Properties

Label 5040.2.t.v.1009.4
Level $5040$
Weight $2$
Character 5040.1009
Analytic conductor $40.245$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5040,2,Mod(1009,5040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5040.1009");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5040.t (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(40.2446026187\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1009.4
Root \(1.45161 - 1.45161i\) of defining polynomial
Character \(\chi\) \(=\) 5040.1009
Dual form 5040.2.t.v.1009.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.311108 + 2.21432i) q^{5} +1.00000i q^{7} +O(q^{10})\) \(q+(-0.311108 + 2.21432i) q^{5} +1.00000i q^{7} +2.00000 q^{11} +6.42864i q^{13} -4.42864i q^{17} -2.42864 q^{19} -1.37778i q^{23} +(-4.80642 - 1.37778i) q^{25} +0.755569 q^{29} -5.18421 q^{31} +(-2.21432 - 0.311108i) q^{35} +7.61285i q^{37} +8.23506 q^{41} +10.1017i q^{43} +2.75557i q^{47} -1.00000 q^{49} +9.18421i q^{53} +(-0.622216 + 4.42864i) q^{55} -14.1017 q^{59} +6.85728 q^{61} +(-14.2351 - 2.00000i) q^{65} -2.75557i q^{67} +2.00000 q^{71} +1.57136i q^{73} +2.00000i q^{77} -4.85728 q^{79} -11.6128i q^{83} +(9.80642 + 1.37778i) q^{85} +4.62222 q^{89} -6.42864 q^{91} +(0.755569 - 5.37778i) q^{95} -11.9398i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{5} + 12 q^{11} + 12 q^{19} - 2 q^{25} + 4 q^{29} - 4 q^{31} - 4 q^{41} - 6 q^{49} - 4 q^{55} - 32 q^{59} - 12 q^{61} - 32 q^{65} + 12 q^{71} + 24 q^{79} + 32 q^{85} + 28 q^{89} - 12 q^{91} + 4 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5040\mathbb{Z}\right)^\times\).

\(n\) \(2017\) \(2801\) \(3151\) \(3601\) \(3781\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.311108 + 2.21432i −0.139132 + 0.990274i
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 6.42864i 1.78298i 0.453037 + 0.891492i \(0.350341\pi\)
−0.453037 + 0.891492i \(0.649659\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.42864i 1.07410i −0.843550 0.537051i \(-0.819538\pi\)
0.843550 0.537051i \(-0.180462\pi\)
\(18\) 0 0
\(19\) −2.42864 −0.557168 −0.278584 0.960412i \(-0.589865\pi\)
−0.278584 + 0.960412i \(0.589865\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.37778i 0.287288i −0.989629 0.143644i \(-0.954118\pi\)
0.989629 0.143644i \(-0.0458820\pi\)
\(24\) 0 0
\(25\) −4.80642 1.37778i −0.961285 0.275557i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.755569 0.140306 0.0701528 0.997536i \(-0.477651\pi\)
0.0701528 + 0.997536i \(0.477651\pi\)
\(30\) 0 0
\(31\) −5.18421 −0.931111 −0.465556 0.885019i \(-0.654145\pi\)
−0.465556 + 0.885019i \(0.654145\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.21432 0.311108i −0.374288 0.0525868i
\(36\) 0 0
\(37\) 7.61285i 1.25154i 0.780006 + 0.625772i \(0.215216\pi\)
−0.780006 + 0.625772i \(0.784784\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.23506 1.28610 0.643050 0.765824i \(-0.277669\pi\)
0.643050 + 0.765824i \(0.277669\pi\)
\(42\) 0 0
\(43\) 10.1017i 1.54050i 0.637744 + 0.770248i \(0.279868\pi\)
−0.637744 + 0.770248i \(0.720132\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.75557i 0.401941i 0.979597 + 0.200971i \(0.0644095\pi\)
−0.979597 + 0.200971i \(0.935590\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 9.18421i 1.26155i 0.775967 + 0.630774i \(0.217263\pi\)
−0.775967 + 0.630774i \(0.782737\pi\)
\(54\) 0 0
\(55\) −0.622216 + 4.42864i −0.0838995 + 0.597158i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −14.1017 −1.83589 −0.917943 0.396712i \(-0.870151\pi\)
−0.917943 + 0.396712i \(0.870151\pi\)
\(60\) 0 0
\(61\) 6.85728 0.877985 0.438992 0.898491i \(-0.355336\pi\)
0.438992 + 0.898491i \(0.355336\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −14.2351 2.00000i −1.76564 0.248069i
\(66\) 0 0
\(67\) 2.75557i 0.336646i −0.985732 0.168323i \(-0.946165\pi\)
0.985732 0.168323i \(-0.0538352\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) 1.57136i 0.183914i 0.995763 + 0.0919569i \(0.0293122\pi\)
−0.995763 + 0.0919569i \(0.970688\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.00000i 0.227921i
\(78\) 0 0
\(79\) −4.85728 −0.546487 −0.273243 0.961945i \(-0.588096\pi\)
−0.273243 + 0.961945i \(0.588096\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 11.6128i 1.27468i −0.770585 0.637338i \(-0.780036\pi\)
0.770585 0.637338i \(-0.219964\pi\)
\(84\) 0 0
