Properties

Label 8010.2.a.bn.1.1
Level $8010$
Weight $2$
Character 8010.1
Self dual yes
Analytic conductor $63.960$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8010,2,Mod(1,8010)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8010, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8010.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8010 = 2 \cdot 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8010.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.9601720190\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 24x^{5} + 34x^{4} + 111x^{3} - 127x^{2} - 20x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 2670)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(5.68936\) of defining polynomial
Character \(\chi\) \(=\) 8010.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -5.07548 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -5.07548 q^{7} +1.00000 q^{8} +1.00000 q^{10} -1.99647 q^{11} -5.82627 q^{13} -5.07548 q^{14} +1.00000 q^{16} +4.54221 q^{17} -7.38225 q^{19} +1.00000 q^{20} -1.99647 q^{22} +1.23986 q^{23} +1.00000 q^{25} -5.82627 q^{26} -5.07548 q^{28} -2.25904 q^{29} -2.32468 q^{31} +1.00000 q^{32} +4.54221 q^{34} -5.07548 q^{35} +8.84545 q^{37} -7.38225 q^{38} +1.00000 q^{40} +10.3743 q^{41} -11.6712 q^{43} -1.99647 q^{44} +1.23986 q^{46} +4.79584 q^{47} +18.7605 q^{49} +1.00000 q^{50} -5.82627 q^{52} +10.0857 q^{53} -1.99647 q^{55} -5.07548 q^{56} -2.25904 q^{58} +3.00090 q^{59} +5.57657 q^{61} -2.32468 q^{62} +1.00000 q^{64} -5.82627 q^{65} +10.6279 q^{67} +4.54221 q^{68} -5.07548 q^{70} +8.65205 q^{71} +1.77313 q^{73} +8.84545 q^{74} -7.38225 q^{76} +10.1330 q^{77} -6.85479 q^{79} +1.00000 q^{80} +10.3743 q^{82} -9.44789 q^{83} +4.54221 q^{85} -11.6712 q^{86} -1.99647 q^{88} -1.00000 q^{89} +29.5711 q^{91} +1.23986 q^{92} +4.79584 q^{94} -7.38225 q^{95} +4.47973 q^{97} +18.7605 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 7 q^{2} + 7 q^{4} + 7 q^{5} + q^{7} + 7 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 7 q^{2} + 7 q^{4} + 7 q^{5} + q^{7} + 7 q^{8} + 7 q^{10} - q^{11} + q^{14} + 7 q^{16} + 9 q^{17} + 9 q^{19} + 7 q^{20} - q^{22} + 8 q^{23} + 7 q^{25} + q^{28} + 4 q^{29} + 16 q^{31} + 7 q^{32} + 9 q^{34} + q^{35} + 2 q^{37} + 9 q^{38} + 7 q^{40} + q^{41} - 10 q^{43} - q^{44} + 8 q^{46} + 13 q^{47} + 26 q^{49} + 7 q^{50} + 24 q^{53} - q^{55} + q^{56} + 4 q^{58} + 6 q^{59} + 23 q^{61} + 16 q^{62} + 7 q^{64} + 5 q^{67} + 9 q^{68} + q^{70} + 8 q^{71} - 2 q^{73} + 2 q^{74} + 9 q^{76} + 6 q^{77} + 7 q^{79} + 7 q^{80} + q^{82} + 7 q^{83} + 9 q^{85} - 10 q^{86} - q^{88} - 7 q^{89} + 48 q^{91} + 8 q^{92} + 13 q^{94} + 9 q^{95} + 30 q^{97} + 26 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −5.07548 −1.91835 −0.959175 0.282813i \(-0.908732\pi\)
−0.959175 + 0.282813i \(0.908732\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −1.99647 −0.601959 −0.300979 0.953631i \(-0.597314\pi\)
−0.300979 + 0.953631i \(0.597314\pi\)
\(12\) 0 0
\(13\) −5.82627 −1.61592 −0.807959 0.589239i \(-0.799428\pi\)
−0.807959 + 0.589239i \(0.799428\pi\)
\(14\) −5.07548 −1.35648
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 4.54221 1.10165 0.550824 0.834622i \(-0.314314\pi\)
0.550824 + 0.834622i \(0.314314\pi\)
\(18\) 0 0
\(19\) −7.38225 −1.69360 −0.846802 0.531908i \(-0.821475\pi\)
−0.846802 + 0.531908i \(0.821475\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −1.99647 −0.425649
\(23\) 1.23986 0.258530 0.129265 0.991610i \(-0.458738\pi\)
0.129265 + 0.991610i \(0.458738\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −5.82627 −1.14263
\(27\) 0 0
\(28\) −5.07548 −0.959175
\(29\) −2.25904 −0.419494 −0.209747 0.977756i \(-0.567264\pi\)
−0.209747 + 0.977756i \(0.567264\pi\)
\(30\) 0 0
\(31\) −2.32468 −0.417525 −0.208763 0.977966i \(-0.566944\pi\)
−0.208763 + 0.977966i \(0.566944\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 4.54221 0.778982
\(35\) −5.07548 −0.857912
\(36\) 0 0
\(37\) 8.84545 1.45418 0.727092 0.686541i \(-0.240872\pi\)
0.727092 + 0.686541i \(0.240872\pi\)
\(38\) −7.38225 −1.19756
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 10.3743 1.62019 0.810096 0.586297i \(-0.199415\pi\)
0.810096 + 0.586297i \(0.199415\pi\)
\(42\) 0 0
\(43\) −11.6712 −1.77985 −0.889923 0.456112i \(-0.849242\pi\)
−0.889923 + 0.456112i \(0.849242\pi\)
\(44\) −1.99647 −0.300979
\(45\) 0 0
\(46\) 1.23986 0.182808
\(47\) 4.79584 0.699545 0.349773 0.936835i \(-0.386259\pi\)
0.349773 + 0.936835i \(0.386259\pi\)
\(48\) 0 0
\(49\) 18.7605 2.68007
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) −5.82627 −0.807959
