Properties

Label 4004.2.e.a.3849.3
Level $4004$
Weight $2$
Character 4004.3849
Analytic conductor $31.972$
Analytic rank $0$
Dimension $48$
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] = [4004,2,Mod(3849,4004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4004, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("4004.3849");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4004 = 2^{2} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4004.e (of order \(2\), degree \(1\), 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: \(31.9721009693\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3849.3
Character \(\chi\) \(=\) 4004.3849
Dual form 4004.2.e.a.3849.46

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.04894i q^{3} +4.30537i q^{5} +(-2.63608 - 0.225969i) q^{7} -6.29604 q^{9} +O(q^{10})\) \(q-3.04894i q^{3} +4.30537i q^{5} +(-2.63608 - 0.225969i) q^{7} -6.29604 q^{9} +(2.23073 + 2.45435i) q^{11} +1.00000 q^{13} +13.1268 q^{15} +6.22792 q^{17} +0.815945 q^{19} +(-0.688966 + 8.03726i) q^{21} -0.626243 q^{23} -13.5362 q^{25} +10.0494i q^{27} +5.50046i q^{29} -1.54277i q^{31} +(7.48318 - 6.80136i) q^{33} +(0.972879 - 11.3493i) q^{35} -5.94206 q^{37} -3.04894i q^{39} -10.8437 q^{41} +2.96511i q^{43} -27.1068i q^{45} -10.6330i q^{47} +(6.89788 + 1.19135i) q^{49} -18.9886i q^{51} -11.1677 q^{53} +(-10.5669 + 9.60410i) q^{55} -2.48777i q^{57} -7.99070i q^{59} -13.4676 q^{61} +(16.5969 + 1.42271i) q^{63} +4.30537i q^{65} +2.69312 q^{67} +1.90938i q^{69} +6.46314 q^{71} +10.5984 q^{73} +41.2710i q^{75} +(-5.32578 - 6.97396i) q^{77} +9.52349i q^{79} +11.7520 q^{81} -8.25806 q^{83} +26.8135i q^{85} +16.7706 q^{87} -1.67918i q^{89} +(-2.63608 - 0.225969i) q^{91} -4.70382 q^{93} +3.51294i q^{95} -3.51534i q^{97} +(-14.0448 - 15.4527i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q - 4 q^{7} - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 48 q - 4 q^{7} - 48 q^{9} + 2 q^{11} + 48 q^{13} + 8 q^{15} + 4 q^{17} + 10 q^{21} + 4 q^{23} - 44 q^{25} + 10 q^{33} - 14 q^{35} - 16 q^{37} + 10 q^{49} - 8 q^{53} - 12 q^{55} + 16 q^{61} + 16 q^{63} + 4 q^{67} - 16 q^{73} + 2 q^{77} + 64 q^{81} - 4 q^{83} - 12 q^{87} - 4 q^{91} + 36 q^{93} - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4004\mathbb{Z}\right)^\times\).

\(n\) \(365\) \(925\) \(2003\) \(3433\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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
\(3\) 3.04894i 1.76031i −0.474689 0.880153i \(-0.657440\pi\)
0.474689 0.880153i \(-0.342560\pi\)
\(4\) 0 0
\(5\) 4.30537i 1.92542i 0.270539 + 0.962709i \(0.412798\pi\)
−0.270539 + 0.962709i \(0.587202\pi\)
\(6\) 0 0
\(7\) −2.63608 0.225969i −0.996346 0.0854082i
\(8\) 0 0
\(9\) −6.29604 −2.09868
\(10\) 0 0
\(11\) 2.23073 + 2.45435i 0.672590 + 0.740016i
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) 0 0
\(15\) 13.1268 3.38933
\(16\) 0 0
\(17\) 6.22792 1.51049 0.755247 0.655441i \(-0.227517\pi\)
0.755247 + 0.655441i \(0.227517\pi\)
\(18\) 0 0
\(19\) 0.815945 0.187191 0.0935953 0.995610i \(-0.470164\pi\)
0.0935953 + 0.995610i \(0.470164\pi\)
\(20\) 0 0
\(21\) −0.688966 + 8.03726i −0.150345 + 1.75387i
\(22\) 0 0
\(23\) −0.626243 −0.130581 −0.0652903 0.997866i \(-0.520797\pi\)
−0.0652903 + 0.997866i \(0.520797\pi\)
\(24\) 0 0
\(25\) −13.5362 −2.70723
\(26\) 0 0
\(27\) 10.0494i 1.93401i
\(28\) 0 0
\(29\) 5.50046i 1.02141i 0.859756 + 0.510705i \(0.170616\pi\)
−0.859756 + 0.510705i \(0.829384\pi\)
\(30\) 0 0
\(31\) 1.54277i 0.277090i −0.990356 0.138545i \(-0.955757\pi\)
0.990356 0.138545i \(-0.0442426\pi\)
\(32\) 0 0
\(33\) 7.48318 6.80136i 1.30265 1.18396i
\(34\) 0 0
\(35\) 0.972879 11.3493i 0.164446 1.91838i
\(36\) 0 0
\(37\) −5.94206 −0.976869 −0.488435 0.872600i \(-0.662432\pi\)
−0.488435 + 0.872600i \(0.662432\pi\)
\(38\) 0 0
\(39\) 3.04894i 0.488221i
\(40\) 0 0
\(41\) −10.8437 −1.69350 −0.846750 0.531991i \(-0.821444\pi\)
−0.846750 + 0.531991i \(0.821444\pi\)
\(42\) 0 0
\(43\) 2.96511i 0.452175i 0.974107 + 0.226087i \(0.0725935\pi\)
−0.974107 + 0.226087i \(0.927407\pi\)
\(44\) 0 0
\(45\) 27.1068i 4.04084i
\(46\) 0 0
\(47\) 10.6330i 1.55098i −0.631359 0.775491i \(-0.717502\pi\)
0.631359 0.775491i \(-0.282498\pi\)
\(48\) 0 0
\(49\) 6.89788 + 1.19135i 0.985411 + 0.170192i
\(50\) 0 0
\(51\) 18.9886i 2.65893i
\(52\) 0 0
\(53\) −11.1677 −1.53400 −0.767000 0.641647i \(-0.778251\pi\)
−0.767000 + 0.641647i \(0.778251\pi\)
\(54\) 0 0
\(55\) −10.5669 + 9.60410i −1.42484 + 1.29502i
\(56\) 0 0
\(57\) 2.48777i 0.329513i
\(58\) 0 0
\(59\) 7.99070i 1.04030i −0.854075 0.520150i \(-0.825876\pi\)
