Properties

Label 2175.2.c.n.349.7
Level $2175$
Weight $2$
Character 2175.349
Analytic conductor $17.367$
Analytic rank $0$
Dimension $8$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2175,2,Mod(349,2175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2175.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2175.c (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: \(17.3674624396\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1267360000.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 37x^{4} + 44x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 435)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 349.7
Root \(0.820249i\) of defining polynomial
Character \(\chi\) \(=\) 2175.349
Dual form 2175.2.c.n.349.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.82025i q^{2} -1.00000i q^{3} -1.31331 q^{4} +1.82025 q^{6} +0.729126i q^{7} +1.24995i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.82025i q^{2} -1.00000i q^{3} -1.31331 q^{4} +1.82025 q^{6} +0.729126i q^{7} +1.24995i q^{8} -1.00000 q^{9} +0.729126 q^{11} +1.31331i q^{12} +3.38351i q^{13} -1.32719 q^{14} -4.90184 q^{16} +5.74301i q^{17} -1.82025i q^{18} -6.11263 q^{19} +0.729126 q^{21} +1.32719i q^{22} -9.48602i q^{23} +1.24995 q^{24} -6.15883 q^{26} +1.00000i q^{27} -0.957567i q^{28} +1.00000 q^{29} +5.48602 q^{31} -6.42266i q^{32} -0.729126i q^{33} -10.4537 q^{34} +1.31331 q^{36} +10.2949i q^{37} -11.1265i q^{38} +3.38351 q^{39} -11.3088 q^{41} +1.32719i q^{42} -10.1404i q^{43} -0.957567 q^{44} +17.2669 q^{46} +1.89749i q^{47} +4.90184i q^{48} +6.46838 q^{49} +5.74301 q^{51} -4.44359i q^{52} +8.14040i q^{53} -1.82025 q^{54} -0.911372 q^{56} +6.11263i q^{57} +1.82025i q^{58} -8.68215 q^{59} -15.5709 q^{61} +9.98592i q^{62} -0.729126i q^{63} +1.88717 q^{64} +1.32719 q^{66} +2.55187i q^{67} -7.54234i q^{68} -9.48602 q^{69} -4.83164 q^{71} -1.24995i q^{72} +6.29488i q^{73} -18.7392 q^{74} +8.02777 q^{76} +0.531625i q^{77} +6.15883i q^{78} +5.39363 q^{79} +1.00000 q^{81} -20.5848i q^{82} +0.0848668i q^{83} -0.957567 q^{84} +18.4581 q^{86} -1.00000i q^{87} +0.911372i q^{88} -4.63674 q^{89} -2.46700 q^{91} +12.4581i q^{92} -5.48602i q^{93} -3.45390 q^{94} -6.42266 q^{96} -1.30377i q^{97} +11.7741i q^{98} -0.729126 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 10 q^{4} + 6 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 10 q^{4} + 6 q^{6} - 8 q^{9} - 4 q^{11} + 6 q^{14} + 22 q^{16} + 4 q^{19} - 4 q^{21} - 24 q^{24} - 14 q^{26} + 8 q^{29} - 8 q^{31} + 2 q^{34} + 10 q^{36} - 16 q^{39} - 24 q^{41} - 18 q^{44} - 16 q^{46} - 12 q^{49} + 20 q^{51} - 6 q^{54} - 4 q^{59} - 52 q^{61} - 68 q^{64} - 6 q^{66} - 24 q^{69} - 20 q^{71} - 96 q^{74} + 32 q^{76} - 44 q^{79} + 8 q^{81} - 18 q^{84} - 8 q^{86} + 8 q^{89} - 16 q^{91} - 78 q^{94} + 34 q^{96} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2175\mathbb{Z}\right)^\times\).

\(n\) \(901\) \(1451\) \(2002\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.82025i 1.28711i 0.765400 + 0.643555i \(0.222541\pi\)
−0.765400 + 0.643555i \(0.777459\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) −1.31331 −0.656654
\(5\) 0 0
\(6\) 1.82025 0.743114
\(7\) 0.729126i 0.275584i 0.990461 + 0.137792i \(0.0440005\pi\)
−0.990461 + 0.137792i \(0.955999\pi\)
\(8\) 1.24995i 0.441925i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 0.729126 0.219840 0.109920 0.993940i \(-0.464941\pi\)
0.109920 + 0.993940i \(0.464941\pi\)
\(12\) 1.31331i 0.379119i
\(13\) 3.38351i 0.938416i 0.883088 + 0.469208i \(0.155461\pi\)
−0.883088 + 0.469208i \(0.844539\pi\)
\(14\) −1.32719 −0.354707
\(15\) 0 0
\(16\) −4.90184 −1.22546
\(17\) 5.74301i 1.39288i 0.717613 + 0.696442i \(0.245235\pi\)
−0.717613 + 0.696442i \(0.754765\pi\)
\(18\) − 1.82025i − 0.429037i
\(19\) −6.11263 −1.40233 −0.701167 0.712997i \(-0.747337\pi\)
−0.701167 + 0.712997i \(0.747337\pi\)
\(20\) 0 0
\(21\) 0.729126 0.159108
\(22\) 1.32719i 0.282958i
\(23\) − 9.48602i − 1.97797i −0.148010 0.988986i \(-0.547287\pi\)
0.148010 0.988986i \(-0.452713\pi\)
\(24\) 1.24995 0.255145
\(25\) 0 0
\(26\) −6.15883 −1.20785
\(27\) 1.00000i 0.192450i
\(28\) − 0.957567i − 0.180963i
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 5.48602 0.985318 0.492659 0.870222i \(-0.336025\pi\)
0.492659 + 0.870222i \(0.336025\pi\)
\(32\) − 6.42266i − 1.13538i
\(33\) − 0.729126i − 0.126925i
\(34\) −10.4537 −1.79280
\(35\) 0 0
\(36\) 1.31331 0.218885
\(37\) 10.2949i 1.69247i 0.532811 + 0.846234i \(0.321135\pi\)
−0.532811 + 0.846234i \(0.678865\pi\)
\(38\) − 11.1265i − 1.80496i
\(39\) 3.38351 0.541795
\(40\) 0 0
\(41\) −11.3088 −1.76613 −0.883066 0.469249i \(-0.844525\pi\)
−0.883066 + 0.469249i \(0.844525\pi\)
\(42\) 1.32719i 0.204790i
\(43\) − 10.1404i − 1.54640i −0.634164 0.773198i \(-0.718656\pi\)
0.634164 0.773198i \(-0.281344\pi\)
\(44\) −0.957567 −0.144359
\(45\) 0 0
\(46\) 17.2669 2.54587
\(47\) 1.89749i 0.276777i 0.990378 + 0.138389i \(0.0441923\pi\)
−0.990378 + 0.138389i \(0.955808\pi\)
\(48\) 4.90184i 0.707519i
\(49\) 6.46838 0.924054
\(50\) 0 0
