Properties

Label 624.4.d.c.287.9
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 287.9
Character \(\chi\) \(=\) 624.287
Dual form 624.4.d.c.287.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.92666 - 4.82576i) q^{3} +18.6497i q^{5} +4.79314i q^{7} +(-19.5759 + 18.5952i) q^{9} +O(q^{10})\) \(q+(-1.92666 - 4.82576i) q^{3} +18.6497i q^{5} +4.79314i q^{7} +(-19.5759 + 18.5952i) q^{9} +18.4244 q^{11} -13.0000 q^{13} +(89.9990 - 35.9317i) q^{15} +42.4664i q^{17} +58.4499i q^{19} +(23.1305 - 9.23476i) q^{21} +20.6202 q^{23} -222.811 q^{25} +(127.452 + 58.6421i) q^{27} -65.8695i q^{29} -129.239i q^{31} +(-35.4975 - 88.9116i) q^{33} -89.3906 q^{35} -280.300 q^{37} +(25.0466 + 62.7349i) q^{39} -49.8512i q^{41} -336.630i q^{43} +(-346.795 - 365.086i) q^{45} -482.061 q^{47} +320.026 q^{49} +(204.933 - 81.8184i) q^{51} -256.465i q^{53} +343.609i q^{55} +(282.065 - 112.613i) q^{57} +96.1273 q^{59} -275.399 q^{61} +(-89.1295 - 93.8302i) q^{63} -242.446i q^{65} +912.032i q^{67} +(-39.7282 - 99.5082i) q^{69} -842.688 q^{71} -851.253 q^{73} +(429.282 + 1075.23i) q^{75} +88.3105i q^{77} -305.206i q^{79} +(37.4352 - 728.038i) q^{81} -394.418 q^{83} -791.986 q^{85} +(-317.870 + 126.908i) q^{87} +1358.54i q^{89} -62.3108i q^{91} +(-623.675 + 248.999i) q^{93} -1090.07 q^{95} +261.640 q^{97} +(-360.675 + 342.605i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 84 q^{9}+O(q^{10}) \) 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
\(3\) −1.92666 4.82576i −0.370786 0.928718i
\(4\) 0 0
\(5\) 18.6497i 1.66808i 0.551704 + 0.834040i \(0.313978\pi\)
−0.551704 + 0.834040i \(0.686022\pi\)
\(6\) 0 0
\(7\) 4.79314i 0.258805i 0.991592 + 0.129403i \(0.0413059\pi\)
−0.991592 + 0.129403i \(0.958694\pi\)
\(8\) 0 0
\(9\) −19.5759 + 18.5952i −0.725035 + 0.688712i
\(10\) 0 0
\(11\) 18.4244 0.505014 0.252507 0.967595i \(-0.418745\pi\)
0.252507 + 0.967595i \(0.418745\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) 0 0
\(15\) 89.9990 35.9317i 1.54918 0.618501i
\(16\) 0 0
\(17\) 42.4664i 0.605860i 0.953013 + 0.302930i \(0.0979648\pi\)
−0.953013 + 0.302930i \(0.902035\pi\)
\(18\) 0 0
\(19\) 58.4499i 0.705754i 0.935670 + 0.352877i \(0.114797\pi\)
−0.935670 + 0.352877i \(0.885203\pi\)
\(20\) 0 0
\(21\) 23.1305 9.23476i 0.240357 0.0959614i
\(22\) 0 0
\(23\) 20.6202 0.186940 0.0934698 0.995622i \(-0.470204\pi\)
0.0934698 + 0.995622i \(0.470204\pi\)
\(24\) 0 0
\(25\) −222.811 −1.78249
\(26\) 0 0
\(27\) 127.452 + 58.6421i 0.908452 + 0.417988i
\(28\) 0 0
\(29\) 65.8695i 0.421781i −0.977510 0.210891i \(-0.932364\pi\)
0.977510 0.210891i \(-0.0676364\pi\)
\(30\) 0 0
\(31\) 129.239i 0.748773i −0.927273 0.374386i \(-0.877853\pi\)
0.927273 0.374386i \(-0.122147\pi\)
\(32\) 0 0
\(33\) −35.4975 88.9116i −0.187252 0.469016i
\(34\) 0 0
\(35\) −89.3906 −0.431708
\(36\) 0 0
\(37\) −280.300 −1.24543 −0.622717 0.782447i \(-0.713971\pi\)
−0.622717 + 0.782447i \(0.713971\pi\)
\(38\) 0 0
\(39\) 25.0466 + 62.7349i 0.102838 + 0.257580i
\(40\) 0 0
\(41\) 49.8512i 0.189889i −0.995483 0.0949446i \(-0.969733\pi\)
0.995483 0.0949446i \(-0.0302674\pi\)
\(42\) 0 0
\(43\) 336.630i 1.19385i −0.802297 0.596925i \(-0.796389\pi\)
0.802297 0.596925i \(-0.203611\pi\)
\(44\) 0 0
\(45\) −346.795 365.086i −1.14883 1.20942i
\(46\) 0 0
\(47\) −482.061 −1.49608 −0.748041 0.663652i \(-0.769005\pi\)
−0.748041 + 0.663652i \(0.769005\pi\)
\(48\) 0 0
\(49\) 320.026 0.933020
\(50\) 0 0
\(51\) 204.933 81.8184i 0.562673 0.224645i
\(52\) 0 0
\(53\) 256.465i 0.664682i −0.943159 0.332341i \(-0.892161\pi\)
0.943159 0.332341i \(-0.107839\pi\)
\(54\) 0 0
\(55\) 343.609i 0.842404i
\(56\) 0 0
\(57\) 282.065 112.613i 0.655447 0.261684i
\(58\) 0 0
\(59\) 96.1273 0.212114 0.106057 0.994360i \(-0.466177\pi\)
0.106057 + 0.994360i \(0.466177\pi\)
\(60\) 0 0
\(61\) −275.399 −0.578053 −0.289027 0.957321i \(-0.593332\pi\)
−0.289027 + 0.957321i \(0.593332\pi\)
\(62\) 0 0
\(63\) −89.1295 93.8302i −0.178242 0.187643i
\(64\) 0 0
\(65\) 242.446i 0.462642i
\(66\) 0 0
\(67\) 912.032i 1.66302i 0.555508 + 0.831511i \(0.312524\pi\)
−0.555508 + 0.831511i \(0.687476\pi\)
\(68\) 0 0
\(69\) −39.7282 99.5082i −0.0693146 0.173614i
\(70\) 0 0
\(71\) −842.688 −1.40857 −0.704286 0.709916i \(-0.748733\pi\)
−0.704286 + 0.709916i \(0.748733\pi\)
\(72\) 0 0
\(73\) −851.253 −1.36482 −0.682408 0.730971i \(-0.739067\pi\)
−0.682408 + 0.730971i \(0.739067\pi\)
\(74\) 0 0
\(75\) 429.282 + 1075.23i 0.660923 + 1.65543i
\(76\) 0 0
\(77\) 88.3105i 0.130700i
\(78\) 0 0
\(79\) 305.206i 0.434663i −0.976098 0.217331i \(-0.930265\pi\)
0.976098 0.217331i \(-0.0697353\pi\)
\(80\) 0 0
\(81\) 37.4352 728.038i 0.0513515 0.998681i
