Properties

Label 1225.2.b.i.99.2
Level $1225$
Weight $2$
Character 1225.99
Analytic conductor $9.782$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.78167424761\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 245)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.2
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1225.99
Dual form 1225.2.b.i.99.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421i q^{2} -0.414214i q^{3} -0.585786 q^{6} -2.82843i q^{8} +2.82843 q^{9} +O(q^{10})\) \(q-1.41421i q^{2} -0.414214i q^{3} -0.585786 q^{6} -2.82843i q^{8} +2.82843 q^{9} -0.171573 q^{11} +4.41421i q^{13} -4.00000 q^{16} -3.24264i q^{17} -4.00000i q^{18} +6.00000 q^{19} +0.242641i q^{22} -7.41421i q^{23} -1.17157 q^{24} +6.24264 q^{26} -2.41421i q^{27} +8.65685 q^{29} -10.2426 q^{31} +0.0710678i q^{33} -4.58579 q^{34} +2.24264i q^{37} -8.48528i q^{38} +1.82843 q^{39} +6.24264 q^{41} -2.00000i q^{43} -10.4853 q^{46} +7.24264i q^{47} +1.65685i q^{48} -1.34315 q^{51} -4.24264i q^{53} -3.41421 q^{54} -2.48528i q^{57} -12.2426i q^{58} -2.24264 q^{59} +2.82843 q^{61} +14.4853i q^{62} -8.00000 q^{64} +0.100505 q^{66} -8.24264i q^{67} -3.07107 q^{69} -3.17157 q^{71} -8.00000i q^{72} -8.48528i q^{73} +3.17157 q^{74} -2.58579i q^{78} -1.48528 q^{79} +7.48528 q^{81} -8.82843i q^{82} -2.82843 q^{86} -3.58579i q^{87} +0.485281i q^{88} -8.00000 q^{89} +4.24264i q^{93} +10.2426 q^{94} -13.2426i q^{97} -0.485281 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 8q^{6} + O(q^{10}) \) \( 4q - 8q^{6} - 12q^{11} - 16q^{16} + 24q^{19} - 16q^{24} + 8q^{26} + 12q^{29} - 24q^{31} - 24q^{34} - 4q^{39} + 8q^{41} - 8q^{46} - 28q^{51} - 8q^{54} + 8q^{59} - 32q^{64} + 40q^{66} + 16q^{69} - 24q^{71} + 24q^{74} + 28q^{79} - 4q^{81} - 32q^{89} + 24q^{94} + 32q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.41421i − 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(3\) − 0.414214i − 0.239146i −0.992825 0.119573i \(-0.961847\pi\)
0.992825 0.119573i \(-0.0381526\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) −0.585786 −0.239146
\(7\) 0 0
\(8\) − 2.82843i − 1.00000i
\(9\) 2.82843 0.942809
\(10\) 0 0
\(11\) −0.171573 −0.0517312 −0.0258656 0.999665i \(-0.508234\pi\)
−0.0258656 + 0.999665i \(0.508234\pi\)
\(12\) 0 0
\(13\) 4.41421i 1.22428i 0.790748 + 0.612141i \(0.209692\pi\)
−0.790748 + 0.612141i \(0.790308\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) − 3.24264i − 0.786456i −0.919441 0.393228i \(-0.871358\pi\)
0.919441 0.393228i \(-0.128642\pi\)
\(18\) − 4.00000i − 0.942809i
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.242641i 0.0517312i
\(23\) − 7.41421i − 1.54597i −0.634424 0.772985i \(-0.718763\pi\)
0.634424 0.772985i \(-0.281237\pi\)
\(24\) −1.17157 −0.239146
\(25\) 0 0
\(26\) 6.24264 1.22428
\(27\) − 2.41421i − 0.464616i
\(28\) 0 0
\(29\) 8.65685 1.60754 0.803769 0.594942i \(-0.202825\pi\)
0.803769 + 0.594942i \(0.202825\pi\)
\(30\) 0 0
\(31\) −10.2426 −1.83963 −0.919816 0.392349i \(-0.871662\pi\)
−0.919816 + 0.392349i \(0.871662\pi\)
\(32\) 0 0
\(33\) 0.0710678i 0.0123713i
\(34\) −4.58579 −0.786456
\(35\) 0 0
\(36\) 0 0
\(37\) 2.24264i 0.368688i 0.982862 + 0.184344i \(0.0590160\pi\)
−0.982862 + 0.184344i \(0.940984\pi\)
\(38\) − 8.48528i − 1.37649i
\(39\) 1.82843 0.292783
\(40\) 0 0
\(41\) 6.24264 0.974937 0.487468 0.873141i \(-0.337920\pi\)
0.487468 + 0.873141i \(0.337920\pi\)
\(42\) 0 0
\(43\) − 2.00000i − 0.304997i −0.988304 0.152499i \(-0.951268\pi\)
0.988304 0.152499i \(-0.0487319\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −10.4853 −1.54597
\(47\) 7.24264i 1.05645i 0.849105 + 0.528224i \(0.177142\pi\)
−0.849105 + 0.528224i \(0.822858\pi\)
\(48\) 1.65685i 0.239146i
\(49\) 0 0
\(50\) 0 0
\(51\) −1.34315 −0.188078
\(52\) 0 0
\(53\) − 4.24264i − 0.582772i −0.956606 0.291386i \(-0.905884\pi\)
0.956606 0.291386i \(-0.0941163\pi\)
\(54\) −3.41421 −0.464616
\(55\) 0 0
\(56\) 0 0
\(57\) − 2.48528i − 0.329184i
\(58\) − 12.2426i − 1.60754i
\(59\) −2.24264 −0.291967 −0.145983 0.989287i \(-0.546635\pi\)
−0.145983 + 0.989287i \(0.546635\pi\)
\(60\) 0 0
\(61\) 2.82843 0.362143 0.181071 0.983470i \(-0.442043\pi\)
0.181071 + 0.983470i \(0.442043\pi\)
\(62\) 14.4853i 1.83963i
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0.100505 0.0123713
\(67\) − 8.24264i − 1.00700i −0.863996 0.503499i \(-0.832046\pi\)
0.863996 0.503499i \(-0.167954\pi\)
\(68\) 0 0
\(69\) −3.07107 −0.369713
\(70\) 0 0
\(71\) −3.17157 −0.376396 −0.188198 0.982131i \(-0.560265\pi\)
−0.188198 + 0.982131i \(0.560265\pi\)
\(72\) − 8.00000i − 0.942809i
