Properties

Label 3850.2.a.bz.1.5
Level $3850$
Weight $2$
Character 3850.1
Self dual yes
Analytic conductor $30.742$
Analytic rank $0$
Dimension $5$
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] = [3850,2,Mod(1,3850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3850, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3850.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3850.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: \(30.7424047782\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.117688.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 5x^{3} + 4x^{2} + 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.97382\) of defining polynomial
Character \(\chi\) \(=\) 3850.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +3.27510 q^{3} +1.00000 q^{4} -3.27510 q^{6} +1.00000 q^{7} -1.00000 q^{8} +7.72630 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +3.27510 q^{3} +1.00000 q^{4} -3.27510 q^{6} +1.00000 q^{7} -1.00000 q^{8} +7.72630 q^{9} +1.00000 q^{11} +3.27510 q^{12} -6.46446 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.38123 q^{17} -7.72630 q^{18} +6.62018 q^{19} +3.27510 q^{21} -1.00000 q^{22} +5.53694 q^{23} -3.27510 q^{24} +6.46446 q^{26} +15.4791 q^{27} +1.00000 q^{28} -1.01326 q^{29} -2.72630 q^{31} -1.00000 q^{32} +3.27510 q^{33} +3.38123 q^{34} +7.72630 q^{36} -2.15571 q^{37} -6.62018 q^{38} -21.1718 q^{39} +2.81064 q^{41} -3.27510 q^{42} +12.4190 q^{43} +1.00000 q^{44} -5.53694 q^{46} -2.82139 q^{47} +3.27510 q^{48} +1.00000 q^{49} -11.0739 q^{51} -6.46446 q^{52} +8.39450 q^{53} -15.4791 q^{54} -1.00000 q^{56} +21.6818 q^{57} +1.01326 q^{58} +11.4308 q^{59} +5.64307 q^{61} +2.72630 q^{62} +7.72630 q^{63} +1.00000 q^{64} -3.27510 q^{66} -3.16897 q^{67} -3.38123 q^{68} +18.1341 q^{69} +4.85755 q^{71} -7.72630 q^{72} -0.831025 q^{73} +2.15571 q^{74} +6.62018 q^{76} +1.00000 q^{77} +21.1718 q^{78} -10.5610 q^{79} +27.5169 q^{81} -2.81064 q^{82} -4.81205 q^{83} +3.27510 q^{84} -12.4190 q^{86} -3.31854 q^{87} -1.00000 q^{88} +8.02653 q^{89} -6.46446 q^{91} +5.53694 q^{92} -8.92893 q^{93} +2.82139 q^{94} -3.27510 q^{96} +11.3961 q^{97} -1.00000 q^{98} +7.72630 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{2} + 5 q^{4} + 5 q^{7} - 5 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{2} + 5 q^{4} + 5 q^{7} - 5 q^{8} + 13 q^{9} + 5 q^{11} - 10 q^{13} - 5 q^{14} + 5 q^{16} - 2 q^{17} - 13 q^{18} + 6 q^{19} - 5 q^{22} + 8 q^{23} + 10 q^{26} + 5 q^{28} + 8 q^{29} + 12 q^{31} - 5 q^{32} + 2 q^{34} + 13 q^{36} - 6 q^{37} - 6 q^{38} - 4 q^{39} + 20 q^{41} + 12 q^{43} + 5 q^{44} - 8 q^{46} - 10 q^{47} + 5 q^{49} - 16 q^{51} - 10 q^{52} + 14 q^{53} - 5 q^{56} + 12 q^{57} - 8 q^{58} + 36 q^{59} + 10 q^{61} - 12 q^{62} + 13 q^{63} + 5 q^{64} + 2 q^{67} - 2 q^{68} + 24 q^{69} + 16 q^{71} - 13 q^{72} - 22 q^{73} + 6 q^{74} + 6 q^{76} + 5 q^{77} + 4 q^{78} - 10 q^{79} + 25 q^{81} - 20 q^{82} + 12 q^{83} - 12 q^{86} - 32 q^{87} - 5 q^{88} + 14 q^{89} - 10 q^{91} + 8 q^{92} + 10 q^{94} - 2 q^{97} - 5 q^{98} + 13 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 3.27510 1.89088 0.945441 0.325793i \(-0.105631\pi\)
0.945441 + 0.325793i \(0.105631\pi\)
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −3.27510 −1.33706
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 7.72630 2.57543
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 3.27510 0.945441
\(13\) −6.46446 −1.79292 −0.896460 0.443125i \(-0.853870\pi\)
−0.896460 + 0.443125i \(0.853870\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −3.38123 −0.820069 −0.410035 0.912070i \(-0.634483\pi\)
−0.410035 + 0.912070i \(0.634483\pi\)
\(18\) −7.72630 −1.82111
\(19\) 6.62018 1.51877 0.759386 0.650640i \(-0.225499\pi\)
0.759386 + 0.650640i \(0.225499\pi\)
\(20\) 0 0
\(21\) 3.27510 0.714686
\(22\) −1.00000 −0.213201
\(23\) 5.53694 1.15453 0.577266 0.816556i \(-0.304119\pi\)
0.577266 + 0.816556i \(0.304119\pi\)
\(24\) −3.27510 −0.668528
\(25\) 0 0
\(26\) 6.46446 1.26779
\(27\) 15.4791 2.97896
\(28\) 1.00000 0.188982
\(29\) −1.01326 −0.188158 −0.0940792 0.995565i \(-0.529991\pi\)
−0.0940792 + 0.995565i \(0.529991\pi\)
\(30\) 0 0
\(31\) −2.72630 −0.489659 −0.244829 0.969566i \(-0.578732\pi\)
−0.244829 + 0.969566i \(0.578732\pi\)
\(32\) −1.00000 −0.176777
\(33\) 3.27510 0.570122
\(34\) 3.38123 0.579877
\(35\) 0 0
\(36\) 7.72630 1.28772
\(37\) −2.15571 −0.354397 −0.177198 0.984175i \(-0.556703\pi\)
−0.177198 + 0.984175i \(0.556703\pi\)
\(38\) −6.62018 −1.07393
\(39\) −21.1718 −3.39020
\(40\) 0 0
\(41\) 2.81064 0.438948 0.219474 0.975618i \(-0.429566\pi\)
0.219474 + 0.975618i \(0.429566\pi\)
\(42\) −3.27510 −0.505359
\(43\) 12.4190 1.89387 0.946937 0.321420i \(-0.104160\pi\)
0.946937 + 0.321420i \(0.104160\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −5.53694 −0.816378
\(47\) −2.82139 −0.411542 −0.205771 0.978600i \(-0.565970\pi\)
