Properties

Label 756.2.e.a.323.14
Level $756$
Weight $2$
Character 756.323
Analytic conductor $6.037$
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] = [756,2,Mod(323,756)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(756, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("756.323");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.e (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: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 323.14
Character \(\chi\) \(=\) 756.323
Dual form 756.2.e.a.323.13

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.310394 + 1.37973i) q^{2} +(-1.80731 + 0.856520i) q^{4} -1.05392i q^{5} -1.00000i q^{7} +(-1.74275 - 2.22774i) q^{8} +O(q^{10})\) \(q+(0.310394 + 1.37973i) q^{2} +(-1.80731 + 0.856520i) q^{4} -1.05392i q^{5} -1.00000i q^{7} +(-1.74275 - 2.22774i) q^{8} +(1.45413 - 0.327131i) q^{10} +2.97924 q^{11} +0.985148 q^{13} +(1.37973 - 0.310394i) q^{14} +(2.53275 - 3.09600i) q^{16} -6.35564i q^{17} +1.26888i q^{19} +(0.902706 + 1.90477i) q^{20} +(0.924740 + 4.11055i) q^{22} +2.66217 q^{23} +3.88925 q^{25} +(0.305784 + 1.35924i) q^{26} +(0.856520 + 1.80731i) q^{28} -5.12873i q^{29} +8.41216i q^{31} +(5.05779 + 2.53353i) q^{32} +(8.76907 - 1.97275i) q^{34} -1.05392 q^{35} +8.91591 q^{37} +(-1.75072 + 0.393854i) q^{38} +(-2.34787 + 1.83672i) q^{40} -4.54486i q^{41} +0.646087i q^{43} +(-5.38442 + 2.55178i) q^{44} +(0.826321 + 3.67307i) q^{46} +4.64851 q^{47} -1.00000 q^{49} +(1.20720 + 5.36611i) q^{50} +(-1.78047 + 0.843799i) q^{52} -2.57717i q^{53} -3.13989i q^{55} +(-2.22774 + 1.74275i) q^{56} +(7.07627 - 1.59193i) q^{58} -13.7573 q^{59} +4.89018 q^{61} +(-11.6065 + 2.61109i) q^{62} +(-1.92568 + 7.76478i) q^{64} -1.03827i q^{65} +8.21439i q^{67} +(5.44374 + 11.4866i) q^{68} +(-0.327131 - 1.45413i) q^{70} +0.583871 q^{71} -12.4131 q^{73} +(2.76745 + 12.3015i) q^{74} +(-1.08683 - 2.29327i) q^{76} -2.97924i q^{77} -15.1046i q^{79} +(-3.26294 - 2.66932i) q^{80} +(6.27068 - 1.41070i) q^{82} +4.73179 q^{83} -6.69835 q^{85} +(-0.891426 + 0.200542i) q^{86} +(-5.19206 - 6.63699i) q^{88} +1.74631i q^{89} -0.985148i q^{91} +(-4.81136 + 2.28020i) q^{92} +(1.44287 + 6.41369i) q^{94} +1.33731 q^{95} +3.00594 q^{97} +(-0.310394 - 1.37973i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 8 q^{4} - 16 q^{10} + 8 q^{16} + 16 q^{22} - 24 q^{25} - 8 q^{28} - 8 q^{34} + 16 q^{37} - 8 q^{40} - 24 q^{49} - 8 q^{52} + 32 q^{58} - 80 q^{61} + 40 q^{64} - 24 q^{70} - 32 q^{82} + 56 q^{85} + 56 q^{88} + 72 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.310394 + 1.37973i 0.219482 + 0.975617i
\(3\) 0 0
\(4\) −1.80731 + 0.856520i −0.903655 + 0.428260i
\(5\) 1.05392i 0.471328i −0.971835 0.235664i \(-0.924273\pi\)
0.971835 0.235664i \(-0.0757266\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) −1.74275 2.22774i −0.616154 0.787626i
\(9\) 0 0
\(10\) 1.45413 0.327131i 0.459836 0.103448i
\(11\) 2.97924 0.898276 0.449138 0.893462i \(-0.351731\pi\)
0.449138 + 0.893462i \(0.351731\pi\)
\(12\) 0 0
\(13\) 0.985148 0.273231 0.136615 0.990624i \(-0.456378\pi\)
0.136615 + 0.990624i \(0.456378\pi\)
\(14\) 1.37973 0.310394i 0.368748 0.0829563i
\(15\) 0 0
\(16\) 2.53275 3.09600i 0.633186 0.773999i
\(17\) 6.35564i 1.54147i −0.637156 0.770735i \(-0.719889\pi\)
0.637156 0.770735i \(-0.280111\pi\)
\(18\) 0 0
\(19\) 1.26888i 0.291102i 0.989351 + 0.145551i \(0.0464955\pi\)
−0.989351 + 0.145551i \(0.953504\pi\)
\(20\) 0.902706 + 1.90477i 0.201851 + 0.425919i
\(21\) 0 0
\(22\) 0.924740 + 4.11055i 0.197155 + 0.876373i
\(23\) 2.66217 0.555100 0.277550 0.960711i \(-0.410478\pi\)
0.277550 + 0.960711i \(0.410478\pi\)
\(24\) 0 0
\(25\) 3.88925 0.777849
\(26\) 0.305784 + 1.35924i 0.0599692 + 0.266569i
\(27\) 0 0
\(28\) 0.856520 + 1.80731i 0.161867 + 0.341550i
\(29\) 5.12873i 0.952382i −0.879342 0.476191i \(-0.842017\pi\)
0.879342 0.476191i \(-0.157983\pi\)
\(30\) 0 0
\(31\) 8.41216i 1.51087i 0.655224 + 0.755435i \(0.272574\pi\)
−0.655224 + 0.755435i \(0.727426\pi\)
\(32\) 5.05779 + 2.53353i 0.894099 + 0.447869i
\(33\) 0 0
\(34\) 8.76907 1.97275i 1.50388 0.338324i
\(35\) −1.05392 −0.178145
\(36\) 0 0
\(37\) 8.91591 1.46577 0.732883 0.680355i \(-0.238174\pi\)
0.732883 + 0.680355i \(0.238174\pi\)
\(38\) −1.75072 + 0.393854i −0.284004 + 0.0638916i
\(39\) 0 0
\(40\) −2.34787 + 1.83672i −0.371231 + 0.290411i
\(41\) 4.54486i 0.709788i −0.934907 0.354894i \(-0.884517\pi\)
0.934907 0.354894i \(-0.115483\pi\)
\(42\) 0 0
\(43\) 0.646087i 0.0985273i 0.998786 + 0.0492637i \(0.0156875\pi\)
−0.998786 + 0.0492637i \(0.984313\pi\)
\(44\) −5.38442 + 2.55178i −0.811732 + 0.384696i
\(45\) 0 0
\(46\) 0.826321 + 3.67307i 0.121834 + 0.541565i
\(47\) 4.64851 0.678055 0.339027 0.940777i \(-0.389902\pi\)
0.339027 + 0.940777i \(0.389902\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 1.20720 + 5.36611i 0.170724 + 0.758883i
\(51\) 0 0
\(52\) −1.78047 + 0.843799i −0.246907 + 0.117014i
\(53\) 2.57717i 0.354002i −0.984211 0.177001i \(-0.943360\pi\)
0.984211 0.177001i \(-0.0566396\pi\)
\(54\) 0 0
\(55\) 3.13989i 0.423383i
\(56\) −2.22774 + 1.74275i −0.297695 + 0.232884i
\(57\) 0 0
\(58\) 7.07627 1.59193i 0.929159 0.209030i
\(59\) −13.7573 −1.79104 −0.895521 0.445019i \(-0.853197\pi\)
−0.895521 + 0.445019i \(0.853197\pi\)
\(60\) 0 0
\(61\) 4.89018 0.626124 0.313062 0.949733i \(-0.398645\pi\)
0.313062 + 0.949733i \(0.398645\pi\)
\(62\) −11.6065 + 2.61109i −1.47403 + 0.331608i
\(63\) 0 0
\(64\) −1.92568 + 7.76478i −0.240709 + 0.970597i
\(65\) 1.03827i 0.128782i
\(66\) 0 0
