Properties

Label 76.4.i.a.9.2
Level $76$
Weight $4$
Character 76.9
Analytic conductor $4.484$
Analytic rank $0$
Dimension $30$
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] = [76,4,Mod(5,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([0, 16]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 76.i (of order \(9\), 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: \(4.48414516044\)
Analytic rank: \(0\)
Dimension: \(30\)
Relative dimension: \(5\) over \(\Q(\zeta_{9})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 9.2
Character \(\chi\) \(=\) 76.9
Dual form 76.4.i.a.17.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.65540 + 1.38904i) q^{3} +(8.99353 + 3.27338i) q^{5} +(-0.0466734 + 0.0808407i) q^{7} +(-3.87760 + 21.9910i) q^{9} +O(q^{10})\) \(q+(-1.65540 + 1.38904i) q^{3} +(8.99353 + 3.27338i) q^{5} +(-0.0466734 + 0.0808407i) q^{7} +(-3.87760 + 21.9910i) q^{9} +(14.3982 + 24.9384i) q^{11} +(61.7744 + 51.8349i) q^{13} +(-19.4347 + 7.07365i) q^{15} +(2.24768 + 12.7472i) q^{17} +(-8.98203 - 82.3306i) q^{19} +(-0.0350282 - 0.198655i) q^{21} +(-31.6580 + 11.5226i) q^{23} +(-25.5870 - 21.4700i) q^{25} +(-53.3005 - 92.3191i) q^{27} +(9.33650 - 52.9499i) q^{29} +(20.3029 - 35.1657i) q^{31} +(-58.4752 - 21.2832i) q^{33} +(-0.684380 + 0.574263i) q^{35} +8.12775 q^{37} -174.262 q^{39} +(196.821 - 165.152i) q^{41} +(-418.967 - 152.492i) q^{43} +(-106.858 + 185.084i) q^{45} +(64.4398 - 365.456i) q^{47} +(171.496 + 297.039i) q^{49} +(-21.4272 - 17.9796i) q^{51} +(423.366 - 154.093i) q^{53} +(47.8578 + 271.415i) q^{55} +(129.229 + 123.813i) q^{57} +(35.0142 + 198.575i) q^{59} +(-209.354 + 76.1986i) q^{61} +(-1.59678 - 1.33986i) q^{63} +(385.895 + 668.389i) q^{65} +(-26.2614 + 148.936i) q^{67} +(36.4012 - 63.0486i) q^{69} +(341.051 + 124.132i) q^{71} +(-2.86543 + 2.40438i) q^{73} +72.1794 q^{75} -2.68805 q^{77} +(-573.251 + 481.015i) q^{79} +(-350.087 - 127.421i) q^{81} +(355.657 - 616.017i) q^{83} +(-21.5119 + 122.000i) q^{85} +(58.0940 + 100.622i) q^{87} +(605.237 + 507.854i) q^{89} +(-7.07358 + 2.57457i) q^{91} +(15.2373 + 86.4148i) q^{93} +(188.719 - 769.844i) q^{95} +(-176.963 - 1003.61i) q^{97} +(-604.250 + 219.929i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 30 q - 3 q^{3} + 6 q^{7} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 30 q - 3 q^{3} + 6 q^{7} + 15 q^{9} + 42 q^{11} - 42 q^{13} + 78 q^{15} + 30 q^{17} + 282 q^{19} + 198 q^{21} - 300 q^{23} - 276 q^{25} + 219 q^{27} + 216 q^{29} + 30 q^{31} - 597 q^{33} - 636 q^{35} + 60 q^{37} - 2172 q^{39} - 63 q^{41} - 246 q^{43} - 882 q^{45} + 762 q^{47} - 525 q^{49} + 2613 q^{51} + 882 q^{53} + 1350 q^{55} + 924 q^{57} + 2085 q^{59} + 1530 q^{61} + 2424 q^{63} + 1530 q^{65} - 3609 q^{67} + 756 q^{69} - 4962 q^{71} - 2394 q^{73} - 3516 q^{77} - 630 q^{79} - 3723 q^{81} - 2382 q^{83} + 3228 q^{85} - 1110 q^{87} + 2196 q^{89} + 6036 q^{91} + 5010 q^{93} + 6204 q^{95} + 6459 q^{97} + 6189 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(e\left(\frac{4}{9}\right)\) \(1\)

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
\(3\) −1.65540 + 1.38904i −0.318581 + 0.267321i −0.788028 0.615639i \(-0.788898\pi\)
0.469447 + 0.882961i \(0.344453\pi\)
\(4\) 0 0
\(5\) 8.99353 + 3.27338i 0.804406 + 0.292780i 0.711311 0.702878i \(-0.248102\pi\)
0.0930948 + 0.995657i \(0.470324\pi\)
\(6\) 0 0
\(7\) −0.0466734 + 0.0808407i −0.00252013 + 0.00436499i −0.867283 0.497816i \(-0.834136\pi\)
0.864763 + 0.502181i \(0.167469\pi\)
\(8\) 0 0
\(9\) −3.87760 + 21.9910i −0.143615 + 0.814481i
\(10\) 0 0
\(11\) 14.3982 + 24.9384i 0.394656 + 0.683564i 0.993057 0.117632i \(-0.0375304\pi\)
−0.598401 + 0.801197i \(0.704197\pi\)
\(12\) 0 0
\(13\) 61.7744 + 51.8349i 1.31793 + 1.10588i 0.986739 + 0.162315i \(0.0518960\pi\)
0.331194 + 0.943563i \(0.392548\pi\)
\(14\) 0 0
\(15\) −19.4347 + 7.07365i −0.334535 + 0.121761i
\(16\) 0 0
\(17\) 2.24768 + 12.7472i 0.0320672 + 0.181862i 0.996634 0.0819754i \(-0.0261229\pi\)
−0.964567 + 0.263837i \(0.915012\pi\)
\(18\) 0 0
\(19\) −8.98203 82.3306i −0.108454 0.994102i
\(20\) 0 0
\(21\) −0.0350282 0.198655i −0.000363989 0.00206428i
\(22\) 0 0
\(23\) −31.6580 + 11.5226i −0.287006 + 0.104462i −0.481512 0.876440i \(-0.659912\pi\)
0.194505 + 0.980901i \(0.437690\pi\)
\(24\) 0 0
\(25\) −25.5870 21.4700i −0.204696 0.171760i
\(26\) 0 0
\(27\) −53.3005 92.3191i −0.379914 0.658030i
\(28\) 0 0
\(29\) 9.33650 52.9499i 0.0597843 0.339053i −0.940215 0.340583i \(-0.889376\pi\)
0.999999 + 0.00152923i \(0.000486771\pi\)
\(30\) 0 0
\(31\) 20.3029 35.1657i 0.117630 0.203740i −0.801198 0.598399i \(-0.795804\pi\)
0.918828 + 0.394659i \(0.129137\pi\)
\(32\) 0 0
\(33\) −58.4752 21.2832i −0.308461 0.112271i
\(34\) 0 0
\(35\) −0.684380 + 0.574263i −0.00330518 + 0.00277338i
\(36\) 0 0
\(37\) 8.12775 0.0361133 0.0180567 0.999837i \(-0.494252\pi\)
0.0180567 + 0.999837i \(0.494252\pi\)
\(38\) 0 0
\(39\) −174.262 −0.715493
\(40\) 0 0
\(41\) 196.821 165.152i 0.749714 0.629085i −0.185713 0.982604i \(-0.559459\pi\)
0.935427 + 0.353519i \(0.115015\pi\)
\(42\) 0 0
\(43\) −418.967 152.492i −1.48586 0.540808i −0.533502 0.845799i \(-0.679124\pi\)
−0.952355 + 0.304991i \(0.901347\pi\)
\(44\) 0 0
\(45\) −106.858 + 185.084i −0.353988 + 0.613125i
\(46\) 0 0
\(47\) 64.4398 365.456i 0.199990 1.13420i −0.705142 0.709066i \(-0.749117\pi\)
0.905132 0.425131i \(-0.139772\pi\)
\(48\) 0 0
\(49\) 171.496 + 297.039i 0.499987 + 0.866003i
\(50\) 0 0
\(51\) −21.4272 17.9796i −0.0588316 0.0493655i
\(52\) 0 0
\(53\) 423.366 154.093i 1.09724 0.399363i 0.270943 0.962596i \(-0.412665\pi\)
0.826299 + 0.563232i \(0.190442\pi\)
