Properties

Label 6003.2.a.k.1.4
Level $6003$
Weight $2$
Character 6003.1
Self dual yes
Analytic conductor $47.934$
Analytic rank $1$
Dimension $10$
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] = [6003,2,Mod(1,6003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6003, 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("6003.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6003.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: \(47.9341963334\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} - 17x^{8} + 23x^{7} + 69x^{6} - 88x^{5} - 106x^{4} + 101x^{3} + 60x^{2} - 23x - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2001)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.92359\) of defining polynomial
Character \(\chi\) \(=\) 6003.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.39479 q^{2} -0.0545700 q^{4} +2.50233 q^{5} +4.61826 q^{7} +2.86569 q^{8} +O(q^{10})\) \(q-1.39479 q^{2} -0.0545700 q^{4} +2.50233 q^{5} +4.61826 q^{7} +2.86569 q^{8} -3.49022 q^{10} -6.41359 q^{11} -4.63148 q^{13} -6.44149 q^{14} -3.88788 q^{16} +0.986447 q^{17} -2.31967 q^{19} -0.136552 q^{20} +8.94559 q^{22} +1.00000 q^{23} +1.26167 q^{25} +6.45993 q^{26} -0.252019 q^{28} -1.00000 q^{29} +7.37989 q^{31} -0.308608 q^{32} -1.37588 q^{34} +11.5564 q^{35} +4.51800 q^{37} +3.23544 q^{38} +7.17090 q^{40} -9.39988 q^{41} +0.107858 q^{43} +0.349990 q^{44} -1.39479 q^{46} -0.665991 q^{47} +14.3284 q^{49} -1.75976 q^{50} +0.252740 q^{52} -9.08358 q^{53} -16.0489 q^{55} +13.2345 q^{56} +1.39479 q^{58} -5.42942 q^{59} +7.43286 q^{61} -10.2934 q^{62} +8.20621 q^{64} -11.5895 q^{65} -10.1253 q^{67} -0.0538304 q^{68} -16.1188 q^{70} -10.8149 q^{71} -1.36129 q^{73} -6.30164 q^{74} +0.126584 q^{76} -29.6196 q^{77} +4.07429 q^{79} -9.72878 q^{80} +13.1108 q^{82} +11.3089 q^{83} +2.46842 q^{85} -0.150439 q^{86} -18.3793 q^{88} -3.27283 q^{89} -21.3894 q^{91} -0.0545700 q^{92} +0.928916 q^{94} -5.80459 q^{95} +10.8603 q^{97} -19.9850 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 3 q^{2} + 17 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8} - 4 q^{10} - 9 q^{11} - 16 q^{13} - 16 q^{14} + 27 q^{16} + q^{19} - 21 q^{20} + 17 q^{22} + 10 q^{23} - 4 q^{25} - 28 q^{26} - 14 q^{28} - 10 q^{29} + 17 q^{31} - 21 q^{32} - 3 q^{34} - 29 q^{35} + q^{37} - 32 q^{38} + 13 q^{40} - 5 q^{43} - 33 q^{44} - 3 q^{46} - 15 q^{47} + 31 q^{49} + 22 q^{50} - 21 q^{52} - 35 q^{53} - 20 q^{55} - 18 q^{56} + 3 q^{58} - 49 q^{59} + 8 q^{61} - 15 q^{62} + 12 q^{64} + 3 q^{65} + 35 q^{67} + 18 q^{68} - 16 q^{70} - 30 q^{71} - 15 q^{73} - 23 q^{74} + 10 q^{76} - 23 q^{77} + 24 q^{79} - 23 q^{80} - 5 q^{82} - q^{83} + 10 q^{86} + 18 q^{88} - 15 q^{89} + 26 q^{91} + 17 q^{92} + 3 q^{94} - 7 q^{95} - 35 q^{97} - 3 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.39479 −0.986263 −0.493132 0.869955i \(-0.664148\pi\)
−0.493132 + 0.869955i \(0.664148\pi\)
\(3\) 0 0
\(4\) −0.0545700 −0.0272850
\(5\) 2.50233 1.11908 0.559539 0.828804i \(-0.310978\pi\)
0.559539 + 0.828804i \(0.310978\pi\)
\(6\) 0 0
\(7\) 4.61826 1.74554 0.872770 0.488132i \(-0.162322\pi\)
0.872770 + 0.488132i \(0.162322\pi\)
\(8\) 2.86569 1.01317
\(9\) 0 0
\(10\) −3.49022 −1.10370
\(11\) −6.41359 −1.93377 −0.966885 0.255213i \(-0.917854\pi\)
−0.966885 + 0.255213i \(0.917854\pi\)
\(12\) 0 0
\(13\) −4.63148 −1.28454 −0.642271 0.766478i \(-0.722008\pi\)
−0.642271 + 0.766478i \(0.722008\pi\)
\(14\) −6.44149 −1.72156
\(15\) 0 0
\(16\) −3.88788 −0.971971
\(17\) 0.986447 0.239249 0.119624 0.992819i \(-0.461831\pi\)
0.119624 + 0.992819i \(0.461831\pi\)
\(18\) 0 0
\(19\) −2.31967 −0.532169 −0.266084 0.963950i \(-0.585730\pi\)
−0.266084 + 0.963950i \(0.585730\pi\)
\(20\) −0.136552 −0.0305340
\(21\) 0 0
\(22\) 8.94559 1.90721
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.26167 0.252334
\(26\) 6.45993 1.26690
\(27\) 0 0
\(28\) −0.252019 −0.0476270
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 7.37989 1.32547 0.662734 0.748855i \(-0.269396\pi\)
0.662734 + 0.748855i \(0.269396\pi\)
\(32\) −0.308608 −0.0545546
\(33\) 0 0
\(34\) −1.37588 −0.235962
\(35\) 11.5564 1.95339
\(36\) 0 0
\(37\) 4.51800 0.742754 0.371377 0.928482i \(-0.378886\pi\)
0.371377 + 0.928482i \(0.378886\pi\)
\(38\) 3.23544 0.524858
\(39\) 0 0
\(40\) 7.17090 1.13382
\(41\) −9.39988 −1.46801 −0.734007 0.679142i \(-0.762352\pi\)
−0.734007 + 0.679142i \(0.762352\pi\)
\(42\) 0 0
\(43\) 0.107858 0.0164483 0.00822413 0.999966i \(-0.497382\pi\)
0.00822413 + 0.999966i \(0.497382\pi\)
\(44\) 0.349990 0.0527629
\(45\) 0 0
\(46\) −1.39479 −0.205650