\(67\) 8.21439i 1.00355i 0.864999 + 0.501774i \(0.167319\pi\)
−0.864999 + 0.501774i \(0.832681\pi\)
\(68\) 5.44374 + 11.4866i 0.660150 + 1.39296i
\(69\) 0 0
\(70\) −0.327131 1.45413i −0.0390997 0.173802i
\(71\) 0.583871 0.0692927 0.0346464 0.999400i \(-0.488970\pi\)
0.0346464 + 0.999400i \(0.488970\pi\)
\(72\) 0 0
\(73\) −12.4131 −1.45284 −0.726422 0.687249i \(-0.758818\pi\)
−0.726422 + 0.687249i \(0.758818\pi\)
\(74\) 2.76745 + 12.3015i 0.321709 + 1.43003i
\(75\) 0 0
\(76\) −1.08683 2.29327i −0.124667 0.263056i
\(77\) 2.97924i 0.339516i
\(78\) 0 0
\(79\) 15.1046i 1.69940i −0.527269 0.849699i \(-0.676784\pi\)
0.527269 0.849699i \(-0.323216\pi\)
\(80\) −3.26294 2.66932i −0.364808 0.298439i
\(81\) 0 0
\(82\) 6.27068 1.41070i 0.692481 0.155786i
\(83\) 4.73179 0.519382 0.259691 0.965692i \(-0.416379\pi\)
0.259691 + 0.965692i \(0.416379\pi\)
\(84\) 0 0
\(85\) −6.69835 −0.726538
\(86\) −0.891426 + 0.200542i −0.0961249 + 0.0216250i
\(87\) 0 0
\(88\) −5.19206 6.63699i −0.553476 0.707505i
\(89\) 1.74631i 0.185108i 0.995708 + 0.0925541i \(0.0295031\pi\)
−0.995708 + 0.0925541i \(0.970497\pi\)
\(90\) 0 0
\(91\) 0.985148i 0.103272i
\(92\) −4.81136 + 2.28020i −0.501619 + 0.237727i
\(93\) 0 0
\(94\) 1.44287 + 6.41369i 0.148821 + 0.661521i
\(95\) 1.33731 0.137205
\(96\) 0 0
\(97\) 3.00594 0.305207 0.152604 0.988287i \(-0.451234\pi\)
0.152604 + 0.988287i \(0.451234\pi\)
\(98\) −0.310394 1.37973i −0.0313545 0.139374i
\(99\) 0 0
\(100\) −7.02908 + 3.33122i −0.702908 + 0.333122i
\(101\) 1.53513i 0.152751i −0.997079 0.0763757i \(-0.975665\pi\)
0.997079 0.0763757i \(-0.0243348\pi\)
\(102\) 0 0
\(103\) 11.6910i 1.15195i −0.817468 0.575974i \(-0.804623\pi\)
0.817468 0.575974i \(-0.195377\pi\)
\(104\) −1.71686 2.19466i −0.168352 0.215204i
\(105\) 0 0
\(106\) 3.55581 0.799940i 0.345370 0.0776970i
\(107\) 15.7149 1.51922 0.759611 0.650378i \(-0.225390\pi\)
0.759611 + 0.650378i \(0.225390\pi\)
\(108\) 0 0
\(109\) −7.34105 −0.703145 −0.351573 0.936161i \(-0.614353\pi\)
−0.351573 + 0.936161i \(0.614353\pi\)
\(110\) 4.33220 0.974604i 0.413059 0.0929249i
\(111\) 0 0
\(112\) −3.09600 2.53275i −0.292544 0.239322i
\(113\) 4.35118i 0.409324i −0.978833 0.204662i \(-0.934390\pi\)
0.978833 0.204662i \(-0.0656096\pi\)
\(114\) 0 0
\(115\) 2.80572i 0.261635i
\(116\) 4.39286 + 9.26921i 0.407867 + 0.860625i
\(117\) 0 0
\(118\) −4.27017 18.9813i −0.393101 1.74737i
\(119\) −6.35564 −0.582621
\(120\) 0 0
\(121\) −2.12410 −0.193100
\(122\) 1.51788 + 6.74713i 0.137423 + 0.610857i
\(123\) 0 0
\(124\) −7.20519 15.2034i −0.647045 1.36531i
\(125\) 9.36858i 0.837951i
\(126\) 0 0
\(127\) 20.4265i 1.81255i 0.422684 + 0.906277i \(0.361088\pi\)
−0.422684 + 0.906277i \(0.638912\pi\)
\(128\) −11.3110 0.246771i −0.999762 0.0218117i
\(129\) 0 0
\(130\) 1.43253 0.322273i 0.125641 0.0282652i
\(131\) −17.1791 −1.50094 −0.750471 0.660903i \(-0.770173\pi\)
−0.750471 + 0.660903i \(0.770173\pi\)
\(132\) 0 0
\(133\) 1.26888 0.110026
\(134\) −11.3336 + 2.54970i −0.979078 + 0.220260i
\(135\) 0 0
\(136\) −14.1587 + 11.0763i −1.21410 + 0.949782i
\(137\) 16.2830i 1.39115i 0.718452 + 0.695577i \(0.244851\pi\)
−0.718452 + 0.695577i \(0.755149\pi\)
\(138\) 0 0
\(139\) 1.05732i 0.0896804i −0.998994 0.0448402i \(-0.985722\pi\)
0.998994 0.0448402i \(-0.0142779\pi\)
\(140\) 1.90477 0.902706i 0.160982 0.0762926i
\(141\) 0 0
\(142\) 0.181230 + 0.805584i 0.0152085 + 0.0676031i
\(143\) 2.93500 0.245437
\(144\) 0 0
\(145\) −5.40529 −0.448885
\(146\) −3.85295 17.1267i −0.318873 1.41742i
\(147\) 0 0
\(148\) −16.1138 + 7.63666i −1.32455 + 0.627729i
\(149\) 6.71787i 0.550349i −0.961394 0.275175i \(-0.911264\pi\)
0.961394 0.275175i \(-0.0887357\pi\)
\(150\) 0 0
\(151\) 4.96946i 0.404409i 0.979343 + 0.202205i \(0.0648106\pi\)
−0.979343 + 0.202205i \(0.935189\pi\)
\(152\) 2.82675 2.21134i 0.229280 0.179364i
\(153\) 0 0
\(154\) 4.11055 0.924740i 0.331238 0.0745177i
\(155\) 8.86577 0.712116
\(156\) 0 0
\(157\) 14.3997 1.14922 0.574609 0.818428i \(-0.305154\pi\)
0.574609 + 0.818428i \(0.305154\pi\)
\(158\) 20.8402 4.68837i 1.65796 0.372987i
\(159\) 0 0
\(160\) 2.67014 5.33052i 0.211093 0.421415i
\(161\) 2.66217i 0.209808i
\(162\) 0 0
\(163\) 2.22595i 0.174350i −0.996193 0.0871749i \(-0.972216\pi\)
0.996193 0.0871749i \(-0.0277839\pi\)
\(164\) 3.89277 + 8.21398i 0.303974 + 0.641404i
\(165\) 0 0
\(166\) 1.46872 + 6.52860i 0.113995 + 0.506717i
\(167\) −12.4976 −0.967094 −0.483547 0.875318i \(-0.660652\pi\)
−0.483547 + 0.875318i \(0.660652\pi\)
\(168\) 0 0
\(169\) −12.0295 −0.925345
\(170\) −2.07913 9.24192i −0.159462 0.708823i
\(171\) 0 0
\(172\) −0.553387 1.16768i −0.0421953 0.0890348i
\(173\) 16.1418i 1.22724i −0.789603 0.613619i \(-0.789713\pi\)
0.789603 0.613619i \(-0.210287\pi\)
\(174\) 0 0
\(175\) 3.88925i 0.293999i
\(176\) 7.54567 9.22373i 0.568776 0.695265i
\(177\) 0 0
\(178\) −2.40943 + 0.542043i −0.180595 + 0.0406279i
\(179\) 14.5759 1.08945 0.544727 0.838613i \(-0.316633\pi\)
0.544727 + 0.838613i \(0.316633\pi\)
\(180\) 0 0
\(181\) 1.95874 0.145592 0.0727960 0.997347i \(-0.476808\pi\)
0.0727960 + 0.997347i \(0.476808\pi\)
\(182\) 1.35924 0.305784i 0.100753 0.0226662i
\(183\) 0 0
\(184\) −4.63948 5.93062i −0.342027 0.437211i
\(185\) 9.39668i 0.690857i
\(186\) 0 0
\(187\) 18.9350i 1.38466i
\(188\) −8.40130 + 3.98154i −0.612728 + 0.290384i
\(189\) 0 0
\(190\) 0.415092 + 1.84512i 0.0301139 + 0.133859i
\(191\) 26.2224 1.89739 0.948694 0.316194i \(-0.102405\pi\)
0.948694 + 0.316194i \(0.102405\pi\)
\(192\) 0 0
\(193\) −23.4000 −1.68437 −0.842186 0.539187i \(-0.818732\pi\)