−0.205771 + 0.978600i \(0.565970\pi\)
\(48\) 3.27510 0.472720
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −11.0739 −1.55065
\(52\) −6.46446 −0.896460
\(53\) 8.39450 1.15307 0.576536 0.817071i \(-0.304404\pi\)
0.576536 + 0.817071i \(0.304404\pi\)
\(54\) −15.4791 −2.10644
\(55\) 0 0
\(56\) −1.00000 −0.133631
\(57\) 21.6818 2.87182
\(58\) 1.01326 0.133048
\(59\) 11.4308 1.48817 0.744083 0.668088i \(-0.232887\pi\)
0.744083 + 0.668088i \(0.232887\pi\)
\(60\) 0 0
\(61\) 5.64307 0.722521 0.361261 0.932465i \(-0.382346\pi\)
0.361261 + 0.932465i \(0.382346\pi\)
\(62\) 2.72630 0.346241
\(63\) 7.72630 0.973423
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −3.27510 −0.403137
\(67\) −3.16897 −0.387152 −0.193576 0.981085i \(-0.562009\pi\)
−0.193576 + 0.981085i \(0.562009\pi\)
\(68\) −3.38123 −0.410035
\(69\) 18.1341 2.18308
\(70\) 0 0
\(71\) 4.85755 0.576485 0.288243 0.957557i \(-0.406929\pi\)
0.288243 + 0.957557i \(0.406929\pi\)
\(72\) −7.72630 −0.910554
\(73\) −0.831025 −0.0972641 −0.0486321 0.998817i \(-0.515486\pi\)
−0.0486321 + 0.998817i \(0.515486\pi\)
\(74\) 2.15571 0.250596
\(75\) 0 0
\(76\) 6.62018 0.759386
\(77\) 1.00000 0.113961
\(78\) 21.1718 2.39723
\(79\) −10.5610 −1.18820 −0.594100 0.804391i \(-0.702492\pi\)
−0.594100 + 0.804391i \(0.702492\pi\)
\(80\) 0 0
\(81\) 27.5169 3.05743
\(82\) −2.81064 −0.310383
\(83\) −4.81205 −0.528191 −0.264095 0.964497i \(-0.585073\pi\)
−0.264095 + 0.964497i \(0.585073\pi\)
\(84\) 3.27510 0.357343
\(85\) 0 0
\(86\) −12.4190 −1.33917
\(87\) −3.31854 −0.355785
\(88\) −1.00000 −0.106600
\(89\) 8.02653 0.850810 0.425405 0.905003i \(-0.360132\pi\)
0.425405 + 0.905003i \(0.360132\pi\)
\(90\) 0 0
\(91\) −6.46446 −0.677660
\(92\) 5.53694 0.577266
\(93\) −8.92893 −0.925887
\(94\) 2.82139 0.291004
\(95\) 0 0
\(96\) −3.27510 −0.334264
\(97\) 11.3961 1.15709 0.578547 0.815649i \(-0.303620\pi\)
0.578547 + 0.815649i \(0.303620\pi\)
\(98\) −1.00000 −0.101015
\(99\) 7.72630 0.776523
\(100\) 0 0
\(101\) −1.47159 −0.146428 −0.0732141 0.997316i \(-0.523326\pi\)
−0.0732141 + 0.997316i \(0.523326\pi\)
\(102\) 11.0739 1.09648
\(103\) −5.49003 −0.540949 −0.270474 0.962727i \(-0.587181\pi\)
−0.270474 + 0.962727i \(0.587181\pi\)
\(104\) 6.46446 0.633893
\(105\) 0 0
\(106\) −8.39450 −0.815346
\(107\) −10.1341 −0.979697 −0.489848 0.871808i \(-0.662948\pi\)
−0.489848 + 0.871808i \(0.662948\pi\)
\(108\) 15.4791 1.48948
\(109\) 11.5635 1.10758 0.553790 0.832656i \(-0.313181\pi\)
0.553790 + 0.832656i \(0.313181\pi\)
\(110\) 0 0
\(111\) −7.06018 −0.670122
\(112\) 1.00000 0.0944911
\(113\) 2.47883 0.233189 0.116594 0.993180i \(-0.462802\pi\)
0.116594 + 0.993180i \(0.462802\pi\)
\(114\) −21.6818 −2.03068
\(115\) 0 0
\(116\) −1.01326 −0.0940792
\(117\) −49.9464 −4.61755
\(118\) −11.4308 −1.05229
\(119\) −3.38123 −0.309957
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −5.64307 −0.510900
\(123\) 9.20514 0.829999
\(124\) −2.72630 −0.244829
\(125\) 0 0
\(126\) −7.72630 −0.688314
\(127\) −9.78648 −0.868410 −0.434205 0.900814i \(-0.642971\pi\)
−0.434205 + 0.900814i \(0.642971\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 40.6734 3.58109
\(130\) 0 0
\(131\) 8.64670 0.755466 0.377733 0.925915i \(-0.376704\pi\)
0.377733 + 0.925915i \(0.376704\pi\)
\(132\) 3.27510 0.285061
\(133\) 6.62018 0.574042
\(134\) 3.16897 0.273758
\(135\) 0 0
\(136\) 3.38123 0.289938
\(137\) −2.95264 −0.252261 −0.126131 0.992014i \(-0.540256\pi\)
−0.126131 + 0.992014i \(0.540256\pi\)
\(138\) −18.1341 −1.54367
\(139\) −17.9063 −1.51879 −0.759397 0.650627i \(-0.774506\pi\)
−0.759397 + 0.650627i \(0.774506\pi\)
\(140\) 0 0
\(141\) −9.24035 −0.778178
\(142\) −4.85755 −0.407637
\(143\) −6.46446 −0.540586
\(144\) 7.72630 0.643859
\(145\) 0 0
\(146\) 0.831025 0.0687761
\(147\) 3.27510 0.270126
\(148\) −2.15571 −0.177198
\(149\) 10.9867 0.900069 0.450034 0.893011i \(-0.351412\pi\)
0.450034 + 0.893011i \(0.351412\pi\)
\(150\) 0 0
\(151\) −7.15320 −0.582119 −0.291060 0.956705i \(-0.594008\pi\)
−0.291060 + 0.956705i \(0.594008\pi\)
\(152\) −6.62018 −0.536967
\(153\) −26.1244 −2.11204
\(154\) −1.00000 −0.0805823
\(155\) 0 0
\(156\) −21.1718 −1.69510
\(157\) −11.8905 −0.948969 −0.474484 0.880264i \(-0.657365\pi\)
−0.474484 + 0.880264i \(0.657365\pi\)
\(158\) 10.5610 0.840185
\(159\) 27.4928 2.18032
\(160\) 0 0
\(161\) 5.53694 0.436372
\(162\) −27.5169 −2.16193
\(163\) 6.47381 0.507068 0.253534 0.967326i \(-0.418407\pi\)
0.253534 + 0.967326i \(0.418407\pi\)
\(164\) 2.81064 0.219474
\(165\) 0 0
\(166\) 4.81205 0.373487
\(167\) −13.6241 −1.05426 −0.527132 0.849783i \(-0.676733\pi\)
−0.527132 + 0.849783i \(0.676733\pi\)
\(168\) −3.27510 −0.252680
\(169\) 28.7893 2.21456
\(170\) 0 0
\(171\) 51.1495 3.91150
\(172\) 12.4190 0.946937
\(173\) −3.05209 −0.232046 −0.116023 0.993247i \(-0.537015\pi\)
−0.116023 + 0.993247i \(0.537015\pi\)
\(174\) 3.31854 0.251578
