Properties

Label 76.4.i.a.5.5
Level $76$
Weight $4$
Character 76.5
Analytic conductor $4.484$
Analytic rank $0$
Dimension $30$
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] = [76,4,Mod(5,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 16]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 76.i (of order \(9\), degree \(6\), 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: \(4.48414516044\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(5\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 5.5
Character \(\chi\) \(=\) 76.5
Dual form 76.4.i.a.61.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.42726 - 8.09439i) q^{3} +(-6.74894 - 5.66304i) q^{5} +(0.266189 + 0.461053i) q^{7} +(-38.1104 - 13.8710i) q^{9} +O(q^{10})\) \(q+(1.42726 - 8.09439i) q^{3} +(-6.74894 - 5.66304i) q^{5} +(0.266189 + 0.461053i) q^{7} +(-38.1104 - 13.8710i) q^{9} +(-18.3647 + 31.8086i) q^{11} +(-8.11500 - 46.0225i) q^{13} +(-55.4713 + 46.5460i) q^{15} +(92.4595 - 33.6525i) q^{17} +(-51.1181 - 65.1609i) q^{19} +(4.11186 - 1.49660i) q^{21} +(81.4195 - 68.3191i) q^{23} +(-8.22776 - 46.6620i) q^{25} +(-55.7110 + 96.4943i) q^{27} +(81.7536 + 29.7559i) q^{29} +(121.465 + 210.384i) q^{31} +(231.260 + 194.050i) q^{33} +(0.814465 - 4.61906i) q^{35} +120.346 q^{37} -384.106 q^{39} +(-56.4094 + 319.913i) q^{41} +(-217.204 - 182.256i) q^{43} +(178.653 + 309.435i) q^{45} +(231.156 + 84.1337i) q^{47} +(171.358 - 296.801i) q^{49} +(-140.433 - 796.434i) q^{51} +(363.284 - 304.832i) q^{53} +(304.076 - 110.675i) q^{55} +(-600.396 + 320.769i) q^{57} +(-500.026 + 181.995i) q^{59} +(418.648 - 351.287i) q^{61} +(-3.74928 - 21.2632i) q^{63} +(-205.859 + 356.559i) q^{65} +(-885.145 - 322.166i) q^{67} +(-436.795 - 756.551i) q^{69} +(355.291 + 298.124i) q^{71} +(-81.3097 + 461.130i) q^{73} -389.443 q^{75} -19.5540 q^{77} +(-139.607 + 791.751i) q^{79} +(-137.284 - 115.195i) q^{81} +(466.579 + 808.139i) q^{83} +(-814.579 - 296.483i) q^{85} +(357.539 - 619.276i) q^{87} +(-192.936 - 1094.19i) q^{89} +(19.0587 - 15.9921i) q^{91} +(1876.29 - 682.915i) q^{93} +(-24.0149 + 729.251i) q^{95} +(899.137 - 327.259i) q^{97} +(1141.11 - 957.501i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q - 3 q^{3} + 6 q^{7} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 30 q - 3 q^{3} + 6 q^{7} + 15 q^{9} + 42 q^{11} - 42 q^{13} + 78 q^{15} + 30 q^{17} + 282 q^{19} + 198 q^{21} - 300 q^{23} - 276 q^{25} + 219 q^{27} + 216 q^{29} + 30 q^{31} - 597 q^{33} - 636 q^{35} + 60 q^{37} - 2172 q^{39} - 63 q^{41} - 246 q^{43} - 882 q^{45} + 762 q^{47} - 525 q^{49} + 2613 q^{51} + 882 q^{53} + 1350 q^{55} + 924 q^{57} + 2085 q^{59} + 1530 q^{61} + 2424 q^{63} + 1530 q^{65} - 3609 q^{67} + 756 q^{69} - 4962 q^{71} - 2394 q^{73} - 3516 q^{77} - 630 q^{79} - 3723 q^{81} - 2382 q^{83} + 3228 q^{85} - 1110 q^{87} + 2196 q^{89} + 6036 q^{91} + 5010 q^{93} + 6204 q^{95} + 6459 q^{97} + 6189 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(e\left(\frac{8}{9}\right)\) \(1\)

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
\(3\) 1.42726 8.09439i 0.274676 1.55777i −0.465312 0.885147i \(-0.654058\pi\)
0.739989 0.672619i \(-0.234831\pi\)
\(4\) 0 0
\(5\) −6.74894 5.66304i −0.603644 0.506517i 0.288971 0.957338i \(-0.406687\pi\)
−0.892615 + 0.450821i \(0.851131\pi\)
\(6\) 0 0
\(7\) 0.266189 + 0.461053i 0.0143729 + 0.0248945i 0.873122 0.487501i \(-0.162091\pi\)
−0.858750 + 0.512396i \(0.828758\pi\)
\(8\) 0 0
\(9\) −38.1104 13.8710i −1.41150 0.513742i
\(10\) 0 0
\(11\) −18.3647 + 31.8086i −0.503379 + 0.871879i 0.496613 + 0.867972i \(0.334577\pi\)
−0.999992 + 0.00390668i \(0.998756\pi\)
\(12\) 0 0
\(13\) −8.11500 46.0225i −0.173131 0.981872i −0.940280 0.340403i \(-0.889436\pi\)
0.767149 0.641469i \(-0.221675\pi\)
\(14\) 0 0
\(15\) −55.4713 + 46.5460i −0.954842 + 0.801208i
\(16\) 0 0
\(17\) 92.4595 33.6525i 1.31910 0.480114i 0.415931 0.909396i \(-0.363456\pi\)
0.903170 + 0.429282i \(0.141233\pi\)
\(18\) 0 0
\(19\) −51.1181 65.1609i −0.617227 0.786786i
\(20\) 0 0
\(21\) 4.11186 1.49660i 0.0427277 0.0155516i
\(22\) 0 0
\(23\) 81.4195 68.3191i 0.738137 0.619370i −0.194200 0.980962i \(-0.562211\pi\)
0.932337 + 0.361592i \(0.117767\pi\)
\(24\) 0 0
\(25\) −8.22776 46.6620i −0.0658221 0.373296i
\(26\) 0 0
\(27\) −55.7110 + 96.4943i −0.397096 + 0.687790i
\(28\) 0 0
\(29\) 81.7536 + 29.7559i 0.523492 + 0.190535i 0.590230 0.807235i \(-0.299037\pi\)
−0.0667383 + 0.997771i \(0.521259\pi\)
\(30\) 0 0
\(31\) 121.465 + 210.384i 0.703736 + 1.21891i 0.967146 + 0.254223i \(0.0818196\pi\)
−0.263410 + 0.964684i \(0.584847\pi\)
\(32\) 0 0
\(33\) 231.260 + 194.050i 1.21992 + 1.02363i
\(34\) 0 0
\(35\) 0.814465 4.61906i 0.00393342 0.0223075i
\(36\) 0 0
\(37\) 120.346 0.534721 0.267361 0.963597i \(-0.413849\pi\)
0.267361 + 0.963597i \(0.413849\pi\)
\(38\) 0 0
\(39\) −384.106 −1.57708
\(40\) 0 0
\(41\) −56.4094 + 319.913i −0.214870 + 1.21859i 0.666263 + 0.745717i \(0.267893\pi\)
−0.881132 + 0.472870i \(0.843218\pi\)
\(42\) 0 0
\(43\) −217.204 182.256i −0.770310 0.646367i 0.170478 0.985361i \(-0.445469\pi\)
−0.940788 + 0.338994i \(0.889913\pi\)
\(44\) 0 0
\(45\) 178.653 + 309.435i 0.591821 + 1.02506i
\(46\) 0 0
\(47\) 231.156 + 84.1337i 0.717394 + 0.261110i 0.674819 0.737983i \(-0.264222\pi\)
0.0425747 + 0.999093i \(0.486444\pi\)
\(48\) 0 0
\(49\) 171.358 296.801i 0.499587 0.865310i
\(50\) 0 0
\(51\) −140.433 796.434i −0.385579 2.18673i
\(52\) 0 0
\(53\) 363.284 304.832i 0.941527 0.790035i −0.0363232 0.999340i \(-0.511565\pi\)
0.977850 + 0.209305i \(0.0671201\pi\)
\(54\) 0 0
