Properties

Label 6525.2.a.be.1.1
Level $6525$
Weight $2$
Character 6525.1
Self dual yes
Analytic conductor $52.102$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6525,2,Mod(1,6525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6525.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6525.a (trivial)

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: yes
Analytic conductor: \(52.1023873189\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 145)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 6525.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.48119 q^{2} +0.193937 q^{4} -1.19394 q^{7} +2.67513 q^{8} +O(q^{10})\) \(q-1.48119 q^{2} +0.193937 q^{4} -1.19394 q^{7} +2.67513 q^{8} -4.15633 q^{11} -2.96239 q^{13} +1.76845 q^{14} -4.35026 q^{16} +5.50659 q^{17} -3.19394 q^{19} +6.15633 q^{22} +1.84367 q^{23} +4.38787 q^{26} -0.231548 q^{28} +1.00000 q^{29} -4.80606 q^{31} +1.09332 q^{32} -8.15633 q^{34} +9.50659 q^{37} +4.73084 q^{38} +11.2750 q^{41} +0.0303172 q^{43} -0.806063 q^{44} -2.73084 q^{46} +4.80606 q^{47} -5.57452 q^{49} -0.574515 q^{52} -1.35026 q^{53} -3.19394 q^{56} -1.48119 q^{58} -13.2750 q^{59} +8.88717 q^{61} +7.11871 q^{62} +7.08110 q^{64} -5.84367 q^{67} +1.06793 q^{68} +1.27504 q^{71} +15.2447 q^{73} -14.0811 q^{74} -0.619421 q^{76} +4.96239 q^{77} -4.93207 q^{79} -16.7005 q^{82} +4.41819 q^{83} -0.0449056 q^{86} -11.1187 q^{88} +3.61213 q^{89} +3.53690 q^{91} +0.357556 q^{92} -7.11871 q^{94} +1.38058 q^{97} +8.25694 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + q^{4} - 4 q^{7} + 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + q^{4} - 4 q^{7} + 3 q^{8} - 2 q^{11} + 2 q^{13} - 6 q^{14} - 3 q^{16} - 4 q^{17} - 10 q^{19} + 8 q^{22} + 16 q^{23} + 14 q^{26} - 12 q^{28} + 3 q^{29} - 14 q^{31} - 3 q^{32} - 14 q^{34} + 8 q^{37} - 8 q^{38} + 2 q^{41} - 2 q^{43} - 2 q^{44} + 14 q^{46} + 14 q^{47} - 5 q^{49} + 10 q^{52} + 6 q^{53} - 10 q^{56} + q^{58} - 8 q^{59} - 6 q^{61} - 11 q^{64} - 28 q^{67} + 12 q^{68} - 28 q^{71} + 16 q^{73} - 10 q^{74} - 14 q^{76} + 4 q^{77} - 6 q^{79} - 30 q^{82} + 12 q^{83} - 24 q^{86} - 12 q^{88} + 10 q^{89} - 12 q^{91} + 4 q^{92} - 8 q^{97} + 21 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.48119 −1.04736 −0.523681 0.851914i \(-0.675442\pi\)
−0.523681 + 0.851914i \(0.675442\pi\)
\(3\) 0 0
\(4\) 0.193937 0.0969683
\(5\) 0 0
\(6\) 0 0
\(7\) −1.19394 −0.451266 −0.225633 0.974212i \(-0.572445\pi\)
−0.225633 + 0.974212i \(0.572445\pi\)
\(8\) 2.67513 0.945802
\(9\) 0 0
\(10\) 0 0
\(11\) −4.15633 −1.25318 −0.626590 0.779349i \(-0.715550\pi\)
−0.626590 + 0.779349i \(0.715550\pi\)
\(12\) 0 0
\(13\) −2.96239 −0.821619 −0.410809 0.911721i \(-0.634754\pi\)
−0.410809 + 0.911721i \(0.634754\pi\)
\(14\) 1.76845 0.472639
\(15\) 0 0
\(16\) −4.35026 −1.08757
\(17\) 5.50659 1.33554 0.667772 0.744366i \(-0.267248\pi\)
0.667772 + 0.744366i \(0.267248\pi\)
\(18\) 0 0
\(19\) −3.19394 −0.732739 −0.366370 0.930469i \(-0.619399\pi\)
−0.366370 + 0.930469i \(0.619399\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 6.15633 1.31253
\(23\) 1.84367 0.384433 0.192216 0.981353i \(-0.438432\pi\)
0.192216 + 0.981353i \(0.438432\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 4.38787 0.860533
\(27\) 0 0
\(28\) −0.231548 −0.0437585
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −4.80606 −0.863194 −0.431597 0.902066i \(-0.642050\pi\)
−0.431597 + 0.902066i \(0.642050\pi\)
\(32\) 1.09332 0.193274
\(33\) 0 0
\(34\) −8.15633 −1.39880
\(35\) 0 0
\(36\) 0 0
\(37\) 9.50659 1.56287 0.781437 0.623985i \(-0.214487\pi\)
0.781437 + 0.623985i \(0.214487\pi\)
\(38\) 4.73084 0.767444
\(39\) 0 0
\(40\) 0 0
\(41\) 11.2750 1.76087 0.880433 0.474171i \(-0.157252\pi\)
0.880433 + 0.474171i \(0.157252\pi\)
\(42\) 0 0
\(43\) 0.0303172 0.00462332 0.00231166 0.999997i \(-0.499264\pi\)
0.00231166 + 0.999997i \(0.499264\pi\)
\(44\) −0.806063 −0.121519
\(45\) 0 0
\(46\) −2.73084 −0.402640
\(47\) 4.80606 0.701036 0.350518 0.936556i \(-0.386005\pi\)
0.350518 + 0.936556i \(0.386005\pi\)
\(48\) 0 0
\(49\) −5.57452 −0.796359
\(50\) 0 0
\(51\) 0 0
\(52\) −0.574515 −0.0796710
\(53\) −1.35026 −0.185473 −0.0927364 0.995691i \(-0.529561\pi\)
