Properties

Label 547.3.b.b.546.1
Level $547$
Weight $3$
Character 547.546
Analytic conductor $14.905$
Analytic rank $0$
Dimension $88$
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] = [547,3,Mod(546,547)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(547, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("547.546");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 547 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 547.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: \(14.9046704605\)
Analytic rank: \(0\)
Dimension: \(88\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 546.1
Character \(\chi\) \(=\) 547.546
Dual form 547.3.b.b.546.88

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.90365i q^{2} +5.00487i q^{3} -11.2385 q^{4} -3.32024i q^{5} +19.5373 q^{6} +0.969581i q^{7} +28.2567i q^{8} -16.0487 q^{9} +O(q^{10})\) \(q-3.90365i q^{2} +5.00487i q^{3} -11.2385 q^{4} -3.32024i q^{5} +19.5373 q^{6} +0.969581i q^{7} +28.2567i q^{8} -16.0487 q^{9} -12.9611 q^{10} +6.84519 q^{11} -56.2473i q^{12} +4.85135 q^{13} +3.78491 q^{14} +16.6174 q^{15} +65.3502 q^{16} -0.703267i q^{17} +62.6485i q^{18} -27.4439 q^{19} +37.3146i q^{20} -4.85262 q^{21} -26.7213i q^{22} -33.5843i q^{23} -141.421 q^{24} +13.9760 q^{25} -18.9380i q^{26} -35.2777i q^{27} -10.8967i q^{28} +5.47585 q^{29} -64.8685i q^{30} -43.4294i q^{31} -142.078i q^{32} +34.2593i q^{33} -2.74531 q^{34} +3.21924 q^{35} +180.363 q^{36} -33.9385i q^{37} +107.132i q^{38} +24.2804i q^{39} +93.8190 q^{40} -29.4683i q^{41} +18.9430i q^{42} -34.3798i q^{43} -76.9298 q^{44} +53.2855i q^{45} -131.101 q^{46} -78.9339 q^{47} +327.069i q^{48} +48.0599 q^{49} -54.5574i q^{50} +3.51976 q^{51} -54.5220 q^{52} -79.8474 q^{53} -137.712 q^{54} -22.7277i q^{55} -27.3971 q^{56} -137.353i q^{57} -21.3758i q^{58} +33.1151i q^{59} -186.755 q^{60} -15.9587i q^{61} -169.533 q^{62} -15.5605i q^{63} -293.223 q^{64} -16.1077i q^{65} +133.736 q^{66} +49.9969 q^{67} +7.90368i q^{68} +168.085 q^{69} -12.5668i q^{70} +80.1855i q^{71} -453.482i q^{72} -25.7088 q^{73} -132.484 q^{74} +69.9480i q^{75} +308.429 q^{76} +6.63697i q^{77} +94.7822 q^{78} -92.2999i q^{79} -216.979i q^{80} +32.1221 q^{81} -115.034 q^{82} -64.5112i q^{83} +54.5363 q^{84} -2.33502 q^{85} -134.207 q^{86} +27.4059i q^{87} +193.422i q^{88} -138.030i q^{89} +208.008 q^{90} +4.70378i q^{91} +377.437i q^{92} +217.358 q^{93} +308.131i q^{94} +91.1205i q^{95} +711.081 q^{96} +164.954 q^{97} -187.609i q^{98} -109.856 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 88 q - 192 q^{4} - 306 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 88 q - 192 q^{4} - 306 q^{9} - 4 q^{10} - 32 q^{11} + 26 q^{13} - 26 q^{14} + 22 q^{15} + 236 q^{16} - 12 q^{19} - 16 q^{21} - 2 q^{24} - 544 q^{25} - 96 q^{29} + 26 q^{34} + 10 q^{35} + 364 q^{36} + 44 q^{40} + 124 q^{44} - 288 q^{46} - 310 q^{47} - 694 q^{49} + 86 q^{51} - 316 q^{52} + 24 q^{53} - 266 q^{54} + 158 q^{56} - 80 q^{60} + 40 q^{62} - 652 q^{64} + 528 q^{66} + 28 q^{67} + 16 q^{69} + 94 q^{73} - 614 q^{74} - 28 q^{76} - 98 q^{78} + 928 q^{81} - 772 q^{82} + 358 q^{84} + 74 q^{85} - 410 q^{86} - 214 q^{90} + 656 q^{93} - 724 q^{96} + 346 q^{97} + 50 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/547\mathbb{Z}\right)^\times\).

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

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.90365i 1.95183i −0.218157 0.975914i \(-0.570005\pi\)
0.218157 0.975914i \(-0.429995\pi\)
\(3\) 5.00487i 1.66829i 0.551546 + 0.834144i \(0.314038\pi\)
−0.551546 + 0.834144i \(0.685962\pi\)
\(4\) −11.2385 −2.80963
\(5\) 3.32024i 0.664049i −0.943271 0.332024i \(-0.892268\pi\)
0.943271 0.332024i \(-0.107732\pi\)
\(6\) 19.5373 3.25621
\(7\) 0.969581i 0.138512i 0.997599 + 0.0692558i \(0.0220625\pi\)
−0.997599 + 0.0692558i \(0.977938\pi\)
\(8\) 28.2567i 3.53208i
\(9\) −16.0487 −1.78319
\(10\) −12.9611 −1.29611
\(11\) 6.84519 0.622290 0.311145 0.950362i \(-0.399287\pi\)
0.311145 + 0.950362i \(0.399287\pi\)
\(12\) 56.2473i 4.68727i
\(13\) 4.85135 0.373181 0.186591 0.982438i \(-0.440256\pi\)
0.186591 + 0.982438i \(0.440256\pi\)
\(14\) 3.78491 0.270351
\(15\) 16.6174 1.10782
\(16\) 65.3502 4.08439
\(17\) 0.703267i 0.0413686i −0.999786 0.0206843i \(-0.993416\pi\)
0.999786 0.0206843i \(-0.00658449\pi\)
\(18\) 62.6485i 3.48047i
\(19\) −27.4439 −1.44442 −0.722209 0.691675i \(-0.756873\pi\)
−0.722209 + 0.691675i \(0.756873\pi\)
\(20\) 37.3146i 1.86573i
\(21\) −4.85262 −0.231077
\(22\) 26.7213i 1.21460i
\(23\) 33.5843i 1.46019i −0.683348 0.730093i \(-0.739477\pi\)
0.683348 0.730093i \(-0.260523\pi\)
\(24\) −141.421 −5.89254
\(25\) 13.9760 0.559040
\(26\) 18.9380i 0.728385i
\(27\) 35.2777i 1.30658i
\(28\) 10.8967i 0.389166i
\(29\) 5.47585 0.188822 0.0944112 0.995533i \(-0.469903\pi\)
0.0944112 + 0.995533i \(0.469903\pi\)
\(30\) 64.8685i 2.16228i
\(31\) 43.4294i 1.40095i −0.713678 0.700474i \(-0.752972\pi\)
0.713678 0.700474i \(-0.247028\pi\)
\(32\) 142.078i 4.43994i
\(33\) 34.2593i 1.03816i
\(34\) −2.74531 −0.0807444
\(35\) 3.21924 0.0919784
\(36\) 180.363 5.01010
\(37\) 33.9385i 0.917256i −0.888628 0.458628i \(-0.848341\pi\)
0.888628 0.458628i \(-0.151659\pi\)
\(38\) 107.132i 2.81925i
\(39\) 24.2804i 0.622574i
\(40\) 93.8190 2.34548