\(85\) 9.80642 + 1.37778i 1.06366 + 0.149442i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.62222 0.489954 0.244977 0.969529i \(-0.421220\pi\)
0.244977 + 0.969529i \(0.421220\pi\)
\(90\) 0 0
\(91\) −6.42864 −0.673905
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.755569 5.37778i 0.0775197 0.551749i
\(96\) 0 0
\(97\) 11.9398i 1.21230i −0.795350 0.606150i \(-0.792713\pi\)
0.795350 0.606150i \(-0.207287\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.47949 −0.147215 −0.0736076 0.997287i \(-0.523451\pi\)
−0.0736076 + 0.997287i \(0.523451\pi\)
\(102\) 0 0
\(103\) 8.85728i 0.872734i −0.899769 0.436367i \(-0.856265\pi\)
0.899769 0.436367i \(-0.143735\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.76494i 0.170623i 0.996354 + 0.0853114i \(0.0271885\pi\)
−0.996354 + 0.0853114i \(0.972811\pi\)
\(108\) 0 0
\(109\) −5.61285 −0.537613 −0.268807 0.963194i \(-0.586629\pi\)
−0.268807 + 0.963194i \(0.586629\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 11.2859i 1.06169i −0.847469 0.530845i \(-0.821875\pi\)
0.847469 0.530845i \(-0.178125\pi\)
\(114\) 0 0
\(115\) 3.05086 + 0.428639i 0.284494 + 0.0399708i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.42864 0.405973
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 4.54617 10.2143i 0.406622 0.913597i
\(126\) 0 0
\(127\) 12.8573i 1.14090i 0.821333 + 0.570450i \(0.193231\pi\)
−0.821333 + 0.570450i \(0.806769\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.10171 −0.183627 −0.0918136 0.995776i \(-0.529266\pi\)
−0.0918136 + 0.995776i \(0.529266\pi\)
\(132\) 0 0
\(133\) 2.42864i 0.210590i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 15.9398i 1.36183i −0.732364 0.680914i \(-0.761583\pi\)
0.732364 0.680914i \(-0.238417\pi\)
\(138\) 0 0
\(139\) 11.6731 0.990097 0.495048 0.868865i \(-0.335150\pi\)
0.495048 + 0.868865i \(0.335150\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 12.8573i 1.07518i
\(144\) 0 0
\(145\) −0.235063 + 1.67307i −0.0195209 + 0.138941i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −21.2257 −1.73888 −0.869438 0.494041i \(-0.835519\pi\)
−0.869438 + 0.494041i \(0.835519\pi\)
\(150\) 0 0
\(151\) −16.8573 −1.37183 −0.685913 0.727684i \(-0.740597\pi\)
−0.685913 + 0.727684i \(0.740597\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.61285 11.4795i 0.129547 0.922055i
\(156\) 0 0
\(157\) 10.4286i 0.832296i −0.909297 0.416148i \(-0.863380\pi\)
0.909297 0.416148i \(-0.136620\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.37778 0.108585
\(162\) 0 0
\(163\) 20.8573i 1.63367i 0.576873 + 0.816834i \(0.304273\pi\)
−0.576873 + 0.816834i \(0.695727\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 15.3461i 1.18752i 0.804642 + 0.593760i \(0.202357\pi\)
−0.804642 + 0.593760i \(0.797643\pi\)
\(168\) 0 0
\(169\) −28.3274 −2.17903
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.06022i 0.156636i 0.996928 + 0.0783179i \(0.0249549\pi\)
−0.996928 + 0.0783179i \(0.975045\pi\)
\(174\) 0 0
\(175\) 1.37778 4.80642i 0.104151 0.363331i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −10.0000 −0.747435 −0.373718 0.927543i \(-0.621917\pi\)
−0.373718 + 0.927543i \(0.621917\pi\)
\(180\) 0 0
\(181\) −12.1017 −0.899513 −0.449757 0.893151i \(-0.648489\pi\)
−0.449757 + 0.893151i \(0.648489\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −16.8573 2.36842i −1.23937 0.174129i
\(186\) 0 0
\(187\) 8.85728i 0.647708i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0.488863 0.0353729 0.0176864 0.999844i \(-0.494370\pi\)
0.0176864 + 0.999844i \(0.494370\pi\)
\(192\) 0 0
\(193\) 22.9590i 1.65262i 0.563212 + 0.826312i \(0.309565\pi\)
−0.563212 + 0.826312i \(0.690435\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.18421i 0.0843713i 0.999110 + 0.0421857i \(0.0134321\pi\)
−0.999110 + 0.0421857i \(0.986568\pi\)
\(198\) 0 0
\(199\) 8.79706 0.623607 0.311803 0.950147i \(-0.399067\pi\)
0.311803 + 0.950147i \(0.399067\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0.755569i 0.0530305i
\(204\) 0 0
\(205\) −2.56199 + 18.2351i −0.178937 + 1.27359i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4.85728 −0.335985
\(210\) 0 0
\(211\) −23.2257 −1.59892 −0.799461 0.600717i \(-0.794882\pi\)
−0.799461 + 0.600717i \(0.794882\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −22.3684 3.14272i −1.52551 0.214332i
\(216\) 0 0
