Properties

Label 336.6.q.a.289.1
Level $336$
Weight $6$
Character 336.289
Analytic conductor $53.889$
Analytic rank $0$
Dimension $2$
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] = [336,6,Mod(193,336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(336, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.193");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 336.q (of order \(3\), degree \(2\), not 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: \(53.8889634572\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 289.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 336.289
Dual form 336.6.q.a.193.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-4.50000 - 7.79423i) q^{3} +(-43.0000 + 74.4782i) q^{5} +(-24.5000 - 127.306i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})\) \(q+(-4.50000 - 7.79423i) q^{3} +(-43.0000 + 74.4782i) q^{5} +(-24.5000 - 127.306i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(17.0000 + 29.4449i) q^{11} -3.00000 q^{13} +774.000 q^{15} +(952.000 + 1648.91i) q^{17} +(-744.500 + 1289.51i) q^{19} +(-882.000 + 763.834i) q^{21} +(-112.000 + 193.990i) q^{23} +(-2135.50 - 3698.79i) q^{25} +729.000 q^{27} -6508.00 q^{29} +(865.500 + 1499.09i) q^{31} +(153.000 - 265.004i) q^{33} +(10535.0 + 3649.43i) q^{35} +(3816.50 - 6610.37i) q^{37} +(13.5000 + 23.3827i) q^{39} +15414.0 q^{41} -18491.0 q^{43} +(-3483.00 - 6032.73i) q^{45} +(9231.00 - 15988.6i) q^{47} +(-15606.5 + 6237.98i) q^{49} +(8568.00 - 14840.2i) q^{51} +(9978.00 + 17282.4i) q^{53} -2924.00 q^{55} +13401.0 q^{57} +(-15914.0 - 27563.9i) q^{59} +(28827.0 - 49929.8i) q^{61} +(9922.50 + 3437.25i) q^{63} +(129.000 - 223.435i) q^{65} +(-30281.5 - 52449.1i) q^{67} +2016.00 q^{69} +44834.0 q^{71} +(-10410.5 - 18031.5i) q^{73} +(-19219.5 + 33289.2i) q^{75} +(3332.00 - 2885.60i) q^{77} +(-15265.5 + 26440.6i) q^{79} +(-3280.50 - 5681.99i) q^{81} -110602. q^{83} -163744. q^{85} +(29286.0 + 50724.8i) q^{87} +(29496.0 - 51088.6i) q^{89} +(73.5000 + 381.917i) q^{91} +(7789.50 - 13491.8i) q^{93} +(-64027.0 - 110898. i) q^{95} -119846. q^{97} -2754.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 9 q^{3} - 86 q^{5} - 49 q^{7} - 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 9 q^{3} - 86 q^{5} - 49 q^{7} - 81 q^{9} + 34 q^{11} - 6 q^{13} + 1548 q^{15} + 1904 q^{17} - 1489 q^{19} - 1764 q^{21} - 224 q^{23} - 4271 q^{25} + 1458 q^{27} - 13016 q^{29} + 1731 q^{31} + 306 q^{33} + 21070 q^{35} + 7633 q^{37} + 27 q^{39} + 30828 q^{41} - 36982 q^{43} - 6966 q^{45} + 18462 q^{47} - 31213 q^{49} + 17136 q^{51} + 19956 q^{53} - 5848 q^{55} + 26802 q^{57} - 31828 q^{59} + 57654 q^{61} + 19845 q^{63} + 258 q^{65} - 60563 q^{67} + 4032 q^{69} + 89668 q^{71} - 20821 q^{73} - 38439 q^{75} + 6664 q^{77} - 30531 q^{79} - 6561 q^{81} - 221204 q^{83} - 327488 q^{85} + 58572 q^{87} + 58992 q^{89} + 147 q^{91} + 15579 q^{93} - 128054 q^{95} - 239692 q^{97} - 5508 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/336\mathbb{Z}\right)^\times\).

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(3\) −4.50000 7.79423i −0.288675 0.500000i
\(4\) 0 0
\(5\) −43.0000 + 74.4782i −0.769207 + 1.33231i 0.168786 + 0.985653i \(0.446015\pi\)
−0.937993 + 0.346654i \(0.887318\pi\)
\(6\) 0 0
\(7\) −24.5000 127.306i −0.188982 0.981981i
\(8\) 0 0
\(9\) −40.5000 + 70.1481i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) 17.0000 + 29.4449i 0.0423611 + 0.0733716i 0.886429 0.462865i \(-0.153179\pi\)
−0.844067 + 0.536237i \(0.819845\pi\)
\(12\) 0 0
\(13\) −3.00000 −0.00492337 −0.00246169 0.999997i \(-0.500784\pi\)
−0.00246169 + 0.999997i \(0.500784\pi\)
\(14\) 0 0
\(15\) 774.000 0.888204
\(16\) 0 0
\(17\) 952.000 + 1648.91i 0.798941 + 1.38381i 0.920306 + 0.391198i \(0.127939\pi\)
−0.121366 + 0.992608i \(0.538727\pi\)
\(18\) 0 0
\(19\) −744.500 + 1289.51i −0.473130 + 0.819486i −0.999527 0.0307534i \(-0.990209\pi\)
0.526397 + 0.850239i \(0.323543\pi\)
\(20\) 0 0
\(21\) −882.000 + 763.834i −0.436436 + 0.377964i
\(22\) 0 0
\(23\) −112.000 + 193.990i −0.0441467 + 0.0764644i −0.887254 0.461280i \(-0.847390\pi\)
0.843108 + 0.537745i \(0.180724\pi\)
\(24\) 0 0
\(25\) −2135.50 3698.79i −0.683360 1.18361i
\(26\) 0 0
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) −6508.00 −1.43699 −0.718493 0.695534i \(-0.755168\pi\)
−0.718493 + 0.695534i \(0.755168\pi\)
\(30\) 0 0
\(31\) 865.500 + 1499.09i 0.161757 + 0.280171i 0.935499 0.353330i \(-0.114951\pi\)
−0.773742 + 0.633501i \(0.781617\pi\)
\(32\) 0 0
\(33\) 153.000 265.004i 0.0244572 0.0423611i
\(34\) 0 0
\(35\) 10535.0 + 3649.43i 1.45367 + 0.503564i
\(36\) 0 0
\(37\) 3816.50 6610.37i 0.458312 0.793819i −0.540560 0.841305i \(-0.681788\pi\)
0.998872 + 0.0474862i \(0.0151210\pi\)
\(38\) 0 0
\(39\) 13.5000 + 23.3827i 0.00142126 + 0.00246169i
\(40\) 0 0
\(41\) 15414.0 1.43204 0.716021 0.698079i \(-0.245961\pi\)
0.716021 + 0.698079i \(0.245961\pi\)
\(42\) 0 0
\(43\) −18491.0 −1.52507 −0.762534 0.646948i \(-0.776045\pi\)
−0.762534 + 0.646948i \(0.776045\pi\)
\(44\) 0 0
\(45\) −3483.00 6032.73i −0.256402 0.444102i
\(46\) 0 0
\(47\) 9231.00 15988.6i 0.609543 1.05576i −0.381773 0.924256i \(-0.624686\pi\)
0.991316 0.131503i \(-0.0419802\pi\)
\(48\) 0 0
\(49\) −15606.5 + 6237.98i −0.928571 + 0.371154i
\(50\) 0 0
\(51\) 8568.00 14840.2i 0.461269 0.798941i
\(52\) 0 0
\(53\) 9978.00 + 17282.4i 0.487926 + 0.845112i 0.999904 0.0138864i \(-0.00442031\pi\)
−0.511978 + 0.858999i \(0.671087\pi\)
\(54\) 0 0
\(55\) −2924.00 −0.130338
\(56\) 0 0