\(55\) 304.076 110.675i 0.745484 0.271334i
\(56\) 0 0
\(57\) −600.396 + 320.769i −1.39517 + 0.745383i
\(58\) 0 0
\(59\) −500.026 + 181.995i −1.10335 + 0.401588i −0.828552 0.559913i \(-0.810835\pi\)
−0.274802 + 0.961501i \(0.588612\pi\)
\(60\) 0 0
\(61\) 418.648 351.287i 0.878727 0.737339i −0.0871901 0.996192i \(-0.527789\pi\)
0.965917 + 0.258852i \(0.0833443\pi\)
\(62\) 0 0
\(63\) −3.74928 21.2632i −0.00749785 0.0425224i
\(64\) 0 0
\(65\) −205.859 + 356.559i −0.392826 + 0.680395i
\(66\) 0 0
\(67\) −885.145 322.166i −1.61399 0.587446i −0.631770 0.775156i \(-0.717671\pi\)
−0.982225 + 0.187710i \(0.939894\pi\)
\(68\) 0 0
\(69\) −436.795 756.551i −0.762085 1.31997i
\(70\) 0 0
\(71\) 355.291 + 298.124i 0.593876 + 0.498322i 0.889471 0.456992i \(-0.151073\pi\)
−0.295594 + 0.955314i \(0.595518\pi\)
\(72\) 0 0
\(73\) −81.3097 + 461.130i −0.130364 + 0.739332i 0.847612 + 0.530616i \(0.178039\pi\)
−0.977976 + 0.208716i \(0.933072\pi\)
\(74\) 0 0
\(75\) −389.443 −0.599587
\(76\) 0 0
\(77\) −19.5540 −0.0289400
\(78\) 0 0
\(79\) −139.607 + 791.751i −0.198823 + 1.12758i 0.708045 + 0.706167i \(0.249577\pi\)
−0.906868 + 0.421415i \(0.861534\pi\)
\(80\) 0 0
\(81\) −137.284 115.195i −0.188318 0.158018i
\(82\) 0 0
\(83\) 466.579 + 808.139i 0.617033 + 1.06873i 0.990024 + 0.140897i \(0.0449987\pi\)
−0.372991 + 0.927835i \(0.621668\pi\)
\(84\) 0 0
\(85\) −814.579 296.483i −1.03945 0.378330i
\(86\) 0 0
\(87\) 357.539 619.276i 0.440600 0.763142i
\(88\) 0 0
\(89\) −192.936 1094.19i −0.229788 1.30319i −0.853317 0.521393i \(-0.825413\pi\)
0.623529 0.781800i \(-0.285698\pi\)
\(90\) 0 0
\(91\) 19.0587 15.9921i 0.0219549 0.0184223i
\(92\) 0 0
\(93\) 1876.29 682.915i 2.09207 0.761452i
\(94\) 0 0
\(95\) −24.0149 + 729.251i −0.0259355 + 0.787574i
\(96\) 0 0
\(97\) 899.137 327.259i 0.941170 0.342558i 0.174542 0.984650i \(-0.444155\pi\)
0.766628 + 0.642092i \(0.221933\pi\)
\(98\) 0 0
\(99\) 1141.11 957.501i 1.15844 0.972046i
\(100\) 0 0
\(101\) 278.121 + 1577.30i 0.274001 + 1.55393i 0.742119 + 0.670269i \(0.233821\pi\)
−0.468118 + 0.883666i \(0.655068\pi\)
\(102\) 0 0
\(103\) −419.331 + 726.302i −0.401144 + 0.694802i −0.993864 0.110606i \(-0.964721\pi\)
0.592720 + 0.805409i \(0.298054\pi\)
\(104\) 0 0
\(105\) −36.2260 13.1852i −0.0336695 0.0122547i
\(106\) 0 0
\(107\) 935.751 + 1620.77i 0.845444 + 1.46435i 0.885235 + 0.465144i \(0.153997\pi\)
−0.0397911 + 0.999208i \(0.512669\pi\)
\(108\) 0 0
\(109\) −1006.92 844.904i −0.884819 0.742451i 0.0823454 0.996604i \(-0.473759\pi\)
−0.967164 + 0.254153i \(0.918203\pi\)
\(110\) 0 0
\(111\) 171.764 974.124i 0.146875 0.832971i
\(112\) 0 0
\(113\) −375.620 −0.312702 −0.156351 0.987702i \(-0.549973\pi\)
−0.156351 + 0.987702i \(0.549973\pi\)
\(114\) 0 0
\(115\) −936.389 −0.759294
\(116\) 0 0
\(117\) −329.114 + 1866.50i −0.260056 + 1.47485i
\(118\) 0 0
\(119\) 40.1273 + 33.6708i 0.0309115 + 0.0259378i
\(120\) 0 0
\(121\) −9.02654 15.6344i −0.00678177 0.0117464i
\(122\) 0 0
\(123\) 2508.99 + 913.199i 1.83925 + 0.669434i
\(124\) 0 0
\(125\) −759.352 + 1315.24i −0.543348 + 0.941107i
\(126\) 0 0
\(127\) −156.370 886.816i −0.109256 0.619624i −0.989435 0.144980i \(-0.953688\pi\)
0.880178 0.474643i \(-0.157423\pi\)
\(128\) 0 0
\(129\) −1785.26 + 1498.01i −1.21847 + 1.02242i
\(130\) 0 0
\(131\) 850.037 309.388i 0.566932 0.206346i −0.0426216 0.999091i \(-0.513571\pi\)
0.609553 + 0.792745i \(0.291349\pi\)
\(132\) 0 0
\(133\) 16.4355 40.9133i 0.0107153 0.0266739i
\(134\) 0 0
\(135\) 922.441 335.741i 0.588082 0.214044i
\(136\) 0 0
\(137\) −939.827 + 788.609i −0.586094 + 0.491791i −0.886942 0.461881i \(-0.847175\pi\)
0.300848 + 0.953672i \(0.402730\pi\)
\(138\) 0 0
\(139\) −231.255 1311.51i −0.141114 0.800296i −0.970406 0.241480i \(-0.922367\pi\)
0.829292 0.558816i \(-0.188744\pi\)
\(140\) 0 0
\(141\) 1010.93 1750.98i 0.603799 1.04581i
\(142\) 0 0
\(143\) 1612.94 + 587.063i 0.943224 + 0.343305i
\(144\) 0 0
\(145\) −383.242 663.794i −0.219493 0.380173i
\(146\) 0 0
\(147\) −2157.85 1810.65i −1.21073 1.01592i
\(148\) 0 0
\(149\) 211.662 1200.40i 0.116376 0.660002i −0.869684 0.493610i \(-0.835677\pi\)
0.986060 0.166392i \(-0.0532117\pi\)
\(150\) 0 0
\(151\) −26.0819 −0.0140564 −0.00702820 0.999975i \(-0.502237\pi\)
−0.00702820 + 0.999975i \(0.502237\pi\)
\(152\) 0 0
\(153\) −3990.46 −2.10856
\(154\) 0 0
\(155\) 371.650 2107.73i 0.192591 1.09224i
\(156\) 0 0
\(157\) 1689.12 + 1417.34i 0.858637 + 0.720482i 0.961674 0.274195i \(-0.0884114\pi\)
−0.103037 + 0.994678i \(0.532856\pi\)
\(158\) 0 0
\(159\) −1948.93 3375.64i −0.972075 1.68368i
\(160\) 0 0
\(161\) 53.1717 + 19.3529i 0.0260281 + 0.00947344i
\(162\) 0 0
\(163\) 81.7353 141.570i 0.0392761 0.0680282i −0.845719 0.533628i \(-0.820828\pi\)
0.884995 + 0.465600i \(0.154161\pi\)
\(164\) 0 0
\(165\) −461.848 2619.27i −0.217908 1.23582i
\(166\) 0 0
\(167\) −1398.72 + 1173.66i −0.648119 + 0.543837i −0.906500 0.422207i \(-0.861256\pi\)
0.258380 + 0.966043i \(0.416811\pi\)
\(168\) 0 0
\(169\) 12.2899 4.47317i 0.00559396 0.00203604i
\(170\) 0 0
\(171\) 1044.28 + 3192.37i 0.467007 + 1.42764i
\(172\) 0 0
\(173\) 3417.90 1244.01i 1.50207 0.546708i 0.545473 0.838128i \(-0.316350\pi\)
0.956595 + 0.291420i \(0.0941277\pi\)
\(174\) 0 0
\(175\) 19.3235 16.2143i 0.00834696 0.00700393i
\(176\) 0 0
\(177\) 759.469 + 4307.16i 0.322515 + 1.82907i
\(178\) 0 0
\(179\) −1154.54 + 1999.73i −0.482092 + 0.835009i −0.999789 0.0205559i \(-0.993456\pi\)
0.517696 + 0.855564i \(0.326790\pi\)
\(180\) 0 0
\(181\) −2914.95 1060.96i −1.19705 0.435692i −0.334858 0.942268i \(-0.608688\pi\)
−0.862195 + 0.506577i \(0.830911\pi\)
