Properties

Label 684.3.t.a.265.9
Level $684$
Weight $3$
Character 684.265
Analytic conductor $18.638$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [684,3,Mod(265,684)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(684, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2, 3]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("684.265");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 684.t (of order \(6\), degree \(2\), 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: \(18.6376500822\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 265.9
Character \(\chi\) \(=\) 684.265
Dual form 684.3.t.a.493.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.57066 + 1.54651i) q^{3} +(1.45280 - 2.51632i) q^{5} +(-0.663126 - 1.14857i) q^{7} +(4.21662 - 7.95111i) q^{9} +O(q^{10})\) \(q+(-2.57066 + 1.54651i) q^{3} +(1.45280 - 2.51632i) q^{5} +(-0.663126 - 1.14857i) q^{7} +(4.21662 - 7.95111i) q^{9} +(-3.65951 - 6.33846i) q^{11} +(16.0381 + 9.25959i) q^{13} +(0.156859 + 8.71539i) q^{15} -16.5415 q^{17} +(-18.9944 + 0.459853i) q^{19} +(3.48095 + 1.92705i) q^{21} +(2.86716 - 4.96607i) q^{23} +(8.27875 + 14.3392i) q^{25} +(1.45697 + 26.9607i) q^{27} +(2.50630 - 1.44701i) q^{29} +(-28.2594 - 16.3156i) q^{31} +(19.2099 + 10.6346i) q^{33} -3.85356 q^{35} +12.1535i q^{37} +(-55.5485 + 0.999762i) q^{39} +(-43.8411 - 25.3117i) q^{41} +(-32.6098 - 56.4819i) q^{43} +(-13.8817 - 22.1617i) q^{45} +(-4.40712 - 7.63335i) q^{47} +(23.6205 - 40.9120i) q^{49} +(42.5226 - 25.5816i) q^{51} +26.3085i q^{53} -21.2662 q^{55} +(48.1171 - 30.5572i) q^{57} +(-44.3022 - 25.5779i) q^{59} +(-4.58788 - 7.94644i) q^{61} +(-11.9285 + 0.429519i) q^{63} +(46.6002 - 26.9047i) q^{65} +(-9.04428 - 5.22172i) q^{67} +(0.309569 + 17.2002i) q^{69} -46.2556i q^{71} -32.7798 q^{73} +(-43.4576 - 24.0581i) q^{75} +(-4.85344 + 8.40640i) q^{77} +(-46.4522 + 26.8192i) q^{79} +(-45.4403 - 67.0535i) q^{81} +(-52.4783 - 90.8950i) q^{83} +(-24.0315 + 41.6238i) q^{85} +(-4.20503 + 7.59580i) q^{87} -58.3307i q^{89} -24.5611i q^{91} +(97.8777 - 1.76160i) q^{93} +(-26.4380 + 48.4642i) q^{95} +(-102.545 + 59.2042i) q^{97} +(-65.8286 + 2.37033i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 80 q + 2 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 80 q + 2 q^{7} + 4 q^{9} + 12 q^{11} - 12 q^{17} - 2 q^{19} - 48 q^{23} - 200 q^{25} - 216 q^{35} + 102 q^{39} + 28 q^{43} + 2 q^{45} - 174 q^{47} - 306 q^{49} + 213 q^{57} + 14 q^{61} + 62 q^{63} + 220 q^{73} - 60 q^{77} + 340 q^{81} + 150 q^{83} - 252 q^{87} - 252 q^{93} + 360 q^{95} + 542 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/684\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(343\) \(533\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) −2.57066 + 1.54651i −0.856888 + 0.515503i
\(4\) 0 0
\(5\) 1.45280 2.51632i 0.290560 0.503265i −0.683382 0.730061i \(-0.739492\pi\)
0.973942 + 0.226796i \(0.0728251\pi\)
\(6\) 0 0
\(7\) −0.663126 1.14857i −0.0947323 0.164081i 0.814765 0.579792i \(-0.196866\pi\)
−0.909497 + 0.415711i \(0.863533\pi\)
\(8\) 0 0
\(9\) 4.21662 7.95111i 0.468513 0.883457i
\(10\) 0 0
\(11\) −3.65951 6.33846i −0.332683 0.576224i 0.650354 0.759631i \(-0.274621\pi\)
−0.983037 + 0.183408i \(0.941287\pi\)
\(12\) 0 0
\(13\) 16.0381 + 9.25959i 1.23370 + 0.712276i 0.967799 0.251725i \(-0.0809977\pi\)
0.265899 + 0.964001i \(0.414331\pi\)
\(14\) 0 0
\(15\) 0.156859 + 8.71539i 0.0104573 + 0.581026i
\(16\) 0 0
\(17\) −16.5415 −0.973030 −0.486515 0.873672i \(-0.661732\pi\)
−0.486515 + 0.873672i \(0.661732\pi\)
\(18\) 0 0
\(19\) −18.9944 + 0.459853i −0.999707 + 0.0242028i
\(20\) 0 0
\(21\) 3.48095 + 1.92705i 0.165759 + 0.0917644i
\(22\) 0 0
\(23\) 2.86716 4.96607i 0.124659 0.215916i −0.796940 0.604058i \(-0.793550\pi\)
0.921600 + 0.388142i \(0.126883\pi\)
\(24\) 0 0
\(25\) 8.27875 + 14.3392i 0.331150 + 0.573568i
\(26\) 0 0
\(27\) 1.45697 + 26.9607i 0.0539619 + 0.998543i
\(28\) 0 0
\(29\) 2.50630 1.44701i 0.0864241 0.0498970i −0.456165 0.889895i \(-0.650777\pi\)
0.542589 + 0.839998i \(0.317444\pi\)
\(30\) 0 0
\(31\) −28.2594 16.3156i −0.911594 0.526309i −0.0306505 0.999530i \(-0.509758\pi\)
−0.880944 + 0.473221i \(0.843091\pi\)
\(32\) 0 0
\(33\) 19.2099 + 10.6346i 0.582117 + 0.322260i
\(34\) 0 0
\(35\) −3.85356 −0.110102
\(36\) 0 0
\(37\) 12.1535i 0.328473i 0.986421 + 0.164237i \(0.0525160\pi\)
−0.986421 + 0.164237i \(0.947484\pi\)
\(38\) 0 0
\(39\) −55.5485 + 0.999762i −1.42432 + 0.0256349i
\(40\) 0 0
\(41\) −43.8411 25.3117i −1.06929 0.617358i −0.141306 0.989966i \(-0.545130\pi\)
−0.927989 + 0.372608i \(0.878464\pi\)
\(42\) 0 0
\(43\) −32.6098 56.4819i −0.758368 1.31353i −0.943682 0.330853i \(-0.892664\pi\)
0.185314 0.982679i \(-0.440670\pi\)
\(44\) 0 0
\(45\) −13.8817 22.1617i −0.308481 0.492483i
\(46\) 0 0
\(47\) −4.40712 7.63335i −0.0937685 0.162412i 0.815325 0.579003i \(-0.196558\pi\)
−0.909094 + 0.416591i \(0.863225\pi\)
\(48\) 0 0
\(49\) 23.6205 40.9120i 0.482052 0.834938i
\(50\) 0 0
\(51\) 42.5226 25.5816i 0.833777 0.501600i
\(52\) 0 0
\(53\) 26.3085i 0.496387i 0.968710 + 0.248194i \(0.0798370\pi\)
−0.968710 + 0.248194i \(0.920163\pi\)
\(54\) 0 0
\(55\) −21.2662 −0.386657
\(56\) 0 0
\(57\) 48.1171 30.5572i 0.844160 0.536091i
\(58\) 0 0
\(59\) −44.3022 25.5779i −0.750885 0.433523i 0.0751288 0.997174i \(-0.476063\pi\)
−0.826013 + 0.563650i \(0.809397\pi\)
\(60\) 0 0
\(61\) −4.58788 7.94644i −0.0752111 0.130269i 0.825967 0.563719i \(-0.190630\pi\)
