Properties

Label 9680.2.a.cq.1.1
Level $9680$
Weight $2$
Character 9680.1
Self dual yes
Analytic conductor $77.295$
Analytic rank $1$
Dimension $4$
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] = [9680,2,Mod(1,9680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("9680.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.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: \(77.2951891566\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.4752.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 3x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4840)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.21969\) of defining polynomial
Character \(\chi\) \(=\) 9680.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.33225 q^{3} -1.00000 q^{5} -0.399804 q^{7} +2.43937 q^{9} +O(q^{10})\) \(q-2.33225 q^{3} -1.00000 q^{5} -0.399804 q^{7} +2.43937 q^{9} +2.43937 q^{13} +2.33225 q^{15} -3.63977 q^{19} +0.932442 q^{21} +1.02473 q^{23} +1.00000 q^{25} +1.30752 q^{27} +4.00000 q^{29} -3.23898 q^{31} +0.399804 q^{35} -4.87875 q^{37} -5.68922 q^{39} +0.0922857 q^{41} -7.14344 q^{43} -2.43937 q^{45} -9.21099 q^{47} -6.84016 q^{49} +9.10387 q^{53} +8.48883 q^{57} +7.68922 q^{59} +6.57221 q^{61} -0.975272 q^{63} -2.43937 q^{65} +11.8755 q^{67} -2.38992 q^{69} -5.28844 q^{71} +8.00000 q^{73} -2.33225 q^{75} +1.67836 q^{79} -10.3676 q^{81} -16.3537 q^{83} -9.32899 q^{87} +10.2795 q^{89} -0.975272 q^{91} +7.55411 q^{93} +3.63977 q^{95} +11.5046 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{3} - 4 q^{5} - 6 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{3} - 4 q^{5} - 6 q^{7} + 4 q^{9} + 4 q^{13} - 2 q^{15} - 12 q^{21} - 4 q^{23} + 4 q^{25} + 2 q^{27} + 16 q^{29} - 16 q^{31} + 6 q^{35} - 8 q^{37} + 8 q^{39} + 8 q^{41} + 10 q^{43} - 4 q^{45} - 14 q^{47} - 4 q^{49} + 8 q^{53} + 12 q^{57} - 4 q^{61} - 12 q^{63} - 4 q^{65} + 2 q^{67} - 20 q^{69} - 8 q^{71} + 32 q^{73} + 2 q^{75} + 4 q^{79} - 8 q^{81} - 12 q^{83} + 8 q^{87} + 12 q^{89} - 12 q^{91} - 32 q^{93}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.33225 −1.34652 −0.673262 0.739404i \(-0.735107\pi\)
−0.673262 + 0.739404i \(0.735107\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −0.399804 −0.151112 −0.0755559 0.997142i \(-0.524073\pi\)
−0.0755559 + 0.997142i \(0.524073\pi\)
\(8\) 0 0
\(9\) 2.43937 0.813125
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) 2.43937 0.676561 0.338280 0.941045i \(-0.390155\pi\)
0.338280 + 0.941045i \(0.390155\pi\)
\(14\) 0 0
\(15\) 2.33225 0.602183
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −3.63977 −0.835019 −0.417510 0.908672i \(-0.637097\pi\)
−0.417510 + 0.908672i \(0.637097\pi\)
\(20\) 0 0
\(21\) 0.932442 0.203476
\(22\) 0 0
\(23\) 1.02473 0.213671 0.106835 0.994277i \(-0.465928\pi\)
0.106835 + 0.994277i \(0.465928\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.30752 0.251632
\(28\) 0 0
\(29\) 4.00000 0.742781 0.371391 0.928477i \(-0.378881\pi\)
0.371391 + 0.928477i \(0.378881\pi\)
\(30\) 0 0
\(31\) −3.23898 −0.581738 −0.290869 0.956763i \(-0.593944\pi\)
−0.290869 + 0.956763i \(0.593944\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.399804 0.0675793
\(36\) 0 0
\(37\) −4.87875 −0.802061 −0.401031 0.916065i \(-0.631348\pi\)
−0.401031 + 0.916065i \(0.631348\pi\)
\(38\) 0 0
\(39\) −5.68922 −0.911004
\(40\) 0 0
\(41\) 0.0922857 0.0144126 0.00720630 0.999974i \(-0.497706\pi\)
0.00720630 + 0.999974i \(0.497706\pi\)
\(42\) 0 0
\(43\) −7.14344 −1.08936 −0.544682 0.838643i \(-0.683350\pi\)
−0.544682 + 0.838643i \(0.683350\pi\)
\(44\) 0 0
\(45\) −2.43937 −0.363640
\(46\) 0 0
\(47\) −9.21099 −1.34356 −0.671781 0.740750i \(-0.734470\pi\)
−0.671781 + 0.740750i \(0.734470\pi\)
\(48\) 0 0
\(49\) −6.84016 −0.977165
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.10387 1.25051 0.625256 0.780420i \(-0.284995\pi\)
0.625256 + 0.780420i \(0.284995\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.48883 1.12437
\(58\) 0 0
\(59\) 7.68922 1.00105 0.500526 0.865722i \(-0.333140\pi\)
0.500526 + 0.865722i \(0.333140\pi\)
\(60\) 0 0
\(61\) 6.57221 0.841485 0.420742 0.907180i \(-0.361770\pi\)
