 0.574576i \(0.805167\pi\)
\(228\) 0 0
\(229\) 1.31371 0.0868123 0.0434062 0.999058i \(-0.486179\pi\)
0.0434062 + 0.999058i \(0.486179\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 3.17157i − 0.208224i
\(233\) 6.97056i 0.456657i 0.973584 + 0.228328i \(0.0733260\pi\)
−0.973584 + 0.228328i \(0.926674\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 14.0000 0.911322
\(237\) 0 0
\(238\) − 8.97056i − 0.581475i
\(239\) 2.00000 0.129369 0.0646846 0.997906i \(-0.479396\pi\)
0.0646846 + 0.997906i \(0.479396\pi\)
\(240\) 0 0
\(241\) 0.343146 0.0221040 0.0110520 0.999939i \(-0.496482\pi\)
0.0110520 + 0.999939i \(0.496482\pi\)
\(242\) 2.89949i 0.186387i
\(243\) 0 0
\(244\) 24.3431 1.55841
\(245\) 0 0
\(246\) 0 0
\(247\) 2.82843i 0.179969i
\(248\) 1.85786i 0.117975i
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) − 8.00000i − 0.502956i
\(254\) 2.34315 0.147022
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) 4.34315i 0.270918i 0.990783 + 0.135459i \(0.0432509\pi\)
−0.990783 + 0.135459i \(0.956749\pi\)
\(258\) 0 0
\(259\) 21.6569 1.34569
\(260\) 0 0
\(261\) 0 0
\(262\) − 3.31371i − 0.204722i
\(263\) 12.0000i 0.739952i 0.929041 + 0.369976i \(0.120634\pi\)
−0.929041 + 0.369976i \(0.879366\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 3.31371 0.203177
\(267\) 0 0
\(268\) 12.4853i 0.762660i
\(269\) 18.0000 1.09748 0.548740 0.835993i \(-0.315108\pi\)
0.548740 + 0.835993i \(0.315108\pi\)
\(270\) 0 0
\(271\) −27.7990 −1.68867 −0.844334 0.535817i \(-0.820004\pi\)
−0.844334 + 0.535817i \(0.820004\pi\)
\(272\) − 22.9706i − 1.39279i
\(273\) 0 0
\(274\) 4.48528 0.270966
\(275\) 0 0
\(276\) 0 0
\(277\) − 2.00000i − 0.120168i −0.998193 0.0600842i \(-0.980863\pi\)
0.998193 0.0600842i \(-0.0191369\pi\)
\(278\) − 3.02944i − 0.181694i
\(279\) 0 0
\(280\) 0 0
\(281\) −21.1716 −1.26299 −0.631495 0.775380i \(-0.717558\pi\)
−0.631495 + 0.775380i \(0.717558\pi\)
\(282\) 0 0
\(283\) − 28.9706i − 1.72212i −0.508502 0.861061i \(-0.669801\pi\)
0.508502 0.861061i \(-0.330199\pi\)
\(284\) −3.65685 −0.216994
\(285\) 0 0
\(286\) 0.828427 0.0489859
\(287\) − 14.6274i − 0.863429i
\(288\) 0 0
\(289\) −41.6274 −2.44867
\(290\) 0 0
\(291\) 0 0
\(292\) − 0.627417i − 0.0367168i
\(293\) 2.14214i 0.125145i 0.998040 + 0.0625724i \(0.0199304\pi\)
−0.998040 + 0.0625724i \(0.980070\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −12.1421 −0.705747
\(297\) 0 0
\(298\) 3.79899i 0.220070i
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) −4.68629 −0.270113
\(302\) 1.45584i 0.0837744i
\(303\) 0 0
\(304\) 8.48528 0.486664
\(305\) 0 0
\(306\) 0 0
\(307\) − 22.8284i − 1.30289i −0.758697 0.651444i \(-0.774164\pi\)
0.758697 0.651444i \(-0.225836\pi\)
\(308\) 10.3431i 0.589355i
\(309\) 0 0
\(310\) 0 0
\(311\) 10.6274 0.602626 0.301313 0.953525i \(-0.402575\pi\)
0.301313 + 0.953525i \(0.402575\pi\)
\(312\) 0 0
\(313\) − 6.00000i − 0.339140i −0.985518 0.169570i \(-0.945762\pi\)
0.985518 0.169570i \(-0.0542379\pi\)
\(314\) −4.14214 −0.233754
\(315\) 0 0
\(316\) 20.6863 1.16369
\(317\) − 8.48528i − 0.476581i −0.971194 0.238290i \(-0.923413\pi\)
0.971194 0.238290i \(-0.0765870\pi\)
\(318\) 0 0
\(319\) 4.00000 0.223957
\(320\) 0 0
\(321\) 0 0
\(322\) − 4.68629i − 0.261157i
\(323\) − 21.6569i − 1.20502i
\(324\) 0 0
\(325\) 0 0
\(326\) −7.79899 −0.431946
\(327\) 0 0
\(328\) 8.20101i 0.452825i
\(329\) −32.9706 −1.81773
\(330\) 0 0
\(331\) 26.1421 1.43690 0.718451 0.695578i \(-0.244852\pi\)