−0.0927364 + 0.995691i \(0.529561\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.19394 −0.426808
\(57\) 0 0
\(58\) −1.48119 −0.194490
\(59\) −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 \(0.307353\pi\)
\(62\) 7.11871 0.904078
\(63\) 0 0
\(64\) 7.08110 0.885138
\(65\) 0 0
\(66\) 0 0
\(67\) −5.84367 −0.713919 −0.356959 0.934120i \(-0.616187\pi\)
−0.356959 + 0.934120i \(0.616187\pi\)
\(68\) 1.06793 0.129505
\(69\) 0 0
\(70\) 0 0
\(71\) 1.27504 0.151319 0.0756596 0.997134i \(-0.475894\pi\)
0.0756596 + 0.997134i \(0.475894\pi\)
\(72\) 0 0
\(73\) 15.2447 1.78426 0.892130 0.451779i \(-0.149210\pi\)
0.892130 + 0.451779i \(0.149210\pi\)
\(74\) −14.0811 −1.63689
\(75\) 0 0
\(76\) −0.619421 −0.0710525
\(77\) 4.96239 0.565517
\(78\) 0 0
\(79\) −4.93207 −0.554901 −0.277451 0.960740i \(-0.589490\pi\)
−0.277451 + 0.960740i \(0.589490\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −16.7005 −1.84426
\(83\) 4.41819 0.484959 0.242480 0.970156i \(-0.422039\pi\)
0.242480 + 0.970156i \(0.422039\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −0.0449056 −0.00484230
\(87\) 0 0
\(88\) −11.1187 −1.18526
\(89\) 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
\(92\) 0.357556 0.0372778
\(93\) 0 0
\(94\) −7.11871 −0.734239
\(95\) 0 0
\(96\) 0 0
\(97\) 1.38058 0.140177 0.0700883 0.997541i \(-0.477672\pi\)
0.0700883 + 0.997541i \(0.477672\pi\)
\(98\) 8.25694 0.834077
\(99\) 0 0
\(100\) 0 0
\(101\) 13.0132 1.29486 0.647430 0.762125i \(-0.275844\pi\)
0.647430 + 0.762125i \(0.275844\pi\)
\(102\) 0 0
\(103\) −5.31994 −0.524190 −0.262095 0.965042i \(-0.584413\pi\)
−0.262095 + 0.965042i \(0.584413\pi\)
\(104\) −7.92478 −0.777088
\(105\) 0 0
\(106\) 2.00000 0.194257
\(107\) −13.8192 −1.33596 −0.667978 0.744181i \(-0.732840\pi\)
−0.667978 + 0.744181i \(0.732840\pi\)
\(108\) 0 0
\(109\) −1.87399 −0.179496 −0.0897479 0.995965i \(-0.528606\pi\)
−0.0897479 + 0.995965i \(0.528606\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 5.19394 0.490781
\(113\) −11.7685 −1.10708 −0.553541 0.832822i \(-0.686724\pi\)
−0.553541 + 0.832822i \(0.686724\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.193937 0.0180066
\(117\) 0 0
\(118\) 19.6629 1.81012
\(119\) −6.57452 −0.602685
\(120\) 0 0
\(121\) 6.27504 0.570458
\(122\) −13.1636 −1.19178
\(123\) 0 0
\(124\) −0.932071 −0.0837025
\(125\) 0 0
\(126\) 0 0
\(127\) −14.2677 −1.26606 −0.633029 0.774128i \(-0.718189\pi\)
−0.633029 + 0.774128i \(0.718189\pi\)
\(128\) −12.6751 −1.12033
\(129\) 0 0
\(130\) 0 0
\(131\) −5.89446 −0.515001 −0.257501 0.966278i \(-0.582899\pi\)
−0.257501 + 0.966278i \(0.582899\pi\)
\(132\) 0 0
\(133\) 3.81336 0.330660
\(134\) 8.65562 0.747731
\(135\) 0 0
\(136\) 14.7308 1.26316
\(137\) 18.2823 1.56197 0.780983 0.624553i \(-0.214719\pi\)
0.780983 + 0.624553i \(0.214719\pi\)
\(138\) 0 0
\(139\) −11.5369 −0.978547 −0.489274 0.872130i \(-0.662738\pi\)
−0.489274 + 0.872130i \(0.662738\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1.88858 −0.158486
\(143\) 12.3127 1.02964
\(144\) 0 0
\(145\) 0 0
\(146\) −22.5804 −1.86877
\(147\) 0 0
\(148\) 1.84367 0.151549
\(149\) −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 \(0.476682\pi\)
\(152\) −8.54420 −0.693026
\(153\) 0 0
\(154\) −7.35026 −0.592301
\(155\) 0 0
\(156\) 0 0
\(157\) −3.76845 −0.300755 −0.150378 0.988629i \(-0.548049\pi\)
−0.150378 + 0.988629i \(0.548049\pi\)
\(158\) 7.30536 0.581183
\(159\) 0 0
\(160\) 0 0
\(161\) −2.20123 −0.173481
\(162\) 0 0
\(163\) −1.64244 −0.128646 −0.0643231 0.997929i \(-0.520489\pi\)
−0.0643231 + 0.997929i \(0.520489\pi\)
\(164\) 2.18664 0.170748
\(165\) 0 0
\(166\) −6.54420 −0.507928
\(167\) 8.08110 0.625334 0.312667 0.949863i \(-0.398778\pi\)
0.312667 + 0.949863i \(0.398778\pi\)
\(168\) 0 0
\(169\) −4.22425 −0.324943
\(170\) 0 0
\(171\) 0 0
\(172\) 0.00587961 0.000448316 0
\(173\) −7.73813 −0.588320 −0.294160 0.955756i \(-0.595040\pi\)
−0.294160 + 0.955756i \(0.595040\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 18.0811 1.36291
\(177\) 0 0
\(178\) −5.35026 −0.401019
\(179\) 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 \(0.307836\pi\)
\(182\) −5.23884 −0.388329
\(183\) 0 0
\(184\) 4.93207 0.363597
\(185\) 0 0
\(186\) 0 0