\(57\) 13401.0 0.546324
\(58\) 0 0
\(59\) −15914.0 27563.9i −0.595181 1.03088i −0.993521 0.113646i \(-0.963747\pi\)
0.398340 0.917238i \(-0.369586\pi\)
\(60\) 0 0
\(61\) 28827.0 49929.8i 0.991916 1.71805i 0.386062 0.922473i \(-0.373835\pi\)
0.605854 0.795576i \(-0.292832\pi\)
\(62\) 0 0
\(63\) 9922.50 + 3437.25i 0.314970 + 0.109109i
\(64\) 0 0
\(65\) 129.000 223.435i 0.00378710 0.00655944i
\(66\) 0 0
\(67\) −30281.5 52449.1i −0.824120 1.42742i −0.902590 0.430501i \(-0.858337\pi\)
0.0784702 0.996916i \(-0.474996\pi\)
\(68\) 0 0
\(69\) 2016.00 0.0509762
\(70\) 0 0
\(71\) 44834.0 1.05551 0.527754 0.849397i \(-0.323034\pi\)
0.527754 + 0.849397i \(0.323034\pi\)
\(72\) 0 0
\(73\) −10410.5 18031.5i −0.228646 0.396027i 0.728761 0.684768i \(-0.240097\pi\)
−0.957407 + 0.288741i \(0.906763\pi\)
\(74\) 0 0
\(75\) −19219.5 + 33289.2i −0.394538 + 0.683360i
\(76\) 0 0
\(77\) 3332.00 2885.60i 0.0640440 0.0554637i
\(78\) 0 0
\(79\) −15265.5 + 26440.6i −0.275197 + 0.476655i −0.970185 0.242367i \(-0.922076\pi\)
0.694988 + 0.719021i \(0.255410\pi\)
\(80\) 0 0
\(81\) −3280.50 5681.99i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) −110602. −1.76225 −0.881125 0.472883i \(-0.843213\pi\)
−0.881125 + 0.472883i \(0.843213\pi\)
\(84\) 0 0
\(85\) −163744. −2.45820
\(86\) 0 0
\(87\) 29286.0 + 50724.8i 0.414822 + 0.718493i
\(88\) 0 0
\(89\) 29496.0 51088.6i 0.394719 0.683673i −0.598346 0.801238i \(-0.704175\pi\)
0.993065 + 0.117564i \(0.0375086\pi\)
\(90\) 0 0
\(91\) 73.5000 + 381.917i 0.000930430 + 0.00483466i
\(92\) 0 0
\(93\) 7789.50 13491.8i 0.0933904 0.161757i
\(94\) 0 0
\(95\) −64027.0 110898.i −0.727871 1.26071i
\(96\) 0 0
\(97\) −119846. −1.29328 −0.646642 0.762793i \(-0.723827\pi\)
−0.646642 + 0.762793i \(0.723827\pi\)
\(98\) 0 0
\(99\) −2754.00 −0.0282407
\(100\) 0 0
\(101\) −50005.0 86611.2i −0.487764 0.844833i 0.512137 0.858904i \(-0.328854\pi\)
−0.999901 + 0.0140714i \(0.995521\pi\)
\(102\) 0 0
\(103\) 60845.5 105387.i 0.565113 0.978805i −0.431926 0.901909i \(-0.642166\pi\)
0.997039 0.0768956i \(-0.0245008\pi\)
\(104\) 0 0
\(105\) −18963.0 98534.6i −0.167855 0.872199i
\(106\) 0 0
\(107\) −24324.0 + 42130.4i −0.205388 + 0.355743i −0.950256 0.311469i \(-0.899179\pi\)
0.744868 + 0.667212i \(0.232512\pi\)
\(108\) 0 0
\(109\) 76037.5 + 131701.i 0.613002 + 1.06175i 0.990732 + 0.135834i \(0.0433714\pi\)
−0.377730 + 0.925916i \(0.623295\pi\)
\(110\) 0 0
\(111\) −68697.0 −0.529213
\(112\) 0 0
\(113\) −60886.0 −0.448561 −0.224280 0.974525i \(-0.572003\pi\)
−0.224280 + 0.974525i \(0.572003\pi\)
\(114\) 0 0
\(115\) −9632.00 16683.1i −0.0679160 0.117634i
\(116\) 0 0
\(117\) 121.500 210.444i 0.000820562 0.00142126i
\(118\) 0 0
\(119\) 186592. 161593.i 1.20789 1.04606i
\(120\) 0 0
\(121\) 79947.5 138473.i 0.496411 0.859809i
\(122\) 0 0
\(123\) −69363.0 120140.i −0.413395 0.716021i
\(124\) 0 0
\(125\) 98556.0 0.564167
\(126\) 0 0
\(127\) 151965. 0.836054 0.418027 0.908435i \(-0.362722\pi\)
0.418027 + 0.908435i \(0.362722\pi\)
\(128\) 0 0
\(129\) 83209.5 + 144123.i 0.440249 + 0.762534i
\(130\) 0 0
\(131\) 117751. 203951.i 0.599496 1.03836i −0.393399 0.919368i \(-0.628701\pi\)
0.992895 0.118990i \(-0.0379657\pi\)
\(132\) 0 0
\(133\) 182402. + 63186.1i 0.894132 + 0.309736i
\(134\) 0 0
\(135\) −31347.0 + 54294.6i −0.148034 + 0.256402i
\(136\) 0 0
\(137\) −162754. 281898.i −0.740850 1.28319i −0.952109 0.305760i \(-0.901090\pi\)
0.211259 0.977430i \(-0.432244\pi\)
\(138\) 0 0
\(139\) −3211.00 −0.0140962 −0.00704812 0.999975i \(-0.502244\pi\)
−0.00704812 + 0.999975i \(0.502244\pi\)
\(140\) 0 0
\(141\) −166158. −0.703839
\(142\) 0 0
\(143\) −51.0000 88.3346i −0.000208560 0.000361236i
\(144\) 0 0
\(145\) 279844. 484704.i 1.10534 1.91451i
\(146\) 0 0
\(147\) 118850. + 93569.7i 0.453632 + 0.357143i
\(148\) 0 0
\(149\) 75942.0 131535.i 0.280231 0.485375i −0.691210 0.722654i \(-0.742922\pi\)
0.971442 + 0.237279i \(0.0762556\pi\)
\(150\) 0 0
\(151\) 38324.0 + 66379.1i 0.136782 + 0.236913i 0.926277 0.376844i \(-0.122991\pi\)
−0.789495 + 0.613757i \(0.789657\pi\)
\(152\) 0 0
\(153\) −154224. −0.532627
\(154\) 0 0
\(155\) −148866. −0.497698
\(156\) 0 0
\(157\) 194855. + 337499.i 0.630903 + 1.09276i 0.987368 + 0.158447i \(0.0506487\pi\)
−0.356465 + 0.934309i \(0.616018\pi\)
\(158\) 0 0
\(159\) 89802.0 155542.i 0.281704 0.487926i
\(160\) 0 0
\(161\) 27440.0 + 9505.49i 0.0834295 + 0.0289008i
\(162\) 0 0
\(163\) −56186.0 + 97317.0i −0.165638 + 0.286893i −0.936882 0.349647i \(-0.886302\pi\)
0.771244 + 0.636540i \(0.219635\pi\)
\(164\) 0 0
\(165\) 13158.0 + 22790.3i 0.0376253 + 0.0651689i
\(166\) 0 0
\(167\) −52550.0 −0.145808 −0.0729040 0.997339i \(-0.523227\pi\)
−0.0729040 + 0.997339i \(0.523227\pi\)
\(168\) 0 0
\(169\) −371284. −0.999976
\(170\) 0 0
\(171\) −60304.5 104450.i −0.157710 0.273162i
\(172\) 0 0
\(173\) 67628.0 117135.i 0.171795 0.297558i −0.767252 0.641345i \(-0.778377\pi\)
0.939048 + 0.343787i \(0.111710\pi\)
\(174\) 0 0
\(175\) −418558. + 362482.i −1.03314 + 0.894728i
\(176\) 0 0
\(177\) −143226. + 248075.i −0.343628 + 0.595181i
\(178\) 0 0
\(179\) 125319. + 217059.i 0.292337 + 0.506343i 0.974362 0.224986i \(-0.0722336\pi\)
−0.682025 + 0.731329i \(0.738900\pi\)
\(180\) 0 0
\(181\) 199233. 0.452027 0.226014 0.974124i \(-0.427431\pi\)
0.226014 + 0.974124i \(0.427431\pi\)
\(182\) 0 0
\(183\) −518886. −1.14537
\(184\) 0 0
\(185\) 328219. + 568492.i 0.705074 + 1.22122i
\(186\) 0 0
\(187\) −32368.0 + 56063.0i −0.0676880 + 0.117239i
\(188\) 0 0
\(189\) −17860.5 92805.9i −0.0363696 0.188982i