0.718451 + 0.695578i \(0.244852\pi\)
\(332\) 6.68629i 0.366958i
\(333\) 0 0
\(334\) 1.51472 0.0828817
\(335\) 0 0
\(336\) 0 0
\(337\) 9.31371i 0.507350i 0.967290 + 0.253675i \(0.0816394\pi\)
−0.967290 + 0.253675i \(0.918361\pi\)
\(338\) 0.414214i 0.0225302i
\(339\) 0 0
\(340\) 0 0
\(341\) −2.34315 −0.126888
\(342\) 0 0
\(343\) 16.9706i 0.916324i
\(344\) 2.62742 0.141661
\(345\) 0 0
\(346\) −4.82843 −0.259578
\(347\) 8.68629i 0.466305i 0.972440 + 0.233152i \(0.0749041\pi\)
−0.972440 + 0.233152i \(0.925096\pi\)
\(348\) 0 0
\(349\) −3.65685 −0.195747 −0.0978735 0.995199i \(-0.531204\pi\)
−0.0978735 + 0.995199i \(0.531204\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 8.82843i − 0.470557i
\(353\) − 33.4558i − 1.78067i −0.455301 0.890337i \(-0.650468\pi\)
0.455301 0.890337i \(-0.349532\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 27.1127 1.43697
\(357\) 0 0
\(358\) 9.65685i 0.510381i
\(359\) 34.9706 1.84568 0.922838 0.385189i \(-0.125864\pi\)
0.922838 + 0.385189i \(0.125864\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) − 5.79899i − 0.304788i
\(363\) 0 0
\(364\) −5.17157 −0.271064
\(365\) 0 0
\(366\) 0 0
\(367\) − 24.0000i − 1.25279i −0.779506 0.626395i \(-0.784530\pi\)
0.779506 0.626395i \(-0.215470\pi\)
\(368\) − 12.0000i − 0.625543i
\(369\) 0 0
\(370\) 0 0
\(371\) 5.65685 0.293689
\(372\) 0 0
\(373\) − 10.0000i − 0.517780i −0.965907 0.258890i \(-0.916643\pi\)
0.965907 0.258890i \(-0.0833568\pi\)
\(374\) −6.34315 −0.327996
\(375\) 0 0
\(376\) 18.4853 0.953306
\(377\) 2.00000i 0.103005i
\(378\) 0 0
\(379\) −0.485281 −0.0249272 −0.0124636 0.999922i \(-0.503967\pi\)
−0.0124636 + 0.999922i \(0.503967\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1.37258i 0.0702275i
\(383\) 30.9706i 1.58252i 0.611479 + 0.791261i \(0.290575\pi\)
−0.611479 + 0.791261i \(0.709425\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.20101 −0.112028
\(387\) 0 0
\(388\) 6.68629i 0.339445i
\(389\) −26.9706 −1.36746 −0.683731 0.729734i \(-0.739644\pi\)
−0.683731 + 0.729734i \(0.739644\pi\)
\(390\) 0 0
\(391\) −30.6274 −1.54890
\(392\) 1.58579i 0.0800943i
\(393\) 0 0
\(394\) −0.201010 −0.0101267
\(395\) 0 0
\(396\) 0 0
\(397\) 30.9706i 1.55437i 0.629273 + 0.777184i \(0.283353\pi\)
−0.629273 + 0.777184i \(0.716647\pi\)
\(398\) 8.97056i 0.449654i
\(399\) 0 0
\(400\) 0 0
\(401\) −26.1421 −1.30548 −0.652738 0.757584i \(-0.726380\pi\)
−0.652738 + 0.757584i \(0.726380\pi\)
\(402\) 0 0
\(403\) − 1.17157i − 0.0583602i
\(404\) −14.0000 −0.696526
\(405\) 0 0
\(406\) 2.34315 0.116288
\(407\) − 15.3137i − 0.759072i
\(408\) 0 0
\(409\) 34.9706 1.72918 0.864592 0.502475i \(-0.167577\pi\)
0.864592 + 0.502475i \(0.167577\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 4.28427i − 0.211071i
\(413\) 21.6569i 1.06566i
\(414\) 0 0
\(415\) 0 0
\(416\) 4.41421 0.216425
\(417\) 0 0
\(418\) − 2.34315i − 0.114607i
\(419\) 14.6274 0.714596 0.357298 0.933990i \(-0.383698\pi\)
0.357298 + 0.933990i \(0.383698\pi\)
\(420\) 0 0
\(421\) 37.3137 1.81856 0.909279 0.416186i \(-0.136634\pi\)
0.909279 + 0.416186i \(0.136634\pi\)
\(422\) 4.97056i 0.241963i
\(423\) 0 0
\(424\) −3.17157 −0.154025
\(425\) 0 0
\(426\) 0 0
\(427\) 37.6569i 1.82234i
\(428\) 20.6863i 0.999910i
\(429\) 0 0
\(430\) 0 0
\(431\) −8.34315 −0.401875 −0.200938 0.979604i \(-0.564399\pi\)
−0.200938 + 0.979604i \(0.564399\pi\)
\(432\) 0 0