\(187\) −22.8872 −1.67368
\(188\) 0.932071 0.0679783
\(189\) 0 0
\(190\) 0 0
\(191\) −3.31994 −0.240223 −0.120111 0.992760i \(-0.538325\pi\)
−0.120111 + 0.992760i \(0.538325\pi\)
\(192\) 0 0
\(193\) 4.88129 0.351363 0.175681 0.984447i \(-0.443787\pi\)
0.175681 + 0.984447i \(0.443787\pi\)
\(194\) −2.04491 −0.146816
\(195\) 0 0
\(196\) −1.08110 −0.0772216
\(197\) −24.2374 −1.72685 −0.863423 0.504481i \(-0.831684\pi\)
−0.863423 + 0.504481i \(0.831684\pi\)
\(198\) 0 0
\(199\) 16.7513 1.18747 0.593734 0.804661i \(-0.297653\pi\)
0.593734 + 0.804661i \(0.297653\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −19.2750 −1.35619
\(203\) −1.19394 −0.0837979
\(204\) 0 0
\(205\) 0 0
\(206\) 7.87987 0.549017
\(207\) 0 0
\(208\) 12.8872 0.893564
\(209\) 13.2750 0.918254
\(210\) 0 0
\(211\) −25.3054 −1.74209 −0.871046 0.491201i \(-0.836558\pi\)
−0.871046 + 0.491201i \(0.836558\pi\)
\(212\) −0.261865 −0.0179850
\(213\) 0 0
\(214\) 20.4690 1.39923
\(215\) 0 0
\(216\) 0 0
\(217\) 5.73813 0.389530
\(218\) 2.77575 0.187997
\(219\) 0 0
\(220\) 0 0
\(221\) −16.3127 −1.09731
\(222\) 0 0
\(223\) −17.6932 −1.18483 −0.592413 0.805634i \(-0.701825\pi\)
−0.592413 + 0.805634i \(0.701825\pi\)
\(224\) −1.30536 −0.0872178
\(225\) 0 0
\(226\) 17.4314 1.15952
\(227\) 26.8423 1.78158 0.890792 0.454412i \(-0.150151\pi\)
0.890792 + 0.454412i \(0.150151\pi\)
\(228\) 0 0
\(229\) −17.2243 −1.13821 −0.569105 0.822265i \(-0.692710\pi\)
−0.569105 + 0.822265i \(0.692710\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.67513 0.175631
\(233\) −9.07381 −0.594445 −0.297222 0.954808i \(-0.596060\pi\)
−0.297222 + 0.954808i \(0.596060\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −2.57452 −0.167587
\(237\) 0 0
\(238\) 9.73813 0.631230
\(239\) −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 \(0.443562\pi\)
\(242\) −9.29455 −0.597476
\(243\) 0 0
\(244\) 1.72355 0.110339
\(245\) 0 0
\(246\) 0 0
\(247\) 9.46168 0.602032
\(248\) −12.8568 −0.816411
\(249\) 0 0
\(250\) 0 0
\(251\) 29.6180 1.86947 0.934736 0.355343i \(-0.115636\pi\)
0.934736 + 0.355343i \(0.115636\pi\)
\(252\) 0 0
\(253\) −7.66291 −0.481763
\(254\) 21.1333 1.32602
\(255\) 0 0
\(256\) 4.61213 0.288258
\(257\) 17.6629 1.10178 0.550891 0.834577i \(-0.314288\pi\)
0.550891 + 0.834577i \(0.314288\pi\)
\(258\) 0 0
\(259\) −11.3503 −0.705271
\(260\) 0 0
\(261\) 0 0
\(262\) 8.73084 0.539393
\(263\) 27.3561 1.68685 0.843426 0.537245i \(-0.180535\pi\)
0.843426 + 0.537245i \(0.180535\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −5.64832 −0.346321
\(267\) 0 0
\(268\) −1.13330 −0.0692275
\(269\) −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 \(0.594363\pi\)
\(272\) −23.9551 −1.45249
\(273\) 0 0
\(274\) −27.0797 −1.63594
\(275\) 0 0
\(276\) 0 0
\(277\) −13.3503 −0.802139 −0.401070 0.916048i \(-0.631362\pi\)
−0.401070 + 0.916048i \(0.631362\pi\)
\(278\) 17.0884 1.02489
\(279\) 0 0
\(280\) 0 0
\(281\) −20.4241 −1.21840 −0.609199 0.793017i \(-0.708509\pi\)
−0.609199 + 0.793017i \(0.708509\pi\)
\(282\) 0 0
\(283\) 8.02047 0.476767 0.238384 0.971171i \(-0.423382\pi\)
0.238384 + 0.971171i \(0.423382\pi\)
\(284\) 0.247277 0.0146732
\(285\) 0 0
\(286\) −18.2374 −1.07840
\(287\) −13.4617 −0.794618
\(288\) 0 0
\(289\) 13.3225 0.783676
\(290\) 0 0
\(291\) 0 0
\(292\) 2.95651 0.173017
\(293\) −23.3054 −1.36151 −0.680757 0.732510i \(-0.738349\pi\)
−0.680757 + 0.732510i \(0.738349\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 25.4314 1.47817
\(297\) 0 0
\(298\) 4.11142 0.238168
\(299\) −5.46168 −0.315857
\(300\) 0 0
\(301\) −0.0361968 −0.00208635
\(302\) −2.66433 −0.153315
\(303\) 0 0
\(304\) 13.8945 0.796902
\(305\) 0 0
\(306\) 0 0
\(307\) 6.73084 0.384149 0.192075 0.981380i \(-0.438478\pi\)
0.192075 + 0.981380i \(0.438478\pi\)
\(308\) 0.962389 0.0548372
\(309\) 0 0
\(310\) 0 0
\(311\) 22.0567 1.25072 0.625359 0.780337i \(-0.284952\pi\)
0.625359 + 0.780337i \(0.284952\pi\)
\(312\) 0 0
\(313\) −5.03761 −0.284743 −0.142371 0.989813i \(-0.545473\pi\)
−0.142371 + 0.989813i \(0.545473\pi\)
\(314\) 5.58181 0.315000
\(315\) 0 0
\(316\) −0.956509 −0.0538078
\(317\) 34.2941 1.92615 0.963074 0.269237i \(-0.0867714\pi\)
0.963074 + 0.269237i \(0.0867714\pi\)
