Properties

Label 725.2.b.e.349.2
Level $725$
Weight $2$
Character 725.349
Analytic conductor $5.789$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [725,2,Mod(349,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("725.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.b (of order \(2\), degree \(1\), 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: \(5.78915414654\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 145)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 349.2
Root \(0.403032 - 0.403032i\) of defining polynomial
Character \(\chi\) \(=\) 725.349
Dual form 725.2.b.e.349.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.48119i q^{2} -0.806063i q^{3} -0.193937 q^{4} -1.19394 q^{6} +1.19394i q^{7} -2.67513i q^{8} +2.35026 q^{9} +O(q^{10})\) \(q-1.48119i q^{2} -0.806063i q^{3} -0.193937 q^{4} -1.19394 q^{6} +1.19394i q^{7} -2.67513i q^{8} +2.35026 q^{9} +4.15633 q^{11} +0.156325i q^{12} -2.96239i q^{13} +1.76845 q^{14} -4.35026 q^{16} +5.50659i q^{17} -3.48119i q^{18} +3.19394 q^{19} +0.962389 q^{21} -6.15633i q^{22} -1.84367i q^{23} -2.15633 q^{24} -4.38787 q^{26} -4.31265i q^{27} -0.231548i q^{28} +1.00000 q^{29} -4.80606 q^{31} +1.09332i q^{32} -3.35026i q^{33} +8.15633 q^{34} -0.455802 q^{36} -9.50659i q^{37} -4.73084i q^{38} -2.38787 q^{39} -11.2750 q^{41} -1.42548i q^{42} +0.0303172i q^{43} -0.806063 q^{44} -2.73084 q^{46} +4.80606i q^{47} +3.50659i q^{48} +5.57452 q^{49} +4.43866 q^{51} +0.574515i q^{52} +1.35026i q^{53} -6.38787 q^{54} +3.19394 q^{56} -2.57452i q^{57} -1.48119i q^{58} -13.2750 q^{59} +8.88717 q^{61} +7.11871i q^{62} +2.80606i q^{63} -7.08110 q^{64} -4.96239 q^{66} +5.84367i q^{67} -1.06793i q^{68} -1.48612 q^{69} -1.27504 q^{71} -6.28726i q^{72} +15.2447i q^{73} -14.0811 q^{74} -0.619421 q^{76} +4.96239i q^{77} +3.53690i q^{78} +4.93207 q^{79} +3.57452 q^{81} +16.7005i q^{82} -4.41819i q^{83} -0.186642 q^{84} +0.0449056 q^{86} -0.806063i q^{87} -11.1187i q^{88} +3.61213 q^{89} +3.53690 q^{91} +0.357556i q^{92} +3.87399i q^{93} +7.11871 q^{94} +0.881286 q^{96} -1.38058i q^{97} -8.25694i q^{98} +9.76845 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{4} - 8 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{4} - 8 q^{6} - 6 q^{9} + 4 q^{11} - 12 q^{14} - 6 q^{16} + 20 q^{19} - 16 q^{21} + 8 q^{24} - 28 q^{26} + 6 q^{29} - 28 q^{31} + 28 q^{34} - 22 q^{36} - 16 q^{39} - 4 q^{41} - 4 q^{44} + 28 q^{46} + 10 q^{49} - 32 q^{51} - 40 q^{54} + 20 q^{56} - 16 q^{59} - 12 q^{61} + 22 q^{64} - 8 q^{66} - 24 q^{69} + 56 q^{71} - 20 q^{74} - 28 q^{76} + 12 q^{79} - 2 q^{81} + 24 q^{84} + 48 q^{86} + 20 q^{89} - 24 q^{91} + 48 q^{96} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/725\mathbb{Z}\right)^\times\).

\(n\) \(176\) \(552\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.48119i − 1.04736i −0.851914 0.523681i \(-0.824558\pi\)
0.851914 0.523681i \(-0.175442\pi\)
\(3\) − 0.806063i − 0.465381i −0.972551 0.232690i \(-0.925247\pi\)
0.972551 0.232690i \(-0.0747529\pi\)
\(4\) −0.193937 −0.0969683
\(5\) 0 0
\(6\) −1.19394 −0.487423
\(7\) 1.19394i 0.451266i 0.974212 + 0.225633i \(0.0724450\pi\)
−0.974212 + 0.225633i \(0.927555\pi\)
\(8\) − 2.67513i − 0.945802i
\(9\) 2.35026 0.783421
\(10\) 0 0
\(11\) 4.15633 1.25318 0.626590 0.779349i \(-0.284450\pi\)
0.626590 + 0.779349i \(0.284450\pi\)
\(12\) 0.156325i 0.0451272i
\(13\) − 2.96239i − 0.821619i −0.911721 0.410809i \(-0.865246\pi\)
0.911721 0.410809i \(-0.134754\pi\)
\(14\) 1.76845 0.472639
\(15\) 0 0
\(16\) −4.35026 −1.08757
\(17\) 5.50659i 1.33554i 0.744366 + 0.667772i \(0.232752\pi\)
−0.744366 + 0.667772i \(0.767248\pi\)
\(18\) − 3.48119i − 0.820525i
\(19\) 3.19394 0.732739 0.366370 0.930469i \(-0.380601\pi\)
0.366370 + 0.930469i \(0.380601\pi\)
\(20\) 0 0
\(21\) 0.962389 0.210010
\(22\) − 6.15633i − 1.31253i
\(23\) − 1.84367i − 0.384433i −0.981353 0.192216i \(-0.938432\pi\)
0.981353 0.192216i \(-0.0615675\pi\)
\(24\) −2.15633 −0.440158
\(25\) 0 0
\(26\) −4.38787 −0.860533
\(27\) − 4.31265i − 0.829970i
\(28\) − 0.231548i − 0.0437585i
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −4.80606 −0.863194 −0.431597 0.902066i \(-0.642050\pi\)
−0.431597 + 0.902066i \(0.642050\pi\)
\(32\) 1.09332i 0.193274i
\(33\) − 3.35026i − 0.583206i
\(34\) 8.15633 1.39880
\(35\) 0 0
\(36\) −0.455802 −0.0759669
\(37\) − 9.50659i − 1.56287i −0.623985 0.781437i \(-0.714487\pi\)
0.623985 0.781437i \(-0.285513\pi\)
\(38\) − 4.73084i − 0.767444i
\(39\) −2.38787 −0.382366
\(40\) 0 0
\(41\) −11.2750 −1.76087 −0.880433 0.474171i \(-0.842748\pi\)
−0.880433 + 0.474171i \(0.842748\pi\)
\(42\) − 1.42548i − 0.219957i
\(43\) 0.0303172i 0.00462332i 0.999997 + 0.00231166i \(0.000735826\pi\)
−0.999997 + 0.00231166i \(0.999264\pi\)
\(44\) −0.806063 −0.121519
\(45\) 0 0
\(46\) −2.73084 −0.402640
\(47\) 4.80606i 0.701036i 0.936556 + 0.350518i \(0.113995\pi\)
−0.936556 + 0.350518i \(0.886005\pi\)
\(48\) 3.50659i 0.506132i
\(49\) 5.57452 0.796359
\(50\) 0 0
\(51\) 4.43866 0.621536
\(52\) 0.574515i 0.0796710i
\(53\) 1.35026i 0.185473i 0.995691 + 0.0927364i \(0.0295614\pi\)
