Properties

Label 756.2.ca.a.173.3
Level $756$
Weight $2$
Character 756.173
Analytic conductor $6.037$
Analytic rank $0$
Dimension $144$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [756,2,Mod(173,756)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(756, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 13, 15]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("756.173");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.ca (of order \(18\), degree \(6\), 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: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(144\)
Relative dimension: \(24\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 173.3
Character \(\chi\) \(=\) 756.173
Dual form 756.2.ca.a.437.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.38241 - 1.04352i) q^{3} +(-2.01034 + 1.68687i) q^{5} +(-2.51700 - 0.815311i) q^{7} +(0.822127 + 2.88515i) q^{9} +O(q^{10})\) \(q+(-1.38241 - 1.04352i) q^{3} +(-2.01034 + 1.68687i) q^{5} +(-2.51700 - 0.815311i) q^{7} +(0.822127 + 2.88515i) q^{9} +(-0.408189 + 0.486461i) q^{11} +(2.83064 + 3.37342i) q^{13} +(4.53941 - 0.234125i) q^{15} +(4.02938 - 6.97910i) q^{17} +(-3.43287 + 1.98197i) q^{19} +(2.62873 + 3.75363i) q^{21} +(1.91284 - 5.25548i) q^{23} +(0.327675 - 1.85834i) q^{25} +(1.87420 - 4.84638i) q^{27} +(4.88838 - 5.82574i) q^{29} +(-2.25819 - 2.69120i) q^{31} +(1.07192 - 0.246535i) q^{33} +(6.43534 - 2.60680i) q^{35} +7.03080 q^{37} +(-0.392870 - 7.61729i) q^{39} +(-4.16466 + 3.49456i) q^{41} +(-0.637408 + 0.231997i) q^{43} +(-6.51964 - 4.41331i) q^{45} +(-2.16030 - 1.81271i) q^{47} +(5.67053 + 4.10427i) q^{49} +(-12.8531 + 5.44324i) q^{51} +(3.98038 - 2.29807i) q^{53} -1.66652i q^{55} +(6.81386 + 0.842375i) q^{57} +(0.191808 + 1.08780i) q^{59} +(1.31335 - 1.56519i) q^{61} +(0.283008 - 7.93221i) q^{63} +(-11.3811 - 2.00679i) q^{65} +(12.5827 + 4.57972i) q^{67} +(-8.12853 + 5.26915i) q^{69} +(6.01970 - 3.47547i) q^{71} -4.66728i q^{73} +(-2.39220 + 2.22705i) q^{75} +(1.42403 - 0.891619i) q^{77} +(13.9442 - 5.07526i) q^{79} +(-7.64821 + 4.74392i) q^{81} +(-0.855255 - 0.717644i) q^{83} +(3.67243 + 20.8274i) q^{85} +(-12.8370 + 2.95245i) q^{87} +(-6.95336 - 12.0436i) q^{89} +(-4.37431 - 10.7987i) q^{91} +(0.313418 + 6.07682i) q^{93} +(3.55790 - 9.77524i) q^{95} +(5.47800 + 15.0507i) q^{97} +(-1.73910 - 0.777756i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 144 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 144 q - 12 q^{9} + 12 q^{11} + 12 q^{15} - 3 q^{21} - 15 q^{23} - 6 q^{29} - 42 q^{39} + 18 q^{45} - 54 q^{47} - 36 q^{49} + 18 q^{51} + 45 q^{53} + 3 q^{57} + 54 q^{61} + 39 q^{63} - 3 q^{65} + 36 q^{69} + 36 q^{71} + 93 q^{77} - 18 q^{79} - 36 q^{81} + 36 q^{85} - 18 q^{91} + 60 q^{93} + 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(e\left(\frac{13}{18}\right)\) \(e\left(\frac{5}{6}\right)\) \(1\)

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
\(3\) −1.38241 1.04352i −0.798136 0.602477i
\(4\) 0 0
\(5\) −2.01034 + 1.68687i −0.899051 + 0.754393i −0.970005 0.243086i \(-0.921840\pi\)
0.0709539 + 0.997480i \(0.477396\pi\)
\(6\) 0 0
\(7\) −2.51700 0.815311i −0.951335 0.308159i
\(8\) 0 0
\(9\) 0.822127 + 2.88515i 0.274042 + 0.961718i
\(10\) 0 0
\(11\) −0.408189 + 0.486461i −0.123074 + 0.146674i −0.824063 0.566498i \(-0.808298\pi\)
0.700989 + 0.713172i \(0.252742\pi\)
\(12\) 0 0
\(13\) 2.83064 + 3.37342i 0.785077 + 0.935619i 0.999151 0.0411989i \(-0.0131177\pi\)
−0.214074 + 0.976817i \(0.568673\pi\)
\(14\) 0 0
\(15\) 4.53941 0.234125i 1.17207 0.0604508i
\(16\) 0 0
\(17\) 4.02938 6.97910i 0.977269 1.69268i 0.305035 0.952341i \(-0.401332\pi\)
0.672234 0.740338i \(-0.265335\pi\)
\(18\) 0 0
\(19\) −3.43287 + 1.98197i −0.787554 + 0.454695i −0.839101 0.543976i \(-0.816918\pi\)
0.0515467 + 0.998671i \(0.483585\pi\)
\(20\) 0 0
\(21\) 2.62873 + 3.75363i 0.573636 + 0.819110i
\(22\) 0 0
\(23\) 1.91284 5.25548i 0.398854 1.09584i −0.563989 0.825782i \(-0.690734\pi\)
0.962843 0.270061i \(-0.0870438\pi\)
\(24\) 0 0
\(25\) 0.327675 1.85834i 0.0655351 0.371668i
\(26\) 0 0
\(27\) 1.87420 4.84638i 0.360690 0.932686i
\(28\) 0 0
\(29\) 4.88838 5.82574i 0.907749 1.08181i −0.0885680 0.996070i \(-0.528229\pi\)
0.996317 0.0857433i \(-0.0273265\pi\)
\(30\) 0 0
\(31\) −2.25819 2.69120i −0.405582 0.483354i 0.524131 0.851637i \(-0.324390\pi\)
−0.929714 + 0.368283i \(0.879946\pi\)
\(32\) 0 0
\(33\) 1.07192 0.246535i 0.186597 0.0429163i
\(34\) 0 0
\(35\) 6.43534 2.60680i 1.08777 0.440630i
\(36\) 0 0
\(37\) 7.03080 1.15586 0.577928 0.816088i \(-0.303861\pi\)
0.577928 + 0.816088i \(0.303861\pi\)
\(38\) 0 0
\(39\) −0.392870 7.61729i −0.0629095 1.21974i
\(40\) 0 0
\(41\) −4.16466 + 3.49456i −0.650410 + 0.545759i −0.907195 0.420710i \(-0.861781\pi\)
0.256785 + 0.966468i \(0.417337\pi\)
\(42\) 0 0
\(43\) −0.637408 + 0.231997i −0.0972038 + 0.0353793i −0.390164 0.920745i \(-0.627582\pi\)
0.292960 + 0.956125i \(0.405360\pi\)
\(44\) 0 0
\(45\) −6.51964 4.41331i −0.971891 0.657897i
\(46\) 0 0
\(47\) −2.16030 1.81271i −0.315112 0.264411i 0.471489 0.881872i \(-0.343717\pi\)
−0.786601 + 0.617461i \(0.788161\pi\)
\(48\) 0 0
\(49\) 5.67053 + 4.10427i 0.810076 + 0.586324i
\(50\) 0 0
\(51\) −12.8531 + 5.44324i −1.79979 + 0.762206i
\(52\) 0 0
\(53\) 3.98038 2.29807i 0.546747 0.315665i −0.201062 0.979579i \(-0.564439\pi\)
0.747809 + 0.663914i \(0.231106\pi\)
\(54\) 0 0
\(55\) 1.66652i 0.224713i
\(56\) 0 0
\(57\) 6.81386 + 0.842375i 0.902518 + 0.111575i
\(58\) 0 0
\(59\) 0.191808 + 1.08780i 0.0249713 + 0.141619i 0.994744 0.102389i \(-0.0326487\pi\)
−0.969773 + 0.244008i \(0.921538\pi\)
