Properties

Label 37.6.b.a.36.3
Level $37$
Weight $6$
Character 37.36
Analytic conductor $5.934$
Analytic rank $0$
Dimension $16$
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] = [37,6,Mod(36,37)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(37, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("37.36");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 37 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 37.b (of order \(2\), degree \(1\), 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: \(5.93420133308\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 390 x^{14} + 60701 x^{12} + 4799932 x^{10} + 203487156 x^{8} + 4519465040 x^{6} + 48993644736 x^{4} + 211923220224 x^{2} + \cdots + 178006118400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 36.3
Root \(-8.69480i\) of defining polynomial
Character \(\chi\) \(=\) 37.36
Dual form 37.6.b.a.36.14

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.69480i q^{2} +28.4744 q^{3} -43.5996 q^{4} -14.1700i q^{5} -247.579i q^{6} +47.5239 q^{7} +100.856i q^{8} +567.789 q^{9} +O(q^{10})\) \(q-8.69480i q^{2} +28.4744 q^{3} -43.5996 q^{4} -14.1700i q^{5} -247.579i q^{6} +47.5239 q^{7} +100.856i q^{8} +567.789 q^{9} -123.205 q^{10} -461.870 q^{11} -1241.47 q^{12} +290.335i q^{13} -413.211i q^{14} -403.482i q^{15} -518.263 q^{16} +1764.19i q^{17} -4936.81i q^{18} -1717.31i q^{19} +617.807i q^{20} +1353.21 q^{21} +4015.87i q^{22} +4414.83i q^{23} +2871.81i q^{24} +2924.21 q^{25} +2524.40 q^{26} +9248.15 q^{27} -2072.02 q^{28} -1387.78i q^{29} -3508.20 q^{30} -6726.60i q^{31} +7733.59i q^{32} -13151.5 q^{33} +15339.3 q^{34} -673.415i q^{35} -24755.3 q^{36} +(3814.02 - 7402.52i) q^{37} -14931.7 q^{38} +8267.10i q^{39} +1429.13 q^{40} +2517.60 q^{41} -11765.9i q^{42} +17522.4i q^{43} +20137.4 q^{44} -8045.57i q^{45} +38386.1 q^{46} -3097.14 q^{47} -14757.2 q^{48} -14548.5 q^{49} -25425.4i q^{50} +50234.2i q^{51} -12658.5i q^{52} -6600.17 q^{53} -80410.8i q^{54} +6544.71i q^{55} +4793.07i q^{56} -48899.4i q^{57} -12066.5 q^{58} +3587.53i q^{59} +17591.6i q^{60} +2461.47i q^{61} -58486.4 q^{62} +26983.5 q^{63} +50657.6 q^{64} +4114.05 q^{65} +114349. i q^{66} +36376.0 q^{67} -76918.0i q^{68} +125710. i q^{69} -5855.21 q^{70} -70209.2 q^{71} +57264.9i q^{72} -49658.6 q^{73} +(-64363.4 - 33162.1i) q^{74} +83265.0 q^{75} +74874.2i q^{76} -21949.9 q^{77} +71880.8 q^{78} -62511.9i q^{79} +7343.80i q^{80} +125362. q^{81} -21890.1i q^{82} -59096.3 q^{83} -58999.5 q^{84} +24998.6 q^{85} +152354. q^{86} -39516.3i q^{87} -46582.4i q^{88} -51391.5i q^{89} -69954.7 q^{90} +13797.9i q^{91} -192485. i q^{92} -191535. i q^{93} +26929.0i q^{94} -24334.4 q^{95} +220209. i q^{96} +100740. i q^{97} +126496. i q^{98} -262245. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 18 q^{3} - 268 q^{4} + 190 q^{7} + 1394 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 18 q^{3} - 268 q^{4} + 190 q^{7} + 1394 q^{9} - 74 q^{10} - 1110 q^{11} + 1402 q^{12} + 2900 q^{16} - 7010 q^{21} - 12052 q^{25} + 4902 q^{26} + 4266 q^{27} - 16824 q^{28} + 19280 q^{30} - 2478 q^{33} + 20556 q^{34} - 51402 q^{36} - 11400 q^{37} + 12108 q^{38} + 16966 q^{40} + 3918 q^{41} + 125394 q^{44} + 17470 q^{46} + 3822 q^{47} - 78034 q^{48} - 32618 q^{49} - 24126 q^{53} - 164718 q^{58} - 81426 q^{62} + 219268 q^{63} + 158076 q^{64} + 98976 q^{65} + 23560 q^{67} - 222404 q^{70} - 50046 q^{71} - 196274 q^{73} + 141216 q^{74} + 214054 q^{75} - 239574 q^{77} - 90822 q^{78} + 317312 q^{81} - 215814 q^{83} + 438572 q^{84} - 346472 q^{85} + 197640 q^{86} - 857612 q^{90} - 132504 q^{95} - 574860 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/37\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.69480i 1.53704i −0.639827 0.768519i \(-0.720994\pi\)
0.639827 0.768519i \(-0.279006\pi\)
\(3\) 28.4744 1.82663 0.913315 0.407253i \(-0.133513\pi\)
0.913315 + 0.407253i \(0.133513\pi\)
\(4\) −43.5996 −1.36249
\(5\) 14.1700i 0.253481i −0.991936 0.126740i \(-0.959548\pi\)
0.991936 0.126740i \(-0.0404515\pi\)
\(6\) 247.579i 2.80760i
\(7\) 47.5239 0.366579 0.183289 0.983059i \(-0.441325\pi\)
0.183289 + 0.983059i \(0.441325\pi\)
\(8\) 100.856i 0.557156i
\(9\) 567.789 2.33658
\(10\) −123.205 −0.389610
\(11\) −461.870 −1.15090 −0.575451 0.817836i \(-0.695173\pi\)
−0.575451 + 0.817836i \(0.695173\pi\)
\(12\) −1241.47 −2.48876
\(13\) 290.335i 0.476476i 0.971207 + 0.238238i \(0.0765698\pi\)
−0.971207 + 0.238238i \(0.923430\pi\)
\(14\) 413.211i 0.563445i
\(15\) 403.482i 0.463016i
\(16\) −518.263 −0.506116
\(17\) 1764.19i 1.48055i 0.672304 + 0.740275i \(0.265305\pi\)
−0.672304 + 0.740275i \(0.734695\pi\)
\(18\) 4936.81i 3.59141i
\(19\) 1717.31i 1.09135i −0.837995 0.545677i \(-0.816272\pi\)
0.837995 0.545677i \(-0.183728\pi\)
\(20\) 617.807i 0.345364i
\(21\) 1353.21 0.669604
\(22\) 4015.87i 1.76898i
\(23\) 4414.83i 1.74018i 0.492890 + 0.870091i \(0.335940\pi\)
−0.492890 + 0.870091i \(0.664060\pi\)
\(24\) 2871.81i 1.01772i
\(25\) 2924.21 0.935747
\(26\) 2524.40 0.732362
\(27\) 9248.15 2.44144
\(28\) −2072.02 −0.499459
\(29\) 1387.78i 0.306427i −0.988193 0.153213i \(-0.951038\pi\)
0.988193 0.153213i \(-0.0489622\pi\)
\(30\) −3508.20 −0.711673
\(31\) 6726.60i 1.25716i −0.777744 0.628581i \(-0.783636\pi\)
0.777744 0.628581i \(-0.216364\pi\)
\(32\) 7733.59i 1.33508i
\(33\) −13151.5 −2.10227
\(34\) 15339.3 2.27566
\(35\) 673.415i 0.0929207i
\(36\) −24755.3 −3.18356
\(37\) 3814.02 7402.52i 0.458013 0.888945i
\(38\) −14931.7 −1.67745
\(39\) 8267.10i 0.870345i
\(40\) 1429.13 0.141228
\(41\) 2517.60 0.233899 0.116949 0.993138i \(-0.462689\pi\)
0.116949 + 0.993138i \(0.462689\pi\)
