Properties

Label 756.2.e.a.323.17
Level $756$
Weight $2$
Character 756.323
Analytic conductor $6.037$
Analytic rank $0$
Dimension $24$
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] = [756,2,Mod(323,756)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(756, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("756.323");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.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: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 323.17
Character \(\chi\) \(=\) 756.323
Dual form 756.2.e.a.323.18

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.575837 - 1.29167i) q^{2} +(-1.33682 - 1.48758i) q^{4} -0.311265i q^{5} +1.00000i q^{7} +(-2.69126 + 0.870128i) q^{8} +O(q^{10})\) \(q+(0.575837 - 1.29167i) q^{2} +(-1.33682 - 1.48758i) q^{4} -0.311265i q^{5} +1.00000i q^{7} +(-2.69126 + 0.870128i) q^{8} +(-0.402052 - 0.179238i) q^{10} -3.59281 q^{11} -6.44348 q^{13} +(1.29167 + 0.575837i) q^{14} +(-0.425810 + 3.97727i) q^{16} -2.96208i q^{17} +0.684831i q^{19} +(-0.463033 + 0.416106i) q^{20} +(-2.06887 + 4.64072i) q^{22} -7.58964 q^{23} +4.90311 q^{25} +(-3.71039 + 8.32285i) q^{26} +(1.48758 - 1.33682i) q^{28} -9.26841i q^{29} +1.91002i q^{31} +(4.89212 + 2.84027i) q^{32} +(-3.82603 - 1.70568i) q^{34} +0.311265 q^{35} -8.40625 q^{37} +(0.884575 + 0.394351i) q^{38} +(0.270841 + 0.837695i) q^{40} -7.50419i q^{41} +9.30454i q^{43} +(4.80295 + 5.34460i) q^{44} +(-4.37040 + 9.80330i) q^{46} +9.82305 q^{47} -1.00000 q^{49} +(2.82340 - 6.33321i) q^{50} +(8.61379 + 9.58521i) q^{52} +5.43690i q^{53} +1.11832i q^{55} +(-0.870128 - 2.69126i) q^{56} +(-11.9717 - 5.33710i) q^{58} -2.29004 q^{59} +0.486233 q^{61} +(2.46712 + 1.09986i) q^{62} +(6.48576 - 4.68348i) q^{64} +2.00563i q^{65} +1.09619i q^{67} +(-4.40635 + 3.95978i) q^{68} +(0.179238 - 0.402052i) q^{70} -1.76422 q^{71} +3.32691 q^{73} +(-4.84063 + 10.8581i) q^{74} +(1.01874 - 0.915497i) q^{76} -3.59281i q^{77} +1.39005i q^{79} +(1.23799 + 0.132540i) q^{80} +(-9.69294 - 4.32119i) q^{82} -0.283103 q^{83} -0.921993 q^{85} +(12.0184 + 5.35790i) q^{86} +(9.66918 - 3.12620i) q^{88} -8.91479i q^{89} -6.44348i q^{91} +(10.1460 + 11.2902i) q^{92} +(5.65648 - 12.6881i) q^{94} +0.213164 q^{95} +0.621922 q^{97} +(-0.575837 + 1.29167i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 8 q^{4} - 16 q^{10} + 8 q^{16} + 16 q^{22} - 24 q^{25} - 8 q^{28} - 8 q^{34} + 16 q^{37} - 8 q^{40} - 24 q^{49} - 8 q^{52} + 32 q^{58} - 80 q^{61} + 40 q^{64} - 24 q^{70} - 32 q^{82} + 56 q^{85} + 56 q^{88} + 72 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.575837 1.29167i 0.407178 0.913349i
\(3\) 0 0
\(4\) −1.33682 1.48758i −0.668411 0.743792i
\(5\) 0.311265i 0.139202i −0.997575 0.0696010i \(-0.977827\pi\)
0.997575 0.0696010i \(-0.0221726\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) −2.69126 + 0.870128i −0.951504 + 0.307637i
\(9\) 0 0
\(10\) −0.402052 0.179238i −0.127140 0.0566801i
\(11\) −3.59281 −1.08327 −0.541636 0.840613i \(-0.682195\pi\)
−0.541636 + 0.840613i \(0.682195\pi\)
\(12\) 0 0
\(13\) −6.44348 −1.78710 −0.893549 0.448965i \(-0.851793\pi\)
−0.893549 + 0.448965i \(0.851793\pi\)
\(14\) 1.29167 + 0.575837i 0.345213 + 0.153899i
\(15\) 0 0
\(16\) −0.425810 + 3.97727i −0.106452 + 0.994318i
\(17\) 2.96208i 0.718411i −0.933259 0.359205i \(-0.883048\pi\)
0.933259 0.359205i \(-0.116952\pi\)
\(18\) 0 0
\(19\) 0.684831i 0.157111i 0.996910 + 0.0785555i \(0.0250308\pi\)
−0.996910 + 0.0785555i \(0.974969\pi\)
\(20\) −0.463033 + 0.416106i −0.103537 + 0.0930442i
\(21\) 0 0
\(22\) −2.06887 + 4.64072i −0.441085 + 0.989406i
\(23\) −7.58964 −1.58255 −0.791274 0.611462i \(-0.790582\pi\)
−0.791274 + 0.611462i \(0.790582\pi\)
\(24\) 0 0
\(25\) 4.90311 0.980623
\(26\) −3.71039 + 8.32285i −0.727668 + 1.63224i
\(27\) 0 0
\(28\) 1.48758 1.33682i 0.281127 0.252636i
\(29\) 9.26841i 1.72110i −0.509365 0.860551i \(-0.670120\pi\)
0.509365 0.860551i \(-0.329880\pi\)
\(30\) 0 0
\(31\) 1.91002i 0.343050i 0.985180 + 0.171525i \(0.0548695\pi\)
−0.985180 + 0.171525i \(0.945130\pi\)
\(32\) 4.89212 + 2.84027i 0.864814 + 0.502093i
\(33\) 0 0
\(34\) −3.82603 1.70568i −0.656159 0.292521i
\(35\) 0.311265 0.0526134
\(36\) 0 0
\(37\) −8.40625 −1.38198 −0.690989 0.722865i \(-0.742825\pi\)
−0.690989 + 0.722865i \(0.742825\pi\)
\(38\) 0.884575 + 0.394351i 0.143497 + 0.0639722i
\(39\) 0 0
\(40\) 0.270841 + 0.837695i 0.0428236 + 0.132451i
\(41\) 7.50419i 1.17196i −0.810326 0.585979i \(-0.800710\pi\)
0.810326 0.585979i \(-0.199290\pi\)
\(42\) 0 0
\(43\) 9.30454i 1.41893i 0.704741 + 0.709464i \(0.251063\pi\)
−0.704741 + 0.709464i \(0.748937\pi\)
\(44\) 4.80295 + 5.34460i 0.724072 + 0.805729i
\(45\) 0 0
\(46\) −4.37040 + 9.80330i −0.644380 + 1.44542i
\(47\) 9.82305 1.43284 0.716419 0.697670i \(-0.245780\pi\)
0.716419 + 0.697670i \(0.245780\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 2.82340 6.33321i 0.399288 0.895650i
\(51\) 0 0
\(52\) 8.61379 + 9.58521i 1.19452 + 1.32923i
\(53\) 5.43690i 0.746816i 0.927667 + 0.373408i \(0.121811\pi\)
−0.927667 + 0.373408i \(0.878189\pi\)
\(54\) 0 0
\(55\) 1.11832i 0.150794i
\(56\) −0.870128 2.69126i −0.116276 0.359635i
\(57\) 0 0
\(58\) −11.9717 5.33710i −1.57197 0.700795i
\(59\) −2.29004 −0.298138 −0.149069 0.988827i \(-0.547628\pi\)
−0.149069 + 0.988827i \(0.547628\pi\)
\(60\) 0 0
\(61\) 0.486233 0.0622557 0.0311278 0.999515i \(-0.490090\pi\)
0.0311278 + 0.999515i \(0.490090\pi\)
\(62\) 2.46712 + 1.09986i 0.313325 + 0.139683i
\(63\) 0 0
\(64\) 6.48576 4.68348i 0.810719 0.585435i
