Properties

Label 1225.2.a.p.1.2
Level $1225$
Weight $2$
Character 1225.1
Self dual yes
Analytic conductor $9.782$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1225.a (trivial)

Newform invariants

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

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1225.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421 q^{2} +0.414214 q^{3} +0.585786 q^{6} -2.82843 q^{8} -2.82843 q^{9} +O(q^{10})\) \(q+1.41421 q^{2} +0.414214 q^{3} +0.585786 q^{6} -2.82843 q^{8} -2.82843 q^{9} -0.171573 q^{11} -4.41421 q^{13} -4.00000 q^{16} -3.24264 q^{17} -4.00000 q^{18} +6.00000 q^{19} -0.242641 q^{22} -7.41421 q^{23} -1.17157 q^{24} -6.24264 q^{26} -2.41421 q^{27} -8.65685 q^{29} +10.2426 q^{31} -0.0710678 q^{33} -4.58579 q^{34} -2.24264 q^{37} +8.48528 q^{38} -1.82843 q^{39} -6.24264 q^{41} -2.00000 q^{43} -10.4853 q^{46} +7.24264 q^{47} -1.65685 q^{48} -1.34315 q^{51} -4.24264 q^{53} -3.41421 q^{54} +2.48528 q^{57} -12.2426 q^{58} -2.24264 q^{59} -2.82843 q^{61} +14.4853 q^{62} +8.00000 q^{64} -0.100505 q^{66} +8.24264 q^{67} -3.07107 q^{69} -3.17157 q^{71} +8.00000 q^{72} +8.48528 q^{73} -3.17157 q^{74} -2.58579 q^{78} +1.48528 q^{79} +7.48528 q^{81} -8.82843 q^{82} -2.82843 q^{86} -3.58579 q^{87} +0.485281 q^{88} -8.00000 q^{89} +4.24264 q^{93} +10.2426 q^{94} -13.2426 q^{97} +0.485281 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 4q^{6} + O(q^{10}) \) \( 2q - 2q^{3} + 4q^{6} - 6q^{11} - 6q^{13} - 8q^{16} + 2q^{17} - 8q^{18} + 12q^{19} + 8q^{22} - 12q^{23} - 8q^{24} - 4q^{26} - 2q^{27} - 6q^{29} + 12q^{31} + 14q^{33} - 12q^{34} + 4q^{37} + 2q^{39} - 4q^{41} - 4q^{43} - 4q^{46} + 6q^{47} + 8q^{48} - 14q^{51} - 4q^{54} - 12q^{57} - 16q^{58} + 4q^{59} + 12q^{62} + 16q^{64} - 20q^{66} + 8q^{67} + 8q^{69} - 12q^{71} + 16q^{72} - 12q^{74} - 8q^{78} - 14q^{79} - 2q^{81} - 12q^{82} - 10q^{87} - 16q^{88} - 16q^{89} + 12q^{94} - 18q^{97} - 16q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(3\) 0.414214 0.239146 0.119573 0.992825i \(-0.461847\pi\)
0.119573 + 0.992825i \(0.461847\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0.585786 0.239146
\(7\) 0 0
\(8\) −2.82843 −1.00000
\(9\) −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
\(13\) −4.41421 −1.22428 −0.612141 0.790748i \(-0.709692\pi\)
−0.612141 + 0.790748i \(0.709692\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −3.24264 −0.786456 −0.393228 0.919441i \(-0.628642\pi\)
−0.393228 + 0.919441i \(0.628642\pi\)
\(18\) −4.00000 −0.942809
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −0.242641 −0.0517312
\(23\) −7.41421 −1.54597 −0.772985 0.634424i \(-0.781237\pi\)
−0.772985 + 0.634424i \(0.781237\pi\)
\(24\) −1.17157 −0.239146
\(25\) 0 0
\(26\) −6.24264 −1.22428
\(27\) −2.41421 −0.464616
\(28\) 0 0
\(29\) −8.65685 −1.60754 −0.803769 0.594942i \(-0.797175\pi\)
−0.803769 + 0.594942i \(0.797175\pi\)
\(30\) 0 0
\(31\) 10.2426 1.83963 0.919816 0.392349i \(-0.128338\pi\)
0.919816 + 0.392349i \(0.128338\pi\)
\(32\) 0 0
\(33\) −0.0710678 −0.0123713
\(34\) −4.58579 −0.786456
\(35\) 0 0
\(36\) 0 0
\(37\) −2.24264 −0.368688 −0.184344 0.982862i \(-0.559016\pi\)
−0.184344 + 0.982862i \(0.559016\pi\)
\(38\) 8.48528 1.37649
\(39\) −1.82843 −0.292783
\(40\) 0 0
\(41\) −6.24264 −0.974937 −0.487468 0.873141i \(-0.662080\pi\)
−0.487468 + 0.873141i \(0.662080\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −10.4853 −1.54597
\(47\) 7.24264 1.05645 0.528224 0.849105i \(-0.322858\pi\)
0.528224 + 0.849105i \(0.322858\pi\)
\(48\) −1.65685 −0.239146
\(49\) 0 0
\(50\) 0 0
\(51\) −1.34315 −0.188078
\(52\) 0 0
\(53\) −4.24264 −0.582772 −0.291386 0.956606i \(-0.594116\pi\)
−0.291386 + 0.956606i \(0.594116\pi\)
\(54\) −3.41421 −0.464616
\(55\) 0 0
\(56\) 0 0
\(57\) 2.48528 0.329184
\(58\) −12.2426 −1.60754
\(59\) −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 \(-0.557957\pi\)
−0.181071 + 0.983470i \(0.557957\pi\)
\(62\) 14.4853 1.83963
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) −0.100505 −0.0123713
\(67\) 8.24264 1.00700 0.503499 0.863996i \(-0.332046\pi\)
0.503499 + 0.863996i \(0.332046\pi\)