\(47\) −0.665991 −0.0971448 −0.0485724 0.998820i \(-0.515467\pi\)
−0.0485724 + 0.998820i \(0.515467\pi\)
\(48\) 0 0
\(49\) 14.3284 2.04691
\(50\) −1.75976 −0.248868
\(51\) 0 0
\(52\) 0.252740 0.0350487
\(53\) −9.08358 −1.24773 −0.623863 0.781534i \(-0.714438\pi\)
−0.623863 + 0.781534i \(0.714438\pi\)
\(54\) 0 0
\(55\) −16.0489 −2.16404
\(56\) 13.2345 1.76853
\(57\) 0 0
\(58\) 1.39479 0.183144
\(59\) −5.42942 −0.706851 −0.353425 0.935463i \(-0.614983\pi\)
−0.353425 + 0.935463i \(0.614983\pi\)
\(60\) 0 0
\(61\) 7.43286 0.951680 0.475840 0.879532i \(-0.342144\pi\)
0.475840 + 0.879532i \(0.342144\pi\)
\(62\) −10.2934 −1.30726
\(63\) 0 0
\(64\) 8.20621 1.02578
\(65\) −11.5895 −1.43750
\(66\) 0 0
\(67\) −10.1253 −1.23700 −0.618499 0.785786i \(-0.712259\pi\)
−0.618499 + 0.785786i \(0.712259\pi\)
\(68\) −0.0538304 −0.00652790
\(69\) 0 0
\(70\) −16.1188 −1.92656
\(71\) −10.8149 −1.28350 −0.641748 0.766915i \(-0.721791\pi\)
−0.641748 + 0.766915i \(0.721791\pi\)
\(72\) 0 0
\(73\) −1.36129 −0.159327 −0.0796636 0.996822i \(-0.525385\pi\)
−0.0796636 + 0.996822i \(0.525385\pi\)
\(74\) −6.30164 −0.732551
\(75\) 0 0
\(76\) 0.126584 0.0145202
\(77\) −29.6196 −3.37547
\(78\) 0 0
\(79\) 4.07429 0.458394 0.229197 0.973380i \(-0.426390\pi\)
0.229197 + 0.973380i \(0.426390\pi\)
\(80\) −9.72878 −1.08771
\(81\) 0 0
\(82\) 13.1108 1.44785
\(83\) 11.3089 1.24131 0.620656 0.784083i \(-0.286866\pi\)
0.620656 + 0.784083i \(0.286866\pi\)
\(84\) 0 0
\(85\) 2.46842 0.267738
\(86\) −0.150439 −0.0162223
\(87\) 0 0
\(88\) −18.3793 −1.95924
\(89\) −3.27283 −0.346919 −0.173459 0.984841i \(-0.555495\pi\)
−0.173459 + 0.984841i \(0.555495\pi\)
\(90\) 0 0
\(91\) −21.3894 −2.24222
\(92\) −0.0545700 −0.00568932
\(93\) 0 0
\(94\) 0.928916 0.0958104
\(95\) −5.80459 −0.595538
\(96\) 0 0
\(97\) 10.8603 1.10270 0.551349 0.834275i \(-0.314113\pi\)
0.551349 + 0.834275i \(0.314113\pi\)
\(98\) −19.9850 −2.01879
\(99\) 0 0
\(100\) −0.0688494 −0.00688494
\(101\) −1.91292 −0.190343 −0.0951713 0.995461i \(-0.530340\pi\)
−0.0951713 + 0.995461i \(0.530340\pi\)
\(102\) 0 0
\(103\) 1.65984 0.163549 0.0817745 0.996651i \(-0.473941\pi\)
0.0817745 + 0.996651i \(0.473941\pi\)
\(104\) −13.2724 −1.30146
\(105\) 0 0
\(106\) 12.6697 1.23059
\(107\) −13.6379 −1.31843 −0.659213 0.751957i \(-0.729110\pi\)
−0.659213 + 0.751957i \(0.729110\pi\)
\(108\) 0 0
\(109\) −19.7417 −1.89091 −0.945456 0.325751i \(-0.894383\pi\)
−0.945456 + 0.325751i \(0.894383\pi\)
\(110\) 22.3848 2.13431
\(111\) 0 0
\(112\) −17.9553 −1.69661
\(113\) 10.1807 0.957718 0.478859 0.877892i \(-0.341051\pi\)
0.478859 + 0.877892i \(0.341051\pi\)
\(114\) 0 0
\(115\) 2.50233 0.233344
\(116\) 0.0545700 0.00506670
\(117\) 0 0
\(118\) 7.57289 0.697141
\(119\) 4.55567 0.417618
\(120\) 0 0
\(121\) 30.1341 2.73946
\(122\) −10.3672 −0.938607
\(123\) 0 0
\(124\) −0.402721 −0.0361654
\(125\) −9.35454 −0.836696
\(126\) 0 0
\(127\) 3.12919 0.277671 0.138835 0.990315i \(-0.455664\pi\)
0.138835 + 0.990315i \(0.455664\pi\)
\(128\) −10.8287 −0.957130
\(129\) 0 0
\(130\) 16.1649 1.41776
\(131\) −15.1334 −1.32221 −0.661106 0.750292i \(-0.729913\pi\)
−0.661106 + 0.750292i \(0.729913\pi\)
\(132\) 0 0
\(133\) −10.7128 −0.928921
\(134\) 14.1226 1.22001
\(135\) 0 0
\(136\) 2.82685 0.242400
\(137\) −10.5892 −0.904701 −0.452350 0.891840i \(-0.649414\pi\)
−0.452350 + 0.891840i \(0.649414\pi\)
\(138\) 0 0
\(139\) −16.6471 −1.41199 −0.705995 0.708217i \(-0.749500\pi\)
−0.705995 + 0.708217i \(0.749500\pi\)
\(140\) −0.630635 −0.0532984
\(141\) 0 0
\(142\) 15.0845 1.26587
\(143\) 29.7044 2.48401
\(144\) 0 0
\(145\) −2.50233 −0.207807
\(146\) 1.89871 0.157138
\(147\) 0 0
\(148\) −0.246547 −0.0202661
\(149\) −23.2503 −1.90474 −0.952370 0.304944i \(-0.901362\pi\)
−0.952370 + 0.304944i \(0.901362\pi\)
\(150\) 0 0
\(151\) −18.0705 −1.47055 −0.735276 0.677768i \(-0.762948\pi\)
−0.735276 + 0.677768i \(0.762948\pi\)
\(152\) −6.64745 −0.539179
\(153\) 0 0
\(154\) 41.3131 3.32910
\(155\) 18.4670 1.48330
\(156\) 0 0
\(157\) −19.8113 −1.58112 −0.790558 0.612387i \(-0.790209\pi\)
−0.790558 + 0.612387i \(0.790209\pi\)
\(158\) −5.68277 −0.452097
\(159\) 0 0
\(160\) −0.772239 −0.0610509
\(161\) 4.61826 0.363970
\(162\) 0 0
\(163\) −1.51109 −0.118358 −0.0591789 0.998247i \(-0.518848\pi\)
−0.0591789 + 0.998247i \(0.518848\pi\)
\(164\) 0.512951 0.0400548
\(165\) 0 0
\(166\) −15.7735 −1.22426
\(167\) 20.9485 1.62104 0.810522 0.585708i \(-0.199184\pi\)
0.810522 + 0.585708i \(0.199184\pi\)
\(168\) 0 0
\(169\) 8.45063 0.650048
\(170\) −3.44292 −0.264060
\(171\) 0 0
\(172\) −0.00588583 −0.000448791 0
\(173\) 14.7018 1.11776 0.558879 0.829249i \(-0.311232\pi\)