\(73\) − 8.48528i − 0.993127i −0.868000 0.496564i \(-0.834595\pi\)
0.868000 0.496564i \(-0.165405\pi\)
\(74\) 3.17157 0.368688
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) − 2.58579i − 0.292783i
\(79\) −1.48528 −0.167107 −0.0835536 0.996503i \(-0.526627\pi\)
−0.0835536 + 0.996503i \(0.526627\pi\)
\(80\) 0 0
\(81\) 7.48528 0.831698
\(82\) − 8.82843i − 0.974937i
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.82843 −0.304997
\(87\) − 3.58579i − 0.384437i
\(88\) 0.485281i 0.0517312i
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.24264i 0.439941i
\(94\) 10.2426 1.05645
\(95\) 0 0
\(96\) 0 0
\(97\) − 13.2426i − 1.34459i −0.740285 0.672293i \(-0.765309\pi\)
0.740285 0.672293i \(-0.234691\pi\)
\(98\) 0 0
\(99\) −0.485281 −0.0487726
\(100\) 0 0
\(101\) −2.48528 −0.247295 −0.123647 0.992326i \(-0.539459\pi\)
−0.123647 + 0.992326i \(0.539459\pi\)
\(102\) 1.89949i 0.188078i
\(103\) 19.2426i 1.89603i 0.318220 + 0.948017i \(0.396915\pi\)
−0.318220 + 0.948017i \(0.603085\pi\)
\(104\) 12.4853 1.22428
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) − 2.48528i − 0.240261i −0.992758 0.120131i \(-0.961669\pi\)
0.992758 0.120131i \(-0.0383313\pi\)
\(108\) 0 0
\(109\) −5.00000 −0.478913 −0.239457 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) 0 0
\(111\) 0.928932 0.0881703
\(112\) 0 0
\(113\) 13.0711i 1.22962i 0.788674 + 0.614811i \(0.210768\pi\)
−0.788674 + 0.614811i \(0.789232\pi\)
\(114\) −3.51472 −0.329184
\(115\) 0 0
\(116\) 0 0
\(117\) 12.4853i 1.15426i
\(118\) 3.17157i 0.291967i
\(119\) 0 0
\(120\) 0 0
\(121\) −10.9706 −0.997324
\(122\) − 4.00000i − 0.362143i
\(123\) − 2.58579i − 0.233153i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 8.24264i 0.731416i 0.930730 + 0.365708i \(0.119173\pi\)
−0.930730 + 0.365708i \(0.880827\pi\)
\(128\) 11.3137i 1.00000i
\(129\) −0.828427 −0.0729389
\(130\) 0 0
\(131\) −12.2426 −1.06964 −0.534822 0.844965i \(-0.679621\pi\)
−0.534822 + 0.844965i \(0.679621\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −11.6569 −1.00700
\(135\) 0 0
\(136\) −9.17157 −0.786456
\(137\) 12.0000i 1.02523i 0.858619 + 0.512615i \(0.171323\pi\)
−0.858619 + 0.512615i \(0.828677\pi\)
\(138\) 4.34315i 0.369713i
\(139\) 7.75736 0.657971 0.328985 0.944335i \(-0.393293\pi\)
0.328985 + 0.944335i \(0.393293\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) 4.48528i 0.376396i
\(143\) − 0.757359i − 0.0633336i
\(144\) −11.3137 −0.942809
\(145\) 0 0
\(146\) −12.0000 −0.993127
\(147\) 0 0
\(148\) 0 0
\(149\) 9.17157 0.751365 0.375682 0.926749i \(-0.377408\pi\)
0.375682 + 0.926749i \(0.377408\pi\)
\(150\) 0 0
\(151\) −7.48528 −0.609144 −0.304572 0.952489i \(-0.598513\pi\)
−0.304572 + 0.952489i \(0.598513\pi\)
\(152\) − 16.9706i − 1.37649i
\(153\) − 9.17157i − 0.741478i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 15.1716i 1.21082i 0.795913 + 0.605412i \(0.206992\pi\)
−0.795913 + 0.605412i \(0.793008\pi\)
\(158\) 2.10051i 0.167107i
\(159\) −1.75736 −0.139368
\(160\) 0 0
\(161\) 0 0
\(162\) − 10.5858i − 0.831698i
\(163\) 10.2426i 0.802266i 0.916020 + 0.401133i \(0.131383\pi\)
−0.916020 + 0.401133i \(0.868617\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 0.757359i − 0.0586062i −0.999571 0.0293031i \(-0.990671\pi\)
0.999571 0.0293031i \(-0.00932881\pi\)
\(168\) 0 0
\(169\) −6.48528 −0.498868
\(170\) 0 0
\(171\) 16.9706 1.29777
\(172\) 0 0
\(173\) − 7.24264i − 0.550648i −0.961352 0.275324i \(-0.911215\pi\)
0.961352 0.275324i \(-0.0887851\pi\)
\(174\) −5.07107 −0.384437
\(175\) 0 0
\(176\) 0.686292 0.0517312
\(177\) 0.928932i 0.0698228i
\(178\) 11.3137i 0.847998i
\(179\) 14.4853 1.08268 0.541340 0.840804i \(-0.317917\pi\)
0.541340 + 0.840804i \(0.317917\pi\)
\(180\) 0 0
\(181\) 18.7279 1.39204 0.696018 0.718025i \(-0.254953\pi\)
0.696018 + 0.718025i \(0.254953\pi\)
\(182\) 0 0
\(183\) − 1.17157i − 0.0866052i
\(184\) −20.9706 −1.54597
\(185\) 0 0
\(186\) 6.00000 0.439941
\(187\) 0.556349i 0.0406843i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 13.9706 1.01087 0.505437 0.862863i \(-0.331331\pi\)
0.505437 + 0.862863i \(0.331331\pi\)
\(192\) 3.31371i 0.239146i
\(193\) 16.0000i 1.15171i 0.817554 + 0.575853i \(0.195330\pi\)
−0.817554 + 0.575853i \(0.804670\pi\)
\(194\) −18.7279 −1.34459
\(195\) 0 0
\(196\) 0 0
\(197\) 13.4142i 0.955723i 0.878435 + 0.477862i \(0.158588\pi\)
−0.878435 + 0.477862i \(0.841412\pi\)
\(198\) 0.686292i 0.0487726i
\(199\) −7.41421 −0.525580 −0.262790 0.964853i \(-0.584643\pi\)
−0.262790 + 0.964853i \(0.584643\pi\)
\(200\) 0 0
\(201\) −3.41421 −0.240820
\(202\) 3.51472i 0.247295i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 27.2132 1.89603
