Properties

Label 128.5.h.a.47.5
Level $128$
Weight $5$
Character 128.47
Analytic conductor $13.231$
Analytic rank $0$
Dimension $60$
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] = [128,5,Mod(15,128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("128.15");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 128.h (of order \(8\), degree \(4\), 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: \(13.2313552747\)
Analytic rank: \(0\)
Dimension: \(60\)
Relative dimension: \(15\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 47.5
Character \(\chi\) \(=\) 128.47
Dual form 128.5.h.a.79.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.03024 + 7.31565i) q^{3} +(12.4171 + 29.9775i) q^{5} +(51.5981 + 51.5981i) q^{7} +(12.9392 + 12.9392i) q^{9} +O(q^{10})\) \(q+(-3.03024 + 7.31565i) q^{3} +(12.4171 + 29.9775i) q^{5} +(51.5981 + 51.5981i) q^{7} +(12.9392 + 12.9392i) q^{9} +(-68.2659 - 164.808i) q^{11} +(25.9560 - 62.6632i) q^{13} -256.932 q^{15} +489.438i q^{17} +(164.710 + 68.2250i) q^{19} +(-533.829 + 221.119i) q^{21} +(-51.6300 + 51.6300i) q^{23} +(-302.523 + 302.523i) q^{25} +(-726.436 + 300.900i) q^{27} +(-779.698 - 322.962i) q^{29} +3.47504i q^{31} +1412.54 q^{33} +(-906.084 + 2187.48i) q^{35} +(-375.072 - 905.504i) q^{37} +(379.770 + 379.770i) q^{39} +(-1135.82 - 1135.82i) q^{41} +(318.285 + 768.407i) q^{43} +(-227.218 + 548.552i) q^{45} +1860.54 q^{47} +2923.74i q^{49} +(-3580.56 - 1483.12i) q^{51} +(2841.49 - 1176.98i) q^{53} +(4092.88 - 4092.88i) q^{55} +(-998.221 + 998.221i) q^{57} +(2613.25 - 1082.44i) q^{59} +(-1511.76 - 626.192i) q^{61} +1335.28i q^{63} +2200.78 q^{65} +(-1325.18 + 3199.26i) q^{67} +(-221.256 - 534.158i) q^{69} +(-5659.31 - 5659.31i) q^{71} +(5143.36 + 5143.36i) q^{73} +(-1296.44 - 3129.87i) q^{75} +(4981.42 - 12026.2i) q^{77} +5180.13 q^{79} -4743.95i q^{81} +(-2002.61 - 829.509i) q^{83} +(-14672.1 + 6077.38i) q^{85} +(4725.35 - 4725.35i) q^{87} +(2938.70 - 2938.70i) q^{89} +(4572.59 - 1894.03i) q^{91} +(-25.4222 - 10.5302i) q^{93} +5784.74i q^{95} +14104.1 q^{97} +(1249.19 - 3015.80i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 60 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 60 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9} + 4 q^{11} - 4 q^{13} + 8 q^{15} + 4 q^{19} - 4 q^{21} + 1156 q^{23} - 4 q^{25} - 3644 q^{27} - 4 q^{29} - 8 q^{33} + 5188 q^{35} - 4 q^{37} + 2692 q^{39} - 4 q^{41} - 5564 q^{43} - 328 q^{45} + 8 q^{47} - 8384 q^{51} + 956 q^{53} + 11780 q^{55} - 4 q^{57} + 13060 q^{59} + 7548 q^{61} - 8 q^{65} - 18876 q^{67} - 19588 q^{69} - 19964 q^{71} - 4 q^{73} + 200 q^{75} + 9404 q^{77} + 50184 q^{79} - 10556 q^{83} + 2496 q^{85} - 49276 q^{87} - 4 q^{89} - 31868 q^{91} + 320 q^{93} - 8 q^{97} + 46920 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(e\left(\frac{3}{8}\right)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.03024 + 7.31565i −0.336694 + 0.812850i 0.661335 + 0.750091i \(0.269990\pi\)
−0.998029 + 0.0627598i \(0.980010\pi\)
\(4\) 0 0
\(5\) 12.4171 + 29.9775i 0.496683 + 1.19910i 0.951260 + 0.308391i \(0.0997905\pi\)
−0.454577 + 0.890708i \(0.650209\pi\)
\(6\) 0 0
\(7\) 51.5981 + 51.5981i 1.05302 + 1.05302i 0.998513 + 0.0545102i \(0.0173597\pi\)
0.0545102 + 0.998513i \(0.482640\pi\)
\(8\) 0 0
\(9\) 12.9392 + 12.9392i 0.159743 + 0.159743i
\(10\) 0 0
\(11\) −68.2659 164.808i −0.564181 1.36205i −0.906395 0.422431i \(-0.861177\pi\)
0.342214 0.939622i \(-0.388823\pi\)
\(12\) 0 0
\(13\) 25.9560 62.6632i 0.153586 0.370788i −0.828294 0.560293i \(-0.810688\pi\)
0.981880 + 0.189505i \(0.0606883\pi\)
\(14\) 0 0
\(15\) −256.932 −1.14192
\(16\) 0 0
\(17\) 489.438i 1.69356i 0.531946 + 0.846778i \(0.321461\pi\)
−0.531946 + 0.846778i \(0.678539\pi\)
\(18\) 0 0
\(19\) 164.710 + 68.2250i 0.456260 + 0.188989i 0.598963 0.800777i \(-0.295580\pi\)
−0.142704 + 0.989765i \(0.545580\pi\)
\(20\) 0 0
\(21\) −533.829 + 221.119i −1.21050 + 0.501404i
\(22\) 0 0
\(23\) −51.6300 + 51.6300i −0.0975992 + 0.0975992i −0.754220 0.656621i \(-0.771985\pi\)
0.656621 + 0.754220i \(0.271985\pi\)
\(24\) 0 0
\(25\) −302.523 + 302.523i −0.484037 + 0.484037i
\(26\) 0 0
\(27\) −726.436 + 300.900i −0.996483 + 0.412757i
\(28\) 0 0
\(29\) −779.698 322.962i −0.927108 0.384021i −0.132528 0.991179i \(-0.542309\pi\)
−0.794581 + 0.607158i \(0.792309\pi\)
\(30\) 0 0
\(31\) 3.47504i 0.00361607i 0.999998 + 0.00180804i \(0.000575516\pi\)
−0.999998 + 0.00180804i \(0.999424\pi\)
\(32\) 0 0
\(33\) 1412.54 1.29710
\(34\) 0 0
\(35\) −906.084 + 2187.48i −0.739660 + 1.78570i
\(36\) 0 0
\(37\) −375.072 905.504i −0.273975 0.661435i 0.725671 0.688042i \(-0.241530\pi\)
−0.999646 + 0.0266075i \(0.991530\pi\)
\(38\) 0 0
\(39\) 379.770 + 379.770i 0.249684 + 0.249684i
\(40\) 0 0
\(41\) −1135.82 1135.82i −0.675681 0.675681i 0.283339 0.959020i \(-0.408558\pi\)
−0.959020 + 0.283339i \(0.908558\pi\)
\(42\) 0 0
\(43\) 318.285 + 768.407i 0.172139 + 0.415580i 0.986279 0.165089i \(-0.0527911\pi\)
−0.814140 + 0.580669i \(0.802791\pi\)
\(44\) 0 0
\(45\) −227.218 + 548.552i −0.112206 + 0.270890i
\(46\) 0 0
\(47\) 1860.54 0.842252 0.421126 0.907002i \(-0.361635\pi\)
0.421126 + 0.907002i \(0.361635\pi\)
\(48\) 0 0
\(49\) 2923.74i 1.21772i
\(50\) 0 0
\(51\) −3580.56 1483.12i −1.37661 0.570210i
\(52\) 0 0
\(53\) 2841.49 1176.98i 1.01157 0.419004i 0.185540 0.982637i \(-0.440597\pi\)