\(53\) 10.0857 1.38538 0.692690 0.721236i \(-0.256425\pi\)
0.692690 + 0.721236i \(0.256425\pi\)
\(54\) 0 0
\(55\) −1.99647 −0.269204
\(56\) −5.07548 −0.678239
\(57\) 0 0
\(58\) −2.25904 −0.296627
\(59\) 3.00090 0.390684 0.195342 0.980735i \(-0.437418\pi\)
0.195342 + 0.980735i \(0.437418\pi\)
\(60\) 0 0
\(61\) 5.57657 0.714007 0.357003 0.934103i \(-0.383798\pi\)
0.357003 + 0.934103i \(0.383798\pi\)
\(62\) −2.32468 −0.295235
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −5.82627 −0.722660
\(66\) 0 0
\(67\) 10.6279 1.29841 0.649204 0.760614i \(-0.275102\pi\)
0.649204 + 0.760614i \(0.275102\pi\)
\(68\) 4.54221 0.550824
\(69\) 0 0
\(70\) −5.07548 −0.606636
\(71\) 8.65205 1.02681 0.513405 0.858147i \(-0.328384\pi\)
0.513405 + 0.858147i \(0.328384\pi\)
\(72\) 0 0
\(73\) 1.77313 0.207530 0.103765 0.994602i \(-0.466911\pi\)
0.103765 + 0.994602i \(0.466911\pi\)
\(74\) 8.84545 1.02826
\(75\) 0 0
\(76\) −7.38225 −0.846802
\(77\) 10.1330 1.15477
\(78\) 0 0
\(79\) −6.85479 −0.771224 −0.385612 0.922661i \(-0.626010\pi\)
−0.385612 + 0.922661i \(0.626010\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) 10.3743 1.14565
\(83\) −9.44789 −1.03704 −0.518520 0.855065i \(-0.673517\pi\)
−0.518520 + 0.855065i \(0.673517\pi\)
\(84\) 0 0
\(85\) 4.54221 0.492672
\(86\) −11.6712 −1.25854
\(87\) 0 0
\(88\) −1.99647 −0.212825
\(89\) −1.00000 −0.106000
\(90\) 0 0
\(91\) 29.5711 3.09990
\(92\) 1.23986 0.129265
\(93\) 0 0
\(94\) 4.79584 0.494653
\(95\) −7.38225 −0.757403
\(96\) 0 0
\(97\) 4.47973 0.454848 0.227424 0.973796i \(-0.426970\pi\)
0.227424 + 0.973796i \(0.426970\pi\)
\(98\) 18.7605 1.89509
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 16.4984 1.64165 0.820826 0.571178i \(-0.193514\pi\)
0.820826 + 0.571178i \(0.193514\pi\)
\(102\) 0 0
\(103\) −0.536798 −0.0528923 −0.0264461 0.999650i \(-0.508419\pi\)
−0.0264461 + 0.999650i \(0.508419\pi\)
\(104\) −5.82627 −0.571313
\(105\) 0 0
\(106\) 10.0857 0.979611
\(107\) −2.47267 −0.239042 −0.119521 0.992832i \(-0.538136\pi\)
−0.119521 + 0.992832i \(0.538136\pi\)
\(108\) 0 0
\(109\) 7.49891 0.718265 0.359133 0.933287i \(-0.383073\pi\)
0.359133 + 0.933287i \(0.383073\pi\)
\(110\) −1.99647 −0.190356
\(111\) 0 0
\(112\) −5.07548 −0.479588
\(113\) 2.99204 0.281468 0.140734 0.990047i \(-0.455054\pi\)
0.140734 + 0.990047i \(0.455054\pi\)
\(114\) 0 0
\(115\) 1.23986 0.115618
\(116\) −2.25904 −0.209747
\(117\) 0 0
\(118\) 3.00090 0.276255
\(119\) −23.0539 −2.11335
\(120\) 0 0
\(121\) −7.01410 −0.637646
\(122\) 5.57657 0.504879
\(123\) 0 0
\(124\) −2.32468 −0.208763
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 6.46370 0.573561 0.286780 0.957996i \(-0.407415\pi\)
0.286780 + 0.957996i \(0.407415\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) −5.82627 −0.510998
\(131\) −4.56974 −0.399260 −0.199630 0.979871i \(-0.563974\pi\)
−0.199630 + 0.979871i \(0.563974\pi\)
\(132\) 0 0
\(133\) 37.4684 3.24893
\(134\) 10.6279 0.918113
\(135\) 0 0
\(136\) 4.54221 0.389491
\(137\) −2.99910 −0.256230 −0.128115 0.991759i \(-0.540893\pi\)
−0.128115 + 0.991759i \(0.540893\pi\)
\(138\) 0 0
\(139\) 10.1580 0.861591 0.430796 0.902449i \(-0.358233\pi\)
0.430796 + 0.902449i \(0.358233\pi\)
\(140\) −5.07548 −0.428956
\(141\) 0 0
\(142\) 8.65205 0.726064
\(143\) 11.6320 0.972715
\(144\) 0 0
\(145\) −2.25904 −0.187603
\(146\) 1.77313 0.146746
\(147\) 0 0
\(148\) 8.84545 0.727092
\(149\) −0.741455 −0.0607424 −0.0303712 0.999539i \(-0.509669\pi\)
−0.0303712 + 0.999539i \(0.509669\pi\)
\(150\) 0 0
\(151\) −3.81921 −0.310803 −0.155402 0.987851i \(-0.549667\pi\)
−0.155402 + 0.987851i \(0.549667\pi\)
\(152\) −7.38225 −0.598779
\(153\) 0 0
\(154\) 10.1330 0.816544
\(155\) −2.32468 −0.186723
\(156\) 0 0
\(157\) −4.38315 −0.349813 −0.174907 0.984585i \(-0.555962\pi\)
−0.174907 + 0.984585i \(0.555962\pi\)
\(158\) −6.85479 −0.545338
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −6.29291 −0.495950
\(162\) 0 0
\(163\) −14.7716 −1.15700 −0.578499 0.815683i \(-0.696361\pi\)
−0.578499 + 0.815683i \(0.696361\pi\)
\(164\) 10.3743 0.810096
\(165\) 0 0
\(166\) −9.44789 −0.733298
\(167\) −1.77037 −0.136996 −0.0684978 0.997651i \(-0.521821\pi\)
−0.0684978 + 0.997651i \(0.521821\pi\)
\(168\) 0 0
\(169\) 20.9455 1.61119
\(170\) 4.54221 0.348371
\(171\) 0 0
\(172\) −11.6712 −0.889923
\(173\) −6.06704 −0.461268 −0.230634 0.973041i \(-0.574080\pi\)
−0.230634 + 0.973041i \(0.574080\pi\)
\(174\) 0 0
\(175\) −5.07548 −0.383670
\(176\) −1.99647 −0.150490
\(177\) 0 0
\(178\) −1.00000 −0.0749532