\(207\) − 20.9706i − 1.45755i
\(208\) − 17.6569i − 1.22428i
\(209\) −1.02944 −0.0712077
\(210\) 0 0
\(211\) 9.00000 0.619586 0.309793 0.950804i \(-0.399740\pi\)
0.309793 + 0.950804i \(0.399740\pi\)
\(212\) 0 0
\(213\) 1.31371i 0.0900138i
\(214\) −3.51472 −0.240261
\(215\) 0 0
\(216\) −6.82843 −0.464616
\(217\) 0 0
\(218\) 7.07107i 0.478913i
\(219\) −3.51472 −0.237503
\(220\) 0 0
\(221\) 14.3137 0.962844
\(222\) − 1.31371i − 0.0881703i
\(223\) − 24.2132i − 1.62144i −0.585437 0.810718i \(-0.699077\pi\)
0.585437 0.810718i \(-0.300923\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 18.4853 1.22962
\(227\) − 15.7279i − 1.04390i −0.852976 0.521949i \(-0.825205\pi\)
0.852976 0.521949i \(-0.174795\pi\)
\(228\) 0 0
\(229\) 18.0416 1.19222 0.596112 0.802901i \(-0.296711\pi\)
0.596112 + 0.802901i \(0.296711\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 24.4853i − 1.60754i
\(233\) 9.17157i 0.600850i 0.953805 + 0.300425i \(0.0971285\pi\)
−0.953805 + 0.300425i \(0.902872\pi\)
\(234\) 17.6569 1.15426
\(235\) 0 0
\(236\) 0 0
\(237\) 0.615224i 0.0399631i
\(238\) 0 0
\(239\) 17.4853 1.13103 0.565514 0.824738i \(-0.308678\pi\)
0.565514 + 0.824738i \(0.308678\pi\)
\(240\) 0 0
\(241\) 0.727922 0.0468896 0.0234448 0.999725i \(-0.492537\pi\)
0.0234448 + 0.999725i \(0.492537\pi\)
\(242\) 15.5147i 0.997324i
\(243\) − 10.3431i − 0.663513i
\(244\) 0 0
\(245\) 0 0
\(246\) −3.65685 −0.233153
\(247\) 26.4853i 1.68522i
\(248\) 28.9706i 1.83963i
\(249\) 0 0
\(250\) 0 0
\(251\) −17.2132 −1.08649 −0.543244 0.839575i \(-0.682804\pi\)
−0.543244 + 0.839575i \(0.682804\pi\)
\(252\) 0 0
\(253\) 1.27208i 0.0799749i
\(254\) 11.6569 0.731416
\(255\) 0 0
\(256\) 0 0
\(257\) 9.51472i 0.593512i 0.954953 + 0.296756i \(0.0959048\pi\)
−0.954953 + 0.296756i \(0.904095\pi\)
\(258\) 1.17157i 0.0729389i
\(259\) 0 0
\(260\) 0 0
\(261\) 24.4853 1.51560
\(262\) 17.3137i 1.06964i
\(263\) 16.6274i 1.02529i 0.858601 + 0.512645i \(0.171334\pi\)
−0.858601 + 0.512645i \(0.828666\pi\)
\(264\) 0.201010 0.0123713
\(265\) 0 0
\(266\) 0 0
\(267\) 3.31371i 0.202796i
\(268\) 0 0
\(269\) −16.2426 −0.990331 −0.495166 0.868799i \(-0.664893\pi\)
−0.495166 + 0.868799i \(0.664893\pi\)
\(270\) 0 0
\(271\) 0.686292 0.0416892 0.0208446 0.999783i \(-0.493364\pi\)
0.0208446 + 0.999783i \(0.493364\pi\)
\(272\) 12.9706i 0.786456i
\(273\) 0 0
\(274\) 16.9706 1.02523
\(275\) 0 0
\(276\) 0 0
\(277\) 15.2132i 0.914073i 0.889448 + 0.457036i \(0.151089\pi\)
−0.889448 + 0.457036i \(0.848911\pi\)
\(278\) − 10.9706i − 0.657971i
\(279\) −28.9706 −1.73442
\(280\) 0 0
\(281\) 2.31371 0.138024 0.0690121 0.997616i \(-0.478015\pi\)
0.0690121 + 0.997616i \(0.478015\pi\)
\(282\) − 4.24264i − 0.252646i
\(283\) 24.5563i 1.45972i 0.683595 + 0.729862i \(0.260416\pi\)
−0.683595 + 0.729862i \(0.739584\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −1.07107 −0.0633336
\(287\) 0 0
\(288\) 0 0
\(289\) 6.48528 0.381487
\(290\) 0 0
\(291\) −5.48528 −0.321553
\(292\) 0 0
\(293\) 25.7279i 1.50304i 0.659710 + 0.751521i \(0.270679\pi\)
−0.659710 + 0.751521i \(0.729321\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 6.34315 0.368688
\(297\) 0.414214i 0.0240351i
\(298\) − 12.9706i − 0.751365i
\(299\) 32.7279 1.89270
\(300\) 0 0
\(301\) 0 0
\(302\) 10.5858i 0.609144i
\(303\) 1.02944i 0.0591396i
\(304\) −24.0000 −1.37649
\(305\) 0 0
\(306\) −12.9706 −0.741478
\(307\) − 30.8995i − 1.76353i −0.471691 0.881764i \(-0.656356\pi\)
0.471691 0.881764i \(-0.343644\pi\)
\(308\) 0 0
\(309\) 7.97056 0.453429
\(310\) 0 0
\(311\) 10.0000 0.567048 0.283524 0.958965i \(-0.408496\pi\)
0.283524 + 0.958965i \(0.408496\pi\)
\(312\) − 5.17157i − 0.292783i
\(313\) − 18.2132i − 1.02947i −0.857349 0.514736i \(-0.827890\pi\)
0.857349 0.514736i \(-0.172110\pi\)
\(314\) 21.4558 1.21082
\(315\) 0 0
\(316\) 0 0
\(317\) − 0.343146i − 0.0192730i −0.999954 0.00963649i \(-0.996933\pi\)
0.999954 0.00963649i \(-0.00306744\pi\)
\(318\) 2.48528i 0.139368i
\(319\) −1.48528 −0.0831598
\(320\) 0 0
\(321\) −1.02944 −0.0574576
\(322\) 0 0
\(323\) − 19.4558i − 1.08255i
\(324\) 0 0
\(325\) 0 0
\(326\) 14.4853 0.802266
\(327\) 2.07107i 0.114530i
\(328\) − 17.6569i − 0.974937i
\(329\) 0 0
\(330\) 0 0
\(331\) −27.4558 −1.50911 −0.754555 0.656237i \(-0.772147\pi\)
−0.754555 + 0.656237i \(0.772147\pi\)
\(332\) 0 0
\(333\) 6.34315i 0.347602i
\(334\) −1.07107 −0.0586062
\(335\) 0 0
\(336\) 0 0
\(337\) 22.2426i 1.21163i 0.795604 + 0.605817i \(0.207154\pi\)
−0.795604 + 0.605817i \(0.792846\pi\)
\(338\) 9.17157i 0.498868i
\(339\) 5.41421 0.294060