\(65\) 2.00563i 0.248768i
\(66\) 0 0
\(67\) 1.09619i 0.133920i 0.997756 + 0.0669602i \(0.0213301\pi\)
−0.997756 + 0.0669602i \(0.978670\pi\)
\(68\) −4.40635 + 3.95978i −0.534348 + 0.480194i
\(69\) 0 0
\(70\) 0.179238 0.402052i 0.0214231 0.0480544i
\(71\) −1.76422 −0.209375 −0.104687 0.994505i \(-0.533384\pi\)
−0.104687 + 0.994505i \(0.533384\pi\)
\(72\) 0 0
\(73\) 3.32691 0.389385 0.194692 0.980864i \(-0.437629\pi\)
0.194692 + 0.980864i \(0.437629\pi\)
\(74\) −4.84063 + 10.8581i −0.562712 + 1.26223i
\(75\) 0 0
\(76\) 1.01874 0.915497i 0.116858 0.105015i
\(77\) 3.59281i 0.409439i
\(78\) 0 0
\(79\) 1.39005i 0.156392i 0.996938 + 0.0781962i \(0.0249161\pi\)
−0.996938 + 0.0781962i \(0.975084\pi\)
\(80\) 1.23799 + 0.132540i 0.138411 + 0.0148184i
\(81\) 0 0
\(82\) −9.69294 4.32119i −1.07041 0.477196i
\(83\) −0.283103 −0.0310746 −0.0155373 0.999879i \(-0.504946\pi\)
−0.0155373 + 0.999879i \(0.504946\pi\)
\(84\) 0 0
\(85\) −0.921993 −0.100004
\(86\) 12.0184 + 5.35790i 1.29598 + 0.577757i
\(87\) 0 0
\(88\) 9.66918 3.12620i 1.03074 0.333254i
\(89\) 8.91479i 0.944966i −0.881340 0.472483i \(-0.843358\pi\)
0.881340 0.472483i \(-0.156642\pi\)
\(90\) 0 0
\(91\) 6.44348i 0.675460i
\(92\) 10.1460 + 11.2902i 1.05779 + 1.17709i
\(93\) 0 0
\(94\) 5.65648 12.6881i 0.583421 1.30868i
\(95\) 0.213164 0.0218702
\(96\) 0 0
\(97\) 0.621922 0.0631466 0.0315733 0.999501i \(-0.489948\pi\)
0.0315733 + 0.999501i \(0.489948\pi\)
\(98\) −0.575837 + 1.29167i −0.0581684 + 0.130478i
\(99\) 0 0
\(100\) −6.55459 7.29379i −0.655459 0.729379i
\(101\) 17.7166i 1.76287i −0.472304 0.881436i \(-0.656578\pi\)
0.472304 0.881436i \(-0.343422\pi\)
\(102\) 0 0
\(103\) 15.5139i 1.52863i −0.644841 0.764317i \(-0.723077\pi\)
0.644841 0.764317i \(-0.276923\pi\)
\(104\) 17.3411 5.60665i 1.70043 0.549777i
\(105\) 0 0
\(106\) 7.02268 + 3.13077i 0.682103 + 0.304087i
\(107\) −4.26969 −0.412767 −0.206383 0.978471i \(-0.566169\pi\)
−0.206383 + 0.978471i \(0.566169\pi\)
\(108\) 0 0
\(109\) 2.03903 0.195304 0.0976519 0.995221i \(-0.468867\pi\)
0.0976519 + 0.995221i \(0.468867\pi\)
\(110\) 1.44450 + 0.643968i 0.137727 + 0.0614000i
\(111\) 0 0
\(112\) −3.97727 0.425810i −0.375817 0.0402352i
\(113\) 8.69512i 0.817968i 0.912541 + 0.408984i \(0.134117\pi\)
−0.912541 + 0.408984i \(0.865883\pi\)
\(114\) 0 0
\(115\) 2.36239i 0.220294i
\(116\) −13.7875 + 12.3902i −1.28014 + 1.15040i
\(117\) 0 0
\(118\) −1.31869 + 2.95798i −0.121395 + 0.272304i
\(119\) 2.96208 0.271534
\(120\) 0 0
\(121\) 1.90828 0.173480
\(122\) 0.279991 0.628052i 0.0253492 0.0568612i
\(123\) 0 0
\(124\) 2.84132 2.55336i 0.255158 0.229299i
\(125\) 3.08250i 0.275707i
\(126\) 0 0
\(127\) 1.66793i 0.148005i −0.997258 0.0740024i \(-0.976423\pi\)
0.997258 0.0740024i \(-0.0235772\pi\)
\(128\) −2.31477 11.0744i −0.204599 0.978846i
\(129\) 0 0
\(130\) 2.59061 + 1.15492i 0.227212 + 0.101293i
\(131\) −19.0156 −1.66140 −0.830700 0.556720i \(-0.812060\pi\)
−0.830700 + 0.556720i \(0.812060\pi\)
\(132\) 0 0
\(133\) −0.684831 −0.0593823
\(134\) 1.41591 + 0.631225i 0.122316 + 0.0545295i
\(135\) 0 0
\(136\) 2.57739 + 7.97173i 0.221009 + 0.683570i
\(137\) 3.52975i 0.301567i −0.988567 0.150784i \(-0.951820\pi\)
0.988567 0.150784i \(-0.0481796\pi\)
\(138\) 0 0
\(139\) 13.7977i 1.17030i −0.810924 0.585152i \(-0.801035\pi\)
0.810924 0.585152i \(-0.198965\pi\)
\(140\) −0.416106 0.463033i −0.0351674 0.0391334i
\(141\) 0 0
\(142\) −1.01590 + 2.27879i −0.0852528 + 0.191232i
\(143\) 23.1502 1.93592
\(144\) 0 0
\(145\) −2.88493 −0.239581
\(146\) 1.91576 4.29726i 0.158549 0.355644i
\(147\) 0 0
\(148\) 11.2377 + 12.5050i 0.923730 + 1.02790i
\(149\) 10.8370i 0.887803i 0.896076 + 0.443901i \(0.146406\pi\)
−0.896076 + 0.443901i \(0.853594\pi\)
\(150\) 0 0
\(151\) 0.421989i 0.0343410i −0.999853 0.0171705i \(-0.994534\pi\)
0.999853 0.0171705i \(-0.00546580\pi\)
\(152\) −0.595890 1.84306i −0.0483331 0.149492i
\(153\) 0 0
\(154\) −4.64072 2.06887i −0.373960 0.166715i
\(155\) 0.594524 0.0477533
\(156\) 0 0
\(157\) 15.0436 1.20061 0.600306 0.799771i \(-0.295046\pi\)
0.600306 + 0.799771i \(0.295046\pi\)
\(158\) 1.79548 + 0.800441i 0.142841 + 0.0636797i
\(159\) 0 0
\(160\) 0.884076 1.52275i 0.0698924 0.120384i
\(161\) 7.58964i 0.598147i
\(162\) 0 0
\(163\) 12.6329i 0.989489i −0.869039 0.494744i \(-0.835262\pi\)
0.869039 0.494744i \(-0.164738\pi\)
\(164\) −11.1631 + 10.0318i −0.871693 + 0.783350i
\(165\) 0 0
\(166\) −0.163021 + 0.365675i −0.0126529 + 0.0283819i
\(167\) −22.1403 −1.71327 −0.856634 0.515925i \(-0.827448\pi\)
−0.856634 + 0.515925i \(0.827448\pi\)
\(168\) 0 0
\(169\) 28.5184 2.19372
\(170\) −0.530918 + 1.19091i −0.0407196 + 0.0913387i
\(171\) 0 0
\(172\) 13.8413 12.4385i 1.05539 0.948428i
\(173\) 18.6961i 1.42144i 0.703478 + 0.710718i \(0.251630\pi\)
−0.703478 + 0.710718i \(0.748370\pi\)
\(174\) 0 0
\(175\) 4.90311i 0.370641i
\(176\) 1.52985 14.2896i 0.115317 1.07712i
\(177\) 0 0
\(178\) −11.5150 5.13347i −0.863083 0.384770i
\(179\) 16.3149 1.21943 0.609716 0.792620i \(-0.291284\pi\)
0.609716 + 0.792620i \(0.291284\pi\)
\(180\) 0 0
\(181\) −7.35019 −0.546335 −0.273168 0.961966i \(-0.588071\pi\)
−0.273168 + 0.961966i \(0.588071\pi\)
\(182\) −8.32285 3.71039i −0.616930 0.275033i
\(183\) 0 0
\(184\) 20.4257 6.60395i 1.50580 0.486850i
\(185\) 2.61657i 0.192374i
\(186\) 0 0
\(187\) 10.6422i 0.778235i
\(188\) −13.1317 14.6126i −0.957726 1.06573i
\(189\) 0 0
\(190\) 0.122748 0.275337i 0.00890506 0.0199751i
\(191\) −20.6900 −1.49708 −0.748538 0.663092i \(-0.769244\pi\)
−0.748538 + 0.663092i \(0.769244\pi\)
\(192\) 0 0
