Properties

Label 169.2.b.b.168.5
Level $169$
Weight $2$
Character 169.168
Analytic conductor $1.349$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [169,2,Mod(168,169)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(169, 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("169.168");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 169 = 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 169.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.34947179416\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 168.5
Root \(1.24698i\) of defining polynomial
Character \(\chi\) \(=\) 169.168
Dual form 169.2.b.b.168.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.801938i q^{2} -2.24698 q^{3} +1.35690 q^{4} +0.246980i q^{5} -1.80194i q^{6} +2.35690i q^{7} +2.69202i q^{8} +2.04892 q^{9} +O(q^{10})\) \(q+0.801938i q^{2} -2.24698 q^{3} +1.35690 q^{4} +0.246980i q^{5} -1.80194i q^{6} +2.35690i q^{7} +2.69202i q^{8} +2.04892 q^{9} -0.198062 q^{10} +4.24698i q^{11} -3.04892 q^{12} -1.89008 q^{14} -0.554958i q^{15} +0.554958 q^{16} -2.15883 q^{17} +1.64310i q^{18} -0.0881460i q^{19} +0.335126i q^{20} -5.29590i q^{21} -3.40581 q^{22} -1.49396 q^{23} -6.04892i q^{24} +4.93900 q^{25} +2.13706 q^{27} +3.19806i q^{28} +4.63102 q^{29} +0.445042 q^{30} -6.63102i q^{31} +5.82908i q^{32} -9.54288i q^{33} -1.73125i q^{34} -0.582105 q^{35} +2.78017 q^{36} -5.69202i q^{37} +0.0706876 q^{38} -0.664874 q^{40} -11.5918i q^{41} +4.24698 q^{42} +0.295897 q^{43} +5.76271i q^{44} +0.506041i q^{45} -1.19806i q^{46} +7.35690i q^{47} -1.24698 q^{48} +1.44504 q^{49} +3.96077i q^{50} +4.85086 q^{51} -10.3937 q^{53} +1.71379i q^{54} -1.04892 q^{55} -6.34481 q^{56} +0.198062i q^{57} +3.71379i q^{58} +6.78017i q^{59} -0.753020i q^{60} +3.47219 q^{61} +5.31767 q^{62} +4.82908i q^{63} -3.56465 q^{64} +7.65279 q^{66} +7.67994i q^{67} -2.92931 q^{68} +3.35690 q^{69} -0.466812i q^{70} -8.66487i q^{71} +5.51573i q^{72} -6.73556i q^{73} +4.56465 q^{74} -11.0978 q^{75} -0.119605i q^{76} -10.0097 q^{77} +9.97046 q^{79} +0.137063i q^{80} -10.9487 q^{81} +9.29590 q^{82} +1.60925i q^{83} -7.18598i q^{84} -0.533188i q^{85} +0.237291i q^{86} -10.4058 q^{87} -11.4330 q^{88} +2.88471i q^{89} -0.405813 q^{90} -2.02715 q^{92} +14.8998i q^{93} -5.89977 q^{94} +0.0217703 q^{95} -13.0978i q^{96} -8.05861i q^{97} +1.15883i q^{98} +8.70171i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{3} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{3} - 6 q^{9} - 10 q^{10} - 10 q^{14} + 4 q^{16} + 4 q^{17} + 6 q^{22} + 10 q^{23} + 10 q^{25} + 2 q^{27} - 2 q^{29} + 2 q^{30} + 8 q^{35} + 14 q^{36} - 24 q^{38} - 6 q^{40} + 16 q^{42} - 26 q^{43} + 2 q^{48} + 8 q^{49} + 2 q^{51} + 2 q^{53} + 12 q^{55} + 8 q^{56} + 8 q^{61} - 2 q^{62} + 22 q^{64} + 10 q^{66} - 42 q^{68} + 12 q^{69} - 16 q^{74} - 30 q^{75} - 16 q^{77} - 10 q^{79} - 2 q^{81} + 28 q^{82} - 36 q^{87} - 30 q^{88} + 24 q^{90} + 10 q^{94} - 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/169\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.801938i 0.567056i 0.958964 + 0.283528i \(0.0915048\pi\)
−0.958964 + 0.283528i \(0.908495\pi\)
\(3\) −2.24698 −1.29729 −0.648647 0.761089i \(-0.724665\pi\)
−0.648647 + 0.761089i \(0.724665\pi\)
\(4\) 1.35690 0.678448
\(5\) 0.246980i 0.110453i 0.998474 + 0.0552263i \(0.0175880\pi\)
−0.998474 + 0.0552263i \(0.982412\pi\)
\(6\) − 1.80194i − 0.735638i
\(7\) 2.35690i 0.890823i 0.895326 + 0.445411i \(0.146943\pi\)
−0.895326 + 0.445411i \(0.853057\pi\)
\(8\) 2.69202i 0.951773i
\(9\) 2.04892 0.682972
\(10\) −0.198062 −0.0626328
\(11\) 4.24698i 1.28051i 0.768161 + 0.640256i \(0.221172\pi\)
−0.768161 + 0.640256i \(0.778828\pi\)
\(12\) −3.04892 −0.880147
\(13\) 0 0
\(14\) −1.89008 −0.505146
\(15\) − 0.554958i − 0.143290i
\(16\) 0.554958 0.138740
\(17\) −2.15883 −0.523594 −0.261797 0.965123i \(-0.584315\pi\)
−0.261797 + 0.965123i \(0.584315\pi\)
\(18\) 1.64310i 0.387283i
\(19\) − 0.0881460i − 0.0202221i −0.999949 0.0101110i \(-0.996782\pi\)
0.999949 0.0101110i \(-0.00321850\pi\)
\(20\) 0.335126i 0.0749364i
\(21\) − 5.29590i − 1.15566i
\(22\) −3.40581 −0.726122
\(23\) −1.49396 −0.311512 −0.155756 0.987796i \(-0.549781\pi\)
−0.155756 + 0.987796i \(0.549781\pi\)
\(24\) − 6.04892i − 1.23473i
\(25\) 4.93900 0.987800
\(26\) 0 0
\(27\) 2.13706 0.411278
\(28\) 3.19806i 0.604377i
\(29\) 4.63102 0.859959 0.429980 0.902839i \(-0.358521\pi\)
0.429980 + 0.902839i \(0.358521\pi\)
\(30\) 0.445042 0.0812532
\(31\) − 6.63102i − 1.19097i −0.803368 0.595483i \(-0.796961\pi\)
0.803368 0.595483i \(-0.203039\pi\)
\(32\) 5.82908i 1.03045i
\(33\) − 9.54288i − 1.66120i
\(34\) − 1.73125i − 0.296907i
\(35\) −0.582105 −0.0983937
\(36\) 2.78017 0.463361
\(37\) − 5.69202i − 0.935763i −0.883791 0.467881i \(-0.845017\pi\)
0.883791 0.467881i \(-0.154983\pi\)
\(38\) 0.0706876 0.0114670
\(39\) 0 0
\(40\) −0.664874 −0.105126
\(41\) − 11.5918i − 1.81033i −0.425056 0.905167i \(-0.639746\pi\)
0.425056 0.905167i \(-0.360254\pi\)
\(42\) 4.24698 0.655323
\(43\) 0.295897 0.0451239 0.0225619 0.999745i \(-0.492818\pi\)
0.0225619 + 0.999745i \(0.492818\pi\)
\(44\) 5.76271i 0.868761i
\(45\) 0.506041i 0.0754361i
\(46\) − 1.19806i − 0.176645i
\(47\) 7.35690i 1.07311i 0.843864 + 0.536557i \(0.180275\pi\)
−0.843864 + 0.536557i \(0.819725\pi\)
\(48\) −1.24698 −0.179986
\(49\) 1.44504 0.206435
\(50\) 3.96077i 0.560138i
\(51\) 4.85086 0.679256
\(52\) 0 0
\(53\) −10.3937 −1.42769 −0.713844 0.700304i \(-0.753048\pi\)
