Properties

Label 1344.4.b.j.895.12
Level $1344$
Weight $4$
Character 1344.895
Analytic conductor $79.299$
Analytic rank $0$
Dimension $24$
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] = [1344,4,Mod(895,1344)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1344.895");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.b (of order \(2\), degree \(1\), not 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: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 672)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 895.12
Character \(\chi\) \(=\) 1344.895
Dual form 1344.4.b.j.895.13

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -2.14586i q^{5} +(17.7853 + 5.16554i) q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -2.14586i q^{5} +(17.7853 + 5.16554i) q^{7} +9.00000 q^{9} +22.4578i q^{11} +45.2827i q^{13} -6.43759i q^{15} +33.9461i q^{17} +43.2405 q^{19} +(53.3559 + 15.4966i) q^{21} -42.1194i q^{23} +120.395 q^{25} +27.0000 q^{27} +65.2787 q^{29} -188.898 q^{31} +67.3734i q^{33} +(11.0845 - 38.1648i) q^{35} -165.117 q^{37} +135.848i q^{39} +210.591i q^{41} +322.736i q^{43} -19.3128i q^{45} -189.974 q^{47} +(289.634 + 183.741i) q^{49} +101.838i q^{51} -511.728 q^{53} +48.1913 q^{55} +129.722 q^{57} -17.9855 q^{59} +472.581i q^{61} +(160.068 + 46.4898i) q^{63} +97.1705 q^{65} +565.144i q^{67} -126.358i q^{69} -595.510i q^{71} -412.203i q^{73} +361.186 q^{75} +(-116.007 + 399.419i) q^{77} -377.133i q^{79} +81.0000 q^{81} +238.431 q^{83} +72.8437 q^{85} +195.836 q^{87} -585.905i q^{89} +(-233.910 + 805.367i) q^{91} -566.694 q^{93} -92.7882i q^{95} -601.047i q^{97} +202.120i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 72 q^{3} - 20 q^{7} + 216 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 72 q^{3} - 20 q^{7} + 216 q^{9} - 56 q^{19} - 60 q^{21} - 432 q^{25} + 648 q^{27} + 464 q^{31} + 568 q^{35} - 504 q^{37} + 560 q^{47} - 128 q^{49} + 784 q^{53} + 424 q^{55} - 168 q^{57} + 800 q^{59} - 180 q^{63} + 560 q^{65} - 1296 q^{75} + 1568 q^{77} + 1944 q^{81} + 1936 q^{83} - 3000 q^{85} - 496 q^{91} + 1392 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 2.14586i 0.191932i −0.995385 0.0959659i \(-0.969406\pi\)
0.995385 0.0959659i \(-0.0305940\pi\)
\(6\) 0 0
\(7\) 17.7853 + 5.16554i 0.960316 + 0.278913i
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 22.4578i 0.615571i 0.951456 + 0.307785i \(0.0995879\pi\)
−0.951456 + 0.307785i \(0.900412\pi\)
\(12\) 0 0
\(13\) 45.2827i 0.966090i 0.875596 + 0.483045i \(0.160469\pi\)
−0.875596 + 0.483045i \(0.839531\pi\)
\(14\) 0 0
\(15\) 6.43759i 0.110812i
\(16\) 0 0
\(17\) 33.9461i 0.484303i 0.970238 + 0.242151i \(0.0778530\pi\)
−0.970238 + 0.242151i \(0.922147\pi\)
\(18\) 0 0
\(19\) 43.2405 0.522108 0.261054 0.965324i \(-0.415930\pi\)
0.261054 + 0.965324i \(0.415930\pi\)
\(20\) 0 0
\(21\) 53.3559 + 15.4966i 0.554439 + 0.161030i
\(22\) 0 0
\(23\) 42.1194i 0.381848i −0.981605 0.190924i \(-0.938852\pi\)
0.981605 0.190924i \(-0.0611483\pi\)
\(24\) 0 0
\(25\) 120.395 0.963162
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 65.2787 0.417998 0.208999 0.977916i \(-0.432979\pi\)
0.208999 + 0.977916i \(0.432979\pi\)
\(30\) 0 0
\(31\) −188.898 −1.09442 −0.547211 0.836994i \(-0.684311\pi\)
−0.547211 + 0.836994i \(0.684311\pi\)
\(32\) 0 0
\(33\) 67.3734i 0.355400i
\(34\) 0 0
\(35\) 11.0845 38.1648i 0.0535322 0.184315i
\(36\) 0 0
\(37\) −165.117 −0.733653 −0.366826 0.930289i \(-0.619556\pi\)
−0.366826 + 0.930289i \(0.619556\pi\)
\(38\) 0 0
\(39\) 135.848i 0.557772i
\(40\) 0 0
\(41\) 210.591i 0.802167i 0.916041 + 0.401084i \(0.131366\pi\)
−0.916041 + 0.401084i \(0.868634\pi\)
\(42\) 0 0
\(43\) 322.736i 1.14458i 0.820053 + 0.572288i \(0.193944\pi\)
−0.820053 + 0.572288i \(0.806056\pi\)
\(44\) 0 0
\(45\) 19.3128i 0.0639773i
\(46\) 0 0
\(47\) −189.974 −0.589586 −0.294793 0.955561i \(-0.595251\pi\)
−0.294793 + 0.955561i \(0.595251\pi\)
\(48\) 0 0
\(49\) 289.634 + 183.741i 0.844415 + 0.535689i
\(50\) 0 0
\(51\) 101.838i 0.279612i
\(52\) 0 0
\(53\) −511.728 −1.32625 −0.663124 0.748509i \(-0.730770\pi\)
−0.663124 + 0.748509i \(0.730770\pi\)
\(54\) 0 0
\(55\) 48.1913 0.118148
\(56\) 0 0
\(57\) 129.722 0.301439
\(58\) 0 0
\(59\) −17.9855 −0.0396866 −0.0198433 0.999803i \(-0.506317\pi\)
−0.0198433 + 0.999803i \(0.506317\pi\)
\(60\) 0 0
\(61\) 472.581i 0.991932i 0.868342 + 0.495966i \(0.165186\pi\)
−0.868342 + 0.495966i \(0.834814\pi\)
\(62\) 0 0
\(63\) 160.068 + 46.4898i 0.320105 + 0.0929709i
\(64\) 0 0
