Properties

Label 912.2.d.b.191.3
Level $912$
Weight $2$
Character 912.191
Analytic conductor $7.282$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [912,2,Mod(191,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("912.191");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 912.d (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(7.28235666434\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 191.3
Character \(\chi\) \(=\) 912.191
Dual form 912.2.d.b.191.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.57777 - 0.714595i) q^{3} -1.85004i q^{5} -0.707624i q^{7} +(1.97871 + 2.25493i) q^{9} +O(q^{10})\) \(q+(-1.57777 - 0.714595i) q^{3} -1.85004i q^{5} -0.707624i q^{7} +(1.97871 + 2.25493i) q^{9} -4.63214 q^{11} -1.78223 q^{13} +(-1.32203 + 2.91894i) q^{15} -1.47660i q^{17} -1.00000i q^{19} +(-0.505665 + 1.11647i) q^{21} -2.68346 q^{23} +1.57733 q^{25} +(-1.51058 - 4.97174i) q^{27} +2.44439i q^{29} +6.63667i q^{31} +(7.30844 + 3.31010i) q^{33} -1.30914 q^{35} -3.04433 q^{37} +(2.81195 + 1.27357i) q^{39} +4.95675i q^{41} +6.80404i q^{43} +(4.17173 - 3.66070i) q^{45} -8.19420 q^{47} +6.49927 q^{49} +(-1.05517 + 2.32973i) q^{51} +3.89976i q^{53} +8.56966i q^{55} +(-0.714595 + 1.57777i) q^{57} -8.14537 q^{59} -11.9620 q^{61} +(1.59564 - 1.40018i) q^{63} +3.29720i q^{65} +4.84424i q^{67} +(4.23388 + 1.91759i) q^{69} +6.09961 q^{71} -15.0559 q^{73} +(-2.48867 - 1.12716i) q^{75} +3.27781i q^{77} -2.21403i q^{79} +(-1.16944 + 8.92370i) q^{81} +2.65489 q^{83} -2.73177 q^{85} +(1.74675 - 3.85669i) q^{87} +10.9884i q^{89} +1.26115i q^{91} +(4.74253 - 10.4711i) q^{93} -1.85004 q^{95} -9.45643 q^{97} +(-9.16564 - 10.4452i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 4 q^{9} - 12 q^{21} - 64 q^{25} + 12 q^{33} + 64 q^{37} - 16 q^{45} - 8 q^{49} - 24 q^{61} - 8 q^{69} - 16 q^{73} - 4 q^{81} - 8 q^{85} + 32 q^{93} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.57777 0.714595i −0.910925 0.412572i
\(4\) 0 0
\(5\) 1.85004i 0.827365i −0.910421 0.413683i \(-0.864242\pi\)
0.910421 0.413683i \(-0.135758\pi\)
\(6\) 0 0
\(7\) 0.707624i 0.267457i −0.991018 0.133728i \(-0.957305\pi\)
0.991018 0.133728i \(-0.0426949\pi\)
\(8\) 0 0
\(9\) 1.97871 + 2.25493i 0.659569 + 0.751644i
\(10\) 0 0
\(11\) −4.63214 −1.39664 −0.698321 0.715785i \(-0.746069\pi\)
−0.698321 + 0.715785i \(0.746069\pi\)
\(12\) 0 0
\(13\) −1.78223 −0.494302 −0.247151 0.968977i \(-0.579494\pi\)
−0.247151 + 0.968977i \(0.579494\pi\)
\(14\) 0 0
\(15\) −1.32203 + 2.91894i −0.341348 + 0.753668i
\(16\) 0 0
\(17\) 1.47660i 0.358128i −0.983837 0.179064i \(-0.942693\pi\)
0.983837 0.179064i \(-0.0573069\pi\)
\(18\) 0 0
\(19\) 1.00000i 0.229416i
\(20\) 0 0
\(21\) −0.505665 + 1.11647i −0.110345 + 0.243633i
\(22\) 0 0
\(23\) −2.68346 −0.559540 −0.279770 0.960067i \(-0.590258\pi\)
−0.279770 + 0.960067i \(0.590258\pi\)
\(24\) 0 0
\(25\) 1.57733 0.315467
\(26\) 0 0
\(27\) −1.51058 4.97174i −0.290711 0.956811i
\(28\) 0 0
\(29\) 2.44439i 0.453913i 0.973905 + 0.226956i \(0.0728775\pi\)
−0.973905 + 0.226956i \(0.927123\pi\)
\(30\) 0 0
\(31\) 6.63667i 1.19198i 0.802991 + 0.595991i \(0.203241\pi\)
−0.802991 + 0.595991i \(0.796759\pi\)
\(32\) 0 0
\(33\) 7.30844 + 3.31010i 1.27224 + 0.576215i
\(34\) 0 0
\(35\) −1.30914 −0.221284
\(36\) 0 0
\(37\) −3.04433 −0.500484 −0.250242 0.968183i \(-0.580510\pi\)
−0.250242 + 0.968183i \(0.580510\pi\)
\(38\) 0 0
\(39\) 2.81195 + 1.27357i 0.450272 + 0.203935i
\(40\) 0 0
\(41\) 4.95675i 0.774114i 0.922056 + 0.387057i \(0.126508\pi\)
−0.922056 + 0.387057i \(0.873492\pi\)
\(42\) 0 0
\(43\) 6.80404i 1.03761i 0.854894 + 0.518803i \(0.173622\pi\)
−0.854894 + 0.518803i \(0.826378\pi\)
\(44\) 0 0
\(45\) 4.17173 3.66070i 0.621884 0.545704i
\(46\) 0 0
\(47\) −8.19420 −1.19525 −0.597624 0.801777i \(-0.703888\pi\)
−0.597624 + 0.801777i \(0.703888\pi\)
\(48\) 0 0
\(49\) 6.49927 0.928467
\(50\) 0 0
\(51\) −1.05517 + 2.32973i −0.147753 + 0.326228i
\(52\) 0 0
\(53\) 3.89976i 0.535674i 0.963464 + 0.267837i \(0.0863088\pi\)
−0.963464 + 0.267837i \(0.913691\pi\)
\(54\) 0 0
\(55\) 8.56966i 1.15553i
\(56\) 0 0
\(57\) −0.714595 + 1.57777i −0.0946505 + 0.208981i
\(58\) 0 0
\(59\) −8.14537 −1.06044 −0.530218 0.847861i \(-0.677890\pi\)
−0.530218 + 0.847861i \(0.677890\pi\)
\(60\) 0 0
\(61\) −11.9620 −1.53158 −0.765792 0.643089i \(-0.777653\pi\)
−0.765792 + 0.643089i \(0.777653\pi\)
\(62\) 0 0
\(63\) 1.59564 1.40018i 0.201032 0.176406i
\(64\) 0 0
\(65\) 3.29720i 0.408968i
\(66\) 0 0
\(67\) 4.84424i 0.591818i 0.955216 + 0.295909i \(0.0956224\pi\)
−0.955216 + 0.295909i \(0.904378\pi\)
\(68\) 0 0
\(69\) 4.23388 + 1.91759i 0.509699 + 0.230850i
\(70\) 0 0
\(71\) 6.09961 0.723891 0.361945 0.932199i \(-0.382113\pi\)
