Properties

Label 624.4.d.c.287.11
Level $624$
Weight $4$
Character 624.287
Analytic conductor $36.817$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [624,4,Mod(287,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.287");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.d (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: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 287.11
Character \(\chi\) \(=\) 624.287
Dual form 624.4.d.c.287.12

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.486351 - 5.17334i) q^{3} +4.15078i q^{5} +6.70503i q^{7} +(-26.5269 + 5.03212i) q^{9} +O(q^{10})\) \(q+(-0.486351 - 5.17334i) q^{3} +4.15078i q^{5} +6.70503i q^{7} +(-26.5269 + 5.03212i) q^{9} +32.5924 q^{11} -13.0000 q^{13} +(21.4734 - 2.01874i) q^{15} -99.7981i q^{17} +73.6834i q^{19} +(34.6874 - 3.26100i) q^{21} -193.704 q^{23} +107.771 q^{25} +(38.9343 + 134.785i) q^{27} +138.975i q^{29} +165.252i q^{31} +(-15.8513 - 168.611i) q^{33} -27.8311 q^{35} -54.1147 q^{37} +(6.32257 + 67.2534i) q^{39} +512.529i q^{41} +336.614i q^{43} +(-20.8872 - 110.107i) q^{45} +495.587 q^{47} +298.043 q^{49} +(-516.289 + 48.5369i) q^{51} -537.479i q^{53} +135.284i q^{55} +(381.189 - 35.8360i) q^{57} +315.255 q^{59} -133.740 q^{61} +(-33.7405 - 177.864i) q^{63} -53.9601i q^{65} +131.721i q^{67} +(94.2081 + 1002.10i) q^{69} -1073.24 q^{71} +1023.67 q^{73} +(-52.4146 - 557.537i) q^{75} +218.533i q^{77} +108.143i q^{79} +(678.355 - 266.974i) q^{81} +1032.79 q^{83} +414.239 q^{85} +(718.964 - 67.5906i) q^{87} +139.135i q^{89} -87.1654i q^{91} +(854.907 - 80.3707i) q^{93} -305.843 q^{95} +1458.52 q^{97} +(-864.575 + 164.009i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 84 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 84 q^{9} - 312 q^{13} - 300 q^{21} - 240 q^{25} + 240 q^{33} - 456 q^{37} + 1836 q^{45} - 1632 q^{49} + 168 q^{57} - 960 q^{61} + 2760 q^{69} + 1248 q^{73} - 468 q^{81} - 7704 q^{85} + 3336 q^{93} - 2496 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(3\) −0.486351 5.17334i −0.0935984 0.995610i
\(4\) 0 0
\(5\) 4.15078i 0.371257i 0.982620 + 0.185628i \(0.0594320\pi\)
−0.982620 + 0.185628i \(0.940568\pi\)
\(6\) 0 0
\(7\) 6.70503i 0.362038i 0.983480 + 0.181019i \(0.0579395\pi\)
−0.983480 + 0.181019i \(0.942061\pi\)
\(8\) 0 0
\(9\) −26.5269 + 5.03212i −0.982479 + 0.186375i
\(10\) 0 0
\(11\) 32.5924 0.893361 0.446680 0.894694i \(-0.352606\pi\)
0.446680 + 0.894694i \(0.352606\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) 0 0
\(15\) 21.4734 2.01874i 0.369627 0.0347490i
\(16\) 0 0
\(17\) 99.7981i 1.42380i −0.702281 0.711900i \(-0.747835\pi\)
0.702281 0.711900i \(-0.252165\pi\)
\(18\) 0 0
\(19\) 73.6834i 0.889691i 0.895607 + 0.444846i \(0.146741\pi\)
−0.895607 + 0.444846i \(0.853259\pi\)
\(20\) 0 0
\(21\) 34.6874 3.26100i 0.360448 0.0338861i
\(22\) 0 0
\(23\) −193.704 −1.75609 −0.878044 0.478580i \(-0.841152\pi\)
−0.878044 + 0.478580i \(0.841152\pi\)
\(24\) 0 0
\(25\) 107.771 0.862168
\(26\) 0 0
\(27\) 38.9343 + 134.785i 0.277515 + 0.960721i
\(28\) 0 0
\(29\) 138.975i 0.889895i 0.895557 + 0.444948i \(0.146778\pi\)
−0.895557 + 0.444948i \(0.853222\pi\)
\(30\) 0 0
\(31\) 165.252i 0.957426i 0.877971 + 0.478713i \(0.158897\pi\)
−0.877971 + 0.478713i \(0.841103\pi\)
\(32\) 0 0
\(33\) −15.8513 168.611i −0.0836171 0.889439i
\(34\) 0 0
\(35\) −27.8311 −0.134409
\(36\) 0 0
\(37\) −54.1147 −0.240443 −0.120222 0.992747i \(-0.538361\pi\)
−0.120222 + 0.992747i \(0.538361\pi\)
\(38\) 0 0
\(39\) 6.32257 + 67.2534i 0.0259595 + 0.276133i
\(40\) 0 0
\(41\) 512.529i 1.95228i 0.217133 + 0.976142i \(0.430329\pi\)
−0.217133 + 0.976142i \(0.569671\pi\)
\(42\) 0 0
\(43\) 336.614i 1.19379i 0.802318 + 0.596897i \(0.203600\pi\)
−0.802318 + 0.596897i \(0.796400\pi\)
\(44\) 0 0
\(45\) −20.8872 110.107i −0.0691929 0.364752i
\(46\) 0 0
\(47\) 495.587 1.53806 0.769029 0.639214i \(-0.220740\pi\)
0.769029 + 0.639214i \(0.220740\pi\)
\(48\) 0 0
\(49\) 298.043 0.868929
\(50\) 0 0
\(51\) −516.289 + 48.5369i −1.41755 + 0.133265i
\(52\) 0 0
\(53\) 537.479i 1.39299i −0.717562 0.696495i \(-0.754742\pi\)
0.717562 0.696495i \(-0.245258\pi\)
\(54\) 0 0
\(55\) 135.284i 0.331666i
\(56\) 0 0
\(57\) 381.189 35.8360i 0.885786 0.0832737i
\(58\) 0 0
\(59\) 315.255 0.695640 0.347820 0.937561i \(-0.386922\pi\)
0.347820 + 0.937561i \(0.386922\pi\)
\(60\) 0 0
\(61\) −133.740 −0.280715 −0.140358 0.990101i \(-0.544825\pi\)
−0.140358 + 0.990101i \(0.544825\pi\)
\(62\) 0 0
\(63\) −33.7405 177.864i −0.0674747 0.355694i
\(64\) 0 0
\(65\) 53.9601i 0.102968i
\(66\) 0 0
\(67\) 131.721i 0.240182i 0.992763 + 0.120091i \(0.0383187\pi\)
−0.992763 + 0.120091i \(0.961681\pi\)
\(68\) 0 0
\(69\) 94.2081 + 1002.10i 0.164367 + 1.74838i