\(318\) 0 0
\(319\) −4.15633 −0.232710
\(320\) 0 0
\(321\) 0 0
\(322\) 3.26045 0.181698
\(323\) −17.5877 −0.978605
\(324\) 0 0
\(325\) 0 0
\(326\) 2.43278 0.134739
\(327\) 0 0
\(328\) 30.1622 1.66543
\(329\) −5.73813 −0.316354
\(330\) 0 0
\(331\) −34.8324 −1.91456 −0.957281 0.289159i \(-0.906625\pi\)
−0.957281 + 0.289159i \(0.906625\pi\)
\(332\) 0.856849 0.0470257
\(333\) 0 0
\(334\) −11.9697 −0.654952
\(335\) 0 0
\(336\) 0 0
\(337\) 17.6326 0.960509 0.480254 0.877129i \(-0.340544\pi\)
0.480254 + 0.877129i \(0.340544\pi\)
\(338\) 6.25694 0.340333
\(339\) 0 0
\(340\) 0 0
\(341\) 19.9756 1.08174
\(342\) 0 0
\(343\) 15.0132 0.810635
\(344\) 0.0811024 0.00437275
\(345\) 0 0
\(346\) 11.4617 0.616184
\(347\) −3.11871 −0.167421 −0.0837107 0.996490i \(-0.526677\pi\)
−0.0837107 + 0.996490i \(0.526677\pi\)
\(348\) 0 0
\(349\) −13.0738 −0.699825 −0.349912 0.936782i \(-0.613789\pi\)
−0.349912 + 0.936782i \(0.613789\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −4.54420 −0.242207
\(353\) 5.19982 0.276758 0.138379 0.990379i \(-0.455811\pi\)
0.138379 + 0.990379i \(0.455811\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0.700523 0.0371277
\(357\) 0 0
\(358\) −31.6991 −1.67535
\(359\) −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
\(362\) −22.6253 −1.18916
\(363\) 0 0
\(364\) 0.685935 0.0359528
\(365\) 0 0
\(366\) 0 0
\(367\) −20.6556 −1.07821 −0.539107 0.842237i \(-0.681238\pi\)
−0.539107 + 0.842237i \(0.681238\pi\)
\(368\) −8.02047 −0.418096
\(369\) 0 0
\(370\) 0 0
\(371\) 1.61213 0.0836975
\(372\) 0 0
\(373\) −11.0884 −0.574135 −0.287068 0.957910i \(-0.592680\pi\)
−0.287068 + 0.957910i \(0.592680\pi\)
\(374\) 33.9003 1.75294
\(375\) 0 0
\(376\) 12.8568 0.663041
\(377\) −2.96239 −0.152571
\(378\) 0 0
\(379\) 10.0811 0.517831 0.258916 0.965900i \(-0.416635\pi\)
0.258916 + 0.965900i \(0.416635\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 4.91748 0.251600
\(383\) 16.3576 0.835832 0.417916 0.908486i \(-0.362761\pi\)
0.417916 + 0.908486i \(0.362761\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −7.23013 −0.368004
\(387\) 0 0
\(388\) 0.267745 0.0135927
\(389\) −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
\(392\) −14.9126 −0.753198
\(393\) 0 0
\(394\) 35.9003 1.80863
\(395\) 0 0
\(396\) 0 0
\(397\) −2.98683 −0.149905 −0.0749523 0.997187i \(-0.523880\pi\)
−0.0749523 + 0.997187i \(0.523880\pi\)
\(398\) −24.8119 −1.24371
\(399\) 0 0
\(400\) 0 0
\(401\) 21.9756 1.09741 0.548704 0.836017i \(-0.315122\pi\)
0.548704 + 0.836017i \(0.315122\pi\)
\(402\) 0 0
\(403\) 14.2374 0.709217
\(404\) 2.52373 0.125560
\(405\) 0 0
\(406\) 1.76845 0.0877668
\(407\) −39.5125 −1.95856
\(408\) 0 0
\(409\) −22.4387 −1.10952 −0.554760 0.832010i \(-0.687190\pi\)
−0.554760 + 0.832010i \(0.687190\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.03173 −0.0508298
\(413\) 15.8496 0.779906
\(414\) 0 0
\(415\) 0 0
\(416\) −3.23884 −0.158797
\(417\) 0 0
\(418\) −19.6629 −0.961744
\(419\) −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 \(0.188676\pi\)
\(422\) 37.4821 1.82460
\(423\) 0 0
\(424\) −3.61213 −0.175420
\(425\) 0 0
\(426\) 0 0
\(427\) −10.6107 −0.513488
\(428\) −2.68006 −0.129545
\(429\) 0 0
\(430\) 0 0
\(431\) −25.7743 −1.24151 −0.620753 0.784006i \(-0.713173\pi\)
−0.620753 + 0.784006i \(0.713173\pi\)
\(432\) 0 0
\(433\) −2.18076 −0.104801 −0.0524004 0.998626i \(-0.516687\pi\)
−0.0524004 + 0.998626i \(0.516687\pi\)
\(434\) −8.49929 −0.407979
\(435\) 0 0
\(436\) −0.363436 −0.0174054
\(437\) −5.88858 −0.281689
\(438\) 0 0
\(439\) −35.5125 −1.69492 −0.847459 0.530861i \(-0.821869\pi\)
−0.847459 + 0.530861i \(0.821869\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 24.1622 1.14928
\(443\) −4.34297 −0.206341 −0.103170 0.994664i \(-0.532899\pi\)
−0.103170 + 0.994664i \(0.532899\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 26.2071 1.24094
\(447\) 0 0
\(448\) −8.45439 −0.399432
\(449\) −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
\(452\) −2.28233 −0.107352
\(453\) 0 0
\(454\) −39.7586 −1.86596
\(455\) 0 0
\(456\) 0 0
\(457\) −34.3488 −1.60677 −0.803386 0.595459i \(-0.796970\pi\)
−0.803386 + 0.595459i \(0.796970\pi\)
\(458\) 25.5125 1.19212
\(459\) 0 0
