Properties

Label 608.4.a.h.1.2
Level $608$
Weight $4$
Character 608.1
Self dual yes
Analytic conductor $35.873$
Analytic rank $1$
Dimension $5$
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] = [608,4,Mod(1,608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(608, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("608.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 608 = 2^{5} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 608.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: \(35.8731612835\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 28x^{3} - 8x^{2} + 73x - 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.31928\) of defining polynomial
Character \(\chi\) \(=\) 608.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.67449 q^{3} -7.12795 q^{5} -0.511313 q^{7} -13.4981 q^{9} +17.5714 q^{11} +52.1247 q^{13} +26.1915 q^{15} +68.1582 q^{17} -19.0000 q^{19} +1.87881 q^{21} +48.8262 q^{23} -74.1924 q^{25} +148.810 q^{27} -87.1627 q^{29} +53.2537 q^{31} -64.5659 q^{33} +3.64461 q^{35} -199.635 q^{37} -191.532 q^{39} -372.432 q^{41} +416.493 q^{43} +96.2140 q^{45} -221.394 q^{47} -342.739 q^{49} -250.446 q^{51} +107.677 q^{53} -125.248 q^{55} +69.8153 q^{57} +155.785 q^{59} -229.061 q^{61} +6.90178 q^{63} -371.542 q^{65} +580.065 q^{67} -179.411 q^{69} -38.2227 q^{71} -803.910 q^{73} +272.619 q^{75} -8.98449 q^{77} -1034.63 q^{79} -182.350 q^{81} -1053.94 q^{83} -485.828 q^{85} +320.278 q^{87} -641.103 q^{89} -26.6521 q^{91} -195.680 q^{93} +135.431 q^{95} +19.9998 q^{97} -237.181 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 6 q^{3} + 5 q^{5} - 7 q^{7} - 5 q^{9} + 13 q^{11} - 72 q^{13} - 72 q^{15} - 59 q^{17} - 95 q^{19} - 224 q^{21} - 52 q^{23} - 86 q^{25} + 54 q^{27} - 128 q^{29} + 110 q^{31} - 68 q^{33} - 45 q^{35}+ \cdots + 601 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.67449 −0.707155 −0.353578 0.935405i \(-0.615035\pi\)
−0.353578 + 0.935405i \(0.615035\pi\)
\(4\) 0 0
\(5\) −7.12795 −0.637543 −0.318771 0.947832i \(-0.603270\pi\)
−0.318771 + 0.947832i \(0.603270\pi\)
\(6\) 0 0
\(7\) −0.511313 −0.0276083 −0.0138042 0.999905i \(-0.504394\pi\)
−0.0138042 + 0.999905i \(0.504394\pi\)
\(8\) 0 0
\(9\) −13.4981 −0.499931
\(10\) 0 0
\(11\) 17.5714 0.481634 0.240817 0.970570i \(-0.422585\pi\)
0.240817 + 0.970570i \(0.422585\pi\)
\(12\) 0 0
\(13\) 52.1247 1.11206 0.556030 0.831162i \(-0.312324\pi\)
0.556030 + 0.831162i \(0.312324\pi\)
\(14\) 0 0
\(15\) 26.1915 0.450842
\(16\) 0 0
\(17\) 68.1582 0.972399 0.486200 0.873848i \(-0.338383\pi\)
0.486200 + 0.873848i \(0.338383\pi\)
\(18\) 0 0
\(19\) −19.0000 −0.229416
\(20\) 0 0
\(21\) 1.87881 0.0195234
\(22\) 0 0
\(23\) 48.8262 0.442651 0.221325 0.975200i \(-0.428962\pi\)
0.221325 + 0.975200i \(0.428962\pi\)
\(24\) 0 0
\(25\) −74.1924 −0.593539
\(26\) 0 0
\(27\) 148.810 1.06068
\(28\) 0 0
\(29\) −87.1627 −0.558128 −0.279064 0.960273i \(-0.590024\pi\)
−0.279064 + 0.960273i \(0.590024\pi\)
\(30\) 0 0
\(31\) 53.2537 0.308537 0.154269 0.988029i \(-0.450698\pi\)
0.154269 + 0.988029i \(0.450698\pi\)
\(32\) 0 0
\(33\) −64.5659 −0.340590
\(34\) 0 0
\(35\) 3.64461 0.0176015
\(36\) 0 0
\(37\) −199.635 −0.887020 −0.443510 0.896269i \(-0.646267\pi\)
−0.443510 + 0.896269i \(0.646267\pi\)
\(38\) 0 0
\(39\) −191.532 −0.786400
\(40\) 0 0
\(41\) −372.432 −1.41864 −0.709318 0.704889i \(-0.750997\pi\)
−0.709318 + 0.704889i \(0.750997\pi\)
\(42\) 0 0
\(43\) 416.493 1.47708 0.738542 0.674207i \(-0.235515\pi\)
0.738542 + 0.674207i \(0.235515\pi\)
\(44\) 0 0
\(45\) 96.2140 0.318728
\(46\) 0 0
\(47\) −221.394 −0.687098 −0.343549 0.939135i \(-0.611629\pi\)
−0.343549 + 0.939135i \(0.611629\pi\)
\(48\) 0 0
\(49\) −342.739 −0.999238
\(50\) 0 0
\(51\) −250.446 −0.687637
\(52\) 0 0
\(53\) 107.677 0.279066 0.139533 0.990217i \(-0.455440\pi\)
0.139533 + 0.990217i \(0.455440\pi\)
\(54\) 0 0
\(55\) −125.248 −0.307063
\(56\) 0 0
\(57\) 69.8153 0.162233
\(58\) 0 0
\(59\) 155.785 0.343753 0.171877 0.985118i \(-0.445017\pi\)
0.171877 + 0.985118i \(0.445017\pi\)
\(60\) 0 0
\(61\) −229.061 −0.480791 −0.240395 0.970675i \(-0.577277\pi\)
−0.240395 + 0.970675i \(0.577277\pi\)
\(62\) 0 0
\(63\) 6.90178 0.0138023
\(64\) 0 0
\(65\) −371.542 −0.708987
