Properties

Label 4600.2.e.v.4049.3
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \(x^{10} + 18 x^{8} + 103 x^{6} + 239 x^{4} + 197 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.3
Root \(-1.64975i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.v.4049.8

$q$-expansion

\(f(q)\) \(=\) \(q-1.51466i q^{3} -3.49880i q^{7} +0.705809 q^{9} +O(q^{10})\) \(q-1.51466i q^{3} -3.49880i q^{7} +0.705809 q^{9} -4.35556 q^{11} -3.67906i q^{13} -5.18556i q^{17} -2.08538 q^{19} -5.29949 q^{21} -1.00000i q^{23} -5.61304i q^{27} +1.24902 q^{29} +4.82185 q^{31} +6.59718i q^{33} +1.04471i q^{37} -5.57253 q^{39} +9.05578 q^{41} -10.4964i q^{43} +12.8597i q^{47} -5.24162 q^{49} -7.85436 q^{51} -9.37352i q^{53} +3.15864i q^{57} -14.1570 q^{59} -9.29949 q^{61} -2.46949i q^{63} +4.44274i q^{67} -1.51466 q^{69} +2.45044 q^{71} +13.2197i q^{73} +15.2392i q^{77} -7.72666 q^{79} -6.38441 q^{81} +2.97086i q^{83} -1.89184i q^{87} +14.6241 q^{89} -12.8723 q^{91} -7.30345i q^{93} +9.37991i q^{97} -3.07419 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 8q^{9} + O(q^{10}) \) \( 10q - 8q^{9} - 8q^{11} - 8q^{19} - 12q^{21} + 22q^{29} + 8q^{31} - 62q^{39} - 16q^{41} + 4q^{49} - 10q^{51} - 46q^{59} - 52q^{61} - 6q^{69} - 4q^{71} - 86q^{79} - 6q^{81} - 30q^{89} - 38q^{91} - 74q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) − 1.51466i − 0.874489i −0.899343 0.437244i \(-0.855954\pi\)
0.899343 0.437244i \(-0.144046\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 3.49880i − 1.32242i −0.750199 0.661212i \(-0.770043\pi\)
0.750199 0.661212i \(-0.229957\pi\)
\(8\) 0 0
\(9\) 0.705809 0.235270
\(10\) 0 0
\(11\) −4.35556 −1.31325 −0.656625 0.754217i \(-0.728016\pi\)
−0.656625 + 0.754217i \(0.728016\pi\)
\(12\) 0 0
\(13\) − 3.67906i − 1.02039i −0.860059 0.510194i \(-0.829573\pi\)
0.860059 0.510194i \(-0.170427\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 5.18556i − 1.25768i −0.777533 0.628842i \(-0.783529\pi\)
0.777533 0.628842i \(-0.216471\pi\)
\(18\) 0 0
\(19\) −2.08538 −0.478419 −0.239209 0.970968i \(-0.576888\pi\)
−0.239209 + 0.970968i \(0.576888\pi\)
\(20\) 0 0
\(21\) −5.29949 −1.15644
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 5.61304i − 1.08023i
\(28\) 0 0
\(29\) 1.24902 0.231937 0.115968 0.993253i \(-0.463003\pi\)
0.115968 + 0.993253i \(0.463003\pi\)
\(30\) 0 0
\(31\) 4.82185 0.866029 0.433015 0.901387i \(-0.357450\pi\)
0.433015 + 0.901387i \(0.357450\pi\)
\(32\) 0 0
\(33\) 6.59718i 1.14842i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.04471i 0.171749i 0.996306 + 0.0858745i \(0.0273684\pi\)
−0.996306 + 0.0858745i \(0.972632\pi\)
\(38\) 0 0
\(39\) −5.57253 −0.892318
\(40\) 0 0
\(41\) 9.05578 1.41427 0.707137 0.707076i \(-0.249986\pi\)
0.707137 + 0.707076i \(0.249986\pi\)
\(42\) 0 0
\(43\) − 10.4964i − 1.60069i −0.599541 0.800344i \(-0.704650\pi\)
0.599541 0.800344i \(-0.295350\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.8597i 1.87577i 0.346940 + 0.937887i \(0.387221\pi\)
−0.346940 + 0.937887i \(0.612779\pi\)
\(48\) 0 0
\(49\) −5.24162 −0.748804
\(50\) 0 0
\(51\) −7.85436 −1.09983
\(52\) 0 0
\(53\) − 9.37352i − 1.28755i −0.765214 0.643776i \(-0.777367\pi\)
0.765214 0.643776i \(-0.222633\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 3.15864i 0.418372i
\(58\) 0 0
\(59\) −14.1570 −1.84309 −0.921543 0.388276i \(-0.873071\pi\)
−0.921543 + 0.388276i \(0.873071\pi\)
\(60\) 0 0
\(61\) −9.29949 −1.19068 −0.595339 0.803475i \(-0.702982\pi\)
−0.595339 + 0.803475i \(0.702982\pi\)
\(62\) 0 0
\(63\) − 2.46949i − 0.311126i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.44274i 0.542767i 0.962471 + 0.271384i \(0.0874812\pi\)
−0.962471 + 0.271384i \(0.912519\pi\)
\(68\) 0 0
\(69\) −1.51466 −0.182343
\(70\) 0 0
\(71\) 2.45044 0.290813 0.145407 0.989372i \(-0.453551\pi\)
0.145407 + 0.989372i \(0.453551\pi\)
\(72\) 0 0
\(73\) 13.2197i 1.54725i 0.633643 + 0.773625i \(0.281559\pi\)
−0.633643 + 0.773625i \(0.718441\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 15.2392i 1.73667i
\(78\) 0 0
\(79\) −7.72666 −0.869318 −0.434659 0.900595i \(-0.643131\pi\)
−0.434659 + 0.900595i \(0.643131\pi\)