\(70\) 0 0
\(71\) −1073.24 −1.79394 −0.896970 0.442091i \(-0.854237\pi\)
−0.896970 + 0.442091i \(0.854237\pi\)
\(72\) 0 0
\(73\) 1023.67 1.64125 0.820626 0.571465i \(-0.193625\pi\)
0.820626 + 0.571465i \(0.193625\pi\)
\(74\) 0 0
\(75\) −52.4146 557.537i −0.0806976 0.858384i
\(76\) 0 0
\(77\) 218.533i 0.323430i
\(78\) 0 0
\(79\) 108.143i 0.154013i 0.997031 + 0.0770067i \(0.0245363\pi\)
−0.997031 + 0.0770067i \(0.975464\pi\)
\(80\) 0 0
\(81\) 678.355 266.974i 0.930529 0.366219i
\(82\) 0 0
\(83\) 1032.79 1.36582 0.682910 0.730503i \(-0.260714\pi\)
0.682910 + 0.730503i \(0.260714\pi\)
\(84\) 0 0
\(85\) 414.239 0.528595
\(86\) 0 0
\(87\) 718.964 67.5906i 0.885989 0.0832927i
\(88\) 0 0
\(89\) 139.135i 0.165711i 0.996562 + 0.0828554i \(0.0264040\pi\)
−0.996562 + 0.0828554i \(0.973596\pi\)
\(90\) 0 0
\(91\) 87.1654i 0.100411i
\(92\) 0 0
\(93\) 854.907 80.3707i 0.953223 0.0896135i
\(94\) 0 0
\(95\) −305.843 −0.330304
\(96\) 0 0
\(97\) 1458.52 1.52670 0.763352 0.645983i \(-0.223552\pi\)
0.763352 + 0.645983i \(0.223552\pi\)
\(98\) 0 0
\(99\) −864.575 + 164.009i −0.877708 + 0.166500i
\(100\) 0 0
\(101\) 1373.84i 1.35349i −0.736220 0.676743i \(-0.763391\pi\)
0.736220 0.676743i \(-0.236609\pi\)
\(102\) 0 0
\(103\) 1505.44i 1.44015i 0.693898 + 0.720073i \(0.255892\pi\)
−0.693898 + 0.720073i \(0.744108\pi\)
\(104\) 0 0
\(105\) 13.5357 + 143.980i 0.0125804 + 0.133819i
\(106\) 0 0
\(107\) −632.092 −0.571090 −0.285545 0.958365i \(-0.592175\pi\)
−0.285545 + 0.958365i \(0.592175\pi\)
\(108\) 0 0
\(109\) 922.022 0.810218 0.405109 0.914268i \(-0.367234\pi\)
0.405109 + 0.914268i \(0.367234\pi\)
\(110\) 0 0
\(111\) 26.3187 + 279.954i 0.0225051 + 0.239388i
\(112\) 0 0
\(113\) 903.296i 0.751990i 0.926622 + 0.375995i \(0.122699\pi\)
−0.926622 + 0.375995i \(0.877301\pi\)
\(114\) 0 0
\(115\) 804.021i 0.651959i
\(116\) 0 0
\(117\) 344.850 65.4176i 0.272491 0.0516911i
\(118\) 0 0
\(119\) 669.149 0.515469
\(120\) 0 0
\(121\) −268.737 −0.201906
\(122\) 0 0
\(123\) 2651.49 249.269i 1.94371 0.182731i
\(124\) 0 0
\(125\) 966.180i 0.691342i
\(126\) 0 0
\(127\) 31.2951i 0.0218660i −0.999940 0.0109330i \(-0.996520\pi\)
0.999940 0.0109330i \(-0.00348016\pi\)
\(128\) 0 0
\(129\) 1741.42 163.713i 1.18855 0.111737i
\(130\) 0 0
\(131\) −306.642 −0.204515 −0.102257 0.994758i \(-0.532607\pi\)
−0.102257 + 0.994758i \(0.532607\pi\)
\(132\) 0 0
\(133\) −494.049 −0.322102
\(134\) 0 0
\(135\) −559.464 + 161.608i −0.356674 + 0.103029i
\(136\) 0 0
\(137\) 70.7897i 0.0441458i −0.999756 0.0220729i \(-0.992973\pi\)
0.999756 0.0220729i \(-0.00702659\pi\)
\(138\) 0 0
\(139\) 725.665i 0.442807i 0.975182 + 0.221403i \(0.0710637\pi\)
−0.975182 + 0.221403i \(0.928936\pi\)
\(140\) 0 0
\(141\) −241.029 2563.84i −0.143960 1.53131i
\(142\) 0 0
\(143\) −423.701 −0.247774
\(144\) 0 0
\(145\) −576.853 −0.330380
\(146\) 0 0
\(147\) −144.953 1541.88i −0.0813303 0.865114i
\(148\) 0 0
\(149\) 2069.01i 1.13759i 0.822481 + 0.568793i \(0.192589\pi\)
−0.822481 + 0.568793i \(0.807411\pi\)
\(150\) 0 0
\(151\) 3034.60i 1.63545i 0.575612 + 0.817723i \(0.304764\pi\)
−0.575612 + 0.817723i \(0.695236\pi\)
\(152\) 0 0
\(153\) 502.196 + 2647.34i 0.265360 + 1.39885i
\(154\) 0 0
\(155\) −685.926 −0.355451
\(156\) 0 0
\(157\) −3473.41 −1.76566 −0.882829 0.469694i \(-0.844364\pi\)
−0.882829 + 0.469694i \(0.844364\pi\)
\(158\) 0 0
\(159\) −2780.56 + 261.404i −1.38688 + 0.130382i
\(160\) 0 0
\(161\) 1298.79i 0.635770i
\(162\) 0 0
\(163\) 2996.56i 1.43993i 0.694010 + 0.719965i \(0.255842\pi\)
−0.694010 + 0.719965i \(0.744158\pi\)
\(164\) 0 0
\(165\) 699.868 65.7954i 0.330210 0.0310434i
\(166\) 0 0
\(167\) 46.8590 0.0217129 0.0108565 0.999941i \(-0.496544\pi\)
0.0108565 + 0.999941i \(0.496544\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) −370.784 1954.59i −0.165816 0.874103i
\(172\) 0 0
\(173\) 2189.42i 0.962189i −0.876669 0.481095i \(-0.840239\pi\)
0.876669 0.481095i \(-0.159761\pi\)
\(174\) 0 0
\(175\) 722.608i 0.312137i
\(176\) 0 0
\(177\) −153.325 1630.92i −0.0651108 0.692586i
\(178\) 0 0
\(179\) 675.873 0.282218 0.141109 0.989994i \(-0.454933\pi\)
0.141109 + 0.989994i \(0.454933\pi\)
\(180\) 0 0
\(181\) −157.711 −0.0647656 −0.0323828 0.999476i \(-0.510310\pi\)
−0.0323828 + 0.999476i \(0.510310\pi\)
\(182\) 0 0
\(183\) 65.0445 + 691.882i 0.0262745 + 0.279483i
\(184\) 0 0
\(185\) 224.618i 0.0892661i
\(186\) 0 0
\(187\) 3252.66i 1.27197i
\(188\) 0 0
\(189\) −903.740 + 261.056i −0.347817 + 0.100471i
\(190\) 0 0
\(191\) 1013.12 0.383806 0.191903 0.981414i \(-0.438534\pi\)
0.191903 + 0.981414i \(0.438534\pi\)
\(192\) 0 0
\(193\) 38.4793 0.0143513 0.00717566 0.999974i \(-0.497716\pi\)
0.00717566 + 0.999974i \(0.497716\pi\)
\(194\) 0 0
