Properties

Label 336.6.q.j.289.4
Level $336$
Weight $6$
Character 336.289
Analytic conductor $53.889$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 98x^{6} + 83x^{5} + 9122x^{4} - 91x^{3} + 28567x^{2} + 2058x + 86436 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{8}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 289.4
Root \(0.895402 + 1.55088i\) of defining polynomial
Character \(\chi\) \(=\) 336.289
Dual form 336.6.q.j.193.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(4.50000 + 7.79423i) q^{3} +(52.0958 - 90.2327i) q^{5} +(7.12980 + 129.446i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})\) \(q+(4.50000 + 7.79423i) q^{3} +(52.0958 - 90.2327i) q^{5} +(7.12980 + 129.446i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(-248.830 - 430.986i) q^{11} -206.551 q^{13} +937.725 q^{15} +(-31.5793 - 54.6969i) q^{17} +(661.977 - 1146.58i) q^{19} +(-976.845 + 638.077i) q^{21} +(-97.2187 + 168.388i) q^{23} +(-3865.45 - 6695.16i) q^{25} -729.000 q^{27} +4323.14 q^{29} +(-3762.66 - 6517.11i) q^{31} +(2239.47 - 3878.87i) q^{33} +(12051.7 + 6100.24i) q^{35} +(-5177.82 + 8968.25i) q^{37} +(-929.481 - 1609.91i) q^{39} -4180.92 q^{41} -5960.87 q^{43} +(4219.76 + 7308.85i) q^{45} +(-2194.87 + 3801.62i) q^{47} +(-16705.3 + 1845.84i) q^{49} +(284.214 - 492.273i) q^{51} +(-8892.39 - 15402.1i) q^{53} -51852.0 q^{55} +11915.6 q^{57} +(1750.23 + 3031.49i) q^{59} +(5316.23 - 9207.98i) q^{61} +(-9369.11 - 4742.41i) q^{63} +(-10760.5 + 18637.7i) q^{65} +(-6637.37 - 11496.3i) q^{67} -1749.94 q^{69} -38811.1 q^{71} +(-15687.8 - 27172.1i) q^{73} +(34789.1 - 60256.5i) q^{75} +(54015.1 - 35282.8i) q^{77} +(19745.7 - 34200.6i) q^{79} +(-3280.50 - 5681.99i) q^{81} +102372. q^{83} -6580.60 q^{85} +(19454.1 + 33695.5i) q^{87} +(56410.5 - 97705.8i) q^{89} +(-1472.67 - 26737.2i) q^{91} +(33863.9 - 58654.0i) q^{93} +(-68972.6 - 119464. i) q^{95} +30334.3 q^{97} +40310.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 36 q^{3} - 258 q^{7} - 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 36 q^{3} - 258 q^{7} - 324 q^{9} + 402 q^{11} + 924 q^{13} - 276 q^{17} + 510 q^{19} - 3564 q^{21} + 6900 q^{23} - 2814 q^{25} - 5832 q^{27} + 1080 q^{29} - 6410 q^{31} - 3618 q^{33} + 33108 q^{35} - 15250 q^{37} + 4158 q^{39} + 8616 q^{41} - 58396 q^{43} - 15060 q^{47} - 64252 q^{49} + 2484 q^{51} - 13692 q^{53} - 146248 q^{55} + 9180 q^{57} + 34830 q^{59} + 5364 q^{61} - 11178 q^{63} - 66864 q^{65} - 5994 q^{67} + 124200 q^{69} - 178536 q^{71} - 59638 q^{73} + 25326 q^{75} - 75660 q^{77} - 44062 q^{79} - 26244 q^{81} + 416892 q^{83} + 72648 q^{85} + 4860 q^{87} + 77520 q^{89} - 104722 q^{91} + 57690 q^{93} - 221376 q^{95} - 377260 q^{97} - 65124 q^{99}+O(q^{100}) \) 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
\(5\) 52.0958 90.2327i 0.931919 1.61413i 0.151881 0.988399i \(-0.451467\pi\)
0.780038 0.625732i \(-0.215200\pi\)
\(6\) 0 0
\(7\) 7.12980 + 129.446i 0.0549961 + 0.998487i
\(8\) 0 0
\(9\) −40.5000 + 70.1481i −0.166667 + 0.288675i
\(10\) 0 0
\(11\) −248.830 430.986i −0.620041 1.07394i −0.989477 0.144687i \(-0.953782\pi\)
0.369436 0.929256i \(-0.379551\pi\)
\(12\) 0 0
\(13\) −206.551 −0.338977 −0.169488 0.985532i \(-0.554211\pi\)
−0.169488 + 0.985532i \(0.554211\pi\)
\(14\) 0 0
\(15\) 937.725 1.07609
\(16\) 0 0
\(17\) −31.5793 54.6969i −0.0265021 0.0459030i 0.852470 0.522776i \(-0.175104\pi\)
−0.878972 + 0.476873i \(0.841770\pi\)
\(18\) 0 0
\(19\) 661.977 1146.58i 0.420687 0.728651i −0.575320 0.817929i \(-0.695122\pi\)
0.996007 + 0.0892772i \(0.0284557\pi\)
\(20\) 0 0
\(21\) −976.845 + 638.077i −0.483367 + 0.315736i
\(22\) 0 0
\(23\) −97.2187 + 168.388i −0.0383204 + 0.0663729i −0.884550 0.466446i \(-0.845534\pi\)
0.846229 + 0.532819i \(0.178867\pi\)
\(24\) 0 0
\(25\) −3865.45 6695.16i −1.23695 2.14245i
\(26\) 0 0
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) 4323.14 0.954562 0.477281 0.878751i \(-0.341622\pi\)
0.477281 + 0.878751i \(0.341622\pi\)
\(30\) 0 0
\(31\) −3762.66 6517.11i −0.703219 1.21801i −0.967331 0.253518i \(-0.918412\pi\)
0.264112 0.964492i \(-0.414921\pi\)
\(32\) 0 0
\(33\) 2239.47 3878.87i 0.357981 0.620041i
\(34\) 0 0
\(35\) 12051.7 + 6100.24i 1.66294 + 0.841738i
\(36\) 0 0
\(37\) −5177.82 + 8968.25i −0.621789 + 1.07697i 0.367364 + 0.930077i \(0.380260\pi\)
−0.989152 + 0.146892i \(0.953073\pi\)
\(38\) 0 0
\(39\) −929.481 1609.91i −0.0978541 0.169488i
\(40\) 0 0
\(41\) −4180.92 −0.388429 −0.194215 0.980959i \(-0.562216\pi\)
−0.194215 + 0.980959i \(0.562216\pi\)
\(42\) 0 0
\(43\) −5960.87 −0.491630 −0.245815 0.969317i \(-0.579056\pi\)
−0.245815 + 0.969317i \(0.579056\pi\)
\(44\) 0 0
\(45\) 4219.76 + 7308.85i 0.310640 + 0.538044i
\(46\) 0 0
\(47\) −2194.87 + 3801.62i −0.144932 + 0.251029i −0.929348 0.369206i \(-0.879630\pi\)
0.784416 + 0.620236i \(0.212963\pi\)
\(48\) 0 0
\(49\) −16705.3 + 1845.84i −0.993951 + 0.109826i
\(50\) 0 0
\(51\) 284.214 492.273i 0.0153010 0.0265021i
\(52\) 0 0
\(53\) −8892.39 15402.1i −0.434839 0.753164i 0.562443 0.826836i \(-0.309862\pi\)
−0.997283 + 0.0736720i \(0.976528\pi\)
\(54\) 0 0
\(55\) −51852.0 −2.31131
\(56\) 0 0
\(57\) 11915.6 0.485768
\(58\) 0 0
\(59\) 1750.23 + 3031.49i 0.0654585 + 0.113377i 0.896897 0.442239i \(-0.145816\pi\)
−0.831439 + 0.555616i \(0.812482\pi\)
\(60\) 0 0
\(61\) 5316.23 9207.98i 0.182928 0.316840i −0.759949 0.649983i \(-0.774776\pi\)
0.942876 + 0.333143i \(0.108109\pi\)
\(62\) 0 0