−0.842186 + 0.539187i \(0.818732\pi\)
\(194\) 0.933027 + 4.14739i 0.0669874 + 0.297765i
\(195\) 0 0
\(196\) 1.80731 0.856520i 0.129094 0.0611800i
\(197\) 16.5377i 1.17826i 0.808036 + 0.589132i \(0.200530\pi\)
−0.808036 + 0.589132i \(0.799470\pi\)
\(198\) 0 0
\(199\) 1.20608i 0.0854964i −0.999086 0.0427482i \(-0.986389\pi\)
0.999086 0.0427482i \(-0.0136113\pi\)
\(200\) −6.77797 8.66424i −0.479275 0.612654i
\(201\) 0 0
\(202\) 2.11807 0.476496i 0.149027 0.0335261i
\(203\) −5.12873 −0.359966
\(204\) 0 0
\(205\) −4.78993 −0.334543
\(206\) 16.1304 3.62881i 1.12386 0.252831i
\(207\) 0 0
\(208\) 2.49513 3.05002i 0.173006 0.211481i
\(209\) 3.78032i 0.261490i
\(210\) 0 0
\(211\) 9.33019i 0.642317i 0.947025 + 0.321159i \(0.104072\pi\)
−0.947025 + 0.321159i \(0.895928\pi\)
\(212\) 2.20740 + 4.65776i 0.151605 + 0.319896i
\(213\) 0 0
\(214\) 4.87783 + 21.6824i 0.333441 + 1.48218i
\(215\) 0.680926 0.0464387
\(216\) 0 0
\(217\) 8.41216 0.571055
\(218\) −2.27862 10.1287i −0.154328 0.686000i
\(219\) 0 0
\(220\) 2.68938 + 5.67476i 0.181318 + 0.382592i
\(221\) 6.26125i 0.421177i
\(222\) 0 0
\(223\) 2.92491i 0.195866i −0.995193 0.0979332i \(-0.968777\pi\)
0.995193 0.0979332i \(-0.0312231\pi\)
\(224\) 2.53353 5.05779i 0.169278 0.337938i
\(225\) 0 0
\(226\) 6.00345 1.35058i 0.399344 0.0898393i
\(227\) −23.0959 −1.53293 −0.766466 0.642285i \(-0.777987\pi\)
−0.766466 + 0.642285i \(0.777987\pi\)
\(228\) 0 0
\(229\) 9.08386 0.600278 0.300139 0.953895i \(-0.402967\pi\)
0.300139 + 0.953895i \(0.402967\pi\)
\(230\) 3.87113 0.870879i 0.255255 0.0574240i
\(231\) 0 0
\(232\) −11.4255 + 8.93807i −0.750120 + 0.586813i
\(233\) 29.4595i 1.92996i 0.262330 + 0.964978i \(0.415509\pi\)
−0.262330 + 0.964978i \(0.584491\pi\)
\(234\) 0 0
\(235\) 4.89917i 0.319586i
\(236\) 24.8636 11.7834i 1.61849 0.767032i
\(237\) 0 0
\(238\) −1.97275 8.76907i −0.127875 0.568414i
\(239\) −17.3719 −1.12369 −0.561846 0.827242i \(-0.689909\pi\)
−0.561846 + 0.827242i \(0.689909\pi\)
\(240\) 0 0
\(241\) −11.8341 −0.762299 −0.381150 0.924513i \(-0.624472\pi\)
−0.381150 + 0.924513i \(0.624472\pi\)
\(242\) −0.659310 2.93069i −0.0423820 0.188392i
\(243\) 0 0
\(244\) −8.83808 + 4.18854i −0.565800 + 0.268144i
\(245\) 1.05392i 0.0673326i
\(246\) 0 0
\(247\) 1.25004i 0.0795381i
\(248\) 18.7401 14.6603i 1.19000 0.930928i
\(249\) 0 0
\(250\) 12.9261 2.90795i 0.817519 0.183915i
\(251\) 22.4110 1.41457 0.707284 0.706930i \(-0.249920\pi\)
0.707284 + 0.706930i \(0.249920\pi\)
\(252\) 0 0
\(253\) 7.93125 0.498633
\(254\) −28.1830 + 6.34025i −1.76836 + 0.397823i
\(255\) 0 0
\(256\) −3.17040 15.6827i −0.198150 0.980172i
\(257\) 20.2382i 1.26242i 0.775611 + 0.631211i \(0.217442\pi\)
−0.775611 + 0.631211i \(0.782558\pi\)
\(258\) 0 0
\(259\) 8.91591i 0.554008i
\(260\) 0.889299 + 1.87648i 0.0551520 + 0.116374i
\(261\) 0 0
\(262\) −5.33228 23.7025i −0.329429 1.46434i
\(263\) −25.1291 −1.54952 −0.774762 0.632253i \(-0.782130\pi\)
−0.774762 + 0.632253i \(0.782130\pi\)
\(264\) 0 0
\(265\) −2.71614 −0.166851
\(266\) 0.393854 + 1.75072i 0.0241488 + 0.107343i
\(267\) 0 0
\(268\) −7.03579 14.8460i −0.429779 0.906861i
\(269\) 4.09135i 0.249454i 0.992191 + 0.124727i \(0.0398055\pi\)
−0.992191 + 0.124727i \(0.960195\pi\)
\(270\) 0 0
\(271\) 2.36584i 0.143715i −0.997415 0.0718573i \(-0.977107\pi\)
0.997415 0.0718573i \(-0.0228926\pi\)
\(272\) −19.6770 16.0972i −1.19310 0.976038i
\(273\) 0 0
\(274\) −22.4662 + 5.05416i −1.35723 + 0.305333i
\(275\) 11.5870 0.698723
\(276\) 0 0
\(277\) 20.9260 1.25732 0.628661 0.777679i \(-0.283603\pi\)
0.628661 + 0.777679i \(0.283603\pi\)
\(278\) 1.45881 0.328185i 0.0874937 0.0196832i
\(279\) 0 0
\(280\) 1.83672 + 2.34787i 0.109765 + 0.140312i
\(281\) 12.7122i 0.758348i −0.925325 0.379174i \(-0.876208\pi\)
0.925325 0.379174i \(-0.123792\pi\)
\(282\) 0 0
\(283\) 18.4751i 1.09823i 0.835747 + 0.549115i \(0.185035\pi\)
−0.835747 + 0.549115i \(0.814965\pi\)
\(284\) −1.05524 + 0.500097i −0.0626168 + 0.0296753i
\(285\) 0 0
\(286\) 0.911006 + 4.04950i 0.0538689 + 0.239452i
\(287\) −4.54486 −0.268275
\(288\) 0 0
\(289\) −23.3942 −1.37613
\(290\) −1.67777 7.45784i −0.0985220 0.437939i
\(291\) 0 0
\(292\) 22.4343 10.6321i 1.31287 0.622195i
\(293\) 26.8597i 1.56916i 0.620028 + 0.784579i \(0.287121\pi\)
−0.620028 + 0.784579i \(0.712879\pi\)
\(294\) 0 0
\(295\) 14.4991i 0.844169i
\(296\) −15.5382 19.8623i −0.903137 1.15448i
\(297\) 0 0
\(298\) 9.26885 2.08519i 0.536930 0.120792i
\(299\) 2.62263 0.151671
\(300\) 0 0
\(301\) 0.646087 0.0372398
\(302\) −6.85652 + 1.54249i −0.394548 + 0.0887605i
\(303\) 0 0
\(304\) 3.92846 + 3.21376i 0.225313 + 0.184322i
\(305\) 5.15387i 0.295110i
\(306\) 0 0
\(307\) 16.6518i 0.950370i −0.879886 0.475185i \(-0.842381\pi\)
0.879886 0.475185i \(-0.157619\pi\)
\(308\) 2.55178 + 5.38442i 0.145401 + 0.306806i
\(309\) 0 0
\(310\) 2.75188 + 12.2324i 0.156296 + 0.694752i
\(311\) −9.01174 −0.511009 −0.255505 0.966808i \(-0.582242\pi\)
−0.255505 + 0.966808i \(0.582242\pi\)
\(312\) 0 0
\(313\) −2.50866 −0.141798 −0.0708988 0.997484i \(-0.522587\pi\)
−0.0708988 + 0.997484i \(0.522587\pi\)
\(314\) 4.46957 + 19.8677i 0.252233 + 1.12120i
\(315\) 0 0
\(316\) 12.9374 + 27.2987i 0.727784 + 1.53567i
\(317\) 22.1219i 1.24249i 0.783616 + 0.621246i \(0.213373\pi\)
−0.783616 + 0.621246i \(0.786627\pi\)
\(318\) 0 0
\(319\) 15.2797i 0.855501i
\(320\) 8.18347 + 2.02951i 0.457470 + 0.113453i
\(321\) 0 0
\(322\) 3.67307 0.826321i 0.204692 0.0460491i
\(323\) 8.06458 0.448725
\(324\) 0 0
\(325\) 3.83148 0.212533