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 37.4371 2.81394
\(178\) −8.02653 −0.601614
\(179\) −17.4417 −1.30365 −0.651827 0.758367i \(-0.725997\pi\)
−0.651827 + 0.758367i \(0.725997\pi\)
\(180\) 0 0
\(181\) −24.9813 −1.85685 −0.928424 0.371523i \(-0.878836\pi\)
−0.928424 + 0.371523i \(0.878836\pi\)
\(182\) 6.46446 0.479178
\(183\) 18.4816 1.36620
\(184\) −5.53694 −0.408189
\(185\) 0 0
\(186\) 8.92893 0.654701
\(187\) −3.38123 −0.247260
\(188\) −2.82139 −0.205771
\(189\) 15.4791 1.12594
\(190\) 0 0
\(191\) 18.6624 1.35036 0.675180 0.737653i \(-0.264066\pi\)
0.675180 + 0.737653i \(0.264066\pi\)
\(192\) 3.27510 0.236360
\(193\) −4.82641 −0.347413 −0.173706 0.984797i \(-0.555574\pi\)
−0.173706 + 0.984797i \(0.555574\pi\)
\(194\) −11.3961 −0.818190
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 13.1677 0.938161 0.469080 0.883155i \(-0.344585\pi\)
0.469080 + 0.883155i \(0.344585\pi\)
\(198\) −7.72630 −0.549085
\(199\) −17.6955 −1.25440 −0.627199 0.778859i \(-0.715799\pi\)
−0.627199 + 0.778859i \(0.715799\pi\)
\(200\) 0 0
\(201\) −10.3787 −0.732059
\(202\) 1.47159 0.103540
\(203\) −1.01326 −0.0711172
\(204\) −11.0739 −0.775327
\(205\) 0 0
\(206\) 5.49003 0.382509
\(207\) 42.7801 2.97342
\(208\) −6.46446 −0.448230
\(209\) 6.62018 0.457927
\(210\) 0 0
\(211\) 11.2669 0.775644 0.387822 0.921734i \(-0.373228\pi\)
0.387822 + 0.921734i \(0.373228\pi\)
\(212\) 8.39450 0.576536
\(213\) 15.9090 1.09007
\(214\) 10.1341 0.692750
\(215\) 0 0
\(216\) −15.4791 −1.05322
\(217\) −2.72630 −0.185074
\(218\) −11.5635 −0.783177
\(219\) −2.72169 −0.183915
\(220\) 0 0
\(221\) 21.8579 1.47032
\(222\) 7.06018 0.473848
\(223\) 24.2768 1.62570 0.812848 0.582476i \(-0.197916\pi\)
0.812848 + 0.582476i \(0.197916\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) −2.47883 −0.164889
\(227\) −19.9074 −1.32130 −0.660652 0.750693i \(-0.729720\pi\)
−0.660652 + 0.750693i \(0.729720\pi\)
\(228\) 21.6818 1.43591
\(229\) −10.9196 −0.721586 −0.360793 0.932646i \(-0.617494\pi\)
−0.360793 + 0.932646i \(0.617494\pi\)
\(230\) 0 0
\(231\) 3.27510 0.215486
\(232\) 1.01326 0.0665240
\(233\) −6.86037 −0.449438 −0.224719 0.974424i \(-0.572146\pi\)
−0.224719 + 0.974424i \(0.572146\pi\)
\(234\) 49.9464 3.26510
\(235\) 0 0
\(236\) 11.4308 0.744083
\(237\) −34.5882 −2.24675
\(238\) 3.38123 0.219173
\(239\) 19.0161 1.23005 0.615024 0.788508i \(-0.289146\pi\)
0.615024 + 0.788508i \(0.289146\pi\)
\(240\) 0 0
\(241\) −9.59461 −0.618043 −0.309022 0.951055i \(-0.600002\pi\)
−0.309022 + 0.951055i \(0.600002\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 43.6832 2.80228
\(244\) 5.64307 0.361261
\(245\) 0 0
\(246\) −9.20514 −0.586898
\(247\) −42.7959 −2.72304
\(248\) 2.72630 0.173120
\(249\) −15.7600 −0.998746
\(250\) 0 0
\(251\) −22.9823 −1.45063 −0.725314 0.688418i \(-0.758305\pi\)
−0.725314 + 0.688418i \(0.758305\pi\)
\(252\) 7.72630 0.486711
\(253\) 5.53694 0.348105
\(254\) 9.78648 0.614058
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −1.18067 −0.0736484 −0.0368242 0.999322i \(-0.511724\pi\)
−0.0368242 + 0.999322i \(0.511724\pi\)
\(258\) −40.6734 −2.53221
\(259\) −2.15571 −0.133949
\(260\) 0 0
\(261\) −7.82878 −0.484590
\(262\) −8.64670 −0.534195
\(263\) −26.9368 −1.66099 −0.830496 0.557024i \(-0.811943\pi\)
−0.830496 + 0.557024i \(0.811943\pi\)
\(264\) −3.27510 −0.201569
\(265\) 0 0
\(266\) −6.62018 −0.405909
\(267\) 26.2877 1.60878
\(268\) −3.16897 −0.193576
\(269\) −20.4439 −1.24649 −0.623244 0.782027i \(-0.714186\pi\)
−0.623244 + 0.782027i \(0.714186\pi\)
\(270\) 0 0
\(271\) 25.6269 1.55672 0.778362 0.627816i \(-0.216051\pi\)
0.778362 + 0.627816i \(0.216051\pi\)
\(272\) −3.38123 −0.205017
\(273\) −21.1718 −1.28138
\(274\) 2.95264 0.178376
\(275\) 0 0
\(276\) 18.1341 1.09154
\(277\) −16.3408 −0.981821 −0.490911 0.871210i \(-0.663336\pi\)
−0.490911 + 0.871210i \(0.663336\pi\)
\(278\) 17.9063 1.07395
\(279\) −21.0643 −1.26108
\(280\) 0 0
\(281\) 4.37872 0.261213 0.130606 0.991434i \(-0.458308\pi\)
0.130606 + 0.991434i \(0.458308\pi\)
\(282\) 9.24035 0.550255
\(283\) −15.1500 −0.900574 −0.450287 0.892884i \(-0.648678\pi\)
−0.450287 + 0.892884i \(0.648678\pi\)
\(284\) 4.85755 0.288243
\(285\) 0 0
\(286\) 6.46446 0.382252
\(287\) 2.81064 0.165907
\(288\) −7.72630 −0.455277
\(289\) −5.56727 −0.327486
\(290\) 0 0
\(291\) 37.3233 2.18793
\(292\) −0.831025 −0.0486321
\(293\) 0.630927 0.0368592 0.0184296 0.999830i \(-0.494133\pi\)
0.0184296 + 0.999830i \(0.494133\pi\)
\(294\) −3.27510 −0.191008
\(295\) 0 0
\(296\) 2.15571 0.125298
\(297\) 15.4791 0.898191
\(298\) −10.9867 −0.636445
\(299\) −35.7934 −2.06998
\(300\) 0 0
\(301\) 12.4190 0.715817
\(302\) 7.15320 0.411620
\(303\) −4.81960 −0.276879
\(304\) 6.62018 0.379693
\(305\) 0 0
\(306\) 26.1244 1.49343
\(307\) −17.5287 −1.00042 −0.500208 0.865905i \(-0.666743\pi\)
−0.500208 + 0.865905i \(0.666743\pi\)
\(308\) 1.00000 0.0569803
