Properties

Label 4014.2.d.a.4013.10
Level $4014$
Weight $2$
Character 4014.4013
Analytic conductor $32.052$
Analytic rank $0$
Dimension $72$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4014,2,Mod(4013,4014)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4014, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4014.4013");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4014.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.0519513713\)
Analytic rank: \(0\)
Dimension: \(72\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4013.10
Character \(\chi\) \(=\) 4014.4013
Dual form 4014.2.d.a.4013.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} +3.59603 q^{5} +4.66381 q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} +3.59603 q^{5} +4.66381 q^{7} -1.00000i q^{8} +3.59603i q^{10} -0.902432 q^{11} +3.90912i q^{13} +4.66381i q^{14} +1.00000 q^{16} -3.88874i q^{17} +2.00606 q^{19} -3.59603 q^{20} -0.902432i q^{22} +9.11811 q^{23} +7.93141 q^{25} -3.90912 q^{26} -4.66381 q^{28} -5.06669i q^{29} -7.64590 q^{31} +1.00000i q^{32} +3.88874 q^{34} +16.7712 q^{35} +2.27714 q^{37} +2.00606i q^{38} -3.59603i q^{40} +1.40849i q^{41} -5.26130 q^{43} +0.902432 q^{44} +9.11811i q^{46} +5.68266i q^{47} +14.7511 q^{49} +7.93141i q^{50} -3.90912i q^{52} +12.3390i q^{53} -3.24517 q^{55} -4.66381i q^{56} +5.06669 q^{58} -1.09050 q^{59} +1.98011i q^{61} -7.64590i q^{62} -1.00000 q^{64} +14.0573i q^{65} -13.9899i q^{67} +3.88874i q^{68} +16.7712i q^{70} -6.23553 q^{71} -15.3293 q^{73} +2.27714i q^{74} -2.00606 q^{76} -4.20877 q^{77} -5.68601i q^{79} +3.59603 q^{80} -1.40849 q^{82} -2.04815i q^{83} -13.9840i q^{85} -5.26130i q^{86} +0.902432i q^{88} -1.95797i q^{89} +18.2314i q^{91} -9.11811 q^{92} -5.68266 q^{94} +7.21384 q^{95} -10.0724i q^{97} +14.7511i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 72 q - 72 q^{4} + 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 72 q - 72 q^{4} + 16 q^{7} + 72 q^{16} - 40 q^{19} + 96 q^{25} - 16 q^{28} - 24 q^{37} - 8 q^{43} + 56 q^{49} + 40 q^{58} - 72 q^{64} - 32 q^{73} + 40 q^{76} + 16 q^{82}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4014\mathbb{Z}\right)^\times\).

\(n\) \(893\) \(2233\)
\(\chi(n)\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 3.59603 1.60819 0.804096 0.594499i \(-0.202650\pi\)
0.804096 + 0.594499i \(0.202650\pi\)
\(6\) 0 0
\(7\) 4.66381 1.76276 0.881378 0.472412i \(-0.156617\pi\)
0.881378 + 0.472412i \(0.156617\pi\)
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 3.59603i 1.13716i
\(11\) −0.902432 −0.272093 −0.136047 0.990702i \(-0.543440\pi\)
−0.136047 + 0.990702i \(0.543440\pi\)
\(12\) 0 0
\(13\) 3.90912i 1.08419i 0.840316 + 0.542097i \(0.182369\pi\)
−0.840316 + 0.542097i \(0.817631\pi\)
\(14\) 4.66381i 1.24646i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.88874i 0.943158i −0.881824 0.471579i \(-0.843684\pi\)
0.881824 0.471579i \(-0.156316\pi\)
\(18\) 0 0
\(19\) 2.00606 0.460222 0.230111 0.973164i \(-0.426091\pi\)
0.230111 + 0.973164i \(0.426091\pi\)
\(20\) −3.59603 −0.804096
\(21\) 0 0
\(22\) 0.902432i 0.192399i
\(23\) 9.11811 1.90126 0.950629 0.310331i \(-0.100440\pi\)
0.950629 + 0.310331i \(0.100440\pi\)
\(24\) 0 0
\(25\) 7.93141 1.58628
\(26\) −3.90912 −0.766641
\(27\) 0 0
\(28\) −4.66381 −0.881378
\(29\) 5.06669i 0.940861i −0.882437 0.470431i \(-0.844099\pi\)
0.882437 0.470431i \(-0.155901\pi\)
\(30\) 0 0
\(31\) −7.64590 −1.37324 −0.686622 0.727014i \(-0.740907\pi\)
−0.686622 + 0.727014i \(0.740907\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 3.88874 0.666914
\(35\) 16.7712 2.83485
\(36\) 0 0
\(37\) 2.27714 0.374359 0.187180 0.982326i \(-0.440065\pi\)
0.187180 + 0.982326i \(0.440065\pi\)
\(38\) 2.00606i 0.325426i
\(39\) 0 0
\(40\) 3.59603i 0.568582i
\(41\) 1.40849i 0.219969i 0.993933 + 0.109984i \(0.0350801\pi\)
−0.993933 + 0.109984i \(0.964920\pi\)
\(42\) 0 0
\(43\) −5.26130 −0.802341 −0.401170 0.916003i \(-0.631396\pi\)
−0.401170 + 0.916003i \(0.631396\pi\)
\(44\) 0.902432 0.136047
\(45\) 0 0
\(46\) 9.11811i 1.34439i
\(47\) 5.68266i 0.828901i 0.910072 + 0.414451i \(0.136026\pi\)
−0.910072 + 0.414451i \(0.863974\pi\)
\(48\) 0 0
\(49\) 14.7511 2.10731
\(50\) 7.93141i 1.12167i
\(51\) 0 0
\(52\) 3.90912i 0.542097i
\(53\) 12.3390i 1.69490i 0.530878 + 0.847448i \(0.321862\pi\)
−0.530878 + 0.847448i \(0.678138\pi\)
\(54\) 0 0
\(55\) −3.24517 −0.437579
\(56\) 4.66381i 0.623228i
\(57\) 0 0
\(58\) 5.06669 0.665289
\(59\) −1.09050 −0.141972 −0.0709858 0.997477i \(-0.522614\pi\)
−0.0709858 + 0.997477i \(0.522614\pi\)
\(60\) 0 0
