Properties

Label 756.4.x.a.125.15
Level $756$
Weight $4$
Character 756.125
Analytic conductor $44.605$
Analytic rank $0$
Dimension $48$
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] = [756,4,Mod(125,756)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(756, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 5, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("756.125");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 756.x (of order \(6\), degree \(2\), not 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: \(44.6054439643\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(24\) over \(\Q(\zeta_{6})\)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 125.15
Character \(\chi\) \(=\) 756.125
Dual form 756.4.x.a.629.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.34269 - 4.05766i) q^{5} +(18.4121 - 1.99859i) q^{7} +O(q^{10})\) \(q+(2.34269 - 4.05766i) q^{5} +(18.4121 - 1.99859i) q^{7} +(-16.1150 + 9.30397i) q^{11} +(-44.1430 - 25.4860i) q^{13} -112.833 q^{17} +111.459i q^{19} +(124.503 + 71.8819i) q^{23} +(51.5236 + 89.2414i) q^{25} +(-206.907 + 119.458i) q^{29} +(179.999 + 103.923i) q^{31} +(35.0243 - 79.3922i) q^{35} -227.815 q^{37} +(133.141 - 230.607i) q^{41} +(-170.287 - 294.946i) q^{43} +(-111.979 - 193.952i) q^{47} +(335.011 - 73.5964i) q^{49} +547.974i q^{53} +87.1854i q^{55} +(-43.9483 + 76.1207i) q^{59} +(312.398 - 180.363i) q^{61} +(-206.827 + 119.412i) q^{65} +(-372.426 + 645.060i) q^{67} +135.948i q^{71} +467.289i q^{73} +(-278.115 + 203.513i) q^{77} +(192.171 + 332.850i) q^{79} +(597.216 + 1034.41i) q^{83} +(-264.332 + 457.836i) q^{85} +1385.63 q^{89} +(-863.702 - 381.027i) q^{91} +(452.262 + 261.114i) q^{95} +(-1070.10 + 617.821i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 48 q + 6 q^{7} + 12 q^{11} + 408 q^{23} - 600 q^{25} + 84 q^{29} + 336 q^{37} + 84 q^{43} + 318 q^{49} - 2964 q^{65} - 588 q^{67} - 2400 q^{77} + 204 q^{79} - 360 q^{85} - 1080 q^{91} - 300 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(-1\) \(1\)

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 0
\(4\) 0 0
\(5\) 2.34269 4.05766i 0.209537 0.362929i −0.742032 0.670365i \(-0.766138\pi\)
0.951569 + 0.307436i \(0.0994710\pi\)
\(6\) 0 0
\(7\) 18.4121 1.99859i 0.994160 0.107914i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −16.1150 + 9.30397i −0.441713 + 0.255023i −0.704324 0.709879i \(-0.748750\pi\)
0.262611 + 0.964902i \(0.415416\pi\)
\(12\) 0 0
\(13\) −44.1430 25.4860i −0.941775 0.543734i −0.0512589 0.998685i \(-0.516323\pi\)
−0.890516 + 0.454951i \(0.849657\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −112.833 −1.60976 −0.804880 0.593438i \(-0.797770\pi\)
−0.804880 + 0.593438i \(0.797770\pi\)
\(18\) 0 0
\(19\) 111.459i 1.34581i 0.739729 + 0.672905i \(0.234954\pi\)
−0.739729 + 0.672905i \(0.765046\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 124.503 + 71.8819i 1.12873 + 0.651671i 0.943614 0.331047i \(-0.107402\pi\)
0.185112 + 0.982717i \(0.440735\pi\)
\(24\) 0 0
\(25\) 51.5236 + 89.2414i 0.412189 + 0.713932i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −206.907 + 119.458i −1.32489 + 0.764924i −0.984504 0.175363i \(-0.943890\pi\)
−0.340383 + 0.940287i \(0.610557\pi\)
\(30\) 0 0
\(31\) 179.999 + 103.923i 1.04287 + 0.602099i 0.920644 0.390404i \(-0.127665\pi\)
0.122222 + 0.992503i \(0.460998\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 35.0243 79.3922i 0.169148 0.383421i
\(36\) 0 0
\(37\) −227.815 −1.01223 −0.506115 0.862466i \(-0.668919\pi\)
−0.506115 + 0.862466i \(0.668919\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 133.141 230.607i 0.507149 0.878407i −0.492817 0.870133i \(-0.664033\pi\)
0.999966 0.00827417i \(-0.00263378\pi\)
\(42\) 0 0
\(43\) −170.287 294.946i −0.603920 1.04602i −0.992221 0.124488i \(-0.960271\pi\)
0.388301 0.921533i \(-0.373062\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −111.979 193.952i −0.347527 0.601934i 0.638283 0.769802i \(-0.279645\pi\)
−0.985809 + 0.167868i \(0.946312\pi\)
\(48\) 0 0
\(49\) 335.011 73.5964i 0.976709 0.214567i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 547.974i 1.42019i 0.704107 + 0.710094i \(0.251348\pi\)
−0.704107 + 0.710094i \(0.748652\pi\)
\(54\) 0 0
\(55\) 87.1854i 0.213747i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −43.9483 + 76.1207i −0.0969760 + 0.167967i −0.910432 0.413660i \(-0.864250\pi\)
0.813456 + 0.581627i \(0.197584\pi\)
\(60\) 0 0
\(61\) 312.398 180.363i 0.655712 0.378575i −0.134929 0.990855i \(-0.543081\pi\)
0.790641 + 0.612280i \(0.209747\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −206.827 + 119.412i −0.394673 + 0.227865i
\(66\) 0 0
\(67\) −372.426 + 645.060i −0.679090 + 1.17622i 0.296165 + 0.955137i \(0.404292\pi\)
