Properties

Label 2496.4.a.v.1.1
Level $2496$
Weight $4$
Character 2496.1
Self dual yes
Analytic conductor $147.269$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2496,4,Mod(1,2496)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2496, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2496.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2496 = 2^{6} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2496.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(147.268767374\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{113}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 312)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(5.81507\) of defining polynomial
Character \(\chi\) \(=\) 2496.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -13.6301 q^{5} +15.6301 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -13.6301 q^{5} +15.6301 q^{7} +9.00000 q^{9} -50.5206 q^{11} -13.0000 q^{13} +40.8904 q^{15} +2.00000 q^{17} +18.1507 q^{19} -46.8904 q^{21} +64.0000 q^{23} +60.7809 q^{25} -27.0000 q^{27} +103.781 q^{29} -34.4522 q^{31} +151.562 q^{33} -213.041 q^{35} +267.343 q^{37} +39.0000 q^{39} -147.411 q^{41} -166.959 q^{43} -122.671 q^{45} +325.781 q^{47} -98.6985 q^{49} -6.00000 q^{51} +111.041 q^{53} +688.603 q^{55} -54.4522 q^{57} +24.3015 q^{59} +640.987 q^{61} +140.671 q^{63} +177.192 q^{65} -382.589 q^{67} -192.000 q^{69} -510.247 q^{71} -291.781 q^{73} -182.343 q^{75} -789.644 q^{77} +794.685 q^{79} +81.0000 q^{81} +945.507 q^{83} -27.2603 q^{85} -311.343 q^{87} +317.329 q^{89} -203.192 q^{91} +103.357 q^{93} -247.397 q^{95} +1572.88 q^{97} -454.685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 6 q^{5} + 10 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 6 q^{5} + 10 q^{7} + 18 q^{9} - 16 q^{11} - 26 q^{13} + 18 q^{15} + 4 q^{17} - 70 q^{19} - 30 q^{21} + 128 q^{23} - 6 q^{25} - 54 q^{27} + 80 q^{29} + 250 q^{31} + 48 q^{33} - 256 q^{35} + 152 q^{37} + 78 q^{39} - 146 q^{41} - 504 q^{43} - 54 q^{45} + 524 q^{47} - 410 q^{49} - 12 q^{51} + 52 q^{53} + 952 q^{55} + 210 q^{57} - 164 q^{59} + 304 q^{61} + 90 q^{63} + 78 q^{65} - 914 q^{67} - 384 q^{69} - 456 q^{73} + 18 q^{75} - 984 q^{77} + 824 q^{79} + 162 q^{81} + 828 q^{83} - 12 q^{85} - 240 q^{87} + 826 q^{89} - 130 q^{91} - 750 q^{93} - 920 q^{95} + 552 q^{97} - 144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) −13.6301 −1.21912 −0.609559 0.792741i \(-0.708653\pi\)
−0.609559 + 0.792741i \(0.708653\pi\)
\(6\) 0 0
\(7\) 15.6301 0.843949 0.421974 0.906608i \(-0.361337\pi\)
0.421974 + 0.906608i \(0.361337\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −50.5206 −1.38478 −0.692388 0.721526i \(-0.743441\pi\)
−0.692388 + 0.721526i \(0.743441\pi\)
\(12\) 0 0
\(13\) −13.0000 −0.277350
\(14\) 0 0
\(15\) 40.8904 0.703858
\(16\) 0 0
\(17\) 2.00000 0.0285336 0.0142668 0.999898i \(-0.495459\pi\)
0.0142668 + 0.999898i \(0.495459\pi\)
\(18\) 0 0
\(19\) 18.1507 0.219161 0.109581 0.993978i \(-0.465049\pi\)
0.109581 + 0.993978i \(0.465049\pi\)
\(20\) 0 0
\(21\) −46.8904 −0.487254
\(22\) 0 0
\(23\) 64.0000 0.580214 0.290107 0.956994i \(-0.406309\pi\)
0.290107 + 0.956994i \(0.406309\pi\)
\(24\) 0 0
\(25\) 60.7809 0.486247
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 103.781 0.664539 0.332269 0.943185i \(-0.392186\pi\)
0.332269 + 0.943185i \(0.392186\pi\)
\(30\) 0 0
\(31\) −34.4522 −0.199606 −0.0998032 0.995007i \(-0.531821\pi\)
−0.0998032 + 0.995007i \(0.531821\pi\)
\(32\) 0 0
\(33\) 151.562 0.799500
\(34\) 0 0
\(35\) −213.041 −1.02887
\(36\) 0 0
\(37\) 267.343 1.18786 0.593930 0.804516i \(-0.297575\pi\)
0.593930 + 0.804516i \(0.297575\pi\)
\(38\) 0 0
\(39\) 39.0000 0.160128
\(40\) 0 0
\(41\) −147.411 −0.561506 −0.280753 0.959780i \(-0.590584\pi\)
−0.280753 + 0.959780i \(0.590584\pi\)
\(42\) 0 0
\(43\) −166.959 −0.592116 −0.296058 0.955170i \(-0.595672\pi\)
−0.296058 + 0.955170i \(0.595672\pi\)
\(44\) 0 0
\(45\) −122.671 −0.406372
\(46\) 0 0
\(47\) 325.781 1.01106 0.505532 0.862808i \(-0.331296\pi\)
0.505532 + 0.862808i \(0.331296\pi\)
\(48\) 0 0
\(49\) −98.6985 −0.287751
\(50\) 0 0
\(51\) −6.00000 −0.0164739
\(52\) 0 0
\(53\) 111.041 0.287786 0.143893 0.989593i \(-0.454038\pi\)
0.143893 + 0.989593i \(0.454038\pi\)
\(54\) 0 0
\(55\) 688.603 1.68820