\(60\) 0 0
\(61\) 1.31335 1.56519i 0.168157 0.200402i −0.675384 0.737466i \(-0.736022\pi\)
0.843541 + 0.537064i \(0.180467\pi\)
\(62\) 0 0
\(63\) 0.283008 7.93221i 0.0356556 0.999364i
\(64\) 0 0
\(65\) −11.3811 2.00679i −1.41165 0.248912i
\(66\) 0 0
\(67\) 12.5827 + 4.57972i 1.53722 + 0.559502i 0.965376 0.260861i \(-0.0840065\pi\)
0.571843 + 0.820363i \(0.306229\pi\)
\(68\) 0 0
\(69\) −8.12853 + 5.26915i −0.978560 + 0.634331i
\(70\) 0 0
\(71\) 6.01970 3.47547i 0.714406 0.412463i −0.0982840 0.995158i \(-0.531335\pi\)
0.812690 + 0.582696i \(0.198002\pi\)
\(72\) 0 0
\(73\) 4.66728i 0.546264i −0.961977 0.273132i \(-0.911940\pi\)
0.961977 0.273132i \(-0.0880596\pi\)
\(74\) 0 0
\(75\) −2.39220 + 2.22705i −0.276227 + 0.257158i
\(76\) 0 0
\(77\) 1.42403 0.891619i 0.162283 0.101609i
\(78\) 0 0
\(79\) 13.9442 5.07526i 1.56884 0.571012i 0.596101 0.802909i \(-0.296716\pi\)
0.972740 + 0.231898i \(0.0744935\pi\)
\(80\) 0 0
\(81\) −7.64821 + 4.74392i −0.849802 + 0.527103i
\(82\) 0 0
\(83\) −0.855255 0.717644i −0.0938764 0.0787717i 0.594642 0.803991i \(-0.297294\pi\)
−0.688518 + 0.725219i \(0.741738\pi\)
\(84\) 0 0
\(85\) 3.67243 + 20.8274i 0.398331 + 2.25905i
\(86\) 0 0
\(87\) −12.8370 + 2.95245i −1.37628 + 0.316536i
\(88\) 0 0
\(89\) −6.95336 12.0436i −0.737055 1.27662i −0.953816 0.300392i \(-0.902883\pi\)
0.216761 0.976225i \(-0.430451\pi\)
\(90\) 0 0
\(91\) −4.37431 10.7987i −0.458552 1.13202i
\(92\) 0 0
\(93\) 0.313418 + 6.07682i 0.0325000 + 0.630136i
\(94\) 0 0
\(95\) 3.55790 9.77524i 0.365033 1.00292i
\(96\) 0 0
\(97\) 5.47800 + 15.0507i 0.556207 + 1.52817i 0.825095 + 0.564995i \(0.191122\pi\)
−0.268887 + 0.963172i \(0.586656\pi\)
\(98\) 0 0
\(99\) −1.73910 0.777756i −0.174786 0.0781674i
\(100\) 0 0
\(101\) 2.95071 1.07397i 0.293607 0.106864i −0.191017 0.981587i \(-0.561179\pi\)
0.484624 + 0.874723i \(0.338956\pi\)
\(102\) 0 0
\(103\) −6.25752 7.45742i −0.616571 0.734801i 0.363905 0.931436i \(-0.381443\pi\)
−0.980477 + 0.196635i \(0.936999\pi\)
\(104\) 0 0
\(105\) −11.6165 3.11174i −1.13366 0.303675i
\(106\) 0 0
\(107\) 11.7794 + 6.80085i 1.13876 + 0.657463i 0.946123 0.323807i \(-0.104963\pi\)
0.192637 + 0.981270i \(0.438296\pi\)
\(108\) 0 0
\(109\) −7.11868 12.3299i −0.681846 1.18099i −0.974417 0.224748i \(-0.927844\pi\)
0.292571 0.956244i \(-0.405489\pi\)
\(110\) 0 0
\(111\) −9.71946 7.33679i −0.922531 0.696377i
\(112\) 0 0
\(113\) −1.74448 + 4.79293i −0.164107 + 0.450881i −0.994303 0.106591i \(-0.966006\pi\)
0.830196 + 0.557472i \(0.188229\pi\)
\(114\) 0 0
\(115\) 5.01988 + 13.7920i 0.468106 + 1.28611i
\(116\) 0 0
\(117\) −7.40569 + 10.9402i −0.684656 + 1.01142i
\(118\) 0 0
\(119\) −15.8321 + 14.2812i −1.45132 + 1.30915i
\(120\) 0 0
\(121\) 1.84010 + 10.4357i 0.167282 + 0.948704i
\(122\) 0 0
\(123\) 9.40392 0.485018i 0.847923 0.0437326i
\(124\) 0 0
\(125\) −4.08473 7.07496i −0.365349 0.632804i
\(126\) 0 0
\(127\) −4.26064 + 7.37965i −0.378071 + 0.654838i −0.990782 0.135469i \(-0.956746\pi\)
0.612711 + 0.790307i \(0.290079\pi\)
\(128\) 0 0
\(129\) 1.12325 + 0.344432i 0.0988970 + 0.0303256i
\(130\) 0 0
\(131\) 17.0232 + 6.19594i 1.48733 + 0.541342i 0.952744 0.303775i \(-0.0982471\pi\)
0.534581 + 0.845117i \(0.320469\pi\)
\(132\) 0 0
\(133\) 10.2564 2.18975i 0.889346 0.189875i
\(134\) 0 0
\(135\) 4.40745 + 12.9044i 0.379333 + 1.11063i
\(136\) 0 0
\(137\) 0.918043 + 0.161876i 0.0784337 + 0.0138300i 0.212727 0.977112i \(-0.431765\pi\)
−0.134294 + 0.990942i \(0.542877\pi\)
\(138\) 0 0
\(139\) 5.49627 0.969140i 0.466187 0.0822014i 0.0643796 0.997925i \(-0.479493\pi\)
0.401808 + 0.915724i \(0.368382\pi\)
\(140\) 0 0
\(141\) 1.09483 + 4.76023i 0.0922011 + 0.400884i
\(142\) 0 0
\(143\) −2.79647 −0.233853
\(144\) 0 0
\(145\) 19.9578i 1.65740i
\(146\) 0 0
\(147\) −3.55612 11.5911i −0.293304 0.956019i
\(148\) 0 0
\(149\) 6.92823 1.22163i 0.567583 0.100080i 0.117510 0.993072i \(-0.462509\pi\)
0.450073 + 0.892992i \(0.351398\pi\)
\(150\) 0 0
\(151\) −11.7862 9.88980i −0.959148 0.804820i 0.0216665 0.999765i \(-0.493103\pi\)
−0.980814 + 0.194945i \(0.937547\pi\)
\(152\) 0 0
\(153\) 23.4484 + 5.88768i 1.89569 + 0.475991i
\(154\) 0 0
\(155\) 9.07944 + 1.60095i 0.729278 + 0.128591i
\(156\) 0 0
\(157\) −17.7713 + 3.13356i −1.41830 + 0.250085i −0.829642 0.558296i \(-0.811455\pi\)
−0.588660 + 0.808381i \(0.700344\pi\)
\(158\) 0 0
\(159\) −7.90062 0.976726i −0.626560 0.0774594i
\(160\) 0 0
\(161\) −9.09946 + 11.6685i −0.717138 + 0.919603i
\(162\) 0 0
\(163\) 2.76364 4.78676i 0.216465 0.374928i −0.737260 0.675609i \(-0.763881\pi\)
0.953725 + 0.300681i \(0.0972140\pi\)
\(164\) 0 0
\(165\) −1.73904 + 2.30381i −0.135384 + 0.179351i
\(166\) 0 0
\(167\) −0.905887 0.329716i −0.0700996 0.0255142i 0.306732 0.951796i \(-0.400764\pi\)
−0.376832 + 0.926282i \(0.622987\pi\)
\(168\) 0 0
\(169\) −1.11004 + 6.29535i −0.0853877 + 0.484258i
\(170\) 0 0
\(171\) −8.54053 8.27492i −0.653111 0.632799i
\(172\) 0 0
\(173\) −0.102770 + 0.582839i −0.00781348 + 0.0443124i −0.988465 0.151448i \(-0.951606\pi\)
0.980652 + 0.195760i \(0.0627175\pi\)
\(174\) 0 0
\(175\) −2.33988 + 4.41027i −0.176878 + 0.333385i
\(176\) 0 0
\(177\) 0.869983 1.70394i 0.0653920 0.128076i
\(178\) 0 0
\(179\) −13.3388 7.70116i −0.996989 0.575612i −0.0896332 0.995975i \(-0.528569\pi\)
−0.907356 + 0.420363i \(0.861903\pi\)
\(180\) 0 0
\(181\) 0.932828 + 0.538569i 0.0693366 + 0.0400315i 0.534267 0.845315i \(-0.320588\pi\)
−0.464931 + 0.885347i \(0.653921\pi\)
\(182\) 0 0
\(183\) −3.44890 + 0.793228i −0.254950 + 0.0586371i
\(184\) 0 0
\(185\) −14.1343 + 11.8601i −1.03917 + 0.871970i
\(186\) 0 0