\(193\) −2.76121 −0.198757 −0.0993783 0.995050i \(-0.531685\pi\)
−0.0993783 + 0.995050i \(0.531685\pi\)
\(194\) 0.358126 0.803318i 0.0257119 0.0576749i
\(195\) 0 0
\(196\) 1.33682 + 1.48758i 0.0954873 + 0.106256i
\(197\) 8.02466i 0.571733i −0.958269 0.285867i \(-0.907719\pi\)
0.958269 0.285867i \(-0.0922815\pi\)
\(198\) 0 0
\(199\) 16.2052i 1.14876i 0.818590 + 0.574378i \(0.194756\pi\)
−0.818590 + 0.574378i \(0.805244\pi\)
\(200\) −13.1956 + 4.26634i −0.933066 + 0.301676i
\(201\) 0 0
\(202\) −22.8841 10.2019i −1.61012 0.717803i
\(203\) 9.26841 0.650515
\(204\) 0 0
\(205\) −2.33579 −0.163139
\(206\) −20.0389 8.93350i −1.39617 0.622426i
\(207\) 0 0
\(208\) 2.74369 25.6275i 0.190241 1.77694i
\(209\) 2.46047i 0.170194i
\(210\) 0 0
\(211\) 20.0235i 1.37848i 0.724535 + 0.689238i \(0.242054\pi\)
−0.724535 + 0.689238i \(0.757946\pi\)
\(212\) 8.08784 7.26817i 0.555475 0.499180i
\(213\) 0 0
\(214\) −2.45865 + 5.51503i −0.168070 + 0.377000i
\(215\) 2.89618 0.197518
\(216\) 0 0
\(217\) −1.91002 −0.129661
\(218\) 1.17415 2.63376i 0.0795235 0.178381i
\(219\) 0 0
\(220\) 1.66359 1.49499i 0.112159 0.100792i
\(221\) 19.0861i 1.28387i
\(222\) 0 0
\(223\) 23.9084i 1.60103i −0.599314 0.800514i \(-0.704560\pi\)
0.599314 0.800514i \(-0.295440\pi\)
\(224\) −2.84027 + 4.89212i −0.189773 + 0.326869i
\(225\) 0 0
\(226\) 11.2312 + 5.00698i 0.747090 + 0.333059i
\(227\) 11.0598 0.734067 0.367034 0.930208i \(-0.380373\pi\)
0.367034 + 0.930208i \(0.380373\pi\)
\(228\) 0 0
\(229\) −23.0407 −1.52257 −0.761287 0.648416i \(-0.775432\pi\)
−0.761287 + 0.648416i \(0.775432\pi\)
\(230\) 3.05143 + 1.36035i 0.201205 + 0.0896990i
\(231\) 0 0
\(232\) 8.06471 + 24.9437i 0.529474 + 1.63763i
\(233\) 5.39550i 0.353471i −0.984258 0.176736i \(-0.943446\pi\)
0.984258 0.176736i \(-0.0565538\pi\)
\(234\) 0 0
\(235\) 3.05757i 0.199454i
\(236\) 3.06138 + 3.40663i 0.199279 + 0.221753i
\(237\) 0 0
\(238\) 1.70568 3.82603i 0.110563 0.248005i
\(239\) −21.5858 −1.39627 −0.698135 0.715966i \(-0.745987\pi\)
−0.698135 + 0.715966i \(0.745987\pi\)
\(240\) 0 0
\(241\) 9.30744 0.599545 0.299772 0.954011i \(-0.403089\pi\)
0.299772 + 0.954011i \(0.403089\pi\)
\(242\) 1.09886 2.46487i 0.0706373 0.158448i
\(243\) 0 0
\(244\) −0.650007 0.723311i −0.0416124 0.0463053i
\(245\) 0.311265i 0.0198860i
\(246\) 0 0
\(247\) 4.41269i 0.280773i
\(248\) −1.66197 5.14037i −0.105535 0.326414i
\(249\) 0 0
\(250\) −3.98157 1.77502i −0.251816 0.112262i
\(251\) 7.57302 0.478005 0.239002 0.971019i \(-0.423180\pi\)
0.239002 + 0.971019i \(0.423180\pi\)
\(252\) 0 0
\(253\) 27.2681 1.71433
\(254\) −2.15441 0.960456i −0.135180 0.0602644i
\(255\) 0 0
\(256\) −15.6374 3.38712i −0.977336 0.211695i
\(257\) 2.44066i 0.152244i −0.997099 0.0761220i \(-0.975746\pi\)
0.997099 0.0761220i \(-0.0242538\pi\)
\(258\) 0 0
\(259\) 8.40625i 0.522339i
\(260\) 2.98354 2.68117i 0.185031 0.166279i
\(261\) 0 0
\(262\) −10.9499 + 24.5619i −0.676487 + 1.51744i
\(263\) −1.25537 −0.0774095 −0.0387048 0.999251i \(-0.512323\pi\)
−0.0387048 + 0.999251i \(0.512323\pi\)
\(264\) 0 0
\(265\) 1.69232 0.103958
\(266\) −0.394351 + 0.884575i −0.0241792 + 0.0542368i
\(267\) 0 0
\(268\) 1.63067 1.46541i 0.0996089 0.0895139i
\(269\) 4.24461i 0.258799i −0.991593 0.129399i \(-0.958695\pi\)
0.991593 0.129399i \(-0.0413049\pi\)
\(270\) 0 0
\(271\) 14.5146i 0.881701i −0.897581 0.440851i \(-0.854677\pi\)
0.897581 0.440851i \(-0.145323\pi\)
\(272\) 11.7810 + 1.26128i 0.714328 + 0.0764765i
\(273\) 0 0
\(274\) −4.55927 2.03256i −0.275436 0.122792i
\(275\) −17.6160 −1.06228
\(276\) 0 0
\(277\) −18.1760 −1.09209 −0.546046 0.837755i \(-0.683868\pi\)
−0.546046 + 0.837755i \(0.683868\pi\)
\(278\) −17.8220 7.94522i −1.06890 0.476523i
\(279\) 0 0
\(280\) −0.837695 + 0.270841i −0.0500619 + 0.0161858i
\(281\) 23.7048i 1.41411i 0.707158 + 0.707055i \(0.249977\pi\)
−0.707158 + 0.707055i \(0.750023\pi\)
\(282\) 0 0
\(283\) 30.4540i 1.81031i −0.425087 0.905153i \(-0.639756\pi\)
0.425087 0.905153i \(-0.360244\pi\)
\(284\) 2.35845 + 2.62443i 0.139948 + 0.155731i
\(285\) 0 0
\(286\) 13.3307 29.9024i 0.788263 1.76817i
\(287\) 7.50419 0.442958
\(288\) 0 0
\(289\) 8.22607 0.483886
\(290\) −1.66125 + 3.72638i −0.0975521 + 0.218821i
\(291\) 0 0
\(292\) −4.44748 4.94905i −0.260269 0.289621i
\(293\) 14.8848i 0.869582i 0.900531 + 0.434791i \(0.143178\pi\)
−0.900531 + 0.434791i \(0.856822\pi\)
\(294\) 0 0
\(295\) 0.712810i 0.0415014i
\(296\) 22.6234 7.31451i 1.31496 0.425147i
\(297\) 0 0
\(298\) 13.9978 + 6.24036i 0.810874 + 0.361494i
\(299\) 48.9036 2.82817
\(300\) 0 0
\(301\) −9.30454 −0.536305
\(302\) −0.545070 0.242997i −0.0313653 0.0139829i
\(303\) 0 0
\(304\) −2.72376 0.291607i −0.156218 0.0167248i
\(305\) 0.151347i 0.00866612i
\(306\) 0 0
\(307\) 18.7491i 1.07007i 0.844831 + 0.535033i \(0.179701\pi\)
−0.844831 + 0.535033i \(0.820299\pi\)
\(308\) −5.34460 + 4.80295i −0.304537 + 0.273673i
\(309\) 0 0
\(310\) 0.342349 0.767929i 0.0194441 0.0436154i
\(311\) 19.1097 1.08361 0.541806 0.840504i \(-0.317741\pi\)
0.541806 + 0.840504i \(0.317741\pi\)
\(312\) 0 0
\(313\) −17.8663 −1.00986 −0.504932 0.863159i \(-0.668483\pi\)
−0.504932 + 0.863159i \(0.668483\pi\)
\(314\) 8.66268 19.4314i 0.488863 1.09658i
\(315\) 0 0
\(316\) 2.06781 1.85825i 0.116323 0.104535i
\(317\) 30.6470i 1.72131i 0.509191 + 0.860653i \(0.329945\pi\)
−0.509191 + 0.860653i \(0.670055\pi\)
\(318\) 0 0
\(319\) 33.2996i 1.86442i
\(320\) −1.45780 2.01879i −0.0814937 0.112854i
\(321\) 0 0
\(322\) −9.80330 4.37040i −0.546317 0.243553i
\(323\) 2.02852 0.112870
\(324\) 0 0
