Properties

Label 84.4.k.c.5.4
Level $84$
Weight $4$
Character 84.5
Analytic conductor $4.956$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [84,4,Mod(5,84)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(84, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("84.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 84.k (of order \(6\), degree \(2\), 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: \(4.95616044048\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 14 x^{10} - 32 x^{9} + 70 x^{8} + 224 x^{7} - 50 x^{6} + 2016 x^{5} + 5670 x^{4} - 23328 x^{3} - 91854 x^{2} + 531441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{7} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 5.4
Root \(-1.67777 + 2.48698i\) of defining polynomial
Character \(\chi\) \(=\) 84.5
Dual form 84.4.k.c.17.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.362863 - 5.18347i) q^{3} +(7.41560 - 12.8442i) q^{5} +(-16.6528 + 8.10467i) q^{7} +(-26.7367 + 3.76177i) q^{9} +O(q^{10})\) \(q+(-0.362863 - 5.18347i) q^{3} +(7.41560 - 12.8442i) q^{5} +(-16.6528 + 8.10467i) q^{7} +(-26.7367 + 3.76177i) q^{9} +(-0.589052 + 0.340089i) q^{11} -52.4373i q^{13} +(-69.2683 - 33.7779i) q^{15} +(-51.6784 - 89.5096i) q^{17} +(86.7417 + 50.0804i) q^{19} +(48.0530 + 83.3781i) q^{21} +(3.63771 + 2.10024i) q^{23} +(-47.4824 - 82.2418i) q^{25} +(29.2008 + 137.224i) q^{27} +53.5367i q^{29} +(235.952 - 136.227i) q^{31} +(1.97659 + 2.92992i) q^{33} +(-19.3922 + 273.992i) q^{35} +(25.2855 - 43.7957i) q^{37} +(-271.807 + 19.0276i) q^{39} +318.804 q^{41} +168.515 q^{43} +(-149.951 + 371.307i) q^{45} +(-313.988 + 543.844i) q^{47} +(211.629 - 269.930i) q^{49} +(-445.218 + 300.353i) q^{51} +(625.481 - 361.121i) q^{53} +10.0879i q^{55} +(228.115 - 467.795i) q^{57} +(-263.093 - 455.691i) q^{59} +(205.162 + 118.450i) q^{61} +(414.751 - 279.336i) q^{63} +(-673.515 - 388.854i) q^{65} +(-149.359 - 258.697i) q^{67} +(9.56651 - 19.6181i) q^{69} -379.893i q^{71} +(-559.469 + 323.010i) q^{73} +(-409.068 + 275.966i) q^{75} +(7.05302 - 10.4375i) q^{77} +(-211.386 + 366.132i) q^{79} +(700.698 - 201.155i) q^{81} -391.295 q^{83} -1532.91 q^{85} +(277.506 - 19.4265i) q^{87} +(-316.431 + 548.075i) q^{89} +(424.987 + 873.226i) q^{91} +(-791.747 - 1173.62i) q^{93} +(1286.48 - 742.752i) q^{95} -1432.13i q^{97} +(14.4699 - 11.3087i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 42 q^{7} - 84 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 42 q^{7} - 84 q^{9} + 132 q^{15} + 204 q^{19} - 378 q^{21} - 444 q^{25} + 1458 q^{31} - 108 q^{33} + 240 q^{37} - 432 q^{39} - 342 q^{45} - 1218 q^{49} - 300 q^{51} + 180 q^{57} + 2148 q^{61} + 1596 q^{63} + 1980 q^{67} - 3084 q^{73} - 3384 q^{75} - 438 q^{79} + 1008 q^{81} - 6144 q^{85} - 2898 q^{87} + 3780 q^{91} + 882 q^{93} + 9216 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(43\) \(73\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{5}{6}\right)\)

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
\(3\) −0.362863 5.18347i −0.0698330 0.997559i
\(4\) 0 0
\(5\) 7.41560 12.8442i 0.663272 1.14882i −0.316479 0.948600i \(-0.602501\pi\)
0.979751 0.200221i \(-0.0641660\pi\)
\(6\) 0 0
\(7\) −16.6528 + 8.10467i −0.899164 + 0.437611i
\(8\) 0 0
\(9\) −26.7367 + 3.76177i −0.990247 + 0.139325i
\(10\) 0 0
\(11\) −0.589052 + 0.340089i −0.0161460 + 0.00932189i −0.508051 0.861327i \(-0.669634\pi\)
0.491905 + 0.870649i \(0.336301\pi\)
\(12\) 0 0
\(13\) 52.4373i 1.11873i −0.828921 0.559365i \(-0.811045\pi\)
0.828921 0.559365i \(-0.188955\pi\)
\(14\) 0 0
\(15\) −69.2683 33.7779i −1.19233 0.581427i
\(16\) 0 0
\(17\) −51.6784 89.5096i −0.737286 1.27702i −0.953713 0.300717i \(-0.902774\pi\)
0.216428 0.976299i \(-0.430559\pi\)
\(18\) 0 0
\(19\) 86.7417 + 50.0804i 1.04736 + 0.604696i 0.921910 0.387404i \(-0.126628\pi\)
0.125454 + 0.992099i \(0.459961\pi\)
\(20\) 0 0
\(21\) 48.0530 + 83.3781i 0.499334 + 0.866410i
\(22\) 0 0
\(23\) 3.63771 + 2.10024i 0.0329790 + 0.0190404i 0.516399 0.856348i \(-0.327272\pi\)
−0.483420 + 0.875389i \(0.660606\pi\)
\(24\) 0 0
\(25\) −47.4824 82.2418i −0.379859 0.657935i
\(26\) 0 0
\(27\) 29.2008 + 137.224i 0.208137 + 0.978100i
\(28\) 0 0
\(29\) 53.5367i 0.342811i 0.985201 + 0.171406i \(0.0548308\pi\)
−0.985201 + 0.171406i \(0.945169\pi\)
\(30\) 0 0
\(31\) 235.952 136.227i 1.36704 0.789261i 0.376492 0.926420i \(-0.377130\pi\)
0.990549 + 0.137158i \(0.0437969\pi\)
\(32\) 0 0
\(33\) 1.97659 + 2.92992i 0.0104266 + 0.0154556i
\(34\) 0 0
\(35\) −19.3922 + 273.992i −0.0936537 + 1.32323i
\(36\) 0 0
\(37\) 25.2855 43.7957i 0.112349 0.194594i −0.804368 0.594131i \(-0.797496\pi\)
0.916717 + 0.399538i \(0.130829\pi\)
\(38\) 0 0
\(39\) −271.807 + 19.0276i −1.11600 + 0.0781243i
\(40\) 0 0
\(41\) 318.804 1.21436 0.607181 0.794563i \(-0.292300\pi\)
0.607181 + 0.794563i \(0.292300\pi\)
\(42\) 0 0
\(43\) 168.515 0.597634 0.298817 0.954310i \(-0.403408\pi\)
0.298817 + 0.954310i \(0.403408\pi\)
\(44\) 0 0
\(45\) −149.951 + 371.307i −0.496743 + 1.23003i
\(46\) 0 0
\(47\) −313.988 + 543.844i −0.974467 + 1.68783i −0.292782 + 0.956179i \(0.594581\pi\)
−0.681684 + 0.731647i \(0.738752\pi\)
\(48\) 0 0
\(49\) 211.629 269.930i 0.616993 0.786969i
\(50\) 0 0
\(51\) −445.218 + 300.353i −1.22241 + 0.824663i
\(52\) 0 0
\(53\) 625.481 361.121i 1.62106 0.935922i 0.634427 0.772982i \(-0.281236\pi\)
0.986636 0.162939i \(-0.0520974\pi\)
\(54\) 0 0
\(55\) 10.0879i 0.0247318i
\(56\) 0 0
\(57\) 228.115 467.795i 0.530079 1.08703i
\(58\) 0 0
\(59\) −263.093 455.691i −0.580539 1.00552i −0.995415 0.0956456i \(-0.969508\pi\)
