Properties

Label 7448.2.a.bq.1.5
Level $7448$
Weight $2$
Character 7448.1
Self dual yes
Analytic conductor $59.473$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7448 = 2^{3} \cdot 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7448.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(59.4725794254\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 15 x^{6} - x^{5} + 66 x^{4} + 4 x^{3} - 76 x^{2} + 8 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 1064)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(0.684189\) of defining polynomial
Character \(\chi\) \(=\) 7448.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.684189 q^{3} -3.65096 q^{5} -2.53189 q^{9} +O(q^{10})\) \(q+0.684189 q^{3} -3.65096 q^{5} -2.53189 q^{9} +2.37730 q^{11} +5.46860 q^{13} -2.49795 q^{15} -4.86063 q^{17} +1.00000 q^{19} -1.77549 q^{23} +8.32954 q^{25} -3.78485 q^{27} -6.04283 q^{29} +1.69856 q^{31} +1.62652 q^{33} +5.62635 q^{37} +3.74155 q^{39} -2.00000 q^{41} +3.10972 q^{43} +9.24382 q^{45} -5.23017 q^{47} -3.32559 q^{51} -6.46513 q^{53} -8.67942 q^{55} +0.684189 q^{57} -7.52266 q^{59} -7.75592 q^{61} -19.9657 q^{65} -2.62090 q^{67} -1.21477 q^{69} +11.1529 q^{71} +14.5305 q^{73} +5.69898 q^{75} +0.961803 q^{79} +5.00610 q^{81} -15.0476 q^{83} +17.7460 q^{85} -4.13444 q^{87} +12.0321 q^{89} +1.16213 q^{93} -3.65096 q^{95} -6.26438 q^{97} -6.01904 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{5} + 6 q^{9} + O(q^{10}) \) \( 8 q - q^{5} + 6 q^{9} + 9 q^{11} + 8 q^{15} + 4 q^{17} + 8 q^{19} + 25 q^{23} + 15 q^{25} + 3 q^{27} + 6 q^{29} + 8 q^{33} + 13 q^{37} - 11 q^{39} - 16 q^{41} + 17 q^{43} + 17 q^{45} - 24 q^{47} + 5 q^{51} + 2 q^{53} + 5 q^{55} + 2 q^{59} - 13 q^{61} - 26 q^{65} + 2 q^{67} - 11 q^{69} + 10 q^{71} + 5 q^{73} - 20 q^{75} + 16 q^{79} - 12 q^{81} - 43 q^{83} + 24 q^{85} + 20 q^{87} + 8 q^{89} + 2 q^{93} - q^{95} - 12 q^{97} + 37 q^{99} + O(q^{100}) \)

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
\(3\) 0.684189 0.395017 0.197508 0.980301i \(-0.436715\pi\)
0.197508 + 0.980301i \(0.436715\pi\)
\(4\) 0 0
\(5\) −3.65096 −1.63276 −0.816380 0.577515i \(-0.804023\pi\)
−0.816380 + 0.577515i \(0.804023\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −2.53189 −0.843962
\(10\) 0 0
\(11\) 2.37730 0.716782 0.358391 0.933572i \(-0.383325\pi\)
0.358391 + 0.933572i \(0.383325\pi\)
\(12\) 0 0
\(13\) 5.46860 1.51672 0.758358 0.651838i \(-0.226002\pi\)
0.758358 + 0.651838i \(0.226002\pi\)
\(14\) 0 0
\(15\) −2.49795 −0.644968
\(16\) 0 0
\(17\) −4.86063 −1.17888 −0.589438 0.807813i \(-0.700651\pi\)
−0.589438 + 0.807813i \(0.700651\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.77549 −0.370216 −0.185108 0.982718i \(-0.559263\pi\)
−0.185108 + 0.982718i \(0.559263\pi\)
\(24\) 0 0
\(25\) 8.32954 1.66591
\(26\) 0 0
\(27\) −3.78485 −0.728396
\(28\) 0 0
\(29\) −6.04283 −1.12213 −0.561063 0.827773i \(-0.689607\pi\)
−0.561063 + 0.827773i \(0.689607\pi\)
\(30\) 0 0
\(31\) 1.69856 0.305070 0.152535 0.988298i \(-0.451256\pi\)
0.152535 + 0.988298i \(0.451256\pi\)
\(32\) 0 0
\(33\) 1.62652 0.283141
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.62635 0.924967 0.462483 0.886628i \(-0.346958\pi\)
0.462483 + 0.886628i \(0.346958\pi\)
\(38\) 0 0
\(39\) 3.74155 0.599128
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 3.10972 0.474227 0.237114 0.971482i \(-0.423799\pi\)
0.237114 + 0.971482i \(0.423799\pi\)
\(44\) 0 0
\(45\) 9.24382 1.37799
\(46\) 0 0
\(47\) −5.23017 −0.762898 −0.381449 0.924390i \(-0.624575\pi\)
−0.381449 + 0.924390i \(0.624575\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −3.32559 −0.465676
\(52\) 0 0
\(53\) −6.46513 −0.888054 −0.444027 0.896013i \(-0.646451\pi\)
−0.444027 + 0.896013i \(0.646451\pi\)
\(54\) 0 0
\(55\) −8.67942 −1.17033
\(56\) 0 0
\(57\) 0.684189 0.0906230
\(58\) 0 0
\(59\) −7.52266 −0.979367 −0.489684 0.871900i \(-0.662888\pi\)
−0.489684 + 0.871900i \(0.662888\pi\)
\(60\) 0 0
\(61\) −7.75592 −0.993044 −0.496522 0.868024i \(-0.665390\pi\)
−0.496522 + 0.868024i \(0.665390\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −19.9657 −2.47644
\(66\) 0 0