\(68\) 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 \(0.560265\pi\)
\(72\) 8.00000 0.942809
\(73\) 8.48528 0.993127 0.496564 0.868000i \(-0.334595\pi\)
0.496564 + 0.868000i \(0.334595\pi\)
\(74\) −3.17157 −0.368688
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) −2.58579 −0.292783
\(79\) 1.48528 0.167107 0.0835536 0.996503i \(-0.473373\pi\)
0.0835536 + 0.996503i \(0.473373\pi\)
\(80\) 0 0
\(81\) 7.48528 0.831698
\(82\) −8.82843 −0.974937
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −2.82843 −0.304997
\(87\) −3.58579 −0.384437
\(88\) 0.485281 0.0517312
\(89\) −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
\(93\) 4.24264 0.439941
\(94\) 10.2426 1.05645
\(95\) 0 0
\(96\) 0 0
\(97\) −13.2426 −1.34459 −0.672293 0.740285i \(-0.734691\pi\)
−0.672293 + 0.740285i \(0.734691\pi\)
\(98\) 0 0
\(99\) 0.485281 0.0487726
\(100\) 0 0
\(101\) 2.48528 0.247295 0.123647 0.992326i \(-0.460541\pi\)
0.123647 + 0.992326i \(0.460541\pi\)
\(102\) −1.89949 −0.188078
\(103\) −19.2426 −1.89603 −0.948017 0.318220i \(-0.896915\pi\)
−0.948017 + 0.318220i \(0.896915\pi\)
\(104\) 12.4853 1.22428
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 2.48528 0.240261 0.120131 0.992758i \(-0.461669\pi\)
0.120131 + 0.992758i \(0.461669\pi\)
\(108\) 0 0
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) 0 0
\(111\) −0.928932 −0.0881703
\(112\) 0 0
\(113\) 13.0711 1.22962 0.614811 0.788674i \(-0.289232\pi\)
0.614811 + 0.788674i \(0.289232\pi\)
\(114\) 3.51472 0.329184
\(115\) 0 0
\(116\) 0 0
\(117\) 12.4853 1.15426
\(118\) −3.17157 −0.291967
\(119\) 0 0
\(120\) 0 0
\(121\) −10.9706 −0.997324
\(122\) −4.00000 −0.362143
\(123\) −2.58579 −0.233153
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −8.24264 −0.731416 −0.365708 0.930730i \(-0.619173\pi\)
−0.365708 + 0.930730i \(0.619173\pi\)
\(128\) 11.3137 1.00000
\(129\) −0.828427 −0.0729389
\(130\) 0 0
\(131\) 12.2426 1.06964 0.534822 0.844965i \(-0.320379\pi\)
0.534822 + 0.844965i \(0.320379\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 11.6569 1.00700
\(135\) 0 0
\(136\) 9.17157 0.786456
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) −4.34315 −0.369713
\(139\) 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
\(142\) −4.48528 −0.376396
\(143\) 0.757359 0.0633336
\(144\) 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 \(-0.622592\pi\)
−0.375682 + 0.926749i \(0.622592\pi\)
\(150\) 0 0
\(151\) −7.48528 −0.609144 −0.304572 0.952489i \(-0.598513\pi\)
−0.304572 + 0.952489i \(0.598513\pi\)
\(152\) −16.9706 −1.37649
\(153\) 9.17157 0.741478
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 15.1716 1.21082 0.605412 0.795913i \(-0.293008\pi\)
0.605412 + 0.795913i \(0.293008\pi\)
\(158\) 2.10051 0.167107
\(159\) −1.75736 −0.139368
\(160\) 0 0
\(161\) 0 0
\(162\) 10.5858 0.831698
\(163\) 10.2426 0.802266 0.401133 0.916020i \(-0.368617\pi\)
0.401133 + 0.916020i \(0.368617\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −0.757359 −0.0586062 −0.0293031 0.999571i \(-0.509329\pi\)
−0.0293031 + 0.999571i \(0.509329\pi\)
\(168\) 0 0
\(169\) 6.48528 0.498868
\(170\) 0 0
\(171\) −16.9706 −1.29777
\(172\) 0 0
\(173\) 7.24264 0.550648 0.275324 0.961352i \(-0.411215\pi\)
0.275324 + 0.961352i \(0.411215\pi\)
\(174\) −5.07107 −0.384437
\(175\) 0 0
\(176\) 0.686292 0.0517312
\(177\) −0.928932 −0.0698228
\(178\) −11.3137 −0.847998
\(179\) −14.4853 −1.08268 −0.541340 0.840804i \(-0.682083\pi\)
−0.541340 + 0.840804i \(0.682083\pi\)
\(180\) 0 0
\(181\) −18.7279 −1.39204 −0.696018 0.718025i \(-0.745047\pi\)
−0.696018 + 0.718025i \(0.745047\pi\)
\(182\) 0 0
\(183\) −1.17157 −0.0866052
\(184\) 20.9706 1.54597
\(185\) 0 0
\(186\) 6.00000 0.439941
\(187\) 0.556349 0.0406843
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 13.9706 1.01087 0.505437 0.862863i \(-0.331331\pi\)
0.505437 + 0.862863i \(0.331331\pi\)
\(192\) 3.31371 0.239146
\(193\) 16.0000 1.15171 0.575853 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) −18.7279 −1.34459
\(195\) 0 0
\(196\) 0 0
\(197\) −13.4142 −0.955723 −0.477862 0.878435i \(-0.658588\pi\)