\(433\) 21.3137i 1.02427i 0.858905 + 0.512136i \(0.171146\pi\)
−0.858905 + 0.512136i \(0.828854\pi\)
\(434\) −1.37258 −0.0658861
\(435\) 0 0
\(436\) −9.71573 −0.465299
\(437\) − 11.3137i − 0.541208i
\(438\) 0 0
\(439\) −16.9706 −0.809961 −0.404980 0.914325i \(-0.632722\pi\)
−0.404980 + 0.914325i \(0.632722\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 3.17157i − 0.150856i
\(443\) − 25.9411i − 1.23250i −0.787551 0.616250i \(-0.788651\pi\)
0.787551 0.616250i \(-0.211349\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 5.17157 0.244881
\(447\) 0 0
\(448\) 11.7990i 0.557450i
\(449\) −31.7990 −1.50069 −0.750344 0.661048i \(-0.770112\pi\)
−0.750344 + 0.661048i \(0.770112\pi\)
\(450\) 0 0
\(451\) −10.3431 −0.487040
\(452\) − 9.71573i − 0.456989i
\(453\) 0 0
\(454\) 7.17157 0.336579
\(455\) 0 0
\(456\) 0 0
\(457\) − 7.65685i − 0.358173i −0.983833 0.179086i \(-0.942686\pi\)
0.983833 0.179086i \(-0.0573141\pi\)
\(458\) − 0.544156i − 0.0254267i
\(459\) 0 0
\(460\) 0 0
\(461\) −5.17157 −0.240864 −0.120432 0.992722i \(-0.538428\pi\)
−0.120432 + 0.992722i \(0.538428\pi\)
\(462\) 0 0
\(463\) 24.4853i 1.13793i 0.822363 + 0.568964i \(0.192656\pi\)
−0.822363 + 0.568964i \(0.807344\pi\)
\(464\) 6.00000 0.278543
\(465\) 0 0
\(466\) 2.88730 0.133752
\(467\) 8.00000i 0.370196i 0.982720 + 0.185098i \(0.0592602\pi\)
−0.982720 + 0.185098i \(0.940740\pi\)
\(468\) 0 0
\(469\) −19.3137 −0.891824
\(470\) 0 0
\(471\) 0 0
\(472\) − 12.1421i − 0.558887i
\(473\) 3.31371i 0.152364i
\(474\) 0 0
\(475\) 0 0
\(476\) 39.5980 1.81497
\(477\) 0 0
\(478\) − 0.828427i − 0.0378914i
\(479\) 25.3137 1.15661 0.578306 0.815820i \(-0.303714\pi\)
0.578306 + 0.815820i \(0.303714\pi\)
\(480\) 0 0
\(481\) 7.65685 0.349123
\(482\) − 0.142136i − 0.00647410i
\(483\) 0 0
\(484\) −12.7990 −0.581772
\(485\) 0 0
\(486\) 0 0
\(487\) − 7.79899i − 0.353406i −0.984264 0.176703i \(-0.943457\pi\)
0.984264 0.176703i \(-0.0565432\pi\)
\(488\) − 21.1127i − 0.955727i
\(489\) 0 0
\(490\) 0 0
\(491\) −30.6274 −1.38220 −0.691098 0.722761i \(-0.742873\pi\)
−0.691098 + 0.722761i \(0.742873\pi\)
\(492\) 0 0
\(493\) − 15.3137i − 0.689695i
\(494\) 1.17157 0.0527116
\(495\) 0 0
\(496\) −3.51472 −0.157816
\(497\) − 5.65685i − 0.253745i
\(498\) 0 0
\(499\) −26.1421 −1.17028 −0.585141 0.810931i \(-0.698961\pi\)
−0.585141 + 0.810931i \(0.698961\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 7.31371i − 0.326102i −0.986618 0.163051i \(-0.947866\pi\)
0.986618 0.163051i \(-0.0521335\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −3.31371 −0.147312
\(507\) 0 0
\(508\) 10.3431i 0.458903i
\(509\) 11.7990 0.522981 0.261491 0.965206i \(-0.415786\pi\)
0.261491 + 0.965206i \(0.415786\pi\)
\(510\) 0 0
\(511\) 0.970563 0.0429352
\(512\) − 22.7574i − 1.00574i
\(513\) 0 0
\(514\) 1.79899 0.0793500
\(515\) 0 0
\(516\) 0 0
\(517\) 23.3137i 1.02534i
\(518\) − 8.97056i − 0.394144i
\(519\) 0 0
\(520\) 0 0
\(521\) −25.3137 −1.10901 −0.554507 0.832179i \(-0.687093\pi\)
−0.554507 + 0.832179i \(0.687093\pi\)
\(522\) 0 0
\(523\) 15.3137i 0.669622i 0.942285 + 0.334811i \(0.108672\pi\)
−0.942285 + 0.334811i \(0.891328\pi\)
\(524\) 14.6274 0.639002
\(525\) 0 0
\(526\) 4.97056 0.216727
\(527\) 8.97056i 0.390764i
\(528\) 0 0
\(529\) 7.00000 0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 14.6274i 0.634179i
\(533\) − 5.17157i − 0.224006i
\(534\) 0 0
\(535\) 0 0
\(536\) 10.8284 0.467717
\(537\) 0 0
\(538\) − 7.45584i − 0.321444i