\(63\) −9369.11 4742.41i −0.297404 0.150538i
\(64\) 0 0
\(65\) −10760.5 + 18637.7i −0.315899 + 0.547153i
\(66\) 0 0
\(67\) −6637.37 11496.3i −0.180638 0.312874i 0.761460 0.648212i \(-0.224483\pi\)
−0.942098 + 0.335338i \(0.891150\pi\)
\(68\) 0 0
\(69\) −1749.94 −0.0442486
\(70\) 0 0
\(71\) −38811.1 −0.913713 −0.456857 0.889540i \(-0.651025\pi\)
−0.456857 + 0.889540i \(0.651025\pi\)
\(72\) 0 0
\(73\) −15687.8 27172.1i −0.344552 0.596782i 0.640720 0.767775i \(-0.278636\pi\)
−0.985272 + 0.170992i \(0.945303\pi\)
\(74\) 0 0
\(75\) 34789.1 60256.5i 0.714151 1.23695i
\(76\) 0 0
\(77\) 54015.1 35282.8i 1.03822 0.678166i
\(78\) 0 0
\(79\) 19745.7 34200.6i 0.355964 0.616547i −0.631319 0.775523i \(-0.717486\pi\)
0.987282 + 0.158976i \(0.0508194\pi\)
\(80\) 0 0
\(81\) −3280.50 5681.99i −0.0555556 0.0962250i
\(82\) 0 0
\(83\) 102372. 1.63112 0.815559 0.578675i \(-0.196430\pi\)
0.815559 + 0.578675i \(0.196430\pi\)
\(84\) 0 0
\(85\) −6580.60 −0.0987912
\(86\) 0 0
\(87\) 19454.1 + 33695.5i 0.275558 + 0.477281i
\(88\) 0 0
\(89\) 56410.5 97705.8i 0.754892 1.30751i −0.190536 0.981680i \(-0.561023\pi\)
0.945428 0.325831i \(-0.105644\pi\)
\(90\) 0 0
\(91\) −1472.67 26737.2i −0.0186424 0.338464i
\(92\) 0 0
\(93\) 33863.9 58654.0i 0.406004 0.703219i
\(94\) 0 0
\(95\) −68972.6 119464.i −0.784092 1.35809i
\(96\) 0 0
\(97\) 30334.3 0.327345 0.163672 0.986515i \(-0.447666\pi\)
0.163672 + 0.986515i \(0.447666\pi\)
\(98\) 0 0
\(99\) 40310.4 0.413361
\(100\) 0 0
\(101\) −53398.2 92488.4i −0.520862 0.902160i −0.999706 0.0242599i \(-0.992277\pi\)
0.478843 0.877900i \(-0.341056\pi\)
\(102\) 0 0
\(103\) 78767.8 136430.i 0.731570 1.26712i −0.224643 0.974441i \(-0.572121\pi\)
0.956212 0.292674i \(-0.0945452\pi\)
\(104\) 0 0
\(105\) 6685.79 + 121384.i 0.0591806 + 1.07446i
\(106\) 0 0
\(107\) −44662.2 + 77357.1i −0.377121 + 0.653192i −0.990642 0.136486i \(-0.956419\pi\)
0.613521 + 0.789678i \(0.289752\pi\)
\(108\) 0 0
\(109\) −83802.3 145150.i −0.675600 1.17017i −0.976293 0.216452i \(-0.930551\pi\)
0.300693 0.953721i \(-0.402782\pi\)
\(110\) 0 0
\(111\) −93200.8 −0.717980
\(112\) 0 0
\(113\) −115794. −0.853079 −0.426539 0.904469i \(-0.640267\pi\)
−0.426539 + 0.904469i \(0.640267\pi\)
\(114\) 0 0
\(115\) 10129.4 + 17544.6i 0.0714230 + 0.123708i
\(116\) 0 0
\(117\) 8365.33 14489.2i 0.0564961 0.0978541i
\(118\) 0 0
\(119\) 6855.13 4477.78i 0.0443760 0.0289865i
\(120\) 0 0
\(121\) −43307.1 + 75010.0i −0.268903 + 0.465753i
\(122\) 0 0
\(123\) −18814.1 32587.0i −0.112130 0.194215i
\(124\) 0 0
\(125\) −479898. −2.74709
\(126\) 0 0
\(127\) −201513. −1.10865 −0.554325 0.832300i \(-0.687023\pi\)
−0.554325 + 0.832300i \(0.687023\pi\)
\(128\) 0 0
\(129\) −26823.9 46460.4i −0.141921 0.245815i
\(130\) 0 0
\(131\) 19234.8 33315.6i 0.0979285 0.169617i −0.812899 0.582405i \(-0.802112\pi\)
0.910827 + 0.412788i \(0.135445\pi\)
\(132\) 0 0
\(133\) 153139. + 77515.2i 0.750685 + 0.379977i
\(134\) 0 0
\(135\) −37977.9 + 65779.6i −0.179348 + 0.310640i
\(136\) 0 0
\(137\) 120861. + 209337.i 0.550155 + 0.952896i 0.998263 + 0.0589167i \(0.0187646\pi\)
−0.448108 + 0.893979i \(0.647902\pi\)
\(138\) 0 0
\(139\) 53112.2 0.233162 0.116581 0.993181i \(-0.462807\pi\)
0.116581 + 0.993181i \(0.462807\pi\)
\(140\) 0 0
\(141\) −39507.6 −0.167353
\(142\) 0 0
\(143\) 51396.1 + 89020.7i 0.210180 + 0.364042i
\(144\) 0 0
\(145\) 225218. 390088.i 0.889574 1.54079i
\(146\) 0 0
\(147\) −89560.9 121899.i −0.341842 0.465271i
\(148\) 0 0
\(149\) −64531.0 + 111771.i −0.238124 + 0.412443i −0.960176 0.279396i \(-0.909866\pi\)
0.722052 + 0.691839i \(0.243199\pi\)
\(150\) 0 0
\(151\) 76603.2 + 132681.i 0.273404 + 0.473549i 0.969731 0.244175i \(-0.0785172\pi\)
−0.696327 + 0.717724i \(0.745184\pi\)
\(152\) 0 0
\(153\) 5115.85 0.0176681
\(154\) 0 0
\(155\) −784075. −2.62137
\(156\) 0 0
\(157\) 75593.9 + 130932.i 0.244758 + 0.423934i 0.962064 0.272825i \(-0.0879581\pi\)
−0.717305 + 0.696759i \(0.754625\pi\)
\(158\) 0 0
\(159\) 80031.5 138619.i 0.251055 0.434839i
\(160\) 0 0
\(161\) −22490.2 11384.0i −0.0683799 0.0346122i
\(162\) 0 0
\(163\) −16458.3 + 28506.7i −0.0485196 + 0.0840384i −0.889265 0.457392i \(-0.848784\pi\)
0.840746 + 0.541430i \(0.182117\pi\)
\(164\) 0 0
\(165\) −233334. 404146.i −0.667219 1.15566i
\(166\) 0 0
\(167\) −217586. −0.603725 −0.301862 0.953352i \(-0.597608\pi\)
−0.301862 + 0.953352i \(0.597608\pi\)
\(168\) 0 0
\(169\) −328630. −0.885095
\(170\) 0 0
\(171\) 53620.2 + 92872.9i 0.140229 + 0.242884i
\(172\) 0 0
\(173\) 210621. 364807.i 0.535041 0.926718i −0.464121 0.885772i \(-0.653630\pi\)
0.999161 0.0409458i \(-0.0130371\pi\)
\(174\) 0 0
\(175\) 839100. 548101.i 2.07118 1.35290i
\(176\) 0 0
\(177\) −15752.1 + 27283.4i −0.0377925 + 0.0654585i
\(178\) 0 0
\(179\) 1747.03 + 3025.94i 0.00407538 + 0.00705876i 0.868056 0.496466i \(-0.165369\pi\)
−0.863981 + 0.503525i \(0.832036\pi\)
\(180\) 0 0
\(181\) 594611. 1.34908 0.674538 0.738240i \(-0.264343\pi\)
0.674538 + 0.738240i \(0.264343\pi\)
\(182\) 0 0
\(183\) 95692.2 0.211227
\(184\) 0 0
\(185\) 539486. + 934418.i 1.15891 + 2.00730i
\(186\) 0 0
\(187\) −15715.7 + 27220.5i −0.0328648 + 0.0569235i
\(188\) 0 0
\(189\) −5197.62 94365.8i −0.0105840 0.192159i
\(190\) 0 0
\(191\) 414322. 717627.i 0.821778 1.42336i −0.0825782 0.996585i \(-0.526315\pi\)
0.904357 0.426777i \(-0.140351\pi\)
\(192\) 0 0
\(193\) −109869. 190299.i −0.212316 0.367743i 0.740123 0.672472i \(-0.234767\pi\)
−0.952439 + 0.304729i \(0.901434\pi\)
\(194\) 0 0