−0.477862 + 0.878435i \(0.658588\pi\)
\(198\) 0.686292 0.0487726
\(199\) −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
\(202\) 3.51472 0.247295
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) −27.2132 −1.89603
\(207\) 20.9706 1.45755
\(208\) 17.6569 1.22428
\(209\) −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
\(213\) −1.31371 −0.0900138
\(214\) 3.51472 0.240261
\(215\) 0 0
\(216\) 6.82843 0.464616
\(217\) 0 0
\(218\) 7.07107 0.478913
\(219\) 3.51472 0.237503
\(220\) 0 0
\(221\) 14.3137 0.962844
\(222\) −1.31371 −0.0881703
\(223\) 24.2132 1.62144 0.810718 0.585437i \(-0.199077\pi\)
0.810718 + 0.585437i \(0.199077\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 18.4853 1.22962
\(227\) −15.7279 −1.04390 −0.521949 0.852976i \(-0.674795\pi\)
−0.521949 + 0.852976i \(0.674795\pi\)
\(228\) 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
\(232\) 24.4853 1.60754
\(233\) 9.17157 0.600850 0.300425 0.953805i \(-0.402872\pi\)
0.300425 + 0.953805i \(0.402872\pi\)
\(234\) 17.6569 1.15426
\(235\) 0 0
\(236\) 0 0
\(237\) 0.615224 0.0399631
\(238\) 0 0
\(239\) −17.4853 −1.13103 −0.565514 0.824738i \(-0.691322\pi\)
−0.565514 + 0.824738i \(0.691322\pi\)
\(240\) 0 0
\(241\) −0.727922 −0.0468896 −0.0234448 0.999725i \(-0.507463\pi\)
−0.0234448 + 0.999725i \(0.507463\pi\)
\(242\) −15.5147 −0.997324
\(243\) 10.3431 0.663513
\(244\) 0 0
\(245\) 0 0
\(246\) −3.65685 −0.233153
\(247\) −26.4853 −1.68522
\(248\) −28.9706 −1.83963
\(249\) 0 0
\(250\) 0 0
\(251\) 17.2132 1.08649 0.543244 0.839575i \(-0.317196\pi\)
0.543244 + 0.839575i \(0.317196\pi\)
\(252\) 0 0
\(253\) 1.27208 0.0799749
\(254\) −11.6569 −0.731416
\(255\) 0 0
\(256\) 0 0
\(257\) 9.51472 0.593512 0.296756 0.954953i \(-0.404095\pi\)
0.296756 + 0.954953i \(0.404095\pi\)
\(258\) −1.17157 −0.0729389
\(259\) 0 0
\(260\) 0 0
\(261\) 24.4853 1.51560
\(262\) 17.3137 1.06964
\(263\) 16.6274 1.02529 0.512645 0.858601i \(-0.328666\pi\)
0.512645 + 0.858601i \(0.328666\pi\)
\(264\) 0.201010 0.0123713
\(265\) 0 0
\(266\) 0 0
\(267\) −3.31371 −0.202796
\(268\) 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 \(-0.506636\pi\)
−0.0208446 + 0.999783i \(0.506636\pi\)
\(272\) 12.9706 0.786456
\(273\) 0 0
\(274\) −16.9706 −1.02523
\(275\) 0 0
\(276\) 0 0
\(277\) −15.2132 −0.914073 −0.457036 0.889448i \(-0.651089\pi\)
−0.457036 + 0.889448i \(0.651089\pi\)
\(278\) 10.9706 0.657971
\(279\) −28.9706 −1.73442
\(280\) 0 0
\(281\) 2.31371 0.138024 0.0690121 0.997616i \(-0.478015\pi\)
0.0690121 + 0.997616i \(0.478015\pi\)
\(282\) 4.24264 0.252646
\(283\) −24.5563 −1.45972 −0.729862 0.683595i \(-0.760416\pi\)
−0.729862 + 0.683595i \(0.760416\pi\)
\(284\) 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
\(293\) −25.7279 −1.50304 −0.751521 0.659710i \(-0.770679\pi\)
−0.751521 + 0.659710i \(0.770679\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 6.34315 0.368688
\(297\) 0.414214 0.0240351
\(298\) −12.9706 −0.751365
\(299\) 32.7279 1.89270
\(300\) 0 0
\(301\) 0 0
\(302\) −10.5858 −0.609144
\(303\) 1.02944 0.0591396
\(304\) −24.0000 −1.37649
\(305\) 0 0
\(306\) 12.9706 0.741478
\(307\) −30.8995 −1.76353 −0.881764 0.471691i \(-0.843644\pi\)
−0.881764 + 0.471691i \(0.843644\pi\)
\(308\) 0 0
\(309\) −7.97056 −0.453429
\(310\) 0 0
\(311\) −10.0000 −0.567048 −0.283524 0.958965i \(-0.591504\pi\)
−0.283524 + 0.958965i \(0.591504\pi\)
\(312\) 5.17157 0.292783
\(313\) 18.2132 1.02947 0.514736 0.857349i \(-0.327890\pi\)
0.514736 + 0.857349i \(0.327890\pi\)
\(314\) 21.4558 1.21082
\(315\) 0 0
\(316\) 0 0
\(317\) 0.343146 0.0192730 0.00963649 0.999954i \(-0.496933\pi\)
0.00963649 + 0.999954i \(0.496933\pi\)
\(318\) −2.48528 −0.139368
\(319\) 1.48528 0.0831598
\(320\) 0 0
\(321\) 1.02944 0.0574576
\(322\) 0 0
\(323\) −19.4558 −1.08255
\(324\) 0 0
\(325\) 0 0
\(326\) 14.4853 0.802266
\(327\) 2.07107 0.114530
\(328\) 17.6569 0.974937
\(329\) 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
\(333\) 6.34315 0.347602
\(334\) −1.07107 −0.0586062
\(335\) 0 0
\(336\) 0 0
\(337\) −22.2426 −1.21163 −0.605817 0.795604i \(-0.707154\pi\)
−0.605817 + 0.795604i \(0.707154\pi\)