−0.995691 + 0.0927364i \(0.970439\pi\)
\(54\) −6.38787 −0.869279
\(55\) 0 0
\(56\) 3.19394 0.426808
\(57\) − 2.57452i − 0.341003i
\(58\) − 1.48119i − 0.194490i
\(59\) −13.2750 −1.72826 −0.864131 0.503266i \(-0.832132\pi\)
−0.864131 + 0.503266i \(0.832132\pi\)
\(60\) 0 0
\(61\) 8.88717 1.13788 0.568942 0.822377i \(-0.307353\pi\)
0.568942 + 0.822377i \(0.307353\pi\)
\(62\) 7.11871i 0.904078i
\(63\) 2.80606i 0.353531i
\(64\) −7.08110 −0.885138
\(65\) 0 0
\(66\) −4.96239 −0.610828
\(67\) 5.84367i 0.713919i 0.934120 + 0.356959i \(0.116187\pi\)
−0.934120 + 0.356959i \(0.883813\pi\)
\(68\) − 1.06793i − 0.129505i
\(69\) −1.48612 −0.178908
\(70\) 0 0
\(71\) −1.27504 −0.151319 −0.0756596 0.997134i \(-0.524106\pi\)
−0.0756596 + 0.997134i \(0.524106\pi\)
\(72\) − 6.28726i − 0.740960i
\(73\) 15.2447i 1.78426i 0.451779 + 0.892130i \(0.350790\pi\)
−0.451779 + 0.892130i \(0.649210\pi\)
\(74\) −14.0811 −1.63689
\(75\) 0 0
\(76\) −0.619421 −0.0710525
\(77\) 4.96239i 0.565517i
\(78\) 3.53690i 0.400476i
\(79\) 4.93207 0.554901 0.277451 0.960740i \(-0.410510\pi\)
0.277451 + 0.960740i \(0.410510\pi\)
\(80\) 0 0
\(81\) 3.57452 0.397168
\(82\) 16.7005i 1.84426i
\(83\) − 4.41819i − 0.484959i −0.970156 0.242480i \(-0.922039\pi\)
0.970156 0.242480i \(-0.0779608\pi\)
\(84\) −0.186642 −0.0203643
\(85\) 0 0
\(86\) 0.0449056 0.00484230
\(87\) − 0.806063i − 0.0864191i
\(88\) − 11.1187i − 1.18526i
\(89\) 3.61213 0.382885 0.191442 0.981504i \(-0.438684\pi\)
0.191442 + 0.981504i \(0.438684\pi\)
\(90\) 0 0
\(91\) 3.53690 0.370768
\(92\) 0.357556i 0.0372778i
\(93\) 3.87399i 0.401714i
\(94\) 7.11871 0.734239
\(95\) 0 0
\(96\) 0.881286 0.0899459
\(97\) − 1.38058i − 0.140177i −0.997541 0.0700883i \(-0.977672\pi\)
0.997541 0.0700883i \(-0.0223281\pi\)
\(98\) − 8.25694i − 0.834077i
\(99\) 9.76845 0.981766
\(100\) 0 0
\(101\) −13.0132 −1.29486 −0.647430 0.762125i \(-0.724156\pi\)
−0.647430 + 0.762125i \(0.724156\pi\)
\(102\) − 6.57452i − 0.650974i
\(103\) − 5.31994i − 0.524190i −0.965042 0.262095i \(-0.915587\pi\)
0.965042 0.262095i \(-0.0844133\pi\)
\(104\) −7.92478 −0.777088
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) − 13.8192i − 1.33596i −0.744181 0.667978i \(-0.767160\pi\)
0.744181 0.667978i \(-0.232840\pi\)
\(108\) 0.836381i 0.0804808i
\(109\) 1.87399 0.179496 0.0897479 0.995965i \(-0.471394\pi\)
0.0897479 + 0.995965i \(0.471394\pi\)
\(110\) 0 0
\(111\) −7.66291 −0.727331
\(112\) − 5.19394i − 0.490781i
\(113\) 11.7685i 1.10708i 0.832822 + 0.553541i \(0.186724\pi\)
−0.832822 + 0.553541i \(0.813276\pi\)
\(114\) −3.81336 −0.357154
\(115\) 0 0
\(116\) −0.193937 −0.0180066
\(117\) − 6.96239i − 0.643673i
\(118\) 19.6629i 1.81012i
\(119\) −6.57452 −0.602685
\(120\) 0 0
\(121\) 6.27504 0.570458
\(122\) − 13.1636i − 1.19178i
\(123\) 9.08840i 0.819473i
\(124\) 0.932071 0.0837025
\(125\) 0 0
\(126\) 4.15633 0.370275
\(127\) 14.2677i 1.26606i 0.774128 + 0.633029i \(0.218189\pi\)
−0.774128 + 0.633029i \(0.781811\pi\)
\(128\) 12.6751i 1.12033i
\(129\) 0.0244376 0.00215161
\(130\) 0 0
\(131\) 5.89446 0.515001 0.257501 0.966278i \(-0.417101\pi\)
0.257501 + 0.966278i \(0.417101\pi\)
\(132\) 0.649738i 0.0565525i
\(133\) 3.81336i 0.330660i
\(134\) 8.65562 0.747731
\(135\) 0 0
\(136\) 14.7308 1.26316
\(137\) 18.2823i 1.56197i 0.624553 + 0.780983i \(0.285281\pi\)
−0.624553 + 0.780983i \(0.714719\pi\)
\(138\) 2.20123i 0.187381i
\(139\) 11.5369 0.978547 0.489274 0.872130i \(-0.337262\pi\)
0.489274 + 0.872130i \(0.337262\pi\)
\(140\) 0 0
\(141\) 3.87399 0.326249
\(142\) 1.88858i 0.158486i
\(143\) − 12.3127i − 1.02964i
\(144\) −10.2243 −0.852021
\(145\) 0 0
\(146\) 22.5804 1.86877
\(147\) − 4.49341i − 0.370610i
\(148\) 1.84367i 0.151549i
\(149\) −2.77575 −0.227398 −0.113699 0.993515i \(-0.536270\pi\)
−0.113699 + 0.993515i \(0.536270\pi\)
\(150\) 0 0
\(151\) 1.79877 0.146382 0.0731909 0.997318i \(-0.476682\pi\)
0.0731909 + 0.997318i \(0.476682\pi\)
\(152\) − 8.54420i − 0.693026i
\(153\) 12.9419i 1.04629i
\(154\) 7.35026 0.592301
\(155\) 0 0
\(156\) 0.463096 0.0370773
\(157\) 3.76845i 0.300755i 0.988629 + 0.150378i \(0.0480489\pi\)
−0.988629 + 0.150378i \(0.951951\pi\)
\(158\) − 7.30536i − 0.581183i
\(159\) 1.08840 0.0863155
\(160\) 0 0
\(161\) 2.20123 0.173481
\(162\) − 5.29455i − 0.415979i
\(163\) − 1.64244i − 0.128646i −0.997929 0.0643231i \(-0.979511\pi\)
0.997929 0.0643231i \(-0.0204888\pi\)
\(164\) 2.18664 0.170748
\(165\) 0 0
\(166\) −6.54420 −0.507928
\(167\) 8.08110i 0.625334i 0.949863 + 0.312667i \(0.101222\pi\)
−0.949863 + 0.312667i \(0.898778\pi\)
\(168\) − 2.57452i − 0.198628i
\(169\) 4.22425 0.324943
\(170\) 0 0
\(171\) 7.50659 0.574043
\(172\) − 0.00587961i 0 0.000448316i
\(173\) 7.73813i 0.588320i 0.955756 + 0.294160i \(0.0950398\pi\)
−0.955756 + 0.294160i \(0.904960\pi\)
\(174\) −1.19394 −0.0905121
\(175\) 0 0
\(176\) −18.0811 −1.36291
\(177\) 10.7005i 0.804301i
\(178\) − 5.35026i − 0.401019i
\(179\) 21.4010 1.59959 0.799795 0.600274i \(-0.204942\pi\)
0.799795 + 0.600274i \(0.204942\pi\)
\(180\) 0 0
\(181\) 15.2750 1.13538 0.567692 0.823241i \(-0.307836\pi\)
0.567692 + 0.823241i \(0.307836\pi\)
