Properties

Label 2850.2.a.bi.1.1
Level $2850$
Weight $2$
Character 2850.1
Self dual yes
Analytic conductor $22.757$
Analytic rank $0$
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: \(0\)
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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -0.732051 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} -0.732051 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.73205 q^{11} +1.00000 q^{12} +2.73205 q^{13} -0.732051 q^{14} +1.00000 q^{16} +6.19615 q^{17} +1.00000 q^{18} -1.00000 q^{19} -0.732051 q^{21} +3.73205 q^{22} -5.92820 q^{23} +1.00000 q^{24} +2.73205 q^{26} +1.00000 q^{27} -0.732051 q^{28} -1.73205 q^{29} -2.46410 q^{31} +1.00000 q^{32} +3.73205 q^{33} +6.19615 q^{34} +1.00000 q^{36} +2.00000 q^{37} -1.00000 q^{38} +2.73205 q^{39} -2.92820 q^{41} -0.732051 q^{42} +8.19615 q^{43} +3.73205 q^{44} -5.92820 q^{46} -3.46410 q^{47} +1.00000 q^{48} -6.46410 q^{49} +6.19615 q^{51} +2.73205 q^{52} +1.73205 q^{53} +1.00000 q^{54} -0.732051 q^{56} -1.00000 q^{57} -1.73205 q^{58} -8.19615 q^{59} +10.6603 q^{61} -2.46410 q^{62} -0.732051 q^{63} +1.00000 q^{64} +3.73205 q^{66} +0.267949 q^{67} +6.19615 q^{68} -5.92820 q^{69} +12.1962 q^{71} +1.00000 q^{72} +2.46410 q^{73} +2.00000 q^{74} -1.00000 q^{76} -2.73205 q^{77} +2.73205 q^{78} -5.53590 q^{79} +1.00000 q^{81} -2.92820 q^{82} -3.73205 q^{83} -0.732051 q^{84} +8.19615 q^{86} -1.73205 q^{87} +3.73205 q^{88} +10.8564 q^{89} -2.00000 q^{91} -5.92820 q^{92} -2.46410 q^{93} -3.46410 q^{94} +1.00000 q^{96} +15.1244 q^{97} -6.46410 q^{98} +3.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} + 4 q^{11} + 2 q^{12} + 2 q^{13} + 2 q^{14} + 2 q^{16} + 2 q^{17} + 2 q^{18} - 2 q^{19} + 2 q^{21} + 4 q^{22} + 2 q^{23} + 2 q^{24} + 2 q^{26} + 2 q^{27} + 2 q^{28} + 2 q^{31} + 2 q^{32} + 4 q^{33} + 2 q^{34} + 2 q^{36} + 4 q^{37} - 2 q^{38} + 2 q^{39} + 8 q^{41} + 2 q^{42} + 6 q^{43} + 4 q^{44} + 2 q^{46} + 2 q^{48} - 6 q^{49} + 2 q^{51} + 2 q^{52} + 2 q^{54} + 2 q^{56} - 2 q^{57} - 6 q^{59} + 4 q^{61} + 2 q^{62} + 2 q^{63} + 2 q^{64} + 4 q^{66} + 4 q^{67} + 2 q^{68} + 2 q^{69} + 14 q^{71} + 2 q^{72} - 2 q^{73} + 4 q^{74} - 2 q^{76} - 2 q^{77} + 2 q^{78} - 18 q^{79} + 2 q^{81} + 8 q^{82} - 4 q^{83} + 2 q^{84} + 6 q^{86} + 4 q^{88} - 6 q^{89} - 4 q^{91} + 2 q^{92} + 2 q^{93} + 2 q^{96} + 6 q^{97} - 6 q^{98} + 4 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) −0.732051 −0.276689 −0.138345 0.990384i \(-0.544178\pi\)
−0.138345 + 0.990384i \(0.544178\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.73205 1.12526 0.562628 0.826710i \(-0.309790\pi\)
0.562628 + 0.826710i \(0.309790\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.73205 0.757735 0.378867 0.925451i \(-0.376314\pi\)
0.378867 + 0.925451i \(0.376314\pi\)
\(14\) −0.732051 −0.195649
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 6.19615 1.50279 0.751394 0.659854i \(-0.229382\pi\)
0.751394 + 0.659854i \(0.229382\pi\)
\(18\) 1.00000 0.235702
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −0.732051 −0.159747
\(22\) 3.73205 0.795676
\(23\) −5.92820 −1.23612 −0.618058 0.786133i \(-0.712080\pi\)
−0.618058 + 0.786133i \(0.712080\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 2.73205 0.535799
\(27\) 1.00000 0.192450
\(28\) −0.732051 −0.138345
\(29\) −1.73205 −0.321634 −0.160817 0.986984i \(-0.551413\pi\)
−0.160817 + 0.986984i \(0.551413\pi\)
\(30\) 0 0
\(31\) −2.46410 −0.442566 −0.221283 0.975210i \(-0.571024\pi\)
−0.221283 + 0.975210i \(0.571024\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.73205 0.649667
\(34\) 6.19615 1.06263
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) −1.00000 −0.162221
\(39\) 2.73205 0.437478
\(40\) 0 0
\(41\) −2.92820 −0.457309 −0.228654 0.973508i \(-0.573433\pi\)
−0.228654 + 0.973508i \(0.573433\pi\)
\(42\) −0.732051 −0.112958
\(43\) 8.19615 1.24990 0.624951 0.780664i \(-0.285119\pi\)
0.624951 + 0.780664i \(0.285119\pi\)
\(44\) 3.73205 0.562628
\(45\) 0 0
\(46\) −5.92820 −0.874066
\(47\) −3.46410 −0.505291 −0.252646 0.967559i \(-0.581301\pi\)
−0.252646 + 0.967559i \(0.581301\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.46410 −0.923443
\(50\) 0 0
\(51\) 6.19615 0.867635
\(52\) 2.73205 0.378867
\(53\) 1.73205 0.237915 0.118958 0.992899i \(-0.462045\pi\)
0.118958 + 0.992899i \(0.462045\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −0.732051 −0.0978244
\(57\) −1.00000 −0.132453
\(58\) −1.73205 −0.227429
\(59\) −8.19615 −1.06705 −0.533524 0.845785i \(-0.679133\pi\)
−0.533524 + 0.845785i \(0.679133\pi\)
\(60\) 0 0
\(61\) 10.6603 1.36491 0.682453 0.730930i \(-0.260913\pi\)
0.682453 + 0.730930i \(0.260913\pi\)
\(62\) −2.46410 −0.312941
\(63\) −0.732051 −0.0922297
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 3.73205 0.459384
