Properties

Label 3525.2.a.bd.1.7
Level $3525$
Weight $2$
Character 3525.1
Self dual yes
Analytic conductor $28.147$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.1472667125\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 3 x^{7} - 7 x^{6} + 24 x^{5} + 8 x^{4} - 47 x^{3} + 8 x^{2} + 13 x + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-1.60641\) of defining polynomial
Character \(\chi\) \(=\) 3525.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.60641 q^{2} +1.00000 q^{3} +0.580562 q^{4} +1.60641 q^{6} -2.35394 q^{7} -2.28020 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.60641 q^{2} +1.00000 q^{3} +0.580562 q^{4} +1.60641 q^{6} -2.35394 q^{7} -2.28020 q^{8} +1.00000 q^{9} +1.21821 q^{11} +0.580562 q^{12} +1.28767 q^{13} -3.78140 q^{14} -4.82407 q^{16} -2.46057 q^{17} +1.60641 q^{18} -0.168892 q^{19} -2.35394 q^{21} +1.95695 q^{22} -8.96026 q^{23} -2.28020 q^{24} +2.06852 q^{26} +1.00000 q^{27} -1.36661 q^{28} -2.39692 q^{29} +6.32953 q^{31} -3.18904 q^{32} +1.21821 q^{33} -3.95270 q^{34} +0.580562 q^{36} -9.44703 q^{37} -0.271311 q^{38} +1.28767 q^{39} -1.59177 q^{41} -3.78140 q^{42} -1.20123 q^{43} +0.707246 q^{44} -14.3939 q^{46} -1.00000 q^{47} -4.82407 q^{48} -1.45896 q^{49} -2.46057 q^{51} +0.747570 q^{52} -5.14255 q^{53} +1.60641 q^{54} +5.36746 q^{56} -0.168892 q^{57} -3.85044 q^{58} -11.9036 q^{59} +14.0274 q^{61} +10.1678 q^{62} -2.35394 q^{63} +4.52522 q^{64} +1.95695 q^{66} -8.76052 q^{67} -1.42852 q^{68} -8.96026 q^{69} -6.83248 q^{71} -2.28020 q^{72} -0.476914 q^{73} -15.1758 q^{74} -0.0980525 q^{76} -2.86759 q^{77} +2.06852 q^{78} -5.18222 q^{79} +1.00000 q^{81} -2.55704 q^{82} +13.6149 q^{83} -1.36661 q^{84} -1.92967 q^{86} -2.39692 q^{87} -2.77776 q^{88} -12.0162 q^{89} -3.03109 q^{91} -5.20199 q^{92} +6.32953 q^{93} -1.60641 q^{94} -3.18904 q^{96} -13.4808 q^{97} -2.34370 q^{98} +1.21821 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 3q^{2} + 8q^{3} + 7q^{4} - 3q^{6} - 8q^{7} - 6q^{8} + 8q^{9} + O(q^{10}) \) \( 8q - 3q^{2} + 8q^{3} + 7q^{4} - 3q^{6} - 8q^{7} - 6q^{8} + 8q^{9} - 8q^{11} + 7q^{12} - 10q^{13} + q^{14} + 5q^{16} - 6q^{17} - 3q^{18} - 2q^{19} - 8q^{21} - 10q^{23} - 6q^{24} - 14q^{26} + 8q^{27} - 44q^{28} - 13q^{29} - 10q^{32} - 8q^{33} + 28q^{34} + 7q^{36} - 3q^{37} - 36q^{38} - 10q^{39} - 16q^{41} + q^{42} - 25q^{43} - 17q^{44} - 5q^{46} - 8q^{47} + 5q^{48} + 16q^{49} - 6q^{51} + 17q^{52} - 4q^{53} - 3q^{54} + 37q^{56} - 2q^{57} - 15q^{58} - 8q^{59} + 15q^{61} - 6q^{62} - 8q^{63} - 14q^{64} - 27q^{67} - 14q^{68} - 10q^{69} + 14q^{71} - 6q^{72} - 28q^{73} - 21q^{74} + 6q^{76} - 4q^{77} - 14q^{78} + 7q^{79} + 8q^{81} + 53q^{82} - 60q^{83} - 44q^{84} - 3q^{86} - 13q^{87} - 54q^{88} - 34q^{89} + 23q^{91} + 43q^{92} + 3q^{94} - 10q^{96} - 7q^{97} - 40q^{98} - 8q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.60641 1.13591 0.567953 0.823061i \(-0.307736\pi\)
0.567953 + 0.823061i \(0.307736\pi\)
\(3\) 1.00000 0.577350
\(4\) 0.580562 0.290281
\(5\) 0 0
\(6\) 1.60641 0.655815
\(7\) −2.35394 −0.889706 −0.444853 0.895604i \(-0.646744\pi\)
−0.444853 + 0.895604i \(0.646744\pi\)
\(8\) −2.28020 −0.806174
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.21821 0.367304 0.183652 0.982991i \(-0.441208\pi\)
0.183652 + 0.982991i \(0.441208\pi\)
\(12\) 0.580562 0.167594
\(13\) 1.28767 0.357134 0.178567 0.983928i \(-0.442854\pi\)
0.178567 + 0.983928i \(0.442854\pi\)
\(14\) −3.78140 −1.01062
\(15\) 0 0
\(16\) −4.82407 −1.20602
\(17\) −2.46057 −0.596777 −0.298388 0.954445i \(-0.596449\pi\)
−0.298388 + 0.954445i \(0.596449\pi\)
\(18\) 1.60641 0.378635
\(19\) −0.168892 −0.0387466 −0.0193733 0.999812i \(-0.506167\pi\)
−0.0193733 + 0.999812i \(0.506167\pi\)
\(20\) 0 0
\(21\) −2.35394 −0.513672
\(22\) 1.95695 0.417222
\(23\) −8.96026 −1.86834 −0.934172 0.356823i \(-0.883860\pi\)
−0.934172 + 0.356823i \(0.883860\pi\)
\(24\) −2.28020 −0.465445
\(25\) 0 0
\(26\) 2.06852 0.405671
\(27\) 1.00000 0.192450
\(28\) −1.36661 −0.258265
\(29\) −2.39692 −0.445097 −0.222548 0.974922i \(-0.571437\pi\)
−0.222548 + 0.974922i \(0.571437\pi\)
\(30\) 0 0
\(31\) 6.32953 1.13682 0.568409 0.822746i \(-0.307559\pi\)
0.568409 + 0.822746i \(0.307559\pi\)
\(32\) −3.18904 −0.563749
\(33\) 1.21821 0.212063
\(34\) −3.95270 −0.677882
\(35\) 0 0
\(36\) 0.580562 0.0967603
\(37\) −9.44703 −1.55308 −0.776541 0.630066i \(-0.783028\pi\)
−0.776541 + 0.630066i \(0.783028\pi\)
\(38\) −0.271311 −0.0440124
\(39\) 1.28767 0.206191
\(40\) 0 0
\(41\) −1.59177 −0.248593 −0.124296 0.992245i \(-0.539667\pi\)
−0.124296 + 0.992245i \(0.539667\pi\)
\(42\) −3.78140 −0.583483
\(43\) −1.20123 −0.183186 −0.0915928 0.995797i \(-0.529196\pi\)
−0.0915928 + 0.995797i \(0.529196\pi\)
\(44\) 0.707246 0.106621
\(45\) 0 0
\(46\) −14.3939 −2.12226
\(47\) −1.00000 −0.145865
\(48\) −4.82407 −0.696295
\(49\) −1.45896 −0.208423
\(50\) 0 0