\(325\) −31.5931 −1.75247
\(326\) −16.3176 7.27452i −0.903748 0.402899i
\(327\) 0 0
\(328\) 6.52961 + 20.1957i 0.360537 + 1.11512i
\(329\) 9.82305i 0.541562i
\(330\) 0 0
\(331\) 11.0261i 0.606051i 0.952982 + 0.303026i \(0.0979968\pi\)
−0.952982 + 0.303026i \(0.902003\pi\)
\(332\) 0.378458 + 0.421139i 0.0207706 + 0.0231130i
\(333\) 0 0
\(334\) −12.7492 + 28.5980i −0.697606 + 1.56481i
\(335\) 0.341204 0.0186420
\(336\) 0 0
\(337\) 2.68990 0.146528 0.0732640 0.997313i \(-0.476658\pi\)
0.0732640 + 0.997313i \(0.476658\pi\)
\(338\) 16.4220 36.8364i 0.893237 2.00363i
\(339\) 0 0
\(340\) 1.23254 + 1.37154i 0.0668440 + 0.0743823i
\(341\) 6.86235i 0.371617i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) −8.09614 25.0409i −0.436514 1.35012i
\(345\) 0 0
\(346\) 24.1491 + 10.7659i 1.29827 + 0.578778i
\(347\) 10.4414 0.560525 0.280262 0.959923i \(-0.409579\pi\)
0.280262 + 0.959923i \(0.409579\pi\)
\(348\) 0 0
\(349\) −2.20342 −0.117947 −0.0589733 0.998260i \(-0.518783\pi\)
−0.0589733 + 0.998260i \(0.518783\pi\)
\(350\) 6.33321 + 2.82340i 0.338524 + 0.150917i
\(351\) 0 0
\(352\) −17.5765 10.2045i −0.936829 0.543904i
\(353\) 12.1792i 0.648234i 0.946017 + 0.324117i \(0.105067\pi\)
−0.946017 + 0.324117i \(0.894933\pi\)
\(354\) 0 0
\(355\) 0.549141i 0.0291454i
\(356\) −13.2615 + 11.9175i −0.702858 + 0.631626i
\(357\) 0 0
\(358\) 9.39472 21.0734i 0.496526 1.11377i
\(359\) −13.7170 −0.723953 −0.361976 0.932187i \(-0.617898\pi\)
−0.361976 + 0.932187i \(0.617898\pi\)
\(360\) 0 0
\(361\) 18.5310 0.975316
\(362\) −4.23251 + 9.49402i −0.222456 + 0.498995i
\(363\) 0 0
\(364\) −9.58521 + 8.61379i −0.502402 + 0.451485i
\(365\) 1.03555i 0.0542032i
\(366\) 0 0
\(367\) 24.1421i 1.26021i 0.776512 + 0.630103i \(0.216987\pi\)
−0.776512 + 0.630103i \(0.783013\pi\)
\(368\) 3.23174 30.1860i 0.168466 1.57356i
\(369\) 0 0
\(370\) 3.37975 + 1.50672i 0.175705 + 0.0783306i
\(371\) −5.43690 −0.282270
\(372\) 0 0
\(373\) 4.44430 0.230117 0.115059 0.993359i \(-0.463294\pi\)
0.115059 + 0.993359i \(0.463294\pi\)
\(374\) 13.7462 + 6.12817i 0.710799 + 0.316880i
\(375\) 0 0
\(376\) −26.4364 + 8.54731i −1.36335 + 0.440794i
\(377\) 59.7208i 3.07578i
\(378\) 0 0
\(379\) 9.03014i 0.463847i 0.972734 + 0.231924i \(0.0745020\pi\)
−0.972734 + 0.231924i \(0.925498\pi\)
\(380\) −0.284962 0.317099i −0.0146183 0.0162668i
\(381\) 0 0
\(382\) −11.9141 + 26.7246i −0.609577 + 1.36735i
\(383\) 20.1601 1.03013 0.515067 0.857150i \(-0.327767\pi\)
0.515067 + 0.857150i \(0.327767\pi\)
\(384\) 0 0
\(385\) −1.11832 −0.0569947
\(386\) −1.59001 + 3.56658i −0.0809294 + 0.181534i
\(387\) 0 0
\(388\) −0.831400 0.925161i −0.0422079 0.0469679i
\(389\) 20.1791i 1.02312i −0.859247 0.511562i \(-0.829067\pi\)
0.859247 0.511562i \(-0.170933\pi\)
\(390\) 0 0
\(391\) 22.4811i 1.13692i
\(392\) 2.69126 0.870128i 0.135929 0.0439481i
\(393\) 0 0
\(394\) −10.3652 4.62090i −0.522192 0.232798i
\(395\) 0.432673 0.0217702
\(396\) 0 0
\(397\) −19.8932 −0.998412 −0.499206 0.866483i \(-0.666375\pi\)
−0.499206 + 0.866483i \(0.666375\pi\)
\(398\) 20.9318 + 9.33156i 1.04922 + 0.467749i
\(399\) 0 0
\(400\) −2.08779 + 19.5010i −0.104390 + 0.975051i
\(401\) 35.2123i 1.75842i 0.476436 + 0.879209i \(0.341928\pi\)
−0.476436 + 0.879209i \(0.658072\pi\)
\(402\) 0 0
\(403\) 12.3072i 0.613065i
\(404\) −26.3550 + 23.6840i −1.31121 + 1.17832i
\(405\) 0 0
\(406\) 5.33710 11.9717i 0.264876 0.594147i
\(407\) 30.2020 1.49706
\(408\) 0 0
\(409\) 18.4098 0.910306 0.455153 0.890413i \(-0.349585\pi\)
0.455153 + 0.890413i \(0.349585\pi\)
\(410\) −1.34504 + 3.01708i −0.0664267 + 0.149003i
\(411\) 0 0
\(412\) −23.0783 + 20.7394i −1.13698 + 1.02176i
\(413\) 2.29004i 0.112686i
\(414\) 0 0
\(415\) 0.0881200i 0.00432564i
\(416\) −31.5223 18.3012i −1.54551 0.897290i
\(417\) 0 0
\(418\) −3.17811 1.41683i −0.155446 0.0692993i
\(419\) −18.5848 −0.907925 −0.453962 0.891021i \(-0.649990\pi\)
−0.453962 + 0.891021i \(0.649990\pi\)
\(420\) 0 0
\(421\) −35.4051 −1.72554 −0.862770 0.505597i \(-0.831272\pi\)
−0.862770 + 0.505597i \(0.831272\pi\)
\(422\) 25.8638 + 11.5303i 1.25903 + 0.561285i
\(423\) 0 0
\(424\) −4.73080 14.6321i −0.229748 0.710598i
\(425\) 14.5234i 0.704490i
\(426\) 0 0
\(427\) 0.486233i 0.0235304i
\(428\) 5.70782 + 6.35152i 0.275898 + 0.307012i
\(429\) 0 0
\(430\) 1.66773 3.74091i 0.0804250 0.180403i
\(431\) 23.0972 1.11255 0.556276 0.830998i \(-0.312230\pi\)
0.556276 + 0.830998i \(0.312230\pi\)
\(432\) 0 0
\(433\) 18.0020 0.865122 0.432561 0.901605i \(-0.357610\pi\)
0.432561 + 0.901605i \(0.357610\pi\)
\(434\) −1.09986 + 2.46712i −0.0527951 + 0.118426i
\(435\) 0 0
\(436\) −2.72582 3.03323i −0.130543 0.145265i
\(437\) 5.19761i 0.248636i
\(438\) 0 0
\(439\) 24.0230i 1.14655i −0.819361 0.573277i \(-0.805672\pi\)
0.819361 0.573277i \(-0.194328\pi\)
\(440\) −0.973078 3.00968i −0.0463897 0.143481i
\(441\) 0 0
\(442\) 24.6530 + 10.9905i 1.17262 + 0.522764i
\(443\) −34.6101 −1.64437 −0.822187 0.569217i \(-0.807246\pi\)
−0.822187 + 0.569217i \(0.807246\pi\)
\(444\) 0 0
\(445\) −2.77486 −0.131541
\(446\) −30.8818 13.7674i −1.46230 0.651904i
\(447\) 0 0
\(448\) 4.68348 + 6.48576i 0.221274 + 0.306423i
\(449\) 19.1474i 0.903620i −0.892114 0.451810i \(-0.850779\pi\)
0.892114 0.451810i \(-0.149221\pi\)
\(450\) 0 0
\(451\) 26.9611i 1.26955i
\(452\) 12.9347 11.6238i 0.608398 0.546739i
\(453\) 0 0
\(454\) 6.36867 14.2857i 0.298896 0.670459i
\(455\) −2.00563 −0.0940254
\(456\) 0 0
\(457\) −14.2205 −0.665208 −0.332604 0.943067i \(-0.607927\pi\)
−0.332604 + 0.943067i \(0.607927\pi\)
\(458\) −13.2677 + 29.7610i −0.619959 + 1.39064i
