Properties

Label 336.5.bh.d.241.2
Level $336$
Weight $5$
Character 336.241
Analytic conductor $34.732$
Analytic rank $0$
Dimension $4$
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] = [336,5,Mod(145,336)]
 
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, 5]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("336.145");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 336.bh (of order \(6\), 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: \(34.7323075962\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: no (minimal twist has level 42)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 241.2
Root \(-0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 336.241
Dual form 336.5.bh.d.145.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(4.50000 + 2.59808i) q^{3} +(-12.2574 + 7.07679i) q^{5} +(-12.1985 - 47.4573i) q^{7} +(13.5000 + 23.3827i) q^{9} +O(q^{10})\) \(q+(4.50000 + 2.59808i) q^{3} +(-12.2574 + 7.07679i) q^{5} +(-12.1985 - 47.4573i) q^{7} +(13.5000 + 23.3827i) q^{9} +(32.0147 - 55.4511i) q^{11} +228.919i q^{13} -73.5442 q^{15} +(-195.250 - 112.728i) q^{17} +(255.250 - 147.369i) q^{19} +(68.4045 - 245.251i) q^{21} +(354.749 + 614.444i) q^{23} +(-212.338 + 367.780i) q^{25} +140.296i q^{27} +740.397 q^{29} +(577.390 + 333.356i) q^{31} +(288.132 - 166.353i) q^{33} +(485.367 + 495.375i) q^{35} +(416.882 + 722.061i) q^{37} +(-594.749 + 1030.14i) q^{39} +2817.60i q^{41} -3066.41 q^{43} +(-330.949 - 191.073i) q^{45} +(531.502 - 306.863i) q^{47} +(-2103.39 + 1157.81i) q^{49} +(-585.749 - 1014.55i) q^{51} +(576.300 - 998.181i) q^{53} +906.246i q^{55} +1531.50 q^{57} +(3024.67 + 1746.30i) q^{59} +(1967.79 - 1136.11i) q^{61} +(945.000 - 925.907i) q^{63} +(-1620.01 - 2805.94i) q^{65} +(-4337.31 + 7512.44i) q^{67} +3686.66i q^{69} +353.591 q^{71} +(3524.68 + 2034.97i) q^{73} +(-1911.04 + 1103.34i) q^{75} +(-3022.09 - 842.913i) q^{77} +(3236.41 + 5605.63i) q^{79} +(-364.500 + 631.333i) q^{81} -8225.83i q^{83} +3191.00 q^{85} +(3331.79 + 1923.61i) q^{87} +(-13456.5 + 7769.13i) q^{89} +(10863.9 - 2792.47i) q^{91} +(1732.17 + 3000.21i) q^{93} +(-2085.79 + 3612.70i) q^{95} -1558.61i q^{97} +1728.79 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 18 q^{3} - 66 q^{5} + 70 q^{7} + 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 18 q^{3} - 66 q^{5} + 70 q^{7} + 54 q^{9} + 162 q^{11} - 396 q^{15} - 204 q^{17} + 444 q^{19} + 630 q^{21} - 312 q^{23} - 476 q^{25} + 2724 q^{29} + 3786 q^{31} + 1458 q^{33} - 672 q^{35} + 1396 q^{37} - 648 q^{39} + 632 q^{43} - 1782 q^{45} + 7896 q^{47} - 98 q^{49} - 612 q^{51} - 1038 q^{53} + 2664 q^{57} + 966 q^{59} + 5088 q^{61} + 3780 q^{63} - 744 q^{65} - 14600 q^{67} + 9696 q^{71} + 22584 q^{73} - 4284 q^{75} - 3654 q^{77} - 3974 q^{79} - 1458 q^{81} + 1224 q^{85} + 12258 q^{87} - 33156 q^{89} + 18984 q^{91} + 11358 q^{93} - 3252 q^{95} + 8748 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(85\) \(113\) \(127\) \(241\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{6}\right)\)

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
\(3\) 4.50000 + 2.59808i 0.500000 + 0.288675i
\(4\) 0 0
\(5\) −12.2574 + 7.07679i −0.490294 + 0.283072i −0.724697 0.689068i \(-0.758020\pi\)
0.234402 + 0.972140i \(0.424687\pi\)
\(6\) 0 0
\(7\) −12.1985 47.4573i −0.248949 0.968517i
\(8\) 0 0
\(9\) 13.5000 + 23.3827i 0.166667 + 0.288675i
\(10\) 0 0
\(11\) 32.0147 55.4511i 0.264584 0.458274i −0.702870 0.711318i \(-0.748099\pi\)
0.967455 + 0.253044i \(0.0814319\pi\)
\(12\) 0 0
\(13\) 228.919i 1.35455i 0.735729 + 0.677276i \(0.236839\pi\)
−0.735729 + 0.677276i \(0.763161\pi\)
\(14\) 0 0
\(15\) −73.5442 −0.326863
\(16\) 0 0
\(17\) −195.250 112.728i −0.675605 0.390061i 0.122592 0.992457i \(-0.460879\pi\)
−0.798197 + 0.602397i \(0.794213\pi\)
\(18\) 0 0
\(19\) 255.250 147.369i 0.707063 0.408223i −0.102910 0.994691i \(-0.532815\pi\)
0.809973 + 0.586468i \(0.199482\pi\)
\(20\) 0 0
\(21\) 68.4045 245.251i 0.155112 0.556124i
\(22\) 0 0
\(23\) 354.749 + 614.444i 0.670604 + 1.16152i 0.977733 + 0.209852i \(0.0672982\pi\)
−0.307129 + 0.951668i \(0.599368\pi\)
\(24\) 0 0
\(25\) −212.338 + 367.780i −0.339741 + 0.588449i
\(26\) 0 0
\(27\) 140.296i 0.192450i
\(28\) 0 0
\(29\) 740.397 0.880377 0.440188 0.897905i \(-0.354912\pi\)
0.440188 + 0.897905i \(0.354912\pi\)
\(30\) 0 0
\(31\) 577.390 + 333.356i 0.600822 + 0.346885i 0.769365 0.638809i \(-0.220573\pi\)
−0.168543 + 0.985694i \(0.553906\pi\)
\(32\) 0 0
\(33\) 288.132 166.353i 0.264584 0.152758i
\(34\) 0 0
\(35\) 485.367 + 495.375i 0.396218 + 0.404388i
\(36\) 0 0
\(37\) 416.882 + 722.061i 0.304516 + 0.527437i 0.977153 0.212535i \(-0.0681720\pi\)
−0.672638 + 0.739972i \(0.734839\pi\)
\(38\) 0 0
\(39\) −594.749 + 1030.14i −0.391025 + 0.677276i
\(40\) 0 0
\(41\) 2817.60i 1.67615i 0.545558 + 0.838073i \(0.316318\pi\)
−0.545558 + 0.838073i \(0.683682\pi\)
\(42\) 0 0
\(43\) −3066.41 −1.65841 −0.829207 0.558942i \(-0.811207\pi\)
−0.829207 + 0.558942i \(0.811207\pi\)
\(44\) 0 0
\(45\) −330.949 191.073i −0.163431 0.0943572i
\(46\) 0 0
\(47\) 531.502 306.863i 0.240608 0.138915i −0.374848 0.927086i \(-0.622305\pi\)
0.615456 + 0.788171i \(0.288972\pi\)
\(48\) 0 0
\(49\) −2103.39 + 1157.81i −0.876049 + 0.482222i
\(50\) 0 0
\(51\) −585.749 1014.55i −0.225202 0.390061i
\(52\) 0 0
\(53\) 576.300 998.181i 0.205162 0.355351i −0.745022 0.667040i \(-0.767561\pi\)