\(51\) −2.46057 −0.344549
\(52\) 0.747570 0.103669
\(53\) −5.14255 −0.706384 −0.353192 0.935551i \(-0.614904\pi\)
−0.353192 + 0.935551i \(0.614904\pi\)
\(54\) 1.60641 0.218605
\(55\) 0 0
\(56\) 5.36746 0.717257
\(57\) −0.168892 −0.0223703
\(58\) −3.85044 −0.505588
\(59\) −11.9036 −1.54972 −0.774861 0.632131i \(-0.782180\pi\)
−0.774861 + 0.632131i \(0.782180\pi\)
\(60\) 0 0
\(61\) 14.0274 1.79602 0.898009 0.439977i \(-0.145013\pi\)
0.898009 + 0.439977i \(0.145013\pi\)
\(62\) 10.1678 1.29132
\(63\) −2.35394 −0.296569
\(64\) 4.52522 0.565653
\(65\) 0 0
\(66\) 1.95695 0.240884
\(67\) −8.76052 −1.07027 −0.535134 0.844767i \(-0.679739\pi\)
−0.535134 + 0.844767i \(0.679739\pi\)
\(68\) −1.42852 −0.173233
\(69\) −8.96026 −1.07869
\(70\) 0 0
\(71\) −6.83248 −0.810866 −0.405433 0.914125i \(-0.632879\pi\)
−0.405433 + 0.914125i \(0.632879\pi\)
\(72\) −2.28020 −0.268725
\(73\) −0.476914 −0.0558185 −0.0279093 0.999610i \(-0.508885\pi\)
−0.0279093 + 0.999610i \(0.508885\pi\)
\(74\) −15.1758 −1.76415
\(75\) 0 0
\(76\) −0.0980525 −0.0112474
\(77\) −2.86759 −0.326792
\(78\) 2.06852 0.234214
\(79\) −5.18222 −0.583045 −0.291523 0.956564i \(-0.594162\pi\)
−0.291523 + 0.956564i \(0.594162\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.55704 −0.282378
\(83\) 13.6149 1.49443 0.747214 0.664583i \(-0.231391\pi\)
0.747214 + 0.664583i \(0.231391\pi\)
\(84\) −1.36661 −0.149109
\(85\) 0 0
\(86\) −1.92967 −0.208082
\(87\) −2.39692 −0.256977
\(88\) −2.77776 −0.296111
\(89\) −12.0162 −1.27371 −0.636855 0.770984i \(-0.719765\pi\)
−0.636855 + 0.770984i \(0.719765\pi\)
\(90\) 0 0
\(91\) −3.03109 −0.317744
\(92\) −5.20199 −0.542345
\(93\) 6.32953 0.656342
\(94\) −1.60641 −0.165689
\(95\) 0 0
\(96\) −3.18904 −0.325480
\(97\) −13.4808 −1.36876 −0.684381 0.729124i \(-0.739928\pi\)
−0.684381 + 0.729124i \(0.739928\pi\)
\(98\) −2.34370 −0.236749
\(99\) 1.21821 0.122435
\(100\) 0 0
\(101\) 5.27825 0.525206 0.262603 0.964904i \(-0.415419\pi\)
0.262603 + 0.964904i \(0.415419\pi\)
\(102\) −3.95270 −0.391375
\(103\) 9.08889 0.895555 0.447778 0.894145i \(-0.352216\pi\)
0.447778 + 0.894145i \(0.352216\pi\)
\(104\) −2.93614 −0.287912
\(105\) 0 0
\(106\) −8.26107 −0.802386
\(107\) 5.74384 0.555278 0.277639 0.960686i \(-0.410448\pi\)
0.277639 + 0.960686i \(0.410448\pi\)
\(108\) 0.580562 0.0558646
\(109\) 5.71806 0.547691 0.273846 0.961774i \(-0.411704\pi\)
0.273846 + 0.961774i \(0.411704\pi\)
\(110\) 0 0
\(111\) −9.44703 −0.896673
\(112\) 11.3556 1.07300
\(113\) 10.9763 1.03257 0.516283 0.856418i \(-0.327315\pi\)
0.516283 + 0.856418i \(0.327315\pi\)
\(114\) −0.271311 −0.0254106
\(115\) 0 0
\(116\) −1.39156 −0.129203
\(117\) 1.28767 0.119045
\(118\) −19.1222 −1.76034
\(119\) 5.79205 0.530956
\(120\) 0 0
\(121\) −9.51597 −0.865088
\(122\) 22.5337 2.04011
\(123\) −1.59177 −0.143525
\(124\) 3.67468 0.329996
\(125\) 0 0
\(126\) −3.78140 −0.336874
\(127\) 3.75661 0.333345 0.166673 0.986012i \(-0.446698\pi\)
0.166673 + 0.986012i \(0.446698\pi\)
\(128\) 13.6475 1.20628
\(129\) −1.20123 −0.105762
\(130\) 0 0
\(131\) 6.15566 0.537823 0.268911 0.963165i \(-0.413336\pi\)
0.268911 + 0.963165i \(0.413336\pi\)
\(132\) 0.707246 0.0615579
\(133\) 0.397563 0.0344731
\(134\) −14.0730 −1.21572
\(135\) 0 0
\(136\) 5.61061 0.481106
\(137\) −1.52211 −0.130043 −0.0650213 0.997884i \(-0.520712\pi\)
−0.0650213 + 0.997884i \(0.520712\pi\)
\(138\) −14.3939 −1.22529
\(139\) 0.241574 0.0204900 0.0102450 0.999948i \(-0.496739\pi\)
0.0102450 + 0.999948i \(0.496739\pi\)
\(140\) 0 0
\(141\) −1.00000 −0.0842152
\(142\) −10.9758 −0.921067
\(143\) 1.56865 0.131177
\(144\) −4.82407 −0.402006
\(145\) 0 0
\(146\) −0.766120 −0.0634046
\(147\) −1.45896 −0.120333
\(148\) −5.48459 −0.450830
\(149\) −6.91605 −0.566585 −0.283292 0.959034i \(-0.591427\pi\)
−0.283292 + 0.959034i \(0.591427\pi\)
\(150\) 0 0
\(151\) −13.1690 −1.07168 −0.535838 0.844321i \(-0.680004\pi\)
−0.535838 + 0.844321i \(0.680004\pi\)
\(152\) 0.385109 0.0312365
\(153\) −2.46057 −0.198926
\(154\) −4.60654 −0.371205
\(155\) 0 0
\(156\) 0.747570 0.0598535
\(157\) 7.97672 0.636612 0.318306 0.947988i \(-0.396886\pi\)
0.318306 + 0.947988i \(0.396886\pi\)
\(158\) −8.32478 −0.662284
\(159\) −5.14255 −0.407831
\(160\) 0 0
\(161\) 21.0919 1.66228
\(162\) 1.60641 0.126212
\(163\) −18.1440 −1.42114 −0.710572 0.703624i \(-0.751564\pi\)
−0.710572 + 0.703624i \(0.751564\pi\)
\(164\) −0.924122 −0.0721618
\(165\) 0 0
\(166\) 21.8711 1.69753
\(167\) −15.5428 −1.20273 −0.601367 0.798973i \(-0.705377\pi\)
−0.601367 + 0.798973i \(0.705377\pi\)
\(168\) 5.36746 0.414109
\(169\) −11.3419 −0.872455
\(170\) 0 0
\(171\) −0.168892 −0.0129155
\(172\) −0.697388 −0.0531753
\(173\) −9.13465 −0.694494 −0.347247 0.937774i \(-0.612884\pi\)
−0.347247 + 0.937774i \(0.612884\pi\)
\(174\) −3.85044 −0.291901
\(175\) 0 0
\(176\) −5.87673 −0.442975
\(177\) −11.9036 −0.894733
\(178\) −19.3029 −1.44681