\(65\) 97.1705 0.185423
\(66\) 0 0
\(67\) 565.144i 1.03050i 0.857041 + 0.515249i \(0.172301\pi\)
−0.857041 + 0.515249i \(0.827699\pi\)
\(68\) 0 0
\(69\) 126.358i 0.220460i
\(70\) 0 0
\(71\) 595.510i 0.995409i −0.867347 0.497704i \(-0.834176\pi\)
0.867347 0.497704i \(-0.165824\pi\)
\(72\) 0 0
\(73\) 412.203i 0.660887i −0.943826 0.330443i \(-0.892802\pi\)
0.943826 0.330443i \(-0.107198\pi\)
\(74\) 0 0
\(75\) 361.186 0.556082
\(76\) 0 0
\(77\) −116.007 + 399.419i −0.171691 + 0.591143i
\(78\) 0 0
\(79\) 377.133i 0.537099i −0.963266 0.268549i \(-0.913456\pi\)
0.963266 0.268549i \(-0.0865442\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 238.431 0.315316 0.157658 0.987494i \(-0.449606\pi\)
0.157658 + 0.987494i \(0.449606\pi\)
\(84\) 0 0
\(85\) 72.8437 0.0929531
\(86\) 0 0
\(87\) 195.836 0.241331
\(88\) 0 0
\(89\) 585.905i 0.697818i −0.937157 0.348909i \(-0.886552\pi\)
0.937157 0.348909i \(-0.113448\pi\)
\(90\) 0 0
\(91\) −233.910 + 805.367i −0.269455 + 0.927752i
\(92\) 0 0
\(93\) −566.694 −0.631865
\(94\) 0 0
\(95\) 92.7882i 0.100209i
\(96\) 0 0
\(97\) 601.047i 0.629146i −0.949233 0.314573i \(-0.898139\pi\)
0.949233 0.314573i \(-0.101861\pi\)
\(98\) 0 0
\(99\) 202.120i 0.205190i
\(100\) 0 0
\(101\) 56.3229i 0.0554885i 0.999615 + 0.0277442i \(0.00883240\pi\)
−0.999615 + 0.0277442i \(0.991168\pi\)
\(102\) 0 0
\(103\) 1434.44 1.37223 0.686113 0.727495i \(-0.259315\pi\)
0.686113 + 0.727495i \(0.259315\pi\)
\(104\) 0 0
\(105\) 33.2536 114.495i 0.0309068 0.106414i
\(106\) 0 0
\(107\) 2180.73i 1.97027i 0.171774 + 0.985136i \(0.445050\pi\)
−0.171774 + 0.985136i \(0.554950\pi\)
\(108\) 0 0
\(109\) −713.382 −0.626877 −0.313439 0.949608i \(-0.601481\pi\)
−0.313439 + 0.949608i \(0.601481\pi\)
\(110\) 0 0
\(111\) −495.352 −0.423575
\(112\) 0 0
\(113\) 1090.97 0.908227 0.454114 0.890944i \(-0.349956\pi\)
0.454114 + 0.890944i \(0.349956\pi\)
\(114\) 0 0
\(115\) −90.3824 −0.0732887
\(116\) 0 0
\(117\) 407.544i 0.322030i
\(118\) 0 0
\(119\) −175.350 + 603.742i −0.135078 + 0.465084i
\(120\) 0 0
\(121\) 826.648 0.621073
\(122\) 0 0
\(123\) 631.774i 0.463131i
\(124\) 0 0
\(125\) 526.585i 0.376793i
\(126\) 0 0
\(127\) 1748.59i 1.22175i 0.791726 + 0.610876i \(0.209183\pi\)
−0.791726 + 0.610876i \(0.790817\pi\)
\(128\) 0 0
\(129\) 968.207i 0.660821i
\(130\) 0 0
\(131\) 1109.66 0.740084 0.370042 0.929015i \(-0.379343\pi\)
0.370042 + 0.929015i \(0.379343\pi\)
\(132\) 0 0
\(133\) 769.046 + 223.360i 0.501389 + 0.145623i
\(134\) 0 0
\(135\) 57.9383i 0.0369373i
\(136\) 0 0
\(137\) 495.842 0.309216 0.154608 0.987976i \(-0.450589\pi\)
0.154608 + 0.987976i \(0.450589\pi\)
\(138\) 0 0
\(139\) 323.217 0.197230 0.0986148 0.995126i \(-0.468559\pi\)
0.0986148 + 0.995126i \(0.468559\pi\)
\(140\) 0 0
\(141\) −569.922 −0.340398
\(142\) 0 0
\(143\) −1016.95 −0.594696
\(144\) 0 0
\(145\) 140.079i 0.0802271i
\(146\) 0 0
\(147\) 868.903 + 551.224i 0.487523 + 0.309280i
\(148\) 0 0
\(149\) −2009.92 −1.10509 −0.552546 0.833482i \(-0.686344\pi\)
−0.552546 + 0.833482i \(0.686344\pi\)
\(150\) 0 0
\(151\) 357.813i 0.192837i −0.995341 0.0964185i \(-0.969261\pi\)
0.995341 0.0964185i \(-0.0307387\pi\)
\(152\) 0 0
\(153\) 305.515i 0.161434i
\(154\) 0 0
\(155\) 405.350i 0.210055i
\(156\) 0 0
\(157\) 2794.84i 1.42071i 0.703841 + 0.710357i \(0.251467\pi\)
−0.703841 + 0.710357i \(0.748533\pi\)
\(158\) 0 0
\(159\) −1535.18 −0.765710
\(160\) 0 0
\(161\) 217.569 749.106i 0.106502 0.366695i
\(162\) 0 0
\(163\) 3205.38i 1.54028i 0.637877 + 0.770138i \(0.279813\pi\)
−0.637877 + 0.770138i \(0.720187\pi\)
\(164\) 0 0
\(165\) 144.574 0.0682125
\(166\) 0 0
\(167\) 146.770 0.0680084 0.0340042 0.999422i \(-0.489174\pi\)
0.0340042 + 0.999422i \(0.489174\pi\)
\(168\) 0 0
\(169\) 146.475 0.0666706
\(170\) 0 0
\(171\) 389.165 0.174036
\(172\) 0 0
\(173\) 4019.32i 1.76638i −0.469019 0.883188i \(-0.655393\pi\)
0.469019 0.883188i \(-0.344607\pi\)
\(174\) 0 0
\(175\) 2141.27 + 621.906i 0.924940 + 0.268638i
\(176\) 0 0
\(177\) −53.9564 −0.0229131
\(178\) 0 0
\(179\) 737.398i 0.307909i 0.988078 + 0.153955i \(0.0492010\pi\)
−0.988078 + 0.153955i \(0.950799\pi\)
\(180\) 0 0
\(181\) 209.250i 0.0859305i 0.999077 + 0.0429653i \(0.0136805\pi\)
−0.999077 + 0.0429653i \(0.986320\pi\)
\(182\) 0 0
\(183\) 1417.74i 0.572692i
\(184\) 0 0
\(185\) 354.319i 0.140811i
\(186\) 0 0
\(187\) −762.355 −0.298122
\(188\) 0 0
\(189\) 480.203 + 139.469i 0.184813 + 0.0536768i
\(190\) 0 0
\(191\) 2950.47i 1.11774i 0.829254 + 0.558871i \(0.188765\pi\)
−0.829254 + 0.558871i \(0.811235\pi\)