0.420742 + 0.907180i \(0.361770\pi\)
\(62\) 0 0
\(63\) −0.975272 −0.122873
\(64\) 0 0
\(65\) −2.43937 −0.302567
\(66\) 0 0
\(67\) 11.8755 1.45082 0.725411 0.688316i \(-0.241650\pi\)
0.725411 + 0.688316i \(0.241650\pi\)
\(68\) 0 0
\(69\) −2.38992 −0.287712
\(70\) 0 0
\(71\) −5.28844 −0.627622 −0.313811 0.949485i \(-0.601606\pi\)
−0.313811 + 0.949485i \(0.601606\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 0 0
\(75\) −2.33225 −0.269305
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.67836 0.188830 0.0944149 0.995533i \(-0.469902\pi\)
0.0944149 + 0.995533i \(0.469902\pi\)
\(80\) 0 0
\(81\) −10.3676 −1.15195
\(82\) 0 0
\(83\) −16.3537 −1.79505 −0.897527 0.440960i \(-0.854638\pi\)
−0.897527 + 0.440960i \(0.854638\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −9.32899 −1.00017
\(88\) 0 0
\(89\) 10.2795 1.08963 0.544814 0.838557i \(-0.316600\pi\)
0.544814 + 0.838557i \(0.316600\pi\)
\(90\) 0 0
\(91\) −0.975272 −0.102236
\(92\) 0 0
\(93\) 7.55411 0.783324
\(94\) 0 0
\(95\) 3.63977 0.373432
\(96\) 0 0
\(97\) 11.5046 1.16812 0.584060 0.811710i \(-0.301463\pi\)
0.584060 + 0.811710i \(0.301463\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 4.39654 0.437472 0.218736 0.975784i \(-0.429807\pi\)
0.218736 + 0.975784i \(0.429807\pi\)
\(102\) 0 0
\(103\) 6.78222 0.668272 0.334136 0.942525i \(-0.391555\pi\)
0.334136 + 0.942525i \(0.391555\pi\)
\(104\) 0 0
\(105\) −0.932442 −0.0909970
\(106\) 0 0
\(107\) −1.83184 −0.177090 −0.0885451 0.996072i \(-0.528222\pi\)
−0.0885451 + 0.996072i \(0.528222\pi\)
\(108\) 0 0
\(109\) −9.58607 −0.918179 −0.459090 0.888390i \(-0.651824\pi\)
−0.459090 + 0.888390i \(0.651824\pi\)
\(110\) 0 0
\(111\) 11.3784 1.07999
\(112\) 0 0
\(113\) −10.4502 −0.983076 −0.491538 0.870856i \(-0.663565\pi\)
−0.491538 + 0.870856i \(0.663565\pi\)
\(114\) 0 0
\(115\) −1.02473 −0.0955564
\(116\) 0 0
\(117\) 5.95054 0.550128
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −0.215233 −0.0194069
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 11.9539 1.06074 0.530369 0.847767i \(-0.322053\pi\)
0.530369 + 0.847767i \(0.322053\pi\)
\(128\) 0 0
\(129\) 16.6603 1.46685
\(130\) 0 0
\(131\) −12.1672 −1.06305 −0.531526 0.847042i \(-0.678381\pi\)
−0.531526 + 0.847042i \(0.678381\pi\)
\(132\) 0 0
\(133\) 1.45519 0.126181
\(134\) 0 0
\(135\) −1.30752 −0.112533
\(136\) 0 0
\(137\) −2.17566 −0.185879 −0.0929397 0.995672i \(-0.529626\pi\)
−0.0929397 + 0.995672i \(0.529626\pi\)
\(138\) 0 0
\(139\) 6.19687 0.525612 0.262806 0.964849i \(-0.415352\pi\)
0.262806 + 0.964849i \(0.415352\pi\)
\(140\) 0 0
\(141\) 21.4823 1.80914
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −4.00000 −0.332182
\(146\) 0 0
\(147\) 15.9529 1.31578
\(148\) 0 0
\(149\) 21.8517 1.79016 0.895082 0.445901i \(-0.147117\pi\)
0.895082 + 0.445901i \(0.147117\pi\)
\(150\) 0 0
\(151\) −5.96141 −0.485133 −0.242566 0.970135i \(-0.577989\pi\)
−0.242566 + 0.970135i \(0.577989\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.23898 0.260161
\(156\) 0 0
\(157\) 19.0865 1.52327 0.761634 0.648008i \(-0.224398\pi\)
0.761634 + 0.648008i \(0.224398\pi\)
\(158\) 0 0
\(159\) −21.2325 −1.68384
\(160\) 0 0
\(161\) −0.409691 −0.0322881
\(162\) 0 0
\(163\) 16.2357 1.27168 0.635840 0.771821i \(-0.280654\pi\)
0.635840 + 0.771821i \(0.280654\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.32801 0.257529 0.128764 0.991675i \(-0.458899\pi\)
0.128764 + 0.991675i \(0.458899\pi\)
\(168\) 0 0
\(169\) −7.04946 −0.542266
\(170\) 0 0
\(171\) −8.87875 −0.678975
\(172\) 0 0
\(173\) 20.9391 1.59197 0.795984 0.605318i \(-0.206954\pi\)
0.795984 + 0.605318i \(0.206954\pi\)
\(174\) 0 0
\(175\) −0.399804 −0.0302224
\(176\) 0 0
\(177\) −17.9332 −1.34794
\(178\) 0 0
\(179\) −5.38496 −0.402491 −0.201246 0.979541i \(-0.564499\pi\)
−0.201246 + 0.979541i \(0.564499\pi\)
\(180\) 0 0
\(181\) 13.1201 0.975211 0.487605 0.873064i \(-0.337871\pi\)
0.487605 + 0.873064i \(0.337871\pi\)
\(182\) 0 0
\(183\) −15.3280 −1.13308
\(184\) 0 0
\(185\) 4.87875 0.358693
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −0.522752 −0.0380246
\(190\) 0 0
\(191\) −0.0965246 −0.00698428 −0.00349214 0.999994i \(-0.501112\pi\)