0.361945 + 0.932199i \(0.382113\pi\)
\(72\) 0 0
\(73\) −15.0559 −1.76215 −0.881077 0.472973i \(-0.843181\pi\)
−0.881077 + 0.472973i \(0.843181\pi\)
\(74\) 0 0
\(75\) −2.48867 1.12716i −0.287367 0.130153i
\(76\) 0 0
\(77\) 3.27781i 0.373541i
\(78\) 0 0
\(79\) 2.21403i 0.249098i −0.992213 0.124549i \(-0.960252\pi\)
0.992213 0.124549i \(-0.0397485\pi\)
\(80\) 0 0
\(81\) −1.16944 + 8.92370i −0.129937 + 0.991522i
\(82\) 0 0
\(83\) 2.65489 0.291412 0.145706 0.989328i \(-0.453455\pi\)
0.145706 + 0.989328i \(0.453455\pi\)
\(84\) 0 0
\(85\) −2.73177 −0.296302
\(86\) 0 0
\(87\) 1.74675 3.85669i 0.187272 0.413480i
\(88\) 0 0
\(89\) 10.9884i 1.16477i 0.812914 + 0.582384i \(0.197880\pi\)
−0.812914 + 0.582384i \(0.802120\pi\)
\(90\) 0 0
\(91\) 1.26115i 0.132204i
\(92\) 0 0
\(93\) 4.74253 10.4711i 0.491778 1.08581i
\(94\) 0 0
\(95\) −1.85004 −0.189811
\(96\) 0 0
\(97\) −9.45643 −0.960155 −0.480077 0.877226i \(-0.659391\pi\)
−0.480077 + 0.877226i \(0.659391\pi\)
\(98\) 0 0
\(99\) −9.16564 10.4452i −0.921182 1.04978i
\(100\) 0 0
\(101\) 3.11881i 0.310333i −0.987888 0.155167i \(-0.950409\pi\)
0.987888 0.155167i \(-0.0495915\pi\)
\(102\) 0 0
\(103\) 15.3329i 1.51080i −0.655264 0.755400i \(-0.727443\pi\)
0.655264 0.755400i \(-0.272557\pi\)
\(104\) 0 0
\(105\) 2.06551 + 0.935502i 0.201573 + 0.0912957i
\(106\) 0 0
\(107\) 13.4792 1.30308 0.651542 0.758613i \(-0.274122\pi\)
0.651542 + 0.758613i \(0.274122\pi\)
\(108\) 0 0
\(109\) 4.05566 0.388461 0.194231 0.980956i \(-0.437779\pi\)
0.194231 + 0.980956i \(0.437779\pi\)
\(110\) 0 0
\(111\) 4.80324 + 2.17546i 0.455904 + 0.206486i
\(112\) 0 0
\(113\) 9.64136i 0.906983i −0.891261 0.453491i \(-0.850178\pi\)
0.891261 0.453491i \(-0.149822\pi\)
\(114\) 0 0
\(115\) 4.96452i 0.462944i
\(116\) 0 0
\(117\) −3.52651 4.01881i −0.326026 0.371539i
\(118\) 0 0
\(119\) −1.04488 −0.0957837
\(120\) 0 0
\(121\) 10.4567 0.950608
\(122\) 0 0
\(123\) 3.54207 7.82060i 0.319378 0.705160i
\(124\) 0 0
\(125\) 12.1684i 1.08837i
\(126\) 0 0
\(127\) 3.70658i 0.328905i −0.986385 0.164453i \(-0.947414\pi\)
0.986385 0.164453i \(-0.0525858\pi\)
\(128\) 0 0
\(129\) 4.86214 10.7352i 0.428087 0.945182i
\(130\) 0 0
\(131\) −0.0121090 −0.00105797 −0.000528983 1.00000i \(-0.500168\pi\)
−0.000528983 1.00000i \(0.500168\pi\)
\(132\) 0 0
\(133\) −0.707624 −0.0613588
\(134\) 0 0
\(135\) −9.19793 + 2.79464i −0.791632 + 0.240524i
\(136\) 0 0
\(137\) 16.5314i 1.41237i 0.708028 + 0.706184i \(0.249585\pi\)
−0.708028 + 0.706184i \(0.750415\pi\)
\(138\) 0 0
\(139\) 4.09796i 0.347585i 0.984782 + 0.173792i \(0.0556022\pi\)
−0.984782 + 0.173792i \(0.944398\pi\)
\(140\) 0 0
\(141\) 12.9286 + 5.85554i 1.08878 + 0.493125i
\(142\) 0 0
\(143\) 8.25553 0.690362
\(144\) 0 0
\(145\) 4.52224 0.375552
\(146\) 0 0
\(147\) −10.2543 4.64435i −0.845764 0.383059i
\(148\) 0 0
\(149\) 21.8009i 1.78600i −0.450059 0.892999i \(-0.648597\pi\)
0.450059 0.892999i \(-0.351403\pi\)
\(150\) 0 0
\(151\) 18.6817i 1.52030i 0.649750 + 0.760148i \(0.274873\pi\)
−0.649750 + 0.760148i \(0.725127\pi\)
\(152\) 0 0
\(153\) 3.32963 2.92176i 0.269185 0.236210i
\(154\) 0 0
\(155\) 12.2781 0.986204
\(156\) 0 0
\(157\) −10.1885 −0.813127 −0.406564 0.913622i \(-0.633273\pi\)
−0.406564 + 0.913622i \(0.633273\pi\)
\(158\) 0 0
\(159\) 2.78675 6.15292i 0.221004 0.487958i
\(160\) 0 0
\(161\) 1.89888i 0.149653i
\(162\) 0 0
\(163\) 5.10007i 0.399469i −0.979850 0.199734i \(-0.935992\pi\)
0.979850 0.199734i \(-0.0640079\pi\)
\(164\) 0 0
\(165\) 6.12384 13.5209i 0.476740 1.05260i
\(166\) 0 0
\(167\) 12.8667 0.995655 0.497827 0.867276i \(-0.334131\pi\)
0.497827 + 0.867276i \(0.334131\pi\)
\(168\) 0 0
\(169\) −9.82366 −0.755666
\(170\) 0 0
\(171\) 2.25493 1.97871i 0.172439 0.151316i
\(172\) 0 0
\(173\) 11.9225i 0.906454i −0.891395 0.453227i \(-0.850273\pi\)
0.891395 0.453227i \(-0.149727\pi\)
\(174\) 0 0
\(175\) 1.11616i 0.0843737i
\(176\) 0 0
\(177\) 12.8515 + 5.82064i 0.965978 + 0.437506i
\(178\) 0 0
\(179\) −1.28974 −0.0963997 −0.0481998 0.998838i \(-0.515348\pi\)
−0.0481998 + 0.998838i \(0.515348\pi\)
\(180\) 0 0
\(181\) −14.7046 −1.09298 −0.546491 0.837465i \(-0.684037\pi\)
−0.546491 + 0.837465i \(0.684037\pi\)
\(182\) 0 0
\(183\) 18.8733 + 8.54802i 1.39516 + 0.631888i
\(184\) 0 0
\(185\) 5.63214i 0.414083i
\(186\) 0 0
\(187\) 6.83981i 0.500176i
\(188\) 0 0
\(189\) −3.51812 + 1.06892i −0.255905 + 0.0777525i
\(190\) 0 0
\(191\) −4.82987 −0.349477 −0.174739 0.984615i \(-0.555908\pi\)
−0.174739 + 0.984615i \(0.555908\pi\)
\(192\) 0 0
\(193\) −2.27363 −0.163659 −0.0818297 0.996646i \(-0.526076\pi\)
−0.0818297 + 0.996646i \(0.526076\pi\)
\(194\) 0 0
\(195\) 2.35617 5.20223i 0.168729 0.372539i
\(196\) 0 0