\(195\) −193688. −0.364768
\(196\) 0 0
\(197\) 475612. 0.873146 0.436573 0.899669i \(-0.356192\pi\)
0.436573 + 0.899669i \(0.356192\pi\)
\(198\) 0 0
\(199\) 313778. + 543479.i 0.561681 + 0.972859i 0.997350 + 0.0727525i \(0.0231783\pi\)
−0.435669 + 0.900107i \(0.643488\pi\)
\(200\) 0 0
\(201\) 59736.4 103466.i 0.104291 0.180638i
\(202\) 0 0
\(203\) 30823.1 + 559611.i 0.0524972 + 0.953117i
\(204\) 0 0
\(205\) −217808. + 377255.i −0.361984 + 0.626975i
\(206\) 0 0
\(207\) −7874.71 13639.4i −0.0127735 0.0221243i
\(208\) 0 0
\(209\) −658879. −1.04337
\(210\) 0 0
\(211\) −570989. −0.882920 −0.441460 0.897281i \(-0.645539\pi\)
−0.441460 + 0.897281i \(0.645539\pi\)
\(212\) 0 0
\(213\) −174650. 302502.i −0.263766 0.456857i
\(214\) 0 0
\(215\) −310536. + 537865.i −0.458159 + 0.793555i
\(216\) 0 0
\(217\) 816785. 533525.i 1.17749 0.769140i
\(218\) 0 0
\(219\) 141190. 244549.i 0.198927 0.344552i
\(220\) 0 0
\(221\) 6522.75 + 11297.7i 0.00898359 + 0.0155600i
\(222\) 0 0
\(223\) −4233.11 −0.00570029 −0.00285015 0.999996i \(-0.500907\pi\)
−0.00285015 + 0.999996i \(0.500907\pi\)
\(224\) 0 0
\(225\) 626204. 0.824630
\(226\) 0 0
\(227\) 564931. + 978490.i 0.727664 + 1.26035i 0.957868 + 0.287209i \(0.0927274\pi\)
−0.230204 + 0.973142i \(0.573939\pi\)
\(228\) 0 0
\(229\) −402322. + 696842.i −0.506973 + 0.878103i 0.492994 + 0.870032i \(0.335902\pi\)
−0.999967 + 0.00807048i \(0.997431\pi\)
\(230\) 0 0
\(231\) 518070. + 262234.i 0.638791 + 0.323339i
\(232\) 0 0
\(233\) −584636. + 1.01262e6i −0.705498 + 1.22196i 0.261013 + 0.965335i \(0.415943\pi\)
−0.966511 + 0.256624i \(0.917390\pi\)
\(234\) 0 0
\(235\) 228687. + 396098.i 0.270130 + 0.467878i
\(236\) 0 0
\(237\) 355423. 0.411031
\(238\) 0 0
\(239\) 1.70554e6 1.93138 0.965689 0.259700i \(-0.0836238\pi\)
0.965689 + 0.259700i \(0.0836238\pi\)
\(240\) 0 0
\(241\) −475598. 823760.i −0.527470 0.913604i −0.999487 0.0320153i \(-0.989807\pi\)
0.472018 0.881589i \(-0.343526\pi\)
\(242\) 0 0
\(243\) 29524.5 51137.9i 0.0320750 0.0555556i
\(244\) 0 0
\(245\) −703723. + 1.60353e6i −0.749008 + 1.70672i
\(246\) 0 0
\(247\) −136732. + 236827.i −0.142603 + 0.246996i
\(248\) 0 0
\(249\) 460673. + 797909.i 0.470863 + 0.815559i
\(250\) 0 0
\(251\) −1.14498e6 −1.14713 −0.573566 0.819159i \(-0.694440\pi\)
−0.573566 + 0.819159i \(0.694440\pi\)
\(252\) 0 0
\(253\) 96763.6 0.0950409
\(254\) 0 0
\(255\) −29612.7 51290.7i −0.0285186 0.0493956i
\(256\) 0 0
\(257\) −591869. + 1.02515e6i −0.558976 + 0.968175i 0.438606 + 0.898679i \(0.355472\pi\)
−0.997582 + 0.0694955i \(0.977861\pi\)
\(258\) 0 0
\(259\) −1.19782e6 606305.i −1.10954 0.561619i
\(260\) 0 0
\(261\) −175087. + 303260.i −0.159094 + 0.275558i
\(262\) 0 0
\(263\) 224158. + 388253.i 0.199832 + 0.346119i 0.948474 0.316856i \(-0.102627\pi\)
−0.748642 + 0.662975i \(0.769294\pi\)
\(264\) 0 0
\(265\) −1.85303e6 −1.62094
\(266\) 0 0
\(267\) 1.01539e6 0.871674
\(268\) 0 0
\(269\) −373737. 647332.i −0.314909 0.545439i 0.664509 0.747280i \(-0.268641\pi\)
−0.979418 + 0.201841i \(0.935307\pi\)
\(270\) 0 0
\(271\) −116319. + 201470.i −0.0962114 + 0.166643i −0.910114 0.414359i \(-0.864006\pi\)
0.813902 + 0.581002i \(0.197339\pi\)
\(272\) 0 0
\(273\) 201769. 131796.i 0.163850 0.107027i
\(274\) 0 0
\(275\) −1.92368e6 + 3.33191e6i −1.53391 + 2.65682i
\(276\) 0 0
\(277\) 1.21462e6 + 2.10379e6i 0.951134 + 1.64741i 0.742977 + 0.669317i \(0.233413\pi\)
0.208157 + 0.978095i \(0.433253\pi\)
\(278\) 0 0
\(279\) 609550. 0.468812
\(280\) 0 0
\(281\) 2.51704e6 1.90163 0.950813 0.309766i \(-0.100251\pi\)
0.950813 + 0.309766i \(0.100251\pi\)
\(282\) 0 0
\(283\) 130497. + 226028.i 0.0968579 + 0.167763i 0.910382 0.413768i \(-0.135787\pi\)
−0.813525 + 0.581530i \(0.802454\pi\)
\(284\) 0 0
\(285\) 620753. 1.07518e6i 0.452696 0.784092i
\(286\) 0 0
\(287\) −29809.1 541201.i −0.0213621 0.387841i
\(288\) 0 0
\(289\) 707934. 1.22618e6i 0.498595 0.863592i
\(290\) 0 0
\(291\) 136505. + 236433.i 0.0944963 + 0.163672i
\(292\) 0 0
\(293\) 65011.7 0.0442408 0.0221204 0.999755i \(-0.492958\pi\)
0.0221204 + 0.999755i \(0.492958\pi\)
\(294\) 0 0
\(295\) 364720. 0.244008
\(296\) 0 0
\(297\) 181397. + 314189.i 0.119327 + 0.206680i
\(298\) 0 0
\(299\) 20080.6 34780.7i 0.0129897 0.0224989i
\(300\) 0 0
\(301\) −42499.8 771608.i −0.0270377 0.490886i
\(302\) 0 0
\(303\) 480584. 832395.i 0.300720 0.520862i
\(304\) 0 0
\(305\) −553907. 959395.i −0.340947 0.590538i
\(306\) 0 0
\(307\) 2.35599e6 1.42668 0.713342 0.700816i \(-0.247181\pi\)
0.713342 + 0.700816i \(0.247181\pi\)
\(308\) 0 0
\(309\) 1.41782e6 0.844744
\(310\) 0 0
\(311\) −1.05903e6 1.83429e6i −0.620878 1.07539i −0.989323 0.145742i \(-0.953443\pi\)
0.368445 0.929650i \(-0.379890\pi\)
\(312\) 0 0
\(313\) 93756.8 162392.i 0.0540931 0.0936920i −0.837711 0.546114i \(-0.816107\pi\)
0.891804 + 0.452422i \(0.149440\pi\)
\(314\) 0 0
\(315\) −916012. + 598340.i −0.520145 + 0.339760i
\(316\) 0 0
\(317\) 502705. 870711.i 0.280974 0.486661i −0.690651 0.723188i \(-0.742676\pi\)
0.971625 + 0.236527i \(0.0760093\pi\)
\(318\) 0 0
\(319\) −1.07573e6 1.86321e6i −0.591868 1.02515i
\(320\) 0 0
\(321\) −803919. −0.435461
\(322\) 0 0
\(323\) −83619.1 −0.0445964
\(324\) 0 0
\(325\) 798415. + 1.38290e6i 0.419296 + 0.726241i
\(326\) 0 0
\(327\) 754220. 1.30635e6i 0.390058 0.675600i
\(328\) 0 0
\(329\) −507753. 257011.i −0.258620 0.130907i
\(330\) 0 0
\(331\) 839920. 1.45478e6i 0.421374 0.729841i −0.574700 0.818364i \(-0.694881\pi\)
0.996074 + 0.0885229i \(0.0282146\pi\)