\(67\) −2.62090 −0.320194 −0.160097 0.987101i \(-0.551181\pi\)
−0.160097 + 0.987101i \(0.551181\pi\)
\(68\) 0 0
\(69\) −1.21477 −0.146241
\(70\) 0 0
\(71\) 11.1529 1.32361 0.661805 0.749676i \(-0.269791\pi\)
0.661805 + 0.749676i \(0.269791\pi\)
\(72\) 0 0
\(73\) 14.5305 1.70067 0.850333 0.526244i \(-0.176400\pi\)
0.850333 + 0.526244i \(0.176400\pi\)
\(74\) 0 0
\(75\) 5.69898 0.658061
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0.961803 0.108211 0.0541057 0.998535i \(-0.482769\pi\)
0.0541057 + 0.998535i \(0.482769\pi\)
\(80\) 0 0
\(81\) 5.00610 0.556234
\(82\) 0 0
\(83\) −15.0476 −1.65169 −0.825844 0.563898i \(-0.809301\pi\)
−0.825844 + 0.563898i \(0.809301\pi\)
\(84\) 0 0
\(85\) 17.7460 1.92482
\(86\) 0 0
\(87\) −4.13444 −0.443258
\(88\) 0 0
\(89\) 12.0321 1.27540 0.637701 0.770284i \(-0.279886\pi\)
0.637701 + 0.770284i \(0.279886\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 1.16213 0.120508
\(94\) 0 0
\(95\) −3.65096 −0.374581
\(96\) 0 0
\(97\) −6.26438 −0.636052 −0.318026 0.948082i \(-0.603020\pi\)
−0.318026 + 0.948082i \(0.603020\pi\)
\(98\) 0 0
\(99\) −6.01904 −0.604937
\(100\) 0 0
\(101\) −5.71595 −0.568759 −0.284379 0.958712i \(-0.591787\pi\)
−0.284379 + 0.958712i \(0.591787\pi\)
\(102\) 0 0
\(103\) −1.69879 −0.167387 −0.0836935 0.996492i \(-0.526672\pi\)
−0.0836935 + 0.996492i \(0.526672\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.25842 0.701698 0.350849 0.936432i \(-0.385893\pi\)
0.350849 + 0.936432i \(0.385893\pi\)
\(108\) 0 0
\(109\) 18.4449 1.76670 0.883351 0.468712i \(-0.155282\pi\)
0.883351 + 0.468712i \(0.155282\pi\)
\(110\) 0 0
\(111\) 3.84949 0.365377
\(112\) 0 0
\(113\) 17.9243 1.68618 0.843090 0.537772i \(-0.180734\pi\)
0.843090 + 0.537772i \(0.180734\pi\)
\(114\) 0 0
\(115\) 6.48226 0.604474
\(116\) 0 0
\(117\) −13.8459 −1.28005
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −5.34846 −0.486224
\(122\) 0 0
\(123\) −1.36838 −0.123382
\(124\) 0 0
\(125\) −12.1560 −1.08727
\(126\) 0 0
\(127\) 16.8044 1.49115 0.745575 0.666421i \(-0.232175\pi\)
0.745575 + 0.666421i \(0.232175\pi\)
\(128\) 0 0
\(129\) 2.12763 0.187328
\(130\) 0 0
\(131\) −10.1064 −0.882999 −0.441500 0.897261i \(-0.645553\pi\)
−0.441500 + 0.897261i \(0.645553\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 13.8184 1.18930
\(136\) 0 0
\(137\) −5.55210 −0.474348 −0.237174 0.971467i \(-0.576221\pi\)
−0.237174 + 0.971467i \(0.576221\pi\)
\(138\) 0 0
\(139\) 4.35122 0.369066 0.184533 0.982826i \(-0.440923\pi\)
0.184533 + 0.982826i \(0.440923\pi\)
\(140\) 0 0
\(141\) −3.57842 −0.301357
\(142\) 0 0
\(143\) 13.0005 1.08716
\(144\) 0 0
\(145\) 22.0622 1.83216
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 12.1443 0.994896 0.497448 0.867494i \(-0.334271\pi\)
0.497448 + 0.867494i \(0.334271\pi\)
\(150\) 0 0
\(151\) −15.7921 −1.28514 −0.642571 0.766226i \(-0.722132\pi\)
−0.642571 + 0.766226i \(0.722132\pi\)
\(152\) 0 0
\(153\) 12.3066 0.994927
\(154\) 0 0
\(155\) −6.20138 −0.498106
\(156\) 0 0
\(157\) 7.83836 0.625569 0.312785 0.949824i \(-0.398738\pi\)
0.312785 + 0.949824i \(0.398738\pi\)
\(158\) 0 0
\(159\) −4.42337 −0.350796
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −4.86466 −0.381030 −0.190515 0.981684i \(-0.561016\pi\)
−0.190515 + 0.981684i \(0.561016\pi\)
\(164\) 0 0
\(165\) −5.93836 −0.462301
\(166\) 0 0
\(167\) −13.6708 −1.05788 −0.528939 0.848660i \(-0.677410\pi\)
−0.528939 + 0.848660i \(0.677410\pi\)
\(168\) 0 0
\(169\) 16.9056 1.30043
\(170\) 0 0
\(171\) −2.53189 −0.193618
\(172\) 0 0
\(173\) −5.34907 −0.406682 −0.203341 0.979108i \(-0.565180\pi\)
−0.203341 + 0.979108i \(0.565180\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −5.14692 −0.386866
\(178\) 0 0
\(179\) −0.687824 −0.0514104 −0.0257052 0.999670i \(-0.508183\pi\)
−0.0257052 + 0.999670i \(0.508183\pi\)
\(180\) 0 0
\(181\) 7.91388 0.588234 0.294117 0.955769i \(-0.404974\pi\)
0.294117 + 0.955769i \(0.404974\pi\)
\(182\) 0 0
\(183\) −5.30652 −0.392269
\(184\) 0 0
\(185\) −20.5416 −1.51025
\(186\) 0 0
\(187\) −11.5552 −0.844998
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 24.0896 1.74306 0.871530 0.490341i \(-0.163128\pi\)