−0.713844 + 0.700304i \(0.753048\pi\)
\(54\) 1.71379i 0.233218i
\(55\) −1.04892 −0.141436
\(56\) −6.34481 −0.847861
\(57\) 0.198062i 0.0262340i
\(58\) 3.71379i 0.487645i
\(59\) 6.78017i 0.882703i 0.897334 + 0.441351i \(0.145501\pi\)
−0.897334 + 0.441351i \(0.854499\pi\)
\(60\) − 0.753020i − 0.0972145i
\(61\) 3.47219 0.444568 0.222284 0.974982i \(-0.428649\pi\)
0.222284 + 0.974982i \(0.428649\pi\)
\(62\) 5.31767 0.675344
\(63\) 4.82908i 0.608407i
\(64\) −3.56465 −0.445581
\(65\) 0 0
\(66\) 7.65279 0.941994
\(67\) 7.67994i 0.938254i 0.883131 + 0.469127i \(0.155431\pi\)
−0.883131 + 0.469127i \(0.844569\pi\)
\(68\) −2.92931 −0.355231
\(69\) 3.35690 0.404123
\(70\) − 0.466812i − 0.0557947i
\(71\) − 8.66487i − 1.02833i −0.857691 0.514166i \(-0.828102\pi\)
0.857691 0.514166i \(-0.171898\pi\)
\(72\) 5.51573i 0.650035i
\(73\) − 6.73556i − 0.788338i −0.919038 0.394169i \(-0.871032\pi\)
0.919038 0.394169i \(-0.128968\pi\)
\(74\) 4.56465 0.530629
\(75\) −11.0978 −1.28147
\(76\) − 0.119605i − 0.0137196i
\(77\) −10.0097 −1.14071
\(78\) 0 0
\(79\) 9.97046 1.12176 0.560882 0.827896i \(-0.310462\pi\)
0.560882 + 0.827896i \(0.310462\pi\)
\(80\) 0.137063i 0.0153241i
\(81\) −10.9487 −1.21652
\(82\) 9.29590 1.02656
\(83\) 1.60925i 0.176638i 0.996092 + 0.0883192i \(0.0281495\pi\)
−0.996092 + 0.0883192i \(0.971850\pi\)
\(84\) − 7.18598i − 0.784055i
\(85\) − 0.533188i − 0.0578323i
\(86\) 0.237291i 0.0255877i
\(87\) −10.4058 −1.11562
\(88\) −11.4330 −1.21876
\(89\) 2.88471i 0.305778i 0.988243 + 0.152889i \(0.0488577\pi\)
−0.988243 + 0.152889i \(0.951142\pi\)
\(90\) −0.405813 −0.0427765
\(91\) 0 0
\(92\) −2.02715 −0.211345
\(93\) 14.8998i 1.54503i
\(94\) −5.89977 −0.608515
\(95\) 0.0217703 0.00223358
\(96\) − 13.0978i − 1.33679i
\(97\) − 8.05861i − 0.818227i −0.912483 0.409114i \(-0.865838\pi\)
0.912483 0.409114i \(-0.134162\pi\)
\(98\) 1.15883i 0.117060i
\(99\) 8.70171i 0.874555i
\(100\) 6.70171 0.670171
\(101\) 13.3545 1.32882 0.664411 0.747367i \(-0.268682\pi\)
0.664411 + 0.747367i \(0.268682\pi\)
\(102\) 3.89008i 0.385176i
\(103\) −1.36227 −0.134229 −0.0671144 0.997745i \(-0.521379\pi\)
−0.0671144 + 0.997745i \(0.521379\pi\)
\(104\) 0 0
\(105\) 1.30798 0.127646
\(106\) − 8.33513i − 0.809579i
\(107\) 3.26875 0.316002 0.158001 0.987439i \(-0.449495\pi\)
0.158001 + 0.987439i \(0.449495\pi\)
\(108\) 2.89977 0.279031
\(109\) 15.7017i 1.50395i 0.659191 + 0.751976i \(0.270899\pi\)
−0.659191 + 0.751976i \(0.729101\pi\)
\(110\) − 0.841166i − 0.0802021i
\(111\) 12.7899i 1.21396i
\(112\) 1.30798i 0.123592i
\(113\) 12.0489 1.13347 0.566733 0.823901i \(-0.308207\pi\)
0.566733 + 0.823901i \(0.308207\pi\)
\(114\) −0.158834 −0.0148761
\(115\) − 0.368977i − 0.0344073i
\(116\) 6.28382 0.583438
\(117\) 0 0
\(118\) −5.43727 −0.500541
\(119\) − 5.08815i − 0.466430i
\(120\) 1.49396 0.136379
\(121\) −7.03684 −0.639712
\(122\) 2.78448i 0.252095i
\(123\) 26.0465i 2.34854i
\(124\) − 8.99761i − 0.808009i
\(125\) 2.45473i 0.219558i
\(126\) −3.87263 −0.345001
\(127\) 9.80731 0.870258 0.435129 0.900368i \(-0.356703\pi\)
0.435129 + 0.900368i \(0.356703\pi\)
\(128\) 8.79954i 0.777777i
\(129\) −0.664874 −0.0585389
\(130\) 0 0
\(131\) −6.57673 −0.574611 −0.287306 0.957839i \(-0.592760\pi\)
−0.287306 + 0.957839i \(0.592760\pi\)
\(132\) − 12.9487i − 1.12704i
\(133\) 0.207751 0.0180143
\(134\) −6.15883 −0.532042
\(135\) 0.527811i 0.0454267i
\(136\) − 5.81163i − 0.498343i
\(137\) − 6.21983i − 0.531396i −0.964056 0.265698i \(-0.914398\pi\)
0.964056 0.265698i \(-0.0856024\pi\)
\(138\) 2.69202i 0.229160i
\(139\) −14.7071 −1.24744 −0.623719 0.781648i \(-0.714379\pi\)
−0.623719 + 0.781648i \(0.714379\pi\)
\(140\) −0.789856 −0.0667550
\(141\) − 16.5308i − 1.39214i
\(142\) 6.94869 0.583121
\(143\) 0 0
\(144\) 1.13706 0.0947553
\(145\) 1.14377i 0.0949848i
\(146\) 5.40150 0.447031
\(147\) −3.24698 −0.267806
\(148\) − 7.72348i − 0.634866i
\(149\) 4.33513i 0.355147i 0.984108 + 0.177574i \(0.0568248\pi\)
−0.984108 + 0.177574i \(0.943175\pi\)
\(150\) − 8.89977i − 0.726663i
\(151\) − 3.94438i − 0.320989i −0.987037 0.160494i \(-0.948691\pi\)
0.987037 0.160494i \(-0.0513089\pi\)
\(152\) 0.237291 0.0192468
\(153\) −4.42327 −0.357600
\(154\) − 8.02715i − 0.646846i
\(155\) 1.63773 0.131545
\(156\) 0 0
\(157\) 4.45473 0.355526 0.177763 0.984073i \(-0.443114\pi\)
0.177763 + 0.984073i \(0.443114\pi\)
\(158\) 7.99569i 0.636103i
\(159\) 23.3545 1.85213
\(160\) −1.43967 −0.113816
\(161\) − 3.52111i − 0.277502i
\(162\) − 8.78017i − 0.689835i
\(163\) − 16.1588i − 1.26566i −0.774292 0.632829i \(-0.781894\pi\)
0.774292 0.632829i \(-0.218106\pi\)
\(164\) − 15.7289i − 1.22822i
\(165\) 2.35690 0.183484
\(166\) −1.29052 −0.100164
\(167\) − 16.1172i − 1.24719i −0.781749 0.623594i \(-0.785672\pi\)
0.781749 0.623594i \(-0.214328\pi\)
\(168\) 14.2567 1.09993
\(169\) 0 0
\(170\) 0.427583 0.0327942
\(171\) − 0.180604i − 0.0138111i
\(172\) 0.401501 0.0306142
\(173\) 21.5362 1.63736 0.818682 0.574247i \(-0.194705\pi\)
0.818682 + 0.574247i \(0.194705\pi\)
\(174\) − 8.34481i − 0.632619i
\(175\) 11.6407i 0.879955i
\(176\) 2.35690i 0.177658i
\(177\) − 15.2349i − 1.14513i
\(178\) −2.31336 −0.173393
\(179\) −11.4330 −0.854540 −0.427270 0.904124i \(-0.640525\pi\)
−0.427270 + 0.904124i \(0.640525\pi\)
\(180\) 0.686645i 0.0511795i
\(181\) −20.9705 −1.55872 −0.779361 0.626575i \(-0.784456\pi\)
−0.779361 + 0.626575i \(0.784456\pi\)
\(182\) 0 0
\(183\) −7.80194 −0.576736
\(184\) − 4.02177i − 0.296489i
\(185\) 1.40581 0.103357
