Properties

Label 4012.2.a.j.1.19
Level $4012$
Weight $2$
Character 4012.1
Self dual yes
Analytic conductor $32.036$
Analytic rank $0$
Dimension $21$
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] = [4012,2,Mod(1,4012)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4012, 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("4012.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4012.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: \(32.0359812909\)
Analytic rank: \(0\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.19
Character \(\chi\) \(=\) 4012.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.11094 q^{3} +3.12050 q^{5} -1.63406 q^{7} +6.67797 q^{9} +O(q^{10})\) \(q+3.11094 q^{3} +3.12050 q^{5} -1.63406 q^{7} +6.67797 q^{9} -1.72167 q^{11} -1.82183 q^{13} +9.70769 q^{15} +1.00000 q^{17} +4.18677 q^{19} -5.08348 q^{21} +5.92515 q^{23} +4.73750 q^{25} +11.4420 q^{27} -0.256534 q^{29} +0.752999 q^{31} -5.35602 q^{33} -5.09909 q^{35} +2.71509 q^{37} -5.66762 q^{39} +1.61114 q^{41} +1.84523 q^{43} +20.8386 q^{45} -11.3816 q^{47} -4.32984 q^{49} +3.11094 q^{51} +8.07848 q^{53} -5.37247 q^{55} +13.0248 q^{57} +1.00000 q^{59} -6.80471 q^{61} -10.9122 q^{63} -5.68503 q^{65} +4.90642 q^{67} +18.4328 q^{69} -3.75393 q^{71} -8.68741 q^{73} +14.7381 q^{75} +2.81332 q^{77} +3.18758 q^{79} +15.5614 q^{81} +1.57902 q^{83} +3.12050 q^{85} -0.798062 q^{87} +13.7968 q^{89} +2.97699 q^{91} +2.34254 q^{93} +13.0648 q^{95} -5.66388 q^{97} -11.4973 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) \) 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
\(3\) 3.11094 1.79610 0.898052 0.439889i \(-0.144982\pi\)
0.898052 + 0.439889i \(0.144982\pi\)
\(4\) 0 0
\(5\) 3.12050 1.39553 0.697764 0.716327i \(-0.254178\pi\)
0.697764 + 0.716327i \(0.254178\pi\)
\(6\) 0 0
\(7\) −1.63406 −0.617618 −0.308809 0.951124i \(-0.599930\pi\)
−0.308809 + 0.951124i \(0.599930\pi\)
\(8\) 0 0
\(9\) 6.67797 2.22599
\(10\) 0 0
\(11\) −1.72167 −0.519103 −0.259552 0.965729i \(-0.583575\pi\)
−0.259552 + 0.965729i \(0.583575\pi\)
\(12\) 0 0
\(13\) −1.82183 −0.505286 −0.252643 0.967560i \(-0.581300\pi\)
−0.252643 + 0.967560i \(0.581300\pi\)
\(14\) 0 0
\(15\) 9.70769 2.50652
\(16\) 0 0
\(17\) 1.00000 0.242536
\(18\) 0 0
\(19\) 4.18677 0.960510 0.480255 0.877129i \(-0.340544\pi\)
0.480255 + 0.877129i \(0.340544\pi\)
\(20\) 0 0
\(21\) −5.08348 −1.10931
\(22\) 0 0
\(23\) 5.92515 1.23548 0.617740 0.786383i \(-0.288049\pi\)
0.617740 + 0.786383i \(0.288049\pi\)
\(24\) 0 0
\(25\) 4.73750 0.947501
\(26\) 0 0
\(27\) 11.4420 2.20200
\(28\) 0 0
\(29\) −0.256534 −0.0476371 −0.0238186 0.999716i \(-0.507582\pi\)
−0.0238186 + 0.999716i \(0.507582\pi\)
\(30\) 0 0
\(31\) 0.752999 0.135243 0.0676213 0.997711i \(-0.478459\pi\)
0.0676213 + 0.997711i \(0.478459\pi\)
\(32\) 0 0
\(33\) −5.35602 −0.932364
\(34\) 0 0
\(35\) −5.09909 −0.861904
\(36\) 0 0
\(37\) 2.71509 0.446359 0.223179 0.974777i \(-0.428356\pi\)
0.223179 + 0.974777i \(0.428356\pi\)
\(38\) 0 0
\(39\) −5.66762 −0.907545
\(40\) 0 0
\(41\) 1.61114 0.251617 0.125809 0.992055i \(-0.459847\pi\)
0.125809 + 0.992055i \(0.459847\pi\)
\(42\) 0 0
\(43\) 1.84523 0.281395 0.140698 0.990053i \(-0.455066\pi\)
0.140698 + 0.990053i \(0.455066\pi\)
\(44\) 0 0
\(45\) 20.8386 3.10643
\(46\) 0 0
\(47\) −11.3816 −1.66017 −0.830086 0.557636i \(-0.811709\pi\)
−0.830086 + 0.557636i \(0.811709\pi\)
\(48\) 0 0
\(49\) −4.32984 −0.618548
\(50\) 0 0
\(51\) 3.11094 0.435619
\(52\) 0 0
\(53\) 8.07848 1.10966 0.554832 0.831962i \(-0.312783\pi\)
0.554832 + 0.831962i \(0.312783\pi\)
\(54\) 0 0
\(55\) −5.37247 −0.724424
\(56\) 0 0
\(57\) 13.0248 1.72518
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) −6.80471 −0.871254 −0.435627 0.900127i \(-0.643473\pi\)
−0.435627 + 0.900127i \(0.643473\pi\)
\(62\) 0 0
\(63\) −10.9122 −1.37481
\(64\) 0 0
\(65\) −5.68503 −0.705141
\(66\) 0 0
\(67\) 4.90642 0.599414 0.299707 0.954031i \(-0.403111\pi\)
0.299707 + 0.954031i \(0.403111\pi\)
\(68\) 0 0
\(69\) 18.4328 2.21905
\(70\) 0 0
\(71\) −3.75393 −0.445509 −0.222755 0.974875i \(-0.571505\pi\)
−0.222755 + 0.974875i \(0.571505\pi\)
