Properties

Label 2352.2.k.i.881.12
Level $2352$
Weight $2$
Character 2352.881
Analytic conductor $18.781$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2352,2,Mod(881,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2352.881");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.k (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: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{15} + 19 x^{14} - 42 x^{13} + 65 x^{12} - 48 x^{11} - 94 x^{10} + 444 x^{9} - 962 x^{8} + 1332 x^{7} - 846 x^{6} - 1296 x^{5} + 5265 x^{4} - 10206 x^{3} + 13851 x^{2} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16} \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 881.12
Root \(0.247636 + 1.71426i\) of defining polynomial
Character \(\chi\) \(=\) 2352.881
Dual form 2352.2.k.i.881.11

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.07159 + 1.36077i) q^{3} +2.57910 q^{5} +(-0.703402 + 2.91637i) q^{9} +O(q^{10})\) \(q+(1.07159 + 1.36077i) q^{3} +2.57910 q^{5} +(-0.703402 + 2.91637i) q^{9} +1.65352i q^{11} +5.71177i q^{13} +(2.76373 + 3.50957i) q^{15} +7.58626 q^{17} -2.99022i q^{19} -0.287913i q^{23} +1.65176 q^{25} +(-4.72227 + 2.16798i) q^{27} +2.05856i q^{29} -6.01840i q^{31} +(-2.25007 + 1.77190i) q^{33} +1.75505 q^{37} +(-7.77242 + 6.12066i) q^{39} -4.28635 q^{41} -2.46537 q^{43} +(-1.81414 + 7.52162i) q^{45} +0.373172 q^{47} +(8.12934 + 10.3232i) q^{51} -7.77418i q^{53} +4.26461i q^{55} +(4.06901 - 3.20429i) q^{57} -9.79219 q^{59} +1.02745i q^{61} +14.7312i q^{65} -2.36561 q^{67} +(0.391784 - 0.308524i) q^{69} +15.6655i q^{71} +3.81247i q^{73} +(1.77000 + 2.24767i) q^{75} +9.12066 q^{79} +(-8.01045 - 4.10276i) q^{81} -6.65166 q^{83} +19.5657 q^{85} +(-2.80123 + 2.20593i) q^{87} +14.5145 q^{89} +(8.18967 - 6.44924i) q^{93} -7.71209i q^{95} -4.43739i q^{97} +(-4.82229 - 1.16309i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{9} - 8 q^{15} + 36 q^{25} + 4 q^{37} - 44 q^{39} - 20 q^{43} + 12 q^{51} - 8 q^{57} + 28 q^{67} + 56 q^{79} - 60 q^{81} + 16 q^{85} - 32 q^{93} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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
\(3\) 1.07159 + 1.36077i 0.618681 + 0.785642i
\(4\) 0 0
\(5\) 2.57910 1.15341 0.576704 0.816953i \(-0.304338\pi\)
0.576704 + 0.816953i \(0.304338\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −0.703402 + 2.91637i −0.234467 + 0.972124i
\(10\) 0 0
\(11\) 1.65352i 0.498556i 0.968432 + 0.249278i \(0.0801934\pi\)
−0.968432 + 0.249278i \(0.919807\pi\)
\(12\) 0 0
\(13\) 5.71177i 1.58416i 0.610418 + 0.792080i \(0.291002\pi\)
−0.610418 + 0.792080i \(0.708998\pi\)
\(14\) 0 0
\(15\) 2.76373 + 3.50957i 0.713592 + 0.906167i
\(16\) 0 0
\(17\) 7.58626 1.83994 0.919970 0.391990i \(-0.128213\pi\)
0.919970 + 0.391990i \(0.128213\pi\)
\(18\) 0 0
\(19\) 2.99022i 0.686004i −0.939335 0.343002i \(-0.888556\pi\)
0.939335 0.343002i \(-0.111444\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.287913i 0.0600341i −0.999549 0.0300170i \(-0.990444\pi\)
0.999549 0.0300170i \(-0.00955615\pi\)
\(24\) 0 0
\(25\) 1.65176 0.330352
\(26\) 0 0
\(27\) −4.72227 + 2.16798i −0.908802 + 0.417227i
\(28\) 0 0
\(29\) 2.05856i 0.382265i 0.981564 + 0.191133i \(0.0612161\pi\)
−0.981564 + 0.191133i \(0.938784\pi\)
\(30\) 0 0
\(31\) 6.01840i 1.08094i −0.841364 0.540468i \(-0.818247\pi\)
0.841364 0.540468i \(-0.181753\pi\)
\(32\) 0 0
\(33\) −2.25007 + 1.77190i −0.391687 + 0.308447i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.75505 0.288528 0.144264 0.989539i \(-0.453919\pi\)
0.144264 + 0.989539i \(0.453919\pi\)
\(38\) 0 0
\(39\) −7.77242 + 6.12066i −1.24458 + 0.980090i
\(40\) 0 0
\(41\) −4.28635 −0.669415 −0.334708 0.942322i \(-0.608638\pi\)
−0.334708 + 0.942322i \(0.608638\pi\)
\(42\) 0 0
\(43\) −2.46537 −0.375965 −0.187982 0.982172i \(-0.560195\pi\)
−0.187982 + 0.982172i \(0.560195\pi\)
\(44\) 0 0
\(45\) −1.81414 + 7.52162i −0.270437 + 1.12126i
\(46\) 0 0
\(47\) 0.373172 0.0544327 0.0272163 0.999630i \(-0.491336\pi\)
0.0272163 + 0.999630i \(0.491336\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 8.12934 + 10.3232i 1.13834 + 1.44553i
\(52\) 0 0
\(53\) 7.77418i 1.06787i −0.845527 0.533933i \(-0.820713\pi\)
0.845527 0.533933i \(-0.179287\pi\)
\(54\) 0 0
\(55\) 4.26461i 0.575039i
\(56\) 0 0
\(57\) 4.06901 3.20429i 0.538954 0.424418i
\(58\) 0 0
\(59\) −9.79219 −1.27483 −0.637417 0.770519i \(-0.719997\pi\)
−0.637417 + 0.770519i \(0.719997\pi\)
\(60\) 0 0
\(61\) 1.02745i 0.131551i 0.997834 + 0.0657755i \(0.0209521\pi\)
−0.997834 + 0.0657755i \(0.979048\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 14.7312i 1.82718i
\(66\) 0 0
\(67\) −2.36561 −0.289005 −0.144503 0.989504i \(-0.546158\pi\)
−0.144503 + 0.989504i \(0.546158\pi\)
