Properties

Label 1840.3.c.b.1151.20
Level $1840$
Weight $3$
Character 1840.1151
Analytic conductor $50.136$
Analytic rank $0$
Dimension $56$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1840,3,Mod(1151,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1840.1151");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1840.c (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: \(50.1363686423\)
Analytic rank: \(0\)
Dimension: \(56\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1151.20
Character \(\chi\) \(=\) 1840.1151
Dual form 1840.3.c.b.1151.37

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.78315i q^{3} +2.23607 q^{5} -11.2760i q^{7} +5.82039 q^{9} +O(q^{10})\) \(q-1.78315i q^{3} +2.23607 q^{5} -11.2760i q^{7} +5.82039 q^{9} -20.5962i q^{11} +19.3818 q^{13} -3.98723i q^{15} -27.1601 q^{17} -29.7144i q^{19} -20.1068 q^{21} +4.79583i q^{23} +5.00000 q^{25} -26.4269i q^{27} +12.1117 q^{29} +5.25345i q^{31} -36.7261 q^{33} -25.2140i q^{35} -2.56136 q^{37} -34.5606i q^{39} +17.2011 q^{41} -7.38042i q^{43} +13.0148 q^{45} +77.4615i q^{47} -78.1487 q^{49} +48.4304i q^{51} +95.1341 q^{53} -46.0546i q^{55} -52.9852 q^{57} +22.2739i q^{59} -23.6344 q^{61} -65.6309i q^{63} +43.3391 q^{65} +63.2193i q^{67} +8.55166 q^{69} -20.7152i q^{71} +114.966 q^{73} -8.91573i q^{75} -232.244 q^{77} +150.947i q^{79} +5.26053 q^{81} +134.051i q^{83} -60.7319 q^{85} -21.5969i q^{87} +61.9897 q^{89} -218.550i q^{91} +9.36767 q^{93} -66.4435i q^{95} -42.1692 q^{97} -119.878i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56 q - 120 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 56 q - 120 q^{9} - 56 q^{13} - 96 q^{17} + 104 q^{21} + 280 q^{25} - 76 q^{29} + 240 q^{33} - 88 q^{37} - 76 q^{41} - 356 q^{49} - 88 q^{53} - 256 q^{57} + 376 q^{61} + 120 q^{65} + 192 q^{73} - 168 q^{77} - 392 q^{81} - 60 q^{85} + 368 q^{89} + 216 q^{93} + 264 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 1.78315i − 0.594382i −0.954818 0.297191i \(-0.903950\pi\)
0.954818 0.297191i \(-0.0960498\pi\)
\(4\) 0 0
\(5\) 2.23607 0.447214
\(6\) 0 0
\(7\) − 11.2760i − 1.61086i −0.592691 0.805430i \(-0.701934\pi\)
0.592691 0.805430i \(-0.298066\pi\)
\(8\) 0 0
\(9\) 5.82039 0.646710
\(10\) 0 0
\(11\) − 20.5962i − 1.87238i −0.351488 0.936192i \(-0.614324\pi\)
0.351488 0.936192i \(-0.385676\pi\)
\(12\) 0 0
\(13\) 19.3818 1.49091 0.745455 0.666556i \(-0.232232\pi\)
0.745455 + 0.666556i \(0.232232\pi\)
\(14\) 0 0
\(15\) − 3.98723i − 0.265816i
\(16\) 0 0
\(17\) −27.1601 −1.59765 −0.798827 0.601560i \(-0.794546\pi\)
−0.798827 + 0.601560i \(0.794546\pi\)
\(18\) 0 0
\(19\) − 29.7144i − 1.56392i −0.623330 0.781959i \(-0.714221\pi\)
0.623330 0.781959i \(-0.285779\pi\)
\(20\) 0 0
\(21\) −20.1068 −0.957466
\(22\) 0 0
\(23\) 4.79583i 0.208514i
\(24\) 0 0
\(25\) 5.00000 0.200000
\(26\) 0 0
\(27\) − 26.4269i − 0.978775i
\(28\) 0 0
\(29\) 12.1117 0.417644 0.208822 0.977954i \(-0.433037\pi\)
0.208822 + 0.977954i \(0.433037\pi\)
\(30\) 0 0
\(31\) 5.25345i 0.169466i 0.996404 + 0.0847331i \(0.0270038\pi\)
−0.996404 + 0.0847331i \(0.972996\pi\)
\(32\) 0 0
\(33\) −36.7261 −1.11291
\(34\) 0 0
\(35\) − 25.2140i − 0.720399i
\(36\) 0 0
\(37\) −2.56136 −0.0692261 −0.0346130 0.999401i \(-0.511020\pi\)
−0.0346130 + 0.999401i \(0.511020\pi\)
\(38\) 0 0
\(39\) − 34.5606i − 0.886170i
\(40\) 0 0
\(41\) 17.2011 0.419539 0.209769 0.977751i \(-0.432729\pi\)
0.209769 + 0.977751i \(0.432729\pi\)
\(42\) 0 0
\(43\) − 7.38042i − 0.171638i −0.996311 0.0858189i \(-0.972649\pi\)
0.996311 0.0858189i \(-0.0273506\pi\)
\(44\) 0 0
\(45\) 13.0148 0.289218
\(46\) 0 0
\(47\) 77.4615i 1.64812i 0.566505 + 0.824058i \(0.308295\pi\)
−0.566505 + 0.824058i \(0.691705\pi\)
\(48\) 0 0
\(49\) −78.1487 −1.59487
\(50\) 0 0
\(51\) 48.4304i 0.949617i
\(52\) 0 0
\(53\) 95.1341 1.79498 0.897492 0.441032i \(-0.145387\pi\)
0.897492 + 0.441032i \(0.145387\pi\)
\(54\) 0 0
\(55\) − 46.0546i − 0.837356i
\(56\) 0 0
\(57\) −52.9852 −0.929564
\(58\) 0 0
\(59\) 22.2739i 0.377524i 0.982023 + 0.188762i \(0.0604474\pi\)
−0.982023 + 0.188762i \(0.939553\pi\)
\(60\) 0 0
\(61\) −23.6344 −0.387449 −0.193724 0.981056i \(-0.562057\pi\)
−0.193724 + 0.981056i \(0.562057\pi\)
\(62\) 0 0
\(63\) − 65.6309i − 1.04176i
\(64\) 0 0
\(65\) 43.3391 0.666755
\(66\) 0 0