\(56\) 0 0
\(57\) −54.4522 −0.126533
\(58\) 0 0
\(59\) 24.3015 0.0536234 0.0268117 0.999641i \(-0.491465\pi\)
0.0268117 + 0.999641i \(0.491465\pi\)
\(60\) 0 0
\(61\) 640.987 1.34541 0.672704 0.739911i \(-0.265133\pi\)
0.672704 + 0.739911i \(0.265133\pi\)
\(62\) 0 0
\(63\) 140.671 0.281316
\(64\) 0 0
\(65\) 177.192 0.338122
\(66\) 0 0
\(67\) −382.589 −0.697622 −0.348811 0.937193i \(-0.613415\pi\)
−0.348811 + 0.937193i \(0.613415\pi\)
\(68\) 0 0
\(69\) −192.000 −0.334987
\(70\) 0 0
\(71\) −510.247 −0.852890 −0.426445 0.904514i \(-0.640234\pi\)
−0.426445 + 0.904514i \(0.640234\pi\)
\(72\) 0 0
\(73\) −291.781 −0.467813 −0.233907 0.972259i \(-0.575151\pi\)
−0.233907 + 0.972259i \(0.575151\pi\)
\(74\) 0 0
\(75\) −182.343 −0.280735
\(76\) 0 0
\(77\) −789.644 −1.16868
\(78\) 0 0
\(79\) 794.685 1.13176 0.565880 0.824487i \(-0.308536\pi\)
0.565880 + 0.824487i \(0.308536\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 945.507 1.25040 0.625198 0.780466i \(-0.285018\pi\)
0.625198 + 0.780466i \(0.285018\pi\)
\(84\) 0 0
\(85\) −27.2603 −0.0347858
\(86\) 0 0
\(87\) −311.343 −0.383672
\(88\) 0 0
\(89\) 317.329 0.377941 0.188971 0.981983i \(-0.439485\pi\)
0.188971 + 0.981983i \(0.439485\pi\)
\(90\) 0 0
\(91\) −203.192 −0.234069
\(92\) 0 0
\(93\) 103.357 0.115243
\(94\) 0 0
\(95\) −247.397 −0.267183
\(96\) 0 0
\(97\) 1572.88 1.64641 0.823204 0.567746i \(-0.192184\pi\)
0.823204 + 0.567746i \(0.192184\pi\)
\(98\) 0 0
\(99\) −454.685 −0.461592
\(100\) 0 0
\(101\) −1820.44 −1.79347 −0.896735 0.442568i \(-0.854068\pi\)
−0.896735 + 0.442568i \(0.854068\pi\)
\(102\) 0 0
\(103\) −859.015 −0.821759 −0.410880 0.911690i \(-0.634778\pi\)
−0.410880 + 0.911690i \(0.634778\pi\)
\(104\) 0 0
\(105\) 639.123 0.594020
\(106\) 0 0
\(107\) −1126.74 −1.01800 −0.509001 0.860766i \(-0.669985\pi\)
−0.509001 + 0.860766i \(0.669985\pi\)
\(108\) 0 0
\(109\) 682.659 0.599879 0.299940 0.953958i \(-0.403033\pi\)
0.299940 + 0.953958i \(0.403033\pi\)
\(110\) 0 0
\(111\) −802.028 −0.685812
\(112\) 0 0
\(113\) 771.946 0.642642 0.321321 0.946970i \(-0.395873\pi\)
0.321321 + 0.946970i \(0.395873\pi\)
\(114\) 0 0
\(115\) −872.329 −0.707349
\(116\) 0 0
\(117\) −117.000 −0.0924500
\(118\) 0 0
\(119\) 31.2603 0.0240809
\(120\) 0 0
\(121\) 1221.33 0.917603
\(122\) 0 0
\(123\) 442.233 0.324185
\(124\) 0 0
\(125\) 875.316 0.626325
\(126\) 0 0
\(127\) 2182.99 1.52527 0.762633 0.646832i \(-0.223906\pi\)
0.762633 + 0.646832i \(0.223906\pi\)
\(128\) 0 0
\(129\) 500.877 0.341858
\(130\) 0 0
\(131\) 881.918 0.588195 0.294097 0.955775i \(-0.404981\pi\)
0.294097 + 0.955775i \(0.404981\pi\)
\(132\) 0 0
\(133\) 283.699 0.184961
\(134\) 0 0
\(135\) 368.014 0.234619
\(136\) 0 0
\(137\) −2764.40 −1.72393 −0.861965 0.506967i \(-0.830767\pi\)
−0.861965 + 0.506967i \(0.830767\pi\)
\(138\) 0 0
\(139\) 1453.40 0.886875 0.443438 0.896305i \(-0.353759\pi\)
0.443438 + 0.896305i \(0.353759\pi\)
\(140\) 0 0
\(141\) −977.343 −0.583738
\(142\) 0 0
\(143\) 656.768 0.384068
\(144\) 0 0
\(145\) −1414.55 −0.810151
\(146\) 0 0
\(147\) 296.096 0.166133
\(148\) 0 0
\(149\) −560.123 −0.307967 −0.153983 0.988073i \(-0.549210\pi\)
−0.153983 + 0.988073i \(0.549210\pi\)
\(150\) 0 0
\(151\) 2962.23 1.59645 0.798223 0.602363i \(-0.205774\pi\)
0.798223 + 0.602363i \(0.205774\pi\)
\(152\) 0 0
\(153\) 18.0000 0.00951120
\(154\) 0 0
\(155\) 469.588 0.243344
\(156\) 0 0
\(157\) −1837.92 −0.934280 −0.467140 0.884183i \(-0.654716\pi\)
−0.467140 + 0.884183i \(0.654716\pi\)
\(158\) 0 0
\(159\) −333.123 −0.166154
\(160\) 0 0
\(161\) 1000.33 0.489671
\(162\) 0 0
\(163\) 1793.52 0.861838 0.430919 0.902391i \(-0.358190\pi\)
0.430919 + 0.902391i \(0.358190\pi\)
\(164\) 0 0
\(165\) −2065.81 −0.974685
\(166\) 0 0
\(167\) −3919.37 −1.81611 −0.908054 0.418853i \(-0.862432\pi\)
−0.908054 + 0.418853i \(0.862432\pi\)
\(168\) 0 0
\(169\) 169.000 0.0769231
\(170\) 0 0
\(171\) 163.357 0.0730537
\(172\) 0 0
\(173\) −3864.60 −1.69838 −0.849192 0.528084i \(-0.822911\pi\)
−0.849192 + 0.528084i \(0.822911\pi\)
\(174\) 0 0
\(175\) 950.014 0.410367
\(176\) 0 0
\(177\) −72.9044 −0.0309595
\(178\) 0 0
\(179\) 2829.04 1.18130 0.590650 0.806928i \(-0.298871\pi\)
0.590650 + 0.806928i \(0.298871\pi\)
\(180\) 0 0
\(181\) −1126.74 −0.462707 −0.231353 0.972870i \(-0.574315\pi\)
