Properties

Label 6422.2.a.bn.1.14
Level $6422$
Weight $2$
Character 6422.1
Self dual yes
Analytic conductor $51.280$
Analytic rank $0$
Dimension $14$
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] = [6422,2,Mod(1,6422)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6422, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6422.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6422.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: \(51.2799281781\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 4 x^{13} - 22 x^{12} + 98 x^{11} + 164 x^{10} - 912 x^{9} - 374 x^{8} + 3996 x^{7} - 817 x^{6} - 8194 x^{5} + 4418 x^{4} + 6430 x^{3} - 4327 x^{2} - 922 x + 358 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 494)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.14
Root \(3.36433\) of defining polynomial
Character \(\chi\) \(=\) 6422.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} +3.36433 q^{3} +1.00000 q^{4} +3.01748 q^{5} +3.36433 q^{6} -1.60598 q^{7} +1.00000 q^{8} +8.31869 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +3.36433 q^{3} +1.00000 q^{4} +3.01748 q^{5} +3.36433 q^{6} -1.60598 q^{7} +1.00000 q^{8} +8.31869 q^{9} +3.01748 q^{10} +4.74435 q^{11} +3.36433 q^{12} -1.60598 q^{14} +10.1518 q^{15} +1.00000 q^{16} -3.15064 q^{17} +8.31869 q^{18} +1.00000 q^{19} +3.01748 q^{20} -5.40303 q^{21} +4.74435 q^{22} -4.43766 q^{23} +3.36433 q^{24} +4.10516 q^{25} +17.8938 q^{27} -1.60598 q^{28} +1.98215 q^{29} +10.1518 q^{30} -3.75696 q^{31} +1.00000 q^{32} +15.9615 q^{33} -3.15064 q^{34} -4.84600 q^{35} +8.31869 q^{36} -8.45099 q^{37} +1.00000 q^{38} +3.01748 q^{40} -6.86672 q^{41} -5.40303 q^{42} +4.39270 q^{43} +4.74435 q^{44} +25.1014 q^{45} -4.43766 q^{46} -12.7582 q^{47} +3.36433 q^{48} -4.42084 q^{49} +4.10516 q^{50} -10.5998 q^{51} -9.81896 q^{53} +17.8938 q^{54} +14.3160 q^{55} -1.60598 q^{56} +3.36433 q^{57} +1.98215 q^{58} -5.56162 q^{59} +10.1518 q^{60} +7.01205 q^{61} -3.75696 q^{62} -13.3596 q^{63} +1.00000 q^{64} +15.9615 q^{66} +4.31241 q^{67} -3.15064 q^{68} -14.9297 q^{69} -4.84600 q^{70} +0.512626 q^{71} +8.31869 q^{72} -0.764448 q^{73} -8.45099 q^{74} +13.8111 q^{75} +1.00000 q^{76} -7.61931 q^{77} -1.84387 q^{79} +3.01748 q^{80} +35.2445 q^{81} -6.86672 q^{82} -15.5237 q^{83} -5.40303 q^{84} -9.50697 q^{85} +4.39270 q^{86} +6.66860 q^{87} +4.74435 q^{88} -6.85279 q^{89} +25.1014 q^{90} -4.43766 q^{92} -12.6397 q^{93} -12.7582 q^{94} +3.01748 q^{95} +3.36433 q^{96} -0.986823 q^{97} -4.42084 q^{98} +39.4667 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 14 q^{2} + 4 q^{3} + 14 q^{4} - 2 q^{5} + 4 q^{6} + 2 q^{7} + 14 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q + 14 q^{2} + 4 q^{3} + 14 q^{4} - 2 q^{5} + 4 q^{6} + 2 q^{7} + 14 q^{8} + 18 q^{9} - 2 q^{10} + 10 q^{11} + 4 q^{12} + 2 q^{14} - 4 q^{15} + 14 q^{16} + 2 q^{17} + 18 q^{18} + 14 q^{19} - 2 q^{20} - 18 q^{21} + 10 q^{22} + 12 q^{23} + 4 q^{24} + 44 q^{25} + 10 q^{27} + 2 q^{28} + 4 q^{29} - 4 q^{30} + 4 q^{31} + 14 q^{32} + 12 q^{33} + 2 q^{34} + 14 q^{35} + 18 q^{36} + 18 q^{37} + 14 q^{38} - 2 q^{40} + 6 q^{41} - 18 q^{42} + 28 q^{43} + 10 q^{44} + 8 q^{45} + 12 q^{46} - 20 q^{47} + 4 q^{48} + 28 q^{49} + 44 q^{50} + 10 q^{51} + 12 q^{53} + 10 q^{54} - 2 q^{55} + 2 q^{56} + 4 q^{57} + 4 q^{58} + 16 q^{59} - 4 q^{60} + 30 q^{61} + 4 q^{62} + 28 q^{63} + 14 q^{64} + 12 q^{66} + 2 q^{67} + 2 q^{68} - 42 q^{69} + 14 q^{70} + 44 q^{71} + 18 q^{72} - 36 q^{73} + 18 q^{74} + 46 q^{75} + 14 q^{76} + 68 q^{77} + 34 q^{79} - 2 q^{80} - 6 q^{81} + 6 q^{82} + 2 q^{83} - 18 q^{84} - 30 q^{85} + 28 q^{86} + 52 q^{87} + 10 q^{88} + 42 q^{89} + 8 q^{90} + 12 q^{92} - 12 q^{93} - 20 q^{94} - 2 q^{95} + 4 q^{96} + 40 q^{97} + 28 q^{98} + 12 q^{99}+O(q^{100}) \) 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
\(3\) 3.36433 1.94239 0.971197 0.238277i \(-0.0765826\pi\)
0.971197 + 0.238277i \(0.0765826\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.01748 1.34946 0.674728 0.738066i \(-0.264261\pi\)
0.674728 + 0.738066i \(0.264261\pi\)
\(6\) 3.36433 1.37348
\(7\) −1.60598 −0.607002 −0.303501 0.952831i \(-0.598156\pi\)
−0.303501 + 0.952831i \(0.598156\pi\)
\(8\) 1.00000 0.353553
\(9\) 8.31869 2.77290
\(10\) 3.01748 0.954210
\(11\) 4.74435 1.43047 0.715237 0.698882i \(-0.246319\pi\)
0.715237 + 0.698882i \(0.246319\pi\)
\(12\) 3.36433 0.971197
\(13\) 0 0
\(14\) −1.60598 −0.429215
\(15\) 10.1518 2.62118
\(16\) 1.00000 0.250000
\(17\) −3.15064 −0.764142 −0.382071 0.924133i \(-0.624789\pi\)
−0.382071 + 0.924133i \(0.624789\pi\)
\(18\) 8.31869 1.96073
\(19\) 1.00000 0.229416
\(20\) 3.01748 0.674728
\(21\) −5.40303 −1.17904
\(22\) 4.74435 1.01150
\(23\) −4.43766 −0.925317 −0.462658 0.886537i \(-0.653104\pi\)
−0.462658 + 0.886537i \(0.653104\pi\)
\(24\) 3.36433 0.686740
\(25\) 4.10516 0.821033
\(26\) 0 0
\(27\) 17.8938 3.44366
\(28\) −1.60598 −0.303501
\(29\) 1.98215 0.368076 0.184038 0.982919i \(-0.441083\pi\)
0.184038 + 0.982919i \(0.441083\pi\)
\(30\) 10.1518 1.85345
\(31\) −3.75696 −0.674771 −0.337385 0.941367i \(-0.609543\pi\)
−0.337385 + 0.941367i \(0.609543\pi\)
\(32\) 1.00000 0.176777
\(33\) 15.9615 2.77855