\(309\) −17.9804 −1.02287
\(310\) 0 0
\(311\) 29.0181 1.64547 0.822734 0.568427i \(-0.192448\pi\)
0.822734 + 0.568427i \(0.192448\pi\)
\(312\) 21.1718 1.19862
\(313\) −26.9475 −1.52316 −0.761582 0.648069i \(-0.775577\pi\)
−0.761582 + 0.648069i \(0.775577\pi\)
\(314\) 11.8905 0.671022
\(315\) 0 0
\(316\) −10.5610 −0.594100
\(317\) −16.7087 −0.938456 −0.469228 0.883077i \(-0.655468\pi\)
−0.469228 + 0.883077i \(0.655468\pi\)
\(318\) −27.4928 −1.54172
\(319\) −1.01326 −0.0567319
\(320\) 0 0
\(321\) −33.1901 −1.85249
\(322\) −5.53694 −0.308562
\(323\) −22.3844 −1.24550
\(324\) 27.5169 1.52871
\(325\) 0 0
\(326\) −6.47381 −0.358551
\(327\) 37.8716 2.09430
\(328\) −2.81064 −0.155192
\(329\) −2.82139 −0.155548
\(330\) 0 0
\(331\) −31.0222 −1.70514 −0.852568 0.522617i \(-0.824956\pi\)
−0.852568 + 0.522617i \(0.824956\pi\)
\(332\) −4.81205 −0.264095
\(333\) −16.6557 −0.912726
\(334\) 13.6241 0.745477
\(335\) 0 0
\(336\) 3.27510 0.178672
\(337\) 14.1958 0.773295 0.386647 0.922228i \(-0.373633\pi\)
0.386647 + 0.922228i \(0.373633\pi\)
\(338\) −28.7893 −1.56593
\(339\) 8.11843 0.440933
\(340\) 0 0
\(341\) −2.72630 −0.147638
\(342\) −51.1495 −2.76585
\(343\) 1.00000 0.0539949
\(344\) −12.4190 −0.669585
\(345\) 0 0
\(346\) 3.05209 0.164082
\(347\) 5.38584 0.289127 0.144564 0.989496i \(-0.453822\pi\)
0.144564 + 0.989496i \(0.453822\pi\)
\(348\) −3.31854 −0.177893
\(349\) −22.9557 −1.22879 −0.614396 0.788997i \(-0.710600\pi\)
−0.614396 + 0.788997i \(0.710600\pi\)
\(350\) 0 0
\(351\) −100.064 −5.34104
\(352\) −1.00000 −0.0533002
\(353\) 2.80195 0.149133 0.0745664 0.997216i \(-0.476243\pi\)
0.0745664 + 0.997216i \(0.476243\pi\)
\(354\) −37.4371 −1.98976
\(355\) 0 0
\(356\) 8.02653 0.425405
\(357\) −11.0739 −0.586092
\(358\) 17.4417 0.921823
\(359\) 9.53192 0.503076 0.251538 0.967847i \(-0.419064\pi\)
0.251538 + 0.967847i \(0.419064\pi\)
\(360\) 0 0
\(361\) 24.8267 1.30667
\(362\) 24.9813 1.31299
\(363\) 3.27510 0.171898
\(364\) −6.46446 −0.338830
\(365\) 0 0
\(366\) −18.4816 −0.966051
\(367\) −11.8716 −0.619691 −0.309845 0.950787i \(-0.600277\pi\)
−0.309845 + 0.950787i \(0.600277\pi\)
\(368\) 5.53694 0.288633
\(369\) 21.7159 1.13048
\(370\) 0 0
\(371\) 8.39450 0.435821
\(372\) −8.92893 −0.462943
\(373\) −12.3379 −0.638835 −0.319417 0.947614i \(-0.603487\pi\)
−0.319417 + 0.947614i \(0.603487\pi\)
\(374\) 3.38123 0.174839
\(375\) 0 0
\(376\) 2.82139 0.145502
\(377\) 6.55021 0.337353
\(378\) −15.4791 −0.796161
\(379\) −23.2669 −1.19514 −0.597570 0.801817i \(-0.703867\pi\)
−0.597570 + 0.801817i \(0.703867\pi\)
\(380\) 0 0
\(381\) −32.0517 −1.64206
\(382\) −18.6624 −0.954849
\(383\) −9.41519 −0.481094 −0.240547 0.970638i \(-0.577327\pi\)
−0.240547 + 0.970638i \(0.577327\pi\)
\(384\) −3.27510 −0.167132
\(385\) 0 0
\(386\) 4.82641 0.245658
\(387\) 95.9526 4.87755
\(388\) 11.3961 0.578547
\(389\) 24.7245 1.25358 0.626791 0.779187i \(-0.284368\pi\)
0.626791 + 0.779187i \(0.284368\pi\)
\(390\) 0 0
\(391\) −18.7217 −0.946797
\(392\) −1.00000 −0.0505076
\(393\) 28.3188 1.42850
\(394\) −13.1677 −0.663380
\(395\) 0 0
\(396\) 7.72630 0.388261
\(397\) 0.894317 0.0448845 0.0224423 0.999748i \(-0.492856\pi\)
0.0224423 + 0.999748i \(0.492856\pi\)
\(398\) 17.6955 0.886994
\(399\) 21.6818 1.08545
\(400\) 0 0
\(401\) 3.60662 0.180106 0.0900531 0.995937i \(-0.471296\pi\)
0.0900531 + 0.995937i \(0.471296\pi\)
\(402\) 10.3787 0.517644
\(403\) 17.6241 0.877919
\(404\) −1.47159 −0.0732141
\(405\) 0 0
\(406\) 1.01326 0.0502874
\(407\) −2.15571 −0.106855
\(408\) 11.0739 0.548239
\(409\) 24.4591 1.20942 0.604711 0.796445i \(-0.293289\pi\)
0.604711 + 0.796445i \(0.293289\pi\)
\(410\) 0 0
\(411\) −9.67020 −0.476996
\(412\) −5.49003 −0.270474
\(413\) 11.4308 0.562474
\(414\) −42.7801 −2.10253
\(415\) 0 0
\(416\) 6.46446 0.316946
\(417\) −58.6451 −2.87186
\(418\) −6.62018 −0.323803
\(419\) 4.70912 0.230056 0.115028 0.993362i \(-0.463304\pi\)
0.115028 + 0.993362i \(0.463304\pi\)
\(420\) 0 0
\(421\) 13.1731 0.642015 0.321008 0.947077i \(-0.395978\pi\)
0.321008 + 0.947077i \(0.395978\pi\)
\(422\) −11.2669 −0.548463
\(423\) −21.7989 −1.05990
\(424\) −8.39450 −0.407673
\(425\) 0 0
\(426\) −15.9090 −0.770793
\(427\) 5.64307 0.273087
\(428\) −10.1341 −0.489848
\(429\) −21.1718 −1.02218
\(430\) 0 0
\(431\) −6.96302 −0.335397 −0.167699 0.985838i \(-0.553634\pi\)
−0.167699 + 0.985838i \(0.553634\pi\)
\(432\) 15.4791 0.744740
\(433\) −3.53318 −0.169794 −0.0848969 0.996390i \(-0.527056\pi\)
−0.0848969 + 0.996390i \(0.527056\pi\)
\(434\) 2.72630 0.130867
\(435\) 0 0
\(436\) 11.5635 0.553790
\(437\) 36.6555 1.75347
\(438\) 2.72169 0.130048
\(439\) 32.6519 1.55839 0.779194 0.626782i \(-0.215628\pi\)
0.779194 + 0.626782i \(0.215628\pi\)
\(440\) 0 0
\(441\) 7.72630 0.367919
\(442\) −21.8579 −1.03967
\(443\) 40.9084 1.94362 0.971808 0.235772i \(-0.0757619\pi\)
0.971808 + 0.235772i \(0.0757619\pi\)