\(179\) −12.1271 −0.906422 −0.453211 0.891403i \(-0.649722\pi\)
−0.453211 + 0.891403i \(0.649722\pi\)
\(180\) 0 0
\(181\) 21.3726 1.58861 0.794305 0.607519i \(-0.207835\pi\)
0.794305 + 0.607519i \(0.207835\pi\)
\(182\) −4.86918 −0.360928
\(183\) 14.0274 1.03693
\(184\) 20.4312 1.50621
\(185\) 0 0
\(186\) 10.1678 0.745542
\(187\) −2.99749 −0.219199
\(188\) −0.580562 −0.0423418
\(189\) −2.35394 −0.171224
\(190\) 0 0
\(191\) 13.4979 0.976673 0.488336 0.872655i \(-0.337604\pi\)
0.488336 + 0.872655i \(0.337604\pi\)
\(192\) 4.52522 0.326580
\(193\) −8.03065 −0.578059 −0.289030 0.957320i \(-0.593333\pi\)
−0.289030 + 0.957320i \(0.593333\pi\)
\(194\) −21.6557 −1.55479
\(195\) 0 0
\(196\) −0.847019 −0.0605013
\(197\) −17.2888 −1.23178 −0.615888 0.787833i \(-0.711203\pi\)
−0.615888 + 0.787833i \(0.711203\pi\)
\(198\) 1.95695 0.139074
\(199\) 5.14079 0.364421 0.182211 0.983260i \(-0.441675\pi\)
0.182211 + 0.983260i \(0.441675\pi\)
\(200\) 0 0
\(201\) −8.76052 −0.617919
\(202\) 8.47905 0.596584
\(203\) 5.64220 0.396005
\(204\) −1.42852 −0.100016
\(205\) 0 0
\(206\) 14.6005 1.01727
\(207\) −8.96026 −0.622781
\(208\) −6.21179 −0.430710
\(209\) −0.205746 −0.0142318
\(210\) 0 0
\(211\) 6.07830 0.418448 0.209224 0.977868i \(-0.432906\pi\)
0.209224 + 0.977868i \(0.432906\pi\)
\(212\) −2.98557 −0.205050
\(213\) −6.83248 −0.468154
\(214\) 9.22697 0.630743
\(215\) 0 0
\(216\) −2.28020 −0.155148
\(217\) −14.8993 −1.01143
\(218\) 9.18557 0.622125
\(219\) −0.476914 −0.0322268
\(220\) 0 0
\(221\) −3.16840 −0.213129
\(222\) −15.1758 −1.01854
\(223\) 12.8780 0.862372 0.431186 0.902263i \(-0.358095\pi\)
0.431186 + 0.902263i \(0.358095\pi\)
\(224\) 7.50682 0.501571
\(225\) 0 0
\(226\) 17.6325 1.17290
\(227\) 13.8756 0.920957 0.460478 0.887671i \(-0.347678\pi\)
0.460478 + 0.887671i \(0.347678\pi\)
\(228\) −0.0980525 −0.00649369
\(229\) 24.1014 1.59266 0.796331 0.604861i \(-0.206771\pi\)
0.796331 + 0.604861i \(0.206771\pi\)
\(230\) 0 0
\(231\) −2.86759 −0.188674
\(232\) 5.46546 0.358825
\(233\) 26.5063 1.73648 0.868241 0.496142i \(-0.165250\pi\)
0.868241 + 0.496142i \(0.165250\pi\)
\(234\) 2.06852 0.135224
\(235\) 0 0
\(236\) −6.91081 −0.449855
\(237\) −5.18222 −0.336621
\(238\) 9.30442 0.603116
\(239\) −5.89635 −0.381403 −0.190702 0.981648i \(-0.561076\pi\)
−0.190702 + 0.981648i \(0.561076\pi\)
\(240\) 0 0
\(241\) 4.50808 0.290391 0.145195 0.989403i \(-0.453619\pi\)
0.145195 + 0.989403i \(0.453619\pi\)
\(242\) −15.2866 −0.982658
\(243\) 1.00000 0.0641500
\(244\) 8.14375 0.521350
\(245\) 0 0
\(246\) −2.55704 −0.163031
\(247\) −0.217477 −0.0138377
\(248\) −14.4326 −0.916472
\(249\) 13.6149 0.862809
\(250\) 0 0
\(251\) 9.71575 0.613253 0.306626 0.951830i \(-0.400800\pi\)
0.306626 + 0.951830i \(0.400800\pi\)
\(252\) −1.36661 −0.0860882
\(253\) −10.9155 −0.686250
\(254\) 6.03466 0.378649
\(255\) 0 0
\(256\) 12.8730 0.804563
\(257\) 22.1299 1.38043 0.690213 0.723607i \(-0.257517\pi\)
0.690213 + 0.723607i \(0.257517\pi\)
\(258\) −1.92967 −0.120136
\(259\) 22.2378 1.38179
\(260\) 0 0
\(261\) −2.39692 −0.148366
\(262\) 9.88854 0.610916
\(263\) −24.7388 −1.52546 −0.762729 0.646719i \(-0.776141\pi\)
−0.762729 + 0.646719i \(0.776141\pi\)
\(264\) −2.77776 −0.170960
\(265\) 0 0
\(266\) 0.638650 0.0391581
\(267\) −12.0162 −0.735377
\(268\) −5.08602 −0.310678
\(269\) 24.4078 1.48817 0.744086 0.668084i \(-0.232885\pi\)
0.744086 + 0.668084i \(0.232885\pi\)
\(270\) 0 0
\(271\) 20.3632 1.23698 0.618488 0.785795i \(-0.287746\pi\)
0.618488 + 0.785795i \(0.287746\pi\)
\(272\) 11.8700 0.719724
\(273\) −3.03109 −0.183450
\(274\) −2.44513 −0.147716
\(275\) 0 0
\(276\) −5.20199 −0.313123
\(277\) 10.2127 0.613621 0.306811 0.951771i \(-0.400738\pi\)
0.306811 + 0.951771i \(0.400738\pi\)
\(278\) 0.388067 0.0232747
\(279\) 6.32953 0.378939
\(280\) 0 0
\(281\) 9.12961 0.544627 0.272314 0.962209i \(-0.412211\pi\)
0.272314 + 0.962209i \(0.412211\pi\)
\(282\) −1.60641 −0.0956605
\(283\) −20.4947 −1.21828 −0.609141 0.793062i \(-0.708486\pi\)
−0.609141 + 0.793062i \(0.708486\pi\)
\(284\) −3.96668 −0.235379
\(285\) 0 0
\(286\) 2.51989 0.149004
\(287\) 3.74694 0.221175
\(288\) −3.18904 −0.187916
\(289\) −10.9456 −0.643857
\(290\) 0 0
\(291\) −13.4808 −0.790256
\(292\) −0.276878 −0.0162031
\(293\) −21.4658 −1.25405 −0.627023 0.779000i \(-0.715727\pi\)
−0.627023 + 0.779000i \(0.715727\pi\)
\(294\) −2.34370 −0.136687
\(295\) 0 0
\(296\) 21.5412 1.25205
\(297\) 1.21821 0.0706877
\(298\) −11.1100 −0.643586
\(299\) −11.5378 −0.667249
\(300\) 0 0
\(301\) 2.82762 0.162981
\(302\) −21.1548 −1.21732
\(303\) 5.27825 0.303228
\(304\) 0.814749 0.0467291
\(305\) 0 0
\(306\) −3.95270 −0.225961
\(307\) −11.4104 −0.651226 −0.325613 0.945503i \(-0.605571\pi\)
−0.325613 + 0.945503i \(0.605571\pi\)
\(308\) −1.66482 −0.0948616
\(309\) 9.08889 0.517049
\(310\) 0 0
\(311\) 7.47198 0.423697 0.211849 0.977303i \(-0.432052\pi\)