\(67\) 63.2193i 0.943572i 0.881713 + 0.471786i \(0.156391\pi\)
−0.881713 + 0.471786i \(0.843609\pi\)
\(68\) 0 0
\(69\) 8.55166 0.123937
\(70\) 0 0
\(71\) − 20.7152i − 0.291764i −0.989302 0.145882i \(-0.953398\pi\)
0.989302 0.145882i \(-0.0466020\pi\)
\(72\) 0 0
\(73\) 114.966 1.57487 0.787436 0.616396i \(-0.211408\pi\)
0.787436 + 0.616396i \(0.211408\pi\)
\(74\) 0 0
\(75\) − 8.91573i − 0.118876i
\(76\) 0 0
\(77\) −232.244 −3.01615
\(78\) 0 0
\(79\) 150.947i 1.91072i 0.295444 + 0.955360i \(0.404532\pi\)
−0.295444 + 0.955360i \(0.595468\pi\)
\(80\) 0 0
\(81\) 5.26053 0.0649448
\(82\) 0 0
\(83\) 134.051i 1.61508i 0.589815 + 0.807538i \(0.299200\pi\)
−0.589815 + 0.807538i \(0.700800\pi\)
\(84\) 0 0
\(85\) −60.7319 −0.714493
\(86\) 0 0
\(87\) − 21.5969i − 0.248240i
\(88\) 0 0
\(89\) 61.9897 0.696513 0.348257 0.937399i \(-0.386774\pi\)
0.348257 + 0.937399i \(0.386774\pi\)
\(90\) 0 0
\(91\) − 218.550i − 2.40165i
\(92\) 0 0
\(93\) 9.36767 0.100728
\(94\) 0 0
\(95\) − 66.4435i − 0.699405i
\(96\) 0 0
\(97\) −42.1692 −0.434734 −0.217367 0.976090i \(-0.569747\pi\)
−0.217367 + 0.976090i \(0.569747\pi\)
\(98\) 0 0
\(99\) − 119.878i − 1.21089i
\(100\) 0 0
\(101\) −16.1829 −0.160227 −0.0801134 0.996786i \(-0.525528\pi\)
−0.0801134 + 0.996786i \(0.525528\pi\)
\(102\) 0 0
\(103\) − 83.8777i − 0.814347i −0.913351 0.407173i \(-0.866514\pi\)
0.913351 0.407173i \(-0.133486\pi\)
\(104\) 0 0
\(105\) −44.9601 −0.428192
\(106\) 0 0
\(107\) − 154.676i − 1.44557i −0.691074 0.722784i \(-0.742862\pi\)
0.691074 0.722784i \(-0.257138\pi\)
\(108\) 0 0
\(109\) −185.589 −1.70266 −0.851328 0.524634i \(-0.824202\pi\)
−0.851328 + 0.524634i \(0.824202\pi\)
\(110\) 0 0
\(111\) 4.56729i 0.0411467i
\(112\) 0 0
\(113\) −3.06892 −0.0271586 −0.0135793 0.999908i \(-0.504323\pi\)
−0.0135793 + 0.999908i \(0.504323\pi\)
\(114\) 0 0
\(115\) 10.7238i 0.0932505i
\(116\) 0 0
\(117\) 112.810 0.964187
\(118\) 0 0
\(119\) 306.258i 2.57360i
\(120\) 0 0
\(121\) −303.205 −2.50582
\(122\) 0 0
\(123\) − 30.6720i − 0.249366i
\(124\) 0 0
\(125\) 11.1803 0.0894427
\(126\) 0 0
\(127\) − 110.645i − 0.871217i −0.900136 0.435608i \(-0.856533\pi\)
0.900136 0.435608i \(-0.143467\pi\)
\(128\) 0 0
\(129\) −13.1604 −0.102018
\(130\) 0 0
\(131\) 215.393i 1.64422i 0.569327 + 0.822111i \(0.307204\pi\)
−0.569327 + 0.822111i \(0.692796\pi\)
\(132\) 0 0
\(133\) −335.061 −2.51925
\(134\) 0 0
\(135\) − 59.0924i − 0.437721i
\(136\) 0 0
\(137\) 72.3393 0.528024 0.264012 0.964519i \(-0.414954\pi\)
0.264012 + 0.964519i \(0.414954\pi\)
\(138\) 0 0
\(139\) − 92.1524i − 0.662967i −0.943461 0.331483i \(-0.892451\pi\)
0.943461 0.331483i \(-0.107549\pi\)
\(140\) 0 0
\(141\) 138.125 0.979611
\(142\) 0 0
\(143\) − 399.193i − 2.79156i
\(144\) 0 0
\(145\) 27.0826 0.186776
\(146\) 0 0
\(147\) 139.350i 0.947962i
\(148\) 0 0
\(149\) −12.0258 −0.0807103 −0.0403552 0.999185i \(-0.512849\pi\)
−0.0403552 + 0.999185i \(0.512849\pi\)
\(150\) 0 0
\(151\) − 45.6425i − 0.302268i −0.988513 0.151134i \(-0.951707\pi\)
0.988513 0.151134i \(-0.0482926\pi\)
\(152\) 0 0
\(153\) −158.083 −1.03322
\(154\) 0 0
\(155\) 11.7471i 0.0757876i
\(156\) 0 0
\(157\) 142.807 0.909600 0.454800 0.890594i \(-0.349711\pi\)
0.454800 + 0.890594i \(0.349711\pi\)
\(158\) 0 0
\(159\) − 169.638i − 1.06691i
\(160\) 0 0
\(161\) 54.0779 0.335888
\(162\) 0 0
\(163\) − 13.6877i − 0.0839733i −0.999118 0.0419867i \(-0.986631\pi\)
0.999118 0.0419867i \(-0.0133687\pi\)
\(164\) 0 0
\(165\) −82.1220 −0.497709
\(166\) 0 0
\(167\) − 100.923i − 0.604327i −0.953256 0.302164i \(-0.902291\pi\)
0.953256 0.302164i \(-0.0977089\pi\)
\(168\) 0 0
\(169\) 206.655 1.22281
\(170\) 0 0
\(171\) − 172.950i − 1.01140i
\(172\) 0 0
\(173\) 218.056 1.26044 0.630219 0.776417i \(-0.282965\pi\)
0.630219 + 0.776417i \(0.282965\pi\)
\(174\) 0 0
\(175\) − 56.3801i − 0.322172i
\(176\) 0 0
\(177\) 39.7176 0.224393
\(178\) 0 0
\(179\) 321.350i 1.79525i 0.440756 + 0.897627i \(0.354710\pi\)
−0.440756 + 0.897627i \(0.645290\pi\)
\(180\) 0 0
\(181\) −272.135 −1.50351 −0.751753 0.659445i \(-0.770792\pi\)
−0.751753 + 0.659445i \(0.770792\pi\)
\(182\) 0 0
\(183\) 42.1435i 0.230292i
\(184\) 0 0
\(185\) −5.72739 −0.0309588
\(186\) 0 0
\(187\) 559.396i 2.99142i
\(188\) 0 0
\(189\) −297.990 −1.57667
\(190\) 0 0
\(191\) 106.689i 0.558582i 0.960207 + 0.279291i \(0.0900994\pi\)