−0.00349214 + 0.999994i \(0.501112\pi\)
\(192\) 0 0
\(193\) −4.69418 −0.337894 −0.168947 0.985625i \(-0.554037\pi\)
−0.168947 + 0.985625i \(0.554037\pi\)
\(194\) 0 0
\(195\) 5.68922 0.407414
\(196\) 0 0
\(197\) −27.0008 −1.92373 −0.961864 0.273527i \(-0.911810\pi\)
−0.961864 + 0.273527i \(0.911810\pi\)
\(198\) 0 0
\(199\) −23.9440 −1.69735 −0.848673 0.528917i \(-0.822598\pi\)
−0.848673 + 0.528917i \(0.822598\pi\)
\(200\) 0 0
\(201\) −27.6966 −1.95356
\(202\) 0 0
\(203\) −1.59922 −0.112243
\(204\) 0 0
\(205\) −0.0922857 −0.00644551
\(206\) 0 0
\(207\) 2.49969 0.173741
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 22.5699 1.55378 0.776889 0.629638i \(-0.216797\pi\)
0.776889 + 0.629638i \(0.216797\pi\)
\(212\) 0 0
\(213\) 12.3339 0.845108
\(214\) 0 0
\(215\) 7.14344 0.487178
\(216\) 0 0
\(217\) 1.29496 0.0879076
\(218\) 0 0
\(219\) −18.6580 −1.26079
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.19870 −0.0802707 −0.0401353 0.999194i \(-0.512779\pi\)
−0.0401353 + 0.999194i \(0.512779\pi\)
\(224\) 0 0
\(225\) 2.43937 0.162625
\(226\) 0 0
\(227\) −19.0643 −1.26534 −0.632671 0.774421i \(-0.718041\pi\)
−0.632671 + 0.774421i \(0.718041\pi\)
\(228\) 0 0
\(229\) −19.8228 −1.30993 −0.654963 0.755661i \(-0.727316\pi\)
−0.654963 + 0.755661i \(0.727316\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.26867 0.0831130 0.0415565 0.999136i \(-0.486768\pi\)
0.0415565 + 0.999136i \(0.486768\pi\)
\(234\) 0 0
\(235\) 9.21099 0.600859
\(236\) 0 0
\(237\) −3.91434 −0.254264
\(238\) 0 0
\(239\) −28.0900 −1.81699 −0.908496 0.417894i \(-0.862768\pi\)
−0.908496 + 0.417894i \(0.862768\pi\)
\(240\) 0 0
\(241\) −20.3495 −1.31082 −0.655412 0.755271i \(-0.727505\pi\)
−0.655412 + 0.755271i \(0.727505\pi\)
\(242\) 0 0
\(243\) 20.2572 1.29950
\(244\) 0 0
\(245\) 6.84016 0.437002
\(246\) 0 0
\(247\) −8.87875 −0.564941
\(248\) 0 0
\(249\) 38.1409 2.41708
\(250\) 0 0
\(251\) 9.11234 0.575166 0.287583 0.957756i \(-0.407148\pi\)
0.287583 + 0.957756i \(0.407148\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2.38340 −0.148672 −0.0743361 0.997233i \(-0.523684\pi\)
−0.0743361 + 0.997233i \(0.523684\pi\)
\(258\) 0 0
\(259\) 1.95054 0.121201
\(260\) 0 0
\(261\) 9.75749 0.603974
\(262\) 0 0
\(263\) 1.42551 0.0879008 0.0439504 0.999034i \(-0.486006\pi\)
0.0439504 + 0.999034i \(0.486006\pi\)
\(264\) 0 0
\(265\) −9.10387 −0.559246
\(266\) 0 0
\(267\) −23.9744 −1.46721
\(268\) 0 0
\(269\) −1.06684 −0.0650465 −0.0325232 0.999471i \(-0.510354\pi\)
−0.0325232 + 0.999471i \(0.510354\pi\)
\(270\) 0 0
\(271\) −7.80695 −0.474238 −0.237119 0.971481i \(-0.576203\pi\)
−0.237119 + 0.971481i \(0.576203\pi\)
\(272\) 0 0
\(273\) 2.27458 0.137664
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 27.6224 1.65967 0.829834 0.558010i \(-0.188435\pi\)
0.829834 + 0.558010i \(0.188435\pi\)
\(278\) 0 0
\(279\) −7.90109 −0.473026
\(280\) 0 0
\(281\) −2.20773 −0.131702 −0.0658512 0.997829i \(-0.520976\pi\)
−0.0658512 + 0.997829i \(0.520976\pi\)
\(282\) 0 0
\(283\) −1.99011 −0.118300 −0.0591500 0.998249i \(-0.518839\pi\)
−0.0591500 + 0.998249i \(0.518839\pi\)
\(284\) 0 0
\(285\) −8.48883 −0.502835
\(286\) 0 0
\(287\) −0.0368962 −0.00217791
\(288\) 0 0
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) −26.8317 −1.57290
\(292\) 0 0
\(293\) −7.13355 −0.416746 −0.208373 0.978049i \(-0.566817\pi\)
−0.208373 + 0.978049i \(0.566817\pi\)
\(294\) 0 0
\(295\) −7.68922 −0.447684
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 2.49969 0.144561
\(300\) 0 0
\(301\) 2.85598 0.164616
\(302\) 0 0
\(303\) −10.2538 −0.589067
\(304\) 0 0
\(305\) −6.57221 −0.376323
\(306\) 0 0
\(307\) 14.2271 0.811982 0.405991 0.913877i \(-0.366926\pi\)
0.405991 + 0.913877i \(0.366926\pi\)
\(308\) 0 0
\(309\) −15.8178 −0.899844
\(310\) 0 0
\(311\) 4.43090 0.251253 0.125627 0.992078i \(-0.459906\pi\)
0.125627 + 0.992078i \(0.459906\pi\)
\(312\) 0 0
\(313\) 4.03207 0.227906 0.113953 0.993486i \(-0.463649\pi\)
0.113953 + 0.993486i \(0.463649\pi\)
\(314\) 0 0
\(315\) 0.975272 0.0549504
\(316\) 0 0
\(317\) −8.65797 −0.486280 −0.243140 0.969991i \(-0.578177\pi\)
