Properties

Label 349.2.b.b.348.16
Level $349$
Weight $2$
Character 349.348
Analytic conductor $2.787$
Analytic rank $0$
Dimension $26$
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] = [349,2,Mod(348,349)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(349, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("349.348");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 349 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 349.b (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: \(2.78677903054\)
Analytic rank: \(0\)
Dimension: \(26\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 348.16
Character \(\chi\) \(=\) 349.348
Dual form 349.2.b.b.348.11

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.980996i q^{2} -2.39151 q^{3} +1.03765 q^{4} -1.59459 q^{5} -2.34607i q^{6} +3.49225i q^{7} +2.97992i q^{8} +2.71933 q^{9} +O(q^{10})\) \(q+0.980996i q^{2} -2.39151 q^{3} +1.03765 q^{4} -1.59459 q^{5} -2.34607i q^{6} +3.49225i q^{7} +2.97992i q^{8} +2.71933 q^{9} -1.56429i q^{10} -5.72888i q^{11} -2.48154 q^{12} +1.18640i q^{13} -3.42589 q^{14} +3.81349 q^{15} -0.847998 q^{16} -7.98398 q^{17} +2.66766i q^{18} -4.33179 q^{19} -1.65462 q^{20} -8.35177i q^{21} +5.62001 q^{22} -4.74560 q^{23} -7.12652i q^{24} -2.45727 q^{25} -1.16385 q^{26} +0.671217 q^{27} +3.62372i q^{28} +3.89895 q^{29} +3.74102i q^{30} -7.04162 q^{31} +5.12796i q^{32} +13.7007i q^{33} -7.83226i q^{34} -5.56872i q^{35} +2.82171 q^{36} +4.56742 q^{37} -4.24947i q^{38} -2.83728i q^{39} -4.75176i q^{40} +6.92928 q^{41} +8.19305 q^{42} +3.20900i q^{43} -5.94456i q^{44} -4.33623 q^{45} -4.65541i q^{46} -9.60778i q^{47} +2.02800 q^{48} -5.19583 q^{49} -2.41058i q^{50} +19.0938 q^{51} +1.23106i q^{52} +4.11388i q^{53} +0.658461i q^{54} +9.13524i q^{55} -10.4066 q^{56} +10.3595 q^{57} +3.82486i q^{58} +10.5501i q^{59} +3.95705 q^{60} +14.5748i q^{61} -6.90780i q^{62} +9.49660i q^{63} -6.72650 q^{64} -1.89182i q^{65} -13.4403 q^{66} -2.31537 q^{67} -8.28455 q^{68} +11.3492 q^{69} +5.46290 q^{70} -2.98972i q^{71} +8.10340i q^{72} +5.97802 q^{73} +4.48062i q^{74} +5.87660 q^{75} -4.49487 q^{76} +20.0067 q^{77} +2.78336 q^{78} +7.16874i q^{79} +1.35221 q^{80} -9.76323 q^{81} +6.79760i q^{82} -15.6039 q^{83} -8.66618i q^{84} +12.7312 q^{85} -3.14802 q^{86} -9.32439 q^{87} +17.0716 q^{88} +13.2842i q^{89} -4.25383i q^{90} -4.14319 q^{91} -4.92425 q^{92} +16.8401 q^{93} +9.42520 q^{94} +6.90744 q^{95} -12.2636i q^{96} -0.486705i q^{97} -5.09709i q^{98} -15.5787i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 26 q - 2 q^{3} - 36 q^{4} - 12 q^{5} + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 26 q - 2 q^{3} - 36 q^{4} - 12 q^{5} + 32 q^{9} + 12 q^{12} + 4 q^{14} + 12 q^{15} + 20 q^{16} - 14 q^{17} - 4 q^{19} - 2 q^{20} - 12 q^{22} - 18 q^{23} + 18 q^{25} + 22 q^{26} + 4 q^{27} - 18 q^{29} + 10 q^{31} - 54 q^{36} + 30 q^{37} - 16 q^{41} - 44 q^{45} - 74 q^{48} - 22 q^{49} + 32 q^{51} - 38 q^{56} - 16 q^{57} - 78 q^{60} - 96 q^{64} + 104 q^{66} + 72 q^{67} + 36 q^{68} - 40 q^{69} + 86 q^{70} + 72 q^{73} - 38 q^{75} + 96 q^{76} - 28 q^{77} - 30 q^{78} + 30 q^{80} - 6 q^{81} - 8 q^{83} - 22 q^{85} + 60 q^{86} + 32 q^{87} + 110 q^{88} - 12 q^{91} + 14 q^{92} + 84 q^{93} + 22 q^{94} - 10 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/349\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.980996i 0.693669i 0.937926 + 0.346835i \(0.112743\pi\)
−0.937926 + 0.346835i \(0.887257\pi\)
\(3\) −2.39151 −1.38074 −0.690370 0.723456i \(-0.742552\pi\)
−0.690370 + 0.723456i \(0.742552\pi\)
\(4\) 1.03765 0.518823
\(5\) −1.59459 −0.713124 −0.356562 0.934272i \(-0.616051\pi\)
−0.356562 + 0.934272i \(0.616051\pi\)
\(6\) 2.34607i 0.957777i
\(7\) 3.49225i 1.31995i 0.751289 + 0.659974i \(0.229433\pi\)
−0.751289 + 0.659974i \(0.770567\pi\)
\(8\) 2.97992i 1.05356i
\(9\) 2.71933 0.906445
\(10\) 1.56429i 0.494672i
\(11\) 5.72888i 1.72732i −0.504072 0.863662i \(-0.668165\pi\)
0.504072 0.863662i \(-0.331835\pi\)
\(12\) −2.48154 −0.716360
\(13\) 1.18640i 0.329047i 0.986373 + 0.164524i \(0.0526086\pi\)
−0.986373 + 0.164524i \(0.947391\pi\)
\(14\) −3.42589 −0.915607
\(15\) 3.81349 0.984639
\(16\) −0.847998 −0.211999
\(17\) −7.98398 −1.93640 −0.968200 0.250177i \(-0.919511\pi\)
−0.968200 + 0.250177i \(0.919511\pi\)
\(18\) 2.66766i 0.628773i
\(19\) −4.33179 −0.993781 −0.496890 0.867813i \(-0.665525\pi\)
−0.496890 + 0.867813i \(0.665525\pi\)
\(20\) −1.65462 −0.369985
\(21\) 8.35177i 1.82251i
\(22\) 5.62001 1.19819
\(23\) −4.74560 −0.989525 −0.494763 0.869028i \(-0.664745\pi\)
−0.494763 + 0.869028i \(0.664745\pi\)
\(24\) 7.12652i 1.45469i
\(25\) −2.45727 −0.491455
\(26\) −1.16385 −0.228250
\(27\) 0.671217 0.129176
\(28\) 3.62372i 0.684819i
\(29\) 3.89895 0.724017 0.362008 0.932175i \(-0.382091\pi\)
0.362008 + 0.932175i \(0.382091\pi\)
\(30\) 3.74102i 0.683014i
\(31\) −7.04162 −1.26471 −0.632356 0.774678i \(-0.717912\pi\)
−0.632356 + 0.774678i \(0.717912\pi\)
\(32\) 5.12796i 0.906503i
\(33\) 13.7007i 2.38499i
\(34\) 7.83226i 1.34322i
\(35\) 5.56872i 0.941286i
\(36\) 2.82171 0.470284
\(37\) 4.56742 0.750879 0.375440 0.926847i \(-0.377492\pi\)
0.375440 + 0.926847i \(0.377492\pi\)
\(38\) 4.24947i 0.689355i
\(39\) 2.83728i 0.454329i
\(40\) 4.75176i 0.751319i
\(41\) 6.92928 1.08217 0.541086 0.840967i \(-0.318013\pi\)