\(51\) 5.74301 0.804182
\(52\) − 4.44359i − 0.616215i
\(53\) 8.14040i 1.11817i 0.829110 + 0.559085i \(0.188848\pi\)
−0.829110 + 0.559085i \(0.811152\pi\)
\(54\) −1.82025 −0.247705
\(55\) 0 0
\(56\) −0.911372 −0.121787
\(57\) 6.11263i 0.809638i
\(58\) 1.82025i 0.239010i
\(59\) −8.68215 −1.13032 −0.565160 0.824981i \(-0.691186\pi\)
−0.565160 + 0.824981i \(0.691186\pi\)
\(60\) 0 0
\(61\) −15.5709 −1.99365 −0.996824 0.0796378i \(-0.974624\pi\)
−0.996824 + 0.0796378i \(0.974624\pi\)
\(62\) 9.98592i 1.26821i
\(63\) − 0.729126i − 0.0918612i
\(64\) 1.88717 0.235897
\(65\) 0 0
\(66\) 1.32719 0.163366
\(67\) 2.55187i 0.311761i 0.987776 + 0.155880i \(0.0498214\pi\)
−0.987776 + 0.155880i \(0.950179\pi\)
\(68\) − 7.54234i − 0.914643i
\(69\) −9.48602 −1.14198
\(70\) 0 0
\(71\) −4.83164 −0.573410 −0.286705 0.958019i \(-0.592560\pi\)
−0.286705 + 0.958019i \(0.592560\pi\)
\(72\) − 1.24995i − 0.147308i
\(73\) 6.29488i 0.736760i 0.929675 + 0.368380i \(0.120087\pi\)
−0.929675 + 0.368380i \(0.879913\pi\)
\(74\) −18.7392 −2.17839
\(75\) 0 0
\(76\) 8.02777 0.920848
\(77\) 0.531625i 0.0605843i
\(78\) 6.15883i 0.697350i
\(79\) 5.39363 0.606831 0.303415 0.952858i \(-0.401873\pi\)
0.303415 + 0.952858i \(0.401873\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 20.5848i − 2.27321i
\(83\) 0.0848668i 0.00931534i 0.999989 + 0.00465767i \(0.00148259\pi\)
−0.999989 + 0.00465767i \(0.998517\pi\)
\(84\) −0.957567 −0.104479
\(85\) 0 0
\(86\) 18.4581 1.99038
\(87\) − 1.00000i − 0.107211i
\(88\) 0.911372i 0.0971526i
\(89\) −4.63674 −0.491493 −0.245747 0.969334i \(-0.579033\pi\)
−0.245747 + 0.969334i \(0.579033\pi\)
\(90\) 0 0
\(91\) −2.46700 −0.258612
\(92\) 12.4581i 1.29884i
\(93\) − 5.48602i − 0.568874i
\(94\) −3.45390 −0.356243
\(95\) 0 0
\(96\) −6.42266 −0.655510
\(97\) − 1.30377i − 0.132378i −0.997807 0.0661891i \(-0.978916\pi\)
0.997807 0.0661891i \(-0.0210840\pi\)
\(98\) 11.7741i 1.18936i
\(99\) −0.729126 −0.0732799
\(100\) 0 0
\(101\) −5.35574 −0.532916 −0.266458 0.963847i \(-0.585853\pi\)
−0.266458 + 0.963847i \(0.585853\pi\)
\(102\) 10.4537i 1.03507i
\(103\) 16.7670i 1.65210i 0.563594 + 0.826052i \(0.309418\pi\)
−0.563594 + 0.826052i \(0.690582\pi\)
\(104\) −4.22922 −0.414709
\(105\) 0 0
\(106\) −14.8176 −1.43921
\(107\) − 7.30876i − 0.706565i −0.935517 0.353282i \(-0.885065\pi\)
0.935517 0.353282i \(-0.114935\pi\)
\(108\) − 1.31331i − 0.126373i
\(109\) 8.21515 0.786868 0.393434 0.919353i \(-0.371287\pi\)
0.393434 + 0.919353i \(0.371287\pi\)
\(110\) 0 0
\(111\) 10.2949 0.977147
\(112\) − 3.57406i − 0.337717i
\(113\) 8.51003i 0.800556i 0.916394 + 0.400278i \(0.131086\pi\)
−0.916394 + 0.400278i \(0.868914\pi\)
\(114\) −11.1265 −1.04209
\(115\) 0 0
\(116\) −1.31331 −0.121938
\(117\) − 3.38351i − 0.312805i
\(118\) − 15.8037i − 1.45485i
\(119\) −4.18738 −0.383856
\(120\) 0 0
\(121\) −10.4684 −0.951670
\(122\) − 28.3429i − 2.56605i
\(123\) 11.3088i 1.01968i
\(124\) −7.20483 −0.647013
\(125\) 0 0
\(126\) 1.32719 0.118236
\(127\) − 4.51398i − 0.400551i −0.979740 0.200275i \(-0.935816\pi\)
0.979740 0.200275i \(-0.0641837\pi\)
\(128\) − 9.41020i − 0.831752i
\(129\) −10.1404 −0.892813
\(130\) 0 0
\(131\) −14.7771 −1.29108 −0.645542 0.763724i \(-0.723369\pi\)
−0.645542 + 0.763724i \(0.723369\pi\)
\(132\) 0.957567i 0.0833455i
\(133\) − 4.45688i − 0.386461i
\(134\) −4.64504 −0.401270
\(135\) 0 0
\(136\) −7.17849 −0.615550
\(137\) 5.19114i 0.443509i 0.975103 + 0.221754i \(0.0711783\pi\)
−0.975103 + 0.221754i \(0.928822\pi\)
\(138\) − 17.2669i − 1.46986i
\(139\) 1.15824 0.0982406 0.0491203 0.998793i \(-0.484358\pi\)
0.0491203 + 0.998793i \(0.484358\pi\)
\(140\) 0 0
\(141\) 1.89749 0.159797
\(142\) − 8.79478i − 0.738042i
\(143\) 2.46700i 0.206301i
\(144\) 4.90184 0.408487
\(145\) 0 0
\(146\) −11.4583 −0.948292
\(147\) − 6.46838i − 0.533503i
\(148\) − 13.5203i − 1.11137i
\(149\) −6.68215 −0.547423 −0.273712 0.961812i \(-0.588251\pi\)
−0.273712 + 0.961812i \(0.588251\pi\)
\(150\) 0 0
\(151\) 20.2253 1.64591 0.822955 0.568107i \(-0.192324\pi\)
0.822955 + 0.568107i \(0.192324\pi\)
\(152\) − 7.64050i − 0.619726i
\(153\) − 5.74301i − 0.464295i
\(154\) −0.967690 −0.0779787
\(155\) 0 0
\(156\) −4.44359 −0.355772
\(157\) 23.8658i 1.90470i 0.305014 + 0.952348i \(0.401339\pi\)
−0.305014 + 0.952348i \(0.598661\pi\)
\(158\) 9.81775i 0.781059i
\(159\) 8.14040 0.645576
\(160\) 0 0
\(161\) 6.91650 0.545097
\(162\) 1.82025i 0.143012i
\(163\) − 1.93538i − 0.151591i −0.997123 0.0757953i \(-0.975850\pi\)
0.997123 0.0757953i \(-0.0241496\pi\)
\(164\) 14.8519 1.15974
\(165\) 0 0
\(166\) −0.154479 −0.0119899
\(167\) 12.4025i 0.959736i 0.877341 + 0.479868i \(0.159315\pi\)
−0.877341 + 0.479868i \(0.840685\pi\)
\(168\) 0.911372i 0.0703139i
\(169\) 1.55187 0.119375
\(170\) 0 0
\(171\) 6.11263 0.467445
\(172\) 13.3175i 1.01545i
\(173\) − 21.7113i − 1.65068i −0.564637 0.825339i \(-0.690984\pi\)
0.564637 0.825339i \(-0.309016\pi\)
\(174\) 1.82025 0.137993
\(175\) 0 0
\(176\) −3.57406 −0.269405
\(177\) 8.68215i 0.652590i
\(178\) − 8.44002i − 0.632606i
\(179\) −6.08487 −0.454804 −0.227402 0.973801i \(-0.573023\pi\)