0.558879 + 0.829249i \(0.311232\pi\)
\(174\) 0 0
\(175\) 5.82673 0.440460
\(176\) 24.9353 1.87957
\(177\) 0 0
\(178\) 4.56490 0.342153
\(179\) −22.2648 −1.66415 −0.832073 0.554666i \(-0.812846\pi\)
−0.832073 + 0.554666i \(0.812846\pi\)
\(180\) 0 0
\(181\) −2.17485 −0.161655 −0.0808277 0.996728i \(-0.525756\pi\)
−0.0808277 + 0.996728i \(0.525756\pi\)
\(182\) 29.8337 2.21142
\(183\) 0 0
\(184\) 2.86569 0.211261
\(185\) 11.3055 0.831200
\(186\) 0 0
\(187\) −6.32667 −0.462652
\(188\) 0.0363431 0.00265060
\(189\) 0 0
\(190\) 8.09616 0.587357
\(191\) 2.72397 0.197099 0.0985497 0.995132i \(-0.468580\pi\)
0.0985497 + 0.995132i \(0.468580\pi\)
\(192\) 0 0
\(193\) 18.5378 1.33438 0.667188 0.744889i \(-0.267498\pi\)
0.667188 + 0.744889i \(0.267498\pi\)
\(194\) −15.1478 −1.08755
\(195\) 0 0
\(196\) −0.781898 −0.0558499
\(197\) 6.76449 0.481950 0.240975 0.970531i \(-0.422533\pi\)
0.240975 + 0.970531i \(0.422533\pi\)
\(198\) 0 0
\(199\) −0.0599479 −0.00424960 −0.00212480 0.999998i \(-0.500676\pi\)
−0.00212480 + 0.999998i \(0.500676\pi\)
\(200\) 3.61556 0.255658
\(201\) 0 0
\(202\) 2.66812 0.187728
\(203\) −4.61826 −0.324139
\(204\) 0 0
\(205\) −23.5216 −1.64282
\(206\) −2.31512 −0.161302
\(207\) 0 0
\(208\) 18.0067 1.24854
\(209\) 14.8774 1.02909
\(210\) 0 0
\(211\) 11.6932 0.804995 0.402497 0.915421i \(-0.368142\pi\)
0.402497 + 0.915421i \(0.368142\pi\)
\(212\) 0.495691 0.0340442
\(213\) 0 0
\(214\) 19.0220 1.30031
\(215\) 0.269898 0.0184069
\(216\) 0 0
\(217\) 34.0823 2.31366
\(218\) 27.5354 1.86494
\(219\) 0 0
\(220\) 0.875790 0.0590458
\(221\) −4.56871 −0.307325
\(222\) 0 0
\(223\) 12.3168 0.824792 0.412396 0.911005i \(-0.364692\pi\)
0.412396 + 0.911005i \(0.364692\pi\)
\(224\) −1.42523 −0.0952273
\(225\) 0 0
\(226\) −14.1999 −0.944562
\(227\) 19.3765 1.28606 0.643032 0.765839i \(-0.277676\pi\)
0.643032 + 0.765839i \(0.277676\pi\)
\(228\) 0 0
\(229\) 5.79473 0.382927 0.191463 0.981500i \(-0.438677\pi\)
0.191463 + 0.981500i \(0.438677\pi\)
\(230\) −3.49022 −0.230138
\(231\) 0 0
\(232\) −2.86569 −0.188142
\(233\) −0.575867 −0.0377263 −0.0188632 0.999822i \(-0.506005\pi\)
−0.0188632 + 0.999822i \(0.506005\pi\)
\(234\) 0 0
\(235\) −1.66653 −0.108713
\(236\) 0.296284 0.0192864
\(237\) 0 0
\(238\) −6.35419 −0.411881
\(239\) −4.15391 −0.268694 −0.134347 0.990934i \(-0.542894\pi\)
−0.134347 + 0.990934i \(0.542894\pi\)
\(240\) 0 0
\(241\) −22.6507 −1.45906 −0.729530 0.683948i \(-0.760261\pi\)
−0.729530 + 0.683948i \(0.760261\pi\)
\(242\) −42.0307 −2.70183
\(243\) 0 0
\(244\) −0.405611 −0.0259666
\(245\) 35.8543 2.29065
\(246\) 0 0
\(247\) 10.7435 0.683593
\(248\) 21.1485 1.34293
\(249\) 0 0
\(250\) 13.0476 0.825202
\(251\) 30.7066 1.93818 0.969092 0.246700i \(-0.0793462\pi\)
0.969092 + 0.246700i \(0.0793462\pi\)
\(252\) 0 0
\(253\) −6.41359 −0.403219
\(254\) −4.36455 −0.273856
\(255\) 0 0
\(256\) −1.30870 −0.0817935
\(257\) 3.85463 0.240445 0.120223 0.992747i \(-0.461639\pi\)
0.120223 + 0.992747i \(0.461639\pi\)
\(258\) 0 0
\(259\) 20.8653 1.29651
\(260\) 0.632440 0.0392222
\(261\) 0 0
\(262\) 21.1079 1.30405
\(263\) −7.92826 −0.488877 −0.244439 0.969665i \(-0.578604\pi\)
−0.244439 + 0.969665i \(0.578604\pi\)
\(264\) 0 0
\(265\) −22.7301 −1.39630
\(266\) 14.9421 0.916161
\(267\) 0 0
\(268\) 0.552536 0.0337515
\(269\) 1.86686 0.113824 0.0569122 0.998379i \(-0.481874\pi\)
0.0569122 + 0.998379i \(0.481874\pi\)
\(270\) 0 0
\(271\) −1.01209 −0.0614802 −0.0307401 0.999527i \(-0.509786\pi\)
−0.0307401 + 0.999527i \(0.509786\pi\)
\(272\) −3.83519 −0.232543
\(273\) 0 0
\(274\) 14.7697 0.892273
\(275\) −8.09184 −0.487956
\(276\) 0 0
\(277\) −12.2498 −0.736018 −0.368009 0.929822i \(-0.619960\pi\)
−0.368009 + 0.929822i \(0.619960\pi\)
\(278\) 23.2192 1.39259
\(279\) 0 0
\(280\) 33.1171 1.97913
\(281\) −7.33020 −0.437283 −0.218641 0.975805i \(-0.570163\pi\)
−0.218641 + 0.975805i \(0.570163\pi\)
\(282\) 0 0
\(283\) −13.3292 −0.792341 −0.396171 0.918177i \(-0.629661\pi\)
−0.396171 + 0.918177i \(0.629661\pi\)
\(284\) 0.590171 0.0350202
\(285\) 0 0
\(286\) −41.4313 −2.44989
\(287\) −43.4111 −2.56248
\(288\) 0 0
\(289\) −16.0269 −0.942760
\(290\) 3.49022 0.204953
\(291\) 0 0
\(292\) 0.0742857 0.00434724
\(293\) 21.2836 1.24340 0.621700 0.783255i \(-0.286442\pi\)
0.621700 + 0.783255i \(0.286442\pi\)
\(294\) 0 0
\(295\) −13.5862 −0.791021
\(296\) 12.9472 0.752539
\(297\) 0 0
\(298\) 32.4292 1.87858
\(299\) −4.63148 −0.267846
\(300\) 0 0
\(301\) 0.498118 0.0287111
\(302\) 25.2044 1.45035
\(303\) 0 0
\(304\) 9.01860 0.517252
\(305\) 18.5995 1.06500
\(306\) 0 0
\(307\) 1.14925 0.0655914 0.0327957 0.999462i \(-0.489559\pi\)