\(42\) 11765.9i 1.02921i
\(43\) 17522.4i 1.44518i 0.691277 + 0.722590i \(0.257048\pi\)
−0.691277 + 0.722590i \(0.742952\pi\)
\(44\) 20137.4 1.56809
\(45\) 8045.57i 0.592278i
\(46\) 38386.1 2.67473
\(47\) −3097.14 −0.204511 −0.102255 0.994758i \(-0.532606\pi\)
−0.102255 + 0.994758i \(0.532606\pi\)
\(48\) −14757.2 −0.924488
\(49\) −14548.5 −0.865620
\(50\) 25425.4i 1.43828i
\(51\) 50234.2i 2.70442i
\(52\) 12658.5i 0.649192i
\(53\) −6600.17 −0.322750 −0.161375 0.986893i \(-0.551593\pi\)
−0.161375 + 0.986893i \(0.551593\pi\)
\(54\) 80410.8i 3.75258i
\(55\) 6544.71i 0.291732i
\(56\) 4793.07i 0.204242i
\(57\) 48899.4i 1.99350i
\(58\) −12066.5 −0.470990
\(59\) 3587.53i 0.134173i 0.997747 + 0.0670866i \(0.0213704\pi\)
−0.997747 + 0.0670866i \(0.978630\pi\)
\(60\) 17591.6i 0.630853i
\(61\) 2461.47i 0.0846973i 0.999103 + 0.0423487i \(0.0134840\pi\)
−0.999103 + 0.0423487i \(0.986516\pi\)
\(62\) −58486.4 −1.93231
\(63\) 26983.5 0.856540
\(64\) 50657.6 1.54595
\(65\) 4114.05 0.120778
\(66\) 114349.i 3.23127i
\(67\) 36376.0 0.989982 0.494991 0.868898i \(-0.335171\pi\)
0.494991 + 0.868898i \(0.335171\pi\)
\(68\) 76918.0i 2.01723i
\(69\) 125710.i 3.17867i
\(70\) −5855.21 −0.142823
\(71\) −70209.2 −1.65291 −0.826453 0.563005i \(-0.809645\pi\)
−0.826453 + 0.563005i \(0.809645\pi\)
\(72\) 57264.9i 1.30184i
\(73\) −49658.6 −1.09065 −0.545327 0.838223i \(-0.683595\pi\)
−0.545327 + 0.838223i \(0.683595\pi\)
\(74\) −64363.4 33162.1i −1.36634 0.703984i
\(75\) 83265.0 1.70926
\(76\) 74874.2i 1.48696i
\(77\) −21949.9 −0.421896
\(78\) 71880.8 1.33775
\(79\) 62511.9i 1.12692i −0.826142 0.563462i \(-0.809469\pi\)
0.826142 0.563462i \(-0.190531\pi\)
\(80\) 7343.80i 0.128291i
\(81\) 125362. 2.12302
\(82\) 21890.1i 0.359511i
\(83\) −59096.3 −0.941598 −0.470799 0.882241i \(-0.656034\pi\)
−0.470799 + 0.882241i \(0.656034\pi\)
\(84\) −58999.5 −0.912326
\(85\) 24998.6 0.375291
\(86\) 152354. 2.22130
\(87\) 39516.3i 0.559729i
\(88\) 46582.4i 0.641232i
\(89\) 51391.5i 0.687728i −0.939020 0.343864i \(-0.888264\pi\)
0.939020 0.343864i \(-0.111736\pi\)
\(90\) −69954.7 −0.910354
\(91\) 13797.9i 0.174666i
\(92\) 192485.i 2.37098i
\(93\) 191535.i 2.29637i
\(94\) 26929.0i 0.314341i
\(95\) −24334.4 −0.276638
\(96\) 220209.i 2.43869i
\(97\) 100740.i 1.08710i 0.839376 + 0.543551i \(0.182921\pi\)
−0.839376 + 0.543551i \(0.817079\pi\)
\(98\) 126496.i 1.33049i
\(99\) −262245. −2.68917
\(100\) −127494. −1.27494
\(101\) −31426.0 −0.306539 −0.153270 0.988184i \(-0.548980\pi\)
−0.153270 + 0.988184i \(0.548980\pi\)
\(102\) 436776. 4.15680
\(103\) 94967.4i 0.882027i −0.897501 0.441013i \(-0.854619\pi\)
0.897501 0.441013i \(-0.145381\pi\)
\(104\) −29282.0 −0.265471
\(105\) 19175.0i 0.169732i
\(106\) 57387.2i 0.496079i
\(107\) 179040. 1.51178 0.755892 0.654696i \(-0.227203\pi\)
0.755892 + 0.654696i \(0.227203\pi\)
\(108\) −403215. −3.32642
\(109\) 55763.1i 0.449553i −0.974410 0.224776i \(-0.927835\pi\)
0.974410 0.224776i \(-0.0721651\pi\)
\(110\) 56904.9 0.448403
\(111\) 108602. 210782.i 0.836621 1.62377i
\(112\) −24629.9 −0.185531
\(113\) 86748.3i 0.639095i −0.947570 0.319547i \(-0.896469\pi\)
0.947570 0.319547i \(-0.103531\pi\)
\(114\) −425171. −3.06409
\(115\) 62558.3 0.441103
\(116\) 60506.8i 0.417503i
\(117\) 164849.i 1.11332i
\(118\) 31192.9 0.206229
\(119\) 83841.3i 0.542738i
\(120\) 40693.6 0.257972
\(121\) 52273.2 0.324575
\(122\) 21402.0 0.130183
\(123\) 71687.1 0.427246
\(124\) 293277.i 1.71287i
\(125\) 85717.4i 0.490675i
\(126\) 234617.i 1.31653i
\(127\) −282739. −1.55552 −0.777760 0.628561i \(-0.783644\pi\)
−0.777760 + 0.628561i \(0.783644\pi\)
\(128\) 192983.i 1.04110i
\(129\) 498938.i 2.63981i
\(130\) 35770.8i 0.185640i
\(131\) 82509.4i 0.420074i −0.977693 0.210037i \(-0.932642\pi\)
0.977693 0.210037i \(-0.0673584\pi\)
\(132\) 573398. 2.86432
\(133\) 81613.5i 0.400067i
\(134\) 316282.i 1.52164i
\(135\) 131046.i 0.618857i
\(136\) −177929. −0.824898
\(137\) 80152.2 0.364850 0.182425 0.983220i \(-0.441605\pi\)
0.182425 + 0.983220i \(0.441605\pi\)
\(138\) 1.09302e6 4.88574
\(139\) 91551.6 0.401910 0.200955 0.979600i \(-0.435595\pi\)
0.200955 + 0.979600i \(0.435595\pi\)
\(140\) 29360.6i 0.126603i
\(141\) −88189.0 −0.373566
\(142\) 610455.i 2.54058i
\(143\) 134097.i 0.548377i
\(144\) −294264. −1.18258
\(145\) −19664.9 −0.0776734
\(146\) 431772.i 1.67638i
\(147\) −414258. −1.58117
\(148\) −166289. + 322747.i −0.624037 + 1.21118i
\(149\) −7264.04 −0.0268048 −0.0134024 0.999910i \(-0.504266\pi\)
−0.0134024 + 0.999910i \(0.504266\pi\)
\(150\) 723973.i 2.62721i
\(151\) −288159. −1.02847 −0.514233 0.857651i \(-0.671923\pi\)
−0.514233 + 0.857651i \(0.671923\pi\)
\(152\) 173202. 0.608055
\(153\) 1.00169e6i 3.45942i
\(154\) 190850.i 0.648470i
\(155\) −95315.9 −0.318666
\(156\) 360442.i 1.18583i
\(157\) 358882. 1.16199 0.580996 0.813906i \(-0.302663\pi\)
0.580996 + 0.813906i \(0.302663\pi\)
\(158\) −543528. −1.73213
\(159\) −187936. −0.589544
\(160\) 109585. 0.338416
\(161\) 209810.i 0.637914i
\(162\) 1.09000e6i 3.26317i
\(163\) 255867.i 0.754302i −0.926152 0.377151i \(-0.876904\pi\)
0.926152 0.377151i \(-0.123096\pi\)
\(164\) −109766. −0.318684
\(165\) 186356.i 0.532886i
\(166\) 513831.i 1.44727i
\(167\) 279041.i 0.774241i 0.922029 + 0.387120i \(0.126530\pi\)
−0.922029 + 0.387120i \(0.873470\pi\)
\(168\) 136480.i 0.373074i
\(169\) 286999. 0.772971
\(170\) 217358.i 0.576837i
\(171\) 975072.i 2.55004i
\(172\) 763968.i 1.96904i
\(173\) −322508. −0.819267 −0.409633 0.912250i \(-0.634343\pi\)
−0.409633 + 0.912250i \(0.634343\pi\)
\(174\) −343586. −0.860325
\(175\) 138970. 0.343025
\(176\) 239370. 0.582490