\(186\) −11.9487 −0.876120
\(187\) − 9.16852i − 0.670469i
\(188\) 9.98254i 0.728052i
\(189\) 5.03684i 0.366376i
\(190\) 0.0174584i 0.00126657i
\(191\) −14.4373 −1.04464 −0.522322 0.852748i \(-0.674934\pi\)
−0.522322 + 0.852748i \(0.674934\pi\)
\(192\) 8.00969 0.578049
\(193\) 13.5797i 0.977489i 0.872427 + 0.488745i \(0.162545\pi\)
−0.872427 + 0.488745i \(0.837455\pi\)
\(194\) 6.46250 0.463980
\(195\) 0 0
\(196\) 1.96077 0.140055
\(197\) − 0.560335i − 0.0399222i −0.999801 0.0199611i \(-0.993646\pi\)
0.999801 0.0199611i \(-0.00635424\pi\)
\(198\) −6.97823 −0.495921
\(199\) −11.4916 −0.814616 −0.407308 0.913291i \(-0.633532\pi\)
−0.407308 + 0.913291i \(0.633532\pi\)
\(200\) 13.2959i 0.940162i
\(201\) − 17.2567i − 1.21719i
\(202\) 10.7095i 0.753516i
\(203\) 10.9148i 0.766071i
\(204\) 6.58211 0.460840
\(205\) 2.86294 0.199956
\(206\) − 1.09246i − 0.0761151i
\(207\) −3.06100 −0.212754
\(208\) 0 0
\(209\) 0.374354 0.0258946
\(210\) 1.04892i 0.0723822i
\(211\) 8.78448 0.604748 0.302374 0.953189i \(-0.402221\pi\)
0.302374 + 0.953189i \(0.402221\pi\)
\(212\) −14.1032 −0.968613
\(213\) 19.4698i 1.33405i
\(214\) 2.62133i 0.179191i
\(215\) 0.0730805i 0.00498405i
\(216\) 5.75302i 0.391443i
\(217\) 15.6286 1.06094
\(218\) −12.5918 −0.852824
\(219\) 15.1347i 1.02271i
\(220\) −1.42327 −0.0959570
\(221\) 0 0
\(222\) −10.2567 −0.688383
\(223\) − 2.25906i − 0.151278i −0.997135 0.0756390i \(-0.975900\pi\)
0.997135 0.0756390i \(-0.0240996\pi\)
\(224\) −13.7385 −0.917945
\(225\) 10.1196 0.674640
\(226\) 9.66248i 0.642739i
\(227\) − 6.96615i − 0.462359i −0.972911 0.231180i \(-0.925741\pi\)
0.972911 0.231180i \(-0.0742585\pi\)
\(228\) 0.268750i 0.0177984i
\(229\) 24.1739i 1.59746i 0.601692 + 0.798728i \(0.294493\pi\)
−0.601692 + 0.798728i \(0.705507\pi\)
\(230\) 0.295897 0.0195109
\(231\) 22.4916 1.47984
\(232\) 12.4668i 0.818486i
\(233\) 3.06100 0.200533 0.100266 0.994961i \(-0.468031\pi\)
0.100266 + 0.994961i \(0.468031\pi\)
\(234\) 0 0
\(235\) −1.81700 −0.118528
\(236\) 9.19998i 0.598868i
\(237\) −22.4034 −1.45526
\(238\) 4.08038 0.264492
\(239\) − 25.1468i − 1.62661i −0.581839 0.813304i \(-0.697667\pi\)
0.581839 0.813304i \(-0.302333\pi\)
\(240\) − 0.307979i − 0.0198799i
\(241\) 20.2664i 1.30547i 0.757586 + 0.652735i \(0.226379\pi\)
−0.757586 + 0.652735i \(0.773621\pi\)
\(242\) − 5.64310i − 0.362752i
\(243\) 18.1903 1.16691
\(244\) 4.71140 0.301616
\(245\) 0.356896i 0.0228012i
\(246\) −20.8877 −1.33175
\(247\) 0 0
\(248\) 17.8509 1.13353
\(249\) − 3.61596i − 0.229152i
\(250\) −1.96854 −0.124501
\(251\) 23.7211 1.49726 0.748631 0.662987i \(-0.230712\pi\)
0.748631 + 0.662987i \(0.230712\pi\)
\(252\) 6.55257i 0.412773i
\(253\) − 6.34481i − 0.398895i
\(254\) 7.86486i 0.493485i
\(255\) 1.19806i 0.0750256i
\(256\) −14.1860 −0.886624
\(257\) −14.2241 −0.887278 −0.443639 0.896206i \(-0.646313\pi\)
−0.443639 + 0.896206i \(0.646313\pi\)
\(258\) − 0.533188i − 0.0331948i
\(259\) 13.4155 0.833599
\(260\) 0 0
\(261\) 9.48858 0.587329
\(262\) − 5.27413i − 0.325837i
\(263\) −17.0954 −1.05415 −0.527075 0.849819i \(-0.676711\pi\)
−0.527075 + 0.849819i \(0.676711\pi\)
\(264\) 25.6896 1.58109
\(265\) − 2.56704i − 0.157692i
\(266\) 0.166603i 0.0102151i
\(267\) − 6.48188i − 0.396684i
\(268\) 10.4209i 0.636556i
\(269\) −6.46681 −0.394288 −0.197144 0.980374i \(-0.563167\pi\)
−0.197144 + 0.980374i \(0.563167\pi\)
\(270\) −0.423272 −0.0257595
\(271\) − 6.44803i − 0.391690i −0.980635 0.195845i \(-0.937255\pi\)
0.980635 0.195845i \(-0.0627449\pi\)
\(272\) −1.19806 −0.0726432
\(273\) 0 0
\(274\) 4.98792 0.301331
\(275\) 20.9758i 1.26489i
\(276\) 4.55496 0.274176
\(277\) −13.4601 −0.808739 −0.404370 0.914596i \(-0.632509\pi\)
−0.404370 + 0.914596i \(0.632509\pi\)
\(278\) − 11.7942i − 0.707367i
\(279\) − 13.5864i − 0.813398i
\(280\) − 1.56704i − 0.0936485i
\(281\) − 5.03684i − 0.300472i −0.988650 0.150236i \(-0.951997\pi\)
0.988650 0.150236i \(-0.0480034\pi\)
\(282\) 13.2567 0.789423
\(283\) −22.1280 −1.31537 −0.657686 0.753293i \(-0.728464\pi\)
−0.657686 + 0.753293i \(0.728464\pi\)
\(284\) − 11.7573i − 0.697669i
\(285\) −0.0489173 −0.00289761
\(286\) 0 0
\(287\) 27.3207 1.61269
\(288\) 11.9433i 0.703766i
\(289\) −12.3394 −0.725849
\(290\) −0.917231 −0.0538616
\(291\) 18.1075i 1.06148i
\(292\) − 9.13946i − 0.534846i
\(293\) − 14.9463i − 0.873172i −0.899663 0.436586i \(-0.856187\pi\)
0.899663 0.436586i \(-0.143813\pi\)
\(294\) − 2.60388i − 0.151861i
\(295\) −1.67456 −0.0974968
\(296\) 15.3230 0.890634
\(297\) 9.07606i 0.526647i
\(298\) −3.47650 −0.201388
\(299\) 0 0
\(300\) −15.0586 −0.869409
\(301\) 0.697398i 0.0401974i
\(302\) 3.16315 0.182019
\(303\) −30.0073 −1.72387
\(304\) − 0.0489173i − 0.00280560i
\(305\) 0.857560i 0.0491037i
\(306\) − 3.54719i − 0.202779i
\(307\) − 19.1293i − 1.09177i −0.837861 0.545883i \(-0.816194\pi\)
0.837861 0.545883i \(-0.183806\pi\)
\(308\) −13.5821 −0.773912
\(309\) 3.06100 0.174134
\(310\) 1.31336i 0.0745936i
\(311\) 0.269815 0.0152998 0.00764990 0.999971i \(-0.497565\pi\)
0.00764990 + 0.999971i \(0.497565\pi\)
\(312\) 0 0
\(313\) −23.3937 −1.32229 −0.661146 0.750257i \(-0.729930\pi\)
−0.661146 + 0.750257i \(0.729930\pi\)
\(314\) 3.57242i 0.201603i
\(315\) −1.19269 −0.0672002
\(316\) 13.5289 0.761059
\(317\) − 13.9952i − 0.786050i −0.919528 0.393025i \(-0.871429\pi\)
0.919528 0.393025i \(-0.128571\pi\)
\(318\) 18.7289i 1.05026i
\(319\) 19.6679i 1.10119i
\(320\) − 0.880395i − 0.0492156i
\(321\) −7.34481 −0.409948