\(182\) 0 0
\(183\) −2245.94 3890.07i −0.907237 1.57138i
\(184\) 0 0
\(185\) −812.205 681.521i −0.322781 0.270846i
\(186\) 0 0
\(187\) −627.553 + 3559.03i −0.245408 + 1.39178i
\(188\) 0 0
\(189\) −59.3186 −0.0228296
\(190\) 0 0
\(191\) 39.7562 0.0150610 0.00753051 0.999972i \(-0.497603\pi\)
0.00753051 + 0.999972i \(0.497603\pi\)
\(192\) 0 0
\(193\) 339.425 1924.98i 0.126593 0.717942i −0.853757 0.520672i \(-0.825681\pi\)
0.980349 0.197270i \(-0.0632076\pi\)
\(194\) 0 0
\(195\) 2592.31 + 2175.21i 0.951996 + 0.798819i
\(196\) 0 0
\(197\) −1277.51 2212.71i −0.462024 0.800249i 0.537038 0.843558i \(-0.319543\pi\)
−0.999062 + 0.0433090i \(0.986210\pi\)
\(198\) 0 0
\(199\) 4978.66 + 1812.08i 1.77351 + 0.645504i 0.999930 + 0.0118012i \(0.00375653\pi\)
0.773577 + 0.633703i \(0.218466\pi\)
\(200\) 0 0
\(201\) −3871.07 + 6704.89i −1.35843 + 2.35287i
\(202\) 0 0
\(203\) 8.04288 + 45.6134i 0.00278078 + 0.0157706i
\(204\) 0 0
\(205\) 2192.38 1839.63i 0.746940 0.626757i
\(206\) 0 0
\(207\) −4050.59 + 1474.29i −1.36007 + 0.495026i
\(208\) 0 0
\(209\) 3011.45 429.337i 0.996681 0.142095i
\(210\) 0 0
\(211\) 4238.15 1542.56i 1.38278 0.503290i 0.459759 0.888043i \(-0.347936\pi\)
0.923020 + 0.384753i \(0.125713\pi\)
\(212\) 0 0
\(213\) 2920.22 2450.36i 0.939392 0.788243i
\(214\) 0 0
\(215\) 433.777 + 2460.07i 0.137597 + 0.780351i
\(216\) 0 0
\(217\) −64.6655 + 112.004i −0.0202294 + 0.0350383i
\(218\) 0 0
\(219\) 3616.52 + 1316.31i 1.11590 + 0.406154i
\(220\) 0 0
\(221\) −2299.08 3982.13i −0.699787 1.21207i
\(222\) 0 0
\(223\) −3384.38 2839.83i −1.01630 0.852778i −0.0271425 0.999632i \(-0.508641\pi\)
−0.989158 + 0.146854i \(0.953085\pi\)
\(224\) 0 0
\(225\) −333.687 + 1892.43i −0.0988702 + 0.560721i
\(226\) 0 0
\(227\) 777.194 0.227243 0.113621 0.993524i \(-0.463755\pi\)
0.113621 + 0.993524i \(0.463755\pi\)
\(228\) 0 0
\(229\) 5918.88 1.70799 0.853997 0.520277i \(-0.174171\pi\)
0.853997 + 0.520277i \(0.174171\pi\)
\(230\) 0 0
\(231\) −27.9086 + 158.277i −0.00794913 + 0.0450817i
\(232\) 0 0
\(233\) −2069.04 1736.13i −0.581749 0.488146i 0.303772 0.952745i \(-0.401754\pi\)
−0.885521 + 0.464599i \(0.846198\pi\)
\(234\) 0 0
\(235\) −1083.60 1876.86i −0.300794 0.520990i
\(236\) 0 0
\(237\) 6209.49 + 2260.07i 1.70190 + 0.619440i
\(238\) 0 0
\(239\) 1058.53 1833.42i 0.286487 0.496210i −0.686482 0.727147i \(-0.740846\pi\)
0.972969 + 0.230937i \(0.0741792\pi\)
\(240\) 0 0
\(241\) 630.358 + 3574.94i 0.168485 + 0.955528i 0.945398 + 0.325919i \(0.105674\pi\)
−0.776912 + 0.629609i \(0.783215\pi\)
\(242\) 0 0
\(243\) −3432.94 + 2880.58i −0.906267 + 0.760448i
\(244\) 0 0
\(245\) −2837.28 + 1032.69i −0.739867 + 0.269290i
\(246\) 0 0
\(247\) −2584.04 + 2881.36i −0.665662 + 0.742254i
\(248\) 0 0
\(249\) 7207.32 2623.25i 1.83432 0.667637i
\(250\) 0 0
\(251\) 3044.21 2554.40i 0.765533 0.642359i −0.174028 0.984741i \(-0.555678\pi\)
0.939561 + 0.342382i \(0.111234\pi\)
\(252\) 0 0
\(253\) 677.890 + 3844.51i 0.168453 + 0.955344i
\(254\) 0 0
\(255\) −3562.46 + 6170.37i −0.874863 + 1.51531i
\(256\) 0 0
\(257\) −1207.97 439.665i −0.293195 0.106714i 0.191235 0.981544i \(-0.438751\pi\)
−0.484430 + 0.874830i \(0.660973\pi\)
\(258\) 0 0
\(259\) 32.0347 + 55.4857i 0.00768547 + 0.0133116i
\(260\) 0 0
\(261\) −2702.91 2268.01i −0.641020 0.537880i
\(262\) 0 0
\(263\) 231.574 1313.32i 0.0542946 0.307920i −0.945551 0.325473i \(-0.894476\pi\)
0.999846 + 0.0175529i \(0.00558756\pi\)
\(264\) 0 0
\(265\) −4178.06 −0.968514
\(266\) 0 0
\(267\) −9132.19 −2.09319
\(268\) 0 0
\(269\) 12.0699 68.4517i 0.00273574 0.0155151i −0.983409 0.181400i \(-0.941937\pi\)
0.986145 + 0.165885i \(0.0530481\pi\)
\(270\) 0 0
\(271\) −2411.95 2023.87i −0.540648 0.453658i 0.331111 0.943592i \(-0.392576\pi\)
−0.871760 + 0.489934i \(0.837021\pi\)
\(272\) 0 0
\(273\) −102.245 177.093i −0.0226672 0.0392607i
\(274\) 0 0
\(275\) 1635.35 + 595.220i 0.358602 + 0.130520i
\(276\) 0 0
\(277\) −1997.82 + 3460.32i −0.433347 + 0.750579i −0.997159 0.0753239i \(-0.976001\pi\)
0.563812 + 0.825903i \(0.309334\pi\)
\(278\) 0 0
\(279\) −1710.84 9702.67i −0.367116 2.08202i
\(280\) 0 0
\(281\) −2105.00 + 1766.31i −0.446883 + 0.374979i −0.838278 0.545244i \(-0.816437\pi\)
0.391395 + 0.920223i \(0.371993\pi\)
\(282\) 0 0
\(283\) −5683.81 + 2068.74i −1.19388 + 0.434536i −0.861084 0.508463i \(-0.830214\pi\)
−0.332795 + 0.942999i \(0.607992\pi\)
\(284\) 0 0
\(285\) 5868.56 + 1235.22i 1.21973 + 0.256729i
\(286\) 0 0
\(287\) −162.513 + 59.1497i −0.0334244 + 0.0121655i
\(288\) 0 0
\(289\) 3652.70 3064.98i 0.743476 0.623850i
\(290\) 0 0
\(291\) −1365.66 7745.05i −0.275108 1.56022i
\(292\) 0 0
\(293\) 367.726 636.921i 0.0733202 0.126994i −0.827034 0.562151i \(-0.809974\pi\)
0.900355 + 0.435157i \(0.143307\pi\)
\(294\) 0 0
\(295\) 4405.29 + 1603.40i 0.869444 + 0.316452i
\(296\) 0 0
\(297\) −2046.23 3544.18i −0.399780 0.692439i
\(298\) 0 0
\(299\) −3804.93 3192.72i −0.735937 0.617524i
\(300\) 0 0
\(301\) 26.2123 148.657i 0.00501944 0.0284666i
\(302\) 0 0
\(303\) 13164.2 2.49593
\(304\) 0 0
\(305\) −4814.78 −0.903913
\(306\) 0 0
\(307\) 432.394 2452.23i 0.0803844 0.455883i −0.917873 0.396874i \(-0.870095\pi\)
0.998257 0.0590087i \(-0.0187940\pi\)
\(308\) 0 0
\(309\) 5280.48 + 4430.85i 0.972155 + 0.815735i
\(310\) 0 0
\(311\) −3468.76 6008.06i −0.632460 1.09545i −0.987047 0.160430i \(-0.948712\pi\)
0.354587 0.935023i \(-0.384622\pi\)
\(312\) 0 0
\(313\) −4478.14 1629.91i −0.808688 0.294338i −0.0956064 0.995419i \(-0.530479\pi\)
−0.713082 + 0.701081i \(0.752701\pi\)
\(314\) 0 0
\(315\) −95.1107 + 164.737i −0.0170123 + 0.0294662i