\(179\) 20.5547 1.53633 0.768165 0.640252i \(-0.221170\pi\)
0.768165 + 0.640252i \(0.221170\pi\)
\(180\) 0 0
\(181\) −25.5112 −1.89623 −0.948116 0.317924i \(-0.897014\pi\)
−0.948116 + 0.317924i \(0.897014\pi\)
\(182\) 29.5711 2.19196
\(183\) 0 0
\(184\) 1.23986 0.0914041
\(185\) 8.84545 0.650330
\(186\) 0 0
\(187\) −9.06839 −0.663146
\(188\) 4.79584 0.349773
\(189\) 0 0
\(190\) −7.38225 −0.535565
\(191\) −12.6042 −0.912008 −0.456004 0.889978i \(-0.650720\pi\)
−0.456004 + 0.889978i \(0.650720\pi\)
\(192\) 0 0
\(193\) 20.2363 1.45664 0.728319 0.685238i \(-0.240302\pi\)
0.728319 + 0.685238i \(0.240302\pi\)
\(194\) 4.47973 0.321626
\(195\) 0 0
\(196\) 18.7605 1.34003
\(197\) −23.2562 −1.65694 −0.828470 0.560034i \(-0.810788\pi\)
−0.828470 + 0.560034i \(0.810788\pi\)
\(198\) 0 0
\(199\) 19.9954 1.41744 0.708719 0.705491i \(-0.249273\pi\)
0.708719 + 0.705491i \(0.249273\pi\)
\(200\) 1.00000 0.0707107
\(201\) 0 0
\(202\) 16.4984 1.16082
\(203\) 11.4657 0.804736
\(204\) 0 0
\(205\) 10.3743 0.724572
\(206\) −0.536798 −0.0374005
\(207\) 0 0
\(208\) −5.82627 −0.403979
\(209\) 14.7384 1.01948
\(210\) 0 0
\(211\) −15.5992 −1.07389 −0.536947 0.843616i \(-0.680422\pi\)
−0.536947 + 0.843616i \(0.680422\pi\)
\(212\) 10.0857 0.692690
\(213\) 0 0
\(214\) −2.47267 −0.169028
\(215\) −11.6712 −0.795971
\(216\) 0 0
\(217\) 11.7989 0.800960
\(218\) 7.49891 0.507890
\(219\) 0 0
\(220\) −1.99647 −0.134602
\(221\) −26.4641 −1.78017
\(222\) 0 0
\(223\) 2.08392 0.139549 0.0697747 0.997563i \(-0.477772\pi\)
0.0697747 + 0.997563i \(0.477772\pi\)
\(224\) −5.07548 −0.339120
\(225\) 0 0
\(226\) 2.99204 0.199028
\(227\) 16.2848 1.08086 0.540429 0.841390i \(-0.318262\pi\)
0.540429 + 0.841390i \(0.318262\pi\)
\(228\) 0 0
\(229\) 16.1214 1.06533 0.532667 0.846325i \(-0.321190\pi\)
0.532667 + 0.846325i \(0.321190\pi\)
\(230\) 1.23986 0.0817543
\(231\) 0 0
\(232\) −2.25904 −0.148313
\(233\) −18.2358 −1.19467 −0.597334 0.801992i \(-0.703773\pi\)
−0.597334 + 0.801992i \(0.703773\pi\)
\(234\) 0 0
\(235\) 4.79584 0.312846
\(236\) 3.00090 0.195342
\(237\) 0 0
\(238\) −23.0539 −1.49436
\(239\) 11.6735 0.755096 0.377548 0.925990i \(-0.376767\pi\)
0.377548 + 0.925990i \(0.376767\pi\)
\(240\) 0 0
\(241\) −1.66417 −0.107198 −0.0535992 0.998563i \(-0.517069\pi\)
−0.0535992 + 0.998563i \(0.517069\pi\)
\(242\) −7.01410 −0.450884
\(243\) 0 0
\(244\) 5.57657 0.357003
\(245\) 18.7605 1.19856
\(246\) 0 0
\(247\) 43.0110 2.73672
\(248\) −2.32468 −0.147617
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −9.80102 −0.618635 −0.309317 0.950959i \(-0.600101\pi\)
−0.309317 + 0.950959i \(0.600101\pi\)
\(252\) 0 0
\(253\) −2.47535 −0.155624
\(254\) 6.46370 0.405569
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −14.6966 −0.916750 −0.458375 0.888759i \(-0.651568\pi\)
−0.458375 + 0.888759i \(0.651568\pi\)
\(258\) 0 0
\(259\) −44.8949 −2.78963
\(260\) −5.82627 −0.361330
\(261\) 0 0
\(262\) −4.56974 −0.282320
\(263\) 22.0723 1.36103 0.680517 0.732733i \(-0.261755\pi\)
0.680517 + 0.732733i \(0.261755\pi\)
\(264\) 0 0
\(265\) 10.0857 0.619561
\(266\) 37.4684 2.29734
\(267\) 0 0
\(268\) 10.6279 0.649204
\(269\) 16.0973 0.981467 0.490733 0.871310i \(-0.336729\pi\)
0.490733 + 0.871310i \(0.336729\pi\)
\(270\) 0 0
\(271\) 8.89944 0.540602 0.270301 0.962776i \(-0.412877\pi\)
0.270301 + 0.962776i \(0.412877\pi\)
\(272\) 4.54221 0.275412
\(273\) 0 0
\(274\) −2.99910 −0.181182
\(275\) −1.99647 −0.120392
\(276\) 0 0
\(277\) 2.86117 0.171911 0.0859554 0.996299i \(-0.472606\pi\)
0.0859554 + 0.996299i \(0.472606\pi\)
\(278\) 10.1580 0.609237
\(279\) 0 0
\(280\) −5.07548 −0.303318
\(281\) −9.20148 −0.548914 −0.274457 0.961599i \(-0.588498\pi\)
−0.274457 + 0.961599i \(0.588498\pi\)
\(282\) 0 0
\(283\) 12.7789 0.759626 0.379813 0.925063i \(-0.375988\pi\)
0.379813 + 0.925063i \(0.375988\pi\)
\(284\) 8.65205 0.513405
\(285\) 0 0
\(286\) 11.6320 0.687814
\(287\) −52.6545 −3.10810
\(288\) 0 0
\(289\) 3.63165 0.213627
\(290\) −2.25904 −0.132656
\(291\) 0 0
\(292\) 1.77313 0.103765
\(293\) 1.01175 0.0591072 0.0295536 0.999563i \(-0.490591\pi\)
0.0295536 + 0.999563i \(0.490591\pi\)
\(294\) 0 0
\(295\) 3.00090 0.174719
\(296\) 8.84545 0.514131
\(297\) 0 0
\(298\) −0.741455 −0.0429513
\(299\) −7.22379 −0.417763
\(300\) 0 0
\(301\) 59.2370 3.41437
\(302\) −3.81921 −0.219771
\(303\) 0 0
\(304\) −7.38225 −0.423401
\(305\) 5.57657 0.319313
\(306\) 0 0
\(307\) 10.8807 0.620995 0.310497 0.950574i \(-0.399504\pi\)
0.310497 + 0.950574i \(0.399504\pi\)
\(308\) 10.1330 0.577384
\(309\) 0 0
\(310\) −2.32468 −0.132033
\(311\) 24.2777 1.37666 0.688331 0.725396i \(-0.258344\pi\)