0.211849 + 0.977303i \(0.432052\pi\)
\(312\) −2.93614 −0.166226
\(313\) 15.3336 0.866707 0.433353 0.901224i \(-0.357330\pi\)
0.433353 + 0.901224i \(0.357330\pi\)
\(314\) 12.8139 0.723131
\(315\) 0 0
\(316\) −3.00860 −0.169247
\(317\) 18.8478 1.05860 0.529298 0.848436i \(-0.322455\pi\)
0.529298 + 0.848436i \(0.322455\pi\)
\(318\) −8.26107 −0.463258
\(319\) −2.91995 −0.163486
\(320\) 0 0
\(321\) 5.74384 0.320590
\(322\) 33.8823 1.88819
\(323\) 0.415572 0.0231231
\(324\) 0.580562 0.0322534
\(325\) 0 0
\(326\) −29.1467 −1.61429
\(327\) 5.71806 0.316210
\(328\) 3.62956 0.200409
\(329\) 2.35394 0.129777
\(330\) 0 0
\(331\) −3.60922 −0.198380 −0.0991902 0.995068i \(-0.531625\pi\)
−0.0991902 + 0.995068i \(0.531625\pi\)
\(332\) 7.90429 0.433804
\(333\) −9.44703 −0.517694
\(334\) −24.9681 −1.36619
\(335\) 0 0
\(336\) 11.3556 0.619498
\(337\) 18.5074 1.00816 0.504082 0.863656i \(-0.331831\pi\)
0.504082 + 0.863656i \(0.331831\pi\)
\(338\) −18.2198 −0.991027
\(339\) 10.9763 0.596152
\(340\) 0 0
\(341\) 7.71069 0.417557
\(342\) −0.271311 −0.0146708
\(343\) 19.9119 1.07514
\(344\) 2.73905 0.147679
\(345\) 0 0
\(346\) −14.6740 −0.788880
\(347\) −1.05799 −0.0567958 −0.0283979 0.999597i \(-0.509041\pi\)
−0.0283979 + 0.999597i \(0.509041\pi\)
\(348\) −1.39156 −0.0745954
\(349\) 27.1342 1.45246 0.726230 0.687452i \(-0.241271\pi\)
0.726230 + 0.687452i \(0.241271\pi\)
\(350\) 0 0
\(351\) 1.28767 0.0687305
\(352\) −3.88492 −0.207067
\(353\) −4.09521 −0.217966 −0.108983 0.994044i \(-0.534759\pi\)
−0.108983 + 0.994044i \(0.534759\pi\)
\(354\) −19.1222 −1.01633
\(355\) 0 0
\(356\) −6.97612 −0.369734
\(357\) 5.79205 0.306548
\(358\) −19.4811 −1.02961
\(359\) 25.4872 1.34516 0.672580 0.740024i \(-0.265186\pi\)
0.672580 + 0.740024i \(0.265186\pi\)
\(360\) 0 0
\(361\) −18.9715 −0.998499
\(362\) 34.3332 1.80451
\(363\) −9.51597 −0.499459
\(364\) −1.75973 −0.0922352
\(365\) 0 0
\(366\) 22.5337 1.17786
\(367\) −18.1282 −0.946284 −0.473142 0.880986i \(-0.656880\pi\)
−0.473142 + 0.880986i \(0.656880\pi\)
\(368\) 43.2249 2.25326
\(369\) −1.59177 −0.0828643
\(370\) 0 0
\(371\) 12.1053 0.628474
\(372\) 3.67468 0.190524
\(373\) −6.22180 −0.322153 −0.161076 0.986942i \(-0.551497\pi\)
−0.161076 + 0.986942i \(0.551497\pi\)
\(374\) −4.81521 −0.248989
\(375\) 0 0
\(376\) 2.28020 0.117593
\(377\) −3.08643 −0.158959
\(378\) −3.78140 −0.194494
\(379\) 5.23729 0.269021 0.134511 0.990912i \(-0.457054\pi\)
0.134511 + 0.990912i \(0.457054\pi\)
\(380\) 0 0
\(381\) 3.75661 0.192457
\(382\) 21.6832 1.10941
\(383\) −11.0477 −0.564512 −0.282256 0.959339i \(-0.591083\pi\)
−0.282256 + 0.959339i \(0.591083\pi\)
\(384\) 13.6475 0.696444
\(385\) 0 0
\(386\) −12.9005 −0.656621
\(387\) −1.20123 −0.0610619
\(388\) −7.82641 −0.397326
\(389\) 9.44478 0.478869 0.239435 0.970913i \(-0.423038\pi\)
0.239435 + 0.970913i \(0.423038\pi\)
\(390\) 0 0
\(391\) 22.0474 1.11498
\(392\) 3.32673 0.168025
\(393\) 6.15566 0.310512
\(394\) −27.7730 −1.39918
\(395\) 0 0
\(396\) 0.707246 0.0355404
\(397\) −3.00343 −0.150738 −0.0753689 0.997156i \(-0.524013\pi\)
−0.0753689 + 0.997156i \(0.524013\pi\)
\(398\) 8.25823 0.413948
\(399\) 0.397563 0.0199030
\(400\) 0 0
\(401\) −29.6585 −1.48107 −0.740537 0.672015i \(-0.765429\pi\)
−0.740537 + 0.672015i \(0.765429\pi\)
\(402\) −14.0730 −0.701898
\(403\) 8.15032 0.405996
\(404\) 3.06435 0.152457
\(405\) 0 0
\(406\) 9.06371 0.449824
\(407\) −11.5085 −0.570453
\(408\) 5.61061 0.277767
\(409\) 10.5918 0.523730 0.261865 0.965105i \(-0.415663\pi\)
0.261865 + 0.965105i \(0.415663\pi\)
\(410\) 0 0
\(411\) −1.52211 −0.0750801
\(412\) 5.27667 0.259963
\(413\) 28.0205 1.37880
\(414\) −14.3939 −0.707421
\(415\) 0 0
\(416\) −4.10642 −0.201334
\(417\) 0.241574 0.0118299
\(418\) −0.330513 −0.0161659
\(419\) −21.2794 −1.03957 −0.519783 0.854298i \(-0.673987\pi\)
−0.519783 + 0.854298i \(0.673987\pi\)
\(420\) 0 0
\(421\) −18.4416 −0.898789 −0.449395 0.893333i \(-0.648360\pi\)
−0.449395 + 0.893333i \(0.648360\pi\)
\(422\) 9.76426 0.475317
\(423\) −1.00000 −0.0486217
\(424\) 11.7261 0.569468
\(425\) 0 0
\(426\) −10.9758 −0.531778
\(427\) −33.0195 −1.59793
\(428\) 3.33465 0.161187
\(429\) 1.56865 0.0757349
\(430\) 0 0
\(431\) −1.32450 −0.0637991 −0.0318995 0.999491i \(-0.510156\pi\)
−0.0318995 + 0.999491i \(0.510156\pi\)
\(432\) −4.82407 −0.232098
\(433\) −27.9318 −1.34232 −0.671159 0.741313i \(-0.734203\pi\)
−0.671159 + 0.741313i \(0.734203\pi\)
\(434\) −23.9345 −1.14889
\(435\) 0 0
\(436\) 3.31969 0.158984
\(437\) 1.51332 0.0723919
\(438\) −0.766120 −0.0366066
\(439\) 18.3491 0.875756 0.437878 0.899034i \(-0.355730\pi\)
0.437878 + 0.899034i \(0.355730\pi\)
\(440\) 0 0
\(441\) −1.45896 −0.0694745
\(442\) −5.08975 −0.242095
\(443\) −13.7771 −0.654569 −0.327285 0.944926i \(-0.606134\pi\)
−0.327285 + 0.944926i \(0.606134\pi\)
\(444\) −5.48459 −0.260287
\(445\) 0 0