\(190\) 0 0
\(191\) −119385. + 206781.i −0.236792 + 0.410135i −0.959792 0.280713i \(-0.909429\pi\)
0.723000 + 0.690848i \(0.242763\pi\)
\(192\) 0 0
\(193\) −42845.5 74210.6i −0.0827965 0.143408i 0.821654 0.569987i \(-0.193052\pi\)
−0.904450 + 0.426579i \(0.859718\pi\)
\(194\) 0 0
\(195\) −2322.00 −0.00437296
\(196\) 0 0
\(197\) −71408.0 −0.131094 −0.0655468 0.997849i \(-0.520879\pi\)
−0.0655468 + 0.997849i \(0.520879\pi\)
\(198\) 0 0
\(199\) −355676. 616049.i −0.636681 1.10276i −0.986156 0.165818i \(-0.946973\pi\)
0.349475 0.936946i \(-0.386360\pi\)
\(200\) 0 0
\(201\) −272534. + 472042.i −0.475806 + 0.824120i
\(202\) 0 0
\(203\) 159446. + 828506.i 0.271565 + 1.41109i
\(204\) 0 0
\(205\) −662802. + 1.14801e6i −1.10154 + 1.90792i
\(206\) 0 0
\(207\) −9072.00 15713.2i −0.0147156 0.0254881i
\(208\) 0 0
\(209\) −50626.0 −0.0801693
\(210\) 0 0
\(211\) 260260. 0.402440 0.201220 0.979546i \(-0.435509\pi\)
0.201220 + 0.979546i \(0.435509\pi\)
\(212\) 0 0
\(213\) −201753. 349446.i −0.304699 0.527754i
\(214\) 0 0
\(215\) 795113. 1.37718e6i 1.17309 2.03186i
\(216\) 0 0
\(217\) 169638. 146911.i 0.244553 0.211790i
\(218\) 0 0
\(219\) −93694.5 + 162284.i −0.132009 + 0.228646i
\(220\) 0 0
\(221\) −2856.00 4946.74i −0.00393349 0.00681300i
\(222\) 0 0
\(223\) −105656. −0.142276 −0.0711381 0.997466i \(-0.522663\pi\)
−0.0711381 + 0.997466i \(0.522663\pi\)
\(224\) 0 0
\(225\) 345951. 0.455573
\(226\) 0 0
\(227\) 327375. + 567030.i 0.421678 + 0.730368i 0.996104 0.0881890i \(-0.0281080\pi\)
−0.574426 + 0.818557i \(0.694775\pi\)
\(228\) 0 0
\(229\) −278856. + 482994.i −0.351392 + 0.608629i −0.986494 0.163800i \(-0.947625\pi\)
0.635101 + 0.772429i \(0.280958\pi\)
\(230\) 0 0
\(231\) −37485.0 12985.2i −0.0462197 0.0160110i
\(232\) 0 0
\(233\) 623797. 1.08045e6i 0.752755 1.30381i −0.193728 0.981055i \(-0.562058\pi\)
0.946483 0.322754i \(-0.104609\pi\)
\(234\) 0 0
\(235\) 793866. + 1.37502e6i 0.937729 + 1.62420i
\(236\) 0 0
\(237\) 274779. 0.317770
\(238\) 0 0
\(239\) 496926. 0.562726 0.281363 0.959601i \(-0.409214\pi\)
0.281363 + 0.959601i \(0.409214\pi\)
\(240\) 0 0
\(241\) 138809. + 240424.i 0.153948 + 0.266646i 0.932676 0.360716i \(-0.117468\pi\)
−0.778727 + 0.627363i \(0.784134\pi\)
\(242\) 0 0
\(243\) −29524.5 + 51137.9i −0.0320750 + 0.0555556i
\(244\) 0 0
\(245\) 206486. 1.43058e6i 0.219774 1.52264i
\(246\) 0 0
\(247\) 2233.50 3868.54i 0.00232940 0.00403463i
\(248\) 0 0
\(249\) 497709. + 862057.i 0.508718 + 0.881125i
\(250\) 0 0
\(251\) 308328. 0.308908 0.154454 0.988000i \(-0.450638\pi\)
0.154454 + 0.988000i \(0.450638\pi\)
\(252\) 0 0
\(253\) −7616.00 −0.00748041
\(254\) 0 0
\(255\) 736848. + 1.27626e6i 0.709623 + 1.22910i
\(256\) 0 0
\(257\) −204381. + 353998.i −0.193022 + 0.334325i −0.946250 0.323435i \(-0.895162\pi\)
0.753228 + 0.657759i \(0.228496\pi\)
\(258\) 0 0
\(259\) −935042. 323908.i −0.866128 0.300035i
\(260\) 0 0
\(261\) 263574. 456524.i 0.239498 0.414822i
\(262\) 0 0
\(263\) 544062. + 942343.i 0.485019 + 0.840078i 0.999852 0.0172127i \(-0.00547924\pi\)
−0.514833 + 0.857291i \(0.672146\pi\)
\(264\) 0 0
\(265\) −1.71622e6 −1.50126
\(266\) 0 0
\(267\) −530928. −0.455782
\(268\) 0 0
\(269\) 334145. + 578756.i 0.281549 + 0.487657i 0.971766 0.235945i \(-0.0758184\pi\)
−0.690217 + 0.723602i \(0.742485\pi\)
\(270\) 0 0
\(271\) −415332. + 719376.i −0.343536 + 0.595022i −0.985087 0.172059i \(-0.944958\pi\)
0.641551 + 0.767081i \(0.278291\pi\)
\(272\) 0 0
\(273\) 2646.00 2291.50i 0.00214874 0.00186086i
\(274\) 0 0
\(275\) 72607.0 125759.i 0.0578958 0.100278i
\(276\) 0 0
\(277\) −462537. 801137.i −0.362198 0.627346i 0.626124 0.779724i \(-0.284640\pi\)
−0.988322 + 0.152377i \(0.951307\pi\)
\(278\) 0 0
\(279\) −140211. −0.107838
\(280\) 0 0
\(281\) 1.33635e6 1.00961 0.504805 0.863233i \(-0.331564\pi\)
0.504805 + 0.863233i \(0.331564\pi\)
\(282\) 0 0
\(283\) −496478. 859926.i −0.368497 0.638256i 0.620833 0.783942i \(-0.286794\pi\)
−0.989331 + 0.145686i \(0.953461\pi\)
\(284\) 0 0
\(285\) −576243. + 998082.i −0.420236 + 0.727871i
\(286\) 0 0
\(287\) −377643. 1.96229e6i −0.270630 1.40624i
\(288\) 0 0
\(289\) −1.10268e6 + 1.90990e6i −0.776613 + 1.34513i
\(290\) 0 0
\(291\) 539307. + 934107.i 0.373339 + 0.646642i
\(292\) 0 0
\(293\) 563544. 0.383494 0.191747 0.981444i \(-0.438585\pi\)
0.191747 + 0.981444i \(0.438585\pi\)
\(294\) 0 0
\(295\) 2.73721e6 1.83127
\(296\) 0 0
\(297\) 12393.0 + 21465.3i 0.00815240 + 0.0141204i
\(298\) 0 0
\(299\) 336.000 581.969i 0.000217351 0.000376463i
\(300\) 0 0
\(301\) 453029. + 2.35401e6i 0.288211 + 1.49759i
\(302\) 0 0
\(303\) −450045. + 779501.i −0.281611 + 0.487764i
\(304\) 0 0
\(305\) 2.47912e6 + 4.29397e6i 1.52598 + 2.64307i
\(306\) 0 0
\(307\) −2.82703e6 −1.71193 −0.855963 0.517037i \(-0.827035\pi\)
−0.855963 + 0.517037i \(0.827035\pi\)
\(308\) 0 0
\(309\) −1.09522e6 −0.652536
\(310\) 0 0
\(311\) −563657. 976283.i −0.330456 0.572367i 0.652145 0.758094i \(-0.273869\pi\)
−0.982601 + 0.185727i \(0.940536\pi\)
\(312\) 0 0
\(313\) 1.18006e6 2.04393e6i 0.680840 1.17925i −0.293885 0.955841i \(-0.594948\pi\)
0.974725 0.223408i \(-0.0717183\pi\)
\(314\) 0 0
\(315\) −682668. + 591208.i −0.387644 + 0.335710i
\(316\) 0 0
\(317\) 1.11210e6 1.92621e6i 0.621578 1.07660i −0.367614 0.929979i \(-0.619825\pi\)
0.989192 0.146626i \(-0.0468415\pi\)
\(318\) 0 0
\(319\) −110636. 191627.i −0.0608723 0.105434i
\(320\) 0 0
\(321\) 437832. 0.237162
\(322\) 0 0
\(323\) −2.83506e6 −1.51201
\(324\) 0 0
\(325\) 6406.50 + 11096.4i 0.00336444 + 0.00582738i