\(340\) 0 0
\(341\) 1.75736 0.0951663
\(342\) − 24.0000i − 1.29777i
\(343\) 0 0
\(344\) −5.65685 −0.304997
\(345\) 0 0
\(346\) −10.2426 −0.550648
\(347\) 13.0711i 0.701692i 0.936433 + 0.350846i \(0.114106\pi\)
−0.936433 + 0.350846i \(0.885894\pi\)
\(348\) 0 0
\(349\) −10.9706 −0.587241 −0.293620 0.955922i \(-0.594860\pi\)
−0.293620 + 0.955922i \(0.594860\pi\)
\(350\) 0 0
\(351\) 10.6569 0.568821
\(352\) 0 0
\(353\) − 6.21320i − 0.330695i −0.986235 0.165348i \(-0.947125\pi\)
0.986235 0.165348i \(-0.0528746\pi\)
\(354\) 1.31371 0.0698228
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) − 20.4853i − 1.08268i
\(359\) 11.3137 0.597115 0.298557 0.954392i \(-0.403495\pi\)
0.298557 + 0.954392i \(0.403495\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) − 26.4853i − 1.39204i
\(363\) 4.54416i 0.238506i
\(364\) 0 0
\(365\) 0 0
\(366\) −1.65685 −0.0866052
\(367\) 23.8701i 1.24601i 0.782219 + 0.623003i \(0.214088\pi\)
−0.782219 + 0.623003i \(0.785912\pi\)
\(368\) 29.6569i 1.54597i
\(369\) 17.6569 0.919179
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 16.4853i 0.853576i 0.904352 + 0.426788i \(0.140355\pi\)
−0.904352 + 0.426788i \(0.859645\pi\)
\(374\) 0.786797 0.0406843
\(375\) 0 0
\(376\) 20.4853 1.05645
\(377\) 38.2132i 1.96808i
\(378\) 0 0
\(379\) −2.00000 −0.102733 −0.0513665 0.998680i \(-0.516358\pi\)
−0.0513665 + 0.998680i \(0.516358\pi\)
\(380\) 0 0
\(381\) 3.41421 0.174915
\(382\) − 19.7574i − 1.01087i
\(383\) 4.48528i 0.229187i 0.993412 + 0.114594i \(0.0365566\pi\)
−0.993412 + 0.114594i \(0.963443\pi\)
\(384\) 4.68629 0.239146
\(385\) 0 0
\(386\) 22.6274 1.15171
\(387\) − 5.65685i − 0.287554i
\(388\) 0 0
\(389\) 6.85786 0.347708 0.173854 0.984771i \(-0.444378\pi\)
0.173854 + 0.984771i \(0.444378\pi\)
\(390\) 0 0
\(391\) −24.0416 −1.21584
\(392\) 0 0
\(393\) 5.07107i 0.255802i
\(394\) 18.9706 0.955723
\(395\) 0 0
\(396\) 0 0
\(397\) − 1.58579i − 0.0795883i −0.999208 0.0397942i \(-0.987330\pi\)
0.999208 0.0397942i \(-0.0126702\pi\)
\(398\) 10.4853i 0.525580i
\(399\) 0 0
\(400\) 0 0
\(401\) 11.8284 0.590683 0.295342 0.955392i \(-0.404566\pi\)
0.295342 + 0.955392i \(0.404566\pi\)
\(402\) 4.82843i 0.240820i
\(403\) − 45.2132i − 2.25223i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 0.384776i − 0.0190727i
\(408\) 3.79899i 0.188078i
\(409\) 2.48528 0.122889 0.0614446 0.998110i \(-0.480429\pi\)
0.0614446 + 0.998110i \(0.480429\pi\)
\(410\) 0 0
\(411\) 4.97056 0.245180
\(412\) 0 0
\(413\) 0 0
\(414\) −29.6569 −1.45755
\(415\) 0 0
\(416\) 0 0
\(417\) − 3.21320i − 0.157351i
\(418\) 1.45584i 0.0712077i
\(419\) 18.7279 0.914919 0.457459 0.889230i \(-0.348759\pi\)
0.457459 + 0.889230i \(0.348759\pi\)
\(420\) 0 0
\(421\) −19.0000 −0.926003 −0.463002 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) − 12.7279i − 0.619586i
\(423\) 20.4853i 0.996028i
\(424\) −12.0000 −0.582772
\(425\) 0 0
\(426\) 1.85786 0.0900138
\(427\) 0 0
\(428\) 0 0
\(429\) −0.313708 −0.0151460
\(430\) 0 0
\(431\) −28.7990 −1.38720 −0.693599 0.720361i \(-0.743976\pi\)
−0.693599 + 0.720361i \(0.743976\pi\)
\(432\) 9.65685i 0.464616i
\(433\) 10.9706i 0.527212i 0.964630 + 0.263606i \(0.0849118\pi\)
−0.964630 + 0.263606i \(0.915088\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 44.4853i − 2.12802i
\(438\) 4.97056i 0.237503i
\(439\) −30.3848 −1.45019 −0.725093 0.688651i \(-0.758203\pi\)
−0.725093 + 0.688651i \(0.758203\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 20.2426i − 0.962844i
\(443\) − 15.1716i − 0.720823i −0.932793 0.360412i \(-0.882636\pi\)
0.932793 0.360412i \(-0.117364\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −34.2426 −1.62144
\(447\) − 3.79899i − 0.179686i
\(448\) 0 0
\(449\) 24.1716 1.14073 0.570364 0.821392i \(-0.306802\pi\)
0.570364 + 0.821392i \(0.306802\pi\)
\(450\) 0 0
\(451\) −1.07107 −0.0504346
\(452\) 0 0
\(453\) 3.10051i 0.145674i
\(454\) −22.2426 −1.04390
\(455\) 0 0
\(456\) −7.02944 −0.329184
\(457\) − 11.7574i − 0.549986i −0.961446 0.274993i \(-0.911324\pi\)
0.961446 0.274993i \(-0.0886756\pi\)
\(458\) − 25.5147i − 1.19222i
\(459\) −7.82843 −0.365400
\(460\) 0 0
\(461\) −3.02944 −0.141095 −0.0705475 0.997508i \(-0.522475\pi\)
−0.0705475 + 0.997508i \(0.522475\pi\)
\(462\) 0 0
\(463\) 21.4558i 0.997138i 0.866850 + 0.498569i \(0.166141\pi\)
−0.866850 + 0.498569i \(0.833859\pi\)
\(464\) −34.6274 −1.60754
\(465\) 0 0
\(466\) 12.9706 0.600850
\(467\) 5.72792i 0.265057i 0.991179 + 0.132528i \(0.0423095\pi\)
−0.991179 + 0.132528i \(0.957690\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 6.28427 0.289564
\(472\) 6.34315i 0.291967i