−0.243140 + 0.969991i \(0.578177\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 4.27229 0.238456
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 2.43937 0.135312
\(326\) 0 0
\(327\) 22.3571 1.23635
\(328\) 0 0
\(329\) 3.68260 0.203028
\(330\) 0 0
\(331\) 6.83168 0.375503 0.187752 0.982217i \(-0.439880\pi\)
0.187752 + 0.982217i \(0.439880\pi\)
\(332\) 0 0
\(333\) −11.9011 −0.652176
\(334\) 0 0
\(335\) −11.8755 −0.648827
\(336\) 0 0
\(337\) 10.6324 0.579185 0.289592 0.957150i \(-0.406480\pi\)
0.289592 + 0.957150i \(0.406480\pi\)
\(338\) 0 0
\(339\) 24.3725 1.32373
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 5.53335 0.298773
\(344\) 0 0
\(345\) 2.38992 0.128669
\(346\) 0 0
\(347\) 29.5292 1.58521 0.792606 0.609734i \(-0.208724\pi\)
0.792606 + 0.609734i \(0.208724\pi\)
\(348\) 0 0
\(349\) 9.77922 0.523470 0.261735 0.965140i \(-0.415705\pi\)
0.261735 + 0.965140i \(0.415705\pi\)
\(350\) 0 0
\(351\) 3.18953 0.170244
\(352\) 0 0
\(353\) 22.4823 1.19661 0.598306 0.801268i \(-0.295841\pi\)
0.598306 + 0.801268i \(0.295841\pi\)
\(354\) 0 0
\(355\) 5.28844 0.280681
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.95945 −0.103416 −0.0517080 0.998662i \(-0.516466\pi\)
−0.0517080 + 0.998662i \(0.516466\pi\)
\(360\) 0 0
\(361\) −5.75211 −0.302743
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −8.00000 −0.418739
\(366\) 0 0
\(367\) 3.66775 0.191455 0.0957276 0.995408i \(-0.469482\pi\)
0.0957276 + 0.995408i \(0.469482\pi\)
\(368\) 0 0
\(369\) 0.225119 0.0117192
\(370\) 0 0
\(371\) −3.63977 −0.188967
\(372\) 0 0
\(373\) −20.8317 −1.07862 −0.539312 0.842106i \(-0.681316\pi\)
−0.539312 + 0.842106i \(0.681316\pi\)
\(374\) 0 0
\(375\) 2.33225 0.120437
\(376\) 0 0
\(377\) 9.75749 0.502537
\(378\) 0 0
\(379\) −37.2730 −1.91459 −0.957293 0.289120i \(-0.906637\pi\)
−0.957293 + 0.289120i \(0.906637\pi\)
\(380\) 0 0
\(381\) −27.8795 −1.42831
\(382\) 0 0
\(383\) 12.4625 0.636806 0.318403 0.947955i \(-0.396854\pi\)
0.318403 + 0.947955i \(0.396854\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −17.4255 −0.885788
\(388\) 0 0
\(389\) 9.67536 0.490560 0.245280 0.969452i \(-0.421120\pi\)
0.245280 + 0.969452i \(0.421120\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 28.3769 1.43142
\(394\) 0 0
\(395\) −1.67836 −0.0844473
\(396\) 0 0
\(397\) −24.0989 −1.20949 −0.604745 0.796419i \(-0.706725\pi\)
−0.604745 + 0.796419i \(0.706725\pi\)
\(398\) 0 0
\(399\) −3.39387 −0.169906
\(400\) 0 0
\(401\) 15.4155 0.769812 0.384906 0.922956i \(-0.374234\pi\)
0.384906 + 0.922956i \(0.374234\pi\)
\(402\) 0 0
\(403\) −7.90109 −0.393581
\(404\) 0 0
\(405\) 10.3676 0.515169
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −23.4444 −1.15925 −0.579626 0.814882i \(-0.696801\pi\)
−0.579626 + 0.814882i \(0.696801\pi\)
\(410\) 0 0
\(411\) 5.07418 0.250291
\(412\) 0 0
\(413\) −3.07418 −0.151271
\(414\) 0 0
\(415\) 16.3537 0.802772
\(416\) 0 0
\(417\) −14.4526 −0.707748
\(418\) 0 0
\(419\) −21.3907 −1.04501 −0.522503 0.852637i \(-0.675002\pi\)
−0.522503 + 0.852637i \(0.675002\pi\)
\(420\) 0 0
\(421\) −8.53238 −0.415843 −0.207921 0.978146i \(-0.566670\pi\)
−0.207921 + 0.978146i \(0.566670\pi\)
\(422\) 0 0
\(423\) −22.4691 −1.09248
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −2.62760 −0.127158
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 37.0568 1.78496 0.892482 0.451084i \(-0.148962\pi\)
0.892482 + 0.451084i \(0.148962\pi\)
\(432\) 0 0
\(433\) −33.0414 −1.58787 −0.793933 0.608005i \(-0.791970\pi\)
−0.793933 + 0.608005i \(0.791970\pi\)
\(434\) 0 0
\(435\) 9.32899 0.447291
\(436\) 0 0
\(437\) −3.72977 −0.178419
\(438\) 0 0
\(439\) 5.17071 0.246784 0.123392 0.992358i \(-0.460623\pi\)
0.123392 + 0.992358i \(0.460623\pi\)
\(440\) 0 0
\(441\) −16.6857 −0.794557
\(442\) 0 0
\(443\) −8.64303 −0.410643 −0.205321 0.978695i \(-0.565824\pi\)
−0.205321 + 0.978695i \(0.565824\pi\)
\(444\) 0 0
\(445\) −10.2795 −0.487296
\(446\) 0 0
\(447\) −50.9636 −2.41050
\(448\) 0 0
\(449\) −32.7336 −1.54479 −0.772397 0.635140i \(-0.780942\pi\)
−0.772397 + 0.635140i \(0.780942\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 13.9035 0.653242
\(454\) 0 0