\(326\) 3.07121 0.690922i 0.170099 0.0382666i
\(327\) 0 0
\(328\) −10.1248 + 7.92054i −0.559047 + 0.437338i
\(329\) 4.64851i 0.256281i
\(330\) 0 0
\(331\) 18.4307i 1.01304i 0.862228 + 0.506520i \(0.169068\pi\)
−0.862228 + 0.506520i \(0.830932\pi\)
\(332\) −8.55182 + 4.05288i −0.469342 + 0.222431i
\(333\) 0 0
\(334\) −3.87918 17.2433i −0.212259 0.943513i
\(335\) 8.65733 0.473001
\(336\) 0 0
\(337\) −24.7708 −1.34935 −0.674675 0.738115i \(-0.735716\pi\)
−0.674675 + 0.738115i \(0.735716\pi\)
\(338\) −3.73388 16.5974i −0.203096 0.902782i
\(339\) 0 0
\(340\) 12.1060 5.73728i 0.656540 0.311147i
\(341\) 25.0619i 1.35718i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 1.43932 1.12597i 0.0776027 0.0607080i
\(345\) 0 0
\(346\) 22.2713 5.01031i 1.19731 0.269356i
\(347\) −27.1358 −1.45673 −0.728364 0.685191i \(-0.759719\pi\)
−0.728364 + 0.685191i \(0.759719\pi\)
\(348\) 0 0
\(349\) 12.5396 0.671232 0.335616 0.941999i \(-0.391056\pi\)
0.335616 + 0.941999i \(0.391056\pi\)
\(350\) 5.36611 1.20720i 0.286831 0.0645275i
\(351\) 0 0
\(352\) 15.0684 + 7.54799i 0.803148 + 0.402309i
\(353\) 24.6724i 1.31318i 0.754247 + 0.656590i \(0.228002\pi\)
−0.754247 + 0.656590i \(0.771998\pi\)
\(354\) 0 0
\(355\) 0.615355i 0.0326596i
\(356\) −1.49575 3.15612i −0.0792744 0.167274i
\(357\) 0 0
\(358\) 4.52427 + 20.1108i 0.239115 + 1.06289i
\(359\) 22.8525 1.20611 0.603054 0.797700i \(-0.293950\pi\)
0.603054 + 0.797700i \(0.293950\pi\)
\(360\) 0 0
\(361\) 17.3899 0.915260
\(362\) 0.607981 + 2.70253i 0.0319548 + 0.142042i
\(363\) 0 0
\(364\) 0.843799 + 1.78047i 0.0442271 + 0.0933219i
\(365\) 13.0824i 0.684767i
\(366\) 0 0
\(367\) 20.3116i 1.06026i 0.847917 + 0.530128i \(0.177856\pi\)
−0.847917 + 0.530128i \(0.822144\pi\)
\(368\) 6.74259 8.24206i 0.351482 0.429647i
\(369\) 0 0
\(370\) 12.9649 2.91667i 0.674012 0.151631i
\(371\) −2.57717 −0.133800
\(372\) 0 0
\(373\) 29.4192 1.52327 0.761635 0.648007i \(-0.224397\pi\)
0.761635 + 0.648007i \(0.224397\pi\)
\(374\) 26.1252 5.87731i 1.35090 0.303909i
\(375\) 0 0
\(376\) −8.10117 10.3557i −0.417786 0.534053i
\(377\) 5.05256i 0.260220i
\(378\) 0 0
\(379\) 18.5441i 0.952546i 0.879298 + 0.476273i \(0.158013\pi\)
−0.879298 + 0.476273i \(0.841987\pi\)
\(380\) −2.41693 + 1.14543i −0.123986 + 0.0587593i
\(381\) 0 0
\(382\) 8.13929 + 36.1799i 0.416442 + 1.85112i
\(383\) −37.8089 −1.93194 −0.965972 0.258648i \(-0.916723\pi\)
−0.965972 + 0.258648i \(0.916723\pi\)
\(384\) 0 0
\(385\) −3.13989 −0.160024
\(386\) −7.26324 32.2857i −0.369689 1.64330i
\(387\) 0 0
\(388\) −5.43267 + 2.57465i −0.275802 + 0.130708i
\(389\) 8.06117i 0.408718i 0.978896 + 0.204359i \(0.0655109\pi\)
−0.978896 + 0.204359i \(0.934489\pi\)
\(390\) 0 0
\(391\) 16.9198i 0.855670i
\(392\) 1.74275 + 2.22774i 0.0880219 + 0.112518i
\(393\) 0 0
\(394\) −22.8176 + 5.13322i −1.14953 + 0.258608i
\(395\) −15.9191 −0.800974
\(396\) 0 0
\(397\) 27.5806 1.38423 0.692115 0.721787i \(-0.256679\pi\)
0.692115 + 0.721787i \(0.256679\pi\)
\(398\) 1.66406 0.374359i 0.0834117 0.0187649i
\(399\) 0 0
\(400\) 9.85048 12.0411i 0.492524 0.602055i
\(401\) 7.24355i 0.361725i −0.983508 0.180863i \(-0.942111\pi\)
0.983508 0.180863i \(-0.0578890\pi\)
\(402\) 0 0
\(403\) 8.28723i 0.412816i
\(404\) 1.31487 + 2.77446i 0.0654173 + 0.138035i
\(405\) 0 0
\(406\) −1.59193 7.07627i −0.0790061 0.351189i
\(407\) 26.5627 1.31666
\(408\) 0 0
\(409\) 26.4380 1.30728 0.653638 0.756807i \(-0.273242\pi\)
0.653638 + 0.756807i \(0.273242\pi\)
\(410\) −1.48677 6.60881i −0.0734262 0.326386i
\(411\) 0 0
\(412\) 10.0136 + 21.1292i 0.493333 + 1.04096i
\(413\) 13.7573i 0.676950i
\(414\) 0 0
\(415\) 4.98694i 0.244799i
\(416\) 4.98267 + 2.49590i 0.244296 + 0.122372i
\(417\) 0 0
\(418\) −5.21582 + 1.17339i −0.255114 + 0.0573923i
\(419\) 20.3362 0.993487 0.496744 0.867897i \(-0.334529\pi\)
0.496744 + 0.867897i \(0.334529\pi\)
\(420\) 0 0
\(421\) 13.6537 0.665442 0.332721 0.943025i \(-0.392033\pi\)
0.332721 + 0.943025i \(0.392033\pi\)
\(422\) −12.8732 + 2.89604i −0.626655 + 0.140977i
\(423\) 0 0
\(424\) −5.74128 + 4.49136i −0.278821 + 0.218120i
\(425\) 24.7187i 1.19903i
\(426\) 0 0
\(427\) 4.89018i 0.236653i
\(428\) −28.4018 + 13.4602i −1.37285 + 0.650622i
\(429\) 0 0
\(430\) 0.211355 + 0.939494i 0.0101925 + 0.0453064i
\(431\) −16.9782 −0.817811 −0.408906 0.912577i \(-0.634089\pi\)
−0.408906 + 0.912577i \(0.634089\pi\)
\(432\) 0 0
\(433\) 0.624418 0.0300076 0.0150038 0.999887i \(-0.495224\pi\)
0.0150038 + 0.999887i \(0.495224\pi\)
\(434\) 2.61109 + 11.6065i 0.125336 + 0.557131i
\(435\) 0 0
\(436\) 13.2676 6.28776i 0.635401 0.301129i
\(437\) 3.37798i 0.161591i
\(438\) 0 0
\(439\) 22.8786i 1.09194i 0.837806 + 0.545969i \(0.183838\pi\)
−0.837806 + 0.545969i \(0.816162\pi\)
\(440\) −6.99487 + 5.47203i −0.333467 + 0.260869i
\(441\) 0 0
\(442\) 8.63883 1.94345i 0.410907 0.0924407i
\(443\) −0.212425 −0.0100926 −0.00504630 0.999987i \(-0.501606\pi\)
−0.00504630 + 0.999987i \(0.501606\pi\)
\(444\) 0 0
\(445\) 1.84047 0.0872467
\(446\) 4.03558 0.907874i 0.191090 0.0429891i
\(447\) 0 0
\(448\) 7.76478 + 1.92568i 0.366851 + 0.0909796i
\(449\) 17.6011i 0.830650i 0.909673 + 0.415325i \(0.136332\pi\)
−0.909673 + 0.415325i \(0.863668\pi\)
\(450\) 0 0
\(451\) 13.5402i 0.637585i
\(452\) 3.72687 + 7.86393i 0.175297 + 0.369888i
\(453\) 0 0
\(454\) −7.16885 31.8662i −0.336451 1.49555i
\(455\) −1.03827 −0.0486748
\(456\) 0 0
\(457\) 8.56351 0.400584 0.200292 0.979736i \(-0.435811\pi\)
0.200292 + 0.979736i \(0.435811\pi\)
\(458\) 2.81958 + 12.5333i 0.131750 + 0.585641i
\(459\) 0 0