\(72\) 0 0
\(73\) −8.68741 −1.01678 −0.508392 0.861126i \(-0.669760\pi\)
−0.508392 + 0.861126i \(0.669760\pi\)
\(74\) 0 0
\(75\) 14.7381 1.70181
\(76\) 0 0
\(77\) 2.81332 0.320608
\(78\) 0 0
\(79\) 3.18758 0.358631 0.179316 0.983792i \(-0.442612\pi\)
0.179316 + 0.983792i \(0.442612\pi\)
\(80\) 0 0
\(81\) 15.5614 1.72904
\(82\) 0 0
\(83\) 1.57902 0.173320 0.0866598 0.996238i \(-0.472381\pi\)
0.0866598 + 0.996238i \(0.472381\pi\)
\(84\) 0 0
\(85\) 3.12050 0.338465
\(86\) 0 0
\(87\) −0.798062 −0.0855612
\(88\) 0 0
\(89\) 13.7968 1.46246 0.731229 0.682132i \(-0.238947\pi\)
0.731229 + 0.682132i \(0.238947\pi\)
\(90\) 0 0
\(91\) 2.97699 0.312073
\(92\) 0 0
\(93\) 2.34254 0.242910
\(94\) 0 0
\(95\) 13.0648 1.34042
\(96\) 0 0
\(97\) −5.66388 −0.575080 −0.287540 0.957769i \(-0.592837\pi\)
−0.287540 + 0.957769i \(0.592837\pi\)
\(98\) 0 0
\(99\) −11.4973 −1.15552
\(100\) 0 0
\(101\) −17.1341 −1.70491 −0.852455 0.522800i \(-0.824887\pi\)
−0.852455 + 0.522800i \(0.824887\pi\)
\(102\) 0 0
\(103\) 12.6944 1.25082 0.625410 0.780296i \(-0.284932\pi\)
0.625410 + 0.780296i \(0.284932\pi\)
\(104\) 0 0
\(105\) −15.8630 −1.54807
\(106\) 0 0
\(107\) 2.43318 0.235224 0.117612 0.993060i \(-0.462476\pi\)
0.117612 + 0.993060i \(0.462476\pi\)
\(108\) 0 0
\(109\) 6.17238 0.591207 0.295603 0.955311i \(-0.404479\pi\)
0.295603 + 0.955311i \(0.404479\pi\)
\(110\) 0 0
\(111\) 8.44650 0.801706
\(112\) 0 0
\(113\) −6.42743 −0.604642 −0.302321 0.953206i \(-0.597761\pi\)
−0.302321 + 0.953206i \(0.597761\pi\)
\(114\) 0 0
\(115\) 18.4894 1.72415
\(116\) 0 0
\(117\) −12.1661 −1.12476
\(118\) 0 0
\(119\) −1.63406 −0.149794
\(120\) 0 0
\(121\) −8.03585 −0.730532
\(122\) 0 0
\(123\) 5.01216 0.451931
\(124\) 0 0
\(125\) −0.819116 −0.0732640
\(126\) 0 0
\(127\) 2.15271 0.191022 0.0955109 0.995428i \(-0.469552\pi\)
0.0955109 + 0.995428i \(0.469552\pi\)
\(128\) 0 0
\(129\) 5.74041 0.505415
\(130\) 0 0
\(131\) 5.40733 0.472440 0.236220 0.971700i \(-0.424091\pi\)
0.236220 + 0.971700i \(0.424091\pi\)
\(132\) 0 0
\(133\) −6.84144 −0.593228
\(134\) 0 0
\(135\) 35.7046 3.07296
\(136\) 0 0
\(137\) 7.15530 0.611319 0.305659 0.952141i \(-0.401123\pi\)
0.305659 + 0.952141i \(0.401123\pi\)
\(138\) 0 0
\(139\) 10.8663 0.921666 0.460833 0.887487i \(-0.347551\pi\)
0.460833 + 0.887487i \(0.347551\pi\)
\(140\) 0 0
\(141\) −35.4074 −2.98184
\(142\) 0 0
\(143\) 3.13660 0.262296
\(144\) 0 0
\(145\) −0.800513 −0.0664790
\(146\) 0 0
\(147\) −13.4699 −1.11098
\(148\) 0 0
\(149\) 14.3498 1.17559 0.587793 0.809012i \(-0.299997\pi\)
0.587793 + 0.809012i \(0.299997\pi\)
\(150\) 0 0
\(151\) −3.14548 −0.255975 −0.127988 0.991776i \(-0.540852\pi\)
−0.127988 + 0.991776i \(0.540852\pi\)
\(152\) 0 0
\(153\) 6.67797 0.539882
\(154\) 0 0
\(155\) 2.34973 0.188735
\(156\) 0 0
\(157\) −19.9883 −1.59524 −0.797620 0.603160i \(-0.793908\pi\)
−0.797620 + 0.603160i \(0.793908\pi\)
\(158\) 0 0
\(159\) 25.1317 1.99307
\(160\) 0 0
\(161\) −9.68207 −0.763054
\(162\) 0 0
\(163\) 5.44363 0.426378 0.213189 0.977011i \(-0.431615\pi\)
0.213189 + 0.977011i \(0.431615\pi\)
\(164\) 0 0
\(165\) −16.7135 −1.30114
\(166\) 0 0
\(167\) −10.4586 −0.809307 −0.404654 0.914470i \(-0.632608\pi\)
−0.404654 + 0.914470i \(0.632608\pi\)
\(168\) 0 0
\(169\) −9.68092 −0.744686
\(170\) 0 0
\(171\) 27.9591 2.13809
\(172\) 0 0
\(173\) −15.2274 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(174\) 0 0
\(175\) −7.74138 −0.585194
\(176\) 0 0
\(177\) 3.11094 0.233833
\(178\) 0 0
\(179\) −12.7205 −0.950778 −0.475389 0.879776i \(-0.657693\pi\)
−0.475389 + 0.879776i \(0.657693\pi\)
\(180\) 0 0
\(181\) 10.3344 0.768146 0.384073 0.923303i \(-0.374521\pi\)
0.384073 + 0.923303i \(0.374521\pi\)
\(182\) 0 0
\(183\) −21.1691 −1.56486
\(184\) 0 0
\(185\) 8.47244 0.622906
\(186\) 0 0
\(187\) −1.72167 −0.125901
\(188\) 0 0
\(189\) −18.6969 −1.36000
\(190\) 0 0
\(191\) −25.5875 −1.85145 −0.925723 0.378203i \(-0.876542\pi\)
−0.925723 + 0.378203i \(0.876542\pi\)
\(192\) 0 0
\(193\) 9.35915 0.673686 0.336843 0.941561i \(-0.390641\pi\)
0.336843 + 0.941561i \(0.390641\pi\)
\(194\) 0 0
\(195\) −17.6858 −1.26651
\(196\) 0 0