\(66\) 0 0
\(67\) 580.065 1.05770 0.528852 0.848714i \(-0.322623\pi\)
0.528852 + 0.848714i \(0.322623\pi\)
\(68\) 0 0
\(69\) −179.411 −0.313023
\(70\) 0 0
\(71\) −38.2227 −0.0638901 −0.0319451 0.999490i \(-0.510170\pi\)
−0.0319451 + 0.999490i \(0.510170\pi\)
\(72\) 0 0
\(73\) −803.910 −1.28891 −0.644456 0.764641i \(-0.722916\pi\)
−0.644456 + 0.764641i \(0.722916\pi\)
\(74\) 0 0
\(75\) 272.619 0.419724
\(76\) 0 0
\(77\) −8.98449 −0.0132971
\(78\) 0 0
\(79\) −1034.63 −1.47348 −0.736739 0.676177i \(-0.763635\pi\)
−0.736739 + 0.676177i \(0.763635\pi\)
\(80\) 0 0
\(81\) −182.350 −0.250137
\(82\) 0 0
\(83\) −1053.94 −1.39379 −0.696897 0.717171i \(-0.745436\pi\)
−0.696897 + 0.717171i \(0.745436\pi\)
\(84\) 0 0
\(85\) −485.828 −0.619946
\(86\) 0 0
\(87\) 320.278 0.394683
\(88\) 0 0
\(89\) −641.103 −0.763560 −0.381780 0.924253i \(-0.624689\pi\)
−0.381780 + 0.924253i \(0.624689\pi\)
\(90\) 0 0
\(91\) −26.6521 −0.0307021
\(92\) 0 0
\(93\) −195.680 −0.218184
\(94\) 0 0
\(95\) 135.431 0.146262
\(96\) 0 0
\(97\) 19.9998 0.0209348 0.0104674 0.999945i \(-0.496668\pi\)
0.0104674 + 0.999945i \(0.496668\pi\)
\(98\) 0 0
\(99\) −237.181 −0.240784
\(100\) 0 0
\(101\) −473.527 −0.466512 −0.233256 0.972415i \(-0.574938\pi\)
−0.233256 + 0.972415i \(0.574938\pi\)
\(102\) 0 0
\(103\) −289.467 −0.276913 −0.138456 0.990369i \(-0.544214\pi\)
−0.138456 + 0.990369i \(0.544214\pi\)
\(104\) 0 0
\(105\) −13.3921 −0.0124470
\(106\) 0 0
\(107\) −390.982 −0.353249 −0.176625 0.984278i \(-0.556518\pi\)
−0.176625 + 0.984278i \(0.556518\pi\)
\(108\) 0 0
\(109\) −1268.65 −1.11481 −0.557407 0.830239i \(-0.688204\pi\)
−0.557407 + 0.830239i \(0.688204\pi\)
\(110\) 0 0
\(111\) 733.555 0.627261
\(112\) 0 0
\(113\) 1100.46 0.916132 0.458066 0.888918i \(-0.348542\pi\)
0.458066 + 0.888918i \(0.348542\pi\)
\(114\) 0 0
\(115\) −348.030 −0.282209
\(116\) 0 0
\(117\) −703.587 −0.555954
\(118\) 0 0
\(119\) −34.8502 −0.0268463
\(120\) 0 0
\(121\) −1022.25 −0.768028
\(122\) 0 0
\(123\) 1368.50 1.00320
\(124\) 0 0
\(125\) 1419.83 1.01595
\(126\) 0 0
\(127\) −2799.28 −1.95587 −0.977935 0.208907i \(-0.933009\pi\)
−0.977935 + 0.208907i \(0.933009\pi\)
\(128\) 0 0
\(129\) −1530.40 −1.04453
\(130\) 0 0
\(131\) 691.265 0.461039 0.230520 0.973068i \(-0.425957\pi\)
0.230520 + 0.973068i \(0.425957\pi\)
\(132\) 0 0
\(133\) 9.71495 0.00633378
\(134\) 0 0
\(135\) −1060.71 −0.676232
\(136\) 0 0
\(137\) 1044.00 0.651057 0.325528 0.945532i \(-0.394458\pi\)
0.325528 + 0.945532i \(0.394458\pi\)
\(138\) 0 0
\(139\) 1663.13 1.01486 0.507429 0.861694i \(-0.330596\pi\)
0.507429 + 0.861694i \(0.330596\pi\)
\(140\) 0 0
\(141\) 813.509 0.485885
\(142\) 0 0
\(143\) 915.904 0.535607
\(144\) 0 0
\(145\) 621.291 0.355830
\(146\) 0 0
\(147\) 1259.39 0.706616
\(148\) 0 0
\(149\) −887.849 −0.488157 −0.244079 0.969755i \(-0.578485\pi\)
−0.244079 + 0.969755i \(0.578485\pi\)
\(150\) 0 0
\(151\) −55.6469 −0.0299899 −0.0149950 0.999888i \(-0.504773\pi\)
−0.0149950 + 0.999888i \(0.504773\pi\)
\(152\) 0 0
\(153\) −920.009 −0.486133
\(154\) 0 0
\(155\) −379.590 −0.196706
\(156\) 0 0
\(157\) −1643.09 −0.835239 −0.417620 0.908622i \(-0.637136\pi\)
−0.417620 + 0.908622i \(0.637136\pi\)
\(158\) 0 0
\(159\) −395.656 −0.197343
\(160\) 0 0
\(161\) −24.9655 −0.0122208
\(162\) 0 0
\(163\) 2241.78 1.07724 0.538619 0.842549i \(-0.318946\pi\)
0.538619 + 0.842549i \(0.318946\pi\)
\(164\) 0 0
\(165\) 460.222 0.217141
\(166\) 0 0
\(167\) −2543.96 −1.17879 −0.589393 0.807847i \(-0.700633\pi\)
−0.589393 + 0.807847i \(0.700633\pi\)
\(168\) 0 0
\(169\) 519.985 0.236680
\(170\) 0 0
\(171\) 256.465 0.114692
\(172\) 0 0
\(173\) −1084.28 −0.476512 −0.238256 0.971202i \(-0.576576\pi\)
−0.238256 + 0.971202i \(0.576576\pi\)
\(174\) 0 0
\(175\) 37.9356 0.0163866
\(176\) 0 0
\(177\) −572.429 −0.243087
\(178\) 0 0
\(179\) 2053.26 0.857362 0.428681 0.903456i \(-0.358978\pi\)
0.428681 + 0.903456i \(0.358978\pi\)
\(180\) 0 0
\(181\) −2066.62 −0.848676 −0.424338 0.905504i \(-0.639493\pi\)
−0.424338 + 0.905504i \(0.639493\pi\)
\(182\) 0 0
\(183\) 841.681 0.339994
\(184\) 0 0
\(185\) 1422.99 0.565513
\(186\) 0 0
\(187\) 1197.64 0.468341
\(188\) 0 0
\(189\) −76.0885 −0.0292837
\(190\) 0 0
\(191\) 1250.64 0.473787 0.236893 0.971536i \(-0.423871\pi\)