\(54\) 0 0
\(55\) 47.8578 + 271.415i 0.117330 + 0.665410i
\(56\) 0 0
\(57\) 129.229 + 123.813i 0.300296 + 0.287710i
\(58\) 0 0
\(59\) 35.0142 + 198.575i 0.0772620 + 0.438175i 0.998760 + 0.0497908i \(0.0158555\pi\)
−0.921498 + 0.388384i \(0.873033\pi\)
\(60\) 0 0
\(61\) −209.354 + 76.1986i −0.439427 + 0.159938i −0.552253 0.833677i \(-0.686232\pi\)
0.112827 + 0.993615i \(0.464010\pi\)
\(62\) 0 0
\(63\) −1.59678 1.33986i −0.00319327 0.00267947i
\(64\) 0 0
\(65\) 385.895 + 668.389i 0.736375 + 1.27544i
\(66\) 0 0
\(67\) −26.2614 + 148.936i −0.0478858 + 0.271574i −0.999344 0.0362028i \(-0.988474\pi\)
0.951459 + 0.307776i \(0.0995849\pi\)
\(68\) 0 0
\(69\) 36.4012 63.0486i 0.0635099 0.110002i
\(70\) 0 0
\(71\) 341.051 + 124.132i 0.570075 + 0.207490i 0.610943 0.791674i \(-0.290790\pi\)
−0.0408687 + 0.999165i \(0.513013\pi\)
\(72\) 0 0
\(73\) −2.86543 + 2.40438i −0.00459416 + 0.00385496i −0.645082 0.764114i \(-0.723177\pi\)
0.640488 + 0.767968i \(0.278732\pi\)
\(74\) 0 0
\(75\) 72.1794 0.111127
\(76\) 0 0
\(77\) −2.68805 −0.00397833
\(78\) 0 0
\(79\) −573.251 + 481.015i −0.816403 + 0.685043i −0.952127 0.305703i \(-0.901108\pi\)
0.135724 + 0.990747i \(0.456664\pi\)
\(80\) 0 0
\(81\) −350.087 127.421i −0.480230 0.174789i
\(82\) 0 0
\(83\) 355.657 616.017i 0.470343 0.814658i −0.529082 0.848571i \(-0.677464\pi\)
0.999425 + 0.0339128i \(0.0107969\pi\)
\(84\) 0 0
\(85\) −21.5119 + 122.000i −0.0274505 + 0.155679i
\(86\) 0 0
\(87\) 58.0940 + 100.622i 0.0715901 + 0.123998i
\(88\) 0 0
\(89\) 605.237 + 507.854i 0.720843 + 0.604859i 0.927618 0.373530i \(-0.121853\pi\)
−0.206776 + 0.978388i \(0.566297\pi\)
\(90\) 0 0
\(91\) −7.07358 + 2.57457i −0.00814850 + 0.00296581i
\(92\) 0 0
\(93\) 15.2373 + 86.4148i 0.0169896 + 0.0963526i
\(94\) 0 0
\(95\) 188.719 769.844i 0.203812 0.831414i
\(96\) 0 0
\(97\) −176.963 1003.61i −0.185236 1.05052i −0.925652 0.378375i \(-0.876483\pi\)
0.740417 0.672148i \(-0.234628\pi\)
\(98\) 0 0
\(99\) −604.250 + 219.929i −0.613429 + 0.223270i
\(100\) 0 0
\(101\) −1282.37 1076.04i −1.26338 1.06010i −0.995314 0.0966975i \(-0.969172\pi\)
−0.268063 0.963401i \(-0.586383\pi\)
\(102\) 0 0
\(103\) 262.718 + 455.041i 0.251324 + 0.435307i 0.963891 0.266298i \(-0.0858006\pi\)
−0.712566 + 0.701605i \(0.752467\pi\)
\(104\) 0 0
\(105\) 0.335244 1.90127i 0.000311586 0.00176709i
\(106\) 0 0
\(107\) 908.450 1573.48i 0.820777 1.42163i −0.0843270 0.996438i \(-0.526874\pi\)
0.905104 0.425190i \(-0.139793\pi\)
\(108\) 0 0
\(109\) −1651.34 601.040i −1.45110 0.528157i −0.508202 0.861238i \(-0.669690\pi\)
−0.942898 + 0.333081i \(0.891912\pi\)
\(110\) 0 0
\(111\) −13.4546 + 11.2898i −0.0115050 + 0.00965386i
\(112\) 0 0
\(113\) 898.636 0.748111 0.374055 0.927406i \(-0.377967\pi\)
0.374055 + 0.927406i \(0.377967\pi\)
\(114\) 0 0
\(115\) −322.435 −0.261454
\(116\) 0 0
\(117\) −1379.44 + 1157.48i −1.08999 + 0.914611i
\(118\) 0 0
\(119\) −1.13540 0.413252i −0.000874638 0.000318342i
\(120\) 0 0
\(121\) 250.884 434.545i 0.188493 0.326480i
\(122\) 0 0
\(123\) −96.4130 + 546.785i −0.0706770 + 0.400829i
\(124\) 0 0
\(125\) −758.007 1312.91i −0.542386 0.939440i
\(126\) 0 0
\(127\) −269.896 226.469i −0.188578 0.158235i 0.543611 0.839337i \(-0.317057\pi\)
−0.732189 + 0.681102i \(0.761501\pi\)
\(128\) 0 0
\(129\) 905.373 329.529i 0.617935 0.224910i
\(130\) 0 0
\(131\) 474.362 + 2690.24i 0.316375 + 1.79425i 0.564402 + 0.825500i \(0.309107\pi\)
−0.248027 + 0.968753i \(0.579782\pi\)
\(132\) 0 0
\(133\) 7.07488 + 3.11653i 0.00461256 + 0.00203186i
\(134\) 0 0
\(135\) −177.164 1004.75i −0.112947 0.640554i
\(136\) 0 0
\(137\) 2083.77 758.431i 1.29948 0.472972i 0.402653 0.915353i \(-0.368088\pi\)
0.896827 + 0.442381i \(0.145866\pi\)
\(138\) 0 0
\(139\) 587.348 + 492.843i 0.358404 + 0.300737i 0.804154 0.594421i \(-0.202619\pi\)
−0.445750 + 0.895157i \(0.647063\pi\)
\(140\) 0 0
\(141\) 400.961 + 694.484i 0.239482 + 0.414795i
\(142\) 0 0
\(143\) −403.239 + 2286.88i −0.235808 + 1.33733i
\(144\) 0 0
\(145\) 257.293 445.645i 0.147359 0.255233i
\(146\) 0 0
\(147\) −696.493 253.503i −0.390788 0.142235i
\(148\) 0 0
\(149\) −1020.29 + 856.123i −0.560975 + 0.470714i −0.878637 0.477490i \(-0.841547\pi\)
0.317662 + 0.948204i \(0.397102\pi\)
\(150\) 0 0
\(151\) 2504.18 1.34959 0.674793 0.738007i \(-0.264233\pi\)
0.674793 + 0.738007i \(0.264233\pi\)
\(152\) 0 0
\(153\) −289.039 −0.152728
\(154\) 0 0
\(155\) 297.706 249.805i 0.154273 0.129450i
\(156\) 0 0
\(157\) −2735.55 995.659i −1.39058 0.506129i −0.465210 0.885200i \(-0.654021\pi\)
−0.925367 + 0.379072i \(0.876243\pi\)
\(158\) 0 0
\(159\) −486.797 + 843.157i −0.242802 + 0.420545i
\(160\) 0 0
\(161\) 0.546093 3.09705i 0.000267318 0.00151603i
\(162\) 0 0
\(163\) 476.727 + 825.716i 0.229081 + 0.396780i 0.957536 0.288314i \(-0.0930947\pi\)
−0.728455 + 0.685093i \(0.759761\pi\)
\(164\) 0 0
\(165\) −456.230 382.822i −0.215257 0.180622i
\(166\) 0 0
\(167\) 3219.48 1171.80i 1.49180 0.542972i 0.537879 0.843022i \(-0.319226\pi\)
0.953923 + 0.300051i \(0.0970036\pi\)
\(168\) 0 0
\(169\) 747.717 + 4240.51i 0.340335 + 1.93014i
\(170\) 0 0
\(171\) 1845.36 + 121.722i 0.825252 + 0.0544345i
\(172\) 0 0
\(173\) −183.470 1040.51i −0.0806299 0.457275i −0.998214 0.0597340i \(-0.980975\pi\)
0.917584 0.397541i \(-0.130136\pi\)
\(174\) 0 0
\(175\) 2.92988 1.06639i 0.00126559 0.000460637i
\(176\) 0 0
\(177\) −333.792 280.085i −0.141748 0.118940i
\(178\) 0 0
\(179\) 1458.78 + 2526.69i 0.609132 + 1.05505i 0.991384 + 0.130990i \(0.0418154\pi\)
−0.382252 + 0.924058i \(0.624851\pi\)
\(180\) 0 0
\(181\) −86.6876 + 491.630i −0.0355991 + 0.201893i −0.997420 0.0717872i \(-0.977130\pi\)