\(217\) 5.18421i 0.351927i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 28.4701 1.91511
\(222\) 0 0
\(223\) 15.2257i 1.01959i −0.860297 0.509794i \(-0.829722\pi\)
0.860297 0.509794i \(-0.170278\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 14.3684i 0.953665i −0.878994 0.476833i \(-0.841785\pi\)
0.878994 0.476833i \(-0.158215\pi\)
\(228\) 0 0
\(229\) 5.61285 0.370907 0.185454 0.982653i \(-0.440625\pi\)
0.185454 + 0.982653i \(0.440625\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 23.2859i 1.52551i 0.646687 + 0.762756i \(0.276154\pi\)
−0.646687 + 0.762756i \(0.723846\pi\)
\(234\) 0 0
\(235\) −6.10171 0.857279i −0.398032 0.0559227i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.48886 0.549099 0.274549 0.961573i \(-0.411471\pi\)
0.274549 + 0.961573i \(0.411471\pi\)
\(240\) 0 0
\(241\) −7.24443 −0.466655 −0.233327 0.972398i \(-0.574961\pi\)
−0.233327 + 0.972398i \(0.574961\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.311108 2.21432i 0.0198759 0.141468i
\(246\) 0 0
\(247\) 15.6128i 0.993422i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 27.6128 1.74291 0.871454 0.490478i \(-0.163178\pi\)
0.871454 + 0.490478i \(0.163178\pi\)
\(252\) 0 0
\(253\) 2.75557i 0.173241i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0.428639i 0.0267378i 0.999911 + 0.0133689i \(0.00425558\pi\)
−0.999911 + 0.0133689i \(0.995744\pi\)
\(258\) 0 0
\(259\) −7.61285 −0.473039
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 9.37778i 0.578259i 0.957290 + 0.289129i \(0.0933658\pi\)
−0.957290 + 0.289129i \(0.906634\pi\)
\(264\) 0 0
\(265\) −20.3368 2.85728i −1.24928 0.175521i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1.74620 0.106468 0.0532339 0.998582i \(-0.483047\pi\)
0.0532339 + 0.998582i \(0.483047\pi\)
\(270\) 0 0
\(271\) 2.69535 0.163731 0.0818653 0.996643i \(-0.473912\pi\)
0.0818653 + 0.996643i \(0.473912\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −9.61285 2.75557i −0.579677 0.166167i
\(276\) 0 0
\(277\) 5.12399i 0.307870i −0.988081 0.153935i \(-0.950805\pi\)
0.988081 0.153935i \(-0.0491947\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −23.9813 −1.43060 −0.715301 0.698816i \(-0.753710\pi\)
−0.715301 + 0.698816i \(0.753710\pi\)
\(282\) 0 0
\(283\) 2.36842i 0.140788i 0.997519 + 0.0703939i \(0.0224256\pi\)
−0.997519 + 0.0703939i \(0.977574\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.23506i 0.486100i
\(288\) 0 0
\(289\) −2.61285 −0.153697
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 8.42864i 0.492406i 0.969218 + 0.246203i \(0.0791831\pi\)
−0.969218 + 0.246203i \(0.920817\pi\)
\(294\) 0 0
\(295\) 4.38715 31.2257i 0.255430 1.81803i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8.85728 0.512230
\(300\) 0 0
\(301\) −10.1017 −0.582253
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.13335 + 15.1842i −0.122155 + 0.869445i
\(306\) 0 0
\(307\) 22.5718i 1.28824i 0.764923 + 0.644121i \(0.222777\pi\)
−0.764923 + 0.644121i \(0.777223\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −24.0830 −1.36562 −0.682810 0.730596i \(-0.739242\pi\)
−0.682810 + 0.730596i \(0.739242\pi\)
\(312\) 0 0
\(313\) 9.65433i 0.545695i −0.962057 0.272848i \(-0.912035\pi\)
0.962057 0.272848i \(-0.0879655\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.04149i 0.339324i 0.985502 + 0.169662i \(0.0542676\pi\)
−0.985502 + 0.169662i \(0.945732\pi\)
\(318\) 0 0
\(319\) 1.51114 0.0846075
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 10.7556i 0.598456i
\(324\) 0 0
\(325\) 8.85728 30.8988i 0.491313 1.71396i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −2.75557 −0.151919
\(330\) 0 0
\(331\) −13.5111 −0.742639 −0.371320 0.928505i \(-0.621095\pi\)
−0.371320 + 0.928505i \(0.621095\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 6.10171 + 0.857279i 0.333372 + 0.0468382i
\(336\) 0 0
\(337\) 10.4889i 0.571365i 0.958324 + 0.285682i \(0.0922202\pi\)
−0.958324 + 0.285682i \(0.907780\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −10.3684 −0.561481
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 16.7239i 0.897787i −0.893585 0.448894i \(-0.851818\pi\)
0.893585 0.448894i \(-0.148182\pi\)
\(348\) 0 0
\(349\) 16.3684 0.876181 0.438091 0.898931i \(-0.355655\pi\)
0.438091 + 0.898931i \(0.355655\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.549086i 0.0292249i 0.999893 + 0.0146124i \(0.00465145\pi\)