−0.960207 + 0.279291i \(0.909901\pi\)
\(192\) 0 0
\(193\) 182.969 0.948023 0.474012 0.880519i \(-0.342805\pi\)
0.474012 + 0.880519i \(0.342805\pi\)
\(194\) 0 0
\(195\) − 77.2799i − 0.396307i
\(196\) 0 0
\(197\) 358.927 1.82197 0.910983 0.412444i \(-0.135325\pi\)
0.910983 + 0.412444i \(0.135325\pi\)
\(198\) 0 0
\(199\) − 260.206i − 1.30757i −0.756681 0.653784i \(-0.773180\pi\)
0.756681 0.653784i \(-0.226820\pi\)
\(200\) 0 0
\(201\) 112.729 0.560842
\(202\) 0 0
\(203\) − 136.572i − 0.672767i
\(204\) 0 0
\(205\) 38.4628 0.187623
\(206\) 0 0
\(207\) 27.9136i 0.134848i
\(208\) 0 0
\(209\) −612.006 −2.92826
\(210\) 0 0
\(211\) 85.8730i 0.406981i 0.979077 + 0.203491i \(0.0652286\pi\)
−0.979077 + 0.203491i \(0.934771\pi\)
\(212\) 0 0
\(213\) −36.9383 −0.173419
\(214\) 0 0
\(215\) − 16.5031i − 0.0767587i
\(216\) 0 0
\(217\) 59.2381 0.272986
\(218\) 0 0
\(219\) − 205.000i − 0.936075i
\(220\) 0 0
\(221\) −526.413 −2.38196
\(222\) 0 0
\(223\) − 7.27585i − 0.0326271i −0.999867 0.0163136i \(-0.994807\pi\)
0.999867 0.0163136i \(-0.00519300\pi\)
\(224\) 0 0
\(225\) 29.1020 0.129342
\(226\) 0 0
\(227\) − 158.670i − 0.698987i −0.936939 0.349493i \(-0.886354\pi\)
0.936939 0.349493i \(-0.113646\pi\)
\(228\) 0 0
\(229\) −170.421 −0.744196 −0.372098 0.928193i \(-0.621362\pi\)
−0.372098 + 0.928193i \(0.621362\pi\)
\(230\) 0 0
\(231\) 414.124i 1.79274i
\(232\) 0 0
\(233\) −441.884 −1.89650 −0.948248 0.317530i \(-0.897147\pi\)
−0.948248 + 0.317530i \(0.897147\pi\)
\(234\) 0 0
\(235\) 173.209i 0.737060i
\(236\) 0 0
\(237\) 269.160 1.13570
\(238\) 0 0
\(239\) 218.029i 0.912257i 0.889914 + 0.456128i \(0.150764\pi\)
−0.889914 + 0.456128i \(0.849236\pi\)
\(240\) 0 0
\(241\) −282.317 −1.17144 −0.585721 0.810513i \(-0.699188\pi\)
−0.585721 + 0.810513i \(0.699188\pi\)
\(242\) 0 0
\(243\) − 247.222i − 1.01738i
\(244\) 0 0
\(245\) −174.746 −0.713248
\(246\) 0 0
\(247\) − 575.920i − 2.33166i
\(248\) 0 0
\(249\) 239.033 0.959972
\(250\) 0 0
\(251\) − 201.439i − 0.802548i −0.915958 0.401274i \(-0.868568\pi\)
0.915958 0.401274i \(-0.131432\pi\)
\(252\) 0 0
\(253\) 98.7761 0.390419
\(254\) 0 0
\(255\) 108.294i 0.424681i
\(256\) 0 0
\(257\) −453.755 −1.76558 −0.882791 0.469766i \(-0.844338\pi\)
−0.882791 + 0.469766i \(0.844338\pi\)
\(258\) 0 0
\(259\) 28.8820i 0.111514i
\(260\) 0 0
\(261\) 70.4948 0.270095
\(262\) 0 0
\(263\) − 107.082i − 0.407155i −0.979059 0.203577i \(-0.934743\pi\)
0.979059 0.203577i \(-0.0652568\pi\)
\(264\) 0 0
\(265\) 212.726 0.802741
\(266\) 0 0
\(267\) − 110.537i − 0.413995i
\(268\) 0 0
\(269\) −278.681 −1.03599 −0.517994 0.855384i \(-0.673321\pi\)
−0.517994 + 0.855384i \(0.673321\pi\)
\(270\) 0 0
\(271\) 306.814i 1.13215i 0.824352 + 0.566077i \(0.191540\pi\)
−0.824352 + 0.566077i \(0.808460\pi\)
\(272\) 0 0
\(273\) −389.706 −1.42750
\(274\) 0 0
\(275\) − 102.981i − 0.374477i
\(276\) 0 0
\(277\) 384.344 1.38752 0.693762 0.720204i \(-0.255952\pi\)
0.693762 + 0.720204i \(0.255952\pi\)
\(278\) 0 0
\(279\) 30.5772i 0.109596i
\(280\) 0 0
\(281\) −95.5712 −0.340111 −0.170055 0.985434i \(-0.554395\pi\)
−0.170055 + 0.985434i \(0.554395\pi\)
\(282\) 0 0
\(283\) 159.227i 0.562639i 0.959614 + 0.281319i \(0.0907721\pi\)
−0.959614 + 0.281319i \(0.909228\pi\)
\(284\) 0 0
\(285\) −118.478 −0.415714
\(286\) 0 0
\(287\) − 193.960i − 0.675818i
\(288\) 0 0
\(289\) 448.672 1.55250
\(290\) 0 0
\(291\) 75.1938i 0.258398i
\(292\) 0 0
\(293\) 68.9553 0.235342 0.117671 0.993053i \(-0.462457\pi\)
0.117671 + 0.993053i \(0.462457\pi\)
\(294\) 0 0
\(295\) 49.8059i 0.168834i
\(296\) 0 0
\(297\) −544.295 −1.83264
\(298\) 0 0
\(299\) 92.9520i 0.310876i
\(300\) 0 0
\(301\) −83.2218 −0.276484
\(302\) 0 0
\(303\) 28.8565i 0.0952359i
\(304\) 0 0
\(305\) −52.8481 −0.173272
\(306\) 0 0
\(307\) − 284.390i − 0.926350i −0.886267 0.463175i \(-0.846710\pi\)
0.886267 0.463175i \(-0.153290\pi\)
\(308\) 0 0
\(309\) −149.566 −0.484033
\(310\) 0 0
\(311\) − 378.124i − 1.21583i −0.794001 0.607917i \(-0.792005\pi\)
0.794001 0.607917i \(-0.207995\pi\)
\(312\) 0 0
\(313\) −33.5344 −0.107139 −0.0535694 0.998564i \(-0.517060\pi\)
−0.0535694 + 0.998564i \(0.517060\pi\)
\(314\) 0 0
\(315\) − 146.755i − 0.465889i
\(316\) 0 0
\(317\) 507.734 1.60168 0.800842 0.598876i \(-0.204386\pi\)
0.800842 + 0.598876i \(0.204386\pi\)
\(318\) 0 0