−0.231353 + 0.972870i \(0.574315\pi\)
\(182\) 0 0
\(183\) −1922.96 −0.776772
\(184\) 0 0
\(185\) −3643.92 −1.44814
\(186\) 0 0
\(187\) −101.041 −0.0395126
\(188\) 0 0
\(189\) −422.014 −0.162418
\(190\) 0 0
\(191\) −662.136 −0.250840 −0.125420 0.992104i \(-0.540028\pi\)
−0.125420 + 0.992104i \(0.540028\pi\)
\(192\) 0 0
\(193\) −1290.49 −0.481305 −0.240652 0.970611i \(-0.577361\pi\)
−0.240652 + 0.970611i \(0.577361\pi\)
\(194\) 0 0
\(195\) −531.576 −0.195215
\(196\) 0 0
\(197\) 1320.01 0.477396 0.238698 0.971094i \(-0.423279\pi\)
0.238698 + 0.971094i \(0.423279\pi\)
\(198\) 0 0
\(199\) 2711.97 0.966064 0.483032 0.875603i \(-0.339535\pi\)
0.483032 + 0.875603i \(0.339535\pi\)
\(200\) 0 0
\(201\) 1147.77 0.402772
\(202\) 0 0
\(203\) 1622.11 0.560837
\(204\) 0 0
\(205\) 2009.23 0.684541
\(206\) 0 0
\(207\) 576.000 0.193405
\(208\) 0 0
\(209\) −916.985 −0.303489
\(210\) 0 0
\(211\) 1612.06 0.525964 0.262982 0.964801i \(-0.415294\pi\)
0.262982 + 0.964801i \(0.415294\pi\)
\(212\) 0 0
\(213\) 1530.74 0.492416
\(214\) 0 0
\(215\) 2275.67 0.721859
\(216\) 0 0
\(217\) −538.493 −0.168457
\(218\) 0 0
\(219\) 875.343 0.270092
\(220\) 0 0
\(221\) −26.0000 −0.00791380
\(222\) 0 0
\(223\) 5602.10 1.68226 0.841130 0.540832i \(-0.181891\pi\)
0.841130 + 0.540832i \(0.181891\pi\)
\(224\) 0 0
\(225\) 547.028 0.162082
\(226\) 0 0
\(227\) −1427.23 −0.417308 −0.208654 0.977990i \(-0.566908\pi\)
−0.208654 + 0.977990i \(0.566908\pi\)
\(228\) 0 0
\(229\) −4826.33 −1.39272 −0.696360 0.717692i \(-0.745198\pi\)
−0.696360 + 0.717692i \(0.745198\pi\)
\(230\) 0 0
\(231\) 2368.93 0.674737
\(232\) 0 0
\(233\) −6244.17 −1.75566 −0.877830 0.478972i \(-0.841010\pi\)
−0.877830 + 0.478972i \(0.841010\pi\)
\(234\) 0 0
\(235\) −4440.44 −1.23261
\(236\) 0 0
\(237\) −2384.06 −0.653422
\(238\) 0 0
\(239\) −6638.69 −1.79674 −0.898370 0.439239i \(-0.855248\pi\)
−0.898370 + 0.439239i \(0.855248\pi\)
\(240\) 0 0
\(241\) −4526.19 −1.20978 −0.604891 0.796308i \(-0.706784\pi\)
−0.604891 + 0.796308i \(0.706784\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 1345.28 0.350802
\(246\) 0 0
\(247\) −235.959 −0.0607844
\(248\) 0 0
\(249\) −2836.52 −0.721917
\(250\) 0 0
\(251\) 2070.36 0.520636 0.260318 0.965523i \(-0.416173\pi\)
0.260318 + 0.965523i \(0.416173\pi\)
\(252\) 0 0
\(253\) −3233.32 −0.803466
\(254\) 0 0
\(255\) 81.7809 0.0200836
\(256\) 0 0
\(257\) −6154.58 −1.49382 −0.746910 0.664925i \(-0.768464\pi\)
−0.746910 + 0.664925i \(0.768464\pi\)
\(258\) 0 0
\(259\) 4178.60 1.00249
\(260\) 0 0
\(261\) 934.028 0.221513
\(262\) 0 0
\(263\) −4111.51 −0.963979 −0.481990 0.876177i \(-0.660086\pi\)
−0.481990 + 0.876177i \(0.660086\pi\)
\(264\) 0 0
\(265\) −1513.51 −0.350845
\(266\) 0 0
\(267\) −951.986 −0.218204
\(268\) 0 0
\(269\) −822.851 −0.186506 −0.0932530 0.995642i \(-0.529727\pi\)
−0.0932530 + 0.995642i \(0.529727\pi\)
\(270\) 0 0
\(271\) −1685.41 −0.377791 −0.188896 0.981997i \(-0.560491\pi\)
−0.188896 + 0.981997i \(0.560491\pi\)
\(272\) 0 0
\(273\) 609.576 0.135140
\(274\) 0 0
\(275\) −3070.69 −0.673343
\(276\) 0 0
\(277\) 185.235 0.0401794 0.0200897 0.999798i \(-0.493605\pi\)
0.0200897 + 0.999798i \(0.493605\pi\)
\(278\) 0 0
\(279\) −310.070 −0.0665354
\(280\) 0 0
\(281\) −1008.34 −0.214066 −0.107033 0.994255i \(-0.534135\pi\)
−0.107033 + 0.994255i \(0.534135\pi\)
\(282\) 0 0
\(283\) −6664.63 −1.39990 −0.699949 0.714193i \(-0.746794\pi\)
−0.699949 + 0.714193i \(0.746794\pi\)
\(284\) 0 0
\(285\) 742.191 0.154258
\(286\) 0 0
\(287\) −2304.06 −0.473882
\(288\) 0 0
\(289\) −4909.00 −0.999186
\(290\) 0 0
\(291\) −4718.63 −0.950554
\(292\) 0 0
\(293\) 7954.02 1.58593 0.792967 0.609265i \(-0.208535\pi\)
0.792967 + 0.609265i \(0.208535\pi\)
\(294\) 0 0
\(295\) −331.232 −0.0653732
\(296\) 0 0
\(297\) 1364.06 0.266500
\(298\) 0 0
\(299\) −832.000 −0.160922
\(300\) 0 0
\(301\) −2609.59 −0.499715
\(302\) 0 0
\(303\) 5461.32 1.03546
\(304\) 0 0
\(305\) −8736.74 −1.64021
\(306\) 0 0
\(307\) 8831.69 1.64186 0.820930 0.571029i \(-0.193456\pi\)
0.820930 + 0.571029i \(0.193456\pi\)
\(308\) 0 0
\(309\) 2577.04 0.474443
\(310\) 0 0
\(311\) −8043.24 −1.46653 −0.733264 0.679944i \(-0.762004\pi\)
−0.733264 + 0.679944i \(0.762004\pi\)
\(312\) 0 0
\(313\) −9939.81 −1.79499 −0.897494 0.441026i \(-0.854615\pi\)