\(68\) 0 0
\(69\) 0.391784 0.308524i 0.0471653 0.0371419i
\(70\) 0 0
\(71\) 15.6655i 1.85915i 0.368631 + 0.929576i \(0.379826\pi\)
−0.368631 + 0.929576i \(0.620174\pi\)
\(72\) 0 0
\(73\) 3.81247i 0.446216i 0.974794 + 0.223108i \(0.0716202\pi\)
−0.974794 + 0.223108i \(0.928380\pi\)
\(74\) 0 0
\(75\) 1.77000 + 2.24767i 0.204382 + 0.259538i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 9.12066 1.02615 0.513077 0.858343i \(-0.328506\pi\)
0.513077 + 0.858343i \(0.328506\pi\)
\(80\) 0 0
\(81\) −8.01045 4.10276i −0.890050 0.455863i
\(82\) 0 0
\(83\) −6.65166 −0.730114 −0.365057 0.930985i \(-0.618951\pi\)
−0.365057 + 0.930985i \(0.618951\pi\)
\(84\) 0 0
\(85\) 19.5657 2.12220
\(86\) 0 0
\(87\) −2.80123 + 2.20593i −0.300324 + 0.236500i
\(88\) 0 0
\(89\) 14.5145 1.53853 0.769265 0.638930i \(-0.220623\pi\)
0.769265 + 0.638930i \(0.220623\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 8.18967 6.44924i 0.849229 0.668755i
\(94\) 0 0
\(95\) 7.71209i 0.791243i
\(96\) 0 0
\(97\) 4.43739i 0.450548i −0.974295 0.225274i \(-0.927672\pi\)
0.974295 0.225274i \(-0.0723278\pi\)
\(98\) 0 0
\(99\) −4.82229 1.16309i −0.484659 0.116895i
\(100\) 0 0
\(101\) −4.07256 −0.405235 −0.202617 0.979258i \(-0.564945\pi\)
−0.202617 + 0.979258i \(0.564945\pi\)
\(102\) 0 0
\(103\) 8.43331i 0.830959i 0.909603 + 0.415479i \(0.136386\pi\)
−0.909603 + 0.415479i \(0.863614\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.5878i 1.41025i −0.709081 0.705127i \(-0.750890\pi\)
0.709081 0.705127i \(-0.249110\pi\)
\(108\) 0 0
\(109\) 17.2999 1.65703 0.828514 0.559969i \(-0.189187\pi\)
0.828514 + 0.559969i \(0.189187\pi\)
\(110\) 0 0
\(111\) 1.88068 + 2.38822i 0.178507 + 0.226680i
\(112\) 0 0
\(113\) 4.00000i 0.376288i 0.982141 + 0.188144i \(0.0602472\pi\)
−0.982141 + 0.188144i \(0.939753\pi\)
\(114\) 0 0
\(115\) 0.742557i 0.0692438i
\(116\) 0 0
\(117\) −16.6576 4.01767i −1.54000 0.371434i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 8.26586 0.751441
\(122\) 0 0
\(123\) −4.59320 5.83274i −0.414155 0.525921i
\(124\) 0 0
\(125\) −8.63545 −0.772378
\(126\) 0 0
\(127\) 16.6481 1.47728 0.738641 0.674099i \(-0.235468\pi\)
0.738641 + 0.674099i \(0.235468\pi\)
\(128\) 0 0
\(129\) −2.64185 3.35480i −0.232602 0.295374i
\(130\) 0 0
\(131\) −16.5949 −1.44990 −0.724951 0.688800i \(-0.758138\pi\)
−0.724951 + 0.688800i \(0.758138\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −12.1792 + 5.59143i −1.04822 + 0.481234i
\(136\) 0 0
\(137\) 9.94987i 0.850075i −0.905176 0.425037i \(-0.860261\pi\)
0.905176 0.425037i \(-0.139739\pi\)
\(138\) 0 0
\(139\) 3.11952i 0.264594i 0.991210 + 0.132297i \(0.0422353\pi\)
−0.991210 + 0.132297i \(0.957765\pi\)
\(140\) 0 0
\(141\) 0.399886 + 0.507801i 0.0336765 + 0.0427646i
\(142\) 0 0
\(143\) −9.44455 −0.789793
\(144\) 0 0
\(145\) 5.30924i 0.440908i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.13979i 0.0933755i −0.998910 0.0466877i \(-0.985133\pi\)
0.998910 0.0466877i \(-0.0148666\pi\)
\(150\) 0 0
\(151\) 12.7724 1.03940 0.519702 0.854347i \(-0.326043\pi\)
0.519702 + 0.854347i \(0.326043\pi\)
\(152\) 0 0
\(153\) −5.33619 + 22.1244i −0.431406 + 1.78865i
\(154\) 0 0
\(155\) 15.5221i 1.24676i
\(156\) 0 0
\(157\) 9.03037i 0.720702i −0.932817 0.360351i \(-0.882657\pi\)
0.932817 0.360351i \(-0.117343\pi\)
\(158\) 0 0
\(159\) 10.5789 8.33071i 0.838961 0.660669i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −0.0997549 −0.00781340 −0.00390670 0.999992i \(-0.501244\pi\)
−0.00390670 + 0.999992i \(0.501244\pi\)
\(164\) 0 0
\(165\) −5.80316 + 4.56990i −0.451775 + 0.355766i
\(166\) 0 0
\(167\) 3.08612 0.238811 0.119406 0.992846i \(-0.461901\pi\)
0.119406 + 0.992846i \(0.461901\pi\)
\(168\) 0 0
\(169\) −19.6243 −1.50956
\(170\) 0 0
\(171\) 8.72060 + 2.10333i 0.666881 + 0.160846i
\(172\) 0 0
\(173\) −7.46075 −0.567231 −0.283615 0.958938i \(-0.591534\pi\)
−0.283615 + 0.958938i \(0.591534\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −10.4932 13.3249i −0.788716 1.00156i
\(178\) 0 0
\(179\) 3.01914i 0.225661i −0.993614 0.112830i \(-0.964008\pi\)
0.993614 0.112830i \(-0.0359917\pi\)
\(180\) 0 0
\(181\) 0.762552i 0.0566801i 0.999598 + 0.0283400i \(0.00902212\pi\)
−0.999598 + 0.0283400i \(0.990978\pi\)
\(182\) 0 0
\(183\) −1.39812 + 1.10100i −0.103352 + 0.0813881i
\(184\) 0 0
\(185\) 4.52644 0.332790
\(186\) 0 0
\(187\) 12.5441i 0.917313i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.22218i 0.0884337i 0.999022 + 0.0442169i \(0.0140793\pi\)
−0.999022 + 0.0442169i \(0.985921\pi\)
\(192\) 0 0
\(193\) −23.5174 −1.69282 −0.846409 0.532534i \(-0.821240\pi\)
−0.846409 + 0.532534i \(0.821240\pi\)
\(194\) 0 0
\(195\) −20.0458 + 15.7858i −1.43551 + 1.13044i
\(196\) 0 0