0.0327957 + 0.999462i \(0.489559\pi\)
\(308\) 1.61634 0.0920997
\(309\) 0 0
\(310\) −25.7575 −1.46293
\(311\) −23.1570 −1.31311 −0.656557 0.754276i \(-0.727988\pi\)
−0.656557 + 0.754276i \(0.727988\pi\)
\(312\) 0 0
\(313\) −17.8633 −1.00970 −0.504848 0.863208i \(-0.668451\pi\)
−0.504848 + 0.863208i \(0.668451\pi\)
\(314\) 27.6326 1.55940
\(315\) 0 0
\(316\) −0.222334 −0.0125073
\(317\) 3.33290 0.187194 0.0935972 0.995610i \(-0.470163\pi\)
0.0935972 + 0.995610i \(0.470163\pi\)
\(318\) 0 0
\(319\) 6.41359 0.359092
\(320\) 20.5347 1.14792
\(321\) 0 0
\(322\) −6.44149 −0.358970
\(323\) −2.28823 −0.127321
\(324\) 0 0
\(325\) −5.84341 −0.324134
\(326\) 2.10765 0.116732
\(327\) 0 0
\(328\) −26.9371 −1.48735
\(329\) −3.07572 −0.169570
\(330\) 0 0
\(331\) −29.0066 −1.59435 −0.797173 0.603751i \(-0.793672\pi\)
−0.797173 + 0.603751i \(0.793672\pi\)
\(332\) −0.617126 −0.0338692
\(333\) 0 0
\(334\) −29.2187 −1.59878
\(335\) −25.3368 −1.38430
\(336\) 0 0
\(337\) 24.8722 1.35488 0.677438 0.735580i \(-0.263090\pi\)
0.677438 + 0.735580i \(0.263090\pi\)
\(338\) −11.7868 −0.641119
\(339\) 0 0
\(340\) −0.134702 −0.00730522
\(341\) −47.3316 −2.56315
\(342\) 0 0
\(343\) 33.8443 1.82742
\(344\) 0.309088 0.0166649
\(345\) 0 0
\(346\) −20.5059 −1.10240
\(347\) −24.8230 −1.33257 −0.666284 0.745698i \(-0.732116\pi\)
−0.666284 + 0.745698i \(0.732116\pi\)
\(348\) 0 0
\(349\) 22.2572 1.19140 0.595701 0.803207i \(-0.296874\pi\)
0.595701 + 0.803207i \(0.296874\pi\)
\(350\) −8.12705 −0.434409
\(351\) 0 0
\(352\) 1.97928 0.105496
\(353\) −1.24400 −0.0662113 −0.0331056 0.999452i \(-0.510540\pi\)
−0.0331056 + 0.999452i \(0.510540\pi\)
\(354\) 0 0
\(355\) −27.0626 −1.43633
\(356\) 0.178598 0.00946568
\(357\) 0 0
\(358\) 31.0546 1.64129
\(359\) −30.0137 −1.58406 −0.792031 0.610481i \(-0.790976\pi\)
−0.792031 + 0.610481i \(0.790976\pi\)
\(360\) 0 0
\(361\) −13.6191 −0.716797
\(362\) 3.03345 0.159435
\(363\) 0 0
\(364\) 1.16722 0.0611789
\(365\) −3.40641 −0.178299
\(366\) 0 0
\(367\) 3.80484 0.198611 0.0993055 0.995057i \(-0.468338\pi\)
0.0993055 + 0.995057i \(0.468338\pi\)
\(368\) −3.88788 −0.202670
\(369\) 0 0
\(370\) −15.7688 −0.819782
\(371\) −41.9504 −2.17795
\(372\) 0 0
\(373\) 12.2575 0.634670 0.317335 0.948313i \(-0.397212\pi\)
0.317335 + 0.948313i \(0.397212\pi\)
\(374\) 8.82435 0.456296
\(375\) 0 0
\(376\) −1.90852 −0.0984245
\(377\) 4.63148 0.238533
\(378\) 0 0
\(379\) −12.7959 −0.657280 −0.328640 0.944455i \(-0.606590\pi\)
−0.328640 + 0.944455i \(0.606590\pi\)
\(380\) 0.316756 0.0162493
\(381\) 0 0
\(382\) −3.79935 −0.194392
\(383\) 21.9083 1.11946 0.559730 0.828675i \(-0.310905\pi\)
0.559730 + 0.828675i \(0.310905\pi\)
\(384\) 0 0
\(385\) −74.1182 −3.77741
\(386\) −25.8562 −1.31605
\(387\) 0 0
\(388\) −0.592647 −0.0300871
\(389\) −16.7327 −0.848381 −0.424191 0.905573i \(-0.639441\pi\)
−0.424191 + 0.905573i \(0.639441\pi\)
\(390\) 0 0
\(391\) 0.986447 0.0498868
\(392\) 41.0606 2.07387
\(393\) 0 0
\(394\) −9.43502 −0.475329
\(395\) 10.1952 0.512978
\(396\) 0 0
\(397\) −18.9814 −0.952650 −0.476325 0.879269i \(-0.658031\pi\)
−0.476325 + 0.879269i \(0.658031\pi\)
\(398\) 0.0836146 0.00419122
\(399\) 0 0
\(400\) −4.90523 −0.245262
\(401\) −8.22424 −0.410699 −0.205349 0.978689i \(-0.565833\pi\)
−0.205349 + 0.978689i \(0.565833\pi\)
\(402\) 0 0
\(403\) −34.1798 −1.70262
\(404\) 0.104388 0.00519350
\(405\) 0 0
\(406\) 6.44149 0.319686
\(407\) −28.9766 −1.43632
\(408\) 0 0
\(409\) −25.9458 −1.28294 −0.641468 0.767150i \(-0.721674\pi\)
−0.641468 + 0.767150i \(0.721674\pi\)
\(410\) 32.8077 1.62025
\(411\) 0 0
\(412\) −0.0905775 −0.00446243
\(413\) −25.0745 −1.23384
\(414\) 0 0
\(415\) 28.2986 1.38912
\(416\) 1.42931 0.0700777
\(417\) 0 0
\(418\) −20.7508 −1.01495
\(419\) −14.0526 −0.686516 −0.343258 0.939241i \(-0.611531\pi\)
−0.343258 + 0.939241i \(0.611531\pi\)
\(420\) 0 0
\(421\) 18.6291 0.907926 0.453963 0.891021i \(-0.350010\pi\)
0.453963 + 0.891021i \(0.350010\pi\)
\(422\) −16.3096 −0.793937
\(423\) 0 0
\(424\) −26.0307 −1.26416
\(425\) 1.24457 0.0603706
\(426\) 0 0
\(427\) 34.3269 1.66119
\(428\) 0.744220 0.0359732
\(429\) 0 0
\(430\) −0.376450 −0.0181540
\(431\) 24.2173 1.16651 0.583253 0.812291i \(-0.301780\pi\)
0.583253 + 0.812291i \(0.301780\pi\)
\(432\) 0 0
\(433\) 24.2063 1.16328 0.581639 0.813447i \(-0.302412\pi\)
0.581639 + 0.813447i \(0.302412\pi\)
\(434\) −47.5375 −2.28187
\(435\) 0 0
\(436\) 1.07730 0.0515935
\(437\) −2.31967 −0.110965
\(438\) 0 0
\(439\) 12.9554 0.618329 0.309165 0.951009i \(-0.399951\pi\)
0.309165 + 0.951009i \(0.399951\pi\)
\(440\) −45.9912 −2.19255
\(441\) 0 0
\(442\) 6.37238 0.303103