\(61\) 1.98011i 0.253527i 0.991933 + 0.126763i \(0.0404589\pi\)
−0.991933 + 0.126763i \(0.959541\pi\)
\(62\) 7.64590i 0.971030i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 14.0573i 1.74359i
\(66\) 0 0
\(67\) 13.9899i 1.70914i −0.519340 0.854568i \(-0.673822\pi\)
0.519340 0.854568i \(-0.326178\pi\)
\(68\) 3.88874i 0.471579i
\(69\) 0 0
\(70\) 16.7712i 2.00454i
\(71\) −6.23553 −0.740021 −0.370010 0.929028i \(-0.620646\pi\)
−0.370010 + 0.929028i \(0.620646\pi\)
\(72\) 0 0
\(73\) −15.3293 −1.79415 −0.897077 0.441875i \(-0.854313\pi\)
−0.897077 + 0.441875i \(0.854313\pi\)
\(74\) 2.27714i 0.264712i
\(75\) 0 0
\(76\) −2.00606 −0.230111
\(77\) −4.20877 −0.479634
\(78\) 0 0
\(79\) 5.68601i 0.639726i −0.947464 0.319863i \(-0.896363\pi\)
0.947464 0.319863i \(-0.103637\pi\)
\(80\) 3.59603 0.402048
\(81\) 0 0
\(82\) −1.40849 −0.155541
\(83\) 2.04815i 0.224813i −0.993662 0.112407i \(-0.964144\pi\)
0.993662 0.112407i \(-0.0358559\pi\)
\(84\) 0 0
\(85\) 13.9840i 1.51678i
\(86\) 5.26130i 0.567341i
\(87\) 0 0
\(88\) 0.902432i 0.0961996i
\(89\) 1.95797i 0.207545i −0.994601 0.103772i \(-0.966909\pi\)
0.994601 0.103772i \(-0.0330914\pi\)
\(90\) 0 0
\(91\) 18.2314i 1.91117i
\(92\) −9.11811 −0.950629
\(93\) 0 0
\(94\) −5.68266 −0.586122
\(95\) 7.21384 0.740125
\(96\) 0 0
\(97\) 10.0724i 1.02270i −0.859374 0.511348i \(-0.829146\pi\)
0.859374 0.511348i \(-0.170854\pi\)
\(98\) 14.7511i 1.49009i
\(99\) 0 0
\(100\) −7.93141 −0.793141
\(101\) 11.9250i 1.18658i −0.804988 0.593291i \(-0.797828\pi\)
0.804988 0.593291i \(-0.202172\pi\)
\(102\) 0 0
\(103\) 15.4768i 1.52498i 0.647003 + 0.762488i \(0.276022\pi\)
−0.647003 + 0.762488i \(0.723978\pi\)
\(104\) 3.90912 0.383320
\(105\) 0 0
\(106\) −12.3390 −1.19847
\(107\) 18.9884 1.83568 0.917841 0.396948i \(-0.129931\pi\)
0.917841 + 0.396948i \(0.129931\pi\)
\(108\) 0 0
\(109\) −4.80760 −0.460484 −0.230242 0.973133i \(-0.573952\pi\)
−0.230242 + 0.973133i \(0.573952\pi\)
\(110\) 3.24517i 0.309415i
\(111\) 0 0
\(112\) 4.66381 0.440689
\(113\) 18.1589 1.70825 0.854125 0.520068i \(-0.174093\pi\)
0.854125 + 0.520068i \(0.174093\pi\)
\(114\) 0 0
\(115\) 32.7890 3.05759
\(116\) 5.06669i 0.470431i
\(117\) 0 0
\(118\) 1.09050i 0.100389i
\(119\) 18.1364i 1.66256i
\(120\) 0 0
\(121\) −10.1856 −0.925965
\(122\) −1.98011 −0.179270
\(123\) 0 0
\(124\) 7.64590 0.686622
\(125\) 10.5414 0.942853
\(126\) 0 0
\(127\) −3.78190 −0.335590 −0.167795 0.985822i \(-0.553665\pi\)
−0.167795 + 0.985822i \(0.553665\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) −14.0573 −1.23291
\(131\) 9.66672i 0.844585i −0.906460 0.422293i \(-0.861225\pi\)
0.906460 0.422293i \(-0.138775\pi\)
\(132\) 0 0
\(133\) 9.35589 0.811258
\(134\) 13.9899 1.20854
\(135\) 0 0
\(136\) −3.88874 −0.333457
\(137\) −0.470346 −0.0401844 −0.0200922 0.999798i \(-0.506396\pi\)
−0.0200922 + 0.999798i \(0.506396\pi\)
\(138\) 0 0
\(139\) −14.4688 −1.22722 −0.613612 0.789607i \(-0.710284\pi\)
−0.613612 + 0.789607i \(0.710284\pi\)
\(140\) −16.7712 −1.41742
\(141\) 0 0
\(142\) 6.23553i 0.523274i
\(143\) 3.52771i 0.295002i
\(144\) 0 0
\(145\) 18.2200i 1.51309i
\(146\) 15.3293i 1.26866i
\(147\) 0 0
\(148\) −2.27714 −0.187180
\(149\) 0.978503 0.0801621 0.0400810 0.999196i \(-0.487238\pi\)
0.0400810 + 0.999196i \(0.487238\pi\)
\(150\) 0 0
\(151\) 3.93932i 0.320577i −0.987070 0.160289i \(-0.948757\pi\)
0.987070 0.160289i \(-0.0512425\pi\)
\(152\) 2.00606i 0.162713i
\(153\) 0 0
\(154\) 4.20877i 0.339153i
\(155\) −27.4949 −2.20844
\(156\) 0 0
\(157\) 11.1663i 0.891167i 0.895240 + 0.445583i \(0.147004\pi\)
−0.895240 + 0.445583i \(0.852996\pi\)
\(158\) 5.68601 0.452355
\(159\) 0 0
\(160\) 3.59603i 0.284291i
\(161\) 42.5251 3.35145
\(162\) 0 0
\(163\) 15.5956i 1.22154i 0.791807 + 0.610771i \(0.209141\pi\)
−0.791807 + 0.610771i \(0.790859\pi\)
\(164\) 1.40849i 0.109984i
\(165\) 0 0
\(166\) 2.04815 0.158967
\(167\) −7.90448 −0.611667 −0.305833 0.952085i \(-0.598935\pi\)
−0.305833 + 0.952085i \(0.598935\pi\)
\(168\) 0 0
\(169\) −2.28120 −0.175477
\(170\) 13.9840 1.07253
\(171\) 0 0
\(172\) 5.26130 0.401170
\(173\) −15.7797 −1.19971 −0.599856 0.800108i \(-0.704775\pi\)
−0.599856 + 0.800108i \(0.704775\pi\)
\(174\) 0 0
\(175\) 36.9906 2.79623
\(176\) −0.902432 −0.0680234
\(177\) 0 0
\(178\) 1.95797 0.146756
\(179\) 4.39415i 0.328434i 0.986424 + 0.164217i \(0.0525098\pi\)
−0.986424 + 0.164217i \(0.947490\pi\)
\(180\) 0 0
\(181\) −5.91690 −0.439800 −0.219900 0.975522i \(-0.570573\pi\)
−0.219900 + 0.975522i \(0.570573\pi\)
\(182\) −18.2314 −1.35140
\(183\) 0 0
\(184\) 9.11811i 0.672196i
\(185\) 8.18865 0.602042
\(186\) 0 0
\(187\) 3.50932i 0.256627i