0.414876 0.909878i \(-0.363825\pi\)
\(60\) 0 0
\(61\) 205.162 + 118.450i 0.430627 + 0.248623i 0.699614 0.714521i \(-0.253355\pi\)
−0.268987 + 0.963144i \(0.586689\pi\)
\(62\) 0 0
\(63\) 414.751 279.336i 0.829424 0.558619i
\(64\) 0 0
\(65\) −673.515 388.854i −1.28522 0.742022i
\(66\) 0 0
\(67\) −149.359 258.697i −0.272344 0.471714i 0.697117 0.716957i \(-0.254466\pi\)
−0.969462 + 0.245243i \(0.921132\pi\)
\(68\) 0 0
\(69\) 9.56651 19.6181i 0.0166909 0.0342281i
\(70\) 0 0
\(71\) 379.893i 0.634999i −0.948258 0.317500i \(-0.897157\pi\)
0.948258 0.317500i \(-0.102843\pi\)
\(72\) 0 0
\(73\) −559.469 + 323.010i −0.896999 + 0.517882i −0.876225 0.481902i \(-0.839946\pi\)
−0.0207735 + 0.999784i \(0.506613\pi\)
\(74\) 0 0
\(75\) −409.068 + 275.966i −0.629802 + 0.424877i
\(76\) 0 0
\(77\) 7.05302 10.4375i 0.0104385 0.0154476i
\(78\) 0 0
\(79\) −211.386 + 366.132i −0.301048 + 0.521431i −0.976374 0.216089i \(-0.930670\pi\)
0.675325 + 0.737520i \(0.264003\pi\)
\(80\) 0 0
\(81\) 700.698 201.155i 0.961177 0.275932i
\(82\) 0 0
\(83\) −391.295 −0.517472 −0.258736 0.965948i \(-0.583306\pi\)
−0.258736 + 0.965948i \(0.583306\pi\)
\(84\) 0 0
\(85\) −1532.91 −1.95608
\(86\) 0 0
\(87\) 277.506 19.4265i 0.341974 0.0239395i
\(88\) 0 0
\(89\) −316.431 + 548.075i −0.376872 + 0.652762i −0.990605 0.136752i \(-0.956334\pi\)
0.613733 + 0.789513i \(0.289667\pi\)
\(90\) 0 0
\(91\) 424.987 + 873.226i 0.489569 + 1.00592i
\(92\) 0 0
\(93\) −791.747 1173.62i −0.882799 1.30859i
\(94\) 0 0
\(95\) 1286.48 742.752i 1.38937 0.802155i
\(96\) 0 0
\(97\) 1432.13i 1.49908i −0.661959 0.749540i \(-0.730275\pi\)
0.661959 0.749540i \(-0.269725\pi\)
\(98\) 0 0
\(99\) 14.4699 11.3087i 0.0146897 0.0114805i
\(100\) 0 0
\(101\) 445.537 + 771.692i 0.438936 + 0.760260i 0.997608 0.0691294i \(-0.0220221\pi\)
−0.558672 + 0.829389i \(0.688689\pi\)
\(102\) 0 0
\(103\) 1067.48 + 616.313i 1.02119 + 0.589583i 0.914448 0.404704i \(-0.132625\pi\)
0.106740 + 0.994287i \(0.465959\pi\)
\(104\) 0 0
\(105\) 1427.27 + 1.09716i 1.32654 + 0.00101973i
\(106\) 0 0
\(107\) −585.738 338.176i −0.529209 0.305539i 0.211485 0.977381i \(-0.432170\pi\)
−0.740694 + 0.671842i \(0.765503\pi\)
\(108\) 0 0
\(109\) 497.209 + 861.191i 0.436917 + 0.756762i 0.997450 0.0713697i \(-0.0227370\pi\)
−0.560533 + 0.828132i \(0.689404\pi\)
\(110\) 0 0
\(111\) −236.189 115.175i −0.201964 0.0984854i
\(112\) 0 0
\(113\) 707.915i 0.589337i 0.955600 + 0.294668i \(0.0952091\pi\)
−0.955600 + 0.294668i \(0.904791\pi\)
\(114\) 0 0
\(115\) 53.9517 31.1490i 0.0437480 0.0252579i
\(116\) 0 0
\(117\) 197.257 + 1402.00i 0.155867 + 1.10782i
\(118\) 0 0
\(119\) 1586.03 + 1071.75i 1.22178 + 0.825603i
\(120\) 0 0
\(121\) −665.269 + 1152.28i −0.499826 + 0.865724i
\(122\) 0 0
\(123\) −115.682 1652.51i −0.0848026 1.21140i
\(124\) 0 0
\(125\) 445.460 0.318745
\(126\) 0 0
\(127\) −885.551 −0.618740 −0.309370 0.950942i \(-0.600118\pi\)
−0.309370 + 0.950942i \(0.600118\pi\)
\(128\) 0 0
\(129\) −61.1477 873.490i −0.0417345 0.596175i
\(130\) 0 0
\(131\) −844.911 + 1463.43i −0.563513 + 0.976034i 0.433673 + 0.901070i \(0.357217\pi\)
−0.997186 + 0.0749634i \(0.976116\pi\)
\(132\) 0 0
\(133\) −1850.37 130.963i −1.20637 0.0853828i
\(134\) 0 0
\(135\) 1979.07 + 642.535i 1.26171 + 0.409634i
\(136\) 0 0
\(137\) −1670.17 + 964.276i −1.04155 + 0.601340i −0.920273 0.391278i \(-0.872033\pi\)
−0.121280 + 0.992618i \(0.538700\pi\)
\(138\) 0 0
\(139\) 402.721i 0.245744i −0.992423 0.122872i \(-0.960790\pi\)
0.992423 0.122872i \(-0.0392104\pi\)
\(140\) 0 0
\(141\) 2932.93 + 1430.21i 1.75176 + 0.854222i
\(142\) 0 0
\(143\) 17.8334 + 30.8883i 0.0104287 + 0.0180630i
\(144\) 0 0
\(145\) 687.637 + 397.007i 0.393828 + 0.227377i
\(146\) 0 0
\(147\) −1475.97 999.022i −0.828134 0.560530i
\(148\) 0 0
\(149\) −1392.62 804.032i −0.765692 0.442073i 0.0656434 0.997843i \(-0.479090\pi\)
−0.831336 + 0.555770i \(0.812423\pi\)
\(150\) 0 0
\(151\) −183.275 317.442i −0.0987731 0.171080i 0.812404 0.583095i \(-0.198158\pi\)
−0.911177 + 0.412015i \(0.864825\pi\)
\(152\) 0 0
\(153\) 1718.42 + 2198.79i 0.908015 + 1.16184i
\(154\) 0 0
\(155\) 4040.82i 2.09398i
\(156\) 0 0
\(157\) 370.608 213.970i 0.188393 0.108769i −0.402837 0.915272i \(-0.631976\pi\)
0.591230 + 0.806503i \(0.298643\pi\)
\(158\) 0 0
\(159\) −2098.82 3111.12i −1.04684 1.55175i
\(160\) 0 0
\(161\) −77.5997 5.49223i −0.0379858 0.00268850i
\(162\) 0 0
\(163\) 116.502 201.787i 0.0559823 0.0969643i −0.836676 0.547698i \(-0.815504\pi\)
0.892658 + 0.450734i \(0.148838\pi\)
\(164\) 0 0
\(165\) 52.2901 3.66051i 0.0246714 0.00172709i
\(166\) 0 0
\(167\) 1013.86 0.469791 0.234896 0.972021i \(-0.424525\pi\)
0.234896 + 0.972021i \(0.424525\pi\)
\(168\) 0 0
\(169\) −552.672 −0.251558
\(170\) 0 0
\(171\) −2507.57 1012.68i −1.12140 0.452874i
\(172\) 0 0
\(173\) 1657.13 2870.23i 0.728262 1.26139i −0.229356 0.973343i \(-0.573662\pi\)
0.957617 0.288044i \(-0.0930048\pi\)
\(174\) 0 0
\(175\) 1457.26 + 984.725i 0.629475 + 0.425361i
\(176\) 0 0
\(177\) −2266.59 + 1529.09i −0.962528 + 0.649341i
\(178\) 0 0
\(179\) −1196.04 + 690.531i −0.499418 + 0.288339i −0.728473 0.685074i \(-0.759770\pi\)
0.229055 + 0.973413i \(0.426436\pi\)
\(180\) 0 0
\(181\) 1558.38i 0.639963i 0.947424 + 0.319981i \(0.103677\pi\)
−0.947424 + 0.319981i \(0.896323\pi\)
\(182\) 0 0
\(183\) 539.537 1106.43i 0.217944 0.446938i
\(184\) 0 0
\(185\) −375.014 649.543i −0.149035 0.258137i
\(186\) 0 0
\(187\) 60.8825 + 35.1505i 0.0238084 + 0.0137458i
\(188\) 0 0
\(189\) −1598.43 2048.49i −0.615176 0.788389i
\(190\) 0 0