\(338\) 9.17157 0.498868
\(339\) 5.41421 0.294060
\(340\) 0 0
\(341\) −1.75736 −0.0951663
\(342\) −24.0000 −1.29777
\(343\) 0 0
\(344\) 5.65685 0.304997
\(345\) 0 0
\(346\) 10.2426 0.550648
\(347\) −13.0711 −0.701692 −0.350846 0.936433i \(-0.614106\pi\)
−0.350846 + 0.936433i \(0.614106\pi\)
\(348\) 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
\(353\) 6.21320 0.330695 0.165348 0.986235i \(-0.447125\pi\)
0.165348 + 0.986235i \(0.447125\pi\)
\(354\) −1.31371 −0.0698228
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −20.4853 −1.08268
\(359\) −11.3137 −0.597115 −0.298557 0.954392i \(-0.596505\pi\)
−0.298557 + 0.954392i \(0.596505\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) −26.4853 −1.39204
\(363\) −4.54416 −0.238506
\(364\) 0 0
\(365\) 0 0
\(366\) −1.65685 −0.0866052
\(367\) 23.8701 1.24601 0.623003 0.782219i \(-0.285912\pi\)
0.623003 + 0.782219i \(0.285912\pi\)
\(368\) 29.6569 1.54597
\(369\) 17.6569 0.919179
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 16.4853 0.853576 0.426788 0.904352i \(-0.359645\pi\)
0.426788 + 0.904352i \(0.359645\pi\)
\(374\) 0.786797 0.0406843
\(375\) 0 0
\(376\) −20.4853 −1.05645
\(377\) 38.2132 1.96808
\(378\) 0 0
\(379\) 2.00000 0.102733 0.0513665 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) 0 0
\(381\) −3.41421 −0.174915
\(382\) 19.7574 1.01087
\(383\) −4.48528 −0.229187 −0.114594 0.993412i \(-0.536557\pi\)
−0.114594 + 0.993412i \(0.536557\pi\)
\(384\) 4.68629 0.239146
\(385\) 0 0
\(386\) 22.6274 1.15171
\(387\) 5.65685 0.287554
\(388\) 0 0
\(389\) −6.85786 −0.347708 −0.173854 0.984771i \(-0.555622\pi\)
−0.173854 + 0.984771i \(0.555622\pi\)
\(390\) 0 0
\(391\) 24.0416 1.21584
\(392\) 0 0
\(393\) 5.07107 0.255802
\(394\) −18.9706 −0.955723
\(395\) 0 0
\(396\) 0 0
\(397\) −1.58579 −0.0795883 −0.0397942 0.999208i \(-0.512670\pi\)
−0.0397942 + 0.999208i \(0.512670\pi\)
\(398\) −10.4853 −0.525580
\(399\) 0 0
\(400\) 0 0
\(401\) 11.8284 0.590683 0.295342 0.955392i \(-0.404566\pi\)
0.295342 + 0.955392i \(0.404566\pi\)
\(402\) 4.82843 0.240820
\(403\) −45.2132 −2.25223
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.384776 0.0190727
\(408\) 3.79899 0.188078
\(409\) 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
\(417\) 3.21320 0.157351
\(418\) −1.45584 −0.0712077
\(419\) 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 \(0.653228\pi\)
\(422\) 12.7279 0.619586
\(423\) −20.4853 −0.996028
\(424\) 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 \(0.743976\pi\)
\(432\) 9.65685 0.464616
\(433\) −10.9706 −0.527212 −0.263606 0.964630i \(-0.584912\pi\)
−0.263606 + 0.964630i \(0.584912\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −44.4853 −2.12802
\(438\) 4.97056 0.237503
\(439\) −30.3848 −1.45019 −0.725093 0.688651i \(-0.758203\pi\)
−0.725093 + 0.688651i \(0.758203\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 20.2426 0.962844
\(443\) −15.1716 −0.720823 −0.360412 0.932793i \(-0.617364\pi\)
−0.360412 + 0.932793i \(0.617364\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 34.2426 1.62144
\(447\) −3.79899 −0.179686
\(448\) 0 0
\(449\) −24.1716 −1.14073 −0.570364 0.821392i \(-0.693198\pi\)
−0.570364 + 0.821392i \(0.693198\pi\)
\(450\) 0 0
\(451\) 1.07107 0.0504346
\(452\) 0 0
\(453\) −3.10051 −0.145674
\(454\) −22.2426 −1.04390
\(455\) 0 0
\(456\) −7.02944 −0.329184
\(457\) 11.7574 0.549986 0.274993 0.961446i \(-0.411324\pi\)
0.274993 + 0.961446i \(0.411324\pi\)
\(458\) 25.5147 1.19222
\(459\) 7.82843 0.365400
\(460\) 0 0
\(461\) 3.02944 0.141095 0.0705475 0.997508i \(-0.477525\pi\)
0.0705475 + 0.997508i \(0.477525\pi\)
\(462\) 0 0
\(463\) 21.4558 0.997138 0.498569 0.866850i \(-0.333859\pi\)
0.498569 + 0.866850i \(0.333859\pi\)
\(464\) 34.6274 1.60754
\(465\) 0 0
\(466\) 12.9706 0.600850
\(467\) 5.72792 0.265057 0.132528 0.991179i \(-0.457690\pi\)
0.132528 + 0.991179i \(0.457690\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 6.28427 0.289564
\(472\) 6.34315 0.291967
\(473\) 0.343146 0.0157779
\(474\) 0.870058 0.0399631
\(475\) 0 0
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) −24.7279 −1.13103