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) 11.5147i 0.494600i
\(543\) 0 0
\(544\) −33.7990 −1.44912
\(545\) 0 0
\(546\) 0 0
\(547\) 23.3137i 0.996822i 0.866941 + 0.498411i \(0.166083\pi\)
−0.866941 + 0.498411i \(0.833917\pi\)
\(548\) 19.7990i 0.845771i
\(549\) 0 0
\(550\) 0 0
\(551\) 5.65685 0.240990
\(552\) 0 0
\(553\) 32.0000i 1.36078i
\(554\) −0.828427 −0.0351965
\(555\) 0 0
\(556\) 13.3726 0.567124
\(557\) 7.79899i 0.330454i 0.986256 + 0.165227i \(0.0528356\pi\)
−0.986256 + 0.165227i \(0.947164\pi\)
\(558\) 0 0
\(559\) −1.65685 −0.0700775
\(560\) 0 0
\(561\) 0 0
\(562\) 8.76955i 0.369921i
\(563\) 4.00000i 0.168580i 0.996441 + 0.0842900i \(0.0268622\pi\)
−0.996441 + 0.0842900i \(0.973138\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −12.0000 −0.504398
\(567\) 0 0
\(568\) 3.17157i 0.133076i
\(569\) −42.9706 −1.80142 −0.900710 0.434421i \(-0.856953\pi\)
−0.900710 + 0.434421i \(0.856953\pi\)
\(570\) 0 0
\(571\) −12.9706 −0.542801 −0.271401 0.962466i \(-0.587487\pi\)
−0.271401 + 0.962466i \(0.587487\pi\)
\(572\) 3.65685i 0.152901i
\(573\) 0 0
\(574\) −6.05887 −0.252893
\(575\) 0 0
\(576\) 0 0
\(577\) − 31.9411i − 1.32973i −0.746965 0.664863i \(-0.768490\pi\)
0.746965 0.664863i \(-0.231510\pi\)
\(578\) 17.2426i 0.717199i
\(579\) 0 0
\(580\) 0 0
\(581\) −10.3431 −0.429106
\(582\) 0 0
\(583\) − 4.00000i − 0.165663i
\(584\) −0.544156 −0.0225173
\(585\) 0 0
\(586\) 0.887302 0.0366541
\(587\) 10.9706i 0.452804i 0.974034 + 0.226402i \(0.0726962\pi\)
−0.974034 + 0.226402i \(0.927304\pi\)
\(588\) 0 0
\(589\) −3.31371 −0.136539
\(590\) 0 0
\(591\) 0 0
\(592\) − 22.9706i − 0.944084i
\(593\) − 20.4853i − 0.841230i −0.907239 0.420615i \(-0.861814\pi\)
0.907239 0.420615i \(-0.138186\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −16.7696 −0.686908
\(597\) 0 0
\(598\) − 1.65685i − 0.0677538i
\(599\) −23.3137 −0.952572 −0.476286 0.879290i \(-0.658017\pi\)
−0.476286 + 0.879290i \(0.658017\pi\)
\(600\) 0 0
\(601\) −0.627417 −0.0255929 −0.0127964 0.999918i \(-0.504073\pi\)
−0.0127964 + 0.999918i \(0.504073\pi\)
\(602\) 1.94113i 0.0791144i
\(603\) 0 0
\(604\) −6.42641 −0.261487
\(605\) 0 0
\(606\) 0 0
\(607\) 41.9411i 1.70234i 0.524892 + 0.851169i \(0.324106\pi\)
−0.524892 + 0.851169i \(0.675894\pi\)
\(608\) − 12.4853i − 0.506345i
\(609\) 0 0
\(610\) 0 0
\(611\) −11.6569 −0.471586
\(612\) 0 0
\(613\) 47.6569i 1.92484i 0.271561 + 0.962421i \(0.412460\pi\)
−0.271561 + 0.962421i \(0.587540\pi\)
\(614\) −9.45584 −0.381607
\(615\) 0 0
\(616\) 8.97056 0.361434
\(617\) 34.8284i 1.40214i 0.713093 + 0.701070i \(0.247294\pi\)
−0.713093 + 0.701070i \(0.752706\pi\)
\(618\) 0 0
\(619\) −23.7990 −0.956562 −0.478281 0.878207i \(-0.658740\pi\)
−0.478281 + 0.878207i \(0.658740\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 4.40202i − 0.176505i
\(623\) 41.9411i 1.68034i
\(624\) 0 0
\(625\) 0 0
\(626\) −2.48528 −0.0993318
\(627\) 0 0
\(628\) − 18.2843i − 0.729622i
\(629\) −58.6274 −2.33763
\(630\) 0 0
\(631\) 43.1127 1.71629 0.858145 0.513408i \(-0.171617\pi\)
0.858145 + 0.513408i \(0.171617\pi\)
\(632\) − 17.9411i − 0.713660i
\(633\) 0 0
\(634\) −3.51472 −0.139587
\(635\) 0 0
\(636\) 0 0
\(637\) − 1.00000i − 0.0396214i
\(638\) − 1.65685i − 0.0655955i
\(639\) 0 0
\(640\) 0 0
\(641\) −30.2843 −1.19616 −0.598078 0.801438i \(-0.704069\pi\)
−0.598078 + 0.801438i \(0.704069\pi\)
\(642\) 0 0
\(643\) − 22.8284i − 0.900265i −0.892962 0.450133i \(-0.851377\pi\)