0.541086 + 0.840967i \(0.318013\pi\)
\(42\) 8.19305 1.26422
\(43\) 3.20900i 0.489368i 0.969603 + 0.244684i \(0.0786842\pi\)
−0.969603 + 0.244684i \(0.921316\pi\)
\(44\) 5.94456i 0.896175i
\(45\) −4.33623 −0.646407
\(46\) 4.65541i 0.686403i
\(47\) 9.60778i 1.40144i −0.713437 0.700720i \(-0.752862\pi\)
0.713437 0.700720i \(-0.247138\pi\)
\(48\) 2.02800 0.292716
\(49\) −5.19583 −0.742261
\(50\) 2.41058i 0.340907i
\(51\) 19.0938 2.67367
\(52\) 1.23106i 0.170717i
\(53\) 4.11388i 0.565085i 0.959255 + 0.282543i \(0.0911779\pi\)
−0.959255 + 0.282543i \(0.908822\pi\)
\(54\) 0.658461i 0.0896052i
\(55\) 9.13524i 1.23180i
\(56\) −10.4066 −1.39064
\(57\) 10.3595 1.37215
\(58\) 3.82486i 0.502228i
\(59\) 10.5501i 1.37350i 0.726893 + 0.686751i \(0.240964\pi\)
−0.726893 + 0.686751i \(0.759036\pi\)
\(60\) 3.95705 0.510853
\(61\) 14.5748i 1.86612i 0.359726 + 0.933058i \(0.382870\pi\)
−0.359726 + 0.933058i \(0.617130\pi\)
\(62\) 6.90780i 0.877291i
\(63\) 9.49660i 1.19646i
\(64\) −6.72650 −0.840813
\(65\) 1.89182i 0.234651i
\(66\) −13.4403 −1.65439
\(67\) −2.31537 −0.282867 −0.141433 0.989948i \(-0.545171\pi\)
−0.141433 + 0.989948i \(0.545171\pi\)
\(68\) −8.28455 −1.00465
\(69\) 11.3492 1.36628
\(70\) 5.46290 0.652941
\(71\) 2.98972i 0.354814i −0.984138 0.177407i \(-0.943229\pi\)
0.984138 0.177407i \(-0.0567709\pi\)
\(72\) 8.10340i 0.954994i
\(73\) 5.97802 0.699674 0.349837 0.936811i \(-0.386237\pi\)
0.349837 + 0.936811i \(0.386237\pi\)
\(74\) 4.48062i 0.520862i
\(75\) 5.87660 0.678571
\(76\) −4.49487 −0.515597
\(77\) 20.0067 2.27998
\(78\) 2.78336 0.315154
\(79\) 7.16874i 0.806546i 0.915080 + 0.403273i \(0.132127\pi\)
−0.915080 + 0.403273i \(0.867873\pi\)
\(80\) 1.35221 0.151182
\(81\) −9.76323 −1.08480
\(82\) 6.79760i 0.750669i
\(83\) −15.6039 −1.71275 −0.856374 0.516356i \(-0.827288\pi\)
−0.856374 + 0.516356i \(0.827288\pi\)
\(84\) 8.66618i 0.945558i
\(85\) 12.7312 1.38089
\(86\) −3.14802 −0.339459
\(87\) −9.32439 −0.999679
\(88\) 17.0716 1.81984
\(89\) 13.2842i 1.40812i 0.710138 + 0.704062i \(0.248632\pi\)
−0.710138 + 0.704062i \(0.751368\pi\)
\(90\) 4.25383i 0.448393i
\(91\) −4.14319 −0.434325
\(92\) −4.92425 −0.513389
\(93\) 16.8401 1.74624
\(94\) 9.42520 0.972135
\(95\) 6.90744 0.708689
\(96\) 12.2636i 1.25165i
\(97\) 0.486705i 0.0494174i −0.999695 0.0247087i \(-0.992134\pi\)
0.999695 0.0247087i \(-0.00786583\pi\)
\(98\) 5.09709i 0.514884i
\(99\) 15.5787i 1.56572i
\(100\) −2.54978 −0.254978
\(101\) 9.78271i 0.973416i −0.873565 0.486708i \(-0.838198\pi\)
0.873565 0.486708i \(-0.161802\pi\)
\(102\) 18.7309i 1.85464i
\(103\) 5.00860i 0.493512i −0.969078 0.246756i \(-0.920635\pi\)
0.969078 0.246756i \(-0.0793646\pi\)
\(104\) −3.53537 −0.346671
\(105\) 13.3177i 1.29967i
\(106\) −4.03570 −0.391982
\(107\) 18.0456i 1.74453i −0.489033 0.872266i \(-0.662650\pi\)
0.489033 0.872266i \(-0.337350\pi\)
\(108\) 0.696486 0.0670194
\(109\) 4.49358 0.430407 0.215204 0.976569i \(-0.430959\pi\)
0.215204 + 0.976569i \(0.430959\pi\)
\(110\) −8.96164 −0.854458
\(111\) −10.9230 −1.03677
\(112\) 2.96142i 0.279828i
\(113\) 5.47325i 0.514880i 0.966294 + 0.257440i \(0.0828790\pi\)
−0.966294 + 0.257440i \(0.917121\pi\)
\(114\) 10.1627i 0.951821i
\(115\) 7.56729 0.705654
\(116\) 4.04573 0.375637
\(117\) 3.22621i 0.298263i
\(118\) −10.3496 −0.952755
\(119\) 27.8821i 2.55595i
\(120\) 11.3639i 1.03738i
\(121\) −21.8201 −1.98365
\(122\) −14.2979 −1.29447
\(123\) −16.5715 −1.49420
\(124\) −7.30671 −0.656162
\(125\) 11.8913 1.06359
\(126\) −9.31613 −0.829947
\(127\) 4.67185i 0.414559i 0.978282 + 0.207280i \(0.0664610\pi\)
−0.978282 + 0.207280i \(0.933539\pi\)
\(128\) 3.65724i 0.323257i
\(129\) 7.67436i 0.675690i
\(130\) 1.85587 0.162770
\(131\) 12.6227i 1.10285i 0.834226 + 0.551423i \(0.185915\pi\)
−0.834226 + 0.551423i \(0.814085\pi\)
\(132\) 14.2165i 1.23739i
\(133\) 15.1277i 1.31174i
\(134\) 2.27137i 0.196216i
\(135\) −1.07032 −0.0921183
\(136\) 23.7916i 2.04012i
\(137\) 17.3749i 1.48444i 0.670158 + 0.742218i \(0.266226\pi\)
−0.670158 + 0.742218i \(0.733774\pi\)
\(138\) 11.1335i 0.947745i
\(139\) 9.56374 0.811185 0.405593 0.914054i \(-0.367065\pi\)
0.405593 + 0.914054i \(0.367065\pi\)
\(140\) 5.77836i 0.488361i
\(141\) 22.9771i 1.93502i
\(142\) 2.93290 0.246124
\(143\) 6.79673 0.568371
\(144\) −2.30599 −0.192166
\(145\) −6.21724 −0.516314
\(146\) 5.86441i 0.485342i
\(147\) 12.4259 1.02487
\(148\) 4.73937 0.389573
\(149\) 9.94243i 0.814515i −0.913313 0.407258i \(-0.866485\pi\)
0.913313 0.407258i \(-0.133515\pi\)
\(150\) 5.76492i 0.470704i
\(151\) −2.12122 −0.172622 −0.0863112 0.996268i \(-0.527508\pi\)
−0.0863112 + 0.996268i \(0.527508\pi\)
\(152\) 12.9084i 1.04701i
\(153\) −21.7111 −1.75524
\(154\) 19.6265i 1.58155i
\(155\) 11.2285 0.901896
\(156\) 2.94409i 0.235716i
\(157\) 5.66729 0.452299 0.226149 0.974093i \(-0.427386\pi\)
0.226149 + 0.974093i \(0.427386\pi\)
\(158\) −7.03250 −0.559476
\(159\) 9.83841i 0.780236i
\(160\) 8.17700i 0.646449i
\(161\) 16.5728i 1.30612i
\(162\) 9.57769i 0.752494i
\(163\) 2.89238i 0.226549i −0.993564 0.113274i \(-0.963866\pi\)
0.993564 0.113274i \(-0.0361339\pi\)
\(164\) 7.19014 0.561456
\(165\) 21.8470i 1.70079i
\(166\) 15.3073i 1.18808i
\(167\) 5.61638i 0.434609i −0.976104 0.217304i \(-0.930274\pi\)
0.976104 0.217304i \(-0.0697264\pi\)
\(168\) 24.8876 1.92012
\(169\) 11.5925 0.891728
\(170\) 12.4893i 0.957883i
