Properties

Label 1225.2.a.r.1.1
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 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1225.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421 q^{2} +2.41421 q^{3} -3.41421 q^{6} +2.82843 q^{8} +2.82843 q^{9} +O(q^{10})\) \(q-1.41421 q^{2} +2.41421 q^{3} -3.41421 q^{6} +2.82843 q^{8} +2.82843 q^{9} -5.82843 q^{11} +1.58579 q^{13} -4.00000 q^{16} -5.24264 q^{17} -4.00000 q^{18} -6.00000 q^{19} +8.24264 q^{22} -4.58579 q^{23} +6.82843 q^{24} -2.24264 q^{26} -0.414214 q^{27} +2.65685 q^{29} -1.75736 q^{31} -14.0711 q^{33} +7.41421 q^{34} +6.24264 q^{37} +8.48528 q^{38} +3.82843 q^{39} -2.24264 q^{41} -2.00000 q^{43} +6.48528 q^{46} +1.24264 q^{47} -9.65685 q^{48} -12.6569 q^{51} +4.24264 q^{53} +0.585786 q^{54} -14.4853 q^{57} -3.75736 q^{58} -6.24264 q^{59} -2.82843 q^{61} +2.48528 q^{62} +8.00000 q^{64} +19.8995 q^{66} -0.242641 q^{67} -11.0711 q^{69} -8.82843 q^{71} +8.00000 q^{72} +8.48528 q^{73} -8.82843 q^{74} -5.41421 q^{78} -15.4853 q^{79} -9.48528 q^{81} +3.17157 q^{82} +2.82843 q^{86} +6.41421 q^{87} -16.4853 q^{88} +8.00000 q^{89} -4.24264 q^{93} -1.75736 q^{94} +4.75736 q^{97} -16.4853 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{3} - 4q^{6} + O(q^{10}) \) \( 2q + 2q^{3} - 4q^{6} - 6q^{11} + 6q^{13} - 8q^{16} - 2q^{17} - 8q^{18} - 12q^{19} + 8q^{22} - 12q^{23} + 8q^{24} + 4q^{26} + 2q^{27} - 6q^{29} - 12q^{31} - 14q^{33} + 12q^{34} + 4q^{37} + 2q^{39} + 4q^{41} - 4q^{43} - 4q^{46} - 6q^{47} - 8q^{48} - 14q^{51} + 4q^{54} - 12q^{57} - 16q^{58} - 4q^{59} - 12q^{62} + 16q^{64} + 20q^{66} + 8q^{67} - 8q^{69} - 12q^{71} + 16q^{72} - 12q^{74} - 8q^{78} - 14q^{79} - 2q^{81} + 12q^{82} + 10q^{87} - 16q^{88} + 16q^{89} - 12q^{94} + 18q^{97} - 16q^{99} + O(q^{100}) \)

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 \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) 2.41421 1.39385 0.696923 0.717146i \(-0.254552\pi\)
0.696923 + 0.717146i \(0.254552\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) −3.41421 −1.39385
\(7\) 0 0
\(8\) 2.82843 1.00000
\(9\) 2.82843 0.942809
\(10\) 0 0
\(11\) −5.82843 −1.75734 −0.878668 0.477432i \(-0.841568\pi\)
−0.878668 + 0.477432i \(0.841568\pi\)
\(12\) 0 0
\(13\) 1.58579 0.439818 0.219909 0.975520i \(-0.429424\pi\)
0.219909 + 0.975520i \(0.429424\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −5.24264 −1.27153 −0.635764 0.771884i \(-0.719315\pi\)
−0.635764 + 0.771884i \(0.719315\pi\)
\(18\) −4.00000 −0.942809
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 8.24264 1.75734
\(23\) −4.58579 −0.956203 −0.478101 0.878305i \(-0.658675\pi\)
−0.478101 + 0.878305i \(0.658675\pi\)
\(24\) 6.82843 1.39385
\(25\) 0 0
\(26\) −2.24264 −0.439818
\(27\) −0.414214 −0.0797154
\(28\) 0 0
\(29\) 2.65685 0.493365 0.246683 0.969096i \(-0.420659\pi\)
0.246683 + 0.969096i \(0.420659\pi\)
\(30\) 0 0
\(31\) −1.75736 −0.315631 −0.157816 0.987469i \(-0.550445\pi\)
−0.157816 + 0.987469i \(0.550445\pi\)
\(32\) 0 0
\(33\) −14.0711 −2.44946
\(34\) 7.41421 1.27153
\(35\) 0 0
\(36\) 0 0
\(37\) 6.24264 1.02628 0.513142 0.858304i \(-0.328481\pi\)
0.513142 + 0.858304i \(0.328481\pi\)
\(38\) 8.48528 1.37649
\(39\) 3.82843 0.613039
\(40\) 0 0
\(41\) −2.24264 −0.350242 −0.175121 0.984547i \(-0.556032\pi\)
−0.175121 + 0.984547i \(0.556032\pi\)
\(42\) 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
\(46\) 6.48528 0.956203
\(47\) 1.24264 0.181258 0.0906289 0.995885i \(-0.471112\pi\)
0.0906289 + 0.995885i \(0.471112\pi\)
\(48\) −9.65685 −1.39385
\(49\) 0 0
\(50\) 0 0
\(51\) −12.6569 −1.77231
\(52\) 0 0
\(53\) 4.24264 0.582772 0.291386 0.956606i \(-0.405884\pi\)
0.291386 + 0.956606i \(0.405884\pi\)
\(54\) 0.585786 0.0797154
\(55\) 0 0
\(56\) 0 0
\(57\) −14.4853 −1.91862
\(58\) −3.75736 −0.493365
\(59\) −6.24264 −0.812723 −0.406361 0.913712i \(-0.633203\pi\)
−0.406361 + 0.913712i \(0.633203\pi\)
\(60\) 0 0
\(61\) −2.82843 −0.362143 −0.181071 0.983470i \(-0.557957\pi\)
−0.181071 + 0.983470i \(0.557957\pi\)
\(62\) 2.48528 0.315631
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 19.8995 2.44946
\(67\) −0.242641 −0.0296433 −0.0148216 0.999890i \(-0.504718\pi\)
−0.0148216 + 0.999890i \(0.504718\pi\)
\(68\) 0 0
\(69\) −11.0711 −1.33280
\(70\) 0 0
\(71\) −8.82843 −1.04774 −0.523871 0.851798i \(-0.675513\pi\)
−0.523871 + 0.851798i \(0.675513\pi\)
\(72\) 8.00000 0.942809
