Properties

Label 8037.2.a.n.1.11
Level $8037$
Weight $2$
Character 8037.1
Self dual yes
Analytic conductor $64.176$
Analytic rank $1$
Dimension $16$
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] = [8037,2,Mod(1,8037)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8037, 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("8037.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8037.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: \(64.1757681045\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 13 x^{14} + 65 x^{13} + 47 x^{12} - 390 x^{11} + 4 x^{10} + 1115 x^{9} - 320 x^{8} - 1639 x^{7} + 618 x^{6} + 1250 x^{5} - 487 x^{4} - 456 x^{3} + 179 x^{2} + \cdots - 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 893)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(1.22346\) of defining polynomial
Character \(\chi\) \(=\) 8037.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.22346 q^{2} -0.503155 q^{4} -1.49488 q^{5} +1.92629 q^{7} -3.06250 q^{8} +O(q^{10})\) \(q+1.22346 q^{2} -0.503155 q^{4} -1.49488 q^{5} +1.92629 q^{7} -3.06250 q^{8} -1.82892 q^{10} +5.98067 q^{11} +0.688389 q^{13} +2.35674 q^{14} -2.74052 q^{16} -1.22062 q^{17} -1.00000 q^{19} +0.752156 q^{20} +7.31709 q^{22} -7.14665 q^{23} -2.76534 q^{25} +0.842214 q^{26} -0.969225 q^{28} -6.31879 q^{29} -3.41339 q^{31} +2.77209 q^{32} -1.49338 q^{34} -2.87958 q^{35} +3.81678 q^{37} -1.22346 q^{38} +4.57807 q^{40} +11.6520 q^{41} -6.30138 q^{43} -3.00920 q^{44} -8.74361 q^{46} +1.00000 q^{47} -3.28939 q^{49} -3.38327 q^{50} -0.346366 q^{52} +4.61374 q^{53} -8.94038 q^{55} -5.89928 q^{56} -7.73076 q^{58} +8.93709 q^{59} +0.297368 q^{61} -4.17613 q^{62} +8.87258 q^{64} -1.02906 q^{65} -14.9319 q^{67} +0.614163 q^{68} -3.52303 q^{70} +8.44664 q^{71} +3.11139 q^{73} +4.66966 q^{74} +0.503155 q^{76} +11.5205 q^{77} -11.6188 q^{79} +4.09675 q^{80} +14.2557 q^{82} -7.52556 q^{83} +1.82468 q^{85} -7.70946 q^{86} -18.3158 q^{88} +7.87193 q^{89} +1.32604 q^{91} +3.59587 q^{92} +1.22346 q^{94} +1.49488 q^{95} +15.4165 q^{97} -4.02443 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} + 10 q^{4} + q^{5} - 9 q^{7} + 9 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{2} + 10 q^{4} + q^{5} - 9 q^{7} + 9 q^{8} - 15 q^{10} - 19 q^{13} + 6 q^{14} + 10 q^{16} + 8 q^{17} - 16 q^{19} + 11 q^{20} - 12 q^{22} + 5 q^{23} - 3 q^{25} - 9 q^{26} - 17 q^{28} + 2 q^{29} - 18 q^{31} - 3 q^{32} - 14 q^{34} + 11 q^{35} - 24 q^{37} - 4 q^{38} - 50 q^{40} + 6 q^{41} - 34 q^{43} + 4 q^{44} - 3 q^{46} + 16 q^{47} + 5 q^{49} - 26 q^{50} - 44 q^{52} + 23 q^{53} - 48 q^{55} + 3 q^{56} - 26 q^{58} + 32 q^{59} - 16 q^{61} - 32 q^{62} + 7 q^{64} + 18 q^{65} - 67 q^{67} + 19 q^{68} + 24 q^{70} - 19 q^{71} - 2 q^{73} + 29 q^{74} - 10 q^{76} - 14 q^{77} - 27 q^{79} - 15 q^{80} - 56 q^{82} + 17 q^{83} + 15 q^{85} + q^{86} - 13 q^{88} - 20 q^{89} - 42 q^{91} - 45 q^{92} + 4 q^{94} - q^{95} - 50 q^{97} - 13 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.22346 0.865114 0.432557 0.901607i \(-0.357611\pi\)
0.432557 + 0.901607i \(0.357611\pi\)
\(3\) 0 0
\(4\) −0.503155 −0.251578
\(5\) −1.49488 −0.668530 −0.334265 0.942479i \(-0.608488\pi\)
−0.334265 + 0.942479i \(0.608488\pi\)
\(6\) 0 0
\(7\) 1.92629 0.728071 0.364035 0.931385i \(-0.381399\pi\)
0.364035 + 0.931385i \(0.381399\pi\)
\(8\) −3.06250 −1.08276
\(9\) 0 0
\(10\) −1.82892 −0.578355
\(11\) 5.98067 1.80324 0.901620 0.432529i \(-0.142379\pi\)
0.901620 + 0.432529i \(0.142379\pi\)
\(12\) 0 0
\(13\) 0.688389 0.190925 0.0954624 0.995433i \(-0.469567\pi\)
0.0954624 + 0.995433i \(0.469567\pi\)
\(14\) 2.35674 0.629864
\(15\) 0 0
\(16\) −2.74052 −0.685131
\(17\) −1.22062 −0.296045 −0.148022 0.988984i \(-0.547291\pi\)
−0.148022 + 0.988984i \(0.547291\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0.752156 0.168187
\(21\) 0 0
\(22\) 7.31709 1.56001
\(23\) −7.14665 −1.49018 −0.745089 0.666965i \(-0.767593\pi\)
−0.745089 + 0.666965i \(0.767593\pi\)
\(24\) 0 0
\(25\) −2.76534 −0.553068
\(26\) 0.842214 0.165172
\(27\) 0 0
\(28\) −0.969225 −0.183166
\(29\) −6.31879 −1.17337 −0.586685 0.809816i \(-0.699567\pi\)
−0.586685 + 0.809816i \(0.699567\pi\)
\(30\) 0 0
\(31\) −3.41339 −0.613063 −0.306532 0.951860i \(-0.599169\pi\)
−0.306532 + 0.951860i \(0.599169\pi\)
\(32\) 2.77209 0.490041
\(33\) 0 0
\(34\) −1.49338 −0.256113
\(35\) −2.87958 −0.486737
\(36\) 0 0
\(37\) 3.81678 0.627475 0.313737 0.949510i \(-0.398419\pi\)
0.313737 + 0.949510i \(0.398419\pi\)
\(38\) −1.22346 −0.198471
\(39\) 0 0
\(40\) 4.57807 0.723856
\(41\) 11.6520 1.81974 0.909869 0.414896i \(-0.136182\pi\)
0.909869 + 0.414896i \(0.136182\pi\)
