Properties

Label 384.6.f.e.191.8
Level $384$
Weight $6$
Character 384.191
Analytic conductor $61.587$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [384,6,Mod(191,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.191");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 384.f (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: \(61.5873868082\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 5192x^{16} + 8441320x^{12} + 4098006217x^{8} + 8949568544x^{4} + 8386816 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{87}\cdot 3^{14} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 191.8
Root \(4.77687 - 4.77687i\) of defining polynomial
Character \(\chi\) \(=\) 384.191
Dual form 384.6.f.e.191.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-6.91183 + 13.9723i) q^{3} -56.7417 q^{5} -130.269i q^{7} +(-147.453 - 193.149i) q^{9} +O(q^{10})\) \(q+(-6.91183 + 13.9723i) q^{3} -56.7417 q^{5} -130.269i q^{7} +(-147.453 - 193.149i) q^{9} +484.591i q^{11} -555.929i q^{13} +(392.189 - 792.815i) q^{15} -1110.46i q^{17} -741.084 q^{19} +(1820.17 + 900.400i) q^{21} -459.270 q^{23} +94.6178 q^{25} +(3717.92 - 725.251i) q^{27} +3061.20 q^{29} -5980.71i q^{31} +(-6770.87 - 3349.41i) q^{33} +7391.70i q^{35} -3466.31i q^{37} +(7767.63 + 3842.49i) q^{39} +14979.6i q^{41} +16818.5 q^{43} +(8366.73 + 10959.6i) q^{45} +19676.9 q^{47} -163.085 q^{49} +(15515.8 + 7675.35i) q^{51} -32218.4 q^{53} -27496.5i q^{55} +(5122.25 - 10354.7i) q^{57} -5250.05i q^{59} +53641.2i q^{61} +(-25161.4 + 19208.6i) q^{63} +31544.3i q^{65} -66623.4 q^{67} +(3174.40 - 6417.08i) q^{69} -33198.9 q^{71} -80330.2 q^{73} +(-653.983 + 1322.03i) q^{75} +63127.3 q^{77} -19463.9i q^{79} +(-15564.2 + 56960.9i) q^{81} -22867.0i q^{83} +63009.6i q^{85} +(-21158.5 + 42772.2i) q^{87} +33539.4i q^{89} -72420.4 q^{91} +(83564.6 + 41337.7i) q^{93} +42050.4 q^{95} -146327. q^{97} +(93598.2 - 71454.4i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 44 q^{9} - 3568 q^{15} + 6112 q^{23} + 15228 q^{25} + 7592 q^{33} - 2800 q^{39} + 26112 q^{47} - 81044 q^{49} - 89296 q^{57} - 14816 q^{63} - 72224 q^{71} - 61256 q^{73} + 89588 q^{81} - 145648 q^{87} + 385504 q^{95} + 92808 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −6.91183 + 13.9723i −0.443394 + 0.896327i
\(4\) 0 0
\(5\) −56.7417 −1.01503 −0.507513 0.861644i \(-0.669435\pi\)
−0.507513 + 0.861644i \(0.669435\pi\)
\(6\) 0 0
\(7\) 130.269i 1.00484i −0.864624 0.502420i \(-0.832443\pi\)
0.864624 0.502420i \(-0.167557\pi\)
\(8\) 0 0
\(9\) −147.453 193.149i −0.606803 0.794852i
\(10\) 0 0
\(11\) 484.591i 1.20752i 0.797167 + 0.603758i \(0.206331\pi\)
−0.797167 + 0.603758i \(0.793669\pi\)
\(12\) 0 0
\(13\) 555.929i 0.912348i −0.889890 0.456174i \(-0.849219\pi\)
0.889890 0.456174i \(-0.150781\pi\)
\(14\) 0 0
\(15\) 392.189 792.815i 0.450057 0.909795i
\(16\) 0 0
\(17\) 1110.46i 0.931928i −0.884804 0.465964i \(-0.845708\pi\)
0.884804 0.465964i \(-0.154292\pi\)
\(18\) 0 0
\(19\) −741.084 −0.470959 −0.235480 0.971879i \(-0.575666\pi\)
−0.235480 + 0.971879i \(0.575666\pi\)
\(20\) 0 0
\(21\) 1820.17 + 900.400i 0.900665 + 0.445540i
\(22\) 0 0
\(23\) −459.270 −0.181029 −0.0905145 0.995895i \(-0.528851\pi\)
−0.0905145 + 0.995895i \(0.528851\pi\)
\(24\) 0 0
\(25\) 94.6178 0.0302777
\(26\) 0 0
\(27\) 3717.92 725.251i 0.981500 0.191460i
\(28\) 0 0
\(29\) 3061.20 0.675923 0.337962 0.941160i \(-0.390263\pi\)
0.337962 + 0.941160i \(0.390263\pi\)
\(30\) 0 0
\(31\) 5980.71i 1.11776i −0.829249 0.558880i \(-0.811231\pi\)
0.829249 0.558880i \(-0.188769\pi\)
\(32\) 0 0
\(33\) −6770.87 3349.41i −1.08233 0.535406i
\(34\) 0 0
\(35\) 7391.70i 1.01994i
\(36\) 0 0
\(37\) 3466.31i 0.416258i −0.978101 0.208129i \(-0.933263\pi\)
0.978101 0.208129i \(-0.0667374\pi\)
\(38\) 0 0
\(39\) 7767.63 + 3842.49i 0.817762 + 0.404530i
\(40\) 0 0
\(41\) 14979.6i 1.39168i 0.718195 + 0.695842i \(0.244969\pi\)
−0.718195 + 0.695842i \(0.755031\pi\)
\(42\) 0 0
\(43\) 16818.5 1.38713 0.693563 0.720396i \(-0.256040\pi\)
0.693563 + 0.720396i \(0.256040\pi\)
\(44\) 0 0
\(45\) 8366.73 + 10959.6i 0.615921 + 0.806796i
\(46\) 0 0
\(47\) 19676.9 1.29931 0.649654 0.760230i \(-0.274914\pi\)
0.649654 + 0.760230i \(0.274914\pi\)
\(48\) 0 0
\(49\) −163.085 −0.00970341
\(50\) 0 0
\(51\) 15515.8 + 7675.35i 0.835312 + 0.413212i
\(52\) 0 0
\(53\) −32218.4 −1.57549 −0.787743 0.616004i \(-0.788751\pi\)
−0.787743 + 0.616004i \(0.788751\pi\)
\(54\) 0 0
\(55\) 27496.5i 1.22566i
\(56\) 0 0
\(57\) 5122.25 10354.7i 0.208821 0.422134i
\(58\) 0 0
\(59\) 5250.05i 0.196351i −0.995169 0.0981756i \(-0.968699\pi\)
0.995169 0.0981756i \(-0.0313007\pi\)
\(60\) 0 0
\(61\) 53641.2i 1.84575i 0.385094 + 0.922877i \(0.374169\pi\)
−0.385094 + 0.922877i \(0.625831\pi\)
\(62\) 0 0
\(63\) −25161.4 + 19208.6i −0.798699 + 0.609740i
\(64\) 0 0
\(65\) 31544.3i 0.926057i
\(66\) 0 0