\(187\) 1.75031 + 4.80893i 0.127995 + 0.351664i
\(188\) 0 0
\(189\) −8.66866 + 10.6703i −0.630552 + 0.776147i
\(190\) 0 0
\(191\) −0.633284 1.73993i −0.0458228 0.125897i 0.914670 0.404200i \(-0.132450\pi\)
−0.960493 + 0.278303i \(0.910228\pi\)
\(192\) 0 0
\(193\) −7.06982 + 5.93228i −0.508897 + 0.427015i −0.860741 0.509044i \(-0.829999\pi\)
0.351844 + 0.936059i \(0.385555\pi\)
\(194\) 0 0
\(195\) 13.6392 + 14.6506i 0.976724 + 1.04915i
\(196\) 0 0
\(197\) 10.0512 + 5.80304i 0.716116 + 0.413450i 0.813321 0.581815i \(-0.197657\pi\)
−0.0972057 + 0.995264i \(0.530990\pi\)
\(198\) 0 0
\(199\) −9.54008 5.50797i −0.676278 0.390449i 0.122173 0.992509i \(-0.461014\pi\)
−0.798451 + 0.602060i \(0.794347\pi\)
\(200\) 0 0
\(201\) −12.6154 19.4614i −0.889823 1.37270i
\(202\) 0 0
\(203\) −17.0538 + 10.6778i −1.19694 + 0.749436i
\(204\) 0 0
\(205\) 2.47748 14.0505i 0.173035 0.981330i
\(206\) 0 0
\(207\) 16.7355 + 1.19816i 1.16319 + 0.0832777i
\(208\) 0 0
\(209\) 0.437110 2.47897i 0.0302355 0.171474i
\(210\) 0 0
\(211\) −19.8285 7.21697i −1.36505 0.496837i −0.447437 0.894316i \(-0.647663\pi\)
−0.917611 + 0.397479i \(0.869885\pi\)
\(212\) 0 0
\(213\) −11.9484 1.47714i −0.818693 0.101212i
\(214\) 0 0
\(215\) 0.890055 1.54162i 0.0607012 0.105138i
\(216\) 0 0
\(217\) 3.48968 + 8.61487i 0.236895 + 0.584815i
\(218\) 0 0
\(219\) −4.87041 + 6.45211i −0.329112 + 0.435993i
\(220\) 0 0
\(221\) 34.9491 6.16248i 2.35093 0.414533i
\(222\) 0 0
\(223\) 23.8990 + 4.21405i 1.60040 + 0.282193i 0.901419 0.432948i \(-0.142527\pi\)
0.698979 + 0.715142i \(0.253638\pi\)
\(224\) 0 0
\(225\) 5.63098 0.582397i 0.375399 0.0388265i
\(226\) 0 0
\(227\) 1.96894 + 1.65214i 0.130683 + 0.109656i 0.705787 0.708425i \(-0.250594\pi\)
−0.575103 + 0.818081i \(0.695038\pi\)
\(228\) 0 0
\(229\) 13.3860 2.36031i 0.884571 0.155974i 0.287135 0.957890i \(-0.407297\pi\)
0.597436 + 0.801917i \(0.296186\pi\)
\(230\) 0 0
\(231\) −2.89902 0.253418i −0.190741 0.0166737i
\(232\) 0 0
\(233\) 15.1229i 0.990735i −0.868683 0.495368i \(-0.835033\pi\)
0.868683 0.495368i \(-0.164967\pi\)
\(234\) 0 0
\(235\) 7.40075 0.482771
\(236\) 0 0
\(237\) −24.5727 7.53493i −1.59617 0.489446i
\(238\) 0 0
\(239\) −19.9067 + 3.51009i −1.28766 + 0.227049i −0.775228 0.631681i \(-0.782365\pi\)
−0.512428 + 0.858730i \(0.671254\pi\)
\(240\) 0 0
\(241\) −14.0899 2.48444i −0.907613 0.160037i −0.299694 0.954035i \(-0.596884\pi\)
−0.607919 + 0.793999i \(0.707996\pi\)
\(242\) 0 0
\(243\) 15.5234 + 1.42301i 0.995825 + 0.0912864i
\(244\) 0 0
\(245\) −18.3231 + 1.31451i −1.17062 + 0.0839807i
\(246\) 0 0
\(247\) −16.4032 5.97028i −1.04371 0.379880i
\(248\) 0 0
\(249\) 0.433438 + 1.88456i 0.0274680 + 0.119429i
\(250\) 0 0
\(251\) 11.6736 20.2193i 0.736831 1.27623i −0.217085 0.976153i \(-0.569655\pi\)
0.953916 0.300075i \(-0.0970118\pi\)
\(252\) 0 0
\(253\) 1.77579 + 3.07575i 0.111643 + 0.193371i
\(254\) 0 0
\(255\) 16.6570 32.6243i 1.04310 2.04301i
\(256\) 0 0
\(257\) −2.08578 11.8291i −0.130108 0.737876i −0.978142 0.207936i \(-0.933325\pi\)
0.848035 0.529940i \(-0.177786\pi\)
\(258\) 0 0
\(259\) −17.6965 5.73229i −1.09961 0.356187i
\(260\) 0 0
\(261\) 20.8270 + 9.31422i 1.28916 + 0.576536i
\(262\) 0 0
\(263\) −10.3465 28.4269i −0.637994 1.75287i −0.657936 0.753074i \(-0.728570\pi\)
0.0199418 0.999801i \(-0.493652\pi\)
\(264\) 0 0
\(265\) −4.12535 + 11.3343i −0.253418 + 0.696261i
\(266\) 0 0
\(267\) −2.95531 + 23.9052i −0.180862 + 1.46297i
\(268\) 0 0
\(269\) −12.1233 20.9981i −0.739170 1.28028i −0.952870 0.303380i \(-0.901885\pi\)
0.213700 0.976899i \(-0.431449\pi\)
\(270\) 0 0
\(271\) 20.5884 + 11.8867i 1.25066 + 0.722067i 0.971240 0.238103i \(-0.0765255\pi\)
0.279417 + 0.960170i \(0.409859\pi\)
\(272\) 0 0
\(273\) −5.22161 + 19.4930i −0.316026 + 1.17977i
\(274\) 0 0
\(275\) 0.770256 + 0.917955i 0.0464482 + 0.0553548i
\(276\) 0 0
\(277\) 18.0535 6.57092i 1.08473 0.394808i 0.263062 0.964779i \(-0.415268\pi\)
0.821665 + 0.569970i \(0.193045\pi\)
\(278\) 0 0
\(279\) 5.90801 8.72772i 0.353703 0.522515i
\(280\) 0 0
\(281\) −4.71932 12.9662i −0.281531 0.773500i −0.997180 0.0750405i \(-0.976091\pi\)
0.715649 0.698460i \(-0.246131\pi\)
\(282\) 0 0
\(283\) −8.47923 + 23.2965i −0.504038 + 1.38483i 0.383262 + 0.923640i \(0.374801\pi\)
−0.887300 + 0.461193i \(0.847422\pi\)
\(284\) 0 0
\(285\) −15.1192 + 9.80067i −0.895581 + 0.580542i
\(286\) 0 0
\(287\) 13.3316 5.40031i 0.786938 0.318770i
\(288\) 0 0
\(289\) −23.9719 41.5205i −1.41011 2.44238i
\(290\) 0 0
\(291\) 8.13286 26.5227i 0.476757 1.55479i
\(292\) 0 0
\(293\) 1.54464 + 8.76011i 0.0902391 + 0.511771i 0.996103 + 0.0882016i \(0.0281120\pi\)
−0.905864 + 0.423570i \(0.860777\pi\)
\(294\) 0 0
\(295\) −2.22058 1.86329i −0.129287 0.108485i
\(296\) 0 0
\(297\) 1.59255 + 2.88996i 0.0924089 + 0.167693i
\(298\) 0 0
\(299\) 23.1435 8.42354i 1.33842 0.487146i
\(300\) 0 0
\(301\) 1.79350 0.0642507i 0.103376 0.00370335i
\(302\) 0 0
\(303\) −5.19981 1.59446i −0.298721 0.0915993i
\(304\) 0 0
\(305\) 5.36201i 0.307028i
\(306\) 0 0
\(307\) −19.1446 + 11.0532i −1.09264 + 0.630837i −0.934279 0.356543i \(-0.883955\pi\)
−0.158364 + 0.987381i \(0.550622\pi\)
\(308\) 0 0
\(309\) 0.868494 + 16.8391i 0.0494069 + 0.957942i
\(310\) 0 0
\(311\) 25.3382 + 9.22235i 1.43680 + 0.522951i 0.938871 0.344268i \(-0.111873\pi\)
0.497926 + 0.867220i \(0.334095\pi\)
\(312\) 0 0
\(313\) 22.5578 + 3.97755i 1.27504 + 0.224825i 0.769874 0.638195i \(-0.220319\pi\)
0.505170 + 0.863020i \(0.331430\pi\)
\(314\) 0 0
\(315\) 12.8117 + 16.4238i 0.721857 + 0.925377i
\(316\) 0 0
\(317\) −2.77225 + 3.30384i −0.155705 + 0.185562i −0.838258 0.545274i \(-0.816425\pi\)