\(41\) 29.4683i 0.718738i −0.933196 0.359369i \(-0.882992\pi\)
0.933196 0.359369i \(-0.117008\pi\)
\(42\) 18.9430i 0.451023i
\(43\) 34.3798i 0.799531i −0.916617 0.399765i \(-0.869092\pi\)
0.916617 0.399765i \(-0.130908\pi\)
\(44\) −76.9298 −1.74841
\(45\) 53.2855i 1.18412i
\(46\) −131.101 −2.85003
\(47\) −78.9339 −1.67944 −0.839722 0.543016i \(-0.817282\pi\)
−0.839722 + 0.543016i \(0.817282\pi\)
\(48\) 327.069i 6.81394i
\(49\) 48.0599 0.980815
\(50\) 54.5574i 1.09115i
\(51\) 3.51976 0.0690148
\(52\) −54.5220 −1.04850
\(53\) −79.8474 −1.50656 −0.753278 0.657703i \(-0.771528\pi\)
−0.753278 + 0.657703i \(0.771528\pi\)
\(54\) −137.712 −2.55022
\(55\) 22.7277i 0.413231i
\(56\) −27.3971 −0.489235
\(57\) 137.353i 2.40971i
\(58\) 21.3758i 0.368549i
\(59\) 33.1151i 0.561272i 0.959814 + 0.280636i \(0.0905454\pi\)
−0.959814 + 0.280636i \(0.909455\pi\)
\(60\) −186.755 −3.11258
\(61\) 15.9587i 0.261617i −0.991408 0.130809i \(-0.958243\pi\)
0.991408 0.130809i \(-0.0417574\pi\)
\(62\) −169.533 −2.73441
\(63\) 15.5605i 0.246992i
\(64\) −293.223 −4.58160
\(65\) 16.1077i 0.247810i
\(66\) 133.736 2.02631
\(67\) 49.9969 0.746222 0.373111 0.927787i \(-0.378291\pi\)
0.373111 + 0.927787i \(0.378291\pi\)
\(68\) 7.90368i 0.116231i
\(69\) 168.085 2.43601
\(70\) 12.5668i 0.179526i
\(71\) 80.1855i 1.12937i 0.825305 + 0.564687i \(0.191003\pi\)
−0.825305 + 0.564687i \(0.808997\pi\)
\(72\) 453.482i 6.29837i
\(73\) −25.7088 −0.352175 −0.176087 0.984375i \(-0.556344\pi\)
−0.176087 + 0.984375i \(0.556344\pi\)
\(74\) −132.484 −1.79032
\(75\) 69.9480i 0.932639i
\(76\) 308.429 4.05828
\(77\) 6.63697i 0.0861944i
\(78\) 94.7822 1.21516
\(79\) 92.2999i 1.16835i −0.811627 0.584177i \(-0.801418\pi\)
0.811627 0.584177i \(-0.198582\pi\)
\(80\) 216.979i 2.71223i
\(81\) 32.1221 0.396570
\(82\) −115.034 −1.40285
\(83\) 64.5112i 0.777243i −0.921398 0.388621i \(-0.872951\pi\)
0.921398 0.388621i \(-0.127049\pi\)
\(84\) 54.5363 0.649242
\(85\) −2.33502 −0.0274708
\(86\) −134.207 −1.56055
\(87\) 27.4059i 0.315010i
\(88\) 193.422i 2.19798i
\(89\) 138.030i 1.55090i −0.631412 0.775448i \(-0.717524\pi\)
0.631412 0.775448i \(-0.282476\pi\)
\(90\) 208.008 2.31120
\(91\) 4.70378i 0.0516899i
\(92\) 377.437i 4.10258i
\(93\) 217.358 2.33718
\(94\) 308.131i 3.27799i
\(95\) 91.1205i 0.959163i
\(96\) 711.081 7.40710
\(97\) 164.954 1.70056 0.850280 0.526330i \(-0.176432\pi\)
0.850280 + 0.526330i \(0.176432\pi\)
\(98\) 187.609i 1.91438i
\(99\) −109.856 −1.10966
\(100\) −157.069 −1.57069
\(101\) 96.2354i 0.952826i −0.879222 0.476413i \(-0.841937\pi\)
0.879222 0.476413i \(-0.158063\pi\)
\(102\) 13.7399i 0.134705i
\(103\) 98.8378i 0.959590i 0.877381 + 0.479795i \(0.159289\pi\)
−0.877381 + 0.479795i \(0.840711\pi\)
\(104\) 137.083i 1.31811i
\(105\) 16.1119i 0.153447i
\(106\) 311.697i 2.94054i
\(107\) 57.3593i 0.536068i 0.963409 + 0.268034i \(0.0863740\pi\)
−0.963409 + 0.268034i \(0.913626\pi\)
\(108\) 396.469i 3.67101i
\(109\) 74.7904i 0.686150i −0.939308 0.343075i \(-0.888531\pi\)
0.939308 0.343075i \(-0.111469\pi\)
\(110\) −88.7211 −0.806556
\(111\) 169.857 1.53025
\(112\) 63.3623i 0.565735i
\(113\) 165.519 1.46477 0.732386 0.680890i \(-0.238407\pi\)
0.732386 + 0.680890i \(0.238407\pi\)
\(114\) −536.180 −4.70333
\(115\) −111.508 −0.969634
\(116\) −61.5404 −0.530521
\(117\) −77.8579 −0.665452
\(118\) 129.270 1.09551
\(119\) 0.681874 0.00573003
\(120\) 469.552i 3.91293i
\(121\) −74.1433 −0.612755
\(122\) −62.2971 −0.510632
\(123\) 147.485 1.19906
\(124\) 488.082i 3.93614i
\(125\) 129.410i 1.03528i
\(126\) −60.7428 −0.482086
\(127\) 84.3990 0.664559 0.332279 0.943181i \(-0.392182\pi\)
0.332279 + 0.943181i \(0.392182\pi\)
\(128\) 576.327i 4.50256i
\(129\) 172.066 1.33385
\(130\) −62.8788 −0.483683
\(131\) −59.8237 −0.456669 −0.228335 0.973583i \(-0.573328\pi\)
−0.228335 + 0.973583i \(0.573328\pi\)
\(132\) 385.024i 2.91685i
\(133\) 26.6091i 0.200069i
\(134\) 195.170i 1.45650i
\(135\) −117.131 −0.867634
\(136\) 19.8720 0.146117
\(137\) 161.651 1.17993 0.589965 0.807428i \(-0.299141\pi\)
0.589965 + 0.807428i \(0.299141\pi\)
\(138\) 656.145i 4.75467i
\(139\) 226.854 1.63204 0.816021 0.578022i \(-0.196175\pi\)
0.816021 + 0.578022i \(0.196175\pi\)
\(140\) −36.1795 −0.258425
\(141\) 395.054i 2.80180i
\(142\) 313.017 2.20434
\(143\) 33.2085 0.232227
\(144\) −1048.79 −7.28323
\(145\) 18.1812i 0.125387i
\(146\) 100.358i 0.687385i
\(147\) 240.533i 1.63628i
\(148\) 381.418i 2.57715i
\(149\) 120.427 0.808232 0.404116 0.914708i \(-0.367579\pi\)
0.404116 + 0.914708i \(0.367579\pi\)
\(150\) 273.053 1.82035
\(151\) 185.662i 1.22955i −0.788701 0.614776i \(-0.789246\pi\)
0.788701 0.614776i \(-0.210754\pi\)
\(152\) 775.474i 5.10181i
\(153\) 11.2865i 0.0737680i
\(154\) 25.9084 0.168237
\(155\) −144.196 −0.930297
\(156\) 272.876i 1.74920i
\(157\) −51.3979 −0.327375 −0.163688 0.986512i \(-0.552339\pi\)
−0.163688 + 0.986512i \(0.552339\pi\)
\(158\) −360.307 −2.28042
\(159\) 399.626i 2.51337i
\(160\) −471.733 −2.94833
\(161\) 32.5627 0.202253
\(162\) 125.394i 0.774035i
\(163\) 289.443i 1.77572i 0.460109 + 0.887862i \(0.347810\pi\)
−0.460109 + 0.887862i \(0.652190\pi\)
\(164\) 331.180i 2.01939i
\(165\) 113.749 0.689389
\(166\) −251.829 −1.51704
\(167\) −280.016 −1.67674 −0.838372 0.545098i \(-0.816492\pi\)
−0.838372 + 0.545098i \(0.816492\pi\)
\(168\) 137.119i 0.816185i