0.871530 + 0.490341i \(0.163128\pi\)
\(192\) 0 0
\(193\) −4.47773 −0.322314 −0.161157 0.986929i \(-0.551523\pi\)
−0.161157 + 0.986929i \(0.551523\pi\)
\(194\) 0 0
\(195\) −13.6603 −0.978233
\(196\) 0 0
\(197\) 15.0277 1.07068 0.535339 0.844637i \(-0.320184\pi\)
0.535339 + 0.844637i \(0.320184\pi\)
\(198\) 0 0
\(199\) −27.3915 −1.94173 −0.970866 0.239623i \(-0.922976\pi\)
−0.970866 + 0.239623i \(0.922976\pi\)
\(200\) 0 0
\(201\) −1.79319 −0.126482
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 7.30193 0.509989
\(206\) 0 0
\(207\) 4.49534 0.312448
\(208\) 0 0
\(209\) 2.37730 0.164441
\(210\) 0 0
\(211\) −1.37592 −0.0947219 −0.0473610 0.998878i \(-0.515081\pi\)
−0.0473610 + 0.998878i \(0.515081\pi\)
\(212\) 0 0
\(213\) 7.63071 0.522848
\(214\) 0 0
\(215\) −11.3535 −0.774300
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 9.94161 0.671792
\(220\) 0 0
\(221\) −26.5809 −1.78802
\(222\) 0 0
\(223\) 27.7782 1.86016 0.930082 0.367352i \(-0.119735\pi\)
0.930082 + 0.367352i \(0.119735\pi\)
\(224\) 0 0
\(225\) −21.0894 −1.40596
\(226\) 0 0
\(227\) 20.5905 1.36664 0.683320 0.730119i \(-0.260535\pi\)
0.683320 + 0.730119i \(0.260535\pi\)
\(228\) 0 0
\(229\) 26.1587 1.72861 0.864307 0.502964i \(-0.167757\pi\)
0.864307 + 0.502964i \(0.167757\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −12.5845 −0.824439 −0.412219 0.911085i \(-0.635246\pi\)
−0.412219 + 0.911085i \(0.635246\pi\)
\(234\) 0 0
\(235\) 19.0952 1.24563
\(236\) 0 0
\(237\) 0.658055 0.0427453
\(238\) 0 0
\(239\) 28.5846 1.84898 0.924492 0.381202i \(-0.124490\pi\)
0.924492 + 0.381202i \(0.124490\pi\)
\(240\) 0 0
\(241\) 4.64546 0.299241 0.149620 0.988744i \(-0.452195\pi\)
0.149620 + 0.988744i \(0.452195\pi\)
\(242\) 0 0
\(243\) 14.7797 0.948117
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 5.46860 0.347959
\(248\) 0 0
\(249\) −10.2954 −0.652444
\(250\) 0 0
\(251\) −12.0953 −0.763446 −0.381723 0.924277i \(-0.624669\pi\)
−0.381723 + 0.924277i \(0.624669\pi\)
\(252\) 0 0
\(253\) −4.22087 −0.265364
\(254\) 0 0
\(255\) 12.1416 0.760337
\(256\) 0 0
\(257\) 1.89239 0.118044 0.0590219 0.998257i \(-0.481202\pi\)
0.0590219 + 0.998257i \(0.481202\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 15.2998 0.947031
\(262\) 0 0
\(263\) 23.1801 1.42935 0.714674 0.699457i \(-0.246575\pi\)
0.714674 + 0.699457i \(0.246575\pi\)
\(264\) 0 0
\(265\) 23.6040 1.44998
\(266\) 0 0
\(267\) 8.23224 0.503805
\(268\) 0 0
\(269\) 24.2982 1.48149 0.740744 0.671787i \(-0.234473\pi\)
0.740744 + 0.671787i \(0.234473\pi\)
\(270\) 0 0
\(271\) 12.4096 0.753831 0.376915 0.926248i \(-0.376985\pi\)
0.376915 + 0.926248i \(0.376985\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 19.8018 1.19409
\(276\) 0 0
\(277\) −2.33798 −0.140475 −0.0702377 0.997530i \(-0.522376\pi\)
−0.0702377 + 0.997530i \(0.522376\pi\)
\(278\) 0 0
\(279\) −4.30056 −0.257468
\(280\) 0 0
\(281\) 3.65566 0.218078 0.109039 0.994037i \(-0.465223\pi\)
0.109039 + 0.994037i \(0.465223\pi\)
\(282\) 0 0
\(283\) 9.58475 0.569754 0.284877 0.958564i \(-0.408047\pi\)
0.284877 + 0.958564i \(0.408047\pi\)
\(284\) 0 0
\(285\) −2.49795 −0.147966
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 6.62575 0.389750
\(290\) 0 0
\(291\) −4.28602 −0.251251
\(292\) 0 0
\(293\) −10.0249 −0.585662 −0.292831 0.956164i \(-0.594597\pi\)
−0.292831 + 0.956164i \(0.594597\pi\)
\(294\) 0 0
\(295\) 27.4650 1.59907
\(296\) 0 0
\(297\) −8.99772 −0.522101
\(298\) 0 0
\(299\) −9.70946 −0.561512
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −3.91079 −0.224669
\(304\) 0 0
\(305\) 28.3166 1.62140
\(306\) 0 0
\(307\) 31.0911 1.77446 0.887230 0.461327i \(-0.152627\pi\)
0.887230 + 0.461327i \(0.152627\pi\)
\(308\) 0 0
\(309\) −1.16230 −0.0661207
\(310\) 0 0
\(311\) −2.59990 −0.147427 −0.0737134 0.997279i \(-0.523485\pi\)
−0.0737134 + 0.997279i \(0.523485\pi\)
\(312\) 0 0
\(313\) −14.5732 −0.823726 −0.411863 0.911246i \(-0.635122\pi\)
−0.411863 + 0.911246i \(0.635122\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 11.3005 0.634700 0.317350 0.948309i \(-0.397207\pi\)
0.317350 + 0.948309i \(0.397207\pi\)
\(318\) 0 0