0.961821 + 0.273680i \(0.0882409\pi\)
\(182\) 0 0
\(183\) 240.720 416.940i 0.0972381 0.168421i
\(184\) 0 0
\(185\) 73.0971 + 26.6052i 0.0290498 + 0.0105733i
\(186\) 0 0
\(187\) −285.533 + 239.590i −0.111659 + 0.0936929i
\(188\) 0 0
\(189\) 9.95085 0.00382972
\(190\) 0 0
\(191\) −3954.16 −1.49797 −0.748987 0.662584i \(-0.769460\pi\)
−0.748987 + 0.662584i \(0.769460\pi\)
\(192\) 0 0
\(193\) −2456.20 + 2060.99i −0.916067 + 0.768671i −0.973263 0.229692i \(-0.926228\pi\)
0.0571967 + 0.998363i \(0.481784\pi\)
\(194\) 0 0
\(195\) −1567.23 570.425i −0.575547 0.209482i
\(196\) 0 0
\(197\) 2385.34 4131.53i 0.862683 1.49421i −0.00664619 0.999978i \(-0.502116\pi\)
0.869329 0.494233i \(-0.164551\pi\)
\(198\) 0 0
\(199\) −154.510 + 876.271i −0.0550399 + 0.312147i −0.999882 0.0153802i \(-0.995104\pi\)
0.944842 + 0.327527i \(0.106215\pi\)
\(200\) 0 0
\(201\) −163.405 283.026i −0.0573419 0.0993191i
\(202\) 0 0
\(203\) 3.84474 + 3.22612i 0.00132930 + 0.00111541i
\(204\) 0 0
\(205\) 2310.72 841.034i 0.787258 0.286538i
\(206\) 0 0
\(207\) −130.635 740.870i −0.0438637 0.248763i
\(208\) 0 0
\(209\) 1923.87 1409.41i 0.636730 0.466463i
\(210\) 0 0
\(211\) −893.500 5067.29i −0.291522 1.65330i −0.681013 0.732271i \(-0.738460\pi\)
0.389491 0.921030i \(-0.372651\pi\)
\(212\) 0 0
\(213\) −736.999 + 268.246i −0.237082 + 0.0862906i
\(214\) 0 0
\(215\) −3268.83 2742.87i −1.03689 0.870058i
\(216\) 0 0
\(217\) 1.89521 + 3.28261i 0.000592882 + 0.00102690i
\(218\) 0 0
\(219\) 1.40364 7.96041i 0.000433100 0.00245623i
\(220\) 0 0
\(221\) −521.901 + 903.959i −0.158855 + 0.275144i
\(222\) 0 0
\(223\) −2658.51 967.619i −0.798328 0.290568i −0.0895346 0.995984i \(-0.528538\pi\)
−0.708794 + 0.705416i \(0.750760\pi\)
\(224\) 0 0
\(225\) 571.363 479.431i 0.169293 0.142054i
\(226\) 0 0
\(227\) −3211.84 −0.939107 −0.469554 0.882904i \(-0.655585\pi\)
−0.469554 + 0.882904i \(0.655585\pi\)
\(228\) 0 0
\(229\) −3296.93 −0.951386 −0.475693 0.879611i \(-0.657803\pi\)
−0.475693 + 0.879611i \(0.657803\pi\)
\(230\) 0 0
\(231\) 4.44978 3.73381i 0.00126742 0.00106349i
\(232\) 0 0
\(233\) −313.608 114.144i −0.0881766 0.0320937i 0.297555 0.954705i \(-0.403829\pi\)
−0.385732 + 0.922611i \(0.626051\pi\)
\(234\) 0 0
\(235\) 1775.82 3075.81i 0.492943 0.853802i
\(236\) 0 0
\(237\) 280.808 1592.54i 0.0769638 0.436484i
\(238\) 0 0
\(239\) 1485.26 + 2572.54i 0.401981 + 0.696251i 0.993965 0.109698i \(-0.0349885\pi\)
−0.591984 + 0.805950i \(0.701655\pi\)
\(240\) 0 0
\(241\) 3859.37 + 3238.39i 1.03155 + 0.865574i 0.991035 0.133605i \(-0.0426554\pi\)
0.0405159 + 0.999179i \(0.487100\pi\)
\(242\) 0 0
\(243\) 3461.17 1259.76i 0.913722 0.332568i
\(244\) 0 0
\(245\) 570.030 + 3232.80i 0.148644 + 0.843004i
\(246\) 0 0
\(247\) 3712.74 5551.50i 0.956420 1.43010i
\(248\) 0 0
\(249\) 266.919 + 1513.77i 0.0679330 + 0.385267i
\(250\) 0 0
\(251\) −6502.67 + 2366.78i −1.63524 + 0.595178i −0.986197 0.165575i \(-0.947052\pi\)
−0.649041 + 0.760753i \(0.724830\pi\)
\(252\) 0 0
\(253\) −743.172 623.595i −0.184675 0.154961i
\(254\) 0 0
\(255\) −133.852 231.839i −0.0328712 0.0569346i
\(256\) 0 0
\(257\) 59.9734 340.126i 0.0145566 0.0825544i −0.976664 0.214773i \(-0.931099\pi\)
0.991221 + 0.132218i \(0.0422100\pi\)
\(258\) 0 0
\(259\) −0.379349 + 0.657052i −9.10101e−5 + 0.000157634i
\(260\) 0 0
\(261\) 1128.22 + 410.638i 0.267567 + 0.0973863i
\(262\) 0 0
\(263\) −3191.15 + 2677.69i −0.748192 + 0.627808i −0.935024 0.354584i \(-0.884623\pi\)
0.186832 + 0.982392i \(0.440178\pi\)
\(264\) 0 0
\(265\) 4311.96 0.999552
\(266\) 0 0
\(267\) −1707.34 −0.391338
\(268\) 0 0
\(269\) −4483.60 + 3762.19i −1.01625 + 0.852731i −0.989151 0.146902i \(-0.953070\pi\)
−0.0270942 + 0.999633i \(0.508625\pi\)
\(270\) 0 0
\(271\) −1408.80 512.760i −0.315787 0.114937i 0.179264 0.983801i \(-0.442628\pi\)
−0.495051 + 0.868864i \(0.664851\pi\)
\(272\) 0 0
\(273\) 8.13339 14.0874i 0.00180313 0.00312312i
\(274\) 0 0
\(275\) 167.022 947.228i 0.0366247 0.207709i
\(276\) 0 0
\(277\) 864.040 + 1496.56i 0.187419 + 0.324620i 0.944389 0.328830i \(-0.106654\pi\)
−0.756970 + 0.653450i \(0.773321\pi\)
\(278\) 0 0
\(279\) 694.602 + 582.840i 0.149049 + 0.125067i
\(280\) 0 0
\(281\) 4756.56 1731.25i 1.00980 0.367536i 0.216442 0.976295i \(-0.430555\pi\)
0.793355 + 0.608760i \(0.208333\pi\)
\(282\) 0 0
\(283\) 1290.93 + 7321.24i 0.271159 + 1.53782i 0.750905 + 0.660410i \(0.229618\pi\)
−0.479746 + 0.877407i \(0.659271\pi\)
\(284\) 0 0
\(285\) 756.941 + 1536.53i 0.157324 + 0.319356i
\(286\) 0 0
\(287\) 4.16473 + 23.6194i 0.000856573 + 0.00485787i
\(288\) 0 0
\(289\) 4459.27 1623.04i 0.907647 0.330357i
\(290\) 0 0
\(291\) 1686.99 + 1415.56i 0.339840 + 0.285159i
\(292\) 0 0
\(293\) 1969.27 + 3410.87i 0.392648 + 0.680087i 0.992798 0.119801i \(-0.0382256\pi\)
−0.600150 + 0.799888i \(0.704892\pi\)
\(294\) 0 0
\(295\) −335.111 + 1900.51i −0.0661387 + 0.375091i
\(296\) 0 0
\(297\) 1534.86 2658.46i 0.299871 0.519391i
\(298\) 0 0
\(299\) −2552.92 929.188i −0.493777 0.179720i
\(300\) 0 0
\(301\) 31.8821 26.7523i 0.00610517 0.00512284i
\(302\) 0 0
\(303\) 3617.50 0.685875
\(304\) 0 0
\(305\) −2132.26 −0.400304
\(306\) 0 0
\(307\) 5529.57 4639.86i 1.02798 0.862577i 0.0373697 0.999302i \(-0.488102\pi\)
0.990609 + 0.136725i \(0.0436576\pi\)
\(308\) 0 0
\(309\) −1066.97 388.347i −0.196434 0.0714961i
\(310\) 0 0
\(311\) −3445.54 + 5967.85i −0.628227 + 1.08812i 0.359680 + 0.933076i \(0.382886\pi\)
−0.987907 + 0.155045i \(0.950448\pi\)
\(312\) 0 0
\(313\) −878.282 + 4980.99i −0.158605 + 0.899495i 0.796810 + 0.604230i \(0.206519\pi\)
−0.955415 + 0.295265i \(0.904592\pi\)
\(314\) 0 0
\(315\) −9.97486 17.2770i −0.00178419 0.00309031i