\(197\) 14.7312i 1.04956i 0.851239 + 0.524778i \(0.175852\pi\)
−0.851239 + 0.524778i \(0.824148\pi\)
\(198\) 0 0
\(199\) 6.88238i 0.487879i 0.969790 + 0.243940i \(0.0784399\pi\)
−0.969790 + 0.243940i \(0.921560\pi\)
\(200\) 0 0
\(201\) −2.53496 3.21906i −0.178802 0.227055i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −11.0549 −0.772109
\(206\) 0 0
\(207\) 0.839662 + 0.202519i 0.0583606 + 0.0140760i
\(208\) 0 0
\(209\) 4.94441 0.342012
\(210\) 0 0
\(211\) 19.0897 1.31419 0.657093 0.753809i \(-0.271786\pi\)
0.657093 + 0.753809i \(0.271786\pi\)
\(212\) 0 0
\(213\) −21.3172 + 16.7869i −1.46063 + 1.15022i
\(214\) 0 0
\(215\) −6.35843 −0.433641
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −5.18790 + 4.08539i −0.350566 + 0.276065i
\(220\) 0 0
\(221\) 43.3310i 2.91476i
\(222\) 0 0
\(223\) 10.9876i 0.735785i −0.929868 0.367892i \(-0.880079\pi\)
0.929868 0.367892i \(-0.119921\pi\)
\(224\) 0 0
\(225\) −1.16185 + 4.81714i −0.0774567 + 0.321143i
\(226\) 0 0
\(227\) 18.9084 1.25499 0.627496 0.778620i \(-0.284080\pi\)
0.627496 + 0.778620i \(0.284080\pi\)
\(228\) 0 0
\(229\) 17.2909i 1.14261i 0.820736 + 0.571307i \(0.193564\pi\)
−0.820736 + 0.571307i \(0.806436\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.13979i 0.336719i −0.985726 0.168360i \(-0.946153\pi\)
0.985726 0.168360i \(-0.0538470\pi\)
\(234\) 0 0
\(235\) 0.962447 0.0627831
\(236\) 0 0
\(237\) 9.77358 + 12.4111i 0.634862 + 0.806190i
\(238\) 0 0
\(239\) 5.67983i 0.367398i −0.982983 0.183699i \(-0.941193\pi\)
0.982983 0.183699i \(-0.0588071\pi\)
\(240\) 0 0
\(241\) 23.2792i 1.49955i −0.661695 0.749773i \(-0.730163\pi\)
0.661695 0.749773i \(-0.269837\pi\)
\(242\) 0 0
\(243\) −3.00097 15.2969i −0.192512 0.981295i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 17.0795 1.08674
\(248\) 0 0
\(249\) −7.12783 9.05139i −0.451708 0.573609i
\(250\) 0 0
\(251\) 21.3799 1.34949 0.674744 0.738052i \(-0.264254\pi\)
0.674744 + 0.738052i \(0.264254\pi\)
\(252\) 0 0
\(253\) 0.476072 0.0299304
\(254\) 0 0
\(255\) 20.9664 + 26.6245i 1.31297 + 1.66729i
\(256\) 0 0
\(257\) −14.1861 −0.884904 −0.442452 0.896792i \(-0.645891\pi\)
−0.442452 + 0.896792i \(0.645891\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −6.00353 1.44800i −0.371609 0.0896288i
\(262\) 0 0
\(263\) 2.20199i 0.135781i 0.997693 + 0.0678904i \(0.0216268\pi\)
−0.997693 + 0.0678904i \(0.978373\pi\)
\(264\) 0 0
\(265\) 20.0504i 1.23169i
\(266\) 0 0
\(267\) 15.5535 + 19.7509i 0.951860 + 1.20873i
\(268\) 0 0
\(269\) −14.6655 −0.894172 −0.447086 0.894491i \(-0.647538\pi\)
−0.447086 + 0.894491i \(0.647538\pi\)
\(270\) 0 0
\(271\) 20.4021i 1.23934i −0.784863 0.619669i \(-0.787267\pi\)
0.784863 0.619669i \(-0.212733\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.73122i 0.164699i
\(276\) 0 0
\(277\) 0.0107055 0.000643231 0.000321615 1.00000i \(-0.499898\pi\)
0.000321615 1.00000i \(0.499898\pi\)
\(278\) 0 0
\(279\) 17.5519 + 4.23335i 1.05080 + 0.253444i
\(280\) 0 0
\(281\) 8.11712i 0.484227i −0.970248 0.242114i \(-0.922159\pi\)
0.970248 0.242114i \(-0.0778406\pi\)
\(282\) 0 0
\(283\) 3.86208i 0.229577i −0.993390 0.114788i \(-0.963381\pi\)
0.993390 0.114788i \(-0.0366190\pi\)
\(284\) 0 0
\(285\) 10.4944 8.26417i 0.621634 0.489527i
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 40.5514 2.38538
\(290\) 0 0
\(291\) 6.03827 4.75505i 0.353970 0.278746i
\(292\) 0 0
\(293\) −5.75351 −0.336123 −0.168062 0.985776i \(-0.553751\pi\)
−0.168062 + 0.985776i \(0.553751\pi\)
\(294\) 0 0
\(295\) −25.2550 −1.47041
\(296\) 0 0
\(297\) −3.58480 7.80840i −0.208011 0.453089i
\(298\) 0 0
\(299\) 1.64449 0.0951035
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −4.36410 5.54182i −0.250711 0.318369i
\(304\) 0 0
\(305\) 2.64989i 0.151732i
\(306\) 0 0
\(307\) 23.9041i 1.36428i −0.731221 0.682140i \(-0.761049\pi\)
0.731221 0.682140i \(-0.238951\pi\)
\(308\) 0 0
\(309\) −11.4758 + 9.03703i −0.652836 + 0.514099i
\(310\) 0 0
\(311\) −21.1578 −1.19975 −0.599874 0.800095i \(-0.704783\pi\)
−0.599874 + 0.800095i \(0.704783\pi\)
\(312\) 0 0
\(313\) 21.1150i 1.19349i −0.802430 0.596746i \(-0.796460\pi\)
0.802430 0.596746i \(-0.203540\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 16.0155i 0.899520i 0.893149 + 0.449760i \(0.148490\pi\)
−0.893149 + 0.449760i \(0.851510\pi\)
\(318\) 0 0
\(319\) −3.40388 −0.190581
\(320\) 0 0
\(321\) 19.8507 15.6321i 1.10796 0.872498i
\(322\) 0 0
\(323\) 22.6846i 1.26221i
\(324\) 0 0
\(325\) 9.43446i 0.523330i
\(326\) 0 0
\(327\) 18.5383 + 23.5412i 1.02517 + 1.30183i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −18.9797 −1.04322 −0.521610 0.853184i \(-0.674668\pi\)
−0.521610 + 0.853184i \(0.674668\pi\)
\(332\) 0 0
\(333\) −1.23450 + 5.11837i −0.0676503 + 0.280485i