\(67\) −66623.4 −1.81317 −0.906587 0.422019i \(-0.861322\pi\)
−0.906587 + 0.422019i \(0.861322\pi\)
\(68\) 0 0
\(69\) 3174.40 6417.08i 0.0802673 0.162261i
\(70\) 0 0
\(71\) −33198.9 −0.781588 −0.390794 0.920478i \(-0.627800\pi\)
−0.390794 + 0.920478i \(0.627800\pi\)
\(72\) 0 0
\(73\) −80330.2 −1.76430 −0.882149 0.470970i \(-0.843904\pi\)
−0.882149 + 0.470970i \(0.843904\pi\)
\(74\) 0 0
\(75\) −653.983 + 1322.03i −0.0134250 + 0.0271387i
\(76\) 0 0
\(77\) 63127.3 1.21336
\(78\) 0 0
\(79\) 19463.9i 0.350883i −0.984490 0.175442i \(-0.943865\pi\)
0.984490 0.175442i \(-0.0561353\pi\)
\(80\) 0 0
\(81\) −15564.2 + 56960.9i −0.263581 + 0.964637i
\(82\) 0 0
\(83\) 22867.0i 0.364345i −0.983267 0.182173i \(-0.941687\pi\)
0.983267 0.182173i \(-0.0583130\pi\)
\(84\) 0 0
\(85\) 63009.6i 0.945931i
\(86\) 0 0
\(87\) −21158.5 + 42772.2i −0.299701 + 0.605848i
\(88\) 0 0
\(89\) 33539.4i 0.448829i 0.974494 + 0.224414i \(0.0720469\pi\)
−0.974494 + 0.224414i \(0.927953\pi\)
\(90\) 0 0
\(91\) −72420.4 −0.916764
\(92\) 0 0
\(93\) 83564.6 + 41337.7i 1.00188 + 0.495608i
\(94\) 0 0
\(95\) 42050.4 0.478036
\(96\) 0 0
\(97\) −146327. −1.57905 −0.789524 0.613719i \(-0.789673\pi\)
−0.789524 + 0.613719i \(0.789673\pi\)
\(98\) 0 0
\(99\) 93598.2 71454.4i 0.959798 0.732725i
\(100\) 0 0
\(101\) 113137. 1.10357 0.551786 0.833986i \(-0.313947\pi\)
0.551786 + 0.833986i \(0.313947\pi\)
\(102\) 0 0
\(103\) 159779.i 1.48398i 0.670411 + 0.741990i \(0.266118\pi\)
−0.670411 + 0.741990i \(0.733882\pi\)
\(104\) 0 0
\(105\) −103279. 51090.2i −0.914198 0.452235i
\(106\) 0 0
\(107\) 16948.9i 0.143114i 0.997437 + 0.0715571i \(0.0227968\pi\)
−0.997437 + 0.0715571i \(0.977203\pi\)
\(108\) 0 0
\(109\) 152539.i 1.22974i 0.788627 + 0.614872i \(0.210792\pi\)
−0.788627 + 0.614872i \(0.789208\pi\)
\(110\) 0 0
\(111\) 48432.4 + 23958.5i 0.373103 + 0.184566i
\(112\) 0 0
\(113\) 145492.i 1.07188i −0.844258 0.535938i \(-0.819958\pi\)
0.844258 0.535938i \(-0.180042\pi\)
\(114\) 0 0
\(115\) 26059.7 0.183749
\(116\) 0 0
\(117\) −107377. + 81973.4i −0.725182 + 0.553616i
\(118\) 0 0
\(119\) −144659. −0.936438
\(120\) 0 0
\(121\) −73777.0 −0.458097
\(122\) 0 0
\(123\) −209300. 103537.i −1.24740 0.617065i
\(124\) 0 0
\(125\) 171949. 0.984293
\(126\) 0 0
\(127\) 90945.7i 0.500349i −0.968201 0.250175i \(-0.919512\pi\)
0.968201 0.250175i \(-0.0804880\pi\)
\(128\) 0 0
\(129\) −116247. + 234994.i −0.615044 + 1.24332i
\(130\) 0 0
\(131\) 40671.9i 0.207070i 0.994626 + 0.103535i \(0.0330153\pi\)
−0.994626 + 0.103535i \(0.966985\pi\)
\(132\) 0 0
\(133\) 96540.5i 0.473239i
\(134\) 0 0
\(135\) −210961. + 41152.0i −0.996248 + 0.194337i
\(136\) 0 0
\(137\) 330626.i 1.50499i 0.658595 + 0.752497i \(0.271151\pi\)
−0.658595 + 0.752497i \(0.728849\pi\)
\(138\) 0 0
\(139\) 176648. 0.775484 0.387742 0.921768i \(-0.373255\pi\)
0.387742 + 0.921768i \(0.373255\pi\)
\(140\) 0 0
\(141\) −136004. + 274933.i −0.576106 + 1.16460i
\(142\) 0 0
\(143\) 269398. 1.10168
\(144\) 0 0
\(145\) −173698. −0.686079
\(146\) 0 0
\(147\) 1127.22 2278.68i 0.00430244 0.00869742i
\(148\) 0 0
\(149\) 56218.4 0.207450 0.103725 0.994606i \(-0.466924\pi\)
0.103725 + 0.994606i \(0.466924\pi\)
\(150\) 0 0
\(151\) 175958.i 0.628010i −0.949421 0.314005i \(-0.898329\pi\)
0.949421 0.314005i \(-0.101671\pi\)
\(152\) 0 0
\(153\) −214485. + 163741.i −0.740745 + 0.565496i
\(154\) 0 0
\(155\) 339355.i 1.13456i
\(156\) 0 0
\(157\) 78738.0i 0.254938i 0.991843 + 0.127469i \(0.0406854\pi\)
−0.991843 + 0.127469i \(0.959315\pi\)
\(158\) 0 0
\(159\) 222688. 450167.i 0.698562 1.41215i
\(160\) 0 0
\(161\) 59828.7i 0.181905i
\(162\) 0 0
\(163\) 127815. 0.376800 0.188400 0.982092i \(-0.439670\pi\)
0.188400 + 0.982092i \(0.439670\pi\)
\(164\) 0 0
\(165\) 384190. + 190051.i 1.09859 + 0.543451i
\(166\) 0 0
\(167\) 307401. 0.852933 0.426466 0.904503i \(-0.359758\pi\)
0.426466 + 0.904503i \(0.359758\pi\)
\(168\) 0 0
\(169\) 62236.2 0.167620
\(170\) 0 0
\(171\) 109275. + 143140.i 0.285780 + 0.374343i
\(172\) 0 0
\(173\) −283267. −0.719582 −0.359791 0.933033i \(-0.617152\pi\)
−0.359791 + 0.933033i \(0.617152\pi\)
\(174\) 0 0
\(175\) 12325.8i 0.0304242i
\(176\) 0 0
\(177\) 73355.6 + 36287.5i 0.175995 + 0.0870611i
\(178\) 0 0
\(179\) 376390.i 0.878023i 0.898481 + 0.439012i \(0.144671\pi\)
−0.898481 + 0.439012i \(0.855329\pi\)
\(180\) 0 0
\(181\) 698692.i 1.58522i 0.609730 + 0.792609i \(0.291278\pi\)
−0.609730 + 0.792609i \(0.708722\pi\)
\(182\) 0 0
\(183\) −749494. 370759.i −1.65440 0.818397i
\(184\) 0 0
\(185\) 196684.i 0.422513i
\(186\) 0 0
\(187\) 538120. 1.12532
\(188\) 0 0
\(189\) −94477.9 484331.i −0.192387 0.986251i
\(190\) 0 0
\(191\) 874587. 1.73468 0.867340 0.497716i \(-0.165828\pi\)
0.867340 + 0.497716i \(0.165828\pi\)
\(192\) 0 0
\(193\) −182483. −0.352637 −0.176319 0.984333i \(-0.556419\pi\)
−0.176319 + 0.984333i \(0.556419\pi\)
\(194\) 0 0
\(195\) −440748. 218029.i −0.830050 0.410609i
