Properties

Label 960.2.h.d.191.2
Level $960$
Weight $2$
Character 960.191
Analytic conductor $7.666$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 960.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.66563859404\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 240)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 191.2
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 960.191
Dual form 960.2.h.d.191.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205 q^{3} +1.00000i q^{5} -3.46410i q^{7} +3.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{3} +1.00000i q^{5} -3.46410i q^{7} +3.00000 q^{9} +3.46410 q^{11} -4.00000 q^{13} -1.73205i q^{15} +6.00000i q^{17} -3.46410i q^{19} +6.00000i q^{21} -3.46410 q^{23} -1.00000 q^{25} -5.19615 q^{27} -6.00000i q^{29} -3.46410i q^{31} -6.00000 q^{33} +3.46410 q^{35} +4.00000 q^{37} +6.92820 q^{39} -12.0000i q^{41} -6.92820i q^{43} +3.00000i q^{45} +3.46410 q^{47} -5.00000 q^{49} -10.3923i q^{51} -6.00000i q^{53} +3.46410i q^{55} +6.00000i q^{57} +3.46410 q^{59} +10.0000 q^{61} -10.3923i q^{63} -4.00000i q^{65} -6.92820i q^{67} +6.00000 q^{69} -13.8564 q^{71} -2.00000 q^{73} +1.73205 q^{75} -12.0000i q^{77} -10.3923i q^{79} +9.00000 q^{81} -10.3923 q^{83} -6.00000 q^{85} +10.3923i q^{87} +13.8564i q^{91} +6.00000i q^{93} +3.46410 q^{95} +10.0000 q^{97} +10.3923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{9} + O(q^{10}) \) \( 4q + 12q^{9} - 16q^{13} - 4q^{25} - 24q^{33} + 16q^{37} - 20q^{49} + 40q^{61} + 24q^{69} - 8q^{73} + 36q^{81} - 24q^{85} + 40q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.73205 −1.00000
\(4\) 0 0
\(5\) 1.00000i 0.447214i
\(6\) 0 0
\(7\) − 3.46410i − 1.30931i −0.755929 0.654654i \(-0.772814\pi\)
0.755929 0.654654i \(-0.227186\pi\)
\(8\) 0 0
\(9\) 3.00000 1.00000
\(10\) 0 0
\(11\) 3.46410 1.04447 0.522233 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) − 1.73205i − 0.447214i
\(16\) 0 0
\(17\) 6.00000i 1.45521i 0.685994 + 0.727607i \(0.259367\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 0 0
\(19\) − 3.46410i − 0.794719i −0.917663 0.397360i \(-0.869927\pi\)
0.917663 0.397360i \(-0.130073\pi\)
\(20\) 0 0
\(21\) 6.00000i 1.30931i
\(22\) 0 0
\(23\) −3.46410 −0.722315 −0.361158 0.932505i \(-0.617618\pi\)
−0.361158 + 0.932505i \(0.617618\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) −5.19615 −1.00000
\(28\) 0 0
\(29\) − 6.00000i − 1.11417i −0.830455 0.557086i \(-0.811919\pi\)
0.830455 0.557086i \(-0.188081\pi\)
\(30\) 0 0
\(31\) − 3.46410i − 0.622171i −0.950382 0.311086i \(-0.899307\pi\)
0.950382 0.311086i \(-0.100693\pi\)
\(32\) 0 0
\(33\) −6.00000 −1.04447
\(34\) 0 0
\(35\) 3.46410 0.585540
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 0 0
\(39\) 6.92820 1.10940
\(40\) 0 0
\(41\) − 12.0000i − 1.87409i −0.349215 0.937043i \(-0.613552\pi\)
0.349215 0.937043i \(-0.386448\pi\)
\(42\) 0 0
\(43\) − 6.92820i − 1.05654i −0.849076 0.528271i \(-0.822841\pi\)
0.849076 0.528271i \(-0.177159\pi\)
\(44\) 0 0
\(45\) 3.00000i 0.447214i
\(46\) 0 0
\(47\) 3.46410 0.505291 0.252646 0.967559i \(-0.418699\pi\)
0.252646 + 0.967559i \(0.418699\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) − 10.3923i − 1.45521i
\(52\) 0 0
\(53\) − 6.00000i − 0.824163i −0.911147 0.412082i \(-0.864802\pi\)
0.911147 0.412082i \(-0.135198\pi\)
\(54\) 0 0
\(55\) 3.46410i 0.467099i
\(56\) 0 0
\(57\) 6.00000i 0.794719i
\(58\) 0 0
\(59\) 3.46410 0.450988 0.225494 0.974245i \(-0.427600\pi\)
0.225494 + 0.974245i \(0.427600\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) − 10.3923i − 1.30931i
\(64\) 0 0
\(65\) − 4.00000i − 0.496139i
\(66\) 0 0
\(67\) − 6.92820i − 0.846415i −0.906033 0.423207i \(-0.860904\pi\)
0.906033 0.423207i \(-0.139096\pi\)
\(68\) 0 0
\(69\) 6.00000 0.722315
\(70\) 0 0
\(71\) −13.8564 −1.64445 −0.822226 0.569160i \(-0.807268\pi\)
−0.822226 + 0.569160i \(0.807268\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 1.73205 0.200000
\(76\) 0 0
\(77\) − 12.0000i − 1.36753i
\(78\) 0 0
\(79\) − 10.3923i − 1.16923i −0.811312 0.584613i \(-0.801246\pi\)