\(192\) 0 0
\(193\) −945.388 −0.352593 −0.176297 0.984337i \(-0.556412\pi\)
−0.176297 + 0.984337i \(0.556412\pi\)
\(194\) 0 0
\(195\) 291.512 0.107054
\(196\) 0 0
\(197\) 2000.29 0.723426 0.361713 0.932290i \(-0.382192\pi\)
0.361713 + 0.932290i \(0.382192\pi\)
\(198\) 0 0
\(199\) 2976.34 1.06024 0.530118 0.847924i \(-0.322148\pi\)
0.530118 + 0.847924i \(0.322148\pi\)
\(200\) 0 0
\(201\) 1695.43i 0.594958i
\(202\) 0 0
\(203\) 1161.00 + 337.199i 0.401410 + 0.116585i
\(204\) 0 0
\(205\) 451.900 0.153961
\(206\) 0 0
\(207\) 379.074i 0.127283i
\(208\) 0 0
\(209\) 971.086i 0.321394i
\(210\) 0 0
\(211\) 1535.80i 0.501085i 0.968106 + 0.250543i \(0.0806091\pi\)
−0.968106 + 0.250543i \(0.919391\pi\)
\(212\) 0 0
\(213\) 1786.53i 0.574699i
\(214\) 0 0
\(215\) 692.547 0.219680
\(216\) 0 0
\(217\) −3359.61 975.760i −1.05099 0.305248i
\(218\) 0 0
\(219\) 1236.61i 0.381563i
\(220\) 0 0
\(221\) −1537.17 −0.467880
\(222\) 0 0
\(223\) −1762.27 −0.529195 −0.264598 0.964359i \(-0.585239\pi\)
−0.264598 + 0.964359i \(0.585239\pi\)
\(224\) 0 0
\(225\) 1083.56 0.321054
\(226\) 0 0
\(227\) −2216.72 −0.648145 −0.324072 0.946032i \(-0.605052\pi\)
−0.324072 + 0.946032i \(0.605052\pi\)
\(228\) 0 0
\(229\) 266.103i 0.0767887i 0.999263 + 0.0383943i \(0.0122243\pi\)
−0.999263 + 0.0383943i \(0.987776\pi\)
\(230\) 0 0
\(231\) −348.020 + 1198.26i −0.0991256 + 0.341296i
\(232\) 0 0
\(233\) −1336.76 −0.375854 −0.187927 0.982183i \(-0.560177\pi\)
−0.187927 + 0.982183i \(0.560177\pi\)
\(234\) 0 0
\(235\) 407.658i 0.113160i
\(236\) 0 0
\(237\) 1131.40i 0.310094i
\(238\) 0 0
\(239\) 5020.39i 1.35875i 0.733790 + 0.679377i \(0.237750\pi\)
−0.733790 + 0.679377i \(0.762250\pi\)
\(240\) 0 0
\(241\) 5225.18i 1.39661i −0.715799 0.698306i \(-0.753937\pi\)
0.715799 0.698306i \(-0.246063\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 394.284 621.516i 0.102816 0.162070i
\(246\) 0 0
\(247\) 1958.05i 0.504403i
\(248\) 0 0
\(249\) 715.293 0.182048
\(250\) 0 0
\(251\) 7419.77 1.86586 0.932932 0.360052i \(-0.117241\pi\)
0.932932 + 0.360052i \(0.117241\pi\)
\(252\) 0 0
\(253\) 945.908 0.235054
\(254\) 0 0
\(255\) 218.531 0.0536665
\(256\) 0 0
\(257\) 2328.28i 0.565112i −0.959251 0.282556i \(-0.908818\pi\)
0.959251 0.282556i \(-0.0911823\pi\)
\(258\) 0 0
\(259\) −2936.67 852.920i −0.704539 0.204625i
\(260\) 0 0
\(261\) 587.508 0.139333
\(262\) 0 0
\(263\) 5075.60i 1.19002i 0.803719 + 0.595009i \(0.202852\pi\)
−0.803719 + 0.595009i \(0.797148\pi\)
\(264\) 0 0
\(265\) 1098.10i 0.254549i
\(266\) 0 0
\(267\) 1757.72i 0.402886i
\(268\) 0 0
\(269\) 3183.37i 0.721536i −0.932655 0.360768i \(-0.882515\pi\)
0.932655 0.360768i \(-0.117485\pi\)
\(270\) 0 0
\(271\) 2852.11 0.639310 0.319655 0.947534i \(-0.396433\pi\)
0.319655 + 0.947534i \(0.396433\pi\)
\(272\) 0 0
\(273\) −701.729 + 2416.10i −0.155570 + 0.535638i
\(274\) 0 0
\(275\) 2703.81i 0.592894i
\(276\) 0 0
\(277\) 5373.67 1.16560 0.582802 0.812614i \(-0.301956\pi\)
0.582802 + 0.812614i \(0.301956\pi\)
\(278\) 0 0
\(279\) −1700.08 −0.364808
\(280\) 0 0
\(281\) −9082.79 −1.92823 −0.964117 0.265479i \(-0.914470\pi\)
−0.964117 + 0.265479i \(0.914470\pi\)
\(282\) 0 0
\(283\) −3175.47 −0.667003 −0.333501 0.942750i \(-0.608230\pi\)
−0.333501 + 0.942750i \(0.608230\pi\)
\(284\) 0 0
\(285\) 278.365i 0.0578558i
\(286\) 0 0
\(287\) −1087.82 + 3745.43i −0.223735 + 0.770334i
\(288\) 0 0
\(289\) 3760.66 0.765451
\(290\) 0 0
\(291\) 1803.14i 0.363237i
\(292\) 0 0
\(293\) 7132.20i 1.42207i −0.703154 0.711037i \(-0.748226\pi\)
0.703154 0.711037i \(-0.251774\pi\)
\(294\) 0 0
\(295\) 38.5944i 0.00761712i
\(296\) 0 0
\(297\) 606.360i 0.118467i
\(298\) 0 0
\(299\) 1907.28 0.368899
\(300\) 0 0
\(301\) −1667.10 + 5739.95i −0.319237 + 1.09915i
\(302\) 0 0
\(303\) 168.969i 0.0320363i
\(304\) 0 0
\(305\) 1014.10 0.190383
\(306\) 0 0
\(307\) −6900.58 −1.28286 −0.641428 0.767183i \(-0.721658\pi\)
−0.641428 + 0.767183i \(0.721658\pi\)
\(308\) 0 0
\(309\) 4303.31 0.792255
\(310\) 0 0
\(311\) 3871.19 0.705837 0.352918 0.935654i \(-0.385189\pi\)
0.352918 + 0.935654i \(0.385189\pi\)
\(312\) 0 0
\(313\) 1658.29i 0.299463i −0.988727 0.149732i \(-0.952159\pi\)
0.988727 0.149732i \(-0.0478410\pi\)
\(314\) 0 0
\(315\) 99.7608 343.484i 0.0178441 0.0614384i
\(316\) 0 0
\(317\) 987.422 0.174950 0.0874750 0.996167i \(-0.472120\pi\)
0.0874750 + 0.996167i \(0.472120\pi\)
\(318\) 0 0
\(319\) 1466.01i 0.257307i
\(320\) 0 0
\(321\) 6542.19i 1.13754i
\(322\) 0 0
\(323\) 1467.85i 0.252858i
\(324\) 0 0