\(82\) 0 0
\(83\) −394.418 −0.521602 −0.260801 0.965393i \(-0.583987\pi\)
−0.260801 + 0.965393i \(0.583987\pi\)
\(84\) 0 0
\(85\) −791.986 −1.01062
\(86\) 0 0
\(87\) −317.870 + 126.908i −0.391716 + 0.156391i
\(88\) 0 0
\(89\) 1358.54i 1.61803i 0.587785 + 0.809017i \(0.300000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(90\) 0 0
\(91\) 62.3108i 0.0717796i
\(92\) 0 0
\(93\) −623.675 + 248.999i −0.695399 + 0.277635i
\(94\) 0 0
\(95\) −1090.07 −1.17725
\(96\) 0 0
\(97\) 261.640 0.273871 0.136936 0.990580i \(-0.456275\pi\)
0.136936 + 0.990580i \(0.456275\pi\)
\(98\) 0 0
\(99\) −360.675 + 342.605i −0.366153 + 0.347809i
\(100\) 0 0
\(101\) 136.738i 0.134712i −0.997729 0.0673562i \(-0.978544\pi\)
0.997729 0.0673562i \(-0.0214564\pi\)
\(102\) 0 0
\(103\) 1253.06i 1.19872i 0.800481 + 0.599359i \(0.204578\pi\)
−0.800481 + 0.599359i \(0.795422\pi\)
\(104\) 0 0
\(105\) 172.225 + 431.378i 0.160071 + 0.400935i
\(106\) 0 0
\(107\) 1030.95 0.931454 0.465727 0.884929i \(-0.345793\pi\)
0.465727 + 0.884929i \(0.345793\pi\)
\(108\) 0 0
\(109\) −1915.84 −1.68353 −0.841764 0.539846i \(-0.818483\pi\)
−0.841764 + 0.539846i \(0.818483\pi\)
\(110\) 0 0
\(111\) 540.044 + 1352.66i 0.461790 + 1.15666i
\(112\) 0 0
\(113\) 27.0817i 0.0225454i −0.999936 0.0112727i \(-0.996412\pi\)
0.999936 0.0112727i \(-0.00358829\pi\)
\(114\) 0 0
\(115\) 384.561i 0.311830i
\(116\) 0 0
\(117\) 254.487 241.738i 0.201089 0.191014i
\(118\) 0 0
\(119\) −203.547 −0.156800
\(120\) 0 0
\(121\) −991.542 −0.744961
\(122\) 0 0
\(123\) −240.570 + 96.0465i −0.176354 + 0.0704083i
\(124\) 0 0
\(125\) 1824.15i 1.30526i
\(126\) 0 0
\(127\) 307.817i 0.215074i −0.994201 0.107537i \(-0.965704\pi\)
0.994201 0.107537i \(-0.0342964\pi\)
\(128\) 0 0
\(129\) −1624.50 + 648.572i −1.10875 + 0.442664i
\(130\) 0 0
\(131\) 1571.57 1.04815 0.524077 0.851671i \(-0.324410\pi\)
0.524077 + 0.851671i \(0.324410\pi\)
\(132\) 0 0
\(133\) −280.158 −0.182653
\(134\) 0 0
\(135\) −1093.66 + 2376.95i −0.697238 + 1.51537i
\(136\) 0 0
\(137\) 2565.67i 1.60000i 0.600001 + 0.799999i \(0.295167\pi\)
−0.600001 + 0.799999i \(0.704833\pi\)
\(138\) 0 0
\(139\) 1134.72i 0.692413i −0.938158 0.346206i \(-0.887470\pi\)
0.938158 0.346206i \(-0.112530\pi\)
\(140\) 0 0
\(141\) 928.769 + 2326.31i 0.554727 + 1.38944i
\(142\) 0 0
\(143\) −239.517 −0.140066
\(144\) 0 0
\(145\) 1228.45 0.703565
\(146\) 0 0
\(147\) −616.582 1544.37i −0.345951 0.866513i
\(148\) 0 0
\(149\) 2873.59i 1.57996i −0.613133 0.789980i \(-0.710091\pi\)
0.613133 0.789980i \(-0.289909\pi\)
\(150\) 0 0
\(151\) 898.495i 0.484228i −0.970248 0.242114i \(-0.922159\pi\)
0.970248 0.242114i \(-0.0778408\pi\)
\(152\) 0 0
\(153\) −789.672 831.320i −0.417263 0.439270i
\(154\) 0 0
\(155\) 2410.26 1.24901
\(156\) 0 0
\(157\) −1045.24 −0.531333 −0.265666 0.964065i \(-0.585592\pi\)
−0.265666 + 0.964065i \(0.585592\pi\)
\(158\) 0 0
\(159\) −1237.64 + 494.121i −0.617303 + 0.246455i
\(160\) 0 0
\(161\) 98.8355i 0.0483809i
\(162\) 0 0
\(163\) 3632.61i 1.74557i −0.488105 0.872785i \(-0.662312\pi\)
0.488105 0.872785i \(-0.337688\pi\)
\(164\) 0 0
\(165\) 1658.18 662.019i 0.782356 0.312352i
\(166\) 0 0
\(167\) 2154.53 0.998339 0.499169 0.866504i \(-0.333639\pi\)
0.499169 + 0.866504i \(0.333639\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) −1086.89 1144.21i −0.486061 0.511697i
\(172\) 0 0
\(173\) 4197.69i 1.84477i −0.386275 0.922384i \(-0.626238\pi\)
0.386275 0.922384i \(-0.373762\pi\)
\(174\) 0 0
\(175\) 1067.97i 0.461318i
\(176\) 0 0
\(177\) −185.205 463.887i −0.0786488 0.196994i
\(178\) 0 0
\(179\) −2570.05 −1.07316 −0.536578 0.843851i \(-0.680283\pi\)
−0.536578 + 0.843851i \(0.680283\pi\)
\(180\) 0 0
\(181\) −1887.31 −0.775043 −0.387522 0.921861i \(-0.626669\pi\)
−0.387522 + 0.921861i \(0.626669\pi\)
\(182\) 0 0
\(183\) 530.601 + 1329.01i 0.214334 + 0.536849i
\(184\) 0 0
\(185\) 5227.52i 2.07748i
\(186\) 0 0
\(187\) 782.417i 0.305968i
\(188\) 0 0
\(189\) −281.080 + 610.897i −0.108177 + 0.235112i
\(190\) 0 0
\(191\) 140.357 0.0531722 0.0265861 0.999647i \(-0.491536\pi\)
0.0265861 + 0.999647i \(0.491536\pi\)
\(192\) 0 0
\(193\) 236.686 0.0882749 0.0441374 0.999025i \(-0.485946\pi\)
0.0441374 + 0.999025i \(0.485946\pi\)
\(194\) 0 0
\(195\) −1169.99 + 467.112i −0.429664 + 0.171541i
\(196\) 0 0
\(197\) 1004.89i 0.363428i 0.983351 + 0.181714i \(0.0581645\pi\)
−0.983351 + 0.181714i \(0.941836\pi\)
\(198\) 0 0
\(199\) 671.172i 0.239086i 0.992829 + 0.119543i \(0.0381430\pi\)
−0.992829 + 0.119543i \(0.961857\pi\)
\(200\) 0 0
\(201\) 4401.25 1757.18i 1.54448 0.616626i
\(202\) 0 0
\(203\) 315.721 0.109159
\(204\) 0 0
\(205\) 929.711 0.316750