\(196\) 0 0
\(197\) 288691. 0.529991 0.264995 0.964250i \(-0.414630\pi\)
0.264995 + 0.964250i \(0.414630\pi\)
\(198\) 0 0
\(199\) 961761.i 1.72161i 0.508935 + 0.860805i \(0.330039\pi\)
−0.508935 + 0.860805i \(0.669961\pi\)
\(200\) 0 0
\(201\) 460490. 930885.i 0.803951 1.62520i
\(202\) 0 0
\(203\) 398781.i 0.679195i
\(204\) 0 0
\(205\) 849968.i 1.41260i
\(206\) 0 0
\(207\) 67720.7 + 88707.6i 0.109849 + 0.143891i
\(208\) 0 0
\(209\) 359122.i 0.568692i
\(210\) 0 0
\(211\) 191585. 0.296248 0.148124 0.988969i \(-0.452677\pi\)
0.148124 + 0.988969i \(0.452677\pi\)
\(212\) 0 0
\(213\) 229465. 463867.i 0.346552 0.700558i
\(214\) 0 0
\(215\) −954309. −1.40797
\(216\) 0 0
\(217\) −779103. −1.12317
\(218\) 0 0
\(219\) 555229. 1.12240e6i 0.782280 1.58139i
\(220\) 0 0
\(221\) −617339. −0.850243
\(222\) 0 0
\(223\) 449203.i 0.604896i −0.953166 0.302448i \(-0.902196\pi\)
0.953166 0.302448i \(-0.0978038\pi\)
\(224\) 0 0
\(225\) −13951.7 18275.4i −0.0183726 0.0240663i
\(226\) 0 0
\(227\) 578361.i 0.744962i 0.928040 + 0.372481i \(0.121493\pi\)
−0.928040 + 0.372481i \(0.878507\pi\)
\(228\) 0 0
\(229\) 342423.i 0.431494i −0.976449 0.215747i \(-0.930781\pi\)
0.976449 0.215747i \(-0.0692186\pi\)
\(230\) 0 0
\(231\) −436325. + 882036.i −0.537998 + 1.08757i
\(232\) 0 0
\(233\) 724022.i 0.873699i −0.899535 0.436849i \(-0.856094\pi\)
0.899535 0.436849i \(-0.143906\pi\)
\(234\) 0 0
\(235\) −1.11650e6 −1.31883
\(236\) 0 0
\(237\) 271957. + 134531.i 0.314506 + 0.155580i
\(238\) 0 0
\(239\) 1.64343e6 1.86104 0.930521 0.366238i \(-0.119355\pi\)
0.930521 + 0.366238i \(0.119355\pi\)
\(240\) 0 0
\(241\) 1.63037e6 1.80819 0.904095 0.427331i \(-0.140546\pi\)
0.904095 + 0.427331i \(0.140546\pi\)
\(242\) 0 0
\(243\) −688300. 611172.i −0.747760 0.663969i
\(244\) 0 0
\(245\) 9253.73 0.00984921
\(246\) 0 0
\(247\) 411990.i 0.429679i
\(248\) 0 0
\(249\) 319505. + 158053.i 0.326572 + 0.161549i
\(250\) 0 0
\(251\) 1.50508e6i 1.50791i 0.656925 + 0.753956i \(0.271857\pi\)
−0.656925 + 0.753956i \(0.728143\pi\)
\(252\) 0 0
\(253\) 222558.i 0.218596i
\(254\) 0 0
\(255\) −880392. 435512.i −0.847863 0.419420i
\(256\) 0 0
\(257\) 1.20632e6i 1.13927i 0.821896 + 0.569637i \(0.192916\pi\)
−0.821896 + 0.569637i \(0.807084\pi\)
\(258\) 0 0
\(259\) −451553. −0.418273
\(260\) 0 0
\(261\) −451384. 591269.i −0.410152 0.537259i
\(262\) 0 0
\(263\) −1.62597e6 −1.44951 −0.724757 0.689005i \(-0.758048\pi\)
−0.724757 + 0.689005i \(0.758048\pi\)
\(264\) 0 0
\(265\) 1.82813e6 1.59916
\(266\) 0 0
\(267\) −468625. 231819.i −0.402297 0.199008i
\(268\) 0 0
\(269\) −137819. −0.116126 −0.0580630 0.998313i \(-0.518492\pi\)
−0.0580630 + 0.998313i \(0.518492\pi\)
\(270\) 0 0
\(271\) 1.48417e6i 1.22761i −0.789457 0.613806i \(-0.789638\pi\)
0.789457 0.613806i \(-0.210362\pi\)
\(272\) 0 0
\(273\) 500558. 1.01188e6i 0.406488 0.821720i
\(274\) 0 0
\(275\) 45850.9i 0.0365608i
\(276\) 0 0
\(277\) 1.61429e6i 1.26410i −0.774927 0.632051i \(-0.782214\pi\)
0.774927 0.632051i \(-0.217786\pi\)
\(278\) 0 0
\(279\) −1.15517e6 + 881874.i −0.888454 + 0.678260i
\(280\) 0 0
\(281\) 110881.i 0.0837704i 0.999122 + 0.0418852i \(0.0133364\pi\)
−0.999122 + 0.0418852i \(0.986664\pi\)
\(282\) 0 0
\(283\) −1.24585e6 −0.924700 −0.462350 0.886698i \(-0.652994\pi\)
−0.462350 + 0.886698i \(0.652994\pi\)
\(284\) 0 0
\(285\) −290645. + 587542.i −0.211959 + 0.428476i
\(286\) 0 0
\(287\) 1.95138e6 1.39842
\(288\) 0 0
\(289\) 186726. 0.131511
\(290\) 0 0
\(291\) 1.01139e6 2.04453e6i 0.700141 1.41534i
\(292\) 0 0
\(293\) −2.78388e6 −1.89444 −0.947222 0.320578i \(-0.896123\pi\)
−0.947222 + 0.320578i \(0.896123\pi\)
\(294\) 0 0
\(295\) 297897.i 0.199302i
\(296\) 0 0
\(297\) 351450. + 1.80167e6i 0.231192 + 1.18518i
\(298\) 0 0
\(299\) 255321.i 0.165162i
\(300\) 0 0
\(301\) 2.19093e6i 1.39384i
\(302\) 0 0
\(303\) −781983. + 1.58079e6i −0.489317 + 0.989161i
\(304\) 0 0
\(305\) 3.04369e6i 1.87349i
\(306\) 0 0
\(307\) 587991. 0.356061 0.178031 0.984025i \(-0.443027\pi\)
0.178031 + 0.984025i \(0.443027\pi\)
\(308\) 0 0
\(309\) −2.23249e6 1.10437e6i −1.33013 0.657988i
\(310\) 0 0
\(311\) 1.26748e6 0.743086 0.371543 0.928416i \(-0.378829\pi\)
0.371543 + 0.928416i \(0.378829\pi\)
\(312\) 0 0
\(313\) 2.70854e6 1.56270 0.781348 0.624096i \(-0.214533\pi\)
0.781348 + 0.624096i \(0.214533\pi\)
\(314\) 0 0
\(315\) 1.42770e6 1.08993e6i 0.810701 0.618902i
\(316\) 0 0
\(317\) −264831. −0.148020 −0.0740100 0.997257i \(-0.523580\pi\)
−0.0740100 + 0.997257i \(0.523580\pi\)
\(318\) 0 0
\(319\) 1.48343e6i 0.816188i
\(320\) 0 0
\(321\) −236816. 117148.i −0.128277 0.0634560i
\(322\) 0 0
\(323\) 822947.i 0.438900i
\(324\) 0 0
\(325\) 52600.8i 0.0276238i
\(326\) 0 0
\(327\) −2.13133e6 1.05432e6i −1.10225 0.545261i
\(328\) 0 0
\(329\) 2.56330e6i 1.30560i
\(330\) 0 0
\(331\) −534781. −0.268291 −0.134145 0.990962i \(-0.542829\pi\)
−0.134145 + 0.990962i \(0.542829\pi\)