\(182\) − 5.23884i − 0.388329i
\(183\) − 7.16362i − 0.529550i
\(184\) −4.93207 −0.363597
\(185\) 0 0
\(186\) 5.73813 0.420740
\(187\) 22.8872i 1.67368i
\(188\) − 0.932071i − 0.0679783i
\(189\) 5.14903 0.374537
\(190\) 0 0
\(191\) 3.31994 0.240223 0.120111 0.992760i \(-0.461675\pi\)
0.120111 + 0.992760i \(0.461675\pi\)
\(192\) 5.70782i 0.411926i
\(193\) 4.88129i 0.351363i 0.984447 + 0.175681i \(0.0562128\pi\)
−0.984447 + 0.175681i \(0.943787\pi\)
\(194\) −2.04491 −0.146816
\(195\) 0 0
\(196\) −1.08110 −0.0772216
\(197\) − 24.2374i − 1.72685i −0.504481 0.863423i \(-0.668316\pi\)
0.504481 0.863423i \(-0.331684\pi\)
\(198\) − 14.4690i − 1.02827i
\(199\) −16.7513 −1.18747 −0.593734 0.804661i \(-0.702347\pi\)
−0.593734 + 0.804661i \(0.702347\pi\)
\(200\) 0 0
\(201\) 4.71037 0.332244
\(202\) 19.2750i 1.35619i
\(203\) 1.19394i 0.0837979i
\(204\) −0.860818 −0.0602693
\(205\) 0 0
\(206\) −7.87987 −0.549017
\(207\) − 4.33312i − 0.301173i
\(208\) 12.8872i 0.893564i
\(209\) 13.2750 0.918254
\(210\) 0 0
\(211\) −25.3054 −1.74209 −0.871046 0.491201i \(-0.836558\pi\)
−0.871046 + 0.491201i \(0.836558\pi\)
\(212\) − 0.261865i − 0.0179850i
\(213\) 1.02776i 0.0704211i
\(214\) −20.4690 −1.39923
\(215\) 0 0
\(216\) −11.5369 −0.784987
\(217\) − 5.73813i − 0.389530i
\(218\) − 2.77575i − 0.187997i
\(219\) 12.2882 0.830360
\(220\) 0 0
\(221\) 16.3127 1.09731
\(222\) 11.3503i 0.761780i
\(223\) − 17.6932i − 1.18483i −0.805634 0.592413i \(-0.798175\pi\)
0.805634 0.592413i \(-0.201825\pi\)
\(224\) −1.30536 −0.0872178
\(225\) 0 0
\(226\) 17.4314 1.15952
\(227\) 26.8423i 1.78158i 0.454412 + 0.890792i \(0.349849\pi\)
−0.454412 + 0.890792i \(0.650151\pi\)
\(228\) 0.499293i 0.0330665i
\(229\) 17.2243 1.13821 0.569105 0.822265i \(-0.307290\pi\)
0.569105 + 0.822265i \(0.307290\pi\)
\(230\) 0 0
\(231\) 4.00000 0.263181
\(232\) − 2.67513i − 0.175631i
\(233\) 9.07381i 0.594445i 0.954808 + 0.297222i \(0.0960603\pi\)
−0.954808 + 0.297222i \(0.903940\pi\)
\(234\) −10.3127 −0.674159
\(235\) 0 0
\(236\) 2.57452 0.167587
\(237\) − 3.97556i − 0.258241i
\(238\) 9.73813i 0.631230i
\(239\) −20.4993 −1.32599 −0.662995 0.748624i \(-0.730715\pi\)
−0.662995 + 0.748624i \(0.730715\pi\)
\(240\) 0 0
\(241\) 5.47627 0.352758 0.176379 0.984322i \(-0.443562\pi\)
0.176379 + 0.984322i \(0.443562\pi\)
\(242\) − 9.29455i − 0.597476i
\(243\) − 15.8192i − 1.01480i
\(244\) −1.72355 −0.110339
\(245\) 0 0
\(246\) 13.4617 0.858285
\(247\) − 9.46168i − 0.602032i
\(248\) 12.8568i 0.816411i
\(249\) −3.56134 −0.225691
\(250\) 0 0
\(251\) −29.6180 −1.86947 −0.934736 0.355343i \(-0.884364\pi\)
−0.934736 + 0.355343i \(0.884364\pi\)
\(252\) − 0.544198i − 0.0342813i
\(253\) − 7.66291i − 0.481763i
\(254\) 21.1333 1.32602
\(255\) 0 0
\(256\) 4.61213 0.288258
\(257\) 17.6629i 1.10178i 0.834577 + 0.550891i \(0.185712\pi\)
−0.834577 + 0.550891i \(0.814288\pi\)
\(258\) − 0.0361968i − 0.00225351i
\(259\) 11.3503 0.705271
\(260\) 0 0
\(261\) 2.35026 0.145478
\(262\) − 8.73084i − 0.539393i
\(263\) − 27.3561i − 1.68685i −0.537245 0.843426i \(-0.680535\pi\)
0.537245 0.843426i \(-0.319465\pi\)
\(264\) −8.96239 −0.551597
\(265\) 0 0
\(266\) 5.64832 0.346321
\(267\) − 2.91160i − 0.178187i
\(268\) − 1.13330i − 0.0692275i
\(269\) −10.4993 −0.640153 −0.320077 0.947392i \(-0.603709\pi\)
−0.320077 + 0.947392i \(0.603709\pi\)
\(270\) 0 0
\(271\) −9.61801 −0.584252 −0.292126 0.956380i \(-0.594363\pi\)
−0.292126 + 0.956380i \(0.594363\pi\)
\(272\) − 23.9551i − 1.45249i
\(273\) − 2.85097i − 0.172548i
\(274\) 27.0797 1.63594
\(275\) 0 0
\(276\) 0.288213 0.0173484
\(277\) 13.3503i 0.802139i 0.916048 + 0.401070i \(0.131362\pi\)
−0.916048 + 0.401070i \(0.868638\pi\)
\(278\) − 17.0884i − 1.02489i
\(279\) −11.2955 −0.676244
\(280\) 0 0
\(281\) 20.4241 1.21840 0.609199 0.793017i \(-0.291491\pi\)
0.609199 + 0.793017i \(0.291491\pi\)
\(282\) − 5.73813i − 0.341701i
\(283\) 8.02047i 0.476767i 0.971171 + 0.238384i \(0.0766176\pi\)
−0.971171 + 0.238384i \(0.923382\pi\)
\(284\) 0.247277 0.0146732
\(285\) 0 0
\(286\) −18.2374 −1.07840
\(287\) − 13.4617i − 0.794618i
\(288\) 2.56959i 0.151415i
\(289\) −13.3225 −0.783676
\(290\) 0 0
\(291\) −1.11283 −0.0652355
\(292\) − 2.95651i − 0.173017i
\(293\) 23.3054i 1.36151i 0.732510 + 0.680757i \(0.238349\pi\)
−0.732510 + 0.680757i \(0.761651\pi\)
\(294\) −6.65562 −0.388164
\(295\) 0 0
\(296\) −25.4314 −1.47817
\(297\) − 17.9248i − 1.04010i
\(298\) 4.11142i 0.238168i
\(299\) −5.46168 −0.315857
\(300\) 0 0
\(301\) −0.0361968 −0.00208635
\(302\) − 2.66433i − 0.153315i
\(303\) 10.4894i 0.602603i
\(304\) −13.8945 −0.796902
\(305\) 0 0
\(306\) 19.1695 1.09585
\(307\) − 6.73084i − 0.384149i −0.981380 0.192075i \(-0.938478\pi\)
0.981380 0.192075i \(-0.0615216\pi\)
\(308\) − 0.962389i − 0.0548372i
\(309\) −4.28821 −0.243948
\(310\) 0 0
\(311\) −22.0567 −1.25072 −0.625359 0.780337i \(-0.715048\pi\)
−0.625359 + 0.780337i \(0.715048\pi\)
\(312\) 6.38787i 0.361642i
\(313\) − 5.03761i − 0.284743i −0.989813 0.142371i \(-0.954527\pi\)
0.989813 0.142371i \(-0.0454727\pi\)
\(314\) 5.58181 0.315000
\(315\) 0 0
\(316\) −0.956509 −0.0538078