−0.227402 + 0.973801i \(0.573023\pi\)
\(180\) 0 0
\(181\) 6.77714 0.503741 0.251870 0.967761i \(-0.418954\pi\)
0.251870 + 0.967761i \(0.418954\pi\)
\(182\) − 4.49056i − 0.332863i
\(183\) 15.5709i 1.15103i
\(184\) 11.8571 0.874115
\(185\) 0 0
\(186\) 9.98592 0.732203
\(187\) 4.18738i 0.306211i
\(188\) − 2.49199i − 0.181747i
\(189\) −0.729126 −0.0530361
\(190\) 0 0
\(191\) −18.9645 −1.37222 −0.686112 0.727496i \(-0.740684\pi\)
−0.686112 + 0.727496i \(0.740684\pi\)
\(192\) − 1.88717i − 0.136195i
\(193\) 3.69760i 0.266159i 0.991105 + 0.133079i \(0.0424865\pi\)
−0.991105 + 0.133079i \(0.957513\pi\)
\(194\) 2.37319 0.170385
\(195\) 0 0
\(196\) −8.49496 −0.606783
\(197\) − 20.4783i − 1.45902i −0.683971 0.729509i \(-0.739748\pi\)
0.683971 0.729509i \(-0.260252\pi\)
\(198\) − 1.32719i − 0.0943194i
\(199\) 10.8418 0.768552 0.384276 0.923218i \(-0.374451\pi\)
0.384276 + 0.923218i \(0.374451\pi\)
\(200\) 0 0
\(201\) 2.55187 0.179995
\(202\) − 9.74878i − 0.685922i
\(203\) 0.729126i 0.0511746i
\(204\) −7.54234 −0.528069
\(205\) 0 0
\(206\) −30.5201 −2.12644
\(207\) 9.48602i 0.659324i
\(208\) − 16.5854i − 1.14999i
\(209\) −4.45688 −0.308289
\(210\) 0 0
\(211\) 2.00000 0.137686 0.0688428 0.997628i \(-0.478069\pi\)
0.0688428 + 0.997628i \(0.478069\pi\)
\(212\) − 10.6908i − 0.734251i
\(213\) 4.83164i 0.331058i
\(214\) 13.3038 0.909427
\(215\) 0 0
\(216\) −1.24995 −0.0850484
\(217\) 4.00000i 0.271538i
\(218\) 14.9536i 1.01279i
\(219\) 6.29488 0.425369
\(220\) 0 0
\(221\) −19.4315 −1.30711
\(222\) 18.7392i 1.25770i
\(223\) − 4.21515i − 0.282267i −0.989991 0.141134i \(-0.954925\pi\)
0.989991 0.141134i \(-0.0450747\pi\)
\(224\) 4.68293 0.312892
\(225\) 0 0
\(226\) −15.4904 −1.03040
\(227\) 15.4214i 1.02355i 0.859118 + 0.511777i \(0.171013\pi\)
−0.859118 + 0.511777i \(0.828987\pi\)
\(228\) − 8.02777i − 0.531652i
\(229\) 24.2530 1.60269 0.801343 0.598205i \(-0.204119\pi\)
0.801343 + 0.598205i \(0.204119\pi\)
\(230\) 0 0
\(231\) 0.531625 0.0349783
\(232\) 1.24995i 0.0820634i
\(233\) − 5.33653i − 0.349608i −0.984603 0.174804i \(-0.944071\pi\)
0.984603 0.174804i \(-0.0559292\pi\)
\(234\) 6.15883 0.402615
\(235\) 0 0
\(236\) 11.4023 0.742229
\(237\) − 5.39363i − 0.350354i
\(238\) − 7.62207i − 0.494066i
\(239\) −16.0202 −1.03626 −0.518132 0.855301i \(-0.673372\pi\)
−0.518132 + 0.855301i \(0.673372\pi\)
\(240\) 0 0
\(241\) −6.60741 −0.425620 −0.212810 0.977094i \(-0.568262\pi\)
−0.212810 + 0.977094i \(0.568262\pi\)
\(242\) − 19.0551i − 1.22491i
\(243\) − 1.00000i − 0.0641500i
\(244\) 20.4494 1.30914
\(245\) 0 0
\(246\) −20.5848 −1.31244
\(247\) − 20.6821i − 1.31597i
\(248\) 6.85726i 0.435436i
\(249\) 0.0848668 0.00537821
\(250\) 0 0
\(251\) 25.6834 1.62112 0.810560 0.585655i \(-0.199163\pi\)
0.810560 + 0.585655i \(0.199163\pi\)
\(252\) 0.957567i 0.0603210i
\(253\) − 6.91650i − 0.434837i
\(254\) 8.21657 0.515553
\(255\) 0 0
\(256\) 20.9033 1.30645
\(257\) 7.93538i 0.494995i 0.968888 + 0.247498i \(0.0796083\pi\)
−0.968888 + 0.247498i \(0.920392\pi\)
\(258\) − 18.4581i − 1.14915i
\(259\) −7.50627 −0.466417
\(260\) 0 0
\(261\) −1.00000 −0.0618984
\(262\) − 26.8981i − 1.66177i
\(263\) 23.5618i 1.45288i 0.687228 + 0.726441i \(0.258827\pi\)
−0.687228 + 0.726441i \(0.741173\pi\)
\(264\) 0.911372 0.0560911
\(265\) 0 0
\(266\) 8.11263 0.497418
\(267\) 4.63674i 0.283764i
\(268\) − 3.35139i − 0.204719i
\(269\) 10.6645 0.650226 0.325113 0.945675i \(-0.394598\pi\)
0.325113 + 0.945675i \(0.394598\pi\)
\(270\) 0 0
\(271\) −3.30876 −0.200993 −0.100497 0.994937i \(-0.532043\pi\)
−0.100497 + 0.994937i \(0.532043\pi\)
\(272\) − 28.1513i − 1.70692i
\(273\) 2.46700i 0.149310i
\(274\) −9.44917 −0.570845
\(275\) 0 0
\(276\) 12.4581 0.749887
\(277\) 0.0469761i 0.00282252i 0.999999 + 0.00141126i \(0.000449218\pi\)
−0.999999 + 0.00141126i \(0.999551\pi\)
\(278\) 2.10828i 0.126447i
\(279\) −5.48602 −0.328439
\(280\) 0 0
\(281\) 29.8037 1.77794 0.888969 0.457967i \(-0.151422\pi\)
0.888969 + 0.457967i \(0.151422\pi\)
\(282\) 3.45390i 0.205677i
\(283\) 5.79498i 0.344476i 0.985055 + 0.172238i \(0.0550998\pi\)
−0.985055 + 0.172238i \(0.944900\pi\)
\(284\) 6.34542 0.376532
\(285\) 0 0
\(286\) −4.49056 −0.265533
\(287\) − 8.24552i − 0.486717i
\(288\) 6.42266i 0.378459i
\(289\) −15.9822 −0.940127
\(290\) 0 0
\(291\) −1.30377 −0.0764286
\(292\) − 8.26711i − 0.483796i
\(293\) 2.99624i 0.175042i 0.996163 + 0.0875211i \(0.0278945\pi\)
−0.996163 + 0.0875211i \(0.972105\pi\)
\(294\) 11.7741 0.686677
\(295\) 0 0
\(296\) −12.8681 −0.747943
\(297\) 0.729126i 0.0423082i
\(298\) − 12.1632i − 0.704594i
\(299\) 32.0960 1.85616
\(300\) 0 0
\(301\) 7.39363 0.426162
\(302\) 36.8150i 2.11847i
\(303\) 5.35574i 0.307679i
\(304\) 29.9631 1.71850
\(305\) 0 0
\(306\) 10.4537 0.597599
\(307\) − 4.05554i − 0.231462i −0.993281 0.115731i \(-0.963079\pi\)
0.993281 0.115731i \(-0.0369210\pi\)
\(308\) − 0.698187i − 0.0397829i
\(309\) 16.7670 0.953842
\(310\) 0 0
\(311\) 13.6734 0.775347 0.387674 0.921797i \(-0.373279\pi\)
0.387674 + 0.921797i \(0.373279\pi\)
\(312\) 4.22922i 0.239433i