\(34\) −3.15064 −0.540330
\(35\) −4.84600 −0.819123
\(36\) 8.31869 1.38645
\(37\) −8.45099 −1.38933 −0.694667 0.719332i \(-0.744448\pi\)
−0.694667 + 0.719332i \(0.744448\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) 3.01748 0.477105
\(41\) −6.86672 −1.07240 −0.536200 0.844091i \(-0.680141\pi\)
−0.536200 + 0.844091i \(0.680141\pi\)
\(42\) −5.40303 −0.833706
\(43\) 4.39270 0.669880 0.334940 0.942239i \(-0.391284\pi\)
0.334940 + 0.942239i \(0.391284\pi\)
\(44\) 4.74435 0.715237
\(45\) 25.1014 3.74190
\(46\) −4.43766 −0.654298
\(47\) −12.7582 −1.86098 −0.930489 0.366320i \(-0.880618\pi\)
−0.930489 + 0.366320i \(0.880618\pi\)
\(48\) 3.36433 0.485599
\(49\) −4.42084 −0.631548
\(50\) 4.10516 0.580558
\(51\) −10.5998 −1.48426
\(52\) 0 0
\(53\) −9.81896 −1.34874 −0.674369 0.738395i \(-0.735584\pi\)
−0.674369 + 0.738395i \(0.735584\pi\)
\(54\) 17.8938 2.43504
\(55\) 14.3160 1.93036
\(56\) −1.60598 −0.214608
\(57\) 3.36433 0.445616
\(58\) 1.98215 0.260269
\(59\) −5.56162 −0.724061 −0.362031 0.932166i \(-0.617916\pi\)
−0.362031 + 0.932166i \(0.617916\pi\)
\(60\) 10.1518 1.31059
\(61\) 7.01205 0.897801 0.448901 0.893582i \(-0.351816\pi\)
0.448901 + 0.893582i \(0.351816\pi\)
\(62\) −3.75696 −0.477135
\(63\) −13.3596 −1.68315
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 15.9615 1.96473
\(67\) 4.31241 0.526844 0.263422 0.964681i \(-0.415149\pi\)
0.263422 + 0.964681i \(0.415149\pi\)
\(68\) −3.15064 −0.382071
\(69\) −14.9297 −1.79733
\(70\) −4.84600 −0.579207
\(71\) 0.512626 0.0608375 0.0304187 0.999537i \(-0.490316\pi\)
0.0304187 + 0.999537i \(0.490316\pi\)
\(72\) 8.31869 0.980367
\(73\) −0.764448 −0.0894719 −0.0447359 0.998999i \(-0.514245\pi\)
−0.0447359 + 0.998999i \(0.514245\pi\)
\(74\) −8.45099 −0.982407
\(75\) 13.8111 1.59477
\(76\) 1.00000 0.114708
\(77\) −7.61931 −0.868301
\(78\) 0 0
\(79\) −1.84387 −0.207452 −0.103726 0.994606i \(-0.533077\pi\)
−0.103726 + 0.994606i \(0.533077\pi\)
\(80\) 3.01748 0.337364
\(81\) 35.2445 3.91606
\(82\) −6.86672 −0.758302
\(83\) −15.5237 −1.70395 −0.851975 0.523582i \(-0.824595\pi\)
−0.851975 + 0.523582i \(0.824595\pi\)
\(84\) −5.40303 −0.589519
\(85\) −9.50697 −1.03118
\(86\) 4.39270 0.473677
\(87\) 6.66860 0.714949
\(88\) 4.74435 0.505749
\(89\) −6.85279 −0.726394 −0.363197 0.931712i \(-0.618315\pi\)
−0.363197 + 0.931712i \(0.618315\pi\)
\(90\) 25.1014 2.64592
\(91\) 0 0
\(92\) −4.43766 −0.462658
\(93\) −12.6397 −1.31067
\(94\) −12.7582 −1.31591
\(95\) 3.01748 0.309587
\(96\) 3.36433 0.343370
\(97\) −0.986823 −0.100197 −0.0500983 0.998744i \(-0.515953\pi\)
−0.0500983 + 0.998744i \(0.515953\pi\)
\(98\) −4.42084 −0.446572
\(99\) 39.4667 3.96656
\(100\) 4.10516 0.410516
\(101\) 5.62489 0.559698 0.279849 0.960044i \(-0.409716\pi\)
0.279849 + 0.960044i \(0.409716\pi\)
\(102\) −10.5998 −1.04953
\(103\) 11.9279 1.17529 0.587646 0.809118i \(-0.300055\pi\)
0.587646 + 0.809118i \(0.300055\pi\)
\(104\) 0 0
\(105\) −16.3035 −1.59106
\(106\) −9.81896 −0.953702
\(107\) −0.978249 −0.0945709 −0.0472854 0.998881i \(-0.515057\pi\)
−0.0472854 + 0.998881i \(0.515057\pi\)
\(108\) 17.8938 1.72183
\(109\) −11.2693 −1.07941 −0.539703 0.841855i \(-0.681464\pi\)
−0.539703 + 0.841855i \(0.681464\pi\)
\(110\) 14.3160 1.36497
\(111\) −28.4319 −2.69863
\(112\) −1.60598 −0.151751
\(113\) 15.2293 1.43266 0.716328 0.697764i \(-0.245821\pi\)
0.716328 + 0.697764i \(0.245821\pi\)
\(114\) 3.36433 0.315098
\(115\) −13.3905 −1.24867
\(116\) 1.98215 0.184038
\(117\) 0 0
\(118\) −5.56162 −0.511989
\(119\) 5.05985 0.463836
\(120\) 10.1518 0.926726
\(121\) 11.5088 1.04626
\(122\) 7.01205 0.634841
\(123\) −23.1019 −2.08303
\(124\) −3.75696 −0.337385
\(125\) −2.70015 −0.241509
\(126\) −13.3596 −1.19017
\(127\) 6.56275 0.582350 0.291175 0.956670i \(-0.405954\pi\)
0.291175 + 0.956670i \(0.405954\pi\)
\(128\) 1.00000 0.0883883
\(129\) 14.7785 1.30117
\(130\) 0 0
\(131\) −16.4259 −1.43513 −0.717567 0.696489i \(-0.754744\pi\)
−0.717567 + 0.696489i \(0.754744\pi\)
\(132\) 15.9615 1.38927
\(133\) −1.60598 −0.139256
\(134\) 4.31241 0.372535
\(135\) 53.9941 4.64708
\(136\) −3.15064 −0.270165
\(137\) 19.7835 1.69022 0.845111 0.534590i \(-0.179534\pi\)
0.845111 + 0.534590i \(0.179534\pi\)
\(138\) −14.9297 −1.27090
\(139\) 17.9507 1.52256 0.761281 0.648423i \(-0.224571\pi\)
0.761281 + 0.648423i \(0.224571\pi\)
\(140\) −4.84600 −0.409562
\(141\) −42.9228 −3.61475
\(142\) 0.512626 0.0430186
\(143\) 0 0
\(144\) 8.31869 0.693224
\(145\) 5.98109 0.496703
\(146\) −0.764448 −0.0632662
\(147\) −14.8731 −1.22672
\(148\) −8.45099 −0.694667
\(149\) −5.34727 −0.438066 −0.219033 0.975717i \(-0.570290\pi\)
−0.219033 + 0.975717i \(0.570290\pi\)
\(150\) 13.8111 1.12767
\(151\) −5.24105 −0.426510 −0.213255 0.976997i \(-0.568407\pi\)
−0.213255 + 0.976997i \(0.568407\pi\)
\(152\) 1.00000 0.0811107
\(153\) −26.2092 −2.11889
\(154\) −7.61931 −0.613982
\(155\) −11.3366 −0.910574
\(156\) 0 0
\(157\) −2.18765 −0.174593 −0.0872967 0.996182i \(-0.527823\pi\)
−0.0872967 + 0.996182i \(0.527823\pi\)
\(158\) −1.84387 −0.146691
\(159\) −33.0342 −2.61978
\(160\) 3.01748 0.238552
\(161\) 7.12678 0.561669
\(162\) 35.2445 2.76907
\(163\) 6.48287 0.507778 0.253889 0.967233i \(-0.418290\pi\)