\(177\) 102153.i 0.245085i
\(178\) −446839. −1.05706
\(179\) 190807.i 0.445105i −0.974921 0.222552i \(-0.928561\pi\)
0.974921 0.222552i \(-0.0714388\pi\)
\(180\) 350784.i 0.806971i
\(181\) 616785. 1.39938 0.699692 0.714444i \(-0.253320\pi\)
0.699692 + 0.714444i \(0.253320\pi\)
\(182\) 119970. 0.268468
\(183\) 70088.7i 0.154711i
\(184\) −445263. −0.969554
\(185\) −104894. 54044.6i −0.225331 0.116098i
\(186\) −1.66536e6 −3.52961
\(187\) 814827.i 1.70397i
\(188\) 135034. 0.278643
\(189\) 439508. 0.894978
\(190\) 211583.i 0.425202i
\(191\) 597918.i 1.18593i −0.805229 0.592964i \(-0.797958\pi\)
0.805229 0.592964i \(-0.202042\pi\)
\(192\) 1.44244e6 2.82387
\(193\) 507806.i 0.981306i 0.871355 + 0.490653i \(0.163242\pi\)
−0.871355 + 0.490653i \(0.836758\pi\)
\(194\) 875910. 1.67092
\(195\) 117145. 0.220616
\(196\) 634307. 1.17940
\(197\) −127024. −0.233196 −0.116598 0.993179i \(-0.537199\pi\)
−0.116598 + 0.993179i \(0.537199\pi\)
\(198\) 2.28017e6i 4.13336i
\(199\) 898525.i 1.60841i −0.594350 0.804207i \(-0.702590\pi\)
0.594350 0.804207i \(-0.297410\pi\)
\(200\) 294924.i 0.521357i
\(201\) 1.03578e6 1.80833
\(202\) 273243.i 0.471162i
\(203\) 65952.9i 0.112330i
\(204\) 2.19019e6i 3.68474i
\(205\) 35674.5i 0.0592888i
\(206\) −825723. −1.35571
\(207\) 2.50669e6i 4.06607i
\(208\) 150470.i 0.241152i
\(209\) 793177.i 1.25604i
\(210\) −166723. −0.260884
\(211\) 494860. 0.765202 0.382601 0.923914i \(-0.375028\pi\)
0.382601 + 0.923914i \(0.375028\pi\)
\(212\) 287765. 0.439742
\(213\) −1.99916e6 −3.01925
\(214\) 1.55671e6i 2.32367i
\(215\) 248292. 0.366325
\(216\) 932731.i 1.36026i
\(217\) 319674.i 0.460849i
\(218\) −484849. −0.690980
\(219\) −1.41400e6 −1.99222
\(220\) 285347.i 0.397481i
\(221\) −512206. −0.705447
\(222\) −1.83271e6 944269.i −2.49580 1.28592i
\(223\) 967287. 1.30255 0.651273 0.758843i \(-0.274235\pi\)
0.651273 + 0.758843i \(0.274235\pi\)
\(224\) 367531.i 0.489411i
\(225\) 1.66033e6 2.18645
\(226\) −754260. −0.982313
\(227\) 1.25464e6i 1.61605i 0.589151 + 0.808023i \(0.299462\pi\)
−0.589151 + 0.808023i \(0.700538\pi\)
\(228\) 2.13199e6i 2.71612i
\(229\) −216765. −0.273149 −0.136575 0.990630i \(-0.543609\pi\)
−0.136575 + 0.990630i \(0.543609\pi\)
\(230\) 543932.i 0.677992i
\(231\) −625009. −0.770648
\(232\) 139966. 0.170728
\(233\) 1.48321e6 1.78983 0.894916 0.446234i \(-0.147235\pi\)
0.894916 + 0.446234i \(0.147235\pi\)
\(234\) 1.43333e6 1.71122
\(235\) 43886.5i 0.0518396i
\(236\) 156415.i 0.182809i
\(237\) 1.77999e6i 2.05847i
\(238\) 728983. 0.834210
\(239\) 13603.1i 0.0154043i −0.999970 0.00770216i \(-0.997548\pi\)
0.999970 0.00770216i \(-0.00245170\pi\)
\(240\) 209110.i 0.234340i
\(241\) 648806.i 0.719568i 0.933036 + 0.359784i \(0.117150\pi\)
−0.933036 + 0.359784i \(0.882850\pi\)
\(242\) 454505.i 0.498885i
\(243\) 1.32231e6 1.43654
\(244\) 107319.i 0.115399i
\(245\) 206152.i 0.219418i
\(246\) 623305.i 0.656694i
\(247\) 498596. 0.520004
\(248\) 678418. 0.700435
\(249\) −1.68273e6 −1.71995
\(250\) −745296. −0.754186
\(251\) 1.63078e6i 1.63385i 0.576745 + 0.816924i \(0.304323\pi\)
−0.576745 + 0.816924i \(0.695677\pi\)
\(252\) −1.17647e6 −1.16702
\(253\) 2.03908e6i 2.00278i
\(254\) 2.45836e6i 2.39090i
\(255\) 711819. 0.685519
\(256\) −56905.4 −0.0542692
\(257\) 680882.i 0.643042i 0.946903 + 0.321521i \(0.104194\pi\)
−0.946903 + 0.321521i \(0.895806\pi\)
\(258\) 4.33817e6 4.05749
\(259\) 181257. 351797.i 0.167898 0.325868i
\(260\) −179371. −0.164558
\(261\) 787968.i 0.715991i
\(262\) −717403. −0.645669
\(263\) −1.05628e6 −0.941649 −0.470825 0.882227i \(-0.656044\pi\)
−0.470825 + 0.882227i \(0.656044\pi\)
\(264\) 1.32640e6i 1.17129i
\(265\) 93524.6i 0.0818109i
\(266\) −709613. −0.614919
\(267\) 1.46334e6i 1.25622i
\(268\) −1.58598e6 −1.34884
\(269\) −566463. −0.477300 −0.238650 0.971106i \(-0.576705\pi\)
−0.238650 + 0.971106i \(0.576705\pi\)
\(270\) −1.13942e6 −0.951207
\(271\) −616867. −0.510233 −0.255116 0.966910i \(-0.582114\pi\)
−0.255116 + 0.966910i \(0.582114\pi\)
\(272\) 914315.i 0.749331i
\(273\) 392885.i 0.319050i
\(274\) 696907.i 0.560788i
\(275\) −1.35061e6 −1.07695
\(276\) 5.48088e6i 4.33090i
\(277\) 569488.i 0.445949i 0.974824 + 0.222975i \(0.0715767\pi\)
−0.974824 + 0.222975i \(0.928423\pi\)
\(278\) 796023.i 0.617751i
\(279\) 3.81928e6i 2.93746i
\(280\) 67917.9 0.0517713
\(281\) 366748.i 0.277078i −0.990357 0.138539i \(-0.955759\pi\)
0.990357 0.138539i \(-0.0442406\pi\)
\(282\) 766786.i 0.574185i
\(283\) 1.60923e6i 1.19441i 0.802089 + 0.597205i \(0.203722\pi\)
−0.802089 + 0.597205i \(0.796278\pi\)
\(284\) 3.06109e6 2.25206
\(285\) −692905. −0.505314
\(286\) −1.16595e6 −0.842876
\(287\) 119646. 0.0857422
\(288\) 4.39104e6i 3.11951i
\(289\) −1.69251e6 −1.19203
\(290\) 170983.i 0.119387i
\(291\) 2.86849e6i 1.98574i
\(292\) 2.16509e6 1.48600
\(293\) 1.19533e6 0.813427 0.406714 0.913556i \(-0.366675\pi\)
0.406714 + 0.913556i \(0.366675\pi\)
\(294\) 3.60189e6i 2.43032i
\(295\) 50835.4 0.0340104
\(296\) 746588. + 384666.i 0.495281 + 0.255185i
\(297\) −4.27144e6 −2.80985
\(298\) 63159.4i 0.0412000i
\(299\) −1.28178e6 −0.829155
\(300\) −3.63032e6 −2.32885
\(301\) 832732.i 0.529772i
\(302\) 2.50548e6i 1.58079i
\(303\) −894835. −0.559934
\(304\) 890021.i 0.552352i
\(305\) 34879.0 0.0214692
\(306\) 8.70948e6 5.31727
\(307\) −95070.0 −0.0575702 −0.0287851 0.999586i \(-0.509164\pi\)
−0.0287851 + 0.999586i \(0.509164\pi\)
\(308\) 957006. 0.574828
\(309\) 2.70414e6i 1.61114i
\(310\) 828753.i 0.489802i
\(311\) 1.06191e6i 0.622571i −0.950317 0.311285i \(-0.899241\pi\)
0.950317 0.311285i \(-0.100759\pi\)
\(312\) −833787. −0.484918
\(313\) 298778.i 0.172380i 0.996279 + 0.0861902i \(0.0274693\pi\)