\(317\) 34.2941i 1.92615i 0.269237 + 0.963074i \(0.413229\pi\)
−0.269237 + 0.963074i \(0.586771\pi\)
\(318\) − 1.61213i − 0.0904036i
\(319\) 4.15633 0.232710
\(320\) 0 0
\(321\) −11.1392 −0.621729
\(322\) − 3.26045i − 0.181698i
\(323\) 17.5877i 0.978605i
\(324\) −0.693229 −0.0385127
\(325\) 0 0
\(326\) −2.43278 −0.134739
\(327\) − 1.51056i − 0.0835340i
\(328\) 30.1622i 1.66543i
\(329\) −5.73813 −0.316354
\(330\) 0 0
\(331\) −34.8324 −1.91456 −0.957281 0.289159i \(-0.906625\pi\)
−0.957281 + 0.289159i \(0.906625\pi\)
\(332\) 0.856849i 0.0470257i
\(333\) − 22.3430i − 1.22439i
\(334\) 11.9697 0.654952
\(335\) 0 0
\(336\) −4.18664 −0.228400
\(337\) − 17.6326i − 0.960509i −0.877129 0.480254i \(-0.840544\pi\)
0.877129 0.480254i \(-0.159456\pi\)
\(338\) − 6.25694i − 0.340333i
\(339\) 9.48612 0.515215
\(340\) 0 0
\(341\) −19.9756 −1.08174
\(342\) − 11.1187i − 0.601231i
\(343\) 15.0132i 0.810635i
\(344\) 0.0811024 0.00437275
\(345\) 0 0
\(346\) 11.4617 0.616184
\(347\) − 3.11871i − 0.167421i −0.996490 0.0837107i \(-0.973323\pi\)
0.996490 0.0837107i \(-0.0266772\pi\)
\(348\) 0.156325i 0.00837991i
\(349\) 13.0738 0.699825 0.349912 0.936782i \(-0.386211\pi\)
0.349912 + 0.936782i \(0.386211\pi\)
\(350\) 0 0
\(351\) −12.7757 −0.681919
\(352\) 4.54420i 0.242207i
\(353\) − 5.19982i − 0.276758i −0.990379 0.138379i \(-0.955811\pi\)
0.990379 0.138379i \(-0.0441893\pi\)
\(354\) 15.8496 0.842394
\(355\) 0 0
\(356\) −0.700523 −0.0371277
\(357\) 5.29948i 0.280478i
\(358\) − 31.6991i − 1.67535i
\(359\) −30.4182 −1.60541 −0.802705 0.596376i \(-0.796607\pi\)
−0.802705 + 0.596376i \(0.796607\pi\)
\(360\) 0 0
\(361\) −8.79877 −0.463093
\(362\) − 22.6253i − 1.18916i
\(363\) − 5.05808i − 0.265480i
\(364\) −0.685935 −0.0359528
\(365\) 0 0
\(366\) −10.6107 −0.554631
\(367\) 20.6556i 1.07821i 0.842237 + 0.539107i \(0.181238\pi\)
−0.842237 + 0.539107i \(0.818762\pi\)
\(368\) 8.02047i 0.418096i
\(369\) −26.4993 −1.37950
\(370\) 0 0
\(371\) −1.61213 −0.0836975
\(372\) − 0.751309i − 0.0389535i
\(373\) − 11.0884i − 0.574135i −0.957910 0.287068i \(-0.907320\pi\)
0.957910 0.287068i \(-0.0926805\pi\)
\(374\) 33.9003 1.75294
\(375\) 0 0
\(376\) 12.8568 0.663041
\(377\) − 2.96239i − 0.152571i
\(378\) − 7.62672i − 0.392276i
\(379\) −10.0811 −0.517831 −0.258916 0.965900i \(-0.583365\pi\)
−0.258916 + 0.965900i \(0.583365\pi\)
\(380\) 0 0
\(381\) 11.5007 0.589199
\(382\) − 4.91748i − 0.251600i
\(383\) − 16.3576i − 0.835832i −0.908486 0.417916i \(-0.862761\pi\)
0.908486 0.417916i \(-0.137239\pi\)
\(384\) 10.2170 0.521382
\(385\) 0 0
\(386\) 7.23013 0.368004
\(387\) 0.0712533i 0.00362201i
\(388\) 0.267745i 0.0135927i
\(389\) −31.9003 −1.61741 −0.808706 0.588213i \(-0.799831\pi\)
−0.808706 + 0.588213i \(0.799831\pi\)
\(390\) 0 0
\(391\) 10.1524 0.513427
\(392\) − 14.9126i − 0.753198i
\(393\) − 4.75131i − 0.239672i
\(394\) −35.9003 −1.80863
\(395\) 0 0
\(396\) −1.89446 −0.0952002
\(397\) 2.98683i 0.149905i 0.997187 + 0.0749523i \(0.0238804\pi\)
−0.997187 + 0.0749523i \(0.976120\pi\)
\(398\) 24.8119i 1.24371i
\(399\) 3.07381 0.153883
\(400\) 0 0
\(401\) −21.9756 −1.09741 −0.548704 0.836017i \(-0.684878\pi\)
−0.548704 + 0.836017i \(0.684878\pi\)
\(402\) − 6.97698i − 0.347980i
\(403\) 14.2374i 0.709217i
\(404\) 2.52373 0.125560
\(405\) 0 0
\(406\) 1.76845 0.0877668
\(407\) − 39.5125i − 1.95856i
\(408\) − 11.8740i − 0.587850i
\(409\) 22.4387 1.10952 0.554760 0.832010i \(-0.312810\pi\)
0.554760 + 0.832010i \(0.312810\pi\)
\(410\) 0 0
\(411\) 14.7367 0.726909
\(412\) 1.03173i 0.0508298i
\(413\) − 15.8496i − 0.779906i
\(414\) −6.41819 −0.315437
\(415\) 0 0
\(416\) 3.23884 0.158797
\(417\) − 9.29948i − 0.455397i
\(418\) − 19.6629i − 0.961744i
\(419\) −10.3634 −0.506287 −0.253143 0.967429i \(-0.581464\pi\)
−0.253143 + 0.967429i \(0.581464\pi\)
\(420\) 0 0
\(421\) 34.0362 1.65882 0.829411 0.558638i \(-0.188676\pi\)
0.829411 + 0.558638i \(0.188676\pi\)
\(422\) 37.4821i 1.82460i
\(423\) 11.2955i 0.549206i
\(424\) 3.61213 0.175420
\(425\) 0 0
\(426\) 1.52232 0.0737564
\(427\) 10.6107i 0.513488i
\(428\) 2.68006i 0.129545i
\(429\) −9.92478 −0.479173
\(430\) 0 0
\(431\) 25.7743 1.24151 0.620753 0.784006i \(-0.286827\pi\)
0.620753 + 0.784006i \(0.286827\pi\)
\(432\) 18.7612i 0.902647i
\(433\) − 2.18076i − 0.104801i −0.998626 0.0524004i \(-0.983313\pi\)
0.998626 0.0524004i \(-0.0166872\pi\)
\(434\) −8.49929 −0.407979
\(435\) 0 0
\(436\) −0.363436 −0.0174054
\(437\) − 5.88858i − 0.281689i
\(438\) − 18.2012i − 0.869688i
\(439\) 35.5125 1.69492 0.847459 0.530861i \(-0.178131\pi\)
0.847459 + 0.530861i \(0.178131\pi\)
\(440\) 0 0
\(441\) 13.1016 0.623884
\(442\) − 24.1622i − 1.14928i
\(443\) 4.34297i 0.206341i 0.994664 + 0.103170i \(0.0328987\pi\)
−0.994664 + 0.103170i \(0.967101\pi\)
\(444\) 1.48612 0.0705281
\(445\) 0 0
\(446\) −26.2071 −1.24094
\(447\) 2.23743i 0.105827i
\(448\) − 8.45439i − 0.399432i
\(449\) −31.3357 −1.47882 −0.739411 0.673254i \(-0.764896\pi\)
−0.739411 + 0.673254i \(0.764896\pi\)
\(450\) 0 0
\(451\) −46.8627 −2.20668
\(452\) − 2.28233i − 0.107352i
\(453\) − 1.44992i − 0.0681233i