0.688331 + 0.725396i \(0.258344\pi\)
\(312\) 0 0
\(313\) 7.58691 0.428837 0.214419 0.976742i \(-0.431214\pi\)
0.214419 + 0.976742i \(0.431214\pi\)
\(314\) −4.38315 −0.247355
\(315\) 0 0
\(316\) −6.85479 −0.385612
\(317\) −27.8079 −1.56185 −0.780923 0.624627i \(-0.785251\pi\)
−0.780923 + 0.624627i \(0.785251\pi\)
\(318\) 0 0
\(319\) 4.51011 0.252518
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) −6.29291 −0.350690
\(323\) −33.5317 −1.86575
\(324\) 0 0
\(325\) −5.82627 −0.323183
\(326\) −14.7716 −0.818121
\(327\) 0 0
\(328\) 10.3743 0.572824
\(329\) −24.3412 −1.34197
\(330\) 0 0
\(331\) −0.340397 −0.0187099 −0.00935496 0.999956i \(-0.502978\pi\)
−0.00935496 + 0.999956i \(0.502978\pi\)
\(332\) −9.44789 −0.518520
\(333\) 0 0
\(334\) −1.77037 −0.0968706
\(335\) 10.6279 0.580666
\(336\) 0 0
\(337\) 1.55121 0.0845000 0.0422500 0.999107i \(-0.486547\pi\)
0.0422500 + 0.999107i \(0.486547\pi\)
\(338\) 20.9455 1.13928
\(339\) 0 0
\(340\) 4.54221 0.246336
\(341\) 4.64116 0.251333
\(342\) 0 0
\(343\) −59.6900 −3.22296
\(344\) −11.6712 −0.629270
\(345\) 0 0
\(346\) −6.06704 −0.326166
\(347\) −16.7549 −0.899450 −0.449725 0.893167i \(-0.648478\pi\)
−0.449725 + 0.893167i \(0.648478\pi\)
\(348\) 0 0
\(349\) −25.9284 −1.38791 −0.693957 0.720016i \(-0.744134\pi\)
−0.693957 + 0.720016i \(0.744134\pi\)
\(350\) −5.07548 −0.271296
\(351\) 0 0
\(352\) −1.99647 −0.106412
\(353\) −21.3394 −1.13578 −0.567890 0.823105i \(-0.692240\pi\)
−0.567890 + 0.823105i \(0.692240\pi\)
\(354\) 0 0
\(355\) 8.65205 0.459203
\(356\) −1.00000 −0.0529999
\(357\) 0 0
\(358\) 20.5547 1.08635
\(359\) 34.3005 1.81031 0.905156 0.425081i \(-0.139754\pi\)
0.905156 + 0.425081i \(0.139754\pi\)
\(360\) 0 0
\(361\) 35.4976 1.86829
\(362\) −25.5112 −1.34084
\(363\) 0 0
\(364\) 29.5711 1.54995
\(365\) 1.77313 0.0928101
\(366\) 0 0
\(367\) −11.1434 −0.581681 −0.290840 0.956772i \(-0.593935\pi\)
−0.290840 + 0.956772i \(0.593935\pi\)
\(368\) 1.23986 0.0646324
\(369\) 0 0
\(370\) 8.84545 0.459853
\(371\) −51.1898 −2.65764
\(372\) 0 0
\(373\) −35.5452 −1.84046 −0.920230 0.391378i \(-0.871999\pi\)
−0.920230 + 0.391378i \(0.871999\pi\)
\(374\) −9.06839 −0.468915
\(375\) 0 0
\(376\) 4.79584 0.247327
\(377\) 13.1618 0.677867
\(378\) 0 0
\(379\) 28.7720 1.47792 0.738960 0.673749i \(-0.235317\pi\)
0.738960 + 0.673749i \(0.235317\pi\)
\(380\) −7.38225 −0.378701
\(381\) 0 0
\(382\) −12.6042 −0.644887
\(383\) 34.0955 1.74220 0.871100 0.491106i \(-0.163407\pi\)
0.871100 + 0.491106i \(0.163407\pi\)
\(384\) 0 0
\(385\) 10.1330 0.516428
\(386\) 20.2363 1.03000
\(387\) 0 0
\(388\) 4.47973 0.227424
\(389\) −36.0768 −1.82917 −0.914584 0.404395i \(-0.867482\pi\)
−0.914584 + 0.404395i \(0.867482\pi\)
\(390\) 0 0
\(391\) 5.63172 0.284809
\(392\) 18.7605 0.947547
\(393\) 0 0
\(394\) −23.2562 −1.17163
\(395\) −6.85479 −0.344902
\(396\) 0 0
\(397\) −28.6401 −1.43741 −0.718703 0.695318i \(-0.755264\pi\)
−0.718703 + 0.695318i \(0.755264\pi\)
\(398\) 19.9954 1.00228
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) 0.299964 0.0149795 0.00748974 0.999972i \(-0.497616\pi\)
0.00748974 + 0.999972i \(0.497616\pi\)
\(402\) 0 0
\(403\) 13.5442 0.674686
\(404\) 16.4984 0.820826
\(405\) 0 0
\(406\) 11.4657 0.569034
\(407\) −17.6597 −0.875358
\(408\) 0 0
\(409\) 16.7289 0.827191 0.413595 0.910461i \(-0.364273\pi\)
0.413595 + 0.910461i \(0.364273\pi\)
\(410\) 10.3743 0.512350
\(411\) 0 0
\(412\) −0.536798 −0.0264461
\(413\) −15.2310 −0.749468
\(414\) 0 0
\(415\) −9.44789 −0.463779
\(416\) −5.82627 −0.285657
\(417\) 0 0
\(418\) 14.7384 0.720881
\(419\) 16.8301 0.822206 0.411103 0.911589i \(-0.365144\pi\)
0.411103 + 0.911589i \(0.365144\pi\)
\(420\) 0 0
\(421\) 26.3008 1.28182 0.640911 0.767615i \(-0.278557\pi\)
0.640911 + 0.767615i \(0.278557\pi\)
\(422\) −15.5992 −0.759358
\(423\) 0 0
\(424\) 10.0857 0.489806
\(425\) 4.54221 0.220329
\(426\) 0 0
\(427\) −28.3038 −1.36971
\(428\) −2.47267 −0.119521
\(429\) 0 0
\(430\) −11.6712 −0.562836
\(431\) 18.2075 0.877025 0.438512 0.898725i \(-0.355506\pi\)
0.438512 + 0.898725i \(0.355506\pi\)
\(432\) 0 0
\(433\) −18.5844 −0.893110 −0.446555 0.894756i \(-0.647349\pi\)
−0.446555 + 0.894756i \(0.647349\pi\)
\(434\) 11.7989 0.566364
\(435\) 0 0
\(436\) 7.49891 0.359133
\(437\) −9.15299 −0.437847
\(438\) 0 0
\(439\) 30.7898 1.46952 0.734759 0.678328i \(-0.237295\pi\)
0.734759 + 0.678328i \(0.237295\pi\)
\(440\) −1.99647 −0.0951780
\(441\) 0 0
\(442\) −26.4641 −1.25877
\(443\) 1.82123 0.0865293 0.0432646 0.999064i \(-0.486224\pi\)
0.0432646 + 0.999064i \(0.486224\pi\)
\(444\) 0 0