\(188\) 5.68266i 0.414451i
\(189\) 0 0
\(190\) 7.21384i 0.523347i
\(191\) 18.1304 1.31187 0.655936 0.754816i \(-0.272274\pi\)
0.655936 + 0.754816i \(0.272274\pi\)
\(192\) 0 0
\(193\) 12.2241i 0.879910i −0.898020 0.439955i \(-0.854994\pi\)
0.898020 0.439955i \(-0.145006\pi\)
\(194\) 10.0724 0.723155
\(195\) 0 0
\(196\) −14.7511 −1.05365
\(197\) 2.60238i 0.185412i −0.995694 0.0927061i \(-0.970448\pi\)
0.995694 0.0927061i \(-0.0295517\pi\)
\(198\) 0 0
\(199\) −14.9269 −1.05814 −0.529071 0.848578i \(-0.677459\pi\)
−0.529071 + 0.848578i \(0.677459\pi\)
\(200\) 7.93141i 0.560835i
\(201\) 0 0
\(202\) 11.9250 0.839040
\(203\) 23.6301i 1.65851i
\(204\) 0 0
\(205\) 5.06496i 0.353752i
\(206\) −15.4768 −1.07832
\(207\) 0 0
\(208\) 3.90912i 0.271049i
\(209\) −1.81033 −0.125223
\(210\) 0 0
\(211\) 16.4855 1.13491 0.567455 0.823405i \(-0.307928\pi\)
0.567455 + 0.823405i \(0.307928\pi\)
\(212\) 12.3390i 0.847448i
\(213\) 0 0
\(214\) 18.9884i 1.29802i
\(215\) −18.9198 −1.29032
\(216\) 0 0
\(217\) −35.6591 −2.42069
\(218\) 4.80760i 0.325612i
\(219\) 0 0
\(220\) 3.24517 0.218789
\(221\) 15.2015 1.02257
\(222\) 0 0
\(223\) 0.326717 + 14.9296i 0.0218786 + 0.999761i
\(224\) 4.66381i 0.311614i
\(225\) 0 0
\(226\) 18.1589i 1.20792i
\(227\) 3.91525i 0.259865i 0.991523 + 0.129932i \(0.0414760\pi\)
−0.991523 + 0.129932i \(0.958524\pi\)
\(228\) 0 0
\(229\) 5.35015i 0.353548i −0.984251 0.176774i \(-0.943434\pi\)
0.984251 0.176774i \(-0.0565662\pi\)
\(230\) 32.7890i 2.16204i
\(231\) 0 0
\(232\) −5.06669 −0.332645
\(233\) 15.0088 0.983258 0.491629 0.870805i \(-0.336402\pi\)
0.491629 + 0.870805i \(0.336402\pi\)
\(234\) 0 0
\(235\) 20.4350i 1.33303i
\(236\) 1.09050 0.0709858
\(237\) 0 0
\(238\) 18.1364 1.17561
\(239\) 22.7374i 1.47076i −0.677656 0.735379i \(-0.737004\pi\)
0.677656 0.735379i \(-0.262996\pi\)
\(240\) 0 0
\(241\) −15.5649 −1.00263 −0.501313 0.865266i \(-0.667150\pi\)
−0.501313 + 0.865266i \(0.667150\pi\)
\(242\) 10.1856i 0.654756i
\(243\) 0 0
\(244\) 1.98011i 0.126763i
\(245\) 53.0455 3.38895
\(246\) 0 0
\(247\) 7.84192i 0.498970i
\(248\) 7.64590i 0.485515i
\(249\) 0 0
\(250\) 10.5414i 0.666698i
\(251\) 25.2944i 1.59657i −0.602280 0.798285i \(-0.705741\pi\)
0.602280 0.798285i \(-0.294259\pi\)
\(252\) 0 0
\(253\) −8.22847 −0.517320
\(254\) 3.78190i 0.237298i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 27.5846i 1.72068i 0.509720 + 0.860340i \(0.329749\pi\)
−0.509720 + 0.860340i \(0.670251\pi\)
\(258\) 0 0
\(259\) 10.6201 0.659904
\(260\) 14.0573i 0.871796i
\(261\) 0 0
\(262\) 9.66672 0.597212
\(263\) 7.69202 0.474310 0.237155 0.971472i \(-0.423785\pi\)
0.237155 + 0.971472i \(0.423785\pi\)
\(264\) 0 0
\(265\) 44.3715i 2.72572i
\(266\) 9.35589i 0.573646i
\(267\) 0 0
\(268\) 13.9899i 0.854568i
\(269\) −31.7587 −1.93636 −0.968181 0.250251i \(-0.919487\pi\)
−0.968181 + 0.250251i \(0.919487\pi\)
\(270\) 0 0
\(271\) 8.67132i 0.526745i −0.964694 0.263373i \(-0.915165\pi\)
0.964694 0.263373i \(-0.0848348\pi\)
\(272\) 3.88874i 0.235790i
\(273\) 0 0
\(274\) 0.470346i 0.0284147i
\(275\) −7.15756 −0.431617
\(276\) 0 0
\(277\) 28.7339i 1.72645i 0.504815 + 0.863227i \(0.331561\pi\)
−0.504815 + 0.863227i \(0.668439\pi\)
\(278\) 14.4688i 0.867779i
\(279\) 0 0
\(280\) 16.7712i 1.00227i
\(281\) 23.8796i 1.42454i 0.701906 + 0.712270i \(0.252333\pi\)
−0.701906 + 0.712270i \(0.747667\pi\)
\(282\) 0 0
\(283\) −18.3009 −1.08787 −0.543937 0.839126i \(-0.683067\pi\)
−0.543937 + 0.839126i \(0.683067\pi\)
\(284\) 6.23553 0.370010
\(285\) 0 0
\(286\) 3.52771 0.208598
\(287\) 6.56892i 0.387751i
\(288\) 0 0
\(289\) 1.87769 0.110452
\(290\) 18.2200 1.06991
\(291\) 0 0
\(292\) 15.3293 0.897077
\(293\) −18.0681 −1.05555 −0.527776 0.849384i \(-0.676974\pi\)
−0.527776 + 0.849384i \(0.676974\pi\)
\(294\) 0 0
\(295\) −3.92148 −0.228317
\(296\) 2.27714i 0.132356i
\(297\) 0 0
\(298\) 0.978503i 0.0566832i
\(299\) 35.6438i 2.06133i
\(300\) 0 0
\(301\) −24.5377 −1.41433
\(302\) 3.93932 0.226682
\(303\) 0 0
\(304\) 2.00606 0.115055
\(305\) 7.12051i 0.407719i
\(306\) 0 0
\(307\) 4.70645i 0.268611i −0.990940 0.134306i \(-0.957120\pi\)
0.990940 0.134306i \(-0.0428804\pi\)
\(308\) 4.20877 0.239817
\(309\) 0 0
\(310\) 27.4949i 1.56160i
\(311\) 15.0235 0.851904 0.425952 0.904746i \(-0.359939\pi\)
0.425952 + 0.904746i \(0.359939\pi\)
\(312\) 0 0
\(313\) 4.54957i 0.257157i −0.991699 0.128578i \(-0.958959\pi\)
0.991699 0.128578i \(-0.0410414\pi\)
\(314\) −11.1663 −0.630150
\(315\) 0 0
\(316\) 5.68601i 0.319863i
\(317\) 18.1436i 1.01905i 0.860457 + 0.509524i \(0.170178\pi\)
−0.860457 + 0.509524i \(0.829822\pi\)
\(318\) 0 0