\(332\) 0 0
\(333\) −419404. 726429.i −0.207263 0.358990i
\(334\) 0 0
\(335\) −1.38312e6 −0.673360
\(336\) 0 0
\(337\) −995036. −0.477270 −0.238635 0.971109i \(-0.576700\pi\)
−0.238635 + 0.971109i \(0.576700\pi\)
\(338\) 0 0
\(339\) −521072. 902523.i −0.246263 0.426539i
\(340\) 0 0
\(341\) −1.87252e6 + 3.24330e6i −0.872049 + 1.51043i
\(342\) 0 0
\(343\) −358042. 2.14927e6i −0.164323 0.986407i
\(344\) 0 0
\(345\) −91164.4 + 157901.i −0.0412361 + 0.0714230i
\(346\) 0 0
\(347\) 2.02833e6 + 3.51317e6i 0.904304 + 1.56630i 0.821849 + 0.569705i \(0.192943\pi\)
0.0824546 + 0.996595i \(0.473724\pi\)
\(348\) 0 0
\(349\) 2.86202e6 1.25779 0.628897 0.777488i \(-0.283507\pi\)
0.628897 + 0.777488i \(0.283507\pi\)
\(350\) 0 0
\(351\) 150576. 0.0652361
\(352\) 0 0
\(353\) −416343. 721126.i −0.177834 0.308017i 0.763305 0.646039i \(-0.223576\pi\)
−0.941138 + 0.338022i \(0.890242\pi\)
\(354\) 0 0
\(355\) −2.02190e6 + 3.50203e6i −0.851507 + 1.47485i
\(356\) 0 0
\(357\) 65748.9 + 33280.4i 0.0273035 + 0.0138203i
\(358\) 0 0
\(359\) −1.34875e6 + 2.33611e6i −0.552327 + 0.956659i 0.445779 + 0.895143i \(0.352927\pi\)
−0.998106 + 0.0615157i \(0.980407\pi\)
\(360\) 0 0
\(361\) 361621. + 626346.i 0.146045 + 0.252957i
\(362\) 0 0
\(363\) −779527. −0.310502
\(364\) 0 0
\(365\) −3.26908e6 −1.28438
\(366\) 0 0
\(367\) 727411. + 1.25991e6i 0.281913 + 0.488287i 0.971856 0.235576i \(-0.0756978\pi\)
−0.689943 + 0.723864i \(0.742364\pi\)
\(368\) 0 0
\(369\) 169327. 293283.i 0.0647382 0.112130i
\(370\) 0 0
\(371\) 1.93033e6 1.26089e6i 0.728110 0.475602i
\(372\) 0 0
\(373\) 1.02344e6 1.77265e6i 0.380883 0.659708i −0.610306 0.792166i \(-0.708953\pi\)
0.991189 + 0.132457i \(0.0422868\pi\)
\(374\) 0 0
\(375\) −2.15954e6 3.74043e6i −0.793018 1.37355i
\(376\) 0 0
\(377\) −892950. −0.323574
\(378\) 0 0
\(379\) 416898. 0.149084 0.0745421 0.997218i \(-0.476250\pi\)
0.0745421 + 0.997218i \(0.476250\pi\)
\(380\) 0 0
\(381\) −906810. 1.57064e6i −0.320040 0.554325i
\(382\) 0 0
\(383\) −1.93813e6 + 3.35694e6i −0.675127 + 1.16935i 0.301305 + 0.953528i \(0.402578\pi\)
−0.976432 + 0.215827i \(0.930755\pi\)
\(384\) 0 0
\(385\) −369694. 6.71201e6i −0.127113 2.30782i
\(386\) 0 0
\(387\) 241415. 418143.i 0.0819383 0.141921i
\(388\) 0 0
\(389\) −1.41901e6 2.45780e6i −0.475458 0.823517i 0.524147 0.851628i \(-0.324384\pi\)
−0.999605 + 0.0281111i \(0.991051\pi\)
\(390\) 0 0
\(391\) 12280.4 0.00406228
\(392\) 0 0
\(393\) 346226. 0.113078
\(394\) 0 0
\(395\) −2.05734e6 3.56342e6i −0.663458 1.14914i
\(396\) 0 0
\(397\) −2.17133e6 + 3.76085e6i −0.691432 + 1.19760i 0.279937 + 0.960018i \(0.409686\pi\)
−0.971369 + 0.237577i \(0.923647\pi\)
\(398\) 0 0
\(399\) 84955.8 + 1.54242e6i 0.0267153 + 0.485032i
\(400\) 0 0
\(401\) −1.70152e6 + 2.94712e6i −0.528417 + 0.915244i 0.471034 + 0.882115i \(0.343881\pi\)
−0.999451 + 0.0331296i \(0.989453\pi\)
\(402\) 0 0
\(403\) 777182. + 1.34612e6i 0.238375 + 0.412877i
\(404\) 0 0
\(405\) −683602. −0.207093
\(406\) 0 0
\(407\) 5.15359e6 1.54214
\(408\) 0 0
\(409\) 2.64716e6 + 4.58501e6i 0.782477 + 1.35529i 0.930495 + 0.366305i \(0.119377\pi\)
−0.148018 + 0.988985i \(0.547289\pi\)
\(410\) 0 0
\(411\) −1.08775e6 + 1.88404e6i −0.317632 + 0.550155i
\(412\) 0 0
\(413\) −379935. + 248174.i −0.109606 + 0.0715947i
\(414\) 0 0
\(415\) 5.33315e6 9.23728e6i 1.52007 2.63284i
\(416\) 0 0
\(417\) 239005. + 413969.i 0.0673080 + 0.116581i
\(418\) 0 0
\(419\) −2.87267e6 −0.799376 −0.399688 0.916651i \(-0.630881\pi\)
−0.399688 + 0.916651i \(0.630881\pi\)
\(420\) 0 0
\(421\) 2.08688e6 0.573843 0.286921 0.957954i \(-0.407368\pi\)
0.286921 + 0.957954i \(0.407368\pi\)
\(422\) 0 0
\(423\) −177784. 307932.i −0.0483106 0.0836765i
\(424\) 0 0
\(425\) −244137. + 422857.i −0.0655633 + 0.113559i
\(426\) 0 0
\(427\) 1.22984e6 + 622512.i 0.326421 + 0.165226i
\(428\) 0 0
\(429\) −462565. + 801187.i −0.121347 + 0.210180i
\(430\) 0 0
\(431\) −1.21432e6 2.10326e6i −0.314876 0.545380i 0.664535 0.747257i \(-0.268629\pi\)
−0.979411 + 0.201876i \(0.935296\pi\)
\(432\) 0 0
\(433\) 956219. 0.245097 0.122548 0.992463i \(-0.460893\pi\)
0.122548 + 0.992463i \(0.460893\pi\)
\(434\) 0 0
\(435\) 4.05392e6 1.02719
\(436\) 0 0
\(437\) 128713. + 222938.i 0.0322418 + 0.0558444i
\(438\) 0 0
\(439\) 1.32102e6 2.28808e6i 0.327152 0.566644i −0.654794 0.755808i \(-0.727244\pi\)
0.981946 + 0.189164i \(0.0605778\pi\)
\(440\) 0 0
\(441\) 547084. 1.24660e6i 0.133955 0.305233i
\(442\) 0 0
\(443\) −1.71983e6 + 2.97883e6i −0.416366 + 0.721168i −0.995571 0.0940147i \(-0.970030\pi\)
0.579205 + 0.815182i \(0.303363\pi\)
\(444\) 0 0
\(445\) −5.87750e6 1.01801e7i −1.40700 2.43699i
\(446\) 0 0
\(447\) −1.16156e6 −0.274962
\(448\) 0 0
\(449\) 4.39903e6 1.02977 0.514886 0.857259i \(-0.327834\pi\)
0.514886 + 0.857259i \(0.327834\pi\)
\(450\) 0 0
\(451\) 1.04034e6 + 1.80192e6i 0.240842 + 0.417151i
\(452\) 0 0
\(453\) −689429. + 1.19413e6i −0.157850 + 0.273404i
\(454\) 0 0
\(455\) −2.48929e6 1.26001e6i −0.563698 0.285329i
\(456\) 0 0
\(457\) 1.12555e6 1.94951e6i 0.252101 0.436652i −0.712003 0.702176i \(-0.752212\pi\)
0.964104 + 0.265524i \(0.0855451\pi\)
\(458\) 0 0
\(459\) 23021.3 + 39874.1i 0.00510033 + 0.00883403i
\(460\) 0 0
\(461\) 1.85307e6 0.406107 0.203053 0.979168i \(-0.434913\pi\)
0.203053 + 0.979168i \(0.434913\pi\)
\(462\) 0 0
\(463\) 3.01089e6 0.652744 0.326372 0.945241i \(-0.394174\pi\)
0.326372 + 0.945241i \(0.394174\pi\)
\(464\) 0 0
\(465\) −3.52834e6 6.11126e6i −0.756725 1.31069i