0.253889 + 0.967233i \(0.418290\pi\)
\(164\) −6.86672 −0.536200
\(165\) 48.1635 3.74953
\(166\) −15.5237 −1.20488
\(167\) 7.51569 0.581581 0.290791 0.956787i \(-0.406082\pi\)
0.290791 + 0.956787i \(0.406082\pi\)
\(168\) −5.40303 −0.416853
\(169\) 0 0
\(170\) −9.50697 −0.729152
\(171\) 8.31869 0.636146
\(172\) 4.39270 0.334940
\(173\) 18.9342 1.43954 0.719769 0.694214i \(-0.244248\pi\)
0.719769 + 0.694214i \(0.244248\pi\)
\(174\) 6.66860 0.505545
\(175\) −6.59280 −0.498369
\(176\) 4.74435 0.357619
\(177\) −18.7111 −1.40641
\(178\) −6.85279 −0.513638
\(179\) −5.62248 −0.420244 −0.210122 0.977675i \(-0.567386\pi\)
−0.210122 + 0.977675i \(0.567386\pi\)
\(180\) 25.1014 1.87095
\(181\) 10.2724 0.763538 0.381769 0.924258i \(-0.375315\pi\)
0.381769 + 0.924258i \(0.375315\pi\)
\(182\) 0 0
\(183\) 23.5908 1.74388
\(184\) −4.43766 −0.327149
\(185\) −25.5007 −1.87485
\(186\) −12.6397 −0.926784
\(187\) −14.9477 −1.09309
\(188\) −12.7582 −0.930489
\(189\) −28.7370 −2.09031
\(190\) 3.01748 0.218911
\(191\) −17.6817 −1.27941 −0.639703 0.768622i \(-0.720943\pi\)
−0.639703 + 0.768622i \(0.720943\pi\)
\(192\) 3.36433 0.242799
\(193\) −1.74232 −0.125415 −0.0627075 0.998032i \(-0.519974\pi\)
−0.0627075 + 0.998032i \(0.519974\pi\)
\(194\) −0.986823 −0.0708497
\(195\) 0 0
\(196\) −4.42084 −0.315774
\(197\) 24.5715 1.75065 0.875324 0.483536i \(-0.160648\pi\)
0.875324 + 0.483536i \(0.160648\pi\)
\(198\) 39.4667 2.80478
\(199\) −15.6179 −1.10712 −0.553561 0.832808i \(-0.686732\pi\)
−0.553561 + 0.832808i \(0.686732\pi\)
\(200\) 4.10516 0.290279
\(201\) 14.5083 1.02334
\(202\) 5.62489 0.395766
\(203\) −3.18329 −0.223423
\(204\) −10.5998 −0.742132
\(205\) −20.7202 −1.44716
\(206\) 11.9279 0.831056
\(207\) −36.9155 −2.56581
\(208\) 0 0
\(209\) 4.74435 0.328173
\(210\) −16.3035 −1.12505
\(211\) 6.76086 0.465437 0.232718 0.972544i \(-0.425238\pi\)
0.232718 + 0.972544i \(0.425238\pi\)
\(212\) −9.81896 −0.674369
\(213\) 1.72464 0.118170
\(214\) −0.978249 −0.0668717
\(215\) 13.2549 0.903974
\(216\) 17.8938 1.21752
\(217\) 6.03360 0.409587
\(218\) −11.2693 −0.763256
\(219\) −2.57185 −0.173790
\(220\) 14.3160 0.965181
\(221\) 0 0
\(222\) −28.4319 −1.90822
\(223\) −1.03472 −0.0692903 −0.0346451 0.999400i \(-0.511030\pi\)
−0.0346451 + 0.999400i \(0.511030\pi\)
\(224\) −1.60598 −0.107304
\(225\) 34.1496 2.27664
\(226\) 15.2293 1.01304
\(227\) 15.5518 1.03221 0.516104 0.856526i \(-0.327382\pi\)
0.516104 + 0.856526i \(0.327382\pi\)
\(228\) 3.36433 0.222808
\(229\) −1.43041 −0.0945243 −0.0472621 0.998883i \(-0.515050\pi\)
−0.0472621 + 0.998883i \(0.515050\pi\)
\(230\) −13.3905 −0.882946
\(231\) −25.6338 −1.68658
\(232\) 1.98215 0.130135
\(233\) 1.48502 0.0972871 0.0486436 0.998816i \(-0.484510\pi\)
0.0486436 + 0.998816i \(0.484510\pi\)
\(234\) 0 0
\(235\) −38.4976 −2.51131
\(236\) −5.56162 −0.362031
\(237\) −6.20340 −0.402954
\(238\) 5.05985 0.327981
\(239\) 7.28803 0.471423 0.235712 0.971823i \(-0.424258\pi\)
0.235712 + 0.971823i \(0.424258\pi\)
\(240\) 10.1518 0.655294
\(241\) 10.0796 0.649282 0.324641 0.945837i \(-0.394757\pi\)
0.324641 + 0.945837i \(0.394757\pi\)
\(242\) 11.5088 0.739815
\(243\) 64.8927 4.16287
\(244\) 7.01205 0.448901
\(245\) −13.3398 −0.852247
\(246\) −23.1019 −1.47292
\(247\) 0 0
\(248\) −3.75696 −0.238568
\(249\) −52.2269 −3.30974
\(250\) −2.70015 −0.170772
\(251\) 19.6753 1.24190 0.620948 0.783852i \(-0.286748\pi\)
0.620948 + 0.783852i \(0.286748\pi\)
\(252\) −13.3596 −0.841577
\(253\) −21.0538 −1.32364
\(254\) 6.56275 0.411783
\(255\) −31.9846 −2.00295
\(256\) 1.00000 0.0625000
\(257\) 17.1766 1.07145 0.535723 0.844394i \(-0.320039\pi\)
0.535723 + 0.844394i \(0.320039\pi\)
\(258\) 14.7785 0.920067
\(259\) 13.5721 0.843329
\(260\) 0 0
\(261\) 16.4889 1.02064
\(262\) −16.4259 −1.01479
\(263\) −7.47064 −0.460660 −0.230330 0.973113i \(-0.573980\pi\)
−0.230330 + 0.973113i \(0.573980\pi\)
\(264\) 15.9615 0.982364
\(265\) −29.6285 −1.82006
\(266\) −1.60598 −0.0984688
\(267\) −23.0550 −1.41094
\(268\) 4.31241 0.263422
\(269\) 12.2533 0.747098 0.373549 0.927610i \(-0.378141\pi\)
0.373549 + 0.927610i \(0.378141\pi\)
\(270\) 53.9941 3.28598
\(271\) −2.46954 −0.150014 −0.0750069 0.997183i \(-0.523898\pi\)
−0.0750069 + 0.997183i \(0.523898\pi\)
\(272\) −3.15064 −0.191035
\(273\) 0 0
\(274\) 19.7835 1.19517
\(275\) 19.4763 1.17447
\(276\) −14.9297 −0.898665
\(277\) 17.7074 1.06393 0.531967 0.846765i \(-0.321453\pi\)
0.531967 + 0.846765i \(0.321453\pi\)
\(278\) 17.9507 1.07661
\(279\) −31.2530 −1.87107
\(280\) −4.84600 −0.289604
\(281\) 29.7058 1.77210 0.886050 0.463589i \(-0.153439\pi\)
0.886050 + 0.463589i \(0.153439\pi\)
\(282\) −42.9228 −2.55602
\(283\) −15.4985 −0.921289 −0.460645 0.887585i \(-0.652382\pi\)
−0.460645 + 0.887585i \(0.652382\pi\)
\(284\) 0.512626 0.0304187
\(285\) 10.1518 0.601339
\(286\) 0 0
\(287\) 11.0278 0.650950
\(288\) 8.31869 0.490183
\(289\) −7.07349 −0.416087
\(290\) 5.98109 0.351222
\(291\) −3.31999 −0.194621
\(292\) −0.764448 −0.0447359
\(293\) −14.1662 −0.827598 −0.413799 0.910368i \(-0.635798\pi\)
−0.413799 + 0.910368i \(0.635798\pi\)
\(294\) −14.8731 −0.867419
\(295\) −16.7821 −0.977089
\(296\) −8.45099 −0.491204