\(459\) 0 0
\(460\) 3.51425 3.15810i 0.163853 0.147247i
\(461\) 28.9424i 1.34798i −0.738740 0.673991i \(-0.764579\pi\)
0.738740 0.673991i \(-0.235421\pi\)
\(462\) 0 0
\(463\) 30.7876i 1.43082i 0.698704 + 0.715411i \(0.253761\pi\)
−0.698704 + 0.715411i \(0.746239\pi\)
\(464\) 36.8630 + 3.94658i 1.71132 + 0.183215i
\(465\) 0 0
\(466\) −6.96921 3.10693i −0.322842 0.143926i
\(467\) 19.6915 0.911215 0.455608 0.890181i \(-0.349422\pi\)
0.455608 + 0.890181i \(0.349422\pi\)
\(468\) 0 0
\(469\) −1.09619 −0.0506172
\(470\) −3.94937 1.76066i −0.182171 0.0812134i
\(471\) 0 0
\(472\) 6.16309 1.99263i 0.283679 0.0917182i
\(473\) 33.4294i 1.53709i
\(474\) 0 0
\(475\) 3.35780i 0.154067i
\(476\) −3.95978 4.40635i −0.181496 0.201964i
\(477\) 0 0
\(478\) −12.4299 + 27.8818i −0.568531 + 1.27528i
\(479\) −12.4606 −0.569341 −0.284671 0.958625i \(-0.591884\pi\)
−0.284671 + 0.958625i \(0.591884\pi\)
\(480\) 0 0
\(481\) 54.1655 2.46973
\(482\) 5.35957 12.0221i 0.244122 0.547593i
\(483\) 0 0
\(484\) −2.55103 2.83872i −0.115956 0.129033i
\(485\) 0.193583i 0.00879014i
\(486\) 0 0
\(487\) 2.92729i 0.132648i 0.997798 + 0.0663240i \(0.0211271\pi\)
−0.997798 + 0.0663240i \(0.978873\pi\)
\(488\) −1.30858 + 0.423084i −0.0592365 + 0.0191521i
\(489\) 0 0
\(490\) 0.402052 + 0.179238i 0.0181629 + 0.00809715i
\(491\) −16.2661 −0.734081 −0.367040 0.930205i \(-0.619629\pi\)
−0.367040 + 0.930205i \(0.619629\pi\)
\(492\) 0 0
\(493\) −27.4538 −1.23646
\(494\) −5.69974 2.54099i −0.256443 0.114325i
\(495\) 0 0
\(496\) −7.59668 0.813307i −0.341101 0.0365185i
\(497\) 1.76422i 0.0791362i
\(498\) 0 0
\(499\) 25.8585i 1.15759i −0.815475 0.578793i \(-0.803524\pi\)
0.815475 0.578793i \(-0.196476\pi\)
\(500\) −4.58547 + 4.12075i −0.205068 + 0.184286i
\(501\) 0 0
\(502\) 4.36083 9.78184i 0.194633 0.436585i
\(503\) −1.22564 −0.0546487 −0.0273244 0.999627i \(-0.508699\pi\)
−0.0273244 + 0.999627i \(0.508699\pi\)
\(504\) 0 0
\(505\) −5.51457 −0.245395
\(506\) 15.7020 35.2214i 0.698039 1.56578i
\(507\) 0 0
\(508\) −2.48118 + 2.22973i −0.110085 + 0.0989281i
\(509\) 10.1313i 0.449062i −0.974467 0.224531i \(-0.927915\pi\)
0.974467 0.224531i \(-0.0720851\pi\)
\(510\) 0 0
\(511\) 3.32691i 0.147174i
\(512\) −13.3796 + 18.2479i −0.591301 + 0.806451i
\(513\) 0 0
\(514\) −3.15252 1.40542i −0.139052 0.0619905i
\(515\) −4.82895 −0.212789
\(516\) 0 0
\(517\) −35.2923 −1.55215
\(518\) −10.8581 4.84063i −0.477077 0.212685i
\(519\) 0 0
\(520\) −1.74515 5.39767i −0.0765301 0.236704i
\(521\) 18.7198i 0.820128i 0.912057 + 0.410064i \(0.134494\pi\)
−0.912057 + 0.410064i \(0.865506\pi\)
\(522\) 0 0
\(523\) 17.2180i 0.752893i 0.926438 + 0.376446i \(0.122854\pi\)
−0.926438 + 0.376446i \(0.877146\pi\)
\(524\) 25.4205 + 28.2873i 1.11050 + 1.23574i
\(525\) 0 0
\(526\) −0.722890 + 1.62153i −0.0315195 + 0.0707019i
\(527\) 5.65765 0.246451
\(528\) 0 0
\(529\) 34.6026 1.50446
\(530\) 0.974500 2.18592i 0.0423296 0.0949502i
\(531\) 0 0
\(532\) 0.915497 + 1.01874i 0.0396918 + 0.0441681i
\(533\) 48.3531i 2.09440i
\(534\) 0 0
\(535\) 1.32901i 0.0574580i
\(536\) −0.953822 2.95012i −0.0411988 0.127426i
\(537\) 0 0
\(538\) −5.48264 2.44421i −0.236373 0.105377i
\(539\) 3.59281 0.154753
\(540\) 0 0
\(541\) −17.0983 −0.735114 −0.367557 0.930001i \(-0.619806\pi\)
−0.367557 + 0.930001i \(0.619806\pi\)
\(542\) −18.7481 8.35807i −0.805300 0.359010i
\(543\) 0 0
\(544\) 8.41310 14.4909i 0.360709 0.621291i
\(545\) 0.634680i 0.0271867i
\(546\) 0 0
\(547\) 15.6670i 0.669875i −0.942240 0.334937i \(-0.891285\pi\)
0.942240 0.334937i \(-0.108715\pi\)
\(548\) −5.25080 + 4.71865i −0.224303 + 0.201571i
\(549\) 0 0
\(550\) −10.1439 + 22.7540i −0.432538 + 0.970234i
\(551\) 6.34729 0.270404
\(552\) 0 0
\(553\) −1.39005 −0.0591108
\(554\) −10.4664 + 23.4774i −0.444677 + 0.997461i
\(555\) 0 0
\(556\) −20.5252 + 18.4451i −0.870462 + 0.782244i
\(557\) 13.2002i 0.559310i 0.960101 + 0.279655i \(0.0902200\pi\)
−0.960101 + 0.279655i \(0.909780\pi\)
\(558\) 0 0
\(559\) 59.9536i 2.53577i
\(560\) −0.132540 + 1.23799i −0.00560082 + 0.0523145i
\(561\) 0 0
\(562\) 30.6188 + 13.6501i 1.29158 + 0.575795i
\(563\) 21.8132 0.919316 0.459658 0.888096i \(-0.347972\pi\)
0.459658 + 0.888096i \(0.347972\pi\)
\(564\) 0 0
\(565\) 2.70649 0.113863
\(566\) −39.3366 17.5366i −1.65344 0.737117i
\(567\) 0 0
\(568\) 4.74798 1.53510i 0.199221 0.0644113i
\(569\) 20.1097i 0.843044i −0.906818 0.421522i \(-0.861496\pi\)
0.906818 0.421522i \(-0.138504\pi\)
\(570\) 0 0
\(571\) 35.0133i 1.46526i −0.680627 0.732630i \(-0.738293\pi\)
0.680627 0.732630i \(-0.261707\pi\)
\(572\) −30.9477 34.4378i −1.29399 1.43992i
\(573\) 0 0
\(574\) 4.32119 9.69294i 0.180363 0.404575i
\(575\) −37.2128 −1.55188
\(576\) 0 0
\(577\) 17.8963 0.745031 0.372515 0.928026i \(-0.378495\pi\)
0.372515 + 0.928026i \(0.378495\pi\)
\(578\) 4.73688 10.6254i 0.197028 0.441957i
\(579\) 0 0
\(580\) 3.85665 + 4.29158i 0.160139 + 0.178198i
\(581\) 0.283103i 0.0117451i
\(582\) 0 0
\(583\) 19.5337i 0.809005i
\(584\) −8.95357 + 2.89483i −0.370501 + 0.119789i
\(585\) 0 0
\(586\) 19.2263 + 8.57125i 0.794232 + 0.354075i
\(587\) −39.4490 −1.62823 −0.814117 0.580701i \(-0.802779\pi\)
−0.814117 + 0.580701i \(0.802779\pi\)
\(588\) 0 0
\(589\) −1.30804 −0.0538970
\(590\) 0.920715 + 0.410463i 0.0379053 + 0.0168985i
\(591\) 0 0
\(592\) 3.57946 33.4339i 0.147115 1.37413i
\(593\) 30.1072i 1.23635i −0.786039 0.618177i \(-0.787872\pi\)
0.786039 0.618177i \(-0.212128\pi\)
\(594\) 0 0
\(595\) 0.921993i 0.0377980i
\(596\) 16.1210 14.4872i 0.660341 0.593418i
\(597\) 0 0