−0.996279 + 0.0861902i \(0.972531\pi\)
\(314\) 3.12041e6i 1.78603i
\(315\) 382357.i 0.217117i
\(316\) 2.72549e6i 1.53542i
\(317\) 766205. 0.428250 0.214125 0.976806i \(-0.431310\pi\)
0.214125 + 0.976806i \(0.431310\pi\)
\(318\) 1.63406e6i 0.906152i
\(319\) 640976.i 0.352667i
\(320\) 717819.i 0.391868i
\(321\) 5.09804e6 2.76147
\(322\) 1.82426e6 0.980498
\(323\) 3.02967e6 1.61581
\(324\) −5.46574e6 −2.89259
\(325\) 849001.i 0.445861i
\(326\) −2.22471e6 −1.15939
\(327\) 1.58782e6i 0.821167i
\(328\) 253915.i 0.130318i
\(329\) −147188. −0.0749693
\(330\) 1.62033e6 0.819066
\(331\) 2.41648e6i 1.21231i −0.795347 0.606154i \(-0.792711\pi\)
0.795347 0.606154i \(-0.207289\pi\)
\(332\) 2.57658e6 1.28291
\(333\) 2.16555e6 4.20306e6i 1.07018 2.07709i
\(334\) 2.42620e6 1.19004
\(335\) 515448.i 0.250942i
\(336\) −701320. −0.338897
\(337\) −552737. −0.265121 −0.132560 0.991175i \(-0.542320\pi\)
−0.132560 + 0.991175i \(0.542320\pi\)
\(338\) 2.49540e6i 1.18809i
\(339\) 2.47010e6i 1.16739i
\(340\) −1.08993e6 −0.511330
\(341\) 3.10681e6i 1.44687i
\(342\) −8.47806e6 −3.91950
\(343\) −1.49014e6 −0.683897
\(344\) −1.76724e6 −0.805190
\(345\) 1.78131e6 0.805732
\(346\) 2.80414e6i 1.25924i
\(347\) 2.33460e6i 1.04085i 0.853907 + 0.520426i \(0.174227\pi\)
−0.853907 + 0.520426i \(0.825773\pi\)
\(348\) 1.72289e6i 0.762623i
\(349\) −1.62677e6 −0.714930 −0.357465 0.933927i \(-0.616359\pi\)
−0.357465 + 0.933927i \(0.616359\pi\)
\(350\) 1.20832e6i 0.527243i
\(351\) 2.68506e6i 1.16329i
\(352\) 3.57192e6i 1.53654i
\(353\) 2.36096e6i 1.00845i 0.863574 + 0.504223i \(0.168221\pi\)
−0.863574 + 0.504223i \(0.831779\pi\)
\(354\) 888198. 0.376705
\(355\) 994866.i 0.418980i
\(356\) 2.24065e6i 0.937020i
\(357\) 2.38733e6i 0.991382i
\(358\) −1.65903e6 −0.684143
\(359\) 3.66248e6 1.49982 0.749911 0.661539i \(-0.230096\pi\)
0.749911 + 0.661539i \(0.230096\pi\)
\(360\) 811445. 0.329991
\(361\) −473070. −0.191055
\(362\) 5.36282e6i 2.15091i
\(363\) 1.48844e6 0.592879
\(364\) 601581.i 0.237980i
\(365\) 703663.i 0.276460i
\(366\) 609408. 0.237796
\(367\) 301840. 0.116980 0.0584899 0.998288i \(-0.481371\pi\)
0.0584899 + 0.998288i \(0.481371\pi\)
\(368\) 2.28805e6i 0.880735i
\(369\) 1.42947e6 0.546522
\(370\) −469907. + 912030.i −0.178446 + 0.346342i
\(371\) −313666. −0.118313
\(372\) 8.35086e6i 3.12877i
\(373\) 2.51494e6 0.935957 0.467979 0.883740i \(-0.344982\pi\)
0.467979 + 0.883740i \(0.344982\pi\)
\(374\) −7.08476e6 −2.61907
\(375\) 2.44075e6i 0.896282i
\(376\) 312365.i 0.113944i
\(377\) 402922. 0.146005
\(378\) 3.82144e6i 1.37562i
\(379\) 3.27388e6 1.17075 0.585376 0.810762i \(-0.300947\pi\)
0.585376 + 0.810762i \(0.300947\pi\)
\(380\) 1.06097e6 0.376915
\(381\) −8.05080e6 −2.84136
\(382\) −5.19878e6 −1.82282
\(383\) 2.28037e6i 0.794345i −0.917744 0.397172i \(-0.869991\pi\)
0.917744 0.397172i \(-0.130009\pi\)
\(384\) 5.49506e6i 1.90171i
\(385\) 311030.i 0.106943i
\(386\) 4.41527e6 1.50831
\(387\) 9.94900e6i 3.37677i
\(388\) 4.39220e6i 1.48116i
\(389\) 2.47732e6i 0.830058i −0.909808 0.415029i \(-0.863771\pi\)
0.909808 0.415029i \(-0.136229\pi\)
\(390\) 1.01855e6i 0.339095i
\(391\) −7.78861e6 −2.57643
\(392\) 1.46730e6i 0.482286i
\(393\) 2.34940e6i 0.767319i
\(394\) 1.10445e6i 0.358430i
\(395\) −885794. −0.285654
\(396\) 1.14338e7 3.66396
\(397\) −42062.8 −0.0133944 −0.00669718 0.999978i \(-0.502132\pi\)
−0.00669718 + 0.999978i \(0.502132\pi\)
\(398\) −7.81250e6 −2.47219
\(399\) 2.32389e6i 0.730775i
\(400\) −1.51551e6 −0.473597
\(401\) 1.70554e6i 0.529665i 0.964294 + 0.264832i \(0.0853166\pi\)
−0.964294 + 0.264832i \(0.914683\pi\)
\(402\) 9.00592e6i 2.77947i
\(403\) 1.95297e6 0.599007
\(404\) 1.37016e6 0.417655
\(405\) 1.77639e6i 0.538145i
\(406\) −573448. −0.172655
\(407\) −1.76158e6 + 3.41900e6i −0.527128 + 1.02309i
\(408\) −5.06642e6 −1.50678
\(409\) 4.37126e6i 1.29211i −0.763292 0.646054i \(-0.776418\pi\)
0.763292 0.646054i \(-0.223582\pi\)
\(410\) −310182. −0.0911292
\(411\) 2.28228e6 0.666445
\(412\) 4.14054e6i 1.20175i
\(413\) 170494.i 0.0491851i
\(414\) 2.17952e7 6.24971
\(415\) 837396.i 0.238677i
\(416\) −2.24533e6 −0.636132
\(417\) 2.60687e6 0.734141
\(418\) 6.89651e6 1.93058
\(419\) 4.55257e6 1.26684 0.633420 0.773808i \(-0.281651\pi\)
0.633420 + 0.773808i \(0.281651\pi\)
\(420\) 836024.i 0.231257i
\(421\) 3.21412e6i 0.883804i 0.897063 + 0.441902i \(0.145696\pi\)
−0.897063 + 0.441902i \(0.854304\pi\)
\(422\) 4.30271e6i 1.17614i
\(423\) −1.75852e6 −0.477855
\(424\) 665667.i 0.179822i
\(425\) 5.15887e6i 1.38542i
\(426\) 1.73823e7i 4.64070i
\(427\) 116979.i 0.0310482i
\(428\) −7.80606e6 −2.05979
\(429\) 3.81833e6i 1.00168i
\(430\) 2.15885e6i 0.563056i
\(431\) 6.25665e6i 1.62237i −0.584792 0.811183i \(-0.698824\pi\)
0.584792 0.811183i \(-0.301176\pi\)
\(432\) −4.79297e6 −1.23565
\(433\) 3.91654e6 1.00388 0.501942 0.864902i \(-0.332619\pi\)
0.501942 + 0.864902i \(0.332619\pi\)
\(434\) −2.77950e6 −0.708342
\(435\) −559946. −0.141881
\(436\) 2.43125e6i 0.612510i
\(437\) 7.58166e6 1.89916
\(438\) 1.22944e7i 3.06212i
\(439\) 3.43312e6i 0.850212i −0.905144 0.425106i \(-0.860237\pi\)
0.905144 0.425106i \(-0.139763\pi\)
\(440\) −660073. −0.162540
\(441\) −8.26046e6 −2.02259
\(442\) 4.45353e6i 1.08430i
\(443\) −5.22540e6 −1.26506 −0.632528 0.774537i \(-0.717983\pi\)
−0.632528 + 0.774537i \(0.717983\pi\)
\(444\) −4.73498e6 + 9.19000e6i −1.13989 + 2.21237i
\(445\) −728219. −0.174326
\(446\) 8.41037e6i 2.00206i
\(447\) −206839. −0.0489625
\(448\) 2.40745e6 0.566711
\(449\) 6.48272e6i 1.51754i 0.651357 + 0.758772i \(0.274200\pi\)
−0.651357 + 0.758772i \(0.725800\pi\)
\(450\) 1.44363e7i 3.36065i
