Properties

Label 3380.2.a.e.1.1
Level $3380$
Weight $2$
Character 3380.1
Self dual yes
Analytic conductor $26.989$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3380.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9894358832\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 260)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3380.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} -5.00000 q^{7} -2.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} -5.00000 q^{7} -2.00000 q^{9} +5.00000 q^{11} -1.00000 q^{15} -1.00000 q^{17} +3.00000 q^{19} +5.00000 q^{21} +3.00000 q^{23} +1.00000 q^{25} +5.00000 q^{27} -1.00000 q^{29} -5.00000 q^{33} -5.00000 q^{35} -7.00000 q^{37} +5.00000 q^{41} +5.00000 q^{43} -2.00000 q^{45} -12.0000 q^{47} +18.0000 q^{49} +1.00000 q^{51} +2.00000 q^{53} +5.00000 q^{55} -3.00000 q^{57} +11.0000 q^{59} -13.0000 q^{61} +10.0000 q^{63} -3.00000 q^{67} -3.00000 q^{69} -13.0000 q^{71} +2.00000 q^{73} -1.00000 q^{75} -25.0000 q^{77} -4.00000 q^{79} +1.00000 q^{81} -12.0000 q^{83} -1.00000 q^{85} +1.00000 q^{87} -7.00000 q^{89} +3.00000 q^{95} -11.0000 q^{97} -10.0000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350 −0.288675 0.957427i \(-0.593215\pi\)
−0.288675 + 0.957427i \(0.593215\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −5.00000 −1.88982 −0.944911 0.327327i \(-0.893852\pi\)
−0.944911 + 0.327327i \(0.893852\pi\)
\(8\) 0 0
\(9\) −2.00000 −0.666667
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −1.00000 −0.242536 −0.121268 0.992620i \(-0.538696\pi\)
−0.121268 + 0.992620i \(0.538696\pi\)
\(18\) 0 0
\(19\) 3.00000 0.688247 0.344124 0.938924i \(-0.388176\pi\)
0.344124 + 0.938924i \(0.388176\pi\)
\(20\) 0 0
\(21\) 5.00000 1.09109
\(22\) 0 0
\(23\) 3.00000 0.625543 0.312772 0.949828i \(-0.398743\pi\)
0.312772 + 0.949828i \(0.398743\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 5.00000 0.962250
\(28\) 0 0
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −5.00000 −0.870388
\(34\) 0 0
\(35\) −5.00000 −0.845154
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 5.00000 0.780869 0.390434 0.920631i \(-0.372325\pi\)
0.390434 + 0.920631i \(0.372325\pi\)
\(42\) 0 0
\(43\) 5.00000 0.762493 0.381246 0.924473i \(-0.375495\pi\)
0.381246 + 0.924473i \(0.375495\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) 18.0000 2.57143
\(50\) 0 0
\(51\) 1.00000 0.140028
\(52\) 0 0
\(53\) 2.00000 0.274721 0.137361 0.990521i \(-0.456138\pi\)
0.137361 + 0.990521i \(0.456138\pi\)
\(54\) 0 0
\(55\) 5.00000 0.674200
\(56\) 0 0
\(57\) −3.00000 −0.397360
\(58\) 0 0
\(59\) 11.0000 1.43208 0.716039 0.698060i \(-0.245953\pi\)
0.716039 + 0.698060i \(0.245953\pi\)
\(60\) 0 0
\(61\) −13.0000 −1.66448 −0.832240 0.554416i \(-0.812942\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) 10.0000 1.25988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.00000 −0.366508 −0.183254 0.983066i \(-0.558663\pi\)
−0.183254 + 0.983066i \(0.558663\pi\)
\(68\) 0 0
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) −13.0000 −1.54282 −0.771408 0.636341i \(-0.780447\pi\)
−0.771408 + 0.636341i \(0.780447\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) −25.0000 −2.84901
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −1.00000 −0.108465
\(86\) 0 0
\(87\) 1.00000 0.107211
\(88\) 0 0
\(89\) −7.00000 −0.741999 −0.370999 0.928633i \(-0.620985\pi\)
−0.370999 + 0.928633i \(0.620985\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 3.00000 0.307794
\(96\) 0 0
\(97\) −11.0000 −1.11688 −0.558440 0.829545i \(-0.688600\pi\)
−0.558440 + 0.829545i \(0.688600\pi\)
\(98\) 0 0
\(99\) −10.0000 −1.00504
\(100\) 0 0
\(101\) −13.0000 −1.29355 −0.646774 0.762682i \(-0.723882\pi\)