0.826025 + 0.563633i \(0.190597\pi\)
\(54\) 0 0
\(55\) 4092.88 4092.88i 1.35302 1.35302i
\(56\) 0 0
\(57\) −998.221 + 998.221i −0.307240 + 0.307240i
\(58\) 0 0
\(59\) 2613.25 1082.44i 0.750717 0.310957i 0.0256828 0.999670i \(-0.491824\pi\)
0.725034 + 0.688713i \(0.241824\pi\)
\(60\) 0 0
\(61\) −1511.76 626.192i −0.406278 0.168286i 0.170179 0.985413i \(-0.445565\pi\)
−0.576458 + 0.817127i \(0.695565\pi\)
\(62\) 0 0
\(63\) 1335.28i 0.336427i
\(64\) 0 0
\(65\) 2200.78 0.520895
\(66\) 0 0
\(67\) −1325.18 + 3199.26i −0.295205 + 0.712688i 0.704790 + 0.709416i \(0.251041\pi\)
−0.999995 + 0.00327157i \(0.998959\pi\)
\(68\) 0 0
\(69\) −221.256 534.158i −0.0464725 0.112195i
\(70\) 0 0
\(71\) −5659.31 5659.31i −1.12266 1.12266i −0.991341 0.131315i \(-0.958080\pi\)
−0.131315 0.991341i \(-0.541920\pi\)
\(72\) 0 0
\(73\) 5143.36 + 5143.36i 0.965163 + 0.965163i 0.999413 0.0342500i \(-0.0109042\pi\)
−0.0342500 + 0.999413i \(0.510904\pi\)
\(74\) 0 0
\(75\) −1296.44 3129.87i −0.230477 0.556422i
\(76\) 0 0
\(77\) 4981.42 12026.2i 0.840178 2.02837i
\(78\) 0 0
\(79\) 5180.13 0.830016 0.415008 0.909818i \(-0.363779\pi\)
0.415008 + 0.909818i \(0.363779\pi\)
\(80\) 0 0
\(81\) 4743.95i 0.723053i
\(82\) 0 0
\(83\) −2002.61 829.509i −0.290697 0.120411i 0.232569 0.972580i \(-0.425287\pi\)
−0.523266 + 0.852169i \(0.675287\pi\)
\(84\) 0 0
\(85\) −14672.1 + 6077.38i −2.03074 + 0.841160i
\(86\) 0 0
\(87\) 4725.35 4725.35i 0.624303 0.624303i
\(88\) 0 0
\(89\) 2938.70 2938.70i 0.371002 0.371002i −0.496840 0.867842i \(-0.665507\pi\)
0.867842 + 0.496840i \(0.165507\pi\)
\(90\) 0 0
\(91\) 4572.59 1894.03i 0.552178 0.228720i
\(92\) 0 0
\(93\) −25.4222 10.5302i −0.00293932 0.00121751i
\(94\) 0 0
\(95\) 5784.74i 0.640968i
\(96\) 0 0
\(97\) 14104.1 1.49900 0.749499 0.662006i \(-0.230295\pi\)
0.749499 + 0.662006i \(0.230295\pi\)
\(98\) 0 0
\(99\) 1249.19 3015.80i 0.127455 0.307703i
\(100\) 0 0
\(101\) 4233.68 + 10221.0i 0.415026 + 1.00196i 0.983768 + 0.179446i \(0.0574305\pi\)
−0.568741 + 0.822516i \(0.692570\pi\)
\(102\) 0 0
\(103\) 135.994 + 135.994i 0.0128188 + 0.0128188i 0.713487 0.700668i \(-0.247115\pi\)
−0.700668 + 0.713487i \(0.747115\pi\)
\(104\) 0 0
\(105\) −13257.2 13257.2i −1.20247 1.20247i
\(106\) 0 0
\(107\) 4453.05 + 10750.6i 0.388947 + 0.939001i 0.990164 + 0.139915i \(0.0446828\pi\)
−0.601217 + 0.799086i \(0.705317\pi\)
\(108\) 0 0
\(109\) 5688.78 13733.9i 0.478813 1.15596i −0.481353 0.876527i \(-0.659854\pi\)
0.960166 0.279431i \(-0.0901457\pi\)
\(110\) 0 0
\(111\) 7760.91 0.629893
\(112\) 0 0
\(113\) 17675.0i 1.38421i 0.721795 + 0.692107i \(0.243317\pi\)
−0.721795 + 0.692107i \(0.756683\pi\)
\(114\) 0 0
\(115\) −2188.83 906.643i −0.165507 0.0685552i
\(116\) 0 0
\(117\) 1146.66 474.964i 0.0837653 0.0346967i
\(118\) 0 0
\(119\) −25254.1 + 25254.1i −1.78335 + 1.78335i
\(120\) 0 0
\(121\) −12148.8 + 12148.8i −0.829782 + 0.829782i
\(122\) 0 0
\(123\) 11751.1 4867.45i 0.776725 0.321730i
\(124\) 0 0
\(125\) 5910.59 + 2448.25i 0.378278 + 0.156688i
\(126\) 0 0
\(127\) 13463.4i 0.834730i 0.908739 + 0.417365i \(0.137046\pi\)
−0.908739 + 0.417365i \(0.862954\pi\)
\(128\) 0 0
\(129\) −6585.88 −0.395762
\(130\) 0 0
\(131\) −2440.14 + 5891.02i −0.142191 + 0.343279i −0.978891 0.204382i \(-0.934482\pi\)
0.836700 + 0.547661i \(0.184482\pi\)
\(132\) 0 0
\(133\) 4978.43 + 12019.0i 0.281442 + 0.679462i
\(134\) 0 0
\(135\) −18040.4 18040.4i −0.989872 0.989872i
\(136\) 0 0
\(137\) −24401.5 24401.5i −1.30010 1.30010i −0.928323 0.371776i \(-0.878749\pi\)
−0.371776 0.928323i \(-0.621251\pi\)
\(138\) 0 0
\(139\) −3377.01 8152.83i −0.174785 0.421967i 0.812074 0.583555i \(-0.198339\pi\)
−0.986858 + 0.161587i \(0.948339\pi\)
\(140\) 0 0
\(141\) −5637.87 + 13611.0i −0.283581 + 0.684625i
\(142\) 0 0
\(143\) −12099.3 −0.591684
\(144\) 0 0
\(145\) 27383.6i 1.30243i
\(146\) 0 0
\(147\) −21389.1 8859.64i −0.989822 0.409998i
\(148\) 0 0
\(149\) −22044.4 + 9131.09i −0.992946 + 0.411292i −0.819206 0.573500i \(-0.805585\pi\)
−0.173740 + 0.984791i \(0.555585\pi\)
\(150\) 0 0
\(151\) 21912.5 21912.5i 0.961031 0.961031i −0.0382374 0.999269i \(-0.512174\pi\)
0.999269 + 0.0382374i \(0.0121743\pi\)
\(152\) 0 0
\(153\) −6332.94 + 6332.94i −0.270535 + 0.270535i
\(154\) 0 0
\(155\) −104.173 + 43.1499i −0.00433603 + 0.00179604i
\(156\) 0 0
\(157\) 26287.6 + 10888.7i 1.06648 + 0.441750i 0.845747 0.533584i \(-0.179155\pi\)
0.220732 + 0.975334i \(0.429155\pi\)
\(158\) 0 0
\(159\) 24353.9i 0.963327i
\(160\) 0 0
\(161\) −5328.02 −0.205548
\(162\) 0 0
\(163\) −4953.52 + 11958.9i −0.186440 + 0.450106i −0.989269 0.146103i \(-0.953327\pi\)
0.802830 + 0.596209i \(0.203327\pi\)
\(164\) 0 0
\(165\) 17539.7 + 42344.5i 0.644248 + 1.55535i
\(166\) 0 0
\(167\) −2732.66 2732.66i −0.0979835 0.0979835i 0.656416 0.754399i \(-0.272072\pi\)
−0.754399 + 0.656416i \(0.772072\pi\)
\(168\) 0 0
\(169\) 16942.7 + 16942.7i 0.593211 + 0.593211i
\(170\) 0 0
\(171\) 1248.44 + 3013.99i 0.0426948 + 0.103074i
\(172\) 0 0
\(173\) −10506.7 + 25365.4i −0.351054 + 0.847519i 0.645437 + 0.763814i \(0.276675\pi\)
−0.996491 + 0.0837050i \(0.973325\pi\)
\(174\) 0 0
\(175\) −31219.2 −1.01940
\(176\) 0 0
\(177\) 22397.7i 0.714918i
\(178\) 0 0
\(179\) 30976.8 + 12831.0i 0.966786 + 0.400456i 0.809515 0.587099i \(-0.199730\pi\)
0.157271 + 0.987555i \(0.449730\pi\)
\(180\) 0 0
\(181\) 31191.2 12919.8i 0.952082 0.394365i 0.148069 0.988977i \(-0.452694\pi\)