\(206\) 0 0
\(207\) −403.660 + 383.437i −0.135538 + 0.128748i
\(208\) 0 0
\(209\) 1076.90i 0.356416i
\(210\) 0 0
\(211\) 4598.85i 1.50046i −0.661174 0.750232i \(-0.729942\pi\)
0.661174 0.750232i \(-0.270058\pi\)
\(212\) 0 0
\(213\) 1623.57 + 4066.61i 0.522279 + 1.30817i
\(214\) 0 0
\(215\) 6278.05 1.99144
\(216\) 0 0
\(217\) 619.459 0.193786
\(218\) 0 0
\(219\) 1640.08 + 4107.94i 0.506055 + 1.26753i
\(220\) 0 0
\(221\) 552.063i 0.168035i
\(222\) 0 0
\(223\) 1793.95i 0.538707i −0.963041 0.269354i \(-0.913190\pi\)
0.963041 0.269354i \(-0.0868101\pi\)
\(224\) 0 0
\(225\) 4361.74 4143.23i 1.29237 1.22762i
\(226\) 0 0
\(227\) −4388.97 −1.28329 −0.641644 0.767003i \(-0.721747\pi\)
−0.641644 + 0.767003i \(0.721747\pi\)
\(228\) 0 0
\(229\) −1744.68 −0.503456 −0.251728 0.967798i \(-0.580999\pi\)
−0.251728 + 0.967798i \(0.580999\pi\)
\(230\) 0 0
\(231\) 426.166 170.145i 0.121384 0.0484619i
\(232\) 0 0
\(233\) 6046.30i 1.70003i 0.526760 + 0.850014i \(0.323407\pi\)
−0.526760 + 0.850014i \(0.676593\pi\)
\(234\) 0 0
\(235\) 8990.30i 2.49558i
\(236\) 0 0
\(237\) −1472.85 + 588.029i −0.403679 + 0.161167i
\(238\) 0 0
\(239\) 2382.04 0.644692 0.322346 0.946622i \(-0.395529\pi\)
0.322346 + 0.946622i \(0.395529\pi\)
\(240\) 0 0
\(241\) −4961.66 −1.32618 −0.663089 0.748540i \(-0.730755\pi\)
−0.663089 + 0.748540i \(0.730755\pi\)
\(242\) 0 0
\(243\) −3585.46 + 1222.03i −0.946533 + 0.322606i
\(244\) 0 0
\(245\) 5968.39i 1.55635i
\(246\) 0 0
\(247\) 759.849i 0.195741i
\(248\) 0 0
\(249\) 759.910 + 1903.37i 0.193403 + 0.484421i
\(250\) 0 0
\(251\) 4132.31 1.03916 0.519579 0.854422i \(-0.326089\pi\)
0.519579 + 0.854422i \(0.326089\pi\)
\(252\) 0 0
\(253\) 379.914 0.0944072
\(254\) 0 0
\(255\) 1525.89 + 3821.93i 0.374725 + 0.938584i
\(256\) 0 0
\(257\) 7269.38i 1.76440i 0.470871 + 0.882202i \(0.343940\pi\)
−0.470871 + 0.882202i \(0.656060\pi\)
\(258\) 0 0
\(259\) 1343.52i 0.322325i
\(260\) 0 0
\(261\) 1224.86 + 1289.46i 0.290486 + 0.305806i
\(262\) 0 0
\(263\) −4002.71 −0.938470 −0.469235 0.883073i \(-0.655470\pi\)
−0.469235 + 0.883073i \(0.655470\pi\)
\(264\) 0 0
\(265\) 4782.99 1.10874
\(266\) 0 0
\(267\) 6556.00 2617.45i 1.50270 0.599945i
\(268\) 0 0
\(269\) 4028.73i 0.913145i −0.889686 0.456572i \(-0.849077\pi\)
0.889686 0.456572i \(-0.150923\pi\)
\(270\) 0 0
\(271\) 3465.52i 0.776810i 0.921489 + 0.388405i \(0.126974\pi\)
−0.921489 + 0.388405i \(0.873026\pi\)
\(272\) 0 0
\(273\) −300.697 + 120.052i −0.0666630 + 0.0266149i
\(274\) 0 0
\(275\) −4105.16 −0.900184
\(276\) 0 0
\(277\) 9176.56 1.99049 0.995245 0.0974014i \(-0.0310531\pi\)
0.995245 + 0.0974014i \(0.0310531\pi\)
\(278\) 0 0
\(279\) 2403.22 + 2529.97i 0.515689 + 0.542886i
\(280\) 0 0
\(281\) 4185.54i 0.888570i 0.895885 + 0.444285i \(0.146542\pi\)
−0.895885 + 0.444285i \(0.853458\pi\)
\(282\) 0 0
\(283\) 1604.62i 0.337049i −0.985697 0.168525i \(-0.946100\pi\)
0.985697 0.168525i \(-0.0539003\pi\)
\(284\) 0 0
\(285\) 2100.20 + 5260.44i 0.436510 + 1.09334i
\(286\) 0 0
\(287\) 238.944 0.0491443
\(288\) 0 0
\(289\) 3109.60 0.632934
\(290\) 0 0
\(291\) −504.092 1262.61i −0.101548 0.254349i
\(292\) 0 0
\(293\) 6509.95i 1.29800i 0.760786 + 0.649002i \(0.224814\pi\)
−0.760786 + 0.649002i \(0.775186\pi\)
\(294\) 0 0
\(295\) 1792.75i 0.353823i
\(296\) 0 0
\(297\) 2348.23 + 1080.44i 0.458782 + 0.211090i
\(298\) 0 0
\(299\) −268.063 −0.0518477
\(300\) 0 0
\(301\) 1613.51 0.308975
\(302\) 0 0
\(303\) −659.865 + 263.448i −0.125110 + 0.0499495i
\(304\) 0 0
\(305\) 5136.11i 0.964239i
\(306\) 0 0
\(307\) 1209.53i 0.224859i 0.993660 + 0.112429i \(0.0358632\pi\)
−0.993660 + 0.112429i \(0.964137\pi\)
\(308\) 0 0
\(309\) 6046.98 2414.23i 1.11327 0.444468i
\(310\) 0 0
\(311\) 5302.65 0.966835 0.483418 0.875390i \(-0.339395\pi\)
0.483418 + 0.875390i \(0.339395\pi\)
\(312\) 0 0
\(313\) −1535.70 −0.277326 −0.138663 0.990340i \(-0.544281\pi\)
−0.138663 + 0.990340i \(0.544281\pi\)
\(314\) 0 0
\(315\) 1749.91 1662.24i 0.313003 0.297322i
\(316\) 0 0
\(317\) 5733.47i 1.01585i 0.861402 + 0.507924i \(0.169587\pi\)
−0.861402 + 0.507924i \(0.830413\pi\)
\(318\) 0 0
\(319\) 1213.60i 0.213006i
\(320\) 0 0
\(321\) −1986.29 4975.11i −0.345370 0.865058i
\(322\) 0 0
\(323\) −2482.16 −0.427588
\(324\) 0 0
\(325\) 2896.55 0.494374
\(326\) 0 0
\(327\) 3691.18 + 9245.40i 0.624229 + 1.56352i
\(328\) 0 0
\(329\) 2310.59i 0.387194i
\(330\) 0 0
\(331\) 4619.03i 0.767024i −0.923536 0.383512i \(-0.874714\pi\)
0.923536 0.383512i \(-0.125286\pi\)
\(332\) 0 0
\(333\) 5487.14 5212.25i 0.902984 0.857746i
\(334\) 0 0
\(335\) −17009.1 −2.77405
\(336\) 0 0
\(337\) −4891.93 −0.790743 −0.395371 0.918521i \(-0.629384\pi\)