−0.999893 + 0.0146124i \(0.995349\pi\)
\(354\) 0 0
\(355\) −0.622216 + 4.42864i −0.0330238 + 0.235048i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0.285442 0.0150651 0.00753253 0.999972i \(-0.497602\pi\)
0.00753253 + 0.999972i \(0.497602\pi\)
\(360\) 0 0
\(361\) −13.1017 −0.689564
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.47949 0.488863i −0.182125 0.0255882i
\(366\) 0 0
\(367\) 1.71456i 0.0894992i −0.998998 0.0447496i \(-0.985751\pi\)
0.998998 0.0447496i \(-0.0142490\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −9.18421 −0.476820
\(372\) 0 0
\(373\) 16.0000i 0.828449i −0.910175 0.414224i \(-0.864053\pi\)
0.910175 0.414224i \(-0.135947\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.85728i 0.250163i
\(378\) 0 0
\(379\) −4.85728 −0.249502 −0.124751 0.992188i \(-0.539813\pi\)
−0.124751 + 0.992188i \(0.539813\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 8.38715i 0.428563i 0.976772 + 0.214282i \(0.0687411\pi\)
−0.976772 + 0.214282i \(0.931259\pi\)
\(384\) 0 0
\(385\) −4.42864 0.622216i −0.225704 0.0317110i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 8.95899 0.454239 0.227119 0.973867i \(-0.427069\pi\)
0.227119 + 0.973867i \(0.427069\pi\)
\(390\) 0 0
\(391\) −6.10171 −0.308577
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.51114 10.7556i 0.0760336 0.541171i
\(396\) 0 0
\(397\) 2.54909i 0.127935i −0.997952 0.0639675i \(-0.979625\pi\)
0.997952 0.0639675i \(-0.0203754\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −0.958989 −0.0478896 −0.0239448 0.999713i \(-0.507623\pi\)
−0.0239448 + 0.999713i \(0.507623\pi\)
\(402\) 0 0
\(403\) 33.3274i 1.66016i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 15.2257i 0.754710i
\(408\) 0 0
\(409\) 31.9813 1.58137 0.790686 0.612222i \(-0.209724\pi\)
0.790686 + 0.612222i \(0.209724\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 14.1017i 0.693900i
\(414\) 0 0
\(415\) 25.7146 + 3.61285i 1.26228 + 0.177348i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0.470127 0.0229672 0.0114836 0.999934i \(-0.496345\pi\)
0.0114836 + 0.999934i \(0.496345\pi\)
\(420\) 0 0
\(421\) −33.6128 −1.63819 −0.819095 0.573658i \(-0.805524\pi\)
−0.819095 + 0.573658i \(0.805524\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −6.10171 + 21.2859i −0.295976 + 1.03252i
\(426\) 0 0
\(427\) 6.85728i 0.331847i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 11.7146 0.564270 0.282135 0.959375i \(-0.408957\pi\)
0.282135 + 0.959375i \(0.408957\pi\)
\(432\) 0 0
\(433\) 0.0602231i 0.00289414i 0.999999 + 0.00144707i \(0.000460616\pi\)
−0.999999 + 0.00144707i \(0.999539\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.34614i 0.160068i
\(438\) 0 0
\(439\) −22.4286 −1.07046 −0.535230 0.844706i \(-0.679775\pi\)
−0.535230 + 0.844706i \(0.679775\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 23.9496i 1.13788i 0.822379 + 0.568940i \(0.192647\pi\)
−0.822379 + 0.568940i \(0.807353\pi\)
\(444\) 0 0
\(445\) −1.43801 + 10.2351i −0.0681681 + 0.485189i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 29.4291 1.38885 0.694423 0.719567i \(-0.255660\pi\)
0.694423 + 0.719567i \(0.255660\pi\)
\(450\) 0 0
\(451\) 16.4701 0.775548
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2.00000 14.2351i 0.0937614 0.667350i
\(456\) 0 0
\(457\) 3.14272i 0.147010i −0.997295 0.0735051i \(-0.976581\pi\)
0.997295 0.0735051i \(-0.0234185\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.37778 0.157319 0.0786596 0.996902i \(-0.474936\pi\)
0.0786596 + 0.996902i \(0.474936\pi\)
\(462\) 0 0
\(463\) 20.8573i 0.969320i 0.874703 + 0.484660i \(0.161057\pi\)
−0.874703 + 0.484660i \(0.838943\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 14.3684i 0.664891i −0.943122 0.332446i \(-0.892126\pi\)
0.943122 0.332446i \(-0.107874\pi\)
\(468\) 0 0
\(469\) 2.75557 0.127240
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 20.2034i 0.928954i
\(474\) 0 0
\(475\) 11.6731 + 3.34614i 0.535597 + 0.153532i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −6.36842 −0.290980 −0.145490 0.989360i \(-0.546476\pi\)
−0.145490 + 0.989360i \(0.546476\pi\)
\(480\) 0 0
\(481\) −48.9403 −2.23148
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 26.4385 + 3.71456i 1.20051 + 0.168669i
\(486\) 0 0
\(487\) 17.3274i 0.785180i −0.919714 0.392590i \(-0.871579\pi\)