0.811312 0.584613i \(-0.198754\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) −10.3923 −1.14070 −0.570352 0.821401i \(-0.693193\pi\)
−0.570352 + 0.821401i \(0.693193\pi\)
\(84\) 0 0
\(85\) −6.00000 −0.650791
\(86\) 0 0
\(87\) 10.3923i 1.11417i
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 13.8564i 1.45255i
\(92\) 0 0
\(93\) 6.00000i 0.622171i
\(94\) 0 0
\(95\) 3.46410 0.355409
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 0 0
\(99\) 10.3923 1.04447
\(100\) 0 0
\(101\) 6.00000i 0.597022i 0.954406 + 0.298511i \(0.0964900\pi\)
−0.954406 + 0.298511i \(0.903510\pi\)
\(102\) 0 0
\(103\) 10.3923i 1.02398i 0.858990 + 0.511992i \(0.171092\pi\)
−0.858990 + 0.511992i \(0.828908\pi\)
\(104\) 0 0
\(105\) −6.00000 −0.585540
\(106\) 0 0
\(107\) −3.46410 −0.334887 −0.167444 0.985882i \(-0.553551\pi\)
−0.167444 + 0.985882i \(0.553551\pi\)
\(108\) 0 0
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −6.92820 −0.657596
\(112\) 0 0
\(113\) − 6.00000i − 0.564433i −0.959351 0.282216i \(-0.908930\pi\)
0.959351 0.282216i \(-0.0910696\pi\)
\(114\) 0 0
\(115\) − 3.46410i − 0.323029i
\(116\) 0 0
\(117\) −12.0000 −1.10940
\(118\) 0 0
\(119\) 20.7846 1.90532
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 20.7846i 1.87409i
\(124\) 0 0
\(125\) − 1.00000i − 0.0894427i
\(126\) 0 0
\(127\) − 10.3923i − 0.922168i −0.887357 0.461084i \(-0.847461\pi\)
0.887357 0.461084i \(-0.152539\pi\)
\(128\) 0 0
\(129\) 12.0000i 1.05654i
\(130\) 0 0
\(131\) −17.3205 −1.51330 −0.756650 0.653820i \(-0.773165\pi\)
−0.756650 + 0.653820i \(0.773165\pi\)
\(132\) 0 0
\(133\) −12.0000 −1.04053
\(134\) 0 0
\(135\) − 5.19615i − 0.447214i
\(136\) 0 0
\(137\) − 18.0000i − 1.53784i −0.639343 0.768922i \(-0.720793\pi\)
0.639343 0.768922i \(-0.279207\pi\)
\(138\) 0 0
\(139\) 10.3923i 0.881464i 0.897639 + 0.440732i \(0.145281\pi\)
−0.897639 + 0.440732i \(0.854719\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) −13.8564 −1.15873
\(144\) 0 0
\(145\) 6.00000 0.498273
\(146\) 0 0
\(147\) 8.66025 0.714286
\(148\) 0 0
\(149\) 6.00000i 0.491539i 0.969328 + 0.245770i \(0.0790407\pi\)
−0.969328 + 0.245770i \(0.920959\pi\)
\(150\) 0 0
\(151\) 3.46410i 0.281905i 0.990016 + 0.140952i \(0.0450164\pi\)
−0.990016 + 0.140952i \(0.954984\pi\)
\(152\) 0 0
\(153\) 18.0000i 1.45521i
\(154\) 0 0
\(155\) 3.46410 0.278243
\(156\) 0 0
\(157\) 16.0000 1.27694 0.638470 0.769647i \(-0.279568\pi\)
0.638470 + 0.769647i \(0.279568\pi\)
\(158\) 0 0
\(159\) 10.3923i 0.824163i
\(160\) 0 0
\(161\) 12.0000i 0.945732i
\(162\) 0 0
\(163\) − 13.8564i − 1.08532i −0.839953 0.542659i \(-0.817418\pi\)
0.839953 0.542659i \(-0.182582\pi\)
\(164\) 0 0
\(165\) − 6.00000i − 0.467099i
\(166\) 0 0
\(167\) 3.46410 0.268060 0.134030 0.990977i \(-0.457208\pi\)
0.134030 + 0.990977i \(0.457208\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) − 10.3923i − 0.794719i
\(172\) 0 0
\(173\) 6.00000i 0.456172i 0.973641 + 0.228086i \(0.0732467\pi\)
−0.973641 + 0.228086i \(0.926753\pi\)
\(174\) 0 0
\(175\) 3.46410i 0.261861i
\(176\) 0 0
\(177\) −6.00000 −0.450988
\(178\) 0 0
\(179\) 24.2487 1.81243 0.906217 0.422813i \(-0.138957\pi\)
0.906217 + 0.422813i \(0.138957\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) −17.3205 −1.28037
\(184\) 0 0
\(185\) 4.00000i 0.294086i
\(186\) 0 0
\(187\) 20.7846i 1.51992i
\(188\) 0 0
\(189\) 18.0000i 1.30931i
\(190\) 0 0
\(191\) 20.7846 1.50392 0.751961 0.659208i \(-0.229108\pi\)
0.751961 + 0.659208i \(0.229108\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) 6.92820i 0.496139i
\(196\) 0 0
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 0 0
\(199\) 24.2487i 1.71895i 0.511182 + 0.859473i \(0.329208\pi\)
−0.511182 + 0.859473i \(0.670792\pi\)
\(200\) 0 0
\(201\) 12.0000i 0.846415i
\(202\) 0 0
\(203\) −20.7846 −1.45879
\(204\) 0 0
\(205\) 12.0000 0.838116
\(206\) 0 0
\(207\) −10.3923 −0.722315
\(208\) 0 0
\(209\) − 12.0000i − 0.830057i
\(210\) 0 0
\(211\) 17.3205i 1.19239i 0.802839 + 0.596196i \(0.203322\pi\)
−0.802839 + 0.596196i \(0.796678\pi\)
\(212\) 0 0
\(213\) 24.0000 1.64445
\(214\) 0 0
\(215\) 6.92820 0.472500
\(216\) 0 0
\(217\) −12.0000 −0.814613
\(218\) 0 0
\(219\) 3.46410 0.234082
\(220\) 0 0