\(73\) 8.48528 0.993127 0.496564 0.868000i \(-0.334595\pi\)
0.496564 + 0.868000i \(0.334595\pi\)
\(74\) −8.82843 −1.02628
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) −5.41421 −0.613039
\(79\) −15.4853 −1.74223 −0.871115 0.491079i \(-0.836603\pi\)
−0.871115 + 0.491079i \(0.836603\pi\)
\(80\) 0 0
\(81\) −9.48528 −1.05392
\(82\) 3.17157 0.350242
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.82843 0.304997
\(87\) 6.41421 0.687676
\(88\) −16.4853 −1.75734
\(89\) 8.00000 0.847998 0.423999 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −4.24264 −0.439941
\(94\) −1.75736 −0.181258
\(95\) 0 0
\(96\) 0 0
\(97\) 4.75736 0.483037 0.241518 0.970396i \(-0.422355\pi\)
0.241518 + 0.970396i \(0.422355\pi\)
\(98\) 0 0
\(99\) −16.4853 −1.65683
\(100\) 0 0
\(101\) 14.4853 1.44134 0.720670 0.693279i \(-0.243834\pi\)
0.720670 + 0.693279i \(0.243834\pi\)
\(102\) 17.8995 1.77231
\(103\) 10.7574 1.05995 0.529977 0.848012i \(-0.322201\pi\)
0.529977 + 0.848012i \(0.322201\pi\)
\(104\) 4.48528 0.439818
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −14.4853 −1.40035 −0.700173 0.713974i \(-0.746894\pi\)
−0.700173 + 0.713974i \(0.746894\pi\)
\(108\) 0 0
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) 0 0
\(111\) 15.0711 1.43048
\(112\) 0 0
\(113\) −1.07107 −0.100758 −0.0503788 0.998730i \(-0.516043\pi\)
−0.0503788 + 0.998730i \(0.516043\pi\)
\(114\) 20.4853 1.91862
\(115\) 0 0
\(116\) 0 0
\(117\) 4.48528 0.414664
\(118\) 8.82843 0.812723
\(119\) 0 0
\(120\) 0 0
\(121\) 22.9706 2.08823
\(122\) 4.00000 0.362143
\(123\) −5.41421 −0.488183
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0.242641 0.0215309 0.0107654 0.999942i \(-0.496573\pi\)
0.0107654 + 0.999942i \(0.496573\pi\)
\(128\) −11.3137 −1.00000
\(129\) −4.82843 −0.425119
\(130\) 0 0
\(131\) −3.75736 −0.328282 −0.164141 0.986437i \(-0.552485\pi\)
−0.164141 + 0.986437i \(0.552485\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0.343146 0.0296433
\(135\) 0 0
\(136\) −14.8284 −1.27153
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 15.6569 1.33280
\(139\) −16.2426 −1.37768 −0.688841 0.724912i \(-0.741880\pi\)
−0.688841 + 0.724912i \(0.741880\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) 12.4853 1.04774
\(143\) −9.24264 −0.772908
\(144\) −11.3137 −0.942809
\(145\) 0 0
\(146\) −12.0000 −0.993127
\(147\) 0 0
\(148\) 0 0
\(149\) −14.8284 −1.21479 −0.607396 0.794399i \(-0.707786\pi\)
−0.607396 + 0.794399i \(0.707786\pi\)
\(150\) 0 0
\(151\) 9.48528 0.771901 0.385951 0.922519i \(-0.373874\pi\)
0.385951 + 0.922519i \(0.373874\pi\)
\(152\) −16.9706 −1.37649
\(153\) −14.8284 −1.19881
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −20.8284 −1.66229 −0.831145 0.556056i \(-0.812314\pi\)
−0.831145 + 0.556056i \(0.812314\pi\)
\(158\) 21.8995 1.74223
\(159\) 10.2426 0.812294
\(160\) 0 0
\(161\) 0 0
\(162\) 13.4142 1.05392
\(163\) 1.75736 0.137647 0.0688235 0.997629i \(-0.478075\pi\)
0.0688235 + 0.997629i \(0.478075\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.24264 0.715217 0.357609 0.933872i \(-0.383592\pi\)
0.357609 + 0.933872i \(0.383592\pi\)
\(168\) 0 0
\(169\) −10.4853 −0.806560
\(170\) 0 0
\(171\) −16.9706 −1.29777
\(172\) 0 0
\(173\) 1.24264 0.0944762 0.0472381 0.998884i \(-0.484958\pi\)
0.0472381 + 0.998884i \(0.484958\pi\)
\(174\) −9.07107 −0.687676
\(175\) 0 0
\(176\) 23.3137 1.75734
\(177\) −15.0711 −1.13281
\(178\) −11.3137 −0.847998
\(179\) 2.48528 0.185759 0.0928793 0.995677i \(-0.470393\pi\)
0.0928793 + 0.995677i \(0.470393\pi\)
\(180\) 0 0
\(181\) −6.72792 −0.500083 −0.250041 0.968235i \(-0.580444\pi\)
−0.250041 + 0.968235i \(0.580444\pi\)
\(182\) 0 0
\(183\) −6.82843 −0.504772
\(184\) −12.9706 −0.956203
\(185\) 0 0
\(186\) 6.00000 0.439941
\(187\) 30.5563 2.23450
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −19.9706 −1.44502 −0.722510 0.691361i \(-0.757011\pi\)
−0.722510 + 0.691361i \(0.757011\pi\)
\(192\) 19.3137 1.39385
\(193\) 16.0000 1.15171 0.575853 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) −6.72792 −0.483037
\(195\) 0 0
\(196\) 0 0
\(197\) −10.5858 −0.754206 −0.377103 0.926171i \(-0.623080\pi\)
−0.377103 + 0.926171i \(0.623080\pi\)
\(198\) 23.3137 1.65683
\(199\) 4.58579 0.325078 0.162539 0.986702i \(-0.448032\pi\)
0.162539 + 0.986702i \(0.448032\pi\)
\(200\) 0 0
\(201\) −0.585786 −0.0413182
\(202\) −20.4853 −1.44134
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) −15.2132 −1.05995