0.804013 + 0.594612i \(0.202694\pi\)
\(182\) 0 0
\(183\) 9162.01 9162.01i 0.273583 0.273583i
\(184\) 0 0
\(185\) 22487.4 22487.4i 0.657047 0.657047i
\(186\) 0 0
\(187\) 80663.5 33411.9i 2.30671 0.955472i
\(188\) 0 0
\(189\) −53008.6 21956.9i −1.48396 0.614677i
\(190\) 0 0
\(191\) 10804.6i 0.296170i 0.988975 + 0.148085i \(0.0473109\pi\)
−0.988975 + 0.148085i \(0.952689\pi\)
\(192\) 0 0
\(193\) 19160.5 0.514388 0.257194 0.966360i \(-0.417202\pi\)
0.257194 + 0.966360i \(0.417202\pi\)
\(194\) 0 0
\(195\) −6668.91 + 16100.2i −0.175382 + 0.423410i
\(196\) 0 0
\(197\) −22479.7 54270.7i −0.579239 1.39841i −0.893497 0.449068i \(-0.851756\pi\)
0.314259 0.949337i \(-0.398244\pi\)
\(198\) 0 0
\(199\) 4494.51 + 4494.51i 0.113495 + 0.113495i 0.761574 0.648079i \(-0.224427\pi\)
−0.648079 + 0.761574i \(0.724427\pi\)
\(200\) 0 0
\(201\) −19389.0 19389.0i −0.479915 0.479915i
\(202\) 0 0
\(203\) −23566.8 56895.2i −0.571884 1.38065i
\(204\) 0 0
\(205\) 19945.4 48152.5i 0.474609 1.14581i
\(206\) 0 0
\(207\) −1336.10 −0.0311817
\(208\) 0 0
\(209\) 31803.0i 0.728074i
\(210\) 0 0
\(211\) 7136.71 + 2956.12i 0.160300 + 0.0663983i 0.461391 0.887197i \(-0.347351\pi\)
−0.301091 + 0.953595i \(0.597351\pi\)
\(212\) 0 0
\(213\) 58550.6 24252.5i 1.29054 0.534560i
\(214\) 0 0
\(215\) −19082.7 + 19082.7i −0.412823 + 0.412823i
\(216\) 0 0
\(217\) −179.306 + 179.306i −0.00380781 + 0.00380781i
\(218\) 0 0
\(219\) −53212.6 + 22041.4i −1.10950 + 0.459569i
\(220\) 0 0
\(221\) 30669.8 + 12703.8i 0.627951 + 0.260106i
\(222\) 0 0
\(223\) 24536.0i 0.493394i −0.969093 0.246697i \(-0.920655\pi\)
0.969093 0.246697i \(-0.0793452\pi\)
\(224\) 0 0
\(225\) −7828.82 −0.154643
\(226\) 0 0
\(227\) 23630.9 57050.0i 0.458594 1.10714i −0.510373 0.859953i \(-0.670493\pi\)
0.968967 0.247190i \(-0.0795072\pi\)
\(228\) 0 0
\(229\) −15539.7 37516.1i −0.296327 0.715396i −0.999988 0.00486046i \(-0.998453\pi\)
0.703662 0.710535i \(-0.251547\pi\)
\(230\) 0 0
\(231\) 72884.6 + 72884.6i 1.36588 + 1.36588i
\(232\) 0 0
\(233\) 1502.52 + 1502.52i 0.0276764 + 0.0276764i 0.720810 0.693133i \(-0.243770\pi\)
−0.693133 + 0.720810i \(0.743770\pi\)
\(234\) 0 0
\(235\) 23102.4 + 55774.1i 0.418332 + 1.00994i
\(236\) 0 0
\(237\) −15697.1 + 37896.0i −0.279461 + 0.674679i
\(238\) 0 0
\(239\) 47803.7 0.836885 0.418442 0.908243i \(-0.362576\pi\)
0.418442 + 0.908243i \(0.362576\pi\)
\(240\) 0 0
\(241\) 28441.1i 0.489679i 0.969564 + 0.244840i \(0.0787353\pi\)
−0.969564 + 0.244840i \(0.921265\pi\)
\(242\) 0 0
\(243\) −24136.2 9997.55i −0.408749 0.169309i
\(244\) 0 0
\(245\) −87646.2 + 36304.3i −1.46016 + 0.604819i
\(246\) 0 0
\(247\) 8550.40 8550.40i 0.140150 0.140150i
\(248\) 0 0
\(249\) 12136.8 12136.8i 0.195752 0.195752i
\(250\) 0 0
\(251\) −104181. + 43153.0i −1.65364 + 0.684958i −0.997565 0.0697419i \(-0.977782\pi\)
−0.656070 + 0.754700i \(0.727782\pi\)
\(252\) 0 0
\(253\) 12033.6 + 4984.49i 0.187999 + 0.0778717i
\(254\) 0 0
\(255\) 125752.i 1.93390i
\(256\) 0 0
\(257\) 58517.2 0.885967 0.442984 0.896530i \(-0.353920\pi\)
0.442984 + 0.896530i \(0.353920\pi\)
\(258\) 0 0
\(259\) 27369.3 66075.3i 0.408004 0.985008i
\(260\) 0 0
\(261\) −5909.82 14267.6i −0.0867547 0.209444i
\(262\) 0 0
\(263\) −72405.5 72405.5i −1.04679 1.04679i −0.998850 0.0479411i \(-0.984734\pi\)
−0.0479411 0.998850i \(-0.515266\pi\)
\(264\) 0 0
\(265\) 70565.9 + 70565.9i 1.00485 + 1.00485i
\(266\) 0 0
\(267\) 12593.6 + 30403.5i 0.176655 + 0.426483i
\(268\) 0 0
\(269\) 3628.21 8759.27i 0.0501404 0.121050i −0.896825 0.442386i \(-0.854132\pi\)
0.946965 + 0.321337i \(0.104132\pi\)
\(270\) 0 0
\(271\) 80438.8 1.09528 0.547642 0.836713i \(-0.315526\pi\)
0.547642 + 0.836713i \(0.315526\pi\)
\(272\) 0 0
\(273\) 39190.8i 0.525847i
\(274\) 0 0
\(275\) 70510.3 + 29206.3i 0.932368 + 0.386199i
\(276\) 0 0
\(277\) 42328.8 17533.2i 0.551666 0.228508i −0.0893965 0.995996i \(-0.528494\pi\)
0.641063 + 0.767488i \(0.278494\pi\)
\(278\) 0 0
\(279\) −44.9644 + 44.9644i −0.000577644 + 0.000577644i
\(280\) 0 0
\(281\) −28015.2 + 28015.2i −0.354798 + 0.354798i −0.861891 0.507093i \(-0.830720\pi\)
0.507093 + 0.861891i \(0.330720\pi\)
\(282\) 0 0
\(283\) −111649. + 46246.3i −1.39406 + 0.577437i −0.948202 0.317669i \(-0.897100\pi\)
−0.445854 + 0.895106i \(0.647100\pi\)
\(284\) 0 0
\(285\) −42319.1 17529.2i −0.521011 0.215810i
\(286\) 0 0
\(287\) 117212.i 1.42302i
\(288\) 0 0
\(289\) −156028. −1.86813
\(290\) 0 0
\(291\) −42738.8 + 103180.i −0.504703 + 1.21846i
\(292\) 0 0
\(293\) −8654.24 20893.2i −0.100808 0.243371i 0.865427 0.501035i \(-0.167047\pi\)
−0.966235 + 0.257664i \(0.917047\pi\)
\(294\) 0 0
\(295\) 64897.7 + 64897.7i 0.745736 + 0.745736i
\(296\) 0 0
\(297\) 99181.6 + 99181.6i 1.12439 + 1.12439i
\(298\) 0 0
\(299\) 1895.20 + 4575.41i 0.0211988 + 0.0511785i
\(300\) 0 0
\(301\) −23225.5 + 56071.3i −0.256349 + 0.618881i
\(302\) 0 0
\(303\) −87602.5 −0.954182
\(304\) 0 0
\(305\) 53094.3i 0.570753i
\(306\) 0 0
\(307\) −11997.1 4969.36i −0.127292 0.0527259i 0.318128 0.948048i \(-0.396946\pi\)
−0.445420 + 0.895322i \(0.646946\pi\)
\(308\) 0 0
\(309\) −1406.98 + 582.791i −0.0147357 + 0.00610374i
\(310\) 0 0
\(311\) 7023.22 7023.22i 0.0726132 0.0726132i −0.669867 0.742481i \(-0.733649\pi\)
0.742481 + 0.669867i \(0.233649\pi\)
\(312\) 0 0
\(313\) 25821.6 25821.6i 0.263569 0.263569i −0.562933 0.826502i \(-0.690327\pi\)
0.826502 + 0.562933i \(0.190327\pi\)