\(221\) − 24.0000i − 1.61441i
\(222\) 0 0
\(223\) 17.3205i 1.15987i 0.814664 + 0.579934i \(0.196921\pi\)
−0.814664 + 0.579934i \(0.803079\pi\)
\(224\) 0 0
\(225\) −3.00000 −0.200000
\(226\) 0 0
\(227\) 3.46410 0.229920 0.114960 0.993370i \(-0.463326\pi\)
0.114960 + 0.993370i \(0.463326\pi\)
\(228\) 0 0
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) 0 0
\(231\) 20.7846i 1.36753i
\(232\) 0 0
\(233\) − 6.00000i − 0.393073i −0.980497 0.196537i \(-0.937031\pi\)
0.980497 0.196537i \(-0.0629694\pi\)
\(234\) 0 0
\(235\) 3.46410i 0.225973i
\(236\) 0 0
\(237\) 18.0000i 1.16923i
\(238\) 0 0
\(239\) 13.8564 0.896296 0.448148 0.893959i \(-0.352084\pi\)
0.448148 + 0.893959i \(0.352084\pi\)
\(240\) 0 0
\(241\) −26.0000 −1.67481 −0.837404 0.546585i \(-0.815928\pi\)
−0.837404 + 0.546585i \(0.815928\pi\)
\(242\) 0 0
\(243\) −15.5885 −1.00000
\(244\) 0 0
\(245\) − 5.00000i − 0.319438i
\(246\) 0 0
\(247\) 13.8564i 0.881662i
\(248\) 0 0
\(249\) 18.0000 1.14070
\(250\) 0 0
\(251\) 3.46410 0.218652 0.109326 0.994006i \(-0.465131\pi\)
0.109326 + 0.994006i \(0.465131\pi\)
\(252\) 0 0
\(253\) −12.0000 −0.754434
\(254\) 0 0
\(255\) 10.3923 0.650791
\(256\) 0 0
\(257\) − 18.0000i − 1.12281i −0.827541 0.561405i \(-0.810261\pi\)
0.827541 0.561405i \(-0.189739\pi\)
\(258\) 0 0
\(259\) − 13.8564i − 0.860995i
\(260\) 0 0
\(261\) − 18.0000i − 1.11417i
\(262\) 0 0
\(263\) −3.46410 −0.213606 −0.106803 0.994280i \(-0.534061\pi\)
−0.106803 + 0.994280i \(0.534061\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 6.00000i − 0.365826i −0.983129 0.182913i \(-0.941447\pi\)
0.983129 0.182913i \(-0.0585527\pi\)
\(270\) 0 0
\(271\) − 10.3923i − 0.631288i −0.948878 0.315644i \(-0.897780\pi\)
0.948878 0.315644i \(-0.102220\pi\)
\(272\) 0 0
\(273\) − 24.0000i − 1.45255i
\(274\) 0 0
\(275\) −3.46410 −0.208893
\(276\) 0 0
\(277\) −4.00000 −0.240337 −0.120168 0.992754i \(-0.538343\pi\)
−0.120168 + 0.992754i \(0.538343\pi\)
\(278\) 0 0
\(279\) − 10.3923i − 0.622171i
\(280\) 0 0
\(281\) 12.0000i 0.715860i 0.933748 + 0.357930i \(0.116517\pi\)
−0.933748 + 0.357930i \(0.883483\pi\)
\(282\) 0 0
\(283\) 6.92820i 0.411839i 0.978569 + 0.205919i \(0.0660185\pi\)
−0.978569 + 0.205919i \(0.933982\pi\)
\(284\) 0 0
\(285\) −6.00000 −0.355409
\(286\) 0 0
\(287\) −41.5692 −2.45375
\(288\) 0 0
\(289\) −19.0000 −1.11765
\(290\) 0 0
\(291\) −17.3205 −1.01535
\(292\) 0 0
\(293\) − 6.00000i − 0.350524i −0.984522 0.175262i \(-0.943923\pi\)
0.984522 0.175262i \(-0.0560772\pi\)
\(294\) 0 0
\(295\) 3.46410i 0.201688i
\(296\) 0 0
\(297\) −18.0000 −1.04447
\(298\) 0 0
\(299\) 13.8564 0.801337
\(300\) 0 0
\(301\) −24.0000 −1.38334
\(302\) 0 0
\(303\) − 10.3923i − 0.597022i
\(304\) 0 0
\(305\) 10.0000i 0.572598i
\(306\) 0 0
\(307\) 6.92820i 0.395413i 0.980261 + 0.197707i \(0.0633494\pi\)
−0.980261 + 0.197707i \(0.936651\pi\)
\(308\) 0 0
\(309\) − 18.0000i − 1.02398i
\(310\) 0 0
\(311\) −13.8564 −0.785725 −0.392862 0.919597i \(-0.628515\pi\)
−0.392862 + 0.919597i \(0.628515\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 0 0
\(315\) 10.3923 0.585540
\(316\) 0 0
\(317\) 18.0000i 1.01098i 0.862832 + 0.505490i \(0.168688\pi\)
−0.862832 + 0.505490i \(0.831312\pi\)
\(318\) 0 0
\(319\) − 20.7846i − 1.16371i
\(320\) 0 0
\(321\) 6.00000 0.334887
\(322\) 0 0
\(323\) 20.7846 1.15649
\(324\) 0 0
\(325\) 4.00000 0.221880
\(326\) 0 0
\(327\) 17.3205 0.957826
\(328\) 0 0
\(329\) − 12.0000i − 0.661581i
\(330\) 0 0
\(331\) − 10.3923i − 0.571213i −0.958347 0.285606i \(-0.907805\pi\)
0.958347 0.285606i \(-0.0921950\pi\)
\(332\) 0 0
\(333\) 12.0000 0.657596
\(334\) 0 0
\(335\) 6.92820 0.378528
\(336\) 0 0
\(337\) 34.0000 1.85210 0.926049 0.377403i \(-0.123183\pi\)
0.926049 + 0.377403i \(0.123183\pi\)
\(338\) 0 0
\(339\) 10.3923i 0.564433i
\(340\) 0 0
\(341\) − 12.0000i − 0.649836i
\(342\) 0 0
\(343\) − 6.92820i − 0.374088i
\(344\) 0 0
\(345\) 6.00000i 0.323029i
\(346\) 0 0
\(347\) −10.3923 −0.557888 −0.278944 0.960307i \(-0.589984\pi\)
−0.278944 + 0.960307i \(0.589984\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) 20.7846 1.10940
\(352\) 0 0
\(353\) 18.0000i 0.958043i 0.877803 + 0.479022i \(0.159008\pi\)