\(195\) −279.154 + 26.2436i −0.102516 + 0.00963764i
\(196\) 0 0
\(197\) 46.5566i 0.0168377i −0.999965 0.00841883i \(-0.997320\pi\)
0.999965 0.00841883i \(-0.00267983\pi\)
\(198\) 0 0
\(199\) 3751.71i 1.33644i 0.743964 + 0.668220i \(0.232944\pi\)
−0.743964 + 0.668220i \(0.767056\pi\)
\(200\) 0 0
\(201\) 681.435 64.0625i 0.239128 0.0224807i
\(202\) 0 0
\(203\) −931.830 −0.322175
\(204\) 0 0
\(205\) −2127.39 −0.724798
\(206\) 0 0
\(207\) 5138.36 974.741i 1.72532 0.327291i
\(208\) 0 0
\(209\) 2401.52i 0.794815i
\(210\) 0 0
\(211\) 3034.14i 0.989948i 0.868908 + 0.494974i \(0.164823\pi\)
−0.868908 + 0.494974i \(0.835177\pi\)
\(212\) 0 0
\(213\) 521.970 + 5552.22i 0.167910 + 1.78606i
\(214\) 0 0
\(215\) −1397.21 −0.443204
\(216\) 0 0
\(217\) −1108.02 −0.346624
\(218\) 0 0
\(219\) −497.863 5295.79i −0.153619 1.63405i
\(220\) 0 0
\(221\) 1297.37i 0.394891i
\(222\) 0 0
\(223\) 502.348i 0.150851i 0.997151 + 0.0754253i \(0.0240314\pi\)
−0.997151 + 0.0754253i \(0.975969\pi\)
\(224\) 0 0
\(225\) −2858.83 + 542.317i −0.847062 + 0.160687i
\(226\) 0 0
\(227\) −4630.13 −1.35380 −0.676899 0.736076i \(-0.736677\pi\)
−0.676899 + 0.736076i \(0.736677\pi\)
\(228\) 0 0
\(229\) 1411.21 0.407228 0.203614 0.979051i \(-0.434731\pi\)
0.203614 + 0.979051i \(0.434731\pi\)
\(230\) 0 0
\(231\) 1130.54 106.284i 0.322010 0.0302725i
\(232\) 0 0
\(233\) 6006.04i 1.68871i −0.535786 0.844354i \(-0.679985\pi\)
0.535786 0.844354i \(-0.320015\pi\)
\(234\) 0 0
\(235\) 2057.07i 0.571014i
\(236\) 0 0
\(237\) 559.462 52.5956i 0.153337 0.0144154i
\(238\) 0 0
\(239\) −588.425 −0.159255 −0.0796277 0.996825i \(-0.525373\pi\)
−0.0796277 + 0.996825i \(0.525373\pi\)
\(240\) 0 0
\(241\) −6547.37 −1.75001 −0.875006 0.484112i \(-0.839143\pi\)
−0.875006 + 0.484112i \(0.839143\pi\)
\(242\) 0 0
\(243\) −1711.06 3379.52i −0.451707 0.892166i
\(244\) 0 0
\(245\) 1237.11i 0.322596i
\(246\) 0 0
\(247\) 957.884i 0.246756i
\(248\) 0 0
\(249\) −502.297 5342.96i −0.127838 1.35982i
\(250\) 0 0
\(251\) 2477.90 0.623122 0.311561 0.950226i \(-0.399148\pi\)
0.311561 + 0.950226i \(0.399148\pi\)
\(252\) 0 0
\(253\) −6313.26 −1.56882
\(254\) 0 0
\(255\) −201.466 2143.00i −0.0494756 0.526274i
\(256\) 0 0
\(257\) 375.601i 0.0911648i −0.998961 0.0455824i \(-0.985486\pi\)
0.998961 0.0455824i \(-0.0145144\pi\)
\(258\) 0 0
\(259\) 362.840i 0.0870494i
\(260\) 0 0
\(261\) −699.338 3686.57i −0.165854 0.874303i
\(262\) 0 0
\(263\) 7331.67 1.71897 0.859487 0.511158i \(-0.170783\pi\)
0.859487 + 0.511158i \(0.170783\pi\)
\(264\) 0 0
\(265\) 2230.96 0.517157
\(266\) 0 0
\(267\) 719.792 67.6684i 0.164983 0.0155103i
\(268\) 0 0
\(269\) 1947.22i 0.441352i −0.975347 0.220676i \(-0.929174\pi\)
0.975347 0.220676i \(-0.0708264\pi\)
\(270\) 0 0
\(271\) 2443.85i 0.547798i −0.961758 0.273899i \(-0.911687\pi\)
0.961758 0.273899i \(-0.0883135\pi\)
\(272\) 0 0
\(273\) −450.936 + 42.3930i −0.0999704 + 0.00939832i
\(274\) 0 0
\(275\) 3512.51 0.770228
\(276\) 0 0
\(277\) −3761.28 −0.815861 −0.407930 0.913013i \(-0.633749\pi\)
−0.407930 + 0.913013i \(0.633749\pi\)
\(278\) 0 0
\(279\) −831.571 4383.64i −0.178440 0.940651i
\(280\) 0 0
\(281\) 5014.21i 1.06449i 0.846590 + 0.532246i \(0.178652\pi\)
−0.846590 + 0.532246i \(0.821348\pi\)
\(282\) 0 0
\(283\) 7204.17i 1.51323i −0.653862 0.756614i \(-0.726852\pi\)
0.653862 0.756614i \(-0.273148\pi\)
\(284\) 0 0
\(285\) 148.747 + 1582.23i 0.0309159 + 0.328854i
\(286\) 0 0
\(287\) −3436.52 −0.706800
\(288\) 0 0
\(289\) −5046.65 −1.02720
\(290\) 0 0
\(291\) −709.353 7545.42i −0.142897 1.52000i
\(292\) 0 0
\(293\) 6452.30i 1.28651i −0.765652 0.643254i \(-0.777584\pi\)
0.765652 0.643254i \(-0.222416\pi\)
\(294\) 0 0
\(295\) 1308.55i 0.258261i
\(296\) 0 0
\(297\) 1268.96 + 4392.98i 0.247921 + 0.858271i
\(298\) 0 0
\(299\) 2518.15 0.487051
\(300\) 0 0
\(301\) −2257.01 −0.432199
\(302\) 0 0
\(303\) −7107.34 + 668.168i −1.34754 + 0.126684i
\(304\) 0 0
\(305\) 555.124i 0.104217i
\(306\) 0 0
\(307\) 3644.73i 0.677576i −0.940863 0.338788i \(-0.889983\pi\)
0.940863 0.338788i \(-0.110017\pi\)
\(308\) 0 0
\(309\) 7788.14 732.171i 1.43382 0.134795i
\(310\) 0 0
\(311\) −4488.18 −0.818332 −0.409166 0.912460i \(-0.634180\pi\)
−0.409166 + 0.912460i \(0.634180\pi\)
\(312\) 0 0
\(313\) −1619.80 −0.292514 −0.146257 0.989247i \(-0.546723\pi\)
−0.146257 + 0.989247i \(0.546723\pi\)
\(314\) 0 0
\(315\) 738.273 140.049i 0.132054 0.0250504i
\(316\) 0 0
\(317\) 3052.76i 0.540884i −0.962736 0.270442i \(-0.912830\pi\)
0.962736 0.270442i \(-0.0871698\pi\)
\(318\) 0 0
\(319\) 4529.52i 0.794998i
\(320\) 0 0
\(321\) 307.419 + 3270.03i 0.0534531 + 0.568583i
\(322\) 0 0
\(323\) 7353.46 1.26674
\(324\) 0 0
\(325\) −1401.02 −0.239123
\(326\) 0 0
\(327\) −448.427 4769.94i −0.0758351 0.806661i