\(322\) 2.82371 0.157359
\(323\) 0.190293i 0.0105882i
\(324\) −14.8562 −0.825346
\(325\) 0 0
\(326\) 12.9584 0.717698
\(327\) − 35.2814i − 1.95107i
\(328\) 31.2054 1.72303
\(329\) −17.3394 −0.955954
\(330\) 1.89008i 0.104046i
\(331\) 17.8213i 0.979548i 0.871849 + 0.489774i \(0.162921\pi\)
−0.871849 + 0.489774i \(0.837079\pi\)
\(332\) 2.18359i 0.119840i
\(333\) − 11.6625i − 0.639100i
\(334\) 12.9250 0.707225
\(335\) −1.89679 −0.103633
\(336\) − 2.93900i − 0.160336i
\(337\) 27.8485 1.51700 0.758501 0.651672i \(-0.225932\pi\)
0.758501 + 0.651672i \(0.225932\pi\)
\(338\) 0 0
\(339\) −27.0737 −1.47044
\(340\) − 0.723480i − 0.0392362i
\(341\) 28.1618 1.52505
\(342\) 0.144833 0.00783167
\(343\) 19.9041i 1.07472i
\(344\) 0.796561i 0.0429477i
\(345\) 0.829085i 0.0446364i
\(346\) 17.2707i 0.928477i
\(347\) 1.50365 0.0807200 0.0403600 0.999185i \(-0.487150\pi\)
0.0403600 + 0.999185i \(0.487150\pi\)
\(348\) −14.1196 −0.756890
\(349\) 14.1860i 0.759358i 0.925118 + 0.379679i \(0.123966\pi\)
−0.925118 + 0.379679i \(0.876034\pi\)
\(350\) −9.33513 −0.498983
\(351\) 0 0
\(352\) −24.7560 −1.31950
\(353\) − 7.16852i − 0.381542i −0.981635 0.190771i \(-0.938901\pi\)
0.981635 0.190771i \(-0.0610988\pi\)
\(354\) 12.2174 0.649350
\(355\) 2.14005 0.113582
\(356\) 3.91425i 0.207455i
\(357\) 11.4330i 0.605096i
\(358\) − 9.16852i − 0.484571i
\(359\) − 19.8853i − 1.04951i −0.851255 0.524753i \(-0.824158\pi\)
0.851255 0.524753i \(-0.175842\pi\)
\(360\) −1.36227 −0.0717981
\(361\) 18.9922 0.999591
\(362\) − 16.8170i − 0.883882i
\(363\) 15.8116 0.829895
\(364\) 0 0
\(365\) 1.66355 0.0870740
\(366\) − 6.25667i − 0.327041i
\(367\) 1.08383 0.0565757 0.0282878 0.999600i \(-0.490994\pi\)
0.0282878 + 0.999600i \(0.490994\pi\)
\(368\) −0.829085 −0.0432190
\(369\) − 23.7506i − 1.23641i
\(370\) 1.12737i 0.0586094i
\(371\) − 24.4969i − 1.27182i
\(372\) 20.2174i 1.04823i
\(373\) −6.13036 −0.317418 −0.158709 0.987325i \(-0.550733\pi\)
−0.158709 + 0.987325i \(0.550733\pi\)
\(374\) 7.35258 0.380193
\(375\) − 5.51573i − 0.284831i
\(376\) −19.8049 −1.02136
\(377\) 0 0
\(378\) −4.03923 −0.207756
\(379\) − 2.40880i − 0.123732i −0.998084 0.0618658i \(-0.980295\pi\)
0.998084 0.0618658i \(-0.0197051\pi\)
\(380\) 0.0295400 0.00151537
\(381\) −22.0368 −1.12898
\(382\) − 11.5778i − 0.592371i
\(383\) − 30.3913i − 1.55292i −0.630164 0.776462i \(-0.717012\pi\)
0.630164 0.776462i \(-0.282988\pi\)
\(384\) − 19.7724i − 1.00901i
\(385\) − 2.47219i − 0.125994i
\(386\) −10.8901 −0.554291
\(387\) 0.606268 0.0308184
\(388\) − 10.9347i − 0.555125i
\(389\) 15.9409 0.808237 0.404118 0.914707i \(-0.367578\pi\)
0.404118 + 0.914707i \(0.367578\pi\)
\(390\) 0 0
\(391\) 3.22521 0.163106
\(392\) 3.89008i 0.196479i
\(393\) 14.7778 0.745440
\(394\) 0.449354 0.0226381
\(395\) 2.46250i 0.123902i
\(396\) 11.8073i 0.593340i
\(397\) − 16.9148i − 0.848931i −0.905444 0.424466i \(-0.860462\pi\)
0.905444 0.424466i \(-0.139538\pi\)
\(398\) − 9.21552i − 0.461932i
\(399\) −0.466812 −0.0233698
\(400\) 2.74094 0.137047
\(401\) − 26.6625i − 1.33146i −0.746192 0.665730i \(-0.768120\pi\)
0.746192 0.665730i \(-0.231880\pi\)
\(402\) 13.8388 0.690215
\(403\) 0 0
\(404\) 18.1207 0.901537
\(405\) − 2.70410i − 0.134368i
\(406\) −8.75302 −0.434405
\(407\) 24.1739 1.19826
\(408\) 13.0586i 0.646497i
\(409\) 28.5163i 1.41004i 0.709187 + 0.705021i \(0.249062\pi\)
−0.709187 + 0.705021i \(0.750938\pi\)
\(410\) 2.29590i 0.113386i
\(411\) 13.9758i 0.689377i
\(412\) −1.84846 −0.0910672
\(413\) −15.9801 −0.786332
\(414\) − 2.45473i − 0.120643i
\(415\) −0.397452 −0.0195102
\(416\) 0 0
\(417\) 33.0465 1.61830
\(418\) 0.300209i 0.0146837i
\(419\) −29.6093 −1.44651 −0.723253 0.690583i \(-0.757354\pi\)
−0.723253 + 0.690583i \(0.757354\pi\)
\(420\) 1.77479 0.0866009
\(421\) − 11.6606i − 0.568301i −0.958780 0.284151i \(-0.908288\pi\)
0.958780 0.284151i \(-0.0917115\pi\)
\(422\) 7.04461i 0.342926i
\(423\) 15.0737i 0.732907i
\(424\) − 27.9801i − 1.35884i
\(425\) −10.6625 −0.517206
\(426\) −15.6136 −0.756480
\(427\) 8.18359i 0.396032i
\(428\) 4.43535 0.214391
\(429\) 0 0
\(430\) −0.0586060 −0.00282623
\(431\) 4.34913i 0.209490i 0.994499 + 0.104745i \(0.0334026\pi\)
−0.994499 + 0.104745i \(0.966597\pi\)
\(432\) 1.18598 0.0570605
\(433\) 14.3884 0.691460 0.345730 0.938334i \(-0.387631\pi\)
0.345730 + 0.938334i \(0.387631\pi\)
\(434\) 12.5332i 0.601612i
\(435\) − 2.57002i − 0.123223i
\(436\) 21.3056i 1.02035i
\(437\) 0.131687i 0.00629942i
\(438\) −12.1371 −0.579931
\(439\) 20.2325 0.965645 0.482822 0.875718i \(-0.339612\pi\)
0.482822 + 0.875718i \(0.339612\pi\)
\(440\) − 2.82371i − 0.134615i
\(441\) 2.96077 0.140989
\(442\) 0 0
\(443\) 8.12200 0.385888 0.192944 0.981210i \(-0.438196\pi\)
0.192944 + 0.981210i \(0.438196\pi\)
\(444\) 17.3545i 0.823608i
\(445\) −0.712464 −0.0337740
\(446\) 1.81163 0.0857830
\(447\) − 9.74094i − 0.460731i
\(448\) − 8.40150i − 0.396934i
\(449\) − 12.4916i − 0.589513i −0.955572 0.294757i \(-0.904761\pi\)
0.955572 0.294757i \(-0.0952386\pi\)
\(450\) 8.11529i 0.382559i
\(451\) 49.2301 2.31816
\(452\) 16.3491 0.768998
\(453\) 8.86294i 0.416417i
\(454\) 5.58642 0.262184
\(455\) 0 0
\(456\) −0.533188 −0.0249688
\(457\) 5.98121i 0.279789i 0.990166 + 0.139895i \(0.0446764\pi\)
−0.990166 + 0.139895i \(0.955324\pi\)
\(458\) −19.3860 −0.905847
\(459\) −4.61356 −0.215343
\(460\) − 0.500664i − 0.0233436i
\(461\) − 2.05669i − 0.0957895i −0.998852 0.0478947i \(-0.984749\pi\)
0.998852 0.0478947i \(-0.0152512\pi\)
\(462\) 18.0368i 0.839150i