\(67\) 0.267949 0.0327352 0.0163676 0.999866i \(-0.494790\pi\)
0.0163676 + 0.999866i \(0.494790\pi\)
\(68\) 6.19615 0.751394
\(69\) −5.92820 −0.713672
\(70\) 0 0
\(71\) 12.1962 1.44742 0.723708 0.690106i \(-0.242436\pi\)
0.723708 + 0.690106i \(0.242436\pi\)
\(72\) 1.00000 0.117851
\(73\) 2.46410 0.288401 0.144201 0.989548i \(-0.453939\pi\)
0.144201 + 0.989548i \(0.453939\pi\)
\(74\) 2.00000 0.232495
\(75\) 0 0
\(76\) −1.00000 −0.114708
\(77\) −2.73205 −0.311346
\(78\) 2.73205 0.309344
\(79\) −5.53590 −0.622837 −0.311419 0.950273i \(-0.600804\pi\)
−0.311419 + 0.950273i \(0.600804\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −2.92820 −0.323366
\(83\) −3.73205 −0.409646 −0.204823 0.978799i \(-0.565662\pi\)
−0.204823 + 0.978799i \(0.565662\pi\)
\(84\) −0.732051 −0.0798733
\(85\) 0 0
\(86\) 8.19615 0.883814
\(87\) −1.73205 −0.185695
\(88\) 3.73205 0.397838
\(89\) 10.8564 1.15078 0.575388 0.817880i \(-0.304851\pi\)
0.575388 + 0.817880i \(0.304851\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) −5.92820 −0.618058
\(93\) −2.46410 −0.255515
\(94\) −3.46410 −0.357295
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) 15.1244 1.53565 0.767823 0.640662i \(-0.221340\pi\)
0.767823 + 0.640662i \(0.221340\pi\)
\(98\) −6.46410 −0.652973
\(99\) 3.73205 0.375085
\(100\) 0 0
\(101\) 9.26795 0.922195 0.461098 0.887349i \(-0.347456\pi\)
0.461098 + 0.887349i \(0.347456\pi\)
\(102\) 6.19615 0.613511
\(103\) 3.53590 0.348402 0.174201 0.984710i \(-0.444266\pi\)
0.174201 + 0.984710i \(0.444266\pi\)
\(104\) 2.73205 0.267900
\(105\) 0 0
\(106\) 1.73205 0.168232
\(107\) −2.19615 −0.212310 −0.106155 0.994350i \(-0.533854\pi\)
−0.106155 + 0.994350i \(0.533854\pi\)
\(108\) 1.00000 0.0962250
\(109\) 2.39230 0.229141 0.114571 0.993415i \(-0.463451\pi\)
0.114571 + 0.993415i \(0.463451\pi\)
\(110\) 0 0
\(111\) 2.00000 0.189832
\(112\) −0.732051 −0.0691723
\(113\) 13.3923 1.25984 0.629921 0.776659i \(-0.283087\pi\)
0.629921 + 0.776659i \(0.283087\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 0 0
\(116\) −1.73205 −0.160817
\(117\) 2.73205 0.252578
\(118\) −8.19615 −0.754517
\(119\) −4.53590 −0.415805
\(120\) 0 0
\(121\) 2.92820 0.266200
\(122\) 10.6603 0.965134
\(123\) −2.92820 −0.264027
\(124\) −2.46410 −0.221283
\(125\) 0 0
\(126\) −0.732051 −0.0652163
\(127\) 5.00000 0.443678 0.221839 0.975083i \(-0.428794\pi\)
0.221839 + 0.975083i \(0.428794\pi\)
\(128\) 1.00000 0.0883883
\(129\) 8.19615 0.721631
\(130\) 0 0
\(131\) 10.2679 0.897115 0.448557 0.893754i \(-0.351938\pi\)
0.448557 + 0.893754i \(0.351938\pi\)
\(132\) 3.73205 0.324833
\(133\) 0.732051 0.0634769
\(134\) 0.267949 0.0231473
\(135\) 0 0
\(136\) 6.19615 0.531316
\(137\) 12.3923 1.05875 0.529373 0.848389i \(-0.322427\pi\)
0.529373 + 0.848389i \(0.322427\pi\)
\(138\) −5.92820 −0.504642
\(139\) −21.1244 −1.79174 −0.895872 0.444312i \(-0.853448\pi\)
−0.895872 + 0.444312i \(0.853448\pi\)
\(140\) 0 0
\(141\) −3.46410 −0.291730
\(142\) 12.1962 1.02348
\(143\) 10.1962 0.852645
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 2.46410 0.203931
\(147\) −6.46410 −0.533150
\(148\) 2.00000 0.164399
\(149\) −21.8564 −1.79055 −0.895273 0.445517i \(-0.853020\pi\)
−0.895273 + 0.445517i \(0.853020\pi\)
\(150\) 0 0
\(151\) −11.4641 −0.932935 −0.466468 0.884538i \(-0.654474\pi\)
−0.466468 + 0.884538i \(0.654474\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 6.19615 0.500929
\(154\) −2.73205 −0.220155
\(155\) 0 0
\(156\) 2.73205 0.218739
\(157\) −13.8564 −1.10586 −0.552931 0.833227i \(-0.686491\pi\)
−0.552931 + 0.833227i \(0.686491\pi\)
\(158\) −5.53590 −0.440412
\(159\) 1.73205 0.137361
\(160\) 0 0
\(161\) 4.33975 0.342020
\(162\) 1.00000 0.0785674
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) −2.92820 −0.228654
\(165\) 0 0
\(166\) −3.73205 −0.289663
\(167\) 8.92820 0.690885 0.345443 0.938440i \(-0.387729\pi\)
0.345443 + 0.938440i \(0.387729\pi\)
\(168\) −0.732051 −0.0564789
\(169\) −5.53590 −0.425838
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 8.19615 0.624951
\(173\) −13.5885 −1.03311 −0.516556 0.856254i \(-0.672786\pi\)
−0.516556 + 0.856254i \(0.672786\pi\)
\(174\) −1.73205 −0.131306
\(175\) 0 0
\(176\) 3.73205 0.281314
\(177\) −8.19615 −0.616061
\(178\) 10.8564 0.813722
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −16.5885 −1.23301 −0.616505 0.787351i \(-0.711452\pi\)
−0.616505 + 0.787351i \(0.711452\pi\)
\(182\) −2.00000 −0.148250
\(183\) 10.6603 0.788029
\(184\) −5.92820 −0.437033
\(185\) 0 0
\(186\) −2.46410 −0.180677
\(187\) 23.1244 1.69102
\(188\) −3.46410 −0.252646
\(189\) −0.732051 −0.0532489
\(190\) 0 0
\(191\) 3.00000 0.217072 0.108536 0.994092i \(-0.465384\pi\)
0.108536 + 0.994092i \(0.465384\pi\)
\(192\) 1.00000 0.0721688
\(193\) −22.0526 −1.58738 −0.793689 0.608324i \(-0.791842\pi\)