\(207\) −12.9706 −0.901516
\(208\) −6.34315 −0.439818
\(209\) 34.9706 2.41896
\(210\) 0 0
\(211\) 9.00000 0.619586 0.309793 0.950804i \(-0.399740\pi\)
0.309793 + 0.950804i \(0.399740\pi\)
\(212\) 0 0
\(213\) −21.3137 −1.46039
\(214\) 20.4853 1.40035
\(215\) 0 0
\(216\) −1.17157 −0.0797154
\(217\) 0 0
\(218\) −7.07107 −0.478913
\(219\) 20.4853 1.38427
\(220\) 0 0
\(221\) −8.31371 −0.559241
\(222\) −21.3137 −1.43048
\(223\) 18.2132 1.21965 0.609823 0.792538i \(-0.291240\pi\)
0.609823 + 0.792538i \(0.291240\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1.51472 0.100758
\(227\) −9.72792 −0.645665 −0.322832 0.946456i \(-0.604635\pi\)
−0.322832 + 0.946456i \(0.604635\pi\)
\(228\) 0 0
\(229\) 30.0416 1.98521 0.992603 0.121402i \(-0.0387390\pi\)
0.992603 + 0.121402i \(0.0387390\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 7.51472 0.493365
\(233\) 14.8284 0.971443 0.485721 0.874114i \(-0.338557\pi\)
0.485721 + 0.874114i \(0.338557\pi\)
\(234\) −6.34315 −0.414664
\(235\) 0 0
\(236\) 0 0
\(237\) −37.3848 −2.42840
\(238\) 0 0
\(239\) −0.514719 −0.0332944 −0.0166472 0.999861i \(-0.505299\pi\)
−0.0166472 + 0.999861i \(0.505299\pi\)
\(240\) 0 0
\(241\) −24.7279 −1.59287 −0.796433 0.604727i \(-0.793282\pi\)
−0.796433 + 0.604727i \(0.793282\pi\)
\(242\) −32.4853 −2.08823
\(243\) −21.6569 −1.38929
\(244\) 0 0
\(245\) 0 0
\(246\) 7.65685 0.488183
\(247\) −9.51472 −0.605407
\(248\) −4.97056 −0.315631
\(249\) 0 0
\(250\) 0 0
\(251\) 25.2132 1.59144 0.795722 0.605663i \(-0.207092\pi\)
0.795722 + 0.605663i \(0.207092\pi\)
\(252\) 0 0
\(253\) 26.7279 1.68037
\(254\) −0.343146 −0.0215309
\(255\) 0 0
\(256\) 0 0
\(257\) −26.4853 −1.65211 −0.826053 0.563592i \(-0.809419\pi\)
−0.826053 + 0.563592i \(0.809419\pi\)
\(258\) 6.82843 0.425119
\(259\) 0 0
\(260\) 0 0
\(261\) 7.51472 0.465149
\(262\) 5.31371 0.328282
\(263\) −28.6274 −1.76524 −0.882621 0.470085i \(-0.844223\pi\)
−0.882621 + 0.470085i \(0.844223\pi\)
\(264\) −39.7990 −2.44946
\(265\) 0 0
\(266\) 0 0
\(267\) 19.3137 1.18198
\(268\) 0 0
\(269\) 7.75736 0.472975 0.236487 0.971635i \(-0.424004\pi\)
0.236487 + 0.971635i \(0.424004\pi\)
\(270\) 0 0
\(271\) 23.3137 1.41621 0.708103 0.706109i \(-0.249551\pi\)
0.708103 + 0.706109i \(0.249551\pi\)
\(272\) 20.9706 1.27153
\(273\) 0 0
\(274\) 16.9706 1.02523
\(275\) 0 0
\(276\) 0 0
\(277\) 27.2132 1.63508 0.817541 0.575870i \(-0.195336\pi\)
0.817541 + 0.575870i \(0.195336\pi\)
\(278\) 22.9706 1.37768
\(279\) −4.97056 −0.297580
\(280\) 0 0
\(281\) −20.3137 −1.21181 −0.605907 0.795535i \(-0.707190\pi\)
−0.605907 + 0.795535i \(0.707190\pi\)
\(282\) −4.24264 −0.252646
\(283\) −6.55635 −0.389735 −0.194867 0.980830i \(-0.562428\pi\)
−0.194867 + 0.980830i \(0.562428\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 13.0711 0.772908
\(287\) 0 0
\(288\) 0 0
\(289\) 10.4853 0.616781
\(290\) 0 0
\(291\) 11.4853 0.673279
\(292\) 0 0
\(293\) 0.272078 0.0158950 0.00794748 0.999968i \(-0.497470\pi\)
0.00794748 + 0.999968i \(0.497470\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 17.6569 1.02628
\(297\) 2.41421 0.140087
\(298\) 20.9706 1.21479
\(299\) −7.27208 −0.420555
\(300\) 0 0
\(301\) 0 0
\(302\) −13.4142 −0.771901
\(303\) 34.9706 2.00901
\(304\) 24.0000 1.37649
\(305\) 0 0
\(306\) 20.9706 1.19881
\(307\) 11.1005 0.633539 0.316770 0.948503i \(-0.397402\pi\)
0.316770 + 0.948503i \(0.397402\pi\)
\(308\) 0 0
\(309\) 25.9706 1.47741
\(310\) 0 0
\(311\) 10.0000 0.567048 0.283524 0.958965i \(-0.408496\pi\)
0.283524 + 0.958965i \(0.408496\pi\)
\(312\) 10.8284 0.613039
\(313\) 24.2132 1.36861 0.684306 0.729195i \(-0.260105\pi\)
0.684306 + 0.729195i \(0.260105\pi\)
\(314\) 29.4558 1.66229
\(315\) 0 0
\(316\) 0 0
\(317\) 11.6569 0.654714 0.327357 0.944901i \(-0.393842\pi\)
0.327357 + 0.944901i \(0.393842\pi\)
\(318\) −14.4853 −0.812294
\(319\) −15.4853 −0.867009
\(320\) 0 0
\(321\) −34.9706 −1.95187
\(322\) 0 0
\(323\) 31.4558 1.75025
\(324\) 0 0
\(325\) 0 0
\(326\) −2.48528 −0.137647
\(327\) 12.0711 0.667532
\(328\) −6.34315 −0.350242
\(329\) 0 0
\(330\) 0 0
\(331\) 23.4558 1.28925 0.644625 0.764499i \(-0.277014\pi\)
0.644625 + 0.764499i \(0.277014\pi\)
\(332\) 0 0
\(333\) 17.6569 0.967590
\(334\) −13.0711 −0.715217
\(335\) 0 0
\(336\) 0 0
\(337\) −13.7574 −0.749411 −0.374706 0.927144i \(-0.622256\pi\)
−0.374706 + 0.927144i \(0.622256\pi\)
\(338\) 14.8284 0.806560