\(463\) 8.44935i 0.392675i 0.980536 + 0.196337i \(0.0629048\pi\)
−0.980536 + 0.196337i \(0.937095\pi\)
\(464\) 2.57002 0.119310
\(465\) −3.67994 −0.170653
\(466\) 2.45473i 0.113713i
\(467\) −33.5139 −1.55084 −0.775420 0.631446i \(-0.782462\pi\)
−0.775420 + 0.631446i \(0.782462\pi\)
\(468\) 0 0
\(469\) −18.1008 −0.835818
\(470\) − 1.45712i − 0.0672121i
\(471\) −10.0097 −0.461222
\(472\) −18.2524 −0.840133
\(473\) 1.25667i 0.0577817i
\(474\) − 17.9661i − 0.825213i
\(475\) − 0.435353i − 0.0199754i
\(476\) − 6.90408i − 0.316448i
\(477\) −21.2959 −0.975072
\(478\) 20.1661 0.922377
\(479\) 24.7313i 1.13000i 0.825091 + 0.565000i \(0.191124\pi\)
−0.825091 + 0.565000i \(0.808876\pi\)
\(480\) 3.23490 0.147652
\(481\) 0 0
\(482\) −16.2524 −0.740275
\(483\) 7.91185i 0.360002i
\(484\) −9.54825 −0.434012
\(485\) 1.99031 0.0903754
\(486\) 14.5875i 0.661702i
\(487\) − 37.7555i − 1.71087i −0.517913 0.855433i \(-0.673291\pi\)
0.517913 0.855433i \(-0.326709\pi\)
\(488\) 9.34721i 0.423128i
\(489\) 36.3086i 1.64193i
\(490\) −0.286208 −0.0129296
\(491\) −31.3110 −1.41304 −0.706522 0.707691i \(-0.749737\pi\)
−0.706522 + 0.707691i \(0.749737\pi\)
\(492\) 35.3424i 1.59336i
\(493\) −9.99761 −0.450270
\(494\) 0 0
\(495\) −2.14914 −0.0965969
\(496\) − 3.67994i − 0.165234i
\(497\) 20.4222 0.916061
\(498\) 2.89977 0.129942
\(499\) 21.4873i 0.961902i 0.876748 + 0.480951i \(0.159708\pi\)
−0.876748 + 0.480951i \(0.840292\pi\)
\(500\) 3.33081i 0.148959i
\(501\) 36.2150i 1.61797i
\(502\) 19.0228i 0.849031i
\(503\) 37.5924 1.67616 0.838081 0.545546i \(-0.183678\pi\)
0.838081 + 0.545546i \(0.183678\pi\)
\(504\) −13.0000 −0.579066
\(505\) 3.29829i 0.146772i
\(506\) 5.08815 0.226196
\(507\) 0 0
\(508\) 13.3075 0.590425
\(509\) − 17.1075i − 0.758278i −0.925340 0.379139i \(-0.876220\pi\)
0.925340 0.379139i \(-0.123780\pi\)
\(510\) −0.960771 −0.0425437
\(511\) 15.8750 0.702269
\(512\) 6.22282i 0.275012i
\(513\) − 0.188374i − 0.00831690i
\(514\) − 11.4069i − 0.503136i
\(515\) − 0.336454i − 0.0148259i
\(516\) −0.902165 −0.0397156
\(517\) −31.2446 −1.37414
\(518\) 10.7584i 0.472697i
\(519\) −48.3913 −2.12414
\(520\) 0 0
\(521\) −19.8465 −0.869493 −0.434746 0.900553i \(-0.643162\pi\)
−0.434746 + 0.900553i \(0.643162\pi\)
\(522\) 7.60925i 0.333048i
\(523\) −11.4300 −0.499798 −0.249899 0.968272i \(-0.580397\pi\)
−0.249899 + 0.968272i \(0.580397\pi\)
\(524\) −8.92394 −0.389844
\(525\) − 26.1564i − 1.14156i
\(526\) − 13.7095i − 0.597762i
\(527\) 14.3153i 0.623583i
\(528\) − 5.29590i − 0.230474i
\(529\) −20.7681 −0.902960
\(530\) 2.05861 0.0894201
\(531\) 13.8920i 0.602862i
\(532\) 0.281896 0.0122218
\(533\) 0 0
\(534\) 5.19806 0.224942
\(535\) 0.807315i 0.0349033i
\(536\) −20.6746 −0.893005
\(537\) 25.6896 1.10859
\(538\) − 5.18598i − 0.223584i
\(539\) 6.13706i 0.264342i
\(540\) 0.716185i 0.0308197i
\(541\) − 16.1884i − 0.695993i −0.937496 0.347996i \(-0.886862\pi\)
0.937496 0.347996i \(-0.113138\pi\)
\(542\) 5.17092 0.222110
\(543\) 47.1202 2.02212
\(544\) − 12.5840i − 0.539536i
\(545\) −3.87800 −0.166115
\(546\) 0 0
\(547\) 5.33081 0.227929 0.113965 0.993485i \(-0.463645\pi\)
0.113965 + 0.993485i \(0.463645\pi\)
\(548\) − 8.43967i − 0.360525i
\(549\) 7.11423 0.303628
\(550\) −16.8213 −0.717263
\(551\) − 0.408206i − 0.0173902i
\(552\) 9.03684i 0.384633i
\(553\) 23.4993i 0.999293i
\(554\) − 10.7942i − 0.458600i
\(555\) −3.15883 −0.134085
\(556\) −19.9560 −0.846322
\(557\) − 7.39075i − 0.313156i −0.987666 0.156578i \(-0.949954\pi\)
0.987666 0.156578i \(-0.0500463\pi\)
\(558\) 10.8955 0.461242
\(559\) 0 0
\(560\) −0.323044 −0.0136511
\(561\) 20.6015i 0.869795i
\(562\) 4.03923 0.170385
\(563\) 9.47889 0.399488 0.199744 0.979848i \(-0.435989\pi\)
0.199744 + 0.979848i \(0.435989\pi\)
\(564\) − 22.4306i − 0.944497i
\(565\) 2.97584i 0.125194i
\(566\) − 17.7453i − 0.745889i
\(567\) − 25.8049i − 1.08370i
\(568\) 23.3260 0.978738
\(569\) 10.1438 0.425249 0.212624 0.977134i \(-0.431799\pi\)
0.212624 + 0.977134i \(0.431799\pi\)
\(570\) − 0.0392287i − 0.00164311i
\(571\) 14.0925 0.589751 0.294876 0.955536i \(-0.404722\pi\)
0.294876 + 0.955536i \(0.404722\pi\)
\(572\) 0 0
\(573\) 32.4403 1.35521
\(574\) 21.9095i 0.914483i
\(575\) −7.37867 −0.307712
\(576\) −7.30367 −0.304319
\(577\) 25.1545i 1.04720i 0.851965 + 0.523598i \(0.175411\pi\)
−0.851965 + 0.523598i \(0.824589\pi\)
\(578\) − 9.89546i − 0.411597i
\(579\) − 30.5133i − 1.26809i
\(580\) 1.55197i 0.0644422i
\(581\) −3.79284 −0.157354
\(582\) −14.5211 −0.601919
\(583\) − 44.1420i − 1.82817i
\(584\) 18.1323 0.750319
\(585\) 0 0
\(586\) 11.9860 0.495137
\(587\) 43.8353i 1.80928i 0.426180 + 0.904639i \(0.359859\pi\)
−0.426180 + 0.904639i \(0.640141\pi\)
\(588\) −4.40581 −0.181693
\(589\) −0.584498 −0.0240838
\(590\) − 1.34290i − 0.0552861i
\(591\) 1.25906i 0.0517909i
\(592\) − 3.15883i − 0.129827i
\(593\) 24.9965i 1.02648i 0.858244 + 0.513242i \(0.171556\pi\)
−0.858244 + 0.513242i \(0.828444\pi\)
\(594\) −7.27844 −0.298638
\(595\) 1.25667 0.0515184
\(596\) 5.88231i 0.240949i
\(597\) 25.8213 1.05680
\(598\) 0 0
\(599\) −6.24027 −0.254971 −0.127485 0.991840i \(-0.540691\pi\)
−0.127485 + 0.991840i \(0.540691\pi\)
\(600\) − 29.8756i − 1.21967i
\(601\) 6.32975 0.258196 0.129098 0.991632i \(-0.458792\pi\)
0.129098 + 0.991632i \(0.458792\pi\)
\(602\) −0.559270 −0.0227941
\(603\) 15.7356i 0.640802i
\(604\) − 5.35211i − 0.217774i
\(605\) − 1.73795i − 0.0706579i
\(606\) − 24.0640i − 0.977532i
\(607\) −43.6480 −1.77162 −0.885809 0.464050i \(-0.846396\pi\)