0.236893 + 0.971536i \(0.423871\pi\)
\(192\) 0 0
\(193\) 2815.01 1.04989 0.524945 0.851136i \(-0.324086\pi\)
0.524945 + 0.851136i \(0.324086\pi\)
\(194\) 0 0
\(195\) 1365.23 0.501364
\(196\) 0 0
\(197\) −593.857 −0.214774 −0.107387 0.994217i \(-0.534248\pi\)
−0.107387 + 0.994217i \(0.534248\pi\)
\(198\) 0 0
\(199\) −15.4805 −0.00551450 −0.00275725 0.999996i \(-0.500878\pi\)
−0.00275725 + 0.999996i \(0.500878\pi\)
\(200\) 0 0
\(201\) −2131.44 −0.747961
\(202\) 0 0
\(203\) 44.5674 0.0154090
\(204\) 0 0
\(205\) 2654.67 0.904441
\(206\) 0 0
\(207\) −659.063 −0.221295
\(208\) 0 0
\(209\) −333.857 −0.110495
\(210\) 0 0
\(211\) −2228.10 −0.726961 −0.363481 0.931602i \(-0.618412\pi\)
−0.363481 + 0.931602i \(0.618412\pi\)
\(212\) 0 0
\(213\) 140.449 0.0451802
\(214\) 0 0
\(215\) −2968.74 −0.941704
\(216\) 0 0
\(217\) −27.2293 −0.00851819
\(218\) 0 0
\(219\) 2953.96 0.911461
\(220\) 0 0
\(221\) 3552.73 1.08137
\(222\) 0 0
\(223\) −2068.17 −0.621054 −0.310527 0.950565i \(-0.600506\pi\)
−0.310527 + 0.950565i \(0.600506\pi\)
\(224\) 0 0
\(225\) 1001.46 0.296729
\(226\) 0 0
\(227\) 591.946 0.173079 0.0865393 0.996248i \(-0.472419\pi\)
0.0865393 + 0.996248i \(0.472419\pi\)
\(228\) 0 0
\(229\) −2519.07 −0.726920 −0.363460 0.931610i \(-0.618405\pi\)
−0.363460 + 0.931610i \(0.618405\pi\)
\(230\) 0 0
\(231\) 33.0134 0.00940313
\(232\) 0 0
\(233\) 3985.94 1.12072 0.560359 0.828250i \(-0.310663\pi\)
0.560359 + 0.828250i \(0.310663\pi\)
\(234\) 0 0
\(235\) 1578.08 0.438054
\(236\) 0 0
\(237\) 3801.73 1.04198
\(238\) 0 0
\(239\) −2371.73 −0.641901 −0.320951 0.947096i \(-0.604002\pi\)
−0.320951 + 0.947096i \(0.604002\pi\)
\(240\) 0 0
\(241\) 5295.59 1.41543 0.707715 0.706498i \(-0.249726\pi\)
0.707715 + 0.706498i \(0.249726\pi\)
\(242\) 0 0
\(243\) −3347.82 −0.883798
\(244\) 0 0
\(245\) 2443.02 0.637057
\(246\) 0 0
\(247\) −990.369 −0.255124
\(248\) 0 0
\(249\) 3872.69 0.985629
\(250\) 0 0
\(251\) 5766.76 1.45018 0.725088 0.688656i \(-0.241799\pi\)
0.725088 + 0.688656i \(0.241799\pi\)
\(252\) 0 0
\(253\) 857.945 0.213196
\(254\) 0 0
\(255\) 1785.17 0.438398
\(256\) 0 0
\(257\) 3631.88 0.881518 0.440759 0.897625i \(-0.354709\pi\)
0.440759 + 0.897625i \(0.354709\pi\)
\(258\) 0 0
\(259\) 102.076 0.0244891
\(260\) 0 0
\(261\) 1176.53 0.279025
\(262\) 0 0
\(263\) −5848.72 −1.37128 −0.685641 0.727940i \(-0.740478\pi\)
−0.685641 + 0.727940i \(0.740478\pi\)
\(264\) 0 0
\(265\) −767.513 −0.177917
\(266\) 0 0
\(267\) 2355.73 0.539955
\(268\) 0 0
\(269\) −4283.31 −0.970848 −0.485424 0.874279i \(-0.661335\pi\)
−0.485424 + 0.874279i \(0.661335\pi\)
\(270\) 0 0
\(271\) −2350.55 −0.526885 −0.263443 0.964675i \(-0.584858\pi\)
−0.263443 + 0.964675i \(0.584858\pi\)
\(272\) 0 0
\(273\) 97.9327 0.0217112
\(274\) 0 0
\(275\) −1303.66 −0.285869
\(276\) 0 0
\(277\) 7901.74 1.71397 0.856985 0.515342i \(-0.172335\pi\)
0.856985 + 0.515342i \(0.172335\pi\)
\(278\) 0 0
\(279\) −718.826 −0.154247
\(280\) 0 0
\(281\) −1555.29 −0.330180 −0.165090 0.986279i \(-0.552791\pi\)
−0.165090 + 0.986279i \(0.552791\pi\)
\(282\) 0 0
\(283\) −3037.74 −0.638073 −0.319036 0.947742i \(-0.603359\pi\)
−0.319036 + 0.947742i \(0.603359\pi\)
\(284\) 0 0
\(285\) −497.639 −0.103430
\(286\) 0 0
\(287\) 190.429 0.0391662
\(288\) 0 0
\(289\) −267.461 −0.0544394
\(290\) 0 0
\(291\) −73.4892 −0.0148042
\(292\) 0 0
\(293\) 4477.08 0.892676 0.446338 0.894864i \(-0.352728\pi\)
0.446338 + 0.894864i \(0.352728\pi\)
\(294\) 0 0
\(295\) −1110.43 −0.219157
\(296\) 0 0
\(297\) 2614.80 0.510862
\(298\) 0 0
\(299\) 2545.05 0.492254
\(300\) 0 0
\(301\) −212.959 −0.0407798
\(302\) 0 0
\(303\) 1739.97 0.329896
\(304\) 0 0
\(305\) 1632.73 0.306525
\(306\) 0 0
\(307\) −1580.87 −0.293893 −0.146947 0.989144i \(-0.546945\pi\)
−0.146947 + 0.989144i \(0.546945\pi\)
\(308\) 0 0
\(309\) 1063.64 0.195820
\(310\) 0 0
\(311\) 3241.23 0.590975 0.295488 0.955347i \(-0.404518\pi\)
0.295488 + 0.955347i \(0.404518\pi\)
\(312\) 0 0
\(313\) −9579.34 −1.72989 −0.864946 0.501865i \(-0.832647\pi\)
−0.864946 + 0.501865i \(0.832647\pi\)
\(314\) 0 0
\(315\) −49.1955 −0.00879954
\(316\) 0 0
\(317\) 5152.39 0.912894 0.456447 0.889751i \(-0.349122\pi\)
0.456447 + 0.889751i \(0.349122\pi\)
\(318\) 0 0