\(454\) 39.7586 1.86596
\(455\) 0 0
\(456\) −6.88717 −0.322521
\(457\) 34.3488i 1.60677i 0.595459 + 0.803386i \(0.296970\pi\)
−0.595459 + 0.803386i \(0.703030\pi\)
\(458\) − 25.5125i − 1.19212i
\(459\) 23.7480 1.10846
\(460\) 0 0
\(461\) 11.8641 0.552568 0.276284 0.961076i \(-0.410897\pi\)
0.276284 + 0.961076i \(0.410897\pi\)
\(462\) − 5.92478i − 0.275646i
\(463\) − 40.4953i − 1.88198i −0.338438 0.940989i \(-0.609899\pi\)
0.338438 0.940989i \(-0.390101\pi\)
\(464\) −4.35026 −0.201956
\(465\) 0 0
\(466\) 13.4401 0.622599
\(467\) − 30.2071i − 1.39782i −0.715210 0.698909i \(-0.753669\pi\)
0.715210 0.698909i \(-0.246331\pi\)
\(468\) 1.35026i 0.0624159i
\(469\) −6.97698 −0.322167
\(470\) 0 0
\(471\) 3.03761 0.139966
\(472\) 35.5125i 1.63459i
\(473\) 0.126008i 0.00579385i
\(474\) −5.88858 −0.270471
\(475\) 0 0
\(476\) 1.27504 0.0584413
\(477\) 3.17347i 0.145303i
\(478\) 30.3634i 1.38879i
\(479\) −0.0547547 −0.00250181 −0.00125090 0.999999i \(-0.500398\pi\)
−0.00125090 + 0.999999i \(0.500398\pi\)
\(480\) 0 0
\(481\) −28.1622 −1.28409
\(482\) − 8.11142i − 0.369465i
\(483\) − 1.77433i − 0.0807349i
\(484\) −1.21696 −0.0553163
\(485\) 0 0
\(486\) −23.4314 −1.06287
\(487\) − 0.881286i − 0.0399349i −0.999801 0.0199674i \(-0.993644\pi\)
0.999801 0.0199674i \(-0.00635626\pi\)
\(488\) − 23.7743i − 1.07621i
\(489\) −1.32391 −0.0598695
\(490\) 0 0
\(491\) 41.0698 1.85346 0.926728 0.375733i \(-0.122609\pi\)
0.926728 + 0.375733i \(0.122609\pi\)
\(492\) − 1.76257i − 0.0794629i
\(493\) 5.50659i 0.248004i
\(494\) −14.0146 −0.630546
\(495\) 0 0
\(496\) 20.9076 0.938780
\(497\) − 1.52232i − 0.0682852i
\(498\) 5.27504i 0.236380i
\(499\) −12.3733 −0.553904 −0.276952 0.960884i \(-0.589324\pi\)
−0.276952 + 0.960884i \(0.589324\pi\)
\(500\) 0 0
\(501\) 6.51388 0.291019
\(502\) 43.8700i 1.95801i
\(503\) 2.26774i 0.101114i 0.998721 + 0.0505569i \(0.0160996\pi\)
−0.998721 + 0.0505569i \(0.983900\pi\)
\(504\) 7.50659 0.334370
\(505\) 0 0
\(506\) −11.3503 −0.504581
\(507\) − 3.40502i − 0.151222i
\(508\) − 2.76704i − 0.122767i
\(509\) 10.9018 0.483212 0.241606 0.970374i \(-0.422326\pi\)
0.241606 + 0.970374i \(0.422326\pi\)
\(510\) 0 0
\(511\) −18.2012 −0.805175
\(512\) 18.5188i 0.818423i
\(513\) − 13.7743i − 0.608152i
\(514\) 26.1622 1.15397
\(515\) 0 0
\(516\) −0.00473934 −0.000208638 0
\(517\) 19.9756i 0.878524i
\(518\) − 16.8119i − 0.738674i
\(519\) 6.23743 0.273793
\(520\) 0 0
\(521\) −4.72496 −0.207004 −0.103502 0.994629i \(-0.533005\pi\)
−0.103502 + 0.994629i \(0.533005\pi\)
\(522\) − 3.48119i − 0.152368i
\(523\) 1.06793i 0.0466973i 0.999727 + 0.0233486i \(0.00743277\pi\)
−0.999727 + 0.0233486i \(0.992567\pi\)
\(524\) −1.14315 −0.0499388
\(525\) 0 0
\(526\) −40.5198 −1.76675
\(527\) − 26.4650i − 1.15283i
\(528\) 14.5745i 0.634274i
\(529\) 19.6009 0.852211
\(530\) 0 0
\(531\) −31.1998 −1.35396
\(532\) − 0.739549i − 0.0320635i
\(533\) 33.4010i 1.44676i
\(534\) −4.31265 −0.186627
\(535\) 0 0
\(536\) 15.6326 0.675225
\(537\) − 17.2506i − 0.744418i
\(538\) 15.5515i 0.670472i
\(539\) 23.1695 0.997981
\(540\) 0 0
\(541\) −7.46168 −0.320803 −0.160401 0.987052i \(-0.551279\pi\)
−0.160401 + 0.987052i \(0.551279\pi\)
\(542\) 14.2461i 0.611924i
\(543\) − 12.3127i − 0.528386i
\(544\) −6.02047 −0.258125
\(545\) 0 0
\(546\) −4.22284 −0.180721
\(547\) − 38.9683i − 1.66616i −0.553150 0.833081i \(-0.686575\pi\)
0.553150 0.833081i \(-0.313425\pi\)
\(548\) − 3.54561i − 0.151461i
\(549\) 20.8872 0.891443
\(550\) 0 0
\(551\) 3.19394 0.136066
\(552\) 3.97556i 0.169211i
\(553\) 5.88858i 0.250408i
\(554\) 19.7743 0.840131
\(555\) 0 0
\(556\) −2.23743 −0.0948881
\(557\) − 22.9986i − 0.974481i −0.873268 0.487241i \(-0.838003\pi\)
0.873268 0.487241i \(-0.161997\pi\)
\(558\) 16.7308i 0.708273i
\(559\) 0.0898112 0.00379861
\(560\) 0 0
\(561\) 18.4485 0.778897
\(562\) − 30.2520i − 1.27610i
\(563\) − 11.6688i − 0.491781i −0.969298 0.245890i \(-0.920920\pi\)
0.969298 0.245890i \(-0.0790804\pi\)
\(564\) −0.751309 −0.0316358
\(565\) 0 0
\(566\) 11.8799 0.499348
\(567\) 4.26774i 0.179228i
\(568\) 3.41090i 0.143118i
\(569\) −11.3357 −0.475216 −0.237608 0.971361i \(-0.576363\pi\)
−0.237608 + 0.971361i \(0.576363\pi\)
\(570\) 0 0
\(571\) 27.1754 1.13725 0.568627 0.822595i \(-0.307475\pi\)
0.568627 + 0.822595i \(0.307475\pi\)
\(572\) 2.38787i 0.0998420i
\(573\) − 2.67609i − 0.111795i
\(574\) −19.9394 −0.832253
\(575\) 0 0
\(576\) −16.6424 −0.693435
\(577\) 22.5950i 0.940641i 0.882496 + 0.470321i \(0.155862\pi\)
−0.882496 + 0.470321i \(0.844138\pi\)
\(578\) 19.7332i 0.820793i
\(579\) 3.93463 0.163517
\(580\) 0 0
\(581\) 5.27504 0.218845
\(582\) 1.64832i 0.0683252i
\(583\) 5.61213i 0.232431i
\(584\) 40.7816 1.68756
\(585\) 0 0
\(586\) 34.5198 1.42600
\(587\) 9.31994i 0.384675i 0.981329 + 0.192338i \(0.0616069\pi\)
−0.981329 + 0.192338i \(0.938393\pi\)
\(588\) 0.871437i 0.0359375i
\(589\) −15.3503 −0.632497
\(590\) 0 0
\(591\) −19.5369 −0.803641
\(592\) 41.3561i 1.69973i
\(593\) 15.1246i 0.621093i 0.950558 + 0.310546i \(0.100512\pi\)
−0.950558 + 0.310546i \(0.899488\pi\)
\(594\) −26.5501 −1.08936
\(595\) 0 0
\(596\) 0.538319 0.0220504