\(466\) 0 0
\(467\) −533627. + 924270.i −0.113226 + 0.196113i −0.917069 0.398728i \(-0.869452\pi\)
0.803843 + 0.594841i \(0.202785\pi\)
\(468\) 0 0
\(469\) 1.44082e6 941145.i 0.302466 0.197572i
\(470\) 0 0
\(471\) −680345. + 1.17839e6i −0.141311 + 0.244758i
\(472\) 0 0
\(473\) 1.48324e6 + 2.56905e6i 0.304831 + 0.527983i
\(474\) 0 0
\(475\) −1.02354e7 −2.08147
\(476\) 0 0
\(477\) 1.44057e6 0.289893
\(478\) 0 0
\(479\) −2.84394e6 4.92585e6i −0.566346 0.980939i −0.996923 0.0783858i \(-0.975023\pi\)
0.430577 0.902554i \(-0.358310\pi\)
\(480\) 0 0
\(481\) 1.06949e6 1.85240e6i 0.210772 0.365068i
\(482\) 0 0
\(483\) −12476.7 226522.i −0.00243350 0.0441816i
\(484\) 0 0
\(485\) 1.58029e6 2.73715e6i 0.305059 0.528377i
\(486\) 0 0
\(487\) −3.42239e6 5.92775e6i −0.653893 1.13258i −0.982170 0.187995i \(-0.939801\pi\)
0.328277 0.944582i \(-0.393532\pi\)
\(488\) 0 0
\(489\) −296250. −0.0560256
\(490\) 0 0
\(491\) −5.60132e6 −1.04854 −0.524272 0.851551i \(-0.675662\pi\)
−0.524272 + 0.851551i \(0.675662\pi\)
\(492\) 0 0
\(493\) −136522. 236462.i −0.0252979 0.0438172i
\(494\) 0 0
\(495\) 2.10001e6 3.63732e6i 0.385219 0.667219i
\(496\) 0 0
\(497\) −276715. 5.02392e6i −0.0502507 0.912330i
\(498\) 0 0
\(499\) −1.16764e6 + 2.02242e6i −0.209923 + 0.363596i −0.951690 0.307061i \(-0.900655\pi\)
0.741767 + 0.670657i \(0.233988\pi\)
\(500\) 0 0
\(501\) −979135. 1.69591e6i −0.174280 0.301862i
\(502\) 0 0
\(503\) 1.08278e7 1.90819 0.954093 0.299510i \(-0.0968233\pi\)
0.954093 + 0.299510i \(0.0968233\pi\)
\(504\) 0 0
\(505\) −1.11273e7 −1.94161
\(506\) 0 0
\(507\) −1.47883e6 2.56141e6i −0.255505 0.442547i
\(508\) 0 0
\(509\) −1.58924e6 + 2.75264e6i −0.271890 + 0.470928i −0.969346 0.245700i \(-0.920982\pi\)
0.697455 + 0.716628i \(0.254316\pi\)
\(510\) 0 0
\(511\) 3.40546e6 2.22445e6i 0.576930 0.376852i
\(512\) 0 0
\(513\) −482582. + 835856.i −0.0809613 + 0.140229i
\(514\) 0 0
\(515\) −8.20695e6 1.42149e7i −1.36353 2.36170i
\(516\) 0 0
\(517\) 2.18460e6 0.359455
\(518\) 0 0
\(519\) 3.79118e6 0.617812
\(520\) 0 0
\(521\) 3.17918e6 + 5.50651e6i 0.513123 + 0.888755i 0.999884 + 0.0152197i \(0.00484477\pi\)
−0.486761 + 0.873535i \(0.661822\pi\)
\(522\) 0 0
\(523\) −3.61094e6 + 6.25433e6i −0.577252 + 0.999830i 0.418541 + 0.908198i \(0.362542\pi\)
−0.995793 + 0.0916321i \(0.970792\pi\)
\(524\) 0 0
\(525\) 8.04798e6 + 4.07368e6i 1.27435 + 0.645043i
\(526\) 0 0
\(527\) −237644. + 411612.i −0.0372735 + 0.0645597i
\(528\) 0 0
\(529\) 3.19927e6 + 5.54130e6i 0.497063 + 0.860939i
\(530\) 0 0
\(531\) −283538. −0.0436390
\(532\) 0 0
\(533\) 863574. 0.131668
\(534\) 0 0
\(535\) 4.65343e6 + 8.05997e6i 0.702892 + 1.21744i
\(536\) 0 0
\(537\) −15723.3 + 27233.5i −0.00235292 + 0.00407538i
\(538\) 0 0
\(539\) 4.95232e6 + 6.74046e6i 0.734237 + 0.999350i
\(540\) 0 0
\(541\) 4.71583e6 8.16805e6i 0.692731 1.19985i −0.278209 0.960521i \(-0.589741\pi\)
0.970940 0.239325i \(-0.0769260\pi\)
\(542\) 0 0
\(543\) 2.67575e6 + 4.63453e6i 0.389444 + 0.674538i
\(544\) 0 0
\(545\) −1.74630e7 −2.51842
\(546\) 0 0
\(547\) −9.91568e6 −1.41695 −0.708474 0.705737i \(-0.750616\pi\)
−0.708474 + 0.705737i \(0.750616\pi\)
\(548\) 0 0
\(549\) 430615. + 745847.i 0.0609759 + 0.105613i
\(550\) 0 0
\(551\) 2.86182e6 4.95682e6i 0.401572 0.695543i
\(552\) 0 0
\(553\) 4.56790e6 + 2.31216e6i 0.635191 + 0.321517i
\(554\) 0 0
\(555\) −4.85538e6 + 8.40976e6i −0.669099 + 1.15891i
\(556\) 0 0
\(557\) −4.82306e6 8.35379e6i −0.658696 1.14089i −0.980953 0.194243i \(-0.937775\pi\)
0.322258 0.946652i \(-0.395558\pi\)
\(558\) 0 0
\(559\) 1.23123e6 0.166651
\(560\) 0 0
\(561\) −282883. −0.0379490
\(562\) 0 0
\(563\) 1.54752e6 + 2.68038e6i 0.205762 + 0.356390i 0.950375 0.311106i \(-0.100699\pi\)
−0.744613 + 0.667496i \(0.767366\pi\)
\(564\) 0 0
\(565\) −6.03238e6 + 1.04484e7i −0.795000 + 1.37698i
\(566\) 0 0
\(567\) 712120. 465158.i 0.0930241 0.0607635i
\(568\) 0 0
\(569\) 6.15927e6 1.06682e7i 0.797533 1.38137i −0.123685 0.992322i \(-0.539471\pi\)
0.921218 0.389046i \(-0.127195\pi\)
\(570\) 0 0
\(571\) −1.61755e6 2.80168e6i −0.207619 0.359607i 0.743345 0.668908i \(-0.233238\pi\)
−0.950964 + 0.309302i \(0.899905\pi\)
\(572\) 0 0
\(573\) 7.45780e6 0.948908
\(574\) 0 0
\(575\) 1.50318e6 0.189601
\(576\) 0 0
\(577\) 4.52525e6 + 7.83796e6i 0.565852 + 0.980084i 0.996970 + 0.0777887i \(0.0247859\pi\)
−0.431118 + 0.902296i \(0.641881\pi\)
\(578\) 0 0
\(579\) 988825. 1.71269e6i 0.122581 0.212316i
\(580\) 0 0
\(581\) 729890. + 1.32516e7i 0.0897051 + 1.62865i
\(582\) 0 0
\(583\) −4.42538e6 + 7.66499e6i −0.539237 + 0.933986i
\(584\) 0 0
\(585\) −871598. 1.50965e6i −0.105300 0.182384i
\(586\) 0 0
\(587\) 2.13170e6 0.255347 0.127673 0.991816i \(-0.459249\pi\)
0.127673 + 0.991816i \(0.459249\pi\)
\(588\) 0 0
\(589\) −9.96318e6 −1.18334
\(590\) 0 0
\(591\) 2.14025e6 + 3.70703e6i 0.252056 + 0.436573i
\(592\) 0 0
\(593\) 3.63467e6 6.29543e6i 0.424451 0.735171i −0.571918 0.820311i \(-0.693800\pi\)
0.996369 + 0.0851397i \(0.0271337\pi\)
\(594\) 0 0
\(595\) −46918.4 851830.i −0.00543313 0.0986417i
\(596\) 0 0
\(597\) −2.82400e6 + 4.89131e6i −0.324286 + 0.561681i
\(598\) 0 0
\(599\) −1.10282e6 1.91015e6i −0.125585 0.217520i 0.796376 0.604802i \(-0.206748\pi\)
−0.921962 + 0.387281i \(0.873414\pi\)
\(600\) 0 0
\(601\) −1.00121e7 −1.13067 −0.565336 0.824860i \(-0.691254\pi\)
−0.565336 + 0.824860i \(0.691254\pi\)
\(602\) 0 0
\(603\) 1.07525e6 0.120425
\(604\) 0 0