\(473\) 0.343146i 0.0157779i
\(474\) 0.870058 0.0399631
\(475\) 0 0
\(476\) 0 0
\(477\) − 12.0000i − 0.549442i
\(478\) − 24.7279i − 1.13103i
\(479\) 11.7574 0.537207 0.268604 0.963251i \(-0.413438\pi\)
0.268604 + 0.963251i \(0.413438\pi\)
\(480\) 0 0
\(481\) −9.89949 −0.451378
\(482\) − 1.02944i − 0.0468896i
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) −14.6274 −0.663513
\(487\) 27.6985i 1.25514i 0.778561 + 0.627569i \(0.215950\pi\)
−0.778561 + 0.627569i \(0.784050\pi\)
\(488\) − 8.00000i − 0.362143i
\(489\) 4.24264 0.191859
\(490\) 0 0
\(491\) −37.2843 −1.68262 −0.841308 0.540556i \(-0.818214\pi\)
−0.841308 + 0.540556i \(0.818214\pi\)
\(492\) 0 0
\(493\) − 28.0711i − 1.26426i
\(494\) 37.4558 1.68522
\(495\) 0 0
\(496\) 40.9706 1.83963
\(497\) 0 0
\(498\) 0 0
\(499\) −3.00000 −0.134298 −0.0671492 0.997743i \(-0.521390\pi\)
−0.0671492 + 0.997743i \(0.521390\pi\)
\(500\) 0 0
\(501\) −0.313708 −0.0140155
\(502\) 24.3431i 1.08649i
\(503\) − 41.2426i − 1.83892i −0.393185 0.919459i \(-0.628627\pi\)
0.393185 0.919459i \(-0.371373\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.79899 0.0799749
\(507\) 2.68629i 0.119302i
\(508\) 0 0
\(509\) −25.2132 −1.11756 −0.558778 0.829317i \(-0.688730\pi\)
−0.558778 + 0.829317i \(0.688730\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 22.6274i 1.00000i
\(513\) − 14.4853i − 0.639541i
\(514\) 13.4558 0.593512
\(515\) 0 0
\(516\) 0 0
\(517\) − 1.24264i − 0.0546513i
\(518\) 0 0
\(519\) −3.00000 −0.131685
\(520\) 0 0
\(521\) −14.9706 −0.655872 −0.327936 0.944700i \(-0.606353\pi\)
−0.327936 + 0.944700i \(0.606353\pi\)
\(522\) − 34.6274i − 1.51560i
\(523\) 32.4853i 1.42048i 0.703959 + 0.710241i \(0.251414\pi\)
−0.703959 + 0.710241i \(0.748586\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 23.5147 1.02529
\(527\) 33.2132i 1.44679i
\(528\) − 0.284271i − 0.0123713i
\(529\) −31.9706 −1.39002
\(530\) 0 0
\(531\) −6.34315 −0.275269
\(532\) 0 0
\(533\) 27.5563i 1.19360i
\(534\) 4.68629 0.202796
\(535\) 0 0
\(536\) −23.3137 −1.00700
\(537\) − 6.00000i − 0.258919i
\(538\) 22.9706i 0.990331i
\(539\) 0 0
\(540\) 0 0
\(541\) −21.9706 −0.944588 −0.472294 0.881441i \(-0.656574\pi\)
−0.472294 + 0.881441i \(0.656574\pi\)
\(542\) − 0.970563i − 0.0416892i
\(543\) − 7.75736i − 0.332900i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 24.4853i − 1.04692i −0.852052 0.523458i \(-0.824642\pi\)
0.852052 0.523458i \(-0.175358\pi\)
\(548\) 0 0
\(549\) 8.00000 0.341432
\(550\) 0 0
\(551\) 51.9411 2.21277
\(552\) 8.68629i 0.369713i
\(553\) 0 0
\(554\) 21.5147 0.914073
\(555\) 0 0
\(556\) 0 0
\(557\) − 7.79899i − 0.330454i −0.986256 0.165227i \(-0.947164\pi\)
0.986256 0.165227i \(-0.0528356\pi\)
\(558\) 40.9706i 1.73442i
\(559\) 8.82843 0.373403
\(560\) 0 0
\(561\) 0.230447 0.00972950
\(562\) − 3.27208i − 0.138024i
\(563\) − 31.9411i − 1.34616i −0.739571 0.673079i \(-0.764971\pi\)
0.739571 0.673079i \(-0.235029\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 34.7279 1.45972
\(567\) 0 0
\(568\) 8.97056i 0.376396i
\(569\) −26.1421 −1.09594 −0.547968 0.836500i \(-0.684598\pi\)
−0.547968 + 0.836500i \(0.684598\pi\)
\(570\) 0 0
\(571\) −17.5147 −0.732968 −0.366484 0.930424i \(-0.619439\pi\)
−0.366484 + 0.930424i \(0.619439\pi\)
\(572\) 0 0
\(573\) − 5.78680i − 0.241747i
\(574\) 0 0
\(575\) 0 0
\(576\) −22.6274 −0.942809
\(577\) − 15.7279i − 0.654762i −0.944892 0.327381i \(-0.893834\pi\)
0.944892 0.327381i \(-0.106166\pi\)
\(578\) − 9.17157i − 0.381487i
\(579\) 6.62742 0.275426
\(580\) 0 0
\(581\) 0 0
\(582\) 7.75736i 0.321553i
\(583\) 0.727922i 0.0301475i
\(584\) −24.0000 −0.993127
\(585\) 0 0
\(586\) 36.3848 1.50304
\(587\) 37.4558i 1.54597i 0.634425 + 0.772984i \(0.281237\pi\)
−0.634425 + 0.772984i \(0.718763\pi\)
\(588\) 0 0
\(589\) −61.4558 −2.53224
\(590\) 0 0
\(591\) 5.55635 0.228558
\(592\) − 8.97056i − 0.368688i
\(593\) 19.2426i 0.790201i 0.918638 + 0.395100i \(0.129290\pi\)
−0.918638 + 0.395100i \(0.870710\pi\)
\(594\) 0.585786 0.0240351
\(595\) 0 0
\(596\) 0 0
\(597\) 3.07107i 0.125690i
\(598\) − 46.2843i − 1.89270i
\(599\) 17.8284 0.728450 0.364225 0.931311i \(-0.381334\pi\)
0.364225 + 0.931311i \(0.381334\pi\)
\(600\) 0 0
\(601\) 10.9706 0.447499 0.223749 0.974647i \(-0.428170\pi\)
0.223749 + 0.974647i \(0.428170\pi\)
\(602\) 0 0
\(603\) − 23.3137i − 0.949408i
\(604\) 0 0
\(605\) 0 0
\(606\) 1.45584 0.0591396
\(607\) 5.10051i 0.207023i 0.994628 + 0.103512i \(0.0330079\pi\)
−0.994628 + 0.103512i \(0.966992\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −31.9706 −1.29339
\(612\) 0 0