0.892962 0.450133i \(-0.148623\pi\)
\(644\) 20.6863 0.815154
\(645\) 0 0
\(646\) −8.97056 −0.352942
\(647\) − 11.3137i − 0.444788i −0.974957 0.222394i \(-0.928613\pi\)
0.974957 0.222394i \(-0.0713871\pi\)
\(648\) 0 0
\(649\) 15.3137 0.601116
\(650\) 0 0
\(651\) 0 0
\(652\) − 34.4264i − 1.34824i
\(653\) − 25.3137i − 0.990602i −0.868721 0.495301i \(-0.835058\pi\)
0.868721 0.495301i \(-0.164942\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −15.5147 −0.605748
\(657\) 0 0
\(658\) 13.6569i 0.532400i
\(659\) −47.3137 −1.84308 −0.921540 0.388283i \(-0.873068\pi\)
−0.921540 + 0.388283i \(0.873068\pi\)
\(660\) 0 0
\(661\) −34.9706 −1.36020 −0.680099 0.733121i \(-0.738063\pi\)
−0.680099 + 0.733121i \(0.738063\pi\)
\(662\) − 10.8284i − 0.420859i
\(663\) 0 0
\(664\) 5.79899 0.225044
\(665\) 0 0
\(666\) 0 0
\(667\) − 8.00000i − 0.309761i
\(668\) 6.68629i 0.258700i
\(669\) 0 0
\(670\) 0 0
\(671\) 26.6274 1.02794
\(672\) 0 0
\(673\) − 16.6274i − 0.640940i −0.947259 0.320470i \(-0.896159\pi\)
0.947259 0.320470i \(-0.103841\pi\)
\(674\) 3.85786 0.148599
\(675\) 0 0
\(676\) −1.82843 −0.0703241
\(677\) − 26.6863i − 1.02564i −0.858497 0.512819i \(-0.828601\pi\)
0.858497 0.512819i \(-0.171399\pi\)
\(678\) 0 0
\(679\) −10.3431 −0.396934
\(680\) 0 0
\(681\) 0 0
\(682\) 0.970563i 0.0371648i
\(683\) 47.9411i 1.83442i 0.398408 + 0.917208i \(0.369563\pi\)
−0.398408 + 0.917208i \(0.630437\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 7.02944 0.268385
\(687\) 0 0
\(688\) 4.97056i 0.189501i
\(689\) 2.00000 0.0761939
\(690\) 0 0
\(691\) −5.85786 −0.222844 −0.111422 0.993773i \(-0.535540\pi\)
−0.111422 + 0.993773i \(0.535540\pi\)
\(692\) − 21.3137i − 0.810226i
\(693\) 0 0
\(694\) 3.59798 0.136577
\(695\) 0 0
\(696\) 0 0
\(697\) 39.5980i 1.49988i
\(698\) 1.51472i 0.0573329i
\(699\) 0 0
\(700\) 0 0
\(701\) −5.02944 −0.189959 −0.0949796 0.995479i \(-0.530279\pi\)
−0.0949796 + 0.995479i \(0.530279\pi\)
\(702\) 0 0
\(703\) − 21.6569i − 0.816804i
\(704\) 8.34315 0.314444
\(705\) 0 0
\(706\) −13.8579 −0.521548
\(707\) − 21.6569i − 0.814490i
\(708\) 0 0
\(709\) 4.62742 0.173786 0.0868931 0.996218i \(-0.472306\pi\)
0.0868931 + 0.996218i \(0.472306\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 23.5147i − 0.881251i
\(713\) 4.68629i 0.175503i
\(714\) 0 0
\(715\) 0 0
\(716\) −42.6274 −1.59306
\(717\) 0 0
\(718\) − 14.4853i − 0.540586i
\(719\) −29.9411 −1.11662 −0.558308 0.829634i \(-0.688549\pi\)
−0.558308 + 0.829634i \(0.688549\pi\)
\(720\) 0 0
\(721\) 6.62742 0.246818
\(722\) 4.55635i 0.169570i
\(723\) 0 0
\(724\) 25.5980 0.951341
\(725\) 0 0
\(726\) 0 0
\(727\) − 10.3431i − 0.383606i −0.981433 0.191803i \(-0.938567\pi\)
0.981433 0.191803i \(-0.0614335\pi\)
\(728\) 4.48528i 0.166236i
\(729\) 0 0
\(730\) 0 0
\(731\) 12.6863 0.469219
\(732\) 0 0
\(733\) 36.6274i 1.35286i 0.736505 + 0.676432i \(0.236475\pi\)
−0.736505 + 0.676432i \(0.763525\pi\)
\(734\) −9.94113 −0.366934
\(735\) 0 0
\(736\) −17.6569 −0.650840
\(737\) 13.6569i 0.503057i
\(738\) 0 0
\(739\) 18.1421 0.667369 0.333685 0.942685i \(-0.391708\pi\)
0.333685 + 0.942685i \(0.391708\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) − 2.34315i − 0.0860196i
\(743\) 2.00000i 0.0733729i 0.999327 + 0.0366864i \(0.0116803\pi\)
−0.999327 + 0.0366864i \(0.988320\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −4.14214 −0.151654
\(747\) 0 0
\(748\) − 28.0000i − 1.02378i
\(749\) −32.0000 −1.16925
