Properties

Label 136.4.b.b.33.7
Level $136$
Weight $4$
Character 136.33
Analytic conductor $8.024$
Analytic rank $0$
Dimension $8$
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] = [136,4,Mod(33,136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(136, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("136.33");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 136 = 2^{3} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 136.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(8.02425976078\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 95x^{6} + 756x^{4} + 1780x^{2} + 1152 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 33.7
Root \(-1.03229i\) of defining polynomial
Character \(\chi\) \(=\) 136.33
Dual form 136.4.b.b.33.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.52350i q^{3} +18.2701i q^{5} +13.8757i q^{7} -45.6501 q^{9} +O(q^{10})\) \(q+8.52350i q^{3} +18.2701i q^{5} +13.8757i q^{7} -45.6501 q^{9} -60.8512i q^{11} +61.8450 q^{13} -155.725 q^{15} +(69.2650 + 10.7408i) q^{17} -40.0349 q^{19} -118.270 q^{21} -4.88830i q^{23} -208.795 q^{25} -158.964i q^{27} +113.141i q^{29} +95.1610i q^{31} +518.665 q^{33} -253.510 q^{35} -273.445i q^{37} +527.136i q^{39} +446.056i q^{41} -274.325 q^{43} -834.029i q^{45} -27.6551 q^{47} +150.464 q^{49} +(-91.5494 + 590.380i) q^{51} +488.605 q^{53} +1111.76 q^{55} -341.237i q^{57} +266.325 q^{59} -502.818i q^{61} -633.428i q^{63} +1129.91i q^{65} -1008.84 q^{67} +41.6654 q^{69} -724.061i q^{71} +188.092i q^{73} -1779.67i q^{75} +844.355 q^{77} +48.3382i q^{79} +122.376 q^{81} +1384.89 q^{83} +(-196.236 + 1265.48i) q^{85} -964.357 q^{87} -50.3364 q^{89} +858.144i q^{91} -811.105 q^{93} -731.440i q^{95} +1285.79i q^{97} +2777.86i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 132 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 132 q^{9} + 44 q^{13} + 24 q^{15} + 28 q^{17} + 48 q^{19} + 308 q^{21} - 520 q^{25} + 812 q^{33} - 1064 q^{35} + 8 q^{43} + 312 q^{47} - 1124 q^{49} + 408 q^{51} + 472 q^{53} + 1416 q^{55} - 72 q^{59} - 624 q^{67} - 180 q^{69} + 1660 q^{77} + 3156 q^{81} + 2472 q^{83} - 2160 q^{85} - 6664 q^{87} + 68 q^{89} - 4036 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/136\mathbb{Z}\right)^\times\).

\(n\) \(69\) \(103\) \(105\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 8.52350i 1.64035i 0.572114 + 0.820174i \(0.306124\pi\)
−0.572114 + 0.820174i \(0.693876\pi\)
\(4\) 0 0
\(5\) 18.2701i 1.63412i 0.576550 + 0.817062i \(0.304399\pi\)
−0.576550 + 0.817062i \(0.695601\pi\)
\(6\) 0 0
\(7\) 13.8757i 0.749219i 0.927183 + 0.374609i \(0.122223\pi\)
−0.927183 + 0.374609i \(0.877777\pi\)
\(8\) 0 0
\(9\) −45.6501 −1.69074
\(10\) 0 0
\(11\) 60.8512i 1.66794i −0.551811 0.833969i \(-0.686063\pi\)
0.551811 0.833969i \(-0.313937\pi\)
\(12\) 0 0
\(13\) 61.8450 1.31944 0.659720 0.751512i \(-0.270675\pi\)
0.659720 + 0.751512i \(0.270675\pi\)
\(14\) 0 0
\(15\) −155.725 −2.68053
\(16\) 0 0
\(17\) 69.2650 + 10.7408i 0.988189 + 0.153237i
\(18\) 0 0
\(19\) −40.0349 −0.483402 −0.241701 0.970351i \(-0.577705\pi\)
−0.241701 + 0.970351i \(0.577705\pi\)
\(20\) 0 0
\(21\) −118.270 −1.22898
\(22\) 0 0
\(23\) 4.88830i 0.0443165i −0.999754 0.0221583i \(-0.992946\pi\)
0.999754 0.0221583i \(-0.00705377\pi\)
\(24\) 0 0
\(25\) −208.795 −1.67036
\(26\) 0 0
\(27\) 158.964i 1.13306i
\(28\) 0 0
\(29\) 113.141i 0.724474i 0.932086 + 0.362237i \(0.117987\pi\)
−0.932086 + 0.362237i \(0.882013\pi\)
\(30\) 0 0
\(31\) 95.1610i 0.551336i 0.961253 + 0.275668i \(0.0888991\pi\)
−0.961253 + 0.275668i \(0.911101\pi\)
\(32\) 0 0
\(33\) 518.665 2.73600
\(34\) 0 0
\(35\) −253.510 −1.22432
\(36\) 0 0
\(37\) 273.445i 1.21497i −0.794330 0.607487i \(-0.792178\pi\)
0.794330 0.607487i \(-0.207822\pi\)
\(38\) 0 0
\(39\) 527.136i 2.16434i
\(40\) 0 0
\(41\) 446.056i 1.69908i 0.527525 + 0.849540i \(0.323120\pi\)
−0.527525 + 0.849540i \(0.676880\pi\)
\(42\) 0 0
\(43\) −274.325 −0.972888 −0.486444 0.873712i \(-0.661706\pi\)
−0.486444 + 0.873712i \(0.661706\pi\)
\(44\) 0 0
\(45\) 834.029i 2.76288i
\(46\) 0 0
\(47\) −27.6551 −0.0858280 −0.0429140 0.999079i \(-0.513664\pi\)
−0.0429140 + 0.999079i \(0.513664\pi\)
\(48\) 0 0
\(49\) 150.464 0.438671
\(50\) 0 0
\(51\) −91.5494 + 590.380i −0.251362 + 1.62097i
\(52\) 0 0
\(53\) 488.605 1.26632 0.633161 0.774020i \(-0.281757\pi\)
0.633161 + 0.774020i \(0.281757\pi\)
\(54\) 0 0