\(598\) 28.1605 63.1674i 1.15157 2.58311i
\(599\) −0.825158 −0.0337151 −0.0168575 0.999858i \(-0.505366\pi\)
−0.0168575 + 0.999858i \(0.505366\pi\)
\(600\) 0 0
\(601\) −24.3800 −0.994481 −0.497240 0.867613i \(-0.665653\pi\)
−0.497240 + 0.867613i \(0.665653\pi\)
\(602\) −5.35790 + 12.0184i −0.218372 + 0.489833i
\(603\) 0 0
\(604\) −0.627744 + 0.564124i −0.0255425 + 0.0229539i
\(605\) 0.593981i 0.0241488i
\(606\) 0 0
\(607\) 26.8238i 1.08874i 0.838844 + 0.544372i \(0.183232\pi\)
−0.838844 + 0.544372i \(0.816768\pi\)
\(608\) −1.94510 + 3.35028i −0.0788843 + 0.135872i
\(609\) 0 0
\(610\) −0.195491 0.0871514i −0.00791519 0.00352866i
\(611\) −63.2946 −2.56062
\(612\) 0 0
\(613\) −1.30874 −0.0528594 −0.0264297 0.999651i \(-0.508414\pi\)
−0.0264297 + 0.999651i \(0.508414\pi\)
\(614\) 24.2176 + 10.7964i 0.977343 + 0.435708i
\(615\) 0 0
\(616\) 3.12620 + 9.66918i 0.125958 + 0.389582i
\(617\) 3.15629i 0.127067i 0.997980 + 0.0635337i \(0.0202370\pi\)
−0.997980 + 0.0635337i \(0.979763\pi\)
\(618\) 0 0
\(619\) 18.9949i 0.763471i −0.924272 0.381735i \(-0.875327\pi\)
0.924272 0.381735i \(-0.124673\pi\)
\(620\) −0.794773 0.884404i −0.0319189 0.0355185i
\(621\) 0 0
\(622\) 11.0041 24.6834i 0.441223 0.989715i
\(623\) 8.91479 0.357164
\(624\) 0 0
\(625\) 23.5561 0.942244
\(626\) −10.2881 + 23.0774i −0.411195 + 0.922358i
\(627\) 0 0
\(628\) −20.1107 22.3786i −0.802502 0.893005i
\(629\) 24.9000i 0.992828i
\(630\) 0 0
\(631\) 37.6706i 1.49964i −0.661639 0.749822i \(-0.730139\pi\)
0.661639 0.749822i \(-0.269861\pi\)
\(632\) −1.20952 3.74098i −0.0481121 0.148808i
\(633\) 0 0
\(634\) 39.5858 + 17.6477i 1.57215 + 0.700879i
\(635\) −0.519168 −0.0206026
\(636\) 0 0
\(637\) 6.44348 0.255300
\(638\) 43.0122 + 19.1752i 1.70287 + 0.759153i
\(639\) 0 0
\(640\) −3.44707 + 0.720508i −0.136257 + 0.0284806i
\(641\) 33.9077i 1.33927i −0.742689 0.669637i \(-0.766450\pi\)
0.742689 0.669637i \(-0.233550\pi\)
\(642\) 0 0
\(643\) 34.4425i 1.35828i 0.734008 + 0.679140i \(0.237647\pi\)
−0.734008 + 0.679140i \(0.762353\pi\)
\(644\) −11.2902 + 10.1460i −0.444897 + 0.399808i
\(645\) 0 0
\(646\) 1.16810 2.62018i 0.0459583 0.103090i
\(647\) 10.9612 0.430931 0.215465 0.976512i \(-0.430873\pi\)
0.215465 + 0.976512i \(0.430873\pi\)
\(648\) 0 0
\(649\) 8.22768 0.322965
\(650\) −18.1925 + 40.8079i −0.713568 + 1.60062i
\(651\) 0 0
\(652\) −18.7926 + 16.8880i −0.735974 + 0.661386i
\(653\) 19.8426i 0.776501i −0.921554 0.388250i \(-0.873080\pi\)
0.921554 0.388250i \(-0.126920\pi\)
\(654\) 0 0
\(655\) 5.91890i 0.231270i
\(656\) 29.8462 + 3.19536i 1.16530 + 0.124758i
\(657\) 0 0
\(658\) 12.6881 + 5.65648i 0.494635 + 0.220512i
\(659\) −0.592911 −0.0230965 −0.0115483 0.999933i \(-0.503676\pi\)
−0.0115483 + 0.999933i \(0.503676\pi\)
\(660\) 0 0
\(661\) 32.1832 1.25178 0.625890 0.779911i \(-0.284736\pi\)
0.625890 + 0.779911i \(0.284736\pi\)
\(662\) 14.2421 + 6.34926i 0.553536 + 0.246771i
\(663\) 0 0
\(664\) 0.761903 0.246336i 0.0295676 0.00955967i
\(665\) 0.213164i 0.00826614i
\(666\) 0 0
\(667\) 70.3439i 2.72373i
\(668\) 29.5976 + 32.9355i 1.14517 + 1.27431i
\(669\) 0 0
\(670\) 0.196478 0.440724i 0.00759062 0.0170266i
\(671\) −1.74694 −0.0674399
\(672\) 0 0
\(673\) 40.2795 1.55266 0.776331 0.630326i \(-0.217079\pi\)
0.776331 + 0.630326i \(0.217079\pi\)
\(674\) 1.54894 3.47446i 0.0596630 0.133831i
\(675\) 0 0
\(676\) −38.1240 42.4235i −1.46631 1.63167i
\(677\) 28.8288i 1.10798i −0.832523 0.553991i \(-0.813104\pi\)
0.832523 0.553991i \(-0.186896\pi\)
\(678\) 0 0
\(679\) 0.621922i 0.0238672i
\(680\) 2.48132 0.802252i 0.0951544 0.0307650i
\(681\) 0 0
\(682\) −8.86390 3.95160i −0.339416 0.151315i
\(683\) 9.31381 0.356383 0.178191 0.983996i \(-0.442975\pi\)
0.178191 + 0.983996i \(0.442975\pi\)
\(684\) 0 0
\(685\) −1.09869 −0.0419787
\(686\) −1.29167 0.575837i −0.0493162 0.0219856i
\(687\) 0 0
\(688\) −37.0067 3.96196i −1.41087 0.151048i
\(689\) 35.0325i 1.33463i
\(690\) 0 0
\(691\) 21.0680i 0.801465i −0.916195 0.400732i \(-0.868756\pi\)
0.916195 0.400732i \(-0.131244\pi\)
\(692\) 27.8119 24.9933i 1.05725 0.950103i
\(693\) 0 0
\(694\) 6.01256 13.4869i 0.228234 0.511955i
\(695\) −4.29474 −0.162909
\(696\) 0 0
\(697\) −22.2280 −0.841947
\(698\) −1.26881 + 2.84610i −0.0480253 + 0.107726i
\(699\) 0 0
\(700\) 7.29379 6.55459i 0.275679 0.247740i
\(701\) 12.1558i 0.459118i −0.973295 0.229559i \(-0.926272\pi\)
0.973295 0.229559i \(-0.0737284\pi\)
\(702\) 0 0
\(703\) 5.75686i 0.217124i
\(704\) −23.3021 + 16.8268i −0.878230 + 0.634186i
\(705\) 0 0
\(706\) 15.7315 + 7.01325i 0.592064 + 0.263947i
\(707\) 17.7166 0.666303
\(708\) 0 0
\(709\) −19.2513 −0.722997 −0.361498 0.932373i \(-0.617735\pi\)
−0.361498 + 0.932373i \(0.617735\pi\)
\(710\) 0.709309 + 0.316216i 0.0266199 + 0.0118674i
\(711\) 0 0
\(712\) 7.75701 + 23.9920i 0.290706 + 0.899139i
\(713\) 14.4964i 0.542894i
\(714\) 0 0
\(715\) 7.20585i 0.269483i
\(716\) −21.8101 24.2697i −0.815082 0.907003i
\(717\) 0 0
\(718\) −7.89873 + 17.7178i −0.294778 + 0.661221i
\(719\) −41.2877 −1.53977 −0.769886 0.638182i \(-0.779687\pi\)
−0.769886 + 0.638182i \(0.779687\pi\)
\(720\) 0 0
\(721\) 15.5139 0.577769
\(722\) 10.6708 23.9359i 0.397128 0.890804i
\(723\) 0 0
\(724\) 9.82590 + 10.9340i 0.365177 + 0.406360i
\(725\) 45.4441i 1.68775i
\(726\) 0 0
\(727\) 22.1628i 0.821973i 0.911641 + 0.410986i \(0.134816\pi\)
−0.911641 + 0.410986i \(0.865184\pi\)
\(728\) 5.60665 + 17.3411i 0.207796 + 0.642703i
\(729\) 0 0
\(730\) −1.33759 0.596308i −0.0495064 0.0220704i
\(731\) 27.5608 1.01937
\(732\) 0 0