\(169\) −145.464 −0.860736
\(170\) 9.11510i 0.0536182i
\(171\) 440.439 2.57567
\(172\) 386.378i 2.24639i
\(173\) 15.8385i 0.0915523i −0.998952 0.0457761i \(-0.985424\pi\)
0.998952 0.0457761i \(-0.0145761\pi\)
\(174\) 106.983 0.614846
\(175\) 13.5509i 0.0774334i
\(176\) 447.335 2.54168
\(177\) −165.736 −0.936364
\(178\) −538.820 −3.02708
\(179\) −177.343 −0.990742 −0.495371 0.868681i \(-0.664968\pi\)
−0.495371 + 0.868681i \(0.664968\pi\)
\(180\) 598.850i 3.32695i
\(181\) −146.264 −0.808087 −0.404044 0.914740i \(-0.632396\pi\)
−0.404044 + 0.914740i \(0.632396\pi\)
\(182\) 18.3619 0.100890
\(183\) 79.8710 0.436453
\(184\) 948.980 5.15750
\(185\) −112.684 −0.609102
\(186\) 848.491i 4.56178i
\(187\) 4.81400i 0.0257433i
\(188\) 887.100 4.71862
\(189\) 34.2046 0.180977
\(190\) 355.703 1.87212
\(191\) 85.2771 0.446477 0.223238 0.974764i \(-0.428337\pi\)
0.223238 + 0.974764i \(0.428337\pi\)
\(192\) 1467.54i 7.64343i
\(193\) −216.985 −1.12427 −0.562137 0.827044i \(-0.690021\pi\)
−0.562137 + 0.827044i \(0.690021\pi\)
\(194\) 643.925i 3.31920i
\(195\) 80.6168 0.413419
\(196\) −540.122 −2.75573
\(197\) 230.341i 1.16924i 0.811306 + 0.584622i \(0.198757\pi\)
−0.811306 + 0.584622i \(0.801243\pi\)
\(198\) 428.841i 2.16587i
\(199\) −105.453 −0.529915 −0.264958 0.964260i \(-0.585358\pi\)
−0.264958 + 0.964260i \(0.585358\pi\)
\(200\) 394.915i 1.97457i
\(201\) 250.228i 1.24491i
\(202\) −375.670 −1.85975
\(203\) 5.30928i 0.0261541i
\(204\) −39.5568 −0.193906
\(205\) −97.8418 −0.477277
\(206\) 385.829 1.87295
\(207\) 538.983i 2.60378i
\(208\) 317.037 1.52422
\(209\) −187.859 −0.898847
\(210\) 62.8952 0.299501
\(211\) 387.259i 1.83535i 0.397334 + 0.917674i \(0.369936\pi\)
−0.397334 + 0.917674i \(0.630064\pi\)
\(212\) 897.367 4.23286
\(213\) −401.318 −1.88412
\(214\) 223.911 1.04631
\(215\) −114.149 −0.530927
\(216\) 996.831 4.61496
\(217\) 42.1083 0.194047
\(218\) −291.956 −1.33925
\(219\) 128.669i 0.587530i
\(220\) 255.426i 1.16103i
\(221\) 3.41180i 0.0154380i
\(222\) 663.065i 2.98678i
\(223\) 11.6351i 0.0521752i −0.999660 0.0260876i \(-0.991695\pi\)
0.999660 0.0260876i \(-0.00830488\pi\)
\(224\) 137.756 0.614983
\(225\) −224.296 −0.996872
\(226\) 646.130i 2.85898i
\(227\) 109.614 0.482881 0.241440 0.970416i \(-0.422380\pi\)
0.241440 + 0.970416i \(0.422380\pi\)
\(228\) 1543.65i 6.77038i
\(229\) 261.170i 1.14048i −0.821478 0.570240i \(-0.806850\pi\)
0.821478 0.570240i \(-0.193150\pi\)
\(230\) 435.288i 1.89256i
\(231\) −33.2171 −0.143797
\(232\) 154.729i 0.666937i
\(233\) 185.781 0.797343 0.398671 0.917094i \(-0.369471\pi\)
0.398671 + 0.917094i \(0.369471\pi\)
\(234\) 303.930i 1.29885i
\(235\) 262.080i 1.11523i
\(236\) 372.164i 1.57697i
\(237\) 461.949 1.94915
\(238\) 2.66180i 0.0111840i
\(239\) −335.028 −1.40179 −0.700896 0.713264i \(-0.747216\pi\)
−0.700896 + 0.713264i \(0.747216\pi\)
\(240\) 1085.95 4.52479
\(241\) 352.978i 1.46464i 0.680962 + 0.732319i \(0.261562\pi\)
−0.680962 + 0.732319i \(0.738438\pi\)
\(242\) 289.430i 1.19599i
\(243\) 156.733i 0.644990i
\(244\) 179.352i 0.735048i
\(245\) 159.571i 0.651308i
\(246\) 575.729i 2.34036i
\(247\) −133.140 −0.539029
\(248\) 1227.17 4.94826
\(249\) 322.870 1.29667
\(250\) −505.171 −2.02068
\(251\) 156.474i 0.623403i −0.950180 0.311702i \(-0.899101\pi\)
0.950180 0.311702i \(-0.100899\pi\)
\(252\) 174.877i 0.693956i
\(253\) 229.891i 0.908659i
\(254\) 329.464i 1.29710i
\(255\) 11.6864i 0.0458292i
\(256\) 1076.89 4.20661
\(257\) 24.9264i 0.0969898i −0.998823 0.0484949i \(-0.984558\pi\)
0.998823 0.0484949i \(-0.0154425\pi\)
\(258\) 671.688i 2.60344i
\(259\) 32.9061 0.127051
\(260\) 181.026i 0.696255i
\(261\) −87.8802 −0.336706
\(262\) 233.531i 0.891339i
\(263\) 21.0838 0.0801666 0.0400833 0.999196i \(-0.487238\pi\)
0.0400833 + 0.999196i \(0.487238\pi\)
\(264\) −968.053 −3.66687
\(265\) 265.113i 1.00043i
\(266\) −103.873 −0.390499
\(267\) 690.820 2.58734
\(268\) −561.891 −2.09661
\(269\) −356.206 −1.32419 −0.662094 0.749421i \(-0.730332\pi\)
−0.662094 + 0.749421i \(0.730332\pi\)
\(270\) 457.237i 1.69347i
\(271\) 191.625i 0.707103i 0.935415 + 0.353552i \(0.115026\pi\)
−0.935415 + 0.353552i \(0.884974\pi\)
\(272\) 45.9586i 0.168966i
\(273\) −23.5418 −0.0862337
\(274\) 631.028i 2.30302i
\(275\) 95.6684 0.347885
\(276\) −1889.02 −6.84429
\(277\) −193.785 −0.699585 −0.349793 0.936827i \(-0.613748\pi\)
−0.349793 + 0.936827i \(0.613748\pi\)
\(278\) 885.559i 3.18546i
\(279\) 696.984i 2.49815i
\(280\) 90.9651i 0.324876i
\(281\) 72.5495i 0.258183i −0.991633 0.129092i \(-0.958794\pi\)
0.991633 0.129092i \(-0.0412061\pi\)
\(282\) −1542.15 −5.46863
\(283\) 405.318i 1.43222i −0.697987 0.716110i \(-0.745921\pi\)
0.697987 0.716110i \(-0.254079\pi\)
\(284\) 901.167i 3.17312i
\(285\) −456.046 −1.60016
\(286\) 129.634i 0.453267i
\(287\) 28.5719 0.0995535
\(288\) 2280.17i 7.91724i
\(289\) 288.505 0.998289
\(290\) −70.9729 −0.244734
\(291\) 825.575i 2.83703i
\(292\) 288.929 0.989481
\(293\) −223.354 −0.762302 −0.381151 0.924513i \(-0.624472\pi\)
−0.381151 + 0.924513i \(0.624472\pi\)
\(294\) 938.959 3.19374
\(295\) 109.950 0.372712
\(296\) 958.988 3.23982
\(297\) 241.483i 0.813074i
\(298\) 470.104i 1.57753i
\(299\) 162.929i 0.544914i
\(300\) 786.111i 2.62037i
\(301\) 33.3340 0.110744
\(302\) −724.762 −2.39987
\(303\) 481.645 1.58959
\(304\) −1793.47 −5.89956