−0.897494 + 0.441026i \(0.854615\pi\)
\(314\) 0 0
\(315\) −1917.37 −0.342957
\(316\) 0 0
\(317\) −6640.48 −1.17655 −0.588275 0.808661i \(-0.700193\pi\)
−0.588275 + 0.808661i \(0.700193\pi\)
\(318\) 0 0
\(319\) −5243.07 −0.920237
\(320\) 0 0
\(321\) 3380.22 0.587743
\(322\) 0 0
\(323\) 36.3015 0.00625346
\(324\) 0 0
\(325\) −790.151 −0.134861
\(326\) 0 0
\(327\) −2047.98 −0.346340
\(328\) 0 0
\(329\) 5092.00 0.853286
\(330\) 0 0
\(331\) −2007.63 −0.333382 −0.166691 0.986009i \(-0.553308\pi\)
−0.166691 + 0.986009i \(0.553308\pi\)
\(332\) 0 0
\(333\) 2406.08 0.395954
\(334\) 0 0
\(335\) 5214.74 0.850483
\(336\) 0 0
\(337\) −4991.43 −0.806826 −0.403413 0.915018i \(-0.632176\pi\)
−0.403413 + 0.915018i \(0.632176\pi\)
\(338\) 0 0
\(339\) −2315.84 −0.371029
\(340\) 0 0
\(341\) 1740.54 0.276410
\(342\) 0 0
\(343\) −6903.81 −1.08680
\(344\) 0 0
\(345\) 2616.99 0.408388
\(346\) 0 0
\(347\) −10913.8 −1.68842 −0.844209 0.536013i \(-0.819930\pi\)
−0.844209 + 0.536013i \(0.819930\pi\)
\(348\) 0 0
\(349\) −9148.00 −1.40310 −0.701549 0.712621i \(-0.747508\pi\)
−0.701549 + 0.712621i \(0.747508\pi\)
\(350\) 0 0
\(351\) 351.000 0.0533761
\(352\) 0 0
\(353\) 1337.64 0.201686 0.100843 0.994902i \(-0.467846\pi\)
0.100843 + 0.994902i \(0.467846\pi\)
\(354\) 0 0
\(355\) 6954.74 1.03977
\(356\) 0 0
\(357\) −93.7809 −0.0139031
\(358\) 0 0
\(359\) −7865.89 −1.15639 −0.578197 0.815897i \(-0.696244\pi\)
−0.578197 + 0.815897i \(0.696244\pi\)
\(360\) 0 0
\(361\) −6529.55 −0.951968
\(362\) 0 0
\(363\) −3663.99 −0.529778
\(364\) 0 0
\(365\) 3977.02 0.570319
\(366\) 0 0
\(367\) −1935.86 −0.275344 −0.137672 0.990478i \(-0.543962\pi\)
−0.137672 + 0.990478i \(0.543962\pi\)
\(368\) 0 0
\(369\) −1326.70 −0.187169
\(370\) 0 0
\(371\) 1735.59 0.242877
\(372\) 0 0
\(373\) 5604.36 0.777970 0.388985 0.921244i \(-0.372826\pi\)
0.388985 + 0.921244i \(0.372826\pi\)
\(374\) 0 0
\(375\) −2625.95 −0.361609
\(376\) 0 0
\(377\) −1349.15 −0.184310
\(378\) 0 0
\(379\) 271.521 0.0367998 0.0183999 0.999831i \(-0.494143\pi\)
0.0183999 + 0.999831i \(0.494143\pi\)
\(380\) 0 0
\(381\) −6548.96 −0.880613
\(382\) 0 0
\(383\) 5763.16 0.768887 0.384443 0.923149i \(-0.374393\pi\)
0.384443 + 0.923149i \(0.374393\pi\)
\(384\) 0 0
\(385\) 10763.0 1.42476
\(386\) 0 0
\(387\) −1502.63 −0.197372
\(388\) 0 0
\(389\) 6596.09 0.859730 0.429865 0.902893i \(-0.358561\pi\)
0.429865 + 0.902893i \(0.358561\pi\)
\(390\) 0 0
\(391\) 128.000 0.0165556
\(392\) 0 0
\(393\) −2645.75 −0.339594
\(394\) 0 0
\(395\) −10831.7 −1.37975
\(396\) 0 0
\(397\) −7173.42 −0.906861 −0.453430 0.891292i \(-0.649800\pi\)
−0.453430 + 0.891292i \(0.649800\pi\)
\(398\) 0 0
\(399\) −851.096 −0.106787
\(400\) 0 0
\(401\) −3759.83 −0.468222 −0.234111 0.972210i \(-0.575218\pi\)
−0.234111 + 0.972210i \(0.575218\pi\)
\(402\) 0 0
\(403\) 447.878 0.0553608
\(404\) 0 0
\(405\) −1104.04 −0.135457
\(406\) 0 0
\(407\) −13506.3 −1.64492
\(408\) 0 0
\(409\) 10857.1 1.31259 0.656295 0.754505i \(-0.272123\pi\)
0.656295 + 0.754505i \(0.272123\pi\)
\(410\) 0 0
\(411\) 8293.20 0.995312
\(412\) 0 0
\(413\) 379.835 0.0452554
\(414\) 0 0
\(415\) −12887.4 −1.52438
\(416\) 0 0
\(417\) −4360.20 −0.512038
\(418\) 0 0
\(419\) 9802.86 1.14296 0.571481 0.820616i \(-0.306369\pi\)
0.571481 + 0.820616i \(0.306369\pi\)
\(420\) 0 0
\(421\) −5004.96 −0.579398 −0.289699 0.957118i \(-0.593555\pi\)
−0.289699 + 0.957118i \(0.593555\pi\)
\(422\) 0 0
\(423\) 2932.03 0.337021
\(424\) 0 0
\(425\) 121.562 0.0138744
\(426\) 0 0
\(427\) 10018.7 1.13546
\(428\) 0 0
\(429\) −1970.30 −0.221742
\(430\) 0 0
\(431\) 9275.81 1.03666 0.518330 0.855181i \(-0.326554\pi\)
0.518330 + 0.855181i \(0.326554\pi\)
\(432\) 0 0
\(433\) 3160.88 0.350813 0.175406 0.984496i \(-0.443876\pi\)
0.175406 + 0.984496i \(0.443876\pi\)
\(434\) 0 0
\(435\) 4243.65 0.467741
\(436\) 0 0
\(437\) 1161.65 0.127160
\(438\) 0 0
\(439\) 14035.4 1.52591 0.762954 0.646453i \(-0.223748\pi\)
0.762954 + 0.646453i \(0.223748\pi\)
\(440\) 0 0
\(441\) −888.287 −0.0959170
\(442\) 0 0
\(443\) 761.046 0.0816217 0.0408108 0.999167i \(-0.487006\pi\)
0.0408108 + 0.999167i \(0.487006\pi\)
\(444\) 0 0
\(445\) −4325.24 −0.460755
\(446\) 0 0
\(447\) 1680.37 0.177805
\(448\) 0 0
\(449\) 8385.22 0.881343 0.440672 0.897668i \(-0.354740\pi\)