\(319\) −14.3656 −0.804319
\(320\) 0 0
\(321\) 4.96613 0.277182
\(322\) 0 0
\(323\) −4.86063 −0.270453
\(324\) 0 0
\(325\) 45.5509 2.52671
\(326\) 0 0
\(327\) 12.6198 0.697877
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −15.3378 −0.843042 −0.421521 0.906819i \(-0.638504\pi\)
−0.421521 + 0.906819i \(0.638504\pi\)
\(332\) 0 0
\(333\) −14.2453 −0.780637
\(334\) 0 0
\(335\) 9.56882 0.522801
\(336\) 0 0
\(337\) 7.62537 0.415380 0.207690 0.978195i \(-0.433405\pi\)
0.207690 + 0.978195i \(0.433405\pi\)
\(338\) 0 0
\(339\) 12.2636 0.666069
\(340\) 0 0
\(341\) 4.03798 0.218669
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 4.43509 0.238777
\(346\) 0 0
\(347\) 24.8757 1.33540 0.667700 0.744431i \(-0.267279\pi\)
0.667700 + 0.744431i \(0.267279\pi\)
\(348\) 0 0
\(349\) 31.0832 1.66384 0.831922 0.554893i \(-0.187241\pi\)
0.831922 + 0.554893i \(0.187241\pi\)
\(350\) 0 0
\(351\) −20.6979 −1.10477
\(352\) 0 0
\(353\) 14.7050 0.782668 0.391334 0.920249i \(-0.372014\pi\)
0.391334 + 0.920249i \(0.372014\pi\)
\(354\) 0 0
\(355\) −40.7189 −2.16114
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.0904 1.06033 0.530165 0.847895i \(-0.322130\pi\)
0.530165 + 0.847895i \(0.322130\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −3.65936 −0.192066
\(364\) 0 0
\(365\) −53.0503 −2.77678
\(366\) 0 0
\(367\) −34.5138 −1.80160 −0.900802 0.434230i \(-0.857020\pi\)
−0.900802 + 0.434230i \(0.857020\pi\)
\(368\) 0 0
\(369\) 5.06377 0.263609
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −27.5909 −1.42860 −0.714301 0.699838i \(-0.753255\pi\)
−0.714301 + 0.699838i \(0.753255\pi\)
\(374\) 0 0
\(375\) −8.31701 −0.429489
\(376\) 0 0
\(377\) −33.0458 −1.70195
\(378\) 0 0
\(379\) 9.40171 0.482933 0.241467 0.970409i \(-0.422372\pi\)
0.241467 + 0.970409i \(0.422372\pi\)
\(380\) 0 0
\(381\) 11.4974 0.589029
\(382\) 0 0
\(383\) 14.3201 0.731723 0.365861 0.930669i \(-0.380774\pi\)
0.365861 + 0.930669i \(0.380774\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −7.87345 −0.400230
\(388\) 0 0
\(389\) −1.63413 −0.0828538 −0.0414269 0.999142i \(-0.513190\pi\)
−0.0414269 + 0.999142i \(0.513190\pi\)
\(390\) 0 0
\(391\) 8.63002 0.436439
\(392\) 0 0
\(393\) −6.91468 −0.348799
\(394\) 0 0
\(395\) −3.51151 −0.176683
\(396\) 0 0
\(397\) 7.30979 0.366868 0.183434 0.983032i \(-0.441279\pi\)
0.183434 + 0.983032i \(0.441279\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 26.9972 1.34818 0.674088 0.738651i \(-0.264537\pi\)
0.674088 + 0.738651i \(0.264537\pi\)
\(402\) 0 0
\(403\) 9.28874 0.462705
\(404\) 0 0
\(405\) −18.2771 −0.908196
\(406\) 0 0
\(407\) 13.3755 0.663000
\(408\) 0 0
\(409\) 25.3639 1.25416 0.627082 0.778953i \(-0.284249\pi\)
0.627082 + 0.778953i \(0.284249\pi\)
\(410\) 0 0
\(411\) −3.79868 −0.187375
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 54.9382 2.69681
\(416\) 0 0
\(417\) 2.97706 0.145787
\(418\) 0 0
\(419\) 8.43567 0.412109 0.206055 0.978540i \(-0.433938\pi\)
0.206055 + 0.978540i \(0.433938\pi\)
\(420\) 0 0
\(421\) 10.5892 0.516087 0.258044 0.966133i \(-0.416922\pi\)
0.258044 + 0.966133i \(0.416922\pi\)
\(422\) 0 0
\(423\) 13.2422 0.643857
\(424\) 0 0
\(425\) −40.4868 −1.96390
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 8.89479 0.429444
\(430\) 0 0
\(431\) −29.0468 −1.39913 −0.699567 0.714567i \(-0.746624\pi\)
−0.699567 + 0.714567i \(0.746624\pi\)
\(432\) 0 0
\(433\) −15.4759 −0.743725 −0.371862 0.928288i \(-0.621281\pi\)
−0.371862 + 0.928288i \(0.621281\pi\)
\(434\) 0 0
\(435\) 15.0947 0.723734
\(436\) 0 0
\(437\) −1.77549 −0.0849333
\(438\) 0 0
\(439\) −24.6189 −1.17500 −0.587499 0.809225i \(-0.699887\pi\)
−0.587499 + 0.809225i \(0.699887\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 8.44795 0.401374 0.200687 0.979655i \(-0.435683\pi\)
0.200687 + 0.979655i \(0.435683\pi\)
\(444\) 0 0
\(445\) −43.9288 −2.08242
\(446\) 0 0
\(447\) 8.30896 0.393000
\(448\) 0 0
\(449\) −12.8063 −0.604368 −0.302184 0.953250i \(-0.597716\pi\)
−0.302184 + 0.953250i \(0.597716\pi\)
\(450\) 0 0
\(451\) −4.75459 −0.223885
\(452\) 0 0
\(453\) −10.8048 −0.507652
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −28.5157 −1.33391 −0.666954 0.745098i \(-0.732402\pi\)