\(460\) 0 0
\(461\) −11.8641 −0.552568 −0.276284 0.961076i \(-0.589103\pi\)
−0.276284 + 0.961076i \(0.589103\pi\)
\(462\) 0 0
\(463\) −40.4953 −1.88198 −0.940989 0.338438i \(-0.890101\pi\)
−0.940989 + 0.338438i \(0.890101\pi\)
\(464\) −4.35026 −0.201956
\(465\) 0 0
\(466\) 13.4401 0.622599
\(467\) −30.2071 −1.39782 −0.698909 0.715210i \(-0.746331\pi\)
−0.698909 + 0.715210i \(0.746331\pi\)
\(468\) 0 0
\(469\) 6.97698 0.322167
\(470\) 0 0
\(471\) 0 0
\(472\) −35.5125 −1.63459
\(473\) −0.126008 −0.00579385
\(474\) 0 0
\(475\) 0 0
\(476\) −1.27504 −0.0584413
\(477\) 0 0
\(478\) 30.3634 1.38879
\(479\) −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
\(482\) −8.11142 −0.369465
\(483\) 0 0
\(484\) 1.21696 0.0553163
\(485\) 0 0
\(486\) 0 0
\(487\) 0.881286 0.0399349 0.0199674 0.999801i \(-0.493644\pi\)
0.0199674 + 0.999801i \(0.493644\pi\)
\(488\) 23.7743 1.07621
\(489\) 0 0
\(490\) 0 0
\(491\) −41.0698 −1.85346 −0.926728 0.375733i \(-0.877391\pi\)
−0.926728 + 0.375733i \(0.877391\pi\)
\(492\) 0 0
\(493\) 5.50659 0.248004
\(494\) −14.0146 −0.630546
\(495\) 0 0
\(496\) 20.9076 0.938780
\(497\) −1.52232 −0.0682852
\(498\) 0 0
\(499\) 12.3733 0.553904 0.276952 0.960884i \(-0.410676\pi\)
0.276952 + 0.960884i \(0.410676\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −43.8700 −1.95801
\(503\) −2.26774 −0.101114 −0.0505569 0.998721i \(-0.516100\pi\)
−0.0505569 + 0.998721i \(0.516100\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 11.3503 0.504581
\(507\) 0 0
\(508\) −2.76704 −0.122767
\(509\) 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
\(512\) 18.5188 0.818423
\(513\) 0 0
\(514\) −26.1622 −1.15397
\(515\) 0 0
\(516\) 0 0
\(517\) −19.9756 −0.878524
\(518\) 16.8119 0.738674
\(519\) 0 0
\(520\) 0 0
\(521\) 4.72496 0.207004 0.103502 0.994629i \(-0.466995\pi\)
0.103502 + 0.994629i \(0.466995\pi\)
\(522\) 0 0
\(523\) 1.06793 0.0466973 0.0233486 0.999727i \(-0.492567\pi\)
0.0233486 + 0.999727i \(0.492567\pi\)
\(524\) −1.14315 −0.0499388
\(525\) 0 0
\(526\) −40.5198 −1.76675
\(527\) −26.4650 −1.15283
\(528\) 0 0
\(529\) −19.6009 −0.852211
\(530\) 0 0
\(531\) 0 0
\(532\) 0.739549 0.0320635
\(533\) −33.4010 −1.44676
\(534\) 0 0
\(535\) 0 0
\(536\) −15.6326 −0.675225
\(537\) 0 0
\(538\) 15.5515 0.670472
\(539\) 23.1695 0.997981
\(540\) 0 0
\(541\) −7.46168 −0.320803 −0.160401 0.987052i \(-0.551279\pi\)
−0.160401 + 0.987052i \(0.551279\pi\)
\(542\) 14.2461 0.611924
\(543\) 0 0
\(544\) 6.02047 0.258125
\(545\) 0 0
\(546\) 0 0
\(547\) 38.9683 1.66616 0.833081 0.553150i \(-0.186575\pi\)
0.833081 + 0.553150i \(0.186575\pi\)
\(548\) 3.54561 0.151461
\(549\) 0 0
\(550\) 0 0
\(551\) −3.19394 −0.136066
\(552\) 0 0
\(553\) 5.88858 0.250408
\(554\) 19.7743 0.840131
\(555\) 0 0
\(556\) −2.23743 −0.0948881
\(557\) −22.9986 −0.974481 −0.487241 0.873268i \(-0.661997\pi\)
−0.487241 + 0.873268i \(0.661997\pi\)
\(558\) 0 0
\(559\) −0.0898112 −0.00379861
\(560\) 0 0
\(561\) 0 0
\(562\) 30.2520 1.27610
\(563\) 11.6688 0.491781 0.245890 0.969298i \(-0.420920\pi\)
0.245890 + 0.969298i \(0.420920\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −11.8799 −0.499348
\(567\) 0 0
\(568\) 3.41090 0.143118
\(569\) −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 \(0.307475\pi\)
\(572\) 2.38787 0.0998420
\(573\) 0 0
\(574\) 19.9394 0.832253
\(575\) 0 0
\(576\) 0 0
\(577\) −22.5950 −0.940641 −0.470321 0.882496i \(-0.655862\pi\)
−0.470321 + 0.882496i \(0.655862\pi\)
\(578\) −19.7332 −0.820793
\(579\) 0 0
\(580\) 0 0
\(581\) −5.27504 −0.218845
\(582\) 0 0
\(583\) 5.61213 0.232431
\(584\) 40.7816 1.68756
\(585\) 0 0
\(586\) 34.5198 1.42600
\(587\) 9.31994 0.384675 0.192338 0.981329i \(-0.438393\pi\)
0.192338 + 0.981329i \(0.438393\pi\)
\(588\) 0 0
\(589\) 15.3503 0.632497
\(590\) 0 0
\(591\) 0 0
\(592\) −41.3561 −1.69973
\(593\) −15.1246 −0.621093 −0.310546 0.950558i \(-0.600512\pi\)
−0.310546 + 0.950558i \(0.600512\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.538319 −0.0220504
\(597\) 0 0
\(598\) 8.08981 0.330817
\(599\) −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 \(0.350232\pi\)
\(602\) 0.0536145 0.00218516
\(603\) 0 0
\(604\) 0.348847 0.0141944
\(605\) 0 0
\(606\) 0 0