\(171\) −11.7796 −0.900807
\(172\) 3.32981i 0.253895i
\(173\) 4.86677i 0.370014i 0.982737 + 0.185007i \(0.0592307\pi\)
−0.982737 + 0.185007i \(0.940769\pi\)
\(174\) 9.14719i 0.693447i
\(175\) 8.58142i 0.648694i
\(176\) 4.85808i 0.366192i
\(177\) 25.2306i 1.89645i
\(178\) −13.0318 −0.976772
\(179\) 13.8165i 1.03269i 0.856381 + 0.516345i \(0.172708\pi\)
−0.856381 + 0.516345i \(0.827292\pi\)
\(180\) −4.49947 −0.335371
\(181\) −2.22392 −0.165303 −0.0826513 0.996579i \(-0.526339\pi\)
−0.0826513 + 0.996579i \(0.526339\pi\)
\(182\) 4.06446i 0.301278i
\(183\) 34.8559i 2.57662i
\(184\) 14.1415i 1.04252i
\(185\) −7.28317 −0.535470
\(186\) 16.5201i 1.21131i
\(187\) 45.7393i 3.34479i
\(188\) 9.96948i 0.727099i
\(189\) 2.34406i 0.170505i
\(190\) 6.77618i 0.491595i
\(191\) −10.5192 −0.761140 −0.380570 0.924752i \(-0.624272\pi\)
−0.380570 + 0.924752i \(0.624272\pi\)
\(192\) 16.0865 1.16094
\(193\) 0.641023i 0.0461418i −0.999734 0.0230709i \(-0.992656\pi\)
0.999734 0.0230709i \(-0.00734435\pi\)
\(194\) 0.477456 0.0342794
\(195\) 4.52431i 0.323993i
\(196\) −5.39143 −0.385102
\(197\) 15.4721i 1.10234i 0.834391 + 0.551172i \(0.185819\pi\)
−0.834391 + 0.551172i \(0.814181\pi\)
\(198\) 15.2827 1.08609
\(199\) 1.38720i 0.0983358i 0.998791 + 0.0491679i \(0.0156569\pi\)
−0.998791 + 0.0491679i \(0.984343\pi\)
\(200\) 7.32248i 0.517777i
\(201\) 5.53723 0.390566
\(202\) 9.59680 0.675229
\(203\) 13.6161i 0.955664i
\(204\) 19.8126 1.38716
\(205\) −11.0494 −0.771722
\(206\) 4.91342 0.342334
\(207\) −12.9049 −0.896950
\(208\) 1.00606i 0.0697578i
\(209\) 24.8163i 1.71658i
\(210\) −13.0646 −0.901542
\(211\) 0.541778i 0.0372975i −0.999826 0.0186488i \(-0.994064\pi\)
0.999826 0.0186488i \(-0.00593643\pi\)
\(212\) 4.26876i 0.293179i
\(213\) 7.14995i 0.489907i
\(214\) 17.7026 1.21013
\(215\) 5.11705i 0.348980i
\(216\) 2.00017i 0.136095i
\(217\) 24.5911i 1.66935i
\(218\) 4.40819i 0.298560i
\(219\) −14.2965 −0.966068
\(220\) 9.47915i 0.639084i
\(221\) 9.47217i 0.637167i
\(222\) 10.7155i 0.719175i
\(223\) −13.5560 −0.907775 −0.453887 0.891059i \(-0.649963\pi\)
−0.453887 + 0.891059i \(0.649963\pi\)
\(224\) −17.9081 −1.19654
\(225\) −6.68215 −0.445476
\(226\) −5.36924 −0.357156
\(227\) −16.5742 −1.10007 −0.550033 0.835143i \(-0.685385\pi\)
−0.550033 + 0.835143i \(0.685385\pi\)
\(228\) 10.7495 0.711905
\(229\) 13.9969i 0.924940i −0.886635 0.462470i \(-0.846963\pi\)
0.886635 0.462470i \(-0.153037\pi\)
\(230\) 7.42349i 0.489490i
\(231\) −47.8463 −3.14806
\(232\) 11.6186i 0.762796i
\(233\) 15.3066 1.00277 0.501386 0.865224i \(-0.332824\pi\)
0.501386 + 0.865224i \(0.332824\pi\)
\(234\) −3.16490 −0.206896
\(235\) 15.3205i 0.999400i
\(236\) 10.9472i 0.712604i
\(237\) 17.1441i 1.11363i
\(238\) 27.3522 1.77298
\(239\) −18.7774 −1.21461 −0.607304 0.794469i \(-0.707749\pi\)
−0.607304 + 0.794469i \(0.707749\pi\)
\(240\) −3.23383 −0.208743
\(241\) −5.02258 −0.323533 −0.161766 0.986829i \(-0.551719\pi\)
−0.161766 + 0.986829i \(0.551719\pi\)
\(242\) 21.4055i 1.37599i
\(243\) 21.3352 1.36866
\(244\) 15.1235i 0.968184i
\(245\) 8.28523 0.529324
\(246\) 16.2565i 1.03648i
\(247\) 5.13922i 0.327001i
\(248\) 20.9834i 1.33245i
\(249\) 37.3169 2.36486
\(250\) 11.6653i 0.737781i
\(251\) 1.98249i 0.125133i 0.998041 + 0.0625667i \(0.0199286\pi\)
−0.998041 + 0.0625667i \(0.980071\pi\)
\(252\) 9.85411i 0.620751i
\(253\) 27.1870i 1.70923i
\(254\) −4.58306 −0.287567
\(255\) −30.4468 −1.90666
\(256\) −17.0407 −1.06505
\(257\) −28.8615 −1.80033 −0.900166 0.435547i \(-0.856555\pi\)
−0.900166 + 0.435547i \(0.856555\pi\)
\(258\) 7.52852 0.468705
\(259\) 15.9506i 0.991121i
\(260\) 1.96304i 0.121742i
\(261\) 10.6025 0.656281
\(262\) −12.3828 −0.765011
\(263\) 0.463991 0.0286109 0.0143054 0.999898i \(-0.495446\pi\)
0.0143054 + 0.999898i \(0.495446\pi\)
\(264\) −40.8270 −2.51273
\(265\) 6.55997i 0.402976i
\(266\) 14.8402 0.909913
\(267\) 31.7694i 1.94425i
\(268\) −2.40253 −0.146758
\(269\) 0.424473 0.0258806 0.0129403 0.999916i \(-0.495881\pi\)
0.0129403 + 0.999916i \(0.495881\pi\)
\(270\) 1.04998i 0.0638996i
\(271\) 22.0294 1.33819 0.669097 0.743176i \(-0.266681\pi\)
0.669097 + 0.743176i \(0.266681\pi\)
\(272\) 6.77040 0.410516
\(273\) 9.90850 0.599690
\(274\) −17.0447 −1.02971
\(275\) 14.0774i 0.848901i
\(276\) 11.7764 0.708856
\(277\) 31.0689i 1.86675i −0.358901 0.933376i \(-0.616848\pi\)
0.358901 0.933376i \(-0.383152\pi\)
\(278\) 9.38199i 0.562694i
\(279\) −19.1485 −1.14639
\(280\) 16.5943 0.991702
\(281\) −16.5425 −0.986842 −0.493421 0.869791i \(-0.664254\pi\)
−0.493421 + 0.869791i \(0.664254\pi\)
\(282\) −22.5405 −1.34227
\(283\) 12.7168 0.755936 0.377968 0.925819i \(-0.376623\pi\)
0.377968 + 0.925819i \(0.376623\pi\)
\(284\) 3.10227i 0.184086i
\(285\) −16.5192 −0.978515
\(286\) 6.66756i 0.394261i
\(287\) 24.1988i 1.42841i
\(288\) 13.9446i 0.821695i
\(289\) 46.7440 2.74965
\(290\) 6.09909i 0.358151i
\(291\) 1.16396i 0.0682327i
\(292\) 6.20307 0.363007
\(293\) 15.0794 0.880947 0.440473 0.897766i \(-0.354811\pi\)
0.440473 + 0.897766i \(0.354811\pi\)
\(294\) 12.1898i 0.710921i
\(295\) 16.8231i 0.979476i
\(296\) 13.6105i 0.791097i
\(297\) 3.84532i 0.223128i
\(298\) 9.75348 0.565004
\(299\) 5.63016i 0.325600i
\(300\) 6.09783 0.352059
\(301\) −11.2066 −0.645940
\(302\) 2.08091i 0.119743i
\(303\) 23.3955i 1.34404i
\(304\) 3.67335 0.210681
\(305\) 23.2409i 1.33077i