−0.877803 + 0.479022i \(0.840992\pi\)
\(354\) 0 0
\(355\) − 13.8564i − 0.735422i
\(356\) 0 0
\(357\) −36.0000 −1.90532
\(358\) 0 0
\(359\) −6.92820 −0.365657 −0.182828 0.983145i \(-0.558525\pi\)
−0.182828 + 0.983145i \(0.558525\pi\)
\(360\) 0 0
\(361\) 7.00000 0.368421
\(362\) 0 0
\(363\) −1.73205 −0.0909091
\(364\) 0 0
\(365\) − 2.00000i − 0.104685i
\(366\) 0 0
\(367\) 3.46410i 0.180825i 0.995904 + 0.0904123i \(0.0288185\pi\)
−0.995904 + 0.0904123i \(0.971182\pi\)
\(368\) 0 0
\(369\) − 36.0000i − 1.87409i
\(370\) 0 0
\(371\) −20.7846 −1.07908
\(372\) 0 0
\(373\) 16.0000 0.828449 0.414224 0.910175i \(-0.364053\pi\)
0.414224 + 0.910175i \(0.364053\pi\)
\(374\) 0 0
\(375\) 1.73205i 0.0894427i
\(376\) 0 0
\(377\) 24.0000i 1.23606i
\(378\) 0 0
\(379\) 24.2487i 1.24557i 0.782392 + 0.622786i \(0.213999\pi\)
−0.782392 + 0.622786i \(0.786001\pi\)
\(380\) 0 0
\(381\) 18.0000i 0.922168i
\(382\) 0 0
\(383\) −17.3205 −0.885037 −0.442518 0.896759i \(-0.645915\pi\)
−0.442518 + 0.896759i \(0.645915\pi\)
\(384\) 0 0
\(385\) 12.0000 0.611577
\(386\) 0 0
\(387\) − 20.7846i − 1.05654i
\(388\) 0 0
\(389\) 6.00000i 0.304212i 0.988364 + 0.152106i \(0.0486055\pi\)
−0.988364 + 0.152106i \(0.951394\pi\)
\(390\) 0 0
\(391\) − 20.7846i − 1.05112i
\(392\) 0 0
\(393\) 30.0000 1.51330
\(394\) 0 0
\(395\) 10.3923 0.522894
\(396\) 0 0
\(397\) 16.0000 0.803017 0.401508 0.915855i \(-0.368486\pi\)
0.401508 + 0.915855i \(0.368486\pi\)
\(398\) 0 0
\(399\) 20.7846 1.04053
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 13.8564i 0.690237i
\(404\) 0 0
\(405\) 9.00000i 0.447214i
\(406\) 0 0
\(407\) 13.8564 0.686837
\(408\) 0 0
\(409\) −14.0000 −0.692255 −0.346128 0.938187i \(-0.612504\pi\)
−0.346128 + 0.938187i \(0.612504\pi\)
\(410\) 0 0
\(411\) 31.1769i 1.53784i
\(412\) 0 0
\(413\) − 12.0000i − 0.590481i
\(414\) 0 0
\(415\) − 10.3923i − 0.510138i
\(416\) 0 0
\(417\) − 18.0000i − 0.881464i
\(418\) 0 0
\(419\) 10.3923 0.507697 0.253849 0.967244i \(-0.418303\pi\)
0.253849 + 0.967244i \(0.418303\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 0 0
\(423\) 10.3923 0.505291
\(424\) 0 0
\(425\) − 6.00000i − 0.291043i
\(426\) 0 0
\(427\) − 34.6410i − 1.67640i
\(428\) 0 0
\(429\) 24.0000 1.15873
\(430\) 0 0
\(431\) −6.92820 −0.333720 −0.166860 0.985981i \(-0.553363\pi\)
−0.166860 + 0.985981i \(0.553363\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) −10.3923 −0.498273
\(436\) 0 0
\(437\) 12.0000i 0.574038i
\(438\) 0 0
\(439\) − 3.46410i − 0.165333i −0.996577 0.0826663i \(-0.973656\pi\)
0.996577 0.0826663i \(-0.0263436\pi\)
\(440\) 0 0
\(441\) −15.0000 −0.714286
\(442\) 0 0
\(443\) −17.3205 −0.822922 −0.411461 0.911427i \(-0.634981\pi\)
−0.411461 + 0.911427i \(0.634981\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 10.3923i − 0.491539i
\(448\) 0 0
\(449\) 36.0000i 1.69895i 0.527633 + 0.849473i \(0.323080\pi\)
−0.527633 + 0.849473i \(0.676920\pi\)
\(450\) 0 0
\(451\) − 41.5692i − 1.95742i
\(452\) 0 0
\(453\) − 6.00000i − 0.281905i
\(454\) 0 0
\(455\) −13.8564 −0.649598
\(456\) 0 0
\(457\) −2.00000 −0.0935561 −0.0467780 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) 0 0
\(459\) − 31.1769i − 1.45521i
\(460\) 0 0
\(461\) 18.0000i 0.838344i 0.907907 + 0.419172i \(0.137680\pi\)
−0.907907 + 0.419172i \(0.862320\pi\)
\(462\) 0 0
\(463\) − 10.3923i − 0.482971i −0.970404 0.241486i \(-0.922365\pi\)
0.970404 0.241486i \(-0.0776347\pi\)
\(464\) 0 0
\(465\) −6.00000 −0.278243
\(466\) 0 0
\(467\) 24.2487 1.12210 0.561048 0.827783i \(-0.310398\pi\)
0.561048 + 0.827783i \(0.310398\pi\)
\(468\) 0 0
\(469\) −24.0000 −1.10822
\(470\) 0 0
\(471\) −27.7128 −1.27694
\(472\) 0 0
\(473\) − 24.0000i − 1.10352i
\(474\) 0 0
\(475\) 3.46410i 0.158944i
\(476\) 0 0
\(477\) − 18.0000i − 0.824163i
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −16.0000 −0.729537
\(482\) 0 0
\(483\) − 20.7846i − 0.945732i
\(484\) 0 0
\(485\) 10.0000i 0.454077i
\(486\) 0 0
\(487\) − 17.3205i − 0.784867i −0.919780 0.392434i \(-0.871633\pi\)
0.919780 0.392434i \(-0.128367\pi\)
\(488\) 0 0
\(489\) 24.0000i 1.08532i
\(490\) 0 0
\(491\) 31.1769 1.40699 0.703497 0.710698i \(-0.251621\pi\)