\(451\) −1.16281e6 −0.269194
\(452\) 3.78219e6i 0.870758i
\(453\) −8.20514e6 −1.87863
\(454\) 1.09088e7 2.48392
\(455\) 195516. 0.0442745
\(456\) 4.93180e6 1.11069
\(457\) 506140.i 0.113365i −0.998392 0.0566827i \(-0.981948\pi\)
0.998392 0.0566827i \(-0.0180523\pi\)
\(458\) 1.88473e6i 0.419841i
\(459\) 1.63155e7i 3.61467i
\(460\) −2.72751e6 −0.600997
\(461\) 600395.i 0.131578i −0.997834 0.0657892i \(-0.979044\pi\)
0.997834 0.0657892i \(-0.0209565\pi\)
\(462\) 5.43433e6i 1.18452i
\(463\) 4.18675e6i 0.907664i 0.891087 + 0.453832i \(0.149943\pi\)
−0.891087 + 0.453832i \(0.850057\pi\)
\(464\) 719238.i 0.155088i
\(465\) −2.71406e6 −0.582086
\(466\) 1.28962e7i 2.75104i
\(467\) 6.20534e6i 1.31666i −0.752730 0.658329i \(-0.771263\pi\)
0.752730 0.658329i \(-0.228737\pi\)
\(468\) 7.18734e6i 1.51689i
\(469\) 1.72873e6 0.362906
\(470\) 381584. 0.0796794
\(471\) 1.02189e7 2.12253
\(472\) −361825. −0.0747555
\(473\) 8.09306e6i 1.66326i
\(474\) −1.54766e7 −3.16395
\(475\) 5.02179e6i 1.02123i
\(476\) 3.65544e6i 0.739474i
\(477\) −3.74750e6 −0.754130
\(478\) −118276. −0.0236770
\(479\) 1.96601e6i 0.391515i 0.980652 + 0.195757i \(0.0627165\pi\)
−0.980652 + 0.195757i \(0.937284\pi\)
\(480\) 3.12036e6 0.618162
\(481\) 2.14921e6 + 1.10734e6i 0.423561 + 0.218232i
\(482\) 5.64124e6 1.10600
\(483\) 5.97421e6i 1.16523i
\(484\) −2.27909e6 −0.442229
\(485\) 1.42748e6 0.275560
\(486\) 1.14972e7i 2.20802i
\(487\) 6.73550e6i 1.28691i 0.765485 + 0.643454i \(0.222499\pi\)
−0.765485 + 0.643454i \(0.777501\pi\)
\(488\) −248254. −0.0471896
\(489\) 7.28565e6i 1.37783i
\(490\) 1.79245e6 0.337254
\(491\) −8.58269e6 −1.60664 −0.803322 0.595545i \(-0.796936\pi\)
−0.803322 + 0.595545i \(0.796936\pi\)
\(492\) −3.12553e6 −0.582117
\(493\) 2.44832e6 0.453681
\(494\) 4.33520e6i 0.799266i
\(495\) 3.71601e6i 0.681654i
\(496\) 3.48615e6i 0.636270i
\(497\) −3.33662e6 −0.605920
\(498\) 1.46310e7i 2.64363i
\(499\) 9.02170e6i 1.62195i −0.585082 0.810974i \(-0.698938\pi\)
0.585082 0.810974i \(-0.301062\pi\)
\(500\) 3.73724e6i 0.668538i
\(501\) 7.94550e6i 1.41425i
\(502\) 1.41793e7 2.51129
\(503\) 1.72194e6i 0.303458i −0.988422 0.151729i \(-0.951516\pi\)
0.988422 0.151729i \(-0.0484840\pi\)
\(504\) 2.72145e6i 0.477227i
\(505\) 445307.i 0.0777018i
\(506\) −1.77294e7 −3.07835
\(507\) 8.17210e6 1.41193
\(508\) 1.23273e7 2.11938
\(509\) 7.97361e6 1.36414 0.682072 0.731285i \(-0.261079\pi\)
0.682072 + 0.731285i \(0.261079\pi\)
\(510\) 6.18913e6i 1.05367i
\(511\) −2.35997e6 −0.399811
\(512\) 5.68067e6i 0.957690i
\(513\) 1.58820e7i 2.66447i
\(514\) 5.92013e6 0.988379
\(515\) −1.34569e6 −0.223577
\(516\) 2.17535e7i 3.59670i
\(517\) 1.43048e6 0.235372
\(518\) −3.05880e6 1.57599e6i −0.500872 0.258066i
\(519\) −9.18321e6 −1.49650
\(520\) 414927.i 0.0672920i
\(521\) −391196. −0.0631394 −0.0315697 0.999502i \(-0.510051\pi\)
−0.0315697 + 0.999502i \(0.510051\pi\)
\(522\) −6.85123e6 −1.10051
\(523\) 1.67343e6i 0.267518i 0.991014 + 0.133759i \(0.0427048\pi\)
−0.991014 + 0.133759i \(0.957295\pi\)
\(524\) 3.59738e6i 0.572345i
\(525\) 3.95708e6 0.626580
\(526\) 9.18414e6i 1.44735i
\(527\) 1.18670e7 1.86129
\(528\) 6.81592e6 1.06399
\(529\) −1.30544e7 −2.02824
\(530\) 813177. 0.125746
\(531\) 2.03696e6i 0.313506i
\(532\) 3.55832e6i 0.545086i
\(533\) 730948.i 0.111447i
\(534\) −1.27235e7 −1.93087
\(535\) 2.53699e6i 0.383208i
\(536\) 3.66873e6i 0.551575i
\(537\) 5.43311e6i 0.813042i
\(538\) 4.92529e6i 0.733628i
\(539\) 6.71951e6 0.996244
\(540\) 5.71357e6i 0.843185i
\(541\) 9.45940e6i 1.38954i 0.719233 + 0.694769i \(0.244494\pi\)
−0.719233 + 0.694769i \(0.755506\pi\)
\(542\) 5.36354e6i 0.784248i
\(543\) 1.75625e7 2.55616
\(544\) −1.36435e7 −1.97665
\(545\) −790164. −0.113953
\(546\) 3.41606e6 0.490392
\(547\) 441408.i 0.0630771i −0.999503 0.0315386i \(-0.989959\pi\)
0.999503 0.0315386i \(-0.0100407\pi\)
\(548\) −3.49460e6 −0.497103
\(549\) 1.39759e6i 0.197902i
\(550\) 1.17433e7i 1.65532i
\(551\) −2.38326e6 −0.334420
\(552\) −1.26786e7 −1.77102
\(553\) 2.97081e6i 0.413106i
\(554\) 4.95159e6 0.685441
\(555\) −2.98678e6 1.53889e6i −0.411596 0.212067i
\(556\) −3.99161e6 −0.547597
\(557\) 3.74743e6i 0.511795i 0.966704 + 0.255897i \(0.0823710\pi\)
−0.966704 + 0.255897i \(0.917629\pi\)
\(558\) −3.32079e7 −4.51498
\(559\) −5.08736e6 −0.688593
\(560\) 349006.i 0.0470287i
\(561\) 2.32017e7i 3.11252i
\(562\) −3.18880e6 −0.425879
\(563\) 2.07672e6i 0.276126i −0.990423 0.138063i \(-0.955912\pi\)
0.990423 0.138063i \(-0.0440877\pi\)
\(564\) 3.84501e6 0.508978
\(565\) −1.22923e6 −0.161998
\(566\) 1.39920e7 1.83585
\(567\) 5.95771e6 0.778254
\(568\) 7.08103e6i 0.920927i
\(569\) 5.04942e6i 0.653824i 0.945055 + 0.326912i \(0.106008\pi\)
−0.945055 + 0.326912i \(0.893992\pi\)
\(570\) 6.02467e6i 0.776688i
\(571\) −4.82893e6 −0.619812 −0.309906 0.950767i \(-0.600298\pi\)
−0.309906 + 0.950767i \(0.600298\pi\)
\(572\) 5.84658e6i 0.747156i
\(573\) 1.70253e7i 2.16625i
\(574\) 1.04030e6i 0.131789i
\(575\) 1.29099e7i 1.62837i
\(576\) 2.87628e7 3.61223
\(577\) 5.22766e6i 0.653684i −0.945079 0.326842i \(-0.894016\pi\)
0.945079 0.326842i \(-0.105984\pi\)
\(578\) 1.47161e7i 1.83220i
\(579\) 1.44594e7i 1.79248i
\(580\) 857382. 0.105829
\(581\) −2.80849e6 −0.345170
\(582\) 2.49410e7 3.05215
\(583\) 3.04842e6 0.371453
\(584\) 5.00837e6i 0.607665i
\(585\) 2.33591e6 0.282206
\(586\) 1.03932e7i 1.25027i
\(587\) 1.82439e6i 0.218536i −0.994012 0.109268i \(-0.965149\pi\)
0.994012 0.109268i \(-0.0348507\pi\)
\(588\) 1.80615e7 2.15432
\(589\) −1.15517e7 −1.37201
\(590\) 442004.i 0.0522752i
\(591\) −3.61693e6 −0.425962
\(592\) −1.97666e6 + 3.83645e6i −0.231808 + 0.449910i