\(325\) 5451.83i 0.930501i
\(326\) 0 0
\(327\) −2140.15 −0.361928
\(328\) 0 0
\(329\) −3378.75 981.317i −0.566189 0.164443i
\(330\) 0 0
\(331\) 2313.90i 0.384240i 0.981371 + 0.192120i \(0.0615362\pi\)
−0.981371 + 0.192120i \(0.938464\pi\)
\(332\) 0 0
\(333\) −1486.06 −0.244551
\(334\) 0 0
\(335\) 1212.72 0.197785
\(336\) 0 0
\(337\) −6416.44 −1.03717 −0.518584 0.855026i \(-0.673541\pi\)
−0.518584 + 0.855026i \(0.673541\pi\)
\(338\) 0 0
\(339\) 3272.90 0.524365
\(340\) 0 0
\(341\) 4242.23i 0.673695i
\(342\) 0 0
\(343\) 4202.12 + 4764.01i 0.661495 + 0.749949i
\(344\) 0 0
\(345\) −271.147 −0.0423133
\(346\) 0 0
\(347\) 3589.22i 0.555272i −0.960686 0.277636i \(-0.910449\pi\)
0.960686 0.277636i \(-0.0895508\pi\)
\(348\) 0 0
\(349\) 5875.58i 0.901183i 0.892730 + 0.450591i \(0.148787\pi\)
−0.892730 + 0.450591i \(0.851213\pi\)
\(350\) 0 0
\(351\) 1222.63i 0.185924i
\(352\) 0 0
\(353\) 5263.01i 0.793546i −0.917917 0.396773i \(-0.870130\pi\)
0.917917 0.396773i \(-0.129870\pi\)
\(354\) 0 0
\(355\) −1277.88 −0.191051
\(356\) 0 0
\(357\) −526.050 + 1811.23i −0.0779874 + 0.268516i
\(358\) 0 0
\(359\) 1668.09i 0.245233i 0.992454 + 0.122616i \(0.0391284\pi\)
−0.992454 + 0.122616i \(0.960872\pi\)
\(360\) 0 0
\(361\) −4989.26 −0.727403
\(362\) 0 0
\(363\) 2479.94 0.358577
\(364\) 0 0
\(365\) −884.531 −0.126845
\(366\) 0 0
\(367\) 9459.64 1.34547 0.672737 0.739882i \(-0.265119\pi\)
0.672737 + 0.739882i \(0.265119\pi\)
\(368\) 0 0
\(369\) 1895.32i 0.267389i
\(370\) 0 0
\(371\) −9101.23 2643.35i −1.27362 0.369908i
\(372\) 0 0
\(373\) 9759.34 1.35474 0.677372 0.735641i \(-0.263119\pi\)
0.677372 + 0.735641i \(0.263119\pi\)
\(374\) 0 0
\(375\) 1579.75i 0.217542i
\(376\) 0 0
\(377\) 2956.00i 0.403824i
\(378\) 0 0
\(379\) 6429.32i 0.871377i −0.900098 0.435688i \(-0.856505\pi\)
0.900098 0.435688i \(-0.143495\pi\)
\(380\) 0 0
\(381\) 5245.78i 0.705379i
\(382\) 0 0
\(383\) −5874.11 −0.783689 −0.391845 0.920031i \(-0.628163\pi\)
−0.391845 + 0.920031i \(0.628163\pi\)
\(384\) 0 0
\(385\) 857.098 + 248.934i 0.113459 + 0.0329529i
\(386\) 0 0
\(387\) 2904.62i 0.381525i
\(388\) 0 0
\(389\) 2826.87 0.368452 0.184226 0.982884i \(-0.441022\pi\)
0.184226 + 0.982884i \(0.441022\pi\)
\(390\) 0 0
\(391\) 1429.79 0.184930
\(392\) 0 0
\(393\) 3328.97 0.427288
\(394\) 0 0
\(395\) −809.276 −0.103086
\(396\) 0 0
\(397\) 4224.05i 0.534002i 0.963696 + 0.267001i \(0.0860328\pi\)
−0.963696 + 0.267001i \(0.913967\pi\)
\(398\) 0 0
\(399\) 2307.14 + 670.081i 0.289477 + 0.0840753i
\(400\) 0 0
\(401\) 3689.69 0.459487 0.229743 0.973251i \(-0.426211\pi\)
0.229743 + 0.973251i \(0.426211\pi\)
\(402\) 0 0
\(403\) 8553.82i 1.05731i
\(404\) 0 0
\(405\) 173.815i 0.0213258i
\(406\) 0 0
\(407\) 3708.17i 0.451615i
\(408\) 0 0
\(409\) 3417.91i 0.413214i 0.978424 + 0.206607i \(0.0662422\pi\)
−0.978424 + 0.206607i \(0.933758\pi\)
\(410\) 0 0
\(411\) 1487.52 0.178526
\(412\) 0 0
\(413\) −319.877 92.9047i −0.0381117 0.0110691i
\(414\) 0 0
\(415\) 511.640i 0.0605191i
\(416\) 0 0
\(417\) 969.651 0.113871
\(418\) 0 0
\(419\) 15173.0 1.76909 0.884547 0.466452i \(-0.154468\pi\)
0.884547 + 0.466452i \(0.154468\pi\)
\(420\) 0 0
\(421\) 3284.80 0.380264 0.190132 0.981759i \(-0.439108\pi\)
0.190132 + 0.981759i \(0.439108\pi\)
\(422\) 0 0
\(423\) −1709.77 −0.196529
\(424\) 0 0
\(425\) 4086.95i 0.466462i
\(426\) 0 0
\(427\) −2441.14 + 8405.01i −0.276663 + 0.952569i
\(428\) 0 0
\(429\) −3050.85 −0.343348
\(430\) 0 0
\(431\) 15962.8i 1.78399i −0.452044 0.891995i \(-0.649305\pi\)
0.452044 0.891995i \(-0.350695\pi\)
\(432\) 0 0
\(433\) 15565.2i 1.72752i 0.503901 + 0.863761i \(0.331897\pi\)
−0.503901 + 0.863761i \(0.668103\pi\)
\(434\) 0 0
\(435\) 420.237i 0.0463191i
\(436\) 0 0
\(437\) 1821.26i 0.199366i
\(438\) 0 0
\(439\) −5475.98 −0.595340 −0.297670 0.954669i \(-0.596209\pi\)
−0.297670 + 0.954669i \(0.596209\pi\)
\(440\) 0 0
\(441\) 2606.71 + 1653.67i 0.281472 + 0.178563i
\(442\) 0 0
\(443\) 7171.60i 0.769149i −0.923094 0.384574i \(-0.874348\pi\)
0.923094 0.384574i \(-0.125652\pi\)
\(444\) 0 0
\(445\) −1257.27 −0.133934
\(446\) 0 0
\(447\) −6029.75 −0.638026
\(448\) 0 0
\(449\) 16309.6 1.71425 0.857123 0.515112i \(-0.172250\pi\)
0.857123 + 0.515112i \(0.172250\pi\)
\(450\) 0 0
\(451\) −4729.42 −0.493791
\(452\) 0 0
\(453\) 1073.44i 0.111334i
\(454\) 0 0
\(455\) 1728.21 + 501.938i 0.178065 + 0.0517169i
\(456\) 0 0
\(457\) 611.848 0.0626281 0.0313140 0.999510i \(-0.490031\pi\)
0.0313140 + 0.999510i \(0.490031\pi\)
\(458\) 0 0
\(459\) 916.545i 0.0932041i