\(316\) 0 0
\(317\) −478.848 2715.68i −0.0848416 0.481160i −0.997391 0.0721932i \(-0.977000\pi\)
0.912549 0.408967i \(-0.134111\pi\)
\(318\) 0 0
\(319\) −2447.88 + 2054.01i −0.429639 + 0.360510i
\(320\) 0 0
\(321\) 14454.7 5261.08i 2.51334 0.914781i
\(322\) 0 0
\(323\) −6919.18 4304.49i −1.19193 0.741511i
\(324\) 0 0
\(325\) −2080.73 + 757.324i −0.355133 + 0.129258i
\(326\) 0 0
\(327\) −8276.12 + 6944.49i −1.39960 + 1.17441i
\(328\) 0 0
\(329\) 22.7410 + 128.970i 0.00381079 + 0.0216121i
\(330\) 0 0
\(331\) −685.049 + 1186.54i −0.113757 + 0.197034i −0.917282 0.398237i \(-0.869622\pi\)
0.803525 + 0.595271i \(0.202955\pi\)
\(332\) 0 0
\(333\) −4586.41 1669.32i −0.754756 0.274709i
\(334\) 0 0
\(335\) 4149.35 + 7186.89i 0.676726 + 1.17212i
\(336\) 0 0
\(337\) 5256.71 + 4410.90i 0.849706 + 0.712988i 0.959725 0.280941i \(-0.0906466\pi\)
−0.110019 + 0.993930i \(0.535091\pi\)
\(338\) 0 0
\(339\) −536.107 + 3040.41i −0.0858919 + 0.487117i
\(340\) 0 0
\(341\) −8922.71 −1.41699
\(342\) 0 0
\(343\) 365.060 0.0574677
\(344\) 0 0
\(345\) −1336.47 + 7579.50i −0.208560 + 1.18280i
\(346\) 0 0
\(347\) 483.123 + 405.388i 0.0747418 + 0.0627158i 0.679393 0.733774i \(-0.262243\pi\)
−0.604651 + 0.796490i \(0.706688\pi\)
\(348\) 0 0
\(349\) 3958.82 + 6856.88i 0.607194 + 1.05169i 0.991701 + 0.128569i \(0.0410384\pi\)
−0.384506 + 0.923122i \(0.625628\pi\)
\(350\) 0 0
\(351\) 4893.00 + 1780.91i 0.744071 + 0.270820i
\(352\) 0 0
\(353\) −2341.51 + 4055.62i −0.353048 + 0.611498i −0.986782 0.162054i \(-0.948188\pi\)
0.633734 + 0.773551i \(0.281522\pi\)
\(354\) 0 0
\(355\) −709.548 4024.05i −0.106081 0.601617i
\(356\) 0 0
\(357\) 329.817 276.749i 0.0488957 0.0410283i
\(358\) 0 0
\(359\) 9258.89 3369.96i 1.36119 0.495431i 0.444765 0.895647i \(-0.353287\pi\)
0.916421 + 0.400216i \(0.131065\pi\)
\(360\) 0 0
\(361\) −1632.87 + 6661.80i −0.238063 + 0.971250i
\(362\) 0 0
\(363\) −139.434 + 50.7500i −0.0201609 + 0.00733797i
\(364\) 0 0
\(365\) 3160.15 2651.68i 0.453178 0.380262i
\(366\) 0 0
\(367\) 2206.46 + 12513.5i 0.313832 + 1.77983i 0.578695 + 0.815544i \(0.303562\pi\)
−0.264863 + 0.964286i \(0.585327\pi\)
\(368\) 0 0
\(369\) 6587.31 11409.6i 0.929327 1.60964i
\(370\) 0 0
\(371\) 237.246 + 86.3504i 0.0332000 + 0.0120838i
\(372\) 0 0
\(373\) −567.039 982.141i −0.0787136 0.136336i 0.823982 0.566617i \(-0.191748\pi\)
−0.902695 + 0.430281i \(0.858415\pi\)
\(374\) 0 0
\(375\) 9562.24 + 8023.68i 1.31678 + 1.10491i
\(376\) 0 0
\(377\) 706.008 4003.97i 0.0964490 0.546989i
\(378\) 0 0
\(379\) −2663.99 −0.361055 −0.180528 0.983570i \(-0.557781\pi\)
−0.180528 + 0.983570i \(0.557781\pi\)
\(380\) 0 0
\(381\) −7401.41 −0.995239
\(382\) 0 0
\(383\) −484.525 + 2747.88i −0.0646425 + 0.366606i 0.935277 + 0.353917i \(0.115150\pi\)
−0.999919 + 0.0126892i \(0.995961\pi\)
\(384\) 0 0
\(385\) 131.969 + 110.735i 0.0174695 + 0.0146586i
\(386\) 0 0
\(387\) 5749.65 + 9958.69i 0.755223 + 1.30808i
\(388\) 0 0
\(389\) −7299.03 2656.63i −0.951351 0.346264i −0.180713 0.983536i \(-0.557840\pi\)
−0.770639 + 0.637272i \(0.780063\pi\)
\(390\) 0 0
\(391\) 5228.90 9056.72i 0.676309 1.17140i
\(392\) 0 0
\(393\) −1291.09 7322.11i −0.165717 0.939826i
\(394\) 0 0
\(395\) 5425.92 4552.88i 0.691158 0.579951i
\(396\) 0 0
\(397\) 7106.69 2586.62i 0.898425 0.327000i 0.148803 0.988867i \(-0.452458\pi\)
0.749621 + 0.661867i \(0.230236\pi\)
\(398\) 0 0
\(399\) −307.710 191.429i −0.0386085 0.0240187i
\(400\) 0 0
\(401\) −2600.31 + 946.436i −0.323824 + 0.117862i −0.498817 0.866707i \(-0.666232\pi\)
0.174993 + 0.984570i \(0.444010\pi\)
\(402\) 0 0
\(403\) 8696.71 7297.40i 1.07497 0.902009i
\(404\) 0 0
\(405\) 274.169 + 1554.89i 0.0336384 + 0.190773i
\(406\) 0 0
\(407\) −2210.11 + 3828.03i −0.269168 + 0.466212i
\(408\) 0 0
\(409\) −4310.73 1568.98i −0.521153 0.189684i 0.0680307 0.997683i \(-0.478328\pi\)
−0.589184 + 0.807999i \(0.700551\pi\)
\(410\) 0 0
\(411\) 5041.93 + 8732.88i 0.605110 + 1.04808i
\(412\) 0 0
\(413\) −217.011 182.094i −0.0258557 0.0216955i
\(414\) 0 0
\(415\) 1427.60 8096.34i 0.168863 0.957671i
\(416\) 0 0
\(417\) −10946.0 −1.28543
\(418\) 0 0
\(419\) 4970.33 0.579514 0.289757 0.957100i \(-0.406425\pi\)
0.289757 + 0.957100i \(0.406425\pi\)
\(420\) 0 0
\(421\) −1247.62 + 7075.59i −0.144430 + 0.819105i 0.823392 + 0.567472i \(0.192079\pi\)
−0.967823 + 0.251633i \(0.919032\pi\)
\(422\) 0 0
\(423\) −7642.40 6412.74i −0.878454 0.737111i
\(424\) 0 0
\(425\) −2331.03 4037.46i −0.266050 0.460813i
\(426\) 0 0
\(427\) 273.401 + 99.5099i 0.0309855 + 0.0112778i
\(428\) 0 0
\(429\) 7054.00 12217.9i 0.793871 1.37502i
\(430\) 0 0
\(431\) 1780.70 + 10098.9i 0.199010 + 1.12864i 0.906591 + 0.422010i \(0.138675\pi\)
−0.707581 + 0.706632i \(0.750214\pi\)
\(432\) 0 0
\(433\) 3021.76 2535.56i 0.335373 0.281411i −0.459512 0.888172i \(-0.651976\pi\)
0.794885 + 0.606760i \(0.207531\pi\)
\(434\) 0 0
\(435\) −5919.99 + 2154.70i −0.652510 + 0.237494i
\(436\) 0 0
\(437\) −8613.75 1813.02i −0.942909 0.198464i
\(438\) 0 0
\(439\) −10816.0 + 3936.69i −1.17590 + 0.427991i −0.854751 0.519038i \(-0.826290\pi\)
−0.321146 + 0.947030i \(0.604068\pi\)
\(440\) 0 0
\(441\) −10647.5 + 8934.29i −1.14971 + 0.964722i
\(442\) 0 0
\(443\) −1353.88 7678.26i −0.145203 0.823488i −0.967204 0.254001i \(-0.918253\pi\)
0.822001 0.569486i \(-0.192858\pi\)
\(444\) 0 0
\(445\) −4894.34 + 8477.24i −0.521380 + 0.903056i
\(446\) 0 0
\(447\) −9414.37 3426.55i −0.996162 0.362573i
\(448\) 0 0
\(449\) 4095.48 + 7093.59i 0.430463 + 0.745584i 0.996913 0.0785124i \(-0.0250170\pi\)
−0.566450 + 0.824096i \(0.691684\pi\)