\(55\) 1111.76 2.72562
\(56\) 0 0
\(57\) 341.237i 0.792947i
\(58\) 0 0
\(59\) 266.325 0.587671 0.293835 0.955856i \(-0.405068\pi\)
0.293835 + 0.955856i \(0.405068\pi\)
\(60\) 0 0
\(61\) 502.818i 1.05540i −0.849432 0.527699i \(-0.823055\pi\)
0.849432 0.527699i \(-0.176945\pi\)
\(62\) 0 0
\(63\) 633.428i 1.26674i
\(64\) 0 0
\(65\) 1129.91i 2.15613i
\(66\) 0 0
\(67\) −1008.84 −1.83955 −0.919776 0.392443i \(-0.871630\pi\)
−0.919776 + 0.392443i \(0.871630\pi\)
\(68\) 0 0
\(69\) 41.6654 0.0726946
\(70\) 0 0
\(71\) 724.061i 1.21029i −0.796117 0.605143i \(-0.793116\pi\)
0.796117 0.605143i \(-0.206884\pi\)
\(72\) 0 0
\(73\) 188.092i 0.301569i 0.988567 + 0.150785i \(0.0481800\pi\)
−0.988567 + 0.150785i \(0.951820\pi\)
\(74\) 0 0
\(75\) 1779.67i 2.73997i
\(76\) 0 0
\(77\) 844.355 1.24965
\(78\) 0 0
\(79\) 48.3382i 0.0688415i 0.999407 + 0.0344207i \(0.0109586\pi\)
−0.999407 + 0.0344207i \(0.989041\pi\)
\(80\) 0 0
\(81\) 122.376 0.167868
\(82\) 0 0
\(83\) 1384.89 1.83146 0.915729 0.401797i \(-0.131614\pi\)
0.915729 + 0.401797i \(0.131614\pi\)
\(84\) 0 0
\(85\) −196.236 + 1265.48i −0.250409 + 1.61482i
\(86\) 0 0
\(87\) −964.357 −1.18839
\(88\) 0 0
\(89\) −50.3364 −0.0599511 −0.0299756 0.999551i \(-0.509543\pi\)
−0.0299756 + 0.999551i \(0.509543\pi\)
\(90\) 0 0
\(91\) 858.144i 0.988549i
\(92\) 0 0
\(93\) −811.105 −0.904384
\(94\) 0 0
\(95\) 731.440i 0.789938i
\(96\) 0 0
\(97\) 1285.79i 1.34589i 0.739691 + 0.672947i \(0.234972\pi\)
−0.739691 + 0.672947i \(0.765028\pi\)
\(98\) 0 0
\(99\) 2777.86i 2.82005i
\(100\) 0 0
\(101\) −932.025 −0.918217 −0.459109 0.888380i \(-0.651831\pi\)
−0.459109 + 0.888380i \(0.651831\pi\)
\(102\) 0 0
\(103\) −283.119 −0.270840 −0.135420 0.990788i \(-0.543238\pi\)
−0.135420 + 0.990788i \(0.543238\pi\)
\(104\) 0 0
\(105\) 2160.80i 2.00831i
\(106\) 0 0
\(107\) 273.739i 0.247321i 0.992325 + 0.123661i \(0.0394634\pi\)
−0.992325 + 0.123661i \(0.960537\pi\)
\(108\) 0 0
\(109\) 834.584i 0.733382i 0.930343 + 0.366691i \(0.119509\pi\)
−0.930343 + 0.366691i \(0.880491\pi\)
\(110\) 0 0
\(111\) 2330.71 1.99298
\(112\) 0 0
\(113\) 193.136i 0.160785i −0.996763 0.0803927i \(-0.974383\pi\)
0.996763 0.0803927i \(-0.0256174\pi\)
\(114\) 0 0
\(115\) 89.3095 0.0724187
\(116\) 0 0
\(117\) −2823.23 −2.23083
\(118\) 0 0
\(119\) −149.037 + 961.102i −0.114808 + 0.740370i
\(120\) 0 0
\(121\) −2371.87 −1.78202
\(122\) 0 0
\(123\) −3801.96 −2.78708
\(124\) 0 0
\(125\) 1530.94i 1.09545i
\(126\) 0 0
\(127\) 1114.62 0.778793 0.389396 0.921070i \(-0.372684\pi\)
0.389396 + 0.921070i \(0.372684\pi\)
\(128\) 0 0
\(129\) 2338.21i 1.59588i
\(130\) 0 0
\(131\) 673.762i 0.449365i −0.974432 0.224683i \(-0.927865\pi\)
0.974432 0.224683i \(-0.0721345\pi\)
\(132\) 0 0
\(133\) 555.513i 0.362174i
\(134\) 0 0
\(135\) 2904.28 1.85156
\(136\) 0 0
\(137\) 1213.64 0.756849 0.378425 0.925632i \(-0.376466\pi\)
0.378425 + 0.925632i \(0.376466\pi\)
\(138\) 0 0
\(139\) 818.573i 0.499500i −0.968310 0.249750i \(-0.919652\pi\)
0.968310 0.249750i \(-0.0803484\pi\)
\(140\) 0 0
\(141\) 235.718i 0.140788i
\(142\) 0 0
\(143\) 3763.34i 2.20074i
\(144\) 0 0
\(145\) −2067.09 −1.18388
\(146\) 0 0
\(147\) 1282.48i 0.719573i
\(148\) 0 0
\(149\) 2397.27 1.31807 0.659033 0.752114i \(-0.270966\pi\)
0.659033 + 0.752114i \(0.270966\pi\)
\(150\) 0 0
\(151\) −1197.87 −0.645573 −0.322787 0.946472i \(-0.604620\pi\)
−0.322787 + 0.946472i \(0.604620\pi\)
\(152\) 0 0
\(153\) −3161.95 490.319i −1.67077 0.259085i
\(154\) 0 0
\(155\) −1738.60 −0.900952
\(156\) 0 0
\(157\) 1926.67 0.979393 0.489696 0.871893i \(-0.337108\pi\)
0.489696 + 0.871893i \(0.337108\pi\)
\(158\) 0 0
\(159\) 4164.63i 2.07721i
\(160\) 0 0
\(161\) 67.8287 0.0332028
\(162\) 0 0
\(163\) 1307.62i 0.628347i −0.949366 0.314174i \(-0.898273\pi\)
0.949366 0.314174i \(-0.101727\pi\)
\(164\) 0 0
\(165\) 9476.04i 4.47096i
\(166\) 0 0
\(167\) 2548.82i 1.18104i 0.807023 + 0.590520i \(0.201077\pi\)
−0.807023 + 0.590520i \(0.798923\pi\)
\(168\) 0 0
\(169\) 1627.80 0.740921
\(170\) 0 0
\(171\) 1827.59 0.817308
\(172\) 0 0
\(173\) 661.688i 0.290793i 0.989373 + 0.145397i \(0.0464458\pi\)
−0.989373 + 0.145397i \(0.953554\pi\)
\(174\) 0 0
\(175\) 2897.18i 1.25147i
\(176\) 0 0
\(177\) 2270.02i 0.963985i
\(178\) 0 0
\(179\) 2927.34 1.22235 0.611173 0.791497i \(-0.290698\pi\)
0.611173 + 0.791497i \(0.290698\pi\)
\(180\) 0 0
\(181\) 209.349i 0.0859711i −0.999076 0.0429856i \(-0.986313\pi\)