\(319\) 4.57235i 0.256002i
\(320\) −3.59603 −0.201024
\(321\) 0 0
\(322\) 42.5251i 2.36983i
\(323\) 7.80105i 0.434062i
\(324\) 0 0
\(325\) 31.0048i 1.71984i
\(326\) −15.5956 −0.863761
\(327\) 0 0
\(328\) 1.40849 0.0777707
\(329\) 26.5029i 1.46115i
\(330\) 0 0
\(331\) 14.3644i 0.789537i 0.918781 + 0.394769i \(0.129175\pi\)
−0.918781 + 0.394769i \(0.870825\pi\)
\(332\) 2.04815i 0.112407i
\(333\) 0 0
\(334\) 7.90448i 0.432514i
\(335\) 50.3080i 2.74862i
\(336\) 0 0
\(337\) 28.9673i 1.57795i 0.614427 + 0.788974i \(0.289387\pi\)
−0.614427 + 0.788974i \(0.710613\pi\)
\(338\) 2.28120i 0.124081i
\(339\) 0 0
\(340\) 13.9840i 0.758390i
\(341\) 6.89991 0.373651
\(342\) 0 0
\(343\) 36.1499 1.95191
\(344\) 5.26130i 0.283670i
\(345\) 0 0
\(346\) 15.7797i 0.848324i
\(347\) 11.7543i 0.631003i −0.948925 0.315501i \(-0.897827\pi\)
0.948925 0.315501i \(-0.102173\pi\)
\(348\) 0 0
\(349\) −1.58365 −0.0847711 −0.0423855 0.999101i \(-0.513496\pi\)
−0.0423855 + 0.999101i \(0.513496\pi\)
\(350\) 36.9906i 1.97723i
\(351\) 0 0
\(352\) 0.902432i 0.0480998i
\(353\) 25.8563i 1.37619i 0.725619 + 0.688096i \(0.241553\pi\)
−0.725619 + 0.688096i \(0.758447\pi\)
\(354\) 0 0
\(355\) −22.4231 −1.19010
\(356\) 1.95797i 0.103772i
\(357\) 0 0
\(358\) −4.39415 −0.232238
\(359\) 28.3204i 1.49469i 0.664434 + 0.747347i \(0.268673\pi\)
−0.664434 + 0.747347i \(0.731327\pi\)
\(360\) 0 0
\(361\) −14.9757 −0.788196
\(362\) 5.91690i 0.310986i
\(363\) 0 0
\(364\) 18.2314i 0.955585i
\(365\) −55.1244 −2.88534
\(366\) 0 0
\(367\) 11.9073 0.621557 0.310778 0.950482i \(-0.399410\pi\)
0.310778 + 0.950482i \(0.399410\pi\)
\(368\) 9.11811 0.475314
\(369\) 0 0
\(370\) 8.18865i 0.425708i
\(371\) 57.5469i 2.98769i
\(372\) 0 0
\(373\) 18.4501i 0.955309i 0.878548 + 0.477655i \(0.158513\pi\)
−0.878548 + 0.477655i \(0.841487\pi\)
\(374\) −3.50932 −0.181463
\(375\) 0 0
\(376\) 5.68266 0.293061
\(377\) 19.8063 1.02008
\(378\) 0 0
\(379\) −3.95014 −0.202905 −0.101453 0.994840i \(-0.532349\pi\)
−0.101453 + 0.994840i \(0.532349\pi\)
\(380\) −7.21384 −0.370062
\(381\) 0 0
\(382\) 18.1304i 0.927634i
\(383\) 2.90672 0.148526 0.0742632 0.997239i \(-0.476340\pi\)
0.0742632 + 0.997239i \(0.476340\pi\)
\(384\) 0 0
\(385\) −15.1349 −0.771344
\(386\) 12.2241 0.622191
\(387\) 0 0
\(388\) 10.0724i 0.511348i
\(389\) 13.6810i 0.693655i −0.937929 0.346827i \(-0.887259\pi\)
0.937929 0.346827i \(-0.112741\pi\)
\(390\) 0 0
\(391\) 35.4580i 1.79319i
\(392\) 14.7511i 0.745045i
\(393\) 0 0
\(394\) 2.60238 0.131106
\(395\) 20.4471i 1.02880i
\(396\) 0 0
\(397\) 26.2699i 1.31845i −0.751946 0.659225i \(-0.770885\pi\)
0.751946 0.659225i \(-0.229115\pi\)
\(398\) 14.9269i 0.748219i
\(399\) 0 0
\(400\) 7.93141 0.396570
\(401\) 1.64967i 0.0823808i −0.999151 0.0411904i \(-0.986885\pi\)
0.999151 0.0411904i \(-0.0131150\pi\)
\(402\) 0 0
\(403\) 29.8887i 1.48886i
\(404\) 11.9250i 0.593291i
\(405\) 0 0
\(406\) 23.6301 1.17274
\(407\) −2.05496 −0.101861
\(408\) 0 0
\(409\) 15.3942i 0.761196i −0.924741 0.380598i \(-0.875718\pi\)
0.924741 0.380598i \(-0.124282\pi\)
\(410\) −5.06496 −0.250141
\(411\) 0 0
\(412\) 15.4768i 0.762488i
\(413\) −5.08591 −0.250261
\(414\) 0 0
\(415\) 7.36519i 0.361543i
\(416\) −3.90912 −0.191660
\(417\) 0 0
\(418\) 1.81033i 0.0885463i
\(419\) 24.8933i 1.21612i −0.793891 0.608060i \(-0.791948\pi\)
0.793891 0.608060i \(-0.208052\pi\)
\(420\) 0 0
\(421\) 1.89567i 0.0923894i 0.998932 + 0.0461947i \(0.0147095\pi\)
−0.998932 + 0.0461947i \(0.985291\pi\)
\(422\) 16.4855i 0.802502i
\(423\) 0 0
\(424\) 12.3390 0.599236
\(425\) 30.8432i 1.49611i
\(426\) 0 0
\(427\) 9.23484i 0.446905i
\(428\) −18.9884 −0.917841
\(429\) 0 0
\(430\) 18.9198i 0.912393i
\(431\) 20.7721 1.00056 0.500279 0.865865i \(-0.333231\pi\)
0.500279 + 0.865865i \(0.333231\pi\)
\(432\) 0 0
\(433\) 23.5538 1.13192 0.565961 0.824432i \(-0.308505\pi\)
0.565961 + 0.824432i \(0.308505\pi\)
\(434\) 35.6591i 1.71169i
\(435\) 0 0
\(436\) 4.80760 0.230242
\(437\) 18.2915 0.875000
\(438\) 0 0
\(439\) 27.5002i 1.31251i −0.754538 0.656256i \(-0.772139\pi\)
0.754538 0.656256i \(-0.227861\pi\)
\(440\) 3.24517i 0.154707i
\(441\) 0 0
\(442\) 15.2015i 0.723064i
\(443\) 24.3687i 1.15779i 0.815401 + 0.578896i \(0.196516\pi\)
−0.815401 + 0.578896i \(0.803484\pi\)
\(444\) 0 0
\(445\) 7.04093i 0.333772i
\(446\) −14.9296 + 0.326717i −0.706938 + 0.0154705i
\(447\) 0 0
\(448\) −4.66381 −0.220344
\(449\) −22.6953 −1.07106 −0.535529 0.844517i \(-0.679888\pi\)
−0.535529 + 0.844517i \(0.679888\pi\)
\(450\) 0 0
\(451\) 1.27106i 0.0598521i
\(452\) −18.1589 −0.854125
\(453\) 0 0
\(454\) −3.91525 −0.183752
\(455\) 65.5606i 3.07353i