\(450\) 0 0
\(451\) −9140.07 7669.43i −0.954299 0.800752i
\(452\) 0 0
\(453\) −37.2257 + 211.117i −0.00386096 + 0.0218966i
\(454\) 0 0
\(455\) −219.190 −0.0225841
\(456\) 0 0
\(457\) 17554.1 1.79682 0.898409 0.439159i \(-0.144724\pi\)
0.898409 + 0.439159i \(0.144724\pi\)
\(458\) 0 0
\(459\) −1903.74 + 10796.6i −0.193592 + 1.09792i
\(460\) 0 0
\(461\) −1482.79 1244.21i −0.149805 0.125702i 0.564805 0.825225i \(-0.308951\pi\)
−0.714610 + 0.699523i \(0.753396\pi\)
\(462\) 0 0
\(463\) 5715.70 + 9899.89i 0.573718 + 0.993708i 0.996180 + 0.0873274i \(0.0278326\pi\)
−0.422462 + 0.906381i \(0.638834\pi\)
\(464\) 0 0
\(465\) −16530.4 6016.56i −1.64855 0.600025i
\(466\) 0 0
\(467\) 4742.62 8214.46i 0.469941 0.813962i −0.529468 0.848330i \(-0.677609\pi\)
0.999409 + 0.0343682i \(0.0109419\pi\)
\(468\) 0 0
\(469\) −87.0801 493.856i −0.00857353 0.0486229i
\(470\) 0 0
\(471\) 13883.3 11649.5i 1.35819 1.13966i
\(472\) 0 0
\(473\) 9786.21 3561.89i 0.951312 0.346249i
\(474\) 0 0
\(475\) −2619.94 + 2921.40i −0.253076 + 0.282196i
\(476\) 0 0
\(477\) −18073.2 + 6578.12i −1.73484 + 0.631429i
\(478\) 0 0
\(479\) 9693.22 8133.58i 0.924623 0.775851i −0.0502211 0.998738i \(-0.515993\pi\)
0.974844 + 0.222887i \(0.0715482\pi\)
\(480\) 0 0
\(481\) −976.605 5538.60i −0.0925766 0.525028i
\(482\) 0 0
\(483\) 232.540 402.771i 0.0219067 0.0379435i
\(484\) 0 0
\(485\) −7921.50 2883.19i −0.741643 0.269936i
\(486\) 0 0
\(487\) −7300.56 12644.9i −0.679301 1.17658i −0.975192 0.221362i \(-0.928950\pi\)
0.295890 0.955222i \(-0.404384\pi\)
\(488\) 0 0
\(489\) −1029.26 863.655i −0.0951838 0.0798687i
\(490\) 0 0
\(491\) −946.574 + 5368.29i −0.0870026 + 0.493416i 0.909904 + 0.414819i \(0.136155\pi\)
−0.996906 + 0.0785970i \(0.974956\pi\)
\(492\) 0 0
\(493\) 8560.26 0.782018
\(494\) 0 0
\(495\) −13123.6 −1.19164
\(496\) 0 0
\(497\) −42.8766 + 243.165i −0.00386977 + 0.0219466i
\(498\) 0 0
\(499\) −10667.1 8950.76i −0.956964 0.802988i 0.0234929 0.999724i \(-0.492521\pi\)
−0.980456 + 0.196736i \(0.936966\pi\)
\(500\) 0 0
\(501\) 7503.75 + 12996.9i 0.669147 + 1.15900i
\(502\) 0 0
\(503\) 7275.70 + 2648.14i 0.644945 + 0.234741i 0.643723 0.765258i \(-0.277389\pi\)
0.00122202 + 0.999999i \(0.499611\pi\)
\(504\) 0 0
\(505\) 7055.29 12220.1i 0.621696 1.07681i
\(506\) 0 0
\(507\) −18.6667 105.864i −0.00163514 0.00927333i
\(508\) 0 0
\(509\) −14923.4 + 12522.3i −1.29955 + 1.09045i −0.309326 + 0.950956i \(0.600103\pi\)
−0.990223 + 0.139494i \(0.955452\pi\)
\(510\) 0 0
\(511\) −234.249 + 85.2598i −0.0202790 + 0.00738096i
\(512\) 0 0
\(513\) 9135.49 1302.43i 0.786241 0.112093i
\(514\) 0 0
\(515\) 6943.11 2527.09i 0.594078 0.216227i
\(516\) 0 0
\(517\) −6921.29 + 5807.65i −0.588777 + 0.494043i
\(518\) 0 0
\(519\) −5191.30 29441.3i −0.439061 2.49004i
\(520\) 0 0
\(521\) 5234.99 9067.26i 0.440209 0.762464i −0.557496 0.830180i \(-0.688238\pi\)
0.997705 + 0.0677155i \(0.0215710\pi\)
\(522\) 0 0
\(523\) 5454.79 + 1985.38i 0.456064 + 0.165994i 0.559829 0.828608i \(-0.310867\pi\)
−0.103765 + 0.994602i \(0.533089\pi\)
\(524\) 0 0
\(525\) −103.665 179.554i −0.00861778 0.0149264i
\(526\) 0 0
\(527\) 18310.6 + 15364.4i 1.51351 + 1.26999i
\(528\) 0 0
\(529\) −151.136 + 857.136i −0.0124218 + 0.0704476i
\(530\) 0 0
\(531\) 21580.6 1.76369
\(532\) 0 0
\(533\) 15181.0 1.23370
\(534\) 0 0
\(535\) 2863.14 16237.7i 0.231373 1.31218i
\(536\) 0 0
\(537\) 14538.7 + 12199.4i 1.16833 + 0.980344i
\(538\) 0 0
\(539\) 6293.90 + 10901.3i 0.502964 + 0.871158i
\(540\) 0 0
\(541\) −15227.2 5542.23i −1.21011 0.440442i −0.343365 0.939202i \(-0.611567\pi\)
−0.866740 + 0.498760i \(0.833789\pi\)
\(542\) 0 0
\(543\) −12748.2 + 22080.5i −1.00751 + 1.74505i
\(544\) 0 0
\(545\) 2010.91 + 11404.4i 0.158051 + 0.896352i
\(546\) 0 0
\(547\) −157.642 + 132.278i −0.0123223 + 0.0103396i −0.648928 0.760850i \(-0.724782\pi\)
0.636606 + 0.771189i \(0.280338\pi\)
\(548\) 0 0
\(549\) −20827.5 + 7580.60i −1.61912 + 0.589312i
\(550\) 0 0
\(551\) −2240.17 6848.20i −0.173202 0.529479i
\(552\) 0 0
\(553\) −402.201 + 146.389i −0.0309283 + 0.0112570i
\(554\) 0 0
\(555\) −6675.72 + 5601.60i −0.510574 + 0.428423i
\(556\) 0 0
\(557\) 3982.46 + 22585.6i 0.302948 + 1.71811i 0.633008 + 0.774145i \(0.281820\pi\)
−0.330060 + 0.943960i \(0.607069\pi\)
\(558\) 0 0
\(559\) −6625.26 + 11475.3i −0.501285 + 0.868252i
\(560\) 0 0
\(561\) 27912.5 + 10159.3i 2.10065 + 0.764576i
\(562\) 0 0
\(563\) 5298.64 + 9177.51i 0.396645 + 0.687009i 0.993310 0.115482i \(-0.0368411\pi\)
−0.596665 + 0.802491i \(0.703508\pi\)
\(564\) 0 0
\(565\) 2535.04 + 2127.15i 0.188761 + 0.158389i
\(566\) 0 0
\(567\) 16.5675 93.9588i 0.00122710 0.00695925i
\(568\) 0 0
\(569\) −10436.4 −0.768925 −0.384463 0.923141i \(-0.625613\pi\)
−0.384463 + 0.923141i \(0.625613\pi\)
\(570\) 0 0
\(571\) 10195.1 0.747200 0.373600 0.927590i \(-0.378123\pi\)
0.373600 + 0.927590i \(0.378123\pi\)
\(572\) 0 0
\(573\) 56.7424 321.802i 0.00413690 0.0234615i
\(574\) 0 0
\(575\) −3857.80 3237.08i −0.279794 0.234775i
\(576\) 0 0
\(577\) −9760.95 16906.5i −0.704252 1.21980i −0.966961 0.254926i \(-0.917949\pi\)
0.262708 0.964875i \(-0.415384\pi\)
\(578\) 0 0
\(579\) −15097.1 5494.88i −1.08361 0.394403i
\(580\) 0 0
\(581\) −248.397 + 430.235i −0.0177370 + 0.0307215i
\(582\) 0 0
\(583\) 3024.67 + 17153.7i 0.214869 + 1.21859i
\(584\) 0 0
\(585\) 12791.2 10733.1i 0.904020 0.758563i
\(586\) 0 0
\(587\) −14354.6 + 5224.63i −1.00933 + 0.367366i −0.793175 0.608994i \(-0.791573\pi\)
−0.216154 + 0.976359i \(0.569351\pi\)
\(588\) 0 0
\(589\) 7499.73 18669.2i 0.524654 1.30603i
\(590\) 0 0