\(593\) −1.01857e7 −1.18947 −0.594736 0.803921i \(-0.702744\pi\)
−0.594736 + 0.803921i \(0.702744\pi\)
\(594\) 3.71394e7i 4.31885i
\(595\) 1.18803e6 0.137574
\(596\) 316709. 0.0365212
\(597\) 2.55849e7i 2.93798i
\(598\) 1.11448e7i 1.27444i
\(599\) −6.34368e6 −0.722395 −0.361197 0.932489i \(-0.617632\pi\)
−0.361197 + 0.932489i \(0.617632\pi\)
\(600\) 8.39778e6i 0.952327i
\(601\) −1.51876e7 −1.71515 −0.857575 0.514359i \(-0.828030\pi\)
−0.857575 + 0.514359i \(0.828030\pi\)
\(602\) 7.24044e6 0.814280
\(603\) 2.06539e7 2.31317
\(604\) 1.25636e7 1.40127
\(605\) 740711.i 0.0822736i
\(606\) 7.78041e6i 0.860639i
\(607\) 9.62724e6i 1.06055i −0.847827 0.530274i \(-0.822089\pi\)
0.847827 0.530274i \(-0.177911\pi\)
\(608\) 1.32810e7 1.45704
\(609\) 1.87797e6i 0.205185i
\(610\) 303266.i 0.0329989i
\(611\) 899208.i 0.0974444i
\(612\) 4.36732e7i 4.71342i
\(613\) −2.24848e6 −0.241678 −0.120839 0.992672i \(-0.538559\pi\)
−0.120839 + 0.992672i \(0.538559\pi\)
\(614\) 826615.i 0.0884876i
\(615\) 1.01581e6i 0.108299i
\(616\) 2.21378e6i 0.235062i
\(617\) −4.22918e6 −0.447243 −0.223622 0.974676i \(-0.571788\pi\)
−0.223622 + 0.974676i \(0.571788\pi\)
\(618\) −2.35119e7 −2.47638
\(619\) −9.60617e6 −1.00768 −0.503841 0.863796i \(-0.668080\pi\)
−0.503841 + 0.863796i \(0.668080\pi\)
\(620\) 4.15573e6 0.434179
\(621\) 4.08290e7i 4.24854i
\(622\) −9.23314e6 −0.956915
\(623\) 2.44233e6i 0.252106i
\(624\) 4.28453e6i 0.440496i
\(625\) 7.92354e6 0.811371
\(626\) 2.59782e6 0.264955
\(627\) 2.25852e7i 2.29432i
\(628\) −1.56471e7 −1.58320
\(629\) 1.30595e7 + 6.72865e6i 1.31613 + 0.678112i
\(630\) −3.32452e6 −0.333716
\(631\) 1.22895e7i 1.22874i −0.789019 0.614369i \(-0.789411\pi\)
0.789019 0.614369i \(-0.210589\pi\)
\(632\) 6.30470e6 0.627873
\(633\) 1.40908e7 1.39774
\(634\) 6.66200e6i 0.658236i
\(635\) 4.00641e6i 0.394295i
\(636\) 8.19392e6 0.803246
\(637\) 4.22393e6i 0.412447i
\(638\) 5.57316e6 0.542063
\(639\) −3.98640e7 −3.86215
\(640\) −2.73457e6 −0.263900
\(641\) 1.31532e7 1.26441 0.632203 0.774803i \(-0.282151\pi\)
0.632203 + 0.774803i \(0.282151\pi\)
\(642\) 4.43264e7i 4.24449i
\(643\) 1.14463e7i 1.09179i 0.837855 + 0.545893i \(0.183809\pi\)
−0.837855 + 0.545893i \(0.816191\pi\)
\(644\) 9.14764e6i 0.869149i
\(645\) 7.06996e6 0.669141
\(646\) 2.63424e7i 2.48356i
\(647\) 1.97923e7i 1.85882i −0.369054 0.929408i \(-0.620318\pi\)
0.369054 0.929408i \(-0.379682\pi\)
\(648\) 1.26435e7i 1.18285i
\(649\) 1.65698e6i 0.154420i
\(650\) 7.38189e6 0.685306
\(651\) 9.10252e6i 0.841800i
\(652\) 1.11557e7i 1.02773i
\(653\) 9.62504e6i 0.883323i 0.897182 + 0.441661i \(0.145611\pi\)
−0.897182 + 0.441661i \(0.854389\pi\)
\(654\) −1.38058e7 −1.26216
\(655\) −1.16916e6 −0.106481
\(656\) −1.30478e6 −0.118380
\(657\) −2.81956e7 −2.54840
\(658\) 1.27977e6i 0.115231i
\(659\) −4.95277e6 −0.444257 −0.222129 0.975017i \(-0.571301\pi\)
−0.222129 + 0.975017i \(0.571301\pi\)
\(660\) 8.12506e6i 0.726050i
\(661\) 1.59938e7i 1.42379i 0.702284 + 0.711897i \(0.252164\pi\)
−0.702284 + 0.711897i \(0.747836\pi\)
\(662\) −2.10108e7 −1.86336
\(663\) −1.45847e7 −1.28859
\(664\) 5.96022e6i 0.524617i
\(665\) −1.15646e6 −0.101409
\(666\) −3.65448e7 1.88291e7i −3.19257 1.64491i
\(667\) 6.12684e6 0.533239
\(668\) 1.21661e7i 1.05489i
\(669\) 2.75429e7 2.37927
\(670\) −4.48172e6 −0.385707
\(671\) 1.13688e6i 0.0974783i
\(672\) 1.04652e7i 0.893972i
\(673\) 6.85466e6 0.583376 0.291688 0.956514i \(-0.405783\pi\)
0.291688 + 0.956514i \(0.405783\pi\)
\(674\) 4.80594e6i 0.407501i
\(675\) 2.70435e7 2.28457
\(676\) −1.25130e7 −1.05316
\(677\) −5.43592e6 −0.455829 −0.227914 0.973681i \(-0.573191\pi\)
−0.227914 + 0.973681i \(0.573191\pi\)
\(678\) −2.14771e7 −1.79432
\(679\) 4.78754e6i 0.398509i
\(680\) 2.52126e6i 0.209096i
\(681\) 3.57250e7i 2.95192i
\(682\) 2.70131e7 2.22389
\(683\) 1.59391e7i 1.30741i −0.756749 0.653706i \(-0.773213\pi\)
0.756749 0.653706i \(-0.226787\pi\)
\(684\) 4.25127e7i 3.47439i
\(685\) 1.13576e6i 0.0924824i
\(686\) 1.29564e7i 1.05118i
\(687\) −6.17224e6 −0.498943
\(688\) 9.08120e6i 0.731429i
\(689\) 1.91626e6i 0.153782i
\(690\) 1.54881e7i 1.23844i
\(691\) −1.36355e6 −0.108636 −0.0543181 0.998524i \(-0.517298\pi\)
−0.0543181 + 0.998524i \(0.517298\pi\)
\(692\) 1.40612e7 1.11624
\(693\) −1.24629e7 −0.985793
\(694\) 2.02989e7 1.59983
\(695\) 1.29729e6i 0.101877i
\(696\) 3.98545e6 0.311856
\(697\) 4.44153e6i 0.346299i
\(698\) 1.41445e7i 1.09887i
\(699\) 4.22334e7 3.26936
\(700\) −6.05903e6 −0.467367
\(701\) 1.03807e7i 0.797867i 0.916980 + 0.398934i \(0.130620\pi\)
−0.916980 + 0.398934i \(0.869380\pi\)
\(702\) 2.33461e7 1.78801
\(703\) −1.27124e7 6.54986e6i −0.970154 0.499855i
\(704\) −2.33972e7 −1.77923
\(705\) 1.24964e6i 0.0946917i
\(706\) 2.05281e7 1.55002
\(707\) −1.49349e6 −0.112371
\(708\) 4.45382e6i 0.333925i
\(709\) 1.48434e7i 1.10896i 0.832196 + 0.554481i \(0.187083\pi\)
−0.832196 + 0.554481i \(0.812917\pi\)
\(710\) 8.65016e6 0.643989
\(711\) 3.54935e7i 2.63315i
\(712\) 5.18315e6 0.383172
\(713\) 2.96968e7 2.18769
\(714\) 2.07573e7 1.52379
\(715\) −1.90016e6 −0.139003
\(716\) 8.31911e6i 0.606449i
\(717\) 387339.i 0.0281380i
\(718\) 3.18446e7i 2.30528i
\(719\) 7.41307e6 0.534781 0.267391 0.963588i \(-0.413839\pi\)
0.267391 + 0.963588i \(0.413839\pi\)
\(720\) 4.16972e6i 0.299762i
\(721\) 4.51323e6i 0.323332i
\(722\) 4.11325e6i 0.293658i
\(723\) 1.84743e7i 1.31439i
\(724\) −2.68915e7 −1.90664
\(725\) 4.05817e6i 0.286738i
\(726\) 1.29417e7i 0.911278i
\(727\) 807655.i 0.0566748i 0.999598 + 0.0283374i \(0.00902128\pi\)
−0.999598 + 0.0283374i \(0.990979\pi\)
\(728\) −1.39160e6 −0.0973162
\(729\) 7.18890e6 0.501007
\(730\) 6.11821e6 0.424930