\(750\) 0 0
\(751\) −0.970563 −0.0354163 −0.0177082 0.999843i \(-0.505637\pi\)
−0.0177082 + 0.999843i \(0.505637\pi\)
\(752\) 34.9706i 1.27525i
\(753\) 0 0
\(754\) 0.828427 0.0301695
\(755\) 0 0
\(756\) 0 0
\(757\) 51.9411i 1.88783i 0.330185 + 0.943916i \(0.392889\pi\)
−0.330185 + 0.943916i \(0.607111\pi\)
\(758\) 0.201010i 0.00730102i
\(759\) 0 0
\(760\) 0 0
\(761\) −32.4853 −1.17759 −0.588795 0.808282i \(-0.700398\pi\)
−0.588795 + 0.808282i \(0.700398\pi\)
\(762\) 0 0
\(763\) − 15.0294i − 0.544102i
\(764\) −6.05887 −0.219202
\(765\) 0 0
\(766\) 12.8284 0.463510
\(767\) 7.65685i 0.276473i
\(768\) 0 0
\(769\) −42.0000 −1.51456 −0.757279 0.653091i \(-0.773472\pi\)
−0.757279 + 0.653091i \(0.773472\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 9.71573i − 0.349677i
\(773\) 34.1421i 1.22801i 0.789303 + 0.614004i \(0.210442\pi\)
−0.789303 + 0.614004i \(0.789558\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 5.79899 0.208172
\(777\) 0 0
\(778\) 11.1716i 0.400520i
\(779\) −14.6274 −0.524082
\(780\) 0 0
\(781\) −4.00000 −0.143131
\(782\) 12.6863i 0.453661i
\(783\) 0 0
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) 0 0
\(787\) − 40.7696i − 1.45328i −0.687020 0.726639i \(-0.741081\pi\)
0.687020 0.726639i \(-0.258919\pi\)
\(788\) − 0.887302i − 0.0316088i
\(789\) 0 0
\(790\) 0 0
\(791\) 15.0294 0.534385
\(792\) 0 0
\(793\) 13.3137i 0.472784i
\(794\) 12.8284 0.455264
\(795\) 0 0
\(796\) −39.5980 −1.40351
\(797\) − 24.3431i − 0.862278i −0.902285 0.431139i \(-0.858112\pi\)
0.902285 0.431139i \(-0.141888\pi\)
\(798\) 0 0
\(799\) 89.2548 3.15761
\(800\) 0 0
\(801\) 0 0
\(802\) 10.8284i 0.382365i
\(803\) − 0.686292i − 0.0242187i
\(804\) 0 0
\(805\) 0 0
\(806\) −0.485281 −0.0170933
\(807\) 0 0
\(808\) 12.1421i 0.427159i
\(809\) 18.6863 0.656975 0.328488 0.944508i \(-0.393461\pi\)
0.328488 + 0.944508i \(0.393461\pi\)
\(810\) 0 0
\(811\) −30.1421 −1.05843 −0.529217 0.848487i \(-0.677514\pi\)
−0.529217 + 0.848487i \(0.677514\pi\)
\(812\) 10.3431i 0.362973i
\(813\) 0 0
\(814\) −6.34315 −0.222327
\(815\) 0 0
\(816\) 0 0
\(817\) 4.68629i 0.163953i
\(818\) − 14.4853i − 0.506466i
\(819\) 0 0
\(820\) 0 0
\(821\) 23.7990 0.830590 0.415295 0.909687i \(-0.363678\pi\)
0.415295 + 0.909687i \(0.363678\pi\)
\(822\) 0 0
\(823\) − 15.0294i − 0.523893i −0.965082 0.261947i \(-0.915636\pi\)
0.965082 0.261947i \(-0.0843644\pi\)
\(824\) −3.71573 −0.129444
\(825\) 0 0
\(826\) 8.97056 0.312126
\(827\) 26.0000i 0.904109i 0.891990 + 0.452054i \(0.149309\pi\)
−0.891990 + 0.452054i \(0.850691\pi\)
\(828\) 0 0
\(829\) 17.3137 0.601330 0.300665 0.953730i \(-0.402791\pi\)
0.300665 + 0.953730i \(0.402791\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 4.17157i 0.144623i
\(833\) 7.65685i 0.265294i
\(834\) 0 0
\(835\) 0 0
\(836\) 10.3431 0.357725
\(837\) 0 0
\(838\) − 6.05887i − 0.209300i
\(839\) 43.2548 1.49332 0.746661 0.665204i \(-0.231656\pi\)
0.746661 + 0.665204i \(0.231656\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) − 15.4558i − 0.532644i
\(843\) 0 0
\(844\) −21.9411 −0.755245
\(845\) 0 0
\(846\) 0 0
\(847\) − 19.7990i − 0.680301i
\(848\) − 6.00000i − 0.206041i
\(849\) 0 0
\(850\) 0 0
\(851\) −30.6274 −1.04989
\(852\) 0 0
\(853\) − 3.65685i − 0.125208i −0.998038 0.0626042i \(-0.980059\pi\)
0.998038 0.0626042i \(-0.0199406\pi\)
\(854\) 15.5980 0.533752
\(855\) 0 0
\(856\) 17.9411 0.613215
\(857\) 49.5980i 1.69423i 0.531406 + 0.847117i \(0.321664\pi\)