\(332\) 0 0
\(333\) −669514. + 511117.i −0.330864 + 0.252586i
\(334\) 0 0
\(335\) 3.78032e6 1.84042
\(336\) 0 0
\(337\) 1.31960e6 0.632946 0.316473 0.948602i \(-0.397501\pi\)
0.316473 + 0.948602i \(0.397501\pi\)
\(338\) 0 0
\(339\) 2.03287e6 + 1.00562e6i 0.960750 + 0.475263i
\(340\) 0 0
\(341\) 2.89819e6 1.34971
\(342\) 0 0
\(343\) 2.16819e6i 0.995090i
\(344\) 0 0
\(345\) −180121. + 364116.i −0.0814734 + 0.164699i
\(346\) 0 0
\(347\) 2.91141e6i 1.29801i 0.760783 + 0.649006i \(0.224815\pi\)
−0.760783 + 0.649006i \(0.775185\pi\)
\(348\) 0 0
\(349\) 1.06388e6i 0.467550i −0.972291 0.233775i \(-0.924892\pi\)
0.972291 0.233775i \(-0.0751080\pi\)
\(350\) 0 0
\(351\) −403188. 2.06690e6i −0.174679 0.895470i
\(352\) 0 0
\(353\) 2.38799e6i 1.01999i 0.860178 + 0.509995i \(0.170353\pi\)
−0.860178 + 0.509995i \(0.829647\pi\)
\(354\) 0 0
\(355\) 1.88376e6 0.793332
\(356\) 0 0
\(357\) 999862. 2.02123e6i 0.415212 0.839355i
\(358\) 0 0
\(359\) 2.68611e6 1.09999 0.549994 0.835169i \(-0.314630\pi\)
0.549994 + 0.835169i \(0.314630\pi\)
\(360\) 0 0
\(361\) −1.92689e6 −0.778197
\(362\) 0 0
\(363\) 509934. 1.03084e6i 0.203118 0.410604i
\(364\) 0 0
\(365\) 4.55807e6 1.79081
\(366\) 0 0
\(367\) 3.87189e6i 1.50058i −0.661111 0.750288i \(-0.729915\pi\)
0.661111 0.750288i \(-0.270085\pi\)
\(368\) 0 0
\(369\) 2.89330e6 2.20879e6i 1.10618 0.844478i
\(370\) 0 0
\(371\) 4.19707e6i 1.58311i
\(372\) 0 0
\(373\) 206921.i 0.0770074i 0.999258 + 0.0385037i \(0.0122591\pi\)
−0.999258 + 0.0385037i \(0.987741\pi\)
\(374\) 0 0
\(375\) −1.18848e6 + 2.40253e6i −0.436430 + 0.882248i
\(376\) 0 0
\(377\) 1.70181e6i 0.616677i
\(378\) 0 0
\(379\) 2.05927e6 0.736404 0.368202 0.929746i \(-0.379973\pi\)
0.368202 + 0.929746i \(0.379973\pi\)
\(380\) 0 0
\(381\) 1.27073e6 + 628602.i 0.448476 + 0.221852i
\(382\) 0 0
\(383\) 545734. 0.190101 0.0950504 0.995472i \(-0.469699\pi\)
0.0950504 + 0.995472i \(0.469699\pi\)
\(384\) 0 0
\(385\) −3.58195e6 −1.23159
\(386\) 0 0
\(387\) −2.47994e6 3.24848e6i −0.841712 1.10256i
\(388\) 0 0
\(389\) −4.56814e6 −1.53061 −0.765307 0.643666i \(-0.777413\pi\)
−0.765307 + 0.643666i \(0.777413\pi\)
\(390\) 0 0
\(391\) 510003.i 0.168706i
\(392\) 0 0
\(393\) −568282. 281118.i −0.185602 0.0918135i
\(394\) 0 0
\(395\) 1.10442e6i 0.356156i
\(396\) 0 0
\(397\) 2.08838e6i 0.665019i 0.943100 + 0.332509i \(0.107895\pi\)
−0.943100 + 0.332509i \(0.892105\pi\)
\(398\) 0 0
\(399\) −1.34890e6 667272.i −0.424177 0.209831i
\(400\) 0 0
\(401\) 2.03905e6i 0.633237i 0.948553 + 0.316619i \(0.102548\pi\)
−0.948553 + 0.316619i \(0.897452\pi\)
\(402\) 0 0
\(403\) −3.32485e6 −1.01979
\(404\) 0 0
\(405\) 883138. 3.23206e6i 0.267541 0.979132i
\(406\) 0 0
\(407\) 1.67974e6 0.502638
\(408\) 0 0
\(409\) −2.09607e6 −0.619580 −0.309790 0.950805i \(-0.600259\pi\)
−0.309790 + 0.950805i \(0.600259\pi\)
\(410\) 0 0
\(411\) −4.61962e6 2.28523e6i −1.34897 0.667306i
\(412\) 0 0
\(413\) −683921. −0.197302
\(414\) 0 0
\(415\) 1.29751e6i 0.369820i
\(416\) 0 0
\(417\) −1.22096e6 + 2.46819e6i −0.343845 + 0.695087i
\(418\) 0 0
\(419\) 549636.i 0.152947i 0.997072 + 0.0764733i \(0.0243660\pi\)
−0.997072 + 0.0764733i \(0.975634\pi\)
\(420\) 0 0
\(421\) 4.43018e6i 1.21819i 0.793096 + 0.609097i \(0.208468\pi\)
−0.793096 + 0.609097i \(0.791532\pi\)
\(422\) 0 0
\(423\) −2.90142e6 3.80058e6i −0.788424 1.03276i
\(424\) 0 0
\(425\) 105070.i 0.0282166i
\(426\) 0 0
\(427\) 6.98780e6 1.85469
\(428\) 0 0
\(429\) −1.86203e6 + 3.76412e6i −0.488477 + 0.987462i
\(430\) 0 0
\(431\) −4.61758e6 −1.19735 −0.598675 0.800992i \(-0.704306\pi\)
−0.598675 + 0.800992i \(0.704306\pi\)
\(432\) 0 0
\(433\) 1.59859e6 0.409747 0.204874 0.978788i \(-0.434322\pi\)
0.204874 + 0.978788i \(0.434322\pi\)
\(434\) 0 0
\(435\) 1.20057e6 2.42697e6i 0.304204 0.614951i
\(436\) 0 0
\(437\) 340358. 0.0852573
\(438\) 0 0
\(439\) 5.04081e6i 1.24836i 0.781281 + 0.624179i \(0.214566\pi\)
−0.781281 + 0.624179i \(0.785434\pi\)
\(440\) 0 0
\(441\) 24047.4 + 31499.8i 0.00588806 + 0.00771278i
\(442\) 0 0
\(443\) 687693.i 0.166489i 0.996529 + 0.0832445i \(0.0265282\pi\)
−0.996529 + 0.0832445i \(0.973472\pi\)
\(444\) 0 0
\(445\) 1.90308e6i 0.455573i
\(446\) 0 0
\(447\) −388572. + 785503.i −0.0919820 + 0.185943i
\(448\) 0 0
\(449\) 2.16418e6i 0.506615i 0.967386 + 0.253307i \(0.0815184\pi\)
−0.967386 + 0.253307i \(0.918482\pi\)
\(450\) 0 0
\(451\) −7.25898e6 −1.68048
\(452\) 0 0
\(453\) 2.45854e6 + 1.21619e6i 0.562902 + 0.278456i
\(454\) 0 0
\(455\) 4.10926e6 0.930539
\(456\) 0 0
\(457\) −128767. −0.0288414 −0.0144207 0.999896i \(-0.504590\pi\)
−0.0144207 + 0.999896i \(0.504590\pi\)
\(458\) 0 0
\(459\) −805365. 4.12862e6i −0.178427 0.914687i
\(460\) 0 0
\(461\) −6.38076e6 −1.39836 −0.699181 0.714944i \(-0.746452\pi\)
−0.699181 + 0.714944i \(0.746452\pi\)
\(462\) 0 0
\(463\) 5.96689e6i 1.29359i 0.762666 + 0.646793i \(0.223890\pi\)
−0.762666 + 0.646793i \(0.776110\pi\)