\(456\) 0 0
\(457\) 20.6720i 0.966996i 0.875345 + 0.483498i \(0.160634\pi\)
−0.875345 + 0.483498i \(0.839366\pi\)
\(458\) 5.35015 0.249996
\(459\) 0 0
\(460\) −32.7890 −1.52879
\(461\) 25.4522i 1.18543i −0.805413 0.592714i \(-0.798057\pi\)
0.805413 0.592714i \(-0.201943\pi\)
\(462\) 0 0
\(463\) 25.3610 1.17863 0.589313 0.807905i \(-0.299399\pi\)
0.589313 + 0.807905i \(0.299399\pi\)
\(464\) 5.06669i 0.235215i
\(465\) 0 0
\(466\) 15.0088i 0.695269i
\(467\) 5.67695 0.262698 0.131349 0.991336i \(-0.458069\pi\)
0.131349 + 0.991336i \(0.458069\pi\)
\(468\) 0 0
\(469\) 65.2462i 3.01279i
\(470\) −20.4350 −0.942596
\(471\) 0 0
\(472\) 1.09050i 0.0501945i
\(473\) 4.74797 0.218312
\(474\) 0 0
\(475\) 15.9109 0.730041
\(476\) 18.1364i 0.831279i
\(477\) 0 0
\(478\) 22.7374 1.03998
\(479\) 7.99218i 0.365172i 0.983190 + 0.182586i \(0.0584468\pi\)
−0.983190 + 0.182586i \(0.941553\pi\)
\(480\) 0 0
\(481\) 8.90160i 0.405878i
\(482\) 15.5649i 0.708964i
\(483\) 0 0
\(484\) 10.1856 0.462983
\(485\) 36.2206i 1.64469i
\(486\) 0 0
\(487\) 17.2751 0.782810 0.391405 0.920219i \(-0.371989\pi\)
0.391405 + 0.920219i \(0.371989\pi\)
\(488\) 1.98011 0.0896352
\(489\) 0 0
\(490\) 53.0455i 2.39635i
\(491\) 6.80166 0.306955 0.153477 0.988152i \(-0.450953\pi\)
0.153477 + 0.988152i \(0.450953\pi\)
\(492\) 0 0
\(493\) −19.7031 −0.887381
\(494\) −7.84192 −0.352825
\(495\) 0 0
\(496\) −7.64590 −0.343311
\(497\) −29.0813 −1.30448
\(498\) 0 0
\(499\) −34.3509 −1.53776 −0.768879 0.639395i \(-0.779185\pi\)
−0.768879 + 0.639395i \(0.779185\pi\)
\(500\) −10.5414 −0.471427
\(501\) 0 0
\(502\) 25.2944 1.12895
\(503\) 38.9506 1.73672 0.868360 0.495935i \(-0.165174\pi\)
0.868360 + 0.495935i \(0.165174\pi\)
\(504\) 0 0
\(505\) 42.8826i 1.90825i
\(506\) 8.22847i 0.365800i
\(507\) 0 0
\(508\) 3.78190 0.167795
\(509\) 7.69102i 0.340898i 0.985366 + 0.170449i \(0.0545219\pi\)
−0.985366 + 0.170449i \(0.945478\pi\)
\(510\) 0 0
\(511\) −71.4928 −3.16265
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) −27.5846 −1.21671
\(515\) 55.6550i 2.45245i
\(516\) 0 0
\(517\) 5.12821i 0.225539i
\(518\) 10.6201i 0.466622i
\(519\) 0 0
\(520\) 14.0573 0.616453
\(521\) 6.39420 0.280135 0.140067 0.990142i \(-0.455268\pi\)
0.140067 + 0.990142i \(0.455268\pi\)
\(522\) 0 0
\(523\) 4.03803i 0.176571i −0.996095 0.0882854i \(-0.971861\pi\)
0.996095 0.0882854i \(-0.0281388\pi\)
\(524\) 9.66672i 0.422293i
\(525\) 0 0
\(526\) 7.69202i 0.335388i
\(527\) 29.7329i 1.29519i
\(528\) 0 0
\(529\) 60.1399 2.61478
\(530\) −44.3715 −1.92737
\(531\) 0 0
\(532\) −9.35589 −0.405629
\(533\) −5.50595 −0.238489
\(534\) 0 0
\(535\) 68.2830 2.95213
\(536\) −13.9899 −0.604271
\(537\) 0 0
\(538\) 31.7587i 1.36921i
\(539\) −13.3119 −0.573384
\(540\) 0 0
\(541\) 7.94748i 0.341689i 0.985298 + 0.170844i \(0.0546495\pi\)
−0.985298 + 0.170844i \(0.945350\pi\)
\(542\) 8.67132 0.372465
\(543\) 0 0
\(544\) 3.88874 0.166728
\(545\) −17.2883 −0.740547
\(546\) 0 0
\(547\) 7.34285 0.313958 0.156979 0.987602i \(-0.449825\pi\)
0.156979 + 0.987602i \(0.449825\pi\)
\(548\) 0.470346 0.0200922
\(549\) 0 0
\(550\) 7.15756i 0.305199i
\(551\) 10.1641i 0.433005i
\(552\) 0 0
\(553\) 26.5185i 1.12768i
\(554\) −28.7339 −1.22079
\(555\) 0 0
\(556\) 14.4688 0.613612
\(557\) −23.3969 −0.991360 −0.495680 0.868505i \(-0.665081\pi\)
−0.495680 + 0.868505i \(0.665081\pi\)
\(558\) 0 0
\(559\) 20.5670i 0.869893i
\(560\) 16.7712 0.708712
\(561\) 0 0
\(562\) −23.8796 −1.00730
\(563\) −9.73178 −0.410146 −0.205073 0.978747i \(-0.565743\pi\)
−0.205073 + 0.978747i \(0.565743\pi\)
\(564\) 0 0
\(565\) 65.3001 2.74719
\(566\) 18.3009i 0.769243i
\(567\) 0 0
\(568\) 6.23553i 0.261637i
\(569\) 4.19768 0.175976 0.0879879 0.996122i \(-0.471956\pi\)
0.0879879 + 0.996122i \(0.471956\pi\)
\(570\) 0 0
\(571\) 35.4089i 1.48182i −0.671606 0.740908i \(-0.734395\pi\)
0.671606 0.740908i \(-0.265605\pi\)
\(572\) 3.52771i 0.147501i
\(573\) 0 0
\(574\) −6.56892 −0.274182
\(575\) 72.3194 3.01593
\(576\) 0 0
\(577\) −24.2588 −1.00991 −0.504953 0.863147i \(-0.668490\pi\)
−0.504953 + 0.863147i \(0.668490\pi\)
\(578\) 1.87769i 0.0781017i
\(579\) 0 0
\(580\) 18.2200i 0.756543i
\(581\) 9.55217i 0.396291i
\(582\) 0 0
\(583\) 11.1351i 0.461170i
\(584\) 15.3293i 0.634329i
\(585\) 0 0
\(586\) 18.0681i 0.746388i
\(587\) 14.2496 0.588146 0.294073 0.955783i \(-0.404989\pi\)
0.294073 + 0.955783i \(0.404989\pi\)
\(588\) 0 0
\(589\) −15.3381 −0.631997
\(590\) 3.92148i 0.161445i
\(591\) 0 0
\(592\) 2.27714 0.0935898
\(593\) −18.9380 −0.777690 −0.388845 0.921303i \(-0.627126\pi\)
−0.388845 + 0.921303i \(0.627126\pi\)
\(594\) 0 0
\(595\) 65.2188i 2.67371i