\(328\) 0 0
\(329\) 3322.92i 0.556835i
\(330\) 0 0
\(331\) 2819.23i 0.468154i −0.972218 0.234077i \(-0.924793\pi\)
0.972218 0.234077i \(-0.0752067\pi\)
\(332\) 0 0
\(333\) 1435.50 272.312i 0.236230 0.0448126i
\(334\) 0 0
\(335\) −546.742 −0.0891693
\(336\) 0 0
\(337\) −6094.96 −0.985203 −0.492601 0.870255i \(-0.663954\pi\)
−0.492601 + 0.870255i \(0.663954\pi\)
\(338\) 0 0
\(339\) 4673.06 439.319i 0.748689 0.0703851i
\(340\) 0 0
\(341\) 5385.97i 0.855327i
\(342\) 0 0
\(343\) 4298.21i 0.676622i
\(344\) 0 0
\(345\) −4159.47 + 391.037i −0.649097 + 0.0610223i
\(346\) 0 0
\(347\) −9811.05 −1.51782 −0.758912 0.651193i \(-0.774269\pi\)
−0.758912 + 0.651193i \(0.774269\pi\)
\(348\) 0 0
\(349\) 3209.72 0.492299 0.246149 0.969232i \(-0.420835\pi\)
0.246149 + 0.969232i \(0.420835\pi\)
\(350\) 0 0
\(351\) −506.146 1752.21i −0.0769689 0.266456i
\(352\) 0 0
\(353\) 8538.15i 1.28736i 0.765293 + 0.643682i \(0.222594\pi\)
−0.765293 + 0.643682i \(0.777406\pi\)
\(354\) 0 0
\(355\) 4454.76i 0.666012i
\(356\) 0 0
\(357\) −325.442 3461.74i −0.0482470 0.513206i
\(358\) 0 0
\(359\) 5268.60 0.774557 0.387278 0.921963i \(-0.373415\pi\)
0.387278 + 0.921963i \(0.373415\pi\)
\(360\) 0 0
\(361\) 1429.76 0.208450
\(362\) 0 0
\(363\) 130.701 + 1390.27i 0.0188981 + 0.201020i
\(364\) 0 0
\(365\) 4249.02i 0.609326i
\(366\) 0 0
\(367\) 11294.8i 1.60649i 0.595647 + 0.803246i \(0.296896\pi\)
−0.595647 + 0.803246i \(0.703104\pi\)
\(368\) 0 0
\(369\) −2579.11 13595.8i −0.363857 1.91808i
\(370\) 0 0
\(371\) 3603.82 0.504315
\(372\) 0 0
\(373\) 10199.9 1.41590 0.707948 0.706265i \(-0.249621\pi\)
0.707948 + 0.706265i \(0.249621\pi\)
\(374\) 0 0
\(375\) 4998.38 469.903i 0.688307 0.0647085i
\(376\) 0 0
\(377\) 1806.67i 0.246813i
\(378\) 0 0
\(379\) 11233.0i 1.52243i −0.648500 0.761215i \(-0.724603\pi\)
0.648500 0.761215i \(-0.275397\pi\)
\(380\) 0 0
\(381\) −161.900 + 15.2204i −0.0217701 + 0.00204663i
\(382\) 0 0
\(383\) 4415.99 0.589155 0.294577 0.955628i \(-0.404821\pi\)
0.294577 + 0.955628i \(0.404821\pi\)
\(384\) 0 0
\(385\) −907.081 −0.120076
\(386\) 0 0
\(387\) −1693.88 8929.34i −0.222493 1.17288i
\(388\) 0 0
\(389\) 4900.14i 0.638681i 0.947640 + 0.319340i \(0.103461\pi\)
−0.947640 + 0.319340i \(0.896539\pi\)
\(390\) 0 0
\(391\) 19331.3i 2.50032i
\(392\) 0 0
\(393\) 149.136 + 1586.36i 0.0191422 + 0.203617i
\(394\) 0 0
\(395\) −448.878 −0.0571785
\(396\) 0 0
\(397\) −13149.3 −1.66233 −0.831165 0.556025i \(-0.812326\pi\)
−0.831165 + 0.556025i \(0.812326\pi\)
\(398\) 0 0
\(399\) 240.282 + 2555.89i 0.0301482 + 0.320688i
\(400\) 0 0
\(401\) 4490.95i 0.559270i −0.960106 0.279635i \(-0.909787\pi\)
0.960106 0.279635i \(-0.0902135\pi\)
\(402\) 0 0
\(403\) 2148.28i 0.265542i
\(404\) 0 0
\(405\) 1108.15 + 2815.70i 0.135961 + 0.345465i
\(406\) 0 0
\(407\) −1763.73 −0.214803
\(408\) 0 0
\(409\) −1405.54 −0.169925 −0.0849627 0.996384i \(-0.527077\pi\)
−0.0849627 + 0.996384i \(0.527077\pi\)
\(410\) 0 0
\(411\) −366.220 + 34.4287i −0.0439520 + 0.00413198i
\(412\) 0 0
\(413\) 2113.80i 0.251848i
\(414\) 0 0
\(415\) 4286.86i 0.507070i
\(416\) 0 0
\(417\) 3754.11 352.928i 0.440863 0.0414460i
\(418\) 0 0
\(419\) −12751.3 −1.48674 −0.743370 0.668880i \(-0.766774\pi\)
−0.743370 + 0.668880i \(0.766774\pi\)
\(420\) 0 0
\(421\) −6483.29 −0.750537 −0.375269 0.926916i \(-0.622450\pi\)
−0.375269 + 0.926916i \(0.622450\pi\)
\(422\) 0 0
\(423\) −13146.4 + 2493.85i −1.51111 + 0.286656i
\(424\) 0 0
\(425\) 10755.3i 1.22755i
\(426\) 0 0
\(427\) 896.729i 0.101629i
\(428\) 0 0
\(429\) 206.067 + 2191.95i 0.0231912 + 0.246686i
\(430\) 0 0
\(431\) −10523.9 −1.17615 −0.588074 0.808807i \(-0.700114\pi\)
−0.588074 + 0.808807i \(0.700114\pi\)
\(432\) 0 0
\(433\) −9569.70 −1.06210 −0.531051 0.847340i \(-0.678203\pi\)
−0.531051 + 0.847340i \(0.678203\pi\)
\(434\) 0 0
\(435\) 280.553 + 2984.26i 0.0309230 + 0.328929i
\(436\) 0 0
\(437\) 14272.8i 1.56238i
\(438\) 0 0
\(439\) 226.844i 0.0246621i 0.999924 + 0.0123310i \(0.00392519\pi\)
−0.999924 + 0.0123310i \(0.996075\pi\)
\(440\) 0 0
\(441\) −7906.15 + 1499.79i −0.853704 + 0.161947i
\(442\) 0 0
\(443\) −5366.00 −0.575499 −0.287750 0.957706i \(-0.592907\pi\)
−0.287750 + 0.957706i \(0.592907\pi\)
\(444\) 0 0
\(445\) −577.518 −0.0615213
\(446\) 0 0
\(447\) 10703.7 1006.27i 1.13259 0.106476i
\(448\) 0 0
\(449\) 1825.33i 0.191854i −0.995388 0.0959272i \(-0.969418\pi\)
0.995388 0.0959272i \(-0.0305816\pi\)
\(450\) 0 0
\(451\) 16704.5i 1.74409i
\(452\) 0 0
\(453\) 15699.0 1475.88i 1.62827 0.153075i
\(454\) 0 0
\(455\) 361.804 0.0372783
\(456\) 0 0
\(457\) 5835.87 0.597353 0.298677 0.954354i \(-0.403455\pi\)
0.298677 + 0.954354i \(0.403455\pi\)
\(458\) 0 0
\(459\) 13451.3 3885.57i 1.36787 0.395126i
\(460\) 0 0