\(314\) 0 0
\(315\) −40028.3 + 16580.3i −0.403409 + 0.167098i
\(316\) 0 0
\(317\) −49717.1 20593.5i −0.494752 0.204933i 0.121334 0.992612i \(-0.461283\pi\)
−0.616086 + 0.787679i \(0.711283\pi\)
\(318\) 0 0
\(319\) 150548.i 1.47943i
\(320\) 0 0
\(321\) −92141.6 −0.894223
\(322\) 0 0
\(323\) −33391.9 + 80615.2i −0.320063 + 0.772701i
\(324\) 0 0
\(325\) 11104.8 + 26809.3i 0.105134 + 0.253816i
\(326\) 0 0
\(327\) 83234.3 + 83234.3i 0.778407 + 0.778407i
\(328\) 0 0
\(329\) 96000.2 + 96000.2i 0.886911 + 0.886911i
\(330\) 0 0
\(331\) 16857.0 + 40696.5i 0.153860 + 0.371450i 0.981949 0.189146i \(-0.0605719\pi\)
−0.828089 + 0.560596i \(0.810572\pi\)
\(332\) 0 0
\(333\) 6863.38 16569.7i 0.0618941 0.149426i
\(334\) 0 0
\(335\) −112360. −1.00121
\(336\) 0 0
\(337\) 39444.9i 0.347321i −0.984806 0.173660i \(-0.944440\pi\)
0.984806 0.173660i \(-0.0555595\pi\)
\(338\) 0 0
\(339\) −129304. 53559.7i −1.12516 0.466056i
\(340\) 0 0
\(341\) 572.717 237.227i 0.00492528 0.00204012i
\(342\) 0 0
\(343\) −26972.3 + 26972.3i −0.229261 + 0.229261i
\(344\) 0 0
\(345\) 13265.4 13265.4i 0.111450 0.111450i
\(346\) 0 0
\(347\) 51393.3 21287.8i 0.426823 0.176796i −0.158922 0.987291i \(-0.550802\pi\)
0.585745 + 0.810495i \(0.300802\pi\)
\(348\) 0 0
\(349\) 44357.7 + 18373.6i 0.364182 + 0.150849i 0.557268 0.830333i \(-0.311850\pi\)
−0.193086 + 0.981182i \(0.561850\pi\)
\(350\) 0 0
\(351\) 53331.0i 0.432878i
\(352\) 0 0
\(353\) −231798. −1.86020 −0.930101 0.367303i \(-0.880281\pi\)
−0.930101 + 0.367303i \(0.880281\pi\)
\(354\) 0 0
\(355\) 99379.7 239924.i 0.788571 1.90378i
\(356\) 0 0
\(357\) −108224. 261276.i −0.849156 2.05004i
\(358\) 0 0
\(359\) −35078.5 35078.5i −0.272177 0.272177i 0.557799 0.829976i \(-0.311646\pi\)
−0.829976 + 0.557799i \(0.811646\pi\)
\(360\) 0 0
\(361\) −69676.2 69676.2i −0.534651 0.534651i
\(362\) 0 0
\(363\) −52062.8 125691.i −0.395106 0.953871i
\(364\) 0 0
\(365\) −90319.3 + 218050.i −0.677946 + 1.63671i
\(366\) 0 0
\(367\) 154694. 1.14853 0.574263 0.818671i \(-0.305289\pi\)
0.574263 + 0.818671i \(0.305289\pi\)
\(368\) 0 0
\(369\) 29393.2i 0.215871i
\(370\) 0 0
\(371\) 207346. + 85885.3i 1.50642 + 0.623981i
\(372\) 0 0
\(373\) 106468. 44100.4i 0.765245 0.316975i 0.0343001 0.999412i \(-0.489080\pi\)
0.730945 + 0.682437i \(0.239080\pi\)
\(374\) 0 0
\(375\) −35821.1 + 35821.1i −0.254728 + 0.254728i
\(376\) 0 0
\(377\) −40475.6 + 40475.6i −0.284781 + 0.284781i
\(378\) 0 0
\(379\) −85313.9 + 35338.2i −0.593939 + 0.246017i −0.659344 0.751841i \(-0.729166\pi\)
0.0654054 + 0.997859i \(0.479166\pi\)
\(380\) 0 0
\(381\) −98493.3 40797.3i −0.678511 0.281048i
\(382\) 0 0
\(383\) 102095.i 0.695994i −0.937496 0.347997i \(-0.886862\pi\)
0.937496 0.347997i \(-0.113138\pi\)
\(384\) 0 0
\(385\) 422370. 2.84952
\(386\) 0 0
\(387\) −5824.23 + 14060.9i −0.0388881 + 0.0938842i
\(388\) 0 0
\(389\) 76247.2 + 184077.i 0.503877 + 1.21647i 0.947356 + 0.320182i \(0.103744\pi\)
−0.443479 + 0.896285i \(0.646256\pi\)
\(390\) 0 0
\(391\) −25269.7 25269.7i −0.165290 0.165290i
\(392\) 0 0
\(393\) −35702.4 35702.4i −0.231160 0.231160i
\(394\) 0 0
\(395\) 64322.0 + 155287.i 0.412255 + 0.995271i
\(396\) 0 0
\(397\) 4363.53 10534.5i 0.0276858 0.0668394i −0.909432 0.415853i \(-0.863483\pi\)
0.937118 + 0.349013i \(0.113483\pi\)
\(398\) 0 0
\(399\) −103013. −0.647061
\(400\) 0 0
\(401\) 79631.3i 0.495217i −0.968860 0.247608i \(-0.920355\pi\)
0.968860 0.247608i \(-0.0796446\pi\)
\(402\) 0 0
\(403\) 217.758 + 90.1981i 0.00134080 + 0.000555376i
\(404\) 0 0
\(405\) 142212. 58905.9i 0.867011 0.359128i
\(406\) 0 0
\(407\) −123630. + 123630.i −0.746338 + 0.746338i
\(408\) 0 0
\(409\) 125746. 125746.i 0.751707 0.751707i −0.223091 0.974798i \(-0.571615\pi\)
0.974798 + 0.223091i \(0.0716146\pi\)
\(410\) 0 0
\(411\) 252456. 104571.i 1.49452 0.619051i
\(412\) 0 0
\(413\) 190691. + 78986.6i 1.11797 + 0.463077i
\(414\) 0 0
\(415\) 70333.3i 0.408380i
\(416\) 0 0
\(417\) 69876.5 0.401845
\(418\) 0 0
\(419\) 70409.1 169983.i 0.401052 0.968225i −0.586359 0.810051i \(-0.699439\pi\)
0.987411 0.158174i \(-0.0505607\pi\)
\(420\) 0 0
\(421\) −15986.4 38594.5i −0.0901956 0.217751i 0.872344 0.488893i \(-0.162599\pi\)
−0.962540 + 0.271141i \(0.912599\pi\)
\(422\) 0 0
\(423\) 24073.9 + 24073.9i 0.134544 + 0.134544i
\(424\) 0 0
\(425\) −148066. 148066.i −0.819743 0.819743i
\(426\) 0 0
\(427\) −45693.8 110315.i −0.250612 0.605030i
\(428\) 0 0
\(429\) 36663.9 88514.6i 0.199216 0.480950i
\(430\) 0 0
\(431\) −246277. −1.32578 −0.662888 0.748719i \(-0.730669\pi\)
−0.662888 + 0.748719i \(0.730669\pi\)
\(432\) 0 0
\(433\) 240681.i 1.28371i 0.766828 + 0.641853i \(0.221834\pi\)
−0.766828 + 0.641853i \(0.778166\pi\)
\(434\) 0 0
\(435\) 200329. + 82979.0i 1.05868 + 0.438520i
\(436\) 0 0
\(437\) −12026.4 + 4981.50i −0.0629758 + 0.0260854i
\(438\) 0 0
\(439\) 267417. 267417.i 1.38759 1.38759i 0.557220 0.830365i \(-0.311868\pi\)
0.830365 0.557220i \(-0.188132\pi\)
\(440\) 0 0
\(441\) −37830.9 + 37830.9i −0.194522 + 0.194522i
\(442\) 0 0
\(443\) 183796. 76130.7i 0.936544 0.387929i 0.138387 0.990378i \(-0.455808\pi\)
0.798157 + 0.602449i \(0.205808\pi\)
\(444\) 0 0
\(445\) 124585. + 51604.8i 0.629138 + 0.260597i
\(446\) 0 0
\(447\) 188939.i 0.945596i
\(448\) 0 0
\(449\) −114159. −0.566260 −0.283130 0.959082i \(-0.591373\pi\)
−0.283130 + 0.959082i \(0.591373\pi\)
\(450\) 0 0
\(451\) −109655. + 264730.i −0.539107 + 1.30152i