\(42\) 0 0
\(43\) −6.30138 −0.960951 −0.480475 0.877008i \(-0.659536\pi\)
−0.480475 + 0.877008i \(0.659536\pi\)
\(44\) −3.00920 −0.453655
\(45\) 0 0
\(46\) −8.74361 −1.28917
\(47\) 1.00000 0.145865
\(48\) 0 0
\(49\) −3.28939 −0.469913
\(50\) −3.38327 −0.478467
\(51\) 0 0
\(52\) −0.346366 −0.0480324
\(53\) 4.61374 0.633746 0.316873 0.948468i \(-0.397367\pi\)
0.316873 + 0.948468i \(0.397367\pi\)
\(54\) 0 0
\(55\) −8.94038 −1.20552
\(56\) −5.89928 −0.788324
\(57\) 0 0
\(58\) −7.73076 −1.01510
\(59\) 8.93709 1.16351 0.581755 0.813364i \(-0.302366\pi\)
0.581755 + 0.813364i \(0.302366\pi\)
\(60\) 0 0
\(61\) 0.297368 0.0380740 0.0190370 0.999819i \(-0.493940\pi\)
0.0190370 + 0.999819i \(0.493940\pi\)
\(62\) −4.17613 −0.530370
\(63\) 0 0
\(64\) 8.87258 1.10907
\(65\) −1.02906 −0.127639
\(66\) 0 0
\(67\) −14.9319 −1.82423 −0.912113 0.409938i \(-0.865550\pi\)
−0.912113 + 0.409938i \(0.865550\pi\)
\(68\) 0.614163 0.0744782
\(69\) 0 0
\(70\) −3.52303 −0.421083
\(71\) 8.44664 1.00243 0.501216 0.865322i \(-0.332886\pi\)
0.501216 + 0.865322i \(0.332886\pi\)
\(72\) 0 0
\(73\) 3.11139 0.364161 0.182081 0.983284i \(-0.441717\pi\)
0.182081 + 0.983284i \(0.441717\pi\)
\(74\) 4.66966 0.542837
\(75\) 0 0
\(76\) 0.503155 0.0577159
\(77\) 11.5205 1.31289
\(78\) 0 0
\(79\) −11.6188 −1.30722 −0.653610 0.756831i \(-0.726746\pi\)
−0.653610 + 0.756831i \(0.726746\pi\)
\(80\) 4.09675 0.458031
\(81\) 0 0
\(82\) 14.2557 1.57428
\(83\) −7.52556 −0.826037 −0.413018 0.910723i \(-0.635526\pi\)
−0.413018 + 0.910723i \(0.635526\pi\)
\(84\) 0 0
\(85\) 1.82468 0.197915
\(86\) −7.70946 −0.831332
\(87\) 0 0
\(88\) −18.3158 −1.95247
\(89\) 7.87193 0.834423 0.417212 0.908809i \(-0.363007\pi\)
0.417212 + 0.908809i \(0.363007\pi\)
\(90\) 0 0
\(91\) 1.32604 0.139007
\(92\) 3.59587 0.374896
\(93\) 0 0
\(94\) 1.22346 0.126190
\(95\) 1.49488 0.153371
\(96\) 0 0
\(97\) 15.4165 1.56531 0.782656 0.622455i \(-0.213865\pi\)
0.782656 + 0.622455i \(0.213865\pi\)
\(98\) −4.02443 −0.406528
\(99\) 0 0
\(100\) 1.39139 0.139139
\(101\) −8.02244 −0.798262 −0.399131 0.916894i \(-0.630688\pi\)
−0.399131 + 0.916894i \(0.630688\pi\)
\(102\) 0 0
\(103\) −15.3108 −1.50862 −0.754310 0.656518i \(-0.772028\pi\)
−0.754310 + 0.656518i \(0.772028\pi\)
\(104\) −2.10819 −0.206725
\(105\) 0 0
\(106\) 5.64471 0.548262
\(107\) −1.39574 −0.134932 −0.0674658 0.997722i \(-0.521491\pi\)
−0.0674658 + 0.997722i \(0.521491\pi\)
\(108\) 0 0
\(109\) −8.40912 −0.805448 −0.402724 0.915322i \(-0.631936\pi\)
−0.402724 + 0.915322i \(0.631936\pi\)
\(110\) −10.9382 −1.04291
\(111\) 0 0
\(112\) −5.27906 −0.498824
\(113\) −6.96791 −0.655486 −0.327743 0.944767i \(-0.606288\pi\)
−0.327743 + 0.944767i \(0.606288\pi\)
\(114\) 0 0
\(115\) 10.6834 0.996229
\(116\) 3.17933 0.295193
\(117\) 0 0
\(118\) 10.9341 1.00657
\(119\) −2.35128 −0.215542
\(120\) 0 0
\(121\) 24.7684 2.25167
\(122\) 0.363816 0.0329384
\(123\) 0 0
\(124\) 1.71747 0.154233
\(125\) 11.6082 1.03827
\(126\) 0 0
\(127\) −21.6731 −1.92317 −0.961586 0.274503i \(-0.911487\pi\)
−0.961586 + 0.274503i \(0.911487\pi\)
\(128\) 5.31103 0.469433
\(129\) 0 0
\(130\) −1.25901 −0.110422
\(131\) −1.73571 −0.151650 −0.0758249 0.997121i \(-0.524159\pi\)
−0.0758249 + 0.997121i \(0.524159\pi\)
\(132\) 0 0
\(133\) −1.92629 −0.167031
\(134\) −18.2686 −1.57816
\(135\) 0 0
\(136\) 3.73816 0.320545
\(137\) −21.5158 −1.83822 −0.919111 0.393998i \(-0.871092\pi\)
−0.919111 + 0.393998i \(0.871092\pi\)
\(138\) 0 0
\(139\) −23.1810 −1.96619 −0.983094 0.183101i \(-0.941387\pi\)
−0.983094 + 0.183101i \(0.941387\pi\)
\(140\) 1.44887 0.122452
\(141\) 0 0
\(142\) 10.3341 0.867218
\(143\) 4.11703 0.344283
\(144\) 0 0
\(145\) 9.44582 0.784433
\(146\) 3.80665 0.315041
\(147\) 0 0
\(148\) −1.92043 −0.157859
\(149\) 4.54134 0.372041 0.186020 0.982546i \(-0.440441\pi\)
0.186020 + 0.982546i \(0.440441\pi\)
\(150\) 0 0
\(151\) 18.6606 1.51858 0.759290 0.650753i \(-0.225547\pi\)
0.759290 + 0.650753i \(0.225547\pi\)
\(152\) 3.06250 0.248402
\(153\) 0 0
\(154\) 14.0949 1.13580
\(155\) 5.10261 0.409851
\(156\) 0 0
\(157\) −19.8978 −1.58802 −0.794009 0.607906i \(-0.792010\pi\)
−0.794009 + 0.607906i \(0.792010\pi\)
\(158\) −14.2151 −1.13089
\(159\) 0 0
\(160\) −4.14394 −0.327607
\(161\) −13.7665 −1.08496
\(162\) 0 0
\(163\) −4.13350 −0.323760 −0.161880 0.986810i \(-0.551756\pi\)
−0.161880 + 0.986810i \(0.551756\pi\)
\(164\) −5.86277 −0.457805
\(165\) 0 0
\(166\) −9.20719 −0.714616
\(167\) −3.16675 −0.245051 −0.122525 0.992465i \(-0.539099\pi\)
−0.122525 + 0.992465i \(0.539099\pi\)
\(168\) 0 0
\(169\) −12.5261 −0.963548
\(170\) 2.23242 0.171219
\(171\) 0 0