\(197\) 18.5158i 1.31919i −0.751619 0.659597i \(-0.770727\pi\)
0.751619 0.659597i \(-0.229273\pi\)
\(198\) 0 0
\(199\) 12.8189i 0.908710i −0.890821 0.454355i \(-0.849870\pi\)
0.890821 0.454355i \(-0.150130\pi\)
\(200\) 0 0
\(201\) 3.46167 7.64308i 0.244167 0.539101i
\(202\) 0 0
\(203\) 1.72971 0.121402
\(204\) 0 0
\(205\) 9.17020 0.640475
\(206\) 0 0
\(207\) −5.30978 6.05102i −0.369055 0.420575i
\(208\) 0 0
\(209\) 4.63214i 0.320412i
\(210\) 0 0
\(211\) 15.1596i 1.04363i −0.853059 0.521815i \(-0.825255\pi\)
0.853059 0.521815i \(-0.174745\pi\)
\(212\) 0 0
\(213\) −9.62378 4.35875i −0.659410 0.298657i
\(214\) 0 0
\(215\) 12.5878 0.858480
\(216\) 0 0
\(217\) 4.69627 0.318803
\(218\) 0 0
\(219\) 23.7546 + 10.7588i 1.60519 + 0.727015i
\(220\) 0 0
\(221\) 2.63164i 0.177023i
\(222\) 0 0
\(223\) 15.0080i 1.00501i −0.864574 0.502505i \(-0.832412\pi\)
0.864574 0.502505i \(-0.167588\pi\)
\(224\) 0 0
\(225\) 3.12108 + 3.55678i 0.208072 + 0.237119i
\(226\) 0 0
\(227\) 19.5512 1.29766 0.648830 0.760933i \(-0.275259\pi\)
0.648830 + 0.760933i \(0.275259\pi\)
\(228\) 0 0
\(229\) −14.3435 −0.947845 −0.473923 0.880566i \(-0.657162\pi\)
−0.473923 + 0.880566i \(0.657162\pi\)
\(230\) 0 0
\(231\) 2.34231 5.17162i 0.154112 0.340268i
\(232\) 0 0
\(233\) 0.893124i 0.0585105i −0.999572 0.0292553i \(-0.990686\pi\)
0.999572 0.0292553i \(-0.00931357\pi\)
\(234\) 0 0
\(235\) 15.1596i 0.988906i
\(236\) 0 0
\(237\) −1.58214 + 3.49323i −0.102771 + 0.226910i
\(238\) 0 0
\(239\) −27.7339 −1.79396 −0.896978 0.442075i \(-0.854243\pi\)
−0.896978 + 0.442075i \(0.854243\pi\)
\(240\) 0 0
\(241\) −25.8414 −1.66459 −0.832294 0.554334i \(-0.812973\pi\)
−0.832294 + 0.554334i \(0.812973\pi\)
\(242\) 0 0
\(243\) 8.22193 13.2439i 0.527437 0.849594i
\(244\) 0 0
\(245\) 12.0239i 0.768181i
\(246\) 0 0
\(247\) 1.78223i 0.113401i
\(248\) 0 0
\(249\) −4.18881 1.89717i −0.265455 0.120229i
\(250\) 0 0
\(251\) 3.39616 0.214363 0.107182 0.994239i \(-0.465817\pi\)
0.107182 + 0.994239i \(0.465817\pi\)
\(252\) 0 0
\(253\) 12.4301 0.781477
\(254\) 0 0
\(255\) 4.31011 + 1.95211i 0.269909 + 0.122246i
\(256\) 0 0
\(257\) 24.6927i 1.54029i −0.637868 0.770145i \(-0.720184\pi\)
0.637868 0.770145i \(-0.279816\pi\)
\(258\) 0 0
\(259\) 2.15424i 0.133858i
\(260\) 0 0
\(261\) −5.51194 + 4.83674i −0.341181 + 0.299387i
\(262\) 0 0
\(263\) −17.4827 −1.07803 −0.539014 0.842297i \(-0.681203\pi\)
−0.539014 + 0.842297i \(0.681203\pi\)
\(264\) 0 0
\(265\) 7.21473 0.443198
\(266\) 0 0
\(267\) 7.85226 17.3372i 0.480551 1.06102i
\(268\) 0 0
\(269\) 22.3903i 1.36516i 0.730809 + 0.682582i \(0.239143\pi\)
−0.730809 + 0.682582i \(0.760857\pi\)
\(270\) 0 0
\(271\) 31.0813i 1.88805i −0.329868 0.944027i \(-0.607004\pi\)
0.329868 0.944027i \(-0.392996\pi\)
\(272\) 0 0
\(273\) 0.901210 1.98980i 0.0545437 0.120428i
\(274\) 0 0
\(275\) −7.30643 −0.440594
\(276\) 0 0
\(277\) −23.4125 −1.40672 −0.703360 0.710834i \(-0.748318\pi\)
−0.703360 + 0.710834i \(0.748318\pi\)
\(278\) 0 0
\(279\) −14.9652 + 13.1320i −0.895946 + 0.786194i
\(280\) 0 0
\(281\) 28.8664i 1.72203i 0.508583 + 0.861013i \(0.330169\pi\)
−0.508583 + 0.861013i \(0.669831\pi\)
\(282\) 0 0
\(283\) 10.8054i 0.642317i 0.947025 + 0.321158i \(0.104072\pi\)
−0.947025 + 0.321158i \(0.895928\pi\)
\(284\) 0 0
\(285\) 2.91894 + 1.32203i 0.172903 + 0.0783105i
\(286\) 0 0
\(287\) 3.50751 0.207042
\(288\) 0 0
\(289\) 14.8197 0.871744
\(290\) 0 0
\(291\) 14.9201 + 6.75752i 0.874629 + 0.396133i
\(292\) 0 0
\(293\) 0.514137i 0.0300362i −0.999887 0.0150181i \(-0.995219\pi\)
0.999887 0.0150181i \(-0.00478059\pi\)
\(294\) 0 0
\(295\) 15.0693i 0.877368i
\(296\) 0 0
\(297\) 6.99720 + 23.0298i 0.406019 + 1.33632i
\(298\) 0 0
\(299\) 4.78254 0.276581
\(300\) 0 0
\(301\) 4.81470 0.277515
\(302\) 0 0
\(303\) −2.22869 + 4.92076i −0.128035 + 0.282690i
\(304\) 0 0
\(305\) 22.1303i 1.26718i
\(306\) 0 0
\(307\) 20.2588i 1.15623i 0.815954 + 0.578116i \(0.196212\pi\)
−0.815954 + 0.578116i \(0.803788\pi\)
\(308\) 0 0
\(309\) −10.9568 + 24.1918i −0.623313 + 1.37623i
\(310\) 0 0
\(311\) 29.4007 1.66716 0.833581 0.552397i \(-0.186287\pi\)
0.833581 + 0.552397i \(0.186287\pi\)
\(312\) 0 0
\(313\) 7.82563 0.442331 0.221165 0.975236i \(-0.429014\pi\)
0.221165 + 0.975236i \(0.429014\pi\)
\(314\) 0 0
\(315\) −2.59040 2.95201i −0.145952 0.166327i
\(316\) 0 0
\(317\) 8.41538i 0.472655i −0.971674 0.236327i \(-0.924056\pi\)
0.971674 0.236327i \(-0.0759438\pi\)
\(318\) 0 0
\(319\) 11.3228i 0.633953i
\(320\) 0 0
\(321\) −21.2671 9.63218i −1.18701 0.537616i
\(322\) 0 0
\(323\) −1.47660 −0.0821602
\(324\) 0 0
\(325\) −2.81117 −0.155936
\(326\) 0 0
\(327\) −6.39889 2.89815i −0.353859 0.160268i
\(328\) 0 0
\(329\) 5.79841i 0.319677i