0.703497 + 0.710698i \(0.251621\pi\)
\(492\) 0 0
\(493\) 36.0000 1.62136
\(494\) 0 0
\(495\) 10.3923i 0.467099i
\(496\) 0 0
\(497\) 48.0000i 2.15309i
\(498\) 0 0
\(499\) − 3.46410i − 0.155074i −0.996989 0.0775372i \(-0.975294\pi\)
0.996989 0.0775372i \(-0.0247057\pi\)
\(500\) 0 0
\(501\) −6.00000 −0.268060
\(502\) 0 0
\(503\) 10.3923 0.463370 0.231685 0.972791i \(-0.425576\pi\)
0.231685 + 0.972791i \(0.425576\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) −5.19615 −0.230769
\(508\) 0 0
\(509\) − 6.00000i − 0.265945i −0.991120 0.132973i \(-0.957548\pi\)
0.991120 0.132973i \(-0.0424523\pi\)
\(510\) 0 0
\(511\) 6.92820i 0.306486i
\(512\) 0 0
\(513\) 18.0000i 0.794719i
\(514\) 0 0
\(515\) −10.3923 −0.457940
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) 0 0
\(519\) − 10.3923i − 0.456172i
\(520\) 0 0
\(521\) − 12.0000i − 0.525730i −0.964833 0.262865i \(-0.915333\pi\)
0.964833 0.262865i \(-0.0846673\pi\)
\(522\) 0 0
\(523\) − 34.6410i − 1.51475i −0.652983 0.757373i \(-0.726483\pi\)
0.652983 0.757373i \(-0.273517\pi\)
\(524\) 0 0
\(525\) − 6.00000i − 0.261861i
\(526\) 0 0
\(527\) 20.7846 0.905392
\(528\) 0 0
\(529\) −11.0000 −0.478261
\(530\) 0 0
\(531\) 10.3923 0.450988
\(532\) 0 0
\(533\) 48.0000i 2.07911i
\(534\) 0 0
\(535\) − 3.46410i − 0.149766i
\(536\) 0 0
\(537\) −42.0000 −1.81243
\(538\) 0 0
\(539\) −17.3205 −0.746047
\(540\) 0 0
\(541\) 2.00000 0.0859867 0.0429934 0.999075i \(-0.486311\pi\)
0.0429934 + 0.999075i \(0.486311\pi\)
\(542\) 0 0
\(543\) −3.46410 −0.148659
\(544\) 0 0
\(545\) − 10.0000i − 0.428353i
\(546\) 0 0
\(547\) 20.7846i 0.888686i 0.895857 + 0.444343i \(0.146563\pi\)
−0.895857 + 0.444343i \(0.853437\pi\)
\(548\) 0 0
\(549\) 30.0000 1.28037
\(550\) 0 0
\(551\) −20.7846 −0.885454
\(552\) 0 0
\(553\) −36.0000 −1.53088
\(554\) 0 0
\(555\) − 6.92820i − 0.294086i
\(556\) 0 0
\(557\) − 18.0000i − 0.762684i −0.924434 0.381342i \(-0.875462\pi\)
0.924434 0.381342i \(-0.124538\pi\)
\(558\) 0 0
\(559\) 27.7128i 1.17213i
\(560\) 0 0
\(561\) − 36.0000i − 1.51992i
\(562\) 0 0
\(563\) −3.46410 −0.145994 −0.0729972 0.997332i \(-0.523256\pi\)
−0.0729972 + 0.997332i \(0.523256\pi\)
\(564\) 0 0
\(565\) 6.00000 0.252422
\(566\) 0 0
\(567\) − 31.1769i − 1.30931i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) − 24.2487i − 1.01478i −0.861717 0.507388i \(-0.830611\pi\)
0.861717 0.507388i \(-0.169389\pi\)
\(572\) 0 0
\(573\) −36.0000 −1.50392
\(574\) 0 0
\(575\) 3.46410 0.144463
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) 0 0
\(579\) −3.46410 −0.143963
\(580\) 0 0
\(581\) 36.0000i 1.49353i
\(582\) 0 0
\(583\) − 20.7846i − 0.860811i
\(584\) 0 0
\(585\) − 12.0000i − 0.496139i
\(586\) 0 0
\(587\) −31.1769 −1.28681 −0.643404 0.765526i \(-0.722479\pi\)
−0.643404 + 0.765526i \(0.722479\pi\)
\(588\) 0 0
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) − 10.3923i − 0.427482i
\(592\) 0 0
\(593\) 42.0000i 1.72473i 0.506284 + 0.862367i \(0.331019\pi\)
−0.506284 + 0.862367i \(0.668981\pi\)
\(594\) 0 0
\(595\) 20.7846i 0.852086i
\(596\) 0 0
\(597\) − 42.0000i − 1.71895i
\(598\) 0 0
\(599\) 20.7846 0.849236 0.424618 0.905373i \(-0.360408\pi\)
0.424618 + 0.905373i \(0.360408\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) − 20.7846i − 0.846415i
\(604\) 0 0
\(605\) 1.00000i 0.0406558i
\(606\) 0 0
\(607\) 31.1769i 1.26543i 0.774384 + 0.632716i \(0.218060\pi\)
−0.774384 + 0.632716i \(0.781940\pi\)
\(608\) 0 0
\(609\) 36.0000 1.45879
\(610\) 0 0
\(611\) −13.8564 −0.560570
\(612\) 0 0
\(613\) −32.0000 −1.29247 −0.646234 0.763139i \(-0.723657\pi\)
−0.646234 + 0.763139i \(0.723657\pi\)
\(614\) 0 0
\(615\) −20.7846 −0.838116
\(616\) 0 0
\(617\) − 18.0000i − 0.724653i −0.932051 0.362326i \(-0.881983\pi\)
0.932051 0.362326i \(-0.118017\pi\)
\(618\) 0 0
\(619\) − 45.0333i − 1.81004i −0.425367 0.905021i \(-0.639855\pi\)
0.425367 0.905021i \(-0.360145\pi\)
\(620\) 0 0
\(621\) 18.0000 0.722315
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 20.7846i 0.830057i
\(628\) 0 0
\(629\) 24.0000i 0.956943i
\(630\) 0 0
\(631\) − 3.46410i − 0.137904i −0.997620 0.0689519i \(-0.978035\pi\)