−0.793689 + 0.608324i \(0.791842\pi\)
\(194\) 15.1244 1.08587
\(195\) 0 0
\(196\) −6.46410 −0.461722
\(197\) −24.5885 −1.75186 −0.875928 0.482443i \(-0.839750\pi\)
−0.875928 + 0.482443i \(0.839750\pi\)
\(198\) 3.73205 0.265225
\(199\) −14.0000 −0.992434 −0.496217 0.868199i \(-0.665278\pi\)
−0.496217 + 0.868199i \(0.665278\pi\)
\(200\) 0 0
\(201\) 0.267949 0.0188997
\(202\) 9.26795 0.652091
\(203\) 1.26795 0.0889926
\(204\) 6.19615 0.433817
\(205\) 0 0
\(206\) 3.53590 0.246358
\(207\) −5.92820 −0.412039
\(208\) 2.73205 0.189434
\(209\) −3.73205 −0.258151
\(210\) 0 0
\(211\) −10.2679 −0.706875 −0.353437 0.935458i \(-0.614987\pi\)
−0.353437 + 0.935458i \(0.614987\pi\)
\(212\) 1.73205 0.118958
\(213\) 12.1962 0.835667
\(214\) −2.19615 −0.150126
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) 1.80385 0.122453
\(218\) 2.39230 0.162027
\(219\) 2.46410 0.166509
\(220\) 0 0
\(221\) 16.9282 1.13871
\(222\) 2.00000 0.134231
\(223\) 3.53590 0.236781 0.118391 0.992967i \(-0.462226\pi\)
0.118391 + 0.992967i \(0.462226\pi\)
\(224\) −0.732051 −0.0489122
\(225\) 0 0
\(226\) 13.3923 0.890843
\(227\) 6.00000 0.398234 0.199117 0.979976i \(-0.436193\pi\)
0.199117 + 0.979976i \(0.436193\pi\)
\(228\) −1.00000 −0.0662266
\(229\) −6.26795 −0.414198 −0.207099 0.978320i \(-0.566402\pi\)
−0.207099 + 0.978320i \(0.566402\pi\)
\(230\) 0 0
\(231\) −2.73205 −0.179756
\(232\) −1.73205 −0.113715
\(233\) 21.1244 1.38390 0.691951 0.721944i \(-0.256751\pi\)
0.691951 + 0.721944i \(0.256751\pi\)
\(234\) 2.73205 0.178600
\(235\) 0 0
\(236\) −8.19615 −0.533524
\(237\) −5.53590 −0.359595
\(238\) −4.53590 −0.294019
\(239\) −14.0000 −0.905585 −0.452792 0.891616i \(-0.649572\pi\)
−0.452792 + 0.891616i \(0.649572\pi\)
\(240\) 0 0
\(241\) −30.1962 −1.94511 −0.972553 0.232683i \(-0.925249\pi\)
−0.972553 + 0.232683i \(0.925249\pi\)
\(242\) 2.92820 0.188232
\(243\) 1.00000 0.0641500
\(244\) 10.6603 0.682453
\(245\) 0 0
\(246\) −2.92820 −0.186695
\(247\) −2.73205 −0.173836
\(248\) −2.46410 −0.156471
\(249\) −3.73205 −0.236509
\(250\) 0 0
\(251\) 11.4641 0.723608 0.361804 0.932254i \(-0.382161\pi\)
0.361804 + 0.932254i \(0.382161\pi\)
\(252\) −0.732051 −0.0461149
\(253\) −22.1244 −1.39095
\(254\) 5.00000 0.313728
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −9.00000 −0.561405 −0.280702 0.959795i \(-0.590567\pi\)
−0.280702 + 0.959795i \(0.590567\pi\)
\(258\) 8.19615 0.510270
\(259\) −1.46410 −0.0909748
\(260\) 0 0
\(261\) −1.73205 −0.107211
\(262\) 10.2679 0.634356
\(263\) −15.0000 −0.924940 −0.462470 0.886635i \(-0.653037\pi\)
−0.462470 + 0.886635i \(0.653037\pi\)
\(264\) 3.73205 0.229692
\(265\) 0 0
\(266\) 0.732051 0.0448849
\(267\) 10.8564 0.664401
\(268\) 0.267949 0.0163676
\(269\) −20.5359 −1.25210 −0.626048 0.779785i \(-0.715329\pi\)
−0.626048 + 0.779785i \(0.715329\pi\)
\(270\) 0 0
\(271\) −15.8038 −0.960015 −0.480008 0.877264i \(-0.659366\pi\)
−0.480008 + 0.877264i \(0.659366\pi\)
\(272\) 6.19615 0.375697
\(273\) −2.00000 −0.121046
\(274\) 12.3923 0.748647
\(275\) 0 0
\(276\) −5.92820 −0.356836
\(277\) −11.1962 −0.672712 −0.336356 0.941735i \(-0.609194\pi\)
−0.336356 + 0.941735i \(0.609194\pi\)
\(278\) −21.1244 −1.26695
\(279\) −2.46410 −0.147522
\(280\) 0 0
\(281\) 17.7846 1.06094 0.530470 0.847703i \(-0.322015\pi\)
0.530470 + 0.847703i \(0.322015\pi\)
\(282\) −3.46410 −0.206284
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 12.1962 0.723708
\(285\) 0 0
\(286\) 10.1962 0.602911
\(287\) 2.14359 0.126532
\(288\) 1.00000 0.0589256
\(289\) 21.3923 1.25837
\(290\) 0 0
\(291\) 15.1244 0.886605
\(292\) 2.46410 0.144201
\(293\) 27.7321 1.62012 0.810062 0.586344i \(-0.199433\pi\)
0.810062 + 0.586344i \(0.199433\pi\)
\(294\) −6.46410 −0.376994
\(295\) 0 0
\(296\) 2.00000 0.116248
\(297\) 3.73205 0.216556
\(298\) −21.8564 −1.26611
\(299\) −16.1962 −0.936648
\(300\) 0 0
\(301\) −6.00000 −0.345834
\(302\) −11.4641 −0.659685
\(303\) 9.26795 0.532430
\(304\) −1.00000 −0.0573539
\(305\) 0 0
\(306\) 6.19615 0.354210
\(307\) 20.2679 1.15675 0.578376 0.815770i \(-0.303686\pi\)
0.578376 + 0.815770i \(0.303686\pi\)
\(308\) −2.73205 −0.155673
\(309\) 3.53590 0.201150
\(310\) 0 0
\(311\) −29.3205 −1.66261 −0.831307 0.555814i \(-0.812407\pi\)
−0.831307 + 0.555814i \(0.812407\pi\)
\(312\) 2.73205 0.154672
\(313\) 3.92820 0.222035 0.111018 0.993818i \(-0.464589\pi\)
0.111018 + 0.993818i \(0.464589\pi\)
\(314\) −13.8564 −0.781962
\(315\) 0 0
\(316\) −5.53590 −0.311419
\(317\) 1.73205 0.0972817 0.0486408 0.998816i \(-0.484511\pi\)
0.0486408 + 0.998816i \(0.484511\pi\)
\(318\) 1.73205 0.0971286
\(319\) −6.46410 −0.361920
\(320\) 0 0
\(321\) −2.19615 −0.122577
\(322\) 4.33975 0.241845
\(323\) −6.19615 −0.344763
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 2.00000 0.110770
\(327\) 2.39230 0.132295