\(446\) 20.6873 0.979573
\(447\) −6.91605 −0.327118
\(448\) −10.6521 −0.503265
\(449\) −9.08241 −0.428625 −0.214313 0.976765i \(-0.568751\pi\)
−0.214313 + 0.976765i \(0.568751\pi\)
\(450\) 0 0
\(451\) −1.93911 −0.0913092
\(452\) 6.37244 0.299734
\(453\) −13.1690 −0.618732
\(454\) 22.2900 1.04612
\(455\) 0 0
\(456\) 0.385109 0.0180344
\(457\) 28.8025 1.34732 0.673662 0.739039i \(-0.264720\pi\)
0.673662 + 0.739039i \(0.264720\pi\)
\(458\) 38.7167 1.80911
\(459\) −2.46057 −0.114850
\(460\) 0 0
\(461\) −29.5937 −1.37831 −0.689157 0.724612i \(-0.742019\pi\)
−0.689157 + 0.724612i \(0.742019\pi\)
\(462\) −4.60654 −0.214315
\(463\) −17.0730 −0.793447 −0.396724 0.917938i \(-0.629853\pi\)
−0.396724 + 0.917938i \(0.629853\pi\)
\(464\) 11.5629 0.536794
\(465\) 0 0
\(466\) 42.5800 1.97248
\(467\) −26.7895 −1.23967 −0.619834 0.784733i \(-0.712800\pi\)
−0.619834 + 0.784733i \(0.712800\pi\)
\(468\) 0.747570 0.0345564
\(469\) 20.6217 0.952223
\(470\) 0 0
\(471\) 7.97672 0.367548
\(472\) 27.1427 1.24935
\(473\) −1.46335 −0.0672848
\(474\) −8.32478 −0.382370
\(475\) 0 0
\(476\) 3.36264 0.154126
\(477\) −5.14255 −0.235461
\(478\) −9.47198 −0.433238
\(479\) −30.0981 −1.37522 −0.687609 0.726081i \(-0.741340\pi\)
−0.687609 + 0.726081i \(0.741340\pi\)
\(480\) 0 0
\(481\) −12.1646 −0.554659
\(482\) 7.24184 0.329857
\(483\) 21.0919 0.959716
\(484\) −5.52461 −0.251119
\(485\) 0 0
\(486\) 1.60641 0.0728684
\(487\) −10.5069 −0.476114 −0.238057 0.971251i \(-0.576510\pi\)
−0.238057 + 0.971251i \(0.576510\pi\)
\(488\) −31.9852 −1.44790
\(489\) −18.1440 −0.820498
\(490\) 0 0
\(491\) 17.5258 0.790928 0.395464 0.918481i \(-0.370584\pi\)
0.395464 + 0.918481i \(0.370584\pi\)
\(492\) −0.924122 −0.0416626
\(493\) 5.89780 0.265623
\(494\) −0.349358 −0.0157183
\(495\) 0 0
\(496\) −30.5341 −1.37102
\(497\) 16.0833 0.721432
\(498\) 21.8711 0.980069
\(499\) 3.44802 0.154354 0.0771772 0.997017i \(-0.475409\pi\)
0.0771772 + 0.997017i \(0.475409\pi\)
\(500\) 0 0
\(501\) −15.5428 −0.694399
\(502\) 15.6075 0.696597
\(503\) 1.65374 0.0737367 0.0368684 0.999320i \(-0.488262\pi\)
0.0368684 + 0.999320i \(0.488262\pi\)
\(504\) 5.36746 0.239086
\(505\) 0 0
\(506\) −17.5348 −0.779515
\(507\) −11.3419 −0.503712
\(508\) 2.18094 0.0967638
\(509\) 23.3010 1.03280 0.516399 0.856348i \(-0.327272\pi\)
0.516399 + 0.856348i \(0.327272\pi\)
\(510\) 0 0
\(511\) 1.12263 0.0496621
\(512\) −6.61555 −0.292369
\(513\) −0.168892 −0.00745678
\(514\) 35.5498 1.56803
\(515\) 0 0
\(516\) −0.697388 −0.0307008
\(517\) −1.21821 −0.0535768
\(518\) 35.7230 1.56958
\(519\) −9.13465 −0.400967
\(520\) 0 0
\(521\) −12.6319 −0.553415 −0.276707 0.960954i \(-0.589243\pi\)
−0.276707 + 0.960954i \(0.589243\pi\)
\(522\) −3.85044 −0.168529
\(523\) −29.1509 −1.27468 −0.637340 0.770583i \(-0.719965\pi\)
−0.637340 + 0.770583i \(0.719965\pi\)
\(524\) 3.57374 0.156120
\(525\) 0 0
\(526\) −39.7407 −1.73278
\(527\) −15.5743 −0.678426
\(528\) −5.87673 −0.255752
\(529\) 57.2863 2.49071
\(530\) 0 0
\(531\) −11.9036 −0.516574
\(532\) 0.230810 0.0100069
\(533\) −2.04967 −0.0887810
\(534\) −19.3029 −0.835319
\(535\) 0 0
\(536\) 19.9758 0.862821
\(537\) −12.1271 −0.523323
\(538\) 39.2091 1.69042
\(539\) −1.77732 −0.0765547
\(540\) 0 0
\(541\) 26.2586 1.12894 0.564472 0.825452i \(-0.309080\pi\)
0.564472 + 0.825452i \(0.309080\pi\)
\(542\) 32.7117 1.40509
\(543\) 21.3726 0.917184
\(544\) 7.84688 0.336432
\(545\) 0 0
\(546\) −4.86918 −0.208382
\(547\) −22.9028 −0.979253 −0.489626 0.871932i \(-0.662867\pi\)
−0.489626 + 0.871932i \(0.662867\pi\)
\(548\) −0.883679 −0.0377489
\(549\) 14.0274 0.598673
\(550\) 0 0
\(551\) 0.404821 0.0172460
\(552\) 20.4312 0.869610
\(553\) 12.1986 0.518739
\(554\) 16.4058 0.697016
\(555\) 0 0
\(556\) 0.140249 0.00594787
\(557\) 5.02429 0.212886 0.106443 0.994319i \(-0.466054\pi\)
0.106443 + 0.994319i \(0.466054\pi\)
\(558\) 10.1678 0.430439
\(559\) −1.54678 −0.0654219
\(560\) 0 0
\(561\) −2.99749 −0.126554
\(562\) 14.6659 0.618645
\(563\) −22.6970 −0.956564 −0.478282 0.878206i \(-0.658740\pi\)
−0.478282 + 0.878206i \(0.658740\pi\)
\(564\) −0.580562 −0.0244461
\(565\) 0 0
\(566\) −32.9229 −1.38385
\(567\) −2.35394 −0.0988562
\(568\) 15.5794 0.653699
\(569\) −17.3367 −0.726791 −0.363395 0.931635i \(-0.618383\pi\)
−0.363395 + 0.931635i \(0.618383\pi\)
\(570\) 0 0
\(571\) −9.39051 −0.392981 −0.196490 0.980506i \(-0.562954\pi\)
−0.196490 + 0.980506i \(0.562954\pi\)
\(572\) 0.910696 0.0380781
\(573\) 13.4979 0.563882
\(574\) 6.01913 0.251233
\(575\) 0 0
\(576\) 4.52522 0.188551
\(577\) 18.8942 0.786576 0.393288 0.919415i \(-0.371337\pi\)
0.393288 + 0.919415i \(0.371337\pi\)
\(578\) −17.5831 −0.731361
\(579\) −8.03065 −0.333743
\(580\) 0 0
\(581\) −32.0487 −1.32960
\(582\) −21.6557 −0.897656
\(583\) −6.26471 −0.259458
\(584\) 1.08746 0.0449994
\(585\) 0 0
\(586\) −34.4830 −1.42448
\(587\) −44.8632 −1.85170 −0.925851 0.377889i \(-0.876650\pi\)