\(444\) −7.06018 −0.335061
\(445\) 0 0
\(446\) −24.2768 −1.14954
\(447\) 35.9827 1.70192
\(448\) 1.00000 0.0472456
\(449\) −29.6135 −1.39755 −0.698774 0.715342i \(-0.746271\pi\)
−0.698774 + 0.715342i \(0.746271\pi\)
\(450\) 0 0
\(451\) 2.81064 0.132348
\(452\) 2.47883 0.116594
\(453\) −23.4275 −1.10072
\(454\) 19.9074 0.934302
\(455\) 0 0
\(456\) −21.6818 −1.01534
\(457\) 15.4719 0.723744 0.361872 0.932228i \(-0.382138\pi\)
0.361872 + 0.932228i \(0.382138\pi\)
\(458\) 10.9196 0.510239
\(459\) −52.3386 −2.44295
\(460\) 0 0
\(461\) 0.231236 0.0107697 0.00538486 0.999986i \(-0.498286\pi\)
0.00538486 + 0.999986i \(0.498286\pi\)
\(462\) −3.27510 −0.152372
\(463\) −0.741666 −0.0344682 −0.0172341 0.999851i \(-0.505486\pi\)
−0.0172341 + 0.999851i \(0.505486\pi\)
\(464\) −1.01326 −0.0470396
\(465\) 0 0
\(466\) 6.86037 0.317800
\(467\) 35.1500 1.62655 0.813274 0.581880i \(-0.197683\pi\)
0.813274 + 0.581880i \(0.197683\pi\)
\(468\) −49.9464 −2.30877
\(469\) −3.16897 −0.146330
\(470\) 0 0
\(471\) −38.9428 −1.79439
\(472\) −11.4308 −0.526146
\(473\) 12.4190 0.571024
\(474\) 34.5882 1.58869
\(475\) 0 0
\(476\) −3.38123 −0.154979
\(477\) 64.8584 2.96966
\(478\) −19.0161 −0.869775
\(479\) 7.82851 0.357694 0.178847 0.983877i \(-0.442763\pi\)
0.178847 + 0.983877i \(0.442763\pi\)
\(480\) 0 0
\(481\) 13.9355 0.635405
\(482\) 9.59461 0.437022
\(483\) 18.1341 0.825128
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) −43.6832 −1.98151
\(487\) −38.4687 −1.74318 −0.871591 0.490233i \(-0.836912\pi\)
−0.871591 + 0.490233i \(0.836912\pi\)
\(488\) −5.64307 −0.255450
\(489\) 21.2024 0.958805
\(490\) 0 0
\(491\) 7.04234 0.317816 0.158908 0.987293i \(-0.449203\pi\)
0.158908 + 0.987293i \(0.449203\pi\)
\(492\) 9.20514 0.415000
\(493\) 3.42608 0.154303
\(494\) 42.7959 1.92548
\(495\) 0 0
\(496\) −2.72630 −0.122415
\(497\) 4.85755 0.217891
\(498\) 15.7600 0.706220
\(499\) −10.7080 −0.479355 −0.239678 0.970853i \(-0.577042\pi\)
−0.239678 + 0.970853i \(0.577042\pi\)
\(500\) 0 0
\(501\) −44.6203 −1.99349
\(502\) 22.9823 1.02575
\(503\) −1.57235 −0.0701078 −0.0350539 0.999385i \(-0.511160\pi\)
−0.0350539 + 0.999385i \(0.511160\pi\)
\(504\) −7.72630 −0.344157
\(505\) 0 0
\(506\) −5.53694 −0.246147
\(507\) 94.2879 4.18747
\(508\) −9.78648 −0.434205
\(509\) 18.1484 0.804415 0.402208 0.915549i \(-0.368243\pi\)
0.402208 + 0.915549i \(0.368243\pi\)
\(510\) 0 0
\(511\) −0.831025 −0.0367624
\(512\) −1.00000 −0.0441942
\(513\) 102.475 4.52436
\(514\) 1.18067 0.0520773
\(515\) 0 0
\(516\) 40.6734 1.79055
\(517\) −2.82139 −0.124085
\(518\) 2.15571 0.0947165
\(519\) −9.99592 −0.438772
\(520\) 0 0
\(521\) 8.77193 0.384305 0.192153 0.981365i \(-0.438453\pi\)
0.192153 + 0.981365i \(0.438453\pi\)
\(522\) 7.82878 0.342657
\(523\) −15.0998 −0.660266 −0.330133 0.943934i \(-0.607094\pi\)
−0.330133 + 0.943934i \(0.607094\pi\)
\(524\) 8.64670 0.377733
\(525\) 0 0
\(526\) 26.9368 1.17450
\(527\) 9.21827 0.401554
\(528\) 3.27510 0.142531
\(529\) 7.65774 0.332945
\(530\) 0 0
\(531\) 88.3179 3.83267
\(532\) 6.62018 0.287021
\(533\) −18.1693 −0.786999
\(534\) −26.2877 −1.13758
\(535\) 0 0
\(536\) 3.16897 0.136879
\(537\) −57.1234 −2.46506
\(538\) 20.4439 0.881401
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 17.8143 0.765896 0.382948 0.923770i \(-0.374909\pi\)
0.382948 + 0.923770i \(0.374909\pi\)
\(542\) −25.6269 −1.10077
\(543\) −81.8164 −3.51108
\(544\) 3.38123 0.144969
\(545\) 0 0
\(546\) 21.1718 0.906069
\(547\) −39.7962 −1.70156 −0.850781 0.525520i \(-0.823871\pi\)
−0.850781 + 0.525520i \(0.823871\pi\)
\(548\) −2.95264 −0.126131
\(549\) 43.6001 1.86081
\(550\) 0 0
\(551\) −6.70798 −0.285770
\(552\) −18.1341 −0.771837
\(553\) −10.5610 −0.449098
\(554\) 16.3408 0.694253
\(555\) 0 0
\(556\) −17.9063 −0.759397
\(557\) −5.61626 −0.237968 −0.118984 0.992896i \(-0.537964\pi\)
−0.118984 + 0.992896i \(0.537964\pi\)
\(558\) 21.0643 0.891721
\(559\) −80.2819 −3.39556
\(560\) 0 0
\(561\) −11.0739 −0.467540
\(562\) −4.37872 −0.184705
\(563\) 27.4346 1.15623 0.578115 0.815955i \(-0.303789\pi\)
0.578115 + 0.815955i \(0.303789\pi\)
\(564\) −9.24035 −0.389089
\(565\) 0 0
\(566\) 15.1500 0.636802
\(567\) 27.5169 1.15560
\(568\) −4.85755 −0.203818
\(569\) 6.02245 0.252474 0.126237 0.992000i \(-0.459710\pi\)
0.126237 + 0.992000i \(0.459710\pi\)
\(570\) 0 0
\(571\) 8.38374 0.350849 0.175424 0.984493i \(-0.443870\pi\)
0.175424 + 0.984493i \(0.443870\pi\)
\(572\) −6.46446 −0.270293
\(573\) 61.1211 2.55337
\(574\) −2.81064 −0.117314
\(575\) 0 0
\(576\) 7.72630 0.321929
\(577\) −1.12762 −0.0469434 −0.0234717 0.999725i \(-0.507472\pi\)
−0.0234717 + 0.999725i \(0.507472\pi\)
\(578\) 5.56727 0.231568
\(579\) −15.8070 −0.656917
\(580\) 0 0
\(581\) −4.81205 −0.199637
\(582\) −37.3233 −1.54710
\(583\) 8.39450 0.347665
\(584\) 0.831025 0.0343881
\(585\) 0 0
\(586\) −0.630927 −0.0260634
\(587\) −11.3187 −0.467172 −0.233586 0.972336i \(-0.575046\pi\)