\(326\) 0 0
\(327\) 684338. 1.18531e6i 0.353917 0.613002i
\(328\) 0 0
\(329\) −2.26160e6 783439.i −1.15193 0.399039i
\(330\) 0 0
\(331\) −1.85152e6 + 3.20693e6i −0.928877 + 1.60886i −0.143673 + 0.989625i \(0.545891\pi\)
−0.785204 + 0.619237i \(0.787442\pi\)
\(332\) 0 0
\(333\) 309136. + 535440.i 0.152771 + 0.264606i
\(334\) 0 0
\(335\) 5.20842e6 2.53568
\(336\) 0 0
\(337\) 1.21432e6 0.582452 0.291226 0.956654i \(-0.405937\pi\)
0.291226 + 0.956654i \(0.405937\pi\)
\(338\) 0 0
\(339\) 273987. + 474559.i 0.129488 + 0.224280i
\(340\) 0 0
\(341\) −29427.0 + 50969.1i −0.0137044 + 0.0237367i
\(342\) 0 0
\(343\) 1.17649e6 + 1.83397e6i 0.539949 + 0.841698i
\(344\) 0 0
\(345\) −86688.0 + 150148.i −0.0392113 + 0.0679160i
\(346\) 0 0
\(347\) −488952. 846890.i −0.217993 0.377575i 0.736201 0.676763i \(-0.236618\pi\)
−0.954194 + 0.299188i \(0.903284\pi\)
\(348\) 0 0
\(349\) 511282. 0.224697 0.112348 0.993669i \(-0.464163\pi\)
0.112348 + 0.993669i \(0.464163\pi\)
\(350\) 0 0
\(351\) −2187.00 −0.000947504
\(352\) 0 0
\(353\) −1.51376e6 2.62191e6i −0.646577 1.11990i −0.983935 0.178528i \(-0.942866\pi\)
0.337357 0.941377i \(-0.390467\pi\)
\(354\) 0 0
\(355\) −1.92786e6 + 3.33915e6i −0.811905 + 1.40626i
\(356\) 0 0
\(357\) −2.09916e6 727170.i −0.871716 0.301971i
\(358\) 0 0
\(359\) 2.29728e6 3.97900e6i 0.940757 1.62944i 0.176725 0.984260i \(-0.443450\pi\)
0.764032 0.645179i \(-0.223217\pi\)
\(360\) 0 0
\(361\) 129489. + 224282.i 0.0522956 + 0.0905786i
\(362\) 0 0
\(363\) −1.43906e6 −0.573206
\(364\) 0 0
\(365\) 1.79061e6 0.703506
\(366\) 0 0
\(367\) −556365. 963653.i −0.215623 0.373470i 0.737842 0.674973i \(-0.235845\pi\)
−0.953465 + 0.301503i \(0.902512\pi\)
\(368\) 0 0
\(369\) −624267. + 1.08126e6i −0.238674 + 0.413395i
\(370\) 0 0
\(371\) 1.95569e6 1.69368e6i 0.737675 0.638845i
\(372\) 0 0
\(373\) 1.22947e6 2.12951e6i 0.457559 0.792516i −0.541272 0.840848i \(-0.682057\pi\)
0.998831 + 0.0483315i \(0.0153904\pi\)
\(374\) 0 0
\(375\) −443502. 768168.i −0.162861 0.282084i
\(376\) 0 0
\(377\) 19524.0 0.00707482
\(378\) 0 0
\(379\) 4.10130e6 1.46664 0.733320 0.679884i \(-0.237970\pi\)
0.733320 + 0.679884i \(0.237970\pi\)
\(380\) 0 0
\(381\) −683842. 1.18445e6i −0.241348 0.418027i
\(382\) 0 0
\(383\) −1.29707e6 + 2.24658e6i −0.451820 + 0.782575i −0.998499 0.0547673i \(-0.982558\pi\)
0.546679 + 0.837342i \(0.315892\pi\)
\(384\) 0 0
\(385\) 71638.0 + 372242.i 0.0246315 + 0.127989i
\(386\) 0 0
\(387\) 748886. 1.29711e6i 0.254178 0.440249i
\(388\) 0 0
\(389\) −1.11703e6 1.93476e6i −0.374276 0.648265i 0.615942 0.787791i \(-0.288775\pi\)
−0.990218 + 0.139526i \(0.955442\pi\)
\(390\) 0 0
\(391\) −426496. −0.141082
\(392\) 0 0
\(393\) −2.11952e6 −0.692238
\(394\) 0 0
\(395\) −1.31283e6 2.27389e6i −0.423367 0.733293i
\(396\) 0 0
\(397\) 1.03200e6 1.78747e6i 0.328627 0.569198i −0.653613 0.756829i \(-0.726748\pi\)
0.982240 + 0.187631i \(0.0600809\pi\)
\(398\) 0 0
\(399\) −328324. 1.70602e6i −0.103245 0.536479i
\(400\) 0 0
\(401\) −576414. + 998378.i −0.179008 + 0.310052i −0.941541 0.336898i \(-0.890622\pi\)
0.762533 + 0.646950i \(0.223956\pi\)
\(402\) 0 0
\(403\) −2596.50 4497.27i −0.000796390 0.00137939i
\(404\) 0 0
\(405\) 564246. 0.170935
\(406\) 0 0
\(407\) 259522. 0.0776583
\(408\) 0 0
\(409\) −2.96706e6 5.13910e6i −0.877038 1.51907i −0.854576 0.519327i \(-0.826183\pi\)
−0.0224623 0.999748i \(-0.507151\pi\)
\(410\) 0 0
\(411\) −1.46479e6 + 2.53708e6i −0.427730 + 0.740850i
\(412\) 0 0
\(413\) −3.11914e6 + 2.70126e6i −0.899830 + 0.779275i
\(414\) 0 0
\(415\) 4.75589e6 8.23744e6i 1.35554 2.34786i
\(416\) 0 0
\(417\) 14449.5 + 25027.3i 0.00406923 + 0.00704812i
\(418\) 0 0
\(419\) −771666. −0.214731 −0.107365 0.994220i \(-0.534241\pi\)
−0.107365 + 0.994220i \(0.534241\pi\)
\(420\) 0 0
\(421\) −2.87542e6 −0.790671 −0.395336 0.918537i \(-0.629372\pi\)
−0.395336 + 0.918537i \(0.629372\pi\)
\(422\) 0 0
\(423\) 747711. + 1.29507e6i 0.203181 + 0.351920i
\(424\) 0 0
\(425\) 4.06599e6 7.04250e6i 1.09193 1.89128i
\(426\) 0 0
\(427\) −7.06261e6 2.44656e6i −1.87454 0.649361i
\(428\) 0 0
\(429\) −459.000 + 795.011i −0.000120412 + 0.000208560i
\(430\) 0 0
\(431\) −68931.0 119392.i −0.0178740 0.0309587i 0.856950 0.515399i \(-0.172356\pi\)
−0.874824 + 0.484441i \(0.839023\pi\)
\(432\) 0 0
\(433\) 1.56526e6 0.401204 0.200602 0.979673i \(-0.435710\pi\)
0.200602 + 0.979673i \(0.435710\pi\)
\(434\) 0 0
\(435\) −5.03719e6 −1.27634
\(436\) 0 0
\(437\) −166768. 288851.i −0.0417743 0.0723552i
\(438\) 0 0
\(439\) 2.44079e6 4.22757e6i 0.604462 1.04696i −0.387675 0.921796i \(-0.626722\pi\)
0.992136 0.125162i \(-0.0399451\pi\)
\(440\) 0 0
\(441\) 194481. 1.34740e6i 0.0476190 0.329914i
\(442\) 0 0
\(443\) −650758. + 1.12715e6i −0.157547 + 0.272879i −0.933984 0.357316i \(-0.883692\pi\)
0.776437 + 0.630195i \(0.217025\pi\)
\(444\) 0 0
\(445\) 2.53666e6 + 4.39362e6i 0.607242 + 1.05177i
\(446\) 0 0
\(447\) −1.36696e6 −0.323583
\(448\) 0 0
\(449\) −3.13141e6 −0.733034 −0.366517 0.930411i \(-0.619450\pi\)
−0.366517 + 0.930411i \(0.619450\pi\)
\(450\) 0 0
\(451\) 262038. + 453863.i 0.0606629 + 0.105071i
\(452\) 0 0
\(453\) 344916. 597412.i 0.0789710 0.136782i
\(454\) 0 0
\(455\) −31605.0 10948.3i −0.00715694 0.00247924i
\(456\) 0 0
\(457\) −3.24634e6 + 5.62283e6i −0.727116 + 1.25940i 0.230981 + 0.972958i \(0.425807\pi\)
−0.958097 + 0.286444i \(0.907527\pi\)
\(458\) 0 0
\(459\) 694008. + 1.20206e6i 0.153756 + 0.266314i
\(460\) 0 0
\(461\) 5.34717e6 1.17185 0.585925 0.810365i \(-0.300731\pi\)