\(605\) 4.51224e6 + 7.81542e6i 0.501191 + 0.868088i
\(606\) 0 0
\(607\) 3.16069e6 5.47448e6i 0.348185 0.603075i −0.637742 0.770250i \(-0.720131\pi\)
0.985927 + 0.167175i \(0.0534646\pi\)
\(608\) 0 0
\(609\) −4.22303e6 + 2.75849e6i −0.461404 + 0.301390i
\(610\) 0 0
\(611\) 453353. 785231.i 0.0491285 0.0850931i
\(612\) 0 0
\(613\) −7.15392e6 1.23910e7i −0.768941 1.33185i −0.938138 0.346263i \(-0.887451\pi\)
0.169196 0.985582i \(-0.445883\pi\)
\(614\) 0 0
\(615\) −3.92055e6 −0.417984
\(616\) 0 0
\(617\) 1.73991e7 1.83999 0.919993 0.391936i \(-0.128194\pi\)
0.919993 + 0.391936i \(0.128194\pi\)
\(618\) 0 0
\(619\) 4.13368e6 + 7.15975e6i 0.433621 + 0.751054i 0.997182 0.0750208i \(-0.0239023\pi\)
−0.563561 + 0.826074i \(0.690569\pi\)
\(620\) 0 0
\(621\) 70872.4 122755.i 0.00737476 0.0127735i
\(622\) 0 0
\(623\) 1.30498e7 + 6.60547e6i 1.34705 + 0.681841i
\(624\) 0 0
\(625\) −1.29211e7 + 2.23800e7i −1.32312 + 2.29172i
\(626\) 0 0
\(627\) −2.96495e6 5.13545e6i −0.301196 0.521687i
\(628\) 0 0
\(629\) 654048. 0.0659148
\(630\) 0 0
\(631\) −2.83238e6 −0.283190 −0.141595 0.989925i \(-0.545223\pi\)
−0.141595 + 0.989925i \(0.545223\pi\)
\(632\) 0 0
\(633\) −2.56945e6 4.45041e6i −0.254877 0.441460i
\(634\) 0 0
\(635\) −1.04980e7 + 1.81831e7i −1.03317 + 1.78951i
\(636\) 0 0
\(637\) 3.45051e6 381261.i 0.336926 0.0372284i
\(638\) 0 0
\(639\) 1.57185e6 2.72252e6i 0.152286 0.263766i
\(640\) 0 0
\(641\) 1.78588e6 + 3.09324e6i 0.171675 + 0.297350i 0.939006 0.343902i \(-0.111749\pi\)
−0.767330 + 0.641252i \(0.778415\pi\)
\(642\) 0 0
\(643\) 5.96911e6 0.569354 0.284677 0.958624i \(-0.408114\pi\)
0.284677 + 0.958624i \(0.408114\pi\)
\(644\) 0 0
\(645\) −5.58966e6 −0.529037
\(646\) 0 0
\(647\) −841379. 1.45731e6i −0.0790189 0.136865i 0.823808 0.566869i \(-0.191845\pi\)
−0.902827 + 0.430004i \(0.858512\pi\)
\(648\) 0 0
\(649\) 871021. 1.50865e6i 0.0811739 0.140597i
\(650\) 0 0
\(651\) 7.83395e6 + 3.96534e6i 0.724483 + 0.366715i
\(652\) 0 0
\(653\) −8.51526e6 + 1.47489e7i −0.781475 + 1.35355i 0.149608 + 0.988745i \(0.452199\pi\)
−0.931082 + 0.364809i \(0.881134\pi\)
\(654\) 0 0
\(655\) −2.00411e6 3.47121e6i −0.182523 0.316139i
\(656\) 0 0
\(657\) 2.54143e6 0.229702
\(658\) 0 0
\(659\) 1.99303e7 1.78772 0.893862 0.448342i \(-0.147985\pi\)
0.893862 + 0.448342i \(0.147985\pi\)
\(660\) 0 0
\(661\) −4.77867e6 8.27691e6i −0.425406 0.736825i 0.571052 0.820914i \(-0.306535\pi\)
−0.996458 + 0.0840888i \(0.973202\pi\)
\(662\) 0 0
\(663\) −58704.7 + 101680.i −0.00518668 + 0.00898359i
\(664\) 0 0
\(665\) 1.49723e7 9.77995e6i 1.31291 0.857595i
\(666\) 0 0
\(667\) −420290. + 727963.i −0.0365792 + 0.0633570i
\(668\) 0 0
\(669\) −19049.0 32993.8i −0.00164553 0.00285015i
\(670\) 0 0
\(671\) −5.29135e6 −0.453691
\(672\) 0 0
\(673\) 1.80397e7 1.53530 0.767648 0.640872i \(-0.221427\pi\)
0.767648 + 0.640872i \(0.221427\pi\)
\(674\) 0 0
\(675\) 2.81792e6 + 4.88077e6i 0.238050 + 0.412315i
\(676\) 0 0
\(677\) 7.23704e6 1.25349e7i 0.606861 1.05111i −0.384894 0.922961i \(-0.625762\pi\)
0.991754 0.128153i \(-0.0409048\pi\)
\(678\) 0 0
\(679\) 216278. + 3.92665e6i 0.0180027 + 0.326849i
\(680\) 0 0
\(681\) −5.08438e6 + 8.80641e6i −0.420117 + 0.727664i
\(682\) 0 0
\(683\) 1.34392e6 + 2.32774e6i 0.110236 + 0.190934i 0.915865 0.401486i \(-0.131506\pi\)
−0.805629 + 0.592420i \(0.798173\pi\)
\(684\) 0 0
\(685\) 2.51854e7 2.05080
\(686\) 0 0
\(687\) −7.24179e6 −0.585402
\(688\) 0 0
\(689\) 1.83674e6 + 3.18132e6i 0.147400 + 0.255305i
\(690\) 0 0
\(691\) 2.33814e6 4.04977e6i 0.186284 0.322653i −0.757725 0.652574i \(-0.773689\pi\)
0.944008 + 0.329922i \(0.107022\pi\)
\(692\) 0 0
\(693\) 287405. + 5.21801e6i 0.0227333 + 0.412735i
\(694\) 0 0
\(695\) 2.76693e6 4.79246e6i 0.217288 0.376354i
\(696\) 0 0
\(697\) 132030. + 228683.i 0.0102942 + 0.0178301i
\(698\) 0 0
\(699\) −1.05235e7 −0.814639
\(700\) 0 0
\(701\) 6.34801e6 0.487913 0.243957 0.969786i \(-0.421555\pi\)
0.243957 + 0.969786i \(0.421555\pi\)
\(702\) 0 0
\(703\) 6.85520e6 + 1.18736e7i 0.523157 + 0.906135i
\(704\) 0 0
\(705\) −2.05818e6 + 3.56488e6i −0.155959 + 0.270130i
\(706\) 0 0
\(707\) 1.15915e7 7.57158e6i 0.872150 0.569690i
\(708\) 0 0
\(709\) 3.20526e6 5.55168e6i 0.239468 0.414771i −0.721094 0.692838i \(-0.756360\pi\)
0.960562 + 0.278067i \(0.0896936\pi\)
\(710\) 0 0
\(711\) 1.59940e6 + 2.77025e6i 0.118655 + 0.205516i
\(712\) 0 0
\(713\) 1.46320e6 0.107790
\(714\) 0 0
\(715\) 1.07101e7 0.783481
\(716\) 0 0
\(717\) 7.67493e6 + 1.32934e7i 0.557541 + 0.965689i
\(718\) 0 0
\(719\) 3.67253e6 6.36100e6i 0.264937 0.458885i −0.702610 0.711575i \(-0.747982\pi\)
0.967547 + 0.252691i \(0.0813155\pi\)
\(720\) 0 0
\(721\) 1.82218e7 + 9.22343e6i 1.30543 + 0.660776i
\(722\) 0 0
\(723\) 4.28038e6 7.41384e6i 0.304535 0.527470i
\(724\) 0 0
\(725\) −1.67109e7 2.89441e7i −1.18074 2.04510i
\(726\) 0 0
\(727\) −1.57839e7 −1.10759 −0.553793 0.832655i \(-0.686820\pi\)
−0.553793 + 0.832655i \(0.686820\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 188240. + 326041.i 0.0130292 + 0.0225673i
\(732\) 0 0
\(733\) −6.76835e6 + 1.17231e7i −0.465289 + 0.805905i −0.999215 0.0396269i \(-0.987383\pi\)
0.533925 + 0.845532i \(0.320716\pi\)
\(734\) 0 0
\(735\) −1.56650e7 + 1.73089e6i −1.06958 + 0.118182i
\(736\) 0 0
\(737\) −3.30315e6 + 5.72123e6i −0.224006 + 0.387990i
\(738\) 0 0
\(739\) −3.66851e6 6.35405e6i −0.247103 0.427996i 0.715617 0.698492i \(-0.246145\pi\)
−0.962721 + 0.270497i \(0.912812\pi\)
\(740\) 0 0
\(741\) −2.46118e6 −0.164664