\(80\) 0 0
\(81\) −6.38441 −0.709379
\(82\) 0 0
\(83\) 2.97086i 0.326095i 0.986618 + 0.163047i \(0.0521323\pi\)
−0.986618 + 0.163047i \(0.947868\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 1.89184i − 0.202826i
\(88\) 0 0
\(89\) 14.6241 1.55015 0.775076 0.631868i \(-0.217712\pi\)
0.775076 + 0.631868i \(0.217712\pi\)
\(90\) 0 0
\(91\) −12.8723 −1.34939
\(92\) 0 0
\(93\) − 7.30345i − 0.757333i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 9.37991i 0.952385i 0.879341 + 0.476193i \(0.157983\pi\)
−0.879341 + 0.476193i \(0.842017\pi\)
\(98\) 0 0
\(99\) −3.07419 −0.308968
\(100\) 0 0
\(101\) −1.02674 −0.102165 −0.0510824 0.998694i \(-0.516267\pi\)
−0.0510824 + 0.998694i \(0.516267\pi\)
\(102\) 0 0
\(103\) 1.71127i 0.168617i 0.996440 + 0.0843084i \(0.0268681\pi\)
−0.996440 + 0.0843084i \(0.973132\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.58418i 0.733191i 0.930380 + 0.366595i \(0.119477\pi\)
−0.930380 + 0.366595i \(0.880523\pi\)
\(108\) 0 0
\(109\) 8.95951 0.858165 0.429083 0.903265i \(-0.358837\pi\)
0.429083 + 0.903265i \(0.358837\pi\)
\(110\) 0 0
\(111\) 1.58238 0.150193
\(112\) 0 0
\(113\) − 18.8103i − 1.76952i −0.466047 0.884760i \(-0.654322\pi\)
0.466047 0.884760i \(-0.345678\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 2.59672i − 0.240066i
\(118\) 0 0
\(119\) −18.1433 −1.66319
\(120\) 0 0
\(121\) 7.97086 0.724624
\(122\) 0 0
\(123\) − 13.7164i − 1.23677i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 15.0385i − 1.33445i −0.744854 0.667227i \(-0.767481\pi\)
0.744854 0.667227i \(-0.232519\pi\)
\(128\) 0 0
\(129\) −15.8985 −1.39978
\(130\) 0 0
\(131\) 20.3503 1.77801 0.889007 0.457893i \(-0.151396\pi\)
0.889007 + 0.457893i \(0.151396\pi\)
\(132\) 0 0
\(133\) 7.29633i 0.632672i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.51494i 0.129430i 0.997904 + 0.0647152i \(0.0206139\pi\)
−0.997904 + 0.0647152i \(0.979386\pi\)
\(138\) 0 0
\(139\) −13.3658 −1.13367 −0.566837 0.823830i \(-0.691833\pi\)
−0.566837 + 0.823830i \(0.691833\pi\)
\(140\) 0 0
\(141\) 19.4780 1.64034
\(142\) 0 0
\(143\) 16.0244i 1.34002i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 7.93927i 0.654820i
\(148\) 0 0
\(149\) 10.6307 0.870901 0.435450 0.900213i \(-0.356589\pi\)
0.435450 + 0.900213i \(0.356589\pi\)
\(150\) 0 0
\(151\) −22.2918 −1.81408 −0.907041 0.421042i \(-0.861664\pi\)
−0.907041 + 0.421042i \(0.861664\pi\)
\(152\) 0 0
\(153\) − 3.66002i − 0.295895i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 2.95590i − 0.235906i −0.993019 0.117953i \(-0.962367\pi\)
0.993019 0.117953i \(-0.0376332\pi\)
\(158\) 0 0
\(159\) −14.1977 −1.12595
\(160\) 0 0
\(161\) −3.49880 −0.275744
\(162\) 0 0
\(163\) − 4.40543i − 0.345060i −0.985004 0.172530i \(-0.944806\pi\)
0.985004 0.172530i \(-0.0551941\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 14.9252i 1.15495i 0.816409 + 0.577474i \(0.195962\pi\)
−0.816409 + 0.577474i \(0.804038\pi\)
\(168\) 0 0
\(169\) −0.535514 −0.0411934
\(170\) 0 0
\(171\) −1.47188 −0.112557
\(172\) 0 0
\(173\) − 18.2696i − 1.38901i −0.719487 0.694506i \(-0.755623\pi\)
0.719487 0.694506i \(-0.244377\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 21.4430i 1.61176i
\(178\) 0 0
\(179\) 6.20356 0.463676 0.231838 0.972754i \(-0.425526\pi\)
0.231838 + 0.972754i \(0.425526\pi\)
\(180\) 0 0
\(181\) 5.74043 0.426683 0.213341 0.976978i \(-0.431565\pi\)
0.213341 + 0.976978i \(0.431565\pi\)
\(182\) 0 0
\(183\) 14.0856i 1.04123i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 22.5860i 1.65165i
\(188\) 0 0
\(189\) −19.6389 −1.42852
\(190\) 0 0
\(191\) −13.4671 −0.974445 −0.487222 0.873278i \(-0.661990\pi\)
−0.487222 + 0.873278i \(0.661990\pi\)
\(192\) 0 0
\(193\) 0.295578i 0.0212762i 0.999943 + 0.0106381i \(0.00338628\pi\)
−0.999943 + 0.0106381i \(0.996614\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 11.6206i 0.827932i 0.910292 + 0.413966i \(0.135857\pi\)
−0.910292 + 0.413966i \(0.864143\pi\)
\(198\) 0 0
\(199\) −4.46949 −0.316833 −0.158417 0.987372i \(-0.550639\pi\)
−0.158417 + 0.987372i \(0.550639\pi\)
\(200\) 0 0
\(201\) 6.72924 0.474644
\(202\) 0 0