\(460\) 0 0
\(461\) 9345.40i 0.944162i −0.881555 0.472081i \(-0.843503\pi\)
0.881555 0.472081i \(-0.156497\pi\)
\(462\) 0 0
\(463\) 4461.44i 0.447820i −0.974610 0.223910i \(-0.928118\pi\)
0.974610 0.223910i \(-0.0718822\pi\)
\(464\) 0 0
\(465\) 1216.05i 0.121275i
\(466\) 0 0
\(467\) −4705.53 −0.466266 −0.233133 0.972445i \(-0.574898\pi\)
−0.233133 + 0.972445i \(0.574898\pi\)
\(468\) 0 0
\(469\) −2919.27 + 10051.3i −0.287419 + 0.989604i
\(470\) 0 0
\(471\) 8384.51i 0.820250i
\(472\) 0 0
\(473\) −7247.93 −0.704567
\(474\) 0 0
\(475\) 5205.95 0.502875
\(476\) 0 0
\(477\) −4605.55 −0.442083
\(478\) 0 0
\(479\) −6801.17 −0.648754 −0.324377 0.945928i \(-0.605155\pi\)
−0.324377 + 0.945928i \(0.605155\pi\)
\(480\) 0 0
\(481\) 7476.97i 0.708774i
\(482\) 0 0
\(483\) 652.707 2247.32i 0.0614891 0.211711i
\(484\) 0 0
\(485\) −1289.77 −0.120753
\(486\) 0 0
\(487\) 9205.87i 0.856587i −0.903640 0.428293i \(-0.859115\pi\)
0.903640 0.428293i \(-0.140885\pi\)
\(488\) 0 0
\(489\) 9616.15i 0.889279i
\(490\) 0 0
\(491\) 9457.66i 0.869283i −0.900603 0.434642i \(-0.856875\pi\)
0.900603 0.434642i \(-0.143125\pi\)
\(492\) 0 0
\(493\) 2215.96i 0.202438i
\(494\) 0 0
\(495\) 433.722 0.0393825
\(496\) 0 0
\(497\) 3076.13 10591.3i 0.277632 0.955907i
\(498\) 0 0
\(499\) 2381.97i 0.213691i 0.994276 + 0.106845i \(0.0340750\pi\)
−0.994276 + 0.106845i \(0.965925\pi\)
\(500\) 0 0
\(501\) 440.310 0.0392647
\(502\) 0 0
\(503\) −20001.0 −1.77297 −0.886483 0.462760i \(-0.846859\pi\)
−0.886483 + 0.462760i \(0.846859\pi\)
\(504\) 0 0
\(505\) 120.861 0.0106500
\(506\) 0 0
\(507\) 439.426 0.0384923
\(508\) 0 0
\(509\) 17010.3i 1.48127i −0.671907 0.740636i \(-0.734524\pi\)
0.671907 0.740636i \(-0.265476\pi\)
\(510\) 0 0
\(511\) 2129.25 7331.16i 0.184330 0.634660i
\(512\) 0 0
\(513\) 1167.49 0.100480
\(514\) 0 0
\(515\) 3078.11i 0.263374i
\(516\) 0 0
\(517\) 4266.39i 0.362932i
\(518\) 0 0
\(519\) 12058.0i 1.01982i
\(520\) 0 0
\(521\) 19855.6i 1.66965i −0.550514 0.834826i \(-0.685568\pi\)
0.550514 0.834826i \(-0.314432\pi\)
\(522\) 0 0
\(523\) 6014.00 0.502818 0.251409 0.967881i \(-0.419106\pi\)
0.251409 + 0.967881i \(0.419106\pi\)
\(524\) 0 0
\(525\) 6423.80 + 1865.72i 0.534015 + 0.155098i
\(526\) 0 0
\(527\) 6412.36i 0.530032i
\(528\) 0 0
\(529\) 10393.0 0.854192
\(530\) 0 0
\(531\) −161.869 −0.0132289
\(532\) 0 0
\(533\) −9536.15 −0.774965
\(534\) 0 0
\(535\) 4679.55 0.378158
\(536\) 0 0
\(537\) 2212.20i 0.177771i
\(538\) 0 0
\(539\) −4126.42 + 6504.55i −0.329754 + 0.519797i
\(540\) 0 0
\(541\) −18092.6 −1.43782 −0.718911 0.695102i \(-0.755359\pi\)
−0.718911 + 0.695102i \(0.755359\pi\)
\(542\) 0 0
\(543\) 627.750i 0.0496120i
\(544\) 0 0
\(545\) 1530.82i 0.120318i
\(546\) 0 0
\(547\) 2206.48i 0.172472i −0.996275 0.0862360i \(-0.972516\pi\)
0.996275 0.0862360i \(-0.0274839\pi\)
\(548\) 0 0
\(549\) 4253.23i 0.330644i
\(550\) 0 0
\(551\) 2822.68 0.218240
\(552\) 0 0
\(553\) 1948.10 6707.43i 0.149804 0.515785i
\(554\) 0 0
\(555\) 1062.96i 0.0812974i
\(556\) 0 0
\(557\) −18990.1 −1.44459 −0.722294 0.691586i \(-0.756912\pi\)
−0.722294 + 0.691586i \(0.756912\pi\)
\(558\) 0 0
\(559\) −14614.4 −1.10576
\(560\) 0 0
\(561\) −2287.06 −0.172121
\(562\) 0 0
\(563\) −7744.41 −0.579730 −0.289865 0.957068i \(-0.593610\pi\)
−0.289865 + 0.957068i \(0.593610\pi\)
\(564\) 0 0
\(565\) 2341.07i 0.174318i
\(566\) 0 0
\(567\) 1440.61 + 418.408i 0.106702 + 0.0309903i
\(568\) 0 0
\(569\) −14284.6 −1.05245 −0.526224 0.850346i \(-0.676393\pi\)
−0.526224 + 0.850346i \(0.676393\pi\)
\(570\) 0 0
\(571\) 1831.70i 0.134246i 0.997745 + 0.0671229i \(0.0213820\pi\)
−0.997745 + 0.0671229i \(0.978618\pi\)
\(572\) 0 0
\(573\) 8851.42i 0.645329i
\(574\) 0 0
\(575\) 5070.97i 0.367781i
\(576\) 0 0
\(577\) 11057.8i 0.797822i −0.916990 0.398911i \(-0.869388\pi\)
0.916990 0.398911i \(-0.130612\pi\)
\(578\) 0 0
\(579\) −2836.17 −0.203570
\(580\) 0 0
\(581\) 4240.57 + 1231.62i 0.302803 + 0.0879456i
\(582\) 0 0
\(583\) 11492.3i 0.816400i
\(584\) 0 0
\(585\) 874.535 0.0618078
\(586\) 0 0
\(587\) −6826.40 −0.479992 −0.239996 0.970774i \(-0.577146\pi\)
−0.239996 + 0.970774i \(0.577146\pi\)
\(588\) 0 0
\(589\) −8168.05 −0.571407
\(590\) 0 0
\(591\) 6000.87 0.417670
\(592\) 0 0
\(593\) 27637.4i 1.91388i −0.290283 0.956941i \(-0.593750\pi\)
0.290283 0.956941i \(-0.406250\pi\)
\(594\) 0 0
\(595\) 1295.55 + 376.277i 0.0892644 + 0.0259258i
\(596\) 0 0
\(597\) 8929.02 0.612128
\(598\) 0 0
\(599\) 21755.3i 1.48397i −0.670415 0.741986i \(-0.733884\pi\)
