Properties

Label 2850.2.a.be.1.1
Level $2850$
Weight $2$
Character 2850.1
Self dual yes
Analytic conductor $22.757$
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] = [2850,2,Mod(1,2850)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2850, 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("2850.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2850.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: \(22.7573645761\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 2850.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.73205 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.73205 q^{7} -1.00000 q^{8} +1.00000 q^{9} +5.19615 q^{11} +1.00000 q^{12} -4.73205 q^{13} +2.73205 q^{14} +1.00000 q^{16} +2.73205 q^{17} -1.00000 q^{18} +1.00000 q^{19} -2.73205 q^{21} -5.19615 q^{22} -8.46410 q^{23} -1.00000 q^{24} +4.73205 q^{26} +1.00000 q^{27} -2.73205 q^{28} -9.19615 q^{29} -5.92820 q^{31} -1.00000 q^{32} +5.19615 q^{33} -2.73205 q^{34} +1.00000 q^{36} +8.92820 q^{37} -1.00000 q^{38} -4.73205 q^{39} +2.73205 q^{42} -8.73205 q^{43} +5.19615 q^{44} +8.46410 q^{46} +3.46410 q^{47} +1.00000 q^{48} +0.464102 q^{49} +2.73205 q^{51} -4.73205 q^{52} +4.66025 q^{53} -1.00000 q^{54} +2.73205 q^{56} +1.00000 q^{57} +9.19615 q^{58} +2.19615 q^{59} +8.26795 q^{61} +5.92820 q^{62} -2.73205 q^{63} +1.00000 q^{64} -5.19615 q^{66} +5.19615 q^{67} +2.73205 q^{68} -8.46410 q^{69} -10.1962 q^{71} -1.00000 q^{72} -14.4641 q^{73} -8.92820 q^{74} +1.00000 q^{76} -14.1962 q^{77} +4.73205 q^{78} +5.92820 q^{79} +1.00000 q^{81} -16.6603 q^{83} -2.73205 q^{84} +8.73205 q^{86} -9.19615 q^{87} -5.19615 q^{88} +3.92820 q^{89} +12.9282 q^{91} -8.46410 q^{92} -5.92820 q^{93} -3.46410 q^{94} -1.00000 q^{96} -17.1244 q^{97} -0.464102 q^{98} +5.19615 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{7} - 2 q^{8} + 2 q^{9} + 2 q^{12} - 6 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{17} - 2 q^{18} + 2 q^{19} - 2 q^{21} - 10 q^{23} - 2 q^{24} + 6 q^{26} + 2 q^{27} - 2 q^{28} - 8 q^{29} + 2 q^{31} - 2 q^{32} - 2 q^{34} + 2 q^{36} + 4 q^{37} - 2 q^{38} - 6 q^{39} + 2 q^{42} - 14 q^{43} + 10 q^{46} + 2 q^{48} - 6 q^{49} + 2 q^{51} - 6 q^{52} - 8 q^{53} - 2 q^{54} + 2 q^{56} + 2 q^{57} + 8 q^{58} - 6 q^{59} + 20 q^{61} - 2 q^{62} - 2 q^{63} + 2 q^{64} + 2 q^{68} - 10 q^{69} - 10 q^{71} - 2 q^{72} - 22 q^{73} - 4 q^{74} + 2 q^{76} - 18 q^{77} + 6 q^{78} - 2 q^{79} + 2 q^{81} - 16 q^{83} - 2 q^{84} + 14 q^{86} - 8 q^{87} - 6 q^{89} + 12 q^{91} - 10 q^{92} + 2 q^{93} - 2 q^{96} - 10 q^{97} + 6 q^{98}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) −2.73205 −1.03262 −0.516309 0.856402i \(-0.672694\pi\)
−0.516309 + 0.856402i \(0.672694\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.19615 1.56670 0.783349 0.621582i \(-0.213510\pi\)
0.783349 + 0.621582i \(0.213510\pi\)
\(12\) 1.00000 0.288675
\(13\) −4.73205 −1.31243 −0.656217 0.754572i \(-0.727845\pi\)
−0.656217 + 0.754572i \(0.727845\pi\)
\(14\) 2.73205 0.730171
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 2.73205 0.662620 0.331310 0.943522i \(-0.392509\pi\)
0.331310 + 0.943522i \(0.392509\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) −2.73205 −0.596182
\(22\) −5.19615 −1.10782
\(23\) −8.46410 −1.76489 −0.882444 0.470418i \(-0.844103\pi\)
−0.882444 + 0.470418i \(0.844103\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 4.73205 0.928032
\(27\) 1.00000 0.192450
\(28\) −2.73205 −0.516309
\(29\) −9.19615 −1.70768 −0.853841 0.520533i \(-0.825733\pi\)
−0.853841 + 0.520533i \(0.825733\pi\)
\(30\) 0 0
\(31\) −5.92820 −1.06474 −0.532368 0.846513i \(-0.678698\pi\)
−0.532368 + 0.846513i \(0.678698\pi\)
\(32\) −1.00000 −0.176777
\(33\) 5.19615 0.904534
\(34\) −2.73205 −0.468543
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 8.92820 1.46779 0.733894 0.679264i \(-0.237701\pi\)
0.733894 + 0.679264i \(0.237701\pi\)
\(38\) −1.00000 −0.162221
\(39\) −4.73205 −0.757735
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 2.73205 0.421565
\(43\) −8.73205 −1.33163 −0.665813 0.746119i \(-0.731915\pi\)
−0.665813 + 0.746119i \(0.731915\pi\)
\(44\) 5.19615 0.783349
\(45\) 0 0
\(46\) 8.46410 1.24796
\(47\) 3.46410 0.505291 0.252646 0.967559i \(-0.418699\pi\)
0.252646 + 0.967559i \(0.418699\pi\)
\(48\) 1.00000 0.144338
\(49\) 0.464102 0.0663002
\(50\) 0 0
\(51\) 2.73205 0.382564
\(52\) −4.73205 −0.656217
\(53\) 4.66025 0.640135 0.320068 0.947395i \(-0.396294\pi\)
0.320068 + 0.947395i \(0.396294\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) 2.73205 0.365086
\(57\) 1.00000 0.132453
\(58\) 9.19615 1.20751
\(59\) 2.19615 0.285915 0.142957 0.989729i \(-0.454339\pi\)
0.142957 + 0.989729i \(0.454339\pi\)
\(60\) 0 0
\(61\) 8.26795 1.05860 0.529301 0.848434i \(-0.322454\pi\)