\(613\) 23.9411i 0.966973i 0.875352 + 0.483486i \(0.160630\pi\)
−0.875352 + 0.483486i \(0.839370\pi\)
\(614\) −43.6985 −1.76353
\(615\) 0 0
\(616\) 0 0
\(617\) 4.58579i 0.184617i 0.995730 + 0.0923084i \(0.0294246\pi\)
−0.995730 + 0.0923084i \(0.970575\pi\)
\(618\) − 11.2721i − 0.453429i
\(619\) 16.9289 0.680431 0.340216 0.940347i \(-0.389500\pi\)
0.340216 + 0.940347i \(0.389500\pi\)
\(620\) 0 0
\(621\) −17.8995 −0.718282
\(622\) − 14.1421i − 0.567048i
\(623\) 0 0
\(624\) −7.31371 −0.292783
\(625\) 0 0
\(626\) −25.7574 −1.02947
\(627\) 0.426407i 0.0170291i
\(628\) 0 0
\(629\) 7.27208 0.289957
\(630\) 0 0
\(631\) −42.4558 −1.69014 −0.845070 0.534655i \(-0.820441\pi\)
−0.845070 + 0.534655i \(0.820441\pi\)
\(632\) 4.20101i 0.167107i
\(633\) − 3.72792i − 0.148172i
\(634\) −0.485281 −0.0192730
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 2.10051i 0.0831598i
\(639\) −8.97056 −0.354870
\(640\) 0 0
\(641\) −23.3137 −0.920836 −0.460418 0.887702i \(-0.652300\pi\)
−0.460418 + 0.887702i \(0.652300\pi\)
\(642\) 1.45584i 0.0574576i
\(643\) 2.27208i 0.0896020i 0.998996 + 0.0448010i \(0.0142654\pi\)
−0.998996 + 0.0448010i \(0.985735\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −27.5147 −1.08255
\(647\) 11.5147i 0.452690i 0.974047 + 0.226345i \(0.0726777\pi\)
−0.974047 + 0.226345i \(0.927322\pi\)
\(648\) − 21.1716i − 0.831698i
\(649\) 0.384776 0.0151038
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 34.9706i − 1.36850i −0.729246 0.684252i \(-0.760129\pi\)
0.729246 0.684252i \(-0.239871\pi\)
\(654\) 2.92893 0.114530
\(655\) 0 0
\(656\) −24.9706 −0.974937
\(657\) − 24.0000i − 0.936329i
\(658\) 0 0
\(659\) −19.9706 −0.777943 −0.388971 0.921250i \(-0.627169\pi\)
−0.388971 + 0.921250i \(0.627169\pi\)
\(660\) 0 0
\(661\) −13.4558 −0.523372 −0.261686 0.965153i \(-0.584278\pi\)
−0.261686 + 0.965153i \(0.584278\pi\)
\(662\) 38.8284i 1.50911i
\(663\) − 5.92893i − 0.230261i
\(664\) 0 0
\(665\) 0 0
\(666\) 8.97056 0.347602
\(667\) − 64.1838i − 2.48521i
\(668\) 0 0
\(669\) −10.0294 −0.387760
\(670\) 0 0
\(671\) −0.485281 −0.0187341
\(672\) 0 0
\(673\) − 3.51472i − 0.135482i −0.997703 0.0677412i \(-0.978421\pi\)
0.997703 0.0677412i \(-0.0215792\pi\)
\(674\) 31.4558 1.21163
\(675\) 0 0
\(676\) 0 0
\(677\) − 44.2132i − 1.69925i −0.527386 0.849626i \(-0.676828\pi\)
0.527386 0.849626i \(-0.323172\pi\)
\(678\) − 7.65685i − 0.294060i
\(679\) 0 0
\(680\) 0 0
\(681\) −6.51472 −0.249645
\(682\) − 2.48528i − 0.0951663i
\(683\) − 31.7990i − 1.21675i −0.793648 0.608377i \(-0.791821\pi\)
0.793648 0.608377i \(-0.208179\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 7.47309i − 0.285116i
\(688\) 8.00000i 0.304997i
\(689\) 18.7279 0.713477
\(690\) 0 0
\(691\) −14.8284 −0.564100 −0.282050 0.959400i \(-0.591014\pi\)
−0.282050 + 0.959400i \(0.591014\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 18.4853 0.701692
\(695\) 0 0
\(696\) −10.1421 −0.384437
\(697\) − 20.2426i − 0.766745i
\(698\) 15.5147i 0.587241i
\(699\) 3.79899 0.143691
\(700\) 0 0
\(701\) 46.4558 1.75461 0.877307 0.479931i \(-0.159338\pi\)
0.877307 + 0.479931i \(0.159338\pi\)
\(702\) − 15.0711i − 0.568821i
\(703\) 13.4558i 0.507497i
\(704\) 1.37258 0.0517312
\(705\) 0 0
\(706\) −8.78680 −0.330695
\(707\) 0 0
\(708\) 0 0
\(709\) −17.0000 −0.638448 −0.319224 0.947679i \(-0.603422\pi\)
−0.319224 + 0.947679i \(0.603422\pi\)
\(710\) 0 0
\(711\) −4.20101 −0.157550
\(712\) 22.6274i 0.847998i
\(713\) 75.9411i 2.84402i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 7.24264i − 0.270481i
\(718\) − 16.0000i − 0.597115i
\(719\) −25.2132 −0.940294 −0.470147 0.882588i \(-0.655799\pi\)
−0.470147 + 0.882588i \(0.655799\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) − 24.0416i − 0.894737i
\(723\) − 0.301515i − 0.0112135i
\(724\) 0 0
\(725\) 0 0
\(726\) 6.42641 0.238506
\(727\) 6.68629i 0.247981i 0.992283 + 0.123990i \(0.0395692\pi\)
−0.992283 + 0.123990i \(0.960431\pi\)
\(728\) 0 0
\(729\) 18.1716 0.673021
\(730\) 0 0
\(731\) −6.48528 −0.239867
\(732\) 0 0
\(733\) 14.6985i 0.542901i 0.962452 + 0.271450i \(0.0875033\pi\)
−0.962452 + 0.271450i \(0.912497\pi\)
\(734\) 33.7574 1.24601
\(735\) 0 0
\(736\) 0 0
\(737\) 1.41421i 0.0520932i
\(738\) − 24.9706i − 0.919179i
\(739\) −29.9706 −1.10248 −0.551242 0.834345i \(-0.685846\pi\)
−0.551242 + 0.834345i \(0.685846\pi\)
\(740\) 0 0
\(741\) 10.9706 0.403014
\(742\) 0 0
\(743\) − 11.2721i − 0.413532i −0.978390 0.206766i \(-0.933706\pi\)
0.978390 0.206766i \(-0.0662940\pi\)
\(744\) 12.0000 0.439941
\(745\) 0 0
\(746\) 23.3137 0.853576
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 29.4853 1.07593 0.537967 0.842966i \(-0.319193\pi\)