0.854075 0.520150i \(-0.174124\pi\)
\(60\) 0 0
\(61\) −13.4676 −1.72435 −0.862176 0.506609i \(-0.830899\pi\)
−0.862176 + 0.506609i \(0.830899\pi\)
\(62\) 0 0
\(63\) 16.5969 + 1.42271i 2.09101 + 0.179245i
\(64\) 0 0
\(65\) 4.30537i 0.534015i
\(66\) 0 0
\(67\) 2.69312 0.329016 0.164508 0.986376i \(-0.447396\pi\)
0.164508 + 0.986376i \(0.447396\pi\)
\(68\) 0 0
\(69\) 1.90938i 0.229862i
\(70\) 0 0
\(71\) 6.46314 0.767033 0.383517 0.923534i \(-0.374713\pi\)
0.383517 + 0.923534i \(0.374713\pi\)
\(72\) 0 0
\(73\) 10.5984 1.24045 0.620226 0.784423i \(-0.287041\pi\)
0.620226 + 0.784423i \(0.287041\pi\)
\(74\) 0 0
\(75\) 41.2710i 4.76556i
\(76\) 0 0
\(77\) −5.32578 6.97396i −0.606929 0.794756i
\(78\) 0 0
\(79\) 9.52349i 1.07148i 0.844384 + 0.535738i \(0.179967\pi\)
−0.844384 + 0.535738i \(0.820033\pi\)
\(80\) 0 0
\(81\) 11.7520 1.30578
\(82\) 0 0
\(83\) −8.25806 −0.906440 −0.453220 0.891399i \(-0.649725\pi\)
−0.453220 + 0.891399i \(0.649725\pi\)
\(84\) 0 0
\(85\) 26.8135i 2.90833i
\(86\) 0 0
\(87\) 16.7706 1.79799
\(88\) 0 0
\(89\) 1.67918i 0.177992i −0.996032 0.0889962i \(-0.971634\pi\)
0.996032 0.0889962i \(-0.0283659\pi\)
\(90\) 0 0
\(91\) −2.63608 0.225969i −0.276337 0.0236880i
\(92\) 0 0
\(93\) −4.70382 −0.487763
\(94\) 0 0
\(95\) 3.51294i 0.360420i
\(96\) 0 0
\(97\) 3.51534i 0.356928i −0.983946 0.178464i \(-0.942887\pi\)
0.983946 0.178464i \(-0.0571129\pi\)
\(98\) 0 0
\(99\) −14.0448 15.4527i −1.41155 1.55306i
\(100\) 0 0
\(101\) −0.569171 −0.0566347 −0.0283173 0.999599i \(-0.509015\pi\)
−0.0283173 + 0.999599i \(0.509015\pi\)
\(102\) 0 0
\(103\) 13.0110i 1.28201i −0.767537 0.641004i \(-0.778518\pi\)
0.767537 0.641004i \(-0.221482\pi\)
\(104\) 0 0
\(105\) −34.6034 2.96625i −3.37694 0.289476i
\(106\) 0 0
\(107\) 5.75874i 0.556718i 0.960477 + 0.278359i \(0.0897906\pi\)
−0.960477 + 0.278359i \(0.910209\pi\)
\(108\) 0 0
\(109\) 9.84093i 0.942590i 0.881976 + 0.471295i \(0.156213\pi\)
−0.881976 + 0.471295i \(0.843787\pi\)
\(110\) 0 0
\(111\) 18.1170i 1.71959i
\(112\) 0 0
\(113\) −9.54776 −0.898178 −0.449089 0.893487i \(-0.648251\pi\)
−0.449089 + 0.893487i \(0.648251\pi\)
\(114\) 0 0
\(115\) 2.69620i 0.251422i
\(116\) 0 0
\(117\) −6.29604 −0.582069
\(118\) 0 0
\(119\) −16.4173 1.40732i −1.50497 0.129009i
\(120\) 0 0
\(121\) −1.04771 + 10.9500i −0.0952466 + 0.995454i
\(122\) 0 0
\(123\) 33.0618i 2.98108i
\(124\) 0 0
\(125\) 36.7514i 3.28714i
\(126\) 0 0
\(127\) 13.9738i 1.23997i 0.784613 + 0.619986i \(0.212862\pi\)
−0.784613 + 0.619986i \(0.787138\pi\)
\(128\) 0 0
\(129\) 9.04044 0.795966
\(130\) 0 0
\(131\) −20.7914 −1.81656 −0.908278 0.418367i \(-0.862603\pi\)
−0.908278 + 0.418367i \(0.862603\pi\)
\(132\) 0 0
\(133\) −2.15090 0.184378i −0.186507 0.0159876i
\(134\) 0 0
\(135\) −43.2665 −3.72379
\(136\) 0 0
\(137\) 9.44527 0.806964 0.403482 0.914988i \(-0.367800\pi\)
0.403482 + 0.914988i \(0.367800\pi\)
\(138\) 0 0
\(139\) 6.48515 0.550063 0.275032 0.961435i \(-0.411312\pi\)
0.275032 + 0.961435i \(0.411312\pi\)
\(140\) 0 0
\(141\) −32.4194 −2.73020
\(142\) 0 0
\(143\) 2.23073 + 2.45435i 0.186543 + 0.205243i
\(144\) 0 0
\(145\) −23.6815 −1.96664
\(146\) 0 0
\(147\) 3.63234 21.0312i 0.299591 1.73463i
\(148\) 0 0
\(149\) 17.7382i 1.45317i 0.687076 + 0.726585i \(0.258894\pi\)
−0.687076 + 0.726585i \(0.741106\pi\)
\(150\) 0 0
\(151\) 2.30475i 0.187558i −0.995593 0.0937791i \(-0.970105\pi\)
0.995593 0.0937791i \(-0.0298948\pi\)
\(152\) 0 0
\(153\) −39.2113 −3.17004
\(154\) 0 0
\(155\) 6.64220 0.533514
\(156\) 0 0
\(157\) 13.2659i 1.05874i 0.848393 + 0.529368i \(0.177571\pi\)
−0.848393 + 0.529368i \(0.822429\pi\)
\(158\) 0 0
\(159\) 34.0496i 2.70031i
\(160\) 0 0
\(161\) 1.65083 + 0.141511i 0.130103 + 0.0111527i
\(162\) 0 0
\(163\) −10.8063 −0.846418 −0.423209 0.906032i \(-0.639096\pi\)
−0.423209 + 0.906032i \(0.639096\pi\)
\(164\) 0 0
\(165\) 29.2823 + 32.2178i 2.27963 + 2.50816i
\(166\) 0 0
\(167\) −0.852279 −0.0659513 −0.0329757 0.999456i \(-0.510498\pi\)
−0.0329757 + 0.999456i \(0.510498\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −5.13722 −0.392853
\(172\) 0 0
\(173\) 3.41091 0.259327 0.129663 0.991558i \(-0.458610\pi\)
0.129663 + 0.991558i \(0.458610\pi\)
\(174\) 0 0
\(175\) 35.6825 + 3.05875i 2.69734 + 0.231220i
\(176\) 0 0
\(177\) −24.3632 −1.83125
\(178\) 0 0
\(179\) −4.78588 −0.357714 −0.178857 0.983875i \(-0.557240\pi\)
−0.178857 + 0.983875i \(0.557240\pi\)
\(180\) 0 0
\(181\) 15.8783i 1.18023i 0.807321 + 0.590113i \(0.200917\pi\)
−0.807321 + 0.590113i \(0.799083\pi\)
\(182\) 0 0
\(183\) 41.0620i 3.03539i
\(184\) 0 0
\(185\) 25.5828i 1.88088i