\(297\) 84.8944 4.92607
\(298\) −5.34727 −0.309759
\(299\) 0 0
\(300\) 13.8111 0.797385
\(301\) −7.05457 −0.406619
\(302\) −5.24105 −0.301588
\(303\) 18.9240 1.08715
\(304\) 1.00000 0.0573539
\(305\) 21.1587 1.21154
\(306\) −26.2092 −1.49828
\(307\) 21.1538 1.20731 0.603655 0.797245i \(-0.293710\pi\)
0.603655 + 0.797245i \(0.293710\pi\)
\(308\) −7.61931 −0.434151
\(309\) 40.1294 2.28288
\(310\) −11.3366 −0.643873
\(311\) 11.3516 0.643690 0.321845 0.946792i \(-0.395697\pi\)
0.321845 + 0.946792i \(0.395697\pi\)
\(312\) 0 0
\(313\) −20.5500 −1.16155 −0.580777 0.814063i \(-0.697251\pi\)
−0.580777 + 0.814063i \(0.697251\pi\)
\(314\) −2.18765 −0.123456
\(315\) −40.3123 −2.27134
\(316\) −1.84387 −0.103726
\(317\) −1.70454 −0.0957367 −0.0478683 0.998854i \(-0.515243\pi\)
−0.0478683 + 0.998854i \(0.515243\pi\)
\(318\) −33.0342 −1.85246
\(319\) 9.40401 0.526523
\(320\) 3.01748 0.168682
\(321\) −3.29115 −0.183694
\(322\) 7.12678 0.397160
\(323\) −3.15064 −0.175306
\(324\) 35.2445 1.95803
\(325\) 0 0
\(326\) 6.48287 0.359053
\(327\) −37.9137 −2.09663
\(328\) −6.86672 −0.379151
\(329\) 20.4894 1.12962
\(330\) 48.1635 2.65132
\(331\) 9.04476 0.497145 0.248573 0.968613i \(-0.420039\pi\)
0.248573 + 0.968613i \(0.420039\pi\)
\(332\) −15.5237 −0.851975
\(333\) −70.3011 −3.85248
\(334\) 7.51569 0.411240
\(335\) 13.0126 0.710954
\(336\) −5.40303 −0.294759
\(337\) 22.5298 1.22727 0.613637 0.789588i \(-0.289706\pi\)
0.613637 + 0.789588i \(0.289706\pi\)
\(338\) 0 0
\(339\) 51.2365 2.78278
\(340\) −9.50697 −0.515588
\(341\) −17.8243 −0.965242
\(342\) 8.31869 0.449823
\(343\) 18.3416 0.990353
\(344\) 4.39270 0.236838
\(345\) −45.0501 −2.42542
\(346\) 18.9342 1.01791
\(347\) 30.5928 1.64231 0.821154 0.570707i \(-0.193331\pi\)
0.821154 + 0.570707i \(0.193331\pi\)
\(348\) 6.66860 0.357475
\(349\) 16.8111 0.899875 0.449937 0.893060i \(-0.351446\pi\)
0.449937 + 0.893060i \(0.351446\pi\)
\(350\) −6.59280 −0.352400
\(351\) 0 0
\(352\) 4.74435 0.252875
\(353\) 12.5945 0.670338 0.335169 0.942158i \(-0.391207\pi\)
0.335169 + 0.942158i \(0.391207\pi\)
\(354\) −18.7111 −0.994484
\(355\) 1.54684 0.0820975
\(356\) −6.85279 −0.363197
\(357\) 17.0230 0.900952
\(358\) −5.62248 −0.297157
\(359\) 17.9825 0.949078 0.474539 0.880234i \(-0.342615\pi\)
0.474539 + 0.880234i \(0.342615\pi\)
\(360\) 25.1014 1.32296
\(361\) 1.00000 0.0526316
\(362\) 10.2724 0.539903
\(363\) 38.7194 2.03224
\(364\) 0 0
\(365\) −2.30670 −0.120738
\(366\) 23.5908 1.23311
\(367\) 24.7904 1.29405 0.647023 0.762470i \(-0.276014\pi\)
0.647023 + 0.762470i \(0.276014\pi\)
\(368\) −4.43766 −0.231329
\(369\) −57.1221 −2.97366
\(370\) −25.5007 −1.32572
\(371\) 15.7690 0.818687
\(372\) −12.6397 −0.655336
\(373\) −11.3778 −0.589118 −0.294559 0.955633i \(-0.595173\pi\)
−0.294559 + 0.955633i \(0.595173\pi\)
\(374\) −14.9477 −0.772928
\(375\) −9.08419 −0.469105
\(376\) −12.7582 −0.657955
\(377\) 0 0
\(378\) −28.7370 −1.47807
\(379\) 17.2668 0.886934 0.443467 0.896291i \(-0.353748\pi\)
0.443467 + 0.896291i \(0.353748\pi\)
\(380\) 3.01748 0.154793
\(381\) 22.0792 1.13115
\(382\) −17.6817 −0.904676
\(383\) −31.1832 −1.59339 −0.796694 0.604383i \(-0.793420\pi\)
−0.796694 + 0.604383i \(0.793420\pi\)
\(384\) 3.36433 0.171685
\(385\) −22.9911 −1.17173
\(386\) −1.74232 −0.0886818
\(387\) 36.5415 1.85751
\(388\) −0.986823 −0.0500983
\(389\) −25.4245 −1.28907 −0.644536 0.764574i \(-0.722949\pi\)
−0.644536 + 0.764574i \(0.722949\pi\)
\(390\) 0 0
\(391\) 13.9815 0.707073
\(392\) −4.42084 −0.223286
\(393\) −55.2620 −2.78760
\(394\) 24.5715 1.23790
\(395\) −5.56385 −0.279948
\(396\) 39.4667 1.98328
\(397\) −17.5256 −0.879586 −0.439793 0.898099i \(-0.644948\pi\)
−0.439793 + 0.898099i \(0.644948\pi\)
\(398\) −15.6179 −0.782854
\(399\) −5.40303 −0.270490
\(400\) 4.10516 0.205258
\(401\) 19.1479 0.956202 0.478101 0.878305i \(-0.341325\pi\)
0.478101 + 0.878305i \(0.341325\pi\)
\(402\) 14.5083 0.723610
\(403\) 0 0
\(404\) 5.62489 0.279849
\(405\) 106.350 5.28455
\(406\) −3.18329 −0.157984
\(407\) −40.0944 −1.98741
\(408\) −10.5998 −0.524767
\(409\) −3.57350 −0.176698 −0.0883490 0.996090i \(-0.528159\pi\)
−0.0883490 + 0.996090i \(0.528159\pi\)
\(410\) −20.7202 −1.02330
\(411\) 66.5583 3.28308
\(412\) 11.9279 0.587646
\(413\) 8.93183 0.439507
\(414\) −36.9155 −1.81430
\(415\) −46.8425 −2.29941
\(416\) 0 0
\(417\) 60.3921 2.95741
\(418\) 4.74435 0.232054
\(419\) 9.65745 0.471797 0.235899 0.971778i \(-0.424197\pi\)
0.235899 + 0.971778i \(0.424197\pi\)
\(420\) −16.3035 −0.795530
\(421\) 11.6457 0.567577 0.283788 0.958887i \(-0.408409\pi\)
0.283788 + 0.958887i \(0.408409\pi\)
\(422\) 6.76086 0.329114
\(423\) −106.132 −5.16030
\(424\) −9.81896 −0.476851
\(425\) −12.9339 −0.627385
\(426\) 1.72464 0.0835591
\(427\) −11.2612 −0.544967
\(428\) −0.978249 −0.0472854
\(429\) 0 0
\(430\) 13.2549 0.639206
\(431\) −35.6809 −1.71869 −0.859344 0.511398i \(-0.829128\pi\)
−0.859344 + 0.511398i \(0.829128\pi\)
\(432\) 17.8938 0.860916
\(433\) −12.4345 −0.597565 −0.298782 0.954321i \(-0.596580\pi\)
−0.298782 + 0.954321i \(0.596580\pi\)
\(434\) 6.03360 0.289622
\(435\) 20.1223 0.964793
\(436\) −11.2693 −0.539703
\(437\) −4.43766 −0.212282
\(438\) −2.57185 −0.122888