\(197\) −17.9219 −1.27688 −0.638442 0.769670i \(-0.720421\pi\)
−0.638442 + 0.769670i \(0.720421\pi\)
\(198\) 0 0
\(199\) 1.06705 0.0756413 0.0378207 0.999285i \(-0.487958\pi\)
0.0378207 + 0.999285i \(0.487958\pi\)
\(200\) 0 0
\(201\) 15.2636 1.07661
\(202\) 0 0
\(203\) 0.419192 0.0294215
\(204\) 0 0
\(205\) 5.02755 0.351139
\(206\) 0 0
\(207\) 39.5680 2.75016
\(208\) 0 0
\(209\) −7.20824 −0.498604
\(210\) 0 0
\(211\) 9.79572 0.674365 0.337183 0.941439i \(-0.390526\pi\)
0.337183 + 0.941439i \(0.390526\pi\)
\(212\) 0 0
\(213\) −11.6783 −0.800181
\(214\) 0 0
\(215\) 5.75804 0.392695
\(216\) 0 0
\(217\) −1.23045 −0.0835283
\(218\) 0 0
\(219\) −27.0260 −1.82625
\(220\) 0 0
\(221\) −1.82183 −0.122550
\(222\) 0 0
\(223\) 23.4168 1.56811 0.784054 0.620693i \(-0.213148\pi\)
0.784054 + 0.620693i \(0.213148\pi\)
\(224\) 0 0
\(225\) 31.6369 2.10913
\(226\) 0 0
\(227\) −1.58755 −0.105369 −0.0526847 0.998611i \(-0.516778\pi\)
−0.0526847 + 0.998611i \(0.516778\pi\)
\(228\) 0 0
\(229\) −3.80626 −0.251525 −0.125762 0.992060i \(-0.540138\pi\)
−0.125762 + 0.992060i \(0.540138\pi\)
\(230\) 0 0
\(231\) 8.75208 0.575845
\(232\) 0 0
\(233\) −0.0179299 −0.00117463 −0.000587313 1.00000i \(-0.500187\pi\)
−0.000587313 1.00000i \(0.500187\pi\)
\(234\) 0 0
\(235\) −35.5161 −2.31682
\(236\) 0 0
\(237\) 9.91639 0.644139
\(238\) 0 0
\(239\) 13.9482 0.902233 0.451116 0.892465i \(-0.351026\pi\)
0.451116 + 0.892465i \(0.351026\pi\)
\(240\) 0 0
\(241\) 3.02119 0.194612 0.0973059 0.995255i \(-0.468977\pi\)
0.0973059 + 0.995255i \(0.468977\pi\)
\(242\) 0 0
\(243\) 14.0846 0.903530
\(244\) 0 0
\(245\) −13.5112 −0.863202
\(246\) 0 0
\(247\) −7.62759 −0.485332
\(248\) 0 0
\(249\) 4.91223 0.311300
\(250\) 0 0
\(251\) 25.4504 1.60642 0.803209 0.595697i \(-0.203124\pi\)
0.803209 + 0.595697i \(0.203124\pi\)
\(252\) 0 0
\(253\) −10.2012 −0.641342
\(254\) 0 0
\(255\) 9.70769 0.607919
\(256\) 0 0
\(257\) −1.91368 −0.119372 −0.0596861 0.998217i \(-0.519010\pi\)
−0.0596861 + 0.998217i \(0.519010\pi\)
\(258\) 0 0
\(259\) −4.43663 −0.275679
\(260\) 0 0
\(261\) −1.71312 −0.106040
\(262\) 0 0
\(263\) −23.0806 −1.42321 −0.711606 0.702579i \(-0.752032\pi\)
−0.711606 + 0.702579i \(0.752032\pi\)
\(264\) 0 0
\(265\) 25.2089 1.54857
\(266\) 0 0
\(267\) 42.9211 2.62673
\(268\) 0 0
\(269\) −8.22239 −0.501328 −0.250664 0.968074i \(-0.580649\pi\)
−0.250664 + 0.968074i \(0.580649\pi\)
\(270\) 0 0
\(271\) 0.743034 0.0451361 0.0225680 0.999745i \(-0.492816\pi\)
0.0225680 + 0.999745i \(0.492816\pi\)
\(272\) 0 0
\(273\) 9.26125 0.560516
\(274\) 0 0
\(275\) −8.15643 −0.491851
\(276\) 0 0
\(277\) −14.4683 −0.869315 −0.434658 0.900596i \(-0.643131\pi\)
−0.434658 + 0.900596i \(0.643131\pi\)
\(278\) 0 0
\(279\) 5.02850 0.301049
\(280\) 0 0
\(281\) −18.7950 −1.12121 −0.560607 0.828082i \(-0.689432\pi\)
−0.560607 + 0.828082i \(0.689432\pi\)
\(282\) 0 0
\(283\) −28.0143 −1.66528 −0.832638 0.553817i \(-0.813171\pi\)
−0.832638 + 0.553817i \(0.813171\pi\)
\(284\) 0 0
\(285\) 40.6438 2.40753
\(286\) 0 0
\(287\) −2.63270 −0.155403
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) −17.6200 −1.03290
\(292\) 0 0
\(293\) −13.7345 −0.802375 −0.401188 0.915996i \(-0.631403\pi\)
−0.401188 + 0.915996i \(0.631403\pi\)
\(294\) 0 0
\(295\) 3.12050 0.181682
\(296\) 0 0
\(297\) −19.6993 −1.14307
\(298\) 0 0
\(299\) −10.7946 −0.624270
\(300\) 0 0
\(301\) −3.01522 −0.173795
\(302\) 0 0
\(303\) −53.3033 −3.06220
\(304\) 0 0
\(305\) −21.2341 −1.21586
\(306\) 0 0
\(307\) −19.0064 −1.08475 −0.542377 0.840135i \(-0.682476\pi\)
−0.542377 + 0.840135i \(0.682476\pi\)
\(308\) 0 0
\(309\) 39.4917 2.24660
\(310\) 0 0
\(311\) −25.2402 −1.43124 −0.715622 0.698488i \(-0.753856\pi\)
−0.715622 + 0.698488i \(0.753856\pi\)
\(312\) 0 0
\(313\) 16.4914 0.932149 0.466075 0.884745i \(-0.345668\pi\)
0.466075 + 0.884745i \(0.345668\pi\)
\(314\) 0 0
\(315\) −34.0516 −1.91859
\(316\) 0 0
\(317\) 1.57805 0.0886320 0.0443160 0.999018i \(-0.485889\pi\)
0.0443160 + 0.999018i \(0.485889\pi\)
\(318\) 0 0
\(319\) 0.441667 0.0247286
\(320\) 0 0
\(321\) 7.56947 0.422487
\(322\) 0 0
\(323\) 4.18677 0.232958
\(324\) 0 0
\(325\) −8.63094 −0.478759
\(326\) 0 0