\(328\) −2.92820 −0.161683
\(329\) 2.53590 0.139809
\(330\) 0 0
\(331\) −22.2679 −1.22396 −0.611979 0.790874i \(-0.709626\pi\)
−0.611979 + 0.790874i \(0.709626\pi\)
\(332\) −3.73205 −0.204823
\(333\) 2.00000 0.109599
\(334\) 8.92820 0.488530
\(335\) 0 0
\(336\) −0.732051 −0.0399366
\(337\) −10.5359 −0.573927 −0.286963 0.957942i \(-0.592646\pi\)
−0.286963 + 0.957942i \(0.592646\pi\)
\(338\) −5.53590 −0.301113
\(339\) 13.3923 0.727370
\(340\) 0 0
\(341\) −9.19615 −0.498000
\(342\) −1.00000 −0.0540738
\(343\) 9.85641 0.532196
\(344\) 8.19615 0.441907
\(345\) 0 0
\(346\) −13.5885 −0.730520
\(347\) 20.7846 1.11578 0.557888 0.829916i \(-0.311612\pi\)
0.557888 + 0.829916i \(0.311612\pi\)
\(348\) −1.73205 −0.0928477
\(349\) −17.0526 −0.912803 −0.456401 0.889774i \(-0.650862\pi\)
−0.456401 + 0.889774i \(0.650862\pi\)
\(350\) 0 0
\(351\) 2.73205 0.145826
\(352\) 3.73205 0.198919
\(353\) −16.0526 −0.854391 −0.427196 0.904159i \(-0.640498\pi\)
−0.427196 + 0.904159i \(0.640498\pi\)
\(354\) −8.19615 −0.435621
\(355\) 0 0
\(356\) 10.8564 0.575388
\(357\) −4.53590 −0.240065
\(358\) −12.0000 −0.634220
\(359\) −16.9282 −0.893436 −0.446718 0.894675i \(-0.647407\pi\)
−0.446718 + 0.894675i \(0.647407\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −16.5885 −0.871870
\(363\) 2.92820 0.153691
\(364\) −2.00000 −0.104828
\(365\) 0 0
\(366\) 10.6603 0.557220
\(367\) 13.2679 0.692581 0.346291 0.938127i \(-0.387441\pi\)
0.346291 + 0.938127i \(0.387441\pi\)
\(368\) −5.92820 −0.309029
\(369\) −2.92820 −0.152436
\(370\) 0 0
\(371\) −1.26795 −0.0658286
\(372\) −2.46410 −0.127758
\(373\) 31.5167 1.63187 0.815935 0.578143i \(-0.196222\pi\)
0.815935 + 0.578143i \(0.196222\pi\)
\(374\) 23.1244 1.19573
\(375\) 0 0
\(376\) −3.46410 −0.178647
\(377\) −4.73205 −0.243713
\(378\) −0.732051 −0.0376526
\(379\) −0.392305 −0.0201513 −0.0100757 0.999949i \(-0.503207\pi\)
−0.0100757 + 0.999949i \(0.503207\pi\)
\(380\) 0 0
\(381\) 5.00000 0.256158
\(382\) 3.00000 0.153493
\(383\) 19.5167 0.997255 0.498627 0.866816i \(-0.333838\pi\)
0.498627 + 0.866816i \(0.333838\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −22.0526 −1.12245
\(387\) 8.19615 0.416634
\(388\) 15.1244 0.767823
\(389\) 21.7128 1.10088 0.550442 0.834874i \(-0.314459\pi\)
0.550442 + 0.834874i \(0.314459\pi\)
\(390\) 0 0
\(391\) −36.7321 −1.85762
\(392\) −6.46410 −0.326486
\(393\) 10.2679 0.517950
\(394\) −24.5885 −1.23875
\(395\) 0 0
\(396\) 3.73205 0.187543
\(397\) −11.7321 −0.588815 −0.294407 0.955680i \(-0.595122\pi\)
−0.294407 + 0.955680i \(0.595122\pi\)
\(398\) −14.0000 −0.701757
\(399\) 0.732051 0.0366484
\(400\) 0 0
\(401\) 16.8564 0.841769 0.420884 0.907114i \(-0.361720\pi\)
0.420884 + 0.907114i \(0.361720\pi\)
\(402\) 0.267949 0.0133641
\(403\) −6.73205 −0.335347
\(404\) 9.26795 0.461098
\(405\) 0 0
\(406\) 1.26795 0.0629273
\(407\) 7.46410 0.369982
\(408\) 6.19615 0.306755
\(409\) 1.60770 0.0794954 0.0397477 0.999210i \(-0.487345\pi\)
0.0397477 + 0.999210i \(0.487345\pi\)
\(410\) 0 0
\(411\) 12.3923 0.611267
\(412\) 3.53590 0.174201
\(413\) 6.00000 0.295241
\(414\) −5.92820 −0.291355
\(415\) 0 0
\(416\) 2.73205 0.133950
\(417\) −21.1244 −1.03446
\(418\) −3.73205 −0.182541
\(419\) 13.8564 0.676930 0.338465 0.940979i \(-0.390092\pi\)
0.338465 + 0.940979i \(0.390092\pi\)
\(420\) 0 0
\(421\) 6.78461 0.330662 0.165331 0.986238i \(-0.447131\pi\)
0.165331 + 0.986238i \(0.447131\pi\)
\(422\) −10.2679 −0.499836
\(423\) −3.46410 −0.168430
\(424\) 1.73205 0.0841158
\(425\) 0 0
\(426\) 12.1962 0.590906
\(427\) −7.80385 −0.377655
\(428\) −2.19615 −0.106155
\(429\) 10.1962 0.492275
\(430\) 0 0
\(431\) −29.3205 −1.41232 −0.706160 0.708053i \(-0.749574\pi\)
−0.706160 + 0.708053i \(0.749574\pi\)
\(432\) 1.00000 0.0481125
\(433\) −24.1962 −1.16279 −0.581396 0.813620i \(-0.697493\pi\)
−0.581396 + 0.813620i \(0.697493\pi\)
\(434\) 1.80385 0.0865875
\(435\) 0 0
\(436\) 2.39230 0.114571
\(437\) 5.92820 0.283584
\(438\) 2.46410 0.117739
\(439\) 25.7846 1.23063 0.615316 0.788280i \(-0.289028\pi\)
0.615316 + 0.788280i \(0.289028\pi\)
\(440\) 0 0
\(441\) −6.46410 −0.307814
\(442\) 16.9282 0.805193
\(443\) −3.33975 −0.158676 −0.0793381 0.996848i \(-0.525281\pi\)
−0.0793381 + 0.996848i \(0.525281\pi\)
\(444\) 2.00000 0.0949158
\(445\) 0 0
\(446\) 3.53590 0.167430
\(447\) −21.8564 −1.03377
\(448\) −0.732051 −0.0345861
\(449\) 24.4641 1.15453 0.577266 0.816556i \(-0.304120\pi\)
0.577266 + 0.816556i \(0.304120\pi\)
\(450\) 0 0
\(451\) −10.9282 −0.514589
\(452\) 13.3923 0.629921
\(453\) −11.4641 −0.538630
\(454\) 6.00000 0.281594
\(455\) 0 0
\(456\) −1.00000 −0.0468293
\(457\) −5.07180 −0.237249 −0.118624 0.992939i \(-0.537848\pi\)
−0.118624 + 0.992939i \(0.537848\pi\)
\(458\) −6.26795 −0.292882
\(459\) 6.19615 0.289212
\(460\) 0 0