\(203\) − 4.37007i − 0.306719i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 0.705809i − 0.0490571i
\(208\) 0 0
\(209\) 9.08299 0.628283
\(210\) 0 0
\(211\) −15.0878 −1.03869 −0.519343 0.854566i \(-0.673823\pi\)
−0.519343 + 0.854566i \(0.673823\pi\)
\(212\) 0 0
\(213\) − 3.71158i − 0.254313i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 16.8707i − 1.14526i
\(218\) 0 0
\(219\) 20.0234 1.35305
\(220\) 0 0
\(221\) −19.0780 −1.28333
\(222\) 0 0
\(223\) 25.2033i 1.68774i 0.536548 + 0.843870i \(0.319728\pi\)
−0.536548 + 0.843870i \(0.680272\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 20.1385i − 1.33664i −0.743875 0.668319i \(-0.767014\pi\)
0.743875 0.668319i \(-0.232986\pi\)
\(228\) 0 0
\(229\) −9.72411 −0.642587 −0.321294 0.946980i \(-0.604118\pi\)
−0.321294 + 0.946980i \(0.604118\pi\)
\(230\) 0 0
\(231\) 23.0822 1.51870
\(232\) 0 0
\(233\) 6.37957i 0.417940i 0.977922 + 0.208970i \(0.0670110\pi\)
−0.977922 + 0.208970i \(0.932989\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 11.7033i 0.760208i
\(238\) 0 0
\(239\) 3.07706 0.199039 0.0995194 0.995036i \(-0.468269\pi\)
0.0995194 + 0.995036i \(0.468269\pi\)
\(240\) 0 0
\(241\) −28.6341 −1.84448 −0.922242 0.386612i \(-0.873645\pi\)
−0.922242 + 0.386612i \(0.873645\pi\)
\(242\) 0 0
\(243\) − 7.16891i − 0.459886i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 7.67225i 0.488173i
\(248\) 0 0
\(249\) 4.49984 0.285166
\(250\) 0 0
\(251\) 0.728308 0.0459704 0.0229852 0.999736i \(-0.492683\pi\)
0.0229852 + 0.999736i \(0.492683\pi\)
\(252\) 0 0
\(253\) 4.35556i 0.273831i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 29.1681i 1.81945i 0.415206 + 0.909727i \(0.363710\pi\)
−0.415206 + 0.909727i \(0.636290\pi\)
\(258\) 0 0
\(259\) 3.65523 0.227125
\(260\) 0 0
\(261\) 0.881568 0.0545677
\(262\) 0 0
\(263\) − 2.79631i − 0.172428i −0.996277 0.0862139i \(-0.972523\pi\)
0.996277 0.0862139i \(-0.0274769\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 22.1505i − 1.35559i
\(268\) 0 0
\(269\) 0.662454 0.0403905 0.0201953 0.999796i \(-0.493571\pi\)
0.0201953 + 0.999796i \(0.493571\pi\)
\(270\) 0 0
\(271\) 14.5266 0.882428 0.441214 0.897402i \(-0.354548\pi\)
0.441214 + 0.897402i \(0.354548\pi\)
\(272\) 0 0
\(273\) 19.4972i 1.18002i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 1.15432i − 0.0693562i −0.999399 0.0346781i \(-0.988959\pi\)
0.999399 0.0346781i \(-0.0110406\pi\)
\(278\) 0 0
\(279\) 3.40330 0.203750
\(280\) 0 0
\(281\) −14.2977 −0.852930 −0.426465 0.904504i \(-0.640241\pi\)
−0.426465 + 0.904504i \(0.640241\pi\)
\(282\) 0 0
\(283\) − 31.2654i − 1.85854i −0.369407 0.929268i \(-0.620439\pi\)
0.369407 0.929268i \(-0.379561\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 31.6844i − 1.87027i
\(288\) 0 0
\(289\) −9.89006 −0.581768
\(290\) 0 0
\(291\) 14.2074 0.832850
\(292\) 0 0
\(293\) − 4.59477i − 0.268429i −0.990952 0.134215i \(-0.957149\pi\)
0.990952 0.134215i \(-0.0428512\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 24.4479i 1.41861i
\(298\) 0 0
\(299\) −3.67906 −0.212766
\(300\) 0 0
\(301\) −36.7249 −2.11679
\(302\) 0 0
\(303\) 1.55517i 0.0893420i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 9.96103i 0.568506i 0.958749 + 0.284253i \(0.0917455\pi\)
−0.958749 + 0.284253i \(0.908254\pi\)
\(308\) 0 0
\(309\) 2.59200 0.147453
\(310\) 0 0
\(311\) −12.7520 −0.723102 −0.361551 0.932352i \(-0.617753\pi\)
−0.361551 + 0.932352i \(0.617753\pi\)
\(312\) 0 0
\(313\) 19.5585i 1.10551i 0.833343 + 0.552756i \(0.186424\pi\)
−0.833343 + 0.552756i \(0.813576\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 32.5493i 1.82815i 0.405542 + 0.914076i \(0.367083\pi\)
−0.405542 + 0.914076i \(0.632917\pi\)
\(318\) 0 0
\(319\) −5.44017 −0.304591
\(320\) 0 0
\(321\) 11.4874 0.641167
\(322\) 0 0
\(323\) 10.8139i 0.601700i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 13.5706i − 0.750456i
\(328\) 0 0
\(329\) 44.9934 2.48057
\(330\) 0 0
\(331\) 23.9704 1.31753 0.658767 0.752347i \(-0.271078\pi\)
0.658767 + 0.752347i \(0.271078\pi\)
\(332\) 0 0
\(333\) 0.737364i 0.0404073i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 5.88246i 0.320438i 0.987082 + 0.160219i \(0.0512200\pi\)