0.997620 0.0689519i \(-0.0219655\pi\)
\(632\) 0 0
\(633\) − 30.0000i − 1.19239i
\(634\) 0 0
\(635\) 10.3923 0.412406
\(636\) 0 0
\(637\) 20.0000 0.792429
\(638\) 0 0
\(639\) −41.5692 −1.64445
\(640\) 0 0
\(641\) − 24.0000i − 0.947943i −0.880540 0.473972i \(-0.842820\pi\)
0.880540 0.473972i \(-0.157180\pi\)
\(642\) 0 0
\(643\) 41.5692i 1.63933i 0.572843 + 0.819665i \(0.305840\pi\)
−0.572843 + 0.819665i \(0.694160\pi\)
\(644\) 0 0
\(645\) −12.0000 −0.472500
\(646\) 0 0
\(647\) 45.0333 1.77044 0.885221 0.465170i \(-0.154007\pi\)
0.885221 + 0.465170i \(0.154007\pi\)
\(648\) 0 0
\(649\) 12.0000 0.471041
\(650\) 0 0
\(651\) 20.7846 0.814613
\(652\) 0 0
\(653\) − 42.0000i − 1.64359i −0.569785 0.821794i \(-0.692974\pi\)
0.569785 0.821794i \(-0.307026\pi\)
\(654\) 0 0
\(655\) − 17.3205i − 0.676768i
\(656\) 0 0
\(657\) −6.00000 −0.234082
\(658\) 0 0
\(659\) −31.1769 −1.21448 −0.607240 0.794518i \(-0.707723\pi\)
−0.607240 + 0.794518i \(0.707723\pi\)
\(660\) 0 0
\(661\) 10.0000 0.388955 0.194477 0.980907i \(-0.437699\pi\)
0.194477 + 0.980907i \(0.437699\pi\)
\(662\) 0 0
\(663\) 41.5692i 1.61441i
\(664\) 0 0
\(665\) − 12.0000i − 0.465340i
\(666\) 0 0
\(667\) 20.7846i 0.804783i
\(668\) 0 0
\(669\) − 30.0000i − 1.15987i
\(670\) 0 0
\(671\) 34.6410 1.33730
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 0 0
\(675\) 5.19615 0.200000
\(676\) 0 0
\(677\) − 30.0000i − 1.15299i −0.817099 0.576497i \(-0.804419\pi\)
0.817099 0.576497i \(-0.195581\pi\)
\(678\) 0 0
\(679\) − 34.6410i − 1.32940i
\(680\) 0 0
\(681\) −6.00000 −0.229920
\(682\) 0 0
\(683\) 17.3205 0.662751 0.331375 0.943499i \(-0.392487\pi\)
0.331375 + 0.943499i \(0.392487\pi\)
\(684\) 0 0
\(685\) 18.0000 0.687745
\(686\) 0 0
\(687\) −24.2487 −0.925146
\(688\) 0 0
\(689\) 24.0000i 0.914327i
\(690\) 0 0
\(691\) 31.1769i 1.18603i 0.805193 + 0.593013i \(0.202062\pi\)
−0.805193 + 0.593013i \(0.797938\pi\)
\(692\) 0 0
\(693\) − 36.0000i − 1.36753i
\(694\) 0 0
\(695\) −10.3923 −0.394203
\(696\) 0 0
\(697\) 72.0000 2.72719
\(698\) 0 0
\(699\) 10.3923i 0.393073i
\(700\) 0 0
\(701\) 42.0000i 1.58632i 0.609015 + 0.793159i \(0.291565\pi\)
−0.609015 + 0.793159i \(0.708435\pi\)
\(702\) 0 0
\(703\) − 13.8564i − 0.522604i
\(704\) 0 0
\(705\) − 6.00000i − 0.225973i
\(706\) 0 0
\(707\) 20.7846 0.781686
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) − 31.1769i − 1.16923i
\(712\) 0 0
\(713\) 12.0000i 0.449404i
\(714\) 0 0
\(715\) − 13.8564i − 0.518200i
\(716\) 0 0
\(717\) −24.0000 −0.896296
\(718\) 0 0
\(719\) −27.7128 −1.03351 −0.516757 0.856132i \(-0.672861\pi\)
−0.516757 + 0.856132i \(0.672861\pi\)
\(720\) 0 0
\(721\) 36.0000 1.34071
\(722\) 0 0
\(723\) 45.0333 1.67481
\(724\) 0 0
\(725\) 6.00000i 0.222834i
\(726\) 0 0
\(727\) − 3.46410i − 0.128476i −0.997935 0.0642382i \(-0.979538\pi\)
0.997935 0.0642382i \(-0.0204617\pi\)
\(728\) 0 0
\(729\) 27.0000 1.00000
\(730\) 0 0
\(731\) 41.5692 1.53749
\(732\) 0 0
\(733\) −28.0000 −1.03420 −0.517102 0.855924i \(-0.672989\pi\)
−0.517102 + 0.855924i \(0.672989\pi\)
\(734\) 0 0
\(735\) 8.66025i 0.319438i
\(736\) 0 0
\(737\) − 24.0000i − 0.884051i
\(738\) 0 0
\(739\) 10.3923i 0.382287i 0.981562 + 0.191144i \(0.0612196\pi\)
−0.981562 + 0.191144i \(0.938780\pi\)
\(740\) 0 0
\(741\) − 24.0000i − 0.881662i
\(742\) 0 0
\(743\) 38.1051 1.39794 0.698971 0.715150i \(-0.253642\pi\)
0.698971 + 0.715150i \(0.253642\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) 0 0
\(747\) −31.1769 −1.14070
\(748\) 0 0
\(749\) 12.0000i 0.438470i
\(750\) 0 0
\(751\) 51.9615i 1.89610i 0.318117 + 0.948051i \(0.396950\pi\)
−0.318117 + 0.948051i \(0.603050\pi\)
\(752\) 0 0
\(753\) −6.00000 −0.218652
\(754\) 0 0
\(755\) −3.46410 −0.126072
\(756\) 0 0
\(757\) 20.0000 0.726912 0.363456 0.931611i \(-0.381597\pi\)
0.363456 + 0.931611i \(0.381597\pi\)
\(758\) 0 0
\(759\) 20.7846 0.754434
\(760\) 0 0
\(761\) − 12.0000i − 0.435000i −0.976060 0.217500i \(-0.930210\pi\)
0.976060 0.217500i \(-0.0697902\pi\)
\(762\) 0 0
\(763\) 34.6410i 1.25409i
\(764\) 0 0
\(765\) −18.0000 −0.650791
\(766\) 0 0
\(767\) −13.8564 −0.500326
\(768\) 0 0
\(769\) −34.0000 −1.22607 −0.613036 0.790055i \(-0.710052\pi\)