\(191\) 1884.48 + 1088.00i 0.713905 + 0.412173i 0.812505 0.582954i \(-0.198103\pi\)
−0.0986002 + 0.995127i \(0.531436\pi\)
\(192\) 0 0
\(193\) −1749.63 3030.45i −0.652546 1.13024i −0.982503 0.186247i \(-0.940368\pi\)
0.329957 0.943996i \(-0.392966\pi\)
\(194\) 0 0
\(195\) −1771.22 + 3632.25i −0.650460 + 1.33390i
\(196\) 0 0
\(197\) 1297.91i 0.469401i 0.972068 + 0.234700i \(0.0754109\pi\)
−0.972068 + 0.234700i \(0.924589\pi\)
\(198\) 0 0
\(199\) 3062.79 1768.30i 1.09103 0.629908i 0.157182 0.987570i \(-0.449759\pi\)
0.933851 + 0.357661i \(0.116426\pi\)
\(200\) 0 0
\(201\) −1286.75 + 868.067i −0.451544 + 0.304621i
\(202\) 0 0
\(203\) −433.898 891.534i −0.150018 0.308244i
\(204\) 0 0
\(205\) 2364.13 4094.79i 0.805452 1.39508i
\(206\) 0 0
\(207\) −105.161 42.4690i −0.0353101 0.0142599i
\(208\) 0 0
\(209\) −68.1271 −0.0225476
\(210\) 0 0
\(211\) −2131.02 −0.695286 −0.347643 0.937627i \(-0.613018\pi\)
−0.347643 + 0.937627i \(0.613018\pi\)
\(212\) 0 0
\(213\) −1969.16 + 137.849i −0.633449 + 0.0443439i
\(214\) 0 0
\(215\) 1249.64 2164.44i 0.396394 0.686574i
\(216\) 0 0
\(217\) −2825.18 + 4180.87i −0.883805 + 1.30791i
\(218\) 0 0
\(219\) 1877.32 + 2782.78i 0.579258 + 0.858644i
\(220\) 0 0
\(221\) −4693.64 + 2709.88i −1.42864 + 0.824824i
\(222\) 0 0
\(223\) 6147.95i 1.84617i −0.384590 0.923087i \(-0.625657\pi\)
0.384590 0.923087i \(-0.374343\pi\)
\(224\) 0 0
\(225\) 1578.89 + 2020.25i 0.467821 + 0.598594i
\(226\) 0 0
\(227\) 1484.97 + 2572.05i 0.434190 + 0.752039i 0.997229 0.0743914i \(-0.0237014\pi\)
−0.563039 + 0.826430i \(0.690368\pi\)
\(228\) 0 0
\(229\) 2479.18 + 1431.35i 0.715409 + 0.413042i 0.813061 0.582179i \(-0.197800\pi\)
−0.0976515 + 0.995221i \(0.531133\pi\)
\(230\) 0 0
\(231\) −56.6617 32.7717i −0.0161388 0.00933429i
\(232\) 0 0
\(233\) 5177.16 + 2989.03i 1.45565 + 0.840421i 0.998793 0.0491180i \(-0.0156410\pi\)
0.456859 + 0.889539i \(0.348974\pi\)
\(234\) 0 0
\(235\) 4656.83 + 8065.86i 1.29267 + 2.23897i
\(236\) 0 0
\(237\) 1974.54 + 962.858i 0.541181 + 0.263900i
\(238\) 0 0
\(239\) 5571.82i 1.50800i 0.656876 + 0.753998i \(0.271877\pi\)
−0.656876 + 0.753998i \(0.728123\pi\)
\(240\) 0 0
\(241\) 3591.11 2073.33i 0.959851 0.554170i 0.0637236 0.997968i \(-0.479702\pi\)
0.896127 + 0.443798i \(0.146369\pi\)
\(242\) 0 0
\(243\) −1296.94 3559.05i −0.342380 0.939561i
\(244\) 0 0
\(245\) −1897.68 4719.90i −0.494852 1.23079i
\(246\) 0 0
\(247\) 2626.08 4548.50i 0.676492 1.17172i
\(248\) 0 0
\(249\) 141.986 + 2028.27i 0.0361366 + 0.516209i
\(250\) 0 0
\(251\) −7611.35 −1.91404 −0.957021 0.290020i \(-0.906338\pi\)
−0.957021 + 0.290020i \(0.906338\pi\)
\(252\) 0 0
\(253\) −2.85707 −0.000709970
\(254\) 0 0
\(255\) 556.235 + 7945.77i 0.136599 + 1.95131i
\(256\) 0 0
\(257\) 1174.76 2034.74i 0.285134 0.493867i −0.687507 0.726177i \(-0.741295\pi\)
0.972642 + 0.232310i \(0.0746284\pi\)
\(258\) 0 0
\(259\) −66.1228 + 934.249i −0.0158636 + 0.224137i
\(260\) 0 0
\(261\) −201.393 1431.39i −0.0477622 0.339468i
\(262\) 0 0
\(263\) 4441.42 2564.25i 1.04133 0.601211i 0.121119 0.992638i \(-0.461352\pi\)
0.920209 + 0.391426i \(0.128018\pi\)
\(264\) 0 0
\(265\) 10711.7i 2.48308i
\(266\) 0 0
\(267\) 2955.75 + 1441.33i 0.677486 + 0.330368i
\(268\) 0 0
\(269\) −20.2532 35.0795i −0.00459055 0.00795107i 0.863721 0.503970i \(-0.168128\pi\)
−0.868312 + 0.496019i \(0.834795\pi\)
\(270\) 0 0
\(271\) 2807.84 + 1621.11i 0.629389 + 0.363378i 0.780515 0.625137i \(-0.214957\pi\)
−0.151127 + 0.988514i \(0.548290\pi\)
\(272\) 0 0
\(273\) 4372.13 2519.77i 0.969279 0.558620i
\(274\) 0 0
\(275\) 55.9391 + 32.2965i 0.0122664 + 0.00708200i
\(276\) 0 0
\(277\) 1107.85 + 1918.85i 0.240304 + 0.416219i 0.960801 0.277239i \(-0.0894195\pi\)
−0.720497 + 0.693458i \(0.756086\pi\)
\(278\) 0 0
\(279\) −5796.12 + 4529.86i −1.24374 + 0.972027i
\(280\) 0 0
\(281\) 5882.33i 1.24879i 0.781108 + 0.624395i \(0.214655\pi\)
−0.781108 + 0.624395i \(0.785345\pi\)
\(282\) 0 0
\(283\) −1323.79 + 764.290i −0.278060 + 0.160538i −0.632545 0.774524i \(-0.717990\pi\)
0.354485 + 0.935062i \(0.384656\pi\)
\(284\) 0 0
\(285\) −4316.85 6398.93i −0.897221 1.32996i
\(286\) 0 0
\(287\) −5308.97 + 2583.80i −1.09191 + 0.531419i
\(288\) 0 0
\(289\) −2884.82 + 4996.65i −0.587180 + 1.01703i
\(290\) 0 0
\(291\) −7423.40 + 519.667i −1.49542 + 0.104685i
\(292\) 0 0
\(293\) 6038.37 1.20398 0.601989 0.798505i \(-0.294375\pi\)
0.601989 + 0.798505i \(0.294375\pi\)
\(294\) 0 0
\(295\) −7803.98 −1.54022
\(296\) 0 0
\(297\) −63.8690 70.9009i −0.0124783 0.0138522i
\(298\) 0 0
\(299\) 110.131 190.752i 0.0213011 0.0368946i
\(300\) 0 0
\(301\) −2806.23 + 1365.76i −0.537371 + 0.261531i
\(302\) 0 0
\(303\) 3838.37 2589.44i 0.727751 0.490956i
\(304\) 0 0
\(305\) 3042.79 1756.76i 0.571246 0.329809i
\(306\) 0 0
\(307\) 7231.98i 1.34447i 0.740340 + 0.672233i \(0.234665\pi\)
−0.740340 + 0.672233i \(0.765335\pi\)
\(308\) 0 0
\(309\) 2807.29 5756.91i 0.516831 1.05987i
\(310\) 0 0
\(311\) −859.000 1487.83i −0.156622 0.271277i 0.777027 0.629468i \(-0.216727\pi\)
−0.933648 + 0.358191i \(0.883394\pi\)
\(312\) 0 0
\(313\) −4787.88 2764.28i −0.864622 0.499190i 0.000935213 1.00000i \(-0.499702\pi\)
−0.865557 + 0.500810i \(0.833036\pi\)
\(314\) 0 0
\(315\) −512.215 7398.59i −0.0916192 1.32338i
\(316\) 0 0
\(317\) 1541.66 + 890.076i 0.273149 + 0.157702i 0.630318 0.776337i \(-0.282925\pi\)
−0.357169 + 0.934040i \(0.616258\pi\)
\(318\) 0 0
\(319\) −18.2073 31.5359i −0.00319565 0.00553502i
\(320\) 0 0
\(321\) −1540.38 + 3158.86i −0.267837 + 0.549254i
\(322\) 0 0
\(323\) 10352.3i 1.78333i