−0.233586 + 0.972336i \(0.575046\pi\)
\(588\) 3.27510 0.135063
\(589\) −18.0486 −0.743680
\(590\) 0 0
\(591\) 43.1256 1.77395
\(592\) −2.15571 −0.0885992
\(593\) −19.0319 −0.781545 −0.390772 0.920487i \(-0.627792\pi\)
−0.390772 + 0.920487i \(0.627792\pi\)
\(594\) −15.4791 −0.635117
\(595\) 0 0
\(596\) 10.9867 0.450034
\(597\) −57.9545 −2.37192
\(598\) 35.7934 1.46370
\(599\) −12.8576 −0.525345 −0.262673 0.964885i \(-0.584604\pi\)
−0.262673 + 0.964885i \(0.584604\pi\)
\(600\) 0 0
\(601\) 3.16565 0.129129 0.0645647 0.997914i \(-0.479434\pi\)
0.0645647 + 0.997914i \(0.479434\pi\)
\(602\) −12.4190 −0.506159
\(603\) −24.4845 −0.997084
\(604\) −7.15320 −0.291060
\(605\) 0 0
\(606\) 4.81960 0.195783
\(607\) −2.18017 −0.0884905 −0.0442453 0.999021i \(-0.514088\pi\)
−0.0442453 + 0.999021i \(0.514088\pi\)
\(608\) −6.62018 −0.268484
\(609\) −3.31854 −0.134474
\(610\) 0 0
\(611\) 18.2388 0.737862
\(612\) −26.1244 −1.05602
\(613\) 32.4835 1.31200 0.655998 0.754762i \(-0.272248\pi\)
0.655998 + 0.754762i \(0.272248\pi\)
\(614\) 17.5287 0.707401
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) 8.85755 0.356592 0.178296 0.983977i \(-0.442942\pi\)
0.178296 + 0.983977i \(0.442942\pi\)
\(618\) 17.9804 0.723279
\(619\) 28.4997 1.14550 0.572749 0.819731i \(-0.305877\pi\)
0.572749 + 0.819731i \(0.305877\pi\)
\(620\) 0 0
\(621\) 85.7071 3.43931
\(622\) −29.0181 −1.16352
\(623\) 8.02653 0.321576
\(624\) −21.1718 −0.847550
\(625\) 0 0
\(626\) 26.9475 1.07704
\(627\) 21.6818 0.865886
\(628\) −11.8905 −0.474484
\(629\) 7.28896 0.290630
\(630\) 0 0
\(631\) −44.8847 −1.78683 −0.893416 0.449231i \(-0.851698\pi\)
−0.893416 + 0.449231i \(0.851698\pi\)
\(632\) 10.5610 0.420092
\(633\) 36.9002 1.46665
\(634\) 16.7087 0.663589
\(635\) 0 0
\(636\) 27.4928 1.09016
\(637\) −6.46446 −0.256131
\(638\) 1.01326 0.0401155
\(639\) 37.5309 1.48470
\(640\) 0 0
\(641\) 38.3641 1.51529 0.757646 0.652666i \(-0.226350\pi\)
0.757646 + 0.652666i \(0.226350\pi\)
\(642\) 33.1901 1.30991
\(643\) −7.61526 −0.300317 −0.150158 0.988662i \(-0.547978\pi\)
−0.150158 + 0.988662i \(0.547978\pi\)
\(644\) 5.53694 0.218186
\(645\) 0 0
\(646\) 22.3844 0.880701
\(647\) 30.7250 1.20793 0.603963 0.797012i \(-0.293587\pi\)
0.603963 + 0.797012i \(0.293587\pi\)
\(648\) −27.5169 −1.08096
\(649\) 11.4308 0.448699
\(650\) 0 0
\(651\) −8.92893 −0.349952
\(652\) 6.47381 0.253534
\(653\) 37.5653 1.47005 0.735023 0.678043i \(-0.237171\pi\)
0.735023 + 0.678043i \(0.237171\pi\)
\(654\) −37.8716 −1.48090
\(655\) 0 0
\(656\) 2.81064 0.109737
\(657\) −6.42075 −0.250497
\(658\) 2.82139 0.109989
\(659\) −39.9380 −1.55576 −0.777882 0.628410i \(-0.783706\pi\)
−0.777882 + 0.628410i \(0.783706\pi\)
\(660\) 0 0
\(661\) −36.3398 −1.41345 −0.706727 0.707486i \(-0.749829\pi\)
−0.706727 + 0.707486i \(0.749829\pi\)
\(662\) 31.0222 1.20571
\(663\) 71.5867 2.78020
\(664\) 4.81205 0.186744
\(665\) 0 0
\(666\) 16.6557 0.645394
\(667\) −5.61038 −0.217235
\(668\) −13.6241 −0.527132
\(669\) 79.5091 3.07400
\(670\) 0 0
\(671\) 5.64307 0.217848
\(672\) −3.27510 −0.126340
\(673\) 33.8212 1.30371 0.651855 0.758344i \(-0.273991\pi\)
0.651855 + 0.758344i \(0.273991\pi\)
\(674\) −14.1958 −0.546802
\(675\) 0 0
\(676\) 28.7893 1.10728
\(677\) −47.8344 −1.83843 −0.919213 0.393760i \(-0.871174\pi\)
−0.919213 + 0.393760i \(0.871174\pi\)
\(678\) −8.11843 −0.311786
\(679\) 11.3961 0.437341
\(680\) 0 0
\(681\) −65.1989 −2.49843
\(682\) 2.72630 0.104396
\(683\) 44.6775 1.70954 0.854768 0.519011i \(-0.173700\pi\)
0.854768 + 0.519011i \(0.173700\pi\)
\(684\) 51.1495 1.95575
\(685\) 0 0
\(686\) −1.00000 −0.0381802
\(687\) −35.7628 −1.36443
\(688\) 12.4190 0.473468
\(689\) −54.2659 −2.06737
\(690\) 0 0
\(691\) −9.08364 −0.345558 −0.172779 0.984961i \(-0.555275\pi\)
−0.172779 + 0.984961i \(0.555275\pi\)
\(692\) −3.05209 −0.116023
\(693\) 7.72630 0.293498
\(694\) −5.38584 −0.204444
\(695\) 0 0
\(696\) 3.31854 0.125789
\(697\) −9.50343 −0.359968
\(698\) 22.9557 0.868888
\(699\) −22.4684 −0.849834
\(700\) 0 0
\(701\) 21.5344 0.813344 0.406672 0.913574i \(-0.366689\pi\)
0.406672 + 0.913574i \(0.366689\pi\)
\(702\) 100.064 3.77668
\(703\) −14.2712 −0.538248
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −2.80195 −0.105453
\(707\) −1.47159 −0.0553447
\(708\) 37.4371 1.40697
\(709\) −0.0726367 −0.00272793 −0.00136396 0.999999i \(-0.500434\pi\)
−0.00136396 + 0.999999i \(0.500434\pi\)
\(710\) 0 0
\(711\) −81.5972 −3.06013
\(712\) −8.02653 −0.300807
\(713\) −15.0954 −0.565327
\(714\) 11.0739 0.414430
\(715\) 0 0
\(716\) −17.4417 −0.651827
\(717\) 62.2796 2.32588
\(718\) −9.53192 −0.355728
\(719\) −14.8626 −0.554281 −0.277140 0.960829i \(-0.589387\pi\)
−0.277140 + 0.960829i \(0.589387\pi\)
\(720\) 0 0
\(721\) −5.49003 −0.204459
\(722\) −24.8267 −0.923955
\(723\) −31.4233 −1.16865
\(724\) −24.9813 −0.928424
\(725\) 0 0
\(726\) −3.27510 −0.121550