−0.885809 + 0.464050i \(0.846396\pi\)
\(608\) 0.513811 0.0208378
\(609\) − 24.5254i − 0.993820i
\(610\) −0.687710 −0.0278445
\(611\) 0 0
\(612\) −6.00192 −0.242613
\(613\) − 25.9541i − 1.04827i −0.851634 0.524137i \(-0.824388\pi\)
0.851634 0.524137i \(-0.175612\pi\)
\(614\) 15.3405 0.619092
\(615\) −6.43296 −0.259402
\(616\) − 26.9463i − 1.08570i
\(617\) 45.9396i 1.84946i 0.380626 + 0.924729i \(0.375709\pi\)
−0.380626 + 0.924729i \(0.624291\pi\)
\(618\) 2.45473i 0.0987437i
\(619\) − 6.73556i − 0.270725i −0.990796 0.135363i \(-0.956780\pi\)
0.990796 0.135363i \(-0.0432199\pi\)
\(620\) 2.22223 0.0892467
\(621\) −3.19269 −0.128118
\(622\) 0.216375i 0.00867583i
\(623\) −6.79895 −0.272394
\(624\) 0 0
\(625\) 24.0887 0.963549
\(626\) − 18.7603i − 0.749813i
\(627\) −0.841166 −0.0335930
\(628\) 6.04461 0.241206
\(629\) 12.2881i 0.489960i
\(630\) − 0.956459i − 0.0381063i
\(631\) − 45.0998i − 1.79539i −0.440614 0.897696i \(-0.645239\pi\)
0.440614 0.897696i \(-0.354761\pi\)
\(632\) 26.8407i 1.06767i
\(633\) −19.7385 −0.784537
\(634\) 11.2233 0.445734
\(635\) 2.42221i 0.0961223i
\(636\) 31.6896 1.25658
\(637\) 0 0
\(638\) −15.7724 −0.624435
\(639\) − 17.7536i − 0.702322i
\(640\) −2.17331 −0.0859075
\(641\) −32.5821 −1.28692 −0.643458 0.765482i \(-0.722501\pi\)
−0.643458 + 0.765482i \(0.722501\pi\)
\(642\) − 5.89008i − 0.232463i
\(643\) 25.5754i 1.00860i 0.863530 + 0.504298i \(0.168249\pi\)
−0.863530 + 0.504298i \(0.831751\pi\)
\(644\) − 4.77777i − 0.188271i
\(645\) − 0.164210i − 0.00646578i
\(646\) −0.152603 −0.00600408
\(647\) 30.1715 1.18616 0.593082 0.805142i \(-0.297911\pi\)
0.593082 + 0.805142i \(0.297911\pi\)
\(648\) − 29.4741i − 1.15785i
\(649\) −28.7952 −1.13031
\(650\) 0 0
\(651\) −35.1172 −1.37635
\(652\) − 21.9259i − 0.858683i
\(653\) 36.9028 1.44412 0.722058 0.691832i \(-0.243196\pi\)
0.722058 + 0.691832i \(0.243196\pi\)
\(654\) 28.2935 1.10636
\(655\) − 1.62432i − 0.0634673i
\(656\) − 6.43296i − 0.251165i
\(657\) − 13.8006i − 0.538413i
\(658\) − 13.9051i − 0.542079i
\(659\) 23.6866 0.922701 0.461350 0.887218i \(-0.347365\pi\)
0.461350 + 0.887218i \(0.347365\pi\)
\(660\) 3.19806 0.124484
\(661\) 31.7590i 1.23528i 0.786460 + 0.617641i \(0.211911\pi\)
−0.786460 + 0.617641i \(0.788089\pi\)
\(662\) −14.2916 −0.555458
\(663\) 0 0
\(664\) −4.33214 −0.168120
\(665\) 0.0513102i 0.00198973i
\(666\) 9.35258 0.362405
\(667\) −6.91856 −0.267888
\(668\) − 21.8694i − 0.846152i
\(669\) 5.07606i 0.196252i
\(670\) − 1.52111i − 0.0587655i
\(671\) 14.7463i 0.569275i
\(672\) 30.8702 1.19085
\(673\) 7.50232 0.289193 0.144597 0.989491i \(-0.453812\pi\)
0.144597 + 0.989491i \(0.453812\pi\)
\(674\) 22.3327i 0.860225i
\(675\) 10.5550 0.406261
\(676\) 0 0
\(677\) −35.0315 −1.34637 −0.673184 0.739475i \(-0.735074\pi\)
−0.673184 + 0.739475i \(0.735074\pi\)
\(678\) − 21.7114i − 0.833821i
\(679\) 18.9933 0.728896
\(680\) 1.43535 0.0550433
\(681\) 15.6528i 0.599816i
\(682\) 22.5840i 0.864787i
\(683\) 24.0834i 0.921524i 0.887524 + 0.460762i \(0.152424\pi\)
−0.887524 + 0.460762i \(0.847576\pi\)
\(684\) − 0.245061i − 0.00937013i
\(685\) 1.53617 0.0586941
\(686\) −15.9618 −0.609426
\(687\) − 54.3183i − 2.07237i
\(688\) 0.164210 0.00626046
\(689\) 0 0
\(690\) −0.664874 −0.0253113
\(691\) 2.01447i 0.0766342i 0.999266 + 0.0383171i \(0.0121997\pi\)
−0.999266 + 0.0383171i \(0.987800\pi\)
\(692\) 29.2223 1.11087
\(693\) −20.5090 −0.779073
\(694\) 1.20583i 0.0457728i
\(695\) − 3.63235i − 0.137783i
\(696\) − 28.0127i − 1.06182i
\(697\) 25.0248i 0.947880i
\(698\) −11.3763 −0.430598
\(699\) −6.87800 −0.260150
\(700\) 15.7952i 0.597004i
\(701\) 48.8189 1.84387 0.921933 0.387350i \(-0.126610\pi\)
0.921933 + 0.387350i \(0.126610\pi\)
\(702\) 0 0
\(703\) −0.501729 −0.0189231
\(704\) − 15.1390i − 0.570572i
\(705\) 4.08277 0.153766
\(706\) 5.74871 0.216355
\(707\) 31.4752i 1.18375i
\(708\) − 20.6722i − 0.776908i
\(709\) 20.8060i 0.781385i 0.920521 + 0.390693i \(0.127764\pi\)
−0.920521 + 0.390693i \(0.872236\pi\)
\(710\) 1.71618i 0.0644073i
\(711\) 20.4286 0.766134
\(712\) −7.76569 −0.291032
\(713\) 9.90648i 0.371000i
\(714\) −9.16852 −0.343123
\(715\) 0 0
\(716\) −15.5133 −0.579761
\(717\) 56.5042i 2.11019i
\(718\) 15.9468 0.595128
\(719\) −21.4306 −0.799225 −0.399613 0.916684i \(-0.630855\pi\)
−0.399613 + 0.916684i \(0.630855\pi\)
\(720\) 0.280831i 0.0104660i
\(721\) − 3.21073i − 0.119574i
\(722\) 15.2306i 0.566824i
\(723\) − 45.5381i − 1.69358i
\(724\) −28.4547 −1.05751
\(725\) 22.8726 0.849468
\(726\) 12.6799i 0.470597i
\(727\) −13.4862 −0.500175 −0.250088 0.968223i \(-0.580459\pi\)
−0.250088 + 0.968223i \(0.580459\pi\)
\(728\) 0 0
\(729\) −8.02715 −0.297302
\(730\) 1.33406i 0.0493758i
\(731\) −0.638792 −0.0236266
\(732\) −10.5864 −0.391285
\(733\) − 43.5424i − 1.60828i −0.594443 0.804138i \(-0.702627\pi\)
0.594443 0.804138i \(-0.297373\pi\)
\(734\) 0.869167i 0.0320816i
\(735\) − 0.801938i − 0.0295799i
\(736\) − 8.70841i − 0.320996i
\(737\) −32.6165 −1.20145
\(738\) 19.0465 0.701112
\(739\) − 20.0543i − 0.737709i −0.929487 0.368855i \(-0.879750\pi\)
0.929487 0.368855i \(-0.120250\pi\)
\(740\) 1.90754 0.0701226
\(741\) 0 0
\(742\) 19.6450 0.721191
\(743\) − 33.1685i − 1.21684i −0.793617 0.608418i \(-0.791805\pi\)
0.793617 0.608418i \(-0.208195\pi\)
\(744\) −40.1105 −1.47052
\(745\) −1.07069 −0.0392270
\(746\) − 4.91617i − 0.179994i
\(747\) 3.29722i 0.120639i
\(748\) − 12.4407i − 0.454878i
\(749\) 7.70410i 0.281502i
\(750\) 4.42327 0.161515