−0.975255 + 0.221082i \(0.929041\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 135.948i 0.227240i 0.993524 + 0.113620i \(0.0362447\pi\)
−0.993524 + 0.113620i \(0.963755\pi\)
\(72\) 0 0
\(73\) 467.289i 0.749206i 0.927185 + 0.374603i \(0.122221\pi\)
−0.927185 + 0.374603i \(0.877779\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −278.115 + 203.513i −0.411613 + 0.301201i
\(78\) 0 0
\(79\) 192.171 + 332.850i 0.273683 + 0.474032i 0.969802 0.243894i \(-0.0784249\pi\)
−0.696119 + 0.717926i \(0.745092\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 597.216 + 1034.41i 0.789795 + 1.36796i 0.926092 + 0.377297i \(0.123146\pi\)
−0.136297 + 0.990668i \(0.543520\pi\)
\(84\) 0 0
\(85\) −264.332 + 457.836i −0.337304 + 0.584228i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1385.63 1.65030 0.825150 0.564914i \(-0.191091\pi\)
0.825150 + 0.564914i \(0.191091\pi\)
\(90\) 0 0
\(91\) −863.702 381.027i −0.994952 0.438929i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 452.262 + 261.114i 0.488433 + 0.281997i
\(96\) 0 0
\(97\) −1070.10 + 617.821i −1.12012 + 0.646704i −0.941432 0.337202i \(-0.890520\pi\)
−0.178691 + 0.983905i \(0.557186\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −94.1707 163.108i −0.0927756 0.160692i 0.815902 0.578190i \(-0.196241\pi\)
−0.908678 + 0.417498i \(0.862907\pi\)
\(102\) 0 0
\(103\) −687.152 396.727i −0.657350 0.379521i 0.133916 0.990993i \(-0.457245\pi\)
−0.791267 + 0.611471i \(0.790578\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1399.55i 1.26448i 0.774773 + 0.632240i \(0.217864\pi\)
−0.774773 + 0.632240i \(0.782136\pi\)
\(108\) 0 0
\(109\) −2137.48 −1.87829 −0.939145 0.343521i \(-0.888380\pi\)
−0.939145 + 0.343521i \(0.888380\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1500.15 + 866.110i 1.24887 + 0.721034i 0.970884 0.239552i \(-0.0770006\pi\)
0.277984 + 0.960586i \(0.410334\pi\)
\(114\) 0 0
\(115\) 583.346 336.795i 0.473020 0.273098i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2077.48 + 225.506i −1.60036 + 0.173715i
\(120\) 0 0
\(121\) −492.372 + 852.814i −0.369926 + 0.640731i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1068.49 0.764549
\(126\) 0 0
\(127\) −770.489 −0.538345 −0.269173 0.963092i \(-0.586750\pi\)
−0.269173 + 0.963092i \(0.586750\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −489.800 + 848.359i −0.326672 + 0.565813i −0.981849 0.189662i \(-0.939261\pi\)
0.655177 + 0.755475i \(0.272594\pi\)
\(132\) 0 0
\(133\) 222.760 + 2052.19i 0.145231 + 1.33795i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 182.458 105.342i 0.113784 0.0656932i −0.442028 0.897001i \(-0.645741\pi\)
0.555812 + 0.831308i \(0.312407\pi\)
\(138\) 0 0
\(139\) 730.515 + 421.763i 0.445766 + 0.257363i 0.706040 0.708172i \(-0.250480\pi\)
−0.260274 + 0.965535i \(0.583813\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 948.484 0.554659
\(144\) 0 0
\(145\) 1119.41i 0.641119i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1137.32 656.632i −0.625321 0.361029i 0.153617 0.988131i \(-0.450908\pi\)
−0.778938 + 0.627101i \(0.784241\pi\)
\(150\) 0 0
\(151\) −1738.76 3011.62i −0.937074 1.62306i −0.770893 0.636965i \(-0.780190\pi\)
−0.166182 0.986095i \(-0.553144\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 843.366 486.918i 0.437038 0.252324i
\(156\) 0 0
\(157\) −499.729 288.519i −0.254030 0.146664i 0.367578 0.929993i \(-0.380187\pi\)
−0.621608 + 0.783328i \(0.713520\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2436.03 + 1074.67i 1.19246 + 0.526060i
\(162\) 0 0
\(163\) 1595.73 0.766794 0.383397 0.923584i \(-0.374754\pi\)
0.383397 + 0.923584i \(0.374754\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 184.918 320.287i 0.0856850 0.148411i −0.819998 0.572366i \(-0.806025\pi\)
0.905683 + 0.423956i \(0.139359\pi\)
\(168\) 0 0
\(169\) 200.572 + 347.402i 0.0912938 + 0.158125i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −973.579 1686.29i −0.427860 0.741076i 0.568822 0.822460i \(-0.307399\pi\)
−0.996683 + 0.0813844i \(0.974066\pi\)
\(174\) 0 0
\(175\) 1127.01 + 1540.15i 0.486824 + 0.665282i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 851.921i 0.355730i −0.984055 0.177865i \(-0.943081\pi\)
0.984055 0.177865i \(-0.0569190\pi\)
\(180\) 0 0
\(181\) 639.259i 0.262518i −0.991348 0.131259i \(-0.958098\pi\)
0.991348 0.131259i \(-0.0419020\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −533.700 + 924.395i −0.212099 + 0.367367i
\(186\) 0 0
\(187\) 1818.29 1049.79i 0.711051 0.410526i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3120.13 + 1801.41i −1.18201 + 0.682435i −0.956479 0.291800i \(-0.905746\pi\)
−0.225533 + 0.974235i \(0.572412\pi\)
\(192\) 0 0
\(193\) −1745.01 + 3022.44i −0.650821 + 1.12725i 0.332103 + 0.943243i \(0.392242\pi\)
−0.982924 + 0.184012i \(0.941092\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1909.86i 0.690719i 0.938470 + 0.345360i \(0.112243\pi\)