−0.987082 + 0.160219i \(0.948780\pi\)
\(338\) 0 0
\(339\) −28.4911 −1.54743
\(340\) 0 0
\(341\) −21.0018 −1.13731
\(342\) 0 0
\(343\) − 6.15221i − 0.332188i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 25.7454i 1.38209i 0.722814 + 0.691043i \(0.242849\pi\)
−0.722814 + 0.691043i \(0.757151\pi\)
\(348\) 0 0
\(349\) 17.7758 0.951517 0.475758 0.879576i \(-0.342174\pi\)
0.475758 + 0.879576i \(0.342174\pi\)
\(350\) 0 0
\(351\) −20.6507 −1.10225
\(352\) 0 0
\(353\) − 20.4426i − 1.08805i −0.839068 0.544026i \(-0.816899\pi\)
0.839068 0.544026i \(-0.183101\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 27.4809i 1.45444i
\(358\) 0 0
\(359\) 14.5323 0.766987 0.383494 0.923543i \(-0.374721\pi\)
0.383494 + 0.923543i \(0.374721\pi\)
\(360\) 0 0
\(361\) −14.6512 −0.771115
\(362\) 0 0
\(363\) − 12.0731i − 0.633675i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 12.2205i − 0.637906i −0.947771 0.318953i \(-0.896669\pi\)
0.947771 0.318953i \(-0.103331\pi\)
\(368\) 0 0
\(369\) 6.39165 0.332736
\(370\) 0 0
\(371\) −32.7961 −1.70269
\(372\) 0 0
\(373\) 1.78846i 0.0926029i 0.998928 + 0.0463015i \(0.0147435\pi\)
−0.998928 + 0.0463015i \(0.985257\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 4.59522i − 0.236666i
\(378\) 0 0
\(379\) −7.02119 −0.360654 −0.180327 0.983607i \(-0.557716\pi\)
−0.180327 + 0.983607i \(0.557716\pi\)
\(380\) 0 0
\(381\) −22.7782 −1.16697
\(382\) 0 0
\(383\) − 13.0896i − 0.668846i −0.942423 0.334423i \(-0.891459\pi\)
0.942423 0.334423i \(-0.108541\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 7.40846i − 0.376593i
\(388\) 0 0
\(389\) 21.5395 1.09210 0.546048 0.837754i \(-0.316132\pi\)
0.546048 + 0.837754i \(0.316132\pi\)
\(390\) 0 0
\(391\) −5.18556 −0.262245
\(392\) 0 0
\(393\) − 30.8238i − 1.55485i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 31.2045i − 1.56611i −0.621953 0.783055i \(-0.713660\pi\)
0.621953 0.783055i \(-0.286340\pi\)
\(398\) 0 0
\(399\) 11.0515 0.553265
\(400\) 0 0
\(401\) 13.7592 0.687100 0.343550 0.939134i \(-0.388371\pi\)
0.343550 + 0.939134i \(0.388371\pi\)
\(402\) 0 0
\(403\) − 17.7399i − 0.883687i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 4.55028i − 0.225549i
\(408\) 0 0
\(409\) −21.9730 −1.08649 −0.543247 0.839573i \(-0.682805\pi\)
−0.543247 + 0.839573i \(0.682805\pi\)
\(410\) 0 0
\(411\) 2.29462 0.113185
\(412\) 0 0
\(413\) 49.5326i 2.43734i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 20.2447i 0.991385i
\(418\) 0 0
\(419\) 13.3024 0.649867 0.324934 0.945737i \(-0.394658\pi\)
0.324934 + 0.945737i \(0.394658\pi\)
\(420\) 0 0
\(421\) −7.32759 −0.357125 −0.178562 0.983929i \(-0.557145\pi\)
−0.178562 + 0.983929i \(0.557145\pi\)
\(422\) 0 0
\(423\) 9.07646i 0.441313i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 32.5371i 1.57458i
\(428\) 0 0
\(429\) 24.2715 1.17184
\(430\) 0 0
\(431\) 29.4301 1.41760 0.708800 0.705409i \(-0.249237\pi\)
0.708800 + 0.705409i \(0.249237\pi\)
\(432\) 0 0
\(433\) − 10.5491i − 0.506960i −0.967341 0.253480i \(-0.918425\pi\)
0.967341 0.253480i \(-0.0815752\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.08538i 0.0997572i
\(438\) 0 0
\(439\) −1.04921 −0.0500761 −0.0250380 0.999686i \(-0.507971\pi\)
−0.0250380 + 0.999686i \(0.507971\pi\)
\(440\) 0 0
\(441\) −3.69958 −0.176171
\(442\) 0 0
\(443\) − 9.57098i − 0.454731i −0.973810 0.227365i \(-0.926989\pi\)
0.973810 0.227365i \(-0.0730112\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 16.1019i − 0.761593i
\(448\) 0 0
\(449\) −8.10079 −0.382300 −0.191150 0.981561i \(-0.561222\pi\)
−0.191150 + 0.981561i \(0.561222\pi\)
\(450\) 0 0
\(451\) −39.4429 −1.85730
\(452\) 0 0
\(453\) 33.7645i 1.58639i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 34.5867i 1.61790i 0.587880 + 0.808948i \(0.299963\pi\)
−0.587880 + 0.808948i \(0.700037\pi\)
\(458\) 0 0
\(459\) −29.1068 −1.35859
\(460\) 0 0
\(461\) −23.3497 −1.08750 −0.543752 0.839246i \(-0.682997\pi\)
−0.543752 + 0.839246i \(0.682997\pi\)
\(462\) 0 0
\(463\) − 36.7512i − 1.70797i −0.520295 0.853987i \(-0.674178\pi\)