0.670415 0.741986i \(-0.266116\pi\)
\(600\) 0 0
\(601\) 18589.8i 1.26172i 0.775896 + 0.630861i \(0.217298\pi\)
−0.775896 + 0.630861i \(0.782702\pi\)
\(602\) 0 0
\(603\) 5086.30i 0.343499i
\(604\) 0 0
\(605\) 1773.87i 0.119204i
\(606\) 0 0
\(607\) −765.817 −0.0512085 −0.0256042 0.999672i \(-0.508151\pi\)
−0.0256042 + 0.999672i \(0.508151\pi\)
\(608\) 0 0
\(609\) 3483.00 + 1011.60i 0.231754 + 0.0673104i
\(610\) 0 0
\(611\) 8602.54i 0.569593i
\(612\) 0 0
\(613\) 7598.18 0.500633 0.250316 0.968164i \(-0.419465\pi\)
0.250316 + 0.968164i \(0.419465\pi\)
\(614\) 0 0
\(615\) 1355.70 0.0888896
\(616\) 0 0
\(617\) −562.905 −0.0367289 −0.0183644 0.999831i \(-0.505846\pi\)
−0.0183644 + 0.999831i \(0.505846\pi\)
\(618\) 0 0
\(619\) −22585.7 −1.46655 −0.733276 0.679931i \(-0.762010\pi\)
−0.733276 + 0.679931i \(0.762010\pi\)
\(620\) 0 0
\(621\) 1137.22i 0.0734866i
\(622\) 0 0
\(623\) 3026.51 10420.5i 0.194630 0.670126i
\(624\) 0 0
\(625\) 13919.4 0.890844
\(626\) 0 0
\(627\) 2913.26i 0.185557i
\(628\) 0 0
\(629\) 5605.10i 0.355310i
\(630\) 0 0
\(631\) 19524.7i 1.23180i −0.787825 0.615899i \(-0.788793\pi\)
0.787825 0.615899i \(-0.211207\pi\)
\(632\) 0 0
\(633\) 4607.41i 0.289302i
\(634\) 0 0
\(635\) 3752.24 0.234493
\(636\) 0 0
\(637\) −8320.31 + 13115.4i −0.517524 + 0.815781i
\(638\) 0 0
\(639\) 5359.59i 0.331803i
\(640\) 0 0
\(641\) −848.601 −0.0522898 −0.0261449 0.999658i \(-0.508323\pi\)
−0.0261449 + 0.999658i \(0.508323\pi\)
\(642\) 0 0
\(643\) 31852.2 1.95354 0.976772 0.214279i \(-0.0687403\pi\)
0.976772 + 0.214279i \(0.0687403\pi\)
\(644\) 0 0
\(645\) 2077.64 0.126833
\(646\) 0 0
\(647\) 18585.6 1.12933 0.564664 0.825321i \(-0.309006\pi\)
0.564664 + 0.825321i \(0.309006\pi\)
\(648\) 0 0
\(649\) 403.914i 0.0244299i
\(650\) 0 0
\(651\) −10078.8 2927.28i −0.606791 0.176235i
\(652\) 0 0
\(653\) −18107.5 −1.08514 −0.542572 0.840009i \(-0.682550\pi\)
−0.542572 + 0.840009i \(0.682550\pi\)
\(654\) 0 0
\(655\) 2381.17i 0.142046i
\(656\) 0 0
\(657\) 3709.83i 0.220296i
\(658\) 0 0
\(659\) 11989.2i 0.708699i 0.935113 + 0.354350i \(0.115298\pi\)
−0.935113 + 0.354350i \(0.884702\pi\)
\(660\) 0 0
\(661\) 10534.7i 0.619896i 0.950754 + 0.309948i \(0.100312\pi\)
−0.950754 + 0.309948i \(0.899688\pi\)
\(662\) 0 0
\(663\) −4611.52 −0.270131
\(664\) 0 0
\(665\) 479.301 1650.27i 0.0279496 0.0962325i
\(666\) 0 0
\(667\) 2749.50i 0.159612i
\(668\) 0 0
\(669\) −5286.82 −0.305531
\(670\) 0 0
\(671\) −10613.1 −0.610604
\(672\) 0 0
\(673\) 18223.8 1.04380 0.521900 0.853007i \(-0.325224\pi\)
0.521900 + 0.853007i \(0.325224\pi\)
\(674\) 0 0
\(675\) 3250.67 0.185361
\(676\) 0 0
\(677\) 30533.9i 1.73340i −0.498830 0.866700i \(-0.666237\pi\)
0.498830 0.866700i \(-0.333763\pi\)
\(678\) 0 0
\(679\) 3104.73 10689.8i 0.175477 0.604179i
\(680\) 0 0
\(681\) −6650.16 −0.374207
\(682\) 0 0
\(683\) 16410.7i 0.919381i −0.888079 0.459690i \(-0.847960\pi\)
0.888079 0.459690i \(-0.152040\pi\)
\(684\) 0 0
\(685\) 1064.01i 0.0593484i
\(686\) 0 0
\(687\) 798.310i 0.0443340i
\(688\) 0 0
\(689\) 23172.4i 1.28128i
\(690\) 0 0
\(691\) 3517.34 0.193641 0.0968205 0.995302i \(-0.469133\pi\)
0.0968205 + 0.995302i \(0.469133\pi\)
\(692\) 0 0
\(693\) −1044.06 + 3594.77i −0.0572302 + 0.197048i
\(694\) 0 0
\(695\) 693.579i 0.0378546i
\(696\) 0 0
\(697\) −7148.76 −0.388492
\(698\) 0 0
\(699\) −4010.28 −0.216999
\(700\) 0 0
\(701\) −7231.67 −0.389638 −0.194819 0.980839i \(-0.562412\pi\)
−0.194819 + 0.980839i \(0.562412\pi\)
\(702\) 0 0
\(703\) −7139.77 −0.383046
\(704\) 0 0
\(705\) 1222.97i 0.0653332i
\(706\) 0 0
\(707\) −290.938 + 1001.72i −0.0154764 + 0.0532865i
\(708\) 0 0
\(709\) 12722.9 0.673933 0.336967 0.941517i \(-0.390599\pi\)
0.336967 + 0.941517i \(0.390599\pi\)
\(710\) 0 0
\(711\) 3394.20i 0.179033i
\(712\) 0 0
\(713\) 7956.27i 0.417903i
\(714\) 0 0
\(715\) 2182.23i 0.114141i
\(716\) 0 0
\(717\) 15061.2i 0.784476i
\(718\) 0 0
\(719\) 1736.00 0.0900442 0.0450221 0.998986i \(-0.485664\pi\)
0.0450221 + 0.998986i \(0.485664\pi\)
\(720\) 0 0
\(721\) 25511.9 + 7409.64i 1.31777 + 0.382731i
\(722\) 0 0
\(723\) 15675.6i 0.806335i
\(724\) 0 0
\(725\) 7859.24 0.402600
\(726\) 0 0
\(727\) −28120.7 −1.43458 −0.717290 0.696774i \(-0.754618\pi\)
−0.717290 + 0.696774i \(0.754618\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −10955.6 −0.554321
\(732\) 0 0
\(733\) 30570.1i 1.54042i −0.637788 0.770212i \(-0.720150\pi\)
0.637788 0.770212i \(-0.279850\pi\)
\(734\) 0 0
\(735\) 1182.85 1864.55i 0.0593607 0.0935713i
\(736\) 0 0
\(737\) −12691.9 −0.634344