−0.646774 + 0.762682i \(0.723882\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 5.00000 0.487950
\(106\) 0 0
\(107\) −9.00000 −0.870063 −0.435031 0.900415i \(-0.643263\pi\)
−0.435031 + 0.900415i \(0.643263\pi\)
\(108\) 0 0
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) 7.00000 0.664411
\(112\) 0 0
\(113\) −1.00000 −0.0940721 −0.0470360 0.998893i \(-0.514978\pi\)
−0.0470360 + 0.998893i \(0.514978\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.00000 0.458349
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) −5.00000 −0.450835
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 7.00000 0.621150 0.310575 0.950549i \(-0.399478\pi\)
0.310575 + 0.950549i \(0.399478\pi\)
\(128\) 0 0
\(129\) −5.00000 −0.440225
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 0 0
\(133\) −15.0000 −1.30066
\(134\) 0 0
\(135\) 5.00000 0.430331
\(136\) 0 0
\(137\) −3.00000 −0.256307 −0.128154 0.991754i \(-0.540905\pi\)
−0.128154 + 0.991754i \(0.540905\pi\)
\(138\) 0 0
\(139\) 13.0000 1.10265 0.551323 0.834292i \(-0.314123\pi\)
0.551323 + 0.834292i \(0.314123\pi\)
\(140\) 0 0
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −1.00000 −0.0830455
\(146\) 0 0
\(147\) −18.0000 −1.48461
\(148\) 0 0
\(149\) −11.0000 −0.901155 −0.450578 0.892737i \(-0.648782\pi\)
−0.450578 + 0.892737i \(0.648782\pi\)
\(150\) 0 0
\(151\) 24.0000 1.95309 0.976546 0.215308i \(-0.0690756\pi\)
0.976546 + 0.215308i \(0.0690756\pi\)
\(152\) 0 0
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 10.0000 0.798087 0.399043 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 0 0
\(159\) −2.00000 −0.158610
\(160\) 0 0
\(161\) −15.0000 −1.18217
\(162\) 0 0
\(163\) −5.00000 −0.391630 −0.195815 0.980641i \(-0.562735\pi\)
−0.195815 + 0.980641i \(0.562735\pi\)
\(164\) 0 0
\(165\) −5.00000 −0.389249
\(166\) 0 0
\(167\) 13.0000 1.00597 0.502985 0.864295i \(-0.332235\pi\)
0.502985 + 0.864295i \(0.332235\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) −6.00000 −0.458831
\(172\) 0 0
\(173\) −17.0000 −1.29249 −0.646243 0.763132i \(-0.723661\pi\)
−0.646243 + 0.763132i \(0.723661\pi\)
\(174\) 0 0
\(175\) −5.00000 −0.377964
\(176\) 0 0
\(177\) −11.0000 −0.826811
\(178\) 0 0
\(179\) 11.0000 0.822179 0.411089 0.911595i \(-0.365148\pi\)
0.411089 + 0.911595i \(0.365148\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 13.0000 0.960988
\(184\) 0 0
\(185\) −7.00000 −0.514650
\(186\) 0 0
\(187\) −5.00000 −0.365636
\(188\) 0 0
\(189\) −25.0000 −1.81848
\(190\) 0 0
\(191\) −15.0000 −1.08536 −0.542681 0.839939i \(-0.682591\pi\)
−0.542681 + 0.839939i \(0.682591\pi\)
\(192\) 0 0
\(193\) −23.0000 −1.65558 −0.827788 0.561041i \(-0.810401\pi\)
−0.827788 + 0.561041i \(0.810401\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −27.0000 −1.92367 −0.961835 0.273629i \(-0.911776\pi\)
−0.961835 + 0.273629i \(0.911776\pi\)
\(198\) 0 0
\(199\) 21.0000 1.48865 0.744325 0.667817i \(-0.232771\pi\)
0.744325 + 0.667817i \(0.232771\pi\)
\(200\) 0 0
\(201\) 3.00000 0.211604
\(202\) 0 0
\(203\) 5.00000 0.350931
\(204\) 0 0
\(205\) 5.00000 0.349215
\(206\) 0 0
\(207\) −6.00000 −0.417029
\(208\) 0 0
\(209\) 15.0000 1.03757
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) 0 0
\(213\) 13.0000 0.890745
\(214\) 0 0
\(215\) 5.00000 0.340997
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −19.0000 −1.27233 −0.636167 0.771551i \(-0.719481\pi\)
−0.636167 + 0.771551i \(0.719481\pi\)
\(224\) 0 0
\(225\) −2.00000 −0.133333
\(226\) 0 0
\(227\) −17.0000 −1.12833 −0.564165 0.825662i \(-0.690802\pi\)
−0.564165 + 0.825662i \(0.690802\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 25.0000 1.64488
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 0 0
\(237\) 4.00000 0.259828
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) −11.0000 −0.708572 −0.354286 0.935137i \(-0.615276\pi\)