0.520295 0.853987i \(-0.325822\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 29.5786i 1.36874i 0.729137 + 0.684368i \(0.239922\pi\)
−0.729137 + 0.684368i \(0.760078\pi\)
\(468\) 0 0
\(469\) 15.5443 0.717768
\(470\) 0 0
\(471\) −4.47717 −0.206297
\(472\) 0 0
\(473\) 45.7177i 2.10210i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 6.61591i − 0.302922i
\(478\) 0 0
\(479\) −7.23409 −0.330534 −0.165267 0.986249i \(-0.552849\pi\)
−0.165267 + 0.986249i \(0.552849\pi\)
\(480\) 0 0
\(481\) 3.84355 0.175251
\(482\) 0 0
\(483\) 5.29949i 0.241135i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 8.99593i 0.407645i 0.979008 + 0.203822i \(0.0653365\pi\)
−0.979008 + 0.203822i \(0.934664\pi\)
\(488\) 0 0
\(489\) −6.67272 −0.301751
\(490\) 0 0
\(491\) 9.71762 0.438550 0.219275 0.975663i \(-0.429631\pi\)
0.219275 + 0.975663i \(0.429631\pi\)
\(492\) 0 0
\(493\) − 6.47686i − 0.291703i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 8.57360i − 0.384578i
\(498\) 0 0
\(499\) −33.3758 −1.49410 −0.747052 0.664765i \(-0.768532\pi\)
−0.747052 + 0.664765i \(0.768532\pi\)
\(500\) 0 0
\(501\) 22.6066 1.00999
\(502\) 0 0
\(503\) − 22.1147i − 0.986046i −0.870016 0.493023i \(-0.835892\pi\)
0.870016 0.493023i \(-0.164108\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.811121i 0.0360232i
\(508\) 0 0
\(509\) −10.8182 −0.479507 −0.239753 0.970834i \(-0.577067\pi\)
−0.239753 + 0.970834i \(0.577067\pi\)
\(510\) 0 0
\(511\) 46.2532 2.04612
\(512\) 0 0
\(513\) 11.7053i 0.516802i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 56.0110i − 2.46336i
\(518\) 0 0
\(519\) −27.6722 −1.21467
\(520\) 0 0
\(521\) −3.20820 −0.140554 −0.0702770 0.997528i \(-0.522388\pi\)
−0.0702770 + 0.997528i \(0.522388\pi\)
\(522\) 0 0
\(523\) − 23.7159i − 1.03702i −0.855071 0.518512i \(-0.826486\pi\)
0.855071 0.518512i \(-0.173514\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 25.0040i − 1.08919i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) −9.99214 −0.433622
\(532\) 0 0
\(533\) − 33.3168i − 1.44311i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 9.39628i − 0.405479i
\(538\) 0 0
\(539\) 22.8302 0.983366
\(540\) 0 0
\(541\) −6.46616 −0.278002 −0.139001 0.990292i \(-0.544389\pi\)
−0.139001 + 0.990292i \(0.544389\pi\)
\(542\) 0 0
\(543\) − 8.69479i − 0.373129i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 7.48526i 0.320047i 0.987113 + 0.160023i \(0.0511570\pi\)
−0.987113 + 0.160023i \(0.948843\pi\)
\(548\) 0 0
\(549\) −6.56366 −0.280130
\(550\) 0 0
\(551\) −2.60468 −0.110963
\(552\) 0 0
\(553\) 27.0341i 1.14961i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 22.2922i − 0.944553i −0.881451 0.472276i \(-0.843432\pi\)
0.881451 0.472276i \(-0.156568\pi\)
\(558\) 0 0
\(559\) −38.6170 −1.63332
\(560\) 0 0
\(561\) 34.2101 1.44435
\(562\) 0 0
\(563\) 9.93402i 0.418669i 0.977844 + 0.209335i \(0.0671298\pi\)
−0.977844 + 0.209335i \(0.932870\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 22.3378i 0.938099i
\(568\) 0 0
\(569\) 26.8239 1.12452 0.562258 0.826962i \(-0.309933\pi\)
0.562258 + 0.826962i \(0.309933\pi\)
\(570\) 0 0
\(571\) 30.0260 1.25655 0.628274 0.777992i \(-0.283762\pi\)
0.628274 + 0.777992i \(0.283762\pi\)
\(572\) 0 0
\(573\) 20.3981i 0.852141i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 23.5058i − 0.978559i −0.872127 0.489279i \(-0.837260\pi\)
0.872127 0.489279i \(-0.162740\pi\)
\(578\) 0 0
\(579\) 0.447701 0.0186058
\(580\) 0 0
\(581\) 10.3945 0.431235
\(582\) 0 0
\(583\) 40.8269i 1.69088i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 5.05879i − 0.208799i −0.994535 0.104399i \(-0.966708\pi\)
0.994535 0.104399i \(-0.0332920\pi\)
\(588\) 0 0
\(589\) −10.0554 −0.414325
\(590\) 0 0
\(591\) 17.6012 0.724017
\(592\) 0 0
\(593\) − 15.0522i − 0.618120i −0.951043 0.309060i \(-0.899986\pi\)
0.951043 0.309060i \(-0.100014\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 6.76975i 0.277067i
\(598\) 0 0
\(599\) 15.9863 0.653181 0.326590 0.945166i \(-0.394100\pi\)
0.326590 + 0.945166i \(0.394100\pi\)
\(600\) 0 0
\(601\) 32.7941 1.33770 0.668851 0.743397i \(-0.266787\pi\)
0.668851 + 0.743397i \(0.266787\pi\)