\(327\) 19.2019 1.06187
\(328\) 0 0
\(329\) 18.5982 1.02535
\(330\) 0 0
\(331\) −5.75710 −0.316439 −0.158219 0.987404i \(-0.550575\pi\)
−0.158219 + 0.987404i \(0.550575\pi\)
\(332\) 0 0
\(333\) 18.1313 0.993589
\(334\) 0 0
\(335\) 15.3105 0.836500
\(336\) 0 0
\(337\) 16.6952 0.909443 0.454722 0.890634i \(-0.349739\pi\)
0.454722 + 0.890634i \(0.349739\pi\)
\(338\) 0 0
\(339\) −19.9954 −1.08600
\(340\) 0 0
\(341\) −1.29642 −0.0702049
\(342\) 0 0
\(343\) 18.5137 0.999644
\(344\) 0 0
\(345\) 57.5195 3.09675
\(346\) 0 0
\(347\) −3.01346 −0.161771 −0.0808854 0.996723i \(-0.525775\pi\)
−0.0808854 + 0.996723i \(0.525775\pi\)
\(348\) 0 0
\(349\) −21.2406 −1.13698 −0.568490 0.822690i \(-0.692472\pi\)
−0.568490 + 0.822690i \(0.692472\pi\)
\(350\) 0 0
\(351\) −20.8453 −1.11264
\(352\) 0 0
\(353\) −16.5532 −0.881036 −0.440518 0.897744i \(-0.645205\pi\)
−0.440518 + 0.897744i \(0.645205\pi\)
\(354\) 0 0
\(355\) −11.7141 −0.621721
\(356\) 0 0
\(357\) −5.08348 −0.269046
\(358\) 0 0
\(359\) −19.6883 −1.03911 −0.519554 0.854438i \(-0.673902\pi\)
−0.519554 + 0.854438i \(0.673902\pi\)
\(360\) 0 0
\(361\) −1.47098 −0.0774198
\(362\) 0 0
\(363\) −24.9991 −1.31211
\(364\) 0 0
\(365\) −27.1090 −1.41895
\(366\) 0 0
\(367\) −6.82074 −0.356040 −0.178020 0.984027i \(-0.556969\pi\)
−0.178020 + 0.984027i \(0.556969\pi\)
\(368\) 0 0
\(369\) 10.7591 0.560098
\(370\) 0 0
\(371\) −13.2007 −0.685348
\(372\) 0 0
\(373\) −26.9808 −1.39701 −0.698507 0.715603i \(-0.746152\pi\)
−0.698507 + 0.715603i \(0.746152\pi\)
\(374\) 0 0
\(375\) −2.54822 −0.131590
\(376\) 0 0
\(377\) 0.467362 0.0240703
\(378\) 0 0
\(379\) 26.3871 1.35542 0.677708 0.735331i \(-0.262973\pi\)
0.677708 + 0.735331i \(0.262973\pi\)
\(380\) 0 0
\(381\) 6.69695 0.343095
\(382\) 0 0
\(383\) 16.8194 0.859431 0.429715 0.902964i \(-0.358614\pi\)
0.429715 + 0.902964i \(0.358614\pi\)
\(384\) 0 0
\(385\) 8.77896 0.447417
\(386\) 0 0
\(387\) 12.3224 0.626382
\(388\) 0 0
\(389\) 30.2679 1.53464 0.767321 0.641263i \(-0.221589\pi\)
0.767321 + 0.641263i \(0.221589\pi\)
\(390\) 0 0
\(391\) 5.92515 0.299648
\(392\) 0 0
\(393\) 16.8219 0.848552
\(394\) 0 0
\(395\) 9.94685 0.500480
\(396\) 0 0
\(397\) 9.85519 0.494618 0.247309 0.968937i \(-0.420454\pi\)
0.247309 + 0.968937i \(0.420454\pi\)
\(398\) 0 0
\(399\) −21.2833 −1.06550
\(400\) 0 0
\(401\) −22.5047 −1.12383 −0.561916 0.827195i \(-0.689935\pi\)
−0.561916 + 0.827195i \(0.689935\pi\)
\(402\) 0 0
\(403\) −1.37184 −0.0683362
\(404\) 0 0
\(405\) 48.5592 2.41292
\(406\) 0 0
\(407\) −4.67450 −0.231706
\(408\) 0 0
\(409\) 8.30743 0.410776 0.205388 0.978681i \(-0.434154\pi\)
0.205388 + 0.978681i \(0.434154\pi\)
\(410\) 0 0
\(411\) 22.2597 1.09799
\(412\) 0 0
\(413\) −1.63406 −0.0804070
\(414\) 0 0
\(415\) 4.92732 0.241872
\(416\) 0 0
\(417\) 33.8044 1.65541
\(418\) 0 0
\(419\) −13.4729 −0.658194 −0.329097 0.944296i \(-0.606744\pi\)
−0.329097 + 0.944296i \(0.606744\pi\)
\(420\) 0 0
\(421\) −15.2479 −0.743137 −0.371568 0.928406i \(-0.621180\pi\)
−0.371568 + 0.928406i \(0.621180\pi\)
\(422\) 0 0
\(423\) −76.0057 −3.69552
\(424\) 0 0
\(425\) 4.73750 0.229803
\(426\) 0 0
\(427\) 11.1193 0.538102
\(428\) 0 0
\(429\) 9.75778 0.471110
\(430\) 0 0
\(431\) −1.19281 −0.0574555 −0.0287278 0.999587i \(-0.509146\pi\)
−0.0287278 + 0.999587i \(0.509146\pi\)
\(432\) 0 0
\(433\) −17.1948 −0.826331 −0.413165 0.910656i \(-0.635577\pi\)
−0.413165 + 0.910656i \(0.635577\pi\)
\(434\) 0 0
\(435\) −2.49035 −0.119403
\(436\) 0 0
\(437\) 24.8072 1.18669
\(438\) 0 0
\(439\) −10.9248 −0.521413 −0.260707 0.965418i \(-0.583956\pi\)
−0.260707 + 0.965418i \(0.583956\pi\)
\(440\) 0 0
\(441\) −28.9145 −1.37688
\(442\) 0 0
\(443\) 13.7355 0.652594 0.326297 0.945267i \(-0.394199\pi\)
0.326297 + 0.945267i \(0.394199\pi\)
\(444\) 0 0
\(445\) 43.0529 2.04090
\(446\) 0 0
\(447\) 44.6416 2.11147
\(448\) 0 0
\(449\) 11.3865 0.537362 0.268681 0.963229i \(-0.413412\pi\)
0.268681 + 0.963229i \(0.413412\pi\)
\(450\) 0 0
\(451\) −2.77385 −0.130615
\(452\) 0 0
\(453\) −9.78540 −0.459758
\(454\) 0 0
\(455\) 9.28969 0.435508
\(456\) 0 0
\(457\) −17.3461 −0.811414 −0.405707 0.914003i \(-0.632975\pi\)