\(316\) 0 0
\(317\) −3261.03 2736.33i −0.577785 0.484819i 0.306434 0.951892i \(-0.400864\pi\)
−0.884219 + 0.467073i \(0.845309\pi\)
\(318\) 0 0
\(319\) 1454.91 529.545i 0.255359 0.0929431i
\(320\) 0 0
\(321\) 681.787 + 3866.61i 0.118547 + 0.672315i
\(322\) 0 0
\(323\) 1029.30 299.548i 0.177311 0.0516016i
\(324\) 0 0
\(325\) −467.724 2652.60i −0.0798298 0.452737i
\(326\) 0 0
\(327\) 3568.49 1298.83i 0.603481 0.219649i
\(328\) 0 0
\(329\) 26.5361 + 22.2664i 0.00444676 + 0.00373127i
\(330\) 0 0
\(331\) −1746.01 3024.18i −0.289938 0.502187i 0.683857 0.729616i \(-0.260301\pi\)
−0.973795 + 0.227429i \(0.926968\pi\)
\(332\) 0 0
\(333\) −31.5162 + 178.737i −0.00518641 + 0.0294136i
\(334\) 0 0
\(335\) −723.707 + 1253.50i −0.118031 + 0.204435i
\(336\) 0 0
\(337\) −4238.55 1542.71i −0.685130 0.249367i −0.0240813 0.999710i \(-0.507666\pi\)
−0.661049 + 0.750343i \(0.729888\pi\)
\(338\) 0 0
\(339\) −1487.60 + 1248.24i −0.238334 + 0.199986i
\(340\) 0 0
\(341\) 1169.30 0.185693
\(342\) 0 0
\(343\) −64.0351 −0.0100804
\(344\) 0 0
\(345\) 533.757 447.875i 0.0832942 0.0698921i
\(346\) 0 0
\(347\) −10159.4 3697.74i −1.57172 0.572060i −0.598339 0.801243i \(-0.704172\pi\)
−0.973383 + 0.229183i \(0.926395\pi\)
\(348\) 0 0
\(349\) −29.1386 + 50.4695i −0.00446921 + 0.00774089i −0.868251 0.496125i \(-0.834756\pi\)
0.863782 + 0.503865i \(0.168089\pi\)
\(350\) 0 0
\(351\) 1492.75 8465.78i 0.227000 1.28738i
\(352\) 0 0
\(353\) −2072.14 3589.05i −0.312433 0.541150i 0.666455 0.745545i \(-0.267811\pi\)
−0.978889 + 0.204395i \(0.934477\pi\)
\(354\) 0 0
\(355\) 2660.92 + 2232.78i 0.397822 + 0.333813i
\(356\) 0 0
\(357\) 2.45356 0.893023i 0.000363743 0.000132392i
\(358\) 0 0
\(359\) −273.150 1549.11i −0.0401568 0.227741i 0.958124 0.286354i \(-0.0924433\pi\)
−0.998281 + 0.0586132i \(0.981332\pi\)
\(360\) 0 0
\(361\) −6697.65 + 1478.99i −0.976476 + 0.215628i
\(362\) 0 0
\(363\) 188.288 + 1067.83i 0.0272246 + 0.154398i
\(364\) 0 0
\(365\) −33.6408 + 12.2443i −0.00482422 + 0.00175587i
\(366\) 0 0
\(367\) −2906.84 2439.13i −0.413449 0.346925i 0.412215 0.911086i \(-0.364755\pi\)
−0.825665 + 0.564161i \(0.809200\pi\)
\(368\) 0 0
\(369\) 2868.67 + 4968.68i 0.404707 + 0.700974i
\(370\) 0 0
\(371\) −7.30297 + 41.4172i −0.00102197 + 0.00579589i
\(372\) 0 0
\(373\) 5536.77 9589.97i 0.768588 1.33123i −0.169741 0.985489i \(-0.554293\pi\)
0.938329 0.345744i \(-0.112373\pi\)
\(374\) 0 0
\(375\) 3078.49 + 1120.48i 0.423926 + 0.154296i
\(376\) 0 0
\(377\) 3321.41 2786.99i 0.453743 0.380736i
\(378\) 0 0
\(379\) −1803.87 −0.244481 −0.122241 0.992500i \(-0.539008\pi\)
−0.122241 + 0.992500i \(0.539008\pi\)
\(380\) 0 0
\(381\) 761.360 0.102377
\(382\) 0 0
\(383\) −7497.92 + 6291.50i −1.00033 + 0.839375i −0.987030 0.160536i \(-0.948678\pi\)
−0.0132985 + 0.999912i \(0.504233\pi\)
\(384\) 0 0
\(385\) −24.1750 8.79899i −0.00320019 0.00116477i
\(386\) 0 0
\(387\) 4978.03 8622.20i 0.653869 1.13253i
\(388\) 0 0
\(389\) −1250.09 + 7089.63i −0.162936 + 0.924058i 0.788230 + 0.615380i \(0.210998\pi\)
−0.951167 + 0.308678i \(0.900114\pi\)
\(390\) 0 0
\(391\) −218.037 377.652i −0.0282011 0.0488457i
\(392\) 0 0
\(393\) −4522.11 3794.50i −0.580433 0.487041i
\(394\) 0 0
\(395\) −6730.10 + 2449.56i −0.857286 + 0.312027i
\(396\) 0 0
\(397\) −1394.61 7909.22i −0.176306 0.999880i −0.936626 0.350331i \(-0.886069\pi\)
0.760320 0.649549i \(-0.225042\pi\)
\(398\) 0 0
\(399\) −16.0407 + 4.66821i −0.00201263 + 0.000585721i
\(400\) 0 0
\(401\) −2306.73 13082.1i −0.287263 1.62915i −0.697087 0.716987i \(-0.745521\pi\)
0.409824 0.912165i \(-0.365590\pi\)
\(402\) 0 0
\(403\) 3077.01 1119.94i 0.380340 0.138432i
\(404\) 0 0
\(405\) −2731.42 2291.94i −0.335125 0.281203i
\(406\) 0 0
\(407\) 117.025 + 202.693i 0.0142523 + 0.0246858i
\(408\) 0 0
\(409\) −1448.04 + 8212.22i −0.175063 + 0.992831i 0.763008 + 0.646389i \(0.223722\pi\)
−0.938071 + 0.346442i \(0.887390\pi\)
\(410\) 0 0
\(411\) −2395.98 + 4149.95i −0.287554 + 0.498058i
\(412\) 0 0
\(413\) −17.6872 6.43761i −0.00210734 0.000767008i
\(414\) 0 0
\(415\) 5215.07 4375.96i 0.616862 0.517609i
\(416\) 0 0
\(417\) −1656.87 −0.194574
\(418\) 0 0
\(419\) 11382.8 1.32718 0.663588 0.748098i \(-0.269033\pi\)
0.663588 + 0.748098i \(0.269033\pi\)
\(420\) 0 0
\(421\) −2016.27 + 1691.85i −0.233413 + 0.195857i −0.751991 0.659174i \(-0.770906\pi\)
0.518578 + 0.855031i \(0.326462\pi\)
\(422\) 0 0
\(423\) 7786.87 + 2834.19i 0.895061 + 0.325775i
\(424\) 0 0
\(425\) 216.172 374.421i 0.0246726 0.0427343i
\(426\) 0 0
\(427\) 3.61131 20.4808i 0.000409282 0.00232115i
\(428\) 0 0
\(429\) −2509.05 4345.81i −0.282374 0.489086i
\(430\) 0 0
\(431\) 6860.32 + 5756.50i 0.766706 + 0.643343i 0.939863 0.341552i \(-0.110952\pi\)
−0.173157 + 0.984894i \(0.555397\pi\)
\(432\) 0 0
\(433\) −13082.0 + 4761.47i −1.45192 + 0.528457i −0.943128 0.332431i \(-0.892131\pi\)
−0.508795 + 0.860887i \(0.669909\pi\)
\(434\) 0 0
\(435\) 193.097 + 1095.11i 0.0212835 + 0.120704i
\(436\) 0 0
\(437\) 1233.01 + 2502.92i 0.134972 + 0.273984i
\(438\) 0 0
\(439\) 2544.55 + 14430.8i 0.276639 + 1.56890i 0.733706 + 0.679467i \(0.237789\pi\)
−0.457067 + 0.889432i \(0.651100\pi\)
\(440\) 0 0
\(441\) −7197.18 + 2619.56i −0.777149 + 0.282859i
\(442\) 0 0
\(443\) 2073.61 + 1739.96i 0.222393 + 0.186610i 0.747176 0.664626i \(-0.231409\pi\)
−0.524783 + 0.851236i \(0.675854\pi\)
\(444\) 0 0
\(445\) 3780.82 + 6548.57i 0.402760 + 0.697600i
\(446\) 0 0
\(447\) 499.789 2834.45i 0.0528842 0.299921i
\(448\) 0 0
\(449\) 1598.46 2768.61i 0.168009 0.291000i −0.769711 0.638393i \(-0.779600\pi\)
0.937720 + 0.347393i \(0.112933\pi\)
\(450\) 0 0