−0.938470 + 0.345360i \(0.887757\pi\)
\(198\) 0 0
\(199\) 713.005i 0.253988i 0.991903 + 0.126994i \(0.0405329\pi\)
−0.991903 + 0.126994i \(0.959467\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −3570.85 + 2612.99i −1.23460 + 0.903430i
\(204\) 0 0
\(205\) −623.816 1080.48i −0.212533 0.368117i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1037.01 1796.15i −0.343213 0.594462i
\(210\) 0 0
\(211\) 788.596 1365.89i 0.257295 0.445648i −0.708221 0.705990i \(-0.750502\pi\)
0.965516 + 0.260343i \(0.0838355\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1595.72 −0.506174
\(216\) 0 0
\(217\) 3521.86 + 1553.69i 1.10175 + 0.486043i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 4980.77 + 2875.65i 1.51603 + 0.875281i
\(222\) 0 0
\(223\) −412.569 + 238.197i −0.123891 + 0.0715284i −0.560665 0.828043i \(-0.689454\pi\)
0.436774 + 0.899571i \(0.356121\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1386.85 2402.10i −0.405501 0.702348i 0.588879 0.808221i \(-0.299569\pi\)
−0.994380 + 0.105873i \(0.966236\pi\)
\(228\) 0 0
\(229\) 1698.53 + 980.646i 0.490139 + 0.282982i 0.724632 0.689136i \(-0.242010\pi\)
−0.234493 + 0.972118i \(0.575343\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4734.94i 1.33132i −0.746257 0.665658i \(-0.768151\pi\)
0.746257 0.665658i \(-0.231849\pi\)
\(234\) 0 0
\(235\) −1049.33 −0.291279
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1055.48 + 609.379i 0.285661 + 0.164927i 0.635984 0.771703i \(-0.280595\pi\)
−0.350322 + 0.936629i \(0.613928\pi\)
\(240\) 0 0
\(241\) 1784.77 1030.43i 0.477041 0.275420i −0.242142 0.970241i \(-0.577850\pi\)
0.719182 + 0.694821i \(0.244517\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 486.199 1531.78i 0.126784 0.399435i
\(246\) 0 0
\(247\) 2840.64 4920.13i 0.731763 1.26745i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −2966.29 −0.745937 −0.372969 0.927844i \(-0.621660\pi\)
−0.372969 + 0.927844i \(0.621660\pi\)
\(252\) 0 0
\(253\) −2675.15 −0.664764
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1133.09 1962.58i 0.275021 0.476351i −0.695119 0.718895i \(-0.744648\pi\)
0.970141 + 0.242543i \(0.0779817\pi\)
\(258\) 0 0
\(259\) −4194.54 + 455.307i −1.00632 + 0.109233i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1580.16 912.308i 0.370483 0.213898i −0.303186 0.952931i \(-0.598050\pi\)
0.673669 + 0.739033i \(0.264717\pi\)
\(264\) 0 0
\(265\) 2223.49 + 1283.73i 0.515427 + 0.297582i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 5616.89 1.27312 0.636558 0.771229i \(-0.280358\pi\)
0.636558 + 0.771229i \(0.280358\pi\)
\(270\) 0 0
\(271\) 5603.22i 1.25598i −0.778220 0.627992i \(-0.783877\pi\)
0.778220 0.627992i \(-0.216123\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1660.60 958.748i −0.364138 0.210235i
\(276\) 0 0
\(277\) 2329.35 + 4034.55i 0.505260 + 0.875137i 0.999981 + 0.00608483i \(0.00193687\pi\)
−0.494721 + 0.869052i \(0.664730\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1939.85 + 1119.97i −0.411820 + 0.237765i −0.691572 0.722308i \(-0.743081\pi\)
0.279751 + 0.960073i \(0.409748\pi\)
\(282\) 0 0
\(283\) −2759.51 1593.21i −0.579633 0.334651i 0.181355 0.983418i \(-0.441952\pi\)
−0.760987 + 0.648767i \(0.775285\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1990.51 4512.05i 0.409395 0.928006i
\(288\) 0 0
\(289\) 7818.18 1.59132
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −4563.57 + 7904.34i −0.909921 + 1.57603i −0.0957489 + 0.995406i \(0.530525\pi\)
−0.814172 + 0.580624i \(0.802809\pi\)
\(294\) 0 0
\(295\) 205.915 + 356.655i 0.0406401 + 0.0703907i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3663.97 6346.18i −0.708671 1.22745i
\(300\) 0 0
\(301\) −3724.82 5090.25i −0.713274 0.974741i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1690.14i 0.317302i
\(306\) 0 0
\(307\) 4744.58i 0.882044i 0.897496 + 0.441022i \(0.145384\pi\)
−0.897496 + 0.441022i \(0.854616\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1089.62 + 1887.29i −0.198672 + 0.344110i −0.948098 0.317978i \(-0.896996\pi\)
0.749426 + 0.662088i \(0.230329\pi\)
\(312\) 0 0
\(313\) 1752.85 1012.01i 0.316540 0.182754i −0.333309 0.942818i \(-0.608165\pi\)
0.649849 + 0.760063i \(0.274832\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −189.948 + 109.666i −0.0336547 + 0.0194305i −0.516733 0.856147i \(-0.672852\pi\)
0.483078 + 0.875577i \(0.339519\pi\)
\(318\) 0 0
\(319\) 2222.87 3850.12i 0.390146 0.675753i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 12576.2i 2.16643i
\(324\) 0 0
\(325\) 5252.52i 0.896484i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −2449.39 3347.27i −0.410454 0.560916i
\(330\) 0 0
\(331\) 1884.36 + 3263.81i 0.312912 + 0.541980i 0.978991 0.203901i \(-0.0653621\pi\)