0.440672 + 0.897668i \(0.354740\pi\)
\(450\) 0 0
\(451\) 7447.29 0.777559
\(452\) 0 0
\(453\) −8886.70 −0.921708
\(454\) 0 0
\(455\) 2769.54 0.285358
\(456\) 0 0
\(457\) 16484.7 1.68736 0.843679 0.536848i \(-0.180385\pi\)
0.843679 + 0.536848i \(0.180385\pi\)
\(458\) 0 0
\(459\) −54.0000 −0.00549129
\(460\) 0 0
\(461\) −2478.45 −0.250396 −0.125198 0.992132i \(-0.539957\pi\)
−0.125198 + 0.992132i \(0.539957\pi\)
\(462\) 0 0
\(463\) 2588.90 0.259863 0.129931 0.991523i \(-0.458524\pi\)
0.129931 + 0.991523i \(0.458524\pi\)
\(464\) 0 0
\(465\) −1408.77 −0.140494
\(466\) 0 0
\(467\) −13670.7 −1.35461 −0.677306 0.735701i \(-0.736853\pi\)
−0.677306 + 0.735701i \(0.736853\pi\)
\(468\) 0 0
\(469\) −5979.92 −0.588757
\(470\) 0 0
\(471\) 5513.76 0.539407
\(472\) 0 0
\(473\) 8434.86 0.819948
\(474\) 0 0
\(475\) 1103.22 0.106566
\(476\) 0 0
\(477\) 999.370 0.0959288
\(478\) 0 0
\(479\) −8034.36 −0.766386 −0.383193 0.923668i \(-0.625176\pi\)
−0.383193 + 0.923668i \(0.625176\pi\)
\(480\) 0 0
\(481\) −3475.45 −0.329453
\(482\) 0 0
\(483\) −3000.99 −0.282712
\(484\) 0 0
\(485\) −21438.6 −2.00716
\(486\) 0 0
\(487\) 7890.43 0.734188 0.367094 0.930184i \(-0.380353\pi\)
0.367094 + 0.930184i \(0.380353\pi\)
\(488\) 0 0
\(489\) −5380.57 −0.497582
\(490\) 0 0
\(491\) 2217.87 0.203852 0.101926 0.994792i \(-0.467500\pi\)
0.101926 + 0.994792i \(0.467500\pi\)
\(492\) 0 0
\(493\) 207.562 0.0189617
\(494\) 0 0
\(495\) 6197.43 0.562735
\(496\) 0 0
\(497\) −7975.23 −0.719795
\(498\) 0 0
\(499\) −6636.90 −0.595407 −0.297704 0.954658i \(-0.596221\pi\)
−0.297704 + 0.954658i \(0.596221\pi\)
\(500\) 0 0
\(501\) 11758.1 1.04853
\(502\) 0 0
\(503\) 9418.80 0.834917 0.417459 0.908696i \(-0.362921\pi\)
0.417459 + 0.908696i \(0.362921\pi\)
\(504\) 0 0
\(505\) 24812.9 2.18645
\(506\) 0 0
\(507\) −507.000 −0.0444116
\(508\) 0 0
\(509\) 11177.6 0.973353 0.486676 0.873582i \(-0.338209\pi\)
0.486676 + 0.873582i \(0.338209\pi\)
\(510\) 0 0
\(511\) −4560.58 −0.394810
\(512\) 0 0
\(513\) −490.070 −0.0421776
\(514\) 0 0
\(515\) 11708.5 1.00182
\(516\) 0 0
\(517\) −16458.6 −1.40010
\(518\) 0 0
\(519\) 11593.8 0.980562
\(520\) 0 0
\(521\) 2993.21 0.251699 0.125849 0.992049i \(-0.459834\pi\)
0.125849 + 0.992049i \(0.459834\pi\)
\(522\) 0 0
\(523\) −15198.3 −1.27070 −0.635351 0.772224i \(-0.719144\pi\)
−0.635351 + 0.772224i \(0.719144\pi\)
\(524\) 0 0
\(525\) −2850.04 −0.236926
\(526\) 0 0
\(527\) −68.9044 −0.00569549
\(528\) 0 0
\(529\) −8071.00 −0.663352
\(530\) 0 0
\(531\) 218.713 0.0178745
\(532\) 0 0
\(533\) 1916.34 0.155734
\(534\) 0 0
\(535\) 15357.6 1.24106
\(536\) 0 0
\(537\) −8487.13 −0.682024
\(538\) 0 0
\(539\) 4986.31 0.398470
\(540\) 0 0
\(541\) 2043.27 0.162379 0.0811894 0.996699i \(-0.474128\pi\)
0.0811894 + 0.996699i \(0.474128\pi\)
\(542\) 0 0
\(543\) 3380.22 0.267144
\(544\) 0 0
\(545\) −9304.74 −0.731323
\(546\) 0 0
\(547\) −4616.69 −0.360869 −0.180434 0.983587i \(-0.557750\pi\)
−0.180434 + 0.983587i \(0.557750\pi\)
\(548\) 0 0
\(549\) 5768.88 0.448470
\(550\) 0 0
\(551\) 1883.70 0.145641
\(552\) 0 0
\(553\) 12421.0 0.955148
\(554\) 0 0
\(555\) 10931.8 0.836085
\(556\) 0 0
\(557\) −18077.9 −1.37520 −0.687600 0.726089i \(-0.741336\pi\)
−0.687600 + 0.726089i \(0.741336\pi\)
\(558\) 0 0
\(559\) 2170.46 0.164223
\(560\) 0 0
\(561\) 303.123 0.0228126
\(562\) 0 0
\(563\) 13561.8 1.01520 0.507602 0.861592i \(-0.330532\pi\)
0.507602 + 0.861592i \(0.330532\pi\)
\(564\) 0 0
\(565\) −10521.7 −0.783456
\(566\) 0 0
\(567\) 1266.04 0.0937721
\(568\) 0 0
\(569\) −7280.83 −0.536429 −0.268215 0.963359i \(-0.586434\pi\)
−0.268215 + 0.963359i \(0.586434\pi\)
\(570\) 0 0
\(571\) −17884.5 −1.31076 −0.655380 0.755299i \(-0.727492\pi\)
−0.655380 + 0.755299i \(0.727492\pi\)
\(572\) 0 0
\(573\) 1986.41 0.144823
\(574\) 0 0
\(575\) 3889.98 0.282127
\(576\) 0 0
\(577\) −12067.2 −0.870645 −0.435323 0.900275i \(-0.643366\pi\)
−0.435323 + 0.900275i \(0.643366\pi\)
\(578\) 0 0
\(579\) 3871.48 0.277881
\(580\) 0 0
\(581\) 14778.4 1.05527
\(582\) 0 0
\(583\) −5609.86 −0.398519
\(584\) 0 0
\(585\) 1594.73 0.112707
\(586\) 0 0
\(587\) −9202.11 −0.647038 −0.323519 0.946222i \(-0.604866\pi\)
−0.323519 + 0.946222i \(0.604866\pi\)
\(588\) 0 0
\(589\) −625.332 −0.0437460
\(590\) 0 0
\(591\) −3960.04 −0.275625
\(592\) 0 0