\(451\) 6952.50 + 2530.50i 0.725899 + 0.264206i
\(452\) 0 0
\(453\) −4145.41 + 3478.41i −0.429952 + 0.360773i
\(454\) 0 0
\(455\) −72.0440 −0.00742303
\(456\) 0 0
\(457\) 6844.40 0.700585 0.350293 0.936640i \(-0.386082\pi\)
0.350293 + 0.936640i \(0.386082\pi\)
\(458\) 0 0
\(459\) 1057.01 886.936i 0.107488 0.0901931i
\(460\) 0 0
\(461\) 13372.5 + 4867.18i 1.35101 + 0.491729i 0.913264 0.407368i \(-0.133553\pi\)
0.437750 + 0.899097i \(0.355775\pi\)
\(462\) 0 0
\(463\) −3164.48 + 5481.04i −0.317637 + 0.550163i −0.979995 0.199024i \(-0.936223\pi\)
0.662357 + 0.749188i \(0.269556\pi\)
\(464\) 0 0
\(465\) −145.831 + 827.051i −0.0145436 + 0.0824808i
\(466\) 0 0
\(467\) 7250.01 + 12557.4i 0.718395 + 1.24430i 0.961636 + 0.274330i \(0.0884562\pi\)
−0.243241 + 0.969966i \(0.578210\pi\)
\(468\) 0 0
\(469\) −10.8144 9.07434i −0.00106474 0.000893420i
\(470\) 0 0
\(471\) 5911.43 2151.58i 0.578311 0.210488i
\(472\) 0 0
\(473\) −2229.47 12644.0i −0.216726 1.22911i
\(474\) 0 0
\(475\) −1537.82 + 2299.44i −0.148547 + 0.222117i
\(476\) 0 0
\(477\) 1747.00 + 9907.74i 0.167693 + 0.951036i
\(478\) 0 0
\(479\) −9427.46 + 3431.31i −0.899273 + 0.327309i −0.749961 0.661482i \(-0.769928\pi\)
−0.149311 + 0.988790i \(0.547706\pi\)
\(480\) 0 0
\(481\) 502.087 + 421.301i 0.0475950 + 0.0399369i
\(482\) 0 0
\(483\) 3.39793 + 5.88539i 0.000320106 + 0.000554440i
\(484\) 0 0
\(485\) 1693.66 9605.22i 0.158567 0.899280i
\(486\) 0 0
\(487\) −1367.77 + 2369.04i −0.127268 + 0.220434i −0.922617 0.385717i \(-0.873954\pi\)
0.795349 + 0.606151i \(0.207287\pi\)
\(488\) 0 0
\(489\) −1936.13 704.692i −0.179048 0.0651683i
\(490\) 0 0
\(491\) 5834.67 4895.87i 0.536283 0.449995i −0.333981 0.942580i \(-0.608392\pi\)
0.870264 + 0.492585i \(0.163948\pi\)
\(492\) 0 0
\(493\) 695.949 0.0635780
\(494\) 0 0
\(495\) −6154.25 −0.558814
\(496\) 0 0
\(497\) −25.9529 + 21.7771i −0.00234235 + 0.00196547i
\(498\) 0 0
\(499\) 1646.96 + 599.444i 0.147752 + 0.0537772i 0.414837 0.909896i \(-0.363838\pi\)
−0.267086 + 0.963673i \(0.586061\pi\)
\(500\) 0 0
\(501\) −3701.84 + 6411.78i −0.330112 + 0.571771i
\(502\) 0 0
\(503\) 1141.90 6476.04i 0.101222 0.574060i −0.891440 0.453139i \(-0.850304\pi\)
0.992662 0.120921i \(-0.0385848\pi\)
\(504\) 0 0
\(505\) −8010.79 13875.1i −0.705892 1.22264i
\(506\) 0 0
\(507\) −7128.02 5981.11i −0.624391 0.523926i
\(508\) 0 0
\(509\) −322.286 + 117.302i −0.0280650 + 0.0102148i −0.356014 0.934480i \(-0.615865\pi\)
0.327950 + 0.944695i \(0.393642\pi\)
\(510\) 0 0
\(511\) −0.0606325 0.343864i −5.24897e−6 2.97684e-5i
\(512\) 0 0
\(513\) −7121.94 + 5217.47i −0.612946 + 0.449039i
\(514\) 0 0
\(515\) 873.243 + 4952.40i 0.0747178 + 0.423746i
\(516\) 0 0
\(517\) 10041.7 3654.88i 0.854224 0.310912i
\(518\) 0 0
\(519\) 1749.03 + 1467.61i 0.147926 + 0.124125i
\(520\) 0 0
\(521\) −2564.96 4442.64i −0.215687 0.373581i 0.737798 0.675022i \(-0.235866\pi\)
−0.953485 + 0.301441i \(0.902532\pi\)
\(522\) 0 0
\(523\) 997.974 5659.79i 0.0834385 0.473203i −0.914244 0.405164i \(-0.867214\pi\)
0.997683 0.0680396i \(-0.0216744\pi\)
\(524\) 0 0
\(525\) −3.36885 + 5.83503i −0.000280055 + 0.000485070i
\(526\) 0 0
\(527\) 493.899 + 179.765i 0.0408247 + 0.0148590i
\(528\) 0 0
\(529\) −8451.00 + 7091.23i −0.694584 + 0.582825i
\(530\) 0 0
\(531\) −4502.64 −0.367981
\(532\) 0 0
\(533\) 20719.2 1.68376
\(534\) 0 0
\(535\) 13320.8 11177.5i 1.07646 0.903259i
\(536\) 0 0
\(537\) −5924.54 2156.36i −0.476095 0.173284i
\(538\) 0 0
\(539\) −4938.45 + 8553.65i −0.394646 + 0.683547i
\(540\) 0 0
\(541\) 926.670 5255.41i 0.0736426 0.417648i −0.925592 0.378523i \(-0.876432\pi\)
0.999234 0.0391247i \(-0.0124570\pi\)
\(542\) 0 0
\(543\) −539.392 934.255i −0.0426290 0.0738356i
\(544\) 0 0
\(545\) −12884.0 10810.9i −1.01264 0.849706i
\(546\) 0 0
\(547\) −3133.63 + 1140.55i −0.244944 + 0.0891522i −0.461575 0.887101i \(-0.652715\pi\)
0.216631 + 0.976254i \(0.430493\pi\)
\(548\) 0 0
\(549\) −863.890 4899.37i −0.0671584 0.380874i
\(550\) 0 0
\(551\) −4443.26 293.082i −0.343537 0.0226601i
\(552\) 0 0
\(553\) −12.1300 68.7926i −0.000932767 0.00528998i
\(554\) 0 0
\(555\) −157.960 + 57.4929i −0.0120812 + 0.00439718i
\(556\) 0 0
\(557\) 14826.8 + 12441.2i 1.12788 + 0.946407i 0.998976 0.0452495i \(-0.0144083\pi\)
0.128908 + 0.991657i \(0.458853\pi\)
\(558\) 0 0
\(559\) −17977.1 31137.2i −1.36019 2.35593i
\(560\) 0 0
\(561\) 139.868 793.233i 0.0105263 0.0596976i
\(562\) 0 0
\(563\) −7945.72 + 13762.4i −0.594800 + 1.03022i 0.398775 + 0.917049i \(0.369435\pi\)
−0.993575 + 0.113175i \(0.963898\pi\)
\(564\) 0 0
\(565\) 8081.90 + 2941.57i 0.601785 + 0.219032i
\(566\) 0 0
\(567\) 26.6406 22.3541i 0.00197319 0.00165570i
\(568\) 0 0
\(569\) 105.009 0.00773674 0.00386837 0.999993i \(-0.498769\pi\)
0.00386837 + 0.999993i \(0.498769\pi\)
\(570\) 0 0
\(571\) −22895.9 −1.67805 −0.839024 0.544095i \(-0.816873\pi\)
−0.839024 + 0.544095i \(0.816873\pi\)
\(572\) 0 0
\(573\) 6545.70 5492.50i 0.477226 0.400440i
\(574\) 0 0
\(575\) 1057.42 + 384.870i 0.0766914 + 0.0279134i
\(576\) 0 0
\(577\) −3092.06 + 5355.61i −0.223092 + 0.386407i −0.955745 0.294195i \(-0.904949\pi\)
0.732653 + 0.680602i \(0.238282\pi\)
\(578\) 0 0
\(579\) 1203.17 6823.52i 0.0863593 0.489768i
\(580\) 0 0
\(581\) 33.1995 + 57.5032i 0.00237065 + 0.00410608i
\(582\) 0 0
\(583\) 9938.52 + 8339.41i 0.706023 + 0.592424i
\(584\) 0 0
\(585\) −16194.9 + 5894.46i −1.14457 + 0.416591i
\(586\) 0 0
\(587\) −3635.15 20616.0i −0.255603 1.44960i −0.794521 0.607237i \(-0.792278\pi\)
0.538918 0.842358i \(-0.318833\pi\)
\(588\) 0 0
\(589\) −3077.57 1355.69i −0.215296 0.0948393i
\(590\) 0 0
\(591\) 1790.19 + 10152.7i 0.124600 + 0.706641i