0.585925 + 0.810365i \(0.300731\pi\)
\(462\) 0 0
\(463\) −3.37285e6 −0.731215 −0.365607 0.930769i \(-0.619139\pi\)
−0.365607 + 0.930769i \(0.619139\pi\)
\(464\) 0 0
\(465\) 669897. + 1.16030e6i 0.143673 + 0.248849i
\(466\) 0 0
\(467\) 1.11726e6 1.93515e6i 0.237062 0.410604i −0.722808 0.691049i \(-0.757149\pi\)
0.959870 + 0.280445i \(0.0904821\pi\)
\(468\) 0 0
\(469\) −5.93517e6 + 5.14001e6i −1.24595 + 1.07903i
\(470\) 0 0
\(471\) 1.75369e6 3.03749e6i 0.364252 0.630903i
\(472\) 0 0
\(473\) −314347. 544465.i −0.0646036 0.111897i
\(474\) 0 0
\(475\) 6.35952e6 1.29327
\(476\) 0 0
\(477\) −1.61644e6 −0.325284
\(478\) 0 0
\(479\) −1.26068e6 2.18357e6i −0.251054 0.434838i 0.712762 0.701406i \(-0.247444\pi\)
−0.963816 + 0.266568i \(0.914110\pi\)
\(480\) 0 0
\(481\) −11449.5 + 19831.1i −0.00225644 + 0.00390827i
\(482\) 0 0
\(483\) −49392.0 256648.i −0.00963360 0.0500577i
\(484\) 0 0
\(485\) 5.15338e6 8.92591e6i 0.994804 1.72305i
\(486\) 0 0
\(487\) 958358. + 1.65993e6i 0.183107 + 0.317151i 0.942937 0.332971i \(-0.108051\pi\)
−0.759830 + 0.650122i \(0.774718\pi\)
\(488\) 0 0
\(489\) 1.01135e6 0.191262
\(490\) 0 0
\(491\) −5.82875e6 −1.09112 −0.545559 0.838073i \(-0.683683\pi\)
−0.545559 + 0.838073i \(0.683683\pi\)
\(492\) 0 0
\(493\) −6.19562e6 1.07311e7i −1.14807 1.98851i
\(494\) 0 0
\(495\) 118422. 205113.i 0.0217230 0.0376253i
\(496\) 0 0
\(497\) −1.09843e6 5.70763e6i −0.199472 1.03649i
\(498\) 0 0
\(499\) −5.00243e6 + 8.66446e6i −0.899352 + 1.55772i −0.0710277 + 0.997474i \(0.522628\pi\)
−0.828324 + 0.560249i \(0.810705\pi\)
\(500\) 0 0
\(501\) 236475. + 409587.i 0.0420912 + 0.0729040i
\(502\) 0 0
\(503\) −1.13666e6 −0.200313 −0.100157 0.994972i \(-0.531934\pi\)
−0.100157 + 0.994972i \(0.531934\pi\)
\(504\) 0 0
\(505\) 8.60086e6 1.50077
\(506\) 0 0
\(507\) 1.67078e6 + 2.89387e6i 0.288668 + 0.499988i
\(508\) 0 0
\(509\) −2.00968e6 + 3.48088e6i −0.343822 + 0.595517i −0.985139 0.171759i \(-0.945055\pi\)
0.641317 + 0.767276i \(0.278388\pi\)
\(510\) 0 0
\(511\) −2.04046e6 + 1.76709e6i −0.345681 + 0.299368i
\(512\) 0 0
\(513\) −542740. + 940054.i −0.0910540 + 0.157710i
\(514\) 0 0
\(515\) 5.23271e6 + 9.06332e6i 0.869378 + 1.50581i
\(516\) 0 0
\(517\) 627708. 0.103284
\(518\) 0 0
\(519\) −1.21730e6 −0.198372
\(520\) 0 0
\(521\) 2.76014e6 + 4.78070e6i 0.445488 + 0.771609i 0.998086 0.0618395i \(-0.0196967\pi\)
−0.552598 + 0.833448i \(0.686363\pi\)
\(522\) 0 0
\(523\) 4.47236e6 7.74636e6i 0.714961 1.23835i −0.248013 0.968757i \(-0.579778\pi\)
0.962974 0.269593i \(-0.0868891\pi\)
\(524\) 0 0
\(525\) 4.70878e6 + 1.63117e6i 0.745607 + 0.258286i
\(526\) 0 0
\(527\) −1.64791e6 + 2.85427e6i −0.258468 + 0.447680i
\(528\) 0 0
\(529\) 3.19308e6 + 5.53058e6i 0.496102 + 0.859274i
\(530\) 0 0
\(531\) 2.57807e6 0.396788
\(532\) 0 0
\(533\) −46242.0 −0.00705048
\(534\) 0 0
\(535\) −2.09186e6 3.62321e6i −0.315972 0.547280i
\(536\) 0 0
\(537\) 1.12787e6 1.95353e6i 0.168781 0.292337i
\(538\) 0 0
\(539\) −448987. 353486.i −0.0665674 0.0524083i
\(540\) 0 0
\(541\) −424528. + 735305.i −0.0623611 + 0.108013i −0.895520 0.445021i \(-0.853196\pi\)
0.833159 + 0.553033i \(0.186530\pi\)
\(542\) 0 0
\(543\) −896548. 1.55287e6i −0.130489 0.226014i
\(544\) 0 0
\(545\) −1.30784e7 −1.88610
\(546\) 0 0
\(547\) −8.61340e6 −1.23085 −0.615426 0.788194i \(-0.711016\pi\)
−0.615426 + 0.788194i \(0.711016\pi\)
\(548\) 0 0
\(549\) 2.33499e6 + 4.04432e6i 0.330639 + 0.572683i
\(550\) 0 0
\(551\) 4.84521e6 8.39214e6i 0.679882 1.17759i
\(552\) 0 0
\(553\) 3.74005e6 + 1.29559e6i 0.520073 + 0.180159i
\(554\) 0 0
\(555\) 2.95397e6 5.11643e6i 0.407074 0.705074i
\(556\) 0 0
\(557\) −3.89940e6 6.75395e6i −0.532549 0.922401i −0.999278 0.0380009i \(-0.987901\pi\)
0.466729 0.884400i \(-0.345432\pi\)
\(558\) 0 0
\(559\) 55473.0 0.00750848
\(560\) 0 0
\(561\) 582624. 0.0781594
\(562\) 0 0
\(563\) 536357. + 928998.i 0.0713153 + 0.123522i 0.899478 0.436966i \(-0.143947\pi\)
−0.828163 + 0.560488i \(0.810614\pi\)
\(564\) 0 0
\(565\) 2.61810e6 4.53468e6i 0.345036 0.597620i
\(566\) 0 0
\(567\) −642978. + 556835.i −0.0839921 + 0.0727393i
\(568\) 0 0
\(569\) 507621. 879225.i 0.0657293 0.113846i −0.831288 0.555842i \(-0.812396\pi\)
0.897017 + 0.441996i \(0.145729\pi\)
\(570\) 0 0
\(571\) −3.39047e6 5.87246e6i −0.435180 0.753754i 0.562130 0.827049i \(-0.309982\pi\)
−0.997310 + 0.0732945i \(0.976649\pi\)
\(572\) 0 0
\(573\) 2.14893e6 0.273423
\(574\) 0 0
\(575\) 956704. 0.120672
\(576\) 0 0
\(577\) 1.65268e6 + 2.86253e6i 0.206657 + 0.357941i 0.950659 0.310236i \(-0.100408\pi\)
−0.744002 + 0.668177i \(0.767075\pi\)
\(578\) 0 0
\(579\) −385610. + 667895.i −0.0478026 + 0.0827965i
\(580\) 0 0
\(581\) 2.70975e6 + 1.40803e7i 0.333034 + 1.73050i
\(582\) 0 0
\(583\) −339252. + 587602.i −0.0413381 + 0.0715998i
\(584\) 0 0
\(585\) 10449.0 + 18098.2i 0.00126237 + 0.00218648i
\(586\) 0 0
\(587\) 1.19833e7 1.43543 0.717715 0.696337i \(-0.245188\pi\)
0.717715 + 0.696337i \(0.245188\pi\)
\(588\) 0 0
\(589\) −2.57746e6 −0.306128
\(590\) 0 0
\(591\) 321336. + 556570.i 0.0378434 + 0.0655468i
\(592\) 0 0
\(593\) 2.65510e6 4.59877e6i 0.310059 0.537038i −0.668316 0.743877i \(-0.732985\pi\)
0.978375 + 0.206840i \(0.0663179\pi\)
\(594\) 0 0
\(595\) 4.01173e6 + 2.08456e7i 0.464557 + 2.41391i
\(596\) 0 0
\(597\) −3.20108e6 + 5.54444e6i −0.367588 + 0.636681i
\(598\) 0 0
\(599\) 1.21118e6 + 2.09782e6i 0.137924 + 0.238892i 0.926711 0.375775i \(-0.122624\pi\)
−0.788786 + 0.614667i \(0.789290\pi\)
\(600\) 0 0
\(601\) −7.10659e6 −0.802556 −0.401278 0.915956i \(-0.631434\pi\)