\(324\) 0 0
\(325\) −4312.54 + 2489.85i −0.736052 + 0.424960i
\(326\) 0 0
\(327\) 4283.53 2889.76i 0.724404 0.488697i
\(328\) 0 0
\(329\) 821.097 11601.3i 0.137594 1.94407i
\(330\) 0 0
\(331\) 3219.65 5576.59i 0.534646 0.926034i −0.464534 0.885555i \(-0.653778\pi\)
0.999180 0.0404789i \(-0.0128884\pi\)
\(332\) 0 0
\(333\) −511.299 + 1266.07i −0.0841412 + 0.208349i
\(334\) 0 0
\(335\) −4430.34 −0.722553
\(336\) 0 0
\(337\) 2095.32 0.338692 0.169346 0.985557i \(-0.445834\pi\)
0.169346 + 0.985557i \(0.445834\pi\)
\(338\) 0 0
\(339\) 3669.45 256.876i 0.587898 0.0411551i
\(340\) 0 0
\(341\) −92.6587 + 160.490i −0.0147148 + 0.0254868i
\(342\) 0 0
\(343\) −1336.50 + 6210.26i −0.210392 + 0.977617i
\(344\) 0 0
\(345\) −181.037 268.354i −0.0282513 0.0418774i
\(346\) 0 0
\(347\) −2115.84 + 1221.58i −0.327332 + 0.188985i −0.654656 0.755927i \(-0.727186\pi\)
0.327324 + 0.944912i \(0.393853\pi\)
\(348\) 0 0
\(349\) 1583.72i 0.242907i 0.992597 + 0.121453i \(0.0387555\pi\)
−0.992597 + 0.121453i \(0.961244\pi\)
\(350\) 0 0
\(351\) 7195.64 1531.21i 1.09423 0.232849i
\(352\) 0 0
\(353\) −4032.35 6984.23i −0.607989 1.05307i −0.991571 0.129561i \(-0.958643\pi\)
0.383583 0.923507i \(-0.374690\pi\)
\(354\) 0 0
\(355\) −4879.42 2817.13i −0.729500 0.421177i
\(356\) 0 0
\(357\) 4979.84 8610.05i 0.738267 1.27645i
\(358\) 0 0
\(359\) 5312.56 + 3067.21i 0.781020 + 0.450922i 0.836792 0.547522i \(-0.184429\pi\)
−0.0557719 + 0.998444i \(0.517762\pi\)
\(360\) 0 0
\(361\) 1586.58 + 2748.04i 0.231314 + 0.400648i
\(362\) 0 0
\(363\) 6214.20 + 3030.28i 0.898515 + 0.438150i
\(364\) 0 0
\(365\) 9581.25i 1.37399i
\(366\) 0 0
\(367\) −3351.77 + 1935.14i −0.476733 + 0.275242i −0.719054 0.694954i \(-0.755425\pi\)
0.242321 + 0.970196i \(0.422091\pi\)
\(368\) 0 0
\(369\) −8523.76 + 1199.27i −1.20252 + 0.169191i
\(370\) 0 0
\(371\) −7489.21 + 11083.0i −1.04803 + 1.55094i
\(372\) 0 0
\(373\) −3644.06 + 6311.70i −0.505851 + 0.876160i 0.494126 + 0.869390i \(0.335488\pi\)
−0.999977 + 0.00676929i \(0.997845\pi\)
\(374\) 0 0
\(375\) −161.641 2309.03i −0.0222589 0.317967i
\(376\) 0 0
\(377\) 2807.32 0.383513
\(378\) 0 0
\(379\) −3203.69 −0.434201 −0.217101 0.976149i \(-0.569660\pi\)
−0.217101 + 0.976149i \(0.569660\pi\)
\(380\) 0 0
\(381\) 321.334 + 4590.23i 0.0432085 + 0.617229i
\(382\) 0 0
\(383\) 2096.49 3631.22i 0.279701 0.484456i −0.691610 0.722272i \(-0.743098\pi\)
0.971310 + 0.237816i \(0.0764314\pi\)
\(384\) 0 0
\(385\) −81.7588 167.991i −0.0108229 0.0222379i
\(386\) 0 0
\(387\) −4505.52 + 633.914i −0.591805 + 0.0832653i
\(388\) 0 0
\(389\) −5025.75 + 2901.62i −0.655053 + 0.378195i −0.790390 0.612605i \(-0.790122\pi\)
0.135336 + 0.990800i \(0.456788\pi\)
\(390\) 0 0
\(391\) 434.147i 0.0561529i
\(392\) 0 0
\(393\) 7892.22 + 3848.55i 1.01300 + 0.493978i
\(394\) 0 0
\(395\) 3135.11 + 5430.18i 0.399354 + 0.691701i
\(396\) 0 0
\(397\) 13668.1 + 7891.30i 1.72792 + 0.997614i 0.898478 + 0.439018i \(0.144673\pi\)
0.829440 + 0.558596i \(0.188660\pi\)
\(398\) 0 0
\(399\) −7.40953 + 9638.87i −0.000929675 + 1.20939i
\(400\) 0 0
\(401\) −6782.69 3915.99i −0.844667 0.487669i 0.0141808 0.999899i \(-0.495486\pi\)
−0.858848 + 0.512231i \(0.828819\pi\)
\(402\) 0 0
\(403\) −7143.38 12372.7i −0.882971 1.52935i
\(404\) 0 0
\(405\) 2612.43 10491.6i 0.320525 1.28724i
\(406\) 0 0
\(407\) 34.3972i 0.00418921i
\(408\) 0 0
\(409\) 11913.3 6878.12i 1.44028 0.831543i 0.442408 0.896814i \(-0.354124\pi\)
0.997868 + 0.0652706i \(0.0207911\pi\)
\(410\) 0 0
\(411\) 5604.34 + 8307.39i 0.672607 + 0.997016i
\(412\) 0 0
\(413\) 8074.45 + 5456.22i 0.962028 + 0.650080i
\(414\) 0 0
\(415\) −2901.69 + 5025.87i −0.343225 + 0.594483i
\(416\) 0 0
\(417\) −2087.49 + 146.133i −0.245144 + 0.0171610i
\(418\) 0 0
\(419\) 9478.94 1.10519 0.552597 0.833448i \(-0.313637\pi\)
0.552597 + 0.833448i \(0.313637\pi\)
\(420\) 0 0
\(421\) −7360.46 −0.852082 −0.426041 0.904704i \(-0.640092\pi\)
−0.426041 + 0.904704i \(0.640092\pi\)
\(422\) 0 0
\(423\) 6349.19 15721.7i 0.729806 1.80713i
\(424\) 0 0
\(425\) −4907.62 + 8500.26i −0.560129 + 0.970172i
\(426\) 0 0
\(427\) −4376.51 309.753i −0.496005 0.0351054i
\(428\) 0 0
\(429\) 153.637 103.647i 0.0172906 0.0116646i
\(430\) 0 0
\(431\) −76.0396 + 43.9015i −0.00849814 + 0.00490641i −0.504243 0.863562i \(-0.668228\pi\)
0.495745 + 0.868468i \(0.334895\pi\)
\(432\) 0 0
\(433\) 1782.09i 0.197787i −0.995098 0.0988936i \(-0.968470\pi\)
0.995098 0.0988936i \(-0.0315303\pi\)
\(434\) 0 0
\(435\) 1808.36 3708.40i 0.199320 0.408745i
\(436\) 0 0
\(437\) 210.361 + 364.356i 0.0230273 + 0.0398845i
\(438\) 0 0
\(439\) 8035.74 + 4639.44i 0.873634 + 0.504393i 0.868554 0.495595i \(-0.165050\pi\)
0.00507951 + 0.999987i \(0.498383\pi\)
\(440\) 0 0
\(441\) −4642.83 + 8013.13i −0.501331 + 0.865256i
\(442\) 0 0
\(443\) 4831.09 + 2789.23i 0.518131 + 0.299143i 0.736170 0.676797i \(-0.236633\pi\)
−0.218039 + 0.975940i \(0.569966\pi\)
\(444\) 0 0
\(445\) 4693.05 + 8128.61i 0.499937 + 0.865917i
\(446\) 0 0
\(447\) −3662.34 + 7510.37i −0.387523 + 0.794694i
\(448\) 0 0
\(449\) 5410.98i 0.568730i −0.958716 0.284365i \(-0.908217\pi\)
0.958716 0.284365i \(-0.0917827\pi\)
\(450\) 0 0
\(451\) −187.792 + 108.422i −0.0196071 + 0.0113201i
\(452\) 0 0
\(453\) −1578.95 + 1065.19i −0.163765 + 0.110479i
\(454\) 0 0
\(455\) 14367.4 + 1016.87i 1.48034 + 0.104773i
\(456\) 0 0
\(457\) 3971.67 6879.13i 0.406536 0.704140i −0.587963 0.808888i \(-0.700070\pi\)
0.994499 + 0.104747i \(0.0334034\pi\)
\(458\) 0 0
\(459\) 10773.8 9705.25i 1.09559 0.986933i
\(460\) 0 0