\(330\) 0 0
\(331\) 26.7185i 1.46858i 0.678836 + 0.734290i \(0.262485\pi\)
−0.678836 + 0.734290i \(0.737515\pi\)
\(332\) 0 0
\(333\) −6.02383 6.86475i −0.330104 0.376186i
\(334\) 0 0
\(335\) 8.96205 0.489649
\(336\) 0 0
\(337\) −14.1242 −0.769395 −0.384697 0.923043i \(-0.625694\pi\)
−0.384697 + 0.923043i \(0.625694\pi\)
\(338\) 0 0
\(339\) −6.88967 + 15.2118i −0.374196 + 0.826194i
\(340\) 0 0
\(341\) 30.7420i 1.66477i
\(342\) 0 0
\(343\) 9.55240i 0.515781i
\(344\) 0 0
\(345\) 3.54762 7.83286i 0.190998 0.421707i
\(346\) 0 0
\(347\) −4.67942 −0.251204 −0.125602 0.992081i \(-0.540086\pi\)
−0.125602 + 0.992081i \(0.540086\pi\)
\(348\) 0 0
\(349\) 1.52611 0.0816911 0.0408455 0.999165i \(-0.486995\pi\)
0.0408455 + 0.999165i \(0.486995\pi\)
\(350\) 0 0
\(351\) 2.69220 + 8.86077i 0.143699 + 0.472953i
\(352\) 0 0
\(353\) 22.6720i 1.20671i −0.797474 0.603354i \(-0.793831\pi\)
0.797474 0.603354i \(-0.206169\pi\)
\(354\) 0 0
\(355\) 11.2846i 0.598922i
\(356\) 0 0
\(357\) 1.64857 + 0.746664i 0.0872517 + 0.0395176i
\(358\) 0 0
\(359\) −19.7942 −1.04470 −0.522350 0.852731i \(-0.674945\pi\)
−0.522350 + 0.852731i \(0.674945\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) −16.4982 7.47230i −0.865932 0.392194i
\(364\) 0 0
\(365\) 27.8540i 1.45794i
\(366\) 0 0
\(367\) 24.7226i 1.29051i −0.763967 0.645255i \(-0.776751\pi\)
0.763967 0.645255i \(-0.223249\pi\)
\(368\) 0 0
\(369\) −11.1771 + 9.80795i −0.581858 + 0.510582i
\(370\) 0 0
\(371\) 2.75956 0.143269
\(372\) 0 0
\(373\) 27.9194 1.44561 0.722805 0.691052i \(-0.242852\pi\)
0.722805 + 0.691052i \(0.242852\pi\)
\(374\) 0 0
\(375\) −8.69545 + 19.1989i −0.449031 + 0.991425i
\(376\) 0 0
\(377\) 4.35647i 0.224370i
\(378\) 0 0
\(379\) 24.4837i 1.25764i 0.777550 + 0.628821i \(0.216462\pi\)
−0.777550 + 0.628821i \(0.783538\pi\)
\(380\) 0 0
\(381\) −2.64870 + 5.84812i −0.135697 + 0.299608i
\(382\) 0 0
\(383\) −13.9416 −0.712382 −0.356191 0.934413i \(-0.615925\pi\)
−0.356191 + 0.934413i \(0.615925\pi\)
\(384\) 0 0
\(385\) 6.06409 0.309055
\(386\) 0 0
\(387\) −15.3427 + 13.4632i −0.779911 + 0.684373i
\(388\) 0 0
\(389\) 9.11256i 0.462025i 0.972951 + 0.231013i \(0.0742039\pi\)
−0.972951 + 0.231013i \(0.925796\pi\)
\(390\) 0 0
\(391\) 3.96239i 0.200387i
\(392\) 0 0
\(393\) 0.0191051 + 0.00865301i 0.000963727 + 0.000436487i
\(394\) 0 0
\(395\) −4.09606 −0.206095
\(396\) 0 0
\(397\) 7.45405 0.374108 0.187054 0.982350i \(-0.440106\pi\)
0.187054 + 0.982350i \(0.440106\pi\)
\(398\) 0 0
\(399\) 1.11647 + 0.505665i 0.0558932 + 0.0253149i
\(400\) 0 0
\(401\) 30.3076i 1.51349i −0.653711 0.756744i \(-0.726789\pi\)
0.653711 0.756744i \(-0.273211\pi\)
\(402\) 0 0
\(403\) 11.8281i 0.589198i
\(404\) 0 0
\(405\) 16.5092 + 2.16351i 0.820351 + 0.107506i
\(406\) 0 0
\(407\) 14.1017 0.698997
\(408\) 0 0
\(409\) 34.7359 1.71758 0.858790 0.512328i \(-0.171217\pi\)
0.858790 + 0.512328i \(0.171217\pi\)
\(410\) 0 0
\(411\) 11.8132 26.0826i 0.582703 1.28656i
\(412\) 0 0
\(413\) 5.76385i 0.283621i
\(414\) 0 0
\(415\) 4.91167i 0.241105i
\(416\) 0 0
\(417\) 2.92839 6.46564i 0.143404 0.316624i
\(418\) 0 0
\(419\) −19.9232 −0.973314 −0.486657 0.873593i \(-0.661784\pi\)
−0.486657 + 0.873593i \(0.661784\pi\)
\(420\) 0 0
\(421\) −9.25156 −0.450893 −0.225447 0.974256i \(-0.572384\pi\)
−0.225447 + 0.974256i \(0.572384\pi\)
\(422\) 0 0
\(423\) −16.2139 18.4774i −0.788348 0.898400i
\(424\) 0 0
\(425\) 2.32909i 0.112977i
\(426\) 0 0
\(427\) 8.46463i 0.409632i
\(428\) 0 0
\(429\) −13.0253 5.89936i −0.628868 0.284824i
\(430\) 0 0
\(431\) −17.6084 −0.848169 −0.424084 0.905623i \(-0.639404\pi\)
−0.424084 + 0.905623i \(0.639404\pi\)
\(432\) 0 0
\(433\) 23.4409 1.12650 0.563248 0.826288i \(-0.309551\pi\)
0.563248 + 0.826288i \(0.309551\pi\)
\(434\) 0 0
\(435\) −7.13505 3.23157i −0.342099 0.154942i
\(436\) 0 0
\(437\) 2.68346i 0.128367i
\(438\) 0 0
\(439\) 10.9049i 0.520461i 0.965547 + 0.260231i \(0.0837986\pi\)
−0.965547 + 0.260231i \(0.916201\pi\)
\(440\) 0 0
\(441\) 12.8601 + 14.6554i 0.612388 + 0.697877i
\(442\) 0 0
\(443\) 4.08004 0.193848 0.0969242 0.995292i \(-0.469100\pi\)
0.0969242 + 0.995292i \(0.469100\pi\)
\(444\) 0 0
\(445\) 20.3290 0.963689
\(446\) 0 0
\(447\) −15.5788 + 34.3967i −0.736852 + 1.62691i
\(448\) 0 0
\(449\) 27.2291i 1.28502i 0.766277 + 0.642511i \(0.222107\pi\)
−0.766277 + 0.642511i \(0.777893\pi\)
\(450\) 0 0
\(451\) 22.9603i 1.08116i
\(452\) 0 0
\(453\) 13.3499 29.4754i 0.627231 1.38487i
\(454\) 0 0
\(455\) 2.33318 0.109381
\(456\) 0 0
\(457\) 6.10668 0.285658 0.142829 0.989747i \(-0.454380\pi\)
0.142829 + 0.989747i \(0.454380\pi\)
\(458\) 0 0
\(459\) −7.34126 + 2.23052i −0.342661 + 0.104112i
\(460\) 0 0
\(461\) 17.8562i 0.831648i 0.909445 + 0.415824i \(0.136507\pi\)