0.919714 0.392590i \(-0.128421\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2.00000 0.0902587 0.0451294 0.998981i \(-0.485630\pi\)
0.0451294 + 0.998981i \(0.485630\pi\)
\(492\) 0 0
\(493\) 3.34614i 0.150703i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.00000i 0.0897123i
\(498\) 0 0
\(499\) 23.3461 1.04512 0.522558 0.852603i \(-0.324978\pi\)
0.522558 + 0.852603i \(0.324978\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0.387152i 0.0172623i −0.999963 0.00863113i \(-0.997253\pi\)
0.999963 0.00863113i \(-0.00274741\pi\)
\(504\) 0 0
\(505\) 0.460282 3.27607i 0.0204823 0.145783i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −29.9496 −1.32749 −0.663747 0.747957i \(-0.731035\pi\)
−0.663747 + 0.747957i \(0.731035\pi\)
\(510\) 0 0
\(511\) −1.57136 −0.0695129
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 19.6128 + 2.75557i 0.864245 + 0.121425i
\(516\) 0 0
\(517\) 5.51114i 0.242380i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 18.5205 0.811398 0.405699 0.914007i \(-0.367028\pi\)
0.405699 + 0.914007i \(0.367028\pi\)
\(522\) 0 0
\(523\) 4.00000i 0.174908i −0.996169 0.0874539i \(-0.972127\pi\)
0.996169 0.0874539i \(-0.0278730\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 22.9590i 1.00011i
\(528\) 0 0
\(529\) 21.1017 0.917466
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 52.9403i 2.29310i
\(534\) 0 0
\(535\) −3.90813 0.549086i −0.168963 0.0237390i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −2.00000 −0.0861461
\(540\) 0 0
\(541\) 14.5906 0.627298 0.313649 0.949539i \(-0.398449\pi\)
0.313649 + 0.949539i \(0.398449\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.74620 12.4286i 0.0747990 0.532384i
\(546\) 0 0
\(547\) 18.7556i 0.801930i 0.916093 + 0.400965i \(0.131325\pi\)
−0.916093 + 0.400965i \(0.868675\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −1.83500 −0.0781738
\(552\) 0 0
\(553\) 4.85728i 0.206553i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 31.8765i 1.35065i −0.737520 0.675325i \(-0.764003\pi\)
0.737520 0.675325i \(-0.235997\pi\)
\(558\) 0 0
\(559\) −64.9403 −2.74668
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2.01874i 0.0850796i 0.999095 + 0.0425398i \(0.0135449\pi\)
−0.999095 + 0.0425398i \(0.986455\pi\)
\(564\) 0 0
\(565\) 24.9906 + 3.51114i 1.05136 + 0.147715i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −28.9590 −1.21402 −0.607012 0.794693i \(-0.707632\pi\)
−0.607012 + 0.794693i \(0.707632\pi\)
\(570\) 0 0
\(571\) −8.97773 −0.375706 −0.187853 0.982197i \(-0.560153\pi\)
−0.187853 + 0.982197i \(0.560153\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.89829 + 6.62222i −0.0791642 + 0.276165i
\(576\) 0 0
\(577\) 28.6766i 1.19382i 0.802307 + 0.596911i \(0.203606\pi\)
−0.802307 + 0.596911i \(0.796394\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 11.6128 0.481782
\(582\) 0 0
\(583\) 18.3684i 0.760742i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 45.2070i 1.86589i 0.360018 + 0.932945i \(0.382771\pi\)
−0.360018 + 0.932945i \(0.617229\pi\)
\(588\) 0 0
\(589\) 12.5906 0.518786
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 18.2636i 0.749998i −0.927025 0.374999i \(-0.877643\pi\)
0.927025 0.374999i \(-0.122357\pi\)
\(594\) 0 0
\(595\) −1.37778 + 9.80642i −0.0564837 + 0.402024i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 22.7368 0.929002 0.464501 0.885573i \(-0.346234\pi\)
0.464501 + 0.885573i \(0.346234\pi\)
\(600\) 0 0
\(601\) 0.488863 0.0199411 0.00997056 0.999950i \(-0.496826\pi\)
0.00997056 + 0.999950i \(0.496826\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2.17775 15.5002i 0.0885383 0.630174i
\(606\) 0 0
\(607\) 20.2034i 0.820032i −0.912078 0.410016i \(-0.865523\pi\)
0.912078 0.410016i \(-0.134477\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −17.7146 −0.716654
\(612\) 0 0
\(613\) 10.3684i 0.418776i 0.977833 + 0.209388i \(0.0671472\pi\)
−0.977833 + 0.209388i \(0.932853\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 39.2859i 1.58159i −0.612080 0.790796i \(-0.709667\pi\)
0.612080 0.790796i \(-0.290333\pi\)
\(618\) 0 0
\(619\) 42.8988 1.72425 0.862123 0.506698i \(-0.169134\pi\)
0.862123 + 0.506698i \(0.169134\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 4.62222i 0.185185i
\(624\) 0 0
\(625\) 21.2034 + 13.2444i 0.848137 + 0.529777i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 33.7146 1.34429
\(630\) 0 0