\(186\) 0 0
\(187\) 13.8928 + 15.2855i 1.01594 + 1.11779i
\(188\) 0 0
\(189\) 2.27086 26.4912i 0.165181 1.92695i
\(190\) 0 0
\(191\) −13.6315 −0.986341 −0.493171 0.869933i \(-0.664162\pi\)
−0.493171 + 0.869933i \(0.664162\pi\)
\(192\) 0 0
\(193\) 20.0156i 1.44075i −0.693582 0.720377i \(-0.743969\pi\)
0.693582 0.720377i \(-0.256031\pi\)
\(194\) 0 0
\(195\) 13.1268 0.940030
\(196\) 0 0
\(197\) 22.7364i 1.61990i −0.586496 0.809952i \(-0.699493\pi\)
0.586496 0.809952i \(-0.300507\pi\)
\(198\) 0 0
\(199\) 7.93627i 0.562587i 0.959622 + 0.281294i \(0.0907635\pi\)
−0.959622 + 0.281294i \(0.909237\pi\)
\(200\) 0 0
\(201\) 8.21115i 0.579170i
\(202\) 0 0
\(203\) 1.24293 14.4997i 0.0872368 1.01768i
\(204\) 0 0
\(205\) 46.6861i 3.26070i
\(206\) 0 0
\(207\) 3.94285 0.274047
\(208\) 0 0
\(209\) 1.82015 + 2.00262i 0.125902 + 0.138524i
\(210\) 0 0
\(211\) 3.89039i 0.267825i −0.990993 0.133913i \(-0.957246\pi\)
0.990993 0.133913i \(-0.0427542\pi\)
\(212\) 0 0
\(213\) 19.7057i 1.35021i
\(214\) 0 0
\(215\) −12.7659 −0.870626
\(216\) 0 0
\(217\) −0.348618 + 4.06688i −0.0236658 + 0.276078i
\(218\) 0 0
\(219\) 32.3140i 2.18358i
\(220\) 0 0
\(221\) 6.22792 0.418935
\(222\) 0 0
\(223\) 20.5418i 1.37558i 0.725908 + 0.687791i \(0.241420\pi\)
−0.725908 + 0.687791i \(0.758580\pi\)
\(224\) 0 0
\(225\) 85.2243 5.68162
\(226\) 0 0
\(227\) −11.2496 −0.746660 −0.373330 0.927699i \(-0.621784\pi\)
−0.373330 + 0.927699i \(0.621784\pi\)
\(228\) 0 0
\(229\) 3.37375i 0.222944i −0.993768 0.111472i \(-0.964443\pi\)
0.993768 0.111472i \(-0.0355565\pi\)
\(230\) 0 0
\(231\) −21.2632 + 16.2380i −1.39902 + 1.06838i
\(232\) 0 0
\(233\) 2.51559i 0.164802i 0.996599 + 0.0824009i \(0.0262588\pi\)
−0.996599 + 0.0824009i \(0.973741\pi\)
\(234\) 0 0
\(235\) 45.7789 2.98629
\(236\) 0 0
\(237\) 29.0366 1.88613
\(238\) 0 0
\(239\) 14.8618i 0.961332i 0.876904 + 0.480666i \(0.159605\pi\)
−0.876904 + 0.480666i \(0.840395\pi\)
\(240\) 0 0
\(241\) −13.0364 −0.839746 −0.419873 0.907583i \(-0.637925\pi\)
−0.419873 + 0.907583i \(0.637925\pi\)
\(242\) 0 0
\(243\) 5.68289i 0.364558i
\(244\) 0 0
\(245\) −5.12918 + 29.6979i −0.327691 + 1.89733i
\(246\) 0 0
\(247\) 0.815945 0.0519173
\(248\) 0 0
\(249\) 25.1783i 1.59561i
\(250\) 0 0
\(251\) 23.8441i 1.50502i −0.658578 0.752512i \(-0.728842\pi\)
0.658578 0.752512i \(-0.271158\pi\)
\(252\) 0 0
\(253\) −1.39698 1.53702i −0.0878271 0.0966317i
\(254\) 0 0
\(255\) 81.7527 5.11956
\(256\) 0 0
\(257\) 15.1277i 0.943642i 0.881694 + 0.471821i \(0.156403\pi\)
−0.881694 + 0.471821i \(0.843597\pi\)
\(258\) 0 0
\(259\) 15.6638 + 1.34272i 0.973300 + 0.0834326i
\(260\) 0 0
\(261\) 34.6311i 2.14361i
\(262\) 0 0
\(263\) 22.2748i 1.37352i 0.726883 + 0.686761i \(0.240968\pi\)
−0.726883 + 0.686761i \(0.759032\pi\)
\(264\) 0 0
\(265\) 48.0810i 2.95359i
\(266\) 0 0
\(267\) −5.11971 −0.313321
\(268\) 0 0
\(269\) 2.95027i 0.179881i −0.995947 0.0899405i \(-0.971332\pi\)
0.995947 0.0899405i \(-0.0286677\pi\)
\(270\) 0 0
\(271\) −5.67235 −0.344571 −0.172285 0.985047i \(-0.555115\pi\)
−0.172285 + 0.985047i \(0.555115\pi\)
\(272\) 0 0
\(273\) −0.688966 + 8.03726i −0.0416981 + 0.486437i
\(274\) 0 0
\(275\) −30.1955 33.2226i −1.82086 2.00340i
\(276\) 0 0
\(277\) 18.7667i 1.12758i −0.825918 0.563791i \(-0.809342\pi\)
0.825918 0.563791i \(-0.190658\pi\)
\(278\) 0 0
\(279\) 9.71336i 0.581523i
\(280\) 0 0
\(281\) 1.31081i 0.0781963i −0.999235 0.0390982i \(-0.987551\pi\)
0.999235 0.0390982i \(-0.0124485\pi\)
\(282\) 0 0
\(283\) −8.13496 −0.483573 −0.241787 0.970329i \(-0.577733\pi\)
−0.241787 + 0.970329i \(0.577733\pi\)
\(284\) 0 0
\(285\) 10.7108 0.634450
\(286\) 0 0
\(287\) 28.5849 + 2.45034i 1.68731 + 0.144639i
\(288\) 0 0
\(289\) 21.7870 1.28159
\(290\) 0 0
\(291\) −10.7181 −0.628303
\(292\) 0 0
\(293\) 21.6265 1.26344 0.631718 0.775199i \(-0.282350\pi\)
0.631718 + 0.775199i \(0.282350\pi\)
\(294\) 0 0
\(295\) 34.4029 2.00301
\(296\) 0 0
\(297\) −24.6649 + 22.4176i −1.43120 + 1.30080i
\(298\) 0 0
\(299\) −0.626243 −0.0362165
\(300\) 0 0
\(301\) 0.670022 7.81627i 0.0386194 0.450523i
\(302\) 0 0
\(303\) 1.73537i 0.0996944i
\(304\) 0 0
\(305\) 57.9830i 3.32010i
\(306\) 0 0
\(307\) −0.544542 −0.0310786 −0.0155393 0.999879i \(-0.504947\pi\)
−0.0155393 + 0.999879i \(0.504947\pi\)
\(308\) 0 0
\(309\) −39.6697 −2.25673
\(310\) 0 0
\(311\) 7.90459i 0.448228i −0.974563 0.224114i \(-0.928051\pi\)
0.974563 0.224114i \(-0.0719488\pi\)
\(312\) 0 0
\(313\) 19.7353i 1.11551i 0.830007 + 0.557754i \(0.188337\pi\)
−0.830007 + 0.557754i \(0.811663\pi\)
\(314\) 0 0
\(315\) −6.12528 + 71.4557i −0.345121 + 4.02607i