\(313\) − 4.25200i − 0.240337i −0.992753 0.120169i \(-0.961656\pi\)
0.992753 0.120169i \(-0.0383435\pi\)
\(314\) −43.4416 −2.45155
\(315\) 0 0
\(316\) −7.08350 −0.398478
\(317\) 4.28476i 0.240656i 0.992734 + 0.120328i \(0.0383946\pi\)
−0.992734 + 0.120328i \(0.961605\pi\)
\(318\) 14.8176i 0.830928i
\(319\) 0.729126 0.0408232
\(320\) 0 0
\(321\) −7.30876 −0.407935
\(322\) 12.5898i 0.701600i
\(323\) − 35.1049i − 1.95329i
\(324\) −1.31331 −0.0729615
\(325\) 0 0
\(326\) 3.52287 0.195114
\(327\) − 8.21515i − 0.454299i
\(328\) − 14.1354i − 0.780497i
\(329\) −1.38351 −0.0762753
\(330\) 0 0
\(331\) 3.61772 0.198848 0.0994240 0.995045i \(-0.468300\pi\)
0.0994240 + 0.995045i \(0.468300\pi\)
\(332\) − 0.111456i − 0.00611695i
\(333\) − 10.2949i − 0.564156i
\(334\) −22.5757 −1.23529
\(335\) 0 0
\(336\) −3.57406 −0.194981
\(337\) − 23.4442i − 1.27709i −0.769586 0.638543i \(-0.779538\pi\)
0.769586 0.638543i \(-0.220462\pi\)
\(338\) 2.82479i 0.153648i
\(339\) 8.51003 0.462201
\(340\) 0 0
\(341\) 4.00000 0.216612
\(342\) 11.1265i 0.601653i
\(343\) 9.82014i 0.530238i
\(344\) 12.6750 0.683391
\(345\) 0 0
\(346\) 39.5200 2.12461
\(347\) − 20.8226i − 1.11781i −0.829231 0.558907i \(-0.811221\pi\)
0.829231 0.558907i \(-0.188779\pi\)
\(348\) 1.31331i 0.0704007i
\(349\) 18.0555 0.966491 0.483245 0.875485i \(-0.339458\pi\)
0.483245 + 0.875485i \(0.339458\pi\)
\(350\) 0 0
\(351\) −3.38351 −0.180598
\(352\) − 4.68293i − 0.249601i
\(353\) − 14.3645i − 0.764545i −0.924050 0.382272i \(-0.875142\pi\)
0.924050 0.382272i \(-0.124858\pi\)
\(354\) −15.8037 −0.839956
\(355\) 0 0
\(356\) 6.08946 0.322741
\(357\) 4.18738i 0.221620i
\(358\) − 11.0760i − 0.585383i
\(359\) −6.20502 −0.327489 −0.163744 0.986503i \(-0.552357\pi\)
−0.163744 + 0.986503i \(0.552357\pi\)
\(360\) 0 0
\(361\) 18.3643 0.966542
\(362\) 12.3361i 0.648370i
\(363\) 10.4684i 0.549447i
\(364\) 3.23993 0.169819
\(365\) 0 0
\(366\) −28.3429 −1.48151
\(367\) − 18.1975i − 0.949902i −0.880012 0.474951i \(-0.842466\pi\)
0.880012 0.474951i \(-0.157534\pi\)
\(368\) 46.4989i 2.42392i
\(369\) 11.3088 0.588711
\(370\) 0 0
\(371\) −5.93538 −0.308150
\(372\) 7.20483i 0.373553i
\(373\) 34.8606i 1.80501i 0.430677 + 0.902506i \(0.358275\pi\)
−0.430677 + 0.902506i \(0.641725\pi\)
\(374\) −7.62207 −0.394128
\(375\) 0 0
\(376\) −2.37177 −0.122315
\(377\) 3.38351i 0.174260i
\(378\) − 1.32719i − 0.0682633i
\(379\) 6.41005 0.329262 0.164631 0.986355i \(-0.447357\pi\)
0.164631 + 0.986355i \(0.447357\pi\)
\(380\) 0 0
\(381\) −4.51398 −0.231258
\(382\) − 34.5201i − 1.76620i
\(383\) 20.7202i 1.05875i 0.848387 + 0.529376i \(0.177574\pi\)
−0.848387 + 0.529376i \(0.822426\pi\)
\(384\) −9.41020 −0.480212
\(385\) 0 0
\(386\) −6.73055 −0.342576
\(387\) 10.1404i 0.515466i
\(388\) 1.71226i 0.0869266i
\(389\) 24.7581 1.25528 0.627642 0.778502i \(-0.284020\pi\)
0.627642 + 0.778502i \(0.284020\pi\)
\(390\) 0 0
\(391\) 54.4783 2.75509
\(392\) 8.08516i 0.408362i
\(393\) 14.7771i 0.745408i
\(394\) 37.2756 1.87792
\(395\) 0 0
\(396\) 0.957567 0.0481195
\(397\) 12.7468i 0.639742i 0.947461 + 0.319871i \(0.103640\pi\)
−0.947461 + 0.319871i \(0.896360\pi\)
\(398\) 19.7347i 0.989211i
\(399\) −4.45688 −0.223123
\(400\) 0 0
\(401\) 28.1959 1.40804 0.704019 0.710181i \(-0.251387\pi\)
0.704019 + 0.710181i \(0.251387\pi\)
\(402\) 4.64504i 0.231674i
\(403\) 18.5620i 0.924639i
\(404\) 7.03373 0.349941
\(405\) 0 0
\(406\) −1.32719 −0.0658674
\(407\) 7.50627i 0.372072i
\(408\) 7.17849i 0.355388i
\(409\) 24.0112 1.18728 0.593638 0.804732i \(-0.297691\pi\)
0.593638 + 0.804732i \(0.297691\pi\)
\(410\) 0 0
\(411\) 5.19114 0.256060
\(412\) − 22.0202i − 1.08486i
\(413\) − 6.33038i − 0.311498i
\(414\) −17.2669 −0.848623
\(415\) 0 0
\(416\) 21.7311 1.06546
\(417\) − 1.15824i − 0.0567192i
\(418\) − 8.11263i − 0.396802i
\(419\) −14.7670 −0.721416 −0.360708 0.932679i \(-0.617465\pi\)
−0.360708 + 0.932679i \(0.617465\pi\)
\(420\) 0 0
\(421\) −12.7302 −0.620430 −0.310215 0.950666i \(-0.600401\pi\)
−0.310215 + 0.950666i \(0.600401\pi\)
\(422\) 3.64050i 0.177217i
\(423\) − 1.89749i − 0.0922591i
\(424\) −10.1751 −0.494147
\(425\) 0 0
\(426\) −8.79478 −0.426109
\(427\) − 11.3531i − 0.549417i
\(428\) 9.59865i 0.463968i
\(429\) 2.46700 0.119108
\(430\) 0 0
\(431\) 9.66327 0.465464 0.232732 0.972541i \(-0.425234\pi\)
0.232732 + 0.972541i \(0.425234\pi\)
\(432\) − 4.90184i − 0.235840i
\(433\) 13.1190i 0.630459i 0.949016 + 0.315229i \(0.102081\pi\)
−0.949016 + 0.315229i \(0.897919\pi\)
\(434\) −7.28100 −0.349499
\(435\) 0 0
\(436\) −10.7890 −0.516700
\(437\) 57.9846i 2.77378i
\(438\) 11.4583i 0.547496i
\(439\) 19.9253 0.950981 0.475490 0.879721i \(-0.342271\pi\)
0.475490 + 0.879721i \(0.342271\pi\)
\(440\) 0 0
\(441\) −6.46838 −0.308018
\(442\) − 35.3702i − 1.68239i
\(443\) 1.94304i 0.0923167i 0.998934 + 0.0461583i \(0.0146979\pi\)
−0.998934 + 0.0461583i \(0.985302\pi\)
\(444\) −13.5203 −0.641647
\(445\) 0 0
\(446\) 7.67262 0.363309
\(447\) 6.68215i 0.316055i
\(448\) 1.37599i 0.0650093i
\(449\) −17.0215 −0.803293 −0.401647 0.915795i \(-0.631562\pi\)