\(460\) 2.40315 + 5.07081i 0.112048 + 0.236428i
\(461\) 33.2233i 1.54736i 0.633575 + 0.773681i \(0.281587\pi\)
−0.633575 + 0.773681i \(0.718413\pi\)
\(462\) 0 0
\(463\) 10.8819i 0.505723i −0.967502 0.252862i \(-0.918628\pi\)
0.967502 0.252862i \(-0.0813717\pi\)
\(464\) −15.8785 12.9898i −0.737143 0.603035i
\(465\) 0 0
\(466\) −40.6462 + 9.14406i −1.88290 + 0.423590i
\(467\) −41.7776 −1.93324 −0.966618 0.256224i \(-0.917522\pi\)
−0.966618 + 0.256224i \(0.917522\pi\)
\(468\) 0 0
\(469\) 8.21439 0.379305
\(470\) 6.75953 1.52067i 0.311794 0.0701434i
\(471\) 0 0
\(472\) 23.9754 + 30.6476i 1.10356 + 1.41067i
\(473\) 1.92485i 0.0885047i
\(474\) 0 0
\(475\) 4.93501i 0.226434i
\(476\) 11.4866 5.44374i 0.526488 0.249513i
\(477\) 0 0
\(478\) −5.39212 23.9685i −0.246630 1.09629i
\(479\) −8.46943 −0.386978 −0.193489 0.981102i \(-0.561980\pi\)
−0.193489 + 0.981102i \(0.561980\pi\)
\(480\) 0 0
\(481\) 8.78349 0.400493
\(482\) −3.67323 16.3278i −0.167311 0.743712i
\(483\) 0 0
\(484\) 3.83892 1.81934i 0.174496 0.0826972i
\(485\) 3.16803i 0.143853i
\(486\) 0 0
\(487\) 23.1496i 1.04901i 0.851408 + 0.524504i \(0.175749\pi\)
−0.851408 + 0.524504i \(0.824251\pi\)
\(488\) −8.52234 10.8941i −0.385788 0.493151i
\(489\) 0 0
\(490\) −1.45413 + 0.327131i −0.0656908 + 0.0147783i
\(491\) 6.91483 0.312062 0.156031 0.987752i \(-0.450130\pi\)
0.156031 + 0.987752i \(0.450130\pi\)
\(492\) 0 0
\(493\) −32.5964 −1.46807
\(494\) −1.72472 + 0.388005i −0.0775987 + 0.0174572i
\(495\) 0 0
\(496\) 26.0440 + 21.3059i 1.16941 + 0.956662i
\(497\) 0.583871i 0.0261902i
\(498\) 0 0
\(499\) 7.58869i 0.339717i 0.985468 + 0.169858i \(0.0543310\pi\)
−0.985468 + 0.169858i \(0.945669\pi\)
\(500\) 8.02438 + 16.9319i 0.358861 + 0.757219i
\(501\) 0 0
\(502\) 6.95623 + 30.9211i 0.310472 + 1.38008i
\(503\) 19.2293 0.857390 0.428695 0.903449i \(-0.358973\pi\)
0.428695 + 0.903449i \(0.358973\pi\)
\(504\) 0 0
\(505\) −1.61791 −0.0719961
\(506\) 2.46181 + 10.9430i 0.109441 + 0.486475i
\(507\) 0 0
\(508\) −17.4957 36.9170i −0.776245 1.63792i
\(509\) 23.2826i 1.03198i −0.856593 0.515992i \(-0.827423\pi\)
0.856593 0.515992i \(-0.172577\pi\)
\(510\) 0 0
\(511\) 12.4131i 0.549123i
\(512\) 20.6539 9.24212i 0.912782 0.408448i
\(513\) 0 0
\(514\) −27.9232 + 6.28181i −1.23164 + 0.277079i
\(515\) −12.3214 −0.542945
\(516\) 0 0
\(517\) 13.8490 0.609080
\(518\) 12.3015 2.76745i 0.540499 0.121595i
\(519\) 0 0
\(520\) −2.31300 + 1.80944i −0.101432 + 0.0793492i
\(521\) 29.5778i 1.29583i −0.761714 0.647914i \(-0.775642\pi\)
0.761714 0.647914i \(-0.224358\pi\)
\(522\) 0 0
\(523\) 35.2794i 1.54266i −0.636434 0.771331i \(-0.719591\pi\)
0.636434 0.771331i \(-0.280409\pi\)
\(524\) 31.0479 14.7142i 1.35633 0.642794i
\(525\) 0 0
\(526\) −7.79991 34.6713i −0.340092 1.51174i
\(527\) 53.4647 2.32896
\(528\) 0 0
\(529\) −15.9129 −0.691864
\(530\) −0.843075 3.74754i −0.0366208 0.162783i
\(531\) 0 0
\(532\) −2.29327 + 1.08683i −0.0994258 + 0.0471199i
\(533\) 4.47736i 0.193936i
\(534\) 0 0
\(535\) 16.5623i 0.716052i
\(536\) 18.2996 14.3156i 0.790420 0.618340i
\(537\) 0 0
\(538\) −5.64496 + 1.26993i −0.243371 + 0.0547506i
\(539\) −2.97924 −0.128325
\(540\) 0 0
\(541\) −12.4892 −0.536955 −0.268477 0.963286i \(-0.586520\pi\)
−0.268477 + 0.963286i \(0.586520\pi\)
\(542\) 3.26422 0.734343i 0.140210 0.0315427i
\(543\) 0 0
\(544\) 16.1022 32.1455i 0.690376 1.37823i
\(545\) 7.73690i 0.331412i
\(546\) 0 0
\(547\) 28.2389i 1.20741i 0.797209 + 0.603703i \(0.206309\pi\)
−0.797209 + 0.603703i \(0.793691\pi\)
\(548\) −13.9468 29.4285i −0.595776 1.25712i
\(549\) 0 0
\(550\) 3.59654 + 15.9870i 0.153357 + 0.681686i
\(551\) 6.50777 0.277240
\(552\) 0 0
\(553\) −15.1046 −0.642312
\(554\) 6.49531 + 28.8723i 0.275959 + 1.22666i
\(555\) 0 0
\(556\) 0.905613 + 1.91090i 0.0384065 + 0.0810402i
\(557\) 44.4589i 1.88378i −0.335918 0.941891i \(-0.609047\pi\)
0.335918 0.941891i \(-0.390953\pi\)
\(558\) 0 0
\(559\) 0.636491i 0.0269207i
\(560\) −2.66932 + 3.26294i −0.112799 + 0.137884i
\(561\) 0 0
\(562\) 17.5395 3.94580i 0.739857 0.166444i
\(563\) 9.21017 0.388162 0.194081 0.980985i \(-0.437827\pi\)
0.194081 + 0.980985i \(0.437827\pi\)
\(564\) 0 0
\(565\) −4.58581 −0.192926
\(566\) −25.4906 + 5.73456i −1.07145 + 0.241041i
\(567\) 0 0
\(568\) −1.01754 1.30071i −0.0426950 0.0545768i
\(569\) 26.5427i 1.11273i −0.830939 0.556363i \(-0.812196\pi\)
0.830939 0.556363i \(-0.187804\pi\)
\(570\) 0 0
\(571\) 16.7288i 0.700077i −0.936735 0.350039i \(-0.886168\pi\)
0.936735 0.350039i \(-0.113832\pi\)
\(572\) −5.30445 + 2.51388i −0.221790 + 0.105111i
\(573\) 0 0
\(574\) −1.41070 6.27068i −0.0588814 0.261733i
\(575\) 10.3538 0.431785
\(576\) 0 0
\(577\) −29.4646 −1.22663 −0.613314 0.789839i \(-0.710164\pi\)
−0.613314 + 0.789839i \(0.710164\pi\)
\(578\) −7.26141 32.2777i −0.302035 1.34257i
\(579\) 0 0
\(580\) 9.76903 4.62974i 0.405637 0.192239i
\(581\) 4.73179i 0.196308i
\(582\) 0 0
\(583\) 7.67803i 0.317992i
\(584\) 21.6329 + 27.6532i 0.895175 + 1.14430i
\(585\) 0 0
\(586\) −37.0591 + 8.33708i −1.53090 + 0.344402i
\(587\) 19.5265 0.805943 0.402972 0.915212i \(-0.367977\pi\)
0.402972 + 0.915212i \(0.367977\pi\)
\(588\) 0 0
\(589\) −10.6741 −0.439817
\(590\) −20.0048 + 4.50043i −0.823585 + 0.185280i
\(591\) 0 0
\(592\) 22.5817 27.6036i 0.928103 1.13450i
\(593\) 1.64809i 0.0676788i −0.999427 0.0338394i \(-0.989227\pi\)
0.999427 0.0338394i \(-0.0107735\pi\)
\(594\) 0 0
\(595\) 6.69835i 0.274606i
\(596\) 5.75399 + 12.1413i 0.235693 + 0.497326i
\(597\) 0 0
\(598\) 0.814049 + 3.61852i 0.0332889 + 0.147972i