\(738\) 0 0
\(739\) 30930.9i 1.53966i −0.638248 0.769831i \(-0.720340\pi\)
0.638248 0.769831i \(-0.279660\pi\)
\(740\) 0 0
\(741\) 5874.15i 0.291217i
\(742\) 0 0
\(743\) 3045.97i 0.150398i −0.997169 0.0751991i \(-0.976041\pi\)
0.997169 0.0751991i \(-0.0239592\pi\)
\(744\) 0 0
\(745\) 4313.01i 0.212102i
\(746\) 0 0
\(747\) 2145.88 0.105105
\(748\) 0 0
\(749\) −11264.6 + 38785.0i −0.549534 + 1.89209i
\(750\) 0 0
\(751\) 6453.80i 0.313585i 0.987632 + 0.156793i \(0.0501155\pi\)
−0.987632 + 0.156793i \(0.949885\pi\)
\(752\) 0 0
\(753\) 22259.3 1.07726
\(754\) 0 0
\(755\) −767.817 −0.0370116
\(756\) 0 0
\(757\) 12161.0 0.583882 0.291941 0.956436i \(-0.405699\pi\)
0.291941 + 0.956436i \(0.405699\pi\)
\(758\) 0 0
\(759\) 2837.72 0.135709
\(760\) 0 0
\(761\) 8701.33i 0.414485i −0.978290 0.207243i \(-0.933551\pi\)
0.978290 0.207243i \(-0.0664489\pi\)
\(762\) 0 0
\(763\) −12687.7 3685.00i −0.602000 0.174844i
\(764\) 0 0
\(765\) 655.594 0.0309844
\(766\) 0 0
\(767\) 814.432i 0.0383408i
\(768\) 0 0
\(769\) 32329.2i 1.51602i 0.652243 + 0.758010i \(0.273828\pi\)
−0.652243 + 0.758010i \(0.726172\pi\)
\(770\) 0 0
\(771\) 6984.83i 0.326268i
\(772\) 0 0
\(773\) 10932.6i 0.508691i −0.967113 0.254345i \(-0.918140\pi\)
0.967113 0.254345i \(-0.0818600\pi\)
\(774\) 0 0
\(775\) −22742.4 −1.05411
\(776\) 0 0
\(777\) −8810.00 2558.76i −0.406766 0.118140i
\(778\) 0 0
\(779\) 9106.08i 0.418818i
\(780\) 0 0
\(781\) 13373.8 0.612744
\(782\) 0 0
\(783\) 1762.52 0.0804438
\(784\) 0 0
\(785\) 5997.34 0.272680
\(786\) 0 0
\(787\) −1491.34 −0.0675484 −0.0337742 0.999429i \(-0.510753\pi\)
−0.0337742 + 0.999429i \(0.510753\pi\)
\(788\) 0 0
\(789\) 15226.8i 0.687058i
\(790\) 0 0
\(791\) 19403.2 + 5635.44i 0.872185 + 0.253316i
\(792\) 0 0
\(793\) −21399.8 −0.958295
\(794\) 0 0
\(795\) 3294.29i 0.146964i
\(796\) 0 0
\(797\) 15439.9i 0.686211i −0.939297 0.343106i \(-0.888521\pi\)
0.939297 0.343106i \(-0.111479\pi\)
\(798\) 0 0
\(799\) 6448.88i 0.285538i
\(800\) 0 0
\(801\) 5273.15i 0.232606i
\(802\) 0 0
\(803\) 9257.17 0.406822
\(804\) 0 0
\(805\) −1607.48 466.874i −0.0703803 0.0204412i
\(806\) 0 0
\(807\) 9550.10i 0.416579i
\(808\) 0 0
\(809\) 3257.49 0.141567 0.0707833 0.997492i \(-0.477450\pi\)
0.0707833 + 0.997492i \(0.477450\pi\)
\(810\) 0 0
\(811\) 12540.0 0.542959 0.271479 0.962444i \(-0.412487\pi\)
0.271479 + 0.962444i \(0.412487\pi\)
\(812\) 0 0
\(813\) 8556.32 0.369106
\(814\) 0 0
\(815\) 6878.31 0.295628
\(816\) 0 0
\(817\) 13955.3i 0.597592i
\(818\) 0 0
\(819\) −2105.19 + 7248.30i −0.0898183 + 0.309251i
\(820\) 0 0
\(821\) 34789.9 1.47890 0.739450 0.673212i \(-0.235086\pi\)
0.739450 + 0.673212i \(0.235086\pi\)
\(822\) 0 0
\(823\) 31049.4i 1.31508i 0.753418 + 0.657542i \(0.228404\pi\)
−0.753418 + 0.657542i \(0.771596\pi\)
\(824\) 0 0
\(825\) 8111.43i 0.342308i
\(826\) 0 0
\(827\) 7640.76i 0.321276i 0.987013 + 0.160638i \(0.0513551\pi\)
−0.987013 + 0.160638i \(0.948645\pi\)
\(828\) 0 0
\(829\) 10612.1i 0.444598i 0.974979 + 0.222299i \(0.0713561\pi\)
−0.974979 + 0.222299i \(0.928644\pi\)
\(830\) 0 0
\(831\) 16121.0 0.672962
\(832\) 0 0
\(833\) −6237.31 + 9831.97i −0.259436 + 0.408953i
\(834\) 0 0
\(835\) 314.948i 0.0130530i
\(836\) 0 0
\(837\) −5100.25 −0.210622
\(838\) 0 0
\(839\) 4130.16 0.169951 0.0849755 0.996383i \(-0.472919\pi\)
0.0849755 + 0.996383i \(0.472919\pi\)
\(840\) 0 0
\(841\) −20127.7 −0.825278
\(842\) 0 0
\(843\) −27248.4 −1.11327
\(844\) 0 0
\(845\) 314.316i 0.0127962i
\(846\) 0 0
\(847\) 14702.2 + 4270.08i 0.596426 + 0.173225i
\(848\) 0 0
\(849\) −9526.40 −0.385094
\(850\) 0 0
\(851\) 6954.64i 0.280143i
\(852\) 0 0
\(853\) 26371.3i 1.05854i −0.848453 0.529270i \(-0.822466\pi\)
0.848453 0.529270i \(-0.177534\pi\)
\(854\) 0 0
\(855\) 835.094i 0.0334031i
\(856\) 0 0
\(857\) 44766.4i 1.78435i −0.451685 0.892177i \(-0.649177\pi\)
0.451685 0.892177i \(-0.350823\pi\)
\(858\) 0 0
\(859\) 8343.09 0.331388 0.165694 0.986177i \(-0.447014\pi\)
0.165694 + 0.986177i \(0.447014\pi\)
\(860\) 0 0
\(861\) −3263.45 + 11236.3i −0.129173 + 0.444753i
\(862\) 0 0
\(863\) 28104.1i 1.10855i 0.832335 + 0.554273i \(0.187004\pi\)
−0.832335 + 0.554273i \(0.812996\pi\)
\(864\) 0 0
\(865\) −8624.90 −0.339024
\(866\) 0 0
\(867\) 11282.0 0.441933
\(868\) 0 0
\(869\) 8469.58 0.330622
\(870\) 0 0
\(871\) −25591.3 −0.995553
\(872\) 0 0
\(873\) 5409.43i 0.209715i
\(874\) 0 0
\(875\) 2720.09 9365.47i 0.105092 0.361841i
\(876\) 0 0
\(877\) −18277.6 −0.703753 −0.351876 0.936047i \(-0.614456\pi\)
−0.351876 + 0.936047i \(0.614456\pi\)