−0.909445 + 0.415824i \(0.863493\pi\)
\(462\) 0 0
\(463\) 14.1434i 0.657299i 0.944452 + 0.328649i \(0.106593\pi\)
−0.944452 + 0.328649i \(0.893407\pi\)
\(464\) 0 0
\(465\) −19.3721 8.77390i −0.898358 0.406880i
\(466\) 0 0
\(467\) −40.7322 −1.88486 −0.942430 0.334404i \(-0.891465\pi\)
−0.942430 + 0.334404i \(0.891465\pi\)
\(468\) 0 0
\(469\) 3.42790 0.158286
\(470\) 0 0
\(471\) 16.0750 + 7.28062i 0.740698 + 0.335473i
\(472\) 0 0
\(473\) 31.5172i 1.44916i
\(474\) 0 0
\(475\) 1.57733i 0.0723730i
\(476\) 0 0
\(477\) −8.79370 + 7.71649i −0.402636 + 0.353314i
\(478\) 0 0
\(479\) 1.94900 0.0890519 0.0445260 0.999008i \(-0.485822\pi\)
0.0445260 + 0.999008i \(0.485822\pi\)
\(480\) 0 0
\(481\) 5.42569 0.247390
\(482\) 0 0
\(483\) 1.35693 2.99599i 0.0617425 0.136322i
\(484\) 0 0
\(485\) 17.4948i 0.794399i
\(486\) 0 0
\(487\) 1.97691i 0.0895824i 0.998996 + 0.0447912i \(0.0142622\pi\)
−0.998996 + 0.0447912i \(0.985738\pi\)
\(488\) 0 0
\(489\) −3.64449 + 8.04674i −0.164809 + 0.363886i
\(490\) 0 0
\(491\) −8.44538 −0.381134 −0.190567 0.981674i \(-0.561033\pi\)
−0.190567 + 0.981674i \(0.561033\pi\)
\(492\) 0 0
\(493\) 3.60939 0.162559
\(494\) 0 0
\(495\) −19.3240 + 16.9568i −0.868549 + 0.762154i
\(496\) 0 0
\(497\) 4.31623i 0.193609i
\(498\) 0 0
\(499\) 17.5057i 0.783664i 0.920037 + 0.391832i \(0.128159\pi\)
−0.920037 + 0.391832i \(0.871841\pi\)
\(500\) 0 0
\(501\) −20.3007 9.19448i −0.906967 0.410779i
\(502\) 0 0
\(503\) −10.6802 −0.476209 −0.238104 0.971240i \(-0.576526\pi\)
−0.238104 + 0.971240i \(0.576526\pi\)
\(504\) 0 0
\(505\) −5.76994 −0.256759
\(506\) 0 0
\(507\) 15.4995 + 7.01994i 0.688355 + 0.311766i
\(508\) 0 0
\(509\) 39.2121i 1.73804i 0.494773 + 0.869022i \(0.335251\pi\)
−0.494773 + 0.869022i \(0.664749\pi\)
\(510\) 0 0
\(511\) 10.6539i 0.471300i
\(512\) 0 0
\(513\) −4.97174 + 1.51058i −0.219507 + 0.0666937i
\(514\) 0 0
\(515\) −28.3666 −1.24998
\(516\) 0 0
\(517\) 37.9567 1.66933
\(518\) 0 0
\(519\) −8.51979 + 18.8110i −0.373977 + 0.825711i
\(520\) 0 0
\(521\) 35.9655i 1.57568i 0.615881 + 0.787839i \(0.288800\pi\)
−0.615881 + 0.787839i \(0.711200\pi\)
\(522\) 0 0
\(523\) 20.0443i 0.876477i −0.898859 0.438238i \(-0.855603\pi\)
0.898859 0.438238i \(-0.144397\pi\)
\(524\) 0 0
\(525\) −0.797602 + 1.76104i −0.0348102 + 0.0768581i
\(526\) 0 0
\(527\) 9.79970 0.426882
\(528\) 0 0
\(529\) −15.7990 −0.686915
\(530\) 0 0
\(531\) −16.1173 18.3672i −0.699431 0.797071i
\(532\) 0 0
\(533\) 8.83406i 0.382646i
\(534\) 0 0
\(535\) 24.9371i 1.07813i
\(536\) 0 0
\(537\) 2.03491 + 0.921642i 0.0878129 + 0.0397718i
\(538\) 0 0
\(539\) −30.1055 −1.29674
\(540\) 0 0
\(541\) −4.27173 −0.183656 −0.0918280 0.995775i \(-0.529271\pi\)
−0.0918280 + 0.995775i \(0.529271\pi\)
\(542\) 0 0
\(543\) 23.2004 + 10.5078i 0.995624 + 0.450933i
\(544\) 0 0
\(545\) 7.50314i 0.321399i
\(546\) 0 0
\(547\) 7.68080i 0.328407i 0.986426 + 0.164204i \(0.0525054\pi\)
−0.986426 + 0.164204i \(0.947495\pi\)
\(548\) 0 0
\(549\) −23.6694 26.9736i −1.01018 1.15121i
\(550\) 0 0
\(551\) 2.44439 0.104135
\(552\) 0 0
\(553\) −1.56670 −0.0666230
\(554\) 0 0
\(555\) 4.02470 8.88621i 0.170839 0.377199i
\(556\) 0 0
\(557\) 44.2479i 1.87484i 0.348197 + 0.937421i \(0.386794\pi\)
−0.348197 + 0.937421i \(0.613206\pi\)
\(558\) 0 0
\(559\) 12.1264i 0.512891i
\(560\) 0 0
\(561\) 4.88769 10.7916i 0.206359 0.455623i
\(562\) 0 0
\(563\) 36.1114 1.52191 0.760956 0.648803i \(-0.224730\pi\)
0.760956 + 0.648803i \(0.224730\pi\)
\(564\) 0 0
\(565\) −17.8370 −0.750406
\(566\) 0 0
\(567\) 6.31462 + 0.827521i 0.265189 + 0.0347526i
\(568\) 0 0
\(569\) 8.27441i 0.346881i −0.984844 0.173441i \(-0.944511\pi\)
0.984844 0.173441i \(-0.0554885\pi\)
\(570\) 0 0
\(571\) 27.3137i 1.14304i −0.820587 0.571521i \(-0.806353\pi\)
0.820587 0.571521i \(-0.193647\pi\)
\(572\) 0 0
\(573\) 7.62042 + 3.45140i 0.318347 + 0.144184i
\(574\) 0 0
\(575\) −4.23271 −0.176516
\(576\) 0 0
\(577\) −11.1555 −0.464408 −0.232204 0.972667i \(-0.574594\pi\)
−0.232204 + 0.972667i \(0.574594\pi\)
\(578\) 0 0
\(579\) 3.58726 + 1.62472i 0.149081 + 0.0675212i
\(580\) 0 0
\(581\) 1.87867i 0.0779402i
\(582\) 0 0
\(583\) 18.0642i 0.748144i
\(584\) 0 0
\(585\) −7.43497 + 6.52420i −0.307398 + 0.269743i
\(586\) 0 0
\(587\) −23.7699 −0.981089 −0.490545 0.871416i \(-0.663202\pi\)
−0.490545 + 0.871416i \(0.663202\pi\)
\(588\) 0 0
\(589\) 6.63667 0.273459
\(590\) 0 0
\(591\) −13.2313 + 29.2136i −0.544262 + 1.20169i
\(592\) 0 0
\(593\) 27.1707i 1.11577i −0.829920 0.557883i \(-0.811614\pi\)
0.829920 0.557883i \(-0.188386\pi\)
\(594\) 0 0
\(595\) 1.93307i 0.0792481i
\(596\) 0 0
\(597\) −9.16035 + 20.2253i −0.374908 + 0.827767i
\(598\) 0 0