\(455\) 0.975272 0.0457215
\(456\) 0 0
\(457\) −37.2641 −1.74314 −0.871571 0.490270i \(-0.836898\pi\)
−0.871571 + 0.490270i \(0.836898\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −16.2035 −0.754672 −0.377336 0.926076i \(-0.623160\pi\)
−0.377336 + 0.926076i \(0.623160\pi\)
\(462\) 0 0
\(463\) −17.0998 −0.794694 −0.397347 0.917668i \(-0.630069\pi\)
−0.397347 + 0.917668i \(0.630069\pi\)
\(464\) 0 0
\(465\) −7.55411 −0.350313
\(466\) 0 0
\(467\) 12.4029 0.573938 0.286969 0.957940i \(-0.407352\pi\)
0.286969 + 0.957940i \(0.407352\pi\)
\(468\) 0 0
\(469\) −4.74787 −0.219236
\(470\) 0 0
\(471\) −44.5144 −2.05111
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −3.63977 −0.167004
\(476\) 0 0
\(477\) 22.2077 1.01682
\(478\) 0 0
\(479\) −14.6560 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(480\) 0 0
\(481\) −11.9011 −0.542643
\(482\) 0 0
\(483\) 0.955500 0.0434767
\(484\) 0 0
\(485\) −11.5046 −0.522399
\(486\) 0 0
\(487\) −28.3467 −1.28451 −0.642255 0.766491i \(-0.722001\pi\)
−0.642255 + 0.766491i \(0.722001\pi\)
\(488\) 0 0
\(489\) −37.8657 −1.71235
\(490\) 0 0
\(491\) 29.0568 1.31131 0.655657 0.755058i \(-0.272392\pi\)
0.655657 + 0.755058i \(0.272392\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.11434 0.0948411
\(498\) 0 0
\(499\) −20.0429 −0.897245 −0.448622 0.893721i \(-0.648085\pi\)
−0.448622 + 0.893721i \(0.648085\pi\)
\(500\) 0 0
\(501\) −7.76173 −0.346769
\(502\) 0 0
\(503\) 2.33296 0.104022 0.0520108 0.998647i \(-0.483437\pi\)
0.0520108 + 0.998647i \(0.483437\pi\)
\(504\) 0 0
\(505\) −4.39654 −0.195644
\(506\) 0 0
\(507\) 16.4411 0.730173
\(508\) 0 0
\(509\) −34.0653 −1.50992 −0.754958 0.655773i \(-0.772343\pi\)
−0.754958 + 0.655773i \(0.772343\pi\)
\(510\) 0 0
\(511\) −3.19843 −0.141490
\(512\) 0 0
\(513\) −4.75906 −0.210118
\(514\) 0 0
\(515\) −6.78222 −0.298860
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −48.8351 −2.14362
\(520\) 0 0
\(521\) −31.3877 −1.37512 −0.687561 0.726127i \(-0.741319\pi\)
−0.687561 + 0.726127i \(0.741319\pi\)
\(522\) 0 0
\(523\) −16.9622 −0.741707 −0.370853 0.928691i \(-0.620935\pi\)
−0.370853 + 0.928691i \(0.620935\pi\)
\(524\) 0 0
\(525\) 0.932442 0.0406951
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −21.9499 −0.954345
\(530\) 0 0
\(531\) 18.7569 0.813979
\(532\) 0 0
\(533\) 0.225119 0.00975100
\(534\) 0 0
\(535\) 1.83184 0.0791972
\(536\) 0 0
\(537\) 12.5591 0.541963
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −34.8962 −1.50031 −0.750153 0.661264i \(-0.770020\pi\)
−0.750153 + 0.661264i \(0.770020\pi\)
\(542\) 0 0
\(543\) −30.5993 −1.31314
\(544\) 0 0
\(545\) 9.58607 0.410622
\(546\) 0 0
\(547\) 9.21073 0.393822 0.196911 0.980421i \(-0.436909\pi\)
0.196911 + 0.980421i \(0.436909\pi\)
\(548\) 0 0
\(549\) 16.0321 0.684232
\(550\) 0 0
\(551\) −14.5591 −0.620237
\(552\) 0 0
\(553\) −0.671014 −0.0285344
\(554\) 0 0
\(555\) −11.3784 −0.482988
\(556\) 0 0
\(557\) −16.8440 −0.713702 −0.356851 0.934161i \(-0.616150\pi\)
−0.356851 + 0.934161i \(0.616150\pi\)
\(558\) 0 0
\(559\) −17.4255 −0.737021
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 47.2669 1.99206 0.996032 0.0889952i \(-0.0283656\pi\)
0.996032 + 0.0889952i \(0.0283656\pi\)
\(564\) 0 0
\(565\) 10.4502 0.439645
\(566\) 0 0
\(567\) 4.14500 0.174074
\(568\) 0 0
\(569\) 17.4213 0.730338 0.365169 0.930941i \(-0.381011\pi\)
0.365169 + 0.930941i \(0.381011\pi\)
\(570\) 0 0
\(571\) −1.32703 −0.0555344 −0.0277672 0.999614i \(-0.508840\pi\)
−0.0277672 + 0.999614i \(0.508840\pi\)
\(572\) 0 0
\(573\) 0.225119 0.00940449
\(574\) 0 0
\(575\) 1.02473 0.0427341
\(576\) 0 0
\(577\) −17.2305 −0.717315 −0.358658 0.933469i \(-0.616765\pi\)
−0.358658 + 0.933469i \(0.616765\pi\)
\(578\) 0 0
\(579\) 10.9480 0.454982
\(580\) 0 0
\(581\) 6.53829 0.271254
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −5.95054 −0.246025
\(586\) 0 0
\(587\) 13.5006 0.557228 0.278614 0.960403i \(-0.410125\pi\)
0.278614 + 0.960403i \(0.410125\pi\)
\(588\) 0 0
\(589\) 11.7891 0.485763
\(590\) 0 0
\(591\) 62.9726 2.59035
\(592\) 0 0
\(593\) 32.5699 1.33749 0.668743 0.743493i \(-0.266833\pi\)