\(461\) −2715.21 −0.274317 −0.137159 0.990549i \(-0.543797\pi\)
−0.137159 + 0.990549i \(0.543797\pi\)
\(462\) 0 0
\(463\) −11587.7 −1.16313 −0.581563 0.813502i \(-0.697558\pi\)
−0.581563 + 0.813502i \(0.697558\pi\)
\(464\) 0 0
\(465\) −20945.5 + 1466.26i −2.08887 + 0.146229i
\(466\) 0 0
\(467\) 1879.76 3255.83i 0.186263 0.322617i −0.757738 0.652558i \(-0.773696\pi\)
0.944001 + 0.329942i \(0.107029\pi\)
\(468\) 0 0
\(469\) 4583.88 + 3097.51i 0.451309 + 0.304968i
\(470\) 0 0
\(471\) −1243.59 1843.39i −0.121659 0.180337i
\(472\) 0 0
\(473\) −99.2639 + 57.3100i −0.00964938 + 0.00557107i
\(474\) 0 0
\(475\) 9511.73i 0.918796i
\(476\) 0 0
\(477\) −15364.8 + 12008.1i −1.47486 + 1.15265i
\(478\) 0 0
\(479\) −2666.10 4617.82i −0.254316 0.440488i 0.710394 0.703804i \(-0.248517\pi\)
−0.964709 + 0.263317i \(0.915184\pi\)
\(480\) 0 0
\(481\) −2296.53 1325.90i −0.217698 0.125688i
\(482\) 0 0
\(483\) −0.310736 + 404.228i −2.92732e−5 + 0.0380808i
\(484\) 0 0
\(485\) −18394.6 10620.1i −1.72217 0.994297i
\(486\) 0 0
\(487\) −521.790 903.766i −0.0485514 0.0840935i 0.840728 0.541457i \(-0.182127\pi\)
−0.889280 + 0.457364i \(0.848794\pi\)
\(488\) 0 0
\(489\) −1088.23 530.662i −0.100637 0.0490744i
\(490\) 0 0
\(491\) 13213.2i 1.21446i −0.794525 0.607232i \(-0.792280\pi\)
0.794525 0.607232i \(-0.207720\pi\)
\(492\) 0 0
\(493\) 4792.05 2766.69i 0.437775 0.252750i
\(494\) 0 0
\(495\) −37.9483 269.716i −0.00344575 0.0244906i
\(496\) 0 0
\(497\) 3078.90 + 6326.26i 0.277883 + 0.570969i
\(498\) 0 0
\(499\) 874.489 1514.66i 0.0784519 0.135883i −0.824130 0.566400i \(-0.808336\pi\)
0.902582 + 0.430518i \(0.141669\pi\)
\(500\) 0 0
\(501\) −367.893 5255.33i −0.0328069 0.468644i
\(502\) 0 0
\(503\) −12215.5 −1.08283 −0.541414 0.840756i \(-0.682111\pi\)
−0.541414 + 0.840756i \(0.682111\pi\)
\(504\) 0 0
\(505\) 13215.7 1.16454
\(506\) 0 0
\(507\) 200.544 + 2864.76i 0.0175670 + 0.250943i
\(508\) 0 0
\(509\) −8483.69 + 14694.2i −0.738768 + 1.27958i 0.214282 + 0.976772i \(0.431259\pi\)
−0.953050 + 0.302812i \(0.902074\pi\)
\(510\) 0 0
\(511\) 6698.82 9913.31i 0.579918 0.858198i
\(512\) 0 0
\(513\) −4339.28 + 13365.4i −0.373458 + 1.15029i
\(514\) 0 0
\(515\) 15832.1 9140.66i 1.35465 0.782108i
\(516\) 0 0
\(517\) 427.136i 0.0363355i
\(518\) 0 0
\(519\) −15479.1 7548.18i −1.30916 0.638397i
\(520\) 0 0
\(521\) 2103.60 + 3643.55i 0.176892 + 0.306385i 0.940814 0.338922i \(-0.110062\pi\)
−0.763923 + 0.645308i \(0.776729\pi\)
\(522\) 0 0
\(523\) −5104.44 2947.05i −0.426772 0.246397i 0.271199 0.962523i \(-0.412580\pi\)
−0.697970 + 0.716127i \(0.745913\pi\)
\(524\) 0 0
\(525\) 4575.50 7910.95i 0.380364 0.657643i
\(526\) 0 0
\(527\) −24387.3 14080.0i −2.01580 1.16382i
\(528\) 0 0
\(529\) −6074.68 10521.7i −0.499275 0.864770i
\(530\) 0 0
\(531\) 8748.44 + 11194.0i 0.714972 + 0.914833i
\(532\) 0 0
\(533\) 16717.2i 1.35854i
\(534\) 0 0
\(535\) −8687.20 + 5015.56i −0.702019 + 0.405311i
\(536\) 0 0
\(537\) 4013.34 + 5949.04i 0.322511 + 0.478064i
\(538\) 0 0
\(539\) −32.8598 + 230.975i −0.00262592 + 0.0184579i
\(540\) 0 0
\(541\) −8059.86 + 13960.1i −0.640518 + 1.10941i 0.344799 + 0.938677i \(0.387947\pi\)
−0.985317 + 0.170734i \(0.945386\pi\)
\(542\) 0 0
\(543\) 8077.79 565.477i 0.638400 0.0446905i
\(544\) 0 0
\(545\) 14748.4 1.15918
\(546\) 0 0
\(547\) 14061.3 1.09912 0.549558 0.835456i \(-0.314796\pi\)
0.549558 + 0.835456i \(0.314796\pi\)
\(548\) 0 0
\(549\) −5930.92 2395.19i −0.461066 0.186201i
\(550\) 0 0
\(551\) −2681.14 + 4643.87i −0.207296 + 0.359048i
\(552\) 0 0
\(553\) 552.786 7810.32i 0.0425079 0.600594i
\(554\) 0 0
\(555\) −3230.81 + 2179.57i −0.247099 + 0.166698i
\(556\) 0 0
\(557\) 7010.50 4047.52i 0.533294 0.307897i −0.209063 0.977902i \(-0.567041\pi\)
0.742357 + 0.670005i \(0.233708\pi\)
\(558\) 0 0
\(559\) 8836.46i 0.668591i
\(560\) 0 0
\(561\) 160.110 328.337i 0.0120496 0.0247102i
\(562\) 0 0
\(563\) 5149.34 + 8918.92i 0.385469 + 0.667651i 0.991834 0.127535i \(-0.0407065\pi\)
−0.606366 + 0.795186i \(0.707373\pi\)
\(564\) 0 0
\(565\) 9092.60 + 5249.62i 0.677042 + 0.390890i
\(566\) 0 0
\(567\) −10038.3 + 9028.71i −0.743505 + 0.668730i
\(568\) 0 0
\(569\) −2507.30 1447.59i −0.184730 0.106654i 0.404783 0.914413i \(-0.367347\pi\)
−0.589513 + 0.807759i \(0.700680\pi\)
\(570\) 0 0
\(571\) 310.198 + 537.278i 0.0227345 + 0.0393772i 0.877169 0.480182i \(-0.159429\pi\)
−0.854434 + 0.519559i \(0.826096\pi\)
\(572\) 0 0
\(573\) 4955.82 10162.9i 0.361313 0.740946i
\(574\) 0 0
\(575\) 398.897i 0.0289307i
\(576\) 0 0
\(577\) −8241.62 + 4758.30i −0.594633 + 0.343312i −0.766927 0.641734i \(-0.778215\pi\)
0.172294 + 0.985046i \(0.444882\pi\)
\(578\) 0 0
\(579\) −15073.4 + 10168.8i −1.08191 + 0.729881i
\(580\) 0 0
\(581\) 6516.14 3171.32i 0.465293 0.226452i
\(582\) 0 0
\(583\) −245.627 + 425.438i −0.0174491 + 0.0302227i
\(584\) 0 0
\(585\) 19470.3 + 7863.05i 1.37607 + 0.555722i
\(586\) 0 0
\(587\) −1093.47 −0.0768861 −0.0384431 0.999261i \(-0.512240\pi\)
−0.0384431 + 0.999261i \(0.512240\pi\)
\(588\) 0 0
\(589\) 27289.2 1.90905
\(590\) 0 0
\(591\) 6727.65 470.962i 0.468255 0.0327796i
\(592\) 0 0
\(593\) −4882.36 + 8456.49i −0.338102 + 0.585610i −0.984076 0.177750i \(-0.943118\pi\)
0.645974 + 0.763360i \(0.276451\pi\)
\(594\) 0 0
\(595\) 25527.1 12423.7i 1.75884 0.856004i
\(596\) 0 0
\(597\) −10277.3 15234.2i −0.704561 1.04438i
\(598\) 0 0
\(599\) −12740.2 + 7355.53i −0.869029 + 0.501734i −0.867026 0.498264i \(-0.833971\pi\)
−0.00200376 + 0.999998i \(0.500638\pi\)
\(600\) 0 0
\(601\) 14047.5i 0.953425i 0.879059 + 0.476713i \(0.158172\pi\)
−0.879059 + 0.476713i \(0.841828\pi\)