\(599\) −5.92633 −0.242143 −0.121072 0.992644i \(-0.538633\pi\)
−0.121072 + 0.992644i \(0.538633\pi\)
\(600\) 0 0
\(601\) −33.0438 −1.34788 −0.673941 0.738785i \(-0.735400\pi\)
−0.673941 + 0.738785i \(0.735400\pi\)
\(602\) 0.200542 + 0.891426i 0.00817346 + 0.0363318i
\(603\) 0 0
\(604\) −4.25645 8.98137i −0.173192 0.365447i
\(605\) 2.23864i 0.0910137i
\(606\) 0 0
\(607\) 17.4880i 0.709815i 0.934901 + 0.354907i \(0.115488\pi\)
−0.934901 + 0.354907i \(0.884512\pi\)
\(608\) −3.21475 + 6.41775i −0.130375 + 0.260274i
\(609\) 0 0
\(610\) 7.11096 1.59973i 0.287914 0.0647713i
\(611\) 4.57947 0.185265
\(612\) 0 0
\(613\) 21.6723 0.875336 0.437668 0.899137i \(-0.355805\pi\)
0.437668 + 0.899137i \(0.355805\pi\)
\(614\) 22.9750 5.16863i 0.927197 0.208589i
\(615\) 0 0
\(616\) −6.63699 + 5.19206i −0.267412 + 0.209194i
\(617\) 6.28958i 0.253209i 0.991953 + 0.126604i \(0.0404079\pi\)
−0.991953 + 0.126604i \(0.959592\pi\)
\(618\) 0 0
\(619\) 35.8504i 1.44095i −0.693481 0.720475i \(-0.743924\pi\)
0.693481 0.720475i \(-0.256076\pi\)
\(620\) −16.0232 + 7.59371i −0.643507 + 0.304971i
\(621\) 0 0
\(622\) −2.79719 12.4338i −0.112157 0.498549i
\(623\) 1.74631 0.0699643
\(624\) 0 0
\(625\) 9.57248 0.382899
\(626\) −0.778673 3.46127i −0.0311220 0.138340i
\(627\) 0 0
\(628\) −26.0247 + 12.3336i −1.03850 + 0.492165i
\(629\) 56.6663i 2.25943i
\(630\) 0 0
\(631\) 30.9763i 1.23315i −0.787298 0.616573i \(-0.788520\pi\)
0.787298 0.616573i \(-0.211480\pi\)
\(632\) −33.6491 + 26.3234i −1.33849 + 1.04709i
\(633\) 0 0
\(634\) −30.5223 + 6.86652i −1.21220 + 0.272704i
\(635\) 21.5279 0.854309
\(636\) 0 0
\(637\) −0.985148 −0.0390330
\(638\) 21.0819 4.74274i 0.834641 0.187767i
\(639\) 0 0
\(640\) −0.260078 + 11.9209i −0.0102805 + 0.471216i
\(641\) 4.95455i 0.195693i 0.995202 + 0.0978466i \(0.0311954\pi\)
−0.995202 + 0.0978466i \(0.968805\pi\)
\(642\) 0 0
\(643\) 4.37251i 0.172435i −0.996276 0.0862175i \(-0.972522\pi\)
0.996276 0.0862175i \(-0.0274780\pi\)
\(644\) 2.28020 + 4.81136i 0.0898525 + 0.189594i
\(645\) 0 0
\(646\) 2.50320 + 11.1269i 0.0984870 + 0.437784i
\(647\) −31.7458 −1.24805 −0.624027 0.781402i \(-0.714505\pi\)
−0.624027 + 0.781402i \(0.714505\pi\)
\(648\) 0 0
\(649\) −40.9862 −1.60885
\(650\) 1.18927 + 5.28641i 0.0466470 + 0.207350i
\(651\) 0 0
\(652\) 1.90657 + 4.02298i 0.0746671 + 0.157552i
\(653\) 0.116763i 0.00456930i −0.999997 0.00228465i \(-0.999273\pi\)
0.999997 0.00228465i \(-0.000727227\pi\)
\(654\) 0 0
\(655\) 18.1054i 0.707437i
\(656\) −14.0709 11.5110i −0.549375 0.449428i
\(657\) 0 0
\(658\) 6.41369 1.44287i 0.250032 0.0562489i
\(659\) 3.95369 0.154014 0.0770069 0.997031i \(-0.475464\pi\)
0.0770069 + 0.997031i \(0.475464\pi\)
\(660\) 0 0
\(661\) 38.7062 1.50550 0.752749 0.658307i \(-0.228727\pi\)
0.752749 + 0.658307i \(0.228727\pi\)
\(662\) −25.4293 + 5.72077i −0.988339 + 0.222344i
\(663\) 0 0
\(664\) −8.24631 10.5412i −0.320019 0.409079i
\(665\) 1.33731i 0.0518585i
\(666\) 0 0
\(667\) 13.6535i 0.528667i
\(668\) 22.5871 10.7045i 0.873920 0.414168i
\(669\) 0 0
\(670\) 2.68719 + 11.9448i 0.103815 + 0.461467i
\(671\) 14.5690 0.562432
\(672\) 0 0
\(673\) −6.45012 −0.248634 −0.124317 0.992243i \(-0.539674\pi\)
−0.124317 + 0.992243i \(0.539674\pi\)
\(674\) −7.68870 34.1770i −0.296158 1.31645i
\(675\) 0 0
\(676\) 21.7410 10.3035i 0.836193 0.396288i
\(677\) 30.4104i 1.16877i −0.811478 0.584384i \(-0.801336\pi\)
0.811478 0.584384i \(-0.198664\pi\)
\(678\) 0 0
\(679\) 3.00594i 0.115358i
\(680\) 11.6735 + 14.9222i 0.447659 + 0.572241i
\(681\) 0 0
\(682\) −34.5786 + 7.77906i −1.32408 + 0.297876i
\(683\) −16.9861 −0.649953 −0.324977 0.945722i \(-0.605356\pi\)
−0.324977 + 0.945722i \(0.605356\pi\)
\(684\) 0 0
\(685\) 17.1611 0.655690
\(686\) −1.37973 + 0.310394i −0.0526783 + 0.0118509i
\(687\) 0 0
\(688\) 2.00028 + 1.63637i 0.0762601 + 0.0623862i
\(689\) 2.53890i 0.0967243i
\(690\) 0 0
\(691\) 2.74292i 0.104346i −0.998638 0.0521729i \(-0.983385\pi\)
0.998638 0.0521729i \(-0.0166147\pi\)
\(692\) 13.8258 + 29.1732i 0.525577 + 1.10900i
\(693\) 0 0
\(694\) −8.42280 37.4401i −0.319725 1.42121i
\(695\) −1.11433 −0.0422689
\(696\) 0 0
\(697\) −28.8855 −1.09412
\(698\) 3.89223 + 17.3013i 0.147323 + 0.654865i
\(699\) 0 0
\(700\) 3.33122 + 7.02908i 0.125908 + 0.265674i
\(701\) 8.97421i 0.338951i 0.985534 + 0.169476i \(0.0542074\pi\)
−0.985534 + 0.169476i \(0.945793\pi\)
\(702\) 0 0
\(703\) 11.3133i 0.426688i
\(704\) −5.73706 + 23.1332i −0.216223 + 0.871864i
\(705\) 0 0
\(706\) −34.0413 + 7.65818i −1.28116 + 0.288219i
\(707\) −1.53513 −0.0577346
\(708\) 0 0
\(709\) 3.13240 0.117640 0.0588200 0.998269i \(-0.481266\pi\)
0.0588200 + 0.998269i \(0.481266\pi\)
\(710\) 0.849023 0.191002i 0.0318633 0.00716820i
\(711\) 0 0
\(712\) 3.89032 3.04337i 0.145796 0.114055i
\(713\) 22.3946i 0.838684i
\(714\) 0 0
\(715\) 3.09326i 0.115681i
\(716\) −26.3432 + 12.4846i −0.984491 + 0.466570i
\(717\) 0 0
\(718\) 7.09328 + 31.5303i 0.264719 + 1.17670i
\(719\) −29.4070 −1.09670 −0.548348 0.836250i \(-0.684743\pi\)
−0.548348 + 0.836250i \(0.684743\pi\)
\(720\) 0 0
\(721\) −11.6910 −0.435395
\(722\) 5.39773 + 23.9934i 0.200883 + 0.892942i
\(723\) 0 0
\(724\) −3.54005 + 1.67770i −0.131565 + 0.0623512i
\(725\) 19.9469i 0.740809i
\(726\) 0 0
\(727\) 11.0639i 0.410338i −0.978727 0.205169i \(-0.934226\pi\)
0.978727 0.205169i \(-0.0657743\pi\)
\(728\) −2.19466 + 1.71686i −0.0813394 + 0.0636312i
\(729\) 0 0
\(730\) −18.0502 + 4.06071i −0.668070 + 0.150294i
\(731\) 4.10630 0.151877
\(732\) 0 0
\(733\) −22.1305 −0.817410 −0.408705 0.912667i \(-0.634019\pi\)