\(452\) 0 0
\(453\) 93904.0 + 226704.i 0.457602 + 1.10475i
\(454\) 0 0
\(455\) 113556. + 113556.i 0.548515 + 0.548515i
\(456\) 0 0
\(457\) −262872. 262872.i −1.25867 1.25867i −0.951728 0.306943i \(-0.900694\pi\)
−0.306943 0.951728i \(-0.599306\pi\)
\(458\) 0 0
\(459\) −147272. 355545.i −0.699027 1.68760i
\(460\) 0 0
\(461\) −33053.9 + 79799.1i −0.155532 + 0.375488i −0.982369 0.186955i \(-0.940138\pi\)
0.826836 + 0.562443i \(0.190138\pi\)
\(462\) 0 0
\(463\) 105487. 0.492080 0.246040 0.969260i \(-0.420870\pi\)
0.246040 + 0.969260i \(0.420870\pi\)
\(464\) 0 0
\(465\) 892.848i 0.00412926i
\(466\) 0 0
\(467\) −179694. 74431.5i −0.823946 0.341290i −0.0694429 0.997586i \(-0.522122\pi\)
−0.754503 + 0.656296i \(0.772122\pi\)
\(468\) 0 0
\(469\) −233452. + 96699.1i −1.06133 + 0.439619i
\(470\) 0 0
\(471\) −159316. + 159316.i −0.718153 + 0.718153i
\(472\) 0 0
\(473\) 104912. 104912.i 0.468924 0.468924i
\(474\) 0 0
\(475\) −70468.1 + 29188.9i −0.312324 + 0.129369i
\(476\) 0 0
\(477\) 51995.9 + 21537.4i 0.228524 + 0.0946578i
\(478\) 0 0
\(479\) 265190.i 1.15581i 0.816105 + 0.577904i \(0.196129\pi\)
−0.816105 + 0.577904i \(0.803871\pi\)
\(480\) 0 0
\(481\) −66477.2 −0.287331
\(482\) 0 0
\(483\) 16145.2 38978.0i 0.0692069 0.167080i
\(484\) 0 0
\(485\) 175131. + 422804.i 0.744526 + 1.79745i
\(486\) 0 0
\(487\) −178867. 178867.i −0.754177 0.754177i 0.221079 0.975256i \(-0.429042\pi\)
−0.975256 + 0.221079i \(0.929042\pi\)
\(488\) 0 0
\(489\) −72476.5 72476.5i −0.303095 0.303095i
\(490\) 0 0
\(491\) −1535.66 3707.42i −0.00636990 0.0153783i 0.920663 0.390359i \(-0.127649\pi\)
−0.927032 + 0.374981i \(0.877649\pi\)
\(492\) 0 0
\(493\) 158070. 381614.i 0.650361 1.57011i
\(494\) 0 0
\(495\) 105917. 0.432271
\(496\) 0 0
\(497\) 584020.i 2.36437i
\(498\) 0 0
\(499\) −94336.4 39075.4i −0.378860 0.156929i 0.185123 0.982715i \(-0.440732\pi\)
−0.563983 + 0.825787i \(0.690732\pi\)
\(500\) 0 0
\(501\) 28271.8 11710.6i 0.112636 0.0466555i
\(502\) 0 0
\(503\) 122210. 122210.i 0.483027 0.483027i −0.423070 0.906097i \(-0.639048\pi\)
0.906097 + 0.423070i \(0.139048\pi\)
\(504\) 0 0
\(505\) −253830. + 253830.i −0.995315 + 0.995315i
\(506\) 0 0
\(507\) −175288. + 72606.5i −0.681923 + 0.282462i
\(508\) 0 0
\(509\) −146281. 60591.4i −0.564613 0.233871i 0.0820732 0.996626i \(-0.473846\pi\)
−0.646687 + 0.762756i \(0.723846\pi\)
\(510\) 0 0
\(511\) 530775.i 2.03268i
\(512\) 0 0
\(513\) −140180. −0.532661
\(514\) 0 0
\(515\) −2388.11 + 5765.41i −0.00900410 + 0.0217378i
\(516\) 0 0
\(517\) −127011. 306632.i −0.475183 1.14719i
\(518\) 0 0
\(519\) −153727. 153727.i −0.570708 0.570708i
\(520\) 0 0
\(521\) 160559. + 160559.i 0.591508 + 0.591508i 0.938039 0.346531i \(-0.112640\pi\)
−0.346531 + 0.938039i \(0.612640\pi\)
\(522\) 0 0
\(523\) 199672. + 482051.i 0.729985 + 1.76234i 0.642648 + 0.766162i \(0.277836\pi\)
0.0873373 + 0.996179i \(0.472164\pi\)
\(524\) 0 0
\(525\) 94601.9 228389.i 0.343227 0.828623i
\(526\) 0 0
\(527\) −1700.82 −0.00612402
\(528\) 0 0
\(529\) 274510.i 0.980949i
\(530\) 0 0
\(531\) 47819.3 + 19807.4i 0.169596 + 0.0702488i
\(532\) 0 0
\(533\) −100655. + 41692.8i −0.354309 + 0.146760i
\(534\) 0 0
\(535\) −266982. + 266982.i −0.932771 + 0.932771i
\(536\) 0 0
\(537\) −187734. + 187734.i −0.651022 + 0.651022i
\(538\) 0 0
\(539\) 481857. 199592.i 1.65859 0.687012i
\(540\) 0 0
\(541\) −11552.6 4785.24i −0.0394717 0.0163497i 0.362860 0.931844i \(-0.381800\pi\)
−0.402332 + 0.915494i \(0.631800\pi\)
\(542\) 0 0
\(543\) 267334.i 0.906681i
\(544\) 0 0
\(545\) 482346. 1.62393
\(546\) 0 0
\(547\) −131230. + 316817.i −0.438589 + 1.05885i 0.537847 + 0.843043i \(0.319238\pi\)
−0.976436 + 0.215806i \(0.930762\pi\)
\(548\) 0 0
\(549\) −11458.6 27663.5i −0.0380177 0.0917829i
\(550\) 0 0
\(551\) −106390. 106390.i −0.350427 0.350427i
\(552\) 0 0
\(553\) 267285. + 267285.i 0.874026 + 0.874026i
\(554\) 0 0
\(555\) 96367.8 + 232653.i 0.312857 + 0.755304i
\(556\) 0 0
\(557\) −98441.9 + 237660.i −0.317300 + 0.766029i 0.682096 + 0.731263i \(0.261069\pi\)
−0.999395 + 0.0347663i \(0.988931\pi\)
\(558\) 0 0
\(559\) 56412.3 0.180530
\(560\) 0 0
\(561\) 691352.i 2.19671i
\(562\) 0 0
\(563\) −215850. 89408.0i −0.680981 0.282072i 0.0152558 0.999884i \(-0.495144\pi\)
−0.696237 + 0.717812i \(0.745144\pi\)
\(564\) 0 0
\(565\) −529853. + 219472.i −1.65981 + 0.687516i
\(566\) 0 0
\(567\) 244779. 244779.i 0.761391 0.761391i
\(568\) 0 0
\(569\) 299019. 299019.i 0.923578 0.923578i −0.0737022 0.997280i \(-0.523481\pi\)
0.997280 + 0.0737022i \(0.0234814\pi\)
\(570\) 0 0
\(571\) 74877.7 31015.4i 0.229657 0.0951272i −0.264887 0.964279i \(-0.585335\pi\)
0.494545 + 0.869152i \(0.335335\pi\)
\(572\) 0 0
\(573\) −79042.5 32740.5i −0.240742 0.0997185i
\(574\) 0 0
\(575\) 31238.5i 0.0944832i
\(576\) 0 0
\(577\) −169930. −0.510410 −0.255205 0.966887i \(-0.582143\pi\)
−0.255205 + 0.966887i \(0.582143\pi\)
\(578\) 0 0
\(579\) −58060.8 + 140171.i −0.173191 + 0.418121i
\(580\) 0 0
\(581\) −60530.0 146132.i −0.179316 0.432906i
\(582\) 0 0
\(583\) −387953. 387953.i −1.14141 1.14141i
\(584\) 0 0
\(585\) 28476.4 + 28476.4i 0.0832096 + 0.0832096i
\(586\) 0 0
\(587\) 69704.8 + 168282.i 0.202296 + 0.488385i 0.992172 0.124882i \(-0.0398552\pi\)
−0.789876 + 0.613266i \(0.789855\pi\)
\(588\) 0 0
\(589\) −237.085 + 572.374i −0.000683397 + 0.00164987i
\(590\) 0 0
\(591\) 465145. 1.33172
\(592\) 0 0