0.537967 + 0.842966i \(0.319193\pi\)
\(752\) − 28.9706i − 1.05645i
\(753\) 7.12994i 0.259830i
\(754\) 54.0416 1.96808
\(755\) 0 0
\(756\) 0 0
\(757\) 0.485281i 0.0176379i 0.999961 + 0.00881893i \(0.00280719\pi\)
−0.999961 + 0.00881893i \(0.997193\pi\)
\(758\) 2.82843i 0.102733i
\(759\) 0.526912 0.0191257
\(760\) 0 0
\(761\) 12.7279 0.461387 0.230693 0.973026i \(-0.425901\pi\)
0.230693 + 0.973026i \(0.425901\pi\)
\(762\) − 4.82843i − 0.174915i
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 6.34315 0.229187
\(767\) − 9.89949i − 0.357450i
\(768\) 0 0
\(769\) −8.82843 −0.318361 −0.159181 0.987249i \(-0.550885\pi\)
−0.159181 + 0.987249i \(0.550885\pi\)
\(770\) 0 0
\(771\) 3.94113 0.141936
\(772\) 0 0
\(773\) − 6.21320i − 0.223473i −0.993738 0.111737i \(-0.964359\pi\)
0.993738 0.111737i \(-0.0356413\pi\)
\(774\) −8.00000 −0.287554
\(775\) 0 0
\(776\) −37.4558 −1.34459
\(777\) 0 0
\(778\) − 9.69848i − 0.347708i
\(779\) 37.4558 1.34199
\(780\) 0 0
\(781\) 0.544156 0.0194714
\(782\) 34.0000i 1.21584i
\(783\) − 20.8995i − 0.746887i
\(784\) 0 0
\(785\) 0 0
\(786\) 7.17157 0.255802
\(787\) − 20.2721i − 0.722622i −0.932445 0.361311i \(-0.882329\pi\)
0.932445 0.361311i \(-0.117671\pi\)
\(788\) 0 0
\(789\) 6.88730 0.245194
\(790\) 0 0
\(791\) 0 0
\(792\) 1.37258i 0.0487726i
\(793\) 12.4853i 0.443365i
\(794\) −2.24264 −0.0795883
\(795\) 0 0
\(796\) 0 0
\(797\) 21.1838i 0.750367i 0.926950 + 0.375184i \(0.122420\pi\)
−0.926950 + 0.375184i \(0.877580\pi\)
\(798\) 0 0
\(799\) 23.4853 0.830850
\(800\) 0 0
\(801\) −22.6274 −0.799500
\(802\) − 16.7279i − 0.590683i
\(803\) 1.45584i 0.0513756i
\(804\) 0 0
\(805\) 0 0
\(806\) −63.9411 −2.25223
\(807\) 6.72792i 0.236834i
\(808\) 7.02944i 0.247295i
\(809\) 48.5980 1.70861 0.854307 0.519769i \(-0.173982\pi\)
0.854307 + 0.519769i \(0.173982\pi\)
\(810\) 0 0
\(811\) −47.3553 −1.66287 −0.831435 0.555621i \(-0.812480\pi\)
−0.831435 + 0.555621i \(0.812480\pi\)
\(812\) 0 0
\(813\) − 0.284271i − 0.00996983i
\(814\) −0.544156 −0.0190727
\(815\) 0 0
\(816\) 5.37258 0.188078
\(817\) − 12.0000i − 0.419827i
\(818\) − 3.51472i − 0.122889i
\(819\) 0 0
\(820\) 0 0
\(821\) −23.4853 −0.819642 −0.409821 0.912166i \(-0.634409\pi\)
−0.409821 + 0.912166i \(0.634409\pi\)
\(822\) − 7.02944i − 0.245180i
\(823\) − 16.7279i − 0.583099i −0.956556 0.291549i \(-0.905829\pi\)
0.956556 0.291549i \(-0.0941708\pi\)
\(824\) 54.4264 1.89603
\(825\) 0 0
\(826\) 0 0
\(827\) 42.0416i 1.46193i 0.682415 + 0.730965i \(0.260930\pi\)
−0.682415 + 0.730965i \(0.739070\pi\)
\(828\) 0 0
\(829\) 5.95837 0.206943 0.103471 0.994632i \(-0.467005\pi\)
0.103471 + 0.994632i \(0.467005\pi\)
\(830\) 0 0
\(831\) 6.30152 0.218597
\(832\) − 35.3137i − 1.22428i
\(833\) 0 0
\(834\) −4.54416 −0.157351
\(835\) 0 0
\(836\) 0 0
\(837\) 24.7279i 0.854722i
\(838\) − 26.4853i − 0.914919i
\(839\) −23.2721 −0.803441 −0.401721 0.915762i \(-0.631588\pi\)
−0.401721 + 0.915762i \(0.631588\pi\)
\(840\) 0 0
\(841\) 45.9411 1.58418
\(842\) 26.8701i 0.926003i
\(843\) − 0.958369i − 0.0330080i
\(844\) 0 0
\(845\) 0 0
\(846\) 28.9706 0.996028
\(847\) 0 0
\(848\) 16.9706i 0.582772i
\(849\) 10.1716 0.349087
\(850\) 0 0
\(851\) 16.6274 0.569981
\(852\) 0 0
\(853\) 10.9706i 0.375625i 0.982205 + 0.187812i \(0.0601397\pi\)
−0.982205 + 0.187812i \(0.939860\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −7.02944 −0.240261
\(857\) 22.4853i 0.768083i 0.923316 + 0.384041i \(0.125468\pi\)
−0.923316 + 0.384041i \(0.874532\pi\)
\(858\) 0.443651i 0.0151460i
\(859\) −24.7696 −0.845126 −0.422563 0.906334i \(-0.638870\pi\)
−0.422563 + 0.906334i \(0.638870\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 40.7279i 1.38720i
\(863\) − 18.3848i − 0.625825i −0.949782 0.312913i \(-0.898695\pi\)
0.949782 0.312913i \(-0.101305\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 15.5147 0.527212
\(867\) − 2.68629i − 0.0912312i
\(868\) 0 0
\(869\) 0.254834 0.00864465
\(870\) 0 0
\(871\) 36.3848 1.23285
\(872\) 14.1421i 0.478913i
\(873\) − 37.4558i − 1.26769i
\(874\) −62.9117 −2.12802
\(875\) 0 0
\(876\) 0 0
\(877\) 13.0294i 0.439973i 0.975503 + 0.219986i \(0.0706013\pi\)
−0.975503 + 0.219986i \(0.929399\pi\)
\(878\) 42.9706i 1.45019i
\(879\) 10.6569 0.359447
\(880\) 0 0
\(881\) 52.9706 1.78462 0.892312 0.451420i \(-0.149082\pi\)
0.892312 + 0.451420i \(0.149082\pi\)
\(882\) 0 0
\(883\) − 31.5147i − 1.06055i −0.847824 0.530277i \(-0.822088\pi\)
0.847824 0.530277i \(-0.177912\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −21.4558 −0.720823
\(887\) − 2.97056i − 0.0997417i −0.998756 0.0498709i \(-0.984119\pi\)