0.999076 0.0429856i \(-0.0136869\pi\)
\(182\) 0 0
\(183\) 4285.77 1.73122
\(184\) 0 0
\(185\) 4995.85 1.98542
\(186\) 0 0
\(187\) 653.592 4214.86i 0.255590 1.64824i
\(188\) 0 0
\(189\) 2205.74 0.848909
\(190\) 0 0
\(191\) 4259.39 1.61361 0.806803 0.590820i \(-0.201196\pi\)
0.806803 + 0.590820i \(0.201196\pi\)
\(192\) 0 0
\(193\) 72.8650i 0.0271758i 0.999908 + 0.0135879i \(0.00432530\pi\)
−0.999908 + 0.0135879i \(0.995675\pi\)
\(194\) 0 0
\(195\) −9630.80 −3.53680
\(196\) 0 0
\(197\) 4200.16i 1.51903i −0.650489 0.759516i \(-0.725436\pi\)
0.650489 0.759516i \(-0.274564\pi\)
\(198\) 0 0
\(199\) 2013.74i 0.717337i 0.933465 + 0.358669i \(0.116769\pi\)
−0.933465 + 0.358669i \(0.883231\pi\)
\(200\) 0 0
\(201\) 8598.89i 3.01751i
\(202\) 0 0
\(203\) −1569.91 −0.542790
\(204\) 0 0
\(205\) −8149.47 −2.77651
\(206\) 0 0
\(207\) 223.151i 0.0749279i
\(208\) 0 0
\(209\) 2436.17i 0.806284i
\(210\) 0 0
\(211\) 1431.34i 0.467001i −0.972357 0.233500i \(-0.924982\pi\)
0.972357 0.233500i \(-0.0750180\pi\)
\(212\) 0 0
\(213\) 6171.54 1.98529
\(214\) 0 0
\(215\) 5011.94i 1.58982i
\(216\) 0 0
\(217\) −1320.43 −0.413072
\(218\) 0 0
\(219\) −1603.21 −0.494679
\(220\) 0 0
\(221\) 4283.69 + 664.266i 1.30386 + 0.202187i
\(222\) 0 0
\(223\) 1220.63 0.366544 0.183272 0.983062i \(-0.441331\pi\)
0.183272 + 0.983062i \(0.441331\pi\)
\(224\) 0 0
\(225\) 9531.51 2.82415
\(226\) 0 0
\(227\) 4338.39i 1.26850i 0.773128 + 0.634250i \(0.218691\pi\)
−0.773128 + 0.634250i \(0.781309\pi\)
\(228\) 0 0
\(229\) −1887.81 −0.544760 −0.272380 0.962190i \(-0.587811\pi\)
−0.272380 + 0.962190i \(0.587811\pi\)
\(230\) 0 0
\(231\) 7196.86i 2.04986i
\(232\) 0 0
\(233\) 4991.46i 1.40344i −0.712453 0.701720i \(-0.752416\pi\)
0.712453 0.701720i \(-0.247584\pi\)
\(234\) 0 0
\(235\) 505.261i 0.140254i
\(236\) 0 0
\(237\) −412.011 −0.112924
\(238\) 0 0
\(239\) −2220.06 −0.600853 −0.300427 0.953805i \(-0.597129\pi\)
−0.300427 + 0.953805i \(0.597129\pi\)
\(240\) 0 0
\(241\) 4989.64i 1.33366i −0.745212 0.666828i \(-0.767652\pi\)
0.745212 0.666828i \(-0.232348\pi\)
\(242\) 0 0
\(243\) 3248.95i 0.857697i
\(244\) 0 0
\(245\) 2748.99i 0.716843i
\(246\) 0 0
\(247\) −2475.96 −0.637819
\(248\) 0 0
\(249\) 11804.1i 3.00423i
\(250\) 0 0
\(251\) −1077.08 −0.270855 −0.135427 0.990787i \(-0.543241\pi\)
−0.135427 + 0.990787i \(0.543241\pi\)
\(252\) 0 0
\(253\) −297.459 −0.0739173
\(254\) 0 0
\(255\) −10786.3 1672.61i −2.64887 0.410757i
\(256\) 0 0
\(257\) −6521.35 −1.58284 −0.791421 0.611271i \(-0.790658\pi\)
−0.791421 + 0.611271i \(0.790658\pi\)
\(258\) 0 0
\(259\) 3794.24 0.910281
\(260\) 0 0
\(261\) 5164.89i 1.22490i
\(262\) 0 0
\(263\) 7879.02 1.84730 0.923652 0.383231i \(-0.125189\pi\)
0.923652 + 0.383231i \(0.125189\pi\)
\(264\) 0 0
\(265\) 8926.85i 2.06933i
\(266\) 0 0
\(267\) 429.043i 0.0983408i
\(268\) 0 0
\(269\) 2539.91i 0.575691i −0.957677 0.287845i \(-0.907061\pi\)
0.957677 0.287845i \(-0.0929390\pi\)
\(270\) 0 0
\(271\) −919.402 −0.206087 −0.103044 0.994677i \(-0.532858\pi\)
−0.103044 + 0.994677i \(0.532858\pi\)
\(272\) 0 0
\(273\) −7314.39 −1.62156
\(274\) 0 0
\(275\) 12705.4i 2.78606i
\(276\) 0 0
\(277\) 6999.15i 1.51819i −0.650981 0.759094i \(-0.725642\pi\)
0.650981 0.759094i \(-0.274358\pi\)
\(278\) 0 0
\(279\) 4344.11i 0.932168i
\(280\) 0 0
\(281\) 1310.63 0.278242 0.139121 0.990275i \(-0.455572\pi\)
0.139121 + 0.990275i \(0.455572\pi\)
\(282\) 0 0
\(283\) 2119.86i 0.445275i 0.974901 + 0.222638i \(0.0714666\pi\)
−0.974901 + 0.222638i \(0.928533\pi\)
\(284\) 0 0
\(285\) 6234.42 1.29577
\(286\) 0 0
\(287\) −6189.35 −1.27298
\(288\) 0 0
\(289\) 4682.27 + 1487.93i 0.953037 + 0.302855i
\(290\) 0 0
\(291\) −10959.4 −2.20773
\(292\) 0 0
\(293\) −4192.22 −0.835877 −0.417939 0.908475i \(-0.637247\pi\)
−0.417939 + 0.908475i \(0.637247\pi\)
\(294\) 0 0
\(295\) 4865.78i 0.960327i
\(296\) 0 0
\(297\) −9673.13 −1.88987
\(298\) 0 0
\(299\) 302.317i 0.0584730i
\(300\) 0 0
\(301\) 3806.46i 0.728906i
\(302\) 0 0
\(303\) 7944.11i 1.50620i
\(304\) 0 0
\(305\) 9186.51 1.72465
\(306\) 0 0
\(307\) 899.329 0.167190 0.0835952 0.996500i \(-0.473360\pi\)
0.0835952 + 0.996500i \(0.473360\pi\)
\(308\) 0 0
\(309\) 2413.17i 0.444273i
\(310\) 0 0
\(311\) 8053.67i 1.46843i 0.678917 + 0.734215i \(0.262450\pi\)
−0.678917 + 0.734215i \(0.737550\pi\)
\(312\) 0 0
\(313\) 3940.63i 0.711622i 0.934558 + 0.355811i \(0.115795\pi\)
−0.934558 + 0.355811i \(0.884205\pi\)
\(314\) 0 0