\(306\) 21.2985i 1.21756i
\(307\) −2.64152 −0.150759 −0.0753797 0.997155i \(-0.524017\pi\)
−0.0753797 + 0.997155i \(0.524017\pi\)
\(308\) 20.7599 1.18290
\(309\) 11.9781i 0.681412i
\(310\) 11.0151i 0.625617i
\(311\) 24.1439i 1.36908i −0.728977 0.684538i \(-0.760004\pi\)
0.728977 0.684538i \(-0.239996\pi\)
\(312\) 8.45487 0.478663
\(313\) −13.0338 −0.736712 −0.368356 0.929685i \(-0.620079\pi\)
−0.368356 + 0.929685i \(0.620079\pi\)
\(314\) 5.55959i 0.313746i
\(315\) 15.1432i 0.853223i
\(316\) 7.43861i 0.418455i
\(317\) 4.71205i 0.264655i 0.991206 + 0.132328i \(0.0422451\pi\)
−0.991206 + 0.132328i \(0.957755\pi\)
\(318\) 9.65144 0.541226
\(319\) 22.3366i 1.25061i
\(320\) 10.7260 0.599603
\(321\) 43.1562i 2.40874i
\(322\) 16.2579 0.906016
\(323\) 34.5849 1.92436
\(324\) −10.1308 −0.562821
\(325\) 2.91530i 0.161712i
\(326\) 2.83741 0.157150
\(327\) −10.7465 −0.594281
\(328\) 20.6487i 1.14013i
\(329\) 33.5528 1.84983
\(330\) 21.4319 1.17979
\(331\) 13.4811i 0.740986i −0.928835 0.370493i \(-0.879189\pi\)
0.928835 0.370493i \(-0.120811\pi\)
\(332\) −16.1913 −0.888613
\(333\) 12.4203 0.680630
\(334\) 5.50965 0.301475
\(335\) 3.69207 0.201719
\(336\) 7.08228i 0.386370i
\(337\) −0.532975 −0.0290330 −0.0145165 0.999895i \(-0.504621\pi\)
−0.0145165 + 0.999895i \(0.504621\pi\)
\(338\) 11.3722i 0.618564i
\(339\) 13.0894i 0.710916i
\(340\) 13.2105 0.716439
\(341\) 40.3406i 2.18457i
\(342\) 11.5557i 0.624862i
\(343\) 6.30062i 0.340201i
\(344\) −9.56256 −0.515579
\(345\) −18.0973 −0.974325
\(346\) −4.77428 −0.256667
\(347\) 4.83614i 0.259617i −0.991539 0.129809i \(-0.958564\pi\)
0.991539 0.129809i \(-0.0414363\pi\)
\(348\) −9.67542 −0.518657
\(349\) 16.2907 9.14396i 0.872023 0.489465i
\(350\) 8.41834 0.449979
\(351\) 0.796329i 0.0425049i
\(352\) 29.3775 1.56582
\(353\) −24.0371 −1.27936 −0.639682 0.768640i \(-0.720934\pi\)
−0.639682 + 0.768640i \(0.720934\pi\)
\(354\) 24.7511 1.31551
\(355\) 4.76738i 0.253026i
\(356\) 13.7843i 0.730568i
\(357\) 66.6804i 3.52910i
\(358\) −13.5539 −0.716345
\(359\) 24.8277i 1.31035i 0.755475 + 0.655177i \(0.227406\pi\)
−0.755475 + 0.655177i \(0.772594\pi\)
\(360\) 12.9216i 0.681029i
\(361\) −0.235592 −0.0123996
\(362\) 2.18166i 0.114665i
\(363\) 52.1831 2.73890
\(364\) −4.29917 −0.225338
\(365\) −9.53250 −0.498954
\(366\) 34.1935 1.78732
\(367\) 22.1347i 1.15542i 0.816242 + 0.577710i \(0.196054\pi\)
−0.816242 + 0.577710i \(0.803946\pi\)
\(368\) 4.02425 0.209779
\(369\) 18.8430 0.980929
\(370\) 7.14477i 0.371439i
\(371\) −14.3667 −0.745883
\(372\) 17.4741 0.905989
\(373\) 26.5860i 1.37657i −0.725441 0.688284i \(-0.758364\pi\)
0.725441 0.688284i \(-0.241636\pi\)
\(374\) −44.8701 −2.32018
\(375\) −28.4382 −1.46854
\(376\) 28.6304 1.47650
\(377\) 4.62570i 0.238236i
\(378\) −2.29951 −0.118274
\(379\) 11.7065i 0.601323i −0.953731 0.300662i \(-0.902793\pi\)
0.953731 0.300662i \(-0.0972075\pi\)
\(380\) 7.16748 0.367684
\(381\) 11.1728i 0.572399i
\(382\) 10.3193i 0.527979i
\(383\) 5.62366i 0.287356i −0.989625 0.143678i \(-0.954107\pi\)
0.989625 0.143678i \(-0.0458929\pi\)
\(384\) 8.74634i 0.446335i
\(385\) −31.9026 −1.62591
\(386\) 0.628841 0.0320072
\(387\) 8.72634i 0.443585i
\(388\) 0.505028i 0.0256389i
\(389\) 22.0957i 1.12030i 0.828392 + 0.560149i \(0.189256\pi\)
−0.828392 + 0.560149i \(0.810744\pi\)
\(390\) −4.43833 −0.224744
\(391\) 37.8888 1.91612
\(392\) 15.4832i 0.782017i
\(393\) 30.1873i 1.52275i
\(394\) −15.1781 −0.764663
\(395\) 11.4312i 0.575167i
\(396\) 16.1652i 0.812333i
\(397\) −0.306913 −0.0154035 −0.00770176 0.999970i \(-0.502452\pi\)
−0.00770176 + 0.999970i \(0.502452\pi\)
\(398\) −1.36084 −0.0682125
\(399\) 36.1781i 1.81117i
\(400\) 2.08376 0.104188
\(401\) 6.58218i 0.328698i −0.986402 0.164349i \(-0.947448\pi\)
0.986402 0.164349i \(-0.0525524\pi\)
\(402\) 5.43200i 0.270924i
\(403\) 8.35415i 0.416150i
\(404\) 10.1510i 0.505031i
\(405\) 15.5684 0.773599
\(406\) −13.3574 −0.662915
\(407\) 26.1662i 1.29701i
\(408\) 56.8980i 2.81687i
\(409\) −5.44638 −0.269306 −0.134653 0.990893i \(-0.542992\pi\)
−0.134653 + 0.990893i \(0.542992\pi\)
\(410\) 10.8394i 0.535320i
\(411\) 41.5523i 2.04962i
\(412\) 5.19715i 0.256045i
\(413\) −36.8435 −1.81295
\(414\) 12.6596i 0.622186i
\(415\) 24.8818 1.22140
\(416\) −6.08379 −0.298282
\(417\) −22.8718 −1.12004
\(418\) −24.3447 −1.19074
\(419\) −30.1276 −1.47183 −0.735915 0.677074i \(-0.763248\pi\)
−0.735915 + 0.677074i \(0.763248\pi\)
\(420\) 13.8190i 0.674300i
\(421\) 14.7654i 0.719619i 0.933026 + 0.359810i \(0.117158\pi\)
−0.933026 + 0.359810i \(0.882842\pi\)
\(422\) 0.531482 0.0258721
\(423\) 26.1268i 1.27033i
\(424\) −12.2590 −0.595352
\(425\) 19.6188 0.951653
\(426\) −7.01408 −0.339833
\(427\) −50.8990 −2.46318
\(428\) 18.7249i 0.905103i
\(429\) −16.2545 −0.784773
\(430\) 5.01980 0.242076
\(431\) 11.0341i 0.531492i −0.964043 0.265746i \(-0.914382\pi\)
0.964043 0.265746i \(-0.0856183\pi\)
\(432\) −0.569190 −0.0273852
\(433\) 30.0172i 1.44254i 0.692657 + 0.721268i \(0.256440\pi\)
−0.692657 + 0.721268i \(0.743560\pi\)
\(434\) 24.1238 1.15798
\(435\) 14.8686 0.712895
\(436\) 4.66275 0.223305
\(437\) 20.5569 0.983371
\(438\) 14.0248i 0.670132i
\(439\) 39.5031i 1.88538i −0.333669 0.942690i \(-0.608287\pi\)
0.333669 0.942690i \(-0.391713\pi\)
\(440\) −27.2223 −1.29777
\(441\) −14.1292 −0.672819
\(442\) 9.29216 0.441983