\(316\) 0 0
\(317\) −30.5219 −1.71428 −0.857140 0.515084i \(-0.827761\pi\)
−0.857140 + 0.515084i \(0.827761\pi\)
\(318\) 0 0
\(319\) −13.5001 + 12.2700i −0.755859 + 0.686990i
\(320\) 0 0
\(321\) 17.5581 0.979995
\(322\) 0 0
\(323\) 5.08164 0.282750
\(324\) 0 0
\(325\) −13.5362 −0.750852
\(326\) 0 0
\(327\) 30.0044 1.65925
\(328\) 0 0
\(329\) −2.40273 + 28.0295i −0.132467 + 1.54531i
\(330\) 0 0
\(331\) −23.4338 −1.28804 −0.644018 0.765010i \(-0.722734\pi\)
−0.644018 + 0.765010i \(0.722734\pi\)
\(332\) 0 0
\(333\) 37.4115 2.05014
\(334\) 0 0
\(335\) 11.5948i 0.633494i
\(336\) 0 0
\(337\) 30.6525i 1.66975i −0.550440 0.834874i \(-0.685540\pi\)
0.550440 0.834874i \(-0.314460\pi\)
\(338\) 0 0
\(339\) 29.1106i 1.58107i
\(340\) 0 0
\(341\) 3.78651 3.44150i 0.205051 0.186368i
\(342\) 0 0
\(343\) −17.9142 4.69919i −0.967274 0.253733i
\(344\) 0 0
\(345\) −8.22056 −0.442580
\(346\) 0 0
\(347\) 8.65156i 0.464440i 0.972663 + 0.232220i \(0.0745989\pi\)
−0.972663 + 0.232220i \(0.925401\pi\)
\(348\) 0 0
\(349\) −33.0707 −1.77024 −0.885118 0.465367i \(-0.845922\pi\)
−0.885118 + 0.465367i \(0.845922\pi\)
\(350\) 0 0
\(351\) 10.0494i 0.536399i
\(352\) 0 0
\(353\) 18.7725i 0.999160i −0.866268 0.499580i \(-0.833488\pi\)
0.866268 0.499580i \(-0.166512\pi\)
\(354\) 0 0
\(355\) 27.8262i 1.47686i
\(356\) 0 0
\(357\) −4.29083 + 50.0555i −0.227095 + 2.64922i
\(358\) 0 0
\(359\) 0.612151i 0.0323081i 0.999870 + 0.0161540i \(0.00514221\pi\)
−0.999870 + 0.0161540i \(0.994858\pi\)
\(360\) 0 0
\(361\) −18.3342 −0.964960
\(362\) 0 0
\(363\) 33.3859 + 3.19441i 1.75230 + 0.167663i
\(364\) 0 0
\(365\) 45.6301i 2.38839i
\(366\) 0 0
\(367\) 18.7775i 0.980180i 0.871672 + 0.490090i \(0.163036\pi\)
−0.871672 + 0.490090i \(0.836964\pi\)
\(368\) 0 0
\(369\) 68.2723 3.55412
\(370\) 0 0
\(371\) 29.4390 + 2.52355i 1.52839 + 0.131016i
\(372\) 0 0
\(373\) 8.97321i 0.464615i 0.972642 + 0.232308i \(0.0746276\pi\)
−0.972642 + 0.232308i \(0.925372\pi\)
\(374\) 0 0
\(375\) −112.053 −5.78638
\(376\) 0 0
\(377\) 5.50046i 0.283288i
\(378\) 0 0
\(379\) −14.6889 −0.754520 −0.377260 0.926107i \(-0.623134\pi\)
−0.377260 + 0.926107i \(0.623134\pi\)
\(380\) 0 0
\(381\) 42.6052 2.18273
\(382\) 0 0
\(383\) 10.4863i 0.535823i 0.963443 + 0.267912i \(0.0863336\pi\)
−0.963443 + 0.267912i \(0.913666\pi\)
\(384\) 0 0
\(385\) 30.0254 22.9294i 1.53024 1.16859i
\(386\) 0 0
\(387\) 18.6684i 0.948970i
\(388\) 0 0
\(389\) −34.0993 −1.72890 −0.864452 0.502715i \(-0.832335\pi\)
−0.864452 + 0.502715i \(0.832335\pi\)
\(390\) 0 0
\(391\) −3.90019 −0.197241
\(392\) 0 0
\(393\) 63.3919i 3.19770i
\(394\) 0 0
\(395\) −41.0021 −2.06304
\(396\) 0 0
\(397\) 19.0340i 0.955290i −0.878553 0.477645i \(-0.841491\pi\)
0.878553 0.477645i \(-0.158509\pi\)
\(398\) 0 0
\(399\) −0.562158 + 6.55796i −0.0281431 + 0.328309i
\(400\) 0 0
\(401\) 8.40109 0.419531 0.209765 0.977752i \(-0.432730\pi\)
0.209765 + 0.977752i \(0.432730\pi\)
\(402\) 0 0
\(403\) 1.54277i 0.0768509i
\(404\) 0 0
\(405\) 50.5967i 2.51417i
\(406\) 0 0
\(407\) −13.2551 14.5839i −0.657032 0.722899i
\(408\) 0 0
\(409\) 15.8928 0.785850 0.392925 0.919571i \(-0.371463\pi\)
0.392925 + 0.919571i \(0.371463\pi\)
\(410\) 0 0
\(411\) 28.7981i 1.42050i
\(412\) 0 0
\(413\) −1.80565 + 21.0641i −0.0888501 + 1.03650i
\(414\) 0 0
\(415\) 35.5540i 1.74528i
\(416\) 0 0
\(417\) 19.7728i 0.968280i
\(418\) 0 0
\(419\) 8.64392i 0.422283i −0.977455 0.211142i \(-0.932282\pi\)
0.977455 0.211142i \(-0.0677181\pi\)
\(420\) 0 0
\(421\) −9.76359 −0.475848 −0.237924 0.971284i \(-0.576467\pi\)
−0.237924 + 0.971284i \(0.576467\pi\)
\(422\) 0 0
\(423\) 66.9458i 3.25502i
\(424\) 0 0
\(425\) −84.3023 −4.08926
\(426\) 0 0
\(427\) 35.5018 + 3.04326i 1.71805 + 0.147274i
\(428\) 0 0
\(429\) 7.48318 6.80136i 0.361291 0.328373i
\(430\) 0 0
\(431\) 15.6870i 0.755617i 0.925884 + 0.377808i \(0.123322\pi\)
−0.925884 + 0.377808i \(0.876678\pi\)
\(432\) 0 0
\(433\) 6.43221i 0.309112i −0.987984 0.154556i \(-0.950605\pi\)
0.987984 0.154556i \(-0.0493947\pi\)
\(434\) 0 0
\(435\) 72.2035i 3.46189i
\(436\) 0 0
\(437\) −0.510979 −0.0244435
\(438\) 0 0
\(439\) 34.9418 1.66768 0.833841 0.552004i \(-0.186137\pi\)
0.833841 + 0.552004i \(0.186137\pi\)
\(440\) 0 0
\(441\) −43.4293 7.50076i −2.06806 0.357179i
\(442\) 0 0
\(443\) −31.4733 −1.49534 −0.747670 0.664070i \(-0.768828\pi\)
−0.747670 + 0.664070i \(0.768828\pi\)
\(444\) 0 0
\(445\) 7.22947 0.342710
\(446\) 0 0
\(447\) 54.0827 2.55803
\(448\) 0 0
\(449\) 1.19601 0.0564433 0.0282216 0.999602i \(-0.491016\pi\)
0.0282216 + 0.999602i \(0.491016\pi\)
\(450\) 0 0