\(599\) −15.2016 −0.621122 −0.310561 0.950554i \(-0.600517\pi\)
−0.310561 + 0.950554i \(0.600517\pi\)
\(600\) 0 0
\(601\) 17.7396 0.723614 0.361807 0.932253i \(-0.382160\pi\)
0.361807 + 0.932253i \(0.382160\pi\)
\(602\) 0 0
\(603\) −10.9234 + 9.58532i −0.444836 + 0.390345i
\(604\) 0 0
\(605\) 19.3453i 0.786500i
\(606\) 0 0
\(607\) 1.00873i 0.0409429i −0.999790 0.0204715i \(-0.993483\pi\)
0.999790 0.0204715i \(-0.00651672\pi\)
\(608\) 0 0
\(609\) −2.72908 1.23604i −0.110588 0.0500870i
\(610\) 0 0
\(611\) 14.6039 0.590813
\(612\) 0 0
\(613\) 45.7234 1.84675 0.923375 0.383899i \(-0.125419\pi\)
0.923375 + 0.383899i \(0.125419\pi\)
\(614\) 0 0
\(615\) −14.4685 6.55299i −0.583425 0.264242i
\(616\) 0 0
\(617\) 7.79096i 0.313652i 0.987626 + 0.156826i \(0.0501262\pi\)
−0.987626 + 0.156826i \(0.949874\pi\)
\(618\) 0 0
\(619\) 18.8993i 0.759628i 0.925063 + 0.379814i \(0.124012\pi\)
−0.925063 + 0.379814i \(0.875988\pi\)
\(620\) 0 0
\(621\) 4.05358 + 13.3415i 0.162664 + 0.535374i
\(622\) 0 0
\(623\) 7.77565 0.311525
\(624\) 0 0
\(625\) −14.6253 −0.585014
\(626\) 0 0
\(627\) 3.31010 7.30844i 0.132193 0.291871i
\(628\) 0 0
\(629\) 4.49525i 0.179237i
\(630\) 0 0
\(631\) 12.4573i 0.495918i −0.968771 0.247959i \(-0.920240\pi\)
0.968771 0.247959i \(-0.0797598\pi\)
\(632\) 0 0
\(633\) −10.8330 + 23.9183i −0.430572 + 0.950668i
\(634\) 0 0
\(635\) −6.85733 −0.272125
\(636\) 0 0
\(637\) −11.5832 −0.458943
\(638\) 0 0
\(639\) 12.0693 + 13.7542i 0.477456 + 0.544108i
\(640\) 0 0
\(641\) 9.64878i 0.381104i 0.981677 + 0.190552i \(0.0610278\pi\)
−0.981677 + 0.190552i \(0.938972\pi\)
\(642\) 0 0
\(643\) 11.1740i 0.440659i −0.975426 0.220329i \(-0.929287\pi\)
0.975426 0.220329i \(-0.0707132\pi\)
\(644\) 0 0
\(645\) −19.8606 8.99517i −0.782011 0.354185i
\(646\) 0 0
\(647\) 38.9594 1.53165 0.765826 0.643048i \(-0.222330\pi\)
0.765826 + 0.643048i \(0.222330\pi\)
\(648\) 0 0
\(649\) 37.7304 1.48105
\(650\) 0 0
\(651\) −7.40962 3.35593i −0.290406 0.131529i
\(652\) 0 0
\(653\) 9.77338i 0.382462i 0.981545 + 0.191231i \(0.0612479\pi\)
−0.981545 + 0.191231i \(0.938752\pi\)
\(654\) 0 0
\(655\) 0.0224021i 0.000875324i
\(656\) 0 0
\(657\) −29.7911 33.9499i −1.16226 1.32451i
\(658\) 0 0
\(659\) 8.05188 0.313657 0.156828 0.987626i \(-0.449873\pi\)
0.156828 + 0.987626i \(0.449873\pi\)
\(660\) 0 0
\(661\) 13.6768 0.531967 0.265984 0.963978i \(-0.414303\pi\)
0.265984 + 0.963978i \(0.414303\pi\)
\(662\) 0 0
\(663\) 1.88056 4.15212i 0.0730348 0.161255i
\(664\) 0 0
\(665\) 1.30914i 0.0507661i
\(666\) 0 0
\(667\) 6.55943i 0.253982i
\(668\) 0 0
\(669\) −10.7247 + 23.6792i −0.414639 + 0.915489i
\(670\) 0 0
\(671\) 55.4098 2.13907
\(672\) 0 0
\(673\) −34.9298 −1.34644 −0.673221 0.739441i \(-0.735090\pi\)
−0.673221 + 0.739441i \(0.735090\pi\)
\(674\) 0 0
\(675\) −2.38269 7.84209i −0.0917096 0.301842i
\(676\) 0 0
\(677\) 43.3133i 1.66466i 0.554277 + 0.832332i \(0.312995\pi\)
−0.554277 + 0.832332i \(0.687005\pi\)
\(678\) 0 0
\(679\) 6.69159i 0.256800i
\(680\) 0 0
\(681\) −30.8473 13.9712i −1.18207 0.535378i
\(682\) 0 0
\(683\) −6.37903 −0.244087 −0.122043 0.992525i \(-0.538945\pi\)
−0.122043 + 0.992525i \(0.538945\pi\)
\(684\) 0 0
\(685\) 30.5837 1.16854
\(686\) 0 0
\(687\) 22.6307 + 10.2498i 0.863416 + 0.391054i
\(688\) 0 0
\(689\) 6.95027i 0.264784i
\(690\) 0 0
\(691\) 32.0940i 1.22091i 0.792050 + 0.610456i \(0.209014\pi\)
−0.792050 + 0.610456i \(0.790986\pi\)
\(692\) 0 0
\(693\) −7.39124 + 6.48582i −0.280770 + 0.246376i
\(694\) 0 0
\(695\) 7.58142 0.287580
\(696\) 0 0
\(697\) 7.31913 0.277232
\(698\) 0 0
\(699\) −0.638222 + 1.40914i −0.0241398 + 0.0532987i
\(700\) 0 0
\(701\) 35.5578i 1.34300i −0.741004 0.671500i \(-0.765650\pi\)
0.741004 0.671500i \(-0.234350\pi\)
\(702\) 0 0
\(703\) 3.04433i 0.114819i
\(704\) 0 0
\(705\) 10.8330 23.9184i 0.407995 0.900819i
\(706\) 0 0
\(707\) −2.20694 −0.0830007
\(708\) 0 0
\(709\) −8.04768 −0.302237 −0.151118 0.988516i \(-0.548287\pi\)
−0.151118 + 0.988516i \(0.548287\pi\)
\(710\) 0 0
\(711\) 4.99250 4.38093i 0.187233 0.164298i
\(712\) 0 0
\(713\) 17.8092i 0.666961i
\(714\) 0 0
\(715\) 15.2731i 0.571182i
\(716\) 0 0
\(717\) 43.7577 + 19.8185i 1.63416 + 0.740136i
\(718\) 0 0
\(719\) −28.8564 −1.07616 −0.538082 0.842893i \(-0.680851\pi\)
−0.538082 + 0.842893i \(0.680851\pi\)
\(720\) 0 0
\(721\) −10.8500 −0.404073
\(722\) 0 0
\(723\) 40.7717 + 18.4661i 1.51632 + 0.686762i
\(724\) 0 0
\(725\) 3.85563i 0.143194i
\(726\) 0 0
\(727\) 33.8395i 1.25504i 0.778602 + 0.627518i \(0.215929\pi\)
−0.778602 + 0.627518i \(0.784071\pi\)
\(728\) 0 0
\(729\) −22.4363 + 15.0204i −0.830974 + 0.556311i
\(730\) 0 0
\(731\) 10.0468 0.371596
\(732\) 0 0
\(733\) −23.6799 −0.874639 −0.437319 0.899306i \(-0.644072\pi\)