0.682552 + 0.730837i \(0.260870\pi\)
\(318\) 0 0
\(319\) 0.838613 + 4.75601i 0.0469533 + 0.266286i
\(320\) 0 0
\(321\) −9.18719 21.6937i −0.512779 1.21082i
\(322\) 0 0
\(323\) 31.9444i 1.77744i
\(324\) 0 0
\(325\) 7.19649 4.15489i 0.399189 0.230472i
\(326\) 0 0
\(327\) −3.02558 + 24.4735i −0.167315 + 1.35339i
\(328\) 0 0
\(329\) 3.95955 + 6.32389i 0.218297 + 0.348648i
\(330\) 0 0
\(331\) 19.7527 + 16.5745i 1.08571 + 0.911018i 0.996382 0.0849845i \(-0.0270841\pi\)
0.0893266 + 0.996002i \(0.471529\pi\)
\(332\) 0 0
\(333\) 5.78021 + 20.2849i 0.316754 + 1.11161i
\(334\) 0 0
\(335\) −33.0209 + 12.0186i −1.80412 + 0.656647i
\(336\) 0 0
\(337\) 21.8311 18.3185i 1.18922 0.997871i 0.189344 0.981911i \(-0.439364\pi\)
0.999873 0.0159600i \(-0.00508043\pi\)
\(338\) 0 0
\(339\) 7.41312 4.80540i 0.402625 0.260994i
\(340\) 0 0
\(341\) 2.23093 0.120812
\(342\) 0 0
\(343\) −10.9265 14.9537i −0.589973 0.807423i
\(344\) 0 0
\(345\) 7.45271 24.3046i 0.401240 1.30852i
\(346\) 0 0
\(347\) 13.2197 + 15.7547i 0.709673 + 0.845756i 0.993584 0.113095i \(-0.0360764\pi\)
−0.283911 + 0.958851i \(0.591632\pi\)
\(348\) 0 0
\(349\) 6.15541 7.33574i 0.329492 0.392673i −0.575711 0.817653i \(-0.695275\pi\)
0.905203 + 0.424980i \(0.139719\pi\)
\(350\) 0 0
\(351\) 21.6540 7.39587i 1.15581 0.394762i
\(352\) 0 0
\(353\) 1.50081 8.51149i 0.0798798 0.453021i −0.918465 0.395503i \(-0.870570\pi\)
0.998344 0.0575179i \(-0.0183186\pi\)
\(354\) 0 0
\(355\) −6.23894 + 17.1413i −0.331129 + 0.909768i
\(356\) 0 0
\(357\) 36.7891 3.22133i 1.94709 0.170491i
\(358\) 0 0
\(359\) −7.80957 + 4.50886i −0.412174 + 0.237969i −0.691723 0.722163i \(-0.743148\pi\)
0.279550 + 0.960131i \(0.409815\pi\)
\(360\) 0 0
\(361\) −1.64361 + 2.84682i −0.0865058 + 0.149832i
\(362\) 0 0
\(363\) 8.34614 16.3467i 0.438059 0.857979i
\(364\) 0 0
\(365\) 7.87312 + 9.38281i 0.412098 + 0.491119i
\(366\) 0 0
\(367\) 13.9181 16.5869i 0.726519 0.865831i −0.268728 0.963216i \(-0.586603\pi\)
0.995247 + 0.0973847i \(0.0310477\pi\)
\(368\) 0 0
\(369\) −13.5062 9.14270i −0.703106 0.475950i
\(370\) 0 0
\(371\) −11.8922 + 2.53899i −0.617415 + 0.131818i
\(372\) 0 0
\(373\) 22.6940 19.0425i 1.17505 0.985984i 0.175051 0.984559i \(-0.443991\pi\)
0.999999 0.00142479i \(-0.000453525\pi\)
\(374\) 0 0
\(375\) −1.73609 + 14.0430i −0.0896513 + 0.725178i
\(376\) 0 0
\(377\) 33.4899 1.72482
\(378\) 0 0
\(379\) −21.5189 −1.10535 −0.552675 0.833397i \(-0.686393\pi\)
−0.552675 + 0.833397i \(0.686393\pi\)
\(380\) 0 0
\(381\) 13.5908 5.75565i 0.696277 0.294871i
\(382\) 0 0
\(383\) 10.0570 8.43886i 0.513891 0.431206i −0.348605 0.937270i \(-0.613345\pi\)
0.862496 + 0.506064i \(0.168900\pi\)
\(384\) 0 0
\(385\) −1.35873 + 4.19461i −0.0692472 + 0.213777i
\(386\) 0 0
\(387\) −1.19338 1.64829i −0.0606628 0.0837871i
\(388\) 0 0
\(389\) −11.4099 + 13.5978i −0.578507 + 0.689438i −0.973354 0.229309i \(-0.926353\pi\)
0.394846 + 0.918747i \(0.370798\pi\)
\(390\) 0 0
\(391\) −28.9709 34.5262i −1.46512 1.74607i
\(392\) 0 0
\(393\) −17.0675 26.3294i −0.860942 1.32814i
\(394\) 0 0
\(395\) −19.4712 + 33.7251i −0.979701 + 1.69689i
\(396\) 0 0
\(397\) −8.16682 + 4.71512i −0.409881 + 0.236645i −0.690739 0.723104i \(-0.742714\pi\)
0.280858 + 0.959749i \(0.409381\pi\)
\(398\) 0 0
\(399\) −16.4637 7.67567i −0.824214 0.384264i
\(400\) 0 0
\(401\) −9.21347 + 25.3138i −0.460099 + 1.26411i 0.465312 + 0.885146i \(0.345942\pi\)
−0.925411 + 0.378964i \(0.876280\pi\)
\(402\) 0 0
\(403\) 2.68645 15.2356i 0.133822 0.758941i
\(404\) 0 0
\(405\) 7.37310 22.4385i 0.366372 1.11498i
\(406\) 0 0
\(407\) −2.86990 + 3.42021i −0.142255 + 0.169533i
\(408\) 0 0
\(409\) −19.8440 23.6491i −0.981222 1.16937i −0.985550 0.169384i \(-0.945822\pi\)
0.00432797 0.999991i \(-0.498622\pi\)
\(410\) 0 0
\(411\) −1.10019 1.18178i −0.0542685 0.0582927i
\(412\) 0 0
\(413\) 0.404114 2.89437i 0.0198852 0.142423i
\(414\) 0 0
\(415\) 2.92993 0.143824
\(416\) 0 0
\(417\) −8.60942 4.39572i −0.421605 0.215259i
\(418\) 0 0
\(419\) 1.30333 1.09362i 0.0636717 0.0534269i −0.610396 0.792096i \(-0.708990\pi\)
0.674068 + 0.738669i \(0.264545\pi\)
\(420\) 0 0
\(421\) −4.04181 + 1.47110i −0.196986 + 0.0716969i −0.438629 0.898668i \(-0.644536\pi\)
0.241643 + 0.970365i \(0.422314\pi\)
\(422\) 0 0
\(423\) 3.45390 7.72307i 0.167934 0.375509i
\(424\) 0 0
\(425\) −11.6492 9.77484i −0.565069 0.474149i
\(426\) 0 0
\(427\) −4.58181 + 2.86878i −0.221729 + 0.138830i
\(428\) 0 0
\(429\) 3.86588 + 2.91818i 0.186646 + 0.140891i
\(430\) 0 0
\(431\) 3.63284 2.09742i 0.174988 0.101029i −0.409948 0.912109i \(-0.634453\pi\)
0.584936 + 0.811080i \(0.301120\pi\)
\(432\) 0 0
\(433\) 8.83087i 0.424385i −0.977228 0.212192i \(-0.931940\pi\)
0.977228 0.212192i \(-0.0680603\pi\)
\(434\) 0 0
\(435\) 20.8264 27.5899i 0.998549 1.32283i
\(436\) 0 0
\(437\) 3.84967 + 21.8325i 0.184155 + 1.04439i
\(438\) 0 0
\(439\) −3.16356 + 3.77018i −0.150988 + 0.179941i −0.836237 0.548368i \(-0.815249\pi\)
0.685248 + 0.728309i \(0.259694\pi\)
\(440\) 0 0
\(441\) −7.17955 + 19.7346i −0.341883 + 0.939742i
\(442\) 0 0
\(443\) −17.1203 3.01878i −0.813411 0.143426i −0.248557 0.968617i \(-0.579956\pi\)
−0.564854 + 0.825191i \(0.691068\pi\)
\(444\) 0 0
\(445\) 34.2946 + 12.4822i 1.62572 + 0.591714i
\(446\) 0 0
\(447\) −10.8525 5.54096i −0.513304 0.262078i
\(448\) 0 0
\(449\) 1.41048 0.814341i 0.0665647 0.0384311i −0.466348 0.884601i \(-0.654431\pi\)
0.532913 + 0.846170i \(0.321097\pi\)
\(450\) 0 0
\(451\) 3.45239i 0.162566i
\(452\) 0 0
\(453\) 5.97318 + 25.9709i 0.280644 + 1.22022i
\(454\) 0 0
\(455\) 27.0100 + 14.3302i 1.26625 + 0.671810i
\(456\) 0 0