\(733\) 0.676700 0.0249945 0.0124972 0.999922i \(-0.496022\pi\)
0.0124972 + 0.999922i \(0.496022\pi\)
\(734\) 31.1836 + 13.9019i 1.15101 + 0.513129i
\(735\) 0 0
\(736\) −37.1294 21.5566i −1.36861 0.794586i
\(737\) 3.93839i 0.145072i
\(738\) 0 0
\(739\) 10.4848i 0.385691i −0.981229 0.192845i \(-0.938228\pi\)
0.981229 0.192845i \(-0.0617716\pi\)
\(740\) 3.89237 3.49789i 0.143086 0.128585i
\(741\) 0 0
\(742\) −3.13077 + 7.02268i −0.114934 + 0.257811i
\(743\) 21.0030 0.770525 0.385263 0.922807i \(-0.374111\pi\)
0.385263 + 0.922807i \(0.374111\pi\)
\(744\) 0 0
\(745\) 3.37319 0.123584
\(746\) 2.55919 5.74057i 0.0936987 0.210177i
\(747\) 0 0
\(748\) 15.8312 14.2267i 0.578844 0.520181i
\(749\) 4.26969i 0.156011i
\(750\) 0 0
\(751\) 15.4787i 0.564825i −0.959293 0.282412i \(-0.908865\pi\)
0.959293 0.282412i \(-0.0911347\pi\)
\(752\) −4.18275 + 39.0689i −0.152529 + 1.42470i
\(753\) 0 0
\(754\) 77.1396 + 34.3895i 2.80926 + 1.25239i
\(755\) −0.131350 −0.00478033
\(756\) 0 0
\(757\) 0.902420 0.0327990 0.0163995 0.999866i \(-0.494780\pi\)
0.0163995 + 0.999866i \(0.494780\pi\)
\(758\) 11.6640 + 5.19989i 0.423654 + 0.188869i
\(759\) 0 0
\(760\) −0.573680 + 0.185480i −0.0208095 + 0.00672806i
\(761\) 21.3378i 0.773493i 0.922186 + 0.386747i \(0.126401\pi\)
−0.922186 + 0.386747i \(0.873599\pi\)
\(762\) 0 0
\(763\) 2.03903i 0.0738179i
\(764\) 27.6588 + 30.7781i 1.00066 + 1.11351i
\(765\) 0 0
\(766\) 11.6090 26.0402i 0.419449 0.940872i
\(767\) 14.7558 0.532802
\(768\) 0 0
\(769\) 24.8199 0.895029 0.447515 0.894277i \(-0.352309\pi\)
0.447515 + 0.894277i \(0.352309\pi\)
\(770\) −0.643968 + 1.44450i −0.0232070 + 0.0520560i
\(771\) 0 0
\(772\) 3.69125 + 4.10754i 0.132851 + 0.147833i
\(773\) 22.8253i 0.820969i 0.911868 + 0.410484i \(0.134640\pi\)
−0.911868 + 0.410484i \(0.865360\pi\)
\(774\) 0 0
\(775\) 9.36507i 0.336403i
\(776\) −1.67375 + 0.541152i −0.0600843 + 0.0194262i
\(777\) 0 0
\(778\) −26.0648 11.6199i −0.934468 0.416594i
\(779\) 5.13910 0.184127
\(780\) 0 0
\(781\) 6.33851 0.226810
\(782\) 29.0382 + 12.9455i 1.03840 + 0.462929i
\(783\) 0 0
\(784\) 0.425810 3.97727i 0.0152075 0.142045i
\(785\) 4.68256i 0.167128i
\(786\) 0 0
\(787\) 41.9789i 1.49639i −0.663481 0.748193i \(-0.730922\pi\)
0.663481 0.748193i \(-0.269078\pi\)
\(788\) −11.9374 + 10.7275i −0.425251 + 0.382153i
\(789\) 0 0
\(790\) 0.249149 0.558871i 0.00886434 0.0198837i
\(791\) −8.69512 −0.309163
\(792\) 0 0
\(793\) −3.13303 −0.111257
\(794\) −11.4553 + 25.6955i −0.406532 + 0.911898i
\(795\) 0 0
\(796\) 24.1066 21.6635i 0.854436 0.767842i
\(797\) 20.0382i 0.709789i −0.934906 0.354894i \(-0.884517\pi\)
0.934906 0.354894i \(-0.115483\pi\)
\(798\) 0 0
\(799\) 29.0967i 1.02937i
\(800\) 23.9866 + 13.9262i 0.848056 + 0.492364i
\(801\) 0 0
\(802\) 45.4827 + 20.2766i 1.60605 + 0.715990i
\(803\) −11.9529 −0.421810
\(804\) 0 0
\(805\) −2.36239 −0.0832633
\(806\) −15.8968 7.08694i −0.559942 0.249627i
\(807\) 0 0
\(808\) 15.4157 + 47.6801i 0.542324 + 1.67738i
\(809\) 8.37127i 0.294318i −0.989113 0.147159i \(-0.952987\pi\)
0.989113 0.147159i \(-0.0470129\pi\)
\(810\) 0 0
\(811\) 9.60694i 0.337345i −0.985672 0.168673i \(-0.946052\pi\)
0.985672 0.168673i \(-0.0539480\pi\)
\(812\) −12.3902 13.7875i −0.434812 0.483848i
\(813\) 0 0
\(814\) 17.3915 39.0111i 0.609571 1.36734i
\(815\) −3.93220 −0.137739
\(816\) 0 0
\(817\) −6.37203 −0.222929
\(818\) 10.6010 23.7794i 0.370657 0.831426i
\(819\) 0 0
\(820\) 3.12254 + 3.47469i 0.109044 + 0.121341i
\(821\) 12.6594i 0.441818i 0.975294 + 0.220909i \(0.0709023\pi\)
−0.975294 + 0.220909i \(0.929098\pi\)
\(822\) 0 0
\(823\) 24.1786i 0.842811i 0.906872 + 0.421406i \(0.138463\pi\)
−0.906872 + 0.421406i \(0.861537\pi\)
\(824\) 13.4991 + 41.7520i 0.470264 + 1.45450i
\(825\) 0 0
\(826\) −2.95798 1.31869i −0.102921 0.0458831i
\(827\) −35.6652 −1.24020 −0.620100 0.784522i \(-0.712908\pi\)
−0.620100 + 0.784522i \(0.712908\pi\)
\(828\) 0 0
\(829\) −0.136683 −0.00474721 −0.00237360 0.999997i \(-0.500756\pi\)
−0.00237360 + 0.999997i \(0.500756\pi\)
\(830\) 0.113822 + 0.0507428i 0.00395082 + 0.00176131i
\(831\) 0 0
\(832\) −41.7908 + 30.1779i −1.44884 + 1.04623i
\(833\) 2.96208i 0.102630i
\(834\) 0 0
\(835\) 6.89150i 0.238490i
\(836\) −3.66015 + 3.28921i −0.126589 + 0.113760i
\(837\) 0 0
\(838\) −10.7018 + 24.0054i −0.369687 + 0.829252i
\(839\) −40.5086 −1.39851 −0.699256 0.714872i \(-0.746485\pi\)
−0.699256 + 0.714872i \(0.746485\pi\)
\(840\) 0 0
\(841\) −56.9035 −1.96219
\(842\) −20.3876 + 45.7317i −0.702603 + 1.57602i
\(843\) 0 0
\(844\) 29.7866 26.7679i 1.02530 0.921388i
\(845\) 8.87678i 0.305371i
\(846\) 0 0
\(847\) 1.90828i 0.0655692i
\(848\) −21.6240 2.31508i −0.742572 0.0795003i
\(849\) 0 0
\(850\) −18.7595 8.36313i −0.643445 0.286853i
\(851\) 63.8004 2.18705
\(852\) 0 0
\(853\) 55.5584 1.90228 0.951141 0.308757i \(-0.0999130\pi\)
0.951141 + 0.308757i \(0.0999130\pi\)
\(854\) 0.628052 + 0.279991i 0.0214915 + 0.00958109i
\(855\) 0 0
\(856\) 11.4908 3.71518i 0.392749 0.126982i
\(857\) 28.1683i 0.962211i −0.876663 0.481105i \(-0.840235\pi\)
0.876663 0.481105i \(-0.159765\pi\)
\(858\) 0 0
\(859\) 37.1189i 1.26648i −0.773955 0.633240i \(-0.781725\pi\)
0.773955 0.633240i \(-0.218275\pi\)
\(860\) −3.87168 4.30831i −0.132023 0.146912i
\(861\) 0 0
\(862\) 13.3002 29.8339i 0.453007 1.01615i
\(863\) 7.83813 0.266813 0.133406 0.991061i \(-0.457408\pi\)
0.133406 + 0.991061i \(0.457408\pi\)
\(864\) 0 0
\(865\) 5.81943 0.197867
\(866\) 10.3662 23.2527i 0.352259 0.790158i
\(867\) 0 0
\(868\) 2.55336 + 2.84132i 0.0866668 + 0.0964407i
\(869\) 4.99417i 0.169416i