−0.901178 + 0.433449i \(0.857296\pi\)
\(62\) 0 0
\(63\) −11.9285 + 0.429519i −0.189342 + 0.00681776i
\(64\) 0 0
\(65\) 46.6002 26.9047i 0.716927 0.413918i
\(66\) 0 0
\(67\) −9.04428 5.22172i −0.134989 0.0779361i 0.430985 0.902359i \(-0.358166\pi\)
−0.565974 + 0.824423i \(0.691500\pi\)
\(68\) 0 0
\(69\) 0.309569 + 17.2002i 0.00448651 + 0.249278i
\(70\) 0 0
\(71\) 46.2556i 0.651487i −0.945458 0.325744i \(-0.894385\pi\)
0.945458 0.325744i \(-0.105615\pi\)
\(72\) 0 0
\(73\) −32.7798 −0.449038 −0.224519 0.974470i \(-0.572081\pi\)
−0.224519 + 0.974470i \(0.572081\pi\)
\(74\) 0 0
\(75\) −43.4576 24.0581i −0.579434 0.320775i
\(76\) 0 0
\(77\) −4.85344 + 8.40640i −0.0630317 + 0.109174i
\(78\) 0 0
\(79\) −46.4522 + 26.8192i −0.588002 + 0.339483i −0.764307 0.644852i \(-0.776919\pi\)
0.176305 + 0.984336i \(0.443586\pi\)
\(80\) 0 0
\(81\) −45.4403 67.0535i −0.560991 0.827822i
\(82\) 0 0
\(83\) −52.4783 90.8950i −0.632268 1.09512i −0.987087 0.160185i \(-0.948791\pi\)
0.354819 0.934935i \(-0.384543\pi\)
\(84\) 0 0
\(85\) −24.0315 + 41.6238i −0.282724 + 0.489692i
\(86\) 0 0
\(87\) −4.20503 + 7.59580i −0.0483337 + 0.0873080i
\(88\) 0 0
\(89\) 58.3307i 0.655402i −0.944782 0.327701i \(-0.893726\pi\)
0.944782 0.327701i \(-0.106274\pi\)
\(90\) 0 0
\(91\) 24.5611i 0.269902i
\(92\) 0 0
\(93\) 97.8777 1.76160i 1.05245 0.0189419i
\(94\) 0 0
\(95\) −26.4380 + 48.4642i −0.278294 + 0.510150i
\(96\) 0 0
\(97\) −102.545 + 59.2042i −1.05716 + 0.610353i −0.924646 0.380828i \(-0.875639\pi\)
−0.132516 + 0.991181i \(0.542306\pi\)
\(98\) 0 0
\(99\) −65.8286 + 2.37033i −0.664935 + 0.0239427i
\(100\) 0 0
\(101\) −64.0887 111.005i −0.634542 1.09906i −0.986612 0.163085i \(-0.947855\pi\)
0.352070 0.935974i \(-0.385478\pi\)
\(102\) 0 0
\(103\) −1.07819 0.622493i −0.0104679 0.00604362i 0.494757 0.869031i \(-0.335257\pi\)
−0.505225 + 0.862988i \(0.668590\pi\)
\(104\) 0 0
\(105\) 9.90620 5.95957i 0.0943448 0.0567578i
\(106\) 0 0
\(107\) 132.982i 1.24282i −0.783485 0.621411i \(-0.786560\pi\)
0.783485 0.621411i \(-0.213440\pi\)
\(108\) 0 0
\(109\) 93.1878i 0.854934i 0.904031 + 0.427467i \(0.140594\pi\)
−0.904031 + 0.427467i \(0.859406\pi\)
\(110\) 0 0
\(111\) −18.7955 31.2426i −0.169329 0.281465i
\(112\) 0 0
\(113\) −46.9613 27.1131i −0.415587 0.239939i 0.277600 0.960697i \(-0.410461\pi\)
−0.693187 + 0.720757i \(0.743794\pi\)
\(114\) 0 0
\(115\) −8.33083 14.4294i −0.0724420 0.125473i
\(116\) 0 0
\(117\) 141.250 88.4764i 1.20727 0.756209i
\(118\) 0 0
\(119\) 10.9691 + 18.9991i 0.0921774 + 0.159656i
\(120\) 0 0
\(121\) 33.7159 58.3977i 0.278644 0.482626i
\(122\) 0 0
\(123\) 151.845 2.73291i 1.23452 0.0222188i
\(124\) 0 0
\(125\) 120.749 0.965995
\(126\) 0 0
\(127\) 225.693i 1.77711i 0.458772 + 0.888554i \(0.348289\pi\)
−0.458772 + 0.888554i \(0.651711\pi\)
\(128\) 0 0
\(129\) 171.179 + 94.7645i 1.32697 + 0.734608i
\(130\) 0 0
\(131\) −24.3115 + 42.1088i −0.185584 + 0.321441i −0.943773 0.330594i \(-0.892751\pi\)
0.758189 + 0.652035i \(0.226084\pi\)
\(132\) 0 0
\(133\) 13.1239 + 21.5115i 0.0986758 + 0.161740i
\(134\) 0 0
\(135\) 69.9584 + 35.5022i 0.518211 + 0.262979i
\(136\) 0 0
\(137\) 11.3479 + 19.6552i 0.0828315 + 0.143468i 0.904465 0.426548i \(-0.140270\pi\)
−0.821634 + 0.570016i \(0.806937\pi\)
\(138\) 0 0
\(139\) −82.3364 + 142.611i −0.592348 + 1.02598i 0.401567 + 0.915830i \(0.368466\pi\)
−0.993915 + 0.110148i \(0.964868\pi\)
\(140\) 0 0
\(141\) 23.1343 + 12.8071i 0.164073 + 0.0908307i
\(142\) 0 0
\(143\) 135.542i 0.947848i
\(144\) 0 0
\(145\) 8.40888i 0.0579922i
\(146\) 0 0
\(147\) 2.55032 + 141.700i 0.0173491 + 0.963947i
\(148\) 0 0
\(149\) −17.6141 + 30.5085i −0.118216 + 0.204755i −0.919061 0.394116i \(-0.871051\pi\)
0.800845 + 0.598872i \(0.204384\pi\)
\(150\) 0 0
\(151\) −210.694 + 121.644i −1.39533 + 0.805592i −0.993898 0.110299i \(-0.964819\pi\)
−0.401428 + 0.915891i \(0.631486\pi\)
\(152\) 0 0
\(153\) −69.7492 + 131.523i −0.455877 + 0.859630i
\(154\) 0 0
\(155\) −82.1106 + 47.4066i −0.529746 + 0.305849i
\(156\) 0 0
\(157\) 99.9191 173.065i 0.636427 1.10232i −0.349783 0.936831i \(-0.613745\pi\)
0.986211 0.165494i \(-0.0529219\pi\)
\(158\) 0 0
\(159\) −40.6864 67.6304i −0.255889 0.425348i
\(160\) 0 0
\(161\) −7.60516 −0.0472370
\(162\) 0 0
\(163\) −58.9658 −0.361753 −0.180877 0.983506i \(-0.557893\pi\)
−0.180877 + 0.983506i \(0.557893\pi\)
\(164\) 0 0
\(165\) 54.6681 32.8883i 0.331322 0.199323i
\(166\) 0 0
\(167\) 71.0960 + 41.0473i 0.425725 + 0.245792i 0.697524 0.716562i \(-0.254285\pi\)
−0.271799 + 0.962354i \(0.587619\pi\)
\(168\) 0 0
\(169\) 86.9800 + 150.654i 0.514674 + 0.891442i
\(170\) 0 0
\(171\) −76.4359 + 152.966i −0.446993 + 0.894537i
\(172\) 0 0
\(173\) −34.2663 + 19.7837i −0.198071 + 0.114356i −0.595755 0.803166i \(-0.703147\pi\)
0.397684 + 0.917522i \(0.369814\pi\)
\(174\) 0 0
\(175\) 10.9797 19.0174i 0.0627412 0.108671i
\(176\) 0 0
\(177\) 153.442 2.76166i 0.866906 0.0156026i
\(178\) 0 0
\(179\) 137.422i 0.767720i −0.923391 0.383860i \(-0.874595\pi\)
0.923391 0.383860i \(-0.125405\pi\)
\(180\) 0 0
\(181\) 140.483i 0.776152i 0.921627 + 0.388076i \(0.126860\pi\)
−0.921627 + 0.388076i \(0.873140\pi\)
\(182\) 0 0
\(183\) 24.0831 + 13.3324i 0.131602 + 0.0728547i
\(184\) 0 0
\(185\) 30.5821 + 17.6566i 0.165309 + 0.0954411i
\(186\) 0 0
\(187\) 60.5339 + 104.848i 0.323710 + 0.560683i
\(188\) 0 0
\(189\) 30.0000 19.5518i 0.158730 0.103448i
\(190\) 0 0
\(191\) 158.930 + 275.275i 0.832096 + 1.44123i 0.896373 + 0.443300i \(0.146192\pi\)