\(457\) 29.3874 10.6961i 1.37468 0.500344i 0.454122 0.890940i \(-0.349953\pi\)
0.920562 + 0.390595i \(0.127731\pi\)
\(458\) 0 0
\(459\) −26.2715 32.6081i −1.22625 1.52202i
\(460\) 0 0
\(461\) 20.4541 + 17.1630i 0.952641 + 0.799361i 0.979740 0.200272i \(-0.0641825\pi\)
−0.0270990 + 0.999633i \(0.508627\pi\)
\(462\) 0 0
\(463\) −1.82238 10.3352i −0.0846931 0.480318i −0.997422 0.0717534i \(-0.977141\pi\)
0.912729 0.408565i \(-0.133971\pi\)
\(464\) 0 0
\(465\) −10.8809 11.6878i −0.504590 0.542007i
\(466\) 0 0
\(467\) 1.84784 + 3.20055i 0.0855078 + 0.148104i 0.905607 0.424117i \(-0.139415\pi\)
−0.820100 + 0.572221i \(0.806082\pi\)
\(468\) 0 0
\(469\) −27.9367 21.7859i −1.28999 1.00598i
\(470\) 0 0
\(471\) 27.8372 + 14.2128i 1.28267 + 0.654893i
\(472\) 0 0
\(473\) 0.147325 0.404773i 0.00677402 0.0186115i
\(474\) 0 0
\(475\) 2.55830 + 7.02887i 0.117383 + 0.322507i
\(476\) 0 0
\(477\) 9.90268 + 9.59470i 0.453412 + 0.439311i
\(478\) 0 0
\(479\) 20.0216 7.28728i 0.914812 0.332964i 0.158639 0.987337i \(-0.449289\pi\)
0.756173 + 0.654372i \(0.227067\pi\)
\(480\) 0 0
\(481\) 19.9016 + 23.7178i 0.907436 + 1.08144i
\(482\) 0 0
\(483\) 24.7555 6.63514i 1.12641 0.301910i
\(484\) 0 0
\(485\) −36.4013 21.0163i −1.65290 0.954300i
\(486\) 0 0
\(487\) −4.17289 7.22766i −0.189092 0.327516i 0.755856 0.654738i \(-0.227221\pi\)
−0.944948 + 0.327222i \(0.893888\pi\)
\(488\) 0 0
\(489\) −8.81557 + 3.73336i −0.398654 + 0.168828i
\(490\) 0 0
\(491\) −2.78557 + 7.65329i −0.125711 + 0.345388i −0.986543 0.163501i \(-0.947721\pi\)
0.860832 + 0.508889i \(0.169944\pi\)
\(492\) 0 0
\(493\) −20.9613 57.5906i −0.944048 2.59375i
\(494\) 0 0
\(495\) 4.80815 1.37009i 0.216110 0.0615808i
\(496\) 0 0
\(497\) −17.9851 + 3.83982i −0.806744 + 0.172240i
\(498\) 0 0
\(499\) 4.77318 + 27.0701i 0.213677 + 1.21182i 0.883188 + 0.469020i \(0.155393\pi\)
−0.669511 + 0.742803i \(0.733496\pi\)
\(500\) 0 0
\(501\) 0.908244 + 1.40112i 0.0405773 + 0.0625972i
\(502\) 0 0
\(503\) −0.341404 0.591329i −0.0152225 0.0263661i 0.858314 0.513125i \(-0.171512\pi\)
−0.873536 + 0.486759i \(0.838179\pi\)
\(504\) 0 0
\(505\) −4.12028 + 7.13653i −0.183350 + 0.317571i
\(506\) 0 0
\(507\) 8.10387 7.54442i 0.359905 0.335059i
\(508\) 0 0
\(509\) −40.6264 14.7868i −1.80073 0.655413i −0.998276 0.0587014i \(-0.981304\pi\)
−0.802456 0.596711i \(-0.796474\pi\)
\(510\) 0 0
\(511\) −3.80529 + 11.7475i −0.168336 + 0.519680i
\(512\) 0 0
\(513\) 3.17148 + 20.3516i 0.140024 + 0.898544i
\(514\) 0 0
\(515\) 25.1595 + 4.43629i 1.10866 + 0.195486i
\(516\) 0 0
\(517\) 1.76362 0.310974i 0.0775640 0.0136766i
\(518\) 0 0
\(519\) 0.750276 0.698481i 0.0329335 0.0306599i
\(520\) 0 0
\(521\) 33.5793 1.47114 0.735568 0.677451i \(-0.236915\pi\)
0.735568 + 0.677451i \(0.236915\pi\)
\(522\) 0 0
\(523\) 15.8523i 0.693173i −0.938018 0.346586i \(-0.887341\pi\)
0.938018 0.346586i \(-0.112659\pi\)
\(524\) 0 0
\(525\) 7.83689 3.65510i 0.342030 0.159522i
\(526\) 0 0
\(527\) −27.8813 + 4.91622i −1.21453 + 0.214154i
\(528\) 0 0
\(529\) −6.34208 5.32164i −0.275743 0.231376i
\(530\) 0 0
\(531\) −2.98078 + 1.44771i −0.129355 + 0.0628251i
\(532\) 0 0
\(533\) −23.5773 4.15731i −1.02124 0.180073i
\(534\) 0 0
\(535\) −35.1528 + 6.19839i −1.51979 + 0.267980i
\(536\) 0 0
\(537\) 10.4034 + 24.5655i 0.448940 + 1.06008i
\(538\) 0 0
\(539\) −4.31122 + 1.08317i −0.185697 + 0.0466556i
\(540\) 0 0
\(541\) −8.71319 + 15.0917i −0.374609 + 0.648842i −0.990268 0.139170i \(-0.955556\pi\)
0.615659 + 0.788012i \(0.288890\pi\)
\(542\) 0 0
\(543\) −0.727545 1.71795i −0.0312220 0.0737243i
\(544\) 0 0
\(545\) 35.1100 + 12.7790i 1.50395 + 0.547392i
\(546\) 0 0
\(547\) 5.05757 28.6829i 0.216246 1.22639i −0.662485 0.749075i \(-0.730498\pi\)
0.878731 0.477317i \(-0.158391\pi\)
\(548\) 0 0
\(549\) 5.59555 + 2.50243i 0.238812 + 0.106801i
\(550\) 0 0
\(551\) −5.23473 + 29.6876i −0.223007 + 1.26474i
\(552\) 0 0
\(553\) −39.2353 + 1.40557i −1.66846 + 0.0597711i
\(554\) 0 0
\(555\) 31.9157 1.64608i 1.35474 0.0698724i
\(556\) 0 0
\(557\) 3.93471 + 2.27170i 0.166719 + 0.0962552i 0.581038 0.813877i \(-0.302647\pi\)
−0.414319 + 0.910132i \(0.635980\pi\)
\(558\) 0 0
\(559\) −2.58689 1.49354i −0.109414 0.0631702i
\(560\) 0 0
\(561\) 2.59857 8.47441i 0.109712 0.357790i
\(562\) 0 0
\(563\) −3.68246 + 3.08995i −0.155197 + 0.130226i −0.717080 0.696991i \(-0.754522\pi\)
0.561883 + 0.827217i \(0.310077\pi\)
\(564\) 0 0
\(565\) −4.57807 12.5781i −0.192601 0.529166i
\(566\) 0 0
\(567\) 23.1183 5.70476i 0.970877 0.239577i
\(568\) 0 0
\(569\) 12.3625 + 33.9657i 0.518263 + 1.42392i 0.872432 + 0.488735i \(0.162542\pi\)
−0.354169 + 0.935181i \(0.615236\pi\)
\(570\) 0 0
\(571\) −9.86718 + 8.27954i −0.412929 + 0.346488i −0.825465 0.564453i \(-0.809087\pi\)
0.412537 + 0.910941i \(0.364643\pi\)
\(572\) 0 0
\(573\) −0.940198 + 3.06615i −0.0392773 + 0.128090i
\(574\) 0 0
\(575\) −9.13967 5.27679i −0.381151 0.220057i
\(576\) 0 0
\(577\) 5.81328 + 3.35630i 0.242010 + 0.139725i 0.616100 0.787668i \(-0.288712\pi\)
−0.374090 + 0.927392i \(0.622045\pi\)
\(578\) 0 0
\(579\) 15.9639 0.823354i 0.663435 0.0342174i
\(580\) 0 0
\(581\) 1.56757 + 2.50361i 0.0650337 + 0.103867i
\(582\) 0 0
\(583\) −0.506825 + 2.87435i −0.0209906 + 0.119043i
\(584\) 0 0
\(585\) −3.56679 34.4860i −0.147469 1.42582i
\(586\) 0 0
\(587\) 2.54009 14.4056i 0.104841 0.594581i −0.886443 0.462837i \(-0.846831\pi\)
0.991284 0.131743i \(-0.0420575\pi\)
\(588\) 0 0
\(589\) 13.0859 + 4.76289i 0.539196 + 0.196251i
\(590\) 0 0
\(591\) −7.83925 18.5108i −0.322464 0.761432i
\(592\) 0 0
\(593\) 20.4263 35.3794i 0.838809 1.45286i −0.0520821 0.998643i \(-0.516586\pi\)