−0.401278 + 0.915956i \(0.631434\pi\)
\(602\) 0 0
\(603\) 4.90560e6 0.549413
\(604\) 0 0
\(605\) 6.87548e6 + 1.19087e7i 0.763686 + 1.32274i
\(606\) 0 0
\(607\) 8.95970e6 1.55187e7i 0.987011 1.70955i 0.354374 0.935104i \(-0.384694\pi\)
0.632636 0.774449i \(-0.281973\pi\)
\(608\) 0 0
\(609\) 5.74006e6 4.97103e6i 0.627152 0.543130i
\(610\) 0 0
\(611\) −27693.0 + 47965.7i −0.00300101 + 0.00519790i
\(612\) 0 0
\(613\) −6.98950e6 1.21062e7i −0.751269 1.30124i −0.947208 0.320619i \(-0.896109\pi\)
0.195940 0.980616i \(-0.437224\pi\)
\(614\) 0 0
\(615\) 1.19304e7 1.27195
\(616\) 0 0
\(617\) −5.25594e6 −0.555824 −0.277912 0.960606i \(-0.589642\pi\)
−0.277912 + 0.960606i \(0.589642\pi\)
\(618\) 0 0
\(619\) 721506. + 1.24969e6i 0.0756857 + 0.131091i 0.901384 0.433020i \(-0.142552\pi\)
−0.825699 + 0.564112i \(0.809219\pi\)
\(620\) 0 0
\(621\) −81648.0 + 141418.i −0.00849604 + 0.0147156i
\(622\) 0 0
\(623\) −7.22652e6 2.50334e6i −0.745949 0.258404i
\(624\) 0 0
\(625\) 2.43553e6 4.21846e6i 0.249398 0.431970i
\(626\) 0 0
\(627\) 227817. + 394591.i 0.0231429 + 0.0400846i
\(628\) 0 0
\(629\) 1.45332e7 1.46466
\(630\) 0 0
\(631\) −1.51723e7 −1.51697 −0.758487 0.651688i \(-0.774061\pi\)
−0.758487 + 0.651688i \(0.774061\pi\)
\(632\) 0 0
\(633\) −1.17117e6 2.02853e6i −0.116174 0.201220i
\(634\) 0 0
\(635\) −6.53450e6 + 1.13181e7i −0.643099 + 1.11388i
\(636\) 0 0
\(637\) 46819.5 18713.9i 0.00457170 0.00182733i
\(638\) 0 0
\(639\) −1.81578e6 + 3.14502e6i −0.175918 + 0.304699i
\(640\) 0 0
\(641\) 72424.0 + 125442.i 0.00696205 + 0.0120586i 0.869485 0.493959i \(-0.164451\pi\)
−0.862523 + 0.506017i \(0.831117\pi\)
\(642\) 0 0
\(643\) 27469.0 0.00262009 0.00131004 0.999999i \(-0.499583\pi\)
0.00131004 + 0.999999i \(0.499583\pi\)
\(644\) 0 0
\(645\) −1.43120e7 −1.35457
\(646\) 0 0
\(647\) 4.27892e6 + 7.41130e6i 0.401858 + 0.696039i 0.993950 0.109831i \(-0.0350311\pi\)
−0.592092 + 0.805870i \(0.701698\pi\)
\(648\) 0 0
\(649\) 541076. 937171.i 0.0504251 0.0873388i
\(650\) 0 0
\(651\) −1.90843e6 661099.i −0.176491 0.0611384i
\(652\) 0 0
\(653\) −6.25152e6 + 1.08279e7i −0.573723 + 0.993718i 0.422456 + 0.906384i \(0.361168\pi\)
−0.996179 + 0.0873343i \(0.972165\pi\)
\(654\) 0 0
\(655\) 1.01266e7 + 1.75398e7i 0.922274 + 1.59742i
\(656\) 0 0
\(657\) 1.68650e6 0.152431
\(658\) 0 0
\(659\) −1.80471e7 −1.61880 −0.809400 0.587258i \(-0.800207\pi\)
−0.809400 + 0.587258i \(0.800207\pi\)
\(660\) 0 0
\(661\) −1.67072e6 2.89377e6i −0.148731 0.257609i 0.782028 0.623243i \(-0.214185\pi\)
−0.930759 + 0.365634i \(0.880852\pi\)
\(662\) 0 0
\(663\) −25704.0 + 44520.6i −0.00227100 + 0.00393349i
\(664\) 0 0
\(665\) −1.25493e7 + 1.08680e7i −1.10044 + 0.953006i
\(666\) 0 0
\(667\) 728896. 1.26248e6i 0.0634382 0.109878i
\(668\) 0 0
\(669\) 475452. + 823507.i 0.0410716 + 0.0711381i
\(670\) 0 0
\(671\) 1.96024e6 0.168075
\(672\) 0 0
\(673\) −8.47066e6 −0.720907 −0.360454 0.932777i \(-0.617378\pi\)
−0.360454 + 0.932777i \(0.617378\pi\)
\(674\) 0 0
\(675\) −1.55678e6 2.69642e6i −0.131513 0.227787i
\(676\) 0 0
\(677\) −7.32764e6 + 1.26918e7i −0.614458 + 1.06427i 0.376021 + 0.926611i \(0.377292\pi\)
−0.990479 + 0.137662i \(0.956041\pi\)
\(678\) 0 0
\(679\) 2.93623e6 + 1.52571e7i 0.244408 + 1.26998i
\(680\) 0 0
\(681\) 2.94638e6 5.10327e6i 0.243456 0.421678i
\(682\) 0 0
\(683\) 9.88081e6 + 1.71141e7i 0.810477 + 1.40379i 0.912531 + 0.409008i \(0.134125\pi\)
−0.102054 + 0.994779i \(0.532541\pi\)
\(684\) 0 0
\(685\) 2.79937e7 2.27947
\(686\) 0 0
\(687\) 5.01942e6 0.405753
\(688\) 0 0
\(689\) −29934.0 51847.2i −0.00240224 0.00416080i
\(690\) 0 0
\(691\) 6.79162e6 1.17634e7i 0.541101 0.937214i −0.457740 0.889086i \(-0.651341\pi\)
0.998841 0.0481282i \(-0.0153256\pi\)
\(692\) 0 0
\(693\) 67473.0 + 350600.i 0.00533700 + 0.0277318i
\(694\) 0 0
\(695\) 138073. 239149.i 0.0108429 0.0187805i
\(696\) 0 0
\(697\) 1.46741e7 + 2.54163e7i 1.14412 + 1.98167i
\(698\) 0 0
\(699\) −1.12283e7 −0.869206
\(700\) 0 0
\(701\) −1.29915e7 −0.998538 −0.499269 0.866447i \(-0.666398\pi\)
−0.499269 + 0.866447i \(0.666398\pi\)
\(702\) 0 0
\(703\) 5.68277e6 + 9.84284e6i 0.433682 + 0.751160i
\(704\) 0 0
\(705\) 7.14479e6 1.23751e7i 0.541398 0.937729i
\(706\) 0 0
\(707\) −9.80098e6 + 8.48790e6i −0.737430 + 0.638633i
\(708\) 0 0
\(709\) −954367. + 1.65301e6i −0.0713017 + 0.123498i −0.899472 0.436978i \(-0.856049\pi\)
0.828170 + 0.560476i \(0.189382\pi\)
\(710\) 0 0
\(711\) −1.23651e6 2.14169e6i −0.0917323 0.158885i
\(712\) 0 0
\(713\) −387744. −0.0285641
\(714\) 0 0
\(715\) 8772.00 0.000641702
\(716\) 0 0
\(717\) −2.23617e6 3.87315e6i −0.162445 0.281363i
\(718\) 0 0
\(719\) −5.55999e6 + 9.63018e6i −0.401099 + 0.694724i −0.993859 0.110655i \(-0.964705\pi\)
0.592760 + 0.805379i \(0.298038\pi\)
\(720\) 0 0
\(721\) −1.49071e7 5.16399e6i −1.06796 0.369953i
\(722\) 0 0
\(723\) 1.24928e6 2.16382e6i 0.0888821 0.153948i
\(724\) 0 0
\(725\) 1.38978e7 + 2.40718e7i 0.981979 + 1.70084i
\(726\) 0 0
\(727\) −8.37406e6 −0.587624 −0.293812 0.955863i \(-0.594924\pi\)
−0.293812 + 0.955863i \(0.594924\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −1.76034e7 3.04900e7i −1.21844 2.11040i
\(732\) 0 0
\(733\) −2.32224e6 + 4.02224e6i −0.159642 + 0.276508i −0.934740 0.355334i \(-0.884367\pi\)
0.775098 + 0.631841i \(0.217701\pi\)
\(734\) 0 0
\(735\) −1.20794e7 + 4.82820e6i −0.824761 + 0.329660i
\(736\) 0 0
\(737\) 1.02957e6 1.78327e6i 0.0698212 0.120934i
\(738\) 0 0
\(739\) −5.53116e6 9.58026e6i −0.372568 0.645307i 0.617392 0.786656i \(-0.288189\pi\)