−0.666954 + 0.745098i \(0.732402\pi\)
\(458\) 0 0
\(459\) 18.3968 0.858688
\(460\) 0 0
\(461\) −37.8648 −1.76354 −0.881771 0.471679i \(-0.843648\pi\)
−0.881771 + 0.471679i \(0.843648\pi\)
\(462\) 0 0
\(463\) 24.6922 1.14754 0.573772 0.819015i \(-0.305480\pi\)
0.573772 + 0.819015i \(0.305480\pi\)
\(464\) 0 0
\(465\) −4.24291 −0.196760
\(466\) 0 0
\(467\) −2.34498 −0.108513 −0.0542563 0.998527i \(-0.517279\pi\)
−0.0542563 + 0.998527i \(0.517279\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 5.36292 0.247110
\(472\) 0 0
\(473\) 7.39272 0.339918
\(474\) 0 0
\(475\) 8.32954 0.382185
\(476\) 0 0
\(477\) 16.3690 0.749484
\(478\) 0 0
\(479\) 29.3129 1.33934 0.669670 0.742659i \(-0.266436\pi\)
0.669670 + 0.742659i \(0.266436\pi\)
\(480\) 0 0
\(481\) 30.7683 1.40291
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 22.8710 1.03852
\(486\) 0 0
\(487\) 14.0645 0.637321 0.318661 0.947869i \(-0.396767\pi\)
0.318661 + 0.947869i \(0.396767\pi\)
\(488\) 0 0
\(489\) −3.32835 −0.150513
\(490\) 0 0
\(491\) 33.8667 1.52838 0.764191 0.644991i \(-0.223139\pi\)
0.764191 + 0.644991i \(0.223139\pi\)
\(492\) 0 0
\(493\) 29.3720 1.32285
\(494\) 0 0
\(495\) 21.9753 0.987717
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 26.2914 1.17696 0.588482 0.808511i \(-0.299726\pi\)
0.588482 + 0.808511i \(0.299726\pi\)
\(500\) 0 0
\(501\) −9.35340 −0.417879
\(502\) 0 0
\(503\) 36.7282 1.63763 0.818815 0.574058i \(-0.194632\pi\)
0.818815 + 0.574058i \(0.194632\pi\)
\(504\) 0 0
\(505\) 20.8687 0.928646
\(506\) 0 0
\(507\) 11.5666 0.513691
\(508\) 0 0
\(509\) 6.77695 0.300383 0.150192 0.988657i \(-0.452011\pi\)
0.150192 + 0.988657i \(0.452011\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −3.78485 −0.167105
\(514\) 0 0
\(515\) 6.20223 0.273303
\(516\) 0 0
\(517\) −12.4337 −0.546832
\(518\) 0 0
\(519\) −3.65977 −0.160646
\(520\) 0 0
\(521\) −31.3361 −1.37286 −0.686431 0.727195i \(-0.740824\pi\)
−0.686431 + 0.727195i \(0.740824\pi\)
\(522\) 0 0
\(523\) −31.8391 −1.39223 −0.696114 0.717931i \(-0.745089\pi\)
−0.696114 + 0.717931i \(0.745089\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −8.25607 −0.359640
\(528\) 0 0
\(529\) −19.8476 −0.862940
\(530\) 0 0
\(531\) 19.0465 0.826549
\(532\) 0 0
\(533\) −10.9372 −0.473743
\(534\) 0 0
\(535\) −26.5002 −1.14570
\(536\) 0 0
\(537\) −0.470602 −0.0203080
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 8.96765 0.385549 0.192775 0.981243i \(-0.438251\pi\)
0.192775 + 0.981243i \(0.438251\pi\)
\(542\) 0 0
\(543\) 5.41459 0.232362
\(544\) 0 0
\(545\) −67.3417 −2.88460
\(546\) 0 0
\(547\) −26.0700 −1.11467 −0.557337 0.830287i \(-0.688177\pi\)
−0.557337 + 0.830287i \(0.688177\pi\)
\(548\) 0 0
\(549\) 19.6371 0.838092
\(550\) 0 0
\(551\) −6.04283 −0.257433
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −14.0543 −0.596574
\(556\) 0 0
\(557\) 24.2751 1.02857 0.514284 0.857620i \(-0.328058\pi\)
0.514284 + 0.857620i \(0.328058\pi\)
\(558\) 0 0
\(559\) 17.0058 0.719268
\(560\) 0 0
\(561\) −7.90592 −0.333788
\(562\) 0 0
\(563\) 31.2151 1.31556 0.657779 0.753211i \(-0.271496\pi\)
0.657779 + 0.753211i \(0.271496\pi\)
\(564\) 0 0
\(565\) −65.4411 −2.75313
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −16.9648 −0.711203 −0.355602 0.934638i \(-0.615724\pi\)
−0.355602 + 0.934638i \(0.615724\pi\)
\(570\) 0 0
\(571\) 6.28139 0.262868 0.131434 0.991325i \(-0.458042\pi\)
0.131434 + 0.991325i \(0.458042\pi\)
\(572\) 0 0
\(573\) 16.4818 0.688538
\(574\) 0 0
\(575\) −14.7890 −0.616745
\(576\) 0 0
\(577\) −16.2518 −0.676573 −0.338287 0.941043i \(-0.609847\pi\)
−0.338287 + 0.941043i \(0.609847\pi\)
\(578\) 0 0
\(579\) −3.06361 −0.127319
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −15.3695 −0.636541
\(584\) 0 0
\(585\) 50.5508 2.09002
\(586\) 0 0
\(587\) −45.7490 −1.88826 −0.944131 0.329571i \(-0.893096\pi\)
−0.944131 + 0.329571i \(0.893096\pi\)
\(588\) 0 0
\(589\) 1.69856 0.0699879
\(590\) 0 0
\(591\) 10.2818 0.422936
\(592\) 0 0
\(593\) 6.94183 0.285067 0.142533 0.989790i \(-0.454475\pi\)
0.142533 + 0.989790i \(0.454475\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −18.7410 −0.767016