\(315\) 11572.8 2.07000
\(316\) 0 0
\(317\) 5790.24i 1.02591i −0.858416 0.512953i \(-0.828551\pi\)
0.858416 0.512953i \(-0.171449\pi\)
\(318\) 0 0
\(319\) 6884.76 1.20838
\(320\) 0 0
\(321\) −2333.22 −0.405693
\(322\) 0 0
\(323\) −2773.01 430.008i −0.477692 0.0740751i
\(324\) 0 0
\(325\) −12912.9 −2.20394
\(326\) 0 0
\(327\) −7113.57 −1.20300
\(328\) 0 0
\(329\) 383.735i 0.0643039i
\(330\) 0 0
\(331\) −6184.41 −1.02697 −0.513484 0.858099i \(-0.671645\pi\)
−0.513484 + 0.858099i \(0.671645\pi\)
\(332\) 0 0
\(333\) 12482.8i 2.05421i
\(334\) 0 0
\(335\) 18431.7i 3.00606i
\(336\) 0 0
\(337\) 2895.51i 0.468038i 0.972232 + 0.234019i \(0.0751877\pi\)
−0.972232 + 0.234019i \(0.924812\pi\)
\(338\) 0 0
\(339\) 1646.20 0.263744
\(340\) 0 0
\(341\) 5790.66 0.919595
\(342\) 0 0
\(343\) 6847.17i 1.07788i
\(344\) 0 0
\(345\) 761.230i 0.118792i
\(346\) 0 0
\(347\) 1555.79i 0.240690i 0.992732 + 0.120345i \(0.0384000\pi\)
−0.992732 + 0.120345i \(0.961600\pi\)
\(348\) 0 0
\(349\) 1103.98 0.169325 0.0846627 0.996410i \(-0.473019\pi\)
0.0846627 + 0.996410i \(0.473019\pi\)
\(350\) 0 0
\(351\) 9831.11i 1.49500i
\(352\) 0 0
\(353\) 9622.07 1.45080 0.725398 0.688330i \(-0.241656\pi\)
0.725398 + 0.688330i \(0.241656\pi\)
\(354\) 0 0
\(355\) 13228.6 1.97776
\(356\) 0 0
\(357\) −8191.95 1270.32i −1.21446 0.188325i
\(358\) 0 0
\(359\) −3095.00 −0.455008 −0.227504 0.973777i \(-0.573056\pi\)
−0.227504 + 0.973777i \(0.573056\pi\)
\(360\) 0 0
\(361\) −5256.21 −0.766323
\(362\) 0 0
\(363\) 20216.6i 2.92313i
\(364\) 0 0
\(365\) −3436.46 −0.492802
\(366\) 0 0
\(367\) 1805.83i 0.256849i 0.991719 + 0.128425i \(0.0409921\pi\)
−0.991719 + 0.128425i \(0.959008\pi\)
\(368\) 0 0
\(369\) 20362.5i 2.87271i
\(370\) 0 0
\(371\) 6779.75i 0.948752i
\(372\) 0 0
\(373\) 6265.65 0.869767 0.434884 0.900487i \(-0.356789\pi\)
0.434884 + 0.900487i \(0.356789\pi\)
\(374\) 0 0
\(375\) 13049.0 1.79693
\(376\) 0 0
\(377\) 6997.20i 0.955900i
\(378\) 0 0
\(379\) 1997.53i 0.270729i 0.990796 + 0.135364i \(0.0432205\pi\)
−0.990796 + 0.135364i \(0.956779\pi\)
\(380\) 0 0
\(381\) 9500.48i 1.27749i
\(382\) 0 0
\(383\) 13879.9 1.85178 0.925889 0.377797i \(-0.123318\pi\)
0.925889 + 0.377797i \(0.123318\pi\)
\(384\) 0 0
\(385\) 15426.4i 2.04208i
\(386\) 0 0
\(387\) 12523.0 1.64490
\(388\) 0 0
\(389\) −7261.69 −0.946483 −0.473242 0.880933i \(-0.656916\pi\)
−0.473242 + 0.880933i \(0.656916\pi\)
\(390\) 0 0
\(391\) 52.5044 338.588i 0.00679095 0.0437931i
\(392\) 0 0
\(393\) 5742.81 0.737115
\(394\) 0 0
\(395\) −883.143 −0.112496
\(396\) 0 0
\(397\) 7324.04i 0.925902i −0.886384 0.462951i \(-0.846791\pi\)
0.886384 0.462951i \(-0.153209\pi\)
\(398\) 0 0
\(399\) 4734.91 0.594091
\(400\) 0 0
\(401\) 2017.73i 0.251274i 0.992076 + 0.125637i \(0.0400975\pi\)
−0.992076 + 0.125637i \(0.959903\pi\)
\(402\) 0 0
\(403\) 5885.23i 0.727455i
\(404\) 0 0
\(405\) 2235.81i 0.274317i
\(406\) 0 0
\(407\) −16639.4 −2.02650
\(408\) 0 0
\(409\) −837.375 −0.101236 −0.0506180 0.998718i \(-0.516119\pi\)
−0.0506180 + 0.998718i \(0.516119\pi\)
\(410\) 0 0
\(411\) 10344.5i 1.24150i
\(412\) 0 0
\(413\) 3695.45i 0.440294i
\(414\) 0 0
\(415\) 25301.9i 2.99283i
\(416\) 0 0
\(417\) 6977.11 0.819353
\(418\) 0 0
\(419\) 6292.34i 0.733653i −0.930289 0.366827i \(-0.880444\pi\)
0.930289 0.366827i \(-0.119556\pi\)
\(420\) 0 0
\(421\) −3436.25 −0.397797 −0.198899 0.980020i \(-0.563736\pi\)
−0.198899 + 0.980020i \(0.563736\pi\)
\(422\) 0 0
\(423\) 1262.46 0.145113
\(424\) 0 0
\(425\) −14462.2 2242.63i −1.65063 0.255962i
\(426\) 0 0
\(427\) 6976.96 0.790724
\(428\) 0 0
\(429\) 32076.8 3.60999
\(430\) 0 0
\(431\) 1198.92i 0.133991i 0.997753 + 0.0669955i \(0.0213413\pi\)
−0.997753 + 0.0669955i \(0.978659\pi\)
\(432\) 0 0
\(433\) −1921.84 −0.213297 −0.106649 0.994297i \(-0.534012\pi\)
−0.106649 + 0.994297i \(0.534012\pi\)
\(434\) 0 0
\(435\) 17618.9i 1.94198i
\(436\) 0 0
\(437\) 195.702i 0.0214227i
\(438\) 0 0
\(439\) 17813.5i 1.93666i −0.249681 0.968328i \(-0.580326\pi\)
0.249681 0.968328i \(-0.419674\pi\)
\(440\) 0 0
\(441\) −6868.70 −0.741680
\(442\) 0 0
\(443\) −10707.1 −1.14833 −0.574165 0.818739i \(-0.694673\pi\)
−0.574165 + 0.818739i \(0.694673\pi\)
\(444\) 0 0
\(445\) 919.650i 0.0979676i
\(446\) 0 0
\(447\) 20433.1i 2.16209i
\(448\) 0 0
\(449\) 9542.98i 1.00303i −0.865149 0.501516i \(-0.832776\pi\)
0.865149 0.501516i \(-0.167224\pi\)
\(450\) 0 0
\(451\) 27143.0 2.83396
\(452\) 0 0