0.529301 + 0.848434i \(0.322454\pi\)
\(62\) 5.92820 0.752883
\(63\) −2.73205 −0.344206
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −5.19615 −0.639602
\(67\) 5.19615 0.634811 0.317406 0.948290i \(-0.397188\pi\)
0.317406 + 0.948290i \(0.397188\pi\)
\(68\) 2.73205 0.331310
\(69\) −8.46410 −1.01896
\(70\) 0 0
\(71\) −10.1962 −1.21006 −0.605030 0.796202i \(-0.706839\pi\)
−0.605030 + 0.796202i \(0.706839\pi\)
\(72\) −1.00000 −0.117851
\(73\) −14.4641 −1.69290 −0.846448 0.532472i \(-0.821263\pi\)
−0.846448 + 0.532472i \(0.821263\pi\)
\(74\) −8.92820 −1.03788
\(75\) 0 0
\(76\) 1.00000 0.114708
\(77\) −14.1962 −1.61780
\(78\) 4.73205 0.535799
\(79\) 5.92820 0.666975 0.333487 0.942755i \(-0.391775\pi\)
0.333487 + 0.942755i \(0.391775\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −16.6603 −1.82870 −0.914350 0.404924i \(-0.867298\pi\)
−0.914350 + 0.404924i \(0.867298\pi\)
\(84\) −2.73205 −0.298091
\(85\) 0 0
\(86\) 8.73205 0.941601
\(87\) −9.19615 −0.985931
\(88\) −5.19615 −0.553912
\(89\) 3.92820 0.416389 0.208194 0.978087i \(-0.433241\pi\)
0.208194 + 0.978087i \(0.433241\pi\)
\(90\) 0 0
\(91\) 12.9282 1.35524
\(92\) −8.46410 −0.882444
\(93\) −5.92820 −0.614726
\(94\) −3.46410 −0.357295
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) −17.1244 −1.73871 −0.869357 0.494184i \(-0.835467\pi\)
−0.869357 + 0.494184i \(0.835467\pi\)
\(98\) −0.464102 −0.0468813
\(99\) 5.19615 0.522233
\(100\) 0 0
\(101\) −11.6603 −1.16024 −0.580119 0.814532i \(-0.696994\pi\)
−0.580119 + 0.814532i \(0.696994\pi\)
\(102\) −2.73205 −0.270513
\(103\) 14.8564 1.46385 0.731923 0.681388i \(-0.238623\pi\)
0.731923 + 0.681388i \(0.238623\pi\)
\(104\) 4.73205 0.464016
\(105\) 0 0
\(106\) −4.66025 −0.452644
\(107\) −4.19615 −0.405657 −0.202829 0.979214i \(-0.565013\pi\)
−0.202829 + 0.979214i \(0.565013\pi\)
\(108\) 1.00000 0.0962250
\(109\) −10.3923 −0.995402 −0.497701 0.867349i \(-0.665822\pi\)
−0.497701 + 0.867349i \(0.665822\pi\)
\(110\) 0 0
\(111\) 8.92820 0.847428
\(112\) −2.73205 −0.258155
\(113\) 14.3205 1.34716 0.673580 0.739114i \(-0.264756\pi\)
0.673580 + 0.739114i \(0.264756\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) −9.19615 −0.853841
\(117\) −4.73205 −0.437478
\(118\) −2.19615 −0.202172
\(119\) −7.46410 −0.684233
\(120\) 0 0
\(121\) 16.0000 1.45455
\(122\) −8.26795 −0.748545
\(123\) 0 0
\(124\) −5.92820 −0.532368
\(125\) 0 0
\(126\) 2.73205 0.243390
\(127\) 9.39230 0.833432 0.416716 0.909037i \(-0.363181\pi\)
0.416716 + 0.909037i \(0.363181\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −8.73205 −0.768814
\(130\) 0 0
\(131\) −11.1962 −0.978212 −0.489106 0.872224i \(-0.662677\pi\)
−0.489106 + 0.872224i \(0.662677\pi\)
\(132\) 5.19615 0.452267
\(133\) −2.73205 −0.236899
\(134\) −5.19615 −0.448879
\(135\) 0 0
\(136\) −2.73205 −0.234271
\(137\) −16.3923 −1.40049 −0.700245 0.713903i \(-0.746926\pi\)
−0.700245 + 0.713903i \(0.746926\pi\)
\(138\) 8.46410 0.720512
\(139\) −3.80385 −0.322638 −0.161319 0.986902i \(-0.551575\pi\)
−0.161319 + 0.986902i \(0.551575\pi\)
\(140\) 0 0
\(141\) 3.46410 0.291730
\(142\) 10.1962 0.855642
\(143\) −24.5885 −2.05619
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 14.4641 1.19706
\(147\) 0.464102 0.0382785
\(148\) 8.92820 0.733894
\(149\) −21.8564 −1.79055 −0.895273 0.445517i \(-0.853020\pi\)
−0.895273 + 0.445517i \(0.853020\pi\)
\(150\) 0 0
\(151\) 10.3923 0.845714 0.422857 0.906196i \(-0.361027\pi\)
0.422857 + 0.906196i \(0.361027\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 2.73205 0.220873
\(154\) 14.1962 1.14396
\(155\) 0 0
\(156\) −4.73205 −0.378867
\(157\) −21.8564 −1.74433 −0.872166 0.489211i \(-0.837285\pi\)
−0.872166 + 0.489211i \(0.837285\pi\)
\(158\) −5.92820 −0.471623
\(159\) 4.66025 0.369582
\(160\) 0 0
\(161\) 23.1244 1.82245
\(162\) −1.00000 −0.0785674
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 16.6603 1.29309
\(167\) −2.00000 −0.154765 −0.0773823 0.997001i \(-0.524656\pi\)
−0.0773823 + 0.997001i \(0.524656\pi\)
\(168\) 2.73205 0.210782
\(169\) 9.39230 0.722485
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) −8.73205 −0.665813
\(173\) −22.6603 −1.72283 −0.861414 0.507904i \(-0.830420\pi\)
−0.861414 + 0.507904i \(0.830420\pi\)
\(174\) 9.19615 0.697159
\(175\) 0 0
\(176\) 5.19615 0.391675
\(177\) 2.19615 0.165073
\(178\) −3.92820 −0.294431
\(179\) −9.85641 −0.736702 −0.368351 0.929687i \(-0.620078\pi\)
−0.368351 + 0.929687i \(0.620078\pi\)
\(180\) 0 0
\(181\) 19.2679 1.43218 0.716088 0.698010i \(-0.245931\pi\)
0.716088 + 0.698010i \(0.245931\pi\)
\(182\) −12.9282 −0.958302
\(183\) 8.26795 0.611184
\(184\) 8.46410 0.623982
\(185\) 0 0
\(186\) 5.92820 0.434677
\(187\) 14.1962 1.03813
\(188\) 3.46410 0.252646
\(189\) −2.73205 −0.198727
\(190\) 0 0