\(461\) −17.8564 −0.831656 −0.415828 0.909443i \(-0.636508\pi\)
−0.415828 + 0.909443i \(0.636508\pi\)
\(462\) −2.73205 −0.127107
\(463\) −35.3205 −1.64148 −0.820742 0.571300i \(-0.806439\pi\)
−0.820742 + 0.571300i \(0.806439\pi\)
\(464\) −1.73205 −0.0804084
\(465\) 0 0
\(466\) 21.1244 0.978567
\(467\) 7.19615 0.332998 0.166499 0.986042i \(-0.446754\pi\)
0.166499 + 0.986042i \(0.446754\pi\)
\(468\) 2.73205 0.126289
\(469\) −0.196152 −0.00905748
\(470\) 0 0
\(471\) −13.8564 −0.638470
\(472\) −8.19615 −0.377258
\(473\) 30.5885 1.40646
\(474\) −5.53590 −0.254272
\(475\) 0 0
\(476\) −4.53590 −0.207903
\(477\) 1.73205 0.0793052
\(478\) −14.0000 −0.640345
\(479\) 3.14359 0.143634 0.0718172 0.997418i \(-0.477120\pi\)
0.0718172 + 0.997418i \(0.477120\pi\)
\(480\) 0 0
\(481\) 5.46410 0.249142
\(482\) −30.1962 −1.37540
\(483\) 4.33975 0.197465
\(484\) 2.92820 0.133100
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 2.14359 0.0971355 0.0485677 0.998820i \(-0.484534\pi\)
0.0485677 + 0.998820i \(0.484534\pi\)
\(488\) 10.6603 0.482567
\(489\) 2.00000 0.0904431
\(490\) 0 0
\(491\) −2.92820 −0.132148 −0.0660740 0.997815i \(-0.521047\pi\)
−0.0660740 + 0.997815i \(0.521047\pi\)
\(492\) −2.92820 −0.132014
\(493\) −10.7321 −0.483347
\(494\) −2.73205 −0.122921
\(495\) 0 0
\(496\) −2.46410 −0.110641
\(497\) −8.92820 −0.400485
\(498\) −3.73205 −0.167237
\(499\) 41.3731 1.85211 0.926056 0.377385i \(-0.123177\pi\)
0.926056 + 0.377385i \(0.123177\pi\)
\(500\) 0 0
\(501\) 8.92820 0.398883
\(502\) 11.4641 0.511668
\(503\) −17.8564 −0.796178 −0.398089 0.917347i \(-0.630326\pi\)
−0.398089 + 0.917347i \(0.630326\pi\)
\(504\) −0.732051 −0.0326081
\(505\) 0 0
\(506\) −22.1244 −0.983548
\(507\) −5.53590 −0.245858
\(508\) 5.00000 0.221839
\(509\) 13.8756 0.615027 0.307514 0.951544i \(-0.400503\pi\)
0.307514 + 0.951544i \(0.400503\pi\)
\(510\) 0 0
\(511\) −1.80385 −0.0797975
\(512\) 1.00000 0.0441942
\(513\) −1.00000 −0.0441511
\(514\) −9.00000 −0.396973
\(515\) 0 0
\(516\) 8.19615 0.360815
\(517\) −12.9282 −0.568582
\(518\) −1.46410 −0.0643289
\(519\) −13.5885 −0.596467
\(520\) 0 0
\(521\) 24.1769 1.05921 0.529605 0.848244i \(-0.322340\pi\)
0.529605 + 0.848244i \(0.322340\pi\)
\(522\) −1.73205 −0.0758098
\(523\) 4.00000 0.174908 0.0874539 0.996169i \(-0.472127\pi\)
0.0874539 + 0.996169i \(0.472127\pi\)
\(524\) 10.2679 0.448557
\(525\) 0 0
\(526\) −15.0000 −0.654031
\(527\) −15.2679 −0.665082
\(528\) 3.73205 0.162417
\(529\) 12.1436 0.527982
\(530\) 0 0
\(531\) −8.19615 −0.355683
\(532\) 0.732051 0.0317384
\(533\) −8.00000 −0.346518
\(534\) 10.8564 0.469803
\(535\) 0 0
\(536\) 0.267949 0.0115736
\(537\) −12.0000 −0.517838
\(538\) −20.5359 −0.885365
\(539\) −24.1244 −1.03911
\(540\) 0 0
\(541\) 17.1962 0.739320 0.369660 0.929167i \(-0.379474\pi\)
0.369660 + 0.929167i \(0.379474\pi\)
\(542\) −15.8038 −0.678833
\(543\) −16.5885 −0.711879
\(544\) 6.19615 0.265658
\(545\) 0 0
\(546\) −2.00000 −0.0855921
\(547\) −43.4449 −1.85757 −0.928784 0.370621i \(-0.879145\pi\)
−0.928784 + 0.370621i \(0.879145\pi\)
\(548\) 12.3923 0.529373
\(549\) 10.6603 0.454969
\(550\) 0 0
\(551\) 1.73205 0.0737878
\(552\) −5.92820 −0.252321
\(553\) 4.05256 0.172332
\(554\) −11.1962 −0.475679
\(555\) 0 0
\(556\) −21.1244 −0.895872
\(557\) −3.80385 −0.161174 −0.0805871 0.996748i \(-0.525680\pi\)
−0.0805871 + 0.996748i \(0.525680\pi\)
\(558\) −2.46410 −0.104314
\(559\) 22.3923 0.947094
\(560\) 0 0
\(561\) 23.1244 0.976311
\(562\) 17.7846 0.750198
\(563\) −7.66025 −0.322841 −0.161421 0.986886i \(-0.551608\pi\)
−0.161421 + 0.986886i \(0.551608\pi\)
\(564\) −3.46410 −0.145865
\(565\) 0 0
\(566\) −4.00000 −0.168133
\(567\) −0.732051 −0.0307432
\(568\) 12.1962 0.511739
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) −6.19615 −0.259301 −0.129650 0.991560i \(-0.541386\pi\)
−0.129650 + 0.991560i \(0.541386\pi\)
\(572\) 10.1962 0.426323
\(573\) 3.00000 0.125327
\(574\) 2.14359 0.0894719
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 17.2487 0.718073 0.359037 0.933323i \(-0.383105\pi\)
0.359037 + 0.933323i \(0.383105\pi\)
\(578\) 21.3923 0.889803
\(579\) −22.0526 −0.916473
\(580\) 0 0
\(581\) 2.73205 0.113345
\(582\) 15.1244 0.626925
\(583\) 6.46410 0.267716
\(584\) 2.46410 0.101965
\(585\) 0 0
\(586\) 27.7321 1.14560
\(587\) 12.2679 0.506352 0.253176 0.967420i \(-0.418525\pi\)
0.253176 + 0.967420i \(0.418525\pi\)
\(588\) −6.46410 −0.266575
\(589\) 2.46410 0.101532
\(590\) 0 0
\(591\) −24.5885 −1.01143
\(592\) 2.00000 0.0821995
\(593\) 27.6077 1.13371 0.566856 0.823817i \(-0.308160\pi\)
0.566856 + 0.823817i \(0.308160\pi\)
\(594\) 3.73205 0.153128
\(595\) 0 0
\(596\) −21.8564 −0.895273
\(597\) −14.0000 −0.572982
\(598\) −16.1962 −0.662310
\(599\) −3.66025 −0.149554 −0.0747770 0.997200i \(-0.523824\pi\)
−0.0747770 + 0.997200i \(0.523824\pi\)