\(334\) 0 0
\(335\) −6.10115 −0.333341
\(336\) 0 0
\(337\) 0.151144 0.00823337 0.00411668 0.999992i \(-0.498690\pi\)
0.00411668 + 0.999992i \(0.498690\pi\)
\(338\) 0 0
\(339\) −5.44309 + 4.28635i −0.295628 + 0.232802i
\(340\) 0 0
\(341\) 9.95157 0.538908
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 1.01045 0.795715i 0.0544009 0.0428398i
\(346\) 0 0
\(347\) 13.3919i 0.718916i −0.933161 0.359458i \(-0.882962\pi\)
0.933161 0.359458i \(-0.117038\pi\)
\(348\) 0 0
\(349\) 13.4025i 0.717421i −0.933449 0.358710i \(-0.883217\pi\)
0.933449 0.358710i \(-0.116783\pi\)
\(350\) 0 0
\(351\) −12.3830 26.9725i −0.660955 1.43969i
\(352\) 0 0
\(353\) 21.4937 1.14400 0.571998 0.820255i \(-0.306169\pi\)
0.571998 + 0.820255i \(0.306169\pi\)
\(354\) 0 0
\(355\) 40.4029i 2.14436i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 28.1947i 1.48806i 0.668146 + 0.744030i \(0.267088\pi\)
−0.668146 + 0.744030i \(0.732912\pi\)
\(360\) 0 0
\(361\) 10.0586 0.529398
\(362\) 0 0
\(363\) 8.85759 + 11.2479i 0.464903 + 0.590364i
\(364\) 0 0
\(365\) 9.83274i 0.514669i
\(366\) 0 0
\(367\) 22.7256i 1.18627i −0.805103 0.593135i \(-0.797890\pi\)
0.805103 0.593135i \(-0.202110\pi\)
\(368\) 0 0
\(369\) 3.01503 12.5006i 0.156956 0.650755i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −13.9140 −0.720438 −0.360219 0.932868i \(-0.617298\pi\)
−0.360219 + 0.932868i \(0.617298\pi\)
\(374\) 0 0
\(375\) −9.25364 11.7509i −0.477856 0.606813i
\(376\) 0 0
\(377\) −11.7580 −0.605569
\(378\) 0 0
\(379\) −20.8656 −1.07179 −0.535897 0.844283i \(-0.680027\pi\)
−0.535897 + 0.844283i \(0.680027\pi\)
\(380\) 0 0
\(381\) 17.8399 + 22.6543i 0.913966 + 1.16061i
\(382\) 0 0
\(383\) 2.47155 0.126290 0.0631451 0.998004i \(-0.479887\pi\)
0.0631451 + 0.998004i \(0.479887\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.73414 7.18992i 0.0881515 0.365484i
\(388\) 0 0
\(389\) 23.5842i 1.19576i −0.801584 0.597882i \(-0.796009\pi\)
0.801584 0.597882i \(-0.203991\pi\)
\(390\) 0 0
\(391\) 2.18419i 0.110459i
\(392\) 0 0
\(393\) −17.7829 22.5819i −0.897027 1.13910i
\(394\) 0 0
\(395\) 23.5231 1.18358
\(396\) 0 0
\(397\) 2.06877i 0.103828i −0.998652 0.0519142i \(-0.983468\pi\)
0.998652 0.0519142i \(-0.0165322\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 7.45996i 0.372533i 0.982499 + 0.186266i \(0.0596387\pi\)
−0.982499 + 0.186266i \(0.940361\pi\)
\(402\) 0 0
\(403\) 34.3757 1.71238
\(404\) 0 0
\(405\) −20.6598 10.5814i −1.02659 0.525796i
\(406\) 0 0
\(407\) 2.90201i 0.143847i
\(408\) 0 0
\(409\) 33.8208i 1.67233i −0.548477 0.836166i \(-0.684792\pi\)
0.548477 0.836166i \(-0.315208\pi\)
\(410\) 0 0
\(411\) 13.5395 10.6622i 0.667854 0.525925i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −17.1553 −0.842120
\(416\) 0 0
\(417\) −4.24495 + 3.34284i −0.207876 + 0.163699i
\(418\) 0 0
\(419\) −15.2980 −0.747358 −0.373679 0.927558i \(-0.621904\pi\)
−0.373679 + 0.927558i \(0.621904\pi\)
\(420\) 0 0
\(421\) 11.8931 0.579633 0.289816 0.957082i \(-0.406406\pi\)
0.289816 + 0.957082i \(0.406406\pi\)
\(422\) 0 0
\(423\) −0.262490 + 1.08831i −0.0127627 + 0.0529153i
\(424\) 0 0
\(425\) 12.5307 0.607827
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −10.1207 12.8519i −0.488630 0.620495i
\(430\) 0 0
\(431\) 17.2222i 0.829563i 0.909921 + 0.414782i \(0.136142\pi\)
−0.909921 + 0.414782i \(0.863858\pi\)
\(432\) 0 0
\(433\) 1.55093i 0.0745329i −0.999305 0.0372664i \(-0.988135\pi\)
0.999305 0.0372664i \(-0.0118650\pi\)
\(434\) 0 0
\(435\) −7.22466 + 5.68931i −0.346396 + 0.272782i
\(436\) 0 0
\(437\) −0.860925 −0.0411836
\(438\) 0 0
\(439\) 19.4310i 0.927392i −0.885994 0.463696i \(-0.846523\pi\)
0.885994 0.463696i \(-0.153477\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3.56761i 0.169502i −0.996402 0.0847510i \(-0.972991\pi\)
0.996402 0.0847510i \(-0.0270095\pi\)
\(444\) 0 0
\(445\) 37.4343 1.77455
\(446\) 0 0
\(447\) 1.55100 1.22139i 0.0733597 0.0577697i
\(448\) 0 0
\(449\) 29.5796i 1.39595i −0.716124 0.697973i \(-0.754085\pi\)
0.716124 0.697973i \(-0.245915\pi\)
\(450\) 0 0
\(451\) 7.08758i 0.333741i
\(452\) 0 0
\(453\) 13.6868 + 17.3803i 0.643060 + 0.816600i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −22.4624 −1.05075 −0.525374 0.850871i \(-0.676075\pi\)
−0.525374 + 0.850871i \(0.676075\pi\)
\(458\) 0 0
\(459\) −35.8244 + 16.4468i −1.67214 + 0.767673i
\(460\) 0 0
\(461\) 9.31904 0.434031 0.217015 0.976168i \(-0.430368\pi\)
0.217015 + 0.976168i \(0.430368\pi\)
\(462\) 0 0
\(463\) 16.6243 0.772597 0.386298 0.922374i \(-0.373754\pi\)
0.386298 + 0.922374i \(0.373754\pi\)
\(464\) 0 0
\(465\) 21.1220 16.6332i 0.979508 0.771348i
\(466\) 0 0
\(467\) 12.1312 0.561365 0.280683 0.959801i \(-0.409439\pi\)