−0.395371 + 0.918521i \(0.629384\pi\)
\(338\) 0 0
\(339\) −130.690 + 52.1773i −0.0209383 + 0.00835954i
\(340\) 0 0
\(341\) 2381.14i 0.378141i
\(342\) 0 0
\(343\) 3177.97i 0.500275i
\(344\) 0 0
\(345\) 1855.80 740.919i 0.289602 0.115622i
\(346\) 0 0
\(347\) −2572.15 −0.397926 −0.198963 0.980007i \(-0.563757\pi\)
−0.198963 + 0.980007i \(0.563757\pi\)
\(348\) 0 0
\(349\) −345.160 −0.0529398 −0.0264699 0.999650i \(-0.508427\pi\)
−0.0264699 + 0.999650i \(0.508427\pi\)
\(350\) 0 0
\(351\) −1656.88 762.348i −0.251959 0.115929i
\(352\) 0 0
\(353\) 2193.51i 0.330733i 0.986232 + 0.165366i \(0.0528807\pi\)
−0.986232 + 0.165366i \(0.947119\pi\)
\(354\) 0 0
\(355\) 15715.9i 2.34961i
\(356\) 0 0
\(357\) 392.167 + 982.271i 0.0581391 + 0.145623i
\(358\) 0 0
\(359\) −9166.56 −1.34761 −0.673805 0.738909i \(-0.735341\pi\)
−0.673805 + 0.738909i \(0.735341\pi\)
\(360\) 0 0
\(361\) 3442.61 0.501911
\(362\) 0 0
\(363\) 1910.37 + 4784.95i 0.276221 + 0.691858i
\(364\) 0 0
\(365\) 15875.6i 2.27662i
\(366\) 0 0
\(367\) 9153.52i 1.30193i −0.759106 0.650967i \(-0.774363\pi\)
0.759106 0.650967i \(-0.225637\pi\)
\(368\) 0 0
\(369\) 926.995 + 975.885i 0.130779 + 0.137676i
\(370\) 0 0
\(371\) 1229.27 0.172023
\(372\) 0 0
\(373\) −7204.23 −1.00006 −0.500028 0.866009i \(-0.666677\pi\)
−0.500028 + 0.866009i \(0.666677\pi\)
\(374\) 0 0
\(375\) −8802.93 + 3514.53i −1.21222 + 0.483972i
\(376\) 0 0
\(377\) 856.303i 0.116981i
\(378\) 0 0
\(379\) 5029.01i 0.681591i −0.940137 0.340795i \(-0.889304\pi\)
0.940137 0.340795i \(-0.110696\pi\)
\(380\) 0 0
\(381\) −1485.45 + 593.060i −0.199743 + 0.0797464i
\(382\) 0 0
\(383\) −361.928 −0.0482864 −0.0241432 0.999709i \(-0.507686\pi\)
−0.0241432 + 0.999709i \(0.507686\pi\)
\(384\) 0 0
\(385\) −1646.97 −0.218019
\(386\) 0 0
\(387\) 6259.71 + 6589.85i 0.822220 + 0.865584i
\(388\) 0 0
\(389\) 6275.67i 0.817966i 0.912542 + 0.408983i \(0.134117\pi\)
−0.912542 + 0.408983i \(0.865883\pi\)
\(390\) 0 0
\(391\) 875.666i 0.113259i
\(392\) 0 0
\(393\) −3027.88 7584.00i −0.388642 0.973441i
\(394\) 0 0
\(395\) 5692.00 0.725053
\(396\) 0 0
\(397\) 6850.54 0.866042 0.433021 0.901384i \(-0.357448\pi\)
0.433021 + 0.901384i \(0.357448\pi\)
\(398\) 0 0
\(399\) 539.771 + 1351.98i 0.0677252 + 0.169633i
\(400\) 0 0
\(401\) 170.197i 0.0211951i 0.999944 + 0.0105976i \(0.00337337\pi\)
−0.999944 + 0.0105976i \(0.996627\pi\)
\(402\) 0 0
\(403\) 1680.10i 0.207672i
\(404\) 0 0
\(405\) 13577.7 + 698.156i 1.66588 + 0.0856584i
\(406\) 0 0
\(407\) −5164.36 −0.628962
\(408\) 0 0
\(409\) 10435.9 1.26167 0.630836 0.775916i \(-0.282712\pi\)
0.630836 + 0.775916i \(0.282712\pi\)
\(410\) 0 0
\(411\) 12381.3 4943.18i 1.48595 0.593258i
\(412\) 0 0
\(413\) 460.751i 0.0548961i
\(414\) 0 0
\(415\) 7355.77i 0.870074i
\(416\) 0 0
\(417\) −5475.87 + 2186.21i −0.643056 + 0.256737i
\(418\) 0 0
\(419\) 16146.5 1.88260 0.941300 0.337572i \(-0.109606\pi\)
0.941300 + 0.337572i \(0.109606\pi\)
\(420\) 0 0
\(421\) −3152.04 −0.364896 −0.182448 0.983216i \(-0.558402\pi\)
−0.182448 + 0.983216i \(0.558402\pi\)
\(422\) 0 0
\(423\) 9436.80 8964.04i 1.08471 1.03037i
\(424\) 0 0
\(425\) 9462.00i 1.07994i
\(426\) 0 0
\(427\) 1320.03i 0.149603i
\(428\) 0 0
\(429\) 461.468 + 1155.85i 0.0519345 + 0.130082i
\(430\) 0 0
\(431\) 3825.08 0.427489 0.213744 0.976890i \(-0.431434\pi\)
0.213744 + 0.976890i \(0.431434\pi\)
\(432\) 0 0
\(433\) −12260.9 −1.36079 −0.680394 0.732846i \(-0.738192\pi\)
−0.680394 + 0.732846i \(0.738192\pi\)
\(434\) 0 0
\(435\) −2366.80 5928.19i −0.260872 0.653413i
\(436\) 0 0
\(437\) 1205.25i 0.131933i
\(438\) 0 0
\(439\) 17157.7i 1.86535i 0.360714 + 0.932677i \(0.382533\pi\)
−0.360714 + 0.932677i \(0.617467\pi\)
\(440\) 0 0
\(441\) −6264.81 + 5950.95i −0.676472 + 0.642582i
\(442\) 0 0
\(443\) 11111.1 1.19166 0.595831 0.803110i \(-0.296823\pi\)
0.595831 + 0.803110i \(0.296823\pi\)
\(444\) 0 0
\(445\) −25336.4 −2.69901
\(446\) 0 0
\(447\) −13867.3 + 5536.44i −1.46734 + 0.585827i
\(448\) 0 0
\(449\) 2795.70i 0.293847i 0.989148 + 0.146924i \(0.0469371\pi\)
−0.989148 + 0.146924i \(0.953063\pi\)
\(450\) 0 0
\(451\) 918.478i 0.0958968i
\(452\) 0 0
\(453\) −4335.92 + 1731.10i −0.449711 + 0.179545i
\(454\) 0 0
\(455\) 1162.08 0.119734
\(456\) 0 0
\(457\) −3631.78 −0.371745 −0.185872 0.982574i \(-0.559511\pi\)
−0.185872 + 0.982574i \(0.559511\pi\)
\(458\) 0 0
\(459\) −2490.32 + 5412.44i −0.253242 + 0.550395i
\(460\) 0 0
\(461\) 12251.8i 1.23780i 0.785471 + 0.618898i \(0.212421\pi\)
−0.785471 + 0.618898i \(0.787579\pi\)
\(462\) 0 0
\(463\) 13122.3i 1.31716i 0.752510 + 0.658581i \(0.228843\pi\)
−0.752510 + 0.658581i \(0.771157\pi\)