\(600\) 0 0
\(601\) −10.0526 −0.410052 −0.205026 0.978756i \(-0.565728\pi\)
−0.205026 + 0.978756i \(0.565728\pi\)
\(602\) −6.00000 −0.244542
\(603\) 0.267949 0.0109117
\(604\) −11.4641 −0.466468
\(605\) 0 0
\(606\) 9.26795 0.376485
\(607\) 11.7846 0.478323 0.239161 0.970980i \(-0.423128\pi\)
0.239161 + 0.970980i \(0.423128\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 1.26795 0.0513799
\(610\) 0 0
\(611\) −9.46410 −0.382877
\(612\) 6.19615 0.250465
\(613\) −38.7846 −1.56650 −0.783248 0.621710i \(-0.786438\pi\)
−0.783248 + 0.621710i \(0.786438\pi\)
\(614\) 20.2679 0.817948
\(615\) 0 0
\(616\) −2.73205 −0.110077
\(617\) −46.3923 −1.86768 −0.933842 0.357686i \(-0.883566\pi\)
−0.933842 + 0.357686i \(0.883566\pi\)
\(618\) 3.53590 0.142235
\(619\) −0.196152 −0.00788403 −0.00394202 0.999992i \(-0.501255\pi\)
−0.00394202 + 0.999992i \(0.501255\pi\)
\(620\) 0 0
\(621\) −5.92820 −0.237891
\(622\) −29.3205 −1.17565
\(623\) −7.94744 −0.318408
\(624\) 2.73205 0.109370
\(625\) 0 0
\(626\) 3.92820 0.157003
\(627\) −3.73205 −0.149044
\(628\) −13.8564 −0.552931
\(629\) 12.3923 0.494114
\(630\) 0 0
\(631\) 2.92820 0.116570 0.0582850 0.998300i \(-0.481437\pi\)
0.0582850 + 0.998300i \(0.481437\pi\)
\(632\) −5.53590 −0.220206
\(633\) −10.2679 −0.408114
\(634\) 1.73205 0.0687885
\(635\) 0 0
\(636\) 1.73205 0.0686803
\(637\) −17.6603 −0.699725
\(638\) −6.46410 −0.255916
\(639\) 12.1962 0.482472
\(640\) 0 0
\(641\) 10.5359 0.416143 0.208071 0.978114i \(-0.433281\pi\)
0.208071 + 0.978114i \(0.433281\pi\)
\(642\) −2.19615 −0.0866752
\(643\) −27.0718 −1.06761 −0.533804 0.845608i \(-0.679238\pi\)
−0.533804 + 0.845608i \(0.679238\pi\)
\(644\) 4.33975 0.171010
\(645\) 0 0
\(646\) −6.19615 −0.243784
\(647\) −9.39230 −0.369250 −0.184625 0.982809i \(-0.559107\pi\)
−0.184625 + 0.982809i \(0.559107\pi\)
\(648\) 1.00000 0.0392837
\(649\) −30.5885 −1.20070
\(650\) 0 0
\(651\) 1.80385 0.0706984
\(652\) 2.00000 0.0783260
\(653\) 17.4641 0.683423 0.341712 0.939805i \(-0.388993\pi\)
0.341712 + 0.939805i \(0.388993\pi\)
\(654\) 2.39230 0.0935465
\(655\) 0 0
\(656\) −2.92820 −0.114327
\(657\) 2.46410 0.0961338
\(658\) 2.53590 0.0988596
\(659\) 7.41154 0.288713 0.144356 0.989526i \(-0.453889\pi\)
0.144356 + 0.989526i \(0.453889\pi\)
\(660\) 0 0
\(661\) −22.6410 −0.880633 −0.440317 0.897843i \(-0.645134\pi\)
−0.440317 + 0.897843i \(0.645134\pi\)
\(662\) −22.2679 −0.865468
\(663\) 16.9282 0.657437
\(664\) −3.73205 −0.144832
\(665\) 0 0
\(666\) 2.00000 0.0774984
\(667\) 10.2679 0.397577
\(668\) 8.92820 0.345443
\(669\) 3.53590 0.136706
\(670\) 0 0
\(671\) 39.7846 1.53587
\(672\) −0.732051 −0.0282395
\(673\) −6.98076 −0.269089 −0.134544 0.990908i \(-0.542957\pi\)
−0.134544 + 0.990908i \(0.542957\pi\)
\(674\) −10.5359 −0.405828
\(675\) 0 0
\(676\) −5.53590 −0.212919
\(677\) 8.66025 0.332841 0.166420 0.986055i \(-0.446779\pi\)
0.166420 + 0.986055i \(0.446779\pi\)
\(678\) 13.3923 0.514328
\(679\) −11.0718 −0.424897
\(680\) 0 0
\(681\) 6.00000 0.229920
\(682\) −9.19615 −0.352139
\(683\) −46.9808 −1.79767 −0.898834 0.438288i \(-0.855585\pi\)
−0.898834 + 0.438288i \(0.855585\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 0 0
\(686\) 9.85641 0.376319
\(687\) −6.26795 −0.239137
\(688\) 8.19615 0.312475
\(689\) 4.73205 0.180277
\(690\) 0 0
\(691\) −22.1962 −0.844381 −0.422191 0.906507i \(-0.638739\pi\)
−0.422191 + 0.906507i \(0.638739\pi\)
\(692\) −13.5885 −0.516556
\(693\) −2.73205 −0.103782
\(694\) 20.7846 0.788973
\(695\) 0 0
\(696\) −1.73205 −0.0656532
\(697\) −18.1436 −0.687238
\(698\) −17.0526 −0.645449
\(699\) 21.1244 0.798997
\(700\) 0 0
\(701\) 3.60770 0.136261 0.0681304 0.997676i \(-0.478297\pi\)
0.0681304 + 0.997676i \(0.478297\pi\)
\(702\) 2.73205 0.103115
\(703\) −2.00000 −0.0754314
\(704\) 3.73205 0.140657
\(705\) 0 0
\(706\) −16.0526 −0.604146
\(707\) −6.78461 −0.255162
\(708\) −8.19615 −0.308030
\(709\) 47.3013 1.77644 0.888218 0.459422i \(-0.151943\pi\)
0.888218 + 0.459422i \(0.151943\pi\)
\(710\) 0 0
\(711\) −5.53590 −0.207612
\(712\) 10.8564 0.406861
\(713\) 14.6077 0.547062
\(714\) −4.53590 −0.169752
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) −14.0000 −0.522840
\(718\) −16.9282 −0.631755
\(719\) 12.1769 0.454122 0.227061 0.973881i \(-0.427088\pi\)
0.227061 + 0.973881i \(0.427088\pi\)
\(720\) 0 0
\(721\) −2.58846 −0.0963992
\(722\) 1.00000 0.0372161
\(723\) −30.1962 −1.12301
\(724\) −16.5885 −0.616505
\(725\) 0 0
\(726\) 2.92820 0.108676
\(727\) 38.1051 1.41324 0.706620 0.707593i \(-0.250219\pi\)
0.706620 + 0.707593i \(0.250219\pi\)
\(728\) −2.00000 −0.0741249
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 50.7846 1.87834
\(732\) 10.6603 0.394014
\(733\) 20.5167 0.757800 0.378900 0.925438i \(-0.376302\pi\)
0.378900 + 0.925438i \(0.376302\pi\)