\(172\) 3.17057 0.241754
\(173\) 7.56877 0.575443 0.287721 0.957714i \(-0.407102\pi\)
0.287721 + 0.957714i \(0.407102\pi\)
\(174\) 0 0
\(175\) −5.32685 −0.402672
\(176\) −16.3902 −1.23546
\(177\) 0 0
\(178\) 9.63096 0.721871
\(179\) 22.3928 1.67371 0.836857 0.547422i \(-0.184391\pi\)
0.836857 + 0.547422i \(0.184391\pi\)
\(180\) 0 0
\(181\) 4.90461 0.364557 0.182279 0.983247i \(-0.441653\pi\)
0.182279 + 0.983247i \(0.441653\pi\)
\(182\) 1.62235 0.120257
\(183\) 0 0
\(184\) 21.8866 1.61350
\(185\) −5.70562 −0.419486
\(186\) 0 0
\(187\) −7.30015 −0.533840
\(188\) −0.503155 −0.0366964
\(189\) 0 0
\(190\) 1.82892 0.132684
\(191\) −9.22858 −0.667757 −0.333878 0.942616i \(-0.608357\pi\)
−0.333878 + 0.942616i \(0.608357\pi\)
\(192\) 0 0
\(193\) −23.6393 −1.70159 −0.850797 0.525495i \(-0.823880\pi\)
−0.850797 + 0.525495i \(0.823880\pi\)
\(194\) 18.8614 1.35417
\(195\) 0 0
\(196\) 1.65507 0.118220
\(197\) 14.1239 1.00629 0.503143 0.864203i \(-0.332177\pi\)
0.503143 + 0.864203i \(0.332177\pi\)
\(198\) 0 0
\(199\) 4.02709 0.285473 0.142736 0.989761i \(-0.454410\pi\)
0.142736 + 0.989761i \(0.454410\pi\)
\(200\) 8.46885 0.598838
\(201\) 0 0
\(202\) −9.81510 −0.690588
\(203\) −12.1718 −0.854296
\(204\) 0 0
\(205\) −17.4183 −1.21655
\(206\) −18.7321 −1.30513
\(207\) 0 0
\(208\) −1.88655 −0.130808
\(209\) −5.98067 −0.413692
\(210\) 0 0
\(211\) −6.89473 −0.474653 −0.237326 0.971430i \(-0.576271\pi\)
−0.237326 + 0.971430i \(0.576271\pi\)
\(212\) −2.32143 −0.159436
\(213\) 0 0
\(214\) −1.70763 −0.116731
\(215\) 9.41979 0.642425
\(216\) 0 0
\(217\) −6.57520 −0.446353
\(218\) −10.2882 −0.696804
\(219\) 0 0
\(220\) 4.49840 0.303282
\(221\) −0.840264 −0.0565223
\(222\) 0 0
\(223\) 15.9566 1.06853 0.534265 0.845317i \(-0.320588\pi\)
0.534265 + 0.845317i \(0.320588\pi\)
\(224\) 5.33986 0.356784
\(225\) 0 0
\(226\) −8.52493 −0.567070
\(227\) −2.18124 −0.144774 −0.0723871 0.997377i \(-0.523062\pi\)
−0.0723871 + 0.997377i \(0.523062\pi\)
\(228\) 0 0
\(229\) 17.0499 1.12669 0.563344 0.826222i \(-0.309514\pi\)
0.563344 + 0.826222i \(0.309514\pi\)
\(230\) 13.0706 0.861852
\(231\) 0 0
\(232\) 19.3513 1.27047
\(233\) −18.3157 −1.19990 −0.599951 0.800037i \(-0.704813\pi\)
−0.599951 + 0.800037i \(0.704813\pi\)
\(234\) 0 0
\(235\) −1.49488 −0.0975151
\(236\) −4.49674 −0.292713
\(237\) 0 0
\(238\) −2.87669 −0.186468
\(239\) −13.2209 −0.855188 −0.427594 0.903971i \(-0.640639\pi\)
−0.427594 + 0.903971i \(0.640639\pi\)
\(240\) 0 0
\(241\) −2.44331 −0.157388 −0.0786938 0.996899i \(-0.525075\pi\)
−0.0786938 + 0.996899i \(0.525075\pi\)
\(242\) 30.3031 1.94795
\(243\) 0 0
\(244\) −0.149622 −0.00957857
\(245\) 4.91724 0.314151
\(246\) 0 0
\(247\) −0.688389 −0.0438011
\(248\) 10.4535 0.663799
\(249\) 0 0
\(250\) 14.2022 0.898224
\(251\) −10.0476 −0.634201 −0.317101 0.948392i \(-0.602709\pi\)
−0.317101 + 0.948392i \(0.602709\pi\)
\(252\) 0 0
\(253\) −42.7417 −2.68715
\(254\) −26.5160 −1.66376
\(255\) 0 0
\(256\) −11.2473 −0.702959
\(257\) −18.6200 −1.16148 −0.580740 0.814089i \(-0.697237\pi\)
−0.580740 + 0.814089i \(0.697237\pi\)
\(258\) 0 0
\(259\) 7.35224 0.456846
\(260\) 0.517776 0.0321111
\(261\) 0 0
\(262\) −2.12357 −0.131194
\(263\) 0.577076 0.0355840 0.0177920 0.999842i \(-0.494336\pi\)
0.0177920 + 0.999842i \(0.494336\pi\)
\(264\) 0 0
\(265\) −6.89698 −0.423678
\(266\) −2.35674 −0.144501
\(267\) 0 0
\(268\) 7.51308 0.458934
\(269\) −19.8758 −1.21185 −0.605924 0.795523i \(-0.707196\pi\)
−0.605924 + 0.795523i \(0.707196\pi\)
\(270\) 0 0
\(271\) 17.5339 1.06511 0.532555 0.846395i \(-0.321232\pi\)
0.532555 + 0.846395i \(0.321232\pi\)
\(272\) 3.34515 0.202829
\(273\) 0 0
\(274\) −26.3237 −1.59027
\(275\) −16.5386 −0.997313
\(276\) 0 0
\(277\) 22.4987 1.35181 0.675907 0.736987i \(-0.263752\pi\)
0.675907 + 0.736987i \(0.263752\pi\)
\(278\) −28.3610 −1.70098
\(279\) 0 0
\(280\) 8.81870 0.527018
\(281\) −22.9312 −1.36796 −0.683981 0.729500i \(-0.739753\pi\)
−0.683981 + 0.729500i \(0.739753\pi\)
\(282\) 0 0
\(283\) −8.03247 −0.477481 −0.238740 0.971083i \(-0.576734\pi\)
−0.238740 + 0.971083i \(0.576734\pi\)
\(284\) −4.24997 −0.252189
\(285\) 0 0
\(286\) 5.03700 0.297844
\(287\) 22.4452 1.32490
\(288\) 0 0
\(289\) −15.5101 −0.912357
\(290\) 11.5565 0.678624
\(291\) 0 0
\(292\) −1.56551 −0.0916148
\(293\) −11.7285 −0.685185 −0.342592 0.939484i \(-0.611305\pi\)
−0.342592 + 0.939484i \(0.611305\pi\)
\(294\) 0 0
\(295\) −13.3599 −0.777842
\(296\) −11.6889 −0.679403
\(297\) 0 0
\(298\) 5.55613 0.321858
\(299\) −4.91967 −0.284512
\(300\) 0 0
\(301\) −12.1383 −0.699640
\(302\) 22.8304 1.31374
\(303\) 0 0
\(304\) 2.74052 0.157180
\(305\) −0.444528 −0.0254536