0.950184 + 0.311688i \(0.100895\pi\)
\(54\) 0 0
\(55\) 906.246i 0.299585i
\(56\) 0 0
\(57\) 1531.50 0.471375
\(58\) 0 0
\(59\) 3024.67 + 1746.30i 0.868909 + 0.501665i 0.866986 0.498333i \(-0.166054\pi\)
0.00192348 + 0.999998i \(0.499388\pi\)
\(60\) 0 0
\(61\) 1967.79 1136.11i 0.528834 0.305323i −0.211707 0.977333i \(-0.567902\pi\)
0.740542 + 0.672010i \(0.234569\pi\)
\(62\) 0 0
\(63\) 945.000 925.907i 0.238095 0.233285i
\(64\) 0 0
\(65\) −1620.01 2805.94i −0.383435 0.664129i
\(66\) 0 0
\(67\) −4337.31 + 7512.44i −0.966208 + 1.67352i −0.259874 + 0.965642i \(0.583681\pi\)
−0.706334 + 0.707879i \(0.749652\pi\)
\(68\) 0 0
\(69\) 3686.66i 0.774346i
\(70\) 0 0
\(71\) 353.591 0.0701431 0.0350715 0.999385i \(-0.488834\pi\)
0.0350715 + 0.999385i \(0.488834\pi\)
\(72\) 0 0
\(73\) 3524.68 + 2034.97i 0.661415 + 0.381868i 0.792816 0.609461i \(-0.208614\pi\)
−0.131401 + 0.991329i \(0.541948\pi\)
\(74\) 0 0
\(75\) −1911.04 + 1103.34i −0.339741 + 0.196150i
\(76\) 0 0
\(77\) −3022.09 842.913i −0.509714 0.142168i
\(78\) 0 0
\(79\) 3236.41 + 5605.63i 0.518573 + 0.898194i 0.999767 + 0.0215805i \(0.00686981\pi\)
−0.481194 + 0.876614i \(0.659797\pi\)
\(80\) 0 0
\(81\) −364.500 + 631.333i −0.0555556 + 0.0962250i
\(82\) 0 0
\(83\) 8225.83i 1.19405i −0.802222 0.597026i \(-0.796349\pi\)
0.802222 0.597026i \(-0.203651\pi\)
\(84\) 0 0
\(85\) 3191.00 0.441660
\(86\) 0 0
\(87\) 3331.79 + 1923.61i 0.440188 + 0.254143i
\(88\) 0 0
\(89\) −13456.5 + 7769.13i −1.69884 + 0.980828i −0.751983 + 0.659183i \(0.770902\pi\)
−0.946860 + 0.321645i \(0.895764\pi\)
\(90\) 0 0
\(91\) 10863.9 2792.47i 1.31191 0.337214i
\(92\) 0 0
\(93\) 1732.17 + 3000.21i 0.200274 + 0.346885i
\(94\) 0 0
\(95\) −2085.79 + 3612.70i −0.231113 + 0.400299i
\(96\) 0 0
\(97\) 1558.61i 0.165651i −0.996564 0.0828254i \(-0.973606\pi\)
0.996564 0.0828254i \(-0.0263944\pi\)
\(98\) 0 0
\(99\) 1728.79 0.176390
\(100\) 0 0
\(101\) 13627.2 + 7867.66i 1.33587 + 0.771264i 0.986192 0.165607i \(-0.0529583\pi\)
0.349676 + 0.936871i \(0.386292\pi\)
\(102\) 0 0
\(103\) −29.2099 + 16.8644i −0.00275332 + 0.00158963i −0.501376 0.865229i \(-0.667173\pi\)
0.498623 + 0.866819i \(0.333839\pi\)
\(104\) 0 0
\(105\) 897.127 + 3490.21i 0.0813721 + 0.316572i
\(106\) 0 0
\(107\) −2723.23 4716.78i −0.237858 0.411981i 0.722242 0.691641i \(-0.243112\pi\)
−0.960099 + 0.279659i \(0.909778\pi\)
\(108\) 0 0
\(109\) −8348.89 + 14460.7i −0.702709 + 1.21713i 0.264803 + 0.964303i \(0.414693\pi\)
−0.967512 + 0.252825i \(0.918640\pi\)
\(110\) 0 0
\(111\) 4332.37i 0.351625i
\(112\) 0 0
\(113\) 9455.64 0.740515 0.370258 0.928929i \(-0.379269\pi\)
0.370258 + 0.928929i \(0.379269\pi\)
\(114\) 0 0
\(115\) −8696.58 5020.97i −0.657586 0.379658i
\(116\) 0 0
\(117\) −5352.74 + 3090.41i −0.391025 + 0.225759i
\(118\) 0 0
\(119\) −2967.99 + 10641.1i −0.209589 + 0.751440i
\(120\) 0 0
\(121\) 5270.62 + 9128.97i 0.359990 + 0.623521i
\(122\) 0 0
\(123\) −7320.35 + 12679.2i −0.483862 + 0.838073i
\(124\) 0 0
\(125\) 14856.7i 0.950827i
\(126\) 0 0
\(127\) 2380.07 0.147564 0.0737822 0.997274i \(-0.476493\pi\)
0.0737822 + 0.997274i \(0.476493\pi\)
\(128\) 0 0
\(129\) −13798.8 7966.76i −0.829207 0.478743i
\(130\) 0 0
\(131\) 3758.37 2169.90i 0.219006 0.126443i −0.386484 0.922296i \(-0.626310\pi\)
0.605490 + 0.795853i \(0.292977\pi\)
\(132\) 0 0
\(133\) −10107.4 10315.8i −0.571393 0.583176i
\(134\) 0 0
\(135\) −992.846 1719.66i −0.0544772 0.0943572i
\(136\) 0 0
\(137\) 15379.6 26638.2i 0.819414 1.41927i −0.0867001 0.996234i \(-0.527632\pi\)
0.906114 0.423033i \(-0.139034\pi\)
\(138\) 0 0
\(139\) 27186.6i 1.40710i 0.710644 + 0.703551i \(0.248404\pi\)
−0.710644 + 0.703551i \(0.751596\pi\)
\(140\) 0 0
\(141\) 3189.01 0.160405
\(142\) 0 0
\(143\) 12693.8 + 7328.78i 0.620755 + 0.358393i
\(144\) 0 0
\(145\) −9075.31 + 5239.63i −0.431644 + 0.249210i
\(146\) 0 0
\(147\) −12473.4 254.611i −0.577230 0.0117827i
\(148\) 0 0
\(149\) 2913.46 + 5046.26i 0.131231 + 0.227299i 0.924151 0.382027i \(-0.124774\pi\)
−0.792920 + 0.609325i \(0.791440\pi\)
\(150\) 0 0
\(151\) 12593.4 21812.4i 0.552317 0.956642i −0.445789 0.895138i \(-0.647077\pi\)
0.998107 0.0615039i \(-0.0195897\pi\)
\(152\) 0 0
\(153\) 6087.29i 0.260040i
\(154\) 0 0
\(155\) −9436.37 −0.392773
\(156\) 0 0
\(157\) 20694.6 + 11948.0i 0.839571 + 0.484726i 0.857118 0.515120i \(-0.172253\pi\)
−0.0175475 + 0.999846i \(0.505586\pi\)
\(158\) 0 0
\(159\) 5186.70 2994.54i 0.205162 0.118450i
\(160\) 0 0
\(161\) 24832.5 24330.7i 0.958005 0.938650i
\(162\) 0 0
\(163\) −21293.0 36880.5i −0.801422 1.38810i −0.918680 0.395001i \(-0.870744\pi\)
0.117259 0.993101i \(-0.462589\pi\)
\(164\) 0 0
\(165\) −2354.50 + 4078.11i −0.0864828 + 0.149793i
\(166\) 0 0
\(167\) 26356.3i 0.945043i −0.881319 0.472521i \(-0.843344\pi\)
0.881319 0.472521i \(-0.156656\pi\)
\(168\) 0 0
\(169\) −23843.0 −0.834809
\(170\) 0 0
\(171\) 6891.74 + 3978.95i 0.235688 + 0.136074i
\(172\) 0 0
\(173\) 24265.3 14009.6i 0.810763 0.468094i −0.0364578 0.999335i \(-0.511607\pi\)
0.847221 + 0.531241i \(0.178274\pi\)
\(174\) 0 0
\(175\) 20044.1 + 5590.63i 0.654500 + 0.182551i
\(176\) 0 0
\(177\) 9074.02 + 15716.7i 0.289636 + 0.501665i
\(178\) 0 0
\(179\) 12699.2 21995.7i 0.396343 0.686487i −0.596928 0.802295i \(-0.703612\pi\)
0.993272 + 0.115808i \(0.0369457\pi\)
\(180\) 0 0
\(181\) 44097.2i 1.34603i −0.739630 0.673014i \(-0.764999\pi\)
0.739630 0.673014i \(-0.235001\pi\)
\(182\) 0 0