−0.925851 + 0.377889i \(0.876650\pi\)
\(588\) −0.847019 −0.0349305
\(589\) −1.06901 −0.0440478
\(590\) 0 0
\(591\) −17.2888 −0.711167
\(592\) 45.5732 1.87305
\(593\) −2.66127 −0.109285 −0.0546425 0.998506i \(-0.517402\pi\)
−0.0546425 + 0.998506i \(0.517402\pi\)
\(594\) 1.95695 0.0802945
\(595\) 0 0
\(596\) −4.01519 −0.164469
\(597\) 5.14079 0.210399
\(598\) −18.5345 −0.757932
\(599\) −12.2072 −0.498771 −0.249386 0.968404i \(-0.580229\pi\)
−0.249386 + 0.968404i \(0.580229\pi\)
\(600\) 0 0
\(601\) −6.16119 −0.251320 −0.125660 0.992073i \(-0.540105\pi\)
−0.125660 + 0.992073i \(0.540105\pi\)
\(602\) 4.54233 0.185131
\(603\) −8.76052 −0.356756
\(604\) −7.64540 −0.311087
\(605\) 0 0
\(606\) 8.47905 0.344438
\(607\) −11.0117 −0.446952 −0.223476 0.974709i \(-0.571740\pi\)
−0.223476 + 0.974709i \(0.571740\pi\)
\(608\) 0.538605 0.0218433
\(609\) 5.64220 0.228634
\(610\) 0 0
\(611\) −1.28767 −0.0520934
\(612\) −1.42852 −0.0577443
\(613\) −24.1631 −0.975938 −0.487969 0.872861i \(-0.662262\pi\)
−0.487969 + 0.872861i \(0.662262\pi\)
\(614\) −18.3298 −0.739731
\(615\) 0 0
\(616\) 6.53869 0.263451
\(617\) −28.8522 −1.16155 −0.580773 0.814066i \(-0.697250\pi\)
−0.580773 + 0.814066i \(0.697250\pi\)
\(618\) 14.6005 0.587319
\(619\) 1.78397 0.0717037 0.0358519 0.999357i \(-0.488586\pi\)
0.0358519 + 0.999357i \(0.488586\pi\)
\(620\) 0 0
\(621\) −8.96026 −0.359563
\(622\) 12.0031 0.481280
\(623\) 28.2853 1.13323
\(624\) −6.21179 −0.248671
\(625\) 0 0
\(626\) 24.6321 0.984497
\(627\) −0.205746 −0.00821671
\(628\) 4.63098 0.184796
\(629\) 23.2451 0.926844
\(630\) 0 0
\(631\) 12.2661 0.488306 0.244153 0.969737i \(-0.421490\pi\)
0.244153 + 0.969737i \(0.421490\pi\)
\(632\) 11.8165 0.470036
\(633\) 6.07830 0.241591
\(634\) 30.2773 1.20246
\(635\) 0 0
\(636\) −2.98557 −0.118386
\(637\) −1.87866 −0.0744351
\(638\) −4.69064 −0.185704
\(639\) −6.83248 −0.270289
\(640\) 0 0
\(641\) −31.0075 −1.22472 −0.612362 0.790578i \(-0.709780\pi\)
−0.612362 + 0.790578i \(0.709780\pi\)
\(642\) 9.22697 0.364160
\(643\) 23.9013 0.942574 0.471287 0.881980i \(-0.343790\pi\)
0.471287 + 0.881980i \(0.343790\pi\)
\(644\) 12.2452 0.482527
\(645\) 0 0
\(646\) 0.667581 0.0262656
\(647\) 28.8735 1.13513 0.567566 0.823328i \(-0.307885\pi\)
0.567566 + 0.823328i \(0.307885\pi\)
\(648\) −2.28020 −0.0895748
\(649\) −14.5011 −0.569219
\(650\) 0 0
\(651\) −14.8993 −0.583951
\(652\) −10.5337 −0.412531
\(653\) 36.1118 1.41316 0.706582 0.707631i \(-0.250236\pi\)
0.706582 + 0.707631i \(0.250236\pi\)
\(654\) 9.18557 0.359184
\(655\) 0 0
\(656\) 7.67882 0.299808
\(657\) −0.476914 −0.0186062
\(658\) 3.78140 0.147414
\(659\) 31.4095 1.22354 0.611770 0.791036i \(-0.290458\pi\)
0.611770 + 0.791036i \(0.290458\pi\)
\(660\) 0 0
\(661\) 10.1647 0.395361 0.197681 0.980266i \(-0.436659\pi\)
0.197681 + 0.980266i \(0.436659\pi\)
\(662\) −5.79789 −0.225341
\(663\) −3.16840 −0.123050
\(664\) −31.0447 −1.20477
\(665\) 0 0
\(666\) −15.1758 −0.588052
\(667\) 21.4770 0.831593
\(668\) −9.02353 −0.349131
\(669\) 12.8780 0.497891
\(670\) 0 0
\(671\) 17.0882 0.659684
\(672\) 7.50682 0.289582
\(673\) 13.4744 0.519400 0.259700 0.965689i \(-0.416376\pi\)
0.259700 + 0.965689i \(0.416376\pi\)
\(674\) 29.7306 1.14518
\(675\) 0 0
\(676\) −6.58469 −0.253257
\(677\) −1.48122 −0.0569281 −0.0284640 0.999595i \(-0.509062\pi\)
−0.0284640 + 0.999595i \(0.509062\pi\)
\(678\) 17.6325 0.677172
\(679\) 31.7329 1.21780
\(680\) 0 0
\(681\) 13.8756 0.531715
\(682\) 12.3866 0.474306
\(683\) 14.5690 0.557469 0.278734 0.960368i \(-0.410085\pi\)
0.278734 + 0.960368i \(0.410085\pi\)
\(684\) −0.0980525 −0.00374913
\(685\) 0 0
\(686\) 31.9867 1.22126
\(687\) 24.1014 0.919524
\(688\) 5.79481 0.220925
\(689\) −6.62189 −0.252274
\(690\) 0 0
\(691\) −25.4543 −0.968327 −0.484163 0.874978i \(-0.660876\pi\)
−0.484163 + 0.874978i \(0.660876\pi\)
\(692\) −5.30323 −0.201599
\(693\) −2.86759 −0.108931
\(694\) −1.69957 −0.0645146
\(695\) 0 0
\(696\) 5.46546 0.207168
\(697\) 3.91667 0.148355
\(698\) 43.5887 1.64986
\(699\) 26.5063 1.00256
\(700\) 0 0
\(701\) 47.1753 1.78179 0.890894 0.454211i \(-0.150079\pi\)
0.890894 + 0.454211i \(0.150079\pi\)
\(702\) 2.06852 0.0780713
\(703\) 1.59553 0.0601766
\(704\) 5.51267 0.207766
\(705\) 0 0
\(706\) −6.57860 −0.247589
\(707\) −12.4247 −0.467279
\(708\) −6.91081 −0.259724
\(709\) −37.8007 −1.41964 −0.709818 0.704385i \(-0.751223\pi\)
−0.709818 + 0.704385i \(0.751223\pi\)
\(710\) 0 0
\(711\) −5.18222 −0.194348
\(712\) 27.3993 1.02683
\(713\) −56.7142 −2.12397
\(714\) 9.30442 0.348209
\(715\) 0 0
\(716\) −7.04053 −0.263117
\(717\) −5.89635 −0.220203
\(718\) 40.9429 1.52797
\(719\) −13.3382 −0.497431 −0.248716 0.968577i \(-0.580008\pi\)
−0.248716 + 0.968577i \(0.580008\pi\)
\(720\) 0 0
\(721\) −21.3947 −0.796781
\(722\) −30.4760 −1.13420
\(723\) 4.50808 0.167657
\(724\) 12.4081 0.461143