\(319\) −1531.57 −0.268813
\(320\) 0 0
\(321\) 1436.66 0.249802
\(322\) 0 0
\(323\) −1295.01 −0.223084
\(324\) 0 0
\(325\) −3867.26 −0.660052
\(326\) 0 0
\(327\) 4661.64 0.788347
\(328\) 0 0
\(329\) 113.202 0.0189696
\(330\) 0 0
\(331\) −4909.62 −0.815278 −0.407639 0.913143i \(-0.633648\pi\)
−0.407639 + 0.913143i \(0.633648\pi\)
\(332\) 0 0
\(333\) 2694.70 0.443449
\(334\) 0 0
\(335\) −4134.67 −0.674332
\(336\) 0 0
\(337\) −5662.83 −0.915353 −0.457677 0.889119i \(-0.651318\pi\)
−0.457677 + 0.889119i \(0.651318\pi\)
\(338\) 0 0
\(339\) −4043.64 −0.647847
\(340\) 0 0
\(341\) 935.743 0.148602
\(342\) 0 0
\(343\) 350.627 0.0551956
\(344\) 0 0
\(345\) 1278.83 0.199565
\(346\) 0 0
\(347\) −12194.6 −1.88658 −0.943289 0.331973i \(-0.892286\pi\)
−0.943289 + 0.331973i \(0.892286\pi\)
\(348\) 0 0
\(349\) 2887.35 0.442854 0.221427 0.975177i \(-0.428929\pi\)
0.221427 + 0.975177i \(0.428929\pi\)
\(350\) 0 0
\(351\) 7756.67 1.17955
\(352\) 0 0
\(353\) 1828.06 0.275632 0.137816 0.990458i \(-0.455992\pi\)
0.137816 + 0.990458i \(0.455992\pi\)
\(354\) 0 0
\(355\) 272.449 0.0407327
\(356\) 0 0
\(357\) 128.057 0.0189845
\(358\) 0 0
\(359\) −2027.89 −0.298128 −0.149064 0.988828i \(-0.547626\pi\)
−0.149064 + 0.988828i \(0.547626\pi\)
\(360\) 0 0
\(361\) 361.000 0.0526316
\(362\) 0 0
\(363\) 3756.23 0.543115
\(364\) 0 0
\(365\) 5730.23 0.821737
\(366\) 0 0
\(367\) 3450.93 0.490836 0.245418 0.969417i \(-0.421075\pi\)
0.245418 + 0.969417i \(0.421075\pi\)
\(368\) 0 0
\(369\) 5027.14 0.709220
\(370\) 0 0
\(371\) −55.0565 −0.00770455
\(372\) 0 0
\(373\) −5713.27 −0.793089 −0.396544 0.918016i \(-0.629791\pi\)
−0.396544 + 0.918016i \(0.629791\pi\)
\(374\) 0 0
\(375\) −5217.16 −0.718434
\(376\) 0 0
\(377\) −4543.33 −0.620672
\(378\) 0 0
\(379\) 7828.71 1.06104 0.530519 0.847673i \(-0.321997\pi\)
0.530519 + 0.847673i \(0.321997\pi\)
\(380\) 0 0
\(381\) 10285.9 1.38310
\(382\) 0 0
\(383\) 11760.3 1.56898 0.784492 0.620139i \(-0.212924\pi\)
0.784492 + 0.620139i \(0.212924\pi\)
\(384\) 0 0
\(385\) 64.0410 0.00847748
\(386\) 0 0
\(387\) −5621.89 −0.738441
\(388\) 0 0
\(389\) −6717.76 −0.875589 −0.437795 0.899075i \(-0.644240\pi\)
−0.437795 + 0.899075i \(0.644240\pi\)
\(390\) 0 0
\(391\) 3327.90 0.430433
\(392\) 0 0
\(393\) −2540.05 −0.326026
\(394\) 0 0
\(395\) 7374.78 0.939406
\(396\) 0 0
\(397\) −12360.2 −1.56257 −0.781287 0.624172i \(-0.785436\pi\)
−0.781287 + 0.624172i \(0.785436\pi\)
\(398\) 0 0
\(399\) −35.6975 −0.00447897
\(400\) 0 0
\(401\) 2006.45 0.249869 0.124934 0.992165i \(-0.460128\pi\)
0.124934 + 0.992165i \(0.460128\pi\)
\(402\) 0 0
\(403\) 2775.83 0.343112
\(404\) 0 0
\(405\) 1299.78 0.159473
\(406\) 0 0
\(407\) −3507.86 −0.427219
\(408\) 0 0
\(409\) 5154.98 0.623221 0.311611 0.950210i \(-0.399132\pi\)
0.311611 + 0.950210i \(0.399132\pi\)
\(410\) 0 0
\(411\) −3836.16 −0.460398
\(412\) 0 0
\(413\) −79.6548 −0.00949045
\(414\) 0 0
\(415\) 7512.42 0.888603
\(416\) 0 0
\(417\) −6111.17 −0.717662
\(418\) 0 0
\(419\) 5037.60 0.587357 0.293679 0.955904i \(-0.405120\pi\)
0.293679 + 0.955904i \(0.405120\pi\)
\(420\) 0 0
\(421\) −4025.51 −0.466013 −0.233006 0.972475i \(-0.574856\pi\)
−0.233006 + 0.972475i \(0.574856\pi\)
\(422\) 0 0
\(423\) 2988.41 0.343502
\(424\) 0 0
\(425\) −5056.82 −0.577157
\(426\) 0 0
\(427\) 117.122 0.0132738
\(428\) 0 0
\(429\) −3365.48 −0.378757
\(430\) 0 0
\(431\) −2863.81 −0.320057 −0.160029 0.987112i \(-0.551159\pi\)
−0.160029 + 0.987112i \(0.551159\pi\)
\(432\) 0 0
\(433\) −6617.66 −0.734468 −0.367234 0.930129i \(-0.619695\pi\)
−0.367234 + 0.930129i \(0.619695\pi\)
\(434\) 0 0
\(435\) −2282.93 −0.251627
\(436\) 0 0
\(437\) −927.698 −0.101551
\(438\) 0 0
\(439\) −8810.03 −0.957813 −0.478907 0.877866i \(-0.658967\pi\)
−0.478907 + 0.877866i \(0.658967\pi\)
\(440\) 0 0
\(441\) 4626.33 0.499550
\(442\) 0 0
\(443\) 202.487 0.0217165 0.0108583 0.999941i \(-0.496544\pi\)
0.0108583 + 0.999941i \(0.496544\pi\)
\(444\) 0 0
\(445\) 4569.75 0.486802
\(446\) 0 0
\(447\) 3262.39 0.345203
\(448\) 0 0
\(449\) 4619.26 0.485516 0.242758 0.970087i \(-0.421948\pi\)
0.242758 + 0.970087i \(0.421948\pi\)
\(450\) 0 0
\(451\) −6544.15 −0.683264
\(452\) 0 0
\(453\) 204.474 0.0212075
\(454\) 0 0
\(455\) 189.974 0.0195739