\(593\) 446782.i 1.27053i −0.772293 0.635267i \(-0.780890\pi\)
0.772293 0.635267i \(-0.219110\pi\)
\(594\) 0 0
\(595\) −1.07064e6 443472.i −3.02418 1.25266i
\(596\) 0 0
\(597\) −46499.8 + 19260.8i −0.130467 + 0.0540414i
\(598\) 0 0
\(599\) −123181. + 123181.i −0.343312 + 0.343312i −0.857611 0.514299i \(-0.828052\pi\)
0.514299 + 0.857611i \(0.328052\pi\)
\(600\) 0 0
\(601\) −268659. + 268659.i −0.743794 + 0.743794i −0.973306 0.229512i \(-0.926287\pi\)
0.229512 + 0.973306i \(0.426287\pi\)
\(602\) 0 0
\(603\) −58542.6 + 24249.1i −0.161004 + 0.0666902i
\(604\) 0 0
\(605\) −515044. 213338.i −1.40713 0.582852i
\(606\) 0 0
\(607\) 374047.i 1.01519i −0.861595 0.507597i \(-0.830534\pi\)
0.861595 0.507597i \(-0.169466\pi\)
\(608\) 0 0
\(609\) 487639. 1.31481
\(610\) 0 0
\(611\) 48292.0 116587.i 0.129358 0.312297i
\(612\) 0 0
\(613\) 160688. + 387935.i 0.427625 + 1.03238i 0.980039 + 0.198807i \(0.0637068\pi\)
−0.552414 + 0.833570i \(0.686293\pi\)
\(614\) 0 0
\(615\) 291828. + 291828.i 0.771572 + 0.771572i
\(616\) 0 0
\(617\) −157953. 157953.i −0.414914 0.414914i 0.468532 0.883446i \(-0.344783\pi\)
−0.883446 + 0.468532i \(0.844783\pi\)
\(618\) 0 0
\(619\) −231148. 558042.i −0.603267 1.45642i −0.870199 0.492700i \(-0.836010\pi\)
0.266932 0.963715i \(-0.413990\pi\)
\(620\) 0 0
\(621\) 21970.4 53041.3i 0.0569712 0.137541i
\(622\) 0 0
\(623\) 303263. 0.781347
\(624\) 0 0
\(625\) 474980.i 1.21595i
\(626\) 0 0
\(627\) 232660. + 96370.8i 0.591815 + 0.245138i
\(628\) 0 0
\(629\) 443188. 183574.i 1.12018 0.463992i
\(630\) 0 0
\(631\) 455364. 455364.i 1.14367 1.14367i 0.155895 0.987774i \(-0.450174\pi\)
0.987774 0.155895i \(-0.0498263\pi\)
\(632\) 0 0
\(633\) −43251.9 + 43251.9i −0.107944 + 0.107944i
\(634\) 0 0
\(635\) −403598. + 167176.i −1.00092 + 0.414596i
\(636\) 0 0
\(637\) 183211. + 75888.4i 0.451515 + 0.187024i
\(638\) 0 0
\(639\) 146454.i 0.358674i
\(640\) 0 0
\(641\) 294774. 0.717418 0.358709 0.933449i \(-0.383217\pi\)
0.358709 + 0.933449i \(0.383217\pi\)
\(642\) 0 0
\(643\) −97454.5 + 235276.i −0.235711 + 0.569057i −0.996830 0.0795549i \(-0.974650\pi\)
0.761119 + 0.648612i \(0.224650\pi\)
\(644\) 0 0
\(645\) −81777.4 197428.i −0.196568 0.474558i
\(646\) 0 0
\(647\) −435797. 435797.i −1.04106 1.04106i −0.999120 0.0419402i \(-0.986646\pi\)
−0.0419402 0.999120i \(-0.513354\pi\)
\(648\) 0 0
\(649\) −356791. 356791.i −0.847080 0.847080i
\(650\) 0 0
\(651\) −768.399 1855.08i −0.00181311 0.00437724i
\(652\) 0 0
\(653\) 212867. 513905.i 0.499208 1.20519i −0.450704 0.892674i \(-0.648827\pi\)
0.949911 0.312520i \(-0.101173\pi\)
\(654\) 0 0
\(655\) −206897. −0.482250
\(656\) 0 0
\(657\) 133102.i 0.308357i
\(658\) 0 0
\(659\) 590015. + 244392.i 1.35860 + 0.562751i 0.938674 0.344806i \(-0.112055\pi\)
0.419928 + 0.907557i \(0.362055\pi\)
\(660\) 0 0
\(661\) −476292. + 197287.i −1.09011 + 0.451538i −0.854045 0.520199i \(-0.825858\pi\)
−0.236065 + 0.971737i \(0.575858\pi\)
\(662\) 0 0
\(663\) −185874. + 185874.i −0.422854 + 0.422854i
\(664\) 0 0
\(665\) −298482. + 298482.i −0.674954 + 0.674954i
\(666\) 0 0
\(667\) 56930.3 23581.3i 0.127965 0.0530049i
\(668\) 0 0
\(669\) 179497. + 74349.9i 0.401055 + 0.166122i
\(670\) 0 0
\(671\) 291899.i 0.648317i
\(672\) 0 0
\(673\) −88432.6 −0.195246 −0.0976230 0.995223i \(-0.531124\pi\)
−0.0976230 + 0.995223i \(0.531124\pi\)
\(674\) 0 0
\(675\) 128735. 310793.i 0.282545 0.682124i
\(676\) 0 0
\(677\) 33417.8 + 80677.8i 0.0729123 + 0.176026i 0.956134 0.292930i \(-0.0946303\pi\)
−0.883222 + 0.468956i \(0.844630\pi\)
\(678\) 0 0
\(679\) 727744. + 727744.i 1.57848 + 1.57848i
\(680\) 0 0
\(681\) 345751. + 345751.i 0.745536 + 0.745536i
\(682\) 0 0
\(683\) 71540.4 + 172714.i 0.153359 + 0.370242i 0.981822 0.189801i \(-0.0607844\pi\)
−0.828463 + 0.560043i \(0.810784\pi\)
\(684\) 0 0
\(685\) 428501. 1.03449e6i 0.913210 2.20468i
\(686\) 0 0
\(687\) 321544. 0.681281
\(688\) 0 0
\(689\) 208606.i 0.439430i
\(690\) 0 0
\(691\) 405341. + 167898.i 0.848915 + 0.351632i 0.764362 0.644787i \(-0.223054\pi\)
0.0845525 + 0.996419i \(0.473054\pi\)
\(692\) 0 0
\(693\) 220065. 91154.1i 0.458232 0.189806i
\(694\) 0 0
\(695\) 202469. 202469.i 0.419168 0.419168i
\(696\) 0 0
\(697\) 555913. 555913.i 1.14430 1.14430i
\(698\) 0 0
\(699\) −15544.9 + 6438.92i −0.0318152 + 0.0131783i
\(700\) 0 0
\(701\) 612430. + 253677.i 1.24629 + 0.516231i 0.905676 0.423971i \(-0.139364\pi\)
0.340617 + 0.940202i \(0.389364\pi\)
\(702\) 0 0
\(703\) 174735.i 0.353564i
\(704\) 0 0
\(705\) −478030. −0.961783
\(706\) 0 0
\(707\) −308935. + 745836.i −0.618057 + 1.49212i
\(708\) 0 0
\(709\) −202534. 488959.i −0.402907 0.972703i −0.986957 0.160985i \(-0.948533\pi\)
0.584050 0.811718i \(-0.301467\pi\)
\(710\) 0 0
\(711\) 67026.8 + 67026.8i 0.132590 + 0.132590i
\(712\) 0 0
\(713\) −179.416 179.416i −0.000352926 0.000352926i
\(714\) 0 0
\(715\) −150238. 362707.i −0.293879 0.709487i
\(716\) 0 0
\(717\) −144857. + 349715.i −0.281774 + 0.680262i
\(718\) 0 0
\(719\) −459937. −0.889694 −0.444847 0.895607i \(-0.646742\pi\)
−0.444847 + 0.895607i \(0.646742\pi\)
\(720\) 0 0
\(721\) 14034.1i 0.0269969i
\(722\) 0 0
\(723\) −208065. 86183.3i −0.398036 0.164872i
\(724\) 0 0
\(725\) 333580. 138173.i 0.634635 0.262874i
\(726\) 0 0
\(727\) 620411. 620411.i 1.17385 1.17385i 0.192560 0.981285i \(-0.438321\pi\)
0.981285 0.192560i \(-0.0616790\pi\)
\(728\) 0 0