\(461\) 3767.02i 0.380581i 0.981728 + 0.190290i \(0.0609429\pi\)
−0.981728 + 0.190290i \(0.939057\pi\)
\(462\) 0 0
\(463\) 8214.87i 0.824573i 0.911054 + 0.412286i \(0.135270\pi\)
−0.911054 + 0.412286i \(0.864730\pi\)
\(464\) 0 0
\(465\) 333.601 + 3548.53i 0.0332696 + 0.353890i
\(466\) 0 0
\(467\) 11066.2 1.09653 0.548267 0.836303i \(-0.315288\pi\)
0.548267 + 0.836303i \(0.315288\pi\)
\(468\) 0 0
\(469\) −883.190 −0.0869551
\(470\) 0 0
\(471\) 1689.30 + 17969.1i 0.165263 + 1.75791i
\(472\) 0 0
\(473\) 10971.1i 1.06649i
\(474\) 0 0
\(475\) 7940.94i 0.767064i
\(476\) 0 0
\(477\) 2704.66 + 14257.7i 0.259618 + 1.36858i
\(478\) 0 0
\(479\) −7515.92 −0.716933 −0.358467 0.933542i \(-0.616700\pi\)
−0.358467 + 0.933542i \(0.616700\pi\)
\(480\) 0 0
\(481\) 703.491 0.0666869
\(482\) 0 0
\(483\) −6719.08 + 631.668i −0.632979 + 0.0595070i
\(484\) 0 0
\(485\) 6053.99i 0.566799i
\(486\) 0 0
\(487\) 2472.60i 0.230071i −0.993361 0.115035i \(-0.963302\pi\)
0.993361 0.115035i \(-0.0366981\pi\)
\(488\) 0 0
\(489\) 15502.2 1457.38i 1.43361 0.134775i
\(490\) 0 0
\(491\) 14395.4 1.32312 0.661562 0.749891i \(-0.269894\pi\)
0.661562 + 0.749891i \(0.269894\pi\)
\(492\) 0 0
\(493\) 13869.4 1.26703
\(494\) 0 0
\(495\) −680.764 3588.66i −0.0618143 0.325855i
\(496\) 0 0
\(497\) 7196.08i 0.649474i
\(498\) 0 0
\(499\) 8887.34i 0.797299i −0.917103 0.398649i \(-0.869479\pi\)
0.917103 0.398649i \(-0.130521\pi\)
\(500\) 0 0
\(501\) −22.7900 242.418i −0.00203230 0.0216176i
\(502\) 0 0
\(503\) 11212.4 0.993911 0.496956 0.867776i \(-0.334451\pi\)
0.496956 + 0.867776i \(0.334451\pi\)
\(504\) 0 0
\(505\) 5702.50 0.502491
\(506\) 0 0
\(507\) −82.1934 874.295i −0.00719987 0.0765854i
\(508\) 0 0
\(509\) 10375.7i 0.903530i −0.892137 0.451765i \(-0.850795\pi\)
0.892137 0.451765i \(-0.149205\pi\)
\(510\) 0 0
\(511\) 6863.73i 0.594195i
\(512\) 0 0
\(513\) −9931.45 + 2868.81i −0.854745 + 0.246903i
\(514\) 0 0
\(515\) −6248.73 −0.534664
\(516\) 0 0
\(517\) 16152.3 1.37404
\(518\) 0 0
\(519\) −11326.6 + 1064.83i −0.957965 + 0.0900593i
\(520\) 0 0
\(521\) 13512.4i 1.13625i −0.822941 0.568127i \(-0.807668\pi\)
0.822941 0.568127i \(-0.192332\pi\)
\(522\) 0 0
\(523\) 11963.6i 1.00025i 0.865952 + 0.500126i \(0.166713\pi\)
−0.865952 + 0.500126i \(0.833287\pi\)
\(524\) 0 0
\(525\) 3738.30 351.441i 0.310767 0.0292156i
\(526\) 0 0
\(527\) 16491.9 1.36318
\(528\) 0 0
\(529\) 25354.1 2.08385
\(530\) 0 0
\(531\) −8362.76 + 1586.40i −0.683451 + 0.129650i
\(532\) 0 0
\(533\) 6662.88i 0.541466i
\(534\) 0 0
\(535\) 2623.67i 0.212021i
\(536\) 0 0
\(537\) −328.712 3496.52i −0.0264152 0.280980i
\(538\) 0 0
\(539\) 9713.91 0.776267
\(540\) 0 0
\(541\) 13732.3 1.09131 0.545654 0.838011i \(-0.316281\pi\)
0.545654 + 0.838011i \(0.316281\pi\)
\(542\) 0 0
\(543\) 76.7030 + 815.893i 0.00606195 + 0.0644813i
\(544\) 0 0
\(545\) 3827.11i 0.300799i
\(546\) 0 0
\(547\) 5437.61i 0.425037i −0.977157 0.212519i \(-0.931833\pi\)
0.977157 0.212519i \(-0.0681666\pi\)
\(548\) 0 0
\(549\) 3547.71 672.995i 0.275797 0.0523183i
\(550\) 0 0
\(551\) −10240.1 −0.791732
\(552\) 0 0
\(553\) −725.104 −0.0557587
\(554\) 0 0
\(555\) −1162.02 + 109.243i −0.0888742 + 0.00835516i
\(556\) 0 0
\(557\) 15910.5i 1.21032i 0.796104 + 0.605159i \(0.206891\pi\)
−0.796104 + 0.605159i \(0.793109\pi\)
\(558\) 0 0
\(559\) 4375.98i 0.331099i
\(560\) 0 0
\(561\) −16827.1 + 1581.93i −1.26638 + 0.119054i
\(562\) 0 0
\(563\) −14833.6 −1.11041 −0.555207 0.831712i \(-0.687361\pi\)
−0.555207 + 0.831712i \(0.687361\pi\)
\(564\) 0 0
\(565\) −3749.38 −0.279181
\(566\) 0 0
\(567\) 1790.07 + 4548.39i 0.132585 + 0.336886i
\(568\) 0 0
\(569\) 11064.8i 0.815224i 0.913155 + 0.407612i \(0.133638\pi\)
−0.913155 + 0.407612i \(0.866362\pi\)
\(570\) 0 0
\(571\) 10460.9i 0.766684i −0.923606 0.383342i \(-0.874773\pi\)
0.923606 0.383342i \(-0.125227\pi\)
\(572\) 0 0
\(573\) −492.734 5241.23i −0.0359237 0.382121i
\(574\) 0 0
\(575\) −20875.7 −1.51404
\(576\) 0 0
\(577\) 17525.4 1.26446 0.632228 0.774782i \(-0.282141\pi\)
0.632228 + 0.774782i \(0.282141\pi\)
\(578\) 0 0
\(579\) −18.7145 199.067i −0.00134326 0.0142883i
\(580\) 0 0
\(581\) 6924.86i 0.494478i
\(582\) 0 0
\(583\) 17517.7i 1.24444i
\(584\) 0 0
\(585\) 271.534 + 1431.40i 0.0191907 + 0.101164i
\(586\) 0 0
\(587\) 8469.95 0.595558 0.297779 0.954635i \(-0.403754\pi\)
0.297779 + 0.954635i \(0.403754\pi\)
\(588\) 0 0
\(589\) −12176.4 −0.851814
\(590\) 0 0
\(591\) −240.853 + 22.6429i −0.0167638 + 0.00157598i
\(592\) 0 0
\(593\) 24461.7i 1.69397i 0.531619 + 0.846984i \(0.321584\pi\)
−0.531619 + 0.846984i \(0.678416\pi\)
\(594\) 0 0
\(595\) 2777.49i 0.191371i
\(596\) 0 0
\(597\) 19408.9 1824.65i 1.33057 0.125089i
\(598\) 0 0