\(191\) −24.4641 −1.77016 −0.885080 0.465439i \(-0.845897\pi\)
−0.885080 + 0.465439i \(0.845897\pi\)
\(192\) 1.00000 0.0721688
\(193\) 10.1962 0.733935 0.366968 0.930234i \(-0.380396\pi\)
0.366968 + 0.930234i \(0.380396\pi\)
\(194\) 17.1244 1.22946
\(195\) 0 0
\(196\) 0.464102 0.0331501
\(197\) −0.339746 −0.0242059 −0.0121029 0.999927i \(-0.503853\pi\)
−0.0121029 + 0.999927i \(0.503853\pi\)
\(198\) −5.19615 −0.369274
\(199\) 3.07180 0.217754 0.108877 0.994055i \(-0.465275\pi\)
0.108877 + 0.994055i \(0.465275\pi\)
\(200\) 0 0
\(201\) 5.19615 0.366508
\(202\) 11.6603 0.820413
\(203\) 25.1244 1.76338
\(204\) 2.73205 0.191282
\(205\) 0 0
\(206\) −14.8564 −1.03509
\(207\) −8.46410 −0.588296
\(208\) −4.73205 −0.328109
\(209\) 5.19615 0.359425
\(210\) 0 0
\(211\) 7.19615 0.495404 0.247702 0.968836i \(-0.420325\pi\)
0.247702 + 0.968836i \(0.420325\pi\)
\(212\) 4.66025 0.320068
\(213\) −10.1962 −0.698629
\(214\) 4.19615 0.286843
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 16.1962 1.09947
\(218\) 10.3923 0.703856
\(219\) −14.4641 −0.977393
\(220\) 0 0
\(221\) −12.9282 −0.869645
\(222\) −8.92820 −0.599222
\(223\) −17.9282 −1.20056 −0.600281 0.799789i \(-0.704944\pi\)
−0.600281 + 0.799789i \(0.704944\pi\)
\(224\) 2.73205 0.182543
\(225\) 0 0
\(226\) −14.3205 −0.952586
\(227\) −2.00000 −0.132745 −0.0663723 0.997795i \(-0.521143\pi\)
−0.0663723 + 0.997795i \(0.521143\pi\)
\(228\) 1.00000 0.0662266
\(229\) 5.19615 0.343371 0.171686 0.985152i \(-0.445079\pi\)
0.171686 + 0.985152i \(0.445079\pi\)
\(230\) 0 0
\(231\) −14.1962 −0.934038
\(232\) 9.19615 0.603757
\(233\) 17.6603 1.15696 0.578481 0.815696i \(-0.303646\pi\)
0.578481 + 0.815696i \(0.303646\pi\)
\(234\) 4.73205 0.309344
\(235\) 0 0
\(236\) 2.19615 0.142957
\(237\) 5.92820 0.385078
\(238\) 7.46410 0.483826
\(239\) −7.07180 −0.457437 −0.228718 0.973493i \(-0.573453\pi\)
−0.228718 + 0.973493i \(0.573453\pi\)
\(240\) 0 0
\(241\) 7.80385 0.502690 0.251345 0.967898i \(-0.419127\pi\)
0.251345 + 0.967898i \(0.419127\pi\)
\(242\) −16.0000 −1.02852
\(243\) 1.00000 0.0641500
\(244\) 8.26795 0.529301
\(245\) 0 0
\(246\) 0 0
\(247\) −4.73205 −0.301093
\(248\) 5.92820 0.376441
\(249\) −16.6603 −1.05580
\(250\) 0 0
\(251\) 3.46410 0.218652 0.109326 0.994006i \(-0.465131\pi\)
0.109326 + 0.994006i \(0.465131\pi\)
\(252\) −2.73205 −0.172103
\(253\) −43.9808 −2.76505
\(254\) −9.39230 −0.589326
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 15.9282 0.993574 0.496787 0.867872i \(-0.334513\pi\)
0.496787 + 0.867872i \(0.334513\pi\)
\(258\) 8.73205 0.543634
\(259\) −24.3923 −1.51566
\(260\) 0 0
\(261\) −9.19615 −0.569228
\(262\) 11.1962 0.691701
\(263\) −12.4641 −0.768569 −0.384285 0.923215i \(-0.625552\pi\)
−0.384285 + 0.923215i \(0.625552\pi\)
\(264\) −5.19615 −0.319801
\(265\) 0 0
\(266\) 2.73205 0.167513
\(267\) 3.92820 0.240402
\(268\) 5.19615 0.317406
\(269\) 15.4641 0.942863 0.471431 0.881903i \(-0.343737\pi\)
0.471431 + 0.881903i \(0.343737\pi\)
\(270\) 0 0
\(271\) −25.1244 −1.52620 −0.763098 0.646283i \(-0.776323\pi\)
−0.763098 + 0.646283i \(0.776323\pi\)
\(272\) 2.73205 0.165655
\(273\) 12.9282 0.782450
\(274\) 16.3923 0.990295
\(275\) 0 0
\(276\) −8.46410 −0.509479
\(277\) 24.5167 1.47306 0.736532 0.676403i \(-0.236462\pi\)
0.736532 + 0.676403i \(0.236462\pi\)
\(278\) 3.80385 0.228140
\(279\) −5.92820 −0.354912
\(280\) 0 0
\(281\) −11.0000 −0.656205 −0.328102 0.944642i \(-0.606409\pi\)
−0.328102 + 0.944642i \(0.606409\pi\)
\(282\) −3.46410 −0.206284
\(283\) 4.00000 0.237775 0.118888 0.992908i \(-0.462067\pi\)
0.118888 + 0.992908i \(0.462067\pi\)
\(284\) −10.1962 −0.605030
\(285\) 0 0
\(286\) 24.5885 1.45395
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −9.53590 −0.560935
\(290\) 0 0
\(291\) −17.1244 −1.00385
\(292\) −14.4641 −0.846448
\(293\) 26.6603 1.55751 0.778754 0.627329i \(-0.215852\pi\)
0.778754 + 0.627329i \(0.215852\pi\)
\(294\) −0.464102 −0.0270670
\(295\) 0 0
\(296\) −8.92820 −0.518941
\(297\) 5.19615 0.301511
\(298\) 21.8564 1.26611
\(299\) 40.0526 2.31630
\(300\) 0 0
\(301\) 23.8564 1.37506
\(302\) −10.3923 −0.598010
\(303\) −11.6603 −0.669864
\(304\) 1.00000 0.0573539
\(305\) 0 0
\(306\) −2.73205 −0.156181
\(307\) −4.66025 −0.265975 −0.132987 0.991118i \(-0.542457\pi\)
−0.132987 + 0.991118i \(0.542457\pi\)
\(308\) −14.1962 −0.808901
\(309\) 14.8564 0.845151
\(310\) 0 0
\(311\) −15.4641 −0.876889 −0.438444 0.898758i \(-0.644470\pi\)
−0.438444 + 0.898758i \(0.644470\pi\)
\(312\) 4.73205 0.267900
\(313\) 4.85641 0.274500 0.137250 0.990536i \(-0.456174\pi\)
0.137250 + 0.990536i \(0.456174\pi\)
\(314\) 21.8564 1.23343
\(315\) 0 0
\(316\) 5.92820 0.333487
\(317\) −23.0526 −1.29476 −0.647380 0.762167i \(-0.724135\pi\)
−0.647380 + 0.762167i \(0.724135\pi\)
\(318\) −4.66025 −0.261334
\(319\) −47.7846 −2.67542