\(751\) −39.2814 −1.43340 −0.716700 0.697382i \(-0.754348\pi\)
−0.716700 + 0.697382i \(0.754348\pi\)
\(752\) 4.08277i 0.148883i
\(753\) −53.3008 −1.94239
\(754\) 0 0
\(755\) 0.974181 0.0354541
\(756\) 6.83446i 0.248567i
\(757\) −46.6426 −1.69526 −0.847628 0.530592i \(-0.821970\pi\)
−0.847628 + 0.530592i \(0.821970\pi\)
\(758\) 1.93171 0.0701627
\(759\) 14.2567i 0.517484i
\(760\) 0.0586060i 0.00212586i
\(761\) 21.8984i 0.793818i 0.917858 + 0.396909i \(0.129917\pi\)
−0.917858 + 0.396909i \(0.870083\pi\)
\(762\) − 17.6722i − 0.640195i
\(763\) −37.0073 −1.33975
\(764\) −19.5899 −0.708737
\(765\) − 1.09246i − 0.0394979i
\(766\) 24.3720 0.880595
\(767\) 0 0
\(768\) 31.8756 1.15021
\(769\) 46.7096i 1.68439i 0.539172 + 0.842196i \(0.318737\pi\)
−0.539172 + 0.842196i \(0.681263\pi\)
\(770\) 1.98254 0.0714458
\(771\) 31.9614 1.15106
\(772\) 18.4263i 0.663175i
\(773\) − 30.2416i − 1.08771i −0.839178 0.543857i \(-0.816963\pi\)
0.839178 0.543857i \(-0.183037\pi\)
\(774\) 0.486189i 0.0174757i
\(775\) − 32.7506i − 1.17644i
\(776\) 21.6939 0.778767
\(777\) −30.1444 −1.08142
\(778\) 12.7836i 0.458315i
\(779\) −1.02177 −0.0366087
\(780\) 0 0
\(781\) 36.7995 1.31679
\(782\) 2.58642i 0.0924901i
\(783\) 9.89679 0.353682
\(784\) 0.801938 0.0286406
\(785\) 1.10023i 0.0392688i
\(786\) 11.8509i 0.422706i
\(787\) − 28.7023i − 1.02313i −0.859246 0.511563i \(-0.829067\pi\)
0.859246 0.511563i \(-0.170933\pi\)
\(788\) − 0.760316i − 0.0270851i
\(789\) 38.4131 1.36754
\(790\) −1.97477 −0.0702592
\(791\) 28.3980i 1.00972i
\(792\) −23.4252 −0.832378
\(793\) 0 0
\(794\) 13.5646 0.481391
\(795\) 5.76809i 0.204573i
\(796\) −15.5929 −0.552674
\(797\) 18.5418 0.656785 0.328392 0.944541i \(-0.393493\pi\)
0.328392 + 0.944541i \(0.393493\pi\)
\(798\) − 0.374354i − 0.0132520i
\(799\) − 15.8823i − 0.561876i
\(800\) 28.7899i 1.01788i
\(801\) 5.91053i 0.208838i
\(802\) 21.3817 0.755012
\(803\) 28.6058 1.00948
\(804\) − 23.4155i − 0.825801i
\(805\) 0.869641 0.0306508
\(806\) 0 0
\(807\) 14.5308 0.511508
\(808\) 35.9506i 1.26474i
\(809\) −10.0677 −0.353962 −0.176981 0.984214i \(-0.556633\pi\)
−0.176981 + 0.984214i \(0.556633\pi\)
\(810\) 2.16852 0.0761941
\(811\) − 10.0285i − 0.352147i −0.984377 0.176074i \(-0.943660\pi\)
0.984377 0.176074i \(-0.0563397\pi\)
\(812\) 14.8103i 0.519740i
\(813\) 14.4886i 0.508137i
\(814\) 19.3860i 0.679478i
\(815\) 3.99090 0.139795
\(816\) 2.69202 0.0942396
\(817\) − 0.0260821i 0 0.000912498i
\(818\) −22.8683 −0.799572
\(819\) 0 0
\(820\) 3.88471 0.135660
\(821\) − 26.1704i − 0.913355i −0.889632 0.456677i \(-0.849039\pi\)
0.889632 0.456677i \(-0.150961\pi\)
\(822\) −11.2078 −0.390915
\(823\) −1.82238 −0.0635242 −0.0317621 0.999495i \(-0.510112\pi\)
−0.0317621 + 0.999495i \(0.510112\pi\)
\(824\) − 3.66727i − 0.127755i
\(825\) − 47.1323i − 1.64094i
\(826\) − 12.8151i − 0.445894i
\(827\) 32.2941i 1.12298i 0.827485 + 0.561488i \(0.189771\pi\)
−0.827485 + 0.561488i \(0.810229\pi\)
\(828\) −4.15346 −0.144343
\(829\) −15.1002 −0.524453 −0.262226 0.965006i \(-0.584457\pi\)
−0.262226 + 0.965006i \(0.584457\pi\)
\(830\) − 0.318732i − 0.0110634i
\(831\) 30.2446 1.04917
\(832\) 0 0
\(833\) −3.11960 −0.108088
\(834\) 26.5013i 0.917663i
\(835\) 3.98062 0.137755
\(836\) 0.507960 0.0175682
\(837\) − 14.1709i − 0.489818i
\(838\) − 23.7448i − 0.820250i
\(839\) 32.9965i 1.13917i 0.821933 + 0.569584i \(0.192896\pi\)
−0.821933 + 0.569584i \(0.807104\pi\)
\(840\) 3.52111i 0.121490i
\(841\) −7.55363 −0.260470
\(842\) 9.35105 0.322258
\(843\) 11.3177i 0.389801i
\(844\) 11.9196 0.410290
\(845\) 0 0
\(846\) −12.0881 −0.415599
\(847\) − 16.5851i − 0.569870i
\(848\) −5.76809 −0.198077
\(849\) 49.7211 1.70642
\(850\) − 8.55065i − 0.293285i
\(851\) 8.50365i 0.291501i
\(852\) 26.4185i 0.905082i
\(853\) 37.7802i 1.29357i 0.762673 + 0.646784i \(0.223887\pi\)
−0.762673 + 0.646784i \(0.776113\pi\)
\(854\) −6.56273 −0.224572
\(855\) 0.0446055 0.00152547
\(856\) 8.79954i 0.300762i
\(857\) −27.3623 −0.934677 −0.467339 0.884078i \(-0.654787\pi\)
−0.467339 + 0.884078i \(0.654787\pi\)
\(858\) 0 0
\(859\) −20.0629 −0.684538 −0.342269 0.939602i \(-0.611195\pi\)
−0.342269 + 0.939602i \(0.611195\pi\)
\(860\) 0.0991626i 0.00338142i
\(861\) −61.3889 −2.09213
\(862\) −3.48773 −0.118792
\(863\) − 6.14483i − 0.209173i −0.994516 0.104586i \(-0.966648\pi\)
0.994516 0.104586i \(-0.0333518\pi\)
\(864\) 12.4571i 0.423800i
\(865\) 5.31900i 0.180851i
\(866\) 11.5386i 0.392096i
\(867\) 27.7265 0.941640
\(868\) 21.2064 0.719793
\(869\) 42.3443i 1.43643i
\(870\) 2.06100 0.0698744
\(871\) 0 0
\(872\) −42.2693 −1.43142
\(873\) − 16.5114i − 0.558827i
\(874\) −0.105604 −0.00357212
\(875\) −5.78554 −0.195587
\(876\) 20.5362i 0.693853i
\(877\) − 13.5077i − 0.456123i −0.973647 0.228061i \(-0.926761\pi\)
0.973647 0.228061i \(-0.0732386\pi\)
\(878\) 16.2252i 0.547574i
\(879\) 33.5840i 1.13276i
\(880\) −0.582105 −0.0196228
\(881\) 5.23431 0.176348 0.0881741 0.996105i \(-0.471897\pi\)
0.0881741 + 0.996105i \(0.471897\pi\)
\(882\) 2.37435i 0.0799487i
\(883\) 4.57301 0.153894 0.0769470 0.997035i \(-0.475483\pi\)
0.0769470 + 0.997035i \(0.475483\pi\)
\(884\) 0 0
\(885\) 3.76271 0.126482
\(886\) 6.51334i 0.218820i
\(887\) −1.64071 −0.0550897 −0.0275448 0.999621i \(-0.508769\pi\)
−0.0275448 + 0.999621i \(0.508769\pi\)
\(888\) −34.4306 −1.15541
\(889\) 23.1148i 0.775246i
\(890\) − 0.571352i − 0.0191517i
\(891\) − 46.4989i − 1.55777i
\(892\) − 3.06531i − 0.102634i
\(893\) 0.648481 0.0217006