\(599\) 4337.40 0.295862 0.147931 0.988998i \(-0.452739\pi\)
0.147931 + 0.988998i \(0.452739\pi\)
\(600\) 0 0
\(601\) 15309.5 1.03908 0.519541 0.854445i \(-0.326103\pi\)
0.519541 + 0.854445i \(0.326103\pi\)
\(602\) 0 0
\(603\) −662.834 3494.14i −0.0447640 0.235974i
\(604\) 0 0
\(605\) 1115.47i 0.0749591i
\(606\) 0 0
\(607\) 2011.88i 0.134530i −0.997735 0.0672651i \(-0.978573\pi\)
0.997735 0.0672651i \(-0.0214273\pi\)
\(608\) 0 0
\(609\) 453.197 + 4820.67i 0.0301551 + 0.320761i
\(610\) 0 0
\(611\) −6442.63 −0.426581
\(612\) 0 0
\(613\) 3790.69 0.249763 0.124881 0.992172i \(-0.460145\pi\)
0.124881 + 0.992172i \(0.460145\pi\)
\(614\) 0 0
\(615\) 1034.66 + 11005.7i 0.0678400 + 0.721617i
\(616\) 0 0
\(617\) 4591.08i 0.299562i −0.988719 0.149781i \(-0.952143\pi\)
0.988719 0.149781i \(-0.0478569\pi\)
\(618\) 0 0
\(619\) 4037.57i 0.262171i −0.991371 0.131085i \(-0.958154\pi\)
0.991371 0.131085i \(-0.0418462\pi\)
\(620\) 0 0
\(621\) −7541.72 26108.4i −0.487341 1.68711i
\(622\) 0 0
\(623\) −932.903 −0.0599935
\(624\) 0 0
\(625\) 9460.98 0.605503
\(626\) 0 0
\(627\) 12423.9 1167.98i 0.791326 0.0743934i
\(628\) 0 0
\(629\) 5400.54i 0.342343i
\(630\) 0 0
\(631\) 6235.96i 0.393423i 0.980461 + 0.196711i \(0.0630262\pi\)
−0.980461 + 0.196711i \(0.936974\pi\)
\(632\) 0 0
\(633\) 15696.7 1475.66i 0.985603 0.0926576i
\(634\) 0 0
\(635\) 129.899 0.00811791
\(636\) 0 0
\(637\) −3874.55 −0.240997
\(638\) 0 0
\(639\) 28469.7 5400.66i 1.76251 0.334345i
\(640\) 0 0
\(641\) 21379.2i 1.31736i −0.752423 0.658681i \(-0.771115\pi\)
0.752423 0.658681i \(-0.228885\pi\)
\(642\) 0 0
\(643\) 5718.32i 0.350713i 0.984505 + 0.175356i \(0.0561078\pi\)
−0.984505 + 0.175356i \(0.943892\pi\)
\(644\) 0 0
\(645\) 679.535 + 7228.24i 0.0414832 + 0.441259i
\(646\) 0 0
\(647\) −12501.6 −0.759643 −0.379822 0.925060i \(-0.624015\pi\)
−0.379822 + 0.925060i \(0.624015\pi\)
\(648\) 0 0
\(649\) 10274.9 0.621458
\(650\) 0 0
\(651\) 538.888 + 5732.18i 0.0324435 + 0.345103i
\(652\) 0 0
\(653\) 19195.3i 1.15034i −0.818035 0.575169i \(-0.804936\pi\)
0.818035 0.575169i \(-0.195064\pi\)
\(654\) 0 0
\(655\) 1272.80i 0.0759274i
\(656\) 0 0
\(657\) −27154.8 + 5151.23i −1.61250 + 0.305888i
\(658\) 0 0
\(659\) 16527.6 0.976972 0.488486 0.872572i \(-0.337549\pi\)
0.488486 + 0.872572i \(0.337549\pi\)
\(660\) 0 0
\(661\) −22315.8 −1.31314 −0.656569 0.754266i \(-0.727993\pi\)
−0.656569 + 0.754266i \(0.727993\pi\)
\(662\) 0 0
\(663\) 6711.76 630.980i 0.393157 0.0369611i
\(664\) 0 0
\(665\) 2050.69i 0.119582i
\(666\) 0 0
\(667\) 26919.9i 1.56273i
\(668\) 0 0
\(669\) 2598.82 244.317i 0.150188 0.0141194i
\(670\) 0 0
\(671\) −4358.90 −0.250780
\(672\) 0 0
\(673\) 12686.7 0.726653 0.363326 0.931662i \(-0.381641\pi\)
0.363326 + 0.931662i \(0.381641\pi\)
\(674\) 0 0
\(675\) 4195.99 + 14526.0i 0.239265 + 0.828304i
\(676\) 0 0
\(677\) 31590.2i 1.79337i 0.442673 + 0.896683i \(0.354030\pi\)
−0.442673 + 0.896683i \(0.645970\pi\)
\(678\) 0 0
\(679\) 9779.42i 0.552724i
\(680\) 0 0
\(681\) 2251.87 + 23953.2i 0.126713 + 1.34786i
\(682\) 0 0
\(683\) −32381.3 −1.81411 −0.907055 0.421013i \(-0.861675\pi\)
−0.907055 + 0.421013i \(0.861675\pi\)
\(684\) 0 0
\(685\) 293.832 0.0163894
\(686\) 0 0
\(687\) −686.343 7300.66i −0.0381159 0.405441i
\(688\) 0 0
\(689\) 6987.23i 0.386346i
\(690\) 0 0
\(691\) 1530.62i 0.0842658i −0.999112 0.0421329i \(-0.986585\pi\)
0.999112 0.0421329i \(-0.0134153\pi\)
\(692\) 0 0
\(693\) −1099.68 5797.00i −0.0602793 0.317763i
\(694\) 0 0
\(695\) −3012.07 −0.164395
\(696\) 0 0
\(697\) 51149.4 2.77966
\(698\) 0 0
\(699\) −31071.3 + 2921.05i −1.68129 + 0.158060i
\(700\) 0 0
\(701\) 8696.96i 0.468587i −0.972166 0.234294i \(-0.924722\pi\)
0.972166 0.234294i \(-0.0752777\pi\)
\(702\) 0 0
\(703\) 3987.35i 0.213920i
\(704\) 0 0
\(705\) 10641.9 1000.46i 0.568508 0.0534460i
\(706\) 0 0
\(707\) 9211.63 0.490013
\(708\) 0 0
\(709\) −31191.9 −1.65224 −0.826120 0.563494i \(-0.809457\pi\)
−0.826120 + 0.563494i \(0.809457\pi\)
\(710\) 0 0
\(711\) −544.190 2868.71i −0.0287043 0.151315i
\(712\) 0 0
\(713\) 32010.0i 1.68132i
\(714\) 0 0
\(715\) 1758.69i 0.0919876i
\(716\) 0 0
\(717\) 286.181 + 3044.12i 0.0149061 + 0.158556i
\(718\) 0 0
\(719\) 25726.2 1.33439 0.667194 0.744884i \(-0.267495\pi\)
0.667194 + 0.744884i \(0.267495\pi\)
\(720\) 0 0
\(721\) −10094.0 −0.521387
\(722\) 0 0
\(723\) 3184.32 + 33871.8i 0.163798 + 1.74233i
\(724\) 0 0
\(725\) 14977.5i 0.767240i
\(726\) 0 0
\(727\) 4644.52i 0.236940i −0.992958 0.118470i \(-0.962201\pi\)
0.992958 0.118470i \(-0.0377990\pi\)
\(728\) 0 0
\(729\) −16651.2 + 10495.6i −0.845971 + 0.533229i
\(730\) 0 0
\(731\) 33593.4 1.69972
\(732\) 0 0
\(733\) −463.154 −0.0233383 −0.0116692 0.999932i \(-0.503714\pi\)
−0.0116692 + 0.999932i \(0.503714\pi\)