\(591\) −19733.9 + 7182.54i −1.37351 + 0.499916i
\(592\) 0 0
\(593\) −11373.9 + 9543.81i −0.787637 + 0.660906i −0.945159 0.326609i \(-0.894094\pi\)
0.157522 + 0.987515i \(0.449649\pi\)
\(594\) 0 0
\(595\) −80.1379 454.485i −0.00552157 0.0313144i
\(596\) 0 0
\(597\) 21773.6 37712.9i 1.49268 2.58540i
\(598\) 0 0
\(599\) 5053.44 + 1839.30i 0.344705 + 0.125462i 0.508570 0.861020i \(-0.330174\pi\)
−0.163866 + 0.986483i \(0.552396\pi\)
\(600\) 0 0
\(601\) 6933.22 + 12008.7i 0.470569 + 0.815049i 0.999433 0.0336573i \(-0.0107155\pi\)
−0.528865 + 0.848706i \(0.677382\pi\)
\(602\) 0 0
\(603\) 29264.4 + 24555.8i 1.97635 + 1.65835i
\(604\) 0 0
\(605\) −27.6187 + 156.633i −0.00185597 + 0.0105257i
\(606\) 0 0
\(607\) −13443.5 −0.898940 −0.449470 0.893295i \(-0.648387\pi\)
−0.449470 + 0.893295i \(0.648387\pi\)
\(608\) 0 0
\(609\) 380.692 0.0253307
\(610\) 0 0
\(611\) 1996.21 11321.1i 0.132174 0.749595i
\(612\) 0 0
\(613\) 217.601 + 182.589i 0.0143374 + 0.0120305i 0.649928 0.759996i \(-0.274799\pi\)
−0.635591 + 0.772026i \(0.719243\pi\)
\(614\) 0 0
\(615\) −11761.6 20371.6i −0.771175 1.33571i
\(616\) 0 0
\(617\) 21426.3 + 7798.54i 1.39804 + 0.508845i 0.927596 0.373585i \(-0.121872\pi\)
0.470444 + 0.882430i \(0.344094\pi\)
\(618\) 0 0
\(619\) −12269.2 + 21250.9i −0.796673 + 1.37988i 0.125099 + 0.992144i \(0.460075\pi\)
−0.921772 + 0.387733i \(0.873258\pi\)
\(620\) 0 0
\(621\) 2056.44 + 11662.6i 0.132886 + 0.753633i
\(622\) 0 0
\(623\) 453.123 380.216i 0.0291396 0.0244511i
\(624\) 0 0
\(625\) 7007.52 2550.53i 0.448481 0.163234i
\(626\) 0 0
\(627\) 822.897 24988.6i 0.0524136 1.59163i
\(628\) 0 0
\(629\) 11127.1 4049.93i 0.705352 0.256727i
\(630\) 0 0
\(631\) 4454.15 3737.47i 0.281009 0.235795i −0.491378 0.870946i \(-0.663507\pi\)
0.772388 + 0.635152i \(0.219062\pi\)
\(632\) 0 0
\(633\) −6437.15 36506.9i −0.404192 2.29229i
\(634\) 0 0
\(635\) −3966.74 + 6870.60i −0.247898 + 0.429372i
\(636\) 0 0
\(637\) −15050.1 5477.79i −0.936117 0.340719i
\(638\) 0 0
\(639\) −9404.96 16289.9i −0.582245 1.00848i
\(640\) 0 0
\(641\) −17383.6 14586.6i −1.07116 0.898809i −0.0760014 0.997108i \(-0.524215\pi\)
−0.995157 + 0.0982991i \(0.968660\pi\)
\(642\) 0 0
\(643\) −775.919 + 4400.45i −0.0475883 + 0.269886i −0.999313 0.0370670i \(-0.988198\pi\)
0.951725 + 0.306953i \(0.0993096\pi\)
\(644\) 0 0
\(645\) 20531.9 1.25340
\(646\) 0 0
\(647\) −2668.71 −0.162161 −0.0810804 0.996708i \(-0.525837\pi\)
−0.0810804 + 0.996708i \(0.525837\pi\)
\(648\) 0 0
\(649\) 3393.84 19247.4i 0.205270 1.16414i
\(650\) 0 0
\(651\) 814.309 + 683.286i 0.0490250 + 0.0411369i
\(652\) 0 0
\(653\) −8339.53 14444.5i −0.499772 0.865631i 0.500228 0.865894i \(-0.333250\pi\)
−1.00000 0.000263124i \(0.999916\pi\)
\(654\) 0 0
\(655\) −7488.93 2725.75i −0.446743 0.162601i
\(656\) 0 0
\(657\) 9495.10 16446.0i 0.563835 0.976590i
\(658\) 0 0
\(659\) 3317.12 + 18812.4i 0.196080 + 1.11203i 0.910872 + 0.412689i \(0.135410\pi\)
−0.714792 + 0.699337i \(0.753479\pi\)
\(660\) 0 0
\(661\) −2932.65 + 2460.79i −0.172567 + 0.144801i −0.724981 0.688769i \(-0.758151\pi\)
0.552413 + 0.833570i \(0.313707\pi\)
\(662\) 0 0
\(663\) −35514.3 + 12926.1i −2.08033 + 0.757179i
\(664\) 0 0
\(665\) −342.616 + 183.046i −0.0199790 + 0.0106740i
\(666\) 0 0
\(667\) 8689.23 3162.62i 0.504421 0.183594i
\(668\) 0 0
\(669\) −27817.1 + 23341.3i −1.60758 + 1.34892i
\(670\) 0 0
\(671\) 3485.61 + 19767.9i 0.200538 + 1.13730i
\(672\) 0 0
\(673\) −14243.5 + 24670.5i −0.815822 + 1.41305i 0.0929140 + 0.995674i \(0.470382\pi\)
−0.908736 + 0.417371i \(0.862951\pi\)
\(674\) 0 0
\(675\) 4960.99 + 1805.65i 0.282887 + 0.102962i
\(676\) 0 0
\(677\) 15257.8 + 26427.3i 0.866181 + 1.50027i 0.865869 + 0.500270i \(0.166766\pi\)
0.000312116 1.00000i \(0.499901\pi\)
\(678\) 0 0
\(679\) 390.224 + 327.437i 0.0220551 + 0.0185064i
\(680\) 0 0
\(681\) 1109.26 6290.91i 0.0624182 0.353991i
\(682\) 0 0
\(683\) −7906.32 −0.442938 −0.221469 0.975167i \(-0.571085\pi\)
−0.221469 + 0.975167i \(0.571085\pi\)
\(684\) 0 0
\(685\) 10808.8 0.602893
\(686\) 0 0
\(687\) 8447.78 47909.7i 0.469145 2.66066i
\(688\) 0 0
\(689\) −16977.2 14245.5i −0.938721 0.787680i
\(690\) 0 0
\(691\) −15436.6 26737.0i −0.849836 1.47196i −0.881354 0.472456i \(-0.843368\pi\)
0.0315186 0.999503i \(-0.489966\pi\)
\(692\) 0 0
\(693\) 745.209 + 271.234i 0.0408487 + 0.0148677i
\(694\) 0 0
\(695\) −5866.42 + 10160.9i −0.320181 + 0.554570i
\(696\) 0 0
\(697\) 5550.31 + 31477.4i 0.301625 + 1.71060i
\(698\) 0 0
\(699\) −17006.0 + 14269.7i −0.920209 + 0.772147i
\(700\) 0 0
\(701\) −32180.3 + 11712.7i −1.73386 + 0.631072i −0.998893 0.0470366i \(-0.985022\pi\)
−0.734962 + 0.678108i \(0.762800\pi\)
\(702\) 0 0
\(703\) −6151.84 7841.82i −0.330044 0.420711i
\(704\) 0 0
\(705\) −16738.6 + 6092.35i −0.894201 + 0.325462i
\(706\) 0 0
\(707\) −653.187 + 548.089i −0.0347463 + 0.0291556i
\(708\) 0 0
\(709\) 2020.41 + 11458.3i 0.107021 + 0.606948i 0.990394 + 0.138276i \(0.0441562\pi\)
−0.883372 + 0.468672i \(0.844733\pi\)
\(710\) 0 0
\(711\) 16302.9 28237.4i 0.859924 1.48943i
\(712\) 0 0
\(713\) 24262.9 + 8830.98i 1.27441 + 0.463847i
\(714\) 0 0
\(715\) −7561.10 13096.2i −0.395481 0.684993i
\(716\) 0 0
\(717\) −13329.6 11184.9i −0.694288 0.582577i
\(718\) 0 0
\(719\) −237.869 + 1349.02i −0.0123380 + 0.0699724i −0.990355 0.138552i \(-0.955755\pi\)
0.978017 + 0.208525i \(0.0668662\pi\)
\(720\) 0 0
\(721\) −446.485 −0.0230624
\(722\) 0 0
\(723\) 29836.6 1.53477
\(724\) 0 0
\(725\) 715.818 4059.61i 0.0366687 0.207959i
\(726\) 0 0
\(727\) 15601.0 + 13090.8i 0.795884 + 0.667826i 0.947194 0.320661i \(-0.103905\pi\)