\(183\) 11806.8 0.352556
\(184\) 0 0
\(185\) −10219.8 5900.38i −0.298605 0.172400i
\(186\) 0 0
\(187\) −12501.7 + 7217.88i −0.357509 + 0.206408i
\(188\) 0 0
\(189\) 6658.08 1711.40i 0.186391 0.0479102i
\(190\) 0 0
\(191\) −32279.6 55909.8i −0.884832 1.53257i −0.845906 0.533332i \(-0.820940\pi\)
−0.0389256 0.999242i \(-0.512394\pi\)
\(192\) 0 0
\(193\) −18337.3 + 31761.1i −0.492290 + 0.852670i −0.999961 0.00888055i \(-0.997173\pi\)
0.507671 + 0.861551i \(0.330507\pi\)
\(194\) 0 0
\(195\) 16835.7i 0.442753i
\(196\) 0 0
\(197\) −73147.0 −1.88480 −0.942398 0.334494i \(-0.891434\pi\)
−0.942398 + 0.334494i \(0.891434\pi\)
\(198\) 0 0
\(199\) 1213.33 + 700.514i 0.0306387 + 0.0176893i 0.515241 0.857045i \(-0.327702\pi\)
−0.484602 + 0.874735i \(0.661036\pi\)
\(200\) 0 0
\(201\) −39035.8 + 22537.3i −0.966208 + 0.557840i
\(202\) 0 0
\(203\) −9031.72 35137.3i −0.219169 0.852660i
\(204\) 0 0
\(205\) −19939.6 34536.4i −0.474469 0.821805i
\(206\) 0 0
\(207\) −9578.23 + 16590.0i −0.223535 + 0.387173i
\(208\) 0 0
\(209\) 18871.8i 0.432038i
\(210\) 0 0
\(211\) −58231.7 −1.30796 −0.653980 0.756512i \(-0.726902\pi\)
−0.653980 + 0.756512i \(0.726902\pi\)
\(212\) 0 0
\(213\) 1591.16 + 918.657i 0.0350715 + 0.0202486i
\(214\) 0 0
\(215\) 37586.1 21700.3i 0.813111 0.469450i
\(216\) 0 0
\(217\) 8776.92 31467.8i 0.186390 0.668263i
\(218\) 0 0
\(219\) 10574.0 + 18314.8i 0.220472 + 0.381868i
\(220\) 0 0
\(221\) 25805.5 44696.4i 0.528357 0.915141i
\(222\) 0 0
\(223\) 61050.9i 1.22767i 0.789434 + 0.613836i \(0.210374\pi\)
−0.789434 + 0.613836i \(0.789626\pi\)
\(224\) 0 0
\(225\) −11466.3 −0.226494
\(226\) 0 0
\(227\) −21645.2 12496.8i −0.420058 0.242520i 0.275044 0.961432i \(-0.411307\pi\)
−0.695102 + 0.718911i \(0.744641\pi\)
\(228\) 0 0
\(229\) −16392.9 + 9464.44i −0.312597 + 0.180478i −0.648088 0.761565i \(-0.724431\pi\)
0.335491 + 0.942043i \(0.391098\pi\)
\(230\) 0 0
\(231\) −11409.5 11644.7i −0.213817 0.218226i
\(232\) 0 0
\(233\) −3342.37 5789.15i −0.0615662 0.106636i 0.833599 0.552369i \(-0.186276\pi\)
−0.895166 + 0.445734i \(0.852943\pi\)
\(234\) 0 0
\(235\) −4343.21 + 7522.66i −0.0786457 + 0.136218i
\(236\) 0 0
\(237\) 33633.8i 0.598796i
\(238\) 0 0
\(239\) 96461.0 1.68871 0.844356 0.535782i \(-0.179983\pi\)
0.844356 + 0.535782i \(0.179983\pi\)
\(240\) 0 0
\(241\) 47371.8 + 27350.1i 0.815616 + 0.470896i 0.848902 0.528550i \(-0.177264\pi\)
−0.0332863 + 0.999446i \(0.510597\pi\)
\(242\) 0 0
\(243\) −3280.50 + 1894.00i −0.0555556 + 0.0320750i
\(244\) 0 0
\(245\) 17588.4 29077.0i 0.293019 0.484415i
\(246\) 0 0
\(247\) 33735.5 + 58431.6i 0.552959 + 0.957753i
\(248\) 0 0
\(249\) 21371.3 37016.2i 0.344693 0.597026i
\(250\) 0 0
\(251\) 108137.i 1.71643i −0.513286 0.858217i \(-0.671572\pi\)
0.513286 0.858217i \(-0.328428\pi\)
\(252\) 0 0
\(253\) 45428.8 0.709725
\(254\) 0 0
\(255\) 14359.5 + 8290.45i 0.220830 + 0.127496i
\(256\) 0 0
\(257\) 31078.3 17943.0i 0.470533 0.271663i −0.245930 0.969288i \(-0.579093\pi\)
0.716463 + 0.697625i \(0.245760\pi\)
\(258\) 0 0
\(259\) 29181.8 28592.2i 0.435023 0.426233i
\(260\) 0 0
\(261\) 9995.36 + 17312.5i 0.146729 + 0.254143i
\(262\) 0 0
\(263\) −32715.8 + 56665.4i −0.472984 + 0.819232i −0.999522 0.0309199i \(-0.990156\pi\)
0.526538 + 0.850151i \(0.323490\pi\)
\(264\) 0 0
\(265\) 16313.4i 0.232302i
\(266\) 0 0
\(267\) −80739.2 −1.13256
\(268\) 0 0
\(269\) −59938.7 34605.6i −0.828329 0.478236i 0.0249511 0.999689i \(-0.492057\pi\)
−0.853280 + 0.521453i \(0.825390\pi\)
\(270\) 0 0
\(271\) −5672.94 + 3275.27i −0.0772449 + 0.0445973i −0.538125 0.842865i \(-0.680867\pi\)
0.460880 + 0.887462i \(0.347534\pi\)
\(272\) 0 0
\(273\) 56142.5 + 15659.1i 0.753298 + 0.210108i
\(274\) 0 0
\(275\) 13595.9 + 23548.8i 0.179780 + 0.311389i
\(276\) 0 0
\(277\) −39551.6 + 68505.4i −0.515471 + 0.892823i 0.484367 + 0.874865i \(0.339050\pi\)
−0.999839 + 0.0179578i \(0.994284\pi\)
\(278\) 0 0
\(279\) 18001.2i 0.231257i
\(280\) 0 0
\(281\) 34363.1 0.435191 0.217596 0.976039i \(-0.430179\pi\)
0.217596 + 0.976039i \(0.430179\pi\)
\(282\) 0 0
\(283\) −79598.4 45956.1i −0.993874 0.573813i −0.0874439 0.996169i \(-0.527870\pi\)
−0.906430 + 0.422356i \(0.861203\pi\)
\(284\) 0 0
\(285\) −18772.1 + 10838.1i −0.231113 + 0.133433i
\(286\) 0 0
\(287\) 133716. 34370.5i 1.62338 0.417274i
\(288\) 0 0
\(289\) −16345.5 28311.3i −0.195705 0.338972i
\(290\) 0 0
\(291\) 4049.38 7013.74i 0.0478193 0.0828254i
\(292\) 0 0
\(293\) 23218.2i 0.270453i −0.990815 0.135227i \(-0.956824\pi\)
0.990815 0.135227i \(-0.0431763\pi\)
\(294\) 0 0
\(295\) −49432.7 −0.568028
\(296\) 0 0
\(297\) 7779.58 + 4491.54i 0.0881948 + 0.0509193i
\(298\) 0 0
\(299\) −140658. + 81208.9i −1.57334 + 0.908367i
\(300\) 0 0
\(301\) 37405.5 + 145523.i 0.412860 + 1.60620i
\(302\) 0 0
\(303\) 40881.6 + 70808.9i 0.445289 + 0.771264i
\(304\) 0 0
\(305\) −16080.0 + 27851.3i −0.172856 + 0.299396i
\(306\) 0 0
\(307\) 66385.9i 0.704367i −0.935931 0.352183i \(-0.885439\pi\)
0.935931 0.352183i \(-0.114561\pi\)
\(308\) 0 0
\(309\) −175.260 −0.00183555
\(310\) 0 0
\(311\) 82879.0 + 47850.2i 0.856888 + 0.494724i 0.862969 0.505257i \(-0.168602\pi\)
−0.00608108 + 0.999982i \(0.501936\pi\)
\(312\) 0 0
\(313\) 41486.1 23952.0i 0.423461 0.244486i −0.273096 0.961987i \(-0.588048\pi\)
0.696557 + 0.717501i \(0.254714\pi\)
\(314\) 0 0
\(315\) −5030.75 + 18036.7i −0.0507005 + 0.181776i
\(316\) 0 0
\(317\) 17064.1 + 29555.9i 0.169811 + 0.294121i 0.938353 0.345678i \(-0.112351\pi\)