\(306\) 0 0
\(307\) 6.18614 0.353062 0.176531 0.984295i \(-0.443512\pi\)
0.176531 + 0.984295i \(0.443512\pi\)
\(308\) −5.79661 −0.330293
\(309\) 0 0
\(310\) 6.24281 0.354568
\(311\) −12.8865 −0.730728 −0.365364 0.930865i \(-0.619056\pi\)
−0.365364 + 0.930865i \(0.619056\pi\)
\(312\) 0 0
\(313\) −10.1220 −0.572128 −0.286064 0.958211i \(-0.592347\pi\)
−0.286064 + 0.958211i \(0.592347\pi\)
\(314\) −24.3441 −1.37382
\(315\) 0 0
\(316\) 5.84607 0.328867
\(317\) 16.2871 0.914774 0.457387 0.889268i \(-0.348785\pi\)
0.457387 + 0.889268i \(0.348785\pi\)
\(318\) 0 0
\(319\) −37.7906 −2.11587
\(320\) −13.2634 −0.741448
\(321\) 0 0
\(322\) −16.8428 −0.938610
\(323\) 1.22062 0.0679173
\(324\) 0 0
\(325\) −1.90363 −0.105594
\(326\) −5.05715 −0.280090
\(327\) 0 0
\(328\) −35.6843 −1.97033
\(329\) 1.92629 0.106200
\(330\) 0 0
\(331\) 7.44340 0.409126 0.204563 0.978853i \(-0.434423\pi\)
0.204563 + 0.978853i \(0.434423\pi\)
\(332\) 3.78652 0.207812
\(333\) 0 0
\(334\) −3.87439 −0.211997
\(335\) 22.3214 1.21955
\(336\) 0 0
\(337\) −17.2307 −0.938617 −0.469308 0.883034i \(-0.655497\pi\)
−0.469308 + 0.883034i \(0.655497\pi\)
\(338\) −15.3252 −0.833579
\(339\) 0 0
\(340\) −0.918099 −0.0497909
\(341\) −20.4144 −1.10550
\(342\) 0 0
\(343\) −19.8204 −1.07020
\(344\) 19.2980 1.04048
\(345\) 0 0
\(346\) 9.26005 0.497824
\(347\) 4.46467 0.239676 0.119838 0.992793i \(-0.461762\pi\)
0.119838 + 0.992793i \(0.461762\pi\)
\(348\) 0 0
\(349\) −21.8361 −1.16886 −0.584431 0.811443i \(-0.698682\pi\)
−0.584431 + 0.811443i \(0.698682\pi\)
\(350\) −6.51717 −0.348357
\(351\) 0 0
\(352\) 16.5790 0.883661
\(353\) −32.7096 −1.74096 −0.870479 0.492205i \(-0.836191\pi\)
−0.870479 + 0.492205i \(0.836191\pi\)
\(354\) 0 0
\(355\) −12.6267 −0.670156
\(356\) −3.96080 −0.209922
\(357\) 0 0
\(358\) 27.3966 1.44795
\(359\) −7.15938 −0.377858 −0.188929 0.981991i \(-0.560502\pi\)
−0.188929 + 0.981991i \(0.560502\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 6.00058 0.315383
\(363\) 0 0
\(364\) −0.667204 −0.0349710
\(365\) −4.65116 −0.243453
\(366\) 0 0
\(367\) 24.5644 1.28225 0.641124 0.767437i \(-0.278468\pi\)
0.641124 + 0.767437i \(0.278468\pi\)
\(368\) 19.5856 1.02097
\(369\) 0 0
\(370\) −6.98058 −0.362903
\(371\) 8.88742 0.461412
\(372\) 0 0
\(373\) −1.26957 −0.0657359 −0.0328680 0.999460i \(-0.510464\pi\)
−0.0328680 + 0.999460i \(0.510464\pi\)
\(374\) −8.93141 −0.461832
\(375\) 0 0
\(376\) −3.06250 −0.157936
\(377\) −4.34978 −0.224025
\(378\) 0 0
\(379\) −5.26839 −0.270619 −0.135310 0.990803i \(-0.543203\pi\)
−0.135310 + 0.990803i \(0.543203\pi\)
\(380\) −0.752156 −0.0385848
\(381\) 0 0
\(382\) −11.2908 −0.577686
\(383\) −2.29974 −0.117511 −0.0587556 0.998272i \(-0.518713\pi\)
−0.0587556 + 0.998272i \(0.518713\pi\)
\(384\) 0 0
\(385\) −17.2218 −0.877704
\(386\) −28.9216 −1.47207
\(387\) 0 0
\(388\) −7.75690 −0.393797
\(389\) 11.3158 0.573732 0.286866 0.957971i \(-0.407387\pi\)
0.286866 + 0.957971i \(0.407387\pi\)
\(390\) 0 0
\(391\) 8.72337 0.441160
\(392\) 10.0738 0.508802
\(393\) 0 0
\(394\) 17.2800 0.870552
\(395\) 17.3687 0.873916
\(396\) 0 0
\(397\) −27.8254 −1.39652 −0.698260 0.715845i \(-0.746042\pi\)
−0.698260 + 0.715845i \(0.746042\pi\)
\(398\) 4.92696 0.246966
\(399\) 0 0
\(400\) 7.57848 0.378924
\(401\) 19.4971 0.973638 0.486819 0.873503i \(-0.338157\pi\)
0.486819 + 0.873503i \(0.338157\pi\)
\(402\) 0 0
\(403\) −2.34974 −0.117049
\(404\) 4.03653 0.200825
\(405\) 0 0
\(406\) −14.8917 −0.739063
\(407\) 22.8269 1.13149
\(408\) 0 0
\(409\) 22.7035 1.12262 0.561308 0.827607i \(-0.310298\pi\)
0.561308 + 0.827607i \(0.310298\pi\)
\(410\) −21.3106 −1.05245
\(411\) 0 0
\(412\) 7.70372 0.379535
\(413\) 17.2155 0.847118
\(414\) 0 0
\(415\) 11.2498 0.552231
\(416\) 1.90828 0.0935609
\(417\) 0 0
\(418\) −7.31709 −0.357890
\(419\) −19.4931 −0.952299 −0.476149 0.879364i \(-0.657968\pi\)
−0.476149 + 0.879364i \(0.657968\pi\)
\(420\) 0 0
\(421\) −0.983589 −0.0479372 −0.0239686 0.999713i \(-0.507630\pi\)
−0.0239686 + 0.999713i \(0.507630\pi\)
\(422\) −8.43540 −0.410629
\(423\) 0 0
\(424\) −14.1296 −0.686193
\(425\) 3.37544 0.163733
\(426\) 0 0
\(427\) 0.572817 0.0277206
\(428\) 0.702275 0.0339458
\(429\) 0 0
\(430\) 11.5247 0.555771
\(431\) 16.4580 0.792755 0.396378 0.918088i \(-0.370267\pi\)
0.396378 + 0.918088i \(0.370267\pi\)
\(432\) 0 0
\(433\) 34.9519 1.67968 0.839840 0.542834i \(-0.182649\pi\)
0.839840 + 0.542834i \(0.182649\pi\)
\(434\) −8.04446 −0.386147
\(435\) 0 0
\(436\) 4.23109 0.202633
\(437\) 7.14665 0.341870
\(438\) 0 0
\(439\) −25.4572 −1.21500 −0.607502 0.794318i \(-0.707828\pi\)
−0.607502 + 0.794318i \(0.707828\pi\)