\(443\) 17.4133 0.827331 0.413666 0.910429i \(-0.364248\pi\)
0.413666 + 0.910429i \(0.364248\pi\)
\(444\) −11.3343 −0.537900
\(445\) 21.1829i 1.00417i
\(446\) 13.2984i 0.629695i
\(447\) 23.7774i 1.12463i
\(448\) 23.4906i 1.10983i
\(449\) −4.08923 −0.192983 −0.0964913 0.995334i \(-0.530762\pi\)
−0.0964913 + 0.995334i \(0.530762\pi\)
\(450\) 6.55516i 0.309013i
\(451\) 39.6971i 1.86926i
\(452\) 5.67930i 0.267132i
\(453\) 5.07292 0.238347
\(454\) 16.2592i 0.763082i
\(455\) 6.60671 0.309727
\(456\) 30.8706i 1.44565i
\(457\) −1.43677 −0.0672094 −0.0336047 0.999435i \(-0.510699\pi\)
−0.0336047 + 0.999435i \(0.510699\pi\)
\(458\) 13.7309 0.641603
\(459\) −5.35898 −0.250136
\(460\) 7.85217 0.366110
\(461\) 16.9247i 0.788263i 0.919054 + 0.394131i \(0.128955\pi\)
−0.919054 + 0.394131i \(0.871045\pi\)
\(462\) 46.9371i 2.18371i
\(463\) 16.7577i 0.778797i 0.921069 + 0.389399i \(0.127317\pi\)
−0.921069 + 0.389399i \(0.872683\pi\)
\(464\) −3.30630 −0.153491
\(465\) −26.8531 −1.24528
\(466\) 15.0158i 0.695591i
\(467\) 8.98624 0.415834 0.207917 0.978147i \(-0.433332\pi\)
0.207917 + 0.978147i \(0.433332\pi\)
\(468\) 3.34766i 0.154746i
\(469\) 8.08584i 0.373370i
\(470\) −15.0294 −0.693253
\(471\) −13.5534 −0.624507
\(472\) −31.4383 −1.44707
\(473\) 18.3840 0.845296
\(474\) 16.8183 0.772491
\(475\) 10.6444 0.488398
\(476\) 28.9317i 1.32608i
\(477\) 11.1870i 0.512219i
\(478\) 18.4206i 0.842537i
\(479\) 8.78802 0.401535 0.200767 0.979639i \(-0.435656\pi\)
0.200767 + 0.979639i \(0.435656\pi\)
\(480\) 19.5554i 0.892578i
\(481\) 5.41877i 0.247075i
\(482\) 4.92713i 0.224425i
\(483\) 39.6341i 1.80341i
\(484\) −22.6416 −1.02916
\(485\) 0.776097i 0.0352407i
\(486\) 20.9298i 0.949394i
\(487\) 10.7584i 0.487508i 0.969837 + 0.243754i \(0.0783790\pi\)
−0.969837 + 0.243754i \(0.921621\pi\)
\(488\) −43.4318 −1.96607
\(489\) 6.91716i 0.312805i
\(490\) 8.12778i 0.367176i
\(491\) 25.6701 1.15847 0.579237 0.815159i \(-0.303350\pi\)
0.579237 + 0.815159i \(0.303350\pi\)
\(492\) −17.1953 −0.775225
\(493\) −31.1292 −1.40199
\(494\) 5.04155 0.226830
\(495\) 24.8418i 1.11655i
\(496\) 5.97127 0.268118
\(497\) 10.4409 0.468336
\(498\) 36.6077i 1.64043i
\(499\) 28.5206i 1.27676i 0.769723 + 0.638378i \(0.220394\pi\)
−0.769723 + 0.638378i \(0.779606\pi\)
\(500\) 12.3390 0.551816
\(501\) 13.4317i 0.600082i
\(502\) −1.94481 −0.0868012
\(503\) 9.51645i 0.424318i 0.977235 + 0.212159i \(0.0680494\pi\)
−0.977235 + 0.212159i \(0.931951\pi\)
\(504\) −28.2991 −1.26054
\(505\) 15.5994i 0.694166i
\(506\) −26.6703 −1.18564
\(507\) −27.7235 −1.23125
\(508\) 4.84772i 0.215083i
\(509\) 29.6614i 1.31472i −0.753577 0.657360i \(-0.771673\pi\)
0.753577 0.657360i \(-0.228327\pi\)
\(510\) 29.8682i 1.32259i
\(511\) 20.8767i 0.923533i
\(512\) 9.40242i 0.415532i
\(513\) −2.90757 −0.128372
\(514\) 28.3130i 1.24883i
\(515\) 7.98667i 0.351935i
\(516\) 7.96327i 0.350564i
\(517\) −55.0419 −2.42074
\(518\) −15.6475 −0.687510
\(519\) 11.6389i 0.510893i
\(520\) 5.63747 0.247219
\(521\) 32.5969i 1.42810i −0.700096 0.714049i \(-0.746859\pi\)
0.700096 0.714049i \(-0.253141\pi\)
\(522\) 10.4011i 0.455242i
\(523\) 34.8552i 1.52411i −0.647510 0.762057i \(-0.724190\pi\)
0.647510 0.762057i \(-0.275810\pi\)
\(524\) 13.0979i 0.572182i
\(525\) 20.5226i 0.895679i
\(526\) 0.455173i 0.0198465i
\(527\) 56.2201 2.44899
\(528\) 11.6182i 0.505616i
\(529\) −0.479320 −0.0208400
\(530\) 6.43531 0.279532
\(531\) 28.6891i 1.24500i
\(532\) 15.6972i 0.680560i
\(533\) 8.22087i 0.356085i
\(534\) 31.1656 1.34867
\(535\) 28.7753i 1.24407i
\(536\) 6.89961i 0.298018i
\(537\) 33.0422i 1.42588i
\(538\) 0.416407i 0.0179526i
\(539\) 29.7663i 1.28213i
\(540\) −1.11061 −0.0477931
\(541\) −2.51302 −0.108043 −0.0540215 0.998540i \(-0.517204\pi\)
−0.0540215 + 0.998540i \(0.517204\pi\)
\(542\) 21.6108i 0.928263i
\(543\) 5.31853 0.228240
\(544\) 40.9415i 1.75535i
\(545\) −7.16543 −0.306933
\(546\) 9.72021i 0.415986i
\(547\) 11.3367 0.484724 0.242362 0.970186i \(-0.422078\pi\)
0.242362 + 0.970186i \(0.422078\pi\)
\(548\) 18.0290i 0.770160i
\(549\) 39.6338i 1.69153i
\(550\) −13.8099 −0.588857
\(551\) −16.8894 −0.719514
\(552\) 33.8196i 1.43946i
\(553\) −25.0350 −1.06460
\(554\) 30.4785 1.29491
\(555\) 17.4178 0.739345
\(556\) 9.92378 0.420862
\(557\) 31.2893i 1.32577i 0.748721 + 0.662885i \(0.230668\pi\)
−0.748721 + 0.662885i \(0.769332\pi\)
\(558\) 18.7846i 0.795216i
\(559\) −3.80714 −0.161025
\(560\) 4.72226i 0.199552i
\(561\) 109.386i 4.61829i
\(562\) 16.2281i 0.684542i
\(563\) 22.0702 0.930148 0.465074 0.885272i \(-0.346028\pi\)
0.465074 + 0.885272i \(0.346028\pi\)
\(564\) 23.8421i 1.00394i
\(565\) 8.72761i 0.367173i
\(566\) 12.4752i 0.524370i
\(567\) 34.0957i 1.43188i
\(568\) 8.90912 0.373818
\(569\) 8.99663i 0.377158i 0.982058 + 0.188579i \(0.0603882\pi\)
−0.982058 + 0.188579i \(0.939612\pi\)
\(570\) 16.2053i 0.678766i
\(571\) 17.9832i 0.752574i 0.926503 + 0.376287i \(0.122799\pi\)
−0.926503 + 0.376287i \(0.877201\pi\)
\(572\) 7.05260 0.294884
\(573\) 25.1567 1.05094
\(574\) −23.7389 −0.990844
\(575\) 11.6612 0.486307
\(576\) −18.2916 −0.762150
\(577\) −12.7766 −0.531897 −0.265949 0.963987i \(-0.585685\pi\)
−0.265949 + 0.963987i \(0.585685\pi\)
\(578\) 45.8557i 1.90735i
\(579\) 1.53301i 0.0637099i
\(580\) −6.45129 −0.267875
\(581\) 54.4927i 2.26074i
\(582\) −1.14184 −0.0473309
\(583\) 23.5680 0.976085