\(598\) 0 0
\(599\) 14.5903 0.596142 0.298071 0.954544i \(-0.403657\pi\)
0.298071 + 0.954544i \(0.403657\pi\)
\(600\) 0 0
\(601\) −43.5586 −1.77679 −0.888396 0.459077i \(-0.848180\pi\)
−0.888396 + 0.459077i \(0.848180\pi\)
\(602\) 0 0
\(603\) 6.63583 0.270232
\(604\) 0 0
\(605\) 19.5270 0.793887
\(606\) 0 0
\(607\) −23.4431 −0.951526 −0.475763 0.879573i \(-0.657828\pi\)
−0.475763 + 0.879573i \(0.657828\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −28.6017 −1.15710
\(612\) 0 0
\(613\) −3.58627 −0.144848 −0.0724240 0.997374i \(-0.523073\pi\)
−0.0724240 + 0.997374i \(0.523073\pi\)
\(614\) 0 0
\(615\) 4.99590 0.201454
\(616\) 0 0
\(617\) 5.93981 0.239128 0.119564 0.992827i \(-0.461850\pi\)
0.119564 + 0.992827i \(0.461850\pi\)
\(618\) 0 0
\(619\) 27.5236 1.10627 0.553134 0.833093i \(-0.313432\pi\)
0.553134 + 0.833093i \(0.313432\pi\)
\(620\) 0 0
\(621\) 6.71998 0.269663
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 2.73349 0.109340
\(626\) 0 0
\(627\) 1.62652 0.0649569
\(628\) 0 0
\(629\) −27.3476 −1.09042
\(630\) 0 0
\(631\) −45.2443 −1.80115 −0.900573 0.434705i \(-0.856853\pi\)
−0.900573 + 0.434705i \(0.856853\pi\)
\(632\) 0 0
\(633\) −0.941386 −0.0374167
\(634\) 0 0
\(635\) −61.3523 −2.43469
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −28.2379 −1.11708
\(640\) 0 0
\(641\) −28.4998 −1.12568 −0.562838 0.826567i \(-0.690290\pi\)
−0.562838 + 0.826567i \(0.690290\pi\)
\(642\) 0 0
\(643\) −22.8611 −0.901553 −0.450776 0.892637i \(-0.648853\pi\)
−0.450776 + 0.892637i \(0.648853\pi\)
\(644\) 0 0
\(645\) −7.76791 −0.305861
\(646\) 0 0
\(647\) −24.0834 −0.946816 −0.473408 0.880843i \(-0.656976\pi\)
−0.473408 + 0.880843i \(0.656976\pi\)
\(648\) 0 0
\(649\) −17.8836 −0.701993
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 4.05259 0.158590 0.0792950 0.996851i \(-0.474733\pi\)
0.0792950 + 0.996851i \(0.474733\pi\)
\(654\) 0 0
\(655\) 36.8980 1.44173
\(656\) 0 0
\(657\) −36.7896 −1.43530
\(658\) 0 0
\(659\) 23.7165 0.923864 0.461932 0.886915i \(-0.347156\pi\)
0.461932 + 0.886915i \(0.347156\pi\)
\(660\) 0 0
\(661\) −17.2174 −0.669681 −0.334840 0.942275i \(-0.608682\pi\)
−0.334840 + 0.942275i \(0.608682\pi\)
\(662\) 0 0
\(663\) −18.1863 −0.706298
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 10.7290 0.415428
\(668\) 0 0
\(669\) 19.0055 0.734796
\(670\) 0 0
\(671\) −18.4381 −0.711796
\(672\) 0 0
\(673\) −20.1890 −0.778229 −0.389115 0.921189i \(-0.627219\pi\)
−0.389115 + 0.921189i \(0.627219\pi\)
\(674\) 0 0
\(675\) −31.5261 −1.21344
\(676\) 0 0
\(677\) 23.4527 0.901362 0.450681 0.892685i \(-0.351181\pi\)
0.450681 + 0.892685i \(0.351181\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 14.0878 0.539846
\(682\) 0 0
\(683\) 25.7238 0.984295 0.492148 0.870512i \(-0.336212\pi\)
0.492148 + 0.870512i \(0.336212\pi\)
\(684\) 0 0
\(685\) 20.2705 0.774496
\(686\) 0 0
\(687\) 17.8975 0.682832
\(688\) 0 0
\(689\) −35.3552 −1.34693
\(690\) 0 0
\(691\) 50.8091 1.93287 0.966434 0.256915i \(-0.0827060\pi\)
0.966434 + 0.256915i \(0.0827060\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −15.8861 −0.602596
\(696\) 0 0
\(697\) 9.72127 0.368219
\(698\) 0 0
\(699\) −8.61018 −0.325667
\(700\) 0 0
\(701\) 46.4663 1.75501 0.877504 0.479570i \(-0.159207\pi\)
0.877504 + 0.479570i \(0.159207\pi\)
\(702\) 0 0
\(703\) 5.62635 0.212202
\(704\) 0 0
\(705\) 13.0647 0.492045
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 2.92358 0.109797 0.0548986 0.998492i \(-0.482516\pi\)
0.0548986 + 0.998492i \(0.482516\pi\)
\(710\) 0 0
\(711\) −2.43518 −0.0913263
\(712\) 0 0
\(713\) −3.01578 −0.112942
\(714\) 0 0
\(715\) −47.4643 −1.77506
\(716\) 0 0
\(717\) 19.5573 0.730379
\(718\) 0 0
\(719\) 29.6006 1.10392 0.551959 0.833871i \(-0.313881\pi\)
0.551959 + 0.833871i \(0.313881\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 3.17837 0.118205
\(724\) 0 0
\(725\) −50.3340 −1.86936
\(726\) 0 0
\(727\) 39.1285 1.45119 0.725597 0.688120i \(-0.241564\pi\)
0.725597 + 0.688120i \(0.241564\pi\)
\(728\) 0 0
\(729\) −4.90622 −0.181712
\(730\) 0 0
\(731\) −15.1152 −0.559055
\(732\) 0 0