\(597\) 13.5026i 0.552625i
\(598\) 8.08981i 0.330817i
\(599\) −4.09569 −0.167345 −0.0836727 0.996493i \(-0.526665\pi\)
−0.0836727 + 0.996493i \(0.526665\pi\)
\(600\) 0 0
\(601\) 22.2276 0.906682 0.453341 0.891337i \(-0.350232\pi\)
0.453341 + 0.891337i \(0.350232\pi\)
\(602\) 0.0536145i 0.00218516i
\(603\) 13.7342i 0.559298i
\(604\) −0.348847 −0.0141944
\(605\) 0 0
\(606\) 15.5369 0.631144
\(607\) − 48.2941i − 1.96020i −0.198512 0.980098i \(-0.563611\pi\)
0.198512 0.980098i \(-0.436389\pi\)
\(608\) 3.49200i 0.141619i
\(609\) 0.962389 0.0389980
\(610\) 0 0
\(611\) 14.2374 0.575985
\(612\) − 2.50991i − 0.101457i
\(613\) 9.74798i 0.393717i 0.980432 + 0.196859i \(0.0630740\pi\)
−0.980432 + 0.196859i \(0.936926\pi\)
\(614\) −9.96968 −0.402344
\(615\) 0 0
\(616\) 13.2750 0.534867
\(617\) 18.2170i 0.733387i 0.930342 + 0.366694i \(0.119510\pi\)
−0.930342 + 0.366694i \(0.880490\pi\)
\(618\) 6.35168i 0.255502i
\(619\) −25.0943 −1.00862 −0.504312 0.863521i \(-0.668254\pi\)
−0.504312 + 0.863521i \(0.668254\pi\)
\(620\) 0 0
\(621\) −7.95112 −0.319068
\(622\) 32.6702i 1.30996i
\(623\) 4.31265i 0.172783i
\(624\) 10.3879 0.415848
\(625\) 0 0
\(626\) −7.46168 −0.298229
\(627\) − 10.7005i − 0.427338i
\(628\) − 0.730841i − 0.0291637i
\(629\) 52.3488 2.08729
\(630\) 0 0
\(631\) 21.4617 0.854376 0.427188 0.904163i \(-0.359504\pi\)
0.427188 + 0.904163i \(0.359504\pi\)
\(632\) − 13.1939i − 0.524827i
\(633\) 20.3977i 0.810737i
\(634\) 50.7962 2.01738
\(635\) 0 0
\(636\) −0.211080 −0.00836986
\(637\) − 16.5139i − 0.654304i
\(638\) − 6.15633i − 0.243731i
\(639\) −2.99668 −0.118547
\(640\) 0 0
\(641\) 3.17347 0.125344 0.0626722 0.998034i \(-0.480038\pi\)
0.0626722 + 0.998034i \(0.480038\pi\)
\(642\) 16.4993i 0.651175i
\(643\) 2.74069i 0.108082i 0.998539 + 0.0540411i \(0.0172102\pi\)
−0.998539 + 0.0540411i \(0.982790\pi\)
\(644\) −0.426899 −0.0168222
\(645\) 0 0
\(646\) 26.0508 1.02495
\(647\) 6.34297i 0.249368i 0.992197 + 0.124684i \(0.0397917\pi\)
−0.992197 + 0.124684i \(0.960208\pi\)
\(648\) − 9.56230i − 0.375642i
\(649\) −55.1754 −2.16582
\(650\) 0 0
\(651\) −4.62530 −0.181280
\(652\) 0.318530i 0.0124746i
\(653\) − 4.08110i − 0.159706i −0.996807 0.0798529i \(-0.974555\pi\)
0.996807 0.0798529i \(-0.0254451\pi\)
\(654\) −2.23743 −0.0874903
\(655\) 0 0
\(656\) 49.0494 1.91506
\(657\) 35.8291i 1.39783i
\(658\) 8.49929i 0.331337i
\(659\) −9.58181 −0.373254 −0.186627 0.982431i \(-0.559756\pi\)
−0.186627 + 0.982431i \(0.559756\pi\)
\(660\) 0 0
\(661\) −27.5271 −1.07068 −0.535339 0.844637i \(-0.679816\pi\)
−0.535339 + 0.844637i \(0.679816\pi\)
\(662\) 51.5936i 2.00524i
\(663\) − 13.1490i − 0.510666i
\(664\) −11.8192 −0.458675
\(665\) 0 0
\(666\) −33.0943 −1.28238
\(667\) − 1.84367i − 0.0713874i
\(668\) − 1.56722i − 0.0606376i
\(669\) −14.2619 −0.551396
\(670\) 0 0
\(671\) 36.9380 1.42597
\(672\) 1.05220i 0.0405895i
\(673\) − 3.13727i − 0.120933i −0.998170 0.0604665i \(-0.980741\pi\)
0.998170 0.0604665i \(-0.0192588\pi\)
\(674\) −26.1173 −1.00600
\(675\) 0 0
\(676\) −0.819237 −0.0315091
\(677\) − 46.2579i − 1.77784i −0.458067 0.888918i \(-0.651458\pi\)
0.458067 0.888918i \(-0.348542\pi\)
\(678\) − 14.0508i − 0.539617i
\(679\) 1.64832 0.0632569
\(680\) 0 0
\(681\) 21.6366 0.829115
\(682\) 29.5877i 1.13297i
\(683\) 9.01905i 0.345104i 0.985000 + 0.172552i \(0.0552014\pi\)
−0.985000 + 0.172552i \(0.944799\pi\)
\(684\) −1.45580 −0.0556640
\(685\) 0 0
\(686\) 22.2374 0.849029
\(687\) − 13.8838i − 0.529702i
\(688\) − 0.131888i − 0.00502817i
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) −50.0625 −1.90447 −0.952234 0.305368i \(-0.901221\pi\)
−0.952234 + 0.305368i \(0.901221\pi\)
\(692\) − 1.50071i − 0.0570483i
\(693\) 11.6629i 0.443037i
\(694\) −4.61942 −0.175351
\(695\) 0 0
\(696\) −2.15633 −0.0817353
\(697\) − 62.0870i − 2.35171i
\(698\) − 19.3649i − 0.732970i
\(699\) 7.31406 0.276643
\(700\) 0 0
\(701\) 45.3014 1.71101 0.855505 0.517795i \(-0.173247\pi\)
0.855505 + 0.517795i \(0.173247\pi\)
\(702\) 18.9234i 0.714216i
\(703\) − 30.3634i − 1.14518i
\(704\) −29.4314 −1.10924
\(705\) 0 0
\(706\) −7.70194 −0.289866
\(707\) − 15.5369i − 0.584325i
\(708\) − 2.07522i − 0.0779916i
\(709\) 3.27504 0.122997 0.0614983 0.998107i \(-0.480412\pi\)
0.0614983 + 0.998107i \(0.480412\pi\)
\(710\) 0 0
\(711\) 11.5917 0.434721
\(712\) − 9.66291i − 0.362133i
\(713\) 8.86082i 0.331840i
\(714\) 7.84955 0.293762
\(715\) 0 0
\(716\) −4.15045 −0.155109
\(717\) 16.5237i 0.617090i
\(718\) 45.0553i 1.68145i
\(719\) −27.7235 −1.03391 −0.516957 0.856011i \(-0.672935\pi\)
−0.516957 + 0.856011i \(0.672935\pi\)
\(720\) 0 0
\(721\) 6.35168 0.236549
\(722\) 13.0327i 0.485026i
\(723\) − 4.41422i − 0.164167i
\(724\) −2.96239 −0.110096
\(725\) 0 0
\(726\) −7.49200 −0.278054
\(727\) − 26.8930i − 0.997408i −0.866772 0.498704i \(-0.833810\pi\)
0.866772 0.498704i \(-0.166190\pi\)
\(728\) − 9.46168i − 0.350673i
\(729\) −2.02776 −0.0751023
\(730\) 0 0
\(731\) −0.166944 −0.00617465
\(732\) 1.38929i 0.0513496i
\(733\) − 3.17935i − 0.117432i −0.998275 0.0587160i \(-0.981299\pi\)
0.998275 0.0587160i \(-0.0187006\pi\)