\(464\) 0 0
\(465\) −4643.76 11631.4i −0.463117 1.15998i
\(466\) 0 0
\(467\) 14522.9 1.43905 0.719526 0.694465i \(-0.244359\pi\)
0.719526 + 0.694465i \(0.244359\pi\)
\(468\) 0 0
\(469\) −4371.50 −0.430398
\(470\) 0 0
\(471\) 2013.83 + 5044.08i 0.197011 + 0.493459i
\(472\) 0 0
\(473\) 6202.20i 0.602912i
\(474\) 0 0
\(475\) 13023.3i 1.25800i
\(476\) 0 0
\(477\) 4769.02 + 5020.54i 0.457775 + 0.481918i
\(478\) 0 0
\(479\) −17609.3 −1.67973 −0.839864 0.542797i \(-0.817365\pi\)
−0.839864 + 0.542797i \(0.817365\pi\)
\(480\) 0 0
\(481\) 3643.90 0.345421
\(482\) 0 0
\(483\) 476.956 190.423i 0.0449322 0.0179390i
\(484\) 0 0
\(485\) 4879.50i 0.456839i
\(486\) 0 0
\(487\) 7159.66i 0.666191i −0.942893 0.333096i \(-0.891907\pi\)
0.942893 0.333096i \(-0.108093\pi\)
\(488\) 0 0
\(489\) −17530.1 + 6998.81i −1.62114 + 0.647233i
\(490\) 0 0
\(491\) 3170.70 0.291429 0.145714 0.989327i \(-0.453452\pi\)
0.145714 + 0.989327i \(0.453452\pi\)
\(492\) 0 0
\(493\) 2797.24 0.255540
\(494\) 0 0
\(495\) −6389.49 6726.47i −0.580174 0.610773i
\(496\) 0 0
\(497\) 4039.12i 0.364546i
\(498\) 0 0
\(499\) 397.678i 0.0356764i −0.999841 0.0178382i \(-0.994322\pi\)
0.999841 0.0178382i \(-0.00567838\pi\)
\(500\) 0 0
\(501\) −4151.05 10397.3i −0.370170 0.927175i
\(502\) 0 0
\(503\) 18268.3 1.61937 0.809684 0.586866i \(-0.199638\pi\)
0.809684 + 0.586866i \(0.199638\pi\)
\(504\) 0 0
\(505\) 2550.12 0.224711
\(506\) 0 0
\(507\) −325.606 815.554i −0.0285220 0.0714399i
\(508\) 0 0
\(509\) 8371.51i 0.728999i 0.931204 + 0.364500i \(0.118760\pi\)
−0.931204 + 0.364500i \(0.881240\pi\)
\(510\) 0 0
\(511\) 4080.17i 0.353221i
\(512\) 0 0
\(513\) −3427.63 + 7449.58i −0.294997 + 0.641144i
\(514\) 0 0
\(515\) −23369.2 −1.99956
\(516\) 0 0
\(517\) −8881.68 −0.755543
\(518\) 0 0
\(519\) −20257.1 + 8087.54i −1.71327 + 0.684015i
\(520\) 0 0
\(521\) 10212.7i 0.858781i −0.903119 0.429391i \(-0.858728\pi\)
0.903119 0.429391i \(-0.141272\pi\)
\(522\) 0 0
\(523\) 8118.39i 0.678761i 0.940649 + 0.339381i \(0.110217\pi\)
−0.940649 + 0.339381i \(0.889783\pi\)
\(524\) 0 0
\(525\) −5153.75 + 2057.61i −0.428434 + 0.171050i
\(526\) 0 0
\(527\) 5488.30 0.453651
\(528\) 0 0
\(529\) −11741.8 −0.965054
\(530\) 0 0
\(531\) −1881.78 + 1787.51i −0.153790 + 0.146085i
\(532\) 0 0
\(533\) 648.066i 0.0526658i
\(534\) 0 0
\(535\) 19226.9i 1.55374i
\(536\) 0 0
\(537\) 4951.63 + 12402.5i 0.397911 + 0.996659i
\(538\) 0 0
\(539\) 5896.28 0.471188
\(540\) 0 0
\(541\) −18079.1 −1.43675 −0.718376 0.695655i \(-0.755114\pi\)
−0.718376 + 0.695655i \(0.755114\pi\)
\(542\) 0 0
\(543\) 3636.21 + 9107.72i 0.287375 + 0.719797i
\(544\) 0 0
\(545\) 35729.9i 2.80826i
\(546\) 0 0
\(547\) 14018.2i 1.09575i 0.836561 + 0.547873i \(0.184562\pi\)
−0.836561 + 0.547873i \(0.815438\pi\)
\(548\) 0 0
\(549\) 5391.20 5121.11i 0.419109 0.398112i
\(550\) 0 0
\(551\) 3850.07 0.297674
\(552\) 0 0
\(553\) 1462.89 0.112493
\(554\) 0 0
\(555\) −25226.7 + 10071.7i −1.92940 + 0.770303i
\(556\) 0 0
\(557\) 11757.3i 0.894386i −0.894437 0.447193i \(-0.852424\pi\)
0.894437 0.447193i \(-0.147576\pi\)
\(558\) 0 0
\(559\) 4376.19i 0.331115i
\(560\) 0 0
\(561\) 3775.76 1507.45i 0.284158 0.113449i
\(562\) 0 0
\(563\) 13391.8 1.00248 0.501241 0.865308i \(-0.332877\pi\)
0.501241 + 0.865308i \(0.332877\pi\)
\(564\) 0 0
\(565\) 505.066 0.0376076
\(566\) 0 0
\(567\) 3489.59 + 179.432i 0.258464 + 0.0132900i
\(568\) 0 0
\(569\) 11097.2i 0.817610i 0.912622 + 0.408805i \(0.134054\pi\)
−0.912622 + 0.408805i \(0.865946\pi\)
\(570\) 0 0
\(571\) 7889.44i 0.578218i 0.957296 + 0.289109i \(0.0933591\pi\)
−0.957296 + 0.289109i \(0.906641\pi\)
\(572\) 0 0
\(573\) −270.421 677.330i −0.0197155 0.0493820i
\(574\) 0 0
\(575\) −4594.42 −0.333218
\(576\) 0 0
\(577\) 4722.33 0.340716 0.170358 0.985382i \(-0.445508\pi\)
0.170358 + 0.985382i \(0.445508\pi\)
\(578\) 0 0
\(579\) −456.015 1142.19i −0.0327311 0.0819825i
\(580\) 0 0
\(581\) 1890.50i 0.134993i
\(582\) 0 0
\(583\) 4725.21i 0.335674i
\(584\) 0 0
\(585\) 4508.34 + 4746.11i 0.318627 + 0.335432i
\(586\) 0 0
\(587\) −22243.9 −1.56406 −0.782030 0.623241i \(-0.785816\pi\)
−0.782030 + 0.623241i \(0.785816\pi\)
\(588\) 0 0
\(589\) 7553.99 0.528450
\(590\) 0 0
\(591\) 4849.35 1936.08i 0.337522 0.134754i
\(592\) 0 0
\(593\) 3564.68i 0.246853i 0.992354 + 0.123426i \(0.0393883\pi\)
−0.992354 + 0.123426i \(0.960612\pi\)
\(594\) 0 0
\(595\) 3796.10i 0.261554i
\(596\) 0 0
\(597\) 3238.92 1293.12i 0.222044 0.0886499i
\(598\) 0 0
\(599\) 8614.13 0.587586 0.293793 0.955869i \(-0.405082\pi\)
0.293793 + 0.955869i \(0.405082\pi\)
\(600\) 0 0
\(601\) −6578.32 −0.446481 −0.223241 0.974763i \(-0.571664\pi\)