\(479\) 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
\(482\) −1.02944 −0.0468896
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 14.6274 0.663513
\(487\) −27.6985 −1.25514 −0.627569 0.778561i \(-0.715950\pi\)
−0.627569 + 0.778561i \(0.715950\pi\)
\(488\) 8.00000 0.362143
\(489\) 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
\(493\) 28.0711 1.26426
\(494\) −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 \(-0.478610\pi\)
0.0671492 + 0.997743i \(0.478610\pi\)
\(500\) 0 0
\(501\) −0.313708 −0.0140155
\(502\) 24.3431 1.08649
\(503\) 41.2426 1.83892 0.919459 0.393185i \(-0.128627\pi\)
0.919459 + 0.393185i \(0.128627\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 1.79899 0.0799749
\(507\) 2.68629 0.119302
\(508\) 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
\(512\) −22.6274 −1.00000
\(513\) −14.4853 −0.639541
\(514\) 13.4558 0.593512
\(515\) 0 0
\(516\) 0 0
\(517\) −1.24264 −0.0546513
\(518\) 0 0
\(519\) 3.00000 0.131685
\(520\) 0 0
\(521\) 14.9706 0.655872 0.327936 0.944700i \(-0.393647\pi\)
0.327936 + 0.944700i \(0.393647\pi\)
\(522\) 34.6274 1.51560
\(523\) −32.4853 −1.42048 −0.710241 0.703959i \(-0.751414\pi\)
−0.710241 + 0.703959i \(0.751414\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 23.5147 1.02529
\(527\) −33.2132 −1.44679
\(528\) 0.284271 0.0123713
\(529\) 31.9706 1.39002
\(530\) 0 0
\(531\) 6.34315 0.275269
\(532\) 0 0
\(533\) 27.5563 1.19360
\(534\) −4.68629 −0.202796
\(535\) 0 0
\(536\) −23.3137 −1.00700
\(537\) −6.00000 −0.258919
\(538\) −22.9706 −0.990331
\(539\) 0 0
\(540\) 0 0
\(541\) −21.9706 −0.944588 −0.472294 0.881441i \(-0.656574\pi\)
−0.472294 + 0.881441i \(0.656574\pi\)
\(542\) −0.970563 −0.0416892
\(543\) −7.75736 −0.332900
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 24.4853 1.04692 0.523458 0.852052i \(-0.324642\pi\)
0.523458 + 0.852052i \(0.324642\pi\)
\(548\) 0 0
\(549\) 8.00000 0.341432
\(550\) 0 0
\(551\) −51.9411 −2.21277
\(552\) 8.68629 0.369713
\(553\) 0 0
\(554\) −21.5147 −0.914073
\(555\) 0 0
\(556\) 0 0
\(557\) 7.79899 0.330454 0.165227 0.986256i \(-0.447164\pi\)
0.165227 + 0.986256i \(0.447164\pi\)
\(558\) −40.9706 −1.73442
\(559\) 8.82843 0.373403
\(560\) 0 0
\(561\) 0.230447 0.00972950
\(562\) 3.27208 0.138024
\(563\) 31.9411 1.34616 0.673079 0.739571i \(-0.264971\pi\)
0.673079 + 0.739571i \(0.264971\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −34.7279 −1.45972
\(567\) 0 0
\(568\) 8.97056 0.376396
\(569\) 26.1421 1.09594 0.547968 0.836500i \(-0.315402\pi\)
0.547968 + 0.836500i \(0.315402\pi\)
\(570\) 0 0
\(571\) −17.5147 −0.732968 −0.366484 0.930424i \(-0.619439\pi\)
−0.366484 + 0.930424i \(0.619439\pi\)
\(572\) 0 0
\(573\) 5.78680 0.241747
\(574\) 0 0
\(575\) 0 0
\(576\) −22.6274 −0.942809
\(577\) −15.7279 −0.654762 −0.327381 0.944892i \(-0.606166\pi\)
−0.327381 + 0.944892i \(0.606166\pi\)
\(578\) −9.17157 −0.381487
\(579\) 6.62742 0.275426
\(580\) 0 0
\(581\) 0 0
\(582\) −7.75736 −0.321553
\(583\) 0.727922 0.0301475
\(584\) −24.0000 −0.993127
\(585\) 0 0
\(586\) −36.3848 −1.50304
\(587\) 37.4558 1.54597 0.772984 0.634425i \(-0.218763\pi\)
0.772984 + 0.634425i \(0.218763\pi\)
\(588\) 0 0
\(589\) 61.4558 2.53224
\(590\) 0 0
\(591\) −5.55635 −0.228558
\(592\) 8.97056 0.368688
\(593\) −19.2426 −0.790201 −0.395100 0.918638i \(-0.629290\pi\)
−0.395100 + 0.918638i \(0.629290\pi\)
\(594\) 0.585786 0.0240351
\(595\) 0 0
\(596\) 0 0
\(597\) −3.07107 −0.125690
\(598\) 46.2843 1.89270
\(599\) −17.8284 −0.728450 −0.364225 0.931311i \(-0.618666\pi\)
−0.364225 + 0.931311i \(0.618666\pi\)
\(600\) 0 0
\(601\) −10.9706 −0.447499 −0.223749 0.974647i \(-0.571830\pi\)
−0.223749 + 0.974647i \(0.571830\pi\)
\(602\) 0 0
\(603\) −23.3137 −0.949408
\(604\) 0 0
\(605\) 0 0
\(606\) 1.45584 0.0591396
\(607\) 5.10051 0.207023 0.103512 0.994628i \(-0.466992\pi\)
0.103512 + 0.994628i \(0.466992\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −31.9706 −1.29339
\(612\) 0 0
\(613\) 23.9411 0.966973 0.483486 0.875352i \(-0.339370\pi\)
0.483486 + 0.875352i \(0.339370\pi\)
\(614\) −43.6985 −1.76353