\(320\) 0 0
\(321\) −4.19615 −0.234206
\(322\) −23.1244 −1.28867
\(323\) 2.73205 0.152015
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −10.0000 −0.553849
\(327\) −10.3923 −0.574696
\(328\) 0 0
\(329\) −9.46410 −0.521773
\(330\) 0 0
\(331\) 19.1962 1.05512 0.527558 0.849519i \(-0.323108\pi\)
0.527558 + 0.849519i \(0.323108\pi\)
\(332\) −16.6603 −0.914350
\(333\) 8.92820 0.489263
\(334\) 2.00000 0.109435
\(335\) 0 0
\(336\) −2.73205 −0.149046
\(337\) −15.3205 −0.834561 −0.417281 0.908778i \(-0.637017\pi\)
−0.417281 + 0.908778i \(0.637017\pi\)
\(338\) −9.39230 −0.510874
\(339\) 14.3205 0.777783
\(340\) 0 0
\(341\) −30.8038 −1.66812
\(342\) −1.00000 −0.0540738
\(343\) 17.8564 0.964155
\(344\) 8.73205 0.470801
\(345\) 0 0
\(346\) 22.6603 1.21822
\(347\) −1.07180 −0.0575371 −0.0287685 0.999586i \(-0.509159\pi\)
−0.0287685 + 0.999586i \(0.509159\pi\)
\(348\) −9.19615 −0.492966
\(349\) −13.5885 −0.727373 −0.363687 0.931521i \(-0.618482\pi\)
−0.363687 + 0.931521i \(0.618482\pi\)
\(350\) 0 0
\(351\) −4.73205 −0.252578
\(352\) −5.19615 −0.276956
\(353\) 3.41154 0.181578 0.0907890 0.995870i \(-0.471061\pi\)
0.0907890 + 0.995870i \(0.471061\pi\)
\(354\) −2.19615 −0.116724
\(355\) 0 0
\(356\) 3.92820 0.208194
\(357\) −7.46410 −0.395042
\(358\) 9.85641 0.520927
\(359\) 19.8564 1.04798 0.523991 0.851724i \(-0.324443\pi\)
0.523991 + 0.851724i \(0.324443\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −19.2679 −1.01270
\(363\) 16.0000 0.839782
\(364\) 12.9282 0.677622
\(365\) 0 0
\(366\) −8.26795 −0.432173
\(367\) −29.5167 −1.54076 −0.770379 0.637587i \(-0.779933\pi\)
−0.770379 + 0.637587i \(0.779933\pi\)
\(368\) −8.46410 −0.441222
\(369\) 0 0
\(370\) 0 0
\(371\) −12.7321 −0.661015
\(372\) −5.92820 −0.307363
\(373\) −27.6603 −1.43219 −0.716097 0.698001i \(-0.754073\pi\)
−0.716097 + 0.698001i \(0.754073\pi\)
\(374\) −14.1962 −0.734066
\(375\) 0 0
\(376\) −3.46410 −0.178647
\(377\) 43.5167 2.24122
\(378\) 2.73205 0.140522
\(379\) −29.1769 −1.49872 −0.749359 0.662164i \(-0.769638\pi\)
−0.749359 + 0.662164i \(0.769638\pi\)
\(380\) 0 0
\(381\) 9.39230 0.481182
\(382\) 24.4641 1.25169
\(383\) −4.05256 −0.207076 −0.103538 0.994625i \(-0.533016\pi\)
−0.103538 + 0.994625i \(0.533016\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −10.1962 −0.518970
\(387\) −8.73205 −0.443875
\(388\) −17.1244 −0.869357
\(389\) 12.9282 0.655486 0.327743 0.944767i \(-0.393712\pi\)
0.327743 + 0.944767i \(0.393712\pi\)
\(390\) 0 0
\(391\) −23.1244 −1.16945
\(392\) −0.464102 −0.0234407
\(393\) −11.1962 −0.564771
\(394\) 0.339746 0.0171162
\(395\) 0 0
\(396\) 5.19615 0.261116
\(397\) 34.1244 1.71265 0.856326 0.516435i \(-0.172741\pi\)
0.856326 + 0.516435i \(0.172741\pi\)
\(398\) −3.07180 −0.153975
\(399\) −2.73205 −0.136774
\(400\) 0 0
\(401\) 18.7128 0.934473 0.467237 0.884132i \(-0.345250\pi\)
0.467237 + 0.884132i \(0.345250\pi\)
\(402\) −5.19615 −0.259161
\(403\) 28.0526 1.39740
\(404\) −11.6603 −0.580119
\(405\) 0 0
\(406\) −25.1244 −1.24690
\(407\) 46.3923 2.29958
\(408\) −2.73205 −0.135257
\(409\) 17.3205 0.856444 0.428222 0.903674i \(-0.359140\pi\)
0.428222 + 0.903674i \(0.359140\pi\)
\(410\) 0 0
\(411\) −16.3923 −0.808573
\(412\) 14.8564 0.731923
\(413\) −6.00000 −0.295241
\(414\) 8.46410 0.415988
\(415\) 0 0
\(416\) 4.73205 0.232008
\(417\) −3.80385 −0.186275
\(418\) −5.19615 −0.254152
\(419\) −5.85641 −0.286104 −0.143052 0.989715i \(-0.545692\pi\)
−0.143052 + 0.989715i \(0.545692\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) −7.19615 −0.350303
\(423\) 3.46410 0.168430
\(424\) −4.66025 −0.226322
\(425\) 0 0
\(426\) 10.1962 0.494005
\(427\) −22.5885 −1.09313
\(428\) −4.19615 −0.202829
\(429\) −24.5885 −1.18714
\(430\) 0 0
\(431\) −1.60770 −0.0774400 −0.0387200 0.999250i \(-0.512328\pi\)
−0.0387200 + 0.999250i \(0.512328\pi\)
\(432\) 1.00000 0.0481125
\(433\) −5.80385 −0.278915 −0.139458 0.990228i \(-0.544536\pi\)
−0.139458 + 0.990228i \(0.544536\pi\)
\(434\) −16.1962 −0.777440
\(435\) 0 0
\(436\) −10.3923 −0.497701
\(437\) −8.46410 −0.404893
\(438\) 14.4641 0.691122
\(439\) 3.39230 0.161906 0.0809529 0.996718i \(-0.474204\pi\)
0.0809529 + 0.996718i \(0.474204\pi\)
\(440\) 0 0
\(441\) 0.464102 0.0221001
\(442\) 12.9282 0.614932
\(443\) 20.5167 0.974776 0.487388 0.873186i \(-0.337950\pi\)
0.487388 + 0.873186i \(0.337950\pi\)
\(444\) 8.92820 0.423714
\(445\) 0 0
\(446\) 17.9282 0.848925
\(447\) −21.8564 −1.03377
\(448\) −2.73205 −0.129077
\(449\) 0.464102 0.0219023 0.0109512 0.999940i \(-0.496514\pi\)
0.0109512 + 0.999940i \(0.496514\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 14.3205 0.673580
\(453\) 10.3923 0.488273
\(454\) 2.00000 0.0938647
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) 32.0000 1.49690 0.748448 0.663193i \(-0.230799\pi\)