\(451\) −24.1893 26.6143i −1.13903 1.25322i
\(452\) 0 0
\(453\) −7.02706 −0.330160
\(454\) 0 0
\(455\) 0.972879 11.3493i 0.0456093 0.532064i
\(456\) 0 0
\(457\) 14.6420i 0.684924i −0.939532 0.342462i \(-0.888739\pi\)
0.939532 0.342462i \(-0.111261\pi\)
\(458\) 0 0
\(459\) 62.5871i 2.92132i
\(460\) 0 0
\(461\) 6.33845 0.295211 0.147606 0.989046i \(-0.452843\pi\)
0.147606 + 0.989046i \(0.452843\pi\)
\(462\) 0 0
\(463\) 4.12809 0.191849 0.0959243 0.995389i \(-0.469419\pi\)
0.0959243 + 0.995389i \(0.469419\pi\)
\(464\) 0 0
\(465\) 20.2517i 0.939149i
\(466\) 0 0
\(467\) 1.77050i 0.0819288i 0.999161 + 0.0409644i \(0.0130430\pi\)
−0.999161 + 0.0409644i \(0.986957\pi\)
\(468\) 0 0
\(469\) −7.09928 0.608560i −0.327814 0.0281007i
\(470\) 0 0
\(471\) 40.4470 1.86370
\(472\) 0 0
\(473\) −7.27743 + 6.61435i −0.334616 + 0.304128i
\(474\) 0 0
\(475\) −11.0448 −0.506769
\(476\) 0 0
\(477\) 70.3122 3.21938
\(478\) 0 0
\(479\) −12.6868 −0.579675 −0.289837 0.957076i \(-0.593601\pi\)
−0.289837 + 0.957076i \(0.593601\pi\)
\(480\) 0 0
\(481\) −5.94206 −0.270935
\(482\) 0 0
\(483\) 0.431460 5.03328i 0.0196321 0.229022i
\(484\) 0 0
\(485\) 15.1348 0.687236
\(486\) 0 0
\(487\) 30.1009 1.36400 0.682002 0.731351i \(-0.261110\pi\)
0.682002 + 0.731351i \(0.261110\pi\)
\(488\) 0 0
\(489\) 32.9479i 1.48996i
\(490\) 0 0
\(491\) 35.0014i 1.57959i −0.613370 0.789795i \(-0.710187\pi\)
0.613370 0.789795i \(-0.289813\pi\)
\(492\) 0 0
\(493\) 34.2564i 1.54283i
\(494\) 0 0
\(495\) 66.5296 60.4678i 2.99028 2.71783i
\(496\) 0 0
\(497\) −17.0374 1.46047i −0.764231 0.0655109i
\(498\) 0 0
\(499\) 6.00138 0.268659 0.134329 0.990937i \(-0.457112\pi\)
0.134329 + 0.990937i \(0.457112\pi\)
\(500\) 0 0
\(501\) 2.59855i 0.116095i
\(502\) 0 0
\(503\) 26.8342 1.19648 0.598238 0.801318i \(-0.295868\pi\)
0.598238 + 0.801318i \(0.295868\pi\)
\(504\) 0 0
\(505\) 2.45049i 0.109045i
\(506\) 0 0
\(507\) 3.04894i 0.135408i
\(508\) 0 0
\(509\) 21.3512i 0.946375i −0.880962 0.473188i \(-0.843103\pi\)
0.880962 0.473188i \(-0.156897\pi\)
\(510\) 0 0
\(511\) −27.9384 2.39492i −1.23592 0.105945i
\(512\) 0 0
\(513\) 8.19979i 0.362029i
\(514\) 0 0
\(515\) 56.0170 2.46840
\(516\) 0 0
\(517\) 26.0971 23.7193i 1.14775 1.04317i
\(518\) 0 0
\(519\) 10.3997i 0.456495i
\(520\) 0 0
\(521\) 13.9400i 0.610721i −0.952237 0.305360i \(-0.901223\pi\)
0.952237 0.305360i \(-0.0987769\pi\)
\(522\) 0 0
\(523\) −5.20406 −0.227558 −0.113779 0.993506i \(-0.536296\pi\)
−0.113779 + 0.993506i \(0.536296\pi\)
\(524\) 0 0
\(525\) 9.32596 108.794i 0.407018 4.74815i
\(526\) 0 0
\(527\) 9.60826i 0.418543i
\(528\) 0 0
\(529\) −22.6078 −0.982949
\(530\) 0 0
\(531\) 50.3097i 2.18326i
\(532\) 0 0
\(533\) −10.8437 −0.469692
\(534\) 0 0
\(535\) −24.7935 −1.07192
\(536\) 0 0
\(537\) 14.5919i 0.629686i
\(538\) 0 0
\(539\) 12.4633 + 19.5874i 0.536832 + 0.843689i
\(540\) 0 0
\(541\) 4.17722i 0.179593i −0.995960 0.0897963i \(-0.971378\pi\)
0.995960 0.0897963i \(-0.0286216\pi\)
\(542\) 0 0
\(543\) 48.4120 2.07756
\(544\) 0 0
\(545\) −42.3688 −1.81488
\(546\) 0 0
\(547\) 26.8440i 1.14777i −0.818937 0.573883i \(-0.805436\pi\)
0.818937 0.573883i \(-0.194564\pi\)
\(548\) 0 0
\(549\) 84.7927 3.61886
\(550\) 0 0
\(551\) 4.48807i 0.191198i
\(552\) 0 0
\(553\) 2.15201 25.1047i 0.0915129 1.06756i
\(554\) 0 0
\(555\) −78.0003 −3.31093
\(556\) 0 0
\(557\) 3.21249i 0.136118i 0.997681 + 0.0680588i \(0.0216805\pi\)
−0.997681 + 0.0680588i \(0.978319\pi\)
\(558\) 0 0
\(559\) 2.96511i 0.125411i
\(560\) 0 0
\(561\) 46.6047 42.3583i 1.96765 1.78837i
\(562\) 0 0
\(563\) 10.5707 0.445504 0.222752 0.974875i \(-0.428496\pi\)
0.222752 + 0.974875i \(0.428496\pi\)
\(564\) 0 0
\(565\) 41.1066i 1.72937i
\(566\) 0 0
\(567\) −30.9793 2.65559i −1.30101 0.111524i
\(568\) 0 0
\(569\) 19.0687i 0.799401i 0.916646 + 0.399700i \(0.130886\pi\)
−0.916646 + 0.399700i \(0.869114\pi\)
\(570\) 0 0
\(571\) 1.87982i 0.0786680i −0.999226 0.0393340i \(-0.987476\pi\)
0.999226 0.0393340i \(-0.0125236\pi\)
\(572\) 0 0
\(573\) 41.5617i 1.73626i
\(574\) 0 0
\(575\) 8.47693 0.353512
\(576\) 0 0
\(577\) 27.7990i 1.15729i 0.815581 + 0.578643i \(0.196417\pi\)
−0.815581 + 0.578643i \(0.803583\pi\)
\(578\) 0 0
\(579\) −61.0264 −2.53617
\(580\) 0 0
\(581\) 21.7689 + 1.86606i 0.903128 + 0.0774174i
\(582\) 0 0
\(583\) −24.9121 27.4095i −1.03175 1.13518i
\(584\) 0 0
\(585\) 27.1068i 1.12073i
\(586\) 0 0
\(587\) 10.4469i 0.431190i −0.976483 0.215595i \(-0.930831\pi\)
0.976483 0.215595i \(-0.0691692\pi\)
\(588\) 0 0
\(589\) 1.25882i 0.0518686i
\(590\) 0 0
\(591\) −69.3221 −2.85153
\(592\) 0 0
\(593\) −9.91927 −0.407336 −0.203668 0.979040i \(-0.565286\pi\)