−0.405707 + 0.914003i \(0.632975\pi\)
\(458\) 0 0
\(459\) 11.4420 0.534065
\(460\) 0 0
\(461\) −4.38993 −0.204460 −0.102230 0.994761i \(-0.532598\pi\)
−0.102230 + 0.994761i \(0.532598\pi\)
\(462\) 0 0
\(463\) 5.45258 0.253403 0.126702 0.991941i \(-0.459561\pi\)
0.126702 + 0.991941i \(0.459561\pi\)
\(464\) 0 0
\(465\) 7.30988 0.338988
\(466\) 0 0
\(467\) 18.5268 0.857317 0.428658 0.903467i \(-0.358986\pi\)
0.428658 + 0.903467i \(0.358986\pi\)
\(468\) 0 0
\(469\) −8.01740 −0.370209
\(470\) 0 0
\(471\) −62.1825 −2.86522
\(472\) 0 0
\(473\) −3.17688 −0.146073
\(474\) 0 0
\(475\) 19.8348 0.910084
\(476\) 0 0
\(477\) 53.9478 2.47010
\(478\) 0 0
\(479\) 2.41160 0.110189 0.0550945 0.998481i \(-0.482454\pi\)
0.0550945 + 0.998481i \(0.482454\pi\)
\(480\) 0 0
\(481\) −4.94645 −0.225539
\(482\) 0 0
\(483\) −30.1204 −1.37052
\(484\) 0 0
\(485\) −17.6741 −0.802541
\(486\) 0 0
\(487\) 39.1865 1.77571 0.887856 0.460122i \(-0.152194\pi\)
0.887856 + 0.460122i \(0.152194\pi\)
\(488\) 0 0
\(489\) 16.9348 0.765819
\(490\) 0 0
\(491\) 30.6693 1.38408 0.692042 0.721857i \(-0.256711\pi\)
0.692042 + 0.721857i \(0.256711\pi\)
\(492\) 0 0
\(493\) −0.256534 −0.0115537
\(494\) 0 0
\(495\) −35.8772 −1.61256
\(496\) 0 0
\(497\) 6.13415 0.275154
\(498\) 0 0
\(499\) −21.0417 −0.941956 −0.470978 0.882145i \(-0.656099\pi\)
−0.470978 + 0.882145i \(0.656099\pi\)
\(500\) 0 0
\(501\) −32.5360 −1.45360
\(502\) 0 0
\(503\) −16.0890 −0.717374 −0.358687 0.933458i \(-0.616775\pi\)
−0.358687 + 0.933458i \(0.616775\pi\)
\(504\) 0 0
\(505\) −53.4670 −2.37925
\(506\) 0 0
\(507\) −30.1168 −1.33753
\(508\) 0 0
\(509\) 21.7446 0.963814 0.481907 0.876222i \(-0.339944\pi\)
0.481907 + 0.876222i \(0.339944\pi\)
\(510\) 0 0
\(511\) 14.1958 0.627984
\(512\) 0 0
\(513\) 47.9048 2.11505
\(514\) 0 0
\(515\) 39.6130 1.74556
\(516\) 0 0
\(517\) 19.5953 0.861801
\(518\) 0 0
\(519\) −47.3717 −2.07938
\(520\) 0 0
\(521\) 22.2214 0.973536 0.486768 0.873531i \(-0.338176\pi\)
0.486768 + 0.873531i \(0.338176\pi\)
\(522\) 0 0
\(523\) 1.66928 0.0729927 0.0364963 0.999334i \(-0.488380\pi\)
0.0364963 + 0.999334i \(0.488380\pi\)
\(524\) 0 0
\(525\) −24.0830 −1.05107
\(526\) 0 0
\(527\) 0.752999 0.0328012
\(528\) 0 0
\(529\) 12.1074 0.526410
\(530\) 0 0
\(531\) 6.67797 0.289799
\(532\) 0 0
\(533\) −2.93522 −0.127139
\(534\) 0 0
\(535\) 7.59272 0.328262
\(536\) 0 0
\(537\) −39.5729 −1.70770
\(538\) 0 0
\(539\) 7.45456 0.321090
\(540\) 0 0
\(541\) −15.7624 −0.677679 −0.338839 0.940844i \(-0.610034\pi\)
−0.338839 + 0.940844i \(0.610034\pi\)
\(542\) 0 0
\(543\) 32.1496 1.37967
\(544\) 0 0
\(545\) 19.2609 0.825046
\(546\) 0 0
\(547\) 6.54160 0.279699 0.139849 0.990173i \(-0.455338\pi\)
0.139849 + 0.990173i \(0.455338\pi\)
\(548\) 0 0
\(549\) −45.4416 −1.93940
\(550\) 0 0
\(551\) −1.07405 −0.0457559
\(552\) 0 0
\(553\) −5.20871 −0.221497
\(554\) 0 0
\(555\) 26.3573 1.11880
\(556\) 0 0
\(557\) 26.6464 1.12904 0.564522 0.825418i \(-0.309061\pi\)
0.564522 + 0.825418i \(0.309061\pi\)
\(558\) 0 0
\(559\) −3.36170 −0.142185
\(560\) 0 0
\(561\) −5.35602 −0.226131
\(562\) 0 0
\(563\) 15.1636 0.639070 0.319535 0.947574i \(-0.396473\pi\)
0.319535 + 0.947574i \(0.396473\pi\)
\(564\) 0 0
\(565\) −20.0568 −0.843795
\(566\) 0 0
\(567\) −25.4282 −1.06789
\(568\) 0 0
\(569\) −12.1517 −0.509427 −0.254714 0.967017i \(-0.581981\pi\)
−0.254714 + 0.967017i \(0.581981\pi\)
\(570\) 0 0
\(571\) −22.1753 −0.928006 −0.464003 0.885834i \(-0.653587\pi\)
−0.464003 + 0.885834i \(0.653587\pi\)
\(572\) 0 0
\(573\) −79.6012 −3.32539
\(574\) 0 0
\(575\) 28.0704 1.17062
\(576\) 0 0
\(577\) −24.5200 −1.02078 −0.510391 0.859942i \(-0.670499\pi\)
−0.510391 + 0.859942i \(0.670499\pi\)
\(578\) 0 0
\(579\) 29.1158 1.21001
\(580\) 0 0
\(581\) −2.58021 −0.107045
\(582\) 0 0
\(583\) −13.9085 −0.576030
\(584\) 0 0
\(585\) −37.9644 −1.56964
\(586\) 0 0
\(587\) 12.7090 0.524558 0.262279 0.964992i \(-0.415526\pi\)
0.262279 + 0.964992i \(0.415526\pi\)
\(588\) 0 0
\(589\) 3.15263 0.129902
\(590\) 0 0
\(591\) −55.7541 −2.29342
\(592\) 0 0
\(593\) 6.68997 0.274724 0.137362 0.990521i \(-0.456138\pi\)
0.137362 + 0.990521i \(0.456138\pi\)
\(594\) 0 0