\(319\) − 249.455i − 0.781991i
\(320\) 0 0
\(321\) −275.810 −0.859220
\(322\) 0 0
\(323\) 807.048i 2.49860i
\(324\) 0 0
\(325\) 96.9092 0.298182
\(326\) 0 0
\(327\) 330.933i 1.01203i
\(328\) 0 0
\(329\) 873.458 2.65489
\(330\) 0 0
\(331\) 501.833i 1.51611i 0.652190 + 0.758056i \(0.273850\pi\)
−0.652190 + 0.758056i \(0.726150\pi\)
\(332\) 0 0
\(333\) −14.9082 −0.0447692
\(334\) 0 0
\(335\) 141.363i 0.421978i
\(336\) 0 0
\(337\) 39.7559 0.117970 0.0589851 0.998259i \(-0.481214\pi\)
0.0589851 + 0.998259i \(0.481214\pi\)
\(338\) 0 0
\(339\) 5.47234i 0.0161426i
\(340\) 0 0
\(341\) 108.201 0.317306
\(342\) 0 0
\(343\) 328.681i 0.958255i
\(344\) 0 0
\(345\) 19.1221 0.0554264
\(346\) 0 0
\(347\) − 52.8521i − 0.152311i −0.997096 0.0761557i \(-0.975735\pi\)
0.997096 0.0761557i \(-0.0242646\pi\)
\(348\) 0 0
\(349\) 469.656 1.34572 0.672860 0.739770i \(-0.265066\pi\)
0.672860 + 0.739770i \(0.265066\pi\)
\(350\) 0 0
\(351\) − 512.202i − 1.45926i
\(352\) 0 0
\(353\) −355.526 −1.00716 −0.503578 0.863950i \(-0.667983\pi\)
−0.503578 + 0.863950i \(0.667983\pi\)
\(354\) 0 0
\(355\) − 46.3207i − 0.130481i
\(356\) 0 0
\(357\) 546.103 1.52970
\(358\) 0 0
\(359\) − 27.0403i − 0.0753211i −0.999291 0.0376606i \(-0.988009\pi\)
0.999291 0.0376606i \(-0.0119906\pi\)
\(360\) 0 0
\(361\) −521.948 −1.44584
\(362\) 0 0
\(363\) 540.658i 1.48942i
\(364\) 0 0
\(365\) 257.071 0.704304
\(366\) 0 0
\(367\) − 468.694i − 1.27710i −0.769582 0.638548i \(-0.779536\pi\)
0.769582 0.638548i \(-0.220464\pi\)
\(368\) 0 0
\(369\) 100.117 0.271320
\(370\) 0 0
\(371\) − 1072.73i − 2.89147i
\(372\) 0 0
\(373\) −139.574 −0.374194 −0.187097 0.982341i \(-0.559908\pi\)
−0.187097 + 0.982341i \(0.559908\pi\)
\(374\) 0 0
\(375\) − 19.9362i − 0.0531631i
\(376\) 0 0
\(377\) 234.747 0.622670
\(378\) 0 0
\(379\) − 517.047i − 1.36424i −0.731240 0.682120i \(-0.761058\pi\)
0.731240 0.682120i \(-0.238942\pi\)
\(380\) 0 0
\(381\) −197.295 −0.517835
\(382\) 0 0
\(383\) 17.7901i 0.0464494i 0.999730 + 0.0232247i \(0.00739332\pi\)
−0.999730 + 0.0232247i \(0.992607\pi\)
\(384\) 0 0
\(385\) −519.312 −1.34886
\(386\) 0 0
\(387\) − 42.9570i − 0.111000i
\(388\) 0 0
\(389\) 48.2931 0.124147 0.0620734 0.998072i \(-0.480229\pi\)
0.0620734 + 0.998072i \(0.480229\pi\)
\(390\) 0 0
\(391\) − 130.255i − 0.333134i
\(392\) 0 0
\(393\) 384.077 0.977295
\(394\) 0 0
\(395\) 337.528i 0.854500i
\(396\) 0 0
\(397\) −343.875 −0.866185 −0.433092 0.901350i \(-0.642578\pi\)
−0.433092 + 0.901350i \(0.642578\pi\)
\(398\) 0 0
\(399\) 597.462i 1.49740i
\(400\) 0 0
\(401\) 266.765 0.665250 0.332625 0.943059i \(-0.392066\pi\)
0.332625 + 0.943059i \(0.392066\pi\)
\(402\) 0 0
\(403\) 101.822i 0.252659i
\(404\) 0 0
\(405\) 11.7629 0.0290442
\(406\) 0 0
\(407\) 52.7545i 0.129618i
\(408\) 0 0
\(409\) 454.087 1.11024 0.555118 0.831771i \(-0.312673\pi\)
0.555118 + 0.831771i \(0.312673\pi\)
\(410\) 0 0
\(411\) − 128.991i − 0.313848i
\(412\) 0 0
\(413\) 251.161 0.608138
\(414\) 0 0
\(415\) 299.748i 0.722284i
\(416\) 0 0
\(417\) −164.321 −0.394055
\(418\) 0 0
\(419\) 635.903i 1.51767i 0.651284 + 0.758834i \(0.274231\pi\)
−0.651284 + 0.758834i \(0.725769\pi\)
\(420\) 0 0
\(421\) 329.846 0.783482 0.391741 0.920076i \(-0.371873\pi\)
0.391741 + 0.920076i \(0.371873\pi\)
\(422\) 0 0
\(423\) 450.856i 1.06585i
\(424\) 0 0
\(425\) −135.801 −0.319531
\(426\) 0 0
\(427\) 266.502i 0.624126i
\(428\) 0 0
\(429\) −711.819 −1.65925
\(430\) 0 0
\(431\) 142.754i 0.331215i 0.986192 + 0.165607i \(0.0529584\pi\)
−0.986192 + 0.165607i \(0.947042\pi\)
\(432\) 0 0
\(433\) 635.368 1.46736 0.733682 0.679493i \(-0.237800\pi\)
0.733682 + 0.679493i \(0.237800\pi\)
\(434\) 0 0
\(435\) − 48.2921i − 0.111016i
\(436\) 0 0
\(437\) 142.505 0.326099
\(438\) 0 0
\(439\) − 154.754i − 0.352515i −0.984344 0.176258i \(-0.943601\pi\)
0.984344 0.176258i \(-0.0563992\pi\)
\(440\) 0 0
\(441\) −454.856 −1.03142
\(442\) 0 0
\(443\) 219.706i 0.495951i 0.968766 + 0.247975i \(0.0797652\pi\)
−0.968766 + 0.247975i \(0.920235\pi\)
\(444\) 0 0
\(445\) 138.613 0.311490
\(446\) 0 0
\(447\) 21.4438i 0.0479727i
\(448\) 0 0
\(449\) 73.0500 0.162695 0.0813475 0.996686i \(-0.474078\pi\)
0.0813475 + 0.996686i \(0.474078\pi\)
\(450\) 0 0
\(451\) − 354.278i − 0.785538i
\(452\) 0 0
\(453\) −81.3873 −0.179663
\(454\) 0 0
\(455\) − 488.693i − 1.07405i