\(734\) 13.2679 0.489729
\(735\) 0 0
\(736\) −5.92820 −0.218516
\(737\) 1.00000 0.0368355
\(738\) −2.92820 −0.107789
\(739\) −53.1769 −1.95614 −0.978072 0.208266i \(-0.933218\pi\)
−0.978072 + 0.208266i \(0.933218\pi\)
\(740\) 0 0
\(741\) −2.73205 −0.100364
\(742\) −1.26795 −0.0465479
\(743\) 5.07180 0.186066 0.0930331 0.995663i \(-0.470344\pi\)
0.0930331 + 0.995663i \(0.470344\pi\)
\(744\) −2.46410 −0.0903383
\(745\) 0 0
\(746\) 31.5167 1.15391
\(747\) −3.73205 −0.136549
\(748\) 23.1244 0.845510
\(749\) 1.60770 0.0587439
\(750\) 0 0
\(751\) 7.46410 0.272369 0.136184 0.990683i \(-0.456516\pi\)
0.136184 + 0.990683i \(0.456516\pi\)
\(752\) −3.46410 −0.126323
\(753\) 11.4641 0.417775
\(754\) −4.73205 −0.172331
\(755\) 0 0
\(756\) −0.732051 −0.0266244
\(757\) 13.5885 0.493881 0.246940 0.969031i \(-0.420575\pi\)
0.246940 + 0.969031i \(0.420575\pi\)
\(758\) −0.392305 −0.0142492
\(759\) −22.1244 −0.803063
\(760\) 0 0
\(761\) 1.26795 0.0459631 0.0229816 0.999736i \(-0.492684\pi\)
0.0229816 + 0.999736i \(0.492684\pi\)
\(762\) 5.00000 0.181131
\(763\) −1.75129 −0.0634009
\(764\) 3.00000 0.108536
\(765\) 0 0
\(766\) 19.5167 0.705166
\(767\) −22.3923 −0.808539
\(768\) 1.00000 0.0360844
\(769\) −34.7128 −1.25178 −0.625888 0.779913i \(-0.715263\pi\)
−0.625888 + 0.779913i \(0.715263\pi\)
\(770\) 0 0
\(771\) −9.00000 −0.324127
\(772\) −22.0526 −0.793689
\(773\) 45.3205 1.63007 0.815033 0.579415i \(-0.196719\pi\)
0.815033 + 0.579415i \(0.196719\pi\)
\(774\) 8.19615 0.294605
\(775\) 0 0
\(776\) 15.1244 0.542933
\(777\) −1.46410 −0.0525244
\(778\) 21.7128 0.778442
\(779\) 2.92820 0.104914
\(780\) 0 0
\(781\) 45.5167 1.62871
\(782\) −36.7321 −1.31354
\(783\) −1.73205 −0.0618984
\(784\) −6.46410 −0.230861
\(785\) 0 0
\(786\) 10.2679 0.366246
\(787\) −2.12436 −0.0757251 −0.0378626 0.999283i \(-0.512055\pi\)
−0.0378626 + 0.999283i \(0.512055\pi\)
\(788\) −24.5885 −0.875928
\(789\) −15.0000 −0.534014
\(790\) 0 0
\(791\) −9.80385 −0.348585
\(792\) 3.73205 0.132613
\(793\) 29.1244 1.03424
\(794\) −11.7321 −0.416355
\(795\) 0 0
\(796\) −14.0000 −0.496217
\(797\) −20.0000 −0.708436 −0.354218 0.935163i \(-0.615253\pi\)
−0.354218 + 0.935163i \(0.615253\pi\)
\(798\) 0.732051 0.0259143
\(799\) −21.4641 −0.759345
\(800\) 0 0
\(801\) 10.8564 0.383592
\(802\) 16.8564 0.595220
\(803\) 9.19615 0.324525
\(804\) 0.267949 0.00944984
\(805\) 0 0
\(806\) −6.73205 −0.237126
\(807\) −20.5359 −0.722898
\(808\) 9.26795 0.326045
\(809\) 37.5167 1.31902 0.659508 0.751698i \(-0.270765\pi\)
0.659508 + 0.751698i \(0.270765\pi\)
\(810\) 0 0
\(811\) 18.3731 0.645166 0.322583 0.946541i \(-0.395449\pi\)
0.322583 + 0.946541i \(0.395449\pi\)
\(812\) 1.26795 0.0444963
\(813\) −15.8038 −0.554265
\(814\) 7.46410 0.261617
\(815\) 0 0
\(816\) 6.19615 0.216909
\(817\) −8.19615 −0.286747
\(818\) 1.60770 0.0562117
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) 2.48334 0.0866691 0.0433346 0.999061i \(-0.486202\pi\)
0.0433346 + 0.999061i \(0.486202\pi\)
\(822\) 12.3923 0.432231
\(823\) 6.78461 0.236497 0.118248 0.992984i \(-0.462272\pi\)
0.118248 + 0.992984i \(0.462272\pi\)
\(824\) 3.53590 0.123179
\(825\) 0 0
\(826\) 6.00000 0.208767
\(827\) −6.73205 −0.234096 −0.117048 0.993126i \(-0.537343\pi\)
−0.117048 + 0.993126i \(0.537343\pi\)
\(828\) −5.92820 −0.206019
\(829\) 6.19615 0.215201 0.107601 0.994194i \(-0.465683\pi\)
0.107601 + 0.994194i \(0.465683\pi\)
\(830\) 0 0
\(831\) −11.1962 −0.388390
\(832\) 2.73205 0.0947168
\(833\) −40.0526 −1.38774
\(834\) −21.1244 −0.731477
\(835\) 0 0
\(836\) −3.73205 −0.129076
\(837\) −2.46410 −0.0851718
\(838\) 13.8564 0.478662
\(839\) −46.5885 −1.60841 −0.804206 0.594351i \(-0.797409\pi\)
−0.804206 + 0.594351i \(0.797409\pi\)
\(840\) 0 0
\(841\) −26.0000 −0.896552
\(842\) 6.78461 0.233813
\(843\) 17.7846 0.612534
\(844\) −10.2679 −0.353437
\(845\) 0 0
\(846\) −3.46410 −0.119098
\(847\) −2.14359 −0.0736547
\(848\) 1.73205 0.0594789
\(849\) −4.00000 −0.137280
\(850\) 0 0
\(851\) −11.8564 −0.406432
\(852\) 12.1962 0.417833
\(853\) 4.53590 0.155306 0.0776531 0.996980i \(-0.475257\pi\)
0.0776531 + 0.996980i \(0.475257\pi\)
\(854\) −7.80385 −0.267042
\(855\) 0 0
\(856\) −2.19615 −0.0750629
\(857\) 42.9282 1.46640 0.733200 0.680013i \(-0.238026\pi\)
0.733200 + 0.680013i \(0.238026\pi\)
\(858\) 10.1962 0.348091
\(859\) 56.9282 1.94237 0.971183 0.238337i \(-0.0766021\pi\)
0.971183 + 0.238337i \(0.0766021\pi\)
\(860\) 0 0
\(861\) 2.14359 0.0730535
\(862\) −29.3205 −0.998660
\(863\) 55.3205 1.88313 0.941566 0.336829i \(-0.109354\pi\)
0.941566 + 0.336829i \(0.109354\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −24.1962 −0.822219
\(867\) 21.3923 0.726521
\(868\) 1.80385 0.0612266
\(869\) −20.6603 −0.700851
\(870\) 0 0
\(871\) 0.732051 0.0248046
\(872\) 2.39230 0.0810137
\(873\) 15.1244 0.511882