\(439\) 22.1563 1.05746 0.528732 0.848789i \(-0.322668\pi\)
0.528732 + 0.848789i \(0.322668\pi\)
\(440\) 14.3160 0.682486
\(441\) −36.7756 −1.75122
\(442\) 0 0
\(443\) −28.3851 −1.34862 −0.674309 0.738449i \(-0.735558\pi\)
−0.674309 + 0.738449i \(0.735558\pi\)
\(444\) −28.4319 −1.34932
\(445\) −20.6781 −0.980237
\(446\) −1.03472 −0.0489956
\(447\) −17.9900 −0.850897
\(448\) −1.60598 −0.0758753
\(449\) −19.4234 −0.916648 −0.458324 0.888785i \(-0.651550\pi\)
−0.458324 + 0.888785i \(0.651550\pi\)
\(450\) 34.1496 1.60983
\(451\) −32.5781 −1.53404
\(452\) 15.2293 0.716328
\(453\) −17.6326 −0.828452
\(454\) 15.5518 0.729882
\(455\) 0 0
\(456\) 3.36433 0.157549
\(457\) −1.69519 −0.0792979 −0.0396489 0.999214i \(-0.512624\pi\)
−0.0396489 + 0.999214i \(0.512624\pi\)
\(458\) −1.43041 −0.0668388
\(459\) −56.3769 −2.63145
\(460\) −13.3905 −0.624337
\(461\) −32.8331 −1.52919 −0.764594 0.644512i \(-0.777060\pi\)
−0.764594 + 0.644512i \(0.777060\pi\)
\(462\) −25.6338 −1.19259
\(463\) 26.2428 1.21960 0.609802 0.792554i \(-0.291249\pi\)
0.609802 + 0.792554i \(0.291249\pi\)
\(464\) 1.98215 0.0920190
\(465\) −38.1399 −1.76869
\(466\) 1.48502 0.0687924
\(467\) −16.6106 −0.768648 −0.384324 0.923198i \(-0.625566\pi\)
−0.384324 + 0.923198i \(0.625566\pi\)
\(468\) 0 0
\(469\) −6.92563 −0.319796
\(470\) −38.4976 −1.77576
\(471\) −7.35996 −0.339129
\(472\) −5.56162 −0.255994
\(473\) 20.8405 0.958246
\(474\) −6.20340 −0.284931
\(475\) 4.10516 0.188358
\(476\) 5.05985 0.231918
\(477\) −81.6809 −3.73991
\(478\) 7.28803 0.333347
\(479\) 22.7182 1.03802 0.519012 0.854767i \(-0.326300\pi\)
0.519012 + 0.854767i \(0.326300\pi\)
\(480\) 10.1518 0.463363
\(481\) 0 0
\(482\) 10.0796 0.459112
\(483\) 23.9768 1.09098
\(484\) 11.5088 0.523128
\(485\) −2.97771 −0.135211
\(486\) 64.8927 2.94359
\(487\) −16.0162 −0.725765 −0.362883 0.931835i \(-0.618207\pi\)
−0.362883 + 0.931835i \(0.618207\pi\)
\(488\) 7.01205 0.317421
\(489\) 21.8105 0.986305
\(490\) −13.3398 −0.602630
\(491\) −10.4778 −0.472856 −0.236428 0.971649i \(-0.575977\pi\)
−0.236428 + 0.971649i \(0.575977\pi\)
\(492\) −23.1019 −1.04151
\(493\) −6.24504 −0.281262
\(494\) 0 0
\(495\) 119.090 5.35270
\(496\) −3.75696 −0.168693
\(497\) −0.823265 −0.0369285
\(498\) −52.2269 −2.34034
\(499\) 12.8864 0.576873 0.288437 0.957499i \(-0.406865\pi\)
0.288437 + 0.957499i \(0.406865\pi\)
\(500\) −2.70015 −0.120754
\(501\) 25.2852 1.12966
\(502\) 19.6753 0.878153
\(503\) 15.6110 0.696060 0.348030 0.937483i \(-0.386851\pi\)
0.348030 + 0.937483i \(0.386851\pi\)
\(504\) −13.3596 −0.595085
\(505\) 16.9730 0.755288
\(506\) −21.0538 −0.935956
\(507\) 0 0
\(508\) 6.56275 0.291175
\(509\) −32.9207 −1.45919 −0.729593 0.683882i \(-0.760290\pi\)
−0.729593 + 0.683882i \(0.760290\pi\)
\(510\) −31.9846 −1.41630
\(511\) 1.22769 0.0543096
\(512\) 1.00000 0.0441942
\(513\) 17.8938 0.790031
\(514\) 17.1766 0.757627
\(515\) 35.9922 1.58600
\(516\) 14.7785 0.650586
\(517\) −60.5294 −2.66208
\(518\) 13.5721 0.596323
\(519\) 63.7007 2.79615
\(520\) 0 0
\(521\) −21.9328 −0.960892 −0.480446 0.877024i \(-0.659525\pi\)
−0.480446 + 0.877024i \(0.659525\pi\)
\(522\) 16.4889 0.721699
\(523\) 25.5453 1.11702 0.558509 0.829498i \(-0.311374\pi\)
0.558509 + 0.829498i \(0.311374\pi\)
\(524\) −16.4259 −0.717567
\(525\) −22.1803 −0.968028
\(526\) −7.47064 −0.325736
\(527\) 11.8368 0.515621
\(528\) 15.9615 0.694636
\(529\) −3.30715 −0.143789
\(530\) −29.6285 −1.28698
\(531\) −46.2654 −2.00775
\(532\) −1.60598 −0.0696279
\(533\) 0 0
\(534\) −23.0550 −0.997688
\(535\) −2.95184 −0.127619
\(536\) 4.31241 0.186268
\(537\) −18.9159 −0.816280
\(538\) 12.2533 0.528278
\(539\) −20.9740 −0.903414
\(540\) 53.9941 2.32354
\(541\) −39.5212 −1.69915 −0.849575 0.527468i \(-0.823142\pi\)
−0.849575 + 0.527468i \(0.823142\pi\)
\(542\) −2.46954 −0.106076
\(543\) 34.5596 1.48309
\(544\) −3.15064 −0.135082
\(545\) −34.0050 −1.45661
\(546\) 0 0
\(547\) 1.72014 0.0735480 0.0367740 0.999324i \(-0.488292\pi\)
0.0367740 + 0.999324i \(0.488292\pi\)
\(548\) 19.7835 0.845111
\(549\) 58.3311 2.48951
\(550\) 19.4763 0.830473
\(551\) 1.98215 0.0844425
\(552\) −14.9297 −0.635452
\(553\) 2.96122 0.125924
\(554\) 17.7074 0.752315
\(555\) −85.7925 −3.64169
\(556\) 17.9507 0.761281
\(557\) 25.0279 1.06046 0.530232 0.847852i \(-0.322105\pi\)
0.530232 + 0.847852i \(0.322105\pi\)
\(558\) −31.2530 −1.32305
\(559\) 0 0
\(560\) −4.84600 −0.204781
\(561\) −50.2890 −2.12320
\(562\) 29.7058 1.25306
\(563\) −0.0275168 −0.00115969 −0.000579847 1.00000i \(-0.500185\pi\)
−0.000579847 1.00000i \(0.500185\pi\)
\(564\) −42.9228 −1.80738
\(565\) 45.9542 1.93331
\(566\) −15.4985 −0.651450
\(567\) −56.6019 −2.37706
\(568\) 0.512626 0.0215093
\(569\) −33.3603 −1.39854 −0.699269 0.714859i \(-0.746491\pi\)
−0.699269 + 0.714859i \(0.746491\pi\)
\(570\) 10.1518 0.425211
\(571\) −0.291209 −0.0121867 −0.00609336 0.999981i \(-0.501940\pi\)
−0.00609336 + 0.999981i \(0.501940\pi\)
\(572\) 0 0
\(573\) −59.4871 −2.48511
\(574\) 11.0278 0.460291
\(575\) −18.2173 −0.759715
\(576\) 8.31869 0.346612
\(577\) 8.64988 0.360099 0.180050 0.983658i \(-0.442374\pi\)
0.180050 + 0.983658i \(0.442374\pi\)
\(578\) −7.07349 −0.294218
\(579\) −5.86174 −0.243605