−0.0642776 + 0.997932i \(0.520474\pi\)
\(192\) 0 0
\(193\) −132.898 76.7287i −0.688591 0.397558i 0.114493 0.993424i \(-0.463476\pi\)
−0.803084 + 0.595866i \(0.796809\pi\)
\(194\) 0 0
\(195\) −78.1852 + 141.231i −0.400950 + 0.724259i
\(196\) 0 0
\(197\) 248.770 1.26279 0.631396 0.775460i \(-0.282482\pi\)
0.631396 + 0.775460i \(0.282482\pi\)
\(198\) 0 0
\(199\) 122.997 0.618076 0.309038 0.951050i \(-0.399993\pi\)
0.309038 + 0.951050i \(0.399993\pi\)
\(200\) 0 0
\(201\) 31.3252 0.563792i 0.155847 0.00280493i
\(202\) 0 0
\(203\) −3.32399 1.91910i −0.0163743 0.00945371i
\(204\) 0 0
\(205\) −127.385 + 73.5456i −0.621389 + 0.358759i
\(206\) 0 0
\(207\) −27.3961 43.7371i −0.132348 0.211291i
\(208\) 0 0
\(209\) 72.4251 + 118.713i 0.346532 + 0.568003i
\(210\) 0 0
\(211\) 166.537 + 96.1504i 0.789276 + 0.455689i 0.839708 0.543039i \(-0.182726\pi\)
−0.0504313 + 0.998728i \(0.516060\pi\)
\(212\) 0 0
\(213\) 71.5347 + 118.907i 0.335844 + 0.558251i
\(214\) 0 0
\(215\) −189.502 −0.881406
\(216\) 0 0
\(217\) 43.2772i 0.199434i
\(218\) 0 0
\(219\) 84.2658 50.6943i 0.384775 0.231481i
\(220\) 0 0
\(221\) −265.294 153.168i −1.20043 0.693066i
\(222\) 0 0
\(223\) −22.7205 + 13.1177i −0.101886 + 0.0588237i −0.550077 0.835114i \(-0.685402\pi\)
0.448191 + 0.893938i \(0.352068\pi\)
\(224\) 0 0
\(225\) 148.921 5.36229i 0.661871 0.0238324i
\(226\) 0 0
\(227\) 363.943 210.123i 1.60327 0.925651i 0.612449 0.790510i \(-0.290185\pi\)
0.990826 0.135141i \(-0.0431487\pi\)
\(228\) 0 0
\(229\) 84.3954 146.177i 0.368539 0.638328i −0.620798 0.783970i \(-0.713191\pi\)
0.989337 + 0.145642i \(0.0465248\pi\)
\(230\) 0 0
\(231\) −0.524028 29.1159i −0.00226852 0.126043i
\(232\) 0 0
\(233\) 403.689 1.73257 0.866284 0.499551i \(-0.166502\pi\)
0.866284 + 0.499551i \(0.166502\pi\)
\(234\) 0 0
\(235\) −25.6106 −0.108981
\(236\) 0 0
\(237\) 77.9368 140.782i 0.328847 0.594016i
\(238\) 0 0
\(239\) −104.337 + 180.717i −0.436557 + 0.756139i −0.997421 0.0717689i \(-0.977136\pi\)
0.560864 + 0.827908i \(0.310469\pi\)
\(240\) 0 0
\(241\) −79.6486 + 45.9851i −0.330492 + 0.190810i −0.656060 0.754709i \(-0.727778\pi\)
0.325567 + 0.945519i \(0.394445\pi\)
\(242\) 0 0
\(243\) 220.511 + 102.098i 0.907451 + 0.420157i
\(244\) 0 0
\(245\) −68.6318 118.874i −0.280130 0.485199i
\(246\) 0 0
\(247\) −308.892 168.506i −1.25058 0.682209i
\(248\) 0 0
\(249\) 275.474 + 152.502i 1.10632 + 0.612459i
\(250\) 0 0
\(251\) −18.4421 −0.0734744 −0.0367372 0.999325i \(-0.511696\pi\)
−0.0367372 + 0.999325i \(0.511696\pi\)
\(252\) 0 0
\(253\) −41.9697 −0.165888
\(254\) 0 0
\(255\) −2.59469 144.166i −0.0101753 0.565356i
\(256\) 0 0
\(257\) −176.738 102.040i −0.687696 0.397042i 0.115052 0.993359i \(-0.463296\pi\)
−0.802748 + 0.596318i \(0.796630\pi\)
\(258\) 0 0
\(259\) 13.9591 8.05931i 0.0538963 0.0311170i
\(260\) 0 0
\(261\) −0.937256 26.0293i −0.00359102 0.0997293i
\(262\) 0 0
\(263\) −243.146 421.141i −0.924508 1.60130i −0.792350 0.610067i \(-0.791143\pi\)
−0.132158 0.991229i \(-0.542191\pi\)
\(264\) 0 0
\(265\) 66.2008 + 38.2210i 0.249814 + 0.144230i
\(266\) 0 0
\(267\) 90.2091 + 149.949i 0.337862 + 0.561606i
\(268\) 0 0
\(269\) 463.171i 1.72183i −0.508752 0.860913i \(-0.669893\pi\)
0.508752 0.860913i \(-0.330107\pi\)
\(270\) 0 0
\(271\) 172.822 0.637719 0.318859 0.947802i \(-0.396700\pi\)
0.318859 + 0.947802i \(0.396700\pi\)
\(272\) 0 0
\(273\) 37.9840 + 63.1383i 0.139136 + 0.231276i
\(274\) 0 0
\(275\) 60.5923 104.949i 0.220336 0.381633i
\(276\) 0 0
\(277\) 144.894 + 250.964i 0.523083 + 0.906007i 0.999639 + 0.0268627i \(0.00855168\pi\)
−0.476556 + 0.879144i \(0.658115\pi\)
\(278\) 0 0
\(279\) −248.886 + 155.897i −0.892065 + 0.558771i
\(280\) 0 0
\(281\) −454.513 + 262.413i −1.61748 + 0.933855i −0.629916 + 0.776663i \(0.716911\pi\)
−0.987568 + 0.157191i \(0.949756\pi\)
\(282\) 0 0
\(283\) −180.446 + 312.542i −0.637620 + 1.10439i 0.348334 + 0.937371i \(0.386748\pi\)
−0.985954 + 0.167019i \(0.946586\pi\)
\(284\) 0 0
\(285\) −6.98725 165.472i −0.0245167 0.580603i
\(286\) 0 0
\(287\) 67.1393i 0.233935i
\(288\) 0 0
\(289\) −15.3785 −0.0532126
\(290\) 0 0
\(291\) 172.048 310.780i 0.591230 1.06797i
\(292\) 0 0
\(293\) −496.429 286.613i −1.69430 0.978203i −0.950974 0.309271i \(-0.899915\pi\)
−0.743323 0.668932i \(-0.766752\pi\)
\(294\) 0 0
\(295\) −128.724 + 74.3191i −0.436354 + 0.251929i
\(296\) 0 0
\(297\) 165.557 107.898i 0.557432 0.363292i
\(298\) 0 0
\(299\) 91.9676 53.0975i 0.307584 0.177584i
\(300\) 0 0
\(301\) −43.2489 + 74.9093i −0.143684 + 0.248868i
\(302\) 0 0
\(303\) 336.421 + 186.242i 1.11030 + 0.614662i
\(304\) 0 0
\(305\) −26.6611 −0.0874134
\(306\) 0 0
\(307\) 354.493i 1.15470i −0.816496 0.577351i \(-0.804087\pi\)
0.816496 0.577351i \(-0.195913\pi\)
\(308\) 0 0
\(309\) 3.73436 0.0672109i 0.0120853 0.000217511i
\(310\) 0 0
\(311\) −222.647 + 385.635i −0.715906 + 1.23998i 0.246704 + 0.969091i \(0.420653\pi\)
−0.962609 + 0.270894i \(0.912681\pi\)
\(312\) 0 0
\(313\) 82.2430 + 142.449i 0.262757 + 0.455109i 0.966974 0.254877i \(-0.0820349\pi\)
−0.704217 + 0.709985i \(0.748702\pi\)
\(314\) 0 0
\(315\) −16.2490 + 30.6401i −0.0515841 + 0.0972701i
\(316\) 0 0
\(317\) 257.031 148.397i 0.810822 0.468128i −0.0364192 0.999337i \(-0.511595\pi\)
0.847241 + 0.531208i \(0.178262\pi\)
\(318\) 0 0
\(319\) −18.3437 10.5907i −0.0575036 0.0331997i
\(320\) 0 0
\(321\) 205.658 + 341.852i 0.640679 + 1.06496i
\(322\) 0 0