−0.401647 + 0.915795i \(0.631562\pi\)
\(450\) 0 0
\(451\) −8.24552 −0.388266
\(452\) − 11.1763i − 0.525688i
\(453\) − 20.2253i − 0.950266i
\(454\) −28.0708 −1.31743
\(455\) 0 0
\(456\) −7.64050 −0.357799
\(457\) − 15.9633i − 0.746731i −0.927684 0.373366i \(-0.878204\pi\)
0.927684 0.373366i \(-0.121796\pi\)
\(458\) 44.1466i 2.06283i
\(459\) −5.74301 −0.268061
\(460\) 0 0
\(461\) −17.6910 −0.823954 −0.411977 0.911194i \(-0.635162\pi\)
−0.411977 + 0.911194i \(0.635162\pi\)
\(462\) 0.967690i 0.0450210i
\(463\) − 2.60741i − 0.121176i −0.998163 0.0605882i \(-0.980702\pi\)
0.998163 0.0605882i \(-0.0192976\pi\)
\(464\) −4.90184 −0.227562
\(465\) 0 0
\(466\) 9.71382 0.449984
\(467\) 8.00000i 0.370196i 0.982720 + 0.185098i \(0.0592602\pi\)
−0.982720 + 0.185098i \(0.940740\pi\)
\(468\) 4.44359i 0.205405i
\(469\) −1.86064 −0.0859162
\(470\) 0 0
\(471\) 23.8658 1.09968
\(472\) − 10.8523i − 0.499516i
\(473\) − 7.39363i − 0.339960i
\(474\) 9.81775 0.450944
\(475\) 0 0
\(476\) 5.49931 0.252061
\(477\) − 8.14040i − 0.372723i
\(478\) − 29.1608i − 1.33379i
\(479\) −20.6833 −0.945045 −0.472523 0.881319i \(-0.656657\pi\)
−0.472523 + 0.881319i \(0.656657\pi\)
\(480\) 0 0
\(481\) −34.8328 −1.58824
\(482\) − 12.0271i − 0.547821i
\(483\) − 6.91650i − 0.314712i
\(484\) 13.7482 0.624918
\(485\) 0 0
\(486\) 1.82025 0.0825682
\(487\) 32.1035i 1.45475i 0.686240 + 0.727375i \(0.259260\pi\)
−0.686240 + 0.727375i \(0.740740\pi\)
\(488\) − 19.4629i − 0.881042i
\(489\) −1.93538 −0.0875209
\(490\) 0 0
\(491\) −40.6933 −1.83646 −0.918232 0.396044i \(-0.870383\pi\)
−0.918232 + 0.396044i \(0.870383\pi\)
\(492\) − 14.8519i − 0.669575i
\(493\) 5.74301i 0.258652i
\(494\) 37.6467 1.69380
\(495\) 0 0
\(496\) −26.8916 −1.20747
\(497\) − 3.52287i − 0.158022i
\(498\) 0.154479i 0.00692236i
\(499\) −10.8418 −0.485344 −0.242672 0.970108i \(-0.578024\pi\)
−0.242672 + 0.970108i \(0.578024\pi\)
\(500\) 0 0
\(501\) 12.4025 0.554104
\(502\) 46.7502i 2.08656i
\(503\) − 2.47727i − 0.110456i −0.998474 0.0552280i \(-0.982411\pi\)
0.998474 0.0552280i \(-0.0175886\pi\)
\(504\) 0.911372 0.0405958
\(505\) 0 0
\(506\) 12.5898 0.559683
\(507\) − 1.55187i − 0.0689210i
\(508\) 5.92824i 0.263023i
\(509\) 24.3101 1.07753 0.538764 0.842457i \(-0.318891\pi\)
0.538764 + 0.842457i \(0.318891\pi\)
\(510\) 0 0
\(511\) −4.58976 −0.203039
\(512\) 19.2287i 0.849798i
\(513\) − 6.11263i − 0.269879i
\(514\) −14.4444 −0.637114
\(515\) 0 0
\(516\) 13.3175 0.586269
\(517\) 1.38351i 0.0608466i
\(518\) − 13.6633i − 0.600330i
\(519\) −21.7113 −0.953020
\(520\) 0 0
\(521\) 25.2997 1.10840 0.554200 0.832384i \(-0.313024\pi\)
0.554200 + 0.832384i \(0.313024\pi\)
\(522\) − 1.82025i − 0.0796701i
\(523\) 39.2277i 1.71531i 0.514228 + 0.857653i \(0.328078\pi\)
−0.514228 + 0.857653i \(0.671922\pi\)
\(524\) 19.4069 0.847796
\(525\) 0 0
\(526\) −42.8884 −1.87002
\(527\) 31.5063i 1.37243i
\(528\) 3.57406i 0.155541i
\(529\) −66.9846 −2.91237
\(530\) 0 0
\(531\) 8.68215 0.376773
\(532\) 5.85325i 0.253771i
\(533\) − 38.2633i − 1.65737i
\(534\) −8.44002 −0.365235
\(535\) 0 0
\(536\) −3.18972 −0.137775
\(537\) 6.08487i 0.262581i
\(538\) 19.4121i 0.836913i
\(539\) 4.71626 0.203144
\(540\) 0 0
\(541\) 6.28080 0.270033 0.135016 0.990843i \(-0.456891\pi\)
0.135016 + 0.990843i \(0.456891\pi\)
\(542\) − 6.02278i − 0.258700i
\(543\) − 6.77714i − 0.290835i
\(544\) 36.8854 1.58145
\(545\) 0 0
\(546\) −4.49056 −0.192178
\(547\) 20.2809i 0.867151i 0.901117 + 0.433575i \(0.142748\pi\)
−0.901117 + 0.433575i \(0.857252\pi\)
\(548\) − 6.81756i − 0.291232i
\(549\) 15.5709 0.664549
\(550\) 0 0
\(551\) −6.11263 −0.260407
\(552\) − 11.8571i − 0.504670i
\(553\) 3.93264i 0.167233i
\(554\) −0.0855081 −0.00363289
\(555\) 0 0
\(556\) −1.52112 −0.0645100
\(557\) − 16.3252i − 0.691720i −0.938286 0.345860i \(-0.887587\pi\)
0.938286 0.345860i \(-0.112413\pi\)
\(558\) − 9.98592i − 0.422738i
\(559\) 34.3101 1.45116
\(560\) 0 0
\(561\) 4.18738 0.176791
\(562\) 54.2501i 2.28840i
\(563\) − 12.8037i − 0.539613i −0.962915 0.269806i \(-0.913040\pi\)
0.962915 0.269806i \(-0.0869597\pi\)
\(564\) −2.49199 −0.104932
\(565\) 0 0
\(566\) −10.5483 −0.443378
\(567\) 0.729126i 0.0306204i
\(568\) − 6.03931i − 0.253404i
\(569\) −9.35574 −0.392213 −0.196107 0.980583i \(-0.562830\pi\)
−0.196107 + 0.980583i \(0.562830\pi\)
\(570\) 0 0
\(571\) −27.0125 −1.13044 −0.565220 0.824940i \(-0.691209\pi\)
−0.565220 + 0.824940i \(0.691209\pi\)
\(572\) − 3.23993i − 0.135468i
\(573\) 18.9645i 0.792254i
\(574\) 15.0089 0.626459
\(575\) 0 0
\(576\) −1.88717 −0.0786322
\(577\) 25.9859i 1.08181i 0.841084 + 0.540904i \(0.181918\pi\)
−0.841084 + 0.540904i \(0.818082\pi\)
\(578\) − 29.0915i − 1.21005i
\(579\) 3.69760 0.153667
\(580\) 0 0
\(581\) −0.0618786 −0.00256716
\(582\) − 2.37319i − 0.0983720i
\(583\) 5.93538i 0.245818i
\(584\) −7.86830 −0.325592
\(585\) 0 0
\(586\) −5.45390 −0.225299
\(587\) − 19.6467i − 0.810905i −0.914116 0.405452i \(-0.867114\pi\)
0.914116 0.405452i \(-0.132886\pi\)
\(588\) 8.49496i 0.350326i
\(589\) −33.5340 −1.38175
\(590\) 0 0
\(591\) −20.4783 −0.842365
\(592\) − 50.4638i − 2.07405i