\(456\) 0 0
\(457\) −8196.37 −0.838972 −0.419486 0.907762i \(-0.637790\pi\)
−0.419486 + 0.907762i \(0.637790\pi\)
\(458\) 0 0
\(459\) 10142.6 1.03141
\(460\) 0 0
\(461\) −1269.48 −0.128255 −0.0641277 0.997942i \(-0.520426\pi\)
−0.0641277 + 0.997942i \(0.520426\pi\)
\(462\) 0 0
\(463\) 8502.91 0.853485 0.426743 0.904373i \(-0.359661\pi\)
0.426743 + 0.904373i \(0.359661\pi\)
\(464\) 0 0
\(465\) 1394.80 0.139101
\(466\) 0 0
\(467\) −8132.55 −0.805845 −0.402923 0.915234i \(-0.632006\pi\)
−0.402923 + 0.915234i \(0.632006\pi\)
\(468\) 0 0
\(469\) −296.595 −0.0292014
\(470\) 0 0
\(471\) 6037.50 0.590644
\(472\) 0 0
\(473\) 7318.37 0.711414
\(474\) 0 0
\(475\) 1409.66 0.136167
\(476\) 0 0
\(477\) −1453.43 −0.139514
\(478\) 0 0
\(479\) 15965.6 1.52293 0.761466 0.648205i \(-0.224480\pi\)
0.761466 + 0.648205i \(0.224480\pi\)
\(480\) 0 0
\(481\) −10405.9 −0.986421
\(482\) 0 0
\(483\) 91.7353 0.00864203
\(484\) 0 0
\(485\) −142.558 −0.0133468
\(486\) 0 0
\(487\) −11358.3 −1.05687 −0.528435 0.848974i \(-0.677221\pi\)
−0.528435 + 0.848974i \(0.677221\pi\)
\(488\) 0 0
\(489\) −8237.40 −0.761775
\(490\) 0 0
\(491\) −5256.31 −0.483124 −0.241562 0.970385i \(-0.577660\pi\)
−0.241562 + 0.970385i \(0.577660\pi\)
\(492\) 0 0
\(493\) −5940.85 −0.542723
\(494\) 0 0
\(495\) 1690.62 0.153510
\(496\) 0 0
\(497\) 19.5438 0.00176390
\(498\) 0 0
\(499\) −11667.7 −1.04673 −0.523364 0.852109i \(-0.675323\pi\)
−0.523364 + 0.852109i \(0.675323\pi\)
\(500\) 0 0
\(501\) 9347.73 0.833585
\(502\) 0 0
\(503\) −16902.2 −1.49828 −0.749139 0.662413i \(-0.769533\pi\)
−0.749139 + 0.662413i \(0.769533\pi\)
\(504\) 0 0
\(505\) 3375.27 0.297421
\(506\) 0 0
\(507\) −1910.68 −0.167369
\(508\) 0 0
\(509\) 2828.32 0.246293 0.123147 0.992388i \(-0.460701\pi\)
0.123147 + 0.992388i \(0.460701\pi\)
\(510\) 0 0
\(511\) 411.050 0.0355847
\(512\) 0 0
\(513\) −2827.39 −0.243338
\(514\) 0 0
\(515\) 2063.30 0.176544
\(516\) 0 0
\(517\) −3890.20 −0.330930
\(518\) 0 0
\(519\) 3984.18 0.336968
\(520\) 0 0
\(521\) 6552.16 0.550970 0.275485 0.961305i \(-0.411162\pi\)
0.275485 + 0.961305i \(0.411162\pi\)
\(522\) 0 0
\(523\) 7136.05 0.596631 0.298315 0.954467i \(-0.403575\pi\)
0.298315 + 0.954467i \(0.403575\pi\)
\(524\) 0 0
\(525\) −139.394 −0.0115879
\(526\) 0 0
\(527\) 3629.68 0.300021
\(528\) 0 0
\(529\) −9783.00 −0.804060
\(530\) 0 0
\(531\) −2102.80 −0.171853
\(532\) 0 0
\(533\) −19412.9 −1.57761
\(534\) 0 0
\(535\) 2786.90 0.225212
\(536\) 0 0
\(537\) −7544.68 −0.606288
\(538\) 0 0
\(539\) −6022.40 −0.481267
\(540\) 0 0
\(541\) −22860.0 −1.81669 −0.908343 0.418227i \(-0.862652\pi\)
−0.908343 + 0.418227i \(0.862652\pi\)
\(542\) 0 0
\(543\) 7593.76 0.600146
\(544\) 0 0
\(545\) 9042.88 0.710742
\(546\) 0 0
\(547\) −9176.33 −0.717279 −0.358639 0.933476i \(-0.616759\pi\)
−0.358639 + 0.933476i \(0.616759\pi\)
\(548\) 0 0
\(549\) 3091.90 0.240362
\(550\) 0 0
\(551\) 1656.09 0.128043
\(552\) 0 0
\(553\) 529.019 0.0406803
\(554\) 0 0
\(555\) −5228.74 −0.399906
\(556\) 0 0
\(557\) −23966.7 −1.82316 −0.911581 0.411120i \(-0.865138\pi\)
−0.911581 + 0.411120i \(0.865138\pi\)
\(558\) 0 0
\(559\) 21709.6 1.64261
\(560\) 0 0
\(561\) −4400.70 −0.331190
\(562\) 0 0
\(563\) −1356.83 −0.101569 −0.0507847 0.998710i \(-0.516172\pi\)
−0.0507847 + 0.998710i \(0.516172\pi\)
\(564\) 0 0
\(565\) −7844.04 −0.584073
\(566\) 0 0
\(567\) 93.2381 0.00690588
\(568\) 0 0
\(569\) −16869.9 −1.24292 −0.621461 0.783445i \(-0.713461\pi\)
−0.621461 + 0.783445i \(0.713461\pi\)
\(570\) 0 0
\(571\) −19377.3 −1.42016 −0.710082 0.704119i \(-0.751342\pi\)
−0.710082 + 0.704119i \(0.751342\pi\)
\(572\) 0 0
\(573\) −4595.47 −0.335041
\(574\) 0 0
\(575\) −3622.53 −0.262730
\(576\) 0 0
\(577\) −16335.8 −1.17863 −0.589313 0.807905i \(-0.700602\pi\)
−0.589313 + 0.807905i \(0.700602\pi\)
\(578\) 0 0
\(579\) −10343.7 −0.742435
\(580\) 0 0
\(581\) 538.893 0.0384803
\(582\) 0 0
\(583\) 1892.03 0.134408
\(584\) 0 0
\(585\) 5015.13 0.354445
\(586\) 0 0
\(587\) −9216.05 −0.648019 −0.324009 0.946054i \(-0.605031\pi\)
−0.324009 + 0.946054i \(0.605031\pi\)
\(588\) 0 0
\(589\) −1011.82 −0.0707833
\(590\) 0 0
\(591\) 2182.12 0.151879
\(592\) 0 0
\(593\) −7176.76 −0.496988 −0.248494 0.968633i \(-0.579936\pi\)