−0.223241 + 0.974763i \(0.571664\pi\)
\(602\) 0 0
\(603\) −16959.4 17853.9i −1.14534 1.20575i
\(604\) 0 0
\(605\) 18492.0i 1.24265i
\(606\) 0 0
\(607\) 26046.8i 1.74169i 0.491556 + 0.870846i \(0.336428\pi\)
−0.491556 + 0.870846i \(0.663572\pi\)
\(608\) 0 0
\(609\) −608.289 1523.60i −0.0404747 0.101378i
\(610\) 0 0
\(611\) 6266.80 0.414939
\(612\) 0 0
\(613\) 10126.5 0.667218 0.333609 0.942712i \(-0.391734\pi\)
0.333609 + 0.942712i \(0.391734\pi\)
\(614\) 0 0
\(615\) −1791.24 4486.56i −0.117447 0.294172i
\(616\) 0 0
\(617\) 4115.13i 0.268507i 0.990947 + 0.134253i \(0.0428636\pi\)
−0.990947 + 0.134253i \(0.957136\pi\)
\(618\) 0 0
\(619\) 7782.81i 0.505359i 0.967550 + 0.252680i \(0.0813119\pi\)
−0.967550 + 0.252680i \(0.918688\pi\)
\(620\) 0 0
\(621\) 2628.09 + 1209.21i 0.169826 + 0.0781385i
\(622\) 0 0
\(623\) −6511.67 −0.418756
\(624\) 0 0
\(625\) 6168.51 0.394784
\(626\) 0 0
\(627\) 5196.88 2074.83i 0.331010 0.132154i
\(628\) 0 0
\(629\) 11903.3i 0.754559i
\(630\) 0 0
\(631\) 19254.9i 1.21478i −0.794404 0.607390i \(-0.792217\pi\)
0.794404 0.607390i \(-0.207783\pi\)
\(632\) 0 0
\(633\) −22193.0 + 8860.43i −1.39351 + 0.556352i
\(634\) 0 0
\(635\) 5740.70 0.358760
\(636\) 0 0
\(637\) −4160.34 −0.258773
\(638\) 0 0
\(639\) 16496.4 15670.0i 1.02126 0.970101i
\(640\) 0 0
\(641\) 16504.5i 1.01699i 0.861066 + 0.508494i \(0.169798\pi\)
−0.861066 + 0.508494i \(0.830202\pi\)
\(642\) 0 0
\(643\) 27292.3i 1.67388i 0.547297 + 0.836938i \(0.315657\pi\)
−0.547297 + 0.836938i \(0.684343\pi\)
\(644\) 0 0
\(645\) −12095.7 30296.4i −0.738398 1.84949i
\(646\) 0 0
\(647\) −30929.0 −1.87936 −0.939680 0.342054i \(-0.888877\pi\)
−0.939680 + 0.342054i \(0.888877\pi\)
\(648\) 0 0
\(649\) 1771.08 0.107120
\(650\) 0 0
\(651\) −1193.49 2989.36i −0.0718533 0.179973i
\(652\) 0 0
\(653\) 2637.30i 0.158048i 0.996873 + 0.0790241i \(0.0251804\pi\)
−0.996873 + 0.0790241i \(0.974820\pi\)
\(654\) 0 0
\(655\) 29309.2i 1.74841i
\(656\) 0 0
\(657\) 16664.1 15829.2i 0.989540 0.939966i
\(658\) 0 0
\(659\) −23789.8 −1.40625 −0.703126 0.711065i \(-0.748213\pi\)
−0.703126 + 0.711065i \(0.748213\pi\)
\(660\) 0 0
\(661\) 19417.3 1.14258 0.571290 0.820748i \(-0.306443\pi\)
0.571290 + 0.820748i \(0.306443\pi\)
\(662\) 0 0
\(663\) −2664.13 + 1063.64i −0.156057 + 0.0623052i
\(664\) 0 0
\(665\) 5224.87i 0.304680i
\(666\) 0 0
\(667\) 1358.24i 0.0788476i
\(668\) 0 0
\(669\) −8657.17 + 3456.33i −0.500307 + 0.199745i
\(670\) 0 0
\(671\) −5074.06 −0.291925
\(672\) 0 0
\(673\) 12145.5 0.695656 0.347828 0.937558i \(-0.386919\pi\)
0.347828 + 0.937558i \(0.386919\pi\)
\(674\) 0 0
\(675\) −28397.8 13066.1i −1.61931 0.745060i
\(676\) 0 0
\(677\) 22646.3i 1.28562i 0.766024 + 0.642812i \(0.222232\pi\)
−0.766024 + 0.642812i \(0.777768\pi\)
\(678\) 0 0
\(679\) 1254.08i 0.0708792i
\(680\) 0 0
\(681\) 8456.06 + 21180.1i 0.475825 + 1.19181i
\(682\) 0 0
\(683\) −33551.7 −1.87968 −0.939840 0.341614i \(-0.889026\pi\)
−0.939840 + 0.341614i \(0.889026\pi\)
\(684\) 0 0
\(685\) −47849.0 −2.66893
\(686\) 0 0
\(687\) 3361.40 + 8419.39i 0.186675 + 0.467569i
\(688\) 0 0
\(689\) 3334.04i 0.184350i
\(690\) 0 0
\(691\) 19643.0i 1.08141i −0.841211 0.540707i \(-0.818157\pi\)
0.841211 0.540707i \(-0.181843\pi\)
\(692\) 0 0
\(693\) −1642.15 1728.76i −0.0900148 0.0947623i
\(694\) 0 0
\(695\) 21162.1 1.15500
\(696\) 0 0
\(697\) 2117.00 0.115046
\(698\) 0 0
\(699\) 29178.0 11649.2i 1.57885 0.630347i
\(700\) 0 0
\(701\) 33965.0i 1.83002i −0.403435 0.915008i \(-0.632184\pi\)
0.403435 0.915008i \(-0.367816\pi\)
\(702\) 0 0
\(703\) 16383.5i 0.878971i
\(704\) 0 0
\(705\) −43385.0 + 17321.3i −2.31770 + 0.925329i
\(706\) 0 0
\(707\) 655.404 0.0348642
\(708\) 0 0
\(709\) 12160.9 0.644166 0.322083 0.946711i \(-0.395617\pi\)
0.322083 + 0.946711i \(0.395617\pi\)
\(710\) 0 0
\(711\) 5675.38 + 5974.70i 0.299358 + 0.315146i
\(712\) 0 0
\(713\) 2664.93i 0.139975i
\(714\) 0 0
\(715\) 4466.92i 0.233641i
\(716\) 0 0
\(717\) −4589.39 11495.2i −0.239043 0.598737i
\(718\) 0 0
\(719\) 30972.1 1.60649 0.803243 0.595652i \(-0.203106\pi\)
0.803243 + 0.595652i \(0.203106\pi\)
\(720\) 0 0
\(721\) −6006.10 −0.310234
\(722\) 0 0
\(723\) 9559.45 + 23943.8i 0.491729 + 1.23165i
\(724\) 0 0
\(725\) 14676.5i 0.751821i
\(726\) 0 0
\(727\) 18943.9i 0.966425i 0.875503 + 0.483212i \(0.160530\pi\)
−0.875503 + 0.483212i \(0.839470\pi\)
\(728\) 0 0
\(729\) 12805.2 + 14948.2i 0.650572 + 0.759445i
\(730\) 0 0
\(731\) 14295.5 0.723306
\(732\) 0 0
\(733\) 5511.95 0.277747 0.138873 0.990310i \(-0.455652\pi\)
0.138873 + 0.990310i \(0.455652\pi\)
\(734\) 0 0
\(735\) 28802.0 11499.1i 1.44541 0.577074i
\(736\) 0 0
\(737\) 16803.6i 0.839850i