\(602\) 0 0
\(603\) 4966.51 + 6354.83i 0.335409 + 0.429169i
\(604\) 0 0
\(605\) 9866.74 + 17089.7i 0.663041 + 1.14842i
\(606\) 0 0
\(607\) −14931.5 8620.73i −0.998439 0.576449i −0.0906530 0.995883i \(-0.528895\pi\)
−0.907786 + 0.419434i \(0.862229\pi\)
\(608\) 0 0
\(609\) −4463.79 + 2572.60i −0.297015 + 0.171177i
\(610\) 0 0
\(611\) 28517.7 + 16464.7i 1.88822 + 1.09017i
\(612\) 0 0
\(613\) −3608.26 6249.69i −0.237742 0.411782i 0.722324 0.691555i \(-0.243074\pi\)
−0.960066 + 0.279773i \(0.909741\pi\)
\(614\) 0 0
\(615\) −22083.0 10768.5i −1.44793 0.706063i
\(616\) 0 0
\(617\) 25662.2i 1.67443i 0.546877 + 0.837213i \(0.315817\pi\)
−0.546877 + 0.837213i \(0.684183\pi\)
\(618\) 0 0
\(619\) −10818.3 + 6245.96i −0.702464 + 0.405568i −0.808265 0.588820i \(-0.799593\pi\)
0.105800 + 0.994387i \(0.466260\pi\)
\(620\) 0 0
\(621\) −181.978 + 560.509i −0.0117593 + 0.0362197i
\(622\) 0 0
\(623\) 827.484 11691.5i 0.0532142 0.751863i
\(624\) 0 0
\(625\) 9238.65 16001.8i 0.591273 1.02412i
\(626\) 0 0
\(627\) 24.7208 + 353.135i 0.00157457 + 0.0224926i
\(628\) 0 0
\(629\) −5226.85 −0.331332
\(630\) 0 0
\(631\) −11952.7 −0.754091 −0.377046 0.926195i \(-0.623060\pi\)
−0.377046 + 0.926195i \(0.623060\pi\)
\(632\) 0 0
\(633\) 773.267 + 11046.1i 0.0485539 + 0.693588i
\(634\) 0 0
\(635\) −6566.90 + 11374.2i −0.410393 + 0.710821i
\(636\) 0 0
\(637\) −14154.4 11097.2i −0.880406 0.690249i
\(638\) 0 0
\(639\) 1429.07 + 10157.1i 0.0884713 + 0.628806i
\(640\) 0 0
\(641\) 12156.4 7018.51i 0.749063 0.432472i −0.0762922 0.997086i \(-0.524308\pi\)
0.825355 + 0.564614i \(0.190975\pi\)
\(642\) 0 0
\(643\) 12089.9i 0.741493i 0.928734 + 0.370747i \(0.120898\pi\)
−0.928734 + 0.370747i \(0.879102\pi\)
\(644\) 0 0
\(645\) −11672.7 5692.06i −0.712579 0.347480i
\(646\) 0 0
\(647\) −2411.27 4176.44i −0.146517 0.253775i 0.783421 0.621492i \(-0.213473\pi\)
−0.929938 + 0.367716i \(0.880140\pi\)
\(648\) 0 0
\(649\) 309.951 + 178.950i 0.0187467 + 0.0108234i
\(650\) 0 0
\(651\) 22696.6 + 13127.1i 1.36643 + 0.790312i
\(652\) 0 0
\(653\) −2592.48 1496.77i −0.155362 0.0896983i 0.420303 0.907384i \(-0.361924\pi\)
−0.575666 + 0.817685i \(0.695257\pi\)
\(654\) 0 0
\(655\) 12531.1 + 21704.4i 0.747525 + 1.29475i
\(656\) 0 0
\(657\) 13743.2 10740.8i 0.816096 0.637806i
\(658\) 0 0
\(659\) 4336.79i 0.256354i 0.991751 + 0.128177i \(0.0409125\pi\)
−0.991751 + 0.128177i \(0.959087\pi\)
\(660\) 0 0
\(661\) 16220.2 9364.73i 0.954451 0.551053i 0.0599904 0.998199i \(-0.480893\pi\)
0.894461 + 0.447146i \(0.147560\pi\)
\(662\) 0 0
\(663\) 15749.7 + 23346.0i 0.922576 + 1.36755i
\(664\) 0 0
\(665\) −15403.7 + 22795.4i −0.898243 + 1.32927i
\(666\) 0 0
\(667\) −112.440 + 194.751i −0.00652726 + 0.0113056i
\(668\) 0 0
\(669\) −31867.7 + 2230.86i −1.84167 + 0.128924i
\(670\) 0 0
\(671\) −161.134 −0.00927053
\(672\) 0 0
\(673\) −19919.4 −1.14092 −0.570459 0.821326i \(-0.693235\pi\)
−0.570459 + 0.821326i \(0.693235\pi\)
\(674\) 0 0
\(675\) 9899.00 8917.22i 0.564463 0.508480i
\(676\) 0 0
\(677\) 15019.2 26014.1i 0.852638 1.47681i −0.0261815 0.999657i \(-0.508335\pi\)
0.878819 0.477155i \(-0.158332\pi\)
\(678\) 0 0
\(679\) 11606.9 + 23848.9i 0.656014 + 1.34792i
\(680\) 0 0
\(681\) 12793.3 8630.60i 0.719882 0.485647i
\(682\) 0 0
\(683\) −3348.00 + 1932.97i −0.187566 + 0.108291i −0.590843 0.806787i \(-0.701205\pi\)
0.403277 + 0.915078i \(0.367871\pi\)
\(684\) 0 0
\(685\) 28602.7i 1.59541i
\(686\) 0 0
\(687\) 6519.78 13370.1i 0.362074 0.742507i
\(688\) 0 0
\(689\) −18936.2 32798.5i −1.04704 1.81353i
\(690\) 0 0
\(691\) −5762.10 3326.75i −0.317222 0.183148i 0.332931 0.942951i \(-0.391962\pi\)
−0.650154 + 0.759803i \(0.725296\pi\)
\(692\) 0 0
\(693\) −149.311 + 305.596i −0.00818449 + 0.0167513i
\(694\) 0 0
\(695\) −5172.64 2986.42i −0.282315 0.162995i
\(696\) 0 0
\(697\) −16475.3 28536.1i −0.895332 1.55076i
\(698\) 0 0
\(699\) 13615.0 27920.2i 0.736717 1.51079i
\(700\) 0 0
\(701\) 21100.3i 1.13687i −0.822728 0.568435i \(-0.807549\pi\)
0.822728 0.568435i \(-0.192451\pi\)
\(702\) 0 0
\(703\) 4386.61 2532.61i 0.235340 0.135874i
\(704\) 0 0
\(705\) 40119.3 27065.3i 2.14324 1.44587i
\(706\) 0 0
\(707\) −13673.7 9239.87i −0.727374 0.491515i
\(708\) 0 0
\(709\) −11715.1 + 20291.1i −0.620548 + 1.07482i 0.368836 + 0.929494i \(0.379756\pi\)
−0.989384 + 0.145326i \(0.953577\pi\)
\(710\) 0 0
\(711\) 4274.46 10584.3i 0.225464 0.558289i
\(712\) 0 0
\(713\) 1144.44 0.0601114
\(714\) 0 0
\(715\) 528.981 0.0276682
\(716\) 0 0
\(717\) 28881.4 2021.81i 1.50432 0.105308i
\(718\) 0 0
\(719\) 4247.60 7357.06i 0.220318 0.381603i −0.734586 0.678515i \(-0.762624\pi\)
0.954905 + 0.296913i \(0.0959571\pi\)
\(720\) 0 0
\(721\) −22771.6 1611.69i −1.17622 0.0832489i
\(722\) 0 0
\(723\) −12050.1 17862.1i −0.619846 0.918808i
\(724\) 0 0
\(725\) 4402.96 2542.05i 0.225547 0.130220i
\(726\) 0 0
\(727\) 2662.18i 0.135811i 0.997692 + 0.0679057i \(0.0216317\pi\)
−0.997692 + 0.0679057i \(0.978368\pi\)
\(728\) 0 0
\(729\) −17977.6 + 8014.07i −0.913358 + 0.407157i
\(730\) 0 0
\(731\) −8708.57 15083.7i −0.440627 0.763188i
\(732\) 0 0
\(733\) −17425.1 10060.4i −0.878052 0.506943i −0.00803608 0.999968i \(-0.502558\pi\)
−0.870015 + 0.493024i \(0.835891\pi\)
\(734\) 0 0
\(735\) −23776.8 + 11549.3i −1.19323 + 0.579593i
\(736\) 0 0
\(737\) 175.960 + 101.590i 0.00879453 + 0.00507752i
\(738\) 0 0
\(739\) 4889.49 + 8468.84i 0.243387 + 0.421558i 0.961677 0.274186i \(-0.0884083\pi\)
−0.718290 + 0.695744i \(0.755075\pi\)
\(740\) 0 0
\(741\) −24529.9 11961.7i −1.21610 0.593015i
\(742\) 0 0