\(440\) 27.3799 1.30529
\(441\) 0 0
\(442\) −1.02803 −0.0488982
\(443\) −23.4363 −1.11349 −0.556747 0.830682i \(-0.687951\pi\)
−0.556747 + 0.830682i \(0.687951\pi\)
\(444\) 0 0
\(445\) −11.7676 −0.557837
\(446\) 19.5221 0.924400
\(447\) 0 0
\(448\) 17.0912 0.807483
\(449\) −35.5270 −1.67662 −0.838311 0.545192i \(-0.816457\pi\)
−0.838311 + 0.545192i \(0.816457\pi\)
\(450\) 0 0
\(451\) 69.6868 3.28142
\(452\) 3.50594 0.164906
\(453\) 0 0
\(454\) −2.66866 −0.125246
\(455\) −1.98227 −0.0929302
\(456\) 0 0
\(457\) −9.39683 −0.439565 −0.219783 0.975549i \(-0.570535\pi\)
−0.219783 + 0.975549i \(0.570535\pi\)
\(458\) 20.8598 0.974714
\(459\) 0 0
\(460\) −5.37539 −0.250629
\(461\) −9.50836 −0.442849 −0.221424 0.975178i \(-0.571071\pi\)
−0.221424 + 0.975178i \(0.571071\pi\)
\(462\) 0 0
\(463\) 3.51358 0.163290 0.0816450 0.996661i \(-0.473983\pi\)
0.0816450 + 0.996661i \(0.473983\pi\)
\(464\) 17.3168 0.803912
\(465\) 0 0
\(466\) −22.4085 −1.03805
\(467\) −18.0939 −0.837285 −0.418642 0.908151i \(-0.637494\pi\)
−0.418642 + 0.908151i \(0.637494\pi\)
\(468\) 0 0
\(469\) −28.7633 −1.32817
\(470\) −1.82892 −0.0843617
\(471\) 0 0
\(472\) −27.3699 −1.25980
\(473\) −37.6865 −1.73282
\(474\) 0 0
\(475\) 2.76534 0.126882
\(476\) 1.18306 0.0542254
\(477\) 0 0
\(478\) −16.1752 −0.739835
\(479\) 29.4690 1.34647 0.673236 0.739428i \(-0.264904\pi\)
0.673236 + 0.739428i \(0.264904\pi\)
\(480\) 0 0
\(481\) 2.62743 0.119800
\(482\) −2.98929 −0.136158
\(483\) 0 0
\(484\) −12.4624 −0.566471
\(485\) −23.0458 −1.04646
\(486\) 0 0
\(487\) −5.48046 −0.248343 −0.124172 0.992261i \(-0.539627\pi\)
−0.124172 + 0.992261i \(0.539627\pi\)
\(488\) −0.910688 −0.0412249
\(489\) 0 0
\(490\) 6.01603 0.271776
\(491\) 24.3494 1.09887 0.549436 0.835536i \(-0.314843\pi\)
0.549436 + 0.835536i \(0.314843\pi\)
\(492\) 0 0
\(493\) 7.71286 0.347370
\(494\) −0.842214 −0.0378930
\(495\) 0 0
\(496\) 9.35448 0.420029
\(497\) 16.2707 0.729841
\(498\) 0 0
\(499\) −26.1326 −1.16986 −0.584928 0.811085i \(-0.698877\pi\)
−0.584928 + 0.811085i \(0.698877\pi\)
\(500\) −5.84074 −0.261206
\(501\) 0 0
\(502\) −12.2928 −0.548657
\(503\) 44.1872 1.97021 0.985105 0.171956i \(-0.0550086\pi\)
0.985105 + 0.171956i \(0.0550086\pi\)
\(504\) 0 0
\(505\) 11.9926 0.533662
\(506\) −52.2926 −2.32469
\(507\) 0 0
\(508\) 10.9049 0.483827
\(509\) 34.7489 1.54022 0.770108 0.637914i \(-0.220202\pi\)
0.770108 + 0.637914i \(0.220202\pi\)
\(510\) 0 0
\(511\) 5.99346 0.265135
\(512\) −24.3827 −1.07757
\(513\) 0 0
\(514\) −22.7807 −1.00481
\(515\) 22.8878 1.00856
\(516\) 0 0
\(517\) 5.98067 0.263030
\(518\) 8.99515 0.395224
\(519\) 0 0
\(520\) 3.15149 0.138202
\(521\) −17.0091 −0.745182 −0.372591 0.927996i \(-0.621530\pi\)
−0.372591 + 0.927996i \(0.621530\pi\)
\(522\) 0 0
\(523\) −40.7876 −1.78352 −0.891759 0.452511i \(-0.850528\pi\)
−0.891759 + 0.452511i \(0.850528\pi\)
\(524\) 0.873332 0.0381517
\(525\) 0 0
\(526\) 0.706027 0.0307842
\(527\) 4.16647 0.181494
\(528\) 0 0
\(529\) 28.0745 1.22063
\(530\) −8.43815 −0.366530
\(531\) 0 0
\(532\) 0.969225 0.0420212
\(533\) 8.02111 0.347433
\(534\) 0 0
\(535\) 2.08647 0.0902058
\(536\) 45.7291 1.97519
\(537\) 0 0
\(538\) −24.3171 −1.04839
\(539\) −19.6728 −0.847366
\(540\) 0 0
\(541\) 1.56934 0.0674714 0.0337357 0.999431i \(-0.489260\pi\)
0.0337357 + 0.999431i \(0.489260\pi\)
\(542\) 21.4520 0.921443
\(543\) 0 0
\(544\) −3.38368 −0.145074
\(545\) 12.5706 0.538466
\(546\) 0 0
\(547\) −8.70011 −0.371990 −0.185995 0.982551i \(-0.559551\pi\)
−0.185995 + 0.982551i \(0.559551\pi\)
\(548\) 10.8258 0.462456
\(549\) 0 0
\(550\) −20.2342 −0.862790
\(551\) 6.31879 0.269189
\(552\) 0 0
\(553\) −22.3813 −0.951749
\(554\) 27.5261 1.16947
\(555\) 0 0
\(556\) 11.6636 0.494649
\(557\) 26.0422 1.10344 0.551722 0.834028i \(-0.313971\pi\)
0.551722 + 0.834028i \(0.313971\pi\)
\(558\) 0 0
\(559\) −4.33780 −0.183469
\(560\) 7.89155 0.333479
\(561\) 0 0
\(562\) −28.0553 −1.18344
\(563\) 0.109118 0.00459876 0.00229938 0.999997i \(-0.499268\pi\)
0.00229938 + 0.999997i \(0.499268\pi\)
\(564\) 0 0
\(565\) 10.4162 0.438212
\(566\) −9.82737 −0.413075
\(567\) 0 0
\(568\) −25.8678 −1.08539
\(569\) 25.2719 1.05945 0.529727 0.848168i \(-0.322294\pi\)
0.529727 + 0.848168i \(0.322294\pi\)
\(570\) 0 0
\(571\) −6.37573 −0.266816 −0.133408 0.991061i \(-0.542592\pi\)
−0.133408 + 0.991061i \(0.542592\pi\)
\(572\) −2.07150 −0.0866139
\(573\) 0 0
\(574\) 27.4607 1.14619
\(575\) 19.7629 0.824169
\(576\) 0 0
\(577\) 15.8745 0.660864 0.330432 0.943830i \(-0.392806\pi\)
0.330432 + 0.943830i \(0.392806\pi\)
\(578\) −18.9759 −0.789293
\(579\) 0 0
\(580\) −4.75271 −0.197346