\(742\) 0 0
\(743\) −1.20844e7 −0.803068 −0.401534 0.915844i \(-0.631523\pi\)
−0.401534 + 0.915844i \(0.631523\pi\)
\(744\) 0 0
\(745\) 6.72360e6 + 1.16456e7i 0.443824 + 0.768726i
\(746\) 0 0
\(747\) −4.14606e6 + 7.18118e6i −0.271853 + 0.470863i
\(748\) 0 0
\(749\) −1.03320e7 5.22978e6i −0.672944 0.340627i
\(750\) 0 0
\(751\) 4.39088e6 7.60523e6i 0.284087 0.492054i −0.688300 0.725426i \(-0.741643\pi\)
0.972387 + 0.233372i \(0.0749761\pi\)
\(752\) 0 0
\(753\) −5.15241e6 8.92423e6i −0.331148 0.573566i
\(754\) 0 0
\(755\) 1.59628e7 1.01916
\(756\) 0 0
\(757\) 465874. 0.0295480 0.0147740 0.999891i \(-0.495297\pi\)
0.0147740 + 0.999891i \(0.495297\pi\)
\(758\) 0 0
\(759\) 435436. + 754198.i 0.0274360 + 0.0475205i
\(760\) 0 0
\(761\) −7.49891e6 + 1.29885e7i −0.469393 + 0.813013i −0.999388 0.0349882i \(-0.988861\pi\)
0.529995 + 0.848001i \(0.322194\pi\)
\(762\) 0 0
\(763\) 1.81915e7 1.18827e7i 1.13125 0.738932i
\(764\) 0 0
\(765\) 266514. 461616.i 0.0164652 0.0285186i
\(766\) 0 0
\(767\) −361513. 626159.i −0.0221889 0.0384323i
\(768\) 0 0
\(769\) −1.85606e7 −1.13181 −0.565907 0.824469i \(-0.691474\pi\)
−0.565907 + 0.824469i \(0.691474\pi\)
\(770\) 0 0
\(771\) −1.06537e7 −0.645450
\(772\) 0 0
\(773\) −6.13802e6 1.06314e7i −0.369470 0.639941i 0.620013 0.784592i \(-0.287128\pi\)
−0.989483 + 0.144651i \(0.953794\pi\)
\(774\) 0 0
\(775\) −2.90888e7 + 5.03832e7i −1.73969 + 3.01323i
\(776\) 0 0
\(777\) −664503. 1.20644e7i −0.0394861 0.716893i
\(778\) 0 0
\(779\) −2.76767e6 + 4.79375e6i −0.163407 + 0.283029i
\(780\) 0 0
\(781\) 9.65736e6 + 1.67270e7i 0.566540 + 0.981276i
\(782\) 0 0
\(783\) −3.15157e6 −0.183706
\(784\) 0 0
\(785\) 1.57525e7 0.912380
\(786\) 0 0
\(787\) 2.63496e6 + 4.56388e6i 0.151648 + 0.262662i 0.931833 0.362886i \(-0.118209\pi\)
−0.780185 + 0.625548i \(0.784875\pi\)
\(788\) 0 0
\(789\) −2.01742e6 + 3.49428e6i −0.115373 + 0.199832i
\(790\) 0 0
\(791\) −825586. 1.49890e7i −0.0469160 0.851788i
\(792\) 0 0
\(793\) −1.09807e6 + 1.90192e6i −0.0620082 + 0.107401i
\(794\) 0 0
\(795\) −8.33862e6 1.44429e7i −0.467925 0.810470i
\(796\) 0 0
\(797\) 1.11889e7 0.623940 0.311970 0.950092i \(-0.399011\pi\)
0.311970 + 0.950092i \(0.399011\pi\)
\(798\) 0 0
\(799\) 277250. 0.0153640
\(800\) 0 0
\(801\) 4.56925e6 + 7.91417e6i 0.251631 + 0.435837i
\(802\) 0 0
\(803\) −7.80719e6 + 1.35225e7i −0.427274 + 0.740059i
\(804\) 0 0
\(805\) −2.19885e6 + 1.43629e6i −0.119593 + 0.0781184i
\(806\) 0 0
\(807\) 3.36363e6 5.82599e6i 0.181813 0.314909i
\(808\) 0 0
\(809\) −3.52526e6 6.10592e6i −0.189374 0.328005i 0.755668 0.654955i \(-0.227312\pi\)
−0.945042 + 0.326950i \(0.893979\pi\)
\(810\) 0 0
\(811\) 2.54873e7 1.36073 0.680364 0.732875i \(-0.261822\pi\)
0.680364 + 0.732875i \(0.261822\pi\)
\(812\) 0 0
\(813\) −2.09374e6 −0.111095
\(814\) 0 0
\(815\) 1.71482e6 + 2.97016e6i 0.0904326 + 0.156634i
\(816\) 0 0
\(817\) −3.94596e6 + 6.83460e6i −0.206822 + 0.358227i
\(818\) 0 0
\(819\) 1.93520e6 + 979550.i 0.100813 + 0.0510290i
\(820\) 0 0
\(821\) 1.17439e7 2.03411e7i 0.608073 1.05321i −0.383485 0.923547i \(-0.625276\pi\)
0.991558 0.129666i \(-0.0413904\pi\)
\(822\) 0 0
\(823\) −6.04517e6 1.04705e7i −0.311107 0.538852i 0.667496 0.744614i \(-0.267366\pi\)
−0.978602 + 0.205761i \(0.934033\pi\)
\(824\) 0 0
\(825\) −3.46263e7 −1.77121
\(826\) 0 0
\(827\) −3.33335e6 −0.169480 −0.0847398 0.996403i \(-0.527006\pi\)
−0.0847398 + 0.996403i \(0.527006\pi\)
\(828\) 0 0
\(829\) 3.85267e6 + 6.67302e6i 0.194704 + 0.337238i 0.946804 0.321812i \(-0.104292\pi\)
−0.752099 + 0.659050i \(0.770959\pi\)
\(830\) 0 0
\(831\) −1.09316e7 + 1.89341e7i −0.549138 + 0.951134i
\(832\) 0 0
\(833\) 628505. + 855440.i 0.0313831 + 0.0427147i
\(834\) 0 0
\(835\) −1.13353e7 + 1.96333e7i −0.562622 + 0.974490i
\(836\) 0 0
\(837\) 2.74298e6 + 4.75098e6i 0.135335 + 0.234406i
\(838\) 0 0
\(839\) −2.40843e7 −1.18121 −0.590607 0.806959i \(-0.701112\pi\)
−0.590607 + 0.806959i \(0.701112\pi\)
\(840\) 0 0
\(841\) −1.82163e6 −0.0888116
\(842\) 0 0
\(843\) 1.13267e7 + 1.96184e7i 0.548952 + 0.950813i
\(844\) 0 0
\(845\) −1.71202e7 + 2.96531e7i −0.824837 + 1.42866i
\(846\) 0 0
\(847\) −1.00185e7 5.07110e6i −0.479837 0.242881i
\(848\) 0 0
\(849\) −1.17447e6 + 2.03425e6i −0.0559209 + 0.0968579i
\(850\) 0 0
\(851\) −1.00676e6 1.74376e6i −0.0476544 0.0825398i
\(852\) 0 0
\(853\) −2.82938e7 −1.33143 −0.665716 0.746206i \(-0.731874\pi\)
−0.665716 + 0.746206i \(0.731874\pi\)
\(854\) 0 0
\(855\) 1.11736e7 0.522728
\(856\) 0 0
\(857\) −1.08088e7 1.87214e7i −0.502719 0.870734i −0.999995 0.00314220i \(-0.999000\pi\)
0.497276 0.867592i \(-0.334334\pi\)
\(858\) 0 0
\(859\) 1.62691e7 2.81789e7i 0.752282 1.30299i −0.194432 0.980916i \(-0.562286\pi\)
0.946714 0.322075i \(-0.104380\pi\)
\(860\) 0 0
\(861\) 4.08411e6 2.66774e6i 0.187754 0.122641i
\(862\) 0 0
\(863\) 4.03192e6 6.98349e6i 0.184283 0.319187i −0.759052 0.651030i \(-0.774337\pi\)
0.943335 + 0.331843i \(0.107670\pi\)
\(864\) 0 0
\(865\) −2.19450e7 3.80098e7i −0.997229 1.72725i
\(866\) 0 0
\(867\) 1.27428e7 0.575728
\(868\) 0 0
\(869\) −1.96533e7 −0.882849
\(870\) 0 0
\(871\) 1.37096e6 + 2.37457e6i 0.0612321 + 0.106057i
\(872\) 0 0
\(873\) −1.22854e6 + 2.12790e6i −0.0545574 + 0.0944963i
\(874\) 0 0
\(875\) −3.42157e6 6.21206e7i −0.151080 2.74294i
\(876\) 0 0
\(877\) −8.51559e6 + 1.47494e7i −0.373866 + 0.647555i −0.990157 0.139964i \(-0.955301\pi\)
0.616291 + 0.787519i \(0.288635\pi\)
\(878\) 0 0
\(879\) 292553. + 506716.i 0.0127712 + 0.0221204i