−0.248494 + 0.968633i \(0.579936\pi\)
\(594\) 0 0
\(595\) 248.410 0.0171157
\(596\) 0 0
\(597\) 56.8830 0.00389961
\(598\) 0 0
\(599\) 3445.62 0.235032 0.117516 0.993071i \(-0.462507\pi\)
0.117516 + 0.993071i \(0.462507\pi\)
\(600\) 0 0
\(601\) 17940.3 1.21764 0.608820 0.793308i \(-0.291643\pi\)
0.608820 + 0.793308i \(0.291643\pi\)
\(602\) 0 0
\(603\) −7829.80 −0.528779
\(604\) 0 0
\(605\) 7286.51 0.489651
\(606\) 0 0
\(607\) 14232.0 0.951664 0.475832 0.879536i \(-0.342147\pi\)
0.475832 + 0.879536i \(0.342147\pi\)
\(608\) 0 0
\(609\) −163.763 −0.0108965
\(610\) 0 0
\(611\) −11540.1 −0.764095
\(612\) 0 0
\(613\) 19440.3 1.28089 0.640446 0.768003i \(-0.278749\pi\)
0.640446 + 0.768003i \(0.278749\pi\)
\(614\) 0 0
\(615\) −9754.56 −0.639580
\(616\) 0 0
\(617\) −359.820 −0.0234778 −0.0117389 0.999931i \(-0.503737\pi\)
−0.0117389 + 0.999931i \(0.503737\pi\)
\(618\) 0 0
\(619\) 18561.5 1.20525 0.602624 0.798026i \(-0.294122\pi\)
0.602624 + 0.798026i \(0.294122\pi\)
\(620\) 0 0
\(621\) 7265.82 0.469513
\(622\) 0 0
\(623\) 327.805 0.0210806
\(624\) 0 0
\(625\) −846.443 −0.0541723
\(626\) 0 0
\(627\) 1226.75 0.0781368
\(628\) 0 0
\(629\) −13606.7 −0.862538
\(630\) 0 0
\(631\) −9684.27 −0.610974 −0.305487 0.952196i \(-0.598819\pi\)
−0.305487 + 0.952196i \(0.598819\pi\)
\(632\) 0 0
\(633\) 8187.13 0.514075
\(634\) 0 0
\(635\) 19953.1 1.24695
\(636\) 0 0
\(637\) −17865.1 −1.11121
\(638\) 0 0
\(639\) 515.935 0.0319407
\(640\) 0 0
\(641\) −4576.97 −0.282027 −0.141014 0.990008i \(-0.545036\pi\)
−0.141014 + 0.990008i \(0.545036\pi\)
\(642\) 0 0
\(643\) −9722.95 −0.596323 −0.298161 0.954515i \(-0.596373\pi\)
−0.298161 + 0.954515i \(0.596373\pi\)
\(644\) 0 0
\(645\) 10908.6 0.665931
\(646\) 0 0
\(647\) −9150.86 −0.556040 −0.278020 0.960575i \(-0.589678\pi\)
−0.278020 + 0.960575i \(0.589678\pi\)
\(648\) 0 0
\(649\) 2737.36 0.165563
\(650\) 0 0
\(651\) 100.054 0.00602369
\(652\) 0 0
\(653\) −9202.67 −0.551498 −0.275749 0.961230i \(-0.588926\pi\)
−0.275749 + 0.961230i \(0.588926\pi\)
\(654\) 0 0
\(655\) −4927.30 −0.293932
\(656\) 0 0
\(657\) 10851.3 0.644367
\(658\) 0 0
\(659\) −5007.55 −0.296004 −0.148002 0.988987i \(-0.547284\pi\)
−0.148002 + 0.988987i \(0.547284\pi\)
\(660\) 0 0
\(661\) −21922.2 −1.28998 −0.644989 0.764192i \(-0.723138\pi\)
−0.644989 + 0.764192i \(0.723138\pi\)
\(662\) 0 0
\(663\) −13054.4 −0.764695
\(664\) 0 0
\(665\) −69.2477 −0.00403806
\(666\) 0 0
\(667\) −4255.82 −0.247056
\(668\) 0 0
\(669\) 7599.47 0.439182
\(670\) 0 0
\(671\) −4024.92 −0.231565
\(672\) 0 0
\(673\) 13736.9 0.786806 0.393403 0.919366i \(-0.371298\pi\)
0.393403 + 0.919366i \(0.371298\pi\)
\(674\) 0 0
\(675\) −11040.6 −0.629558
\(676\) 0 0
\(677\) 20015.0 1.13625 0.568123 0.822944i \(-0.307670\pi\)
0.568123 + 0.822944i \(0.307670\pi\)
\(678\) 0 0
\(679\) −10.2262 −0.000577975 0
\(680\) 0 0
\(681\) −2175.10 −0.122393
\(682\) 0 0
\(683\) 23674.0 1.32630 0.663148 0.748488i \(-0.269220\pi\)
0.663148 + 0.748488i \(0.269220\pi\)
\(684\) 0 0
\(685\) −7441.56 −0.415077
\(686\) 0 0
\(687\) 9256.29 0.514046
\(688\) 0 0
\(689\) 5612.61 0.310339
\(690\) 0 0
\(691\) 2796.62 0.153963 0.0769815 0.997033i \(-0.475472\pi\)
0.0769815 + 0.997033i \(0.475472\pi\)
\(692\) 0 0
\(693\) 121.274 0.00664765
\(694\) 0 0
\(695\) −11854.7 −0.647015
\(696\) 0 0
\(697\) −25384.3 −1.37948
\(698\) 0 0
\(699\) −14646.3 −0.792522
\(700\) 0 0
\(701\) −222.627 −0.0119950 −0.00599750 0.999982i \(-0.501909\pi\)
−0.00599750 + 0.999982i \(0.501909\pi\)
\(702\) 0 0
\(703\) 3793.06 0.203496
\(704\) 0 0
\(705\) −5798.65 −0.309773
\(706\) 0 0
\(707\) 242.121 0.0128796
\(708\) 0 0
\(709\) −1068.75 −0.0566119 −0.0283059 0.999599i \(-0.509011\pi\)
−0.0283059 + 0.999599i \(0.509011\pi\)
\(710\) 0 0
\(711\) 13965.6 0.736638
\(712\) 0 0
\(713\) 2600.18 0.136574
\(714\) 0 0
\(715\) −6528.52 −0.341472
\(716\) 0 0
\(717\) 8714.89 0.453924
\(718\) 0 0
\(719\) 27102.1 1.40575 0.702876 0.711312i \(-0.251899\pi\)
0.702876 + 0.711312i \(0.251899\pi\)
\(720\) 0 0
\(721\) 148.008 0.00764509
\(722\) 0 0
\(723\) −19458.6 −1.00093
\(724\) 0 0
\(725\) 6466.81 0.331271
\(726\) 0 0
\(727\) 6285.65 0.320663 0.160331 0.987063i \(-0.448744\pi\)
0.160331 + 0.987063i \(0.448744\pi\)
\(728\) 0 0
\(729\) 17225.0 0.875120