\(593\) 26.1682i 1.07460i 0.843392 + 0.537299i \(0.180555\pi\)
−0.843392 + 0.537299i \(0.819445\pi\)
\(594\) −1.32719 −0.0544553
\(595\) 0 0
\(596\) 8.77572 0.359467
\(597\) − 10.8418i − 0.443724i
\(598\) 58.4228i 2.38908i
\(599\) 3.45565 0.141194 0.0705970 0.997505i \(-0.477510\pi\)
0.0705970 + 0.997505i \(0.477510\pi\)
\(600\) 0 0
\(601\) 13.5063 0.550932 0.275466 0.961311i \(-0.411168\pi\)
0.275466 + 0.961311i \(0.411168\pi\)
\(602\) 13.4583i 0.548517i
\(603\) − 2.55187i − 0.103920i
\(604\) −26.5620 −1.08079
\(605\) 0 0
\(606\) −9.74878 −0.396017
\(607\) − 12.0773i − 0.490204i −0.969497 0.245102i \(-0.921178\pi\)
0.969497 0.245102i \(-0.0788215\pi\)
\(608\) 39.2594i 1.59218i
\(609\) 0.729126 0.0295457
\(610\) 0 0
\(611\) −6.42017 −0.259732
\(612\) 7.54234i 0.304881i
\(613\) − 19.9935i − 0.807531i −0.914863 0.403765i \(-0.867701\pi\)
0.914863 0.403765i \(-0.132299\pi\)
\(614\) 7.38209 0.297917
\(615\) 0 0
\(616\) −0.664505 −0.0267737
\(617\) 31.9784i 1.28740i 0.765277 + 0.643701i \(0.222602\pi\)
−0.765277 + 0.643701i \(0.777398\pi\)
\(618\) 30.5201i 1.22770i
\(619\) 33.0862 1.32985 0.664924 0.746911i \(-0.268464\pi\)
0.664924 + 0.746911i \(0.268464\pi\)
\(620\) 0 0
\(621\) 9.48602 0.380661
\(622\) 24.8890i 0.997958i
\(623\) − 3.38077i − 0.135448i
\(624\) −16.5854 −0.663948
\(625\) 0 0
\(626\) 7.73970 0.309341
\(627\) 4.45688i 0.177991i
\(628\) − 31.3431i − 1.25073i
\(629\) −59.1236 −2.35741
\(630\) 0 0
\(631\) −3.02654 −0.120485 −0.0602423 0.998184i \(-0.519187\pi\)
−0.0602423 + 0.998184i \(0.519187\pi\)
\(632\) 6.74178i 0.268174i
\(633\) − 2.00000i − 0.0794929i
\(634\) −7.79933 −0.309751
\(635\) 0 0
\(636\) −10.6908 −0.423920
\(637\) 21.8858i 0.867147i
\(638\) 1.32719i 0.0525440i
\(639\) 4.83164 0.191137
\(640\) 0 0
\(641\) 14.0747 0.555919 0.277959 0.960593i \(-0.410342\pi\)
0.277959 + 0.960593i \(0.410342\pi\)
\(642\) − 13.3038i − 0.525058i
\(643\) 4.25044i 0.167621i 0.996482 + 0.0838104i \(0.0267090\pi\)
−0.996482 + 0.0838104i \(0.973291\pi\)
\(644\) −9.08350 −0.357940
\(645\) 0 0
\(646\) 63.8997 2.51410
\(647\) 43.0185i 1.69123i 0.533792 + 0.845616i \(0.320766\pi\)
−0.533792 + 0.845616i \(0.679234\pi\)
\(648\) 1.24995i 0.0491027i
\(649\) −6.33038 −0.248489
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) 2.54175i 0.0995425i
\(653\) 18.5175i 0.724648i 0.932052 + 0.362324i \(0.118017\pi\)
−0.932052 + 0.362324i \(0.881983\pi\)
\(654\) 14.9536 0.584733
\(655\) 0 0
\(656\) 55.4337 2.16432
\(657\) − 6.29488i − 0.245587i
\(658\) − 2.51833i − 0.0981747i
\(659\) −6.19736 −0.241415 −0.120707 0.992688i \(-0.538516\pi\)
−0.120707 + 0.992688i \(0.538516\pi\)
\(660\) 0 0
\(661\) −29.0936 −1.13161 −0.565805 0.824539i \(-0.691435\pi\)
−0.565805 + 0.824539i \(0.691435\pi\)
\(662\) 6.58516i 0.255939i
\(663\) 19.4315i 0.754658i
\(664\) −0.106079 −0.00411668
\(665\) 0 0
\(666\) 18.7392 0.726131
\(667\) − 9.48602i − 0.367300i
\(668\) − 16.2883i − 0.630214i
\(669\) −4.21515 −0.162967
\(670\) 0 0
\(671\) −11.3531 −0.438283
\(672\) − 4.68293i − 0.180648i
\(673\) − 36.6288i − 1.41194i −0.708243 0.705969i \(-0.750512\pi\)
0.708243 0.705969i \(-0.249488\pi\)
\(674\) 42.6742 1.64375
\(675\) 0 0
\(676\) −2.03808 −0.0783878
\(677\) − 25.9330i − 0.996686i −0.866980 0.498343i \(-0.833942\pi\)
0.866980 0.498343i \(-0.166058\pi\)
\(678\) 15.4904i 0.594904i
\(679\) 0.950615 0.0364813
\(680\) 0 0
\(681\) 15.4214 0.590949
\(682\) 7.28100i 0.278804i
\(683\) 22.4403i 0.858653i 0.903149 + 0.429327i \(0.141249\pi\)
−0.903149 + 0.429327i \(0.858751\pi\)
\(684\) −8.02777 −0.306949
\(685\) 0 0
\(686\) −17.8751 −0.682475
\(687\) − 24.2530i − 0.925311i
\(688\) 49.7066i 1.89505i
\(689\) −27.5431 −1.04931
\(690\) 0 0
\(691\) 16.3000 0.620082 0.310041 0.950723i \(-0.399657\pi\)
0.310041 + 0.950723i \(0.399657\pi\)
\(692\) 28.5136i 1.08392i
\(693\) − 0.531625i − 0.0201948i
\(694\) 37.9022 1.43875
\(695\) 0 0
\(696\) 1.24995 0.0473793
\(697\) − 64.9463i − 2.46002i
\(698\) 32.8656i 1.24398i
\(699\) −5.33653 −0.201846
\(700\) 0 0
\(701\) −10.9811 −0.414751 −0.207376 0.978261i \(-0.566492\pi\)
−0.207376 + 0.978261i \(0.566492\pi\)
\(702\) − 6.15883i − 0.232450i
\(703\) − 62.9288i − 2.37341i
\(704\) 1.37599 0.0518595
\(705\) 0 0
\(706\) 26.1470 0.984054
\(707\) − 3.90501i − 0.146863i
\(708\) − 11.4023i − 0.428526i
\(709\) 33.0681 1.24190 0.620949 0.783851i \(-0.286748\pi\)
0.620949 + 0.783851i \(0.286748\pi\)
\(710\) 0 0
\(711\) −5.39363 −0.202277
\(712\) − 5.79570i − 0.217203i
\(713\) − 52.0405i − 1.94893i
\(714\) −7.62207 −0.285249
\(715\) 0 0
\(716\) 7.99130 0.298649
\(717\) 16.0202i 0.598287i
\(718\) − 11.2947i − 0.421514i
\(719\) 44.0289 1.64200 0.821001 0.570926i \(-0.193416\pi\)
0.821001 + 0.570926i \(0.193416\pi\)
\(720\) 0 0
\(721\) −12.2253 −0.455293
\(722\) 33.4276i 1.24405i
\(723\) 6.60741i 0.245732i
\(724\) −8.90047 −0.330783
\(725\) 0 0
\(726\) −19.0551 −0.707199
\(727\) − 20.2708i − 0.751803i −0.926660 0.375902i \(-0.877333\pi\)
0.926660 0.375902i \(-0.122667\pi\)
\(728\) − 3.08364i − 0.114287i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 58.2364 2.15395