\(339\) −2.58579 −0.140441
\(340\) 0 0
\(341\) 10.2426 0.554670
\(342\) 24.0000 1.29777
\(343\) 0 0
\(344\) −5.65685 −0.304997
\(345\) 0 0
\(346\) −1.75736 −0.0944762
\(347\) 1.07107 0.0574979 0.0287490 0.999587i \(-0.490848\pi\)
0.0287490 + 0.999587i \(0.490848\pi\)
\(348\) 0 0
\(349\) −22.9706 −1.22959 −0.614793 0.788688i \(-0.710760\pi\)
−0.614793 + 0.788688i \(0.710760\pi\)
\(350\) 0 0
\(351\) −0.656854 −0.0350603
\(352\) 0 0
\(353\) 36.2132 1.92743 0.963717 0.266925i \(-0.0860078\pi\)
0.963717 + 0.266925i \(0.0860078\pi\)
\(354\) 21.3137 1.13281
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −3.51472 −0.185759
\(359\) 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
\(362\) 9.51472 0.500083
\(363\) 55.4558 2.91068
\(364\) 0 0
\(365\) 0 0
\(366\) 9.65685 0.504772
\(367\) 29.8701 1.55920 0.779602 0.626275i \(-0.215421\pi\)
0.779602 + 0.626275i \(0.215421\pi\)
\(368\) 18.3431 0.956203
\(369\) −6.34315 −0.330211
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −0.485281 −0.0251269 −0.0125635 0.999921i \(-0.503999\pi\)
−0.0125635 + 0.999921i \(0.503999\pi\)
\(374\) −43.2132 −2.23450
\(375\) 0 0
\(376\) 3.51472 0.181258
\(377\) 4.21320 0.216991
\(378\) 0 0
\(379\) 2.00000 0.102733 0.0513665 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) 0 0
\(381\) 0.585786 0.0300107
\(382\) 28.2426 1.44502
\(383\) −12.4853 −0.637968 −0.318984 0.947760i \(-0.603342\pi\)
−0.318984 + 0.947760i \(0.603342\pi\)
\(384\) −27.3137 −1.39385
\(385\) 0 0
\(386\) −22.6274 −1.15171
\(387\) −5.65685 −0.287554
\(388\) 0 0
\(389\) −35.1421 −1.78178 −0.890889 0.454222i \(-0.849917\pi\)
−0.890889 + 0.454222i \(0.849917\pi\)
\(390\) 0 0
\(391\) 24.0416 1.21584
\(392\) 0 0
\(393\) −9.07107 −0.457575
\(394\) 14.9706 0.754206
\(395\) 0 0
\(396\) 0 0
\(397\) 4.41421 0.221543 0.110772 0.993846i \(-0.464668\pi\)
0.110772 + 0.993846i \(0.464668\pi\)
\(398\) −6.48528 −0.325078
\(399\) 0 0
\(400\) 0 0
\(401\) 6.17157 0.308194 0.154097 0.988056i \(-0.450753\pi\)
0.154097 + 0.988056i \(0.450753\pi\)
\(402\) 0.828427 0.0413182
\(403\) −2.78680 −0.138820
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −36.3848 −1.80353
\(408\) −35.7990 −1.77231
\(409\) 14.4853 0.716251 0.358126 0.933673i \(-0.383416\pi\)
0.358126 + 0.933673i \(0.383416\pi\)
\(410\) 0 0
\(411\) −28.9706 −1.42901
\(412\) 0 0
\(413\) 0 0
\(414\) 18.3431 0.901516
\(415\) 0 0
\(416\) 0 0
\(417\) −39.2132 −1.92028
\(418\) −49.4558 −2.41896
\(419\) 6.72792 0.328681 0.164340 0.986404i \(-0.447450\pi\)
0.164340 + 0.986404i \(0.447450\pi\)
\(420\) 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
\(423\) 3.51472 0.170891
\(424\) 12.0000 0.582772
\(425\) 0 0
\(426\) 30.1421 1.46039
\(427\) 0 0
\(428\) 0 0
\(429\) −22.3137 −1.07732
\(430\) 0 0
\(431\) 10.7990 0.520169 0.260085 0.965586i \(-0.416250\pi\)
0.260085 + 0.965586i \(0.416250\pi\)
\(432\) 1.65685 0.0797154
\(433\) −22.9706 −1.10389 −0.551947 0.833879i \(-0.686115\pi\)
−0.551947 + 0.833879i \(0.686115\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 27.5147 1.31621
\(438\) −28.9706 −1.38427
\(439\) −6.38478 −0.304729 −0.152364 0.988324i \(-0.548689\pi\)
−0.152364 + 0.988324i \(0.548689\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 11.7574 0.559241
\(443\) −20.8284 −0.989588 −0.494794 0.869010i \(-0.664757\pi\)
−0.494794 + 0.869010i \(0.664757\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −25.7574 −1.21965
\(447\) −35.7990 −1.69323
\(448\) 0 0
\(449\) −29.8284 −1.40769 −0.703845 0.710353i \(-0.748535\pi\)
−0.703845 + 0.710353i \(0.748535\pi\)
\(450\) 0 0
\(451\) 13.0711 0.615493
\(452\) 0 0
\(453\) 22.8995 1.07591
\(454\) 13.7574 0.645665
\(455\) 0 0
\(456\) −40.9706 −1.91862
\(457\) 20.2426 0.946911 0.473455 0.880818i \(-0.343007\pi\)
0.473455 + 0.880818i \(0.343007\pi\)
\(458\) −42.4853 −1.98521
\(459\) 2.17157 0.101360
\(460\) 0 0
\(461\) −36.9706 −1.72189 −0.860945 0.508697i \(-0.830127\pi\)
−0.860945 + 0.508697i \(0.830127\pi\)
\(462\) 0 0
\(463\) −29.4558 −1.36893 −0.684465 0.729046i \(-0.739964\pi\)
−0.684465 + 0.729046i \(0.739964\pi\)
\(464\) −10.6274 −0.493365
\(465\) 0 0
\(466\) −20.9706 −0.971443
\(467\) 19.7279 0.912899 0.456450 0.889749i \(-0.349121\pi\)
0.456450 + 0.889749i \(0.349121\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −50.2843 −2.31698
\(472\) −17.6569 −0.812723
\(473\) 11.6569 0.535983
\(474\) 52.8701 2.42840