0.890891 0.454217i \(-0.150081\pi\)
\(594\) 0 0
\(595\) 7.73732 55.4167i 0.317199 2.27186i
\(596\) 0 0
\(597\) 7.44064 + 17.5695i 0.304525 + 0.719074i
\(598\) 0 0
\(599\) −32.9819 + 5.81559i −1.34760 + 0.237619i −0.800442 0.599410i \(-0.795402\pi\)
−0.547160 + 0.837028i \(0.684291\pi\)
\(600\) 0 0
\(601\) −39.7569 7.01021i −1.62172 0.285953i −0.712312 0.701863i \(-0.752352\pi\)
−0.909406 + 0.415911i \(0.863463\pi\)
\(602\) 0 0
\(603\) −2.86863 + 40.0681i −0.116820 + 1.63170i
\(604\) 0 0
\(605\) −21.3030 17.8754i −0.866091 0.726737i
\(606\) 0 0
\(607\) 19.2678 3.39743i 0.782055 0.137897i 0.231654 0.972798i \(-0.425586\pi\)
0.550400 + 0.834901i \(0.314475\pi\)
\(608\) 0 0
\(609\) 34.7179 + 3.03488i 1.40684 + 0.122979i
\(610\) 0 0
\(611\) 12.4187i 0.502408i
\(612\) 0 0
\(613\) 21.0322 0.849484 0.424742 0.905314i \(-0.360365\pi\)
0.424742 + 0.905314i \(0.360365\pi\)
\(614\) 0 0
\(615\) −18.0869 + 16.8383i −0.729334 + 0.678985i
\(616\) 0 0
\(617\) −16.9370 + 2.98646i −0.681859 + 0.120230i −0.503840 0.863797i \(-0.668080\pi\)
−0.178019 + 0.984027i \(0.556969\pi\)
\(618\) 0 0
\(619\) 9.72838 + 1.71538i 0.391017 + 0.0689468i 0.365700 0.930733i \(-0.380830\pi\)
0.0253164 + 0.999679i \(0.491941\pi\)
\(620\) 0 0
\(621\) −21.8850 19.1201i −0.878215 0.767265i
\(622\) 0 0
\(623\) 7.68232 + 35.9828i 0.307786 + 1.44162i
\(624\) 0 0
\(625\) 29.0123 + 10.5596i 1.16049 + 0.422384i
\(626\) 0 0
\(627\) −3.19113 + 2.97083i −0.127441 + 0.118644i
\(628\) 0 0
\(629\) 28.3298 49.0686i 1.12958 1.95649i
\(630\) 0 0
\(631\) 13.3688 + 23.1555i 0.532204 + 0.921804i 0.999293 + 0.0375939i \(0.0119693\pi\)
−0.467089 + 0.884210i \(0.654697\pi\)
\(632\) 0 0
\(633\) 19.8801 + 30.6683i 0.790161 + 1.21895i
\(634\) 0 0
\(635\) −3.88321 22.0228i −0.154100 0.873946i
\(636\) 0 0
\(637\) 2.20579 + 30.7468i 0.0873965 + 1.21823i
\(638\) 0 0
\(639\) 14.9762 + 14.5105i 0.592450 + 0.574025i
\(640\) 0 0
\(641\) −6.68607 18.3698i −0.264084 0.725564i −0.998882 0.0472781i \(-0.984945\pi\)
0.734798 0.678286i \(-0.237277\pi\)
\(642\) 0 0
\(643\) −8.14572 + 22.3802i −0.321236 + 0.882588i 0.669010 + 0.743254i \(0.266719\pi\)
−0.990245 + 0.139334i \(0.955504\pi\)
\(644\) 0 0
\(645\) −2.83914 + 1.20236i −0.111791 + 0.0473430i
\(646\) 0 0
\(647\) 5.30690 + 9.19182i 0.208636 + 0.361368i 0.951285 0.308313i \(-0.0997644\pi\)
−0.742649 + 0.669680i \(0.766431\pi\)
\(648\) 0 0
\(649\) −0.607466 0.350721i −0.0238451 0.0137670i
\(650\) 0 0
\(651\) 4.16562 15.5509i 0.163264 0.609486i
\(652\) 0 0
\(653\) −18.0356 21.4940i −0.705787 0.841125i 0.287381 0.957816i \(-0.407215\pi\)
−0.993168 + 0.116692i \(0.962771\pi\)
\(654\) 0 0
\(655\) −44.6742 + 16.2601i −1.74557 + 0.635334i
\(656\) 0 0
\(657\) 13.4658 3.83710i 0.525352 0.149699i
\(658\) 0 0
\(659\) 14.7119 + 40.4206i 0.573094 + 1.57456i 0.799587 + 0.600551i \(0.205052\pi\)
−0.226492 + 0.974013i \(0.572726\pi\)
\(660\) 0 0
\(661\) −9.41490 + 25.8672i −0.366197 + 1.00612i 0.610598 + 0.791941i \(0.290929\pi\)
−0.976795 + 0.214177i \(0.931293\pi\)
\(662\) 0 0
\(663\) −54.7448 27.9511i −2.12611 1.08553i
\(664\) 0 0
\(665\) −16.9251 + 21.7035i −0.656326 + 0.841624i
\(666\) 0 0
\(667\) −21.2664 36.8345i −0.823438 1.42624i
\(668\) 0 0
\(669\) −28.6409 30.7647i −1.10732 1.18943i
\(670\) 0 0
\(671\) 0.225308 + 1.27779i 0.00869793 + 0.0493284i
\(672\) 0 0
\(673\) −11.0731 9.29143i −0.426836 0.358158i 0.403920 0.914794i \(-0.367647\pi\)
−0.830757 + 0.556636i \(0.812092\pi\)
\(674\) 0 0
\(675\) −8.39208 5.07094i −0.323011 0.195180i
\(676\) 0 0
\(677\) 13.6558 4.97032i 0.524836 0.191025i −0.0659950 0.997820i \(-0.521022\pi\)
0.590831 + 0.806795i \(0.298800\pi\)
\(678\) 0 0
\(679\) −1.51711 42.3488i −0.0582214 1.62520i
\(680\) 0 0
\(681\) −0.997846 4.33856i −0.0382376 0.166254i
\(682\) 0 0
\(683\) 43.5520i 1.66647i 0.552917 + 0.833236i \(0.313515\pi\)
−0.552917 + 0.833236i \(0.686485\pi\)
\(684\) 0 0
\(685\) −2.11864 + 1.22320i −0.0809491 + 0.0467360i
\(686\) 0 0
\(687\) −20.9680 10.7056i −0.799978 0.408445i
\(688\) 0 0
\(689\) 19.0194 + 6.92249i 0.724581 + 0.263726i
\(690\) 0 0
\(691\) 4.21794 + 0.743737i 0.160458 + 0.0282931i 0.253300 0.967388i \(-0.418484\pi\)
−0.0928420 + 0.995681i \(0.529595\pi\)
\(692\) 0 0
\(693\) 3.74319 + 3.37551i 0.142192 + 0.128225i
\(694\) 0 0
\(695\) −9.41454 + 11.2198i −0.357114 + 0.425592i
\(696\) 0 0
\(697\) 7.60789 + 43.1465i 0.288169 + 1.63429i
\(698\) 0 0
\(699\) −15.7811 + 20.9061i −0.596895 + 0.790741i
\(700\) 0 0
\(701\) 18.9199i 0.714595i 0.933991 + 0.357297i \(0.116302\pi\)
−0.933991 + 0.357297i \(0.883698\pi\)
\(702\) 0 0
\(703\) −24.1358 + 13.9348i −0.910299 + 0.525562i
\(704\) 0 0
\(705\) −10.2309 7.72283i −0.385317 0.290859i
\(706\) 0 0
\(707\) −8.30255 + 0.297432i −0.312250 + 0.0111861i
\(708\) 0 0
\(709\) 2.31877 + 1.94568i 0.0870831 + 0.0730714i 0.685290 0.728270i \(-0.259675\pi\)
−0.598207 + 0.801342i \(0.704120\pi\)
\(710\) 0 0
\(711\) 26.1068 + 36.0585i 0.979081 + 1.35230i
\(712\) 0 0
\(713\) −18.4631 + 6.72002i −0.691449 + 0.251667i
\(714\) 0 0
\(715\) 5.62186 4.71730i 0.210246 0.176417i
\(716\) 0 0
\(717\) 31.1821 + 15.9207i 1.16452 + 0.594568i
\(718\) 0 0
\(719\) 0.543044 0.0202521 0.0101261 0.999949i \(-0.496777\pi\)
0.0101261 + 0.999949i \(0.496777\pi\)
\(720\) 0 0
\(721\) 9.67002 + 23.8721i 0.360131 + 0.889044i
\(722\) 0 0
\(723\) 16.8855 + 18.1377i 0.627980 + 0.674547i
\(724\) 0 0
\(725\) −9.22440 10.9932i −0.342586 0.408278i
\(726\) 0 0
\(727\) −24.2349 + 28.8820i −0.898821 + 1.07117i 0.0982857 + 0.995158i \(0.468664\pi\)
−0.997107 + 0.0760149i \(0.975780\pi\)
\(728\) 0 0
\(729\) −19.9748 18.1662i −0.739806 0.672821i