\(453\) 10210.1i 1.05896i
\(454\) 0 0
\(455\) −15678.4 −1.61541
\(456\) 0 0
\(457\) −17234.9 −1.76415 −0.882075 0.471109i \(-0.843854\pi\)
−0.882075 + 0.471109i \(0.843854\pi\)
\(458\) 0 0
\(459\) 1707.40 11010.6i 0.173627 1.11968i
\(460\) 0 0
\(461\) 11358.1 1.14751 0.573755 0.819027i \(-0.305486\pi\)
0.573755 + 0.819027i \(0.305486\pi\)
\(462\) 0 0
\(463\) 5407.62 0.542793 0.271397 0.962468i \(-0.412514\pi\)
0.271397 + 0.962468i \(0.412514\pi\)
\(464\) 0 0
\(465\) 14818.9i 1.47787i
\(466\) 0 0
\(467\) −5059.61 −0.501350 −0.250675 0.968071i \(-0.580653\pi\)
−0.250675 + 0.968071i \(0.580653\pi\)
\(468\) 0 0
\(469\) 13998.5i 1.37823i
\(470\) 0 0
\(471\) 16421.9i 1.60655i
\(472\) 0 0
\(473\) 16693.0i 1.62272i
\(474\) 0 0
\(475\) 8359.09 0.807455
\(476\) 0 0
\(477\) −22304.8 −2.14102
\(478\) 0 0
\(479\) 2151.09i 0.205190i 0.994723 + 0.102595i \(0.0327146\pi\)
−0.994723 + 0.102595i \(0.967285\pi\)
\(480\) 0 0
\(481\) 16911.2i 1.60308i
\(482\) 0 0
\(483\) 578.138i 0.0544641i
\(484\) 0 0
\(485\) −23491.4 −2.19936
\(486\) 0 0
\(487\) 12035.6i 1.11988i −0.828532 0.559942i \(-0.810823\pi\)
0.828532 0.559942i \(-0.189177\pi\)
\(488\) 0 0
\(489\) 11145.5 1.03071
\(490\) 0 0
\(491\) 16672.1 1.53238 0.766191 0.642613i \(-0.222150\pi\)
0.766191 + 0.642613i \(0.222150\pi\)
\(492\) 0 0
\(493\) −1215.23 + 7836.70i −0.111016 + 0.715918i
\(494\) 0 0
\(495\) −50751.7 −4.60832
\(496\) 0 0
\(497\) 10046.9 0.906769
\(498\) 0 0
\(499\) 6097.31i 0.547000i −0.961872 0.273500i \(-0.911819\pi\)
0.961872 0.273500i \(-0.0881814\pi\)
\(500\) 0 0
\(501\) −21724.9 −1.93732
\(502\) 0 0
\(503\) 4307.43i 0.381827i 0.981607 + 0.190913i \(0.0611450\pi\)
−0.981607 + 0.190913i \(0.938855\pi\)
\(504\) 0 0
\(505\) 17028.2i 1.50048i
\(506\) 0 0
\(507\) 13874.6i 1.21537i
\(508\) 0 0
\(509\) 12433.5 1.08272 0.541362 0.840790i \(-0.317909\pi\)
0.541362 + 0.840790i \(0.317909\pi\)
\(510\) 0 0
\(511\) −2609.92 −0.225941
\(512\) 0 0
\(513\) 6364.09i 0.547722i
\(514\) 0 0
\(515\) 5172.60i 0.442587i
\(516\) 0 0
\(517\) 1682.85i 0.143156i
\(518\) 0 0
\(519\) −5639.90 −0.477002
\(520\) 0 0
\(521\) 628.628i 0.0528612i −0.999651 0.0264306i \(-0.991586\pi\)
0.999651 0.0264306i \(-0.00841410\pi\)
\(522\) 0 0
\(523\) −10333.5 −0.863963 −0.431981 0.901883i \(-0.642185\pi\)
−0.431981 + 0.901883i \(0.642185\pi\)
\(524\) 0 0
\(525\) 24694.2 2.05284
\(526\) 0 0
\(527\) −1022.11 + 6591.32i −0.0844852 + 0.544825i
\(528\) 0 0
\(529\) 12143.1 0.998036
\(530\) 0 0
\(531\) −12157.8 −0.993600
\(532\) 0 0
\(533\) 27586.3i 2.24183i
\(534\) 0 0
\(535\) −5001.24 −0.404154
\(536\) 0 0
\(537\) 24951.2i 2.00507i
\(538\) 0 0
\(539\) 9155.92i 0.731676i
\(540\) 0 0
\(541\) 4122.30i 0.327599i 0.986494 + 0.163800i \(0.0523751\pi\)
−0.986494 + 0.163800i \(0.947625\pi\)
\(542\) 0 0
\(543\) 1784.38 0.141023
\(544\) 0 0
\(545\) −15247.9 −1.19844
\(546\) 0 0
\(547\) 21268.5i 1.66248i −0.555913 0.831241i \(-0.687631\pi\)
0.555913 0.831241i \(-0.312369\pi\)
\(548\) 0 0
\(549\) 22953.7i 1.78441i
\(550\) 0 0
\(551\) 4529.58i 0.350212i
\(552\) 0 0
\(553\) −670.728 −0.0515773
\(554\) 0 0
\(555\) 42582.1i 3.25678i
\(556\) 0 0
\(557\) 8773.96 0.667441 0.333721 0.942672i \(-0.391696\pi\)
0.333721 + 0.942672i \(0.391696\pi\)
\(558\) 0 0
\(559\) −16965.6 −1.28367
\(560\) 0 0
\(561\) 35925.3 + 5570.89i 2.70369 + 0.419257i
\(562\) 0 0
\(563\) −5507.46 −0.412276 −0.206138 0.978523i \(-0.566090\pi\)
−0.206138 + 0.978523i \(0.566090\pi\)
\(564\) 0 0
\(565\) 3528.62 0.262743
\(566\) 0 0
\(567\) 1698.05i 0.125770i
\(568\) 0 0
\(569\) −13023.4 −0.959528 −0.479764 0.877398i \(-0.659278\pi\)
−0.479764 + 0.877398i \(0.659278\pi\)
\(570\) 0 0
\(571\) 13591.0i 0.996089i −0.867152 0.498044i \(-0.834052\pi\)
0.867152 0.498044i \(-0.165948\pi\)
\(572\) 0 0
\(573\) 36304.9i 2.64688i
\(574\) 0 0
\(575\) 1020.65i 0.0740246i
\(576\) 0 0
\(577\) 2171.91 0.156703 0.0783517 0.996926i \(-0.475034\pi\)
0.0783517 + 0.996926i \(0.475034\pi\)
\(578\) 0 0
\(579\) −621.065 −0.0445778
\(580\) 0 0
\(581\) 19216.3i 1.37216i
\(582\) 0 0
\(583\) 29732.2i 2.11215i
\(584\) 0 0
\(585\) 51580.5i 3.64546i
\(586\) 0 0
\(587\) −25902.3 −1.82130 −0.910651 0.413176i \(-0.864419\pi\)
−0.910651 + 0.413176i \(0.864419\pi\)
\(588\) 0 0
\(589\) 3809.76i 0.266517i
\(590\) 0 0
\(591\) 35800.1 2.49174
\(592\) 0 0
\(593\) −24167.0 −1.67356 −0.836780 0.547539i \(-0.815565\pi\)
−0.836780 + 0.547539i \(0.815565\pi\)