−0.203668 + 0.979040i \(0.565286\pi\)
\(594\) 0 0
\(595\) 6.05901 70.6826i 0.248395 2.89770i
\(596\) 0 0
\(597\) 24.1972 0.990326
\(598\) 0 0
\(599\) 25.3821 1.03708 0.518542 0.855052i \(-0.326475\pi\)
0.518542 + 0.855052i \(0.326475\pi\)
\(600\) 0 0
\(601\) 22.3480 0.911592 0.455796 0.890084i \(-0.349355\pi\)
0.455796 + 0.890084i \(0.349355\pi\)
\(602\) 0 0
\(603\) −16.9560 −0.690500
\(604\) 0 0
\(605\) −47.1437 4.51079i −1.91666 0.183390i
\(606\) 0 0
\(607\) 12.7769 0.518599 0.259300 0.965797i \(-0.416508\pi\)
0.259300 + 0.965797i \(0.416508\pi\)
\(608\) 0 0
\(609\) −44.2087 3.78963i −1.79142 0.153563i
\(610\) 0 0
\(611\) 10.6330i 0.430165i
\(612\) 0 0
\(613\) 43.0433i 1.73850i 0.494371 + 0.869251i \(0.335398\pi\)
−0.494371 + 0.869251i \(0.664602\pi\)
\(614\) 0 0
\(615\) −142.343 −5.73982
\(616\) 0 0
\(617\) 28.1068 1.13154 0.565768 0.824565i \(-0.308580\pi\)
0.565768 + 0.824565i \(0.308580\pi\)
\(618\) 0 0
\(619\) 15.8211i 0.635902i 0.948107 + 0.317951i \(0.102995\pi\)
−0.948107 + 0.317951i \(0.897005\pi\)
\(620\) 0 0
\(621\) 6.29338i 0.252545i
\(622\) 0 0
\(623\) −0.379442 + 4.42645i −0.0152020 + 0.177342i
\(624\) 0 0
\(625\) 90.5472 3.62189
\(626\) 0 0
\(627\) 6.10586 5.54953i 0.243845 0.221627i
\(628\) 0 0
\(629\) −37.0067 −1.47555
\(630\) 0 0
\(631\) 1.82300 0.0725725 0.0362863 0.999341i \(-0.488447\pi\)
0.0362863 + 0.999341i \(0.488447\pi\)
\(632\) 0 0
\(633\) −11.8616 −0.471455
\(634\) 0 0
\(635\) −60.1622 −2.38746
\(636\) 0 0
\(637\) 6.89788 + 1.19135i 0.273304 + 0.0472028i
\(638\) 0 0
\(639\) −40.6922 −1.60976
\(640\) 0 0
\(641\) 26.8720 1.06138 0.530690 0.847566i \(-0.321933\pi\)
0.530690 + 0.847566i \(0.321933\pi\)
\(642\) 0 0
\(643\) 2.22230i 0.0876391i −0.999039 0.0438195i \(-0.986047\pi\)
0.999039 0.0438195i \(-0.0139527\pi\)
\(644\) 0 0
\(645\) 38.9224i 1.53257i
\(646\) 0 0
\(647\) 31.7005i 1.24628i −0.782112 0.623138i \(-0.785857\pi\)
0.782112 0.623138i \(-0.214143\pi\)
\(648\) 0 0
\(649\) 19.6120 17.8251i 0.769838 0.699695i
\(650\) 0 0
\(651\) 12.3997 + 1.06292i 0.485981 + 0.0416590i
\(652\) 0 0
\(653\) −8.35326 −0.326888 −0.163444 0.986553i \(-0.552260\pi\)
−0.163444 + 0.986553i \(0.552260\pi\)
\(654\) 0 0
\(655\) 89.5147i 3.49763i
\(656\) 0 0
\(657\) −66.7282 −2.60331
\(658\) 0 0
\(659\) 24.2630i 0.945153i 0.881290 + 0.472577i \(0.156676\pi\)
−0.881290 + 0.472577i \(0.843324\pi\)
\(660\) 0 0
\(661\) 27.7909i 1.08094i 0.841363 + 0.540471i \(0.181754\pi\)
−0.841363 + 0.540471i \(0.818246\pi\)
\(662\) 0 0
\(663\) 18.9886i 0.737455i
\(664\) 0 0
\(665\) 0.793815 9.26041i 0.0307828 0.359103i
\(666\) 0 0
\(667\) 3.44462i 0.133376i
\(668\) 0 0
\(669\) 62.6308 2.42145
\(670\) 0 0
\(671\) −30.0426 33.0543i −1.15978 1.27605i
\(672\) 0 0
\(673\) 47.3618i 1.82566i 0.408334 + 0.912832i \(0.366110\pi\)
−0.408334 + 0.912832i \(0.633890\pi\)
\(674\) 0 0
\(675\) 136.031i 5.23583i
\(676\) 0 0
\(677\) 8.49513 0.326494 0.163247 0.986585i \(-0.447803\pi\)
0.163247 + 0.986585i \(0.447803\pi\)
\(678\) 0 0
\(679\) −0.794357 + 9.26672i −0.0304846 + 0.355624i
\(680\) 0 0
\(681\) 34.2992i 1.31435i
\(682\) 0 0
\(683\) 35.3789 1.35374 0.676869 0.736104i \(-0.263337\pi\)
0.676869 + 0.736104i \(0.263337\pi\)
\(684\) 0 0
\(685\) 40.6654i 1.55374i
\(686\) 0 0
\(687\) −10.2864 −0.392450
\(688\) 0 0
\(689\) −11.1677 −0.425455
\(690\) 0 0
\(691\) 4.72419i 0.179717i −0.995955 0.0898583i \(-0.971359\pi\)
0.995955 0.0898583i \(-0.0286414\pi\)
\(692\) 0 0
\(693\) 33.5313 + 43.9083i 1.27375 + 1.66794i
\(694\) 0 0
\(695\) 27.9209i 1.05910i
\(696\) 0 0
\(697\) −67.5337 −2.55802
\(698\) 0 0
\(699\) 7.66989 0.290102
\(700\) 0 0
\(701\) 35.5468i 1.34258i −0.741193 0.671292i \(-0.765740\pi\)
0.741193 0.671292i \(-0.234260\pi\)
\(702\) 0 0
\(703\) −4.84840 −0.182861
\(704\) 0 0
\(705\) 139.577i 5.25678i
\(706\) 0 0
\(707\) 1.50038 + 0.128615i 0.0564277 + 0.00483706i
\(708\) 0 0
\(709\) −6.38729 −0.239880 −0.119940 0.992781i \(-0.538270\pi\)
−0.119940 + 0.992781i \(0.538270\pi\)
\(710\) 0 0
\(711\) 59.9603i 2.24869i
\(712\) 0 0
\(713\) 0.966149i 0.0361826i
\(714\) 0 0
\(715\) −10.5669 + 9.60410i −0.395179 + 0.359173i
\(716\) 0 0
\(717\) 45.3129 1.69224
\(718\) 0 0
\(719\) 5.32828i 0.198711i 0.995052 + 0.0993556i \(0.0316781\pi\)
−0.995052 + 0.0993556i \(0.968322\pi\)
\(720\) 0 0
\(721\) −2.94007 + 34.2980i −0.109494 + 1.27732i
\(722\) 0 0
\(723\) 39.7471i 1.47821i
\(724\) 0 0
\(725\) 74.4552i 2.76520i
\(726\) 0 0
\(727\) 17.2630i 0.640248i 0.947376 + 0.320124i \(0.103725\pi\)
−0.947376 + 0.320124i \(0.896275\pi\)
\(728\) 0 0
\(729\) 17.9292 0.664046
\(730\) 0 0