−0.666079 + 0.745881i \(0.732029\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1744.96 + 3022.36i 0.284589 + 0.492922i
\(336\) 0 0
\(337\) 3056.60 5294.18i 0.494076 0.855764i −0.505901 0.862592i \(-0.668840\pi\)
0.999977 + 0.00682736i \(0.00217323\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −3867.57 −0.614196
\(342\) 0 0
\(343\) 6021.17 2024.61i 0.947851 0.318714i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1143.36 660.117i −0.176883 0.102124i 0.408944 0.912559i \(-0.365897\pi\)
−0.585828 + 0.810436i \(0.699230\pi\)
\(348\) 0 0
\(349\) 2309.07 1333.14i 0.354159 0.204474i −0.312357 0.949965i \(-0.601118\pi\)
0.666515 + 0.745491i \(0.267785\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −4725.05 8184.03i −0.712434 1.23397i −0.963941 0.266116i \(-0.914260\pi\)
0.251508 0.967855i \(-0.419074\pi\)
\(354\) 0 0
\(355\) 551.632 + 318.485i 0.0824720 + 0.0476152i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 6846.88i 1.00659i −0.864116 0.503293i \(-0.832122\pi\)
0.864116 0.503293i \(-0.167878\pi\)
\(360\) 0 0
\(361\) −5564.07 −0.811206
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1896.10 + 1094.71i 0.271908 + 0.156986i
\(366\) 0 0
\(367\) −5440.63 + 3141.15i −0.773838 + 0.446775i −0.834242 0.551399i \(-0.814094\pi\)
0.0604042 + 0.998174i \(0.480761\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1095.17 + 10089.4i 0.153258 + 1.41189i
\(372\) 0 0
\(373\) −5253.75 + 9099.76i −0.729300 + 1.26318i 0.227880 + 0.973689i \(0.426821\pi\)
−0.957179 + 0.289495i \(0.906513\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12178.0 1.66366
\(378\) 0 0
\(379\) 5904.89 0.800301 0.400150 0.916450i \(-0.368958\pi\)
0.400150 + 0.916450i \(0.368958\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1175.26 + 2035.61i −0.156797 + 0.271580i −0.933712 0.358026i \(-0.883450\pi\)
0.776915 + 0.629605i \(0.216783\pi\)
\(384\) 0 0
\(385\) 174.248 + 1605.27i 0.0230662 + 0.212499i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 11034.6 6370.81i 1.43824 0.830368i 0.440511 0.897747i \(-0.354797\pi\)
0.997727 + 0.0673794i \(0.0214638\pi\)
\(390\) 0 0
\(391\) −14048.0 8110.62i −1.81698 1.04903i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1800.79 0.229386
\(396\) 0 0
\(397\) 13326.3i 1.68470i −0.538930 0.842350i \(-0.681171\pi\)
0.538930 0.842350i \(-0.318829\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 8090.75 + 4671.20i 1.00756 + 0.581717i 0.910477 0.413559i \(-0.135715\pi\)
0.0970856 + 0.995276i \(0.469048\pi\)
\(402\) 0 0
\(403\) −5297.15 9174.92i −0.654763 1.13408i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3671.22 2119.58i 0.447115 0.258142i
\(408\) 0 0
\(409\) −268.033 154.749i −0.0324044 0.0187087i 0.483710 0.875228i \(-0.339289\pi\)
−0.516115 + 0.856519i \(0.672622\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −657.047 + 1489.38i −0.0782837 + 0.177452i
\(414\) 0 0
\(415\) 5596.38 0.661965
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −3008.72 + 5211.26i −0.350801 + 0.607605i −0.986390 0.164422i \(-0.947424\pi\)
0.635589 + 0.772028i \(0.280757\pi\)
\(420\) 0 0
\(421\) 1920.18 + 3325.85i 0.222289 + 0.385017i 0.955503 0.294982i \(-0.0953137\pi\)
−0.733213 + 0.679999i \(0.761980\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −5813.53 10069.3i −0.663524 1.14926i
\(426\) 0 0
\(427\) 5391.43 3945.21i 0.611029 0.447125i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4510.80i 0.504125i −0.967711 0.252062i \(-0.918891\pi\)
0.967711 0.252062i \(-0.0811088\pi\)
\(432\) 0 0
\(433\) 7012.84i 0.778328i 0.921169 + 0.389164i \(0.127236\pi\)
−0.921169 + 0.389164i \(0.872764\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −8011.88 + 13877.0i −0.877025 + 1.51905i
\(438\) 0 0
\(439\) 3220.45 1859.33i 0.350122 0.202143i −0.314617 0.949219i \(-0.601876\pi\)
0.664739 + 0.747075i \(0.268543\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −5177.61 + 2989.29i −0.555295 + 0.320599i −0.751255 0.660012i \(-0.770551\pi\)
0.195960 + 0.980612i \(0.437218\pi\)
\(444\) 0 0
\(445\) 3246.11 5622.43i 0.345799 0.598941i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1188.91i 0.124963i −0.998046 0.0624813i \(-0.980099\pi\)
0.998046 0.0624813i \(-0.0199014\pi\)
\(450\) 0 0
\(451\) 4954.95i 0.517338i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −3569.47 + 2611.98i −0.367779 + 0.269125i
\(456\) 0 0
\(457\) 2039.30 + 3532.17i 0.208741 + 0.361549i 0.951318 0.308211i \(-0.0997302\pi\)
−0.742577 + 0.669760i \(0.766397\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −3283.79 5687.69i −0.331760 0.574625i 0.651097 0.758995i \(-0.274309\pi\)
−0.982857 + 0.184369i \(0.940976\pi\)
\(462\) 0 0
\(463\) −4120.60 + 7137.09i −0.413608 + 0.716390i −0.995281 0.0970322i \(-0.969065\pi\)