−0.768543 + 0.639799i \(0.779018\pi\)
\(318\) 0 0
\(319\) 23703.6 41055.8i 0.232934 0.403454i
\(320\) 0 0
\(321\) 28300.7i 0.274654i
\(322\) 0 0
\(323\) −66450.0 −0.636927
\(324\) 0 0
\(325\) −84192.0 48608.3i −0.797084 0.460196i
\(326\) 0 0
\(327\) −75140.0 + 43382.1i −0.702709 + 0.405709i
\(328\) 0 0
\(329\) −21046.4 21480.4i −0.194440 0.198450i
\(330\) 0 0
\(331\) 66642.6 + 115428.i 0.608270 + 1.05355i 0.991526 + 0.129912i \(0.0414694\pi\)
−0.383256 + 0.923642i \(0.625197\pi\)
\(332\) 0 0
\(333\) −11255.8 + 19495.7i −0.101505 + 0.175812i
\(334\) 0 0
\(335\) 122777.i 1.09402i
\(336\) 0 0
\(337\) −49734.4 −0.437922 −0.218961 0.975734i \(-0.570267\pi\)
−0.218961 + 0.975734i \(0.570267\pi\)
\(338\) 0 0
\(339\) 42550.4 + 24566.5i 0.370258 + 0.213768i
\(340\) 0 0
\(341\) 36970.0 21344.6i 0.317936 0.183561i
\(342\) 0 0
\(343\) 80605.0 + 85697.8i 0.685131 + 0.728420i
\(344\) 0 0
\(345\) −26089.7 45188.8i −0.219195 0.379658i
\(346\) 0 0
\(347\) 9384.77 16254.9i 0.0779408 0.134997i −0.824421 0.565978i \(-0.808499\pi\)
0.902361 + 0.430980i \(0.141832\pi\)
\(348\) 0 0
\(349\) 4574.17i 0.0375545i 0.999824 + 0.0187772i \(0.00597733\pi\)
−0.999824 + 0.0187772i \(0.994023\pi\)
\(350\) 0 0
\(351\) −32116.5 −0.260683
\(352\) 0 0
\(353\) 52734.9 + 30446.5i 0.423203 + 0.244336i 0.696447 0.717608i \(-0.254763\pi\)
−0.273244 + 0.961945i \(0.588097\pi\)
\(354\) 0 0
\(355\) −4334.10 + 2502.29i −0.0343908 + 0.0198555i
\(356\) 0 0
\(357\) −41002.5 + 40174.0i −0.321717 + 0.315217i
\(358\) 0 0
\(359\) 116059. + 201020.i 0.900513 + 1.55973i 0.826830 + 0.562452i \(0.190142\pi\)
0.0736826 + 0.997282i \(0.476525\pi\)
\(360\) 0 0
\(361\) −21725.5 + 37629.7i −0.166708 + 0.288746i
\(362\) 0 0
\(363\) 54773.8i 0.415681i
\(364\) 0 0
\(365\) −57604.4 −0.432384
\(366\) 0 0
\(367\) 125329. + 72358.9i 0.930509 + 0.537229i 0.886972 0.461822i \(-0.152804\pi\)
0.0435362 + 0.999052i \(0.486138\pi\)
\(368\) 0 0
\(369\) −65883.1 + 38037.6i −0.483862 + 0.279358i
\(370\) 0 0
\(371\) −54401.0 15173.4i −0.395238 0.110239i
\(372\) 0 0
\(373\) 74441.6 + 128937.i 0.535054 + 0.926741i 0.999161 + 0.0409616i \(0.0130421\pi\)
−0.464107 + 0.885779i \(0.653625\pi\)
\(374\) 0 0
\(375\) 38598.8 66855.0i 0.274480 0.475414i
\(376\) 0 0
\(377\) 169491.i 1.19252i
\(378\) 0 0
\(379\) −140667. −0.979299 −0.489649 0.871919i \(-0.662875\pi\)
−0.489649 + 0.871919i \(0.662875\pi\)
\(380\) 0 0
\(381\) 10710.3 + 6183.59i 0.0737822 + 0.0425982i
\(382\) 0 0
\(383\) −25393.9 + 14661.2i −0.173114 + 0.0999473i −0.584053 0.811715i \(-0.698534\pi\)
0.410939 + 0.911663i \(0.365201\pi\)
\(384\) 0 0
\(385\) 43008.0 11054.8i 0.290153 0.0745814i
\(386\) 0 0
\(387\) −41396.5 71700.8i −0.276402 0.478743i
\(388\) 0 0
\(389\) −66209.9 + 114679.i −0.437546 + 0.757852i −0.997500 0.0706723i \(-0.977486\pi\)
0.559954 + 0.828524i \(0.310819\pi\)
\(390\) 0 0
\(391\) 159960.i 1.04630i
\(392\) 0 0
\(393\) 22550.2 0.146004
\(394\) 0 0
\(395\) −79339.7 45806.8i −0.508507 0.293586i
\(396\) 0 0
\(397\) 29642.0 17113.8i 0.188073 0.108584i −0.403007 0.915197i \(-0.632035\pi\)
0.591080 + 0.806613i \(0.298702\pi\)
\(398\) 0 0
\(399\) −18682.0 72680.8i −0.117348 0.456535i
\(400\) 0 0
\(401\) 89540.6 + 155089.i 0.556841 + 0.964477i 0.997758 + 0.0669288i \(0.0213200\pi\)
−0.440917 + 0.897548i \(0.645347\pi\)
\(402\) 0 0
\(403\) −76311.7 + 132176.i −0.469873 + 0.813845i
\(404\) 0 0
\(405\) 10318.0i 0.0629048i
\(406\) 0 0
\(407\) 53385.5 0.322281
\(408\) 0 0
\(409\) 43099.3 + 24883.4i 0.257646 + 0.148752i 0.623260 0.782014i \(-0.285808\pi\)
−0.365614 + 0.930767i \(0.619141\pi\)
\(410\) 0 0
\(411\) 138416. 79914.7i 0.819414 0.473089i
\(412\) 0 0
\(413\) 45978.1 164845.i 0.269557 0.966442i
\(414\) 0 0
\(415\) 58212.4 + 100827.i 0.338002 + 0.585437i
\(416\) 0 0
\(417\) −70633.0 + 122340.i −0.406196 + 0.703551i
\(418\) 0 0
\(419\) 43951.2i 0.250347i −0.992135 0.125174i \(-0.960051\pi\)
0.992135 0.125174i \(-0.0399488\pi\)
\(420\) 0 0
\(421\) −218257. −1.23141 −0.615707 0.787975i \(-0.711129\pi\)
−0.615707 + 0.787975i \(0.711129\pi\)
\(422\) 0 0
\(423\) 14350.6 + 8285.30i 0.0802025 + 0.0463050i
\(424\) 0 0
\(425\) 82917.9 47872.7i 0.459061 0.265039i
\(426\) 0 0
\(427\) −77920.6 79527.4i −0.427363 0.436175i
\(428\) 0 0
\(429\) 38081.5 + 65959.0i 0.206918 + 0.358393i
\(430\) 0 0
\(431\) −38369.8 + 66458.5i −0.206555 + 0.357763i −0.950627 0.310336i \(-0.899558\pi\)
0.744072 + 0.668099i \(0.232892\pi\)
\(432\) 0 0
\(433\) 216713.i 1.15587i −0.816083 0.577935i \(-0.803859\pi\)
0.816083 0.577935i \(-0.196141\pi\)
\(434\) 0 0
\(435\) −54451.9 −0.287763
\(436\) 0 0
\(437\) 181099. + 104558.i 0.948318 + 0.547512i
\(438\) 0 0
\(439\) −20578.6 + 11881.0i −0.106779 + 0.0616489i −0.552438 0.833554i \(-0.686303\pi\)
0.445659 + 0.895203i \(0.352969\pi\)
\(440\) 0 0
\(441\) −55468.6 33552.5i −0.285214 0.172523i
\(442\) 0 0
\(443\) 87432.4 + 151437.i 0.445517 + 0.771659i 0.998088 0.0618072i \(-0.0196864\pi\)
−0.552571 + 0.833466i \(0.686353\pi\)
\(444\) 0 0
\(445\) 109961. 190458.i 0.555289 0.961788i
\(446\) 0 0
\(447\) 30277.6i 0.151532i
\(448\) 0 0
\(449\) −195154. −0.968023 −0.484011 0.875062i \(-0.660821\pi\)
−0.484011 + 0.875062i \(0.660821\pi\)
\(450\) 0 0
\(451\) 156239. + 90204.7i 0.768134 + 0.443482i
\(452\) 0 0
\(453\) 113340. 65437.2i 0.552317 0.318881i
\(454\) 0 0
\(455\) −113401. + 111110.i −0.547764 + 0.536697i