\(584\) 17.8140i 0.737149i
\(585\) 5.14449i 0.212698i
\(586\) 14.7928i 0.611085i
\(587\) 27.0706 1.11732 0.558662 0.829396i \(-0.311315\pi\)
0.558662 + 0.829396i \(0.311315\pi\)
\(588\) 12.8937 0.531727
\(589\) 30.5028 1.25685
\(590\) 16.5034 0.679432
\(591\) 37.0018i 1.52205i
\(592\) −3.87316 −0.159186
\(593\) 12.3372i 0.506628i 0.967384 + 0.253314i \(0.0815205\pi\)
−0.967384 + 0.253314i \(0.918479\pi\)
\(594\) 3.77225 0.154777
\(595\) 44.4606i 1.82271i
\(596\) 10.3167i 0.422589i
\(597\) 3.31750i 0.135776i
\(598\) 5.52316 0.225859
\(599\) 11.2676i 0.460382i −0.973145 0.230191i \(-0.926065\pi\)
0.973145 0.230191i \(-0.0739352\pi\)
\(600\) 17.5118i 0.714916i
\(601\) 8.67363i 0.353805i −0.984228 0.176902i \(-0.943392\pi\)
0.984228 0.176902i \(-0.0566077\pi\)
\(602\) 10.9937i 0.448068i
\(603\) −6.29625 −0.256403
\(604\) −2.20108 −0.0895605
\(605\) 34.7942 1.41459
\(606\) −22.9509 −0.932316
\(607\) −19.8405 −0.805299 −0.402650 0.915354i \(-0.631911\pi\)
−0.402650 + 0.915354i \(0.631911\pi\)
\(608\) 22.2132i 0.900866i
\(609\) 32.5631i 1.31952i
\(610\) 22.7993 0.923115
\(611\) 11.3986 0.461139
\(612\) −22.5285 −0.910659
\(613\) −45.1114 −1.82203 −0.911017 0.412369i \(-0.864701\pi\)
−0.911017 + 0.412369i \(0.864701\pi\)
\(614\) 2.59132i 0.104577i
\(615\) 26.4247 1.06555
\(616\) 59.6184i 2.40209i
\(617\) −18.6381 −0.750340 −0.375170 0.926956i \(-0.622416\pi\)
−0.375170 + 0.926956i \(0.622416\pi\)
\(618\) −11.7505 −0.472674
\(619\) 11.8408i 0.475923i 0.971275 + 0.237961i \(0.0764792\pi\)
−0.971275 + 0.237961i \(0.923521\pi\)
\(620\) 11.6512 0.467924
\(621\) −3.18532 −0.127823
\(622\) 23.6851 0.949685
\(623\) −46.3919 −1.85865
\(624\) 2.40601i 0.0963174i
\(625\) −6.67544 −0.267018
\(626\) 12.7861i 0.511035i
\(627\) 59.3486i 2.37015i
\(628\) 5.88064 0.234663
\(629\) −36.4662 −1.45400
\(630\) 14.8554 0.591855
\(631\) −36.4135 −1.44960 −0.724800 0.688960i \(-0.758068\pi\)
−0.724800 + 0.688960i \(0.758068\pi\)
\(632\) −21.3623 −0.849745
\(633\) 1.29567i 0.0514982i
\(634\) −4.62251 −0.183583
\(635\) 7.44969i 0.295632i
\(636\) 10.2088i 0.404805i
\(637\) 6.16431i 0.244239i
\(638\) 21.9122 0.867511
\(639\) 8.13004i 0.321620i
\(640\) 5.83181i 0.230523i
\(641\) −20.0689 −0.792674 −0.396337 0.918105i \(-0.629719\pi\)
−0.396337 + 0.918105i \(0.629719\pi\)
\(642\) −42.3361 −1.67087
\(643\) 1.91957i 0.0757005i −0.999283 0.0378502i \(-0.987949\pi\)
0.999283 0.0378502i \(-0.0120510\pi\)
\(644\) 17.1967i 0.677646i
\(645\) 12.2375i 0.481850i
\(646\) 33.9277i 1.33487i
\(647\) −5.25465 −0.206582 −0.103291 0.994651i \(-0.532937\pi\)
−0.103291 + 0.994651i \(0.532937\pi\)
\(648\) 29.0936i 1.14291i
\(649\) 60.4401 2.37248
\(650\) 2.85990 0.112174
\(651\) 58.8099i 2.30494i
\(652\) 3.00127i 0.117539i
\(653\) −47.0966 −1.84303 −0.921516 0.388339i \(-0.873049\pi\)
−0.921516 + 0.388339i \(0.873049\pi\)
\(654\) 10.5422i 0.412234i
\(655\) 20.1280i 0.786466i
\(656\) −5.87601 −0.229420
\(657\) 16.2562 0.634216
\(658\) 32.9152i 1.28317i
\(659\) 30.4873i 1.18762i −0.804607 0.593808i \(-0.797624\pi\)
0.804607 0.593808i \(-0.202376\pi\)
\(660\) 22.6695i 0.882409i
\(661\) 7.38284 0.287159 0.143580 0.989639i \(-0.454139\pi\)
0.143580 + 0.989639i \(0.454139\pi\)
\(662\) 13.2249 0.513999
\(663\) 22.6528i 0.879762i
\(664\) 46.4983i 1.80448i
\(665\) 24.1225i 0.935432i
\(666\) 12.1843i 0.472132i
\(667\) −18.5028 −0.716433
\(668\) 5.82782i 0.225485i
\(669\) 32.4193 1.25340
\(670\) 3.62190i 0.139926i
\(671\) 83.4975 3.22339
\(672\) 42.8275 1.65211
\(673\) 30.5443 1.17739 0.588697 0.808353i \(-0.299641\pi\)
0.588697 + 0.808353i \(0.299641\pi\)
\(674\) 0.522847i 0.0201393i
\(675\) −1.64936 −0.0634840
\(676\) 12.0289 0.462649
\(677\) 23.8157i 0.915311i 0.889129 + 0.457656i \(0.151311\pi\)
−0.889129 + 0.457656i \(0.848689\pi\)
\(678\) 12.8406 0.493140
\(679\) 1.69970 0.0652284
\(680\) 37.9380i 1.45485i
\(681\) 39.6373 1.51891
\(682\) −39.5740 −1.51537
\(683\) −4.65270 −0.178031 −0.0890153 0.996030i \(-0.528372\pi\)
−0.0890153 + 0.996030i \(0.528372\pi\)
\(684\) −12.2230 −0.467360
\(685\) 27.7059i 1.05859i
\(686\) −6.18088 −0.235987
\(687\) 33.4737i 1.27710i
\(688\) 2.72122i 0.103746i
\(689\) −4.88070 −0.185940
\(690\) 17.7534i 0.675859i
\(691\) 7.84795i 0.298550i −0.988796 0.149275i \(-0.952306\pi\)
0.988796 0.149275i \(-0.0476940\pi\)
\(692\) 5.04998i 0.191972i
\(693\) 54.4049 2.06667
\(694\) 4.74423 0.180089
\(695\) −15.2503 −0.578476
\(696\) 27.7859i 1.05322i
\(697\) −55.3233 −2.09552
\(698\) 8.97019 + 15.9811i 0.339527 + 0.604895i
\(699\) −36.6060 −1.38457
\(700\) 8.90448i 0.336558i
\(701\) −24.1372 −0.911649 −0.455825 0.890070i \(-0.650656\pi\)
−0.455825 + 0.890070i \(0.650656\pi\)
\(702\) −0.781196 −0.0294843
\(703\) −19.7851 −0.746209
\(704\) 38.5354i 1.45236i
\(705\) 36.6392i 1.37991i
\(706\) 23.5803i 0.887455i
\(707\) 34.1637 1.28486
\(708\) 26.1804i 0.983921i
\(709\) 30.6567i 1.15134i 0.817683 + 0.575669i \(0.195258\pi\)
−0.817683 + 0.575669i \(0.804742\pi\)
\(710\) −4.67679 −0.175517
\(711\) 19.4942i 0.731089i
\(712\) −39.5859 −1.48354
\(713\) 33.4167 1.25146
\(714\) −65.4132 −2.44803
\(715\) −10.8380 −0.405319
\(716\) 14.3366i 0.535784i
\(717\) 44.9064 1.67706
\(718\) −24.3559 −0.908953
\(719\) 18.4804i 0.689201i 0.938749 + 0.344600i \(0.111986\pi\)
−0.938749 + 0.344600i \(0.888014\pi\)
\(720\) 3.67711 0.137038
\(721\) 17.4913 0.651410