0.668743 + 0.743493i \(0.266833\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 55.8434 2.28552
\(598\) 0 0
\(599\) −5.71747 −0.233610 −0.116805 0.993155i \(-0.537265\pi\)
−0.116805 + 0.993155i \(0.537265\pi\)
\(600\) 0 0
\(601\) −13.0718 −0.533210 −0.266605 0.963806i \(-0.585902\pi\)
−0.266605 + 0.963806i \(0.585902\pi\)
\(602\) 0 0
\(603\) 28.9688 1.17970
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 10.8317 0.439644 0.219822 0.975540i \(-0.429452\pi\)
0.219822 + 0.975540i \(0.429452\pi\)
\(608\) 0 0
\(609\) 3.72977 0.151138
\(610\) 0 0
\(611\) −22.4691 −0.909001
\(612\) 0 0
\(613\) 6.37058 0.257305 0.128653 0.991690i \(-0.458935\pi\)
0.128653 + 0.991690i \(0.458935\pi\)
\(614\) 0 0
\(615\) 0.215233 0.00867903
\(616\) 0 0
\(617\) 13.6807 0.550766 0.275383 0.961335i \(-0.411195\pi\)
0.275383 + 0.961335i \(0.411195\pi\)
\(618\) 0 0
\(619\) −25.8564 −1.03926 −0.519628 0.854392i \(-0.673930\pi\)
−0.519628 + 0.854392i \(0.673930\pi\)
\(620\) 0 0
\(621\) 1.33985 0.0537664
\(622\) 0 0
\(623\) −4.10980 −0.164656
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −7.19843 −0.286565 −0.143283 0.989682i \(-0.545766\pi\)
−0.143283 + 0.989682i \(0.545766\pi\)
\(632\) 0 0
\(633\) −52.6386 −2.09220
\(634\) 0 0
\(635\) −11.9539 −0.474376
\(636\) 0 0
\(637\) −16.6857 −0.661111
\(638\) 0 0
\(639\) −12.9005 −0.510335
\(640\) 0 0
\(641\) −43.9205 −1.73476 −0.867379 0.497649i \(-0.834197\pi\)
−0.867379 + 0.497649i \(0.834197\pi\)
\(642\) 0 0
\(643\) 15.9655 0.629617 0.314809 0.949155i \(-0.398060\pi\)
0.314809 + 0.949155i \(0.398060\pi\)
\(644\) 0 0
\(645\) −16.6603 −0.655997
\(646\) 0 0
\(647\) −2.70292 −0.106263 −0.0531313 0.998588i \(-0.516920\pi\)
−0.0531313 + 0.998588i \(0.516920\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −3.02016 −0.118370
\(652\) 0 0
\(653\) −18.3513 −0.718143 −0.359071 0.933310i \(-0.616907\pi\)
−0.359071 + 0.933310i \(0.616907\pi\)
\(654\) 0 0
\(655\) 12.1672 0.475411
\(656\) 0 0
\(657\) 19.5150 0.761352
\(658\) 0 0
\(659\) −18.9480 −0.738108 −0.369054 0.929408i \(-0.620318\pi\)
−0.369054 + 0.929408i \(0.620318\pi\)
\(660\) 0 0
\(661\) −9.30230 −0.361818 −0.180909 0.983500i \(-0.557904\pi\)
−0.180909 + 0.983500i \(0.557904\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.45519 −0.0564300
\(666\) 0 0
\(667\) 4.09891 0.158710
\(668\) 0 0
\(669\) 2.79566 0.108086
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 21.4750 0.827799 0.413900 0.910323i \(-0.364166\pi\)
0.413900 + 0.910323i \(0.364166\pi\)
\(674\) 0 0
\(675\) 1.30752 0.0503264
\(676\) 0 0
\(677\) 33.6634 1.29379 0.646894 0.762580i \(-0.276068\pi\)
0.646894 + 0.762580i \(0.276068\pi\)
\(678\) 0 0
\(679\) −4.59961 −0.176517
\(680\) 0 0
\(681\) 44.4626 1.70381
\(682\) 0 0
\(683\) −14.2504 −0.545277 −0.272638 0.962117i \(-0.587896\pi\)
−0.272638 + 0.962117i \(0.587896\pi\)
\(684\) 0 0
\(685\) 2.17566 0.0831278
\(686\) 0 0
\(687\) 46.2316 1.76385
\(688\) 0 0
\(689\) 22.2077 0.846047
\(690\) 0 0
\(691\) −14.6386 −0.556880 −0.278440 0.960454i \(-0.589817\pi\)
−0.278440 + 0.960454i \(0.589817\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −6.19687 −0.235061
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −2.95884 −0.111914
\(700\) 0 0
\(701\) 28.8440 1.08942 0.544711 0.838624i \(-0.316639\pi\)
0.544711 + 0.838624i \(0.316639\pi\)
\(702\) 0 0
\(703\) 17.7575 0.669737
\(704\) 0 0
\(705\) −21.4823 −0.809071
\(706\) 0 0
\(707\) −1.75776 −0.0661073
\(708\) 0 0
\(709\) −40.0483 −1.50405 −0.752023 0.659137i \(-0.770922\pi\)
−0.752023 + 0.659137i \(0.770922\pi\)
\(710\) 0 0
\(711\) 4.09414 0.153542
\(712\) 0 0
\(713\) −3.31908 −0.124300
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 65.5128 2.44662
\(718\) 0 0
\(719\) −21.5244 −0.802725 −0.401363 0.915919i \(-0.631463\pi\)
−0.401363 + 0.915919i \(0.631463\pi\)
\(720\) 0 0
\(721\) −2.71156 −0.100984
\(722\) 0 0
\(723\) 47.4600 1.76506
\(724\) 0 0
\(725\) 4.00000 0.148556
\(726\) 0 0
\(727\) −50.1455 −1.85979 −0.929897 0.367821i \(-0.880104\pi\)
−0.929897 + 0.367821i \(0.880104\pi\)
\(728\) 0 0
\(729\) −16.1420 −0.597853
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −27.3606 −1.01059 −0.505294 0.862947i \(-0.668616\pi\)