\(593\) −16985.1 −1.17621 −0.588106 0.808784i \(-0.700126\pi\)
−0.588106 + 0.808784i \(0.700126\pi\)
\(594\) 0 0
\(595\) −426.082 −0.0293574
\(596\) 0 0
\(597\) −8135.92 −0.557757
\(598\) 0 0
\(599\) −541.705 −0.0369507 −0.0184753 0.999829i \(-0.505881\pi\)
−0.0184753 + 0.999829i \(0.505881\pi\)
\(600\) 0 0
\(601\) −24039.4 −1.63160 −0.815798 0.578337i \(-0.803702\pi\)
−0.815798 + 0.578337i \(0.803702\pi\)
\(602\) 0 0
\(603\) −3443.30 −0.232541
\(604\) 0 0
\(605\) −16646.9 −1.11867
\(606\) 0 0
\(607\) −22951.8 −1.53474 −0.767369 0.641206i \(-0.778435\pi\)
−0.767369 + 0.641206i \(0.778435\pi\)
\(608\) 0 0
\(609\) −4866.33 −0.323799
\(610\) 0 0
\(611\) −4235.15 −0.280419
\(612\) 0 0
\(613\) 19479.1 1.28345 0.641725 0.766935i \(-0.278219\pi\)
0.641725 + 0.766935i \(0.278219\pi\)
\(614\) 0 0
\(615\) −6027.70 −0.395220
\(616\) 0 0
\(617\) 12586.6 0.821258 0.410629 0.911803i \(-0.365309\pi\)
0.410629 + 0.911803i \(0.365309\pi\)
\(618\) 0 0
\(619\) 10589.4 0.687598 0.343799 0.939043i \(-0.388286\pi\)
0.343799 + 0.939043i \(0.388286\pi\)
\(620\) 0 0
\(621\) −1728.00 −0.111662
\(622\) 0 0
\(623\) 4959.89 0.318963
\(624\) 0 0
\(625\) −19528.3 −1.24981
\(626\) 0 0
\(627\) 2750.96 0.175219
\(628\) 0 0
\(629\) 534.685 0.0338940
\(630\) 0 0
\(631\) 11104.0 0.700547 0.350273 0.936647i \(-0.386089\pi\)
0.350273 + 0.936647i \(0.386089\pi\)
\(632\) 0 0
\(633\) −4836.17 −0.303666
\(634\) 0 0
\(635\) −29754.4 −1.85948
\(636\) 0 0
\(637\) 1283.08 0.0798077
\(638\) 0 0
\(639\) −4592.22 −0.284297
\(640\) 0 0
\(641\) 15811.9 0.974311 0.487155 0.873315i \(-0.338035\pi\)
0.487155 + 0.873315i \(0.338035\pi\)
\(642\) 0 0
\(643\) 1406.62 0.0862704 0.0431352 0.999069i \(-0.486265\pi\)
0.0431352 + 0.999069i \(0.486265\pi\)
\(644\) 0 0
\(645\) −6827.02 −0.416765
\(646\) 0 0
\(647\) 11661.8 0.708613 0.354307 0.935129i \(-0.384717\pi\)
0.354307 + 0.935129i \(0.384717\pi\)
\(648\) 0 0
\(649\) −1227.72 −0.0742564
\(650\) 0 0
\(651\) 1615.48 0.0972590
\(652\) 0 0
\(653\) 5545.04 0.332303 0.166152 0.986100i \(-0.446866\pi\)
0.166152 + 0.986100i \(0.446866\pi\)
\(654\) 0 0
\(655\) −12020.7 −0.717078
\(656\) 0 0
\(657\) −2626.03 −0.155938
\(658\) 0 0
\(659\) 10950.0 0.647272 0.323636 0.946182i \(-0.395095\pi\)
0.323636 + 0.946182i \(0.395095\pi\)
\(660\) 0 0
\(661\) −18488.8 −1.08795 −0.543973 0.839103i \(-0.683081\pi\)
−0.543973 + 0.839103i \(0.683081\pi\)
\(662\) 0 0
\(663\) 78.0000 0.00456903
\(664\) 0 0
\(665\) −3866.85 −0.225489
\(666\) 0 0
\(667\) 6641.98 0.385575
\(668\) 0 0
\(669\) −16806.3 −0.971254
\(670\) 0 0
\(671\) −32383.0 −1.86309
\(672\) 0 0
\(673\) −5865.80 −0.335973 −0.167987 0.985789i \(-0.553727\pi\)
−0.167987 + 0.985789i \(0.553727\pi\)
\(674\) 0 0
\(675\) −1641.08 −0.0935783
\(676\) 0 0
\(677\) 10258.6 0.582376 0.291188 0.956666i \(-0.405950\pi\)
0.291188 + 0.956666i \(0.405950\pi\)
\(678\) 0 0
\(679\) 24584.3 1.38948
\(680\) 0 0
\(681\) 4281.70 0.240933
\(682\) 0 0
\(683\) −26215.4 −1.46867 −0.734337 0.678785i \(-0.762507\pi\)
−0.734337 + 0.678785i \(0.762507\pi\)
\(684\) 0 0
\(685\) 37679.2 2.10167
\(686\) 0 0
\(687\) 14479.0 0.804088
\(688\) 0 0
\(689\) −1443.54 −0.0798176
\(690\) 0 0
\(691\) −7290.15 −0.401346 −0.200673 0.979658i \(-0.564313\pi\)
−0.200673 + 0.979658i \(0.564313\pi\)
\(692\) 0 0
\(693\) −7106.80 −0.389560
\(694\) 0 0
\(695\) −19810.0 −1.08120
\(696\) 0 0
\(697\) −294.822 −0.0160218
\(698\) 0 0
\(699\) 18732.5 1.01363
\(700\) 0 0
\(701\) −15472.1 −0.833628 −0.416814 0.908992i \(-0.636853\pi\)
−0.416814 + 0.908992i \(0.636853\pi\)
\(702\) 0 0
\(703\) 4852.46 0.260333
\(704\) 0 0
\(705\) 13321.3 0.711645
\(706\) 0 0
\(707\) −28453.7 −1.51360
\(708\) 0 0
\(709\) −17594.7 −0.931994 −0.465997 0.884786i \(-0.654304\pi\)
−0.465997 + 0.884786i \(0.654304\pi\)
\(710\) 0 0
\(711\) 7152.17 0.377253
\(712\) 0 0
\(713\) −2204.94 −0.115814
\(714\) 0 0
\(715\) −8951.84 −0.468223
\(716\) 0 0
\(717\) 19916.1 1.03735
\(718\) 0 0
\(719\) 12944.2 0.671400 0.335700 0.941969i \(-0.391027\pi\)
0.335700 + 0.941969i \(0.391027\pi\)
\(720\) 0 0
\(721\) −13426.5 −0.693523
\(722\) 0 0
\(723\) 13578.6 0.698469
\(724\) 0 0
\(725\) 6307.89 0.323130
\(726\) 0 0
\(727\) −20231.8 −1.03213 −0.516063 0.856551i \(-0.672603\pi\)
−0.516063 + 0.856551i \(0.672603\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −333.918 −0.0168952