\(732\) − 20.4494i − 0.755830i
\(733\) 45.9138i 1.69586i 0.530105 + 0.847932i \(0.322153\pi\)
−0.530105 + 0.847932i \(0.677847\pi\)
\(734\) 33.1240 1.22263
\(735\) 0 0
\(736\) −60.9255 −2.24574
\(737\) 1.86064i 0.0685374i
\(738\) 20.5848i 0.757736i
\(739\) −38.1884 −1.40478 −0.702392 0.711791i \(-0.747885\pi\)
−0.702392 + 0.711791i \(0.747885\pi\)
\(740\) 0 0
\(741\) −20.6821 −0.759778
\(742\) − 10.8039i − 0.396623i
\(743\) − 34.4773i − 1.26485i −0.774622 0.632424i \(-0.782060\pi\)
0.774622 0.632424i \(-0.217940\pi\)
\(744\) 6.85726 0.251399
\(745\) 0 0
\(746\) −63.4550 −2.32325
\(747\) − 0.0848668i − 0.00310511i
\(748\) − 5.49931i − 0.201075i
\(749\) 5.32901 0.194718
\(750\) 0 0
\(751\) 41.7113 1.52207 0.761033 0.648713i \(-0.224692\pi\)
0.761033 + 0.648713i \(0.224692\pi\)
\(752\) − 9.30118i − 0.339179i
\(753\) − 25.6834i − 0.935954i
\(754\) −6.15883 −0.224291
\(755\) 0 0
\(756\) 0.957567 0.0348264
\(757\) 13.5659i 0.493061i 0.969135 + 0.246530i \(0.0792905\pi\)
−0.969135 + 0.246530i \(0.920709\pi\)
\(758\) 11.6679i 0.423797i
\(759\) −6.91650 −0.251053
\(760\) 0 0
\(761\) 3.27982 0.118893 0.0594467 0.998231i \(-0.481066\pi\)
0.0594467 + 0.998231i \(0.481066\pi\)
\(762\) − 8.21657i − 0.297655i
\(763\) 5.98988i 0.216848i
\(764\) 24.9062 0.901076
\(765\) 0 0
\(766\) −37.7159 −1.36273
\(767\) − 29.3761i − 1.06071i
\(768\) − 20.9033i − 0.754281i
\(769\) 35.5527 1.28206 0.641032 0.767514i \(-0.278507\pi\)
0.641032 + 0.767514i \(0.278507\pi\)
\(770\) 0 0
\(771\) 7.93538 0.285786
\(772\) − 4.85608i − 0.174774i
\(773\) − 0.965870i − 0.0347399i −0.999849 0.0173700i \(-0.994471\pi\)
0.999849 0.0173700i \(-0.00552931\pi\)
\(774\) −18.4581 −0.663461
\(775\) 0 0
\(776\) 1.62965 0.0585012
\(777\) 7.50627i 0.269286i
\(778\) 45.0659i 1.61569i
\(779\) 69.1263 2.47671
\(780\) 0 0
\(781\) −3.52287 −0.126058
\(782\) 99.1641i 3.54610i
\(783\) 1.00000i 0.0357371i
\(784\) −31.7069 −1.13239
\(785\) 0 0
\(786\) −26.8981 −0.959423
\(787\) − 15.0756i − 0.537387i −0.963226 0.268693i \(-0.913408\pi\)
0.963226 0.268693i \(-0.0865918\pi\)
\(788\) 26.8943i 0.958070i
\(789\) 23.5618 0.838822
\(790\) 0 0
\(791\) −6.20488 −0.220620
\(792\) − 0.911372i − 0.0323842i
\(793\) − 52.6842i − 1.87087i
\(794\) −23.2023 −0.823419
\(795\) 0 0
\(796\) −14.2386 −0.504673
\(797\) − 27.9884i − 0.991399i −0.868494 0.495700i \(-0.834912\pi\)
0.868494 0.495700i \(-0.165088\pi\)
\(798\) − 8.11263i − 0.287184i
\(799\) −10.8973 −0.385519
\(800\) 0 0
\(801\) 4.63674 0.163831
\(802\) 51.3236i 1.81230i
\(803\) 4.58976i 0.161969i
\(804\) −3.35139 −0.118194
\(805\) 0 0
\(806\) −33.7875 −1.19011
\(807\) − 10.6645i − 0.375408i
\(808\) − 6.69442i − 0.235509i
\(809\) −10.0950 −0.354921 −0.177460 0.984128i \(-0.556788\pi\)
−0.177460 + 0.984128i \(0.556788\pi\)
\(810\) 0 0
\(811\) 16.3555 0.574321 0.287160 0.957882i \(-0.407289\pi\)
0.287160 + 0.957882i \(0.407289\pi\)
\(812\) − 0.957567i − 0.0336040i
\(813\) 3.30876i 0.116043i
\(814\) −13.6633 −0.478898
\(815\) 0 0
\(816\) −28.1513 −0.985493
\(817\) 61.9846i 2.16857i
\(818\) 43.7063i 1.52815i
\(819\) 2.46700 0.0862041
\(820\) 0 0
\(821\) 20.7468 0.724067 0.362034 0.932165i \(-0.382083\pi\)
0.362034 + 0.932165i \(0.382083\pi\)
\(822\) 9.44917i 0.329578i
\(823\) 33.4619i 1.16641i 0.812326 + 0.583204i \(0.198201\pi\)
−0.812326 + 0.583204i \(0.801799\pi\)
\(824\) −20.9580 −0.730105
\(825\) 0 0
\(826\) 11.5229 0.400932
\(827\) 20.3363i 0.707164i 0.935404 + 0.353582i \(0.115036\pi\)
−0.935404 + 0.353582i \(0.884964\pi\)
\(828\) − 12.4581i − 0.432947i
\(829\) −18.4858 −0.642039 −0.321020 0.947073i \(-0.604026\pi\)
−0.321020 + 0.947073i \(0.604026\pi\)
\(830\) 0 0
\(831\) 0.0469761 0.00162958
\(832\) 6.38526i 0.221369i
\(833\) 37.1479i 1.28710i
\(834\) 2.10828 0.0730039
\(835\) 0 0
\(836\) 5.85325 0.202439
\(837\) 5.48602i 0.189625i
\(838\) − 26.8797i − 0.928542i
\(839\) 22.3571 0.771853 0.385927 0.922529i \(-0.373882\pi\)
0.385927 + 0.922529i \(0.373882\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) − 23.1721i − 0.798562i
\(843\) − 29.8037i − 1.02649i
\(844\) −2.62661 −0.0904118
\(845\) 0 0
\(846\) 3.45390 0.118748
\(847\) − 7.63277i − 0.262265i
\(848\) − 39.9029i − 1.37027i
\(849\) 5.79498 0.198883
\(850\) 0 0
\(851\) 97.6574 3.34765
\(852\) − 6.34542i − 0.217391i
\(853\) 16.1950i 0.554505i 0.960797 + 0.277253i \(0.0894239\pi\)
−0.960797 + 0.277253i \(0.910576\pi\)
\(854\) 20.6655 0.707160
\(855\) 0 0
\(856\) 9.13560 0.312249
\(857\) − 11.4785i − 0.392098i −0.980594 0.196049i \(-0.937189\pi\)
0.980594 0.196049i \(-0.0628112\pi\)
\(858\) 4.49056i 0.153305i
\(859\) 21.6744 0.739522 0.369761 0.929127i \(-0.379440\pi\)
0.369761 + 0.929127i \(0.379440\pi\)
\(860\) 0 0
\(861\) −8.24552 −0.281006
\(862\) 17.5896i 0.599103i
\(863\) − 30.7226i − 1.04581i −0.852391 0.522905i \(-0.824848\pi\)
0.852391 0.522905i \(-0.175152\pi\)
\(864\) 6.42266 0.218503
\(865\) 0 0
\(866\) −23.8798 −0.811470
\(867\) 15.9822i 0.542783i
\(868\) − 5.25323i − 0.178306i
\(869\) 3.93264 0.133406
\(870\) 0 0
\(871\) −8.63428 −0.292561