\(734\) 0 0
\(735\) 6399.98 601.669i 0.321179 0.0301944i
\(736\) 0 0
\(737\) 4293.08i 0.214570i
\(738\) 0 0
\(739\) 2346.21i 0.116789i 0.998294 + 0.0583943i \(0.0185981\pi\)
−0.998294 + 0.0583943i \(0.981402\pi\)
\(740\) 0 0
\(741\) −4955.46 + 465.868i −0.245673 + 0.0230960i
\(742\) 0 0
\(743\) 14287.4 0.705454 0.352727 0.935726i \(-0.385254\pi\)
0.352727 + 0.935726i \(0.385254\pi\)
\(744\) 0 0
\(745\) −8588.01 −0.422336
\(746\) 0 0
\(747\) −27396.6 + 5197.11i −1.34189 + 0.254555i
\(748\) 0 0
\(749\) 4238.20i 0.206756i
\(750\) 0 0
\(751\) 5991.43i 0.291119i −0.989350 0.145559i \(-0.953502\pi\)
0.989350 0.145559i \(-0.0464982\pi\)
\(752\) 0 0
\(753\) −1205.13 12819.0i −0.0583232 0.620387i
\(754\) 0 0
\(755\) −12595.9 −0.607170
\(756\) 0 0
\(757\) 15089.4 0.724481 0.362240 0.932085i \(-0.382012\pi\)
0.362240 + 0.932085i \(0.382012\pi\)
\(758\) 0 0
\(759\) 3070.47 + 32660.7i 0.146839 + 1.56193i
\(760\) 0 0
\(761\) 26872.4i 1.28006i 0.768352 + 0.640028i \(0.221077\pi\)
−0.768352 + 0.640028i \(0.778923\pi\)
\(762\) 0 0
\(763\) 6182.19i 0.293329i
\(764\) 0 0
\(765\) −10988.5 + 2084.50i −0.519333 + 0.0985168i
\(766\) 0 0
\(767\) −4098.32 −0.192936
\(768\) 0 0
\(769\) −23951.3 −1.12316 −0.561578 0.827424i \(-0.689806\pi\)
−0.561578 + 0.827424i \(0.689806\pi\)
\(770\) 0 0
\(771\) −1943.11 + 182.674i −0.0907646 + 0.00853287i
\(772\) 0 0
\(773\) 26706.8i 1.24266i −0.783548 0.621331i \(-0.786592\pi\)
0.783548 0.621331i \(-0.213408\pi\)
\(774\) 0 0
\(775\) 17809.4i 0.825463i
\(776\) 0 0
\(777\) −1877.10 + 176.468i −0.0866673 + 0.00814769i
\(778\) 0 0
\(779\) −37764.9 −1.73693
\(780\) 0 0
\(781\) −34979.3 −1.60264
\(782\) 0 0
\(783\) −18731.8 + 5410.88i −0.854941 + 0.246959i
\(784\) 0 0
\(785\) 14417.3i 0.655512i
\(786\) 0 0
\(787\) 29838.7i 1.35151i −0.737128 0.675753i \(-0.763819\pi\)
0.737128 0.675753i \(-0.236181\pi\)
\(788\) 0 0
\(789\) −3565.77 37929.2i −0.160893 1.71143i
\(790\) 0 0
\(791\) −6056.62 −0.272249
\(792\) 0 0
\(793\) 1738.62 0.0778564
\(794\) 0 0
\(795\) −1085.03 11541.5i −0.0484050 0.514887i
\(796\) 0 0
\(797\) 9775.89i 0.434479i −0.976118 0.217240i \(-0.930295\pi\)
0.976118 0.217240i \(-0.0697053\pi\)
\(798\) 0 0
\(799\) 49458.6i 2.18989i
\(800\) 0 0
\(801\) −700.144 3690.82i −0.0308844 0.162807i
\(802\) 0 0
\(803\) 33363.8 1.46623
\(804\) 0 0
\(805\) 5390.98 0.236034
\(806\) 0 0
\(807\) −10073.6 + 947.031i −0.439415 + 0.0413099i
\(808\) 0 0
\(809\) 20854.3i 0.906303i 0.891433 + 0.453152i \(0.149700\pi\)
−0.891433 + 0.453152i \(0.850300\pi\)
\(810\) 0 0
\(811\) 5752.41i 0.249068i −0.992215 0.124534i \(-0.960256\pi\)
0.992215 0.124534i \(-0.0397436\pi\)
\(812\) 0 0
\(813\) −12642.9 + 1188.57i −0.545393 + 0.0512730i
\(814\) 0 0
\(815\) −12438.0 −0.534584
\(816\) 0 0
\(817\) −24802.9 −1.06211
\(818\) 0 0
\(819\) 438.627 + 2312.23i 0.0187141 + 0.0986518i
\(820\) 0 0
\(821\) 6934.64i 0.294788i 0.989078 + 0.147394i \(0.0470885\pi\)
−0.989078 + 0.147394i \(0.952912\pi\)
\(822\) 0 0
\(823\) 26056.7i 1.10362i −0.833970 0.551810i \(-0.813937\pi\)
0.833970 0.551810i \(-0.186063\pi\)
\(824\) 0 0
\(825\) −1708.32 18171.4i −0.0720921 0.766846i
\(826\) 0 0
\(827\) 19162.5 0.805740 0.402870 0.915257i \(-0.368013\pi\)
0.402870 + 0.915257i \(0.368013\pi\)
\(828\) 0 0
\(829\) −36255.5 −1.51895 −0.759473 0.650539i \(-0.774543\pi\)
−0.759473 + 0.650539i \(0.774543\pi\)
\(830\) 0 0
\(831\) 1829.30 + 19458.4i 0.0763632 + 0.812279i
\(832\) 0 0
\(833\) 29744.1i 1.23718i
\(834\) 0 0
\(835\) 194.501i 0.00806107i
\(836\) 0 0
\(837\) −22273.6 + 6433.99i −0.919820 + 0.265700i
\(838\) 0 0
\(839\) 14044.0 0.577894 0.288947 0.957345i \(-0.406695\pi\)
0.288947 + 0.957345i \(0.406695\pi\)
\(840\) 0 0
\(841\) 5075.02 0.208087
\(842\) 0 0
\(843\) 25940.2 2438.67i 1.05982 0.0996348i
\(844\) 0 0
\(845\) 701.481i 0.0285582i
\(846\) 0 0
\(847\) 1801.89i 0.0730977i
\(848\) 0 0
\(849\) −37269.6 + 3503.76i −1.50658 + 0.141636i
\(850\) 0 0
\(851\) 10482.2 0.422239
\(852\) 0 0
\(853\) 33205.3 1.33286 0.666428 0.745569i \(-0.267822\pi\)
0.666428 + 0.745569i \(0.267822\pi\)
\(854\) 0 0
\(855\) 8113.08 1539.04i 0.324516 0.0615604i
\(856\) 0 0
\(857\) 24286.2i 0.968028i 0.875060 + 0.484014i \(0.160822\pi\)
−0.875060 + 0.484014i \(0.839178\pi\)
\(858\) 0 0
\(859\) 28543.6i 1.13375i −0.823803 0.566876i \(-0.808152\pi\)
0.823803 0.566876i \(-0.191848\pi\)
\(860\) 0 0
\(861\) 1671.36 + 17778.3i 0.0661553 + 0.703697i
\(862\) 0 0
\(863\) −5973.84 −0.235634 −0.117817 0.993035i \(-0.537590\pi\)
−0.117817 + 0.993035i \(0.537590\pi\)
\(864\) 0 0
\(865\) 9087.80 0.357219
\(866\) 0 0
\(867\) 2454.45 + 26108.1i 0.0961446 + 1.02269i
\(868\) 0 0
\(869\) 3524.64i 0.137590i
\(870\) 0 0
\(871\) 1712.37i 0.0666146i
\(872\) 0 0