\(730\) 0 0
\(731\) −0.949227 + 5.38334i −0.0351084 + 0.199110i
\(732\) 0 0
\(733\) 3.08763 8.48319i 0.114044 0.313334i −0.869519 0.493900i \(-0.835571\pi\)
0.983563 + 0.180566i \(0.0577931\pi\)
\(734\) 0 0
\(735\) 26.7018 + 17.3033i 0.984910 + 0.638243i
\(736\) 0 0
\(737\) −7.36397 + 4.25159i −0.271255 + 0.156609i
\(738\) 0 0
\(739\) −9.82993 + 17.0259i −0.361600 + 0.626309i −0.988224 0.153012i \(-0.951103\pi\)
0.626624 + 0.779321i \(0.284436\pi\)
\(740\) 0 0
\(741\) 16.4459 + 25.3705i 0.604155 + 0.932008i
\(742\) 0 0
\(743\) −7.89524 9.40918i −0.289648 0.345189i 0.601524 0.798855i \(-0.294561\pi\)
−0.891172 + 0.453665i \(0.850116\pi\)
\(744\) 0 0
\(745\) −11.8673 + 14.1430i −0.434786 + 0.518158i
\(746\) 0 0
\(747\) 1.36738 3.05754i 0.0500300 0.111869i
\(748\) 0 0
\(749\) −24.1039 26.7216i −0.880739 0.976387i
\(750\) 0 0
\(751\) −11.7750 + 9.88037i −0.429674 + 0.360540i −0.831829 0.555032i \(-0.812706\pi\)
0.402154 + 0.915572i \(0.368262\pi\)
\(752\) 0 0
\(753\) −37.2369 + 15.7697i −1.35699 + 0.574680i
\(754\) 0 0
\(755\) 40.3771 1.46947
\(756\) 0 0
\(757\) 19.2152 0.698390 0.349195 0.937050i \(-0.386455\pi\)
0.349195 + 0.937050i \(0.386455\pi\)
\(758\) 0 0
\(759\) 0.754743 6.10502i 0.0273955 0.221598i
\(760\) 0 0
\(761\) −25.5250 + 21.4180i −0.925279 + 0.776401i −0.974964 0.222364i \(-0.928623\pi\)
0.0496848 + 0.998765i \(0.484178\pi\)
\(762\) 0 0
\(763\) 7.86497 + 36.8383i 0.284731 + 1.33364i
\(764\) 0 0
\(765\) −57.0711 + 27.7183i −2.06341 + 1.00216i
\(766\) 0 0
\(767\) −3.12667 + 3.72621i −0.112897 + 0.134546i
\(768\) 0 0
\(769\) −1.96014 2.33600i −0.0706843 0.0842383i 0.729543 0.683934i \(-0.239733\pi\)
−0.800228 + 0.599696i \(0.795288\pi\)
\(770\) 0 0
\(771\) −9.46046 + 18.5292i −0.340710 + 0.667313i
\(772\) 0 0
\(773\) −11.9954 + 20.7767i −0.431445 + 0.747285i −0.996998 0.0774272i \(-0.975329\pi\)
0.565553 + 0.824712i \(0.308663\pi\)
\(774\) 0 0
\(775\) −5.74112 + 3.31464i −0.206227 + 0.119065i
\(776\) 0 0
\(777\) 18.4821 + 26.3911i 0.663041 + 0.946774i
\(778\) 0 0
\(779\) 7.37061 20.2506i 0.264080 0.725553i
\(780\) 0 0
\(781\) −0.766493 + 4.34700i −0.0274273 + 0.155548i
\(782\) 0 0
\(783\) −19.0720 34.6095i −0.681576 1.23684i
\(784\) 0 0
\(785\) 30.4404 36.2774i 1.08646 1.29480i
\(786\) 0 0
\(787\) 9.36700 + 11.1632i 0.333897 + 0.397924i 0.906704 0.421767i \(-0.138590\pi\)
−0.572807 + 0.819690i \(0.694145\pi\)
\(788\) 0 0
\(789\) −15.3609 + 50.0945i −0.546861 + 1.78341i
\(790\) 0 0
\(791\) 8.29859 10.6415i 0.295064 0.378368i
\(792\) 0 0
\(793\) 8.99765 0.319516
\(794\) 0 0
\(795\) 17.5305 11.3638i 0.621744 0.403032i
\(796\) 0 0
\(797\) −32.9880 + 27.6802i −1.16850 + 0.980484i −0.999986 0.00521632i \(-0.998340\pi\)
−0.168509 + 0.985700i \(0.553895\pi\)
\(798\) 0 0
\(799\) −21.3557 + 7.77285i −0.755512 + 0.274984i
\(800\) 0 0
\(801\) 29.0310 29.9629i 1.02576 1.05869i
\(802\) 0 0
\(803\) 2.27045 + 1.90513i 0.0801224 + 0.0672307i
\(804\) 0 0
\(805\) −1.39024 38.8072i −0.0489994 1.36777i
\(806\) 0 0
\(807\) −5.15263 + 41.6790i −0.181381 + 1.46717i
\(808\) 0 0
\(809\) 11.1160 6.41785i 0.390819 0.225640i −0.291696 0.956511i \(-0.594219\pi\)
0.682515 + 0.730871i \(0.260886\pi\)
\(810\) 0 0
\(811\) 17.1111i 0.600851i 0.953805 + 0.300425i \(0.0971286\pi\)
−0.953805 + 0.300425i \(0.902871\pi\)
\(812\) 0 0
\(813\) −16.0576 37.9168i −0.563165 1.32980i
\(814\) 0 0
\(815\) 2.51882 + 14.2849i 0.0882303 + 0.500379i
\(816\) 0 0
\(817\) 1.72833 2.05974i 0.0604664 0.0720611i
\(818\) 0 0
\(819\) 27.5598 21.4985i 0.963016 0.751218i
\(820\) 0 0
\(821\) −35.1941 6.20568i −1.22828 0.216580i −0.478396 0.878144i \(-0.658782\pi\)
−0.749888 + 0.661565i \(0.769893\pi\)
\(822\) 0 0
\(823\) −13.5867 4.94517i −0.473604 0.172378i 0.0941802 0.995555i \(-0.469977\pi\)
−0.567784 + 0.823177i \(0.692199\pi\)
\(824\) 0 0
\(825\) −0.106905 2.07277i −0.00372197 0.0721646i
\(826\) 0 0
\(827\) −11.6145 + 6.70561i −0.403874 + 0.233177i −0.688154 0.725564i \(-0.741579\pi\)
0.284280 + 0.958741i \(0.408245\pi\)
\(828\) 0 0
\(829\) 48.7434i 1.69293i 0.532446 + 0.846464i \(0.321273\pi\)
−0.532446 + 0.846464i \(0.678727\pi\)
\(830\) 0 0
\(831\) −31.8142 9.75545i −1.10362 0.338413i
\(832\) 0 0
\(833\) 51.4929 23.0375i 1.78412 0.798203i
\(834\) 0 0
\(835\) 2.37733 0.865277i 0.0822709 0.0299441i
\(836\) 0 0
\(837\) −17.2749 + 5.90018i −0.597107 + 0.203940i
\(838\) 0 0
\(839\) −12.0259 10.0909i −0.415179 0.348377i 0.411146 0.911569i \(-0.365129\pi\)
−0.826326 + 0.563192i \(0.809573\pi\)
\(840\) 0 0
\(841\) −5.00724 28.3975i −0.172663 0.979223i
\(842\) 0 0
\(843\) −7.00649 + 22.8494i −0.241316 + 0.786974i
\(844\) 0 0
\(845\) −8.38791 14.5283i −0.288553 0.499788i
\(846\) 0 0
\(847\) 3.87685 27.7670i 0.133210 0.954085i
\(848\) 0 0
\(849\) 36.0322 23.3571i 1.23662 0.801613i
\(850\) 0 0
\(851\) 13.4488 36.9502i 0.461018 1.26664i
\(852\) 0 0
\(853\) −7.29133 20.0328i −0.249650 0.685909i −0.999699 0.0245263i \(-0.992192\pi\)
0.750049 0.661382i \(-0.230030\pi\)
\(854\) 0 0
\(855\) 31.1281 + 2.22859i 1.06456 + 0.0762160i
\(856\) 0 0
\(857\) −4.05119 + 1.47451i −0.138386 + 0.0503683i −0.410285 0.911958i \(-0.634571\pi\)
0.271899 + 0.962326i \(0.412348\pi\)
\(858\) 0 0
\(859\) 20.1834 + 24.0537i 0.688650 + 0.820701i 0.991192 0.132435i \(-0.0422797\pi\)
−0.302542 + 0.953136i \(0.597835\pi\)
\(860\) 0 0
\(861\) −24.0651 6.44634i −0.820135 0.219691i
\(862\) 0 0
\(863\) −22.4488 12.9608i −0.764165 0.441191i 0.0666239 0.997778i \(-0.478777\pi\)
−0.830789 + 0.556587i \(0.812111\pi\)
\(864\) 0 0
\(865\) −0.776573 1.34506i −0.0264043 0.0457336i
\(866\) 0 0
\(867\) −10.1885 + 82.4136i −0.346020 + 2.79891i