\(595\) −5.09909 −0.209042
\(596\) 0 0
\(597\) 3.31954 0.135860
\(598\) 0 0
\(599\) −34.3707 −1.40435 −0.702175 0.712004i \(-0.747788\pi\)
−0.702175 + 0.712004i \(0.747788\pi\)
\(600\) 0 0
\(601\) 30.2340 1.23327 0.616635 0.787249i \(-0.288496\pi\)
0.616635 + 0.787249i \(0.288496\pi\)
\(602\) 0 0
\(603\) 32.7649 1.33429
\(604\) 0 0
\(605\) −25.0758 −1.01948
\(606\) 0 0
\(607\) −22.7544 −0.923571 −0.461785 0.886992i \(-0.652791\pi\)
−0.461785 + 0.886992i \(0.652791\pi\)
\(608\) 0 0
\(609\) 1.30408 0.0528441
\(610\) 0 0
\(611\) 20.7353 0.838861
\(612\) 0 0
\(613\) 1.83408 0.0740779 0.0370389 0.999314i \(-0.488207\pi\)
0.0370389 + 0.999314i \(0.488207\pi\)
\(614\) 0 0
\(615\) 15.6404 0.630683
\(616\) 0 0
\(617\) 24.2013 0.974306 0.487153 0.873317i \(-0.338035\pi\)
0.487153 + 0.873317i \(0.338035\pi\)
\(618\) 0 0
\(619\) 41.7256 1.67709 0.838546 0.544831i \(-0.183406\pi\)
0.838546 + 0.544831i \(0.183406\pi\)
\(620\) 0 0
\(621\) 67.7953 2.72053
\(622\) 0 0
\(623\) −22.5449 −0.903240
\(624\) 0 0
\(625\) −26.2436 −1.04974
\(626\) 0 0
\(627\) −22.4244 −0.895545
\(628\) 0 0
\(629\) 2.71509 0.108258
\(630\) 0 0
\(631\) −29.8422 −1.18800 −0.593999 0.804466i \(-0.702452\pi\)
−0.593999 + 0.804466i \(0.702452\pi\)
\(632\) 0 0
\(633\) 30.4739 1.21123
\(634\) 0 0
\(635\) 6.71751 0.266576
\(636\) 0 0
\(637\) 7.88824 0.312543
\(638\) 0 0
\(639\) −25.0686 −0.991699
\(640\) 0 0
\(641\) 4.39563 0.173617 0.0868084 0.996225i \(-0.472333\pi\)
0.0868084 + 0.996225i \(0.472333\pi\)
\(642\) 0 0
\(643\) 26.5052 1.04526 0.522631 0.852559i \(-0.324950\pi\)
0.522631 + 0.852559i \(0.324950\pi\)
\(644\) 0 0
\(645\) 17.9129 0.705321
\(646\) 0 0
\(647\) 32.2232 1.26682 0.633412 0.773814i \(-0.281654\pi\)
0.633412 + 0.773814i \(0.281654\pi\)
\(648\) 0 0
\(649\) −1.72167 −0.0675815
\(650\) 0 0
\(651\) −3.82785 −0.150025
\(652\) 0 0
\(653\) −13.2258 −0.517564 −0.258782 0.965936i \(-0.583321\pi\)
−0.258782 + 0.965936i \(0.583321\pi\)
\(654\) 0 0
\(655\) 16.8735 0.659304
\(656\) 0 0
\(657\) −58.0143 −2.26335
\(658\) 0 0
\(659\) −44.7479 −1.74313 −0.871564 0.490281i \(-0.836894\pi\)
−0.871564 + 0.490281i \(0.836894\pi\)
\(660\) 0 0
\(661\) 16.2194 0.630861 0.315430 0.948949i \(-0.397851\pi\)
0.315430 + 0.948949i \(0.397851\pi\)
\(662\) 0 0
\(663\) −5.66762 −0.220112
\(664\) 0 0
\(665\) −21.3487 −0.827867
\(666\) 0 0
\(667\) −1.52000 −0.0588547
\(668\) 0 0
\(669\) 72.8485 2.81649
\(670\) 0 0
\(671\) 11.7155 0.452271
\(672\) 0 0
\(673\) −28.4518 −1.09674 −0.548369 0.836237i \(-0.684751\pi\)
−0.548369 + 0.836237i \(0.684751\pi\)
\(674\) 0 0
\(675\) 54.2063 2.08640
\(676\) 0 0
\(677\) 2.14609 0.0824808 0.0412404 0.999149i \(-0.486869\pi\)
0.0412404 + 0.999149i \(0.486869\pi\)
\(678\) 0 0
\(679\) 9.25514 0.355180
\(680\) 0 0
\(681\) −4.93878 −0.189254
\(682\) 0 0
\(683\) 29.8845 1.14350 0.571749 0.820428i \(-0.306265\pi\)
0.571749 + 0.820428i \(0.306265\pi\)
\(684\) 0 0
\(685\) 22.3281 0.853113
\(686\) 0 0
\(687\) −11.8410 −0.451764
\(688\) 0 0
\(689\) −14.7176 −0.560697
\(690\) 0 0
\(691\) 4.57599 0.174079 0.0870394 0.996205i \(-0.472259\pi\)
0.0870394 + 0.996205i \(0.472259\pi\)
\(692\) 0 0
\(693\) 18.7873 0.713669
\(694\) 0 0
\(695\) 33.9082 1.28621
\(696\) 0 0
\(697\) 1.61114 0.0610262
\(698\) 0 0
\(699\) −0.0557788 −0.00210975
\(700\) 0 0
\(701\) 17.0624 0.644438 0.322219 0.946665i \(-0.395571\pi\)
0.322219 + 0.946665i \(0.395571\pi\)
\(702\) 0 0
\(703\) 11.3675 0.428732
\(704\) 0 0
\(705\) −110.489 −4.16124
\(706\) 0 0
\(707\) 27.9983 1.05298
\(708\) 0 0
\(709\) 0.879464 0.0330290 0.0165145 0.999864i \(-0.494743\pi\)
0.0165145 + 0.999864i \(0.494743\pi\)
\(710\) 0 0
\(711\) 21.2866 0.798309
\(712\) 0 0
\(713\) 4.46163 0.167090
\(714\) 0 0
\(715\) 9.78775 0.366041
\(716\) 0 0
\(717\) 43.3920 1.62050
\(718\) 0 0
\(719\) −10.2871 −0.383644 −0.191822 0.981430i \(-0.561440\pi\)
−0.191822 + 0.981430i \(0.561440\pi\)
\(720\) 0 0
\(721\) −20.7435 −0.772529
\(722\) 0 0
\(723\) 9.39874 0.349543
\(724\) 0 0
\(725\) −1.21533 −0.0451362
\(726\) 0 0
\(727\) 49.3441 1.83007 0.915036 0.403373i \(-0.132162\pi\)
0.915036 + 0.403373i \(0.132162\pi\)
\(728\) 0 0