\(475\) 0 0
\(476\) 0 0
\(477\) 12.0000 0.549442
\(478\) 0.727922 0.0332944
\(479\) −20.2426 −0.924910 −0.462455 0.886643i \(-0.653031\pi\)
−0.462455 + 0.886643i \(0.653031\pi\)
\(480\) 0 0
\(481\) 9.89949 0.451378
\(482\) 34.9706 1.59287
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 30.6274 1.38929
\(487\) 31.6985 1.43640 0.718198 0.695839i \(-0.244967\pi\)
0.718198 + 0.695839i \(0.244967\pi\)
\(488\) −8.00000 −0.362143
\(489\) 4.24264 0.191859
\(490\) 0 0
\(491\) 19.2843 0.870287 0.435143 0.900361i \(-0.356698\pi\)
0.435143 + 0.900361i \(0.356698\pi\)
\(492\) 0 0
\(493\) −13.9289 −0.627328
\(494\) 13.4558 0.605407
\(495\) 0 0
\(496\) 7.02944 0.315631
\(497\) 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
\(501\) 22.3137 0.996903
\(502\) −35.6569 −1.59144
\(503\) −32.7574 −1.46058 −0.730289 0.683138i \(-0.760615\pi\)
−0.730289 + 0.683138i \(0.760615\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −37.7990 −1.68037
\(507\) −25.3137 −1.12422
\(508\) 0 0
\(509\) −17.2132 −0.762962 −0.381481 0.924377i \(-0.624586\pi\)
−0.381481 + 0.924377i \(0.624586\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 22.6274 1.00000
\(513\) 2.48528 0.109728
\(514\) 37.4558 1.65211
\(515\) 0 0
\(516\) 0 0
\(517\) −7.24264 −0.318531
\(518\) 0 0
\(519\) 3.00000 0.131685
\(520\) 0 0
\(521\) 18.9706 0.831115 0.415558 0.909567i \(-0.363586\pi\)
0.415558 + 0.909567i \(0.363586\pi\)
\(522\) −10.6274 −0.465149
\(523\) 15.5147 0.678411 0.339206 0.940712i \(-0.389842\pi\)
0.339206 + 0.940712i \(0.389842\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 40.4853 1.76524
\(527\) 9.21320 0.401333
\(528\) 56.2843 2.44946
\(529\) −1.97056 −0.0856766
\(530\) 0 0
\(531\) −17.6569 −0.766242
\(532\) 0 0
\(533\) −3.55635 −0.154043
\(534\) −27.3137 −1.18198
\(535\) 0 0
\(536\) −0.686292 −0.0296433
\(537\) 6.00000 0.258919
\(538\) −10.9706 −0.472975
\(539\) 0 0
\(540\) 0 0
\(541\) 11.9706 0.514655 0.257327 0.966324i \(-0.417158\pi\)
0.257327 + 0.966324i \(0.417158\pi\)
\(542\) −32.9706 −1.41621
\(543\) −16.2426 −0.697038
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 7.51472 0.321306 0.160653 0.987011i \(-0.448640\pi\)
0.160653 + 0.987011i \(0.448640\pi\)
\(548\) 0 0
\(549\) −8.00000 −0.341432
\(550\) 0 0
\(551\) −15.9411 −0.679115
\(552\) −31.3137 −1.33280
\(553\) 0 0
\(554\) −38.4853 −1.63508
\(555\) 0 0
\(556\) 0 0
\(557\) −31.7990 −1.34737 −0.673683 0.739020i \(-0.735289\pi\)
−0.673683 + 0.739020i \(0.735289\pi\)
\(558\) 7.02944 0.297580
\(559\) −3.17157 −0.134143
\(560\) 0 0
\(561\) 73.7696 3.11455
\(562\) 28.7279 1.21181
\(563\) 35.9411 1.51474 0.757369 0.652987i \(-0.226484\pi\)
0.757369 + 0.652987i \(0.226484\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 9.27208 0.389735
\(567\) 0 0
\(568\) −24.9706 −1.04774
\(569\) −2.14214 −0.0898030 −0.0449015 0.998991i \(-0.514297\pi\)
−0.0449015 + 0.998991i \(0.514297\pi\)
\(570\) 0 0
\(571\) −34.4853 −1.44316 −0.721582 0.692329i \(-0.756585\pi\)
−0.721582 + 0.692329i \(0.756585\pi\)
\(572\) 0 0
\(573\) −48.2132 −2.01414
\(574\) 0 0
\(575\) 0 0
\(576\) 22.6274 0.942809
\(577\) −9.72792 −0.404979 −0.202489 0.979284i \(-0.564903\pi\)
−0.202489 + 0.979284i \(0.564903\pi\)
\(578\) −14.8284 −0.616781
\(579\) 38.6274 1.60530
\(580\) 0 0
\(581\) 0 0
\(582\) −16.2426 −0.673279
\(583\) −24.7279 −1.02413
\(584\) 24.0000 0.993127
\(585\) 0 0
\(586\) −0.384776 −0.0158950
\(587\) 13.4558 0.555382 0.277691 0.960670i \(-0.410431\pi\)
0.277691 + 0.960670i \(0.410431\pi\)
\(588\) 0 0
\(589\) 10.5442 0.434464
\(590\) 0 0
\(591\) −25.5563 −1.05125
\(592\) −24.9706 −1.02628
\(593\) 10.7574 0.441752 0.220876 0.975302i \(-0.429108\pi\)
0.220876 + 0.975302i \(0.429108\pi\)
\(594\) −3.41421 −0.140087
\(595\) 0 0
\(596\) 0 0
\(597\) 11.0711 0.453109
\(598\) 10.2843 0.420555
\(599\) −12.1716 −0.497317 −0.248658 0.968591i \(-0.579990\pi\)
−0.248658 + 0.968591i \(0.579990\pi\)
\(600\) 0 0
\(601\) −22.9706 −0.936989 −0.468494 0.883466i \(-0.655203\pi\)
−0.468494 + 0.883466i \(0.655203\pi\)
\(602\) 0 0
\(603\) −0.686292 −0.0279480
\(604\) 0 0
\(605\) 0 0
\(606\) −49.4558 −2.00901
\(607\) −24.8995 −1.01064 −0.505320 0.862932i \(-0.668625\pi\)
−0.505320 + 0.862932i \(0.668625\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.97056 0.0797204
\(612\) 0 0
\(613\) −43.9411 −1.77477 −0.887383 0.461034i \(-0.847479\pi\)
−0.887383 + 0.461034i \(0.847479\pi\)