\(731\) −3.09128e7 −2.13966
\(732\) 3.05584e6i 0.210791i
\(733\) 1.29072e7 0.887306 0.443653 0.896199i \(-0.353682\pi\)
0.443653 + 0.896199i \(0.353682\pi\)
\(734\) 2.62444e6i 0.179803i
\(735\) 5.87005e6i 0.400796i
\(736\) −3.41425e7 −2.32328
\(737\) −1.68010e7 −1.13937
\(738\) 1.24289e7i 0.840026i
\(739\) −5.36335e6 −0.361265 −0.180632 0.983551i \(-0.557814\pi\)
−0.180632 + 0.983551i \(0.557814\pi\)
\(740\) 4.57332e6 + 2.35632e6i 0.307010 + 0.158181i
\(741\) 1.41972e7 0.949855
\(742\) 2.72727e6i 0.181852i
\(743\) −1.71956e7 −1.14274 −0.571369 0.820693i \(-0.693587\pi\)
−0.571369 + 0.820693i \(0.693587\pi\)
\(744\) 1.93175e7 1.27944
\(745\) 102932.i 0.00679450i
\(746\) 2.18669e7i 1.43860i
\(747\) −3.35542e7 −2.20012
\(748\) 3.55261e7i 2.32164i
\(749\) 8.50867e6 0.554188
\(750\) −2.12218e7 −1.37762
\(751\) 1.95976e7 1.26795 0.633976 0.773353i \(-0.281422\pi\)
0.633976 + 0.773353i \(0.281422\pi\)
\(752\) 1.60513e6 0.103506
\(753\) 4.64355e7i 2.98444i
\(754\) 3.50333e6i 0.224415i
\(755\) 4.08321e6i 0.260696i
\(756\) −1.91624e7 −1.21940
\(757\) 2.38291e7i 1.51136i 0.654941 + 0.755680i \(0.272694\pi\)
−0.654941 + 0.755680i \(0.727306\pi\)
\(758\) 2.84658e7i 1.79949i
\(759\) 5.80615e7i 3.65834i
\(760\) 2.45427e6i 0.154130i
\(761\) 1.78221e7 1.11557 0.557784 0.829986i \(-0.311652\pi\)
0.557784 + 0.829986i \(0.311652\pi\)
\(762\) 7.00001e7i 4.36728i
\(763\) 2.65008e6i 0.164796i
\(764\) 2.60690e7i 1.61581i
\(765\) 1.41939e7 0.876898
\(766\) −1.98274e7 −1.22094
\(767\) −1.04159e6 −0.0639303
\(768\) −1.62034e6 −0.0991297
\(769\) 2.34202e7i 1.42815i −0.700069 0.714075i \(-0.746848\pi\)
0.700069 0.714075i \(-0.253152\pi\)
\(770\) 2.70435e6 0.164375
\(771\) 1.93877e7i 1.17460i
\(772\) 2.21401e7i 1.33702i
\(773\) −3.01305e7 −1.81367 −0.906834 0.421488i \(-0.861508\pi\)
−0.906834 + 0.421488i \(0.861508\pi\)
\(774\) 8.65046e7 5.19023
\(775\) 1.96700e7i 1.17639i
\(776\) −1.01602e7 −0.605686
\(777\) 5.16117e6 1.00172e7i 0.306687 0.595241i
\(778\) −2.15398e7 −1.27583
\(779\) 4.32351e6i 0.255266i
\(780\) −5.10747e6 −0.300586
\(781\) 3.24276e7 1.90233
\(782\) 6.77204e7i 3.96007i
\(783\) 1.28344e7i 0.748122i
\(784\) 7.53994e6 0.438105
\(785\) 5.08537e6i 0.294543i
\(786\) −2.04276e7 −1.17940
\(787\) 1.11009e7 0.638884 0.319442 0.947606i \(-0.396505\pi\)
0.319442 + 0.947606i \(0.396505\pi\)
\(788\) 5.53819e6 0.317726
\(789\) −3.00769e7 −1.72005
\(790\) 7.70180e6i 0.439061i
\(791\) 4.12262e6i 0.234278i
\(792\) 2.64490e7i 1.49829i
\(793\) −714650. −0.0403562
\(794\) 365728.i 0.0205877i
\(795\) 2.66305e6i 0.149438i
\(796\) 3.91753e7i 2.19144i
\(797\) 1.27525e7i 0.711130i 0.934652 + 0.355565i \(0.115712\pi\)
−0.934652 + 0.355565i \(0.884288\pi\)
\(798\) −2.02058e7 −1.12323
\(799\) 5.46395e6i 0.302789i
\(800\) 2.26146e7i 1.24929i
\(801\) 2.91795e7i 1.60693i
\(802\) 1.48293e7 0.814115
\(803\) 2.29358e7 1.25524
\(804\) −4.51596e7 −2.46383
\(805\) 2.97301e6 0.161699
\(806\) 1.69806e7i 0.920697i
\(807\) −1.61297e7 −0.871850
\(808\) 3.16950e6i 0.170790i
\(809\) 1.44917e6i 0.0778480i 0.999242 + 0.0389240i \(0.0123930\pi\)
−0.999242 + 0.0389240i \(0.987607\pi\)
\(810\) −1.54453e7 −0.827150
\(811\) 1.18481e7 0.632554 0.316277 0.948667i \(-0.397567\pi\)
0.316277 + 0.948667i \(0.397567\pi\)
\(812\) 2.87552e6i 0.153048i
\(813\) −1.75649e7 −0.932007
\(814\) 2.97275e7 + 1.53166e7i 1.57253 + 0.810217i
\(815\) −3.62564e6 −0.191201
\(816\) 2.60345e7i 1.36875i
\(817\) 3.00914e7 1.57720
\(818\) −3.80073e7 −1.98602
\(819\) 7.83427e6i 0.408121i
\(820\) 1.55539e6i 0.0807802i
\(821\) −1.26759e7 −0.656328 −0.328164 0.944621i \(-0.606430\pi\)
−0.328164 + 0.944621i \(0.606430\pi\)
\(822\) 1.98440e7i 1.02435i
\(823\) 8.00608e6 0.412022 0.206011 0.978550i \(-0.433952\pi\)
0.206011 + 0.978550i \(0.433952\pi\)
\(824\) 9.57804e6 0.491427
\(825\) −3.84576e7 −1.96720
\(826\) 1.48241e6 0.0755993
\(827\) 2.11433e6i 0.107500i −0.998554 0.0537501i \(-0.982883\pi\)
0.998554 0.0537501i \(-0.0171174\pi\)
\(828\) 1.09291e8i 5.53997i
\(829\) 3.78313e6i 0.191190i 0.995420 + 0.0955949i \(0.0304753\pi\)
−0.995420 + 0.0955949i \(0.969525\pi\)
\(830\) 7.28099e6 0.366856
\(831\) 1.62158e7i 0.814585i
\(832\) 1.47077e7i 0.736607i
\(833\) 2.56663e7i 1.28159i
\(834\) 2.26662e7i 1.12840i
\(835\) 3.95401e6 0.196255
\(836\) 3.45822e7i 1.71134i
\(837\) 6.22085e7i 3.06928i
\(838\) 3.95837e7i 1.94718i
\(839\) −2.05721e7 −1.00896 −0.504481 0.863423i \(-0.668316\pi\)
−0.504481 + 0.863423i \(0.668316\pi\)
\(840\) 1.93392e6 0.0945671
\(841\) 1.85852e7 0.906103
\(842\) 2.79461e7 1.35844
\(843\) 1.04429e7i 0.506119i
\(844\) −2.15757e7 −1.04258
\(845\) 4.06677e6i 0.195933i
\(846\) 1.52900e7i 0.734482i
\(847\) 2.48423e6 0.118982
\(848\) 3.42063e6 0.163349
\(849\) 4.58219e7i 2.18174i
\(850\) 4.48553e7 2.12945
\(851\) 3.26809e7 + 1.68382e7i 1.54693 + 0.797027i
\(852\) 8.71626e7 4.11369
\(853\) 2.89651e7i 1.36302i −0.731809 0.681509i \(-0.761324\pi\)
0.731809 0.681509i \(-0.238676\pi\)
\(854\) 1.01711e6 0.0477223
\(855\) −1.38168e7 −0.646385
\(856\) 1.80572e7i 0.842300i
\(857\) 3.44820e7i 1.60376i 0.597483 + 0.801881i \(0.296167\pi\)
−0.597483 + 0.801881i \(0.703833\pi\)
\(858\) −3.31996e7 −1.53962
\(859\) 2.20350e7i 1.01890i 0.860501 + 0.509449i \(0.170151\pi\)
−0.860501 + 0.509449i \(0.829849\pi\)
\(860\) −1.08254e7 −0.499113
\(861\) 3.40685e6 0.156619
\(862\) −5.44004e7 −2.49364
\(863\) 1.42747e7 0.652439 0.326219 0.945294i \(-0.394225\pi\)
0.326219 + 0.945294i \(0.394225\pi\)
\(864\) 7.15214e7i 3.25950i
\(865\) 4.56994e6i 0.207669i
\(866\) 3.40536e7i 1.54301i
\(867\) −4.81932e7 −2.17740
\(868\) 1.39377e7i 0.627900i
\(869\) 2.88724e7i 1.29698i