\(729\) −2.86754 −0.106205
\(730\) 0 0
\(731\) 1.84523 0.0682483
\(732\) 0 0
\(733\) 29.3806 1.08520 0.542598 0.839993i \(-0.317441\pi\)
0.542598 + 0.839993i \(0.317441\pi\)
\(734\) 0 0
\(735\) −42.0327 −1.55040
\(736\) 0 0
\(737\) −8.44724 −0.311158
\(738\) 0 0
\(739\) 31.2950 1.15120 0.575602 0.817730i \(-0.304768\pi\)
0.575602 + 0.817730i \(0.304768\pi\)
\(740\) 0 0
\(741\) −23.7290 −0.871707
\(742\) 0 0
\(743\) −39.6344 −1.45405 −0.727023 0.686613i \(-0.759097\pi\)
−0.727023 + 0.686613i \(0.759097\pi\)
\(744\) 0 0
\(745\) 44.7787 1.64056
\(746\) 0 0
\(747\) 10.5446 0.385808
\(748\) 0 0
\(749\) −3.97596 −0.145279
\(750\) 0 0
\(751\) −7.03957 −0.256878 −0.128439 0.991717i \(-0.540997\pi\)
−0.128439 + 0.991717i \(0.540997\pi\)
\(752\) 0 0
\(753\) 79.1749 2.88529
\(754\) 0 0
\(755\) −9.81545 −0.357221
\(756\) 0 0
\(757\) 11.4060 0.414558 0.207279 0.978282i \(-0.433539\pi\)
0.207279 + 0.978282i \(0.433539\pi\)
\(758\) 0 0
\(759\) −31.7352 −1.15192
\(760\) 0 0
\(761\) 41.5123 1.50482 0.752409 0.658696i \(-0.228892\pi\)
0.752409 + 0.658696i \(0.228892\pi\)
\(762\) 0 0
\(763\) −10.0861 −0.365140
\(764\) 0 0
\(765\) 20.8386 0.753421
\(766\) 0 0
\(767\) −1.82183 −0.0657826
\(768\) 0 0
\(769\) 6.74890 0.243372 0.121686 0.992569i \(-0.461170\pi\)
0.121686 + 0.992569i \(0.461170\pi\)
\(770\) 0 0
\(771\) −5.95336 −0.214405
\(772\) 0 0
\(773\) 15.9701 0.574404 0.287202 0.957870i \(-0.407275\pi\)
0.287202 + 0.957870i \(0.407275\pi\)
\(774\) 0 0
\(775\) 3.56734 0.128143
\(776\) 0 0
\(777\) −13.8021 −0.495148
\(778\) 0 0
\(779\) 6.74546 0.241681
\(780\) 0 0
\(781\) 6.46303 0.231265
\(782\) 0 0
\(783\) −2.93525 −0.104897
\(784\) 0 0
\(785\) −62.3735 −2.22620
\(786\) 0 0
\(787\) −9.60301 −0.342310 −0.171155 0.985244i \(-0.554750\pi\)
−0.171155 + 0.985244i \(0.554750\pi\)
\(788\) 0 0
\(789\) −71.8025 −2.55624
\(790\) 0 0
\(791\) 10.5028 0.373438
\(792\) 0 0
\(793\) 12.3970 0.440232
\(794\) 0 0
\(795\) 78.4233 2.78139
\(796\) 0 0
\(797\) 46.8184 1.65839 0.829197 0.558956i \(-0.188798\pi\)
0.829197 + 0.558956i \(0.188798\pi\)
\(798\) 0 0
\(799\) −11.3816 −0.402651
\(800\) 0 0
\(801\) 92.1346 3.25542
\(802\) 0 0
\(803\) 14.9569 0.527816
\(804\) 0 0
\(805\) −30.2129 −1.06486
\(806\) 0 0
\(807\) −25.5794 −0.900437
\(808\) 0 0
\(809\) −37.0221 −1.30163 −0.650814 0.759237i \(-0.725572\pi\)
−0.650814 + 0.759237i \(0.725572\pi\)
\(810\) 0 0
\(811\) −8.88874 −0.312126 −0.156063 0.987747i \(-0.549880\pi\)
−0.156063 + 0.987747i \(0.549880\pi\)
\(812\) 0 0
\(813\) 2.31154 0.0810691
\(814\) 0 0
\(815\) 16.9868 0.595023
\(816\) 0 0
\(817\) 7.72555 0.270283
\(818\) 0 0
\(819\) 19.8803 0.694672
\(820\) 0 0
\(821\) −49.4325 −1.72521 −0.862603 0.505881i \(-0.831168\pi\)
−0.862603 + 0.505881i \(0.831168\pi\)
\(822\) 0 0
\(823\) −8.70614 −0.303477 −0.151738 0.988421i \(-0.548487\pi\)
−0.151738 + 0.988421i \(0.548487\pi\)
\(824\) 0 0
\(825\) −25.3742 −0.883416
\(826\) 0 0
\(827\) 27.8666 0.969018 0.484509 0.874786i \(-0.338998\pi\)
0.484509 + 0.874786i \(0.338998\pi\)
\(828\) 0 0
\(829\) 6.13383 0.213037 0.106518 0.994311i \(-0.466030\pi\)
0.106518 + 0.994311i \(0.466030\pi\)
\(830\) 0 0
\(831\) −45.0100 −1.56138
\(832\) 0 0
\(833\) −4.32984 −0.150020
\(834\) 0 0
\(835\) −32.6359 −1.12941
\(836\) 0 0
\(837\) 8.61578 0.297805
\(838\) 0 0
\(839\) 49.0553 1.69358 0.846789 0.531930i \(-0.178533\pi\)
0.846789 + 0.531930i \(0.178533\pi\)
\(840\) 0 0
\(841\) −28.9342 −0.997731
\(842\) 0 0
\(843\) −58.4701 −2.01382
\(844\) 0 0
\(845\) −30.2093 −1.03923
\(846\) 0 0
\(847\) 13.1311 0.451189
\(848\) 0 0
\(849\) −87.1509 −2.99101
\(850\) 0 0
\(851\) 16.0873 0.551467
\(852\) 0 0
\(853\) −46.2840 −1.58473 −0.792367 0.610044i \(-0.791152\pi\)
−0.792367 + 0.610044i \(0.791152\pi\)
\(854\) 0 0
\(855\) 87.2463 2.98376
\(856\) 0 0
\(857\) −9.25611 −0.316183 −0.158091 0.987424i \(-0.550534\pi\)
−0.158091 + 0.987424i \(0.550534\pi\)
\(858\) 0 0
\(859\) 10.2033 0.348130 0.174065 0.984734i \(-0.444310\pi\)
0.174065 + 0.984734i \(0.444310\pi\)
\(860\) 0 0
\(861\) −8.19018 −0.279121
\(862\) 0 0
\(863\) −2.27365 −0.0773960 −0.0386980 0.999251i \(-0.512321\pi\)
−0.0386980 + 0.999251i \(0.512321\pi\)