\(596\) −0.978503 −0.0400810
\(597\) 0 0
\(598\) −35.6438 −1.45758
\(599\) 45.9774i 1.87858i −0.343118 0.939292i \(-0.611483\pi\)
0.343118 0.939292i \(-0.388517\pi\)
\(600\) 0 0
\(601\) 19.7024i 0.803677i −0.915711 0.401839i \(-0.868371\pi\)
0.915711 0.401839i \(-0.131629\pi\)
\(602\) 24.5377i 1.00008i
\(603\) 0 0
\(604\) 3.93932i 0.160289i
\(605\) −36.6277 −1.48913
\(606\) 0 0
\(607\) 17.8892i 0.726099i 0.931770 + 0.363049i \(0.118264\pi\)
−0.931770 + 0.363049i \(0.881736\pi\)
\(608\) 2.00606i 0.0813565i
\(609\) 0 0
\(610\) −7.12051 −0.288301
\(611\) −22.2142 −0.898690
\(612\) 0 0
\(613\) 1.04687i 0.0422828i −0.999776 0.0211414i \(-0.993270\pi\)
0.999776 0.0211414i \(-0.00673001\pi\)
\(614\) 4.70645 0.189937
\(615\) 0 0
\(616\) 4.20877i 0.169576i
\(617\) 2.67707i 0.107775i −0.998547 0.0538873i \(-0.982839\pi\)
0.998547 0.0538873i \(-0.0171612\pi\)
\(618\) 0 0
\(619\) 31.8865i 1.28163i 0.767696 + 0.640814i \(0.221403\pi\)
−0.767696 + 0.640814i \(0.778597\pi\)
\(620\) 27.4949 1.10422
\(621\) 0 0
\(622\) 15.0235i 0.602387i
\(623\) 9.13162i 0.365851i
\(624\) 0 0
\(625\) −1.74981 −0.0699925
\(626\) 4.54957 0.181837
\(627\) 0 0
\(628\) 11.1663i 0.445583i
\(629\) 8.85520i 0.353080i
\(630\) 0 0
\(631\) 46.9308i 1.86828i 0.356901 + 0.934142i \(0.383833\pi\)
−0.356901 + 0.934142i \(0.616167\pi\)
\(632\) −5.68601 −0.226177
\(633\) 0 0
\(634\) −18.1436 −0.720576
\(635\) −13.5998 −0.539693
\(636\) 0 0
\(637\) 57.6640i 2.28473i
\(638\) −4.57235 −0.181021
\(639\) 0 0
\(640\) 3.59603i 0.142145i
\(641\) −10.6710 −0.421480 −0.210740 0.977542i \(-0.567587\pi\)
−0.210740 + 0.977542i \(0.567587\pi\)
\(642\) 0 0
\(643\) −2.07541 −0.0818463 −0.0409231 0.999162i \(-0.513030\pi\)
−0.0409231 + 0.999162i \(0.513030\pi\)
\(644\) −42.5251 −1.67573
\(645\) 0 0
\(646\) 7.80105 0.306928
\(647\) 25.0950i 0.986587i −0.869863 0.493294i \(-0.835793\pi\)
0.869863 0.493294i \(-0.164207\pi\)
\(648\) 0 0
\(649\) 0.984106 0.0386295
\(650\) −31.0048 −1.21611
\(651\) 0 0
\(652\) 15.5956i 0.610771i
\(653\) −13.3767 −0.523470 −0.261735 0.965140i \(-0.584295\pi\)
−0.261735 + 0.965140i \(0.584295\pi\)
\(654\) 0 0
\(655\) 34.7618i 1.35826i
\(656\) 1.40849i 0.0549922i
\(657\) 0 0
\(658\) −26.5029 −1.03319
\(659\) 44.7250i 1.74224i −0.491072 0.871119i \(-0.663395\pi\)
0.491072 0.871119i \(-0.336605\pi\)
\(660\) 0 0
\(661\) 32.3751i 1.25924i −0.776901 0.629622i \(-0.783210\pi\)
0.776901 0.629622i \(-0.216790\pi\)
\(662\) −14.3644 −0.558287
\(663\) 0 0
\(664\) −2.04815 −0.0794835
\(665\) 33.6440 1.30466
\(666\) 0 0
\(667\) 46.1987i 1.78882i
\(668\) 7.90448 0.305833
\(669\) 0 0
\(670\) 50.3080 1.94357
\(671\) 1.78691i 0.0689829i
\(672\) 0 0
\(673\) −44.4949 −1.71515 −0.857575 0.514358i \(-0.828030\pi\)
−0.857575 + 0.514358i \(0.828030\pi\)
\(674\) −28.9673 −1.11578
\(675\) 0 0
\(676\) 2.28120 0.0877384
\(677\) 5.75638i 0.221236i −0.993863 0.110618i \(-0.964717\pi\)
0.993863 0.110618i \(-0.0352830\pi\)
\(678\) 0 0
\(679\) 46.9757i 1.80276i
\(680\) −13.9840 −0.536263
\(681\) 0 0
\(682\) 6.89991i 0.264211i
\(683\) 36.0279i 1.37857i 0.724490 + 0.689285i \(0.242075\pi\)
−0.724490 + 0.689285i \(0.757925\pi\)
\(684\) 0 0
\(685\) −1.69138 −0.0646242
\(686\) 36.1499i 1.38021i
\(687\) 0 0
\(688\) −5.26130 −0.200585
\(689\) −48.2347 −1.83760
\(690\) 0 0
\(691\) 31.5987i 1.20207i 0.799222 + 0.601036i \(0.205245\pi\)
−0.799222 + 0.601036i \(0.794755\pi\)
\(692\) 15.7797 0.599856
\(693\) 0 0
\(694\) 11.7543 0.446186
\(695\) −52.0301 −1.97361
\(696\) 0 0
\(697\) 5.47725 0.207465
\(698\) 1.58365i 0.0599422i
\(699\) 0 0
\(700\) −36.9906 −1.39811
\(701\) 24.6638i 0.931538i −0.884906 0.465769i \(-0.845778\pi\)
0.884906 0.465769i \(-0.154222\pi\)
\(702\) 0 0
\(703\) 4.56808 0.172288
\(704\) 0.902432 0.0340117
\(705\) 0 0
\(706\) −25.8563 −0.973115
\(707\) 55.6160i 2.09165i
\(708\) 0 0
\(709\) 14.3003i 0.537060i −0.963271 0.268530i \(-0.913462\pi\)
0.963271 0.268530i \(-0.0865378\pi\)
\(710\) 22.4231i 0.841525i
\(711\) 0 0
\(712\) −1.95797 −0.0733782
\(713\) −69.7162 −2.61089
\(714\) 0 0
\(715\) 12.6857i 0.474420i
\(716\) 4.39415i 0.164217i
\(717\) 0 0
\(718\) −28.3204 −1.05691
\(719\) 3.36040i 0.125322i −0.998035 0.0626609i \(-0.980041\pi\)
0.998035 0.0626609i \(-0.0199586\pi\)
\(720\) 0 0
\(721\) 72.1809i 2.68816i
\(722\) 14.9757i 0.557339i
\(723\) 0 0
\(724\) 5.91690 0.219900
\(725\) 40.1860i 1.49247i
\(726\) 0 0
\(727\) 15.6392 0.580025 0.290013 0.957023i \(-0.406340\pi\)
0.290013 + 0.957023i \(0.406340\pi\)
\(728\) 18.2314 0.675700
\(729\) 0 0
\(730\) 55.1244i 2.04025i
\(731\) 20.4598i 0.756734i
\(732\) 0 0