\(607\) 48.2941 1.96020 0.980098 0.198512i \(-0.0636110\pi\)
0.980098 + 0.198512i \(0.0636110\pi\)
\(608\) −3.49200 −0.141619
\(609\) 0 0
\(610\) 0 0
\(611\) −14.2374 −0.575985
\(612\) 0 0
\(613\) 9.74798 0.393717 0.196859 0.980432i \(-0.436926\pi\)
0.196859 + 0.980432i \(0.436926\pi\)
\(614\) −9.96968 −0.402344
\(615\) 0 0
\(616\) 13.2750 0.534867
\(617\) 18.2170 0.733387 0.366694 0.930342i \(-0.380490\pi\)
0.366694 + 0.930342i \(0.380490\pi\)
\(618\) 0 0
\(619\) 25.0943 1.00862 0.504312 0.863521i \(-0.331746\pi\)
0.504312 + 0.863521i \(0.331746\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −32.6702 −1.30996
\(623\) −4.31265 −0.172783
\(624\) 0 0
\(625\) 0 0
\(626\) 7.46168 0.298229
\(627\) 0 0
\(628\) −0.730841 −0.0291637
\(629\) 52.3488 2.08729
\(630\) 0 0
\(631\) 21.4617 0.854376 0.427188 0.904163i \(-0.359504\pi\)
0.427188 + 0.904163i \(0.359504\pi\)
\(632\) −13.1939 −0.524827
\(633\) 0 0
\(634\) −50.7962 −2.01738
\(635\) 0 0
\(636\) 0 0
\(637\) 16.5139 0.654304
\(638\) 6.15633 0.243731
\(639\) 0 0
\(640\) 0 0
\(641\) −3.17347 −0.125344 −0.0626722 0.998034i \(-0.519962\pi\)
−0.0626722 + 0.998034i \(0.519962\pi\)
\(642\) 0 0
\(643\) 2.74069 0.108082 0.0540411 0.998539i \(-0.482790\pi\)
0.0540411 + 0.998539i \(0.482790\pi\)
\(644\) −0.426899 −0.0168222
\(645\) 0 0
\(646\) 26.0508 1.02495
\(647\) 6.34297 0.249368 0.124684 0.992197i \(-0.460208\pi\)
0.124684 + 0.992197i \(0.460208\pi\)
\(648\) 0 0
\(649\) 55.1754 2.16582
\(650\) 0 0
\(651\) 0 0
\(652\) −0.318530 −0.0124746
\(653\) 4.08110 0.159706 0.0798529 0.996807i \(-0.474555\pi\)
0.0798529 + 0.996807i \(0.474555\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −49.0494 −1.91506
\(657\) 0 0
\(658\) 8.49929 0.331337
\(659\) −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 \(0.679816\pi\)
\(662\) 51.5936 2.00524
\(663\) 0 0
\(664\) 11.8192 0.458675
\(665\) 0 0
\(666\) 0 0
\(667\) 1.84367 0.0713874
\(668\) 1.56722 0.0606376
\(669\) 0 0
\(670\) 0 0
\(671\) −36.9380 −1.42597
\(672\) 0 0
\(673\) −3.13727 −0.120933 −0.0604665 0.998170i \(-0.519259\pi\)
−0.0604665 + 0.998170i \(0.519259\pi\)
\(674\) −26.1173 −1.00600
\(675\) 0 0
\(676\) −0.819237 −0.0315091
\(677\) −46.2579 −1.77784 −0.888918 0.458067i \(-0.848542\pi\)
−0.888918 + 0.458067i \(0.848542\pi\)
\(678\) 0 0
\(679\) −1.64832 −0.0632569
\(680\) 0 0
\(681\) 0 0
\(682\) −29.5877 −1.13297
\(683\) −9.01905 −0.345104 −0.172552 0.985000i \(-0.555201\pi\)
−0.172552 + 0.985000i \(0.555201\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −22.2374 −0.849029
\(687\) 0 0
\(688\) −0.131888 −0.00502817
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) −50.0625 −1.90447 −0.952234 0.305368i \(-0.901221\pi\)
−0.952234 + 0.305368i \(0.901221\pi\)
\(692\) −1.50071 −0.0570483
\(693\) 0 0
\(694\) 4.61942 0.175351
\(695\) 0 0
\(696\) 0 0
\(697\) 62.0870 2.35171
\(698\) 19.3649 0.732970
\(699\) 0 0
\(700\) 0 0
\(701\) −45.3014 −1.71101 −0.855505 0.517795i \(-0.826753\pi\)
−0.855505 + 0.517795i \(0.826753\pi\)
\(702\) 0 0
\(703\) −30.3634 −1.14518
\(704\) −29.4314 −1.10924
\(705\) 0 0
\(706\) −7.70194 −0.289866
\(707\) −15.5369 −0.584325
\(708\) 0 0
\(709\) −3.27504 −0.122997 −0.0614983 0.998107i \(-0.519588\pi\)
−0.0614983 + 0.998107i \(0.519588\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 9.66291 0.362133
\(713\) −8.86082 −0.331840
\(714\) 0 0
\(715\) 0 0
\(716\) 4.15045 0.155109
\(717\) 0 0
\(718\) 45.0553 1.68145
\(719\) −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
\(722\) 13.0327 0.485026
\(723\) 0 0
\(724\) 2.96239 0.110096
\(725\) 0 0
\(726\) 0 0
\(727\) 26.8930 0.997408 0.498704 0.866772i \(-0.333810\pi\)
0.498704 + 0.866772i \(0.333810\pi\)
\(728\) 9.46168 0.350673
\(729\) 0 0
\(730\) 0 0
\(731\) 0.166944 0.00617465
\(732\) 0 0
\(733\) −3.17935 −0.117432 −0.0587160 0.998275i \(-0.518701\pi\)
−0.0587160 + 0.998275i \(0.518701\pi\)
\(734\) 30.5950 1.12928
\(735\) 0 0
\(736\) 2.01573 0.0743007
\(737\) 24.2882 0.894668
\(738\) 0 0
\(739\) −29.7440 −1.09415 −0.547076 0.837083i \(-0.684259\pi\)
−0.547076 + 0.837083i \(0.684259\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2.38787 −0.0876616
\(743\) −4.34297 −0.159328 −0.0796640 0.996822i \(-0.525385\pi\)