−0.989960 + 0.141349i \(0.954856\pi\)
\(740\) 0 0
\(741\) −40203.0 −0.00268976
\(742\) 0 0
\(743\) −1.97245e6 −0.131079 −0.0655395 0.997850i \(-0.520877\pi\)
−0.0655395 + 0.997850i \(0.520877\pi\)
\(744\) 0 0
\(745\) 6.53101e6 + 1.13120e7i 0.431112 + 0.746707i
\(746\) 0 0
\(747\) 4.47938e6 7.75852e6i 0.293708 0.508718i
\(748\) 0 0
\(749\) 5.95938e6 + 2.06439e6i 0.388147 + 0.134458i
\(750\) 0 0
\(751\) 7.51232e6 1.30117e7i 0.486042 0.841850i −0.513829 0.857893i \(-0.671773\pi\)
0.999871 + 0.0160425i \(0.00510670\pi\)
\(752\) 0 0
\(753\) −1.38748e6 2.40318e6i −0.0891740 0.154454i
\(754\) 0 0
\(755\) −6.59173e6 −0.420854
\(756\) 0 0
\(757\) −4.72426e6 −0.299636 −0.149818 0.988714i \(-0.547869\pi\)
−0.149818 + 0.988714i \(0.547869\pi\)
\(758\) 0 0
\(759\) 34272.0 + 59360.8i 0.00215941 + 0.00374021i
\(760\) 0 0
\(761\) 4.28918e6 7.42907e6i 0.268480 0.465021i −0.699989 0.714153i \(-0.746812\pi\)
0.968470 + 0.249132i \(0.0801453\pi\)
\(762\) 0 0
\(763\) 1.49034e7 1.29067e7i 0.926771 0.802607i
\(764\) 0 0
\(765\) 6.63163e6 1.14863e7i 0.409701 0.709623i
\(766\) 0 0
\(767\) 47742.0 + 82691.6i 0.00293030 + 0.00507543i
\(768\) 0 0
\(769\) 1.76168e7 1.07426 0.537131 0.843499i \(-0.319508\pi\)
0.537131 + 0.843499i \(0.319508\pi\)
\(770\) 0 0
\(771\) 3.67886e6 0.222883
\(772\) 0 0
\(773\) 627101. + 1.08617e6i 0.0377475 + 0.0653807i 0.884282 0.466953i \(-0.154648\pi\)
−0.846534 + 0.532334i \(0.821315\pi\)
\(774\) 0 0
\(775\) 3.69655e6 6.40261e6i 0.221076 0.382916i
\(776\) 0 0
\(777\) 1.68308e6 + 8.74552e6i 0.100012 + 0.519677i
\(778\) 0 0
\(779\) −1.14757e7 + 1.98765e7i −0.677542 + 1.17354i
\(780\) 0 0
\(781\) 762178. + 1.32013e6i 0.0447125 + 0.0774443i
\(782\) 0 0
\(783\) −4.74433e6 −0.276548
\(784\) 0 0
\(785\) −3.35151e7 −1.94118
\(786\) 0 0
\(787\) 3.54560e6 + 6.14117e6i 0.204058 + 0.353439i 0.949832 0.312760i \(-0.101254\pi\)
−0.745774 + 0.666199i \(0.767920\pi\)
\(788\) 0 0
\(789\) 4.89656e6 8.48109e6i 0.280026 0.485019i
\(790\) 0 0
\(791\) 1.49171e6 + 7.75114e6i 0.0847700 + 0.440478i
\(792\) 0 0
\(793\) −86481.0 + 149789.i −0.00488357 + 0.00845860i
\(794\) 0 0
\(795\) 7.72297e6 + 1.33766e7i 0.433378 + 0.750632i
\(796\) 0 0
\(797\) −2.71630e6 −0.151472 −0.0757358 0.997128i \(-0.524131\pi\)
−0.0757358 + 0.997128i \(0.524131\pi\)
\(798\) 0 0
\(799\) 3.51516e7 1.94795
\(800\) 0 0
\(801\) 2.38918e6 + 4.13817e6i 0.131573 + 0.227891i
\(802\) 0 0
\(803\) 353957. 613072.i 0.0193714 0.0335523i
\(804\) 0 0
\(805\) −1.88787e6 + 1.63495e6i −0.102679 + 0.0889229i
\(806\) 0 0
\(807\) 3.00730e6 5.20881e6i 0.162552 0.281549i
\(808\) 0 0
\(809\) 1.09370e7 + 1.89434e7i 0.587524 + 1.01762i 0.994556 + 0.104207i \(0.0332304\pi\)
−0.407032 + 0.913414i \(0.633436\pi\)
\(810\) 0 0
\(811\) 1.72352e7 0.920164 0.460082 0.887876i \(-0.347820\pi\)
0.460082 + 0.887876i \(0.347820\pi\)
\(812\) 0 0
\(813\) 7.47598e6 0.396681
\(814\) 0 0
\(815\) −4.83200e6 8.36926e6i −0.254819 0.441360i
\(816\) 0 0
\(817\) 1.37665e7 2.38444e7i 0.721556 1.24977i
\(818\) 0 0
\(819\) −29767.5 10311.8i −0.00155072 0.000537184i
\(820\) 0 0
\(821\) 1.13096e7 1.95887e7i 0.585582 1.01426i −0.409220 0.912436i \(-0.634199\pi\)
0.994803 0.101823i \(-0.0324674\pi\)
\(822\) 0 0
\(823\) 813164. + 1.40844e6i 0.0418484 + 0.0724835i 0.886191 0.463320i \(-0.153342\pi\)
−0.844343 + 0.535804i \(0.820009\pi\)
\(824\) 0 0
\(825\) −1.30693e6 −0.0668523
\(826\) 0 0
\(827\) 2.28304e7 1.16078 0.580391 0.814338i \(-0.302900\pi\)
0.580391 + 0.814338i \(0.302900\pi\)
\(828\) 0 0
\(829\) −1.29297e7 2.23948e7i −0.653432 1.13178i −0.982284 0.187397i \(-0.939995\pi\)
0.328852 0.944382i \(-0.393338\pi\)
\(830\) 0 0
\(831\) −4.16283e6 + 7.21023e6i −0.209115 + 0.362198i
\(832\) 0 0
\(833\) −2.51433e7 1.97952e7i −1.25548 0.988433i
\(834\) 0 0
\(835\) 2.25965e6 3.91383e6i 0.112157 0.194261i
\(836\) 0 0
\(837\) 630950. + 1.09284e6i 0.0311301 + 0.0539190i
\(838\) 0 0
\(839\) 1.23061e7 0.603554 0.301777 0.953379i \(-0.402420\pi\)
0.301777 + 0.953379i \(0.402420\pi\)
\(840\) 0 0
\(841\) 2.18429e7 1.06493
\(842\) 0 0
\(843\) −6.01357e6 1.04158e7i −0.291449 0.504805i
\(844\) 0 0
\(845\) 1.59652e7 2.76526e7i 0.769189 1.33227i
\(846\) 0 0
\(847\) −1.95871e7 6.78518e6i −0.938129 0.324977i
\(848\) 0 0
\(849\) −4.46831e6 + 7.73933e6i −0.212752 + 0.368497i
\(850\) 0 0
\(851\) 854896. + 1.48072e6i 0.0404659 + 0.0700890i
\(852\) 0 0
\(853\) −1.91416e7 −0.900753 −0.450377 0.892839i \(-0.648710\pi\)
−0.450377 + 0.892839i \(0.648710\pi\)
\(854\) 0 0
\(855\) 1.03724e7 0.485247
\(856\) 0 0
\(857\) 2.37754e6 + 4.11801e6i 0.110580 + 0.191529i 0.916004 0.401169i \(-0.131396\pi\)
−0.805424 + 0.592698i \(0.798063\pi\)
\(858\) 0 0
\(859\) −4.59938e6 + 7.96636e6i −0.212675 + 0.368364i −0.952551 0.304380i \(-0.901551\pi\)
0.739876 + 0.672743i \(0.234884\pi\)
\(860\) 0 0
\(861\) −1.35951e7 + 1.17737e7i −0.624994 + 0.541261i
\(862\) 0 0
\(863\) −8.67659e6 + 1.50283e7i −0.396572 + 0.686883i −0.993300 0.115560i \(-0.963134\pi\)
0.596728 + 0.802443i \(0.296467\pi\)
\(864\) 0 0
\(865\) 5.81601e6 + 1.00736e7i 0.264292 + 0.457768i
\(866\) 0 0
\(867\) 1.98482e7 0.896756
\(868\) 0 0
\(869\) −1.03805e6 −0.0466305
\(870\) 0 0
\(871\) 90844.5 + 157347.i 0.00405745 + 0.00702771i
\(872\) 0 0
\(873\) 4.85376e6 8.40696e6i 0.215547 0.373339i
\(874\) 0 0
\(875\) −2.41462e6 1.25467e7i −0.106618 0.554001i
\(876\) 0 0
\(877\) 1.77811e7 3.07978e7i 0.780655 1.35213i −0.150905 0.988548i \(-0.548219\pi\)
0.931560 0.363587i \(-0.118448\pi\)
\(878\) 0 0
\(879\) −2.53595e6 4.39239e6i −0.110705 0.191747i