\(734\) 30.5950 1.12928
\(735\) 0 0
\(736\) 2.01573 0.0743007
\(737\) 24.2882i 0.894668i
\(738\) 39.2506i 1.44483i
\(739\) 29.7440 1.09415 0.547076 0.837083i \(-0.315741\pi\)
0.547076 + 0.837083i \(0.315741\pi\)
\(740\) 0 0
\(741\) −7.62672 −0.280174
\(742\) 2.38787i 0.0876616i
\(743\) 4.34297i 0.159328i 0.996822 + 0.0796640i \(0.0253847\pi\)
−0.996822 + 0.0796640i \(0.974615\pi\)
\(744\) 10.3634 0.379942
\(745\) 0 0
\(746\) −16.4241 −0.601328
\(747\) − 10.3839i − 0.379927i
\(748\) − 4.43866i − 0.162293i
\(749\) 16.4993 0.602871
\(750\) 0 0
\(751\) 22.5804 0.823970 0.411985 0.911191i \(-0.364836\pi\)
0.411985 + 0.911191i \(0.364836\pi\)
\(752\) − 20.9076i − 0.762423i
\(753\) 23.8740i 0.870017i
\(754\) −4.38787 −0.159797
\(755\) 0 0
\(756\) −0.998585 −0.0363182
\(757\) 9.88461i 0.359262i 0.983734 + 0.179631i \(0.0574904\pi\)
−0.983734 + 0.179631i \(0.942510\pi\)
\(758\) 14.9321i 0.542357i
\(759\) −6.17679 −0.224203
\(760\) 0 0
\(761\) 13.6991 0.496592 0.248296 0.968684i \(-0.420129\pi\)
0.248296 + 0.968684i \(0.420129\pi\)
\(762\) − 17.0348i − 0.617105i
\(763\) 2.23743i 0.0810003i
\(764\) −0.643859 −0.0232940
\(765\) 0 0
\(766\) −24.2287 −0.875419
\(767\) 39.3258i 1.41997i
\(768\) − 3.71767i − 0.134150i
\(769\) −25.0132 −0.901998 −0.450999 0.892524i \(-0.648932\pi\)
−0.450999 + 0.892524i \(0.648932\pi\)
\(770\) 0 0
\(771\) 14.2374 0.512748
\(772\) − 0.946660i − 0.0340710i
\(773\) 35.9062i 1.29146i 0.763567 + 0.645728i \(0.223446\pi\)
−0.763567 + 0.645728i \(0.776554\pi\)
\(774\) 0.105540 0.00379356
\(775\) 0 0
\(776\) −3.69323 −0.132579
\(777\) − 9.14903i − 0.328220i
\(778\) 47.2506i 1.69402i
\(779\) −36.0118 −1.29026
\(780\) 0 0
\(781\) −5.29948 −0.189630
\(782\) − 15.0376i − 0.537744i
\(783\) − 4.31265i − 0.154122i
\(784\) −24.2506 −0.866093
\(785\) 0 0
\(786\) −7.03761 −0.251023
\(787\) 50.3839i 1.79599i 0.440003 + 0.897996i \(0.354977\pi\)
−0.440003 + 0.897996i \(0.645023\pi\)
\(788\) 4.70052i 0.167449i
\(789\) −22.0508 −0.785029
\(790\) 0 0
\(791\) −14.0508 −0.499588
\(792\) − 26.1319i − 0.928556i
\(793\) − 26.3272i − 0.934908i
\(794\) 4.42407 0.157004
\(795\) 0 0
\(796\) 3.24869 0.115147
\(797\) − 5.69323i − 0.201665i −0.994903 0.100832i \(-0.967849\pi\)
0.994903 0.100832i \(-0.0321505\pi\)
\(798\) − 4.55291i − 0.161171i
\(799\) −26.4650 −0.936265
\(800\) 0 0
\(801\) 8.48944 0.299960
\(802\) 32.5501i 1.14938i
\(803\) 63.3620i 2.23600i
\(804\) −0.913513 −0.0322171
\(805\) 0 0
\(806\) 21.0884 0.742807
\(807\) 8.46310i 0.297915i
\(808\) 34.8119i 1.22468i
\(809\) 7.76257 0.272918 0.136459 0.990646i \(-0.456428\pi\)
0.136459 + 0.990646i \(0.456428\pi\)
\(810\) 0 0
\(811\) −26.4894 −0.930170 −0.465085 0.885266i \(-0.653976\pi\)
−0.465085 + 0.885266i \(0.653976\pi\)
\(812\) − 0.231548i − 0.00812574i
\(813\) 7.75272i 0.271900i
\(814\) −58.5256 −2.05132
\(815\) 0 0
\(816\) −19.3093 −0.675962
\(817\) 0.0968311i 0.00338769i
\(818\) − 33.2360i − 1.16207i
\(819\) 8.31265 0.290468
\(820\) 0 0
\(821\) 25.4763 0.889128 0.444564 0.895747i \(-0.353359\pi\)
0.444564 + 0.895747i \(0.353359\pi\)
\(822\) − 21.8279i − 0.761337i
\(823\) 9.22028i 0.321399i 0.987003 + 0.160699i \(0.0513750\pi\)
−0.987003 + 0.160699i \(0.948625\pi\)
\(824\) −14.2315 −0.495779
\(825\) 0 0
\(826\) −23.4763 −0.816844
\(827\) 24.5343i 0.853143i 0.904454 + 0.426571i \(0.140279\pi\)
−0.904454 + 0.426571i \(0.859721\pi\)
\(828\) 0.840350i 0.0292042i
\(829\) −0.201231 −0.00698903 −0.00349452 0.999994i \(-0.501112\pi\)
−0.00349452 + 0.999994i \(0.501112\pi\)
\(830\) 0 0
\(831\) 10.7612 0.373300
\(832\) 20.9770i 0.727246i
\(833\) 30.6966i 1.06357i
\(834\) −13.7743 −0.476966
\(835\) 0 0
\(836\) −2.57452 −0.0890415
\(837\) 20.7269i 0.716425i
\(838\) 15.3503i 0.530266i
\(839\) −1.45580 −0.0502599 −0.0251299 0.999684i \(-0.508000\pi\)
−0.0251299 + 0.999684i \(0.508000\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) − 50.4142i − 1.73739i
\(843\) − 16.4631i − 0.567019i
\(844\) 4.90763 0.168928
\(845\) 0 0
\(846\) 16.7308 0.575218
\(847\) 7.49200i 0.257428i
\(848\) − 5.87399i − 0.201714i
\(849\) 6.46501 0.221878
\(850\) 0 0
\(851\) −17.5271 −0.600820
\(852\) − 0.199321i − 0.00682861i
\(853\) − 43.1793i − 1.47843i −0.673468 0.739216i \(-0.735196\pi\)
0.673468 0.739216i \(-0.264804\pi\)
\(854\) 15.7165 0.537808
\(855\) 0 0
\(856\) −36.9683 −1.26355
\(857\) − 20.9887i − 0.716962i −0.933537 0.358481i \(-0.883295\pi\)
0.933537 0.358481i \(-0.116705\pi\)
\(858\) 14.7005i 0.501868i
\(859\) 49.4069 1.68574 0.842871 0.538115i \(-0.180863\pi\)
0.842871 + 0.538115i \(0.180863\pi\)
\(860\) 0 0
\(861\) −10.8510 −0.369800
\(862\) − 38.1768i − 1.30031i
\(863\) − 56.6820i − 1.92948i −0.263211 0.964738i \(-0.584782\pi\)
0.263211 0.964738i \(-0.415218\pi\)
\(864\) 4.71511 0.160411
\(865\) 0 0
\(866\) −3.23013 −0.109764
\(867\) 10.7388i 0.364708i
\(868\) 1.11283i 0.0377721i
\(869\) 20.4993 0.695391
\(870\) 0 0
\(871\) 17.3112 0.586569
\(872\) − 5.01317i − 0.169767i
\(873\) − 3.24472i − 0.109817i
\(874\) −8.72213 −0.295031
\(875\) 0 0
\(876\) −2.38313 −0.0805186
\(877\) − 13.1998i − 0.445726i −0.974850 0.222863i \(-0.928460\pi\)