\(445\) −1.00000 −0.0474045
\(446\) 2.08392 0.0986764
\(447\) 0 0
\(448\) −5.07548 −0.239794
\(449\) −15.6167 −0.736997 −0.368498 0.929628i \(-0.620128\pi\)
−0.368498 + 0.929628i \(0.620128\pi\)
\(450\) 0 0
\(451\) −20.7120 −0.975289
\(452\) 2.99204 0.140734
\(453\) 0 0
\(454\) 16.2848 0.764282
\(455\) 29.5711 1.38632
\(456\) 0 0
\(457\) 41.5922 1.94560 0.972799 0.231649i \(-0.0744121\pi\)
0.972799 + 0.231649i \(0.0744121\pi\)
\(458\) 16.1214 0.753305
\(459\) 0 0
\(460\) 1.23986 0.0578090
\(461\) 10.9806 0.511417 0.255708 0.966754i \(-0.417691\pi\)
0.255708 + 0.966754i \(0.417691\pi\)
\(462\) 0 0
\(463\) −36.8145 −1.71091 −0.855457 0.517874i \(-0.826724\pi\)
−0.855457 + 0.517874i \(0.826724\pi\)
\(464\) −2.25904 −0.104873
\(465\) 0 0
\(466\) −18.2358 −0.844758
\(467\) 32.3762 1.49819 0.749096 0.662462i \(-0.230488\pi\)
0.749096 + 0.662462i \(0.230488\pi\)
\(468\) 0 0
\(469\) −53.9418 −2.49080
\(470\) 4.79584 0.221216
\(471\) 0 0
\(472\) 3.00090 0.138128
\(473\) 23.3013 1.07139
\(474\) 0 0
\(475\) −7.38225 −0.338721
\(476\) −23.0539 −1.05667
\(477\) 0 0
\(478\) 11.6735 0.533934
\(479\) 0.901550 0.0411929 0.0205964 0.999788i \(-0.493443\pi\)
0.0205964 + 0.999788i \(0.493443\pi\)
\(480\) 0 0
\(481\) −51.5360 −2.34984
\(482\) −1.66417 −0.0758007
\(483\) 0 0
\(484\) −7.01410 −0.318823
\(485\) 4.47973 0.203414
\(486\) 0 0
\(487\) −42.0001 −1.90320 −0.951602 0.307333i \(-0.900564\pi\)
−0.951602 + 0.307333i \(0.900564\pi\)
\(488\) 5.57657 0.252439
\(489\) 0 0
\(490\) 18.7605 0.847512
\(491\) 19.7629 0.891885 0.445943 0.895061i \(-0.352868\pi\)
0.445943 + 0.895061i \(0.352868\pi\)
\(492\) 0 0
\(493\) −10.2610 −0.462134
\(494\) 43.0110 1.93516
\(495\) 0 0
\(496\) −2.32468 −0.104381
\(497\) −43.9133 −1.96978
\(498\) 0 0
\(499\) −9.29093 −0.415919 −0.207959 0.978137i \(-0.566682\pi\)
−0.207959 + 0.978137i \(0.566682\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −9.80102 −0.437441
\(503\) 16.3646 0.729661 0.364830 0.931074i \(-0.381127\pi\)
0.364830 + 0.931074i \(0.381127\pi\)
\(504\) 0 0
\(505\) 16.4984 0.734169
\(506\) −2.47535 −0.110043
\(507\) 0 0
\(508\) 6.46370 0.286780
\(509\) 6.05293 0.268292 0.134146 0.990962i \(-0.457171\pi\)
0.134146 + 0.990962i \(0.457171\pi\)
\(510\) 0 0
\(511\) −8.99950 −0.398115
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −14.6966 −0.648240
\(515\) −0.536798 −0.0236542
\(516\) 0 0
\(517\) −9.57476 −0.421097
\(518\) −44.8949 −1.97257
\(519\) 0 0
\(520\) −5.82627 −0.255499
\(521\) 25.2209 1.10495 0.552475 0.833530i \(-0.313684\pi\)
0.552475 + 0.833530i \(0.313684\pi\)
\(522\) 0 0
\(523\) −44.2331 −1.93418 −0.967090 0.254434i \(-0.918111\pi\)
−0.967090 + 0.254434i \(0.918111\pi\)
\(524\) −4.56974 −0.199630
\(525\) 0 0
\(526\) 22.0723 0.962396
\(527\) −10.5592 −0.459966
\(528\) 0 0
\(529\) −21.4627 −0.933162
\(530\) 10.0857 0.438096
\(531\) 0 0
\(532\) 37.4684 1.62446
\(533\) −60.4435 −2.61810
\(534\) 0 0
\(535\) −2.47267 −0.106903
\(536\) 10.6279 0.459056
\(537\) 0 0
\(538\) 16.0973 0.694002
\(539\) −37.4547 −1.61329
\(540\) 0 0
\(541\) −12.6888 −0.545536 −0.272768 0.962080i \(-0.587939\pi\)
−0.272768 + 0.962080i \(0.587939\pi\)
\(542\) 8.89944 0.382263
\(543\) 0 0
\(544\) 4.54221 0.194746
\(545\) 7.49891 0.321218
\(546\) 0 0
\(547\) 21.1619 0.904819 0.452409 0.891810i \(-0.350565\pi\)
0.452409 + 0.891810i \(0.350565\pi\)
\(548\) −2.99910 −0.128115
\(549\) 0 0
\(550\) −1.99647 −0.0851298
\(551\) 16.6768 0.710456
\(552\) 0 0
\(553\) 34.7913 1.47948
\(554\) 2.86117 0.121559
\(555\) 0 0
\(556\) 10.1580 0.430796
\(557\) 36.4757 1.54553 0.772763 0.634694i \(-0.218874\pi\)
0.772763 + 0.634694i \(0.218874\pi\)
\(558\) 0 0
\(559\) 67.9997 2.87608
\(560\) −5.07548 −0.214478
\(561\) 0 0
\(562\) −9.20148 −0.388141
\(563\) −26.1875 −1.10367 −0.551836 0.833953i \(-0.686072\pi\)
−0.551836 + 0.833953i \(0.686072\pi\)
\(564\) 0 0
\(565\) 2.99204 0.125876
\(566\) 12.7789 0.537136
\(567\) 0 0
\(568\) 8.65205 0.363032
\(569\) 29.5139 1.23729 0.618643 0.785672i \(-0.287683\pi\)
0.618643 + 0.785672i \(0.287683\pi\)
\(570\) 0 0
\(571\) −21.8378 −0.913883 −0.456941 0.889497i \(-0.651055\pi\)
−0.456941 + 0.889497i \(0.651055\pi\)
\(572\) 11.6320 0.486358
\(573\) 0 0
\(574\) −52.6545 −2.19776
\(575\) 1.23986 0.0517059
\(576\) 0 0
\(577\) 19.2920 0.803138 0.401569 0.915829i \(-0.368465\pi\)
0.401569 + 0.915829i \(0.368465\pi\)
\(578\) 3.63165 0.151057
\(579\) 0 0
\(580\) −2.25904 −0.0938017
\(581\) 47.9525 1.98941
\(582\) 0 0
\(583\) −20.1358 −0.833941
\(584\) 1.77313 0.0733728
\(585\) 0 0
\(586\) 1.01175 0.0417951