\(868\) 0 0
\(869\) −3.22294 + 8.85496i −0.109331 + 0.300384i
\(870\) 0 0
\(871\) 20.1677 + 55.4102i 0.683355 + 1.87750i
\(872\) 0 0
\(873\) −38.9199 + 28.1785i −1.31724 + 0.953696i
\(874\) 0 0
\(875\) 4.51295 + 21.1380i 0.152566 + 0.714594i
\(876\) 0 0
\(877\) −8.75989 49.6798i −0.295800 1.67757i −0.663931 0.747794i \(-0.731113\pi\)
0.368131 0.929774i \(-0.379998\pi\)
\(878\) 0 0
\(879\) 7.00603 13.7220i 0.236307 0.462830i
\(880\) 0 0
\(881\) 13.2598 + 22.9666i 0.446734 + 0.773766i 0.998171 0.0604506i \(-0.0192538\pi\)
−0.551437 + 0.834216i \(0.685920\pi\)
\(882\) 0 0
\(883\) 17.7664 30.7722i 0.597885 1.03557i −0.395247 0.918575i \(-0.629341\pi\)
0.993133 0.116993i \(-0.0373255\pi\)
\(884\) 0 0
\(885\) 1.12538 + 4.89306i 0.0378291 + 0.164478i
\(886\) 0 0
\(887\) −9.20249 3.34943i −0.308989 0.112463i 0.182871 0.983137i \(-0.441461\pi\)
−0.491861 + 0.870674i \(0.663683\pi\)
\(888\) 0 0
\(889\) 16.7407 15.1008i 0.561466 0.506464i
\(890\) 0 0
\(891\) 0.814184 5.65698i 0.0272762 0.189516i
\(892\) 0 0
\(893\) 11.0088 + 1.94114i 0.368394 + 0.0649578i
\(894\) 0 0
\(895\) 39.8064 7.01895i 1.33058 0.234617i
\(896\) 0 0
\(897\) −40.7840 12.5059i −1.36174 0.417560i
\(898\) 0 0
\(899\) −26.7171 −0.891066
\(900\) 0 0
\(901\) 37.0393i 1.23396i
\(902\) 0 0
\(903\) −2.54641 1.78274i −0.0847391 0.0593258i
\(904\) 0 0
\(905\) −2.78380 + 0.490859i −0.0925366 + 0.0163167i
\(906\) 0 0
\(907\) 9.00516 + 7.55623i 0.299011 + 0.250900i 0.779933 0.625863i \(-0.215253\pi\)
−0.480921 + 0.876764i \(0.659698\pi\)
\(908\) 0 0
\(909\) 5.52443 + 7.63031i 0.183234 + 0.253082i
\(910\) 0 0
\(911\) 56.5365 + 9.96892i 1.87314 + 0.330285i 0.990250 0.139298i \(-0.0444846\pi\)
0.882888 + 0.469583i \(0.155596\pi\)
\(912\) 0 0
\(913\) 0.698212 0.123114i 0.0231074 0.00407446i
\(914\) 0 0
\(915\) 5.59538 7.41251i 0.184977 0.245050i
\(916\) 0 0
\(917\) −37.7957 29.4744i −1.24813 0.973330i
\(918\) 0 0
\(919\) 6.33126 10.9661i 0.208849 0.361737i −0.742503 0.669843i \(-0.766362\pi\)
0.951352 + 0.308105i \(0.0996949\pi\)
\(920\) 0 0
\(921\) 38.0000 + 4.69781i 1.25214 + 0.154798i
\(922\) 0 0
\(923\) 28.7638 + 10.4692i 0.946772 + 0.344597i
\(924\) 0 0
\(925\) 2.30382 13.0656i 0.0757491 0.429595i
\(926\) 0 0
\(927\) 16.3713 24.1848i 0.537705 0.794334i
\(928\) 0 0
\(929\) 7.28069 41.2909i 0.238872 1.35471i −0.595433 0.803405i \(-0.703019\pi\)
0.834304 0.551304i \(-0.185870\pi\)
\(930\) 0 0
\(931\) −27.6007 2.85061i −0.904577 0.0934248i
\(932\) 0 0
\(933\) −25.4041 39.1900i −0.831693 1.28302i
\(934\) 0 0
\(935\) −11.6308 6.71503i −0.380367 0.219605i
\(936\) 0 0
\(937\) 32.3294 + 18.6654i 1.05616 + 0.609772i 0.924367 0.381505i \(-0.124594\pi\)
0.131790 + 0.991278i \(0.457928\pi\)
\(938\) 0 0
\(939\) −27.0336 29.0382i −0.882207 0.947626i
\(940\) 0 0
\(941\) 19.1474 16.0666i 0.624187 0.523755i −0.274930 0.961464i \(-0.588655\pi\)
0.899116 + 0.437710i \(0.144210\pi\)
\(942\) 0 0
\(943\) 10.3993 + 28.5718i 0.338647 + 0.930426i
\(944\) 0 0
\(945\) −0.572440 36.0738i −0.0186215 1.17348i
\(946\) 0 0
\(947\) 8.35293 + 22.9495i 0.271434 + 0.745758i 0.998262 + 0.0589382i \(0.0187715\pi\)
−0.726828 + 0.686819i \(0.759006\pi\)
\(948\) 0 0
\(949\) 15.7447 13.2114i 0.511095 0.428859i
\(950\) 0 0
\(951\) 7.28003 1.67437i 0.236071 0.0542951i
\(952\) 0 0
\(953\) 21.6346 + 12.4907i 0.700812 + 0.404614i 0.807650 0.589662i \(-0.200739\pi\)
−0.106838 + 0.994276i \(0.534073\pi\)
\(954\) 0 0
\(955\) 4.20816 + 2.42958i 0.136173 + 0.0786195i
\(956\) 0 0
\(957\) 3.80369 7.44988i 0.122956 0.240820i
\(958\) 0 0
\(959\) −2.17873 1.15593i −0.0703549 0.0373270i
\(960\) 0 0
\(961\) 3.23993 18.3746i 0.104514 0.592728i
\(962\) 0 0
\(963\) −9.93732 + 39.5766i −0.320226 + 1.27534i
\(964\) 0 0
\(965\) 4.20571 23.8518i 0.135387 0.767816i
\(966\) 0 0
\(967\) −2.06564 0.751830i −0.0664263 0.0241772i 0.308593 0.951194i \(-0.400142\pi\)
−0.375020 + 0.927017i \(0.622364\pi\)
\(968\) 0 0
\(969\) 33.3347 44.1604i 1.07086 1.41864i
\(970\) 0 0
\(971\) −13.9368 + 24.1393i −0.447253 + 0.774666i −0.998206 0.0598708i \(-0.980931\pi\)
0.550953 + 0.834536i \(0.314264\pi\)
\(972\) 0 0
\(973\) −14.6242 2.04185i −0.468831 0.0654586i
\(974\) 0 0
\(975\) −14.2842 1.76591i −0.457462 0.0565544i
\(976\) 0 0
\(977\) −7.56835 + 1.33451i −0.242133 + 0.0426946i −0.293398 0.955991i \(-0.594786\pi\)
0.0512645 + 0.998685i \(0.483675\pi\)
\(978\) 0 0
\(979\) 8.69702 + 1.53352i 0.277958 + 0.0490115i
\(980\) 0 0
\(981\) 29.7212 30.6752i 0.948926 0.979385i
\(982\) 0 0
\(983\) 47.7893 + 40.1000i 1.52424 + 1.27899i 0.827095 + 0.562062i \(0.189992\pi\)
0.697147 + 0.716929i \(0.254453\pi\)
\(984\) 0 0
\(985\) −29.9952 + 5.28897i −0.955728 + 0.168521i
\(986\) 0 0
\(987\) 1.12539 12.8741i 0.0358216 0.409787i
\(988\) 0 0
\(989\) 3.79366i 0.120631i
\(990\) 0 0
\(991\) −17.6865 −0.561829 −0.280914 0.959733i \(-0.590638\pi\)
−0.280914 + 0.959733i \(0.590638\pi\)
\(992\) 0 0
\(993\) −10.0106 43.5252i −0.317676 1.38123i
\(994\) 0 0
\(995\) 28.4700 5.02003i 0.902561 0.159146i
\(996\) 0 0
\(997\) 18.3435 + 3.23445i 0.580943 + 0.102436i 0.456394 0.889778i \(-0.349141\pi\)
0.124549 + 0.992213i \(0.460252\pi\)
\(998\) 0 0
\(999\) 13.1771 34.0739i 0.416906 1.07805i
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 756.2.ca.a.173.3 144
7.3 odd 6 756.2.ck.a.605.13 yes 144
27.5 odd 18 756.2.ck.a.5.13 yes 144
189.59 even 18 inner 756.2.ca.a.437.3 yes 144
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
756.2.ca.a.173.3 144 1.1 even 1 trivial
756.2.ca.a.437.3 yes 144 189.59 even 18 inner
756.2.ck.a.5.13 yes 144 27.5 odd 18
756.2.ck.a.605.13 yes 144 7.3 odd 6