0.280683 + 0.959801i \(0.409439\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 12.2883 9.67682i 0.566214 0.445885i
\(472\) 0 0
\(473\) 4.07654i 0.187440i
\(474\) 0 0
\(475\) 4.93913i 0.226623i
\(476\) 0 0
\(477\) 22.6724 + 5.46838i 1.03810 + 0.250380i
\(478\) 0 0
\(479\) 26.5188 1.21168 0.605839 0.795588i \(-0.292838\pi\)
0.605839 + 0.795588i \(0.292838\pi\)
\(480\) 0 0
\(481\) 10.0244i 0.457074i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 11.4445i 0.519666i
\(486\) 0 0
\(487\) 35.1972 1.59494 0.797469 0.603360i \(-0.206172\pi\)
0.797469 + 0.603360i \(0.206172\pi\)
\(488\) 0 0
\(489\) −0.106896 0.135744i −0.00483401 0.00613854i
\(490\) 0 0
\(491\) 32.5795i 1.47029i 0.677910 + 0.735145i \(0.262886\pi\)
−0.677910 + 0.735145i \(0.737114\pi\)
\(492\) 0 0
\(493\) 15.6168i 0.703345i
\(494\) 0 0
\(495\) −12.4372 2.99973i −0.559010 0.134828i
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −4.93790 −0.221051 −0.110525 0.993873i \(-0.535253\pi\)
−0.110525 + 0.993873i \(0.535253\pi\)
\(500\) 0 0
\(501\) 3.30705 + 4.19951i 0.147748 + 0.187620i
\(502\) 0 0
\(503\) −16.7907 −0.748661 −0.374331 0.927295i \(-0.622127\pi\)
−0.374331 + 0.927295i \(0.622127\pi\)
\(504\) 0 0
\(505\) −10.5035 −0.467401
\(506\) 0 0
\(507\) −21.0291 26.7042i −0.933937 1.18597i
\(508\) 0 0
\(509\) −1.26298 −0.0559806 −0.0279903 0.999608i \(-0.508911\pi\)
−0.0279903 + 0.999608i \(0.508911\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 6.48274 + 14.1207i 0.286220 + 0.623442i
\(514\) 0 0
\(515\) 21.7504i 0.958435i
\(516\) 0 0
\(517\) 0.617048i 0.0271378i
\(518\) 0 0
\(519\) −7.99485 10.1524i −0.350935 0.445640i
\(520\) 0 0
\(521\) −29.9891 −1.31384 −0.656922 0.753958i \(-0.728142\pi\)
−0.656922 + 0.753958i \(0.728142\pi\)
\(522\) 0 0
\(523\) 35.5171i 1.55306i −0.630083 0.776528i \(-0.716979\pi\)
0.630083 0.776528i \(-0.283021\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 45.6572i 1.98886i
\(528\) 0 0
\(529\) 22.9171 0.996396
\(530\) 0 0
\(531\) 6.88785 28.5577i 0.298907 1.23930i
\(532\) 0 0
\(533\) 24.4826i 1.06046i
\(534\) 0 0
\(535\) 37.6234i 1.62660i
\(536\) 0 0
\(537\) 4.10836 3.23527i 0.177289 0.139612i
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −23.8316 −1.02460 −0.512300 0.858807i \(-0.671206\pi\)
−0.512300 + 0.858807i \(0.671206\pi\)
\(542\) 0 0
\(543\) −1.03766 + 0.817141i −0.0445302 + 0.0350669i
\(544\) 0 0
\(545\) 44.6181 1.91123
\(546\) 0 0
\(547\) −21.1040 −0.902342 −0.451171 0.892437i \(-0.648994\pi\)
−0.451171 + 0.892437i \(0.648994\pi\)
\(548\) 0 0
\(549\) −2.99641 0.722707i −0.127884 0.0308444i
\(550\) 0 0
\(551\) 6.15556 0.262236
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 4.85047 + 6.15945i 0.205891 + 0.261454i
\(556\) 0 0
\(557\) 22.2880i 0.944374i 0.881499 + 0.472187i \(0.156535\pi\)
−0.881499 + 0.472187i \(0.843465\pi\)
\(558\) 0 0
\(559\) 14.0816i 0.595588i
\(560\) 0 0
\(561\) −17.0696 + 13.4421i −0.720680 + 0.567525i
\(562\) 0 0
\(563\) −40.4393 −1.70431 −0.852157 0.523286i \(-0.824706\pi\)
−0.852157 + 0.523286i \(0.824706\pi\)
\(564\) 0 0
\(565\) 10.3164i 0.434014i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 25.1398i 1.05391i −0.849892 0.526957i \(-0.823333\pi\)
0.849892 0.526957i \(-0.176667\pi\)
\(570\) 0 0
\(571\) −1.31069 −0.0548506 −0.0274253 0.999624i \(-0.508731\pi\)
−0.0274253 + 0.999624i \(0.508731\pi\)
\(572\) 0 0
\(573\) −1.66311 + 1.30967i −0.0694773 + 0.0547123i
\(574\) 0 0
\(575\) 0.475563i 0.0198324i
\(576\) 0 0
\(577\) 6.02452i 0.250804i −0.992106 0.125402i \(-0.959978\pi\)
0.992106 0.125402i \(-0.0400221\pi\)
\(578\) 0 0
\(579\) −25.2009 32.0018i −1.04731 1.32995i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 12.8548 0.532391
\(584\) 0 0
\(585\) −42.9617 10.3620i −1.77625 0.428415i
\(586\) 0 0
\(587\) 39.5131 1.63088 0.815439 0.578843i \(-0.196496\pi\)
0.815439 + 0.578843i \(0.196496\pi\)
\(588\) 0 0
\(589\) −17.9964 −0.741527
\(590\) 0 0
\(591\) −20.0458 + 15.7858i −0.824576 + 0.649341i
\(592\) 0 0
\(593\) −13.5171 −0.555081 −0.277540 0.960714i \(-0.589519\pi\)
−0.277540 + 0.960714i \(0.589519\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −9.36536 + 7.37507i −0.383299 + 0.301842i
\(598\) 0 0
\(599\) 6.56750i 0.268341i 0.990958 + 0.134170i \(0.0428369\pi\)
−0.990958 + 0.134170i \(0.957163\pi\)
\(600\) 0 0
\(601\) 10.0499i 0.409946i 0.978768 + 0.204973i \(0.0657106\pi\)
−0.978768 + 0.204973i \(0.934289\pi\)
\(602\) 0 0
\(603\) 1.66398 6.89900i 0.0677623 0.280949i
\(604\) 0 0
\(605\) 21.3185 0.866719
\(606\) 0 0
\(607\) 0.778176i 0.0315852i 0.999875 + 0.0157926i \(0.00502715\pi\)
−0.999875 + 0.0157926i \(0.994973\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.13147i 0.0862300i
\(612\) 0 0
\(613\) 38.6697 1.56186 0.780928 0.624621i \(-0.214747\pi\)