0.748448 + 0.663193i \(0.230799\pi\)
\(458\) −5.19615 −0.242800
\(459\) 2.73205 0.127521
\(460\) 0 0
\(461\) −1.85641 −0.0864615 −0.0432307 0.999065i \(-0.513765\pi\)
−0.0432307 + 0.999065i \(0.513765\pi\)
\(462\) 14.1962 0.660465
\(463\) −11.6077 −0.539455 −0.269728 0.962937i \(-0.586934\pi\)
−0.269728 + 0.962937i \(0.586934\pi\)
\(464\) −9.19615 −0.426921
\(465\) 0 0
\(466\) −17.6603 −0.818095
\(467\) −16.6603 −0.770945 −0.385472 0.922719i \(-0.625961\pi\)
−0.385472 + 0.922719i \(0.625961\pi\)
\(468\) −4.73205 −0.218739
\(469\) −14.1962 −0.655517
\(470\) 0 0
\(471\) −21.8564 −1.00709
\(472\) −2.19615 −0.101086
\(473\) −45.3731 −2.08626
\(474\) −5.92820 −0.272291
\(475\) 0 0
\(476\) −7.46410 −0.342117
\(477\) 4.66025 0.213378
\(478\) 7.07180 0.323456
\(479\) −16.6077 −0.758825 −0.379412 0.925228i \(-0.623874\pi\)
−0.379412 + 0.925228i \(0.623874\pi\)
\(480\) 0 0
\(481\) −42.2487 −1.92638
\(482\) −7.80385 −0.355456
\(483\) 23.1244 1.05219
\(484\) 16.0000 0.727273
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 13.0718 0.592340 0.296170 0.955135i \(-0.404291\pi\)
0.296170 + 0.955135i \(0.404291\pi\)
\(488\) −8.26795 −0.374272
\(489\) 10.0000 0.452216
\(490\) 0 0
\(491\) −34.9282 −1.57629 −0.788144 0.615491i \(-0.788958\pi\)
−0.788144 + 0.615491i \(0.788958\pi\)
\(492\) 0 0
\(493\) −25.1244 −1.13154
\(494\) 4.73205 0.212905
\(495\) 0 0
\(496\) −5.92820 −0.266184
\(497\) 27.8564 1.24953
\(498\) 16.6603 0.746564
\(499\) 13.1244 0.587527 0.293763 0.955878i \(-0.405092\pi\)
0.293763 + 0.955878i \(0.405092\pi\)
\(500\) 0 0
\(501\) −2.00000 −0.0893534
\(502\) −3.46410 −0.154610
\(503\) 12.7846 0.570038 0.285019 0.958522i \(-0.408000\pi\)
0.285019 + 0.958522i \(0.408000\pi\)
\(504\) 2.73205 0.121695
\(505\) 0 0
\(506\) 43.9808 1.95518
\(507\) 9.39230 0.417127
\(508\) 9.39230 0.416716
\(509\) 0.267949 0.0118766 0.00593832 0.999982i \(-0.498110\pi\)
0.00593832 + 0.999982i \(0.498110\pi\)
\(510\) 0 0
\(511\) 39.5167 1.74811
\(512\) −1.00000 −0.0441942
\(513\) 1.00000 0.0441511
\(514\) −15.9282 −0.702563
\(515\) 0 0
\(516\) −8.73205 −0.384407
\(517\) 18.0000 0.791639
\(518\) 24.3923 1.07174
\(519\) −22.6603 −0.994675
\(520\) 0 0
\(521\) 3.67949 0.161201 0.0806007 0.996746i \(-0.474316\pi\)
0.0806007 + 0.996746i \(0.474316\pi\)
\(522\) 9.19615 0.402505
\(523\) −17.8564 −0.780806 −0.390403 0.920644i \(-0.627664\pi\)
−0.390403 + 0.920644i \(0.627664\pi\)
\(524\) −11.1962 −0.489106
\(525\) 0 0
\(526\) 12.4641 0.543461
\(527\) −16.1962 −0.705515
\(528\) 5.19615 0.226134
\(529\) 48.6410 2.11483
\(530\) 0 0
\(531\) 2.19615 0.0953049
\(532\) −2.73205 −0.118449
\(533\) 0 0
\(534\) −3.92820 −0.169990
\(535\) 0 0
\(536\) −5.19615 −0.224440
\(537\) −9.85641 −0.425335
\(538\) −15.4641 −0.666705
\(539\) 2.41154 0.103872
\(540\) 0 0
\(541\) 8.66025 0.372333 0.186167 0.982518i \(-0.440394\pi\)
0.186167 + 0.982518i \(0.440394\pi\)
\(542\) 25.1244 1.07918
\(543\) 19.2679 0.826867
\(544\) −2.73205 −0.117136
\(545\) 0 0
\(546\) −12.9282 −0.553276
\(547\) −3.58846 −0.153431 −0.0767157 0.997053i \(-0.524443\pi\)
−0.0767157 + 0.997053i \(0.524443\pi\)
\(548\) −16.3923 −0.700245
\(549\) 8.26795 0.352867
\(550\) 0 0
\(551\) −9.19615 −0.391769
\(552\) 8.46410 0.360256
\(553\) −16.1962 −0.688730
\(554\) −24.5167 −1.04161
\(555\) 0 0
\(556\) −3.80385 −0.161319
\(557\) 38.5885 1.63505 0.817523 0.575896i \(-0.195347\pi\)
0.817523 + 0.575896i \(0.195347\pi\)
\(558\) 5.92820 0.250961
\(559\) 41.3205 1.74767
\(560\) 0 0
\(561\) 14.1962 0.599362
\(562\) 11.0000 0.464007
\(563\) 32.1962 1.35691 0.678453 0.734644i \(-0.262651\pi\)
0.678453 + 0.734644i \(0.262651\pi\)
\(564\) 3.46410 0.145865
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) −2.73205 −0.114735
\(568\) 10.1962 0.427821
\(569\) 13.7128 0.574871 0.287436 0.957800i \(-0.407197\pi\)
0.287436 + 0.957800i \(0.407197\pi\)
\(570\) 0 0
\(571\) −10.7321 −0.449122 −0.224561 0.974460i \(-0.572095\pi\)
−0.224561 + 0.974460i \(0.572095\pi\)
\(572\) −24.5885 −1.02810
\(573\) −24.4641 −1.02200
\(574\) 0 0
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 20.3205 0.845954 0.422977 0.906140i \(-0.360985\pi\)
0.422977 + 0.906140i \(0.360985\pi\)
\(578\) 9.53590 0.396641
\(579\) 10.1962 0.423738
\(580\) 0 0
\(581\) 45.5167 1.88835
\(582\) 17.1244 0.709827
\(583\) 24.2154 1.00290
\(584\) 14.4641 0.598529
\(585\) 0 0
\(586\) −26.6603 −1.10132
\(587\) 18.2679 0.753999 0.376999 0.926213i \(-0.376956\pi\)
0.376999 + 0.926213i \(0.376956\pi\)
\(588\) 0.464102 0.0191392
\(589\) −5.92820 −0.244267
\(590\) 0 0
\(591\) −0.339746 −0.0139753
\(592\) 8.92820 0.366947
\(593\) 30.2487 1.24217 0.621083 0.783745i \(-0.286693\pi\)
0.621083 + 0.783745i \(0.286693\pi\)
\(594\) −5.19615 −0.213201
\(595\) 0 0
\(596\) −21.8564 −0.895273