\(464\) 0 0
\(465\) −4.74159e6 2.34557e6i −1.01693 0.503055i
\(466\) 0 0
\(467\) 3.67725e6i 0.780245i 0.920763 + 0.390122i \(0.127567\pi\)
−0.920763 + 0.390122i \(0.872433\pi\)
\(468\) 0 0
\(469\) 8.67898e6i 1.82195i
\(470\) 0 0
\(471\) −1.10015e6 544224.i −0.228508 0.113038i
\(472\) 0 0
\(473\) 8.15008e6i 1.67498i
\(474\) 0 0
\(475\) −70119.8 −0.0142596
\(476\) 0 0
\(477\) 4.75071e6 + 6.22296e6i 0.956010 + 1.25228i
\(478\) 0 0
\(479\) 408276. 0.0813045 0.0406523 0.999173i \(-0.487056\pi\)
0.0406523 + 0.999173i \(0.487056\pi\)
\(480\) 0 0
\(481\) −1.92702e6 −0.379772
\(482\) 0 0
\(483\) −835948. 413526.i −0.163046 0.0806558i
\(484\) 0 0
\(485\) 8.30285e6 1.60278
\(486\) 0 0
\(487\) 3.14606e6i 0.601097i −0.953767 0.300548i \(-0.902830\pi\)
0.953767 0.300548i \(-0.0971697\pi\)
\(488\) 0 0
\(489\) −883433. + 1.78587e6i −0.167071 + 0.337736i
\(490\) 0 0
\(491\) 566933.i 0.106127i −0.998591 0.0530637i \(-0.983101\pi\)
0.998591 0.0530637i \(-0.0168986\pi\)
\(492\) 0 0
\(493\) 3.39936e6i 0.629912i
\(494\) 0 0
\(495\) −5.31092e6 + 4.05444e6i −0.974220 + 0.743734i
\(496\) 0 0
\(497\) 4.32480e6i 0.785371i
\(498\) 0 0
\(499\) 9.53698e6 1.71459 0.857293 0.514829i \(-0.172144\pi\)
0.857293 + 0.514829i \(0.172144\pi\)
\(500\) 0 0
\(501\) −2.12471e6 + 4.29512e6i −0.378186 + 0.764506i
\(502\) 0 0
\(503\) 9.64354e6 1.69948 0.849741 0.527200i \(-0.176758\pi\)
0.849741 + 0.527200i \(0.176758\pi\)
\(504\) 0 0
\(505\) −6.41957e6 −1.12015
\(506\) 0 0
\(507\) −430167. + 869586.i −0.0743219 + 0.150243i
\(508\) 0 0
\(509\) −3.21318e6 −0.549719 −0.274860 0.961484i \(-0.588631\pi\)
−0.274860 + 0.961484i \(0.588631\pi\)
\(510\) 0 0
\(511\) 1.04646e7i 1.77284i
\(512\) 0 0
\(513\) −2.75529e6 + 537472.i −0.462247 + 0.0901701i
\(514\) 0 0
\(515\) 9.06615e6i 1.50628i
\(516\) 0 0
\(517\) 9.53524e6i 1.56894i
\(518\) 0 0
\(519\) 1.95789e6 3.95790e6i 0.319059 0.644981i
\(520\) 0 0
\(521\) 3.10693e6i 0.501461i 0.968057 + 0.250730i \(0.0806708\pi\)
−0.968057 + 0.250730i \(0.919329\pi\)
\(522\) 0 0
\(523\) −8.71972e6 −1.39395 −0.696976 0.717094i \(-0.745472\pi\)
−0.696976 + 0.717094i \(0.745472\pi\)
\(524\) 0 0
\(525\) 172220. + 85193.9i 0.0272701 + 0.0134899i
\(526\) 0 0
\(527\) −6.64136e6 −1.04167
\(528\) 0 0
\(529\) −6.22541e6 −0.967228
\(530\) 0 0
\(531\) −1.01404e6 + 774137.i −0.156070 + 0.119147i
\(532\) 0 0
\(533\) 8.32760e6 1.26970
\(534\) 0 0
\(535\) 961710.i 0.145265i
\(536\) 0 0
\(537\) −5.25906e6 2.60155e6i −0.786996 0.389311i
\(538\) 0 0
\(539\) 79029.5i 0.0117170i
\(540\) 0 0
\(541\) 9.33788e6i 1.37169i −0.727749 0.685843i \(-0.759433\pi\)
0.727749 0.685843i \(-0.240567\pi\)
\(542\) 0 0
\(543\) −9.76236e6 4.82924e6i −1.42087 0.702877i
\(544\) 0 0
\(545\) 8.65532e6i 1.24822i
\(546\) 0 0
\(547\) −2.94115e6 −0.420290 −0.210145 0.977670i \(-0.567394\pi\)
−0.210145 + 0.977670i \(0.567394\pi\)
\(548\) 0 0
\(549\) 1.03608e7 7.90956e6i 1.46710 1.12001i
\(550\) 0 0
\(551\) −2.26861e6 −0.318332
\(552\) 0 0
\(553\) −2.53555e6 −0.352581
\(554\) 0 0
\(555\) −2.74814e6 1.35945e6i −0.378709 0.187340i
\(556\) 0 0
\(557\) 9.13152e6 1.24711 0.623555 0.781779i \(-0.285688\pi\)
0.623555 + 0.781779i \(0.285688\pi\)
\(558\) 0 0
\(559\) 9.34988e6i 1.26554i
\(560\) 0 0
\(561\) −3.71940e6 + 7.51881e6i −0.498960 + 1.00865i
\(562\) 0 0
\(563\) 1.23038e7i 1.63595i 0.575254 + 0.817975i \(0.304903\pi\)
−0.575254 + 0.817975i \(0.695097\pi\)
\(564\) 0 0
\(565\) 8.25548e6i 1.08798i
\(566\) 0 0
\(567\) 7.42025e6 + 2.02753e6i 0.969306 + 0.264856i
\(568\) 0 0
\(569\) 482588.i 0.0624879i −0.999512 0.0312440i \(-0.990053\pi\)
0.999512 0.0312440i \(-0.00994688\pi\)
\(570\) 0 0
\(571\) −226475. −0.0290689 −0.0145345 0.999894i \(-0.504627\pi\)
−0.0145345 + 0.999894i \(0.504627\pi\)
\(572\) 0 0
\(573\) −6.04500e6 + 1.22200e7i −0.769148 + 1.55484i
\(574\) 0 0
\(575\) −43455.1 −0.00548114
\(576\) 0 0
\(577\) 2.54105e6 0.317741 0.158871 0.987299i \(-0.449215\pi\)
0.158871 + 0.987299i \(0.449215\pi\)
\(578\) 0 0
\(579\) 1.26129e6 2.54971e6i 0.156357 0.316078i
\(580\) 0 0
\(581\) −2.97886e6 −0.366109
\(582\) 0 0
\(583\) 1.56127e7i 1.90243i
\(584\) 0 0
\(585\) 6.09276e6 4.65131e6i 0.736079 0.561934i
\(586\) 0 0
\(587\) 6.14425e6i 0.735993i 0.929827 + 0.367997i \(0.119956\pi\)
−0.929827 + 0.367997i \(0.880044\pi\)
\(588\) 0 0
\(589\) 4.43221e6i 0.526420i
\(590\) 0 0
\(591\) −1.99539e6 + 4.03370e6i −0.234995 + 0.475045i
\(592\) 0 0
\(593\) 1.36071e7i 1.58901i −0.607255 0.794507i \(-0.707729\pi\)
0.607255 0.794507i \(-0.292271\pi\)
\(594\) 0 0
\(595\) 8.20822e6 0.950509
\(596\) 0 0
\(597\) −1.34381e7 6.64754e6i −1.54312 0.763352i
\(598\) 0 0
\(599\) 6.20455e6 0.706551 0.353275 0.935519i \(-0.385068\pi\)
0.353275 + 0.935519i \(0.385068\pi\)
\(600\) 0 0
\(601\) 228374. 0.0257906 0.0128953 0.999917i \(-0.495895\pi\)
0.0128953 + 0.999917i \(0.495895\pi\)
\(602\) 0 0
\(603\) 9.82382e6 + 1.28682e7i 1.10024 + 1.44121i