−0.408705 + 0.912667i \(0.634019\pi\)
\(734\) −28.0245 + 6.30460i −1.03440 + 0.232707i
\(735\) 0 0
\(736\) 13.4647 + 6.74467i 0.496315 + 0.248612i
\(737\) 24.4727i 0.901463i
\(738\) 0 0
\(739\) 5.76011i 0.211889i −0.994372 0.105944i \(-0.966213\pi\)
0.994372 0.105944i \(-0.0337866\pi\)
\(740\) 8.04844 + 16.9827i 0.295867 + 0.624297i
\(741\) 0 0
\(742\) −0.799940 3.55581i −0.0293667 0.130538i
\(743\) 51.9086 1.90434 0.952170 0.305567i \(-0.0988461\pi\)
0.952170 + 0.305567i \(0.0988461\pi\)
\(744\) 0 0
\(745\) −7.08011 −0.259395
\(746\) 9.13155 + 40.5906i 0.334330 + 1.48613i
\(747\) 0 0
\(748\) 16.2182 + 34.2214i 0.592997 + 1.25126i
\(749\) 15.7149i 0.574212i
\(750\) 0 0
\(751\) 36.7888i 1.34244i −0.741257 0.671221i \(-0.765770\pi\)
0.741257 0.671221i \(-0.234230\pi\)
\(752\) 11.7735 14.3918i 0.429335 0.524814i
\(753\) 0 0
\(754\) 6.97117 1.56828i 0.253875 0.0571136i
\(755\) 5.23743 0.190610
\(756\) 0 0
\(757\) 32.2356 1.17162 0.585812 0.810447i \(-0.300776\pi\)
0.585812 + 0.810447i \(0.300776\pi\)
\(758\) −25.5858 + 5.75597i −0.929319 + 0.209066i
\(759\) 0 0
\(760\) −2.33058 2.97917i −0.0845392 0.108066i
\(761\) 13.8150i 0.500793i −0.968143 0.250396i \(-0.919439\pi\)
0.968143 0.250396i \(-0.0805609\pi\)
\(762\) 0 0
\(763\) 7.34105i 0.265764i
\(764\) −47.3921 + 22.4600i −1.71459 + 0.812576i
\(765\) 0 0
\(766\) −11.7357 52.1660i −0.424026 1.88484i
\(767\) −13.5529 −0.489368
\(768\) 0 0
\(769\) −47.3420 −1.70720 −0.853599 0.520931i \(-0.825585\pi\)
−0.853599 + 0.520931i \(0.825585\pi\)
\(770\) −0.974604 4.33220i −0.0351223 0.156122i
\(771\) 0 0
\(772\) 42.2912 20.0426i 1.52209 0.721349i
\(773\) 12.7984i 0.460325i 0.973152 + 0.230163i \(0.0739259\pi\)
−0.973152 + 0.230163i \(0.926074\pi\)
\(774\) 0 0
\(775\) 32.7170i 1.17523i
\(776\) −5.23859 6.69647i −0.188055 0.240389i
\(777\) 0 0
\(778\) −11.1222 + 2.50214i −0.398752 + 0.0897061i
\(779\) 5.76690 0.206621
\(780\) 0 0
\(781\) 1.73949 0.0622440
\(782\) 23.3447 5.25180i 0.834806 0.187804i
\(783\) 0 0
\(784\) −2.53275 + 3.09600i −0.0904552 + 0.110571i
\(785\) 15.1761i 0.541660i
\(786\) 0 0
\(787\) 35.5078i 1.26572i 0.774267 + 0.632858i \(0.218119\pi\)
−0.774267 + 0.632858i \(0.781881\pi\)
\(788\) −14.1649 29.8888i −0.504604 1.06475i
\(789\) 0 0
\(790\) −4.94118 21.9640i −0.175799 0.781444i
\(791\) −4.35118 −0.154710
\(792\) 0 0
\(793\) 4.81755 0.171076
\(794\) 8.56086 + 38.0538i 0.303813 + 1.35048i
\(795\) 0 0
\(796\) 1.03303 + 2.17975i 0.0366147 + 0.0772593i
\(797\) 30.3922i 1.07655i 0.842770 + 0.538273i \(0.180923\pi\)
−0.842770 + 0.538273i \(0.819077\pi\)
\(798\) 0 0
\(799\) 29.5442i 1.04520i
\(800\) 19.6710 + 9.85351i 0.695475 + 0.348374i
\(801\) 0 0
\(802\) 9.99414 2.24835i 0.352905 0.0793922i
\(803\) −36.9817 −1.30505
\(804\) 0 0
\(805\) −2.80572 −0.0988886
\(806\) −11.4341 + 2.57231i −0.402750 + 0.0906056i
\(807\) 0 0
\(808\) −3.41988 + 2.67535i −0.120311 + 0.0941183i
\(809\) 41.0141i 1.44198i −0.692946 0.720989i \(-0.743688\pi\)
0.692946 0.720989i \(-0.256312\pi\)
\(810\) 0 0
\(811\) 46.2839i 1.62525i 0.582788 + 0.812624i \(0.301962\pi\)
−0.582788 + 0.812624i \(0.698038\pi\)
\(812\) 9.26921 4.39286i 0.325286 0.154159i
\(813\) 0 0
\(814\) 8.24489 + 36.6493i 0.288983 + 1.28456i
\(815\) −2.34598 −0.0821760
\(816\) 0 0
\(817\) −0.819810 −0.0286815
\(818\) 8.20621 + 36.4773i 0.286923 + 1.27540i
\(819\) 0 0
\(820\) 8.65690 4.10267i 0.302312 0.143272i
\(821\) 4.65424i 0.162434i −0.996696 0.0812170i \(-0.974119\pi\)
0.996696 0.0812170i \(-0.0258807\pi\)
\(822\) 0 0
\(823\) 24.1740i 0.842654i −0.906909 0.421327i \(-0.861564\pi\)
0.906909 0.421327i \(-0.138436\pi\)
\(824\) −26.0445 + 20.3744i −0.907303 + 0.709776i
\(825\) 0 0
\(826\) −18.9813 + 4.27017i −0.660444 + 0.148578i
\(827\) −36.6008 −1.27273 −0.636367 0.771386i \(-0.719564\pi\)
−0.636367 + 0.771386i \(0.719564\pi\)
\(828\) 0 0
\(829\) −37.1578 −1.29054 −0.645272 0.763953i \(-0.723256\pi\)
−0.645272 + 0.763953i \(0.723256\pi\)
\(830\) 6.88064 1.54792i 0.238830 0.0537290i
\(831\) 0 0
\(832\) −1.89708 + 7.64946i −0.0657693 + 0.265197i
\(833\) 6.35564i 0.220210i
\(834\) 0 0
\(835\) 13.1715i 0.455819i
\(836\) −3.23792 6.83221i −0.111986 0.236297i
\(837\) 0 0
\(838\) 6.31223 + 28.0584i 0.218052 + 0.969263i
\(839\) −3.38158 −0.116745 −0.0583726 0.998295i \(-0.518591\pi\)
−0.0583726 + 0.998295i \(0.518591\pi\)
\(840\) 0 0
\(841\) 2.69611 0.0929694
\(842\) 4.23803 + 18.8384i 0.146052 + 0.649216i
\(843\) 0 0
\(844\) −7.99150 16.8626i −0.275079 0.580433i
\(845\) 12.6781i 0.436141i
\(846\) 0 0
\(847\) 2.12410i 0.0729851i
\(848\) −7.97893 6.52733i −0.273997 0.224149i
\(849\) 0 0
\(850\) 34.1051 7.67253i 1.16979 0.263166i
\(851\) 23.7356 0.813647
\(852\) 0 0
\(853\) 28.2168 0.966124 0.483062 0.875586i \(-0.339525\pi\)
0.483062 + 0.875586i \(0.339525\pi\)
\(854\) 6.74713 1.51788i 0.230882 0.0519409i
\(855\) 0 0
\(856\) −27.3872 35.0089i −0.936074 1.19658i
\(857\) 17.3707i 0.593373i −0.954975 0.296686i \(-0.904118\pi\)
0.954975 0.296686i \(-0.0958816\pi\)
\(858\) 0 0
\(859\) 45.9940i 1.56930i 0.619941 + 0.784648i \(0.287156\pi\)
−0.619941 + 0.784648i \(0.712844\pi\)
\(860\) −1.23064 + 0.583227i −0.0419646 + 0.0198879i
\(861\) 0 0
\(862\) −5.26993 23.4253i −0.179495 0.797870i
\(863\) −1.05635 −0.0359586 −0.0179793 0.999838i \(-0.505723\pi\)
−0.0179793 + 0.999838i \(0.505723\pi\)
\(864\) 0 0
\(865\) −17.0122 −0.578432
\(866\) 0.193816 + 0.861528i 0.00658613 + 0.0292759i
\(867\) 0 0
\(868\) −15.2034 + 7.20519i −0.516037 + 0.244560i
\(869\) 45.0002i 1.52653i