\(592\) 0 0
\(593\) 6568.39 2390.70i 0.454859 0.165555i −0.104422 0.994533i \(-0.533299\pi\)
0.559281 + 0.828978i \(0.311077\pi\)
\(594\) 0 0
\(595\) −8.85852 7.43318i −0.000610360 0.000512153i
\(596\) 0 0
\(597\) −961.402 1665.20i −0.0659088 0.114157i
\(598\) 0 0
\(599\) 4674.76 26511.9i 0.318874 1.80843i −0.230749 0.973013i \(-0.574118\pi\)
0.549624 0.835412i \(-0.314771\pi\)
\(600\) 0 0
\(601\) −4059.07 + 7030.52i −0.275496 + 0.477173i −0.970260 0.242065i \(-0.922175\pi\)
0.694764 + 0.719238i \(0.255509\pi\)
\(602\) 0 0
\(603\) −3173.42 1155.03i −0.214314 0.0780041i
\(604\) 0 0
\(605\) 3678.76 3086.85i 0.247212 0.207435i
\(606\) 0 0
\(607\) −26841.4 −1.79482 −0.897411 0.441195i \(-0.854555\pi\)
−0.897411 + 0.441195i \(0.854555\pi\)
\(608\) 0 0
\(609\) −10.8458 −0.000721664
\(610\) 0 0
\(611\) 22924.1 19235.6i 1.51786 1.27363i
\(612\) 0 0
\(613\) 11582.1 + 4215.55i 0.763128 + 0.277756i 0.694119 0.719860i \(-0.255794\pi\)
0.0690088 + 0.997616i \(0.478016\pi\)
\(614\) 0 0
\(615\) −2656.93 + 4601.93i −0.174208 + 0.301736i
\(616\) 0 0
\(617\) −1470.10 + 8337.33i −0.0959220 + 0.544000i 0.898539 + 0.438894i \(0.144630\pi\)
−0.994461 + 0.105107i \(0.966482\pi\)
\(618\) 0 0
\(619\) 10519.2 + 18219.8i 0.683040 + 1.18306i 0.974048 + 0.226340i \(0.0726760\pi\)
−0.291008 + 0.956721i \(0.593991\pi\)
\(620\) 0 0
\(621\) 2751.14 + 2308.48i 0.177777 + 0.149172i
\(622\) 0 0
\(623\) −69.3037 + 25.2245i −0.00445681 + 0.00162215i
\(624\) 0 0
\(625\) −1794.51 10177.2i −0.114849 0.651338i
\(626\) 0 0
\(627\) −1227.03 + 5005.46i −0.0781547 + 0.318818i
\(628\) 0 0
\(629\) 18.2686 + 103.606i 0.00115805 + 0.00656764i
\(630\) 0 0
\(631\) −3881.74 + 1412.84i −0.244897 + 0.0891351i −0.461552 0.887113i \(-0.652707\pi\)
0.216656 + 0.976248i \(0.430485\pi\)
\(632\) 0 0
\(633\) 8517.77 + 7147.26i 0.534836 + 0.448781i
\(634\) 0 0
\(635\) −1686.00 2920.23i −0.105365 0.182497i
\(636\) 0 0
\(637\) −4802.95 + 27238.9i −0.298744 + 1.69426i
\(638\) 0 0
\(639\) −4052.25 + 7018.71i −0.250868 + 0.434516i
\(640\) 0 0
\(641\) 23825.3 + 8671.68i 1.46808 + 0.534338i 0.947579 0.319521i \(-0.103522\pi\)
0.520503 + 0.853860i \(0.325744\pi\)
\(642\) 0 0
\(643\) 14347.8 12039.3i 0.879975 0.738386i −0.0861993 0.996278i \(-0.527472\pi\)
0.966174 + 0.257892i \(0.0830277\pi\)
\(644\) 0 0
\(645\) 9221.17 0.562920
\(646\) 0 0
\(647\) 10979.6 0.667160 0.333580 0.942722i \(-0.391743\pi\)
0.333580 + 0.942722i \(0.391743\pi\)
\(648\) 0 0
\(649\) −4448.01 + 3732.32i −0.269029 + 0.225742i
\(650\) 0 0
\(651\) −7.69700 2.80148i −0.000463394 0.000168662i
\(652\) 0 0
\(653\) 6995.50 12116.6i 0.419227 0.726122i −0.576635 0.817002i \(-0.695635\pi\)
0.995862 + 0.0908798i \(0.0289679\pi\)
\(654\) 0 0
\(655\) −4539.98 + 25747.5i −0.270827 + 1.53594i
\(656\) 0 0
\(657\) −41.7637 72.3369i −0.00248000 0.00429548i
\(658\) 0 0
\(659\) 6586.19 + 5526.47i 0.389319 + 0.326678i 0.816348 0.577560i \(-0.195995\pi\)
−0.427029 + 0.904238i \(0.640440\pi\)
\(660\) 0 0
\(661\) −30322.9 + 11036.6i −1.78430 + 0.649433i −0.784742 + 0.619822i \(0.787205\pi\)
−0.999561 + 0.0296112i \(0.990573\pi\)
\(662\) 0 0
\(663\) −391.684 2221.35i −0.0229438 0.130121i
\(664\) 0 0
\(665\) 53.4265 + 51.1874i 0.00311548 + 0.00298490i
\(666\) 0 0
\(667\) 314.544 + 1783.87i 0.0182597 + 0.103556i
\(668\) 0 0
\(669\) 5744.95 2090.99i 0.332007 0.120841i
\(670\) 0 0
\(671\) −4914.59 4123.83i −0.282750 0.237256i
\(672\) 0 0
\(673\) −5398.45 9350.38i −0.309205 0.535558i 0.668984 0.743277i \(-0.266729\pi\)
−0.978189 + 0.207719i \(0.933396\pi\)
\(674\) 0 0
\(675\) −618.296 + 3506.53i −0.0352566 + 0.199950i
\(676\) 0 0
\(677\) −8537.64 + 14787.6i −0.484680 + 0.839490i −0.999845 0.0176007i \(-0.994397\pi\)
0.515165 + 0.857091i \(0.327731\pi\)
\(678\) 0 0
\(679\) 89.3916 + 32.5359i 0.00505234 + 0.00183890i
\(680\) 0 0
\(681\) 5316.87 4461.38i 0.299182 0.251043i
\(682\) 0 0
\(683\) −23053.3 −1.29152 −0.645762 0.763538i \(-0.723460\pi\)
−0.645762 + 0.763538i \(0.723460\pi\)
\(684\) 0 0
\(685\) 21223.1 1.18379
\(686\) 0 0
\(687\) 5457.72 4579.57i 0.303093 0.254326i
\(688\) 0 0
\(689\) 34140.5 + 12426.1i 1.88774 + 0.687080i
\(690\) 0 0
\(691\) −1222.32 + 2117.13i −0.0672929 + 0.116555i −0.897709 0.440589i \(-0.854770\pi\)
0.830416 + 0.557144i \(0.188103\pi\)
\(692\) 0 0
\(693\) 10.4232 59.1128i 0.000571348 0.00324027i
\(694\) 0 0
\(695\) 3669.07 + 6355.01i 0.200253 + 0.346848i
\(696\) 0 0
\(697\) 2547.62 + 2137.71i 0.138448 + 0.116172i
\(698\) 0 0
\(699\) 677.696 246.661i 0.0366707 0.0133470i
\(700\) 0 0
\(701\) −1013.54 5748.05i −0.0546088 0.309702i 0.945253 0.326339i \(-0.105815\pi\)
−0.999862 + 0.0166372i \(0.994704\pi\)
\(702\) 0 0
\(703\) −73.0036 669.162i −0.00391662 0.0359003i
\(704\) 0 0
\(705\) 1332.74 + 7558.36i 0.0711972 + 0.403779i
\(706\) 0 0
\(707\) 146.841 53.4456i 0.00781118 0.00284304i
\(708\) 0 0
\(709\) 9627.24 + 8078.21i 0.509956 + 0.427903i 0.861114 0.508413i \(-0.169768\pi\)
−0.351158 + 0.936316i \(0.614212\pi\)
\(710\) 0 0
\(711\) −8355.15 14471.5i −0.440707 0.763327i
\(712\) 0 0
\(713\) −237.551 + 1347.22i −0.0124773 + 0.0707625i
\(714\) 0 0
\(715\) −11112.4 + 19247.2i −0.581229 + 1.00672i
\(716\) 0 0
\(717\) −6032.06 2195.49i −0.314186 0.114354i
\(718\) 0 0
\(719\) −12046.4 + 10108.1i −0.624832 + 0.524296i −0.899318 0.437295i \(-0.855937\pi\)
0.274486 + 0.961591i \(0.411492\pi\)
\(720\) 0 0
\(721\) −49.0478 −0.00253348
\(722\) 0 0
\(723\) −10887.0 −0.560019
\(724\) 0 0
\(725\) −1375.73 + 1154.37i −0.0704735 + 0.0591343i
\(726\) 0 0
\(727\) 14496.9 + 5276.45i 0.739562 + 0.269178i 0.684207 0.729288i \(-0.260149\pi\)
0.0553552 + 0.998467i \(0.482371\pi\)