−0.354286 + 0.935137i \(0.615276\pi\)
\(242\) 0 0
\(243\) −16.0000 −1.02640
\(244\) 0 0
\(245\) 18.0000 1.14998
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) −15.0000 −0.946792 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(252\) 0 0
\(253\) 15.0000 0.943042
\(254\) 0 0
\(255\) 1.00000 0.0626224
\(256\) 0 0
\(257\) 15.0000 0.935674 0.467837 0.883815i \(-0.345033\pi\)
0.467837 + 0.883815i \(0.345033\pi\)
\(258\) 0 0
\(259\) 35.0000 2.17479
\(260\) 0 0
\(261\) 2.00000 0.123797
\(262\) 0 0
\(263\) 11.0000 0.678289 0.339145 0.940734i \(-0.389862\pi\)
0.339145 + 0.940734i \(0.389862\pi\)
\(264\) 0 0
\(265\) 2.00000 0.122859
\(266\) 0 0
\(267\) 7.00000 0.428393
\(268\) 0 0
\(269\) −9.00000 −0.548740 −0.274370 0.961624i \(-0.588469\pi\)
−0.274370 + 0.961624i \(0.588469\pi\)
\(270\) 0 0
\(271\) −7.00000 −0.425220 −0.212610 0.977137i \(-0.568196\pi\)
−0.212610 + 0.977137i \(0.568196\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.00000 0.301511
\(276\) 0 0
\(277\) −13.0000 −0.781094 −0.390547 0.920583i \(-0.627714\pi\)
−0.390547 + 0.920583i \(0.627714\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −30.0000 −1.78965 −0.894825 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) 23.0000 1.36721 0.683604 0.729853i \(-0.260412\pi\)
0.683604 + 0.729853i \(0.260412\pi\)
\(284\) 0 0
\(285\) −3.00000 −0.177705
\(286\) 0 0
\(287\) −25.0000 −1.47570
\(288\) 0 0
\(289\) −16.0000 −0.941176
\(290\) 0 0
\(291\) 11.0000 0.644831
\(292\) 0 0
\(293\) −7.00000 −0.408944 −0.204472 0.978872i \(-0.565548\pi\)
−0.204472 + 0.978872i \(0.565548\pi\)
\(294\) 0 0
\(295\) 11.0000 0.640445
\(296\) 0 0
\(297\) 25.0000 1.45065
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −25.0000 −1.44098
\(302\) 0 0
\(303\) 13.0000 0.746830
\(304\) 0 0
\(305\) −13.0000 −0.744378
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 0 0
\(315\) 10.0000 0.563436
\(316\) 0 0
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) −5.00000 −0.279946
\(320\) 0 0
\(321\) 9.00000 0.502331
\(322\) 0 0
\(323\) −3.00000 −0.166924
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 18.0000 0.995402
\(328\) 0 0
\(329\) 60.0000 3.30791
\(330\) 0 0
\(331\) −1.00000 −0.0549650 −0.0274825 0.999622i \(-0.508749\pi\)
−0.0274825 + 0.999622i \(0.508749\pi\)
\(332\) 0 0
\(333\) 14.0000 0.767195
\(334\) 0 0
\(335\) −3.00000 −0.163908
\(336\) 0 0
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) 1.00000 0.0543125
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −55.0000 −2.96972
\(344\) 0 0
\(345\) −3.00000 −0.161515
\(346\) 0 0
\(347\) −27.0000 −1.44944 −0.724718 0.689046i \(-0.758030\pi\)
−0.724718 + 0.689046i \(0.758030\pi\)
\(348\) 0 0
\(349\) −35.0000 −1.87351 −0.936754 0.349990i \(-0.886185\pi\)
−0.936754 + 0.349990i \(0.886185\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 5.00000 0.266123 0.133062 0.991108i \(-0.457519\pi\)
0.133062 + 0.991108i \(0.457519\pi\)
\(354\) 0 0
\(355\) −13.0000 −0.689968
\(356\) 0 0
\(357\) −5.00000 −0.264628
\(358\) 0 0
\(359\) 24.0000 1.26667 0.633336 0.773877i \(-0.281685\pi\)
0.633336 + 0.773877i \(0.281685\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 0 0
\(363\) −14.0000 −0.734809
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 0 0
\(367\) 3.00000 0.156599 0.0782994 0.996930i \(-0.475051\pi\)
0.0782994 + 0.996930i \(0.475051\pi\)
\(368\) 0 0
\(369\) −10.0000 −0.520579
\(370\) 0 0
\(371\) −10.0000 −0.519174
\(372\) 0 0
\(373\) 19.0000 0.983783 0.491891 0.870657i \(-0.336306\pi\)
0.491891 + 0.870657i \(0.336306\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 21.0000 1.07870 0.539349 0.842082i \(-0.318670\pi\)
0.539349 + 0.842082i \(0.318670\pi\)
\(380\) 0 0