\(443\) 13.2017 0.627234 0.313617 0.949550i \(-0.398459\pi\)
0.313617 + 0.949550i \(0.398459\pi\)
\(444\) 0 0
\(445\) −8.18970 −0.388229
\(446\) −17.1793 −0.813462
\(447\) 0 0
\(448\) 37.8984 1.79053
\(449\) −25.9082 −1.22269 −0.611343 0.791366i \(-0.709370\pi\)
−0.611343 + 0.791366i \(0.709370\pi\)
\(450\) 0 0
\(451\) 60.2870 2.83880
\(452\) −0.555560 −0.0261313
\(453\) 0 0
\(454\) −27.0261 −1.26840
\(455\) −53.5234 −2.50922
\(456\) 0 0
\(457\) −21.7732 −1.01851 −0.509254 0.860616i \(-0.670078\pi\)
−0.509254 + 0.860616i \(0.670078\pi\)
\(458\) −8.08242 −0.377667
\(459\) 0 0
\(460\) −0.136552 −0.00636679
\(461\) −11.2064 −0.521934 −0.260967 0.965348i \(-0.584041\pi\)
−0.260967 + 0.965348i \(0.584041\pi\)
\(462\) 0 0
\(463\) −12.2954 −0.571415 −0.285707 0.958317i \(-0.592229\pi\)
−0.285707 + 0.958317i \(0.592229\pi\)
\(464\) 3.88788 0.180490
\(465\) 0 0
\(466\) 0.803212 0.0372081
\(467\) −31.0609 −1.43733 −0.718663 0.695358i \(-0.755246\pi\)
−0.718663 + 0.695358i \(0.755246\pi\)
\(468\) 0 0
\(469\) −46.7611 −2.15923
\(470\) 2.32446 0.107219
\(471\) 0 0
\(472\) −15.5590 −0.716162
\(473\) −0.691759 −0.0318071
\(474\) 0 0
\(475\) −2.92666 −0.134284
\(476\) −0.248603 −0.0113947
\(477\) 0 0
\(478\) 5.79382 0.265003
\(479\) 31.7397 1.45022 0.725111 0.688632i \(-0.241788\pi\)
0.725111 + 0.688632i \(0.241788\pi\)
\(480\) 0 0
\(481\) −20.9250 −0.954099
\(482\) 31.5929 1.43902
\(483\) 0 0
\(484\) −1.64442 −0.0747463
\(485\) 27.1761 1.23400
\(486\) 0 0
\(487\) 14.5756 0.660482 0.330241 0.943897i \(-0.392870\pi\)
0.330241 + 0.943897i \(0.392870\pi\)
\(488\) 21.3002 0.964217
\(489\) 0 0
\(490\) −50.0091 −2.25918
\(491\) 9.94122 0.448641 0.224321 0.974515i \(-0.427984\pi\)
0.224321 + 0.974515i \(0.427984\pi\)
\(492\) 0 0
\(493\) −0.986447 −0.0444273
\(494\) −14.9849 −0.674203
\(495\) 0 0
\(496\) −28.6922 −1.28832
\(497\) −49.9462 −2.24039
\(498\) 0 0
\(499\) −4.63611 −0.207541 −0.103770 0.994601i \(-0.533091\pi\)
−0.103770 + 0.994601i \(0.533091\pi\)
\(500\) 0.510477 0.0228292
\(501\) 0 0
\(502\) −42.8292 −1.91156
\(503\) −7.14673 −0.318657 −0.159329 0.987226i \(-0.550933\pi\)
−0.159329 + 0.987226i \(0.550933\pi\)
\(504\) 0 0
\(505\) −4.78676 −0.213008
\(506\) 8.94559 0.397680
\(507\) 0 0
\(508\) −0.170760 −0.00757625
\(509\) −38.5325 −1.70792 −0.853962 0.520336i \(-0.825807\pi\)
−0.853962 + 0.520336i \(0.825807\pi\)
\(510\) 0 0
\(511\) −6.28680 −0.278112
\(512\) 23.4827 1.03780
\(513\) 0 0
\(514\) −5.37639 −0.237143
\(515\) 4.15347 0.183024
\(516\) 0 0
\(517\) 4.27139 0.187856
\(518\) −29.1027 −1.27870
\(519\) 0 0
\(520\) −33.2119 −1.45644
\(521\) −7.23574 −0.317004 −0.158502 0.987359i \(-0.550666\pi\)
−0.158502 + 0.987359i \(0.550666\pi\)
\(522\) 0 0
\(523\) 2.87399 0.125671 0.0628355 0.998024i \(-0.479986\pi\)
0.0628355 + 0.998024i \(0.479986\pi\)
\(524\) 0.825830 0.0360766
\(525\) 0 0
\(526\) 11.0582 0.482162
\(527\) 7.27988 0.317116
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 31.7037 1.37712
\(531\) 0 0
\(532\) 0.584600 0.0253456
\(533\) 43.5354 1.88573
\(534\) 0 0
\(535\) −34.1266 −1.47542
\(536\) −29.0158 −1.25329
\(537\) 0 0
\(538\) −2.60387 −0.112261
\(539\) −91.8962 −3.95825
\(540\) 0 0
\(541\) −0.761604 −0.0327439 −0.0163720 0.999866i \(-0.505212\pi\)
−0.0163720 + 0.999866i \(0.505212\pi\)
\(542\) 1.41165 0.0606356
\(543\) 0 0
\(544\) −0.304425 −0.0130521
\(545\) −49.4003 −2.11608
\(546\) 0 0
\(547\) −21.4518 −0.917211 −0.458606 0.888640i \(-0.651651\pi\)
−0.458606 + 0.888640i \(0.651651\pi\)
\(548\) 0.577855 0.0246848
\(549\) 0 0
\(550\) 11.2864 0.481254
\(551\) 2.31967 0.0988212
\(552\) 0 0
\(553\) 18.8162 0.800144
\(554\) 17.0858 0.725908
\(555\) 0 0
\(556\) 0.908433 0.0385261
\(557\) −9.40382 −0.398452 −0.199226 0.979954i \(-0.563843\pi\)
−0.199226 + 0.979954i \(0.563843\pi\)
\(558\) 0 0
\(559\) −0.499544 −0.0211285
\(560\) −44.9301 −1.89864
\(561\) 0 0
\(562\) 10.2241 0.431276
\(563\) 22.7588 0.959170 0.479585 0.877496i \(-0.340787\pi\)
0.479585 + 0.877496i \(0.340787\pi\)
\(564\) 0 0
\(565\) 25.4754 1.07176
\(566\) 18.5915 0.781457
\(567\) 0 0
\(568\) −30.9922 −1.30040
\(569\) 29.8392 1.25092 0.625461 0.780255i \(-0.284911\pi\)
0.625461 + 0.780255i \(0.284911\pi\)
\(570\) 0 0
\(571\) −21.3918 −0.895220 −0.447610 0.894229i \(-0.647725\pi\)
−0.447610 + 0.894229i \(0.647725\pi\)
\(572\) −1.62097 −0.0677762
\(573\) 0 0
\(574\) 60.5493 2.52728
\(575\) 1.26167 0.0526153
\(576\) 0 0
\(577\) −12.4877 −0.519871 −0.259935 0.965626i \(-0.583701\pi\)
−0.259935 + 0.965626i \(0.583701\pi\)
\(578\) 22.3541 0.929810
\(579\) 0 0
\(580\) 0.136552 0.00567003
\(581\) 52.2274 2.16676
\(582\) 0 0