\(580\) 5.98109 0.248351
\(581\) 24.9308 1.03430
\(582\) −3.31999 −0.137618
\(583\) −46.5845 −1.92933
\(584\) −0.764448 −0.0316331
\(585\) 0 0
\(586\) −14.1662 −0.585200
\(587\) 34.8523 1.43851 0.719254 0.694747i \(-0.244484\pi\)
0.719254 + 0.694747i \(0.244484\pi\)
\(588\) −14.8731 −0.613358
\(589\) −3.75696 −0.154803
\(590\) −16.7821 −0.690906
\(591\) 82.6666 3.40045
\(592\) −8.45099 −0.347333
\(593\) −29.5726 −1.21440 −0.607200 0.794549i \(-0.707707\pi\)
−0.607200 + 0.794549i \(0.707707\pi\)
\(594\) 84.8944 3.48326
\(595\) 15.2680 0.625926
\(596\) −5.34727 −0.219033
\(597\) −52.5437 −2.15047
\(598\) 0 0
\(599\) 10.2670 0.419497 0.209748 0.977755i \(-0.432736\pi\)
0.209748 + 0.977755i \(0.432736\pi\)
\(600\) 13.8111 0.563836
\(601\) −25.1219 −1.02474 −0.512372 0.858764i \(-0.671233\pi\)
−0.512372 + 0.858764i \(0.671233\pi\)
\(602\) −7.05457 −0.287523
\(603\) 35.8736 1.46089
\(604\) −5.24105 −0.213255
\(605\) 34.7276 1.41188
\(606\) 18.9240 0.768734
\(607\) 1.68508 0.0683953 0.0341977 0.999415i \(-0.489112\pi\)
0.0341977 + 0.999415i \(0.489112\pi\)
\(608\) 1.00000 0.0405554
\(609\) −10.7096 −0.433976
\(610\) 21.1587 0.856691
\(611\) 0 0
\(612\) −26.2092 −1.05944
\(613\) −4.01859 −0.162309 −0.0811547 0.996702i \(-0.525861\pi\)
−0.0811547 + 0.996702i \(0.525861\pi\)
\(614\) 21.1538 0.853697
\(615\) −69.7094 −2.81095
\(616\) −7.61931 −0.306991
\(617\) 13.5199 0.544293 0.272146 0.962256i \(-0.412267\pi\)
0.272146 + 0.962256i \(0.412267\pi\)
\(618\) 40.1294 1.61424
\(619\) 27.5059 1.10555 0.552777 0.833329i \(-0.313568\pi\)
0.552777 + 0.833329i \(0.313568\pi\)
\(620\) −11.3366 −0.455287
\(621\) −79.4067 −3.18648
\(622\) 11.3516 0.455157
\(623\) 11.0054 0.440923
\(624\) 0 0
\(625\) −28.6735 −1.14694
\(626\) −20.5500 −0.821342
\(627\) 15.9615 0.637442
\(628\) −2.18765 −0.0872967
\(629\) 26.6260 1.06165
\(630\) −40.3123 −1.60608
\(631\) 9.68396 0.385513 0.192756 0.981247i \(-0.438257\pi\)
0.192756 + 0.981247i \(0.438257\pi\)
\(632\) −1.84387 −0.0733454
\(633\) 22.7457 0.904062
\(634\) −1.70454 −0.0676961
\(635\) 19.8029 0.785856
\(636\) −33.0342 −1.30989
\(637\) 0 0
\(638\) 9.40401 0.372308
\(639\) 4.26437 0.168696
\(640\) 3.01748 0.119276
\(641\) 14.2237 0.561802 0.280901 0.959737i \(-0.409367\pi\)
0.280901 + 0.959737i \(0.409367\pi\)
\(642\) −3.29115 −0.129891
\(643\) −30.6473 −1.20861 −0.604306 0.796752i \(-0.706550\pi\)
−0.604306 + 0.796752i \(0.706550\pi\)
\(644\) 7.12678 0.280835
\(645\) 44.5937 1.75587
\(646\) −3.15064 −0.123960
\(647\) −0.270247 −0.0106245 −0.00531226 0.999986i \(-0.501691\pi\)
−0.00531226 + 0.999986i \(0.501691\pi\)
\(648\) 35.2445 1.38454
\(649\) −26.3863 −1.03575
\(650\) 0 0
\(651\) 20.2990 0.795580
\(652\) 6.48287 0.253889
\(653\) 27.9086 1.09215 0.546075 0.837736i \(-0.316121\pi\)
0.546075 + 0.837736i \(0.316121\pi\)
\(654\) −37.9137 −1.48254
\(655\) −49.5647 −1.93665
\(656\) −6.86672 −0.268100
\(657\) −6.35921 −0.248096
\(658\) 20.4894 0.798760
\(659\) 33.3056 1.29740 0.648701 0.761043i \(-0.275313\pi\)
0.648701 + 0.761043i \(0.275313\pi\)
\(660\) 48.1635 1.87476
\(661\) −10.9624 −0.426389 −0.213195 0.977010i \(-0.568387\pi\)
−0.213195 + 0.977010i \(0.568387\pi\)
\(662\) 9.04476 0.351535
\(663\) 0 0
\(664\) −15.5237 −0.602438
\(665\) −4.84600 −0.187920
\(666\) −70.3011 −2.72411
\(667\) −8.79611 −0.340587
\(668\) 7.51569 0.290791
\(669\) −3.48115 −0.134589
\(670\) 13.0126 0.502720
\(671\) 33.2676 1.28428
\(672\) −5.40303 −0.208426
\(673\) 5.63771 0.217318 0.108659 0.994079i \(-0.465344\pi\)
0.108659 + 0.994079i \(0.465344\pi\)
\(674\) 22.5298 0.867814
\(675\) 73.4570 2.82736
\(676\) 0 0
\(677\) 27.4262 1.05407 0.527037 0.849842i \(-0.323303\pi\)
0.527037 + 0.849842i \(0.323303\pi\)
\(678\) 51.2365 1.96772
\(679\) 1.58481 0.0608196
\(680\) −9.50697 −0.364576
\(681\) 52.3213 2.00496
\(682\) −17.8243 −0.682529
\(683\) −12.4989 −0.478258 −0.239129 0.970988i \(-0.576862\pi\)
−0.239129 + 0.970988i \(0.576862\pi\)
\(684\) 8.31869 0.318073
\(685\) 59.6964 2.28088
\(686\) 18.3416 0.700286
\(687\) −4.81237 −0.183603
\(688\) 4.39270 0.167470
\(689\) 0 0
\(690\) −45.0501 −1.71503
\(691\) −49.5378 −1.88451 −0.942254 0.334900i \(-0.891298\pi\)
−0.942254 + 0.334900i \(0.891298\pi\)
\(692\) 18.9342 0.719769
\(693\) −63.3827 −2.40771
\(694\) 30.5928 1.16129
\(695\) 54.1659 2.05463
\(696\) 6.66860 0.252773
\(697\) 21.6345 0.819466
\(698\) 16.8111 0.636308
\(699\) 4.99610 0.188970
\(700\) −6.59280 −0.249184
\(701\) −30.4261 −1.14918 −0.574589 0.818442i \(-0.694838\pi\)
−0.574589 + 0.818442i \(0.694838\pi\)
\(702\) 0 0
\(703\) −8.45099 −0.318735
\(704\) 4.74435 0.178809
\(705\) −129.519 −4.87795
\(706\) 12.5945 0.474001
\(707\) −9.03345 −0.339738
\(708\) −18.7111 −0.703206
\(709\) −34.3500 −1.29004 −0.645021 0.764165i \(-0.723151\pi\)
−0.645021 + 0.764165i \(0.723151\pi\)
\(710\) 1.54684 0.0580517
\(711\) −15.3386 −0.575243
\(712\) −6.85279 −0.256819
\(713\) 16.6721 0.624377
\(714\) 17.0230 0.637069
\(715\) 0 0
\(716\) −5.62248 −0.210122
\(717\) 24.5193 0.915690
\(718\) 17.9825 0.671100
\(719\) −20.4893 −0.764121 −0.382060 0.924137i \(-0.624785\pi\)
−0.382060 + 0.924137i \(0.624785\pi\)
\(720\) 25.1014 0.935476