\(381\) −7.00000 −0.358621
\(382\) 0 0
\(383\) 3.00000 0.153293 0.0766464 0.997058i \(-0.475579\pi\)
0.0766464 + 0.997058i \(0.475579\pi\)
\(384\) 0 0
\(385\) −25.0000 −1.27412
\(386\) 0 0
\(387\) −10.0000 −0.508329
\(388\) 0 0
\(389\) −10.0000 −0.507020 −0.253510 0.967333i \(-0.581585\pi\)
−0.253510 + 0.967333i \(0.581585\pi\)
\(390\) 0 0
\(391\) −3.00000 −0.151717
\(392\) 0 0
\(393\) −4.00000 −0.201773
\(394\) 0 0
\(395\) −4.00000 −0.201262
\(396\) 0 0
\(397\) 13.0000 0.652451 0.326226 0.945292i \(-0.394223\pi\)
0.326226 + 0.945292i \(0.394223\pi\)
\(398\) 0 0
\(399\) 15.0000 0.750939
\(400\) 0 0
\(401\) −27.0000 −1.34832 −0.674158 0.738587i \(-0.735493\pi\)
−0.674158 + 0.738587i \(0.735493\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) −35.0000 −1.73489
\(408\) 0 0
\(409\) −19.0000 −0.939490 −0.469745 0.882802i \(-0.655654\pi\)
−0.469745 + 0.882802i \(0.655654\pi\)
\(410\) 0 0
\(411\) 3.00000 0.147979
\(412\) 0 0
\(413\) −55.0000 −2.70637
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) −13.0000 −0.636613
\(418\) 0 0
\(419\) −17.0000 −0.830504 −0.415252 0.909706i \(-0.636307\pi\)
−0.415252 + 0.909706i \(0.636307\pi\)
\(420\) 0 0
\(421\) −10.0000 −0.487370 −0.243685 0.969854i \(-0.578356\pi\)
−0.243685 + 0.969854i \(0.578356\pi\)
\(422\) 0 0
\(423\) 24.0000 1.16692
\(424\) 0 0
\(425\) −1.00000 −0.0485071
\(426\) 0 0
\(427\) 65.0000 3.14557
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 21.0000 1.01153 0.505767 0.862670i \(-0.331209\pi\)
0.505767 + 0.862670i \(0.331209\pi\)
\(432\) 0 0
\(433\) 7.00000 0.336399 0.168199 0.985753i \(-0.446205\pi\)
0.168199 + 0.985753i \(0.446205\pi\)
\(434\) 0 0
\(435\) 1.00000 0.0479463
\(436\) 0 0
\(437\) 9.00000 0.430528
\(438\) 0 0
\(439\) −29.0000 −1.38409 −0.692047 0.721852i \(-0.743291\pi\)
−0.692047 + 0.721852i \(0.743291\pi\)
\(440\) 0 0
\(441\) −36.0000 −1.71429
\(442\) 0 0
\(443\) −20.0000 −0.950229 −0.475114 0.879924i \(-0.657593\pi\)
−0.475114 + 0.879924i \(0.657593\pi\)
\(444\) 0 0
\(445\) −7.00000 −0.331832
\(446\) 0 0
\(447\) 11.0000 0.520282
\(448\) 0 0
\(449\) 21.0000 0.991051 0.495526 0.868593i \(-0.334975\pi\)
0.495526 + 0.868593i \(0.334975\pi\)
\(450\) 0 0
\(451\) 25.0000 1.17720
\(452\) 0 0
\(453\) −24.0000 −1.12762
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −11.0000 −0.514558 −0.257279 0.966337i \(-0.582826\pi\)
−0.257279 + 0.966337i \(0.582826\pi\)
\(458\) 0 0
\(459\) −5.00000 −0.233380
\(460\) 0 0
\(461\) 33.0000 1.53696 0.768482 0.639872i \(-0.221013\pi\)
0.768482 + 0.639872i \(0.221013\pi\)
\(462\) 0 0
\(463\) 8.00000 0.371792 0.185896 0.982569i \(-0.440481\pi\)
0.185896 + 0.982569i \(0.440481\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −8.00000 −0.370196 −0.185098 0.982720i \(-0.559260\pi\)
−0.185098 + 0.982720i \(0.559260\pi\)
\(468\) 0 0
\(469\) 15.0000 0.692636
\(470\) 0 0
\(471\) −10.0000 −0.460776
\(472\) 0 0
\(473\) 25.0000 1.14950
\(474\) 0 0
\(475\) 3.00000 0.137649
\(476\) 0 0
\(477\) −4.00000 −0.183147
\(478\) 0 0
\(479\) −11.0000 −0.502603 −0.251301 0.967909i \(-0.580859\pi\)
−0.251301 + 0.967909i \(0.580859\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 15.0000 0.682524
\(484\) 0 0
\(485\) −11.0000 −0.499484
\(486\) 0 0
\(487\) −17.0000 −0.770344 −0.385172 0.922845i \(-0.625858\pi\)
−0.385172 + 0.922845i \(0.625858\pi\)
\(488\) 0 0
\(489\) 5.00000 0.226108
\(490\) 0 0
\(491\) 15.0000 0.676941 0.338470 0.940977i \(-0.390091\pi\)
0.338470 + 0.940977i \(0.390091\pi\)
\(492\) 0 0
\(493\) 1.00000 0.0450377
\(494\) 0 0
\(495\) −10.0000 −0.449467
\(496\) 0 0
\(497\) 65.0000 2.91565
\(498\) 0 0
\(499\) 4.00000 0.179065 0.0895323 0.995984i \(-0.471463\pi\)
0.0895323 + 0.995984i \(0.471463\pi\)
\(500\) 0 0
\(501\) −13.0000 −0.580797