\(874\) 5.92820 0.200524
\(875\) 0 0
\(876\) 2.46410 0.0832543
\(877\) −31.1769 −1.05277 −0.526385 0.850246i \(-0.676453\pi\)
−0.526385 + 0.850246i \(0.676453\pi\)
\(878\) 25.7846 0.870188
\(879\) 27.7321 0.935379
\(880\) 0 0
\(881\) −26.5359 −0.894017 −0.447009 0.894530i \(-0.647511\pi\)
−0.447009 + 0.894530i \(0.647511\pi\)
\(882\) −6.46410 −0.217658
\(883\) −27.2679 −0.917640 −0.458820 0.888529i \(-0.651728\pi\)
−0.458820 + 0.888529i \(0.651728\pi\)
\(884\) 16.9282 0.569357
\(885\) 0 0
\(886\) −3.33975 −0.112201
\(887\) 7.60770 0.255441 0.127721 0.991810i \(-0.459234\pi\)
0.127721 + 0.991810i \(0.459234\pi\)
\(888\) 2.00000 0.0671156
\(889\) −3.66025 −0.122761
\(890\) 0 0
\(891\) 3.73205 0.125028
\(892\) 3.53590 0.118391
\(893\) 3.46410 0.115922
\(894\) −21.8564 −0.730988
\(895\) 0 0
\(896\) −0.732051 −0.0244561
\(897\) −16.1962 −0.540774
\(898\) 24.4641 0.816378
\(899\) 4.26795 0.142344
\(900\) 0 0
\(901\) 10.7321 0.357536
\(902\) −10.9282 −0.363869
\(903\) −6.00000 −0.199667
\(904\) 13.3923 0.445421
\(905\) 0 0
\(906\) −11.4641 −0.380869
\(907\) −49.3205 −1.63766 −0.818830 0.574036i \(-0.805377\pi\)
−0.818830 + 0.574036i \(0.805377\pi\)
\(908\) 6.00000 0.199117
\(909\) 9.26795 0.307398
\(910\) 0 0
\(911\) 34.1051 1.12995 0.564976 0.825107i \(-0.308885\pi\)
0.564976 + 0.825107i \(0.308885\pi\)
\(912\) −1.00000 −0.0331133
\(913\) −13.9282 −0.460956
\(914\) −5.07180 −0.167760
\(915\) 0 0
\(916\) −6.26795 −0.207099
\(917\) −7.51666 −0.248222
\(918\) 6.19615 0.204504
\(919\) −4.48334 −0.147892 −0.0739459 0.997262i \(-0.523559\pi\)
−0.0739459 + 0.997262i \(0.523559\pi\)
\(920\) 0 0
\(921\) 20.2679 0.667852
\(922\) −17.8564 −0.588069
\(923\) 33.3205 1.09676
\(924\) −2.73205 −0.0898779
\(925\) 0 0
\(926\) −35.3205 −1.16070
\(927\) 3.53590 0.116134
\(928\) −1.73205 −0.0568574
\(929\) 42.9282 1.40843 0.704214 0.709987i \(-0.251299\pi\)
0.704214 + 0.709987i \(0.251299\pi\)
\(930\) 0 0
\(931\) 6.46410 0.211852
\(932\) 21.1244 0.691951
\(933\) −29.3205 −0.959910
\(934\) 7.19615 0.235465
\(935\) 0 0
\(936\) 2.73205 0.0892999
\(937\) 13.2154 0.431728 0.215864 0.976423i \(-0.430743\pi\)
0.215864 + 0.976423i \(0.430743\pi\)
\(938\) −0.196152 −0.00640460
\(939\) 3.92820 0.128192
\(940\) 0 0
\(941\) −12.2679 −0.399924 −0.199962 0.979804i \(-0.564082\pi\)
−0.199962 + 0.979804i \(0.564082\pi\)
\(942\) −13.8564 −0.451466
\(943\) 17.3590 0.565286
\(944\) −8.19615 −0.266762
\(945\) 0 0
\(946\) 30.5885 0.994517
\(947\) −5.85641 −0.190308 −0.0951538 0.995463i \(-0.530334\pi\)
−0.0951538 + 0.995463i \(0.530334\pi\)
\(948\) −5.53590 −0.179798
\(949\) 6.73205 0.218532
\(950\) 0 0
\(951\) 1.73205 0.0561656
\(952\) −4.53590 −0.147009
\(953\) −5.24871 −0.170022 −0.0850112 0.996380i \(-0.527093\pi\)
−0.0850112 + 0.996380i \(0.527093\pi\)
\(954\) 1.73205 0.0560772
\(955\) 0 0
\(956\) −14.0000 −0.452792
\(957\) −6.46410 −0.208955
\(958\) 3.14359 0.101565
\(959\) −9.07180 −0.292944
\(960\) 0 0
\(961\) −24.9282 −0.804136
\(962\) 5.46410 0.176170
\(963\) −2.19615 −0.0707700
\(964\) −30.1962 −0.972553
\(965\) 0 0
\(966\) 4.33975 0.139629
\(967\) 18.1051 0.582221 0.291111 0.956689i \(-0.405975\pi\)
0.291111 + 0.956689i \(0.405975\pi\)
\(968\) 2.92820 0.0941160
\(969\) −6.19615 −0.199049
\(970\) 0 0
\(971\) −37.7654 −1.21195 −0.605974 0.795484i \(-0.707217\pi\)
−0.605974 + 0.795484i \(0.707217\pi\)
\(972\) 1.00000 0.0320750
\(973\) 15.4641 0.495756
\(974\) 2.14359 0.0686852
\(975\) 0 0
\(976\) 10.6603 0.341226
\(977\) −1.46410 −0.0468408 −0.0234204 0.999726i \(-0.507456\pi\)
−0.0234204 + 0.999726i \(0.507456\pi\)
\(978\) 2.00000 0.0639529
\(979\) 40.5167 1.29492
\(980\) 0 0
\(981\) 2.39230 0.0763804
\(982\) −2.92820 −0.0934427
\(983\) −30.3397 −0.967688 −0.483844 0.875154i \(-0.660760\pi\)
−0.483844 + 0.875154i \(0.660760\pi\)
\(984\) −2.92820 −0.0933477
\(985\) 0 0
\(986\) −10.7321 −0.341778
\(987\) 2.53590 0.0807185
\(988\) −2.73205 −0.0869181
\(989\) −48.5885 −1.54502
\(990\) 0 0
\(991\) −40.3205 −1.28082 −0.640412 0.768032i \(-0.721236\pi\)
−0.640412 + 0.768032i \(0.721236\pi\)
\(992\) −2.46410 −0.0782353
\(993\) −22.2679 −0.706652
\(994\) −8.92820 −0.283185
\(995\) 0 0
\(996\) −3.73205 −0.118255
\(997\) −63.0526 −1.99689 −0.998447 0.0557047i \(-0.982259\pi\)
−0.998447 + 0.0557047i \(0.982259\pi\)
\(998\) 41.3731 1.30964
\(999\) 2.00000 0.0632772
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.bi.1.1 yes 2
3.2 odd 2 8550.2.a.bs.1.1 2
5.2 odd 4 2850.2.d.x.799.3 4
5.3 odd 4 2850.2.d.x.799.2 4
5.4 even 2 2850.2.a.bd.1.2 2
15.14 odd 2 8550.2.a.by.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2850.2.a.bd.1.2 2 5.4 even 2
2850.2.a.bi.1.1 yes 2 1.1 even 1 trivial
2850.2.d.x.799.2 4 5.3 odd 4
2850.2.d.x.799.3 4 5.2 odd 4
8550.2.a.bs.1.1 2 3.2 odd 2
8550.2.a.by.1.2 2 15.14 odd 2