0.780928 + 0.624621i \(0.214747\pi\)
\(614\) 0 0
\(615\) −11.8463 15.0432i −0.477689 0.606602i
\(616\) 0 0
\(617\) 7.83523i 0.315434i −0.987484 0.157717i \(-0.949587\pi\)
0.987484 0.157717i \(-0.0504134\pi\)
\(618\) 0 0
\(619\) 20.6963i 0.831854i 0.909398 + 0.415927i \(0.136543\pi\)
−0.909398 + 0.415927i \(0.863457\pi\)
\(620\) 0 0
\(621\) 0.624189 + 1.35961i 0.0250479 + 0.0545591i
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −30.5305 −1.22122
\(626\) 0 0
\(627\) 5.29836 + 6.72821i 0.211596 + 0.268699i
\(628\) 0 0
\(629\) 13.3142 0.530874
\(630\) 0 0
\(631\) 7.21022 0.287034 0.143517 0.989648i \(-0.454159\pi\)
0.143517 + 0.989648i \(0.454159\pi\)
\(632\) 0 0
\(633\) 20.4562 + 25.9767i 0.813062 + 1.03248i
\(634\) 0 0
\(635\) 42.9372 1.70391
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −45.6864 11.0191i −1.80733 0.435910i
\(640\) 0 0
\(641\) 36.8529i 1.45560i −0.685788 0.727802i \(-0.740542\pi\)
0.685788 0.727802i \(-0.259458\pi\)
\(642\) 0 0
\(643\) 10.5183i 0.414801i −0.978256 0.207400i \(-0.933500\pi\)
0.978256 0.207400i \(-0.0665003\pi\)
\(644\) 0 0
\(645\) −6.81361 8.65237i −0.268286 0.340687i
\(646\) 0 0
\(647\) 20.4114 0.802456 0.401228 0.915978i \(-0.368583\pi\)
0.401228 + 0.915978i \(0.368583\pi\)
\(648\) 0 0
\(649\) 16.1916i 0.635577i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 33.1840i 1.29859i −0.760536 0.649295i \(-0.775064\pi\)
0.760536 0.649295i \(-0.224936\pi\)
\(654\) 0 0
\(655\) −42.7999 −1.67233
\(656\) 0 0
\(657\) −11.1186 2.68170i −0.433777 0.104623i
\(658\) 0 0
\(659\) 7.18286i 0.279804i −0.990165 0.139902i \(-0.955321\pi\)
0.990165 0.139902i \(-0.0446788\pi\)
\(660\) 0 0
\(661\) 21.0571i 0.819025i 0.912304 + 0.409513i \(0.134301\pi\)
−0.912304 + 0.409513i \(0.865699\pi\)
\(662\) 0 0
\(663\) −58.9636 + 46.4329i −2.28996 + 1.80331i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0.592687 0.0229489
\(668\) 0 0
\(669\) 14.9516 11.7742i 0.578064 0.455216i
\(670\) 0 0
\(671\) −1.69891 −0.0655856
\(672\) 0 0
\(673\) −21.5441 −0.830464 −0.415232 0.909715i \(-0.636300\pi\)
−0.415232 + 0.909715i \(0.636300\pi\)
\(674\) 0 0
\(675\) −7.80006 + 3.58097i −0.300224 + 0.137832i
\(676\) 0 0
\(677\) 5.39752 0.207444 0.103722 0.994606i \(-0.466925\pi\)
0.103722 + 0.994606i \(0.466925\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 20.2620 + 25.7300i 0.776440 + 0.985975i
\(682\) 0 0
\(683\) 33.3716i 1.27693i 0.769651 + 0.638465i \(0.220430\pi\)
−0.769651 + 0.638465i \(0.779570\pi\)
\(684\) 0 0
\(685\) 25.6617i 0.980483i
\(686\) 0 0
\(687\) −23.5290 + 18.5287i −0.897686 + 0.706914i
\(688\) 0 0
\(689\) 44.4043 1.69167
\(690\) 0 0
\(691\) 39.8020i 1.51414i 0.653334 + 0.757070i \(0.273370\pi\)
−0.653334 + 0.757070i \(0.726630\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 8.04555i 0.305185i
\(696\) 0 0
\(697\) −32.5174 −1.23168
\(698\) 0 0
\(699\) 6.99409 5.50774i 0.264541 0.208322i
\(700\) 0 0
\(701\) 10.6583i 0.402559i 0.979534 + 0.201280i \(0.0645100\pi\)
−0.979534 + 0.201280i \(0.935490\pi\)
\(702\) 0 0
\(703\) 5.24798i 0.197931i
\(704\) 0 0
\(705\) 1.03135 + 1.30967i 0.0388427 + 0.0493251i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 35.1453 1.31991 0.659955 0.751305i \(-0.270575\pi\)
0.659955 + 0.751305i \(0.270575\pi\)
\(710\) 0 0
\(711\) −6.41549 + 26.5992i −0.240600 + 0.997549i
\(712\) 0 0
\(713\) −1.73278 −0.0648930
\(714\) 0 0
\(715\) −24.3584 −0.910954
\(716\) 0 0
\(717\) 7.72896 6.08644i 0.288643 0.227302i
\(718\) 0 0
\(719\) −31.2617 −1.16586 −0.582932 0.812521i \(-0.698095\pi\)
−0.582932 + 0.812521i \(0.698095\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 31.6777 24.9457i 1.17811 0.927741i
\(724\) 0 0
\(725\) 3.40025i 0.126282i
\(726\) 0 0
\(727\) 39.7975i 1.47601i 0.674797 + 0.738003i \(0.264231\pi\)
−0.674797 + 0.738003i \(0.735769\pi\)
\(728\) 0 0
\(729\) 17.5998 20.4756i 0.651843 0.758354i
\(730\) 0 0
\(731\) −18.7029 −0.691752
\(732\) 0 0
\(733\) 12.2258i 0.451570i 0.974177 + 0.225785i \(0.0724947\pi\)
−0.974177 + 0.225785i \(0.927505\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 3.91160i 0.144086i
\(738\) 0 0
\(739\) −29.0001 −1.06679 −0.533393 0.845868i \(-0.679083\pi\)
−0.533393 + 0.845868i \(0.679083\pi\)
\(740\) 0 0
\(741\) 18.3021 + 23.2413i 0.672346 + 0.853789i
\(742\) 0 0
\(743\) 33.4864i 1.22850i −0.789113 0.614248i \(-0.789459\pi\)
0.789113 0.614248i \(-0.210541\pi\)
\(744\) 0 0
\(745\) 2.93964i 0.107700i
\(746\) 0 0
\(747\) 4.67879 19.3987i 0.171188 0.709762i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −11.7204 −0.427684 −0.213842 0.976868i \(-0.568598\pi\)
−0.213842 + 0.976868i \(0.568598\pi\)
\(752\) 0 0
\(753\) 22.9104 + 29.0932i 0.834903 + 1.06021i
\(754\) 0 0