\(323\) 314.197 7.60666i 0.972745 0.0235500i
\(324\) 0 0
\(325\) 306.631i 0.943480i
\(326\) 0 0
\(327\) −144.116 239.554i −0.440721 0.732582i
\(328\) 0 0
\(329\) −5.84495 + 10.1238i −0.0177658 + 0.0307713i
\(330\) 0 0
\(331\) 293.764 169.605i 0.887504 0.512401i 0.0143790 0.999897i \(-0.495423\pi\)
0.873125 + 0.487496i \(0.162090\pi\)
\(332\) 0 0
\(333\) 96.6338 + 51.2467i 0.290192 + 0.153894i
\(334\) 0 0
\(335\) −26.2791 + 15.1722i −0.0784450 + 0.0452902i
\(336\) 0 0
\(337\) 207.414 + 119.750i 0.615472 + 0.355343i 0.775104 0.631834i \(-0.217698\pi\)
−0.159632 + 0.987177i \(0.551031\pi\)
\(338\) 0 0
\(339\) 162.652 2.92742i 0.479801 0.00863545i
\(340\) 0 0
\(341\) 238.828i 0.700376i
\(342\) 0 0
\(343\) −127.640 −0.372128
\(344\) 0 0
\(345\) 43.7310 + 24.2095i 0.126756 + 0.0701723i
\(346\) 0 0
\(347\) −124.105 + 214.957i −0.357652 + 0.619472i −0.987568 0.157191i \(-0.949756\pi\)
0.629916 + 0.776664i \(0.283089\pi\)
\(348\) 0 0
\(349\) −300.432 520.364i −0.860838 1.49101i −0.871122 0.491068i \(-0.836607\pi\)
0.0102838 0.999947i \(-0.496727\pi\)
\(350\) 0 0
\(351\) −226.278 + 445.888i −0.644666 + 1.27034i
\(352\) 0 0
\(353\) −103.500 179.267i −0.293200 0.507837i 0.681365 0.731944i \(-0.261387\pi\)
−0.974565 + 0.224107i \(0.928054\pi\)
\(354\) 0 0
\(355\) −116.394 67.2001i −0.327870 0.189296i
\(356\) 0 0
\(357\) −57.5801 31.8763i −0.161289 0.0892895i
\(358\) 0 0
\(359\) 218.156 0.607676 0.303838 0.952724i \(-0.401732\pi\)
0.303838 + 0.952724i \(0.401732\pi\)
\(360\) 0 0
\(361\) 360.577 17.4693i 0.998828 0.0483914i
\(362\) 0 0
\(363\) 3.64033 + 202.263i 0.0100284 + 0.557198i
\(364\) 0 0
\(365\) −47.6225 + 82.4845i −0.130473 + 0.225985i
\(366\) 0 0
\(367\) −100.949 174.848i −0.275064 0.476425i 0.695087 0.718925i \(-0.255366\pi\)
−0.970151 + 0.242500i \(0.922032\pi\)
\(368\) 0 0
\(369\) −386.117 + 241.856i −1.04639 + 0.655436i
\(370\) 0 0
\(371\) 30.2172 17.4459i 0.0814479 0.0470239i
\(372\) 0 0
\(373\) 210.083 + 121.292i 0.563226 + 0.325179i 0.754439 0.656370i \(-0.227909\pi\)
−0.191213 + 0.981549i \(0.561242\pi\)
\(374\) 0 0
\(375\) −310.406 + 186.740i −0.827750 + 0.497974i
\(376\) 0 0
\(377\) 53.5949 0.142162
\(378\) 0 0
\(379\) 89.6768i 0.236614i 0.992977 + 0.118307i \(0.0377467\pi\)
−0.992977 + 0.118307i \(0.962253\pi\)
\(380\) 0 0
\(381\) −349.036 580.180i −0.916105 1.52278i
\(382\) 0 0
\(383\) 424.451 + 245.057i 1.10823 + 0.639835i 0.938369 0.345634i \(-0.112336\pi\)
0.169857 + 0.985469i \(0.445669\pi\)
\(384\) 0 0
\(385\) 14.1021 + 24.4256i 0.0366290 + 0.0634432i
\(386\) 0 0
\(387\) −586.597 + 21.1220i −1.51575 + 0.0545788i
\(388\) 0 0
\(389\) −36.5170 63.2493i −0.0938740 0.162594i 0.815264 0.579089i \(-0.196592\pi\)
−0.909138 + 0.416495i \(0.863258\pi\)
\(390\) 0 0
\(391\) −47.4272 + 82.1463i −0.121297 + 0.210093i
\(392\) 0 0
\(393\) −2.62493 145.846i −0.00667920 0.371108i
\(394\) 0 0
\(395\) 155.852i 0.394561i
\(396\) 0 0
\(397\) 310.040 0.780958 0.390479 0.920612i \(-0.372309\pi\)
0.390479 + 0.920612i \(0.372309\pi\)
\(398\) 0 0
\(399\) −67.0048 35.0025i −0.167932 0.0877256i
\(400\) 0 0
\(401\) 162.090 + 93.5826i 0.404214 + 0.233373i 0.688301 0.725425i \(-0.258357\pi\)
−0.284087 + 0.958799i \(0.591690\pi\)
\(402\) 0 0
\(403\) −302.151 523.341i −0.749755 1.29861i
\(404\) 0 0
\(405\) −234.744 + 16.9271i −0.579615 + 0.0417953i
\(406\) 0 0
\(407\) 77.0345 44.4759i 0.189274 0.109277i
\(408\) 0 0
\(409\) 60.9834 + 35.2088i 0.149104 + 0.0860850i 0.572696 0.819768i \(-0.305898\pi\)
−0.423592 + 0.905853i \(0.639231\pi\)
\(410\) 0 0
\(411\) −59.5686 32.9772i −0.144936 0.0802364i
\(412\) 0 0
\(413\) 67.8455i 0.164275i
\(414\) 0 0
\(415\) −304.962 −0.734847
\(416\) 0 0
\(417\) −8.88990 493.939i −0.0213187 1.18450i
\(418\) 0 0
\(419\) 115.222 199.571i 0.274994 0.476303i −0.695140 0.718875i \(-0.744657\pi\)
0.970134 + 0.242571i \(0.0779908\pi\)
\(420\) 0 0
\(421\) −107.934 + 62.3157i −0.256375 + 0.148018i −0.622680 0.782477i \(-0.713956\pi\)
0.366305 + 0.930495i \(0.380623\pi\)
\(422\) 0 0
\(423\) −79.2767 + 2.85457i −0.187415 + 0.00674839i
\(424\) 0 0
\(425\) −136.943 237.192i −0.322219 0.558099i
\(426\) 0 0
\(427\) −6.08469 + 10.5390i −0.0142499 + 0.0246815i
\(428\) 0 0
\(429\) 209.618 + 348.434i 0.488619 + 0.812200i
\(430\) 0 0
\(431\) 684.497i 1.58816i 0.607813 + 0.794080i \(0.292047\pi\)
−0.607813 + 0.794080i \(0.707953\pi\)
\(432\) 0 0
\(433\) 250.418i 0.578333i 0.957279 + 0.289166i \(0.0933781\pi\)
−0.957279 + 0.289166i \(0.906622\pi\)
\(434\) 0 0
\(435\) 13.0044 + 21.6164i 0.0298952 + 0.0496928i
\(436\) 0 0
\(437\) −52.1765 + 95.6462i −0.119397 + 0.218870i
\(438\) 0 0
\(439\) 45.8879 26.4934i 0.104528 0.0603495i −0.446824 0.894622i \(-0.647445\pi\)
0.551353 + 0.834272i \(0.314112\pi\)
\(440\) 0 0
\(441\) −225.697 360.319i −0.511784 0.817051i
\(442\) 0 0
\(443\) −261.160 452.343i −0.589527 1.02109i −0.994294 0.106671i \(-0.965981\pi\)
0.404768 0.914419i \(-0.367352\pi\)
\(444\) 0 0
\(445\) −146.779 84.7429i −0.329840 0.190433i
\(446\) 0 0
\(447\) −1.90180 105.668i −0.00425460 0.236393i
\(448\) 0 0
\(449\) 552.299i 1.23006i −0.788502 0.615032i \(-0.789143\pi\)
0.788502 0.615032i \(-0.210857\pi\)
\(450\) 0 0
\(451\) 370.513i 0.821537i
\(452\) 0 0
\(453\) 353.500 638.547i 0.780352 1.40960i
\(454\) 0 0
\(455\) −61.8037 35.6824i −0.135832 0.0784228i
\(456\) 0 0
\(457\) −176.969 306.519i −0.387240 0.670719i 0.604837 0.796349i \(-0.293238\pi\)
−0.992077 + 0.125630i \(0.959905\pi\)
\(458\) 0 0