\(722\) 0.231115i 0.00860120i
\(723\) 12.0116 0.446715
\(724\) −2.30764 −0.0857628
\(725\) −9.58079 −0.355821
\(726\) 51.1914i 1.89989i
\(727\) −1.95848 −0.0726360 −0.0363180 0.999340i \(-0.511563\pi\)
−0.0363180 + 0.999340i \(0.511563\pi\)
\(728\) 12.3464i 0.457588i
\(729\) −21.7338 −0.804955
\(730\) 9.35135i 0.346109i
\(731\) 25.6206i 0.947612i
\(732\) 36.1681i 1.33681i
\(733\) 29.1103i 1.07521i −0.843196 0.537606i \(-0.819329\pi\)
0.843196 0.537606i \(-0.180671\pi\)
\(734\) −21.7140 −0.801479
\(735\) −19.8142 −0.730859
\(736\) 24.3352i 0.897008i
\(737\) 13.2645i 0.488603i
\(738\) 18.4849i 0.680440i
\(739\) 29.5061 1.08540 0.542699 0.839927i \(-0.317402\pi\)
0.542699 + 0.839927i \(0.317402\pi\)
\(740\) −7.55736 −0.277814
\(741\) 12.2905i 0.451503i
\(742\) 14.0937i 0.517396i
\(743\) 1.59648 0.0585691 0.0292846 0.999571i \(-0.490677\pi\)
0.0292846 + 0.999571i \(0.490677\pi\)
\(744\) 50.1822i 1.83977i
\(745\) 15.8541i 0.580850i
\(746\) 26.0807 0.954883
\(747\) −42.4322 −1.55251
\(748\) 47.4612i 1.73535i
\(749\) 63.0197 2.30269
\(750\) 27.8978i 1.01868i
\(751\) 34.3164i 1.25222i −0.779734 0.626112i \(-0.784645\pi\)
0.779734 0.626112i \(-0.215355\pi\)
\(752\) 8.14738i 0.297104i
\(753\) 4.74114i 0.172777i
\(754\) −4.53779 −0.165257
\(755\) 3.38248 0.123101
\(756\) 2.43230i 0.0884621i
\(757\) 7.39838i 0.268899i −0.990920 0.134449i \(-0.957073\pi\)
0.990920 0.134449i \(-0.0429265\pi\)
\(758\) 11.4840 0.417119
\(759\) 65.0180i 2.36000i
\(760\) 20.5836i 0.746647i
\(761\) 18.4892i 0.670235i 0.942176 + 0.335117i \(0.108776\pi\)
−0.942176 + 0.335117i \(0.891224\pi\)
\(762\) 10.9605 0.397055
\(763\) 15.6927i 0.568115i
\(764\) −10.9152 −0.394897
\(765\) 34.6204 1.25170
\(766\) 5.51679 0.199330
\(767\) −12.5166 −0.451946
\(768\) 40.7532 1.47055
\(769\) 2.49935i 0.0901289i −0.998984 0.0450644i \(-0.985651\pi\)
0.998984 0.0450644i \(-0.0143493\pi\)
\(770\) 31.2963i 1.12784i
\(771\) 69.0227 2.48579
\(772\) 0.665155i 0.0239395i
\(773\) 47.4026 1.70495 0.852476 0.522766i \(-0.175100\pi\)
0.852476 + 0.522766i \(0.175100\pi\)
\(774\) −8.56051 −0.307701
\(775\) 17.3032 0.621548
\(776\) 1.45034 0.0520643
\(777\) 38.1460i 1.36848i
\(778\) −21.6758 −0.777117
\(779\) −30.0162 −1.07544
\(780\) 4.69463i 0.168095i
\(781\) −17.1278 −0.612879
\(782\) 37.1687i 1.32915i
\(783\) 2.61704 0.0935254
\(784\) 4.40605 0.157359
\(785\) −9.03701 −0.322545
\(786\) 29.6136 1.05628
\(787\) 13.2402i 0.471961i 0.971758 + 0.235981i \(0.0758302\pi\)
−0.971758 + 0.235981i \(0.924170\pi\)
\(788\) 16.0546i 0.571922i
\(789\) −1.10964 −0.0395042
\(790\) 11.2140 0.398975
\(791\) −19.1140 −0.679615
\(792\) 46.4234 1.64958
\(793\) −17.2915 −0.614040
\(794\) 0.301080i 0.0106849i
\(795\) 15.6883i 0.556405i
\(796\) 1.43942i 0.0510189i
\(797\) 28.5449i 1.01111i 0.862794 + 0.505556i \(0.168713\pi\)
−0.862794 + 0.505556i \(0.831287\pi\)
\(798\) −35.4906 −1.25635
\(799\) 76.7084i 2.71375i
\(800\) 12.6008i 0.445505i
\(801\) 36.1242i 1.27639i
\(802\) 6.45709 0.228008
\(803\) 34.2474i 1.20856i
\(804\) 5.74568 0.202635
\(805\) 26.4269i 0.931426i
\(806\) 8.19539 0.288670
\(807\) −1.01513 −0.0357344
\(808\) 29.1517 1.02555
\(809\) 33.7152 1.18536 0.592681 0.805437i \(-0.298070\pi\)
0.592681 + 0.805437i \(0.298070\pi\)
\(810\) 15.2725i 0.536621i
\(811\) 0.888225i 0.0311898i 0.999878 + 0.0155949i \(0.00496421\pi\)
−0.999878 + 0.0155949i \(0.995036\pi\)
\(812\) 14.1287i 0.495821i
\(813\) −52.6837 −1.84770
\(814\) 25.6690 0.899697
\(815\) 4.61217i 0.161557i
\(816\) −16.1915 −0.566816
\(817\) 13.9007i 0.486324i
\(818\) 5.34288i 0.186809i
\(819\) −11.2667 −0.393691
\(820\) −11.4654 −0.400387
\(821\) 20.6766 0.721619 0.360810 0.932639i \(-0.382500\pi\)
0.360810 + 0.932639i \(0.382500\pi\)
\(822\) 40.7626 1.42176
\(823\) 5.34483 0.186309 0.0931545 0.995652i \(-0.470305\pi\)
0.0931545 + 0.995652i \(0.470305\pi\)
\(824\) 14.9252 0.519945
\(825\) 33.6664i 1.17211i
\(826\) 36.1433i 1.25759i
\(827\) 22.5308i 0.783472i 0.920078 + 0.391736i \(0.128125\pi\)
−0.920078 + 0.391736i \(0.871875\pi\)
\(828\) −13.3907 −0.465358
\(829\) 2.34189i 0.0813372i −0.999173 0.0406686i \(-0.987051\pi\)
0.999173 0.0406686i \(-0.0129488\pi\)
\(830\) 24.4090i 0.847248i
\(831\) 74.3017i 2.57750i
\(832\) 7.98030i 0.276667i
\(833\) 41.4834 1.43732
\(834\) 22.4371i 0.776935i
\(835\) 8.95585i 0.309930i
\(836\) 25.7506i 0.890602i
\(837\) −4.72645 −0.163370
\(838\) 29.5551i 1.02096i
\(839\) 53.7752i 1.85653i 0.371925 + 0.928263i \(0.378698\pi\)
−0.371925 + 0.928263i \(0.621302\pi\)
\(840\) −39.6856 −1.36928
\(841\) −13.7982 −0.475800
\(842\) −14.4848 −0.499178
\(843\) 39.5616 1.36257
\(844\) 0.562174i 0.0193508i
\(845\) −18.4853 −0.635912
\(846\) 25.6303 0.881187
\(847\) 76.2014i 2.61831i
\(848\) 3.48856i 0.119798i
\(849\) −30.4124 −1.04375
\(850\) 19.2460i 0.660132i
\(851\) −21.6751 −0.743014
\(852\) 7.41912i 0.254175i
\(853\) 20.4467 0.700080 0.350040 0.936735i \(-0.386168\pi\)
0.350040 + 0.936735i \(0.386168\pi\)
\(854\) 49.9317i 1.70863i
\(855\) 18.7836 0.642387
\(856\) 53.7744 1.83797
\(857\) 43.2339i 1.47684i −0.674340 0.738421i \(-0.735572\pi\)
0.674340 0.738421i \(-0.264428\pi\)
\(858\) 15.9456i 0.544373i
\(859\) 35.0119i 1.19459i −0.802021 0.597295i \(-0.796242\pi\)
0.802021 0.597295i \(-0.203758\pi\)
\(860\) 5.30969i 0.181059i
\(861\) 57.8717i 1.97226i
\(862\) 10.8244 0.368680