\(305\) −52.9866 −0.173727
\(306\) 44.0586 0.143982
\(307\) 186.097i 0.606178i 0.952962 + 0.303089i \(0.0980180\pi\)
−0.952962 + 0.303089i \(0.901982\pi\)
\(308\) 74.5897i 0.242174i
\(309\) −494.670 −1.60087
\(310\) 562.891i 1.81578i
\(311\) 109.411 0.351803 0.175902 0.984408i \(-0.443716\pi\)
0.175902 + 0.984408i \(0.443716\pi\)
\(312\) −686.083 −2.19898
\(313\) 484.583 1.54819 0.774095 0.633070i \(-0.218205\pi\)
0.774095 + 0.633070i \(0.218205\pi\)
\(314\) 200.640i 0.638980i
\(315\) −51.6646 −0.164015
\(316\) 1037.31i 3.28264i
\(317\) −225.140 −0.710221 −0.355111 0.934824i \(-0.615557\pi\)
−0.355111 + 0.934824i \(0.615557\pi\)
\(318\) −1560.00 −4.90566
\(319\) 37.4833 0.117502
\(320\) 973.570i 3.04241i
\(321\) −287.076 −0.894316
\(322\) 127.113i 0.394762i
\(323\) 19.3004i 0.0597536i
\(324\) −361.005 −1.11421
\(325\) 67.8025 0.208623
\(326\) 1129.89 3.46591
\(327\) 374.316 1.14470
\(328\) 832.675 2.53864
\(329\) 76.5328i 0.232623i
\(330\) 444.037i 1.34557i
\(331\) 551.970i 1.66758i 0.552080 + 0.833791i \(0.313834\pi\)
−0.552080 + 0.833791i \(0.686166\pi\)
\(332\) 725.010i 2.18376i
\(333\) 544.668i 1.63564i
\(334\) 1093.09i 3.27272i
\(335\) 166.002i 0.495527i
\(336\) −317.120 −0.943810
\(337\) 474.094i 1.40681i −0.710790 0.703404i \(-0.751663\pi\)
0.710790 0.703404i \(-0.248337\pi\)
\(338\) 567.843i 1.68001i
\(339\) 828.401i 2.44366i
\(340\) 26.2421 0.0771827
\(341\) 297.282i 0.871796i
\(342\) 1719.32i 5.02726i
\(343\) 94.1075i 0.274366i
\(344\) 971.460 2.82401
\(345\) 558.082i 1.61763i
\(346\) −61.8282 −0.178694
\(347\) −207.655 −0.598430 −0.299215 0.954186i \(-0.596725\pi\)
−0.299215 + 0.954186i \(0.596725\pi\)
\(348\) 308.002i 0.885062i
\(349\) −448.334 −1.28462 −0.642312 0.766443i \(-0.722025\pi\)
−0.642312 + 0.766443i \(0.722025\pi\)
\(350\) 52.8978 0.151137
\(351\) 171.145i 0.487592i
\(352\) 972.552i 2.76293i
\(353\) 129.045 0.365567 0.182783 0.983153i \(-0.441489\pi\)
0.182783 + 0.983153i \(0.441489\pi\)
\(354\) 646.978i 1.82762i
\(355\) 266.235 0.749959
\(356\) 1551.25i 4.35744i
\(357\) 3.41269i 0.00955935i
\(358\) 692.285i 1.93376i
\(359\) 286.030i 0.796742i −0.917224 0.398371i \(-0.869576\pi\)
0.917224 0.398371i \(-0.130424\pi\)
\(360\) −1505.67 −4.18242
\(361\) 392.170 1.08634
\(362\) 570.963i 1.57725i
\(363\) 371.077i 1.02225i
\(364\) 52.8635i 0.145230i
\(365\) 85.3594i 0.233861i
\(366\) 311.789i 0.851882i
\(367\) −153.171 −0.417359 −0.208680 0.977984i \(-0.566917\pi\)
−0.208680 + 0.977984i \(0.566917\pi\)
\(368\) 2194.74i 5.96397i
\(369\) 472.927i 1.28164i
\(370\) 439.879i 1.18886i
\(371\) 77.4186i 0.208675i
\(372\) −2442.78 −6.56662
\(373\) 422.859i 1.13367i 0.823832 + 0.566835i \(0.191832\pi\)
−0.823832 + 0.566835i \(0.808168\pi\)
\(374\) −18.7922 −0.0502465
\(375\) 647.678 1.72714
\(376\) 2230.41i 5.93194i
\(377\) 26.5653 0.0704650
\(378\) 133.523i 0.353235i
\(379\) −106.386 −0.280702 −0.140351 0.990102i \(-0.544823\pi\)
−0.140351 + 0.990102i \(0.544823\pi\)
\(380\) 1024.06i 2.69489i
\(381\) 422.406i 1.10868i
\(382\) 332.892i 0.871446i
\(383\) −213.762 −0.558125 −0.279062 0.960273i \(-0.590024\pi\)
−0.279062 + 0.960273i \(0.590024\pi\)
\(384\) −2884.44 −7.51157
\(385\) 22.0364 0.0572373
\(386\) 847.034i 2.19439i
\(387\) 551.751i 1.42571i
\(388\) −1853.84 −4.77795
\(389\) 251.111i 0.645529i −0.946479 0.322764i \(-0.895388\pi\)
0.946479 0.322764i \(-0.104612\pi\)
\(390\) 314.700i 0.806923i
\(391\) −23.6187 −0.0604059
\(392\) 1358.01i 3.46432i
\(393\) 299.409i 0.761856i
\(394\) 899.171 2.28216
\(395\) −306.458 −0.775843
\(396\) 1234.62 3.11773
\(397\) 377.872i 0.951819i −0.879494 0.475910i \(-0.842119\pi\)
0.879494 0.475910i \(-0.157881\pi\)
\(398\) 411.653i 1.03430i
\(399\) 133.175 0.333772
\(400\) 913.334 2.28333
\(401\) −5.96869 −0.0148845 −0.00744225 0.999972i \(-0.502369\pi\)
−0.00744225 + 0.999972i \(0.502369\pi\)
\(402\) 976.802 2.42986
\(403\) 210.691i 0.522807i
\(404\) 1081.54i 2.67709i
\(405\) 106.653i 0.263341i
\(406\) 20.7256 0.0510483
\(407\) 232.315i 0.570799i
\(408\) 99.4566i 0.243766i
\(409\) 362.692 0.886777 0.443389 0.896329i \(-0.353776\pi\)
0.443389 + 0.896329i \(0.353776\pi\)
\(410\) 381.940i 0.931562i
\(411\) 809.039i 1.96847i
\(412\) 1110.79i 2.69609i
\(413\) −32.1077 −0.0777427
\(414\) 2104.00 5.08214
\(415\) −214.193 −0.516127
\(416\) 689.271i 1.65690i
\(417\) 1135.37i 2.72272i
\(418\) 733.337i 1.75439i
\(419\) −249.142 −0.594611 −0.297305 0.954782i \(-0.596088\pi\)
−0.297305 + 0.954782i \(0.596088\pi\)
\(420\) 181.074i 0.431128i
\(421\) 590.667i 1.40301i −0.712664 0.701505i \(-0.752512\pi\)
0.712664 0.701505i \(-0.247488\pi\)
\(422\) 1511.72 3.58228
\(423\) 1266.79 2.99476
\(424\) 2256.22i 5.32128i
\(425\) 9.82885i 0.0231267i
\(426\) 1566.61i 3.67748i
\(427\) 15.4732 0.0362370
\(428\) 644.633i 1.50615i
\(429\) 166.204i 0.387422i
\(430\) 445.600i 1.03628i
\(431\) 588.476i 1.36537i 0.730711 + 0.682687i \(0.239189\pi\)
−0.730711 + 0.682687i \(0.760811\pi\)
\(432\) 2305.41i 5.33659i
\(433\) 704.427i 1.62685i −0.581668 0.813426i \(-0.697600\pi\)
0.581668 0.813426i \(-0.302400\pi\)
\(434\) 164.376i 0.378747i
\(435\) 90.9942 0.209182
\(436\) 840.533i 1.92783i
\(437\) 921.684i 2.10912i
\(438\) −502.279 −1.14676
\(439\) −14.7433 −0.0335839 −0.0167919 0.999859i \(-0.505345\pi\)
−0.0167919 + 0.999859i \(0.505345\pi\)