\(729\) 417990. 417990.i 0.786522 0.786522i
\(730\) 0 0
\(731\) −376087. + 155781.i −0.703808 + 0.291527i
\(732\) 0 0
\(733\) −434533. 179989.i −0.808751 0.334996i −0.0602955 0.998181i \(-0.519204\pi\)
−0.748456 + 0.663185i \(0.769204\pi\)
\(734\) 0 0
\(735\) 751200.i 1.39053i
\(736\) 0 0
\(737\) 617729. 1.13727
\(738\) 0 0
\(739\) 195326. 471558.i 0.357660 0.863469i −0.637968 0.770063i \(-0.720225\pi\)
0.995628 0.0934055i \(-0.0297753\pi\)
\(740\) 0 0
\(741\) 36642.0 + 88461.6i 0.0667333 + 0.161108i
\(742\) 0 0
\(743\) 126962. + 126962.i 0.229982 + 0.229982i 0.812685 0.582703i \(-0.198005\pi\)
−0.582703 + 0.812685i \(0.698005\pi\)
\(744\) 0 0
\(745\) −547454. 547454.i −0.986359 0.986359i
\(746\) 0 0
\(747\) −15179.0 36645.5i −0.0272021 0.0656718i
\(748\) 0 0
\(749\) −324943. + 784481.i −0.579220 + 1.39836i
\(750\) 0 0
\(751\) −849371. −1.50597 −0.752987 0.658036i \(-0.771388\pi\)
−0.752987 + 0.658036i \(0.771388\pi\)
\(752\) 0 0
\(753\) 892914.i 1.57478i
\(754\) 0 0
\(755\) 928969. + 384792.i 1.62970 + 0.675043i
\(756\) 0 0
\(757\) −199956. + 82824.7i −0.348934 + 0.144533i −0.550265 0.834990i \(-0.685473\pi\)
0.201331 + 0.979523i \(0.435473\pi\)
\(758\) 0 0
\(759\) −72929.6 + 72929.6i −0.126596 + 0.126596i
\(760\) 0 0
\(761\) −183793. + 183793.i −0.317366 + 0.317366i −0.847755 0.530389i \(-0.822046\pi\)
0.530389 + 0.847755i \(0.322046\pi\)
\(762\) 0 0
\(763\) 1.00218e6 415115.i 1.72145 0.713049i
\(764\) 0 0
\(765\) −268482. 111209.i −0.458768 0.190028i
\(766\) 0 0
\(767\) 191850.i 0.326116i
\(768\) 0 0
\(769\) −267282. −0.451979 −0.225989 0.974130i \(-0.572561\pi\)
−0.225989 + 0.974130i \(0.572561\pi\)
\(770\) 0 0
\(771\) −177321. + 428092.i −0.298300 + 0.720159i
\(772\) 0 0
\(773\) 41754.1 + 100803.i 0.0698779 + 0.168700i 0.954960 0.296735i \(-0.0958978\pi\)
−0.885082 + 0.465435i \(0.845898\pi\)
\(774\) 0 0
\(775\) −1051.28 1051.28i −0.00175031 0.00175031i
\(776\) 0 0
\(777\) 400449. + 400449.i 0.663292 + 0.663292i
\(778\) 0 0
\(779\) −109589. 264572.i −0.180590 0.435982i
\(780\) 0 0
\(781\) −546364. + 1.31904e6i −0.895736 + 2.16250i
\(782\) 0 0
\(783\) 663580. 1.08235
\(784\) 0 0
\(785\) 923242.i 1.49822i
\(786\) 0 0
\(787\) −993710. 411608.i −1.60439 0.664561i −0.612363 0.790577i \(-0.709781\pi\)
−0.992028 + 0.126016i \(0.959781\pi\)
\(788\) 0 0
\(789\) 749100. 310287.i 1.20333 0.498437i
\(790\) 0 0
\(791\) −911999. + 911999.i −1.45761 + 1.45761i
\(792\) 0 0
\(793\) −78478.5 + 78478.5i −0.124797 + 0.124797i
\(794\) 0 0
\(795\) −730067. + 302404.i −1.15512 + 0.478468i
\(796\) 0 0
\(797\) −700909. 290326.i −1.10343 0.457056i −0.244760 0.969584i \(-0.578709\pi\)
−0.858671 + 0.512528i \(0.828709\pi\)
\(798\) 0 0
\(799\) 910616.i 1.42640i
\(800\) 0 0
\(801\) 76049.1 0.118530
\(802\) 0 0
\(803\) 496553. 1.19878e6i 0.770077 1.85913i
\(804\) 0 0
\(805\) −66158.4 159721.i −0.102092 0.246473i
\(806\) 0 0
\(807\) 53085.4 + 53085.4i 0.0815133 + 0.0815133i
\(808\) 0 0
\(809\) 237940. + 237940.i 0.363555 + 0.363555i 0.865120 0.501565i \(-0.167242\pi\)
−0.501565 + 0.865120i \(0.667242\pi\)
\(810\) 0 0
\(811\) 245980. + 593849.i 0.373989 + 0.902889i 0.993066 + 0.117559i \(0.0375069\pi\)
−0.619077 + 0.785330i \(0.712493\pi\)
\(812\) 0 0
\(813\) −243749. + 588462.i −0.368775 + 0.890302i
\(814\) 0 0
\(815\) −420004. −0.632322
\(816\) 0 0
\(817\) 148279.i 0.222145i
\(818\) 0 0
\(819\) 83673.0 + 34658.5i 0.124743 + 0.0516704i
\(820\) 0 0
\(821\) 867491. 359327.i 1.28700 0.533093i 0.368910 0.929465i \(-0.379731\pi\)
0.918090 + 0.396372i \(0.129731\pi\)
\(822\) 0 0
\(823\) −891926. + 891926.i −1.31683 + 1.31683i −0.400557 + 0.916272i \(0.631183\pi\)
−0.916272 + 0.400557i \(0.868817\pi\)
\(824\) 0 0
\(825\) −427327. + 427327.i −0.627845 + 0.627845i
\(826\) 0 0
\(827\) 401413. 166271.i 0.586922 0.243111i −0.0694040 0.997589i \(-0.522110\pi\)
0.656326 + 0.754478i \(0.272110\pi\)
\(828\) 0 0
\(829\) 654720. + 271194.i 0.952678 + 0.394612i 0.804237 0.594309i \(-0.202574\pi\)
0.148441 + 0.988921i \(0.452574\pi\)
\(830\) 0 0
\(831\) 362793.i 0.525359i
\(832\) 0 0
\(833\) −1.43099e6 −2.06227
\(834\) 0 0
\(835\) 47986.6 115850.i 0.0688251 0.166159i
\(836\) 0 0
\(837\) −1045.64 2524.40i −0.00149256 0.00360335i
\(838\) 0 0
\(839\) −231028. 231028.i −0.328202 0.328202i 0.523700 0.851903i \(-0.324551\pi\)
−0.851903 + 0.523700i \(0.824551\pi\)
\(840\) 0 0
\(841\) 3501.86 + 3501.86i 0.00495116 + 0.00495116i
\(842\) 0 0
\(843\) −120057. 289843.i −0.168940 0.407857i
\(844\) 0 0
\(845\) −297521. + 718278.i −0.416681 + 1.00596i
\(846\) 0 0
\(847\) −1.25371e6 −1.74756
\(848\) 0 0
\(849\) 956919.i 1.32758i
\(850\) 0 0
\(851\) 66116.1 + 27386.2i 0.0912952 + 0.0378157i
\(852\) 0 0
\(853\) −293641. + 121630.i −0.403570 + 0.167164i −0.575229 0.817992i \(-0.695087\pi\)
0.171659 + 0.985156i \(0.445087\pi\)
\(854\) 0 0
\(855\) −74850.0 + 74850.0i −0.102390 + 0.102390i
\(856\) 0 0
\(857\) 579802. 579802.i 0.789438 0.789438i −0.191964 0.981402i \(-0.561486\pi\)
0.981402 + 0.191964i \(0.0614857\pi\)
\(858\) 0 0
\(859\) −52706.4 + 21831.7i −0.0714294 + 0.0295870i −0.418112 0.908395i \(-0.637308\pi\)
0.346683 + 0.937982i \(0.387308\pi\)
\(860\) 0 0
\(861\) 857485. + 355182.i 1.15670 + 0.479120i
\(862\) 0 0
\(863\) 1.04826e6i 1.40750i −0.710449 0.703748i \(-0.751508\pi\)
0.710449 0.703748i \(-0.248492\pi\)
\(864\) 0 0
\(865\) −890852. −1.19062
\(866\) 0 0
\(867\) 472804. 1.14145e6i 0.628988 1.51851i