\(870\) 0 0
\(871\) 7.06325i 0.239329i
\(872\) −5.48756 + 1.77422i −0.185832 + 0.0600826i
\(873\) 0 0
\(874\) −6.71360 2.99298i −0.227091 0.101239i
\(875\) 3.08250 0.104207
\(876\) 0 0
\(877\) −17.4403 −0.588918 −0.294459 0.955664i \(-0.595139\pi\)
−0.294459 + 0.955664i \(0.595139\pi\)
\(878\) −31.0298 13.8333i −1.04720 0.466852i
\(879\) 0 0
\(880\) −4.44785 0.476190i −0.149937 0.0160524i
\(881\) 37.5232i 1.26419i 0.774891 + 0.632095i \(0.217805\pi\)
−0.774891 + 0.632095i \(0.782195\pi\)
\(882\) 0 0
\(883\) 0.149324i 0.00502515i −0.999997 0.00251258i \(-0.999200\pi\)
0.999997 0.00251258i \(-0.000799779\pi\)
\(884\) 28.3922 25.5147i 0.954932 0.858154i
\(885\) 0 0
\(886\) −19.9298 + 44.7048i −0.669554 + 1.50189i
\(887\) 34.2533 1.15011 0.575057 0.818113i \(-0.304980\pi\)
0.575057 + 0.818113i \(0.304980\pi\)
\(888\) 0 0
\(889\) 1.66793 0.0559406
\(890\) −1.59787 + 3.58421i −0.0535607 + 0.120143i
\(891\) 0 0
\(892\) −35.5658 + 31.9614i −1.19083 + 1.07015i
\(893\) 6.72712i 0.225115i
\(894\) 0 0
\(895\) 5.07825i 0.169747i
\(896\) 11.0744 2.31477i 0.369969 0.0773311i
\(897\) 0 0
\(898\) −24.7321 11.0258i −0.825320 0.367934i
\(899\) 17.7029 0.590425
\(900\) 0 0
\(901\) 16.1045 0.536520
\(902\) 34.8249 + 15.5252i 1.15954 + 0.516933i
\(903\) 0 0
\(904\) −7.56587 23.4008i −0.251637 0.778300i
\(905\) 2.28786i 0.0760510i
\(906\) 0 0
\(907\) 8.77393i 0.291334i 0.989334 + 0.145667i \(0.0465328\pi\)
−0.989334 + 0.145667i \(0.953467\pi\)
\(908\) −14.7850 16.4524i −0.490659 0.545993i
\(909\) 0 0
\(910\) −1.15492 + 2.59061i −0.0382851 + 0.0858780i
\(911\) −22.3335 −0.739942 −0.369971 0.929043i \(-0.620632\pi\)
−0.369971 + 0.929043i \(0.620632\pi\)
\(912\) 0 0
\(913\) 1.01713 0.0336622
\(914\) −8.18871 + 18.3682i −0.270859 + 0.607567i
\(915\) 0 0
\(916\) 30.8013 + 34.2750i 1.01771 + 1.13248i
\(917\) 19.0156i 0.627951i
\(918\) 0 0
\(919\) 20.3537i 0.671406i −0.941968 0.335703i \(-0.891026\pi\)
0.941968 0.335703i \(-0.108974\pi\)
\(920\) −2.05558 6.35780i −0.0677705 0.209611i
\(921\) 0 0
\(922\) −37.3840 16.6661i −1.23118 0.548869i
\(923\) 11.3677 0.374173
\(924\) 0 0
\(925\) −41.2168 −1.35520
\(926\) 39.7674 + 17.7287i 1.30684 + 0.582600i
\(927\) 0 0
\(928\) 26.3248 45.3422i 0.864153 1.48843i
\(929\) 39.6973i 1.30242i 0.758896 + 0.651212i \(0.225739\pi\)
−0.758896 + 0.651212i \(0.774261\pi\)
\(930\) 0 0
\(931\) 0.684831i 0.0224444i
\(932\) −8.02626 + 7.21283i −0.262909 + 0.236264i
\(933\) 0 0
\(934\) 11.3391 25.4350i 0.371027 0.832257i
\(935\) 3.31255 0.108332
\(936\) 0 0
\(937\) 25.3076 0.826762 0.413381 0.910558i \(-0.364348\pi\)
0.413381 + 0.910558i \(0.364348\pi\)
\(938\) −0.631225 + 1.41591i −0.0206102 + 0.0462311i
\(939\) 0 0
\(940\) −4.54839 + 4.08743i −0.148352 + 0.133317i
\(941\) 8.43714i 0.275043i −0.990499 0.137521i \(-0.956086\pi\)
0.990499 0.137521i \(-0.0439136\pi\)
\(942\) 0 0
\(943\) 56.9541i 1.85468i
\(944\) 0.975121 9.10811i 0.0317375 0.296444i
\(945\) 0 0
\(946\) −43.1798 19.2499i −1.40390 0.625869i
\(947\) 23.3318 0.758180 0.379090 0.925360i \(-0.376237\pi\)
0.379090 + 0.925360i \(0.376237\pi\)
\(948\) 0 0
\(949\) −21.4368 −0.695869
\(950\) 4.33717 + 1.93355i 0.140716 + 0.0627326i
\(951\) 0 0
\(952\) −7.97173 + 2.57739i −0.258365 + 0.0835337i
\(953\) 27.5785i 0.893355i −0.894695 0.446678i \(-0.852607\pi\)
0.894695 0.446678i \(-0.147393\pi\)
\(954\) 0 0
\(955\) 6.44007i 0.208396i
\(956\) 28.8564 + 32.1107i 0.933283 + 1.03853i
\(957\) 0 0
\(958\) −7.17530 + 16.0950i −0.231823 + 0.520007i
\(959\) 3.52975 0.113982
\(960\) 0 0
\(961\) 27.3518 0.882316
\(962\) 31.1905 69.9639i 1.00562 2.25573i
\(963\) 0 0
\(964\) −12.4424 13.8456i −0.400743 0.445936i
\(965\) 0.859470i 0.0276673i
\(966\) 0 0
\(967\) 3.59047i 0.115462i −0.998332 0.0577308i \(-0.981613\pi\)
0.998332 0.0577308i \(-0.0183865\pi\)
\(968\) −5.13567 + 1.66045i −0.165067 + 0.0533688i
\(969\) 0 0
\(970\) −0.250045 0.111472i −0.00802846 0.00357916i
\(971\) −4.89788 −0.157180 −0.0785902 0.996907i \(-0.525042\pi\)
−0.0785902 + 0.996907i \(0.525042\pi\)
\(972\) 0 0
\(973\) 13.7977 0.442333
\(974\) 3.78109 + 1.68564i 0.121154 + 0.0540114i
\(975\) 0 0
\(976\) −0.207042 + 1.93388i −0.00662727 + 0.0619019i
\(977\) 6.72581i 0.215178i 0.994195 + 0.107589i \(0.0343130\pi\)
−0.994195 + 0.107589i \(0.965687\pi\)
\(978\) 0 0
\(979\) 32.0291i 1.02366i
\(980\) 0.463033 0.416106i 0.0147910 0.0132920i
\(981\) 0 0
\(982\) −9.36665 + 21.0105i −0.298902 + 0.670472i
\(983\) −34.3802 −1.09656 −0.548279 0.836295i \(-0.684717\pi\)
−0.548279 + 0.836295i \(0.684717\pi\)
\(984\) 0 0
\(985\) −2.49780 −0.0795865
\(986\) −15.8089 + 35.4613i −0.503459 + 1.12932i
\(987\) 0 0
\(988\) −6.56425 + 5.89899i −0.208836 + 0.187672i
\(989\) 70.6180i 2.24552i
\(990\) 0 0
\(991\) 35.9495i 1.14197i −0.820960 0.570986i \(-0.806561\pi\)
0.820960 0.570986i \(-0.193439\pi\)
\(992\) −5.42498 + 9.34408i −0.172243 + 0.296675i
\(993\) 0 0
\(994\) −2.27879 1.01590i −0.0722789 0.0322225i
\(995\) 5.04412 0.159909
\(996\) 0 0
\(997\) −30.3636 −0.961625 −0.480812 0.876824i \(-0.659658\pi\)
−0.480812 + 0.876824i \(0.659658\pi\)
\(998\) −33.4007 14.8903i −1.05728 0.471344i
\(999\) 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 756.2.e.a.323.17 yes 24
3.2 odd 2 inner 756.2.e.a.323.8 yes 24
4.3 odd 2 inner 756.2.e.a.323.7 24
12.11 even 2 inner 756.2.e.a.323.18 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
756.2.e.a.323.7 24 4.3 odd 2 inner
756.2.e.a.323.8 yes 24 3.2 odd 2 inner
756.2.e.a.323.17 yes 24 1.1 even 1 trivial
756.2.e.a.323.18 yes 24 12.11 even 2 inner