\(615\) 0 0
\(616\) 0 0
\(617\) −4.58579 −0.184617 −0.0923084 0.995730i \(-0.529425\pi\)
−0.0923084 + 0.995730i \(0.529425\pi\)
\(618\) −11.2721 −0.453429
\(619\) 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
\(622\) −14.1421 −0.567048
\(623\) 0 0
\(624\) 7.31371 0.292783
\(625\) 0 0
\(626\) 25.7574 1.02947
\(627\) −0.426407 −0.0170291
\(628\) 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 \(0.820441\pi\)
\(632\) −4.20101 −0.167107
\(633\) 3.72792 0.148172
\(634\) 0.485281 0.0192730
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 2.10051 0.0831598
\(639\) 8.97056 0.354870
\(640\) 0 0
\(641\) −23.3137 −0.920836 −0.460418 0.887702i \(-0.652300\pi\)
−0.460418 + 0.887702i \(0.652300\pi\)
\(642\) 1.45584 0.0574576
\(643\) −2.27208 −0.0896020 −0.0448010 0.998996i \(-0.514265\pi\)
−0.0448010 + 0.998996i \(0.514265\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −27.5147 −1.08255
\(647\) 11.5147 0.452690 0.226345 0.974047i \(-0.427322\pi\)
0.226345 + 0.974047i \(0.427322\pi\)
\(648\) −21.1716 −0.831698
\(649\) 0.384776 0.0151038
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −34.9706 −1.36850 −0.684252 0.729246i \(-0.739871\pi\)
−0.684252 + 0.729246i \(0.739871\pi\)
\(654\) 2.92893 0.114530
\(655\) 0 0
\(656\) 24.9706 0.974937
\(657\) −24.0000 −0.936329
\(658\) 0 0
\(659\) 19.9706 0.777943 0.388971 0.921250i \(-0.372831\pi\)
0.388971 + 0.921250i \(0.372831\pi\)
\(660\) 0 0
\(661\) 13.4558 0.523372 0.261686 0.965153i \(-0.415722\pi\)
0.261686 + 0.965153i \(0.415722\pi\)
\(662\) −38.8284 −1.50911
\(663\) 5.92893 0.230261
\(664\) 0 0
\(665\) 0 0
\(666\) 8.97056 0.347602
\(667\) 64.1838 2.48521
\(668\) 0 0
\(669\) 10.0294 0.387760
\(670\) 0 0
\(671\) 0.485281 0.0187341
\(672\) 0 0
\(673\) −3.51472 −0.135482 −0.0677412 0.997703i \(-0.521579\pi\)
−0.0677412 + 0.997703i \(0.521579\pi\)
\(674\) −31.4558 −1.21163
\(675\) 0 0
\(676\) 0 0
\(677\) −44.2132 −1.69925 −0.849626 0.527386i \(-0.823172\pi\)
−0.849626 + 0.527386i \(0.823172\pi\)
\(678\) 7.65685 0.294060
\(679\) 0 0
\(680\) 0 0
\(681\) −6.51472 −0.249645
\(682\) −2.48528 −0.0951663
\(683\) −31.7990 −1.21675 −0.608377 0.793648i \(-0.708179\pi\)
−0.608377 + 0.793648i \(0.708179\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 7.47309 0.285116
\(688\) 8.00000 0.304997
\(689\) 18.7279 0.713477
\(690\) 0 0
\(691\) 14.8284 0.564100 0.282050 0.959400i \(-0.408986\pi\)
0.282050 + 0.959400i \(0.408986\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −18.4853 −0.701692
\(695\) 0 0
\(696\) 10.1421 0.384437
\(697\) 20.2426 0.766745
\(698\) −15.5147 −0.587241
\(699\) 3.79899 0.143691
\(700\) 0 0
\(701\) 46.4558 1.75461 0.877307 0.479931i \(-0.159338\pi\)
0.877307 + 0.479931i \(0.159338\pi\)
\(702\) 15.0711 0.568821
\(703\) −13.4558 −0.507497
\(704\) −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 \(-0.396578\pi\)
0.319224 + 0.947679i \(0.396578\pi\)
\(710\) 0 0
\(711\) −4.20101 −0.157550
\(712\) 22.6274 0.847998
\(713\) −75.9411 −2.84402
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −7.24264 −0.270481
\(718\) −16.0000 −0.597115
\(719\) −25.2132 −0.940294 −0.470147 0.882588i \(-0.655799\pi\)
−0.470147 + 0.882588i \(0.655799\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 24.0416 0.894737
\(723\) −0.301515 −0.0112135
\(724\) 0 0
\(725\) 0 0
\(726\) −6.42641 −0.238506
\(727\) 6.68629 0.247981 0.123990 0.992283i \(-0.460431\pi\)
0.123990 + 0.992283i \(0.460431\pi\)
\(728\) 0 0
\(729\) −18.1716 −0.673021
\(730\) 0 0
\(731\) 6.48528 0.239867
\(732\) 0 0
\(733\) −14.6985 −0.542901 −0.271450 0.962452i \(-0.587503\pi\)
−0.271450 + 0.962452i \(0.587503\pi\)
\(734\) 33.7574 1.24601
\(735\) 0 0
\(736\) 0 0
\(737\) −1.41421 −0.0520932
\(738\) 24.9706 0.919179
\(739\) 29.9706 1.10248 0.551242 0.834345i \(-0.314154\pi\)
0.551242 + 0.834345i \(0.314154\pi\)
\(740\) 0 0
\(741\) −10.9706 −0.403014
\(742\) 0 0
\(743\) −11.2721 −0.413532 −0.206766 0.978390i \(-0.566294\pi\)
−0.206766 + 0.978390i \(0.566294\pi\)
\(744\) −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 \(0.319193\pi\)
\(752\) −28.9706 −1.05645