\(502\) 0 0
\(503\) −11.0000 −0.490466 −0.245233 0.969464i \(-0.578864\pi\)
−0.245233 + 0.969464i \(0.578864\pi\)
\(504\) 0 0
\(505\) −13.0000 −0.578492
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −3.00000 −0.132973 −0.0664863 0.997787i \(-0.521179\pi\)
−0.0664863 + 0.997787i \(0.521179\pi\)
\(510\) 0 0
\(511\) −10.0000 −0.442374
\(512\) 0 0
\(513\) 15.0000 0.662266
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −60.0000 −2.63880
\(518\) 0 0
\(519\) 17.0000 0.746217
\(520\) 0 0
\(521\) 22.0000 0.963837 0.481919 0.876216i \(-0.339940\pi\)
0.481919 + 0.876216i \(0.339940\pi\)
\(522\) 0 0
\(523\) −21.0000 −0.918266 −0.459133 0.888368i \(-0.651840\pi\)
−0.459133 + 0.888368i \(0.651840\pi\)
\(524\) 0 0
\(525\) 5.00000 0.218218
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) −22.0000 −0.954719
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −9.00000 −0.389104
\(536\) 0 0
\(537\) −11.0000 −0.474685
\(538\) 0 0
\(539\) 90.0000 3.87657
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 0 0
\(543\) −10.0000 −0.429141
\(544\) 0 0
\(545\) −18.0000 −0.771035
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) 26.0000 1.10965
\(550\) 0 0
\(551\) −3.00000 −0.127804
\(552\) 0 0
\(553\) 20.0000 0.850487
\(554\) 0 0
\(555\) 7.00000 0.297133
\(556\) 0 0
\(557\) 13.0000 0.550828 0.275414 0.961326i \(-0.411185\pi\)
0.275414 + 0.961326i \(0.411185\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 5.00000 0.211100
\(562\) 0 0
\(563\) 9.00000 0.379305 0.189652 0.981851i \(-0.439264\pi\)
0.189652 + 0.981851i \(0.439264\pi\)
\(564\) 0 0
\(565\) −1.00000 −0.0420703
\(566\) 0 0
\(567\) −5.00000 −0.209980
\(568\) 0 0
\(569\) 39.0000 1.63497 0.817483 0.575953i \(-0.195369\pi\)
0.817483 + 0.575953i \(0.195369\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) 15.0000 0.626634
\(574\) 0 0
\(575\) 3.00000 0.125109
\(576\) 0 0
\(577\) 42.0000 1.74848 0.874241 0.485491i \(-0.161359\pi\)
0.874241 + 0.485491i \(0.161359\pi\)
\(578\) 0 0
\(579\) 23.0000 0.955847
\(580\) 0 0
\(581\) 60.0000 2.48922
\(582\) 0 0
\(583\) 10.0000 0.414158
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −3.00000 −0.123823 −0.0619116 0.998082i \(-0.519720\pi\)
−0.0619116 + 0.998082i \(0.519720\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 27.0000 1.11063
\(592\) 0 0
\(593\) 2.00000 0.0821302 0.0410651 0.999156i \(-0.486925\pi\)
0.0410651 + 0.999156i \(0.486925\pi\)
\(594\) 0 0
\(595\) 5.00000 0.204980
\(596\) 0 0
\(597\) −21.0000 −0.859473
\(598\) 0 0
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) −5.00000 −0.203954 −0.101977 0.994787i \(-0.532517\pi\)
−0.101977 + 0.994787i \(0.532517\pi\)
\(602\) 0 0
\(603\) 6.00000 0.244339
\(604\) 0 0
\(605\) 14.0000 0.569181
\(606\) 0 0
\(607\) −31.0000 −1.25825 −0.629126 0.777304i \(-0.716587\pi\)
−0.629126 + 0.777304i \(0.716587\pi\)
\(608\) 0 0
\(609\) −5.00000 −0.202610
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 25.0000 1.00974 0.504870 0.863195i \(-0.331540\pi\)
0.504870 + 0.863195i \(0.331540\pi\)
\(614\) 0 0
\(615\) −5.00000 −0.201619
\(616\) 0 0
\(617\) −27.0000 −1.08698 −0.543490 0.839416i \(-0.682897\pi\)
−0.543490 + 0.839416i \(0.682897\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 15.0000 0.601929
\(622\) 0 0
\(623\) 35.0000 1.40225
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −15.0000 −0.599042
\(628\) 0 0
\(629\) 7.00000 0.279108
\(630\) 0 0
\(631\) 27.0000 1.07485 0.537427 0.843311i \(-0.319397\pi\)
0.537427 + 0.843311i \(0.319397\pi\)
\(632\) 0 0
\(633\) 5.00000 0.198732
\(634\) 0 0
\(635\) 7.00000 0.277787
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 26.0000 1.02854
\(640\) 0 0
\(641\) 27.0000 1.06644 0.533218 0.845978i \(-0.320983\pi\)