\(873\) −38690.0 + 7339.45i −1.49995 + 0.284539i
\(874\) 0 0
\(875\) −6478.27 −0.250292
\(876\) 0 0
\(877\) −18301.3 −0.704665 −0.352332 0.935875i \(-0.614611\pi\)
−0.352332 + 0.935875i \(0.614611\pi\)
\(878\) 0 0
\(879\) −33379.9 + 3138.08i −1.28086 + 0.120415i
\(880\) 0 0
\(881\) 40284.2i 1.54053i −0.637722 0.770266i \(-0.720123\pi\)
0.637722 0.770266i \(-0.279877\pi\)
\(882\) 0 0
\(883\) 37019.3i 1.41087i 0.708774 + 0.705435i \(0.249248\pi\)
−0.708774 + 0.705435i \(0.750752\pi\)
\(884\) 0 0
\(885\) 6769.60 636.417i 0.257127 0.0241728i
\(886\) 0 0
\(887\) −5439.74 −0.205917 −0.102959 0.994686i \(-0.532831\pi\)
−0.102959 + 0.994686i \(0.532831\pi\)
\(888\) 0 0
\(889\) 209.834 0.00791633
\(890\) 0 0
\(891\) 22109.2 8701.30i 0.831298 0.327166i
\(892\) 0 0
\(893\) 36516.5i 1.36840i
\(894\) 0 0
\(895\) 2805.40i 0.104775i
\(896\) 0 0
\(897\) −1224.71 13027.2i −0.0455872 0.484913i
\(898\) 0 0
\(899\) −22965.9 −0.852009
\(900\) 0 0
\(901\) −53639.4 −1.98334
\(902\) 0 0
\(903\) 1097.70 + 11676.3i 0.0404531 + 0.430301i
\(904\) 0 0
\(905\) 654.623i 0.0240447i
\(906\) 0 0
\(907\) 14617.8i 0.535144i −0.963538 0.267572i \(-0.913779\pi\)
0.963538 0.267572i \(-0.0862213\pi\)
\(908\) 0 0
\(909\) 6913.33 + 36443.7i 0.252256 + 1.32977i
\(910\) 0 0
\(911\) 21082.5 0.766735 0.383368 0.923596i \(-0.374764\pi\)
0.383368 + 0.923596i \(0.374764\pi\)
\(912\) 0 0
\(913\) 33661.0 1.22017
\(914\) 0 0
\(915\) −2871.85 + 269.985i −0.103760 + 0.00975458i
\(916\) 0 0
\(917\) 2056.04i 0.0740420i
\(918\) 0 0
\(919\) 23894.1i 0.857666i 0.903384 + 0.428833i \(0.141075\pi\)
−0.903384 + 0.428833i \(0.858925\pi\)
\(920\) 0 0
\(921\) −18855.4 + 1772.62i −0.674602 + 0.0634200i
\(922\) 0 0
\(923\) 13952.1 0.497549
\(924\) 0 0
\(925\) −5832.00 −0.207303
\(926\) 0 0
\(927\) −7575.54 39934.6i −0.268407 1.41491i
\(928\) 0 0
\(929\) 51500.7i 1.81882i 0.415903 + 0.909409i \(0.363466\pi\)
−0.415903 + 0.909409i \(0.636534\pi\)
\(930\) 0 0
\(931\) 21960.8i 0.773078i
\(932\) 0 0
\(933\) 2182.83 + 23218.9i 0.0765946 + 0.814740i
\(934\) 0 0
\(935\) 13501.0 0.472226
\(936\) 0 0
\(937\) 2862.58 0.0998041 0.0499021 0.998754i \(-0.484109\pi\)
0.0499021 + 0.998754i \(0.484109\pi\)
\(938\) 0 0
\(939\) 787.794 + 8379.80i 0.0273788 + 0.291230i
\(940\) 0 0
\(941\) 27932.6i 0.967671i −0.875159 0.483835i \(-0.839243\pi\)
0.875159 0.483835i \(-0.160757\pi\)
\(942\) 0 0
\(943\) 99278.9i 3.42838i
\(944\) 0 0
\(945\) −1083.58 3751.22i −0.0373005 0.129129i
\(946\) 0 0
\(947\) −32402.3 −1.11186 −0.555930 0.831229i \(-0.687638\pi\)
−0.555930 + 0.831229i \(0.687638\pi\)
\(948\) 0 0
\(949\) −13307.7 −0.455202
\(950\) 0 0
\(951\) −15793.0 + 1484.72i −0.538509 + 0.0506258i
\(952\) 0 0
\(953\) 15804.2i 0.537197i 0.963252 + 0.268598i \(0.0865605\pi\)
−0.963252 + 0.268598i \(0.913440\pi\)
\(954\) 0 0
\(955\) 4205.25i 0.142491i
\(956\) 0 0
\(957\) 23432.7 2202.94i 0.791508 0.0744105i
\(958\) 0 0
\(959\) 474.647 0.0159824
\(960\) 0 0
\(961\) 2482.64 0.0833353
\(962\) 0 0
\(963\) 16767.5 3180.77i 0.561084 0.106437i
\(964\) 0 0
\(965\) 159.719i 0.00532802i
\(966\) 0 0
\(967\) 39370.7i 1.30928i −0.755939 0.654642i \(-0.772820\pi\)
0.755939 0.654642i \(-0.227180\pi\)
\(968\) 0 0
\(969\) −3576.37 38042.0i −0.118565 1.26118i
\(970\) 0 0
\(971\) −28551.2 −0.943616 −0.471808 0.881701i \(-0.656398\pi\)
−0.471808 + 0.881701i \(0.656398\pi\)
\(972\) 0 0
\(973\) −4865.61 −0.160313
\(974\) 0 0
\(975\) 681.390 + 7247.97i 0.0223815 + 0.238073i
\(976\) 0 0
\(977\) 39675.1i 1.29920i −0.760275 0.649601i \(-0.774936\pi\)
0.760275 0.649601i \(-0.225064\pi\)
\(978\) 0 0
\(979\) 4534.73i 0.148040i
\(980\) 0 0
\(981\) −24458.4 + 4639.73i −0.796022 + 0.151004i
\(982\) 0 0
\(983\) −7940.67 −0.257648 −0.128824 0.991667i \(-0.541120\pi\)
−0.128824 + 0.991667i \(0.541120\pi\)
\(984\) 0 0
\(985\) 193.246 0.00625110
\(986\) 0 0
\(987\) 17190.6 1616.11i 0.554390 0.0521188i
\(988\) 0 0
\(989\) 65203.4i 2.09641i
\(990\) 0 0
\(991\) 46308.7i 1.48440i −0.670177 0.742201i \(-0.733782\pi\)
0.670177 0.742201i \(-0.266218\pi\)
\(992\) 0 0
\(993\) −14584.8 + 1371.14i −0.466099 + 0.0438184i
\(994\) 0 0
\(995\) −15572.5 −0.496162
\(996\) 0 0
\(997\) 46001.3 1.46126 0.730630 0.682774i \(-0.239226\pi\)
0.730630 + 0.682774i \(0.239226\pi\)
\(998\) 0 0
\(999\) −2106.92 7293.87i −0.0667266 0.230999i
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 624.4.d.c.287.11 24
3.2 odd 2 inner 624.4.d.c.287.13 yes 24
4.3 odd 2 inner 624.4.d.c.287.14 yes 24
12.11 even 2 inner 624.4.d.c.287.12 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
624.4.d.c.287.11 24 1.1 even 1 trivial
624.4.d.c.287.12 yes 24 12.11 even 2 inner
624.4.d.c.287.13 yes 24 3.2 odd 2 inner
624.4.d.c.287.14 yes 24 4.3 odd 2 inner