\(597\) 3.07180 0.125720
\(598\) −40.0526 −1.63787
\(599\) −16.9808 −0.693815 −0.346908 0.937899i \(-0.612768\pi\)
−0.346908 + 0.937899i \(0.612768\pi\)
\(600\) 0 0
\(601\) 25.8038 1.05256 0.526280 0.850311i \(-0.323586\pi\)
0.526280 + 0.850311i \(0.323586\pi\)
\(602\) −23.8564 −0.972315
\(603\) 5.19615 0.211604
\(604\) 10.3923 0.422857
\(605\) 0 0
\(606\) 11.6603 0.473665
\(607\) 35.1051 1.42487 0.712436 0.701737i \(-0.247592\pi\)
0.712436 + 0.701737i \(0.247592\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 25.1244 1.01809
\(610\) 0 0
\(611\) −16.3923 −0.663162
\(612\) 2.73205 0.110437
\(613\) −7.07180 −0.285627 −0.142814 0.989750i \(-0.545615\pi\)
−0.142814 + 0.989750i \(0.545615\pi\)
\(614\) 4.66025 0.188073
\(615\) 0 0
\(616\) 14.1962 0.571979
\(617\) −1.32051 −0.0531617 −0.0265808 0.999647i \(-0.508462\pi\)
−0.0265808 + 0.999647i \(0.508462\pi\)
\(618\) −14.8564 −0.597612
\(619\) 26.9808 1.08445 0.542224 0.840234i \(-0.317582\pi\)
0.542224 + 0.840234i \(0.317582\pi\)
\(620\) 0 0
\(621\) −8.46410 −0.339653
\(622\) 15.4641 0.620054
\(623\) −10.7321 −0.429971
\(624\) −4.73205 −0.189434
\(625\) 0 0
\(626\) −4.85641 −0.194101
\(627\) 5.19615 0.207514
\(628\) −21.8564 −0.872166
\(629\) 24.3923 0.972585
\(630\) 0 0
\(631\) −2.92820 −0.116570 −0.0582850 0.998300i \(-0.518563\pi\)
−0.0582850 + 0.998300i \(0.518563\pi\)
\(632\) −5.92820 −0.235811
\(633\) 7.19615 0.286021
\(634\) 23.0526 0.915534
\(635\) 0 0
\(636\) 4.66025 0.184791
\(637\) −2.19615 −0.0870147
\(638\) 47.7846 1.89181
\(639\) −10.1962 −0.403354
\(640\) 0 0
\(641\) −12.3923 −0.489467 −0.244733 0.969590i \(-0.578700\pi\)
−0.244733 + 0.969590i \(0.578700\pi\)
\(642\) 4.19615 0.165609
\(643\) −32.9282 −1.29856 −0.649281 0.760549i \(-0.724930\pi\)
−0.649281 + 0.760549i \(0.724930\pi\)
\(644\) 23.1244 0.911227
\(645\) 0 0
\(646\) −2.73205 −0.107491
\(647\) 13.9282 0.547574 0.273787 0.961790i \(-0.411724\pi\)
0.273787 + 0.961790i \(0.411724\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 11.4115 0.447942
\(650\) 0 0
\(651\) 16.1962 0.634777
\(652\) 10.0000 0.391630
\(653\) −25.1769 −0.985249 −0.492624 0.870242i \(-0.663962\pi\)
−0.492624 + 0.870242i \(0.663962\pi\)
\(654\) 10.3923 0.406371
\(655\) 0 0
\(656\) 0 0
\(657\) −14.4641 −0.564298
\(658\) 9.46410 0.368949
\(659\) −7.26795 −0.283119 −0.141560 0.989930i \(-0.545212\pi\)
−0.141560 + 0.989930i \(0.545212\pi\)
\(660\) 0 0
\(661\) 26.9282 1.04739 0.523693 0.851907i \(-0.324554\pi\)
0.523693 + 0.851907i \(0.324554\pi\)
\(662\) −19.1962 −0.746080
\(663\) −12.9282 −0.502090
\(664\) 16.6603 0.646543
\(665\) 0 0
\(666\) −8.92820 −0.345961
\(667\) 77.8372 3.01387
\(668\) −2.00000 −0.0773823
\(669\) −17.9282 −0.693144
\(670\) 0 0
\(671\) 42.9615 1.65851
\(672\) 2.73205 0.105391
\(673\) 29.2679 1.12820 0.564098 0.825708i \(-0.309224\pi\)
0.564098 + 0.825708i \(0.309224\pi\)
\(674\) 15.3205 0.590124
\(675\) 0 0
\(676\) 9.39230 0.361242
\(677\) 23.8756 0.917616 0.458808 0.888536i \(-0.348277\pi\)
0.458808 + 0.888536i \(0.348277\pi\)
\(678\) −14.3205 −0.549976
\(679\) 46.7846 1.79543
\(680\) 0 0
\(681\) −2.00000 −0.0766402
\(682\) 30.8038 1.17954
\(683\) 20.8756 0.798784 0.399392 0.916780i \(-0.369221\pi\)
0.399392 + 0.916780i \(0.369221\pi\)
\(684\) 1.00000 0.0382360
\(685\) 0 0
\(686\) −17.8564 −0.681761
\(687\) 5.19615 0.198246
\(688\) −8.73205 −0.332906
\(689\) −22.0526 −0.840136
\(690\) 0 0
\(691\) 19.1244 0.727525 0.363762 0.931492i \(-0.381492\pi\)
0.363762 + 0.931492i \(0.381492\pi\)
\(692\) −22.6603 −0.861414
\(693\) −14.1962 −0.539267
\(694\) 1.07180 0.0406848
\(695\) 0 0
\(696\) 9.19615 0.348579
\(697\) 0 0
\(698\) 13.5885 0.514331
\(699\) 17.6603 0.667972
\(700\) 0 0
\(701\) 43.3205 1.63619 0.818097 0.575081i \(-0.195029\pi\)
0.818097 + 0.575081i \(0.195029\pi\)
\(702\) 4.73205 0.178600
\(703\) 8.92820 0.336734
\(704\) 5.19615 0.195837
\(705\) 0 0
\(706\) −3.41154 −0.128395
\(707\) 31.8564 1.19808
\(708\) 2.19615 0.0825365
\(709\) −29.4449 −1.10583 −0.552913 0.833239i \(-0.686484\pi\)
−0.552913 + 0.833239i \(0.686484\pi\)
\(710\) 0 0
\(711\) 5.92820 0.222325
\(712\) −3.92820 −0.147216
\(713\) 50.1769 1.87914
\(714\) 7.46410 0.279337
\(715\) 0 0
\(716\) −9.85641 −0.368351
\(717\) −7.07180 −0.264101
\(718\) −19.8564 −0.741035
\(719\) −40.5692 −1.51298 −0.756488 0.654007i \(-0.773087\pi\)
−0.756488 + 0.654007i \(0.773087\pi\)
\(720\) 0 0
\(721\) −40.5885 −1.51159
\(722\) −1.00000 −0.0372161
\(723\) 7.80385 0.290228
\(724\) 19.2679 0.716088
\(725\) 0 0
\(726\) −16.0000 −0.593816
\(727\) 3.46410 0.128476 0.0642382 0.997935i \(-0.479538\pi\)
0.0642382 + 0.997935i \(0.479538\pi\)
\(728\) −12.9282 −0.479151
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −23.8564 −0.882361
\(732\) 8.26795 0.305592