0.581673 + 0.813423i \(0.302398\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 11330.9 1.12276 0.561382 0.827557i \(-0.310270\pi\)
0.561382 + 0.827557i \(0.310270\pi\)
\(468\) 0 0
\(469\) −5567.93 + 12621.2i −0.548194 + 1.24263i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 5488.35 + 3168.70i 0.533519 + 0.308027i
\(474\) 0 0
\(475\) −9946.74 + 5742.76i −0.960817 + 0.554728i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −4151.12 7189.94i −0.395969 0.685839i 0.597255 0.802051i \(-0.296258\pi\)
−0.993224 + 0.116212i \(0.962925\pi\)
\(480\) 0 0
\(481\) 10056.4 + 5806.08i 0.953292 + 0.550384i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 5789.46i 0.542033i
\(486\) 0 0
\(487\) −76.4693 −0.00711530 −0.00355765 0.999994i \(-0.501132\pi\)
−0.00355765 + 0.999994i \(0.501132\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −6951.74 4013.59i −0.638957 0.368902i 0.145256 0.989394i \(-0.453599\pi\)
−0.784213 + 0.620492i \(0.786933\pi\)
\(492\) 0 0
\(493\) 23345.9 13478.7i 2.13275 1.23134i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 271.704 + 2503.09i 0.0245223 + 0.225913i
\(498\) 0 0
\(499\) 8035.68 13918.2i 0.720895 1.24863i −0.239747 0.970836i \(-0.577064\pi\)
0.960641 0.277791i \(-0.0896023\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 7450.97 0.660481 0.330241 0.943897i \(-0.392870\pi\)
0.330241 + 0.943897i \(0.392870\pi\)
\(504\) 0 0
\(505\) −882.452 −0.0777596
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 6060.49 10497.1i 0.527753 0.914095i −0.471724 0.881746i \(-0.656368\pi\)
0.999477 0.0323486i \(-0.0102987\pi\)
\(510\) 0 0
\(511\) 933.918 + 8603.77i 0.0808495 + 0.744830i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −3219.57 + 1858.82i −0.275478 + 0.159047i
\(516\) 0 0
\(517\) 3609.06 + 2083.69i 0.307014 + 0.177255i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −10433.7 −0.877371 −0.438685 0.898641i \(-0.644556\pi\)
−0.438685 + 0.898641i \(0.644556\pi\)
\(522\) 0 0
\(523\) 22346.3i 1.86833i −0.356846 0.934163i \(-0.616148\pi\)
0.356846 0.934163i \(-0.383852\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −20309.8 11725.9i −1.67876 0.969234i
\(528\) 0 0
\(529\) 4250.53 + 7362.13i 0.349349 + 0.605090i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −11754.5 + 6786.45i −0.955240 + 0.551508i
\(534\) 0 0
\(535\) 5678.89 + 3278.71i 0.458916 + 0.264955i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −4713.95 + 4302.94i −0.376706 + 0.343860i
\(540\) 0 0
\(541\) −13690.0 −1.08794 −0.543972 0.839103i \(-0.683080\pi\)
−0.543972 + 0.839103i \(0.683080\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −5007.46 + 8673.18i −0.393571 + 0.681685i
\(546\) 0 0
\(547\) −8042.35 13929.8i −0.628640 1.08884i −0.987825 0.155570i \(-0.950279\pi\)
0.359185 0.933266i \(-0.383055\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −13314.6 23061.6i −1.02944 1.78305i
\(552\) 0 0
\(553\) 4203.50 + 5744.40i 0.323239 + 0.441730i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9207.28i 0.700404i 0.936674 + 0.350202i \(0.113887\pi\)
−0.936674 + 0.350202i \(0.886113\pi\)
\(558\) 0 0
\(559\) 17359.8i 1.31349i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 10095.6 17486.0i 0.755732 1.30897i −0.189278 0.981924i \(-0.560615\pi\)
0.945010 0.327042i \(-0.106052\pi\)
\(564\) 0 0
\(565\) 7028.77 4058.06i 0.523367 0.302166i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3118.35 1800.38i 0.229750 0.132646i −0.380707 0.924696i \(-0.624319\pi\)
0.610457 + 0.792050i \(0.290986\pi\)
\(570\) 0 0
\(571\) −1111.32 + 1924.86i −0.0814489 + 0.141074i −0.903873 0.427802i \(-0.859288\pi\)
0.822424 + 0.568876i \(0.192621\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 14814.5i 1.07444i
\(576\) 0 0
\(577\) 8818.52i 0.636256i 0.948048 + 0.318128i \(0.103054\pi\)
−0.948048 + 0.318128i \(0.896946\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 13063.4 + 17852.0i 0.932805 + 1.27475i
\(582\) 0 0
\(583\) −5098.33 8830.57i −0.362181 0.627316i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 9540.65 + 16524.9i 0.670843 + 1.16193i 0.977665 + 0.210167i \(0.0674008\pi\)
−0.306823 + 0.951767i \(0.599266\pi\)
\(588\) 0 0
\(589\) −11583.1 + 20062.5i −0.810311 + 1.40350i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −7386.65 −0.511523 −0.255762 0.966740i \(-0.582326\pi\)
−0.255762 + 0.966740i \(0.582326\pi\)
\(594\) 0 0
\(595\) −3951.88 + 8958.02i −0.272288 + 0.617215i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −5442.40 3142.17i −0.371236 0.214333i 0.302762 0.953066i \(-0.402091\pi\)
−0.673998 + 0.738733i \(0.735425\pi\)
\(600\) 0 0
\(601\) −23587.5 + 13618.2i −1.60092 + 0.924291i −0.609615 + 0.792697i \(0.708676\pi\)
−0.991304 + 0.131594i \(0.957991\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2306.95 + 3995.76i 0.155026 + 0.268514i