\(581\) −14.4964 −0.601413
\(582\) 0 0
\(583\) 27.5933 1.14280
\(584\) −9.52865 −0.394298
\(585\) 0 0
\(586\) −14.3493 −0.592763
\(587\) 24.3827 1.00638 0.503190 0.864176i \(-0.332160\pi\)
0.503190 + 0.864176i \(0.332160\pi\)
\(588\) 0 0
\(589\) 3.41339 0.140646
\(590\) −16.3452 −0.672922
\(591\) 0 0
\(592\) −10.4600 −0.429903
\(593\) −47.0711 −1.93298 −0.966489 0.256709i \(-0.917362\pi\)
−0.966489 + 0.256709i \(0.917362\pi\)
\(594\) 0 0
\(595\) 3.51488 0.144096
\(596\) −2.28500 −0.0935971
\(597\) 0 0
\(598\) −6.01900 −0.246135
\(599\) −18.6311 −0.761245 −0.380623 0.924730i \(-0.624290\pi\)
−0.380623 + 0.924730i \(0.624290\pi\)
\(600\) 0 0
\(601\) 24.0775 0.982141 0.491071 0.871120i \(-0.336606\pi\)
0.491071 + 0.871120i \(0.336606\pi\)
\(602\) −14.8507 −0.605269
\(603\) 0 0
\(604\) −9.38919 −0.382041
\(605\) −37.0258 −1.50531
\(606\) 0 0
\(607\) −19.3991 −0.787385 −0.393693 0.919242i \(-0.628803\pi\)
−0.393693 + 0.919242i \(0.628803\pi\)
\(608\) −2.77209 −0.112423
\(609\) 0 0
\(610\) −0.543861 −0.0220203
\(611\) 0.688389 0.0278492
\(612\) 0 0
\(613\) 48.2484 1.94873 0.974367 0.224965i \(-0.0722268\pi\)
0.974367 + 0.224965i \(0.0722268\pi\)
\(614\) 7.56847 0.305439
\(615\) 0 0
\(616\) −35.2816 −1.42154
\(617\) −10.5954 −0.426555 −0.213277 0.976992i \(-0.568414\pi\)
−0.213277 + 0.976992i \(0.568414\pi\)
\(618\) 0 0
\(619\) −29.9503 −1.20380 −0.601902 0.798570i \(-0.705590\pi\)
−0.601902 + 0.798570i \(0.705590\pi\)
\(620\) −2.56740 −0.103109
\(621\) 0 0
\(622\) −15.7661 −0.632163
\(623\) 15.1637 0.607519
\(624\) 0 0
\(625\) −3.52622 −0.141049
\(626\) −12.3838 −0.494956
\(627\) 0 0
\(628\) 10.0117 0.399510
\(629\) −4.65885 −0.185761
\(630\) 0 0
\(631\) 26.2272 1.04409 0.522043 0.852919i \(-0.325170\pi\)
0.522043 + 0.852919i \(0.325170\pi\)
\(632\) 35.5827 1.41540
\(633\) 0 0
\(634\) 19.9265 0.791384
\(635\) 32.3986 1.28570
\(636\) 0 0
\(637\) −2.26438 −0.0897180
\(638\) −46.2351 −1.83047
\(639\) 0 0
\(640\) −7.93935 −0.313830
\(641\) 14.4853 0.572135 0.286067 0.958210i \(-0.407652\pi\)
0.286067 + 0.958210i \(0.407652\pi\)
\(642\) 0 0
\(643\) −2.63994 −0.104109 −0.0520545 0.998644i \(-0.516577\pi\)
−0.0520545 + 0.998644i \(0.516577\pi\)
\(644\) 6.92671 0.272950
\(645\) 0 0
\(646\) 1.49338 0.0587562
\(647\) 8.18381 0.321739 0.160869 0.986976i \(-0.448570\pi\)
0.160869 + 0.986976i \(0.448570\pi\)
\(648\) 0 0
\(649\) 53.4498 2.09809
\(650\) −2.32900 −0.0913511
\(651\) 0 0
\(652\) 2.07979 0.0814509
\(653\) −32.2292 −1.26123 −0.630613 0.776098i \(-0.717196\pi\)
−0.630613 + 0.776098i \(0.717196\pi\)
\(654\) 0 0
\(655\) 2.59468 0.101382
\(656\) −31.9326 −1.24676
\(657\) 0 0
\(658\) 2.35674 0.0918751
\(659\) 3.30985 0.128934 0.0644668 0.997920i \(-0.479465\pi\)
0.0644668 + 0.997920i \(0.479465\pi\)
\(660\) 0 0
\(661\) −19.8264 −0.771157 −0.385579 0.922675i \(-0.625998\pi\)
−0.385579 + 0.922675i \(0.625998\pi\)
\(662\) 9.10667 0.353941
\(663\) 0 0
\(664\) 23.0470 0.894398
\(665\) 2.87958 0.111665
\(666\) 0 0
\(667\) 45.1581 1.74853
\(668\) 1.59337 0.0616493
\(669\) 0 0
\(670\) 27.3093 1.05505
\(671\) 1.77846 0.0686566
\(672\) 0 0
\(673\) 4.12655 0.159067 0.0795333 0.996832i \(-0.474657\pi\)
0.0795333 + 0.996832i \(0.474657\pi\)
\(674\) −21.0810 −0.812011
\(675\) 0 0
\(676\) 6.30258 0.242407
\(677\) 13.2120 0.507779 0.253890 0.967233i \(-0.418290\pi\)
0.253890 + 0.967233i \(0.418290\pi\)
\(678\) 0 0
\(679\) 29.6968 1.13966
\(680\) −5.58810 −0.214294
\(681\) 0 0
\(682\) −24.9761 −0.956384
\(683\) 21.0203 0.804319 0.402159 0.915570i \(-0.368260\pi\)
0.402159 + 0.915570i \(0.368260\pi\)
\(684\) 0 0
\(685\) 32.1636 1.22891
\(686\) −24.2494 −0.925846
\(687\) 0 0
\(688\) 17.2691 0.658377
\(689\) 3.17605 0.120998
\(690\) 0 0
\(691\) −19.0482 −0.724630 −0.362315 0.932056i \(-0.618013\pi\)
−0.362315 + 0.932056i \(0.618013\pi\)
\(692\) −3.80826 −0.144768
\(693\) 0 0
\(694\) 5.46233 0.207347
\(695\) 34.6528 1.31446
\(696\) 0 0
\(697\) −14.2227 −0.538724
\(698\) −26.7156 −1.01120
\(699\) 0 0
\(700\) 2.68023 0.101303
\(701\) 44.2411 1.67096 0.835481 0.549519i \(-0.185189\pi\)
0.835481 + 0.549519i \(0.185189\pi\)
\(702\) 0 0
\(703\) −3.81678 −0.143953
\(704\) 53.0640 1.99992
\(705\) 0 0
\(706\) −40.0188 −1.50613
\(707\) −15.4536 −0.581191
\(708\) 0 0
\(709\) 33.7579 1.26780 0.633902 0.773414i \(-0.281452\pi\)
0.633902 + 0.773414i \(0.281452\pi\)
\(710\) −15.4482 −0.579761
\(711\) 0 0
\(712\) −24.1078 −0.903478
\(713\) 24.3943 0.913574
\(714\) 0 0
\(715\) −6.15446 −0.230164
\(716\) −11.2670 −0.421069
\(717\) 0 0
\(718\) −8.75919 −0.326890
\(719\) 37.0118 1.38031 0.690153 0.723664i \(-0.257543\pi\)
0.690153 + 0.723664i \(0.257543\pi\)