\(614\) −15.6985 −0.633539
\(615\) 0 0
\(616\) 0 0
\(617\) −7.41421 −0.298485 −0.149242 0.988801i \(-0.547684\pi\)
−0.149242 + 0.988801i \(0.547684\pi\)
\(618\) −36.7279 −1.47741
\(619\) −31.0711 −1.24885 −0.624426 0.781084i \(-0.714667\pi\)
−0.624426 + 0.781084i \(0.714667\pi\)
\(620\) 0 0
\(621\) 1.89949 0.0762241
\(622\) −14.1421 −0.567048
\(623\) 0 0
\(624\) −15.3137 −0.613039
\(625\) 0 0
\(626\) −34.2426 −1.36861
\(627\) 84.4264 3.37167
\(628\) 0 0
\(629\) −32.7279 −1.30495
\(630\) 0 0
\(631\) 8.45584 0.336622 0.168311 0.985734i \(-0.446169\pi\)
0.168311 + 0.985734i \(0.446169\pi\)
\(632\) −43.7990 −1.74223
\(633\) 21.7279 0.863607
\(634\) −16.4853 −0.654714
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 21.8995 0.867009
\(639\) −24.9706 −0.987820
\(640\) 0 0
\(641\) −0.686292 −0.0271069 −0.0135534 0.999908i \(-0.504314\pi\)
−0.0135534 + 0.999908i \(0.504314\pi\)
\(642\) 49.4558 1.95187
\(643\) 27.7279 1.09348 0.546741 0.837302i \(-0.315868\pi\)
0.546741 + 0.837302i \(0.315868\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −44.4853 −1.75025
\(647\) −28.4853 −1.11987 −0.559936 0.828536i \(-0.689174\pi\)
−0.559936 + 0.828536i \(0.689174\pi\)
\(648\) −26.8284 −1.05392
\(649\) 36.3848 1.42823
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.02944 −0.0402850 −0.0201425 0.999797i \(-0.506412\pi\)
−0.0201425 + 0.999797i \(0.506412\pi\)
\(654\) −17.0711 −0.667532
\(655\) 0 0
\(656\) 8.97056 0.350242
\(657\) 24.0000 0.936329
\(658\) 0 0
\(659\) −13.9706 −0.544216 −0.272108 0.962267i \(-0.587721\pi\)
−0.272108 + 0.962267i \(0.587721\pi\)
\(660\) 0 0
\(661\) 37.4558 1.45686 0.728432 0.685118i \(-0.240250\pi\)
0.728432 + 0.685118i \(0.240250\pi\)
\(662\) −33.1716 −1.28925
\(663\) −20.0711 −0.779496
\(664\) 0 0
\(665\) 0 0
\(666\) −24.9706 −0.967590
\(667\) −12.1838 −0.471757
\(668\) 0 0
\(669\) 43.9706 1.70000
\(670\) 0 0
\(671\) 16.4853 0.636407
\(672\) 0 0
\(673\) −20.4853 −0.789650 −0.394825 0.918756i \(-0.629195\pi\)
−0.394825 + 0.918756i \(0.629195\pi\)
\(674\) 19.4558 0.749411
\(675\) 0 0
\(676\) 0 0
\(677\) 1.78680 0.0686722 0.0343361 0.999410i \(-0.489068\pi\)
0.0343361 + 0.999410i \(0.489068\pi\)
\(678\) 3.65685 0.140441
\(679\) 0 0
\(680\) 0 0
\(681\) −23.4853 −0.899958
\(682\) −14.4853 −0.554670
\(683\) 7.79899 0.298420 0.149210 0.988806i \(-0.452327\pi\)
0.149210 + 0.988806i \(0.452327\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 72.5269 2.76707
\(688\) 8.00000 0.304997
\(689\) 6.72792 0.256313
\(690\) 0 0
\(691\) −9.17157 −0.348903 −0.174452 0.984666i \(-0.555815\pi\)
−0.174452 + 0.984666i \(0.555815\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −1.51472 −0.0574979
\(695\) 0 0
\(696\) 18.1421 0.687676
\(697\) 11.7574 0.445342
\(698\) 32.4853 1.22959
\(699\) 35.7990 1.35404
\(700\) 0 0
\(701\) −4.45584 −0.168295 −0.0841475 0.996453i \(-0.526817\pi\)
−0.0841475 + 0.996453i \(0.526817\pi\)
\(702\) 0.928932 0.0350603
\(703\) −37.4558 −1.41267
\(704\) −46.6274 −1.75734
\(705\) 0 0
\(706\) −51.2132 −1.92743
\(707\) 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
\(711\) −43.7990 −1.64259
\(712\) 22.6274 0.847998
\(713\) 8.05887 0.301807
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −1.24264 −0.0464073
\(718\) −16.0000 −0.597115
\(719\) −17.2132 −0.641944 −0.320972 0.947089i \(-0.604010\pi\)
−0.320972 + 0.947089i \(0.604010\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −24.0416 −0.894737
\(723\) −59.6985 −2.22021
\(724\) 0 0
\(725\) 0 0
\(726\) −78.4264 −2.91068
\(727\) −29.3137 −1.08719 −0.543593 0.839349i \(-0.682936\pi\)
−0.543593 + 0.839349i \(0.682936\pi\)
\(728\) 0 0
\(729\) −23.8284 −0.882534
\(730\) 0 0
\(731\) 10.4853 0.387812
\(732\) 0 0
\(733\) −44.6985 −1.65098 −0.825488 0.564420i \(-0.809100\pi\)
−0.825488 + 0.564420i \(0.809100\pi\)
\(734\) −42.2426 −1.55920
\(735\) 0 0
\(736\) 0 0
\(737\) 1.41421 0.0520932
\(738\) 8.97056 0.330211
\(739\) −3.97056 −0.146060 −0.0730298 0.997330i \(-0.523267\pi\)
−0.0730298 + 0.997330i \(0.523267\pi\)
\(740\) 0 0
\(741\) −22.9706 −0.843845
\(742\) 0 0
\(743\) −36.7279 −1.34742 −0.673708 0.738997i \(-0.735300\pi\)
−0.673708 + 0.738997i \(0.735300\pi\)
\(744\) −12.0000 −0.439941
\(745\) 0 0
\(746\) 0.686292 0.0251269
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 12.5147 0.456669 0.228334 0.973583i \(-0.426672\pi\)
0.228334 + 0.973583i \(0.426672\pi\)
\(752\) −4.97056 −0.181258