\(594\) 0 0
\(595\) −17559.4 2722.91i −1.20986 0.187611i
\(596\) 0 0
\(597\) −17164.1 −1.17668
\(598\) 0 0
\(599\) −14936.3 −1.01883 −0.509415 0.860521i \(-0.670138\pi\)
−0.509415 + 0.860521i \(0.670138\pi\)
\(600\) 0 0
\(601\) 15669.9i 1.06354i 0.846889 + 0.531770i \(0.178473\pi\)
−0.846889 + 0.531770i \(0.821527\pi\)
\(602\) 0 0
\(603\) 46053.8 3.11021
\(604\) 0 0
\(605\) 43334.2i 2.91204i
\(606\) 0 0
\(607\) 16641.8i 1.11280i 0.830914 + 0.556401i \(0.187818\pi\)
−0.830914 + 0.556401i \(0.812182\pi\)
\(608\) 0 0
\(609\) 13381.2i 0.890364i
\(610\) 0 0
\(611\) −1710.33 −0.113245
\(612\) 0 0
\(613\) 17114.6 1.12766 0.563828 0.825892i \(-0.309328\pi\)
0.563828 + 0.825892i \(0.309328\pi\)
\(614\) 0 0
\(615\) 69462.0i 4.55444i
\(616\) 0 0
\(617\) 21652.6i 1.41280i −0.707811 0.706402i \(-0.750317\pi\)
0.707811 0.706402i \(-0.249683\pi\)
\(618\) 0 0
\(619\) 17883.9i 1.16125i 0.814171 + 0.580625i \(0.197192\pi\)
−0.814171 + 0.580625i \(0.802808\pi\)
\(620\) 0 0
\(621\) −777.062 −0.0502132
\(622\) 0 0
\(623\) 698.455i 0.0449165i
\(624\) 0 0
\(625\) 1871.02 0.119745
\(626\) 0 0
\(627\) −20764.7 −1.32259
\(628\) 0 0
\(629\) 2937.02 18940.1i 0.186179 1.20062i
\(630\) 0 0
\(631\) 18222.4 1.14964 0.574820 0.818280i \(-0.305072\pi\)
0.574820 + 0.818280i \(0.305072\pi\)
\(632\) 0 0
\(633\) 12200.0 0.766044
\(634\) 0 0
\(635\) 20364.2i 1.27264i
\(636\) 0 0
\(637\) 9305.46 0.578800
\(638\) 0 0
\(639\) 33053.4i 2.04628i
\(640\) 0 0
\(641\) 17105.1i 1.05399i 0.849868 + 0.526996i \(0.176682\pi\)
−0.849868 + 0.526996i \(0.823318\pi\)
\(642\) 0 0
\(643\) 155.386i 0.00953008i 0.999989 + 0.00476504i \(0.00151677\pi\)
−0.999989 + 0.00476504i \(0.998483\pi\)
\(644\) 0 0
\(645\) 42719.2 2.60786
\(646\) 0 0
\(647\) 17013.9 1.03383 0.516914 0.856037i \(-0.327081\pi\)
0.516914 + 0.856037i \(0.327081\pi\)
\(648\) 0 0
\(649\) 16206.2i 0.980199i
\(650\) 0 0
\(651\) 11254.7i 0.677581i
\(652\) 0 0
\(653\) 31766.8i 1.90373i −0.306524 0.951863i \(-0.599166\pi\)
0.306524 0.951863i \(-0.400834\pi\)
\(654\) 0 0
\(655\) 12309.7 0.734318
\(656\) 0 0
\(657\) 8586.43i 0.509876i
\(658\) 0 0
\(659\) 5545.66 0.327812 0.163906 0.986476i \(-0.447591\pi\)
0.163906 + 0.986476i \(0.447591\pi\)
\(660\) 0 0
\(661\) 12808.2 0.753676 0.376838 0.926279i \(-0.377011\pi\)
0.376838 + 0.926279i \(0.377011\pi\)
\(662\) 0 0
\(663\) −5661.87 + 36512.0i −0.331658 + 2.13878i
\(664\) 0 0
\(665\) 10149.3 0.591836
\(666\) 0 0
\(667\) 553.067 0.0321062
\(668\) 0 0
\(669\) 10404.0i 0.601260i
\(670\) 0 0
\(671\) −30597.1 −1.76034
\(672\) 0 0
\(673\) 13921.9i 0.797399i 0.917082 + 0.398699i \(0.130538\pi\)
−0.917082 + 0.398699i \(0.869462\pi\)
\(674\) 0 0
\(675\) 33190.9i 1.89262i
\(676\) 0 0
\(677\) 15570.9i 0.883954i −0.897026 0.441977i \(-0.854277\pi\)
0.897026 0.441977i \(-0.145723\pi\)
\(678\) 0 0
\(679\) −17841.2 −1.00837
\(680\) 0 0
\(681\) −36978.3 −2.08078
\(682\) 0 0
\(683\) 6547.76i 0.366827i 0.983036 + 0.183414i \(0.0587148\pi\)
−0.983036 + 0.183414i \(0.941285\pi\)
\(684\) 0 0
\(685\) 22173.3i 1.23679i
\(686\) 0 0
\(687\) 16090.8i 0.893597i
\(688\) 0 0
\(689\) 30217.8 1.67084
\(690\) 0 0
\(691\) 12241.5i 0.673936i 0.941516 + 0.336968i \(0.109401\pi\)
−0.941516 + 0.336968i \(0.890599\pi\)
\(692\) 0 0
\(693\) −38544.8 −2.11284
\(694\) 0 0
\(695\) 14955.4 0.816244
\(696\) 0 0
\(697\) −4791.01 + 30896.1i −0.260362 + 1.67901i
\(698\) 0 0
\(699\) 42544.7 2.30213
\(700\) 0 0
\(701\) −20002.0 −1.07770 −0.538849 0.842403i \(-0.681141\pi\)
−0.538849 + 0.842403i \(0.681141\pi\)
\(702\) 0 0
\(703\) 10947.3i 0.587320i
\(704\) 0 0
\(705\) 4306.59 0.230065
\(706\) 0 0
\(707\) 12932.5i 0.687946i
\(708\) 0 0
\(709\) 781.933i 0.0414191i 0.999786 + 0.0207095i \(0.00659252\pi\)
−0.999786 + 0.0207095i \(0.993407\pi\)
\(710\) 0 0
\(711\) 2206.64i 0.116393i
\(712\) 0 0
\(713\) 465.175 0.0244333
\(714\) 0 0
\(715\) 68756.5 3.59629
\(716\) 0 0
\(717\) 18922.7i 0.985609i
\(718\) 0 0
\(719\) 3489.63i 0.181003i 0.995896 + 0.0905017i \(0.0288470\pi\)
−0.995896 + 0.0905017i \(0.971153\pi\)
\(720\) 0 0
\(721\) 3928.48i 0.202919i
\(722\) 0 0
\(723\) 42529.2 2.18766
\(724\) 0 0
\(725\) 23623.3i 1.21013i
\(726\) 0 0
\(727\) −4894.00 −0.249667 −0.124834 0.992178i \(-0.539840\pi\)
−0.124834 + 0.992178i \(0.539840\pi\)
\(728\) 0 0
\(729\) 30996.6 1.57479
\(730\) 0 0
\(731\) −19001.1 2946.48i −0.961398 0.149083i
\(732\) 0 0
\(733\) −5187.08 −0.261377 −0.130688 0.991423i \(-0.541719\pi\)