\(721\) −19.1559 −0.713404
\(722\) 1.00000 0.0372161
\(723\) 33.9109 1.26116
\(724\) 10.2724 0.381769
\(725\) 8.13705 0.302202
\(726\) 38.7194 1.43701
\(727\) 8.97463 0.332851 0.166425 0.986054i \(-0.446778\pi\)
0.166425 + 0.986054i \(0.446778\pi\)
\(728\) 0 0
\(729\) 112.586 4.16987
\(730\) −2.30670 −0.0853749
\(731\) −13.8398 −0.511883
\(732\) 23.5908 0.871942
\(733\) 30.7616 1.13621 0.568103 0.822958i \(-0.307678\pi\)
0.568103 + 0.822958i \(0.307678\pi\)
\(734\) 24.7904 0.915029
\(735\) −44.8793 −1.65540
\(736\) −4.43766 −0.163574
\(737\) 20.4596 0.753637
\(738\) −57.1221 −2.10269
\(739\) −25.8774 −0.951914 −0.475957 0.879469i \(-0.657898\pi\)
−0.475957 + 0.879469i \(0.657898\pi\)
\(740\) −25.5007 −0.937423
\(741\) 0 0
\(742\) 15.7690 0.578899
\(743\) −3.70756 −0.136017 −0.0680086 0.997685i \(-0.521665\pi\)
−0.0680086 + 0.997685i \(0.521665\pi\)
\(744\) −12.6397 −0.463392
\(745\) −16.1353 −0.591151
\(746\) −11.3778 −0.416569
\(747\) −129.137 −4.72488
\(748\) −14.9477 −0.546543
\(749\) 1.57105 0.0574047
\(750\) −9.08419 −0.331708
\(751\) 26.1045 0.952566 0.476283 0.879292i \(-0.341984\pi\)
0.476283 + 0.879292i \(0.341984\pi\)
\(752\) −12.7582 −0.465245
\(753\) 66.1943 2.41225
\(754\) 0 0
\(755\) −15.8147 −0.575557
\(756\) −28.7370 −1.04516
\(757\) −0.521137 −0.0189411 −0.00947053 0.999955i \(-0.503015\pi\)
−0.00947053 + 0.999955i \(0.503015\pi\)
\(758\) 17.2668 0.627157
\(759\) −70.8319 −2.57103
\(760\) 3.01748 0.109455
\(761\) −48.2179 −1.74790 −0.873949 0.486018i \(-0.838449\pi\)
−0.873949 + 0.486018i \(0.838449\pi\)
\(762\) 22.0792 0.799846
\(763\) 18.0983 0.655202
\(764\) −17.6817 −0.639703
\(765\) −79.0856 −2.85934
\(766\) −31.1832 −1.12670
\(767\) 0 0
\(768\) 3.36433 0.121400
\(769\) −11.6926 −0.421646 −0.210823 0.977524i \(-0.567614\pi\)
−0.210823 + 0.977524i \(0.567614\pi\)
\(770\) −22.9911 −0.828541
\(771\) 57.7876 2.08117
\(772\) −1.74232 −0.0627075
\(773\) −1.49281 −0.0536927 −0.0268463 0.999640i \(-0.508546\pi\)
−0.0268463 + 0.999640i \(0.508546\pi\)
\(774\) 36.5415 1.31346
\(775\) −15.4230 −0.554009
\(776\) −0.986823 −0.0354249
\(777\) 45.6609 1.63808
\(778\) −25.4245 −0.911512
\(779\) −6.86672 −0.246026
\(780\) 0 0
\(781\) 2.43207 0.0870264
\(782\) 13.9815 0.499976
\(783\) 35.4682 1.26753
\(784\) −4.42084 −0.157887
\(785\) −6.60118 −0.235606
\(786\) −55.2620 −1.97113
\(787\) 23.6691 0.843713 0.421856 0.906663i \(-0.361379\pi\)
0.421856 + 0.906663i \(0.361379\pi\)
\(788\) 24.5715 0.875324
\(789\) −25.1337 −0.894783
\(790\) −5.56385 −0.197953
\(791\) −24.4580 −0.869625
\(792\) 39.4667 1.40239
\(793\) 0 0
\(794\) −17.5256 −0.621961
\(795\) −99.6798 −3.53528
\(796\) −15.6179 −0.553561
\(797\) −36.5242 −1.29376 −0.646878 0.762594i \(-0.723926\pi\)
−0.646878 + 0.762594i \(0.723926\pi\)
\(798\) −5.40303 −0.191265
\(799\) 40.1965 1.42205
\(800\) 4.10516 0.145139
\(801\) −57.0062 −2.01422
\(802\) 19.1479 0.676137
\(803\) −3.62681 −0.127987
\(804\) 14.5083 0.511670
\(805\) 21.5049 0.757948
\(806\) 0 0
\(807\) 41.2242 1.45116
\(808\) 5.62489 0.197883
\(809\) 39.8942 1.40261 0.701303 0.712863i \(-0.252602\pi\)
0.701303 + 0.712863i \(0.252602\pi\)
\(810\) 106.350 3.73674
\(811\) −14.6962 −0.516052 −0.258026 0.966138i \(-0.583072\pi\)
−0.258026 + 0.966138i \(0.583072\pi\)
\(812\) −3.18329 −0.111712
\(813\) −8.30833 −0.291386
\(814\) −40.0944 −1.40531
\(815\) 19.5619 0.685224
\(816\) −10.5998 −0.371066
\(817\) 4.39270 0.153681
\(818\) −3.57350 −0.124944
\(819\) 0 0
\(820\) −20.7202 −0.723579
\(821\) 20.5084 0.715749 0.357874 0.933770i \(-0.383502\pi\)
0.357874 + 0.933770i \(0.383502\pi\)
\(822\) 66.5583 2.32149
\(823\) 26.9934 0.940931 0.470465 0.882418i \(-0.344086\pi\)
0.470465 + 0.882418i \(0.344086\pi\)
\(824\) 11.9279 0.415528
\(825\) 65.5247 2.28128
\(826\) 8.93183 0.310778
\(827\) 10.9795 0.381796 0.190898 0.981610i \(-0.438860\pi\)
0.190898 + 0.981610i \(0.438860\pi\)
\(828\) −36.9155 −1.28290
\(829\) 9.08917 0.315680 0.157840 0.987465i \(-0.449547\pi\)
0.157840 + 0.987465i \(0.449547\pi\)
\(830\) −46.8425 −1.62593
\(831\) 59.5734 2.06658
\(832\) 0 0
\(833\) 13.9285 0.482592
\(834\) 60.3921 2.09121
\(835\) 22.6784 0.784818
\(836\) 4.74435 0.164087
\(837\) −67.2264 −2.32368
\(838\) 9.65745 0.333611
\(839\) −1.83497 −0.0633502 −0.0316751 0.999498i \(-0.510084\pi\)
−0.0316751 + 0.999498i \(0.510084\pi\)
\(840\) −16.3035 −0.562525
\(841\) −25.0711 −0.864520
\(842\) 11.6457 0.401338
\(843\) 99.9401 3.44212
\(844\) 6.76086 0.232718
\(845\) 0 0
\(846\) −106.132 −3.64888
\(847\) −18.4829 −0.635080
\(848\) −9.81896 −0.337184
\(849\) −52.1420 −1.78951
\(850\) −12.9339 −0.443628
\(851\) 37.5026 1.28557
\(852\) 1.72464 0.0590852
\(853\) −0.647190 −0.0221594 −0.0110797 0.999939i \(-0.503527\pi\)
−0.0110797 + 0.999939i \(0.503527\pi\)
\(854\) −11.2612 −0.385350
\(855\) 25.1014 0.858451
\(856\) −0.978249 −0.0334359
\(857\) 34.2606 1.17032 0.585161 0.810917i \(-0.301031\pi\)
0.585161 + 0.810917i \(0.301031\pi\)
\(858\) 0 0
\(859\) −55.5935 −1.89683 −0.948414 0.317036i \(-0.897313\pi\)
−0.948414 + 0.317036i \(0.897313\pi\)
\(860\) 13.2549 0.451987
\(861\) 37.1011 1.26440
\(862\) −35.6809 −1.21530
\(863\) −3.48869 −0.118756 −0.0593781 0.998236i \(-0.518912\pi\)