\(728\) 0 0
\(729\) 1049.75 1818.22i 0.0533328 0.0923752i
\(730\) 0 0
\(731\) 1002.14 5683.41i 0.0507051 0.287563i
\(732\) 0 0
\(733\) −12280.5 21270.5i −0.618814 1.07182i −0.989702 0.143140i \(-0.954280\pi\)
0.370888 0.928678i \(-0.379053\pi\)
\(734\) 0 0
\(735\) −5434.12 4559.77i −0.272708 0.228829i
\(736\) 0 0
\(737\) −4092.34 + 1489.49i −0.204537 + 0.0744452i
\(738\) 0 0
\(739\) −6302.25 35741.8i −0.313711 1.77914i −0.579361 0.815071i \(-0.696698\pi\)
0.265650 0.964070i \(-0.414413\pi\)
\(740\) 0 0
\(741\) 1565.22 + 14347.1i 0.0775978 + 0.711273i
\(742\) 0 0
\(743\) 798.473 + 4528.37i 0.0394255 + 0.223593i 0.998154 0.0607291i \(-0.0193426\pi\)
−0.958729 + 0.284322i \(0.908231\pi\)
\(744\) 0 0
\(745\) −11978.4 + 4359.78i −0.589067 + 0.214403i
\(746\) 0 0
\(747\) 12167.7 + 10209.9i 0.595975 + 0.500082i
\(748\) 0 0
\(749\) 84.8009 + 146.879i 0.00413692 + 0.00716536i
\(750\) 0 0
\(751\) 6559.85 37202.7i 0.318738 1.80765i −0.231712 0.972785i \(-0.574433\pi\)
0.550450 0.834868i \(-0.314456\pi\)
\(752\) 0 0
\(753\) 7476.94 12950.4i 0.361852 0.626746i
\(754\) 0 0
\(755\) 22521.4 + 8197.13i 1.08561 + 0.395131i
\(756\) 0 0
\(757\) 3645.48 3058.92i 0.175029 0.146867i −0.551065 0.834463i \(-0.685778\pi\)
0.726094 + 0.687596i \(0.241334\pi\)
\(758\) 0 0
\(759\) 2096.44 0.100258
\(760\) 0 0
\(761\) 16224.0 0.772825 0.386413 0.922326i \(-0.373714\pi\)
0.386413 + 0.922326i \(0.373714\pi\)
\(762\) 0 0
\(763\) 125.662 105.443i 0.00596235 0.00500301i
\(764\) 0 0
\(765\) −2599.48 946.135i −0.122856 0.0447158i
\(766\) 0 0
\(767\) −8130.15 + 14081.8i −0.382741 + 0.662927i
\(768\) 0 0
\(769\) 4203.82 23841.1i 0.197131 1.11799i −0.712220 0.701956i \(-0.752310\pi\)
0.909351 0.416029i \(-0.136579\pi\)
\(770\) 0 0
\(771\) 373.170 + 646.349i 0.0174311 + 0.0301916i
\(772\) 0 0
\(773\) −32244.3 27056.2i −1.50032 1.25892i −0.880382 0.474265i \(-0.842714\pi\)
−0.619937 0.784652i \(-0.712842\pi\)
\(774\) 0 0
\(775\) −1274.50 + 463.880i −0.0590728 + 0.0215007i
\(776\) 0 0
\(777\) −0.284700 1.61461i −1.31449e−5 7.45482e-5i
\(778\) 0 0
\(779\) −15365.0 14721.0i −0.706684 0.677066i
\(780\) 0 0
\(781\) 1814.85 + 10292.5i 0.0831505 + 0.471570i
\(782\) 0 0
\(783\) −5385.93 + 1960.32i −0.245820 + 0.0894713i
\(784\) 0 0
\(785\) −21343.1 17909.0i −0.970404 0.814266i
\(786\) 0 0
\(787\) 1059.61 + 1835.30i 0.0479937 + 0.0831275i 0.889024 0.457860i \(-0.151384\pi\)
−0.841031 + 0.540988i \(0.818051\pi\)
\(788\) 0 0
\(789\) 1563.19 8865.28i 0.0705335 0.400015i
\(790\) 0 0
\(791\) −41.9424 + 72.6463i −0.00188533 + 0.00326549i
\(792\) 0 0
\(793\) −16882.5 6144.71i −0.756007 0.275164i
\(794\) 0 0
\(795\) −7137.99 + 5989.49i −0.318438 + 0.267202i
\(796\) 0 0
\(797\) −15388.2 −0.683911 −0.341955 0.939716i \(-0.611089\pi\)
−0.341955 + 0.939716i \(0.611089\pi\)
\(798\) 0 0
\(799\) 4803.39 0.212681
\(800\) 0 0
\(801\) −13515.1 + 11340.5i −0.596170 + 0.500246i
\(802\) 0 0
\(803\) −101.218 36.8405i −0.00444822 0.00161902i
\(804\) 0 0
\(805\) 15.0491 26.0658i 0.000658896 0.00114124i
\(806\) 0 0
\(807\) 2196.30 12455.8i 0.0958034 0.543328i
\(808\) 0 0
\(809\) −21524.0 37280.7i −0.935408 1.62017i −0.773905 0.633301i \(-0.781699\pi\)
−0.161502 0.986872i \(-0.551634\pi\)
\(810\) 0 0
\(811\) −15389.8 12913.6i −0.666349 0.559133i 0.245633 0.969363i \(-0.421004\pi\)
−0.911982 + 0.410230i \(0.865449\pi\)
\(812\) 0 0
\(813\) 3044.36 1108.06i 0.131329 0.0477998i
\(814\) 0 0
\(815\) 1584.58 + 8986.61i 0.0681049 + 0.386242i
\(816\) 0 0
\(817\) −8791.54 + 35863.5i −0.376471 + 1.53575i
\(818\) 0 0
\(819\) −29.1889 165.538i −0.00124535 0.00706273i
\(820\) 0 0
\(821\) 26070.6 9488.93i 1.10825 0.403369i 0.277897 0.960611i \(-0.410363\pi\)
0.830350 + 0.557242i \(0.188140\pi\)
\(822\) 0 0
\(823\) 22760.5 + 19098.3i 0.964011 + 0.808902i 0.981601 0.190944i \(-0.0611548\pi\)
−0.0175896 + 0.999845i \(0.505599\pi\)
\(824\) 0 0
\(825\) 1039.25 + 1800.04i 0.0438571 + 0.0759627i
\(826\) 0 0
\(827\) −3476.25 + 19714.8i −0.146168 + 0.828962i 0.820253 + 0.572000i \(0.193832\pi\)
−0.966422 + 0.256961i \(0.917279\pi\)
\(828\) 0 0
\(829\) 13793.1 23890.3i 0.577869 1.00090i −0.417855 0.908514i \(-0.637218\pi\)
0.995723 0.0923842i \(-0.0294488\pi\)
\(830\) 0 0
\(831\) −3509.11 1277.21i −0.146486 0.0533165i
\(832\) 0 0
\(833\) −3400.95 + 2853.74i −0.141460 + 0.118699i
\(834\) 0 0
\(835\) 32790.2 1.35899
\(836\) 0 0
\(837\) −4328.62 −0.178756
\(838\) 0 0
\(839\) 19091.1 16019.3i 0.785575 0.659176i −0.159071 0.987267i \(-0.550850\pi\)
0.944646 + 0.328092i \(0.106405\pi\)
\(840\) 0 0
\(841\) 20201.6 + 7352.80i 0.828310 + 0.301480i
\(842\) 0 0
\(843\) −5469.22 + 9472.97i −0.223452 + 0.387030i
\(844\) 0 0
\(845\) −7156.18 + 40584.7i −0.291337 + 1.65226i
\(846\) 0 0
\(847\) 23.4192 + 40.5633i 0.000950053 + 0.00164554i
\(848\) 0 0
\(849\) −12306.5 10326.4i −0.497477 0.417433i
\(850\) 0 0
\(851\) −257.308 + 93.6524i −0.0103648 + 0.00377246i
\(852\) 0 0
\(853\) −1123.74 6373.04i −0.0451068 0.255813i 0.953913 0.300084i \(-0.0970147\pi\)
−0.999020 + 0.0442706i \(0.985904\pi\)
\(854\) 0 0
\(855\) 16197.8 + 7135.26i 0.647900 + 0.285404i
\(856\) 0 0
\(857\) −830.583 4710.47i −0.0331064 0.187756i 0.963770 0.266735i \(-0.0859449\pi\)
−0.996876 + 0.0789795i \(0.974834\pi\)
\(858\) 0 0
\(859\) −2874.73 + 1046.32i −0.114185 + 0.0415598i −0.398481 0.917177i \(-0.630462\pi\)
0.284296 + 0.958737i \(0.408240\pi\)
\(860\) 0 0
\(861\) −39.7026 33.3144i −0.00157150 0.00131864i
\(862\) 0 0
\(863\) 8221.71 + 14240.4i 0.324299 + 0.561703i 0.981370 0.192126i \(-0.0615382\pi\)
−0.657071 + 0.753829i \(0.728205\pi\)
\(864\) 0 0
\(865\) 1755.94 9958.43i 0.0690217 0.391441i