\(738\) 0 0
\(739\) 13044.0i 0.649298i 0.945835 + 0.324649i \(0.105246\pi\)
−0.945835 + 0.324649i \(0.894754\pi\)
\(740\) 0 0
\(741\) −3666.85 + 1463.97i −0.181788 + 0.0725781i
\(742\) 0 0
\(743\) −12314.1 −0.608021 −0.304010 0.952669i \(-0.598326\pi\)
−0.304010 + 0.952669i \(0.598326\pi\)
\(744\) 0 0
\(745\) 53591.7 2.63550
\(746\) 0 0
\(747\) 7721.10 7334.29i 0.378180 0.359234i
\(748\) 0 0
\(749\) 4941.48i 0.241065i
\(750\) 0 0
\(751\) 18037.7i 0.876436i −0.898869 0.438218i \(-0.855610\pi\)
0.898869 0.438218i \(-0.144390\pi\)
\(752\) 0 0
\(753\) −7961.56 19941.5i −0.385306 0.965086i
\(754\) 0 0
\(755\) 16756.7 0.807731
\(756\) 0 0
\(757\) −8852.23 −0.425020 −0.212510 0.977159i \(-0.568164\pi\)
−0.212510 + 0.977159i \(0.568164\pi\)
\(758\) 0 0
\(759\) −731.967 1833.38i −0.0350049 0.0876776i
\(760\) 0 0
\(761\) 15773.5i 0.751365i 0.926748 + 0.375683i \(0.122592\pi\)
−0.926748 + 0.375683i \(0.877408\pi\)
\(762\) 0 0
\(763\) 9182.90i 0.435706i
\(764\) 0 0
\(765\) 15503.9 14727.2i 0.732737 0.696028i
\(766\) 0 0
\(767\) −1249.65 −0.0588297
\(768\) 0 0
\(769\) 15893.5 0.745296 0.372648 0.927973i \(-0.378450\pi\)
0.372648 + 0.927973i \(0.378450\pi\)
\(770\) 0 0
\(771\) 35080.3 14005.6i 1.63863 0.654217i
\(772\) 0 0
\(773\) 12767.1i 0.594049i 0.954870 + 0.297025i \(0.0959943\pi\)
−0.954870 + 0.297025i \(0.904006\pi\)
\(774\) 0 0
\(775\) 28795.9i 1.33468i
\(776\) 0 0
\(777\) −6483.50 + 2588.50i −0.299349 + 0.119514i
\(778\) 0 0
\(779\) 2913.80 0.134015
\(780\) 0 0
\(781\) −15526.0 −0.711349
\(782\) 0 0
\(783\) 3862.73 8395.22i 0.176300 0.383168i
\(784\) 0 0
\(785\) 19493.4i 0.886306i
\(786\) 0 0
\(787\) 1198.97i 0.0543057i 0.999631 + 0.0271529i \(0.00864408\pi\)
−0.999631 + 0.0271529i \(0.991356\pi\)
\(788\) 0 0
\(789\) 7711.87 + 19316.1i 0.347972 + 0.871574i
\(790\) 0 0
\(791\) 129.806 0.00583487
\(792\) 0 0
\(793\) 3580.19 0.160323
\(794\) 0 0
\(795\) −9215.22 23081.6i −0.411107 1.02971i
\(796\) 0 0
\(797\) 40863.5i 1.81613i 0.418825 + 0.908067i \(0.362442\pi\)
−0.418825 + 0.908067i \(0.637558\pi\)
\(798\) 0 0
\(799\) 20471.4i 0.906416i
\(800\) 0 0
\(801\) −25262.4 26594.7i −1.11436 1.17313i
\(802\) 0 0
\(803\) −15683.8 −0.689252
\(804\) 0 0
\(805\) −1843.25 −0.0807032
\(806\) 0 0
\(807\) −19441.7 + 7762.00i −0.848054 + 0.338582i
\(808\) 0 0
\(809\) 6162.20i 0.267801i 0.990995 + 0.133901i \(0.0427503\pi\)
−0.990995 + 0.133901i \(0.957250\pi\)
\(810\) 0 0
\(811\) 30637.4i 1.32654i −0.748381 0.663269i \(-0.769168\pi\)
0.748381 0.663269i \(-0.230832\pi\)
\(812\) 0 0
\(813\) 16723.8 6676.89i 0.721437 0.288030i
\(814\) 0 0
\(815\) 67747.1 2.91175
\(816\) 0 0
\(817\) 19676.0 0.842565
\(818\) 0 0
\(819\) 1158.68 + 1219.79i 0.0494355 + 0.0520427i
\(820\) 0 0
\(821\) 41698.5i 1.77258i 0.463131 + 0.886290i \(0.346726\pi\)
−0.463131 + 0.886290i \(0.653274\pi\)
\(822\) 0 0
\(823\) 15087.6i 0.639029i 0.947581 + 0.319514i \(0.103520\pi\)
−0.947581 + 0.319514i \(0.896480\pi\)
\(824\) 0 0
\(825\) 7909.26 + 19810.5i 0.333776 + 0.836017i
\(826\) 0 0
\(827\) −22253.3 −0.935701 −0.467850 0.883808i \(-0.654971\pi\)
−0.467850 + 0.883808i \(0.654971\pi\)
\(828\) 0 0
\(829\) −6763.95 −0.283380 −0.141690 0.989911i \(-0.545254\pi\)
−0.141690 + 0.989911i \(0.545254\pi\)
\(830\) 0 0
\(831\) −17680.1 44283.9i −0.738047 1.84860i
\(832\) 0 0
\(833\) 13590.3i 0.565279i
\(834\) 0 0
\(835\) 40181.4i 1.66531i
\(836\) 0 0
\(837\) 7578.83 16471.8i 0.312978 0.680224i
\(838\) 0 0
\(839\) −35137.5 −1.44586 −0.722932 0.690919i \(-0.757206\pi\)
−0.722932 + 0.690919i \(0.757206\pi\)
\(840\) 0 0
\(841\) 20050.2 0.822101
\(842\) 0 0
\(843\) 20198.4 8064.12i 0.825232 0.329470i
\(844\) 0 0
\(845\) 3151.80i 0.128314i
\(846\) 0 0
\(847\) 4752.60i 0.192800i
\(848\) 0 0
\(849\) −7743.53 + 3091.57i −0.313024 + 0.124973i
\(850\) 0 0
\(851\) −5779.85 −0.232821
\(852\) 0 0
\(853\) −46756.8 −1.87681 −0.938407 0.345531i \(-0.887699\pi\)
−0.938407 + 0.345531i \(0.887699\pi\)
\(854\) 0 0
\(855\) 21339.2 20270.2i 0.853551 0.810790i
\(856\) 0 0
\(857\) 46686.1i 1.86087i −0.366454 0.930436i \(-0.619428\pi\)
0.366454 0.930436i \(-0.380572\pi\)
\(858\) 0 0
\(859\) 32547.1i 1.29278i −0.763009 0.646388i \(-0.776279\pi\)
0.763009 0.646388i \(-0.223721\pi\)
\(860\) 0 0
\(861\) −460.364 1153.09i −0.0182220 0.0456412i
\(862\) 0 0
\(863\) 19498.4 0.769102 0.384551 0.923104i \(-0.374356\pi\)
0.384551 + 0.923104i \(0.374356\pi\)
\(864\) 0 0
\(865\) 78285.8 3.07722
\(866\) 0 0
\(867\) −5991.16 15006.2i −0.234683 0.587817i
\(868\) 0 0
\(869\) 5623.23i 0.219511i
\(870\) 0 0
\(871\) 11856.4i 0.461239i
\(872\) 0 0
\(873\) −5121.85 + 4865.25i −0.198566 + 0.188618i