\(456\) 0 0
\(457\) 348.797 0.763232 0.381616 0.924321i \(-0.375368\pi\)
0.381616 + 0.924321i \(0.375368\pi\)
\(458\) 0 0
\(459\) 717.758i 1.56374i
\(460\) 0 0
\(461\) 384.410 0.833860 0.416930 0.908939i \(-0.363106\pi\)
0.416930 + 0.908939i \(0.363106\pi\)
\(462\) 0 0
\(463\) 180.334i 0.389490i 0.980854 + 0.194745i \(0.0623879\pi\)
−0.980854 + 0.194745i \(0.937612\pi\)
\(464\) 0 0
\(465\) 20.9467 0.0450468
\(466\) 0 0
\(467\) 52.8643i 0.113200i 0.998397 + 0.0565998i \(0.0180259\pi\)
−0.998397 + 0.0565998i \(0.981974\pi\)
\(468\) 0 0
\(469\) 712.863 1.51996
\(470\) 0 0
\(471\) − 254.646i − 0.540649i
\(472\) 0 0
\(473\) −152.009 −0.321372
\(474\) 0 0
\(475\) − 148.572i − 0.312784i
\(476\) 0 0
\(477\) 553.718 1.16083
\(478\) 0 0
\(479\) − 189.099i − 0.394780i −0.980325 0.197390i \(-0.936754\pi\)
0.980325 0.197390i \(-0.0632465\pi\)
\(480\) 0 0
\(481\) −49.6439 −0.103210
\(482\) 0 0
\(483\) − 96.4288i − 0.199645i
\(484\) 0 0
\(485\) −94.2932 −0.194419
\(486\) 0 0
\(487\) 104.030i 0.213615i 0.994280 + 0.106807i \(0.0340628\pi\)
−0.994280 + 0.106807i \(0.965937\pi\)
\(488\) 0 0
\(489\) −24.4071 −0.0499122
\(490\) 0 0
\(491\) − 396.964i − 0.808480i −0.914653 0.404240i \(-0.867536\pi\)
0.914653 0.404240i \(-0.132464\pi\)
\(492\) 0 0
\(493\) −328.955 −0.667251
\(494\) 0 0
\(495\) − 268.056i − 0.541527i
\(496\) 0 0
\(497\) −233.586 −0.469991
\(498\) 0 0
\(499\) − 626.983i − 1.25648i −0.778020 0.628239i \(-0.783776\pi\)
0.778020 0.628239i \(-0.216224\pi\)
\(500\) 0 0
\(501\) −179.960 −0.359201
\(502\) 0 0
\(503\) − 359.980i − 0.715667i −0.933785 0.357833i \(-0.883516\pi\)
0.933785 0.357833i \(-0.116484\pi\)
\(504\) 0 0
\(505\) −36.1861 −0.0716556
\(506\) 0 0
\(507\) − 368.497i − 0.726818i
\(508\) 0 0
\(509\) 395.425 0.776867 0.388433 0.921477i \(-0.373016\pi\)
0.388433 + 0.921477i \(0.373016\pi\)
\(510\) 0 0
\(511\) − 1296.36i − 2.53690i
\(512\) 0 0
\(513\) −785.261 −1.53072
\(514\) 0 0
\(515\) − 187.556i − 0.364187i
\(516\) 0 0
\(517\) 1595.42 3.08591
\(518\) 0 0
\(519\) − 388.825i − 0.749182i
\(520\) 0 0
\(521\) −220.949 −0.424086 −0.212043 0.977260i \(-0.568012\pi\)
−0.212043 + 0.977260i \(0.568012\pi\)
\(522\) 0 0
\(523\) 619.464i 1.18444i 0.805775 + 0.592222i \(0.201749\pi\)
−0.805775 + 0.592222i \(0.798251\pi\)
\(524\) 0 0
\(525\) −100.534 −0.191493
\(526\) 0 0
\(527\) − 142.684i − 0.270749i
\(528\) 0 0
\(529\) −23.0000 −0.0434783
\(530\) 0 0
\(531\) 129.643i 0.244148i
\(532\) 0 0
\(533\) 333.389 0.625494
\(534\) 0 0
\(535\) − 345.866i − 0.646478i
\(536\) 0 0
\(537\) 573.015 1.06707
\(538\) 0 0
\(539\) 1609.57i 2.98621i
\(540\) 0 0
\(541\) −80.8928 −0.149525 −0.0747623 0.997201i \(-0.523820\pi\)
−0.0747623 + 0.997201i \(0.523820\pi\)
\(542\) 0 0
\(543\) 485.256i 0.893657i
\(544\) 0 0
\(545\) −414.991 −0.761451
\(546\) 0 0
\(547\) 902.721i 1.65031i 0.564905 + 0.825156i \(0.308913\pi\)
−0.564905 + 0.825156i \(0.691087\pi\)
\(548\) 0 0
\(549\) −137.561 −0.250567
\(550\) 0 0
\(551\) − 359.892i − 0.653162i
\(552\) 0 0
\(553\) 1702.08 3.07790
\(554\) 0 0
\(555\) 10.2128i 0.0184014i
\(556\) 0 0
\(557\) −352.533 −0.632914 −0.316457 0.948607i \(-0.602493\pi\)
−0.316457 + 0.948607i \(0.602493\pi\)
\(558\) 0 0
\(559\) − 143.046i − 0.255896i
\(560\) 0 0
\(561\) 997.485 1.77805
\(562\) 0 0
\(563\) − 292.609i − 0.519731i −0.965645 0.259866i \(-0.916322\pi\)
0.965645 0.259866i \(-0.0836782\pi\)
\(564\) 0 0
\(565\) −6.86232 −0.0121457
\(566\) 0 0
\(567\) − 59.3178i − 0.104617i
\(568\) 0 0
\(569\) 190.490 0.334781 0.167390 0.985891i \(-0.446466\pi\)
0.167390 + 0.985891i \(0.446466\pi\)
\(570\) 0 0
\(571\) − 57.4750i − 0.100657i −0.998733 0.0503284i \(-0.983973\pi\)
0.998733 0.0503284i \(-0.0160268\pi\)
\(572\) 0 0
\(573\) 190.242 0.332011
\(574\) 0 0
\(575\) 23.9792i 0.0417029i
\(576\) 0 0
\(577\) 47.5032 0.0823280 0.0411640 0.999152i \(-0.486893\pi\)
0.0411640 + 0.999152i \(0.486893\pi\)
\(578\) 0 0
\(579\) − 326.259i − 0.563488i
\(580\) 0 0
\(581\) 1511.57 2.60166
\(582\) 0 0
\(583\) − 1959.40i − 3.36090i
\(584\) 0 0
\(585\) 252.251 0.431198
\(586\) 0 0
\(587\) 1030.76i 1.75599i 0.478672 + 0.877994i \(0.341118\pi\)
−0.478672 + 0.877994i \(0.658882\pi\)
\(588\) 0 0
\(589\) 156.103 0.265031
\(590\) 0 0
\(591\) − 640.019i − 1.08294i
\(592\) 0 0
\(593\) 911.094 1.53641 0.768207 0.640201i \(-0.221149\pi\)