\(631\) −15.3461 −0.610920 −0.305460 0.952205i \(-0.598810\pi\)
−0.305460 + 0.952205i \(0.598810\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −28.4701 4.00000i −1.12980 0.158735i
\(636\) 0 0
\(637\) 6.42864i 0.254712i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −30.6735 −1.21153 −0.605766 0.795643i \(-0.707133\pi\)
−0.605766 + 0.795643i \(0.707133\pi\)
\(642\) 0 0
\(643\) 49.0607i 1.93477i 0.253320 + 0.967383i \(0.418477\pi\)
−0.253320 + 0.967383i \(0.581523\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 15.3461i 0.603319i 0.953416 + 0.301660i \(0.0975406\pi\)
−0.953416 + 0.301660i \(0.902459\pi\)
\(648\) 0 0
\(649\) −28.2034 −1.10708
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 19.4697i 0.761906i 0.924594 + 0.380953i \(0.124404\pi\)
−0.924594 + 0.380953i \(0.875596\pi\)
\(654\) 0 0
\(655\) 0.653858 4.65386i 0.0255484 0.181841i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −30.9403 −1.20526 −0.602631 0.798020i \(-0.705881\pi\)
−0.602631 + 0.798020i \(0.705881\pi\)
\(660\) 0 0
\(661\) 47.7975 1.85911 0.929554 0.368685i \(-0.120192\pi\)
0.929554 + 0.368685i \(0.120192\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 5.37778 + 0.755569i 0.208542 + 0.0292997i
\(666\) 0 0
\(667\) 1.04101i 0.0403081i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 13.7146 0.529445
\(672\) 0 0
\(673\) 27.8163i 1.07224i 0.844142 + 0.536119i \(0.180110\pi\)
−0.844142 + 0.536119i \(0.819890\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 19.0005i 0.730248i −0.930959 0.365124i \(-0.881027\pi\)
0.930959 0.365124i \(-0.118973\pi\)
\(678\) 0 0
\(679\) 11.9398 0.458207
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 4.52051i 0.172972i 0.996253 + 0.0864862i \(0.0275638\pi\)
−0.996253 + 0.0864862i \(0.972436\pi\)
\(684\) 0 0
\(685\) 35.2958 + 4.95899i 1.34858 + 0.189473i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −59.0420 −2.24932
\(690\) 0 0
\(691\) 1.18421 0.0450494 0.0225247 0.999746i \(-0.492830\pi\)
0.0225247 + 0.999746i \(0.492830\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3.63158 + 25.8479i −0.137754 + 0.980467i
\(696\) 0 0
\(697\) 36.4701i 1.38140i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 26.6735 1.00745 0.503723 0.863865i \(-0.331963\pi\)
0.503723 + 0.863865i \(0.331963\pi\)
\(702\) 0 0
\(703\) 18.4889i 0.697321i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.47949i 0.0556421i
\(708\) 0 0
\(709\) 18.2034 0.683644 0.341822 0.939765i \(-0.388956\pi\)
0.341822 + 0.939765i \(0.388956\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 7.14272i 0.267497i
\(714\) 0 0
\(715\) −28.4701 4.00000i −1.06472 0.149592i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −4.85728 −0.181146 −0.0905730 0.995890i \(-0.528870\pi\)
−0.0905730 + 0.995890i \(0.528870\pi\)
\(720\) 0 0
\(721\) 8.85728 0.329862
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3.63158 1.04101i −0.134874 0.0386622i
\(726\) 0 0
\(727\) 21.0607i 0.781098i 0.920582 + 0.390549i \(0.127715\pi\)
−0.920582 + 0.390549i \(0.872285\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 44.7368 1.65465
\(732\) 0 0
\(733\) 9.45091i 0.349077i 0.984650 + 0.174539i \(0.0558434\pi\)
−0.984650 + 0.174539i \(0.944157\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.51114i 0.203005i
\(738\) 0 0
\(739\) −8.20342 −0.301768 −0.150884 0.988551i \(-0.548212\pi\)
−0.150884 + 0.988551i \(0.548212\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 8.33677i 0.305847i 0.988238 + 0.152923i \(0.0488687\pi\)
−0.988238 + 0.152923i \(0.951131\pi\)
\(744\) 0 0
\(745\) 6.60348 47.0005i 0.241933 1.72196i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −1.76494 −0.0644894
\(750\) 0 0
\(751\) 25.9180 0.945760 0.472880 0.881127i \(-0.343214\pi\)
0.472880 + 0.881127i \(0.343214\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 5.24443 37.3274i 0.190864 1.35848i
\(756\) 0 0
\(757\) 8.94025i 0.324939i −0.986714 0.162470i \(-0.948054\pi\)
0.986714 0.162470i \(-0.0519459\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0.825636 0.0299293 0.0149646 0.999888i \(-0.495236\pi\)
0.0149646 + 0.999888i \(0.495236\pi\)
\(762\) 0 0
\(763\) 5.61285i 0.203199i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 90.6548i 3.27336i
\(768\) 0 0
\(769\) −21.2257 −0.765418 −0.382709 0.923869i \(-0.625009\pi\)