\(753\) 7.12994 0.259830
\(754\) 54.0416 1.96808
\(755\) 0 0
\(756\) 0 0
\(757\) −0.485281 −0.0176379 −0.00881893 0.999961i \(-0.502807\pi\)
−0.00881893 + 0.999961i \(0.502807\pi\)
\(758\) 2.82843 0.102733
\(759\) 0.526912 0.0191257
\(760\) 0 0
\(761\) −12.7279 −0.461387 −0.230693 0.973026i \(-0.574099\pi\)
−0.230693 + 0.973026i \(0.574099\pi\)
\(762\) −4.82843 −0.174915
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −6.34315 −0.229187
\(767\) 9.89949 0.357450
\(768\) 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
\(773\) 6.21320 0.223473 0.111737 0.993738i \(-0.464359\pi\)
0.111737 + 0.993738i \(0.464359\pi\)
\(774\) 8.00000 0.287554
\(775\) 0 0
\(776\) 37.4558 1.34459
\(777\) 0 0
\(778\) −9.69848 −0.347708
\(779\) −37.4558 −1.34199
\(780\) 0 0
\(781\) 0.544156 0.0194714
\(782\) 34.0000 1.21584
\(783\) 20.8995 0.746887
\(784\) 0 0
\(785\) 0 0
\(786\) 7.17157 0.255802
\(787\) −20.2721 −0.722622 −0.361311 0.932445i \(-0.617671\pi\)
−0.361311 + 0.932445i \(0.617671\pi\)
\(788\) 0 0
\(789\) 6.88730 0.245194
\(790\) 0 0
\(791\) 0 0
\(792\) −1.37258 −0.0487726
\(793\) 12.4853 0.443365
\(794\) −2.24264 −0.0795883
\(795\) 0 0
\(796\) 0 0
\(797\) 21.1838 0.750367 0.375184 0.926950i \(-0.377580\pi\)
0.375184 + 0.926950i \(0.377580\pi\)
\(798\) 0 0
\(799\) −23.4853 −0.830850
\(800\) 0 0
\(801\) 22.6274 0.799500
\(802\) 16.7279 0.590683
\(803\) −1.45584 −0.0513756
\(804\) 0 0
\(805\) 0 0
\(806\) −63.9411 −2.25223
\(807\) −6.72792 −0.236834
\(808\) −7.02944 −0.247295
\(809\) −48.5980 −1.70861 −0.854307 0.519769i \(-0.826018\pi\)
−0.854307 + 0.519769i \(0.826018\pi\)
\(810\) 0 0
\(811\) 47.3553 1.66287 0.831435 0.555621i \(-0.187520\pi\)
0.831435 + 0.555621i \(0.187520\pi\)
\(812\) 0 0
\(813\) −0.284271 −0.00996983
\(814\) 0.544156 0.0190727
\(815\) 0 0
\(816\) 5.37258 0.188078
\(817\) −12.0000 −0.419827
\(818\) 3.51472 0.122889
\(819\) 0 0
\(820\) 0 0
\(821\) −23.4853 −0.819642 −0.409821 0.912166i \(-0.634409\pi\)
−0.409821 + 0.912166i \(0.634409\pi\)
\(822\) −7.02944 −0.245180
\(823\) −16.7279 −0.583099 −0.291549 0.956556i \(-0.594171\pi\)
−0.291549 + 0.956556i \(0.594171\pi\)
\(824\) 54.4264 1.89603
\(825\) 0 0
\(826\) 0 0
\(827\) −42.0416 −1.46193 −0.730965 0.682415i \(-0.760930\pi\)
−0.730965 + 0.682415i \(0.760930\pi\)
\(828\) 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
\(832\) −35.3137 −1.22428
\(833\) 0 0
\(834\) 4.54416 0.157351
\(835\) 0 0
\(836\) 0 0
\(837\) −24.7279 −0.854722
\(838\) 26.4853 0.914919
\(839\) −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
\(842\) −26.8701 −0.926003
\(843\) 0.958369 0.0330080
\(844\) 0 0
\(845\) 0 0
\(846\) −28.9706 −0.996028
\(847\) 0 0
\(848\) 16.9706 0.582772
\(849\) −10.1716 −0.349087
\(850\) 0 0
\(851\) 16.6274 0.569981
\(852\) 0 0
\(853\) −10.9706 −0.375625 −0.187812 0.982205i \(-0.560140\pi\)
−0.187812 + 0.982205i \(0.560140\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −7.02944 −0.240261
\(857\) 22.4853 0.768083 0.384041 0.923316i \(-0.374532\pi\)
0.384041 + 0.923316i \(0.374532\pi\)
\(858\) 0.443651 0.0151460
\(859\) −24.7696 −0.845126 −0.422563 0.906334i \(-0.638870\pi\)
−0.422563 + 0.906334i \(0.638870\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −40.7279 −1.38720
\(863\) −18.3848 −0.625825 −0.312913 0.949782i \(-0.601305\pi\)
−0.312913 + 0.949782i \(0.601305\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −15.5147 −0.527212
\(867\) −2.68629 −0.0912312
\(868\) 0 0
\(869\) −0.254834 −0.00864465
\(870\) 0 0
\(871\) −36.3848 −1.23285
\(872\) −14.1421 −0.478913
\(873\) 37.4558 1.26769
\(874\) −62.9117 −2.12802
\(875\) 0 0
\(876\) 0 0
\(877\) −13.0294 −0.439973 −0.219986 0.975503i \(-0.570601\pi\)
−0.219986 + 0.975503i \(0.570601\pi\)
\(878\) −42.9706 −1.45019
\(879\) −10.6569 −0.359447
\(880\) 0 0
\(881\) −52.9706 −1.78462 −0.892312 0.451420i \(-0.850918\pi\)
−0.892312 + 0.451420i \(0.850918\pi\)
\(882\) 0 0
\(883\) −31.5147 −1.06055 −0.530277 0.847824i \(-0.677912\pi\)
−0.530277 + 0.847824i \(0.677912\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −21.4558 −0.720823
\(887\) −2.97056 −0.0997417 −0.0498709 0.998756i \(-0.515881\pi\)