0.768207 + 0.640201i \(0.221149\pi\)
\(594\) 0 0
\(595\) 684.814i 1.15095i
\(596\) 0 0
\(597\) −463.985 −0.777195
\(598\) 0 0
\(599\) 911.862i 1.52231i 0.648572 + 0.761153i \(0.275367\pi\)
−0.648572 + 0.761153i \(0.724633\pi\)
\(600\) 0 0
\(601\) −227.152 −0.377957 −0.188978 0.981981i \(-0.560518\pi\)
−0.188978 + 0.981981i \(0.560518\pi\)
\(602\) 0 0
\(603\) 367.961i 0.610218i
\(604\) 0 0
\(605\) −677.987 −1.12064
\(606\) 0 0
\(607\) 155.710i 0.256524i 0.991740 + 0.128262i \(0.0409399\pi\)
−0.991740 + 0.128262i \(0.959060\pi\)
\(608\) 0 0
\(609\) −243.527 −0.399880
\(610\) 0 0
\(611\) 1501.35i 2.45719i
\(612\) 0 0
\(613\) −131.088 −0.213847 −0.106924 0.994267i \(-0.534100\pi\)
−0.106924 + 0.994267i \(0.534100\pi\)
\(614\) 0 0
\(615\) − 68.5847i − 0.111520i
\(616\) 0 0
\(617\) 540.179 0.875492 0.437746 0.899099i \(-0.355777\pi\)
0.437746 + 0.899099i \(0.355777\pi\)
\(618\) 0 0
\(619\) 1050.32i 1.69680i 0.529353 + 0.848402i \(0.322435\pi\)
−0.529353 + 0.848402i \(0.677565\pi\)
\(620\) 0 0
\(621\) 126.739 0.204089
\(622\) 0 0
\(623\) − 698.997i − 1.12199i
\(624\) 0 0
\(625\) 25.0000 0.0400000
\(626\) 0 0
\(627\) 1091.29i 1.74050i
\(628\) 0 0
\(629\) 69.5670 0.110599
\(630\) 0 0
\(631\) − 339.913i − 0.538689i −0.963044 0.269344i \(-0.913193\pi\)
0.963044 0.269344i \(-0.0868070\pi\)
\(632\) 0 0
\(633\) 153.124 0.241902
\(634\) 0 0
\(635\) − 247.409i − 0.389620i
\(636\) 0 0
\(637\) −1514.67 −2.37781
\(638\) 0 0
\(639\) − 120.571i − 0.188687i
\(640\) 0 0
\(641\) −1064.68 −1.66096 −0.830480 0.557048i \(-0.811934\pi\)
−0.830480 + 0.557048i \(0.811934\pi\)
\(642\) 0 0
\(643\) 472.734i 0.735201i 0.929984 + 0.367600i \(0.119821\pi\)
−0.929984 + 0.367600i \(0.880179\pi\)
\(644\) 0 0
\(645\) −29.4275 −0.0456240
\(646\) 0 0
\(647\) 661.303i 1.02211i 0.859549 + 0.511054i \(0.170745\pi\)
−0.859549 + 0.511054i \(0.829255\pi\)
\(648\) 0 0
\(649\) 458.758 0.706869
\(650\) 0 0
\(651\) − 105.630i − 0.162258i
\(652\) 0 0
\(653\) −91.5511 −0.140201 −0.0701004 0.997540i \(-0.522332\pi\)
−0.0701004 + 0.997540i \(0.522332\pi\)
\(654\) 0 0
\(655\) 481.634i 0.735318i
\(656\) 0 0
\(657\) 669.145 1.01849
\(658\) 0 0
\(659\) 803.989i 1.22001i 0.792396 + 0.610007i \(0.208833\pi\)
−0.792396 + 0.610007i \(0.791167\pi\)
\(660\) 0 0
\(661\) −882.772 −1.33551 −0.667755 0.744381i \(-0.732745\pi\)
−0.667755 + 0.744381i \(0.732745\pi\)
\(662\) 0 0
\(663\) 938.671i 1.41579i
\(664\) 0 0
\(665\) −749.219 −1.12664
\(666\) 0 0
\(667\) 58.0856i 0.0870849i
\(668\) 0 0
\(669\) −12.9739 −0.0193930
\(670\) 0 0
\(671\) 486.779i 0.725453i
\(672\) 0 0
\(673\) −174.407 −0.259149 −0.129574 0.991570i \(-0.541361\pi\)
−0.129574 + 0.991570i \(0.541361\pi\)
\(674\) 0 0
\(675\) − 132.135i − 0.195755i
\(676\) 0 0
\(677\) 323.903 0.478439 0.239219 0.970966i \(-0.423109\pi\)
0.239219 + 0.970966i \(0.423109\pi\)
\(678\) 0 0
\(679\) 475.501i 0.700296i
\(680\) 0 0
\(681\) −282.932 −0.415465
\(682\) 0 0
\(683\) 16.7421i 0.0245126i 0.999925 + 0.0122563i \(0.00390140\pi\)
−0.999925 + 0.0122563i \(0.996099\pi\)
\(684\) 0 0
\(685\) 161.755 0.236139
\(686\) 0 0
\(687\) 303.885i 0.442337i
\(688\) 0 0
\(689\) 1843.87 2.67616
\(690\) 0 0
\(691\) − 1107.77i − 1.60314i −0.597904 0.801568i \(-0.704000\pi\)
0.597904 0.801568i \(-0.296000\pi\)
\(692\) 0 0
\(693\) −1351.75 −1.95058
\(694\) 0 0
\(695\) − 206.059i − 0.296488i
\(696\) 0 0
\(697\) −467.184 −0.670278
\(698\) 0 0
\(699\) 787.943i 1.12724i
\(700\) 0 0
\(701\) 638.357 0.910637 0.455319 0.890329i \(-0.349525\pi\)
0.455319 + 0.890329i \(0.349525\pi\)
\(702\) 0 0
\(703\) 76.1095i 0.108264i
\(704\) 0 0
\(705\) 308.857 0.438095
\(706\) 0 0
\(707\) 182.479i 0.258103i
\(708\) 0 0
\(709\) −855.687 −1.20689 −0.603446 0.797404i \(-0.706206\pi\)
−0.603446 + 0.797404i \(0.706206\pi\)
\(710\) 0 0
\(711\) 878.570i 1.23568i
\(712\) 0 0
\(713\) −25.1947 −0.0353362
\(714\) 0 0
\(715\) − 892.622i − 1.24842i
\(716\) 0 0
\(717\) 388.778 0.542229
\(718\) 0 0
\(719\) − 359.613i − 0.500158i −0.968225 0.250079i \(-0.919543\pi\)
0.968225 0.250079i \(-0.0804566\pi\)
\(720\) 0 0
\(721\) −945.807 −1.31180
\(722\) 0 0
\(723\) 503.413i 0.696283i
\(724\) 0 0
\(725\) 60.5584 0.0835289
\(726\) 0 0
\(727\) 106.361i 0.146301i 0.997321 + 0.0731505i \(0.0233053\pi\)
−0.997321 + 0.0731505i \(0.976695\pi\)
\(728\) 0 0