\(440\) 642.209 1.45957
\(441\) −771.298 −1.74898
\(442\) −13.3185 −0.0301323
\(443\) 153.885 0.347369 0.173685 0.984801i \(-0.444433\pi\)
0.173685 + 0.984801i \(0.444433\pi\)
\(444\) −1908.95 −4.29943
\(445\) −458.292 −1.02987
\(446\) −45.4193 −0.101837
\(447\) 602.719i 1.34836i
\(448\) 284.303i 0.634605i
\(449\) 739.862 1.64780 0.823900 0.566735i \(-0.191794\pi\)
0.823900 + 0.566735i \(0.191794\pi\)
\(450\) 875.575i 1.94572i
\(451\) 201.716i 0.447264i
\(452\) −1860.19 −4.11546
\(453\) 929.216 2.05125
\(454\) 427.895i 0.942499i
\(455\) 15.6177 0.0343246
\(456\) 3881.15 8.51128
\(457\) 631.065i 1.38089i 0.723387 + 0.690443i \(0.242584\pi\)
−0.723387 + 0.690443i \(0.757416\pi\)
\(458\) −1019.52 −2.22602
\(459\) −24.8097 −0.0540515
\(460\) 1253.18 2.72431
\(461\) 231.719i 0.502645i −0.967903 0.251322i \(-0.919135\pi\)
0.967903 0.251322i \(-0.0808654\pi\)
\(462\) 129.668i 0.280667i
\(463\) 453.210i 0.978855i −0.872044 0.489427i \(-0.837206\pi\)
0.872044 0.489427i \(-0.162794\pi\)
\(464\) 357.848 0.771224
\(465\) 721.682i 1.55200i
\(466\) 725.224i 1.55628i
\(467\) −802.518 −1.71845 −0.859227 0.511595i \(-0.829055\pi\)
−0.859227 + 0.511595i \(0.829055\pi\)
\(468\) 875.007 1.86967
\(469\) 48.4760i 0.103360i
\(470\) 1023.07 2.17674
\(471\) 257.240i 0.546157i
\(472\) −935.721 −1.98246
\(473\) 235.337i 0.497540i
\(474\) 1803.29i 3.80440i
\(475\) −383.556 −0.807487
\(476\) −7.66325 −0.0160993
\(477\) 1281.45 2.68647
\(478\) 1307.83i 2.73606i
\(479\) 550.111 1.14846 0.574228 0.818695i \(-0.305302\pi\)
0.574228 + 0.818695i \(0.305302\pi\)
\(480\) 2360.96i 4.91867i
\(481\) 164.648i 0.342303i
\(482\) 1377.90 2.85872
\(483\) 162.972i 0.337416i
\(484\) 833.261 1.72161
\(485\) 547.689i 1.12925i
\(486\) −611.830 −1.25891
\(487\) 400.505i 0.822392i 0.911547 + 0.411196i \(0.134889\pi\)
−0.911547 + 0.411196i \(0.865111\pi\)
\(488\) 450.939 0.924055
\(489\) −1448.62 −2.96242
\(490\) −622.908 −1.27124
\(491\) 233.743i 0.476056i 0.971258 + 0.238028i \(0.0765009\pi\)
−0.971258 + 0.238028i \(0.923499\pi\)
\(492\) −1657.51 −3.36892
\(493\) 3.85098i 0.00781132i
\(494\) 519.734i 1.05209i
\(495\) 364.750i 0.736868i
\(496\) 2838.12i 5.72201i
\(497\) −77.7464 −0.156431
\(498\) 1260.37i 2.53087i
\(499\) −496.829 −0.995648 −0.497824 0.867278i \(-0.665868\pi\)
−0.497824 + 0.867278i \(0.665868\pi\)
\(500\) 1454.37i 2.90875i
\(501\) 1401.44i 2.79729i
\(502\) −610.821 −1.21678
\(503\) 713.303i 1.41810i −0.705160 0.709048i \(-0.749125\pi\)
0.705160 0.709048i \(-0.250875\pi\)
\(504\) 439.688 0.872397
\(505\) −319.525 −0.632723
\(506\) −897.414 −1.77355
\(507\) 728.030i 1.43596i
\(508\) −948.519 −1.86716
\(509\) 364.570 0.716248 0.358124 0.933674i \(-0.383417\pi\)
0.358124 + 0.933674i \(0.383417\pi\)
\(510\) −45.6198 −0.0894507
\(511\) 24.9267i 0.0487803i
\(512\) 1898.51i 3.70803i
\(513\) 968.160i 1.88725i
\(514\) −97.3039 −0.189307
\(515\) 328.165 0.637215
\(516\) −1933.77 −3.74762
\(517\) −540.318 −1.04510
\(518\) 128.454i 0.247981i
\(519\) 79.2698 0.152736
\(520\) 455.149 0.875287
\(521\) 919.984 1.76580 0.882902 0.469557i \(-0.155586\pi\)
0.882902 + 0.469557i \(0.155586\pi\)
\(522\) 343.054i 0.657191i
\(523\) 234.101i 0.447611i −0.974634 0.223806i \(-0.928152\pi\)
0.974634 0.223806i \(-0.0718481\pi\)
\(524\) 672.329 1.28307
\(525\) −67.8202 −0.129181
\(526\) 82.3039i 0.156471i
\(527\) −30.5424 −0.0579553
\(528\) 2238.85i 4.24025i
\(529\) −598.903 −1.13214
\(530\) 1034.91 1.95266
\(531\) 531.453i 1.00085i
\(532\) 299.047i 0.562119i
\(533\) 142.961i 0.268219i
\(534\) 2696.72i 5.05004i
\(535\) 190.447 0.355975
\(536\) 1412.74i 2.63572i
\(537\) 887.577i 1.65284i
\(538\) 1390.51i 2.58459i
\(539\) 328.979 0.610351
\(540\) 1316.37 2.43773
\(541\) 599.817i 1.10872i 0.832277 + 0.554360i \(0.187037\pi\)
−0.832277 + 0.554360i \(0.812963\pi\)
\(542\) 748.038 1.38014
\(543\) 732.031i 1.34812i
\(544\) −99.9187 −0.183674
\(545\) −248.322 −0.455637
\(546\) 91.8990i 0.168313i
\(547\) 530.243 134.356i 0.969365 0.245624i
\(548\) −1816.71 −3.31517
\(549\) 256.116i 0.466513i
\(550\) 373.456i 0.679011i
\(551\) −150.279 −0.272738
\(552\) 4749.52i 8.60420i
\(553\) 89.4922 0.161830
\(554\) 756.470i 1.36547i
\(555\) 563.968i 1.01616i
\(556\) −2549.50 −4.58543
\(557\) −224.433 −0.402932 −0.201466 0.979496i \(-0.564570\pi\)
−0.201466 + 0.979496i \(0.564570\pi\)
\(558\) 2720.79 4.87596
\(559\) 166.789i 0.298370i
\(560\) 210.378 0.375676
\(561\) 24.0934 0.0429473
\(562\) −283.208 −0.503929
\(563\) −467.590 −0.830533 −0.415266 0.909700i \(-0.636312\pi\)
−0.415266 + 0.909700i \(0.636312\pi\)
\(564\) 4439.82i 7.87202i
\(565\) 549.564i 0.972679i
\(566\) −1582.22 −2.79545
\(567\) 31.1450i 0.0549295i
\(568\) −2265.78 −3.98904
\(569\) 218.850i 0.384622i 0.981334 + 0.192311i \(0.0615982\pi\)
−0.981334 + 0.192311i \(0.938402\pi\)
\(570\) 1780.25i 3.12324i
\(571\) 500.613 0.876730 0.438365 0.898797i \(-0.355558\pi\)
0.438365 + 0.898797i \(0.355558\pi\)
\(572\) −373.214 −0.652472
\(573\) 426.800i 0.744852i
\(574\) 111.535i 0.194311i
\(575\) 469.373i 0.816301i
\(576\) 4705.84 8.16985
\(577\) 286.318i 0.496218i 0.968732 + 0.248109i \(0.0798091\pi\)
−0.968732 + 0.248109i \(0.920191\pi\)
\(578\) 1126.23i 1.94849i
\(579\) 1085.98i 1.87561i
\(580\) 204.329i 0.352292i
\(581\) 62.5488 0.107657
\(582\) 3222.76 5.53738