\(602\) 0 0
\(603\) 3.13573i 0.127697i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 38.4884i − 1.56220i −0.624409 0.781098i \(-0.714660\pi\)
0.624409 0.781098i \(-0.285340\pi\)
\(608\) 0 0
\(609\) −6.61916 −0.268222
\(610\) 0 0
\(611\) 47.3115 1.91402
\(612\) 0 0
\(613\) − 6.47626i − 0.261574i −0.991411 0.130787i \(-0.958250\pi\)
0.991411 0.130787i \(-0.0417503\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 33.8193i 1.36152i 0.732509 + 0.680758i \(0.238349\pi\)
−0.732509 + 0.680758i \(0.761651\pi\)
\(618\) 0 0
\(619\) −2.02686 −0.0814663 −0.0407331 0.999170i \(-0.512969\pi\)
−0.0407331 + 0.999170i \(0.512969\pi\)
\(620\) 0 0
\(621\) −5.61304 −0.225243
\(622\) 0 0
\(623\) − 51.1669i − 2.04996i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 13.7576i − 0.549427i
\(628\) 0 0
\(629\) 5.41740 0.216006
\(630\) 0 0
\(631\) −3.45649 −0.137601 −0.0688003 0.997630i \(-0.521917\pi\)
−0.0688003 + 0.997630i \(0.521917\pi\)
\(632\) 0 0
\(633\) 22.8529i 0.908319i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 19.2843i 0.764071i
\(638\) 0 0
\(639\) 1.72954 0.0684195
\(640\) 0 0
\(641\) 47.2630 1.86678 0.933388 0.358869i \(-0.116837\pi\)
0.933388 + 0.358869i \(0.116837\pi\)
\(642\) 0 0
\(643\) 23.6052i 0.930899i 0.885075 + 0.465449i \(0.154107\pi\)
−0.885075 + 0.465449i \(0.845893\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 20.3090i − 0.798430i −0.916857 0.399215i \(-0.869283\pi\)
0.916857 0.399215i \(-0.130717\pi\)
\(648\) 0 0
\(649\) 61.6617 2.42043
\(650\) 0 0
\(651\) −25.5533 −1.00151
\(652\) 0 0
\(653\) 15.8982i 0.622146i 0.950386 + 0.311073i \(0.100688\pi\)
−0.950386 + 0.311073i \(0.899312\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 9.33059i 0.364021i
\(658\) 0 0
\(659\) −13.6170 −0.530441 −0.265221 0.964188i \(-0.585445\pi\)
−0.265221 + 0.964188i \(0.585445\pi\)
\(660\) 0 0
\(661\) 27.8049 1.08148 0.540742 0.841188i \(-0.318143\pi\)
0.540742 + 0.841188i \(0.318143\pi\)
\(662\) 0 0
\(663\) 28.8967i 1.12225i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 1.24902i − 0.0483622i
\(668\) 0 0
\(669\) 38.1744 1.47591
\(670\) 0 0
\(671\) 40.5045 1.56366
\(672\) 0 0
\(673\) 4.92071i 0.189679i 0.995493 + 0.0948396i \(0.0302338\pi\)
−0.995493 + 0.0948396i \(0.969766\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 48.4096i − 1.86053i −0.366886 0.930266i \(-0.619576\pi\)
0.366886 0.930266i \(-0.380424\pi\)
\(678\) 0 0
\(679\) 32.8184 1.25946
\(680\) 0 0
\(681\) −30.5029 −1.16887
\(682\) 0 0
\(683\) 36.1788i 1.38434i 0.721732 + 0.692172i \(0.243346\pi\)
−0.721732 + 0.692172i \(0.756654\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 14.7287i 0.561935i
\(688\) 0 0
\(689\) −34.4858 −1.31380
\(690\) 0 0
\(691\) 32.1568 1.22330 0.611651 0.791127i \(-0.290506\pi\)
0.611651 + 0.791127i \(0.290506\pi\)
\(692\) 0 0
\(693\) 10.7560i 0.408586i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 46.9593i − 1.77871i
\(698\) 0 0
\(699\) 9.66287 0.365483
\(700\) 0 0
\(701\) 17.9408 0.677614 0.338807 0.940856i \(-0.389977\pi\)
0.338807 + 0.940856i \(0.389977\pi\)
\(702\) 0 0
\(703\) − 2.17861i − 0.0821680i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3.59238i 0.135105i
\(708\) 0 0
\(709\) −45.4153 −1.70561 −0.852804 0.522231i \(-0.825100\pi\)
−0.852804 + 0.522231i \(0.825100\pi\)
\(710\) 0 0
\(711\) −5.45355 −0.204524
\(712\) 0 0
\(713\) − 4.82185i − 0.180580i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 4.66070i − 0.174057i
\(718\) 0 0
\(719\) −5.90426 −0.220192 −0.110096 0.993921i \(-0.535116\pi\)
−0.110096 + 0.993921i \(0.535116\pi\)
\(720\) 0 0
\(721\) 5.98741 0.222983
\(722\) 0 0
\(723\) 43.3709i 1.61298i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 34.8037i − 1.29080i −0.763846 0.645398i \(-0.776691\pi\)
0.763846 0.645398i \(-0.223309\pi\)
\(728\) 0 0
\(729\) −30.0117 −1.11154
\(730\) 0 0
\(731\) −54.4298 −2.01316
\(732\) 0 0
\(733\) 21.6346i 0.799093i 0.916713 + 0.399547i \(0.130832\pi\)
−0.916713 + 0.399547i \(0.869168\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 19.3506i − 0.712789i
\(738\) 0 0
\(739\) 4.53418 0.166792 0.0833962 0.996516i \(-0.473423\pi\)