−0.151310 + 0.988486i \(0.548349\pi\)
\(728\) 0 0
\(729\) 15997.5 + 27708.4i 0.812755 + 1.40773i
\(730\) 0 0
\(731\) −26216.0 9541.83i −1.32645 0.482787i
\(732\) 0 0
\(733\) 13283.0 23006.8i 0.669330 1.15931i −0.308762 0.951139i \(-0.599915\pi\)
0.978092 0.208174i \(-0.0667520\pi\)
\(734\) 0 0
\(735\) 4309.43 + 24440.0i 0.216266 + 1.22651i
\(736\) 0 0
\(737\) 26503.1 22238.8i 1.32463 1.11150i
\(738\) 0 0
\(739\) −5877.58 + 2139.26i −0.292571 + 0.106487i −0.484136 0.874993i \(-0.660866\pi\)
0.191565 + 0.981480i \(0.438644\pi\)
\(740\) 0 0
\(741\) 19634.8 + 25028.7i 0.973417 + 1.24083i
\(742\) 0 0
\(743\) −2905.50 + 1057.52i −0.143462 + 0.0522160i −0.412753 0.910843i \(-0.635433\pi\)
0.269291 + 0.963059i \(0.413211\pi\)
\(744\) 0 0
\(745\) −8226.38 + 6902.75i −0.404552 + 0.339459i
\(746\) 0 0
\(747\) −6571.78 37270.4i −0.321886 1.82551i
\(748\) 0 0
\(749\) −498.173 + 862.862i −0.0243029 + 0.0420938i
\(750\) 0 0
\(751\) 2007.09 + 730.521i 0.0975229 + 0.0354954i 0.390321 0.920679i \(-0.372364\pi\)
−0.292798 + 0.956174i \(0.594586\pi\)
\(752\) 0 0
\(753\) −16331.4 28286.8i −0.790371 1.36896i
\(754\) 0 0
\(755\) 176.025 + 147.703i 0.00848506 + 0.00711981i
\(756\) 0 0
\(757\) 2248.44 12751.5i 0.107954 0.612236i −0.882046 0.471164i \(-0.843834\pi\)
0.989999 0.141072i \(-0.0450549\pi\)
\(758\) 0 0
\(759\) 32086.5 1.53447
\(760\) 0 0
\(761\) 31461.5 1.49866 0.749329 0.662198i \(-0.230376\pi\)
0.749329 + 0.662198i \(0.230376\pi\)
\(762\) 0 0
\(763\) 121.515 689.147i 0.00576559 0.0326983i
\(764\) 0 0
\(765\) 26931.4 + 22598.1i 1.27282 + 1.06802i
\(766\) 0 0
\(767\) 12433.6 + 21535.6i 0.585332 + 1.01383i
\(768\) 0 0
\(769\) 8870.73 + 3228.68i 0.415978 + 0.151403i 0.541526 0.840684i \(-0.317847\pi\)
−0.125548 + 0.992088i \(0.540069\pi\)
\(770\) 0 0
\(771\) −5282.91 + 9150.27i −0.246770 + 0.427418i
\(772\) 0 0
\(773\) 2460.15 + 13952.2i 0.114470 + 0.649192i 0.987011 + 0.160652i \(0.0513596\pi\)
−0.872541 + 0.488541i \(0.837529\pi\)
\(774\) 0 0
\(775\) 8817.55 7398.80i 0.408691 0.342933i
\(776\) 0 0
\(777\) 494.844 180.109i 0.0228474 0.00831578i
\(778\) 0 0
\(779\) 23729.4 12677.7i 1.09139 0.583088i
\(780\) 0 0
\(781\) −16007.7 + 5826.34i −0.733421 + 0.266943i
\(782\) 0 0
\(783\) −7425.85 + 6231.03i −0.338925 + 0.284392i
\(784\) 0 0
\(785\) −3373.32 19131.0i −0.153374 0.869830i
\(786\) 0 0
\(787\) 761.140 1318.33i 0.0344748 0.0597122i −0.848273 0.529559i \(-0.822357\pi\)
0.882748 + 0.469847i \(0.155691\pi\)
\(788\) 0 0
\(789\) −10300.0 3748.91i −0.464754 0.169157i
\(790\) 0 0
\(791\) −99.9859 173.181i −0.00449442 0.00778457i
\(792\) 0 0
\(793\) −19564.4 16416.5i −0.876107 0.735141i
\(794\) 0 0
\(795\) −5963.17 + 33818.8i −0.266028 + 1.50872i
\(796\) 0 0
\(797\) −22894.6 −1.01753 −0.508764 0.860906i \(-0.669897\pi\)
−0.508764 + 0.860906i \(0.669897\pi\)
\(798\) 0 0
\(799\) 24203.8 1.07168
\(800\) 0 0
\(801\) −7824.74 + 44376.3i −0.345160 + 1.95750i
\(802\) 0 0
\(803\) −13174.7 11054.9i −0.578985 0.485826i
\(804\) 0 0
\(805\) −249.257 431.725i −0.0109132 0.0189022i
\(806\) 0 0
\(807\) −536.848 195.397i −0.0234175 0.00852328i
\(808\) 0 0
\(809\) −4454.63 + 7715.65i −0.193593 + 0.335312i −0.946438 0.322885i \(-0.895347\pi\)
0.752846 + 0.658197i \(0.228681\pi\)
\(810\) 0 0
\(811\) −3986.15 22606.6i −0.172593 0.978823i −0.940886 0.338724i \(-0.890005\pi\)
0.768293 0.640098i \(-0.221106\pi\)
\(812\) 0 0
\(813\) −19824.5 + 16634.7i −0.855196 + 0.717594i
\(814\) 0 0
\(815\) −1353.34 + 492.576i −0.0581663 + 0.0211708i
\(816\) 0 0
\(817\) −772.881 + 23469.8i −0.0330963 + 1.00502i
\(818\) 0 0
\(819\) −948.160 + 345.102i −0.0404535 + 0.0147239i
\(820\) 0 0
\(821\) 12939.7 10857.7i 0.550061 0.461556i −0.324901 0.945748i \(-0.605331\pi\)
0.874962 + 0.484192i \(0.160887\pi\)
\(822\) 0 0
\(823\) 4495.45 + 25494.9i 0.190403 + 1.07983i 0.918815 + 0.394688i \(0.129147\pi\)
−0.728412 + 0.685139i \(0.759741\pi\)
\(824\) 0 0
\(825\) 7152.02 12387.7i 0.301820 0.522767i
\(826\) 0 0
\(827\) −3873.96 1410.00i −0.162891 0.0592874i 0.259288 0.965800i \(-0.416512\pi\)
−0.422178 + 0.906513i \(0.638734\pi\)
\(828\) 0 0
\(829\) 16132.1 + 27941.6i 0.675864 + 1.17063i 0.976215 + 0.216803i \(0.0695628\pi\)
−0.300351 + 0.953829i \(0.597104\pi\)
\(830\) 0 0
\(831\) 25157.8 + 21109.9i 1.05020 + 0.881220i
\(832\) 0 0
\(833\) 5855.60 33208.7i 0.243559 1.38129i
\(834\) 0 0
\(835\) 16086.3 0.666696
\(836\) 0 0
\(837\) −27067.8 −1.11780
\(838\) 0 0
\(839\) −5326.50 + 30208.1i −0.219179 + 1.24303i 0.654327 + 0.756212i \(0.272952\pi\)
−0.873506 + 0.486814i \(0.838159\pi\)
\(840\) 0 0
\(841\) −12884.8 10811.6i −0.528305 0.443300i
\(842\) 0 0
\(843\) 11292.8 + 19559.7i 0.461382 + 0.799136i
\(844\) 0 0
\(845\) −108.276 39.4092i −0.00440805 0.00160440i
\(846\) 0 0
\(847\) 4.80553 8.32342i 0.000194947 0.000337658i
\(848\) 0 0
\(849\) 8632.90 + 48959.6i 0.348976 + 1.97914i
\(850\) 0 0
\(851\) 9798.48 8221.90i 0.394697 0.331190i
\(852\) 0 0
\(853\) 15883.2 5781.01i 0.637550 0.232049i −0.00296407 0.999996i \(-0.500943\pi\)
0.640514 + 0.767946i \(0.278721\pi\)
\(854\) 0 0
\(855\) 11030.7 27458.9i 0.441218 1.09833i
\(856\) 0 0
\(857\) 14641.4 5329.04i 0.583596 0.212411i −0.0333147 0.999445i \(-0.510606\pi\)
0.616910 + 0.787033i \(0.288384\pi\)
\(858\) 0 0
\(859\) 589.608 494.740i 0.0234193 0.0196511i −0.631003 0.775780i \(-0.717356\pi\)
0.654422 + 0.756129i \(0.272912\pi\)
\(860\) 0 0
\(861\) 246.833 + 1399.86i 0.00977010 + 0.0554090i
\(862\) 0 0
\(863\) 663.275 1148.83i 0.0261624 0.0453146i −0.852648 0.522486i \(-0.825005\pi\)
0.878810 + 0.477172i \(0.158338\pi\)
\(864\) 0 0