\(731\) 18.4665i 0.683007i
\(732\) 0 0
\(733\) 6.57837 0.242978 0.121489 0.992593i \(-0.461233\pi\)
0.121489 + 0.992593i \(0.461233\pi\)
\(734\) 0 0
\(735\) 90.5471 + 15.6386i 3.33988 + 0.576837i
\(736\) 0 0
\(737\) 6.00761 + 6.60986i 0.221293 + 0.243477i
\(738\) 0 0
\(739\) 0.0295860i 0.00108834i −1.00000 0.000544169i \(-0.999827\pi\)
1.00000 0.000544169i \(-0.000173215\pi\)
\(740\) 0 0
\(741\) 2.48777i 0.0913904i
\(742\) 0 0
\(743\) 46.2496i 1.69673i −0.529409 0.848367i \(-0.677586\pi\)
0.529409 0.848367i \(-0.322414\pi\)
\(744\) 0 0
\(745\) −76.3695 −2.79796
\(746\) 0 0
\(747\) 51.9931 1.90233
\(748\) 0 0
\(749\) 1.30130 15.1805i 0.0475483 0.554684i
\(750\) 0 0
\(751\) 12.5975 0.459688 0.229844 0.973227i \(-0.426178\pi\)
0.229844 + 0.973227i \(0.426178\pi\)
\(752\) 0 0
\(753\) −72.6992 −2.64931
\(754\) 0 0
\(755\) 9.92281 0.361128
\(756\) 0 0
\(757\) 47.3686 1.72164 0.860821 0.508908i \(-0.169951\pi\)
0.860821 + 0.508908i \(0.169951\pi\)
\(758\) 0 0
\(759\) −4.68629 + 4.25930i −0.170101 + 0.154603i
\(760\) 0 0
\(761\) 34.3236 1.24423 0.622115 0.782926i \(-0.286274\pi\)
0.622115 + 0.782926i \(0.286274\pi\)
\(762\) 0 0
\(763\) 2.22374 25.9415i 0.0805049 0.939146i
\(764\) 0 0
\(765\) 168.819i 6.10366i
\(766\) 0 0
\(767\) 7.99070i 0.288527i
\(768\) 0 0
\(769\) −19.4091 −0.699910 −0.349955 0.936767i \(-0.613803\pi\)
−0.349955 + 0.936767i \(0.613803\pi\)
\(770\) 0 0
\(771\) 46.1236 1.66110
\(772\) 0 0
\(773\) 31.6784i 1.13939i 0.821855 + 0.569697i \(0.192940\pi\)
−0.821855 + 0.569697i \(0.807060\pi\)
\(774\) 0 0
\(775\) 20.8832i 0.750148i
\(776\) 0 0
\(777\) 4.09388 47.7579i 0.146867 1.71331i
\(778\) 0 0
\(779\) −8.84785 −0.317007
\(780\) 0 0
\(781\) 14.4175 + 15.8628i 0.515899 + 0.567617i
\(782\) 0 0
\(783\) −55.2765 −1.97542
\(784\) 0 0
\(785\) −57.1146 −2.03851
\(786\) 0 0
\(787\) 45.0390 1.60547 0.802733 0.596339i \(-0.203379\pi\)
0.802733 + 0.596339i \(0.203379\pi\)
\(788\) 0 0
\(789\) 67.9145 2.41782
\(790\) 0 0
\(791\) 25.1687 + 2.15750i 0.894896 + 0.0767118i
\(792\) 0 0
\(793\) −13.4676 −0.478249
\(794\) 0 0
\(795\) −146.596 −5.19923
\(796\) 0 0
\(797\) 31.7427i 1.12438i 0.827006 + 0.562192i \(0.190042\pi\)
−0.827006 + 0.562192i \(0.809958\pi\)
\(798\) 0 0
\(799\) 66.2215i 2.34275i
\(800\) 0 0
\(801\) 10.5722i 0.373549i
\(802\) 0 0
\(803\) 23.6422 + 26.0123i 0.834316 + 0.917955i
\(804\) 0 0
\(805\) −0.609258 + 7.10742i −0.0214735 + 0.250504i
\(806\) 0 0
\(807\) −8.99519 −0.316646
\(808\) 0 0
\(809\) 5.08147i 0.178655i 0.996002 + 0.0893275i \(0.0284718\pi\)
−0.996002 + 0.0893275i \(0.971528\pi\)
\(810\) 0 0
\(811\) −29.6887 −1.04251 −0.521255 0.853401i \(-0.674536\pi\)
−0.521255 + 0.853401i \(0.674536\pi\)
\(812\) 0 0
\(813\) 17.2947i 0.606550i
\(814\) 0 0
\(815\) 46.5252i 1.62971i
\(816\) 0 0
\(817\) 2.41937i 0.0846429i
\(818\) 0 0
\(819\) 16.5969 + 1.42271i 0.579942 + 0.0497135i
\(820\) 0 0
\(821\) 37.1933i 1.29805i 0.760765 + 0.649027i \(0.224824\pi\)
−0.760765 + 0.649027i \(0.775176\pi\)
\(822\) 0 0
\(823\) 23.6722 0.825160 0.412580 0.910921i \(-0.364628\pi\)
0.412580 + 0.910921i \(0.364628\pi\)
\(824\) 0 0
\(825\) −101.294 + 92.0643i −3.52659 + 3.20527i
\(826\) 0 0
\(827\) 11.9003i 0.413814i −0.978361 0.206907i \(-0.933660\pi\)
0.978361 0.206907i \(-0.0663398\pi\)
\(828\) 0 0
\(829\) 11.1597i 0.387593i −0.981042 0.193796i \(-0.937920\pi\)
0.981042 0.193796i \(-0.0620801\pi\)
\(830\) 0 0
\(831\) −57.2185 −1.98489
\(832\) 0 0
\(833\) 42.9594 + 7.41961i 1.48846 + 0.257074i
\(834\) 0 0
\(835\) 3.66937i 0.126984i
\(836\) 0 0
\(837\) 15.5040 0.535896
\(838\) 0 0
\(839\) 56.8859i 1.96392i −0.189086 0.981960i \(-0.560553\pi\)
0.189086 0.981960i \(-0.439447\pi\)
\(840\) 0 0
\(841\) −1.25506 −0.0432779
\(842\) 0 0
\(843\) −3.99658 −0.137650
\(844\) 0 0
\(845\) 4.30537i 0.148109i
\(846\) 0 0
\(847\) 5.23622 28.6283i 0.179918 0.983682i
\(848\) 0 0
\(849\) 24.8030i 0.851237i
\(850\) 0 0
\(851\) 3.72117 0.127560
\(852\) 0 0
\(853\) −23.5307 −0.805676 −0.402838 0.915271i \(-0.631976\pi\)
−0.402838 + 0.915271i \(0.631976\pi\)
\(854\) 0 0
\(855\) 22.1176i 0.756407i
\(856\) 0 0
\(857\) −26.3336 −0.899538 −0.449769 0.893145i \(-0.648494\pi\)
−0.449769 + 0.893145i \(0.648494\pi\)
\(858\) 0 0
\(859\) 25.3084i 0.863512i −0.901990 0.431756i \(-0.857894\pi\)
0.901990 0.431756i \(-0.142106\pi\)
\(860\) 0 0
\(861\) 7.47093 87.1536i 0.254609 2.97019i
\(862\) 0 0
\(863\) 7.09350 0.241466 0.120733 0.992685i \(-0.461476\pi\)
0.120733 + 0.992685i \(0.461476\pi\)
\(864\) 0 0
\(865\) 14.6852i 0.499312i
\(866\) 0 0
\(867\) 66.4274i 2.25599i
\(868\) 0 0