\(755\) 32.9413 1.19886
\(756\) 0 0
\(757\) 11.2688 0.409571 0.204785 0.978807i \(-0.434350\pi\)
0.204785 + 0.978807i \(0.434350\pi\)
\(758\) 0 0
\(759\) 0.510152 + 0.647825i 0.0185174 + 0.0235146i
\(760\) 0 0
\(761\) −20.1265 −0.729587 −0.364793 0.931088i \(-0.618860\pi\)
−0.364793 + 0.931088i \(0.618860\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −13.7626 + 57.0610i −0.497587 + 2.06304i
\(766\) 0 0
\(767\) 55.9307i 2.01954i
\(768\) 0 0
\(769\) 7.95157i 0.286741i −0.989669 0.143370i \(-0.954206\pi\)
0.989669 0.143370i \(-0.0457940\pi\)
\(770\) 0 0
\(771\) −15.2016 19.3040i −0.547473 0.695218i
\(772\) 0 0
\(773\) −31.7854 −1.14324 −0.571620 0.820518i \(-0.693685\pi\)
−0.571620 + 0.820518i \(0.693685\pi\)
\(774\) 0 0
\(775\) 9.94094i 0.357089i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 12.8171i 0.459222i
\(780\) 0 0
\(781\) −25.9033 −0.926892
\(782\) 0 0
\(783\) −4.46291 9.72110i −0.159492 0.347404i
\(784\) 0 0
\(785\) 23.2902i 0.831264i
\(786\) 0 0
\(787\) 30.4344i 1.08487i 0.840099 + 0.542434i \(0.182497\pi\)
−0.840099 + 0.542434i \(0.817503\pi\)
\(788\) 0 0
\(789\) −2.99641 + 2.35963i −0.106675 + 0.0840050i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −5.86853 −0.208398
\(794\) 0 0
\(795\) 27.2840 21.4857i 0.967664 0.762021i
\(796\) 0 0
\(797\) 40.0924 1.42015 0.710074 0.704127i \(-0.248662\pi\)
0.710074 + 0.704127i \(0.248662\pi\)
\(798\) 0 0
\(799\) 2.83098 0.100153
\(800\) 0 0
\(801\) −10.2095 + 42.3296i −0.360735 + 1.49564i
\(802\) 0 0
\(803\) −6.30401 −0.222464
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −15.7154 19.9564i −0.553207 0.702499i
\(808\) 0 0
\(809\) 39.2952i 1.38155i −0.723072 0.690773i \(-0.757271\pi\)
0.723072 0.690773i \(-0.242729\pi\)
\(810\) 0 0
\(811\) 23.6789i 0.831480i −0.909484 0.415740i \(-0.863523\pi\)
0.909484 0.415740i \(-0.136477\pi\)
\(812\) 0 0
\(813\) 27.7626 21.8626i 0.973676 0.766755i
\(814\) 0 0
\(815\) −0.257278 −0.00901205
\(816\) 0 0
\(817\) 7.37200i 0.257914i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 43.2749i 1.51030i −0.655550 0.755152i \(-0.727563\pi\)
0.655550 0.755152i \(-0.272437\pi\)
\(822\) 0 0
\(823\) 0.969513 0.0337951 0.0168975 0.999857i \(-0.494621\pi\)
0.0168975 + 0.999857i \(0.494621\pi\)
\(824\) 0 0
\(825\) −3.71657 + 2.92674i −0.129394 + 0.101896i
\(826\) 0 0
\(827\) 43.9510i 1.52833i 0.645023 + 0.764163i \(0.276848\pi\)
−0.645023 + 0.764163i \(0.723152\pi\)
\(828\) 0 0
\(829\) 7.58776i 0.263534i 0.991281 + 0.131767i \(0.0420650\pi\)
−0.991281 + 0.131767i \(0.957935\pi\)
\(830\) 0 0
\(831\) 0.0114719 + 0.0145677i 0.000397955 + 0.000505349i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 7.95942 0.275447
\(836\) 0 0
\(837\) 13.0478 + 28.4205i 0.450996 + 0.982357i
\(838\) 0 0
\(839\) 8.87477 0.306391 0.153196 0.988196i \(-0.451044\pi\)
0.153196 + 0.988196i \(0.451044\pi\)
\(840\) 0 0
\(841\) 24.7623 0.853873
\(842\) 0 0
\(843\) 11.0456 8.69821i 0.380429 0.299582i
\(844\) 0 0
\(845\) −50.6130 −1.74114
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 5.25541 4.13855i 0.180365 0.142035i
\(850\) 0 0
\(851\) 0.505301i 0.0173215i
\(852\) 0 0
\(853\) 0.208510i 0.00713924i 0.999994 + 0.00356962i \(0.00113625\pi\)
−0.999994 + 0.00356962i \(0.998864\pi\)
\(854\) 0 0
\(855\) 22.4913 + 5.42470i 0.769187 + 0.185521i
\(856\) 0 0
\(857\) −29.9891 −1.02441 −0.512204 0.858864i \(-0.671171\pi\)
−0.512204 + 0.858864i \(0.671171\pi\)
\(858\) 0 0
\(859\) 20.6953i 0.706115i −0.935602 0.353057i \(-0.885142\pi\)
0.935602 0.353057i \(-0.114858\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 17.1051i 0.582263i 0.956683 + 0.291131i \(0.0940317\pi\)
−0.956683 + 0.291131i \(0.905968\pi\)
\(864\) 0 0
\(865\) −19.2420 −0.654249
\(866\) 0 0
\(867\) 43.4543 + 55.1812i 1.47579 + 1.87405i
\(868\) 0 0
\(869\) 15.0812i 0.511596i
\(870\) 0 0
\(871\) 13.5118i 0.457831i
\(872\) 0 0
\(873\) 12.9411 + 3.12127i 0.437989 + 0.105639i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 19.4343 0.656248 0.328124 0.944635i \(-0.393584\pi\)
0.328124 + 0.944635i \(0.393584\pi\)
\(878\) 0 0
\(879\) −6.16538 7.82921i −0.207953 0.264073i
\(880\) 0 0
\(881\) −30.0526 −1.01250 −0.506249 0.862387i \(-0.668968\pi\)
−0.506249 + 0.862387i \(0.668968\pi\)
\(882\) 0 0
\(883\) 16.8382 0.566649 0.283324 0.959024i \(-0.408563\pi\)
0.283324 + 0.959024i \(0.408563\pi\)
\(884\) 0 0
\(885\) −27.0630 34.3664i −0.909712 1.15521i
\(886\) 0 0
\(887\) 26.6566 0.895041 0.447520 0.894274i \(-0.352307\pi\)
0.447520 + 0.894274i \(0.352307\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 6.78402 13.2455i 0.227273 0.443740i
\(892\) 0 0
\(893\) 1.11587i 0.0373410i
\(894\) 0 0
\(895\) 7.78665i 0.260279i
\(896\) 0 0
\(897\) 1.76222 + 2.23778i 0.0588388 + 0.0747174i