\(874\) 0 0
\(875\) 8743.42 0.337808
\(876\) 0 0
\(877\) 34603.0 1.33234 0.666170 0.745800i \(-0.267933\pi\)
0.666170 + 0.745800i \(0.267933\pi\)
\(878\) 0 0
\(879\) 31415.5 12542.5i 1.20548 0.481283i
\(880\) 0 0
\(881\) 21333.9i 0.815843i 0.913017 + 0.407922i \(0.133746\pi\)
−0.913017 + 0.407922i \(0.866254\pi\)
\(882\) 0 0
\(883\) 28520.0i 1.08695i 0.839426 + 0.543474i \(0.182892\pi\)
−0.839426 + 0.543474i \(0.817108\pi\)
\(884\) 0 0
\(885\) 8651.36 3454.01i 0.328601 0.131193i
\(886\) 0 0
\(887\) −43303.3 −1.63921 −0.819606 0.572928i \(-0.805808\pi\)
−0.819606 + 0.572928i \(0.805808\pi\)
\(888\) 0 0
\(889\) 1475.41 0.0556622
\(890\) 0 0
\(891\) 689.720 13413.6i 0.0259332 0.504348i
\(892\) 0 0
\(893\) 28176.4i 1.05587i
\(894\) 0 0
\(895\) 47930.7i 1.79011i
\(896\) 0 0
\(897\) 516.466 + 1293.61i 0.0192244 + 0.0481519i
\(898\) 0 0
\(899\) −8512.89 −0.315818
\(900\) 0 0
\(901\) 10891.1 0.402704
\(902\) 0 0
\(903\) −3108.70 7786.43i −0.114564 0.286950i
\(904\) 0 0
\(905\) 35197.8i 1.29283i
\(906\) 0 0
\(907\) 41510.0i 1.51965i 0.650130 + 0.759823i \(0.274714\pi\)
−0.650130 + 0.759823i \(0.725286\pi\)
\(908\) 0 0
\(909\) 2542.68 + 2676.78i 0.0927780 + 0.0976712i
\(910\) 0 0
\(911\) 7424.35 0.270010 0.135005 0.990845i \(-0.456895\pi\)
0.135005 + 0.990845i \(0.456895\pi\)
\(912\) 0 0
\(913\) −7266.90 −0.263417
\(914\) 0 0
\(915\) −24785.7 + 9895.56i −0.895507 + 0.357527i
\(916\) 0 0
\(917\) 7532.73i 0.271268i
\(918\) 0 0
\(919\) 17469.0i 0.627039i 0.949582 + 0.313519i \(0.101508\pi\)
−0.949582 + 0.313519i \(0.898492\pi\)
\(920\) 0 0
\(921\) 5836.91 2330.36i 0.208830 0.0833745i
\(922\) 0 0
\(923\) 10954.9 0.390668
\(924\) 0 0
\(925\) 62454.1 2.21998
\(926\) 0 0
\(927\) −23301.0 24529.9i −0.825571 0.869112i
\(928\) 0 0
\(929\) 44640.1i 1.57653i 0.615336 + 0.788265i \(0.289020\pi\)
−0.615336 + 0.788265i \(0.710980\pi\)
\(930\) 0 0
\(931\) 18705.5i 0.658483i
\(932\) 0 0
\(933\) −10216.4 25589.3i −0.358489 0.897918i
\(934\) 0 0
\(935\) −14591.8 −0.510379
\(936\) 0 0
\(937\) 24243.0 0.845235 0.422618 0.906308i \(-0.361111\pi\)
0.422618 + 0.906308i \(0.361111\pi\)
\(938\) 0 0
\(939\) 2958.78 + 7410.94i 0.102829 + 0.257558i
\(940\) 0 0
\(941\) 45703.7i 1.58331i −0.610966 0.791657i \(-0.709219\pi\)
0.610966 0.791657i \(-0.290781\pi\)
\(942\) 0 0
\(943\) 1027.94i 0.0354978i
\(944\) 0 0
\(945\) −11393.0 5242.05i −0.392186 0.180449i
\(946\) 0 0
\(947\) 33165.0 1.13803 0.569017 0.822326i \(-0.307324\pi\)
0.569017 + 0.822326i \(0.307324\pi\)
\(948\) 0 0
\(949\) 11066.3 0.378532
\(950\) 0 0
\(951\) 27668.3 11046.5i 0.943436 0.376662i
\(952\) 0 0
\(953\) 9676.78i 0.328921i −0.986384 0.164460i \(-0.947412\pi\)
0.986384 0.164460i \(-0.0525883\pi\)
\(954\) 0 0
\(955\) 2617.62i 0.0886954i
\(956\) 0 0
\(957\) −5856.56 + 2338.20i −0.197822 + 0.0789795i
\(958\) 0 0
\(959\) −12297.6 −0.414088
\(960\) 0 0
\(961\) 13088.4 0.439339
\(962\) 0 0
\(963\) −20181.8 + 19170.7i −0.675336 + 0.641503i
\(964\) 0 0
\(965\) 4414.13i 0.147250i
\(966\) 0 0
\(967\) 30085.9i 1.00051i −0.865877 0.500257i \(-0.833239\pi\)
0.865877 0.500257i \(-0.166761\pi\)
\(968\) 0 0
\(969\) 4782.28 + 11978.3i 0.158544 + 0.397109i
\(970\) 0 0
\(971\) −23229.4 −0.767732 −0.383866 0.923389i \(-0.625408\pi\)
−0.383866 + 0.923389i \(0.625408\pi\)
\(972\) 0 0
\(973\) 5438.85 0.179200
\(974\) 0 0
\(975\) −5580.67 13978.1i −0.183307 0.459134i
\(976\) 0 0
\(977\) 10271.5i 0.336351i −0.985757 0.168176i \(-0.946212\pi\)
0.985757 0.168176i \(-0.0537876\pi\)
\(978\) 0 0
\(979\) 25030.3i 0.817131i
\(980\) 0 0
\(981\) 37504.5 35625.5i 1.22062 1.15947i
\(982\) 0 0
\(983\) −26200.0 −0.850103 −0.425052 0.905169i \(-0.639744\pi\)
−0.425052 + 0.905169i \(0.639744\pi\)
\(984\) 0 0
\(985\) −18740.9 −0.606227
\(986\) 0 0
\(987\) −11150.3 + 4451.72i −0.359594 + 0.143566i
\(988\) 0 0
\(989\) 6941.38i 0.223178i
\(990\) 0 0
\(991\) 33380.1i 1.06998i 0.844858 + 0.534991i \(0.179685\pi\)
−0.844858 + 0.534991i \(0.820315\pi\)
\(992\) 0 0
\(993\) −22290.4 + 8899.32i −0.712349 + 0.284402i
\(994\) 0 0
\(995\) −12517.2 −0.398815
\(996\) 0 0
\(997\) −39447.8 −1.25308 −0.626541 0.779388i \(-0.715530\pi\)
−0.626541 + 0.779388i \(0.715530\pi\)
\(998\) 0 0
\(999\) −35724.9 16437.4i −1.13142 0.520577i
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.287.9 24
3.2 odd 2 inner 624.4.d.c.287.15 yes 24
4.3 odd 2 inner 624.4.d.c.287.16 yes 24
12.11 even 2 inner 624.4.d.c.287.10 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
624.4.d.c.287.9 24 1.1 even 1 trivial
624.4.d.c.287.10 yes 24 12.11 even 2 inner
624.4.d.c.287.15 yes 24 3.2 odd 2 inner
624.4.d.c.287.16 yes 24 4.3 odd 2 inner