0.533218 + 0.845978i \(0.320983\pi\)
\(642\) 0 0
\(643\) −5.00000 −0.197181 −0.0985904 0.995128i \(-0.531433\pi\)
−0.0985904 + 0.995128i \(0.531433\pi\)
\(644\) 0 0
\(645\) −5.00000 −0.196875
\(646\) 0 0
\(647\) −9.00000 −0.353827 −0.176913 0.984226i \(-0.556611\pi\)
−0.176913 + 0.984226i \(0.556611\pi\)
\(648\) 0 0
\(649\) 55.0000 2.15894
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 39.0000 1.52619 0.763094 0.646288i \(-0.223679\pi\)
0.763094 + 0.646288i \(0.223679\pi\)
\(654\) 0 0
\(655\) 4.00000 0.156293
\(656\) 0 0
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) 17.0000 0.662226 0.331113 0.943591i \(-0.392576\pi\)
0.331113 + 0.943591i \(0.392576\pi\)
\(660\) 0 0
\(661\) −3.00000 −0.116686 −0.0583432 0.998297i \(-0.518582\pi\)
−0.0583432 + 0.998297i \(0.518582\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −15.0000 −0.581675
\(666\) 0 0
\(667\) −3.00000 −0.116160
\(668\) 0 0
\(669\) 19.0000 0.734582
\(670\) 0 0
\(671\) −65.0000 −2.50930
\(672\) 0 0
\(673\) 11.0000 0.424019 0.212009 0.977268i \(-0.431999\pi\)
0.212009 + 0.977268i \(0.431999\pi\)
\(674\) 0 0
\(675\) 5.00000 0.192450
\(676\) 0 0
\(677\) 42.0000 1.61419 0.807096 0.590421i \(-0.201038\pi\)
0.807096 + 0.590421i \(0.201038\pi\)
\(678\) 0 0
\(679\) 55.0000 2.11071
\(680\) 0 0
\(681\) 17.0000 0.651441
\(682\) 0 0
\(683\) −49.0000 −1.87493 −0.937466 0.348076i \(-0.886835\pi\)
−0.937466 + 0.348076i \(0.886835\pi\)
\(684\) 0 0
\(685\) −3.00000 −0.114624
\(686\) 0 0
\(687\) 10.0000 0.381524
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 5.00000 0.190209 0.0951045 0.995467i \(-0.469681\pi\)
0.0951045 + 0.995467i \(0.469681\pi\)
\(692\) 0 0
\(693\) 50.0000 1.89934
\(694\) 0 0
\(695\) 13.0000 0.493118
\(696\) 0 0
\(697\) −5.00000 −0.189389
\(698\) 0 0
\(699\) −6.00000 −0.226941
\(700\) 0 0
\(701\) 26.0000 0.982006 0.491003 0.871158i \(-0.336630\pi\)
0.491003 + 0.871158i \(0.336630\pi\)
\(702\) 0 0
\(703\) −21.0000 −0.792030
\(704\) 0 0
\(705\) 12.0000 0.451946
\(706\) 0 0
\(707\) 65.0000 2.44458
\(708\) 0 0
\(709\) −23.0000 −0.863783 −0.431892 0.901926i \(-0.642154\pi\)
−0.431892 + 0.901926i \(0.642154\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −8.00000 −0.298765
\(718\) 0 0
\(719\) 33.0000 1.23069 0.615346 0.788257i \(-0.289016\pi\)
0.615346 + 0.788257i \(0.289016\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 11.0000 0.409094
\(724\) 0 0
\(725\) −1.00000 −0.0371391
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −5.00000 −0.184932
\(732\) 0 0
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) 0 0
\(735\) −18.0000 −0.663940
\(736\) 0 0
\(737\) −15.0000 −0.552532
\(738\) 0 0
\(739\) −15.0000 −0.551784 −0.275892 0.961189i \(-0.588973\pi\)
−0.275892 + 0.961189i \(0.588973\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 9.00000 0.330178 0.165089 0.986279i \(-0.447209\pi\)
0.165089 + 0.986279i \(0.447209\pi\)
\(744\) 0 0
\(745\) −11.0000 −0.403009
\(746\) 0 0
\(747\) 24.0000 0.878114
\(748\) 0 0
\(749\) 45.0000 1.64426
\(750\) 0 0
\(751\) −25.0000 −0.912263 −0.456131 0.889912i \(-0.650765\pi\)
−0.456131 + 0.889912i \(0.650765\pi\)
\(752\) 0 0
\(753\) 15.0000 0.546630
\(754\) 0 0
\(755\) 24.0000 0.873449
\(756\) 0 0
\(757\) −9.00000 −0.327111 −0.163555 0.986534i \(-0.552296\pi\)
−0.163555 + 0.986534i \(0.552296\pi\)
\(758\) 0 0
\(759\) −15.0000 −0.544466
\(760\) 0 0
\(761\) −19.0000 −0.688749 −0.344375 0.938832i \(-0.611909\pi\)
−0.344375 + 0.938832i \(0.611909\pi\)
\(762\) 0 0
\(763\) 90.0000 3.25822
\(764\) 0 0
\(765\) 2.00000 0.0723102
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −19.0000 −0.685158 −0.342579 0.939489i \(-0.611300\pi\)
−0.342579 + 0.939489i \(0.611300\pi\)
\(770\) 0 0
\(771\) −15.0000 −0.540212
\(772\) 0 0