\(604\) 0 0
\(605\) 4.18623e6 0.464980
\(606\) 0 0
\(607\) 1.17119e6i 0.129020i −0.997917 0.0645101i \(-0.979452\pi\)
0.997917 0.0645101i \(-0.0205485\pi\)
\(608\) 0 0
\(609\) 5.57191e6 + 2.75631e6i 0.608780 + 0.301151i
\(610\) 0 0
\(611\) 1.09390e7i 1.18542i
\(612\) 0 0
\(613\) 526490.i 0.0565899i −0.999600 0.0282950i \(-0.990992\pi\)
0.999600 0.0282950i \(-0.00900777\pi\)
\(614\) 0 0
\(615\) 1.18761e7 + 5.87484e6i 1.26615 + 0.626337i
\(616\) 0 0
\(617\) 2.34265e6i 0.247739i 0.992299 + 0.123869i \(0.0395304\pi\)
−0.992299 + 0.123869i \(0.960470\pi\)
\(618\) 0 0
\(619\) −881741. −0.0924942 −0.0462471 0.998930i \(-0.514726\pi\)
−0.0462471 + 0.998930i \(0.514726\pi\)
\(620\) 0 0
\(621\) −1.70753e6 + 333086.i −0.177680 + 0.0346599i
\(622\) 0 0
\(623\) 4.36916e6 0.451001
\(624\) 0 0
\(625\) −1.00524e7 −1.02936
\(626\) 0 0
\(627\) 5.01778e6 + 2.48219e6i 0.509733 + 0.252155i
\(628\) 0 0
\(629\) −3.84921e6 −0.387922
\(630\) 0 0
\(631\) 6.37410e6i 0.637302i −0.947872 0.318651i \(-0.896770\pi\)
0.947872 0.318651i \(-0.103230\pi\)
\(632\) 0 0
\(633\) −1.32420e6 + 2.67689e6i −0.131355 + 0.265535i
\(634\) 0 0
\(635\) 5.16041e6i 0.507867i
\(636\) 0 0
\(637\) 90663.7i 0.00885289i
\(638\) 0 0
\(639\) 4.89528e6 + 6.41234e6i 0.474270 + 0.621247i
\(640\) 0 0
\(641\) 7.98600e6i 0.767687i 0.923398 + 0.383844i \(0.125400\pi\)
−0.923398 + 0.383844i \(0.874600\pi\)
\(642\) 0 0
\(643\) 7.81890e6 0.745793 0.372897 0.927873i \(-0.378365\pi\)
0.372897 + 0.927873i \(0.378365\pi\)
\(644\) 0 0
\(645\) 6.59603e6 1.33339e7i 0.624285 1.26200i
\(646\) 0 0
\(647\) 8.38044e6 0.787057 0.393528 0.919313i \(-0.371254\pi\)
0.393528 + 0.919313i \(0.371254\pi\)
\(648\) 0 0
\(649\) 2.54413e6 0.237097
\(650\) 0 0
\(651\) 5.38503e6 1.08859e7i 0.498007 1.00673i
\(652\) 0 0
\(653\) −3.01923e6 −0.277085 −0.138542 0.990357i \(-0.544242\pi\)
−0.138542 + 0.990357i \(0.544242\pi\)
\(654\) 0 0
\(655\) 2.30779e6i 0.210181i
\(656\) 0 0
\(657\) 1.18449e7 + 1.55157e7i 1.07058 + 1.40236i
\(658\) 0 0
\(659\) 7.78900e6i 0.698664i 0.936999 + 0.349332i \(0.113591\pi\)
−0.936999 + 0.349332i \(0.886409\pi\)
\(660\) 0 0
\(661\) 5.90709e6i 0.525860i 0.964815 + 0.262930i \(0.0846888\pi\)
−0.964815 + 0.262930i \(0.915311\pi\)
\(662\) 0 0
\(663\) 4.26694e6 8.62568e6i 0.376993 0.762095i
\(664\) 0 0
\(665\) 5.47787e6i 0.480350i
\(666\) 0 0
\(667\) −1.40592e6 −0.122362
\(668\) 0 0
\(669\) 6.27642e6 + 3.10482e6i 0.542184 + 0.268207i
\(670\) 0 0
\(671\) −2.59940e7 −2.22878
\(672\) 0 0
\(673\) 8.71964e6 0.742097 0.371049 0.928613i \(-0.378998\pi\)
0.371049 + 0.928613i \(0.378998\pi\)
\(674\) 0 0
\(675\) 351781. 68621.7i 0.0297176 0.00579698i
\(676\) 0 0
\(677\) −1.44508e7 −1.21177 −0.605887 0.795551i \(-0.707181\pi\)
−0.605887 + 0.795551i \(0.707181\pi\)
\(678\) 0 0
\(679\) 1.90619e7i 1.58669i
\(680\) 0 0
\(681\) −8.08106e6 3.99754e6i −0.667730 0.330312i
\(682\) 0 0
\(683\) 1.40378e7i 1.15145i −0.817642 0.575727i \(-0.804719\pi\)
0.817642 0.575727i \(-0.195281\pi\)
\(684\) 0 0
\(685\) 1.87602e7i 1.52761i
\(686\) 0 0
\(687\) 4.78446e6 + 2.36677e6i 0.386759 + 0.191322i
\(688\) 0 0
\(689\) 1.79112e7i 1.43739i
\(690\) 0 0
\(691\) −1.42291e7 −1.13366 −0.566828 0.823836i \(-0.691830\pi\)
−0.566828 + 0.823836i \(0.691830\pi\)
\(692\) 0 0
\(693\) −9.30831e6 1.21930e7i −0.736271 0.964443i
\(694\) 0 0
\(695\) −1.00233e7 −0.787136
\(696\) 0 0
\(697\) 1.66343e7 1.29695
\(698\) 0 0
\(699\) 1.01163e7 + 5.00432e6i 0.783119 + 0.387393i
\(700\) 0 0
\(701\) 2.47356e6 0.190120 0.0950598 0.995472i \(-0.469696\pi\)
0.0950598 + 0.995472i \(0.469696\pi\)
\(702\) 0 0
\(703\) 2.56882e6i 0.196041i
\(704\) 0 0
\(705\) 7.71707e6 1.56001e7i 0.584763 1.18210i
\(706\) 0 0
\(707\) 1.47383e7i 1.10891i
\(708\) 0 0
\(709\) 3.27275e6i 0.244510i 0.992499 + 0.122255i \(0.0390126\pi\)
−0.992499 + 0.122255i \(0.960987\pi\)
\(710\) 0 0
\(711\) −3.75944e6 + 2.87001e6i −0.278900 + 0.212917i
\(712\) 0 0
\(713\) 2.74676e6i 0.202347i
\(714\) 0 0
\(715\) −1.52861e7 −1.11823
\(716\) 0 0
\(717\) −1.13591e7 + 2.29626e7i −0.825176 + 1.66810i
\(718\) 0 0
\(719\) −1.25182e7 −0.903067 −0.451534 0.892254i \(-0.649123\pi\)
−0.451534 + 0.892254i \(0.649123\pi\)
\(720\) 0 0
\(721\) 2.08144e7 1.49116
\(722\) 0 0
\(723\) −1.12689e7 + 2.27801e7i −0.801741 + 1.62073i
\(724\) 0 0
\(725\) 289644. 0.0204654
\(726\) 0 0
\(727\) 2.09674e7i 1.47132i −0.677349 0.735661i \(-0.736871\pi\)
0.677349 0.735661i \(-0.263129\pi\)
\(728\) 0 0
\(729\) 1.32969e7 5.39285e6i 0.926686 0.375837i
\(730\) 0 0
\(731\) 1.86763e7i 1.29270i
\(732\) 0 0
\(733\) 2.37515e7i 1.63279i −0.577491 0.816397i \(-0.695968\pi\)
0.577491 0.816397i \(-0.304032\pi\)
\(734\) 0 0
\(735\) −63960.2 + 129296.i −0.00436709 + 0.00882811i
\(736\) 0 0
\(737\) 3.22850e7i 2.18944i
\(738\) 0 0
\(739\) −1.98341e7 −1.33598 −0.667991 0.744170i \(-0.732845\pi\)
−0.667991 + 0.744170i \(0.732845\pi\)