\(733\) −13.7550 −0.508051 −0.254025 0.967198i \(-0.581755\pi\)
−0.254025 + 0.967198i \(0.581755\pi\)
\(734\) 11.9073i 0.439507i
\(735\) 0 0
\(736\) 9.11811i 0.336098i
\(737\) 12.6249i 0.465045i
\(738\) 0 0
\(739\) 22.3492i 0.822127i −0.911607 0.411064i \(-0.865157\pi\)
0.911607 0.411064i \(-0.134843\pi\)
\(740\) −8.18865 −0.301021
\(741\) 0 0
\(742\) −57.5469 −2.11261
\(743\) 43.1880i 1.58441i −0.610252 0.792207i \(-0.708932\pi\)
0.610252 0.792207i \(-0.291068\pi\)
\(744\) 0 0
\(745\) 3.51872 0.128916
\(746\) −18.4501 −0.675506
\(747\) 0 0
\(748\) 3.50932i 0.128314i
\(749\) 88.5586 3.23586
\(750\) 0 0
\(751\) −27.0187 −0.985925 −0.492962 0.870051i \(-0.664086\pi\)
−0.492962 + 0.870051i \(0.664086\pi\)
\(752\) 5.68266i 0.207225i
\(753\) 0 0
\(754\) 19.8063i 0.721303i
\(755\) 14.1659i 0.515550i
\(756\) 0 0
\(757\) 32.1030i 1.16680i 0.812184 + 0.583402i \(0.198279\pi\)
−0.812184 + 0.583402i \(0.801721\pi\)
\(758\) 3.95014i 0.143476i
\(759\) 0 0
\(760\) 7.21384i 0.261674i
\(761\) 22.6216 0.820031 0.410016 0.912078i \(-0.365523\pi\)
0.410016 + 0.912078i \(0.365523\pi\)
\(762\) 0 0
\(763\) −22.4217 −0.811721
\(764\) −18.1304 −0.655936
\(765\) 0 0
\(766\) 2.90672i 0.105024i
\(767\) 4.26291i 0.153925i
\(768\) 0 0
\(769\) −38.3830 −1.38413 −0.692063 0.721837i \(-0.743298\pi\)
−0.692063 + 0.721837i \(0.743298\pi\)
\(770\) 15.1349i 0.545423i
\(771\) 0 0
\(772\) 12.2241i 0.439955i
\(773\) −41.3477 −1.48717 −0.743586 0.668640i \(-0.766877\pi\)
−0.743586 + 0.668640i \(0.766877\pi\)
\(774\) 0 0
\(775\) −60.6428 −2.17835
\(776\) −10.0724 −0.361577
\(777\) 0 0
\(778\) 13.6810 0.490488
\(779\) 2.82551i 0.101234i
\(780\) 0 0
\(781\) 5.62714 0.201355
\(782\) 35.4580 1.26797
\(783\) 0 0
\(784\) 14.7511 0.526827
\(785\) 40.1543i 1.43317i
\(786\) 0 0
\(787\) 30.4917i 1.08691i 0.839437 + 0.543456i \(0.182885\pi\)
−0.839437 + 0.543456i \(0.817115\pi\)
\(788\) 2.60238i 0.0927061i
\(789\) 0 0
\(790\) 20.4471 0.727473
\(791\) 84.6899 3.01123
\(792\) 0 0
\(793\) −7.74047 −0.274872
\(794\) 26.2699 0.932284
\(795\) 0 0
\(796\) 14.9269 0.529071
\(797\) 1.28996i 0.0456928i −0.999739 0.0228464i \(-0.992727\pi\)
0.999739 0.0228464i \(-0.00727286\pi\)
\(798\) 0 0
\(799\) 22.0984 0.781785
\(800\) 7.93141i 0.280418i
\(801\) 0 0
\(802\) 1.64967 0.0582520
\(803\) 13.8336 0.488177
\(804\) 0 0
\(805\) 152.922 5.38978
\(806\) 29.8887 1.05279
\(807\) 0 0
\(808\) −11.9250 −0.419520
\(809\) 41.9767 1.47582 0.737911 0.674898i \(-0.235813\pi\)
0.737911 + 0.674898i \(0.235813\pi\)
\(810\) 0 0
\(811\) 2.73208i 0.0959365i 0.998849 + 0.0479682i \(0.0152746\pi\)
−0.998849 + 0.0479682i \(0.984725\pi\)
\(812\) 23.6301i 0.829254i
\(813\) 0 0
\(814\) 2.05496i 0.0720264i
\(815\) 56.0822i 1.96448i
\(816\) 0 0
\(817\) −10.5545 −0.369255
\(818\) 15.3942 0.538247
\(819\) 0 0
\(820\) 5.06496i 0.176876i
\(821\) 52.4879i 1.83184i 0.401359 + 0.915921i \(0.368538\pi\)
−0.401359 + 0.915921i \(0.631462\pi\)
\(822\) 0 0
\(823\) 48.4730i 1.68966i −0.535032 0.844832i \(-0.679701\pi\)
0.535032 0.844832i \(-0.320299\pi\)
\(824\) 15.4768 0.539160
\(825\) 0 0
\(826\) 5.08591i 0.176961i
\(827\) 19.6698 0.683987 0.341994 0.939702i \(-0.388898\pi\)
0.341994 + 0.939702i \(0.388898\pi\)
\(828\) 0 0
\(829\) 0.791876i 0.0275030i −0.999905 0.0137515i \(-0.995623\pi\)
0.999905 0.0137515i \(-0.00437737\pi\)
\(830\) 7.36519 0.255649
\(831\) 0 0
\(832\) 3.90912i 0.135524i
\(833\) 57.3634i 1.98752i
\(834\) 0 0
\(835\) −28.4247 −0.983678
\(836\) 1.81033 0.0626117
\(837\) 0 0
\(838\) 24.8933 0.859926
\(839\) −21.2090 −0.732215 −0.366108 0.930573i \(-0.619310\pi\)
−0.366108 + 0.930573i \(0.619310\pi\)
\(840\) 0 0
\(841\) 3.32862 0.114780
\(842\) −1.89567 −0.0653292
\(843\) 0 0
\(844\) −16.4855 −0.567455
\(845\) −8.20325 −0.282200
\(846\) 0 0
\(847\) −47.5038 −1.63225
\(848\) 12.3390i 0.423724i
\(849\) 0 0
\(850\) 30.8432 1.05791
\(851\) 20.7632 0.711753
\(852\) 0 0
\(853\) 12.6616i 0.433525i −0.976224 0.216762i \(-0.930450\pi\)
0.976224 0.216762i \(-0.0695497\pi\)
\(854\) −9.23484 −0.316010
\(855\) 0 0
\(856\) 18.9884i 0.649012i
\(857\) 27.3655i 0.934786i 0.884050 + 0.467393i \(0.154807\pi\)
−0.884050 + 0.467393i \(0.845193\pi\)
\(858\) 0 0
\(859\) 29.0721i 0.991927i 0.868343 + 0.495963i \(0.165185\pi\)
−0.868343 + 0.495963i \(0.834815\pi\)
\(860\) 18.9198 0.645159
\(861\) 0 0
\(862\) 20.7721i 0.707501i
\(863\) 26.9203 0.916379 0.458190 0.888854i \(-0.348498\pi\)
0.458190 + 0.888854i \(0.348498\pi\)
\(864\) 0 0
\(865\) −56.7444 −1.92937
\(866\) 23.5538i 0.800389i
\(867\) 0 0
\(868\) 35.6591 1.21035
\(869\) 5.13124i 0.174065i
\(870\) 0 0
\(871\) 54.6881 1.85303