\(729\) −393.489 −0.539765
\(730\) 0 0
\(731\) 200.453i 0.274218i
\(732\) 0 0
\(733\) −714.794 −0.975162 −0.487581 0.873078i \(-0.662121\pi\)
−0.487581 + 0.873078i \(0.662121\pi\)
\(734\) 0 0
\(735\) 311.597i 0.423942i
\(736\) 0 0
\(737\) 1302.08 1.76673
\(738\) 0 0
\(739\) − 1131.01i − 1.53046i −0.643757 0.765230i \(-0.722625\pi\)
0.643757 0.765230i \(-0.277375\pi\)
\(740\) 0 0
\(741\) −1026.95 −1.38590
\(742\) 0 0
\(743\) − 316.289i − 0.425692i −0.977086 0.212846i \(-0.931727\pi\)
0.977086 0.212846i \(-0.0682732\pi\)
\(744\) 0 0
\(745\) −26.8906 −0.0360947
\(746\) 0 0
\(747\) 780.231i 1.04449i
\(748\) 0 0
\(749\) −1744.13 −2.32861
\(750\) 0 0
\(751\) − 249.612i − 0.332373i −0.986094 0.166187i \(-0.946855\pi\)
0.986094 0.166187i \(-0.0531454\pi\)
\(752\) 0 0
\(753\) −359.196 −0.477020
\(754\) 0 0
\(755\) − 102.060i − 0.135179i
\(756\) 0 0
\(757\) 78.0177 0.103062 0.0515308 0.998671i \(-0.483590\pi\)
0.0515308 + 0.998671i \(0.483590\pi\)
\(758\) 0 0
\(759\) − 176.132i − 0.232058i
\(760\) 0 0
\(761\) 380.576 0.500099 0.250050 0.968233i \(-0.419553\pi\)
0.250050 + 0.968233i \(0.419553\pi\)
\(762\) 0 0
\(763\) 2092.71i 2.74274i
\(764\) 0 0
\(765\) −353.484 −0.462070
\(766\) 0 0
\(767\) 431.709i 0.562854i
\(768\) 0 0
\(769\) 410.236 0.533467 0.266734 0.963770i \(-0.414056\pi\)
0.266734 + 0.963770i \(0.414056\pi\)
\(770\) 0 0
\(771\) 809.110i 1.04943i
\(772\) 0 0
\(773\) 560.359 0.724914 0.362457 0.932000i \(-0.381938\pi\)
0.362457 + 0.932000i \(0.381938\pi\)
\(774\) 0 0
\(775\) 26.2673i 0.0338932i
\(776\) 0 0
\(777\) 51.5008 0.0662816
\(778\) 0 0
\(779\) − 511.121i − 0.656124i
\(780\) 0 0
\(781\) −426.656 −0.546294
\(782\) 0 0
\(783\) − 320.074i − 0.408780i
\(784\) 0 0
\(785\) 319.327 0.406785
\(786\) 0 0
\(787\) − 649.620i − 0.825439i −0.910858 0.412719i \(-0.864579\pi\)
0.910858 0.412719i \(-0.135421\pi\)
\(788\) 0 0
\(789\) −190.942 −0.242005
\(790\) 0 0
\(791\) 34.6053i 0.0437487i
\(792\) 0 0
\(793\) −458.077 −0.577651
\(794\) 0 0
\(795\) − 379.322i − 0.477134i
\(796\) 0 0
\(797\) 1116.25 1.40056 0.700282 0.713867i \(-0.253058\pi\)
0.700282 + 0.713867i \(0.253058\pi\)
\(798\) 0 0
\(799\) − 2103.86i − 2.63312i
\(800\) 0 0
\(801\) 360.804 0.450442
\(802\) 0 0
\(803\) − 2367.86i − 2.94877i
\(804\) 0 0
\(805\) 120.922 0.150214
\(806\) 0 0
\(807\) 496.928i 0.615773i
\(808\) 0 0
\(809\) 349.204 0.431649 0.215824 0.976432i \(-0.430756\pi\)
0.215824 + 0.976432i \(0.430756\pi\)
\(810\) 0 0
\(811\) 602.463i 0.742864i 0.928460 + 0.371432i \(0.121133\pi\)
−0.928460 + 0.371432i \(0.878867\pi\)
\(812\) 0 0
\(813\) 547.093 0.672932
\(814\) 0 0
\(815\) − 30.6065i − 0.0375540i
\(816\) 0 0
\(817\) −219.305 −0.268427
\(818\) 0 0
\(819\) − 1272.05i − 1.55317i
\(820\) 0 0
\(821\) 207.306 0.252504 0.126252 0.991998i \(-0.459705\pi\)
0.126252 + 0.991998i \(0.459705\pi\)
\(822\) 0 0
\(823\) − 388.053i − 0.471510i −0.971813 0.235755i \(-0.924244\pi\)
0.971813 0.235755i \(-0.0757563\pi\)
\(824\) 0 0
\(825\) −183.630 −0.222582
\(826\) 0 0
\(827\) 177.374i 0.214479i 0.994233 + 0.107240i \(0.0342012\pi\)
−0.994233 + 0.107240i \(0.965799\pi\)
\(828\) 0 0
\(829\) −1521.76 −1.83566 −0.917828 0.396979i \(-0.870059\pi\)
−0.917828 + 0.396979i \(0.870059\pi\)
\(830\) 0 0
\(831\) − 685.341i − 0.824719i
\(832\) 0 0
\(833\) 2122.53 2.54805
\(834\) 0 0
\(835\) − 225.670i − 0.270263i
\(836\) 0 0
\(837\) 138.833 0.165869
\(838\) 0 0
\(839\) − 253.074i − 0.301637i −0.988561 0.150819i \(-0.951809\pi\)
0.988561 0.150819i \(-0.0481910\pi\)
\(840\) 0 0
\(841\) −694.307 −0.825573
\(842\) 0 0
\(843\) 170.417i 0.202156i
\(844\) 0 0
\(845\) 462.096 0.546859
\(846\) 0 0
\(847\) 3418.94i 4.03653i
\(848\) 0 0
\(849\) 283.925 0.334422
\(850\) 0 0
\(851\) − 12.2839i − 0.0144346i
\(852\) 0 0
\(853\) −1324.50 −1.55275 −0.776376 0.630270i \(-0.782944\pi\)
−0.776376 + 0.630270i \(0.782944\pi\)
\(854\) 0 0
\(855\) − 386.727i − 0.452313i
\(856\) 0 0
\(857\) 652.655 0.761558 0.380779 0.924666i \(-0.375656\pi\)
0.380779 + 0.924666i \(0.375656\pi\)
\(858\) 0 0
\(859\) 557.810i 0.649372i 0.945822 + 0.324686i \(0.105259\pi\)
−0.945822 + 0.324686i \(0.894741\pi\)
\(860\) 0 0
\(861\) −345.859 −0.401694
\(862\) 0 0
\(863\) 613.258i 0.710612i 0.934750 + 0.355306i \(0.115623\pi\)
−0.934750 + 0.355306i \(0.884377\pi\)
\(864\) 0 0