−0.382709 + 0.923869i \(0.625009\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 29.4893i 1.06066i −0.847792 0.530329i \(-0.822068\pi\)
0.847792 0.530329i \(-0.177932\pi\)
\(774\) 0 0
\(775\) 24.9175 + 7.14272i 0.895063 + 0.256574i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −20.0000 −0.716574
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 23.0923 + 3.24443i 0.824201 + 0.115799i
\(786\) 0 0
\(787\) 34.4514i 1.22806i −0.789283 0.614030i \(-0.789547\pi\)
0.789283 0.614030i \(-0.210453\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 11.2859 0.401281
\(792\) 0 0
\(793\) 44.0830i 1.56543i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 18.9175i 0.670092i 0.942202 + 0.335046i \(0.108752\pi\)
−0.942202 + 0.335046i \(0.891248\pi\)
\(798\) 0 0
\(799\) 12.2034 0.431726
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 3.14272i 0.110904i
\(804\) 0 0
\(805\) −0.428639 + 3.05086i −0.0151076 + 0.107529i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 21.2257 0.746256 0.373128 0.927780i \(-0.378285\pi\)
0.373128 + 0.927780i \(0.378285\pi\)
\(810\) 0 0
\(811\) 21.5081 0.755251 0.377625 0.925958i \(-0.376741\pi\)
0.377625 + 0.925958i \(0.376741\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −46.1847 6.48886i −1.61778 0.227295i
\(816\) 0 0
\(817\) 24.5334i 0.858315i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 46.2034 1.61251 0.806255 0.591568i \(-0.201491\pi\)
0.806255 + 0.591568i \(0.201491\pi\)
\(822\) 0 0
\(823\) 17.8350i 0.621689i 0.950461 + 0.310845i \(0.100612\pi\)
−0.950461 + 0.310845i \(0.899388\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 35.2128i 1.22447i −0.790676 0.612234i \(-0.790271\pi\)
0.790676 0.612234i \(-0.209729\pi\)
\(828\) 0 0
\(829\) −14.3872 −0.499686 −0.249843 0.968286i \(-0.580379\pi\)
−0.249843 + 0.968286i \(0.580379\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 4.42864i 0.153443i
\(834\) 0 0
\(835\) −33.9813 4.77430i −1.17597 0.165222i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −1.51114 −0.0521703 −0.0260851 0.999660i \(-0.508304\pi\)
−0.0260851 + 0.999660i \(0.508304\pi\)
\(840\) 0 0
\(841\) −28.4291 −0.980314
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 8.81288 62.7259i 0.303172 2.15784i
\(846\) 0 0
\(847\) 7.00000i 0.240523i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 10.4889 0.359554
\(852\) 0 0
\(853\) 15.4064i 0.527504i −0.964591 0.263752i \(-0.915040\pi\)
0.964591 0.263752i \(-0.0849600\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 19.8578i 0.678328i 0.940727 + 0.339164i \(0.110144\pi\)
−0.940727 + 0.339164i \(0.889856\pi\)
\(858\) 0 0
\(859\) 2.42864 0.0828641 0.0414321 0.999141i \(-0.486808\pi\)
0.0414321 + 0.999141i \(0.486808\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 39.2958i 1.33764i −0.743423 0.668822i \(-0.766799\pi\)
0.743423 0.668822i \(-0.233201\pi\)
\(864\) 0 0
\(865\) −4.56199 0.640951i −0.155112 0.0217930i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −9.71456 −0.329544
\(870\) 0 0
\(871\) 17.7146 0.600235
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 10.2143 + 4.54617i 0.345307 + 0.153689i
\(876\) 0 0
\(877\) 56.2864i 1.90066i 0.311249 + 0.950328i \(0.399253\pi\)
−0.311249 + 0.950328i \(0.600747\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 2.33677 0.0787279 0.0393640 0.999225i \(-0.487467\pi\)
0.0393640 + 0.999225i \(0.487467\pi\)
\(882\) 0 0
\(883\) 33.7146i 1.13459i −0.823516 0.567293i \(-0.807991\pi\)
0.823516 0.567293i \(-0.192009\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 47.8992i 1.60830i −0.594427 0.804150i \(-0.702621\pi\)
0.594427 0.804150i \(-0.297379\pi\)
\(888\) 0 0
\(889\) −12.8573 −0.431219
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 6.69228i 0.223949i
\(894\) 0 0
\(895\) 3.11108 22.1432i 0.103992 0.740165i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −3.91703 −0.130640
\(900\) 0 0
\(901\) 40.6735 1.35503
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 3.76494 26.7971i 0.125151 0.890764i
\(906\) 0 0
\(907\) 23.7591i 0.788908i 0.918916 + 0.394454i \(0.129066\pi\)
−0.918916 + 0.394454i \(0.870934\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 22.9403 0.760045 0.380022 0.924977i \(-0.375916\pi\)
0.380022 + 0.924977i \(0.375916\pi\)
\(912\) 0 0
\(913\) 23.2257i 0.768658i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.10171i 0.0694046i
\(918\) 0 0