\(456\) 0 0
\(457\) −29236.1 50638.5i −0.139987 0.242464i 0.787505 0.616309i \(-0.211373\pi\)
−0.927491 + 0.373844i \(0.878039\pi\)
\(458\) 0 0
\(459\) 15815.2 27392.8i 0.0750672 0.130020i
\(460\) 0 0
\(461\) 307243.i 1.44571i −0.691002 0.722853i \(-0.742831\pi\)
0.691002 0.722853i \(-0.257169\pi\)
\(462\) 0 0
\(463\) 17772.2 0.0829049 0.0414524 0.999140i \(-0.486801\pi\)
0.0414524 + 0.999140i \(0.486801\pi\)
\(464\) 0 0
\(465\) −42463.7 24516.4i −0.196387 0.113384i
\(466\) 0 0
\(467\) −56261.4 + 32482.5i −0.257974 + 0.148942i −0.623410 0.781895i \(-0.714253\pi\)
0.365436 + 0.930837i \(0.380920\pi\)
\(468\) 0 0
\(469\) 409429. + 114197.i 1.86137 + 0.519168i
\(470\) 0 0
\(471\) 62083.7 + 107532.i 0.279857 + 0.484726i
\(472\) 0 0
\(473\) −98170.2 + 170036.i −0.438790 + 0.760007i
\(474\) 0 0
\(475\) 125168.i 0.554760i
\(476\) 0 0
\(477\) 31120.2 0.136775
\(478\) 0 0
\(479\) −182677. 105468.i −0.796181 0.459675i 0.0459530 0.998944i \(-0.485368\pi\)
−0.842134 + 0.539268i \(0.818701\pi\)
\(480\) 0 0
\(481\) −165294. + 95432.3i −0.714440 + 0.412482i
\(482\) 0 0
\(483\) 174959. 44971.7i 0.749967 0.192773i
\(484\) 0 0
\(485\) 11029.9 + 19104.4i 0.0468910 + 0.0812177i
\(486\) 0 0
\(487\) −114444. + 198223.i −0.482542 + 0.835787i −0.999799 0.0200431i \(-0.993620\pi\)
0.517257 + 0.855830i \(0.326953\pi\)
\(488\) 0 0
\(489\) 221283.i 0.925402i
\(490\) 0 0
\(491\) −140350. −0.582169 −0.291085 0.956697i \(-0.594016\pi\)
−0.291085 + 0.956697i \(0.594016\pi\)
\(492\) 0 0
\(493\) −144562. 83463.1i −0.594787 0.343400i
\(494\) 0 0
\(495\) −21190.5 + 12234.3i −0.0864828 + 0.0499309i
\(496\) 0 0
\(497\) −4313.28 16780.5i −0.0174620 0.0679348i
\(498\) 0 0
\(499\) −172663. 299062.i −0.693424 1.20105i −0.970709 0.240258i \(-0.922768\pi\)
0.277285 0.960788i \(-0.410566\pi\)
\(500\) 0 0
\(501\) 68475.7 118603.i 0.272810 0.472521i
\(502\) 0 0
\(503\) 58979.0i 0.233110i −0.993184 0.116555i \(-0.962815\pi\)
0.993184 0.116555i \(-0.0371851\pi\)
\(504\) 0 0
\(505\) −222711. −0.873291
\(506\) 0 0
\(507\) −107293. 61945.9i −0.417404 0.240988i
\(508\) 0 0
\(509\) 20653.6 11924.3i 0.0797186 0.0460255i −0.459611 0.888120i \(-0.652011\pi\)
0.539329 + 0.842095i \(0.318678\pi\)
\(510\) 0 0
\(511\) 53578.7 192095.i 0.205187 0.735657i
\(512\) 0 0
\(513\) 20675.2 + 35810.6i 0.0785626 + 0.136074i
\(514\) 0 0
\(515\) 238.691 413.425i 0.000899957 0.00155877i
\(516\) 0 0
\(517\) 39296.5i 0.147019i
\(518\) 0 0
\(519\) 145592. 0.540509
\(520\) 0 0
\(521\) 417171. + 240854.i 1.53688 + 0.887315i 0.999019 + 0.0442788i \(0.0140990\pi\)
0.537856 + 0.843037i \(0.319234\pi\)
\(522\) 0 0
\(523\) 399593. 230705.i 1.46088 0.843438i 0.461826 0.886970i \(-0.347194\pi\)
0.999052 + 0.0435320i \(0.0138610\pi\)
\(524\) 0 0
\(525\) 75673.4 + 77233.9i 0.274552 + 0.280214i
\(526\) 0 0
\(527\) −75156.9 130176.i −0.270612 0.468714i
\(528\) 0 0
\(529\) −111774. + 193598.i −0.399419 + 0.691813i
\(530\) 0 0
\(531\) 94299.9i 0.334443i
\(532\) 0 0
\(533\) −645003. −2.27043
\(534\) 0 0
\(535\) 66759.3 + 38543.5i 0.233240 + 0.134661i
\(536\) 0 0
\(537\) 114293. 65987.2i 0.396343 0.228829i
\(538\) 0 0
\(539\) −3137.44 + 153703.i −0.0107994 + 0.529059i
\(540\) 0 0
\(541\) 254098. + 440111.i 0.868174 + 1.50372i 0.863861 + 0.503731i \(0.168040\pi\)
0.00431302 + 0.999991i \(0.498627\pi\)
\(542\) 0 0
\(543\) 114568. 198438.i 0.388565 0.673014i
\(544\) 0 0
\(545\) 236333.i 0.795668i
\(546\) 0 0
\(547\) −40170.8 −0.134257 −0.0671283 0.997744i \(-0.521384\pi\)
−0.0671283 + 0.997744i \(0.521384\pi\)
\(548\) 0 0
\(549\) 53130.4 + 30674.9i 0.176278 + 0.101774i
\(550\) 0 0
\(551\) 188986. 109111.i 0.622482 0.359390i
\(552\) 0 0
\(553\) 226549. 221972.i 0.740818 0.725851i
\(554\) 0 0
\(555\) −30659.3 53103.4i −0.0995350 0.172400i
\(556\) 0 0
\(557\) 39230.2 67948.7i 0.126447 0.219013i −0.795850 0.605493i \(-0.792976\pi\)
0.922298 + 0.386480i \(0.126309\pi\)
\(558\) 0 0
\(559\) 701959.i 2.24641i
\(560\) 0 0
\(561\) −75010.4 −0.238339
\(562\) 0 0
\(563\) 355612. + 205312.i 1.12191 + 0.647737i 0.941889 0.335926i \(-0.109049\pi\)
0.180024 + 0.983662i \(0.442382\pi\)
\(564\) 0 0
\(565\) −115901. + 66915.6i −0.363071 + 0.209619i
\(566\) 0 0
\(567\) 34407.7 + 9596.89i 0.107026 + 0.0298514i
\(568\) 0 0
\(569\) 74404.3 + 128872.i 0.229812 + 0.398047i 0.957752 0.287594i \(-0.0928554\pi\)
−0.727940 + 0.685641i \(0.759522\pi\)
\(570\) 0 0
\(571\) 36146.3 62607.3i 0.110864 0.192023i −0.805255 0.592929i \(-0.797971\pi\)
0.916119 + 0.400906i \(0.131305\pi\)
\(572\) 0 0
\(573\) 335459.i 1.02172i
\(574\) 0 0
\(575\) −301307. −0.911326
\(576\) 0 0
\(577\) −482666. 278667.i −1.44976 0.837018i −0.451290 0.892377i \(-0.649036\pi\)
−0.998466 + 0.0553596i \(0.982369\pi\)
\(578\) 0 0
\(579\) −165036. + 95283.4i −0.492290 + 0.284223i
\(580\) 0 0
\(581\) −390376. + 100343.i −1.15646 + 0.297258i
\(582\) 0 0
\(583\) −36900.2 63913.0i −0.108565 0.188041i
\(584\) 0 0
\(585\) 43740.3 75760.5i 0.127812 0.221376i
\(586\) 0 0
\(587\) 308119.i 0.894217i −0.894480 0.447109i \(-0.852454\pi\)
0.894480 0.447109i \(-0.147546\pi\)
\(588\) 0 0
\(589\) 196505. 0.566426
\(590\) 0 0
\(591\) −329162. 190042.i −0.942398 0.544094i
\(592\) 0 0
\(593\) −200805. + 115935.i −0.571038 + 0.329689i −0.757564 0.652761i \(-0.773611\pi\)
0.186526 + 0.982450i \(0.440277\pi\)
\(594\) 0 0
\(595\) −38925.3 151436.i −0.109951 0.427755i
\(596\) 0 0
\(597\) 3639.98 + 6304.62i 0.0102129 + 0.0176893i