\(733\) 14.6944 0.542750 0.271375 0.962474i \(-0.412522\pi\)
0.271375 + 0.962474i \(0.412522\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −6.23067 −0.229510
\(738\) 0 0
\(739\) 16.1624 0.594542 0.297271 0.954793i \(-0.403924\pi\)
0.297271 + 0.954793i \(0.403924\pi\)
\(740\) 0 0
\(741\) 3.74155 0.137449
\(742\) 0 0
\(743\) 21.2412 0.779265 0.389632 0.920971i \(-0.372602\pi\)
0.389632 + 0.920971i \(0.372602\pi\)
\(744\) 0 0
\(745\) −44.3382 −1.62443
\(746\) 0 0
\(747\) 38.0988 1.39396
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −52.0353 −1.89880 −0.949398 0.314075i \(-0.898306\pi\)
−0.949398 + 0.314075i \(0.898306\pi\)
\(752\) 0 0
\(753\) −8.27544 −0.301574
\(754\) 0 0
\(755\) 57.6563 2.09833
\(756\) 0 0
\(757\) 30.6379 1.11355 0.556777 0.830662i \(-0.312038\pi\)
0.556777 + 0.830662i \(0.312038\pi\)
\(758\) 0 0
\(759\) −2.88787 −0.104823
\(760\) 0 0
\(761\) −19.5926 −0.710230 −0.355115 0.934823i \(-0.615558\pi\)
−0.355115 + 0.934823i \(0.615558\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −44.9308 −1.62448
\(766\) 0 0
\(767\) −41.1384 −1.48542
\(768\) 0 0
\(769\) −6.70349 −0.241734 −0.120867 0.992669i \(-0.538567\pi\)
−0.120867 + 0.992669i \(0.538567\pi\)
\(770\) 0 0
\(771\) 1.29475 0.0466293
\(772\) 0 0
\(773\) −13.5509 −0.487394 −0.243697 0.969851i \(-0.578360\pi\)
−0.243697 + 0.969851i \(0.578360\pi\)
\(774\) 0 0
\(775\) 14.1482 0.508219
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2.00000 −0.0716574
\(780\) 0 0
\(781\) 26.5138 0.948739
\(782\) 0 0
\(783\) 22.8712 0.817351
\(784\) 0 0
\(785\) −28.6176 −1.02140
\(786\) 0 0
\(787\) −36.3319 −1.29509 −0.647547 0.762026i \(-0.724205\pi\)
−0.647547 + 0.762026i \(0.724205\pi\)
\(788\) 0 0
\(789\) 15.8596 0.564616
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −42.4141 −1.50617
\(794\) 0 0
\(795\) 16.1496 0.572766
\(796\) 0 0
\(797\) −55.0232 −1.94902 −0.974511 0.224338i \(-0.927978\pi\)
−0.974511 + 0.224338i \(0.927978\pi\)
\(798\) 0 0
\(799\) 25.4219 0.899363
\(800\) 0 0
\(801\) −30.4639 −1.07639
\(802\) 0 0
\(803\) 34.5433 1.21901
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 16.6246 0.585213
\(808\) 0 0
\(809\) −27.1162 −0.953354 −0.476677 0.879079i \(-0.658159\pi\)
−0.476677 + 0.879079i \(0.658159\pi\)
\(810\) 0 0
\(811\) −20.7862 −0.729903 −0.364952 0.931027i \(-0.618914\pi\)
−0.364952 + 0.931027i \(0.618914\pi\)
\(812\) 0 0
\(813\) 8.49052 0.297776
\(814\) 0 0
\(815\) 17.7607 0.622130
\(816\) 0 0
\(817\) 3.10972 0.108795
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 42.1674 1.47165 0.735826 0.677170i \(-0.236794\pi\)
0.735826 + 0.677170i \(0.236794\pi\)
\(822\) 0 0
\(823\) 27.9529 0.974377 0.487189 0.873297i \(-0.338022\pi\)
0.487189 + 0.873297i \(0.338022\pi\)
\(824\) 0 0
\(825\) 13.5482 0.471686
\(826\) 0 0
\(827\) −46.6553 −1.62236 −0.811182 0.584794i \(-0.801175\pi\)
−0.811182 + 0.584794i \(0.801175\pi\)
\(828\) 0 0
\(829\) −29.5816 −1.02741 −0.513706 0.857966i \(-0.671728\pi\)
−0.513706 + 0.857966i \(0.671728\pi\)
\(830\) 0 0
\(831\) −1.59962 −0.0554901
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 49.9116 1.72726
\(836\) 0 0
\(837\) −6.42880 −0.222212
\(838\) 0 0
\(839\) −27.0614 −0.934263 −0.467132 0.884188i \(-0.654713\pi\)
−0.467132 + 0.884188i \(0.654713\pi\)
\(840\) 0 0
\(841\) 7.51580 0.259166
\(842\) 0 0
\(843\) 2.50116 0.0861446
\(844\) 0 0
\(845\) −61.7217 −2.12329
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 6.55778 0.225062
\(850\) 0 0
\(851\) −9.98955 −0.342437
\(852\) 0 0
\(853\) −1.19742 −0.0409989 −0.0204995 0.999790i \(-0.506526\pi\)
−0.0204995 + 0.999790i \(0.506526\pi\)
\(854\) 0 0
\(855\) 9.24382 0.316132
\(856\) 0 0
\(857\) 34.4417 1.17651 0.588253 0.808677i \(-0.299816\pi\)
0.588253 + 0.808677i \(0.299816\pi\)
\(858\) 0 0
\(859\) −3.74549 −0.127794 −0.0638972 0.997956i \(-0.520353\pi\)
−0.0638972 + 0.997956i \(0.520353\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −29.8671 −1.01669 −0.508343 0.861154i \(-0.669742\pi\)
−0.508343 + 0.861154i \(0.669742\pi\)
\(864\) 0 0
\(865\) 19.5293 0.664015
\(866\) 0 0
\(867\) 4.53327 0.153958
\(868\) 0 0
\(869\) 2.28649 0.0775639