\(732\) 0 0
\(733\) 26581.0 1.33942 0.669708 0.742624i \(-0.266419\pi\)
0.669708 + 0.742624i \(0.266419\pi\)
\(734\) 0 0
\(735\) −4035.83 −0.202536
\(736\) 0 0
\(737\) 19328.6 0.966050
\(738\) 0 0
\(739\) −1018.30 −0.0506886 −0.0253443 0.999679i \(-0.508068\pi\)
−0.0253443 + 0.999679i \(0.508068\pi\)
\(740\) 0 0
\(741\) 707.878 0.0350939
\(742\) 0 0
\(743\) 31784.2 1.56938 0.784691 0.619888i \(-0.212822\pi\)
0.784691 + 0.619888i \(0.212822\pi\)
\(744\) 0 0
\(745\) 7634.56 0.375448
\(746\) 0 0
\(747\) 8509.57 0.416799
\(748\) 0 0
\(749\) −17611.1 −0.859141
\(750\) 0 0
\(751\) 3549.69 0.172477 0.0862385 0.996275i \(-0.472515\pi\)
0.0862385 + 0.996275i \(0.472515\pi\)
\(752\) 0 0
\(753\) −6211.07 −0.300589
\(754\) 0 0
\(755\) −40375.7 −1.94625
\(756\) 0 0
\(757\) 18852.3 0.905151 0.452576 0.891726i \(-0.350505\pi\)
0.452576 + 0.891726i \(0.350505\pi\)
\(758\) 0 0
\(759\) 9699.95 0.463881
\(760\) 0 0
\(761\) −8131.76 −0.387353 −0.193677 0.981065i \(-0.562041\pi\)
−0.193677 + 0.981065i \(0.562041\pi\)
\(762\) 0 0
\(763\) 10670.1 0.506267
\(764\) 0 0
\(765\) −245.343 −0.0115953
\(766\) 0 0
\(767\) −315.919 −0.0148725
\(768\) 0 0
\(769\) −15773.3 −0.739662 −0.369831 0.929099i \(-0.620584\pi\)
−0.369831 + 0.929099i \(0.620584\pi\)
\(770\) 0 0
\(771\) 18463.7 0.862458
\(772\) 0 0
\(773\) −23061.4 −1.07304 −0.536521 0.843887i \(-0.680262\pi\)
−0.536521 + 0.843887i \(0.680262\pi\)
\(774\) 0 0
\(775\) −2094.03 −0.0970580
\(776\) 0 0
\(777\) −12535.8 −0.578790
\(778\) 0 0
\(779\) −2675.62 −0.123060
\(780\) 0 0
\(781\) 25778.0 1.18106
\(782\) 0 0
\(783\) −2802.08 −0.127891
\(784\) 0 0
\(785\) 25051.1 1.13900
\(786\) 0 0
\(787\) −38874.7 −1.76078 −0.880390 0.474251i \(-0.842719\pi\)
−0.880390 + 0.474251i \(0.842719\pi\)
\(788\) 0 0
\(789\) 12334.5 0.556554
\(790\) 0 0
\(791\) 12065.6 0.542357
\(792\) 0 0
\(793\) −8332.83 −0.373149
\(794\) 0 0
\(795\) 4540.52 0.202561
\(796\) 0 0
\(797\) −8956.95 −0.398082 −0.199041 0.979991i \(-0.563783\pi\)
−0.199041 + 0.979991i \(0.563783\pi\)
\(798\) 0 0
\(799\) 651.562 0.0288493
\(800\) 0 0
\(801\) 2855.96 0.125980
\(802\) 0 0
\(803\) 14740.9 0.647816
\(804\) 0 0
\(805\) −13634.6 −0.596966
\(806\) 0 0
\(807\) 2468.55 0.107679
\(808\) 0 0
\(809\) −21956.3 −0.954195 −0.477098 0.878850i \(-0.658311\pi\)
−0.477098 + 0.878850i \(0.658311\pi\)
\(810\) 0 0
\(811\) 14652.2 0.634414 0.317207 0.948356i \(-0.397255\pi\)
0.317207 + 0.948356i \(0.397255\pi\)
\(812\) 0 0
\(813\) 5056.23 0.218118
\(814\) 0 0
\(815\) −24446.0 −1.05068
\(816\) 0 0
\(817\) −3030.42 −0.129769
\(818\) 0 0
\(819\) −1828.73 −0.0780231
\(820\) 0 0
\(821\) −12172.5 −0.517444 −0.258722 0.965952i \(-0.583301\pi\)
−0.258722 + 0.965952i \(0.583301\pi\)
\(822\) 0 0
\(823\) 22022.1 0.932738 0.466369 0.884590i \(-0.345562\pi\)
0.466369 + 0.884590i \(0.345562\pi\)
\(824\) 0 0
\(825\) 9212.06 0.388755
\(826\) 0 0
\(827\) 7620.60 0.320428 0.160214 0.987082i \(-0.448781\pi\)
0.160214 + 0.987082i \(0.448781\pi\)
\(828\) 0 0
\(829\) −18727.6 −0.784603 −0.392302 0.919837i \(-0.628321\pi\)
−0.392302 + 0.919837i \(0.628321\pi\)
\(830\) 0 0
\(831\) −555.705 −0.0231976
\(832\) 0 0
\(833\) −197.397 −0.00821057
\(834\) 0 0
\(835\) 53421.6 2.21405
\(836\) 0 0
\(837\) 930.209 0.0384143
\(838\) 0 0
\(839\) −6853.37 −0.282008 −0.141004 0.990009i \(-0.545033\pi\)
−0.141004 + 0.990009i \(0.545033\pi\)
\(840\) 0 0
\(841\) −13618.5 −0.558388
\(842\) 0 0
\(843\) 3025.02 0.123591
\(844\) 0 0
\(845\) −2303.49 −0.0937783
\(846\) 0 0
\(847\) 19089.6 0.774410
\(848\) 0 0
\(849\) 19993.9 0.808232
\(850\) 0 0
\(851\) 17109.9 0.689214
\(852\) 0 0
\(853\) 16744.0 0.672102 0.336051 0.941844i \(-0.390909\pi\)
0.336051 + 0.941844i \(0.390909\pi\)
\(854\) 0 0
\(855\) −2226.57 −0.0890611
\(856\) 0 0
\(857\) 28271.8 1.12689 0.563446 0.826153i \(-0.309475\pi\)
0.563446 + 0.826153i \(0.309475\pi\)
\(858\) 0 0
\(859\) 40681.8 1.61589 0.807943 0.589260i \(-0.200581\pi\)
0.807943 + 0.589260i \(0.200581\pi\)
\(860\) 0 0
\(861\) 6912.17 0.273596
\(862\) 0 0
\(863\) −16589.8 −0.654371 −0.327186 0.944960i \(-0.606100\pi\)
−0.327186 + 0.944960i \(0.606100\pi\)
\(864\) 0 0
\(865\) 52675.1 2.07053
\(866\) 0 0
\(867\) 14727.0 0.576880
\(868\) 0 0
\(869\) −40148.0 −1.56723