−0.130688 + 0.991423i \(0.541719\pi\)
\(734\) 0 0
\(735\) −23431.0 −1.17587
\(736\) 0 0
\(737\) 61389.4i 3.06826i
\(738\) 0 0
\(739\) 10603.0 0.527793 0.263896 0.964551i \(-0.414992\pi\)
0.263896 + 0.964551i \(0.414992\pi\)
\(740\) 0 0
\(741\) 21103.8i 1.04625i
\(742\) 0 0
\(743\) 4784.54i 0.236242i −0.992999 0.118121i \(-0.962313\pi\)
0.992999 0.118121i \(-0.0376871\pi\)
\(744\) 0 0
\(745\) 43798.2i 2.15388i
\(746\) 0 0
\(747\) −63220.1 −3.09652
\(748\) 0 0
\(749\) −3798.33 −0.185298
\(750\) 0 0
\(751\) 6361.82i 0.309116i −0.987984 0.154558i \(-0.950605\pi\)
0.987984 0.154558i \(-0.0493953\pi\)
\(752\) 0 0
\(753\) 9180.47i 0.444296i
\(754\) 0 0
\(755\) 21885.2i 1.05495i
\(756\) 0 0
\(757\) −28528.8 −1.36975 −0.684873 0.728662i \(-0.740142\pi\)
−0.684873 + 0.728662i \(0.740142\pi\)
\(758\) 0 0
\(759\) 2535.39i 0.121250i
\(760\) 0 0
\(761\) −18097.5 −0.862069 −0.431035 0.902335i \(-0.641851\pi\)
−0.431035 + 0.902335i \(0.641851\pi\)
\(762\) 0 0
\(763\) −11580.5 −0.549463
\(764\) 0 0
\(765\) 8958.16 57769.0i 0.423377 2.73025i
\(766\) 0 0
\(767\) 16470.9 0.775396
\(768\) 0 0
\(769\) 7189.93 0.337159 0.168580 0.985688i \(-0.446082\pi\)
0.168580 + 0.985688i \(0.446082\pi\)
\(770\) 0 0
\(771\) 55584.7i 2.59641i
\(772\) 0 0
\(773\) −5316.50 −0.247375 −0.123688 0.992321i \(-0.539472\pi\)
−0.123688 + 0.992321i \(0.539472\pi\)
\(774\) 0 0
\(775\) 19869.2i 0.920931i
\(776\) 0 0
\(777\) 32340.2i 1.49318i
\(778\) 0 0
\(779\) 17857.8i 0.821338i
\(780\) 0 0
\(781\) −44060.0 −2.01868
\(782\) 0 0
\(783\) 17985.3 0.820871
\(784\) 0 0
\(785\) 35200.3i 1.60045i
\(786\) 0 0
\(787\) 8498.14i 0.384912i −0.981306 0.192456i \(-0.938355\pi\)
0.981306 0.192456i \(-0.0616453\pi\)
\(788\) 0 0
\(789\) 67156.8i 3.03022i
\(790\) 0 0
\(791\) 2679.91 0.120463
\(792\) 0 0
\(793\) 31096.8i 1.39253i
\(794\) 0 0
\(795\) −76088.0 −3.39442
\(796\) 0 0
\(797\) −19009.1 −0.844841 −0.422420 0.906400i \(-0.638819\pi\)
−0.422420 + 0.906400i \(0.638819\pi\)
\(798\) 0 0
\(799\) −1915.53 297.039i −0.0848143 0.0131520i
\(800\) 0 0
\(801\) 2297.86 0.101362
\(802\) 0 0
\(803\) 11445.7 0.502999
\(804\) 0 0
\(805\) 1239.23i 0.0542575i
\(806\) 0 0
\(807\) 21648.9 0.944334
\(808\) 0 0
\(809\) 16963.6i 0.737215i −0.929585 0.368608i \(-0.879835\pi\)
0.929585 0.368608i \(-0.120165\pi\)
\(810\) 0 0
\(811\) 24943.1i 1.07999i 0.841669 + 0.539994i \(0.181574\pi\)
−0.841669 + 0.539994i \(0.818426\pi\)
\(812\) 0 0
\(813\) 7836.52i 0.338055i
\(814\) 0 0
\(815\) 23890.3 1.02680
\(816\) 0 0
\(817\) 10982.6 0.470296
\(818\) 0 0
\(819\) 39174.3i 1.67138i
\(820\) 0 0
\(821\) 24317.6i 1.03373i 0.856068 + 0.516863i \(0.172900\pi\)
−0.856068 + 0.516863i \(0.827100\pi\)
\(822\) 0 0
\(823\) 4969.93i 0.210499i 0.994446 + 0.105250i \(0.0335642\pi\)
−0.994446 + 0.105250i \(0.966436\pi\)
\(824\) 0 0
\(825\) −108295. −4.57011
\(826\) 0 0
\(827\) 13711.0i 0.576516i 0.957553 + 0.288258i \(0.0930761\pi\)
−0.957553 + 0.288258i \(0.906924\pi\)
\(828\) 0 0
\(829\) 35645.2 1.49338 0.746689 0.665173i \(-0.231642\pi\)
0.746689 + 0.665173i \(0.231642\pi\)
\(830\) 0 0
\(831\) 59657.3 2.49036
\(832\) 0 0
\(833\) 10421.9 + 1616.11i 0.433490 + 0.0672207i
\(834\) 0 0
\(835\) −46567.1 −1.92996
\(836\) 0 0
\(837\) 15127.1 0.624696
\(838\) 0 0
\(839\) 39704.8i 1.63380i 0.576777 + 0.816902i \(0.304310\pi\)
−0.576777 + 0.816902i \(0.695690\pi\)
\(840\) 0 0
\(841\) 11588.1 0.475137
\(842\) 0 0
\(843\) 11171.2i 0.456413i
\(844\) 0 0
\(845\) 29740.1i 1.21076i
\(846\) 0 0
\(847\) 32911.4i 1.33512i
\(848\) 0 0
\(849\) −18068.7 −0.730406
\(850\) 0 0
\(851\) −1336.68 −0.0538434
\(852\) 0 0
\(853\) 4378.77i 0.175763i −0.996131 0.0878817i \(-0.971990\pi\)
0.996131 0.0878817i \(-0.0280097\pi\)
\(854\) 0 0
\(855\) 33390.3i 1.33558i
\(856\) 0 0
\(857\) 14706.1i 0.586173i −0.956086 0.293086i \(-0.905318\pi\)
0.956086 0.293086i \(-0.0946824\pi\)
\(858\) 0 0
\(859\) −22056.6 −0.876092 −0.438046 0.898953i \(-0.644329\pi\)
−0.438046 + 0.898953i \(0.644329\pi\)
\(860\) 0 0
\(861\) 52755.0i 2.08813i
\(862\) 0 0
\(863\) 13451.2 0.530571 0.265286 0.964170i \(-0.414534\pi\)
0.265286 + 0.964170i \(0.414534\pi\)
\(864\) 0 0
\(865\) −12089.1 −0.475192
\(866\) 0 0
\(867\) −12682.3 + 39909.3i −0.496787 + 1.56331i
\(868\) 0 0
\(869\) 2941.44 0.114823
\(870\) 0 0
\(871\) −62392.0 −2.42718
\(872\) 0 0
\(873\) 58696.2i 2.27556i
\(874\) 0 0
\(875\) 21242.9 0.820734
\(876\) 0 0