\(878\) 0 0
\(879\) 21396.6i 0.821035i
\(880\) 0 0
\(881\) 23503.1i 0.898796i 0.893332 + 0.449398i \(0.148362\pi\)
−0.893332 + 0.449398i \(0.851638\pi\)
\(882\) 0 0
\(883\) 14588.2i 0.555982i −0.960584 0.277991i \(-0.910332\pi\)
0.960584 0.277991i \(-0.0896685\pi\)
\(884\) 0 0
\(885\) 115.783i 0.00439775i
\(886\) 0 0
\(887\) −9514.11 −0.360149 −0.180075 0.983653i \(-0.557634\pi\)
−0.180075 + 0.983653i \(0.557634\pi\)
\(888\) 0 0
\(889\) −9032.42 + 31099.3i −0.340762 + 1.17327i
\(890\) 0 0
\(891\) 1819.08i 0.0683967i
\(892\) 0 0
\(893\) −8214.57 −0.307828
\(894\) 0 0
\(895\) 1582.36 0.0590976
\(896\) 0 0
\(897\) 5721.84 0.212984
\(898\) 0 0
\(899\) −12331.0 −0.457467
\(900\) 0 0
\(901\) 17371.2i 0.642306i
\(902\) 0 0
\(903\) −5001.31 + 17219.9i −0.184311 + 0.634597i
\(904\) 0 0
\(905\) 449.022 0.0164928
\(906\) 0 0
\(907\) 36132.0i 1.32276i −0.750051 0.661381i \(-0.769971\pi\)
0.750051 0.661381i \(-0.230029\pi\)
\(908\) 0 0
\(909\) 506.906i 0.0184962i
\(910\) 0 0
\(911\) 9475.15i 0.344595i 0.985045 + 0.172297i \(0.0551190\pi\)
−0.985045 + 0.172297i \(0.944881\pi\)
\(912\) 0 0
\(913\) 5354.63i 0.194099i
\(914\) 0 0
\(915\) 3042.29 0.109918
\(916\) 0 0
\(917\) 19735.6 + 5731.96i 0.710715 + 0.206419i
\(918\) 0 0
\(919\) 6230.48i 0.223639i 0.993729 + 0.111820i \(0.0356679\pi\)
−0.993729 + 0.111820i \(0.964332\pi\)
\(920\) 0 0
\(921\) −20701.7 −0.740657
\(922\) 0 0
\(923\) 26966.3 0.961654
\(924\) 0 0
\(925\) −19879.4 −0.706626
\(926\) 0 0
\(927\) 12909.9 0.457409
\(928\) 0 0
\(929\) 47363.5i 1.67271i −0.548189 0.836354i \(-0.684683\pi\)
0.548189 0.836354i \(-0.315317\pi\)
\(930\) 0 0
\(931\) 12523.9 + 7945.07i 0.440876 + 0.279688i
\(932\) 0 0
\(933\) 11613.6 0.407515
\(934\) 0 0
\(935\) 1635.91i 0.0572192i
\(936\) 0 0
\(937\) 42614.5i 1.48576i 0.669427 + 0.742878i \(0.266540\pi\)
−0.669427 + 0.742878i \(0.733460\pi\)
\(938\) 0 0
\(939\) 4974.87i 0.172895i
\(940\) 0 0
\(941\) 1348.67i 0.0467219i −0.999727 0.0233609i \(-0.992563\pi\)
0.999727 0.0233609i \(-0.00743669\pi\)
\(942\) 0 0
\(943\) 8869.98 0.306306
\(944\) 0 0
\(945\) 299.282 1030.45i 0.0103023 0.0354715i
\(946\) 0 0
\(947\) 9182.47i 0.315090i −0.987512 0.157545i \(-0.949642\pi\)
0.987512 0.157545i \(-0.0503579\pi\)
\(948\) 0 0
\(949\) 18665.7 0.638476
\(950\) 0 0
\(951\) 2962.27 0.101007
\(952\) 0 0
\(953\) −56268.6 −1.91261 −0.956306 0.292367i \(-0.905557\pi\)
−0.956306 + 0.292367i \(0.905557\pi\)
\(954\) 0 0
\(955\) 6331.31 0.214530
\(956\) 0 0
\(957\) 4398.04i 0.148556i
\(958\) 0 0
\(959\) 8818.70 + 2561.29i 0.296945 + 0.0862443i
\(960\) 0 0
\(961\) 5891.51 0.197761
\(962\) 0 0
\(963\) 19626.6i 0.656758i
\(964\) 0 0
\(965\) 2028.67i 0.0676739i
\(966\) 0 0
\(967\) 57083.0i 1.89831i 0.314812 + 0.949154i \(0.398059\pi\)
−0.314812 + 0.949154i \(0.601941\pi\)
\(968\) 0 0
\(969\) 4403.54i 0.145988i
\(970\) 0 0
\(971\) −10206.5 −0.337325 −0.168663 0.985674i \(-0.553945\pi\)
−0.168663 + 0.985674i \(0.553945\pi\)
\(972\) 0 0
\(973\) 5748.51 + 1669.59i 0.189403 + 0.0550098i
\(974\) 0 0
\(975\) 16355.5i 0.537225i
\(976\) 0 0
\(977\) 29142.9 0.954313 0.477156 0.878818i \(-0.341668\pi\)
0.477156 + 0.878818i \(0.341668\pi\)
\(978\) 0 0
\(979\) 13158.1 0.429556
\(980\) 0 0
\(981\) −6420.44 −0.208959
\(982\) 0 0
\(983\) 48767.5 1.58234 0.791171 0.611595i \(-0.209472\pi\)
0.791171 + 0.611595i \(0.209472\pi\)
\(984\) 0 0
\(985\) 4292.35i 0.138848i
\(986\) 0 0
\(987\) −10136.2 2943.95i −0.326890 0.0949413i
\(988\) 0 0
\(989\) 13593.4 0.437053
\(990\) 0 0
\(991\) 37756.7i 1.21027i 0.796122 + 0.605137i \(0.206882\pi\)
−0.796122 + 0.605137i \(0.793118\pi\)
\(992\) 0 0
\(993\) 6941.69i 0.221841i
\(994\) 0 0
\(995\) 6386.81i 0.203493i
\(996\) 0 0
\(997\) 12731.7i 0.404429i −0.979341 0.202215i \(-0.935186\pi\)
0.979341 0.202215i \(-0.0648139\pi\)
\(998\) 0 0
\(999\) −4458.17 −0.141192
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 1344.4.b.j.895.12 24
4.3 odd 2 1344.4.b.i.895.12 24
7.6 odd 2 1344.4.b.i.895.13 24
8.3 odd 2 672.4.b.b.223.13 yes 24
8.5 even 2 672.4.b.a.223.13 yes 24
28.27 even 2 inner 1344.4.b.j.895.13 24
56.13 odd 2 672.4.b.b.223.12 yes 24
56.27 even 2 672.4.b.a.223.12 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
672.4.b.a.223.12 24 56.27 even 2
672.4.b.a.223.13 yes 24 8.5 even 2
672.4.b.b.223.12 yes 24 56.13 odd 2
672.4.b.b.223.13 yes 24 8.3 odd 2
1344.4.b.i.895.12 24 4.3 odd 2
1344.4.b.i.895.13 24 7.6 odd 2
1344.4.b.j.895.12 24 1.1 even 1 trivial
1344.4.b.j.895.13 24 28.27 even 2 inner