−0.437319 + 0.899306i \(0.644072\pi\)
\(734\) 0 0
\(735\) −8.59225 + 18.9710i −0.316930 + 0.699756i
\(736\) 0 0
\(737\) 22.4392i 0.826557i
\(738\) 0 0
\(739\) 37.9341i 1.39543i 0.716375 + 0.697715i \(0.245800\pi\)
−0.716375 + 0.697715i \(0.754200\pi\)
\(740\) 0 0
\(741\) 1.27357 2.81195i 0.0467859 0.103299i
\(742\) 0 0
\(743\) 7.47556 0.274252 0.137126 0.990554i \(-0.456214\pi\)
0.137126 + 0.990554i \(0.456214\pi\)
\(744\) 0 0
\(745\) −40.3326 −1.47767
\(746\) 0 0
\(747\) 5.25326 + 5.98660i 0.192207 + 0.219038i
\(748\) 0 0
\(749\) 9.53820i 0.348518i
\(750\) 0 0
\(751\) 52.8763i 1.92948i −0.263198 0.964742i \(-0.584777\pi\)
0.263198 0.964742i \(-0.415223\pi\)
\(752\) 0 0
\(753\) −5.35835 2.42688i −0.195269 0.0884403i
\(754\) 0 0
\(755\) 34.5620 1.25784
\(756\) 0 0
\(757\) 3.02847 0.110071 0.0550357 0.998484i \(-0.482473\pi\)
0.0550357 + 0.998484i \(0.482473\pi\)
\(758\) 0 0
\(759\) −19.6119 8.88253i −0.711867 0.322415i
\(760\) 0 0
\(761\) 12.3323i 0.447046i 0.974699 + 0.223523i \(0.0717558\pi\)
−0.974699 + 0.223523i \(0.928244\pi\)
\(762\) 0 0
\(763\) 2.86988i 0.103897i
\(764\) 0 0
\(765\) −5.40538 6.15996i −0.195432 0.222714i
\(766\) 0 0
\(767\) 14.5169 0.524175
\(768\) 0 0
\(769\) 35.7181 1.28803 0.644015 0.765013i \(-0.277268\pi\)
0.644015 + 0.765013i \(0.277268\pi\)
\(770\) 0 0
\(771\) −17.6453 + 38.9594i −0.635481 + 1.40309i
\(772\) 0 0
\(773\) 40.3015i 1.44954i −0.688989 0.724771i \(-0.741945\pi\)
0.688989 0.724771i \(-0.258055\pi\)
\(774\) 0 0
\(775\) 10.4682i 0.376031i
\(776\) 0 0
\(777\) 1.53941 3.39889i 0.0552260 0.121934i
\(778\) 0 0
\(779\) 4.95675 0.177594
\(780\) 0 0
\(781\) −28.2542 −1.01102
\(782\) 0 0
\(783\) 12.1529 3.69245i 0.434309 0.131957i
\(784\) 0 0
\(785\) 18.8491i 0.672753i
\(786\) 0 0
\(787\) 26.5594i 0.946742i −0.880863 0.473371i \(-0.843037\pi\)
0.880863 0.473371i \(-0.156963\pi\)
\(788\) 0 0
\(789\) 27.5836 + 12.4930i 0.982004 + 0.444764i
\(790\) 0 0
\(791\) −6.82246 −0.242579
\(792\) 0 0
\(793\) 21.3191 0.757064
\(794\) 0 0
\(795\) −11.3832 5.15562i −0.403720 0.182851i
\(796\) 0 0
\(797\) 27.4101i 0.970915i −0.874260 0.485457i \(-0.838653\pi\)
0.874260 0.485457i \(-0.161347\pi\)
\(798\) 0 0
\(799\) 12.0995i 0.428051i
\(800\) 0 0
\(801\) −24.7781 + 21.7428i −0.875491 + 0.768245i
\(802\) 0 0
\(803\) 69.7407 2.46110
\(804\) 0 0
\(805\) 3.51301 0.123817
\(806\) 0 0
\(807\) 16.0000 35.3268i 0.563228 1.24356i
\(808\) 0 0
\(809\) 27.2725i 0.958849i −0.877583 0.479425i \(-0.840845\pi\)
0.877583 0.479425i \(-0.159155\pi\)
\(810\) 0 0
\(811\) 18.6173i 0.653740i −0.945069 0.326870i \(-0.894006\pi\)
0.945069 0.326870i \(-0.105994\pi\)
\(812\) 0 0
\(813\) −22.2105 + 49.0391i −0.778958 + 1.71988i
\(814\) 0 0
\(815\) −9.43536 −0.330506
\(816\) 0 0
\(817\) 6.80404 0.238043
\(818\) 0 0
\(819\) −2.84380 + 2.49544i −0.0993705 + 0.0871978i
\(820\) 0 0
\(821\) 13.9352i 0.486341i −0.969984 0.243171i \(-0.921812\pi\)
0.969984 0.243171i \(-0.0781875\pi\)
\(822\) 0 0
\(823\) 49.9501i 1.74115i −0.492035 0.870575i \(-0.663747\pi\)
0.492035 0.870575i \(-0.336253\pi\)
\(824\) 0 0
\(825\) 11.5278 + 5.22114i 0.401348 + 0.181777i
\(826\) 0 0
\(827\) −12.1270 −0.421697 −0.210849 0.977519i \(-0.567623\pi\)
−0.210849 + 0.977519i \(0.567623\pi\)
\(828\) 0 0
\(829\) 18.3614 0.637717 0.318859 0.947802i \(-0.396700\pi\)
0.318859 + 0.947802i \(0.396700\pi\)
\(830\) 0 0
\(831\) 36.9395 + 16.7305i 1.28142 + 0.580373i
\(832\) 0 0
\(833\) 9.59681i 0.332510i
\(834\) 0 0
\(835\) 23.8040i 0.823770i
\(836\) 0 0
\(837\) 32.9958 10.0252i 1.14050 0.346522i
\(838\) 0 0
\(839\) −54.5183 −1.88218 −0.941090 0.338156i \(-0.890197\pi\)
−0.941090 + 0.338156i \(0.890197\pi\)
\(840\) 0 0
\(841\) 23.0249 0.793963
\(842\) 0 0
\(843\) 20.6278 45.5445i 0.710459 1.56864i
\(844\) 0 0
\(845\) 18.1742i 0.625212i
\(846\) 0 0
\(847\) 7.39940i 0.254246i
\(848\) 0 0
\(849\) 7.72152 17.0485i 0.265002 0.585103i
\(850\) 0 0
\(851\) 8.16933 0.280041
\(852\) 0 0
\(853\) 3.62135 0.123993 0.0619964 0.998076i \(-0.480253\pi\)
0.0619964 + 0.998076i \(0.480253\pi\)
\(854\) 0 0
\(855\) −3.66070 4.17173i −0.125193 0.142670i
\(856\) 0 0
\(857\) 58.1816i 1.98745i −0.111869 0.993723i \(-0.535684\pi\)
0.111869 0.993723i \(-0.464316\pi\)
\(858\) 0 0
\(859\) 11.0241i 0.376136i 0.982156 + 0.188068i \(0.0602226\pi\)
−0.982156 + 0.188068i \(0.939777\pi\)
\(860\) 0 0
\(861\) −5.53404 2.50645i −0.188600 0.0854196i
\(862\) 0 0
\(863\) 55.9330 1.90398 0.951991 0.306126i \(-0.0990330\pi\)
0.951991 + 0.306126i \(0.0990330\pi\)
\(864\) 0 0
\(865\) −22.0572 −0.749968
\(866\) 0 0
\(867\) −23.3820 10.5901i −0.794094 0.359657i
\(868\) 0 0
\(869\) 10.2557i 0.347901i
\(870\) 0 0
\(871\) 8.63354i 0.292536i
\(872\) 0 0