\(866\) 0 0
\(867\) −5127.38 + 8880.89i −0.200848 + 0.347879i
\(868\) 0 0
\(869\) −20249.5 7370.22i −0.790470 0.287707i
\(870\) 0 0
\(871\) −9342.37 + 7839.18i −0.363437 + 0.304960i
\(872\) 0 0
\(873\) 22756.5 0.882234
\(874\) 0 0
\(875\) 141.515 0.00546752
\(876\) 0 0
\(877\) −19700.2 + 16530.4i −0.758526 + 0.636479i −0.937743 0.347331i \(-0.887088\pi\)
0.179217 + 0.983810i \(0.442644\pi\)
\(878\) 0 0
\(879\) −7997.77 2910.95i −0.306892 0.111700i
\(880\) 0 0
\(881\) −7182.33 + 12440.2i −0.274664 + 0.475731i −0.970050 0.242904i \(-0.921900\pi\)
0.695387 + 0.718636i \(0.255233\pi\)
\(882\) 0 0
\(883\) 2856.53 16200.2i 0.108867 0.617417i −0.880738 0.473605i \(-0.842953\pi\)
0.989605 0.143813i \(-0.0459363\pi\)
\(884\) 0 0
\(885\) −2085.14 3611.58i −0.0791993 0.137177i
\(886\) 0 0
\(887\) 34311.4 + 28790.7i 1.29883 + 1.08985i 0.990346 + 0.138620i \(0.0442668\pi\)
0.308486 + 0.951229i \(0.400178\pi\)
\(888\) 0 0
\(889\) 30.9049 11.2485i 0.00116594 0.000424366i
\(890\) 0 0
\(891\) −1862.94 10565.3i −0.0700458 0.397250i
\(892\) 0 0
\(893\) −30667.0 2022.83i −1.14920 0.0758022i
\(894\) 0 0
\(895\) 4848.81 + 27499.0i 0.181093 + 1.02703i
\(896\) 0 0
\(897\) 5516.78 2007.94i 0.205351 0.0747416i
\(898\) 0 0
\(899\) −1672.46 1403.36i −0.0620465 0.0520632i
\(900\) 0 0
\(901\) 2915.84 + 5050.39i 0.107814 + 0.186740i
\(902\) 0 0
\(903\) −15.6175 + 88.5712i −0.000575546 + 0.00326408i
\(904\) 0 0
\(905\) −2388.92 + 4137.73i −0.0877462 + 0.151981i
\(906\) 0 0
\(907\) 42023.3 + 15295.2i 1.53844 + 0.559945i 0.965669 0.259774i \(-0.0836481\pi\)
0.572766 + 0.819719i \(0.305870\pi\)
\(908\) 0 0
\(909\) 28635.7 24028.2i 1.04487 0.876750i
\(910\) 0 0
\(911\) −29208.6 −1.06227 −0.531133 0.847288i \(-0.678234\pi\)
−0.531133 + 0.847288i \(0.678234\pi\)
\(912\) 0 0
\(913\) 20483.3 0.742495
\(914\) 0 0
\(915\) 3529.73 2961.79i 0.127529 0.107010i
\(916\) 0 0
\(917\) −239.621 87.2148i −0.00862919 0.00314077i
\(918\) 0 0
\(919\) 18170.8 31472.8i 0.652231 1.12970i −0.330349 0.943859i \(-0.607166\pi\)
0.982580 0.185839i \(-0.0595002\pi\)
\(920\) 0 0
\(921\) −2708.67 + 15361.6i −0.0969095 + 0.549601i
\(922\) 0 0
\(923\) 14633.8 + 25346.5i 0.521862 + 0.903891i
\(924\) 0 0
\(925\) −207.965 174.503i −0.00739225 0.00620284i
\(926\) 0 0
\(927\) −11025.5 + 4012.96i −0.390643 + 0.142182i
\(928\) 0 0
\(929\) 3470.02 + 19679.4i 0.122549 + 0.695007i 0.982734 + 0.185025i \(0.0592367\pi\)
−0.860185 + 0.509982i \(0.829652\pi\)
\(930\) 0 0
\(931\) 22915.0 16787.3i 0.806670 0.590959i
\(932\) 0 0
\(933\) −2585.86 14665.1i −0.0907366 0.514593i
\(934\) 0 0
\(935\) −3352.21 + 1220.11i −0.117250 + 0.0426757i
\(936\) 0 0
\(937\) −7274.24 6103.81i −0.253617 0.212810i 0.507111 0.861881i \(-0.330713\pi\)
−0.760728 + 0.649071i \(0.775158\pi\)
\(938\) 0 0
\(939\) −5464.89 9465.47i −0.189925 0.328961i
\(940\) 0 0
\(941\) −6026.29 + 34176.8i −0.208769 + 1.18399i 0.682629 + 0.730765i \(0.260836\pi\)
−0.891398 + 0.453221i \(0.850275\pi\)
\(942\) 0 0
\(943\) −4327.98 + 7496.28i −0.149457 + 0.258868i
\(944\) 0 0
\(945\) 89.4933 + 32.5729i 0.00308065 + 0.00112127i
\(946\) 0 0
\(947\) −13221.9 + 11094.5i −0.453699 + 0.380699i −0.840806 0.541336i \(-0.817919\pi\)
0.387107 + 0.922035i \(0.373474\pi\)
\(948\) 0 0
\(949\) −301.641 −0.0103179
\(950\) 0 0
\(951\) 9199.18 0.313674
\(952\) 0 0
\(953\) −1948.33 + 1634.84i −0.0662250 + 0.0555694i −0.675299 0.737544i \(-0.735986\pi\)
0.609074 + 0.793113i \(0.291541\pi\)
\(954\) 0 0
\(955\) −35561.9 12943.5i −1.20498 0.438577i
\(956\) 0 0
\(957\) −1672.90 + 2897.54i −0.0565069 + 0.0978728i
\(958\) 0 0
\(959\) −35.9446 + 203.852i −0.00121034 + 0.00686416i
\(960\) 0 0
\(961\) 14071.1 + 24371.8i 0.472327 + 0.818094i
\(962\) 0 0
\(963\) 31079.8 + 26079.0i 1.04001 + 0.872674i
\(964\) 0 0
\(965\) −28836.3 + 10495.5i −0.961941 + 0.350118i
\(966\) 0 0
\(967\) 3948.75 + 22394.5i 0.131317 + 0.744733i 0.977354 + 0.211610i \(0.0678707\pi\)
−0.846038 + 0.533123i \(0.821018\pi\)
\(968\) 0 0
\(969\) −1287.81 + 1925.61i −0.0426939 + 0.0638384i
\(970\) 0 0
\(971\) −5503.23 31210.4i −0.181882 1.03150i −0.929897 0.367820i \(-0.880104\pi\)
0.748015 0.663681i \(-0.231007\pi\)
\(972\) 0 0
\(973\) −67.2553 + 24.4789i −0.00221594 + 0.000806535i
\(974\) 0 0
\(975\) 4458.84 + 3741.41i 0.146459 + 0.122893i
\(976\) 0 0
\(977\) 14690.8 + 25445.3i 0.481066 + 0.833231i 0.999764 0.0217266i \(-0.00691634\pi\)
−0.518698 + 0.854958i \(0.673583\pi\)
\(978\) 0 0
\(979\) −3950.75 + 22405.8i −0.128975 + 0.731454i
\(980\) 0 0
\(981\) 19620.7 33984.1i 0.638574 1.10604i
\(982\) 0 0
\(983\) −7117.62 2590.60i −0.230943 0.0840564i 0.223957 0.974599i \(-0.428103\pi\)
−0.454900 + 0.890543i \(0.650325\pi\)
\(984\) 0 0
\(985\) 34976.7 29348.9i 1.13142 0.949376i
\(986\) 0 0
\(987\) −74.8568 −0.00241410
\(988\) 0 0
\(989\) 15020.7 0.482944
\(990\) 0 0
\(991\) −9195.90 + 7716.28i −0.294770 + 0.247342i −0.778164 0.628062i \(-0.783849\pi\)
0.483393 + 0.875403i \(0.339404\pi\)
\(992\) 0 0
\(993\) 7091.05 + 2580.93i 0.226614 + 0.0824807i
\(994\) 0 0
\(995\) −4257.96 + 7375.00i −0.135665 + 0.234978i
\(996\) 0 0
\(997\) 34.3793 194.974i 0.00109208 0.00619348i −0.984257 0.176744i \(-0.943444\pi\)
0.985349 + 0.170551i \(0.0545546\pi\)
\(998\) 0 0
\(999\) −433.213 750.346i −0.0137200 0.0237637i
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 76.4.i.a.9.2 30
19.6 even 9 1444.4.a.j.1.6 15
19.13 odd 18 1444.4.a.k.1.10 15
19.17 even 9 inner 76.4.i.a.17.2 yes 30
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.4.i.a.9.2 30 1.1 even 1 trivial
76.4.i.a.17.2 yes 30 19.17 even 9 inner
1444.4.a.j.1.6 15 19.6 even 9
1444.4.a.k.1.10 15 19.13 odd 18