\(583\) 58.2583 2.41281
\(584\) −3.90104 −0.161426
\(585\) 0 0
\(586\) −29.6861 −1.22632
\(587\) 22.4462 0.926454 0.463227 0.886240i \(-0.346691\pi\)
0.463227 + 0.886240i \(0.346691\pi\)
\(588\) 0 0
\(589\) −17.1189 −0.705372
\(590\) 18.9499 0.780154
\(591\) 0 0
\(592\) −17.5654 −0.721935
\(593\) 18.2624 0.749947 0.374973 0.927036i \(-0.377652\pi\)
0.374973 + 0.927036i \(0.377652\pi\)
\(594\) 0 0
\(595\) 11.3998 0.467347
\(596\) 1.26877 0.0519708
\(597\) 0 0
\(598\) 6.45993 0.264166
\(599\) −23.6299 −0.965490 −0.482745 0.875761i \(-0.660360\pi\)
−0.482745 + 0.875761i \(0.660360\pi\)
\(600\) 0 0
\(601\) −46.9123 −1.91359 −0.956797 0.290756i \(-0.906093\pi\)
−0.956797 + 0.290756i \(0.906093\pi\)
\(602\) −0.694769 −0.0283167
\(603\) 0 0
\(604\) 0.986105 0.0401240
\(605\) 75.4056 3.06567
\(606\) 0 0
\(607\) 35.9639 1.45973 0.729864 0.683592i \(-0.239583\pi\)
0.729864 + 0.683592i \(0.239583\pi\)
\(608\) 0.715867 0.0290323
\(609\) 0 0
\(610\) −25.9423 −1.05037
\(611\) 3.08453 0.124787
\(612\) 0 0
\(613\) 20.7712 0.838941 0.419470 0.907769i \(-0.362216\pi\)
0.419470 + 0.907769i \(0.362216\pi\)
\(614\) −1.60296 −0.0646904
\(615\) 0 0
\(616\) −84.8806 −3.41994
\(617\) −37.6453 −1.51554 −0.757772 0.652519i \(-0.773712\pi\)
−0.757772 + 0.652519i \(0.773712\pi\)
\(618\) 0 0
\(619\) 21.4753 0.863165 0.431583 0.902073i \(-0.357955\pi\)
0.431583 + 0.902073i \(0.357955\pi\)
\(620\) −1.00774 −0.0404719
\(621\) 0 0
\(622\) 32.2991 1.29508
\(623\) −15.1148 −0.605561
\(624\) 0 0
\(625\) −29.7165 −1.18866
\(626\) 24.9156 0.995826
\(627\) 0 0
\(628\) 1.08110 0.0431408
\(629\) 4.45677 0.177703
\(630\) 0 0
\(631\) 20.0076 0.796491 0.398245 0.917279i \(-0.369619\pi\)
0.398245 + 0.917279i \(0.369619\pi\)
\(632\) 11.6756 0.464432
\(633\) 0 0
\(634\) −4.64869 −0.184623
\(635\) 7.83028 0.310735
\(636\) 0 0
\(637\) −66.3615 −2.62934
\(638\) −8.94559 −0.354159
\(639\) 0 0
\(640\) −27.0970 −1.07110
\(641\) −13.3358 −0.526732 −0.263366 0.964696i \(-0.584833\pi\)
−0.263366 + 0.964696i \(0.584833\pi\)
\(642\) 0 0
\(643\) 46.7029 1.84178 0.920892 0.389819i \(-0.127462\pi\)
0.920892 + 0.389819i \(0.127462\pi\)
\(644\) −0.252019 −0.00993093
\(645\) 0 0
\(646\) 3.19159 0.125572
\(647\) −1.75480 −0.0689882 −0.0344941 0.999405i \(-0.510982\pi\)
−0.0344941 + 0.999405i \(0.510982\pi\)
\(648\) 0 0
\(649\) 34.8221 1.36689
\(650\) 8.15031 0.319682
\(651\) 0 0
\(652\) 0.0824602 0.00322939
\(653\) 27.3833 1.07159 0.535796 0.844348i \(-0.320012\pi\)
0.535796 + 0.844348i \(0.320012\pi\)
\(654\) 0 0
\(655\) −37.8688 −1.47966
\(656\) 36.5456 1.42687
\(657\) 0 0
\(658\) 4.28998 0.167241
\(659\) 43.6629 1.70087 0.850433 0.526084i \(-0.176340\pi\)
0.850433 + 0.526084i \(0.176340\pi\)
\(660\) 0 0
\(661\) 31.1204 1.21044 0.605221 0.796058i \(-0.293085\pi\)
0.605221 + 0.796058i \(0.293085\pi\)
\(662\) 40.4580 1.57244
\(663\) 0 0
\(664\) 32.4077 1.25766
\(665\) −26.8071 −1.03953
\(666\) 0 0
\(667\) −1.00000 −0.0387202
\(668\) −1.14316 −0.0442302
\(669\) 0 0
\(670\) 35.3394 1.36528
\(671\) −47.6713 −1.84033
\(672\) 0 0
\(673\) 17.5655 0.677099 0.338549 0.940949i \(-0.390064\pi\)
0.338549 + 0.940949i \(0.390064\pi\)
\(674\) −34.6914 −1.33626
\(675\) 0 0
\(676\) −0.461151 −0.0177366
\(677\) 45.3371 1.74245 0.871223 0.490887i \(-0.163327\pi\)
0.871223 + 0.490887i \(0.163327\pi\)
\(678\) 0 0
\(679\) 50.1558 1.92480
\(680\) 7.07372 0.271265
\(681\) 0 0
\(682\) 66.0175 2.52794
\(683\) 16.6077 0.635475 0.317738 0.948179i \(-0.397077\pi\)
0.317738 + 0.948179i \(0.397077\pi\)
\(684\) 0 0
\(685\) −26.4978 −1.01243
\(686\) −47.2055 −1.80232
\(687\) 0 0
\(688\) −0.419341 −0.0159872
\(689\) 42.0704 1.60276
\(690\) 0 0
\(691\) 17.5575 0.667917 0.333959 0.942588i \(-0.391615\pi\)
0.333959 + 0.942588i \(0.391615\pi\)
\(692\) −0.802278 −0.0304980
\(693\) 0 0
\(694\) 34.6228 1.31426
\(695\) −41.6566 −1.58013
\(696\) 0 0
\(697\) −9.27248 −0.351220
\(698\) −31.0441 −1.17504
\(699\) 0 0
\(700\) −0.317965 −0.0120179
\(701\) −12.5055 −0.472325 −0.236162 0.971714i \(-0.575890\pi\)
−0.236162 + 0.971714i \(0.575890\pi\)
\(702\) 0 0
\(703\) −10.4803 −0.395271
\(704\) −52.6312 −1.98361
\(705\) 0 0
\(706\) 1.73511 0.0653018
\(707\) −8.83437 −0.332251
\(708\) 0 0
\(709\) −36.5122 −1.37125 −0.685623 0.727957i \(-0.740470\pi\)
−0.685623 + 0.727957i \(0.740470\pi\)
\(710\) 37.7465 1.41660
\(711\) 0 0
\(712\) −9.37890 −0.351489
\(713\) 7.37989 0.276379
\(714\) 0 0
\(715\) 74.3304 2.77980
\(716\) 1.21499 0.0454062
\(717\) 0 0
\(718\) 41.8627 1.56230
\(719\) 27.1961 1.01424 0.507122 0.861874i \(-0.330709\pi\)
0.507122 + 0.861874i \(0.330709\pi\)
\(720\) 0 0
\(721\) 7.66558 0.285481