\(727\) −14.3413 −0.531889 −0.265945 0.963988i \(-0.585684\pi\)
−0.265945 + 0.963988i \(0.585684\pi\)
\(728\) 6.46446 0.239589
\(729\) 60.5163 2.24135
\(730\) 0 0
\(731\) −41.9914 −1.55311
\(732\) 18.4816 0.683101
\(733\) 38.0104 1.40394 0.701972 0.712204i \(-0.252303\pi\)
0.701972 + 0.712204i \(0.252303\pi\)
\(734\) 11.8716 0.438188
\(735\) 0 0
\(736\) −5.53694 −0.204094
\(737\) −3.16897 −0.116731
\(738\) −21.7159 −0.799372
\(739\) 8.83792 0.325108 0.162554 0.986700i \(-0.448027\pi\)
0.162554 + 0.986700i \(0.448027\pi\)
\(740\) 0 0
\(741\) −140.161 −5.14894
\(742\) −8.39450 −0.308172
\(743\) −32.2049 −1.18148 −0.590742 0.806861i \(-0.701165\pi\)
−0.590742 + 0.806861i \(0.701165\pi\)
\(744\) 8.92893 0.327350
\(745\) 0 0
\(746\) 12.3379 0.451724
\(747\) −37.1793 −1.36032
\(748\) −3.38123 −0.123630
\(749\) −10.1341 −0.370291
\(750\) 0 0
\(751\) −37.1493 −1.35560 −0.677799 0.735247i \(-0.737066\pi\)
−0.677799 + 0.735247i \(0.737066\pi\)
\(752\) −2.82139 −0.102886
\(753\) −75.2693 −2.74297
\(754\) −6.55021 −0.238544
\(755\) 0 0
\(756\) 15.4791 0.562971
\(757\) 3.34714 0.121654 0.0608269 0.998148i \(-0.480626\pi\)
0.0608269 + 0.998148i \(0.480626\pi\)
\(758\) 23.2669 0.845091
\(759\) 18.1341 0.658225
\(760\) 0 0
\(761\) 24.7682 0.897848 0.448924 0.893570i \(-0.351807\pi\)
0.448924 + 0.893570i \(0.351807\pi\)
\(762\) 32.0517 1.16111
\(763\) 11.5635 0.418626
\(764\) 18.6624 0.675180
\(765\) 0 0
\(766\) 9.41519 0.340184
\(767\) −73.8941 −2.66816
\(768\) 3.27510 0.118180
\(769\) −13.2016 −0.476063 −0.238032 0.971257i \(-0.576502\pi\)
−0.238032 + 0.971257i \(0.576502\pi\)
\(770\) 0 0
\(771\) −3.86683 −0.139260
\(772\) −4.82641 −0.173706
\(773\) −5.00683 −0.180083 −0.0900416 0.995938i \(-0.528700\pi\)
−0.0900416 + 0.995938i \(0.528700\pi\)
\(774\) −95.9526 −3.44895
\(775\) 0 0
\(776\) −11.3961 −0.409095
\(777\) −7.06018 −0.253282
\(778\) −24.7245 −0.886417
\(779\) 18.6069 0.666662
\(780\) 0 0
\(781\) 4.85755 0.173817
\(782\) 18.7217 0.669486
\(783\) −15.6844 −0.560516
\(784\) 1.00000 0.0357143
\(785\) 0 0
\(786\) −28.3188 −1.01010
\(787\) 20.5280 0.731746 0.365873 0.930665i \(-0.380770\pi\)
0.365873 + 0.930665i \(0.380770\pi\)
\(788\) 13.1677 0.469080
\(789\) −88.2207 −3.14074
\(790\) 0 0
\(791\) 2.47883 0.0881371
\(792\) −7.72630 −0.274542
\(793\) −36.4794 −1.29542
\(794\) −0.894317 −0.0317381
\(795\) 0 0
\(796\) −17.6955 −0.627199
\(797\) 4.72846 0.167491 0.0837454 0.996487i \(-0.473312\pi\)
0.0837454 + 0.996487i \(0.473312\pi\)
\(798\) −21.6818 −0.767526
\(799\) 9.53978 0.337493
\(800\) 0 0
\(801\) 62.0154 2.19121
\(802\) −3.60662 −0.127354
\(803\) −0.831025 −0.0293262
\(804\) −10.3787 −0.366029
\(805\) 0 0
\(806\) −17.6241 −0.620782
\(807\) −66.9560 −2.35696
\(808\) 1.47159 0.0517702
\(809\) −31.1911 −1.09662 −0.548310 0.836275i \(-0.684728\pi\)
−0.548310 + 0.836275i \(0.684728\pi\)
\(810\) 0 0
\(811\) 3.74209 0.131403 0.0657013 0.997839i \(-0.479072\pi\)
0.0657013 + 0.997839i \(0.479072\pi\)
\(812\) −1.01326 −0.0355586
\(813\) 83.9308 2.94358
\(814\) 2.15571 0.0755576
\(815\) 0 0
\(816\) −11.0739 −0.387664
\(817\) 82.2157 2.87636
\(818\) −24.4591 −0.855191
\(819\) −49.9464 −1.74527
\(820\) 0 0
\(821\) 16.1553 0.563825 0.281913 0.959440i \(-0.409031\pi\)
0.281913 + 0.959440i \(0.409031\pi\)
\(822\) 9.67020 0.337287
\(823\) −1.62701 −0.0567139 −0.0283570 0.999598i \(-0.509028\pi\)
−0.0283570 + 0.999598i \(0.509028\pi\)
\(824\) 5.49003 0.191254
\(825\) 0 0
\(826\) −11.4308 −0.397729
\(827\) 28.5580 0.993061 0.496530 0.868019i \(-0.334607\pi\)
0.496530 + 0.868019i \(0.334607\pi\)
\(828\) 42.7801 1.48671
\(829\) 18.7101 0.649830 0.324915 0.945743i \(-0.394664\pi\)
0.324915 + 0.945743i \(0.394664\pi\)
\(830\) 0 0
\(831\) −53.5177 −1.85651
\(832\) −6.46446 −0.224115
\(833\) −3.38123 −0.117153
\(834\) 58.6451 2.03071
\(835\) 0 0
\(836\) 6.62018 0.228964
\(837\) −42.2008 −1.45867
\(838\) −4.70912 −0.162674
\(839\) 36.4414 1.25810 0.629049 0.777366i \(-0.283445\pi\)
0.629049 + 0.777366i \(0.283445\pi\)
\(840\) 0 0
\(841\) −27.9733 −0.964596
\(842\) −13.1731 −0.453973
\(843\) 14.3408 0.493922
\(844\) 11.2669 0.387822
\(845\) 0 0
\(846\) 21.7989 0.749463
\(847\) 1.00000 0.0343604
\(848\) 8.39450 0.288268
\(849\) −49.6178 −1.70288
\(850\) 0 0
\(851\) −11.9361 −0.409163
\(852\) 15.9090 0.545033
\(853\) −25.9043 −0.886945 −0.443472 0.896288i \(-0.646254\pi\)
−0.443472 + 0.896288i \(0.646254\pi\)
\(854\) −5.64307 −0.193102
\(855\) 0 0
\(856\) 10.1341 0.346375
\(857\) 15.6452 0.534431 0.267215 0.963637i \(-0.413896\pi\)
0.267215 + 0.963637i \(0.413896\pi\)
\(858\) 21.1718 0.722793
\(859\) −31.7666 −1.08386 −0.541930 0.840423i \(-0.682306\pi\)
−0.541930 + 0.840423i \(0.682306\pi\)
\(860\) 0 0
\(861\) 9.20514 0.313710
\(862\) 6.96302 0.237162
\(863\) 7.07738 0.240917 0.120458 0.992718i \(-0.461564\pi\)
0.120458 + 0.992718i \(0.461564\pi\)
\(864\) −15.4791 −0.526611