\(870\) 0 0
\(871\) 8.09239i 0.274200i
\(872\) 12.7936 + 16.3540i 0.433246 + 0.553816i
\(873\) 0 0
\(874\) −4.66071 + 1.04851i −0.157651 + 0.0354663i
\(875\) −9.36858 −0.316716
\(876\) 0 0
\(877\) 4.34738 0.146801 0.0734003 0.997303i \(-0.476615\pi\)
0.0734003 + 0.997303i \(0.476615\pi\)
\(878\) −31.5663 + 7.10139i −1.06531 + 0.239660i
\(879\) 0 0
\(880\) −9.72110 7.95255i −0.327698 0.268080i
\(881\) 58.0353i 1.95526i −0.210331 0.977630i \(-0.567454\pi\)
0.210331 0.977630i \(-0.432546\pi\)
\(882\) 0 0
\(883\) 37.8799i 1.27476i −0.770550 0.637379i \(-0.780018\pi\)
0.770550 0.637379i \(-0.219982\pi\)
\(884\) 5.36289 + 11.3160i 0.180373 + 0.380599i
\(885\) 0 0
\(886\) −0.0659353 0.293089i −0.00221514 0.00984650i
\(887\) −4.92085 −0.165226 −0.0826130 0.996582i \(-0.526327\pi\)
−0.0826130 + 0.996582i \(0.526327\pi\)
\(888\) 0 0
\(889\) 20.4265 0.685081
\(890\) 0.571272 + 2.53935i 0.0191491 + 0.0851194i
\(891\) 0 0
\(892\) 2.50524 + 5.28622i 0.0838818 + 0.176996i
\(893\) 5.89842i 0.197383i
\(894\) 0 0
\(895\) 15.3619i 0.513491i
\(896\) −0.246771 + 11.3110i −0.00824404 + 0.377875i
\(897\) 0 0
\(898\) −24.2848 + 5.46329i −0.810396 + 0.182312i
\(899\) 43.1437 1.43892
\(900\) 0 0
\(901\) −16.3796 −0.545684
\(902\) 18.6819 4.20281i 0.622039 0.139938i
\(903\) 0 0
\(904\) −9.69331 + 7.58300i −0.322395 + 0.252207i
\(905\) 2.06436i 0.0686216i
\(906\) 0 0
\(907\) 40.3942i 1.34127i −0.741789 0.670633i \(-0.766022\pi\)
0.741789 0.670633i \(-0.233978\pi\)
\(908\) 41.7416 19.7821i 1.38524 0.656494i
\(909\) 0 0
\(910\) −0.322273 1.43253i −0.0106832 0.0474880i
\(911\) −24.4159 −0.808934 −0.404467 0.914553i \(-0.632543\pi\)
−0.404467 + 0.914553i \(0.632543\pi\)
\(912\) 0 0
\(913\) 14.0972 0.466548
\(914\) 2.65806 + 11.8153i 0.0879209 + 0.390817i
\(915\) 0 0
\(916\) −16.4174 + 7.78051i −0.542445 + 0.257075i
\(917\) 17.1791i 0.567303i
\(918\) 0 0
\(919\) 0.922920i 0.0304443i 0.999884 + 0.0152222i \(0.00484555\pi\)
−0.999884 + 0.0152222i \(0.995154\pi\)
\(920\) −6.25042 + 4.88965i −0.206070 + 0.161207i
\(921\) 0 0
\(922\) −45.8392 + 10.3123i −1.50963 + 0.339618i
\(923\) 0.575199 0.0189329
\(924\) 0 0
\(925\) 34.6762 1.14015
\(926\) 15.0140 3.37767i 0.493392 0.110997i
\(927\) 0 0
\(928\) 12.9938 25.9400i 0.426542 0.851524i
\(929\) 18.1210i 0.594530i −0.954795 0.297265i \(-0.903926\pi\)
0.954795 0.297265i \(-0.0960744\pi\)
\(930\) 0 0
\(931\) 1.26888i 0.0415860i
\(932\) −25.2327 53.2425i −0.826523 1.74402i
\(933\) 0 0
\(934\) −12.9675 57.6418i −0.424310 1.88610i
\(935\) −19.9560 −0.652632
\(936\) 0 0
\(937\) 58.1414 1.89940 0.949699 0.313165i \(-0.101389\pi\)
0.949699 + 0.313165i \(0.101389\pi\)
\(938\) 2.54970 + 11.3336i 0.0832506 + 0.370057i
\(939\) 0 0
\(940\) 4.19624 + 8.85432i 0.136866 + 0.288796i
\(941\) 11.4151i 0.372121i −0.982538 0.186061i \(-0.940428\pi\)
0.982538 0.186061i \(-0.0595721\pi\)
\(942\) 0 0
\(943\) 12.0992i 0.394004i
\(944\) −34.8436 + 42.5924i −1.13406 + 1.38627i
\(945\) 0 0
\(946\) −2.65577 + 0.597462i −0.0863467 + 0.0194252i
\(947\) 53.1805 1.72813 0.864067 0.503377i \(-0.167909\pi\)
0.864067 + 0.503377i \(0.167909\pi\)
\(948\) 0 0
\(949\) −12.2287 −0.396962
\(950\) −6.80898 + 1.53180i −0.220912 + 0.0496981i
\(951\) 0 0
\(952\) 11.0763 + 14.1587i 0.358984 + 0.458887i
\(953\) 32.5256i 1.05361i 0.849987 + 0.526803i \(0.176609\pi\)
−0.849987 + 0.526803i \(0.823391\pi\)
\(954\) 0 0
\(955\) 27.6364i 0.894293i
\(956\) 31.3963 14.8793i 1.01543 0.481232i
\(957\) 0 0
\(958\) −2.62886 11.6855i −0.0849347 0.377542i
\(959\) 16.2830 0.525807
\(960\) 0 0
\(961\) −39.7645 −1.28273
\(962\) 2.72634 + 12.1188i 0.0879008 + 0.390727i
\(963\) 0 0
\(964\) 21.3878 10.1361i 0.688856 0.326462i
\(965\) 24.6618i 0.793892i
\(966\) 0 0
\(967\) 34.1888i 1.09944i −0.835350 0.549719i \(-0.814735\pi\)
0.835350 0.549719i \(-0.185265\pi\)
\(968\) 3.70177 + 4.73196i 0.118980 + 0.152091i
\(969\) 0 0
\(970\) 4.37103 0.983338i 0.140345 0.0315731i
\(971\) 14.1172 0.453041 0.226521 0.974006i \(-0.427265\pi\)
0.226521 + 0.974006i \(0.427265\pi\)
\(972\) 0 0
\(973\) −1.05732 −0.0338960
\(974\) −31.9402 + 7.18549i −1.02343 + 0.230238i
\(975\) 0 0
\(976\) 12.3856 15.1400i 0.396453 0.484619i
\(977\) 23.3492i 0.747007i 0.927629 + 0.373504i \(0.121844\pi\)
−0.927629 + 0.373504i \(0.878156\pi\)
\(978\) 0 0
\(979\) 5.20267i 0.166278i
\(980\) −0.902706 1.90477i −0.0288359 0.0608455i
\(981\) 0 0
\(982\) 2.14632 + 9.54060i 0.0684919 + 0.304453i
\(983\) −27.3413 −0.872052 −0.436026 0.899934i \(-0.643614\pi\)
−0.436026 + 0.899934i \(0.643614\pi\)
\(984\) 0 0
\(985\) 17.4295 0.555350
\(986\) −10.1177 44.9742i −0.322214 1.43227i
\(987\) 0 0
\(988\) −1.07068 2.25921i −0.0340630 0.0718750i
\(989\) 1.71999i 0.0546926i
\(990\) 0 0
\(991\) 27.2385i 0.865259i 0.901572 + 0.432629i \(0.142414\pi\)
−0.901572 + 0.432629i \(0.857586\pi\)
\(992\) −21.3124 + 42.5470i −0.676671 + 1.35087i
\(993\) 0 0
\(994\) 0.805584 0.181230i 0.0255516 0.00574827i
\(995\) −1.27111 −0.0402969
\(996\) 0 0
\(997\) 56.6550 1.79428 0.897141 0.441744i \(-0.145640\pi\)
0.897141 + 0.441744i \(0.145640\pi\)
\(998\) −10.4704 + 2.35549i −0.331433 + 0.0745616i
\(999\) 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 756.2.e.a.323.14 yes 24
3.2 odd 2 inner 756.2.e.a.323.11 24
4.3 odd 2 inner 756.2.e.a.323.12 yes 24
12.11 even 2 inner 756.2.e.a.323.13 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
756.2.e.a.323.11 24 3.2 odd 2 inner
756.2.e.a.323.12 yes 24 4.3 odd 2 inner
756.2.e.a.323.13 yes 24 12.11 even 2 inner
756.2.e.a.323.14 yes 24 1.1 even 1 trivial