\(459\) −24.1005 445.970i −0.0525066 0.971612i
\(460\) 0 0
\(461\) 49.3548 + 85.4849i 0.107060 + 0.185434i 0.914578 0.404409i \(-0.132523\pi\)
−0.807518 + 0.589843i \(0.799190\pi\)
\(462\) 0 0
\(463\) 61.1161 105.856i 0.132000 0.228631i −0.792447 0.609940i \(-0.791193\pi\)
0.924448 + 0.381309i \(0.124527\pi\)
\(464\) 0 0
\(465\) 137.764 248.851i 0.296266 0.535164i
\(466\) 0 0
\(467\) −180.368 −0.386228 −0.193114 0.981176i \(-0.561859\pi\)
−0.193114 + 0.981176i \(0.561859\pi\)
\(468\) 0 0
\(469\) 13.8506i 0.0295323i
\(470\) 0 0
\(471\) 10.7883 + 599.418i 0.0229051 + 1.27265i
\(472\) 0 0
\(473\) −238.672 + 413.392i −0.504592 + 0.873980i
\(474\) 0 0
\(475\) −163.844 268.558i −0.344935 0.565386i
\(476\) 0 0
\(477\) 209.182 + 110.933i 0.438537 + 0.232564i
\(478\) 0 0
\(479\) −335.479 581.067i −0.700374 1.21308i −0.968335 0.249654i \(-0.919683\pi\)
0.267961 0.963430i \(-0.413650\pi\)
\(480\) 0 0
\(481\) −112.536 + 194.919i −0.233964 + 0.405237i
\(482\) 0 0
\(483\) 19.5503 11.7615i 0.0404768 0.0243509i
\(484\) 0 0
\(485\) 344.047i 0.709376i
\(486\) 0 0
\(487\) 415.292i 0.852755i −0.904545 0.426378i \(-0.859790\pi\)
0.904545 0.426378i \(-0.140210\pi\)
\(488\) 0 0
\(489\) 151.581 91.1912i 0.309982 0.186485i
\(490\) 0 0
\(491\) −120.766 + 209.174i −0.245960 + 0.426015i −0.962401 0.271632i \(-0.912437\pi\)
0.716441 + 0.697648i \(0.245770\pi\)
\(492\) 0 0
\(493\) −41.4580 + 23.9358i −0.0840932 + 0.0485512i
\(494\) 0 0
\(495\) −89.6712 + 169.090i −0.181154 + 0.341595i
\(496\) 0 0
\(497\) −53.1277 + 30.6733i −0.106897 + 0.0617169i
\(498\) 0 0
\(499\) −40.5201 + 70.1828i −0.0812025 + 0.140647i −0.903767 0.428025i \(-0.859209\pi\)
0.822564 + 0.568672i \(0.192543\pi\)
\(500\) 0 0
\(501\) −246.244 + 4.43190i −0.491505 + 0.00884610i
\(502\) 0 0
\(503\) 611.143 1.21500 0.607498 0.794321i \(-0.292173\pi\)
0.607498 + 0.794321i \(0.292173\pi\)
\(504\) 0 0
\(505\) −372.432 −0.737490
\(506\) 0 0
\(507\) −456.584 252.765i −0.900560 0.498550i
\(508\) 0 0
\(509\) 647.042 + 373.570i 1.27120 + 0.733929i 0.975214 0.221262i \(-0.0710177\pi\)
0.295988 + 0.955192i \(0.404351\pi\)
\(510\) 0 0
\(511\) 21.7371 + 37.6498i 0.0425384 + 0.0736787i
\(512\) 0 0
\(513\) −40.0723 511.433i −0.0781136 0.996944i
\(514\) 0 0
\(515\) −3.13279 + 1.80872i −0.00608309 + 0.00351207i
\(516\) 0 0
\(517\) −32.2558 + 55.8687i −0.0623903 + 0.108063i
\(518\) 0 0
\(519\) 57.4915 103.850i 0.110774 0.200097i
\(520\) 0 0
\(521\) 412.618i 0.791974i 0.918256 + 0.395987i \(0.129597\pi\)
−0.918256 + 0.395987i \(0.870403\pi\)
\(522\) 0 0
\(523\) 732.398i 1.40038i −0.713957 0.700190i \(-0.753099\pi\)
0.713957 0.700190i \(-0.246901\pi\)
\(524\) 0 0
\(525\) 1.18548 + 65.8676i 0.00225807 + 0.125462i
\(526\) 0 0
\(527\) 467.453 + 269.884i 0.887008 + 0.512115i
\(528\) 0 0
\(529\) 248.059 + 429.650i 0.468920 + 0.812194i
\(530\) 0 0
\(531\) −390.178 + 244.400i −0.734798 + 0.460263i
\(532\) 0 0
\(533\) −468.751 811.901i −0.879458 1.52327i
\(534\) 0 0
\(535\) −334.626 193.196i −0.625468 0.361114i
\(536\) 0 0
\(537\) 212.524 + 353.265i 0.395762 + 0.657850i
\(538\) 0 0
\(539\) −345.758 −0.641481
\(540\) 0 0
\(541\) 256.682 0.474458 0.237229 0.971454i \(-0.423761\pi\)
0.237229 + 0.971454i \(0.423761\pi\)
\(542\) 0 0
\(543\) −217.259 361.136i −0.400109 0.665075i
\(544\) 0 0
\(545\) 234.491 + 135.383i 0.430258 + 0.248409i
\(546\) 0 0
\(547\) 779.393 449.983i 1.42485 0.822638i 0.428143 0.903711i \(-0.359168\pi\)
0.996708 + 0.0810730i \(0.0258347\pi\)
\(548\) 0 0
\(549\) −82.5283 + 2.97165i −0.150325 + 0.00541284i
\(550\) 0 0
\(551\) −46.9403 + 28.6377i −0.0851911 + 0.0519741i
\(552\) 0 0
\(553\) 61.6073 + 35.5690i 0.111406 + 0.0643201i
\(554\) 0 0
\(555\) −105.922 + 1.90639i −0.190851 + 0.00343494i
\(556\) 0 0
\(557\) 18.5151 0.0332407 0.0166204 0.999862i \(-0.494709\pi\)
0.0166204 + 0.999862i \(0.494709\pi\)
\(558\) 0 0
\(559\) 1207.81i 2.16067i
\(560\) 0 0
\(561\) −317.760 175.912i −0.566417 0.313569i
\(562\) 0 0
\(563\) −217.520 125.585i −0.386359 0.223065i 0.294222 0.955737i \(-0.404939\pi\)
−0.680581 + 0.732672i \(0.738273\pi\)
\(564\) 0 0
\(565\) −136.451 + 78.7799i −0.241506 + 0.139433i
\(566\) 0 0
\(567\) −46.8829 + 96.6563i −0.0826860 + 0.170470i
\(568\) 0 0
\(569\) −376.237 + 217.220i −0.661225 + 0.381758i −0.792744 0.609555i \(-0.791348\pi\)
0.131519 + 0.991314i \(0.458015\pi\)
\(570\) 0 0
\(571\) 46.6699 80.8346i 0.0817335 0.141567i −0.822261 0.569110i \(-0.807288\pi\)
0.903995 + 0.427544i \(0.140621\pi\)
\(572\) 0 0
\(573\) −834.272 461.853i −1.45597 0.806026i
\(574\) 0 0
\(575\) 94.9460 0.165124
\(576\) 0 0
\(577\) −553.366 −0.959040 −0.479520 0.877531i \(-0.659189\pi\)
−0.479520 + 0.877531i \(0.659189\pi\)
\(578\) 0 0
\(579\) 460.298 8.28443i 0.794987 0.0143082i
\(580\) 0 0
\(581\) −69.5994 + 120.550i −0.119792 + 0.207487i
\(582\) 0 0
\(583\) 166.756 96.2764i 0.286030 0.165140i
\(584\) 0 0
\(585\) −17.4266 483.970i −0.0297891 0.827300i
\(586\) 0 0
\(587\) 87.0791 + 150.825i 0.148346 + 0.256943i 0.930616 0.365996i \(-0.119272\pi\)
−0.782270 + 0.622939i \(0.785938\pi\)
\(588\) 0 0
\(589\) 544.274 + 296.910i 0.924065 + 0.504092i
\(590\) 0 0
\(591\) −639.504 + 384.725i −1.08207 + 0.650973i
\(592\) 0 0
\(593\) 206.615 0.348423 0.174212 0.984708i \(-0.444262\pi\)
0.174212 + 0.984708i \(0.444262\pi\)
\(594\) 0 0
\(595\) 63.7437 0.107132
\(596\) 0 0
\(597\) −316.184 + 190.216i −0.529622 + 0.318620i
\(598\) 0 0
\(599\) −30.4371 17.5729i −0.0508132 0.0293370i 0.474378 0.880321i \(-0.342673\pi\)