\(870\) 0 0
\(871\) −14.3327 −0.485644
\(872\) 0 0
\(873\) 15.8607 0.536803
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −9.48210 −0.320188 −0.160094 0.987102i \(-0.551180\pi\)
−0.160094 + 0.987102i \(0.551180\pi\)
\(878\) 0 0
\(879\) −6.85894 −0.231346
\(880\) 0 0
\(881\) 33.1936 1.11832 0.559160 0.829060i \(-0.311124\pi\)
0.559160 + 0.829060i \(0.311124\pi\)
\(882\) 0 0
\(883\) −27.9645 −0.941079 −0.470540 0.882379i \(-0.655941\pi\)
−0.470540 + 0.882379i \(0.655941\pi\)
\(884\) 0 0
\(885\) 18.7912 0.631660
\(886\) 0 0
\(887\) 31.3624 1.05305 0.526523 0.850161i \(-0.323495\pi\)
0.526523 + 0.850161i \(0.323495\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 11.9010 0.398698
\(892\) 0 0
\(893\) −5.23017 −0.175021
\(894\) 0 0
\(895\) 2.51122 0.0839409
\(896\) 0 0
\(897\) −6.64310 −0.221807
\(898\) 0 0
\(899\) −10.2641 −0.342327
\(900\) 0 0
\(901\) 31.4246 1.04691
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −28.8933 −0.960446
\(906\) 0 0
\(907\) 8.73561 0.290061 0.145031 0.989427i \(-0.453672\pi\)
0.145031 + 0.989427i \(0.453672\pi\)
\(908\) 0 0
\(909\) 14.4721 0.480011
\(910\) 0 0
\(911\) 6.50534 0.215532 0.107766 0.994176i \(-0.465630\pi\)
0.107766 + 0.994176i \(0.465630\pi\)
\(912\) 0 0
\(913\) −35.7726 −1.18390
\(914\) 0 0
\(915\) 19.3739 0.640481
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 32.4502 1.07043 0.535216 0.844715i \(-0.320230\pi\)
0.535216 + 0.844715i \(0.320230\pi\)
\(920\) 0 0
\(921\) 21.2722 0.700941
\(922\) 0 0
\(923\) 60.9909 2.00754
\(924\) 0 0
\(925\) 46.8649 1.54091
\(926\) 0 0
\(927\) 4.30115 0.141268
\(928\) 0 0
\(929\) −21.9512 −0.720196 −0.360098 0.932914i \(-0.617257\pi\)
−0.360098 + 0.932914i \(0.617257\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −1.77882 −0.0582360
\(934\) 0 0
\(935\) 42.1875 1.37968
\(936\) 0 0
\(937\) −20.0361 −0.654552 −0.327276 0.944929i \(-0.606131\pi\)
−0.327276 + 0.944929i \(0.606131\pi\)
\(938\) 0 0
\(939\) −9.97083 −0.325386
\(940\) 0 0
\(941\) −6.51229 −0.212295 −0.106147 0.994350i \(-0.533852\pi\)
−0.106147 + 0.994350i \(0.533852\pi\)
\(942\) 0 0
\(943\) 3.55098 0.115636
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 56.1061 1.82320 0.911602 0.411073i \(-0.134846\pi\)
0.911602 + 0.411073i \(0.134846\pi\)
\(948\) 0 0
\(949\) 79.4615 2.57943
\(950\) 0 0
\(951\) 7.73168 0.250717
\(952\) 0 0
\(953\) 23.9841 0.776920 0.388460 0.921466i \(-0.373007\pi\)
0.388460 + 0.921466i \(0.373007\pi\)
\(954\) 0 0
\(955\) −87.9502 −2.84600
\(956\) 0 0
\(957\) −9.82878 −0.317719
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −28.1149 −0.906932
\(962\) 0 0
\(963\) −18.3775 −0.592206
\(964\) 0 0
\(965\) 16.3480 0.526261
\(966\) 0 0
\(967\) 25.6539 0.824974 0.412487 0.910963i \(-0.364660\pi\)
0.412487 + 0.910963i \(0.364660\pi\)
\(968\) 0 0
\(969\) −3.32559 −0.106833
\(970\) 0 0
\(971\) 7.50346 0.240797 0.120399 0.992726i \(-0.461583\pi\)
0.120399 + 0.992726i \(0.461583\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 31.1654 0.998092
\(976\) 0 0
\(977\) −58.6541 −1.87651 −0.938255 0.345943i \(-0.887559\pi\)
−0.938255 + 0.345943i \(0.887559\pi\)
\(978\) 0 0
\(979\) 28.6039 0.914185
\(980\) 0 0
\(981\) −46.7004 −1.49103
\(982\) 0 0
\(983\) −16.3863 −0.522640 −0.261320 0.965252i \(-0.584158\pi\)
−0.261320 + 0.965252i \(0.584158\pi\)
\(984\) 0 0
\(985\) −54.8655 −1.74816
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −5.52128 −0.175566
\(990\) 0 0
\(991\) −45.0420 −1.43080 −0.715402 0.698713i \(-0.753757\pi\)
−0.715402 + 0.698713i \(0.753757\pi\)
\(992\) 0 0
\(993\) −10.4940 −0.333015
\(994\) 0 0
\(995\) 100.005 3.17038
\(996\) 0 0
\(997\) −3.40836 −0.107944 −0.0539719 0.998542i \(-0.517188\pi\)
−0.0539719 + 0.998542i \(0.517188\pi\)
\(998\) 0 0
\(999\) −21.2949 −0.673742
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 7448.2.a.bq.1.5 8
7.2 even 3 1064.2.q.n.305.4 16
7.4 even 3 1064.2.q.n.457.4 yes 16
7.6 odd 2 7448.2.a.br.1.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1064.2.q.n.305.4 16 7.2 even 3
1064.2.q.n.457.4 yes 16 7.4 even 3
7448.2.a.bq.1.5 8 1.1 even 1 trivial
7448.2.a.br.1.4 8 7.6 odd 2