\(587\) −21.4839 −0.886734 −0.443367 0.896340i \(-0.646216\pi\)
−0.443367 + 0.896340i \(0.646216\pi\)
\(588\) 0 0
\(589\) 17.1614 0.707122
\(590\) 3.00090 0.123545
\(591\) 0 0
\(592\) 8.84545 0.363546
\(593\) 35.9315 1.47553 0.737766 0.675057i \(-0.235881\pi\)
0.737766 + 0.675057i \(0.235881\pi\)
\(594\) 0 0
\(595\) −23.0539 −0.945117
\(596\) −0.741455 −0.0303712
\(597\) 0 0
\(598\) −7.22379 −0.295403
\(599\) 29.7968 1.21746 0.608732 0.793376i \(-0.291678\pi\)
0.608732 + 0.793376i \(0.291678\pi\)
\(600\) 0 0
\(601\) 30.4573 1.24238 0.621190 0.783660i \(-0.286649\pi\)
0.621190 + 0.783660i \(0.286649\pi\)
\(602\) 59.2370 2.41432
\(603\) 0 0
\(604\) −3.81921 −0.155402
\(605\) −7.01410 −0.285164
\(606\) 0 0
\(607\) −25.6738 −1.04207 −0.521033 0.853537i \(-0.674453\pi\)
−0.521033 + 0.853537i \(0.674453\pi\)
\(608\) −7.38225 −0.299390
\(609\) 0 0
\(610\) 5.57657 0.225789
\(611\) −27.9419 −1.13041
\(612\) 0 0
\(613\) 2.04363 0.0825416 0.0412708 0.999148i \(-0.486859\pi\)
0.0412708 + 0.999148i \(0.486859\pi\)
\(614\) 10.8807 0.439110
\(615\) 0 0
\(616\) 10.1330 0.408272
\(617\) 18.1550 0.730894 0.365447 0.930832i \(-0.380916\pi\)
0.365447 + 0.930832i \(0.380916\pi\)
\(618\) 0 0
\(619\) −0.644800 −0.0259167 −0.0129584 0.999916i \(-0.504125\pi\)
−0.0129584 + 0.999916i \(0.504125\pi\)
\(620\) −2.32468 −0.0933615
\(621\) 0 0
\(622\) 24.2777 0.973447
\(623\) 5.07548 0.203345
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 7.58691 0.303234
\(627\) 0 0
\(628\) −4.38315 −0.174907
\(629\) 40.1779 1.60200
\(630\) 0 0
\(631\) 16.4511 0.654907 0.327453 0.944867i \(-0.393810\pi\)
0.327453 + 0.944867i \(0.393810\pi\)
\(632\) −6.85479 −0.272669
\(633\) 0 0
\(634\) −27.8079 −1.10439
\(635\) 6.46370 0.256504
\(636\) 0 0
\(637\) −109.304 −4.33077
\(638\) 4.51011 0.178557
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) −23.1359 −0.913812 −0.456906 0.889515i \(-0.651042\pi\)
−0.456906 + 0.889515i \(0.651042\pi\)
\(642\) 0 0
\(643\) −27.6516 −1.09047 −0.545236 0.838283i \(-0.683560\pi\)
−0.545236 + 0.838283i \(0.683560\pi\)
\(644\) −6.29291 −0.247975
\(645\) 0 0
\(646\) −33.5317 −1.31929
\(647\) 23.9588 0.941918 0.470959 0.882155i \(-0.343908\pi\)
0.470959 + 0.882155i \(0.343908\pi\)
\(648\) 0 0
\(649\) −5.99121 −0.235176
\(650\) −5.82627 −0.228525
\(651\) 0 0
\(652\) −14.7716 −0.578499
\(653\) 47.2481 1.84896 0.924480 0.381229i \(-0.124499\pi\)
0.924480 + 0.381229i \(0.124499\pi\)
\(654\) 0 0
\(655\) −4.56974 −0.178555
\(656\) 10.3743 0.405048
\(657\) 0 0
\(658\) −24.3412 −0.948918
\(659\) 35.5901 1.38639 0.693196 0.720749i \(-0.256202\pi\)
0.693196 + 0.720749i \(0.256202\pi\)
\(660\) 0 0
\(661\) 17.5722 0.683479 0.341739 0.939795i \(-0.388984\pi\)
0.341739 + 0.939795i \(0.388984\pi\)
\(662\) −0.340397 −0.0132299
\(663\) 0 0
\(664\) −9.44789 −0.366649
\(665\) 37.4684 1.45296
\(666\) 0 0
\(667\) −2.80091 −0.108452
\(668\) −1.77037 −0.0684978
\(669\) 0 0
\(670\) 10.6279 0.410593
\(671\) −11.1335 −0.429802
\(672\) 0 0
\(673\) −29.5686 −1.13978 −0.569892 0.821719i \(-0.693015\pi\)
−0.569892 + 0.821719i \(0.693015\pi\)
\(674\) 1.55121 0.0597505
\(675\) 0 0
\(676\) 20.9455 0.805594
\(677\) 13.6656 0.525213 0.262607 0.964903i \(-0.415418\pi\)
0.262607 + 0.964903i \(0.415418\pi\)
\(678\) 0 0
\(679\) −22.7368 −0.872557
\(680\) 4.54221 0.174186
\(681\) 0 0
\(682\) 4.64116 0.177719
\(683\) −6.12490 −0.234363 −0.117181 0.993111i \(-0.537386\pi\)
−0.117181 + 0.993111i \(0.537386\pi\)
\(684\) 0 0
\(685\) −2.99910 −0.114590
\(686\) −59.6900 −2.27897
\(687\) 0 0
\(688\) −11.6712 −0.444961
\(689\) −58.7621 −2.23866
\(690\) 0 0
\(691\) 18.5414 0.705348 0.352674 0.935746i \(-0.385272\pi\)
0.352674 + 0.935746i \(0.385272\pi\)
\(692\) −6.06704 −0.230634
\(693\) 0 0
\(694\) −16.7549 −0.636008
\(695\) 10.1580 0.385315
\(696\) 0 0
\(697\) 47.1222 1.78488
\(698\) −25.9284 −0.981403
\(699\) 0 0
\(700\) −5.07548 −0.191835
\(701\) 3.09528 0.116907 0.0584536 0.998290i \(-0.481383\pi\)
0.0584536 + 0.998290i \(0.481383\pi\)
\(702\) 0 0
\(703\) −65.2993 −2.46281
\(704\) −1.99647 −0.0752448
\(705\) 0 0
\(706\) −21.3394 −0.803118
\(707\) −83.7372 −3.14926
\(708\) 0 0
\(709\) 16.6815 0.626486 0.313243 0.949673i \(-0.398585\pi\)
0.313243 + 0.949673i \(0.398585\pi\)
\(710\) 8.65205 0.324706
\(711\) 0 0
\(712\) −1.00000 −0.0374766
\(713\) −2.88229 −0.107943
\(714\) 0 0
\(715\) 11.6320 0.435012
\(716\) 20.5547 0.768165
\(717\) 0 0
\(718\) 34.3005 1.28008
\(719\) 38.5013 1.43586 0.717928 0.696118i \(-0.245091\pi\)
0.717928 + 0.696118i \(0.245091\pi\)
\(720\) 0 0
\(721\) 2.72451 0.101466
\(722\) 35.4976 1.32108
\(723\) 0 0