\(753\) 60.8701 2.21823
\(754\) −5.95837 −0.216991
\(755\) 0 0
\(756\) 0 0
\(757\) 16.4853 0.599168 0.299584 0.954070i \(-0.403152\pi\)
0.299584 + 0.954070i \(0.403152\pi\)
\(758\) −2.82843 −0.102733
\(759\) 64.5269 2.34218
\(760\) 0 0
\(761\) −12.7279 −0.461387 −0.230693 0.973026i \(-0.574099\pi\)
−0.230693 + 0.973026i \(0.574099\pi\)
\(762\) −0.828427 −0.0300107
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 17.6569 0.637968
\(767\) −9.89949 −0.357450
\(768\) 0 0
\(769\) 3.17157 0.114370 0.0571849 0.998364i \(-0.481788\pi\)
0.0571849 + 0.998364i \(0.481788\pi\)
\(770\) 0 0
\(771\) −63.9411 −2.30278
\(772\) 0 0
\(773\) 36.2132 1.30250 0.651249 0.758864i \(-0.274245\pi\)
0.651249 + 0.758864i \(0.274245\pi\)
\(774\) 8.00000 0.287554
\(775\) 0 0
\(776\) 13.4558 0.483037
\(777\) 0 0
\(778\) 49.6985 1.78178
\(779\) 13.4558 0.482106
\(780\) 0 0
\(781\) 51.4558 1.84123
\(782\) −34.0000 −1.21584
\(783\) −1.10051 −0.0393288
\(784\) 0 0
\(785\) 0 0
\(786\) 12.8284 0.457575
\(787\) 45.7279 1.63002 0.815012 0.579444i \(-0.196730\pi\)
0.815012 + 0.579444i \(0.196730\pi\)
\(788\) 0 0
\(789\) −69.1127 −2.46048
\(790\) 0 0
\(791\) 0 0
\(792\) −46.6274 −1.65683
\(793\) −4.48528 −0.159277
\(794\) −6.24264 −0.221543
\(795\) 0 0
\(796\) 0 0
\(797\) 55.1838 1.95471 0.977355 0.211608i \(-0.0678700\pi\)
0.977355 + 0.211608i \(0.0678700\pi\)
\(798\) 0 0
\(799\) −6.51472 −0.230474
\(800\) 0 0
\(801\) 22.6274 0.799500
\(802\) −8.72792 −0.308194
\(803\) −49.4558 −1.74526
\(804\) 0 0
\(805\) 0 0
\(806\) 3.94113 0.138820
\(807\) 18.7279 0.659254
\(808\) 40.9706 1.44134
\(809\) 30.5980 1.07577 0.537884 0.843019i \(-0.319224\pi\)
0.537884 + 0.843019i \(0.319224\pi\)
\(810\) 0 0
\(811\) 23.3553 0.820117 0.410058 0.912059i \(-0.365508\pi\)
0.410058 + 0.912059i \(0.365508\pi\)
\(812\) 0 0
\(813\) 56.2843 1.97398
\(814\) 51.4558 1.80353
\(815\) 0 0
\(816\) 50.6274 1.77231
\(817\) 12.0000 0.419827
\(818\) −20.4853 −0.716251
\(819\) 0 0
\(820\) 0 0
\(821\) −6.51472 −0.227365 −0.113683 0.993517i \(-0.536265\pi\)
−0.113683 + 0.993517i \(0.536265\pi\)
\(822\) 40.9706 1.42901
\(823\) 8.72792 0.304236 0.152118 0.988362i \(-0.451391\pi\)
0.152118 + 0.988362i \(0.451391\pi\)
\(824\) 30.4264 1.05995
\(825\) 0 0
\(826\) 0 0
\(827\) 6.04163 0.210088 0.105044 0.994468i \(-0.466502\pi\)
0.105044 + 0.994468i \(0.466502\pi\)
\(828\) 0 0
\(829\) −54.0416 −1.87694 −0.938472 0.345356i \(-0.887758\pi\)
−0.938472 + 0.345356i \(0.887758\pi\)
\(830\) 0 0
\(831\) 65.6985 2.27906
\(832\) 12.6863 0.439818
\(833\) 0 0
\(834\) 55.4558 1.92028
\(835\) 0 0
\(836\) 0 0
\(837\) 0.727922 0.0251607
\(838\) −9.51472 −0.328681
\(839\) 48.7279 1.68227 0.841137 0.540822i \(-0.181887\pi\)
0.841137 + 0.540822i \(0.181887\pi\)
\(840\) 0 0
\(841\) −21.9411 −0.756591
\(842\) 26.8701 0.926003
\(843\) −49.0416 −1.68908
\(844\) 0 0
\(845\) 0 0
\(846\) −4.97056 −0.170891
\(847\) 0 0
\(848\) −16.9706 −0.582772
\(849\) −15.8284 −0.543230
\(850\) 0 0
\(851\) −28.6274 −0.981335
\(852\) 0 0
\(853\) −22.9706 −0.786497 −0.393249 0.919432i \(-0.628649\pi\)
−0.393249 + 0.919432i \(0.628649\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −40.9706 −1.40035
\(857\) −5.51472 −0.188379 −0.0941896 0.995554i \(-0.530026\pi\)
−0.0941896 + 0.995554i \(0.530026\pi\)
\(858\) 31.5563 1.07732
\(859\) −48.7696 −1.66400 −0.831998 0.554779i \(-0.812803\pi\)
−0.831998 + 0.554779i \(0.812803\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −15.2721 −0.520169
\(863\) 18.3848 0.625825 0.312913 0.949782i \(-0.398695\pi\)
0.312913 + 0.949782i \(0.398695\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 32.4853 1.10389
\(867\) 25.3137 0.859699
\(868\) 0 0
\(869\) 90.2548 3.06169
\(870\) 0 0
\(871\) −0.384776 −0.0130376
\(872\) 14.1421 0.478913
\(873\) 13.4558 0.455411
\(874\) −38.9117 −1.31621
\(875\) 0 0
\(876\) 0 0
\(877\) −46.9706 −1.58608 −0.793042 0.609167i \(-0.791504\pi\)
−0.793042 + 0.609167i \(0.791504\pi\)
\(878\) 9.02944 0.304729
\(879\) 0.656854 0.0221551
\(880\) 0 0
\(881\) 19.0294 0.641118 0.320559 0.947229i \(-0.396129\pi\)
0.320559 + 0.947229i \(0.396129\pi\)
\(882\) 0 0
\(883\) −48.4853 −1.63166 −0.815830 0.578292i \(-0.803719\pi\)
−0.815830 + 0.578292i \(0.803719\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 29.4558 0.989588
\(887\) −30.9706 −1.03989 −0.519945 0.854200i \(-0.674048\pi\)
−0.519945 + 0.854200i \(0.674048\pi\)