−0.0593781 + 0.998236i \(0.518912\pi\)
\(864\) 17.8938 0.608760
\(865\) 57.1334 1.94259
\(866\) −12.4345 −0.422542
\(867\) −23.7975 −0.808206
\(868\) 6.03360 0.204794
\(869\) −8.74798 −0.296755
\(870\) 20.1223 0.682211
\(871\) 0 0
\(872\) −11.2693 −0.381628
\(873\) −8.20907 −0.277835
\(874\) −4.43766 −0.150106
\(875\) 4.33638 0.146596
\(876\) −2.57185 −0.0868948
\(877\) 8.78684 0.296710 0.148355 0.988934i \(-0.452602\pi\)
0.148355 + 0.988934i \(0.452602\pi\)
\(878\) 22.1563 0.747739
\(879\) −47.6597 −1.60752
\(880\) 14.3160 0.482591
\(881\) 8.39000 0.282666 0.141333 0.989962i \(-0.454861\pi\)
0.141333 + 0.989962i \(0.454861\pi\)
\(882\) −36.7756 −1.23830
\(883\) −26.2560 −0.883586 −0.441793 0.897117i \(-0.645658\pi\)
−0.441793 + 0.897117i \(0.645658\pi\)
\(884\) 0 0
\(885\) −56.4603 −1.89789
\(886\) −28.3851 −0.953617
\(887\) 57.8010 1.94077 0.970384 0.241567i \(-0.0776615\pi\)
0.970384 + 0.241567i \(0.0776615\pi\)
\(888\) −28.4319 −0.954111
\(889\) −10.5396 −0.353488
\(890\) −20.6781 −0.693132
\(891\) 167.212 5.60182
\(892\) −1.03472 −0.0346451
\(893\) −12.7582 −0.426938
\(894\) −17.9900 −0.601675
\(895\) −16.9657 −0.567101
\(896\) −1.60598 −0.0536519
\(897\) 0 0
\(898\) −19.4234 −0.648168
\(899\) −7.44687 −0.248367
\(900\) 34.1496 1.13832
\(901\) 30.9360 1.03063
\(902\) −32.5781 −1.08473
\(903\) −23.7339 −0.789814
\(904\) 15.2293 0.506520
\(905\) 30.9966 1.03036
\(906\) −17.6326 −0.585804
\(907\) −7.29698 −0.242292 −0.121146 0.992635i \(-0.538657\pi\)
−0.121146 + 0.992635i \(0.538657\pi\)
\(908\) 15.5518 0.516104
\(909\) 46.7917 1.55198
\(910\) 0 0
\(911\) 25.2188 0.835536 0.417768 0.908554i \(-0.362813\pi\)
0.417768 + 0.908554i \(0.362813\pi\)
\(912\) 3.36433 0.111404
\(913\) −73.6500 −2.43746
\(914\) −1.69519 −0.0560721
\(915\) 71.1848 2.35330
\(916\) −1.43041 −0.0472621
\(917\) 26.3796 0.871130
\(918\) −56.3769 −1.86071
\(919\) −28.1990 −0.930200 −0.465100 0.885258i \(-0.653982\pi\)
−0.465100 + 0.885258i \(0.653982\pi\)
\(920\) −13.3905 −0.441473
\(921\) 71.1682 2.34507
\(922\) −32.8331 −1.08130
\(923\) 0 0
\(924\) −25.6338 −0.843292
\(925\) −34.6927 −1.14069
\(926\) 26.2428 0.862391
\(927\) 99.2245 3.25896
\(928\) 1.98215 0.0650673
\(929\) −7.39808 −0.242723 −0.121362 0.992608i \(-0.538726\pi\)
−0.121362 + 0.992608i \(0.538726\pi\)
\(930\) −38.1399 −1.25066
\(931\) −4.42084 −0.144887
\(932\) 1.48502 0.0486436
\(933\) 38.1904 1.25030
\(934\) −16.6106 −0.543516
\(935\) −45.1044 −1.47507
\(936\) 0 0
\(937\) −27.5732 −0.900777 −0.450389 0.892833i \(-0.648715\pi\)
−0.450389 + 0.892833i \(0.648715\pi\)
\(938\) −6.92563 −0.226130
\(939\) −69.1368 −2.25619
\(940\) −38.4976 −1.25565
\(941\) −38.8296 −1.26581 −0.632904 0.774231i \(-0.718137\pi\)
−0.632904 + 0.774231i \(0.718137\pi\)
\(942\) −7.35996 −0.239800
\(943\) 30.4722 0.992310
\(944\) −5.56162 −0.181015
\(945\) −86.7133 −2.82078
\(946\) 20.8405 0.677582
\(947\) 26.7230 0.868380 0.434190 0.900821i \(-0.357035\pi\)
0.434190 + 0.900821i \(0.357035\pi\)
\(948\) −6.20340 −0.201477
\(949\) 0 0
\(950\) 4.10516 0.133189
\(951\) −5.73464 −0.185958
\(952\) 5.05985 0.163991
\(953\) 44.4391 1.43952 0.719761 0.694222i \(-0.244251\pi\)
0.719761 + 0.694222i \(0.244251\pi\)
\(954\) −81.6809 −2.64452
\(955\) −53.3542 −1.72650
\(956\) 7.28803 0.235712
\(957\) 31.6382 1.02272
\(958\) 22.7182 0.733993
\(959\) −31.7719 −1.02597
\(960\) 10.1518 0.327647
\(961\) −16.8852 −0.544684
\(962\) 0 0
\(963\) −8.13775 −0.262235
\(964\) 10.0796 0.324641
\(965\) −5.25741 −0.169242
\(966\) 23.9768 0.771442
\(967\) 15.9816 0.513933 0.256966 0.966420i \(-0.417277\pi\)
0.256966 + 0.966420i \(0.417277\pi\)
\(968\) 11.5088 0.369908
\(969\) −10.5998 −0.340514
\(970\) −2.97771 −0.0956086
\(971\) 36.6172 1.17510 0.587551 0.809187i \(-0.300092\pi\)
0.587551 + 0.809187i \(0.300092\pi\)
\(972\) 64.8927 2.08143
\(973\) −28.8285 −0.924198
\(974\) −16.0162 −0.513193
\(975\) 0 0
\(976\) 7.01205 0.224450
\(977\) 24.9648 0.798695 0.399347 0.916800i \(-0.369237\pi\)
0.399347 + 0.916800i \(0.369237\pi\)
\(978\) 21.8105 0.697423
\(979\) −32.5120 −1.03909
\(980\) −13.3398 −0.426123
\(981\) −93.7461 −2.99308
\(982\) −10.4778 −0.334360
\(983\) −16.5472 −0.527774 −0.263887 0.964554i \(-0.585005\pi\)
−0.263887 + 0.964554i \(0.585005\pi\)
\(984\) −23.1019 −0.736461
\(985\) 74.1440 2.36242
\(986\) −6.24504 −0.198883
\(987\) 68.9331 2.19416
\(988\) 0 0
\(989\) −19.4933 −0.619851
\(990\) 119.090 3.78493
\(991\) −34.7410 −1.10358 −0.551792 0.833982i \(-0.686056\pi\)
−0.551792 + 0.833982i \(0.686056\pi\)
\(992\) −3.75696 −0.119284
\(993\) 30.4295 0.965652
\(994\) −0.823265 −0.0261124
\(995\) −47.1266 −1.49401
\(996\) −52.2269 −1.65487
\(997\) 14.9452 0.473318 0.236659 0.971593i \(-0.423948\pi\)
0.236659 + 0.971593i \(0.423948\pi\)
\(998\) 12.8864 0.407911
\(999\) −151.220 −4.78440
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 6422.2.a.bn.1.14 14
13.2 odd 12 494.2.m.b.381.8 yes 28
13.7 odd 12 494.2.m.b.153.8 28
13.12 even 2 6422.2.a.bm.1.14 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
494.2.m.b.153.8 28 13.7 odd 12
494.2.m.b.381.8 yes 28 13.2 odd 12
6422.2.a.bm.1.14 14 13.12 even 2
6422.2.a.bn.1.14 14 1.1 even 1 trivial