\(870\) 4.86862e6i 0.218076i
\(871\) 1.05612e7i 0.471703i
\(872\) 5.62404e6 0.250471
\(873\) 5.71988e7i 2.54010i
\(874\) 6.59210e7i 2.91908i
\(875\) 4.07363e6i 0.179871i
\(876\) 6.16497e7 2.71438
\(877\) −2.53596e7 −1.11338 −0.556689 0.830721i \(-0.687928\pi\)
−0.556689 + 0.830721i \(0.687928\pi\)
\(878\) −2.98503e7 −1.30681
\(879\) 3.40362e7 1.48583
\(880\) 3.39188e6i 0.147650i
\(881\) 4.97673e6 0.216025 0.108012 0.994150i \(-0.465551\pi\)
0.108012 + 0.994150i \(0.465551\pi\)
\(882\) 7.18231e7i 3.10880i
\(883\) 1.36078e7i 0.587337i −0.955907 0.293669i \(-0.905124\pi\)
0.955907 0.293669i \(-0.0948762\pi\)
\(884\) 2.23320e7 0.961162
\(885\) 1.44751e6 0.0621244
\(886\) 4.54338e7i 1.94444i
\(887\) −6.45258e6 −0.275375 −0.137688 0.990476i \(-0.543967\pi\)
−0.137688 + 0.990476i \(0.543967\pi\)
\(888\) 2.12586e7 + 1.09531e7i 0.904696 + 0.466129i
\(889\) −1.34368e7 −0.570221
\(890\) 6.33172e6i 0.267945i
\(891\) −5.79011e7 −2.44339
\(892\) −4.21733e7 −1.77470
\(893\) 5.31876e6i 0.223194i
\(894\) 1.79842e6i 0.0752572i
\(895\) −2.70374e6 −0.112826
\(896\) 9.17131e6i 0.381646i
\(897\) −3.64979e7 −1.51456
\(898\) 5.63659e7 2.33252
\(899\) −9.33506e6 −0.385228
\(900\) −7.23898e7 −2.97901
\(901\) 1.16440e7i 0.477847i
\(902\) 1.01104e7i 0.413762i
\(903\) 2.37115e7i 0.967697i
\(904\) 8.74910e6 0.356075
\(905\) 8.73985e6i 0.354717i
\(906\) 7.13420e7i 2.88752i
\(907\) 4.02195e6i 0.162337i 0.996700 + 0.0811687i \(0.0258653\pi\)
−0.996700 + 0.0811687i \(0.974135\pi\)
\(908\) 5.47017e7i 2.20184i
\(909\) −1.78433e7 −0.716253
\(910\) 1.69997e6i 0.0680516i
\(911\) 3.83319e7i 1.53026i −0.643876 0.765129i \(-0.722675\pi\)
0.643876 0.765129i \(-0.277325\pi\)
\(912\) 2.53428e7i 1.00894i
\(913\) 2.72948e7 1.08369
\(914\) −4.40079e6 −0.174247
\(915\) 993158. 0.0392162
\(916\) 9.45085e6 0.372162
\(917\) 3.92117e6i 0.153990i
\(918\) 1.41860e8 5.55589
\(919\) 4.19419e6i 0.163817i 0.996640 + 0.0819085i \(0.0261015\pi\)
−0.996640 + 0.0819085i \(0.973898\pi\)
\(920\) 6.30938e6i 0.245763i
\(921\) −2.70706e6 −0.105159
\(922\) −5.22032e6 −0.202241
\(923\) 2.03842e7i 0.787570i
\(924\) 2.72501e7 1.05000
\(925\) 1.11530e7 2.16465e7i 0.428585 0.831828i
\(926\) 3.64030e7 1.39511
\(927\) 5.39214e7i 2.06092i
\(928\) 1.07326e7 0.409103
\(929\) −922581. −0.0350724 −0.0175362 0.999846i \(-0.505582\pi\)
−0.0175362 + 0.999846i \(0.505582\pi\)
\(930\) 2.35982e7i 0.894688i
\(931\) 2.49843e7i 0.944698i
\(932\) −6.46673e7 −2.43862
\(933\) 3.02373e7i 1.13721i
\(934\) −5.39542e7 −2.02376
\(935\) −1.15461e7 −0.431924
\(936\) −1.66260e7 −0.620295
\(937\) −5.51331e6 −0.205146 −0.102573 0.994725i \(-0.532708\pi\)
−0.102573 + 0.994725i \(0.532708\pi\)
\(938\) 1.50309e7i 0.557801i
\(939\) 8.50751e6i 0.314875i
\(940\) 1.91343e6i 0.0706307i
\(941\) 4.10357e7 1.51073 0.755366 0.655303i \(-0.227459\pi\)
0.755366 + 0.655303i \(0.227459\pi\)
\(942\) 8.88517e7i 3.26241i
\(943\) 1.11148e7i 0.407026i
\(944\) 1.85929e6i 0.0679073i
\(945\) 6.22784e6i 0.226860i
\(946\) −7.03676e7 −2.55649
\(947\) 2.34797e7i 0.850781i −0.905010 0.425390i \(-0.860137\pi\)
0.905010 0.425390i \(-0.139863\pi\)
\(948\) 7.76066e7i 2.80464i
\(949\) 1.44176e7i 0.519671i
\(950\) −4.36635e7 −1.56967
\(951\) 2.18172e7 0.782254
\(952\) −8.45590e6 −0.302390
\(953\) −2.94523e7 −1.05048 −0.525238 0.850955i \(-0.676024\pi\)
−0.525238 + 0.850955i \(0.676024\pi\)
\(954\) 3.25838e7i 1.15913i
\(955\) −8.47251e6 −0.300610
\(956\) 593088.i 0.0209882i
\(957\) 1.82514e7i 0.644193i
\(958\) 1.70941e7 0.601773
\(959\) 3.80915e6 0.133746
\(960\) 2.04394e7i 0.715798i
\(961\) −1.66179e7 −0.580455
\(962\) 9.62812e6 1.86869e7i 0.335431 0.651030i
\(963\) 1.01657e8 3.53240
\(964\) 2.82877e7i 0.980403i
\(965\) 7.19562e6 0.248742
\(966\) 5.19446e7 1.79101
\(967\) 2.02726e7i 0.697179i 0.937276 + 0.348589i \(0.113339\pi\)
−0.937276 + 0.348589i \(0.886661\pi\)
\(968\) 5.27206e6i 0.180839i
\(969\) 8.62679e7 2.95148
\(970\) 1.24117e7i 0.423546i
\(971\) 8.07665e6 0.274905 0.137453 0.990508i \(-0.456109\pi\)
0.137453 + 0.990508i \(0.456109\pi\)
\(972\) −5.76522e7 −1.95727
\(973\) 4.35089e6 0.147332
\(974\) 5.85638e7 1.97803
\(975\) 2.41747e7i 0.814423i
\(976\) 1.27569e6i 0.0428667i
\(977\) 1.45675e7i 0.488257i 0.969743 + 0.244129i \(0.0785019\pi\)
−0.969743 + 0.244129i \(0.921498\pi\)
\(978\) −6.33473e7 −2.11778
\(979\) 2.37362e7i 0.791507i
\(980\) 8.98814e6i 0.298954i
\(981\) 3.16616e7i 1.05042i
\(982\) 7.46248e7i 2.46947i
\(983\) −2.94115e7 −0.970808 −0.485404 0.874290i \(-0.661327\pi\)
−0.485404 + 0.874290i \(0.661327\pi\)
\(984\) 7.23008e6i 0.238043i
\(985\) 1.79993e6i 0.0591106i
\(986\) 2.12876e7i 0.697325i
\(987\) −4.19109e6 −0.136941
\(988\) −2.17386e7 −0.708499
\(989\) −7.73584e7 −2.51488
\(990\) 3.23100e7 1.04773
\(991\) 5.95225e7i 1.92529i 0.270762 + 0.962646i \(0.412724\pi\)
−0.270762 + 0.962646i \(0.587276\pi\)
\(992\) 5.20207e7 1.67841
\(993\) 6.88077e7i 2.21444i
\(994\) 2.90112e7i 0.931323i
\(995\) −1.27321e7 −0.407702
\(996\) 7.33663e7 2.34341
\(997\) 3.48734e6i 0.111111i −0.998456 0.0555553i \(-0.982307\pi\)
0.998456 0.0555553i \(-0.0176929\pi\)
\(998\) −7.84419e7 −2.49300
\(999\) 3.52726e7 6.84596e7i 1.11821 2.17030i
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 37.6.b.a.36.3 16
3.2 odd 2 333.6.c.d.73.14 16
4.3 odd 2 592.6.g.c.369.1 16
37.36 even 2 inner 37.6.b.a.36.14 yes 16
111.110 odd 2 333.6.c.d.73.3 16
148.147 odd 2 592.6.g.c.369.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
37.6.b.a.36.3 16 1.1 even 1 trivial
37.6.b.a.36.14 yes 16 37.36 even 2 inner
333.6.c.d.73.3 16 111.110 odd 2
333.6.c.d.73.14 16 3.2 odd 2
592.6.g.c.369.1 16 4.3 odd 2
592.6.g.c.369.2 16 148.147 odd 2