\(869\) −23.3740 + 21.2443i −0.792910 + 0.720664i
\(870\) 0 0
\(871\) 2.69312 0.0912527
\(872\) 0 0
\(873\) 22.1327i 0.749079i
\(874\) 0 0
\(875\) −8.30466 + 96.8797i −0.280749 + 3.27513i
\(876\) 0 0
\(877\) 18.6667i 0.630331i 0.949037 + 0.315165i \(0.102060\pi\)
−0.949037 + 0.315165i \(0.897940\pi\)
\(878\) 0 0
\(879\) 65.9380i 2.22403i
\(880\) 0 0
\(881\) 33.2091i 1.11884i −0.828883 0.559422i \(-0.811023\pi\)
0.828883 0.559422i \(-0.188977\pi\)
\(882\) 0 0
\(883\) −0.221802 −0.00746423 −0.00373212 0.999993i \(-0.501188\pi\)
−0.00373212 + 0.999993i \(0.501188\pi\)
\(884\) 0 0
\(885\) 104.892i 3.52592i
\(886\) 0 0
\(887\) −20.8697 −0.700736 −0.350368 0.936612i \(-0.613943\pi\)
−0.350368 + 0.936612i \(0.613943\pi\)
\(888\) 0 0
\(889\) 3.15764 36.8360i 0.105904 1.23544i
\(890\) 0 0
\(891\) 26.2155 + 28.8436i 0.878254 + 0.966297i
\(892\) 0 0
\(893\) 8.67594i 0.290329i
\(894\) 0 0
\(895\) 20.6050i 0.688749i
\(896\) 0 0
\(897\) 1.90938i 0.0637522i
\(898\) 0 0
\(899\) 8.48595 0.283022
\(900\) 0 0
\(901\) −69.5515 −2.31710
\(902\) 0 0
\(903\) −23.8314 2.04286i −0.793058 0.0679821i
\(904\) 0 0
\(905\) −68.3619 −2.27243
\(906\) 0 0
\(907\) −50.3950 −1.67334 −0.836669 0.547708i \(-0.815500\pi\)
−0.836669 + 0.547708i \(0.815500\pi\)
\(908\) 0 0
\(909\) 3.58353 0.118858
\(910\) 0 0
\(911\) −39.7787 −1.31793 −0.658964 0.752174i \(-0.729005\pi\)
−0.658964 + 0.752174i \(0.729005\pi\)
\(912\) 0 0
\(913\) −18.4215 20.2682i −0.609662 0.670780i
\(914\) 0 0
\(915\) −176.787 −5.84439
\(916\) 0 0
\(917\) 54.8080 + 4.69822i 1.80992 + 0.155149i
\(918\) 0 0
\(919\) 17.4011i 0.574009i 0.957929 + 0.287004i \(0.0926594\pi\)
−0.957929 + 0.287004i \(0.907341\pi\)
\(920\) 0 0
\(921\) 1.66028i 0.0547079i
\(922\) 0 0
\(923\) 6.46314 0.212737
\(924\) 0 0
\(925\) 80.4328 2.64461
\(926\) 0 0
\(927\) 81.9176i 2.69053i
\(928\) 0 0
\(929\) 26.7888i 0.878912i 0.898264 + 0.439456i \(0.144829\pi\)
−0.898264 + 0.439456i \(0.855171\pi\)
\(930\) 0 0
\(931\) 5.62829 + 0.972072i 0.184460 + 0.0318584i
\(932\) 0 0
\(933\) −24.1006 −0.789019
\(934\) 0 0
\(935\) −65.8098 + 59.8136i −2.15221 + 1.95611i
\(936\) 0 0
\(937\) −31.5445 −1.03051 −0.515256 0.857036i \(-0.672303\pi\)
−0.515256 + 0.857036i \(0.672303\pi\)
\(938\) 0 0
\(939\) 60.1719 1.96364
\(940\) 0 0
\(941\) 55.3278 1.80364 0.901818 0.432116i \(-0.142233\pi\)
0.901818 + 0.432116i \(0.142233\pi\)
\(942\) 0 0
\(943\) 6.79078 0.221138
\(944\) 0 0
\(945\) 114.054 + 9.77688i 3.71018 + 0.318042i
\(946\) 0 0
\(947\) 52.6400 1.71057 0.855285 0.518157i \(-0.173382\pi\)
0.855285 + 0.518157i \(0.173382\pi\)
\(948\) 0 0
\(949\) 10.5984 0.344040
\(950\) 0 0
\(951\) 93.0594i 3.01766i
\(952\) 0 0
\(953\) 5.60473i 0.181555i −0.995871 0.0907775i \(-0.971065\pi\)
0.995871 0.0907775i \(-0.0289352\pi\)
\(954\) 0 0
\(955\) 58.6886i 1.89912i
\(956\) 0 0
\(957\) 37.4106 + 41.1609i 1.20931 + 1.33054i
\(958\) 0 0
\(959\) −24.8985 2.13434i −0.804016 0.0689214i
\(960\) 0 0
\(961\) 28.6199 0.923221
\(962\) 0 0
\(963\) 36.2573i 1.16837i
\(964\) 0 0
\(965\) 86.1745 2.77405
\(966\) 0 0
\(967\) 33.5560i 1.07909i 0.841957 + 0.539544i \(0.181403\pi\)
−0.841957 + 0.539544i \(0.818597\pi\)
\(968\) 0 0
\(969\) 15.4936i 0.497727i
\(970\) 0 0
\(971\) 4.83687i 0.155223i −0.996984 0.0776113i \(-0.975271\pi\)
0.996984 0.0776113i \(-0.0247293\pi\)
\(972\) 0 0
\(973\) −17.0954 1.46544i −0.548053 0.0469799i
\(974\) 0 0
\(975\) 41.2710i 1.32173i
\(976\) 0 0
\(977\) 27.7375 0.887402 0.443701 0.896175i \(-0.353665\pi\)
0.443701 + 0.896175i \(0.353665\pi\)
\(978\) 0 0
\(979\) 4.12130 3.74579i 0.131717 0.119716i
\(980\) 0 0
\(981\) 61.9589i 1.97820i
\(982\) 0 0
\(983\) 53.3096i 1.70031i −0.526531 0.850156i \(-0.676508\pi\)
0.526531 0.850156i \(-0.323492\pi\)
\(984\) 0 0
\(985\) 97.8887 3.11899
\(986\) 0 0
\(987\) 85.4602 + 7.32577i 2.72023 + 0.233182i
\(988\) 0 0
\(989\) 1.85688i 0.0590453i
\(990\) 0 0
\(991\) 43.4003 1.37866 0.689328 0.724449i \(-0.257906\pi\)
0.689328 + 0.724449i \(0.257906\pi\)
\(992\) 0 0
\(993\) 71.4482i 2.26734i
\(994\) 0 0
\(995\) −34.1685 −1.08322
\(996\) 0 0
\(997\) 2.47459 0.0783711 0.0391855 0.999232i \(-0.487524\pi\)
0.0391855 + 0.999232i \(0.487524\pi\)
\(998\) 0 0
\(999\) 59.7144i 1.88928i
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 4004.2.e.a.3849.3 48
7.6 odd 2 4004.2.e.b.3849.46 yes 48
11.10 odd 2 4004.2.e.b.3849.3 yes 48
77.76 even 2 inner 4004.2.e.a.3849.46 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4004.2.e.a.3849.3 48 1.1 even 1 trivial
4004.2.e.a.3849.46 yes 48 77.76 even 2 inner
4004.2.e.b.3849.3 yes 48 11.10 odd 2
4004.2.e.b.3849.46 yes 48 7.6 odd 2