\(598\) 0 0
\(599\) −167042. + 289325.i −0.465555 + 0.806365i −0.999226 0.0393269i \(-0.987479\pi\)
0.533671 + 0.845692i \(0.320812\pi\)
\(600\) 0 0
\(601\) 645072.i 1.78591i −0.450147 0.892955i \(-0.648628\pi\)
0.450147 0.892955i \(-0.351372\pi\)
\(602\) 0 0
\(603\) −234215. −0.644139
\(604\) 0 0
\(605\) −129208. 74598.1i −0.353002 0.203806i
\(606\) 0 0
\(607\) −120643. + 69653.0i −0.327433 + 0.189044i −0.654701 0.755888i \(-0.727205\pi\)
0.327268 + 0.944932i \(0.393872\pi\)
\(608\) 0 0
\(609\) 50646.5 181583.i 0.136557 0.489598i
\(610\) 0 0
\(611\) 70246.8 + 121671.i 0.188167 + 0.325915i
\(612\) 0 0
\(613\) 203665. 352759.i 0.541997 0.938765i −0.456793 0.889573i \(-0.651002\pi\)
0.998789 0.0491924i \(-0.0156647\pi\)
\(614\) 0 0
\(615\) 207218.i 0.547870i
\(616\) 0 0
\(617\) 276504. 0.726325 0.363163 0.931726i \(-0.381697\pi\)
0.363163 + 0.931726i \(0.381697\pi\)
\(618\) 0 0
\(619\) −193904. 111950.i −0.506064 0.292176i 0.225150 0.974324i \(-0.427713\pi\)
−0.731214 + 0.682148i \(0.761046\pi\)
\(620\) 0 0
\(621\) −86204.1 + 49770.0i −0.223535 + 0.129058i
\(622\) 0 0
\(623\) 532852. + 543839.i 1.37287 + 1.40118i
\(624\) 0 0
\(625\) −27573.7 47759.1i −0.0705888 0.122263i
\(626\) 0 0
\(627\) 49030.5 84923.3i 0.124719 0.216019i
\(628\) 0 0
\(629\) 187976.i 0.475119i
\(630\) 0 0
\(631\) −299528. −0.752278 −0.376139 0.926563i \(-0.622749\pi\)
−0.376139 + 0.926563i \(0.622749\pi\)
\(632\) 0 0
\(633\) −262042. 151290.i −0.653980 0.377575i
\(634\) 0 0
\(635\) −29173.3 + 16843.2i −0.0723500 + 0.0417713i
\(636\) 0 0
\(637\) −265046. 481507.i −0.653194 1.18665i
\(638\) 0 0
\(639\) 4773.48 + 8267.92i 0.0116905 + 0.0202486i
\(640\) 0 0
\(641\) −288942. + 500462.i −0.703226 + 1.21802i 0.264102 + 0.964495i \(0.414924\pi\)
−0.967328 + 0.253528i \(0.918409\pi\)
\(642\) 0 0
\(643\) 135320.i 0.327295i −0.986519 0.163647i \(-0.947674\pi\)
0.986519 0.163647i \(-0.0523259\pi\)
\(644\) 0 0
\(645\) 225516. 0.542074
\(646\) 0 0
\(647\) −188880. 109050.i −0.451209 0.260506i 0.257132 0.966376i \(-0.417223\pi\)
−0.708341 + 0.705871i \(0.750556\pi\)
\(648\) 0 0
\(649\) 193668. 111814.i 0.459800 0.265465i
\(650\) 0 0
\(651\) 121252. 118802.i 0.286106 0.280325i
\(652\) 0 0
\(653\) −72127.2 124928.i −0.169150 0.292977i 0.768971 0.639284i \(-0.220769\pi\)
−0.938121 + 0.346307i \(0.887436\pi\)
\(654\) 0 0
\(655\) −30711.8 + 53194.4i −0.0715851 + 0.123989i
\(656\) 0 0
\(657\) 109889.i 0.254579i
\(658\) 0 0
\(659\) 159392. 0.367024 0.183512 0.983017i \(-0.441253\pi\)
0.183512 + 0.983017i \(0.441253\pi\)
\(660\) 0 0
\(661\) 305674. + 176481.i 0.699609 + 0.403919i 0.807202 0.590276i \(-0.200981\pi\)
−0.107593 + 0.994195i \(0.534314\pi\)
\(662\) 0 0
\(663\) 232249. 134089.i 0.528357 0.305047i
\(664\) 0 0
\(665\) 196892. + 54916.7i 0.445231 + 0.124183i
\(666\) 0 0
\(667\) 262655. + 454932.i 0.590384 + 1.02258i
\(668\) 0 0
\(669\) −158615. + 274729.i −0.354398 + 0.613836i
\(670\) 0 0
\(671\) 145488.i 0.323135i
\(672\) 0 0
\(673\) 504858. 1.11465 0.557326 0.830294i \(-0.311827\pi\)
0.557326 + 0.830294i \(0.311827\pi\)
\(674\) 0 0
\(675\) −51598.2 29790.2i −0.113247 0.0653832i
\(676\) 0 0
\(677\) −169486. + 97853.0i −0.369792 + 0.213500i −0.673368 0.739308i \(-0.735153\pi\)
0.303576 + 0.952807i \(0.401820\pi\)
\(678\) 0 0
\(679\) −73967.4 + 19012.7i −0.160436 + 0.0412385i
\(680\) 0 0
\(681\) −64935.5 112472.i −0.140019 0.242520i
\(682\) 0 0
\(683\) −117221. + 203032.i −0.251283 + 0.435235i −0.963879 0.266340i \(-0.914186\pi\)
0.712596 + 0.701574i \(0.247519\pi\)
\(684\) 0 0
\(685\) 435352.i 0.927812i
\(686\) 0 0
\(687\) −98357.4 −0.208398
\(688\) 0 0
\(689\) 228503. + 131926.i 0.481341 + 0.277902i
\(690\) 0 0
\(691\) 77344.0 44654.6i 0.161983 0.0935212i −0.416817 0.908990i \(-0.636854\pi\)
0.578800 + 0.815469i \(0.303521\pi\)
\(692\) 0 0
\(693\) −21088.7 82044.0i −0.0439120 0.170836i
\(694\) 0 0
\(695\) −192394. 333236.i −0.398311 0.689895i
\(696\) 0 0
\(697\) 317621. 550136.i 0.653799 1.13241i
\(698\) 0 0
\(699\) 34734.9i 0.0710905i
\(700\) 0 0
\(701\) 122213. 0.248704 0.124352 0.992238i \(-0.460315\pi\)
0.124352 + 0.992238i \(0.460315\pi\)
\(702\) 0 0
\(703\) 212818. + 122871.i 0.430624 + 0.248621i
\(704\) 0 0
\(705\) −39088.9 + 22568.0i −0.0786457 + 0.0454061i
\(706\) 0 0
\(707\) 207147. 742683.i 0.414419 1.48582i
\(708\) 0 0
\(709\) −42009.9 72763.3i −0.0835717 0.144750i 0.821210 0.570626i \(-0.193299\pi\)
−0.904782 + 0.425876i \(0.859966\pi\)
\(710\) 0 0
\(711\) −87383.1 + 151352.i −0.172858 + 0.299398i
\(712\) 0 0
\(713\) 473032.i 0.930489i
\(714\) 0 0
\(715\) −207457. −0.405804
\(716\) 0 0
\(717\) 434074. + 250613.i 0.844356 + 0.487489i
\(718\) 0 0
\(719\) −657416. + 379560.i −1.27169 + 0.734213i −0.975307 0.220855i \(-0.929115\pi\)
−0.296388 + 0.955068i \(0.595782\pi\)
\(720\) 0 0
\(721\) 1156.65 + 1180.51i 0.00222502 + 0.00227090i
\(722\) 0 0
\(723\) 142115. + 246151.i 0.271872 + 0.470896i
\(724\) 0 0
\(725\) −157214. + 272303.i −0.299100 + 0.518057i
\(726\) 0 0
\(727\) 92384.1i 0.174795i 0.996174 + 0.0873974i \(0.0278550\pi\)
−0.996174 + 0.0873974i \(0.972145\pi\)
\(728\) 0 0
\(729\) −19683.0 −0.0370370
\(730\) 0 0
\(731\) 598715. + 345668.i 1.12043 + 0.646882i
\(732\) 0 0
\(733\) 2903.50 1676.34i 0.00540399 0.00312000i −0.497296 0.867581i \(-0.665674\pi\)
0.502700 + 0.864461i \(0.332340\pi\)
\(734\) 0 0
\(735\) 154692. 85150.5i 0.286348 0.157620i
\(736\) 0 0