−0.525191 + 0.850984i \(0.676006\pi\)
\(600\) 0 0
\(601\) 53.9451 31.1452i 0.0897590 0.0518224i −0.454449 0.890773i \(-0.650164\pi\)
0.544208 + 0.838951i \(0.316830\pi\)
\(602\) 0 0
\(603\) −79.6547 + 49.8941i −0.132097 + 0.0827431i
\(604\) 0 0
\(605\) −97.9650 169.680i −0.161926 0.280463i
\(606\) 0 0
\(607\) −408.713 235.970i −0.673333 0.388749i 0.124006 0.992282i \(-0.460426\pi\)
−0.797338 + 0.603533i \(0.793759\pi\)
\(608\) 0 0
\(609\) 11.5128 0.207207i 0.0189044 0.000340241i
\(610\) 0 0
\(611\) 163.232i 0.267156i
\(612\) 0 0
\(613\) −259.476 −0.423289 −0.211645 0.977347i \(-0.567882\pi\)
−0.211645 + 0.977347i \(0.567882\pi\)
\(614\) 0 0
\(615\) 213.724 386.062i 0.347519 0.627744i
\(616\) 0 0
\(617\) −447.580 + 775.232i −0.725414 + 1.25645i 0.233390 + 0.972383i \(0.425018\pi\)
−0.958803 + 0.284070i \(0.908315\pi\)
\(618\) 0 0
\(619\) 249.739 + 432.560i 0.403455 + 0.698804i 0.994140 0.108098i \(-0.0344759\pi\)
−0.590686 + 0.806902i \(0.701143\pi\)
\(620\) 0 0
\(621\) 138.066 + 70.0652i 0.222328 + 0.112826i
\(622\) 0 0
\(623\) −66.9969 + 38.6807i −0.107539 + 0.0620877i
\(624\) 0 0
\(625\) −31.5439 + 54.6356i −0.0504702 + 0.0874170i
\(626\) 0 0
\(627\) −369.771 193.164i −0.589746 0.308077i
\(628\) 0 0
\(629\) 201.037i 0.319614i
\(630\) 0 0
\(631\) −46.8556 −0.0742561 −0.0371280 0.999311i \(-0.511821\pi\)
−0.0371280 + 0.999311i \(0.511821\pi\)
\(632\) 0 0
\(633\) −576.809 + 10.3814i −0.911230 + 0.0164003i
\(634\) 0 0
\(635\) 567.916 + 327.886i 0.894355 + 0.516356i
\(636\) 0 0
\(637\) 757.656 437.433i 1.18941 0.686708i
\(638\) 0 0
\(639\) −367.783 195.042i −0.575561 0.305230i
\(640\) 0 0
\(641\) −235.482 + 135.955i −0.367366 + 0.212099i −0.672307 0.740272i \(-0.734697\pi\)
0.304941 + 0.952371i \(0.401363\pi\)
\(642\) 0 0
\(643\) 404.629 700.837i 0.629283 1.08995i −0.358413 0.933563i \(-0.616682\pi\)
0.987696 0.156386i \(-0.0499845\pi\)
\(644\) 0 0
\(645\) 487.146 293.067i 0.755266 0.454368i
\(646\) 0 0
\(647\) 46.2587 0.0714972 0.0357486 0.999361i \(-0.488618\pi\)
0.0357486 + 0.999361i \(0.488618\pi\)
\(648\) 0 0
\(649\) 374.410i 0.576903i
\(650\) 0 0
\(651\) −66.9286 111.251i −0.102809 0.170893i
\(652\) 0 0
\(653\) 349.601 605.526i 0.535376 0.927298i −0.463769 0.885956i \(-0.653503\pi\)
0.999145 0.0413423i \(-0.0131634\pi\)
\(654\) 0 0
\(655\) 70.6396 + 122.351i 0.107847 + 0.186796i
\(656\) 0 0
\(657\) −138.220 + 260.636i −0.210380 + 0.396706i
\(658\) 0 0
\(659\) 226.326 130.670i 0.343439 0.198285i −0.318353 0.947972i \(-0.603130\pi\)
0.661792 + 0.749688i \(0.269796\pi\)
\(660\) 0 0
\(661\) 321.288 + 185.496i 0.486063 + 0.280629i 0.722940 0.690911i \(-0.242790\pi\)
−0.236877 + 0.971540i \(0.576124\pi\)
\(662\) 0 0
\(663\) 918.857 16.5376i 1.38591 0.0249436i
\(664\) 0 0
\(665\) 73.1962 1.77207i 0.110069 0.00266477i
\(666\) 0 0
\(667\) 16.5953i 0.0248805i
\(668\) 0 0
\(669\) 38.1201 68.8586i 0.0569807 0.102928i
\(670\) 0 0
\(671\) −33.5788 + 58.1602i −0.0500429 + 0.0866769i
\(672\) 0 0
\(673\) 951.502 549.350i 1.41382 0.816270i 0.418076 0.908412i \(-0.362705\pi\)
0.995746 + 0.0921420i \(0.0293714\pi\)
\(674\) 0 0
\(675\) −374.533 + 244.092i −0.554863 + 0.361618i
\(676\) 0 0
\(677\) −182.351 + 105.281i −0.269352 + 0.155510i −0.628593 0.777734i \(-0.716369\pi\)
0.359241 + 0.933245i \(0.383036\pi\)
\(678\) 0 0
\(679\) 136.000 + 78.5197i 0.200295 + 0.115640i
\(680\) 0 0
\(681\) −610.619 + 1103.00i −0.896650 + 1.61967i
\(682\) 0 0
\(683\) 739.534i 1.08277i 0.840774 + 0.541386i \(0.182100\pi\)
−0.840774 + 0.541386i \(0.817900\pi\)
\(684\) 0 0
\(685\) 65.9450 0.0962701
\(686\) 0 0
\(687\) 9.11221 + 506.291i 0.0132638 + 0.736959i
\(688\) 0 0
\(689\) −243.606 + 421.938i −0.353565 + 0.612392i
\(690\) 0 0
\(691\) 294.555 + 510.184i 0.426273 + 0.738326i 0.996538 0.0831341i \(-0.0264930\pi\)
−0.570265 + 0.821460i \(0.693160\pi\)
\(692\) 0 0
\(693\) 46.3751 + 74.0368i 0.0669194 + 0.106835i
\(694\) 0 0
\(695\) 239.237 + 414.370i 0.344225 + 0.596216i
\(696\) 0 0
\(697\) 725.198 + 418.693i 1.04046 + 0.600708i
\(698\) 0 0
\(699\) −1037.75 + 624.308i −1.48462 + 0.893145i
\(700\) 0 0
\(701\) 1159.94 1.65469 0.827345 0.561694i \(-0.189850\pi\)
0.827345 + 0.561694i \(0.189850\pi\)
\(702\) 0 0
\(703\) −5.58882 230.849i −0.00794996 0.328377i
\(704\) 0 0
\(705\) 65.8363 39.6071i 0.0933849 0.0561803i
\(706\) 0 0
\(707\) −84.9979 + 147.221i −0.120223 + 0.208233i
\(708\) 0 0
\(709\) −544.899 943.793i −0.768546 1.33116i −0.938351 0.345683i \(-0.887647\pi\)
0.169806 0.985478i \(-0.445686\pi\)
\(710\) 0 0
\(711\) 17.3713 + 482.433i 0.0244322 + 0.678527i
\(712\) 0 0
\(713\) −162.049 + 93.5589i −0.227277 + 0.131219i
\(714\) 0 0
\(715\) −341.068 196.916i −0.477019 0.275407i
\(716\) 0 0
\(717\) −11.2653 625.921i −0.0157118 0.872973i
\(718\) 0 0
\(719\) 788.886 1.09720 0.548599 0.836085i \(-0.315161\pi\)
0.548599 + 0.836085i \(0.315161\pi\)
\(720\) 0 0
\(721\) 1.65117i 0.00229011i
\(722\) 0 0
\(723\) 133.633 241.390i 0.184832 0.333872i
\(724\) 0 0
\(725\) 41.4980 + 23.9589i 0.0572386 + 0.0330467i
\(726\) 0 0
\(727\) 538.278 + 932.324i 0.740409 + 1.28243i 0.952309 + 0.305135i \(0.0987016\pi\)
−0.211900 + 0.977291i \(0.567965\pi\)
\(728\) 0 0
\(729\) −724.754 + 78.5619i −0.994176 + 0.107767i
\(730\) 0 0
\(731\) 539.416 + 934.296i 0.737915 + 1.27811i
\(732\) 0 0
\(733\) 538.967 933.518i 0.735289 1.27356i −0.219307 0.975656i \(-0.570380\pi\)
0.954596 0.297903i \(-0.0962871\pi\)
\(734\) 0 0