0.974850 0.222863i \(-0.0715403\pi\)
\(878\) − 52.6009i − 1.77519i
\(879\) 18.7856 0.633622
\(880\) 0 0
\(881\) 6.37802 0.214881 0.107441 0.994212i \(-0.465734\pi\)
0.107441 + 0.994212i \(0.465734\pi\)
\(882\) − 19.4060i − 0.653433i
\(883\) − 48.6213i − 1.63624i −0.575049 0.818119i \(-0.695017\pi\)
0.575049 0.818119i \(-0.304983\pi\)
\(884\) −3.16362 −0.106404
\(885\) 0 0
\(886\) 6.43278 0.216113
\(887\) − 15.0317i − 0.504716i −0.967634 0.252358i \(-0.918794\pi\)
0.967634 0.252358i \(-0.0812061\pi\)
\(888\) 20.4993i 0.687911i
\(889\) −17.0348 −0.571328
\(890\) 0 0
\(891\) 14.8568 0.497723
\(892\) 3.43136i 0.114891i
\(893\) 15.3503i 0.513677i
\(894\) 3.31406 0.110839
\(895\) 0 0
\(896\) −15.1333 −0.505568
\(897\) 4.40246i 0.146994i
\(898\) 46.4142i 1.54886i
\(899\) −4.80606 −0.160291
\(900\) 0 0
\(901\) −7.43533 −0.247707
\(902\) 69.4128i 2.31119i
\(903\) 0.0291769i 0 0.000970946i
\(904\) 31.4821 1.04708
\(905\) 0 0
\(906\) −2.14762 −0.0713498
\(907\) − 0.342968i − 0.0113880i −0.999984 0.00569402i \(-0.998188\pi\)
0.999984 0.00569402i \(-0.00181247\pi\)
\(908\) − 5.20570i − 0.172757i
\(909\) −30.5844 −1.01442
\(910\) 0 0
\(911\) 20.9076 0.692701 0.346350 0.938105i \(-0.387421\pi\)
0.346350 + 0.938105i \(0.387421\pi\)
\(912\) 11.1998i 0.370863i
\(913\) − 18.3634i − 0.607741i
\(914\) 50.8773 1.68287
\(915\) 0 0
\(916\) −3.34041 −0.110370
\(917\) 7.03761i 0.232402i
\(918\) − 35.1754i − 1.16096i
\(919\) 1.90034 0.0626864 0.0313432 0.999509i \(-0.490022\pi\)
0.0313432 + 0.999509i \(0.490022\pi\)
\(920\) 0 0
\(921\) −5.42548 −0.178776
\(922\) − 17.5731i − 0.578739i
\(923\) 3.77716i 0.124327i
\(924\) −0.775746 −0.0255202
\(925\) 0 0
\(926\) −59.9814 −1.97111
\(927\) − 12.5033i − 0.410661i
\(928\) 1.09332i 0.0358900i
\(929\) −39.3522 −1.29110 −0.645551 0.763717i \(-0.723372\pi\)
−0.645551 + 0.763717i \(0.723372\pi\)
\(930\) 0 0
\(931\) 17.8046 0.583524
\(932\) − 1.75974i − 0.0576423i
\(933\) 17.7791i 0.582061i
\(934\) −44.7426 −1.46402
\(935\) 0 0
\(936\) −18.6253 −0.608787
\(937\) − 6.37802i − 0.208361i −0.994558 0.104180i \(-0.966778\pi\)
0.994558 0.104180i \(-0.0332220\pi\)
\(938\) 10.3343i 0.337426i
\(939\) −4.06063 −0.132514
\(940\) 0 0
\(941\) −26.6253 −0.867960 −0.433980 0.900923i \(-0.642891\pi\)
−0.433980 + 0.900923i \(0.642891\pi\)
\(942\) − 4.49929i − 0.146595i
\(943\) 20.7875i 0.676934i
\(944\) 57.7499 1.87960
\(945\) 0 0
\(946\) 0.186642 0.00606827
\(947\) − 12.2823i − 0.399122i −0.979885 0.199561i \(-0.936048\pi\)
0.979885 0.199561i \(-0.0639516\pi\)
\(948\) 0.771007i 0.0250411i
\(949\) 45.1608 1.46598
\(950\) 0 0
\(951\) 27.6432 0.896393
\(952\) 17.5877i 0.570020i
\(953\) − 0.821792i − 0.0266205i −0.999911 0.0133102i \(-0.995763\pi\)
0.999911 0.0133102i \(-0.00423690\pi\)
\(954\) 4.70052 0.152185
\(955\) 0 0
\(956\) 3.97556 0.128579
\(957\) − 3.35026i − 0.108299i
\(958\) 0.0811024i 0.00262030i
\(959\) −21.8279 −0.704861
\(960\) 0 0
\(961\) −7.90175 −0.254895
\(962\) 41.7137i 1.34490i
\(963\) − 32.4788i − 1.04662i
\(964\) −1.06205 −0.0342063
\(965\) 0 0
\(966\) −2.62813 −0.0845587
\(967\) − 37.4314i − 1.20371i −0.798605 0.601856i \(-0.794428\pi\)
0.798605 0.601856i \(-0.205572\pi\)
\(968\) − 16.7866i − 0.539540i
\(969\) 14.1768 0.455424
\(970\) 0 0
\(971\) −8.71625 −0.279718 −0.139859 0.990171i \(-0.544665\pi\)
−0.139859 + 0.990171i \(0.544665\pi\)
\(972\) 3.06793i 0.0984039i
\(973\) 13.7743i 0.441585i
\(974\) −1.30536 −0.0418263
\(975\) 0 0
\(976\) −38.6615 −1.23752
\(977\) − 33.7645i − 1.08022i −0.841594 0.540111i \(-0.818382\pi\)
0.841594 0.540111i \(-0.181618\pi\)
\(978\) 1.96097i 0.0627050i
\(979\) 15.0132 0.479823
\(980\) 0 0
\(981\) 4.40437 0.140621
\(982\) − 60.8324i − 1.94124i
\(983\) − 43.6082i − 1.39088i −0.718582 0.695442i \(-0.755209\pi\)
0.718582 0.695442i \(-0.244791\pi\)
\(984\) 24.3127 0.775059
\(985\) 0 0
\(986\) 8.15633 0.259750
\(987\) 4.62530i 0.147225i
\(988\) 1.83497i 0.0583780i
\(989\) 0.0558950 0.00177736
\(990\) 0 0
\(991\) −52.9741 −1.68278 −0.841390 0.540429i \(-0.818262\pi\)
−0.841390 + 0.540429i \(0.818262\pi\)
\(992\) − 5.25457i − 0.166833i
\(993\) 28.0771i 0.891001i
\(994\) −2.25485 −0.0715193
\(995\) 0 0
\(996\) 0.690674 0.0218849
\(997\) 13.6326i 0.431749i 0.976421 + 0.215874i \(0.0692601\pi\)
−0.976421 + 0.215874i \(0.930740\pi\)
\(998\) 18.3272i 0.580139i
\(999\) −40.9986 −1.29714
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 725.2.b.e.349.2 6
5.2 odd 4 725.2.a.e.1.3 3
5.3 odd 4 145.2.a.c.1.1 3
5.4 even 2 inner 725.2.b.e.349.5 6
15.2 even 4 6525.2.a.be.1.1 3
15.8 even 4 1305.2.a.p.1.3 3
20.3 even 4 2320.2.a.n.1.2 3
35.13 even 4 7105.2.a.o.1.1 3
40.3 even 4 9280.2.a.br.1.2 3
40.13 odd 4 9280.2.a.bj.1.2 3
145.28 odd 4 4205.2.a.f.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.c.1.1 3 5.3 odd 4
725.2.a.e.1.3 3 5.2 odd 4
725.2.b.e.349.2 6 1.1 even 1 trivial
725.2.b.e.349.5 6 5.4 even 2 inner
1305.2.a.p.1.3 3 15.8 even 4
2320.2.a.n.1.2 3 20.3 even 4
4205.2.a.f.1.3 3 145.28 odd 4
6525.2.a.be.1.1 3 15.2 even 4
7105.2.a.o.1.1 3 35.13 even 4
9280.2.a.bj.1.2 3 40.13 odd 4
9280.2.a.br.1.2 3 40.3 even 4