\(740\) 0 0
\(741\) −5.75647e6 2.84761e6i −0.385133 0.190517i
\(742\) 0 0
\(743\) 8.45507e6 0.561882 0.280941 0.959725i \(-0.409353\pi\)
0.280941 + 0.959725i \(0.409353\pi\)
\(744\) 0 0
\(745\) −3.18992e6 −0.210567
\(746\) 0 0
\(747\) −4.41673e6 + 3.37180e6i −0.289601 + 0.221086i
\(748\) 0 0
\(749\) 2.20792e6 0.143807
\(750\) 0 0
\(751\) 2.69261e7i 1.74210i 0.491193 + 0.871051i \(0.336561\pi\)
−0.491193 + 0.871051i \(0.663439\pi\)
\(752\) 0 0
\(753\) −2.10295e7 1.04029e7i −1.35158 0.668599i
\(754\) 0 0
\(755\) 9.98414e6i 0.637446i
\(756\) 0 0
\(757\) 2.12014e7i 1.34470i −0.740234 0.672350i \(-0.765285\pi\)
0.740234 0.672350i \(-0.234715\pi\)
\(758\) 0 0
\(759\) 3.10965e6 + 1.53828e6i 0.195933 + 0.0969241i
\(760\) 0 0
\(761\) 1.80031e7i 1.12690i −0.826151 0.563449i \(-0.809474\pi\)
0.826151 0.563449i \(-0.190526\pi\)
\(762\) 0 0
\(763\) 1.98711e7 1.23570
\(764\) 0 0
\(765\) 1.21703e7 9.29096e6i 0.751875 0.573994i
\(766\) 0 0
\(767\) −2.91866e6 −0.179141
\(768\) 0 0
\(769\) 7.91227e6 0.482486 0.241243 0.970465i \(-0.422445\pi\)
0.241243 + 0.970465i \(0.422445\pi\)
\(770\) 0 0
\(771\) −1.68551e7 8.33786e6i −1.02116 0.505148i
\(772\) 0 0
\(773\) −2.31768e7 −1.39510 −0.697548 0.716538i \(-0.745726\pi\)
−0.697548 + 0.716538i \(0.745726\pi\)
\(774\) 0 0
\(775\) 565882.i 0.0338432i
\(776\) 0 0
\(777\) 3.12106e6 6.30926e6i 0.185460 0.374909i
\(778\) 0 0
\(779\) 1.11012e7i 0.655427i
\(780\) 0 0
\(781\) 1.60879e7i 0.943781i
\(782\) 0 0
\(783\) 1.13813e7 2.22014e6i 0.663419 0.129412i
\(784\) 0 0
\(785\) 4.46773e6i 0.258769i
\(786\) 0 0
\(787\) 2.36874e7 1.36327 0.681634 0.731693i \(-0.261270\pi\)
0.681634 + 0.731693i \(0.261270\pi\)
\(788\) 0 0
\(789\) 1.12384e7 2.27186e7i 0.642706 1.29924i
\(790\) 0 0
\(791\) −1.89532e7 −1.07706
\(792\) 0 0
\(793\) 2.98207e7 1.68397
\(794\) 0 0
\(795\) −1.26357e7 + 2.55432e7i −0.709058 + 1.43337i
\(796\) 0 0
\(797\) −1.45597e7 −0.811907 −0.405953 0.913894i \(-0.633060\pi\)
−0.405953 + 0.913894i \(0.633060\pi\)
\(798\) 0 0
\(799\) 2.18505e7i 1.21086i
\(800\) 0 0
\(801\) 6.47811e6 4.94549e6i 0.356753 0.272351i
\(802\) 0 0
\(803\) 3.89273e7i 2.13042i
\(804\) 0 0
\(805\) 3.39478e6i 0.184639i
\(806\) 0 0
\(807\) 952584. 1.92566e6i 0.0514896 0.104087i
\(808\) 0 0
\(809\) 3.22475e7i 1.73231i 0.499779 + 0.866153i \(0.333415\pi\)
−0.499779 + 0.866153i \(0.666585\pi\)
\(810\) 0 0
\(811\) −1.06545e7 −0.568828 −0.284414 0.958702i \(-0.591799\pi\)
−0.284414 + 0.958702i \(0.591799\pi\)
\(812\) 0 0
\(813\) 2.07374e7 + 1.02584e7i 1.10034 + 0.544316i
\(814\) 0 0
\(815\) −7.25241e6 −0.382462
\(816\) 0 0
\(817\) −1.24639e7 −0.653280
\(818\) 0 0
\(819\) 1.06786e7 + 1.39879e7i 0.556295 + 0.728692i
\(820\) 0 0
\(821\) 1.14818e7 0.594498 0.297249 0.954800i \(-0.403931\pi\)
0.297249 + 0.954800i \(0.403931\pi\)
\(822\) 0 0
\(823\) 2.52972e7i 1.30189i 0.759126 + 0.650944i \(0.225627\pi\)
−0.759126 + 0.650944i \(0.774373\pi\)
\(824\) 0 0
\(825\) −640645. 316914.i −0.0327704 0.0162109i
\(826\) 0 0
\(827\) 8.83375e6i 0.449140i 0.974458 + 0.224570i \(0.0720977\pi\)
−0.974458 + 0.224570i \(0.927902\pi\)
\(828\) 0 0
\(829\) 4.86456e6i 0.245843i −0.992416 0.122921i \(-0.960774\pi\)
0.992416 0.122921i \(-0.0392263\pi\)
\(830\) 0 0
\(831\) 2.25554e7 + 1.11577e7i 1.13305 + 0.560495i
\(832\) 0 0
\(833\) 181100.i 0.00904288i
\(834\) 0 0
\(835\) −1.74425e7 −0.865749
\(836\) 0 0
\(837\) −4.33752e6 2.22358e7i −0.214007 1.09708i
\(838\) 0 0
\(839\) −2.36695e7 −1.16087 −0.580436 0.814306i \(-0.697118\pi\)
−0.580436 + 0.814306i \(0.697118\pi\)
\(840\) 0 0
\(841\) −1.11402e7 −0.543128
\(842\) 0 0
\(843\) −1.54927e6 766390.i −0.0750857 0.0371433i
\(844\) 0 0
\(845\) −3.53139e6 −0.170139
\(846\) 0 0
\(847\) 9.61087e6i 0.460314i
\(848\) 0 0
\(849\) 8.61113e6 1.74075e7i 0.410007 0.828833i
\(850\) 0 0
\(851\) 1.59197e6i 0.0753548i
\(852\) 0 0
\(853\) 1.69049e7i 0.795498i 0.917494 + 0.397749i \(0.130209\pi\)
−0.917494 + 0.397749i \(0.869791\pi\)
\(854\) 0 0
\(855\) −6.20046e6 8.12199e6i −0.290074 0.379968i
\(856\) 0 0
\(857\) 2.33438e7i 1.08572i −0.839822 0.542861i \(-0.817341\pi\)
0.839822 0.542861i \(-0.182659\pi\)
\(858\) 0 0
\(859\) 1.96996e7 0.910908 0.455454 0.890259i \(-0.349477\pi\)
0.455454 + 0.890259i \(0.349477\pi\)
\(860\) 0 0
\(861\) −1.34876e7 + 2.72654e7i −0.620052 + 1.25344i
\(862\) 0 0
\(863\) 3.42226e6 0.156418 0.0782088 0.996937i \(-0.475080\pi\)
0.0782088 + 0.996937i \(0.475080\pi\)
\(864\) 0 0
\(865\) 1.60730e7 0.730395
\(866\) 0 0
\(867\) −1.29062e6 + 2.60900e6i −0.0583111 + 0.117876i
\(868\) 0 0
\(869\) 9.43203e6 0.423697
\(870\) 0 0
\(871\) 3.70378e7i 1.65425i
\(872\) 0 0
\(873\) 2.15764e7 + 2.82630e7i 0.958171 + 1.25511i
\(874\) 0 0
\(875\) 2.23997e7i 0.989057i
\(876\) 0 0
\(877\) 1.82299e7i 0.800361i −0.916436 0.400181i \(-0.868947\pi\)
0.916436 0.400181i \(-0.131053\pi\)
\(878\) 0 0
\(879\) 1.92417e7 3.88974e7i 0.839986 1.69804i