\(737\) 277715. + 481017.i 0.511287 + 0.885575i
\(738\) 0 0
\(739\) −13592.8 + 23543.4i −0.0248897 + 0.0431103i −0.878202 0.478290i \(-0.841257\pi\)
0.853312 + 0.521400i \(0.174590\pi\)
\(740\) 0 0
\(741\) 350589.i 0.638502i
\(742\) 0 0
\(743\) −773801. −1.40169 −0.700845 0.713314i \(-0.747193\pi\)
−0.700845 + 0.713314i \(0.747193\pi\)
\(744\) 0 0
\(745\) −71422.6 41235.9i −0.128684 0.0742955i
\(746\) 0 0
\(747\) 192342. 111049.i 0.344693 0.199009i
\(748\) 0 0
\(749\) −190626. + 186775.i −0.339797 + 0.332931i
\(750\) 0 0
\(751\) 98922.2 + 171338.i 0.175394 + 0.303791i 0.940297 0.340354i \(-0.110547\pi\)
−0.764904 + 0.644145i \(0.777214\pi\)
\(752\) 0 0
\(753\) 280948. 486617.i 0.495492 0.858217i
\(754\) 0 0
\(755\) 356483.i 0.625381i
\(756\) 0 0
\(757\) −770706. −1.34492 −0.672461 0.740132i \(-0.734763\pi\)
−0.672461 + 0.740132i \(0.734763\pi\)
\(758\) 0 0
\(759\) 204430. + 118027.i 0.354863 + 0.204880i
\(760\) 0 0
\(761\) −131757. + 76069.9i −0.227512 + 0.131354i −0.609424 0.792845i \(-0.708599\pi\)
0.381912 + 0.924199i \(0.375266\pi\)
\(762\) 0 0
\(763\) 788109. + 219817.i 1.35375 + 0.377583i
\(764\) 0 0
\(765\) 43078.4 + 74614.0i 0.0736100 + 0.127496i
\(766\) 0 0
\(767\) −399760. + 692405.i −0.679531 + 1.17698i
\(768\) 0 0
\(769\) 961897.i 1.62658i 0.581857 + 0.813291i \(0.302326\pi\)
−0.581857 + 0.813291i \(0.697674\pi\)
\(770\) 0 0
\(771\) 186470. 0.313689
\(772\) 0 0
\(773\) −130829. 75534.4i −0.218951 0.126411i 0.386514 0.922284i \(-0.373679\pi\)
−0.605464 + 0.795872i \(0.707013\pi\)
\(774\) 0 0
\(775\) −245204. + 141569.i −0.408248 + 0.235702i
\(776\) 0 0
\(777\) 205603. 52848.3i 0.340554 0.0875365i
\(778\) 0 0
\(779\) 415226. + 719192.i 0.684242 + 1.18514i
\(780\) 0 0
\(781\) 11320.1 19607.0i 0.0185588 0.0321447i
\(782\) 0 0
\(783\) 103875.i 0.169429i
\(784\) 0 0
\(785\) −338215. −0.548849
\(786\) 0 0
\(787\) 172244. + 99445.4i 0.278097 + 0.160559i 0.632561 0.774510i \(-0.282004\pi\)
−0.354465 + 0.935069i \(0.615337\pi\)
\(788\) 0 0
\(789\) −294442. + 169996.i −0.472984 + 0.273077i
\(790\) 0 0
\(791\) −115344. 448739.i −0.184350 0.717202i
\(792\) 0 0
\(793\) 260076. + 450465.i 0.413575 + 0.716333i
\(794\) 0 0
\(795\) −42383.5 + 73410.4i −0.0670599 + 0.116151i
\(796\) 0 0
\(797\) 990756.i 1.55973i −0.625947 0.779866i \(-0.715287\pi\)
0.625947 0.779866i \(-0.284713\pi\)
\(798\) 0 0
\(799\) −138368. −0.216741
\(800\) 0 0
\(801\) −363326. 209767.i −0.566281 0.326943i
\(802\) 0 0
\(803\) 225683. 130298.i 0.350000 0.202073i
\(804\) 0 0
\(805\) −132197. + 473965.i −0.204000 + 0.731399i
\(806\) 0 0
\(807\) −179816. 311451.i −0.276110 0.478236i
\(808\) 0 0
\(809\) 46144.3 79924.3i 0.0705052 0.122119i −0.828618 0.559815i \(-0.810872\pi\)
0.899123 + 0.437696i \(0.144206\pi\)
\(810\) 0 0
\(811\) 617125.i 0.938277i 0.883125 + 0.469139i \(0.155436\pi\)
−0.883125 + 0.469139i \(0.844564\pi\)
\(812\) 0 0
\(813\) −34037.6 −0.0514966
\(814\) 0 0
\(815\) 521991. + 301372.i 0.785865 + 0.453719i
\(816\) 0 0
\(817\) −782700. + 451892.i −1.17260 + 0.677003i
\(818\) 0 0
\(819\) 211958. + 216329.i 0.315996 + 0.322512i
\(820\) 0 0
\(821\) −153533. 265927.i −0.227780 0.394526i 0.729370 0.684119i \(-0.239813\pi\)
−0.957150 + 0.289593i \(0.906480\pi\)
\(822\) 0 0
\(823\) 423429. 733401.i 0.625146 1.08278i −0.363367 0.931646i \(-0.618373\pi\)
0.988513 0.151138i \(-0.0482937\pi\)
\(824\) 0 0
\(825\) 141293.i 0.207592i
\(826\) 0 0
\(827\) −294059. −0.429956 −0.214978 0.976619i \(-0.568968\pi\)
−0.214978 + 0.976619i \(0.568968\pi\)
\(828\) 0 0
\(829\) 895803. + 517192.i 1.30348 + 0.752563i 0.980999 0.194013i \(-0.0621506\pi\)
0.322479 + 0.946577i \(0.395484\pi\)
\(830\) 0 0
\(831\) −355964. + 205516.i −0.515471 + 0.297608i
\(832\) 0 0
\(833\) 541205. + 11047.3i 0.779959 + 0.0159208i
\(834\) 0 0
\(835\) 186518. + 323059.i 0.267515 + 0.463349i
\(836\) 0 0
\(837\) −46768.6 + 81005.6i −0.0667580 + 0.115628i
\(838\) 0 0
\(839\) 307258.i 0.436495i −0.975893 0.218248i \(-0.929966\pi\)
0.975893 0.218248i \(-0.0700340\pi\)
\(840\) 0 0
\(841\) −159093. −0.224937
\(842\) 0 0
\(843\) 154634. + 89278.1i 0.217596 + 0.125629i
\(844\) 0 0
\(845\) 292252. 168732.i 0.409302 0.236311i
\(846\) 0 0
\(847\) 368943. 361489.i 0.514272 0.503881i
\(848\) 0 0
\(849\) −238795. 413605.i −0.331291 0.573813i
\(850\) 0 0
\(851\) −295777. + 512302.i −0.408419 + 0.707402i
\(852\) 0 0
\(853\) 70737.4i 0.0972190i 0.998818 + 0.0486095i \(0.0154790\pi\)
−0.998818 + 0.0486095i \(0.984521\pi\)
\(854\) 0 0
\(855\) −112633. −0.154075
\(856\) 0 0
\(857\) 285670. + 164932.i 0.388958 + 0.224565i 0.681709 0.731624i \(-0.261237\pi\)
−0.292751 + 0.956189i \(0.594571\pi\)
\(858\) 0 0
\(859\) 60589.3 34981.2i 0.0821125 0.0474077i −0.458382 0.888756i \(-0.651571\pi\)
0.540494 + 0.841348i \(0.318237\pi\)
\(860\) 0 0
\(861\) 691018. + 192737.i 0.932145 + 0.259991i
\(862\) 0 0
\(863\) −386112. 668765.i −0.518431 0.897949i −0.999771 0.0214151i \(-0.993183\pi\)
0.481339 0.876534i \(-0.340150\pi\)
\(864\) 0 0
\(865\) −198286. + 343441.i −0.265008 + 0.459008i
\(866\) 0 0
\(867\) 169868.i 0.225981i
\(868\) 0 0
\(869\) 414451. 0.548825
\(870\) 0 0
\(871\) −1.71974e6 992893.i −2.26687 1.30878i
\(872\) 0 0
\(873\) 36444.4 21041.2i 0.0478193 0.0276085i
\(874\) 0 0
\(875\) −705058. + 181229.i −0.920892 + 0.236707i
\(876\) 0 0
\(877\) −121260. 210028.i −0.157659 0.273073i 0.776365 0.630283i \(-0.217061\pi\)
−0.934024 + 0.357211i \(0.883728\pi\)