\(865\) −30112.1 10959.9i −1.18363 0.430807i
\(866\) 0 0
\(867\) −19595.8 33940.9i −0.767597 1.32952i
\(868\) 0 0
\(869\) −22620.7 18981.0i −0.883031 0.740951i
\(870\) 0 0
\(871\) −7643.94 + 43350.9i −0.297365 + 1.68644i
\(872\) 0 0
\(873\) −38805.9 −1.50444
\(874\) 0 0
\(875\) −808.525 −0.0312379
\(876\) 0 0
\(877\) 6156.67 34916.2i 0.237054 1.34440i −0.601192 0.799105i \(-0.705307\pi\)
0.838245 0.545293i \(-0.183582\pi\)
\(878\) 0 0
\(879\) −4630.64 3885.57i −0.177688 0.149098i
\(880\) 0 0
\(881\) −16247.0 28140.6i −0.621311 1.07614i −0.989242 0.146289i \(-0.953267\pi\)
0.367930 0.929853i \(-0.380066\pi\)
\(882\) 0 0
\(883\) −23145.7 8424.36i −0.882125 0.321067i −0.139058 0.990284i \(-0.544407\pi\)
−0.743067 + 0.669217i \(0.766630\pi\)
\(884\) 0 0
\(885\) 19266.0 33369.7i 0.731773 1.26747i
\(886\) 0 0
\(887\) −6915.72 39221.0i −0.261789 1.48468i −0.778024 0.628235i \(-0.783778\pi\)
0.516235 0.856447i \(-0.327333\pi\)
\(888\) 0 0
\(889\) 367.245 308.155i 0.0138549 0.0116256i
\(890\) 0 0
\(891\) 6185.38 2251.29i 0.232568 0.0846477i
\(892\) 0 0
\(893\) −6334.01 19363.1i −0.237357 0.725599i
\(894\) 0 0
\(895\) 19116.5 6957.82i 0.713958 0.259860i
\(896\) 0 0
\(897\) −31273.7 + 26241.8i −1.16410 + 0.976798i
\(898\) 0 0
\(899\) 3670.06 + 20814.0i 0.136155 + 0.772174i
\(900\) 0 0
\(901\) 23330.7 40410.0i 0.862663 1.49418i
\(902\) 0 0
\(903\) −1165.88 424.345i −0.0429656 0.0156382i
\(904\) 0 0
\(905\) 13664.6 + 23667.8i 0.501908 + 0.869331i
\(906\) 0 0
\(907\) 31913.9 + 26778.9i 1.16834 + 0.980352i 0.999986 0.00537469i \(-0.00171083\pi\)
0.168353 + 0.985727i \(0.446155\pi\)
\(908\) 0 0
\(909\) 11279.5 63969.4i 0.411571 2.33414i
\(910\) 0 0
\(911\) −29821.5 −1.08455 −0.542277 0.840200i \(-0.682438\pi\)
−0.542277 + 0.840200i \(0.682438\pi\)
\(912\) 0 0
\(913\) −34274.4 −1.24241
\(914\) 0 0
\(915\) −6871.94 + 38972.7i −0.248283 + 1.40809i
\(916\) 0 0
\(917\) 368.915 + 309.556i 0.0132853 + 0.0111477i
\(918\) 0 0
\(919\) 8810.07 + 15259.5i 0.316232 + 0.547730i 0.979699 0.200476i \(-0.0642488\pi\)
−0.663467 + 0.748206i \(0.730916\pi\)
\(920\) 0 0
\(921\) −19232.1 6999.93i −0.688079 0.250440i
\(922\) 0 0
\(923\) 10837.2 18770.6i 0.386470 0.669385i
\(924\) 0 0
\(925\) −990.174 5615.56i −0.0351965 0.199609i
\(926\) 0 0
\(927\) 26055.4 21863.1i 0.923163 0.774625i
\(928\) 0 0
\(929\) 30747.5 11191.2i 1.08589 0.395232i 0.263793 0.964579i \(-0.415026\pi\)
0.822097 + 0.569348i \(0.192804\pi\)
\(930\) 0 0
\(931\) −28099.3 + 4006.07i −0.989171 + 0.141024i
\(932\) 0 0
\(933\) −53582.4 + 19502.4i −1.88018 + 0.684330i
\(934\) 0 0
\(935\) 24390.2 20465.8i 0.853098 0.715834i
\(936\) 0 0
\(937\) −6534.42 37058.6i −0.227823 1.29205i −0.857214 0.514960i \(-0.827807\pi\)
0.629391 0.777089i \(-0.283305\pi\)
\(938\) 0 0
\(939\) −19584.6 + 33921.5i −0.680638 + 1.17890i
\(940\) 0 0
\(941\) −11394.5 4147.26i −0.394740 0.143674i 0.137021 0.990568i \(-0.456247\pi\)
−0.531761 + 0.846894i \(0.678470\pi\)
\(942\) 0 0
\(943\) 17263.4 + 29901.0i 0.596153 + 1.03257i
\(944\) 0 0
\(945\) 400.338 + 335.924i 0.0137809 + 0.0115636i
\(946\) 0 0
\(947\) 6684.28 37908.4i 0.229366 1.30080i −0.624794 0.780790i \(-0.714817\pi\)
0.854160 0.520011i \(-0.174072\pi\)
\(948\) 0 0
\(949\) 21882.2 0.748500
\(950\) 0 0
\(951\) −22665.2 −0.772839
\(952\) 0 0
\(953\) 2043.83 11591.1i 0.0694713 0.393992i −0.930168 0.367134i \(-0.880339\pi\)
0.999639 0.0268572i \(-0.00854995\pi\)
\(954\) 0 0
\(955\) −268.312 225.141i −0.00909149 0.00762867i
\(956\) 0 0
\(957\) 13132.2 + 22745.7i 0.443578 + 0.768300i
\(958\) 0 0
\(959\) −613.762 223.391i −0.0206667 0.00752208i
\(960\) 0 0
\(961\) −14612.2 + 25309.0i −0.490489 + 0.849552i
\(962\) 0 0
\(963\) −13180.1 74748.0i −0.441041 2.50127i
\(964\) 0 0
\(965\) −13192.0 + 11069.4i −0.440067 + 0.369260i
\(966\) 0 0
\(967\) 32805.5 11940.2i 1.09095 0.397075i 0.266980 0.963702i \(-0.413974\pi\)
0.823974 + 0.566627i \(0.191752\pi\)
\(968\) 0 0
\(969\) −44717.7 + 49863.0i −1.48250 + 1.65307i
\(970\) 0 0
\(971\) −25214.5 + 9177.33i −0.833339 + 0.303311i −0.723229 0.690609i \(-0.757343\pi\)
−0.110111 + 0.993919i \(0.535120\pi\)
\(972\) 0 0
\(973\) 543.120 455.732i 0.0178948 0.0150155i
\(974\) 0 0
\(975\) 3160.33 + 17923.1i 0.103807 + 0.588718i
\(976\) 0 0
\(977\) −19774.5 + 34250.4i −0.647536 + 1.12156i 0.336174 + 0.941800i \(0.390867\pi\)
−0.983710 + 0.179765i \(0.942466\pi\)
\(978\) 0 0
\(979\) 38348.0 + 13957.5i 1.25190 + 0.455653i
\(980\) 0 0
\(981\) 26654.3 + 46166.6i 0.867489 + 1.50253i
\(982\) 0 0
\(983\) 15096.8 + 12667.8i 0.489842 + 0.411026i 0.853970 0.520323i \(-0.174188\pi\)
−0.364128 + 0.931349i \(0.618633\pi\)
\(984\) 0 0
\(985\) −3908.82 + 22168.0i −0.126442 + 0.717089i
\(986\) 0 0
\(987\) 1076.39 0.0347133
\(988\) 0 0
\(989\) −30136.2 −0.968935
\(990\) 0 0
\(991\) 1956.42 11095.4i 0.0627121 0.355658i −0.937263 0.348623i \(-0.886649\pi\)
0.999975 0.00703523i \(-0.00223940\pi\)
\(992\) 0 0
\(993\) 8626.57 + 7238.56i 0.275686 + 0.231328i
\(994\) 0 0
\(995\) −23338.8 40424.0i −0.743608 1.28797i
\(996\) 0 0
\(997\) 27936.4 + 10168.0i 0.887418 + 0.322994i 0.745199 0.666842i \(-0.232354\pi\)
0.142218 + 0.989835i \(0.454577\pi\)
\(998\) 0 0
\(999\) −6704.57 + 11612.7i −0.212336 + 0.367776i
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 76.4.i.a.5.5 30
19.2 odd 18 1444.4.a.k.1.2 15
19.4 even 9 inner 76.4.i.a.61.5 yes 30
19.17 even 9 1444.4.a.j.1.14 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.4.i.a.5.5 30 1.1 even 1 trivial
76.4.i.a.61.5 yes 30 19.4 even 9 inner
1444.4.a.j.1.14 15 19.17 even 9
1444.4.a.k.1.2 15 19.2 odd 18