−0.531406 + 0.847117i \(0.678336\pi\)
\(858\) 0 0
\(859\) 0.686292 0.0234160 0.0117080 0.999931i \(-0.496273\pi\)
0.0117080 + 0.999931i \(0.496273\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 3.45584i 0.117707i
\(863\) − 28.3431i − 0.964812i −0.875948 0.482406i \(-0.839763\pi\)
0.875948 0.482406i \(-0.160237\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 8.82843 0.300002
\(867\) 0 0
\(868\) − 6.05887i − 0.205652i
\(869\) 22.6274 0.767583
\(870\) 0 0
\(871\) −6.82843 −0.231372
\(872\) 8.42641i 0.285354i
\(873\) 0 0
\(874\) −4.68629 −0.158516
\(875\) 0 0
\(876\) 0 0
\(877\) 42.2843i 1.42784i 0.700228 + 0.713919i \(0.253082\pi\)
−0.700228 + 0.713919i \(0.746918\pi\)
\(878\) 7.02944i 0.237232i
\(879\) 0 0
\(880\) 0 0
\(881\) 25.5980 0.862418 0.431209 0.902252i \(-0.358087\pi\)
0.431209 + 0.902252i \(0.358087\pi\)
\(882\) 0 0
\(883\) − 27.5980i − 0.928746i −0.885640 0.464373i \(-0.846280\pi\)
0.885640 0.464373i \(-0.153720\pi\)
\(884\) 14.0000 0.470871
\(885\) 0 0
\(886\) −10.7452 −0.360991
\(887\) 8.00000i 0.268614i 0.990940 + 0.134307i \(0.0428808\pi\)
−0.990940 + 0.134307i \(0.957119\pi\)
\(888\) 0 0
\(889\) −16.0000 −0.536623
\(890\) 0 0
\(891\) 0 0
\(892\) 22.8284i 0.764352i
\(893\) 32.9706i 1.10332i
\(894\) 0 0
\(895\) 0 0
\(896\) 29.8579 0.997481
\(897\) 0 0
\(898\) 13.1716i 0.439541i
\(899\) −2.34315 −0.0781483
\(900\) 0 0
\(901\) −15.3137 −0.510174
\(902\) 4.28427i 0.142651i
\(903\) 0 0
\(904\) −8.42641 −0.280258
\(905\) 0 0
\(906\) 0 0
\(907\) − 12.9706i − 0.430680i −0.976539 0.215340i \(-0.930914\pi\)
0.976539 0.215340i \(-0.0690860\pi\)
\(908\) 31.6569i 1.05057i
\(909\) 0 0
\(910\) 0 0
\(911\) 40.0000 1.32526 0.662630 0.748947i \(-0.269440\pi\)
0.662630 + 0.748947i \(0.269440\pi\)
\(912\) 0 0
\(913\) 7.31371i 0.242048i
\(914\) −3.17157 −0.104906
\(915\) 0 0
\(916\) 2.40202 0.0793650
\(917\) 22.6274i 0.747223i
\(918\) 0 0
\(919\) −3.31371 −0.109309 −0.0546546 0.998505i \(-0.517406\pi\)
−0.0546546 + 0.998505i \(0.517406\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 2.14214i 0.0705475i
\(923\) − 2.00000i − 0.0658308i
\(924\) 0 0
\(925\) 0 0
\(926\) 10.1421 0.333291
\(927\) 0 0
\(928\) − 8.82843i − 0.289807i
\(929\) 11.7990 0.387112 0.193556 0.981089i \(-0.437998\pi\)
0.193556 + 0.981089i \(0.437998\pi\)
\(930\) 0 0
\(931\) −2.82843 −0.0926980
\(932\) 12.7452i 0.417482i
\(933\) 0 0
\(934\) 3.31371 0.108428
\(935\) 0 0
\(936\) 0 0
\(937\) − 21.3137i − 0.696289i −0.937441 0.348144i \(-0.886812\pi\)
0.937441 0.348144i \(-0.113188\pi\)
\(938\) 8.00000i 0.261209i
\(939\) 0 0
\(940\) 0 0
\(941\) −34.1421 −1.11300 −0.556501 0.830847i \(-0.687856\pi\)
−0.556501 + 0.830847i \(0.687856\pi\)
\(942\) 0 0
\(943\) 20.6863i 0.673638i
\(944\) 22.9706 0.747628
\(945\) 0 0
\(946\) 1.37258 0.0446265
\(947\) 21.0294i 0.683365i 0.939815 + 0.341682i \(0.110997\pi\)
−0.939815 + 0.341682i \(0.889003\pi\)
\(948\) 0 0
\(949\) 0.343146 0.0111390
\(950\) 0 0
\(951\) 0 0
\(952\) − 34.3431i − 1.11307i
\(953\) 40.3431i 1.30684i 0.756994 + 0.653421i \(0.226667\pi\)
−0.756994 + 0.653421i \(0.773333\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 3.65685 0.118271
\(957\) 0 0
\(958\) − 10.4853i − 0.338764i
\(959\) −30.6274 −0.989011
\(960\) 0 0
\(961\) −29.6274 −0.955723
\(962\) − 3.17157i − 0.102256i
\(963\) 0 0
\(964\) 0.627417 0.0202077
\(965\) 0 0
\(966\) 0 0
\(967\) 18.1421i 0.583412i 0.956508 + 0.291706i \(0.0942228\pi\)