\(583\) −546.571 −0.937515
\(584\) 726.444i 1.24391i
\(585\) 258.507i 0.441892i
\(586\) 871.898i 1.48788i
\(587\) −434.405 −0.740042 −0.370021 0.929023i \(-0.620650\pi\)
−0.370021 + 0.929023i \(0.620650\pi\)
\(588\) 2703.24i 4.59735i
\(589\) 1191.87i 2.02355i
\(590\) 429.207i 0.727469i
\(591\) −1152.83 −1.95063
\(592\) 2217.89i 3.74643i
\(593\) 998.651 1.68407 0.842033 0.539426i \(-0.181359\pi\)
0.842033 + 0.539426i \(0.181359\pi\)
\(594\) −942.666 −1.58698
\(595\) 2.26399i 0.00380502i
\(596\) −1353.42 −2.27083
\(597\) 527.779i 0.884052i
\(598\) −636.019 −1.06358
\(599\) −98.2919 −0.164093 −0.0820467 0.996628i \(-0.526146\pi\)
−0.0820467 + 0.996628i \(0.526146\pi\)
\(600\) −1976.50 −3.29416
\(601\) −1067.75 −1.77661 −0.888307 0.459250i \(-0.848118\pi\)
−0.888307 + 0.459250i \(0.848118\pi\)
\(602\) 130.125i 0.216154i
\(603\) −802.384 −1.33065
\(604\) 2086.57i 3.45459i
\(605\) 246.174i 0.406899i
\(606\) 1880.18i 3.10260i
\(607\) 809.497 1.33360 0.666802 0.745235i \(-0.267663\pi\)
0.666802 + 0.745235i \(0.267663\pi\)
\(608\) 3899.18i 6.41312i
\(609\) −26.5722 −0.0436326
\(610\) 206.842i 0.339084i
\(611\) −382.936 −0.626737
\(612\) 126.844i 0.207261i
\(613\) 174.436 0.284560 0.142280 0.989826i \(-0.454557\pi\)
0.142280 + 0.989826i \(0.454557\pi\)
\(614\) 726.457 1.18316
\(615\) 489.685i 0.796236i
\(616\) −187.539 −0.304446
\(617\) 578.735i 0.937982i −0.883203 0.468991i \(-0.844618\pi\)
0.883203 0.468991i \(-0.155382\pi\)
\(618\) 1931.02i 3.12463i
\(619\) 580.916i 0.938475i 0.883072 + 0.469238i \(0.155471\pi\)
−0.883072 + 0.469238i \(0.844529\pi\)
\(620\) 1620.55 2.61379
\(621\) −1184.78 −1.90785
\(622\) 427.102i 0.686659i
\(623\) 133.831 0.214817
\(624\) 1586.73i 2.54283i
\(625\) −80.2720 −0.128435
\(626\) 1891.65i 3.02180i
\(627\) 940.210i 1.49954i
\(628\) 577.637 0.919804
\(629\) −23.8678 −0.0379456
\(630\) 201.681i 0.320128i
\(631\) −454.380 −0.720096 −0.360048 0.932934i \(-0.617240\pi\)
−0.360048 + 0.932934i \(0.617240\pi\)
\(632\) 2608.09 4.12672
\(633\) −1938.18 −3.06189
\(634\) 878.869i 1.38623i
\(635\) 280.225i 0.441299i
\(636\) 4491.20i 7.06164i
\(637\) 233.156 0.366022
\(638\) 146.322i 0.229344i
\(639\) 1286.87i 2.01388i
\(640\) 1913.55 2.98992
\(641\) 366.675i 0.572035i 0.958224 + 0.286018i \(0.0923316\pi\)
−0.958224 + 0.286018i \(0.907668\pi\)
\(642\) 1120.64i 1.74555i
\(643\) 1101.06 1.71239 0.856193 0.516657i \(-0.172824\pi\)
0.856193 + 0.516657i \(0.172824\pi\)
\(644\) −365.956 −0.568255
\(645\) 571.302i 0.885740i
\(646\) 75.3421 0.116629
\(647\) −471.671 −0.729012 −0.364506 0.931201i \(-0.618762\pi\)
−0.364506 + 0.931201i \(0.618762\pi\)
\(648\) 907.665i 1.40072i
\(649\) 226.679i 0.349274i
\(650\) 264.677i 0.407196i
\(651\) 210.746i 0.323727i
\(652\) 3252.91i 4.98913i
\(653\) 1097.71i 1.68103i 0.541791 + 0.840513i \(0.317746\pi\)
−0.541791 + 0.840513i \(0.682254\pi\)
\(654\) 1461.20i 2.23425i
\(655\) 198.629i 0.303250i
\(656\) 1925.76i 2.93561i
\(657\) 412.592 0.627994
\(658\) −298.758 −0.454039
\(659\) 406.433i 0.616742i −0.951266 0.308371i \(-0.900216\pi\)
0.951266 0.308371i \(-0.0997839\pi\)
\(660\) −1278.37 −1.93693
\(661\) −594.616 −0.899570 −0.449785 0.893137i \(-0.648499\pi\)
−0.449785 + 0.893137i \(0.648499\pi\)
\(662\) 2154.70 3.25483
\(663\) 17.0756 0.0257550
\(664\) 1822.87 2.74529
\(665\) −88.3487 −0.132855
\(666\) 2126.19 3.19248
\(667\) 183.902i 0.275716i
\(668\) 3146.97 4.71103
\(669\) 58.2320 0.0870433
\(670\) −648.013 −0.967184
\(671\) 109.240i 0.162802i
\(672\) 689.451i 1.02597i
\(673\) 664.918 0.987992 0.493996 0.869464i \(-0.335536\pi\)
0.493996 + 0.869464i \(0.335536\pi\)
\(674\) −1850.70 −2.74585
\(675\) 493.041i 0.730431i
\(676\) 1634.80 2.41835
\(677\) −892.607 −1.31847 −0.659237 0.751935i \(-0.729121\pi\)
−0.659237 + 0.751935i \(0.729121\pi\)
\(678\) 3233.79 4.76961
\(679\) 159.937i 0.235547i
\(680\) 65.9798i 0.0970291i
\(681\) 548.603i 0.805584i
\(682\) −1160.49 −1.70160
\(683\) 364.405 0.533536 0.266768 0.963761i \(-0.414044\pi\)
0.266768 + 0.963761i \(0.414044\pi\)
\(684\) −4949.88 −7.23667
\(685\) 536.719i 0.783531i
\(686\) 367.363 0.535515
\(687\) 1307.12 1.90265
\(688\) 2246.73i 3.26560i
\(689\) −387.368 −0.562218
\(690\) −2178.56 −3.15733
\(691\) 229.110 0.331564 0.165782 0.986162i \(-0.446985\pi\)
0.165782 + 0.986162i \(0.446985\pi\)
\(692\) 178.002i 0.257228i
\(693\) 106.515i 0.153701i
\(694\) 810.614i 1.16803i
\(695\) 753.210i 1.08375i
\(696\) −774.400 −1.11264
\(697\) −20.7240 −0.0297332
\(698\) 1750.14i 2.50736i
\(699\) 929.808i 1.33020i
\(700\) 152.292i 0.217559i
\(701\) −157.823 −0.225139 −0.112570 0.993644i \(-0.535908\pi\)
−0.112570 + 0.993644i \(0.535908\pi\)
\(702\) −668.090 −0.951695
\(703\) 931.405i 1.32490i
\(704\) −2007.17 −2.85109
\(705\) −1311.67 −1.86053
\(706\) 503.747i 0.713523i
\(707\) 93.3080 0.131977
\(708\) 1862.63 2.63084
\(709\) 961.391i 1.35598i 0.735070 + 0.677991i \(0.237149\pi\)
−0.735070 + 0.677991i \(0.762851\pi\)
\(710\) 1039.29i 1.46379i
\(711\) 1481.29i 2.08339i
\(712\) 3900.26 5.47789
\(713\) −1458.54 −2.04564
\(714\) 13.3220 0.0186582
\(715\) 110.260i 0.154210i
\(716\) 1993.07 2.78362
\(717\) 1676.77i 2.33859i
\(718\) −1116.56 −1.55510
\(719\) 42.2639i 0.0587815i −0.999568 0.0293907i \(-0.990643\pi\)
0.999568 0.0293907i \(-0.00935671\pi\)
\(720\) 3482.22i 4.83642i