0.998756 0.0498709i \(-0.0158810\pi\)
\(888\) − 2.62742i − 0.0881703i
\(889\) 0 0
\(890\) 0 0
\(891\) −1.28427 −0.0430247
\(892\) 0 0
\(893\) 43.4558i 1.45419i
\(894\) −5.37258 −0.179686
\(895\) 0 0
\(896\) 0 0
\(897\) − 13.5563i − 0.452633i
\(898\) − 34.1838i − 1.14073i
\(899\) −88.6690 −2.95728
\(900\) 0 0
\(901\) −13.7574 −0.458324
\(902\) 1.51472i 0.0504346i
\(903\) 0 0
\(904\) 36.9706 1.22962
\(905\) 0 0
\(906\) 4.38478 0.145674
\(907\) − 44.1838i − 1.46710i −0.679637 0.733549i \(-0.737863\pi\)
0.679637 0.733549i \(-0.262137\pi\)
\(908\) 0 0
\(909\) −7.02944 −0.233152
\(910\) 0 0
\(911\) 5.65685 0.187420 0.0937100 0.995600i \(-0.470127\pi\)
0.0937100 + 0.995600i \(0.470127\pi\)
\(912\) 9.94113i 0.329184i
\(913\) 0 0
\(914\) −16.6274 −0.549986
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 11.0711i 0.365400i
\(919\) −12.4558 −0.410880 −0.205440 0.978670i \(-0.565863\pi\)
−0.205440 + 0.978670i \(0.565863\pi\)
\(920\) 0 0
\(921\) −12.7990 −0.421741
\(922\) 4.28427i 0.141095i
\(923\) − 14.0000i − 0.460816i
\(924\) 0 0
\(925\) 0 0
\(926\) 30.3431 0.997138
\(927\) 54.4264i 1.78760i
\(928\) 0 0
\(929\) 29.2721 0.960386 0.480193 0.877163i \(-0.340567\pi\)
0.480193 + 0.877163i \(0.340567\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) − 4.14214i − 0.135607i
\(934\) 8.10051 0.265057
\(935\) 0 0
\(936\) 35.3137 1.15426
\(937\) 48.5563i 1.58627i 0.609048 + 0.793133i \(0.291552\pi\)
−0.609048 + 0.793133i \(0.708448\pi\)
\(938\) 0 0
\(939\) −7.54416 −0.246194
\(940\) 0 0
\(941\) −28.9706 −0.944413 −0.472207 0.881488i \(-0.656542\pi\)
−0.472207 + 0.881488i \(0.656542\pi\)
\(942\) − 8.88730i − 0.289564i
\(943\) − 46.2843i − 1.50722i
\(944\) 8.97056 0.291967
\(945\) 0 0
\(946\) 0.485281 0.0157779
\(947\) − 52.2426i − 1.69766i −0.528668 0.848829i \(-0.677308\pi\)
0.528668 0.848829i \(-0.322692\pi\)
\(948\) 0 0
\(949\) 37.4558 1.21587
\(950\) 0 0
\(951\) −0.142136 −0.00460906
\(952\) 0 0
\(953\) − 53.0122i − 1.71723i −0.512618 0.858617i \(-0.671324\pi\)
0.512618 0.858617i \(-0.328676\pi\)
\(954\) −16.9706 −0.549442
\(955\) 0 0
\(956\) 0 0
\(957\) 0.615224i 0.0198874i
\(958\) − 16.6274i − 0.537207i
\(959\) 0 0
\(960\) 0 0
\(961\) 73.9117 2.38425
\(962\) 14.0000i 0.451378i
\(963\) − 7.02944i − 0.226520i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 60.4264i − 1.94318i −0.236666 0.971591i \(-0.576055\pi\)
0.236666 0.971591i \(-0.423945\pi\)
\(968\) 31.0294i 0.997324i
\(969\) −8.05887 −0.258888
\(970\) 0 0
\(971\) 17.2721 0.554287 0.277144 0.960828i \(-0.410612\pi\)
0.277144 + 0.960828i \(0.410612\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 39.1716 1.25514
\(975\) 0 0
\(976\) −11.3137 −0.362143
\(977\) 42.7696i 1.36832i 0.729332 + 0.684160i \(0.239831\pi\)
−0.729332 + 0.684160i \(0.760169\pi\)
\(978\) − 6.00000i − 0.191859i
\(979\) 1.37258 0.0438679
\(980\) 0 0
\(981\) −14.1421 −0.451524
\(982\) 52.7279i 1.68262i
\(983\) 0.213203i 0.00680013i 0.999994 + 0.00340007i \(0.00108228\pi\)
−0.999994 + 0.00340007i \(0.998918\pi\)
\(984\) −7.31371 −0.233153
\(985\) 0 0
\(986\) −39.6985 −1.26426
\(987\) 0 0
\(988\) 0 0
\(989\) −14.8284 −0.471517
\(990\) 0 0
\(991\) 19.9411 0.633451 0.316725 0.948517i \(-0.397417\pi\)
0.316725 + 0.948517i \(0.397417\pi\)
\(992\) 0 0
\(993\) 11.3726i 0.360898i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 21.7279i 0.688130i 0.938946 + 0.344065i \(0.111804\pi\)
−0.938946 + 0.344065i \(0.888196\pi\)
\(998\) 4.24264i 0.134298i
\(999\) 5.41421 0.171298
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 1225.2.b.i.99.2 4
5.2 odd 4 1225.2.a.r.1.2 2
5.3 odd 4 245.2.a.e.1.1 2
5.4 even 2 inner 1225.2.b.i.99.3 4
7.6 odd 2 1225.2.b.j.99.1 4
15.8 even 4 2205.2.a.v.1.2 2
20.3 even 4 3920.2.a.bw.1.1 2
35.3 even 12 245.2.e.f.226.2 4
35.13 even 4 245.2.a.f.1.1 yes 2
35.18 odd 12 245.2.e.g.226.2 4
35.23 odd 12 245.2.e.g.116.2 4
35.27 even 4 1225.2.a.p.1.2 2
35.33 even 12 245.2.e.f.116.2 4
35.34 odd 2 1225.2.b.j.99.4 4
105.83 odd 4 2205.2.a.t.1.2 2
140.83 odd 4 3920.2.a.br.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.2.a.e.1.1 2 5.3 odd 4
245.2.a.f.1.1 yes 2 35.13 even 4
245.2.e.f.116.2 4 35.33 even 12
245.2.e.f.226.2 4 35.3 even 12
245.2.e.g.116.2 4 35.23 odd 12
245.2.e.g.226.2 4 35.18 odd 12
1225.2.a.p.1.2 2 35.27 even 4
1225.2.a.r.1.2 2 5.2 odd 4
1225.2.b.i.99.2 4 1.1 even 1 trivial
1225.2.b.i.99.3 4 5.4 even 2 inner
1225.2.b.j.99.1 4 7.6 odd 2
1225.2.b.j.99.4 4 35.34 odd 2
2205.2.a.t.1.2 2 105.83 odd 4
2205.2.a.v.1.2 2 15.8 even 4
3920.2.a.br.1.2 2 140.83 odd 4
3920.2.a.bw.1.1 2 20.3 even 4