−0.613036 + 0.790055i \(0.710052\pi\)
\(770\) 0 0
\(771\) 31.1769i 1.12281i
\(772\) 0 0
\(773\) 54.0000i 1.94225i 0.238581 + 0.971123i \(0.423318\pi\)
−0.238581 + 0.971123i \(0.576682\pi\)
\(774\) 0 0
\(775\) 3.46410i 0.124434i
\(776\) 0 0
\(777\) 24.0000i 0.860995i
\(778\) 0 0
\(779\) −41.5692 −1.48937
\(780\) 0 0
\(781\) −48.0000 −1.71758
\(782\) 0 0
\(783\) 31.1769i 1.11417i
\(784\) 0 0
\(785\) 16.0000i 0.571064i
\(786\) 0 0
\(787\) − 34.6410i − 1.23482i −0.786642 0.617409i \(-0.788182\pi\)
0.786642 0.617409i \(-0.211818\pi\)
\(788\) 0 0
\(789\) 6.00000 0.213606
\(790\) 0 0
\(791\) −20.7846 −0.739016
\(792\) 0 0
\(793\) −40.0000 −1.42044
\(794\) 0 0
\(795\) −10.3923 −0.368577
\(796\) 0 0
\(797\) − 30.0000i − 1.06265i −0.847167 0.531327i \(-0.821693\pi\)
0.847167 0.531327i \(-0.178307\pi\)
\(798\) 0 0
\(799\) 20.7846i 0.735307i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −6.92820 −0.244491
\(804\) 0 0
\(805\) −12.0000 −0.422944
\(806\) 0 0
\(807\) 10.3923i 0.365826i
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) − 24.2487i − 0.851487i −0.904844 0.425744i \(-0.860013\pi\)
0.904844 0.425744i \(-0.139987\pi\)
\(812\) 0 0
\(813\) 18.0000i 0.631288i
\(814\) 0 0
\(815\) 13.8564 0.485369
\(816\) 0 0
\(817\) −24.0000 −0.839654
\(818\) 0 0
\(819\) 41.5692i 1.45255i
\(820\) 0 0
\(821\) 6.00000i 0.209401i 0.994504 + 0.104701i \(0.0333885\pi\)
−0.994504 + 0.104701i \(0.966612\pi\)
\(822\) 0 0
\(823\) − 45.0333i − 1.56976i −0.619646 0.784881i \(-0.712724\pi\)
0.619646 0.784881i \(-0.287276\pi\)
\(824\) 0 0
\(825\) 6.00000 0.208893
\(826\) 0 0
\(827\) −38.1051 −1.32504 −0.662522 0.749042i \(-0.730514\pi\)
−0.662522 + 0.749042i \(0.730514\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) 0 0
\(831\) 6.92820 0.240337
\(832\) 0 0
\(833\) − 30.0000i − 1.03944i
\(834\) 0 0
\(835\) 3.46410i 0.119880i
\(836\) 0 0
\(837\) 18.0000i 0.622171i
\(838\) 0 0
\(839\) −48.4974 −1.67432 −0.837158 0.546960i \(-0.815785\pi\)
−0.837158 + 0.546960i \(0.815785\pi\)
\(840\) 0 0
\(841\) −7.00000 −0.241379
\(842\) 0 0
\(843\) − 20.7846i − 0.715860i
\(844\) 0 0
\(845\) 3.00000i 0.103203i
\(846\) 0 0
\(847\) − 3.46410i − 0.119028i
\(848\) 0 0
\(849\) − 12.0000i − 0.411839i
\(850\) 0 0
\(851\) −13.8564 −0.474991
\(852\) 0 0
\(853\) −8.00000 −0.273915 −0.136957 0.990577i \(-0.543732\pi\)
−0.136957 + 0.990577i \(0.543732\pi\)
\(854\) 0 0
\(855\) 10.3923 0.355409
\(856\) 0 0
\(857\) 6.00000i 0.204956i 0.994735 + 0.102478i \(0.0326771\pi\)
−0.994735 + 0.102478i \(0.967323\pi\)
\(858\) 0 0
\(859\) 51.9615i 1.77290i 0.462820 + 0.886452i \(0.346838\pi\)
−0.462820 + 0.886452i \(0.653162\pi\)
\(860\) 0 0
\(861\) 72.0000 2.45375
\(862\) 0 0
\(863\) 51.9615 1.76879 0.884395 0.466738i \(-0.154571\pi\)
0.884395 + 0.466738i \(0.154571\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 0 0
\(867\) 32.9090 1.11765
\(868\) 0 0
\(869\) − 36.0000i − 1.22122i
\(870\) 0 0
\(871\) 27.7128i 0.939013i
\(872\) 0 0
\(873\) 30.0000 1.01535
\(874\) 0 0
\(875\) −3.46410 −0.117108
\(876\) 0 0
\(877\) −32.0000 −1.08056 −0.540282 0.841484i \(-0.681682\pi\)
−0.540282 + 0.841484i \(0.681682\pi\)
\(878\) 0 0
\(879\) 10.3923i 0.350524i
\(880\) 0 0
\(881\) 48.0000i 1.61716i 0.588386 + 0.808581i \(0.299764\pi\)
−0.588386 + 0.808581i \(0.700236\pi\)
\(882\) 0 0
\(883\) 13.8564i 0.466305i 0.972440 + 0.233153i \(0.0749042\pi\)
−0.972440 + 0.233153i \(0.925096\pi\)
\(884\) 0 0
\(885\) − 6.00000i − 0.201688i
\(886\) 0 0
\(887\) −24.2487 −0.814192 −0.407096 0.913385i \(-0.633459\pi\)
−0.407096 + 0.913385i \(0.633459\pi\)
\(888\) 0 0
\(889\) −36.0000 −1.20740
\(890\) 0 0
\(891\) 31.1769 1.04447
\(892\) 0 0
\(893\) − 12.0000i − 0.401565i
\(894\) 0 0
\(895\) 24.2487i 0.810545i
\(896\) 0 0
\(897\) −24.0000 −0.801337
\(898\) 0 0
\(899\) −20.7846 −0.693206
\(900\) 0 0
\(901\) 36.0000 1.19933
\(902\) 0 0
\(903\) 41.5692 1.38334
\(904\) 0 0
\(905\) 2.00000i 0.0664822i
\(906\) 0 0
\(907\) − 41.5692i − 1.38028i −0.723674 0.690142i \(-0.757548\pi\)
0.723674 0.690142i \(-0.242452\pi\)
\(908\) 0 0
\(909\) 18.0000i 0.597022i
\(910\) 0 0