−0.505294 + 0.862947i \(0.668616\pi\)
\(734\) 0 0
\(735\) −15.9529 −0.588433
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −45.3734 −1.66909 −0.834543 0.550943i \(-0.814268\pi\)
−0.834543 + 0.550943i \(0.814268\pi\)
\(740\) 0 0
\(741\) 20.7074 0.760706
\(742\) 0 0
\(743\) −32.6396 −1.19743 −0.598716 0.800962i \(-0.704322\pi\)
−0.598716 + 0.800962i \(0.704322\pi\)
\(744\) 0 0
\(745\) −21.8517 −0.800586
\(746\) 0 0
\(747\) −39.8928 −1.45960
\(748\) 0 0
\(749\) 0.732376 0.0267604
\(750\) 0 0
\(751\) −1.56020 −0.0569324 −0.0284662 0.999595i \(-0.509062\pi\)
−0.0284662 + 0.999595i \(0.509062\pi\)
\(752\) 0 0
\(753\) −21.2522 −0.774474
\(754\) 0 0
\(755\) 5.96141 0.216958
\(756\) 0 0
\(757\) −49.7943 −1.80981 −0.904903 0.425618i \(-0.860057\pi\)
−0.904903 + 0.425618i \(0.860057\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.65572 −0.0600197 −0.0300099 0.999550i \(-0.509554\pi\)
−0.0300099 + 0.999550i \(0.509554\pi\)
\(762\) 0 0
\(763\) 3.83255 0.138748
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 18.7569 0.677272
\(768\) 0 0
\(769\) 17.0370 0.614371 0.307185 0.951650i \(-0.400613\pi\)
0.307185 + 0.951650i \(0.400613\pi\)
\(770\) 0 0
\(771\) 5.55867 0.200191
\(772\) 0 0
\(773\) 19.0267 0.684342 0.342171 0.939638i \(-0.388838\pi\)
0.342171 + 0.939638i \(0.388838\pi\)
\(774\) 0 0
\(775\) −3.23898 −0.116348
\(776\) 0 0
\(777\) −4.54915 −0.163200
\(778\) 0 0
\(779\) −0.335898 −0.0120348
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 5.23007 0.186908
\(784\) 0 0
\(785\) −19.0865 −0.681226
\(786\) 0 0
\(787\) 35.6732 1.27161 0.635807 0.771848i \(-0.280667\pi\)
0.635807 + 0.771848i \(0.280667\pi\)
\(788\) 0 0
\(789\) −3.32464 −0.118360
\(790\) 0 0
\(791\) 4.17805 0.148554
\(792\) 0 0
\(793\) 16.0321 0.569315
\(794\) 0 0
\(795\) 21.2325 0.753038
\(796\) 0 0
\(797\) −25.8390 −0.915265 −0.457633 0.889141i \(-0.651302\pi\)
−0.457633 + 0.889141i \(0.651302\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 25.0756 0.886003
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0.409691 0.0144397
\(806\) 0 0
\(807\) 2.48814 0.0875866
\(808\) 0 0
\(809\) −3.45562 −0.121493 −0.0607466 0.998153i \(-0.519348\pi\)
−0.0607466 + 0.998153i \(0.519348\pi\)
\(810\) 0 0
\(811\) 7.32517 0.257221 0.128611 0.991695i \(-0.458948\pi\)
0.128611 + 0.991695i \(0.458948\pi\)
\(812\) 0 0
\(813\) 18.2077 0.638573
\(814\) 0 0
\(815\) −16.2357 −0.568713
\(816\) 0 0
\(817\) 26.0004 0.909640
\(818\) 0 0
\(819\) −2.37905 −0.0831308
\(820\) 0 0
\(821\) −48.0086 −1.67551 −0.837756 0.546045i \(-0.816133\pi\)
−0.837756 + 0.546045i \(0.816133\pi\)
\(822\) 0 0
\(823\) −46.5400 −1.62228 −0.811141 0.584851i \(-0.801153\pi\)
−0.811141 + 0.584851i \(0.801153\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −10.9847 −0.381976 −0.190988 0.981592i \(-0.561169\pi\)
−0.190988 + 0.981592i \(0.561169\pi\)
\(828\) 0 0
\(829\) −4.52161 −0.157042 −0.0785209 0.996912i \(-0.525020\pi\)
−0.0785209 + 0.996912i \(0.525020\pi\)
\(830\) 0 0
\(831\) −64.4222 −2.23478
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −3.32801 −0.115170
\(836\) 0 0
\(837\) −4.23503 −0.146384
\(838\) 0 0
\(839\) −4.33012 −0.149492 −0.0747462 0.997203i \(-0.523815\pi\)
−0.0747462 + 0.997203i \(0.523815\pi\)
\(840\) 0 0
\(841\) −13.0000 −0.448276
\(842\) 0 0
\(843\) 5.14898 0.177340
\(844\) 0 0
\(845\) 7.04946 0.242509
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 4.64144 0.159294
\(850\) 0 0
\(851\) −4.99939 −0.171377
\(852\) 0 0
\(853\) 27.1082 0.928167 0.464084 0.885791i \(-0.346384\pi\)
0.464084 + 0.885791i \(0.346384\pi\)
\(854\) 0 0
\(855\) 8.87875 0.303647
\(856\) 0 0
\(857\) 20.1668 0.688883 0.344442 0.938808i \(-0.388068\pi\)
0.344442 + 0.938808i \(0.388068\pi\)
\(858\) 0 0
\(859\) −51.0541 −1.74194 −0.870972 0.491333i \(-0.836510\pi\)
−0.870972 + 0.491333i \(0.836510\pi\)
\(860\) 0 0
\(861\) 0.0860511 0.00293261
\(862\) 0 0
\(863\) 51.7782 1.76255 0.881276 0.472603i \(-0.156685\pi\)
0.881276 + 0.472603i \(0.156685\pi\)
\(864\) 0 0
\(865\) −20.9391 −0.711950
\(866\) 0 0
\(867\) 39.6482 1.34652
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 28.9688 0.981569