\(730\) 0 0
\(731\) 28387.4 1.43632
\(732\) 0 0
\(733\) 1594.38 0.0803406 0.0401703 0.999193i \(-0.487210\pi\)
0.0401703 + 0.999193i \(0.487210\pi\)
\(734\) 0 0
\(735\) −8976.85 −0.450498
\(736\) 0 0
\(737\) 10192.6 0.509427
\(738\) 0 0
\(739\) −11152.4 −0.555141 −0.277571 0.960705i \(-0.589529\pi\)
−0.277571 + 0.960705i \(0.589529\pi\)
\(740\) 0 0
\(741\) 3639.10 0.180413
\(742\) 0 0
\(743\) 35565.1 1.75606 0.878032 0.478601i \(-0.158856\pi\)
0.878032 + 0.478601i \(0.158856\pi\)
\(744\) 0 0
\(745\) 6328.54 0.311221
\(746\) 0 0
\(747\) 14226.2 0.696801
\(748\) 0 0
\(749\) 199.915 0.00975263
\(750\) 0 0
\(751\) 33922.2 1.64825 0.824126 0.566406i \(-0.191667\pi\)
0.824126 + 0.566406i \(0.191667\pi\)
\(752\) 0 0
\(753\) −21189.9 −1.02550
\(754\) 0 0
\(755\) 396.648 0.0191199
\(756\) 0 0
\(757\) −18859.2 −0.905479 −0.452740 0.891643i \(-0.649553\pi\)
−0.452740 + 0.891643i \(0.649553\pi\)
\(758\) 0 0
\(759\) −3152.51 −0.150763
\(760\) 0 0
\(761\) 15707.4 0.748217 0.374109 0.927385i \(-0.377949\pi\)
0.374109 + 0.927385i \(0.377949\pi\)
\(762\) 0 0
\(763\) 648.679 0.0307782
\(764\) 0 0
\(765\) 6557.78 0.309931
\(766\) 0 0
\(767\) 8120.23 0.382275
\(768\) 0 0
\(769\) 1292.73 0.0606202 0.0303101 0.999541i \(-0.490351\pi\)
0.0303101 + 0.999541i \(0.490351\pi\)
\(770\) 0 0
\(771\) −13345.3 −0.623370
\(772\) 0 0
\(773\) 38483.0 1.79060 0.895301 0.445461i \(-0.146960\pi\)
0.895301 + 0.445461i \(0.146960\pi\)
\(774\) 0 0
\(775\) −3951.02 −0.183129
\(776\) 0 0
\(777\) −375.077 −0.0173176
\(778\) 0 0
\(779\) 7076.20 0.325457
\(780\) 0 0
\(781\) −671.626 −0.0307717
\(782\) 0 0
\(783\) −12970.7 −0.591997
\(784\) 0 0
\(785\) 11711.8 0.532501
\(786\) 0 0
\(787\) −18911.7 −0.856579 −0.428290 0.903641i \(-0.640884\pi\)
−0.428290 + 0.903641i \(0.640884\pi\)
\(788\) 0 0
\(789\) 21491.0 0.969710
\(790\) 0 0
\(791\) −562.682 −0.0252929
\(792\) 0 0
\(793\) −11939.7 −0.534668
\(794\) 0 0
\(795\) 2820.22 0.125815
\(796\) 0 0
\(797\) 35376.1 1.57225 0.786127 0.618065i \(-0.212083\pi\)
0.786127 + 0.618065i \(0.212083\pi\)
\(798\) 0 0
\(799\) −15089.8 −0.668134
\(800\) 0 0
\(801\) 8653.71 0.381727
\(802\) 0 0
\(803\) −14125.8 −0.620784
\(804\) 0 0
\(805\) 177.953 0.00779131
\(806\) 0 0
\(807\) 15739.0 0.686541
\(808\) 0 0
\(809\) 29057.9 1.26282 0.631411 0.775448i \(-0.282476\pi\)
0.631411 + 0.775448i \(0.282476\pi\)
\(810\) 0 0
\(811\) 25476.2 1.10307 0.551534 0.834152i \(-0.314043\pi\)
0.551534 + 0.834152i \(0.314043\pi\)
\(812\) 0 0
\(813\) 8637.07 0.372590
\(814\) 0 0
\(815\) −15979.3 −0.686786
\(816\) 0 0
\(817\) −7913.37 −0.338866
\(818\) 0 0
\(819\) 359.753 0.0153490
\(820\) 0 0
\(821\) 41186.3 1.75081 0.875403 0.483394i \(-0.160596\pi\)
0.875403 + 0.483394i \(0.160596\pi\)
\(822\) 0 0
\(823\) −9004.24 −0.381371 −0.190685 0.981651i \(-0.561071\pi\)
−0.190685 + 0.981651i \(0.561071\pi\)
\(824\) 0 0
\(825\) 4790.30 0.202154
\(826\) 0 0
\(827\) 11943.7 0.502204 0.251102 0.967961i \(-0.419207\pi\)
0.251102 + 0.967961i \(0.419207\pi\)
\(828\) 0 0
\(829\) 869.327 0.0364209 0.0182105 0.999834i \(-0.494203\pi\)
0.0182105 + 0.999834i \(0.494203\pi\)
\(830\) 0 0
\(831\) −29034.8 −1.21204
\(832\) 0 0
\(833\) −23360.4 −0.971658
\(834\) 0 0
\(835\) 18133.2 0.751526
\(836\) 0 0
\(837\) 7924.68 0.327261
\(838\) 0 0
\(839\) −1265.71 −0.0520824 −0.0260412 0.999661i \(-0.508290\pi\)
−0.0260412 + 0.999661i \(0.508290\pi\)
\(840\) 0 0
\(841\) −16791.7 −0.688493
\(842\) 0 0
\(843\) 5714.88 0.233489
\(844\) 0 0
\(845\) −3706.43 −0.150893
\(846\) 0 0
\(847\) 522.688 0.0212040
\(848\) 0 0
\(849\) 11162.1 0.451217
\(850\) 0 0
\(851\) −9747.40 −0.392640
\(852\) 0 0
\(853\) 15823.2 0.635143 0.317572 0.948234i \(-0.397133\pi\)
0.317572 + 0.948234i \(0.397133\pi\)
\(854\) 0 0
\(855\) −1828.07 −0.0731211
\(856\) 0 0
\(857\) 36570.0 1.45765 0.728825 0.684700i \(-0.240067\pi\)
0.728825 + 0.684700i \(0.240067\pi\)
\(858\) 0 0
\(859\) 18900.5 0.750731 0.375366 0.926877i \(-0.377517\pi\)
0.375366 + 0.926877i \(0.377517\pi\)
\(860\) 0 0
\(861\) −699.730 −0.0276966
\(862\) 0 0
\(863\) 30643.8 1.20872 0.604361 0.796710i \(-0.293428\pi\)
0.604361 + 0.796710i \(0.293428\pi\)
\(864\) 0 0
\(865\) 7728.71 0.303797
\(866\) 0 0
\(867\) 982.780 0.0384971
\(868\) 0 0