\(870\) 0 0
\(871\) 4973.66 0.193486
\(872\) 0 0
\(873\) 14155.9 0.548803
\(874\) 0 0
\(875\) 13681.3 0.528586
\(876\) 0 0
\(877\) 41833.7 1.61074 0.805372 0.592769i \(-0.201965\pi\)
0.805372 + 0.592769i \(0.201965\pi\)
\(878\) 0 0
\(879\) −23862.0 −0.915639
\(880\) 0 0
\(881\) −24676.3 −0.943663 −0.471831 0.881689i \(-0.656407\pi\)
−0.471831 + 0.881689i \(0.656407\pi\)
\(882\) 0 0
\(883\) 30567.6 1.16499 0.582493 0.812836i \(-0.302077\pi\)
0.582493 + 0.812836i \(0.302077\pi\)
\(884\) 0 0
\(885\) 993.697 0.0377432
\(886\) 0 0
\(887\) −19020.9 −0.720023 −0.360012 0.932948i \(-0.617227\pi\)
−0.360012 + 0.932948i \(0.617227\pi\)
\(888\) 0 0
\(889\) 34120.4 1.28725
\(890\) 0 0
\(891\) −4092.17 −0.153864
\(892\) 0 0
\(893\) 5913.16 0.221586
\(894\) 0 0
\(895\) −38560.3 −1.44014
\(896\) 0 0
\(897\) 2496.00 0.0929086
\(898\) 0 0
\(899\) −3575.48 −0.132646
\(900\) 0 0
\(901\) 222.082 0.00821158
\(902\) 0 0
\(903\) 7828.77 0.288511
\(904\) 0 0
\(905\) 15357.6 0.564094
\(906\) 0 0
\(907\) −39186.2 −1.43457 −0.717287 0.696778i \(-0.754616\pi\)
−0.717287 + 0.696778i \(0.754616\pi\)
\(908\) 0 0
\(909\) −16384.0 −0.597823
\(910\) 0 0
\(911\) 1303.25 0.0473969 0.0236985 0.999719i \(-0.492456\pi\)
0.0236985 + 0.999719i \(0.492456\pi\)
\(912\) 0 0
\(913\) −47767.6 −1.73152
\(914\) 0 0
\(915\) 26210.2 0.946976
\(916\) 0 0
\(917\) 13784.5 0.496406
\(918\) 0 0
\(919\) −11710.2 −0.420330 −0.210165 0.977666i \(-0.567400\pi\)
−0.210165 + 0.977666i \(0.567400\pi\)
\(920\) 0 0
\(921\) −26495.1 −0.947928
\(922\) 0 0
\(923\) 6633.21 0.236549
\(924\) 0 0
\(925\) 16249.3 0.577594
\(926\) 0 0
\(927\) −7731.13 −0.273920
\(928\) 0 0
\(929\) 34967.9 1.23494 0.617470 0.786595i \(-0.288158\pi\)
0.617470 + 0.786595i \(0.288158\pi\)
\(930\) 0 0
\(931\) −1791.45 −0.0630638
\(932\) 0 0
\(933\) 24129.7 0.846700
\(934\) 0 0
\(935\) 1377.21 0.0481705
\(936\) 0 0
\(937\) −6538.36 −0.227960 −0.113980 0.993483i \(-0.536360\pi\)
−0.113980 + 0.993483i \(0.536360\pi\)
\(938\) 0 0
\(939\) 29819.4 1.03634
\(940\) 0 0
\(941\) 26190.2 0.907309 0.453654 0.891178i \(-0.350120\pi\)
0.453654 + 0.891178i \(0.350120\pi\)
\(942\) 0 0
\(943\) −9434.31 −0.325793
\(944\) 0 0
\(945\) 5752.11 0.198007
\(946\) 0 0
\(947\) 44316.2 1.52068 0.760340 0.649526i \(-0.225032\pi\)
0.760340 + 0.649526i \(0.225032\pi\)
\(948\) 0 0
\(949\) 3793.15 0.129748
\(950\) 0 0
\(951\) 19921.4 0.679282
\(952\) 0 0
\(953\) −10052.0 −0.341676 −0.170838 0.985299i \(-0.554647\pi\)
−0.170838 + 0.985299i \(0.554647\pi\)
\(954\) 0 0
\(955\) 9025.00 0.305803
\(956\) 0 0
\(957\) 15729.2 0.531299
\(958\) 0 0
\(959\) −43208.0 −1.45491
\(960\) 0 0
\(961\) −28604.0 −0.960157
\(962\) 0 0
\(963\) −10140.7 −0.339334
\(964\) 0 0
\(965\) 17589.6 0.586767
\(966\) 0 0
\(967\) 10809.8 0.359482 0.179741 0.983714i \(-0.442474\pi\)
0.179741 + 0.983714i \(0.442474\pi\)
\(968\) 0 0
\(969\) −108.904 −0.00361044
\(970\) 0 0
\(971\) 16280.1 0.538056 0.269028 0.963132i \(-0.413298\pi\)
0.269028 + 0.963132i \(0.413298\pi\)
\(972\) 0 0
\(973\) 22716.8 0.748477
\(974\) 0 0
\(975\) 2370.45 0.0778618
\(976\) 0 0
\(977\) 56327.0 1.84448 0.922242 0.386613i \(-0.126355\pi\)
0.922242 + 0.386613i \(0.126355\pi\)
\(978\) 0 0
\(979\) −16031.6 −0.523364
\(980\) 0 0
\(981\) 6143.93 0.199960
\(982\) 0 0
\(983\) −32733.6 −1.06210 −0.531048 0.847342i \(-0.678202\pi\)
−0.531048 + 0.847342i \(0.678202\pi\)
\(984\) 0 0
\(985\) −17992.0 −0.582002
\(986\) 0 0
\(987\) −15276.0 −0.492645
\(988\) 0 0
\(989\) −10685.4 −0.343554
\(990\) 0 0
\(991\) 2430.76 0.0779168 0.0389584 0.999241i \(-0.487596\pi\)
0.0389584 + 0.999241i \(0.487596\pi\)
\(992\) 0 0
\(993\) 6022.90 0.192478
\(994\) 0 0
\(995\) −36964.6 −1.17775
\(996\) 0 0
\(997\) 38619.5 1.22677 0.613387 0.789783i \(-0.289807\pi\)
0.613387 + 0.789783i \(0.289807\pi\)
\(998\) 0 0
\(999\) −7218.25 −0.228604
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 2496.4.a.v.1.1 2
4.3 odd 2 2496.4.a.be.1.1 2
8.3 odd 2 312.4.a.c.1.2 2
8.5 even 2 624.4.a.q.1.2 2
24.5 odd 2 1872.4.a.x.1.1 2
24.11 even 2 936.4.a.d.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
312.4.a.c.1.2 2 8.3 odd 2
624.4.a.q.1.2 2 8.5 even 2
936.4.a.d.1.1 2 24.11 even 2
1872.4.a.x.1.1 2 24.5 odd 2
2496.4.a.v.1.1 2 1.1 even 1 trivial
2496.4.a.be.1.1 2 4.3 odd 2