\(735\) 360.269 + 199.445i 0.490161 + 0.271353i
\(736\) 0 0
\(737\) 76.4358i 0.103712i
\(738\) 0 0
\(739\) 825.434 1.11696 0.558480 0.829518i \(-0.311385\pi\)
0.558480 + 0.829518i \(0.311385\pi\)
\(740\) 0 0
\(741\) 1054.65 44.5341i 1.42328 0.0601000i
\(742\) 0 0
\(743\) −64.5430 37.2639i −0.0868681 0.0501533i 0.455937 0.890012i \(-0.349304\pi\)
−0.542805 + 0.839859i \(0.682638\pi\)
\(744\) 0 0
\(745\) 51.1796 + 88.6456i 0.0686974 + 0.118987i
\(746\) 0 0
\(747\) −943.997 + 33.9911i −1.26372 + 0.0455035i
\(748\) 0 0
\(749\) −152.739 + 88.1839i −0.203924 + 0.117735i
\(750\) 0 0
\(751\) −462.816 267.207i −0.616266 0.355801i 0.159148 0.987255i \(-0.449125\pi\)
−0.775414 + 0.631453i \(0.782459\pi\)
\(752\) 0 0
\(753\) 47.4084 28.5208i 0.0629593 0.0378763i
\(754\) 0 0
\(755\) 706.900i 0.936291i
\(756\) 0 0
\(757\) −372.882 −0.492579 −0.246289 0.969196i \(-0.579211\pi\)
−0.246289 + 0.969196i \(0.579211\pi\)
\(758\) 0 0
\(759\) 107.890 64.9065i 0.142147 0.0855158i
\(760\) 0 0
\(761\) −437.900 + 758.465i −0.575427 + 0.996669i 0.420568 + 0.907261i \(0.361831\pi\)
−0.995995 + 0.0894081i \(0.971502\pi\)
\(762\) 0 0
\(763\) 107.033 61.7953i 0.140279 0.0809899i
\(764\) 0 0
\(765\) 229.624 + 366.589i 0.300162 + 0.479201i
\(766\) 0 0
\(767\) −473.681 820.440i −0.617577 1.06967i
\(768\) 0 0
\(769\) −343.462 + 594.894i −0.446635 + 0.773594i −0.998164 0.0605610i \(-0.980711\pi\)
0.551530 + 0.834155i \(0.314044\pi\)
\(770\) 0 0
\(771\) 612.139 11.0173i 0.793954 0.0142896i
\(772\) 0 0
\(773\) 1020.13i 1.31970i 0.751398 + 0.659850i \(0.229380\pi\)
−0.751398 + 0.659850i \(0.770620\pi\)
\(774\) 0 0
\(775\) 540.290i 0.697149i
\(776\) 0 0
\(777\) −23.4204 + 42.3057i −0.0301421 + 0.0544475i
\(778\) 0 0
\(779\) 844.376 + 460.620i 1.08392 + 0.591297i
\(780\) 0 0
\(781\) −293.189 + 169.273i −0.375402 + 0.216739i
\(782\) 0 0
\(783\) 42.6640 + 65.4632i 0.0544879 + 0.0836056i
\(784\) 0 0
\(785\) −290.325 502.858i −0.369841 0.640583i
\(786\) 0 0
\(787\) 428.852 + 247.598i 0.544920 + 0.314609i 0.747070 0.664745i \(-0.231460\pi\)
−0.202151 + 0.979354i \(0.564793\pi\)
\(788\) 0 0
\(789\) 1276.34 + 706.584i 1.61767 + 0.895543i
\(790\) 0 0
\(791\) 71.9178i 0.0909200i
\(792\) 0 0
\(793\) 169.927i 0.214284i
\(794\) 0 0
\(795\) −229.289 + 4.12674i −0.288414 + 0.00519087i
\(796\) 0 0
\(797\) −1145.16 661.160i −1.43684 0.829561i −0.439212 0.898383i \(-0.644742\pi\)
−0.997629 + 0.0688225i \(0.978076\pi\)
\(798\) 0 0
\(799\) 72.9004 + 126.267i 0.0912395 + 0.158031i
\(800\) 0 0
\(801\) −463.794 245.958i −0.579019 0.307064i
\(802\) 0 0
\(803\) 119.958 + 207.773i 0.149387 + 0.258746i
\(804\) 0 0
\(805\) −11.0488 + 19.1371i −0.0137252 + 0.0237727i
\(806\) 0 0
\(807\) 716.299 + 1190.66i 0.887607 + 1.47541i
\(808\) 0 0
\(809\) 1170.22 1.44650 0.723250 0.690586i \(-0.242647\pi\)
0.723250 + 0.690586i \(0.242647\pi\)
\(810\) 0 0
\(811\) 951.145i 1.17281i −0.810020 0.586403i \(-0.800544\pi\)
0.810020 0.586403i \(-0.199456\pi\)
\(812\) 0 0
\(813\) −444.267 + 267.271i −0.546453 + 0.328746i
\(814\) 0 0
\(815\) −85.6655 + 148.377i −0.105111 + 0.182058i
\(816\) 0 0
\(817\) 645.379 + 1057.85i 0.789937 + 1.29479i
\(818\) 0 0
\(819\) −195.288 103.565i −0.238447 0.126453i
\(820\) 0 0
\(821\) −522.954 905.783i −0.636972 1.10327i −0.986094 0.166190i \(-0.946853\pi\)
0.349122 0.937077i \(-0.386480\pi\)
\(822\) 0 0
\(823\) 221.025 382.827i 0.268560 0.465160i −0.699930 0.714212i \(-0.746785\pi\)
0.968490 + 0.249051i \(0.0801188\pi\)
\(824\) 0 0
\(825\) 6.54218 + 363.495i 0.00792992 + 0.440600i
\(826\) 0 0
\(827\) 837.366i 1.01253i 0.862377 + 0.506267i \(0.168975\pi\)
−0.862377 + 0.506267i \(0.831025\pi\)
\(828\) 0 0
\(829\) 333.104i 0.401814i 0.979610 + 0.200907i \(0.0643890\pi\)
−0.979610 + 0.200907i \(0.935611\pi\)
\(830\) 0 0
\(831\) −760.592 421.064i −0.915273 0.506695i
\(832\) 0 0
\(833\) −390.719 + 676.745i −0.469051 + 0.812420i
\(834\) 0 0
\(835\) 206.577 119.267i 0.247397 0.142835i
\(836\) 0 0
\(837\) 398.706 785.664i 0.476351 0.938667i
\(838\) 0 0
\(839\) 410.267 236.868i 0.488995 0.282322i −0.235162 0.971956i \(-0.575562\pi\)
0.724158 + 0.689634i \(0.242229\pi\)
\(840\) 0 0
\(841\) −416.312 + 721.074i −0.495021 + 0.857401i
\(842\) 0 0
\(843\) 762.575 1377.48i 0.904597 1.63403i
\(844\) 0 0
\(845\) 505.458 0.598175
\(846\) 0 0
\(847\) −89.4317 −0.105586
\(848\) 0 0
\(849\) −19.4829 1082.50i −0.0229480 1.27503i
\(850\) 0 0
\(851\) 60.3552 + 34.8461i 0.0709226 + 0.0409472i
\(852\) 0 0
\(853\) 29.6335 + 51.3267i 0.0347403 + 0.0601720i 0.882873 0.469612i \(-0.155606\pi\)
−0.848132 + 0.529784i \(0.822273\pi\)
\(854\) 0 0
\(855\) 273.865 + 414.566i 0.320311 + 0.484873i
\(856\) 0 0
\(857\) −565.033 + 326.222i −0.659316 + 0.380656i −0.792016 0.610500i \(-0.790968\pi\)
0.132701 + 0.991156i \(0.457635\pi\)
\(858\) 0 0
\(859\) −124.886 + 216.309i −0.145385 + 0.251815i −0.929517 0.368780i \(-0.879776\pi\)
0.784131 + 0.620595i \(0.213109\pi\)
\(860\) 0 0
\(861\) −103.832 172.593i −0.120594 0.200456i
\(862\) 0 0
\(863\) 1259.81i 1.45980i 0.683553 + 0.729901i \(0.260434\pi\)
−0.683553 + 0.729901i \(0.739566\pi\)
\(864\) 0 0
\(865\) 114.967i 0.132910i
\(866\) 0 0
\(867\) 39.5328 23.7829i 0.0455972 0.0274313i
\(868\) 0 0
\(869\) 339.985 + 196.290i 0.391237 + 0.225881i
\(870\) 0 0
\(871\) −96.7020 167.493i −0.111024 0.192299i
\(872\) 0 0
\(873\) 38.3476 + 1064.99i 0.0439263 + 1.21991i
\(874\) 0 0
\(875\) −80.0721 138.689i −0.0915110 0.158502i
\(876\) 0 0