0.0833962 + 0.996516i \(0.473423\pi\)
\(740\) 0 0
\(741\) 11.6208 0.426902
\(742\) 0 0
\(743\) − 17.9137i − 0.657192i −0.944471 0.328596i \(-0.893425\pi\)
0.944471 0.328596i \(-0.106575\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 2.09686i 0.0767201i
\(748\) 0 0
\(749\) 26.5356 0.969588
\(750\) 0 0
\(751\) −25.1812 −0.918875 −0.459438 0.888210i \(-0.651949\pi\)
−0.459438 + 0.888210i \(0.651949\pi\)
\(752\) 0 0
\(753\) − 1.10314i − 0.0402006i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 30.7079i 1.11610i 0.829809 + 0.558048i \(0.188450\pi\)
−0.829809 + 0.558048i \(0.811550\pi\)
\(758\) 0 0
\(759\) 6.59718 0.239462
\(760\) 0 0
\(761\) 9.58720 0.347536 0.173768 0.984787i \(-0.444406\pi\)
0.173768 + 0.984787i \(0.444406\pi\)
\(762\) 0 0
\(763\) − 31.3476i − 1.13486i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 52.0846i 1.88066i
\(768\) 0 0
\(769\) 6.31547 0.227742 0.113871 0.993496i \(-0.463675\pi\)
0.113871 + 0.993496i \(0.463675\pi\)
\(770\) 0 0
\(771\) 44.1797 1.59109
\(772\) 0 0
\(773\) − 47.4787i − 1.70769i −0.520528 0.853845i \(-0.674265\pi\)
0.520528 0.853845i \(-0.325735\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 5.53642i − 0.198618i
\(778\) 0 0
\(779\) −18.8847 −0.676616
\(780\) 0 0
\(781\) −10.6730 −0.381910
\(782\) 0 0
\(783\) − 7.01078i − 0.250545i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 47.0588i − 1.67747i −0.544543 0.838733i \(-0.683297\pi\)
0.544543 0.838733i \(-0.316703\pi\)
\(788\) 0 0
\(789\) −4.23546 −0.150786
\(790\) 0 0
\(791\) −65.8134 −2.34005
\(792\) 0 0
\(793\) 34.2134i 1.21495i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 22.8819i − 0.810517i −0.914202 0.405259i \(-0.867181\pi\)
0.914202 0.405259i \(-0.132819\pi\)
\(798\) 0 0
\(799\) 66.6846 2.35913
\(800\) 0 0
\(801\) 10.3218 0.364704
\(802\) 0 0
\(803\) − 57.5792i − 2.03193i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 1.00339i − 0.0353211i
\(808\) 0 0
\(809\) −21.6918 −0.762643 −0.381322 0.924442i \(-0.624531\pi\)
−0.381322 + 0.924442i \(0.624531\pi\)
\(810\) 0 0
\(811\) 5.84992 0.205418 0.102709 0.994711i \(-0.467249\pi\)
0.102709 + 0.994711i \(0.467249\pi\)
\(812\) 0 0
\(813\) − 22.0028i − 0.771674i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 21.8890i 0.765799i
\(818\) 0 0
\(819\) −9.08540 −0.317469
\(820\) 0 0
\(821\) 5.98613 0.208917 0.104459 0.994529i \(-0.466689\pi\)
0.104459 + 0.994529i \(0.466689\pi\)
\(822\) 0 0
\(823\) − 27.4906i − 0.958261i −0.877744 0.479131i \(-0.840952\pi\)
0.877744 0.479131i \(-0.159048\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 2.85519i − 0.0992848i −0.998767 0.0496424i \(-0.984192\pi\)
0.998767 0.0496424i \(-0.0158082\pi\)
\(828\) 0 0
\(829\) 7.63982 0.265342 0.132671 0.991160i \(-0.457645\pi\)
0.132671 + 0.991160i \(0.457645\pi\)
\(830\) 0 0
\(831\) −1.74840 −0.0606512
\(832\) 0 0
\(833\) 27.1808i 0.941758i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 27.0652i − 0.935510i
\(838\) 0 0
\(839\) 8.41078 0.290372 0.145186 0.989404i \(-0.453622\pi\)
0.145186 + 0.989404i \(0.453622\pi\)
\(840\) 0 0
\(841\) −27.4400 −0.946205
\(842\) 0 0
\(843\) 21.6561i 0.745877i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 27.8885i − 0.958260i
\(848\) 0 0
\(849\) −47.3564 −1.62527
\(850\) 0 0
\(851\) 1.04471 0.0358121
\(852\) 0 0
\(853\) − 35.6430i − 1.22039i −0.792250 0.610197i \(-0.791090\pi\)
0.792250 0.610197i \(-0.208910\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2.66046i 0.0908794i 0.998967 + 0.0454397i \(0.0144689\pi\)
−0.998967 + 0.0454397i \(0.985531\pi\)
\(858\) 0 0
\(859\) −14.0909 −0.480774 −0.240387 0.970677i \(-0.577274\pi\)
−0.240387 + 0.970677i \(0.577274\pi\)
\(860\) 0 0
\(861\) −47.9910 −1.63553
\(862\) 0 0
\(863\) − 48.1228i − 1.63812i −0.573708 0.819060i \(-0.694496\pi\)
0.573708 0.819060i \(-0.305504\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 14.9801i 0.508750i
\(868\) 0 0
\(869\) 33.6539 1.14163
\(870\) 0 0
\(871\) 16.3451 0.553834
\(872\) 0 0
\(873\) 6.62042i 0.224067i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 26.3172i − 0.888668i −0.895861 0.444334i \(-0.853440\pi\)