\(721\) −95.8313 −0.132914
\(722\) 1530.89i 2.12035i
\(723\) −1766.61 −2.44344
\(724\) 1643.79 2.27043
\(725\) 76.5304 0.105559
\(726\) −1448.56 −1.99526
\(727\) 1128.71i 1.55256i −0.630387 0.776281i \(-0.717104\pi\)
0.630387 0.776281i \(-0.282896\pi\)
\(728\) −132.913 −0.182573
\(729\) 1073.52 1.47260
\(730\) 333.213 0.456457
\(731\) −24.1782 −0.0330755
\(732\) −897.631 −1.22627
\(733\) 1014.33i 1.38381i −0.721988 0.691905i \(-0.756772\pi\)
0.721988 0.691905i \(-0.243228\pi\)
\(734\) 597.926i 0.814613i
\(735\) 798.629 1.08657
\(736\) −4771.59 −6.48313
\(737\) 342.238 0.464367
\(738\) 1846.14 2.50155
\(739\) 1091.99i 1.47766i 0.673894 + 0.738828i \(0.264620\pi\)
−0.673894 + 0.738828i \(0.735380\pi\)
\(740\) 1266.40 1.71135
\(741\) 666.349i 0.899257i
\(742\) −302.215 −0.407298
\(743\) 351.284 0.472791 0.236396 0.971657i \(-0.424034\pi\)
0.236396 + 0.971657i \(0.424034\pi\)
\(744\) 6141.82i 8.25513i
\(745\) 399.845i 0.536705i
\(746\) 1650.69 2.21273
\(747\) 1035.32i 1.38597i
\(748\) 54.1022i 0.0723291i
\(749\) −55.6145 −0.0742516
\(750\) 2528.31i 3.37108i
\(751\) 143.419 0.190971 0.0954855 0.995431i \(-0.469560\pi\)
0.0954855 + 0.995431i \(0.469560\pi\)
\(752\) −5158.35 −6.85950
\(753\) 783.132 1.04002
\(754\) 103.702i 0.137535i
\(755\) −616.444 −0.816483
\(756\) −384.409 −0.508478
\(757\) 808.819 1.06845 0.534226 0.845341i \(-0.320603\pi\)
0.534226 + 0.845341i \(0.320603\pi\)
\(758\) 415.294i 0.547881i
\(759\) 1150.57 1.51591
\(760\) −2574.76 −3.38785
\(761\) 36.9250 0.0485217 0.0242608 0.999706i \(-0.492277\pi\)
0.0242608 + 0.999706i \(0.492277\pi\)
\(762\) 1648.93 2.16394
\(763\) 72.5153 0.0950397
\(764\) −958.388 −1.25443
\(765\) 37.4739 0.0489855
\(766\) 834.452i 1.08936i
\(767\) 160.653i 0.209456i
\(768\) 5389.71i 7.01785i
\(769\) 1010.47i 1.31400i 0.753890 + 0.657001i \(0.228175\pi\)
−0.753890 + 0.657001i \(0.771825\pi\)
\(770\) 86.0223i 0.111717i
\(771\) 124.753 0.161807
\(772\) 2438.59 3.15879
\(773\) 688.071i 0.890130i −0.895498 0.445065i \(-0.853181\pi\)
0.895498 0.445065i \(-0.146819\pi\)
\(774\) 2153.85 2.78275
\(775\) 606.968i 0.783185i
\(776\) 4661.06i 6.00652i
\(777\) 164.691i 0.211957i
\(778\) −980.249 −1.25996
\(779\) 808.725i 1.03816i
\(780\) −906.013 −1.16156
\(781\) 548.886i 0.702798i
\(782\) 92.1992i 0.117902i
\(783\) 193.176i 0.246712i
\(784\) 3140.73 4.00603
\(785\) 170.654i 0.217393i
\(786\) −1168.79 −1.48701
\(787\) 132.330 0.168145 0.0840726 0.996460i \(-0.473207\pi\)
0.0840726 + 0.996460i \(0.473207\pi\)
\(788\) 2588.69i 3.28514i
\(789\) 105.522i 0.133741i
\(790\) 1196.31i 1.51431i
\(791\) 160.484i 0.202888i
\(792\) 3104.18i 3.91941i
\(793\) 77.4211i 0.0976307i
\(794\) −1475.08 −1.85779
\(795\) −1326.85 −1.66900
\(796\) 1185.14 1.48887
\(797\) −693.629 −0.870300 −0.435150 0.900358i \(-0.643305\pi\)
−0.435150 + 0.900358i \(0.643305\pi\)
\(798\) 519.870i 0.651466i
\(799\) 55.5116i 0.0694763i
\(800\) 1985.68i 2.48210i
\(801\) 2215.20i 2.76554i
\(802\) 23.2997i 0.0290520i
\(803\) −175.982 −0.219155
\(804\) 2812.19i 3.49775i
\(805\) 108.116i 0.134306i
\(806\) −822.466 −1.02043
\(807\) 1782.77i 2.20913i
\(808\) 2719.29 3.36546
\(809\) 1579.45i 1.95235i −0.216974 0.976177i \(-0.569619\pi\)
0.216974 0.976177i \(-0.430381\pi\)
\(810\) −416.338 −0.513997
\(811\) 308.993 0.381003 0.190501 0.981687i \(-0.438989\pi\)
0.190501 + 0.981687i \(0.438989\pi\)
\(812\) 59.6684i 0.0734833i
\(813\) −959.058 −1.17965
\(814\) −906.879 −1.11410
\(815\) 961.021 1.17917
\(816\) 230.017 0.281883
\(817\) 943.518i 1.15486i
\(818\) 1415.82i 1.73084i
\(819\) 75.4895i 0.0921728i
\(820\) 1099.60 1.34097
\(821\) 720.263i 0.877300i −0.898658 0.438650i \(-0.855457\pi\)
0.898658 0.438650i \(-0.144543\pi\)
\(822\) 3158.21 3.84210
\(823\) −295.522 −0.359078 −0.179539 0.983751i \(-0.557461\pi\)
−0.179539 + 0.983751i \(0.557461\pi\)
\(824\) −2792.83 −3.38935
\(825\) 478.807i 0.580373i
\(826\) 125.337i 0.151740i
\(827\) 487.477i 0.589452i −0.955582 0.294726i \(-0.904772\pi\)
0.955582 0.294726i \(-0.0952284\pi\)
\(828\) 6057.37i 7.31567i
\(829\) 135.412 0.163344 0.0816718 0.996659i \(-0.473974\pi\)
0.0816718 + 0.996659i \(0.473974\pi\)
\(830\) 836.134i 1.00739i
\(831\) 969.868i 1.16711i
\(832\) −1422.53 −1.70977
\(833\) 33.7989i 0.0405750i
\(834\) 4432.10 5.31427
\(835\) 929.722i 1.11344i
\(836\) 2111.26 2.52543
\(837\) −1532.09 −1.83045
\(838\) 972.564i 1.16058i
\(839\) −827.593 −0.986405 −0.493202 0.869915i \(-0.664174\pi\)
−0.493202 + 0.869915i \(0.664174\pi\)
\(840\) −455.268 −0.541986
\(841\) −811.015 −0.964346
\(842\) −2305.76 −2.73843
\(843\) 363.100 0.430724
\(844\) 4352.21i 5.15665i
\(845\) 482.977i 0.571570i
\(846\) 4945.09i 5.84526i
\(847\) 71.8880i 0.0848736i
\(848\) −5218.05 −6.15336
\(849\) 2028.56 2.38936
\(850\) −38.3684 −0.0451393
\(851\) −1139.80 −1.33936
\(852\) 4510.22 5.29368
\(853\) 1611.25 1.88893 0.944463 0.328617i \(-0.106583\pi\)
0.944463 + 0.328617i \(0.106583\pi\)
\(854\) 60.4021i 0.0707284i
\(855\) 1462.36i 1.71037i
\(856\) −1620.78 −1.89344
\(857\) 727.225i 0.848571i −0.905528 0.424286i \(-0.860525\pi\)
0.905528 0.424286i \(-0.139475\pi\)
\(858\) 648.803 0.756180
\(859\) 1207.94 1.40622 0.703108 0.711083i \(-0.251795\pi\)
0.703108 + 0.711083i \(0.251795\pi\)
\(860\) 1282.87 1.49171
\(861\) 142.998i 0.166084i
\(862\) 2297.21 2.66497