\(911\) 34.6410 1.14771 0.573854 0.818958i \(-0.305448\pi\)
0.573854 + 0.818958i \(0.305448\pi\)
\(912\) 0 0
\(913\) −36.0000 −1.19143
\(914\) 0 0
\(915\) − 17.3205i − 0.572598i
\(916\) 0 0
\(917\) 60.0000i 1.98137i
\(918\) 0 0
\(919\) 45.0333i 1.48551i 0.669562 + 0.742756i \(0.266482\pi\)
−0.669562 + 0.742756i \(0.733518\pi\)
\(920\) 0 0
\(921\) − 12.0000i − 0.395413i
\(922\) 0 0
\(923\) 55.4256 1.82436
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 0 0
\(927\) 31.1769i 1.02398i
\(928\) 0 0
\(929\) − 12.0000i − 0.393707i −0.980433 0.196854i \(-0.936928\pi\)
0.980433 0.196854i \(-0.0630724\pi\)
\(930\) 0 0
\(931\) 17.3205i 0.567657i
\(932\) 0 0
\(933\) 24.0000 0.785725
\(934\) 0 0
\(935\) −20.7846 −0.679729
\(936\) 0 0
\(937\) 38.0000 1.24141 0.620703 0.784046i \(-0.286847\pi\)
0.620703 + 0.784046i \(0.286847\pi\)
\(938\) 0 0
\(939\) −24.2487 −0.791327
\(940\) 0 0
\(941\) − 54.0000i − 1.76035i −0.474650 0.880175i \(-0.657425\pi\)
0.474650 0.880175i \(-0.342575\pi\)
\(942\) 0 0
\(943\) 41.5692i 1.35368i
\(944\) 0 0
\(945\) −18.0000 −0.585540
\(946\) 0 0
\(947\) 31.1769 1.01311 0.506557 0.862207i \(-0.330918\pi\)
0.506557 + 0.862207i \(0.330918\pi\)
\(948\) 0 0
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) − 31.1769i − 1.01098i
\(952\) 0 0
\(953\) 18.0000i 0.583077i 0.956559 + 0.291539i \(0.0941672\pi\)
−0.956559 + 0.291539i \(0.905833\pi\)
\(954\) 0 0
\(955\) 20.7846i 0.672574i
\(956\) 0 0
\(957\) 36.0000i 1.16371i
\(958\) 0 0
\(959\) −62.3538 −2.01351
\(960\) 0 0
\(961\) 19.0000 0.612903
\(962\) 0 0
\(963\) −10.3923 −0.334887
\(964\) 0 0
\(965\) 2.00000i 0.0643823i
\(966\) 0 0
\(967\) 51.9615i 1.67097i 0.549513 + 0.835485i \(0.314813\pi\)
−0.549513 + 0.835485i \(0.685187\pi\)
\(968\) 0 0
\(969\) −36.0000 −1.15649
\(970\) 0 0
\(971\) 3.46410 0.111168 0.0555842 0.998454i \(-0.482298\pi\)
0.0555842 + 0.998454i \(0.482298\pi\)
\(972\) 0 0
\(973\) 36.0000 1.15411
\(974\) 0 0
\(975\) −6.92820 −0.221880
\(976\) 0 0
\(977\) 30.0000i 0.959785i 0.877327 + 0.479893i \(0.159324\pi\)
−0.877327 + 0.479893i \(0.840676\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −30.0000 −0.957826
\(982\) 0 0
\(983\) −58.8897 −1.87829 −0.939145 0.343520i \(-0.888381\pi\)
−0.939145 + 0.343520i \(0.888381\pi\)
\(984\) 0 0
\(985\) −6.00000 −0.191176
\(986\) 0 0
\(987\) 20.7846i 0.661581i
\(988\) 0 0
\(989\) 24.0000i 0.763156i
\(990\) 0 0
\(991\) 17.3205i 0.550204i 0.961415 + 0.275102i \(0.0887116\pi\)
−0.961415 + 0.275102i \(0.911288\pi\)
\(992\) 0 0
\(993\) 18.0000i 0.571213i
\(994\) 0 0
\(995\) −24.2487 −0.768736
\(996\) 0 0
\(997\) 52.0000 1.64686 0.823428 0.567420i \(-0.192059\pi\)
0.823428 + 0.567420i \(0.192059\pi\)
\(998\) 0 0
\(999\) −20.7846 −0.657596
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 960.2.h.d.191.2 4
3.2 odd 2 inner 960.2.h.d.191.3 4
4.3 odd 2 inner 960.2.h.d.191.4 4
8.3 odd 2 240.2.h.b.191.1 4
8.5 even 2 240.2.h.b.191.3 yes 4
12.11 even 2 inner 960.2.h.d.191.1 4
24.5 odd 2 240.2.h.b.191.2 yes 4
24.11 even 2 240.2.h.b.191.4 yes 4
40.3 even 4 1200.2.o.b.1199.2 4
40.13 odd 4 1200.2.o.b.1199.3 4
40.19 odd 2 1200.2.h.m.1151.3 4
40.27 even 4 1200.2.o.a.1199.3 4
40.29 even 2 1200.2.h.m.1151.2 4
40.37 odd 4 1200.2.o.a.1199.2 4
120.29 odd 2 1200.2.h.m.1151.4 4
120.53 even 4 1200.2.o.a.1199.1 4
120.59 even 2 1200.2.h.m.1151.1 4
120.77 even 4 1200.2.o.b.1199.4 4
120.83 odd 4 1200.2.o.a.1199.4 4
120.107 odd 4 1200.2.o.b.1199.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
240.2.h.b.191.1 4 8.3 odd 2
240.2.h.b.191.2 yes 4 24.5 odd 2
240.2.h.b.191.3 yes 4 8.5 even 2
240.2.h.b.191.4 yes 4 24.11 even 2
960.2.h.d.191.1 4 12.11 even 2 inner
960.2.h.d.191.2 4 1.1 even 1 trivial
960.2.h.d.191.3 4 3.2 odd 2 inner
960.2.h.d.191.4 4 4.3 odd 2 inner
1200.2.h.m.1151.1 4 120.59 even 2
1200.2.h.m.1151.2 4 40.29 even 2
1200.2.h.m.1151.3 4 40.19 odd 2
1200.2.h.m.1151.4 4 120.29 odd 2
1200.2.o.a.1199.1 4 120.53 even 4
1200.2.o.a.1199.2 4 40.37 odd 4
1200.2.o.a.1199.3 4 40.27 even 4
1200.2.o.a.1199.4 4 120.83 odd 4
1200.2.o.b.1199.1 4 120.107 odd 4
1200.2.o.b.1199.2 4 40.3 even 4
1200.2.o.b.1199.3 4 40.13 odd 4
1200.2.o.b.1199.4 4 120.77 even 4