−0.956508 + 0.291706i \(0.905777\pi\)
\(968\) 11.1005i 0.356784i
\(969\) 0 0
\(970\) 0 0
\(971\) 15.3137 0.491440 0.245720 0.969341i \(-0.420976\pi\)
0.245720 + 0.969341i \(0.420976\pi\)
\(972\) 0 0
\(973\) 20.6863i 0.663172i
\(974\) −3.23045 −0.103510
\(975\) 0 0
\(976\) 39.9411 1.27848
\(977\) − 42.1421i − 1.34825i −0.738619 0.674123i \(-0.764522\pi\)
0.738619 0.674123i \(-0.235478\pi\)
\(978\) 0 0
\(979\) 29.6569 0.947837
\(980\) 0 0
\(981\) 0 0
\(982\) 12.6863i 0.404836i
\(983\) 25.3137i 0.807382i 0.914895 + 0.403691i \(0.132273\pi\)
−0.914895 + 0.403691i \(0.867727\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −6.34315 −0.202007
\(987\) 0 0
\(988\) 5.17157i 0.164530i
\(989\) 6.62742 0.210740
\(990\) 0 0
\(991\) 4.68629 0.148865 0.0744325 0.997226i \(-0.476285\pi\)
0.0744325 + 0.997226i \(0.476285\pi\)
\(992\) 5.17157i 0.164198i
\(993\) 0 0
\(994\) −2.34315 −0.0743201
\(995\) 0 0
\(996\) 0 0
\(997\) − 39.2548i − 1.24321i −0.783330 0.621607i \(-0.786480\pi\)
0.783330 0.621607i \(-0.213520\pi\)
\(998\) 10.8284i 0.342768i
\(999\) 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 2925.2.c.u.2224.2 4
3.2 odd 2 975.2.c.h.274.3 4
5.2 odd 4 2925.2.a.v.1.2 2
5.3 odd 4 117.2.a.c.1.1 2
5.4 even 2 inner 2925.2.c.u.2224.3 4
15.2 even 4 975.2.a.l.1.1 2
15.8 even 4 39.2.a.b.1.2 2
15.14 odd 2 975.2.c.h.274.2 4
20.3 even 4 1872.2.a.w.1.2 2
35.13 even 4 5733.2.a.u.1.1 2
40.3 even 4 7488.2.a.co.1.1 2
40.13 odd 4 7488.2.a.cl.1.1 2
45.13 odd 12 1053.2.e.e.703.2 4
45.23 even 12 1053.2.e.m.703.1 4
45.38 even 12 1053.2.e.m.352.1 4
45.43 odd 12 1053.2.e.e.352.2 4
60.23 odd 4 624.2.a.k.1.1 2
65.8 even 4 1521.2.b.j.1351.2 4
65.18 even 4 1521.2.b.j.1351.3 4
65.38 odd 4 1521.2.a.f.1.2 2
105.83 odd 4 1911.2.a.h.1.2 2
120.53 even 4 2496.2.a.bf.1.2 2
120.83 odd 4 2496.2.a.bi.1.2 2
165.98 odd 4 4719.2.a.p.1.1 2
195.8 odd 4 507.2.b.e.337.3 4
195.23 even 12 507.2.e.d.22.2 4
195.38 even 4 507.2.a.h.1.1 2
195.68 even 12 507.2.e.h.22.1 4
195.83 odd 4 507.2.b.e.337.2 4
195.98 odd 12 507.2.j.f.361.3 8
195.113 even 12 507.2.e.h.484.1 4
195.128 odd 12 507.2.j.f.316.2 8
195.158 odd 12 507.2.j.f.316.3 8
195.173 even 12 507.2.e.d.484.2 4
195.188 odd 12 507.2.j.f.361.2 8
780.623 odd 4 8112.2.a.bm.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.a.b.1.2 2 15.8 even 4
117.2.a.c.1.1 2 5.3 odd 4
507.2.a.h.1.1 2 195.38 even 4
507.2.b.e.337.2 4 195.83 odd 4
507.2.b.e.337.3 4 195.8 odd 4
507.2.e.d.22.2 4 195.23 even 12
507.2.e.d.484.2 4 195.173 even 12
507.2.e.h.22.1 4 195.68 even 12
507.2.e.h.484.1 4 195.113 even 12
507.2.j.f.316.2 8 195.128 odd 12
507.2.j.f.316.3 8 195.158 odd 12
507.2.j.f.361.2 8 195.188 odd 12
507.2.j.f.361.3 8 195.98 odd 12
624.2.a.k.1.1 2 60.23 odd 4
975.2.a.l.1.1 2 15.2 even 4
975.2.c.h.274.2 4 15.14 odd 2
975.2.c.h.274.3 4 3.2 odd 2
1053.2.e.e.352.2 4 45.43 odd 12
1053.2.e.e.703.2 4 45.13 odd 12
1053.2.e.m.352.1 4 45.38 even 12
1053.2.e.m.703.1 4 45.23 even 12
1521.2.a.f.1.2 2 65.38 odd 4
1521.2.b.j.1351.2 4 65.8 even 4
1521.2.b.j.1351.3 4 65.18 even 4
1872.2.a.w.1.2 2 20.3 even 4
1911.2.a.h.1.2 2 105.83 odd 4
2496.2.a.bf.1.2 2 120.53 even 4
2496.2.a.bi.1.2 2 120.83 odd 4
2925.2.a.v.1.2 2 5.2 odd 4
2925.2.c.u.2224.2 4 1.1 even 1 trivial
2925.2.c.u.2224.3 4 5.4 even 2 inner
4719.2.a.p.1.1 2 165.98 odd 4
5733.2.a.u.1.1 2 35.13 even 4
7488.2.a.cl.1.1 2 40.13 odd 4
7488.2.a.co.1.1 2 40.3 even 4
8112.2.a.bm.1.2 2 780.623 odd 4