\(880\) 0 0
\(881\) 1.24740e7 0.541459 0.270729 0.962655i \(-0.412735\pi\)
0.270729 + 0.962655i \(0.412735\pi\)
\(882\) 0 0
\(883\) −7.28955e6 −0.314629 −0.157315 0.987549i \(-0.550284\pi\)
−0.157315 + 0.987549i \(0.550284\pi\)
\(884\) 0 0
\(885\) 1.64124e6 + 2.84271e6i 0.0704390 + 0.122004i
\(886\) 0 0
\(887\) 9.38377e6 1.62532e7i 0.400469 0.693632i −0.593314 0.804971i \(-0.702181\pi\)
0.993782 + 0.111339i \(0.0355140\pi\)
\(888\) 0 0
\(889\) −1.43675e6 2.60850e7i −0.0609714 1.10697i
\(890\) 0 0
\(891\) −1.63257e6 + 2.82770e6i −0.0688935 + 0.119327i
\(892\) 0 0
\(893\) 2.90591e6 + 5.03318e6i 0.121942 + 0.211210i
\(894\) 0 0
\(895\) 364052. 0.0151917
\(896\) 0 0
\(897\) 361452. 0.0149992
\(898\) 0 0
\(899\) −1.62665e7 2.81744e7i −0.671266 1.16267i
\(900\) 0 0
\(901\) −561631. + 972773.i −0.0230483 + 0.0399208i
\(902\) 0 0
\(903\) 5.82284e6 3.80349e6i 0.237638 0.155225i
\(904\) 0 0
\(905\) 3.09767e7 5.36533e7i 1.25723 2.17758i
\(906\) 0 0
\(907\) 631833. + 1.09437e6i 0.0255026 + 0.0441718i 0.878495 0.477751i \(-0.158548\pi\)
−0.852992 + 0.521923i \(0.825215\pi\)
\(908\) 0 0
\(909\) 8.65051e6 0.347242
\(910\) 0 0
\(911\) 2.47718e7 0.988921 0.494461 0.869200i \(-0.335366\pi\)
0.494461 + 0.869200i \(0.335366\pi\)
\(912\) 0 0
\(913\) −2.54732e7 4.41208e7i −1.01136 1.75173i
\(914\) 0 0
\(915\) 4.98516e6 8.63456e6i 0.196846 0.340947i
\(916\) 0 0
\(917\) 4.44970e6 + 2.25233e6i 0.174746 + 0.0884520i
\(918\) 0 0
\(919\) −4.68338e6 + 8.11185e6i −0.182924 + 0.316834i −0.942875 0.333147i \(-0.891890\pi\)
0.759951 + 0.649980i \(0.225223\pi\)
\(920\) 0 0
\(921\) 1.06020e7 + 1.83631e7i 0.411848 + 0.713342i
\(922\) 0 0
\(923\) 8.01648e6 0.309727
\(924\) 0 0
\(925\) 8.00586e7 3.07648
\(926\) 0 0
\(927\) 6.38019e6 + 1.10508e7i 0.243857 + 0.422372i
\(928\) 0 0
\(929\) 3.43675e6 5.95263e6i 0.130650 0.226292i −0.793277 0.608860i \(-0.791627\pi\)
0.923927 + 0.382568i \(0.124960\pi\)
\(930\) 0 0
\(931\) −8.94215e6 + 2.03759e7i −0.338118 + 0.770446i
\(932\) 0 0
\(933\) 9.53125e6 1.65086e7i 0.358464 0.620878i
\(934\) 0 0
\(935\) 1.63745e6 + 2.83615e6i 0.0612546 + 0.106096i
\(936\) 0 0
\(937\) −4.26197e7 −1.58585 −0.792923 0.609322i \(-0.791442\pi\)
−0.792923 + 0.609322i \(0.791442\pi\)
\(938\) 0 0
\(939\) 1.68762e6 0.0624614
\(940\) 0 0
\(941\) −2.25113e6 3.89907e6i −0.0828755 0.143545i 0.821608 0.570052i \(-0.193077\pi\)
−0.904484 + 0.426508i \(0.859744\pi\)
\(942\) 0 0
\(943\) 406463. 704015.i 0.0148848 0.0257812i
\(944\) 0 0
\(945\) −8.78566e6 4.44707e6i −0.320033 0.161992i
\(946\) 0 0
\(947\) 1.68863e7 2.92479e7i 0.611870 1.05979i −0.379055 0.925374i \(-0.623751\pi\)
0.990925 0.134415i \(-0.0429156\pi\)
\(948\) 0 0
\(949\) 3.24034e6 + 5.61243e6i 0.116795 + 0.202295i
\(950\) 0 0
\(951\) 9.04870e6 0.324440
\(952\) 0 0
\(953\) 4.81813e7 1.71849 0.859244 0.511566i \(-0.170934\pi\)
0.859244 + 0.511566i \(0.170934\pi\)
\(954\) 0 0
\(955\) −4.31689e7 7.47708e7i −1.53166 2.65292i
\(956\) 0 0
\(957\) 9.68153e6 1.67689e7i 0.341715 0.591868i
\(958\) 0 0
\(959\) −2.62361e7 + 1.71375e7i −0.921198 + 0.601728i
\(960\) 0 0
\(961\) −1.40006e7 + 2.42498e7i −0.489033 + 0.847030i
\(962\) 0 0
\(963\) −3.61763e6 6.26593e6i −0.125707 0.217731i
\(964\) 0 0
\(965\) −2.28950e7 −0.791446
\(966\) 0 0
\(967\) 3.54640e7 1.21961 0.609805 0.792551i \(-0.291248\pi\)
0.609805 + 0.792551i \(0.291248\pi\)
\(968\) 0 0
\(969\) −376286. 651747.i −0.0128739 0.0222982i
\(970\) 0 0
\(971\) −3.92148e6 + 6.79221e6i −0.133476 + 0.231187i −0.925014 0.379933i \(-0.875947\pi\)
0.791538 + 0.611119i \(0.209280\pi\)
\(972\) 0 0
\(973\) 378679. + 6.87514e6i 0.0128230 + 0.232809i
\(974\) 0 0
\(975\) −7.18573e6 + 1.24461e7i −0.242080 + 0.419296i
\(976\) 0 0
\(977\) 1.23767e7 + 2.14371e7i 0.414829 + 0.718505i 0.995410 0.0956975i \(-0.0305081\pi\)
−0.580582 + 0.814202i \(0.697175\pi\)
\(978\) 0 0
\(979\) −5.61464e7 −1.87226
\(980\) 0 0
\(981\) 1.35760e7 0.450400
\(982\) 0 0
\(983\) 2.26564e7 + 3.92421e7i 0.747838 + 1.29529i 0.948857 + 0.315707i \(0.102242\pi\)
−0.201018 + 0.979587i \(0.564425\pi\)
\(984\) 0 0
\(985\) 2.47774e7 4.29157e7i 0.813701 1.40937i
\(986\) 0 0
\(987\) −281681. 5.11409e6i −0.00920376 0.167100i
\(988\) 0 0
\(989\) 579508. 1.00374e6i 0.0188395 0.0326309i
\(990\) 0 0
\(991\) −1.87071e7 3.24017e7i −0.605095 1.04805i −0.992037 0.125951i \(-0.959802\pi\)
0.386942 0.922104i \(-0.373531\pi\)
\(992\) 0 0
\(993\) 1.51186e7 0.486561
\(994\) 0 0
\(995\) 6.53861e7 2.09376
\(996\) 0 0
\(997\) −2.32639e7 4.02943e7i −0.741216 1.28382i −0.951942 0.306279i \(-0.900916\pi\)
0.210726 0.977545i \(-0.432417\pi\)
\(998\) 0 0
\(999\) 3.77463e6 6.53786e6i 0.119663 0.207263i
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.j.289.4 8
4.3 odd 2 21.6.e.c.16.3 yes 8
7.4 even 3 inner 336.6.q.j.193.4 8
12.11 even 2 63.6.e.e.37.2 8
28.3 even 6 147.6.e.o.67.3 8
28.11 odd 6 21.6.e.c.4.3 8
28.19 even 6 147.6.a.l.1.2 4
28.23 odd 6 147.6.a.m.1.2 4
28.27 even 2 147.6.e.o.79.3 8
84.11 even 6 63.6.e.e.46.2 8
84.23 even 6 441.6.a.w.1.3 4
84.47 odd 6 441.6.a.v.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.c.4.3 8 28.11 odd 6
21.6.e.c.16.3 yes 8 4.3 odd 2
63.6.e.e.37.2 8 12.11 even 2
63.6.e.e.46.2 8 84.11 even 6
147.6.a.l.1.2 4 28.19 even 6
147.6.a.m.1.2 4 28.23 odd 6
147.6.e.o.67.3 8 28.3 even 6
147.6.e.o.79.3 8 28.27 even 2
336.6.q.j.193.4 8 7.4 even 3 inner
336.6.q.j.289.4 8 1.1 even 1 trivial
441.6.a.v.1.3 4 84.47 odd 6
441.6.a.w.1.3 4 84.23 even 6