−0.0796640 + 0.996822i \(0.525385\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 16.4241 0.601328
\(747\) 0 0
\(748\) −4.43866 −0.162293
\(749\) 16.4993 0.602871
\(750\) 0 0
\(751\) 22.5804 0.823970 0.411985 0.911191i \(-0.364836\pi\)
0.411985 + 0.911191i \(0.364836\pi\)
\(752\) −20.9076 −0.762423
\(753\) 0 0
\(754\) 4.38787 0.159797
\(755\) 0 0
\(756\) 0 0
\(757\) −9.88461 −0.359262 −0.179631 0.983734i \(-0.557490\pi\)
−0.179631 + 0.983734i \(0.557490\pi\)
\(758\) −14.9321 −0.542357
\(759\) 0 0
\(760\) 0 0
\(761\) −13.6991 −0.496592 −0.248296 0.968684i \(-0.579871\pi\)
−0.248296 + 0.968684i \(0.579871\pi\)
\(762\) 0 0
\(763\) 2.23743 0.0810003
\(764\) −0.643859 −0.0232940
\(765\) 0 0
\(766\) −24.2287 −0.875419
\(767\) 39.3258 1.41997
\(768\) 0 0
\(769\) 25.0132 0.901998 0.450999 0.892524i \(-0.351068\pi\)
0.450999 + 0.892524i \(0.351068\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0.946660 0.0340710
\(773\) −35.9062 −1.29146 −0.645728 0.763567i \(-0.723446\pi\)
−0.645728 + 0.763567i \(0.723446\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 3.69323 0.132579
\(777\) 0 0
\(778\) 47.2506 1.69402
\(779\) −36.0118 −1.29026
\(780\) 0 0
\(781\) −5.29948 −0.189630
\(782\) −15.0376 −0.537744
\(783\) 0 0
\(784\) 24.2506 0.866093
\(785\) 0 0
\(786\) 0 0
\(787\) −50.3839 −1.79599 −0.897996 0.440003i \(-0.854977\pi\)
−0.897996 + 0.440003i \(0.854977\pi\)
\(788\) −4.70052 −0.167449
\(789\) 0 0
\(790\) 0 0
\(791\) 14.0508 0.499588
\(792\) 0 0
\(793\) −26.3272 −0.934908
\(794\) 4.42407 0.157004
\(795\) 0 0
\(796\) 3.24869 0.115147
\(797\) −5.69323 −0.201665 −0.100832 0.994903i \(-0.532151\pi\)
−0.100832 + 0.994903i \(0.532151\pi\)
\(798\) 0 0
\(799\) 26.4650 0.936265
\(800\) 0 0
\(801\) 0 0
\(802\) −32.5501 −1.14938
\(803\) −63.3620 −2.23600
\(804\) 0 0
\(805\) 0 0
\(806\) −21.0884 −0.742807
\(807\) 0 0
\(808\) 34.8119 1.22468
\(809\) 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 \(0.653976\pi\)
\(812\) −0.231548 −0.00812574
\(813\) 0 0
\(814\) 58.5256 2.05132
\(815\) 0 0
\(816\) 0 0
\(817\) −0.0968311 −0.00338769
\(818\) 33.2360 1.16207
\(819\) 0 0
\(820\) 0 0
\(821\) −25.4763 −0.889128 −0.444564 0.895747i \(-0.646641\pi\)
−0.444564 + 0.895747i \(0.646641\pi\)
\(822\) 0 0
\(823\) 9.22028 0.321399 0.160699 0.987003i \(-0.448625\pi\)
0.160699 + 0.987003i \(0.448625\pi\)
\(824\) −14.2315 −0.495779
\(825\) 0 0
\(826\) −23.4763 −0.816844
\(827\) 24.5343 0.853143 0.426571 0.904454i \(-0.359721\pi\)
0.426571 + 0.904454i \(0.359721\pi\)
\(828\) 0 0
\(829\) 0.201231 0.00698903 0.00349452 0.999994i \(-0.498888\pi\)
0.00349452 + 0.999994i \(0.498888\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −20.9770 −0.727246
\(833\) −30.6966 −1.06357
\(834\) 0 0
\(835\) 0 0
\(836\) 2.57452 0.0890415
\(837\) 0 0
\(838\) 15.3503 0.530266
\(839\) −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
\(842\) −50.4142 −1.73739
\(843\) 0 0
\(844\) −4.90763 −0.168928
\(845\) 0 0
\(846\) 0 0
\(847\) −7.49200 −0.257428
\(848\) 5.87399 0.201714
\(849\) 0 0
\(850\) 0 0
\(851\) 17.5271 0.600820
\(852\) 0 0
\(853\) −43.1793 −1.47843 −0.739216 0.673468i \(-0.764804\pi\)
−0.739216 + 0.673468i \(0.764804\pi\)
\(854\) 15.7165 0.537808
\(855\) 0 0
\(856\) −36.9683 −1.26355
\(857\) −20.9887 −0.716962 −0.358481 0.933537i \(-0.616705\pi\)
−0.358481 + 0.933537i \(0.616705\pi\)
\(858\) 0 0
\(859\) −49.4069 −1.68574 −0.842871 0.538115i \(-0.819137\pi\)
−0.842871 + 0.538115i \(0.819137\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 38.1768 1.30031
\(863\) 56.6820 1.92948 0.964738 0.263211i \(-0.0847816\pi\)
0.964738 + 0.263211i \(0.0847816\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 3.23013 0.109764
\(867\) 0 0
\(868\) 1.11283 0.0377721
\(869\) 20.4993 0.695391
\(870\) 0 0
\(871\) 17.3112 0.586569
\(872\) −5.01317 −0.169767
\(873\) 0 0
\(874\) 8.72213 0.295031
\(875\) 0 0
\(876\) 0 0
\(877\) 13.1998 0.445726 0.222863 0.974850i \(-0.428460\pi\)
0.222863 + 0.974850i \(0.428460\pi\)
\(878\) 52.6009 1.77519
\(879\) 0 0
\(880\) 0 0
\(881\) −6.37802 −0.214881 −0.107441 0.994212i \(-0.534266\pi\)
−0.107441 + 0.994212i \(0.534266\pi\)
\(882\) 0 0
\(883\) −48.6213 −1.63624 −0.818119 0.575049i \(-0.804983\pi\)