\(733\) 11.7321 0.433333 0.216667 0.976246i \(-0.430482\pi\)
0.216667 + 0.976246i \(0.430482\pi\)
\(734\) 29.5167 1.08948
\(735\) 0 0
\(736\) 8.46410 0.311991
\(737\) 27.0000 0.994558
\(738\) 0 0
\(739\) −22.2487 −0.818432 −0.409216 0.912438i \(-0.634198\pi\)
−0.409216 + 0.912438i \(0.634198\pi\)
\(740\) 0 0
\(741\) −4.73205 −0.173836
\(742\) 12.7321 0.467408
\(743\) 16.7846 0.615768 0.307884 0.951424i \(-0.400379\pi\)
0.307884 + 0.951424i \(0.400379\pi\)
\(744\) 5.92820 0.217338
\(745\) 0 0
\(746\) 27.6603 1.01271
\(747\) −16.6603 −0.609567
\(748\) 14.1962 0.519063
\(749\) 11.4641 0.418889
\(750\) 0 0
\(751\) 31.4641 1.14814 0.574071 0.818806i \(-0.305363\pi\)
0.574071 + 0.818806i \(0.305363\pi\)
\(752\) 3.46410 0.126323
\(753\) 3.46410 0.126239
\(754\) −43.5167 −1.58478
\(755\) 0 0
\(756\) −2.73205 −0.0993637
\(757\) −11.9808 −0.435448 −0.217724 0.976010i \(-0.569863\pi\)
−0.217724 + 0.976010i \(0.569863\pi\)
\(758\) 29.1769 1.05975
\(759\) −43.9808 −1.59640
\(760\) 0 0
\(761\) −21.5167 −0.779978 −0.389989 0.920819i \(-0.627521\pi\)
−0.389989 + 0.920819i \(0.627521\pi\)
\(762\) −9.39230 −0.340247
\(763\) 28.3923 1.02787
\(764\) −24.4641 −0.885080
\(765\) 0 0
\(766\) 4.05256 0.146425
\(767\) −10.3923 −0.375244
\(768\) 1.00000 0.0360844
\(769\) 51.9282 1.87258 0.936289 0.351229i \(-0.114236\pi\)
0.936289 + 0.351229i \(0.114236\pi\)
\(770\) 0 0
\(771\) 15.9282 0.573640
\(772\) 10.1962 0.366968
\(773\) 42.3923 1.52475 0.762373 0.647138i \(-0.224034\pi\)
0.762373 + 0.647138i \(0.224034\pi\)
\(774\) 8.73205 0.313867
\(775\) 0 0
\(776\) 17.1244 0.614729
\(777\) −24.3923 −0.875069
\(778\) −12.9282 −0.463499
\(779\) 0 0
\(780\) 0 0
\(781\) −52.9808 −1.89580
\(782\) 23.1244 0.826925
\(783\) −9.19615 −0.328644
\(784\) 0.464102 0.0165751
\(785\) 0 0
\(786\) 11.1962 0.399354
\(787\) −41.9808 −1.49645 −0.748226 0.663444i \(-0.769094\pi\)
−0.748226 + 0.663444i \(0.769094\pi\)
\(788\) −0.339746 −0.0121029
\(789\) −12.4641 −0.443734
\(790\) 0 0
\(791\) −39.1244 −1.39110
\(792\) −5.19615 −0.184637
\(793\) −39.1244 −1.38935
\(794\) −34.1244 −1.21103
\(795\) 0 0
\(796\) 3.07180 0.108877
\(797\) −29.5692 −1.04740 −0.523698 0.851904i \(-0.675448\pi\)
−0.523698 + 0.851904i \(0.675448\pi\)
\(798\) 2.73205 0.0967136
\(799\) 9.46410 0.334816
\(800\) 0 0
\(801\) 3.92820 0.138796
\(802\) −18.7128 −0.660772
\(803\) −75.1577 −2.65226
\(804\) 5.19615 0.183254
\(805\) 0 0
\(806\) −28.0526 −0.988109
\(807\) 15.4641 0.544362
\(808\) 11.6603 0.410206
\(809\) −19.1244 −0.672377 −0.336188 0.941795i \(-0.609138\pi\)
−0.336188 + 0.941795i \(0.609138\pi\)
\(810\) 0 0
\(811\) −18.5167 −0.650208 −0.325104 0.945678i \(-0.605399\pi\)
−0.325104 + 0.945678i \(0.605399\pi\)
\(812\) 25.1244 0.881692
\(813\) −25.1244 −0.881150
\(814\) −46.3923 −1.62605
\(815\) 0 0
\(816\) 2.73205 0.0956409
\(817\) −8.73205 −0.305496
\(818\) −17.3205 −0.605597
\(819\) 12.9282 0.451748
\(820\) 0 0
\(821\) 36.1962 1.26325 0.631627 0.775272i \(-0.282387\pi\)
0.631627 + 0.775272i \(0.282387\pi\)
\(822\) 16.3923 0.571747
\(823\) 25.7128 0.896292 0.448146 0.893960i \(-0.352084\pi\)
0.448146 + 0.893960i \(0.352084\pi\)
\(824\) −14.8564 −0.517547
\(825\) 0 0
\(826\) 6.00000 0.208767
\(827\) −12.7321 −0.442737 −0.221368 0.975190i \(-0.571052\pi\)
−0.221368 + 0.975190i \(0.571052\pi\)
\(828\) −8.46410 −0.294148
\(829\) −6.73205 −0.233814 −0.116907 0.993143i \(-0.537298\pi\)
−0.116907 + 0.993143i \(0.537298\pi\)
\(830\) 0 0
\(831\) 24.5167 0.850474
\(832\) −4.73205 −0.164054
\(833\) 1.26795 0.0439318
\(834\) 3.80385 0.131716
\(835\) 0 0
\(836\) 5.19615 0.179713
\(837\) −5.92820 −0.204909
\(838\) 5.85641 0.202306
\(839\) −35.9090 −1.23972 −0.619858 0.784714i \(-0.712810\pi\)
−0.619858 + 0.784714i \(0.712810\pi\)
\(840\) 0 0
\(841\) 55.5692 1.91618
\(842\) 2.00000 0.0689246
\(843\) −11.0000 −0.378860
\(844\) 7.19615 0.247702
\(845\) 0 0
\(846\) −3.46410 −0.119098
\(847\) −43.7128 −1.50199
\(848\) 4.66025 0.160034
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) −75.5692 −2.59048
\(852\) −10.1962 −0.349314
\(853\) 11.1769 0.382690 0.191345 0.981523i \(-0.438715\pi\)
0.191345 + 0.981523i \(0.438715\pi\)
\(854\) 22.5885 0.772961
\(855\) 0 0
\(856\) 4.19615 0.143422
\(857\) −13.8564 −0.473326 −0.236663 0.971592i \(-0.576054\pi\)
−0.236663 + 0.971592i \(0.576054\pi\)
\(858\) 24.5885 0.839436
\(859\) −30.7846 −1.05036 −0.525179 0.850992i \(-0.676002\pi\)
−0.525179 + 0.850992i \(0.676002\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 1.60770 0.0547583
\(863\) 23.6077 0.803615 0.401808 0.915724i \(-0.368382\pi\)
0.401808 + 0.915724i \(0.368382\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) 5.80385 0.197223
\(867\) −9.53590 −0.323856
\(868\) 16.1962 0.549733
\(869\) 30.8038 1.04495
\(870\) 0 0
\(871\) −24.5885 −0.833148