\(865\) 487.588 0.563685
\(866\) 0 0
\(867\) − 800.048i − 0.922778i
\(868\) 0 0
\(869\) 3108.94 3.57760
\(870\) 0 0
\(871\) 1225.31i 1.40678i
\(872\) 0 0
\(873\) −245.441 −0.281147
\(874\) 0 0
\(875\) − 126.070i − 0.144080i
\(876\) 0 0
\(877\) −817.583 −0.932249 −0.466125 0.884719i \(-0.654350\pi\)
−0.466125 + 0.884719i \(0.654350\pi\)
\(878\) 0 0
\(879\) − 122.957i − 0.139883i
\(880\) 0 0
\(881\) 1211.48 1.37512 0.687558 0.726129i \(-0.258683\pi\)
0.687558 + 0.726129i \(0.258683\pi\)
\(882\) 0 0
\(883\) − 1591.37i − 1.80224i −0.433574 0.901118i \(-0.642748\pi\)
0.433574 0.901118i \(-0.357252\pi\)
\(884\) 0 0
\(885\) 88.8112 0.100352
\(886\) 0 0
\(887\) − 609.859i − 0.687552i −0.939052 0.343776i \(-0.888294\pi\)
0.939052 0.343776i \(-0.111706\pi\)
\(888\) 0 0
\(889\) −1247.63 −1.40341
\(890\) 0 0
\(891\) − 108.347i − 0.121602i
\(892\) 0 0
\(893\) 2301.73 2.57752
\(894\) 0 0
\(895\) 718.561i 0.802862i
\(896\) 0 0
\(897\) 165.747 0.184779
\(898\) 0 0
\(899\) 63.6282i 0.0707766i
\(900\) 0 0
\(901\) −2583.85 −2.86776
\(902\) 0 0
\(903\) 148.397i 0.164337i
\(904\) 0 0
\(905\) −608.512 −0.672389
\(906\) 0 0
\(907\) 73.5050i 0.0810418i 0.999179 + 0.0405209i \(0.0129017\pi\)
−0.999179 + 0.0405209i \(0.987098\pi\)
\(908\) 0 0
\(909\) −94.1909 −0.103620
\(910\) 0 0
\(911\) − 1275.16i − 1.39973i −0.714273 0.699867i \(-0.753243\pi\)
0.714273 0.699867i \(-0.246757\pi\)
\(912\) 0 0
\(913\) 2760.95 3.02404
\(914\) 0 0
\(915\) 94.2357i 0.102990i
\(916\) 0 0
\(917\) 2428.78 2.64861
\(918\) 0 0
\(919\) 310.848i 0.338246i 0.985595 + 0.169123i \(0.0540935\pi\)
−0.985595 + 0.169123i \(0.945907\pi\)
\(920\) 0 0
\(921\) −507.108 −0.550606
\(922\) 0 0
\(923\) − 401.499i − 0.434994i
\(924\) 0 0
\(925\) −12.8068 −0.0138452
\(926\) 0 0
\(927\) − 488.201i − 0.526646i
\(928\) 0 0
\(929\) −591.042 −0.636213 −0.318107 0.948055i \(-0.603047\pi\)
−0.318107 + 0.948055i \(0.603047\pi\)
\(930\) 0 0
\(931\) 2322.15i 2.49425i
\(932\) 0 0
\(933\) −674.250 −0.722669
\(934\) 0 0
\(935\) 1250.85i 1.33781i
\(936\) 0 0
\(937\) 1458.34 1.55639 0.778197 0.628020i \(-0.216135\pi\)
0.778197 + 0.628020i \(0.216135\pi\)
\(938\) 0 0
\(939\) 59.7967i 0.0636813i
\(940\) 0 0
\(941\) −1212.25 −1.28826 −0.644128 0.764918i \(-0.722780\pi\)
−0.644128 + 0.764918i \(0.722780\pi\)
\(942\) 0 0
\(943\) 82.4935i 0.0874799i
\(944\) 0 0
\(945\) −666.327 −0.705108
\(946\) 0 0
\(947\) − 422.808i − 0.446471i −0.974764 0.223236i \(-0.928338\pi\)
0.974764 0.223236i \(-0.0716620\pi\)
\(948\) 0 0
\(949\) 2228.25 2.34799
\(950\) 0 0
\(951\) − 905.363i − 0.952012i
\(952\) 0 0
\(953\) 352.540 0.369927 0.184963 0.982745i \(-0.440783\pi\)
0.184963 + 0.982745i \(0.440783\pi\)
\(954\) 0 0
\(955\) 238.564i 0.249805i
\(956\) 0 0
\(957\) −444.815 −0.464801
\(958\) 0 0
\(959\) − 815.699i − 0.850573i
\(960\) 0 0
\(961\) 933.401 0.971281
\(962\) 0 0
\(963\) − 900.274i − 0.934864i
\(964\) 0 0
\(965\) 409.130 0.423969
\(966\) 0 0
\(967\) 297.029i 0.307166i 0.988136 + 0.153583i \(0.0490812\pi\)
−0.988136 + 0.153583i \(0.950919\pi\)
\(968\) 0 0
\(969\) 1439.08 1.48512
\(970\) 0 0
\(971\) 950.451i 0.978837i 0.872049 + 0.489419i \(0.162791\pi\)
−0.872049 + 0.489419i \(0.837209\pi\)
\(972\) 0 0
\(973\) −1039.11 −1.06795
\(974\) 0 0
\(975\) − 172.803i − 0.177234i
\(976\) 0 0
\(977\) −723.725 −0.740762 −0.370381 0.928880i \(-0.620773\pi\)
−0.370381 + 0.928880i \(0.620773\pi\)
\(978\) 0 0
\(979\) − 1276.75i − 1.30414i
\(980\) 0 0
\(981\) −1080.20 −1.10113
\(982\) 0 0
\(983\) 103.447i 0.105236i 0.998615 + 0.0526181i \(0.0167566\pi\)
−0.998615 + 0.0526181i \(0.983243\pi\)
\(984\) 0 0
\(985\) 802.586 0.814808
\(986\) 0 0
\(987\) − 1557.50i − 1.57802i
\(988\) 0 0
\(989\) 35.3953 0.0357889
\(990\) 0 0
\(991\) − 1451.61i − 1.46479i −0.680880 0.732395i \(-0.738402\pi\)
0.680880 0.732395i \(-0.261598\pi\)
\(992\) 0 0
\(993\) 894.841 0.901149
\(994\) 0 0
\(995\) − 581.838i − 0.584762i
\(996\) 0 0
\(997\) −541.612 −0.543242 −0.271621 0.962404i \(-0.587560\pi\)
−0.271621 + 0.962404i \(0.587560\pi\)
\(998\) 0 0
\(999\) 67.6890i 0.0677567i
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 1840.3.c.b.1151.20 56
4.3 odd 2 inner 1840.3.c.b.1151.37 yes 56
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1840.3.c.b.1151.20 56 1.1 even 1 trivial
1840.3.c.b.1151.37 yes 56 4.3 odd 2 inner