\(880\) 0 0
\(881\) −2.63056e7 −1.14185 −0.570923 0.821003i \(-0.693415\pi\)
−0.570923 + 0.821003i \(0.693415\pi\)
\(882\) 0 0
\(883\) −1.20394e7 −0.519642 −0.259821 0.965657i \(-0.583664\pi\)
−0.259821 + 0.965657i \(0.583664\pi\)
\(884\) 0 0
\(885\) −1.23174e7 2.13344e7i −0.528643 0.915636i
\(886\) 0 0
\(887\) −66795.0 + 115692.i −0.00285059 + 0.00493737i −0.867447 0.497529i \(-0.834241\pi\)
0.864597 + 0.502467i \(0.167574\pi\)
\(888\) 0 0
\(889\) −3.72314e6 1.93460e7i −0.157999 0.820989i
\(890\) 0 0
\(891\) 111537. 193188.i 0.00470679 0.00815240i
\(892\) 0 0
\(893\) 1.37450e7 + 2.38070e7i 0.576786 + 0.999023i
\(894\) 0 0
\(895\) −2.15549e7 −0.899472
\(896\) 0 0
\(897\) −6048.00 −0.000250975
\(898\) 0 0
\(899\) −5.63267e6 9.75608e6i −0.232442 0.402602i
\(900\) 0 0
\(901\) −1.89981e7 + 3.29057e7i −0.779648 + 1.35039i
\(902\) 0 0
\(903\) 1.63091e7 1.41241e7i 0.665594 0.576422i
\(904\) 0 0
\(905\) −8.56702e6 + 1.48385e7i −0.347703 + 0.602239i
\(906\) 0 0
\(907\) −6.37379e6 1.10397e7i −0.257264 0.445595i 0.708244 0.705968i \(-0.249488\pi\)
−0.965508 + 0.260373i \(0.916154\pi\)
\(908\) 0 0
\(909\) 8.10081e6 0.325176
\(910\) 0 0
\(911\) −2.43253e7 −0.971095 −0.485548 0.874210i \(-0.661380\pi\)
−0.485548 + 0.874210i \(0.661380\pi\)
\(912\) 0 0
\(913\) −1.88023e6 3.25666e6i −0.0746509 0.129299i
\(914\) 0 0
\(915\) 2.23121e7 3.86457e7i 0.881024 1.52598i
\(916\) 0 0
\(917\) −2.88490e7 9.99359e6i −1.13294 0.392462i
\(918\) 0 0
\(919\) −1.92402e7 + 3.33251e7i −0.751487 + 1.30161i 0.195615 + 0.980681i \(0.437330\pi\)
−0.947102 + 0.320933i \(0.896004\pi\)
\(920\) 0 0
\(921\) 1.27216e7 + 2.20345e7i 0.494190 + 0.855963i
\(922\) 0 0
\(923\) −134502. −0.00519666
\(924\) 0 0
\(925\) −3.26005e7 −1.25277
\(926\) 0 0
\(927\) 4.92849e6 + 8.53639e6i 0.188371 + 0.326268i
\(928\) 0 0
\(929\) −1.98339e7 + 3.43533e7i −0.753996 + 1.30596i 0.191876 + 0.981419i \(0.438543\pi\)
−0.945872 + 0.324540i \(0.894791\pi\)
\(930\) 0 0
\(931\) 3.57509e6 2.47689e7i 0.135180 0.936555i
\(932\) 0 0
\(933\) −5.07291e6 + 8.78654e6i −0.190789 + 0.330456i
\(934\) 0 0
\(935\) −2.78365e6 4.82142e6i −0.104132 0.180362i
\(936\) 0 0
\(937\) 1.48428e7 0.552288 0.276144 0.961116i \(-0.410943\pi\)
0.276144 + 0.961116i \(0.410943\pi\)
\(938\) 0 0
\(939\) −2.12412e7 −0.786166
\(940\) 0 0
\(941\) −2.33532e7 4.04488e7i −0.859748 1.48913i −0.872169 0.489205i \(-0.837287\pi\)
0.0124203 0.999923i \(-0.496046\pi\)
\(942\) 0 0
\(943\) −1.72637e6 + 2.99016e6i −0.0632199 + 0.109500i
\(944\) 0 0
\(945\) 7.68001e6 + 2.66044e6i 0.279758 + 0.0969110i
\(946\) 0 0
\(947\) 2.01190e7 3.48471e7i 0.729006 1.26267i −0.228298 0.973591i \(-0.573316\pi\)
0.957304 0.289083i \(-0.0933505\pi\)
\(948\) 0 0
\(949\) 31231.5 + 54094.5i 0.00112571 + 0.00194979i
\(950\) 0 0
\(951\) −2.00178e7 −0.717737
\(952\) 0 0
\(953\) 2.45579e7 0.875908 0.437954 0.898997i \(-0.355703\pi\)
0.437954 + 0.898997i \(0.355703\pi\)
\(954\) 0 0
\(955\) −1.02671e7 1.77832e7i −0.364284 0.630958i
\(956\) 0 0
\(957\) −995724. + 1.72464e6i −0.0351446 + 0.0608723i
\(958\) 0 0
\(959\) −3.18998e7 + 2.76260e7i −1.12006 + 0.970000i
\(960\) 0 0
\(961\) 1.28164e7 2.21986e7i 0.447669 0.775386i
\(962\) 0 0
\(963\) −1.97024e6 3.41256e6i −0.0684627 0.118581i
\(964\) 0 0
\(965\) 7.36943e6 0.254751
\(966\) 0 0
\(967\) −5.34313e7 −1.83751 −0.918754 0.394830i \(-0.870803\pi\)
−0.918754 + 0.394830i \(0.870803\pi\)
\(968\) 0 0
\(969\) 1.27578e7 + 2.20971e7i 0.436480 + 0.756006i
\(970\) 0 0
\(971\) −1.40743e6 + 2.43773e6i −0.0479046 + 0.0829733i −0.888983 0.457939i \(-0.848588\pi\)
0.841079 + 0.540913i \(0.181921\pi\)
\(972\) 0 0
\(973\) 78669.5 + 408779.i 0.00266394 + 0.0138422i
\(974\) 0 0
\(975\) 57658.5 99867.5i 0.00194246 0.00336444i
\(976\) 0 0
\(977\) −1.27356e7 2.20588e7i −0.426859 0.739341i 0.569733 0.821830i \(-0.307047\pi\)
−0.996592 + 0.0824885i \(0.973713\pi\)
\(978\) 0 0
\(979\) 2.00573e6 0.0668829
\(980\) 0 0
\(981\) −1.23181e7 −0.408668
\(982\) 0 0
\(983\) −1.45067e6 2.51264e6i −0.0478835 0.0829367i 0.841090 0.540895i \(-0.181914\pi\)
−0.888974 + 0.457958i \(0.848581\pi\)
\(984\) 0 0
\(985\) 3.07054e6 5.31834e6i 0.100838 0.174657i
\(986\) 0 0
\(987\) 4.07087e6 + 2.11529e7i 0.133013 + 0.691156i
\(988\) 0 0
\(989\) 2.07099e6 3.58706e6i 0.0673268 0.116613i
\(990\) 0 0
\(991\) 1.36892e6 + 2.37103e6i 0.0442785 + 0.0766927i 0.887315 0.461163i \(-0.152568\pi\)
−0.843037 + 0.537856i \(0.819234\pi\)
\(992\) 0 0
\(993\) 3.33274e7 1.07258
\(994\) 0 0
\(995\) 6.11763e7 1.95896
\(996\) 0 0
\(997\) 1.37040e7 + 2.37359e7i 0.436624 + 0.756256i 0.997427 0.0716942i \(-0.0228406\pi\)
−0.560802 + 0.827950i \(0.689507\pi\)
\(998\) 0 0
\(999\) 2.78223e6 4.81896e6i 0.0882021 0.152771i
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 336.6.q.a.289.1 2
4.3 odd 2 42.6.e.b.37.1 yes 2
7.4 even 3 inner 336.6.q.a.193.1 2
12.11 even 2 126.6.g.b.37.1 2
28.3 even 6 294.6.e.m.67.1 2
28.11 odd 6 42.6.e.b.25.1 2
28.19 even 6 294.6.a.e.1.1 1
28.23 odd 6 294.6.a.d.1.1 1
28.27 even 2 294.6.e.m.79.1 2
84.11 even 6 126.6.g.b.109.1 2
84.23 even 6 882.6.a.m.1.1 1
84.47 odd 6 882.6.a.y.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.6.e.b.25.1 2 28.11 odd 6
42.6.e.b.37.1 yes 2 4.3 odd 2
126.6.g.b.37.1 2 12.11 even 2
126.6.g.b.109.1 2 84.11 even 6
294.6.a.d.1.1 1 28.23 odd 6
294.6.a.e.1.1 1 28.19 even 6
294.6.e.m.67.1 2 28.3 even 6
294.6.e.m.79.1 2 28.27 even 2
336.6.q.a.193.1 2 7.4 even 3 inner
336.6.q.a.289.1 2 1.1 even 1 trivial
882.6.a.m.1.1 1 84.23 even 6
882.6.a.y.1.1 1 84.47 odd 6