\(606\) 0 0
\(607\) 2294.69 + 1324.84i 0.153441 + 0.0885891i 0.574755 0.818326i \(-0.305098\pi\)
−0.421314 + 0.906915i \(0.638431\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 11415.5i 0.755848i
\(612\) 0 0
\(613\) 19866.5 1.30897 0.654486 0.756074i \(-0.272885\pi\)
0.654486 + 0.756074i \(0.272885\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 21498.0 + 12411.9i 1.40272 + 0.809858i 0.994671 0.103104i \(-0.0328775\pi\)
0.408044 + 0.912962i \(0.366211\pi\)
\(618\) 0 0
\(619\) −6535.31 + 3773.16i −0.424356 + 0.245002i −0.696939 0.717130i \(-0.745455\pi\)
0.272583 + 0.962132i \(0.412122\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 25512.4 2769.31i 1.64066 0.178090i
\(624\) 0 0
\(625\) −3937.30 + 6819.61i −0.251987 + 0.436455i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 25704.9 1.62944
\(630\) 0 0
\(631\) −10950.1 −0.690837 −0.345418 0.938449i \(-0.612263\pi\)
−0.345418 + 0.938449i \(0.612263\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1805.02 + 3126.39i −0.112803 + 0.195381i
\(636\) 0 0
\(637\) −16664.1 5289.33i −1.03651 0.328996i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −27715.4 + 16001.5i −1.70779 + 0.985994i −0.770504 + 0.637435i \(0.779996\pi\)
−0.937287 + 0.348559i \(0.886671\pi\)
\(642\) 0 0
\(643\) 4359.15 + 2516.76i 0.267353 + 0.154356i 0.627684 0.778468i \(-0.284003\pi\)
−0.360331 + 0.932824i \(0.617336\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −5406.97 −0.328547 −0.164274 0.986415i \(-0.552528\pi\)
−0.164274 + 0.986415i \(0.552528\pi\)
\(648\) 0 0
\(649\) 1635.58i 0.0989245i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 22480.8 + 12979.3i 1.34723 + 0.777826i 0.987857 0.155367i \(-0.0496561\pi\)
0.359377 + 0.933193i \(0.382989\pi\)
\(654\) 0 0
\(655\) 2294.90 + 3974.89i 0.136900 + 0.237117i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −22849.6 + 13192.2i −1.35067 + 0.779811i −0.988344 0.152240i \(-0.951351\pi\)
−0.362328 + 0.932051i \(0.618018\pi\)
\(660\) 0 0
\(661\) −4642.84 2680.54i −0.273200 0.157732i 0.357141 0.934051i \(-0.383752\pi\)
−0.630341 + 0.776318i \(0.717085\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 8848.96 + 3903.77i 0.516012 + 0.227642i
\(666\) 0 0
\(667\) −34347.5 −1.99391
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −3356.18 + 5813.08i −0.193091 + 0.334443i
\(672\) 0 0
\(673\) 3378.73 + 5852.13i 0.193522 + 0.335190i 0.946415 0.322953i \(-0.104676\pi\)
−0.752893 + 0.658143i \(0.771342\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −2962.04 5130.41i −0.168154 0.291252i 0.769617 0.638506i \(-0.220447\pi\)
−0.937771 + 0.347254i \(0.887114\pi\)
\(678\) 0 0
\(679\) −18468.0 + 13514.1i −1.04379 + 0.763804i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 8838.52i 0.495163i 0.968867 + 0.247582i \(0.0796358\pi\)
−0.968867 + 0.247582i \(0.920364\pi\)
\(684\) 0 0
\(685\) 987.135i 0.0550606i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 13965.7 24189.2i 0.772205 1.33750i
\(690\) 0 0
\(691\) 25392.1 14660.1i 1.39792 0.807087i 0.403741 0.914873i \(-0.367710\pi\)
0.994174 + 0.107787i \(0.0343763\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3422.75 1976.12i 0.186809 0.107854i
\(696\) 0 0
\(697\) −15022.6 + 26019.9i −0.816387 + 1.41402i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 15890.7i 0.856184i −0.903735 0.428092i \(-0.859186\pi\)
0.903735 0.428092i \(-0.140814\pi\)
\(702\) 0 0
\(703\) 25391.9i 1.36227i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2059.87 2814.96i −0.109575 0.149742i
\(708\) 0 0
\(709\) 1060.77 + 1837.31i 0.0561890 + 0.0973222i 0.892752 0.450549i \(-0.148772\pi\)
−0.836563 + 0.547871i \(0.815438\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 14940.3 + 25877.4i 0.784740 + 1.35921i
\(714\) 0 0
\(715\) 2222.01 3848.63i 0.116222 0.201302i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −23897.3 −1.23953 −0.619764 0.784789i \(-0.712772\pi\)
−0.619764 + 0.784789i \(0.712772\pi\)
\(720\) 0 0
\(721\) −13444.8 5931.25i −0.694467 0.306368i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −21321.2 12309.8i −1.09221 0.630586i
\(726\) 0 0
\(727\) 10755.1 6209.44i 0.548670 0.316775i −0.199915 0.979813i \(-0.564067\pi\)
0.748586 + 0.663038i \(0.230733\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 19213.9 + 33279.5i 0.972166 + 1.68384i
\(732\) 0 0
\(733\) −199.317 115.076i −0.0100436 0.00579865i 0.494970 0.868910i \(-0.335179\pi\)
−0.505013 + 0.863112i \(0.668512\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 13860.2i 0.692734i
\(738\) 0 0
\(739\) 6421.89 0.319666 0.159833 0.987144i \(-0.448904\pi\)
0.159833 + 0.987144i \(0.448904\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 693.090 + 400.156i 0.0342221 + 0.0197581i 0.517013 0.855977i \(-0.327044\pi\)