\(725\) 0 0
\(726\) −15.2866 −0.567338
\(727\) 0.000763187 0 2.83050e−5 0 1.41525e−5 1.00000i \(-0.499995\pi\)
1.41525e−5 1.00000i \(0.499995\pi\)
\(728\) 6.91150 0.256157
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 2.95571 0.109321
\(732\) 8.14375 0.301002
\(733\) 39.7135 1.46685 0.733425 0.679770i \(-0.237920\pi\)
0.733425 + 0.679770i \(0.237920\pi\)
\(734\) −29.1214 −1.07489
\(735\) 0 0
\(736\) 28.5747 1.05328
\(737\) −10.6721 −0.393113
\(738\) −2.55704 −0.0941260
\(739\) 20.0290 0.736777 0.368389 0.929672i \(-0.379910\pi\)
0.368389 + 0.929672i \(0.379910\pi\)
\(740\) 0 0
\(741\) −0.217477 −0.00798921
\(742\) 19.4461 0.713887
\(743\) 12.4716 0.457539 0.228769 0.973481i \(-0.426530\pi\)
0.228769 + 0.973481i \(0.426530\pi\)
\(744\) −14.4326 −0.529125
\(745\) 0 0
\(746\) −9.99478 −0.365935
\(747\) 13.6149 0.498143
\(748\) −1.74023 −0.0636292
\(749\) −13.5207 −0.494034
\(750\) 0 0
\(751\) −21.9589 −0.801291 −0.400646 0.916233i \(-0.631214\pi\)
−0.400646 + 0.916233i \(0.631214\pi\)
\(752\) 4.82407 0.175916
\(753\) 9.71575 0.354062
\(754\) −4.95808 −0.180563
\(755\) 0 0
\(756\) −1.36661 −0.0497031
\(757\) −26.4162 −0.960113 −0.480056 0.877238i \(-0.659384\pi\)
−0.480056 + 0.877238i \(0.659384\pi\)
\(758\) 8.41325 0.305583
\(759\) −10.9155 −0.396207
\(760\) 0 0
\(761\) −36.2026 −1.31234 −0.656172 0.754611i \(-0.727826\pi\)
−0.656172 + 0.754611i \(0.727826\pi\)
\(762\) 6.03466 0.218613
\(763\) −13.4600 −0.487284
\(764\) 7.83636 0.283510
\(765\) 0 0
\(766\) −17.7472 −0.641232
\(767\) −15.3279 −0.553459
\(768\) 12.8730 0.464515
\(769\) 34.8533 1.25684 0.628422 0.777873i \(-0.283701\pi\)
0.628422 + 0.777873i \(0.283701\pi\)
\(770\) 0 0
\(771\) 22.1299 0.796989
\(772\) −4.66229 −0.167800
\(773\) 9.23266 0.332076 0.166038 0.986119i \(-0.446903\pi\)
0.166038 + 0.986119i \(0.446903\pi\)
\(774\) −1.92967 −0.0693605
\(775\) 0 0
\(776\) 30.7389 1.10346
\(777\) 22.2378 0.797775
\(778\) 15.1722 0.543950
\(779\) 0.268838 0.00963212
\(780\) 0 0
\(781\) −8.32339 −0.297834
\(782\) 35.4172 1.26652
\(783\) −2.39692 −0.0856589
\(784\) 7.03814 0.251362
\(785\) 0 0
\(786\) 9.88854 0.352712
\(787\) −17.2217 −0.613886 −0.306943 0.951728i \(-0.599306\pi\)
−0.306943 + 0.951728i \(0.599306\pi\)
\(788\) −10.0372 −0.357561
\(789\) −24.7388 −0.880723
\(790\) 0 0
\(791\) −25.8376 −0.918680
\(792\) −2.77776 −0.0987036
\(793\) 18.0625 0.641419
\(794\) −4.82475 −0.171224
\(795\) 0 0
\(796\) 2.98455 0.105785
\(797\) −2.93899 −0.104105 −0.0520523 0.998644i \(-0.516576\pi\)
−0.0520523 + 0.998644i \(0.516576\pi\)
\(798\) 0.638650 0.0226080
\(799\) 2.46057 0.0870489
\(800\) 0 0
\(801\) −12.0162 −0.424570
\(802\) −47.6438 −1.68236
\(803\) −0.580981 −0.0205024
\(804\) −5.08602 −0.179370
\(805\) 0 0
\(806\) 13.0928 0.461173
\(807\) 24.4078 0.859196
\(808\) −12.0355 −0.423407
\(809\) −47.0667 −1.65478 −0.827389 0.561629i \(-0.810175\pi\)
−0.827389 + 0.561629i \(0.810175\pi\)
\(810\) 0 0
\(811\) 12.1936 0.428176 0.214088 0.976814i \(-0.431322\pi\)
0.214088 + 0.976814i \(0.431322\pi\)
\(812\) 3.27565 0.114953
\(813\) 20.3632 0.714168
\(814\) −18.4873 −0.647981
\(815\) 0 0
\(816\) 11.8700 0.415533
\(817\) 0.202878 0.00709782
\(818\) 17.0148 0.594907
\(819\) −3.03109 −0.105915
\(820\) 0 0
\(821\) −27.8694 −0.972647 −0.486324 0.873779i \(-0.661662\pi\)
−0.486324 + 0.873779i \(0.661662\pi\)
\(822\) −2.44513 −0.0852839
\(823\) 26.1727 0.912323 0.456161 0.889897i \(-0.349224\pi\)
0.456161 + 0.889897i \(0.349224\pi\)
\(824\) −20.7245 −0.721973
\(825\) 0 0
\(826\) 45.0125 1.56618
\(827\) 6.25176 0.217395 0.108697 0.994075i \(-0.465332\pi\)
0.108697 + 0.994075i \(0.465332\pi\)
\(828\) −5.20199 −0.180782
\(829\) −2.54642 −0.0884409 −0.0442205 0.999022i \(-0.514080\pi\)
−0.0442205 + 0.999022i \(0.514080\pi\)
\(830\) 0 0
\(831\) 10.2127 0.354274
\(832\) 5.82697 0.202014
\(833\) 3.58989 0.124382
\(834\) 0.388067 0.0134377
\(835\) 0 0
\(836\) −0.119448 −0.00413121
\(837\) 6.32953 0.218781
\(838\) −34.1835 −1.18085
\(839\) 6.20012 0.214052 0.107026 0.994256i \(-0.465867\pi\)
0.107026 + 0.994256i \(0.465867\pi\)
\(840\) 0 0
\(841\) −23.2548 −0.801889
\(842\) −29.6248 −1.02094
\(843\) 9.12961 0.314441
\(844\) 3.52883 0.121467
\(845\) 0 0
\(846\) −1.60641 −0.0552296
\(847\) 22.4000 0.769674
\(848\) 24.8081 0.851912
\(849\) −20.4947 −0.703375
\(850\) 0 0
\(851\) 84.6479 2.90169
\(852\) −3.96668 −0.135896
\(853\) −30.5276 −1.04525 −0.522623 0.852564i \(-0.675047\pi\)
−0.522623 + 0.852564i \(0.675047\pi\)
\(854\) −53.0430 −1.81509
\(855\) 0 0
\(856\) −13.0971 −0.447650
\(857\) −18.2826 −0.624522 −0.312261 0.949996i \(-0.601086\pi\)
−0.312261 + 0.949996i \(0.601086\pi\)
\(858\) 2.51989 0.0860277
\(859\) 42.6716 1.45594 0.727969 0.685610i \(-0.240465\pi\)
0.727969 + 0.685610i \(0.240465\pi\)
\(860\) 0 0
\(861\) 3.74694 0.127695
\(862\) −2.12770 −0.0724697