\(724\) −25.5112 −0.948116
\(725\) −2.25904 −0.0838987
\(726\) 0 0
\(727\) 49.8143 1.84751 0.923755 0.382983i \(-0.125103\pi\)
0.923755 + 0.382983i \(0.125103\pi\)
\(728\) 29.5711 1.09598
\(729\) 0 0
\(730\) 1.77313 0.0656266
\(731\) −53.0131 −1.96076
\(732\) 0 0
\(733\) 4.99504 0.184496 0.0922480 0.995736i \(-0.470595\pi\)
0.0922480 + 0.995736i \(0.470595\pi\)
\(734\) −11.1434 −0.411310
\(735\) 0 0
\(736\) 1.23986 0.0457020
\(737\) −21.2183 −0.781588
\(738\) 0 0
\(739\) 23.9949 0.882667 0.441333 0.897343i \(-0.354506\pi\)
0.441333 + 0.897343i \(0.354506\pi\)
\(740\) 8.84545 0.325165
\(741\) 0 0
\(742\) −51.1898 −1.87924
\(743\) −19.3354 −0.709347 −0.354674 0.934990i \(-0.615408\pi\)
−0.354674 + 0.934990i \(0.615408\pi\)
\(744\) 0 0
\(745\) −0.741455 −0.0271648
\(746\) −35.5452 −1.30140
\(747\) 0 0
\(748\) −9.06839 −0.331573
\(749\) 12.5500 0.458567
\(750\) 0 0
\(751\) 29.8005 1.08744 0.543718 0.839268i \(-0.317016\pi\)
0.543718 + 0.839268i \(0.317016\pi\)
\(752\) 4.79584 0.174886
\(753\) 0 0
\(754\) 13.1618 0.479324
\(755\) −3.81921 −0.138995
\(756\) 0 0
\(757\) 20.7274 0.753349 0.376675 0.926346i \(-0.377068\pi\)
0.376675 + 0.926346i \(0.377068\pi\)
\(758\) 28.7720 1.04505
\(759\) 0 0
\(760\) −7.38225 −0.267782
\(761\) 19.5659 0.709265 0.354632 0.935006i \(-0.384606\pi\)
0.354632 + 0.935006i \(0.384606\pi\)
\(762\) 0 0
\(763\) −38.0605 −1.37788
\(764\) −12.6042 −0.456004
\(765\) 0 0
\(766\) 34.0955 1.23192
\(767\) −17.4841 −0.631313
\(768\) 0 0
\(769\) 31.6008 1.13955 0.569777 0.821799i \(-0.307030\pi\)
0.569777 + 0.821799i \(0.307030\pi\)
\(770\) 10.1330 0.365170
\(771\) 0 0
\(772\) 20.2363 0.728319
\(773\) 10.1917 0.366568 0.183284 0.983060i \(-0.441327\pi\)
0.183284 + 0.983060i \(0.441327\pi\)
\(774\) 0 0
\(775\) −2.32468 −0.0835050
\(776\) 4.47973 0.160813
\(777\) 0 0
\(778\) −36.0768 −1.29342
\(779\) −76.5856 −2.74396
\(780\) 0 0
\(781\) −17.2736 −0.618097
\(782\) 5.63172 0.201390
\(783\) 0 0
\(784\) 18.7605 0.670017
\(785\) −4.38315 −0.156441
\(786\) 0 0
\(787\) −20.1985 −0.719999 −0.359999 0.932953i \(-0.617223\pi\)
−0.359999 + 0.932953i \(0.617223\pi\)
\(788\) −23.2562 −0.828470
\(789\) 0 0
\(790\) −6.85479 −0.243883
\(791\) −15.1860 −0.539954
\(792\) 0 0
\(793\) −32.4906 −1.15378
\(794\) −28.6401 −1.01640
\(795\) 0 0
\(796\) 19.9954 0.708719
\(797\) 21.4681 0.760440 0.380220 0.924896i \(-0.375848\pi\)
0.380220 + 0.924896i \(0.375848\pi\)
\(798\) 0 0
\(799\) 21.7837 0.770652
\(800\) 1.00000 0.0353553
\(801\) 0 0
\(802\) 0.299964 0.0105921
\(803\) −3.54001 −0.124924
\(804\) 0 0
\(805\) −6.29291 −0.221796
\(806\) 13.5442 0.477075
\(807\) 0 0
\(808\) 16.4984 0.580412
\(809\) −11.9849 −0.421366 −0.210683 0.977554i \(-0.567569\pi\)
−0.210683 + 0.977554i \(0.567569\pi\)
\(810\) 0 0
\(811\) 29.0621 1.02051 0.510254 0.860023i \(-0.329551\pi\)
0.510254 + 0.860023i \(0.329551\pi\)
\(812\) 11.4657 0.402368
\(813\) 0 0
\(814\) −17.6597 −0.618972
\(815\) −14.7716 −0.517425
\(816\) 0 0
\(817\) 86.1599 3.01435
\(818\) 16.7289 0.584912
\(819\) 0 0
\(820\) 10.3743 0.362286
\(821\) −21.7919 −0.760544 −0.380272 0.924875i \(-0.624170\pi\)
−0.380272 + 0.924875i \(0.624170\pi\)
\(822\) 0 0
\(823\) 24.9556 0.869898 0.434949 0.900455i \(-0.356766\pi\)
0.434949 + 0.900455i \(0.356766\pi\)
\(824\) −0.536798 −0.0187002
\(825\) 0 0
\(826\) −15.2310 −0.529954
\(827\) 32.2356 1.12094 0.560471 0.828174i \(-0.310620\pi\)
0.560471 + 0.828174i \(0.310620\pi\)
\(828\) 0 0
\(829\) 0.997690 0.0346512 0.0173256 0.999850i \(-0.494485\pi\)
0.0173256 + 0.999850i \(0.494485\pi\)
\(830\) −9.44789 −0.327941
\(831\) 0 0
\(832\) −5.82627 −0.201990
\(833\) 85.2140 2.95249
\(834\) 0 0
\(835\) −1.77037 −0.0612663
\(836\) 14.7384 0.509740
\(837\) 0 0
\(838\) 16.8301 0.581387
\(839\) −33.1776 −1.14542 −0.572709 0.819759i \(-0.694107\pi\)
−0.572709 + 0.819759i \(0.694107\pi\)
\(840\) 0 0
\(841\) −23.8967 −0.824025
\(842\) 26.3008 0.906385
\(843\) 0 0
\(844\) −15.5992 −0.536947
\(845\) 20.9455 0.720545
\(846\) 0 0
\(847\) 35.5999 1.22323
\(848\) 10.0857 0.346345
\(849\) 0 0
\(850\) 4.54221 0.155796
\(851\) 10.9672 0.375950
\(852\) 0 0
\(853\) 9.71562 0.332657 0.166328 0.986070i \(-0.446809\pi\)
0.166328 + 0.986070i \(0.446809\pi\)
\(854\) −28.3038 −0.968534
\(855\) 0 0
\(856\) −2.47267 −0.0845142
\(857\) −20.3458 −0.695001 −0.347500 0.937680i \(-0.612969\pi\)
−0.347500 + 0.937680i \(0.612969\pi\)
\(858\) 0 0
\(859\) −37.1046 −1.26599 −0.632996 0.774155i \(-0.718175\pi\)
−0.632996 + 0.774155i \(0.718175\pi\)
\(860\) −11.6712 −0.397985
\(861\) 0 0
\(862\) 18.2075 0.620150