\(919\) 16.9777 0.560043 0.280022 0.959994i \(-0.409658\pi\)
0.280022 + 0.959994i \(0.409658\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 12.8573i 0.423202i
\(924\) 0 0
\(925\) 10.4889 36.5906i 0.344872 1.20309i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 39.3403 1.29071 0.645357 0.763881i \(-0.276709\pi\)
0.645357 + 0.763881i \(0.276709\pi\)
\(930\) 0 0
\(931\) 2.42864 0.0795954
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 19.6128 + 2.75557i 0.641409 + 0.0901167i
\(936\) 0 0
\(937\) 17.7748i 0.580677i 0.956924 + 0.290338i \(0.0937679\pi\)
−0.956924 + 0.290338i \(0.906232\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −35.5812 −1.15991 −0.579957 0.814647i \(-0.696931\pi\)
−0.579957 + 0.814647i \(0.696931\pi\)
\(942\) 0 0
\(943\) 11.3461i 0.369481i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 30.5018i 0.991174i −0.868558 0.495587i \(-0.834953\pi\)
0.868558 0.495587i \(-0.165047\pi\)
\(948\) 0 0
\(949\) −10.1017 −0.327915
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 51.1655i 1.65741i 0.559684 + 0.828706i \(0.310923\pi\)
−0.559684 + 0.828706i \(0.689077\pi\)
\(954\) 0 0
\(955\) −0.152089 + 1.08250i −0.00492148 + 0.0350288i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 15.9398 0.514722
\(960\) 0 0
\(961\) −4.12399 −0.133032
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −50.8385 7.14272i −1.63655 0.229932i
\(966\) 0 0
\(967\) 47.8992i 1.54034i 0.637841 + 0.770168i \(0.279828\pi\)
−0.637841 + 0.770168i \(0.720172\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 40.6735 1.30528 0.652638 0.757670i \(-0.273662\pi\)
0.652638 + 0.757670i \(0.273662\pi\)
\(972\) 0 0
\(973\) 11.6731i 0.374221i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 27.4893i 0.879462i −0.898130 0.439731i \(-0.855074\pi\)
0.898130 0.439731i \(-0.144926\pi\)
\(978\) 0 0
\(979\) 9.24443 0.295453
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 24.0000i 0.765481i 0.923856 + 0.382741i \(0.125020\pi\)
−0.923856 + 0.382741i \(0.874980\pi\)
\(984\) 0 0
\(985\) −2.62222 0.368416i −0.0835507 0.0117387i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 13.9180 0.442566
\(990\) 0 0
\(991\) 34.6923 1.10204 0.551018 0.834493i \(-0.314239\pi\)
0.551018 + 0.834493i \(0.314239\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −2.73683 + 19.4795i −0.0867634 + 0.617541i
\(996\) 0 0
\(997\) 28.6766i 0.908197i 0.890951 + 0.454099i \(0.150039\pi\)
−0.890951 + 0.454099i \(0.849961\pi\)
\(998\) 0 0
\(999\) 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 5040.2.t.v.1009.4 6
3.2 odd 2 1680.2.t.k.1009.2 6
4.3 odd 2 315.2.d.e.64.5 6
5.4 even 2 inner 5040.2.t.v.1009.3 6
12.11 even 2 105.2.d.b.64.2 6
15.2 even 4 8400.2.a.dg.1.3 3
15.8 even 4 8400.2.a.dj.1.1 3
15.14 odd 2 1680.2.t.k.1009.5 6
20.3 even 4 1575.2.a.w.1.3 3
20.7 even 4 1575.2.a.x.1.1 3
20.19 odd 2 315.2.d.e.64.2 6
28.27 even 2 2205.2.d.l.1324.5 6
60.23 odd 4 525.2.a.k.1.1 3
60.47 odd 4 525.2.a.j.1.3 3
60.59 even 2 105.2.d.b.64.5 yes 6
84.11 even 6 735.2.q.e.79.5 12
84.23 even 6 735.2.q.e.214.2 12
84.47 odd 6 735.2.q.f.214.2 12
84.59 odd 6 735.2.q.f.79.5 12
84.83 odd 2 735.2.d.b.589.2 6
140.139 even 2 2205.2.d.l.1324.2 6
420.59 odd 6 735.2.q.f.79.2 12
420.83 even 4 3675.2.a.bj.1.1 3
420.167 even 4 3675.2.a.bi.1.3 3
420.179 even 6 735.2.q.e.79.2 12
420.299 odd 6 735.2.q.f.214.5 12
420.359 even 6 735.2.q.e.214.5 12
420.419 odd 2 735.2.d.b.589.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.d.b.64.2 6 12.11 even 2
105.2.d.b.64.5 yes 6 60.59 even 2
315.2.d.e.64.2 6 20.19 odd 2
315.2.d.e.64.5 6 4.3 odd 2
525.2.a.j.1.3 3 60.47 odd 4
525.2.a.k.1.1 3 60.23 odd 4
735.2.d.b.589.2 6 84.83 odd 2
735.2.d.b.589.5 6 420.419 odd 2
735.2.q.e.79.2 12 420.179 even 6
735.2.q.e.79.5 12 84.11 even 6
735.2.q.e.214.2 12 84.23 even 6
735.2.q.e.214.5 12 420.359 even 6
735.2.q.f.79.2 12 420.59 odd 6
735.2.q.f.79.5 12 84.59 odd 6
735.2.q.f.214.2 12 84.47 odd 6
735.2.q.f.214.5 12 420.299 odd 6
1575.2.a.w.1.3 3 20.3 even 4
1575.2.a.x.1.1 3 20.7 even 4
1680.2.t.k.1009.2 6 3.2 odd 2
1680.2.t.k.1009.5 6 15.14 odd 2
2205.2.d.l.1324.2 6 140.139 even 2
2205.2.d.l.1324.5 6 28.27 even 2
3675.2.a.bi.1.3 3 420.167 even 4
3675.2.a.bj.1.1 3 420.83 even 4
5040.2.t.v.1009.3 6 5.4 even 2 inner
5040.2.t.v.1009.4 6 1.1 even 1 trivial
8400.2.a.dg.1.3 3 15.2 even 4
8400.2.a.dj.1.1 3 15.8 even 4