\(863\) 503.549i 0.583486i 0.956497 + 0.291743i \(0.0942353\pi\)
−0.956497 + 0.291743i \(0.905765\pi\)
\(864\) −5012.19 −5.80114
\(865\) −52.5878 −0.0607951
\(866\) −2749.84 −3.17533
\(867\) 1443.93i 1.66543i
\(868\) −473.235 −0.545201
\(869\) 631.811i 0.727055i
\(870\) 355.210i 0.408287i
\(871\) 242.552 0.278476
\(872\) 2113.33 2.42354
\(873\) −2647.30 −3.03242
\(874\) 3597.94 4.11663
\(875\) 125.473 0.143398
\(876\) 1446.05i 1.65074i
\(877\) 1097.38i 1.25129i −0.780109 0.625643i \(-0.784837\pi\)
0.780109 0.625643i \(-0.215163\pi\)
\(878\) 57.5528i 0.0655499i
\(879\) 1117.86i 1.27174i
\(880\) 1485.26i 1.68780i
\(881\) 1117.59i 1.26855i −0.773108 0.634274i \(-0.781299\pi\)
0.773108 0.634274i \(-0.218701\pi\)
\(882\) 3010.88i 3.41370i
\(883\) −533.431 −0.604112 −0.302056 0.953290i \(-0.597673\pi\)
−0.302056 + 0.953290i \(0.597673\pi\)
\(884\) 38.3435i 0.0433750i
\(885\) 550.285i 0.621791i
\(886\) 600.713i 0.678005i
\(887\) 417.592 0.470791 0.235396 0.971900i \(-0.424361\pi\)
0.235396 + 0.971900i \(0.424361\pi\)
\(888\) 4799.61i 5.40496i
\(889\) 81.8316i 0.0920491i
\(890\) 1789.01i 2.01013i
\(891\) 219.882 0.246781
\(892\) 130.761i 0.146593i
\(893\) 2166.26 2.42582
\(894\) 2352.81 2.63177
\(895\) 588.821i 0.657901i
\(896\) −558.796 −0.623656
\(897\) 815.439 0.909073
\(898\) 2888.17i 3.21622i
\(899\) 237.813i 0.264530i
\(900\) 2520.76 2.80084
\(901\) 56.1540i 0.0623241i
\(902\) −787.429 −0.872982
\(903\) 166.832i 0.184753i
\(904\) 4677.02i 5.17370i
\(905\) 485.631i 0.536609i
\(906\) 3627.34i 4.00368i
\(907\) 1770.66 1.95221 0.976107 0.217291i \(-0.0697222\pi\)
0.976107 + 0.217291i \(0.0697222\pi\)
\(908\) −1231.90 −1.35672
\(909\) 1544.45i 1.69907i
\(910\) 60.9661i 0.0669957i
\(911\) 307.188i 0.337199i 0.985685 + 0.168599i \(0.0539244\pi\)
−0.985685 + 0.168599i \(0.946076\pi\)
\(912\) 8976.06i 9.84218i
\(913\) 441.591i 0.483671i
\(914\) 2463.46 2.69525
\(915\) 265.191i 0.289826i
\(916\) 2935.16i 3.20433i
\(917\) 58.0039i 0.0632540i
\(918\) 96.8483i 0.105499i
\(919\) 134.435 0.146284 0.0731420 0.997322i \(-0.476697\pi\)
0.0731420 + 0.997322i \(0.476697\pi\)
\(920\) 3150.84i 3.42483i
\(921\) −931.389 −1.01128
\(922\) −904.551 −0.981075
\(923\) 389.008i 0.421461i
\(924\) 373.312 0.404017
\(925\) 474.323i 0.512782i
\(926\) −1769.17 −1.91056
\(927\) 1586.22i 1.71113i
\(928\) 777.998i 0.838360i
\(929\) 1004.28i 1.08103i 0.841333 + 0.540517i \(0.181771\pi\)
−0.841333 + 0.540517i \(0.818229\pi\)
\(930\) −2817.20 −3.02924
\(931\) −1318.95 −1.41671
\(932\) −2087.90 −2.24024
\(933\) 547.586i 0.586909i
\(934\) 3132.75i 3.35412i
\(935\) −15.9836 −0.0170948
\(936\) 2200.00i 2.35043i
\(937\) 721.788i 0.770318i 0.922850 + 0.385159i \(0.125853\pi\)
−0.922850 + 0.385159i \(0.874147\pi\)
\(938\) 189.234 0.201742
\(939\) 2425.28i 2.58283i
\(940\) 2945.39i 3.13339i
\(941\) 732.588 0.778521 0.389260 0.921128i \(-0.372731\pi\)
0.389260 + 0.921128i \(0.372731\pi\)
\(942\) −1004.18 −1.06600
\(943\) −989.670 −1.04949
\(944\) 2164.08i 2.29245i
\(945\) 113.568i 0.120177i
\(946\) −918.673 −0.971113
\(947\) 898.921 0.949230 0.474615 0.880193i \(-0.342587\pi\)
0.474615 + 0.880193i \(0.342587\pi\)
\(948\) −5191.62 −5.47639
\(949\) −124.722 −0.131425
\(950\) 1497.27i 1.57607i
\(951\) 1126.80i 1.18485i
\(952\) 19.2675i 0.0202390i
\(953\) −736.249 −0.772559 −0.386280 0.922382i \(-0.626240\pi\)
−0.386280 + 0.922382i \(0.626240\pi\)
\(954\) 5002.32i 5.24353i
\(955\) 283.141i 0.296482i
\(956\) 3765.22 3.93852
\(957\) 187.599i 0.196028i
\(958\) 2147.44i 2.24159i
\(959\) 156.733i 0.163434i
\(960\) −4872.59 −5.07561
\(961\) −925.109 −0.962653
\(962\) −642.727 −0.668115
\(963\) 920.541i 0.955910i
\(964\) 3966.95i 4.11509i
\(965\) 720.442i 0.746572i
\(966\) 636.186 0.658577
\(967\) 1378.98i 1.42604i 0.701142 + 0.713022i \(0.252674\pi\)
−0.701142 + 0.713022i \(0.747326\pi\)
\(968\) 2095.04i 2.16430i
\(969\) −96.5960 −0.0996862
\(970\) −2137.99 −2.20411
\(971\) 125.504i 0.129253i −0.997910 0.0646264i \(-0.979414\pi\)
0.997910 0.0646264i \(-0.0205856\pi\)
\(972\) 1761.44i 1.81218i
\(973\) 219.953i 0.226057i
\(974\) 1563.43 1.60517
\(975\) 339.342i 0.348043i
\(976\) 1042.90i 1.06855i
\(977\) 735.470i 0.752784i 0.926461 + 0.376392i \(0.122835\pi\)
−0.926461 + 0.376392i \(0.877165\pi\)
\(978\) 5654.93i 5.78213i
\(979\) 944.840i 0.965108i
\(980\) 1793.34i 1.82994i
\(981\) 1200.29i 1.22353i
\(982\) 912.453 0.929178
\(983\) 678.521i 0.690256i 0.938556 + 0.345128i \(0.112164\pi\)
−0.938556 + 0.345128i \(0.887836\pi\)
\(984\) 4167.43i 4.23519i
\(985\) 764.788 0.776434
\(986\) −15.0329 −0.0152464
\(987\) 383.036 0.388082
\(988\) 1496.30 1.51447
\(989\) −1154.62 −1.16746
\(990\) 1423.86 1.43824
\(991\) −1158.52 −1.16905 −0.584523 0.811377i \(-0.698718\pi\)
−0.584523 + 0.811377i \(0.698718\pi\)
\(992\) −6170.36 −6.22012
\(993\) −2762.54 −2.78201
\(994\) 303.495i 0.305327i
\(995\) 350.130i 0.351889i
\(996\) −3628.58 −3.64315
\(997\) 819.959i 0.822426i −0.911539 0.411213i \(-0.865105\pi\)
0.911539 0.411213i \(-0.134895\pi\)
\(998\) 1939.45i 1.94333i
\(999\) −1197.27 −1.19847
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 547.3.b.b.546.1 88
547.546 odd 2 inner 547.3.b.b.546.88 yes 88
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
547.3.b.b.546.1 88 1.1 even 1 trivial
547.3.b.b.546.88 yes 88 547.546 odd 2 inner