\(743\) 35045.8i 1.73043i 0.501404 + 0.865213i \(0.332817\pi\)
−0.501404 + 0.865213i \(0.667183\pi\)
\(744\) 0 0
\(745\) −20654.3 + 11924.8i −1.01572 + 0.586429i
\(746\) 0 0
\(747\) 10461.9 1471.96i 0.512425 0.0720969i
\(748\) 0 0
\(749\) 12495.0 + 884.348i 0.609554 + 0.0431420i
\(750\) 0 0
\(751\) −10043.6 + 17396.0i −0.488011 + 0.845260i −0.999905 0.0137887i \(-0.995611\pi\)
0.511894 + 0.859049i \(0.328944\pi\)
\(752\) 0 0
\(753\) 2761.88 + 39453.2i 0.133663 + 1.90937i
\(754\) 0 0
\(755\) −5436.39 −0.262054
\(756\) 0 0
\(757\) −2813.68 −0.135092 −0.0675462 0.997716i \(-0.521517\pi\)
−0.0675462 + 0.997716i \(0.521517\pi\)
\(758\) 0 0
\(759\) 1.03672 + 14.8095i 4.95793e−5 + 0.000708237i
\(760\) 0 0
\(761\) 10857.5 18805.7i 0.517191 0.895801i −0.482610 0.875836i \(-0.660311\pi\)
0.999801 0.0199656i \(-0.00635567\pi\)
\(762\) 0 0
\(763\) −15259.6 10311.5i −0.724028 0.489254i
\(764\) 0 0
\(765\) 40984.8 5766.45i 1.93700 0.272531i
\(766\) 0 0
\(767\) −23895.2 + 13795.9i −1.12491 + 0.649467i
\(768\) 0 0
\(769\) 15956.6i 0.748257i 0.927377 + 0.374128i \(0.122058\pi\)
−0.927377 + 0.374128i \(0.877942\pi\)
\(770\) 0 0
\(771\) −10973.3 5351.00i −0.512573 0.249950i
\(772\) 0 0
\(773\) −3910.77 6773.65i −0.181967 0.315176i 0.760583 0.649240i \(-0.224913\pi\)
−0.942550 + 0.334064i \(0.891580\pi\)
\(774\) 0 0
\(775\) −22407.1 12936.8i −1.03857 0.599616i
\(776\) 0 0
\(777\) 4866.64 + 3.74105i 0.224697 + 0.000172728i
\(778\) 0 0
\(779\) 27653.6 + 15965.8i 1.27188 + 0.734320i
\(780\) 0 0
\(781\) 129.197 + 223.776i 0.00591939 + 0.0102527i
\(782\) 0 0
\(783\) −7346.50 + 1563.31i −0.335304 + 0.0713516i
\(784\) 0 0
\(785\) 6346.88i 0.288573i
\(786\) 0 0
\(787\) −21587.3 + 12463.4i −0.977767 + 0.564514i −0.901595 0.432581i \(-0.857603\pi\)
−0.0761716 + 0.997095i \(0.524270\pi\)
\(788\) 0 0
\(789\) −14903.3 22091.5i −0.672463 0.996802i
\(790\) 0 0
\(791\) −5737.42 11788.7i −0.257900 0.529910i
\(792\) 0 0
\(793\) 6211.21 10758.1i 0.278142 0.481756i
\(794\) 0 0
\(795\) −55523.9 + 3886.89i −2.47702 + 0.173401i
\(796\) 0 0
\(797\) 16452.6 0.731217 0.365609 0.930769i \(-0.380861\pi\)
0.365609 + 0.930769i \(0.380861\pi\)
\(798\) 0 0
\(799\) 64905.7 2.87384
\(800\) 0 0
\(801\) 6398.58 15844.0i 0.282250 0.698903i
\(802\) 0 0
\(803\) 219.704 380.539i 0.00965528 0.0167234i
\(804\) 0 0
\(805\) −645.992 + 955.978i −0.0282835 + 0.0418557i
\(806\) 0 0
\(807\) −174.485 + 117.711i −0.00761109 + 0.00513459i
\(808\) 0 0
\(809\) 8105.17 4679.52i 0.352240 0.203366i −0.313431 0.949611i \(-0.601479\pi\)
0.665672 + 0.746245i \(0.268145\pi\)
\(810\) 0 0
\(811\) 6805.42i 0.294662i −0.989087 0.147331i \(-0.952932\pi\)
0.989087 0.147331i \(-0.0470682\pi\)
\(812\) 0 0
\(813\) 7384.10 15142.6i 0.318539 0.653228i
\(814\) 0 0
\(815\) −1727.86 2992.74i −0.0742630 0.128627i
\(816\) 0 0
\(817\) 14617.3 + 8439.27i 0.625940 + 0.361387i
\(818\) 0 0
\(819\) −14647.6 21748.4i −0.624944 0.927902i
\(820\) 0 0
\(821\) 23251.5 + 13424.3i 0.988407 + 0.570657i 0.904798 0.425841i \(-0.140022\pi\)
0.0836095 + 0.996499i \(0.473355\pi\)
\(822\) 0 0
\(823\) 7846.13 + 13589.9i 0.332319 + 0.575594i 0.982966 0.183786i \(-0.0588355\pi\)
−0.650647 + 0.759381i \(0.725502\pi\)
\(824\) 0 0
\(825\) 147.109 301.678i 0.00620811 0.0127310i
\(826\) 0 0
\(827\) 17201.9i 0.723301i 0.932314 + 0.361650i \(0.117787\pi\)
−0.932314 + 0.361650i \(0.882213\pi\)
\(828\) 0 0
\(829\) 16605.1 9586.94i 0.695679 0.401650i −0.110057 0.993925i \(-0.535103\pi\)
0.805736 + 0.592275i \(0.201770\pi\)
\(830\) 0 0
\(831\) 9544.31 6438.78i 0.398422 0.268783i
\(832\) 0 0
\(833\) −35098.0 4993.23i −1.45987 0.207689i
\(834\) 0 0
\(835\) 7518.41 13022.3i 0.311599 0.539706i
\(836\) 0 0
\(837\) 25583.6 + 28400.3i 1.05651 + 1.17283i
\(838\) 0 0
\(839\) 10848.7 0.446410 0.223205 0.974771i \(-0.428348\pi\)
0.223205 + 0.974771i \(0.428348\pi\)
\(840\) 0 0
\(841\) 21522.8 0.882481
\(842\) 0 0
\(843\) 30490.8 2134.48i 1.24574 0.0872068i
\(844\) 0 0
\(845\) −4098.40 + 7098.63i −0.166851 + 0.288994i
\(846\) 0 0
\(847\) 1739.71 24580.4i 0.0705753 0.997158i
\(848\) 0 0
\(849\) 4442.02 + 6584.48i 0.179564 + 0.266171i
\(850\) 0 0
\(851\) 183.963 106.211i 0.00741029 0.00427833i
\(852\) 0 0
\(853\) 42970.3i 1.72482i 0.506207 + 0.862412i \(0.331047\pi\)
−0.506207 + 0.862412i \(0.668953\pi\)
\(854\) 0 0
\(855\) −31602.2 + 24698.2i −1.26406 + 0.987906i
\(856\) 0 0
\(857\) 18302.7 + 31701.2i 0.729532 + 1.26359i 0.957081 + 0.289820i \(0.0935955\pi\)
−0.227549 + 0.973767i \(0.573071\pi\)
\(858\) 0 0
\(859\) −8607.79 4969.71i −0.341902 0.197397i 0.319211 0.947684i \(-0.396582\pi\)
−0.661113 + 0.750286i \(0.729916\pi\)
\(860\) 0 0
\(861\) 15319.5 + 26581.3i 0.606373 + 1.05214i
\(862\) 0 0
\(863\) 11425.0 + 6596.25i 0.450652 + 0.260184i 0.708106 0.706106i \(-0.249550\pi\)
−0.257453 + 0.966291i \(0.582883\pi\)
\(864\) 0 0
\(865\) −24577.2 42569.0i −0.966071 1.67328i
\(866\) 0 0
\(867\) 26946.7 + 13140.2i 1.05555 + 0.514725i
\(868\) 0 0
\(869\) 287.561i 0.0112254i
\(870\) 0 0
\(871\) −13565.4 + 7831.97i −0.527721 + 0.304680i
\(872\) 0 0
\(873\) 5387.35 + 38290.4i 0.208859 + 1.48446i
\(874\) 0 0
\(875\) −7418.13 + 3610.30i −0.286604 + 0.139486i
\(876\) 0 0
\(877\) −4666.37 + 8082.39i −0.179672 + 0.311201i −0.941768 0.336263i \(-0.890837\pi\)
0.762096 + 0.647464i \(0.224170\pi\)
\(878\) 0 0
\(879\) −2191.10 31299.7i −0.0840773 1.20104i
\(880\) 0 0
\(881\) −22655.3 −0.866376 −0.433188 0.901304i \(-0.642611\pi\)
−0.433188 + 0.901304i \(0.642611\pi\)
\(882\) 0 0
\(883\) 32700.9 1.24629 0.623145 0.782107i \(-0.285855\pi\)