\(865\) 0 0
\(866\) 3.53318 0.120062
\(867\) −18.2334 −0.619238
\(868\) −2.72630 −0.0925368
\(869\) −10.5610 −0.358256
\(870\) 0 0
\(871\) 20.4857 0.694132
\(872\) −11.5635 −0.391589
\(873\) 88.0494 2.98002
\(874\) −36.6555 −1.23989
\(875\) 0 0
\(876\) −2.72169 −0.0919575
\(877\) −13.7101 −0.462957 −0.231478 0.972840i \(-0.574356\pi\)
−0.231478 + 0.972840i \(0.574356\pi\)
\(878\) −32.6519 −1.10195
\(879\) 2.06635 0.0696963
\(880\) 0 0
\(881\) −15.9858 −0.538574 −0.269287 0.963060i \(-0.586788\pi\)
−0.269287 + 0.963060i \(0.586788\pi\)
\(882\) −7.72630 −0.260158
\(883\) −44.0546 −1.48256 −0.741278 0.671198i \(-0.765780\pi\)
−0.741278 + 0.671198i \(0.765780\pi\)
\(884\) 21.8579 0.735159
\(885\) 0 0
\(886\) −40.9084 −1.37434
\(887\) −51.9962 −1.74586 −0.872930 0.487845i \(-0.837783\pi\)
−0.872930 + 0.487845i \(0.837783\pi\)
\(888\) 7.06018 0.236924
\(889\) −9.78648 −0.328228
\(890\) 0 0
\(891\) 27.5169 0.921850
\(892\) 24.2768 0.812848
\(893\) −18.6781 −0.625039
\(894\) −35.9827 −1.20344
\(895\) 0 0
\(896\) −1.00000 −0.0334077
\(897\) −117.227 −3.91410
\(898\) 29.6135 0.988216
\(899\) 2.76247 0.0921334
\(900\) 0 0
\(901\) −28.3837 −0.945600
\(902\) −2.81064 −0.0935841
\(903\) 40.6734 1.35353
\(904\) −2.47883 −0.0824447
\(905\) 0 0
\(906\) 23.4275 0.778326
\(907\) 25.8158 0.857200 0.428600 0.903494i \(-0.359007\pi\)
0.428600 + 0.903494i \(0.359007\pi\)
\(908\) −19.9074 −0.660652
\(909\) −11.3699 −0.377116
\(910\) 0 0
\(911\) 56.2763 1.86452 0.932258 0.361794i \(-0.117836\pi\)
0.932258 + 0.361794i \(0.117836\pi\)
\(912\) 21.6818 0.717955
\(913\) −4.81205 −0.159256
\(914\) −15.4719 −0.511764
\(915\) 0 0
\(916\) −10.9196 −0.360793
\(917\) 8.64670 0.285539
\(918\) 52.3386 1.72743
\(919\) −21.0335 −0.693831 −0.346916 0.937896i \(-0.612771\pi\)
−0.346916 + 0.937896i \(0.612771\pi\)
\(920\) 0 0
\(921\) −57.4084 −1.89167
\(922\) −0.231236 −0.00761534
\(923\) −31.4015 −1.03359
\(924\) 3.27510 0.107743
\(925\) 0 0
\(926\) 0.741666 0.0243727
\(927\) −42.4176 −1.39318
\(928\) 1.01326 0.0332620
\(929\) −17.0767 −0.560268 −0.280134 0.959961i \(-0.590379\pi\)
−0.280134 + 0.959961i \(0.590379\pi\)
\(930\) 0 0
\(931\) 6.62018 0.216967
\(932\) −6.86037 −0.224719
\(933\) 95.0374 3.11139
\(934\) −35.1500 −1.15014
\(935\) 0 0
\(936\) 49.9464 1.63255
\(937\) 4.00689 0.130899 0.0654497 0.997856i \(-0.479152\pi\)
0.0654497 + 0.997856i \(0.479152\pi\)
\(938\) 3.16897 0.103471
\(939\) −88.2559 −2.88012
\(940\) 0 0
\(941\) −14.4810 −0.472067 −0.236033 0.971745i \(-0.575847\pi\)
−0.236033 + 0.971745i \(0.575847\pi\)
\(942\) 38.9428 1.26882
\(943\) 15.5624 0.506780
\(944\) 11.4308 0.372041
\(945\) 0 0
\(946\) −12.4190 −0.403775
\(947\) −34.1301 −1.10908 −0.554539 0.832158i \(-0.687105\pi\)
−0.554539 + 0.832158i \(0.687105\pi\)
\(948\) −34.5882 −1.12337
\(949\) 5.37213 0.174387
\(950\) 0 0
\(951\) −54.7228 −1.77451
\(952\) 3.38123 0.109586
\(953\) −6.31885 −0.204688 −0.102344 0.994749i \(-0.532634\pi\)
−0.102344 + 0.994749i \(0.532634\pi\)
\(954\) −64.8584 −2.09987
\(955\) 0 0
\(956\) 19.0161 0.615024
\(957\) −3.31854 −0.107273
\(958\) −7.82851 −0.252928
\(959\) −2.95264 −0.0953457
\(960\) 0 0
\(961\) −23.5673 −0.760234
\(962\) −13.9355 −0.449299
\(963\) −78.2989 −2.52315
\(964\) −9.59461 −0.309022
\(965\) 0 0
\(966\) −18.1341 −0.583454
\(967\) 32.6668 1.05049 0.525247 0.850950i \(-0.323973\pi\)
0.525247 + 0.850950i \(0.323973\pi\)
\(968\) −1.00000 −0.0321412
\(969\) −73.3111 −2.35509
\(970\) 0 0
\(971\) −32.9801 −1.05838 −0.529190 0.848503i \(-0.677504\pi\)
−0.529190 + 0.848503i \(0.677504\pi\)
\(972\) 43.6832 1.40114
\(973\) −17.9063 −0.574050
\(974\) 38.4687 1.23262
\(975\) 0 0
\(976\) 5.64307 0.180630
\(977\) −5.50566 −0.176142 −0.0880709 0.996114i \(-0.528070\pi\)
−0.0880709 + 0.996114i \(0.528070\pi\)
\(978\) −21.2024 −0.677978
\(979\) 8.02653 0.256529
\(980\) 0 0
\(981\) 89.3429 2.85250
\(982\) −7.04234 −0.224730
\(983\) 15.5437 0.495767 0.247883 0.968790i \(-0.420265\pi\)
0.247883 + 0.968790i \(0.420265\pi\)
\(984\) −9.20514 −0.293449
\(985\) 0 0
\(986\) −3.42608 −0.109109
\(987\) −9.24035 −0.294124
\(988\) −42.7959 −1.36152
\(989\) 68.7631 2.18654
\(990\) 0 0
\(991\) −2.47470 −0.0786115 −0.0393057 0.999227i \(-0.512515\pi\)
−0.0393057 + 0.999227i \(0.512515\pi\)
\(992\) 2.72630 0.0865602
\(993\) −101.601 −3.22421
\(994\) −4.85755 −0.154072
\(995\) 0 0
\(996\) −15.7600 −0.499373
\(997\) 13.9735 0.442544 0.221272 0.975212i \(-0.428979\pi\)
0.221272 + 0.975212i \(0.428979\pi\)
\(998\) 10.7080 0.338955
\(999\) −33.3685 −1.05573
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 3850.2.a.bz.1.5 5
5.2 odd 4 770.2.c.f.309.1 10
5.3 odd 4 770.2.c.f.309.10 yes 10
5.4 even 2 3850.2.a.ca.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.c.f.309.1 10 5.2 odd 4
770.2.c.f.309.10 yes 10 5.3 odd 4
3850.2.a.bz.1.5 5 1.1 even 1 trivial
3850.2.a.ca.1.1 5 5.4 even 2