\(894\) 7.81163 0.261260
\(895\) − 2.82371i − 0.0943861i
\(896\) −20.7396 −0.692862
\(897\) 0 0
\(898\) 10.0175 0.334287
\(899\) − 30.7084i − 1.02418i
\(900\) 13.7313 0.457708
\(901\) 22.4383 0.747529
\(902\) 39.4795i 1.31452i
\(903\) − 1.56704i − 0.0521478i
\(904\) 32.4359i 1.07880i
\(905\) − 5.17928i − 0.172165i
\(906\) −7.10752 −0.236132
\(907\) −8.10215 −0.269027 −0.134514 0.990912i \(-0.542947\pi\)
−0.134514 + 0.990912i \(0.542947\pi\)
\(908\) − 9.45234i − 0.313687i
\(909\) 27.3623 0.907549
\(910\) 0 0
\(911\) −9.18119 −0.304187 −0.152093 0.988366i \(-0.548601\pi\)
−0.152093 + 0.988366i \(0.548601\pi\)
\(912\) 0.109916i 0.00363969i
\(913\) −6.83446 −0.226188
\(914\) −4.79656 −0.158656
\(915\) − 1.92692i − 0.0637020i
\(916\) 32.8015i 1.08379i
\(917\) − 15.5007i − 0.511877i
\(918\) − 3.69979i − 0.122111i
\(919\) 27.5036 0.907262 0.453631 0.891190i \(-0.350128\pi\)
0.453631 + 0.891190i \(0.350128\pi\)
\(920\) 0.993295 0.0327480
\(921\) 42.9831i 1.41634i
\(922\) 1.64933 0.0543180
\(923\) 0 0
\(924\) 30.5187 1.00399
\(925\) − 28.1129i − 0.924346i
\(926\) −6.77586 −0.222668
\(927\) −2.79118 −0.0916745
\(928\) 26.9946i 0.886142i
\(929\) 24.2131i 0.794407i 0.917731 + 0.397203i \(0.130019\pi\)
−0.917731 + 0.397203i \(0.869981\pi\)
\(930\) − 2.95108i − 0.0967698i
\(931\) − 0.127375i − 0.00417454i
\(932\) 4.15346 0.136051
\(933\) −0.606268 −0.0198483
\(934\) − 26.8761i − 0.879412i
\(935\) 2.26444 0.0740550
\(936\) 0 0
\(937\) 11.1830 0.365333 0.182666 0.983175i \(-0.441527\pi\)
0.182666 + 0.983175i \(0.441527\pi\)
\(938\) − 14.5157i − 0.473955i
\(939\) 52.5652 1.71540
\(940\) −2.46548 −0.0804152
\(941\) − 15.9638i − 0.520404i −0.965554 0.260202i \(-0.916211\pi\)
0.965554 0.260202i \(-0.0837891\pi\)
\(942\) − 8.02715i − 0.261539i
\(943\) 17.3177i 0.563941i
\(944\) 3.76271i 0.122466i
\(945\) −1.24400 −0.0404672
\(946\) −1.00777 −0.0327654
\(947\) − 6.51466i − 0.211698i −0.994382 0.105849i \(-0.966244\pi\)
0.994382 0.105849i \(-0.0337560\pi\)
\(948\) −30.3991 −0.987317
\(949\) 0 0
\(950\) 0.349126 0.0113271
\(951\) 31.4470i 1.01974i
\(952\) 13.6974 0.443935
\(953\) −47.6469 −1.54344 −0.771718 0.635965i \(-0.780602\pi\)
−0.771718 + 0.635965i \(0.780602\pi\)
\(954\) − 17.0780i − 0.552920i
\(955\) − 3.56571i − 0.115384i
\(956\) − 34.1215i − 1.10357i
\(957\) − 44.1933i − 1.42857i
\(958\) −19.8329 −0.640773
\(959\) 14.6595 0.473380
\(960\) 1.97823i 0.0638471i
\(961\) −12.9705 −0.418402
\(962\) 0 0
\(963\) 6.69740 0.215821
\(964\) 27.4993i 0.885694i
\(965\) −3.35391 −0.107966
\(966\) −6.34481 −0.204141
\(967\) − 43.8122i − 1.40891i −0.709751 0.704453i \(-0.751192\pi\)
0.709751 0.704453i \(-0.248808\pi\)
\(968\) − 18.9433i − 0.608861i
\(969\) − 0.427583i − 0.0137360i
\(970\) 1.59611i 0.0512479i
\(971\) 4.29483 0.137828 0.0689139 0.997623i \(-0.478047\pi\)
0.0689139 + 0.997623i \(0.478047\pi\)
\(972\) 24.6823 0.791686
\(973\) − 34.6631i − 1.11125i
\(974\) 30.2776 0.970156
\(975\) 0 0
\(976\) 1.92692 0.0616792
\(977\) 26.8019i 0.857470i 0.903430 + 0.428735i \(0.141041\pi\)
−0.903430 + 0.428735i \(0.858959\pi\)
\(978\) −29.1172 −0.931066
\(979\) −12.2513 −0.391553
\(980\) 0.484271i 0.0154695i
\(981\) 32.1715i 1.02716i
\(982\) − 25.1094i − 0.801275i
\(983\) − 27.2495i − 0.869124i −0.900642 0.434562i \(-0.856903\pi\)
0.900642 0.434562i \(-0.143097\pi\)
\(984\) −70.1178 −2.23527
\(985\) 0.138391 0.00440951
\(986\) − 8.01746i − 0.255328i
\(987\) 38.9614 1.24015
\(988\) 0 0
\(989\) −0.442058 −0.0140566
\(990\) − 1.72348i − 0.0547758i
\(991\) 24.3889 0.774740 0.387370 0.921924i \(-0.373384\pi\)
0.387370 + 0.921924i \(0.373384\pi\)
\(992\) 38.6528 1.22723
\(993\) − 40.0441i − 1.27076i
\(994\) 16.3773i 0.519458i
\(995\) − 2.83818i − 0.0899764i
\(996\) − 4.90648i − 0.155468i
\(997\) 31.3207 0.991935 0.495967 0.868341i \(-0.334814\pi\)
0.495967 + 0.868341i \(0.334814\pi\)
\(998\) −17.2314 −0.545452
\(999\) − 12.1642i − 0.384859i
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 169.2.b.b.168.5 6
3.2 odd 2 1521.2.b.l.1351.2 6
4.3 odd 2 2704.2.f.o.337.6 6
13.2 odd 12 169.2.c.c.22.1 6
13.3 even 3 169.2.e.b.147.2 12
13.4 even 6 169.2.e.b.23.2 12
13.5 odd 4 169.2.a.b.1.3 3
13.6 odd 12 169.2.c.c.146.1 6
13.7 odd 12 169.2.c.b.146.3 6
13.8 odd 4 169.2.a.c.1.1 yes 3
13.9 even 3 169.2.e.b.23.5 12
13.10 even 6 169.2.e.b.147.5 12
13.11 odd 12 169.2.c.b.22.3 6
13.12 even 2 inner 169.2.b.b.168.2 6
39.5 even 4 1521.2.a.r.1.1 3
39.8 even 4 1521.2.a.o.1.3 3
39.38 odd 2 1521.2.b.l.1351.5 6
52.31 even 4 2704.2.a.z.1.3 3
52.47 even 4 2704.2.a.ba.1.3 3
52.51 odd 2 2704.2.f.o.337.5 6
65.34 odd 4 4225.2.a.bb.1.3 3
65.44 odd 4 4225.2.a.bg.1.1 3
91.34 even 4 8281.2.a.bj.1.1 3
91.83 even 4 8281.2.a.bf.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.3 3 13.5 odd 4
169.2.a.c.1.1 yes 3 13.8 odd 4
169.2.b.b.168.2 6 13.12 even 2 inner
169.2.b.b.168.5 6 1.1 even 1 trivial
169.2.c.b.22.3 6 13.11 odd 12
169.2.c.b.146.3 6 13.7 odd 12
169.2.c.c.22.1 6 13.2 odd 12
169.2.c.c.146.1 6 13.6 odd 12
169.2.e.b.23.2 12 13.4 even 6
169.2.e.b.23.5 12 13.9 even 3
169.2.e.b.147.2 12 13.3 even 3
169.2.e.b.147.5 12 13.10 even 6
1521.2.a.o.1.3 3 39.8 even 4
1521.2.a.r.1.1 3 39.5 even 4
1521.2.b.l.1351.2 6 3.2 odd 2
1521.2.b.l.1351.5 6 39.38 odd 2
2704.2.a.z.1.3 3 52.31 even 4
2704.2.a.ba.1.3 3 52.47 even 4
2704.2.f.o.337.5 6 52.51 odd 2
2704.2.f.o.337.6 6 4.3 odd 2
4225.2.a.bb.1.3 3 65.34 odd 4
4225.2.a.bg.1.1 3 65.44 odd 4
8281.2.a.bf.1.3 3 91.83 even 4
8281.2.a.bj.1.1 3 91.34 even 4