\(877\) 36821.0i 1.41774i −0.705339 0.708870i \(-0.749205\pi\)
0.705339 0.708870i \(-0.250795\pi\)
\(878\) 0 0
\(879\) 35732.4i 1.37113i
\(880\) 0 0
\(881\) 20929.4i 0.800372i 0.916434 + 0.400186i \(0.131055\pi\)
−0.916434 + 0.400186i \(0.868945\pi\)
\(882\) 0 0
\(883\) 14362.9 0.547394 0.273697 0.961816i \(-0.411753\pi\)
0.273697 + 0.961816i \(0.411753\pi\)
\(884\) 0 0
\(885\) −41473.4 −1.57527
\(886\) 0 0
\(887\) 15832.4i 0.599323i 0.954046 + 0.299662i \(0.0968738\pi\)
−0.954046 + 0.299662i \(0.903126\pi\)
\(888\) 0 0
\(889\) 15466.2i 0.583486i
\(890\) 0 0
\(891\) 7446.71i 0.279993i
\(892\) 0 0
\(893\) 1107.17 0.0414894
\(894\) 0 0
\(895\) 53482.7i 1.99746i
\(896\) 0 0
\(897\) 2576.80 0.0959161
\(898\) 0 0
\(899\) −10766.6 −0.399429
\(900\) 0 0
\(901\) 33843.2 + 5248.02i 1.25137 + 0.194048i
\(902\) 0 0
\(903\) 32444.4 1.19566
\(904\) 0 0
\(905\) 3824.82 0.140487
\(906\) 0 0
\(907\) 34058.7i 1.24686i −0.781880 0.623429i \(-0.785739\pi\)
0.781880 0.623429i \(-0.214261\pi\)
\(908\) 0 0
\(909\) 42547.0 1.55247
\(910\) 0 0
\(911\) 19317.1i 0.702531i 0.936276 + 0.351265i \(0.114248\pi\)
−0.936276 + 0.351265i \(0.885752\pi\)
\(912\) 0 0
\(913\) 84271.9i 3.05476i
\(914\) 0 0
\(915\) 78301.2i 2.82903i
\(916\) 0 0
\(917\) 9348.93 0.336673
\(918\) 0 0
\(919\) −19221.2 −0.689934 −0.344967 0.938615i \(-0.612110\pi\)
−0.344967 + 0.938615i \(0.612110\pi\)
\(920\) 0 0
\(921\) 7665.43i 0.274250i
\(922\) 0 0
\(923\) 44779.6i 1.59690i
\(924\) 0 0
\(925\) 57093.9i 2.02945i
\(926\) 0 0
\(927\) 12924.4 0.457921
\(928\) 0 0
\(929\) 34753.9i 1.22738i 0.789546 + 0.613692i \(0.210316\pi\)
−0.789546 + 0.613692i \(0.789684\pi\)
\(930\) 0 0
\(931\) −6023.81 −0.212054
\(932\) 0 0
\(933\) −68645.5 −2.40874
\(934\) 0 0
\(935\) 77005.7 + 11941.2i 2.69343 + 0.417666i
\(936\) 0 0
\(937\) 649.069 0.0226298 0.0113149 0.999936i \(-0.496398\pi\)
0.0113149 + 0.999936i \(0.496398\pi\)
\(938\) 0 0
\(939\) −33588.0 −1.16731
\(940\) 0 0
\(941\) 13716.6i 0.475183i 0.971365 + 0.237591i \(0.0763579\pi\)
−0.971365 + 0.237591i \(0.923642\pi\)
\(942\) 0 0
\(943\) 2180.46 0.0752973
\(944\) 0 0
\(945\) 40299.0i 1.38722i
\(946\) 0 0
\(947\) 43131.8i 1.48004i −0.672586 0.740019i \(-0.734816\pi\)
0.672586 0.740019i \(-0.265184\pi\)
\(948\) 0 0
\(949\) 11632.6i 0.397903i
\(950\) 0 0
\(951\) 49353.1 1.68284
\(952\) 0 0
\(953\) −4974.39 −0.169083 −0.0845415 0.996420i \(-0.526943\pi\)
−0.0845415 + 0.996420i \(0.526943\pi\)
\(954\) 0 0
\(955\) 77819.4i 2.63683i
\(956\) 0 0
\(957\) 58682.3i 1.98216i
\(958\) 0 0
\(959\) 16840.2i 0.567046i
\(960\) 0 0
\(961\) 20735.4 0.696028
\(962\) 0 0
\(963\) 12496.2i 0.418157i
\(964\) 0 0
\(965\) −1331.25 −0.0444087
\(966\) 0 0
\(967\) −83.3000 −0.00277016 −0.00138508 0.999999i \(-0.500441\pi\)
−0.00138508 + 0.999999i \(0.500441\pi\)
\(968\) 0 0
\(969\) 3665.17 23635.8i 0.121509 0.783582i
\(970\) 0 0
\(971\) −1577.72 −0.0521438 −0.0260719 0.999660i \(-0.508300\pi\)
−0.0260719 + 0.999660i \(0.508300\pi\)
\(972\) 0 0
\(973\) 11358.3 0.374235
\(974\) 0 0
\(975\) 110063.i 3.61523i
\(976\) 0 0
\(977\) 30161.6 0.987672 0.493836 0.869555i \(-0.335594\pi\)
0.493836 + 0.869555i \(0.335594\pi\)
\(978\) 0 0
\(979\) 3063.03i 0.0999948i
\(980\) 0 0
\(981\) 38098.8i 1.23996i
\(982\) 0 0
\(983\) 3072.86i 0.0997040i −0.998757 0.0498520i \(-0.984125\pi\)
0.998757 0.0498520i \(-0.0158750\pi\)
\(984\) 0 0
\(985\) 76737.3 2.48229
\(986\) 0 0
\(987\) 3270.77 0.105481
\(988\) 0 0
\(989\) 1340.98i 0.0431150i
\(990\) 0 0
\(991\) 22582.4i 0.723869i −0.932204 0.361934i \(-0.882116\pi\)
0.932204 0.361934i \(-0.117884\pi\)
\(992\) 0 0
\(993\) 52712.8i 1.68458i
\(994\) 0 0
\(995\) −36791.1 −1.17222
\(996\) 0 0
\(997\) 17226.0i 0.547195i −0.961844 0.273597i \(-0.911786\pi\)
0.961844 0.273597i \(-0.0882136\pi\)
\(998\) 0 0
\(999\) −43467.8 −1.37664
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 136.4.b.b.33.7 yes 8
3.2 odd 2 1224.4.c.e.577.1 8
4.3 odd 2 272.4.b.f.33.2 8
17.4 even 4 2312.4.a.k.1.7 8
17.13 even 4 2312.4.a.k.1.2 8
17.16 even 2 inner 136.4.b.b.33.2 8
51.50 odd 2 1224.4.c.e.577.8 8
68.67 odd 2 272.4.b.f.33.7 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
136.4.b.b.33.2 8 17.16 even 2 inner
136.4.b.b.33.7 yes 8 1.1 even 1 trivial
272.4.b.f.33.2 8 4.3 odd 2
272.4.b.f.33.7 8 68.67 odd 2
1224.4.c.e.577.1 8 3.2 odd 2
1224.4.c.e.577.8 8 51.50 odd 2
2312.4.a.k.1.2 8 17.13 even 4
2312.4.a.k.1.7 8 17.4 even 4