\(773\) 37.0000 1.33080 0.665399 0.746488i \(-0.268262\pi\)
0.665399 + 0.746488i \(0.268262\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −35.0000 −1.25562
\(778\) 0 0
\(779\) 15.0000 0.537431
\(780\) 0 0
\(781\) −65.0000 −2.32588
\(782\) 0 0
\(783\) −5.00000 −0.178685
\(784\) 0 0
\(785\) 10.0000 0.356915
\(786\) 0 0
\(787\) 11.0000 0.392108 0.196054 0.980593i \(-0.437187\pi\)
0.196054 + 0.980593i \(0.437187\pi\)
\(788\) 0 0
\(789\) −11.0000 −0.391610
\(790\) 0 0
\(791\) 5.00000 0.177780
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) −2.00000 −0.0709327
\(796\) 0 0
\(797\) 23.0000 0.814702 0.407351 0.913272i \(-0.366453\pi\)
0.407351 + 0.913272i \(0.366453\pi\)
\(798\) 0 0
\(799\) 12.0000 0.424529
\(800\) 0 0
\(801\) 14.0000 0.494666
\(802\) 0 0
\(803\) 10.0000 0.352892
\(804\) 0 0
\(805\) −15.0000 −0.528681
\(806\) 0 0
\(807\) 9.00000 0.316815
\(808\) 0 0
\(809\) −37.0000 −1.30085 −0.650425 0.759570i \(-0.725409\pi\)
−0.650425 + 0.759570i \(0.725409\pi\)
\(810\) 0 0
\(811\) 8.00000 0.280918 0.140459 0.990086i \(-0.455142\pi\)
0.140459 + 0.990086i \(0.455142\pi\)
\(812\) 0 0
\(813\) 7.00000 0.245501
\(814\) 0 0
\(815\) −5.00000 −0.175142
\(816\) 0 0
\(817\) 15.0000 0.524784
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 25.0000 0.872506 0.436253 0.899824i \(-0.356305\pi\)
0.436253 + 0.899824i \(0.356305\pi\)
\(822\) 0 0
\(823\) −31.0000 −1.08059 −0.540296 0.841475i \(-0.681688\pi\)
−0.540296 + 0.841475i \(0.681688\pi\)
\(824\) 0 0
\(825\) −5.00000 −0.174078
\(826\) 0 0
\(827\) −12.0000 −0.417281 −0.208640 0.977992i \(-0.566904\pi\)
−0.208640 + 0.977992i \(0.566904\pi\)
\(828\) 0 0
\(829\) −25.0000 −0.868286 −0.434143 0.900844i \(-0.642949\pi\)
−0.434143 + 0.900844i \(0.642949\pi\)
\(830\) 0 0
\(831\) 13.0000 0.450965
\(832\) 0 0
\(833\) −18.0000 −0.623663
\(834\) 0 0
\(835\) 13.0000 0.449884
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 43.0000 1.48452 0.742262 0.670109i \(-0.233753\pi\)
0.742262 + 0.670109i \(0.233753\pi\)
\(840\) 0 0
\(841\) −28.0000 −0.965517
\(842\) 0 0
\(843\) 30.0000 1.03325
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −70.0000 −2.40523
\(848\) 0 0
\(849\) −23.0000 −0.789358
\(850\) 0 0
\(851\) −21.0000 −0.719871
\(852\) 0 0
\(853\) −34.0000 −1.16414 −0.582069 0.813139i \(-0.697757\pi\)
−0.582069 + 0.813139i \(0.697757\pi\)
\(854\) 0 0
\(855\) −6.00000 −0.205196
\(856\) 0 0
\(857\) −26.0000 −0.888143 −0.444072 0.895991i \(-0.646466\pi\)
−0.444072 + 0.895991i \(0.646466\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 0 0
\(861\) 25.0000 0.851998
\(862\) 0 0
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) −17.0000 −0.578017
\(866\) 0 0
\(867\) 16.0000 0.543388
\(868\) 0 0
\(869\) −20.0000 −0.678454
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 22.0000 0.744587
\(874\) 0 0
\(875\) −5.00000 −0.169031
\(876\) 0 0
\(877\) −31.0000 −1.04680 −0.523398 0.852088i \(-0.675336\pi\)
−0.523398 + 0.852088i \(0.675336\pi\)
\(878\) 0 0
\(879\) 7.00000 0.236104
\(880\) 0 0
\(881\) −45.0000 −1.51609 −0.758044 0.652203i \(-0.773845\pi\)
−0.758044 + 0.652203i \(0.773845\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 0 0
\(885\) −11.0000 −0.369761
\(886\) 0 0
\(887\) 31.0000 1.04088 0.520439 0.853899i \(-0.325768\pi\)
0.520439 + 0.853899i \(0.325768\pi\)
\(888\) 0 0
\(889\) −35.0000 −1.17386
\(890\) 0 0
\(891\) 5.00000 0.167506
\(892\) 0 0
\(893\) −36.0000 −1.20469
\(894\) 0 0
\(895\) 11.0000 0.367689
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −2.00000 −0.0666297
\(902\) 0 0
\(903\) 25.0000 0.831948
\(904\) 0 0
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) −17.0000 −0.564476 −0.282238 0.959344i \(-0.591077\pi\)
−0.282238 + 0.959344i \(0.591077\pi\)
\(908\) 0 0