\(872\) 0 0
\(873\) 28.0641 0.949827
\(874\) 0 0
\(875\) 0.399804 0.0135159
\(876\) 0 0
\(877\) 8.40317 0.283755 0.141877 0.989884i \(-0.454686\pi\)
0.141877 + 0.989884i \(0.454686\pi\)
\(878\) 0 0
\(879\) 16.6372 0.561159
\(880\) 0 0
\(881\) 36.7336 1.23759 0.618793 0.785554i \(-0.287622\pi\)
0.618793 + 0.785554i \(0.287622\pi\)
\(882\) 0 0
\(883\) −29.7104 −0.999835 −0.499918 0.866073i \(-0.666636\pi\)
−0.499918 + 0.866073i \(0.666636\pi\)
\(884\) 0 0
\(885\) 17.9332 0.602817
\(886\) 0 0
\(887\) −46.6697 −1.56702 −0.783508 0.621382i \(-0.786572\pi\)
−0.783508 + 0.621382i \(0.786572\pi\)
\(888\) 0 0
\(889\) −4.77922 −0.160290
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 33.5259 1.12190
\(894\) 0 0
\(895\) 5.38496 0.179999
\(896\) 0 0
\(897\) −5.82990 −0.194655
\(898\) 0 0
\(899\) −12.9559 −0.432104
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −6.66084 −0.221659
\(904\) 0 0
\(905\) −13.1201 −0.436127
\(906\) 0 0
\(907\) 22.5206 0.747785 0.373893 0.927472i \(-0.378023\pi\)
0.373893 + 0.927472i \(0.378023\pi\)
\(908\) 0 0
\(909\) 10.7248 0.355720
\(910\) 0 0
\(911\) 1.81347 0.0600830 0.0300415 0.999549i \(-0.490436\pi\)
0.0300415 + 0.999549i \(0.490436\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 15.3280 0.506728
\(916\) 0 0
\(917\) 4.86449 0.160640
\(918\) 0 0
\(919\) 25.9416 0.855736 0.427868 0.903841i \(-0.359265\pi\)
0.427868 + 0.903841i \(0.359265\pi\)
\(920\) 0 0
\(921\) −33.1810 −1.09335
\(922\) 0 0
\(923\) −12.9005 −0.424624
\(924\) 0 0
\(925\) −4.87875 −0.160412
\(926\) 0 0
\(927\) 16.5444 0.543389
\(928\) 0 0
\(929\) −28.5243 −0.935852 −0.467926 0.883768i \(-0.654999\pi\)
−0.467926 + 0.883768i \(0.654999\pi\)
\(930\) 0 0
\(931\) 24.8966 0.815952
\(932\) 0 0
\(933\) −10.3339 −0.338318
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −15.7213 −0.513592 −0.256796 0.966466i \(-0.582667\pi\)
−0.256796 + 0.966466i \(0.582667\pi\)
\(938\) 0 0
\(939\) −9.40378 −0.306881
\(940\) 0 0
\(941\) −8.89767 −0.290056 −0.145028 0.989428i \(-0.546327\pi\)
−0.145028 + 0.989428i \(0.546327\pi\)
\(942\) 0 0
\(943\) 0.0945677 0.00307955
\(944\) 0 0
\(945\) 0.522752 0.0170051
\(946\) 0 0
\(947\) −1.78761 −0.0580895 −0.0290447 0.999578i \(-0.509247\pi\)
−0.0290447 + 0.999578i \(0.509247\pi\)
\(948\) 0 0
\(949\) 19.5150 0.633483
\(950\) 0 0
\(951\) 20.1925 0.654787
\(952\) 0 0
\(953\) −1.07180 −0.0347189 −0.0173595 0.999849i \(-0.505526\pi\)
−0.0173595 + 0.999849i \(0.505526\pi\)
\(954\) 0 0
\(955\) 0.0965246 0.00312346
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0.869840 0.0280886
\(960\) 0 0
\(961\) −20.5090 −0.661580
\(962\) 0 0
\(963\) −4.46853 −0.143996
\(964\) 0 0
\(965\) 4.69418 0.151111
\(966\) 0 0
\(967\) −17.3661 −0.558458 −0.279229 0.960225i \(-0.590079\pi\)
−0.279229 + 0.960225i \(0.590079\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 45.2773 1.45302 0.726508 0.687158i \(-0.241142\pi\)
0.726508 + 0.687158i \(0.241142\pi\)
\(972\) 0 0
\(973\) −2.47753 −0.0794261
\(974\) 0 0
\(975\) −5.68922 −0.182201
\(976\) 0 0
\(977\) −24.5916 −0.786754 −0.393377 0.919377i \(-0.628693\pi\)
−0.393377 + 0.919377i \(0.628693\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −23.3840 −0.746594
\(982\) 0 0
\(983\) −21.4860 −0.685297 −0.342648 0.939464i \(-0.611324\pi\)
−0.342648 + 0.939464i \(0.611324\pi\)
\(984\) 0 0
\(985\) 27.0008 0.860318
\(986\) 0 0
\(987\) −8.58872 −0.273382
\(988\) 0 0
\(989\) −7.32008 −0.232765
\(990\) 0 0
\(991\) 38.4495 1.22139 0.610694 0.791867i \(-0.290890\pi\)
0.610694 + 0.791867i \(0.290890\pi\)
\(992\) 0 0
\(993\) −15.9332 −0.505624
\(994\) 0 0
\(995\) 23.9440 0.759077
\(996\) 0 0
\(997\) −36.5530 −1.15764 −0.578822 0.815454i \(-0.696487\pi\)
−0.578822 + 0.815454i \(0.696487\pi\)
\(998\) 0 0
\(999\) −6.37905 −0.201824
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 9680.2.a.cq.1.1 4
4.3 odd 2 4840.2.a.x.1.4 yes 4
11.10 odd 2 9680.2.a.cr.1.1 4
44.43 even 2 4840.2.a.w.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.w.1.4 4 44.43 even 2
4840.2.a.x.1.4 yes 4 4.3 odd 2
9680.2.a.cq.1.1 4 1.1 even 1 trivial
9680.2.a.cr.1.1 4 11.10 odd 2