−0.482791 + 0.875735i \(0.660377\pi\)
\(744\) 0 0
\(745\) −5328.78 + 3076.58i −0.262056 + 0.151298i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2797.12 + 25768.6i 0.136455 + 1.25710i
\(750\) 0 0
\(751\) −971.585 + 1682.83i −0.0472086 + 0.0817676i −0.888664 0.458559i \(-0.848366\pi\)
0.841456 + 0.540326i \(0.181699\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −16293.5 −0.785406
\(756\) 0 0
\(757\) 16016.6 0.768999 0.384500 0.923125i \(-0.374374\pi\)
0.384500 + 0.923125i \(0.374374\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 20615.4 35706.8i 0.982005 1.70088i 0.327450 0.944869i \(-0.393811\pi\)
0.654555 0.756014i \(-0.272856\pi\)
\(762\) 0 0
\(763\) −39355.5 + 4271.94i −1.86732 + 0.202693i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3880.03 2240.13i 0.182659 0.105458i
\(768\) 0 0
\(769\) 4147.70 + 2394.68i 0.194499 + 0.112294i 0.594087 0.804401i \(-0.297513\pi\)
−0.399588 + 0.916695i \(0.630847\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 12950.9 0.602601 0.301301 0.953529i \(-0.402579\pi\)
0.301301 + 0.953529i \(0.402579\pi\)
\(774\) 0 0
\(775\) 21417.9i 0.992713i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 25703.1 + 14839.7i 1.18217 + 0.682526i
\(780\) 0 0
\(781\) −1264.86 2190.80i −0.0579515 0.100375i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2341.42 + 1351.82i −0.106457 + 0.0614632i
\(786\) 0 0
\(787\) −2678.82 1546.62i −0.121334 0.0700521i 0.438105 0.898924i \(-0.355650\pi\)
−0.559439 + 0.828872i \(0.688983\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 29351.9 + 12948.7i 1.31938 + 0.582053i
\(792\) 0 0
\(793\) −18386.9 −0.823377
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −5377.34 + 9313.83i −0.238990 + 0.413943i −0.960425 0.278539i \(-0.910150\pi\)
0.721434 + 0.692483i \(0.243483\pi\)
\(798\) 0 0
\(799\) 12634.8 + 21884.1i 0.559434 + 0.968968i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −4347.64 7530.34i −0.191065 0.330934i
\(804\) 0 0
\(805\) 10067.5 7366.97i 0.440786 0.322548i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 29409.3i 1.27809i 0.769169 + 0.639045i \(0.220670\pi\)
−0.769169 + 0.639045i \(0.779330\pi\)
\(810\) 0 0
\(811\) 8850.91i 0.383228i 0.981470 + 0.191614i \(0.0613721\pi\)
−0.981470 + 0.191614i \(0.938628\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 3738.31 6474.95i 0.160672 0.278291i
\(816\) 0 0
\(817\) 32874.4 18980.0i 1.40775 0.812763i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 23750.3 13712.2i 1.00961 0.582899i 0.0985331 0.995134i \(-0.468585\pi\)
0.911078 + 0.412235i \(0.135252\pi\)
\(822\) 0 0
\(823\) 12157.4 21057.2i 0.514921 0.891869i −0.484929 0.874554i \(-0.661155\pi\)
0.999850 0.0173159i \(-0.00551209\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 14415.0i 0.606119i −0.952972 0.303059i \(-0.901992\pi\)
0.952972 0.303059i \(-0.0980081\pi\)
\(828\) 0 0
\(829\) 8609.61i 0.360704i 0.983602 + 0.180352i \(0.0577238\pi\)
−0.983602 + 0.180352i \(0.942276\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −37800.2 + 8304.07i −1.57227 + 0.345401i
\(834\) 0 0
\(835\) −866.413 1500.67i −0.0359083 0.0621950i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 3728.54 + 6458.02i 0.153425 + 0.265740i 0.932484 0.361210i \(-0.117636\pi\)
−0.779059 + 0.626950i \(0.784303\pi\)
\(840\) 0 0
\(841\) 16345.9 28311.9i 0.670216 1.16085i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1879.52 0.0765177
\(846\) 0 0
\(847\) −7361.18 + 16686.1i −0.298623 + 0.676910i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −28363.6 16375.8i −1.14253 0.659640i
\(852\) 0 0
\(853\) −29109.8 + 16806.6i −1.16847 + 0.674615i −0.953318 0.301967i \(-0.902357\pi\)
−0.215148 + 0.976581i \(0.569024\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −20047.3 34723.0i −0.799071 1.38403i −0.920222 0.391397i \(-0.871992\pi\)
0.121151 0.992634i \(-0.461341\pi\)
\(858\) 0 0
\(859\) −15246.8 8802.77i −0.605606 0.349647i 0.165638 0.986187i \(-0.447032\pi\)
−0.771244 + 0.636540i \(0.780365\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 36465.5i 1.43835i 0.694827 + 0.719177i \(0.255481\pi\)
−0.694827 + 0.719177i \(0.744519\pi\)
\(864\) 0 0
\(865\) −9123.19 −0.358610
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −6193.65 3575.91i −0.241778 0.139591i
\(870\) 0 0
\(871\) 32880.0 18983.3i 1.27910 0.738489i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 19673.1 2135.47i 0.760084 0.0825052i
\(876\) 0 0
\(877\) −16637.8 + 28817.5i −0.640614 + 1.10958i 0.344682 + 0.938719i \(0.387987\pi\)
−0.985296 + 0.170856i \(0.945347\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −11874.3 −0.454091 −0.227045 0.973884i \(-0.572907\pi\)
−0.227045 + 0.973884i \(0.572907\pi\)
\(882\) 0 0
\(883\) −17571.9 −0.669697 −0.334848 0.942272i \(-0.608685\pi\)