\(872\) 4.80760i 0.162806i
\(873\) 0 0
\(874\) 18.2915i 0.618718i
\(875\) 49.1632 1.66202
\(876\) 0 0
\(877\) 51.7787i 1.74844i 0.485529 + 0.874220i \(0.338627\pi\)
−0.485529 + 0.874220i \(0.661373\pi\)
\(878\) 27.5002 0.928086
\(879\) 0 0
\(880\) −3.24517 −0.109395
\(881\) 4.09113i 0.137834i 0.997622 + 0.0689168i \(0.0219543\pi\)
−0.997622 + 0.0689168i \(0.978046\pi\)
\(882\) 0 0
\(883\) 3.63618i 0.122367i 0.998127 + 0.0611837i \(0.0194875\pi\)
−0.998127 + 0.0611837i \(0.980512\pi\)
\(884\) −15.2015 −0.511283
\(885\) 0 0
\(886\) −24.3687 −0.818683
\(887\) 7.58846i 0.254796i 0.991852 + 0.127398i \(0.0406625\pi\)
−0.991852 + 0.127398i \(0.959338\pi\)
\(888\) 0 0
\(889\) −17.6381 −0.591563
\(890\) 7.04093 0.236012
\(891\) 0 0
\(892\) −0.326717 14.9296i −0.0109393 0.499880i
\(893\) 11.3998i 0.381478i
\(894\) 0 0
\(895\) 15.8015i 0.528186i
\(896\) 4.66381i 0.155807i
\(897\) 0 0
\(898\) 22.6953i 0.757352i
\(899\) 38.7394i 1.29203i
\(900\) 0 0
\(901\) 47.9833 1.59856
\(902\) 1.27106 0.0423218
\(903\) 0 0
\(904\) 18.1589i 0.603958i
\(905\) −21.2773 −0.707283
\(906\) 0 0
\(907\) −32.2633 −1.07129 −0.535643 0.844444i \(-0.679931\pi\)
−0.535643 + 0.844444i \(0.679931\pi\)
\(908\) 3.91525i 0.129932i
\(909\) 0 0
\(910\) −65.5606 −2.17331
\(911\) 18.1569i 0.601564i −0.953693 0.300782i \(-0.902752\pi\)
0.953693 0.300782i \(-0.0972477\pi\)
\(912\) 0 0
\(913\) 1.84831i 0.0611702i
\(914\) −20.6720 −0.683769
\(915\) 0 0
\(916\) 5.35015i 0.176774i
\(917\) 45.0838i 1.48880i
\(918\) 0 0
\(919\) 49.5535i 1.63462i −0.576200 0.817309i \(-0.695465\pi\)
0.576200 0.817309i \(-0.304535\pi\)
\(920\) 32.7890i 1.08102i
\(921\) 0 0
\(922\) 25.4522 0.838224
\(923\) 24.3754i 0.802326i
\(924\) 0 0
\(925\) 18.0609 0.593839
\(926\) 25.3610i 0.833414i
\(927\) 0 0
\(928\) 5.06669 0.166322
\(929\) 24.2071i 0.794208i −0.917774 0.397104i \(-0.870015\pi\)
0.917774 0.397104i \(-0.129985\pi\)
\(930\) 0 0
\(931\) 29.5917 0.969828
\(932\) −15.0088 −0.491629
\(933\) 0 0
\(934\) 5.67695i 0.185756i
\(935\) 12.6196i 0.412706i
\(936\) 0 0
\(937\) 22.3246i 0.729314i −0.931142 0.364657i \(-0.881186\pi\)
0.931142 0.364657i \(-0.118814\pi\)
\(938\) 65.2462 2.13036
\(939\) 0 0
\(940\) 20.4350i 0.666516i
\(941\) 17.3826i 0.566657i 0.959023 + 0.283328i \(0.0914386\pi\)
−0.959023 + 0.283328i \(0.908561\pi\)
\(942\) 0 0
\(943\) 12.8427i 0.418217i
\(944\) −1.09050 −0.0354929
\(945\) 0 0
\(946\) 4.74797i 0.154370i
\(947\) 13.8285i 0.449365i −0.974432 0.224682i \(-0.927866\pi\)
0.974432 0.224682i \(-0.0721344\pi\)
\(948\) 0 0
\(949\) 59.9238i 1.94521i
\(950\) 15.9109i 0.516217i
\(951\) 0 0
\(952\) −18.1364 −0.587803
\(953\) −23.3803 −0.757363 −0.378682 0.925527i \(-0.623622\pi\)
−0.378682 + 0.925527i \(0.623622\pi\)
\(954\) 0 0
\(955\) 65.1975 2.10974
\(956\) 22.7374i 0.735379i
\(957\) 0 0
\(958\) −7.99218 −0.258216
\(959\) −2.19361 −0.0708353
\(960\) 0 0
\(961\) 27.4598 0.885800
\(962\) −8.90160 −0.286999
\(963\) 0 0
\(964\) 15.5649 0.501313
\(965\) 43.9582i 1.41506i
\(966\) 0 0
\(967\) 31.7180i 1.01998i −0.860179 0.509992i \(-0.829649\pi\)
0.860179 0.509992i \(-0.170351\pi\)
\(968\) 10.1856i 0.327378i
\(969\) 0 0
\(970\) 36.2206 1.16297
\(971\) −43.6042 −1.39933 −0.699663 0.714473i \(-0.746667\pi\)
−0.699663 + 0.714473i \(0.746667\pi\)
\(972\) 0 0
\(973\) −67.4796 −2.16330
\(974\) 17.2751i 0.553530i
\(975\) 0 0
\(976\) 1.98011i 0.0633816i
\(977\) −35.3637 −1.13139 −0.565693 0.824616i \(-0.691391\pi\)
−0.565693 + 0.824616i \(0.691391\pi\)
\(978\) 0 0
\(979\) 1.76694i 0.0564716i
\(980\) −53.0455 −1.69448
\(981\) 0 0
\(982\) 6.80166i 0.217050i
\(983\) 42.9170 1.36884 0.684420 0.729088i \(-0.260056\pi\)
0.684420 + 0.729088i \(0.260056\pi\)
\(984\) 0 0
\(985\) 9.35825i 0.298178i
\(986\) 19.7031i 0.627473i
\(987\) 0 0
\(988\) 7.84192i 0.249485i
\(989\) −47.9731 −1.52546
\(990\) 0 0
\(991\) 31.2085i 0.991371i −0.868502 0.495685i \(-0.834917\pi\)
0.868502 0.495685i \(-0.165083\pi\)
\(992\) 7.64590i 0.242758i
\(993\) 0 0
\(994\) 29.0813i 0.922404i
\(995\) −53.6776 −1.70170
\(996\) 0 0
\(997\) −17.6931 −0.560347 −0.280174 0.959949i \(-0.590392\pi\)
−0.280174 + 0.959949i \(0.590392\pi\)
\(998\) 34.3509i 1.08736i
\(999\) 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 4014.2.d.a.4013.10 yes 72
3.2 odd 2 inner 4014.2.d.a.4013.37 yes 72
223.222 odd 2 inner 4014.2.d.a.4013.38 yes 72
669.668 even 2 inner 4014.2.d.a.4013.9 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4014.2.d.a.4013.9 72 669.668 even 2 inner
4014.2.d.a.4013.10 yes 72 1.1 even 1 trivial
4014.2.d.a.4013.37 yes 72 3.2 odd 2 inner
4014.2.d.a.4013.38 yes 72 223.222 odd 2 inner