\(722\) 18.9958 0.706950
\(723\) 0 0
\(724\) 0.118682 0.00441077
\(725\) −1.26167 −0.0468573
\(726\) 0 0
\(727\) 0.266634 0.00988892 0.00494446 0.999988i \(-0.498426\pi\)
0.00494446 + 0.999988i \(0.498426\pi\)
\(728\) −61.2953 −2.27176
\(729\) 0 0
\(730\) 4.75121 0.175850
\(731\) 0.106397 0.00393522
\(732\) 0 0
\(733\) −32.7767 −1.21063 −0.605317 0.795985i \(-0.706954\pi\)
−0.605317 + 0.795985i \(0.706954\pi\)
\(734\) −5.30694 −0.195883
\(735\) 0 0
\(736\) −0.308608 −0.0113754
\(737\) 64.9393 2.39207
\(738\) 0 0
\(739\) −39.4205 −1.45011 −0.725053 0.688693i \(-0.758185\pi\)
−0.725053 + 0.688693i \(0.758185\pi\)
\(740\) −0.616943 −0.0226793
\(741\) 0 0
\(742\) 58.5118 2.14804
\(743\) −35.0382 −1.28543 −0.642714 0.766107i \(-0.722191\pi\)
−0.642714 + 0.766107i \(0.722191\pi\)
\(744\) 0 0
\(745\) −58.1800 −2.13155
\(746\) −17.0966 −0.625952
\(747\) 0 0
\(748\) 0.345246 0.0126235
\(749\) −62.9834 −2.30136
\(750\) 0 0
\(751\) −20.1565 −0.735523 −0.367761 0.929920i \(-0.619876\pi\)
−0.367761 + 0.929920i \(0.619876\pi\)
\(752\) 2.58930 0.0944219
\(753\) 0 0
\(754\) −6.45993 −0.235257
\(755\) −45.2183 −1.64566
\(756\) 0 0
\(757\) −35.0219 −1.27289 −0.636446 0.771321i \(-0.719596\pi\)
−0.636446 + 0.771321i \(0.719596\pi\)
\(758\) 17.8475 0.648251
\(759\) 0 0
\(760\) −16.6341 −0.603383
\(761\) −7.20416 −0.261151 −0.130575 0.991438i \(-0.541682\pi\)
−0.130575 + 0.991438i \(0.541682\pi\)
\(762\) 0 0
\(763\) −91.1723 −3.30066
\(764\) −0.148647 −0.00537786
\(765\) 0 0
\(766\) −30.5574 −1.10408
\(767\) 25.1463 0.907979
\(768\) 0 0
\(769\) 26.2557 0.946806 0.473403 0.880846i \(-0.343025\pi\)
0.473403 + 0.880846i \(0.343025\pi\)
\(770\) 103.379 3.72552
\(771\) 0 0
\(772\) −1.01161 −0.0364085
\(773\) −29.5688 −1.06352 −0.531758 0.846896i \(-0.678468\pi\)
−0.531758 + 0.846896i \(0.678468\pi\)
\(774\) 0 0
\(775\) 9.31100 0.334461
\(776\) 31.1222 1.11722
\(777\) 0 0
\(778\) 23.3385 0.836727
\(779\) 21.8046 0.781231
\(780\) 0 0
\(781\) 69.3625 2.48199
\(782\) −1.37588 −0.0492015
\(783\) 0 0
\(784\) −55.7070 −1.98953
\(785\) −49.5745 −1.76939
\(786\) 0 0
\(787\) −6.24715 −0.222687 −0.111343 0.993782i \(-0.535515\pi\)
−0.111343 + 0.993782i \(0.535515\pi\)
\(788\) −0.369138 −0.0131500
\(789\) 0 0
\(790\) −14.2202 −0.505931
\(791\) 47.0170 1.67173
\(792\) 0 0
\(793\) −34.4251 −1.22247
\(794\) 26.4750 0.939563
\(795\) 0 0
\(796\) 0.00327136 0.000115950 0
\(797\) −40.4338 −1.43224 −0.716119 0.697978i \(-0.754083\pi\)
−0.716119 + 0.697978i \(0.754083\pi\)
\(798\) 0 0
\(799\) −0.656965 −0.0232418
\(800\) −0.389361 −0.0137660
\(801\) 0 0
\(802\) 11.4711 0.405057
\(803\) 8.73076 0.308102
\(804\) 0 0
\(805\) 11.5564 0.407311
\(806\) 47.6736 1.67923
\(807\) 0 0
\(808\) −5.48183 −0.192850
\(809\) 54.6396 1.92103 0.960513 0.278237i \(-0.0897500\pi\)
0.960513 + 0.278237i \(0.0897500\pi\)
\(810\) 0 0
\(811\) −5.20954 −0.182932 −0.0914659 0.995808i \(-0.529155\pi\)
−0.0914659 + 0.995808i \(0.529155\pi\)
\(812\) 0.252019 0.00884412
\(813\) 0 0
\(814\) 40.4162 1.41659
\(815\) −3.78125 −0.132452
\(816\) 0 0
\(817\) −0.250196 −0.00875324
\(818\) 36.1888 1.26531
\(819\) 0 0
\(820\) 1.28358 0.0448244
\(821\) 48.3765 1.68835 0.844176 0.536066i \(-0.180090\pi\)
0.844176 + 0.536066i \(0.180090\pi\)
\(822\) 0 0
\(823\) −14.9775 −0.522083 −0.261042 0.965328i \(-0.584066\pi\)
−0.261042 + 0.965328i \(0.584066\pi\)
\(824\) 4.75658 0.165703
\(825\) 0 0
\(826\) 34.9736 1.21689
\(827\) 34.3322 1.19385 0.596924 0.802298i \(-0.296390\pi\)
0.596924 + 0.802298i \(0.296390\pi\)
\(828\) 0 0
\(829\) −3.10314 −0.107777 −0.0538883 0.998547i \(-0.517161\pi\)
−0.0538883 + 0.998547i \(0.517161\pi\)
\(830\) −39.4705 −1.37004
\(831\) 0 0
\(832\) −38.0069 −1.31765
\(833\) 14.1342 0.489720
\(834\) 0 0
\(835\) 52.4201 1.81407
\(836\) −0.811860 −0.0280788
\(837\) 0 0
\(838\) 19.6004 0.677085
\(839\) 34.6315 1.19561 0.597807 0.801640i \(-0.296039\pi\)
0.597807 + 0.801640i \(0.296039\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −25.9836 −0.895454
\(843\) 0 0
\(844\) −0.638099 −0.0219643
\(845\) 21.1463 0.727455
\(846\) 0 0
\(847\) 139.167 4.78184
\(848\) 35.3159 1.21275
\(849\) 0 0
\(850\) −1.73591 −0.0595413
\(851\) 4.51800 0.154875
\(852\) 0 0
\(853\) 3.13665 0.107397 0.0536985 0.998557i \(-0.482899\pi\)
0.0536985 + 0.998557i \(0.482899\pi\)
\(854\) −47.8787 −1.63837
\(855\) 0 0
\(856\) −39.0819 −1.33579
\(857\) −22.7250 −0.776272 −0.388136 0.921602i \(-0.626881\pi\)
−0.388136 + 0.921602i \(0.626881\pi\)
\(858\) 0 0
\(859\) −38.9334 −1.32839 −0.664196 0.747559i \(-0.731226\pi\)
−0.664196 + 0.747559i \(0.731226\pi\)
\(860\) −0.0147283 −0.000502231 0