−0.0498709 + 0.998756i \(0.515881\pi\)
\(888\) 2.62742 0.0881703
\(889\) 0 0
\(890\) 0 0
\(891\) −1.28427 −0.0430247
\(892\) 0 0
\(893\) 43.4558 1.45419
\(894\) −5.37258 −0.179686
\(895\) 0 0
\(896\) 0 0
\(897\) 13.5563 0.452633
\(898\) −34.1838 −1.14073
\(899\) −88.6690 −2.95728
\(900\) 0 0
\(901\) 13.7574 0.458324
\(902\) 1.51472 0.0504346
\(903\) 0 0
\(904\) −36.9706 −1.22962
\(905\) 0 0
\(906\) −4.38478 −0.145674
\(907\) 44.1838 1.46710 0.733549 0.679637i \(-0.237863\pi\)
0.733549 + 0.679637i \(0.237863\pi\)
\(908\) 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 \(0.470127\pi\)
\(912\) −9.94113 −0.329184
\(913\) 0 0
\(914\) 16.6274 0.549986
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 11.0711 0.365400
\(919\) 12.4558 0.410880 0.205440 0.978670i \(-0.434137\pi\)
0.205440 + 0.978670i \(0.434137\pi\)
\(920\) 0 0
\(921\) −12.7990 −0.421741
\(922\) 4.28427 0.141095
\(923\) 14.0000 0.460816
\(924\) 0 0
\(925\) 0 0
\(926\) 30.3431 0.997138
\(927\) 54.4264 1.78760
\(928\) 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
\(933\) −4.14214 −0.135607
\(934\) 8.10051 0.265057
\(935\) 0 0
\(936\) −35.3137 −1.15426
\(937\) 48.5563 1.58627 0.793133 0.609048i \(-0.208448\pi\)
0.793133 + 0.609048i \(0.208448\pi\)
\(938\) 0 0
\(939\) 7.54416 0.246194
\(940\) 0 0
\(941\) 28.9706 0.944413 0.472207 0.881488i \(-0.343458\pi\)
0.472207 + 0.881488i \(0.343458\pi\)
\(942\) 8.88730 0.289564
\(943\) 46.2843 1.50722
\(944\) 8.97056 0.291967
\(945\) 0 0
\(946\) 0.485281 0.0157779
\(947\) 52.2426 1.69766 0.848829 0.528668i \(-0.177308\pi\)
0.848829 + 0.528668i \(0.177308\pi\)
\(948\) 0 0
\(949\) −37.4558 −1.21587
\(950\) 0 0
\(951\) 0.142136 0.00460906
\(952\) 0 0
\(953\) −53.0122 −1.71723 −0.858617 0.512618i \(-0.828676\pi\)
−0.858617 + 0.512618i \(0.828676\pi\)
\(954\) 16.9706 0.549442
\(955\) 0 0
\(956\) 0 0
\(957\) 0.615224 0.0198874
\(958\) 16.6274 0.537207
\(959\) 0 0
\(960\) 0 0
\(961\) 73.9117 2.38425
\(962\) 14.0000 0.451378
\(963\) −7.02944 −0.226520
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 60.4264 1.94318 0.971591 0.236666i \(-0.0760546\pi\)
0.971591 + 0.236666i \(0.0760546\pi\)
\(968\) 31.0294 0.997324
\(969\) −8.05887 −0.258888
\(970\) 0 0
\(971\) −17.2721 −0.554287 −0.277144 0.960828i \(-0.589388\pi\)
−0.277144 + 0.960828i \(0.589388\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −39.1716 −1.25514
\(975\) 0 0
\(976\) 11.3137 0.362143
\(977\) −42.7696 −1.36832 −0.684160 0.729332i \(-0.739831\pi\)
−0.684160 + 0.729332i \(0.739831\pi\)
\(978\) 6.00000 0.191859
\(979\) 1.37258 0.0438679
\(980\) 0 0
\(981\) −14.1421 −0.451524
\(982\) −52.7279 −1.68262
\(983\) −0.213203 −0.00680013 −0.00340007 0.999994i \(-0.501082\pi\)
−0.00340007 + 0.999994i \(0.501082\pi\)
\(984\) 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
\(993\) −11.3726 −0.360898
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 21.7279 0.688130 0.344065 0.938946i \(-0.388196\pi\)
0.344065 + 0.938946i \(0.388196\pi\)
\(998\) 4.24264 0.134298
\(999\) 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.a.p.1.2 2
5.2 odd 4 1225.2.b.j.99.4 4
5.3 odd 4 1225.2.b.j.99.1 4
5.4 even 2 245.2.a.f.1.1 yes 2
7.6 odd 2 1225.2.a.r.1.2 2
15.14 odd 2 2205.2.a.t.1.2 2
20.19 odd 2 3920.2.a.br.1.2 2
35.4 even 6 245.2.e.f.226.2 4
35.9 even 6 245.2.e.f.116.2 4
35.13 even 4 1225.2.b.i.99.2 4
35.19 odd 6 245.2.e.g.116.2 4
35.24 odd 6 245.2.e.g.226.2 4
35.27 even 4 1225.2.b.i.99.3 4
35.34 odd 2 245.2.a.e.1.1 2
105.104 even 2 2205.2.a.v.1.2 2
140.139 even 2 3920.2.a.bw.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.2.a.e.1.1 2 35.34 odd 2
245.2.a.f.1.1 yes 2 5.4 even 2
245.2.e.f.116.2 4 35.9 even 6
245.2.e.f.226.2 4 35.4 even 6
245.2.e.g.116.2 4 35.19 odd 6
245.2.e.g.226.2 4 35.24 odd 6
1225.2.a.p.1.2 2 1.1 even 1 trivial
1225.2.a.r.1.2 2 7.6 odd 2
1225.2.b.i.99.2 4 35.13 even 4
1225.2.b.i.99.3 4 35.27 even 4
1225.2.b.j.99.1 4 5.3 odd 4
1225.2.b.j.99.4 4 5.2 odd 4
2205.2.a.t.1.2 2 15.14 odd 2
2205.2.a.v.1.2 2 105.104 even 2
3920.2.a.br.1.2 2 20.19 odd 2
3920.2.a.bw.1.1 2 140.139 even 2