\(869\) −18179.9 −0.709678
\(870\) 0 0
\(871\) 30235.7 1.17623
\(872\) 0 0
\(873\) −269.961 −0.0104660
\(874\) 0 0
\(875\) −725.979 −0.0280487
\(876\) 0 0
\(877\) −33839.9 −1.30295 −0.651477 0.758668i \(-0.725850\pi\)
−0.651477 + 0.758668i \(0.725850\pi\)
\(878\) 0 0
\(879\) −16451.0 −0.631261
\(880\) 0 0
\(881\) −3158.38 −0.120781 −0.0603907 0.998175i \(-0.519235\pi\)
−0.0603907 + 0.998175i \(0.519235\pi\)
\(882\) 0 0
\(883\) 39820.3 1.51762 0.758810 0.651312i \(-0.225781\pi\)
0.758810 + 0.651312i \(0.225781\pi\)
\(884\) 0 0
\(885\) 4080.24 0.154978
\(886\) 0 0
\(887\) −50176.9 −1.89941 −0.949705 0.313146i \(-0.898617\pi\)
−0.949705 + 0.313146i \(0.898617\pi\)
\(888\) 0 0
\(889\) 1431.31 0.0539983
\(890\) 0 0
\(891\) −3204.15 −0.120475
\(892\) 0 0
\(893\) 4206.48 0.157631
\(894\) 0 0
\(895\) −14635.5 −0.546605
\(896\) 0 0
\(897\) −9351.76 −0.348100
\(898\) 0 0
\(899\) −4641.74 −0.172203
\(900\) 0 0
\(901\) 7339.04 0.271364
\(902\) 0 0
\(903\) 782.513 0.0288377
\(904\) 0 0
\(905\) 14730.7 0.541068
\(906\) 0 0
\(907\) −42737.3 −1.56457 −0.782287 0.622918i \(-0.785947\pi\)
−0.782287 + 0.622918i \(0.785947\pi\)
\(908\) 0 0
\(909\) 6391.74 0.233224
\(910\) 0 0
\(911\) −47034.3 −1.71056 −0.855278 0.518170i \(-0.826614\pi\)
−0.855278 + 0.518170i \(0.826614\pi\)
\(912\) 0 0
\(913\) −18519.2 −0.671299
\(914\) 0 0
\(915\) −5999.46 −0.216761
\(916\) 0 0
\(917\) −353.453 −0.0127285
\(918\) 0 0
\(919\) 35256.7 1.26552 0.632759 0.774349i \(-0.281922\pi\)
0.632759 + 0.774349i \(0.281922\pi\)
\(920\) 0 0
\(921\) 5808.90 0.207828
\(922\) 0 0
\(923\) −1992.35 −0.0710497
\(924\) 0 0
\(925\) 14811.4 0.526481
\(926\) 0 0
\(927\) 3907.26 0.138437
\(928\) 0 0
\(929\) 3588.82 0.126744 0.0633720 0.997990i \(-0.479815\pi\)
0.0633720 + 0.997990i \(0.479815\pi\)
\(930\) 0 0
\(931\) 6512.03 0.229241
\(932\) 0 0
\(933\) −11909.9 −0.417911
\(934\) 0 0
\(935\) −8536.68 −0.298587
\(936\) 0 0
\(937\) −13741.6 −0.479101 −0.239551 0.970884i \(-0.577000\pi\)
−0.239551 + 0.970884i \(0.577000\pi\)
\(938\) 0 0
\(939\) 35199.1 1.22330
\(940\) 0 0
\(941\) 31893.3 1.10488 0.552440 0.833553i \(-0.313697\pi\)
0.552440 + 0.833553i \(0.313697\pi\)
\(942\) 0 0
\(943\) −18184.4 −0.627960
\(944\) 0 0
\(945\) 542.355 0.0186696
\(946\) 0 0
\(947\) 22540.6 0.773465 0.386733 0.922192i \(-0.373604\pi\)
0.386733 + 0.922192i \(0.373604\pi\)
\(948\) 0 0
\(949\) −41903.6 −1.43335
\(950\) 0 0
\(951\) −18932.4 −0.645558
\(952\) 0 0
\(953\) −2204.23 −0.0749233 −0.0374617 0.999298i \(-0.511927\pi\)
−0.0374617 + 0.999298i \(0.511927\pi\)
\(954\) 0 0
\(955\) −8914.51 −0.302060
\(956\) 0 0
\(957\) 5627.74 0.190093
\(958\) 0 0
\(959\) −533.810 −0.0179746
\(960\) 0 0
\(961\) −26955.0 −0.904805
\(962\) 0 0
\(963\) 5277.54 0.176600
\(964\) 0 0
\(965\) −20065.2 −0.669350
\(966\) 0 0
\(967\) −56851.1 −1.89060 −0.945299 0.326205i \(-0.894230\pi\)
−0.945299 + 0.326205i \(0.894230\pi\)
\(968\) 0 0
\(969\) 4758.48 0.157755
\(970\) 0 0
\(971\) 48267.9 1.59525 0.797627 0.603151i \(-0.206088\pi\)
0.797627 + 0.603151i \(0.206088\pi\)
\(972\) 0 0
\(973\) −850.383 −0.0280185
\(974\) 0 0
\(975\) 14210.2 0.466759
\(976\) 0 0
\(977\) 32914.4 1.07781 0.538907 0.842366i \(-0.318838\pi\)
0.538907 + 0.842366i \(0.318838\pi\)
\(978\) 0 0
\(979\) −11265.1 −0.367757
\(980\) 0 0
\(981\) 17124.4 0.557331
\(982\) 0 0
\(983\) −29352.6 −0.952394 −0.476197 0.879338i \(-0.657985\pi\)
−0.476197 + 0.879338i \(0.657985\pi\)
\(984\) 0 0
\(985\) 4232.98 0.136928
\(986\) 0 0
\(987\) −415.958 −0.0134145
\(988\) 0 0
\(989\) 20335.8 0.653832
\(990\) 0 0
\(991\) −2460.63 −0.0788744 −0.0394372 0.999222i \(-0.512557\pi\)
−0.0394372 + 0.999222i \(0.512557\pi\)
\(992\) 0 0
\(993\) 18040.3 0.576528
\(994\) 0 0
\(995\) 110.344 0.00351573
\(996\) 0 0
\(997\) 971.621 0.0308641 0.0154321 0.999881i \(-0.495088\pi\)
0.0154321 + 0.999881i \(0.495088\pi\)
\(998\) 0 0
\(999\) −29707.6 −0.940849
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 608.4.a.h.1.2 yes 5
4.3 odd 2 608.4.a.e.1.4 5
8.3 odd 2 1216.4.a.bd.1.2 5
8.5 even 2 1216.4.a.y.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
608.4.a.e.1.4 5 4.3 odd 2
608.4.a.h.1.2 yes 5 1.1 even 1 trivial
1216.4.a.y.1.4 5 8.5 even 2
1216.4.a.bd.1.2 5 8.3 odd 2