\(863\) 2.88320i 0.0981454i 0.998795 + 0.0490727i \(0.0156266\pi\)
−0.998795 + 0.0490727i \(0.984373\pi\)
\(864\) 3.44197i 0.117098i
\(865\) 7.76051i 0.263865i
\(866\) −29.4468 −1.00064
\(867\) −111.789 −3.79655
\(868\) 25.5169i 0.866099i
\(869\) 41.0689 1.39317
\(870\) 14.5860i 0.494513i
\(871\) 2.74694i 0.0930766i
\(872\) 13.3905i 0.453460i
\(873\) 1.32351i 0.0447942i
\(874\) 20.1663i 0.682134i
\(875\) 41.5275i 1.40389i
\(876\) −14.8347 −0.501219
\(877\) 32.5115i 1.09783i 0.835877 + 0.548917i \(0.184960\pi\)
−0.835877 + 0.548917i \(0.815040\pi\)
\(878\) 38.7524 1.30783
\(879\) −36.0625 −1.21636
\(880\) 7.74666i 0.261140i
\(881\) 16.7339i 0.563780i −0.959447 0.281890i \(-0.909039\pi\)
0.959447 0.281890i \(-0.0909613\pi\)
\(882\) 13.8607i 0.466714i
\(883\) −42.2003 −1.42015 −0.710076 0.704125i \(-0.751340\pi\)
−0.710076 + 0.704125i \(0.751340\pi\)
\(884\) 9.82876i 0.330577i
\(885\) 40.2326i 1.35240i
\(886\) 17.0824i 0.573894i
\(887\) 4.10824i 0.137941i −0.997619 0.0689705i \(-0.978029\pi\)
0.997619 0.0689705i \(-0.0219714\pi\)
\(888\) 32.5498i 1.09230i
\(889\) −16.3153 −0.547196
\(890\) 20.7804 0.696560
\(891\) 55.9324i 1.87381i
\(892\) −14.0663 −0.470975
\(893\) 41.6189i 1.39272i
\(894\) −23.3256 −0.780124
\(895\) 22.0316i 0.736436i
\(896\) −12.7720 −0.426683
\(897\) 13.4646i 0.449570i
\(898\) 4.01152i 0.133866i
\(899\) −27.4549 −0.915672
\(900\) −6.93370 −0.231123
\(901\) 32.8452i 1.09423i
\(902\) 38.9427 1.29665
\(903\) 26.8008 0.891875
\(904\) −16.3098 −0.542458
\(905\) 3.54625 0.117881
\(906\) 4.97652i 0.165334i
\(907\) 10.1189i 0.335993i −0.985788 0.167997i \(-0.946270\pi\)
0.985788 0.167997i \(-0.0537298\pi\)
\(908\) −17.1981 −0.570740
\(909\) 26.6025i 0.882348i
\(910\) 6.48116i 0.214848i
\(911\) 15.1869i 0.503164i 0.967836 + 0.251582i \(0.0809507\pi\)
−0.967836 + 0.251582i \(0.919049\pi\)
\(912\) −8.78486 −0.290896
\(913\) 89.3928i 2.95847i
\(914\) 1.40947i 0.0466211i
\(915\) 55.5810i 1.83745i
\(916\) 14.5238i 0.479880i
\(917\) −44.0815 −1.45570
\(918\) 5.25714i 0.173512i
\(919\) 41.2261i 1.35992i 0.733247 + 0.679962i \(0.238004\pi\)
−0.733247 + 0.679962i \(0.761996\pi\)
\(920\) 22.5499i 0.743449i
\(921\) 6.31722 0.208159
\(922\) −16.6031 −0.546794
\(923\) 3.54699 0.116751
\(924\) −49.6475 −1.63328
\(925\) −11.2234 −0.369023
\(926\) −16.4393 −0.540228
\(927\) 13.6200i 0.447341i
\(928\) 19.9936i 0.656324i
\(929\) 35.9091 1.17814 0.589070 0.808082i \(-0.299494\pi\)
0.589070 + 0.808082i \(0.299494\pi\)
\(930\) 26.3428i 0.863815i
\(931\) 22.5072 0.737645
\(932\) 15.8829 0.520261
\(933\) 57.7405i 1.89034i
\(934\) 8.81547i 0.288451i
\(935\) 72.9356i 2.38525i
\(936\) −9.61384 −0.314238
\(937\) 13.4405 0.439081 0.219541 0.975603i \(-0.429544\pi\)
0.219541 + 0.975603i \(0.429544\pi\)
\(938\) 7.93218 0.258995
\(939\) 31.1704 1.01721
\(940\) 15.8973i 0.518512i
\(941\) 34.0899 1.11130 0.555649 0.831417i \(-0.312470\pi\)
0.555649 + 0.831417i \(0.312470\pi\)
\(942\) 13.2958i 0.433201i
\(943\) −32.8836 −1.07084
\(944\) 8.94643i 0.291181i
\(945\) 3.73782i 0.121591i
\(946\) 18.0346i 0.586356i
\(947\) 25.4684 0.827611 0.413805 0.910365i \(-0.364199\pi\)
0.413805 + 0.910365i \(0.364199\pi\)
\(948\) 17.7895i 0.577777i
\(949\) 7.09230i 0.230226i
\(950\) 10.4421i 0.338787i
\(951\) 11.2689i 0.365420i
\(952\) 83.0864 2.69285
\(953\) −12.5906 −0.407849 −0.203924 0.978987i \(-0.565370\pi\)
−0.203924 + 0.978987i \(0.565370\pi\)
\(954\) −10.9744 −0.355310
\(955\) 16.7738 0.542787
\(956\) −19.4843 −0.630167
\(957\) 53.4183i 1.72677i
\(958\) 8.62101i 0.278532i
\(959\) −60.6775 −1.95938
\(960\) −25.6514 −0.827897
\(961\) 18.5844 0.599495
\(962\) −5.31579 −0.171388
\(963\) 49.0719i 1.58132i
\(964\) −5.21166 −0.167856
\(965\) 1.02217i 0.0329048i
\(966\) −38.8809 −1.25097
\(967\) 24.0220 0.772496 0.386248 0.922395i \(-0.373771\pi\)
0.386248 + 0.922395i \(0.373771\pi\)
\(968\) 65.0222i 2.08989i
\(969\) −82.7103 −2.65704
\(970\) −0.761348 −0.0244454
\(971\) 35.4625 1.13805 0.569024 0.822321i \(-0.307321\pi\)
0.569024 + 0.822321i \(0.307321\pi\)
\(972\) 22.1384 0.710090
\(973\) 33.3990i 1.07072i
\(974\) −10.5539 −0.338169
\(975\) 6.97198i 0.223282i
\(976\) 12.3594i 0.395615i
\(977\) 35.7682 1.14433 0.572163 0.820140i \(-0.306104\pi\)
0.572163 + 0.820140i \(0.306104\pi\)
\(978\) −6.78571 −0.216983
\(979\) 76.1038 2.43229
\(980\) 8.59714 0.274626
\(981\) 12.2195 0.390140
\(982\) 25.1823i 0.803598i
\(983\) 24.8100 0.791316 0.395658 0.918398i \(-0.370517\pi\)
0.395658 + 0.918398i \(0.370517\pi\)
\(984\) 49.3816i 1.57423i
\(985\) 24.6718i 0.786108i
\(986\) 30.5376i 0.972515i
\(987\) −80.2420 −2.55413
\(988\) 5.33269i 0.169656i
\(989\) 15.2286i 0.484242i
\(990\) −24.3697 −0.774519
\(991\) 6.88373 0.218669 0.109335 0.994005i \(-0.465128\pi\)
0.109335 + 0.994005i \(0.465128\pi\)
\(992\) 36.1091i 1.14647i
\(993\) 32.2401i 1.02311i
\(994\) 10.2424i 0.324870i
\(995\) 2.21202i 0.0701256i
\(996\) 38.7217 1.22694
\(997\) 13.6028i 0.430806i −0.976525 0.215403i \(-0.930893\pi\)
0.976525 0.215403i \(-0.0691065\pi\)
\(998\) −27.9786 −0.885647
\(999\) 3.06573 0.0969954
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 349.2.b.b.348.16 yes 26
349.348 even 2 inner 349.2.b.b.348.11 26
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
349.2.b.b.348.11 26 349.348 even 2 inner
349.2.b.b.348.16 yes 26 1.1 even 1 trivial