0.623145 + 0.782107i \(0.285855\pi\)
\(884\) 0 0
\(885\) 2831.77 + 40451.7i 0.107558 + 1.53646i
\(886\) 0 0
\(887\) 12678.9 21960.5i 0.479950 0.831299i −0.519785 0.854297i \(-0.673988\pi\)
0.999735 + 0.0229985i \(0.00732131\pi\)
\(888\) 0 0
\(889\) 14746.9 7177.10i 0.556349 0.270768i
\(890\) 0 0
\(891\) −344.337 + 356.790i −0.0129469 + 0.0134152i
\(892\) 0 0
\(893\) −54471.8 + 31449.3i −2.04124 + 1.17851i
\(894\) 0 0
\(895\) 20482.8i 0.764989i
\(896\) 0 0
\(897\) −1028.72 501.642i −0.0382920 0.0186726i
\(898\) 0 0
\(899\) 7293.15 + 12632.1i 0.270568 + 0.468637i
\(900\) 0 0
\(901\) −64647.7 37324.4i −2.39037 1.38008i
\(902\) 0 0
\(903\) 8097.63 + 14050.4i 0.298419 + 0.517796i
\(904\) 0 0
\(905\) 20016.1 + 11556.3i 0.735202 + 0.424469i
\(906\) 0 0
\(907\) 22286.5 + 38601.3i 0.815888 + 1.41316i 0.908688 + 0.417475i \(0.137085\pi\)
−0.0928000 + 0.995685i \(0.529582\pi\)
\(908\) 0 0
\(909\) −14815.1 18956.5i −0.540578 0.691690i
\(910\) 0 0
\(911\) 581.992i 0.0211660i 0.999944 + 0.0105830i \(0.00336874\pi\)
−0.999944 + 0.0105830i \(0.996631\pi\)
\(912\) 0 0
\(913\) 230.493 133.075i 0.00835510 0.00482382i
\(914\) 0 0
\(915\) −10210.2 15134.8i −0.368895 0.546819i
\(916\) 0 0
\(917\) 2209.49 31217.9i 0.0795679 1.12421i
\(918\) 0 0
\(919\) 20983.4 36344.4i 0.753188 1.30456i −0.193082 0.981183i \(-0.561848\pi\)
0.946270 0.323377i \(-0.104818\pi\)
\(920\) 0 0
\(921\) 37486.7 2624.22i 1.34118 0.0938880i
\(922\) 0 0
\(923\) −19920.5 −0.710393
\(924\) 0 0
\(925\) −4802.45 −0.170707
\(926\) 0 0
\(927\) −30859.4 12462.5i −1.09337 0.441556i
\(928\) 0 0
\(929\) 35.0320 60.6772i 0.00123720 0.00214290i −0.865406 0.501071i \(-0.832940\pi\)
0.866643 + 0.498928i \(0.166273\pi\)
\(930\) 0 0
\(931\) 31875.2 12815.8i 1.12209 0.451149i
\(932\) 0 0
\(933\) −7400.43 + 4992.48i −0.259677 + 0.175184i
\(934\) 0 0
\(935\) 902.961 521.325i 0.0315829 0.0182344i
\(936\) 0 0
\(937\) 26725.4i 0.931783i −0.884842 0.465892i \(-0.845734\pi\)
0.884842 0.465892i \(-0.154266\pi\)
\(938\) 0 0
\(939\) −12591.2 + 25820.9i −0.437592 + 0.897371i
\(940\) 0 0
\(941\) 19532.8 + 33831.7i 0.676673 + 1.17203i 0.975977 + 0.217875i \(0.0699124\pi\)
−0.299303 + 0.954158i \(0.596754\pi\)
\(942\) 0 0
\(943\) 1159.72 + 669.564i 0.0400484 + 0.0231220i
\(944\) 0 0
\(945\) −38164.5 + 5339.72i −1.31375 + 0.183811i
\(946\) 0 0
\(947\) −25718.0 14848.3i −0.882495 0.509509i −0.0110151 0.999939i \(-0.503506\pi\)
−0.871480 + 0.490430i \(0.836840\pi\)
\(948\) 0 0
\(949\) 16937.8 + 29337.1i 0.579371 + 1.00350i
\(950\) 0 0
\(951\) 4054.27 8314.11i 0.138243 0.283495i
\(952\) 0 0
\(953\) 29039.6i 0.987079i 0.869723 + 0.493539i \(0.164297\pi\)
−0.869723 + 0.493539i \(0.835703\pi\)
\(954\) 0 0
\(955\) 27949.0 16136.4i 0.947026 0.546766i
\(956\) 0 0
\(957\) −156.859 + 105.820i −0.00529835 + 0.00357437i
\(958\) 0 0
\(959\) 19997.9 29594.1i 0.673373 0.996499i
\(960\) 0 0
\(961\) 22220.1 38486.4i 0.745867 1.29188i
\(962\) 0 0
\(963\) 16932.8 + 6838.28i 0.566617 + 0.228827i
\(964\) 0 0
\(965\) −51898.3 −1.73126
\(966\) 0 0
\(967\) −10669.9 −0.354831 −0.177416 0.984136i \(-0.556774\pi\)
−0.177416 + 0.984136i \(0.556774\pi\)
\(968\) 0 0
\(969\) −53660.8 + 3756.46i −1.77898 + 0.124536i
\(970\) 0 0
\(971\) 20680.1 35819.0i 0.683478 1.18382i −0.290435 0.956895i \(-0.593800\pi\)
0.973913 0.226923i \(-0.0728667\pi\)
\(972\) 0 0
\(973\) 3263.92 + 6706.42i 0.107540 + 0.220964i
\(974\) 0 0
\(975\) 14470.9 + 21450.4i 0.475323 + 0.704578i
\(976\) 0 0
\(977\) 31547.9 18214.2i 1.03307 0.596443i 0.115207 0.993342i \(-0.463247\pi\)
0.917862 + 0.396899i \(0.129914\pi\)
\(978\) 0 0
\(979\) 430.459i 0.0140526i
\(980\) 0 0
\(981\) −16533.3 21155.0i −0.538092 0.688508i
\(982\) 0 0
\(983\) 362.416 + 627.723i 0.0117592 + 0.0203675i 0.871845 0.489782i \(-0.162924\pi\)
−0.860086 + 0.510149i \(0.829590\pi\)
\(984\) 0 0
\(985\) 16670.6 + 9624.75i 0.539257 + 0.311340i
\(986\) 0 0
\(987\) −60432.8 46.4555i −1.94893 0.00149817i
\(988\) 0 0
\(989\) 613.008 + 353.921i 0.0197093 + 0.0113792i
\(990\) 0 0
\(991\) −14449.0 25026.5i −0.463157 0.802212i 0.535959 0.844244i \(-0.319950\pi\)
−0.999116 + 0.0420324i \(0.986617\pi\)
\(992\) 0 0
\(993\) −30074.4 14665.4i −0.961109 0.468673i
\(994\) 0 0
\(995\) 52452.2i 1.67120i
\(996\) 0 0
\(997\) −8836.00 + 5101.46i −0.280681 + 0.162051i −0.633732 0.773553i \(-0.718478\pi\)
0.353051 + 0.935604i \(0.385144\pi\)
\(998\) 0 0
\(999\) 6748.16 + 2190.89i 0.213716 + 0.0693862i
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 84.4.k.c.5.4 12
3.2 odd 2 inner 84.4.k.c.5.6 yes 12
4.3 odd 2 336.4.bc.c.257.3 12
7.2 even 3 588.4.f.c.293.10 12
7.3 odd 6 inner 84.4.k.c.17.6 yes 12
7.4 even 3 588.4.k.c.521.1 12
7.5 odd 6 588.4.f.c.293.3 12
7.6 odd 2 588.4.k.c.509.3 12
12.11 even 2 336.4.bc.c.257.1 12
21.2 odd 6 588.4.f.c.293.4 12
21.5 even 6 588.4.f.c.293.9 12
21.11 odd 6 588.4.k.c.521.3 12
21.17 even 6 inner 84.4.k.c.17.4 yes 12
21.20 even 2 588.4.k.c.509.1 12
28.3 even 6 336.4.bc.c.17.1 12
84.59 odd 6 336.4.bc.c.17.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.4.k.c.5.4 12 1.1 even 1 trivial
84.4.k.c.5.6 yes 12 3.2 odd 2 inner
84.4.k.c.17.4 yes 12 21.17 even 6 inner
84.4.k.c.17.6 yes 12 7.3 odd 6 inner
336.4.bc.c.17.1 12 28.3 even 6
336.4.bc.c.17.3 12 84.59 odd 6
336.4.bc.c.257.1 12 12.11 even 2
336.4.bc.c.257.3 12 4.3 odd 2
588.4.f.c.293.3 12 7.5 odd 6
588.4.f.c.293.4 12 21.2 odd 6
588.4.f.c.293.9 12 21.5 even 6
588.4.f.c.293.10 12 7.2 even 3
588.4.k.c.509.1 12 21.20 even 2
588.4.k.c.509.3 12 7.6 odd 2
588.4.k.c.521.1 12 7.4 even 3
588.4.k.c.521.3 12 21.11 odd 6