−0.334848 + 0.942272i \(0.608685\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 14706.9 25473.1i 0.556719 0.964266i −0.441048 0.897483i \(-0.645393\pi\)
0.997768 0.0667825i \(-0.0212734\pi\)
\(888\) 0 0
\(889\) −14186.3 + 1539.89i −0.535202 + 0.0580948i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 21617.7 12481.0i 0.810089 0.467705i
\(894\) 0 0
\(895\) −3456.81 1995.79i −0.129104 0.0745385i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −49657.5 −1.84224
\(900\) 0 0
\(901\) 61829.3i 2.28616i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2593.90 1497.59i −0.0952753 0.0550072i
\(906\) 0 0
\(907\) 6950.77 + 12039.1i 0.254462 + 0.440740i 0.964749 0.263171i \(-0.0847684\pi\)
−0.710288 + 0.703912i \(0.751435\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 40292.9 23263.1i 1.46538 0.846038i 0.466129 0.884717i \(-0.345648\pi\)
0.999252 + 0.0386789i \(0.0123149\pi\)
\(912\) 0 0
\(913\) −19248.2 11113.0i −0.697725 0.402832i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −7322.74 + 16599.0i −0.263706 + 0.597761i
\(918\) 0 0
\(919\) 48733.4 1.74926 0.874628 0.484794i \(-0.161106\pi\)
0.874628 + 0.484794i \(0.161106\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 3464.77 6001.16i 0.123558 0.214009i
\(924\) 0 0
\(925\) −11737.8 20330.5i −0.417229 0.722662i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −24705.0 42790.3i −0.872491 1.51120i −0.859412 0.511284i \(-0.829170\pi\)
−0.0130789 0.999914i \(-0.504163\pi\)
\(930\) 0 0
\(931\) 8202.97 + 37340.0i 0.288766 + 1.31447i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 9837.35i 0.344081i
\(936\) 0 0
\(937\) 37596.2i 1.31080i 0.755284 + 0.655398i \(0.227499\pi\)
−0.755284 + 0.655398i \(0.772501\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −4437.16 + 7685.39i −0.153717 + 0.266245i −0.932591 0.360935i \(-0.882458\pi\)
0.778874 + 0.627180i \(0.215791\pi\)
\(942\) 0 0
\(943\) 33152.9 19140.8i 1.14486 0.660987i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −40026.5 + 23109.3i −1.37348 + 0.792979i −0.991364 0.131136i \(-0.958138\pi\)
−0.382115 + 0.924115i \(0.624804\pi\)
\(948\) 0 0
\(949\) 11909.3 20627.5i 0.407369 0.705583i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 23772.8i 0.808054i 0.914747 + 0.404027i \(0.132390\pi\)
−0.914747 + 0.404027i \(0.867610\pi\)
\(954\) 0 0
\(955\) 16880.6i 0.571981i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 3148.89 2304.22i 0.106030 0.0775884i
\(960\) 0 0
\(961\) 6704.33 + 11612.2i 0.225046 + 0.389790i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 8176.04 + 14161.3i 0.272742 + 0.472403i
\(966\) 0 0
\(967\) 15698.7 27190.9i 0.522064 0.904242i −0.477606 0.878574i \(-0.658495\pi\)
0.999671 0.0256679i \(-0.00817125\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −42214.9 −1.39520 −0.697600 0.716487i \(-0.745749\pi\)
−0.697600 + 0.716487i \(0.745749\pi\)
\(972\) 0 0
\(973\) 14293.2 + 6305.55i 0.470936 + 0.207756i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −39073.6 22559.2i −1.27950 0.738722i −0.302747 0.953071i \(-0.597904\pi\)
−0.976757 + 0.214349i \(0.931237\pi\)
\(978\) 0 0
\(979\) −22329.4 + 12891.9i −0.728959 + 0.420865i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 17928.1 + 31052.5i 0.581708 + 1.00755i 0.995277 + 0.0970751i \(0.0309487\pi\)
−0.413569 + 0.910473i \(0.635718\pi\)
\(984\) 0 0
\(985\) 7749.56 + 4474.21i 0.250682 + 0.144731i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 48962.3i 1.57423i
\(990\) 0 0
\(991\) 41167.8 1.31961 0.659807 0.751435i \(-0.270638\pi\)
0.659807 + 0.751435i \(0.270638\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 2893.14 + 1670.35i 0.0921795 + 0.0532199i
\(996\) 0 0
\(997\) −30797.6 + 17781.0i −0.978304 + 0.564824i −0.901758 0.432242i \(-0.857723\pi\)
−0.0765462 + 0.997066i \(0.524389\pi\)
\(998\) 0 0
\(999\) 0 0
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 756.4.x.a.125.15 48
3.2 odd 2 252.4.x.a.41.24 yes 48
7.6 odd 2 inner 756.4.x.a.125.10 48
9.2 odd 6 inner 756.4.x.a.629.10 48
9.4 even 3 2268.4.f.a.1133.20 48
9.5 odd 6 2268.4.f.a.1133.29 48
9.7 even 3 252.4.x.a.209.1 yes 48
21.20 even 2 252.4.x.a.41.1 48
63.13 odd 6 2268.4.f.a.1133.30 48
63.20 even 6 inner 756.4.x.a.629.15 48
63.34 odd 6 252.4.x.a.209.24 yes 48
63.41 even 6 2268.4.f.a.1133.19 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.4.x.a.41.1 48 21.20 even 2
252.4.x.a.41.24 yes 48 3.2 odd 2
252.4.x.a.209.1 yes 48 9.7 even 3
252.4.x.a.209.24 yes 48 63.34 odd 6
756.4.x.a.125.10 48 7.6 odd 2 inner
756.4.x.a.125.15 48 1.1 even 1 trivial
756.4.x.a.629.10 48 9.2 odd 6 inner
756.4.x.a.629.15 48 63.20 even 6 inner
2268.4.f.a.1133.19 48 63.41 even 6
2268.4.f.a.1133.20 48 9.4 even 3
2268.4.f.a.1133.29 48 9.5 odd 6
2268.4.f.a.1133.30 48 63.13 odd 6