0.895861 0.444334i \(-0.146560\pi\)
\(878\) 0 0
\(879\) −6.95951 −0.234738
\(880\) 0 0
\(881\) 7.59556 0.255901 0.127951 0.991781i \(-0.459160\pi\)
0.127951 + 0.991781i \(0.459160\pi\)
\(882\) 0 0
\(883\) − 11.8772i − 0.399699i −0.979827 0.199849i \(-0.935955\pi\)
0.979827 0.199849i \(-0.0640453\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 35.9421i − 1.20682i −0.797432 0.603409i \(-0.793809\pi\)
0.797432 0.603409i \(-0.206191\pi\)
\(888\) 0 0
\(889\) −52.6169 −1.76471
\(890\) 0 0
\(891\) 27.8076 0.931591
\(892\) 0 0
\(893\) − 26.8173i − 0.897406i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 5.57253i 0.186061i
\(898\) 0 0
\(899\) 6.02258 0.200864
\(900\) 0 0
\(901\) −48.6070 −1.61933
\(902\) 0 0
\(903\) 55.6257i 1.85111i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 56.9912i 1.89236i 0.323637 + 0.946181i \(0.395094\pi\)
−0.323637 + 0.946181i \(0.604906\pi\)
\(908\) 0 0
\(909\) −0.724685 −0.0240363
\(910\) 0 0
\(911\) −36.5339 −1.21042 −0.605212 0.796065i \(-0.706912\pi\)
−0.605212 + 0.796065i \(0.706912\pi\)
\(912\) 0 0
\(913\) − 12.9398i − 0.428243i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 71.2017i − 2.35129i
\(918\) 0 0
\(919\) 1.65041 0.0544419 0.0272210 0.999629i \(-0.491334\pi\)
0.0272210 + 0.999629i \(0.491334\pi\)
\(920\) 0 0
\(921\) 15.0876 0.497152
\(922\) 0 0
\(923\) − 9.01531i − 0.296743i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1.20783i 0.0396704i
\(928\) 0 0
\(929\) −3.24062 −0.106321 −0.0531606 0.998586i \(-0.516930\pi\)
−0.0531606 + 0.998586i \(0.516930\pi\)
\(930\) 0 0
\(931\) 10.9308 0.358242
\(932\) 0 0
\(933\) 19.3150i 0.632344i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 4.26930i − 0.139472i −0.997565 0.0697360i \(-0.977784\pi\)
0.997565 0.0697360i \(-0.0222157\pi\)
\(938\) 0 0
\(939\) 29.6244 0.966757
\(940\) 0 0
\(941\) −57.0889 −1.86105 −0.930523 0.366234i \(-0.880647\pi\)
−0.930523 + 0.366234i \(0.880647\pi\)
\(942\) 0 0
\(943\) − 9.05578i − 0.294897i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 50.1371i − 1.62924i −0.579997 0.814618i \(-0.696946\pi\)
0.579997 0.814618i \(-0.303054\pi\)
\(948\) 0 0
\(949\) 48.6362 1.57880
\(950\) 0 0
\(951\) 49.3011 1.59870
\(952\) 0 0
\(953\) − 6.17257i − 0.199949i −0.994990 0.0999745i \(-0.968124\pi\)
0.994990 0.0999745i \(-0.0318761\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 8.24000i 0.266361i
\(958\) 0 0
\(959\) 5.30049 0.171162
\(960\) 0 0
\(961\) −7.74979 −0.249993
\(962\) 0 0
\(963\) 5.35298i 0.172497i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 7.04066i 0.226412i 0.993572 + 0.113206i \(0.0361121\pi\)
−0.993572 + 0.113206i \(0.963888\pi\)
\(968\) 0 0
\(969\) 16.3793 0.526180
\(970\) 0 0
\(971\) −5.91622 −0.189861 −0.0949303 0.995484i \(-0.530263\pi\)
−0.0949303 + 0.995484i \(0.530263\pi\)
\(972\) 0 0
\(973\) 46.7644i 1.49920i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 14.7695i − 0.472517i −0.971690 0.236259i \(-0.924079\pi\)
0.971690 0.236259i \(-0.0759213\pi\)
\(978\) 0 0
\(979\) −63.6961 −2.03574
\(980\) 0 0
\(981\) 6.32370 0.201900
\(982\) 0 0
\(983\) 22.8892i 0.730054i 0.930997 + 0.365027i \(0.118940\pi\)
−0.930997 + 0.365027i \(0.881060\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 68.1497i − 2.16923i
\(988\) 0 0
\(989\) −10.4964 −0.333766
\(990\) 0 0
\(991\) 22.4611 0.713500 0.356750 0.934200i \(-0.383885\pi\)
0.356750 + 0.934200i \(0.383885\pi\)
\(992\) 0 0
\(993\) − 36.3070i − 1.15217i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 43.9293i − 1.39125i −0.718403 0.695627i \(-0.755127\pi\)
0.718403 0.695627i \(-0.244873\pi\)
\(998\) 0 0
\(999\) 5.86398 0.185528
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 4600.2.e.v.4049.3 10
5.2 odd 4 4600.2.a.bc.1.2 5
5.3 odd 4 4600.2.a.bg.1.4 yes 5
5.4 even 2 inner 4600.2.e.v.4049.8 10
20.3 even 4 9200.2.a.cs.1.2 5
20.7 even 4 9200.2.a.cw.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.bc.1.2 5 5.2 odd 4
4600.2.a.bg.1.4 yes 5 5.3 odd 4
4600.2.e.v.4049.3 10 1.1 even 1 trivial
4600.2.e.v.4049.8 10 5.4 even 2 inner
9200.2.a.cs.1.2 5 20.3 even 4
9200.2.a.cw.1.4 5 20.7 even 4