\(861\) 0 0
\(862\) −33.7780 −1.15048
\(863\) −8.80848 −0.299844 −0.149922 0.988698i \(-0.547902\pi\)
−0.149922 + 0.988698i \(0.547902\pi\)
\(864\) 0 0
\(865\) 36.7888 1.25086
\(866\) −33.7626 −1.14730
\(867\) 0 0
\(868\) −1.85987 −0.0631281
\(869\) −26.1308 −0.886428
\(870\) 0 0
\(871\) 46.8950 1.58898
\(872\) −56.5735 −1.91582
\(873\) 0 0
\(874\) 3.23544 0.109441
\(875\) −43.2017 −1.46049
\(876\) 0 0
\(877\) 37.9186 1.28042 0.640211 0.768199i \(-0.278847\pi\)
0.640211 + 0.768199i \(0.278847\pi\)
\(878\) −18.0701 −0.609835
\(879\) 0 0
\(880\) 62.3964 2.10338
\(881\) −39.8855 −1.34378 −0.671889 0.740652i \(-0.734517\pi\)
−0.671889 + 0.740652i \(0.734517\pi\)
\(882\) 0 0
\(883\) 38.3760 1.29145 0.645727 0.763568i \(-0.276554\pi\)
0.645727 + 0.763568i \(0.276554\pi\)
\(884\) 0.249315 0.00838536
\(885\) 0 0
\(886\) −18.4136 −0.618618
\(887\) 37.1535 1.24749 0.623746 0.781627i \(-0.285610\pi\)
0.623746 + 0.781627i \(0.285610\pi\)
\(888\) 0 0
\(889\) 14.4514 0.484685
\(890\) 11.4229 0.382896
\(891\) 0 0
\(892\) −0.672126 −0.0225045
\(893\) 1.54488 0.0516974
\(894\) 0 0
\(895\) −55.7139 −1.86231
\(896\) −50.0098 −1.67071
\(897\) 0 0
\(898\) 36.1365 1.20589
\(899\) −7.37989 −0.246133
\(900\) 0 0
\(901\) −8.96047 −0.298517
\(902\) −84.0874 −2.79981
\(903\) 0 0
\(904\) 29.1746 0.970334
\(905\) −5.44220 −0.180905
\(906\) 0 0
\(907\) 45.7862 1.52031 0.760154 0.649743i \(-0.225124\pi\)
0.760154 + 0.649743i \(0.225124\pi\)
\(908\) −1.05738 −0.0350903
\(909\) 0 0
\(910\) 74.6538 2.47475
\(911\) 4.87510 0.161519 0.0807597 0.996734i \(-0.474265\pi\)
0.0807597 + 0.996734i \(0.474265\pi\)
\(912\) 0 0
\(913\) −72.5305 −2.40041
\(914\) 30.3690 1.00452
\(915\) 0 0
\(916\) −0.316219 −0.0104482
\(917\) −69.8901 −2.30797
\(918\) 0 0
\(919\) −51.1956 −1.68879 −0.844393 0.535724i \(-0.820039\pi\)
−0.844393 + 0.535724i \(0.820039\pi\)
\(920\) 7.17090 0.236418
\(921\) 0 0
\(922\) 15.6305 0.514764
\(923\) 50.0892 1.64871
\(924\) 0 0
\(925\) 5.70023 0.187422
\(926\) 17.1494 0.563565
\(927\) 0 0
\(928\) 0.308608 0.0101305
\(929\) 21.7566 0.713811 0.356905 0.934141i \(-0.383832\pi\)
0.356905 + 0.934141i \(0.383832\pi\)
\(930\) 0 0
\(931\) −33.2370 −1.08930
\(932\) 0.0314251 0.00102936
\(933\) 0 0
\(934\) 43.3233 1.41758
\(935\) −15.8314 −0.517743
\(936\) 0 0
\(937\) −50.8293 −1.66052 −0.830260 0.557376i \(-0.811808\pi\)
−0.830260 + 0.557376i \(0.811808\pi\)
\(938\) 65.2218 2.12957
\(939\) 0 0
\(940\) 0.0909427 0.00296622
\(941\) 6.67920 0.217736 0.108868 0.994056i \(-0.465277\pi\)
0.108868 + 0.994056i \(0.465277\pi\)
\(942\) 0 0
\(943\) −9.39988 −0.306102
\(944\) 21.1090 0.687038
\(945\) 0 0
\(946\) 0.964857 0.0313702
\(947\) 11.3759 0.369669 0.184834 0.982770i \(-0.440825\pi\)
0.184834 + 0.982770i \(0.440825\pi\)
\(948\) 0 0
\(949\) 6.30480 0.204662
\(950\) 4.08207 0.132440
\(951\) 0 0
\(952\) 13.0551 0.423119
\(953\) −52.7421 −1.70848 −0.854242 0.519876i \(-0.825978\pi\)
−0.854242 + 0.519876i \(0.825978\pi\)
\(954\) 0 0
\(955\) 6.81627 0.220569
\(956\) 0.226679 0.00733132
\(957\) 0 0
\(958\) −44.2700 −1.43030
\(959\) −48.9039 −1.57919
\(960\) 0 0
\(961\) 23.4628 0.756866
\(962\) 29.1860 0.940993
\(963\) 0 0
\(964\) 1.23605 0.0398105
\(965\) 46.3876 1.49327
\(966\) 0 0
\(967\) 26.2173 0.843091 0.421545 0.906807i \(-0.361488\pi\)
0.421545 + 0.906807i \(0.361488\pi\)
\(968\) 86.3549 2.77555
\(969\) 0 0
\(970\) −37.9049 −1.21705
\(971\) 33.1078 1.06248 0.531239 0.847222i \(-0.321727\pi\)
0.531239 + 0.847222i \(0.321727\pi\)
\(972\) 0 0
\(973\) −76.8807 −2.46468
\(974\) −20.3298 −0.651409
\(975\) 0 0
\(976\) −28.8981 −0.925005
\(977\) −24.2127 −0.774632 −0.387316 0.921947i \(-0.626598\pi\)
−0.387316 + 0.921947i \(0.626598\pi\)
\(978\) 0 0
\(979\) 20.9906 0.670861
\(980\) −1.95657 −0.0625003
\(981\) 0 0
\(982\) −13.8659 −0.442478
\(983\) −26.3114 −0.839203 −0.419601 0.907708i \(-0.637830\pi\)
−0.419601 + 0.907708i \(0.637830\pi\)
\(984\) 0 0
\(985\) 16.9270 0.539339
\(986\) 1.37588 0.0438171
\(987\) 0 0
\(988\) −0.586273 −0.0186518
\(989\) 0.107858 0.00342970
\(990\) 0 0
\(991\) 44.4800 1.41295 0.706477 0.707736i \(-0.250284\pi\)
0.706477 + 0.707736i \(0.250284\pi\)
\(992\) −2.27749 −0.0723104
\(993\) 0 0
\(994\) 69.6643 2.20962
\(995\) −0.150010 −0.00475563
\(996\) 0 0
\(997\) −4.43615 −0.140494 −0.0702471 0.997530i \(-0.522379\pi\)
−0.0702471 + 0.997530i \(0.522379\pi\)
\(998\) 6.46639 0.204690
\(999\) 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 6003.2.a.k.1.4 10
3.2 odd 2 2001.2.a.k.1.7 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.k.1.7 10 3.2 odd 2
6003.2.a.k.1.4 10 1.1 even 1 trivial