\(872\) 10.3923 0.351928
\(873\) −17.1244 −0.579572
\(874\) 8.46410 0.286303
\(875\) 0 0
\(876\) −14.4641 −0.488697
\(877\) −51.4641 −1.73782 −0.868910 0.494971i \(-0.835179\pi\)
−0.868910 + 0.494971i \(0.835179\pi\)
\(878\) −3.39230 −0.114485
\(879\) 26.6603 0.899228
\(880\) 0 0
\(881\) 19.6077 0.660600 0.330300 0.943876i \(-0.392850\pi\)
0.330300 + 0.943876i \(0.392850\pi\)
\(882\) −0.464102 −0.0156271
\(883\) −43.9090 −1.47765 −0.738827 0.673895i \(-0.764620\pi\)
−0.738827 + 0.673895i \(0.764620\pi\)
\(884\) −12.9282 −0.434823
\(885\) 0 0
\(886\) −20.5167 −0.689271
\(887\) 1.46410 0.0491597 0.0245799 0.999698i \(-0.492175\pi\)
0.0245799 + 0.999698i \(0.492175\pi\)
\(888\) −8.92820 −0.299611
\(889\) −25.6603 −0.860617
\(890\) 0 0
\(891\) 5.19615 0.174078
\(892\) −17.9282 −0.600281
\(893\) 3.46410 0.115922
\(894\) 21.8564 0.730988
\(895\) 0 0
\(896\) 2.73205 0.0912714
\(897\) 40.0526 1.33732
\(898\) −0.464102 −0.0154873
\(899\) 54.5167 1.81823
\(900\) 0 0
\(901\) 12.7321 0.424166
\(902\) 0 0
\(903\) 23.8564 0.793891
\(904\) −14.3205 −0.476293
\(905\) 0 0
\(906\) −10.3923 −0.345261
\(907\) −6.67949 −0.221789 −0.110894 0.993832i \(-0.535372\pi\)
−0.110894 + 0.993832i \(0.535372\pi\)
\(908\) −2.00000 −0.0663723
\(909\) −11.6603 −0.386746
\(910\) 0 0
\(911\) −17.6077 −0.583369 −0.291684 0.956515i \(-0.594216\pi\)
−0.291684 + 0.956515i \(0.594216\pi\)
\(912\) 1.00000 0.0331133
\(913\) −86.5692 −2.86502
\(914\) −32.0000 −1.05847
\(915\) 0 0
\(916\) 5.19615 0.171686
\(917\) 30.5885 1.01012
\(918\) −2.73205 −0.0901711
\(919\) 36.8372 1.21515 0.607573 0.794264i \(-0.292143\pi\)
0.607573 + 0.794264i \(0.292143\pi\)
\(920\) 0 0
\(921\) −4.66025 −0.153561
\(922\) 1.85641 0.0611375
\(923\) 48.2487 1.58813
\(924\) −14.1962 −0.467019
\(925\) 0 0
\(926\) 11.6077 0.381453
\(927\) 14.8564 0.487948
\(928\) 9.19615 0.301878
\(929\) 34.9282 1.14596 0.572979 0.819570i \(-0.305788\pi\)
0.572979 + 0.819570i \(0.305788\pi\)
\(930\) 0 0
\(931\) 0.464102 0.0152103
\(932\) 17.6603 0.578481
\(933\) −15.4641 −0.506272
\(934\) 16.6603 0.545140
\(935\) 0 0
\(936\) 4.73205 0.154672
\(937\) −5.21539 −0.170379 −0.0851897 0.996365i \(-0.527150\pi\)
−0.0851897 + 0.996365i \(0.527150\pi\)
\(938\) 14.1962 0.463521
\(939\) 4.85641 0.158483
\(940\) 0 0
\(941\) −7.73205 −0.252058 −0.126029 0.992027i \(-0.540223\pi\)
−0.126029 + 0.992027i \(0.540223\pi\)
\(942\) 21.8564 0.712120
\(943\) 0 0
\(944\) 2.19615 0.0714787
\(945\) 0 0
\(946\) 45.3731 1.47521
\(947\) 29.8564 0.970203 0.485101 0.874458i \(-0.338783\pi\)
0.485101 + 0.874458i \(0.338783\pi\)
\(948\) 5.92820 0.192539
\(949\) 68.4449 2.22181
\(950\) 0 0
\(951\) −23.0526 −0.747530
\(952\) 7.46410 0.241913
\(953\) 7.39230 0.239460 0.119730 0.992806i \(-0.461797\pi\)
0.119730 + 0.992806i \(0.461797\pi\)
\(954\) −4.66025 −0.150881
\(955\) 0 0
\(956\) −7.07180 −0.228718
\(957\) −47.7846 −1.54466
\(958\) 16.6077 0.536570
\(959\) 44.7846 1.44617
\(960\) 0 0
\(961\) 4.14359 0.133664
\(962\) 42.2487 1.36215
\(963\) −4.19615 −0.135219
\(964\) 7.80385 0.251345
\(965\) 0 0
\(966\) −23.1244 −0.744014
\(967\) 49.0333 1.57681 0.788403 0.615160i \(-0.210908\pi\)
0.788403 + 0.615160i \(0.210908\pi\)
\(968\) −16.0000 −0.514259
\(969\) 2.73205 0.0877661
\(970\) 0 0
\(971\) −33.8038 −1.08482 −0.542409 0.840115i \(-0.682488\pi\)
−0.542409 + 0.840115i \(0.682488\pi\)
\(972\) 1.00000 0.0320750
\(973\) 10.3923 0.333162
\(974\) −13.0718 −0.418847
\(975\) 0 0
\(976\) 8.26795 0.264651
\(977\) 2.53590 0.0811306 0.0405653 0.999177i \(-0.487084\pi\)
0.0405653 + 0.999177i \(0.487084\pi\)
\(978\) −10.0000 −0.319765
\(979\) 20.4115 0.652356
\(980\) 0 0
\(981\) −10.3923 −0.331801
\(982\) 34.9282 1.11460
\(983\) 11.6603 0.371904 0.185952 0.982559i \(-0.440463\pi\)
0.185952 + 0.982559i \(0.440463\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 25.1244 0.800122
\(987\) −9.46410 −0.301246
\(988\) −4.73205 −0.150547
\(989\) 73.9090 2.35017
\(990\) 0 0
\(991\) 13.7846 0.437883 0.218941 0.975738i \(-0.429740\pi\)
0.218941 + 0.975738i \(0.429740\pi\)
\(992\) 5.92820 0.188221
\(993\) 19.1962 0.609171
\(994\) −27.8564 −0.883552
\(995\) 0 0
\(996\) −16.6603 −0.527900
\(997\) −23.3397 −0.739177 −0.369589 0.929196i \(-0.620501\pi\)
−0.369589 + 0.929196i \(0.620501\pi\)
\(998\) −13.1244 −0.415444
\(999\) 8.92820 0.282476
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 2850.2.a.be.1.1 2
3.2 odd 2 8550.2.a.bz.1.1 2
5.2 odd 4 2850.2.d.u.799.1 4
5.3 odd 4 2850.2.d.u.799.4 4
5.4 even 2 2850.2.a.bh.1.2 yes 2
15.14 odd 2 8550.2.a.bt.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2850.2.a.be.1.1 2 1.1 even 1 trivial
2850.2.a.bh.1.2 yes 2 5.4 even 2
2850.2.d.u.799.1 4 5.2 odd 4
2850.2.d.u.799.4 4 5.3 odd 4
8550.2.a.bt.1.2 2 15.14 odd 2
8550.2.a.bz.1.1 2 3.2 odd 2