\(909\) 26.0000 0.862366
\(910\) 0 0
\(911\) −8.00000 −0.265052 −0.132526 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) 0 0
\(913\) −60.0000 −1.98571
\(914\) 0 0
\(915\) 13.0000 0.429767
\(916\) 0 0
\(917\) −20.0000 −0.660458
\(918\) 0 0
\(919\) 49.0000 1.61636 0.808180 0.588935i \(-0.200453\pi\)
0.808180 + 0.588935i \(0.200453\pi\)
\(920\) 0 0
\(921\) −12.0000 −0.395413
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −7.00000 −0.230159
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 45.0000 1.47640 0.738201 0.674581i \(-0.235676\pi\)
0.738201 + 0.674581i \(0.235676\pi\)
\(930\) 0 0
\(931\) 54.0000 1.76978
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −5.00000 −0.163517
\(936\) 0 0
\(937\) −18.0000 −0.588034 −0.294017 0.955800i \(-0.594992\pi\)
−0.294017 + 0.955800i \(0.594992\pi\)
\(938\) 0 0
\(939\) −6.00000 −0.195803
\(940\) 0 0
\(941\) −14.0000 −0.456387 −0.228193 0.973616i \(-0.573282\pi\)
−0.228193 + 0.973616i \(0.573282\pi\)
\(942\) 0 0
\(943\) 15.0000 0.488467
\(944\) 0 0
\(945\) −25.0000 −0.813250
\(946\) 0 0
\(947\) −31.0000 −1.00736 −0.503682 0.863889i \(-0.668022\pi\)
−0.503682 + 0.863889i \(0.668022\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) 3.00000 0.0971795 0.0485898 0.998819i \(-0.484527\pi\)
0.0485898 + 0.998819i \(0.484527\pi\)
\(954\) 0 0
\(955\) −15.0000 −0.485389
\(956\) 0 0
\(957\) 5.00000 0.161627
\(958\) 0 0
\(959\) 15.0000 0.484375
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 18.0000 0.580042
\(964\) 0 0
\(965\) −23.0000 −0.740396
\(966\) 0 0
\(967\) −16.0000 −0.514525 −0.257263 0.966342i \(-0.582821\pi\)
−0.257263 + 0.966342i \(0.582821\pi\)
\(968\) 0 0
\(969\) 3.00000 0.0963739
\(970\) 0 0
\(971\) −7.00000 −0.224641 −0.112320 0.993672i \(-0.535828\pi\)
−0.112320 + 0.993672i \(0.535828\pi\)
\(972\) 0 0
\(973\) −65.0000 −2.08380
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 45.0000 1.43968 0.719839 0.694141i \(-0.244216\pi\)
0.719839 + 0.694141i \(0.244216\pi\)
\(978\) 0 0
\(979\) −35.0000 −1.11860
\(980\) 0 0
\(981\) 36.0000 1.14939
\(982\) 0 0
\(983\) 16.0000 0.510321 0.255160 0.966899i \(-0.417872\pi\)
0.255160 + 0.966899i \(0.417872\pi\)
\(984\) 0 0
\(985\) −27.0000 −0.860292
\(986\) 0 0
\(987\) −60.0000 −1.90982
\(988\) 0 0
\(989\) 15.0000 0.476972
\(990\) 0 0
\(991\) 3.00000 0.0952981 0.0476491 0.998864i \(-0.484827\pi\)
0.0476491 + 0.998864i \(0.484827\pi\)
\(992\) 0 0
\(993\) 1.00000 0.0317340
\(994\) 0 0
\(995\) 21.0000 0.665745
\(996\) 0 0
\(997\) 7.00000 0.221692 0.110846 0.993838i \(-0.464644\pi\)
0.110846 + 0.993838i \(0.464644\pi\)
\(998\) 0 0
\(999\) −35.0000 −1.10735
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 3380.2.a.e.1.1 1
13.4 even 6 260.2.i.c.81.1 yes 2
13.5 odd 4 3380.2.f.c.3041.1 2
13.8 odd 4 3380.2.f.c.3041.2 2
13.10 even 6 260.2.i.c.61.1 2
13.12 even 2 3380.2.a.d.1.1 1
39.17 odd 6 2340.2.q.c.2161.1 2
39.23 odd 6 2340.2.q.c.1621.1 2
52.23 odd 6 1040.2.q.f.321.1 2
52.43 odd 6 1040.2.q.f.81.1 2
65.4 even 6 1300.2.i.c.601.1 2
65.17 odd 12 1300.2.bb.c.549.2 4
65.23 odd 12 1300.2.bb.c.1049.2 4
65.43 odd 12 1300.2.bb.c.549.1 4
65.49 even 6 1300.2.i.c.1101.1 2
65.62 odd 12 1300.2.bb.c.1049.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
260.2.i.c.61.1 2 13.10 even 6
260.2.i.c.81.1 yes 2 13.4 even 6
1040.2.q.f.81.1 2 52.43 odd 6
1040.2.q.f.321.1 2 52.23 odd 6
1300.2.i.c.601.1 2 65.4 even 6
1300.2.i.c.1101.1 2 65.49 even 6
1300.2.bb.c.549.1 4 65.43 odd 12
1300.2.bb.c.549.2 4 65.17 odd 12
1300.2.bb.c.1049.1 4 65.62 odd 12
1300.2.bb.c.1049.2 4 65.23 odd 12
2340.2.q.c.1621.1 2 39.23 odd 6
2340.2.q.c.2161.1 2 39.17 odd 6
3380.2.a.d.1.1 1 13.12 even 2
3380.2.a.e.1.1 1 1.1 even 1 trivial
3380.2.f.c.3041.1 2 13.5 odd 4
3380.2.f.c.3041.2 2 13.8 odd 4