\(888\) 42.6274 1.43048
\(889\) 0 0
\(890\) 0 0
\(891\) 55.2843 1.85209
\(892\) 0 0
\(893\) −7.45584 −0.249500
\(894\) 50.6274 1.69323
\(895\) 0 0
\(896\) 0 0
\(897\) −17.5563 −0.586189
\(898\) 42.1838 1.40769
\(899\) −4.66905 −0.155721
\(900\) 0 0
\(901\) −22.2426 −0.741010
\(902\) −18.4853 −0.615493
\(903\) 0 0
\(904\) −3.02944 −0.100758
\(905\) 0 0
\(906\) −32.3848 −1.07591
\(907\) −32.1838 −1.06864 −0.534322 0.845281i \(-0.679433\pi\)
−0.534322 + 0.845281i \(0.679433\pi\)
\(908\) 0 0
\(909\) 40.9706 1.35891
\(910\) 0 0
\(911\) −5.65685 −0.187420 −0.0937100 0.995600i \(-0.529873\pi\)
−0.0937100 + 0.995600i \(0.529873\pi\)
\(912\) 57.9411 1.91862
\(913\) 0 0
\(914\) −28.6274 −0.946911
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) −3.07107 −0.101360
\(919\) −38.4558 −1.26854 −0.634271 0.773111i \(-0.718699\pi\)
−0.634271 + 0.773111i \(0.718699\pi\)
\(920\) 0 0
\(921\) 26.7990 0.883057
\(922\) 52.2843 1.72189
\(923\) −14.0000 −0.460816
\(924\) 0 0
\(925\) 0 0
\(926\) 41.6569 1.36893
\(927\) 30.4264 0.999334
\(928\) 0 0
\(929\) −54.7279 −1.79556 −0.897782 0.440439i \(-0.854823\pi\)
−0.897782 + 0.440439i \(0.854823\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 24.1421 0.790378
\(934\) −27.8995 −0.912899
\(935\) 0 0
\(936\) 12.6863 0.414664
\(937\) −17.4437 −0.569859 −0.284930 0.958548i \(-0.591970\pi\)
−0.284930 + 0.958548i \(0.591970\pi\)
\(938\) 0 0
\(939\) 58.4558 1.90763
\(940\) 0 0
\(941\) 4.97056 0.162036 0.0810179 0.996713i \(-0.474183\pi\)
0.0810179 + 0.996713i \(0.474183\pi\)
\(942\) 71.1127 2.31698
\(943\) 10.2843 0.334902
\(944\) 24.9706 0.812723
\(945\) 0 0
\(946\) −16.4853 −0.535983
\(947\) 43.7574 1.42192 0.710962 0.703231i \(-0.248260\pi\)
0.710962 + 0.703231i \(0.248260\pi\)
\(948\) 0 0
\(949\) 13.4558 0.436795
\(950\) 0 0
\(951\) 28.1421 0.912571
\(952\) 0 0
\(953\) 29.0122 0.939797 0.469899 0.882720i \(-0.344290\pi\)
0.469899 + 0.882720i \(0.344290\pi\)
\(954\) −16.9706 −0.549442
\(955\) 0 0
\(956\) 0 0
\(957\) −37.3848 −1.20848
\(958\) 28.6274 0.924910
\(959\) 0 0
\(960\) 0 0
\(961\) −27.9117 −0.900377
\(962\) −14.0000 −0.451378
\(963\) −40.9706 −1.32026
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −24.4264 −0.785500 −0.392750 0.919645i \(-0.628476\pi\)
−0.392750 + 0.919645i \(0.628476\pi\)
\(968\) 64.9706 2.08823
\(969\) 75.9411 2.43958
\(970\) 0 0
\(971\) 42.7279 1.37120 0.685602 0.727976i \(-0.259539\pi\)
0.685602 + 0.727976i \(0.259539\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −44.8284 −1.43640
\(975\) 0 0
\(976\) 11.3137 0.362143
\(977\) 30.7696 0.984405 0.492203 0.870481i \(-0.336192\pi\)
0.492203 + 0.870481i \(0.336192\pi\)
\(978\) −6.00000 −0.191859
\(979\) −46.6274 −1.49022
\(980\) 0 0
\(981\) 14.1421 0.451524
\(982\) −27.2721 −0.870287
\(983\) −42.2132 −1.34639 −0.673196 0.739464i \(-0.735079\pi\)
−0.673196 + 0.739464i \(0.735079\pi\)
\(984\) −15.3137 −0.488183
\(985\) 0 0
\(986\) 19.6985 0.627328
\(987\) 0 0
\(988\) 0 0
\(989\) 9.17157 0.291639
\(990\) 0 0
\(991\) −47.9411 −1.52290 −0.761450 0.648224i \(-0.775512\pi\)
−0.761450 + 0.648224i \(0.775512\pi\)
\(992\) 0 0
\(993\) 56.6274 1.79702
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 3.72792 0.118064 0.0590322 0.998256i \(-0.481199\pi\)
0.0590322 + 0.998256i \(0.481199\pi\)
\(998\) −4.24264 −0.134298
\(999\) −2.58579 −0.0818107
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.r.1.1 2
5.2 odd 4 1225.2.b.i.99.1 4
5.3 odd 4 1225.2.b.i.99.4 4
5.4 even 2 245.2.a.e.1.2 2
7.6 odd 2 1225.2.a.p.1.1 2
15.14 odd 2 2205.2.a.v.1.1 2
20.19 odd 2 3920.2.a.bw.1.2 2
35.4 even 6 245.2.e.g.226.1 4
35.9 even 6 245.2.e.g.116.1 4
35.13 even 4 1225.2.b.j.99.3 4
35.19 odd 6 245.2.e.f.116.1 4
35.24 odd 6 245.2.e.f.226.1 4
35.27 even 4 1225.2.b.j.99.2 4
35.34 odd 2 245.2.a.f.1.2 yes 2
105.104 even 2 2205.2.a.t.1.1 2
140.139 even 2 3920.2.a.br.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.2.a.e.1.2 2 5.4 even 2
245.2.a.f.1.2 yes 2 35.34 odd 2
245.2.e.f.116.1 4 35.19 odd 6
245.2.e.f.226.1 4 35.24 odd 6
245.2.e.g.116.1 4 35.9 even 6
245.2.e.g.226.1 4 35.4 even 6
1225.2.a.p.1.1 2 7.6 odd 2
1225.2.a.r.1.1 2 1.1 even 1 trivial
1225.2.b.i.99.1 4 5.2 odd 4
1225.2.b.i.99.4 4 5.3 odd 4
1225.2.b.j.99.2 4 35.27 even 4
1225.2.b.j.99.3 4 35.13 even 4
2205.2.a.t.1.1 2 105.104 even 2
2205.2.a.v.1.1 2 15.14 odd 2
3920.2.a.br.1.1 2 140.139 even 2
3920.2.a.bw.1.2 2 20.19 odd 2