\(863\) −21.3660 −0.727306 −0.363653 0.931534i \(-0.618471\pi\)
−0.363653 + 0.931534i \(0.618471\pi\)
\(864\) 0 0
\(865\) −6.06704 −0.206285
\(866\) −18.5844 −0.631524
\(867\) 0 0
\(868\) 11.7989 0.400480
\(869\) 13.6854 0.464245
\(870\) 0 0
\(871\) −61.9212 −2.09812
\(872\) 7.49891 0.253945
\(873\) 0 0
\(874\) −9.15299 −0.309605
\(875\) −5.07548 −0.171582
\(876\) 0 0
\(877\) −13.9356 −0.470571 −0.235285 0.971926i \(-0.575603\pi\)
−0.235285 + 0.971926i \(0.575603\pi\)
\(878\) 30.7898 1.03911
\(879\) 0 0
\(880\) −1.99647 −0.0673010
\(881\) −18.3276 −0.617474 −0.308737 0.951147i \(-0.599906\pi\)
−0.308737 + 0.951147i \(0.599906\pi\)
\(882\) 0 0
\(883\) 13.2023 0.444293 0.222147 0.975013i \(-0.428694\pi\)
0.222147 + 0.975013i \(0.428694\pi\)
\(884\) −26.4641 −0.890085
\(885\) 0 0
\(886\) 1.82123 0.0611854
\(887\) 7.40324 0.248577 0.124288 0.992246i \(-0.460335\pi\)
0.124288 + 0.992246i \(0.460335\pi\)
\(888\) 0 0
\(889\) −32.8064 −1.10029
\(890\) −1.00000 −0.0335201
\(891\) 0 0
\(892\) 2.08392 0.0697747
\(893\) −35.4041 −1.18475
\(894\) 0 0
\(895\) 20.5547 0.687068
\(896\) −5.07548 −0.169560
\(897\) 0 0
\(898\) −15.6167 −0.521135
\(899\) 5.25156 0.175149
\(900\) 0 0
\(901\) 45.8114 1.52620
\(902\) −20.7120 −0.689633
\(903\) 0 0
\(904\) 2.99204 0.0995139
\(905\) −25.5112 −0.848021
\(906\) 0 0
\(907\) 30.5955 1.01591 0.507954 0.861384i \(-0.330402\pi\)
0.507954 + 0.861384i \(0.330402\pi\)
\(908\) 16.2848 0.540429
\(909\) 0 0
\(910\) 29.5711 0.980273
\(911\) −34.5117 −1.14342 −0.571711 0.820455i \(-0.693720\pi\)
−0.571711 + 0.820455i \(0.693720\pi\)
\(912\) 0 0
\(913\) 18.8624 0.624255
\(914\) 41.5922 1.37575
\(915\) 0 0
\(916\) 16.1214 0.532667
\(917\) 23.1936 0.765921
\(918\) 0 0
\(919\) 3.15305 0.104010 0.0520048 0.998647i \(-0.483439\pi\)
0.0520048 + 0.998647i \(0.483439\pi\)
\(920\) 1.23986 0.0408771
\(921\) 0 0
\(922\) 10.9806 0.361626
\(923\) −50.4092 −1.65924
\(924\) 0 0
\(925\) 8.84545 0.290837
\(926\) −36.8145 −1.20980
\(927\) 0 0
\(928\) −2.25904 −0.0741567
\(929\) −37.5164 −1.23087 −0.615436 0.788187i \(-0.711020\pi\)
−0.615436 + 0.788187i \(0.711020\pi\)
\(930\) 0 0
\(931\) −138.494 −4.53897
\(932\) −18.2358 −0.597334
\(933\) 0 0
\(934\) 32.3762 1.05938
\(935\) −9.06839 −0.296568
\(936\) 0 0
\(937\) −41.8150 −1.36604 −0.683018 0.730401i \(-0.739333\pi\)
−0.683018 + 0.730401i \(0.739333\pi\)
\(938\) −53.9418 −1.76126
\(939\) 0 0
\(940\) 4.79584 0.156423
\(941\) 28.9525 0.943824 0.471912 0.881646i \(-0.343564\pi\)
0.471912 + 0.881646i \(0.343564\pi\)
\(942\) 0 0
\(943\) 12.8627 0.418868
\(944\) 3.00090 0.0976709
\(945\) 0 0
\(946\) 23.3013 0.757589
\(947\) −46.7288 −1.51848 −0.759241 0.650809i \(-0.774430\pi\)
−0.759241 + 0.650809i \(0.774430\pi\)
\(948\) 0 0
\(949\) −10.3308 −0.335351
\(950\) −7.38225 −0.239512
\(951\) 0 0
\(952\) −23.0539 −0.747180
\(953\) −1.34233 −0.0434823 −0.0217411 0.999764i \(-0.506921\pi\)
−0.0217411 + 0.999764i \(0.506921\pi\)
\(954\) 0 0
\(955\) −12.6042 −0.407862
\(956\) 11.6735 0.377548
\(957\) 0 0
\(958\) 0.901550 0.0291278
\(959\) 15.2219 0.491540
\(960\) 0 0
\(961\) −25.5959 −0.825673
\(962\) −51.5360 −1.66159
\(963\) 0 0
\(964\) −1.66417 −0.0535992
\(965\) 20.2363 0.651429
\(966\) 0 0
\(967\) 41.9277 1.34830 0.674152 0.738593i \(-0.264509\pi\)
0.674152 + 0.738593i \(0.264509\pi\)
\(968\) −7.01410 −0.225442
\(969\) 0 0
\(970\) 4.47973 0.143835
\(971\) −12.4573 −0.399775 −0.199887 0.979819i \(-0.564058\pi\)
−0.199887 + 0.979819i \(0.564058\pi\)
\(972\) 0 0
\(973\) −51.5568 −1.65283
\(974\) −42.0001 −1.34577
\(975\) 0 0
\(976\) 5.57657 0.178502
\(977\) −7.40272 −0.236834 −0.118417 0.992964i \(-0.537782\pi\)
−0.118417 + 0.992964i \(0.537782\pi\)
\(978\) 0 0
\(979\) 1.99647 0.0638075
\(980\) 18.7605 0.599281
\(981\) 0 0
\(982\) 19.7629 0.630658
\(983\) 55.2136 1.76104 0.880521 0.474008i \(-0.157193\pi\)
0.880521 + 0.474008i \(0.157193\pi\)
\(984\) 0 0
\(985\) −23.2562 −0.741006
\(986\) −10.2610 −0.326778
\(987\) 0 0
\(988\) 43.0110 1.36836
\(989\) −14.4707 −0.460143
\(990\) 0 0
\(991\) −16.5854 −0.526853 −0.263426 0.964680i \(-0.584853\pi\)
−0.263426 + 0.964680i \(0.584853\pi\)
\(992\) −2.32468 −0.0738087
\(993\) 0 0
\(994\) −43.9133 −1.39284
\(995\) 19.9954 0.633898
\(996\) 0 0
\(997\) 21.2925 0.674341 0.337171 0.941444i \(-0.390530\pi\)
0.337171 + 0.941444i \(0.390530\pi\)
\(998\) −9.29093 −0.294099
\(999\) 0 0
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 8010.2.a.bn.1.1 7
3.2 odd 2 2670.2.a.t.1.1 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2670.2.a.t.1.1 7 3.2 odd 2
8010.2.a.bn.1.1 7 1.1 even 1 trivial