\(863\) −3.21332 −0.109383 −0.0546913 0.998503i \(-0.517417\pi\)
−0.0546913 + 0.998503i \(0.517417\pi\)
\(864\) −3.18904 −0.108493
\(865\) 0 0
\(866\) −44.8701 −1.52475
\(867\) −10.9456 −0.371731
\(868\) −8.64999 −0.293600
\(869\) −6.31303 −0.214155
\(870\) 0 0
\(871\) −11.2806 −0.382229
\(872\) −13.0383 −0.441534
\(873\) −13.4808 −0.456254
\(874\) 2.43102 0.0822304
\(875\) 0 0
\(876\) −0.276878 −0.00935484
\(877\) 48.8397 1.64920 0.824600 0.565717i \(-0.191400\pi\)
0.824600 + 0.565717i \(0.191400\pi\)
\(878\) 29.4763 0.994776
\(879\) −21.4658 −0.724024
\(880\) 0 0
\(881\) −29.2473 −0.985367 −0.492684 0.870209i \(-0.663984\pi\)
−0.492684 + 0.870209i \(0.663984\pi\)
\(882\) −2.34370 −0.0789164
\(883\) −37.5839 −1.26480 −0.632400 0.774642i \(-0.717930\pi\)
−0.632400 + 0.774642i \(0.717930\pi\)
\(884\) −1.83945 −0.0618674
\(885\) 0 0
\(886\) −22.1317 −0.743529
\(887\) 10.7876 0.362211 0.181106 0.983464i \(-0.442032\pi\)
0.181106 + 0.983464i \(0.442032\pi\)
\(888\) 21.5412 0.722874
\(889\) −8.84283 −0.296579
\(890\) 0 0
\(891\) 1.21821 0.0408115
\(892\) 7.47645 0.250330
\(893\) 0.168892 0.00565177
\(894\) −11.1100 −0.371575
\(895\) 0 0
\(896\) −32.1253 −1.07323
\(897\) −11.5378 −0.385237
\(898\) −14.5901 −0.486878
\(899\) −15.1714 −0.505993
\(900\) 0 0
\(901\) 12.6536 0.421554
\(902\) −3.11501 −0.103719
\(903\) 2.82762 0.0940973
\(904\) −25.0282 −0.832427
\(905\) 0 0
\(906\) −21.1548 −0.702821
\(907\) 35.2429 1.17022 0.585111 0.810953i \(-0.301051\pi\)
0.585111 + 0.810953i \(0.301051\pi\)
\(908\) 8.05565 0.267336
\(909\) 5.27825 0.175069
\(910\) 0 0
\(911\) −11.1069 −0.367987 −0.183993 0.982927i \(-0.558903\pi\)
−0.183993 + 0.982927i \(0.558903\pi\)
\(912\) 0.814749 0.0269790
\(913\) 16.5858 0.548910
\(914\) 46.2687 1.53043
\(915\) 0 0
\(916\) 13.9923 0.462320
\(917\) −14.4901 −0.478504
\(918\) −3.95270 −0.130458
\(919\) −47.4809 −1.56625 −0.783126 0.621863i \(-0.786376\pi\)
−0.783126 + 0.621863i \(0.786376\pi\)
\(920\) 0 0
\(921\) −11.4104 −0.375985
\(922\) −47.5396 −1.56563
\(923\) −8.79795 −0.289588
\(924\) −1.66482 −0.0547684
\(925\) 0 0
\(926\) −27.4262 −0.901281
\(927\) 9.08889 0.298518
\(928\) 7.64388 0.250923
\(929\) 32.0236 1.05066 0.525331 0.850898i \(-0.323942\pi\)
0.525331 + 0.850898i \(0.323942\pi\)
\(930\) 0 0
\(931\) 0.246408 0.00807569
\(932\) 15.3885 0.504068
\(933\) 7.47198 0.244622
\(934\) −43.0350 −1.40815
\(935\) 0 0
\(936\) −2.93614 −0.0959707
\(937\) −29.3965 −0.960342 −0.480171 0.877175i \(-0.659425\pi\)
−0.480171 + 0.877175i \(0.659425\pi\)
\(938\) 33.1270 1.08164
\(939\) 15.3336 0.500394
\(940\) 0 0
\(941\) −27.5767 −0.898976 −0.449488 0.893286i \(-0.648393\pi\)
−0.449488 + 0.893286i \(0.648393\pi\)
\(942\) 12.8139 0.417500
\(943\) 14.2627 0.464457
\(944\) 57.4240 1.86899
\(945\) 0 0
\(946\) −2.35074 −0.0764292
\(947\) −6.49455 −0.211044 −0.105522 0.994417i \(-0.533651\pi\)
−0.105522 + 0.994417i \(0.533651\pi\)
\(948\) −3.00860 −0.0977147
\(949\) −0.614105 −0.0199347
\(950\) 0 0
\(951\) 18.8478 0.611181
\(952\) −13.2070 −0.428043
\(953\) −54.1799 −1.75506 −0.877530 0.479522i \(-0.840810\pi\)
−0.877530 + 0.479522i \(0.840810\pi\)
\(954\) −8.26107 −0.267462
\(955\) 0 0
\(956\) −3.42320 −0.110714
\(957\) −2.91995 −0.0943885
\(958\) −48.3500 −1.56212
\(959\) 3.58295 0.115700
\(960\) 0 0
\(961\) 9.06294 0.292353
\(962\) −19.5414 −0.630040
\(963\) 5.74384 0.185093
\(964\) 2.61722 0.0842950
\(965\) 0 0
\(966\) 33.8823 1.09015
\(967\) −49.7977 −1.60139 −0.800693 0.599075i \(-0.795535\pi\)
−0.800693 + 0.599075i \(0.795535\pi\)
\(968\) 21.6983 0.697411
\(969\) 0.415572 0.0133501
\(970\) 0 0
\(971\) −43.9456 −1.41028 −0.705140 0.709068i \(-0.749116\pi\)
−0.705140 + 0.709068i \(0.749116\pi\)
\(972\) 0.580562 0.0186215
\(973\) −0.568651 −0.0182301
\(974\) −16.8784 −0.540820
\(975\) 0 0
\(976\) −67.6689 −2.16603
\(977\) −23.1730 −0.741371 −0.370685 0.928758i \(-0.620877\pi\)
−0.370685 + 0.928758i \(0.620877\pi\)
\(978\) −29.1467 −0.932008
\(979\) −14.6382 −0.467839
\(980\) 0 0
\(981\) 5.71806 0.182564
\(982\) 28.1537 0.898420
\(983\) −61.2466 −1.95346 −0.976732 0.214465i \(-0.931199\pi\)
−0.976732 + 0.214465i \(0.931199\pi\)
\(984\) 3.62956 0.115706
\(985\) 0 0
\(986\) 9.47429 0.301723
\(987\) 2.35394 0.0749268
\(988\) −0.126259 −0.00401683
\(989\) 10.7633 0.342254
\(990\) 0 0
\(991\) 57.7571 1.83471 0.917357 0.398064i \(-0.130318\pi\)
0.917357 + 0.398064i \(0.130318\pi\)
\(992\) −20.1851 −0.640879
\(993\) −3.60922 −0.114535
\(994\) 25.8363 0.819479
\(995\) 0 0
\(996\) 7.90429 0.250457
\(997\) 37.2895 1.18097 0.590486 0.807048i \(-0.298936\pi\)
0.590486 + 0.807048i \(0.298936\pi\)
\(998\) 5.53894 0.175332
\(999\) −9.44703 −0.298891
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 3525.2.a.bd.1.7 8
5.4 even 2 3525.2.a.be.1.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3525.2.a.bd.1.7 8 1.1 even 1 trivial
3525.2.a.be.1.2 yes 8 5.4 even 2