\(868\) 0 0
\(869\) −353626. 853729.i −0.468279 1.13053i
\(870\) 0 0
\(871\) 166080. + 166080.i 0.218917 + 0.218917i
\(872\) 0 0
\(873\) 182496. + 182496.i 0.239455 + 0.239455i
\(874\) 0 0
\(875\) 178651. + 431301.i 0.233340 + 0.563332i
\(876\) 0 0
\(877\) 367736. 887793.i 0.478120 1.15428i −0.482369 0.875968i \(-0.660224\pi\)
0.960490 0.278316i \(-0.0897763\pi\)
\(878\) 0 0
\(879\) 179072. 0.231766
\(880\) 0 0
\(881\) 723917.i 0.932689i −0.884603 0.466344i \(-0.845571\pi\)
0.884603 0.466344i \(-0.154429\pi\)
\(882\) 0 0
\(883\) 412295. + 170778.i 0.528794 + 0.219034i 0.631075 0.775722i \(-0.282614\pi\)
−0.102281 + 0.994756i \(0.532614\pi\)
\(884\) 0 0
\(885\) −671425. + 278113.i −0.857257 + 0.355087i
\(886\) 0 0
\(887\) 252860. 252860.i 0.321390 0.321390i −0.527910 0.849300i \(-0.677024\pi\)
0.849300 + 0.527910i \(0.177024\pi\)
\(888\) 0 0
\(889\) −694685. + 694685.i −0.878991 + 0.878991i
\(890\) 0 0
\(891\) −781843. + 323850.i −0.984836 + 0.407932i
\(892\) 0 0
\(893\) 306448. + 126935.i 0.384286 + 0.159176i
\(894\) 0 0
\(895\) 1.08793e6i 1.35817i
\(896\) 0 0
\(897\) −39215.0 −0.0487380
\(898\) 0 0
\(899\) 1122.31 2709.49i 0.00138865 0.00335249i
\(900\) 0 0
\(901\) 576059. + 1.39073e6i 0.709607 + 1.71314i
\(902\) 0 0
\(903\) −339819. 339819.i −0.416747 0.416747i
\(904\) 0 0
\(905\) 774606. + 774606.i 0.945766 + 0.945766i
\(906\) 0 0
\(907\) 24975.7 + 60296.7i 0.0303601 + 0.0732958i 0.938332 0.345734i \(-0.112370\pi\)
−0.907972 + 0.419030i \(0.862370\pi\)
\(908\) 0 0
\(909\) −77471.4 + 187033.i −0.0937592 + 0.226355i
\(910\) 0 0
\(911\) 620126. 0.747211 0.373606 0.927588i \(-0.378121\pi\)
0.373606 + 0.927588i \(0.378121\pi\)
\(912\) 0 0
\(913\) 386675.i 0.463878i
\(914\) 0 0
\(915\) 388419. + 160889.i 0.463937 + 0.192169i
\(916\) 0 0
\(917\) −429872. + 178059.i −0.511212 + 0.211751i
\(918\) 0 0
\(919\) −456978. + 456978.i −0.541084 + 0.541084i −0.923847 0.382763i \(-0.874973\pi\)
0.382763 + 0.923847i \(0.374973\pi\)
\(920\) 0 0
\(921\) 72708.3 72708.3i 0.0857165 0.0857165i
\(922\) 0 0
\(923\) −501523. + 207738.i −0.588692 + 0.243844i
\(924\) 0 0
\(925\) 387404. + 160468.i 0.452773 + 0.187545i
\(926\) 0 0
\(927\) 3519.32i 0.00409543i
\(928\) 0 0
\(929\) −1.29711e6 −1.50296 −0.751479 0.659757i \(-0.770659\pi\)
−0.751479 + 0.659757i \(0.770659\pi\)
\(930\) 0 0
\(931\) −199472. + 481568.i −0.230135 + 0.555595i
\(932\) 0 0
\(933\) 30097.4 + 72661.5i 0.0345753 + 0.0834721i
\(934\) 0 0
\(935\) 2.00321e6 + 2.00321e6i 2.29141 + 2.29141i
\(936\) 0 0
\(937\) 170416. + 170416.i 0.194102 + 0.194102i 0.797466 0.603364i \(-0.206173\pi\)
−0.603364 + 0.797466i \(0.706173\pi\)
\(938\) 0 0
\(939\) 110656. + 267147.i 0.125500 + 0.302984i
\(940\) 0 0
\(941\) −105038. + 253584.i −0.118622 + 0.286380i −0.972027 0.234869i \(-0.924534\pi\)
0.853405 + 0.521249i \(0.174534\pi\)
\(942\) 0 0
\(943\) 117285. 0.131892
\(944\) 0 0
\(945\) 1.86170e6i 2.08472i
\(946\) 0 0
\(947\) 1.58561e6 + 656779.i 1.76805 + 0.732351i 0.995211 + 0.0977520i \(0.0311652\pi\)
0.772841 + 0.634599i \(0.218835\pi\)
\(948\) 0 0
\(949\) 455800. 188799.i 0.506107 0.209636i
\(950\) 0 0
\(951\) 301310. 301310.i 0.333160 0.333160i
\(952\) 0 0
\(953\) 776606. 776606.i 0.855096 0.855096i −0.135659 0.990756i \(-0.543315\pi\)
0.990756 + 0.135659i \(0.0433153\pi\)
\(954\) 0 0
\(955\) −323894. + 134161.i −0.355137 + 0.147102i
\(956\) 0 0
\(957\) −1.10136e6 456197.i −1.20255 0.498114i
\(958\) 0 0
\(959\) 2.51815e6i 2.73807i
\(960\) 0 0
\(961\) 923509. 0.999987
\(962\) 0 0
\(963\) −81485.6 + 196724.i −0.0878675 + 0.212131i
\(964\) 0 0
\(965\) 237917. + 574382.i 0.255488 + 0.616802i
\(966\) 0 0
\(967\) −1.23673e6 1.23673e6i −1.32258 1.32258i −0.911679 0.410902i \(-0.865214\pi\)
−0.410902 0.911679i \(-0.634786\pi\)
\(968\) 0 0
\(969\) −488567. 488567.i −0.520327 0.520327i
\(970\) 0 0
\(971\) −407205. 983079.i −0.431891 1.04268i −0.978677 0.205405i \(-0.934149\pi\)
0.546786 0.837273i \(-0.315851\pi\)
\(972\) 0 0
\(973\) 246423. 594919.i 0.260289 0.628394i
\(974\) 0 0
\(975\) −229778. −0.241713
\(976\) 0 0
\(977\) 1.17141e6i 1.22721i 0.789614 + 0.613604i \(0.210281\pi\)
−0.789614 + 0.613604i \(0.789719\pi\)
\(978\) 0 0
\(979\) −684936. 283710.i −0.714636 0.296012i
\(980\) 0 0
\(981\) 251315. 104098.i 0.261144 0.108169i
\(982\) 0 0
\(983\) −518176. + 518176.i −0.536254 + 0.536254i −0.922427 0.386173i \(-0.873797\pi\)
0.386173 + 0.922427i \(0.373797\pi\)
\(984\) 0 0
\(985\) 1.34777e6 1.34777e6i 1.38913 1.38913i
\(986\) 0 0
\(987\) −993208. + 411400.i −1.01954 + 0.422309i
\(988\) 0 0
\(989\) −56105.9 23239.8i −0.0573609 0.0237596i
\(990\) 0 0
\(991\) 522347.i 0.531878i −0.963990 0.265939i \(-0.914318\pi\)
0.963990 0.265939i \(-0.0856820\pi\)
\(992\) 0 0
\(993\) −348802. −0.353737
\(994\) 0 0
\(995\) −78925.4 + 190543.i −0.0797206 + 0.192463i
\(996\) 0 0
\(997\) −646593. 1.56101e6i −0.650490 1.57042i −0.812068 0.583563i \(-0.801658\pi\)
0.161578 0.986860i \(-0.448342\pi\)
\(998\) 0 0
\(999\) 544932. + 544932.i 0.546023 + 0.546023i
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 128.5.h.a.47.5 60
4.3 odd 2 32.5.h.a.3.12 60
32.11 odd 8 inner 128.5.h.a.79.5 60
32.21 even 8 32.5.h.a.11.12 yes 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.5.h.a.3.12 60 4.3 odd 2
32.5.h.a.11.12 yes 60 32.21 even 8
128.5.h.a.47.5 60 1.1 even 1 trivial
128.5.h.a.79.5 60 32.11 odd 8 inner