\(880\) 0 0
\(881\) 3.79378e7i 1.64677i 0.567483 + 0.823385i \(0.307917\pi\)
−0.567483 + 0.823385i \(0.692083\pi\)
\(882\) 0 0
\(883\) 3.47312e7 1.49906 0.749529 0.661972i \(-0.230280\pi\)
0.749529 + 0.661972i \(0.230280\pi\)
\(884\) 0 0
\(885\) −4.16232e6 2.05901e6i −0.178639 0.0883692i
\(886\) 0 0
\(887\) −8.89149e6 −0.379459 −0.189730 0.981836i \(-0.560761\pi\)
−0.189730 + 0.981836i \(0.560761\pi\)
\(888\) 0 0
\(889\) −1.18474e7 −0.502771
\(890\) 0 0
\(891\) −2.76027e7 7.54225e6i −1.16482 0.318278i
\(892\) 0 0
\(893\) −1.45822e7 −0.611922
\(894\) 0 0
\(895\) 2.13570e7i 0.891216i
\(896\) 0 0
\(897\) −3.56744e6 1.76474e6i −0.148039 0.0732317i
\(898\) 0 0
\(899\) 1.83082e7i 0.755520i
\(900\) 0 0
\(901\) 3.57774e7i 1.46824i
\(902\) 0 0
\(903\) 3.06125e7 + 1.51434e7i 1.24934 + 0.618021i
\(904\) 0 0
\(905\) 3.96449e7i 1.60904i
\(906\) 0 0
\(907\) 4.17482e7 1.68508 0.842539 0.538636i \(-0.181060\pi\)
0.842539 + 0.538636i \(0.181060\pi\)
\(908\) 0 0
\(909\) −1.66824e7 2.18523e7i −0.669650 0.877176i
\(910\) 0 0
\(911\) 4.26866e6 0.170410 0.0852052 0.996363i \(-0.472845\pi\)
0.0852052 + 0.996363i \(0.472845\pi\)
\(912\) 0 0
\(913\) 1.10811e7 0.439953
\(914\) 0 0
\(915\) 4.25275e7 + 2.10375e7i 1.67926 + 0.830695i
\(916\) 0 0
\(917\) 5.29830e6 0.208072
\(918\) 0 0
\(919\) 1.88729e7i 0.737138i −0.929600 0.368569i \(-0.879848\pi\)
0.929600 0.368569i \(-0.120152\pi\)
\(920\) 0 0
\(921\) −4.06410e6 + 8.21562e6i −0.157876 + 0.319147i
\(922\) 0 0
\(923\) 1.84562e7i 0.713081i
\(924\) 0 0
\(925\) 327974.i 0.0126033i
\(926\) 0 0
\(927\) 3.08613e7 2.35600e7i 1.17954 0.900483i
\(928\) 0 0
\(929\) 1.24738e7i 0.474197i 0.971486 + 0.237098i \(0.0761964\pi\)
−0.971486 + 0.237098i \(0.923804\pi\)
\(930\) 0 0
\(931\) 120860. 0.00456991
\(932\) 0 0
\(933\) −8.76059e6 + 1.77096e7i −0.329480 + 0.666048i
\(934\) 0 0
\(935\) −3.05339e7 −1.14223
\(936\) 0 0
\(937\) −2.08166e7 −0.774570 −0.387285 0.921960i \(-0.626587\pi\)
−0.387285 + 0.921960i \(0.626587\pi\)
\(938\) 0 0
\(939\) −1.87210e7 + 3.78447e7i −0.692890 + 1.40069i
\(940\) 0 0
\(941\) −1.20794e6 −0.0444703 −0.0222351 0.999753i \(-0.507078\pi\)
−0.0222351 + 0.999753i \(0.507078\pi\)
\(942\) 0 0
\(943\) 6.87968e6i 0.251935i
\(944\) 0 0
\(945\) 5.36084e6 + 2.74817e7i 0.195278 + 1.00107i
\(946\) 0 0
\(947\) 4.75552e7i 1.72315i 0.507631 + 0.861574i \(0.330521\pi\)
−0.507631 + 0.861574i \(0.669479\pi\)
\(948\) 0 0
\(949\) 4.46579e7i 1.60965i
\(950\) 0 0
\(951\) 1.83047e6 3.70031e6i 0.0656313 0.132674i
\(952\) 0 0
\(953\) 2.18909e7i 0.780784i 0.920649 + 0.390392i \(0.127661\pi\)
−0.920649 + 0.390392i \(0.872339\pi\)
\(954\) 0 0
\(955\) −4.96255e7 −1.76075
\(956\) 0 0
\(957\) −2.07270e7 1.02532e7i −0.731571 0.361893i
\(958\) 0 0
\(959\) 4.30704e7 1.51228
\(960\) 0 0
\(961\) −7.13973e6 −0.249387
\(962\) 0 0
\(963\) 3.27367e6 2.49917e6i 0.113755 0.0868421i
\(964\) 0 0
\(965\) 1.03544e7 0.357936
\(966\) 0 0
\(967\) 3.28485e7i 1.12967i −0.825205 0.564833i \(-0.808941\pi\)
0.825205 0.564833i \(-0.191059\pi\)
\(968\) 0 0
\(969\) −1.14985e7 5.68808e6i −0.393398 0.194606i
\(970\) 0 0
\(971\) 1.25929e7i 0.428624i −0.976765 0.214312i \(-0.931249\pi\)
0.976765 0.214312i \(-0.0687509\pi\)
\(972\) 0 0
\(973\) 2.30119e7i 0.779237i
\(974\) 0 0
\(975\) 734956. + 363568.i 0.0247600 + 0.0122482i
\(976\) 0 0
\(977\) 3.98034e6i 0.133408i 0.997773 + 0.0667042i \(0.0212484\pi\)
−0.997773 + 0.0667042i \(0.978752\pi\)
\(978\) 0 0
\(979\) −1.62529e7 −0.541968
\(980\) 0 0
\(981\) 2.94628e7 2.24923e7i 0.977465 0.746212i
\(982\) 0 0
\(983\) −5.24394e7 −1.73091 −0.865453 0.500989i \(-0.832970\pi\)
−0.865453 + 0.500989i \(0.832970\pi\)
\(984\) 0 0
\(985\) −1.63808e7 −0.537954
\(986\) 0 0
\(987\) 3.58153e7 + 1.77171e7i 1.17024 + 0.578895i
\(988\) 0 0
\(989\) −7.72422e6 −0.251110
\(990\) 0 0
\(991\) 1.65620e7i 0.535708i −0.963459 0.267854i \(-0.913685\pi\)
0.963459 0.267854i \(-0.0863146\pi\)
\(992\) 0 0
\(993\) 3.69632e6 7.47214e6i 0.118959 0.240476i
\(994\) 0 0
\(995\) 5.45720e7i 1.74748i
\(996\) 0 0
\(997\) 1.73906e7i 0.554087i 0.960857 + 0.277043i \(0.0893545\pi\)
−0.960857 + 0.277043i \(0.910645\pi\)
\(998\) 0 0
\(999\) −2.51394e6 1.28874e7i −0.0796969 0.408557i
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 384.6.f.e.191.8 yes 20
3.2 odd 2 384.6.f.f.191.7 yes 20
4.3 odd 2 384.6.f.f.191.13 yes 20
8.3 odd 2 384.6.f.f.191.8 yes 20
8.5 even 2 inner 384.6.f.e.191.13 yes 20
12.11 even 2 inner 384.6.f.e.191.14 yes 20
24.5 odd 2 384.6.f.f.191.14 yes 20
24.11 even 2 inner 384.6.f.e.191.7 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.6.f.e.191.7 20 24.11 even 2 inner
384.6.f.e.191.8 yes 20 1.1 even 1 trivial
384.6.f.e.191.13 yes 20 8.5 even 2 inner
384.6.f.e.191.14 yes 20 12.11 even 2 inner
384.6.f.f.191.7 yes 20 3.2 odd 2
384.6.f.f.191.8 yes 20 8.3 odd 2
384.6.f.f.191.13 yes 20 4.3 odd 2
384.6.f.f.191.14 yes 20 24.5 odd 2