\(873\) −18.7115 21.3236i −0.633288 0.721694i
\(874\) 0 0
\(875\) −8.61062 −0.291092
\(876\) 0 0
\(877\) 33.6467 1.13617 0.568084 0.822971i \(-0.307685\pi\)
0.568084 + 0.822971i \(0.307685\pi\)
\(878\) 0 0
\(879\) −0.367400 + 0.811190i −0.0123921 + 0.0273608i
\(880\) 0 0
\(881\) 50.7744i 1.71063i −0.518105 0.855317i \(-0.673362\pi\)
0.518105 0.855317i \(-0.326638\pi\)
\(882\) 0 0
\(883\) 15.6602i 0.527007i 0.964658 + 0.263504i \(0.0848781\pi\)
−0.964658 + 0.263504i \(0.915122\pi\)
\(884\) 0 0
\(885\) 10.7684 23.7759i 0.361977 0.799217i
\(886\) 0 0
\(887\) −7.98064 −0.267964 −0.133982 0.990984i \(-0.542776\pi\)
−0.133982 + 0.990984i \(0.542776\pi\)
\(888\) 0 0
\(889\) −2.62286 −0.0879679
\(890\) 0 0
\(891\) 5.41699 41.3358i 0.181476 1.38480i
\(892\) 0 0
\(893\) 8.19420i 0.274208i
\(894\) 0 0
\(895\) 2.38608i 0.0797577i
\(896\) 0 0
\(897\) −7.54574 3.41758i −0.251945 0.114110i
\(898\) 0 0
\(899\) −16.2226 −0.541055
\(900\) 0 0
\(901\) 5.75838 0.191840
\(902\) 0 0
\(903\) −7.59648 3.44056i −0.252795 0.114495i
\(904\) 0 0
\(905\) 27.2041i 0.904295i
\(906\) 0 0
\(907\) 21.9105i 0.727527i −0.931491 0.363764i \(-0.881492\pi\)
0.931491 0.363764i \(-0.118508\pi\)
\(908\) 0 0
\(909\) 7.03271 6.17121i 0.233260 0.204686i
\(910\) 0 0
\(911\) −42.5173 −1.40866 −0.704330 0.709873i \(-0.748752\pi\)
−0.704330 + 0.709873i \(0.748752\pi\)
\(912\) 0 0
\(913\) −12.2978 −0.406999
\(914\) 0 0
\(915\) 15.8142 34.9165i 0.522802 1.15430i
\(916\) 0 0
\(917\) 0.00856859i 0.000282960i
\(918\) 0 0
\(919\) 20.2272i 0.667234i −0.942709 0.333617i \(-0.891731\pi\)
0.942709 0.333617i \(-0.108269\pi\)
\(920\) 0 0
\(921\) 14.4769 31.9637i 0.477029 1.05324i
\(922\) 0 0
\(923\) −10.8709 −0.357820
\(924\) 0 0
\(925\) −4.80192 −0.157886
\(926\) 0 0
\(927\) 34.5747 30.3394i 1.13558 0.996476i
\(928\) 0 0
\(929\) 48.3458i 1.58617i −0.609109 0.793086i \(-0.708473\pi\)
0.609109 0.793086i \(-0.291527\pi\)
\(930\) 0 0
\(931\) 6.49927i 0.213005i
\(932\) 0 0
\(933\) −46.3875 21.0096i −1.51866 0.687824i
\(934\) 0 0
\(935\) 12.6539 0.413828
\(936\) 0 0
\(937\) −49.7559 −1.62545 −0.812727 0.582645i \(-0.802018\pi\)
−0.812727 + 0.582645i \(0.802018\pi\)
\(938\) 0 0
\(939\) −12.3470 5.59216i −0.402930 0.182493i
\(940\) 0 0
\(941\) 24.6221i 0.802657i 0.915934 + 0.401328i \(0.131451\pi\)
−0.915934 + 0.401328i \(0.868549\pi\)
\(942\) 0 0
\(943\) 13.3012i 0.433148i
\(944\) 0 0
\(945\) 1.97755 + 6.50868i 0.0643298 + 0.211727i
\(946\) 0 0
\(947\) −8.29796 −0.269647 −0.134824 0.990870i \(-0.543047\pi\)
−0.134824 + 0.990870i \(0.543047\pi\)
\(948\) 0 0
\(949\) 26.8330 0.871035
\(950\) 0 0
\(951\) −6.01359 + 13.2775i −0.195004 + 0.430553i
\(952\) 0 0
\(953\) 10.5290i 0.341069i −0.985352 0.170534i \(-0.945451\pi\)
0.985352 0.170534i \(-0.0545494\pi\)
\(954\) 0 0
\(955\) 8.93548i 0.289145i
\(956\) 0 0
\(957\) −8.09120 + 17.8647i −0.261551 + 0.577484i
\(958\) 0 0
\(959\) 11.6980 0.377747
\(960\) 0 0
\(961\) −13.0454 −0.420820
\(962\) 0 0
\(963\) 26.6714 + 30.3947i 0.859474 + 0.979455i
\(964\) 0 0
\(965\) 4.20631i 0.135406i
\(966\) 0 0
\(967\) 49.6937i 1.59804i 0.601302 + 0.799022i \(0.294649\pi\)
−0.601302 + 0.799022i \(0.705351\pi\)
\(968\) 0 0
\(969\) 2.32973 + 1.05517i 0.0748417 + 0.0338970i
\(970\) 0 0
\(971\) 40.7513 1.30777 0.653886 0.756593i \(-0.273137\pi\)
0.653886 + 0.756593i \(0.273137\pi\)
\(972\) 0 0
\(973\) 2.89982 0.0929639
\(974\) 0 0
\(975\) 4.43538 + 2.00885i 0.142046 + 0.0643347i
\(976\) 0 0
\(977\) 49.7877i 1.59285i 0.604737 + 0.796425i \(0.293278\pi\)
−0.604737 + 0.796425i \(0.706722\pi\)
\(978\) 0 0
\(979\) 50.8998i 1.62676i
\(980\) 0 0
\(981\) 8.02495 + 9.14523i 0.256217 + 0.291985i
\(982\) 0 0
\(983\) 20.1050 0.641250 0.320625 0.947206i \(-0.396107\pi\)
0.320625 + 0.947206i \(0.396107\pi\)
\(984\) 0 0
\(985\) −34.2550 −1.09146
\(986\) 0 0
\(987\) 4.14352 9.14855i 0.131890 0.291202i
\(988\) 0 0
\(989\) 18.2584i 0.580582i
\(990\) 0 0
\(991\) 4.64260i 0.147477i −0.997278 0.0737384i \(-0.976507\pi\)
0.997278 0.0737384i \(-0.0234930\pi\)
\(992\) 0 0
\(993\) 19.0929 42.1556i 0.605895 1.33777i
\(994\) 0 0
\(995\) −23.7156 −0.751835
\(996\) 0 0
\(997\) −46.1056 −1.46018 −0.730089 0.683352i \(-0.760521\pi\)
−0.730089 + 0.683352i \(0.760521\pi\)
\(998\) 0 0
\(999\) 4.59869 + 15.1356i 0.145496 + 0.478869i
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 912.2.d.b.191.3 24
3.2 odd 2 inner 912.2.d.b.191.21 yes 24
4.3 odd 2 inner 912.2.d.b.191.22 yes 24
12.11 even 2 inner 912.2.d.b.191.4 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
912.2.d.b.191.3 24 1.1 even 1 trivial
912.2.d.b.191.4 yes 24 12.11 even 2 inner
912.2.d.b.191.21 yes 24 3.2 odd 2 inner
912.2.d.b.191.22 yes 24 4.3 odd 2 inner