−0.818119 + 0.575049i \(0.804983\pi\)
\(884\) −3.16362 −0.106404
\(885\) 0 0
\(886\) 6.43278 0.216113
\(887\) −15.0317 −0.504716 −0.252358 0.967634i \(-0.581206\pi\)
−0.252358 + 0.967634i \(0.581206\pi\)
\(888\) 0 0
\(889\) 17.0348 0.571328
\(890\) 0 0
\(891\) 0 0
\(892\) −3.43136 −0.114891
\(893\) −15.3503 −0.513677
\(894\) 0 0
\(895\) 0 0
\(896\) 15.1333 0.505568
\(897\) 0 0
\(898\) 46.4142 1.54886
\(899\) −4.80606 −0.160291
\(900\) 0 0
\(901\) −7.43533 −0.247707
\(902\) 69.4128 2.31119
\(903\) 0 0
\(904\) −31.4821 −1.04708
\(905\) 0 0
\(906\) 0 0
\(907\) 0.342968 0.0113880 0.00569402 0.999984i \(-0.498188\pi\)
0.00569402 + 0.999984i \(0.498188\pi\)
\(908\) 5.20570 0.172757
\(909\) 0 0
\(910\) 0 0
\(911\) −20.9076 −0.692701 −0.346350 0.938105i \(-0.612579\pi\)
−0.346350 + 0.938105i \(0.612579\pi\)
\(912\) 0 0
\(913\) −18.3634 −0.607741
\(914\) 50.8773 1.68287
\(915\) 0 0
\(916\) −3.34041 −0.110370
\(917\) 7.03761 0.232402
\(918\) 0 0
\(919\) −1.90034 −0.0626864 −0.0313432 0.999509i \(-0.509978\pi\)
−0.0313432 + 0.999509i \(0.509978\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 17.5731 0.578739
\(923\) −3.77716 −0.124327
\(924\) 0 0
\(925\) 0 0
\(926\) 59.9814 1.97111
\(927\) 0 0
\(928\) 1.09332 0.0358900
\(929\) −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
\(932\) −1.75974 −0.0576423
\(933\) 0 0
\(934\) 44.7426 1.46402
\(935\) 0 0
\(936\) 0 0
\(937\) 6.37802 0.208361 0.104180 0.994558i \(-0.466778\pi\)
0.104180 + 0.994558i \(0.466778\pi\)
\(938\) −10.3343 −0.337426
\(939\) 0 0
\(940\) 0 0
\(941\) 26.6253 0.867960 0.433980 0.900923i \(-0.357109\pi\)
0.433980 + 0.900923i \(0.357109\pi\)
\(942\) 0 0
\(943\) 20.7875 0.676934
\(944\) 57.7499 1.87960
\(945\) 0 0
\(946\) 0.186642 0.00606827
\(947\) −12.2823 −0.399122 −0.199561 0.979885i \(-0.563952\pi\)
−0.199561 + 0.979885i \(0.563952\pi\)
\(948\) 0 0
\(949\) −45.1608 −1.46598
\(950\) 0 0
\(951\) 0 0
\(952\) −17.5877 −0.570020
\(953\) 0.821792 0.0266205 0.0133102 0.999911i \(-0.495763\pi\)
0.0133102 + 0.999911i \(0.495763\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −3.97556 −0.128579
\(957\) 0 0
\(958\) 0.0811024 0.00262030
\(959\) −21.8279 −0.704861
\(960\) 0 0
\(961\) −7.90175 −0.254895
\(962\) 41.7137 1.34490
\(963\) 0 0
\(964\) 1.06205 0.0342063
\(965\) 0 0
\(966\) 0 0
\(967\) 37.4314 1.20371 0.601856 0.798605i \(-0.294428\pi\)
0.601856 + 0.798605i \(0.294428\pi\)
\(968\) 16.7866 0.539540
\(969\) 0 0
\(970\) 0 0
\(971\) 8.71625 0.279718 0.139859 0.990171i \(-0.455335\pi\)
0.139859 + 0.990171i \(0.455335\pi\)
\(972\) 0 0
\(973\) 13.7743 0.441585
\(974\) −1.30536 −0.0418263
\(975\) 0 0
\(976\) −38.6615 −1.23752
\(977\) −33.7645 −1.08022 −0.540111 0.841594i \(-0.681618\pi\)
−0.540111 + 0.841594i \(0.681618\pi\)
\(978\) 0 0
\(979\) −15.0132 −0.479823
\(980\) 0 0
\(981\) 0 0
\(982\) 60.8324 1.94124
\(983\) 43.6082 1.39088 0.695442 0.718582i \(-0.255209\pi\)
0.695442 + 0.718582i \(0.255209\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −8.15633 −0.259750
\(987\) 0 0
\(988\) 1.83497 0.0583780
\(989\) 0.0558950 0.00177736
\(990\) 0 0
\(991\) −52.9741 −1.68278 −0.841390 0.540429i \(-0.818262\pi\)
−0.841390 + 0.540429i \(0.818262\pi\)
\(992\) −5.25457 −0.166833
\(993\) 0 0
\(994\) 2.25485 0.0715193
\(995\) 0 0
\(996\) 0 0
\(997\) −13.6326 −0.431749 −0.215874 0.976421i \(-0.569260\pi\)
−0.215874 + 0.976421i \(0.569260\pi\)
\(998\) −18.3272 −0.580139
\(999\) 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 6525.2.a.be.1.1 3
3.2 odd 2 725.2.a.e.1.3 3
5.4 even 2 1305.2.a.p.1.3 3
15.2 even 4 725.2.b.e.349.5 6
15.8 even 4 725.2.b.e.349.2 6
15.14 odd 2 145.2.a.c.1.1 3
60.59 even 2 2320.2.a.n.1.2 3
105.104 even 2 7105.2.a.o.1.1 3
120.29 odd 2 9280.2.a.bj.1.2 3
120.59 even 2 9280.2.a.br.1.2 3
435.434 odd 2 4205.2.a.f.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.c.1.1 3 15.14 odd 2
725.2.a.e.1.3 3 3.2 odd 2
725.2.b.e.349.2 6 15.8 even 4
725.2.b.e.349.5 6 15.2 even 4
1305.2.a.p.1.3 3 5.4 even 2
2320.2.a.n.1.2 3 60.59 even 2
4205.2.a.f.1.3 3 435.434 odd 2
6525.2.a.be.1.1 3 1.1 even 1 trivial
7105.2.a.o.1.1 3 105.104 even 2
9280.2.a.bj.1.2 3 120.29 odd 2
9280.2.a.br.1.2 3 120.59 even 2