\(864\) 0 0
\(865\) −47.5171 −1.61563
\(866\) 0 0
\(867\) 3.11094 0.105653
\(868\) 0 0
\(869\) −5.48797 −0.186167
\(870\) 0 0
\(871\) −8.93867 −0.302875
\(872\) 0 0
\(873\) −37.8232 −1.28012
\(874\) 0 0
\(875\) 1.33849 0.0452491
\(876\) 0 0
\(877\) −24.7172 −0.834641 −0.417320 0.908759i \(-0.637031\pi\)
−0.417320 + 0.908759i \(0.637031\pi\)
\(878\) 0 0
\(879\) −42.7271 −1.44115
\(880\) 0 0
\(881\) 40.6557 1.36972 0.684862 0.728673i \(-0.259862\pi\)
0.684862 + 0.728673i \(0.259862\pi\)
\(882\) 0 0
\(883\) 8.59522 0.289252 0.144626 0.989486i \(-0.453802\pi\)
0.144626 + 0.989486i \(0.453802\pi\)
\(884\) 0 0
\(885\) 9.70769 0.326320
\(886\) 0 0
\(887\) −24.7662 −0.831566 −0.415783 0.909464i \(-0.636493\pi\)
−0.415783 + 0.909464i \(0.636493\pi\)
\(888\) 0 0
\(889\) −3.51766 −0.117978
\(890\) 0 0
\(891\) −26.7915 −0.897550
\(892\) 0 0
\(893\) −47.6520 −1.59461
\(894\) 0 0
\(895\) −39.6944 −1.32684
\(896\) 0 0
\(897\) −33.5815 −1.12125
\(898\) 0 0
\(899\) −0.193170 −0.00644257
\(900\) 0 0
\(901\) 8.07848 0.269133
\(902\) 0 0
\(903\) −9.38019 −0.312153
\(904\) 0 0
\(905\) 32.2483 1.07197
\(906\) 0 0
\(907\) 15.0365 0.499279 0.249640 0.968339i \(-0.419688\pi\)
0.249640 + 0.968339i \(0.419688\pi\)
\(908\) 0 0
\(909\) −114.421 −3.79511
\(910\) 0 0
\(911\) 25.0833 0.831046 0.415523 0.909583i \(-0.363599\pi\)
0.415523 + 0.909583i \(0.363599\pi\)
\(912\) 0 0
\(913\) −2.71855 −0.0899708
\(914\) 0 0
\(915\) −66.0580 −2.18381
\(916\) 0 0
\(917\) −8.83591 −0.291788
\(918\) 0 0
\(919\) −21.0181 −0.693322 −0.346661 0.937991i \(-0.612685\pi\)
−0.346661 + 0.937991i \(0.612685\pi\)
\(920\) 0 0
\(921\) −59.1279 −1.94833
\(922\) 0 0
\(923\) 6.83903 0.225109
\(924\) 0 0
\(925\) 12.8628 0.422925
\(926\) 0 0
\(927\) 84.7730 2.78431
\(928\) 0 0
\(929\) −8.39790 −0.275526 −0.137763 0.990465i \(-0.543991\pi\)
−0.137763 + 0.990465i \(0.543991\pi\)
\(930\) 0 0
\(931\) −18.1280 −0.594122
\(932\) 0 0
\(933\) −78.5210 −2.57066
\(934\) 0 0
\(935\) −5.37247 −0.175699
\(936\) 0 0
\(937\) −34.9004 −1.14015 −0.570073 0.821594i \(-0.693085\pi\)
−0.570073 + 0.821594i \(0.693085\pi\)
\(938\) 0 0
\(939\) 51.3038 1.67424
\(940\) 0 0
\(941\) −40.2151 −1.31097 −0.655487 0.755206i \(-0.727537\pi\)
−0.655487 + 0.755206i \(0.727537\pi\)
\(942\) 0 0
\(943\) 9.54623 0.310868
\(944\) 0 0
\(945\) −58.3435 −1.89792
\(946\) 0 0
\(947\) 4.15888 0.135145 0.0675727 0.997714i \(-0.478475\pi\)
0.0675727 + 0.997714i \(0.478475\pi\)
\(948\) 0 0
\(949\) 15.8270 0.513767
\(950\) 0 0
\(951\) 4.90922 0.159192
\(952\) 0 0
\(953\) 46.8759 1.51846 0.759229 0.650823i \(-0.225576\pi\)
0.759229 + 0.650823i \(0.225576\pi\)
\(954\) 0 0
\(955\) −79.8457 −2.58375
\(956\) 0 0
\(957\) 1.37400 0.0444151
\(958\) 0 0
\(959\) −11.6922 −0.377561
\(960\) 0 0
\(961\) −30.4330 −0.981709
\(962\) 0 0
\(963\) 16.2487 0.523606
\(964\) 0 0
\(965\) 29.2052 0.940149
\(966\) 0 0
\(967\) 38.0541 1.22374 0.611869 0.790959i \(-0.290418\pi\)
0.611869 + 0.790959i \(0.290418\pi\)
\(968\) 0 0
\(969\) 13.0248 0.418417
\(970\) 0 0
\(971\) −26.8137 −0.860493 −0.430246 0.902711i \(-0.641573\pi\)
−0.430246 + 0.902711i \(0.641573\pi\)
\(972\) 0 0
\(973\) −17.7562 −0.569238
\(974\) 0 0
\(975\) −26.8504 −0.859900
\(976\) 0 0
\(977\) 21.4495 0.686229 0.343114 0.939294i \(-0.388518\pi\)
0.343114 + 0.939294i \(0.388518\pi\)
\(978\) 0 0
\(979\) −23.7536 −0.759167
\(980\) 0 0
\(981\) 41.2190 1.31602
\(982\) 0 0
\(983\) 17.5036 0.558279 0.279139 0.960251i \(-0.409951\pi\)
0.279139 + 0.960251i \(0.409951\pi\)
\(984\) 0 0
\(985\) −55.9253 −1.78193
\(986\) 0 0
\(987\) 57.8579 1.84164
\(988\) 0 0
\(989\) 10.9333 0.347658
\(990\) 0 0
\(991\) −2.55287 −0.0810946 −0.0405473 0.999178i \(-0.512910\pi\)
−0.0405473 + 0.999178i \(0.512910\pi\)
\(992\) 0 0
\(993\) −17.9100 −0.568357
\(994\) 0 0
\(995\) 3.32973 0.105560
\(996\) 0 0
\(997\) −49.7604 −1.57593 −0.787964 0.615722i \(-0.788864\pi\)
−0.787964 + 0.615722i \(0.788864\pi\)
\(998\) 0 0
\(999\) 31.0660 0.982884
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 4012.2.a.j.1.19 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4012.2.a.j.1.19 21 1.1 even 1 trivial