\(720\) 0 0
\(721\) −29.4931 −1.09838
\(722\) 1.22346 0.0455323
\(723\) 0 0
\(724\) −2.46778 −0.0917144
\(725\) 17.4736 0.648952
\(726\) 0 0
\(727\) −4.84394 −0.179652 −0.0898260 0.995957i \(-0.528631\pi\)
−0.0898260 + 0.995957i \(0.528631\pi\)
\(728\) −4.06100 −0.150511
\(729\) 0 0
\(730\) −5.69049 −0.210614
\(731\) 7.69161 0.284484
\(732\) 0 0
\(733\) 16.7224 0.617655 0.308828 0.951118i \(-0.400063\pi\)
0.308828 + 0.951118i \(0.400063\pi\)
\(734\) 30.0534 1.10929
\(735\) 0 0
\(736\) −19.8111 −0.730248
\(737\) −89.3030 −3.28952
\(738\) 0 0
\(739\) 24.0263 0.883821 0.441910 0.897059i \(-0.354301\pi\)
0.441910 + 0.897059i \(0.354301\pi\)
\(740\) 2.87081 0.105533
\(741\) 0 0
\(742\) 10.8734 0.399174
\(743\) 25.6863 0.942340 0.471170 0.882042i \(-0.343832\pi\)
0.471170 + 0.882042i \(0.343832\pi\)
\(744\) 0 0
\(745\) −6.78875 −0.248720
\(746\) −1.55327 −0.0568691
\(747\) 0 0
\(748\) 3.67311 0.134302
\(749\) −2.68861 −0.0982397
\(750\) 0 0
\(751\) −4.49816 −0.164140 −0.0820701 0.996627i \(-0.526153\pi\)
−0.0820701 + 0.996627i \(0.526153\pi\)
\(752\) −2.74052 −0.0999367
\(753\) 0 0
\(754\) −5.32177 −0.193807
\(755\) −27.8954 −1.01522
\(756\) 0 0
\(757\) 24.3153 0.883754 0.441877 0.897076i \(-0.354313\pi\)
0.441877 + 0.897076i \(0.354313\pi\)
\(758\) −6.44565 −0.234116
\(759\) 0 0
\(760\) −4.57807 −0.166064
\(761\) −16.0639 −0.582317 −0.291158 0.956675i \(-0.594041\pi\)
−0.291158 + 0.956675i \(0.594041\pi\)
\(762\) 0 0
\(763\) −16.1984 −0.586423
\(764\) 4.64341 0.167993
\(765\) 0 0
\(766\) −2.81363 −0.101661
\(767\) 6.15220 0.222143
\(768\) 0 0
\(769\) 9.39454 0.338776 0.169388 0.985549i \(-0.445821\pi\)
0.169388 + 0.985549i \(0.445821\pi\)
\(770\) −21.0701 −0.759314
\(771\) 0 0
\(772\) 11.8942 0.428083
\(773\) 19.5949 0.704779 0.352390 0.935853i \(-0.385369\pi\)
0.352390 + 0.935853i \(0.385369\pi\)
\(774\) 0 0
\(775\) 9.43918 0.339065
\(776\) −47.2131 −1.69485
\(777\) 0 0
\(778\) 13.8443 0.496343
\(779\) −11.6520 −0.417476
\(780\) 0 0
\(781\) 50.5166 1.80762
\(782\) 10.6727 0.381653
\(783\) 0 0
\(784\) 9.01466 0.321952
\(785\) 29.7448 1.06164
\(786\) 0 0
\(787\) 0.0212478 0.000757403 0 0.000378701 1.00000i \(-0.499879\pi\)
0.000378701 1.00000i \(0.499879\pi\)
\(788\) −7.10651 −0.253159
\(789\) 0 0
\(790\) 21.2499 0.756037
\(791\) −13.4222 −0.477240
\(792\) 0 0
\(793\) 0.204705 0.00726927
\(794\) −34.0432 −1.20815
\(795\) 0 0
\(796\) −2.02625 −0.0718185
\(797\) −23.4986 −0.832365 −0.416182 0.909281i \(-0.636632\pi\)
−0.416182 + 0.909281i \(0.636632\pi\)
\(798\) 0 0
\(799\) −1.22062 −0.0431826
\(800\) −7.66576 −0.271026
\(801\) 0 0
\(802\) 23.8538 0.842308
\(803\) 18.6082 0.656670
\(804\) 0 0
\(805\) 20.5793 0.725325
\(806\) −2.87480 −0.101261
\(807\) 0 0
\(808\) 24.5687 0.864325
\(809\) 43.8069 1.54017 0.770085 0.637942i \(-0.220214\pi\)
0.770085 + 0.637942i \(0.220214\pi\)
\(810\) 0 0
\(811\) 50.0957 1.75910 0.879549 0.475808i \(-0.157844\pi\)
0.879549 + 0.475808i \(0.157844\pi\)
\(812\) 6.12432 0.214922
\(813\) 0 0
\(814\) 27.9277 0.978866
\(815\) 6.17908 0.216444
\(816\) 0 0
\(817\) 6.30138 0.220457
\(818\) 27.7767 0.971191
\(819\) 0 0
\(820\) 8.76412 0.306057
\(821\) 13.9525 0.486947 0.243473 0.969908i \(-0.421713\pi\)
0.243473 + 0.969908i \(0.421713\pi\)
\(822\) 0 0
\(823\) 0.555562 0.0193657 0.00968284 0.999953i \(-0.496918\pi\)
0.00968284 + 0.999953i \(0.496918\pi\)
\(824\) 46.8894 1.63347
\(825\) 0 0
\(826\) 21.0624 0.732854
\(827\) −53.0942 −1.84627 −0.923134 0.384479i \(-0.874381\pi\)
−0.923134 + 0.384479i \(0.874381\pi\)
\(828\) 0 0
\(829\) −51.5139 −1.78915 −0.894575 0.446918i \(-0.852522\pi\)
−0.894575 + 0.446918i \(0.852522\pi\)
\(830\) 13.7636 0.477742
\(831\) 0 0
\(832\) 6.10778 0.211749
\(833\) 4.01511 0.139115
\(834\) 0 0
\(835\) 4.73391 0.163824
\(836\) 3.00920 0.104076
\(837\) 0 0
\(838\) −23.8489 −0.823847
\(839\) −23.7392 −0.819568 −0.409784 0.912183i \(-0.634396\pi\)
−0.409784 + 0.912183i \(0.634396\pi\)
\(840\) 0 0
\(841\) 10.9271 0.376795
\(842\) −1.20338 −0.0414711
\(843\) 0 0
\(844\) 3.46912 0.119412
\(845\) 18.7250 0.644161
\(846\) 0 0
\(847\) 47.7112 1.63938
\(848\) −12.6441 −0.434199
\(849\) 0 0
\(850\) 4.12970 0.141648
\(851\) −27.2772 −0.935050
\(852\) 0 0
\(853\) −3.82861 −0.131089 −0.0655446 0.997850i \(-0.520878\pi\)
−0.0655446 + 0.997850i \(0.520878\pi\)
\(854\) 0.700817 0.0239815
\(855\) 0 0
\(856\) 4.27446 0.146098
\(857\) −0.182682 −0.00624031 −0.00312016 0.999995i \(-0.500993\pi\)
−0.00312016 + 0.999995i \(0.500993\pi\)
\(858\) 0 0
\(859\) 28.5064 0.972625 0.486313 0.873785i \(-0.338342\pi\)
0.486313 + 0.873785i \(0.338342\pi\)
\(860\) −4.73962 −0.161620