\(877\) 543.869 + 314.003i 0.620147 + 0.358042i 0.776926 0.629591i \(-0.216778\pi\)
−0.156779 + 0.987634i \(0.550111\pi\)
\(878\) 0 0
\(879\) 1719.40 30.9458i 1.95609 0.0352057i
\(880\) 0 0
\(881\) −1307.34 −1.48393 −0.741963 0.670441i \(-0.766105\pi\)
−0.741963 + 0.670441i \(0.766105\pi\)
\(882\) 0 0
\(883\) 1325.22 1.50081 0.750405 0.660978i \(-0.229858\pi\)
0.750405 + 0.660978i \(0.229858\pi\)
\(884\) 0 0
\(885\) 215.972 390.123i 0.244036 0.440817i
\(886\) 0 0
\(887\) 664.056 + 383.393i 0.748654 + 0.432236i 0.825207 0.564830i \(-0.191058\pi\)
−0.0765534 + 0.997065i \(0.524392\pi\)
\(888\) 0 0
\(889\) 259.224 149.663i 0.291590 0.168350i
\(890\) 0 0
\(891\) −258.727 + 533.405i −0.290378 + 0.598659i
\(892\) 0 0
\(893\) 87.2209 + 142.965i 0.0976718 + 0.160095i
\(894\) 0 0
\(895\) −345.798 199.646i −0.386366 0.223069i
\(896\) 0 0
\(897\) −154.302 + 278.724i −0.172020 + 0.310730i
\(898\) 0 0
\(899\) −94.4354 −0.105045
\(900\) 0 0
\(901\) 435.183i 0.483000i
\(902\) 0 0
\(903\) −4.66960 259.451i −0.00517121 0.287322i
\(904\) 0 0
\(905\) 353.502 + 204.094i 0.390610 + 0.225519i
\(906\) 0 0
\(907\) 501.802 289.716i 0.553255 0.319422i −0.197179 0.980368i \(-0.563178\pi\)
0.750434 + 0.660946i \(0.229845\pi\)
\(908\) 0 0
\(909\) −1152.85 + 41.5114i −1.26826 + 0.0456671i
\(910\) 0 0
\(911\) 1480.53 854.782i 1.62517 0.938290i 0.639658 0.768659i \(-0.279076\pi\)
0.985508 0.169631i \(-0.0542575\pi\)
\(912\) 0 0
\(913\) −384.090 + 665.263i −0.420690 + 0.728656i
\(914\) 0 0
\(915\) 68.5366 41.2316i 0.0749034 0.0450619i
\(916\) 0 0
\(917\) 64.4865 0.0703233
\(918\) 0 0
\(919\) −517.810 −0.563449 −0.281725 0.959495i \(-0.590906\pi\)
−0.281725 + 0.959495i \(0.590906\pi\)
\(920\) 0 0
\(921\) 548.227 + 911.283i 0.595252 + 0.989449i
\(922\) 0 0
\(923\) 428.308 741.851i 0.464039 0.803738i
\(924\) 0 0
\(925\) −174.272 + 100.616i −0.188402 + 0.108774i
\(926\) 0 0
\(927\) −9.49583 + 5.94799i −0.0102436 + 0.00641639i
\(928\) 0 0
\(929\) 355.354 + 615.492i 0.382513 + 0.662531i 0.991421 0.130709i \(-0.0417255\pi\)
−0.608908 + 0.793241i \(0.708392\pi\)
\(930\) 0 0
\(931\) −429.845 + 787.961i −0.461703 + 0.846360i
\(932\) 0 0
\(933\) −24.0393 1335.66i −0.0257656 1.43158i
\(934\) 0 0
\(935\) 351.774 0.376229
\(936\) 0 0
\(937\) −742.921 −0.792872 −0.396436 0.918062i \(-0.629753\pi\)
−0.396436 + 0.918062i \(0.629753\pi\)
\(938\) 0 0
\(939\) −431.718 238.999i −0.459763 0.254525i
\(940\) 0 0
\(941\) −1363.82 787.404i −1.44933 0.836773i −0.450892 0.892578i \(-0.648894\pi\)
−0.998442 + 0.0558053i \(0.982227\pi\)
\(942\) 0 0
\(943\) −251.399 + 145.145i −0.266595 + 0.153919i
\(944\) 0 0
\(945\) −5.61453 103.895i −0.00594130 0.109941i
\(946\) 0 0
\(947\) 525.446 + 910.099i 0.554853 + 0.961034i 0.997915 + 0.0645429i \(0.0205589\pi\)
−0.443062 + 0.896491i \(0.646108\pi\)
\(948\) 0 0
\(949\) −525.725 303.527i −0.553978 0.319839i
\(950\) 0 0
\(951\) −431.242 + 778.978i −0.453462 + 0.819115i
\(952\) 0 0
\(953\) 485.193i 0.509122i −0.967057 0.254561i \(-0.918069\pi\)
0.967057 0.254561i \(-0.0819310\pi\)
\(954\) 0 0
\(955\) 923.576 0.967095
\(956\) 0 0
\(957\) 63.5340 1.14348i 0.0663887 0.00119486i
\(958\) 0 0
\(959\) 15.0502 26.0677i 0.0156936 0.0271822i
\(960\) 0 0
\(961\) 51.8965 + 89.8874i 0.0540026 + 0.0935353i
\(962\) 0 0
\(963\) −1057.35 560.734i −1.09798 0.582278i
\(964\) 0 0
\(965\) −386.148 + 222.943i −0.400154 + 0.231029i
\(966\) 0 0
\(967\) 400.703 694.038i 0.414377 0.717723i −0.580985 0.813914i \(-0.697333\pi\)
0.995363 + 0.0961912i \(0.0306660\pi\)
\(968\) 0 0
\(969\) −795.930 + 505.462i −0.821393 + 0.521633i
\(970\) 0 0
\(971\) 1333.51i 1.37333i −0.726972 0.686667i \(-0.759073\pi\)
0.726972 0.686667i \(-0.240927\pi\)
\(972\) 0 0
\(973\) 218.398 0.224458
\(974\) 0 0
\(975\) −474.208 788.245i −0.486367 0.808457i
\(976\) 0 0
\(977\) −663.222 382.911i −0.678835 0.391926i 0.120581 0.992704i \(-0.461524\pi\)
−0.799416 + 0.600778i \(0.794858\pi\)
\(978\) 0 0
\(979\) −369.727 + 213.462i −0.377658 + 0.218041i
\(980\) 0 0
\(981\) 740.946 + 392.937i 0.755297 + 0.400547i
\(982\) 0 0
\(983\) 999.494 577.058i 1.01678 0.587038i 0.103610 0.994618i \(-0.466961\pi\)
0.913169 + 0.407580i \(0.133627\pi\)
\(984\) 0 0
\(985\) 361.413 625.986i 0.366917 0.635519i
\(986\) 0 0
\(987\) −0.631082 35.0640i −0.000639394 0.0355259i
\(988\) 0 0
\(989\) −373.991 −0.378150
\(990\) 0 0
\(991\) 486.461i 0.490879i 0.969412 + 0.245440i \(0.0789323\pi\)
−0.969412 + 0.245440i \(0.921068\pi\)
\(992\) 0 0
\(993\) −492.873 + 890.305i −0.496347 + 0.896581i
\(994\) 0 0
\(995\) 178.690 309.501i 0.179588 0.311056i
\(996\) 0 0
\(997\) 533.786 + 924.544i 0.535392 + 0.927326i 0.999144 + 0.0413613i \(0.0131695\pi\)
−0.463752 + 0.885965i \(0.653497\pi\)
\(998\) 0 0
\(999\) −327.666 + 17.7073i −0.327994 + 0.0177250i
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 684.3.t.a.265.9 80
3.2 odd 2 2052.3.t.a.37.13 80
9.2 odd 6 2052.3.t.a.721.14 80
9.7 even 3 inner 684.3.t.a.493.32 yes 80
19.18 odd 2 inner 684.3.t.a.265.32 yes 80
57.56 even 2 2052.3.t.a.37.14 80
171.56 even 6 2052.3.t.a.721.13 80
171.151 odd 6 inner 684.3.t.a.493.9 yes 80
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
684.3.t.a.265.9 80 1.1 even 1 trivial
684.3.t.a.265.32 yes 80 19.18 odd 2 inner
684.3.t.a.493.9 yes 80 171.151 odd 6 inner
684.3.t.a.493.32 yes 80 9.7 even 3 inner
2052.3.t.a.37.13 80 3.2 odd 2
2052.3.t.a.37.14 80 57.56 even 2
2052.3.t.a.721.13 80 171.56 even 6
2052.3.t.a.721.14 80 9.2 odd 6