\(878\) 0 0
\(879\) 60322.5 104482.i 0.0780732 0.135227i
\(880\) 0 0
\(881\) 646528.i 0.832982i 0.909140 + 0.416491i \(0.136740\pi\)
−0.909140 + 0.416491i \(0.863260\pi\)
\(882\) 0 0
\(883\) 461877. 0.592387 0.296193 0.955128i \(-0.404283\pi\)
0.296193 + 0.955128i \(0.404283\pi\)
\(884\) 0 0
\(885\) −222447. 128430.i −0.284014 0.163976i
\(886\) 0 0
\(887\) 186391. 107613.i 0.236907 0.136778i −0.376847 0.926275i \(-0.622992\pi\)
0.613754 + 0.789497i \(0.289658\pi\)
\(888\) 0 0
\(889\) −29033.2 112952.i −0.0367359 0.142919i
\(890\) 0 0
\(891\) 23338.7 + 40423.9i 0.0293983 + 0.0509193i
\(892\) 0 0
\(893\) 90443.9 156653.i 0.113417 0.196443i
\(894\) 0 0
\(895\) 359479.i 0.448774i
\(896\) 0 0
\(897\) −843948. −1.04889
\(898\) 0 0
\(899\) 427498. + 246816.i 0.528950 + 0.305389i
\(900\) 0 0
\(901\) −225045. + 129930.i −0.277217 + 0.160051i
\(902\) 0 0
\(903\) −209756. + 752038.i −0.257240 + 0.922283i
\(904\) 0 0
\(905\) 312067. + 540516.i 0.381022 + 0.659950i
\(906\) 0 0
\(907\) 549134. 951128.i 0.667519 1.15618i −0.311076 0.950385i \(-0.600689\pi\)
0.978596 0.205793i \(-0.0659773\pi\)
\(908\) 0 0
\(909\) 424854.i 0.514176i
\(910\) 0 0
\(911\) 1.32312e6 1.59427 0.797133 0.603803i \(-0.206349\pi\)
0.797133 + 0.603803i \(0.206349\pi\)
\(912\) 0 0
\(913\) −456131. 263348.i −0.547203 0.315928i
\(914\) 0 0
\(915\) −144720. + 83553.9i −0.172856 + 0.0997987i
\(916\) 0 0
\(917\) −148824. 151893.i −0.176984 0.180633i
\(918\) 0 0
\(919\) 258264. + 447326.i 0.305797 + 0.529656i 0.977438 0.211221i \(-0.0677438\pi\)
−0.671642 + 0.740876i \(0.734411\pi\)
\(920\) 0 0
\(921\) 172476. 298736.i 0.203333 0.352183i
\(922\) 0 0
\(923\) 80943.8i 0.0950124i
\(924\) 0 0
\(925\) −354080. −0.413826
\(926\) 0 0
\(927\) −788.669 455.338i −0.000917773 0.000529876i
\(928\) 0 0
\(929\) 1.07335e6 619697.i 1.24368 0.718039i 0.273838 0.961776i \(-0.411707\pi\)
0.969841 + 0.243737i \(0.0783734\pi\)
\(930\) 0 0
\(931\) −366265. + 605506.i −0.422568 + 0.698585i
\(932\) 0 0
\(933\) 248637. + 430652.i 0.285629 + 0.494724i
\(934\) 0 0
\(935\) 102159. 176944.i 0.116856 0.202401i
\(936\) 0 0
\(937\) 1.15260e6i 1.31281i −0.754411 0.656403i \(-0.772077\pi\)
0.754411 0.656403i \(-0.227923\pi\)
\(938\) 0 0
\(939\) 248917. 0.282308
\(940\) 0 0
\(941\) 93104.2 + 53753.7i 0.105145 + 0.0607057i 0.551650 0.834075i \(-0.313998\pi\)
−0.446505 + 0.894781i \(0.647332\pi\)
\(942\) 0 0
\(943\) −1.73126e6 + 999543.i −1.94688 + 1.12403i
\(944\) 0 0
\(945\) −69499.2 + 68095.1i −0.0778245 + 0.0762521i
\(946\) 0 0
\(947\) −206307. 357335.i −0.230046 0.398451i 0.727775 0.685815i \(-0.240554\pi\)
−0.957821 + 0.287364i \(0.907221\pi\)
\(948\) 0 0
\(949\) −465845. + 806867.i −0.517260 + 0.895920i
\(950\) 0 0
\(951\) 177335.i 0.196080i
\(952\) 0 0
\(953\) 1.43052e6 1.57510 0.787550 0.616251i \(-0.211349\pi\)
0.787550 + 0.616251i \(0.211349\pi\)
\(954\) 0 0
\(955\) 791324. + 456871.i 0.867656 + 0.500942i
\(956\) 0 0
\(957\) 213332. 123168.i 0.232934 0.134485i
\(958\) 0 0
\(959\) −1.45179e6 404928.i −1.57858 0.440292i
\(960\) 0 0
\(961\) −239507. 414839.i −0.259342 0.449193i
\(962\) 0 0
\(963\) 73527.3 127353.i 0.0792859 0.137327i
\(964\) 0 0
\(965\) 519077.i 0.557413i
\(966\) 0 0
\(967\) 344533. 0.368449 0.184225 0.982884i \(-0.441023\pi\)
0.184225 + 0.982884i \(0.441023\pi\)
\(968\) 0 0
\(969\) −299025. 172642.i −0.318463 0.183865i
\(970\) 0 0
\(971\) 1.25848e6 726581.i 1.33477 0.770630i 0.348744 0.937218i \(-0.386608\pi\)
0.986027 + 0.166588i \(0.0532751\pi\)
\(972\) 0 0
\(973\) 1.29020e6 331636.i 1.36280 0.350296i
\(974\) 0 0
\(975\) −252576. 437474.i −0.265695 0.460196i
\(976\) 0 0
\(977\) −772853. + 1.33862e6i −0.809669 + 1.40239i 0.103424 + 0.994637i \(0.467020\pi\)
−0.913093 + 0.407751i \(0.866313\pi\)
\(978\) 0 0
\(979\) 994907.i 1.03805i
\(980\) 0 0
\(981\) −450840. −0.468473
\(982\) 0 0
\(983\) −116280. 67134.0i −0.120336 0.0694761i 0.438624 0.898671i \(-0.355466\pi\)
−0.558960 + 0.829195i \(0.688799\pi\)
\(984\) 0 0
\(985\) 896589. 517646.i 0.924105 0.533532i
\(986\) 0 0
\(987\) −38901.1 151342.i −0.0399326 0.155355i
\(988\) 0 0
\(989\) −1.08781e6 1.88414e6i −1.11214 1.92628i
\(990\) 0 0
\(991\) −395596. + 685192.i −0.402814 + 0.697694i −0.994064 0.108794i \(-0.965301\pi\)
0.591251 + 0.806488i \(0.298634\pi\)
\(992\) 0 0
\(993\) 692571.i 0.702369i
\(994\) 0 0
\(995\) −19829.5 −0.0200293
\(996\) 0 0
\(997\) 582624. + 336378.i 0.586136 + 0.338406i 0.763568 0.645727i \(-0.223446\pi\)
−0.177432 + 0.984133i \(0.556779\pi\)
\(998\) 0 0
\(999\) −101302. + 58487.0i −0.101505 + 0.0586041i
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 336.5.bh.d.241.2 4
4.3 odd 2 42.5.g.a.31.1 yes 4
7.5 odd 6 inner 336.5.bh.d.145.2 4
12.11 even 2 126.5.n.b.73.2 4
28.3 even 6 294.5.c.a.97.3 4
28.11 odd 6 294.5.c.a.97.4 4
28.19 even 6 42.5.g.a.19.1 4
28.23 odd 6 294.5.g.c.19.1 4
28.27 even 2 294.5.g.c.31.1 4
84.11 even 6 882.5.c.a.685.2 4
84.47 odd 6 126.5.n.b.19.2 4
84.59 odd 6 882.5.c.a.685.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.5.g.a.19.1 4 28.19 even 6
42.5.g.a.31.1 yes 4 4.3 odd 2
126.5.n.b.19.2 4 84.47 odd 6
126.5.n.b.73.2 4 12.11 even 2
294.5.c.a.97.3 4 28.3 even 6
294.5.c.a.97.4 4 28.11 odd 6
294.5.g.c.19.1 4 28.23 odd 6
294.5.g.c.31.1 4 28.27 even 2
336.5.bh.d.145.2 4 7.5 odd 6 inner
336.5.bh.d.241.2 4 1.1 even 1 trivial
882.5.c.a.685.1 4 84.59 odd 6
882.5.c.a.685.2 4 84.11 even 6