\(861\) 0 0
\(862\) 20.1357 0.685824
\(863\) −5.32537 −0.181278 −0.0906389 0.995884i \(-0.528891\pi\)
−0.0906389 + 0.995884i \(0.528891\pi\)
\(864\) 0 0
\(865\) −11.3144 −0.384701
\(866\) 42.7621 1.45312
\(867\) 0 0
\(868\) 3.30834 0.112292
\(869\) −69.4884 −2.35723
\(870\) 0 0
\(871\) −10.2790 −0.348290
\(872\) 25.7529 0.872104
\(873\) 0 0
\(874\) 8.74361 0.295757
\(875\) 22.3609 0.755936
\(876\) 0 0
\(877\) 38.7758 1.30937 0.654684 0.755903i \(-0.272802\pi\)
0.654684 + 0.755903i \(0.272802\pi\)
\(878\) −31.1457 −1.05112
\(879\) 0 0
\(880\) 24.5013 0.825939
\(881\) 50.1536 1.68972 0.844859 0.534989i \(-0.179684\pi\)
0.844859 + 0.534989i \(0.179684\pi\)
\(882\) 0 0
\(883\) 41.8316 1.40774 0.703872 0.710326i \(-0.251453\pi\)
0.703872 + 0.710326i \(0.251453\pi\)
\(884\) 0.422783 0.0142197
\(885\) 0 0
\(886\) −28.6733 −0.963299
\(887\) 11.0837 0.372153 0.186076 0.982535i \(-0.440423\pi\)
0.186076 + 0.982535i \(0.440423\pi\)
\(888\) 0 0
\(889\) −41.7487 −1.40021
\(890\) −14.3971 −0.482593
\(891\) 0 0
\(892\) −8.02862 −0.268818
\(893\) −1.00000 −0.0334637
\(894\) 0 0
\(895\) −33.4745 −1.11893
\(896\) 10.2306 0.341781
\(897\) 0 0
\(898\) −43.4657 −1.45047
\(899\) 21.5685 0.719349
\(900\) 0 0
\(901\) −5.63164 −0.187617
\(902\) 85.2587 2.83881
\(903\) 0 0
\(904\) 21.3392 0.709732
\(905\) −7.33180 −0.243717
\(906\) 0 0
\(907\) 48.3842 1.60657 0.803286 0.595594i \(-0.203083\pi\)
0.803286 + 0.595594i \(0.203083\pi\)
\(908\) 1.09750 0.0364219
\(909\) 0 0
\(910\) −2.42522 −0.0803952
\(911\) −16.9006 −0.559943 −0.279971 0.960008i \(-0.590325\pi\)
−0.279971 + 0.960008i \(0.590325\pi\)
\(912\) 0 0
\(913\) −45.0079 −1.48954
\(914\) −11.4966 −0.380274
\(915\) 0 0
\(916\) −8.57874 −0.283450
\(917\) −3.34349 −0.110412
\(918\) 0 0
\(919\) 30.7264 1.01357 0.506785 0.862073i \(-0.330834\pi\)
0.506785 + 0.862073i \(0.330834\pi\)
\(920\) −32.7178 −1.07867
\(921\) 0 0
\(922\) −11.6331 −0.383115
\(923\) 5.81457 0.191389
\(924\) 0 0
\(925\) −10.5547 −0.347036
\(926\) 4.29871 0.141264
\(927\) 0 0
\(928\) −17.5162 −0.574999
\(929\) −8.50427 −0.279016 −0.139508 0.990221i \(-0.544552\pi\)
−0.139508 + 0.990221i \(0.544552\pi\)
\(930\) 0 0
\(931\) 3.28939 0.107805
\(932\) 9.21564 0.301868
\(933\) 0 0
\(934\) −22.1371 −0.724347
\(935\) 10.9128 0.356888
\(936\) 0 0
\(937\) −21.8894 −0.715096 −0.357548 0.933895i \(-0.616387\pi\)
−0.357548 + 0.933895i \(0.616387\pi\)
\(938\) −35.1906 −1.14902
\(939\) 0 0
\(940\) 0.752156 0.0245326
\(941\) −26.7202 −0.871055 −0.435528 0.900175i \(-0.643438\pi\)
−0.435528 + 0.900175i \(0.643438\pi\)
\(942\) 0 0
\(943\) −83.2728 −2.71173
\(944\) −24.4923 −0.797157
\(945\) 0 0
\(946\) −46.1077 −1.49909
\(947\) −52.6629 −1.71131 −0.855657 0.517544i \(-0.826846\pi\)
−0.855657 + 0.517544i \(0.826846\pi\)
\(948\) 0 0
\(949\) 2.14185 0.0695274
\(950\) 3.38327 0.109768
\(951\) 0 0
\(952\) 7.20080 0.233379
\(953\) 40.1413 1.30030 0.650152 0.759804i \(-0.274705\pi\)
0.650152 + 0.759804i \(0.274705\pi\)
\(954\) 0 0
\(955\) 13.7956 0.446415
\(956\) 6.65216 0.215146
\(957\) 0 0
\(958\) 36.0540 1.16485
\(959\) −41.4458 −1.33836
\(960\) 0 0
\(961\) −19.3488 −0.624154
\(962\) 3.21454 0.103641
\(963\) 0 0
\(964\) 1.22937 0.0395952
\(965\) 35.3379 1.13757
\(966\) 0 0
\(967\) 25.6891 0.826107 0.413053 0.910707i \(-0.364462\pi\)
0.413053 + 0.910707i \(0.364462\pi\)
\(968\) −75.8533 −2.43802
\(969\) 0 0
\(970\) −28.1956 −0.905305
\(971\) 4.88601 0.156800 0.0783998 0.996922i \(-0.475019\pi\)
0.0783998 + 0.996922i \(0.475019\pi\)
\(972\) 0 0
\(973\) −44.6535 −1.43152
\(974\) −6.70510 −0.214845
\(975\) 0 0
\(976\) −0.814943 −0.0260857
\(977\) 8.61035 0.275469 0.137735 0.990469i \(-0.456018\pi\)
0.137735 + 0.990469i \(0.456018\pi\)
\(978\) 0 0
\(979\) 47.0794 1.50467
\(980\) −2.47413 −0.0790333
\(981\) 0 0
\(982\) 29.7904 0.950649
\(983\) −32.6483 −1.04132 −0.520660 0.853764i \(-0.674314\pi\)
−0.520660 + 0.853764i \(0.674314\pi\)
\(984\) 0 0
\(985\) −21.1135 −0.672732
\(986\) 9.43635 0.300515
\(987\) 0 0
\(988\) 0.346366 0.0110194
\(989\) 45.0337 1.43199
\(990\) 0 0
\(991\) 13.4086 0.425938 0.212969 0.977059i \(-0.431687\pi\)
0.212969 + 0.977059i \(0.431687\pi\)
\(992\) −9.46223 −0.300426
\(993\) 0 0
\(994\) 19.9065 0.631396
\(995\) −6.02001 −0.190847
\(996\) 0 0
\(997\) −24.1372 −0.764432 −0.382216 0.924073i \(-0.624839\pi\)
−0.382216 + 0.924073i \(0.624839\pi\)
\(998\) −31.9721 −1.01206
\(999\) 0 0
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 8037.2.a.n.1.11 16
3.2 odd 2 893.2.a.b.1.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
893.2.a.b.1.6 16 3.2 odd 2
8037.2.a.n.1.11 16 1.1 even 1 trivial