\(898\) 0 0
\(899\) 12.3892 0.413205
\(900\) 0 0
\(901\) 58.9770i 1.96481i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.96670i 0.0653753i
\(906\) 0 0
\(907\) −8.45506 −0.280746 −0.140373 0.990099i \(-0.544830\pi\)
−0.140373 + 0.990099i \(0.544830\pi\)
\(908\) 0 0
\(909\) 2.86464 11.8771i 0.0950143 0.393938i
\(910\) 0 0
\(911\) 15.4171i 0.510792i 0.966837 + 0.255396i \(0.0822058\pi\)
−0.966837 + 0.255396i \(0.917794\pi\)
\(912\) 0 0
\(913\) 10.9987i 0.364003i
\(914\) 0 0
\(915\) −3.60589 + 2.83958i −0.119207 + 0.0938737i
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 57.5865 1.89960 0.949802 0.312851i \(-0.101284\pi\)
0.949802 + 0.312851i \(0.101284\pi\)
\(920\) 0 0
\(921\) 32.5281 25.6154i 1.07184 0.844055i
\(922\) 0 0
\(923\) −89.4776 −2.94519
\(924\) 0 0
\(925\) 2.89891 0.0953156
\(926\) 0 0
\(927\) −24.5947 5.93201i −0.807795 0.194833i
\(928\) 0 0
\(929\) 47.3759 1.55435 0.777176 0.629283i \(-0.216652\pi\)
0.777176 + 0.629283i \(0.216652\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −22.6724 28.7909i −0.742261 0.942572i
\(934\) 0 0
\(935\) 32.3524i 1.05804i
\(936\) 0 0
\(937\) 40.6136i 1.32679i −0.748270 0.663394i \(-0.769115\pi\)
0.748270 0.663394i \(-0.230885\pi\)
\(938\) 0 0
\(939\) 28.7327 22.6266i 0.937657 0.738391i
\(940\) 0 0
\(941\) −29.8284 −0.972379 −0.486189 0.873854i \(-0.661613\pi\)
−0.486189 + 0.873854i \(0.661613\pi\)
\(942\) 0 0
\(943\) 1.23410i 0.0401877i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 40.5461i 1.31757i 0.752331 + 0.658785i \(0.228929\pi\)
−0.752331 + 0.658785i \(0.771071\pi\)
\(948\) 0 0
\(949\) −21.7759 −0.706877
\(950\) 0 0
\(951\) −21.7934 + 17.1620i −0.706701 + 0.556516i
\(952\) 0 0
\(953\) 36.7169i 1.18938i 0.803956 + 0.594688i \(0.202724\pi\)
−0.803956 + 0.594688i \(0.797276\pi\)
\(954\) 0 0
\(955\) 3.15212i 0.102000i
\(956\) 0 0
\(957\) −3.64756 4.63191i −0.117909 0.149728i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −5.22113 −0.168424
\(962\) 0 0
\(963\) 42.5434 + 10.2611i 1.37094 + 0.330659i
\(964\) 0 0
\(965\) −60.6536 −1.95251
\(966\) 0 0
\(967\) −30.1106 −0.968290 −0.484145 0.874988i \(-0.660869\pi\)
−0.484145 + 0.874988i \(0.660869\pi\)
\(968\) 0 0
\(969\) 30.8686 24.3086i 0.991642 0.780903i
\(970\) 0 0
\(971\) 22.2580 0.714294 0.357147 0.934048i \(-0.383750\pi\)
0.357147 + 0.934048i \(0.383750\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −12.8382 + 10.1098i −0.411150 + 0.323774i
\(976\) 0 0
\(977\) 55.8781i 1.78770i 0.448368 + 0.893849i \(0.352005\pi\)
−0.448368 + 0.893849i \(0.647995\pi\)
\(978\) 0 0
\(979\) 24.0000i 0.767044i
\(980\) 0 0
\(981\) −12.1688 + 50.4529i −0.388519 + 1.61084i
\(982\) 0 0
\(983\) 43.7757 1.39623 0.698114 0.715987i \(-0.254023\pi\)
0.698114 + 0.715987i \(0.254023\pi\)
\(984\) 0 0
\(985\) 37.9933i 1.21057i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0.709812i 0.0225707i
\(990\) 0 0
\(991\) 23.5471 0.747999 0.374000 0.927429i \(-0.377986\pi\)
0.374000 + 0.927429i \(0.377986\pi\)
\(992\) 0 0
\(993\) −20.3384 25.8271i −0.645420 0.819597i
\(994\) 0 0
\(995\) 17.7504i 0.562724i
\(996\) 0 0
\(997\) 19.0437i 0.603120i 0.953447 + 0.301560i \(0.0975074\pi\)
−0.953447 + 0.301560i \(0.902493\pi\)
\(998\) 0 0
\(999\) −8.28781 + 3.80490i −0.262215 + 0.120382i
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 2352.2.k.i.881.12 16
3.2 odd 2 inner 2352.2.k.i.881.6 16
4.3 odd 2 1176.2.k.a.881.5 16
7.2 even 3 336.2.bc.f.17.1 16
7.3 odd 6 336.2.bc.f.257.3 16
7.6 odd 2 inner 2352.2.k.i.881.5 16
12.11 even 2 1176.2.k.a.881.11 16
21.2 odd 6 336.2.bc.f.17.3 16
21.17 even 6 336.2.bc.f.257.1 16
21.20 even 2 inner 2352.2.k.i.881.11 16
28.3 even 6 168.2.u.a.89.6 yes 16
28.11 odd 6 1176.2.u.b.1097.3 16
28.19 even 6 1176.2.u.b.521.1 16
28.23 odd 6 168.2.u.a.17.8 yes 16
28.27 even 2 1176.2.k.a.881.12 16
84.11 even 6 1176.2.u.b.1097.1 16
84.23 even 6 168.2.u.a.17.6 16
84.47 odd 6 1176.2.u.b.521.3 16
84.59 odd 6 168.2.u.a.89.8 yes 16
84.83 odd 2 1176.2.k.a.881.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.2.u.a.17.6 16 84.23 even 6
168.2.u.a.17.8 yes 16 28.23 odd 6
168.2.u.a.89.6 yes 16 28.3 even 6
168.2.u.a.89.8 yes 16 84.59 odd 6
336.2.bc.f.17.1 16 7.2 even 3
336.2.bc.f.17.3 16 21.2 odd 6
336.2.bc.f.257.1 16 21.17 even 6
336.2.bc.f.257.3 16 7.3 odd 6
1176.2.k.a.881.5 16 4.3 odd 2
1176.2.k.a.881.6 16 84.83 odd 2
1176.2.k.a.881.11 16 12.11 even 2
1176.2.k.a.881.12 16 28.27 even 2
1176.2.u.b.521.1 16 28.19 even 6
1176.2.u.b.521.3 16 84.47 odd 6
1176.2.u.b.1097.1 16 84.11 even 6
1176.2.u.b.1097.3 16 28.11 odd 6
2352.2.k.i.881.5 16 7.6 odd 2 inner
2352.2.k.i.881.6 16 3.2 odd 2 inner
2352.2.k.i.881.11 16 21.20 even 2 inner
2352.2.k.i.881.12 16 1.1 even 1 trivial