\(872\) 10.2685i 0.347737i
\(873\) 1.30377i 0.0441260i
\(874\) −105.546 −3.57016
\(875\) 0 0
\(876\) −8.26711 −0.279320
\(877\) − 34.2908i − 1.15792i −0.815357 0.578959i \(-0.803459\pi\)
0.815357 0.578959i \(-0.196541\pi\)
\(878\) 36.2689i 1.22402i
\(879\) 2.99624 0.101061
\(880\) 0 0
\(881\) 12.8620 0.433332 0.216666 0.976246i \(-0.430482\pi\)
0.216666 + 0.976246i \(0.430482\pi\)
\(882\) − 11.7741i − 0.396453i
\(883\) 1.37495i 0.0462707i 0.999732 + 0.0231354i \(0.00736487\pi\)
−0.999732 + 0.0231354i \(0.992635\pi\)
\(884\) 25.5196 0.858316
\(885\) 0 0
\(886\) −3.53682 −0.118822
\(887\) 27.1710i 0.912312i 0.889900 + 0.456156i \(0.150774\pi\)
−0.889900 + 0.456156i \(0.849226\pi\)
\(888\) 12.8681i 0.431825i
\(889\) 3.29126 0.110385
\(890\) 0 0
\(891\) 0.729126 0.0244266
\(892\) 5.53578i 0.185352i
\(893\) − 11.5987i − 0.388134i
\(894\) −12.1632 −0.406798
\(895\) 0 0
\(896\) 6.86122 0.229217
\(897\) − 32.0960i − 1.07166i
\(898\) − 30.9833i − 1.03393i
\(899\) 5.48602 0.182969
\(900\) 0 0
\(901\) −46.7504 −1.55748
\(902\) − 15.0089i − 0.499741i
\(903\) − 7.39363i − 0.246045i
\(904\) −10.6371 −0.353785
\(905\) 0 0
\(906\) 36.8150 1.22310
\(907\) − 0.327640i − 0.0108791i −0.999985 0.00543955i \(-0.998269\pi\)
0.999985 0.00543955i \(-0.00173147\pi\)
\(908\) − 20.2530i − 0.672121i
\(909\) 5.35574 0.177639
\(910\) 0 0
\(911\) 43.0377 1.42590 0.712951 0.701214i \(-0.247358\pi\)
0.712951 + 0.701214i \(0.247358\pi\)
\(912\) − 29.9631i − 0.992179i
\(913\) 0.0618786i 0.00204788i
\(914\) 29.0572 0.961125
\(915\) 0 0
\(916\) −31.8517 −1.05241
\(917\) − 10.7744i − 0.355802i
\(918\) − 10.4537i − 0.345024i
\(919\) 28.2061 0.930432 0.465216 0.885197i \(-0.345977\pi\)
0.465216 + 0.885197i \(0.345977\pi\)
\(920\) 0 0
\(921\) −4.05554 −0.133634
\(922\) − 32.2021i − 1.06052i
\(923\) − 16.3479i − 0.538097i
\(924\) −0.698187 −0.0229687
\(925\) 0 0
\(926\) 4.74613 0.155967
\(927\) − 16.7670i − 0.550701i
\(928\) − 6.42266i − 0.210834i
\(929\) 14.4505 0.474107 0.237053 0.971497i \(-0.423818\pi\)
0.237053 + 0.971497i \(0.423818\pi\)
\(930\) 0 0
\(931\) −39.5388 −1.29583
\(932\) 7.00851i 0.229571i
\(933\) − 13.6734i − 0.447647i
\(934\) −14.5620 −0.476483
\(935\) 0 0
\(936\) 4.22922 0.138236
\(937\) − 13.9657i − 0.456241i −0.973633 0.228121i \(-0.926742\pi\)
0.973633 0.228121i \(-0.0732580\pi\)
\(938\) − 3.38682i − 0.110584i
\(939\) −4.25200 −0.138759
\(940\) 0 0
\(941\) −40.6262 −1.32438 −0.662189 0.749337i \(-0.730372\pi\)
−0.662189 + 0.749337i \(0.730372\pi\)
\(942\) 43.4416i 1.41541i
\(943\) 107.275i 3.49336i
\(944\) 42.5585 1.38516
\(945\) 0 0
\(946\) 13.4583 0.437566
\(947\) 20.4495i 0.664519i 0.943188 + 0.332260i \(0.107811\pi\)
−0.943188 + 0.332260i \(0.892189\pi\)
\(948\) 7.08350i 0.230061i
\(949\) −21.2988 −0.691388
\(950\) 0 0
\(951\) 4.28476 0.138943
\(952\) − 5.23402i − 0.169636i
\(953\) − 51.5527i − 1.66996i −0.550283 0.834978i \(-0.685480\pi\)
0.550283 0.834978i \(-0.314520\pi\)
\(954\) 14.8176 0.479736
\(955\) 0 0
\(956\) 21.0395 0.680466
\(957\) − 0.729126i − 0.0235693i
\(958\) − 37.6488i − 1.21638i
\(959\) −3.78499 −0.122224
\(960\) 0 0
\(961\) −0.903587 −0.0291480
\(962\) − 63.4044i − 2.04424i
\(963\) 7.30876i 0.235522i
\(964\) 8.67756 0.279485
\(965\) 0 0
\(966\) 12.5898 0.405069
\(967\) − 16.5429i − 0.531985i −0.963975 0.265992i \(-0.914300\pi\)
0.963975 0.265992i \(-0.0856996\pi\)
\(968\) − 13.0850i − 0.420567i
\(969\) −35.1049 −1.12773
\(970\) 0 0
\(971\) 32.5417 1.04431 0.522157 0.852849i \(-0.325127\pi\)
0.522157 + 0.852849i \(0.325127\pi\)
\(972\) 1.31331i 0.0421244i
\(973\) 0.844503i 0.0270735i
\(974\) −58.4365 −1.87242
\(975\) 0 0
\(976\) 76.3260 2.44313
\(977\) 38.7111i 1.23848i 0.785203 + 0.619239i \(0.212559\pi\)
−0.785203 + 0.619239i \(0.787441\pi\)
\(978\) − 3.52287i − 0.112649i
\(979\) −3.38077 −0.108050
\(980\) 0 0
\(981\) −8.21515 −0.262289
\(982\) − 74.0720i − 2.36373i
\(983\) − 31.7315i − 1.01208i −0.862510 0.506039i \(-0.831109\pi\)
0.862510 0.506039i \(-0.168891\pi\)
\(984\) −14.1354 −0.450620
\(985\) 0 0
\(986\) −10.4537 −0.332914
\(987\) 1.38351i 0.0440376i
\(988\) 27.1620i 0.864139i
\(989\) −96.1921 −3.05873
\(990\) 0 0
\(991\) −22.8620 −0.726236 −0.363118 0.931743i \(-0.618288\pi\)
−0.363118 + 0.931743i \(0.618288\pi\)
\(992\) − 35.2349i − 1.11871i
\(993\) − 3.61772i − 0.114805i
\(994\) 6.41251 0.203392
\(995\) 0 0
\(996\) −0.111456 −0.00353162
\(997\) − 40.9024i − 1.29539i −0.761898 0.647696i \(-0.775733\pi\)
0.761898 0.647696i \(-0.224267\pi\)
\(998\) − 19.7347i − 0.624691i
\(999\) −10.2949 −0.325716
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 2175.2.c.n.349.7 8
5.2 odd 4 435.2.a.j.1.2 4
5.3 odd 4 2175.2.a.v.1.3 4
5.4 even 2 inner 2175.2.c.n.349.2 8
15.2 even 4 1305.2.a.r.1.3 4
15.8 even 4 6525.2.a.bi.1.2 4
20.7 even 4 6960.2.a.co.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.2.a.j.1.2 4 5.2 odd 4
1305.2.a.r.1.3 4 15.2 even 4
2175.2.a.v.1.3 4 5.3 odd 4
2175.2.c.n.349.2 8 5.4 even 2 inner
2175.2.c.n.349.7 8 1.1 even 1 trivial
6525.2.a.bi.1.2 4 15.8 even 4
6960.2.a.co.1.3 4 20.7 even 4